Datasets:

hkust-nlp

/

llm-compression

like 5

--- abstract: | We propose a numerical method to obtain an adequate value for the upper bound on the rank for the tensor completion problem on the variety of third-order tensors of bounded tensor-train rank. The method is inspired by the parametrization of the tangent cone derived by Kutschan (2018). A proof of the adequacy of the upper bound for a related low-rank tensor approximation problem is given and an estimated rank is defined to extend the result to the low-rank tensor completion problem. Some experiments on synthetic data illustrate the approach and show that the method is very robust, e.g., to noise on the data. author: - - - - bibliography: - references.bib title: Rank Estimation for Third-Order Tensor Completion in the Tensor-Train Format --- tensor-train, tensor completion, rank estimation, tangent cone # Introduction We consider the *low-rank tensor completion problem (LRTCP)* formulated as a least squares optimization problem on the algebraic variety $\mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}$ [@kutschan2018 Definition 1.4] of $n_1 \times n_2 \times n_3$ real third-order tensors of tensor-train (TT) rank at most $(k_1,k_2)$: $$\label{eq:min_completion_TT} \min_{X \in \mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}} \underbrace{\frac{1}{2} \lVert X_{\Omega} - A_{\Omega} \rVert^2}_{=:f_\Omega(X)},$$ where $A \in \mathbb{R}^{n_1 \times n_2 \times n_3}$, $\Omega \subseteq \lbrace 1, \dots, n_1 \rbrace \times \lbrace 1, \dots, n_2 \rbrace \times \lbrace 1, \dots, n_3 \rbrace$ is called the sampling set, $$Z_{\Omega}\left(i_1,i_2,i_3\right) := \begin{cases} Z\left(i_1,i_2,i_3 \right) & \text{if }\left(i_1,i_2,i_3 \right) \in \Omega, \\ 0 & \text{otherwise}, \\ \end{cases}$$ for all $Z \in \mathbb{R}^{n_1 \times n_2 \times n_3}$, and the norm is induced by the inner product $$\label{eq:inner_product} \langle Y ,X \rangle = \langle \mathrm{vec}(Y) ,\mathrm{vec}(X) \rangle, \quad \forall X,Y \in \mathbb{R}^{n_1 \times n_2 \times n_3}.$$ A *tensor-train decomposition* (TTD) of a third-order tensor $X \in \mathbb{R}^{n_1 \times n_2 \times n_3}$ is a factorization $X = X_1 \cdot X_2 \cdot X_3$, where $X_1 \in \mathbb{R}^{n_1 \times r_1}$, $X_2 \in \mathbb{R}^{r_1 \times n_2 \times r_2}$, and $X_3 \in \mathbb{R}^{r_2 \times n_3}$ [@Oseledets2011]. The '$\cdot$' indicates the contraction between a matrix and a tensor. They interact with the left and right unfolding of $X_2$, $$\begin{split} X_2^\mathrm{L} &:= \left[ X_2 \right]^{r_1 \times n_2 r_2}:= \mathrm{reshape}\left( X_2, r_1 \times n_2 r_2 \right), \\ X_2^\mathrm{R} &:= \left[ X_2 \right]^{r_1 n_2 \times r_2} := \mathrm{reshape}\left( X_2, r_1 n_2 \times r_2 \right), \end{split}$$ in the following way: $$\begin{split} X_1 \cdot X_2 = \left[ X_1 X_2^\mathrm{L} \right]^{n_1 \times n_2 \times r_2}, X_2 \cdot X_3= \left[ X_2^\mathrm{R} X_3 \right]^{r_1 \times n_2 \times n_3}.\\ \end{split}$$ The minimal $r_1$ and $r_2$ for which a TTD of $X$ exists, is called the *TT-rank* of $X$. For second-order tensors (matrices), the TT-rank reduces to the usual matrix rank, and since no other definition of tensor rank is used in this paper, it is simply denoted by $\mathop{\mathrm{rank}}X$ and can be determined as $$\label{eq:def_rank_TT} \mathop{\mathrm{rank}}X = \left (\mathop{\mathrm{rank}}X^\mathrm{L}, \mathop{\mathrm{rank}}X^\mathrm{R} \right)=: (r_1,r_2).$$ The minimal rank decomposition can be obtained by computing successive singular value decompositions (SVDs) of the unfoldings [@Oseledets2011 Algorithm 1]. The low-rank variety can then be defined as $$\label{eq:variety} \mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3} := \{X \in \mathbb{R}^{n_1 \times n_2 \times n_3} \mid \mathop{\mathrm{rank}}X \le (k_1, k_2)\}$$ and the fixed-rank smooth manifold [@Holtz_manifolds_2012] as $$\label{eq:fixed_rank_manifold} \mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3} := \{X \in \mathbb{R}^{n_1 \times n_2 \times n_3} \mid \mathop{\mathrm{rank}}X = (r_1, r_2)\}.$$ In practical LRTCPs, the rank of $A$ is not known or $A$ has full rank due to noise. This is notably the case for movie rating recommendation systems [@Movielens_summary], where, e.g., the ratings of different users over time (or any other variable of interest) form samples of a large third-order tensor. Evaluating a solution to [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"} in elements outside the set $\Omega$ allows us to recommend movies with a high estimated rating. Note that evaluating one element of a third-order TTD corresponds to performing a vector-matrix-vector multiplication and can be done efficiently in $\mathcal{O}\left( r_1r_2\right)$ operations. When $k_1$ and $k_2$ are set too high however, the complexity of an algorithm to solve [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"} is unnecessarily high and furthermore overfitting can occur, i.e., $X$ approximates $A_\Omega$ well but not the full tensor $A$. To detect overfitting, usually a test data set $\Gamma$ is used [@Steinl_high_dim_TT_compl_2016]. When the error on this test set increases during optimization while the error of [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"} decreases overfitting has occurred and the algorithm should be stopped or the rank decreased. On the other hand, when $k_1$ or $k_2$ are set too low, the search space may not contain a sufficiently good approximation of $A$. It is thus important to choose adequate values for $k_1$ and $k_2$. Intuitively the smaller $\lvert \Omega \rvert$, the more difficult it is to recover $A$ from $A_{\Omega}$ by solving [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"}. However, the minimal number of samples needed is not known [@budzinskiy2021tensor]. In this work, a method to estimate the rank of $A$ from $A_\Omega$ is proposed. When $A$ is not exactly low rank, a good value for a low-rank approximation is obtained. This method can then be used, e.g., in a rank-adaptive optimization algorithm to solve [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"}. The paper is organized as follows. First, in [2](#sec:TT){reference-type="ref" reference="sec:TT"}, the preliminaries for [4](#sec:tangent_cone){reference-type="ref" reference="sec:tangent_cone"} and [5](#sec:rank_est){reference-type="ref" reference="sec:rank_est"} are given. This includes some basic facts concerning orthogonal projections onto vector spaces and arithmetic rules for TTDs. For a more extensive overview of properties of the TTD, we refer to the original paper [@Oseledets2011] and the notation introduced in [@LR_tensor_methods_Steinlechner_2014], which is also used in [@Steinl_high_dim_TT_compl_2016; @Steinlechner_thesis2016]. In [3](#sec:low_rank_approx){reference-type="ref" reference="sec:low_rank_approx"}, the auxiliary low-rank tensor approximation problem (LRTAP) is defined. In [4](#sec:tangent_cone){reference-type="ref" reference="sec:tangent_cone"}, the parametrization of the tangent cone to the low-rank variety is given [@kutschan2018]. New orthogonality conditions are derived to ensure that in no matrix inverse is needed, which improves the stability of the proposed method, and makes the proofs in the rest of the paper easier. In [5](#sec:rank_est){reference-type="ref" reference="sec:rank_est"}, the main proposition is derived, and an estimated rank is defined to extend our result to the LRTCP. Lastly, in [6](#sec:experiments){reference-type="ref" reference="sec:experiments"}, some experiments illustrate the use and advantages of the proposed rank estimation method for the LRTCP. # Preliminaries {#sec:TT} A TTD is not unique. Orthogonality conditions can be enforced, which can improve the stability of algorithms working with TTDs. Those used in this paper are introduced in [4](#sec:tangent_cone){reference-type="ref" reference="sec:tangent_cone"}. Given $n, p \in \mathbb{N}$ with $n \ge p$, we let $\mathrm{St}(p, n) := \{U \in \mathbb{R}^{n \times p} \mid U^\top U = I_p\}$ denote the Stiefel manifold. For every $U \in \mathrm{St}(p, n)$, we let $P_U := UU^\top$ and $P_U^\perp := I_n-P_U$ denote the orthogonal projections onto the range of $U$ and its orthogonal complement, respectively. A tensor is said to be *left-orthogonal* if $n_1 \leq n_2n_3$ and $\left( X^\mathrm{L} \right)^\top \in \mathrm{St}(n_1,n_2n_3)$, and *right-orthogonal* if $n_3 \leq n_1 n_2$ and $X^\mathrm{R} \in \mathrm{St}(n_3,n_1n_2)$. The following properties and arithmetic rules are used frequently in the rest of the paper. For all matrices $Y$ and $Z$, it holds that $Y \cdot (X_1 \cdot {X}_2 \cdot X_3) \cdot Z = YX_1 \cdot {X}_2 \cdot X_3 Z$. The left and right unfoldings of $X=X_1 \cdot X_2 \cdot X_3$ can be rewritten as: $$\label{eq:LR_TTD} \begin{split} &X^\mathrm{L} = X_1 \left( X_2 \cdot X_3 \right)^\mathrm{L} = X_1 X_2^\mathrm{L} \left(X_3 \otimes I_{n_2} \right),\\ &X^\mathrm{R} =\left( X_1 \cdot X_2 \right)^\mathrm{R} X_3 = \left( I_{n_2} \otimes X_1 \right) X_2^\mathrm{R} X_3, \end{split}$$ where '$\otimes$' denotes the Kronecker product. From [\[eq:def_rank_TT\]](#eq:def_rank_TT){reference-type="eqref" reference="eq:def_rank_TT"} and [\[eq:LR_TTD\]](#eq:LR_TTD){reference-type="eqref" reference="eq:LR_TTD"} it can be deduced that $$\label{eq:rank_factors_TTD1} \begin{split} \mathop{\mathrm{rank}}(X_1) &= \mathop{\mathrm{rank}}\left( X_2^\mathrm{L} \left(X_3 \otimes I_{n_2} \right) \right) = r_1, \\ \mathop{\mathrm{rank}}(X_3) &= \mathop{\mathrm{rank}}\left(\left( I_{n_2} \otimes X_1 \right) X_2^\mathrm{R} \right) = r_2, \\ \end{split}$$ and because the ranks of $I_{n_2} \otimes X_1$ and $X_3 \otimes I_{n_2}$ are $n_2 r_1$ and $r_2 n_2$, respectively, [\[eq:rank_factors_TTD1\]](#eq:rank_factors_TTD1){reference-type="eqref" reference="eq:rank_factors_TTD1"} can be simplified to $\mathop{\mathrm{rank}}\left( X_2^\mathrm{L} \right) = r_1$ and $\mathop{\mathrm{rank}}\left( X_2^\mathrm{R} \right) = r_2.$ Orthogonality between TTDs is exploited frequently in the parametrization of the tangent cone in [4](#sec:tangent_cone){reference-type="ref" reference="sec:tangent_cone"} and the proofs in [5](#sec:rank_est){reference-type="ref" reference="sec:rank_est"}. If $Y = Y_1 \cdot Y_2 \cdot Y_3$ and $Z = Z_1 \cdot Z_2 \cdot Z_3$, then by using [\[eq:LR_TTD\]](#eq:LR_TTD){reference-type="eqref" reference="eq:LR_TTD"}, the inner product $\langle Y,Z \rangle$ is zero if at least one of the following equalities holds: $$\begin{aligned} \label{eq:inner_prod_zero} Y_1^\top Z_1 &= 0, && &\left( Y_2 \cdot Y_3 \right)^\mathrm{L} \big( \left( Z_2 \cdot Z_3 \right)^\mathrm{L} \big)^\top &= 0, & \nonumber \\ Y_3 Z_3^\top &= 0, && &\big( \left( Y_1 \cdot Y_2 \right)^\mathrm{R} \big)^\top \left( Z_1 \cdot Z_2 \right)^\mathrm{R} &= 0. & \end{aligned}$$ # Low-Rank Tensor Approximation {#sec:low_rank_approx} The *low-rank tensor approximation problem* (LRTAP) is defined as: $$\label{eq:low-rank_approx} \min_{X \in \mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}} \underbrace{\frac{1}{2} \lVert X - A \rVert^2}_{=: f(X)}.$$ This problem is related to the LRTCP [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"} because $f_\Omega(X) = f(X)$ for $\Omega = \lbrace 1, \dots, n_1 \rbrace \times \lbrace 1, \dots, n_2 \rbrace \times \lbrace 1, \dots, n_3 \rbrace$. Remark that, as for [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"}, a global minimizer is, in general, not unique because $\mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}$ is nonconvex and NP-hard to obtain [@hillar2013most]. This problem is used in [5](#sec:rank_est){reference-type="ref" reference="sec:rank_est"}. # Tangent Cone {#sec:tangent_cone} The set of all tangent vectors to $\mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}$ at $X = X_1'\cdot X_2' \cdot X_3 \in \mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}$, where $X_1' \in \mathrm{St}(r_1,n_1)$, ${X_2'}^\mathrm{R} \in \mathrm{St}(r_2,r_1n_2)$, and $k_1 \ge r_1$, $k_2 \ge r_2$, is a closed cone called the tangent cone to $\mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}$ at $X$ and denoted by $T_X \mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}$. By [@kutschan2018 Theorem 2.6], $T_X\mathbb{R}_{\le (k_1, k_2)}^{n_1 \times n_2 \times n_3}$ is the set of all $G \in \mathbb{R}^{n_1 \times n_2 \times n_3}$ that can be decomposed as $$%\label{eq:tangent_cone_TT_3D} \begin{split} G &= \begin{bmatrix} X_1' & U_1 & W_1 \end{bmatrix} \cdot \begin{bmatrix} X_2' & U_2 & W_2 \\ 0 & Z_2 & V_2 \\ 0 & 0 & X_2' \\ \end{bmatrix} \cdot \begin{bmatrix} W_3 \\ V_3 \\ X_3 \end{bmatrix}, \\ \end{split}$$ where $U_1 \in \mathbb{R}^{n_1 \times s_1}$, $W_1 \in \mathbb{R}^{n_1 \times r_1}$, $U_2 \in \mathbb{R}^{r_1 \times n_2 \times s_2}$, $W_2 \in \mathbb{R}^{r_1 \times n_2 \times r_2}$, $Z_2 \in \mathbb{R}^{s_1 \times n_2 \times s_2}$, $V_2 \in \mathbb{R}^{s_1 \times n_2 \times r_2}$, $W_3 \in \mathbb{R}^{r_2 \times n_3}$, $V_3 \in \mathbb{R}^{s_2 \times n_3}$, and $s_i=k_i - r_i$. As for the TTD itself, this parametrization is not unique. In [@kutschan2018], the following orthogonality conditions are derived: $$\begin{aligned} {U}_1^\top X_1' &= 0, & {W}_1^\top X_1' &= 0, \nonumber\\ \left({U}_2^{\mathrm{R}} \right)^\top X_2'^{\mathrm{R}} &= 0, & \left( {V}_2 \cdot X_3 \right)^{\mathrm{L}} \left( \left( X_{2}' \cdot X_3 \right)^\mathrm{L} \right)^\top &= 0, \\ \left({W}_2^{\mathrm{R}} \right)^\top X_2'^{\mathrm{R}}&=0, & {V}_3 X_{3}^\top &= 0. \nonumber\end{aligned}$$ We change these orthogonality conditions slightly to make the proofs in the rest of this paper easier and the computations in the experiments more stable. To do so, we notice that $$\begin{split} G = &\begin{bmatrix} X_1' & U_1 & \dot{W}_1 \end{bmatrix} \cdot \begin{bmatrix} X_2' & U_2 & \dot{W}_2\\ 0 & Z_2 & \dot{V}_2 \\ 0 & 0 & X_2'' \\ \end{bmatrix} \cdot \begin{bmatrix} W_3 \\ V_3 \\ X_3'' \end{bmatrix}, \\ \end{split}$$ where we have defined $\dot{W}_1 := W_1 R^{-1}$, $\dot{W}_2 := W_2 \cdot C$, $\dot{V}_2 := V_2 \cdot C$, $X_2'' := R \cdot X_2' \cdot C$, $X_3'' := C^{-1} X_3$, and $R \in \mathbb{R}^{r_1 \times r_1}$ and $C \in \mathbb{R}^{r_2 \times r_2}$ are chosen such that $X_2''$ and $X_3''$ are left-orthogonal. Thus, we can also define $X_1:= X_1' R^{-1}$ and $X_2: = X_2' \cdot C$, such that $X = X_1 \cdot X_2'' \cdot X_3'' = X_1' \cdot X_2 \cdot X_3''$. Additionally, $W_3$ can be decomposed as $W_3 = W_3 X_3''^\top X_3'' + \hat{W}_3$. The two terms involving $W_3$ and $\dot{W}_2$ can then be regrouped as $$\begin{split} X_1' \cdot \left( X_2' \cdot W_3 X_3''^\top + \dot{W}_2\right) \cdot X_3''+ X_1' \cdot X_2' \cdot \hat{W}_3. \end{split}$$ Defining $\tilde{W}_2 := X_2' \cdot W_3 X_3''^\top + \dot{W}_2$, we obtain $$\label{eq:paran_g_final} \begin{split} G = &\begin{bmatrix} X_1' & U_1 & \dot{W}_1 \end{bmatrix} \cdot \begin{bmatrix} X_2' & U_2 & \tilde{W}_2\\ 0 & Z_2 & \dot{V}_2 \\ 0 & 0 & X_2'' \\ \end{bmatrix} \cdot \begin{bmatrix} \hat{W}_3 \\ V_3 \\ X_3'' \end{bmatrix}, \\ \end{split}$$ with the modified orthogonality conditions $$\begin{aligned} \label{eq:orth_cond_K_modified} {U}_1^\top X_1' &= 0, & \dot{W}_1^\top X_1' &= 0, & \left({U}_2^{\mathrm{R}} \right)^\top X_2'^{\mathrm{R}} &= 0, \nonumber\\ \hat{W}_3 X_3''^{\top}&=0, & {V}_3 X_{3}''^\top &= 0, & \dot{V}_2^{\mathrm{L}} \left( X_{2}''^{\mathrm{L}} \right)^\top &= 0. \end{aligned}$$ Expanding [\[eq:paran_g\_final\]](#eq:paran_g_final){reference-type="eqref" reference="eq:paran_g_final"}, a sum of 6 mutually orthogonal TTDs is obtained because of [\[eq:inner_prod_zero\]](#eq:inner_prod_zero){reference-type="eqref" reference="eq:inner_prod_zero"} and [\[eq:orth_cond_K\_modified\]](#eq:orth_cond_K_modified){reference-type="eqref" reference="eq:orth_cond_K_modified"}. The projection onto the closed cone $T_X \mathbb{R}_{\leq (k_1 , k_2)}^{n_1 \times n_2 \times n_3}$ is not known and, in general, difficult to obtain because it is nonlinear and nonconvex. However, in what follows we show that any tensor $Y \in \mathbb{R}^{n_1 \times n_2 \times n_3}$ is an element of $T_X \mathbb{R}_{\leq (n_1, n_3)}^{n_1 \times n_2 \times n_3}$. Straightforward computations show that $$\label{eq:Y_6terms} \begin{split} Y = ~&P_{X_1'} \cdot Y \cdot P_{X_3''^\top} + P_{X_1'}^\perp \cdot Y \cdot P_{X_3''^\top} \\ + ~&P_{X_1'} \cdot Y \cdot P_{X_3''^\top}^\perp + P_{X_1'}^\perp \cdot Y \cdot P_{X_3''^\top}^\perp\\ = ~&P_{X_1'} \cdot Y \cdot P_{X_3''^\top} \\ + ~&\left[ P_{X_1'}^\perp \left( Y \cdot X_3''^\top \right)^\mathrm{L} P_{\left( X_2''^{\mathrm{L}}\right)^\top} \right]^{n_1 \times n_2 \times r_2} \cdot X_3''\\ + ~& \left[ P_{X_1'}^\perp \left( Y \cdot X_3''^\top \right)^\mathrm{L} P_{\left( X_2''^{\mathrm{L}}\right)^\top}^\perp \right]^{n_1 \times n_2 \times r_2} \cdot X_3''\\ + ~&X_1' \cdot \left[ P_{X_2'^{\mathrm{R}}} \left( X_1'^\top \cdot Y \right)^{\mathrm{R}} P_{X_3''^\top}^\perp \right]^{r_1 \times n_2 \times n_3} \\ + ~&X_1' \cdot \Big[ P_{X_2'^{\mathrm{R}}}^\perp \left( X_1'^\top \cdot Y \right)^{\mathrm{R}} P_{X_3''^\top}^\perp \Big]^{r_1 \times n_2 \times n_3}\\ + ~&P_{X_1'}^\perp \cdot Y \cdot P_{X_3''^\top}^\perp. \\ \end{split}$$ Thus, $Y=Y_1 \cdot Y_2 \cdot Y_3 \in \mathbb{R}_{\leq (n_1, n_3)}^{n_1 \times n_2 \times n_3}$ can be parameterized as in [\[eq:paran_g\_final\]](#eq:paran_g_final){reference-type="eqref" reference="eq:paran_g_final"}, with $$\label{eq:param_Y_TC} \begin{split} U_1 &= P_{X_1'}^\perp Y_1, \\ U_2 &= \Big[ P_{X_2'^{\mathrm{R}}}^\perp \left( X_1'^\top \cdot Y_1 \cdot Y_2 \right)^{\mathrm{R}} \Big]^{r_1 \times n_2 \times n_3}, \\ \dot{V}_2 &= \Big[ \left( Y_2 \cdot Y_3 \cdot X_3''^\top \right)^\mathrm{L} P_{\left( X_2''^{\mathrm{L}}\right)^\top}^\perp \Big]^{n_1 \times n_2 \times r_2}, \\ {V}_3 &= Y_3 P_{X_3''^\top}^\perp, \\ \dot{W}_1& = \left( P_{X_1'}^\perp \cdot Y \cdot X_3''^\top \right)^\mathrm{L} \left( X_2''^{\mathrm{L}}\right)^\top,\\ \tilde{W}_2 &= X_1'^{\top} \cdot Y \cdot X_3''^{\top}, \\ \hat{W}_3 &= \left( X_2'^{\mathrm{R}}\right)^\top \left( X_1'^\top \cdot Y \cdot P_{X_3''^\top}^\perp \right)^{\mathrm{R}}, \quad \\ Z_2 &= Y_2, \\ \end{split}$$ which satisfies [\[eq:orth_cond_K\_modified\]](#eq:orth_cond_K_modified){reference-type="eqref" reference="eq:orth_cond_K_modified"}. Furthermore, $U_1$ and $V_3$ have rank at most $n_1 - r_1$ and $n_3 - r_2$, respectively, and thus the matrix $U_1 \dot{V}_2^\mathrm{L}$ has rank $n_1 - r_1$, and $U_1$ and $\dot{V}_2^\mathrm{L}$ can be reduced to size $n_1 \times \left( n_1 - r_1 \right)$ and $\left( n_1 - r_1 \right) \times n_2 r_2$, respectively, e.g., by computing the SVD of $U_1 \dot{V}_2^\mathrm{L}$. Similarly, this can be done for $U_2^\mathrm{R} V_3$ to obtain $U_2^\mathrm{R}$ and $V_3$ of size $r_1 n_2 \times \left( n_3 - r_2 \right)$ and $\left( n_3 - r_2 \right) \times n_3$, respectively. Then, $Z_2$ can be changed accordingly to $U_1^\top \cdot Y_2 \cdot V_3^\top$, which is the same result as would be obtained by the TT-rounding algorithm [@Oseledets2011 Algorithm 2], except for the orthogonality conditions. Thus, $Y$ can be written in the form [\[eq:paran_g\_final\]](#eq:paran_g_final){reference-type="eqref" reference="eq:paran_g_final"} with $s_1= n_1 - r_1$, $s_2 = n_3 - r_2$ and hence by definition $Y \in T_X \mathbb{R}_{\leq (n_1, n_3)}^{n_1 \times n_2 \times n_3}$. The Riemannian gradient of [\[eq:low-rank_approx\]](#eq:low-rank_approx){reference-type="eqref" reference="eq:low-rank_approx"} at $X \in \mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}$ is defined as the projection of the Euclidean gradient $\nabla f(X) = X-A$ onto the tangent space [@Holtz_manifolds_2012]: $$\begin{split} &T_X\mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3} := \\ & \left\lbrace\begin{array}{l} X_1' \cdot X_2' \cdot \hat{W}_3 + X_1' \cdot \tilde{W}_2 \cdot X_3'' + \dot{W}_1 \cdot X_2'' \cdot X_3'' \end{array}\right\rbrace. \end{split}$$ By replacing $Y$ by $X-A$ in [\[eq:param_Y\_TC\]](#eq:param_Y_TC){reference-type="eqref" reference="eq:param_Y_TC"}, the parameters of $\mathcal{P}_{T_X\mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}} \nabla f(X)$ are: $$\label{eq:proj_tangent_space_param} \begin{split} \dot{W}_1 &= - \left( P_{X_1'}^\perp \cdot A \cdot X_3''^\top \right)^\mathrm{L} \left( X_2''^{\mathrm{L}}\right)^\top, \\ \tilde{W}_2 &= {X}_2 - X_1'^{\top} \cdot A \cdot X_3''^{\top}, \\ \hat{W}_3 &= -\left( X_2'^{\mathrm{R}}\right)^\top \left( X_1'^\top \cdot A \cdot P_{X_3''^\top}^\perp \right)^{\mathrm{R}}. \end{split}$$ A similar projection onto the tangent space was used in [@Steinl_high_dim_TT_compl_2016] and [@Lubich_TT_time_int_2015], but with different orthogonality conditions. # Rank Estimation {#sec:rank_est} states the main result for the LRTAP [\[eq:low-rank_approx\]](#eq:low-rank_approx){reference-type="eqref" reference="eq:low-rank_approx"}. Afterwards, the estimated rank [\[eq:num_rank_increase\]](#eq:num_rank_increase){reference-type="eqref" reference="eq:num_rank_increase"} is defined to extend this result to the LRTCP [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"}. To prove , the following auxiliary lemma is used. **Lemma 1**. *Let $X = X_1' \cdot {X}_2 \cdot X_3'' = X_1' \cdot {X}_2' \cdot X_3 = X_1 \cdot {X}_2'' \cdot X_3'' \in \mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}$, with $X_1' \in \mathrm{St}(r_1,n_1)$, ${X_2'}^\mathrm{R} \in \mathrm{St}(r_2,r_1 n_2)$, $\left({X_2''}^\mathrm{L}\right)^\top \in \mathrm{St}(r_1,n_2 r_2)$, and ${X_3''}^\top \in \mathrm{St}(r_2,n_3)$, and let $A = A_1' \cdot {A}_2 \cdot A_3'' \in \mathbb{R}_{(r_1', r_2')}^{n_1 \times n_2 \times n_3}$, with $A_1' \in \mathrm{St}(r_1',n_1)$ and ${A_3''}^\top \in \mathrm{St}(r_2',n_3)$. If $\mathcal{P}_{T_{X}\mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}} \nabla f \left(X \right)=0$, then $$\begin{aligned} X_1'= A_1' B_1, && X_3'' = B_3 A_3'', && {X}_2 = B_1^\top \cdot {A}_2 \cdot B_3^\top,\end{aligned}$$ for some $B_1 \in \mathrm{St}(r_1,r'_1)$ and $B_3^\top \in \mathrm{St}(r_2,r_2')$.* *Proof.* From [\[eq:proj_tangent_space_param\]](#eq:proj_tangent_space_param){reference-type="eqref" reference="eq:proj_tangent_space_param"}, $\mathcal{P}_{T_{X}\mathbb{R}_{(r_1,r_2)}^{n_1 \times n_2 \times n_3}}\nabla f \left(X \right)=0$ if and only if $\dot{W}_1 = 0$, $\tilde{W}_2 = 0$, and $\hat{W}_3 = 0$. From the second equation in [\[eq:proj_tangent_space_param\]](#eq:proj_tangent_space_param){reference-type="eqref" reference="eq:proj_tangent_space_param"}, it is clear that $\tilde{W}_2$ can only be zero if $X_2 = X_1'^{\top} \cdot A \cdot X_3''^{\top}$. The matrices $X_1'$ and $X_3''$ are decomposed as $$\begin{aligned} \label{eq:X1_X3_ifv_A1_A3} &X_1' = \begin{bmatrix} A_1' & A_1'^{\perp} \end{bmatrix} \begin{bmatrix} B_1 \\ B_2 \end{bmatrix} = A_1' B_1 + A_1'^{\perp} B_2,\\ &X_3'' =\begin{bmatrix} B_3 & B_4 \end{bmatrix} \begin{bmatrix} A_3'' \\ A_3''^{\perp} \end{bmatrix} = B_3 A_3'' + B_4 A_3''^{\perp} \nonumber,\end{aligned}$$ where $\begin{bmatrix} A_1' & A_1'^{\perp} \end{bmatrix} \in \mathrm{St}(n_1, n_1)$, $\begin{bmatrix} {A_3''} \\ A_3''^{\perp} \end{bmatrix} \in \mathrm{St}(n_3, n_3)$, $\begin{bmatrix} B_1 \\ B_2 \end{bmatrix} \in \mathrm{St}(r_1, r_1 + r_1')$, and $\begin{bmatrix} B_3^\top \\ B_4^\top \end{bmatrix} \in \mathrm{St}(r_2, r_2 + r_2')$. Substituting [\[eq:X1_X3_ifv_A1_A3\]](#eq:X1_X3_ifv_A1_A3){reference-type="eqref" reference="eq:X1_X3_ifv_A1_A3"} into the equation for $X_2$, we obtain $X_2 = B_1^\top \cdot A_2 \cdot B_3^\top$. Substituting [\[eq:X1_X3_ifv_A1_A3\]](#eq:X1_X3_ifv_A1_A3){reference-type="eqref" reference="eq:X1_X3_ifv_A1_A3"} into the equation for $\dot{W}_1$ in [\[eq:proj_tangent_space_param\]](#eq:proj_tangent_space_param){reference-type="eqref" reference="eq:proj_tangent_space_param"}, we get $$\begin{split} &\dot{W}_1 = 0 \Leftrightarrow \Big( \Big( I_{n_1} - A_1' B_1 B_1^{\top} A_1'^{\top} - A_1'^{\perp} B_2 B_1^{\top} A_1'^{\top} - \\ & A_1' B_1 B_2^{\top} \left( A_1'^\perp \right)^{\top} -A_1'^{\perp} B_2 B_2^{\top} \left( A_1'^\perp \right)^{\top} \Big)A_1' \cdot A_2 \cdot A_3'' \\ & \left( A_3''^{\top} B_3^{\top} + \left(A_3''^\perp \right)^{\top} B_4^{\top} \right) \Big)^\mathrm{L} \left( {X_2''}^{\mathrm{L}}\right)^{\top} = 0 \Leftrightarrow\\ &\left( \left( A_1' - A_1' B_1 B_1^{\top} - A_1'^{\perp} B_2 B_1^{\top} \right) \cdot A_2 \cdot B_3^{\top} \right)^{\mathrm{L}} \big( {X_2''}^{\mathrm{L}}\big)^{\top} = 0. \\ \end{split}$$ Multiplying both sides by $\left(A_1'^\perp \right)^\top$, we obtain $$\begin{split} \dot{W}_1 = 0 &\Rightarrow B_2 B_1^{\top} A_2^{\mathrm{L}} \left( B_3^{\top} \otimes I_{n_2} \right) \left( {X_2''}^{\mathrm{L}}\right)^{\top} = 0 \\ &\Leftrightarrow B_2 {X_2}^{\mathrm{L}} \left( {X_2''}^{\mathrm{L}}\right)^{\top} = 0 \\ &\Leftrightarrow B_2 R^{-1} {X_2''}^{\mathrm{L}} \left( {X_2''}^{\mathrm{L}}\right)^{\top} = 0 \Leftrightarrow B_2 = 0. \end{split}$$ Thus, it holds that $X_1' = A_1' B_1$ with $B_1 \in \mathrm{St}(r_1,r_1')$. From $\hat{W}_3 =0$ in [\[eq:proj_tangent_space_param\]](#eq:proj_tangent_space_param){reference-type="eqref" reference="eq:proj_tangent_space_param"}, it can similarly be derived that $B_4 = 0$. Hence, $X_3'' = B_3 A_3''$ and $B_3^\top \in \mathrm{St}(r_2,r_2')$. ◻ The two equalities in enable to deduce the TT-rank of $A$---and thus a value of $(k_1, k_2)$ for which the optimum of LRTAP [\[eq:low-rank_approx\]](#eq:low-rank_approx){reference-type="eqref" reference="eq:low-rank_approx"} is zero---from any stationary point of $\min_{X \in \mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}} f(X)$. **Proposition 2**. *If the same conditions as in hold, then $\nabla f(X) \in T_{X}\mathbb{R}_{\le (r_1', r_2')}^{n_1 \times n_2 \times n_3}$, and $$\begin{split} &\left( \mathop{\mathrm{rank}}\left( \left( \nabla f(X) \cdot X_3''^\top \right)^\mathrm{L} \right),\mathop{\mathrm{rank}}\left( \left( X_1'^\top \cdot \nabla f(X) \right)^R \right)\right) \\ &= \left( r'_1-r_1, r'_2-r_2 \right). \end{split}$$* *Proof.* By decomposing $\nabla f(X)$ as $Y$ in [\[eq:Y_6terms\]](#eq:Y_6terms){reference-type="eqref" reference="eq:Y_6terms"} and by setting $\dot{W}_1$, $\tilde{W}_2$, and $\hat{W}_3$ in [\[eq:param_Y\_TC\]](#eq:param_Y_TC){reference-type="eqref" reference="eq:param_Y_TC"} to zero because $\mathcal{P}_{T_X \mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}} \nabla f(X) = 0$, we obtain $$\begin{split} &\nabla f(X) =~ P_{X_1'}^\perp \cdot \nabla f(X) \cdot P_{X_3''^\top}^\perp \\ + ~ & X_1' \cdot \Big[ P_{X_2'^{\mathrm{R}}}^\perp \left( X_1'^\top \cdot \nabla f(X) \right)^{\mathrm{R}}\Big]^{r_1 \times n_2 \times n_3} \cdot P_{X_3''^\top}^\perp \\ + ~ & P_{X_1'}^\perp \cdot \Big[ \left( \nabla f(X) \cdot X_3''^\top \right)^\mathrm{L} P_{\left( X_2''^{\mathrm{L}}\right)^\top}^\perp \Big]^{n_1 \times n_2 \times r_2} \cdot X_3''. \end{split}$$ Multiplying both sides on the right by $X_3''^\top$, we obtain $$\begin{split} & \nabla f(X) \cdot X_3''^\top =\\ & ~ P_{X_1'}^\perp \cdot \Big[ \left( \nabla f(X) \cdot X_3''^\top \right)^\mathrm{L} P_{\left( X_2''^{\mathrm{L}}\right)^\top}^\perp \Big]^{n_1 \times n_2 \times r_2} = U_1 \cdot \dot{V}_2, \end{split}$$ with $U_1$ and $\dot{V}_2$ as in [\[eq:param_Y\_TC\]](#eq:param_Y_TC){reference-type="eqref" reference="eq:param_Y_TC"} with $Y$ replaced by $\nabla f(X)$. Furthermore, from , we know that $X_1' = A_1'B_1$, $X_3'' = B_3 A_3''$, and $X_2=B_1^\top \cdot A_2 \cdot B_3^\top$, for some $B_1 \in \mathrm{St}(r_1,r_1')$ and $B_3^\top \in \mathrm{St}(r_2,r_2')$. Therefore, $$\begin{split} &\mathop{\mathrm{rank}}\left( \left( \nabla f(X) \cdot X_3''^\top \right)^\mathrm{L} \right) \\ &= \mathop{\mathrm{rank}}\left( \left( X_1' \cdot X_2 - A_1' \cdot A_2 \cdot A_3'' \cdot X_3''^\top \right)^\mathrm{L} \right) \\ &= \mathop{\mathrm{rank}}\left( \left( A_1' B_1 \cdot B_1^\top \cdot A_2 \cdot B_3^\top - A_1' \cdot A_2 \cdot B_3^\top \right)^\mathrm{L} \right) \\ &= \mathop{\mathrm{rank}}\left( A_1' \left( B_1 B_1^\top - I_{r_1'} \right) \left( A_2 \cdot B_3^\top \right) ^\mathrm{L} \right) \\ &= \mathrm{rank} \left( I_{r'_1} - B_1 B_1^\top \right) = r'_1 - r_1 \end{split}$$ because $\left( A_2 \cdot B_3^\top \right)^\mathrm{L}$ has full rank $r_1'$, knowing that $X_2^\mathrm{L} = B_1^\top \left(A_2 \cdot B_3^\top \right)^\mathrm{L}$ has rank $r_1$ and using the Sylvester rank inequality. Thus, $\mathop{\mathrm{rank}}\left( U_1 \dot{V}_2^{\mathrm{L}} \right) = r'_1 - r_1$. A similar derivation can be made for $\left( X_1'^\top \cdot \nabla f(X) \right)^\mathrm{R} = U_2^\mathrm{R} V_3$. Hence, $\nabla f(X)$ can be parameterized as in [\[eq:paran_g\_final\]](#eq:paran_g_final){reference-type="eqref" reference="eq:paran_g_final"} with $s_1= r'_1-r_1$, $s_2 = r'_2-r_2)$, and thus by definition $\nabla f(X) \in T_{X} \mathbb{R}_{\le (r_1', r_2')}^{n_1 \times n_2 \times n_3}$. ◻ We propose to exploit the two equalities from in the context of LRTCP [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"} by using the *estimated rank* of $B \in \mathbb{R}^{n \times m}$ which is inspired by [@gao2022riemannian] and defined as: $$\label{eq:num_rank_increase} \begin{split} &\tilde{r}_s \left( B \right) := \begin{cases} 0 & \text{if } B = 0, \\ \mathop{\mathrm{argmax}}_{\substack{j\le s}} \frac{\sigma_j\left( B\right) - \sigma_{j+1}\left( B \right)}{\sigma_j \left( B\right)}& \text{otherwise}, \end{cases} \end{split}$$ where $\sigma_{j}\left( B \right)$, $j= 1 \dots \mathop{\mathrm{rank}}(B)$, denote the singular values of $B$ in decreasing order, i.e., $\sigma_i(B) \geq \sigma_j(B)$ for $i \leq j$, and $s < \mathrm{rank} \left( B \right)$. The upper bound $s$ prevents the estimated rank from being too high and should be chosen by the user. Thus, we propose $\left(r_1 + \tilde{r}_s \big( \left( \nabla f_{\Omega}(X) \cdot X_3''^\top \right)^\mathrm{L} \big), r_2 + \tilde{r}_s \big( \left( X_1'^\top \cdot \nabla f_{\Omega}(X) \right)^R \big)\right)$ as an adequate value for $(k_1,k_2)$ in [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"}, where $X=X_1' \cdot X_2 \cdot X_3''$ has been obtained by running a Riemannian optimization algorithm on $$\label{eq:min_completion_TT_fixed} \min_{X \in \mathbb{R}_{ (r_1, r_2)}^{n_1 \times n_2 \times n_3}} {\frac{1}{2} \lVert X_{\Omega} - A_{\Omega} \rVert^2}.$$ # Experiments {#sec:experiments} In this section, three experiments are generated where we have optimized [\[eq:min_completion_TT_fixed\]](#eq:min_completion_TT_fixed){reference-type="eqref" reference="eq:min_completion_TT_fixed"} for $(r_1,r_2) := (2,2)$, $n_1:=n_2:=n_3:=100$, and $(r_1',r_2'):= (6,6)$, using a Riemannian conjugate gradient (CG) algorithm [@Steinl_high_dim_TT_compl_2016; @manopt]. The tensor $A$ and the starting point $X_0$ given to the optimization algorithm are generated as follows: $$\label{eq:A_randn} \begin{split} A = \mbox{\tt{randn}} \left( n_1,r_1' \right) \cdot \mbox{\tt{randn}} \left( r_1', n_2, r_2' \right) \cdot \mbox{\tt{randn}} \left( r_2', n_3 \right), \\ X_0 = \mbox{\tt{randn}} \left( n_1,r_1 \right) \cdot \mbox{\tt{randn}} \left( r_1, n_2, r_2 \right) \cdot \mbox{\tt{randn}} \left( r_2, n_3 \right), \end{split}$$ where $\mbox{\tt{randn}}$ is a built-in [Matlab]{.smallcaps} function to generate pseudo-random numbers. It can be shown that the elements of $A$, generated in this way, have standard deviation $\sqrt{r'_1 r'_2}=6$. To obtain $A_\Omega$, $4 \cdot 10^4$ random samples of this tensor were generated. An illustration of how the estimated rank of $\left( \nabla f_{\Omega }(X) \cdot X_3''^\top \right)^\mathrm{L}$ can be used to estimate a good value for $(k_1,k_2)$ in [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"} is given in . The first $20$ singular values of $\left( \nabla f_{\Omega }(X) \cdot X_3''^\top \right)^\mathrm{L}$ are shown in the left upper subfigure. The squared norm of the Riemannian gradient that is obtained at $X$ is approximately $10^{-8}$. There were 200 iterations needed to obtain this accuracy. Based on the upper right subfigure, where the relative gap between the singular values is shown, it can be seen that the estimated rank equals $r'_1 - r_1$. In the lower two subfigures, the first $20$ singular values of $\nabla f_\Omega (X)^\mathrm{L}$ are shown. The estimated rank of $\nabla f_\Omega (X)^\mathrm{L}$ equals $3$, and thus cannot be used to estimate the rank of $A$. {#fig:svd_grad_Omega_vs_proj_rA6_6_local_opt_ni_100_Omega_40000 width="\\linewidth"} In , it is shown that in practice the norm of the Riemannian gradient does not need to be very small for the estimated rank of $\left( \nabla f_{\Omega }(X) \cdot X_3''^\top \right)^\mathrm{L}$ to equal $r'_1 - r_1$. In this experiment, only 10 iterations of the Riemannian CG algorithm where used, such that the squared norm of the Riemannian gradient was approximately $504$. However the estimated rank still equals $r_1'-r_1$. {#fig:svd_grad_proj_rx2_n100_3D_r6_Omega_4_10+4_gradR2_10_60it width="\\linewidth"} In a last experiment, another advantage of the proposed method is illustrated. For this experiment noise with $\eta = 10$ is added to the low-rank tensor as follows: $$\label{eq:A_noise} \begin{split} A_{\eta} = A + \eta \mbox{\tt{ randn}} \left( n_1, n_2, n_3 \right). \end{split}$$ This means that the noise has the same magnitude as $A$ but the estimated rank of $\left( \nabla f_{\Omega }(X) \cdot X_3''^\top \right)^\mathrm{L}$ still equals $r_1'-r_1=4$ after 120 iterations of the CG algorithm, as shown in . The squared norm of the Riemannian gradient equals 0.9. ![The first 20 singular values of $\left( \nabla f_{\Omega }(X) \cdot X_3''^\top \right)^\mathrm{L}$ (left) and their relative gap (right), for $\lVert \mathcal{P}_{T_X \mathbb{R}_{(r_1,r_2)}^{n_1 \times n_2 \times n_3}} \nabla f_{\Omega}(X) \rVert^2 = 0.9$, obtained after 120 iterations, and with noise added to the data as in [\[eq:A_noise\]](#eq:A_noise){reference-type="eqref" reference="eq:A_noise"} with $\eta = 10$.](Figures/svd_gradL_vs_gradL_X3T_noise_10_n100_rA_6_r_2_Omega_4000_100it_ngradr2_09.eps){#fig:svd_grad_proj_rx2_n100_3D_r6_Omega_gradR2_100_20it_Omega_8_10+4_noise_10+0 width="\\linewidth"} # Conclusion The two equalities given in enable to compute the TT-rank of $A$ based on a stationary point of LRTAP on the fixed-rank manifold, i.e., $\min_{X \in \mathbb{R}_{(r_1, r_2)}^{n_1 \times n_2 \times n_3}} \frac{1}{2}\|X-A\|^2$, which can be obtained using classic Riemannian optimization. Moreover, numerical experiments indicate that, for LRTCP [\[eq:min_completion_TT\]](#eq:min_completion_TT){reference-type="eqref" reference="eq:min_completion_TT"}, using these equalities with the rank replaced by the estimated rank [\[eq:num_rank_increase\]](#eq:num_rank_increase){reference-type="eqref" reference="eq:num_rank_increase"} provides a plausible estimation of the TT-rank of $A$ which can be used as an adequate value for $(k_1, k_2)$. We are working on a Riemannian rank-adaptive method using this rank estimation method on the LRTCP and additionally on an extension of this method to higher dimensions. # Acknowledgment {#acknowledgment .unnumbered} This work was supported by the Fonds de la Recherche Scientifique -- FNRS and the Fonds Wetenschappelijk Onderzoek -- Vlaanderen under EOS Project no 30468160, and by the Fonds de la Recherche Scientifique -- FNRS under Grant no T.0001.23.

arxiv_math

--- abstract: | A Ramanujan-type series satisfies $$\frac{1}{\pi} = \sum_{n=0}^{\infty} \frac{\left( \frac{1}{2} \right)_{n} \left( \frac{1}{s} \right)_{n} \left(1 - \frac{1}{s} \right)_{n} }{ \left( 1 \right)_{n}^{3} } z^{n} (a + b n),$$ where $s \in \{ 2, 3, 4, 6 \}$, and where $a$, $b$, and $z$ are real algebraic numbers. The level $3$ case whereby $s = 3$ has been considered as the most mysterious and the most challenging, out of all possible values for $s$, and this motivates the development of new techniques for constructing Ramanujan-type series of level $3$. Chan and Liaw introduced an alternating analogue of the Borwein brothers' identity for Ramanujan-type series of level $3$; subsequently, Chan, Liaw, and Tian formulated another proof of the Chan--Liaw identity, via the use of Ramanujan's class invariant. Using the elliptic lambda function and the elliptic alpha function, we prove, using a limiting case of the Kummer--Goursat transformation, a new identity for evaluating $z$, $a$, and $b$ for Ramanujan-type series such that $s = 3$ and $z < 0$, and we apply this new identity to prove three conjectured formulas for quadratic-irrational, Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud. We also apply our identity to prove a new Ramanujan-type series of level $3$ with quartic values for $z < 0$, $a$, and $b$. address: Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 4R2, Canada author: - John M. Campbell title: Proofs of conjectures on Ramanujan-type series of level 3 --- # Introduction {#sectionIntroduction} In his first letter to Hardy in January 1913, Ramanujan [@BR1995 p. 25, Sect. (5), eq. (3)] presented the following remarkable series for $1/\pi$: $$\label{slowestRamanujan} \frac{1}{\pi} = \sum_{n = 0}^{\infty} \frac{ \left( \frac{1}{2} \right)_{n}^3 }{ \left( 1 \right)_{n}^3} (-1)^{n} \big(\tfrac{1}{2}+2n\big).$$ Soon after, what are referred to as Ramanujan's seventeen series for $1/\pi$ were introduced in [@Ramanujan1914] (see also [@Ramanujan2000 pp. 36--38]), and have gone on, over the decades, to have a great influence on many different areas in mathematics. This motivates the development of mathematical topics based on the application of generalizations and variants of Ramanujan's seventeen series. Formally, we say a *Ramanujan-type series* is of the form $$\label{maindefinition} \frac{1}{\pi} = \sum_{n = 0}^{\infty} \frac{\left( \frac{1}{2} \right)_{n} \left( \frac{1}{s} \right)_{n} \left(1 - \frac{1}{s} \right)_{n} }{ \left( 1 \right)_{n}^{3} } z^n (a + b n),$$ where $s \in \{ 2, 3, 4, 6 \}$ and where the parameters $a$, $b$, and $z$ are real algebraic numbers. This definition broadly agrees with relevant references that have inspired our work, as in [@CooperZudilin2019; @Guillera2020; @Guillera2013; @Guillera2006Exp; @Guillera2021; @Guillera2019; @Wan2014]. Historically, Ramanujan-type series may be classified by their *level* $\ell$, under the mapping $\ell:=4\sin^2\frac{\pi}{s}$, so as to reach a correspondence with the levels of the modular forms that parameterize Ramanujan-type series. See, for instance, [@Guillera2021]. The work of Guillera [@Guillerafastest2021] on the level 3 case of [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"} has led us to investigate new techniques for proving closed-form evaluations, in the $s = 3$ case of [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"}, for the expressions $z$, $a$, and $b$ in [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"}. The alternative bases and modular relations associated with the level $3$ case of [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"} have been considered as the most important [@BBG1995] and most interesting [@ChanGeeTan2003] out of all possible levels for Ramanujan-type series, which motivates the results and techniques we introduce concerning Ramanujan-type series of level $3$. Subsequent to the seminal monograph *Pi and the AGM* [@BorweinBorwein1987text] by the Borwein brothers and to further groundbreaking works by the Borweins on Ramanujan-type series [@BorweinBorwein1993; @BorweinBorwein1988; @BorweinBorwein1987rational; @BorweinBorwein1992], an especially notable reference that introduced a large number of Ramanujan-type series is due to Baruah and Berndt [@BaruahBerndt2010], who derived their results using techniques based on Eisenstein series. In [@BaruahBerndt2010], Baruah and Berndt's Ramanujan-type formulas are restricted to the scenarios where $s$ is among $\{ 2, 4, 6 \}$. This greatly motivates the problem as to how new Ramanujan-type series in the vein of the results from [@BaruahBerndt2010], in which the $z$- and $a$- and $b$-values in [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"} are often quadratic or quartic, could be determined for the $s = 3$ case. This forms a main purpose of our article, in which we: 1. Prove a new identity, in terms of the elliptic lambda and elliptic alpha functions, for evaluating $z$, $a$, and $b$ in [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"}, for the case whereby $s = 3$ and $z < 0$, with our techniques being fundamentally different compared to the work of Chan and Liaw [@ChanLiaw2000] and of Chan, Liaw, and Tan [@ChanLiawTan2001] on the $s = 3$ and $z < 0$ case of [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"}; 2. Apply our identity for evaluating $z < 0$, $a$, and $b$ to prove three conjectured formulas on quadratic-irrational, Ramanujan-type series that had been discovered experimentally via numerical experiments with Maple by Aldawoud in 2012 [@Aldawoud2012]; and 3. Apply our identity to prove an evaluation for a new Ramanujan-type series of level $3$, with quartic values for $z < 0$, $a$, and $b$. # Background and preliminaries The *Pochhammer symbol* is defined by $$\begin{aligned} (x)_n:=\begin{cases} 1, & n=0,\\ x(x+1) \cdots (x + n - 1), & n\in\mathbb{N}. \end{cases}\end{aligned}$$ We let ${}_{2}F_{1}$-series be denoted and defined so that $${}_{2}F_{1}\!\!\left[ \begin{matrix} a, b \vspace{1mm}\\ c \end{matrix} \ \Bigg| \ x \right] = \sum_{n=0}^{\infty} \frac{ \left( a \right)_{n} \left( b \right)_{n} }{ \left( c \right)_{n} } \frac{x^n}{n!}.$$ A cubic series due to the Borwein brothers was explored by Chan and Liaw in [@ChanLiaw2000], and what may be regarded as an alternating analogue of this cubic series due to Chan and Liaw [@ChanLiaw2000] was reproved in [@ChanLiawTan2001]. Following [@ChanLiawTan2001], we set $$\label{mquotient} m = \frac{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ \alpha \right] }{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ \beta \right]},$$ and we are letting $\alpha$ and $\beta$ be such that $$\frac{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ 1 - \beta \right] }{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ \beta \right] } = n \frac{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ 1 - \alpha \right] }{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ \alpha \right] }.$$ In something of a similar fashion, the *cubic singular modulus* is the unique value $\alpha_{n}$ such that $$\label{cubicalpha} \frac{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ 1 - \alpha_{n} \right] }{ {}_{2}F_{1}\!\!\left[ \begin{matrix} \frac{1}{3}, \frac{2}{3} \vspace{1mm}\\ 1 \end{matrix} \ \Bigg| \ \alpha_{n} \right] } = \sqrt{n}.$$ The Borwein brothers' family of Ramanujan-type series of level 3 is such that the convergence rate $\mathcal{H}_{n}$ satisfies $\mathcal{H}_{n} = 4 \alpha_{n}(1- \alpha_{n})$, referring to [@ChanLiawTan2001] for details. From [\[cubicalpha\]](#cubicalpha){reference-type="eqref" reference="cubicalpha"}, this convergence rate is necessarily positive for $n > 0$. This motivates us to determine an explicit way of evaluating the $z$- and $a$- and $b$-expressions in [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"} for the $s = 3$ and $z < 0$ case, and with the use of the elliptic lambda and elliptic alpha functions given in the *Pi and the AGM* text [@BorweinBorwein1987text]. The infinite family of Ramanujan-type series under consideration in [@ChanLiawTan2001] is such that $$\label{CLTfamily} \frac{1}{2\pi} \sqrt{\frac{3}{n}} \sum_{k=0}^{\infty} (a_{n} + b_{n} k) \frac{ \left( \frac{1}{2} \right)_{k} \left( \frac{1}{3} \right)_{k} \left( \frac{2}{3} \right)_{k} }{ \left( k! \right)^{3} } \mathcal{H}_{n}^{k},$$ where $$\label{bada} a_{n} = \frac{ \alpha_{n} (1-\alpha_{n}) }{\sqrt{n}} \frac{dm}{d\alpha}(1 - \alpha_{n}, \alpha_{n})$$ and $b_{n} = 1 - 2 \alpha_{n}$ and $\mathcal{H}_{n} = 4 \alpha_{n} (1 - \alpha_{n})$; see also [@Guillerafastest2021] for a family of Ramanujan-type series that is similarly formulated in terms of a transformation of modular origin, as in the modular function in [\[mquotient\]](#mquotient){reference-type="eqref" reference="mquotient"}. This family of Ramanujan-type series given by Guillera [@Guillerafastest2021] was derived using material due to Guillera in [@Guillera2020] and Wan in [@Wan2014]. As expressed in [@ChanLiaw2000], the evaluation of expressions as in [\[bada\]](#bada){reference-type="eqref" reference="bada"} is very nontrivial. The main difference between previously known formulas for evaluating summands as in [\[CLTfamily\]](#CLTfamily){reference-type="eqref" reference="CLTfamily"} and our new technique for determining closed forms for $z$, $a$, and $b$ in [\[maindefinition\]](#maindefinition){reference-type="eqref" reference="maindefinition"} for the case such that $z < 0$ and $s = 3$ is given by how we do not require the derivative of the $m$-function shown in [\[bada\]](#bada){reference-type="eqref" reference="bada"}. This has an advantage given by how we may evaluate the equivalent of the $a_{n}$-expression in [\[bada\]](#bada){reference-type="eqref" reference="bada"} in a more explicit way in terms of the elliptic alpha function introduced by the Borwein brothers [@BorweinBorwein1987text], and this, in turn, has the advantage of allowing us to apply previously known values for the elliptic alpha function. Furthermore, our method gives us a more explicit way of expressing the cubic singular modulus in [\[cubicalpha\]](#cubicalpha){reference-type="eqref" reference="cubicalpha"}, and in a way that allows us to apply previously known closed forms for the elliptic lambda function $\lambda^{\ast}$. This leads us toward our review, as below, of the complete elliptic integrals and of the elliptic lambda and elliptic alpha functions. The *complete elliptic integrals* of the first and second kinds [@BorweinBorwein1987text Sect. 1.3] are, respectively, defined by $$\begin{aligned} \mathbf{K}(k) := \int_{0}^{\pi/2} \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}} \ \ \ \text{and} \ \ \ \mathbf{E}(k) := \int_{0}^{\pi/2} \sqrt{1 - k^2 \sin^2 \theta} \, d\theta. \end{aligned}$$ The argument $k$ indicated above is referred to as the *modulus*, and it is often convenient to write $k' := \sqrt{1-k^2}$ to denote the *complementary modulus*, so that we may write, accordingly, $\mathbf{K}'(k) := \mathbf{K}(k')$ and $\mathbf{E}'(k) := \mathbf{E}(k')$. By virtue of *Jacobi's imaginary quadratic transformations*, the two complete elliptic integrals satisfy the following modular equations [@BorweinBorwein1987text p. 72, Exs. 7--8]: $$\begin{aligned} \label{maincomplex} \mathbf{K}\left(i\frac{k}{k'}\right) = k'\mathbf{K}(k) \ \ \ \text{and} \ \ \ \mathbf{E}\left(i\frac{k}{k'}\right) = \frac{1}{k'}\mathbf{E}(k). \end{aligned}$$ Furthermore, the differential equation such that $$\label{mainODE} \frac{d\mathbf{K}}{dk} = \frac{ \mathbf{E} - \left( k' \right)^{2} \mathbf{K} }{ k \left( k' \right)^{2} }$$ is to be used in a key way in our work. Next, for a positive argument $r$, the *elliptic lambda function* $\lambda^{\ast}(r)$ is defined so that $\lambda^{\ast}(r):= k_r$, where $0 < k_{r} < 1$ is such that $\frac{ \mathbf{K}' }{\mathbf{K}}(k_{r}) = \sqrt{r}$ [@BorweinBorwein1987text p. 67, eq. (3.2.2)]. The function $\lambda^{\ast}$ may equivalently be defined through the use of Ramanujan's $G$-function [@Ramanujan1914] such that $$\label{202310067777372787A7M1A} \prod_{k=0}^{\infty} \left( 1 + e^{-(2k+1) \pi \sqrt{n}} \right) = 2^{1/4} e^{-\pi \sqrt{n}/24} G_{n},$$ and the $G$-function in [\[202310067777372787A7M1A\]](#202310067777372787A7M1A){reference-type="eqref" reference="202310067777372787A7M1A"} is to be heavily involved in our proofs of Aldawoud's conjectures. The *elliptic alpha function* is such that $\alpha(r) := \frac{ \mathbf{E}' }{\mathbf{K}} - \frac{\pi}{4 \mathbf{K}^2}$ [@BorweinBorwein1987text p. 152, eq. (5.1.1)], under the modulus $k = \lambda^{\ast}(r)$, and we may rewrite this as [@BorweinBorwein1987text p. 152, eq. (5.1.2)]: $$\label{mainalpha} \alpha(r) = \frac{\pi}{4 \mathbf{K}^{2}} - \sqrt{r} \left( \frac{\mathbf{E}}{\mathbf{K}} - 1 \right).$$ We adopt a notational convention from [@BaruahBerndt2010] by setting $$\begin{aligned} A_n := \frac{ \left( \frac{1}{2} \right)_{n}^3 }{ \left( 1 \right)_{n}^3},\qquad B_n:=\frac{ \left( \frac{1}{4} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{3}{4} \right)_{n} }{ \left( 1 \right)_{n}^3 },\qquad C_n:=\frac{ \left( \frac{1}{6} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{5}{6} \right)_{n} }{ \left( 1 \right)_{n}^3 }.\end{aligned}$$ We also set $D_n:=\frac{ \left( \frac{1}{3} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{2}{3} \right)_{n} }{ \left( 1 \right)_{n}^3 }$. A hypergeometric identity of fundamental importance in the study of Ramanujan-type series is such that $$\sum_{n=0}^{\infty} A_{n} x^{n} = \frac{4 \mathbf{K}^2 \left( \sqrt{\frac{1 - \sqrt{1-x}}{2}} \right)}{\pi ^2}.$$ This relation is highlighted in a different form in [@BorweinBorwein1987text p. 180, Theorem 5.7(a), eq. (i)]. The generating function of $B_n$ is of great importance in the quartic theory of elliptic functions, and we recall the following equivalent formulation of [@BorweinBorwein1987text p. 181, Theorem 5.7(b), eq. (iv)]: $$\label{Bgenerating} \sum_{n=0}^{\infty} B_{n} x^{n} = \frac{4 \sqrt{2}\, \mathbf{K}^2\left(\sqrt{\frac{1}{2} - \frac{\sqrt{\frac{1-\sqrt{1-x}}{x}}}{\sqrt{2}}}\right)}{\pi ^2 \sqrt[4]{2 \sqrt{1-x}+2-x}}.$$ According to [@BorweinBorwein1987text p. 181, Theorem 5.7(c), eq. (vi)], *Bailey's cubic transformation* can be stated as $$\label{Baileycubic} \sum_{n=0}^{\infty} A_{n} x^{n} = \frac{2}{\sqrt{4-x}} \sum_{n=0}^{\infty} C_{n} \left( \frac{27 x^2}{(4-x)^3} \right)^{n}.$$ Another specialization of Bailey's transformation [@BorweinBorwein1987text p. 181, eq. (5.5.9)] tells us that $$\label{Baileycubicanother} \sum_{n=0}^{\infty} A_{n} x^{n} = \frac{1}{\sqrt{1-4x}} \sum_{n=0}^{\infty} C_{n} \left( -\frac{27 x}{(1-4x)^3} \right)^{n}.$$ Furthermore, a limiting case of the *Kummer--Goursat transform*, as given by squaring [@Vid2009 eq. (25)], provides a connection between the generating functions of $C_n$ and $D_n$: $$\begin{aligned} \sum_{n=0}^{\infty} D_n \big(4 y (1 - y)\big)^{n} = \frac{3}{\sqrt{9 - 8 y}} \sum_{n=0}^{\infty} C_n \left( \frac{ 64 y^3(1-y) }{(9 - 8 y)^3} \right)^{n}. \label{20230219259AM2A} \end{aligned}$$ # New applications of modular relations {#sectionmodular} Our main identity, which we are to apply in Section [3.1](#subsectionProofs){reference-type="ref" reference="subsectionProofs"} to prove conjectures due to Aldawoud [@Aldawoud2012], and which we are to apply in Section [3.2](#subsectionnewRamanujan){reference-type="ref" reference="subsectionnewRamanujan"} to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients, is given as Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} below. The key relation in [\[mainobstacle\]](#mainobstacle){reference-type="eqref" reference="mainobstacle"} can be shown to be equivalent to the relation $$\frac{ (9 - 8 \alpha_{n})^{3} }{ 64 \alpha_{n}^{3}(1 - \alpha_{n}) } = \frac{ ( 4 G_{3n}^{24} - 1 )^3 }{ 27 G_{3n}^{24} }$$ used by the Borwein brothers for evaluating $\alpha_{n}$ and later considered by Chan and Liaw in [@ChanLiaw2000]. Our formula for the $a$-value given in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} below appears to be new, and this is to be applied to solve open problems given by Aldawoud in [@Aldawoud2012]. **Theorem 1**. *We set $$\label{xinmain} x = 4 \left( \left(\lambda^{\ast}(r)\right)^2 - \left(\lambda^{\ast}(r)\right)^4\right).$$ We further set $y$ as the unique solution in $(-\frac{1}{2},0)$, if such a solution exists, such that $$\label{mainobstacle} \frac{64 (y-1) y^3}{(8 y-9)^3} = -\frac{27 x}{(1-4 x)^3}.$$ Then the following Ramanujan-type series holds true: $$\label{2999909293919090969390989A9M91A} \frac{1}{\pi} = \sum_{n = 0}^{\infty} \frac{ \left( \frac{1}{3} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{2}{3} \right)_{n} }{ \left( 1 \right)_{n}^3} z^{n} (a+bn),$$ if we choose $$\begin{aligned} z &=4 y(1-y), \nonumber \\ a &= \frac{\sqrt{9-8 y}} {6 \sqrt{(1 - 4x)^{3}} \big(27 - 36 y+8 y^2\big)}\cdot \Big[ 2 \big(1-4x\big) \big(27 - 36 y + 8 y^2\big) \cdot\alpha (r) \nonumber \\ &\quad+ 4x\Big(\big(27 - 36 y+8 y^2\big) -\big(16y-16y^2\big)\sqrt{1 - x}\Big)\cdot \sqrt{r} \nonumber \\ &\quad- \Big(\big(27 - 36 y + 8 y^2\big)-\big(27 - 44 y + 16 y^2\big)\sqrt{1-x}\Big)\cdot \sqrt{r}\Big], \nonumber \\ b & = \frac{1}{3} \sqrt{r} (1-2 y), \label{20231006311000000000000000000005AM1A} \end{aligned}$$ provided that $|z|<1$ (cf. [@BorweinBorwein1987text pp. 185--186]) .* *Proof.* For $x$ as in [\[xinmain\]](#xinmain){reference-type="eqref" reference="xinmain"}, we may obtain, from [\[Baileycubicanother\]](#Baileycubicanother){reference-type="eqref" reference="Baileycubicanother"}, that $$\sum_{n=0}^{\infty} C_{n} \left( -\frac{27x}{(1-4x)^3} \right)^{n} = \frac{ 4 \text{{\bf K}}^{2}\left( \lambda^{\ast}(r) \right) }{\pi^2} \sqrt{1-4x}$$ and, via [\[mainODE\]](#mainODE){reference-type="eqref" reference="mainODE"}, that $$\begin{aligned} & \sum_{n=0}^{\infty} C_{n} \left( -\frac{27x}{(1-4x)^3} \right)^{n} n = \frac{ \text{{\bf K}}( \lambda^{\ast}(r) }{\pi^2} \frac{2 \sqrt{1-4 x} ) }{ \sqrt{1-x} (8 x+1)} \\ & \big( (2-8 x) \text{{\bf E}}( \lambda^{\ast} (r)) - \left(-4 x+\sqrt{1-x}+1\right) \text{{\bf K}}( \lambda^{\ast} (r)) \big). \end{aligned}$$ For $y$ satisfying [\[mainobstacle\]](#mainobstacle){reference-type="eqref" reference="mainobstacle"} in the specified range, we may obtain, from [\[20230219259AM2A\]](#20230219259AM2A){reference-type="eqref" reference="20230219259AM2A"}, that $$\label{202307216217A7M71A} \sum_{n=0}^{\infty} D_{n} \left( 4(1-y)y \right)^{n} = \frac{12 \sqrt{1-4 x}}{ \sqrt{9-8 y}} \frac{ \text{{\bf K}}^{2}\left( \lambda^{\ast}(r) \right)}{\pi^2},$$ and, via [\[mainODE\]](#mainODE){reference-type="eqref" reference="mainODE"}, that $$\begin{aligned} & \sum_{n=0}^{\infty} (4(1-y)y)^{n} D_{n} n = \frac{48 \sqrt{1-4 x} (y-1) y}{ (9-8 y)^{3/2} (2 y-1)} \frac{ \text{{\bf K}}^{2}\left( \lambda^{\ast}(r)\right)\textbf{}}{\pi^2} + \\ & \frac{6 \sqrt{1-4 x} (4 y (2 y-9)+27)}{ \sqrt{1-x} (8 x+1) (9-8 y)^{3/2} (2 y-1)} \frac{\text{{\bf K}}( \lambda^{\ast}(r))}{\pi^2} \times \\ & \left((8 x-2) \text{{\bf E}}(\lambda^{\ast}(r))+\left(-4 x+\sqrt{1 -x}+1\right) \text{{\bf K}}(\lambda^{\ast}(r))\right).\end{aligned}$$ By rewriting $\text{{\bf E}}\left( \lambda^{\ast}(r) \right)$ according to the elliptic alpha function identity in [\[mainalpha\]](#mainalpha){reference-type="eqref" reference="mainalpha"}, we may express $\sum_{n=0}^{\infty} (4(1-y)y)^{n} D_{n} n$ as a linear combination of an algebraic multiple of $\frac{1}{\pi}$ and an algebraic multiple of $\frac{\text{{\bf K}}^{2}\left( \lambda^{\ast}(r) \right)}{\pi^2}$. The desired result can then be shown to follow by taking an appropriate linear combination of $\sum_{n=0}^{\infty} (4(1-y)y)^{n} D_{n} n$ and the left-hand side of [\[202307216217A7M71A\]](#202307216217A7M71A){reference-type="eqref" reference="202307216217A7M71A"}. More explicitly, we would obtain that the $b$-value in [\[2999909293919090969390989A9M91A\]](#2999909293919090969390989A9M91A){reference-type="eqref" reference="2999909293919090969390989A9M91A"}, for $x$ and $y$ as specified in [\[xinmain\]](#xinmain){reference-type="eqref" reference="xinmain"} and [\[mainobstacle\]](#mainobstacle){reference-type="eqref" reference="mainobstacle"}, is such that $$\label{20231006316AM1111111111111111A} b = \frac{(1+8x) \sqrt{1-x}\cdot (1-2 y) \sqrt{(9-8 y)^{3}} \cdot \sqrt{r}}{3 \sqrt{(1-4x)^{3}} \big(27 - 36 y+8 y^2\big)},$$ with the right-hand side of [\[20231006316AM1111111111111111A\]](#20231006316AM1111111111111111A){reference-type="eqref" reference="20231006316AM1111111111111111A"} reducing to the specified value in [\[20231006311000000000000000000005AM1A\]](#20231006311000000000000000000005AM1A){reference-type="eqref" reference="20231006311000000000000000000005AM1A"}. ◻ ## Proofs of Aldawoud's conjectures {#subsectionProofs} Aldawoud [@Aldawoud2012 p. 14] provided the formula $$\sqrt{1-108x} \sum_{k=0}^{\infty} \frac{ (3k)! (2k)! }{ \left( k! \right)^{5} } \left( k + \lambda \right) x^{k} = \frac{\sqrt{3/N}}{\pi}$$ as a special case of a formula given as $$\label{20231003777273727A7M1A} \sqrt{1-4ax} \sum_{k=0}^{\infty} \binom{2k}{k} s(k) (k + \lambda) x^{k} = \frac{1}{\rho}$$ in [@Aldawoud2012 p. 10], referring to [@Aldawoud2012] for details. As stated in [@Aldawoud2012 p. 10], the evaluation of the parameter $\lambda$ in [\[20231003777273727A7M1A\]](#20231003777273727A7M1A){reference-type="eqref" reference="20231003777273727A7M1A"} depends on a result due to Chan, Chan, and Lui [@ChanChanLiu2004] whereby $$\label{CCLderivative} \lambda = \frac{x}{2N} \frac{dM}{dx} \Big|_{q = e^{-2\pi/\sqrt{N \ell}}},$$ where $$\label{20231003302A77777777777777777777M772A} M(q) = \frac{Z(q)}{Z\left( q^{N} \right) }$$ and $$\label{ZqAldawoud} Z(q) = \sum_{k=0}^{\infty} h(k) x^{k}(q),$$ referring to [@Aldawoud2012; @ChanChanLiu2004] for details. However, actually finding and proving closed forms for the required values in [\[CCLderivative\]](#CCLderivative){reference-type="eqref" reference="CCLderivative"}--[\[ZqAldawoud\]](#ZqAldawoud){reference-type="eqref" reference="ZqAldawoud"} is, typically, of a very challenging nature. As described in [@Aldawoud2012 §3], conjectured values for $x$ and $\lambda$ were obtained experimentally, via numerical experiments with Maple. In particular, the Maple `identify` command was applied in [@Aldawoud2012 §3] to obtain conjectured closed forms. However, proofs for the experimentally discovered values for $x$ and $\lambda$ are not given in [@Aldawoud2012], and proving these conjectured valuations is very difficult. A conjectured Ramanujan-type series given in Table 3.7 in [@Aldawoud2012 p. 28] is such that $$\label{input19} \sqrt{1 - 108x} \sum_{k=0}^{\infty} \frac{ \left( \frac{1}{3} \right)_{k} \left( \frac{1}{2} \right)_{k} \left( \frac{2}{3} \right)_{k} }{ \left( k! \right)^{3} } (k + \lambda) (108 x)^{k} = \frac{\sqrt{3/N}}{\pi},$$ where $$\begin{aligned} & N = 19, \label{Nequals19} \\ & x = -\frac{4261}{22781250}-\frac{3913}{91125000} \sqrt{19}, \\ & \lambda = \frac{49}{510} + \frac{73}{9690} \sqrt{19}. \label{twoafterN19} \end{aligned}$$ As below, we are to prove that the conjectured Ramanujan-type series in [\[input19\]](#input19){reference-type="eqref" reference="input19"}, for the specified values in [\[Nequals19\]](#Nequals19){reference-type="eqref" reference="Nequals19"}--[\[twoafterN19\]](#twoafterN19){reference-type="eqref" reference="twoafterN19"}, agrees with the $r = 57$ case of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}. A series for $\frac{1}{\pi}$ is said to be *quadratic-irrational* if an algebraic number times this series yields an infinite sum of quadratic irrational numbers [@Aldawoud2012 p. 7]. Given the huge amount of interest in *rational* Ramanujan-type series, which refer to Ramanujan-type series for $\frac{1}{\pi}$ that yield a rational expansion after being multiplied by an algebraic value, this motivates our proofs of Aldawoud's conjectured and experimentally discovered results on quadratic-irrational Ramanujan-type series, which may be seen as a natural "next step" after rational Ramanujan-type series. **Theorem 2**. *Aldawoud's conjectured Ramanujan-type series formula [@Aldawoud2012 p. 28] $$\begin{aligned} \begin{split} \frac{1}{\pi} = & \sum_{n = 0}^{\infty} \frac{ \left( \frac{1}{3} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{2}{3} \right)_{n} }{ \left( 1 \right)_{n}^{3} } \Bigg( \frac{5719+13 \sqrt{19}}{2250} n + \frac{1654+133 \sqrt{19}}{6750} \Bigg) \times \\ & \ \ \ \ \ \left( \frac{-17044-3913 \sqrt{19}}{843750} \right)^{n} \end{split}\label{202310011111111110100000485707PM1A} \end{aligned}$$ holds true.* *Proof.* Our strategy is to make use of the $r = 57$ case of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} together with closed forms for $\lambda^{\ast}(57)$ and $\alpha(57)$. In this regard, the explicit evaluation $$G_{57}^{-6} = \left( \frac{ 3 \sqrt{19} - 13 }{\sqrt{2}} \right) \left( \frac{\sqrt{3}-1}{\sqrt{2}} \right)^{3}$$ is given by Borwein and Borwein in [@BorweinBorwein1987text p. 148]. We thus obtain that $$\label{202302221202024282432AAA} \lambda^{\ast}\left( 57 \right) = \frac{1}{2} \left( \sqrt{1 + G_{57}^{-12}} - \sqrt{1 - G_{57}^{-12}} \right),$$ according to an equivalent definition for the elliptic lambda function [@BorweinBorwein1987text p. 161]. According to Theorem 5.3 from Borwein and Borwein's text [@BorweinBorwein1987text p. 158], we have that $$\label{20777777273717070747874727PM1A} \alpha(p) = \sqrt{p} \frac{1 + \left( \lambda^{\ast}(p) \right)^{2}}{3} - \frac{\sigma(p)}{6},$$ for $\sigma(p) := R_{p}(k', k)$ as defined in [@BorweinBorwein1987text §5.2], with $k := e^{-\pi \sqrt{p}}$. The formula $$\sigma(57) = 3 G_{57}^{-6} \sqrt{ \frac{2 \sqrt{19} + 5 \sqrt{3} }{2} } \left( 5 \sqrt{57} + 13 \sqrt{19} + 49 \sqrt{3} + 19 \right)$$ is given in the Borwein brothers' text [@BorweinBorwein1987text p. 167], so that the closed form obtained from [\[202302221202024282432AAA\]](#202302221202024282432AAA){reference-type="eqref" reference="202302221202024282432AAA"}, in conjunction with the $p = 57$ case of [\[20777777273717070747874727PM1A\]](#20777777273717070747874727PM1A){reference-type="eqref" reference="20777777273717070747874727PM1A"}, give us a closed form for $\alpha(57)$. As in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, we write $x = 4 \big( \big(\lambda^{\ast}(r)\big)^2 - \big(\lambda^{\ast}(r)\big)^4\big)$, and we may verify that the value $$\label{20231006332A11111111222222222222aaAA2M22} y = \frac{1}{750} \left(375-\sqrt{\frac{1}{6} \left(860794+3913 \sqrt{19}\right)}\right)$$ is such that [\[mainobstacle\]](#mainobstacle){reference-type="eqref" reference="mainobstacle"} holds. It is thus a matter of routine to demonstrate how the values of $z$, $a$, and $b$ in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, subject to the closed forms for $\lambda^{\ast}(57)$ and $\alpha(57)$ and $y$, reduce to the corresponding values in [\[202310011111111110100000485707PM1A\]](#202310011111111110100000485707PM1A){reference-type="eqref" reference="202310011111111110100000485707PM1A"}. ◻ **Remark 1**. As a byproduct of our method, the $y$-values involved in our proofs, as in [\[20231006332A11111111222222222222aaAA2M22\]](#20231006332A11111111222222222222aaAA2M22){reference-type="eqref" reference="20231006332A11111111222222222222aaAA2M22"}, give us new, closed-form evaluations for the cubic singular modulus function defined in [\[cubicalpha\]](#cubicalpha){reference-type="eqref" reference="cubicalpha"}. The fourth convergent series conjectured by Aldawoud and listed in Table 3.7 in [@Aldawoud2012 p. 28] is given in an equivalent form in the below Theorem. **Theorem 3**. *Aldawoud's conjectured Ramanujan-type series formula [@Aldawoud2012 p. 28] $$\begin{aligned} \frac{1}{\pi} = & \sum_{n=0}^{\infty} \frac{ \left( \frac{1}{3} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{2}{3} \right)_{n} }{ \left( 1 \right)_{n}^{3} } \left( \frac{217 + 35113 \sqrt{31}}{60750} n + \frac{14662 + 7843 \sqrt{31} }{182250} \right) \times \\ & \ \ \ \ \ \left( \frac{-1368394-245791 \sqrt{31}}{615093750} \right)^{n}\end{aligned}$$ holds true.* *Proof.* Our strategy is to make use of the $r = 93$ case of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}. The explicit evaluation $$\label{G93power} G_{93}^{-6} = \left( \frac{39 - 7 \sqrt{31}}{\sqrt{2}} \right) \left( \frac{\sqrt{31} - 3 \sqrt{3}}{2} \right)^{3/2}$$ is given in Borwein and Borwein's text [@BorweinBorwein1987text p. 148], and the explicit evaluation $$\label{207237177007787170797A7c7o7mm7a77M71A} \sigma(93) = 6 G_{93}^{-6} \left( \frac{\sqrt{3} + 1}{2} \right)^{3} (15 \sqrt{93} + 13 \sqrt{31} + 201 \sqrt{3} + 217)$$ is given in the same text [@BorweinBorwein1987text p. 168]. The closed form in [\[G93power\]](#G93power){reference-type="eqref" reference="G93power"} leads us to the closed form $$\label{20727371707078771718781878A878M81A} \lambda^{\ast}\left( 93 \right) = \frac{1}{2} \left( \sqrt{1 + G_{93}^{-12}} - \sqrt{1 - G_{93}^{-12}} \right),$$ and the closed form in [\[207237177007787170797A7c7o7mm7a77M71A\]](#207237177007787170797A7c7o7mm7a77M71A){reference-type="eqref" reference="207237177007787170797A7c7o7mm7a77M71A"} gives us, according to the $p = 93$ case of [\[20777777273717070747874727PM1A\]](#20777777273717070747874727PM1A){reference-type="eqref" reference="20777777273717070747874727PM1A"}, a closed form for $\alpha(93)$. The closed form in [\[20727371707078771718781878A878M81A\]](#20727371707078771718781878A878M81A){reference-type="eqref" reference="20727371707078771718781878A878M81A"} leads us to a closed form for $x = 4 \left( \left(\lambda^{\ast}(93)\right)^2 - \left(\lambda^{\ast}(93)\right)^4\right)$, and we may verify that $x$ and $$y = \frac{-1368394-245791 \sqrt{31}}{121500 \left(10125+\sqrt{\frac{1}{6} \left(616462144+245791 \sqrt{31}\right)}\right)}$$ satisfy the desired relation in [\[mainobstacle\]](#mainobstacle){reference-type="eqref" reference="mainobstacle"}, with $y$ in the specified interval. So, from the specified values for $r$, $x$, $y$, and $\alpha(r)$, it is a matter of routine to verify, with the use of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, our closed forms for $z$, $a$, and $b$. ◻ The below proof of one of Aldawoud's conjectures from [@Aldawoud2012] is especially remarkable due to the recursive approach used in this proof and due to the extremely nontrivial computations required and due to its "computer proof" nature. **Theorem 4**. *Aldawoud's conjectured Ramanujan-type series formula [@Aldawoud2012 p. 28] $$\begin{aligned} \begin{split} \frac{1}{\pi} = & \sum_{n=0}^{\infty} \frac{ \left( \frac{1}{3} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{2}{3} \right)_{n} }{ \left( 1 \right)_{n}^{3} } \left( \frac{297 \sqrt{3}-91 \sqrt{11}}{64} n + \frac{75 \sqrt{3}-33 \sqrt{11}}{64} \right) \times \\ & \ \ \ \ \ \left( \frac{ 2457 \sqrt{33} - 14121 }{2048} \right)^{n} \end{split}\label{2023180808888175727A7M1A}\end{aligned}$$ holds true.* *Proof.* Our strategy is to use the $r = 99$ case of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}. Let $$\begin{aligned} \label{eq:rho-value} \rho=\tfrac{1}{6}\big({\textstyle{-\sqrt[3]{16}}+\sqrt[3]{38-6\sqrt{33}}+\sqrt[3]{38+6\sqrt{33}}}\big) \end{aligned}$$ be the unique real solution to the cubic equation $$\begin{aligned} \label{eq:rho-cubic} 2\rho^3+2^{\frac{4}{3}}\rho^2-1=0. \end{aligned}$$ Then $$\begin{aligned} \lambda^*(11) &= \sqrt{\tfrac{1-\sqrt{1-4\rho^{12}}}{2}},\label{eq:lambda-11}\\ \alpha(11) &= \tfrac{\left(3-\sqrt{1-4\rho^{12}}\right)\sqrt{11}}{6} - \tfrac{3\rho^6+2\rho^3+2}{3}.\label{eq:alpha-11} \end{aligned}$$ The value for $\lambda^{\ast}(11)$ is given, in an equivalent form in [@BorweinBorwein1987text p. 162]. The value for $\alpha(11)$ is given in an equivalent way in [@BorweinBorwein1987text p. 158]. According to the recursion $$\label{GntoG9n} 9 = \left( 1 + 2 \sqrt{2} \frac{ G_{9n}^{3} }{ G_{n}^{9} } \right) \left( 1 + 2 \sqrt{2} \frac{ G_{n}^{3} }{ G_{9n}^{9} } \right)$$ given in [@BorweinBorwein1987text p. 145] together with the recursion $$\alpha(9r) = s^2(r) \alpha(r) - \frac{ \sqrt{r} (s^2(r) + 2 s(r) - 3) }{2}$$ given in [@BorweinBorwein1987text p. 160], where $$s(r) := \sqrt{1 + 4 \frac{ (k k')^{3/4} }{ (ll')^{1/4} }},$$ and where $l:= \lambda^{\ast}(r)$ and $k:= \lambda^{\ast}(9r)$, we may derive closed forms for $\lambda^{\ast}(99)$ and $\alpha(99)$, as below, letting $$G_{11} = \frac{1}{\sqrt[12]{2} \sqrt[24]{\frac{1}{2} \left(1-\sqrt{1-4 \rho ^{12}}\right)-\frac{1}{4} \left(1-\sqrt{1-4 \rho ^{12}}\right)^2}}.$$ Inputting Solve[(64 (y - 1) y^3)/(8 y - 9)^3 == -((27 x)/(1 - 4 x)^3) , y] into Mathematica, we may verify that the corresponding output exhausts the possible ways of expressing $y$ in terms of $x$, subject to the desired relation in [\[mainobstacle\]](#mainobstacle){reference-type="eqref" reference="mainobstacle"}. Out of all of the possible ways of expressing $y$ in terms of $x$, we may verify that: If $x = 4 \big( \big( \lambda^{\ast}(99) \big)^{2} - \big( \lambda^{\ast}(99) \big)^{4} \big)$, then it follows that $y$ is equal to the value denoted as "`y`" in the Mathematica input shown below. We have that $G_{99}$, according to the recursion in [\[GntoG9n\]](#GntoG9n){reference-type="eqref" reference="GntoG9n"}, is equal to the expression denoted as "`G99`" in the Mathematica input shown below. This allows us to determine a closed form for $$\label{202302221202024282qqq4qqq3q2qAqAA} \lambda^{\ast}\left( 99 \right) = \frac{1}{2} \left( \sqrt{1 + G_{99}^{-12}} - \sqrt{1 - G_{99}^{-12}} \right).$$ Our strategy, at this point, is to show that $4y(1-y)$, for the value of $y$ specified above, is equal to the convergence rate in [\[2023180808888175727A7M1A\]](#2023180808888175727A7M1A){reference-type="eqref" reference="2023180808888175727A7M1A"}. This, would give us an explicit closed form for the $z$-value involved in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, for the $r = 99$ case. So, to show that $$\label{strategyPolynomial} 4 y(1-y) - \frac{2457 \sqrt{33} - 14121}{2048}$$ vanishes, our strategy is to use the `MinimalPolynomial` command in the Wolfram language, i.e., to show that the minimal polynomial for [\[strategyPolynomial\]](#strategyPolynomial){reference-type="eqref" reference="strategyPolynomial"} is given by a polynomial of degree $1$ with $0$ as its constant term. \[Rho] = (-2^(4/3) + (38 - 6*Sqrt[33])^(1/3) + (38 + 6*Sqrt[33])^ (1/3))/6; G11 = 1/(2^(1/12)*(Sqrt[(1 - Sqrt[1 - 4*\[Rho]^12])/2]^2 - Sqrt[(1 - Sqrt[1 - 4*\[Rho]^12])/2]^4)^(1/24)); G99 = (G11^9/Sqrt[2] + Sqrt[G11^2 + G11^10 + G11^18]/Sqrt[2] + Sqrt[-G11^2 - G11^10 + 2*G11^18 - (2*G11^3)/Sqrt[G11^2 + G11^10 + G11^18] + (2*G11^27)/Sqrt[G11^2 + G11^10 + G11^18]]/Sqrt[2])^(1/3); lambda99 = (1/2)*(Sqrt[1 + G99^(-12)] - Sqrt[1 - G99^(-12)]); x = 4*(lambda99^2 - lambda99^4); y = -((1 - 228*x + 48*x^2 - 64*x^3)/(4*(-1 + 4*x)^3)) + (1/2)* Sqrt[-((729*x)/(-1 + 4*x)^3) + (1 - 228*x + 48*x^2 - 64*x^3)^2/(4* (-1 + 4*x)^6) + (243*x)/(-1 + 12*x - 48*x^2 + 64*x^3) + (81*(-x + 12*x^2 - 48*x^3 + 64*x^4))/(2*2^(2/3)*(-1 + 4*x)^3*(x - 51*x^2 + 564*x^3 - 2576*x^4 + 5568*x^5 - 6144*x^6 + 4096*x^7 + Sqrt[x^2 + 6*x^3 - 159*x^4 - 472*x^5 + 11376*x^6 - 2304*x^7 - 383232*x^8 + 1124352*x^9 + 3575808*x^10 - 25821184*x^11 + 55050240*x^12 - 50331648*x^13 + 16777216*x^14])^(1/3)) + (1/(4*2^(1/3)*(-1 + 4*x)^ 3))*27*(x - 51*x^2 + 564*x^3 - 2576*x^4 + 5568*x^5 - 6144*x^6 + 4096*x^7 + Sqrt[x^2 + 6*x^3 - 159*x^4 - 472*x^5 + 11376*x^6 - 2304*x^7 - 383232*x^8 + 1124352*x^9 + 3575808*x^10 - 25821184*x^ 11 + 55050240*x^12 - 50331648*x^13 + 16777216*x^14])^(1/3)] - (1/ 2)*Sqrt[-((729*x)/(-1 + 4*x)^3) + (1 - 228*x + 48*x^2 - 64*x^3)^2/ (2*(-1 + 4*x)^6) - (243*x)/(-1 + 12*x - 48*x^2 + 64*x^3) - (81* (-x + 12*x^2 - 48*x^3 + 64*x^4))/(2*2^(2/3)*(-1 + 4*x)^3*(x - 51* x^2 + 564*x^3 - 2576*x^4 + 5568*x^5 - 6144*x^6 + 4096*x^7 + Sqrt[x^2 + 6*x^3 - 159*x^4 - 472*x^5 + 11376*x^6 - 2304*x^7 - 383232*x^8 + 1124352*x^9 + 3575808*x^10 - 25821184*x^11 + 55050240*x^12 - 50331648*x^13 + 16777216*x^14])^(1/3)) - (1/(4*2^ (1/3)*(-1 + 4*x)^3))*27*(x - 51*x^2 + 564*x^3 - 2576*x^4 + 5568*x^ 5 - 6144*x^6 + 4096*x^7 + Sqrt[x^2 + 6*x^3 - 159*x^4 - 472*x^5 + 11376*x^6 - 2304*x^7 - 383232*x^8 + 1124352*x^9 + 3575808*x^10 - 25821184*x^11 + 55050240*x^12 - 50331648*x^13 + 16777216*x^14])^ (1/3) + ((6561*x)/(-1 + 4*x)^3 + (2916*x*(1 - 228*x + 48*x^2 - 64*x^3))/(-1 + 4*x)^6 - (1 - 228*x + 48*x^2 - 64*x^3)^3/(-1 + 4* x)^9)/(4*Sqrt[-((729*x)/(-1 + 4*x)^3) + (1 - 228*x + 48*x^2 - 64* x^3)^2/(4*(-1 + 4*x)^6) + (243*x)/(-1 + 12*x - 48*x^2 + 64*x^3) + (81*(-x + 12*x^2 - 48*x^3 + 64*x^4))/(2*2^(2/3)*(-1 + 4*x)^3*(x - 51*x^2 + 564*x^3 - 2576*x^4 + 5568*x^5 - 6144*x^6 + 4096*x^7 + Sqrt[x^2 + 6*x^3 - 159*x^4 - 472*x^5 + 11376*x^6 - 2304*x^7 - 383232*x^8 + 1124352*x^9 + 3575808*x^10 - 25821184*x^11 + 55050240*x^12 - 50331648*x^13 + 16777216*x^14])^(1/3)) + (1/(4*2^ (1/3)*(-1 + 4*x)^3))*27*(x - 51*x^2 + 564*x^3 - 2576*x^4 + 5568*x^ 5 - 6144*x^6 + 4096*x^7 + Sqrt[x^2 + 6*x^3 - 159*x^4 - 472*x^5 + 11376*x^6 - 2304*x^7 - 383232*x^8 + 1124352*x^9 + 3575808*x^10 - 25821184*x^11 + 55050240*x^12 - 50331648*x^13 + 16777216*x^14])^ (1/3)])] MinimalPolynomial[4*y*(1 - y) - (2457*Sqrt[33] - 14121)/2048, w] The output gives us that it is a matter of routine to verify the desired vanishing of [\[strategyPolynomial\]](#strategyPolynomial){reference-type="eqref" reference="strategyPolynomial"}. This gives us that $$\label{yafterstrategy} y = \frac{1}{128} \left(155-27 \sqrt{33}\right).$$ Resetting the inputted value for `y` in Mathematica as y = (1/128)*(155 - 27*Sqrt[33]) and inputting r = 99; lambda11 = Sqrt[(1 - Sqrt[1 - 4*\[Rho]^12])/2]; s11 = Sqrt[1 + 4*((lambda99*Sqrt[1 - lambda99^2])^(3/4)/(lambda11* Sqrt[1 - lambda11^2])^(1/4))]; alpha11 = ((3 - Sqrt[1 - 4*\[Rho]^12])*Sqrt[11])/6 - (3*\[Rho]^6 + 2*\[Rho]^3 + 2)/3; alpha99 = s11^2*alpha11 - (Sqrt[11]*(s11^2 + 2*s11 - 3))/2; a = (Sqrt[9 - 8*y]/(6*Sqrt[(1 - 4*x)^3]*(27 - 36*y + 8*y^2)))*(2* (1 - 4*x)*(27 - 36*y + 8*y^2)*alpha99 + 4*x*((27 - 36*y + 8*y^ 2) - (16*y - 16*y^2)*Sqrt[1 - x])*Sqrt[r] - ((27 - 36*y + 8*y^2) - (27 - 44*y + 16*y^2)*Sqrt[1 - x])*Sqrt[r]); MinimalPolynomial[a - (75*Sqrt[3] - 33*Sqrt[11])/64, w] into Mathematica, we find that it is a matter of routine to verify that the $a$-value in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} reduces so that $a = \frac{1}{64} \left(75 \sqrt{3}-33 \sqrt{11}\right)$, as desired. From the closed form for $y$ in [\[yafterstrategy\]](#yafterstrategy){reference-type="eqref" reference="yafterstrategy"}, it is immediate that the desired $b$-value holds, according to the relation $b = \frac{1}{3} \sqrt{r} (1-2 y)$ specified in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}. ◻ ## A new Ramanujan-type series {#subsectionnewRamanujan} To the best of our knowledge, the only previously published and previously proved Ramanujan-type series of level 3 with negative convergence rates are listed below, apart from Aldawoud's conjectures given in [@Aldawoud2012]. The Ramanujan-type series among [\[9290929390929199 727PM1A\]](#9290929390929199 727PM1A){reference-type="eqref" reference="9290929390929199 727PM1A"}--[\[20230302110111112111111P1M1A\]](#20230302110111112111111P1M1A){reference-type="eqref" reference="20230302110111112111111P1M1A"} are due to Chan, Liaw, $\&$ Tan [@ChanLiawTan2001], and the Ramanujan-type series in [\[202303021108P200addM1A\]](#202303021108P200addM1A){reference-type="eqref" reference="202303021108P200addM1A"}--[\[lastknown\]](#lastknown){reference-type="eqref" reference="lastknown"} is due to Berkovich, Chan, $\&$ Schlosser [@BerkovichChanSchlosser2018]. Each of the following results can be shown to be a special case of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}. $$\begin{aligned} & \sum_{n=0}^{\infty} \left( -\frac{9}{16} \right)^{n} \frac{ \left( \frac{1}{2} \right)_{n} \left( \frac{1}{3} \right)_{n} \left( \frac{2}{3} \right)_{n} }{ (n!)^3 } (5n+1) = \frac{4}{\pi \sqrt{3}}, \label{9290929390929199 727PM1A} \\ & \sum_{n=0}^{\infty} \left( -\frac{1}{16} \right)^{n} \frac{ \left( \frac{1}{2} \right)_{n} \left( \frac{1}{3} \right)_{n} \left( \frac{2}{3} \right)_{n} }{(n!)^{3}} (51n+7) = \frac{12 \sqrt{3}}{\pi}, \label{202303021101001001007PM1A} \\ & \sum_{n=0}^{\infty} \left( -\frac{1}{80} \right)^{n} \frac{ \left( \frac{1}{2} \right)_{n} \left( \frac{1}{3} \right)_{n} \left( \frac{2}{3} \right)_{n} }{(n!)^3} (9n+1) = \frac{4 \sqrt{3}}{\pi \sqrt{5}}, \\ & \sum_{n=0}^{\infty} \left( -\frac{1}{1024} \right)^{n} \frac{ \left( \frac{1}{2} \right)_{n} \left( \frac{1}{3} \right)_{n} \left( \frac{2}{3} \right)_{n} }{(n!)^3} (615 n + 53) = \frac{96 \sqrt{3}}{\pi}, \\ & \sum_{n=0}^{\infty} \left( -\frac{1}{3024} \right)^{n} \frac{ \left( \frac{1}{2} \right)_{n} \left( \frac{1}{3} \right)_{n} \left( \frac{2}{3} \right)_{n} }{(n!)^3} (165n+13) = \frac{2^2 \cdot 3^3}{\pi \sqrt{7}}, \\ & \sum_{n=0}^{\infty} \frac{(-1)^n}{500^{2n}} \frac{ \left( \frac{1}{2} \right)_{n} \left( \frac{1}{3} \right)_{n} \left( \frac{2}{3} \right)_{n} }{(n!)^{3}} (14151n + 827) = \frac{1500\sqrt{3}}{\pi}, \label{20230302110111112111111P1M1A} \\ & \sum_{n=0}^{\infty} \left( -\frac{194}{1331} - \frac{225 \sqrt{3}}{2662} \right)^{n} \frac{ \left( \frac{1}{2} \right)_{n} \left( \frac{1}{3} \right)_{n} \left( \frac{2}{3} \right)_{n} }{(n!)^3} \cdot \label{202303021108P200addM1A} \\ & \left( \left( \frac{45 \sqrt{3}}{22} + \frac{5}{22} \right) n + \frac{13}{66} + \frac{3 \sqrt{3}}{11} \right) = \frac{\sqrt{3}}{\pi}. \label{lastknown} \end{aligned}$$ The Berkovich--Chan--Schlosser formula reproduced in [\[202303021108P200addM1A\]](#202303021108P200addM1A){reference-type="eqref" reference="202303021108P200addM1A"}--[\[lastknown\]](#lastknown){reference-type="eqref" reference="lastknown"} [@BerkovichChanSchlosser2018] is the same as the Ramanujan-type series corresponding to the $N = 11$ case of Table 3.7 in [@Aldawoud2012 p. 28]. So, the proof via Wronskians of theta functions of [\[202303021108P200addM1A\]](#202303021108P200addM1A){reference-type="eqref" reference="202303021108P200addM1A"}--[\[lastknown\]](#lastknown){reference-type="eqref" reference="lastknown"} due to Berkovich et al. [@BerkovichChanSchlosser2018] actually solves an open problem from 2012 [@Aldawoud2012], but the authors of [@BerkovichChanSchlosser2018] were unaware of this. The same formula in [\[202303021108P200addM1A\]](#202303021108P200addM1A){reference-type="eqref" reference="202303021108P200addM1A"}--[\[lastknown\]](#lastknown){reference-type="eqref" reference="lastknown"} can also be proved using our main identity given in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, according to the $r = 33$ case, and with the use of known values for $\lambda^{\ast}(33)$ and $\alpha(33)$ that may be obtained from the *Pi and the AGM* text [@BorweinBorwein1987text]. This has led us to consider the $r = 30$ case of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, as below. It appears that the Ramanujan-type series highlighted in Theorem [\[newspecialcasetheorem\]](#newspecialcasetheorem){reference-type="eqref" reference="newspecialcasetheorem"} below is new. **Theorem 5**. *The Ramanujan-type series formula $$\begin{aligned} \frac{1}{\pi} = & \sum_{n=0}^{\infty} \frac{ \left( \frac{1}{3} \right)_{n} \left( \frac{1}{2} \right)_{n} \left( \frac{2}{3} \right)_{n} }{ \left( 1 \right)_{n}^{3} } \left( \frac{22617 \sqrt{15}+27314 \sqrt{10}-36036 \sqrt{6}-87049}{3375} \right)^{n} \times \\ & \Bigg( \frac{54 \sqrt{15}+68 \sqrt{10}-72 \sqrt{6}-193}{135} + \frac{63 \sqrt{15}+71 \sqrt{10}-63 \sqrt{6}-217}{45} n \Bigg). \end{aligned}$$ holds true.* *Proof.* This can be shown to follow in a direct way from Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, for the $r = 30$ case, with the use of known closed forms such that $$\lambda^{\ast}(30) = \left( \sqrt{3} - \sqrt{2} \right)^{2} \left( 2 - \sqrt{3} \right) \left( \sqrt{6} - \sqrt{5} \right) \left( 4 - \sqrt{15} \right)$$ and $$\begin{aligned} \alpha\left( 30 \right) = & \frac{1}{2} \Big\{ \sqrt{30} - \left( 2 + \sqrt{5} \right)^{2} \left( 3 + \sqrt{10} \right) ^{2} \times \\ & \left( - 6 - 5 \sqrt{2} - 3 \sqrt{5} - 2 \sqrt{10} + \sqrt{6} \sqrt{57 + 40 \sqrt{2}} \right) \times \\ & \left[ 56 + 38 \sqrt{2} + \sqrt{30} \left( 2 + \sqrt{5} \right) \left( 3 + \sqrt{10} \right) \right] \Big\}.\end{aligned}$$ Using these closed forms, it is a matter of routine to show that the unique $y$-value in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} is such that $$y = \frac{1}{2} \left(1-\sqrt{1+\frac{87049+36036 \sqrt{6}-6750 \sqrt{\frac{3026695259}{9112500}+\frac{34320041 \sqrt{\frac{2}{3}}}{84375}}}{3375}}\right),$$ and that the the evaluation of the $z$- and $a$- and $b$-values reduce in the specified manner. ◻ The new Ramanujan-type series highlighted above is inspired by Baruah and Berndt's Ramanujan-type series with summands involving quartic expressions, including the following results introduced in [@BaruahBerndt2010]: $$\begin{aligned} \frac{3\sqrt{2} + \sqrt{5} + 2}{\pi} & = \sum_{n=0}^{\infty} \frac{ \left( \frac{1}{2} \right)_{n}^3 }{ \left( n! \right)^3} \left( (3+\sqrt{5}) (2 + \sqrt{5}) (3\sqrt{2} - \sqrt{5} - 2) \right)^{2n}\notag\\ & \ \ \ \ \ \big( 2 \sqrt{10} - 3 \sqrt{5} + 5 \sqrt{2} - 4 + (15 \sqrt{2} + 6 \sqrt{10} - 6 \sqrt{5}) n\big), \\ \frac{\sqrt{6}+\sqrt{2}+1}{\pi } & = \sum_{n=0}^{\infty} \frac{ \left( \frac{1}{2} \right)_{n}^{3} }{ \left( n! \right)^{3} } \Big( \left( 6 \sqrt{3}+3 \sqrt{6}-6 \right) n + 2 \sqrt{3}+\sqrt{6}-3-\sqrt{2} \Big) \\ & \ \ \ \ \ \left(8 \left(\sqrt{2}+1\right)^2 \left(\sqrt{3}-\sqrt{2}\right)^3 \left(2-\sqrt{3}\right)^3\right)^n, \\ \frac{8}{\pi } & = \sum_{n = 0}^{\infty} \frac{ \left( \frac{1}{2} \right)_{n}^{3} }{(n!)^3} \left( \frac{1 - \sqrt{5}}{4} \right)^{3 n} \left(\frac{\sqrt{5}+1}{2} - \sqrt{\frac{\sqrt{5}+1}{2} }\right)^{6 n} \\ & \ \ \ \ \ \Bigg(2 \left(\left(15+5 \sqrt{5}\right) \sqrt{\sqrt{5}+1}-7 \sqrt{10}-5 \sqrt{2}\right) n + \\ & \ \ \ \ \ \left(9+3 \sqrt{5}\right) \sqrt{\sqrt{5}+1}-7 \sqrt{2}-5 \sqrt{10}\Bigg). \end{aligned}$$ # Conclusion We conclude with two open problems concerning values shown in Table [1](#sqrtqqtqabqlqe){reference-type="ref" reference="sqrtqqtqabqlqe"}. $N$ $r$ ----- ----- 11 33 13 39 19 57 31 93 33 99 59 177 : The $N$-indices for the alternating Ramanujan-type series of level 3 conjectured by Aldawoud [@Aldawoud2012] and given in Tables 3.7 and 3.8 in [@Aldawoud2012], along with the corresponding $r$-values according to Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}.   The Ramanujan-type series correpsonding to the $N = 13$ case of Table 3.7 in [@Aldawoud2012 p. 28] can be proved via the $r = 39$ case of Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}, but this would require closed forms for $\lambda^{\ast}(39)$ and $\alpha(39)$. How can the values of $\lambda^{\ast}(39)$ and $\alpha(39)$ be evaluated in closed form? We would also require closed forms for $\lambda^{\ast}(177)$ and $\alpha(177)$, to apply Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} to prove Aldawoud's conjecture concerning the $N = 59$ case shown in Table 3.7 in [@Aldawoud2012 p. 28]. How can the values of $\lambda^{\ast}(177)$ and $\alpha(177)$ be evaluated in closed form? ## Acknowledgements {#acknowledgements .unnumbered} The author is grateful to acknowledge support from a Killam Postdoctoral Fellowship and is thankful to Shane Chern, who has provided the author with extremely useful comments and insights concerning the material in this article. The author also wants to thank Christophe Vignat and Tanay Wakhare for useful and engaging discussions. 99 A. M. Aldawoud, *Ramanujan-type Series for $\frac{1}{\pi}$ with Quadratic Irrationals*, M.Sc. Thesis, Massey University, 2012. N. D. Baruah and B. C. Berndt, Eisenstein series and Ramanujan-type series for $1 / \pi$, *Ramanujan J.* **23** (2010), no. 1-3, 17--44. A. Berkovich, H. H. Chan, and M. J. Schlosser, Wronskians of theta functions and series for $1/\pi$, *Adv. Math.* **338** (2018), 266--304. B. C. Berndt, S. Bhargava, and F. G. Garvan, Ramanujan's theories of elliptic functions to alternative bases, *Trans. Amer. Math. Soc.* **347** (1995), no. 11, 4163--4244. B. C. Berndt and R. A. Rankin, *Ramanujan: Letters and Commentary*, American Mathematical Society, Providence, RI, 1995. J. M. Borwein and P. B. Borwein, *Pi and the AGM. A study in analytic number theory and computational complexity*, John Wiley & Sons, Inc., New York, 1987. J. M. Borwein and P. B. Borwein, Ramanujan's rational and algebraic series for $1/\pi$, *J. Indian Math. Soc. (N.S.)* **51** (1987), 147--160. J. M. Borwein and P. B. Borwein, More Ramanujan-type series for 1/$\pi$, In: *Ramanujan Revisited*, 359--374, Academic Press, Boston, MA, 1988. J. M. Borwein and P. B. Borwein, Some observations on computer aided analysis, *Notices Amer. Math. Soc.* **39** (1992), no. 8, 825--829. J. M. Borwein and P. B. Borwein, Class number three Ramanujan type series for $1/\pi$, *J. Comput. Appl. Math.* **46** (1993), no. 1-2, 281--290. H. H. Chan, S. H. Chan, and Z. Liu, Domb's numbers and Ramanujan-Sato type series for $1/\pi$, *Adv. Math.* **186** (2004), no. 2, 396--410. H. H. Chan, A. Gee, and V. Tan, Cubic singular moduli, Ramanujan's class invariants $\lambda_n$ and the explicit Shimura reciprocity law, *Pacific J. Math.* **208** (2003), no. 1, 23--37. H. H. Chan and W.-C. Liaw, Cubic modular equations and new Ramanujan-type series for $1/\pi$, *Pacific J. Math.* **192** (2000), no. 2, 219--238. H. H. Chan, W.-C. Liaw, and V. Tan, Ramanujan's class invariant $\lambda_ n$ and a new class of series for $1/\pi$, *J. London Math. Soc. (2)* **64** (2001), no. 1, 93--106. S. Cooper and W. Zudilin, Hypergeometric modular equations, *J. Aust. Math. Soc.* **107** (2019), no. 3, 338--366. J. Guillera, A new method to obtain series for $1/\pi$ and $1/\pi^2$, *Exp. Math.* **15** (2006), no. 1, 83--89. J. Guillera, More hypergeometric identities related to Ramanujan-type series, *Ramanujan J.* **32** (2013), no. 1, 5--22. J. Guillera, Ramanujan series with a shift, *J. Aust. Math. Soc.* **107** (2019), no. 3, 367--380. J. Guillera, A method for proving Ramanujan's series for $1/\pi$, *Ramanujan J.* **52** (2020), no. 2, 421--431. J. Guillera, Proof of a rational Ramanujan-type series for $1/\pi$. The fastest one in level $3$, *Int. J. Number Theory* **17** (2021), no. 2, 473--477. J. Guillera, Proof of Chudnovskys' hypergeometric series for $1/\pi$ using Weber modular polynomials, In: *Transcendence in Algebra, Combinatorics, Geometry and Number Theory*, 341--354, Springer Proc. Math. Stat., **373**, Springer, Cham, 2021. S. Ramanujan, Modular equations and approximations to $\pi$, *Quart. J. Math.* **45** (1914), 350--372. S. Ramanujan, *Collected Papers*, AMS Chelsea Publishing, Providence, RI, 2000. R. Vidūnas, Algebraic transformations of Gauss hypergeometric functions, *Funkcial. Ekvac.* **52** (2009), no. 2, 139--180. J. G. Wan, Series for $1/\pi$ using Legendre's relation, *Integral Transforms Spec. Funct.* **25** (2014), no. 1, 1--14.

arxiv_math

arxiv_math

--- abstract: | A conjecture of Jackson from 1981 states that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large $n$. In fact we prove a more general result that for all $\alpha>0$, there exists $n_0=n_0(\alpha)$ such that every $d$-regular digraph on $n\geq n_0$ vertices with $d \ge \alpha n$ can be covered by at most $n/(d+1)$ vertex-disjoint cycles, and moreover that if $G$ is an oriented graph, then at most $n/(2d+1)$ cycles suffice. **Keywords:** cycle cover, Hamilton cycle, regular, oriented graph author: - "Allan Lo[^1] Viresh Patel[^2] Mehmet Akif Yıldız[^3]" title: Cycle Partitions in Dense Regular Digraphs and Oriented Graphs --- # Introduction A *Hamilton cycle* in a (directed) graph is a (directed) cycle that visits every vertex. Hamilton cycles are one of the most intensely studied structures in graph theory and there are numerous results that establish (best-possible) conditions guaranteeing their existence. The seminal result of Dirac [@Dirac] states that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ is Hamiltonian. Ghouila-Houri [@GhouilaHouri] showed the corresponding version in directed graphs (*digraph* for short), that is, every digraph on $n\ge 3$ vertices with minimum semi-degree at least $n/2$ (i.e. every vertex has in- and outdegree at least $n/2$) is Hamiltonian. These bounds are tight by taking e.g. the disjoint union of two cliques (a regular extremal example) or a slightly imbalanced complete bipartite (di)graph (an irregular extremal example). Recall that an oriented graph is a digraph that can have at most one edge between each pair of vertices (whereas a digraph can have up to two, one in each direction). For oriented graphs, a more recent result of Keevash, Kühn, and Osthus [@OrientedExactMinDegree] established a (tight) minimum semi-degree threshold of $\lceil (3n - 4)/8 \rceil$ for Hamiltonicity. In contrast to graphs and digraphs, there are no *regular* extremal examples for this result. Jackson [@JacksonConjecture] conjectured in 1981 that regularity actually reduces the degree threshold significantly for oriented graphs: **Conjecture 1** (Jackson [@JacksonConjecture]). *For each $d > 2$, every $d$-regular oriented graph (i.e. every vertex has $d$ in- and outneighbours) on $n\leq 4d+1$ vertices has a Hamilton cycle.* The disjoint union of two regular tournaments shows that Jackson's conjecture is best possible. This example works for $n\equiv 2\pmod{4}$ (since regular tournaments require an odd number of vertices), but similar examples exist for $n\equiv 0,1,3\pmod{4}$ (see Section [1.2](#sec:examples+proofs){reference-type="ref" reference="sec:examples+proofs"}). We note that an approximate version of Jackson's conjecture was recently verified by the current authors in [@LPY], that is, for every $\varepsilon>0$, there exists $n_0(\varepsilon)$ such that every $d$-regular oriented graph on $n\ge n_0(\varepsilon)$ vertices with $d\ge (1/4+\varepsilon)n$ is Hamiltonian. Here, we verify the exact version for large $n$. **Theorem 2**. *There exists an integer $n_0$ such that every $d$-regular oriented graph on $n\ge n_0$ vertices with $n\leq 4d+1$ has a Hamilton cycle.* Generalizing questions about Hamilton cycles, one can consider the question of covering the vertices of a (di)graph with as few vertex-disjoint cycles as possible. Indeed, we prove Theorem [Theorem 2](#thm:Jackson){reference-type="ref" reference="thm:Jackson"} by showing a more general result about covering regular digraphs and oriented graphs with few vertex-disjoint cycles. **Theorem 3**. *For all $\alpha >0$, there exists $n_0 = n_0(\alpha)$ such that every $d$-regular digraph $G$ on $n\geq n_0$ vertices with $d \ge \alpha n$ can be covered by at most $n/(d+1)$ vertex-disjoint cycles. Moreover if $G$ is an oriented graph, then at most $n/(2d+1)$ cycles suffice.* This is best possible by considering the disjoint union of complete digraphs of order $d+1$ for digraphs and the disjoint union of regular tournaments of order $2d+1$ for oriented graphs. Notice that we have $n/(2d+1)<2$ when $n\leq 4d+1$, so that Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} implies Theorem [Theorem 2](#thm:Jackson){reference-type="ref" reference="thm:Jackson"}. We also note that Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} generalizes the following result of Gruslys and Letzter [@GruslysLetzter] from regular graphs to regular digraphs and oriented graphs. **Theorem 4** (Gruslys and Letzter [@GruslysLetzter]). *For all $\alpha >0$, there exists $n_0 = n_0(\alpha)$ such that every $d$-regular graph on $n \ge n_0$ vertices with $d \ge \alpha n$ can be covered by at most $n/(d+1)$ vertex-disjoint cycles.* Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} implies Theorem [Theorem 4](#thm:GruslysLetzter){reference-type="ref" reference="thm:GruslysLetzter"} by making every edge into a directed $2$-cycle. ## Related work Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} also has connections with several well-studied problems in extremal graph theory: here we mention some of them. ### Path cover A weaker version of cycle cover is path cover. The *path cover number* of a (di)graph $G$, denoted by $\pi(G)$, is the minimum number of vertex-disjoint (directed) paths needed to cover $V(G)$. This was introduced by Ore [@ORE-PathCover], and he showed that $\pi(G)\leq n-\sigma_2(G)$ holds where $\sigma_2(G)$ denotes the minimum sum of the degrees over all non-adjacent vertices. Magnant and Martin [@PathCover] conjectured that *regularity* significantly reduces the upper bound for $\pi(G)$: **Conjecture 5** (Magnant and Martin [@PathCover]). *If $G$ is a $d$-regular graph on $n$ vertices, then $\pi(G)\leq n/(d+1)$.* It is known that Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"} holds for small values of $d$ (see [@PathCover] for $d\leq 5$ and see [@Feige-Fuchs] for $d=6$). Han [@Han-Path-Cover] showed that, for dense graphs, it is enough to use $1+n/(d+1)$ paths to cover almost all vertices. Also, Theorem [Theorem 4](#thm:GruslysLetzter){reference-type="ref" reference="thm:GruslysLetzter"} verifies Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"} in the dense case. It is worth noting that the Linear Arboricity Conjecture [@linear-arboricity] implies Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"} for odd values of $d$, and gives $\pi(G)\leq 2n/(d+2)$ for general $d$ (see [@Feige-Fuchs] for a detailed discussion). For digraphs, the classical result of Gallai and Milgram [@GAL-MIL] states that $\pi(G)$ can be bounded above by the size of the maximum independent set (and Dilworth's [@DILWORTH] theorem says that equality holds for the special case of posets). As our Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} generalizes Theorem [Theorem 4](#thm:GruslysLetzter){reference-type="ref" reference="thm:GruslysLetzter"} from graphs to digraphs and oriented graphs, we believe the following stronger version of Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"} holds, which Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} establishes in the dense case. **Conjecture 6**. *If $G$ is a $d$-regular digraph on $n$ vertices, then $\pi(G)\leq n/(d+1)$. Moreover, $\pi(G)\leq n/(2d+1)$ holds if $G$ is oriented.* Also, Conjecture [Conjecture 6](#conj:DIRECTED-PATH-COVER){reference-type="ref" reference="conj:DIRECTED-PATH-COVER"} implies Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"} by making every edge into a directed $2$-cycle. ### Extending perfect matchings Gruslys and Letzter [@GruslysLetzter], as well as proving Theorem [Theorem 4](#thm:GruslysLetzter){reference-type="ref" reference="thm:GruslysLetzter"}, proved that every large $d$-regular *bipartite graph* $G$ on $n$ vertices with $d$ linear in $n$ can be covered by at most $n/2d$ vertex-disjoint paths. They mentioned that one should be able to replace paths by cycles. Indeed, as a corollary of Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, the result below shows that those cycles can be found in such a way that they even contain any prescribed perfect matching. **Corollary 7**. *For all $\alpha >0$, there exists $n_1 = n_1(\alpha)$ such that, for every $d$-regular bipartite graph on $n \ge n_1$ vertices with $d \ge \alpha n$, any perfect matching can be extended to vertex-disjoint cycles covering all vertices with at most $n/2d$ cycles.* *Proof of Corollary $\ref{thm:reg-bibpartite}$.* Let $\alpha > 0$ and $n_1 = 2 n_0(\alpha)$, where $n_0$ is the function given in Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. Let $G$ be a $d$-regular bipartite graph on $n \ge n_1$ vertices with $d \ge \alpha n$ and vertex classes $X$ and $Y$. Since $G$ is bipartite and regular, $n$ is even and $|X| = |Y| = n/2$. Let $M$ be any perfect matching of $G$. Let $X = \{ x_1, \dots, x_{n/2}\}$ and $Y = \{y_1, \dots, y_{n/2}\}$ be such that $x_iy_i \in E(M)$ for all $i$. Define the digraph $H$ on $X$ such that $x_ix_j \in E(H)$ if and only if $x_i y_j \in E(G)$. Note that $H$ is $(d-1)$-regular on $n/2$ vertices. By Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, $H$ can be covered by at most $n/2d$ vertex-disjoint cycles. Note that a (directed) cycle $x_{i_1} x_{i_2}\dots x_{i_\ell}$ in $H$ corresponds to a cycle $x_{i_1} y_{i_2 } x_{i_2} y_{i_3}\dots x_{i_\ell} y_{i_1}$ in $G$. Therefore $G$ can be covered by at most $n/2d$ vertex-disjoint cycles that contain all the edges of $M$. ◻ Note that Corollary [Corollary 7](#thm:reg-bibpartite){reference-type="ref" reference="thm:reg-bibpartite"} is tight by considering the disjoint union of $n/2d$ many $K_{d,d}$'s. It also shows that $d$-regular bipartite graphs on $n$ (sufficiently large) vertices with $d>n/4$ are examples of graphs in which every perfect matching can be extended into a Hamilton cycle. This property is called the PMH-property in [@Line-Graphs-PMH]. Häggkvist [@Haggvist-F-Hamiltonian] initiated the study of sufficient conditions for the PMH-property (using the name $F$-Hamiltonian where $F$ is a perfect matching) by showing $\sigma_2(G)\geq n+1$ is sufficient. Las Vergnas [@las-vergnas] proved a similar condition for bipartite graphs, which also (almost) implies Corollary [Corollary 7](#thm:reg-bibpartite){reference-type="ref" reference="thm:reg-bibpartite"} in the case $d>n/4$, and Yang [@Yang-F-Hamiltonian] gave minimum edge density conditions to guarantee the PMH property in graphs and bipartite graphs. In the sparse setting, as a special case of a conjecture of Ruskey and Savage [@HYPERCUBE-MATCHING], Fink [@FINK-HYPERCUBE] proved that the hypercube has the PMH-property. ## Extremal examples for Conjecture [Conjecture 1](#conj:Jackson){reference-type="ref" reference="conj:Jackson"} {#sec:examples+proofs} We end the introduction with the deferred examples from earlier. Note that we give a general overview of the paper at the end of the next section. Recall that the disjoint union of two regular tournaments on $n/2$ vertices shows that Jackson's conjecture (Conjecture [Conjecture 1](#conj:Jackson){reference-type="ref" reference="conj:Jackson"}) is best possible. This example works when $n \equiv 2 \pmod{4}$ and similar examples for other values of $n$ are constructed as follows. When $n \equiv 0 \pmod{4}$, we take the disjoint union of two complete graphs on $n/2$ vertices, remove a perfect matching, and orient the edges so that the resulting oriented graph is regular. When $n \equiv 3 \pmod{4}$, we take a disjoint union of a regular tournament on $(n-1)/2$ vertices and a regular orientation of the complete graph minus a perfect matching on $(n+1)/2$ vertices. When $n \equiv 1 \pmod{4}$, we take a regular orientation of a disjoint union of a clique minus a perfect matching on $(n-1)/2$ vertices and a clique minus a Hamilton cycle on $(n+1)/2$ vertices. # Notation and preliminaries Throughout the paper, we use standard graph theory notation and terminology. For $k\in\mathbb{N}$, we sometimes denote the set $\{1,2,\ldots,k\}$ by $[k]$. For a digraph $G$, we denote its vertex set by $V(G)$ and its edge set $E(G)$, and sometimes write $|G| := |V(G)|$ and $e(G) := |E(G)|$. For $a,b \in V(G)$, we write $ab$ for the directed edge from $a$ to $b$. We write $H \subseteq G$ to mean $H$ is a subdigraph of $G$, i.e. $V(H) \subseteq V(G)$ and $E(H) \subseteq E(G)$. We sometimes think of $F \subseteq E(G)$ as a subdigraph of $G$ with vertex set consisting of those vertices incident to edges in $F$ and with edge set $F$. We write $G-F$ for the digraph obtained from $G$ by deleting the edges in $F$. For $S \subseteq V(G)$, we write $G[S]$ for the subdigraph of $G$ induced by $S$ and $G-S$ for the digraph $G[V(G) \setminus S]$. For $A,B \subseteq V(G)$ not necessarily disjoint, we define $E_G(A,B):= \{ab \in E(G): a \in A, \; b \in B \}$ and we write $e_G(A,B) := |E_G(A,B)|$[^4]. We often drop subscripts if these are clear from context. For two digraphs $H_1$ and $H_2$, the union $H_1 \cup H_2$ is the digraph with vertex set $V(H_1) \cup V(H_2)$ and edge set $E(H_1) \cup E(H_2)$. We say an undirected graph $G$ is bipartite with bipartition $A,B$ if $V(G)=A\cup B$ and $E(G) \subseteq \{ab: a \in A, \; b \in B \}$. For a digraph $G$ and $v\in V(G)$, we denote the set of outneighbours and inneighbours of $v$ by $N_G^{+}(v)$ and $N_G^{-}(v)$, respectively, and we write $d_G^{+}(v):=|N_G^{+}(v)|$ and $d_G^{-}(v):=|N_G^{-}(v)|$ for the out- and indegree of $v$, respectively. For $S\subseteq V(G)$ we write $d^{-}(v,S):= |N_G^-(v) \cap S|$ and $d^{+}(v,S):= |N_G^+(v) \cap S|$. We write $\delta^+(G)$ and $\delta^-(G)$, respectively, for the minimum out- and indegree of $G$ and $\delta^{0}(G):=\min\{\delta^{+}(G),\delta^{-}(G)\}$ for the minimum semi-degree. Similarly, the maximum semi-degree $\Delta^{0}(G)$ of $G$ is defined by $\Delta^{0}(G):=\max\{\Delta^{+}(G),\Delta^{-}(G)\}$ where $\Delta^{+}(G)$ and $\Delta^{-}(G)$ denote the maximum out- and maximum indegree of $G$, respectively. A digraph is called $d$-regular if each vertex has exactly $d$ outneighbours and $d$ inneighbours. The notation above extends to undirected graphs in the obvious ways. In particular for an undirected graphs $G$, we write $\Delta(G)$ and $\delta(G)$, respectively for the maximum degree and the minimum degree. A graph is called $d$-regular if each vertex has exactly $d$ neighbours. For a vertex $v \in V(G)$ and subset $S \subseteq V(G)$, we write $d_G(v,S):=|N_G(v) \cap S|$. A directed path $Q$ in a digraph $G$ is a subdigraph of $G$ such that $V(Q) = \{v_1, \ldots, v_k \}$ for some $k \in \mathbb{N}$ and $E(Q) = \{v_1v_2, v_2v_3, \ldots, v_{k-1}v_k \}$. We denote such a directed path by its vertices in order, i.e. we write $Q = v_1v_2 \cdots v_k$. A directed cycle in $G$ is exactly the same except that it also includes the edge $v_kv_1$. Sometimes we identify paths with their edge sets. A set of vertex-disjoint directed paths $\mathcal{Q}=\{Q_1,Q_2,\ldots\}$ in a digraph $G$ is called a path system in $G$. We interchangeably think of $\mathcal{Q}$ as a set of vertex-disjoint directed paths in $G$ and as a subdigraph of $G$ with vertex set $V(\mathcal{Q}) = \bigcup_{i} V(Q_i)$ and edge set $E(\mathcal{Q}) = \bigcup_{i} E(Q_i)$. We sometimes call this subdigraph the graph induced by $\mathcal{Q}$. A matching $M$ in a digraph (or undirected graph) $G$ is a set of edges $M \subseteq E(G)$ such that every vertex of $G$ is incident to at most one edge in $M$. If $M = \{a_ib_i: i \in[m]\}$ is a matching in a digraph, then we write $V^+(M):= \{a_i : i \in [m]\}$ and $V^-(M) = \{ b_i : i \in [m] \}$. A $1$-factor in a digraph $G$ is a set of vertex-disjoint directed cycles whose union has the same vertices as $G$. Throughout the paper, we will work with partitions of a vertex set $V$ of the form $\{V_{ij}: i,j \in [k]\}$ for some $k \in \mathbb{N}$ (so a partition into $k^2$ parts). For each $i$, we write $V_{i*} := V_{i1} \cup \cdots \cup V_{ik}$ and $V_{*i} := V_{1i} \cup \cdots \cup V_{ki}$. Note that $\{V_{i*}: i \in [k]\}$ is a partition of $V$ as is $\{V_{*i}: i \in [k]\}$. Note also that $V_{ij} = V_{i*} \cap V_{*j}$. For two sets $A$ and $B$, the symmetric difference of $A$ and $B$ is the set $A\triangle B := (A\setminus B) \cup (B\setminus A)$. For $x, y \in (0,1]$, we often use the notation $x \ll y$ to mean that $x$ is sufficiently small as a function of $y$ i.e. $x \leq f(y)$ for some implicitly given non-decreasing function $f:(0,1] \rightarrow (0,1]$. We implicitly assume all constants in such hierarchies are positive, and we omit floors and ceilings whenever this does not affect the argument. We use the following non-standard notation. Let $G$ be a directed graph and let $U,W \subseteq V(G)$ not necessarily disjoint. We define $G[U,W]$ to be the undirected bipartite graph with bipartition $U , W$ where, for each $u \in U$ and $w \in W$, $uv$ is an (undirected) edge of $G[U,W]$ if and only if $uv$ is a directed edge in $E(G)$. So $G[U,W]$ has $|U| + |W|$ vertices and $e_G(U,W)$ edges. ## Sketch proof of Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} {#sketch-proof-of-theorem-thmmain} Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} is proved in several steps. One of the key ingredients is a structural result for directed graphs (Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"}) that we derive from a result of Kühn, Lo, Osthus and Staden [@IntoTwoBipartiteExpander] about partitioning dense regular undirected graphs into robust expanders. Robust expansion is a notion introduced and used by Kühn and Osthus together with several coauthors to obtain a number of breakthrough results on (di)graph decompositions and Hamiltonicity (see [@RobustExpanderHamilton; @On-Kelly-Conjecture; @IntoTwoBipartiteExpander]). In Section [3](#sec:Robust){reference-type="ref" reference="sec:Robust"}, we give the necessary background on robust expansion before proving Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"}, which we informally describe below. Informally, robust expanders are dense (di)graphs that are highly connected in a certain sense, and one of their key properties is that they are Hamiltonian under suitable (mild) degree conditions (see Theorem [Theorem 14](#thm:RobustExpanderImpliesHamilton){reference-type="ref" reference="thm:RobustExpanderImpliesHamilton"}). Moreover, they are robust to small alterations (see Lemma [Lemma 8](#lma:birobexpdifferent){reference-type="ref" reference="lma:birobexpdifferent"}). If we could show that every $d$-regular digraph (resp. oriented graph) can be partitioned into at most $n/(d+1)$ (resp. $n/(2d+1)$) robust expanders, it would be enough to prove Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. Such a partition does not exist in general, but our structural result, Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"}, gives us a starting point. Roughly, it says that for any $d$-regular $n$-vertex digraph $G$ with $d$ linear in $n$ and $n$ sufficiently large, there exist two vertex partitions $V(G)=V_{1*} \cup \cdots \cup V_{k*}$ and $V(G)=V_{*1} \cup \cdots \cup V_{*k}$ with $k\leq 1+n/(d+1)$ such that for each $i\in[k]$, $G[V_{i*}, V_{*i}]$ is a bipartite robust expander and $|V_{i*}|\approx |V_{*i}|$. Note here that $k$, the number of parts in each partition, is at most one more than the number of cycles we desire (in the case of digraphs). Writing $V_{ij} = V_{i*} \cap V_{*j}$ for all $i,j\in[k]$, we can think of our two partitions as a single $k^2$-partition $\mathcal{P}=\{V_{ij}: i,j \in [k]\}$ of $V(G)$. If the partition $\mathcal{P}$ described above is balanced, meaning that $|V_{i*}| = |V_{*i}|$ for all $i$, then it turns out that $G$ can be covered by $k$ vertex-disjoint cycles.[^5] So to prove Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, we would like to ensure our partition is balanced, and when $k = 1+ \lfloor n/(d+1) \rfloor$, we would like to slightly improve on the number of cycles. For the latter, in Section [4](#sec:longcycles){reference-type="ref" reference="sec:longcycles"}, we define an auxiliary graph $S(\mathcal{P})$ for the partition $\mathcal{P}$, which has vertex set $[k]$ and $ij\in E(S(\mathcal{P}))$ if and only if $V_{ij}\cup V_{ji}\neq \emptyset$. For each connected component $I \subseteq [k]$ of $S(\mathcal{P})$, Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"} shows how to find a cycle whose vertices are exactly $\bigcup_{i \in I} V_{i*} \cup V_{*i}$ (provided $\mathcal{P}$ is balanced). These cycles are necessarily disjoint for different connected components of $S(\mathcal{P})$, so we obtain a cycle partition where the number of cycles is exactly the number of connected components of $S(\mathcal{P})$. For digraphs, if $S(\mathcal{P})$ has at least one edge (so has at most $k-1 \leq n/(d+1)$ connected components), then Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} follows. Similarly, for oriented graphs, one can show (using Proposition [Proposition 24](#prop:orientedcomponent){reference-type="ref" reference="prop:orientedcomponent"}) that at least one edge in $S(\mathcal{P})$ is enough to prove Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} for oriented graphs. Thus, it is enough if our partition $\mathcal{P}$ is balanced and $S(\mathcal{P})$ has at least one edge. To ensure our partition is balanced, i.e. that $|V_{i*}| = |V_{*i}|$ for all $i$, note first that, as mentioned earlier, Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"} already guarantees that $|V_{i*}|\approx |V_{*i}|$ for all $i$. The idea will be to find and contract a suitable collection of vertex-disjoint paths $\mathcal{Q}$. Here suitable simply means that the numbers of edges of $\mathcal{Q}$ between the various parts of $\mathcal{P}$ are related in the right way, in which case we call $\mathcal{Q}$ a $\mathcal{P}$-balanced path system (introduced in Section [5](#sec:Path-Contraction){reference-type="ref" reference="sec:Path-Contraction"}). If we can find such a $\mathcal{P}$-balanced path system with few edges, then contracting the paths will result in a graph with an adjusted partition $\mathcal{P}'$ that is balanced and has similar properties as $\mathcal{P}$ (Proposition [Proposition 18](#prop:contracting){reference-type="ref" reference="prop:contracting"}), and any cycle partition in the contracted graph can be lifted to a cycle partition in the original graph (Propositions [Proposition 16](#prop:1-fact){reference-type="ref" reference="prop:1-fact"}). If the path system $\mathcal{Q}$ satisfies some further properties then we can also guarantee that $S(\mathcal{P}')$ has at least one edge and this will be enough to prove Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} says that we can always find a $\mathcal{P}$-balanced path system with the required properties, and so we reduce the task of proving Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} to the task of proving Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}. In Section [6](#sec:pfbalancingpathsystem){reference-type="ref" reference="sec:pfbalancingpathsystem"} we reduce the proof of Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} to two lemmas, namely Lemmas [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} and [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}. Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} allows us to find a $\mathcal{P}$-balanced path system with the desired properties under most circumstances, while Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"} (together with Proposition [Proposition 21](#prop:balancepartition){reference-type="ref" reference="prop:balancepartition"}) allows us to find such a path system in the remaining "extremal" circumstances when $G$ is close to the disjoint union of cliques. Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} is proved in Section [7](#sec:pf_balancedpath){reference-type="ref" reference="sec:pf_balancedpath"} using a flow argument and Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"} is proved in Section [8](#sec:pf_regdigraph){reference-type="ref" reference="sec:pf_regdigraph"}. These last two lemmas are the most technical parts of the paper, so we defer their sketch proofs to their respective sections. # Robust expanders {#sec:Robust} We first define robust expansion for graphs. Let $0 < \nu \le \tau < 1$ and let $G$ be a graph on $n$ vertices. For $S \subseteq V(G)$, the *$\nu$-robust neighbourhood* $RN_{\nu,G}(S)$ is the set of all those vertices with at least $\nu n$ neighbours in $S$. We say that $G$ is a robust $(\nu, \tau)$-expander if every $S \subseteq V(G)$ with $\tau n \le |S| \le (1 - \tau) n$ satisfies $|RN_{\nu,G} (S)| \ge |S| + \nu n$. In fact we will mainly be concerned with bipartite robust expanders. Let $G$ be a bipartite graph with bipartition $A, B$. We say that $G$ is a *bipartite robust $(\nu, \tau)$-expander with bipartition $A, B$* if every $S \subseteq A$ with $\tau |A| \le |S| \le (1 - \tau) |A|$ satisfies $|RN_{\nu,G} (S) \cap B| \ge |S| + \nu n$. Note that the order of $A$ and $B$ matters here. Our first lemma says that bipartite robust expansion is robust to small alterations; the lemma can easily be derived from the definition of robust expansion. **Lemma 8** ([@StadenThesis Lemma 3.4.9]). *Let $0 < 1/n \ll \nu \le \tau \ll 1$ with $\nu \le 1/2$. Let $G$ be a bipartite graph with $U \subseteq V(G)$. Suppose $G[U]$ is a bipartite robust $(\nu, \tau)$-expander on $n$ vertices with bipartition $A,B$ and that $A',B' \subseteq V(G)$ are sets satisfying $|A\triangle A'|+|B \triangle B'|\leq \nu |A|/4$. Then $G[ A' \cup B' ]$ is a bipartite robust $(\nu/2, 2\tau)$-expander with bipartition $A',B'$.* Next we give a structural result due to Kühn, Lo, Osthus and Staden [@IntoTwoBipartiteExpander] which states that any regular graph of linear minimum degree has a vertex partition into a small number of parts where each part induces a robust expander or a bipartite robust expander. In fact we state the special case of this result for bipartite graphs, which is all we require. **Theorem 9** (Bipartite special case of [@IntoTwoBipartiteExpander Theorem 3.1]). *For all $\alpha, \tau > 0$ and every non-decreasing function $f: (0, 1) \rightarrow (0, 1)$, there exists $n_0$ such that the following holds. For all $d$-regular bipartite graphs $G$ on $2n \geq n_0$ vertices with bipartition $A,B$ and $d \ge \alpha n$, there exist $\rho, \nu$ with $$\begin{aligned} 1/n_0 \le \rho \le \nu \le \tau; \qquad \rho \le f(\nu); \qquad \text{and} \qquad 1/n_0 \le f(\rho) \end{aligned}$$ such that there is a partition of $V(G)$ into sets $A_1,\dots, A_k,B_1, \dots, B_k$ with the following properties for all $i \in [k]$:* 1. *[\[itm:structure1\]]{#itm:structure1 label="itm:structure1"} $G[A_i \cup B_i]$ is a bipartite robust $(\nu, \tau)$-expander with bipartition $A_i,B_i$;* 2. *[\[itm:structure2\]]{#itm:structure2 label="itm:structure2"} for all $x \in A_i \cup B_i$ and $j \in [k]$, $d(x, A_i \cup B_i) \ge d(x, A_j \cup B_j)$, so in particular, $\delta( G[ A_i \cup B_i ] ) \ge d/k$;* 3. *[\[itm:structure3\]]{#itm:structure3 label="itm:structure3"} $\left| |A_i| - |B_i| \right| \le 2 \rho n$;* 4. *[\[itm:structure4\]]{#itm:structure4 label="itm:structure4"} all but at most $2 \rho n$ vertices $x \in A_i \cup B_i$ satisfy $d(x, A_i \cup B_i) \ge d - 2 \rho n$;* 5. *[\[itm:structure5\]]{#itm:structure5 label="itm:structure5"} $k \le n / (d-2 \rho n)$;* 6. *[\[itm:structure6\]]{#itm:structure6 label="itm:structure6"} $A_1, \ldots, A_k$ is a partition of $A$ and $B_1, \ldots, B_k$ is a partition of $B$.* **Remark 10**. *Note that [\[itm:structure1\]](#itm:structure1){reference-type="ref" reference="itm:structure1"}--[\[itm:structure4\]](#itm:structure4){reference-type="ref" reference="itm:structure4"} follow directly from the statement of [@IntoTwoBipartiteExpander Theorem 3.1] and that [\[itm:structure5\]](#itm:structure5){reference-type="ref" reference="itm:structure5"} follows from [\[itm:structure2\]](#itm:structure2){reference-type="ref" reference="itm:structure2"} and [\[itm:structure4\]](#itm:structure4){reference-type="ref" reference="itm:structure4"}. While [\[itm:structure6\]](#itm:structure6){reference-type="ref" reference="itm:structure6"} cannot be deduced immediately from the statement of [@IntoTwoBipartiteExpander Theorem 3.1], it follows immediately from the proof.[^6]* We now define robust expansion for digraphs. Let $0 < \nu \le \tau < 1$ and let $G$ be a digraph on $n$ vertices. For $S \subseteq V(G)$, the *$\nu$-robust outneighbourhood* $RN^+_{\nu,G}(S)$ is the set of all those vertices with at least $\nu n$ inneighbours in $S$. We say that $G$ is a robust $(\nu, \tau)$-outexpander if every $S \subseteq V(G)$ with $\tau n \le |S| \le (1 - \tau) n$ satisfies $|RN^+_{\nu,G} (S)| \ge |S| + \nu n$. **Proposition 11**. *Let $0 < 1/m \ll \nu \le \tau \ll 1$ with $\nu \le 1/2$. Suppose $G$ is a bipartite graph with bipartition $A,B$, where $A=\{a_1, \ldots, a_m\}$ and $B=\{b_1, \ldots, b_m\}$. If $G$ is a bipartite robust $(\nu, \tau)$-expander with bipartition $A,B$, then the digraph $H$ with vertex set $A$ and edge set $E(H) = \{a_ia_j: a_ib_j \in E(G), i \not= j\}$ is a robust $(\nu/2, 2\tau)$-outexpander.* The proposition above follows immediately from the definitions; we crudely replace $(\nu, \tau)$ with $(\nu/2, 2\tau)$ to account for the loss of any edges of the form $a_ib_i$. We now state and prove a structure lemma for regular digraphs that is derived from Theorem [Theorem 9](#thm:structure){reference-type="ref" reference="thm:structure"}. This will be one of the key ingredients in the proof of Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. It says that any dense regular digraph $G$ has two vertex partitions $V(G)=V_{1*} \cup \cdots \cup V_{k*}$ and $V(G)=V_{*1} \cup \cdots \cup V_{*k}$ with $k$ relatively small such that for each $i$, the (undirected) bipartite graph $G[V_{i*}, V_{*i}]$ is a bipartite robust expander (with bipartition $V_{i*}, V_{*i}$). Various other degree and size conditions relating to the partition are also given. Note that in the theorem below, we actually give a partition $\{V_{ij}: i,j \in [k]\}$ and recall that for each $i\in [k]$, we write $V_{i*} := V_{i1} \cup \cdots \cup V_{ik}$ and $V_{*i} := V_{1i} \cup \cdots \cup V_{ki}$. Thus $V_{ij} = V_{i*} \cap V_{*j}$. **Theorem 12**. *For all $\alpha, \tau > 0$ and every non-decreasing function $f : (0, 1) \rightarrow (0, 1)$, there exists $n_0$ such that the following holds. For all $d$-regular digraphs $G$ on $n \ge n_0$ vertices with $d \ge \alpha n$, there exist $\rho, \nu$ with $$\begin{aligned} 1/n_0 \le \rho \le \nu \le \tau; \qquad \rho \le f(\nu); \qquad \text{and} \qquad 1/n_0 \le f(\rho) \end{aligned}$$ such that there is a partition $\mathcal{P} = \{ V_{ij}: i,j \in [k]\}$ of $V(G)$ satisfying, for all $i,j \in [k]$,* 1. *[\[itm:di1\]]{#itm:di1 label="itm:di1"}$G[V_{i*},V_{*i}]$ is a bipartite robust $(\nu, \tau)$-expander with bipartition $V_{i*},V_{*i}$;* 2. *[\[itm:di2\]]{#itm:di2 label="itm:di2"} for all $x \in V_{ij}$ and $i',j' \in [k]$, $d^+(x, V_{*i}) \ge d^+(x, V_{*i'})$ and $d^-(x, V_{j*}) \ge d^-(x, V_{j'*})$, so in particular, $\delta( G [ V_{i*},V_{*i} ] ) \ge d/k$;* 3. *[\[itm:di3\]]{#itm:di3 label="itm:di3"} $\left| |V_{i*}| - |V_{*i}| \right| \le 2\rho n$;* 4. *[\[itm:di4\]]{#itm:di4 label="itm:di4"} all but at most $2\rho n$ vertices $x \in V_{ij}$ satisfy $d^+(x, V_{*i}), d^-(x, V_{j*}) \ge d - 2\rho n$;* 5. *[\[itm:di5\]]{#itm:di5 label="itm:di5"} $k \le n / (d-2\rho n)$.* *Proof.* We simply apply Theorem [Theorem 9](#thm:structure){reference-type="ref" reference="thm:structure"} to the obvious bipartite graph obtained from the directed graph $G$, with the natural correspondence in parameters, as follows. For any $d$-regular digraph $G$ on $n$ vertices, let $H_G$ be the $d$-regular bipartite undirected graph on $2n$ vertices defined as follows. Let $V(H_G) = A \cup B$, where $A$ and $B$ are disjoint copies of $V(G)$ and for $a \in A$ and $b \in B$ let $ab \in E(H_G)$ if and only if $ab \in E(G)$. Now applying Theorem [Theorem 9](#thm:structure){reference-type="ref" reference="thm:structure"} with $H_G$ playing the role of $G$, we obtain the following statement. For all $\alpha, \tau > 0$ and every non-decreasing function $f: (0, 1) \rightarrow (0, 1)$, there exists $n_0$ such that the following holds. For all $d$-regular digraphs $G$ on $n \geq n_0$ vertices and $d \ge \alpha n$, taking $H_G$ to be the corresponding bipartite $d$-regular graph on $2n \geq n_0$ vertices there exist $\rho, \nu$ with $$\begin{aligned} 1/n_0 \le \rho \le \nu \le \tau; \qquad \rho \le f(\nu) ;\qquad \text{and} \qquad 1/n_0 \le f(\rho) \end{aligned}$$ such that there is a partition of $V(H_G)$ into sets $A_1,\dots, A_{k}, B_1, \dots, B_{k}$ with the following properties for all $i \in [k]$: 1. [\[itm:1\'\]]{#itm:1' label="itm:1'"} $H_G[A_i \cup B_i]$ is a bipartite robust $(\nu, \tau)$-expander with bipartition $A_i,B_i$; 2. [\[itm:2\'\]]{#itm:2' label="itm:2'"} for all $x \in A_i \cup B_i$ and $j \in [k]$, $d_{H_G}(x, A_i \cup B_i) \ge d_{H_G}(x, A_j \cup B_j)$, so in particular, $\delta( H_G [ A_i \cup B_i ] ) \ge d/k$; 3. [\[itm:3\'\]]{#itm:3' label="itm:3'"} $\left| |A_i| - |B_i| \right| \le 2 \rho n$; 4. [\[itm:4\'\]]{#itm:4' label="itm:4'"} all but at most $2 \rho n$ vertices $x \in A_i \cup B_i$ satisfy $d_{H_G}(x, A_i \cup B_i) \ge d - 2 \rho n$; 5. [\[itm:5\'\]]{#itm:5' label="itm:5'"} $k \le n / (d-2 \rho n)$; 6. [\[itm:6\'\]]{#itm:6' label="itm:6'"} $A_1, \ldots, A_k$ is a partition of $A$ and $B_1, \ldots, B_k$ is a partition of $B$. For $i,j \in [k]$, define $V_{ij} = A_i \cap B_j$ (where we think of $A_i$ and $B_j$ as sets of vertices of the digraph $G$). First note that, by [\[itm:6\'\]](#itm:6'){reference-type="ref" reference="itm:6'"}, $\mathcal{P} = \{V_{ij} : i,j \in [k] \}$ is a partition of $V(G)$, and $A_i = V_{i*}$ and $B_i = V_{*i}$. Now the natural correspondence between $H_G$ and $G$ means that [\[itm:1\'\]](#itm:1'){reference-type="ref" reference="itm:1'"}--[\[itm:5\'\]](#itm:5'){reference-type="ref" reference="itm:5'"} imply [\[itm:di1\]](#itm:di1){reference-type="ref" reference="itm:di1"}--[\[itm:di5\]](#itm:di5){reference-type="ref" reference="itm:di5"}, respectively. ◻ # Finding long cycles {#sec:longcycles} Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"} from the previous section shows that every (dense) regular digraph has a vertex partition with some useful properties. In this section we show how properties [\[itm:di1\]](#itm:di1){reference-type="ref" reference="itm:di1"} and [\[itm:di2\]](#itm:di2){reference-type="ref" reference="itm:di2"} from Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"} together with a simple balancing condition on the partition allows us to construct few long cycles that can cover all the vertices of several parts in the partition. Let $G$ be a digraph on $V$ and let $\mathcal{P} =\{V_{ij}:i,j \in [k]\}$ be a partition of $V$. We say that $\mathcal{P}$ is *balanced* if $|V_{i*}| = |V_{*i}|$ for all $i \in [k]$. For $i, j \in [k]$, let $G_{ij}$ be the subdigraph of $G$ on $V_{i*} \cup V_{*j}$ with edges from $V_{i*}$ to $V_{*j}$, that is, $E(G_{ij}) = E_G(V_{i*},V_{*j}) = E(G) \cap (V_{i*} \times V_{*j})$. Define $S(\mathcal{P})$ to be the graph (without loops) on $[k]$ such that $ij \in E(S(\mathcal{P}))$ if and only if $V_{ij} \cup V_{ji} \ne \emptyset$ for all $i\neq j$. The connected components of $S(\mathcal{P})$ will determine which parts of $\mathcal{P}$ can be covered by one long cycle. We remark that, by definition, $\bigcup_{i\in I}(V_{i*}\cup V_{*i})$ and $\bigcup_{j\in J}(V_{j*}\cup V_{*j})$ are disjoint for two distinct connected components $I$ and $J$ of $S(\mathcal{P})$. The aim of this section is to prove the following lemma. **Lemma 13**. *Let $1/m \ll \nu \leq \tau \ll \alpha <1$. Let $G$ be a digraph with a balanced vertex partition $\mathcal{P} = \{V_{ij} : i,j \in [k]\}$. Let $I$ be a connected component in $S(\mathcal{P})$. Suppose that, for all $i \in I$, $G[V_{i*},V_{*i}]$ is a bipartite robust $(\nu, \tau)$-expander with bipartition $V_{i*},V_{*i}$ such that $|V_{i*}| \ge m$ and $\delta( G[V_{i*},V_{*i}] ) \ge \alpha |V_{i*}|$. Then there exists a cycle $C$ in $G$ with $V(C) = \bigcup_{i \in I} \left( V_{i*} \cup V_{*i} \right)$.* The lemma above will be used as follows. Suppose we have a dense $d$-regular digraph $G$ with a vertex partition $\mathcal{P}$ such as that given by Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"} but with the additional property that $\mathcal{P}$ is balanced. Then Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"} applied to each connected component of $S(\mathcal{P})$ gives us a collection of $s$ vertex-disjoint cycles that cover $V(G)$, where $s$ is the number of connected components in $S(\mathcal{P})$. So in later sections we will be interested in obtaining balanced partitions $\mathcal{P}$ where the number of connected components of $S(\mathcal{P})$ is "small". We need the following theorem, which states that a robust outexpander with linear minimum degree contains a (directed) Hamilton cycle. **Theorem 14** ([@RobustExpanderHamilton]; see also [@LoPatel]). *Let $1/n \ll \nu\leq \tau\ll\gamma<1$. Let $G$ be a robust $(\nu,\tau)$-outexpander on $n$ vertices with $\delta^{0}(G)\geq \gamma n$. Then $G$ contains a Hamilton cycle.* The next lemma shows that, under the conditions of Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"}, $V_{i*} \cup V_{*i}$ can be covered either by vertex-disjoint paths from $V_{i*} \setminus V_{ii}$ to $V_{*i} \setminus V_{ii}$ or by a cycle, and it is used inductively to prove Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"}. **Lemma 15**. *Let $1/m \ll \nu \leq \tau \ll \alpha <1$. Let $G$ be a digraph with a vertex partition $\mathcal{P} = \{V_{ij} : i,j \in [k]\}$. Let $i \in [k]$. Suppose that $G[V_{i*},V_{*i}]$ is a bipartite robust $(\nu, \tau)$-expander with bipartition $V_{i*},V_{*i}$ such that $|V_{i*}| = |V_{*i}| = m$ and $\delta( G[V_{i*},V_{*i}] ) \ge \alpha m$. Let $\phi: V_{i*} \setminus V_{ii} \rightarrow V_{*i} \setminus V_{ii}$ be a bijection. Then there exists a path system $\mathcal{Q}$ in $G_{ii}$ such that $\mathcal{Q} \cup \{\phi(v)v : v \in V_{i*} \setminus V_{ii} \}$ is a cycle with vertex set $V_{i*} \cup V_{*i}$. In the special case that $V_{i*} = V_{*i} = V_{ii}$, we have that $\phi$ is an empty function and $\mathcal{Q}$ is a cycle (rather than a path system) with vertex set $V_{ii}$.* *Proof.* Let $H$ be the digraph on $V_{i*}$ obtained from $G_{ii}$ by identifying $v$ with $\phi(v)$ for all $v \in V_{i*} \setminus V_{ii}$ and deleting any loops. Note that $\delta^0(H) \ge \delta( G[V_{i*},V_{*i}] ) - 1 \ge \alpha m /2$ and $H$ is a robust $(\nu/2, 2\tau)$-outexpander by Proposition [Proposition 11](#pr:contractbipartite){reference-type="ref" reference="pr:contractbipartite"} applied to the undirected bipartite graph $G[V_{i*},V_{*i}]$. By Theorem [Theorem 14](#thm:RobustExpanderImpliesHamilton){reference-type="ref" reference="thm:RobustExpanderImpliesHamilton"}, $H$ has a Hamilton cycle $C$. Note that if $V_{i*}=V_{ii}=V_{*i}$, then $H$ coincides with $G[V_{ii}]$, proving the lemma in this case. If not, let $V_{i*}\setminus V_{ii} = \{ x_1, \dots, x_{\ell}\}$ be such that $C$ visits $x_1, \ldots, x_{\ell}$ in that order. Thus, $C$ can be decomposed into paths $P_1, \dots, P_{\ell}$ such that $P_j$ is a path from $x_j$ to $x_{j+1}$, where we take $x_{\ell+1} = x_1$. Let $Q_j$ be obtained from $P_j$ by replacing $x_{j+1}$ with $\phi(x_{j+1})$. Note that $Q_j$ is a path in $G_{ii}$ from $x_j$ to $\phi(x_{j+1})$. The result follows by setting $\mathcal{Q} = \{Q_1, \ldots, Q_{\ell}\}$. ◻ We now prove Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"}. For a connected component $I$ in $S(\mathcal{P})$, the idea is to apply Lemma [Lemma 15](#lma:longcycle1){reference-type="ref" reference="lma:longcycle1"} to each $i \in I$. Some care is needed to ensure that the union of path systems forms only one cycle. *Proof of Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"}.* If $I$ consists of a single vertex say $i$, then the result follows by Lemma [Lemma 15](#lma:longcycle1){reference-type="ref" reference="lma:longcycle1"} (since in that case $V_{i*} = V_{*i} = V_{ii}$). Now assume $|I|\geq 2$, so $V_{i*}\cup V_{*i}\neq V_{ii}$ for all $i\in I$. Without loss of generality, let $I = [\ell]$ and order the indices of $I$ such that for each $j \in [\ell-1]$, there is a $j' \in [\ell] \setminus [j]$ with $j j' \in E(S(\mathcal{P}))$. This can be achieved since $I$ is connected, e.g. by taking a reverse breadth-first search ordering. Note that $V_{ij}=V_{ji}=\emptyset$ for all $i\in[\ell]$ and $j\notin [\ell]$. For $i\in[\ell]$, let $W^+_i=\bigcup_{i'\in [i],\, j'\notin[i]}V_{i'j'} = \bigcup_{i' \in [i]}V_{i'*} \setminus \bigcup_{i',j'\in [i]}V_{i'j'}$ and $W^-_i=\bigcup_{i'\notin [i],\, j'\in[i]}V_{i'j'} = \bigcup_{j' \in [i]}V_{*j'} \setminus \bigcup_{i',j'\in [i]}V_{i'j'}$. Since $\mathcal{P}$ is balanced, $|W^+_i| = |W^-_i|$ for each $i \in [\ell]$. Also, by our ordering of the indices, we have that $W_i^+,W_i^-\neq \emptyset$ for $i\in[\ell-1]$ and that $W_{\ell}^+=W_{\ell}^+= \emptyset$. Let $\mathcal{Q}_0=W_0^+=W_0^-=\emptyset$ and suppose for some $i \in [\ell]$, we have already found a path system $\mathcal{Q}_{i-1}$ such that $V(\mathcal{Q}_{i-1}) = \bigcup_{i' \in [i-1]} \left( V_{i'*} \cup V_{*i'} \right)$ and $\mathcal{Q}_{i-1}$ consists of precisely $|W^+_{i-1}|$ paths from $W^+_{i-1}$ to $W^-_{i-1}$. We now construct $\mathcal{Q}_{i}$ as follows. Let $\phi: V_{i*} \setminus V_{ii} \rightarrow V_{*i} \setminus V_{ii}$ be any bijection such that if there is a path in $\mathcal{Q}_{i-1}$ from $v_+ \in V_{*i}\setminus V_{ii}$ to $v_- \in V_{i*}\setminus V_{ii}$, then $\phi(v_-) = v_+$. Apply Lemma [Lemma 15](#lma:longcycle1){reference-type="ref" reference="lma:longcycle1"} and obtain a path system $\mathcal{Q}_i'$ in $G_{ii}$ such that $\mathcal{Q}_i' \cup \{\phi(v)v : v \in V_{i*}\setminus V_{ii}\}$ is a cycle with vertex set $V_{i*} \cup V_{*i}$. We set $\mathcal{Q}_{i} = \mathcal{Q}_{i-1} \cup \mathcal{Q}_i'$. Suppose $i = \ell$. Note that $W^-_{\ell-1} = V_{\ell*} \setminus V_{\ell\ell}$ and $W^+_{\ell-1} = V_{*\ell} \setminus V_{\ell\ell}$. Hence $\mathcal{Q}_{\ell-1}$ consists of paths from $\phi(v) \in W_{\ell-1}^+$ to $v \in W_{\ell-1}^-$, we have $V(\mathcal{Q}_{\ell-1}) = \bigcup_{i' \in [\ell-1]} \left( V_{i'*} \cup V_{*i'} \right)= \bigcup_{i' \in [\ell]} \left( V_{i'*} \cup V_{*i'} \right) \setminus V_{\ell \ell}$. Since $\mathcal{Q}_{\ell}' \cup \{\phi(v)v : v \in V_{\ell *} \setminus V_{\ell \ell}\}$ is a cycle with vertex set $V_{\ell *} \cup V_{* \ell}$, we deduce that $\mathcal{Q}_{\ell} = \mathcal{Q}_{\ell}' \cup \mathcal{Q}_{\ell-1}$ is a cycle with vertex set $\bigcup_{i' \in [\ell]} \left( V_{i'*} \cup V_{*i'} \right)$ as required. Suppose $i \in [\ell - 1]$. Note that $V(\mathcal{Q}_{i}) = \bigcup_{i' \in [i]} \left( V_{i'*} \cup V_{*i'} \right)$. It remains to check that $\mathcal{Q}_i$ is a path system consisting of precisely $|W_i^+|$ paths from $W_i^+$ to $W_i^-$. Note first that paths in $\mathcal{Q}'_i \subseteq G_{ii}$ start in $V_{i*} \setminus V_{ii}$ and end in $V_{*i} \setminus V_{ii}$ and have all internal vertices in $V_{ii}$. On the other hand a path in $\mathcal{Q}_{i-1}$ can only intersect $V_{*i}$ at its start point and $V_{i*}$ at its end point and it avoids $V_{ii}$. Therefore in $\mathcal{Q}_{i} = \mathcal{Q}_{i-1} \cup \mathcal{Q}'_i$, all in-degrees and outdegrees are at most $1$. A vertex has in-degree zero in $\mathcal{Q}_{i-1}\cup \mathcal{Q}_i'$ if and only if it is a start point of some path in $\mathcal{Q}_{i-1}\cup \mathcal{Q}_i'$ but not an endpoint of any path in $\mathcal{Q}_{i-1}\cup \mathcal{Q}_i'$, i.e. the set of vertices of in-degree zero is $$\begin{aligned} \left(W_{i-1}^+\cup (V_{i*}\setminus V_{ii})\right)\setminus \left(W_{i-1}^-\cup (V_{*i}\setminus V_{ii})\right)=\left(W_{i-1}^+\cup V_{i*}\right)\setminus V_{*i} =W_i^+\end{aligned}$$ and similarly the set of vertices of out-degree zero is $W_i^-$. Finally it remains to check that there are no cycles in $\mathcal{Q}_{i-1}\cup \mathcal{Q}_i'$. By our choice of $\phi$, if there is a cycle, it spans all the vertices of $V(\mathcal{Q}_{i-1}\cup \mathcal{Q}_i')$, but this cannot happen because vertices in $W_i^+\neq \emptyset$ have in-degree zero. ◻ # Balanced path systems and path contraction {#sec:Path-Contraction} As mentioned in the previous section, in order to apply Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"} to obtain a suitable cycle partition of a dense regular digraph $G$, we will require a vertex partition $\mathcal{P}$ of $G$ such as that given by Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"} but with the additional properties that $\mathcal{P}$ is balanced and $S(\mathcal{P})$ has few connected components. In this section, we state a result (Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}) that allows us to adjust a partition $\mathcal{P}$ to have these additional properties. In particular, if $\mathcal{P}$ is not balanced, Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} guarantees us a so-called $\mathcal{P}$-balanced path system in $G$ whose "contraction" makes $\mathcal{P}$ balanced in a suitable way. At the end of the section we show how Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} implies our main result, Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}. Let $\mathcal{P} =\{V_{ij}:i,j \in [k]\}$ be a partition of a vertex set $V$ and let $G$ be a digraph on $V$. Recall the definition of $G_{ij}$ at the beginning of Section [4](#sec:longcycles){reference-type="ref" reference="sec:longcycles"}. Write $G_{i*} = \bigcup_{j\neq i} G_{ij}$ and $G_{*j} = \bigcup_{i \ne j} G_{ij}$. Let $\mathcal{B}(G, \mathcal{P}) = G - \bigcup_{i \in [k]} G_{ii}$. Note that $\mathcal{B}(G, \mathcal{P}) = \bigcup_{i \in [k]} G_{i*} = \bigcup_{j \in [k]} G_{*j}$. We say that a digraph $H$ on $V$ (where $H$ is usually a path system in $G$) is *$\mathcal{P}$-balanced* if, for all $i \in [k]$, $$\begin{aligned} |V_{i*}|-|V_{*i}| = e_H(V_{i*},V) - e_H(V,V_{*i}) = \sum_{j \in [k]} e(H_{ij}) - \sum_{j \in [k]} e(H_{ji}) = e(H_{i*}) - e(H_{*i}).\end{aligned}$$ The above gives three equivalent conditions for $H$ to be $\mathcal{P}$-balanced. Note that if $H$ is regular, then it is $\mathcal{P}$-balanced for any $\mathcal{P}$. Let $Q$ be a path in $G$ from $v_+ \in V_{i_+j_+}$ to $v_- \in V_{i_-j_-}$. We define the *$Q$-contracted subgraph $G'$ of $G$* to be $G \setminus V(Q)$ together with a new vertex $w$ such that $N^+_{G'}(w) = N^+_G(v_-) \setminus V(Q)$ and $N^-_{G'}(w) = N^-_G(v_+) \setminus V(Q)$. We call $\mathcal{P}'=\{V_{ij}':i,j \in [k]\}$ the *$Q$-contracted partition of $\mathcal{P}$*, where $$\begin{aligned} V_{ij}' = \begin{cases} ( V_{ij} \setminus V(Q) ) \cup \{w\} & \text{if $(i,j) = (i_-,j_+)$};\\ V_{ij} \setminus V(Q) &\text{otherwise}. \end{cases}\end{aligned}$$ Let $\mathcal{Q}$ be a path system in $G$. The *$\mathcal{Q}$-contracted subgraph of $G$* (and *$\mathcal{Q}$-contracted partition of $\mathcal{P}$*) is obtained by successively contracting each $Q \in \mathcal{Q}$ for $G$ (and $\mathcal{P}$, respectively). The following two propositions follow from the definition of $\mathcal{Q}$-contraction. In fact the definitions are chosen precisely so that these propositions hold. **Proposition 16**. *Let $G$ be a digraph and $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$. Let $\mathcal{Q}$ be a path system in $G$. Suppose that the $\mathcal{Q}$-contracted subgraph of $G$ contains a $1$-factor with $\ell$ cycles. Then $G$ also contains a $1$-factor containing $\mathcal{Q}$ with $\ell$ cycles.* **Proposition 17**. *Let $G$ be a digraph on $n$ vertices and let $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$. Let $\mathcal{Q}$ be a path system in $G$ such that $e(\mathcal{Q}) < |V_{i*}|, |V_{*i}|$ for all $i \in [k]$. Let $G'$ and $\mathcal{P}' = \{ V'_{ij} : i,j \in [k]\}$ be the $\mathcal{Q}$-contracted subgraph of $G$ and $\mathcal{Q}$-contracted partition of $\mathcal{P}$, respectively. Then for all $i \in [k]$, $\delta(G'[V'_{i*},V'_{*i}] )\geq \delta(G[V_{i*},V_{*i}] ) - 2 e(\mathcal{Q})$.[^7]* **Proposition 18**. *Let $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of a vertex set $V$ and $\mathcal{Q}$ be a $\mathcal{P}$-balanced path system on $V$. Then the $\mathcal{Q}$-contracted partition of $\mathcal{P}$ is balanced.* *Proof.* Let $Q \in \mathcal{Q}$ be a path from $v_+ \in V_{i_+j_+}$ to $v_- \in V_{i_-j_-}$. For each $i,j \in [k]$, let $q_{ij} = e(Q_{ij})$, i.e. the number of edges of $Q$ from $V_{i*}$ to $V_{*j}$. Let $\mathcal{P}' = \{ V_{ij}' : i,j \in [k] \}$ be the $Q$-contracted partition of $\mathcal{P}$. Consider $i \in [k]$. Note that $$\begin{aligned} |V'_{i*}| &= |V_{i*}| - |V_{i*} \cap V(Q) | + \mathbbm{1}({i = i_-}) = |V_{i*}| - \sum_{j \in [k]} q_{ij}, \\ |V'_{*i}| &= |V_{*i}| - |V_{*i} \cap V(Q) | + \mathbbm{1}({i = j_+}) = |V_{*i}| - \sum_{j \in [k]} q_{ji}.\end{aligned}$$ Therefore, $|V'_{i*}| - |V'_{*i}| = |V_{i*}| - |V_{*i}| - \sum_{j \in [k]} (q_{ij} - q_{ji}) = |V_{i*}| - |V_{*i}| - ( e(Q_{i*}) - e(Q_{*i}) )$. A similar statement holds for the $\mathcal{Q}$-contracted partition $\mathcal{P}^* = \{ V_{ij}^* : i,j \in [k] \}$ of $\mathcal{P}$, and the result follows as $\mathcal{Q}$ is $\mathcal{P}$-balanced (i.e. for each $i \in [k]$, $|V^*_{i*}| - |V^*_{*i}|=|V_{i*}| - |V_{*i}| - ( e(\mathcal{Q}_{i*}) - e(\mathcal{Q}_{*i}) )$). ◻ The next lemma says that one can adjust the vertex partition $\mathcal{P}$ found in Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"} by a small amount, and find a $\mathcal{P}$-balanced path system $\mathcal{Q}$ such that $S(\mathcal{P}^*)$ has few connected components, where $\mathcal{P}^*$ is the $\mathcal{Q}$-contracted partition of $\mathcal{P}$. It is exactly what we need to prove Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}, as we shall see. In Section [6](#sec:pfbalancingpathsystem){reference-type="ref" reference="sec:pfbalancingpathsystem"}, we break this lemma down into two further lemmas. **Lemma 19**. *Let $1/n \ll \gamma \ll \alpha, 1/k$. Let $G$ be a $d$-regular digraph on $n$ vertices with $d \ge \alpha n$ and $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$ such that, for all $i \in [k]$,* 1. *$\delta( G [ V_{i*},V_{*i} ] ) \ge d/k$;* 2. *$\left| |V_{i*}| - |V_{*i}| \right| \le \gamma n$;* 3. *$|V_{i*}|, |V_{*i}| \ge d - \gamma n$. [\[itm:balancingpathsystemcombined:5\]]{#itm:balancingpathsystemcombined:5 label="itm:balancingpathsystemcombined:5"}* *Then there exists a partition $\mathcal{P}' = \{ V'_{ij} : i,j \in [k]\}$ of $V(G)$ and a $\mathcal{P}'$-balanced path system $\mathcal{Q}$ in $G$ such that* 1. *for all $i, j \in [k]$, $| V_{ij} \triangle V'_{ij}| \le \gamma^{1/2} n$; [\[itm:balancingpathsystemcombined:a\]]{#itm:balancingpathsystemcombined:a label="itm:balancingpathsystemcombined:a"}* 2. *for all $i \in [k], \delta(G[V_{i*}', V_{*i}']) \geq (d/k) - \gamma^{1/2}n$; [\[itm:balancingpathsystemcombined:b\]]{#itm:balancingpathsystemcombined:b label="itm:balancingpathsystemcombined:b"}* 3. *$e(\mathcal{Q}) \le \gamma^{1/2} n$; [\[itm:balancingpathsystemcombined:c\]]{#itm:balancingpathsystemcombined:c label="itm:balancingpathsystemcombined:c"}* 4. *$S(\mathcal{P}^*)$ has at most $n/(qd+1)$ connected components, where $\mathcal{P}^*$ is the $\mathcal{Q}$-contracted partition of $\mathcal{P}'$, and we set $q = 2$ if $G$ is an oriented graph and $q=1$ otherwise. [\[itm:balancingpathsystemcombined:d\]]{#itm:balancingpathsystemcombined:d label="itm:balancingpathsystemcombined:d"}* ## Proof of Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} {#proof-of-theorem-thmmain} We now prove Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} assuming Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}. *Proof of Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}.* Choose a non-decreasing function $g : (0, 1) \rightarrow (0, 1)$ such that the requirements of Lemmas [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"} (with $m = \alpha n/2$) and [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} are satisfied whenever $n, \gamma, \nu, \tau, \alpha$ satisfy $$\begin{aligned} 1/n \ll_g \gamma \ll_g \nu \leq \tau \ll_g \alpha, \end{aligned}$$ where we write $a \ll_g b$ to mean $a \leq g(b)$. Set $f(x) := \min(x, g(x^2/4))$. Fix $\tau \leq f(\alpha)$ and apply Theorem [Theorem 12](#thm:digraphstructure){reference-type="ref" reference="thm:digraphstructure"} to obtain $n_0$ such that for any $d$-regular digraph $G$ on $n \geq n_0$ vertices with $d \ge \alpha n$, there exists $\gamma$ and $\nu$ such that $$\begin{aligned} 1/n \ll_f \gamma \ll_f \nu \leq \tau \ll_f \alpha\end{aligned}$$ and a partition $\mathcal{P} = \{ V_{ij}: i,j \in [k]\}$ of $V(G)$ satisfying, for all $i,j \in [k]$, 1. $G[V_{i*},V_{*i}]$ is a bipartite robust $(\nu, \tau)$-expander with partition $V_{i*},V_{*i}$; [\[item:bip-rob-exp\]]{#item:bip-rob-exp label="item:bip-rob-exp"} 2. for all $x \in V_{ij}$ and $i',j' \in [k]$, $d^+(x, V_{*i}) \ge d^+(x, V_{*i'})$ and $d^-(x, V_{j*}) \ge d^-(x, V_{j'*})$, so in particular, $\delta( G [ V_{i*},V_{*i} ] ) \ge d/k$; [\[item:large-good-degree\]]{#item:large-good-degree label="item:large-good-degree"} 3. $\left| |V_{i*}| - |V_{*i}| \right| \le \gamma n$; [\[item:almost-balanced\]]{#item:almost-balanced label="item:almost-balanced"} 4. all but at most $\gamma n$ vertices $x \in V_{ij}$ satisfy $d^+(x, V_{*i}), d^-(x, V_{j*}) \ge d - \gamma n$; [\[item:small-bad-edges\]]{#item:small-bad-edges label="item:small-bad-edges"} 5. $k \le n / (d-\gamma n)$. [\[item:number-of-parts\]]{#item:number-of-parts label="item:number-of-parts"} Furthermore, [\[item:large-good-degree\]](#item:large-good-degree){reference-type="ref" reference="item:large-good-degree"} and [\[item:small-bad-edges\]](#item:small-bad-edges){reference-type="ref" reference="item:small-bad-edges"} imply that 1. for all $i \in [k]$, $|V_{i*}|,|V_{*i}|\ge d-\gamma n$. [\[itm:each-part-is-at-least-d\]]{#itm:each-part-is-at-least-d label="itm:each-part-is-at-least-d"} Let $q=2$ if $G$ is oriented, and $q=1$ otherwise. Apply Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} and obtain a partition $\mathcal{P}' = \{ V'_{ij} : i,j \in [k]\}$ of $V(G)$ and a $\mathcal{P}'$-balanced path system $\mathcal{Q}$ in $G$ such that 1. for all $i, j \in [k]$, $| V_{ij} \triangle V'_{ij}| \le \gamma^{1/2} n$; [\[itm:b_1\]]{#itm:b_1 label="itm:b_1"} 2. for all $i \in [k], \delta(G[V_{i*}', V_{*i}']) \geq (d/k) - \gamma^{1/2}n$; [\[itm:b_deg\]]{#itm:b_deg label="itm:b_deg"} 3. $e(\mathcal{Q}) \le \gamma^{1/2} n$; [\[itm:b_2\]]{#itm:b_2 label="itm:b_2"} 4. $S(\mathcal{P}^*)$ has at most $n/(qd+1)$ connected components, where $\mathcal{P}^*$ is the $\mathcal{Q}$-contracted partition of $\mathcal{P}'$. [\[itm:b_3\]]{#itm:b_3 label="itm:b_3"} Let $\mathcal{P}^* = \{V^*_{ij}:i,j \in [k]\}$ and let $G^*$ be the $\mathcal{Q}$-contracted subgraph of $G$. We will check that, for all $i, j \in [k]$, 1. $\mathcal{P}^*$ is balanced; [\[itm:c_1\]]{#itm:c_1 label="itm:c_1"} 2. $| V_{ij} \triangle V^*_{ij}| \le 7 \gamma^{1/2} n$; [\[itm:c_2\]]{#itm:c_2 label="itm:c_2"} 3. $|V_{i*}^*|,|V_{*i}^*|\ge d- 8 k\gamma^{1/2} n \ge d/2$; [\[itm:c_3\]]{#itm:c_3 label="itm:c_3"} 4. $G^*[V^*_{i*},V^*_{*i}]$ is a bipartite robust $(\nu/2, 2\tau)$-expander with bipartition $V^*_{i*},V^*_{*i}$; [\[itm:c_4\]]{#itm:c_4 label="itm:c_4"} 5. $\delta( G^*[V^*_{i*},V^*_{*i}] ) \ge \alpha^2 |V^*_{i*}|/2$. [\[itm:c_5\]]{#itm:c_5 label="itm:c_5"} Note that [\[itm:c_1\]](#itm:c_1){reference-type="ref" reference="itm:c_1"} holds by Proposition [Proposition 18](#prop:contracting){reference-type="ref" reference="prop:contracting"}. Consider $i,j \in [k]$. Note that $$\begin{aligned} |V^*_{ij} \triangle V_{ij}| \le |V^*_{ij} \triangle V'_{ij}| + |V'_{ij} \triangle V_{ij}| \le 3 |V (\mathcal{Q}) |+|V'_{ij} \triangle V_{ij}| \overset{\mathclap{\text{\ref{itm:b_1},\ref{itm:b_2}}}}{\le} 6\gamma^{1/2} n +\gamma^{1/2} n = 7 \gamma^{1/2}n\end{aligned}$$ implying [\[itm:c_2\]](#itm:c_2){reference-type="ref" reference="itm:c_2"}. Hence [\[itm:c_3\]](#itm:c_3){reference-type="ref" reference="itm:c_3"} follows from [\[itm:each-part-is-at-least-d\]](#itm:each-part-is-at-least-d){reference-type="ref" reference="itm:each-part-is-at-least-d"} and [\[itm:c_2\]](#itm:c_2){reference-type="ref" reference="itm:c_2"} as $\gamma \ll 1/k$. By [\[item:bip-rob-exp\]](#item:bip-rob-exp){reference-type="ref" reference="item:bip-rob-exp"}, [\[itm:c_2\]](#itm:c_2){reference-type="ref" reference="itm:c_2"}, [\[itm:c_3\]](#itm:c_3){reference-type="ref" reference="itm:c_3"}, and $\gamma \ll \nu$, Lemma [Lemma 8](#lma:birobexpdifferent){reference-type="ref" reference="lma:birobexpdifferent"} (applied to the bipartite graph $G^*[V^*_{i*},V^*_{*i}]\cup G[V_{i*},V_{*i}]$) implies that [\[itm:c_4\]](#itm:c_4){reference-type="ref" reference="itm:c_4"} holds. By Proposition [Proposition 17](#prop:min-degree-contraction2){reference-type="ref" reference="prop:min-degree-contraction2"}, $$\begin{aligned} \delta( G^*[V^*_{i*},V^*_{*i}] ) & \ge \delta( G [ V'_{i*},V'_{*i} ] ) - 2e(\mathcal{Q}) \overset{\mathclap{\text{\ref{itm:b_deg},\ref{itm:b_2}}}}{\ge} (d/k) - 3\gamma^{1/2}n \geq \alpha^2 |V^*_{i*}|/4.\end{aligned}$$ where the last inequality holds as $\gamma \ll \alpha, 1/k$ and $k (\alpha - \gamma) <1$ by [\[item:number-of-parts\]](#item:number-of-parts){reference-type="ref" reference="item:number-of-parts"}. Hence [\[itm:c_5\]](#itm:c_5){reference-type="ref" reference="itm:c_5"} holds. Apply Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"} (with $G^*$, $\mathcal{P}^*$, $d/2$, $\nu/2$, $2\tau$, $\alpha^2/2$ playing the roles of $G$, $\mathcal{P}$, $m$, $\nu$, $\tau$, $\alpha$, respectively) to each connected component in $S(\mathcal{P}^*)$. Thus, by [\[itm:b_3\]](#itm:b_3){reference-type="ref" reference="itm:b_3"}, $G^*$ can be covered with at most $n/(qd + 1)$ vertex-disjoint cycles. By Proposition [Proposition 16](#prop:1-fact){reference-type="ref" reference="prop:1-fact"}, $G$ can be covered with at most $n/(qd+1)$ vertex-disjoint cycles. ◻ # Proof of Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} {#sec:pfbalancingpathsystem} In the last section we reduced the task of proving Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} to the task of proving Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}. In this section we reduce this further to the task of proving Lemmas [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} and [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}. We start by introducing the notion of a *non-trivial path system*, which will help us control the number of connected components of $S(\mathcal{P}^*)$ (as required in the conclusion of Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}). Recall the definitions at the beginning of Section [5](#sec:Path-Contraction){reference-type="ref" reference="sec:Path-Contraction"}. Let $\mathcal{P} =\{V_{ij}:i,j \in [k]\}$ be a partition of a vertex set $V$. We call a path system $\mathcal{Q}$ on $V$ *non-trivial* if there exists $i \ne j$ such that $V_{ij} \setminus V(\mathcal{Q}) \ne \emptyset$ or $\mathcal{Q}$ contains a path from $V_{*j}$ to $V_{i*}$. Otherwise, we call $\mathcal{Q}$ a *trivial* path system. Note that if $\mathcal{Q}$ is non-trivial, then the $\mathcal{Q}$-contracted partition $\mathcal{P}'$ of $\mathcal{P}$ has the property that $S(\mathcal{P}')$ has at least one edge so that its number of connected components is strictly less than its number of vertices. Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} below shows that, under the same hypothesis as Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}, there exists a $\mathcal{P}$-balanced path system with few edges, and moreover that this path system is non-trivial if further assumptions are made. We shall see at the end of the section that the existence of such a non-trivial $\mathcal{P}$-balanced path system is enough to prove Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}. In fact, a $\mathcal{P}$-balanced path system alone (without the condition of being non-trivial) is enough to prove Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} except in the extremal cases when $d \approx n/k$ if $G$ is digraph and when $d \approx n/ 2k$ if $G$ is an oriented graph. **Lemma 20**. *Let $1/n \ll \gamma \ll 1/k, \alpha$. Let $G^0$ be a $d$-regular digraph on $n$ vertices with $d \ge \alpha n$ and $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G^0)$ such that, for all $i \in [k]$,* 1. *$\delta( G^0 [ V_{i*},V_{*i} ] ) \ge d/k$; [\[item:good-degree-condition\]]{#item:good-degree-condition label="item:good-degree-condition"}* 2. *$\left| |V_{i*}| - |V_{*i}| \right| \le \gamma n$; [\[itm:balancedpath3\]]{#itm:balancedpath3 label="itm:balancedpath3"}* 3. *$|V_{i*}|, |V_{*i}| \ge d - \gamma n$.* *Then $G^0$ contains a $\mathcal{P}$-balanced path system $\mathcal{Q}$ such that $e(\mathcal{Q}) \le k^2 \gamma n$. Moreover, if one of the following holds* 1. *$\sum_{i,j \in [k] \colon i \ne j} |V_{ij}| > k^2\gamma n$, or [\[item:size-condition-V-ij\]]{#item:size-condition-V-ij label="item:size-condition-V-ij"}* 2. *there exists $v_0 \in V_{i_0j_0}$ for some $i_0 \ne j_0$ such that $d^{+}(v_0,V_{*i_0})-d^{-}(v_0,V_{i_0*})\ge 100 k^{12} \gamma^{1/3}d$, [\[item:degree-condition-v0\]]{#item:degree-condition-v0 label="item:degree-condition-v0"}* *then we may assume that $\mathcal{Q}$ is non-trivial.* In case Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} gives a $\mathcal{P}$-balanced but trivial path system (i.e. when both [\[item:size-condition-V-ij\]](#item:size-condition-V-ij){reference-type="ref" reference="item:size-condition-V-ij"} and [\[item:degree-condition-v0\]](#item:degree-condition-v0){reference-type="ref" reference="item:degree-condition-v0"} fail), we use the next proposition to slightly modify $\mathcal{P}$ and Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"} to find the desired non-trivial, $\mathcal{P}$-balanced path system. **Proposition 21**. *Let $1/n \ll \gamma \ll 1/k, \alpha$. Let $G^0$ be a $d$-regular digraph on $n$ vertices with $d \ge \alpha n$ and $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G^0)$ such that, for all distinct $i,j \in [k]$,* 1. *$\delta( G^0 [ V_{i*},V_{*i} ] ) \ge d/k$; [\[itm:balance1\]]{#itm:balance1 label="itm:balance1"}* 2. *$\sum_{i,j \in [k] \colon i \ne j} |V_{ij}| \le \gamma n$; [\[itm:balance4\]]{#itm:balance4 label="itm:balance4"}* 3. *for all $v \in V_{ij}$, we have $d^+(v,V_{*i}) - d^-(v, V_{i *}) \le \gamma n$. [\[itm:balance5\]]{#itm:balance5 label="itm:balance5"}* *Then there exists a partition $\mathcal{P}' = \{V_{ii}':i \in [k]\}$ of $V(G^0)$ such that, for all $i \in [k]$,* 1. *$|V_{ii} \triangle V'_{ii}| \le \gamma n$; [\[itm:balance\'2\]]{#itm:balance'2 label="itm:balance'2"}* 2. *$\delta^0( G^0 [ V_{ii}' ] ) \ge d/k - \gamma n$. [\[itm:balance\'1\]]{#itm:balance'1 label="itm:balance'1"}* *Proof.* For each $i \in [k]$, let $V_{ii}' =V_{i*}$. Clearly, $\mathcal{P}' = \{V_{ii}':i \in [k]\}$ is a partition of $V(G^0)$. Note that [\[itm:balance4\]](#itm:balance4){reference-type="ref" reference="itm:balance4"} implies [\[itm:balance\'2\]](#itm:balance'2){reference-type="ref" reference="itm:balance'2"}. For $v \in V_{ij}$ (possibly $i=j$), note that $$\begin{aligned} d^{+}(v, V_{ii}')= d^{+}(v, V_{i*})\ge d^{+}(v, V_{ii}) \overset{\mathclap{\text{\ref{itm:balance4}}}}{\ge} d^{+}(v, V_{*i})-\gamma n \overset{\mathclap{\text{\ref{itm:balance1}}}}{\ge} d/k-\gamma n. \end{aligned}$$ If $i\neq j$, similarly, we have $$\begin{aligned} d^{-}(v, V_{ii}')= d^{-}(v, V_{i*})\overset{\mathclap{\text{\ref{itm:balance5}}}}{\ge}d^{+}(v, V_{*i})-\gamma n \overset{\mathclap{\text{\ref{itm:balance1}}}}{\ge}d/k-\gamma n.\end{aligned}$$ Also, for all $v\in V_{ii}$, [\[itm:balance1\]](#itm:balance1){reference-type="ref" reference="itm:balance1"} implies $d^{-}(v, V_{ii}')= d^{-}(v, V_{i*})\ge d/k$. Hence [\[itm:balance\'1\]](#itm:balance'1){reference-type="ref" reference="itm:balance'1"} holds. ◻ **Lemma 22**. *Let $d,k \in \mathbb{N}$ be such that $k\geq2$ and $d > 165k^5$. Let $G$ be a $d$-regular digraph on $n$ vertices and $\mathcal{P}=\{V_{ii}: i \in [k]\}$ be a partition of $V(G)$ such that, for all $i \in [k]$,* 1. *$|V_{ii}|\ge d/2$; [\[item:each-part-is-at-least-d-over-2\]]{#item:each-part-is-at-least-d-over-2 label="item:each-part-is-at-least-d-over-2"}* 2. *$n<(2d+1)k$, and $n<(d+1)k$ if $G$ is not oriented;[\[item:relation-n-d-k\]]{#item:relation-n-d-k label="item:relation-n-d-k"}* 3. *for all $v \in V_{ii}$, $d^+(v,V_{ii})+d^-(v,V_{ii})\ge d/k$.[\[item:degree-is-large-inside-diagonal\]]{#item:degree-is-large-inside-diagonal label="item:degree-is-large-inside-diagonal"}* *Then $G$ contains a non-trivial $\mathcal{P}$-balanced path system $\mathcal{Q}$ with $e (\mathcal{Q}) \le k$.* Before we can prove Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}, we need a good lower bound in oriented graphs for $|\bigcup_{i \in I} \left( V_{i*} \cup V_{*i} \right)|$ for any connected component $I$ in $S(\mathcal{P})$; see Proposition [Proposition 24](#prop:orientedcomponent){reference-type="ref" reference="prop:orientedcomponent"}. For this, we use the following result on the minimum semi-degree threshold for an oriented graph to be a robust outexpander. Recall that the definition of robust outexpander is given immediately after Remark [Remark 10](#rem:remark1){reference-type="ref" reference="rem:remark1"}. **Lemma 23** ([@On-Kelly-Conjecture Lemma 13.1]). *Let $1/n \ll \nu \ll \tau \leq \varepsilon/2 \leq 1$. Then every oriented graph $G$ on $n$ vertices with $\delta^0(G)\ge (3/8+\varepsilon)n$ is a robust $(\nu,\tau)$-outexpander.* **Proposition 24**. *Let $1/ n \ll \gamma \ll \tau \ll \alpha <1$. Let $G$ be an oriented graph on $n$ vertices with $\Delta^0(G)\le d$ where $d \ge \alpha n$. Let $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$ such that $|V_{i*}| \ge 2\tau n$ for all $i \in [k]$. Suppose that, for all $i,j \in [k]$, all but at most $\gamma n$ vertices $x \in V_{i*}$ and all but at most $\gamma n$ vertices $y \in V_{*j}$ satisfy $d^+(x, V_{*i}), d^-(y, V_{j*}) \ge d - \gamma n$. Let $I \subseteq S(\mathcal{P})$ be a connected component. Then $$\begin{aligned} \left| \bigcup_{i \in I } (V_{i*} \cup V_{*i}) \right| \ge \begin{cases} 2(d - \gamma n) & \text{if $|I| =1$,}\\ 9(d - 5\gamma n)/4 & \text{otherwise.} \end{cases}\end{aligned}$$* *Proof.* Without loss of generality, let $I = [\ell]$. If $\ell \ge 3$, then, for each $i \in [\ell]$, we can find vertices $v_i \in V_{i*}$ such that $d^+(v_i, V_{*i})\ge d - \gamma n$. Hence, $$\begin{aligned} \left|\bigcup_{i \in I } (V_{i*} \cup V_{*i})\right| \ge \sum_{i \in [\ell] } | V_{*i}| \ge \sum_{i \in [\ell]} d^+(v_i, V_{*i}) \ge \ell (d - \gamma n) \ge 3(d-\gamma n).\end{aligned}$$ If $\ell = 1$, then $V_{11} = V_{1*} = V_{*1}$. There exists a vertex $v \in V_{11}$ such that $d^+(v, V_{11}), d^-(v, V_{11}) \ge d - \gamma n$. Since $G$ is an oriented graph, we have $|V_{11}| \ge d^+(v, V_{11}) + d^-(v, V_{11}) \ge 2(d - \gamma n)$. If $\ell = 2$, then $\bigcup_{i \in [2] } V_{i*} = \bigcup_{i \in [2] } V_{*i}$ and $V_{ij} = V_{ji} = \emptyset$ for all $(i,j) \in [2] \times ([k] \setminus [2])$. Let $V_I = \bigcup_{i \in [2] } V_{i*}$. There exists a vertex subset $W \subseteq V_I$ such that $$\begin{aligned} |W| \ge |V_I| - 4 \gamma n \ge 2(d-\gamma n)-4\gamma n \ge \alpha n,\end{aligned}$$ and such that, writing $W_{ij} = W \cap V_{ij}$ for $i,j \in [2]$, we have for all $i,j \in [2]$ that $$\begin{aligned} \label{eq:degW} d^+(x, W_{*i}), d^-(y, W_{j*}) \ge d - 5\gamma n \text{ for all } x \in W_{i*} \text{ and } y \in W_{*j}.\end{aligned}$$ Hence $\delta^0( G[W] ) \ge d - 5\gamma n$. Next we show that $G[W]$ is not a robust $( \gamma^{1/3} , \tau)$-outexpander. For $i \in [2]$, [\[eq:degW\]](#eq:degW){reference-type="eqref" reference="eq:degW"} implies that $|W_{i*}| \geq d - 5 \gamma n \ge \tau n \ge \tau |W|$ and so $\tau |W| \leq |W_{i*}| \leq (1 - \tau)|W|$. Since $W_{1*} \cup W_{2*} = W_{*1} \cup W_{*2}$, we may assume without loss of generality that $|W_{1*}| \geq |W_{*1}|$. Recall that $\Delta^0(G) \le d$. By [\[eq:degW\]](#eq:degW){reference-type="eqref" reference="eq:degW"}, each vertex in $|W_{1*}|$ has at most $5\gamma n$ outneighbours in $W_{*2}$, so $$\begin{aligned} |{\rm RN}_{\gamma^{1/3}}^+(W_{1*}) \cap W_{*2}| \leq \frac{e_G(W_{1*}, W_{*2})}{ \gamma^{1/3} |W|} \le \frac{5 \gamma n^2} {\alpha \gamma^{1/3} n } < \gamma^{1/3} \alpha n \le \gamma^{1/3} |W|,\end{aligned}$$ where the penultimate inequality holds as $\gamma \ll \alpha$. This implies that $$\begin{aligned} |{\rm RN}_{\gamma^{1/3}}^+(W_{1*})| \le |W_{*1}| + |{\rm RN}_{\gamma^{1/3}}^+(W_{1*}) \cap W_{*2}| < |W_{1*}| + \gamma^{1/3} |W|.\end{aligned}$$ Hence $G[W]$ is not a robust $( \gamma^{1/3}, \tau)$-outexpander as claimed. Lemma [Lemma 23](#lem:oriented-3-over-8-robust-expander){reference-type="ref" reference="lem:oriented-3-over-8-robust-expander"} with $\varepsilon=5/72$ implies $\delta^0( G[W] ) \leq (3/8 + \varepsilon )|W| = 4|W|/9$. Then $4|W|/9 \geq d - 5\gamma n$, and the result follows. ◻ We now prove Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} assuming Lemmas [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} and [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}. *Proof of Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}.* Let $G$ and $\mathcal{P} = \{V_{ij}: i,j \in [k]\}$ be as in the statement of Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}. We apply Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} (with $G^0 = G$) and obtain a $\mathcal{P}$-balanced path system $\mathcal{Q}^0$ such that $e(\mathcal{Q}^0) \le k^2 \gamma n$. Let $G^{0}$ and $\mathcal{P}^{0}$ be the $\mathcal{Q}^0$-contracted subgraph of $G$ and $\mathcal{Q}^0$-contracted partition of $\mathcal{P}$, respectively. Let $k^*$ be the smallest integer larger than $n/(qd+1)$, that is, $k^* = \lfloor n/(qd+1) \rfloor +1$. If $S(\mathcal{P}^{0})$ has at most $n/(qd+1)$ connected components, then we are done by setting $\mathcal{P}' = \mathcal{P}$ and $\mathcal{Q}= \mathcal{Q}^0$ since $e(\mathcal{Q}^0) \le k^2 \gamma n\le \gamma^{1/2} n$. Hence we may assume that $S(\mathcal{P}^{0})$ has at least $k^*$ connected components. **Claim 25**. *We have $k^* = k$ and $S(\mathcal{P}^{0})$ is an empty graph on $[k^*]$.* *Proof of claim.* Since $\gamma \ll \alpha$ and $d \ge \alpha n$, we have that $$\begin{aligned} \left\lfloor \frac{n}{ q (d-\gamma n ) } \right\rfloor \le \left\lfloor \frac{n}{q d+1} \right\rfloor +1 \label{eqn:k^*}\end{aligned}$$ First suppose that $G$ is not oriented. Recall [\[itm:balancingpathsystemcombined:5\]](#itm:balancingpathsystemcombined:5){reference-type="ref" reference="itm:balancingpathsystemcombined:5"} that $|V_{i*}| \ge d - \gamma n$ for all $i \in [k]$. Together with [\[eqn:k\^\*\]](#eqn:k^*){reference-type="eqref" reference="eqn:k^*"}, we have $k \le \left\lfloor n/ (d-\gamma n) \right\rfloor \le k^*$. On the other hand, since $S(\mathcal{P}^{0})$ has at least $k^*$ connected components, we have $k = | V( S(\mathcal{P}^{0}) ) | \ge k^*$. Thus $k = k^*$. Furthermore, $S(\mathcal{P}^{0})$ is an empty graph on $[k^*]$ (or else $S(\mathcal{P}^{0})$ would have fewer than $k^*$ connected components). Next, suppose that $G$ is oriented. Let $I_1 , \dots , I_{a+b}$ be connected components of $S(\mathcal{P}^{0})$ such that $|I_j| = 1$ if and only if $j \in [a]$. Note that $a+b \ge k^*$ and $\{ \bigcup_{i \in I_j } (V_{i*} \cup V_{*i}) : j \in [a+b] \}$ is a partition of $V(G)$. By Proposition [Proposition 24](#prop:orientedcomponent){reference-type="ref" reference="prop:orientedcomponent"}, we have $$\begin{aligned} n = \sum_{j \in [a+b]} \left| \bigcup_{i \in I_j } (V_{i*} \cup V_{*i}) \right| \ge 2(d-\gamma n) a + \frac{9(d-5\gamma n)}4 b \ge 2(d-\gamma n) k^* + \frac{bd}8,\end{aligned}$$ where the last inequality holds as $\gamma \ll \alpha$. Hence [\[eqn:k\^\*\]](#eqn:k^*){reference-type="eqref" reference="eqn:k^*"} implies that $b = 0$ (since $1/n\ll \gamma \ll \alpha$) and $a = k^*$, and the result follows. ◻ Since $S(\mathcal{P}^{0})$ is an empty graph on $[k^*]$, we deduce that $\mathcal{Q}^0$ is trivial as defined at the start of the section. By the moreover statement of Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"}, we have 1. $\sum_{i,j \in [k] \colon i \ne j} |V_{ij}| \le k^2\gamma n \le \gamma^{1/2} n$ and [\[itm:m1(int)\]]{#itm:m1(int) label="itm:m1(int)"} 2. for all distinct $i,j \in [k]$ and $v \in V_{ij}$, we have $d^+(v,V_{*i}) - d^-(v, V_{i *}) \le 100k^{12} \gamma^{1/3}d \le \gamma^{1/2} n$. [\[itm:m2(int)\]]{#itm:m2(int) label="itm:m2(int)"} By Proposition [Proposition 21](#prop:balancepartition){reference-type="ref" reference="prop:balancepartition"} (with $\gamma^{1/2}$ playing the role of $\gamma$), there exists a partition $\mathcal{P}' = \{V_{ii}':i \in [k]\}$ of $V(G)$ (and we take $V_{ij}' = \emptyset$ for $i \not= j$) such that, for all $i \in [k]$, 1. $|V_{ii} \triangle V'_{ii}| \le \gamma^{1/2} n$; [\[itm:symdiff\]]{#itm:symdiff label="itm:symdiff"} 2. $\delta^0( G [ V_{ii}' ] ) \ge (d/k) - \gamma^{1/2} n$. [\[itm:deg\]]{#itm:deg label="itm:deg"} Finally, apply Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"} (with $\mathcal{P}'$ playing the role of $\mathcal{P}$) by noting that property [\[item:each-part-is-at-least-d-over-2\]](#item:each-part-is-at-least-d-over-2){reference-type="ref" reference="item:each-part-is-at-least-d-over-2"} of Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"} holds by property [\[itm:balancingpathsystemcombined:5\]](#itm:balancingpathsystemcombined:5){reference-type="ref" reference="itm:balancingpathsystemcombined:5"} of Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"} together with [\[itm:m1(int)\]](#itm:m1(int)){reference-type="ref" reference="itm:m1(int)"} and [\[itm:symdiff\]](#itm:symdiff){reference-type="ref" reference="itm:symdiff"} above, and properties [\[item:relation-n-d-k\]](#item:relation-n-d-k){reference-type="ref" reference="item:relation-n-d-k"} and [\[item:degree-is-large-inside-diagonal\]](#item:degree-is-large-inside-diagonal){reference-type="ref" reference="item:degree-is-large-inside-diagonal"} of Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"} hold by Claim [Claim 25](#clm:S(P)-empty){reference-type="ref" reference="clm:S(P)-empty"} and [\[itm:deg\]](#itm:deg){reference-type="ref" reference="itm:deg"} above, respectively. Thus, in $G$, we obtain a non-trivial $\mathcal{P}'$-balanced path system $\mathcal{Q}$ with $e (\mathcal{Q}) \le k \leq \gamma^{1/2}n$. Also, if $\mathcal{P}^*$ is the $\mathcal{Q}$-contracted partition of $\mathcal{P}'$, then $S(\mathcal{P}^*)$ has at most $k^* - 1 \leq n/(qd + 1)$ connected components since $\mathcal{Q}$ is non-trivial. This gives [\[itm:balancingpathsystemcombined:c\]](#itm:balancingpathsystemcombined:c){reference-type="ref" reference="itm:balancingpathsystemcombined:c"} and [\[itm:balancingpathsystemcombined:d\]](#itm:balancingpathsystemcombined:d){reference-type="ref" reference="itm:balancingpathsystemcombined:d"} in Lemma [Lemma 19](#lma:balancingpathsystemcombined){reference-type="ref" reference="lma:balancingpathsystemcombined"}, while [\[itm:m1(int)\]](#itm:m1(int)){reference-type="ref" reference="itm:m1(int)"} and [\[itm:symdiff\]](#itm:symdiff){reference-type="ref" reference="itm:symdiff"} imply [\[itm:balancingpathsystemcombined:a\]](#itm:balancingpathsystemcombined:a){reference-type="ref" reference="itm:balancingpathsystemcombined:a"}, and [\[itm:deg\]](#itm:deg){reference-type="ref" reference="itm:deg"} implies [\[itm:balancingpathsystemcombined:b\]](#itm:balancingpathsystemcombined:b){reference-type="ref" reference="itm:balancingpathsystemcombined:b"}. ◻ # Proof of Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} {#sec:pf_balancedpath} Recall that in Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"}, we are given a $d$-regular digraph $G^0$ on $n$ vertices together with a partition $\mathcal{P}=\{V_{ij}:i,j\in[k]\}$ of $V(G^0)$ satisfying certain size and degree conditions, and we must find a $\mathcal{P}$-balanced path system $\mathcal{Q}$ with few edges. Moreover, we require that $\mathcal{Q}$ is non-trivial under certain circumstances. We begin with a sketch of our proof approach for this lemma (ignoring the moreover part for now). Recall the definitions at the start of Section [5](#sec:Path-Contraction){reference-type="ref" reference="sec:Path-Contraction"}. Let $V^+$ and $V^-$ be two disjoint copies of $V(G^0)$. Let $V_{1*}^+, \dots, V_{k*}^+$ to be the partition of $V^+$ corresponding to $V_{1*}, \dots, V_{k*}$ in $V(G^0)$. Similarly, let $V_{*1}^-, \dots, V_{*k}^-$ to be the partition of $V^-$ corresponding to $V_{*1}, \dots, V_{*k}$ in $V(G^0)$. Let $H$ be the bipartite digraph with bipartition $V^+, V^-$ such that for each $x \in V^+_{i*}$ and $y \in V^-_{*j}$, we have $xy \in E(H)$ if and only if $xy \in E(G^0)$ and $i \ne j$. So there is a bijection between $E(H)$ and $E(\mathcal{B}( G^0, \mathcal{P}))$. A naive approach is to find, for each $i, j \in [k]$ with $i \ne j$, a matching $M_{ij}$ in $H[V_{i*}^+, V^-_{*j}]$ with $e(M_{ij}) = e(G^0_{ij})/d$. If $Q_{ij}$ is the subgraph in $G^0$ corresponding to $M_{ij}$, then $Q = \bigcup_{i,j \in [k]} Q_{ij}$ is $\mathcal{P}$-balanced by construction since $G^0$ is $d$-regular. It is relatively easy to guarantee $Q$ has few edges by first passing from $G^0$ to a suitable subdigraph $G$ (see Proposition [Proposition 31](#prop:subgraph){reference-type="ref" reference="prop:subgraph"}). However, there are three problems with this approach: (i) $\Delta^0(Q)$ might be greater than one (meaning that $Q$ is not a path system), (ii) $e(G^0_{ij})/d$ may not be an integer, and (iii) $Q$ may contain cycles, even if $\Delta^0(Q) \leq 1$. To overcome (i) and (ii), we consider a suitable flow problem by converting $H$ to a network $\mathcal{F}^* = (F^*,w, s^*, t^*)$ as follows (the formal definition of a network and flow are stated in Section [7.1](#subsect:flows){reference-type="ref" reference="subsect:flows"}). Starting with the graph $H$ (with all edges of $H$ having capacity 1), we add new vertices $s^*, s_1, \dots, s_k, t^*, t_1, \dots, t_k$, where $s^*$ and $t^*$ are the source and sink respectively and where the $s_i$ and $t_j$ are viewed as 'local' sources and sinks, respectively. For each $i,j \in [k]$ and each $x^+ \in V^+_{i*}$, $y^- \in V^-_{*j}$, we add the edges $s_i x^+$ and $y^- t_j$, each of capacity $1$. For each $i \in [k]$, we add the edge $s^* s_i$ of capacity $\max \{|V_{i*}| - |V_{*i}|, 0\}$, the edge $t_i t^*$ of capacity $\max \{|V_{*i}| - |V_{i*}|, 0\}$ and the edge $t_i s_i$ of infinite capacity. This gives the network $\mathcal{F}^*$; see Figure [\[fig:F\*\]](#fig:F*){reference-type="ref" reference="fig:F*"}. Consider a maximum integer flow $f$ for $\mathcal{F}^*$. Define $Q$ to be the subdigraph of $\mathcal{B}(G^0, \mathcal{P})$ such that $xy \in E(Q)$ if and only if $x^+ y^-$ has a flow of one in $f$, where $x^+$ and $y^-$ are the corresponding copies of $x$ and $y$ in $V^+$ and $V^-$, respectively. Note that $\Delta^0(Q) \le 1$ by construction. Also it is easy to check that $Q$ is $\mathcal{P}$-balanced provided all edges at $s^*$ (and hence also at $t^*$) are saturated by $f$. To guarantee this latter condition on $f$, it turns out that it is enough to find a fractional flow $f^*$ such that $f_{ij}^*$, the total amount of flow in $f^*$ through the edges in $H[V_{i*},V_{*j}]$, is roughly $e(G^0_{ij})/d$ for all $i\neq j$. Such a fractional flow is almost a maximum flow, and we can use the max-flow min-cut theorem to convert this fractional flow into an integer flow that saturates the edges at $s^*$ (and $t^*$); see Claim [Claim 35](#clm:fractionalflow){reference-type="ref" reference="clm:fractionalflow"}. This also addresses (ii) above. To deal with (iii), suppose we could find a a matching $M^0$ of $G^0$ such that $e(M^0[V_{i*},V_{*j}]) \ge e(G^0_{ij})/d$ for all $i,j \in [k]$. Then defining $H$ (and $\mathcal{F}^*$) using $M^0$ instead of $G^0$, we could apply the same argument as before to obtain a $\mathcal{P}$-balanced subgraph $Q \subseteq M^0$ (which is therefore necessarily a path system as required). However, it is not always possible to find such a matching, so instead, for each $i,j$, we find a large "extendable mathcing" in $G^0_{ij}$ which consists of a matching $M_{ij} \subseteq G^0_{ij}$ together with sets of vertices $X_{ij}^+ \subseteq V_{i*}$ and $X_{ij}^- \subseteq V_{*j}$ where each vertex $v^+ \in X_{ij}^+$ has high outdegree in $G^0_{ij}$ and each vertex $v^- \in X_{ij}^-$ has high indegree in $G^0_{ij}$, and where $\bigcup_{i,j} M_{ij}$ is a matching (see Lemma [Lemma 33](#lem:matching){reference-type="ref" reference="lem:matching"}). Here the very large degrees of vertices in $X_{ij} = X_{ij}^+ \cup X_{ij}^-$ allow us to greedily extend $M_{ij}$ into a suitably large path system[^8] (see Proposition [Proposition 34](#prop:extendingmatching){reference-type="ref" reference="prop:extendingmatching"}). These path systems would be disjoint (as required) if the $X_{ij}$ are disjoint, but a priori the $X_{ij}$ will not be disjoint. Therefore, we must modify the $X_{ij}$ so that each vertex is assigned to at most one $X_{ij}^+$ and at most one $X_{ij}^-$. It turns out that this assignment problem is not too hard to incorporate into the flow problem described earlier (we simply add an edge of capacity $1$ from $v^+ \in V^+$ to $t_j$ if $v^+ \in X_{ij}^+$ and similarly if $v^- \in X_{ij}^-$). Solving the flow problem gives us the *disjoint* "extendable matchings" we seek (meaning the $X_{ij}$'s are disjoint), which can greedily be extended to give the desired path system. ## Flows {#subsect:flows} We recall some common definitions and facts about flow networks. We note that flows are only used in the proof of Theorem [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} (and not in any preliminary results). A *flow network* is a tuple $\mathcal{F} = (F,w,S,T)$, where $F$ is a digraph, $w\colon E(F) \to \mathbb{R}_{\geq 0}$ is the *capacity* function, and $S \subset V$ is a set of *sources* (i.e. each $s \in S$ only has outedges incident to it) and $T \subset V$ is a set of *sinks* (i.e. each $t \in T$ only has inedges incident to it). A *flow* for the flow network $\mathcal{F}$ is a function $f \colon E(F) \to \mathbb{R}_{\geq 0}$ such that, for all $e\in E$, we have $f(e)\leq w(e)$ and, for all $v\in V\setminus (S \cup T)$, we have $\sum_{u\in N^-_F(v)}f(uv)=\sum_{u\in N^+_F(v)}f(vu)$. We define the *value* of $f$ as $val(f)\coloneqq \sum_{s\in S}\sum_{v\in N^+_F(s)} f(sv)$. A *maximum flow* in a given flow network is a flow $f$ that maximises $val(f)$. We say that $f$ is an *integer flow* if $f(e)$ is an integer for all $e \in E$, and to emphasize the contrast, we sometimes refer to a flow that is not necessarily an integer flow as a *fractional flow*. For any set of edges $E' \subseteq E$, we write $f(E'):= \sum_{e \in E'}f(e)$. If $S = \{s\}$ and $T = \{t\}$, then we simply write $(F,w,s,t)$. There are variants of the above notions that we will use. In particular, as well as having edge capacities, a flow network can also have vertex capacities (which restricts the amount of flow that can pass through a vertex) so that now $w: E(F) \cup V(F) \to \mathbb{R}^+$, and for each $v \in V(F) \setminus (S \cup T)$ a flow is defined as before with the added restriction that $f(v) \leq w(v)$, where $f(v):=\sum_{u\in N^-_F(v)}f(uv)=\sum_{u\in N^+_F(v)}f(vu)$. One can easily reduce this to the situation of just edge capacities by replacing each vertex $v$ with a directed edge $v^-v^+$ of capacity $w(v)$ and each directed edge $uv$ (or $vw$) with $uv^-$ (or $v^+w$, respectively) where new edges inherit the capacities of their old counterparts. We say that a flow $f$ *saturates* an edge $e$ (or a vertex $v$) if $f(e) = w(e)$ (or $f(v) = w(v)$, respectively). Let $(F,w,s,t)$ be a flow network. For a partition $(U,W)$ of $V(F)$ with $s\in U$, $t\in W$ the edge set $E_F(U,W)$ is called a *cut*. (Recall that $E_F(U,W)$ is the set of edges in $F$ from $U$ to $W$.) The *capacity* of a cut $E_F(U,W)$ is the sum of the capacities of its edges, i.e. $w(E_F(U,W)) \coloneqq \sum_{e \in E_F(U,W)}w(e)$. A *minimum cut* of the given flow network is a cut of minimum capacity. We make use of the following well-known theorem. **Theorem 26** (Max-flow min-cut [@DantzigFukerson]). *Let $(F,w,s,t)$ be a flow network.* 1. *[\[itm:maxflow1\]]{#itm:maxflow1 label="itm:maxflow1"} If $f$ is a flow and $E_F(U,W)$ is a cut then $\mathit{val}(f) \leq \mathit{w}(E_F(U,W))$.* 2. *[\[itm:maxflow2\]]{#itm:maxflow2 label="itm:maxflow2"} We have that $f$ is a maximum flow and $E_F(U,W)$ is a minimum cut if and only if $\mathit{val}(f)=\mathit{w}(E_F(U,W))$ and in that case $f$ saturates every edge in $E_F(U,W)$.* 3. *[\[itm:maxflow3\]]{#itm:maxflow3 label="itm:maxflow3"} If all capacities are non-negative integers, i.e. $w(e) \in \mathbb{Z}_{\geq 0}$ for all $e \in E(F)$, then there exists a maximum flow $f$ that is an integer flow.* We also use the following whose proof is omitted as it is straightforward. **Proposition 27**. *Let $\mathcal{F} = (F,w,s,t)$ be a flow network and $f$ a flow for $\mathcal{F}$. Then, for any edge $e\in E(F)$, there exists a flow $f'$ for $\mathcal{F}$ such that $f'(e)=0$ and $val(f') \ge val(f)-f(e)$.* ## Preliminaries We will require Vizing's theorem for multigraphs in the proof of Lemma [Lemma 33](#lem:matching){reference-type="ref" reference="lem:matching"}. Let $H$ be an (undirected) multigraph (without loops). The multiplicity $\mu(H)$ of $H$ is the maximum number of edges between two vertices of $H$, and, as usual, $\Delta(H)$ is the maximum degree of $H$. A proper $k$-edge-colouring of $H$ is an assignment of $k$ colours to the edges of $H$ such that incident edges receive different colours. **Theorem 28** ([@Vizing]; see e.g. [@Diestel]). *Any multigraph $H$ has a proper $k$-edge colouring with $k = \Delta(H) + \mu(H)$ colours. In particular, by taking the largest colour class, there is a matching in $H$ of size at least $e(H)/(\Delta(G)+\mu(G))$.* We will also require the following simple proposition about decomposing acyclic digraphs into paths. **Proposition 29** ([@LPST Proposition 2.6]). *Let $G$ be an acyclic digraph. Then the edges of $G$ can be partitioned into $\sum_{v \in V(G)}\left| d^+(v) - d^-(v) \right|/2$ directed paths.* We will need the following property of regular digraphs; its proof is a simple exercise by considering $\sum_{v \in V_{i*}} d^+(v)$ and $\sum_{v \in V_{*i}} d^-(v)$. **Proposition 30** ([@LPY Proposition 3.2]). *Let $G$ be a $d$-regular digraph, $k\in \mathbb{N}$, and $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$. Then, for all $i \in [k]$, we have $d(|V_{i*}|-|V_{*i}|)= e( G_{i*} ) - e(G_{*i})$.* The next proposition shows that we can consider a simpler subgraph $G$ in $\mathcal{B}( G^0 , \mathcal{P} )$. **Proposition 31**. *Let $G^0$ be a digraph, $k,d \in \mathbb{N}$ and $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G^0)$ such that for all $i \in [k]$, $e( G^0_{i*} ) - e(G^0_{*i}) = d ( |V_{i*}|-|V_{*i}| )$. Then, by reordering $[k]$ if necessary, there exists a subdigraph $G$ of $\mathcal{B}(G^0, \mathcal{P})$ such that $$\begin{aligned} e(G) & \le d (k-1) \sum_{i \in [k]}\left| |V_{i*}|-|V_{*i}| \right|/2,\\ e( G_{i*} ) - e(G_{*i}) & = d(|V_{i*}|-|V_{*i}|) \text{ for all $i \in [k]$,}\end{aligned}$$ and for all $i,j \in [k]$ with $i \le j$, $e(G_{ji}) = 0$.* *Proof.* Define an auxiliary multidigraph $H^0$ on $[k]$ (with loops) such that there are precisely $e(G^0_{ij})$ many $ij$ edges in $H^0$. There is a natural bijection between $E(G^0)$ and $E(H^0)$. Let $H$ be an acyclic subdigraph of $H^0$ obtained by successively removing all edges of a directed cycle. Note that, for all $i \in [k]$, $$\begin{aligned} d^+_{H}(i) - d^-_{H}(i) = d^+_{H^0}(i) - d^-_{H^0}(i) = e( G^0_{i*} ) - e(G^0_{*i}) = d(|V_{i*}|-|V_{*i}|). \label{eqn:prop1}\end{aligned}$$ Since $H$ is acyclic, by relabelling if necessary, we may assume that there are no edges $ji$ with $i \le j$. By Proposition [Proposition 29](#prop:path-decomposition){reference-type="ref" reference="prop:path-decomposition"}, $E(H)$ can be decomposed into $\sum_{i \in [k]}\left| d^+_{H}(i) - d^-_{H}(i) \right|/2$ directed paths. Since each path can have length at most $k-1$ (as $H$ has $k$ vertices), we have $$\begin{aligned} e(H) & \le (k-1) \sum_{i \in [k]}\left| d^+_{H}(i) - d^-_{H}(i) \right| / 2 \overset{\mathclap{\text{\eqref{eqn:prop1}}}}{=} d (k-1) \sum_{i \in [k]}\left| |V_{i*}|-|V_{*i}| \right|/2.\end{aligned}$$ The result follows by setting $G$ to be the subdigraph of $G^0$ corresponding to $H$. ◻ We need the following lemma from our previous work [@LPY]. It states that given a set of matchings of low total maximum degree, one can select a relatively large number of the edges from each matching so that the union of selected edges is also a matching. **Lemma 32** ([@LPY Lemma 4.2]). *Let $k,\ell\in\mathbb{N}$ and $M_1,M_2,\ldots,M_{\ell}$ be matchings with $\Delta\left(\bigcup_{i \in [\ell]} M_i\right)\leq k$. Suppose $e(M_i)> 8k^6\ln \ell$ for all $i \in [\ell]$. Then, there exists a matching $M \subseteq \bigcup_{i \in [\ell]}M_i$ with $|M\cap M_i|\geq e(M_i)/2k^2$ for all $i \in [\ell]$.* Any graph (or digraph) either has a large matching or has all its edges incident to a small set of vertices (so these vertices have relatively large degree). The following lemma allows us to interpolate between these extremes and moreover does it simultaneously for all $G_{ij}$. This corresponds to the "extendable matchings" described in the sketch proof. **Lemma 33**. *Let $G$ be a digraph, $k,d\in \mathbb{N}$, and $\mathcal{P} = \{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$. Let $\theta \in(0,1)$ with $\theta d \ge 2$. For all $i,j \in [k]$ and $v \in V(G)$, let $X_{ij}^+ = \{v \in V(G) : d_{G_{ij}}^+(v) \ge \theta d \}$ and define $X_{ij}^-$ similarly. Then, for each $i,j \in [k]$, there exists a matching $M_{ij}$ of $G_{ij}$ such that* 1. *$e(G_{ij}) \le \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) + (6 k^2 \theta d) e( M_{ij} ) + 50 k^{10} \theta d$;* 2. *for all $uv \in M_{ij}$, $u \notin X_{ij}^+$ and $v \notin X_{ij}^-$;* 3. *$\bigcup_{i,j \in [k]} M_{ij}$ is a matching.* *Proof.* Consider $i,j \in [k]$. Let $H_{ij}$ be the multigraph obtained from $G_{ij}$ by deleting all the edges $uv$ with $u\in X_{ij}^+$ or $v \in X_{ij}^-$, and by making all the edges undirected. Note that we have $\Delta(H_{ij}) + \mu(H_{ij}) \leq 2\theta d+2$ and $$\begin{aligned} e(H_{ij}) \geq e(G_{ij}) - \left( \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) \right). \label{eqn:e(H_{ij})} \end{aligned}$$ Then, by Vizing's theorem for multigraphs (Theorem [Theorem 28](#thm:Vizing){reference-type="ref" reference="thm:Vizing"}), there exists a matching $M_{ij}^H$ in $H_{ij}$ of size at least $e(H_{ij})/(2\theta d+2) \ge e(H_{ij})/3\theta d$. Let $M_{ij}^0$ be the corresponding matching in $G_{ij}$. If $e(M_{ij}^0) \le 16 k^{10}$, then we set $M_{ij}^0$ to be empty. Thus, together with [\[eqn:e(H\_{ij})\]](#eqn:e(H_{ij})){reference-type="eqref" reference="eqn:e(H_{ij})"} we have $$\begin{aligned} e(G_{ij}) \le \left( \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) \right) + (3 \theta d) e( M_{ij}^0 ) + 50 k^{10} \theta d .\end{aligned}$$ Observe that $\Delta\left(\bigcup_{i,j \in [k]} M_{ij}^0\right)\leq k$. Apply Lemma [Lemma 32](#lem:matchinglemma){reference-type="ref" reference="lem:matchinglemma"} for nonempty matchings $M_{ij}^0$ (with $\ell\le k^2$) to obtain $M_{ij} \subseteq M_{ij}^0$ (set $M_{ij}=\emptyset$ if $M_{ij}^0=\emptyset$) such that $\bigcup_{i,j \in [k]}M_{ij}$ is a matching and, for all $i,j \in [k]$, $e(M_{ij}) \ge e(M_{ij}^0) / 2k^2$. The result follows. ◻ Recall that for any directed matching $M$, $V^+(M)$ and $V^-(M)$ are the sets of starting and ending vertices of the directed edges in $M$, respectively. Formally, $V^+(M) =\{v\in V(M): vw\in E(M) \text{ for some }w\in V(M) \}$ and similarly for $V^-(M)$. **Proposition 34**. *Let $G$ be a digraph, $k\in \mathbb{N}$, and $\mathcal{P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$. Let $W \subseteq V(G)$. Suppose that, for each $i,j \in [k]$ with $i \ne j$, there exist $Y_{ij}^+ \subseteq V_{i*}$, $Y_{ij}^- \subseteq V_{*j}$ and $M_{ij} \subseteq G_{ij}$ such that* 1. *$M = \bigcup_{i, j \in [k]} M_{ij}$ is a matching;* 2. *both of $\{Y_{ij}^+, V^+(M_{ij}) : i,j \in [k] \}$ and $\{ Y_{ij}^-, V^-(M_{ij}) : i,j \in [k] \}$ are sets of disjoint sets;* 3. *[\[itm:extendingmatchingc\]]{#itm:extendingmatchingc label="itm:extendingmatchingc"} for all $y^+ \in Y_{ij}^+$ and $y^- \in Y_{ij}^-$, $d_{G_{ij}}^+(y^+),d_{G_{ij}}^-(y^-) \ge 2\sum_{i,j \in [k]} ( |Y_{ij}^+| + |Y_{ij}^-| + e(M_{ij}) ) + |W|+1$.* *Then $\mathcal{B}(G, \mathcal{P})$ contains a path system $\mathcal{Q}$ such that the following hold:* 1. *[\[itm:extendingmatchingi\]]{#itm:extendingmatchingi label="itm:extendingmatchingi"} $e(\mathcal{Q}_{ij}) = |Y_{ij}^+| + |Y_{ij}^-| + e(M_{ij})$ for all $i,j \in [k]$, where $\mathcal{Q}_{ij}=\mathcal{Q}\cap E(G_{ij})$;* 2. *[\[itm:extendingmatchingii\]]{#itm:extendingmatchingii label="itm:extendingmatchingii"} $V(\mathcal{Q})\cap W \subseteq \bigcup_{i,j \in [k]} (Y_{ij}^+ \cup Y_{ij}^- \cup V(M_{ij}))$;* 3. *[\[itm:extendingmatchingiii\]]{#itm:extendingmatchingiii label="itm:extendingmatchingiii"} if $y^+\in Y_{ij}^+\setminus V^-(M)$, then there exists $u\in V_{*j}\setminus W$ such that the single edge $y^+u$ is a path in $\mathcal{Q}$, and a similar statement holds for $y^-\in Y_{ij}^-\setminus V^+(M)$.* *Proof.* Start by setting $\mathcal{Q}=M$ and then for each $y \in Y_{ij}^+$ (and $y \in Y_{ij}^-$), greedily add an edge $yv$ (and $vy$, respectively) in $G_{ij} \setminus W$ such that $v$ avoids all current vertices in $\mathcal{Q}$ and all vertices in $\bigcup_{i,j}(Y_{ij}^+\cup Y_{ij}^-)$, which is possible by [\[itm:extendingmatchingc\]](#itm:extendingmatchingc){reference-type="ref" reference="itm:extendingmatchingc"}. It is clear that we always maintain a path system, and that [\[itm:extendingmatchingi\]](#itm:extendingmatchingi){reference-type="ref" reference="itm:extendingmatchingi"}--[\[itm:extendingmatchingiii\]](#itm:extendingmatchingiii){reference-type="ref" reference="itm:extendingmatchingiii"} hold by construction. ◻ ## Proof of Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"} {#proof-of-lemma-lembalancedpath} We now prove Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"}. *Proof of Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"}.* We split the proof into several steps. **Step 1: Defining $G$.** By Proposition [Proposition 30](#prop:partition-regular-graphs){reference-type="ref" reference="prop:partition-regular-graphs"}, $e( G^0_{i*} ) - e(G^0_{*i}) = d(|V_{i*}|-|V_{*i}|)$ for all $i \in [k]$. Apply Proposition [Proposition 31](#prop:subgraph){reference-type="ref" reference="prop:subgraph"} and, without loss of generality, obtain a subdigraph $G$ of $\mathcal{B}(G^0, \mathcal{P})$ such that $$\begin{aligned} e( G ) & \le (k-1)d \sum_{i \in [k]} | |V_{i*}|-|V_{*i}| |/2 \overset{\mathclap{\text{\ref{itm:balancedpath3}}}}{\le} k^2 \gamma d n. \label{eqn:subgraph2}\\ e( G_{i*} ) - e(G_{*i}) & = d(|V_{i*}|-|V_{*i}|) \text{ for all $i \in [k]$,} \label{eqn:subgraph1}\end{aligned}$$ and, for all $i,j \in [k]$ with $i \le j$, $e(G_{ji}) = 0$. **Step 2: Finding $X_{ij}^+$, $X_{ij}^-$ and $M_{ij}$.** For all $i,j \in [k]$, let $$\begin{aligned} X_{ij}^+ = \{v \in V(G) : d_{G_{ij}}^+(v) \ge \gamma^{1/3} d \} \,\text{ and }\, X_{ij}^- = \{v \in V(G) : d_{G_{ij}}^-(v) \ge \gamma^{1/3} d \}. \label{eqn:DEFINITION-X-ij}\end{aligned}$$ Observe that $X_{ij}^+\subseteq V_{i*}$ and $X_{ij}^-\subseteq V_{*j}$ for all $i,j\in[k]$. Note that $\gamma^{1/3} d \ge \gamma^{1/3} \alpha n \ge 2$ as $1/n\ll\gamma\ll \alpha$. Apply Lemma [Lemma 33](#lem:matching){reference-type="ref" reference="lem:matching"} to $G$ (with $\theta = \gamma^{1/3}$) and obtain a matching $M_{ij}$ of $G_{ij}$ for $i,j \in [k]$, such that 1. $e(G_{ij}) \le \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) + (6 k^2 \gamma^{1/3} d) e( M_{ij} ) + 50 k^{10}\gamma^{1/3} d$; [\[itm:Msize\]]{#itm:Msize label="itm:Msize"} 2. for all $uv \in M_{ij}$, $u \notin X_{ij}^+$ and $v \notin X_{ij}^-$; 3. $\bigcup_{i,j \in [k]} M_{ij}$ is a matching. [\[itm:M\]]{#itm:M label="itm:M"} By deleting vertices in $X_{ij}^+ \cup X_{ij}^-$ and edges of $M_{ij}$ if necessary, we may assume that the RHS of [\[itm:Msize\]](#itm:Msize){reference-type="ref" reference="itm:Msize"} is bounded above by $e(G_{ij})+d$. Hence $$\begin{aligned} \gamma^{1/3} d ( |X_{ij}^+|+ |X_{ij}^-|+ e( M_{ij} ) ) & \le \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^- ( x^- ) + (6 k^2 \gamma^{1/3} d ) e( M_{ij} ) + 50 k^{10}\gamma^{1/3} d \\ & \le e(G_{ij})+ d \overset{\text{\eqref{eqn:subgraph2}}}{\le} k^2 \gamma d n +d \le \gamma^{2/3} k^{-2} d^2 /3,\end{aligned}$$ where the last inequality holds as $d \ge \alpha n$ and $1/n \ll \gamma \ll \alpha, 1/k$. Therefore, 1. $\sum_{i,j \in [k]} \left ( |X_{ij}^+| + |X_{ij}^-| + e(M_{ij}) \right) \le \gamma^{1/3} d / 3$.[\[itm:size\]]{#itm:size label="itm:size"} **Step 3: Defining flow networks.** For $i,j \in [k]$, let $$\begin{aligned} M =\bigcup_{i,j \in [k]} M_{ij}, \quad X^+_{i*} = \bigcup_{j \in [k]} \left( X_{ij}^+ \cup V^+(M_{ij}) \right), \text{ and } X^-_{*j} = \bigcup_{i \in [k]} \left( X_{ij}^- \cup V^-(M_{ij}) \right).\end{aligned}$$ Note that $X^+_{i*} \subseteq V_{i*}$ and $X^-_{*j} \subseteq V_{*j}$. Let $X^+ = \bigcup_{i \in [k]} X^+_{i*}$ and $X^- = \bigcup_{j \in [k]} X^-_{*j}$. We now define a flow network $\mathcal{F} = (F,w,S,T)$ with multiple sources and sinks as follows. Let $S= \{s_i: i \in [k]\}$ and $T= \{t_i: i \in [k]\}$. Let $V(F) = S \cup T \cup X^+ \cup X^-$. Here we treat $X^+$ and $X^-$ as disjoint, i.e. both $X^+$ and $X^-$ contain a distinct copy of $v$ for any vertex $v\in X^+\cap X^-$. For $v \in V(G)$, we write $v^+$ (and $v^-$) for the copy of $v$ belonging to $X^+$ (and $X^-$, respectively). We define $E(F)$ as the union of edge-disjoint paths as follows: for each $i,j \in [k]$ with $i < j$, - for each $x^+ \in X_{ij}^+$, we add the directed path $s_{i} x^+ t_{j}$; - for each $x^- \in X_{ij}^-$, we add the directed path $s_{i} x^- t_{j}$; - for each $e = uv \in M_{ij}$, we add the directed path $s_{i} u^+ v^- t_{j}$; - every edge and vertex has capacity $1$, i.e. $w(e) = 1$ for all $e \in E(F)$ and $w(v) = 1$ for all $v \in V(F) \setminus (S \cup T) = X^+ \cup X^-$. We modify the flow network slightly in the moreover statement. Let $$\begin{aligned} F_0=\begin{cases} F - \{v_0^+ \} - \{s_{i_0} v_0^-\} - E_F(X^+,v_0^-)&\text{ if }v_0\in V_{i_0j_0}\text{ exists satisfying \ref{item:degree-condition-v0}},\\ F&\text{ otherwise.} \end{cases}\end{aligned}$$ We further remove $v_0^-$ from $F_0$ if $v_0^-$ has no inneighbour in $F_0$. Write $w_0$ for the capacities on $F_0$ inherited from $F$ and define $\mathcal{F}_0 = (F_0,w_0,S,T)$. By the definition of $\mathcal{F}_0$, note for later that $$\begin{aligned} \text{if there is any flow through }v_0^-\text{, then it must be via some edge }s_iv_0^-\text{ with }i\ne i_0. \label{fact:incoming-edges-v0}\end{aligned}$$ Given a flow $f$ on $\mathcal{F}$ or $\mathcal{F}_0$, for each $i,j \in [k]$, let $f_{ij}$ be the sum of flow over all edges from $s_i \cup X^+_{i*}$ to $t_j \cup X^-_{*j}$, that is, $$\begin{aligned} f_{ij} = \sum_{e \in E_F(s_i \cup X^+_{i*}, t_j \cup X^-_{*j})} f(e).\label{eqn:definition-of-f-ij} \end{aligned}$$ Note that the total amount of flow going out of $s_i$ (all going to $X^+$) is $\sum_{j\in[k]}f_{ij}$, and the total amount of flow into $t_j$ is $\sum_{i\in[k]}f_{ij}$. We now reduce the lemma to the following claim about fractional flows. **Claim 35**. *To prove the lemma, it suffices to find a fractional flow $f$ on $\mathcal{F}_0$ such that $$\begin{aligned} \sum_{i,j \in [k]} \max\{ e(G_{ij})/d - f_{ij}, 0 \} < 1. \label{eqn:flow}\end{aligned}$$* *Proof of claim.* Assume $f$ is as given in the claim and we wish to prove the lemma. We may assume that $f_{ij} \le e(G_{ij})/d$ for all $i,j \in [k]$.[^9] We define a new flow network $\mathcal{F}^* = (F^*, w^*, s^*,t^*)$ with a single source $s^*$ and sink $t^*$ as follows. We obtain $\mathcal{F}^*$ from $\mathcal{F}_0$ by adding the new vertices $s^*$ and $t^*$ to $V(F_0)$ and, for each $i \in [k]$, - adding the edge $s^* s_i$ of capacity $\max \{|V_{i*}| - |V_{*i}|, 0\}$; - adding the edge $t_i t^*$ of capacity $\max \{|V_{*i}| - |V_{i*}|, 0\}$; - adding an edge $t_i s_i$ of infinite capacity; - giving the vertices in $S \cup T$ infinite capacity. We will use $f$ to find an integer flow in $\mathcal{F}^*$ of value $\sum_{i \in [k]} \left| |V_{i*}| - |V_{*i}|\right|/2$. Then, by looking at the vertices through which there is non-zero flow, we will define the sets $Y_{ij}^+$, $Y_{ij}^+$, and $M_{ij}^*$ satisfying the properties for Proposition [Proposition 34](#prop:extendingmatching){reference-type="ref" reference="prop:extendingmatching"}, which will give the desired path system $\mathcal{Q}$. First we find a fractional flow on $\mathcal{F}^*$ as follows. Let $\mathcal{F}^+ = (F^+, w^+,s^*,t^*)$ be obtained from $\mathcal{F}^*$ by adding, for each $i,j \in [k]$, an edge $s_i t_j$ with edge capacity $1$. Define a flow $f^+$ on $\mathcal{F}^+$ such that, for all $i,j \in [k]$, - $f^+(e) = f(e)$ for all $e \in E(F_0)$; - $f^+(s^* s_i) = \max \{|V_{i*}| - |V_{*i}|, 0\}$; - $f^+(t_j t^*) = \max \{|V_{*j}| - |V_{j*}|, 0\}$; - $f^+(t_i s_i) = \min \{ e(G_{i*})/d, e(G_{*i})/d\}$; - $f^+(s_i t_j) = e(G_{ij})/d - f_{ij}\ge 0$. One can check that $f^+$ is indeed a fractional flow[^10] on $\mathcal{F}^+$ with value $\sum_{i \in [k]} \left| |V_{i*}| - |V_{*i}|\right|/2$. By applying Proposition [Proposition 27](#prop:flow-reduction){reference-type="ref" reference="prop:flow-reduction"} to reduce the flow in $s_it_j$ to $0$ for all $i,j\in[k]$, we obtain a fractional flow on $\mathcal{F}^*$ of value $$\begin{aligned} r & = \frac12\sum_{i \in [k]} \left| |V_{i*}| - |V_{*i}|\right| - \sum_{i,j \in [k]} f^+(s_it_j) = \frac12 \sum_{i \in [k]} \left| |V_{i*}| - |V_{*i}|\right| - \sum_{i,j \in [k]} \left( \frac{e(G_{ij})}d - f_{ij} \right)\\ & \overset{\text{\eqref{eqn:flow}}}{>} \frac12\left(\sum_{i \in [k]} \left| |V_{i*}| - |V_{*i}|\right|\right)-1. \end{aligned}$$ Note that all edge capacities are integral, so Theorem [Theorem 26](#thm:maxflow){reference-type="ref" reference="thm:maxflow"} implies there exists an integral flow $f^*$ on $\mathcal{F}^*$ of value at least $\lceil r \rceil = \sum_{i \in [k]} \left| |V_{i*}| - |V_{*i}|\right|/2$. Moreover, since the cuts $E_{F^*}(s^*, V(F^*) \setminus \{s^*\})$ and $E_{F^*}( V(F^*) \setminus \{ t^*\}, t^*)$ have capacity $\sum_{i \in [k]} \left| |V_{i*}| - |V_{*i}|\right|/2$, Theorem [Theorem 26](#thm:maxflow){reference-type="ref" reference="thm:maxflow"} implies that $f^*$ saturates all the edges $s^*s_i$ and $t_jt^*$ for all $i,j \in [k]$. We now define $Y_{ij}^+ \subseteq X_{ij}^+$, $Y_{ij}^-\subseteq X_{ij}^-$ and $M^*_{ij} \subseteq M_{ij}$ as follows. Initially, set $Y_{ij}^+ = Y_{ij}^- = M_{ij}= \emptyset$ for all $i,j \in [k]$. For each $x^+\in X^+$ with $f^*(x^+) = 1$, since $f^*$ is an integral flow, there exist unique $i,j\in[k]$ such that either $f^*(s_i x^+)=f^*(x^+t_j)=1$ or $f^*(s_i x^+)=f^*(x^+z)=f^*(zt_j)=1$ for some (unique) $z\in X^-$. In the former case we add $x^+$ to $Y_{ij}^+$ while we add $x^+z$ into $M_{ij}^*$ for the latter. In a similar way, for $x^- \in X^-$ with $f^*(x^-) = 1$, we either add $x^-$ to the set $Y_{ij}^-$ or add an edge ending at $x^-$ into $M_{ij}^*$. Note that $$\begin{aligned} |Y_{ij}^+| + |Y_{ij}^-| + e (M^*_{ij}) =\sum_{e\in E_{F^*}(s_i\cup X_{i*}^+,t_j\cup X_{*j}^-)}f^*(e) := f^*_{ij}. \label{eqn:flow2}\end{aligned}$$ Clearly $M^* : = \bigcup_{i,j \in[k]} M^*_{ij} \subseteq M$ is a matching by [\[itm:M\]](#itm:M){reference-type="ref" reference="itm:M"}. Both of $\{Y_{ij}^+, V^+(M_{ij}^*) : i,j \in [k] \}$ and $\{ Y_{ij}^-, V^-(M_{ij}^*) : i,j \in [k] \}$ are sets of disjoint sets as each $x \in X^+ \cup X^-$ has vertex capacity of one. Set $$\begin{aligned} W = \begin{cases} \bigcup_{i,j \in [k] \colon i \ne j} V_{ij} & \text{if $\sum_{i,j \in [k] \colon i \ne j} |V_{ij}| \le k^2 \gamma n$,}\\ \emptyset & \text{otherwise.} \end{cases}\end{aligned}$$ For all $y^+ \in Y_{ij}^+ \subseteq X_{ij}^+$, we have $$\begin{aligned} d_{G_{ij}}^+(y^+) & \overset{\mathclap{\text{\eqref{eqn:DEFINITION-X-ij}}}}{\ge} \gamma^{1/3} d \overset{\mathclap{\text{\ref{itm:size}}}}{\ge} 2\sum_{i,j \in [k]} \left( |X_{ij}^+| + |X_{ij}^-| + e(M_{ij}) \right)+ \gamma^{1/3} d/3 \\ & \ge 2\sum_{i,j \in [k]} \left( |Y_{ij}^+| + |Y_{ij}^-| + e(M^*_{ij}) \right)+ |W|+1\end{aligned}$$ since $k^2\gamma n+1\le 2k^2\gamma n\le \gamma^{1/3}d/3$ as $\gamma \ll \alpha \le d/n$, and a similar inequality holds for all $y^- \in Y_{ij}^-$. We apply Proposition [Proposition 34](#prop:extendingmatching){reference-type="ref" reference="prop:extendingmatching"} and obtain a path system $\mathcal{Q}$ of $G$ such that 1. [\[itm:flow3\]]{#itm:flow3 label="itm:flow3"} $e(\mathcal{Q}_{ij}) = |Y_{ij}^+| + |Y_{ij}^-| + e(M^*_{ij}) \overset{\mathclap{\text{\eqref{eqn:flow2}}}}{=} f^*_{ij}$ for all $i,j \in [k]$; 2. $V(\mathcal{Q})\cap W \subseteq \bigcup_{i,j \in [k]} (Y_{ij}^+ \cup Y_{ij}^- \cup V(M^*_{ij}))$; [\[item:path-system-2\]]{#item:path-system-2 label="item:path-system-2"} 3. if $y^+\in Y_{ij}^+\setminus V^-(M^*)$, then there exists $u\in V_{*j}\setminus W$ such that the single edge $y^+u$ is a path in $\mathcal{Q}$, and a similar statement for $y^-\in Y_{ij}^-\setminus V^+(M^*)$.[\[item:path-system-3\]]{#item:path-system-3 label="item:path-system-3"} Consider $i \in [k]$. Recall that $N^-_{F^*}(s_i) = \{s^*, t_i\}$, $N^+_{F^*}(s_i) \subseteq X^+_{i*} \cup X^-$, and each vertex in $X^+$ has capacity one. We have $$\begin{aligned} f^*(s^*s_i) + f^*(t_is_i) & = \sum_{u \in N^-_{F^*}(s_i)} f^*( u s_i) = \sum_{v \in N^+_{F^*}(s_i)} f^*( s_i v )\\ &= \sum_{v \in X^+_{i*} \cup X^-} f^*( s_i v ) = \sum_{j \in [k]} \sum_{e \in E_{F^*} (s_i \cup X^+_{i*}, t_j \cup X^-_{*j})} f^*(e) = \sum_{j \in [k]} f^*_{ij} \overset{\mathclap{\text{\ref{itm:flow3}}}}{=} \sum_{j \in [k]} e(\mathcal{Q}_{ij}) \end{aligned}$$ and similarly we have $f^*(t_it^*) + f^*(t_is_i) = \sum_{j \in [k]} e(\mathcal{Q}_{ji})$. Since $f^*$ saturates the edges $s^*s_i$ and $t_it^*$, we deduce that, $$\begin{aligned} \sum_{j \in [k]} e(\mathcal{Q}_{ij}) - \sum_{j \in [k]} e(\mathcal{Q}_{ji}) &=f^*(s^*s_i)-f^*(t_it^*) = w^*(s^*s_i)-w^*(t_it^*) \\ &=\max \{|V_{i*}| - |V_{*i}|, 0\}-\max \{|V_{*i}| - |V_{i*}|, 0\} = |V_{i*}|-|V_{*i}|\end{aligned}$$ implying that $\mathcal{Q}$ is $\mathcal{P}$-balanced, as required. By applying Proposition [Proposition 31](#prop:subgraph){reference-type="ref" reference="prop:subgraph"} (with $d = 1$ and $G^0=\mathcal{Q}$), we may further assume that $$\begin{aligned} e(\mathcal{Q}) \le (k-1) \sum_{i \in [k]} | |V_{i*}|-|V_{*i}| |/2 \overset{\mathclap{\text{\ref{itm:balancedpath3}}}}{\le} k^2 \gamma n . \label{eqn:flow4}\end{aligned}$$ We now check the moreover statement of the Lemma [Lemma 20](#lem:balancedpath){reference-type="ref" reference="lem:balancedpath"}. If $\sum_{i,j \in [k] \colon i \ne j} |V_{ij}| > k^2\gamma n$ (that is, [\[item:size-condition-V-ij\]](#item:size-condition-V-ij){reference-type="ref" reference="item:size-condition-V-ij"} holds), then [\[eqn:flow4\]](#eqn:flow4){reference-type="eqref" reference="eqn:flow4"} implies that $\bigcup_{i,j \in [k] \colon i \ne j} V_{ij} \setminus V(\mathcal{Q}) \ne \emptyset$, so $\mathcal{Q}$ is non-trivial. Suppose that $\sum_{i,j \in [k] \colon i \ne j} |V_{ij}| \le k^2 \gamma n$ and there exists $v_0\in V_{i_0j_0}$ satisfying [\[item:degree-condition-v0\]](#item:degree-condition-v0){reference-type="ref" reference="item:degree-condition-v0"}. If $v_0 \notin V(\mathcal{Q})$, then $\mathcal{Q}$ is non-trivial. So suppose that $v_0 \in V(\mathcal{Q})$. Since $v_0 \in W$, by [\[item:path-system-2\]](#item:path-system-2){reference-type="ref" reference="item:path-system-2"} we must have $v_0 \in Y_{ij}^+ \cup Y_{ij}^- \cup V(M_{ij}^*)$ for some $i,j \in [k]$. This means that there is a flow through $v_0^-$ as $v_0^+ \notin V(F^*)$. By [\[fact:incoming-edges-v0\]](#fact:incoming-edges-v0){reference-type="eqref" reference="fact:incoming-edges-v0"}, $v_0 \in Y_{ij_0}^-$ for some $i \ne i_0$. Also, $v_0^+\notin V(F^*)$ implies that $v_0\notin V^+(M^*)$. By [\[item:path-system-3\]](#item:path-system-3){reference-type="ref" reference="item:path-system-3"}, we deduce that there exists $u \in V_{i*} \setminus W = V_{ii}$ such that $uv_0$ is a path in $\mathcal{Q}$. Note that $uv_0$ is a path from $V_{ii} \subseteq V_{*i}$ to $V_{i_0j_0} \subseteq V_{i_0 *}$. Hence $\mathcal{Q}$ is non-trivial, as required. ◻ **Step 4: Define a distribution function on $X^+ \cup X^-$.** For each $i \in [k]$, $x^+ \in X^+_{i*}$ and $x^- \in X^-_{*i}$, $j \in [k] \setminus \{ i\}$, we set $$\begin{aligned} p_j({x^+}) = \max \{ d_{G_{ij}}^+(x^+)/d , 6 k^2 \gamma^{1/3}\} \,\,\text{ and }\,\, p_j({x^-}) = \max \{d_{G_{ji}}^-(x^-)/d , 6 k^2 \gamma^{1/3}\}.\end{aligned}$$ Given $x^+ \in X^+_{i*}$, we 'view' $p_j({x^+})$ to be flow through $x^+$ from $s_i$ to $t_j$, and similarly for $p_j({x^-})$. **Claim 36**. *For each $i \in [k]$ and $x^+ \in X^+_{i*}$, $$\begin{aligned} \sum_{j \in [k] \setminus \{i\} } p_j({x^+}) \le 1 - \frac{d_{G^0}^+(x,V_{*i})}{d} + 6 k^3 \gamma^{1/3} < 1\end{aligned}$$ and a similar statement holds for each $j \in [k]$ and $x^- \in X^-_{*j}$.* *Proof of claim.* We will only consider the case when $x^+ \in X^+_{i*}$ as the other case when $x^- \in X^-_{*j}$ can be proved similarly. Since $G$ is a subgraph of $G^0$, we have $$\begin{aligned} \sum_{j \in [k]\setminus\{i\} } d_{G_{ij}}^+(x^+) \le \sum_{j \in [k]\setminus\{i\} } d_{G_{ij}^0}^+(x^+) = d-d_{G^0}^+(x,V_{*i}) .\end{aligned}$$ Hence $$\begin{aligned} \sum_{j \in [k]\setminus \{i\} } p_j({x^+}) & \le \sum_{j \in [k]\setminus \{i\} } \left( \frac{d_{G_{ij}}^+(x^+)}d + 6 k^2 \gamma^{1/3} \right) \le \frac{\sum_{j \in [k]\setminus \{i\}} d_{G_{ij}}^+(x^+)} d + 6 k^3 \gamma^{1/3}\\ & \le 1 - \frac{d_{G^0}^+(x,V_{*i})}{d} + 6 k^3 \gamma^{1/3} <1,\end{aligned}$$ where the last inequality holds as $d_{G^0}^+(x,V_{*i})\ge d/k$ by [\[item:good-degree-condition\]](#item:good-degree-condition){reference-type="ref" reference="item:good-degree-condition"} and $\gamma\ll 1/k$. ◻ **Step 5: Defining fractional flow on $\mathcal{F}_0$.** We first define the fractional flow $f$ on $\mathcal{F}$ as follows. For each $i,j \in [k]$, we do the following - for each $x^+ \in X_{ij}^+ \subseteq V_{i*}$, add a flow through $s_{i} x^+ t_{j}$ of value $p_j(x^+)$; - for each $x^- \in X_{ij}^- \subseteq V_{*j}$, add a flow through $s_{i} x^- t_{j}$ of value $p_i(x^-)$; - for each edge $e=uv \in M_{ij}$, add a flow through $s_{i} u^+ v^- t_{j}$ of value $6 k^2 \gamma^{1/3}$. Our construction and Claim [Claim 36](#clm:probability){reference-type="ref" reference="clm:probability"} imply that each vertex $v \in X^+ \cup X^-$ has a flow of value at most $1$ through it. Thus $f$ is indeed a fractional flow on $\mathcal{F}$. For each $i,j \in [k]$, $$\begin{aligned} d f_{ij} & \overset{\mathclap{\text{\eqref{eqn:definition-of-f-ij}}}}{=} d \left( \sum_{x^+ \in X_{ij}^+} p_j(x^+) + \sum_{x^- \in X_{ij}^-} p_i(x^-) + \sum_{uv \in M_{ij}} 6 k^2 \gamma^{1/3} \right) \nonumber \\ & \ge \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) + (6 k^2 \gamma^{1/3} d) e( M_{ij} ) \overset{\mathclap{\text{\ref{itm:Msize}}}}{\ge} e(G_{ij}) - 50 k^{10} \gamma^{1/3}d. \nonumber \end{aligned}$$ Therefore, $$\begin{aligned} \sum_{i,j \in [k]} \max\{ e(G_{ij})/d - f_{ij}, 0 \} \le 50 k^{12} \gamma^{1/3} <1 .\end{aligned}$$ If there is no $v_0$ satisfying [\[item:degree-condition-v0\]](#item:degree-condition-v0){reference-type="ref" reference="item:degree-condition-v0"}, then $\mathcal{F}_0 = \mathcal{F}$ and we are done by Claim [Claim 35](#clm:fractionalflow){reference-type="ref" reference="clm:fractionalflow"}. If such a $v_0$ exists, then recall that $F_0$ is obtained from $F$ removing the vertex $v_0^+$, the edge $s_{i_0} v_0^-$ and edges in $E_F(X^+,v_0^-)$. By repeated application of Proposition [Proposition 27](#prop:flow-reduction){reference-type="ref" reference="prop:flow-reduction"}, we obtain a flow in $\mathcal{F}_0$ whose value is lower than that of $f$ by at most $$\begin{aligned} & \sum_{j \in [k]\setminus \{ i_0 \} } p_j(v_0^+) + p_{i_0}(v_0^-) + (k-1) 6 k^2 \gamma^{1/3}\\ & \overset{\mathclap{\text{Claim~\ref{clm:probability}}}}{\le} 1 - \frac{d_{G^0}^+(v_0,V_{*i_0})}{d} + 6 k^3 \gamma^{1/3} + \left( \frac{d^-_{G_{i_0j_0}}(v_0)}{d} + 6 k^2 \gamma^{1/3} \right) + 6 k^2 (k-1) \gamma^{1/3} \\ &\le 1 - \frac{d_{G^0}^+(v_0,V_{*i_0})}{d}+\frac{d^-_{G^0}(v_0,V_{i_0*})}{d} +12k^3\gamma^{1/3} \\ & \overset{\mathclap{\text{\ref{item:degree-condition-v0}}}}{\le} 1 - 100 k^{12} \gamma^{1/3}+12k^3\gamma^{1/3}< 1-50k^{12}\gamma^{1/3}.\end{aligned}$$ Therefore, $\sum_{i,j \in [k]} \max\{ e(G_{ij})/d - f_{ij}, 0 \}$ is still less than $1$. Hence we are done by Claim [Claim 35](#clm:fractionalflow){reference-type="ref" reference="clm:fractionalflow"}. ◻ # Proof of Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"} {#sec:pf_regdigraph} We begin with an outline of the proof of the Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}. Let $G$ and $\mathcal{P}$ be the digraph and vertex partition as in the statement of Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}. Suppose, contrary to the lemma, that there is no non-trivial $\mathcal{P}$-balanced path system. Consider distinct $i,j \in [k]$. Note that if there exist two vertex-disjoint edges $e_1 \in E(G_{ij})$ and $e_2 \in E(G_{ji})$, then $\{e_1,e_2\}$ forms a non-trivial $\mathcal{P}$-balanced path system. Thus, if $e(G_{ij}), e(G_{ji}) \ge 3$, then we may assume that there exists a vertex $w_{ij} \in V_{ii} \cup V_{jj}$ that is contained in all edges of $G_{ij} \cup G_{ji}$. Let $H$ be the oriented graph on $[k]$ where $ij \in E(H)$ if $e(G_{ij}), e(G_{ji}) \ge 3$ and $w_{ij} \in V_{jj}$; we give weight $w(ij):=|E(G_{ij}\cup G_{ji})|$ to $ij$.[^11] It turns out that the underlying undirected graph of $H$ is acyclic (see Claim [Claim 37](#clm:cyclesize){reference-type="ref" reference="clm:cyclesize"}) and that total weight of edges in $H$ is almost equal to $|\mathcal{B}(G, \mathcal{P})|$ (see [\[eqn:e(G\*)\]](#eqn:e(G*)){reference-type="eqref" reference="eqn:e(G*)"}). Set $w(i)=\sum_{j\in [k]}(w(ij)-w(ji))$. We focus on the $V_{ii}$ that are relatively small: let $c_i=qd+1-|V_{ii}|$ (where we set $q=2$ if $G$ is an oriented graph and $q=1$ otherwise) and after relabelling indices assume $|\{i \in [k]: c_i\geq 0\}| = [k_0]$. We will lower and upper bound $\sum_{i\in[k_0]}w(i)$. For the lower bound, we note that if $c_i>0$ then every vertex of $V_{ii}$ has at least one edge from outside $V_{ii}$, and so we can lower bound $w(i)$ in terms of $c_i$ for each $i\in [k_0]$ (see [\[eqn:e(G\[V\'\_i,barV_i\])\]](#eqn:e(G[V'_i,barV_i])){reference-type="eqref" reference="eqn:e(G[V'_i,barV_i])"}). For the upper bound, we note that $w(ij)$ can be upper bounded using [\[item:degree-is-large-inside-diagonal\]](#item:degree-is-large-inside-diagonal){reference-type="ref" reference="item:degree-is-large-inside-diagonal"} (see [\[eqn:e(Gij)+e(Gji)\]](#eqn:e(Gij)+e(Gji)){reference-type="eqref" reference="eqn:e(Gij)+e(Gji)"}). Then, as $H$ is acyclic (so has few edges), we are able to find a good upper bound for $\sum_{i\in[k_0]}w(i)$ by considering connected components (i.e. trees) in $H[[k_0]]$. Combining the lower and upper bounds, we obtain $\sum_{i\in[k_0]}c_i\leq k-k_0-1$ (see [\[eqn:sum-of-components-vertices-k0\]](#eqn:sum-of-components-vertices-k0){reference-type="eqref" reference="eqn:sum-of-components-vertices-k0"}) whereas [\[item:relation-n-d-k\]](#item:relation-n-d-k){reference-type="ref" reference="item:relation-n-d-k"} implies $\sum_{i\in[k_0]}c_i>k-k_0$, a contradiction. Recall that for a digraph $G$ and $A,B \subseteq V(G)$ not necessarily disjoint, we write $E_G(A,B):= \{ab \in E(G): a \in A, \; b \in B \}$ and $e_G(A,B) := |E_G(A,B)|$. We write $E_G(A)$ and $e_G(A)$ for $E_G(A,A)$ and $e_G(A,A)$, respectively. *Proof of Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}.* Assume $G$ and $\mathcal{P}$ satisfy [\[item:each-part-is-at-least-d-over-2\]](#item:each-part-is-at-least-d-over-2){reference-type="ref" reference="item:each-part-is-at-least-d-over-2"}--[\[item:degree-is-large-inside-diagonal\]](#item:degree-is-large-inside-diagonal){reference-type="ref" reference="item:degree-is-large-inside-diagonal"} in the statement of Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}. Let $q =2$ if $G$ is an oriented graph and $q =1$ otherwise, so [\[item:relation-n-d-k\]](#item:relation-n-d-k){reference-type="ref" reference="item:relation-n-d-k"} says that $n<(qd + 1)k$. We write $V_i$ instead of $V_{ii}$ for $i \in [k]$, so that, for all distinct $i,j\in[k]$, $G_{ij}$ becomes the bipartite digraph with vertex classes $V_i$ and $V_j$ and edges given by the edges of $G$ from $V_{i}$ to $V_{j}$. In this context (where $V_{ij}=\emptyset$ for all distinct $i,j\in[k]$), $\mathcal{Q}$ is a $\mathcal{P}$-balanced path system if, for each $i\in[k]$, the number of edges of $\mathcal{Q}$ going into $V_i$ is equal to the number of edges of $\mathcal{Q}$ leaving $V_i$. Also, a $\mathcal{P}$-balanced path system is non-trivial if it has at least one path whose endpoints lie in distinct parts of the partition $\mathcal{P}=\{V_i:i\in[k]\}$. Now suppose for a contradiction that $G$ does not contain any non-trivial $\mathcal{P}$-balanced path system with at most $k$ edges. **Claim 37**. *Let $H_0$ be a digraph on $[k]$ such that $ij \in E(H_0)$ if $e(G_{ij}) \ge 3$ and $i\neq j$. Then $H_0$ does not have any directed cycle of length at least $3$.* *Proof of claim.* Suppose to the contrary that $C$ is such a cycle in $H_0$. For each $ij \in E(C)$, we pick an edge $e_{ij}\in E(G_{ij})$ with $ij \in E(C)$ and call the resulting subdigraph $G'$. Note that $G'$ is $\mathcal{P}$-balanced and has at most $k$ edges; moreover it is either a cycle or a path system. If $G'$ is a cycle, then pick an $ij\in E(C)$, and find $e_{ij}'\in E(G_{ij})$ with $e_{ij}'\neq e_{ij}$. Write $G''=(G'-\{e_{ij}\})\cup \{e_{ij}'\}$ if $G'$ is a cycle and $G''=G'$ otherwise. Note that $G''$ is a $\mathcal{P}$-balanced path system. If $G''$ is a single path, then there exists $t\in [k]$ such that the endpoints of $G''$ lies in $V_{t}$. Then, pick an $i'j'\in E(C)$ with $i',j'\neq t$, and find $e_{i'j'}'\in E(G_{i'j'})$ with $e_{i'j'}'\neq e_{ij}, e_{i'j'}$. Write $G'''=(G''-\{e_{i'j'}\})\cup \{e_{i'j'}'\}$ if $G''$ is a single path and $G'''=G''$ otherwise. Note that $G'''$ consists of at least two vertex-disjoint paths and that for any path in $G'''$, the endpoints lie in distinct parts of the partition $\mathcal{P}=\{V_{i}:i\in[k]\}$, so $G'''$ is a non-trivial $\mathcal{P}$-balanced path system with at most $k$ edges, a contradiction. ◻ **Claim 38**. *For all distinct $i,j \in [k]$, if $e(G_{ij})\le 2$, then $e(G_{ji})\le 2k^2$.* *Proof of claim.* Suppose to the contrary and without loss of generality that $e(G_{21})\le 2$ and $e(G_{12})\ge 2k^2+1$. Let $H_0$ be as defined in Claim [Claim 37](#clm:cyclesize){reference-type="ref" reference="clm:cyclesize"}, so $12 \in E(H_0)$ and $21\notin E(H_0)$. Let $A$ be the set of $i\in [k]$ such that $H_0$ contains a directed path from $1$ to $i$ starting with the edge $12$, where we a priori allow $1 \in A$ (i.e. resulting from a closed path). Let $B=[k]\setminus A$. It is clear from the definition that $2\in A$, and by Claim [Claim 37](#clm:cyclesize){reference-type="ref" reference="clm:cyclesize"} and since $21\notin E(H_0)$ we have $1\in B$. Note that by the definition of $A$, there are no edges in $E_{H_0}(A,B)$ so that $e(G_{ij})\le 2$ for all $ij\in A\times B$. Together with the regularity of $G$, we have $$\begin{aligned} 2k^2\ge \sum_{ij\in A\times B}e(G_{ij})=\sum_{ij\in B\times A} e(G_{ij}) \ge e(G_{12})\ge 2k^2+1, \end{aligned}$$ which is a contradiction. ◻ Let $H^*$ be the graph on $[k]$ such that $ij \in E(H^*)$ if $e(G_{ij}), e(G_{ji}) \ge 3$, i.e. $H^*$ is the graph obtained from $H_0$ by deleting all the edges which do not lie in a directed $2$-cycle and by making each directed $2$-cycle into an undirected edge. By Claim [Claim 37](#clm:cyclesize){reference-type="ref" reference="clm:cyclesize"}, $H^*$ is acyclic. Consider distinct $i,j \in [k]$. Note that there do not exist disjoint edges $e_1, e_2$ with $e_1 \in E(G_{ij})$ and $e_2 \in E(G_{ji})$ because otherwise $\{e_1,e_2\}$ would form a non-trivial $\mathcal{P}$-balanced path system. Thus if $ij \in E(H^*)$ (that is, $e(G_{ij}) , e(G_{ji})\ge 3$), then there exists a vertex $w_{ij} = w_{ij} \in V_{i} \cup V_j$ that is contained in all edges of $G_{ij} \cup G_{ji}$, and $$\begin{aligned} e(G_{ij}) +e(G_{ji}) = d^+_{G_{ij} \cup G_{ji}} (w_{ij}) + d^-_{G_{ij} \cup G_{ji}}(w_{ij}) \overset{\mathclap{\text{\ref{item:degree-is-large-inside-diagonal} }}}{\le } ( 2-1/k ) d . \end{aligned}$$ If $ij \not\in E(H^*)$, then $\min\{e(G_{ij}), e(G_{ji})\}\le 2$ and by Claim [Claim 38](#clm:sizedifference){reference-type="ref" reference="clm:sizedifference"}, we obtain $$\begin{aligned} e(G_{ij}) +e(G_{ji})\leq 2k^2+2\leq ( 2-1/k)d \label{eqn:e(Gij)+e(Gji):2} \end{aligned}$$ as $k\ge 2$ and $d>165k^5$. Hence we have $$\begin{aligned} e(G_{ij}) +e(G_{ji}) \le ( 2-1/k) d \text{ for all distinct $i ,j \in[k]$} . \label{eqn:e(Gij)+e(Gji)} \end{aligned}$$ Recall that $q =2$ if $G$ is an oriented graph, and $q =1$ otherwise. Note that, if $|V_i|\le qd+1$ for some $i\in[k]$, then $$\begin{aligned} 2kd & \overset{\mathclap{\text{\eqref{eqn:e(Gij)+e(Gji)}}}}{\ge } \sum_{j \in [k] \setminus \{i\}}e(G_{ij}) = d|V_i| - e(G_{ii}) \ge d|V_i| - \frac{|V_i| (|V_i|-1) }{q} \nonumber \\ & = \frac{q d +1 - |V_i|}{q} |V_i| \overset{\mathclap{\text{ \ref{item:each-part-is-at-least-d-over-2} }}}{\ge } \frac{d ( q d +1 - |V_i|)}{2q} \ge \frac{d ( q d +1 - |V_i|)}{4} . \nonumber\end{aligned}$$ Hence, for all $i\in [k]$, $$\begin{aligned} |V_i| & \ge qd+1-8k. \label{eqn:sizeV_i}\end{aligned}$$ We now orient $H^*$ to obtain an oriented graph $H$ on $[k]$ as follows. Recall that if $ij \in E(H^*)$, then there exists a vertex $w_{ij} \in V_i \cup V_j$ contained in all edges of $E(G_{ij}) \cup E(G_{ji})$. We orient from $i$ to $j$ if $w_{ij} \in V_j$. Let $W$ be the set of $w_{ij}$ and $W_i = W \cap V_i$ for all $i \in [k]$. Let $V_i'=V_i\setminus W_i$. Note that $$\begin{aligned} |W_i| \le k-1 \text{ and so } |V_i'|= |V_i|-|W_i|\overset{\mathclap{\text{\eqref{eqn:sizeV_i}}}}{\ge }qd-9k. \label{eqn:|W_i|}\end{aligned}$$ Recall that $\mathcal{B}(G, \mathcal{P})=\bigcup_{i,j\in[k]:\,i\neq j}G_{ij}$. Let $G^* = \bigcup_{i,j \in [k]}G^*_{ij}$ where $G^*_{ij}$ is the subdigraph of $\mathcal{B}(G, \mathcal{P})$ consisting of the edges (in both directions) between $V_i'$ and $w_{ij}$ if $ij\in E(H)$ and $G^*_{ij} = \emptyset$ otherwise.[^12] Note that $G^*_{ij}$ is a subdigraph of $G_{ij}\cup G_{ji}$ whenever $ij\in E(H)$ and $G^*_{ij}=\emptyset$ otherwise. Therefore, $$\begin{aligned} e( G^*_{ij} ) \begin{cases} \le e(G_{ij} \cup G_{ji}) & \text{ if $ij \in E(H)$;} \\ = 0 & \text{ if $ij \notin E(H)$.} \end{cases} \label{eqn:H_ij}\end{aligned}$$ Hence, we deduce that $$\begin{aligned} e (\mathcal{B}(G, \mathcal{P}) )- e(G^*) & \le \sum_{ij \notin E(H^*)} \left( e(G_{ij})+ e(G_{ji}) \right) + \sum_{i,j \in [k] : i \ne j } |W_i||W_j| \nonumber \\ & \overset{\mathclap{\text{\eqref{eqn:e(Gij)+e(Gji):2},\eqref{eqn:|W_i|}}}}{\le} \quad\binom{k}2 (2k^2+2) + k (k-1)^3 < 3 k^4 . \label{eqn:e(G*)}\end{aligned}$$ For $i \in [k]$, let $\overline{V_i} = V(G) \setminus V_i$. Since all edges in $G^*$ contain a vertex in $W$ and $G^*[W] \cup \bigcup_{i \in [k]}G^*_{ii}$ is empty, we have for all $i \in [k]$, $$\begin{aligned} \sum_{j\in[k]}e(G^*_{ij})=\sum_{j\in[k],\,ij\in E(H)}\bigg(e_{G^*}(V_i',w_{ij})+e_{G^*}(w_{ij},V_i')\bigg)=e_{G^*}(V_i',\overline{V_i})+e_{G^*}(\overline{V_i},V_i') \label{eqn:G^*[V_i', W_j]NEW} \end{aligned}$$ and, for all $j \in [k]$, $$\begin{aligned} \sum_{i\in[k]}e(G^*_{ij})=\sum_{i\in[k],\,ij\in E(H)}\bigg(e_{G^*}(V_i',w_{ij})+e_{G^*}(w_{ij},V_i')\bigg)= e_{G^*}(\overline{V_j},W_j)+e_{G^*}(W_j,\overline{V_j}). \label{eqn:G^*[V_i', W_j]2NEW} \end{aligned}$$ We also need the following inequality $$\begin{aligned} e_G( V_i',V_i) = e_G (V_i')+\sum_{w \in W_i } d^-_G(w, V'_{i} ) \le e_G (V_i')+\sum_{w \in W_i } d^-_G(w, V_{i} ). \label{eqn:k_0eq1}\end{aligned}$$ Now, let $k_0 = |\{i \in [k]: |V_i| \le q d+1\}|$. Note that $k_0>0$ since $n<(qd+1)k$ by [\[item:relation-n-d-k\]](#item:relation-n-d-k){reference-type="ref" reference="item:relation-n-d-k"}. Without loss of generality, $$\begin{aligned} |V_i| \le q d+1 \text{ if and only if } i \in [k_0]. \label{eqn:[k_0}\end{aligned}$$ For $i \in [k_0]$, $$\begin{aligned} d |V'_i| - e_{G}( V_i' ) & \ge d |V'_i| - \frac{|V'_i|(|V'_i|-1)}{q} = \frac{|V_i'|}{q} \left( q d +1 - |V'_i| \right) = \frac{|V_i'|}{q} \left( q d +1 - |V_i| \right) + \frac{|V_i'||W_i|}{q} \nonumber \\ & \overset{\mathclap{\text{\eqref{eqn:|W_i|}}}}{\ge } \frac{qd-9k}{q} \left( q d +1 - |V_i| \right) + \frac{|V_i'||W_i|}{q} \nonumber \\ & = d \left( q d +1 - |V_i| \right) - \frac{9k \left( q d +1 - |V_i| \right)}{q} + \frac{|V_i'||W_i|}{q} \nonumber \\ & \overset{\mathclap{\text{\eqref{eqn:sizeV_i}}}}{\ge } d \left( q d +1 - |V_i| \right) - 72k^2 + \frac{|V_i'||W_i|}{q} . \label{eqn:k_0eq2}\end{aligned}$$ Then, $$\begin{aligned} e_{G} ( V_i', \overline{V_i} ) &= \sum_{v \in V_i' } d^+_G(v, \overline{V_i} ) = \sum_{v \in V_i' } \left( d - d^+_G(v, V_{i} ) \right) = d |V'_i| - e_{G}(V_i',V_i) \nonumber \\ & \overset{\mathclap{\text{\eqref{eqn:k_0eq1}}}}{\ge } d |V'_i| - e_{G}( V_i' ) - \sum_{w \in W_i } d^-_G(w, V_{i} ) = d |V'_i| - e_{G}( V_i' ) - \sum_{w \in W_i } \left( d - d^-_G(w, \overline{V_{i}} ) \right) \nonumber \\ & = d |V'_i| - e_{G}( V_i' ) - d |W_i| + e_G ( \overline{V_{i}},W_i ) \nonumber \\ & \overset{\mathclap{\text{\eqref{eqn:k_0eq2}}}}{\ge } d \left( q d +1 - |V_i| \right) - 72k^2 - \frac{|W_i|}{q} (q d - |V_i'|) + e_G ( \overline{V_{i}},W_i ) \nonumber \\ & \overset{\mathclap{\text{\eqref{eqn:|W_i|}}}}{\ge } d \left( q d +1 - |V_i| \right) - 81k^2 + e_G ( \overline{V_{i}},W_i ) \nonumber\end{aligned}$$ and a similar inequality holds for $e_G ( \overline{V_{i}},V_i' )$. Hence, for $i \in [k_0]$, $$\begin{aligned} e_{G} ( V_i', \overline{V_i} ) + e_G ( \overline{V_{i}},V_i' ) \ge 2d \left( q d +1 - |V_i| \right) - 162k^2 + e_G ( \overline{V_{i}},W_i ) + e_G ( W_i,\overline{V_{i}}) . \label{eqn:e(G[V'_i,barV_i])}\end{aligned}$$ Let $\mathcal{I}$ be the set of all connected components $I$ in $H^* \left[ [k_0] \right]$ with $\sum_{ i \in I} (qd +1 - |V_i| ) >0$.[^13] Such $\mathcal{I}$ is non empty as $n < (qd +1) k$ by [\[item:relation-n-d-k\]](#item:relation-n-d-k){reference-type="ref" reference="item:relation-n-d-k"}. We have that $$\begin{aligned} \sum_{i \in [k_0]} (qd +1 - |V_i| ) = \sum_{I\in\mathcal{I}} \sum_{ i \in I} (qd +1 - |V_i| ) .\label{eqn:sum_k_0}\end{aligned}$$ Recall that $H^*$ is acyclic and $H$ is an orientation of $H^*$. Note that for each $I\in\mathcal{I}$, $H[I]$ is an oriented tree and $E_H ( I, [k]\setminus [k_0] ) = E_H ( I,[k] ) \setminus E_H (I)$. Therefore, for each $I\in\mathcal{I}$, $$\begin{aligned} & ( 2-1/k ) d \cdot e_H ( I, [k]\setminus [k_0] ) = (2-1/k) d \cdot |E_H ( I,[k] ) \setminus E_H (I)| \\ & \overset{\mathclap{\eqref{eqn:e(Gij)+e(Gji)}}}{\ge} \sum_{ ij\in E(H) \colon i\in I,\, j\in[k]\setminus I} e(G_{ij} \cup G_{ji}) \overset{\mathclap{\eqref{eqn:H_ij}}}{\ge} \sum_{ij\in E(H) \colon i\in I,\, j\in[k]\setminus I} e(G^*_{ij}) \\ &= \sum_{ij\in E(H)\colon i\in I,\, j\in[k]} e(G^*_{ij}) - \sum_{ij\in E(H) \colon i,j\in I,\, i\neq j} e(G^*_{ij}) \\ &\overset{\mathclap{\eqref{eqn:H_ij}}}{\ge} \sum_{ij\in I\times[k]} e(G^*_{ij}) - \sum_{ ij\in [k]\times I} e(G^*_{ij}) \\ & \overset{\mathclap{\eqref{eqn:G^*[V_i', W_j]NEW}, \eqref{eqn:G^*[V_i', W_j]2NEW}}}{=}\quad\,\, \sum_{i\in I} \bigg(e_{G^*}(V_i', \overline{V_i})+ e_{G^*}( \overline{V_i},V_i') \bigg)- \sum_{ j\in I } \bigg(e_{G^*}(\overline{V_j}, W_j)+e_{G^*}(W_j,\overline{V_j})\bigg) \\ & \overset{\mathclap{\eqref{eqn:e(G*)}}}{\ge} \,\,\sum_{i\in I} \bigg(e_{G}(V_i', \overline{V_i})+ e_{G}( \overline{V_i},V_i') \bigg)- \sum_{ j\in I } \bigg(e_{G}(\overline{V_j}, W_j)+e_{G}(W_j,\overline{V_j})\bigg) - 3k^4 \\ & = \sum_{i\in I} \Bigg(\bigg(e_{G}(V_i', \overline{V_i})+ e_{G}( \overline{V_i},V_i') \bigg)- \bigg(e_{G}(\overline{V_i}, W_i)+e_{G}(W_i,\overline{V_i})\bigg) \Bigg)- 3k^4 \\ & \overset{\mathclap{\eqref{eqn:e(G[V'_i,barV_i])}}}{\ge} \,\,\, 2d \sum_{ i \in I} (qd +1 - |V_i| ) - 162|I| k^2 - 3k^4 \ge 2d \sum_{ i \in I} (qd +1 - |V_i| ) - 165 k^4.\end{aligned}$$ After rearranging, for each $I\in \mathcal{I}$, we have $$\begin{aligned} e_H(I,[k]\setminus [k_0] ) &\ge \frac{2}{2-1/k} \sum_{ i \in I} (qd +1 - |V_i| ) - \frac{165 k^4}{(2-1/k)d} \\ &= \sum_{ i \in I}(qd +1 - |V_i| )+\frac{1}{2k-1}\sum_{ i \in I}(qd +1 - |V_i| ) - \frac{165 k^5}{(2k-1)d}\\ &\ge \sum_{ i \in I} (qd +1 - |V_i| )+\frac{1}{2k-1}-\frac{165 k^5}{(2k-1)d}>\sum_{ i \in I} (qd +1 - |V_i| )\end{aligned}$$ as $\sum_{ i \in I} (qd +1 - |V_i| )\ge 1$ and $d>165k^5$. Hence, we have $$\begin{aligned} \sum_{I\in\mathcal{I}} e_H(I,[k]\setminus [k_0] ) &\ge \sum_{I\in\mathcal{I}} \left( 1 + \sum_{ i \in I} (qd +1 - |V_i| ) \right) = |\mathcal{I}|+ \sum_{I\in\mathcal{I}} \sum_{ i \in I} (qd +1 - |V_i| ) \nonumber \\ & \overset{\mathclap{\text{\eqref{eqn:sum_k_0}}}}{=} |\mathcal{I}|+ \sum_{i \in [k_0]} (qd +1 - |V_i| ) . \label{eqn:k-k_0}\end{aligned}$$ Recall that $H$ is an oriented forest and so $H[I]$ is an oriented tree for all $I \in \mathcal{I}$. Then, $$\begin{aligned} \sum_{I\in \mathcal{I}} e_H (I,[k]\setminus [k_0] ) & \le e_{H} \left( \left([k]\setminus[k_0]\right)\cup \bigcup_{I\in \mathcal{I}} I \right) - \sum_{I\in \mathcal{I}}e_H(I) \nonumber \\ &\le \left( \left(k-k_0 +\sum_{I\in \mathcal{I}}|I|\right) - 1 \right)-\sum_{I\in \mathcal{I}}\left(|I|-1\right) = k-k_0+|\mathcal{I}|-1. \nonumber\end{aligned}$$ Together with [\[eqn:k-k_0\]](#eqn:k-k_0){reference-type="eqref" reference="eqn:k-k_0"}, we have $$\begin{aligned} \sum_{ i \in [k_0]} (qd +1 - |V_i| ) \le k-k_0-1. \label{eqn:sum-of-components-vertices-k0}\end{aligned}$$ Finally $$\begin{aligned} n & = \sum_{i \in [k]} |V_i| \overset{\mathclap{\eqref{eqn:[k_0}}}{\ge} \sum_{i \in [k_0]} |V_i| + (qd+2) (k-k_0)\\ & = (qd+1)k +( k-k_0 ) - \sum_{i \in [k_0]} (qd +1 - |V_i| ) \overset{\mathclap{\eqref{eqn:sum-of-components-vertices-k0}}}{ \ge} (q d+1)k +1,\end{aligned}$$ a contradiction to property [\[item:relation-n-d-k\]](#item:relation-n-d-k){reference-type="ref" reference="item:relation-n-d-k"} of Lemma [Lemma 22](#lma:regdigraph){reference-type="ref" reference="lma:regdigraph"}. This completes the proof of the lemma. ◻ # Conclusion {#sec:conlusion} ## Path cover for (non-regular) graphs Magnant, Wang and Yuan [@PathCover2] gave a stronger version of Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"}. Recall that $\pi(G)$ is the minimum number of vertex-disjoint paths needed to cover $G$. **Conjecture 39** (Magnant, Wang and Yuan [@PathCover2]). *If $G$ is a graph on $n$ vertices with $\Delta(G) = \Delta$ and $\delta(G) = \delta$, then $\pi(G)\leq \max \{ n/(\delta+1) , (\Delta - \delta) n / (\Delta + \delta) \}$.* The bound is tight by considering a disjoint union of $K_{\delta+1}$ or a disjoint union of $K_{\delta, \Delta}$. The conjecture holds if $\delta \le 2$ [@PathCover2] and when $\Delta \ge 2 \delta$ [@KouiderZamime]. Naturally, one can ask for the directed or oriented analogues. ## Edge-disjoint cycles In a weaker version of the problem we have considered, one is interested in finding *edge-disjoint* cycles whose union covers all the vertices. As a generalization of Dirac's theorem, it was conjectured [@ENO-KA-TU] that if a graph $G$ on $n$ vertices has minimum degree at least $n/k$, then $V(G)$ can be covered by $k-1$ edge-disjoint cycles. The case $k=3$ was also proved in [@ENO-KA-TU]. The conjecture was proved for $2$-connected graphs [@Edge-Disjoint-2-connected], and has been completely resolved in [@Edge-Disjoint-Full]. Later, Balogh, Mousset and Skokan [@Edge-Disjoint-Stability] obtained a stability result, showing that every graph on $n$ vertices with minimum degree nearly $n/k$ has a special structure if it does not have $k-1$ edge-disjoint cycles covering all vertices. One can ask the same question for digraphs as a generalization of Ghoulia-Houri's theorem, and Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} answers it affirmatively for regular digraphs: **Conjecture 40**. *Let $k\in\mathbb{N}$ with $k\geq 2$. If $G$ is a digraph on $n$ vertices with minimum semi-degree at least $n/k$, then $V(G)$ can be covered by $k-1$ edge-disjoint cycles.* ## Connectivity and regularity Jackson's conjecture states that imposing regularity on an oriented graph reduces the degree threshold for Hamiltonicity. One might hope that imposing connectivity on regular oriented graph to reduce the degree threshold for Hamiltonicity. We refer the readers to [@LPY Sections 1 and 7] for history and conjectures on Hamiltonicity in regular (directed or oriented) graphs with given connectivity. Similarly, in the setting of cycle partitions, one might hope that connectivity in addition to the regularity might help to reduce the upper bound in Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"}. In the sparse setting, Reed [@Reed-Connected] proved that every $3$-regular connected $n$-vertex graph can be covered by at most $\lceil n/9 \rceil$ vertex-disjoint paths, and conjectured that it suffices to use $\lceil n/10 \rceil$ paths if connectivity is replaced by $2$-connectivity. Recall that Conjecture [Conjecture 5](#conj:PATH-COVER){reference-type="ref" reference="conj:PATH-COVER"} gives the upper bound of $n/4$ for $3$-regular graphs (that are not necessarily connected). Yu [@Yu-Verifies-Reed] recently verified Reed's conjecture, and gave an example of (2-connected) $d$-regular graph on $n$ vertices which require at least $\approx n/(d+4)$ paths for $d\ge 13$. It would be interesting to investigate the general relationship between the degree and connectivity of a regular (di)graph or oriented graph that guarantees a small number of vertex-disjoint cycles that cover all the vertices. 1 M. Abreu, J.B. Gauci, D. Labbate, G. Mazzuoccolo, and J. P. Zerafa, "Extending perfect matchings to Hamiltonian cycles in line graphs\", *Electron. J. Comb.*, **28(1)** (2021), Paper P1.7. J. Akiyama, G. Exoo, and F. Harary, "Covering and packing in graphs IV: Linear arboricity\", *Networks*, **11(1)**, (2006) 69--72. J. Balogh, F. Mousset, and J. Skokan, "Stability for vertex cycle covers\", *Electron. J. Comb.*, **24** (2017), Paper P3.56. G.B. Dantzig and D.R. Fulkerson, "On the Max Flow MinCut Theorem of Networks", in *Linear inequalities and related systems*, Ann. Math. Studies, **38** (1956), 215--221. R. Diestel, "Graph Theory (4th Edition)\", *Graduate texts in mathematics 173, Springer* (2012). P. Dilworth, "A decomposition theorem for partially ordered sets\", *Ann. of Math.*, **51(1)**, (1950), 161--166. G.A. Dirac, "Some theorems on abstract graphs\", *Proc. Lond. Math. Soc.*, **2** (1952), 69---81. H. Enomoto, A. Kaneko, and Z. Tuza, "$P_3$-factors and covering cycles in graphs of minimum degree $n/3$\", *Combinatorics (Proc. Coll. Math. Soc. János Bolyai Eger, 1987)*, North Holland (1988), 213--220. U. Feige and E. Fuchs, "On the path partition number of $6$‐regular graphs\", *J Graph Theory*, **101**, 345-378. J. Fink, "Perfect matchings extend to Hamilton cycles in hypercubes\", *J. Combin. Theory. Ser. B*, **97** (2007) 1074-1076. T. Gallai and A.N. Milgram, "Verallgemeinerung eines graphentheoretischen satzes von Redei\", *Acta Sci. Math*, **21**, (1960), 181-186. A. Ghouila-Houri, "Une condition suffisante d'existence d'un circuit hamiltonien\", *C.R. Acad. Sci. Paris* **25** (1960), 495--497. V. Gruslys and S. Letzter, "Cycle partitions of regular graphs", *Comb. Probab. Comput.*, **30(4)** (2021), 526--549. R. Häggkvist, "On $F$-Hamiltonian graphs\", In *Graph Theory and Related Topics* by J.A. Bondy and U.S.R. Murty, Academic Press, New York, (1979), 219--231. J. Han, "On vertex-disjoint paths in regular graphs\", *Electron. J. Comb.*, **25** (2018), Paper P2.12. B. Jackson, "Long paths and cycles in oriented graphs\", *J Graph Theory*, **5** (1981), 245--252. P. Keevash, D. Kühn, and D. Osthus, "An exact minimum degree condition for Hamilton cycles in oriented graphs\", *J. Lond. Math. Soc.*, **79** (2009), 144--166. M. Kouider, "Covering vertices by cycles\", *J Graph Theory*, **18** (1994) 757--776. M. Kouider and Z. Lonc, "Covering cycles and k-term degree sums\", *Combinatorica*, **16** (1996), 407--412. M. Kouider and M. Zamime, "On the path partition of graphs", arXiv:2212.12793 (2022). D. Kühn, A. Lo, D. Osthus, and K. Staden, "The robust component structure of dense regular graphs and applications\", *Proc. Lond. Math. Soc.*, **110(1)** (2015), 19--56. D. Kühn and D. Osthus, "Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments\", *Adv. Math.*, **237** (2013), 62--146. D. Kühn, D. Osthus, and A. Treglown, "Hamiltonian degree sequences in digraphs\", *J. Combin. Theory. Ser. B*, **100** (2010), 367--380. M. Las Vergnas, "Problèmes de couplages et problèmes hamiltoniens en Théorie des graphes\", Ph.D. Thesis, University of Paris, (1972). A. Lo and V. Patel, "Hamilton cycles in sparse robustly expanding digraphs", *Electron. J. Comb.*, **25(3)** (2018), P3.44 A. Lo., V. Patel, J. Skokan, and J. Talbot, "Decomposing tournaments into paths", *Proc. Lond. Math. Soc.*, **121(2)** (2020), 426---461. A. Lo, V. Patel, and M.A. Yıldız, "Hamilton cycles on dense regular digraphs and oriented Graphs", *J. Combin. Theory. Ser. B*, to appear. C. Magnant and D. Martin, "A note on the path cover number of regular graphs\", *Australas. J. Comb.* , **43** (2009), 211--217. C. Magnant, H. Wang, and S. Yuan, "Path partitions of almost regular graphs\", *Australas. J. Comb.* , **64** (2016), 334--340. O. Ore, "Arc coverings of graphs\", *Annali di Matematica Pura ed Applicata*, **55** (1961), 315--321. B. Reed, "Paths, stars and the number three\", *Comb. Probab. Comput.*, **3** (1996), 277---295. F. Ruskey and C.D. Savage, "Hamilton cycles that extend transposition matchings in Cayley graphs of $S_n$\", *SIAM J. Discrete Math*, **6(1)** (1993), 152--166. K. Staden, "Robust expansion and Hamiltonicity", PhD Thesis, University of Birmingham(2015). V.G. Vizing, "On an estimate of the chromatic class of a p-graph", *Diskret. Analiz.*, **3** (1964), 25--30. Z. Yang, "On $F$-Hamiltonian graphs\", *Discrete Math.*, **196** (1999), 281--286. G. Yu, "Covering 2-connected 3-regular graphs with disjoint paths\" *J Graph Theory*, **88** (2018), 385---401. [^1]: School of Mathematics, University of Birmingham, United Kingdom, Email: s.a.lo\@bham.ac.uk. A. Lo was partially supported by EPSRC, grant no. EP/V002279/1 and EP/V048287/1. There are no additional data beyond that contained within the main manuscript. [^2]: School of Mathematical Sciences, Queen Mary University of London, United Kingdom, Email: viresh.patel\@qmul.ac.uk. V. Patel was supported by the Netherlands Organisation for Scientific Research (NWO) through the Gravitation Programme Networks (024.002.003). [^3]: Korteweg de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, The Netherlands. Email: m.a.yildiz\@uva.nl. M.A. Yıldız was supported by a Marie Skłodowska-Curie Action from the EC (COFUND grant no. 945045 ) and by the NWO Gravitation project NETWORKS (grant no. 024.002.003). [^4]: We DO NOT write $G[A,B]$ for the graph with vertex set $A \cup B$ and edge set $E_G(A,B)$, but instead for a bipartite undirected graph; see below. [^5]: This follows essentially from Lemma [Lemma 13](#lma:longcycle2){reference-type="ref" reference="lma:longcycle2"} although we do not show it explicitly in the paper. [^6]: *The idea of the proof is to successively refine partitions of $V(G) = A \cup B$ as follows. Assume we have obtained a partition $\mathcal{U} = \{U_1, \ldots, U_r\}$ of $V(G)$ such that there are few edges leaving or entering $U$ for every $U \in \mathcal{U}$. If for some $U \in \mathcal{U}$ we have that $G[U]$ is not a bipartite robust expander (with bipartition $U \cap A, U \cap B$), then there is some $S \subseteq U \cap A$ whose robust neighbourhood $R \subseteq U \cap B$ is not much larger than $S$. Writing $U' = S \cup R$ and $U'' = U \setminus U'$, we let $\mathcal{U}' = \mathcal{U} \setminus \{U\} \cup \{U', U''\}$. It is not too hard to show (using the fact that $G$ is regular) that, as with $\mathcal{U}$, there are not many edges entering or leaving each $U \in \mathcal{U}'$. We continue refining the partition in this way until we obtain a partition $\mathcal{U}^*$ where every $U \in \mathcal{U}^*$ satisfies that $G[U]$ is a bipartite robust expander with bipartition $U \cap A, U \cap B$. The process of refining the partition must eventually stop because each $U \in \mathcal{U}$ cannot be much smaller than $d$ (the degree of $G$) since not many edges enter of leave $U$. This essentially shows [\[itm:structure1\]](#itm:structure1){reference-type="ref" reference="itm:structure1"}, [\[itm:structure3\]](#itm:structure3){reference-type="ref" reference="itm:structure3"}, [\[itm:structure5\]](#itm:structure5){reference-type="ref" reference="itm:structure5"} and [\[itm:structure6\]](#itm:structure6){reference-type="ref" reference="itm:structure6"}, while [\[itm:structure2\]](#itm:structure2){reference-type="ref" reference="itm:structure2"} and [\[itm:structure4\]](#itm:structure4){reference-type="ref" reference="itm:structure4"} are obtained by making slight adjustments to the final partition.* [^7]: *Note that, in the definition of contraction, the new vertices created are placed in exactly the right parts to ensure the degree condition in the proposition.* [^8]: When doing the greedy extension we obtain a path system (with paths of length at most 3) rather than a matching because we allow the start (resp. end) point of an edge in $M_{ij}$ to be in $X_{ij}^-$ (resp. $X_{ij}^+$). [^9]: Otherwise we could suitably decrease flow along paths $P$ with positive flow from any source to any sink that uses an edge from $E_F(s_i\cup X_{i*}^+, t_j\cup X_{*j}^-)$, and this can be done independently for each $i,j\in[k]$, since the $E_F(s_i\cup X_{i*}^+, t_j\cup X_{*j}^-)$ are disjoint over all $i,j\in[k]$. [^10]: Since $f$ is a fractional flow on $\mathcal{F}_0$, it suffices to examine only $s_i$ for $i\in[k]$. For $s_i$, the amount of flow going out of $s_i$ is $\sum_{j\in[k]}f_{ij}$ (from $s_i$ to $X^+$) and $\sum_{j\in[k]}\left(e(G_{ij})/d-f_{ij}\right)$ (from $s_i$ to $T$), and the flow going into $s_i$ is $\min\{e(G_{i*})/d,e(G_{*i})/d\}$ (from $t_i$ to $s_i$) and $\max\{|V_{i*}|-|V_{*i}|,0\}$ (from $s^*$ to $s_i$). By noting that [\[eqn:subgraph1\]](#eqn:subgraph1){reference-type="eqref" reference="eqn:subgraph1"} gives $e(G_{i*})/d-e(G_{*i})/d=|V_{i*}|-|V_{*i}|$, the total contribution is 0. For $t_i$, the calculation is similar. [^11]: Note that it is convenient to introduce the weight $w(ij)$ (and later $w(i)$) for the sketch proof; these are not used in the actual proof. [^12]: i.e. $G^*_{ij}=G_{ij}[V_i'\cup w_{ij}]\cup G_{ji}[V_i'\cup w_{ij}]$ if $ij\in E(H)$ and $G^*_{ij}=\emptyset$ otherwise. [^13]: Here we identify the connected component with its vertex set.

arxiv_math

--- abstract: | A novel distributed algorithm is proposed for finite-time converging to a feasible consensus solution satisfying global optimality to a certain accuracy of the distributed robust convex optimization problem (DRCO) subject to bounded uncertainty under a uniformly strongly connected network. Firstly, a distributed lower bounding procedure is developed, which is based on an outer iterative approximation of the DRCO through the discretization of the compact uncertainty set into a finite number of points. Secondly, a distributed upper bounding procedure is proposed, which is based on iteratively approximating the DRCO by restricting the constraints right-hand side with a proper positive parameter and enforcing the compact uncertainty set at finitely many points. The lower and upper bounds of the global optimal objective for the DRCO are obtained from these two procedures. Thirdly, two distributed termination methods are proposed to make all agents stop updating simultaneously by exploring whether the gap between the upper and the lower bounds reaches the certain accuracy. Fourthly, it is proved that all the agents finite-time converges to a feasible consensus solution that satisfies global optimality within a certain accuracy. Finally, a numerical case study is included to illustrate the effectiveness of the distributed algorithm. address: State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China author: - Xunhao Wu - Jun Fu title: Distributed robust optimization for multi-agent systems with guaranteed finite-time convergence --- [^1] , Distributed robust convex optimization; Bounded uncertainty; Uniformly strongly connected network; Finite-time convergence. # Introduction Multi-agent systems are network systems consisting of multiple decision-making agents, each possessing computational, communicative, learning, perceptual, and executive capabilities [@ferber1999multi]. Such systems have been used in a wide variety of fields, such as wireless networks [@kotary2020distributed; @ghosal2020distributed], power systems [@yin2022distributed; @ufa2022review], and robotics [@zhou2023racer; @9851519]. To minimize the global cost by designing some suitable distributed controllers for the agents, distributed optimization for multi-agent systems has been extensively studied, leading to significant advancements in both theoretical and computational aspects [@yang2019survey; @nedic2018distributed; @zheng2022review]. Constrained distributed optimization is one of the important categories since there may be various constraints, such as local constraints, global inequality and equality constraints in practical applications [@yang2019survey]. On the constrained distributed optimization, there are a considerable number of algorithms have been proposed, e.g., [@5404774; @9335004; @chen2020distributed; @chen2021fixed; @8262781; @bastianello2022novel; @xie2017distributed; @xie2018distributed]. However, to the best of our knowledge, most of the existing constrained distributed optimization algorithms can only be applied to bi-directional (or undirected) and weight-balanced communication networks, except for literature [@xie2017distributed; @xie2018distributed] which can be used to time-varying unbalanced directed graphs under the assumption of uniformly strong connectivity. Furthermore, these algorithms were designed for the case where the local data of all agents are completely accurate. However, these data of real-world optimization problems tend to be uncertain as a result of measurement/estimation errors and implementation errors [@ben2009robust]. Hence, the main focus of this article is to solve a distributed robust convex optimization problem (DRCO) with bounded uncertainty under the weakest assumption of network communication: uniformly strong connectivity [@burger2013polyhedral]. Recently, some distributed algorithms for dealing with the DRCO were developed in [@yang2008distributed; @wang2016distributed; @6461383; @lee2015asynchronous; @8022966; @falsone2020scenario; @you2018distributed; @carlone2014distributed; @chamanbaz2017randomized; @chamanbaz2017; @burger2012distributed; @burger2013polyhedral; @yang2014distributed], which can be categorized into four groups according to the treatment of uncertainty. Firstly, inspired by the robust counterpart approach in [@ben2009robust], some algorithms were proposed in [@yang2008distributed; @wang2016distributed], which make all the agents asymptotically converge to a feasible optimal solution of the DRCO by transforming the DRCO into a robust counterpart problem and then doing parallel computation with a constrained distributed optimization algorithm. However, these algorithms are confined to special constraint structures. Secondly, in [@6461383; @lee2015asynchronous], some random projection algorithms were designed that almost surely converge to a feasible optimal solution of the DRCO, yet the local feasibility of the solutions of all the agents cannot be guaranteed. Thirdly, some scenario-based algorithms were developed in [@8022966; @falsone2020scenario; @you2018distributed; @carlone2014distributed; @chamanbaz2017randomized; @chamanbaz2017] by sampling a large number of scenarios from the uncertainty set to approximate the DRCO, which asymptotically converge to a probabilistically feasible approximate optimal solution. However, these algorithms can not converge to the feasible optimal solution of the DRCO. Fourthly, some most relevant algorithms to our article were presented in [@burger2013polyhedral; @burger2012distributed; @yang2014distributed; @xunhao]. These algorithms are based on iteratively approximating the DRCO by populating the cutting-planes/cutting-surfaces into the existing finite sets of constraints. The algorithms given in [@burger2013polyhedral; @burger2012distributed; @yang2014distributed] asymptotically converge to a feasible optimal solution, while the algorithm in [@xunhao] enables all agents to finite-time converge to feasible and approximately optimal solutions. With the exception of [@xunhao], to our best knowledge, none of the existing algorithms can guarantee the finite-time convergence and local feasibility of the solutions for all agents. However, the solutions in [@xunhao] only satisfy the zero-order optimality conditions and cannot provide specific accuracy assurance of global optimality. Therefore, the motivation of this article is to *propose a novel distributed algorithm for locating a feasible consensus solution satisfying global optimality to a certain accuracy of the DRCO under a uniformly strongly connected network within a finite number of iterations*. In this paper, the DRCO is studied for uniformly strongly connected multi-agent systems with the strictly convex global objective function. To solve this problem, based on the right-hand restriction approach [@Mitsos], a distributed robust convex optimization algorithm is proposed, which has three parts. The first part is the distributed lower bounding procedure, which is based on iteratively approximating the DRCO by enforcing the compact uncertain sets at finitely many points. The second part is the distributed upper bounding procedure, which is developed by successively reducing the restriction parameters of the right-hand constraints and tightening the discretization of the compact uncertain sets. Both procedures above guarantee that each agent converges to the optimal solution of the DRCO. Moreover, at each iteration, the sum of the local objectives of all agents in the two procedures constitutes the lower and upper bounds of the global optimal value of the DRCO, respectively. The third part is an adaptation of the finite-time consensus algorithm proposed in [@xie2017stop], which make all agents terminate simultaneously when the gap between the lower and the upper bounds reaches the certain accuracy. The main contribution of this paper is threefold: 1. A distributed robust convex optimization algorithm is proposed to locate a feasible consensus solution of the DRCO satisfying global optimality to a certain accuracy under the assumption of a uniformly strongly connected network. 2. Two distributed termination methods are proposed to ensure finite-time convergence of the distributed robust convex optimization algorithm, and the performance of these two methods is compared. 3. It is mathematically proven that the distributed robust convex optimization algorithm terminates within a finite number of iterations. The remaining sections of the paper are organized as follows. In Section 2, the problem formulation and some fundamental assumptions are given. In Section 3, two approximate problems of the DRCO are presented, and the distributed lower bounding procedure and distributed upper bounding procedure are designed. In Section 4, the distributed robust convex optimization algorithm for solving the DRCO is described, and the proof of finite-time convergence of the algorithm is also presented. In Section 5, a numerical case study is conducted to validate the proposed algorithm, and comparisons are provided between the proposed algorithm and some related algorithms. Finally, conclusions and an outlook on future work are drawn in Section 6. # Problem Statement We consider a multi-agent system consisting of a set of agents $\mathcal{V}=\left\{1,... ,m\right\}$, in which each agent stores its local constraint, local cost function, identifier, and other private information. In a distributed optimization task, all the agents locate a feasible consensus solution for minimizing the global objective function based on agent communication and local computation. However, in real-world multi-agent systems, there exist perturbation errors in the parameters of each agent due to erroneous inputs, such as in the estimation and implementation. In order to guarantee the safety and stability of the system, we consider a distributed robust convex optimization problem with bounded uncertainty in the local objective functions and constraints of the following form to find a robust optimal consensus solution for the multi-agent system. [\[DRO\]]{#DRO label="DRO"} &\_x & & F(x)=\^m\_i=1 f_i(x,\_i)\ & s.t.& & xX= \_i=1\^m \_\_i_i X_i(\_i), where $x\in\mathbb{R}^n$ is a common decision vector of agents. $\delta_i\in \Delta_i$ and $\gamma_i\in \Gamma_i$ represent the uncertain parameters of local objective function and constraint set for agent $i$, respectively. $\Delta_i$ and $\Gamma_i$ are non-empty and compact sets. For each $i=1,...,m$, $f_i(\cdot):\mathbb{R}^n\times\Delta_i\rightarrow\mathbb{R}$ is the local objective function of agent $i$, and $X_i={\bigcap}_{\gamma_i\in \Gamma_i} X_i(\gamma_i)\subseteq\mathbb{R}^n$ is its constraint set. Suppose that $f_i$ is a convex function on $x$, and the set $X_i$ is convex and compact for all $i\in\mathcal{V}$. Problem ([\[DRO\]](#DRO){reference-type="ref" reference="DRO"}) is the general form of a distributed robust convex optimization problem, where the robust optimal point $x^*\in X$ minimizes the global objective function $F$ considering the worst-case uncertainty $\overline\delta_i=\arg{\max}_{\delta_i\in\Delta_i}f_i$ for all $i\in\mathcal{V}$. To simplify the problem ([\[DRO\]](#DRO){reference-type="ref" reference="DRO"}), we adopt the epigraphic reformulation technique to transform ([\[DRO\]](#DRO){reference-type="ref" reference="DRO"}) into a standard distributed robust convex optimization problem. This technique ensures that the uncertainty is confined to the constraints, while the local objective functions remain unaffected [@ben2009robust]. The following is the standard form of a distributed robust convex optimization problem ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). [\[DRCO\]]{#DRCO label="DRCO"} &\_x & & F(x)=\^m\_i=1f_i(x)\ & s.t. & & g_i(x,y_i), y_iY_i,i=1,2,\...m, where $x\in\mathbb{R}^n$ is a common decision vector of agents, and $y_i \in Y_i$ is an uncertain vector of agent $i$. $Y_i\subseteq\mathbb{R}^{n_y}$ is a non-empty and compact set. For each $i=1,...,m$, $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ is the local objective function, and $g_i:\mathbb{R}^n\times Y_i\rightarrow \mathbb{R}$ is the local constraint function, where $g_i$ is a semi-infinite constraint consisting of finite-dimensional decision variables and an infinite number of inequality constraints. For all agents $i$, assume that $f_i$ and $g_i$ are convex functions concerning $x$ and that $g_i$ is continuous with respect to $y_i$. Let the feasible region of agent $i$ be $X_i=\left\{x|g_i(x,y_i)\le 0\ \forall y_i\in Y_i\right\}$. **Assumption 1** (Solvability and Uniqueness). *The global feasible region of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) $X=\bigcap^m_{i=1} X_i$ is not empty, and the global objective function $F(x)$ is strictly convex on $x\in \mathbb{R}^n$, i.e., for any two point $u\neq v\in \mathbb{R}^n$, any $0<\theta<1$, there is* *F(u+(1-) v)\<F(u)+(1-)F(v).* Due to Assumption [Assumption 1](#Assumption 1){reference-type="ref" reference="Assumption 1"}, the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) is solvable and has a unique optimal consensus solution $x^*$, i.e. for any $x\in X\verb|\| \left\{x^*\right\}$, it follows that $F^*=F(x^*)< F(x)$. Note that for the case where the global objective function is not strictly convex but convex, a tie-break rule can be used to ensure the uniqueness of the solution. For further interpretation of the tie-break rule, the reader is referred to [@burger2013polyhedral; @Calafiore2006]. The communication among agents can be characterized in graph theory [@yang2019survey; @west2001introduction]. A graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ can be used to represent the information sharing relationships among agents, where $\mathcal{V}=\left\{1,...,m\right\}$ denotes the vertex set, and $\mathcal{E}=\mathcal{V}\times\mathcal{V}$ is the edge set. A directed edge $(i,j)\in \mathcal{E}$ represents that agent $j$ can directly obtain information from agent $i$. $(i,i)$ indicates the self-loop of agent $i$. For agent $i$, the set of its in-neighbors is $N_i^{in}=\left\{j|(j,i)\in \mathcal{E}\right\}$, and the set of its out-neighbors is $N_i^{out}=\left\{j|(i,j)\in \mathcal{E}\right\}$. If for graph $\mathcal{G}$, $(j, i)\in\mathcal{E}$ if and only if $(i,j)\in\mathcal{E}$, then $\mathcal{G}$ is an undirected graph; otherwise, it is a directed graph. Define the weight matrix $A=\left\{a_{ij}\right\}\in \mathbb{R}^{m\times m}$, which satisfies that $a_{ij}>0$ if $j\in N_i^{in}$ and $a_{ij}=0$, otherwise. If $\sum^m_{i=1} a_{ij} =\sum^m_{i=1} a_{ji}$ for any $j\in \mathcal{V}$, then the graph $\mathcal{G}$ is weight-balanced; otherwise, $\mathcal{G}$ is weight-imbalanced. A time-invariant network is one in which the edge set remains unchanged over time slots, while a time-varying network experiences changes in the edge set due to unexpected loss of communication links. A time-invariant communication graph $\mathcal{G}$ is said to be strongly connected if and only if every agent in the graph is reachable from all other agents. For the time-variant network, the following assumption is commonly made [@yang2019survey; @6930814]. **Assumption 2** (Uniformly Strong Connectivity). *The graph sequence $\left\{\mathcal G(t)\right\}$ is uniformly strongly connected, i.e. for all $t\ge 0$, there exists an interger $T>0$ that makes $G(t:t+T)$ strongly connected. $$\mathcal G(t:t+T)=([m],\mathcal G(t)\cup...\cup \mathcal G(t+T-1)).$$* The assumption of uniformly strong connectivity is seen as the weakest assumption in network communication [@burger2013polyhedral]. The main focus of this article is to propose a distributed optimization algorithm that converges to an optimal consensus solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) in a finite number of iterations under the uniformly strongly connected assumption in the communication network while ensuring the feasibility of the solution at each agent for local constraints. # Approximation Problems Since the constraints of all agents are semi-infinite constraints with finite-dimensional decision variables and an infinite number of inequality constraints, solving the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) is NP-hard [@ben2009robust]. This section introduces the distributed lower bounding problem and designs a distributed lower bounding procedure based on the approach of successively tighter discretization of the compact sets $Y_i$. Then, this section presents the distributed upper bounding problem and illustrates that this problem is neither a relaxation nor a restriction of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). Moreover, a distributed upper bounding procedure is developed by successively reducing the restriction parameters of the right-hand constraints and tightening the discretization of the compact sets $Y_i$. ## Distributed Lower Bounding Procedure {#sec:Distributed Lower Bounding Procedure} A distributed lower bounding problem is introduced by discretizing the compact sets $Y_i$ into finite sets $\widetilde Y^k_i$. [\[dlbd\]]{#dlbd label="dlbd"} &\_x & & F(x)=\^m\_i=1f_i(x)\ & s.t. & & g_i(x,y_i), y_iY\^k_i,i=1,2,\...m. Let $\widetilde X_i^k=\left\{x\in\mathbb{R}^n|g_i(x,y_i)\le0,\ \forall y_i\in \widetilde Y_i^k\right\}$ be the feasible set of the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) for agent $i\in\mathcal{V}$. The global feasible set is $\widetilde X^k={\bigcap}_{i=1}^m\ \widetilde X^k_i$. For any finite set $\widetilde Y^{k+1}_i$, there is $\widetilde X_i^k\supset X_i$. Therefore, the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) is a relaxation of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) and $\widetilde X^k\supset X$. Under Assumption 1, for any finite sets $\widetilde Y^{k+1}_i$ of all agents, the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) is a solvable constrained distributed convex optimization problem, where the global optimal solution satisfies uniqueness. Motivated by the strategy proposed in [@1976Infinitely; @Mitsos; @burger2013polyhedral], we develop the first algorithmic primitive by successively tightening the discretization of the compact sets $Y_i$ of all agents, *distributed lower bounding procedure*. For any $k\ge0$, during the $(k+1)$-th iteration, each agent initially acquires the optimal point $x_i^{k+1}$ for ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) within finite time slots by agent communication and local computation. Each agent then assigns the value of $x^{k+1}_i$ to $\widetilde x^{k+1}_i$ for the stopping criterion detection in Section [4.2](#sec:Finite-time Convergence){reference-type="ref" reference="sec:Finite-time Convergence"}. Subsequently, each agent updates the finite set $\widetilde Y_i^{k+1}$ through the distributed lower bounding (DLBD) Oracle. - **DLBD Oracle** $\widetilde Y_i^{k+1}=LORC({x}^{k+1}_i,Y_i)$: verify the feasibility of a given point ${x}^{k+1}_i$ for agent $i$ by solving a lower level problem (LLP) to global optimality. $$\label{LLP} \tag{LLP} g^{\max}_i({x}^{k+1}_i)=\mathop{\max}\limits_{y_i\in Y_i} g_i({x}^{k+1}_i,y_i)$$ - **If** $(\romannumeral 1)$ $g^{\max}_i({x}^{k+1}_i)>0$, i.e. ${x}^{k+1}_i\notin X_i$ then it populates the point $\hat{y}_i=\arg \max_{y_i\in Y_i} g_i({x}^{k+1}_i,y_i)$ into the finite set: $\widetilde Y_i^{k+1}=\widetilde Y_i^{k}\cup\left\{\hat{y}_i\right\}$, separating ${x}^{k+1}_i$ and $X_i$, **otherwise** $(\romannumeral 2)$ it indicates that ${x}^{k+1}_i\in X_i$ and remains $\widetilde Y_i^{k+1}=\widetilde Y_i^{k}$. $\square$ ## Distributed Upper Bounding Procedure {#sec:Distributed Upper Bounding Procedure} A distributed upper bounding problem is constructed by discretizing the compact sets $Y_i$ into finite sets $\overline Y_i^k$ and restricting the rights-hand constraints with proper positive parameters $\varepsilon^k_i$. [\[dubd\]]{#dubd label="dubd"} &\_x & & F(x)=\^m\_i=1f_i(x)\ & s.t. & & g_i(x,y_i)-\_i\^k, y_iY_i\^k,i=1,2,\...m. Set $\overline X_i^k=\left\{x\in\mathbb{R}^n|g_i(x,y_i)\le-\varepsilon_i^k,\ \forall y_i\in \overline Y_i^k\right\}$ as the feasible set of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) for agent $i\in\mathcal{V}$. The global feasible set is $\overline X^k={\bigcap}_{i=1}^m\ \overline X^k_i$. To elucidate the relations between the set $\overline X^k_i$ and the feasible set $X_i$ of the agent $i$ in the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), we present the subsequent illustrations. *Example 1*: Consider a distributed system where the constraint of the agent $i\in\mathcal{V}$ is a semi-infinite constraint as shown below: $$\label{example} \begin{aligned} & h_i(x,y_i)=[x(1)^2-2 x(1)]\times e^{-{x(1)}^2+{y_i}^2-2 x(1) y_i},\\ & g_i(x,y_i)=x(2)+{h_i(x,y_i)}\le 0, \ \ \forall y_i\in Y_i, \end{aligned} %g(x,y)=x(2)+[x(1)^2-2\times x(1)]\times e^{-{x(1)}^2+y^2-2 x(2) y}\le 0 \forall y\in Y$$ where $x=[x(1),x(2)]^\top\in \mathcal{F}_i=[0,2]\times[0,1]$, $Y_i=[0,2]$. Since $g_i(x,y_i)$ is a concave function on the uncertain parameter $y_i$, we can obtain that the semi-infinite constraint ([\[example\]](#example){reference-type="ref" reference="example"}) results in the feasible set of agents $i$: $X_i=\left\{x\in \mathcal{F}_i|x(2)+[x(1)^2-2\times x(1)]\times e^{-2{x(1)}^2}\le 0\right\}$. The corresponding inequality constraint constructed from the semi-infinite constraint is: $$\label{inequality} g_i(x,y_i)=x(2)+{h_i(x,y_i)}\le -{\varepsilon^k_i}, \ \ \forall y_i\in \overline Y_i^k,$$ where ${\varepsilon_i^k}\ge 0$, $\overline Y_i^k\subset Y_i$, and $\overline X_i^k$ is the feasible set of ([\[inequality\]](#inequality){reference-type="ref" reference="inequality"}). Figure [\[neither\]](#neither){reference-type="ref" reference="neither"} illustrates the relations between $X_i$ and $\overline X_i^k$ for different combinations of $\overline Y_i^k$ and $\varepsilon_i^k$. +:--------------------------------------------------------------:+:-:+ | {width="3in"} | | | | | | {width="3in"} | | +----------------------------------------------------------------+---+ | {width="3in"} | | | | | | {width="3in"} | | +----------------------------------------------------------------+---+ a) For $\overline Y_i^k=Y_i$ and $\varepsilon_i^k>0$, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is a restriction of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}): the feasible set of the agent $i$ satisfies $\overline X_i^k\subset X_i$; b) For the finite set $\overline Y_i^k\subset Y_i^k$ and $\varepsilon_i^k=0$, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is a relaxation of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}): the feasible sets of the agent $i$ satisfy $\overline X_i^k\supset X_i$; c) For $\varepsilon_i^k=0.4$ and $\overline Y_i^k=\left\{1\right\}$, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is a restriction of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}); d) For a smaller value $\varepsilon_i^k=0.1$ of the restriction parameter and the same finite set $\overline Y_i^k=\left\{1\right\}$, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is neither a relaxation nor a restriction of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}): whether the optimal point of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is feasible for ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) relies on the global objective function $F$; e) For $\varepsilon_i^k=0.1$ and $\overline Y_i^k=\left\{1,0.5\right\}$, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is again a restriction of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). This case is a tighter restriction compared to the case c). Based on the above example, it can be concluded that for any finite sets $\overline Y_i^k\subset Y_i^k$ and positive restriction parameters $\varepsilon_i^k$ of all agents, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) cannot be regarded as a relaxation or a restriction of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). However, by gradually populating some points in the finite set $\overline Y_i^k$ and proportionally reducing the restriction parameter $\varepsilon_i^k$ for all agents, the feasible region of ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) can gradually approach that of ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), which can make the optimal solution of ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) converge to that of ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). Following this idea and the centralized right-hand restriction strategy [@Mitsos], we propose the second algorithmic primitive: *distributed upper bounding procedure*. For any $k\ge 0$, during the $(k+1)$-th iteration, each agent firstly obtains the optimal point $z_i^{k+1}$ of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) in a finite number of time slots by exchanging local information with neighboring agents and performing local calculations. Then, each agent executes the distributed upper bouding (DUBD) Oracle via internal computation. Here, the vector $\overline x_i^{k+1}$ is chosen for the stopping criterion detection introduced in Section [4.2](#sec:Finite-time Convergence){reference-type="ref" reference="sec:Finite-time Convergence"}. - **DUBD Oracle** $[\overline x_i^{k+1},\overline Y_i^{k+1},\varepsilon^{k+1}_i]=UORC$\ $(z_i^{k+1}, Y_i)$: queried at a given point $z_i^{k+1}$ for the compact set $Y_i$ by solving a lower level problem (LLP) to global optimality. $$\tag{LLP} g^{\max}_i(z_i^{k+1})=\mathop{\max}\limits_{y_i\in Y_i} g_i(z_i^{k+1},y_i)$$ - **If** $(\romannumeral 1)$ $g^{\max}_i(z_i^{k+1})>0$, i.e. $z_i^{k+1}\notin X_i$ then it populates the point $\hat{y}_i=\arg \max_{y_i\in Y_i} g_i(z_i^{k+1},y_i)$ into the finite set: $\overline Y_i^{k+1}=\overline Y_i^k\cup\left\{\hat{y}_i\right\}$, separating $z_i^{k+1}$ and $X_i$. Assign a value to $\overline x^{k+1}_i$ that satisfies $f_i(\overline x^{k+1}_i)\rightarrow+\infty$, and let $\varepsilon_i^{k+1}=\varepsilon_i^k$, **otherwise** $(\romannumeral 2)$ it reduces the restriction parameter proportionally, i.e. $\varepsilon_i^{k+1}\leftarrow\varepsilon_i^k/r$, where $r>1$ is a reduction parameter, and let $\overline Y_i^{k+1}=\overline Y_i^{k}$. Additionally, we assign the value of vector $z_i^{k+1}$ to $\overline x^{k+1}_i$. $\square$ There may be an unsolvable case of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) in the distributed upper bouding procedure, i.e. $\overline X^k=\emptyset$. We make the following assumption to exclude this case. **Assumption 3** (Interior Point). *The feasibile set of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) $X=\bigcap_{i=1}^m X_i$ contains at least one interior point $x_0$, i.e. $$g_i(x_0,y_i)<0,\ \forall y_i\in Y_i, \quad i=1,...,m.$$* **Lemma 1**. *Under Assumption [Assumption 3](#Assumption 3){reference-type="ref" reference="Assumption 3"}, for any agent $i\in\mathcal{V}$, there exists an initial value $\varepsilon^{0}_i>0$ for the restriction parameter $\varepsilon_i^k$ such that the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is solvable in any iteration $k+1>0$ of the distributed upper bounding procedure.* **Proof.** Since the feasible set of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) exists at least an interior point $x_0$ that satisfies $$\nonumber x_0\in \bigcap^m_{i=1} \left\{x\in \mathbb{R}^n|g_i(x,y_i)<0,\forall y_i\in Y_i\right\}.$$ Hence, there are a set of $\varepsilon^{0}_i$ that satisfies $$\nonumber g_i(x_0,y_i)+\varepsilon^{0}_i\le0,\ \forall y_i\in Y_i, \quad i=1,...,m.$$ We set $\hat{X}=\left\{x\in\mathbb{R}^n|g_i({x},y_i)\le-\varepsilon^{0}_i,\ \forall y_i\in Y_i,\ \forall i\in\mathcal{V}\right\}$. The set $\overline{X}^k$ is the global feasible set of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}), i.e. $$\overline{X}^{k}=\left\{x\in\mathbb{R}^n|g_i({x},y_i)\le-\varepsilon^{k}_i,\ \forall y_i\in \overline Y_i^k,\ \forall i\in\mathcal{V}\right\}.$$ As $\varepsilon^{0}_i\ge\varepsilon_i^k$ and $\overline Y_i^k\subset Y_i$ hold for any agent $i$, we have $\hat X \subseteq \overline{X}^k$ and $\overline{X}^k\neq\emptyset$. Therefore, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is solvable in any iteration of the distributed upper bounding procedure. $\square$ Therefore, under Assumptions [Assumption 1](#Assumption 1){reference-type="ref" reference="Assumption 1"} and [Assumption 3](#Assumption 3){reference-type="ref" reference="Assumption 3"}, take proper values of $\varepsilon_i^0$ for all agents, the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is a solvable constrained distributed convex optimization problem in any iteration $k+1>0$ of the distributed upper bounding procedure, where the global optimal solution meets uniqueness. Overall, this section presents two approximation problems for the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) and proposes a distributed lower bounding procedure and a distributed upper bounding procedure. The convergence of these procedures is given in Section [4](#sec:Algorithm Design and Convergence Analysis){reference-type="ref" reference="sec:Algorithm Design and Convergence Analysis"}. # Algorithm Design and Convergence Analysis {#sec:Algorithm Design and Convergence Analysis} In this section, based on the aforementioned procedures, a distributed robust convex optimization algorithm to locate a feasible consensus solution satisfying global optimality to a certain accuracy is described, and its finite-time convergence is established. ## Distributed Robust Convex Optimization Algorithm {#{sec:Distributed Robust Convex Optimization Algorithm}} The algorithm for solving the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) is as follows: **Input:** for each agent $i\in\mathcal{V}$: initial restriction parameter $\varepsilon_i^{0}>0$; two finite or empty subsets of $Y_i$, namely $\widetilde Y_i^{0}$ and $\overline Y_i^{0}$; reduction parameter $r>1$; termination parameter $\epsilon^f$; iteration counter $k=0$.\ **Repeat:** - ***Distributed lower bounding procedure*** - - Solve the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) to optimality: each agent obtains the optimal solution $x_i^{k+1}$, and set $\widetilde x^{k+1}_i\leftarrow x^{k+1}_i$. - Call the DLBD Oracle for the compact set $Y_i$ at the query point $x^{k+1}_i$, i.e. $\widetilde Y_i^{k+1}=LORC(x^{k+1}_i,Y_i)$. - ***Distributed upper bounding procedure*** - - Solve the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) to optimality: each agent obtains the optimal solution $z_i^{k+1}$. - Call the DUBD Oracle for the compact set $Y_i$ at the query point $z_i^{k+1}$, i.e. $[\overline x_i^{k+1},\overline Y_i^{k+1},\varepsilon^{k+1}_i]=UORC(z_i^{k+1},Y_i)$. - ***Finite-time termination*** - - Check whether the stopping criterion is satisfied (see: Section [4.2](#sec:Finite-time Convergence){reference-type="ref" reference="sec:Finite-time Convergence"}). - **if** the stopping criterion is satisfied **then** - - Set the optimal solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) $x^{opt}_i\leftarrow \overline x^{k+1}_i$, **Terminate**. - **end if** - **Set** $k\leftarrow k+1$ **Remark 1**. *In Steps 1 and 3 of the Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}, we require to solve the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) and ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) to optimality within a finite number of time slots. In the literature, many approaches have been proposed to finite-time/fixed-time converges to a consensus solution in the discrete-time setting [@rikos2022finite; @chen2023privacy; @jiang2022fast; @yao2018distributed; @prakash2019distributed; @manitara2014distributed; @mai2017local; @carlone2014distributed; @xie2017stop] or the continuous-time setting [@wang2020distributed; @shi2020continuous; @shi2022finite; @garg2020fixed; @song2021fixed; @wu2021designing; @shi2022finite; @firouzbahrami2022cooperative]. Since the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) and the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) are distributed convex optimization problems with local inequality constraints under uniformly strongly connected networks, we present two practical strategies for such problems to make all agents converge to their consensus optimal solutions within a finite number of time slots. The first strategy involves the D-RFP algorithm [@xie2018distributed] equipped with the finite-time consensus algorithm [@xie2017stop]. The second strategy entails transforming the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) (or the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"})) into a distributed optimization problem with identical local objective functions via epigraphic reformulation (see: Example 2 in [@you2018distributed]), and then adopts the approach by exchanging the parameters of the local constraint function with neighboring agents to make all agents converge to an optimal consensus solution in a fixed number of time slots [@carlone2014distributed].* **Assumption 4**. *For distributed convex optimization problems with a finite number of inequality constraints under uniformly strongly connected networks, the decision variables of all agents $i\in\mathcal{V}$ can converge to an optimal consensus solution within finite time slots.* **Remark 2**. *In Steps 2 and 4, it is necessary to globally solve the LLP problems. For the case that the constraint function $g_i$ is a differentiable and concave function with respect to $y_i$ for any agent $i\in\mathcal{V}$. According to the optimality condition of convex problems (see: Literature [@boyd2004convex] p267), we can obtain the optimal solution of the LLP problems by finding points that satisfy the Karush-Kuhn-Tucker (KKT) conditions. Literature [@ben2009robust; @burger2012distributed] concludes some results of the solution to convex LLP problems under specific uncertain sets $Y_i$ and constraint functions $g_i$. For the case where the LLP is a nonconvex optimization problem, there is no direct method to find the global optimal solution. The two main indirect methods are the discretization method [@chen2005inequality] and the $\alpha BB$ method [@stein2012adaptive], respectively. The former is based on iteratively approximating the LLP problems by successively discretizing the set $Y_i$. The latter focuses on adaptively constructing convex relaxations of the LLP problems.* **Assumption 5**. *For $i\in\mathcal{V}$, at any iteration $k+1>0$, the LLP is globally solved for the query point $\hat{x}_i$ either establishing $\max_{y_i\in Y_i}g_i(\hat{x}_i,y_i)\le 0$, or furnishing a point $\hat{y}_i$ such that $g_i(\hat{x}_i,\hat{y}_i)> 0$.* **Lemma 2**. *For any $i\in\mathcal{V}$, take any $\widetilde Y^{0}_i\subset Y_i$. Under Assumptions [Assumption 1](#Assumption 1){reference-type="ref" reference="Assumption 1"}-[Assumption 2](#Assumption 2){reference-type="ref" reference="Assumption 2"} and [Assumption 4](#Assumption 4){reference-type="ref" reference="Assumption 4"}-[Assumption 5](#Assumption 5){reference-type="ref" reference="Assumption 5"}, suppose that ${x}^{k+1}$ is the optimal consensus solution of the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) in the $(k+1)$-th iteration of the distributed lower bounding procedure, where ${x}_i^{k+1}=x^{k+1},\ \forall i\in\mathcal{V}$. Let $F^*$ be the optimal objective of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). Then,* - *$F(x_i^{k+1})\le F^*$ for all $i\in\mathcal{V}$ and all $k\ge 0$;* - *$F(x_i^{k+1})\le F(x_i^{k+2})$ for all $i\in\mathcal{V}$ and all $k\ge 0$;* - *There exists a point $p_i\in\mathbb{R}^n$ that the sequence $\left\{x_i^{k+1}\right\}_k$ converges to it, i.e. $\mathop{\lim}\limits_{k\rightarrow +\infty} x_i^{k+1}=p_i$ for all $i\in\mathcal{V}$.* - *The limit point $p_i$ is feasible for the agent $i\in\mathcal{V}$ in the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), i.e. $p_i \in X_i$.* **Proof.** *Proof of ):* The feasible region $X$ of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) satisfies $$X=\mathop{\bigcap}\limits_{i=1}^m X_i\subset X_i.$$ For any agent $i\in\mathcal{V}$ and $k\ge 0$, it follows that $$\widetilde X_i^k=\left\{x|g_i(x,y_i)\le 0, \ \forall y_i\in \widetilde Y_i^k\right\}\supset X_i.$$ Since $\widetilde X^k=\bigcap_{i=1}^m \widetilde X_i^k\supset X$ for all $k\ge 0$, it can be concluded that $F(x^{k+1})\le F^*$. Therefore, it is satisfied that $F(x_i^{k+1})\le F^*$ for any $i\in\mathcal{V}$ and $k\ge 0$. *Proof of ):* For any $k\ge 0$ and any $i\in\mathcal{V}$, after the step of the DLBD Oracle, there is $$\widetilde Y_i^k\subset \widetilde Y_i^{k+1}.$$ Therefore, the feasible domain of ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) in two iterations satisfies the following relation: $$\widetilde X_i^k\supset \widetilde X_i^{k+1},\ \ \forall i\in\mathcal{V}.$$ $x_i^{k+1}$ and $x_i^{k+2}$ are the solutions obtained from these two iterations of the distributed lower bouding procedure respectively. Therefore, we can conclude that $F(x_i^{k+1})\le F(x_i^{k+2})$ for any $i\in\mathcal{V}$ and $k\ge 0$. *Proof of ):* According to \[Lemma [Lemma 2](#DLBD_lemma){reference-type="ref" reference="DLBD_lemma"}, )\] and \[Lemma [Lemma 2](#DLBD_lemma){reference-type="ref" reference="DLBD_lemma"}, )\], we can conclude that the sequence $\left\{F(x_i^{k+1})\right\}_k$ is bounded and non-decreasing. It can be followed that the sequence $\left\{F(x_i^{k+1})\right\}_k$ is convergent, i.e. $$\mathop{\lim}\limits_{k\rightarrow +\infty} \vert F(x_i^k)-F(p_i)\vert= 0.$$ Due to strict convexity of global objective function $F$, the sequence $\left\{x_i^{k+1}\right\}_k$ converges to a point $p_i$. *Proof of ):* Refering to the proof idea in [@Mitsos]. we prove the feasibility of the limit point $p_i$ for the agent $i$. For $i\in\mathcal{V}$ and $k\ge 0$, consider the corresponding solution of the ([\[LLP\]](#LLP){reference-type="ref" reference="LLP"}) $\hat y_i$ in the DLBD Oracle for which $g_i(x_i^{k+1},\hat y_i)>0$. By reconstruction of the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) we have $$\begin{aligned} \nonumber g_i(x_i^l,\hat y_i)\le 0,\quad \forall l>k+1> 0. \end{aligned}$$ Since $g_i$ is a continuous function on $x$, for any $\epsilon>0$, there is a positive parameter $\delta$ satisfying: $$\begin{aligned} \label{lemma1_3_1} g_i(x,\hat y_i)<\epsilon,\quad \Vert x-x_i^l\Vert<\delta,\ \forall l>k+1> 0. \end{aligned}$$ Due to the convergence of $\left\{x_i^{k+1}\right\}_k$ \[Lemma [Lemma 2](#DLBD_lemma){reference-type="ref" reference="DLBD_lemma"} )\], we have for any $\delta>0$, $$\label{lemma1_3_2} \exists K:\ \Vert x_i^l-x_i^{k+1}\Vert<\delta,\quad \forall l>k+1\ge K.$$ Combining the results of ([\[lemma1_3\_1\]](#lemma1_3_1){reference-type="ref" reference="lemma1_3_1"}) and ([\[lemma1_3\_2\]](#lemma1_3_2){reference-type="ref" reference="lemma1_3_2"}), we can obtain that for any $\epsilon>0$, $$\begin{aligned} \nonumber \exists K:\ g_i(x_i^{k+1},\hat y_i)<\epsilon,\quad \forall k+1\ge K. \end{aligned}$$ Since $g_i(x_i^{k+1},\hat y_i)>0$, we have $g_i(x_i^{k+1},\hat y_i)\rightarrow 0$. Therefore, the limit point $p_i$ is feasible: $$\nonumber \mathop{\max}\limits_{y_i\in Y_i} g_i(p_i,y_i)=\mathop{\lim}\limits_{k\rightarrow \infty} g_i({x}_i^{k+1},\hat y_i)=0. \vspace{-8mm}$$ $\square$ **Proposition 1**. *For any $i\in\mathcal{V}$, take any $\widetilde Y^{0}_i\subset Y_i$. Suppose that Assumptions 1-2 and 4-5 hold. Then, the distributed lower bounding procedure is convergent, i.e. ${\sum}^m_{i=1}f_i({\widetilde x^{k+1}_i})\rightarrow F^*$. Moreover, for any $k\ge0$, it is satisfied that ${\sum}^m_{i=1}f_i({\widetilde x^{k+1}_i})\le F^*$.* **Proof.** Under Assumption [Assumption 4](#Assumption 4){reference-type="ref" reference="Assumption 4"}, the following relation holds for any agent $i\in\mathcal{V}$: $\widetilde x^{k+1}_i=x^{k+1}_i=x^{k+1}$, owing to the consensus of the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) solution. Combining the result of \[Lemma [Lemma 2](#DLBD_lemma){reference-type="ref" reference="DLBD_lemma"} )\], we have $$\mathop{\sum}\limits^m_{i=1}f_i(\widetilde x_i^{k+1})=\mathop{\sum}\limits^m_{i=1}f_i(x_j^{k+1})=F(x_j^{k+1})\le F^{*},\ \forall k\ge0.$$ where $j$ is the identify of an arbitrary agent satisfying $j\in\mathcal{V}$.\ According to \[Lemma [Lemma 2](#DLBD_lemma){reference-type="ref" reference="DLBD_lemma"}, )\] and \[Lemma [Lemma 2](#DLBD_lemma){reference-type="ref" reference="DLBD_lemma"}, )\], we can conclude that the sequence $\left\{\widetilde x_i^{k+1}\right\}_k$ converges to a consensus point $p$ for any agent $i\in\mathcal{V}$, where $p$ is a feasible point for the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), i.e. $p \in{\bigcap}_{i=1}^m X_i=X$. It follws that $$\nonumber F(p)\ge F^{*}.$$ Since $$F(p)=\lim_{k\rightarrow\infty}\mathop{\sum}\limits^m_{i=1}f_i(\widetilde x_i^{k+1})\le F^{*},$$ we can obtain $F(p)= F^{*}$. To sum up, the sequence $\left\{{\sum}^m_{i=1}f_i(\widetilde x_i^{k+1})\right\}_k$ converges to $F^*$. $\square$ **Lemma 3**. *For all agents $i\in\mathcal{V}$, take any $\overline Y^0_i\subset Y_i$ and proper restriction parameter $\varepsilon^0_i$. Suppose that Assumptions [Assumption 1](#Assumption 1){reference-type="ref" reference="Assumption 1"}-[Assumption 5](#Assumption 5){reference-type="ref" reference="Assumption 5"} hold. Let ${z}^{k+1}$ be the optimal consensus solution of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) in the $(k+1)$-th iteration of the distributed upper bounding procedure, in which ${z}_i^{k+1}=z^{k+1},\ \forall i\in\mathcal{V}$. Then,* - *there exists a point $q_i$ that the sequence $\left\{z_i^{k+1}\right\}_k$ converges to it, i.e. $\mathop{\lim}\limits_{k\rightarrow+\infty}z_i^{k+1}=q_i$ for all $i\in\mathcal{V}$.* - *any agent $i$ can obtain a locally feasible point $\hat{x}_i$ for the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) (i.e. $\hat{x}_i\in X_i$) within a finite number of iterations through the distributed upper bounding procedure.* **Proof.** *proof of ):* Before proving this result, we make the following settings: [\[8\]]{#8 label="8"} &X\^k=\_i=1\^m{x\^n\|g_i(x,y_i)-\^k_i,  y_iY_i\^k},\ & A\^k=\_i=1\^m{x\^n\|g_i(x,y_i)-\_i\^k,  y_iY_i},\ & B\^k=\_i=1\^m{x\^n\|g_i(x,y_i),  y_iY\^k_i}. Based on the above settings, we can easily derive the following result: for any $k\ge 0$, A\^kX\^kB\^k. As the number of iterations of the distributed upper bounding procedure increases, the value of $\varepsilon_i^k$ decreases proportionally, while the number of elements in the set $\overline Y^k_i$ continues to increase. It follows that [\[10\]]{#10 label="10"} A\^k\^k+1\^+=X,\ B\^k\^k+1\^+=X. According to the squeeze theorem, it can be concluded that $$\mathop{\lim}\limits_{k\rightarrow \infty} \overline X^k\rightarrow X.$$ Since the global objective function is strictly convex and the global feasible region in the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) converges to that of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), the optimal point sequence $\left\{z_i^{k+1}\right\}_k$ of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is convergent for any agent $i\in\mathcal{V}$. *proof of ):* Since the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is neither a restriction nor a relaxation of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), the solution of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) may not lie in the feasible domain of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). Referring to the proof idea in [@Mitsos], we prove that the solution of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) satisfying the constraints in the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) can be obtained within a finite number of iterations of the distributed upper bounding procedure. For any agent $i\in\mathcal{V}$, suppose that at the $(k+1)$-th iteration of the distributed upper bounding procedure, the optimal point of agent $i$ obtained by solving the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) does not satisfy its local semi-infinite constraint, i.e. $z_i^{k+1}\notin X_i$, and let $\hat{y}_i$ be the corresponding maximum constraint violation point obtained through the DUBD Oracle. It follows that $$g_i(z_i^{k+1},\hat{y}_i)>0.$$ For any $l>k+1> 0$, there exists a positive parameter $\epsilon$ that satisfies $$g_i(z_i^l,\hat{y}_i)\le -\epsilon<0.$$ Given that $X_i$ and $Y_i$ are both compact sets and the constraint function $g_i(x,y_i)$ is continuous, we can conclude that for any $l>k+1> 0$ $$\exists \delta>0,\ \ g_i(x,\hat{y}_i)\le -\epsilon/2<0,\ \ \Vert x-z_i^l\Vert<\delta.$$ Since the sequence $\left\{z_i^{k+1}\right\}_k$ is convergent \[Lemma [Lemma 3](#DUBD_lemma){reference-type="ref" reference="DUBD_lemma"} )\], it is satisfied that for any $\delta>0$ $$\exists K:\ \Vert z_i^l-z_i^{k+1}\Vert<\delta,\ \ \forall l,k:\ l>k+1\ge K.$$ Based on (12) and (13), we can infer that $$\nonumber \exists K: \ g_i(z_i^{k+1},\hat{y}_i)\le -\epsilon/2<0,\ \ \forall k:\ k+1\ge K.$$ Therefore, it can be inferred that for any agent $i\in\mathcal{V}$, there exists a finite number of iterations $K$ in the distributed upper bounding procedure, such that the optimal point obtained by solving the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) at the $K$-th iteration satisfies its local semi-infinite constraint in the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). $\square$ **Proposition 2**. *For any $i\in\mathcal{V}$, take any $\overline Y^{0}_i\subset Y_i$ and proper restriction parameter $\varepsilon^0_i$. Suppose that Assumptions 1-5 hold. Then, the distributed upper bounding procedure is convergent, i.e. ${\sum}^m_{i=1}f_i({z^{k+1}_i})\rightarrow F^*$.* **Proof.** Our another paper [@xunhao] has proved the convergence of the distributed upper bounding procedure, for the sake of completeness a proof is given here. Under Assumption [Assumption 4](#Assumption 4){reference-type="ref" reference="Assumption 4"}, we have $z^{k+1}_i=z^{k+1}$ for all agents $i\in\mathcal{V}$.\ Let $a^{k+1}$ and $b^{k+1}$ be the optimal solutions of the optimization problems with $A^k$ and $B^k$ as feasible sets, respectively (see: formula ([\[8\]](#8){reference-type="ref" reference="8"})). It follows that $F(b^{k+1})\le F(z^{k+1})\le F(a^{k+1})$. According to the result of formula ([\[10\]](#10){reference-type="ref" reference="10"}), taking the limit on both sides, we have $$\!\mathop{\lim}\limits_{k\rightarrow+\infty}\mathop{\sum}\limits^m_{i=1}\!f_i({z^{k+1}_i})\!=\!\mathop{\lim}\limits_{k\rightarrow+\infty}\!\!F(z^{k+1})\!\ge\! \mathop{\lim}\limits_{k\rightarrow+\infty}\!\!F(b^{k+1})\!=\!F^*\!,$$ $$\!\mathop{\lim}\limits_{k\rightarrow+\infty}\mathop{\sum}\limits^m_{i=1}\!f_i({z^{k+1}_i})\!=\!\mathop{\lim}\limits_{k\rightarrow+\infty}\!\!F(z^{k+1})\!\le\! \mathop{\lim}\limits_{k\rightarrow+\infty}\!\!F(a^{k+1})\!=\!F^*\!.$$ Therefore, the sequence $\left\{{\sum}^m_{i=1}f_i({z^{k+1}_i})\right\}_k$ converges to $F^*$. $\square$ **Proposition 3**. *For all agents $i\in\mathcal{V}$, take any $\overline Y^0_i\subset Y_i$ and proper restriction parameters $\varepsilon^0_i$. Suppose that Assumptions 1-5 hold. Then, for any iteration $k+1> 0$, it is satisfied that ${\sum}^m_{i=1}f_i({\overline x^{k+1}_i})\ge F^*$.* **Proof.** Based on \[Lemma [Lemma 3](#DUBD_lemma){reference-type="ref" reference="DUBD_lemma"} )\], the optimal solution $z_i^{k+1}$ of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) for agent $i$ satisfies its semi-infinite constraint in the $(k+1)$-th iteration, i.e. $z_i^{k+1}\in X_i$. According to the assumption that the solution of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) for each agent satisfies consensus, we can get $$z_j^{k+1}\in X_i,\ \ \forall j\in\mathcal{V}.$$ Based on the above formula, we can infer that within a finite number of iterations $K$ of distributed upper bounding procedure, we can obtain a consensus point $z^{K}$ that satisfies $z^{K}\in\bigcap_{i=1}^m X_i$.\ Due to the feasibility of $z^{K}$ for the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), we can conclude that at the K-th iteration, $$\mathop{\sum}\limits^m_{i=1}f_i({\overline x^{K}_i})=\mathop{\sum}\limits^m_{i=1}f_i({z^{K}_i})=\mathop{\sum}\limits^m_{i=1}f_i({ z^{K}})\ge F^*.$$ In other iterations, it is satisfied that $z^{k+1}\notin\bigcap_{i=1}^m X_i$. For this case, there exists at least one agent $i$ satisfying $f_i(\overline x^{k+1}_i)\rightarrow +\infty$ (see: DUBD Oracle in Section [3.2](#sec:Distributed Upper Bounding Procedure){reference-type="ref" reference="sec:Distributed Upper Bounding Procedure"}). It follows that ${\sum}^m_{i=1}f_i({\overline x^{k+1}_i})\ge F^*$.\ Overall, for any iteration $k\ge 0$, it is satisfied that ${\sum}^m_{i=1}f_i({\overline x^{k+1}_i})\ge F^*$. $\square$ ## Finite-time Convergence {#sec:Finite-time Convergence} Based on the results from Propositions 1-3, it can be concluded that both the distributed upper bounding procedure and the distributed lower bounding procedure of Algorithm 1 converge to the optimal solution of the (DRCO) while satisfying the following condition: for any $k\ge0$ and $i\in\mathcal{V}$, $$\mathop{\sum}\limits^m_{i=1}f_i({\widetilde x^{k+1}_i})\le F^*\le \mathop{\sum}\limits^m_{i=1}f_i({\overline x^{k+1}_i}).$$ where $\widetilde x^{k+1}_i$ is the solution of agent $i$ obtained by solving the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}), and $\overline x^{k+1}_i$ is the solution output by the DUBD Oracle in the $(k+1)$-th iteration of Algorithm 1. Moreover, from \[Lemma [Lemma 3](#DUBD_lemma){reference-type="ref" reference="DUBD_lemma"} )\], it follows that for a finite number of iterations $K$, we can definitely obtain a set of $\overline x^{K}_i$ satisfying ${\sum}^m_{i=1}f_i({\overline x^{K}_i})<+\infty$. For a centralized system, as a result of the above results, suppose that we set the formula ([\[stopping criterion\]](#stopping criterion){reference-type="ref" reference="stopping criterion"}) as a stopping criterion for Algorithm 1. $$\label{stopping criterion} \vert \mathop{\sum}\limits_{i=1}^{m} f_i(\overline x^{k+1}_i)-\mathop{\sum}\limits_{i=1}^{m}f_i(\widetilde x^{k+1}_i)\vert \le \epsilon^f, %{\sum}_{i=1}^{m}[f_i(\overline x^{k+1}_i)-f_i(\widetilde x^{k+1}_i)]\le \hat\epsilon^f=\epsilon^f$$ where $\epsilon^f>0$ is a termination parameter. When the stopping criterion ([\[stopping criterion\]](#stopping criterion){reference-type="ref" reference="stopping criterion"}) is satisfied, we have $$\label{approximate optimality} \begin{aligned} &\hspace{-1mm}\vert\mathop{\sum}\limits_{i=1}^{m}\! f_i(\overline x^{k+1}_i)\!-\!F^*\vert\! \le\! \vert \mathop{\sum}\limits_{i=1}^{m}\! f_i(\overline x^{k+1}_i)\!-\!\mathop{\sum}\limits_{i=1}^{m}\!f_i(\widetilde x^{k+1}_i)\vert\! \le\! \epsilon^f\!,\\ &\vert F^*\!-\!\mathop{\sum}\limits_{i=1}^{m}\!f_i(\widetilde x^{k+1}_i)\vert\! \le\! \vert \mathop{\sum}\limits_{i=1}^{m}\! f_i(\overline x^{k+1}_i)\!-\!\mathop{\sum}\limits_{i=1}^{m}\!f_i(\widetilde x^{k+1}_i)\vert\! \le\! \epsilon^f\!.\\ %&\vert F^*-\mathop{\sum}\limits_{i=1}^{m}f_i(\widetilde x^{k+1}_i)\vert\le \vert\mathop{\sum}\limits_{i=1}^{m} f_i(\overline x^{k+1}_i)-\mathop{\sum}\limits_{i=1}^{m}f_i(\widetilde x^{k+1}_i)\vert \le \epsilon^f \end{aligned}$$ Let $\hat \epsilon^f$ be the accuracy of the approximate optimal solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) for the centralized case. Hence, we can obtain an $\hat \epsilon^f$-approximate optimal solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) by Algorithm 1, in which $\hat\epsilon^f=\epsilon^f$. However, compared to centralized systems, distributed systems lack a central processing unit that has access to global network information. Instead, each agent can only gather local information about its in-neighbors through agent communication. Consequently, it is not practical to directly sum the local objective function values of all agents to determine the stopping criterion ([\[stopping criterion\]](#stopping criterion){reference-type="ref" reference="stopping criterion"}). In order to ensure that Algorithm 1 is finite-time convergent, motivated by the finite-time consensus algorithm [@xie2017stop], this subsection presents two termination methods specifically designed for uniformly strongly connected graphs. Furthermore, the effectiveness of these two methods is compared. ***1) Method :*** The objective of the first method is to make all agents stop updating information when the local objective values obtained by the distributed upper bounding procedure and distributed lower bounding procedure satisfy the following stopping criterion: [\[stopping_criterion_1\]]{#stopping_criterion_1 label="stopping_criterion_1"} f_i(x_i\^k+1)-f_i(x_i\^k+1)\^f,i. To this purpose, our idea is to adapt the minimum-consensus algorithm in [@xie2017stop] to propose an internal iterative method of Algorithm 1 executed in a distributed way, so that each agent can track the minimum number that consecutively satisfies the following conditions. f_i(x_i\^k+1)-f_i(x_i\^k+1)\^f,i_i\^in(t){i}. where the time slot $t\ge 0$ represents the number of internal iterations, and the graph sequence $\left\{\mathcal{G}(t)\right\}$ varies with $t$. Firstly, each agent sends two-bit data $[h_i(t), c_i(t)]$ to its out-neighbors between the time slots $t$ and $t+1$. Then, each agent calculates ([\[19\]](#19){reference-type="ref" reference="19"})-([\[20\]](#20){reference-type="ref" reference="20"}) according to its and its in-neighbors' information at time slot $t+1$. [\[19\]]{#19 label="19"} & h_i(t+1)=\_jN\^in\_i(t){i}{h_i(t),c_i(t)}+1,& [\[20\]]{#20 label="20"} & c_i(t+1)= c_i(t)+1,f_j(x_j\^k+1)-f_j(x_j\^k+1)\^f,\     jN\^in\_i(t){i},\ 0,otherwise, & where $h_i(0)=0$, $c_i(0)=0$. In the following, we show how to use $h_i$ to check whether the solutions obtained by Algorithm 1 satisfies the stopping criterion ([\[stopping_criterion_1\]](#stopping_criterion_1){reference-type="ref" reference="stopping_criterion_1"}). **Proposition 4**. *Under Assumption [Assumption 2](#Assumption 2){reference-type="ref" reference="Assumption 2"}, the calculation is performed according to ([\[19\]](#19){reference-type="ref" reference="19"})-([\[20\]](#20){reference-type="ref" reference="20"}). If at the $[T(m-1)+1]$-th time slot, there is an agent $i\in\mathcal{V}$ satisfying $h_i(t)\ge T(m-1)+1$, the uniformly strongly connected network reaches the stopping criterion of ([\[stopping_criterion_1\]](#stopping_criterion_1){reference-type="ref" reference="stopping_criterion_1"}).* **Proof.** We refer to the proof idea in [@xie2017stop] to prove this proposition. From Assumption [Assumption 2](#Assumption 2){reference-type="ref" reference="Assumption 2"} that $\mathcal G(t:t+T)$ is strongly connected, there exists a directed path $(i,i_1),(i_1,i_2),...,(i_d,j)$ from $i$ to $j$ with $d\le T(m-1)-1$ for any $i\neq j\in\mathcal{V}$.\ Suppose that there is an agent $j\in\mathcal{V}$ satisfying $h_j(t)\ge T(m-1)+1$ at time slot t, then at time slot $(t-1)$, it follows that h\_i\_d(t-1)T(m-1),\ c\_i\_d(t-1)T(m-1). Similarly, when the time slot is $(t-2)$, it follows that h\_i\_d-1(t-2)T(m-1)-1,\ c\_i\_d-1(t-2)T(m-1)-1. Repeat the same steps, it follows that h\_i(t-d-1)T(m-1)-d,\ c\_i(t-d-1)T(m-1)-d. Therefore, the stopping criterion ([\[stopping_criterion_1\]](#stopping_criterion_1){reference-type="ref" reference="stopping_criterion_1"}) is reached at time slot $(t-d-1)$. $\square$ Assuming that there is an agent $i$ satisfying $h_i\ge T(m-1)+1$, this agent will issue an exit command to stop updating its information. Since the values of $\widetilde x^{k+1}_i$ and $\overline x^{k+1}_i$ do not change with time slot $t$ in this termination method, it is obvious that all other agents also satisfy $h_j\ge T(m-1)+1,\ i\neq j\in\mathcal{V}$. Overall, all agents can stop updating information simultaneously by their own exit commands, in contrast to the Literature [@xie2017stop] where they cannot terminate simultaneously and need to broadcast the exit command to their out-neighbors. **Theorem 1**. *Under Assumptions 1-5, Algorithm 1 terminates finitely and generates a feasible $\widetilde \epsilon^f$-approximate optimal consensus solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), where the accuracy of approximate optimality is $\widetilde \epsilon^f=m\epsilon^f$. This holds for any reduciton parameter $r\ge 1$, any finite sets $\widetilde Y_i^0\subset Y_i$, $\overline Y_i^0\subset Y_i$, and proper restriction parameter $\varepsilon_i^0>0$.* **Proof.** From Propositions 1-4, it is straightforward to derive the finite-time convergence of Algorithm 1. Assume that Algorithm 1 terminates at the $k1$-th iteration. During the $k1$-th iteration, the $\overline x^{k1}_i$ generated by DUBD Oracle satisfies the following conditions: $$x^{k1}_i\in X_i,\ \forall i\in\mathcal{V},$$ and it is assigned to $x^{opt}_i$. Under Assumption 5, we can conclude that the solution obtained by Algorithm 1 satisfies the feasibility of the $(\ref{DRCO})$. Therefore, Algorithm 1 converges to a feasible consensus solution in a finite number of iterations. Next, we demonstrate the approximate accuracy of the solution obtained by Algorithm 1 to the optimal solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}).\ Due to the property of triangle inequality, it follows that \_i=1\^m f_i(x\^k+1\_i)-\_i=1\^mf_i(x\^k+1\_i) &= \_i=1\^m \[f_i(x\^k+1\_i)-f_i(x\^k+1\_i)\]\ &\_i=1\^m f_i(x\^k+1\_i)-f_i(x\^k+1\_i)\ &m \^f=\^f. Combining the inequality relations of ([\[approximate optimality\]](#approximate optimality){reference-type="ref" reference="approximate optimality"}), the accuracy of the approximate optimal solution obtained by Algorithm 1 is $\widetilde \epsilon^f=m\epsilon^f$. $\square$ Note that the accuracy of the approximate optimal solution is only related to the number of agents $m$ and the value of the termination parameter. There is a drawback to this termination method: when the number of agents in the multi-agent system is very large, an extremely small value of the termination parameter is required to guarantee the accuracy of the approximate optimal solution, which imposes high demands on the computational accuracy during the numerical solution, and greatly increases the number of iterations $k$ of Algorithm 1. ***2) Method :*** To address this issue, we make an improvement to the aforementioned method. The following is the stopping criterion: [\[stopping_criterion_2\]]{#stopping_criterion_2 label="stopping_criterion_2"} \_jN\^in\_i(t){i}f_j(x_j\^k+1)-f_j(x_j\^k+1)\^f,i. Our primary idea is to compute the value of $e_i=\vert f_i(\overline x_i^{k+1})-f_i(\widetilde x_i^{k+1})\vert$ for each agent and to transmit the value $e_i$ to its out-neighbors after the calculation of the distributed lower bounding procedure and distributed upper bounding procedure of Algorithm 1. Then, we adapt the Method so that each agent can keep track of the minimum number that consecutively satisfies the following conditions: \_jN\^in\_i(t){i} f_j(x_j\^k+1)-f_j(x_j\^k+1)\^f. The adapted termination method is shown below: firstly, each agent sends two-bit data $[h_i(t), c_i(t)]$ to its out-neighbors between the time slots t and t+1. Then, each agent calculates ([\[23\]](#23){reference-type="ref" reference="23"})-([\[24\]](#24){reference-type="ref" reference="24"}) according to its own and in-neighbors' information at time slot $t+1$. [\[23\]]{#23 label="23"} & h_i(t+1)=\_jN\^in\_i(t){i}{h_i(t),c_i(t)}+1,& [\[24\]]{#24 label="24"} & c_i(t+1)= c_i(t)+1,\_jN\^in\_i(t){i} e_j\^f,\ 0,otherwise, & where $h_i(0)=0$, $c_i(0)=0$, $e_j=\vert f_j(\overline x_j^{k+1})-f_j(\widetilde x_j^{k+1})\vert$. Similar to the Method , we show how to check whether the solutions of Algorithm 1 satisfy the stopping criterion ([\[stopping_criterion_2\]](#stopping_criterion_2){reference-type="ref" reference="stopping_criterion_2"}) by using $h_i$. **Proposition 5**. *Under Assumption [Assumption 2](#Assumption 2){reference-type="ref" reference="Assumption 2"}, the calculation is performed according to ([\[23\]](#23){reference-type="ref" reference="23"})-([\[24\]](#24){reference-type="ref" reference="24"}). If at the $[T(m-1)+1]$-th time slot, there is an agent $i\in\mathcal{V}$ satisfying $h_i(t)\ge T(m-1)+1$, the network reaches the stopping criterion of ([\[stopping_criterion_2\]](#stopping_criterion_2){reference-type="ref" reference="stopping_criterion_2"}).* **Proof.** Similar to the proof of Proposition 4, the above result is straightforward. $\square$ **Theorem 2**. *Under Assumptions 1-5, suppose that the graph sequence $\mathcal{G}(t)$ is known. For any reduciton parameter $r\ge 1$, any finite sets $\widetilde Y_i^0\subset Y_i$, $\overline Y_i^0\subset Y_i$, and proper restriction parameter $\varepsilon_i^0>0$, Algorithm 1 terminates finitely and generates a feasible $\overline \epsilon^f$-approximate optimal consensus solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), where the accuracy of approximate optimality $\overline \epsilon^f$ is an optimal objective of the following linear program:* *[\[linear program\]]{#linear program label="linear program"}* *\^f=& & & \^m\_i=1e_i\ &s.t. &&\^m\_i=1\^T\_t=1\_jN\^in(t)\_i{i}e_j m T\^f,\ &&& 0e_i\^f,i=1,\...,m.* **Proof.** Similar to the proof of Theorem 1, we can easily establish that Algorithm 1 terminates within a finite number of iterations and yields a feasible approximate optimal consensus solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) based on Propositions 1-3 and 5. Next, we provide a proof of the accuracy of the approximate optimal solution.\ When Algorithm 1 has reached the stopping criterion ([\[stopping_criterion_2\]](#stopping_criterion_2){reference-type="ref" reference="stopping_criterion_2"}), there is $$\label{naive_constraint} 0\le e_i\le \mathop{\sum}\limits_{j\in N^{in}_i(t)\cup\left\{i\right\}} e_j\le \epsilon^f, \ \forall i\in\mathcal{V},\ \forall 1\le t\le T,$$ where the graph $\mathcal{G}(1:T)$ is strongly connected.\ By relaxing the constraint ([\[naive_constraint\]](#naive_constraint){reference-type="ref" reference="naive_constraint"}), we can obtain the constraints shown in ([\[linear program\]](#linear program){reference-type="ref" reference="linear program"}).\ Therefore, the accuracy of the approximate optimal solution obtained by Algorithm 1 is the maximum value of $\sum_{i=1}^m e_i$ subject to the relaxed constraints, as shown in ([\[linear program\]](#linear program){reference-type="ref" reference="linear program"}). $\square$ {#fig3 width="\\columnwidth"} {#fig3 width="\\columnwidth"} Based on the results of Theorems 1 and 2, it can be established that Algorithm 1 can finite-time converge to a feasible consensus solution satisfying global optimality to a certain accuracy of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). Furthermore, the result of Theorem 2 is significantly less conservative than that of Theorem 1 since the optimal objective of the linear program (25) satisfies $\overline \epsilon^f\le m\epsilon^f$, whereas this $\overline\epsilon^f$ is related to network graphs. This can be observed through a numerical example in Fig. 3, where we study how $\hat \epsilon^f$, $\widetilde \epsilon^f$ and $\overline \epsilon^f$ change as a function of the number of agents $m$ in three types of graphs. Algorithm 1 is designed with a termination parameter $\epsilon^f=0.01$. The approximate accuracy $\hat\epsilon^f$ (represented by the red dashed line) in centralized systems is independent of the number of agents $m$ and graphs, and therefore remains constant as $m$ increases. In the case of the Method , the approximate accuracy $\widetilde\epsilon^f$ (represented by the green dotted-dashed line) linearly increases with $m$ as $\widetilde\epsilon^f=m\epsilon^f$, but independent of graphs. However, there is a different pattern in the case of the Method , influenced by both the number of agents and the network structures. For directed cyclic graphs, the approximate accuracy $\overline\epsilon^f_{(a)}$ (represented by the black solid line) increases linearly with the number of agents at $m\ge 2$, where the rate of growth is less than that of the Method . The accuracy of the approximate optimal solution for complete graphs can attain levels comparable to those achieved by the centralized system, i.e., $\overline\epsilon^f_{(c)}=\epsilon^f$ (see the grey solid line). For customized graphs, the approximate accuracy $\overline\epsilon^f_{(b)}$ (represented by the blue solid lines) increases moderately with $m$. Note that here for customized graphs, we consider a specific example of a time-invariant strongly connected network composed of $m$ agents. Among these $m$ agents, $(m-1)$ agents are fully connected to each other, while the remaining one agent is only connected to the $(m-1)$-th agent. Overall, This Method provides a less conservative result compared to the Method while still allowing for distributed information, which contrasts the centralized systems. # Numerical Case Studies In order to verify the effectiveness of the proposed Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}, we consider a distributed robust convex optimization problem with bounded uncertainty, as shown below. &\_x F(x)=\^m\_i=1x-u_i\^2\ & s.t. g_i(x,y_i)=(x(1)-v_i)\^2+2y_ix(2)-y_i\^2-1,\ &  y_i,i=1,\...,m, where the decision variable $x=[x(1),x(2)]^\top$, the global constraint $\mathcal{F}\!=\!\left\{\!x\in\mathbb{R}^2|\!-2\!\le\! x(1)\!\le\!2, \!-1\!\le\! x(2)\!\le\!1\right\}\!$. $u_i\in \mathbb{R}^2$ and $v_i\in \mathbb{R}$ are the local objective function vector and local constraint parameter for agent $i$, respectively, where the corresponding values are shown in Table [\[Table1\]](#Table1){reference-type="ref" reference="Table1"}. Furthermore, $y_i\in [-1,1]$ is the uncertain parameter for agent $i$. We adopt three types of network graphs, see Fig. 2, to verify that the distributed robust convex optimization algorithm (see: Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}) terminates in a finite number of iterations and to illustrate that the solutions of all agents are feasible with respect to their own local constraints in ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). The implementation is carried out in MATLAB Version 9.5.0.944444 (R2018b, win64) and runs on an Intel(R) Core (TM) i7-7700HQ CPU @ 2.80GHz, 256GB terminal server. In addition, this section compares Algorithm 1 with the existing related algorithms. ## Effectiveness of the distributed robust convex optimization algorithm We initialize the parameters of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} as follows: the initial restriction parameters $\varepsilon_i^0 = 0.01$, reduction parameter $r=2$, two uncertainty sets $\widetilde Y_i^0 =\emptyset$, $\overline Y_i^0 =\emptyset$, and termination parameter $\epsilon^f = 0.01$. It is worth noting that for the initial values of the restriction parameters $\varepsilon_i^0$, we ensure that there exists at least one point $\hat x=[0,0]^\top$ satisfying $$g_i(\hat x,y_i)+\varepsilon^{0}_i\le0,\ \forall y_i\in Y_i, \quad i=1,...,m.$$ Therefore, the solvability of the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"}) is guaranteed for each iteration of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}. Then, we implement Algorithm 1 to solve the above numerical case. In solving the two problems (([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) and ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"})), we use a strategy by combining the distributed random-fixed projection algorithm [@xie2018distributed] and the finite-time consensus algorithm [@xie2017stop] to obtain the optimal consensus solutions of the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) (or the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"})) within a finite number of time slots. Note that in order to ensure the consensus of the solutions obtained by solving the ([\[dlbd\]](#dlbd){reference-type="ref" reference="dlbd"}) (or the ([\[dubd\]](#dubd){reference-type="ref" reference="dubd"})) for all agents, we set a pretty high consensus accuracy $10^{-4}$ on the finite-time consensus algorithm. In addition, since the local constraint functions of the agents are concave with respect to the uncertain parameters $y_i$, the solutions of the LLPs are rigorously solved by the analytical method. {#fig4 width="\\columnwidth"} \(a\) {#fig4 width="\\columnwidth"} \(b\) \|c\|c\|c\|c\|c\|c\|c\|c\|c\| & Agent 1 & Agent 2 & Agent 3 & Agent 4 & Agent 5 & Agent 6\ \* &\* & Optimal point &\[-0.0000;0.6597\]&\[+0.0001;0.6597\]&\[+0.0000;0.6598\]&\[+0.0001;0.6597\]&\[+0.0001;0.6597\]&\[+0.0001;0.6598\]\  &  &Local feasibility &$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$ &$\checkmark$\  &  &Objective value & &\  &\* & Solution &\[-0.0000;0.6610\]&\[-0.0000;0.6610\]&\[-0.0000;0.6611\]&\[+0.0001;0.6611\]&\[+0.0001;0.6610\]&\[+0.0000;0.6611\]\  &  &Local feasibility &$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$ &$\checkmark$\  &  &Objective value & &\  &\* & Solution &\[-0.0001;0.6598\]&\[-0.0000;0.6598\]&\[-0.0001;0.6598\]&\[-0.0001;0.6599\]&\[+0.0000;0.6599\]&\[+0.0000;0.6598\]\  &  &Local feasibility &$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$ &$\checkmark$\  &  &Objective value & &\ \* &\* & Solution &\[-0.0000;0.6611\]&\[-0.0000;0.6612\]&\[+0.0000;0.6612\]&\[-0.0001;0.6612\]&\[+0.0000;0.6611\]&\[+0.0000;0.6612\]\  &  &Local feasibility &$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$ &$\checkmark$\  &  &Objective value & &\  &\* & Solution &\[-0.0000;0.6612\]&\[-0.0000;0.6612\]&\[-0.0000;0.6611\]&\[-0.0000;0.6612\]&\[-0.0001;0.6611\]&\[+0.0000;0.6612\]\  &  &Local feasibility &$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$&$\checkmark$ &$\checkmark$\  &  &Objective value & &\  &\* & Solution &\[-0.0000;0.6610\]&\[+0.0000;0.6610\]&\[-0.0001;0.6611\]&\[-0.0000;0.6610\]&\[-0.0001;0.6611\]&\[+0.0000;0.6611\]\  &  &Local feasibility &$\checkmark$ & $\checkmark$ & $\checkmark$&$\checkmark$& $\checkmark$ &$\checkmark$\  &  &Objective value & &\ The graphical results of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} are shown in Fig. [4](#fig4){reference-type="ref" reference="fig4"}, where Fig. [4](#fig4){reference-type="ref" reference="fig4"} (a) presents the convergence process of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} with Method as the termination method, while Fig. [4](#fig4){reference-type="ref" reference="fig4"} (b) illustrates the convergence process of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} employing Method as the termination method. Furthermore, we considered three types of network graphs, where the black solid lines indicate the convergence process of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} under a directed cycle graph, the blue dotted lines represent the convergence process of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} over the customized graph, and the green dotted-dashed lines refer to the case of a complete graph. Note that to better present the convergence process of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"}, for the case of $\sum_{i=1}^m f_i(\overline x^k_i)=\infty$, we assign a value of $39$ to $\sum_{i=1}^m f_i(\overline x^k_i)$ in this iteration. Based on the Fig. [4](#fig4){reference-type="ref" reference="fig4"}, we can draw the following results: 1. Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} converges to the global optimal value within a finite number of iterations. 2. The solutions obtained from the distributed upper and lower bounding procedures serve as upper and lower bounds for the global optimal value, respectively. 3. Compared to Method , using Method as the termination method leads to a higher number of iterations for Algorithm 1 to terminate. This further indicates that the algorithm with Method as the termination method can obtain a higher accuracy solution than Method . In addition, the effect of different network graphs on the number of iterations of Algorithm 1 is considered. For Algorithm 1 with Method I as the termination method, the number of iterations is independent of the network graphs. Conversely, when adopting Method II as the termination method in Algorithm 1, the number of iterations is related to the network graphs. The main reason is that, for identical termination parameter $\epsilon^f$ and the number of agents $m$, the approximation accuracy of the solutions obtained by Algorithm 1 with Method II as the termination method is different in terms of different network graphs (see: Fig. [2](#fig3){reference-type="ref" reference="fig3"}). The complete graph corresponds to the highest approximation accuracy and requires Algorithm 1 to perform a greater number of iterations. On the contrary, the directed cycle graph has the lowest approximation accuracy, resulting in the fewest iterations of Algorithm 1. However, since the number of agents considered in this numerical case is small, the effect of the network graph on the number of iterations of Algorithm 1 is not significant. The numerical results of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} are illustrated in Table [\[Table2\]](#Table2){reference-type="ref" reference="Table2"}, where all the data provided have been rounded to four decimal places. The global optimal solution of the numerical case is $x^*=[0,\sqrt{7}/4]^\top$, and the corresponding global optimal value is $F^*\approx 38.687746$. According to Table [\[Table2\]](#Table2){reference-type="ref" reference="Table2"}, we can conclude that each agent can obtain a locally feasible solution satisfying global optimality to a certain accuracy when Algorithm 1 terminates finitely. ## Comparison with Relevant Algorithms {width="1.6\\columnwidth"} This subsection compares Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} with some related algorithms. To clearly illustrate the differences between Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} and the related algorithms, we consider the simplest case: a system composed of only one agent. Our goal is to find the minimum value of the objective function under the set constraints (the area enclosed by the two dotted lines and coordinate axes) and semi-infinite constraints (the area below the arc). Fig. [\[fig5\]](#fig5){reference-type="ref" reference="fig5"} graphically illustrates different strategies for solving this ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}). Note: both Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} and the related algorithms construct approximation problems for ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) at each iteration. In the case that there are an infinite number of solutions to the approximation problem, we select the one with the smallest value of the horizontal coordinate as the optimal solution for the approximation problem. The distributed cutting-plane consensus algorithm in [@burger2012distributed; @burger2013polyhedral] is based on iteratively approximating the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) by successively populating the cutting-planes into the existing finite set of constraints (see: blue solid lines in Fig. [\[fig5\]](#fig5){reference-type="ref" reference="fig5"} (a)). The black points in Fig. [\[fig5\]](#fig5){reference-type="ref" reference="fig5"} (a) indicate the asymptotic convergence process of the agent. However, the agent cannot obtain a solution with guaranteed feasibility within a finite number of iterations. The distributed cutting-surface consensus algorithm in [@xunhao] is based on iteratively approximating the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) by successively reducing the restriction parameters of the right-hand constraints and populating the cutting-surfaces into the existing finite set of constraints. As shown in Fig. [\[fig5\]](#fig5){reference-type="ref" reference="fig5"} (b), (c), and (d), the iterative update of the blue points indicates the iterative process of this algorithm. When the decision variable of the agent lies outside the feasible domain, a cutting-surface is populated into the existing finite set of constraints (see: the green lines in Fig. [\[fig5\]](#fig5){reference-type="ref" reference="fig5"} (b) and (d)). Conversely, when the decision variable lies within the feasible domain, the original constraints in the constraint set (see: the grey lines in Fig. 5 (c)) move toward the feasible domain boundary (see: the green lines in Fig. 5 (c)) due to the reduction of the restriction parameters. This algorithm asymptotically converges to the optimal solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}), and the agent can obtain a solution that meets the feasibility in a finite number of iterations. However, the above algorithms cannot locate the global optimal solution for the certain accuracy in a finite number of iterations. Literature [@yang2014distributed] presents a distributed cutting-plane primal-dual algorithm by adding projection operation to the cutting-plane consensus algorithm, which converges to the optimal solution of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) from the outer and inner directions of the feasible region (see: the black and blue points in Fig. [\[fig5\]](#fig5){reference-type="ref" reference="fig5"} (a)). This algorithm, together with the distributed termination method in our article, can get the global optimal solution with a certain accuracy. Nonetheless, this method is confined to the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) with special constraint structures [@yang2014distributed]. The algorithm in this article combines the advantages of the above algorithms so that each agent can obtain a feasible consensus solution satisfying global optimality to a certain accuracy of the ([\[DRCO\]](#DRCO){reference-type="ref" reference="DRCO"}) within a finite number of iterations. The iterative convergence process of Algorithm [\[alg:1\]](#alg:1){reference-type="ref" reference="alg:1"} is shown in the black and blue points in Figures (b), (c), and (d). # Conclusions and future work Based on the right-hand restriction approach proposed in [@Mitsos], a distributed robust convex optimization algorithm is proposed for locating a feasible solution for each agent satisfying global optimality to a certain accuracy of the DRCO within a finite number of iterations. In addition, two distributed termination algorithms, namely the Method and the Method , are proposed, which ensure all the agents terminate simultaneously when approximate optimal solutions for a certain accuracy are obtained. The Method is less conservative than the Method in terms of the accuracy guarantee of the global optimality, but the accuracy of the solution obtained by the Method is related to the network structures. Direct extensions may lie in the following two aspects. In the proposed algorithm, finite-time convergence is proved. We could consider analyzing the convergence rate of the algorithm by thorough theoretical and computational analysis. In addition, this paper mainly considers the DRCO, where the global cost function is strictly convex, and the decision variables are all continuous. Future research would consider extending the proposed algorithm to the case where the local cost function is nonconvex, or some of the decision variables are constrained to integer values. 10 [ Bastianello, N., Schenato, L., & Carli, R. (2022). A novel bound on the convergence rate of ADMM for distributed optimization. *Automatica*, 142, Article 110403.](https://www.sciencedirect.com/science/article/pii/S0005109822002564) [ Ben-Tal, A., Ghaoui, L. E., & Nemirovski, A. (2009). *Robust optimization*. Princeton, NJ, USA: Princeton University Press.](https://www2.isye.gatech.edu/~nemirovs/FullBookDec11.pdf) [ Blankenship, J. W., & Falk, J. E. (1976). Infinitely constrained optimization problems. *Journal of Optimization Theory and Applications*, 19(2), 261-281.](https://link.springer.com/article/10.1007/BF00934096) [ Boyd, S., & Vandenberghe, L. (2004). *Convex optimization*. Cambridge, U.K.: Cambridge University Press.](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf) [ Bürger, M., Notarstefano, G., & Allgöwer, F. (2012). Distributed robust optimization via cutting-plane consensus. In *Proceedings of 51st IEEE Conference on Decision and Control* (pp. 7457-7463).](https://ieeexplore.ieee.org/abstract/document/6426782) [ Bürger, M., Notarstefano, G., & Allgöwer, F. (2014). A polyhedral approximation framework for convex and robust distributed optimization. *IEEE Transactions on Automatic Control*, 59(2), 384-395.](https://ieeexplore.ieee.org/abstract/document/6600797) [ Calafiore, G. C., & Campi, M. C. (2006). The scenario approach to robust control design. *IEEE Transactions on Automatic Control*, 51(5), 742-753.](https://ieeexplore.ieee.org/abstract/document/1632303) [ Carlone, L., Srivastava, V., Bullo, F., & Calafiore, G. C. (2014). Distributed random convex programming via constraints consensus. *SIAM Journal on Control and Optimization*, 52(1), 629-662.](https://epubs.siam.org/doi/abs/10.1137/120885796) [ Carnevale, G., Camisa, A., & Notarstefano, G. (2023). Distributed online aggregative optimization for dynamic multirobot coordination. *IEEE Transactions on Automatic Control*, 68(6), 3736-3743.](https://ieeexplore.ieee.org/abstract/document/9851519) [ Chamanbaz, M., Notarstefano, G., & Bouffanais, R. (2017). Randomized constraints consensus for distributed robust linear programming. *IFAC-PapersOnline*, 50(1), 4973-4978.](https://www.sciencedirect.com/science/article/pii/S2405896317312120) [ Chamanbaz, M., Notarstefano, G., & Bouffanais, R. (2017). A randomized distributed ellipsoid algorithm for uncertain feasibility problems. In *Proceedings of IEEE 56th Annual Conference on Decision and Control* (pp. 1305-1310).](https://ieeexplore.ieee.org/abstract/document/8263835) [ Chen, G., Yang, Q., Song, Y., & Lewis, F. L. (2020). A distributed continuous-time algorithm for nonsmooth constrained optimization. *IEEE Transactions on Automatic Control*, 65(11), 4914-4921.](https://ieeexplore.ieee.org/abstract/document/8957281) [ Chen, G., Yang, Q., Song, Y., & Lewis, F. L. (2021). Fixed-time projection algorithm for distributed constrained optimization on time-varying digraphs. *IEEE Transactions on Automatic Control*, 67(1), 390-397.](https://ieeexplore.ieee.org/abstract/document/9345459) [ Chen, T. W., & Vassiliadis, V. S. (2005). Inequality path constraints in optimal control: a finite iteration $\varepsilon$-convergent scheme based on pointwise discretization. *Journal of Process Control*, 15(3), 353-362.](https://www.sciencedirect.com/science/article/pii/S0959152404000496) Chen, X., Jiang, W., Charalambous, T., & Shi, L. (2023). A privacy-preserving finite-time push-sum based gradient method for distributed optimization over digraphs. arXiv preprint [arXiv:2305.15202](https://arxiv.org/abs/2305.15202). [ Falsone, A., Margellos, K., Prandini, M., & Garatti, S. (2020). A scenario-based approach to multi-agent optimization with distributed information. *IFAC-PapersOnline*, 53(2), 20-25.](https://www.sciencedirect.com/science/article/pii/S2405896320302834) [ Ferber, J., & Weiss, G. (1999). *Multi-agent systems: an introduction to distributed artificial intelligence*. Harlow, U.K.: Addison Wesley Longman.](https://www.jasss.org/4/2/reviews/rouchier.html) [ Firouzbahrami, M., & Nobakhti, A. (2022). Cooperative fixed-time/finite-time distributed robust optimization of multi-agent systems. *Automatica*, 142, Article 110358.](https://www.sciencedirect.com/science/article/pii/S0005109822002084) Fu, J., & Wu, X. (2023). A cutting-surface consensus approach for distributed robust optimization of multi-agent systems. Submitted in *IEEE Transactions on Automatic Control*. [ Garg, K., Baranwal, M., & Panagou, D. (2020). A fixed-time convergent distributed algorithm for strongly convex functions in a time-varying network. In *Proceedings of 59th IEEE Conference on Decision and Control* (pp. 4405-4410).](https://ieeexplore.ieee.org/abstract/document/9303778) [ Ghosal, A., Halder, S., & Das, S. K. (2020). Distributed on-demand clustering algorithm for lifetime optimization in wireless sensor networks. *Journal of Parallel and Distributed Computing*, 141, 129-142.](https://www.sciencedirect.com/science/article/pii/S0743731518305793) [ Hamedani, E. Y., & Aybat, N. S. (2017). Multi-agent constrained optimization of a strongly convex function over time-varying directed networks. In *Proceedings of 55th Annual Allerton Conference on Communication, Control, Computing* (pp. 518-525).](https://ieeexplore.ieee.org/abstract/document/8262781) [ Jiang, W., & Charalambous, T. (2022). A fast finite-time consensus based gradient method for distributed optimization over digraphs. In *Proceedings of 61st IEEE Conference on Decision and Control* (pp. 6848-6854).](https://ieeexplore.ieee.org/abstract/document/9992389) [ Kotary, D. K., & Nanda, S. J. (2020). Distributed robust data clustering in wireless sensor networks using diffusion moth flame optimization. *Engineering Applications of Artificial Intelligence*, 87, Article 103342.](https://www.sciencedirect.com/science/article/pii/S0952197619302830) [ Lee, S., & Nedić, A. (2013). Distributed random projection algorithm for convex optimization. *IEEE Journal of Selected Topics in Signal Processing*, 7(2), 221-229.](https://ieeexplore.ieee.org/abstract/document/6461383) [ Lee, S., & Nedić, A. (2016). Asynchronous gossip-based random projection algorithms over networks. *IEEE Transactions on Automatic Control*, 61(4), 953-968.](https://ieeexplore.ieee.org/abstract/document/7165636) [ Lin, P., Xu, J., Ren, W., Yang, C., & Gui, W. (2021). Angle-based analysis approach for distributed constrained optimization. *IEEE Transactions on Automatic Control*, 66(11), 5569-5576.](https://ieeexplore.ieee.org/abstract/document/9335004) [ Mai, V. S., & Abed, E. H. (2018). Local prediction for enhanced convergence of distributed optimization algorithms. *IEEE Transactions on Control of Network Systems*, 5(4), 1962-1975.](https://ieeexplore.ieee.org/abstract/document/8115297) [ Manitara, N. E., & Hadjicostis, C. N. (2014). Distributed stopping for average consensus in directed graphs via a randomized event-triggered strategy. In *Proceedings of 6th International Symposium on Communications, Control and Signal Processing* (pp. 483-486).](https://ieeexplore.ieee.org/abstract/document/6877918) [ Margellos, K., Falsone, A., Garatti, S., & Prandini, M. (2018). Distributed constrained optimization and consensus in uncertain networks via proximal minimization. *IEEE Transactions on Automatic Control*, 63(5), 1372-1387.](https://ieeexplore.ieee.org/abstract/document/8022966) [ Mitsos, A. (2011). Global optimization of semi-infinite programs via restriction of the right-hand side. *Optimization*, 60(10-11), 1291-1308.](https://www.tandfonline.com/doi/abs/10.1080/02331934.2010.527970) [ Nedić, A., & Olshevsky, A. (2015). Distributed optimization over time-varying directed graphs. *IEEE Transactions on Automatic Control*, 60(3), 601-615.](https://ieeexplore.ieee.org/abstract/document/6930814) [ Nedić, A., & Liu, J. (2018). Distributed optimization for control. *Annual Review of Control, Robotics, and Autonomous Systems*, 1(1), 77-103.](https://www.annualreviews.org/doi/abs/10.1146/annurev-control-060117-105131) [ Nedić, A., Ozdaglar, A., & Parrilo, P. A. (2010). Constrained consensus and optimization in multi-agent networks. *IEEE Transactions on Automatic Control*, 55(4), 922-938.](https://ieeexplore.ieee.org/abstract/document/5404774) [ Prakash, M., Talukdar, S., Attree, S., Yadav, V., & Salapaka, M. V. (2020). Distributed stopping criterion for consensus in the presence of delays. *IEEE Transactions on Control of Network Systems*, 7(1), 85-95.](https://ieeexplore.ieee.org/abstract/document/8715396) Rikos, A. I., Jiang, W., Charalambous, T., & Johansson, K. H. (2022). Finite-time distributed optimization with quantized gradient descent. arXiv preprint [arXiv:2211.10855](https://arxiv.org/abs/2211.10855). Shi, X., Wen, G., & Yu, X. (2022). Finite-time convergent algorithms for time-varying distributed optimization. arXiv preprint [arXiv:2210.03947](https://arxiv.org/abs/2210.03947). [ Shi, X., Yu, X., Cao, J., & Wen, G. (2020). Continuous distributed algorithms for solving linear equations in finite time. *Automatica*, 113, Article 108755.](https://www.sciencedirect.com/science/article/pii/S0005109819306181) [ Song, Y., Cao, J., & Rutkowski, L. (2021). A fixed-time distributed optimization algorithm based on event-triggered strategy. *IEEE Transactions on Network Science and Engineering*, 9(3), 1154-1162.](https://ieeexplore.ieee.org/abstract/document/9640505) [ Stein, O., & Steuermann, P. (2012). The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets. *Mathematical Programming*, 136(1), 183-207.](https://link.springer.com/article/10.1007/s10107-012-0556-5) [ Ufa, R. A., Malkova, Y. Y., Rudnik, V. E., Andreev, M. V., & Borisov, V. A. (2022). A review on distributed generation impacts on electric power system. *International Journal of Hydrogen Energy*, 47(47), 20347-20361.](https://www.sciencedirect.com/science/article/pii/S0360319922017153) [ Wang, S., & Li, C. (2017). Distributed robust optimization in networked system. *IEEE Transactions on Cybernetics*, 47(8), 2321-2333.](https://ieeexplore.ieee.org/abstract/document/7588087) [ Wang, X., Wang, G., & Li, S. (2020). A distributed fixed-time optimization algorithm for multi-agent systems. *Automatica*, 122, Article 109289.](https://www.sciencedirect.com/science/article/pii/S000510982030488X) [ West, D. B. (2001). *Introduction to graph theory*. Upper Saddle River, NJ, USA: Prentice Hall.](https://www.maths.ed.ac.uk/~v1ranick/papers/wilsongraph.pdf) [ Wu, Z., Li, Z., & Yu, J. (2021). Designing zero-gradient-sum protocols for finite-time distributed optimization problem. *IEEE Transactions on Systems, Man, and Cybernetics: Systems*, 52(7), 4569-4577.](https://ieeexplore.ieee.org/abstract/document/9501947) [ Xie, P., You, K., Song, S., & Wu, C. (2017). Distributed random-fixed projected algorithm for constrained optimization over digraphs. *IFAC-PapersOnline*, 50(1), 14436-14441.](https://www.sciencedirect.com/science/article/pii/S2405896317330951) [ Xie, P., You, K., Tempo, R., Song, S., & Wu, C. (2018). Distributed convex optimization with inequality constraints over time-varying unbalanced digraphs. *IEEE Transactions on Automatic Control*, 63(12), 4331-4337.](https://ieeexplore.ieee.org/abstract/document/8316938) [ Xie, P., You, K., & Wu, C. (2017). How to stop consensus algorithms, locally? In *Proceedings of IEEE 56th Annual Conference on Decision and Control* (pp. 4544-4549).](https://ieeexplore.ieee.org/abstract/document/8264330) [ Yang, K., Huang, J., Wu, Y., Wang, X., & Chiang, M. (2014). Distributed robust optimization (DRO), part I: framework and example. *Optimization and Engineering*, 15(1), 35-67.](https://link.springer.com/article/10.1007/s11081-012-9198-y) [ Yang, K., Wu, Y., Huang, J., Wang, X., & Verdú, S. (2008). Distributed robust optimization for communication networks. In *Proceedings of IEEE INFOCOM* (pp. 1157-1165).](https://ieeexplore.ieee.org/abstract/document/4509766) [ Yang, T., Yi, X., Wu, J., Yuan, Y., Wu, D., Meng, Z., et al. (2019). A survey of distributed optimization. *Annual Reviews in Control*, 47, 278-305.](https://www.sciencedirect.com/science/article/pii/S1367578819300082) [ Yao, L., Yuan, Y., Sundaram, S., & Yang, T. (2018). Distributed finite-time optimization. In *Proceedings of IEEE 14th International Conference on Control and Automation* (pp. 147-154).](https://ieeexplore.ieee.org/abstract/document/8444343) [ Yin, L., & Sun, Z. (2022). Distributed multi-objective grey wolf optimizer for distributed multi-objective economic dispatch of multi-area interconnected power systems. *Applied Soft Computing*, 117, Article 108345.](https://www.sciencedirect.com/science/article/pii/S1568494621011297) [ You, K., Tempo, R., & Xie, P. (2019). Distributed algorithms for robust convex optimization via the scenario approach. *IEEE Transactions on Automatic Control*, 64(3), 880-895.](https://ieeexplore.ieee.org/abstract/document/8340797) [ Zheng, Y., & Liu, Q. (2022). A review of distributed optimization: Problems, models and algorithms. *Neurocomputing*, 483, 446-459.](https://www.sciencedirect.com/science/article/pii/S0925231221016040) [ Zhou, B., Xu, H., & Shen, S. (2023). RACER: Rapid collaborative exploration with a decentralized multi-UAV system. *IEEE Transactions on Robotics*, 39(3), 1816-1835.](https://ieeexplore.ieee.org/abstract/document/10038280)  received the B.S. degree in automation from Northeastern University, Shenyang, China, in 2021, where he is currently pursuing the Ph.D. degree in control theory and control engineering with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China. His current research interests cover stochastic optimization, robust optimization and their applications in multi-agent systems.  He was a Postdoctoral Researcher with the Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, from 2010 to 2014. He is a Full Professor with Northeastern University, Shenyang, China. His current research is on dynamic optimization, optimal control, switched systems and their applications. Dr. Fu received the 2018 Young Scientist Award in Science issued by the Ministry of Education of China (the first awardee in Chinese Control Community). He is currently an Associate Editor for the Control Engineering Practice, the IEEE Transactions on Industrial Informatics, and the IEEE Transactions on Neural Networks and Learning Systems. [^1]: This paper was supported in part by the National Nature Science Foundation of China under Grant 61825301. Corresponding author Jun Fu.

arxiv_math

--- abstract: | For the usual action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \geq 1$) denote by $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ the $C^*$-algebra acting on the weighted Bergman space $\mathcal{A}^2_\lambda(D_{n+1})$ ($\lambda > -1$) generated by Toeplitz operators whose symbols belong to $L^\infty(D_{n+1})^{\mathbb{H}_n}$ (essentially bounded and $\mathbb{H}_n$-invariant). We prove that $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ is commutative and isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$ (very slowly oscillating functions on $\mathbb{R}_+$), for every $\lambda > -1$ and $n \geq 1$. We achieve this by Lie theory and symplectic geometry methods. We also compute the moment map of the $\mathbb{H}_n$-action and some of its subgroups. A decomposition of the Bergman spaces $\mathcal{A}^2_\lambda(D_{n+1})$ as direct integrals of Fock spaces is also obtained and used to explain our main results. address: - Centro de Investigación en Matemáticas, Guanajuato, México - Centro de Investigación en Matemáticas, Guanajuato, México author: - Julio A. Barrera-Reyes - Raúl Quiroga-Barranco title: Toeplitz operators on the Siegel domain and the Heisenberg group --- # Introduction {#sec:intro} Analytic function spaces and their operators is a current and very active research topic. Among the classical examples of these we have the weighted Bergman spaces over the Siegel domain, the natural unbounded realization of the complex unit ball. Equally important are the Toeplitz operators acting on such Bergman spaces. The two of them together yield concrete examples of Hilbert spaces and $C^*$-algebras of operators acting on them. In fact, it has been very fruitful to study these objects towards a better understanding of both complex analysis and operator theory. A Toeplitz operator comes from the choice of a (usually essentially bounded) measurable function called a symbol. Choosing a family of such symbols then yields a $C^*$-algebra: the one generated by the corresponding Toeplitz operators. To understand the structure of these $C^*$-algebras in terms of the choice of symbols is at the core of the study of Toeplitz operators. For example, on strictly pseudoconvex bounded domains with smooth boundary the $C^*$-algebra generated by Toeplitz operators whose symbols are continuous up to the boundary is essentially commutative (see [@UpmeierToepBook]). It is by now well known that geometric restrictions on the families of symbols yield Toeplitz operators that generate actually commutative $C^*$-algebras. On the unit disk, it was proved in [@KorenblumZhu1995] that radial symbols (depending only on the radial variable of polar coordinates) yield commutative $C^*$-algebras. More recent works have shown that this phenomenon is part of a more general principle: on bounded symmetric domains, families of symbols invariant under suitable biholomorphism groups yield Toeplitz operators that generate commutative $C^*$-algebras. Beyond the seminal work [@KorenblumZhu1995] this was proved on the upper half-plane for dilations and translations (see [@GKVHyperbolic; @GKVParabolic]) and for the so-called maximal Abelian subgroups of biholomorphisms on the unit ball and the Siegel domain (see [@QVUnitBall1]). Later on, a large family of biholomorphism groups acting on bounded symmetric domains that yield commutative $C^*$-algebras through invariant symbols were exhibited in [@DOQJFA]. This line of work is also related to the understanding of the properties of commutators and semicommutators of Toeplitz operators. For this we refer to [@AppuhamyLe2016; @AxlerCuckovicRao2000; @ChoeKooLee2004; @CuckovicLouhichi2008; @Le2017] as examples. Within the scope of this setup, we consider for $n \geq 1$ the $n+1$-dimensional Siegel domain $D_{n+1}$ and the biholomorphic action of the Heisenberg group $\mathbb{H}_n$ on $D_{n+1}$. We study the symbols on $D_{n+1}$ that are $\mathbb{H}_n$-invariant and the $C^*$-algebras generated by the corresponding Toeplitz operators on the weighted Bergman spaces. We will denote such $C^*$-algebras by $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$, where $\lambda > -1$. We refer to Definition [Definition 24](#def:two-types-symbols){reference-type="ref" reference="def:two-types-symbols"} and subsection [4.3](#subsec:Toeplitz-HHn-invariant){reference-type="ref" reference="subsec:Toeplitz-HHn-invariant"} where this notation is introduced. With respect to the Toeplitz operators with $\mathbb{H}_n$-invariant symbols we obtain in Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} simultaneous diagonalizing expressions given by multiplier operators acting on $L^2(\mathbb{R}_+)$. In particular we have a proof, by explicit diagonalization, of the commutativity of the $C^*$-algebras $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$, for every $\lambda > -1$ and $n \geq 1$. A striking fact of the diagonalizing formulas obtained in Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} is that, although they depend on the weight $\lambda$ as expected, they do not depend on $n$ and so they are the same for all Siegel domains $D_{n+1}$ where $n \geq 1$. We state this fact in Theorem [Theorem 36](#thm:Heisenberg-Toeplitz-C*-commutative){reference-type="ref" reference="thm:Heisenberg-Toeplitz-C*-commutative"} which yields another of our main results: for all $\lambda > -1$, the $C^*$-algebras $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ are all isomorphic, independently of $n$, to a fixed $C^*$-algebra acting on $L^2(\mathbb{R}_+)$. As for the weight $\lambda > -1$, we prove that the isomorphism class of the $C^*$-algebras $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ is independent of $\lambda$ as well. Our Theorem [Theorem 39](#thm:Toeplitz-Hn-VSO){reference-type="ref" reference="thm:Toeplitz-Hn-VSO"} proves that the $C^*$-algebras $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ are all isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$, the algebra of very slowly oscillating functions on $\mathbb{R}_+$ (see Definition [Definition 38](#def:VSO){reference-type="ref" reference="def:VSO"}), acting on $L^2(\mathbb{R}_+)$. The reason is that the diagonalizing formulas [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"} obtained in Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} yield exactly the same family of functions found in [@HHM2014VerticalWeighted; @HMV2013Vertical] for Toeplitz operators with vertical symbols acting on the weighted Bergman spaces over the upper half-plane. Since the $C^*$-algebras generated by the latter sort of Toeplitz operators have been proved in [@HHM2014VerticalWeighted] to be isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$ acting on $L^2(\mathbb{R}_+)$, we thus conclude that $\mathcal{T}^{(\lambda)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ have such isomorphism class as well for every $\lambda > -1$ and $n \geq 1$. For further details on this reasoning and its relevance we refer to Remarks [Remark 37](#rmk:Heisenberg-Toeplitz-C*-commutative){reference-type="ref" reference="rmk:Heisenberg-Toeplitz-C*-commutative"} and [Remark 40](#rmk:Toeplitz-Hn-VSO){reference-type="ref" reference="rmk:Toeplitz-Hn-VSO"}. The latter prove that the most natural generalization of the horizontal translations on the upper half-plane is the action of the Heisenberg group $\mathbb{H}_n$ on $D_{n+1}$; this is so from the viewpoint of Toeplitz operators with suitably invariant symbols and the $C^*$-algebras that they generate. To obtain the theorems described above we prove in Theorem [Theorem 34](#thm:Rlambda-Ulambda){reference-type="ref" reference="thm:Rlambda-Ulambda"} a result which is interesting on its own: the existence, for every $\lambda > -1$ and $n \geq 1$, of a unitary map $$U_\lambda : \mathcal{A}^2_\lambda(D_{n+1}) \longrightarrow \int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi,$$ that decomposes the weighted Bergman spaces $\mathcal{A}^2_\lambda(D_{n+1})$ as a direct integral of Fock spaces $\mathcal{F}^2_{2\xi}(\mathbb{C}^n)$, for $\xi \in \mathbb{R}_+$. This decomposition is the consequence of our construction of a Segal-Bargmann type transform on the $L^2$-spaces containing the weighted Bergman spaces. This sort of construction usually requires a careful choice of coordinates on the domain in question. For this we resort to Lie theory and symplectic geometry, which are natural tools given the action of the Heisenberg group. We consider in Section [3](#sec:sympgeom-Siegel){reference-type="ref" reference="sec:sympgeom-Siegel"} the notion of moment maps for symplectic actions on Kähler manifolds and compute the moment map for the action of the Heisenberg group $\mathbb{H}_n$ on $D_{n+1}$, as well as for some of its subgroups which we classify for the connected case. Next, we use this to define moment map symbols (see Definition [Definition 20](#def:moment-map-symbol){reference-type="ref" reference="def:moment-map-symbol"} and [@QSMomentMapJFA]). It is proved in Proposition [Proposition 22](#prop:Hn-orbits-moment-map-center){reference-type="ref" reference="prop:Hn-orbits-moment-map-center"} that the moment map symbols corresponding to the center of $\mathbb{H}_n$ are precisely the $\mathbb{H}_n$-invariant symbols. We define in subsection [4.2](#subsec:group-moment-coordinates){reference-type="ref" reference="subsec:group-moment-coordinates"} a set of coordinates on $D_{n+1}$ that are very well adapted to the action of $\mathbb{H}_n$. Furthermore, these coordinates are defined using both the action of $\mathbb{H}_n$ and the moment map of the center of $\mathbb{H}_n$. They turn out to be quite natural since they greatly simplify the construction of our Segal-Bargmann type transform. For example, we need to take Fourier transform on a single variable only: the central coordinate of $\mathbb{H}_n$. As a comparison the Segal-Bargmann transform obtained in [@QVUnitBall1] for the nilpotent MASG (maximal Abelian subgroup) required to take Fourier transform in as many coordinates as the dimension of the Siegel domain. Also to compare previous techniques with ours we show in subsection [4.4](#subsec:nilpotent-MASG){reference-type="ref" reference="subsec:nilpotent-MASG"} how to obtain the diagonalizing formulas [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"} using the nilpotent MASG. This is possible due to the fact that for symbols $\mathbb{H}_n$-invariance implies invariance under the nilpotent MASG (see Corollary [Corollary 41](#cor:Heisenberg-inv-nilpMASG-inv){reference-type="ref" reference="cor:Heisenberg-inv-nilpMASG-inv"}). As noted in Remark [Remark 42](#rmk:Heisenberg-vs-nilpotentMASG){reference-type="ref" reference="rmk:Heisenberg-vs-nilpotentMASG"} our methods are more simple and clear than going through the nilpotent MASG. Most importantly, our Lie theoretic and symplectic methods provide more information and insight of the Toeplitz operators with $\mathbb{H}_n$-invariant symbols. # Analysis on the Siegel domain {#sec:Siegel} ## Bergman spaces and Toeplitz operators In the rest of this work we will consider the complex vector space $\mathbb{C}^{n+1}$ and assume that $n \geq 1$. Furthermore, any vector $z \in \mathbb{C}^{n+1}$ will be decomposed as $z = (z',z_{n+1})$ where $z' \in \mathbb{C}^n$ and $z_{n+1} \in \mathbb{C}$. The Siegel domain in $\mathbb{C}^{n+1}$ is defined by $$D_{n+1} = \{ z \in \mathbb{C}^{n+1} \mid \operatorname{Im}(z_{n+1}) > |z'|^2 \},$$ which is well known to be biholomorphically equivalent to the $(n+1)$-dimensional unit ball through the Cayley transform. In particular, $D_{n+1}$ is a symmetric domain that yields the unbounded realization of the $(n+1)$-dimensional unit ball. We will denote by $\dif z$ the Lebesgue measure on $\mathbb{C}^{n+1}$, and for every $\lambda > -1$ we will consider the measure $v_\lambda$ on $D_{n+1}$ given by $$\dif v_\lambda(z) = \frac{c_\lambda}{4} \big(\operatorname{Im}(z_{n+1}) - |z'|\big)^\lambda \dif z,$$ where the constant $c_\lambda$ is defined as $$c_\lambda = \frac{\Gamma(\lambda + n + 2)}{\pi^{n+1} \Gamma(\lambda + 1)}.$$ The weighted Bergman space on the Siegel domain $D_{n+1}$ with weight $\lambda > -1$ is denoted by $\mathcal{A}^2_\lambda(D_{n+1})$ and consists of the holomorphic functions on $D_{n+1}$ that belong to $L^2(D_{n+1}, v_\lambda)$. It is well known that $\mathcal{A}^2_\lambda(D_{n+1})$ is a closed subspace of $L^2(D_{n+1},v_\lambda)$ whose orthogonal projection $B_{n+1,\lambda} : L^2(D_{n+1},v_\lambda) \rightarrow \mathcal{A}^2_\lambda(D_{n+1})$ satisfies $$B_{n+1,\lambda}(f)(z) = \int_{D_{n+1}} f(w) K_{n+1,\lambda}(z,w) \dif v_\lambda(w),$$ where the function $K_{n+1,\lambda} : D_{n+1} \times D_{n+1} \rightarrow \mathbb{C}$ is given by $$K_{n+1,\lambda}(z,w) = \frac{1}{\Big(\displaystyle\frac{z_{n+1} - \overline{w}_{n+1}}{2i} - z' \cdot \overline{w'}\Big)^{\lambda + n + 2}}.$$ The projection $B_{n+1,\lambda}$ and the function $K_{n+1,\lambda}$ are called the Bergman projection and the Bergman kernel of $D_{n+1}$, respectively. For every $a \in L^\infty(D_{n+1})$, we will denote by $T^{(\lambda)}_a = T_a$ the Toeplitz operator acting on the Bergman space $\mathcal{A}^2_\lambda(D_{n+1})$, which is defined as the compression of the multiplier operator operator $M_a$. In other words, we have $$T^{(\lambda)}_a = B_{n+1,\lambda} M_a|_{\mathcal{A}^2_\lambda(D_{n+1})}.$$ ## The Heisenberg group In this section we describe our main Lie theory tools: the Heisenberg group and its action on the Siegel domain. This can be considered as a particular case of the quasi-translations studied in [@UpmeierToepBook]. Here we state some properties that will be used latter on, we prove some of them for the sake of completeness and refer to [@UpmeierToepBook] for further details. We note that our Siegel domain may be seen as a generalized upper half-space, while [@UpmeierToepBook] considers what may be described as a left-hand side half-space. Nevertheless, a simply linear biholomorphism relates both realizations. Let us consider in $\mathbb{C}^n \times \mathbb{R}$ the product given by $$(z',s) \cdot (w', t) = (z'+ w', s + t + 2 \operatorname{Im}(z'\cdot \overline{w'})),$$ where $z',w' \in \mathbb{C}^n$ and $s,t \in \mathbb{R}$. It is straightforward to check that $\mathbb{C}^n \times \mathbb{R}$ endowed with this operation becomes a Lie group, which we will denote by $\mathbb{H}_n$ and will be called the **Heisenberg group**. Note that $(0,0)$ is the identity element and that we also have $(z',s)^{-1} = (-z',-s)$ for this product. The next result collects the very basic properties of the Heisenberg group. **Proposition 1**. *For every $n \geq 1$, the Heisenberg group $\mathbb{H}_n$ has Lie algebra given by $\mathfrak{h}_n = \mathbb{C}^n \times \mathbb{R}$ carrying the Lie brackets $$[(w',t),(z',s)] = (0, 4\operatorname{Im}(w'\cdot \overline{z'})).$$ The exponential map $\exp : \mathfrak{h}_n \rightarrow \mathbb{H}_n$ is the identity map, and the adjoint representation $\mathrm{Ad}_{\mathbb{H}_n} : \mathbb{H}_n \rightarrow \mathrm{GL}(\mathfrak{h}_n)$ is given by $$\mathrm{Ad}_{\mathbb{H}_n}(w',t)(z',s) = (z', s + 4 \operatorname{Im}(w'\cdot \overline{z'})).$$ Furthermore, the centers $Z(\mathbb{H}_n)$ and $Z(\mathfrak{h}_n)$ of $\mathbb{H}_n$ and $\mathfrak{h}_n$ are both given by $\{0\} \times \mathbb{R}\subset \mathbb{C}^n \times \mathbb{R}$, considered as a subgroup and as a subspace, respectively.* *Proof.* Since $\mathbb{H}_n$ has the manifold structure given by a real vector space, its tangent space at the identity $(0,0)$ is $\mathbb{C}^n \times \mathbb{R}$. Hence, the latter yields the underlying vector space of the Lie algebra $\mathfrak{h}_n$. From the definition of the product in $\mathbb{H}_n$ it follows immediately that for every fixed $(z',s) \in \mathbb{H}_n$, the map $\mathbb{R}\rightarrow \mathbb{H}_n$ given by $r \mapsto (rz',rs)$ is a homomorphism of Lie groups. By definition of the exponential map of Lie groups, it follows that the exponential map of $\mathbb{H}_n$ is indeed the identity map. On the other hand, for any $(z',s), (w',t) \in \mathbb{H}_n$ we have $$(rw',rt) \cdot (rz',rs) = (r(w' + z'), r(t + s) + 2r^2\operatorname{Im}(w'\cdot \overline{z'}))$$ for every $r \in \mathbb{R}$. Hence, it follows from our computation of the exponential map and [@Helgason Chapter II Lemma 1.8] or the Campbell-Baker-Hausdorff formula (see [@KnappBeyond2nd]) that the Lie brackets in $\mathfrak{h}_n$ are given by $$[(w',t),(z',s)] = (0, 4\operatorname{Im}(w'\cdot \overline{z'})).$$ In particular, the Lie algebra adjoint representation $\mathrm{ad}: \mathfrak{h}_n \rightarrow \mathfrak{gl}(\mathfrak{h}_n)$ satisfies $\mathrm{ad}(X)^2 = 0$ for every $X \in \mathfrak{h}_n$. The previous remarks allow us to compute the Lie group adjoint representation $\mathrm{Ad}_{\mathbb{H}_n} : \mathbb{H}_n \rightarrow \mathrm{GL}(\mathfrak{h}_n)$ as follows $$\begin{aligned} \mathrm{Ad}_{\mathbb{H}_n}(w',t) &= \mathrm{Ad}_{\mathbb{H}_n}(\exp(w',t)) = e^{\mathrm{ad}(w',t)} \\ &= I_{\mathfrak{h}_n} + \mathrm{ad}(w',t) = I_{\mathfrak{h}_n} + [(w',t), \cdot], \end{aligned}$$ where the second identity follows from elementary Lie theory. This yields the required formula for $\mathrm{Ad}_{\mathbb{H}_n}$. Finally, the claims on the centers of $\mathbb{H}_n$ and $\mathfrak{h}_n$ are clear from the formulas obtained. ◻ We will consider the $\mathbb{H}_n$-action on $\mathbb{C}^{n+1}$ obtained from the next elementary result, whose proof we present for the sake of completeness. **Proposition 2**. *The assignment $\mathbb{H}_n \times \mathbb{C}^{n+1} \rightarrow \mathbb{C}^{n+1}$ defined by $$(w',t)\cdot z = (z' + w', z_{n+1} + t + 2i z'\cdot \overline{w'} + i |w'|^2).$$ is a free $\mathbb{H}_n$-action which is holomorphic and preserves the map on $\mathbb{C}^{n+1}$ given by $z \mapsto \operatorname{Im}(z_{n+1}) - |z'|^2$. In particular, this $\mathbb{H}_n$-action yields a holomorphic action on $D_{n+1}$.* *Proof.* A straightforward computation using the product law in $\mathbb{H}_n$ shows that the mapping in the statements yields indeed an action (see also [@UpmeierToepBook]). It is also clear that such action yields biholomorphisms of $\mathbb{C}^{n+1}$. On the other hand, we have for every $(w',t) \in \mathbb{H}_n$ and $z \in \mathbb{C}^{n+1}$ $$\begin{aligned} \operatorname{Im}\big(\big((w',t)\cdot z\big)_{n+1}\big) &- \big|\big((w',t)\cdot z\big)'\big|^2 = \\ &= \operatorname{Im}(z_{n+1}) + 2 \operatorname{Im}(iz'\cdot \overline{w'}) + |w'|^2 - |z' + w'|^2 \\ &= \operatorname{Im}(z_{n+1}) + 2 \operatorname{Re}(z'\cdot \overline{w'}) + |w'|^2 - |z' + w'|^2 \\ &= \operatorname{Im}(z_{n+1}) - |z'|^2, \end{aligned}$$ thus proving the $\mathbb{H}_n$-invariance of the given function. In particular, the $\mathbb{H}_n$-action preserves $D_{n+1}$ and so defines a biholomorphic action on it. ◻ **Remark 3**. It is clear that the subset $\mathbb{R}^{n+1} = \mathbb{R}^n \times \mathbb{R}\subset \mathbb{H}_n$ is a closed Lie subgroup. Moreover, the corresponding $\mathbb{R}^{n+1}$-action on $D_{n+1}$ yields precisely one of the maximal Abelian subgroups (MASGs) of biholomorphisms studied in [@QVUnitBall1; @QVUnitBall2], namely the so-called nilpotent MASG. # Symplectic geometry on the Siegel domain {#sec:sympgeom-Siegel} ## Symplectic manifolds and moment maps Our main geometric tool will be symplectic geometry and the computation of moment maps of symplectic actions. Hence, we review some definitions and properties. Further details can be found in [@QSMomentMapJFA] and the references therein. A symplectic manifold is a manifold $M$ carrying a non-degenerate closed $2$-form $\omega$. In this case, $\omega$ is called the symplectic form of $M$, and the pair $(M,\omega)$ will be used to denote the symplectic manifold when emphasis on the $2$-form is required. The geometric symmetries of a symplectic manifold are given by the so-called symplectomorphisms. These are diffeomorphisms $\varphi : M \rightarrow M$ that preserve $\omega$. More precisely, we require that $\omega_z(u,v) = \omega_{\varphi(z)}(\dif\varphi_z(u), \dif\varphi_z(v))$, for every $z \in M$ and $u, v \in T_z M$. For a given smooth action of a Lie group $H$ on a manifold $M$ we have a natural map from $\mathfrak{h}$, the Lie algebra of $H$, to the Lie algebra of smooth vector fields on $M$. More precisely, for any $X \in \mathfrak{h}$ we will denote by $X^\sharp$ the vector field given by $$X^\sharp_z = \frac{\dif}{\dif r}\sVert[2]_{r=0} \exp(rX)\cdot z,$$ for every $z \in M$. It is easily seen that this map defines an anti-homomorphism of Lie algebras (see [@Helgason]). If $H$ is a Lie group acting smoothly on a symplectic manifold $M$, then we will say that the $H$-action is symplectic when every element of $H$ yields a symplectomorphism of $M$. When such an action is given we have the fundamental notion of moment map. **Definition 4**. Let $H$ be a Lie group with Lie algebra $\mathfrak{h}$ and denote by $\mathfrak{h}^*$ the dual vector space of the latter. For a symplectic $H$-action on a symplectic manifold $(M,\omega)$, a moment map for the $H$-action is a smooth map $\mu : M \rightarrow \mathfrak{h}^*$ for which the following properties hold. 1. For every $X \in \mathfrak{h}$, the function $\mu_X : M \rightarrow \mathbb{R}$ given by $\mu_X(z) = \langle \mu(z), X\rangle$ satisfies $$\dif \mu_X = \omega(X^\sharp, \cdot).$$ 2. The map $\mu : M \rightarrow \mathfrak{h}^*$ is $H$-equivariant for the dual action of the adjoint representation on the target. In other words, we have $$\mu(h\cdot z) = \mathrm{Ad}_H^*(h)(\mu(z)),$$ for every $z \in M$ and $h \in H$, where $\mathrm{Ad}_H^*(h) = \mathrm{Ad}_H(h^{-1})^\top$ is the dual map of $\mathrm{Ad}_H(h^{-1})$ and $\mathrm{Ad}_H : H \rightarrow \mathrm{GL}(\mathfrak{h})$ denotes the adjoint representation of $H$. **Remark 5**. With the notation from Definition [Definition 4](#def:moment-map){reference-type="ref" reference="def:moment-map"}, if the group $H$ is Abelian, then the adjoint representation is trivial and the $H$-equivariance condition reduces to an $H$-invariance condition: $\mu(h\cdot z) = \mu(z)$ for every $z \in M$ and $h \in H$. But, for a general Lie group $H$ it is important to consider both the adjoint representation and its dual representation. On the other hand, to simplify computations it is useful to introduce some sort of identification between $\mathfrak{h}$ and $\mathfrak{h}^*$. When $\mathfrak{h}$ is considered just as a vector space, this is straightforward: any inner product on $\mathfrak{h}$ implements one such identification. However, one has to carry over the adjoint representation and so not every inner product will be adequate. The first alternative is to look for an inner product in $\mathfrak{h}$, positive definite or not, which is $\mathrm{Ad}_H(H)$-invariant. Then, the corresponding identification between $\mathfrak{h}$ and $\mathfrak{h}^*$ also realizes an identification between $\mathrm{Ad}_H$ and $\mathrm{Ad}_H^*$. This allows to replace in Definition [Definition 4](#def:moment-map){reference-type="ref" reference="def:moment-map"} $\mathfrak{h}^*$ and $\mathrm{Ad}_H^*$ with $\mathfrak{h}$ and $\mathrm{Ad}_H$, respectively. Nevertheless, there are Lie groups $H$ for which there does not exist $\mathrm{Ad}_H(H)$-invariant inner products on $\mathfrak{h}$: the Heisenberg group is one of them. In this situation, the best alternative is to choose some "natural" inner product, identify $\mathfrak{h}$ and $\mathfrak{h}^*$ and compute the representation on $\mathfrak{h}$ that corresponds to $\mathrm{Ad}_H^*$ on $\mathfrak{h}^*$. We will follow this alternative for the Heisenberg group and its subgroups. To achieve this we consider the next general result. **Lemma 6**. *Let $H$ be a Lie group with Lie algebra $\mathfrak{h}$, let $\langle\cdot,\cdot\rangle$ be an inner product on $\mathfrak{h}$ and consider the identification $\mathfrak{h}\simeq \mathfrak{h}^*$ given by $\langle\cdot,\cdot\rangle$. Then, the map $\rho : H \rightarrow \mathrm{GL}(\mathfrak{h})$ defined by the condition $\langle\rho(h)(X), Y\rangle = \langle X, \mathrm{Ad}_H(h^{-1})(Y)\rangle$, for every $h \in H$, $X,Y \in \mathfrak{h}$, where $\mathrm{Ad}_H$ is the adjoint representation of $H$, is the representation corresponding to $\mathrm{Ad}_H^* : H \rightarrow \mathrm{GL}(\mathfrak{h}^*)$ for the identification $\mathfrak{h}\simeq \mathfrak{h}^*$. Furthermore, if $\langle\cdot,\cdot\rangle$ is $\mathrm{Ad}_H(H)$-invariant, then $\rho = \mathrm{Ad}_H$.* *Proof.* If we consider the transpose operation with respect to the given inner product as well as our identification $\mathfrak{h}\simeq \mathfrak{h}^*$, then we clearly have $\rho(h) = \mathrm{Ad}_H(h^{-1})^\top$, for every $h \in H$. From this it follows that $\rho$ is a representation of $H$ on $\mathfrak{h}$. Furthermore, the identification $\mathfrak{h}\simeq \mathfrak{h}^*$ is given by the map $X \mapsto \langle X, \cdot \rangle$, and so the definition of $\rho$ is equivalent to the commutativity of the following diagram $$\xymatrix{ \mathfrak{h}\ar[rr]^{\rho(h)} \ar[d] & & \mathfrak{h}\ar[d] \\ \mathfrak{h}^* \ar[rr]^{\mathrm{Ad}_H^*(h)}& & \mathfrak{h}^* }$$ for every $h \in H$. This shows that $\rho$ corresponds to $\mathrm{Ad}_H^*$ under the identification $\mathfrak{h}\simeq \mathfrak{h}^*$ under consideration. The last claim is now a clear consequence. ◻ **Remark 7**. In the rest of this work, and in the notation of Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"}, we will say that $\rho$ is the representation of $H$ on its Lie algebra $\mathfrak{h}$ transpose to $\mathrm{Ad}_H$. For these notation and construction to be considered we assume the choice of some inner product in $\mathfrak{h}$. The upshot of Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"} is that $\rho$, the transpose of $\mathrm{Ad}_H$, is the same as $\mathrm{Ad}_H^*$, the dual of $\mathrm{Ad}_H$, up to the identification $\mathfrak{h}\simeq \mathfrak{h}^*$ given by the chosen inner product. Definition [Definition 4](#def:moment-map){reference-type="ref" reference="def:moment-map"} and Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"} yield the next consequence. We will use it as alternative definition of moment map in the rest of this work, since it will provide a more convenient way to compute moment maps for the symplectic actions that we consider. **Corollary 8**. *Let $H$ be a Lie group acting by symplectomorphisms on a symplectic manifold $(M, \omega)$ and let us consider the identification $\mathfrak{h}\simeq \mathfrak{h}^*$ obtained from a given inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak{h}$. With respect to such identification, a map $\mu : M \rightarrow \mathfrak{h}$ is a moment map for the $H$-action on $M$ if and only if the following conditions are satisfied.* 1. *For every $X \in \mathfrak{h}$, the function $\mu_X : M \rightarrow \mathbb{R}$ given by $\mu_X(z) = \langle \mu(z), X\rangle$ satisfies $\dif \mu_X = \omega(X^\sharp, \cdot)$.* 2. *The map $\mu : M \rightarrow \mathfrak{h}$ is $H$-equivariant for the representation $\rho : H \rightarrow \mathrm{GL}(\mathfrak{h})$ transpose to $\mathrm{Ad}_H$ and defined in Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"}.* *Proof.* We note that in condition (1) from Definition [Definition 4](#def:moment-map){reference-type="ref" reference="def:moment-map"} the expression $\mu_X(z) = \langle \mu(z), X\rangle$ considers the angled brackets as evaluation of a linear functional on a vector. Under the identification $\mathfrak{h}\simeq \mathfrak{h}^*$ given by the inner product such evaluation corresponds to the inner product itself. Hence, both conditions (1), from our statement and from Definition [Definition 4](#def:moment-map){reference-type="ref" reference="def:moment-map"}, are equivalent under the identification $\mathfrak{h}\simeq \mathfrak{h}^*$. Similarly, the equivariance required in conditions (2) from the definition and this corollary are equivalent by Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"} and the commutative diagram found in its proof. ◻ It is also useful to relate the moment maps for actions of subgroups to those of the action of the ambient group. We consider this problem within the setup of the previous results. **Proposition 9**. *Let $H$ be a Lie group acting by symplectomorphisms on a symplectic manifold $(M,\omega)$ and let $G \subset H$ be a Lie subgroup with their Lie algebras denoted by $\mathfrak{g}$ and $\mathfrak{h}$, respectively. Let us also fix an inner product in $\mathfrak{h}$ and the corresponding identification $\mathfrak{h}\simeq \mathfrak{h}^*$. With the notation from Corollary [Corollary 8](#cor:moment-map-fh){reference-type="ref" reference="cor:moment-map-fh"}, if $\mu : M \rightarrow \mathfrak{h}$ is a moment map for the $H$-action and $\pi : \mathfrak{h}\rightarrow \mathfrak{g}$ is the orthogonal projection, then $\pi \circ \mu : M \rightarrow \mathfrak{g}$ is a moment map for the $G$-action.* *Proof.* We will prove that $\pi \circ \mu$ satisfies the properties stated in Corollary [Corollary 8](#cor:moment-map-fh){reference-type="ref" reference="cor:moment-map-fh"} that characterize a moment map for the $G$-action. Hence, we will use the notation of that result. We note that for $X \in \mathfrak{g}$ $$\big(\pi\circ\mu\big)_X (z) = \langle\pi\circ\mu(z),X\rangle = \langle\mu(z),X\rangle = \mu_X(z),$$ for every $z \in M$. Hence, the map $\pi \circ \mu$ satisfies Corollary [Corollary 8](#cor:moment-map-fh){reference-type="ref" reference="cor:moment-map-fh"}(1) for the $G$-action as a consequence of $\mu$ satisfying the same property. Let us now consider the map $\rho_H : H \rightarrow \mathrm{GL}(\mathfrak{h})$ given by $\rho_H(h) = \mathrm{Ad}_H(h^{-1})^\top$, which is the representation transpose to $\mathrm{Ad}_H$. We will use the notation where the group associated to a representation is indicated as a subindex. We claim that $\rho_H(h)(\mathfrak{g}^\perp) = \mathfrak{g}^\perp$ for every $h \in G$. This follows from the fact that, for every $h \in G$, we have $\mathrm{Ad}_G(h^{-1}) = \mathrm{Ad}_H(h^{-1})|_{\mathfrak{g}}$, while the former leaves invariant $\mathfrak{g}$. As a consequence we have $$\pi \circ \rho_H(h) = \pi \circ \rho_H(h) \circ \pi$$ for every $h \in G$. This identity is obtained by noting that it holds for elements of both $\mathfrak{g}$ and $\mathfrak{g}^\perp$. We thus conclude that for $h \in G$ and $X,Y \in \mathfrak{g}$ we have $$\begin{aligned} \langle\pi\circ\rho_H(h)\circ\pi(X),Y\rangle &= \langle\mathrm{Ad}_H(h^{-1})^\top(X), Y \rangle = \langle X, \mathrm{Ad}_H(h^{-1})(Y) \rangle \\ &= \langle X, \mathrm{Ad}_G(h^{-1})(Y) \rangle = \langle \mathrm{Ad}_G(h^{-1})^\top(X), Y \rangle, \end{aligned}$$ which yields the identity $$\rho_G(h) = \mathrm{Ad}_G(h^{-1})^\top = \pi \circ \rho_H(h) \circ \pi |_{\mathfrak{g}},$$ for every $h \in G$, where $\rho_{G}$ is the representation transpose to $\mathrm{Ad}_G$. Once, we have proved these properties we can compute as follows for every $z \in M$ and $h \in G$ $$(\pi\circ\mu)(hz) = \pi(\rho_H(h)(\mu(z))) = (\pi \circ \rho_H(h) \circ \pi) (\pi\circ\mu(z)) \\ = \rho_{G}(h)(\pi\circ\mu(z)),$$ where we have used the $H$-equivariance of $\mu$ in the first identity. This proves that $\pi\circ\mu$ satisfies Corollary [Corollary 8](#cor:moment-map-fh){reference-type="ref" reference="cor:moment-map-fh"}(2) for the $G$-action. We conclude from this corollary that $\pi\circ\mu$ is a moment map for the $G$-action on $M$. ◻ ## Kähler manifolds We recall here some well known facts and refer to [@KNII; @Mok] for further details. Let us consider a complex manifold $M$ and let us denote by $J$ its complex structure as a tensor acting on its tangent bundle. A Riemannian metric $g$ on $M$ is called Hermitian when it satisfies $$g_z(J_z u, J_z v) = g_z(u,v),$$ for every $z \in M$ and $u, v \in T_zM$. In this case, the tensor $\omega$ defined by $$\omega_z(u,v) = g_z(J_z u, v),$$ for every $z \in M$ and $u, v \in T_zM$ as well, is a non-degenerate $2$-form. With this notation, we say that the pair $(M,g)$ is a Hermitian manifold with associated $2$-form $\omega$. If the $2$-form $\omega$ is closed, then $M$ is called a Kähler manifold and $\omega$ is referred as its associated Kähler form. In particular, every Kähler manifold is a symplectic manifold whose symplectic form is precisely the associated Kähler form. Our main example is given by the description of the Siegel domain as a Kähler manifold. From now on we will freely use Wirtinger partial derivatives and their dual $1$-forms. For this and the next result we refer to [@KNII; @Mok; @QSMomentMapJFA]. **Proposition 10**. *The Siegel domain $D_{n+1}$ has a Kähler structure whose Riemannian metric and Kähler form $g$ and $\omega$, respectively, are given by the following expressions. $$\begin{aligned} g_z =&\; \frac{1}{(\operatorname{Im}(z_{n+1}) - |z'|^2)^2} \bigg( (\operatorname{Im}(z_{n+1}) - |z'|^2) \sum_{j=1}^n \dif z_j \otimes \dif \overline{z}_j + \sum_{j,k=1}^n \overline{z}_j z_k \dif z_j \otimes \dif \overline{z}_k \\ &+ \frac{1}{2i} \sum_{j=1}^n \big( \overline{z}_j \dif z_j \otimes \dif \overline{z}_{n+1} -z_j \dif z_{n+1} \otimes \dif \overline{z}_j \big) + \frac{1}{4} \dif z_{n+1} \otimes \dif \overline{z}_{n+1} \bigg) \\ \omega_z =&\; \frac{i}{(\operatorname{Im}(z_{n+1}) - |z'|^2)^2} \bigg( (\operatorname{Im}(z_{n+1}) - |z'|^2) \sum_{j=1}^n \dif z_j \wedge \dif \overline{z}_j + \sum_{j,k=1}^n \overline{z}_j z_k \dif z_j \wedge \dif \overline{z}_k \\ &+ \frac{1}{2i} \sum_{j=1}^n \big( \overline{z}_j \dif z_j \wedge \dif \overline{z}_{n+1} -z_j \dif z_{n+1} \wedge \dif \overline{z}_j \big) + \frac{1}{4} \dif z_{n+1} \wedge \dif \overline{z}_{n+1} \bigg). \end{aligned}$$ Furthermore, every biholomorphism of $D_{n+1}$ preserves both tensors and so the group of automorphisms $D_{n+1}$ yields an action which is both isometric and symplectic.* In the rest of this work we will consider $D_{n+1}$ as a Kähler manifold for the structure described in Proposition [Proposition 10](#prop:Siegel-Kahler){reference-type="ref" reference="prop:Siegel-Kahler"}. As an immediate consequence of the latter and Proposition [Proposition 2](#prop:Hn-action){reference-type="ref" reference="prop:Hn-action"} we obtain the next result. **Corollary 11**. *The action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ is symplectic.* ## Moment maps for the Heisenberg group and its subgroups We will now compute a moment map for the $\mathbb{H}_n$-action on $D_{n+1}$. We start with the next simple lemma that computes the vector fields on $D_{n+1}$ induced from the Lie algebra $\mathfrak{h}_n$. **Lemma 12**. *For the $\mathbb{H}_n$-action on $D_{n+1}$ and for every $X = (w',t) \in \mathfrak{h}_n$ we have $$\begin{aligned} X^\sharp_z &= (w', t + 2i z'\cdot \overline{w'}) \\ &= \sum_{j=1}^n\bigg( w_j \frac{\partial}{\partial z_j} + \overline{w}_j \frac{\partial}{\partial \overline{z}_j} \bigg) + \big(t + 2i z'\cdot \overline{w'} \big) \frac{\partial}{\partial z_{n+1}} + \overline{\big(t + 2i z'\cdot \overline{w'} \big)} \frac{\partial}{\partial \overline{z}_{n+1}} \end{aligned}$$ for every $z \in D_{n+1}$, where the first expression belongs to $\mathbb{C}^{n+1} = T_z D_{n+1}$ and the second expression corresponds to the vector field as a linear combination of the Wirtinger operators.* *Proof.* For the given objects, Propositions [Proposition 1](#prop:Heisenberg-properties){reference-type="ref" reference="prop:Heisenberg-properties"} and [Proposition 2](#prop:Hn-action){reference-type="ref" reference="prop:Hn-action"} imply that for every $r \in \mathbb{R}$ we have $$\exp(rX)\cdot z = (z' + rw', z_{n+1} + rt + 2i r z'\cdot \overline{w'} + i r^2 |w'|^2),$$ whose derivative at $r = 0$ yields the first expression of the statement. On the other hand, if $Y = (f_1, \dots, f_{n+1})$ is a vector field on $D_{n+1}$ written as a $\mathbb{C}^{n+1}$-valued function, then $Y$ in terms of the Wirtinger operators is given by (see [@QSMomentMapJFA]) $$Y = \sum_{j=1}^{n+1} \bigg( f_j \frac{\partial}{\partial z_j} + \overline{f}_j \frac{\partial}{\partial \overline{z}_j} \bigg),$$ which yields the second expression of the statement. ◻ In the next result we obtain an expression for the representation transpose to the adjoint representation of $\mathbb{H}_n$ on $\mathfrak{h}_n$ (see Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"} and Remark [Remark 7](#rmk:rho-from-Ad*){reference-type="ref" reference="rmk:rho-from-Ad*"}). Recall that such transpose representation corresponds to the transpose of the adjoint representation. **Lemma 13**. *Consider the natural isomorphism $\mathfrak{h}_n = \mathbb{C}^n \times \mathbb{R}\simeq \mathbb{R}^{2n+1}$ and the corresponding inner product $\langle\cdot,\cdot\rangle$ induced on $\mathfrak{h}_n$ from the canonical inner product in $\mathbb{R}^{2n+1}$. Then, in the notation of Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"}, the representation $\rho : \mathbb{H}_n \rightarrow \mathrm{GL}(\mathfrak{h}_n)$ transpose to $\mathrm{Ad}_{\mathbb{H}_n}$ is given by $\rho(w',t)(z',s) = (z' + 4is w', s)$.* *Proof.* For $X = (z',s), Y = (z_1',s_1)$ belonging to $\mathfrak{h}_n$ and $h = (w',t) \in \mathbb{H}_n$ we have $$\begin{aligned} \langle \rho(w',t)(z',s), Y \rangle &= \langle \rho(h)(X), Y \rangle \\ &= \langle X, \mathrm{Ad}_{\mathbb{H}_n}(h^{-1})(Y)\rangle \\ &= \langle (z',s), (z_1', s_1 - 4\operatorname{Im}(w'\cdot \overline{z_1'}) \rangle \\ &= \operatorname{Re}(z'\cdot \overline{z_1'}) + s s_1 - 4 \operatorname{Im}(s w' \cdot \overline{z_1'}) \\ &= \operatorname{Re}(z'\cdot \overline{z_1'} + 4i s w'\cdot \overline{z_1'}) + s s_1 \\ &= \operatorname{Re}((z' + 4i s w')\cdot \overline{z_1'}) + s s_1 \\ &= \langle (z' + 4is w', s), Y\rangle, \end{aligned}$$ where we have used Proposition [Proposition 1](#prop:Heisenberg-properties){reference-type="ref" reference="prop:Heisenberg-properties"} in the third identity. This yields the claimed expression. ◻ **Remark 14**. In the rest of this work, we will assume $\mathfrak{h}_n$ endowed with the inner product considered in Lemma [Lemma 13](#lem:rho-for-Heisenberg){reference-type="ref" reference="lem:rho-for-Heisenberg"}. This fixes the identification between $\mathfrak{h}_n$ and $\mathfrak{h}_n^*$ required by Lemma [Lemma 6](#lem:rho-from-Ad*){reference-type="ref" reference="lem:rho-from-Ad*"}, and so yields a well-defined representation transpose to $\mathrm{Ad}_{\mathbb{H}_n}$, whose expression has been obtained in Lemma [Lemma 13](#lem:rho-for-Heisenberg){reference-type="ref" reference="lem:rho-for-Heisenberg"}. We now obtain the main result of this section. Recall that we have agreed to use the definition of moment map provided by Corollary [Corollary 8](#cor:moment-map-fh){reference-type="ref" reference="cor:moment-map-fh"}. **Theorem 15**. *The map $\mu^{\mathbb{H}_n} : D_{n+1} \rightarrow \mathfrak{h}_n$ given by $$\mu^{\mathbb{H}_n}(z) = -\frac{(4iz', 1)}{2(\operatorname{Im}(z_{n+1}) - |z'|^2)},$$ is a moment map for the $\mathbb{H}_n$-action on $D_{n+1}$.* *Proof.* For simplicity we will denote $\mu = \mu^{\mathbb{H}_n}$ in this proof. Following Corollary [Corollary 8](#cor:moment-map-fh){reference-type="ref" reference="cor:moment-map-fh"} and the remarks from its proof, for every $X \in \mathfrak{h}_n$, the function $\mu_X : D_{n+1} \rightarrow \mathbb{R}$ is given by $$\mu_X(z) = \langle\mu(z),X\rangle$$ where $\langle\cdot,\cdot\rangle$ denotes the inner product considered in $\mathfrak{h}_n$. In particular, for every $X = (w',t) \in \mathfrak{h}_n$ we have $$\mu_X(z) = \frac{4\operatorname{Im}(z'\cdot \overline{w'}) - t}{2(\operatorname{Im}(z_{n+1}) - |z'|^2)} = \frac{2i(z'\cdot \overline{w'} - \overline{z'}\cdot w') + t}% {i(z_{n+1} - \overline{z}_{n+1}) + 2z'\cdot \overline{z'}}.$$ We compute the differential of this function using Wirtinger derivatives to obtain $$\begin{aligned} \dif \big(\mu_X\big)_z &= \frac{1}{(\operatorname{Im}(z_{n+1}) - |z'|^2)^2} \times \\ \times \Bigg(& \sum_{j=1}^n \Big( 2\operatorname{Im}(z'\cdot \overline{w'}) \overline{z}_j - i(\operatorname{Im}(z_{n+1}) - |z'|^2) \overline{w}_j - \frac{1}{2}t\overline{z}_j \Big) \dif z_j \\ &+ \sum_{j=1}^n \Big( 2\operatorname{Im}(z'\cdot \overline{w'}) z_j + i(\operatorname{Im}(z_{n+1}) - |z'|^2) w_j - \frac{1}{2}tz_j \Big) \dif \overline{z}_j \\ &+ i \Big(\operatorname{Im}(z'\cdot\overline{w'}) - \frac{1}{4} t \Big) \dif z_{n+1} - i \Big(\operatorname{Im}(z'\cdot\overline{w'}) - \frac{1}{4} t \Big) \dif \overline{z}_{n+1} \Bigg) \end{aligned}$$ for every $z \in D_{n+1}$. On the other hand, using Proposition [Proposition 10](#prop:Siegel-Kahler){reference-type="ref" reference="prop:Siegel-Kahler"} and Lemma [Lemma 12](#lem:X-sharp){reference-type="ref" reference="lem:X-sharp"} we obtain the following after some manipulations $$\begin{aligned} \omega_z(X^\sharp_z,&\cdot) = \frac{1}{(\operatorname{Im}(z_{n+1}) - |z'|^2)^2} \times \\ \times \Bigg(& -\Big(i(z'\cdot\overline{w'} - \overline{z'}\cdot w') + \frac{1}{2}t\Big) \sum_{j=1}^n \overline{z}_j \dif z_j - i(\operatorname{Im}(z_{n+1}) - |z'|^2) \sum_{j=1}^n \overline{w}_j \dif z_j \\ &-\Big(i(z'\cdot\overline{w'} - \overline{z'}\cdot w') + \frac{1}{2}t\Big) \sum_{j=1}^n z_j \dif \overline{z}_j + i(\operatorname{Im}(z_{n+1}) - |z'|^2) \sum_{j=1}^n w_j \dif \overline{z}_j \\ &+ i\Big(\frac{1}{2i}(z'\cdot\overline{w'} - \overline{z'}\cdot w') - \frac{1}{4} t \Big) \dif z_{n+1} - i\Big(\frac{1}{2i}(z'\cdot\overline{w'} - \overline{z'}\cdot w') - \frac{1}{4} t \Big) \dif \overline{z}_{n+1} \Bigg). \end{aligned}$$ Comparing both of these expressions we easily conclude that $$\dif \big(\mu_X\big)_z = \omega_z(X^\sharp_z,\cdot),$$ for every $X \in \mathfrak{h}_n$ at every $z \in D_{n+1}$. It remains to prove the $\mathbb{H}_n$-equivariance of $\mu$ with respect to the representation $\rho$ transpose to $\mathrm{Ad}_{\mathbb{H}_n}$. Since the map $z \mapsto \operatorname{Im}(z_{n+1}) - |z'|^2$ is already $\mathbb{H}_n$-invariant (see Proposition [Proposition 2](#prop:Hn-action){reference-type="ref" reference="prop:Hn-action"}), it is enough to prove that the map $\mu_0 : D_{n+1} \rightarrow \mathfrak{h}_n$ defined by $\mu_0(z) = (4iz',1)$ is $\mathbb{H}_n$-equivariant. Hence, we compute for $h = (w',t) \in \mathbb{H}_n$ $$\begin{aligned} \mu_0(h\cdot z) &= \mu_0(z' + w', z_{n+1} + t + 2i z'\cdot \overline{w'} + i |w'|^2) \\ &= (4i(z'+w'),1) = (4iz' + 4iw', 1) = \rho(w',t)(4iz',1) \\ &= \rho(w',t)(\mu_0(z)), \end{aligned}$$ which holds for every $z \in D_{n+1}$. This completes the proof that $\mu$ is a moment map for the $\mathbb{H}_n$-action. ◻ We now obtain moment maps for the subgroups of $\mathbb{H}_n$. Theorem [Theorem 15](#thm:Heisenberg-moment-map){reference-type="ref" reference="thm:Heisenberg-moment-map"} and Proposition [Proposition 9](#prop:moment-map-subgroups){reference-type="ref" reference="prop:moment-map-subgroups"} will provide the main tools to achieve this. First we consider all such subgroups. **Proposition 16**. *The following is a complete list of the connected Lie subgroups of $\mathbb{H}_n$.* 1. *$V \times \mathbb{R}$, where $V \subset \mathbb{C}^n$ is a real subspace.* 2. *$H_{V,f} = \mathrm{Graph}(f)$, the graph of a real linear functional $f : V \rightarrow \mathbb{R}$, where $V \subset \mathbb{C}^n$ is an isotropic (real) subspace for the canonical symplectic structure of $\mathbb{C}^n$.* *Proof.* Recall that the canonical symplectic structure of $\mathbb{C}^n$ is (a constant multiple) of $(z',w') \mapsto \operatorname{Im}(z'\cdot \overline{w'})$. Hence, the sets listed in the statement are easily seen to be connected subgroups of $\mathbb{H}_n$. To prove that we have listed all connected subgroups of $\mathbb{H}_n$, it is enough to classify all Lie subalgebras of $\mathfrak{h}_n$ and consider the groups generated by their images under the exponential map of $\mathbb{H}_n$, which in this case is the identity. Let $\mathfrak{h}\subset \mathfrak{h}_n$ be a Lie subalgebra, and let us denote by $V$ and $V_0$ the projections of $\mathfrak{h}$ onto $\mathbb{C}^n$ and $\mathbb{R}$. Recall that $\mathfrak{h}_n = \mathbb{C}^n \times \mathbb{R}$ as a vector space. Hence, we have $\mathfrak{h}\subset V \times \mathbb{R}$ and $$\dim V \leq \dim \mathfrak{h}\leq \dim V + 1.$$ If $\dim \mathfrak{h}= \dim V + 1$, then we necessarily have $\mathfrak{h}= V \times \mathbb{R}$, which is clearly a Lie subalgebra of $\mathfrak{h}_n$. After applying the exponential map (the identity map) we obtain a subgroup of the form (1) from the statement. Let us now assume that $\dim \mathfrak{h}= \dim V$. Since $\mathfrak{h}$ is a subspace of $V \times \mathbb{R}$ whose projection on the first factor is $V$, we conclude that there exists $f : V \rightarrow \mathbb{R}$ linear such that $\mathfrak{h}= \mathrm{Graph}(f)$. If $V$ is not an isotropic subspace of $\mathbb{C}^n$, then there exist $(z',f(z')), (w',f(w')) \in \mathfrak{h}$ such that $\operatorname{Im}(z'\cdot \overline{w'}) \not= 0$. By Proposition [Proposition 1](#prop:Heisenberg-properties){reference-type="ref" reference="prop:Heisenberg-properties"} we have $(0,4\operatorname{Im}(z'\cdot \overline{w'})) \in \mathfrak{h}$, whose last coordinate is non-zero. This implies that $V_0 = \mathbb{R}$ and also that $\{0\} \times \mathbb{R}\subset \mathfrak{h}$. Since $\mathfrak{h}$ is a vector space, this implies that $V\times \{0\} \subset \mathfrak{h}$ and so that we have $\mathfrak{h}= V \times \mathbb{R}$. This contradicts our current assumption that $\dim \mathfrak{h}= \dim V$. Hence, $V$ is necessarily an isotropic subspace of $\mathbb{C}^n$. The image of $\mathfrak{h}= \mathrm{Graph}(f)$ under the exponential map (the identity as noted above) is now a subgroup of the form (2) from the statement. ◻ **Remark 17**. In the list of subgroups obtained in Proposition [Proposition 16](#prop:Heisenberg-subgroups){reference-type="ref" reference="prop:Heisenberg-subgroups"}, case (1) allows $V$ to be either isotropic or non-isotropic. However, in both cases (1) and (2), the latter alternative determines whether the subgroup is Abelian or not. More precisely, it is easy to check that for both types of subgroups considered in Proposition [Proposition 16](#prop:Heisenberg-subgroups){reference-type="ref" reference="prop:Heisenberg-subgroups"} the subgroup is Abelian if and only if $V$ is an isotropic subspace of $\mathbb{C}^n$. As a particular case, $\mathbb{R}^n \times \mathbb{R}$ yields the subgroup whose action on $D_{n+1}$ corresponds to the nilpotent MASG considered in [@QVUnitBall1; @QVUnitBall2]. See also Remark [Remark 3](#rmk:Hn-NilpotentAction){reference-type="ref" reference="rmk:Hn-NilpotentAction"}. The next result is a consequence of Theorem [Theorem 15](#thm:Heisenberg-moment-map){reference-type="ref" reference="thm:Heisenberg-moment-map"} and Proposition [Proposition 9](#prop:moment-map-subgroups){reference-type="ref" reference="prop:moment-map-subgroups"}. Also note that the particular cases can be considered because of Proposition [Proposition 16](#prop:Heisenberg-subgroups){reference-type="ref" reference="prop:Heisenberg-subgroups"} and are obtained from an easy computation. In the rest of this work for every integer $1 \leq \ell \leq n-1$, inducing the decomposition $\mathbb{C}^n = \mathbb{C}^{n-\ell} \times \mathbb{C}^\ell$, we will write every $z' \in \mathbb{C}^n$ as $z' = (z_{(1)}, z_{(2)})$ where $z_{(1)} \in \mathbb{C}^{n-\ell}$ and $z_{(2)} \in \mathbb{C}^\ell$. **Corollary 18**. *Let $H \subset \mathbb{H}_n$ be a connected subgroup. If $\pi_{\mathfrak{h}} : \mathfrak{h}_n \rightarrow \mathfrak{h}$ is the orthogonal projection for the natural inner product of $\mathfrak{h}_n = \mathbb{C}^n \times \mathbb{R}$, then the map $\mu^H = \pi_{\mathfrak{h}} \circ \mu^{\mathbb{H}_n} : D_{n+1} \rightarrow \mathfrak{h}$ is a moment map for the $H$-action on $D_{n+1}$. In particular, we have the following moment maps.* 1. *For the center $Z(\mathbb{H}_n)$: $$\mu^{Z(\mathbb{H}_n)}(z) = -\frac{1}{2(\operatorname{Im}(z_{n+1}) - |z'|^2)}.$$* 2. *For $H_{\mathbb{R}} = \mathbb{R}^{n+1}$: $$\mu^{H_{\mathbb{R}}}(z) = -\frac{(-4\operatorname{Im}(z'), 1)}{2(\operatorname{Im}(z_{n+1}) - |z'|^2)}.$$* 3. *For $H_{i\mathbb{R}} = i\mathbb{R}^n \times \mathbb{R}$: $$\mu^{H_{i\mathbb{R}}}(z) = -\frac{(4\operatorname{Re}(z'), 1)}{2(\operatorname{Im}(z_{n+1}) - |z'|^2)}.$$* 4. *For $1 \leq \ell \leq n-1$ and $H_{\ell,\mathbb{R}} = \mathbb{C}^{n-\ell} \times \mathbb{R}^\ell \times \mathbb{R}$: $$\mu^{H_{\ell,\mathbb{R}}}(z) = -\frac{\big(4z_{(1)}, -4\operatorname{Im}(z_{(2)}), 1\big)}% {2(\operatorname{Im}(z_{n+1}) - |z'|^2)}.$$* 5. *For $1 \leq \ell \leq n-1$ and $H_{\ell,i\mathbb{R}} = \mathbb{C}^{n-\ell} \times i\mathbb{R}^\ell \times \mathbb{R}$: $$\mu^{H_{\ell,i\mathbb{R}}}(z) = -\frac{\big(4z_{(1)}, 4\operatorname{Re}(z_{(2)}), 1\big)}% {2(\operatorname{Im}(z_{n+1}) - |z'|^2)}.$$* **Remark 19**. As a consequence of Corollary [Corollary 18](#cor:subgroups-moment-map){reference-type="ref" reference="cor:subgroups-moment-map"} we observe that for every connected Lie subgroup $H \subset \mathbb{H}_n$ there always exists a moment map for the $H$-action on $D_{n+1}$. # Toeplitz operators and moment maps {#sec:Toeplitz-moment} ## Moment map symbols We recall and apply to our particular setup the following notion already considered in [@QSMomentMapJFA]. **Definition 20**. Let $H \subset \mathbb{H}_n$ be a connected Lie subgroup with a moment map $\mu^H : D_{n+1} \rightarrow \mathfrak{h}$. A symbol $a \in L^\infty(D_{n+1})$ will be called a moment map symbol for the $H$-action or a $\mu^H$-symbol if there exists a measurable function $f$ defined on the image of $\mu^H$ such that $a = f \circ \mu^H$. **Remark 21**. Theorem [Theorem 15](#thm:Heisenberg-moment-map){reference-type="ref" reference="thm:Heisenberg-moment-map"} and Corollary [Corollary 18](#cor:subgroups-moment-map){reference-type="ref" reference="cor:subgroups-moment-map"} yield explicit formulas for moment maps for the Heisenberg group $\mathbb{H}_n$ and several of its subgroups. In the rest of this work, we will use the notation introduced in Corollary [Corollary 18](#cor:subgroups-moment-map){reference-type="ref" reference="cor:subgroups-moment-map"} for the subgroups considered therein. It is interesting to note that some simple coordinates changes in the target of moment maps allow to consider equivalent characterizations of the corresponding symbols. For example, a function $a \in L^\infty(D_{n+1})$ is a $\mu^{Z(\mathbb{H}_n)}$-symbol if and only if there is some essentially bounded measurable function $\widetilde{a}$ such that $a(z) = \widetilde{a}(\operatorname{Im}(z_{n+1}) - |z'|^2)$ for almost every $z \in D_{n+1}$. Similarly, such a function $a$ is a $\mu^{H_{\ell,\mathbb{R}}}$-symbol if and only if there is an essentially bounded measurable function $\widetilde{a}$ such that $a(z) = \widetilde{a}\big(z_{(1)}, \operatorname{Im}(z_{(2)}), \operatorname{Im}(z_{n+1}) - |z'|^2\big)$ for almost every $z \in D_{n+1}$. We will freely use this simplified alternative expressions whenever it is useful while being careful to clarify any non-trivial claim. An interesting feature of moment maps is that, in some cases, they satisfy invariance conditions stronger than those required in its definition. The next result will be an important example. **Proposition 22**. *The moment map $\mu^{Z(\mathbb{H}_n)}$ computed in Corollary [Corollary 18](#cor:subgroups-moment-map){reference-type="ref" reference="cor:subgroups-moment-map"} is $\mathbb{H}_n$-invariant. Furthermore, the sets of $\mathbb{H}_n$-orbits in $D_{n+1}$ and the level sets of the moment map $\mu^{Z(\mathbb{H}_n)}$ for the $\mathbb{H}_n$-action are the same. More precisely, for every $X \in \mu^{Z(\mathbb{H}_n)}(D_{n+1})$ there exists $z \in D_{n+1}$ such that $$\big(\mu^{Z(\mathbb{H}_n)}\big)^{-1}(X) = \mathbb{H}_n z,$$ the $\mathbb{H}_n$-orbit of $z$, and this exhaust the family of all $\mathbb{H}_n$-orbits. In particular, for any given function $a \in L^\infty(D_{n+1})$ the following conditions are equivalent.* 1. *$a$ is a $\mu^{Z(\mathbb{H}_n)}$-symbol.* 2. *$a$ is $\mathbb{H}_n$-invariant.* *Proof.* By the formula for $\mu^{Z({\mathbb{H}_n})}$ from Corollary [Corollary 18](#cor:subgroups-moment-map){reference-type="ref" reference="cor:subgroups-moment-map"} and by Proposition [Proposition 2](#prop:Hn-action){reference-type="ref" reference="prop:Hn-action"} its expression is $\mathbb{H}_n$-invariant, and so it follows that every level set of this function is a union of $\mathbb{H}_n$-orbits and that every $\mathbb{H}_n$-orbit is contained in a level set. Hence, it is enough to prove that every level set of $\mu^{Z(\mathbb{H}_n)}$ consists of a single $\mathbb{H}_n$-orbit. Let $z, w \in D_{n+1}$ be such that $\mu^{Z(\mathbb{H}_n)}(z) = \mu^{Z(\mathbb{H}_n)}(w)$, which is equivalent to $$\operatorname{Im}(z_{n+1}) - |z'|^2 = \operatorname{Im}(w_{n+1}) - |w'|^2.$$ If we choose $(\zeta,t) \in \mathbb{H}_n$ where $$\zeta = z' - w', \quad t = \operatorname{Re}(z_{n+1}) - \operatorname{Re}(w_{n+1}) + 2 \operatorname{Im}(w'\cdot \overline{(z' - w')}),$$ then we can compute as follows $$\begin{aligned} (\zeta,t)\cdot w =&\; \big(z', w_{n+1} + \operatorname{Re}(z_{n+1}) - \operatorname{Re}(w_{n+1}) + 2 \operatorname{Im}(w'\cdot \overline{(z' - w')}) \\ &+ 2i w'\cdot \overline{(z' - w')} + i |z' - w'|^2\big) \\ =&\; \big(z', \operatorname{Re}(z_{n+1}) + i\operatorname{Im}(w_{n+1}) + 2 \operatorname{Im}(w'\cdot \overline{z'}) \\ &+ 2i w'\cdot \overline{z'} - 2i|w'|^2 + i|z'|^2 + i|w'|^2 - 2i\operatorname{Re}(w'\cdot\overline{z'}) \big) \\ =&\; \big(z', \operatorname{Re}(z_{n+1}) + i(\operatorname{Im}(w_{n+1}) - |w'|^2) + i|z'|^2 \big) \\ =&\; \big(z', \operatorname{Re}(z_{n+1}) + i \operatorname{Im}(z_{n+1}) \big) = z, \end{aligned}$$ where we have used in the second to last line the identity that comes from the fact that $z$ and $w$ lie in the same level set. This completes the proof of our claim on level sets and orbits. The last part of the statement is now clear. ◻ **Remark 23**. The proof of Proposition [Proposition 22](#prop:Hn-orbits-moment-map-center){reference-type="ref" reference="prop:Hn-orbits-moment-map-center"} allows to improve the conclusions of Proposition [Proposition 2](#prop:Hn-action){reference-type="ref" reference="prop:Hn-action"}. From the former it follows that the $\mathbb{H}_n$-orbits in $\mathbb{C}^{n+1}$ are precisely the level sets of the map $z \mapsto \operatorname{Im}(z_{n+1}) - |z'|^2$ defined on $\mathbb{C}^{n+1}$. Since the $\mathbb{H}_n$-action clearly has no fixed points it follows that the group $\mathbb{H}_n$ can be identified with any such orbit. We also note that the $\mathbb{H}_n$-orbit defined by the condition $\operatorname{Im}(z_{n+1}) = |z'|^2$ is precisely the topological boundary of $D_{n+1}$. For this reason the Heisenberg group $\mathbb{H}_n$ is sometimes thought of as the topological boundary of the Siegel domain $D_{n+1}$. Through the examples considered above, we observe that there are two types of symbols that are useful. We fix a notation for them in the next definition. **Definition 24**. Let $H \subset \mathbb{H}_n$ be a connected Lie subgroup. We will denote by $L^\infty(D_{n+1})^H$ the space of $H$-invariant symbols and by $L^\infty(D_{n+1})^{\mu^H}$ the space of $\mu^H$-symbols, where $\mu^H$ is a moment map for the $H$-action on $D_{n+1}$. **Remark 25**. With the notation introduced in Definition [Definition 24](#def:two-types-symbols){reference-type="ref" reference="def:two-types-symbols"}, the second claim from Proposition [Proposition 22](#prop:Hn-orbits-moment-map-center){reference-type="ref" reference="prop:Hn-orbits-moment-map-center"} can be subsumed in the following expression $$L^\infty(D_{n+1})^{\mathbb{H}_n} = L^\infty(D_{n+1})^{\mu^{Z(\mathbb{H}_n)}}.$$ However, it is not known to the authors whether similar formulas hold for other subgroups $H \subset \mathbb{H}_n$. More precisely and within our setup, we can ask whether a given invariance condition is equivalent to being a function of a moment map (for a possible different group) and conversely. Such problems, although interesting, are beyond the scope of this work. ## Group-moment coordinates for $\mathbb{H}_n$ {#subsec:group-moment-coordinates} We will now introduce a set of coordinates on $D_{n+1}$ obtained from the $\mathbb{H}_n$-action and (a subset of) its moment map. This will turn out to be very useful in the description of Toeplitz operators with $\mathbb{H}_n$-invariant symbols acting on $\mathcal{A}^2_\lambda(D_{n+1})$. From now on, we will consider the function $H : D_{n+1} \rightarrow \mathbb{R}_+$ defined by $$H(z) = \frac{1}{\operatorname{Im}(z_{n+1}) - |z'|^2}.$$ It follows from Theorem [Theorem 15](#thm:Heisenberg-moment-map){reference-type="ref" reference="thm:Heisenberg-moment-map"} and Corollary [Corollary 18](#cor:subgroups-moment-map){reference-type="ref" reference="cor:subgroups-moment-map"} that $H$ is, up to a constant, the last coordinate of the moment map for the $\mathbb{H}_n$-action and the moment map of the $Z(\mathbb{H}_n)$-action. Since the function $H$ has as level sets the $\mathbb{H}_n$-orbits (see Proposition [Proposition 22](#prop:Hn-orbits-moment-map-center){reference-type="ref" reference="prop:Hn-orbits-moment-map-center"}), the values of $H$ and the $\mathbb{H}_n$-orbits can be considered as complementary values that may allow to build coordinates for $D_{n+1}$. To achieve this we will introduce maps $\sigma : \mathbb{R}_+ \rightarrow D_{n+1}$ and $\rho : D_{n+1} \rightarrow \mathbb{H}_n$, so that $\mathbb{R}_+$ can be seen as a coordinate submanifold of $D_{n+1}$ and so that the elements of $D_{n+1}$ have $\mathbb{H}_n$-coordinates. For this to hold, we impose the conditions $$H(\sigma(r)) = r, \quad z = \rho(z)\cdot \sigma(H(z)),$$ for all $r \in \mathbb{R}_+$ and $z \in D_{n+1}$, in order to make $H$ and $\sigma$ compatible as coordinates and involve the $\mathbb{H}_n$-action, respectively. It is easy to very that the functions defined by $$\sigma(r) = \Big(0',\frac{i}{r}\Big), \quad \rho(z) = (z',\operatorname{Re}(z_{n+1})),$$ satisfy the conditions required above. From now on, we will consider $\sigma$ and $\rho$ so defined. The next result shows that these functions can be put together to obtain a coordinate system for $D_{n+1}$. It turns out that the weighted measures on $D_{n+1}$ are particularly simple in these coordinates. **Proposition 26**. *The functions given by $$\begin{aligned} \kappa : \mathbb{H}_n \times \mathbb{R}_+ &\longrightarrow D_{n+1} & \tau : D_{n+1} &\longrightarrow \mathbb{H}_n \times \mathbb{R}_+ \\ \kappa(w',t,r) &= (w',t)\cdot \sigma(r) & \tau(z) &= (\rho(z), H(z)) \\ &= \Big(w', t + \frac{i}{r} + i|w'|^2\Big) & &= \Big(z', \operatorname{Re}(z_{n+1}), \frac{1}{\operatorname{Im}(z_{n+1}) - |z'|^2}\Big), \end{aligned}$$ are smooth and inverse of each other. In particular, they are both diffeomorphisms. Furthermore, the push-forward under $\tau$ of the weighted measure $v_\lambda$ is the measure on $\mathbb{H}_n \times \mathbb{R}_+$ given by $$\dif \nu_\lambda(w',t,r) = \frac{c_\lambda}{4r^{\lambda + 2}} \dif w' \dif t \dif r,$$ for every $\lambda > -1$.* *Proof.* A straightforward computation shows that both functions are indeed inverses of each other, and so they yield a set of coordinates. It is also easy to show using rectangular coordinates that we have $$|\det(\dif \tau_{(x,y)})| = \frac{1}{(y_{n+1} - |x'|^2 - |y'|^2)^2} = r^2,$$ where $z' = x' + iy'$ and $z_{n+1} = x_{n+1} + i y_{n+1}$. Note that for the last identity we have used the change of coordinates corresponding to $\kappa$ and $\tau$. We conclude that the Lebesgue measure $\dif z$ on $D_{n+1}$ in the coordinates $(w',t,r)$ is given by $$\begin{aligned} \dif z &= |\det(\dif \kappa_{(w',t,r)})| \dif w' \dif t \dif r \\ &= |\det(\dif \tau_{\kappa(w',t,r)})|^{-1} \dif w' \dif t \dif r = \frac{1}{r^2} \dif w' \dif t \dif r. \end{aligned}$$ On the other hand, in terms of the coordinates $(w',t,r)$, the weight applied to the Lebesgue measure to obtain $v_\lambda$ is given by $$\frac{c_\lambda}{4} (\operatorname{Im}(z_{n+1}) - |z'|^2)^\lambda = \frac{c_\lambda}{4r^\lambda}.$$ Hence, the formula in the statement is now clear. ◻ **Remark 27**. The coordinates on $D_{n+1}$ given by $\kappa$ and $\tau$ in Proposition [Proposition 26](#prop:group-moment-coordinates){reference-type="ref" reference="prop:group-moment-coordinates"} are particularly well-suited to the $\mathbb{H}_n$-action and a moment map of the $Z(\mathbb{H}_n)$-action. By its very definition, the map $\kappa$ is $\mathbb{H}_n$-equivariant for the action on the domain on the first factor only and the last coordinate of $\tau$ is basically a moment map for the $Z(\mathbb{H}_n)$-action, the center of $\mathbb{H}_n$. For this reason, in the rest of these work we will refer to the coordinates defined by $\kappa$ and $\tau$ as the **group-moment coordinates** associated to the $\mathbb{H}_n$-action on $D_{n+1}$. From now on and with the notation of Proposition [Proposition 26](#prop:group-moment-coordinates){reference-type="ref" reference="prop:group-moment-coordinates"}, we will denote by $U_0$ the map given by the assignment $$f \longmapsto f\circ \kappa,$$ that takes complex-valued functions defined on $D_{n+1}$ onto corresponding ones defined on $\mathbb{H}_n \times \mathbb{R}_+$. In other words, $U_0$ is the change of coordinates, from $D_{n+1}$ to $\mathbb{H}_n \times \mathbb{R}_+$, corresponding to the group-moment coordinates associated to $\mathbb{H}_n$. We observe that this assignment does not require any further assumption on the functions involved and that its inverse $U_0^{-1}$ is given by $$f \longmapsto f\circ \tau.$$ Both maps have some important properties. Firstly, they map smooth functions onto smooth functions. Secondly, the restriction of $U_0$ to $v_\lambda$-square-integrable functions yields a unitary map $L^2(D_{n+1}, v_\lambda) \rightarrow L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda)$ that we will denote with the same symbol. Both of these claims are consequence of Proposition [Proposition 26](#prop:group-moment-coordinates){reference-type="ref" reference="prop:group-moment-coordinates"}. The next result computes the Cauchy-Riemann equations on $D_{n+1}$ with respect to the group-moment coordinates associated to $\mathbb{H}_n$. **Lemma 28**. *For a smooth function $\varphi : \mathbb{H}_n \times \mathbb{R}_+ \rightarrow \mathbb{C}$, and where $\mathbb{H}_n \times \mathbb{R}_+$ has coordinates denoted by $(w',t,r)$, the following sets of equations are equivalent.* 1. *For every $j = 1, \dots, n+1$: $$U_0 \frac{\partial}{\partial \overline{z}_j} U_0^{-1} \varphi = 0.$$* 2. *For every $j = 1, \dots, n$: $$\bigg(\frac{\partial}{\partial \overline{w}_j} + w_j r^2 \frac{\partial}{\partial r}\bigg) \varphi = 0$$ and $$\bigg(i r^2 \frac{\partial}{\partial r} - \frac{\partial}{\partial t} \bigg) \varphi = 0.$$* *Proof.* Let us consider $\mathbb{H}_n \times \mathbb{R}_+$ as an open subset of $\mathbb{C}^{n+1}$ through the assignment $$(w',t,r) \longmapsto (w', t+ir).$$ Then, a straightforward application of the chain rule for the Wirtinger partial derivatives allows us to obtain the following formulas $$\begin{aligned} \bigg(\frac{\partial}{\partial \overline{z}_j} U_0^{-1} \varphi\bigg)(z) =&\; \frac{\partial \varphi}{\partial \overline{w}_j}(\tau(z)) \\ &+ \frac{iz_j}{(\operatorname{Im}(z_{n+1})-|z'|^2)^2} \bigg( \frac{\partial \varphi}{\partial w_{n+1}}(\tau(z)) - \frac{\partial \varphi}{\partial \overline{w}_{n+1}} (\tau(z)) \bigg) \\ \bigg(\frac{\partial}{\partial \overline{z}_{n+1}} U_0^{-1} \varphi\bigg)(z) =&\; \frac{1}{2}\bigg( \frac{\partial \varphi}{\partial w_{n+1}}(\tau(z)) + \frac{\partial \varphi}{\partial \overline{w}_{n+1}}(\tau(z)) \bigg) \\ &+ \frac{1}{2(\operatorname{Im}(z_{n+1})-|z'|^2)^2}\bigg( \frac{\partial \varphi}{\partial w_{n+1}}(\tau(z)) - \frac{\partial \varphi}{\partial \overline{w}_{n+1}}(\tau(z)) \bigg), \end{aligned}$$ where $j = 1, \dots, n$. The equivalence is now obtained applying $U_0$. ◻ We now consider the usual Fourier transform $$\mathfrak{F}(f)(\xi) = \frac{1}{\sqrt{2\pi}} \int_\mathbb{R}f(t) e^{-i\xi t} \dif t,$$ and define the unitary map $U_1 = I \otimes \mathfrak{F}\otimes I$ acting on $L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda)$ that applies the transform $\mathfrak{F}$ on the variable $t$. We recall that $\mathbb{H}_n \times \mathbb{R}_+$ has coordinates denoted by $(w',t,r)$. Then, by the usual properties of the Fourier transform the set of equations from Lemma [Lemma 28](#lem:CR-U0){reference-type="ref" reference="lem:CR-U0"} are equivalent to $$\begin{aligned} \bigg(\frac{\partial}{\partial \overline{w}_j} + w_j r^2 \frac{\partial}{\partial r}\bigg) \varphi &= 0 \quad j=1, \dots, n, \label{eq:CR-U1U0} \\ \bigg(r^2 \frac{\partial}{\partial r} -\xi\bigg) \varphi &= 0. \notag\end{aligned}$$ We now describe the general solution of these equations. **Lemma 29**. *A function $\varphi : \mathbb{H}_n \times \mathbb{R}_+ \rightarrow \mathbb{C}$ satisfies equations [\[eq:CR-U1U0\]](#eq:CR-U1U0){reference-type="eqref" reference="eq:CR-U1U0"} if and only if there is a function $\psi : \mathbb{C}^n \times \mathbb{R}\rightarrow \mathbb{C}$ holomorphic in its variable $w' \in \mathbb{C}^n$ such that $$\varphi(w',\xi,r) = e^{-\xi|w'|^2} e^{-\frac{\xi}{r}} \psi(w',\xi),$$ for every $(w',\xi,r) \in \mathbb{H}_n \times \mathbb{R}_+$.* *Proof.* If $\varphi$ has the expression in terms of $\psi$ as in the statement, then by Leibniz rule we have $$\begin{aligned} \frac{\partial \varphi}{\partial \overline{w}_j}(w',\xi,r) &= -\xi w_j \varphi(w',\xi,r) \\ \frac{\partial \varphi}{\partial r}(w',\xi,r) &= \frac{\xi}{r^2} \varphi(w',\xi,r), \end{aligned}$$ where $j = 1, \dots, n$. And these are clearly equivalent to equations [\[eq:CR-U1U0\]](#eq:CR-U1U0){reference-type="eqref" reference="eq:CR-U1U0"}. Conversely, let us assume that $\varphi$ is a solution to [\[eq:CR-U1U0\]](#eq:CR-U1U0){reference-type="eqref" reference="eq:CR-U1U0"} and let us consider the function $\psi : \mathbb{H}_n \times \mathbb{R}\rightarrow \mathbb{C}$ given by $$\psi(w',\xi,r) = \varphi(w',\xi,r) e^{\xi|w'|^2} e^{\frac{\xi}{r}}.$$ It is enough to prove that $\psi$ is independent of $r$ and holomorphic in $w'$. For this we compute $$\frac{\partial \psi}{\partial r}(w',\xi,r) = \frac{\partial \varphi}{\partial r}(w',\xi,r) e^{\xi|w'|^2} e^{\frac{\xi}{r}} - \frac{\xi}{r^2} \varphi(w',\xi,r) e^{\xi|w'|^2} e^{\frac{\xi}{r}} = 0,$$ which vanishes because of the last equation from [\[eq:CR-U1U0\]](#eq:CR-U1U0){reference-type="eqref" reference="eq:CR-U1U0"}. We also have $$\begin{aligned} \frac{\partial \psi}{\partial \overline{w}_j}(w',\xi,r) &= \frac{\partial \varphi}{\partial \overline{w}_j}(w',\xi,r) e^{\xi|w'|^2} e^{\frac{\xi}{r}} + \xi w_j \varphi(w',\xi,r) e^{\xi|w'|^2} e^{\frac{\xi}{r}} \\ &= -w_j r^2 \frac{\partial \varphi}{\partial r}(w',\xi,r) e^{\xi|w'|^2} e^{\frac{\xi}{r}} + \xi w_j \varphi(w',\xi,r) e^{\xi|w'|^2} e^{\frac{\xi}{r}} = 0, \end{aligned}$$ where we have used again equations [\[eq:CR-U1U0\]](#eq:CR-U1U0){reference-type="eqref" reference="eq:CR-U1U0"}. ◻ **Remark 30**. In this subsection we have introduced two important constructions. Firstly, the change of variable given by the group-moment coordinates. Secondly, a Fourier transform on the (last) real variable of the Heisenberg group. These lead us to obtain a characterization of holomorphic functions on $D_{n+1}$ as suitable functions on $\mathbb{H}_n \times \mathbb{R}_+$. Lemma [Lemma 29](#lem:CR-U1){reference-type="ref" reference="lem:CR-U1"} yields the main result with this respect. ## Toeplitz operators with $\mathbb{H}_n$-invariant symbols {#subsec:Toeplitz-HHn-invariant} We will now consider Toeplitz operators whose symbols are $\mathbb{H}_n$-invariant. Recall that Proposition [Proposition 22](#prop:Hn-orbits-moment-map-center){reference-type="ref" reference="prop:Hn-orbits-moment-map-center"} shows that such symbols may equivalently be described as $\mu^{\mathbb{Z}(\mathbb{H}_n)}$-symbols (see also Remark [Remark 25](#rmk:two-types-symbols){reference-type="ref" reference="rmk:two-types-symbols"}). Building from the results in the previous subsection, we need to consider the square-integrability of the solutions to the equations [\[eq:CR-U1U0\]](#eq:CR-U1U0){reference-type="eqref" reference="eq:CR-U1U0"} obtained in Lemma [Lemma 29](#lem:CR-U1){reference-type="ref" reference="lem:CR-U1"}. To achieve this, we introduce a functional renormalization in the next result. **Lemma 31**. *Let us consider the measure given by $$\dif \eta (w',\xi) = \bigg(\frac{2\xi}{\pi}\bigg)^n e^{-2\xi|w'|^2}\dif w' \dif \xi,$$ on $\mathbb{C}^n \times \mathbb{R}_+$ and the measure $\nu_\lambda$ on $\mathbb{H}_n \times \mathbb{R}_+$ defined in Proposition [Proposition 26](#prop:group-moment-coordinates){reference-type="ref" reference="prop:group-moment-coordinates"}. Then, the map $$\begin{aligned} V_\lambda : L^2(\mathbb{C}^n \times \mathbb{R}_+, \eta) &\longrightarrow L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda) \\ (V_\lambda\psi)(w',\xi,r) &= 2\sqrt{\frac{\pi(2\xi)^{\lambda+n+1}}% {\Gamma(\lambda + n + 2)}} e^{-\xi|w'|^2} e^{-\frac{\xi}{r}} \chi_{\mathbb{R}_+}(\xi) \psi(w',\xi), \end{aligned}$$ is a well-defined isometry whose adjoint operator is given by $$(V_\lambda^*\varphi)(w',\xi) = \frac{\sqrt{\pi(2\xi)^{\lambda-n+1}\Gamma(\lambda+n+2)}}% {2\pi\Gamma(\lambda + 1)} e^{\xi|w'|^2} \int_0^\infty \frac{\varphi(w',\xi,r) e^{-\frac{\xi}{r}} \dif r}{r^{\lambda+2}}.$$* *Proof.* For a given $\psi \in L^2(\mathbb{C}^n \times \mathbb{R}_+, \eta)$ we compute as follows. $$\begin{aligned} \|V_\lambda\psi\|^2 =&\; \int_{\mathbb{C}^n \times \mathbb{R}_+^2} \frac{4\pi (2\xi)^{\lambda+n+1}}% {\Gamma(\lambda + n + 2)} e^{-2\xi|w'|^2} e^{-\frac{2\xi}{r}} |\psi(w',\xi)|^2 \frac{c_\lambda}{4r^{\lambda + 2}} \dif w' \dif \xi \dif r \\ =&\; \frac{\pi c_\lambda}% {\Gamma(\lambda + n + 2)} \int_{\mathbb{C}^n \times \mathbb{R}_+} (2\xi)^{\lambda+n+1} e^{-2\xi|w'|^2} |\psi(w',\xi)|^2 \times \\ &\times \bigg(\int_0^\infty e^{-\frac{2\xi}{r}} \frac{1}{r^{\lambda + 2}} \dif r \bigg) \dif w' \dif \xi \\ =&\; \frac{1}{\pi^n \Gamma(\lambda + 1)} \int_{\mathbb{C}^n \times \mathbb{R}_+} (2\xi)^{\lambda+n+1} e^{-2\xi|w'|^2} |\psi(w',\xi)|^2 \frac{\Gamma(\lambda + 1)}{(2\xi)^{\lambda + 1}} \dif w' \dif \xi \\ =&\; \int_0^\infty \bigg( \int_{\mathbb{C}^n} |\psi(w',\xi)|^2 \bigg(\frac{2\xi}{\pi}\bigg)^n e^{-2\xi|w'|^2} \dif w' \bigg) \dif \xi, \end{aligned}$$ and so the first claim has been proved. If we now also choose $\varphi \in L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda)$, then we have $$\begin{aligned} \langle V_\lambda \psi, &\varphi \rangle = \\ =&\; \int_{\mathbb{C}^n \times \mathbb{R}_+^2} 2\sqrt{ \frac{\pi(2\xi)^{\lambda+n+1}}% {\Gamma(\lambda+n+2)} } e^{-\xi|w'|^2} e^{-\frac{\xi}{r}} \psi(w',\xi) \overline{\varphi(w',\xi,r)} \frac{c_\lambda}{4r^{\lambda + 2}} \dif w' \dif \xi \dif r \\ =&\; \int_{\mathbb{C}^n \times \mathbb{R}_+} \psi(w',\xi) \bigg( \sqrt{ \frac{\pi(2\xi)^{\lambda+n+1}}{\Gamma(\lambda+n+2)} } e^{\xi|w'|^2} \int_0^\infty \frac{\overline{\varphi(w',\xi,r)} e^{-\frac{\xi}{r}} \dif r}{r^{\lambda+2}} \bigg) \times \\ &\times \frac{\Gamma(\lambda + n + 2)}{2\pi^{n+1}\Gamma(\lambda + 1)} e^{-2\xi|w'|^2} \dif w' \dif \xi \\ =&\; \int_{\mathbb{C}^n \times \mathbb{R}_+} \psi(w',\xi) \bigg( \frac{\sqrt{\pi(2\xi)^{\lambda-n+1}\Gamma(\lambda+n+2)}}% {2\pi\Gamma(\lambda + 1)} e^{\xi|w'|^2} \int_0^\infty \frac{\overline{\varphi(w',\xi,r)} e^{-\frac{\xi}{r}} \dif r}{r^{\lambda+2}} \bigg) \times \\ &\times \bigg(\frac{2\xi}{\pi}\bigg)^n e^{-2\xi|w'|^2} \dif w' \dif \xi, \\ \end{aligned}$$ which yields the required expression for the adjoint operator $V_\lambda^*$. ◻ **Remark 32**. We observe that the map $V_\lambda$ from Lemma [Lemma 31](#lem:Vlambda){reference-type="ref" reference="lem:Vlambda"} is defined using functions of the form considered in Lemma [Lemma 29](#lem:CR-U1){reference-type="ref" reference="lem:CR-U1"}, except for the holomorphicity condition on the variable $w' \in \mathbb{C}^n$. In other words, Lemma [Lemma 31](#lem:Vlambda){reference-type="ref" reference="lem:Vlambda"} and the map $V_\lambda$ take into account only the last equation from [\[eq:CR-U1U0\]](#eq:CR-U1U0){reference-type="eqref" reference="eq:CR-U1U0"}. The role of Lemma [Lemma 31](#lem:Vlambda){reference-type="ref" reference="lem:Vlambda"} is to renormalize the functions with respect to the variable $\xi$ to obtain square-integrable functions and an isometric assignment from them. The next step is to consider the holomorphicity condition on the variable $w' \in \mathbb{C}^n$. For this we will make use of direct integrals of Hilbert spaces, specifically weighted Fock spaces. For the basic properties and definitions associated to the former we refer to [@KRvolII]. As for the latter, $\mathcal{F}^2_\xi(\mathbb{C}^n)$ will denote the Fock space over $\mathbb{C}^n$ with weight $\xi > 0$, which is defined as the (closed) subspace of holomorphic functions that belong to $L^2(\mathbb{C}^n, (\xi/\pi)^n e^{-\xi|w'|^2} \dif w')$. **Lemma 33**. *The map $$W_\lambda : \int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi \longrightarrow L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda),$$ defined as the restriction of $V_\lambda$ given in Lemma [Lemma 31](#lem:Vlambda){reference-type="ref" reference="lem:Vlambda"} (to the direct integral of the family $(\mathcal{F}^2_{2\xi}(\mathbb{C}^n))_{\xi \in \mathbb{R}_+}$ over $\mathbb{R}_+$ with the Lebesgue measure) is an isometry with image $\big(U_1 U_0 \big)\big(\mathcal{A}^2_\lambda(D_{n+1})\big)$.* *Proof.* We observe that the subspace of functions $\psi \in L^2(\mathbb{C}^n \times \mathbb{R}_+, \eta)$ holomorphic in the variable $w' \in \mathbb{C}^n$ is a closed subspace and (the) direct integral of the family $(\mathcal{F}^2_{2\xi}(\mathbb{C}^n))_{\xi \in \mathbb{R}_+}$ over $\mathbb{R}_+$ with the Lebesgue measure. To see this, we note first that for such a function $\psi$, Fubini's theorem yields $$\int_{\mathbb{R}_+} \bigg( \int_{\mathbb{C}^n} |\psi(w',\xi)|^2 \bigg(\frac{2\xi}{\pi}\bigg)^n e^{-2\xi|w'|^2} \dif w' \bigg) \dif \xi = \int_{\mathbb{C}^n \times \mathbb{R}_+} |\psi|^2 \dif \eta < \infty$$ which implies that $\psi(\cdot,\xi) \in \mathcal{F}^2_{2\xi}(\mathbb{C}^n)$, for almost every $\xi \in \mathbb{R}_+$. This same formula and elementary convergence theorems can be used to prove that the space of functions $\psi$, as described above, is closed. It is also a simple exercise to prove that such space is in fact the direct integral claimed to be. On the other hand, the (closed) subspace $\big(U_1 U_0 \big)\big(\mathcal{A}^2_\lambda(D_{n+1})\big)$ of the Hilbert space $L^2(\mathbb{H}_n\times \mathbb{R}_+, \nu_\lambda)$ consists precisely of functions on $\mathbb{H}_n \times \mathbb{R}_+$ of the form $$(w',\xi,r) \longmapsto e^{-\xi|w'|^2} e^{-\frac{\xi}{r}} \psi(w',\xi)$$ such that $\psi$ is holomorphic in $w' \in \mathbb{C}^n$ and square-integrable with respect to $\eta$. This claim is a consequence of the definitions of $U_0$ and $U_1$ as well as Lemmas [Lemma 28](#lem:CR-U0){reference-type="ref" reference="lem:CR-U0"} and [Lemma 29](#lem:CR-U1){reference-type="ref" reference="lem:CR-U1"}. Also note that the computations from Lemma [Lemma 31](#lem:Vlambda){reference-type="ref" reference="lem:Vlambda"} show that one must restrict those functions in the variable $\xi$ from $\mathbb{R}$ to $\mathbb{R}_+$ to ensure the square-integrability. The operator $V_\lambda$ from Lemma [Lemma 31](#lem:Vlambda){reference-type="ref" reference="lem:Vlambda"} has considered such restriction in its definition. Hence, $V_\lambda$ yields the same family of functions since it just introduces a renormalization on the variable $\xi$. The only condition not considered by $V_\lambda$ is the holomorphicity in the variable $w' \in \mathbb{C}^n$, which is however achieved by $W_\lambda$. Hence we conclude that indeed the image of $W_\lambda$ is $\big(U_1 U_0 \big)\big(\mathcal{A}^2_\lambda(D_{n+1})\big)$. This discussion can be summarized in the following diagram $$\xymatrix{ L^2(D_{n+1},v_\lambda) \ar[rrr]^{U_1 U_0} &&& L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda) & L^2(\mathbb{C}^n \times \mathbb{R}_+, \eta) \ar[l]_{V_\lambda} \\ \mathcal{A}^2_\lambda(D_{n+1}) \ar[u] \ar[rrr]^{U_1 U_0|_{\mathcal{A}^2_\lambda(D_{n+1})}} &&& L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda) \ar@{=}[u] & \displaystyle\int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi \ar[u] \ar[l]_{W_\lambda} }$$ where the extreme left and right vertical arrows are inclusions and the horizontal arrows in the second row are the restriction of the corresponding ones in the first row. We have proved above that in the diagram the operators $U_1 U_0|_{\mathcal{A}^2_\lambda(D_{n+1})}$ and $W_\lambda$ have the same image. Finally, $W_\lambda$ is the restriction of an isometry and so it is itself an isometry. ◻ The next result provides a description of the weighted Bergman spaces that will allow us to diagonalize Toeplitz operator with $\mathbb{H}_n$-invariant symbols. **Theorem 34**. *With the operators $U_0$, $U_1$ and $W_\lambda$ considered above, let us define the operator $$R_\lambda = W_\lambda^* U_1 U_0 : L^2(D_{n+1}, v_\lambda) \longrightarrow \int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi.$$ Then, $R_\lambda$ is a surjective partial isometry with initial space $\mathcal{A}^2_\lambda(D_{n+1})$. In other words, we have $$R_\lambda R_\lambda^* = I, \quad R_\lambda^* R_\lambda = B_{n+1,\lambda},$$ the identity on the direct integral above and the Bergman projection from $L^2(D_{n+1},v_\lambda)$ onto $\mathcal{A}^2_\lambda(D_{n+1})$, respectively. In particular, the restriction $$U_\lambda = R_\lambda|_{\mathcal{A}^2_\lambda(D_{n+1})} : \mathcal{A}^2_\lambda(D_{n+1}) \longrightarrow \int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi,$$ is a unitary map.* *Proof.* From Lemma [Lemma 33](#lem:Wlambda-directint-Fock){reference-type="ref" reference="lem:Wlambda-directint-Fock"} we know that $W_\lambda$ is an isometry and so it is a partial isometry with initial space its domain and final space its image. Hence, $W_\lambda^*$ is a surjective partial isometry and $R_\lambda$ is a surjective partial isometry as well. Note that the latter claim follows from the fact that $U_0$ and $U_1$ are unitary maps. The surjectivity of $R_\lambda$ shows that its final space is the whole target, and this implies that $R_\lambda R_\lambda^* = I$, the identity on the direct integral from the statement. Let us now consider the following maps $$\xymatrix{ L^2(D_{n+1},v_\lambda) \ar[rr]^{U_1 U_0} & & L^2(\mathbb{H}_n \times \mathbb{R}_+,\nu_\lambda) \ar@/_/[rr]_{W_\lambda^*} & & \displaystyle\int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi \ar@/_/[ll]_{W_\lambda} }.$$ From this diagram and Lemma [Lemma 33](#lem:Wlambda-directint-Fock){reference-type="ref" reference="lem:Wlambda-directint-Fock"} we conclude that $R_\lambda$ restricted to $\mathcal{A}^2_\lambda(D_{n+1})$ is a unitary map onto the target direct integral. This implies that the initial space of $R_\lambda$ as a partial isometry is precisely $\mathcal{A}^2_\lambda(D_{n+1})$, and so we conclude that $R_\lambda^* R_\lambda = B_{n+1,\lambda}$, the projection onto such initial space. In particular, $U_\lambda$ is a unitary map. ◻ We now obtain the diagonalization of Toeplitz operators with $\mathbb{H}_n$-invariant symbols using the partial isometries constructed above from the group-moment coordinates. **Theorem 35**. *Let us consider $a \in L^\infty(D_{n+1})$ an $\mathbb{H}_n$-invariant symbol and let $\widetilde{a} \in L^\infty(\mathbb{R}_+)$ be a function such that $a(z) = \widetilde{a}(\operatorname{Im}(z_{n+1}) - |z'|^2)$, for almost every $z \in D_{n+1}$. Then, for the unitary map $U_\lambda$ defined in Theorem [Theorem 34](#thm:Rlambda-Ulambda){reference-type="ref" reference="thm:Rlambda-Ulambda"} we have $$U_\lambda T^{(\lambda)}_a U_\lambda^* = M_{\gamma_{\widetilde{a},\lambda}},$$ where $M_{\gamma_{\widetilde{a},\lambda}}$ is the multiplier operator acting on the direct integral of the family of Fock spaces $(\mathcal{F}^2_{2\xi}(\mathbb{C}^n))_{\xi \in \mathbb{R}_+}$ over $\mathbb{R}_+$ with the Lebesgue measure and $\gamma_{\widetilde{a},\lambda} \in L^\infty(\mathbb{R}_+)$ is the function given by $$\label{eq:gamma-for-Heisenberg} \gamma_{\widetilde{a},\lambda}(\xi) = \frac{(2\xi)^{\lambda+1}}{\Gamma(\lambda +1)} \int_0^\infty\widetilde{a}(r) e^{-2\xi r} r^\lambda \dif r = \frac{\xi^{\lambda+1}}{\Gamma(\lambda +1)} \int_0^\infty\widetilde{a}\bigg(\frac{s}{2}\bigg) e^{-\xi s} s^\lambda \dif s,$$ for almost every $\xi \in \mathbb{R}_+$.* *Proof.* From the properties established in Theorem [Theorem 34](#thm:Rlambda-Ulambda){reference-type="ref" reference="thm:Rlambda-Ulambda"} we obtain $$\begin{aligned} U_\lambda T^{(\lambda)}_a U_\lambda^* &= R_\lambda B_{n+1,\lambda} M_a B_{n+1,\lambda} R_\lambda^* = R_\lambda R_\lambda^* R_\lambda M_a R_\lambda^* R_\lambda R_\lambda^* \\ &= R_\lambda M_a R_\lambda^* = W_\lambda^* U_1 U_0 M_a U_0^* U_1^* W_\lambda. \end{aligned}$$ We observe that the definition of $U_0$ yields $$U_0 M_a U_0^* = M_{a \circ \kappa},$$ acting on $L^2(\mathbb{H}_n \times \mathbb{R}_+, \nu_\lambda)$ and we can compute $$a \circ \kappa(w',t,r) = a((w',t)\cdot \sigma(r)) = a(\sigma(r)) = a(0',ir^{-1}) = \widetilde{a}(r^{-1}),$$ for almost every $(w',t,r) \in \mathbb{H}_n \times \mathbb{R}_+$. We have used the $\mathbb{H}_n$-invariance of $a$ in the second identity. Note that the latter function does not depend on the variables $(w',t) \in \mathbb{H}_n$ and so the multiplier operator $M_{a \circ \kappa} = M_{\widetilde{a}}$ commutes with $U_1 = I \otimes \mathfrak{F}\otimes I$. The reason is that the Fourier transform is applied in the variable $t$ to obtain a new variable that we have denoted by $\xi$. Hence, we arrive to $$U_\lambda T^{(\lambda)}_a U_\lambda^* = W_\lambda^* M_{\widetilde{a}} W_\lambda,$$ that we now proceed to compute using Lemmas [Lemma 31](#lem:Vlambda){reference-type="ref" reference="lem:Vlambda"} and [Lemma 33](#lem:Wlambda-directint-Fock){reference-type="ref" reference="lem:Wlambda-directint-Fock"}. For any given $\psi$ in the direct integral of the family $(\mathcal{F}^2_{2\xi}(\mathbb{C}^n))_{\xi \in \mathbb{R}_+}$ over $\mathbb{R}_+$ with the Lebesgue measure we have for every $(w',\xi) \in \mathbb{C}^n \times \mathbb{R}_+$ $$\begin{aligned} (W_\lambda^* &M_{\widetilde{a}} W_\lambda)\psi(w',\xi) = \\ =&\; \frac{\sqrt{\pi(2\xi)^{\lambda-n+1}\Gamma(\lambda+n+2)}}% {2\pi\Gamma(\lambda + 1)} e^{\xi|w'|^2} \times \\ &\times \int_{\mathbb{R}_+} \widetilde{a}(r^{-1}) 2\sqrt{\frac{\pi(2\xi)^{\lambda+n+1}}% {\Gamma(\lambda + n + 2)}} e^{-\xi|w'|^2} e^{-\frac{\xi}{r}} \chi_{\mathbb{R}_+}(\xi) \psi(w',\xi) \frac{e^{-\frac{\xi}{r}} \dif r}{r^{\lambda+2}} \\ =&\; \frac{(2\xi)^{\lambda + 1}}{\Gamma(\lambda+1)} \bigg( \int_{\mathbb{R}_+} \frac{\widetilde{a}(r^{-1}) e^{-\frac{2\xi}{r}}}% {r^{\lambda+2}} \dif r \bigg) \psi(w',\xi), \end{aligned}$$ and the result now follows from some simple changes of variable. ◻ From now on, if $\mathcal{S}\subset L^\infty(D_{n+1})$ is a family of symbols, then we will denote by $\mathcal{T}^{(\lambda)}(\mathcal{S})$ the $C^*$-algebra generated by Toeplitz operators, acting on $\mathcal{A}^2_\lambda(D_{n+1})$ where $\lambda > -1$, with symbols belonging to $\mathcal{S}$. Let us now apply Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} to describe the structure of the $C^*$-algebra generated by Toeplitz operators with $\mathbb{H}_n$-invariant symbols. **Theorem 36**. *The $C^*$-algebra $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ is commutative. Furthermore, there exists a unitary map $$U_\lambda : \mathcal{A}^2_\lambda(D_{n+1}) \longrightarrow \int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi,$$ that satisfies $$U_\lambda \{ T^{(\lambda)}_a : a \in L^\infty(D_{n+1})^{\mathbb{H}_n} \} U_\lambda^* = \{ M_{\gamma_{\widetilde{a},\lambda}} : \widetilde{a} \in L^\infty(\mathbb{R}_+) \},$$ where $\gamma_{\widetilde{a},\lambda}$ is defined by equation [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}. Hence, $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ is isomorphic to the $C^*$-subalgebra of $L^\infty(\mathbb{R}_+)$ generated by the functions $\gamma_{\widetilde{a},\lambda}$ given by equation [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}, where $\widetilde{a} \in L^\infty(\mathbb{R}_+)$. In particular, the isomorphism class of $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ does not depend on the dimension of the Siegel domain $D_{n+1}$.* *Proof.* The relationship between the Toeplitz operators operators $T^{(\lambda)}_a$, with $a$ an $\mathbb{H}_n$-invariant symbol, and the multiplier operators $M_{\gamma_{\widetilde{a},\lambda}}$, with $\gamma_{\widetilde{a},\lambda}$ as in equation [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}, has already been established in Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} for the unitary map $U_\lambda$ from its statement. From this it follows immediately that $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ is commutative. For the same unitary map $U_\lambda$, this also proves that $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ corresponds to a $C^*$-subalgebra of the von Neumann algebra of diagonalizable operators for the direct integral decomposition in the statement. Such von Neumann algebra is canonically isomorphic to $L^\infty(\mathbb{R}_+)$ through the map $\gamma \mapsto M_\gamma$ since the direct integral is taken over $\mathbb{R}_+$ (see [@KRvolII]). Within this setup, Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} implies that $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ is isomorphic to the $C^*$-subalgebra of $L^\infty(\mathbb{R}_+)$ generated by the functions $\gamma_{\widetilde{a},\lambda}$ is defined by equation [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}. This completes the proof of the theorem. ◻ **Remark 37**. We recall that our results are stated for $D_{n+1}$ and that we have assumed $n \geq 1$, so that the lowest dimensional Siegel domain considered in this work is $D_2$ with the action of the ($3$-dimensional) Heisenberg group $\mathbb{H}_1$. In particular, we have not considered the upper half-plane given by $D_1 = \{z \in \mathbb{C}: \operatorname{Im}(z) > 0\}$ with the action of $\mathbb{R}$ by the horizontal translations $z \mapsto z+t$, defined for $t \in \mathbb{R}$. This $1$-dimensional case has already been studied in [@GKVParabolic; @HMV2013Vertical; @HHM2014VerticalWeighted], listed in chronological order. Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"}, and a straightforward comparison with these references (see for example [@GKVParabolic Theorem 2.5]), shows that the exact same family of functions $\gamma_{\widetilde{a},\lambda}$ given by equation [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"} is obtained for all dimensions, including the $1$-dimensional case of the $\mathbb{R}$-action on $D_1$. The previous observations lead to a number of important conclusions. Firstly, from the viewpoint of $C^*$-algebras generated by Toeplitz operators, the most natural generalization of the $\mathbb{R}$-action on the upper half-plane $D_1$ is the action of the Heisenberg group $\mathbb{H}_n$ on $D_{n+1}$. The reason is that in all such cases, and by considering Toeplitz operators with invariant symbols, we generate the same $C^*$-algebra up to isomorphism. With this respect, a similar situation was observed in [@BHVRadial2014; @GMVRadial] for the case of the $C^*$-algebra (weighted case and weightless case, respectively) generated by radial Toeplitz operators acting on the $n$-dimensional ball. It was proved on those references that such $C^*$-algebra is independent of $n$. However, in our case we have proved that not only is the $C^*$-algebra $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ independent of $n$, up to isomorphism, but we have also proved that their natural generators, the Toeplitz operators which correspond to the multiplier operators given by [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}, do not actually depend on $n$ either. To the best of our knowledge, such result has not been achieved for radial Toeplitz operators on the unit ball. Secondly, thanks to the functions obtained in Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"}, the currently known structure of the $C^*$-algebras involved for the $1$-dimensional case (see [@HHM2014VerticalWeighted]) allows us to obtain results for arbitrary dimension. We are able to provide below a precise description of the $C^*$-algebra $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ for every $n \geq 1$. We recall the notion of continuity that allow us to describe the $C^*$-algebras obtained from $\mathbb{H}_n$-invariant symbols. **Definition 38**. A function $f : \mathbb{R}_+ \rightarrow \mathbb{C}$ is called very slowly oscillating if it is uniformly continuous for the logarithmic metric $d(x,y) = |\log(x) - \log(y)|$ on its domain $\mathbb{R}_+$. The set of all very slowly oscillating functions is denoted by $\mathrm{VSO}(\mathbb{R}_+)$. It is a well known fact that $\mathrm{VSO}(\mathbb{R}_+)$ is $C^*$-subalgebra of $C_b(\mathbb{R}_+)$. It turns out that the $C^*$-algebra $\mathrm{VSO}(\mathbb{R}_+)$ yields the description of the $C^*$-algebras generated by Toeplitz operators with $\mathbb{H}_n$-invariant symbols, independently of $n$ or the weight considered. **Theorem 39**. *For every $n \geq 1$ and for every weight $\lambda > -1$, the $C^*$-algebra $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ is isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$.* *Proof.* On one hand, Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} implies that the $C^*$-algebra $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ is isomorphic to the $C^*$-subalgebra of $L^\infty(\mathbb{R}_+)$ generated by the family of functions $$\mathbb{F}_\lambda = \{ \gamma_{\widetilde{a},\lambda} : \widetilde{a} \in L^\infty(\mathbb{R}_+) \},$$ where $\gamma_{\widetilde{a},\lambda}$ is given by equation [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}. On the other hand, [@HHM2014VerticalWeighted Theorem 2] proves that the same set $\mathbb{F}_\lambda$ of functions is dense in $\mathrm{VSO}(\mathbb{R}_+)$. The result follows from these two claims. ◻ **Remark 40**. Building from the observations of Remark [Remark 37](#rmk:Heisenberg-Toeplitz-C*-commutative){reference-type="ref" reference="rmk:Heisenberg-Toeplitz-C*-commutative"}, we note that the proof of Theorem [Theorem 39](#thm:Toeplitz-Hn-VSO){reference-type="ref" reference="thm:Toeplitz-Hn-VSO"} implies the following interesting, almost canonical, properties satisfied by the family $\mathbb{F}_\lambda$ of functions $\gamma_{\widetilde{a},\lambda}$ defined by [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}. 1. $\mathbb{F}_\lambda$ generates $\mathrm{VSO}(\mathbb{R}_+)$ for every $\lambda > -1$. This is due to [@HHM2014VerticalWeighted Theorem 2]. 2. $\mathbb{F}_\lambda$ correspond to the natural generators of $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$: the Toeplitz operators acting on $\mathcal{A}^2_\lambda(D_{n+1})$ with $\mathbb{H}_n$-invariant symbols. This follows from Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"}. In particular, not only has Theorem [Theorem 39](#thm:Toeplitz-Hn-VSO){reference-type="ref" reference="thm:Toeplitz-Hn-VSO"} established that $\mathcal{T}^{(\lambda)}\big(L^\infty(D_{n+1})^{\mathbb{H}_n}\big)$ and $\mathrm{VSO}(\mathbb{R}_+)$ are isomorphic for all $n \geq 1$ and $\lambda > -1$. It has also proved that they have corresponding generating sets, one of which is naturally associated to the former. ## A proof using the nilpotent MASG {#subsec:nilpotent-MASG} Let us consider the Abelian subgroup $\mathbb{R}^{n+1} \subset \mathbb{H}_n \times \mathbb{R}$, which corresponds to the so-called nilpotent MASG (see Remark [Remark 3](#rmk:Hn-NilpotentAction){reference-type="ref" reference="rmk:Hn-NilpotentAction"}). Then, we have the following obvious result. **Corollary 41**. *For every $n \geq 1$ we have $L^\infty(D_{n+1})^{\mathbb{H}_n} \subset L^\infty(D_{n+1})^{\mathbb{R}^{n+1}}$. In other words, every $\mathbb{H}_n$-invariant symbol is symbol invariant under the nilpotent MASG acting on $D_{n+1}$.* It was proved in [@QSMomentMapJFA; @QVUnitBall1; @QVUnitBall2] the mutual commutativity of Toeplitz operators with nilpotent symbols, i.e. with $\mathbb{R}^{n+1}$-invariant symbols. Furthermore, these previous works ensure the existence of unitary maps with respect to which the corresponding Toeplitz operators transform into multiplier operators. We now use these previous results to obtain a second proof of the diagonalizing formulas for Toeplitz operators with $\mathbb{H}_n$-invariant symbols. Let us fix a symbol $a \in L^\infty(D_{n+1})^{\mathbb{H}_n}$ and consider a function $\widetilde{a}$ such that $a(z) = \widetilde{a}(\operatorname{Im}(z_{n+1}) - |z'|^2)$, for almost every $z \in D_{n+1}$. Let us now define the function $f : \mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{C}$ by the expression $$f(u',t) = \widetilde{a}\bigg(\frac{1}{2t}\bigg).$$ In particular, we also have $$\widetilde{a}(t) = f\bigg(u',\frac{1}{2t}\bigg)$$ for almost every $u' \in \mathbb{R}^n$ and $t \in \mathbb{R}_+$. From this we conclude that $$\begin{aligned} a(z) &= \widetilde{a}(\operatorname{Im}(z_{n+1}) - |z'|^2) \\ &= f\bigg(-\frac{2\operatorname{Im}(z')}{\operatorname{Im}(z_{n+1}) - |z'|^2}, \frac{1}{2(\operatorname{Im}(z_{n+1}) - |z'|^2)} \bigg),\end{aligned}$$ for almost every $z \in D_{n+1}$. Note that the set of the first $n$ variables of $f$, which belongs to $\mathbb{R}^n$, can be chosen arbitrarily since this function does not depend on it. Once this expression has been established we can apply [@QSMomentMapJFA Theorem 7.8] to conclude the existence of a unitary map $U : \mathcal{A}^2_\lambda(D_{n+1}) \rightarrow L^2(\mathbb{R}^n) \otimes L^2(\mathbb{R}_+)$, independent of the symbol under consideration, such that we have $U T^{(\lambda)}_a U^* = M_{\widehat{\gamma}_{f,\lambda}}$ where the function $\widehat{\gamma}_{f,\lambda}$ is given by $$\begin{aligned} \widehat{\gamma}_{f,\lambda}&(y',\xi) = \\ &=\frac{\xi^{\lambda + \frac{n}{2} + 1}}% {2^n \pi^{\frac{n}{2}} \Gamma(\lambda + 1)} \int_{\mathbb{R}^n \times \mathbb{R}_+} \frac{f(u',t) e^{-\frac{\xi}{t} -\sVert[1] \frac{\sqrt{\xi} u'}{2t} - y' \sVert[1]^2}}% {t^{\lambda + n + 2}} \dif u' \dif t.\end{aligned}$$ Next, we replace our choice of $f$ in terms of $\widetilde{a}$ to obtain after some computations the following $$\begin{aligned} \widehat{\gamma}_{f,\lambda}&(y',\xi) = \\ &=\frac{\xi^{\lambda + \frac{n}{2} + 1}}% {2^n \pi^{\frac{n}{2}} \Gamma(\lambda + 1)} \int_{\mathbb{R}^n \times \mathbb{R}_+} \frac{\widetilde{a}(1/(2t)) e^{-\frac{\xi}{t} -\sVert[1] \frac{\sqrt{\xi} u'}{2t} - y' \sVert[1]^2}}% {t^{\lambda + n + 2}} \dif u' \dif t \\ &=\frac{\xi^{\lambda + \frac{n}{2} + 1}}% {2^n \pi^{\frac{n}{2}} \Gamma(\lambda + 1)} \int_0^\infty\frac{\widetilde{a}(1/(2t)) e^{-\frac{\xi}{t}}}% {t^{\lambda + n + 2}} \bigg( \int_{\mathbb{R}^n} e^{-\sVert[1] \frac{\sqrt{\xi} u'}{2t} - y' \sVert[1]^2} \dif u' \bigg) \dif t \\ &=\frac{\xi^{\lambda + \frac{n}{2} + 1}}% {2^n \pi^{\frac{n}{2}} \Gamma(\lambda + 1)} \int_0^\infty\frac{\widetilde{a}(1/(2t)) e^{-\frac{\xi}{t}}}% {t^{\lambda + n + 2}} \bigg( \int_{\mathbb{R}^n} e^{-\sVert[1] \frac{\sqrt{\xi} u'}{2t} \sVert[1]^2} \dif u' \bigg) \dif t \\ &=\frac{\xi^{\lambda + \frac{n}{2} + 1}}% {2^n \pi^{\frac{n}{2}} \Gamma(\lambda + 1)} \int_0^\infty\frac{\widetilde{a}(1/(2t)) e^{-\frac{\xi}{t}}}% {t^{\lambda + n + 2}} \bigg( \int_{\mathbb{R}^n} e^{-\frac{\xi}{4t^2} |u'|^2} \dif u' \bigg) \dif t \\ &=\frac{\xi^{\lambda + \frac{n}{2} + 1}}% {2^n \pi^{\frac{n}{2}} \Gamma(\lambda + 1)} \int_0^\infty\frac{\widetilde{a}(1/(2t)) e^{-\frac{\xi}{t}}}% {t^{\lambda + n + 2}} \bigg( \frac{\pi^\frac{n}{2} (2t)^n}{\xi^\frac{n}{2}} \bigg) \dif t \\ &=\frac{\xi^{\lambda + 1}}% {\Gamma(\lambda + 1)} \int_0^\infty\frac{\widetilde{a}(1/(2t)) e^{-\frac{\xi}{t}}}% {t^{\lambda + 2}} \dif t =\frac{\xi^{\lambda + 1}}% {\Gamma(\lambda + 1)} \int_0^\infty \widetilde{a}\bigg(\frac{s}{2}\bigg) e^{-\xi s} s^{\lambda} \dif s.\end{aligned}$$ We note that this function is independent of $y' \in \mathbb{R}^n$, and so once we rename it to $\gamma_{\widetilde{a},\lambda}$ to emphasize its dependence on $\widetilde{a}$ we obtain the same expression given by equation [\[eq:gamma-for-Heisenberg\]](#eq:gamma-for-Heisenberg){reference-type="eqref" reference="eq:gamma-for-Heisenberg"}. **Remark 42**. Although the diagonalizing formulas from Theorem [Theorem 35](#thm:Heisenberg-Toeplitz-multiplier){reference-type="ref" reference="thm:Heisenberg-Toeplitz-multiplier"} can be obtained from the corresponding ones for the nilpotent MASG acting on $D_{n+1}$, there are a number of advantages using our techniques. - Our computations using the Heisenberg group are more direct and short, even after omitting the computations needed to obtain the result for the nilpotent MASG. - To translate the Cauchy-Riemann equations on $D_{n+1}$ to corresponding ones after changes of coordinates using the nilpotent MASG as found in [@QSMomentMapJFA; @QVUnitBall1; @QVUnitBall2] required to apply Fourier transform on $n+1$ (real) variables. It is remarkable that our method using the Heisenberg group required to apply Fourier transform only over the (last) real variable of $\mathbb{H}_n$: in some sense, the rest of Cauchy-Riemann equations simply went through since we obtained Fock spaces. - The use of the Heisenberg group and the group-moment coordinates provided a unitary map $U_\lambda$ that uncovers an important fact: the weigthed Bergman spaces on $D_{n+1}$ are the direct integral of the family of all Fock spaces $$\mathcal{A}^2_\lambda(D_{n+1}) \simeq \int_{\mathbb{R}_+}^\oplus \mathcal{F}^2_{2\xi}(\mathbb{C}^n) \dif \xi.$$ This fact was already known (see for example [@SNVNilpotent]). However, this work unveils the source of this unitary equivalence: the action of the Heisenberg group on $D_{n+1}$. This is actually related to the fact that, as noted in the previous item, the first $n$ Cauchy-Riemann equations went through. In conclusion, the use of group-moment coordinates associated to $\mathbb{H}_n$ yield interesting formulas and facts while explaining their source. ## Acknowledgement {#acknowledgement .unnumbered} This research was partially supported by a Conahcyt scholarship held by the first author, by SNI-Conahcyt and by Conahcyt Grants 280732 and 61517. XX Appuhamy, Amila and Le, Trieu: *Commutants of Toeplitz operators with separately radial polynomial symbols*. Complex Anal. Oper. Theory 10 (2016), no. 1, 1--12. Axler, Sheldon; Čučković, Željko and Rao, N. V.: *Commutants of analytic Toeplitz operators on the Bergman space*. Proc. Amer. Math. Soc. 128 (2000), no. 7, 1951--1953. Bauer, Wolfram; Herrera Yañez, Crispin and Vasilevski, Nikolai: *Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball*. Integral Equations Operator Theory 78 (2014), no.2, 271--300. Choe, Boo Rim; Koo, Hyungwoon and Lee, Young Joo: *Commuting Toeplitz operators on the polydisk*. Trans. Amer. Math. Soc. 356 (2004), no. 5, 1727--1749. Čučković, Željko and Louhichi, Issam: *Finite rank commutators and semicommutators of quasihomogeneous Toeplitz operators*. Complex Anal. Oper. Theory 2 (2008), no. 3, 429--439. Dawson, Matthew; Ólafsson, Gestur and Quiroga-Barranco, Raul: *Commuting Toeplitz operators on bounded symmetric domains and multiplicity-free restrictions of holomorphic discrete series*. J. Funct. Anal. 268 (2015), no. 7, 1711--1732. Grudsky, S.; Karapetyants, A. and Vasilevski, N.: *Toeplitz operators on the unit ball in $\mathbb{C}^n$ with radial symbols*. J. Operator Theory 49 (2003), no. 2, 325--346. Grudsky, S., Karapetyants, A. and Vasilevski, N.: *Dynamics of properties of Toeplitz operators on the upper half-plane: hyperbolic case*. Bol. Soc. Mat. Mexicana (3) 10 (2004), no. 1, 119--138. Grudsky, S., Karapetyants, A. and Vasilevski, N.: *Dynamics of properties of Toeplitz operators on the upper half-plane: parabolic case*. J. Operator Theory 52 (2004), no. 1, 185--214. Grudsky, Sergei M.; Maximenko, Egor A. and Vasilevski, Nikolai L.: *Radial Toeplitz operators on the unit ball and slowly oscillating sequences*. Commun. Math. Anal. 14 (2013), no. 2, 77--94. Helgason, Sigurdur: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, 2001. Herrera Yañez, Crispin; Hutník, Ondrej and Maximenko, Egor A.: *Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions*. C. R. Math. Acad. Sci. Paris 352 (2014), no. 2, 129--132. Herrera Yañez, Crispin; Maximenko, Egor A. and Vasilevski, Nikolai: *Vertical Toeplitz operators on the upper half-plane and very slowly oscillating functions*. Integral Equations Operator Theory 77 (2013), no. 2, 149--166. Kadison, Richard V. and Ringrose, John R.: Fundamentals of the theory of operator algebras. Vol. I. Elementary theory. Reprint of the 1983 original. Graduate Studies in Mathematics, 15. American Mathematical Society, Providence, RI, 1997. Kadison, Richard V. and Ringrose, John R.: Fundamentals of the theory of operator algebras. Vol. II. Advanced theory. Corrected reprint of the 1986 original. Graduate Studies in Mathematics, 16. American Mathematical Society, Providence, RI, 1997. Knapp, Anthony W.: Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 2002. Kobayashi, Shoshichi and Nomizu, Katsumi: Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. Korenblum, Boris and Zhu, Ke He: *An application of Tauberian theorems to Toeplitz operators*. J. Operator Theory 33 (1995), no. 2, 353--361. Le, Trieu: *Commutants of separately radial Toeplitz operators in several variables*. J. Math. Anal. Appl. 453 (2017), no. 1, 48--63. Mok, Ngaiming: Metric rigidity theorems on Hermitian locally symmetric manifolds. Series in Pure Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. Quiroga-Barranco, Raul and Sanchez-Nungaray, Armando: *Moment maps of Abelian groups and commuting Toeplitz operators acting on the unit ball*. J. Funct. Anal. 281 (2021), no. 3, Paper No. 109039, 50 pp. Quiroga-Barranco, Raul and Vasilevski, Nikolai: *Commutative $C^*$-algebras of Toeplitz operators on the unit ball. I. Bargmann-type transforms and spectral representations of Toeplitz operators.* Integral Equations Operator Theory 59 (2007), no. 3, 379--419. Quiroga-Barranco, Raul and Vasilevski, Nikolai: *Commutative $C^*$-algebras of Toeplitz operators on the unit ball. II. Geometry of the level sets of symbols.* Integral Equations Operator Theory 60 (2008), no. 1, 89--132. Sanchez-Nungaray, Armando and Vasilevski, Nikolai: *Commutative algebras of Toeplitz operators on a Siegel domain associated with the nilpotent group of its biholomorphisms*. Recent trends in operator theory and partial differential equations, 275--300. Operator Theory: Advances and Applications, 258. Birkhäuser/Springer, Cham, 2017. Upmeier, Harald: Toeplitz operators and index theory in several complex variables. Operator Theory: Advances and Applications, 81. Birkhäuser Verlag, Basel, 1996.

arxiv_math

--- abstract: | The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are adjacent if and only if either $u=v^m$ or $v=u^n$ for some positive integers $m$, $n$. Forbidden subgraph has a significant role in graph theory. In our previous work [@cmm], we consider five important classes of forbidden subgraphs of power graph which include perfect graphs, cographs, chordal graphs, split graphs and threshold graphs. In this communication, we go even further in that way. This study, inspired by the articles [@celmmp; @dong; @ck], examines additional $4$ significant forbidden classes, including chain graphs, diamond-free graphs, $\{P_{5}, \overline{P_{5}}\}$-free graphs and $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free graph. The finite groups whose power graphs are chain graphs, diamond-free graphs, and $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free graphs have been successfully identified in this work. In case of $\{P_{5}, \overline{P_{5}}\}$-free graphs, we completely determine all the nilpotent groups, direct product of two groups, finite simple groups whose power graph is $\{P_{5}, \overline{P_{5}}\}$-free. author: - "Santanu Mandal[^1]" - "Pallabi Manna[^2]" title: Further study on forbidden subgraphs of power graph --- **AMS Subject Classification:** 05C25.\ **Keywords:** Power graphs, nilpotent groups, direct product, induced subgraphs, chain graphs, diamond graphs. # Introduction Graphs defined on various algebraic structures like groups, rings, vector spaces become very popular among the researchers from last few decades. Power graph is one such major graph representation of semigroups, groups. In 2002, Kelarev and Quinn introduced the directed power graph of semigroups (see [@Kelarev]). The directed power graph of a semigroup $S$, denoted by $\overrightarrow{P}(S)$), is a graph whose vertex set is $S$ and there is an arc $u\rightarrow v$ (where, $u\neq v$) if $v=u^{m}$ for some positive integer $m$. The corresponding underlying graph is called the undirected power graph, which is denoted by $P(S)$. Chakrabarty et al. [@Chakrabarty] introduced the idea of this graph in 2009. Throughout the paper we consider the power graph means the undirected power graph of finite group. In particular if we remove the identity element of the group $G$ from its original power graph $P(G)$ then the remaining graph is known as the reduced power graph or a proper power graph which is denoted by the symbol $P^*(G)$. In [@Doostabadi], the authors introduced the proper power graph of a group. For more existing results regarding power graph we refer the articles [@Abawajy; @Chakrabarty].\ Forbidden subgraph has an extensive role in graph theory. There are several graph classes that can be represent in terms of forbidden subgraphs. In [@Br], Brandst et al. discussed about various graph classes which is represented by the forbidden subgraphs. A graph is said to be $H$-free if it does not contain $H$ as its induced subgraph. In graph theory there are plenty of research articles are available in which the researchers deal with any NP-complete problem or any structural properties of a particular type of forbidden subgraph class. In this direction we refer few of such articles, namely [@celmmp; @dong; @am; @ck; @ck1; @ck2; @hk; @ckmm].\ In our previous work [@cmm], we consider several classes of forbidden subgraphs like perfect graphs, cographs, chordal graphs, split graphs and threshold graphs. Moreover in [@Mehatari], we discussed about the direct product of two groups, simple groups of Lie type whose power graph is a cograph.\ Motivated by the above mentioned articles we consider the graph classes like chain graph, $\{P_{5}, \overline{P_{5}}\}$-free graph, $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free graph and diamond free graph in case of power graph of finite groups. These graph classes are one of the important forbidden graph class because the class $\{P_{5}, \overline{P_{5}}\}$-free is a very large class of graph that contains cographs, even-hole (of length more than $4$) free graphs, odd-hole (of length more than $5$) free graphs, threshold graphs, complete graphs etc. On the other hand, the classes like complete graphs, complete bipartite graphs, cluster graphs, perfect graphs, even-hole (of length more than $6$) etc. are the subclass of a $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free graph. Furthermore, this approach provides the benefits for handling several open NP-complete problems in the context of power graphs. The problems with power graphs of any arbitrary finite group, such as clique number, chromatic number, Hamiltonicity, clique-width, graph partition problem, etc. that are difficult to solve, in such cases, by taking into consideration some forbidden subgraph classes of power graphs, we can at least partially come to a conclusion regarding these problems.\ The paper is organized following this manner: in section $2$ we recall some basic definitions, theorems which we use in this study and the notainal conventions of this paper. In section $3$ we conclude that the proper power graph of a finite group $G$ is a chain graph if and only if $G$ is either a) $C_{3}$ or b) a $2$-group of exponent $2$ or c) a EPO group $C_{3}\rtimes P$, where $P$ is a non-cyclic $2$-group of exponent $2$ or d) $S_{3}$. A group is called EPPO if every non-identity elements are of prime power order; whereas if every non-identity element of a group are of prime order then it is called an EPO group.\ Section $4$ is devoted to $\{P_{5}, \overline{P_{5}}\}$-free graph. Here we conclude the necessary and sufficient condition of a $\{P_{5}, \overline{P_{5}}\}$-free graph in case of nilpotent groups and direct product of two groups and obtain the following results. **Theorem 1**. *Let $G$ be a finite nilpotent group. Then $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if $G$ is either a) a $p$ group or b) a cyclic group $C_{p^{a}q}$, where $p, q$ are distinct primes and $a \geq 1$.* **Theorem 2**. *Let $G, H$ be two finite groups. Then $P(G\times H)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if $G, H$ take one of the following forms:\ a) both $G, H$ are power of same prime;\ b) one of $G, H$ is the cyclic group $C_{p^k}$ and the other one is $C_{q}$, where $p, q$ are distinct primes;\ c) one of $G, H$ is a cyclic group $C_{q^m}$ then we have the other is the group (i) or (ii).\ i) $C_{p^r}\rtimes Q$ ($r \geq 1$) if $m=1$.\ ii) $C_{p}\times Q$ if $m>1$.\ Provided $p, q$ are distinct prime divisors of $o(G\times H)$ and $Q$ is Sylow $q$-subgroup of $H$.* We classify the low dimensional simple groups of Lie type whose power graph is $\{P_{5}, \overline{P_{5}}\}$-free. We obtain the following result: **Theorem 3**. *Let $G$ be a finite simple group of Lie type except the Ree group ${}^{2}{G_{2}}(q)$ (where, $q=3^{2e+1}$). Then $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if either of the followings hold:\ I) $G\cong A_{n}$ with $n \leq 6$;\ II) $G\cong PSL(2, q)$ such that conditions a) or b) occurs: a) the numbers $(q\pm 1)/2$ are either a prime or product of some prime and a prime power if $q$ odd;\ b) $q\pm 1$ are either a prime or product of some prime and a prime power if $q$ even;\ III) $G={}^{2}{B_{2}}(q)=Sz(q)$, where $q=2^{2e+1}$ with the numbers $q-1, q\pm\sqrt{2q}+1$ are either a prime or product of some prime and a prime power;\ IV) $G\cong PSL(3, 4)$.* Additionally, we show that there is no sporadic simple groups whose power graph is $\{P_{5}, \overline{P_{5}}\}$-free.\ In section $5$ we find a necessay and sufficient condition for a nilpotent group as well as a non-nilpotent group whose power graph is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free. And the final section i.e., section $6$ determines the finite groups having diamond-free power graph. # Preliminaries We use the notations $K_{n}, P_{n}, C_{n}, 2K_{2}, \overline{\Gamma}, \Gamma_{1}\cup \Gamma_{2}$ to indicate a complete graph of order $n$, a path on $n$-vertices, a cycle of length $n$, the complement of $C_{4}$, complement of the graph $\Gamma$, disjoint unions of two graphs $\Gamma_{1}$ and $\Gamma_{2}$ respectively. From group theory we use the standard notations like $o(G), o(a), C_{n}, S_{n}, A_{n}, G\times H$ and $G\rtimes H$ to mean the order of the group $G$, the order of the element $a$ of a group, a cyclic group of order $n$, a symmetric group on $n$-symbols, an alternating group on $n$-symbols, the direct product of two groups $G, H$ and the semi-direct product of $G, H$. We use the same notation $C_{n}$ for both the cyclic group of order $n$ and a cycle of length. This will be clear from the context which we intend. The notation $\pi(G)$ stands for the set of all distinct prime divisors of $o(G)$, and $|\pi(G)|$ is the cardinality of $\pi(G)$.\ We now want to recall the definition of nilpotent group. A group is nilpotent if it is the direct products of its Sylow subgroups. Power graph has one important property that for a given group $G$, the power graph of any subgroup of $G$ is an induced subgraph of $P(G)$. This property helps us to determine a group whose power graph is whether lies in the classes of graph considered in this paper.\ In [@cmm], we completely characterized finite nilpotent power-cograph groups. We proved the following theorem: **Theorem 4** ([@cmm], Theorem 3.2). *Let $G$ be a finite nilpotent group. Then $P(G)$ is a cograph if and only if either $|G|$ is a prime power, or $G$ is cyclic of order $pq$ for distinct primes $p$ and $q$.* For a given group $G$ its prime graph is the graph whose vertex set is the distinct prime divisors of $o(G)$ and there is an edge between any two distinct primes if $G$ has an element of order product of these two distinct primes. Earlier (see [@cmm]) we proved that: **Theorem 5**. *Let $G$ be a group whose prime graph is a null graph. Then $P(G)$ is a cograph.* Moreover if the prime graph of a group is a null graph (or in other words, the group is an EPPO group) then any two adjacent vertices must belong to the same cyclic subgroup of prime power order. So, in that case $P(G)$ contains neither an induced path of length $2$ and above nor any induced cycle of length more than $3$. A graph is chordal if it contains no induced cycles of length greater than $3$. We recall a theorem from [@cmm] that determines the chordality of the power graph of a nilpotent group. **Theorem 6**. *Let $G$ be a finite nilpotent group. Then $P(G)$ is chordal if and only if $G$ is either a group of prime power order or $|G|$ has two prime divisors, one of the two Sylow subgroups is cyclic, and the other has prime exponent.* # Chain Graph A graph is called chain graph if it forbids $\{C_{3}, C_{5}, 2K_{2}\}$. It is obvious that if we consider the power graph of any finite group $G$ then $P(G)$ is a chain graph if and only if $G$ is a $2$-group of exponent $2$. Thus, in this section we consider the proper power graph and determine the groups whose proper power graph is a chain graph. **Theorem 7**. *For any finite group $G$, $P^{*}(G)$ is a chain graph if and only if $G$ is either a) $C_{3}$ or b) a $2$-group of exponent $2$ or c) a EPO group $C_{3}\rtimes P$, where $P$ is a non-cyclic $2$-group of exponent $2$ or d) $S_{3}$.* *Proof.* Let, $P^*(G)$ be a chain graph.\ Since $P^{*}(G)$ is $C_{3}$-free, so it does not contain an element of order $\geq 4$. Clearly, $o(G)$ has at most two distinct prime divisors. Otherwise, $o(G)$ has at least one odd prime divisor say $p\geq 5$. Then there exists an element $a$ of order $p$ and $\{a, a^{2}, a^{-1}\}$ generate a $C_{3}$ in $P^{*}(G)$.\ Now, if $G$ is a $p$-group then $G$ must be either a $2$-group of exponent $2$ or a $3$-group of exponent $3$. But if $G$ is a $3$-group of exponent $3$ then $P^{*}(G)$ is the disjoint union of multiple copies of $K_{2}$. So $P^*(G)$ contains $2K_{2}$ unless $G$ will be $C_{3}$. Therefore $G$ is either a) $C_{3}$ or b) a $2$-group of exponent $2$.\ Next consider $o(G)$ has two distinct prime divisor. Since, $G$ cannot have any element of order $\geq 4$ so $\pi(G)=\{2, 3\}$ and $G$ is an EPO-group. Again, we observe that the Sylow $3$-subgroup must be normal and cyclic; elsewhere there exist two elements, say $a, b$, of order $3$ in $G$ such that the pairs $\{a, a^2\}, \{b, b^2\}$ form $2K_{2}$. Thus $G\cong C_{3}\rtimes P$ with $P$ is a $2$-group of exponent $2$.\ In particular, if $P$ is cyclic then $G \cong S_{3}$.\ Converse:\ a) If $G\cong C_{3}$ then $P^*(C_{3})$ is a complete graph $K_{2}$. Thus, $P^*(G)$ is a chain graph.\ b) Let $G$ be a $2$-group of exponent $2$ then $P^{*}(G)$ is the disjoint union of isoated vertices. Thus, $P^{*}(G)$ is a chain graph.\ c) Let $G$ be a EPO group $C_{3}\rtimes P$, where $P$ is a $2$-group of exponent $2$. Clearly, $P^*(G)$ is the disjoint union of $K_{2}$ and some isolated vertices. This implies that $P^*(G)$ is $\{C_{3}, C_{5}, 2K_{2}\}$-free. Hence $P^*(G)$ is a chain graph.\ d) If $G\cong S_{3}$ then $P^*(G)$ is $K_{2}\cup 3K_{1}$. So it is a chain graph. ◻ # $\{P_{5}, \overline{P_{5}}\}$-free In this section we consider the finite nilpotent groups, simple groups of Lie type and sporadic simple groups, and explore those groups whose power graph is $\{P_{5}, \overline{P_{5}}\}$-free. Moreover, we also find the structures of two finite groups $G$ and $H$ such that $P(G\times H)$ is $\{P_{5}, \overline{P_{5}}\}$-free. ## Nilpotent group, Direct product of two groups **Theorem 8**. *Let $G$ be a finite nilpotent group. Then $P(G)$ is $P_{5}$-free if and only if $G$ is either a $p$-group or a cyclic group $C_{p^{a}q}$, where $p, q$ are distinct primes and $a\geq 1$.* *Proof.* Let $G$ be a finite nilpotent group such that $P(G)$ is $P_{5}$-free.\ **[Claim 1.]{.ul}** $o(G)$ has at most two distinct prime divisors.\ **Proof of Claim 1:** Suppose, $p, q, r$ are $3$ distinct primes divide $o(G)$. Let $a, b, c$ be the elements of order $p, q, r$ respectively. Then $P(G)$ contains a path $a\sim ab \sim b \sim bc\sim c$.\ According to the above claim we have either $G$ is a $p$-group or $o(G)=p^{a}q^{b}$, where $p, q$ are distinct primes and $a, b \geq 1$. We now consider the following cases.\ **Case 1.** Let $G$ be a $p$-group.\ By Theorem [Theorem 4](#pre_th_1){reference-type="ref" reference="pre_th_1"}, $P(G)$ is a cograph implies $P(G)$ is $P_{5}$-free.\ **Case 2.** Let $o(G)$ has two distinct prime divisors say $p, q$.\ Suppose, $o(G)=p^{a}q^{b}$ with $a, b \geq 1$. Let $P, Q$ be the Sylow $p$- and Sylow $q$-subgroups of $G$.\ **[Claim 2.]{.ul}** We claim that both Sylow subgroups must be cyclic.\ For the sake of contradiction, let the Sylow $p$-subgroup $P$ be non-cyclic. Then there exist elements say $a, b$ of order $p$ such that $a \nsim b$ in $P(G)$. In that case, $P(G)$ contains a path $a\sim ac\sim c\sim bc \sim b$, where $c \in Q$. Therefore, $G \cong C_{p^{a}q^{b}}$ with $a, b \geq 1$.\ If both $a, b >1$ then again the path $a \sim b \sim c \sim d \sim e$ is contained in $P(G)$, where $o(a)=p^{a}, o(b)=p, o(c)=pq, o(d)=q, o(e)=q^{b}$. Thus one of $a$ or $b$ must be $1$ and hence $G$ is the cyclic group $C_{p^{a}q}$ with $a\geq 1$.\ Converse part is obvious. ◻ **Theorem 9**. *Let $G$ be a finite nilpotent group. Then $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if $G$ is either i) a $p$ group or ii) a cyclic group $C_{p^{a}q}$, where $p, q$ are distinct primes and $a \geq 1$.* *Proof.* Let $G$ be a finite nilpotent group such that $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free. Since, $P(G)$ is $P_{5}$-free so by Theorem [Theorem 8](#th_p5){reference-type="ref" reference="th_p5"} $G$ must be the groups i) or ii).\ [Converse Part:]{.ul}\ i) Let $G$ be a $p$-group. If $P(G)$ contains $\overline{P_{5}}$ then $P(G)$ must comprises a $4$-vertex induced path. This contradicts the Theorem [Theorem 4](#pre_th_1){reference-type="ref" reference="pre_th_1"}, that is $P(G)$ is a cograph.\ ii) On the other hand, let $G$ be the cyclic group $C_{p^{a}q}$ ($a, b \geq 1$). Suppose $P(G)$ has induced subgraph $\overline{P_{5}}$. We choose $4$ consecutive vertices of orders $p^{a}, p, pq, q$. Then the fifth vertex must be of order $p^{i}q$, which is adjacent to both the second and third vertices. This leads to a contradiction that $\overline{P_{5}}$ is an induced subgraph.\ Thus, in any cases $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free. ◻ **Theorem 10**. *Let $G, H$ be two finite groups. Then $P(G\times H)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if $G, H$ take one of the following forms:\ a) both $G, H$ are power of same prime;\ b) one of $G, H$ is the cyclic group $C_{p^k}$ and the other one is $C_{q}$, where $p, q$ are distinct primes;\ c) one of $G, H$ is a cyclic group $C_{q^m}$ then we have the other is the group either (i) or (ii).\ i) $C_{p^r}\rtimes Q$ ($r \geq 1$) if $m=1$.\ ii) $C_{p}\times Q$ if $m>1$.\ Provided $p, q$ are distinct prime divisors of $o(G\times H)$ and $Q$ is Sylow $q$-subgroup of $H$.* *Proof.* Suppose $P(G\times H)$ is $\{P_{5}, \overline{P_{5}}\}$-free.\ First observe that $o(G\times H)$ has at most two distinct prime divisors. Elsewhere there exists $3$ primes say $p, q, r$ such that $p, q|o(G)$ and $r|o(H)$. Then $P(G\times H)$ contains a path $a\sim ac\sim c \sim bc \sim b$, where $o(a)=p, o(b)=q$ and $o(c)=r$.\ If $G$ and $H$ are both power of same prime then $G\times H$ is a $p$-group. So, there is nothing to prove (see Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}).\ Let $o(G\times H)$ has precisely two distinct prime divisors say $p, q$.\ If both $G, H$ are abelian then $G, H$ have the structure in b).\ Let us assume that both $G, H$ can not be abelian.\ **[Claim:]{.ul}** one of $o(G), o(H)$ must be of prime power order.\ **Proof of the claim** Let $p, q$ be two primes such that $pq|o(G), o(H)$ both. Consider the Sylow $p$- and Sylow $q$-subgroups of both $G, H$ are $P_{G}, Q_{G}$ and $P_{H}, Q_{H}$ respectively. Now $P_{G}\times Q_{H}$ is nilpotent and $\{P_{5}, \overline{P_{5}}\}$-free. Without loss of generality let $P_{G}\cong C_{p^a}$ and $Q_{H}\cong C_{q}$ ( by Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}). Similarly, $Q_{G}\times P_{H}$ is nilpotent and $\{P_{5}, \overline{P_{5}}\}$-free implies $Q_{G}\cong C_{q^b}$ ($b\geq 1$) and $P_{H}\cong C_{p}$ or $Q_{G}\cong C_{q}$ and $P_{H}\cong C_{p^k}$ ($k\geq 1$). Also, none of $G, H$ contains any abelian subgroup say $M$ of order $pq$; otherwise $G\times H$ contains one of the four nilpotent subgroups $P_{G}\times M$, $Q_{G}\times M$, $M\times P_{H}$, $M\times Q_{H}$, whose power graphs comprise a $P_{5}$ \[see Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}\].\ Since, $G,~H$ can not have abelian subgroup of order $pq$ so they are not nilpotent. So, one of the two Sylow subgroups of them must be not normal. Without loss of generality, let the Sylow $q$-subgroups of $G$ be not normal. Then the Sylow $p$-subgroup $P_{H}$ must be normal as otherwise $P(G\times H)$ has $P_{5}$. Using the similar argument we can say that the Sylow $q$-subgroup of $H$ ($Q_{H}$) is not normal and the Sylow $p$-subgroup of $G$ ($P_{G}$) is normal (since both $G, H$ are not nilpotent). Hence $G\cong C_{p^a}\rtimes C_{q}$ and $H \cong C_{p^k}\rtimes C_{q}$ or $G \cong C_{p^a}\rtimes C_{q^b}$ and $H\cong C_{p}\rtimes C_{q}$. But in any of the cases we obtain that $G$ contains two elements of order $q$ (say, $a, b$) such that $a\nsim b$ in $P(G\times H)$ and $H$ contains an element of order $p$, say $c$, for which $P(G\times H)$ has a path $a\sim ac\sim c\sim bc\sim b$.\ Therefore, one of $G, H$ must be the group of prime power order. Let $G$ be the group with $o(G)=q^{m}$ and $pq|o(H)$.\ Clearly, $G$ must be cyclic elsewhere $G$ contains two elements of order $q$ that are non-adjacent in $P(G\times H)$ and $H$ has an element of order $p$ such that $P(G\times H)$ comprises a $5$-vertex induced path.\ Let $P_{H}, Q$ be Sylow $p$- and Sylow $q$-subgroups of $H$. Now, $G\times P_{H}$ is nilpotent gives either $G\cong C_{q}$ and $P_{H}\cong C_{p^r}$ or $G\cong C_{q^m}$ and $P_{H}\cong C_{p}$. But clearly $H$ does not consist any abelian subgroup $K$ of order $pq$ as otherwise $G\times H$ contains a nilpotent subgroup $G\times K$ whose power graph has $P_{5}$ (by Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}). This implies that either $G\cong C_{q}$ and $H\cong C_{p^r}\rtimes Q$ or $G\cong C_{q^m}$ and $H\cong C_{p}\rtimes Q$, where $Q$ is the Sylow $q$-subgroup of $H$. Thus we get the structures of $G, H$ which take the form as given in c).\ [Converse Part:]{.ul}\ If $G$ and $H$ are either a) or b) then $P(G\times H)$ is $\{P_{5}, \overline{P_{5}}\}$-free by Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}.\ Let $G$ and $H$ have the form as in c). Firstly, we prove that $P(G)$ is $P_{5}$-free. If possible let $P(G)$ contain a $5$-vertex induced path $P_{5}$. Let $x_{1}\rightarrow x_{2}\leftarrow x_{3}\rightarrow x_{4}\leftarrow x_{5}\rightarrow \cdots$ be such a path, where $x_{i}=(g_{i}, h_{i})$.\ If $o(h_{3})=1$, then $x_{2}\sim x_{4}$ as $G$ is cyclic.\ If $o(g_{3})=1$, then again $x_{2}\sim x_{4}$ since $H$ contains only elements of order power of either $p$ or $q$.\ Let $o(g_{3})$ and $o(h_{3})$ both be a power of $q$. Then we obtain that $x_{2}\sim x_{4}$.\ Suppose $o(g_{3})$ is a power of $q$ and $o(h_{3})$ is a power of $p$. Let $g_{3}=a, h_{3}=b$. Without loss of generality, we assume that $g_{2}=1, h_{2}=b^{q^k}$ and $g_{4}=a^{p^l}, h_{4}=1$. Then as $x_{1}\sim x_{2}$ and $G$ is cyclic so $x_{1}\sim x_{3}$. Similarly, if we reverse $(g_{2}, h_{2})$ and $(g_{4}, h_{4})$ then we have either $x_{1}\sim x_{3}$ or $x_{3}\sim x_{5}$. Thus, $P(G)$ does not contain $P_{5}$.\ Next we prove that $P(G)$ is $\overline{P_{5}}$-free. Clearly, if a graph contains $\overline{P_{5}}$ then the graph must have an induced $4$-vertex cycle. Suppose, $P(G)$ carries $\overline{P_{5}}$. Then $P(G)$ has a $4$-vertex induced cycle say $A\sim B\sim C\sim D\sim A$ with $C=(a, b)$. If one of $a$ or $b$ is $1$ then $B\sim D$. On the other hand, if $a, b$ are both power of $q$ then also $B\sim D$; whereas for the case $o(a)=q^{k}, o(b)=p^{l}$ we obtain $A\sim C$. Thus, $P(G)$ does not contain any $4$-vertex induced cycle and hence $P(G)$ is $\overline{P_{5}}$-free. ◻ **Theorem 11**. *$P(S_{n})$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if $n \leq 5$.* *Proof.* If $n \geq 6$ then $P(S_{n})$ contains a path $(5~6)\sim (1~2 ~3)(5~6) \sim (1~2~3)\sim (1~2~3)(4~5)\sim (4~5)\sim (4~5)(1~2~6)$. So, $n \leq 5$.\ If $n \leq 5$ then the maximal cyclic subgroups of $P(S_{n})$ intersect in the identity, and their orders are in the set $\{2\}$ (for $n=2$), $\{2, 3\}$ (for $n=3$), $\{2, 3, 4\}$ (for $n=4$), or $\{4, 5, 6\}$ (for $n=5$). As for each $n\leq 5$, the power graph of the maximal cyclic subgroups of $S_{n}$ does not contain any path of length $3$ and above so $P(S_{n})$ is $P_{5}$-free and $\overline{P_{5}}$-free (since, $\overline{P_{5}}$ contains induced $P_{4}$). ◻ ## Simple Groups of Lie type and Sporadic smple group **Theorem 12**. *If $G$ is a sporadic simple group then $P(G)$ is never $\{P_{5}, \overline{P_{5}}\}$-free.* *Proof.* Firstly consider the Mathieu group $M_{11}$. It contains $165$ elements of order $2$, $440$ elements of order $3$, $1320$ elements of order $6$ and $990$ elements of order $4$. So, there exist elements $a, b, c, d, e$ of orders $4, 2, 6, 3, 6$ resp. such that $a^{2}=b=c^{3}, c^{2}=d=e^{2}$ with $c^{3}\neq e^{3}$. Thus, $P(M_{11})$ contains $P_{5}$.\ Since, $M_{11}$ is contained as a subgroup in every sporadic group except $J_{1}, J_{2}, J_{3}, M_{22}, He, Ru$ and $Th$ so their power graph contains $P_{5}$.\ But $M_{22}$ contains $A_{7}$, $J_{1}, J_{2}, J_{3}$ contain $D_{3}\times D_{5}, A_{4}\times A_{5}, C_{3}\times A_{6}$ respectively, $He, Ru$ contain $S_{7}, A_{8}$ respectively. By Theorems [Theorem 11](#S_{n}){reference-type="ref" reference="S_{n}"}, [Theorem 14](#A_{n}){reference-type="ref" reference="A_{n}"}, [Theorem 10](#th_p5_direct_product){reference-type="ref" reference="th_p5_direct_product"} the power graphs of these subgroups contains either $P_{5}$ or its complement. Hence, the power graphs of these groups are not $\{P_{5}, \overline{P_{5}}\}$-free. From the information in $\mathbb{ATLAS}$ [@Conway], we observe that $Th$ contains elements $x, y, z, w, u$ whose orders are $3, 6, 2, 10, 5$ respectively. Additionally these elements satify the conditions $y^{2}=x, y^{3}=z=w^{5}$ and $u=w^{2}$. Thus, $P(Th)$ contains a $5$ vertex induced path $x\sim y\sim z\sim w\sim u$.\ This completes the proof of the theorem. ◻ **Theorem 13**. *Let $G$ be a finite simple group of Lie type except the Ree group ${}^{2}{G_{2}}(q)$ (where, $q=3^{2e+1}$). Then $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if either of the followings hold:\ I) $G\cong A_{n}$ with $n \leq 6$;\ II) $G\cong PSL(2, q)$ such that conditions a) or b) occurs: a) the numbers $(q\pm 1)/2$ are either a prime power or product of some prime and a prime power if $q$ odd;\ b) $q\pm 1$ are either a prime or product of some prime and a prime power if $q$ even;\ III) $G={}^{2}{B_{2}}(q)=Sz(q)$, where $q=2^{2e+1}$ with the numbers $q-1, q\pm\sqrt{2q}+1$ are either a prime power or product of some prime and a prime power;\ IV) $G\cong PSL(3, 4)$.* We prove this theorem by proving the following subsequent theorems. **Theorem 14**. *$P(A_{n})$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if $n \leq 6$.* *Proof.* For $n \geq 7$, $P(A_{n})$ contains a path $(1~2~3~4)(5~6)\sim (1~3)(2~4)\sim (1~3)(2~4)(5~6~7)\sim (5~6~7)\sim (1~2)(3~4)(5~6~7)$. Thus, $n \leq 6$.\ For $n=4, 5, 6$ then prime graph of $A_{n}$ is a null graph; so $P(A_{n})$ must be $P_{5}$-free. Otherwise, $P(A_{n})$ contains a $4$-vertex induced path which contradicts that $P(A_{n})$ is a cograph (see Theorem [Theorem 5](#pre_th_2){reference-type="ref" reference="pre_th_2"}). Since, $\overline{P_{5}}$ contains induced path $P_{4}$ and $P(A_{n})$ (where $n=4, 5, 6$) is $P_{4}$-free so $P(A_{n})$ is also $\overline{P_{5}}$-free.\ If $n=3$ then $P(A_{3})$ is a complete graph (as $A_{3}$ is a cyclic group of order $3$). This implies that $P(A_{3})$ is $\{P_{5}, \overline{P_{5}}\}$-free. ◻ **Theorem 15**. *Let $G\cong PSL(2, q)$. Then $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if the followings hold:\ a) the numbers $(q\pm 1)/2$ are either a prime power or product of some prime and a prime power if $q$ odd;\ b) $q\pm 1$ are either a prime power or product of some prime and a prime power if $q$ even.* *Proof.* Let $q$ be a power of some odd prime. Now, $P^{*}(G)$ is the disjoint union of $P^{*}(C_{(q\pm1)/2})$ along with some isolated vertices. Thus if $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free then $P^{*}(C_{(q\pm1)/2})$ is also $\{P_{5}, \overline{P_{5}}\}$-free. This implies the condition in a) according to the Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}.\ Suppose $q$ is a power of $2$. Then $P^*(G)$ is the disjoint union of $P^{*}(C_{(q\pm1)})$ along with some isolated vertices. Then by Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"} the numbers $q\pm 1$ satisfy the conditions in b). ◻ **Theorem 16**. *Let $G={}^{2}{B_{2}}(q)=Sz(q)$, where $q=2^{2e+1}$. Then $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if the numbers $q-1, q\pm\sqrt{2q}+1$ are either a prime power or product of some prime and a prime power.* *Proof.* Here $G$ has $4$ maximal cyclic subgroups of orders $4, q-1, q\pm\sqrt{2q}+1$. Since these $4$ numbers are pairwise coprime so any edge in $P(G)$ must lie in a maximal cyclic subgroup. Thus if $P(G)$ contains either $P_{5}$ and its complement then it must be contained in a maximal cyclic subgroup. Now the power graph of a cyclic group of order $4$ is a complete graph. Therefore, $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if the numbers $q-1, q\pm\sqrt{2q}+1$ satify the stated condition according to the Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}. ◻ **Theorem 17**. *If $q (\geq 4)$ is a power of $2$ then $P(PSU(3, q))$ is never $\{P_{5}, \overline{P_{5}}\}$-free.* *Proof.* Let $\beta$ be a generator of the multiplicative group of $GF(q^{2})$. So $o(\beta)=q+1$. Let $p (>3)$ be a prime factor of $q+1$. Set $d=(q+1)/p$. Now $\alpha=\beta^{d(q-1)}$ has order $p$. Then $\overline{\alpha}=\beta^{d(q^{2}-1)}$ and $\alpha\overline{\alpha}=1$ in $GF(q^{2})$. Choose $3$ matrices $g, h, k$ as follows:\ $$g=\begin{bmatrix} 0 & 1& 0 \\ 1& 0 & 0 \\ 0& 0& 1 \end{bmatrix},~~h=\begin{bmatrix} \alpha & 0& 0 \\ 0 & \alpha & 0 \\ 0& 0& \alpha^{-2} \end{bmatrix}~~\text{and} ~~k=\begin{bmatrix} 0 & \alpha & 0 \\ 1& 0 & 0 \\ 0& 0& \alpha^{-1} \end{bmatrix}$$ Then $o(g)=2, o(h)=p, o(k)=o(gh)=o(hk)=2p$ and $k^{2}=h$, $gh=hg$, $hk=kh$. So, $P(SU(3, q))$ contains the induced path $g\sim gh\sim h\sim k\sim k^{p}$. Since $g, h, k\in SU(3, q)\setminus Z$ so choosing $a=gZ, b=hZ, c=kZ$; then an induced path $a\sim ab \sim b \sim c \sim c^{p}$ is contained in $P(PSU(3, q))$.\ But this argument is not valid when $q=8$. In that case, $P(PSU(3, 8))$ contains a subgroup $C_{3}\times PSL(2, 8)$ whose power graph is not $\{P_{5}, \overline{P_{5}}\}$-free (by Theorem [Theorem 10](#th_p5_direct_product){reference-type="ref" reference="th_p5_direct_product"}). ◻ **Theorem 18**. *If $q$ is a power of an odd prime then $P(PSU(3, q))$ is never $\{P_{5}, \overline{P_{5}}\}$-free.* *Proof.* If $q$ is odd, then $PSU(3, q)$ contains a cyclic subgroup of order $(q^{2}-1)/gcd(q+ 1, 3)$. Since $q$ is odd, both the numbers $q -1$ and $(q+1)/gcd(q +1, 3)$ are even. Thus $P(PSU(3, q))$ is $\{P_{5}, \overline{P_{5}}\}$-free if $(q - 1)(q + 1)/gcd(q + 1, 3)$ is either a power of $2$ or of the form $2^{k}p'$, where $p'$ is an odd prime.\ First suppose that both $(q + 1)/gcd(q + 1, 3)$ and $q - 1$ are powers of $2$. As only one of $q + 1$ and $q - 1$ is divisible by $4$, so the pair $(q - 1, q + 1)$ is either $(2, 4)$ or $(4, 6)$. Hence $q = 3$ or $5$.\ Next, suppose that $(q - 1)(q + 1)/gcd(q + 1, 3) = 2^{k}p'$. Then one of $q - 1, (q + 1)/gcd(q + 1, 3)$ is a power of $2$. Without loss of generality, we assume that $q - 1$ is a power of $2$. Now if $q \neq 3, 9$, then $q + 1$ is either of the forms $2p'$ or $6p'$ for some odd prime $p'$. If $q + 1 = 6p'$, then $PSU(3, q)$ contains a subgroup $C_{2^k}\times C_{2p}$ whose power graph is not $\{P_{5}, \overline{P_{5}}\}$-free (by Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}). Again, if $q + 1 = 2p'$, then $PSU(3, q)$ contains the subgroup $C_{q+1}\times C_{q+1/gcd(q+1,3)}$. By Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}, $P(C_{2p'} \times C_{2p'})$ is not $\{P_{5}, \overline{P_{5}}\}$-free. So either $q = 3$ or $9$ in this case.\ But if $q=9$ then $C_{8} \times C_{10}$ is contained in $PSU(3, q)$. For $q=5$, $PSU(3, 5)$ contains $A_{7}$. The power graph of none of these subgroups are $\{P_{5}, \overline{P_{5}}\}$-free \[see Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}\]. On the other hand, if $q=3$ then $PSU(3, 3)$ contains $4\cdot S_{4}$. Now, $4\cdot S_{4}$ is given by $⟨a, b, c, d, e|a^4 = d^3 = 1, b^2 = c^2 = e^2 = a^2, ab = ba, ac = ca, ad = da, eae^{-1} = a^{-1}, cbc^{-1} = a^{2}b, dbd^{-1} = a^{2}bc, ebe^{-1} = bc, dcd^{-1} = b, ece^{-1} = a^{2}c, ede^{-1} = d^{-1}⟩$. Then $P(PSU(3, 5))$ contains the induced path $a\sim ad\sim d\sim ed\sim e$. Thus, in any of the case $q=3, 5, 9$, the power graph of $PSU(3, q)$ is not $\{P_{5}, \overline{P_{5}}\}$-free. ◻ *Remark 1*. Let $G$ be the Ree group ${}^{2}{G_{2}}(q) = R_1(q)$, where $q = 3^{2e+1}$. We observe that $C_2 \times PSL(2, q)$ is the centralizer of an involution in the group $G$. The group $C_2 \times PSL(2, q)$ (see [@wilson]) carries the subgroups $C_2 \times C_{(q\pm1)/2}$. Therefore, by Theorem [Theorem 9](#th_p5_1){reference-type="ref" reference="th_p5_1"}, $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if $(q \pm1)/2$ is either a power of $2$ or a power of an odd prime or of the form $2p^r$.\ If both $q \pm 1$ are powers of $2$, then we get a solution of Catalan's conjecture which contradicts Mihailescu's theorem (see [@Cohn Section 6.11]).\ Let $q + 1 = 2p_{1}^{r_{1}}$ and $q - 1 = 2p_{2}^{r_2}$, where $r_1$ and $r_2$ are odd primes. Then the diophantine equation $p_{1}^{r_1}-p_{2}^{r_2}=1$ has a solution. This leads to a contradiction to Mihailescu's theorem as both $p_1$ and $p_2$ are odd.\ Similarly, both $q \pm 1$ can not be of the form $4p^{k}$ as the diophantine equation $2x - 2y = 1$ has no solution.\ Again, if any one of $q + 1$ or $q -1$ is $4p_{1}^{s}$ and the other one is $2p_{2}^{t}$ then, for $x = p_{1}^{s}$ and $y = p_{2}^{t}$, the corresponding diophantine equation is either $2x - y = 1$ or $x - 2y = 1$. One can check that the solutions exist for infinitely many values of $x, y$. So in this case, the question arise :\ **Problem 1:** Does there exist infinitely many values of $q$ for which $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free? **Theorem 19**. *Let $q$ be power of an odd prime. Then $P(PSL(3, q))$ is never $\{P_{5}, \overline{P_{5}}\}$-free.* *Proof.* Consider $3$ matrices $g, h, x$ in $SL(3, q)$ as follows:\ $$g=\begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix},~~h=\begin{bmatrix} 0 & -1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}~~\text{and}~~x=\begin{bmatrix} 0 & 1 & 0\\ -1 & -1 & 0\\ 0 & 0 & -1 \end{bmatrix}$$ Here, $h^{2}=x^{2}, h^{3}=g^{2}$ and $o(h)=o(x)=6, o(g)=4$. Then the induced path $x \sim h^{2}\sim h\sim g^{2}\sim g$ is contained in $P(SL(3, q))$. Since, $g, h, x \in SL(3, q)\setminus Z$ so set $a=gZ, b=hZ, c=xZ$ and $P(PSL(3, q))$ contains the path $c\sim b^{2}\sim b \sim a^{2}\sim a$. This completes the proof of the theorem. ◻ **Theorem 20**. *$P(PSL(3, q))$ (where, $q (\geq 4)$ is a power of $2$) is $\{P_{5}, \overline{P_{5}}\}$-free if $q=2, 4$.* *Proof.* Firstly, let $q$ be a power of $2$ with $q>4$. If $q$ is an odd power of $2$ then $q-1$ is not divisible by $3$; whereas if $q$ is an even power of $2$ then $q-1$ is not a power of $3$ (by the solution of Catalan's conjecture (see [@Cohn Section 6.11]). In that case $q-1$ must have a large prime divisor. Let $\alpha$ be an element in the multiplicative group of $GF(q)$ such that $o(\alpha)=p>3$. Choose the matrices $h~\text{and}~ k$ as follows:\ $$h=\begin{bmatrix} \alpha & 0 & 0\\ 0 & \alpha & 0\\ 0 & 0 & \alpha^{-2} \end{bmatrix}~~\text{and}~~k=\begin{bmatrix} 0 & \alpha & 0\\ 1 & 0 & 0\\ 0 & 0 & \alpha^{-1} \end{bmatrix}$$ Set $x=-k$. Then $k^{2}=h=x^{2}, o(k)=o(x)=2p, o(h)=p$ and $o(hk^{p})=2p$; hence $P(SL(3, q))$ contains a $5$-vertex induced path $k^{p}\sim hk^{p}\sim h \sim x \sim x^{p}$. Since the matrices are not scalar so $P(PSL(3, q))$ carries the induced path $P_{5}$. Now the remaining cases are $q=2, 4$.\ If $q=2, 4$ then the power graphs of $PSL(3, 2), PSL(3, 4)$ have prime graph which is a null graph so their power graphs are cograph (by Theorem [Theorem 5](#pre_th_2){reference-type="ref" reference="pre_th_2"}) so they are also $\{P_{5}, \overline{P_{5}}\}$-free (as $\overline{P_{5}}, P_{5}$ has $P_{4}$ as an induced subgraph). ◻ **Theorem 21**. *$P(PSp(4, q))$ is never $\{P_{5}, \overline{P_{5}}\}$-free.* *Proof.* First, suppose that $q$ is a power of $2$. Then $PSp(4, q)$ contains $PSL(2, q)\times PSL(2, q)$, and so it contains $C_{(q\pm1)} \times C_{(q\pm1)}$. Now $(q +1, q -1) = 1$ and $3$ divides one of them. Thus, $P(G)$ is $\{P_{5}, \overline{P_{5}}\}$-free if and only if one of $q \pm 1$ is a prime and the other is a power of another prime. So we must have $q = 2, 4$ or $8$. Next, suppose that $q$ is a power of an odd prime. Then $G$ contains the central product of two copies of $SL(2, q)$, and hence it contains $C_{(q\pm1)} \times C_{(q\pm1)/2}$. Thus if $PSp(4, q)$ is $\{P_{5}, \overline{P_{5}}\}$-free, then both $(q \pm 1)/2$ have to be prime powers (as $C_{(q\pm1)}\times C_{(q\pm1)/2}$ is contained in $PSp(4, q))$. Now one of $(q -1)/2$ or $(q +1)/2$ is even, so one of $(q\pm1)/2$ must be a power of $2$. This implies one of $C_{(q\pm1)}$ must be $4$, or else $PSp(4, q)$ contains a subgroup whose power graph is $\{P_{5}, \overline{P_{5}}\}$. Thus the possible values of $q$ are $3$ and $5$.\ If $q = 2$, then $PSp(4, 2)$ is isomorphic to $S_{6}$. By Theorem [Theorem 11](#S_{n}){reference-type="ref" reference="S_{n}"}, $P(PSp(4, 2))$ is not $\{P_{5}, \overline{P_{5}}\}$-free.\ If $q = 3$ or $4$, then $PSp(4, q)$ contains $S_6$, and so its power graph is not $\{P_{5}, \overline{P_{5}}\}$-free.\ Again, $PSp(4, 5)$ contains the subgroup $S_{3}\times S_{5}$ whose power graph is not $\{P_{5}, \overline{P_{5}}\}$-free.\ graph. The group $PSp(4, 8)$ contains $PSp(4, 2)$ (see Mitchell Theorem [@King]), and so $P(PSp(4, 8))$ is not $\{P_{5}, \overline{P_{5}}\}$-free. Thus, we get our conclusion. ◻ **Theorem 22**. *The power graph of $G_{2}(q)$ is never $\{P_{5}, \overline{P_{5}}\}$-free.* *Proof.* $G_{2}(q)$ comprises $SL(3, q)$ and $SU(3, q)$ as the subgroups (see [@cooperstein; @kleidman2]). Now $SL(3, q) \cong PSL(3, q)$ if $q \not \equiv 1(mod~3)$ and $SU(3, q)\cong PSU(3, q)$ if $q \not \equiv -1(mod~3)$. Thus, for any $q$, either $PSL(3, q)$ or $PSU(3, q)$ is contained in $G_{2}(q)$. Now, $PSL(3, q)$ is $\{P_{5}, \overline{P_{5}}\}$-free if $q = 2, 4$, whereas $PSU(3, q)$ is not $\{P_{5}, \overline{P_{5}}\}$-free for every $q \neq 2$. So only remaining case is $q = 2$. The group $G_{2}(2)$ is not simple, and it contains $PSU(3, 3)$ as a subgroup. Hence, the power graph of $G_{2}(q)$ is not $\{P_{5}, \overline{P_{5}}\}$-free. So we arrive at the conclusion. ◻ We now consider the other simple groups of Lie type of rank $2$.\ 1) Let $G = PSU(4, q)$. If $q > 2$, then $G$ contains $PSp(4, q)$. Again, if $q = 2$, then $G$ isomorphic to $PSp(4, 3)$. Hence, $P(PSU(4, q))$ is not $\{P_{5}, \overline{P_{5}}\}$-free.\ 2) Let $G = PSU(5, q)$. If $q = 2$, then $PSU(5, q)$ contains $PSU(4, 2)$ as a subgroup. Again, if $q > 2$, then $PSU(5, q)$ contains $SL(2, q)\times SL(2, q)$ (see [@wilson]). Now, by Theorem [Theorem 10](#th_p5_direct_product){reference-type="ref" reference="th_p5_direct_product"}, $P(SL(2, q)\times SL(2, q))$ is not $\{P_{5}, \overline{P_{5}}\}$-free, and so $P(PSU(5, q))$ is not $\{P_{5}, \overline{P_{5}}\}$-free.\ 3) The group ${}^{2}{F_{4}}(2^d)$ contains ${}^{2}{F_{4}}(2)$ for all odd $d$ (see [@malle]), and so it contains $PSL(3, 3)$. Thus, ${}^{2}{F_{4}}(2^d)$ is not $\{P_{5}, \overline{P_{5}}\}$-free.\ 4) The group ${}^{3}{D_{4}}(q)$ contains $G_{2}(q)$ (see [@kleidman]); so its power graph is also a not $\{P_{5}, \overline{P_{5}}\}$-free. **Theorem 23**. *Let $G$ be a Lie type simple groups of rank more than $2$. Then $P(G)$ is never $\{P_{5}, \overline{P_{5}}\}$-free.* *Proof.* Let $G$ be a simple group of Lie type of rank greater than $2$. Now, as the Dynkin diagram of G has a single bond in each case, $G$ contains $PSL(3, q)$ as a subgroup. But the only values for which the power graph of $PSL(3, q)$ is $\{P_{5}, \overline{P_{5}}\}$-free are $q = 2, 4$. Hence, we have to check only for those simple groups whose underlying fields are finite fields with $2$ and $4$ elements, respectively. Now $PSL(4, 2)$ is isomorphic to $A_8$. So its power graph is not $\{P_{5}, \overline{P_{5}}\}$-free. The group $PSL(4, 4)$ contains $PSL(4, 2)$, whose power graph is not $\{P_{5}, \overline{P_{5}}\}$-free.\ On the other hand, $PSp(6, 2)$ contains $S_8$, whose power graph is not $\{P_{5}, \overline{P_{5}}\}$-free. Again, $PSp(6, 4)$ contains $PSp(6, 2)$. So, in this case, we get both $PSp(6, 2), PSp(6, 4)$ whose power graphs are not $\{P_{5}, \overline{P_{5}}\}$-free.\ If $q = 2, 4$, then the orthogonal and unitary groups of Lie type of rank $3$ contain $PSp(4, q)$. Hence their power graphs are not $\{P_{5}, \overline{P_{5}}\}$-free. ◻ # $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free **Theorem 24**. *For any finite nilpotent group $G$, $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free if and only if $G$ is either a $p$-group or a cyclic group $C_{p^{a}q}$ (where $a \geq 1$) or $P\times C_{q^b}$, where $P$ is a non-cyclic $2$-group of exponent $2$ and $b \geq 1$.* *Proof.* Let $G$ be a finite nilpotent group such that $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free.\ We claim that $o(G)$ must have at most $2$ distinct prime divisors. If possible let $o(G)$ be divisible by $3$ distinct primes say $p<q< r$ with the corresponding elements $a, b, c$. Then the vertices $\{c, c^{-1}\}\cup \{a, ab, b\}$ form $P_{2}\cup P_{3}$.Thus $o(G)$ is either a prime power or of the form $p^{a}q^{b}$(where, $a, b \geq 1$).\ Let $o(G)=p^{a}q^{b}$ (with $p<q$) and $P, Q$ be the Sylow $p$- and Sylow $q$-subgroups of $G$. Here we consider two cases based on $p$.\ **Case 1.** $p$ odd\ If any one of two Sylow subgroups is non-cyclic, say $P$, then $P(G)$ contains $P_{2}\cup P_{3}$ by the vertices $\{a, a^{-1}\}\cup \{b, bc, c\}$ (where, $o(a)=o(b)=p, o(c)=q, a\nsim b$). Therefore in this case $G\cong C_{p^{a}q^{b}}$. But if both $a, b>1$ then $P(G)$ contains $\overline{P_{2}\cup P_{3}}$. Thus in this case $G\cong C_{p^{a}q}$.\ **Case 2.** $p=2$\ Clearly, the Sylow $q$-subgroup must be cyclic; otherwise we get $P_{2}\cup P_{3}$ in $P(G)$.\ If the Sylow $2$-subgroup $P$ of $G$ is cyclic then $G\cong C_{2^{a}q^{b}}$, where $a, b \geq 1$. Again, we have either $a$ or $b$ must be $1$; otherwise $P(G)$ contains $\overline{P_{2}\cup P_{3}}$. Thus $G\cong C_{2^{a}q}$.\ Let the Sylow $2$-subgroup $P$ of $G$ be non-cyclic. If $P$ has two distinct elements of order power of $2$ that are non-adjacent in $P(G)$ then $P(G)$ carries $P_{2}\cup P_{3}$. Thus, either $P$ is cyclic or a non-cyclic $2$-group of exponent $2$.\ Therefore $G$ is either a $p$-group or a cyclic group $C_{p^{a}q}$ with $a \geq 1$ or $P\times C_{q^b}$, where $b \geq 1$ and $P$ is a non-cyclic $2$-group of exponent $2$.\ [Converse Part:]{.ul}\ Let $G$ be a $p$-group. Then any $3$ adjacent vertices form a triangle in $P(G)$. Thus $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free.\ If $G \cong C_{p^{a}q}$ (where $a, b \geq 1$), then as $G$ has unique subgroup of each orders dividing $o(G)$, so it is easy to confirm that $P(G)$ is $\{P_{2}\cup P_{3}\}$-free. On the other hand, if $P(G)$ contains $\overline{P_{2}\cup P_{3}}$ then $P(G)$ has induced $4$-cycle. But since $P(G)$ is chordal by Theorem [Theorem 6](#pre_th_3){reference-type="ref" reference="pre_th_3"}, so such $C_{4}$ never exists. Hence $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free.\ Next, suppose $G \cong P\times C_{q^b}$, where $P$ is a non-cyclic $2$-group of exponent $2$ and $b \geq 1$. Then it is easy to check that $P(G)$ is $P_{2}\cup P_{3}$-free. But suppose $P(G)$ contains $\overline{P_{2}\cup P_{3}}$. Then $P(G)$ comprises a $4$-vertex induced cycle, which contradicts the Theorem [Theorem 6](#pre_th_3){reference-type="ref" reference="pre_th_3"} that is $P(G)$ is a chordal graph. Hence $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free in this case. ◻ **Theorem 25**. *Let $G$ be a finite non-nilpotent group. Then $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free if and only if $G$ has one of the following possibilities:\ (a) $|\pi(G)|\geq 4$ and $G$ is an EPPO group;\ (b) for $|\pi(G)|=3$, $G$ is either an EPPO group or the group $C_{q^{a}r}\rtimes P$ of order $2^{a}q^{b}r^{c}$, where $2<q<r$ with $P, Q, R$ are the Sylow $2$-, Sylow $q$- and Sylow $r$-subgroups of $G$ respectively and $P$ must be a $2$-group of exponent $2$;\ (c) if $\pi(G)=\{p, q\}$ then $G$ is either an EPPO group or a group of order $2^{a}q^{b}$ ($q$ odd) such that the Sylow $q$-subgroup of $G$ must be cyclic as well as normal and the Sylow $2$-subgroups of $G$ must be of exponent $2$.* *Proof.* Let $G$ be a non-nilpotent group such that $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free.\ If $o(G)$ has $4$ and more divisors then we claim that $G$ must be an EPPO group. Otherwise, $o(G)$ has at least $3$ distinct odd prime divisors say $p, q, r$. Since $G$ is non-EPPO so contains an element whose order is product of two distinct primes. Thus $P(G)$ contains $P_{3}$. Without loss of generality, we suppose that $a\sim b \sim c$ are $3$ vertices of $P_{3}$ with orders $p, pq, q$. There exists an element in $G$ of order $r$, say $c$ and then the pair $\{c, c^{-1}\}$ form $P_{2}$. This implies that $P(G)$ contains $\{P_{2}\cup P_{3}\}$. Thus, in this case $G$ must be an EPPO group.\ Suppose $o(G)$ has $3$ distinct prime divisors say $p<q<r$.\ Here we consider two cases depending on $p$:\ **Case 1.** $p$ is odd prime\ Here we claim that $G$ must be an EPPO group; otherwise $G$ contains an element of order product of two distinct primes. As $p, q, r$ all are odd primes so $P(G)$ has $\{P_{2}\cup P_{3}\}$.\ **Case 2.** $p=2$\ In this case $G$ may be an EPPO group.\ Now let $G$ be a non-EPPO group. Obviously, $G$ cannot contain any element of order $2q$ or $2r$. In that case if $G$ has an element say $x$ of order $2q$ then $P(G)$ carries the path $x^{q}, x, x^{2}$ along with the $P_{2}$ by the vertices $\{c, c^{-1}\}$, where $o(c)=r$. Thus, $G$ can contains an element of order $qr$ only. Additionally, we observe that the Sylow $q$-, Sylow $r$-subgroup must be cyclic as well as normal and Sylow $2$-subgroup must be of exponent $2$. Elsewhere $\{P_{2}\cup P_{3}\}$ is contained in $P(G)$. Thus $G$ has a normal subgroup $C_{q^{a}r^{b}}$ (as Sylow $q$-and $r$-subgroups are normal so their product is also normal) and all the elements outside of the normal subgroup must be of order $2$. Moreover, we observe that $P(C_{q^{a}r^{b}})$ carries $\overline{P_{2}\cup P_{3}}$ if both $a, b>1$. So any one of two Sylow $q$-subgroup or Sylow $r$-subgroup must be of prime order. Let $Q\cong C_{q^a}, R\cong C_{r}$. Therefore, $G$ is the group $C_{q^{a}r}\rtimes P$ where $P, Q, R$ are the Sylow $2$-, Sylow $q$- and Sylow $r$-subgroups of $G$ respectively and $P$ must be a $2$-group of exponent $2$.\ Suppose that $o(G)$ has exactly two distinct prime divisors say $p, q$. If both $p, q$ are odd primes then $G$ must be an EPPO group; elsewhere both the Sylow subgroups of $G$ must be cyclic as well as normal which forces $G$ to be a nilpotent group. This leads to a contradiction as $G$ is not nilpotent. Now consider $o(G)=2^{k}q^{m}$. If $G$ is a non-EPPO group then the Sylow $q$-subgroup of $G$ must be cyclic as well as normal as otherwise $P(G)$ comprises $\{P_{2}\cup P_{3}\}$. Moreover, the Sylow $2$-subgroups of $G$ must be of exponent $2$ since $P(G)$ is $\{P_{2}\cup P_{3}\}$-free.\ [Converse Part:]{.ul}\ (1) Let $G$ be an EPPO group. Then any $3$ consecutive adjacent vertices must belong to the same cyclic subgroup of prime power order. So they form a triangle. Hence, $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free.\ (2) Let $G\cong C_{q^{a}r}\rtimes P$ along with the prescribed condition in (b). Clearly $P(G)$ never contains $P_{2}\cup P_{3}$. Every elements (non-identity) are of order power of $q$, or $r$, or $2$, or of the form $q^{i}r$. If $P(G)$ contains $\overline{P_{2}\cup P_{3}}$ then an induced $4$-cycle is contained in $P(G)$. So, to prove that $P(G)$ is $\overline{P_{2}\cup P_{3}}$-free it is enough to show that $P(G)$ is induced cycle $C_{4}$-free. Clearly in all the vertices of this cycle must belong to the cyclic subgroup $C_{q^{a}r}$. As, $P(C_{q^{a}r})$ is a chordal graph by Theorem [Theorem 6](#pre_th_3){reference-type="ref" reference="pre_th_3"} so $P(G)$ is $C_{4}$-free. Hence, $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free.\ (3) If $G$ is a group of order $2^{a}q^{b}$ ($q$ odd) such that the conditions in (c) hold. One can easily observe that $P(G)$ is $P_{2}\cup P_{3}$-free. Also, using the same argument as done in (2) we can conclude that $P(G)$ is $\overline{P_{2}\cup P_{3}}$-free. Hence, in this case $P(G)$ is $\{P_{2}\cup P_{3}, \overline{P_{2}\cup P_{3}}\}$-free. ◻ # Diamond-free The diamond graph is a planar, undirected and simple graph with $4$ vertices and $5$ edges. It consists of a complete graph $K_{4}$ with an edge deletion. It looks like: The complement of a diamond graph is called a co-diamond graph. Here we identify the finite groups having diamond-free as well co-diamond free power graph. **Lemma 1**. *Let $G$ be a finite group. Then $P(G)$ is a diamond-free graph if and only if $G$ is either a $p$-group or an EPPO group.* *Proof.* First suppose, $G$ is a finite group for which $P(G)$ is diamond-free.\ If possible let, $o(G)$ have $2$ or more distinct prime divisors. Let $p, q$ be two distinct prime divisors of $o(G)$. We claim that $G$ must be an EPPO group. If not, then there exist at least two elements of order $pq$. We choose the consecutive $4$ vertices as $a, b, c, d$ of order $p, pq, q, pq$ with $d=b^{-1}$. Then they form a diamond in $P(G)$. This gives $G$ must be an EPPO group.\ Therefore, $G$ is either a $p$-group or an EPPO group.\ [Converse Part:]{.ul}\ Let $G$ be a $p$-group. For the sake of contradiction, suppose $P(G)$ contains a diamond with the $4$ consecutive vertices as $a, b, c, d$. Clearly, as any $3$ consecutive adjacent vertices belong to same cyclic $p$-subgroup so they form a triangle in $P(G)$. This implies $a, b, c, d$ must be the complete graph $K_{4}$, which contradicts that $a, b, c, d$ is a diamond.\ Again, let $G$ be an EPPO group. If possible let $P(G)$ contain a diamond where the $4$ consecutive vertices as $a, b, c, d$. Suppose, $degree(b)=degree(d)=3$ and $degree(a)=degree(c)=2$. Now as $a\sim b \sim c$ and $G$ is EPPO group so $a, b, c$ belong to same cyclic subgroup of prime power order. In that case $a$ must adjacent to $c$. This contradicts the $a, b, c, d$ form a diamond.\ Thus in any cases $P(G)$ is diamond-free. ◻ **Theorem 26**. *Let $G$ be a finite group. Then $P(G)$ is $\{even-hole, diamond\}$-free if and only if $G$ is either a $p$-group or an EPPO group.* *Proof.* Firstly, let $G$ be a group with $P(G)$ is $\{even-hole, diamond\}$-free. Since, $P(G)$ is diamond-free so $G$ is either a $p$-group or an EPPO group.\ [Converse Part:]{.ul}\ Let $G$ be either a $p$-group or an EPPO group. Then $P(G)$ is a chordal graph \[see Theorem [Theorem 6](#pre_th_3){reference-type="ref" reference="pre_th_3"}\]. So, $P(G)$ does not contain any even-hole. Also, by Lemma [Lemma 1](#lm_dia){reference-type="ref" reference="lm_dia"} $P(G)$ is diamond-free. Hence $P(G)$ is $\{even-hole, diamond\}$-free. ◻ In the next result we use the Kulakoff theorem from group theory which states that: **Theorem 27** (Kulakoff Theorem). *Let $G$ be a p-group of order $p^\alpha$. Then* - *the number of subgroup of prime power order is congruent to $1$ (mod $p$).* - *if $G$ has unique subgroup of order $p^{\beta}$ for all $\beta$ with $1<\beta \leq \alpha$, then $G$ is cyclic or $\beta =1$ and $p=2$, $G$ is the generalized quaternion group $Q_{2^{\alpha}}$.* **Theorem 28**. *Let $G$ be a finite group. Then $P(G)$ is $\{diamond, co-diamond\}$-free if and only if $G$ is either a cyclic group of prime power order or a $2$-group of exponent $2$.* *Proof.* For any finite group $G$, let $P(G)$ be $\{diamond, co-diamond\}$-free. As, $P(G)$ is diamond-free so $G$ is either a $p$-group or an EPPO group.\ But if $G$ is an EPPO group then $o(G)$ can have exactly two distinct prime divisors. Otherwise, $P(G)$ contains a co-diamond. Additionally, we observe that both the Sylow subgroup of $G$ must be cyclic as well as normal; elsewhere if Sylow $p$-subgroup is either non-cyclic or not normal then $G$ contains $3$ elements say $a, b$ of order $p$ with $a\nsim b$ in $P(G)$ and $c\in G$ of order $q$. Clearly, $a,b, c, c^{-1}$ form a co-diamond. Next, let $p=2$ such that the Sylow $2$-subgroup is $C_{2}$ and normal and Sylow $q$-subgroups are either non-cyclic or not normal. Then $G$ contains two elements $x, y$ of order $q$ with $x \nsim y$ and $a\in P$. So, $a, x, x^{-1}, y$ form a co-diamond in $P(G)$. Therefore, in any cases, $o(G)$ has exactly two distinct prime divisors with all Sylow subgroups are cyclic and normal. This implies $G$ must be the group $C_{p^{r}q^{s}}$, which contradicts that $G$ is an EPPO group. Hence $G$ must be a $p$-group.\ If $G$ is a $p$-group we claim that $G$ is either a cyclic group of prime power order or a $2$-group of exponent $2$.\ Let $p$ be an odd prime. If $G$ is non-cyclic then (by Kulakoff theorem) there exist at least $3$ distinct subgroups of order $p$. Thus $P(G)$ contains a co-diamond. Hence in this case $G$ must be a cyclic group of prime power order.\ Now let $p=2$. If $G$ is cyclic then $P(G)$ is complete and so $\{diamond, co-diamond\}$-free. But if $G$ is non-cyclic then either $G$ is generalized quaternion group or $G$ has at least $3$ distinct minimal subgroups. For the latter case, $G$ does not contain any element of order $4$ and above, because otherwise we get a co-diamond. This gives $G$ must be a $2$-group of exponent $2$.\ Next consider $G$ as generalized quaternion. Since in this case $G$ has at least $3$ distinct elements of order $4$ so $P(G)$ carries a co-diamond. Therefore, if $G$ is a $p$-group then $G$ is either a cyclic group of prime power order or a $2$-group of exponent $2$.\ Converse part is obvious. ◻ # Acknowledgement The author Pallabi Manna is supported by Department of Atomic Energy (DAE), India and Santanu Mandal acknowledges VIT Bhopal University, India, for providing the infrastructure. # Statements and Declarations **Competing Interests:** The authors made no mention of any potential conflicts of interest. # Data Availability Data sharing is not applicable to this article as no data were created or analyzed in this study. 99 J. Abawajy, A. Kelarev, M. Chowdhury, Power graphs: A survey, *Electron. J. Graph Theory Appl.,* **1** (2013), 125--147. A. Brandstädt, V. B. Le and J. P. Spinrad, Graph Classes: A Survey, (SIAM Monographs on Discrete Mathematics and Applications, 1999), DOI:10.1137/1.9780898719796. I. Chakrabarty, S. Ghosh, M. K. Sen, Undirected power graphs of semigroups, *Semigroup Forum,* **78** (2009), 410--426. A. Doostabadi , M. Farrokhi and D. Ghouchan, On the connectivity of proper power graph of finite group, *Comm. Algebra,* **43** (2015), 4305--4319. P. J. Cameron, P. Manna and R. Mehatari, On finite groups whose power graph is a cograph, *J. Algebra,* **591** (2021), 59-74. H. Cohn, *Number Theory*, Graduate Texts in Mathematics **240**, Springer, New York, 2007. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, $\mathbb{ATLAS}$ of Finite Groups, Clarendon Press, Oxford, 1985. Bruce N. Cooperstein, Maximal subgroups of $G_2(2^n)$, *J. Algebra,* **70** (1981), 23--36. The GAP group, GAP-groups, algorithms and programming, Version 4.11.1, 2021. Available at: http://www.gap-system.org. A. Kelarev and S.J. Quinn, Directed graphs and combinatorial properties of semigroups, *J. Algebra,* **251** (2002), 16--26. O.H. King, The subgroup structure of finite classical groups in terms of geometric configurations, Surveys in combinatorics 2005, 29-56, *London Math. Soc. Lecture Note Ser.,* **327**, Cambridge Univ. Press, Cambridge, 2005. Peter B. Kleidman, The maximal subgroups of the Steinberg triality groups ${}^3D_4(q)$ and of their automorphism groups, *J. Algebra,* **115** (1988), 182--199. Peter B. Kleidman, The maximal subgroups of the Chevalley groups $G_2(q)$ with $q$ odd, the Ree groups ${}^2G_2(q)$ and of their automorphism groups, *J. Algebra,* **117** (1988), 30--71. Gunter Malle, The maximal subgroups of ${}^2F_4(q^2)$, *J. Algebra*, **139** (1991), 39--61. Pallabi Manna, Peter J. Cameron and Ranjit Mehatari, Forbidden subgraphs of power graphs, *Electron. J. Combin.*, **28(3)** (2021), P3.4, 14pp. The Sage Developers. SageMath, the Sage Mathematics Software System, Version 9.3, 2022. Available at:https://www.sagemath.org. Michio Suzuki, Finite groups with nilpotent centralizers, *Trans. Amer. Math. Soc.,* **99** (1961), 425--470. Michio Suzuki, On a class of doubly transitive groups, *Ann. Math.,* **75** (1962), 105--145. Robert A. Wilson, *The Finite Simple Groups*, Springer, London, 2009. M. Chudnovsky, L. Esperet, L. Lemoine, P. Maceli, F. Maffray and I. Penev, Graphs with no induced $P_{5}$ or $\overline{P_{5}}$, *J. Graph Theory*, **84(3)** (2017), 221-232. W. Dong, B. Xu and Y. Xu, On the chromatic number of some $P_{5}$-free graphs, *Discrete Math.,* **10** (2022), 113004. C. Arbib and R. Mosca, On $(P_{5}, diamond)$-free graphs, *Discrete Math.,* **250** (2002), 1-22. A. Char and T. Karthick, Optimal chromatic bound for $(P_{2}+P_{3}, P_{2}\mp P_{3})$-free graphs, *J. Graph Theory*, *https://doi.org/10.1002/jgt.23009*, (2023). A. Char and T. Karthick, Coloring of $(P_{5},$`<!-- -->`{=html}4$-wheel)$-free graphs, *Discrete Math.,* **345(5):112795** (2022). A. Char and T. Karthick, Improved bounds on the chromatic number of $(P_{5}, flag)$-free graphs, *Discrete Math.*, **346(9):113501** (2023). S. Huang and T. Karthick, On graphs with no induced five-vertex path or paraglider, *J. Graph Theory*, **97:2** (2021), 305-323. M. Chudnovsky, T. Karthick, P. Maceli and F. Maffray, Coloring graphs with no induced five-vertex path or gem, *J. Graph Theory*, **95:4** (2020), 527-542. [^1]: santanu.vumath\@gmail.com [^2]: mannapallabimath001\@gmail.com

arxiv_math

--- abstract: | This paper investigates the polynomial stability of solutions to abstract thermoelastic systems with Cattaneo's law on Hilbert space. The study includes two cases: one with an inertial term and the other without it. For the case with an inertial term, we partition the region of parameters for non-exponential stability into three sub-regions and obtain distinct polynomial decay rates for each sub-region. Meanwhile, for the case without an inertial term, we explore two areas and obtain dissimilar polynomial decay rates in these areas. Moreover, we demonstrate that all the given polynomial decay rates are sharp. Finally, we give some examples to explain our conclusion.   35Q74, 74F05 author: - "Chenxi Deng[^1] ,Zhong-Jie Han[^2],Zhaobin Kuang[^3], Qiong Zhang[^4] [^5]" title: | Stability of Abstract Thermoelastic Systems with Cattaneo's Law and Inertial Term [^6] --- thermoelastic system; Cattaneo's law; inertial term; polynomial stability. # Introduction {#sec:intro} Let $H$ be a complex Hilbert space with inner product $(\cdot,\,\cdot)$ and the induced norm $\|\cdot\|$, $A$ be a self-adjoint, positive definite operator with compact resolvent on $H$. We consider the following abstract coupled thermoelastic system: $$\label{101} \left\{ \begin{array}{l} u_{tt} +mA^\gamma u_{tt} +\sigma A u -A^\alpha\theta =0, \\ \noalign{\medskip} \displaystyle \theta_t -A^{\frac{\beta}{2}}q +A^\alpha u_t =0, \\ \noalign{\medskip} \displaystyle \tau q_t+q+A^{\frac{\beta}{2}}\theta=0, \\ \noalign{\medskip} \displaystyle u(0)= u_0, \;\; u_t(0) = u_1,\;\; \theta(0) =\theta_0,\;\; q(0) = q_0, \end{array} \right.$$ where the constants $m\geq 0, \; \sigma>0,\; \tau>0$ correspond to the inertial term, wave speed and relaxation parameter, respectively. The parameters $(\alpha,\;\beta,\;\gamma) \in E := [0,1]\times[0,1]\times (0,1]$ represent the coupling order, damping order, and inertial order, respectively. The system with $\gamma=0$, corresponding to the system without inertial term, can be contained in the case $m=0$, which will be investigated in Section [4](#8.7){reference-type="ref" reference="8.7"}. If $\tau = 0$, the heat conduction in the system follows the classical Fourier's law. However, if $\tau>0$, the system follows the Cattaneo's law, which is used to describe the finite speed of heat propagation in a medium. This eliminates the paradox of infinite speed of heat transfer in Fourier's law and describes the wave-like motion of heat, also known as the second sound in physics. The objective of this work is to investigate the long-time behavior of the coupled system [\[101\]](#101){reference-type="eqref" reference="101"} with Cattaneo's law. In particular, we demonstrate the impact of thermoelastic parameters $\alpha, \; \beta$ and the inertial term (determined by $m$) on the stability of this abstract coupled system [\[101\]](#101){reference-type="eqref" reference="101"}. The natural energy of system [\[101\]](#101){reference-type="eqref" reference="101"} is $$\label{diss} E(t)=\frac{1}{2}\big(\|u_t\|^2+m\|A^{\frac{\gamma}{2}}u_t\|^2+\sigma\|A^{\frac{1}{2}} u\|^2+\|\theta\|^2+\tau\|q\|^2\big).$$ A direct computation gives $${d\over dt} E(t)=-\|q\|^2 \leq 0.$$ Hence, the system is dissipative. There is a significant body of work that studies the asymptotic behavior of thermoelastic systems ([\[101\]](#101){reference-type="ref" reference="101"}) under various heat conduction mechanisms, including Fourier's law ($\tau=0$) and Cattaneo's law ($\tau\ne0$), with ($m\ne0$) or without ($m=0$) inertial terms. In [@russell], Russell developed an elastic system with indirect damping that encompasses thermoelastic string, beam, and plate equations. It is precisely the system ([\[101\]](#101){reference-type="ref" reference="101"}) with $\tau=m=0,~\sigma=1$. Later, Rivera and Racke studied the regularity and exponential stability of the system in [@rivera1], where they also provided specific examples of the system ([\[101\]](#101){reference-type="ref" reference="101"}). Ammar-Khodja *et al.* [@Ammar] identified the regions of exponential stability for the system ([\[101\]](#101){reference-type="ref" reference="101"}) with $\tau=m=0$. Hao and Liu [@haoliu1] investigated the polynomial stability of the system outside the regions of exponential stability in [@Ammar] and presented optimal polynomial decay rates. Further details on the case $\tau=m=0$ can be found in [@Avalos; @haoliu2; @Kim; @Lasiecka; @LiuLiu] and the references therein. Regarding the case $\tau=0,~m\ne0$, Dell'Oro *et al.* [@rivera2] investigated the stability of the system ([\[101\]](#101){reference-type="ref" reference="101"}) for the special case $\beta=\gamma=\frac{1}{2}$, $0\leq\alpha\leq\frac{3}{4}$, which, to the best of the authors' knowledge, was the first result that considered the inertial term. In [@liure], the authors provided a comprehensive overview of the regions of exponential stability and polynomial stability for parameters $(\alpha,\beta)\in[0,1]\times [0,1]$, $\gamma\in(0,1]$. The authors in [@kuang] pointed out that the polynomial decay rates in [@liure] for the system with Fourier's law are sharp. The study of thermoelastic systems with Fourier's law is relatively complete, but other heat conduction mechanisms, such as Cattaneo's law and Vernotte's law, are also interesting to study, see [@Cattaneo; @Cattaneo1; @lord; @vernotte] for more details. Liu *et al.* studied the thermoelastic model with Cattaneo's law in [@liure]. They claimed that "the change from Fourier's law to Cattaneo's law leads to a loss of exponential stability in most coupled systems." Later, Han, Kuang and Zhang [@han] showed that if the system has the same wave speed, that is, $\sigma\tau=1$, the region of exponential stability shown in [@liure] can be larger. Moreover, they characterized the polynomial decay rates of the system ([\[101\]](#101){reference-type="ref" reference="101"}) with two special choices of $\alpha$. In this paper, we aim to address the question posed in the article by Han *et al.* [@han] regarding the stability of the system ([\[101\]](#101){reference-type="ref" reference="101"}) on Hilbert space with Cattaneo's law. While Han *et al.* focused on exponentially stable regions of the system, we investigate the polynomial stability of the system both with and without an inertial term outside those exponentially stable regions. For the system with an inertial term, we divide the non-exponentially stable region into three parts and get different polynomial decay rates for each part; whereas in the case without an inertial term, we only need to consider two regions and also obtain different polynomial decay rates on these regions. Furthermore, we prove that the polynomial decay rates above are all optimal. Our main analysis method is the frequency domain method developed by Borichev and Tomilov [@bobtov1], along with interpolation inequalities and detailed spectral analysis to support our proofs. The paper is structured as follows. In Section [2](#sec3){reference-type="ref" reference="sec3"}, we describe the problem and show our main results. We demonstrate the proof of polynomial stability of the system with and without an inertial term in and , respectively. Then we show the decay rate in each area is optimal in Section [5](#8.6){reference-type="ref" reference="8.6"}. In Section [6](#8.5){reference-type="ref" reference="8.5"}, the polynomial stability of some Rayleigh beam equations with thermal damping is given using our results. Finally, we give a conclusion of our work in Section [7](#8.4){reference-type="ref" reference="8.4"}. # Preliminaries and Main Results {#sec3} In this section, we state some preliminaries and the main results of this paper. To describe system [\[101\]](#101){reference-type="eqref" reference="101"} as an abstract first-order evolution equation, we define a Hilbert space $$\mathcal{H}:=\mathcal{D}(A^{\frac{1}{2}})\times \mathcal{D}(A^{\frac{\gamma}{2}})\times H\times H,$$ which is equipped with the inner product $$\langle U_1,U_2\rangle_{\mathcal{H}}=\sigma(A^{\frac{1}{2}}u_1,A^{\frac{1}{2}}u_2)+m(A^{\frac{\gamma}{2}}v_1, A^{\frac{\gamma}{2}}v_2)+(v_1,v_2)+(\theta_1, \theta_2)+\tau (q_1,q_2),$$ where $U_{i}=(u_{i},v_{i},\theta_{i},q_{i})\in\mathcal{H},\;i=1,2$. Let $t>0$, introduce a new variable $v(t) := u_t(t)$, define $U(t):=(u(t), v(t), \theta(t), q(t))^\top$, $U_0:=(u_0, u_1, \theta_0 , q_0)^\top \in \mathcal{H}$, then the system [\[101\]](#101){reference-type="eqref" reference="101"} can be written as an evolution equation in $\mathcal{H}$: $$\label{105} \left\{ \begin{array}{l} \displaystyle {d\over dt} U(t)=\mathcal{A} U(t) ,\quad t>0,\\ \noalign{\medskip} \displaystyle U(0)=U_0, \end{array} \right.$$ where the operator $\mathcal{A}:{\cal D}({\cal A}) \subset \mathcal{H}\to \mathcal{H}$ is defined by $$\mathcal{A}\left[\begin{array}{c} u\\ v\\ \theta\\ q \end{array} \right]=\left(\begin{array}{c} v\\ -(I+mA^{\gamma})^{-1}(\sigma A u-A^{\alpha}\theta)\\ -A^\alpha v+A^{\frac{\beta}{2}}q\\ \displaystyle {1\over \tau}(-q-A^{\frac{\beta}{2}}\theta) \end{array}\right),$$ with domain $$\mathcal{D}(\mathcal{A})=\left\{ (u, v, \theta, q)\in \mathcal{H}\,\left|\, %\begin{array}{l} v\in \mathcal{D}(A^{1\over2}),\; \sigma A u-A^\alpha\theta \in \mathcal{D}(A^{ -\frac{\gamma}{2}}),\; -A^\alpha v+A^{\frac{\beta}{2}}q\in H,\; q+A^{\frac{\beta}{2}}\theta\in H % \theta \in \mathcal{D}( A^{\frac{\beta}{2}}) %\end{array} \right.\right\}.$$ It is clear that $$\mathcal{D}(\mathcal{A}) \subset\left\{ (u, v, \theta, q)\in \mathcal{D}(\mathcal{A})\,\left|\,v\in \mathcal{D}(A^{\alpha}),\; q\in \mathcal{D}( A^{\frac{\beta}{2}})\right.\right\} \quad \; \mbox{when } \; 0\le \alpha\leq {1\over2},$$ and $$\mathcal{D}(\mathcal{A}) \subset\left\{ (u, v, \theta, q)\in \mathcal{D}(\mathcal{A})\,\left|\,u\in \mathcal{D}(A^{1-\alpha})\right.\right\} \quad \; \mbox{when } \; \frac{1}{2}<\alpha\leq 1.$$ The well-posedness of the system [\[105\]](#105){reference-type="eqref" reference="105"} is stated as follows. **Lemma 1**. *Let $\mathcal{A}$ and $\mathcal{H}$ be defined as above and $(\alpha, \beta, \gamma)\in {[0,1]\times[0,1]\times(0,1]}$. If $\beta\geq 2\alpha-1$, then $0\in \rho(\mathcal{A})$ and $\mathcal{A}$ generates a $C_{0}$ semigroup $e^{{\cal A}t }$ of contractions on $\mathcal{H}$.* *Proof.* It is easy to see that $\mathcal{A}$ is a densely defined operator on $\mathcal{H}$ and $${\cal R}e \langle\mathcal{A}U,U \rangle_{\mathcal{H}}=-\|q\|^2\leq 0,\;\forall \; U\in\mathcal{D}(\mathcal{A}),$$ then $\mathcal{A}$ is dissipative in $\mathcal{H}$. We claim that $\mathcal{A}$ is a bijection. On the one hand, for any given $G:=(g_1,g_2,g_3,g_4)^{\top}\in \mathcal{H}$, let $U=(u,v,\theta,q)^{\top}$, where $$\begin{aligned} v =g_1, \quad \theta =-A^{-\frac{\beta}{2}}(\tau g_4+A^{-\frac{\beta}{2}}(g_3+A^{\alpha}g_1)), \quad q =A^{-\frac{\beta}{2}}(g_3+A^{\alpha}g_1),\end{aligned}$$ and $$u =\sigma^{-1}A^{-1}(A^{\alpha}\theta-(I+mA^{\gamma})g_2).$$ Then $U \in \mathcal{D}(\mathcal{A})$ and $\mathcal{A} U=G$. On the other hand, it is clear that ${\cal A}$ is an injective. Thus $\mathcal{A}^{-1}$ exists. Now we show $\mathcal{A}^{-1}$ is bounded. Note that $\mathcal{D}(A^{\frac{1}{2}})\subset \mathcal{D}(A^{\frac{\gamma}{2}})$, we have $$\label{202} \|v\|_{\mathcal{D}(A^{\frac{\gamma}{2}})}\leq\|g_1\|_{\mathcal{D}(A^{\frac{1}{2}})}\leq C\|G\|_{\mathcal{H}}.$$ Recalling $\beta\ge2\alpha-1$, we can also get $$\label{203} \|q\|\leq C(\|g_3\|+\|A^{\alpha-\frac{\beta}{2}}g_1\|) \leq C(\|g_3\|+\|g_1\|_{\mathcal{D}(A^{\frac{1}{2}})})\leq C\|G\|_{\mathcal{H}},$$ $$\label{204} \|\theta\|\leq C(\|g_4\|+\|q\|)\leq C\|G\|_{\mathcal{H}},$$ and $$\label{205} \begin{array}{lcl} &\leq& C\big(\|A^{-\frac{1}{2}+\alpha-\frac{\beta}{2}}g_4\|+ \|A^{-\frac{1}{2}+\alpha-\beta}g_3\| + \|A^{2\alpha-\beta-\frac{1}{2}}g_1\| + \|A^{-\frac{1}{2}}(I+mA^\gamma)g_2\|\big) \cr &\leq& C\|G\|_{\mathcal{H}}\quad (\mbox{Using the assumption}\; \beta\geq 2\alpha-1). \end{array}$$ Consequently, according to ([\[202\]](#202){reference-type="ref" reference="202"})-([\[205\]](#205){reference-type="ref" reference="205"}), one has that $\mathcal{A}^{-1}$ is bounded. By the Lumer-Phillips Theorem [@pazy], $\mathcal{A}$ generates a $C_{0}$ semigroup $e^{{\cal A}t }$ of contractions on $\mathcal{H}$. The proof is completed. ◻ Our interest is the stability properties of system [\[101\]](#101){reference-type="eqref" reference="101"}, especially the polynomial stability when $\alpha, \;\beta,\; \gamma$ satisfy certain conditions. Let us recall the corresponding definition. **Definition 1**. *Let $\mathcal{A}\,:\,\mathcal{D}(\mathcal{A}) \subset \mathcal{H} \to \mathcal{H}$ generate a $C_0$ semigroup $e^{\mathcal{A}t}$ of contractions on a Hilbert space $\mathcal{H}$.* 1. *$e^{\mathcal{A}t}$ is said to be **strongly stable** if $\underset{t\to\infty}{\lim}\|e^{\mathcal{A}t}x_0 \|_{\mathcal{H}}=0, \quad \forall \;x_0\in \mathcal{H};$* 2. *$e^{\mathcal{A}t}$ is said to be **exponentially stable** with decay rate $\omega>0$, if there exists a constant $C>0$ such that $$\label{def1} \|e^{\mathcal{A}t} \|_{{\cal L}(\mathcal{H})}\le C e^{-\omega t}, \quad \forall \;t\ge 0;$$* 3. *$e^{\mathcal{A}t}$ is said to be **polynomially stable** of order $k>0$ if there exists a constant $C>0$ such that $$\label{def2} \|e^{\mathcal{A}t}\mathcal{A}^{-1}\|_{{\cal L}(\mathcal{H})} \le C t^{-k}, \quad \forall \;t>0.$$* Following the result in [@Arendt; @huang], the strong stability of the semigroup $e^{t\mathcal{A}}$ is equivalent to $i {\mathbb R}\subset \rho(\mathcal{A})$ since $0\in \rho(\mathcal{A})$ and $\mathcal{A}^{-1}$ is compact ([@han]). The proof of the following lemma will be given in Appendix. **Lemma 2**. *Assume that $(\alpha, \beta,\gamma)\in E$ satisfies $\beta\geq 2\alpha-1$. Then $i {\mathbb R}\subset \rho(\mathcal{A})$ for $m\ge0$, and consequently, the semigroup $e^{t\mathcal{A}}$ is strongly stable.* We further discuss the explicit decay rates of the system with $(\alpha,\beta,\gamma)\in E$. To begin with, we divide the parameter space $E$ into the following regions according to the computation in Section [5](#8.6){reference-type="ref" reference="8.6"}. **Definition 2**. *The parameter space $E = \left\{( \alpha,\beta,\gamma) \mid \alpha \in [0,1], \beta \in [0,1], \gamma \in (0,1] \right\}$ can be partitioned into the following non-overlapping regions $$\begin{aligned} T_1 =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \frac{1}{2}<\alpha <\frac{\beta +1}{2},\;0<\beta \leq 1,\; 0 < \gamma <2 \alpha -\beta\right\} , \\ T_2 =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; 0 \leq \alpha <\frac{1}{2},\;0<\beta <1,\; 0 < \gamma <1-\beta\right\}, \\ T_3 =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; 0\leq \alpha <\frac{\beta +\gamma }{2},\;1-\gamma <\beta \leq 1,\;0<\gamma \le 1\right\}, \\ T_4 =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \frac{\beta +1}{2}<\alpha \leq 1,\;0\leq \beta <1,\;0 < \gamma \leq 1\right\}, \\ F_{12} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \alpha =\frac{1}{2},\;0<\beta <1-\gamma ,\;0 < \gamma <1\right\}, \\ F_{13} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \alpha =\frac{\beta +\gamma }{2},\;1-\gamma <\beta \leq 1,\; 0 < \gamma <1\right\}, \\ F_{14} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \alpha =\frac{\beta +1}{2},\;0<\beta \leq 1,\;0 < \gamma <1\right\}, \\ F_{2} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; 0\leq \alpha <\frac{1}{2},\;\beta =0,\;0 < \gamma <1\right\}, \\ F_{23} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; 0\leq \alpha <\frac{1}{2},\;\beta =1-\gamma ,\;0<\gamma <1\right\}, \\ L_{123} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \alpha =\frac{1}{2},\;\ \beta =1-\gamma,\;\ 0<\gamma <1\right\}, \\ L_{124} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \alpha =\frac{1}{2},\ \beta =0,\ 0 < \gamma <1\right\},\; \\ L_2 =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; 0\leq \alpha <\frac{1}{2},\;\beta =0,\;\gamma =1\right\},\; \\ L_{34} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \alpha =\frac{\beta +1}{2},0<\beta \leq 1,\;\gamma =1\right\}, \\ P_{234} =& \left\{(\alpha,\beta,\gamma) \in E \;\Big|\; \alpha =\frac{1}{2},\;\beta =0,\;\gamma =1\right\}.\end{aligned}$$* **Remark 1**. *From the definition, it is clear that planes $F_{12},\;F_{13},\;F_{14}, \;F_{23}$ are (parts) of the intersections of $T_i$ and $T_j$, lines $L_{123},\;L_{124}$ are intersections of $T_i,\;T_j,\; T_k$. Furthermore, plane $F_2$ is a part of the boundary of $T_2$, $L_2$ is a special part of the boundary of $F_2$, $L_{34}$ is a part of the boundary of $F_{14}$, $P_{234}$ is a point. The regions in are plotted in Figure [\[fig:E\]](#fig:E){reference-type="ref" reference="fig:E"}.* *It is known from [@han Theorem 2.1] that on the plane $F_{13}$, the lines $L_{123}$, $L_{34}$ and the point $P_{234}$, the semigroup is exponentially stable. On the region $T_{4},$ $0$ is a spectrum of $\mathcal{A}$. Moreover, the asymptotic behavior on $L_2$ is not clear now (see computation in Section [5](#8.6){reference-type="ref" reference="8.6"}). In this paper, we ignore these cases.* {#m>0 width="6.5cm"}     \     \ According to the computation in Section [5](#8.6){reference-type="ref" reference="8.6"}, we shall analyze the polynomial stability of system [\[101\]](#101){reference-type="eqref" reference="101"} with $m\not=0$ when parameters $(\alpha,\,\beta,\,\gamma)$ belong to the following three regions: $$\label{par1} \begin{array}{l} \displaystyle V_1:=T_1\cup F_{12}\cup F_{14}\cup L_{124}, \\ \noalign{\medskip} \displaystyle V_2:=T_2\cup F_{2}, \\ \noalign{\medskip} \displaystyle V_3:=T_{3}\cup F_{23}. \end{array}$$ The regions $V_i, \; i=1,2,3$ are shown in Figure [1](#m>0){reference-type="ref" reference="m>0"}. The first main result of this paper read as follows. **Theorem 1**. *Let $m>0$ and $V_i,i=1,2,3$ be defined as in [\[par1\]](#par1){reference-type="eqref" reference="par1"}. Then semigroup $e^{\mathcal{A}t}$ associated with ([\[105\]](#105){reference-type="ref" reference="105"}) has the following stability properties:* 1. *It is polynomially stable of order $k_1=\displaystyle \frac{2\alpha-\gamma}{2(2\alpha-\beta-\gamma)}$ in $V_1$;* 2. *It is polynomially stable of order $k_2=\displaystyle \frac{1-\gamma}{2(-2\alpha-\beta-\gamma+2)}$ in $V_2$;* 3. *It is polynomially stable of order $k_3=\displaystyle \frac{1-\gamma}{2(-2\alpha+\beta+\gamma)}$ in $V_3$.* We also consider the case when there is no inertial term in system [\[105\]](#105){reference-type="eqref" reference="105"}. Similarly, we partition the parameter space $E^*:=[0,1]^2$ as follows. **Definition 3**. *The parameter space $E^* = \left\{( \alpha,\beta) \mid \alpha \in [0,1], \beta \in [0,1] \right\}$ can be partitioned into the following non-overlapping regions $$\begin{aligned} T_1^* =& \left\{(\alpha,\beta) \in E \Big| \frac{1}{2}<\alpha <\frac{\beta +1}{2},0<\beta \leq 1\right\} ,\\ T_2^* =& \left\{(\alpha,\beta) \in E \Big| 0\leq \alpha <\frac{1}{2},0<\beta <1\right\},\\ T_4^* =& \left\{(\alpha,\beta) \in E \Big| \frac{\beta +1}{2}<\alpha \leq 1,0\leq \beta <1\right\},\\ F^*_{12} =& \left\{(\alpha,\beta) \in E \Big| \alpha =\frac{1}{2},0<\beta <1\right\},\\ F^*_{14} =& \left\{(\alpha,\beta) \in E \Big| \alpha =\frac{\beta +1}{2},0<\beta \leq 1\right\},\\ F^*_{2} =& \left\{(\alpha,\beta) \in E \Big| 0\leq \alpha <\frac{1}{2},\beta =0\right\},\\ L^*_{124} =& \left\{(\alpha,\beta) \in E \Big| \alpha =\frac{1}{2},\beta =0\right\},\\ L^*_{23} =& \left\{(\alpha,\beta) \in E \Big| 0\leq \alpha <\frac{1}{2},\beta =1\right\},\\ P^*_{123} =& \left\{(\alpha,\beta) \in E \Big| \alpha =\frac{1}{2},\beta =1\right\}.\end{aligned}$$* We plot the regions given in in Figure [\[8131\]](#8131){reference-type="ref" reference="8131"}. The lines $F^*_{12}, F^*_{14}$ denote the boundary of $T^*_1$, $T^*_2$ and $T^*_1$, $T^*_4$, respectively. The line $F^*_2$ and $L_{23}$ are the boundary of $T^*_2$. Point $L^*_{124}$ is the intersection of $T_i^*, i=1,2,4$. The region $T^*_4$ is the area where $0$ is a spectrum of $\mathcal{A}$. It is known from [@han Theorem 2.2] that the system is exponentially stable on the point $P_{123}^*$. We consider the decay rate of the solution when parameters belonging to the remaining area $E^*\backslash (T_4^*\cup P^*_{123})$ and partition the region into the following two parts according to the computation in Section 5. $$\label{par2} \begin{array}{l}\displaystyle V_1^*:=T_1^*\cup F^*_{12}\cup F^*_{14}\cup L^*_{124}, \\ \noalign{\medskip} \displaystyle V_2^*:=T_2^*\cup F^*_2\cup L^*_{23}. \end{array}$$ {#8133} {#8133} The regions of $V_i^*$ are shown in Figure [3](#8133){reference-type="ref" reference="8133"}. The second main result is as follows. **Theorem 2**. *Let $m=0$ and $V_i^*,i=1,2$ be defined as in[\[par2\]](#par2){reference-type="eqref" reference="par2"}. Then semigroup $e^{\mathcal{A}t}$ associated with ([\[105\]](#105){reference-type="ref" reference="105"}) satisfies* 1. *It is polynomially stable of order $\displaystyle k_1^*=\frac{\alpha}{2\alpha-\beta}$ in $V_1^*$;* 2. *It is polynomially stable of order $\displaystyle k_2^*=\frac{1}{2(2-2\alpha-\beta)}$ in $V_2^*$.* Furthermore, through a detailed and creative spectral analysis, we can show the sharpness of the decay rates obtained in the above two theorems. **Theorem 3**. *The orders of polynomial decay in and are all optimal.* In Theorem 2.1 and 2.2, we provide a complete polynomial stability analysis for system [\[101\]](#101){reference-type="eqref" reference="101"}. When $m>0$, i.e., the system [\[101\]](#101){reference-type="eqref" reference="101"} is with an inertial term, we divide the parameter region into three parts, where the semigroup decays polynomially at different rates. In the case where there is no inertial term (i.e., $m=0$), we also establish a relationship between the parameters $\alpha, \; \beta$ and the order of polynomial stability. Our results complement the stability analysis of the system previously discussed in [@han]. On the other hand, it is known that the decay rates, as obtained in Theorem [Theorem 2](#m00){reference-type="ref" reference="m00"}, are slower when compared to the findings presented in [@haoliu2]. The works in [@haoliu2] focused on a thermoelastic system with Fourier's law while neglecting the inertial term. This observation reveals that the thermal damping of Cattaneo's type is weaker than the Fourier's law. Our subsequent findings on polynomial stability reply on the following lemmas, which provide necessary and sufficient conditions for the polynomial stability of $C_0$ semigroup (see [@bobtov1; @liurao] or [@batty; @Rozendaal] for more general case). **Lemma 3**. *Assume that $\mathcal{A}$ generates a $C_0$ contraction semigroup $e^{\mathcal{A}t}$ on a Hilbert space $\mathcal{H}$ and satisfies $i\mathbb{R}\subset \rho(\mathcal{A})$. Then, $e^{\mathcal{A}t}$ is polynomially stable with order $k$, if and only if $$\label{e2} \limsup_{|s|\to\infty}\limits s^{-{1\over k}}\|(is-\mathcal{A})^{-1}\|_{\mathcal{L}(\mathcal{H})}<\infty.$$* The following interpolation theorem will play a crucial role in our proof. **Lemma 4**. *Let $A \,:\, {\cal D}(A) \subset H$ be self-adjoint and positive definite, $r,\;p\;,q\in \mathbb{R}$. Then $$\begin{aligned} \label{4.41} \|A^p x\| \le \|A^q x\|^{p-r\over q-r} \|A^r x\|^{q-p\over q-r}, \quad \forall \; r\le p \le q, \; x\in {\cal D}(A^q). \end{aligned}$$* *Proof.* The case $0\leq r\le p \le q$ can be found in [@liuzheng]. If $r\leq 0$ and $0\le p \le q$, we let $y\in D(A^{q-r}) ,\; x = A^{-r}y.$ It follows from ([\[4.41\]](#4.41){reference-type="ref" reference="4.41"}) that $$\|A^p x\| = \|A^{p-r}y\| \le \|A^{q-r}y\|^{{p-r\over q-r}}\| y\|^{q-p\over q-r}=\|A^q x\|^{p-r\over q-r} \|A^r x\|^{q-p\over q-r}.$$ The other cases can be proved by the same argument. ◻ # Polynomial Stability of the System with an Inertial Term (Proof of ) {#8.8} This section is devoted to analyzing the polynomial stability of system [\[101\]](#101){reference-type="eqref" reference="101"} with $m>0$. Due to Lemma [Lemma 2](#l202){reference-type="ref" reference="l202"} and [Lemma 3](#l-2-1){reference-type="ref" reference="l-2-1"}, it is sufficient to show that there exists a constant $r > 0$ such that $$\label{huang2} \inf \big\{ \lambda^{k}\|i\,\lambda U- {\cal A} U\|_{\cal H} \,\big|\quad \|U\|_{\cal H}=1,\: \lambda\in{\mathbb R} \big\} \geqslant r$$ holds for some $k>0$. Suppose by contradiction that [\[huang2\]](#huang2){reference-type="eqref" reference="huang2"} fails. Then there exists a sequence $\{ \lambda_{n}, U_{n} \}_{n=1}^\infty$, where $\lambda_{n}\in {\mathbb R} ,\; U_n:=(u_{n},\, {v}_{n},\, \theta_{n},\, q_{n})\in D({\cal A})$ such that $$\label{unitnorm} \|U_{n}\|_{\cal H} =\big\| (u_{n},\, {v}_{n},\, \theta_{n},\, q_{n})\big\|_{ \cal H } =1,$$ and as $n\to\infty$, $$\begin{aligned} & \lambda_{n}\to\infty,\nonumber \\ \noalign{\medskip} & \displaystyle \label{331} \lambda_{n}^{k}\big( i\, \lambda_{n} U_{n}- {\cal A} U_{n} \big) =o(1) \quad \; \mbox{ in } \; {\cal H} ,\end{aligned}$$ i.e., $$\begin{aligned} &&\lambda_n^k(i\lambda_nA^{\frac{1}{2}}u_n- A^{\frac{1}{2}}v_n)=o(1)\quad \; \mbox{ in } \; H,\label{s1+}\\ \noalign{\medskip} \displaystyle &&\lambda_n^kA^{-\frac{\gamma}{2}}\left(i\lambda_nv_n+i\lambda_n m A^{\gamma}v_n+\sigma A u_n-A^{\alpha}\theta_n\right)=o(1)\quad \; \mbox{ in } \; H,\label{s2+}\\\noalign{\medskip} \displaystyle &&\lambda_n^k(i\lambda_n\theta_n-A^{\frac{\beta}{2}}q_n+A^\alpha v_n)=o(1)\quad \; \mbox{ in } \; H,\label{s3+}\\\noalign{\medskip} \displaystyle &&\lambda_n^k(i\lambda_n\tau q_n+ q_n+A^{\frac{\beta}{2}}\theta_n)=o(1)\quad \; \mbox{ in } \; H.\label{s4+}\end{aligned}$$ **Lemma 5**. *For $m\ge 0$, one has $$\begin{aligned} \label{v1_0} && \|q_n\|= \lambda_n^{-\frac{k}{2}} o(1), \\ \noalign{\medskip} \displaystyle \label{v1_8} && \| A^{\frac{\beta}{2}}\theta_n\|=\lambda_n^{1-\frac{k}{2}}o(1), \\ \noalign{\medskip} \displaystyle \label{+211} && \|A^{\frac{1}{2}}u_n\|,\; \|v_n\|,\; \|\theta_n\|,\; \|\lambda_n^{-1}A^{\frac{1}{2}}v_n\| =O(1).\end{aligned}$$ Furthermore, when $m>0,$ it holds $$\label{+211-1} \|A^{\gamma \over2}v_n\|,\;\;\|\lambda_n A^{\gamma \over2}u_n\| =O(1) .$$* *Proof.* We can obtain [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"} by the dissipativeness of $\mathcal{A}$ and [\[331\]](#331){reference-type="eqref" reference="331"}. Then, combining [\[s4+\]](#s4+){reference-type="eqref" reference="s4+"} and [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"} yields [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}. [\[+211\]](#+211){reference-type="eqref" reference="+211"} and [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"} are clear from [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"} and [\[s1+\]](#s1+){reference-type="eqref" reference="s1+"}. ◻ Now, we proceed to show (i)--(iii) in , respectively. **(i)** Assume $(\alpha, \, \beta,\, \gamma) \in V_1$ and let $k = \frac{2( 2\alpha-\beta-\gamma)}{2\alpha-\gamma}$. From [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} and the assumption $- \alpha+ \frac{\gamma}{2}<-{ \beta\over 2}\leq 0$, we have that $$i\lambda_nA^{-\alpha+\frac{\gamma}{2}}\theta_n-A^{-\alpha+\frac{\gamma}{2}+\frac{\beta}{2}}q_n+A^{\frac{\gamma}{2}} v_n =o(1).$$ Combining these with [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"} yields $$\label{v1_1} \|A^{-\alpha+\frac{\gamma}{2}}\theta_n\|=\lambda_n^{-1}O(1).$$ By using Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, [\[v1_1\]](#v1_1){reference-type="eqref" reference="v1_1"} and the choice of $k$ given above, one has $$\label{v1_2} \left\| \theta_n\right\| \leq \|A^{\frac{\beta}{2}} \theta_n\|^{\frac{2\alpha-\gamma}{2\alpha+\beta-\gamma}} \| A^{-\alpha+\frac{\gamma}{2}} \theta_n \| ^{\frac{\beta}{2\alpha+\beta-\gamma}}=o(1).$$ Note that $A^{-\alpha+\frac{\gamma}{2}}\theta_{n}$ and $A^{-\alpha}(I+mA^{\gamma})v_n$ are bounded due to $-\alpha+\frac{\gamma}{2}<0$ and [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}. Then, taking the inner product of [\[s2+\]](#s2+){reference-type="eqref" reference="s2+"} with $\lambda_n^{-k}A^{-\alpha+\frac{\gamma}{2}}\theta_{n}$, [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} with $\lambda_n^{-k}A^{-\alpha}(I+mA^{\gamma})v_n$, respectively, we have $$\label{v1_3} (\theta_{n},i\lambda_nA^{-\alpha}(I+{m}A^{\gamma})v_n)+\sigma(A^{1-\alpha}\theta_n,u_n)-(\theta_{n},\theta_{n})=o(1),$$ $$\label{v1_4} (i\lambda_n\theta_n, A^{-\alpha}(I+mA^{\gamma})v_n)-(A^{\frac{\beta}{2}-\alpha}(I+{m}A^{\gamma})q_n, v_n)+((I+mA^{\gamma})v_n, v_n)=o(1).$$ Adding the above two equations, we get $$\label{v1_5} \sigma(A^{1-\alpha}\theta_n,\, u_n)-(A^{\frac{\beta}{2}-\alpha}(I+mA^{\gamma})q_n,\, v_n) -\|\theta_{n}\|^2+\|v_n\|^2+m\|A^{\frac{\gamma}{2}}v_n\|^2=o(1).$$ Combining $\alpha>{1\over2} ,\;\alpha> {\beta+\gamma \over 2}$ and [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}, [\[v1_2\]](#v1_2){reference-type="eqref" reference="v1_2"}, one has $$\label{vq} (A^{{1\over2}-\alpha}\theta_n,\, A^{1\over2} u_n),\;\; (A^{\frac{\beta+\gamma}{2}-\alpha}q_n,\, A^{-{\gamma\over 2}}(I+mA^{\gamma}) v_n) = o(1).$$ Substituting [\[vq\]](#vq){reference-type="eqref" reference="vq"} into [\[v1_5\]](#v1_5){reference-type="eqref" reference="v1_5"}, along with ([\[v1_2\]](#v1_2){reference-type="ref" reference="v1_2"}), yields $$\label{v1_10} \|v_n\|,~~\|A^{\frac{\gamma}{2}}v_n\| =o(1).$$ Now, taking the inner product of [\[s2+\]](#s2+){reference-type="eqref" reference="s2+"} with ${\lambda_n^{-k}}A^{\frac{\gamma}{2}}u_n$ on $H$, we get $$\label{v1_6} (i\lambda_nv_n, u_n) +(i\lambda_n{m}A^{\gamma}v_n, u_n) +\sigma (A u_n, u_n) -(A^{\alpha}\theta_n, u_n)\to 0,$$ which together with [\[s1+\]](#s1+){reference-type="eqref" reference="s1+"}, one can conclude that $$\label{316-} -\|v_n\|^2-m\|A^{\frac{\gamma}{2}}v_n\|^2 +\sigma\|A^{\frac{1}{2}} u_n\|^2 -(A^{\alpha}\theta_n, u_n) = o(1).$$ Thanks to $\alpha -{\beta\over2} \le {1\over2}$, $\eqref{s1+}$ and [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, we get $$\label{m0191} (A^{\alpha}\theta_n,\, u_n) = (A^{\frac{\beta}{2}}\theta_n,\, (i\lambda_n)^{-1}A^{\alpha-\frac{\beta}{2}} v_n + \lambda_n^{-1-k}o(1)) = (A^{\frac{\beta}{2}}\theta_n,\, (i\lambda_n)^{-1}A^{\alpha-\frac{\beta}{2}} v_n) +{\lambda_n^{-\frac{3k}{2}}}o(1).$$ By $\eqref{s3+}$, we know $$\label{m019} A^{\alpha-\frac{\beta}{2}} v_n= -i\lambda_nA^{-\frac{\beta}{2}}\theta_n+ q_n+ \lambda_n^{ -k}o(1).$$ Thus, by substituting [\[m019\]](#m019){reference-type="eqref" reference="m019"} into [\[m0191\]](#m0191){reference-type="eqref" reference="m0191"} and using [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, [\[v1_2\]](#v1_2){reference-type="eqref" reference="v1_2"}, we conclude that $$\label{m0192} (A^{\alpha}\theta_n,\, u_n)=o(1).$$ Combining [\[v1_10\]](#v1_10){reference-type="eqref" reference="v1_10"}, [\[316-\]](#316-){reference-type="eqref" reference="316-"} and [\[m0192\]](#m0192){reference-type="eqref" reference="m0192"} yields $$\label{m018} \|A^{\frac{1}{2}}u_n\|={o(1)}.$$ In summary, by [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[v1_2\]](#v1_2){reference-type="eqref" reference="v1_2"}, [\[v1_10\]](#v1_10){reference-type="eqref" reference="v1_10"} and [\[m018\]](#m018){reference-type="eqref" reference="m018"}, we obtain that $\|U_n\|_{\mathcal{H}}=o(1)$, which contradicts the assumption [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"}. The proof of (i) is completed. **(ii)** Assume $(\alpha,\; \beta,\; \gamma)\in V_2$, and let $k= \frac{2(-2\alpha-\beta-\gamma+2)}{1-\gamma}$. We first show that for $\alpha <{1\over2}$, $$\label{vv3} \|A^{-\frac{1}{2}}(I+mA^{\gamma})v_n\| =\lambda_n^{-1}O(1).$$ In fact, it follows from [\[s2+\]](#s2+){reference-type="eqref" reference="s2+"} that $$\label{vv2} \|\lambda_nA^{-\frac{1}{2}}(I+mA^{\gamma})v_n\| \le \sigma \|A^{1\over2} u_n\| +\|A^{-{1\over2}+\alpha}\theta_n\| + \lambda_n^{-k}o(1).$$ Thus, [\[vv3\]](#vv3){reference-type="eqref" reference="vv3"} follows from [\[+211\]](#+211){reference-type="eqref" reference="+211"} and [\[vv2\]](#vv2){reference-type="eqref" reference="vv2"}. Taking the inner product of [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} with $\lambda_n^{-1-k}\theta_n$ on $H$ yields $$\label{v2_0} i\|\theta_n\|^2 -\lambda_n^{-1}(A^{\frac{\beta}{2}}q_n,\theta_{n})+\lambda_n^{-1}(A^{\alpha}v_n, \theta_{n})=\lambda_n^{-1-k}o(1).$$ From [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"} and [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, it is easy to see that $$\label{qth} | \lambda_n^{-1}(A^{\frac{\beta}{2}}q_n, \theta_n)| \leq\|q_n\|\|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_n\|=\lambda_n^{-k}o(1).$$ To deal with the third term of [\[v2_0\]](#v2_0){reference-type="eqref" reference="v2_0"}, we use the Cauchy-Schwarz inequality and [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"} to get $$\label{a} |\lambda_n^{-1} (A^{\alpha}v_n,\theta_{n})| \leq \|A^{\alpha-\frac{\beta}{2}}v_n\|\|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_{n}\| ={\lambda_n}^{-\frac{k}{2}}\|A^{\alpha-\frac{\beta}{2}}v_n\|o(1) .$$ Furthermore, by Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}, [\[vv3\]](#vv3){reference-type="eqref" reference="vv3"} and $\beta+\gamma<1$, we deduce that $$\label{vv} \|A^{\alpha-\frac{\beta}{2}}v_n\| \leq \left\{ \begin{array}{ll} O(1) & \; \mbox{when} \; \; \displaystyle 0\leq \alpha\leq\frac{ \gamma}{2}, \\ \noalign{\medskip} \displaystyle {\|A^{\gamma-\frac{1}{2}}v_n\|^{1- \frac{2\alpha-\beta-2\gamma+1}{2(1-\gamma)}}\|A^{\frac{1}{2}}v_n\|^{\frac{2\alpha-\beta-2\gamma+1}{2(1-\gamma)}} =\lambda_n^{\frac{2\alpha-\beta-\gamma}{1-\gamma}}O(1)} & \; \mbox{when} \; \; \displaystyle \frac{ \gamma}{2}<\alpha<{1\over2}. \end{array} \right.$$ Substituting [\[qth\]](#qth){reference-type="eqref" reference="qth"}-[\[vv\]](#vv){reference-type="eqref" reference="vv"} into [\[v2_0\]](#v2_0){reference-type="eqref" reference="v2_0"} and noticing the choice of $k$, we have $$\label{th1} \|\theta_n\| = \left\{ \begin{array}{ll} \lambda_n^{-{k\over4}}o(1) & \; \mbox{when} \; \; \displaystyle 0\leq \alpha\leq\frac{ \gamma}{2}, \\ \noalign{\medskip} \displaystyle \lambda_n^{\frac{2\alpha-1}{1-\gamma}}o(1) & \; \mbox{when} \; \; \displaystyle \frac{ \gamma}{2}<\alpha<\frac{1}{2}. \end{array} \right.$$ Consequently, one has $$\label{th2} \|\theta_n\| = o(1).$$ Now we proceed to prove that $$\label{vv1} \|v_n\|, \;\; \|A^{\frac{\gamma}{2}}v_n\| =o(1).$$ Based on the estimation [\[th1\]](#th1){reference-type="eqref" reference="th1"}, we shall prove [\[vv1\]](#vv1){reference-type="eqref" reference="vv1"} by considering the following two cases, respectively (Case ii-A and B). **Case ii-A.** Assume $\alpha\leq\frac{\gamma}{2}$. Note that $\|A^{\frac{\beta-1}{2}}(I+ mA^{\gamma})v_n\| \le C\|A^{\frac{\beta+\gamma-1}{2}} A^{\gamma\over2}v_n\| =O(1)$ due to $\beta+\gamma<1$ and [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}. Then, we can take the inner product of [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} with ${\lambda_n^{-k}}A^{\frac{\beta-1}{2}}(I+mA^{\gamma})v_n$ to get $$\label{v2_3} \begin{array}{l} -(A^{\frac{\beta}{2}}\theta_n,i\lambda_nA^{-\frac{1}{2}}(I+mA^{\gamma})v_n) -( q_n,A^{\frac{2\beta-1}{2}}(I+mA^{\gamma})v_n) \\ \noalign{\medskip} \displaystyle +\|A^{\frac{2\alpha+\beta-1}{4}} v_n\|^2+m\|A^{\frac{2\alpha+2\gamma+\beta-1}{4}} v_n\|^2=\lambda_n^{-k}o(1). \end{array}$$ By [\[+211\]](#+211){reference-type="eqref" reference="+211"} and $\beta+\gamma<1$, we have $$\|A^{\frac{2\beta-1}{2}}(I+mA^{\gamma})v_n\| \le C \|A^{ \beta+ \gamma -1} A^{{1\over2}}v_n\| =\lambda_n O(1).$$ Then, substituting the above and [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, [\[vv3\]](#vv3){reference-type="eqref" reference="vv3"} into [\[v2_3\]](#v2_3){reference-type="eqref" reference="v2_3"}, we have $$\label{v2_4} \|A^{\frac{2\alpha+\beta-1}{4}} v_n\|,~~\|A^{\frac{2\alpha+2\gamma+\beta-1}{4}} v_n\|=\lambda_n^{\frac{1}{2}-\frac{k}{4}}o(1)=\lambda_n^{\frac{2\alpha+\beta-1}{2(1-\gamma)}}o(1).$$ Thanks to $\alpha\leq \frac{\gamma}{2}$, we easily obtain $\frac{2\alpha+2\gamma+\beta-1}{4}<\frac{\gamma}{2}<\frac{1}{2}$, and thus, by interpolation, one gets $$\label{v2_5} \|A^{\frac{\gamma}{2}}v_n\|\leq\|A^{\frac{2\alpha+2\gamma+\beta-1}{4}} v_n\|^{1-\frac{1-\beta-2\alpha}{3-\beta-2\alpha-2\gamma}}\|A^{\frac{1}{2}}v_n\|^{\frac{1-\beta-2\alpha}{3-\beta-2\alpha-2\gamma}}.$$ Therefore, we obtain [\[vv1\]](#vv1){reference-type="eqref" reference="vv1"} by combining [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[v2_4\]](#v2_4){reference-type="eqref" reference="v2_4"} and [\[v2_5\]](#v2_5){reference-type="eqref" reference="v2_5"}. **Case ii-B.** Assume $\frac{\gamma}{2}<\alpha<{1\over2}.$ Using Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"} and [\[vv3\]](#vv3){reference-type="eqref" reference="vv3"} yields $$\label{v2_8} \|A^{-\alpha}(I+mA^{\gamma})v_n\|\leq \|A^{-\frac{1}{2}}(I+mA^{\gamma})v_n\|^{\frac{2\alpha-\gamma}{1-\gamma}} \|A^{-\frac{\gamma}{2}}(I+mA^{\gamma})v_n\|^{1-\frac{2\alpha-\gamma}{1-\gamma}}=\lambda_n^{\frac{-2\alpha+\gamma}{1-\gamma}}O(1).$$ Taking the inner product of [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} with $\lambda_n^{-k} A^{-\alpha}(I+mA^{\gamma})v_n$, we get $$\label{v2_7} (i\lambda_n\theta_n, A^{-\alpha}(I+mA^{\gamma})v_n)-(q_n, A^{\frac{\beta}{2}-\alpha}(I+mA^{\gamma})v_n)+((I+mA^{\gamma})v_n, v_n)= o(1).$$ Combining [\[v2_8\]](#v2_8){reference-type="eqref" reference="v2_8"} and [\[th1\]](#th1){reference-type="eqref" reference="th1"}, one has $$\label{th3}(i\lambda_n\theta_n, A^{-\alpha}(I+mA^{\gamma})v_n)=o(1).$$ To estimate the second term of [\[v2_7\]](#v2_7){reference-type="eqref" reference="v2_7"}, we notice that if $\frac{\gamma}{2}<\alpha\leq\frac{\beta+\gamma}{2}$, $$\label{vv4} \|A^{-\alpha+\frac{\beta}{2}+\gamma}v_n\|\leq \|A^{\frac{\gamma}{2}}v_n\|^{1-\frac{-2\alpha+\beta+\gamma}{1-\gamma}} \|A^{\frac{1}{2}}v_n\|^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}} =\lambda_n^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}}O(1),$$ where we have used Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, [Lemma 5](#lemma-1){reference-type="ref" reference="lemma-1"} and $\beta+\gamma<1$. Therefore, from [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[+211\]](#+211){reference-type="eqref" reference="+211"} and [\[vv4\]](#vv4){reference-type="eqref" reference="vv4"}, we can deduce that $$\label{vv5} |(q_n, A^{\frac{\beta}{2}-\alpha}(I+mA^{\gamma})v_n)| \le \left\{ \begin{array}{ll} C\|q_n\|\|A^{-\alpha+\frac{\beta}{2}+\gamma} v_n\|=\lambda_n^{2(\beta+\gamma-1)\over 1-\gamma}o(1), & \displaystyle {\gamma\over 2}<\alpha\leq\frac{\beta+\gamma}{2}, \\ \noalign{\medskip} \displaystyle \|A^{-\alpha+\frac{\beta+\gamma}{2}}q_n\|\|A^{-\frac{ \gamma}{2}}(I+mA^{ \gamma})v_n\|=\lambda_n^{-{k\over 2}}o(1), & \displaystyle \frac{\beta+\gamma}{2} < \alpha <{1\over2}. \end{array}\right.$$ Therefore, we get [\[vv1\]](#vv1){reference-type="eqref" reference="vv1"} by substituting [\[th3\]](#th3){reference-type="eqref" reference="th3"} and [\[vv5\]](#vv5){reference-type="eqref" reference="vv5"} into [\[v2_7\]](#v2_7){reference-type="eqref" reference="v2_7"}. Finally, similar to (i), we can show that the fourth term in [\[316-\]](#316-){reference-type="eqref" reference="316-"} tends to zero. Indeed, we have $|(A^{\alpha} \theta_n, u_n)| \le \|A^{\alpha-\frac{1}{2}}\theta_n\| \|A^{\frac{1}{2}}u_n\|=o(1)$ thanks to [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[th2\]](#th2){reference-type="eqref" reference="th2"} and $\alpha<\frac{1}{2}$. Thus, by ([\[316-\]](#316-){reference-type="ref" reference="316-"}), along with ([\[vv1\]](#vv1){reference-type="ref" reference="vv1"}), we get $\|A^{\frac{1}{2}}u_n\| =o(1)$. Therefore, combining this with [\[th2\]](#th2){reference-type="eqref" reference="th2"}, [\[vv1\]](#vv1){reference-type="eqref" reference="vv1"}, one can arrive at the contradiction $\|U_n\|_{\mathcal{H}}=o(1)$. **(iii)** Assume $(\alpha,\beta,\gamma)\in V_3$, and let $k=\displaystyle \frac{2(-2\alpha+\beta+\gamma)}{1-\gamma}$. Taking the inner product of [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} with $\lambda_n^{-k}\theta_n$ yields $$\label{v3_0} i\|\theta_n\|^2 -\lambda_n^{-1}(A^{\frac{\beta}{2}}q_n,\theta_{n})+\lambda_n^{-1}(A^{\alpha}v_n, \theta_{n})=\lambda_n^{-1-k}o(1).$$ From Lemma [Lemma 5](#lemma-1){reference-type="ref" reference="lemma-1"}, it is easy to get $$\label{vv61} |(q_n, \lambda_n^{-1}A^{\frac{\beta}{2}}\theta_{n})|\leq\|q_n\| \|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_{n}\|={\lambda_n^{-k}}o(1),$$ and thanks to $\alpha<\frac{\beta+\gamma}{2}$, we have $$\label{vv6} |(A^{\alpha}v_n, \lambda_n^{-1}\theta_{n})|\leq \|A^{\alpha-\frac{\beta}{2}}v_n\|\|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_{n}\|={\lambda_n^{-\frac{k}{2}}}o(1).$$ Thus, from [\[v3_0\]](#v3_0){reference-type="eqref" reference="v3_0"}-[\[vv6\]](#vv6){reference-type="eqref" reference="vv6"}, we obtain $$\label{v3_1} \|\theta_{n}\|={\lambda_n}^{-\frac{k}{4}}o(1).$$ Now, let us divide $V_3$ into four regions (see the 2-D picture Figure [4](#8134){reference-type="ref" reference="8134"}), and we will prove $\|v_n\|,$ $\|A^{\frac{\gamma}{2}}v_n\|,$ $\|A^{\frac{1}{2}}u_n\|=o(1)$ in each region, respectively (Case iii-A, B, C, D). {#8134 width="6cm"} **Case iii-A.** Let $(\alpha, \beta, \gamma) \in V_3$ and $0\leq \alpha \leq \frac{\gamma}{2}$. Since $\beta+\gamma\ge 1$ and [\[s2+\]](#s2+){reference-type="eqref" reference="s2+"}, [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}, it is easy to get $$\label{vv7} \| A^{-\frac{\beta+\gamma}{2}}\left(I+mA^{\gamma}\right)v_n\| \le \lambda_n^{-1}(\sigma \|A^{1-\frac{\beta+\gamma}{2}} u_n\| +\|A^{\alpha-\frac{\beta+\gamma}{2}}\theta_n\|+ \lambda_n^{-k}o(1)) = \lambda_n^{-1}O(1).$$ Consequently, we have $$\label{vv8} \| A^{{1\over2} - \beta- \gamma}\left(I+mA^{\gamma}\right)v_n\|=\| A^{{1\over2}-\frac{\beta+\gamma}{2}}A^{-\frac{\beta+\gamma}{2}}\left(I+mA^{\gamma}\right)v_n\| = \lambda_n^{-1}O(1).$$ Taking the inner product of ([\[s3+\]](#s3+){reference-type="ref" reference="s3+"}) with $\lambda_n^{-k}A^{ \frac{1}{2}-\frac{\beta}{2}-\gamma}(I+mA^{\gamma})v_n$, we have $$\label{vv9} \begin{array}{l} i\lambda_n(A^{\frac{\beta}{2}}\theta_n, A^{\frac{1}{2}-\beta-\gamma}(I+mA^{\gamma})v_n) -( q_n,A^{{1\over2} - \gamma}(I+mA^{\gamma})v_n) \\ \noalign{\medskip} \displaystyle +\|A^{{1\over2}( \alpha-\frac{\beta}{2}-\gamma+\frac{1}{2})} v_n\|^2+m\|A^{{1\over2} ( \alpha-\frac{\beta}{2}+\frac{1}{2}) } v_n\|^2=\lambda_n^{-k}o(1). \end{array}$$ We deduce from [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}-[\[+211\]](#+211){reference-type="eqref" reference="+211"} and [\[vv8\]](#vv8){reference-type="eqref" reference="vv8"} that $$\label{v9} i\lambda_n(A^{\frac{\beta}{2}}\theta_n, A^{\frac{1}{2}-\beta-\gamma}(I+mA^{\gamma})v_n),\;\; ( q_n,A^{{1\over2} - \gamma}(I+mA^{\gamma})v_n) = \lambda_n^{1-\frac{k}{2}}o(1).$$ Combining [\[vv9\]](#vv9){reference-type="eqref" reference="vv9"}-[\[v9\]](#v9){reference-type="eqref" reference="v9"} yields $$\label{v3_3} \|A^{{1\over2}( \alpha-\frac{\beta}{2}-\gamma+\frac{1}{2})} v_n\|, \quad \|A^{{1\over2} ( \alpha-\frac{\beta}{2}+\frac{1}{2}) } v_n\| =\lambda_n^{\frac{1}{2}-\frac{k}{4}}o(1).$$ Note that ${1\over2}(\alpha-\frac{\beta}{2}+\frac{1}{2})\le \frac{\gamma}{2}<\frac{1}{2}$ due to $\alpha \le {\gamma\over2}$ and $\beta+\gamma\ge1.$ Then using Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, along with [\[+211\]](#+211){reference-type="eqref" reference="+211"} and [\[v3_3\]](#v3_3){reference-type="eqref" reference="v3_3"}, one has $$\|A^{\frac{\gamma}{2}}v_n\|\leq\|A^{{1\over2}(\alpha-\frac{\beta}{2}+\frac{1}{2})} v_n\|^{1-\frac{\beta+2\gamma-2\alpha-1}{1-2\alpha+\beta}} \|A^{\frac{1}{2}}v_n\|^{\frac{\beta+2\gamma-2\alpha-1}{1-2\alpha+\beta}} = o(1).$$ Thus, [\[vv1\]](#vv1){reference-type="eqref" reference="vv1"} is proved. It is clear that $|(A^{\alpha}\theta_n, u_n)| \le \| \theta_n\| \|A^{\alpha} u_n\| = o(1)$ by $\alpha\le {\gamma\over2} \leq{1\over2}$, [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"} and [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}. Therefore, by ([\[316-\]](#316-){reference-type="ref" reference="316-"}) along with ([\[vv1\]](#vv1){reference-type="ref" reference="vv1"}), we obtain $\|A^{1\over2} u_n\| = o(1)$, and consequently, $\|U_n\|_{\cal H} = o(1)$ by [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[316-\]](#316-){reference-type="eqref" reference="316-"}, [\[vv1\]](#vv1){reference-type="eqref" reference="vv1"} and [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}. **Case iii-B.** Let $(\alpha, \beta, \gamma) \in V_3,\;\frac{\gamma}{2}<\alpha\leq\frac{1}{2}$ and $-\alpha+\frac{\beta}{2}+\gamma\le \frac{1}{2}$. Similar to (i), one has that [\[v1_5\]](#v1_5){reference-type="eqref" reference="v1_5"} still holds since $\alpha >{\gamma \over2}.$ Let us estimate the first two terms of [\[v1_5\]](#v1_5){reference-type="eqref" reference="v1_5"}. First, note that $0<1-\alpha-\frac{\beta}{2}\leq\frac{1}{2}$ since $\frac{\gamma}{2}<\alpha\leq\frac{1}{2}$ and $\gamma+\beta \ge1.$ From [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}, we can deduce that $$\label{vv10} \|A^{1-\alpha-\frac{\beta}{2}}u_n\| \le \left\{\begin{array}{ll} \lambda_n^{-1}O(1), & 0< 1-\alpha-\frac{\beta}{2}\le {\gamma\over2}, \\ \noalign{\medskip} \displaystyle \|A^{\frac{\gamma}{2}}u_n\|^{1-\frac{2-2\alpha-\beta-\gamma}{1-\gamma}}\|A^{\frac{1}{2}}u_n\|^{\frac{2-2\alpha-\beta-\gamma}{1-\gamma}} =\lambda_n^{\frac{1-2\alpha-\beta}{1-\gamma}}O(1), & {\gamma\over2} < 1-\alpha-\frac{\beta}{2}\le {1\over2}. \end{array}\right.$$ Consequently, by [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"} and [\[vv10\]](#vv10){reference-type="eqref" reference="vv10"}, $$\label{vv11} |(A^{{\beta\over2}}\theta_n, A^{1-\alpha-\frac{\beta}{2}}u_n)| \le \|A^{{\beta\over2}}\theta_n\| \|A^{1-\alpha-\frac{\beta}{2}}u_n\|= \left\{\begin{array}{ll} \lambda_n^{-{k\over2}}o(1), & 0< 1-\alpha-\frac{\beta}{2}\le {\gamma\over2}, \\ \noalign{\medskip} \displaystyle \lambda_n^{\frac{2(1-\beta-\gamma)}{1-\gamma}}o(1), & {\gamma\over2} < 1-\alpha-\frac{\beta}{2}\le {1\over2}. \end{array}\right.$$ Note that $\beta+\gamma\ge 1$, it turns out that $$\label{vv12} |(A^{1-\alpha}\theta_n, u_n)| = o(1).$$ By $\frac{\gamma}{2}<-\alpha+\frac{\beta}{2}+\gamma\le \frac{1}{2}$, we have the following estimation by Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}, $$\label{vv13} \|A^{-\alpha+\frac{\beta}{2}+\gamma}v_n\| \leq\|A^{\frac{\gamma}{2}}v_n\|^{1-\frac{-2\alpha+\beta+\gamma}{1-\gamma}} \|A^{\frac{1}{2}}v_n\|^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}} =\lambda_n^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}}O(1).$$ This combined with [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"} yields $$\label{vv14} |(q_n, A^{-\alpha+\frac{\beta}{2}}(I+m A^{\gamma}) v_n)| \le C \|q_n\| \| A^{-\alpha+\frac{\beta}{2}+\gamma } v_n\|= o(1).$$ Substituting [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}, [\[vv12\]](#vv12){reference-type="eqref" reference="vv12"}, [\[vv14\]](#vv14){reference-type="eqref" reference="vv14"} into [\[v1_5\]](#v1_5){reference-type="eqref" reference="v1_5"}, we get $\|v_n\|, \|A^{\gamma\over 2} v_n\| = o(1),$ i.e., [\[vv1\]](#vv1){reference-type="eqref" reference="vv1"} holds. Then, similar to the discussion in **Case iii-A**, we can get $|(A^{\alpha}\theta_n, u_n)| \le \| \theta_n\| \|A^{\alpha} u_n\| = o(1)$ due to $\alpha\le {1\over2}$ and [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}, which together with ([\[316-\]](#316-){reference-type="ref" reference="316-"}) and ([\[vv1\]](#vv1){reference-type="ref" reference="vv1"}) implies $\|A^{1\over2} u_n\| = o(1)$. Finally, combining this with [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}, we get the contradiction $\|U_n\|_{\cal H} = o(1).$ **Case iii-C.** Let $(\alpha, \beta, \gamma) \in V_3,\;\frac{\gamma}{2}<\alpha\leq\frac{1}{2},$ and $-\alpha+\frac{\beta}{2}+\gamma> \frac{1}{2}$. It follows from [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} that $$\label{200} \|A^{\frac{\beta}{2}}q_n\| \le \|\lambda_n\theta_n\|+ \|A^\alpha v_n\| + \lambda_n^{-k}o(1).$$ From [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}, by interpolation [\[4.41\]](#4.41){reference-type="eqref" reference="4.41"}, we get $$\label{vv15} \|A^\alpha v_n\| \le \|A^{\gamma \over2} v_n\|^{1-\frac{2\alpha-\gamma}{1-\gamma}} \|A^{1\over2} v_n\|^{\frac{2\alpha-\gamma}{1-\gamma}} = \lambda_n^{\frac{2\alpha-\gamma}{1-\gamma}}O(1).$$ We shall show that $$\label{3.31} \|\lambda_n\theta_n\|=\left\{ \begin{array}{ll} \displaystyle \lambda_n^{1-\frac{k}{4}}o(1),\quad &\mbox{if}~~~ 1-\frac{k}{4}\leq{\frac{2\alpha-\gamma}{1-\gamma}},\\ \noalign{\medskip} \displaystyle \lambda_n^{\frac{2\alpha-\gamma}{1-\gamma}}o(1), \quad &\mbox{if}~~~ 1-\frac{k}{4}>{\frac{2\alpha-\gamma}{1-\gamma}}. \end{array}\right.$$ Therefore, combining [\[200\]](#200){reference-type="eqref" reference="200"}-[\[3.31\]](#3.31){reference-type="eqref" reference="3.31"}, we see $$\label{3.32} \|A^{\frac{\beta}{2}}q_n\|=\lambda_n^{\frac{2\alpha-\gamma}{1-\gamma}}O(1).$$ Thanks to $0<-\alpha+\frac{\beta}{2}+\frac{\gamma}{2}<\frac{\beta}{2}$, Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, ([\[+211-1\]](#+211-1){reference-type="ref" reference="+211-1"}) and [\[3.32\]](#3.32){reference-type="eqref" reference="3.32"}, we obtain $$\begin{array}{ll} |(q_n, A^{-\alpha+\frac{\beta}{2}}(I+m A^{\gamma}) v_n)|&\le C\|A^{-\alpha+\frac{\beta}{2}+\frac{\gamma}{2}}q_n\|\|A^\frac{\gamma}{2}v_n\|\\ \noalign{\medskip} &\le C\|A^{-\alpha+\frac{\beta}{2}+\frac{\gamma}{2}}q_n\| \\ \displaystyle & \le C\|q_n\|^{\frac{2\alpha-\gamma}{\beta}}\|A^{\frac{\beta}{2}}q_n\|^{1-\frac{2\alpha-\gamma}{\beta}}=o(1). \end{array}$$ Then, using the same argument after [\[vv14\]](#vv14){reference-type="eqref" reference="vv14"}, one can also arrive at the contradiction $\|U_n\|_{\cal H} = o(1).$ Now we are the place to prove [\[3.31\]](#3.31){reference-type="eqref" reference="3.31"}. By [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}, it is obvious that $$\label{vv17} \|\lambda_n \theta_{n}\|=\lambda_n^{1-\frac{k}{4}}o(1) = \lambda_n^{\frac{2+2\alpha-\beta-3\gamma}{ 2(1-\gamma)} } o(1).$$ On the other hand, if $1-\frac{k}{4}>{\frac{2\alpha-\gamma}{1-\gamma}}$, i.e., $\alpha+\frac{\beta+\gamma}{2}<1$, one can get a better decay rate than [\[vv17\]](#vv17){reference-type="eqref" reference="vv17"}. In fact, recall that $\frac{1}{2}-\beta-\frac{\gamma}{2}\leq 0$ due to $\beta+\gamma \ge 1$. Then, it follows from ([\[s2+\]](#s2+){reference-type="ref" reference="s2+"}) that $$\label{vv18} i\lambda_nA^{\frac{1}{2}-\beta}(mI+A^{-\gamma})v_n+\sigma A^{\frac{3}{2}-\beta-\gamma}u_n-A^{\frac{1}{2}+\alpha-\beta-\gamma}\theta_{n}=o(1).$$ By [\[+211\]](#+211){reference-type="eqref" reference="+211"}, [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}, [\[vv18\]](#vv18){reference-type="eqref" reference="vv18"}, $\frac{3}{2}-\beta-\gamma\leq \frac{1}{2}$ and $\frac{1}{2}+\alpha-\beta-\gamma\leq 0$, one has $$\label{3.3} \|A^{\frac{1}{2}-\beta}v_n\|=\lambda_n^{-1}O(1).$$ Note that $\frac{1}{2}-\beta\le\alpha-\frac{\beta}{2} <-\beta-\frac{\gamma}{2}+1$ where we use $\alpha+\frac{\beta+\gamma}{2}<1$ and $\beta+\gamma\ge 1$. Combining [\[+211-1\]](#+211-1){reference-type="eqref" reference="+211-1"}, [\[3.3\]](#3.3){reference-type="eqref" reference="3.3"} and interpolation, we have $$\label{vv19} \|A^{\alpha-\frac{\beta}{2}}v_n\|\leq \|A^{\frac{1}{2}-\beta}v_n \|^{\frac{2-2\alpha-\beta-\gamma}{1-\gamma}} \|A^{-\beta-\frac{\gamma}{2}+1}v_n \|^{1-\frac{2-2\alpha-\beta-\gamma}{1-\gamma}} =\lambda_n^{-\frac{2-2\alpha-\beta-\gamma}{1-\gamma}}O(1),$$ where $\|A^{-\beta-\frac{\gamma}{2}+1}v_n\|$ is bounded since $-\beta-\frac{\gamma}{2}+1\le\frac{\gamma}{2}$. Using [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"} and [\[vv19\]](#vv19){reference-type="eqref" reference="vv19"} yields $$\label{vv20} |\lambda_n^{-1}(A^{\alpha}v_n, \theta_{n})| \leq\|A^{\alpha-\frac{\beta}{2}}v_n\| \|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_{n}\| =\lambda_n^{\frac{4\alpha-2}{1-\gamma}}o(1).$$ Note that $-k\leq \frac{4\alpha-2}{1-\gamma}$ due to $\beta+\gamma\ge 1$. Thus, substituting [\[vv61\]](#vv61){reference-type="eqref" reference="vv61"} and [\[vv20\]](#vv20){reference-type="eqref" reference="vv20"} into ([\[v3_0\]](#v3_0){reference-type="ref" reference="v3_0"}) yields $\|\theta_{n}\|=\lambda_n^{\frac{2\alpha-1}{1-\gamma}}o(1)$. Consequently, the second part of [\[3.31\]](#3.31){reference-type="eqref" reference="3.31"} is obtained. **Case iii-D.** Let $\frac{1}{2}<\alpha\leq\frac{\beta+\gamma}{2}$. Note that by ([\[s2+\]](#s2+){reference-type="ref" reference="s2+"}), ([\[+211\]](#+211){reference-type="ref" reference="+211"}) and ([\[v3_1\]](#v3_1){reference-type="ref" reference="v3_1"}), along with $\alpha>\frac{1}{2}$, we see that $$\label{111} \|\lambda_nA^{-\alpha}(I+mA^{\gamma})v_n\|=O(1).$$ Thus, taking the inner product of [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"} with $\lambda_n^{-k}A^{-\alpha}(I+mA^{\gamma})v_n$ in $H$, we have $$\label{v3_5} (i\lambda_n\theta_n, A^{-\alpha}(I+mA^{\gamma})v_n)-(A^{\frac{\beta+\gamma}{2}-\alpha}q_n, A^{-\frac{\gamma}{2}}(I+mA^{\gamma})v_n)+\|v_n\|^2+m\|A^{\frac{\gamma}{2}}v_n\|^2 = o(1).$$ Recalling that $\alpha>\frac{1}{2}$ and [\[s3+\]](#s3+){reference-type="eqref" reference="s3+"}, one has $$\|i\lambda_nA^{\frac{1}{2}-\alpha}\theta_n-A^{\frac{1}{2}-\alpha+\frac{\beta}{2}}q_n+A^{\frac{1}{2}} v_n\|=\lambda_n^{-k}o(1).$$ Thus, we deduce from [\[+211\]](#+211){reference-type="eqref" reference="+211"} and [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"} that $$\label{vv21} \|A^{\frac{\beta+1-2\alpha}{2}}q_n\|=\lambda_nO(1).$$ Thus, by interpolation [\[4.41\]](#4.41){reference-type="eqref" reference="4.41"}, along with [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"} and [\[vv21\]](#vv21){reference-type="eqref" reference="vv21"}, we get $$\label{vv22} \|A^{\frac{\beta+\gamma}{2}-\alpha}q_n\|\leq\|q_n\|^{1-{\frac{\beta+\gamma-2\alpha}{\beta+1-2\alpha}}}\|A^{\frac{\beta+1-2\alpha}{2}}q_n\|^{\frac{\beta+\gamma-2\alpha}{\beta+1-2\alpha}}=o(1).$$ Note that $\|A^{-\frac{\gamma}{2}}(I+mA^{\gamma})v_n\|\leq \| A^{-\frac{\gamma}{2}}v_n\|+ \|mA^{\frac{\gamma}{2}}v_n\|=O(1)$. Substituting [\[v3_1\]](#v3_1){reference-type="eqref" reference="v3_1"}, ([\[111\]](#111){reference-type="ref" reference="111"}) and [\[vv22\]](#vv22){reference-type="eqref" reference="vv22"} into [\[v3_5\]](#v3_5){reference-type="eqref" reference="v3_5"} yields $$\label{222} \|v_n\|,\;\|A^{\frac{\gamma}{2}}v_n\| = o(1).$$ Finally, due to $0<\alpha-\frac{\beta}{2}<\frac{\gamma}{2}$ and [\[s1+\]](#s1+){reference-type="eqref" reference="s1+"}, [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, it is clear that $$\label{v23} |(A^{\alpha}\theta_n, u_n)|\leq \| \lambda_n^{-1}A^{\frac{\beta}{2}}\theta_n\| \| \lambda_nA^{\alpha-\frac{\beta}{2}}u_n\|\leq\|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_n\|\|A^{\alpha-\frac{\beta}{2}}v_n\|=\lambda_n^{-\frac{k}{2}}o(1).$$ Then, by [\[316-\]](#316-){reference-type="eqref" reference="316-"}, along with ([\[222\]](#222){reference-type="ref" reference="222"}) and ([\[v23\]](#v23){reference-type="ref" reference="v23"}), we obtain $$\label{333} \|A^{\frac{1}{2}}u_n\|=o(1).$$ In summary, by ([\[v1_0\]](#v1_0){reference-type="ref" reference="v1_0"}), ([\[v3_1\]](#v3_1){reference-type="ref" reference="v3_1"}), ([\[222\]](#222){reference-type="ref" reference="222"}) and ([\[333\]](#333){reference-type="ref" reference="333"}), we have arrived at the contradiction $\|U_n\|_{\mathcal{H}}=o(1)$. # Polynomial Stability of the System without an Inertial Term (Proof of ) {#8.7} This section is devoted to considering the stability of the system ([\[101\]](#101){reference-type="ref" reference="101"}) without an inertial term, i.e., $m=0$. Note that by Lemma [Lemma 2](#l202){reference-type="ref" reference="l202"}, we have known that the corresponding semigroup $e^{t{\cal A}}$ is strongly stable. We shall further estimate the polynomial decay rates of the solutions to the system when parameters $(\alpha,\;\beta)\in V_i^*,\; i=1,2,$ respectively. Similar to the argument given in Section [3](#8.8){reference-type="ref" reference="8.8"}, we still employ the proof by contradiction to show . Specifically, suppose ([\[huang2\]](#huang2){reference-type="ref" reference="huang2"}) fails. Then, there at least exists a sequence $\{ \lambda_{n}, U_{n} \}_{n=1}^\infty \subset {\mathbb R} \times D({\cal A})$ such that [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"}-[\[331\]](#331){reference-type="eqref" reference="331"} hold with $m=0$, that is $$\begin{aligned} &&\lambda_n^k(i\lambda_nA^{\frac{1}{2}}u_n- A^{\frac{1}{2}}v_n)=o(1)\quad \; \mbox{ in } \; H,\label{s1+1}\\ \noalign{\medskip} \displaystyle &&\lambda_n^k \left(i\lambda_nv_n +\sigma A u_n-A^{\alpha}\theta_n\right)=o(1)\quad \; \mbox{ in } \; H,\label{s2+1}\\\noalign{\medskip} \displaystyle &&\lambda_n^k(i\lambda_n\theta_n-A^{\frac{\beta}{2}}q_n+A^\alpha v_n)=o(1)\quad \; \mbox{ in } \; H,\label{s3+1}\\\noalign{\medskip} \displaystyle &&\lambda_n^k(i\lambda_n\tau q_n+ q_n+A^{\frac{\beta}{2}}\theta_n)=o(1)\quad \; \mbox{ in } \; H.\label{s4+1}\end{aligned}$$ Consequently, [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}-[\[+211\]](#+211){reference-type="eqref" reference="+211"} remain true. Now, we proceed to show (i)-(ii) in , respectively. **(i)** Let $(\alpha,\;\beta)\in V_1^*$(see ([\[par2\]](#par2){reference-type="ref" reference="par2"})) and $k= \frac{2\alpha-\beta}{\alpha}$. Thanks to $-\alpha+\frac{\beta}{2}\le 0$ and [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"}, we can deduce from [\[s3+1\]](#s3+1){reference-type="eqref" reference="s3+1"} that $$\label{mv2} \|\lambda_nA^{-\alpha}\theta_n\|=O(1).$$ Then, using ([\[v1_8\]](#v1_8){reference-type="ref" reference="v1_8"}), ([\[mv2\]](#mv2){reference-type="ref" reference="mv2"}) and the interpolation inequality ([\[4.41\]](#4.41){reference-type="ref" reference="4.41"}) in Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, we obtain $$\label{m05} \left\| \theta_n\right\| \leq \left\| A^{-\alpha} \theta_n\right\| ^{\frac{\beta}{2\alpha+\beta}}\|A^{\frac{\beta}{2}} \theta_n\|^{\frac{2\alpha}{2\alpha+\beta}}=o(1).$$ We take the inner product of ([\[s3+1\]](#s3+1){reference-type="ref" reference="s3+1"}) with $\lambda_n^{-k}A^{-\alpha}v_{n}$ to get $$\label{m06} (i\lambda_n\theta_n, A^{-\alpha}v_n)-(A^{\frac{\beta}{2}-\alpha}q_n, v_n)+\|v_n\|^2 =o(1).$$ By ([\[s2+1\]](#s2+1){reference-type="ref" reference="s2+1"}), it is clear that $\|i \lambda_nA^{-\alpha} v_n +\sigma A^{1-\alpha}u_n - \theta_n \|=\lambda_n^{-k}o(1)$. Combining this with [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"}, along with $\alpha\geq \frac{1}{2}$, yields $$\label{mv1} \|A^{-\alpha}v_n\|=\lambda_n^{-1}O(1).$$ Substituting [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"}, [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[m05\]](#m05){reference-type="eqref" reference="m05"}, [\[mv1\]](#mv1){reference-type="eqref" reference="mv1"} into [\[m06\]](#m06){reference-type="eqref" reference="m06"}, along with $\frac{\beta}{2}-\alpha\leq 0$, we obtain $$\label{m07} \|v_n\|= o(1).$$ Furthermore, similar to ([\[v1_6\]](#v1_6){reference-type="ref" reference="v1_6"})-[\[316-\]](#316-){reference-type="eqref" reference="316-"}, we can deduce from [\[s1+1\]](#s1+1){reference-type="eqref" reference="s1+1"} and [\[s2+1\]](#s2+1){reference-type="eqref" reference="s2+1"} that $$\label{m09} -\|v_n\|^2+\sigma\|A^{\frac{1}{2}} u_n\|^2 -i(\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_n, A^{\alpha-\frac{\beta}{2}}v_n) =o(1).$$ It follows from [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, [\[s3+1\]](#s3+1){reference-type="eqref" reference="s3+1"} and [\[m05\]](#m05){reference-type="eqref" reference="m05"} that $$\label{mv3}%\begin{array}{ll} (\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_n, \, A^{\alpha-\frac{\beta}{2}}v_n) = (\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_n, \, q_n-i\lambda_nA^{-\frac{\beta}{2}}\theta_n) + \lambda_n^{-1-k}o(1) = o(1). %\end{array}$$ Combining this with [\[m07\]](#m07){reference-type="eqref" reference="m07"} and [\[m09\]](#m09){reference-type="eqref" reference="m09"} yields $$\label{m010} \|A^{\frac{1}{2}}u_n\|=o(1).$$ Consequently, we arrive at the contradiction $\|U_n\|_{\mathcal{H}}=o(1)$ by [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"}, [\[m05\]](#m05){reference-type="eqref" reference="m05"}, [\[m07\]](#m07){reference-type="eqref" reference="m07"} and [\[m010\]](#m010){reference-type="eqref" reference="m010"}.\ **(ii)** Let $(\alpha,\;\beta)\in V_2^*$ and $k= 2(2-2\alpha-\beta)$. Taking the inner product of [\[s3+1\]](#s3+1){reference-type="eqref" reference="s3+1"} with $\lambda_n^{-k-1}\theta_n$, one has $$\label{m011} (i\lambda_n\theta_n, \lambda_n^{-1}\theta_{n})-(A^{\frac{\beta}{2}}q_n, \lambda_n^{-1}\theta_{n})+(A^{\alpha}v_n, \lambda_n^{-1}\theta_{n})=\lambda_n^{-1-k}o(1).$$ By [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"} and [\[v1_8\]](#v1_8){reference-type="eqref" reference="v1_8"}, we get $$\label{mv4} |(A^{\frac{\beta}{2}}q_n, \lambda_n^{-1}\theta_n)| \leq\|q_n\|\|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_n\|=\lambda^{-k}o(1).$$ Moreover, by [\[s2+1\]](#s2+1){reference-type="eqref" reference="s2+1"} and [\[+211\]](#+211){reference-type="eqref" reference="+211"}, we have $$\begin{aligned} \label{4.40} \|\lambda_n A^{-\frac{1}{2}}v_n\|=O(1).\end{aligned}$$ Recalling $0\leq\alpha<{1\over2}$, we deduce from [\[4.41\]](#4.41){reference-type="eqref" reference="4.41"}, [\[+211\]](#+211){reference-type="eqref" reference="+211"} and [\[4.40\]](#4.40){reference-type="eqref" reference="4.40"} that $$\label{mv5} \|A^{\alpha-\frac{\beta}{2}}v_n\|\leq \|A^{-\frac{1}{2}}v_n\|^{{1\over2}-\alpha+{\beta\over 2}} \|A^{\frac{1}{2}}v_n\|^{{1\over2}+\alpha-{\beta\over 2}} =\lambda_n^{2\alpha-\beta}O(1).$$ This implies that $$\label{mv7} |(A^{\alpha}v_n, \lambda_n^{-1}\theta_{n})|\leq \|A^{\alpha-\frac{\beta}{2}}v_n\| \|\lambda_n^{-1}A^{\frac{\beta}{2}}\theta_{n}\| =\lambda^{4\alpha-2}o(1).$$ Thus, substituting [\[mv4\]](#mv4){reference-type="eqref" reference="mv4"} and [\[mv7\]](#mv7){reference-type="eqref" reference="mv7"} into [\[m011\]](#m011){reference-type="eqref" reference="m011"}, along with $-k\leq 4\alpha-2$, yields $$\label{m015} \|\theta_{n}\|=\lambda^{2\alpha-1}o(1).$$ We claim that $$\label{m013} \|v_n\| = o(1).$$ Indeed, note that ([\[m06\]](#m06){reference-type="ref" reference="m06"}) still holds for this case. Then, in order to show ([\[m013\]](#m013){reference-type="ref" reference="m013"}), it suffices to prove that the first and second terms in ([\[m06\]](#m06){reference-type="ref" reference="m06"}) are both $o(1)$. From [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"} and [\[4.40\]](#4.40){reference-type="eqref" reference="4.40"}, it is easy to get $$\label{mvv1} \|A^{-\alpha}v_n\|\leq\|A^{-\frac{1}{2}}v_n\|^{2\alpha}\|v_n\|^{1-2\alpha}=\lambda_n^{-2\alpha}O(1).$$ Combining this with [\[m015\]](#m015){reference-type="eqref" reference="m015"} gives that the first term in ([\[m06\]](#m06){reference-type="ref" reference="m06"}) satisfies $$\label{mvv11} |(i\lambda_n\theta_n, A^{-\alpha}v_n)|\leq\|\theta_{n}\|\|\lambda_nA^{-\alpha}v_n\|=o(1).$$ We proceed to show the second term in [\[m06\]](#m06){reference-type="eqref" reference="m06"} is also $o(1)$. In fact, if $0\leq\alpha\leq\frac{\beta}{2}$, by interpolation inequality ([\[4.41\]](#4.41){reference-type="ref" reference="4.41"}), along with [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"} and [\[s1+1\]](#s1+1){reference-type="eqref" reference="s1+1"}, we have $$\label{m012} \|A^{\frac{\beta}{2}-\alpha}v_n\|\leq\|A^{\frac{1}{2}}v_n\|^{\beta-2\alpha}\|v_n\|^{1-\beta+2\alpha}=\lambda_n^{\beta-2\alpha}O(1).$$ Thus, by ([\[v1_0\]](#v1_0){reference-type="ref" reference="v1_0"}) and ([\[m012\]](#m012){reference-type="ref" reference="m012"}), we get $$\label{mvv2} |(q_n, A^{\frac{\beta}{2}-\alpha}v_n )| \leq\|q_n\|\|A^{\frac{\beta}{2}-\alpha}v_n\|=\lambda_n^{2\beta-2}o(1) =o(1).$$ If $\frac{\beta}{2}<\alpha < \frac{1}{2}$, it is clear from [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"} and [\[v1_0\]](#v1_0){reference-type="eqref" reference="v1_0"} that $$\label{444} |(q_n, A^{\frac{\beta}{2}-\alpha}v_n )| \le \|q_n\| \|A^{\frac{\beta}{2}-\alpha}v_n\|= \lambda_n^{-{k\over2}}o(1).$$ By ([\[mvv2\]](#mvv2){reference-type="ref" reference="mvv2"}) and ([\[444\]](#444){reference-type="ref" reference="444"}), we have proved that the second term in ([\[m06\]](#m06){reference-type="ref" reference="m06"}) is $o(1)$, and hence, ([\[m013\]](#m013){reference-type="ref" reference="m013"}) holds. Moreover, it is clear that $|(\theta_n, A^{\alpha}u_n)| \le \|\theta_n\|\|A^{\alpha}u_n\|= \lambda_n^{2\alpha-1}o(1)$ by [\[unitnorm\]](#unitnorm){reference-type="eqref" reference="unitnorm"}, [\[m015\]](#m015){reference-type="eqref" reference="m015"} and $\alpha < {1\over2}$. This along with [\[m09\]](#m09){reference-type="eqref" reference="m09"} and [\[m013\]](#m013){reference-type="eqref" reference="m013"}, yields $$\label{555} \|A^{\frac{1}{2}} u_n\|=o(1).$$ In summary, by ([\[v1_0\]](#v1_0){reference-type="ref" reference="v1_0"}), ([\[m015\]](#m015){reference-type="ref" reference="m015"}), ([\[m013\]](#m013){reference-type="ref" reference="m013"}) and ([\[555\]](#555){reference-type="ref" reference="555"}), we have arrived at the contradiction $\|U_n\|_{\mathcal{H}}=o(1)$. The desired result follows. # Proof of Optimality of Decay Rates () {#8.6} In this section, we are going to prove , which shows the orders of polynomial decay achieved in and are optimal. To this end, we analyze the eigenvalues $\lambda$ of the operator $\mathcal{A}$ when $m \ne 0$ (with an inertial term) and when $m = 0$ (without an inertial term). In addition, we assume that the system has different wave speeds, that is, $\sigma \tau \ne 1$. In what follows, we first describe the characteristic equation associated with $\mathcal{A}$ (). We then describe the solutions to the characteristic equation in an asymptotic setting in when the system is with an inertial term () and without an inertial term (), respectively. Finally, we show that these eigenvalues indicate the optimality of the polynomial decay rates described in and (). ## Characteristic Equations {#sec:characteristic-equation} We describe the characteristic equations associated with the operator $\mathcal{A}$ when the system is with an inertial term ($m \ne 0$) and without an inertial term ($m=0$). For this purpose, recall that $A$ is a self-adjoint, positive-definite operator with compact resolvent. Thus, there exists a sequence of eigenvalues $\{\mu_n\}_{n\ge1}$ of $A$ such that $$0<\mu_1\le\mu_2\le\cdots\le\mu_n\le\cdots,~~\qquad\lim_{n\to\infty}\mu_n=\infty.$$ As such, the eigenvalues $\lambda$ of operator ${\cal A}$ satisfies the following quartic equation: $$\label{f=0} f(\lambda,\mu_n) \equiv (m\tau \cdot \mu_n^\gamma + \tau ) \lambda^4 + (m \cdot \mu_n^\gamma +1) \lambda^3 + (\tau \cdot \mu_n^{2\alpha} + m \cdot \mu_n^{\beta+\gamma} + \sigma \tau \cdot \mu_n + \mu_n^\beta) \lambda^2 + (\mu_n^{2\alpha}+\sigma \cdot \mu_n) \lambda + \sigma \cdot \mu_n^{1+\beta}=0.$$ Because the system has different wave speeds, we take $\sigma=2$ and $\tau=1$ so that $\sigma \tau = 2 \ne 1$ without loss of generality. In addition, by taking $m=1$ and $m=0$, we can derive the characteristic equation of the system from ([\[f=0\]](#f=0){reference-type="ref" reference="f=0"}) when it is with and without an inertial term respectively as follows: $$\label{eq:characteristic-with-inertial} (\mu_n^\gamma + 1 ) \lambda_n^4 + ( \mu_n^\gamma +1) \lambda_n^3 + ( \mu_n^{2\alpha} + \mu_n^{\beta+\gamma} + 2 \mu_n + \mu_n^\beta) \lambda_n^2 + (\mu_n^{2\alpha}+2 \mu_n) \lambda_n + 2 \mu_n^{1+\beta}=0.$$ $$\label{eq:characteristic-without-inertial} \lambda_n^4 + \lambda_n^3 + ( \mu_n^{2\alpha} + 2 \mu_n + \mu_n^\beta) \lambda_n^2 + (\mu_n^{2\alpha}+ 2 \mu_n) \lambda_n + 2 \mu_n^{1+\beta}=0.$$ In ([\[eq:characteristic-with-inertial\]](#eq:characteristic-with-inertial){reference-type="ref" reference="eq:characteristic-with-inertial"}) and ([\[eq:characteristic-without-inertial\]](#eq:characteristic-without-inertial){reference-type="ref" reference="eq:characteristic-without-inertial"}), $\lambda_n$'s are sequences of eigenvalues of $\mathcal{A}$ associated with $\mu_n \rightarrow \infty$ as $n \rightarrow \infty$. We seek to solve [\[eq:characteristic-with-inertial\]](#eq:characteristic-with-inertial){reference-type="eqref" reference="eq:characteristic-with-inertial"} and [\[eq:characteristic-without-inertial\]](#eq:characteristic-without-inertial){reference-type="eqref" reference="eq:characteristic-without-inertial"} for $\lambda_n$ in an asymptotic setting, that is, in the setting as $\mu_n \rightarrow \infty$. Several existing works [@han; @haoliu2; @kuang] have described the applications of a procedure that can also be used to solve ([\[eq:characteristic-with-inertial\]](#eq:characteristic-with-inertial){reference-type="ref" reference="eq:characteristic-with-inertial"}) and ([\[eq:characteristic-without-inertial\]](#eq:characteristic-without-inertial){reference-type="ref" reference="eq:characteristic-without-inertial"}). In this paper, we apply this procedure to solve ([\[eq:characteristic-with-inertial\]](#eq:characteristic-with-inertial){reference-type="ref" reference="eq:characteristic-with-inertial"}) and ([\[eq:characteristic-without-inertial\]](#eq:characteristic-without-inertial){reference-type="ref" reference="eq:characteristic-without-inertial"}) and omit the discussion of its details. We refer interested readers to the aforementioned works for details of this procedure. The solutions to ([\[eq:characteristic-with-inertial\]](#eq:characteristic-with-inertial){reference-type="ref" reference="eq:characteristic-with-inertial"}) and ([\[eq:characteristic-without-inertial\]](#eq:characteristic-without-inertial){reference-type="ref" reference="eq:characteristic-without-inertial"}) are summarized in Table [\[tab:sigma-2\]](#tab:sigma-2){reference-type="ref" reference="tab:sigma-2"} and [\[tab:m-0-sigma-2\]](#tab:m-0-sigma-2){reference-type="ref" reference="tab:m-0-sigma-2"}, respectively. ## Eigenvalues of the System {#sec:with-inertial} First, we consider the solutions to ([\[eq:characteristic-with-inertial\]](#eq:characteristic-with-inertial){reference-type="ref" reference="eq:characteristic-with-inertial"}), which are eigenvalues of $\mathcal{A}$ with an inertial term, in each regions given in . It should be noticed that the partition in is similar to that reported in [@han], where the exponential stability was proved when parameters $(\alpha,\beta,\gamma) \in F_{13},\;L_{123},\; L_{34}$. More precisely, the solutions to [\[eq:characteristic-with-inertial\]](#eq:characteristic-with-inertial){reference-type="eqref" reference="eq:characteristic-with-inertial"} are given in . As a result, one can conclude that ([@l-rao]) in region $V_1$, although the coefficients for the real and imaginary parts of the eigenvalues on $T_1, F_{12}, F_{14},$ and $L_{124}$ are different, there always exists a sequence of eigenvalues on each region satisfying ${\cal R}e \lambda_{n} = | {\cal I}m \lambda_{n} |^{-{1\over k_1}},$ where $k_1$ is defined in . Therefore, the decay properties of the corresponding semigroup are uniform in $V_1$. By a similar argument, we can obtain that the long-time behavior of the system [\[101\]](#101){reference-type="eqref" reference="101"} is the same when parameters are in $V_2,\; V_3$ or $V_1^*,\; V_2^*$, respectively. Note that to characterize the long-time behavior in $V_1^*$ and $V_2^*$, we have used the solutions to [\[eq:characteristic-without-inertial\]](#eq:characteristic-without-inertial){reference-type="eqref" reference="eq:characteristic-without-inertial"} given in . [\[tab:sigma-2\]]{#tab:sigma-2 label="tab:sigma-2"} [\[tab:m-0-sigma-2\]]{#tab:m-0-sigma-2 label="tab:m-0-sigma-2"} ## Optimality of the Polynomial Decay Rates {#sec:optimality-remark} With the eigenvalues of $\mathcal{A}$ as reported in and , we can show that the polynomial decay rates given in and are optimal. We first show the rates achieved in are optimal. From , we can deduce that there exists a sequence of eigenvalues satisfying $$\label{o1} {\cal R}e \lambda_{n} = | {\cal I}m \lambda_{n} |^{-{1\over k_i}}, \quad \; \mbox{ when } \; (\alpha, \; \beta,\; \gamma) \in V_i,\; i=1,2,3,$$ where $k_i$ and $V_i$ are defined in . Then by the same argument as given in [@haoliu2 Corollary 4.7], we can obtain $$\label{o2} \varlimsup\limits_{\lambda\in {\mathbb R}, |\lambda|\to \infty} |\lambda |^{-{1\over k_i}}\| (i\lambda I - {\cal A})^{-1}\|_{{\cal L}({\cal H})}\ge C>0,\quad \; \mbox{ when } \; (\alpha, \; \beta,\; \gamma) \in V_i,\; i=1,2,3.$$ Therefore, the obtained decay rates in are optimal. Following the similar discussion as given above, we can also derive that the obtained decay rates given in are also optimal based on the expression of eigenvalues in . # Examples {#8.5} This section is devoted to presenting some examples. From these examples, we can find that the decay rates of the semigroups associated with the coupled systems can be easily identified based on our main results and . Assume $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with boundary $\partial \Omega = \Gamma_0\cup\Gamma_1$. $\Gamma_0$ and $\Gamma_1$ are smooth. Let $A=\Delta^2$ be the bi-Laplace operator on $\Omega$ with domain $D(A) = \{u\in H^4(\Omega)\,|\, u |_{ \Gamma_0}=\partial_\nu u |_{\Gamma_0}=\Delta u |_{\Gamma_1}=\partial_\nu \Delta u |_{\Gamma_1} =0 \},\; H= L^2(\Omega),$ where $\nu$ is the outward unit normal vector to the boundary. If $m>0$ and $\gamma = {1\over2},\; \alpha=1,\; \beta = 1$, the system [\[101\]](#101){reference-type="eqref" reference="101"} is as follows. $$\label{601} \left\{\begin{array}{ll} u_{tt} - m \Delta u_{tt} + \sigma \Delta^2 u - \Delta^2 \theta=0, & x\in\Omega, t>0,\\ \theta_{t} - \Delta q + \Delta^2 u_{t} = 0,& x\in\Omega, t>0, \\ \tau q_t + q + \Delta \theta =0 , & x\in\Omega, t>0,\\ u =\partial_\nu u = \theta =\partial_\nu \theta= q =\partial_\nu q=0 , & x\in\Gamma_0, t>0,\\ \Delta u =\partial_\nu \Delta u =0, & x\in\Gamma_1, t>0,\\ u(0) = u_0, \; u_t(0)=u_1, \; \theta(0)=\theta_0, \; q(0)=q_0, & x\in\Omega. \end{array}\right.$$ The model [\[601\]](#601){reference-type="eqref" reference="601"} describes the vibration of Rayleigh beam with thermal damping of Cattaneo type. By , we obtain that $(\alpha, \; \beta,\; \gamma) \in V_1$ and the semigroup associated with system ([\[601\]](#601){reference-type="ref" reference="601"}) decays polynomially with optimal order $t^{-\frac{3}{2}}$. Moreover, it was known from [@han] that the energy of system [\[101\]](#101){reference-type="eqref" reference="101"} decays exponentially when $\gamma = {1\over2},\;\alpha = {3\over4}, \; \beta = 1$, which corresponds to the Rayleigh beam with Cattaneo thermal conduction and weaker coupling. Comparing this with the long-time behavior of system [\[601\]](#601){reference-type="eqref" reference="601"}, one can conclude that the coupling order should coordinate with the damping order to get a better decay rate. Consider the following one-dimensional thermo-elastic Rayleigh beam system with Cattaneo's law. $$\label{602} \left\{ \begin{array}{l} u_{tt} - m u_{ttxx} + \sigma u_{xxxx} - \theta=0, \quad x\in(0,1),\; t>0,\\ \theta_{t} - q_x + u_{t} = 0,\quad x\in(0,1), \;t>0, \\ \tau q_t + q + \theta_{x} =0 ,\quad x\in(0,1),\; t>0,\\ u (0) = u_x(0) = u_{xx}(1) = u_{xxx}(1) = \theta(0) = \theta(1) = q(0)=q(1) =0 , \quad t>0,\\ u(0) = u_0, \; u_t(0)=u_1,\; \theta(0)=\theta_0, \; q(0)=q_0,\quad x\in(0,1). \end{array}\right.$$ To apply the abstract results in and [Theorem 2](#m00){reference-type="ref" reference="m00"} to this equation, we let operator $A=\partial_{xxxx}$ on $H = L^2(0,1)$ with domain $D(A) = \{u\in H^4(0,1)\,|\,u (0) = u_x(0) = u_{xx}(1) = u_{xxx}(1) =0 \}.$ Note that system ([\[602\]](#602){reference-type="ref" reference="602"}) is corresponding to the abstract system ([\[101\]](#101){reference-type="ref" reference="101"}) with $(\alpha, \, \beta,\,\gamma) = (0,\;{1\over2}, \;{1\over2})$ when $m>0.$ Thus, we easily see that the semigroup corresponding to [\[602\]](#602){reference-type="eqref" reference="602"} is polynomially stable with order $t^{-{1\over 4}}$ by choosing parameters $(\alpha, \, \beta,\,\gamma) = (0,\;{1\over2}, \;{1\over2}) \in V_3$ in . While for the case $m=0$, the decay rate of the semigroup is $t^{-{1\over 3}}$ due to with parameters $(\alpha, \, \beta ) = (0,\;{1\over2 }) \in V_2^*$. # Conclusions {#8.4} In this work, we focus on investigating the polynomial stability of abstract thermoelastic systems with Cattaneo's law and mainly obtained the following results: For the system with an inertial term ($m>0$), based on system parameter choices $(\alpha,\beta,\gamma)$, we classified the "non-exponential stability\" area into three subregions, and obtained explicit polynomial decay rates dependent on these parameters for each subregion. For the system without an inertial term ($m=0$), we also divided the "non-exponential stability\" region into two subregions, and estimated its polynomial decay rates at each subregion, respectively. Upon comparing the stability outcomes between systems with and without an inertial term, it is evident that this term plays a substantial role in determining stability for such coupled PDEs' systems. Furthermore, we conducted a detailed asymptotic spectral analysis on the system operator ($m>0$ or $m=0$), which led us to reveal the eigenvalues asymptotics of the system operator. The optimality of the above decay rates was verified, based on these results, respectively. # Appendix **Proof of Lemma [Lemma 2](#l202){reference-type="ref" reference="l202"}.** Define $$\widetilde{\lambda}:=\sup \big\{R>0\,|\, [-iR,\, iR]\subset \rho(\mathcal{A})\big\}.$$ It is sufficient to show $\widetilde{\lambda}=\infty$ since $0\in\rho({\mathcal{A}})$ (see [@han]). By contradiction, if $0<\widetilde{\lambda}<\infty$, there exists a sequence $\lambda_n \to \widetilde{\lambda}$ and a sequence $Z_n=(u_{n}, v_{n}, \theta_{n}, q_{n})\in\mathcal{D}(\mathcal{A})$ with $\|Z_n\|_\mathcal{H}=1$ such that $$\label{+331-} \Vert (i\lambda_n I-\mathcal{A}) Z_n\Vert_\mathcal{H} =o(1),\quad n\to\infty,$$ i.e., $$\begin{aligned} &&i\lambda_nA^{\frac{1}{2}}u_n- A^{\frac{1}{2}}v_n:=f_{1n}=o(1) \quad \mbox{in } \; H,\label{+g1}\\ &&A^{-\frac{\gamma}{2}}\left(i\lambda_nv_n+i\lambda_n mA^{\gamma}v_n+A^{\alpha}(\sigma A^{1-\alpha} u_n-\theta_n)\right):= f_{2n}=o(1)\quad \mbox{in } \; H,\label{+g2}\\ &&i\lambda_n\theta_n+A^\alpha v_n-A^{\frac{\beta}{2}}q_n:= f_{3n}=o(1)\quad \mbox{in } \; H,\label{+g3}\\ &&i\lambda_n\tau q_n+q_n+A^{\frac{\beta}{2}}\theta_n:=f_{4n}=o(1)\quad \mbox{in } \; H.\label{+g4} \end{aligned}$$ It follows from ${\|Z_n\|_{\mathcal{H}}}=1$ that $$\label{+2111} \|A^{\frac{1}{2}}u_n\|,\; \|A^{\frac{\gamma}{2}}v_n\|,\; \|v_n\|, \; \|\theta_n\|,\; \|q_n\|\; \; \mbox{are bounded.}$$ Moreover, [\[+g1\]](#+g1){reference-type="eqref" reference="+g1"}, [\[+2111\]](#+2111){reference-type="eqref" reference="+2111"} and the fact $0<\widetilde{\lambda}<\infty$ imply that $\|A^{\frac{1}{2}}v_n\|$ is bounded. By the dissipativeness of $\mathcal{A}$, we get $$\label{+k1-} {\cal R}e \langle(i\lambda_n I-\mathcal{A}) Z_n, Z_n\rangle_\mathcal{H}=-{\cal R}e \langle\mathcal{A}{Z_n},{Z_n}\rangle_{\mathcal{H}}=\|q_n\|^2=o(1).$$ Thus, by [\[+g4\]](#+g4){reference-type="eqref" reference="+g4"} and [\[+k1-\]](#+k1-){reference-type="eqref" reference="+k1-"}, we get $$\label{+209+} \|A^{\frac{\beta}{2}}\theta_n\|=o(1),$$ and hence, $$\label{+210+} \|\theta_n\|=o(1).$$ By taking the inner product of [\[+g3\]](#+g3){reference-type="eqref" reference="+g3"} with $A^{-\alpha} v_n$ on $H$, we obtain $$\label{+211+} (i\lambda_n\theta_n,A^{-\alpha} v_n)+(A^\alpha v_n,A^{-\alpha} v_n)-(A^{\frac{\beta}{2}}q_n,A^{-\alpha} v_n)=o(1).$$ It is clear that the first term in [\[+211+\]](#+211+){reference-type="eqref" reference="+211+"} converges to $0$ due to the boundedness of $\|A^{-\alpha} v_n\|$ and [\[+210+\]](#+210+){reference-type="eqref" reference="+210+"}. Moreover, due to the boundedness of $\|A^{\frac{1}{2}}v_n\|$ and [\[+k1-\]](#+k1-){reference-type="eqref" reference="+k1-"}, one has $$|(A^{\frac{\beta}{2}}q_n,A^{-\alpha} v_n)|=|(q_n,A^{\frac{\beta}{2}-\alpha} v_n)|\leq \|q_n\|\|A^{\frac{\beta}{2}-\alpha} v_n\|\leq \|q_n\|\|A^{\frac{1}{2}} v_n\| =o(1).$$ Therefore, it follows from [\[+211+\]](#+211+){reference-type="eqref" reference="+211+"} that $$\label{+212+} \|v_n\| =o(1).$$ By the interpolation inequality in Lemma [Lemma 4](#lemma-inter){reference-type="ref" reference="lemma-inter"}, along with [\[+212+\]](#+212+){reference-type="eqref" reference="+212+"} and the boundedness of $\|A^{\frac{1}{2}}v_n\|$, we have $$\label{+216} {\|A^{\frac{\gamma}{2}}v_n\|\leq \|v_n\|^{1-\gamma}\|A^{\frac{1}{2}}v_n\|^{\gamma}=o(1).}$$ Note that $\|A^{\frac{\gamma}{2}}u_n\|$ is bounded from [\[+2111\]](#+2111){reference-type="eqref" reference="+2111"} and $0\le \gamma\le 1$. Thus, taking the inner product of [\[+g2\]](#+g2){reference-type="eqref" reference="+g2"} with $A^{\frac{\gamma}{2}}u_n$ on $H$, substituting [\[+g1\]](#+g1){reference-type="eqref" reference="+g1"} into it, we get $$\label{+217} -(v_n,v_n)- m(A^{\frac{\gamma}{2}}v_n, A^{\frac{\gamma}{2}}v_n)+\sigma(A^{\frac{1}{2}}u_n, A^{\frac{1}{2}}u_n)-(A^\alpha\theta_n,u_n)=o(1), \quad m\ge0.$$ One has that $\|A^{-\frac{\beta}{2}+\alpha}u_n\|$ is bounded thanks to [\[+2111\]](#+2111){reference-type="eqref" reference="+2111"} and the fact that $\beta\geq 2\alpha-1$. Recalling [\[+209+\]](#+209+){reference-type="eqref" reference="+209+"}, we have $$\label{217+} (A^\alpha\theta_n,u_n)=(A^{\frac{\beta}{2}}\theta_n, A^{-\frac{\beta}{2}+\alpha}u_n)\leq\|A^{\frac{\beta}{2}}\theta_n\|\|A^{-\frac{\beta}{2}+\alpha}u_n\|=o(1).$$ Hence, substituting [\[+212+\]](#+212+){reference-type="eqref" reference="+212+"}, [\[+216\]](#+216){reference-type="eqref" reference="+216"} and [\[217+\]](#217+){reference-type="eqref" reference="217+"} into [\[+217\]](#+217){reference-type="eqref" reference="+217"} yields $$\label{+214+} \|A^{\frac{1}{2}}u_n\| =o(1).$$ In summary, by [\[+k1-\]](#+k1-){reference-type="eqref" reference="+k1-"}, [\[+210+\]](#+210+){reference-type="eqref" reference="+210+"}, [\[+212+\]](#+212+){reference-type="eqref" reference="+212+"}, [\[+216\]](#+216){reference-type="eqref" reference="+216"} and [\[+214+\]](#+214+){reference-type="eqref" reference="+214+"}, we arrive at $\|Z_n\|_{\mathcal{H}}=\|(u_{n}, v_{n}, \theta_{n}, q_{n})\|_{\mathcal{H}}=o(1)$, which contradicts the assumption that $\|Z_n\|_\mathcal{H}=1$. The desired result follows. abc99xyz F. Ammar-Khodja, A. Bader, A. Benabdallah, Dynamic stabilization of systems via decoupling techniques, ESAIM Control Optim. Calc. Var. 4 (1999) 577--593. W. Arendt, C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans Amer Math Soc. 306 (1988) 837--852. G. Avalos, I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, R nd. Ist. Mat. Univ. Trieste Suppl. 28 (1997) 1--28. C.J.K. Batty, R. Chill, Y. Tomilov, Fine scales of decays of operator semigroups, J. Eur. Math. Soc. 18 (2016) 853--929. A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010) 455--478. C. Cattaneo, Sulla conduzione del calore, Atti. Sem. Mat. Fis. Univ. 3 (1948) 83--101. C. Cattaneo, Sur une forme de l quation de la chaleur liminant le paradoxe dune propagation instantane, Comp. Rend. Hebd. S ances Acad. Sci. 247 (1958) 431--433. F. Dell'Oro, J. E. Muñoz Rivera, V. Pata, Stability properties of an abstract system with applications to linear thermoelastic plates, J. Evol. Equ. 13 (2013) 777--794. H. D. Fernández Sare, Z. Liu, R. Racke, Stability of abstract thermoelastic systems with inertial terms, J. Differ. Equ. 267 (2019) 7085--7134. Z. J. Han, Z. Kuang, Q. Zhang, Stability analysis for abstract theomoelastic systems with Cattaneo's law and inertial terms. Math. Control. Relat. F. 13 (2023) 1639--1673. J. Hao, Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys. 64 (2013) 1145--1159. J. Hao, Z. Liu, J. Yong, Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations, J. Differ. Equ. 259 (2015) 4763--4798. F. Huang, Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Differ. Equ. 104 (1993) 307--324. J.U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (1992) 889--899. Z. Kuang, Z. Liu, H. D. Fernández Sare, Regularity analysis for an abstract thermoelastic system with inertial term, ESAIM Control Optim. Calc. 27 (2021) S24. I. Lasiecka, R. Triggiani, Two direct proofs on the analyticity of the S.C. semigroup arising in abstract thermoelastic equations, Adv. Differ. Equ. 3 (1998) 387--416. K. Liu, Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys. 48 (1997) 885--904. Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys. 56 (2005) 630--644. Z. Liu, S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999. H. W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids. 15 (1967) 299--309. P. Loreti, B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation, SIAM J. Control Optim. 45 (2006) 1612--1632. J. E. Muñoz Rivera, R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems, J. Differ. Equ. 127 (1996) 454--483. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. J. Rozendaal, D. Seifert, and R. Stahn, Optimal rates of decay for operator semigroups on Hilbert spaces, Adv. Math. 346 (2019) 359--388. D. L. Russell, A general framework for the study of indirect mechanisms in elastic systems, J. Math. Anal. Appl. 173 (1993) 339--358. P. Vernotte, Les paradoxes de la thorie continue de l quation de la chaleur, Comp. Rend. Hebd. Sances Acad. Sci. 246 (1958) 3154--3155. [^1]: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China (email: 3120215730\@bit.edu.cn) [^2]: School of Mathematics, BIIT Lab, Tianjin University, Tianjin 300354, China (email: zjhan\@tju.edu.cn) [^3]: Computer Science Department, Stanford University, Stanford 94305, U.S.A.(email: zhaobin.kuang\@gmail.com) [^4]: School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China (email: zhangqiong\@bit.edu.cn) [^5]: Corresponding author. [^6]: CHXD was supported by the China Scholarship Council, ZJH was supported by the National Natural Science Foundation of China (grants No. 62073236). QZ was supported by the National Natural Science Foundation of China (grants No. 12271035, 12131008) and Beijing Municipal Natural Science Foundation (grant No. 1232018).

arxiv_math

--- abstract: | A reverse Hölder inequality is established on the space of Kähler metrics in the first Chern class of a Fano manifold $X$ endowed with Darvas' $L^{p}-$Finsler metrics. The inequality holds under a uniform bound on a twisted Ricci potential and extends to the general setup of Fano varieties. Its proof leverages a "hidden" log-concavity. An application to destabilizing geodesic rays is provided, which yields a reverse Hölder inequality for the speed of the geodesic. In the case of Aubin's continuity path the constant in the corresponding Hölder bound is shown to be universal when $X$ is K-unstable. Some intruiging relations to Harnack bounds and the partial $C^{0}-$estimate are pointed out. author: - Robert J. Berman title: Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and log-concavity --- # Introduction Let $X$ be an $n-$dimensional compact connected Kähler manifold. Consider the space $\mathcal{H}$ of all Kähler metrics on $X,$ in a fixed cohomology class in $H^{2}(X,\R).$ Assuming that $\mathcal{H}$ contains some Kähler metric $\omega$ the space $\mathcal{H}$ may be identified with the quotient space $\mathcal{H}(X,\omega)/\R,$ where $\mathcal{H}(X,\omega)$ denotes the space of all Kähler potentials (relative to $\omega):$ $$\mathcal{H}(X,\omega):=\left\{ u\in C^{\infty}(X):\,\omega_{u}:=\omega+dd^{c}u>0\right\} \,\,\,\,\left(dd^{c}u:=\frac{i}{2\pi}\partial\bar{\partial}u\right).$$ A canonical Riemannanian metric on $\mathcal{H}(X,\omega)$ was introduced in [@ma; @se; @do], turning $\mathcal{H}$ into an infinite dimensional Riemannian symmetric space of constant non-negative sectional curvature. More generally, a canonical $L^{p}-$Finsler metric on $\mathcal{H}(X,\omega)$ was put forth in [@da], defined by $$\left\Vert \frac{du_{t}}{dt}\right\Vert _{p}:=\left(\int_{X}\left|\frac{du_{t}}{dt}\right|^{p}\frac{\omega_{u_{t}}^{n}}{V}\right)^{1/p},\,\,\,V:=\int_{X}\omega^{n}$$ This Finsler metric induces a bona fide distance function (metric) $d_{p}$ on $\mathcal{H}(X,\omega),$ as shown in [@da] (generalizing the case $p=2$ established in [@ch]). From the point of view of quantization, the metric space $(\mathcal{H}(X,\omega),d_{p})$ can be viewed as a limit of the finite dimensional spaces $GL(N,\C)/U(N),$ endowed with an $L^{p}-$Finsler metric induced by the standard $l^{p}-$norm on $\R^{N},$ as $N\rightarrow\infty$ [@d-l-r]. An important motivation for allowing $p\neq2$ comes from the existence problem for canonical metrics on $X,$ where $p=1$ plays a privileged role. For example, assuming for simplicity that $X$ admits no non-trivial holomorphic vector fields, there exists, by [@c-cII; @b-d-l2], a constant scalar curvature Kähler (CSCK) metric in $\mathcal{H}$ iff the Mabuchi functional $\mathcal{M}$ on $\mathcal{H},$ introduced in [@ma], admits a minimizer (namely the CSCK metric) iff $\mathcal{M}$ is coercive with respect to $d_{1},$ i.e. iff $$\mathcal{M}(u)\geq Cd_{1}(u,0)-C$$ for some constant $C$ (where $\mathcal{H}$ has been identified with the space $\mathcal{H}(X,\omega)_{0}$ of all Kähler potentials $u$ satisfying $\sup_{X}u=0).$ Moreover, in the *Fano case -* i.e when $\omega$ is in the first Chern class $c_{1}(X)$ of $X$ - the coercivity in question is equivalent to (uniform) K-stability, by the solution of the Yau-Tian-Donaldson conjecture in this case [@c-d-s; @d-s; @b-b-j; @l-x-z]. By Hölder's inequality, applied to $\left\Vert \frac{du_{t}}{dt}\right\Vert _{p},$ $$d_{1}(u,0)\leq d_{p}(u,0)$$ However, in general, $d_{p}(u,0)$ can not be controlled by $d_{1}(u,0),$ since the metric completions $\overline{(\mathcal{H}(X,\omega),d_{p})}$ are strictly decreasing with respect to $p.$ This is illustrated by the toric case, where $d_{p}$ may be identified with the standard $L^{p}-$norm for the space of convex functions on the moment polytope of $X$ [@gu]. In contrast, a reverse Hölder inequality does hold in $\R^{N}$ (by the compactness of the unit-sphere), $$\left\Vert \cdot\right\Vert _{p}\leq C_{N}\left\Vert \cdot\right\Vert _{1},\label{eq:reverse H with N}$$ but the constant $C_{N}$ diverges as $N\rightarrow\infty.$ Still, the following result shows that in the Fano case a reverse Hölder type inequality holds on $\mathcal{H}(X,\omega),$ under the assumption that the twisted Ricci potential $\rho_{u,\gamma}$ of $u$ is uniformly bounded, for some $\gamma\in]0,1[.$ ** 1**. *Let $X$ be an $n-$dimensional Fano manifold. Given a Kähler form $\omega$ in $c_{1}(X),$ the following inequality holds on $\mathcal{H}(X,\omega),$ for any $p\in[1,\infty[$ and $\gamma\in]0,1[$* *$$d_{p}(u,0)\leq Ad_{1}(u,0)+B,$$ with $$A=A_{p}e^{2\left\Vert \rho_{u,\gamma}\right\Vert _{L^{\infty}}},\,\,\,B=B_{p}\left(\gamma^{-1}+(1-\gamma)^{-1}\right)e^{\left\Vert \rho_{u,\gamma}\right\Vert _{L^{\infty}}}$$ where $A_{p}$ only depends on $(p,n)$ and $B_{p}$ also depends on $(X,$$\omega).$ Moreover, if $\sup_{X}u=0$ then $B_{p}$ only depends on $(p,n).$* We recall that the twisted Ricci potential $\rho_{u,\gamma},$ which depends on $(\omega_{u},\omega,\gamma),$ may be defined by the equations $$dd^{c}\rho_{u,\gamma}=\text{Ric}\ensuremath{\omega_{u}}-\gamma\omega_{u}-(1-\gamma)\omega,\,\,\,\int_{X}e^{\rho_{u,\gamma}}\frac{\omega_{u}^{n}}{V}=1.\label{eq:def of twisted Ricci intro}$$ In particular, $h_{u,t}$ vanishes identically along *Aubin's continuity path $\omega_{t},$* defined by the following equations [@au]: $$\text{Ric}\,\ensuremath{\omega_{t}}=t\ensuremath{\omega_{t}}+(1-t)\omega,\label{eq:Aubin intr}$$ which for $t=1$ is the Kähler Einstein equation. By [@b-b-j; @c-r-z], the sup over all $t\in[0,1[$ for which the equations are solvable coincides with $\min\{\delta(X),1\},$ where $\delta(X)$ is the algebro-geometric invariant introduced in [@f-o]. This invariant - known as the *delta-invariant* or the *stability threshold* of $X$ - has the property that $X$ is (uniformly) K-stable if and only if $\delta(X)>1$ and K-semistable iff $\delta(X)\geq1$ [@bl-j; @l-x-z] ## Application to destabilizing geodesic rays There is a range of deformation methods in Kähler geometry that - given an initial Kähler form $\omega_{0}$ in $c_{1}(X)$ - produce a path $\omega_{t}$ of Kähler metrics along which the Mabuchi functional $\mathcal{M}$ decreases. A notable example - apart from Aubin's continuity path $\omega_{t}$ - is the Kähler-Ricci flow. For these two examples it is well-known that, as $t$ is increased, the $d_{1}-$distance $d_{1}(u_{t},0)$ at the level of Kähler potentials $u_{t}$ stays bounded iff $X$ admits a Kähler-Einstein metric, in which case $u_{t}$ converges to a Kähler-Einstein potential. On the other hand, if $d_{1}(u_{t},0)\rightarrow\infty$ then $u_{t}$ is weakly asymptotic to a geodesic ray in the metric completion $\overline{(\mathcal{H}(X,\omega),d_{1})}$ along which $\mathcal{M}$ decreases. In fact, the existence of such a $d_{1}-$geodesic only uses that $\mathcal{M}(u_{t})$ is decreasing (this result is implicit in [@d-h]). However, for special deformations one should obtain geodesic rays $v_{t}$ with advantageous properties. In the light of the Yau-Tian-Donaldson conjecture and its ramifications the best one can hope for is that the ray $v_{t}$ be induced by a *test configuration* for $X$ [@c-d-s; @d-s; @c-s-w] (as we shall come back to below). In particular, such a ray is a $d_{p}-$geodesic ray in the metric completions $\overline{(\mathcal{H}(X,\omega),d_{p})}$ for *any* $p\geq1.$ Here we show that the latter property holds under a uniform bound on the twisted Ricci potentials of $u_{t}:$ ** 2**. *Let $X$ be an $n-$dimensional Fano manifold and $u_{j}$ a sequence in $\mathcal{H}(X,\omega)$ such that* *$$(i)\,d_{1}(u_{j},0)\rightarrow\infty,\,\,\,(ii)\,\mathcal{M}(u_{j})\leq C,\,\,\,\ensuremath{(iii)\,\left\Vert \rho_{u,\gamma}\right\Vert _{L^{\infty}(X)}\leq R}$$ for some sequence $\gamma_{j}$ contained in a compact subset of $]0,1[.$ Then* - *$u_{j}$ is weakly asymptotic to a ray $v_{t}$ which is a $d_{p}-$geodesic ray in $\overline{(\mathcal{H}(X,\omega),d_{p})}$ for any $p\in[1,\infty[$ and $t\mapsto\mathcal{M}(v_{t})$ is decreasing.* - *the $d_{p}-$ speed $\left\Vert \dot{v}\right\Vert _{p}$ of the geodesic $v_{t}$ satisfies $$\left\Vert \dot{v}\right\Vert _{p}\leq A\left\Vert \dot{v}\right\Vert _{1},\label{eq:rever Holder for geod}$$ for a constant $A$ of the form $A_{p}e^{2R}$ where $A_{p}$ only depends on $(n,p).$* When the bound on the Ricci potential is replaced by a uniform Harnack bound, the first item above was established in [@d-h] and applied to the Kähler-Ricci flow. The proof in [@d-h] uses the Harnack bound in [@ru], which also holds for Aubin's continuity path [@t1]. However, in general, Harnack bounds tend to require rather detailed control on $\omega_{u},$ such as lower bounds on the Ricci curvature or uniform Sobelev constants (as discussed in Section [\[subsec:Comparison-with-Harnack\]](#subsec:Comparison-with-Harnack){reference-type="ref" reference="subsec:Comparison-with-Harnack"}). Accordingly, one advantage of Theorem [ 1](#thm:main intro){reference-type="ref" reference="thm:main intro"} and its corollary is that they generalize to singular setups, in particular the case when $X$ is a singular Fano variety. By the solution of the Yau-Tian-Donaldson conjecture for such varieties [@l-t-w; @li1; @l-x-z], $X$ admits a Kähler-Einstein metrics if and only if it is K-polystable. For non-singular $X$ this was originally shown in [@c-d-s] using a singular version of Aubin's continuity path and then in [@d-s] using Aubin's original continuity path $\omega_{t}.$ The proof is based on the partial $C^{0}-$estimate, which yields a detailed description of the blow-up behaviour of $\omega_{t}$ (as discussed below). However, for singular $X$ the only proof of the Yau-Tian-Donaldson conjecture is variational [@li1; @l-t-w; @l-x-z], building on [@bbegz; @b-b-j]. In general, given a positive $(1,1)-$current $\omega$ in $c_{1}(X),$ with locally bounded potentials the variational approach in [@bbegz] shows that there exists a solution $\omega_{t}$ to Aubin's continuity equation [ 3](#cor:Aubin intro){reference-type="ref" reference="cor:Aubin intro"} for some $t>0$ (in the weak sense of pluripotential theory) iff $X$ has log terminal singularities. The following result appears to give the first description of the blow-up behaviour of $\omega_{t}$ for singular $X.$ ** 3**. *Assume that $X$ is an $n-$dimensional K-unstable Fano variety with log terminal singularities, i.e. $\delta(X)\in]0,1[.$ Given a positive $(1,1)-$current $\omega$ in $c_{1}(X)$ with locally bounded potentials, denote by $\omega_{t}$ the corresponding solutions to Aubin's continuity equation [\[eq:Aubin intr\]](#eq:Aubin intr){reference-type="ref" reference="eq:Aubin intr"}, defined for $t\in[0,\delta(X)[.$ Then the curve $u_{t}$ of the corresponding sup-normalized potentials $u_{t}$ is weakly asymptotic - as $t$ increases towards $\delta(X)$ - to a non-trivial asymptotic ray $v_{t}$, which is a $d_{p}-$geodesic ray in $\overline{(\mathcal{H}(X,\omega),d_{p})}$ for any $p\in[1,\infty[$ and $t\mapsto\mathcal{M}(v_{t})$ is decreasing. Moreover, the $d_{p}-$ speed $\left\Vert \dot{v}\right\Vert _{p}$ of the geodesic $v_{t}$ satisfies $$\left\Vert \dot{v}\right\Vert _{p}\leq A_{p}\left\Vert \dot{v}\right\Vert _{1},\label{eq:rever Holder for geod-1}$$ for a constant $A_{p}$ only depending on $(p,n).$* Finally, we point out some intriguing relations between the inequality [\[eq:rever Holder for geod-1\]](#eq:rever Holder for geod-1){reference-type="ref" reference="eq:rever Holder for geod-1"} and the partial $C^{0}-$estimate along Aubin's continuity path conjectured in [@t2] and established in [@sz], when $X$ is non-singular. The partial $C^{0}-$estimate says that the Kähler potential $u_{t}$ of $\omega_{t}$ is of the form $$u_{t}=\varphi_{t}+O(1),\label{eq:partial}$$ where $O(1)$ is uniformly bounded in $L^{\infty}(X)$ and $\varphi_{t}$ is a family of Bergman metrics associated to a some tensor power $K_{X}^{*\otimes k}\rightarrow X.$ Embedding $X$ in the projectivization of $H^{0}(X,K_{X}^{*\otimes k}),$ identified with $\P^{N_{k}-1},$ this means that the corresponding Kähler forms $\omega_{\varphi_{t}}$ are the restriction to $X$ of $A_{t}^{*}\omega_{\text{FS}},$ for some curve $A_{t}\in\text{GL (}N_{k},\C),$ where $\omega_{\text{FS }}$denotes the Fubini-Study metric on $\P^{N_{k}-1}.$ Since the space of Bergman metrics at level $k$ - which is parameterized by $\text{GL (}N_{k},\C)/U(N_{k})$ - is finite dimensional it seems thus natural to expect that the inequality [\[eq:rever Holder for geod-1\]](#eq:rever Holder for geod-1){reference-type="ref" reference="eq:rever Holder for geod-1"} could be deduced from the standard reverse Hölder inequality in $\R^{N_{k}}$(formula [\[eq:reverse H with N\]](#eq:reverse H with N){reference-type="ref" reference="eq:reverse H with N"}). This can be made more precise as follows. As shown in [@c-d-s III], a subsequence of a family $A_{t}$ in $\text{GL (}N_{k},\C)$ - satisfying appropriate assumptions - is asymptotic to a one-parameter subgroup of $\text{GL (}N_{k},\C),$ induced by a *special test configuration* $\mathcal{X}$ for $X.$ If these assumption would apply, the previous corollary would follow from the fact that geodesic rays associated to a special test configurations also satisfy a universal reverse Hölder inequality of the form [\[eq:rever Holder for geod-1\]](#eq:rever Holder for geod-1){reference-type="ref" reference="eq:rever Holder for geod-1"} (as observed in Section [\[subsec:Test-configurations-and\]](#subsec:Test-configurations-and){reference-type="ref" reference="subsec:Test-configurations-and"}). However, as discussed in [@d-s Section 3.1], the assumptions in question have not yet been established for Aubin's original continuity method. But they do hold in the singular setup considered in [@c-d-s], where the Kähler form $\omega$ is replaced by the positive current defined by an appropriate anti-canonical divisor $\Delta$ on $X.$ Anyhow, the partial $C^{0}-$estimate appears to be widely open for singular Fano varieties $X$ (since its proof requires, in particular, a uniform bound on the Sobolev constants; cf. [@zh0]). In the light of the previous discussion it is natural to conjecture that the geodesic ray $v_{t}$ appearing in the previous corollary is induced by a special test configuration $\mathcal{X}$ and that $\mathcal{X}$ computes the stability threshold $\delta(X)$ in the sense of [@l-x-z Thm 1.2] (a related conjecture is proposed in [@x]). ## Proof via log-concavity In a nutshell, the idea of the proof of Theorem [ 1](#thm:main intro){reference-type="ref" reference="thm:main intro"} is to relate $d_{p}(u,0)$ to the moments of a somewhat hidden log-concave measure on $\R$ and use the reverse Hölder inequality for random variables with log-concave distribution [@m-s App.III][@la], known as the Kahane-Khinchin inequality. In general, $d_{p}(u,0)$ can also expressed directly as the $p-$th moment of a probability measure on $\R,$ introduced in [@bern2]. However, as pointed out in Section [\[subsec:Failure-of-log-concavity\]](#subsec:Failure-of-log-concavity){reference-type="ref" reference="subsec:Failure-of-log-concavity"}, this measure is not log-concave, nor is the analogous measure associated to a test configuration, studied in [@wi; @hi; @bhj], in general. ## Acknowledgments I am grateful to Bo Berndtsson for countless discussions over the years on the complex Brunn-Minkowski theory developed in [@bern1; @bern1b]. Also thanks also for comments on a draft of the present paper from Rolf Andreasson, Sébastien Boucksom, Tamas Darvas, Mattias Jonsson and Gabor Székelyhidi. This work was supported by a grant from the Knut and Alice Wallenberg foundation. # Preliminaries Throughout the paper the precise value of the constants $C$ etc appearing in the inequalities may change from line to line. ## [\[subsec:The-space-of\]]{#subsec:The-space-of label="subsec:The-space-of"}The space of Kähler potentials Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$ and denote by $\mathcal{H}(X,\omega)$ the corresponding space of all Kähler potentials $u$ on $X:$ $$\mathcal{H}(X,\omega):=\left\{ u\in C^{\infty}(X):\,\omega_{u}:=\omega+dd^{c}u>0\right\} \,\,\,\,\left(dd^{c}u:=\frac{i}{2\pi}\partial\bar{\partial}u\right).$$ Following [@da] the Finsler $L^{p}-$metric on $\mathcal{H}(X,\omega)$ defined by $$\left(\int_{X}\left|\frac{du_{t}}{dt}\right|^{p}\frac{\omega_{u_{t}}^{n}}{V}\right)^{1/p},\,\,\,V:=\int_{X}\omega^{n}$$ induces a metric $d_{p}$ on $\mathcal{H}(X,\omega)$ satisfying the following inequalities, for constants $c_{n,p}$ only depending on $n$ and $p$ (see [@da Thm 3]): $$c_{n,p}^{-1}\left(\int|u_{0}-u_{1}|^{p}\left(\frac{\omega_{u_{0}}^{n}}{V}+\frac{\omega_{u_{1}}^{n}}{V}\right)\right)^{1/p}\leq d_{p}(u_{0},u_{1})\leq c_{n,p}\left(\int|u_{0}-u_{1}|^{p}\left(\frac{\omega_{u_{0}}^{n}}{V}+\frac{\omega_{u_{1}}^{n}}{V}\right)\right)^{1/p}\label{eq:Darvas}$$ The first inequality, applied to $(u_{0},u_{1})=(0,u),$ implies that $$\left(\int|u|^{p}\frac{\omega^{n}}{V}\right)^{1/p}\leq c_{n,p}d_{p}(u,0)\label{eq:L p norm leq d p}$$ and, as a consequence, $$\left|\sup_{X}u\right|\leq c_{n,1}d_{1}(u,0)+C_{\omega}\label{eq:bounds of sup in terms of d one}$$ (see [@da Cor 4]). We will denote by $\mathcal{H}(X,\omega)_{0}$ the subspace $\mathcal{H}(X,\omega)$ of all *sup-normalized* Kähler potentials: $$\mathcal{H}(X,\omega)_{0}:=\left\{ u\in\mathcal{H}(X,\omega):\,\sup_{X}u=0\right\} .$$ ### [\[subsec:Metric-completions-and\]]{#subsec:Metric-completions-and label="subsec:Metric-completions-and"}Metric completions, finite energy spaces and geodesics Denote by $\text{PSH }(X,\omega)$ the subspace of $L^{1}(X)$ consisting of all *$\omega-$plurisubharmonic ($\omega-$psh)* functions on $X,$ i.e. all strongly upper-semicontinuous functions $u$ such that $\omega_{u}\geq0$ holds in the sense of currents [@g-z]. We will denote by $\omega_{u}^{n}$ the *non-pluripolar Monge-Ampère measure* of $u$ [@g-z2]. As shown in [@da] - answering a conjecture of Guedjs when $p=2$ - the metric completion $\overline{(\mathcal{H}(X,\omega),d_{p})}$ of the metric space $(\mathcal{H}(X,\omega),d_{p})$ may be identified with the finite energy space $$\mathcal{E}^{p}(X,\omega):=\left\{ u\in\text{PSH }(X,\omega):\,\int_{X}\omega_{u}^{n}=\int_{X}\omega^{n},\,\,\int_{X}|u|^{p}\omega_{u}^{n}<\infty\right\} ,$$ introduced in [@g-z2] and the inequalities [\[eq:Darvas\]](#eq:Darvas){reference-type="ref" reference="eq:Darvas"} hold on all of $\mathcal{E}^{p}(X,\omega).$ Moreover, the spaces $\mathcal{E}^{p}(X,\omega)$ are strictly decreasing wrt $p.$ Any two elements $u_{0},u_{1}$ in $\mathcal{E}^{1}(X,\omega)$ can be connected by a canonical path $u_{t}$ called a *finite energy geodesic* in [@da; @d-l] and a *psh geodesic* in [@b-b-j], defined by the following envelope: $$u_{t}(x):=\sup\left\{ v_{t}(x):v_{t}\,\,\text{is a subgeodesic}\,\,\,\limsup_{t\rightarrow0}v_{t}\leq u_{0},\,\limsup_{t\rightarrow1}v_{t}\leq u_{0}\right\} .\label{eq:psh geodes}$$ We recall that a *subgeodesic* $v_{t}$ is defined as a curve in $\text{PSH }(X,\omega)$ with the following property: complexifying $t$ the corresponding $i\R-$invariant function $V(x,t):=v_{\text{\ensuremath{\Re}}t}(x)$ on $X\times]0,1[\times i\R$ is in $\text{PSH}\left(X\times]0,1[\times i\R,\omega\right),$ using the same notation $\omega$ for the pull-back of $\omega$ to the product $X\times(]0,1[\times i\R).$ Recall that, given a metric space $(M,d),$ a curve $u_{t}$ connecting two given points $u_{0}$ and $u_{1}$ in $M$ is said to be a *$d-$geodesic* (also known as a *constant speed geodesic*) if $$d(u_{t},u_{0})=td(u_{1},u_{0}),\,\,\text{when \ensuremath{t\in[0,1]}}.$$ As shown in [@da], building on [@bern2], the psh geodesic $u_{t}$ connecting any two given elements $u_{0},u_{1}$ in $\mathcal{E}^{p}(X,\omega)$ is a $d_{p}-$geodesic. When $p>1$ this is the unique $d_{p}-$geodesic connecting $u_{0},u_{1}$ (by [@d-l Thm 3.3]). Given a $d_{p}-$geodesic $v_{t}$ we will use the notation $$\left\Vert \dot{v}\right\Vert _{p}:=d_{p}(v_{t},v_{0})/t,\,\,t>0$$ which is independent of $t.$ In the terminology of [@d-l] $\left\Vert \dot{v}\right\Vert _{p}$ is the distance between the $d_{p}-$geodesic $v_{t}$ and the constant geodesic $0$ wrt the cordal metric on the space of $d_{p}-$geodesic rays, introduced in [@d-l]. ## [\[subsec:The-Fano-setup\]]{#subsec:The-Fano-setup label="subsec:The-Fano-setup"}The Fano setup Henceforth $X$ will be assumed to be a compact *Fano manifold*. This means that the anti-canonical line bundle $K_{X}^{*}$ is ample. Equivalently, $X$ admits a volume form $dV_{X}$ with positive Ricci curvature, i.e. $$\omega:=\text{Ric \ensuremath{dV_{X}}}\,\,\,(:=-dd^{c}\log dV_{X})$$ defines a Kähler form on $X$ in the first Chern class $c_{1}(X)$ of $X.$ Conversely, any Kähler form $\omega$ in $c_{1}(X)$ may be expressed as in the previous equation and the corresponding volume form $dV_{X}$ is uniquely determined by $\omega$ under the normalization condition $\int_{X}dV_{X}=1,$ which will henceforth be assumed. Denote by $V$ the *volume* of $X:$ $$V:=c_{1}(X)^{n}\left(=\int_{X}\omega^{n}\right)$$ To $u$ in $\mathcal{H}(X,\omega)$ and $\gamma\in]0,\infty[$ we attach the following probability measure on $X:$ $$\mu_{\gamma u}:=\frac{e^{-\gamma u}dV_{X}}{\int_{X}e^{-\gamma u}dV_{X}}.$$ (which only depends on the Kähler form $\omega+dd^{c}u$ when $\gamma=1).$ In terms of this measure, the *twisted Ricci potential* $\rho_{u,\gamma}$ of a Kähler potential $u$ (definined in formula[\[eq:def of twisted Ricci intro\]](#eq:def of twisted Ricci intro){reference-type="ref" reference="eq:def of twisted Ricci intro"}) may, alternatively, be defined by the relation $$e^{\rho_{u,\gamma}}\frac{\omega_{u}^{n}}{V}=\mu_{\gamma u}.\label{eq:def of twisted Ricci pot}$$ $\rho_{u,1}$ is independent of $\omega$ and is usually simply called the *Ricci potential of $\omega_{u}.$* More generally, the twisted Ricci potential $\rho_{u,\gamma}$ is well-defined for any $u\in\text{PSH }(X,\omega)$ satisfying - the non-pluripolar Monge-Ampère measure $\omega_{u}^{n}$ is, locally, absolutely continious wrt Lesbegue measure - $e^{-\gamma u}dV_{X}$ has finite total mass. # [\[subsec:Lifting-to-the\]]{#subsec:Lifting-to-the label="subsec:Lifting-to-the"}Log-concavity Fix $u\in\mathcal{H}(X,\omega)_{0}$ and consider the following function on $\R:$ $$Z_{X}(\gamma):=\int_{X}e^{-\gamma u}dV_{X}.$$ The key analytic ingredient in the proofs in the coming sections is the following result, expressing $Z_{X}(\gamma)$ in terms of the Laplace transform of a *log concave measure* $\nu_{0}$ on $\R.$ That is to say that $\nu_{0}$ is either a Dirac mass or $\nu_{0}$ is absolutely continuous wrt Lesbegue measure on $\R$ and the logarithm of its density is concave. ** 4**. *For any $u\in\mathcal{H}(X,\omega)_{0}$ and $\gamma\in]0,1[$ $$\int_{X}e^{-\gamma u}dV_{X}=G(\gamma)\int_{\R}e^{-t\gamma}\nu_{0}\label{eq:Z in terms of nu 1}$$ for a log-concave measure $\nu_{0}$ on $\R$ (depending on $u)$ and a smooth function $G(\gamma)$ on $[0,1[$ (independent of $u)$.* It should be stressed that the previous theorem does not hold with a *constant* function $G(\gamma),$ as can be checked in simple examples. This means that the measure on $\R$ defined as the push-forward of $dV_{X}$ under $u$ is not log-concave, in general. Still it would be interesting to know if $G(\gamma)$ can be taken to be bounded as $\gamma\rightarrow1$? If so, this would eliminate the diverging factor $(1-\gamma)^{-1}$ appearing in Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"}. ## Preparations for the proof of Thm [ 4](#Thm:Z as log concave){reference-type="ref" reference="Thm:Z as log concave"} {#preparations-for-the-proof-of-thm-thmz-as-log-concave} The proof is based on a lifting argument where $X$ is replaced by the total space of $K_{X}\rightarrow X.$ We start by setting up some notation. Given a Fano manfold $X$ denote by $Y$ the total space of the canonical line bundle $K_{X}\rightarrow X$ and by $Y^{*}$ the space obtained by deleting the zero-section of $K_{X}\rightarrow X.$ There is a canonical $\C^{*}-$action on $Y,$ induced by the linear structure on the fibers of the line bundle $K_{X}\rightarrow X.$ Moreover, the space $Y$ comes with a canonical holomorphic top form $\Omega,$ which is one-homogeneous wrt the $\C^{*}-$action on $Y.$ Indeed, $\Omega$ can be constructed as follows. Fix local holomorphic coordinates $z\in\C^{n}$ on $X$ and trivialize, locally, $K_{X}\rightarrow X$ by the corresponding holomorphic section $dz(:=dz_{1}\wedge\cdots\wedge dz_{n}).$ This induces local holomorphic coordinates $(z,w)\in\C^{n+1}$ on $Y$ and the top form $\Omega$ defined by $dz\wedge dw$ glues to a globally well-defined holomorphic top form on $Y.$ Recall that we have fixed a volume form $dV_{X}$ on $X$ with positive Ricci curvature, denoted by $\omega$ (Section [\[subsec:The-Fano-setup\]](#subsec:The-Fano-setup){reference-type="ref" reference="subsec:The-Fano-setup"}). It induces a function $r$ on $Y$ determined by $$i^{(n+1)}\Omega\wedge\bar{\Omega}=rdr\wedge dV_{X}\wedge d\theta,\,\,\text{on \ensuremath{Y^{*}}}\label{eq:polar coordinates}$$ where $dV_{X}\wedge d\theta$ denotes the fiber product of the measure $dV_{X}$ on the base of the fibration $Y^{*}\rightarrow X$ with the family of standard $S^{1}-$invariant measures $d\theta$ on the fibers of the fibration. The function $r$ is one-homogeneous wrt the $\R_{>0}-$action on $Y$ (since $\Omega$ is) and plurisubharmonic (since, by assumption, $\text{Ric \ensuremath{dV_{X}\geq0}). }$ Moreover, $r$ extends to a psh function on $Y$ (since it is bounded from above in a neighourhood of the zero-section in $K_{X}).$ To any given $u\in\text{PSH}(X,\omega)$ we associate the following psh function on $Y:$ $$\psi:=u+\log r^{2}.\label{eq:psi as u}$$ The psh functions $\psi$ on $Y$ that can be be expressed in this way are precisely the ones which are *log-homogeneous* under the $\C^{*}-$action on $Y;$ $\psi(\lambda\cdot)=\log(|\lambda|^{2})+\psi$ for any $\lambda\in\C^{*}.$ ** 5**. *Assume that $\psi_{0}=f(\log r^{2}),$ where $f$ is an increasing convex function on $]-\infty,\infty[,$ assumed bounded from below and that $\psi_{1}$ is psh and log-homogeneous on $Y.$ Then the following function is concave on $\R:$ $$V(t):=\log\int_{\left\{ \psi_{1}\leq t\right\} }e^{-\psi_{0}}i^{(n+1)}\Omega\wedge\bar{\Omega},$$ if it is finite for all $t.$ More generally, the result holds when $Y$ is a (possibly singular) Fano variety and $r$ is the one-homogeneous function corresponding to a (possibly singular) metric on $K_{X}^{*}$ with positive curvature current, assuming that the corresponding measure $dV_{X}$ on $X$ gives finite total volume to $X.$* *Proof.* In the case when $Y=\C^{m}$ (which, after passing to a finite cover, corresponds to the case when $X=\P^{n})$ the result follows from [@ber1 Prop 6.5], which is a slight generalization of [@bern1 Thm 1.2] and [@b-b Thm 2.3]. The proof first uses the subharmonicity result for Bergman kernels established in [@bern1 Thm 1.1]. Then the $S^{1}-$symmetry of $\psi_{0}$ and $\psi_{1}$ is used. In the case when $X$ is smooth one could, presumably, proceed in a similar manner. Indeed, the proof of [@bern1 Thm 1.1] is based on $\overline{\partial}-$estimates for $(n+1,0)-$forms on the pseudoconvex domain $\{\left\{ \psi_{1}\leq t\right\} \}$ with weight $e^{-\psi_{0}},$ which still apply when $Y$ is a complex manifold. But one technical difficulty in the proof in [@bern1] stems from the fact that, in general, the domain $\left\{ \psi_{1}\leq t\right\}$ may be not have a smooth boundary, which is bypassed using an approximation procedure. Since this seems to lead to major technical difficulties when $X$ (and thus also $Y)$ is singular we will instead give a more direct proof, which has the virtue that is also applies to singular $X.$ First assume that $dV_{X}$ is a smooth volume form. Then $r$ is a smooth coordinate on $Y^{*}.$ Hence, using the factorization [\[eq:polar coordinates\]](#eq:polar coordinates){reference-type="ref" reference="eq:polar coordinates"} and first performing the integration over $r,$ we can express $$V(t)=\int_{0}^{e^{-u_{t}}}e^{-\psi_{0}(r)}rdr\int_{X}dV_{X},\,\,\,u_{t}:=u-t.\label{eq:fubini}$$ The change of variables $s=\log(r^{2})$ yields $$\frac{1}{2}\int_{0}^{e^{-u_{t}}}e^{-\psi_{0}(r)}rdr=\int_{\{s<-u\}}e^{-\phi(s)},\,\,\,\phi(s):=f(s)-s$$ Now, given a real variable $u,$ define $\chi(u)$ by $$\chi(u)=-\log\int_{\{s<-u\}}e^{-\phi(s)}.$$ Let us show that $$(i)\,\,\chi''(u)\geq0,\,\,\,(ii)\,0\leq\chi'(u)\leq1.\label{eq:chi deriv}$$ Since $\phi(s)$ is convex the first item follows directly from Prekopa's theorem [@p] (or the Brunn-Minkoski inequality). Next, by definition, $$\chi'(u):=\frac{e^{-\phi(-u)}}{\int_{\{s<-u\}}e^{-\phi(s)}},$$ which is manifestly non-negative. In order to show that $\chi'(u)\leq1$ it is - since $\chi(u)$ is convex, by $(i)$ - enough to consider the limit when $u\rightarrow\infty.$ By assumption, we have that $\phi(s)=f(-\infty)-s+o(1),$ uniformly as $s\rightarrow-\infty.$ Hence, in the limit where $u\rightarrow\infty$ the quotient above coincides with $$\lim_{u\rightarrow\infty}\frac{e^{-u}}{\int_{\{s<-u\}}e^{s}}=1,$$ using that the denominater equals $e^{-u}.$ This proves formula [\[eq:chi deriv\]](#eq:chi deriv){reference-type="ref" reference="eq:chi deriv"}. Next note that, if $\chi$ satisfies the conditions in formula [\[eq:chi deriv\]](#eq:chi deriv){reference-type="ref" reference="eq:chi deriv"} , then $$u\in\text{PSH }(X,\omega),\implies\chi(u)\in\text{PSH }(X,\omega).\label{eq:implication}$$ Indeed, $$dd^{c}\chi(u)=\chi''du\wedge d^{c}u+\chi'dd^{c}u\geq\chi'\omega_{u}-\chi'\omega\geq-\omega.$$ Now complexify $t$ and consider the function $U:=u-\Re t$ on $X\times(\R\times i\R).$ Then $dd^{c}U\geq-\omega,$ where we have identified $\omega$ with its pull-back to $X\times(\R\times i\R).$ Applying the implication [\[eq:implication\]](#eq:implication){reference-type="ref" reference="eq:implication"} with $u$ replaced by $U$ thus reveals that $\chi(U)\in\text{PSH }(X\times(\R\times i\R),\omega).$ This means that $v_{t}:=\chi(u_{t})$ is a subgeodesic in $\text{PSH}(X,\omega)$ (as defined in Section [\[subsec:Metric-completions-and\]](#subsec:Metric-completions-and){reference-type="ref" reference="subsec:Metric-completions-and"}). Hence, the proposition follows from the following complex generalization of the Prekopa theorem for convex functions established in [@bern1b]: the function $$t\mapsto-\log\int_{X}e^{-v_{t}}dV_{X}\label{eq:complex prekopa}$$ is convex for any subgeodesic $v_{t}.$ Next, consider the case when $X$ is non-singular, but $dV_{X}$ corresponds to a (possibly singular) metric on $K_{X}^{*}.$ This means that the local density of $dV_{X}$ is of the form $e^{-\phi}$ where $\phi$ locally represents the metric on $K_{X}^{*}$ in question, in additive notation. In the case that $\phi$ is locally bounded the same proof as in the case when $dV_{X}$ is a volume form (i.e. $\phi$ is smooth) still applies. In the general case we can express $\phi$ as a decreasing limit of locally bounded metrics $\phi_{j}$ on $K_{X}^{*}$ with positive curvature current and conclude using the monotone convergence theorem. Finally, when $X$ is non-singular we proceed in essentially the same way, using that the convexity of the function [\[eq:complex prekopa\]](#eq:complex prekopa){reference-type="ref" reference="eq:complex prekopa"} still holds, as observed in [@bbegz] (and the decomposition [\[eq:fubini\]](#eq:fubini){reference-type="ref" reference="eq:fubini"} still holds, since it can be applied on the regular locus of $X).$ ◻ ## Conclusion of the proof of Thm [ 4](#Thm:Z as log concave){reference-type="ref" reference="Thm:Z as log concave"} {#conclusion-of-the-proof-of-thm-thmz-as-log-concave} We will apply Prop [ 5](#prop:concavity){reference-type="ref" reference="prop:concavity"} with $\psi_{0}=r^{2}.$ Expressing $u$ in terms of $\psi$ on $Y$ we can write $$\int_{X}e^{-\gamma u}dV_{X}=\frac{1}{\int_{0}^{\infty}r^{-2\gamma}e^{-r^{2}}rdr}\int_{Y}e^{-\gamma\psi}dV_{Y},\,\,\,\,dV_{Y}:=e^{-\psi_{0}}i^{(n+1)}\Omega\wedge\bar{\Omega}\label{eq:def of dV Y}$$ Hence, pushing forward the integration over $Y$ to $\R$ and using the factorization [\[eq:polar coordinates\]](#eq:polar coordinates){reference-type="ref" reference="eq:polar coordinates"} gives $$Z_{X}(\gamma)=\frac{1}{\int_{0}^{\infty}r^{-2\gamma}e^{-r^{2}}rdr}\int_{\R}e^{-\gamma t}\psi_{*}dV_{Y},\label{eq:Z in terms of push}$$ for any $\gamma.$ Next, note that for $\gamma>0$ $$\int_{\R}e^{-\gamma t}\psi_{*}dV_{Y}=\gamma\int_{\R}e^{-t\gamma}\nu_{0},\,\,\,\nu_{0}=V(t)dt,\,\,V(t):=dV_{Y}\left(\left\{ \psi<t\right\} \right).\label{eq:integrtion by parts}$$ Indeed, since $\psi_{*}dV_{Y}=V'(t)dt$ (in the sense of distributions) integrating by parts reveals that the previous formula holds if $$\lim_{t\rightarrow\pm\infty}e^{-t\gamma}V(t)=0,\,\,\,\,(\gamma>0)$$ But, by Chebyshev's inequality, $V(t)\leq C_{\epsilon}e^{\epsilon t}$ for any $\epsilon\in]0,1[$ (with $C_{\epsilon}=\int_{Y}e^{-\epsilon\psi}dV_{Y}$). The vanishing in the previous equation thus follows when $t\rightarrow\infty$ and $t\rightarrow-\infty$ by taking $\epsilon$ sufficiently close to $0$ and $1,$ respectively. This concludes the proof of formula [\[eq:integrtion by parts\]](#eq:integrtion by parts){reference-type="ref" reference="eq:integrtion by parts"} and shows that formula in the Theorem holds with $$G(\gamma):=\frac{\gamma}{\int_{0}^{\infty}r^{-2\gamma}e^{-r^{2}}rdr}(=\frac{\gamma}{\Gamma(1-\gamma)/2}),$$ where $\Gamma(s)$ is the classical Gamma-function. Finally, the log-concavity of $\nu_{0}$ follows from Prop [ 5](#prop:concavity){reference-type="ref" reference="prop:concavity"}, applied to the functions $(\psi_{0},\psi_{1})=(r^{2},\psi)$ on $Y.$ # Moment bounds on Fano manifolds Let $X$ be a Fano manifold. Using the notation introduced in the previous sections we will, in this section, prove the following general "moment bounds" (from which Theorem [ 1](#thm:main intro){reference-type="ref" reference="thm:main intro"} will be deduced in the next section). ** 6**. *Given $p\in[1,\infty[$ and $\gamma\in]0,1[$ the following inequality holds for any $u$ in $\mathcal{H}(X,\omega)$ such that $\sup_{X}u\leq0:$* *$$\left(\int_{X}(-u)^{p}\mu_{\gamma u}\right)^{1/p}\leq A_{p}\int_{X}(-u)\mu_{\gamma u}+B_{p}\left(\gamma^{-1}+(1-\gamma)^{-1}\right)$$ where the constants $A_{p}$ and $B_{p}$ only depend on $p$. More generally, given $\gamma\in]0,1[$ the inequality holds for any $u\in\text{PSH }(X,\omega)$ such that $\sup_{X}u\leq0,$ if $\int e^{-\gamma u}dV_{X}<\infty.$* * 7*. The inequality above does not hold with $A_{p}$ and $B_{p}$ independent of $p.$ Indeed, otherwise one could, by letting $p\rightarrow\infty,$ replace the lhs in the inequality with $\left\Vert u\right\Vert _{L^{\infty}(X)}.$ But this contradicts the fact that there exist $u\in\text{PSH \ensuremath{(X,\omega)}}$ which are unbounded, while $\int_{X}(-u)\mu_{\gamma u}$ is finite for any $\gamma$ (for example, any unbounded $u\in\text{PSH \ensuremath{(X,\omega)}}$ with vanishing Lelong numbers). The idea of the proof is to combine Thm [ 4](#Thm:Z as log concave){reference-type="ref" reference="Thm:Z as log concave"} with the following well-known "reverse" Hölder inequality that holds for any log-concave probability measure $\nu$ on $\R$ and $p\geq1:$ $$\left(\int|t|^{p}d\nu\right)^{1/p}\leq C_{p}\left(\int|t|d\nu\right)\label{eq:reversed H=0000F6lder log concave}$$ with a universal constant $C_{p}$ (only depending on $p)$ [@m-s App.III][@la]. In order to use this inequality we will apply the following elementary ** 8**. *Let $\sigma$ be a measure on a space $S$ and $f$ a measurable function on $(S,\sigma).$ Given $\gamma\in\R$ denote by $\left\langle \cdot\right\rangle _{\gamma}$ integration wrt the probability measure $e^{\gamma f}\sigma/\int_{S}e^{\gamma f}\sigma,$ assuming that $\int_{S}e^{(\gamma+\epsilon)f}\sigma<\infty$ for all sufficiently small $\epsilon.$ Then there exist universal coefficents $a_{j_{1},...,j_{p-1}}\in\R$ such that* *$$\left\langle f^{p}\right\rangle _{\gamma}=\frac{d^{p}}{d\gamma^{p}}\log\left\langle e^{f\gamma}\right\rangle _{\gamma}+\sum_{j_{1},...,j_{p-1}}a_{j_{1},...,j_{p-1}}\left\langle f^{j_{1}}\right\rangle _{\gamma}\left\langle f^{j_{2}}\right\rangle _{\gamma}\cdots\left\langle f^{j_{p-1}}\right\rangle _{\gamma},$$ where the sum ranges over all non-negative integer indices $(j_{1},...,j_{p-1})$ such that $j_{i}<p,$ $j_{1}+...+j_{p-1}=p.$* *Proof.* This can be shown using direct differentiation, by rewriting $e^{\gamma f}\sigma/\int_{S}e^{\gamma f}\sigma=e^{\gamma f+\log\int_{S}e^{\gamma f}\sigma}\sigma.$ Alternatively, the formula follows from the well-known general result expressing $a_{j_{1},...,j_{p-1}}$ in terms of partial Bell polynomials [@w-n]. ◻ We will prove Theorem [ 6](#thm:moment bounds){reference-type="ref" reference="thm:moment bounds"} by induction over $p.$ We thus assume that $p\geq2$ and that Theorem [ 6](#thm:moment bounds){reference-type="ref" reference="thm:moment bounds"} holds for $p-1.$ By the formula in Prop[ 4](#Thm:Z as log concave){reference-type="ref" reference="Thm:Z as log concave"} we can express $$\log Z_{X}(\gamma)=\log\int_{\R}e^{-t\gamma}\nu_{0}+\log G(\gamma),$$ for a log-concave measure $\nu_{0}.$ Applying the previous lemma to $(S,\sigma,f)$ given by $(X,dV_{X},-u)$ and using the assumption that the theorem holds for $p-1$ gives $$\left\langle (-u)^{p}\right\rangle _{\gamma}\leq\frac{d^{p}}{d\gamma^{p}}\log\left\langle e^{-u\gamma}\right\rangle +\sum_{j_{1},...,j_{p-1}}\left|a_{j_{1},...,j_{p-1}}\right|\left\langle (-u)\right\rangle _{\gamma}^{j_{1}+...+j_{p-1}}\leq$$ $$\leq\frac{d^{p}\gamma}{d\gamma}\log\left\langle e^{t\gamma}\right\rangle _{\gamma}+\left(A_{p-1}\left\langle (-u)\right\rangle _{\gamma}+c_{p-1}\right)^{p}+C_{\gamma},$$ where $c_{p-1}$ is the additive constant appearing in the theorem for $p-1$ and $C_{\gamma}$ is the absolute value of the $p-$th derivative of the smooth function $\log G(\gamma),$ which satisfies $C_{\gamma}\leq C\left(\gamma^{-1}+(1-\gamma)^{-1}\right)^{p}$ (since $\Gamma(s)$ has a simple pole at $s=0$). Next, applying Lemma [ 8](#lem:element){reference-type="ref" reference="lem:element"} to $(\R,\nu_{0},t),$ combined with the "reverse" Hölder inequality [\[eq:reversed H=0000F6lder log concave\]](#eq:reversed H=0000F6lder log concave){reference-type="ref" reference="eq:reversed H=0000F6lder log concave"} for the log-concave probability measures $\nu_{\gamma}$ on $\R$ defined by $$\nu_{\gamma}:=\frac{e^{-t\gamma}\nu_{0}}{\int_{\R}e^{-t\gamma}\nu_{0}},$$ gives $$\frac{d^{p}\gamma}{d\gamma}\log\left\langle e^{t\gamma}\right\rangle _{\gamma}\leq C_{p}\left(\int|t|d\nu\right)^{p}$$ The induction step is thus conclude by combining the following lemma with the standard reverse Hölder inequality in $\R^{3}$ (appearing in formula [\[eq:reverse H with N\]](#eq:reverse H with N){reference-type="ref" reference="eq:reverse H with N"}). ** 9**. *The following inequality holds when $\gamma\in]0,1[:$ $$\int_{\R}|t|d\nu_{\gamma}\leq\int_{X}(-u)\mu_{\gamma u}+C\left(\gamma^{-1}+(1-\gamma)^{-1}\right)$$* *Proof.* Let us first show that $$\int_{\R}|t|d\nu_{\gamma}\leq\int_{Y}|\psi|\frac{e^{-\gamma\psi}dV_{Y}}{\int_{Y}e^{-\gamma\psi}dV_{Y}}+\gamma^{-1}.\label{eq:bound on abs t}$$ First, integrating by parts (just as in the proof of formula [\[eq:integrtion by parts\]](#eq:integrtion by parts){reference-type="ref" reference="eq:integrtion by parts"}) yields, with $\sigma(t)$ denoting the $L^{\infty}-$function defined by the sign of $t$, which is the distributional derivative of $|t|:$ $$\gamma\int_{\R}|t|e^{-\gamma t}V(t)dt=\int_{\R}e^{-\gamma t}\left(|t|V(t)\right)'dt=\int_{\R}e^{-\gamma t}\left(|t|V'(t)+\sigma(t)V(t)\right)dt$$ Hence, $$\int_{\R}|t|e^{-\gamma t}V(t)dt\leq\gamma^{-1}\int_{\R}e^{-\gamma t}|t|V'(t)dt+\gamma^{-1}\int_{\R}e^{-\gamma t}V(t)dt.$$ Dividing both sides with $\int e^{-\gamma t}V(t)dt$ and invoking formulas [\[eq:Z in terms of nu 1\]](#eq:Z in terms of nu 1){reference-type="ref" reference="eq:Z in terms of nu 1"}, [\[eq:integrtion by parts\]](#eq:integrtion by parts){reference-type="ref" reference="eq:integrtion by parts"}, thus proves formula [\[eq:bound on abs t\]](#eq:bound on abs t){reference-type="ref" reference="eq:bound on abs t"}. Next, using that that $|\psi|\leq|u|+|\log r^{2}|$ the integral over $Y$ in formula [\[eq:bound on abs t\]](#eq:bound on abs t){reference-type="ref" reference="eq:bound on abs t"} may be estimated by $$\int_{Y}|u|\frac{e^{-\gamma\psi}dV_{Y}}{\int_{Y}e^{-\gamma\psi}dV_{Y}}+\int_{Y}|\log r^{2}|\frac{e^{-\gamma\psi}dV_{Y}}{\int_{Y}e^{-\gamma\psi}dV_{Y}},$$ where $dV_{Y}$ was defined in formula [\[eq:def of dV Y\]](#eq:def of dV Y){reference-type="ref" reference="eq:def of dV Y"}. Expressing $dV_{Y}$ in terms of $dV_{X},$ the first integral equals $\int_{Y}|u|\mu_{\gamma}$ and the second one is equal to the following constant, only depending on $\gamma,$ $$C_{\gamma}:=\frac{\int_{0}^{\infty}e^{-r^{2}}|\log r^{2}|r^{-2\gamma}rdr}{\int_{0}^{\infty}e^{-r^{2}}r^{-2\gamma}rdr},$$ which satisfies $C_{\gamma}\leq C(1-\gamma)^{-1}$ (using that $C_{\gamma}$ is comparable to the first derivative of $\Gamma(1-\gamma)$ at $\gamma=1).$ ◻ ## The proof for general $u$ and openness Finally, given $\gamma\in]0,1[$ we will show that the inequality in the theorem holds in the general case where $u\in\text{PSH }(X,\omega)$ and $\int e^{-\gamma u}dV_{X}<\infty$ (assuming that $\sup_{X}u\leq0,$ as before). First observe that both sides in the inequality to be shown are still finite in the general case. Indeed, by the resolution of Demailly-Kollar's openness conjecture [@bern3; @g-zh] there exists $\epsilon>0$ such that $$\int e^{-(\gamma+\epsilon)u}dV_{X}<\infty.\label{eq:integrability}$$ Incidently, as next observed, an alternative simple proof of this result can be given in the present case, where $\gamma<1,$ by exploting log-concavity (which also applies when $X$ is a singular Fano variety). First observe that the formula in Thm [ 4](#Thm:Z as log concave){reference-type="ref" reference="Thm:Z as log concave"} still applies to $u.$ Indeed, when $u\in\text{PSH }(X,\omega)\cap L^{\infty}(X)$ the same proof applies and the general case then follows by writing $u$ is a decreasing limit of $u_{j}\in\text{PSH }(X,\omega)\cap L^{\infty}(X)$ and letting $j\rightarrow\infty.$ The integrability property [\[eq:integrability\]](#eq:integrability){reference-type="ref" reference="eq:integrability"} then follows from basic properties of convex functions. Indeed, expressing $e^{-t\gamma}\nu_{0}=e^{-\psi(t)}dt$ for a convex function $\psi$ on $\R$ we have, by assumption, that $\int e^{-\psi(t)}dt<\infty.$ But then $\int e^{\epsilon t-\psi(t)}dt<\infty$ for any sufficently small $\epsilon.$ Indeed, when $\psi$ is convex $\int e^{-\psi(t)}dt<\infty$ iff the asymptotic slope of $\psi(t)$ as $t\rightarrow\infty$ ($-\infty)$ is strictly positive (negative). Now, given the integrability [\[eq:integrability\]](#eq:integrability){reference-type="ref" reference="eq:integrability"} the proof of the theorem proceeds exactly as in the previous case. # Reverse Hölder inequalities on the space of Kähler metrics We next turn to the proof of the following slightly more general formulation of Theorem [ 1](#thm:main intro){reference-type="ref" reference="thm:main intro"}, stated in the introduction: ** 10**. *Given a Kähler form $\omega$ in $c_{1}(X)$ and $p\in[1,\infty[$ the following inequality holds for any $u\in\text{PSH }(X,\omega)$ such that $\omega_{u}^{n}$ is absolutely continuous wrt Lesbesgue measure and any $\gamma\in]0,1[$ such that $\int e^{-\gamma u}dV_{X}<\infty:$* *$$d_{p}(u,0)\leq Ad_{1}(u,0)+B,$$ with $$A=A_{p}e^{2\left\Vert \rho_{u,\gamma}\right\Vert _{L^{\infty}}},\,\,\,B=B_{p}\left(\gamma^{-1}+(1-\gamma)^{-1}\right)e^{\left\Vert \rho_{u,\gamma}\right\Vert _{L^{\infty}}},$$ where $A_{p}$ only depends on $(p,n)$ and $B_{p}$ also depends on $(X,$$\omega).$ Moreover, if $\sup_{X}u=0$ then $B_{p}$ only depends on $(p,n).$* First note that combining the upper bound [\[eq:L p norm leq d p\]](#eq:L p norm leq d p){reference-type="ref" reference="eq:L p norm leq d p"} with the standard reverse Hölder inequality in $\R^{2}$ (appearing in formula [\[eq:reverse H with N\]](#eq:reverse H with N){reference-type="ref" reference="eq:reverse H with N"}), yields $$\left(\int_{X}(-u)^{p}(\frac{\omega_{u}^{n}}{V}+\frac{\omega^{n}}{V})\right)^{1/p}\leq\left(\int_{X}(-u)^{p}\omega_{u}^{n}+(c_{n,p}d_{1}(u,0))^{p}\right)^{1/p}\leq$$ $$\leq C_{2}\left(\left(\int_{X}(-u)^{p}\omega_{u}^{n}\right)^{1/p}+c_{n,p}d_{1}(u,0)\right).$$ Now assume that $\sup_{X}u=0,$ i.e $u\in\text{PSH}(X,\omega)_{0}.$ Then Theorem [ 6](#thm:moment bounds){reference-type="ref" reference="thm:moment bounds"} implies, by the very definition of the twisted Ricci potential (definition [\[eq:def of twisted Ricci pot\]](#eq:def of twisted Ricci pot){reference-type="ref" reference="eq:def of twisted Ricci pot"}), that $$\left(\int_{X}(-u)^{p}\frac{\omega_{u}^{n}}{V}\right)^{1/p}\leq A_{p}e^{\sup_{X}\rho_{u,\gamma}-p^{-1}\inf_{X}\rho_{u,\gamma}}\int_{X}(-u)\frac{\omega_{u}^{n}}{V}+Be^{-p^{-1}\inf_{X}\rho_{u,\gamma}},\label{eq:bound on p energy}$$ where $B_{p}\left(\gamma^{-1}+(1-\gamma)^{-1}\right).$ Invoking the inequalities [\[eq:Darvas\]](#eq:Darvas){reference-type="ref" reference="eq:Darvas"} thus concludes the proof when $\sup_{X}u=0$. Finally, in the general case we may decompose $u=\tilde{u}+\sup_{X}u$ where $\sup_{X}\tilde{u}=0.$ Combining the previous case with the triangle inequality for $d_{p}$ yields $$d_{p}(u,0)\leq d_{p}(\tilde{u},0)+\left|\sup_{X}u\right|\leq Ad_{1}(\tilde{u},0)+B+\left|\sup_{X}u\right|\leq Ad_{1}(u,0)+B+2\left|\sup_{X}u\right|.$$ The proof is thus concluded by invoking the inequality [\[eq:bounds of sup in terms of d one\]](#eq:bounds of sup in terms of d one){reference-type="ref" reference="eq:bounds of sup in terms of d one"} for $p=1.$ ## [\[subsec:Application-to-destabilizing\]]{#subsec:Application-to-destabilizing label="subsec:Application-to-destabilizing"}Application to destablizing geodesic rays We next deduce the following slighltly more general formulation of Cor [ 2](#cor:intro){reference-type="ref" reference="cor:intro"}. ** 11**. *Let $u_{j}$ be a sequence in $\mathcal{E}^{1}(X,\omega)$ such that $\omega_{u_{j}}^{n}$ is absolutely continuous wrt Lesbesgue measure and $\int e^{-\gamma_{j}u}dV_{X}<\infty$ for some sequence $\gamma_{j}$ contained in a compact subset of $]0,1[.$ Assume that* *$$(i)\,d_{1}(u_{j},0)\rightarrow\infty,\,\,\,(ii)\,\mathcal{M}(u_{j})\leq C,\,\,\,\ensuremath{(iii)\,\left|\rho_{u_{j,\gamma_{j}}}\right|\leq R}$$ Then* - *$u_{j}$ is weakly asymptotic to a ray $v_{t}$ which is a $d_{p}-$geodesic ray in $\overline{(\mathcal{H}(X,\omega),d_{p})}$ for any $p\in[1,\infty[$ and $t\mapsto\mathcal{M}(v_{t})$ is decreasing.* - *the $d_{p}-$ speed $\left\Vert \dot{v}\right\Vert _{p}$ of the geodesic $v_{t}$ satisfies $$\left\Vert \dot{v}\right\Vert _{p}\leq A\left\Vert \dot{v}\right\Vert _{1},\label{eq:rever Holder for geod-2}$$ for a constant $A$ of the form $A_{p}e^{2R}$ where $A_{p}$ only depends on $(n,p).$* Given Theorem [ 1](#thm:main intro){reference-type="ref" reference="thm:main intro"} the proof is similar to the proof of [@d-h Thm 3.2], using also some convergence results in [@b-b-j; @bdl] (see also [@x] for general results in Hadamard spaces covering the case $p=2).$ Let $v_{j}(t)$ be the psh geodesic $$\rightarrow\mathcal{E}^{1}(X,\omega),\,\,\,t\mapsto v_{j}(t)\label{eq:map defined by geodec}$$ coinciding with $0$ and $u_{j}$ at $t=0$ and at $t=d_{1}(u_{j},0),$ respectively (the parametrization has been made so that $v_{j}(t)$ has unit $d_{1}-$speed). We will first show (assuming $(i)$ and $(ii)$) that there exists a geodesic ray $v(t)$ in $\mathcal{E}^{1}(X,\omega)$ such that $v_{j}(t)\rightarrow v(t)$ in $\mathcal{E}^{1}(X,\omega)$ uniformly on $[0,T],$ for any given finite $T$ - after perhaps passing to a subsequence. Moreover, if the Ricci potential of $u_{j}$ is uniformly bounded, then the construction will show that $v_{t}$ is a $d_{p}-$geodesic ray for any $p\in[1,\infty[.$ To this end fix $T>0$ and consider $v_{j}(t)$ on $[0,T].$ By construction $$t\in[0,T]\implies d_{1}(v_{j}(t),0)=t\leq T\,\,\,\,(\implies\left|\sup v_{j}(t)\right|\leq C_{T})\label{eq:smallet than T}$$ Moreover, by assumption $(ii)$ $$\mathcal{M}(v_{j}(t))\leq C_{0},\,\,\,\,C_{0}:=\max\{C,\text{\emph{\ensuremath{\mathcal{M}}(0)\}}}\label{eq:bound on Mab}$$ Indeed, by assumption the estimate holds at $t=T_{j}$ and hence it holds for $t\leq T_{j}$ by the convexity of $\mathcal{M}$ on $\mathcal{E}^{1}(X,\omega)$ [@bdl]. The previous two estimates imply, by the compactness theorem in [@bbegz], that the maps [\[eq:map defined by geodec\]](#eq:map defined by geodec){reference-type="ref" reference="eq:map defined by geodec"}, defined by the geodesic $v_{j}(t),$ when restricted to $[0,T],$ take values in a fixed compact subset $K_{T}$ of $\mathcal{E}^{1}(X,\omega).$ Since the maps are $1-$Lipschitz (by the very definition of $d_{1}-$geodesics) it thus follows from the Arzela-Ascoli theorem in metric spaces that $v_{j}(t)$ converges uniformly on $[0,T]$ to a curve $v(t)$ in $\mathcal{E}^{1}(X,\omega),$ after perhaps passing to a subsequence. As a consequence, by [@b-b-j Prop 1.11], $v(t)$ is a *psh geodesic* and, in particular, a $d_{1}-$geodesic in $\mathcal{E}^{1}(X,\omega).$ Using a diagonal argument this yields a $d_{1}-$geodesic ray $v(t)$ with the required properties. Next, assume a uniform bound $R$ on the twisted Ricci $\rho_{u_{j},\gamma_{j}}$ potentials of $u_{j}$ and that $\gamma_{j}$ is contained in a compact subset $K$ of $]0,1[.$ Then, by Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"}, the bound on $d_{1}(v_{j}(t),0)$ in formula [\[eq:smallet than T\]](#eq:smallet than T){reference-type="ref" reference="eq:smallet than T"} yields, since any psh geodesic is a $d_{p}-$geodesic, $$d_{p}(v_{j}(t),0)=t\frac{d_{p}(u_{j},0)}{d_{1}(u_{j},0)}\leq TA+\frac{B}{d_{1}(u_{j},0)})\label{eq:d p bound in pf geodesic}$$ (where $A$ and $B$ depend on $R$ and $p$ and $B$ also depends on the compact subset $K).$ Hence, by [@bdl Prop 2.7], the $d_{1}-$limit point $v(t)$ is in $\mathcal{E}^{p}(X,\omega)$ for any $p\geq1.$ Moreover, $$d_{p}(v(t),0)\leq C_{p,n}\liminf_{j\rightarrow\infty}\left(\int_{X}\left|v_{j}(t)\right|^{p}\omega_{v_{j}}^{n}\right)^{1/p}$$ Indeed, as explained in the proof of [@bdl Prop 2.7], there exists a decreasing sequence $w_{j}(t)$ satisfying $v_{j}(t)\leq w_{j}(t)$ such that $w_{j}(t)$ $d_{p}-$converges towards $v(t).$ Observe that we may as well (by subtracting a constant) assume that $\sup_{X}w_{j}(t)=0.$ Thus, by [@g-z2 Lemma 3.5], $$\int_{X}\left|v_{j}(t)\right|^{p}\omega_{v_{j}}^{n}=\int_{X}(-v_{j}(t))^{p}\omega_{v_{j}}^{n}\geq(p+1)^{-n}\int_{X}(-w_{j}(t))^{p}\omega_{w_{j}}^{n}\label{eq:lsc ine}$$ The integral in the rhs above may (thanks to the inequalities [\[eq:Darvas\]](#eq:Darvas){reference-type="ref" reference="eq:Darvas"}) be replaced by $d_{p}(w_{j},0)^{p},$ which converges towards $d_{p}(v,0)^{p}$ (since $w_{j}(t)$ $d_{p}-$converges towards $v(t)).$ This proves the inequality [\[eq:lsc ine\]](#eq:lsc ine){reference-type="ref" reference="eq:lsc ine"}. The proof of Theorem [ 1](#thm:main intro){reference-type="ref" reference="thm:main intro"} thus reveals that $$d_{p}(v(t),0)\leq Ad(v(t),0)+B$$ for some constants $A$ and $B.$ Dividing both sides by $t$ and letting $t\rightarrow\infty$ then concludes the proof of the inequality [\[eq:rever Holder for geod\]](#eq:rever Holder for geod){reference-type="ref" reference="eq:rever Holder for geod"}. Finally, note that $v_{t}$ defines a $d_{p}-$geodesic for any $p\geq1.$ Indeed, as explained above $v(t)$ is a *psh geodesic* and any psh geodesic in $\mathcal{E}^{p}(X,\omega)$ is, in fact, a $d_{p}-$geodesic (as recalled in Section [\[subsec:Metric-completions-and\]](#subsec:Metric-completions-and){reference-type="ref" reference="subsec:Metric-completions-and"}). Moreover, the bound [\[eq:bound on Mab\]](#eq:bound on Mab){reference-type="ref" reference="eq:bound on Mab"} on $\mathcal{M}(v_{j}(t))$ implies, since $\mathcal{M}$ is $d_{1}-$lower semi continuous [@bdl], that $\mathcal{M}(v(t))\leq C_{0}.$ It thus follows from the convexity of $t\mapsto\mathcal{M}(v(t))$ that $\mathcal{M}(v(t))$ is decreasing in $t.$ # Generalization to singular Fano varieties In this section we explain how to extend the previous results to singular Fano varieties and prove Corollary [ 3](#cor:Aubin intro){reference-type="ref" reference="cor:Aubin intro"}. The results also extend - with essentially the same proofs - to the more general twisted setting (considered in [@b-b-j] when $X$ is smooth). We thus let $X$ be a singular Fano variety. This means that $X$ is a normal complex variety such that $K_{X}^{*}$ is defined as an ample $\Q-$line bundle over $X$ (see [@bbegz; @d-g; @l-t-w; @li1]). Fix a measure $dV_{X}$ on $X$ corresponding to a locally bounded metric on $K_{X}^{*}$ with positive curvature current, denoted by $\omega.$ We will assume that $dV_{X}$ has finite total mass. As is well-known [@bbegz] this is equivalent to $X$ having *log terminal singularities* which will henceforth be assumed. We recall that a Kähler form on a normal complex variety $X$ is, by definition, locally the restriction to $X$ of a Kähler form on $\C^{M}$ under a local embedding of $X$ as a variety in $\C^{M}.$ The space $\mathcal{H}(X,\omega)$ of Kähler potentials (relative to $\omega)$ is defined exactly as in the case when $X$ is singular. Theorem [ 6](#thm:moment bounds){reference-type="ref" reference="thm:moment bounds"} now extends directly to singular setup, using that Prop [ 5](#prop:concavity){reference-type="ref" reference="prop:concavity"} applies to singular Fano varieties. By [@d-g], all the results in Section [\[subsec:The-space-of\]](#subsec:The-space-of){reference-type="ref" reference="subsec:The-space-of"} on Darvas' $L^{p}-$distances extend to singular normal varieties. Hence, Theorem [ 6](#thm:moment bounds){reference-type="ref" reference="thm:moment bounds"} implies, precisely as before, that Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"} holds for any singular Fano variety with log terminal singularities. In turn, just as before, the latter theorem implies that Corollary [ 11](#cor:geodesic rays text){reference-type="ref" reference="cor:geodesic rays text"} applies to singular Fano varieties (using the results on singular Fano varieties in [@d-g; @bbegz]). ### Proof of Corollary [ 3](#cor:Aubin intro){reference-type="ref" reference="cor:Aubin intro"} {#proof-of-corollary-coraubin-intro} Consider now Aubin's equations [\[eq:Aubin intr\]](#eq:Aubin intr){reference-type="ref" reference="eq:Aubin intr"} on $X,$ defined in the weak sense of pluripotential theory [@bbegz], i.e. the potential $u_{t}$ of $\omega_{t},$ which is in $\text{PSH \ensuremath{(X,\omega)\cap L^{\infty}(X)} },$ satisfies $$\omega_{u_{t}}^{n}/V=\mu_{u_{t},t}\label{eq:MA on sing Fano}$$ By results in [@bbegz] this means, equivalently, that $u_{t}$ minimizes a twisted Mabuchi functional that we shall denote by $\mathcal{M}_{t}.$ Moreover, it follows from results in [@l-t-w; @li1] (or more precisely the proof of the main results) that for any $\{1,\delta(X)\}$ is the sup over all $t\in[0,1[$ for which the equation [\[eq:MA on sing Fano\]](#eq:MA on sing Fano){reference-type="ref" reference="eq:MA on sing Fano"} admits a solution $u_{t}$ in $\mathcal{E}^{1}(X,\omega)$ (the assumption that $X$ has log terminal singularities ensures that $\delta(X)>0$). The equation [\[eq:MA on sing Fano\]](#eq:MA on sing Fano){reference-type="ref" reference="eq:MA on sing Fano"} says, in particular, that $\omega_{u_{t}}^{n}$ is, locally on the regular locus of $X,$ absolutely continuous wrt Lebesgue measure, $\int_{X}e^{-tu_{t}}dV_{X}<\infty$ and $\rho_{u,t}\equiv0.$ Hence, Cor [ 3](#cor:Aubin intro){reference-type="ref" reference="cor:Aubin intro"} follows from Cor [ 11](#cor:geodesic rays text){reference-type="ref" reference="cor:geodesic rays text"} on Fano varieties, once we have verified that $$d_{1}(u_{t},0)\rightarrow\infty\label{eq:divergence}$$ as $t\rightarrow\delta(X)$ (since $\sup_{X}u_{t}=0$ the corresponding geodesic ray $v_{t}$ is then non-trivial in the sense that $v_{t}$ is not of the form $ct$ for any constant $c).$ Assume, in order to get a contractiction, that $d_{1}(u_{t},0)\leq C,$ , after passing to a subsequence. Then it follows from results in [@bbegz; @d-g] that, after passing to a subsequence, $u_{t}$ $d_{1}-$converges to $u_{\delta}$ in $\mathcal{E}^{1}(X,\omega),$ minimizing $\mathcal{M}_{\delta(X)}$ and thus $u_{\delta}$ satisfies the equation [\[eq:MA on sing Fano\]](#eq:MA on sing Fano){reference-type="ref" reference="eq:MA on sing Fano"} for $t=\delta(X).$ By, assumption, $\delta(X)<1$ and, as a consequence, any solution of the equation [\[eq:MA on sing Fano\]](#eq:MA on sing Fano){reference-type="ref" reference="eq:MA on sing Fano"} for $t=\delta(X)$ is uniquely determined. Indeed, since $\omega$ is a positive current with locally bounded potentials [@bbegz Thm 11.1] implies, just as in the case of $t=1$ considered in[@bbegz Thm 5.1], that $u_{\delta}$ is uniquely determined modulo the flow of a holomorphic vector field $W$ on $X,$ preserving $\omega.$ But this can only happen if $W$ vanishes identically, as follows from [@bern1b Prop 8.2] applied to any non-singular resolution of $X.$ Thus $u_{\delta}$ is uniquely determined. As a consequence - just as in the case that $t=1$ and there are no holomorphic vector fields, considered in [@d-g] - $\mathcal{M}_{t}$ is coercive on $\mathcal{E}^{1}(X,\omega)$ when $t=\delta(X).$ Since coercivity is preserved when $t$ is replaced by $t+\epsilon$ for any sufficently small number $\epsilon$ this means that $\mathcal{M}_{\delta(X)+\epsilon}$ is coercive for any sufficently small positive number $\epsilon.$ It thus follows from [@bbegz] that $\mathcal{M}_{\delta(X)+\epsilon}$ has a minimizer, which satisfies the equation [\[eq:MA on sing Fano\]](#eq:MA on sing Fano){reference-type="ref" reference="eq:MA on sing Fano"} for $t=\delta(X)+\epsilon.$ But this contradicts the fact that $\min\{1,\delta(X)\}$ is the sup over all $t\in[0,1[$ for which the equation [\[eq:MA on sing Fano\]](#eq:MA on sing Fano){reference-type="ref" reference="eq:MA on sing Fano"} is solvable. This proves the divergence [\[eq:divergence\]](#eq:divergence){reference-type="ref" reference="eq:divergence"}. # [\[subsec:Comparison-with-Harnack\]]{#subsec:Comparison-with-Harnack label="subsec:Comparison-with-Harnack"}Comparison with Harnack type bounds The following analog of Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"} follows from the Harnack type bound in [@t1]: ** 12**. *Let $X$ be a Fano manifold of dimension $n$ and fix $\omega\in c^{1}(X).$ Given $\delta>0$ consider $u\in\mathcal{H}(X,\omega)$ such that $$\text{Ric \ensuremath{\omega}}_{u}\geq\delta\omega_{u}.$$ Given $p\in[1,\infty[,$ exists a constants $A$ and $B$ such that $$d_{p}(u,0)\leq Ad_{1}(u,0)+B,$$ where $A$ only depends on $(n,\delta,p)$ and $B$ also depends on $(X,\omega).$* *Proof.* In order to prove the proposition it will - just as in the beginning of the proof of Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"} - be enough to consider the case when $\sup_{X}u=0.$ Then, as shown in [@t1] (in the course of the proof of [@t1 Thm 2.1]) Moser iteration yields, $$\left\Vert u\right\Vert _{L^{\infty}(X)}\leq C\int_{X}(-u)\frac{\omega_{u}^{n}}{V}+C\label{eq:tian}$$ for a constant $C$ only depending on $(n,\delta)$ (which is an $L^{\infty}-$analog of the $L^{p}-$estimate [\[eq:bound on p energy\]](#eq:bound on p energy){reference-type="ref" reference="eq:bound on p energy"}, but with a constant depending on $n$). As a consequence, $$\left\Vert u\right\Vert _{L^{\infty}(X)}\leq C(-\mathcal{E}(u)+1)$$ (after perhaps increasing the constant $C),$ where $$\mathcal{E}(u):=\frac{1}{V(n+1)}\int_{X}\sum_{j=0}^{n}(-u)\omega_{u}^{j}\wedge\omega^{n-j}\label{eq:primitive appears}$$ (which is the unique primitive of the one-form on $\mathcal{H}(X,\omega)$ defined by $u\mapsto\omega_{u}^{n}/V$ satisfying $\mathcal{E}(u)=0).$ Now, in order to prove the proposition it will - just as in the beginning of the proof of Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"} - be enough to consider the case when $\sup_{X}u=0.$ Since $-\mathcal{E}(u)=d_{1}(u,0),$ when $\sup_{X}u\leq0,$ by [@da Cor 4.14] we deduce that $$\left\Vert u\right\Vert _{L^{\infty}(X)}\leq Cd_{1}(u,0)+C$$ Invoking the upper bound [\[eq:Darvas\]](#eq:Darvas){reference-type="ref" reference="eq:Darvas"} on $d_{p}(u,0)$ thus concludes the proof. ◻ * 13*. The proof of the inequality [\[eq:tian\]](#eq:tian){reference-type="ref" reference="eq:tian"} only uses that a strict lower bound on the Ricci curvature of $\omega_{u}$ implies a uniform upper bound on the Sobolev constant and the Poincaré constant of $(X,\omega_{u}).$ As shown in [@ru] the latter upper bounds hold along the Kähler-Ricci flow$.$ As a consequence, so does the inequality [\[eq:tian\]](#eq:tian){reference-type="ref" reference="eq:tian"} (see Step 3 in the proof of the main result in [@ru]). This means, by the proof of the previous proposition, that the reversed Hölder inequality holds along the Kähler-Ricci flow. This was first shown in [@d-h] for a particular normalization of the potentials, using the uniform Harnack bound, discussed below. Now consider $\omega_{u_{t}}$ satisfying Aubin's continuity equation. Then the Ricci curvature bound in the previous proposition automatically holds with for $t\geq\delta>0.$ The reversed Hölder bound in the previous proposition thus follows for $u_{t}$ when $t\geq\delta>0.$ While the potential $u_{t}$ is only determined up to an additive constant (depending on $t),$ the constant is often fixed by demanding that $$\int_{X}e^{-tu_{t}}dV_{X}=1,\label{eq:exp integral is one}$$ (where, as before, $\text{Ric \ensuremath{dV_{X}:=\omega}}$). Equivalently, this means that $u_{t}$ is the unique solution to Aubin's Monge-Ampère equation [@au]: $$\frac{\omega_{u_{t}}^{n}}{V}=e^{-tu_{t}}dV_{X}.\label{eq:Aubins Monge}$$ Under this normalization, the bound [\[eq:tian\]](#eq:tian){reference-type="ref" reference="eq:tian"} implies that $u_{t}$ satisfies the following *uniform Harnack bound*: $$-\inf_{X}u\leq A\sup_{X}u+B\label{eq:uniform Har}$$ for $t\geq\delta>0$ (after, possibly, increasing the constants $A$ and $B)$, as follows from the bound [\[eq:primitive leq C\]](#eq:primitive leq C){reference-type="ref" reference="eq:primitive leq C"} below. In general, as shown [@d-h] (in the context of the Kähler-Ricci flow) a uniform Harnack bound implies that $$d_{p}(u,0)\leq A_{p,\delta}\left|\sup_{X}u\right|+B_{p,\delta},\label{eq:d p distance smaller than sup}$$ using the inequalities [\[eq:Darvas\]](#eq:Darvas){reference-type="ref" reference="eq:Darvas"}; see [@d-h Thm 3.1] and its proof. However, for singular $X$ the uniform Harnack bound appears to be widely open along Aubin's complex Monge-Ampère equation. Still, Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"} implies that the inequality [\[eq:d p distance smaller than sup\]](#eq:d p distance smaller than sup){reference-type="ref" reference="eq:d p distance smaller than sup"} does hold, assuming that $\delta(X)<1:$ ** 14**. *Let $X$ be Fano variety with log terminal singularities and let $u_{t}\in L^{\infty}(X)\cap\text{PSH \ensuremath{(X,\omega)}}$ be the solution to Aubin's Monge-Ampère equation [\[eq:Aubins Monge\]](#eq:Aubins Monge){reference-type="ref" reference="eq:Aubins Monge"}, for $t\in]0,\delta(X)[.$ Then, for any given $p\in[1,\infty[$ $$B^{-1}\left|\sup_{X}u_{t}\right|-B\leq d_{p}(u_{t},0)\leq A\left|\sup_{X}u_{t}\right|+B\left(t^{-1}+(1-t)^{-1}\right)$$ for a constant $A$ only depending on $(n,p)$ and a constant $B$ also depending on $(X,\omega).$* *Proof.* First observe that applying Theorem [ 10](#thm:main in text){reference-type="ref" reference="thm:main in text"} to $u_{t}-\sup_{X}u_{t}$ and using the triangle inequality for $d_{p}$ yields $$d_{p}(u_{t},0)\leq A_{p}d_{1}(u_{t},0)+B_{p}+2\left|\sup_{X}u_{t}\right|$$ for constants $A_{p}$ and $B_{p}$ only depending on $p.$ Hence, it will be enough to show that $$d_{1}(u_{t},0)\leq\left|\sup_{X}u_{t}\right|+C\label{eq:d one along Aubin}$$ for a constant $C$ only depending on $(X,\omega).$ To this end first note that for any $u\in L^{\infty}(X)\cap\text{PSH \ensuremath{(X,\omega)}}$ $$d_{1}(u,0)\leq\left|\sup_{X}u\right|-\mathcal{E}(u),$$ where $-\mathcal{E}(u)$ is the functional appearing in formula [\[eq:primitive appears\]](#eq:primitive appears){reference-type="ref" reference="eq:primitive appears"}. Indeed, in the general setup of normal varieties $-\mathcal{E}(u)=d_{1}(u,0),$ when $\sup_{X}u=0,$ by [@d-g Prop 3.12]. The general case then follows form the triangle inequality for $d_{1}.$ Hence, it will be enough to show that $$-\mathcal{E}(u_{t})\leq C.\label{eq:primitive leq C}$$ To this end consider the twisted Ding functional defined by $$\mathcal{D}_{t}(u):=-\mathcal{E}(u)-t^{-1}\log\int_{X}e^{-tu}dV_{X}$$ for a given $t\in[0,1[$ We define $\mathcal{D}_{0}(u)$ as the limit of $\mathcal{D}_{t}(u)$ as $t$ decreases to $0,$ which amounts to replacing the second term in the definition of $\mathcal{D}_{t}(u)$ by $\int_{X}udV_{X}.$ Note that $\mathcal{D}_{t}(u)$ is decreasing in $t,$ as follows directly from Hölder's inequality. Moreover, $u_{t}$ minimizes $\mathcal{D}_{t}(u)$ (see [@bbegz]). Hence, $$\mathcal{D}_{1}(u_{t})\leq\mathcal{D}_{t}(u_{t})\leq\mathcal{D}_{t}(u_{0})\leq C:=\mathcal{D}_{0}(u_{0})$$ By formula [\[eq:exp integral is one\]](#eq:exp integral is one){reference-type="ref" reference="eq:exp integral is one"}, this proves the bound [\[eq:primitive leq C\]](#eq:primitive leq C){reference-type="ref" reference="eq:primitive leq C"}. ◻ Let us conclude this section by pointing out some intruiging connections between the proof of the uniform Harnack bound [\[eq:uniform Har\]](#eq:uniform Har){reference-type="ref" reference="eq:uniform Har"} and the upper bound in Prop [ 14](#prop:Aubin's MA){reference-type="ref" reference="prop:Aubin's MA"}. As discussed in Remark [ 13](#rem:Sob){reference-type="ref" reference="rem:Sob"} the Harnack bound follows from bounds on the Sobolev and Poincaré constants, which in turn follow from a strictly positive uniform lower bound on the Ricci curvature of $(X,\omega_{u}).$ The latter bounds have been extended to complete metric spaces $(X,d)$ using various generalized notions of Ricci curvature, defined in terms of the convexity of the entropy functional on the space of all probability measures on $X,$ endowed with the $L^{2}-$Wasserstein metric [@v Section 30][@pr Prop 3.3]. When $X$ is a singular Fano variety and $\omega$ is taken to be a Kähler form the positive current $\omega_{u_{t}}$ defines a Kähler metric on the regular locus $X_{\text{reg}}$ of $X.$ However, $(X_{\text{reg }},\omega_{t})$ is not complete and its seems to be unknown whether the metric completion of $(X_{\text{reg }},\omega_{t})$ satisfies any kind of Ricci curvature bound in the sense of metric spaces. From this point of view the key advantage of the method of proof of Prop [ 14](#prop:Aubin's MA){reference-type="ref" reference="prop:Aubin's MA"} is that it is based on the convexity of the functional appearing in formula [\[eq:complex prekopa\]](#eq:complex prekopa){reference-type="ref" reference="eq:complex prekopa"}, which holds when $\text{Ric}dV\geq0$ in the general setup of singular Fano varieties. Incidently, this functional is precisely the Legendre-Fenchel tranform of the entropy functional appearing in the definition of Ricci curvature of metric spaces. # Comparison with Duistermaat-Heckman type measures and log-concavity ## [\[subsec:Test-configurations-and\]]{#subsec:Test-configurations-and label="subsec:Test-configurations-and"}Test configurations and K-stability Let $L$ be an ample line over a compact complex manifold $X.$ Recall that a *test configuration* $(\mathcal{X},\mathcal{L})$ for $(X,L)$ (as appearing the definition of K-stability, discussed below) may be defined as a $\C^{*}-$equivariant embedding $$(X\times\C^{*},L)\hookrightarrow(\mathcal{X},\mathcal{L})$$ of the polarized trivial fibration $(X\times\C^{*},L)$ over $\C^{*}$ into a normal variety $\mathcal{X},$ fibered over $\C,$ endowed with a relatively ample $\Q-$line bundle *$\mathcal{L}$* and a $\C^{*}-$action on $(\mathcal{X},\mathcal{L})$ covering the standard $\C^{*}-$action on $\C.$ In particular, there is a $\C^{*}-$action on the scheme defined by the central fiber $(\mathcal{X}_{0},\mathcal{L}_{0}).$ Given $\omega\in c_{1}(L)$ a test configuration induces, by [@p-s], a psh geodesic ray $u_{t}$ (emanating from $0)$ which is in $\mathcal{E}^{p}(X,\omega)$ and thus defines a $d_{p}-$geodesic, for any $p.$ Indeed, $u_{t}$ may be defined as an envelope (as in formula [\[eq:psh geodes\]](#eq:psh geodes){reference-type="ref" reference="eq:psh geodes"}) over all $v_{t}$ such that the corresponding $\omega-$psh function $V(x,\tau)$ on $X\times\C_{\tau}^{*},$ where $\tau:=e^{-t},$ has the property that $dd^{c}V+\omega$ is the restriction to $X\times\C^{*}$ of the curvature form of a locally bounded and positively curved metric on $\mathcal{L}\rightarrow\mathcal{X}.$ Set $$\dot{u}:=\frac{du_{t}}{dt}|_{t=0},\,\,\,\,\mu:=\dot{u}_{*}(\frac{\omega^{n}}{V}),$$ By [@c-t-w], $\dot{u}\in L^{\infty}(X)$, which ensures that the push-forward $\dot{u}_{*}(\frac{\omega^{n}}{V})$ is well-defined. As in [@bern2], the $d_{p}-$speed of the geodesic $u_{t}$ may be expressed as $$\left\Vert \dot{u}\right\Vert _{p}:=d_{p}(u_{1},0)=\left(\int_{X}\left|\dot{u}\right|^{p}\frac{\omega^{n}}{V}\right)^{1/p}=\left(\int_{\R}|t|^{p}d\mu\right)^{1/p}.\label{eq:speed for test}$$ By the main result of [@hi] - proving a conjecture in [@wi] - the probability measure $\mu$ on $\R$ can be expressed as the following weak limit of weight measures: $$\mu=\lim_{k\rightarrow\infty}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\delta_{\lambda_{i}^{(k)}/k}$$ where the real numbers $\lambda_{1}^{(k)},...,\lambda_{N_{k}}^{(k)}$ are the weights of the $\C^{*}-$action on the complex vector space $H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{\otimes k}).$ This limit is called the *Duistermaat-Heckman measure* of $(\mathcal{X},\mathcal{L})$ in [@bhj]. In the terminology of [@do2; @bhj] $\left\Vert \dot{u}\right\Vert _{p}$ thus coincides with the $L^{p}-$norm $\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{L^{p}}$ of the test configuration $(\mathcal{X},\mathcal{L}).$ When the central fiber of $\mathcal{X}_{0}$ is reduced and irreducible it follows directly from the main result of [@ok], concerning weight measures, that the probability measure $\mu$ on $\R$ is log-concave. Thus, by the reverse Hölder inequality [\[eq:reversed H=0000F6lder log concave\]](#eq:reversed H=0000F6lder log concave){reference-type="ref" reference="eq:reversed H=0000F6lder log concave"} for log-concave measures, there exists, for any given $p\in[1,\infty[,$ a constant $C_{p}$ (only depending on $p)$ such that $$\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{L^{p}}\leq C_{p}\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{L^{1}}\label{eq:reversed test}$$ This inequality does not seem to have been observed before, but as pointed out the author by Sébastien Boucksom and Mattias Jonsson, it is closely related to the inequality in [@b-j Lemma 3.14]. Indeed, the latter inequality yields a stronger $L^{\infty}-$bound, but with a constant $C$ depending on $n.$ Moreover, similar inequalities also appear in [@zh], as pointed out the the author by Tamas Darvas. It should be stressed that the bound [\[eq:reversed test\]](#eq:reversed test){reference-type="ref" reference="eq:reversed test"} does not hold for *all* test configurations. Indeed, by the example in [@bhj Prop 8.5], the inequality fails for $p>n/(n-1)$ when $\mathcal{X}$ is taken to be the deformation to the normal cone of a given point $x$ on $X$, i.e. $p:\mathcal{X}\rightarrow X\times\C$ is the blow-up of $\{x\}\times\{0\}$ in $X\times\C$ and $\mathcal{L}_{\epsilon}:=p^{*}L-\epsilon E,$ where $E$ denotes the exceptional divisor over $p$ and $\epsilon$ is a given sufficiently small positive rational number. In particular, in this example (where$\mathcal{X}_{0}$ is reduced, but has two components) $\mu$ can not be log-concave. ### Relations to K-stability Recall that, in the context of the Yau-Tian-Donaldson conjecture [@c-d-s; @d-s; @b-b-j], a Fano manifold $(X,K_{X}^{*})$ is said to be *K-stable* if the Donaldson-Futaki invariant $\text{DF }(\mathcal{X},\mathcal{L})$ is strictly positive for all non-trivial test configurations and *uniformly K-stable* if there exists a constant $\epsilon>0$ such that\| $$\text{DF }(\mathcal{X},\mathcal{L})\geq\epsilon\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{1},$$ where $\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{p}$ is defined, in terms of the Duistermaat-Heckman $\mu,$ discussed above, as $$\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{p}:=\left(\int_{\R}|t-c|^{p}\mu\right)^{1/p},\,\,\,c:=\int_{\R}\mu.$$ By [@l-x-z], $(X,K_{X}^{*})$ is, in fact, K-stable iff it is uniformly K-stable. Moreover, by [@l-x], one may when testing (uniform) K-stability restrict to *special* test konfigurations $(\mathcal{X},\mathcal{L}),$ i.e. such that $\mathcal{X}_{0}$ has log terminal singularities (and, in particular, is reduced and irreducible). But in this case $\mu$ is, as pointed out above, log-concave and, as a consequence, so is the translation of $\mu$ by $c.$ Hence, as explained above, for any $p\in[1,\infty[$ there exists a constant $C_{p}$ such that $$\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{p}\leq C_{p}\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{1}$$ for all special test configurations. All in all, this means that $$(X,K_{X}^{*})\,\text{is K-stable \ensuremath{\iff\forall p\in[1,\infty[\,\exists\epsilon_{p}\in]0,\infty[:\,\text{DF }(\mathcal{X},\mathcal{L})\geq\epsilon_{p}\left\Vert (\mathcal{X},\mathcal{L})\right\Vert _{p}} }$$ for all non-trivial special test configurations ## [\[subsec:Failure-of-log-concavity\]]{#subsec:Failure-of-log-concavity label="subsec:Failure-of-log-concavity"}Failure of log-concavity for general geodesic segments Given $u\in\mathcal{H}(X,\omega),$ let $u_{t}$ be the psh geodesic coinciding with $0$ and $u$ at $t=0$ and $t=1,$ respectively. Then $$d_{p}(u,0)=\left(\int_{X}\left|\dot{u}\right|^{p}\frac{\omega^{n}}{V}\right)^{1/p}=\left(\int_{\R}|t|^{p}d\mu\right)^{1/p}$$ using that $\dot{u}\in L^{\infty}(X)$ [@bern2]. However, in general, $\mu$ is *not* log-concave. Indeed, assume in order to get a contraction that $\mu$ is log-concave. Then the reverse Hölder inequality [\[eq:reversed H=0000F6lder log concave\]](#eq:reversed H=0000F6lder log concave){reference-type="ref" reference="eq:reversed H=0000F6lder log concave"} for log-concave measures implies that $$d_{p}(u,0)\leq Cd_{1}(u,0)$$ for a universal constant $C.$ But such a reverse Hölder inequality does not hold, in general, as stressed in the introduction of the paper. 10 T. Aubin, Réduction du cas positif de l'équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d'une inégalité, J. Funct. Anal., 57 (1984), 143--153. Berman, R.J: A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics*.* Advances in Math. 1254. Volume: 248. 2013 Berman, R.J: Ensemble Equivalence for Mean Field Models and Plurisubharmonicity. Archive for Rational Mechanics and Analysis. 246, pages 771--822 (2022) Berman, R.J., Boucksom, S., Guedj, V., Zeriahi, A.: A variational approach to complex Monge-Ampere equations. Publications Math. de l'IHÉS, 117(1), 179--245 (2013) Berman, R.J., Berndtsson, B.: Symmetrization of Plurisubharmonic and Convex Functions. Indiana Univ. Math. J. 63(2), 345--365, 2014 R.J. Berman, S. Boucksom, M. Jonsson, A variational approach to the Yau--Tian--Donaldson conjecture, J. Am. Math. Soc. (2021). R.J. Berman, T. Darvas, C. Lu, Convexity of the extended K-energy and the large time behavior of the weak Calabi flow, Geom. Topol. 21(5) (2017) 2945--2988. R.J Berman, T Darvas, CH Lu: Regularity of weak minimizers of the K-energy and applications to properness and K-stabiliy. des Annales Scientifiques de l'École Normale Supérieure. 53, fasc. 2 (2020). R.J.Berman*;* Eyssidieux, P: S. Boucksom, V. Guedj, A. Zeriahi: Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties. Journal fur die Reine und Angewandte Mathematik (published on-line 2016). Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann Inst Fourier 56, 1633--1662, 2006 B. Berndtsson: A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kahler geometry. Invent. Math 200 (2015), no. 1, 149--200. B. Berndtsson, Probability measures associated to geodesics in the space of Kähler metrics, in: Algebraic and Analytic Microlocal Analysis, in: Springer Proc. Math. Stat., vol.269, Springer, Cham, 2018, pp.395--419 (https://arxiv.org/abs/0907.1806) B. Berndtsson.The openness conjecture and complex Brunn-Minkowski inequalities. Complex geometry and dynamics, 29--44, Abel Symp.,10, Springer, Cham, 2015 Blum, H.; Jonsson, M.; Thresholds, valuations, and K-stability. Adv. Math. 365 (2020). S.Boucksom, T.Hisamoto, and M.Jonsson: Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743--841. S Boucksom, M Jonsson: A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations. preprint arXiv:2107.11221, 2021 - arxiv.org I. A. Cheltsov, Y. A. Rubinstein and K. Zhang. Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces. Selecta Math. (N.S.) 25 (2019), no. 2, Art. 34, 36 pp. X.X. Chen, The space of Kahler metrics, J. Differential Geom. 56 (2000), no. 2, 189--234. X.X. Chen and J. Cheng. On the constant scalar curvature K̈ahler metrics (II)---Existence results. J. Amer. Math. Soc. 34 (2021), 937--1009 X.X Chen, S. Donaldson, S. Sun. Kahler-Einstein metrics on Fano manifolds, I, II, III. J. Amer. Math. Soc. 28 (2015). X. Chen, S. Sun, and B. Wang, Kahler-Ricci flow, Kahler-Einstein metric, and Kstability, Geom. Topol. 22 (2018), no. 6, 3145--3173. J. Chu, V. Tosatti and B. Weinkove, C1,1 regularity for degenerate complex Monge- Amp'ere equations and geodesic rays. Comm. Partial Differential Equations 43 (2018), 292-312. V. Datar and G. Szekelyhidi. Kahler-Einstein metrics along the smooth continuity method. Geom. Funct. Anal., 26(4):975--1010, 2016. T Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015) 182--219 T Darvas, The Mabuchi completion of the space of Kähler potentials, Amer. J. Math. 139 (2017) 1275--1313 MR Darvas, T., He, W.: Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds. Trans. Amer. Math. Soc. 369 (2017) 5069--5085 MR T. Darvas, C.H. Lu, Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry, Geom. Topol. 24(4) (2020) 1907--1967. T. Darvas, C. H. Lu, and Y. A. Rubinstein. Quantization in geometric pluripotential theory. Comm. Pure Appl. Math., 73(5):1100--1138, 2020. T.Darvas, K.Zhang: Twisted Kähler-Einstein metrics in big classes. arXiv preprint arXiv:2208.08324, 2022 - arxiv.org Di Nezza, E., Guedj, V.: Geometry and topology of the space of Kähler metrics on singular varieties. Compos. Math. 154, 1593--1632 (2018) S.K. Donaldson: Symmetric spaces, Kahler geometry and Hamiltonian dynamics, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence RI, 1999, 13--33. S.K. Donaldson: Lower bounds on the Calabi functional. J. Differ. Geom. 70 (2005), no. 3, p. 453-472. Fujita, Kento.J; Odaka, Y: On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J. (2) 70 (2018), no. 4, 511--521. Q. Guan, X. Zhou.A proof of Demailly's strong openness conjecture. Ann. of Math. (2)182(2015), no.2, 605--616 V. Guedj, The metric completion of the Riemannian space of Kahler metrics, arXiv:1401.7857. V.Guedj; A.Zeriahi: Intrinsic capacities on compact Kähler manifolds. The Journal of Geometric Analysis volume 15, pages 607--639 (2005) V. Guedj, A. Zeriahi, The weighted Monge-Amp'ere energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), no. 2, 442--482 T. Hisamoto, "On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold", J. Reine Angew. Math. 713 (2016), p. 129-148. Ilias, S: Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes. Annales de l'institut Fourier, 33 no. 2 (1983), p. 151-165 J. L. Kazdan; F. W. Warner: Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974) 1447. R. Latała: On the equivalence between geometric and arithmetic means for logconcave measures. Convex Geometric Analysis (Berkeley, CA, 1996), 123--127;Math. Sci. Res. Inst. Publ. 34, CambridgeUniversity Press, Cambridge, 1999. C. Li;C. Xu: Special test configuration and K-stability of Fano varieties. Ann. of Math.180(2014), no. 1, 197--232 Y Liu, C Xu, Z Zhuang: Finite generation for valuations computing stability thresholds and applications to K-stability. Annals of Mathematics (2), 196(2):507--566, 2022 C Li, G Tian, F Wang: The uniform version of Yau--Tian--Donaldson conjecture for singular Fano varieties. Peking Mathematical Journal, 2019 - Springer Li, C: G-uniform stability and Kähler-Einstein metrics on Fano varieties. Invent. Math. 227 (2022), no. 2, 661--744. T. Mabuchi, Some symplectic geometry on compact Kahler manifolds I, Osaka J. Math. 24, 1987, 227--252. Vitali D. Milman, Gideon Schechtman: Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture notes in math. vol 1200. Springer (1986) Nicholas McCleerey, Valentino Tosatti: Pluricomplex Green's functions and Fano manifolds. Épijournal de Géométrie Algébrique, Volume 3 (June 21, 2019) epiga:4706 A. Prekopa: On logarithmic concave measures and functions, Acad. Sci. Math. (Szeged) 34 (1973), p. 335-343 A. Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), 405-411. D. H. Phong and J. Sturm: Test congurations for K-stability and geodesic rays. J. Symplectic Geom. 5 (2007), no. 2, 221{247. A Profeta, The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds, Potential Anal. 43 (2015) 513--529 Y. Rubinstein, On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow, Trans. Amer. Math. Soc. 361 (2009), 5839-5850. S. Semmes, Complex Monge-Amp'ere and symplectic manifolds, Amer. J. Math. 114 (1992), 495--550. G. Székelyhidi, The partial C0-estimate along the continuity method, J. Amer. Math. Soc. 29 (2016), no. 2, 537--560. G. Tian, On Kahler-Einstein metrics on certain Kahler manifolds with C1(M) \> 0, Invent. Math. 89 (1987), no. 2, 225--246. G. Tian: On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101--172. G. Tian, X. Zhu: Convergence of Kahler-Ricci flow. J. Amer. Math. Soc. 20 (2007), no. 3, 675--699. Villani, C.: Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin (2009) C.S. Withers; S.Nadarajah: Moments from cumulants and vice versa. International Journal of Mathematical Education in Science and Technology Volume 40, 2009 - Issue 6 D. Witt Nyström, Test configurations and Okounkov bodies, Compos. Math. 148 (2012), 1736--1756. M. Xia, On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows. Anal. PDE 14 (2021), no. 6, 1951--1976. Qi S. Zhang, A uniform Sobolev inequality under Ricci flow, International Mathe-matics Research Notices (2007), Article ID rnm056. Erratum to: A uniform Sobolev inequality under Ricci flow (2007), Article ID rnm096. K. Zhang. Some refinements of the partial C0 estimate. Analysis & PDE 14.7 (2021), pp. 2307--2326. K. Zhang. Valuative invariants with higher moments. J. Geom. Anal. 32 (2022), no.1, Paper No. 10, 33 pp.

arxiv_math

--- abstract: | We show that a complete connected locally compact median space of finite rank which admits a transitive action is isometric to $\mathbb{R}^n$ endowed with the $\ell^1$-metric. In parallel, we give a characterization of the compactness in complete median spaces of finite rank by the combinatorics of their halfspaces. Under a mild assumptions on the isometry group, we show also that all the orbits of a complete connected median space of finite rank are discrete. bibliography: - bibli.bib nocite: "[@*]" --- **Isometric actions on locally compact finite rank median spaces**\ [Mohamed-Lamine Messaci]{.smallcaps} \ Laboratoire J.-A. Dieudonné\ Université Côte d'Azur\ Parc Valrose, 06108 Nice Cedex 02, France\ [messaci\@unice.fr ](messaci@unice.fr )\ # Introduction A median space is a metric space such that for any three points, the three intervals relating each two ponts intersect in a unique point. The interval between two point $a,b\in (X,d)$, denoted by $[a,b]$, is the set of point $x\in X$ such that $d(a,b)=d(a,x)+d(x,b)$. First examples of median space goes back to [@Birkhoff_median] and are given by metric distributive lattices, see [@Birkhoff_lattice] Ch. V, §9 for a definition of metric lattices. A more geometric examples of median spaces are given by real trees. In this case, the intervals coincide with the geodesics. In fact, it is the only case where it occurs. Zero skeleton of CAT$(0)$ cube complexes when endowed with the combinatoric metric constitute another important class of median spaces of finite dimension. In fact many results of isometric group action on real trees and CAT$(0)$ cube complexes extend to the case of finite \"dimensional\" median spaces. There is a suitable and practical way of speaking of dimension in the case of median spaces, given by the notion of the rank. Loosely speaking, this detects the highest dimension of discrete cubes, endowed with the $\ell^1$ metric, that can be isometrically embedded into the space, see Subsection [2.2](#halfspaces){reference-type="ref" reference="halfspaces"} for a definition. Median spaces of finite rank generalize finite dimensional CAT($0$) cube complexes the same way $\mathbb{R}$-trees generalize simplicial trees. Any geometric action, i.e. properly discontinuous cocompact action, on an $\mathbb{R}$-tree gives rise to a geometric action on a simplicial tree. It is unknown whether any geometric action on a finite rank median space gives rise to a geometric action on a finite dimensional CAT($0$) cube complex, see [@ChaD_hyperbolic] subsection 1.b. It is false in the infinite rank case, it was shown in [@ChaD_hyperbolic] that $SO(n,1)$ acts properly cocompactly on an infinite rank median space. In the other hand, we know by results given in [@Fior_superrigidity] that any action of an irreducible lattice in a product of $SO(n,1)$ on a median space of finite rank has finite orbit. Many evidence for an affirmative answer to the latter question in the finite rank case are given by works of Fioravanti which extended many results concerning action on CAT($0$) cube complexes to the case of finite rank median spaces. In [@Shalom], the author obtains as a consequence of his superrigidity result that irreducible lattices in higher rank lie groups, except few cases, have the fixed point property for their isometric actions on trees. In the case of CAT$(0)$ cube complexes, it was done in the Appendix [@ChaIF]. A similar result follows in the case of complete median spaces of finite rank as it was shown in [@Fior_superrigidity]. A version of Tits alternative for groups acting on CAT$(0)$ cube complexes was shown in [@SagW], [@SagC]. The same results hold in the case of median spaces of finite rank and it was done in [@Fior_tits] by extending the machinery used in [@SagC] to the latter case. It is a well known fact that isometry group of real trees are semi simple. The same hold for CAT$(0)$ cube complexes as it was shown in [@Hag]. The result was extended to the case of finite rank median spaces in [@Fior_auto]. In this paper, we will be investigating isometric action on complete locally compact median space of finite rank. We first show, assuming mild conditions on the isometry group of the median space, that orbits are discrete: **Theorem 1**. *Let $X$ be an irreducible complete connected locally compact median space of finite rank. Let us assume that the action of $G:=Isom(X)$ on $X$ is Roller non elementary and Roller minimal. Then any $G$-orbit is discrete.* The argument of the proof relies in an essential way on the machinery developed in [@Fior_superrigidity]. Assuming that the median space is locally compact imposes a certain configuration on the halfspaces which are transverse to a ball. A halfspace is a convex subset such that its complementary is also convex. We give the following characterization of compact subsets by the combinatoric of the halfspaces which are transverse to the subset: **Theorem 2**. *Let $X$ be a complete connected median space of rank $n$. Let $C$ be a closed bounded subset of $X$. Then the following are equivalent:* 1. *The subset $C$ is compact.* 2. *For any $x_0\in C$ and $\epsilon >0$, there exist $x_1,...,x_{k_\epsilon}\in C$ such that for any $x\in C$ we have $d(x,[x_0,x_i])\leq \epsilon$ for some $i\in\{1,..,k_\epsilon\}$.* 3. *For any $\epsilon>0$, if $\mathcal{H}_\epsilon$ is a family of pairwise disjoint halfspaces transverse to $C$ and of depth bigger than $\epsilon$ in the convex hull of $C$, then it is finite.* Where the depth of a subset $H\subset X$ inside another subset $C$ is defined as $depth_C(H):=sup(\{d(x,H^c) / x\in C \}$. Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"} falls within the framework of the duality between the category of median spaces and the category of pointed measured partially ordered sets with inverse operation (see subsections [2.2](#halfspaces){reference-type="ref" reference="halfspaces"}, [2.3](#duality){reference-type="ref" reference="duality"} for definitions and [@Roller], [@ChaDH_median], [@Fior_median_property] for a reference). It characterizes the subcategory of the latter category which is dual to the subcategory of complete compact connected median space of finite rank. Having Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"} in hand, we prove the following classification of complete connected locally compact median space of finite rank which admits a transitive action: **Theorem 3**. *Let $X$ be a connected locally compact median space of finite rank which admits a transitive action, then $X$ is isometric to $(\mathbb{R}^n,l^1)$.* The argument consists of considering the set of halfspaces which are \"branched\" at an arbitrary point. A halfspace is a convex subset such that its complementary is also convex. The set $\mathcal{H}_x(X)$ of halfspaces branched at a point $x\in X$ in a complete median space of finite rank is the set of halfspaces $\mathfrak{h}\subset X$ such that $x\in \bar{\mathfrak{h}}\cap \bar{\mathfrak{h}^c}$. The set $\mathcal{H}_x(X)$ can be seen as the extension of the notion of the valency at a point $x$ in an $\mathbb{R}$-tree to the case of complete median space of finite rank. Theorem [Theorem 3](#rigidity){reference-type="ref" reference="rigidity"} is obtained then as a consequence of the following results: **Theorem 4**. *Let $X$ be a complete connected median space of rank $n$ which admits a transitive action. If for some or equivalently any $x\in X$:* 1. *The set $\mathcal{H}_x$ contains no triple of pairwise disjoint halfspaces then the space $X$ is isomorphic to $(\mathbb{R}^n,l^1)$.* 2. *The set $\mathcal{H}_x$ contains three halfspaces which are pairwise disjoint then the space $X$ is not locally compact.* We note that the result of Theorem [Theorem 3](#rigidity){reference-type="ref" reference="rigidity"} still holds when we assume that the action of the group of isometry is topologically transitive, although we do not give a proof. It is tempting to try to extract from the discrete orbits ensured by Theorem [Theorem 1](#Theorem_discrete_orbit){reference-type="ref" reference="Theorem_discrete_orbit"} an isometric action on a CAT$(0)$ cube complex, but until now it is still unclear, at least for the author, how to do so. ## Structure of the paper Section 2 is a quick overview of the geometry of median spaces, the structure of their halfspaces and the duality between them. Section 3 is devoted to the proof of an embedding lemma of the convex hull between two convex subsets. The lemma will be needed in the proof of Theorem [Theorem 3](#rigidity){reference-type="ref" reference="rigidity"}. In section 4, we recall results about the compactness of the convex hull before proving Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"} about the characterization of compact subsets. Section 5 is devoted to prove Theorem [Theorem 3](#rigidity){reference-type="ref" reference="rigidity"} about the transitivity of action and the local compactness. Finally, we recall in the last section definitions and results from the machinery developed in [@Fior_superrigidity], in order to prove Theorem [Theorem 1](#Theorem_discrete_orbit){reference-type="ref" reference="Theorem_discrete_orbit"}.\ ***Acknowledgments:***  The author expresses his sincere gratitude to his PhD advisor Professor Indira Chatterji for insightful discussions and unwavering encouragements throughout the preparation of this work. # Preliminaries ## Median spaces and algebras A ***median space*** is a metric space $(X,d)$ such that for any triple of points $a,b,c\in X$, there exists a unique point $m\in X$ such that $$[a,b]\cap [b,c]\cap [a,c] =\{ m\}$$ where $[a,b]=\{x\in X | d(a,b)=d(a,x)+d(x,b)\}$ is the ***interval*** between $a$ and $b$. Let us denote by $m:X\times X\times X:\rightarrow X$ the ternary operation that associates to each triple $(x,y,z)$ their unique median point $m(x,y,z)$. We call the latter the ***median operation***. We say that a map $f$ between two median spaces is a ***median morphism*** if it respects the median operation, i.e. $f(m_X(x,y,z))=m_Y(f(x),f(y),f(z))$. Note that any isometry between two median spaces is a median morphism. A subset $C\subseteq X$ is ***convex*** if for any $a,b\in C$, the interval $[a,b]$ between $a$ and $b$ is contained in $C$. Each convex subset is naturally endowed with a median structure induced from the ambient space. In a complete median space, the nearest point projection to a closed convex subset $C$, that we denote by $\pi_C$, is a $1$-Lipschitz median morphism and verifies the following property: For any $x\in X$ and $c\in C$, we have $\pi_{C}(x)\in [c,x]$ (see Lemma 2.13 [@ChaDH_median]). In the particular case where the convex subset is an interval $[a,b]$, the nearest point projection coincide with the morphism $m(a,b,*)$ and the interval $[a,b]$ correspond to the fixed point of the latter application. Hence, the notion of convexity can be described by the ternary operation $m$, in fact, most of the interesting geometric feature of median spaces are encoded algebraically in the ternary operation. More precisely, the given ternary operation $m$ endow $X$ with a structure of a median algebra, which is defined as follow: **Definition 5**. *A ***median algebra*** $(M,m)$ is set $M$ endowed with a ternary operation $m:M\times M\times M\rightarrow M$ which verifies the following equations: $$\begin{aligned} m(x,x,y)&=&x\\ m(x,y,z)&=&m(y,x,z)\ =\ m(x,z,y)\\ m(m(x,y,z),u,v)&=&m(x,m(y,u,v),m(z,u,v))\end{aligned}$$* Let $M$ be a median algebra, and let $a,b\in M$, then the interval between $a,b$ is defined as the fixed points of the projection map $m(a,b,*)$, i.e. $[a,b]:=\{x\in M | m(a,b,x)=x\}$. The convexity is defined with respect to this notion of interval. Conversely, if $(M,m)$ is a median algebra and there exists a metric $d$ on $M$ such that the intervals corresponding to the metric $d$ coincide with the intervals corresponding to the ternary operation $m$, then the metric space $(M,d)$ is a median space. Let $X$ be a median space and let $\tilde{X}$ be its metric completion. The median operation being $1$-lipschitz, where $X^3$ is endowed with the $\ell^1$-product metric (Corollary 2.15 in [@ChaDH_median]), it extends to $\tilde{X}$. The equation arising in the axiom of the median algebra are defined by continuous functions with respect to $\tilde{X}$. Hence, the set of solution are closed in $\tilde{X}$ and contains the dense subset $X$, therefore they are verified by all points of $\tilde{X}$. We conclude that the metric completion is also a median space (see Proposition 2.21 in [@ChaDH_median]). A well known fact, due to Eduard Helly (see [@Helly]), is that the intersection of a finite family of convex subsets in the euclidean space $\mathbb{R}^n$ is empty if and only if the intersection of some subfamily of cardinal less then or equal $n$ is empty. In the particular case of the real line, the intersection of a finite family of convex subsets is empty if and only if there exist two convex subsets of the family which are disjoint. The same holds for median algebras (see Theorem 2.2 [@Roller]): **Theorem 6** (Helly's Theorem). *Let $X$ be a median space and let $C_1,..,C_n\in X$ be a family of pairwise intersecting convex subsets. Then their intersection is not empty.* ## Halfspaces and poc sets {#halfspaces} **Definition 7**. *A convex subset of a median algebra is a ***halfspace*** if its complementary is also convex. We denote by $\mathcal{H}(X)$ the set of all halfspaces of the median algebra $X$.* Assuming Zorn's lemma, such halfspaces do exist. Moreover, any two convex subsets in a median algebra are separated by a halfspace as stated in the following theorem (see Theorem 2.8 [@Roller]): **Theorem 8**. *Let $X$ be a median algebra, then for any two convex subsets $C_1,C_2\subseteq X$ such that $C_1\cap C_2= \emptyset$, there exists a halfspace $\mathfrak{h}\in \mathcal{H}(X)$ such that $C_1\subseteq \mathfrak{h}$ and $C_2\subseteq \mathfrak{h}^c$.* **Definition 9**. *Two halfspaces $\mathfrak{h}_1,\mathfrak{h}_2\in\mathcal{H}(X)$ are called ***transverse*** if the following intersections $\mathfrak{h}_1\cap \mathfrak{h}_2, \mathfrak{h}_1^c\cap \mathfrak{h}_2, \mathfrak{h}_1\cap \mathfrak{h}_2^c, \mathfrak{h}_1^c\cap \mathfrak{h}_2^c$ are not empty. We say that a median space is of ***rank*** $n$ if there exists a family of pairwise transverse halfspaces $\mathfrak{h}_1,..,\mathfrak{h}_n\in\mathcal{H}(X)$ where $n$ is maximal. In the case of connected median space, the rank coincide with the maximum of topological dimension over compact subsets[^1].* *A ***wall*** in $X$ is a couple $(\mathfrak{h},\mathfrak{h}^c)$ where $\mathfrak{h}\in\mathcal{H}(X)$ is a halfspace of $X$. For any $A,B\subseteq X$, we denote by $\mathcal{H}(A,B)$ the set of halfspaces which separate $B$ from $A$ $i.e. \mathcal{H}(A,B):=\{\mathfrak{h}\in\mathcal{H}(X) | B\subseteq \mathfrak{h}, A\subseteq \mathfrak{h}^c \}$. In the same vein, we define the walls interval between $A$ and $B$ as $\mathcal{W}:=\{(\mathfrak{h},\mathfrak{h}^c) | \mathfrak{h}\in \mathcal{H}(A,B)\}$. When $A$ and $B$ are singletons, we simply write $\mathcal{H}(x,y)$ and $\mathcal{W}(x,y)$. A halfspace $\mathfrak{h}\subseteq X$ is said to be ***transverse*** to a subset $C\subseteq X$ if both $\mathfrak{h}\cap C$ and $\mathfrak{h}^c\cap C$ are non empty. In that case, we say also that the wall $(\mathfrak{h},\mathfrak{h}^c)$ is transverse to $C$. We denote by $\mathcal{H}(C)$ (respectively $\mathcal{W}(C)$) the set of halfspaces (respectively walls) which are transverse to $C$.* A convex subset in a complete median space is naturally endowed with a median structure, induced from the ambient space. The following proposition states that the halfspaces of a closed convex subset are induced by the halfspaces of the ambient space: **Proposition 10** (See Proposition 2.3 [@Fior_median_property]). *Let $X$ be a complete median space and let $C\in X$ be a closed convex subset. We have then the following: $$\mathcal{H}(C)=\{\mathfrak{h}\cap C | \mathfrak{h}\in\mathcal{H}(X)\}$$* In [@Bowd_conv] and [@Fior_median_property], the notion of hyperplanes arising in CAT$(0)$ cube complexes was introduced and investigated in the case of complete connected topological median space and median space of finite rank respectively. In a complete connected median space of finite rank $X$, the ***hyperplane*** $\hat{\mathfrak{h}}\subseteq X$ bounding the halfspace $\mathfrak{h}\subseteq X$ is the convex subset obtained by taking the intersection of the closure of $\mathfrak{h}$ with the closure of its complementary, i.e. $\hat{\mathfrak{h}}=\bar{\mathfrak{h}}\cap\bar{\mathfrak{h}^c}$. It is a closed convex subset of rank less or equal $n-1$. These hyperplanes appear to be useful. For instance, they are practical to prove properties about median space of finite rank using an argument by induction on the rank of the space $X$. For a complete connected median space of finite rank, we set $\mathcal{H}_{x}:=\{\mathfrak{h}\in\mathcal{H}(X) | x\in\overline{\mathfrak{h}}\cap\overline{\mathfrak{h}^c}\}$. It consists of the set of halfspaces which are \"branched at the point $x$. It is the natural generalization of the valency from $\mathbb{R}$-trees to the higher rank case. **Remark 11**. *In a complete finite rank median space, halfspaces are either open, closed or possibly both. The latter case occurs when the median space is not connected (see [@Fior_median_property] Corollary 2.23). Hence, any halfspace $\mathfrak{h}\in \mathcal{H}_x$ which contains $x$ is necessarily closed.* The set of halfspaces $\mathcal{H}(X)$ of a median algebra $X$ is naturally endowed with a partial order relation given by the inclusion and a complementary operation which associates to each halfspaces its complementary in $X$, thus inversing the partial order relation. This provides $\mathcal{H}(X)$ with a natural structure of a poc set. A ***poc set*** $(P,\ \leq,\ ^*,0)$ is a partially ordered set $(P,\ \leq)$ with a minimal element $0$ and an involution $^*$ on $P$, called the complementary operation, which inverse the order and such that the minimal element $0$ is the unique element in $P$ which is smaller than its complementary $0^*$. ## Duality The set of halfspaces of a median algebra $X$, seen as an abstract poc set, carries all the informations about the median structure of $X$. In [@Isbell], the author extended the Stone duality between boolean algebra and stone space to the case of median algebras and poc sets, see Theorem 6.13 therein and [@Roller] for more detail. In the metric case, one obtain a duality between median spaces and measurable poc sets. This was investigated in [@ChaDH_median] and [@Fior_median_property]. Let us describe quickly the duality in question and cite a result from [@Fior_median_property] that we will be using later in the paper. Let us fix a poc set $(P,\ \leq,\ ^*,0)$. A set $\mathfrak{u}\in\mathcal{P}(P)$ of subsets of $P$ is an ***ultrafilter*** over $P$ if it verifies the following: - For any $p,q\in \mathfrak{u}$ we do not have $p<q*$. - For any $p\in P$ we have either $p\in\mathfrak{u}$ or $p*\in\mathfrak{u}$ The first condition is an adaptation of the condition that the intersection between $p$ and $q$ being not empty when the poc set structure of $P$ comes from a set of subsets. In [@ChaDH_median], the authors used instead the language of admissible sections. A ***pointed measured poc set*** is a quadruple $(P,\mathcal{D},\mu,\mathfrak{u}_0)$ where $P$ is a poc set, $\mathcal{D}$ a $\sigma$-algebra over $P$, $\mu$ a measure defined on $\mathcal{D}$ and $\mathfrak{u}_0$ a fixed ultrafiter over $P$. To each such pointed measured poc set, we associates a median space as follow : we consider the sets of ultrafilters $\mathfrak{u}$ over $P$ such that the symmetric difference $\mathfrak{u}\Delta\mathfrak{u}_0$ lies in $\mathcal{D}$. We identify the ultrafilters which have symmetric difference of zero measure. The set of such classes is denoted by $\mathcal{M}(P)$ and the metric given by the measure of the symmetric difference endow it with a structure of a median space. Conversely to each median space $X$, its set of halfspaces comes with a natural structure of a measured poc set $(\mathcal{H}(X),\mathcal{D},\mu,\mathfrak{u}_{a})$ where $\mathcal{D}$ is the $\sigma$-algebra generated by the walls intervals, $\mu$ a measure defined on $\mathcal{D}$ such that $\mu(\mathcal{W}(x,y))=d(x,y)$ and $\mathfrak{u}_{a}$is the principal ultrafilter associated to a base point point $a\in X$ $i.e.$ the set of halfspaces containing $a$.The functors $\mathcal{H}$ and $\mathcal{M}$ verify the following contravariance property: **Proposition 12** (Proposition 2.10 [@Fior_tits]). *Let $(X_1,d_1),(X_2,d_2)$ be two median spaces and let $(P_1,\mu_1,p_1)$,$(P_1,\mu_1,p_1)$ be two pointed measured poc set. Then we have:* - *The median space $\mathcal{M}(P_1 \sqcup P_2)$ is isometric to $(\mathcal{M}(P_1)\times \mathcal{M}(P_2),d_{\ell^1})$.* - *The measured poc set $\mathcal{H}(X_1 \times X_2)$ is isomorphic to $\mathcal{H}(X_1)\sqcup \mathcal{H}(X_2)$.* We have the following theorem: **Theorem 13** (Theorem A [@Fior_median_property]). *Let $X$ be a complete locally convex median space. Then $\mathcal{M}(\mathcal{H}(X))$ is isometric to $X$.* # An embedding of the convex hull The aim of this section is to prove an embedding lemma of the convex hull between two convex subsets which do not share a transverse halfspace. We first recall some definitions and properties regarding the convex hull. ## Description of the convex hull Let $X$ be a median algebra, the ***convex hull*** of a subset $A\subseteq X$, that we denote by $Conv(A)$ is the intersection of all convex subsets containing $A$. The convex hull of the union of subsets which form a directed sets is the union of the convex hull of each sets. Therefore, we get the following proposition (Corollary 2.5 [@Roller]): **Proposition 14**. *Let $X$ be a median algebra, then for any $A\subseteq X$ we have: $$Conv(A)=\displaystyle{\bigcup_{x_1,..,x_n\in A}Conv(\{x_1,..,x_n\})}$$* The ***join*** $[A,B]$ between two subsets $A,B$ is the union of all intervals having endpoints in $A$ and $B$. The ***$n$-iterated*** join $A^n$ of a subset $A$ is defined recursively by $A^n=[A^{n-1},A^{n-1}]$ and $A^1=A$. We have then the following: **Proposition 15**. *Let $C_1,C_2\subseteq X$ be two closed convex subsets. Then any point $x\in Conv(C_1,C_2)$ lies in the interval $[\pi_{C_1}(x),\pi_{C_2}(x)]$. In particular, we have: $$Conv(C_1,C_2)=\displaystyle{\bigcup_{\substack{x\in C_1\\ y\in C_2}}[x,y]}$$* *Proof.* Any halfspace separating $x$ from $\pi_{C_i}(x)$ separates $x$ from $C_i$. Hence, any halfspace which separates $x$ from $[\pi_{C_1}(x),\pi_{C_2}(x)]$ separates $x$ from $C_1$ and $C_2$, thus from $Conv(C_1,C_2)$. Therefore, if a point $x\in X$ lies outside $[\pi_{C_1}(x),\pi_{C_2}(x)]$ then it does not belongs to $Conv(C_1,C_2)$. ◻ We have the following constraint on the projection onto closed convex subset : **Lemma 16**. *Let $C$ be a closed convex subset and $A$ a convex set such that $A\cap C\neq \emptyset$. Then the projection of $A$ into $C$ lies in $A\cap C\neq \emptyset$.* *Proof.* Let us fix $c\in A\cap C$ and consider any $a\in A$. We have $\pi_C(a)\in[a,c]\subseteq A$. Hence $\pi_{C}(A)\subseteq A\cap C$. ◻ **Remark 17**. *Let $X$ be a complete median space and let $C_1,C_2\subseteq X$ be two closed convex subsets. Then there is no halfspace which is transverse to both $C_1$ and $C_2$ if and only if $\pi_{C_1}(C_2)$ and $\pi_{C_2}(C_1)$ are singletons. The first implication is due to Lemma [Lemma 16](#constraint_on_the_projection){reference-type="ref" reference="constraint_on_the_projection"}. The other direction is given by the fact that the image under a projection into a closed convex subset is also convex and that the inverse image of any halfspace by such projections, which are median morphisms, is a halfspace of the ambient space (See Proposition [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"}).* Before closing this subsection, we take a brief look at the convex hull of balls in median spaces of finite rank. In a median space, the balls are convex if and only if the median space is of rank $1$, that is, it is tree like. In the general setting, median spaces are not necessarily locally convex. However, in the finite rank case, the median space is locally convex and the convex hull of any subset is obtained after iterating a finite number of time its join: **Proposition 18** (Lemma 6.4 [@Bowd_conv]). *Let $X$ be a median space of rank $n$. Then for any $A\subseteq X$ we have $Conv(A)=A^n$.* One may remark from the latter Proposition that the convex hull of the ball of radius $r$ lies in a ball of radius $2^{n-1}r$. In the following proposition we show that in a median space of rank $n$, the convex hull of the ball of radius $R>0$ lies in a ball of radius $nr$: **Proposition 19**. *Let $X$ be a median space of rank $n$. Then for any $a\in X$ and $r>0$, we have: $$Conv(B(a,r))\subseteq B(a,nr)$$* Before proving the statement, we will be needing some lemmas. The following lemma is a strengthening of the separation Theorem [Theorem 8](#separation_theorem){reference-type="ref" reference="separation_theorem"} in the case of complete median space of finite rank: **Lemma 20**. *Let $X$ be a complete connected median space of finite rank. Then for any $a,b\in X$, there exists a halfspace $\mathfrak{h}\in\mathcal{H}(a,b)$ such that $d(a,\mathfrak{h})=0$.\ In particular, we have $\displaystyle{\bigcap_{\substack{\mathfrak{h}\in \mathcal{H}_a(X) \\ a \in \mathfrak{h}}}}=\{a\}$* *Proof.* Let us consider two distinct points $a,b\in X$. The median space $X$ being complete and connected, there exists a midpoint $b_1\in [a,b]$, i.e. $d(a,b_1)=d(b_1,b)=\frac{d(a,b)}{2}$. Let $\mathfrak{h}_1\in \mathcal{H}(a,b_1)$ be a halfspace separating $b_1$ from $a$. Let $b_2$ be a midpoint of $[a,\pi_{\hat{\mathfrak{h}}_1}(a)]$ and let $\mathfrak{h}_2\in\mathcal{H}(a,b_2)$ be a halfspace separating $b_2$ from $a$. The halfspace $\mathfrak{h}_2$ separates $\hat{\mathfrak{h}_1}$ from $a$, hence it contains the halfspace $\mathfrak{h}_1$. Proceeding by iteration, we obtain an ascending sequence of halfspaces $(\mathfrak{h}_n)_{n\in\mathbb{N}*}$ separating $b$ from $a$ and such that $\displaystyle{\lim_{n\rightarrow\infty}d(\mathfrak{h}_n,a)=0}$. The subset $\mathfrak{h}:=\displaystyle{\bigcup_{n\in\mathbb{N}^*}\mathfrak{h}_n}$ is our desired halfspace. ◻ **Remark 21**. *Lemma [Lemma 20](#strenghtening_separation_theorem){reference-type="ref" reference="strenghtening_separation_theorem"} above remains true in the case of a complete connected locally convex median space. One may adapt the argument given in [@Roller] by looking at a maximal element in the family of convex subsets which contains $b$ in their interior and separates it from $a$, and show that such a maximal element is a halfspace which contains $a$ on its closure.* We deduce the following lemma: **Lemma 22**. *Let $X$ be a median space of rank $n$, and $a,b\in X$. Then there exists a halfspace $\mathfrak{h}\in\mathcal{H}(a,b)$ such that $d(a,\mathfrak{h})=0$ and $d(b,\mathfrak{h}^c)\geq \frac{d(a,b)}{n}$.* *Proof.* We proceed by induction on the rank of the space $X$. The lemma is trivial for a connected $\mathbb{R}$-tree. Let us assume that the lemma is true for median spaces of rank $n-1$. Let $X$ be a median space of rank $n$, and let $a,b\in X$. Let us take a halfspace $\mathfrak{h}\in\mathcal{H}(a,b)$ such that $d(a,\mathfrak{h})=0$, Lemma [Lemma 20](#strenghtening_separation_theorem){reference-type="ref" reference="strenghtening_separation_theorem"} ensures the existence of such halfspace. Let us assume that $\mathfrak{h}$ is not our desired halfspace, that is $d(b,\mathfrak{h}^c)< \frac{d(a,b)}{n}$. Let us consider then the projections $\tilde{b}:=\pi_{\hat{\mathfrak{h}}}(b)$, $\pi_{\hat{\mathfrak{h}}}(a)=a$. We have then: $$d(a,b)=d(a,\tilde{b})+d(\tilde{b},b)$$ We deduce the following: $$\begin{aligned} d(a,\tilde{b})&=&d(a,b)-d(b,\tilde{b})\\ \ \ \ \ \ \ &\geq & d(a,b)-\frac{d(a,b)}{n}\\ &\geq & \frac{(n-1)d(a,b)}{n}\end{aligned}$$ The interval $[a,\tilde{b}]$ lies in the hyperplane $\hat{\mathfrak{h}}$, which is a median space of rank $n-1$. Hence, there exists a halfspace $\tilde{\mathfrak{h}}\in\mathcal{H}(a,\tilde{b})$ such that $d(a,\tilde{\mathfrak{h}})=0$ and $d(\tilde{b},\tilde{\mathfrak{h}}^c)\geq \frac{d(a,\tilde{b})}{n-1}$. The projection into a convex subset being $1$-lipschitz, we get $d(b,\tilde{\mathfrak{h}}^c)\geq d(\tilde{b},\tilde{\mathfrak{h}}^c)$. We conclude then: $$\begin{aligned} d(b,\tilde{\mathfrak{h}}^c)&\geq & d(\tilde{b},\tilde{\mathfrak{h}}^c) \geq \frac{d(\tilde{a},b)}{n-1}\\ &\geq & \frac{\frac{(n-1)d(a,b)}{n}}{n-1}\geq \frac{d(a,b)}{n} \end{aligned}$$ Therefore $\tilde{\mathfrak{h}}$ is our desired halfspace. ◻ *Proof of Proposition [Proposition 19](#convex_hull_ball){reference-type="ref" reference="convex_hull_ball"}.* Note that there is no loss of generality if we assume $X$ to be complete. Let $b\in X$ such that $d(a,b)>nr$. By Lemma [Lemma 22](#lemma_local_convexity){reference-type="ref" reference="lemma_local_convexity"}, there exists a halfspace $\mathfrak{h}\in\mathcal{H}(a,b)$ such that $d(a,\mathfrak{h})=0$ and $d(b,\mathfrak{h}^c)\geq\frac{d(a,b)}{n}$. Thus, the ball $B(b,r)$ is contained in the halfspace $\mathfrak{h}$. Hence, the convex hull of $B(b,r)$ also lies in $\mathfrak{h}$. We conclude that any point which is at distance bigger then $nr$ from the point $b$ lies outside the convex hull of $B(b,r)$. ◻ ## Embedding lemma The following embedding property of the convex hull of two closed convex subsets with no transverse halfspace in common will be a key ingredient in the proof of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"} : **Proposition 23**. *Let $X$ be a complete median space of finite rank and let $C_1,C_2\in X$ be closed convex subsets such that there is no half-space which is transverse to both. We have then an isometric embedding of the convex hull $Conv(C_1,C_2)$ into the $\ell^1$-product $C_1\times C_2 \times [c_1,c_2]$ where $c_1=\pi_{C_1}(C_2)$ and $c_2=\pi_{C_2}(C_1)$ (see Remark [Remark 17](#strong_separation_and_singleton_bridge){reference-type="ref" reference="strong_separation_and_singleton_bridge"}).* For the proof of Proposition [Proposition 23](#embed_lemm){reference-type="ref" reference="embed_lemm"}, we need the following lemma: **Lemma 24**. *Let $X$ be a complete median space of finite rank and let $C_1,C_2\subseteq X$ such that there is no halfspace which is transverse to both. After setting $c_1=\pi_{C_1}(C_2)$, $c_2=\pi_{C_2}(C_1)$ and considering any $x,y\in Conv(C_1,C_2)$ we have: $$\mathcal{W} (x,y)= \mathcal{W} (\pi_{C_1}(x),\pi_{C_1}(y)) \sqcup \mathcal{W} (\pi_{C_2}(x),\pi_{C_2}(y)) \sqcup \mathcal{W} (\pi_{[c_1,c_2]}(x),\pi_{[c_1,c_2]}(y))$$* *Proof.* For any closed convex subset $C\subseteq X$ and any $x,y\in X$, we have: $$\mathcal{W}(\pi_C(x),\pi_C(y))\subseteq \mathcal{W}(x,y)$$ This come from the fact that for any $c\in C$, the interval $[c,x]$ contains $\pi_C(x)$. Thus, we have the inclusion of the right hand of the equality $(1)$ into the left side. For the other inclusion, let us consider $x,y\in Conv(C_1,C_2)$ and $\mathfrak{h}\in\mathcal{H}(x,y)$. Let us assume that $\mathfrak{h}$ does not separates $\pi_{C_1}(y)$ from $\pi_{C_1}(x)$ and $\pi_{C_2}(y)$ from $\pi_{C_2}(x)$. By Proposition [Proposition 15](#properties_of_convex_hull){reference-type="ref" reference="properties_of_convex_hull"}, if the projections $\pi_{C_1}(x)$, $\pi_{C_1}(y)$, $\pi_{C_2}(x)$ and $\pi_{C_2}(y)$ lie in a halfspace $\mathfrak{h}$, then so do $x$ and $y$. As the halfspace $\mathfrak{h}$ separates $y$ from $x$ there is no loss of generality if we assume that $\pi_{C_1}(x),\pi_{C_1}(y)$ belong to $\mathfrak{h}^c$ and $\pi_{C_2}(x),\pi_{C_2}(y)$ to $\mathfrak{h}$. As $\pi_{C_1}(y)\in[c_1,y]$ and $\pi_{C_2}(x)\in[c_2,x]$, we necessarily get that $c_1\in\mathfrak{h}^c$ and $c_2\in\mathfrak{h}$. We conclude that $\pi_{[c_1,c_2]}(x)\in[c_1,x]\subseteq\mathfrak{h}^c$ and $\pi_{[c_1,c_2]}(y)\in[c_2,y]\subseteq\mathfrak{h}$ (see the Figure [1](#example1){reference-type="ref" reference="example1"} below). Therefore, the halfspace $\mathfrak{h}$ lies in $\mathcal{H}(\pi_{[c_1,c_2]}(x),\pi_{[c_1,c_2]}(y))$. It is left to show that the sets arising in the right hand of the equality are indeed disjoint. Under our assumtpion that the convex subsets $C_1$ and $C_2$ being strongly separated, we already have the disjointness of $\mathcal{W} (\pi_{C_1}(x),\pi_{C_1}(y))$ with $\mathcal{W} (\pi_{C_2}(x),\pi_{C_2}(y))$. A wall which separates two points of the interval $[c_1,c_2]$ must separate $c_1$ and $c_2$. The point $c_1$ being contained in any interval connecting $C_1$ to $C_2$, we deduce that any wall in $\mathcal{W} (\pi_{[c_1,c_2]}(x),\pi_{[c_1,c_2]}(y))$ must separates $C_1$ and $C_2$. Hence, such wall cannot be in $\mathcal{W} (\pi_{C_1}(x),\pi_{C_1}(y))$ nor in $\mathcal{W} (\pi_{C_2}(x),\pi_{C_2}(y))$. ◻ ![Any halfspace which separates $x$ and $y$ is either transverse to $C_1$, to $C_2$ or to the interval $[c_1,c_2]$.](Example1.jpeg){#example1} *Proof of Proposition [Proposition 23](#embed_lemm){reference-type="ref" reference="embed_lemm"}.* As there is no halfspace which is transverse to both $C_1$ and $C_2$, the projection of $C_1$ (resp $C_2$) into $C_2$ (resp $C_1$) is a singleton, according to Remark [Remark 17](#strong_separation_and_singleton_bridge){reference-type="ref" reference="strong_separation_and_singleton_bridge"}. Let us set $c_1=\pi_{C_1}(C_2)$ and $c_2=\pi_{C_2}(C_1)$ and consider the following map: $$\begin{aligned} \ \ \ \ \ \ f:Conv(C_1,C_2)&\rightarrow & C_1\times C_2 \times [c_1,c_2]\\ \ \ \ \ \ \ x & \mapsto & f(x)=(\pi_{C_1}(x),\pi_{C_2}(x),\pi_{[c_1,c_2]}(x)) \end{aligned}$$ where $\pi_C$ denote the projection onto the closed convex subset $C$. By Lemma [Lemma 24](#decomposition_of_separating_halfpsace){reference-type="ref" reference="decomposition_of_separating_halfpsace"}, we get the following: $$\mathcal{W} (x,y)= \mathcal{W} (\pi_{C_1}(x),\pi_{C_1}(y)) \sqcup \mathcal{W} (\pi_{C_2}(x),\pi_{C_2}(y)) \sqcup \mathcal{W} (\pi_{[c_1,c_2]}(x),\pi_{[c_1,c_2]}(y))$$ We deduce then the following: $$\begin{aligned} \mu (\mathcal{W} (x,y)) &=& \mu (\mathcal{W} (\pi_{C_1}(x),\pi_{C_1}(y)))+\mu (\mathcal{W} (\pi_{C_2}(x),\pi_{C_2}(y)))+\mu (\mathcal{W} (\pi_{[c_1,c_2]}(x),\pi_{[c_1,c_2]}(y)))\\ d(x,y)&=& d(\pi_{C_1}(x),\pi_{C_1}(y))+d(\pi_{C_2}(x),\pi_{C_2}(y))+d(\pi_{[c_1,c_2]}(x),\pi_{[c_1,c_2]}(y))\\ &=& d_{\ell ^1}(f(x),f(y))\end{aligned}$$ where $\mu$ is the measure described in Subsection [2.3](#duality){reference-type="ref" reference="duality"}. ◻ By considering a stronger assumption in Proposition [Proposition 23](#embed_lemm){reference-type="ref" reference="embed_lemm"}, we get a local version of the first part of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"}: **Proposition 25**. *Let $X$ be a complete median space of finite rank and let $C_1,C_2\subseteq X$ be two closed convex subsets such that $C_1\cap C_2=\{x_0\}$. Then, for any $x\in Conv(C_1,C_2)$, the interval $[x,x_0]$ is isometric to the $\ell^1$ product of $[\pi_{C_1}(x),x_0]$ and $[\pi_{C_2}(x),x_0]$.* *Proof.* For any $x\in Conv(C_1,C_2)$, we have: $$\mathcal{H}([x,x_0])\ =\ \mathcal{W}(x,x_0)\ =\footnote{This equality comes from Lemma \ref{decomposition_of_separating_halfpsace}}\ \mathcal{W}(\pi_{C_1}(x),x_0)\sqcup\mathcal{W}(\pi_{C_2}(x),x_0)$$ where any halfspace in $\mathcal{H}(\pi_{C_1}(x),x_0)$ is transverse to any halfspace in $\mathcal{H}(\pi_{C_2}(x),x_0)$. Hence, by Proposition [Proposition 12](#contravariance){reference-type="ref" reference="contravariance"} the median space $\mathcal{M}(\mathcal{H}([x,x_0]))$ is isometric to the $\ell^1$- product of $\mathcal{M}(\mathcal{H}([\pi_{C_1}(x),x_0]))$ with $\mathcal{M}(\mathcal{H}([\pi_{C_2}(x),x_0]))$. The intervals being closed subsets and the median space being complete, we conclude by Theorem [Theorem 13](#duality_thm){reference-type="ref" reference="duality_thm"} that the interval $[x,x_0]$ is isomorphic to $\mathcal{M}(\mathcal{H}([x,x_0]))$ and that the latter is isomorphic to the $\ell^1$-product of the two intervals $[\pi_{C_1}(x),x_0],[\pi_{C_2}(x),x_0]$, by Proposition [Proposition 12](#contravariance){reference-type="ref" reference="contravariance"}. ◻ Median space are in general far from being CAT(0) spaces or even geodesic spaces. Nevertheless, convexity in median space being defined by mean of intervals, convex subsets in median spaces are rigid enough to share many properties that are featured in CAT$(0)$ spaces. Let $X$ be a complete median space and let $C_1,C_2\subseteq X$ be two closed convex subsets. Then, their respective convex subsets $\pi_{C_1}(C_2)$ and $\pi_{C_2}(C_1)$ are isometric and the isometry is given by the gate projection. Moreover, the following median version of the Sandwich lemma ( [@Brid_Haef] Exercise II.2.12) holds: **Proposition 26** (Proposition 2.21 [@Fior_superrigidity]). *Let $X$ be a median space and let $C_1,C_2\subseteq X$ be two closed convex subset. Then $Conv(\pi_{C_1}(C_2),\pi_{C_2}(C_1))$ is isometric to $\pi_{C_1}(C_2)\times [x,\pi_{C_2}(x)]$ where $x$ is any point in $\pi_{C_1}(C_2)$.* We call the convex subset $Conv(\pi_{C_1}(C_2),\pi_{C_2}(C_1))$ the ***bridge*** between $C_1$ and $C_2$, let us denote it by $\mathcal{B}(C_1,C_2)$. **Remark 27**. *Proposition [Proposition 23](#embed_lemm){reference-type="ref" reference="embed_lemm"} can be extended to the case where $C_1$ and $C_2$ admits a common transverse halfspace by taking the projection of the convex hull between $C_1$ and $C_2$ into the product $C_1\times C_2\times \mathcal{B}(C_1,C_2)$ endowed with the $\ell^1$ product metric. The map is not necessarily an isometry, it is a 2-lipschitz embedding. For any two points in $Conv(C_1\cup C_2)$ separated by halfspaces which are transverse to both $C_1$ and $C_2$, the horizontal distance with respect to $C_1$ and $C_2$ is counted twice in $C_1\times C_2\times \mathcal{B}(C_1,C_2)$.* # Characterization of compactness by mean of halfspaces In the first subsection, we recall some results about the convex hull of compact subsets in a median space. The next subsection is devoted to the proof of Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"}. ## Convex hull of compact subsets It was shown in [@Fior_median_property] that any interval in a median space of rank $n$ embeds isometrically into $\mathbb{R}^n$, see Proposition 2.19 therein. A direct consequence is that the convex hull of finite subsets in a finite rank median space are compact. More generally, the convex hull of a compact subset in a complete finite rank median space is also compact, see Lemma 13.2.11 in [@Bowd_median-algebras]. In a complete median space, the convex hull of a compact subset is not necessarily compact. In general, even the interval are not necessarily compact, consider for instance intervals in $L^1(\mathbb{R})$. However, under the assumption that the intervals of $X$ are compact, the convex hull between any two convex compact subsets is also compact. Before giving a proof, we will be needing some results: **Lemma 28**. *Let $C\subseteq X$ be a convex subset and a point $x\in [a,b]\subseteq X$, we have: $$d(x,C)\leq d(a,C)+d(b,C)$$* *Proof.* Any halfspace which separates $C$ from $x$ must separates it either from $a$ or from $b$ (or from both). Thus we get $$d(x,C)=\mu(\mathcal{W}(x,C))\leq \mu(\mathcal{W}(a,C))+ \mu(\mathcal{W}(b,C))=d(a,C)+d(b,C)$$ ◻ We deduce the following lemma: **Lemma 29**. *Let $X$ be a complete median space. Then the join between any two closed convex subsets is closed.* *Proof.* Let us consider two convex subsets $C_1,C_2\subseteq X$ and let $(x_n)_{n\in\mathbb{N}}\subseteq [C_1,C_2]$ be a sequence of points converging to $x\in X$. Note that each $x_n$ lies in the interval $[\pi_{C_1}(x_n),\pi_{C_2}(x_n)]$. As gate projections are $1$-lipschitz, the sequences $(\pi_{C_1}(x_n))_{n\in\mathbb{N}}$ and $(\pi_{C_2}(x_n))_{n\in\mathbb{N}}$ are Cauchy sequences. Thus they converge to $a\in C_1$ and $b\in C_2$ respectively. In the other hand, we have $$\begin{aligned} d(x,[a,b])=d(x,m(x,a,b)) &\leq & d(x,x_n) + d(x_n,m(x_n,a,b)) + d(m(x_n,a,b),m(x,a,b))\\ &=& d(x,x_n) + d(m(x_n,a,b),m(x,a,b)) + d(x_n,[a,b])\end{aligned}$$ Where the right side tend to zero when $n$ goes to infinity by the continuity of the projection and Lemma [Lemma 28](#distance_interval_from_convex){reference-type="ref" reference="distance_interval_from_convex"}. ◻ **Remark 30**. *The join between two closed subset of a complete median space of finite rank is not necessarily closed, even if we assume that the subsets are bounded. Take for instance the product of the closed segment of the real line with a star like simplicial tree with infinite edges of length $1$. One may consider then a sequence of points such that their projections into the star like simplicial tree run injectively through its vertices and their projections into the closed segment of the real line accumulate around $0$ but never attain it.* **Proposition 31**. *Let $X$ be a complete median space which have compact intervals. Then the convex hull between any two compact convex subsets is also compact.* *Proof.* By Proposition [Proposition 23](#embed_lemm){reference-type="ref" reference="embed_lemm"} and Lemma [Lemma 29](#join_between_closed_convex_is_closed){reference-type="ref" reference="join_between_closed_convex_is_closed"}, for any closed convex subset $C_1,C_2\subseteq X$, their convex hull embeds as a closed subsets into the $\ell^1$-product of $C_1$, $C_2$ and $\mathcal{B}(C_1,C_2)$, where the latter, by Proposition [Proposition 26](#description_gate){reference-type="ref" reference="description_gate"}, is isometric to an interval and a closed convex subset of $C_1$, which is compact. ◻ In particular, we have the following: **Corollary 32**. *Let $X$ be a complete medians space with compact intervals. Then the convex hull of any finite subset is compact.* ## Proof of Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"} {#proof-of-theorem-local_compactness} **Definition 33**. *Let $X$ be a complete median space of finite rank and let $\mathfrak{h}\in\mathcal{H}(X)$ be a halfspace. We call the ***depth of*** $\mathfrak{h}$ in $A\subseteq X$, that we denote by $depth_A(H)$, the maximum distance between points lying in $\mathfrak{h}\cap A$ and the hyperplane $\hat{\mathfrak{h}}$ bounding $\mathfrak{h}$, i.e. $depth_A(H):=sup\{d(x,\hat{\mathfrak{h}}) / x\in\mathfrak{h}\cap A \}$.* Before proving Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"}, we will be needing some lemmas. The following lemma is a strengthening of Lemma [Lemma 22](#lemma_local_convexity){reference-type="ref" reference="lemma_local_convexity"}: **Lemma 34**. *Let $X$ be a complete connected median space of rank $n$ and let $a,b\in X$. Then for any small $\epsilon>0$ which is smaller then $\frac{d(a,b)}{n}$, there exist a pairwise disjoint halfspaces $\mathfrak{h}_1,..,\mathfrak{h}_k\in\mathcal{H}(a,b)$, where $k\leq n$, such that for all $i\in\{1,..,k\}$ we have:* - *$d(\mathfrak{h}_i^c,b)\geq\epsilon$.* - *$d(a,\displaystyle{\bigcap_{i=1}^k\mathfrak{h}_i})\geq d(a,b)-\frac{n(n+1)}{2}\epsilon$.* *Proof.* Let us proceed by induction on the rank of $X$. The lemma is trivially true for complete connected median space of rank $1$, that is, in the case of $\mathbb{R}$-trees. Let us assume then that the lemma is true for complete connected median space of rank $n-1$. Let us fix $a,b\in X$ and $0<\epsilon\leq\frac{d(a,b)}{n}$. Let us consider $x\in[a,b]$ such that $d(x,b)=n\epsilon$, such point $x$ exists as the space $X$ is connected. By Lemma [Lemma 22](#lemma_local_convexity){reference-type="ref" reference="lemma_local_convexity"}, there exists a halfspace $\mathfrak{h}\in\Delta(x,b)$ such that $d(\mathfrak{h}^c,b)\geq\frac{d(x,b)}{n}\geq \epsilon$ and $d(x,\mathfrak{h})=0$. For simplicity let us set $\tilde{a}:=\pi_{\hat{\mathfrak{h}}}(a)$. The hyperplane $\hat{\mathfrak{h}}$ being a median space of rank less than $n$, there exists then a family of pairwise transverse halfspaces $\mathfrak{h}_1,..,\mathfrak{h}_k\in\mathcal{H}(\tilde{a},x)$, where $k\leq n-1$ and such that for any $i\in\{1,..,k\}$, we have $d(\mathfrak{h}_i^c,x)\geq\epsilon$ and $d(\tilde{a},\displaystyle{\bigcap_{i=1}^k\mathfrak{h}_i})\geq d(\tilde{a},x)-\frac{n(n-1)}{2}\epsilon$. Note that each halfspace $\mathfrak{h}_i$ is transverse to $\mathfrak{h}$ as it is, with its complementary, the lift of a halfspace of $\hat{\mathfrak{h}}$ with non empty interior (see Proposition [Proposition 39](#lift_and_transversality){reference-type="ref" reference="lift_and_transversality"} below). As the point $x$ belongs to the interval $[a,b]$, any halfspace which separates $x$ from $\mathfrak{h}_i^c$ must also separates $b$ from $\mathfrak{h}_i^c$. Hence each $\mathfrak{h}_i^c$ is at distance greater than $\epsilon>0$ from $b$. It last to show that the intersection of $\mathfrak{h}$ with the $\mathfrak{h}_i$'s is at distance greater than $d(a,b)-\frac{n(n+1)}{2}\epsilon$ from $a$. For any $y\in\displaystyle{(\bigcap_{i=1}^k\mathfrak{h}_i)\cap\mathfrak{h}}$, the point $\tilde{a}=\pi_{\hat{\mathfrak{h}}}(a)=\pi_{\overline{\mathfrak{h}}}(a)$belongs to the interval $[a,y]$. Thus we have: $$\label{inequality_transverse_halfspaces_at_given_distance} d(a,y)= d(a,\tilde{a})+d(\tilde{a},y)\geq d(a,\tilde{a})+d(\tilde{a},x)-\frac{n(n-1)}{2}\epsilon$$ The equality $d(\tilde{a},x)+d(a,\tilde{a})=d(a,x)=d(a,b)-n\epsilon$ combined with Inequality ([\[inequality_transverse_halfspaces_at_given_distance\]](#inequality_transverse_halfspaces_at_given_distance){reference-type="ref" reference="inequality_transverse_halfspaces_at_given_distance"}) above yields $$d(a,y)\geq d(a,b)-n\epsilon-\frac{n(n-1)}{2}\epsilon=d(a,b)-\frac{n(n+1)}{2}\epsilon$$ ◻ **Lemma 35**. *Let $C$ be a complete connected median space of finite rank which is bounded. Let $\mathfrak{h}\subset X$ be a halfspace, then for any $\epsilon>0$ and $a\in \mathfrak{h}$ such that $d(a,\hat{\mathfrak{h}})\geq depth_c(\mathfrak{h})-\epsilon$, the convex hull $Conv(\{a\}\cup\hat{\mathfrak{h}})$ it at Gromov Hausdorff distance less than $(n(n+1)+1)\epsilon$ from $\mathfrak{h}$, where $n$ is the rank of $C$.* *Proof.* Let us choose a point $a\in\mathfrak{h}\cap C$ such that $d(a,\hat{\mathfrak{h}})\geq depth_C(\mathfrak{h})-\epsilon$. We set $C_{\mathfrak{h}}=Conv(\hat{\mathfrak{h}},a)$ and take a point $x\in\mathfrak{h}\cap C$ lying outside $C_{\mathfrak{h}}$. We consider its projections into $C_{\mathfrak{h}}$ and $[a,\pi_{\hat{\mathfrak{h}}}(a)]$ that we denote by $x_{C_{\mathfrak{h}}}:=\pi_{C_{\mathfrak{h}}}(x)$ and $\tilde{x}:=m(x,a,\pi_{\hat{\mathfrak{h}}}(a))$ respectively (See Figure [2](#Figure_compactness_theorem){reference-type="ref" reference="Figure_compactness_theorem"}) .\ {reference-type="ref" reference="depth_hyperplane"}](proof_theorem_compactness.jpeg){#Figure_compactness_theorem} Let us first show that $d(\tilde{x},\hat{\mathfrak{h}})=d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}})$. As the interval $[\pi_{\hat{\mathfrak{h}}}(a),a]$ lies in $C_{\mathfrak{h}}$, we have: $$\tilde{x}=m(x,a,\pi_{\hat{\mathfrak{h}}}(a))=\pi_{[\pi_{\hat{\mathfrak{h}}}(a),a]}(x)=\pi_{[\pi_{\hat{\mathfrak{h}}}(a),a]}(\pi_{C_{\mathfrak{h}}}(x))=\pi_{[\pi_{\hat{\mathfrak{h}}}(a),a]}(x_{C_{\mathfrak{h}}})$$ Hence, any halfspace separating $\tilde{x}$ from $\hat{\mathfrak{h}}$, separates $a$ from $\pi_{\hat{\mathfrak{h}}}(a)$, therefore it must separates $x_{C_{\mathfrak{h}}}$ from $\hat{\mathfrak{h}}$ as well by Lemma [Lemma 16](#constraint_on_the_projection){reference-type="ref" reference="constraint_on_the_projection"}. In the other hand, note that any halfspace separating any point in $C_{\mathfrak{h}}$ from $\hat{\mathfrak{h}}$ must separates the point $a$ from $\hat{\mathfrak{h}}$ as the convex subset $C_{\mathfrak{h}}$ is the convex hull of $\hat{\mathfrak{h}}\cup\{a\}$. Hence by Lemma [Lemma 16](#constraint_on_the_projection){reference-type="ref" reference="constraint_on_the_projection"}, any halfspace separating $x_{C_{\mathfrak{h}}}$ from $\hat{\mathfrak{h}}$ separates also $\tilde{x}$ from $\hat{\mathfrak{h}}$. Therefore, the two walls intervals $\mathcal{W} (\tilde{x},\hat{\mathfrak{h}})$ and $\mathcal{W} (x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}})$ coincide, which implies the equality $d(\tilde{x},\hat{\mathfrak{h}})=d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}})$. We deduce then the following: $$\begin{aligned} d(x,x_{C_{\mathfrak{h}}})&=& d(x,\hat{\mathfrak{h}})-d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}})\\ &\leq & depth_C(H)-d(\tilde{x},\hat{\mathfrak{h}})\\ &\leq & d(a,\hat{\mathfrak{h}})+\epsilon -d(\tilde{x},\hat{\mathfrak{h}})\end{aligned}$$ As $\tilde{x}$ lies in the interval $[a,\pi_{\mathfrak{h}}(a)]$, then its projection into $\hat{\mathfrak{h}}$ is precisely the point $\pi_\mathfrak{h}(a)$. Hence $d(a,\hat{\mathfrak{h}})=d(\tilde{x},\pi_{\mathfrak{h}}(a))$. Replacing the latter in the inequality above, we get: $$\label{distance_x_from_C_H} d(x,x_{C_{\mathfrak{h}}})\leq d(a,\tilde{x})+\epsilon$$ If the distance between $x$ and $C_{\mathfrak{h}}$ is less than $(n+1)\epsilon$, then there is nothing to show. Let us assume then that $d(x,C_{\mathfrak{h}})\geq (n+1)\epsilon$, which by Inequality ([\[distance_x\_from_C\_H\]](#distance_x_from_C_H){reference-type="ref" reference="distance_x_from_C_H"}) above, implies also that $d(a,\tilde{x})\geq n\epsilon$. Hence by Lemma [Lemma 34](#transverse_halfspace_at_given_distance){reference-type="ref" reference="transverse_halfspace_at_given_distance"}, there exist two families of pairwise transverse halfspaces $\{\mathfrak{h}_1,..,\mathfrak{h}_p\}\subseteq \mathcal{H}(\tilde{x},a)$ and $\{\mathfrak{h}_1',..,\mathfrak{h}_q'\}\subseteq \mathcal{H}(x_{C_{\mathfrak{h}}},x)$ such that the halfspaces $\mathfrak{h}_i^c$ and $\mathfrak{h}_j'^c$ are of depth bigger than $\epsilon$ and verify the following: $$d(\tilde{x},\displaystyle{\bigcap_{i=1}^p\mathfrak{h}_i})\geq d(\tilde{x},a)-\frac{n(n+1)}{2}\epsilon \ \ \ \text{and}\ \ \ d(x_{C_{\mathfrak{h}}},\displaystyle{\bigcap_{i=1}^q\mathfrak{h}_i})\geq d(x_{C_{\mathfrak{h}}},x)-\frac{n(n+1)}{2}\epsilon.$$ By the maximality of the set $\mathcal{H}_\epsilon$, the halfspaces $\mathfrak{h}_i^c$ and $\mathfrak{h}_j'^c$ are not disjoint for any $i\in\{1,..p\}$ and $j\in\{1,..,q\}$. In the other hand, any halfspace $\mathfrak{h}$ in $\Delta(\hat{\mathfrak{h}},\tilde{x})=\Delta(\hat{\mathfrak{h}},x_{C_\mathfrak{h}})$ contains the points $a$ and $x$ (note that they do not not necessarily contain the halfspaces $\mathfrak{h}_i$ and $\mathfrak{h}_j$). Hence the intersection $\displaystyle{(\bigcap_{\mathfrak{h}'\in\Delta(\hat{\mathfrak{h}},\tilde{x})}\mathfrak{h}')}$ contains the interval $[a,x]$. Therefore by Helly's Theorem [Theorem 6](#helly_theorem){reference-type="ref" reference="helly_theorem"}, the intersection $\displaystyle{(\bigcap_{\mathfrak{h}'\in\Delta(\hat{\mathfrak{h}},\tilde{x})}\mathfrak{h}')\cap(\bigcap_{i=1}^p\mathfrak{h}_i^c)\cap(\bigcap_{j=1}^q\mathfrak{h}_j'^c) }$ is not empty. Let us consider a point $y$ in the latter intersection and let $y_1$ and $y_2$ be its projections into the interval $[x_{C_{\mathfrak{h}}},x]$ and $[\tilde{x},a]$ respectively. We claim the following: $$\label{second_inequality} d(y,\hat{\mathfrak{h}})\geq d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}}) + d(x_{C_{\mathfrak{h}}},y_1) + d(\tilde{x},y_2)$$ Indeed, by construction we have the following inclusion: $$\mathcal{W}(y_1,x_{C_{\mathfrak{h}}})\cup\mathcal{W}(y_2,\tilde{x})\cup \mathcal{W}(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}})\subseteq \mathcal{W}(y,\hat{\mathfrak{h}})$$ In the other hand, all the wall intervals arising on the left hand of the inclusion are disjoints, therefore we get: $$\begin{aligned} \mu(\mathcal{W}(y_1,x_{C_{\mathfrak{h}}})\cup\mathcal{W}(y_2,\tilde{x})\cup \mathcal{W}(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}}))&=& \mu(\mathcal{W}(y_1,x_{C_{\mathfrak{h}}}))+\mu(\mathcal{W}(y_2,\tilde{x}))+ \mu(\mathcal{W}(x_{C_{\mathfrak{h}}}),\hat{\mathfrak{h}}))\\ &=& d(y_1,x_{C_{\mathfrak{h}}})+d(y_2,\tilde{x})+d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}})\\ &\leq & \mu(\mathcal{W}(y,\hat{\mathfrak{h}}))=d(y,\hat{\mathfrak{h}})\\ \end{aligned}$$ Having the inequality [\[second_inequality\]](#second_inequality){reference-type="ref" reference="second_inequality"} in hand, we get: $$\begin{aligned} depth_C(H)&\geq & d(y,\hat{\mathfrak{h}})\\ &\geq & d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}}) + d(y_1,x_{C_{\mathfrak{h}}})+d(y_2,\tilde{x})\\ &\geq & d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}}) + d(x_{C_{\mathfrak{h}}},x) -\frac{n(n+1)}{2}\epsilon + d(\tilde{x},a)-\frac{n(n+1)}{2}\epsilon \\\end{aligned}$$ As $d(x_{C_{\mathfrak{h}}},\hat{\mathfrak{h}})=d(\tilde{x},\hat{\mathfrak{h}})$, we get: $$depth_C(H) \geq d(\tilde{x},\hat{\mathfrak{h}})+d(\tilde{x},a)+d(x_{C_{\mathfrak{h}}},x)-n(n+1)\epsilon=d(\hat{\mathfrak{h}},a)+d(x_{C_{\mathfrak{h}}},x)-n(n+1)\epsilon$$ We deduce then the following: $$d(x_{C_{\mathfrak{h}}},x)\leq depth_C(H)-d(\hat{\mathfrak{h}},a) +n(n+1)\epsilon \leq (n(n+1)+1)\epsilon$$ Which finishes the proof. ◻ *Proof of Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"}.* The implication $1 \Rightarrow 2$ is obvious. Let us first show the implication $2 \Rightarrow 3$. For a fixed $\epsilon>0$ there exist $x_0,x_1,...,x_{n_\epsilon}\in C$ such that the subset $C$ is at Gromov-Hausdorff distance less than $\epsilon$ from $\displaystyle{\bigcup_{i=1}^{n_\epsilon}[x_0,x_i]}$. Let $\mathfrak{h}\in \mathcal{H}(X)$ be a halfspace transverse to $C$. If $\mathfrak{h}$ does not contain any of the $x_i$, then it must be of depth less than $\epsilon$ in $C$. Thus any halfspace transverse to $C$ of depth bigger than $\epsilon$ must separate $x_0$ from some $x_i$. Therefore there is only finitely many pairwise disjoint halfspaces transverse to $C$ and of depth bigger than $\epsilon$. We now prove the implication $3\Rightarrow 1$. The idea is to show that, under the assumptions of Statement $(3)$, the set $C$ is the Gromov-Hausdorff limit of a sequence of compact subsets. By Proposition [Proposition 19](#convex_hull_ball){reference-type="ref" reference="convex_hull_ball"}, the convex hull of $C$ is bounded and Condition $(3)$ remains true as a halfspace is transverse to a set if and only if it is transverse to its convex hull. Hence, without loss of generality we can assume that $C$ is convex. Let us fix $\epsilon$ and consider a maximal family $\mathcal{H}_\epsilon$ of pairwise disjoint halfspaces transverse to $C$ and of depth bigger than $\epsilon$ in $C$. As we are considering only halfspaces which are transverse to the convex subset $C$, we may forget about the ambient space $X$ and assume that each halfspace is a halfspace of $C$, up to taking the intersection with the latter (see Propositions [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"} and [Proposition 38](#induced_hyperplane){reference-type="ref" reference="induced_hyperplane"}). Here we mean also by the maximality of $\mathcal{H}_\epsilon$ that there is no two disjoint halfspaces of depth bigger than $\epsilon$ in $C$ contained inside some halfspace in $\mathcal{H}_\epsilon$. By assumption Condition $(3)$ holds, hence this family of halfspaces is finite. Let us set $C_{\epsilon}=Conv(\displaystyle{\bigcup_{\mathfrak{h}\in \mathcal{H}_\epsilon} \hat{\mathfrak{h}}} )$ and first show that it is at Gromov-Hausdorff distance less than $n\epsilon$ from $C \ \backslash \displaystyle{\bigcup_{\mathfrak{h}\in\mathcal{H}_\epsilon}\mathfrak{h}}$. Let $x\in C$ be a point lying outside all of the halfspace $\mathfrak{h}\in\mathcal{H}_\epsilon$. Note then that any halfspace separating $x$ from $C_\epsilon$ is disjoint from any halfspace in $\mathcal{H}_\epsilon$. Hence by the maximality of the family $\mathcal{H}_\epsilon$, any halfspace separating $x$ from $C_\epsilon$ is of depth less than $\epsilon$ in $C$. Therefore, we conclude by Lemma [Lemma 22](#lemma_local_convexity){reference-type="ref" reference="lemma_local_convexity"} that the point $x$ is at distance at most $n\epsilon$ from $C_\epsilon$. For each $\mathfrak{h}\in \mathcal{H}_\epsilon$, we choose a point $a_\mathfrak{h}\in \mathfrak{h}$ such that $d(a_\mathfrak{h},\hat{\mathfrak{h}})\geq depth_C(\mathfrak{h})-\epsilon$. We set $C_\mathfrak{h}:=Conv(\{a_\mathfrak{h}\}\cup \hat{\mathfrak{h}})$ and use Lemma [Lemma 35](#depth_hyperplane){reference-type="ref" reference="depth_hyperplane"} to conclude that it is at Gromov Hausdorff distance less than $(n(n+1)+1)\epsilon$ from $\mathfrak{h}$. Let us set $\tilde{C}_\epsilon=\displaystyle{\bigcup_{\mathfrak{h}\in\mathcal{H}_\epsilon}C_{\mathfrak{h}}}\cup C_\epsilon$. We have shown that $\tilde{C}_\epsilon$ is at Gromov-Hausdorff distance less than $(n(n+1)+1)\epsilon$ from $C$. It last to show that it is compact. We proceed by induction on the rank of $C$. Note that when the rank of $C$ is $1$, then the hyperplane correspond to a point. Hence by Proposition [Proposition 31](#convex_hull_compact_convex){reference-type="ref" reference="convex_hull_compact_convex"}, the subset $\tilde{C}$ is compact as it is a finite union of the convex hull of compact subsets. Let us assume now that the rank of $C$ is equal $n$ and that the implication $3\Rightarrow 1$ is true for median space of rank less or equal $n-1$. As we assumed that the condition $(3)$ to be true, it is verified by each hyperplane $\hat{\mathfrak{h}}$. Therefore each hyperplane $\hat{\mathfrak{h}}$ is compact. Therefore we conclude by Proposition [Proposition 31](#convex_hull_compact_convex){reference-type="ref" reference="convex_hull_compact_convex"} that $\tilde{C}_\epsilon$ is compact, which finishes the proof. ◻ For the general case when the rank is infinite, it is harder to manipulate halfspaces as they may all be dense in the space and even if it is the case, one can no longer use an argument by induction on the rank of the space. Let us give a criterion of local compactness in the infinite rank case: **Proposition 36**. *Let $X$ be a complete median space with compact intervals and let $K\subseteq X$ be a closed subset. If the outer measure of the set of transverse halfspaces to $K$ is finite,\ $i.e\ \bar{\mu}(\mathcal{H}(K))<+\infty$, where $\mu$ is the canonical measure associated to $\mathcal{H}(X)$, then $K$ is compact.* Let us first make the following remarks: **Remark 37**. - *If $X$ be a complete median space and $K\subseteq X$ a subset such that $\bar{\mu}(\mathcal{H}(K))<+\infty$, then the convex hull of $K$ is bounded. This is first due to the fact that the set of halfspaces which are transverse to $K$ is the same as the set of halfspaces which are transverse to the convex hull of $K$. In the other hand, having a sequence of points which is unbounded give rise to a sequence of wall interval with an arbitrarily big measure.* - *The converse of Proposition [Remark 37](#finite_measure_convex_hull){reference-type="ref" reference="finite_measure_convex_hull"} is false, even in the finite rank case. One may consider a star like tree obtained from the concatenation of the intervals $[0,\frac{1}{n}]$ at $\{0\}$.* *Proof of Proposition [Proposition 36](#local_compactness_criterion_infinite_rank_case){reference-type="ref" reference="local_compactness_criterion_infinite_rank_case"}.* Let $K\subseteq X$ be a closed subset such that $\bar{\mu}(\mathcal{H}(K))=M$. By Remark [Remark 37](#finite_measure_convex_hull){reference-type="ref" reference="finite_measure_convex_hull"}, there is no loss of generality if we consider the closure of the convex hull of $K$. Let us first remark that for any $x,y\in X$, the set of halfspaces which separates $x$ from $y$ is exactly the same as the set of halfspaces separating $\pi_{K}(x)$ from $\pi_{K}(y)$. Hence, for any $\epsilon >0$, there exist $x_1,y_1,..,x_{n_\epsilon},y_{n_\epsilon}\in K$ such that $\mathcal{W}(x_i,y_i)$ and $\mathcal{W}(x_j,y_j)$ are disjoint for any $i\neq j$ and : $$\mu(\displaystyle{\bigcup_{i=1}^{n_\epsilon} \mathcal{W} (x_i,y_i)})=\mu(\displaystyle{\sum_{i=1}^{n_\epsilon}d(x_i,y_i)})\geq M - \epsilon$$ Let consider the convex hull of all the point $C_\epsilon=Conv(\{x_1,y_1,..,x_{n_\epsilon},y_{n_\epsilon}\})$ and a point $x\in K$. Then any halfspace separates $x$ from $C_\epsilon$ if and only if it separates $x$ from all the points $x_i$ and $y_i$. Thus, we must have $\mu(\mathcal{W} (x,C_\epsilon)) \leq \epsilon$. Let us then consider a sequence $C_{\frac{1}{n}}$ defined as above. We may assume that the sequence $(C_{\frac{1}{n}})_{n\in\mathbb{N}}$ is ascending with respect to the inclusion. By Proposition [Corollary 32](#convex_hull_finite_subset_compact){reference-type="ref" reference="convex_hull_finite_subset_compact"}, each subset $C_\frac{1}{n}$ is compact. In the other hand, the sequence $(C_{\frac{1}{n}})_{n\in\mathbb{N}}$ converges with respect to the Gromov-Hausdorff metric to $C=\displaystyle{\bigcup_{n\in\mathbb{N}}^i C_\frac{1}{n}}$. As each $C_\frac{1}{n}$ is compact, the subset $C$ is totally bounded. Thus its closure is a compact which contains the closed subset $K$. We conclude that the subset $K$ is also compact. ◻ # Transitive actions on median spaces of finite rank and local compactness ## Trace of halfspaces on convex sets Throughout this section $X$ is a complete connected median space of finite rank. Let us denote the set of hyperplanes of $X$ by $\hat{\mathcal{H}}(X)$ and deduce from Proposition [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"} that any hyperplane in a closed convex subset is induced from a hyperplane of the ambient space as stated in the following: **Proposition 38**. *Let $C\subseteq X$ be a closed convex subset. We have then: $$\hat{\mathcal{H}} (C)=\{\hat{\mathfrak{h}}\cap C \ |\ \ \ \hat{\mathfrak{h}}\in\hat{\mathcal{H}} (X) \ \text{and separates two points of} \ C\}$$* *Proof.* By Proposition [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"}, it is enough to show that for any halfspace $\mathfrak{h}$ which is transverse to $C$ we have $\overline{\mathfrak{h}\cap C}=\bar{\mathfrak{h}}\cap C$. The inclusion $\overline{\mathfrak{h}\cap C}\subseteq\bar{\mathfrak{h}}\cap C$ is obvious. For the converse, let us take a sequence $(x_n)_{n\in\mathbb{N}}\subseteq\mathfrak{h}$ which converges to a point $x\in C$. We denote by $y_n:=\pi_{C}(x_n)$ their projection into the closed convex set $C$. As $\mathfrak{h}\cap C$ is not empty, we get that $(y_n)_{n\in\mathbb{N}}\subseteq\mathfrak{h}\cap C$. Due to the continuity of the projection, the sequence $(y_n)_{n\in\mathbb{N}}\subseteq\mathfrak{h}$ converges to $\pi_{C}(x)=x$. ◻ Let $C\subseteq X$ be a convex subset and $\mathfrak{h}\subseteq X$ a halfspace. We call the ***trace*** of the hyperplane $\hat{\mathfrak{h}}$ on $C$ the intersection $\hat{\mathfrak{h}}\cap C$. **Proposition 39**. *Let $C\subseteq X$ be a closed convex subset and $\mathfrak{h}\in\mathcal{H}(X)$ be a halfspace. Then the lift of any halfspace $T$ of $\hat{\mathfrak{h}}\cap C$ to the ambient space $X$ is a halfspace transverse to $\mathfrak{h}$ assuming that $T$ and its complementary are of non empty interior inside $\hat{\mathfrak{h}}\cap C$.* *Proof.* Let $T\in\mathcal{H}(\hat{\mathfrak{h}}\cap C)$ be an halfpsace of the trace of the hyperplane $\hat{\mathfrak{h}}$ on the convex subset $C$. To show that the lift of $T$ to $X$ is transverse to $\mathfrak{h}$, it is enough to show that there exist points in $\mathfrak{h}$ and $\mathfrak{h}^c$ which projects into $T$ and another ones which projects into $T^c\cap C\cap \hat{\mathfrak{h}}$. The halfspace $T$ being inside the trace of the hyperplane $\hat{\mathfrak{h}}$, for any point $x$ in the interior of $T$ or its complementary inside $\hat{\mathfrak{h}}\cap C$, there exists a sequence $(x_n)_{n\in\mathbb{N}}$ inside $\mathfrak{h}\cap C$ which converge to $x$. Taking the index $n$ big enough, the point $x_n$ projects inside a small neighborhood of $x$ in $T$. ◻ **Remark 40**. *The proposition will no longer be true if we drop the assumption on $T$ and its complementary being both of non empty interior. One may consider for instance the following subspace of $(\mathbb{R}^2,\ell^1)$: $$X=\{(x,y)\in \mathbb{R}^2 | x\leq 0\}\cup \{(x,y)\in \mathbb{R}^2 | y-x\geq 0\}$$ We take the convex subset $C$ to be the half line $x=0$ and $T$, a halfspace of $C$, to be the trace of the halfspace $\mathfrak{h}:=\{(x,y)\in X\ | \ y>0\}$. The convex subset $C$ is also the hyperplane which bounds the halfspace defined by the inequality $x>0$. The lift of $(0,0)$, the complement of $T$ inside $C$, is not transverse to $\mathfrak{h}$ as it is disjoint from it.* **Proposition 41**. *Let $C\subseteq X$ be a convex subset which isometric to an $n$-cube $([-\epsilon,\epsilon],\ell^1)$ where $n$ is the rank of X. Let $\mathfrak{h}\in \mathcal{H}(X)$ be a halfspace in $X$ which is disjoint from $C$. If the intersection $\hat{\mathfrak{h}}\cap C$ is not empty and does not lie in the corner of $C$, then there exist $\mathfrak{h}_1,\mathfrak{h}_2\in \mathcal{H}(X)$ such that they constitute with the halfspace $\mathfrak{h}$ a facing triple in $X$ and $\hat{\mathfrak{h}}\cap \hat{\mathfrak{h}_1}\cap \hat{\mathfrak{h}_2}\neq \emptyset$.* *Proof.* Let us identify $C$ with $[-\epsilon,\epsilon]^n$. By Proposition [Proposition 39](#lift_and_transversality){reference-type="ref" reference="lift_and_transversality"}, the rank of $\hat{\mathfrak{h}}\cap C$ is smaller than $n$. Hence the latter is contained in a hyperplane of $C$, let us say the hyperplane given by the equation $x_1=0$. Then the lift to the ambient space of the two halfspaces $\mathfrak{ H_1}=\{(x_1,..,x_n)\in C | x_1>0 \}$ and $\mathfrak{ H_2}=\{(x_1,..,x_n)\in C | x_1<0 \}$ give us the desired halfspaces. Indeed, Lemma [Lemma 16](#constraint_on_the_projection){reference-type="ref" reference="constraint_on_the_projection"} tell us that $\overline{\mathfrak{h}}$ projects into $\overline{\mathfrak{h}}\cap C=\hat{\mathfrak{h}}\cap C$, which is outside $\mathfrak{ H_1}$ and $\mathfrak{ H_2}$. ◻ Finally we deduce the following corollary in the case where the median space admits a transitive action: **Corollary 42**. *Let $X$ be a complete connected median space of rank $n$ which admits a transitive action. Let $x\in X$ and let $([-\epsilon,\epsilon]^n,\ell^1)\cong C\subseteq X$ be an isometrically embedded $n$-cube centred at $x$. If $\mathcal{H}_{x}(X)$ does not contain a facing triple, then it coincides with $\mathcal{H}_x(C)$.* *Proof.* By Proposition [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"}, it is enough to show that any halfspace in $\mathcal{H}_x(X)$ is transverse to the $n$-cube $C$. Let us consider a $\mathfrak{h}\in \mathcal{H}_x(X)$ branched at $x$. The set $\mathcal{H}_x(X)$ does not contain a facing triple and the intersection $\hat{\mathfrak{h}}\cap C$ is not empty as both contain the point $x$, hence Proposition [Proposition 41](#trace_and_facing_triple){reference-type="ref" reference="trace_and_facing_triple"} implies then that the halfspace $\mathfrak{h}$ is necessarily transverse to $C$. ◻ ## Proof of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"} {#section_theorem_principal} #### Proof of the first claim of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"} : {#proof-of-the-first-claim-of-theorem-thm_principal} Let $X$ be a complete connected median space of finite rank which admits a transitive action. Let us fix a maximal pairwise transverse family of halfspaces $\mathcal{H}=\{\mathfrak{h}_1,..,\mathfrak{h}_n\}$ in $\mathcal{H}(X)$. Let us set the following $$\label{equality_lines} D_i:=\bigcap_{j\neq i}\hat{\mathfrak{h}}_j$$ Let us show that each $D_i$ is a strongly convex isometric embedding of an $\mathbb{R}$-tree. **Proposition 43**. *Each $D_i$, endowed with the induced metric of $X$, is a complete connected median space of rank $1$.* Let us first show the following lemma: **Lemma 44**. *Let $X$ be a complete connected median space of finite rank. We assume that there exist two transverse halfspaces $\mathfrak{h}_1,\mathfrak{h}_2\subseteq X$. Then there exist two transverse halfspaces such that both them and their complements are of non empty interior inside $X$.* *Proof.* Let us consider two transverse halfspaces $\mathfrak{h}_1,\mathfrak{h}_2\subseteq X$. By Remark [Remark 11](#closed_branched_halfspaces){reference-type="ref" reference="closed_branched_halfspaces"} and up to considering the complement, we may assume that they are both open in $X$. Let us consider a point $x\in \mathfrak{h}_1\cap \mathfrak{h}_2$. Let us set the following $x_i:=\pi_{\mathfrak{h}_i^c}(x)$ and $x_0:=\pi_{\mathfrak{h}_1^c\cap\mathfrak{h}_2^c}(x)$. Note that $x_0=\pi_{\mathfrak{h}_1^c}(x_2)=\pi_{\mathfrak{h}_2^c}(x_1)$ by Lemma [Lemma 16](#constraint_on_the_projection){reference-type="ref" reference="constraint_on_the_projection"}. The halfspaces $\mathfrak{h}_1$ and $\mathfrak{h}_2$ being both open, the points $x_1$ and $x_2$ are distinct from $x_0$. Let us set $C:=Conv([x_0,x_1]\cup[x_0,x_2])$ and set $\tilde{x}:=\pi_C(x)$. Note that $\pi_{\mathfrak{h}_i^c}(\tilde{x})=x_i$. As $[x_0,x_1]\cap[x_0,x_2]=\{x_0\}$, Proposition [Proposition 25](#ncube){reference-type="ref" reference="ncube"} implies that the interval $[x,x_0]$ is isometric to the $\ell^1$-product $[x,x_1]\times[x,x_2]$. Hence, the lift to $X$ of any halfspaces $\mathfrak{H}_1\in \mathcal{H}([x,x_1]$ and $\mathfrak{H}_2\in \mathcal{H}([x,x_2]$, such that $\mathfrak{H}_i$ and $\mathfrak{H}_i^c$ are of non empty interior in $[x,x_i]$, yields two transverse halfspaces such that both them and their complements are of non empty interior in $X$. ◻ *Proof of Proposition [Proposition 43](#Di_rank_1){reference-type="ref" reference="Di_rank_1"}.* By Helly's Theorem [Theorem 6](#helly_theorem){reference-type="ref" reference="helly_theorem"}, each $D_i$ is a non empty closed convex subset of $X$ and it intersects both $\mathfrak{h}_i$ and $\mathfrak{h}_i^c$. Hence, each $D_i$ is of rank bigger then or equal one. It is left to show that it is of rank smaller then two. If there exist two transverse halfspaces in $D_i$, Lemma [Lemma 44](#halfspace_non_empty_interior){reference-type="ref" reference="halfspace_non_empty_interior"} ensures that there is no loss of generality if we assume them to be with their complement inside $D_i$ of non empty interior. In the other hand, Proposition [Proposition 39](#lift_and_transversality){reference-type="ref" reference="lift_and_transversality"} implies that the lift of such halfspaces to the ambient space $X$ yields halfspaces which are transverse to each $\mathfrak{h}_j$ where $j\neq i$, which would contradict the maximality of the family $(\mathfrak{h}_i)$. ◻ Following the same argument of the proof of Proposition [Proposition 43](#Di_rank_1){reference-type="ref" reference="Di_rank_1"}, we note that for $i\neq j$, we have $D_i\cap D_j=\displaystyle{\bigcap_{k=1}^n \hat{\mathfrak{h}}_k}=\{a_0\}$ for some $a_0\in X$. In the following, we show that any point is the center of an isometrically embedded $n$-cube. **Lemma 45**. *Under the assumptions that the median space $X$ is complete and admits a transitive action, there exists $\epsilon>0$ such that any $x\in X$ is the center of an isometrically embedded convex $n$-cube $([-\epsilon,\epsilon]^n,\ell^1)$ centred at $x$.* *Proof.* By the transitivity assumption of the isometry group of $X$, it is enough to show the existence of an isometrically embedded $n$-cube in $X$. By Helly's Theorem [Theorem 6](#helly_theorem){reference-type="ref" reference="helly_theorem"}, the intersection $\displaystyle{\bigcap_{i=1}^n\mathfrak{h}_i}$ is not empty. Let us consider a point $a$ in the latter intersection. Again by Helly's Theorem, the intersections $\mathfrak{h}_i\cap D_i$ are not empty and do not contain $a_0$, hence the projection of $a$ into $D_i$ avoid $a_0$. Let us set $a_i:=\pi_{D_i}(a)$. By Proposition [Proposition 43](#Di_rank_1){reference-type="ref" reference="Di_rank_1"}, each interval $[a_0,a_i]$ is isometric to closed interval of $\mathbb{R}$. Let us denote by $C$ the convex hull between the $\mathbb{R}$-trees $D_i$. By Lemma [Proposition 25](#ncube){reference-type="ref" reference="ncube"}, the interval $[a_0,\pi_C(a)]$ is isometric to the $\ell^1$ product of the intervals $[a_0,a_i]$, thus completing the argument. ◻ Let us assume now that the set $\mathcal{H}_{a_0}(X)$ does not contain a pairwise disjoint triple of halfspaces. Remark that the transitivity assumption implies then that for any $x\in X$ the set $\mathcal{H}_x(X)$ does not contain such triple. **Proposition 46**. *Each $D_i$, as defined in the beginning of Subsction [5.2](#section_theorem_principal){reference-type="ref" reference="section_theorem_principal"} Equality ([\[equality_lines\]](#equality_lines){reference-type="ref" reference="equality_lines"}), is isometric to the real line.* *Proof.* By Proposition [Proposition 43](#Di_rank_1){reference-type="ref" reference="Di_rank_1"}, we already know that each $D_i$ is a closed convex subset of rank $1$, hence it is isometric to a complete connected $\mathbb{R}$-tree. Under the assumption that the set $\mathcal{H}_{a_0}(X)$ does not contain any facing triple, Proposition [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"} and Helly's Theorem [Theorem 6](#helly_theorem){reference-type="ref" reference="helly_theorem"} imply that no $D_i$ can contain a facing triple. Hence, each $D_i$ is isometric to an interval of the real line. To conclude that it is isometric to the real line, it is enough to show that any point in $D_i$ lies in the interior of an interval in $D_i$. Let us consider a point $x\in D_i$. By Lemma [Lemma 45](#embedded_cube){reference-type="ref" reference="embedded_cube"}, there exists an embedded $n$-cube $C\cong ]-\epsilon,\epsilon[^n$ in $X$ centred at the point $x$. Under the assumption that there is no facing triple in $\mathcal{H}_{x}(X)$, Corollary [Corollary 42](#no_facing_triple){reference-type="ref" reference="no_facing_triple"} implies that the halfspace $\mathfrak{h}_j$, for any $j\neq i$, are transverse to the $n$-cube. We have then $\displaystyle{(\bigcap_{j\neq i}\hat{\mathfrak{h}}_j)\cap C}\cong ]-\epsilon,\epsilon[ \subseteq D_i$. As $D_i$ identifies with an interval of $\mathbb{R}$, the latter intersection intersection is open in $D_i$, which finish the argument. ◻ Now, we have all the ingredients needed to prove the first part of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"}: **Proposition 47**. *The convex hull of the lines $D_i$ contains $X$ and is isometric to $(\mathbb{R}^n,\ell^1)$.* *Proof.* We set $C:=Conv(D_1\cup ...\cup D_n)$. The set $C$ is a closed subset of the complete median space $X$, as it is the convex hull of finitely many closed convex subsets, hence it is also complete. By Proposition [Proposition 23](#embed_lemm){reference-type="ref" reference="embed_lemm"}, it embeds isometrically, through the projections onto the $D_i$'s, as a closed subset of $\displaystyle{\prod_{i=1}^n D_i \cong \mathbb{R}^n}$. It is enough to show that the embedding is open to conclude that it is surjective. In our way proving that, we prove also that $C$ contains $X$. Let us take a point $x\in X$ and consider the family $\mathfrak{h}_{i,l}:=\pi_{D_i}^{-1}(]-\infty,\pi_{D_i}(\widetilde{x})[),\mathfrak{h}_{i,r}:=\pi_{D_i}^{-1}([\pi_{D_i}(\widetilde{x}),+\infty[)$, where $\widetilde{x}:=\pi_C(x)$ and each $D_i$ is identified with $\mathbb{R}$. By Proposition [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"}, each $\mathfrak{h}_{i,l}$ and $\mathfrak{h}_{i,r}$ is a halfspace of $X$. By Lemma [Lemma 45](#embedded_cube){reference-type="ref" reference="embedded_cube"}, there exists an $n$-cube centered at $\tilde{x}$. Thus by Corollary [Corollary 42](#no_facing_triple){reference-type="ref" reference="no_facing_triple"}, the family of halfspaces $\{\mathfrak{h}_{1,l},\mathfrak{h}_{1,r},..,\mathfrak{h}_{n,l},\mathfrak{h}_{n,r}\}$ and their complementary in $X$ constitutes all the elements of $\mathcal{H}_{\tilde{x}}$. In one hand, this implies that the projection map $(\pi_{D_1},..,\pi_{D_n})$ is open. In the other hand, by Lemma [Lemma 20](#strenghtening_separation_theorem){reference-type="ref" reference="strenghtening_separation_theorem"} we get : $$\displaystyle{\bigcap_{i=1}^n (\mathfrak{h}_{i,l}\cap\mathfrak{h}_{i,r})}\ =\ \{\widetilde{x}\}$$ As the projections onto each $D_i$ factor through the projection onto $C$, that is $\pi_{D_i}(x)=\pi_{D_i}(\pi_C(x))=\pi_{D_i}(\tilde{x})$, the point $x$ lie in $(\mathfrak{h}_{i,l}\cap\mathfrak{h}_{i,r})$ for any $i\in\{1,..,n\}$. Hence, we get $x=\tilde{x}$, which proves that $C=X$ and complete the proof. ◻ #### Proof of the second part of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"} {#proof-of-the-second-part-of-theorem-thm_principal} The idea of the proof is to show that under the assumption of the existence of a facing triple in $\mathcal{H}_{a_0}$ and a transitive action on $X$, there exist infinitely many pairwise disjoint halfspaces with depth uniformly bounded below inside any ball centred at $a_0$. Let us first show that any halfpaces in $\mathcal{H}_{a_0}$ is of positive depth inside any ball centred at $a_0$. **Proposition 48**. *Let $X$ be a complete connected median space of finite rank which admits a transitive action. Then for any halfspace $\mathfrak{h}\in\mathcal{H}_{a_0}$ and $r>0$, we have $depth_{B(a_0,r)}(\mathfrak{h})>0$.* *Proof.* Let $\mathfrak{h}\in\mathcal{H}_{a_0}$ be a halfspace containing the point $a_0$ in its hyperplane. If the halfspace $\mathfrak{h}$ is open, its depth inside any ball centered at $a_0$ is positive. Let us assume then that the halfspace $\mathfrak{h}$ is closed. By Proposition [Proposition 25](#ncube){reference-type="ref" reference="ncube"}, there exists an isometrically embedded $n$-cube $C\cong[-\epsilon,\epsilon]^n$ where $n$ is the rank of the space $X$. The action being transitive, we can assume that the $n$-cube is centred at $a_0$. If the halfspace $\mathfrak{h}$ is transverse to the $n$-cube $C$, then it gives rise to a halfspace of $C$ which contains $a_0$ and which is of positive depth inside $C$. If $\mathfrak{h}$ is not transverse to $C$, then it will contain it. Let us consider then the trace of the hyperplane $\hat{\mathfrak{h}}$ which bounds the halfspace $\mathfrak{h}$, on the $n$-cube, that is, its intersection with the latter. Let us denote it by $\hat{C}$. It is a convex subset which contains $a_0$. The rank of $X$ being $n$, Proposition [Proposition 39](#lift_and_transversality){reference-type="ref" reference="lift_and_transversality"} implies that the convex subset $\hat{C}$ is of rank less than $n-1$. Again, there exist then points inside $C$ which are at positive distance from $\hat{C}$, hence from $\hat{\mathfrak{h}}$. ◻ *Proof of the second part of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"}.* Let us fix a point $a_0\in X$ and show that under the assumptions of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"}, any neighbourhood of $a_0$ contains infinitely many disjoint halfspaces of depth bigger than some $\epsilon>0$ inside the latter neighbourhood. We will conclude then by Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"} that the space is not locally compact. The action of $Isom(X)$ being transitive, by Lemma [Lemma 45](#embedded_cube){reference-type="ref" reference="embedded_cube"}, there exists an isometrically embedded $n$-cube $C\cong([-\eta,\eta]^n,\ell^1)$ centred at $a_0$. Let us parametrize the $n$-cube by $x_1,..,x_n$ and identify $a_0$ with $(0,..,0)$. Let $\mathfrak{h}_1,\mathfrak{h}_2,\mathfrak{h}_3\in\mathcal{H}_{a_0}$ be a facing triple. By Proposition [Proposition 48](#depth_transitive_assumption){reference-type="ref" reference="depth_transitive_assumption"}, each of the $\mathfrak{h}_i$ is of positive depth inside any ball centred at $x_0$. For any point $x$ inside the $n$-cube, there exists an isometry $g\in Isom(X)$ which maps $a_0$ to $x$. At least, the image of one of the $\mathfrak{h}_i$'s by the isometry $g$ is disjoint from the $n$-cube. Hence, for any $r\in]0,\eta[$ there exists a halfspace $\mathfrak{h}_r\in\mathcal{H}_{(r,..,r)}$ which is disjoint from the $n$-cube and of depth bigger than some uniform $\epsilon$. The trace of the hyperplane $\hat{\mathfrak{h}}_r$ on the $n$-cube $C$ is a convex subset of rank less than $n$ containing the point $(r,..,r)$. Hence, it is contained in a hyperplane of the $n$-cube $C$ given by an equation of the form $x_{i_r}=r$. Thus, there exists an infinite subset $I\subseteq ]0,\eta[$ such that for any $r_1,r_2\in I$, the trace of the hyperplane $\hat{\mathfrak{h}}_{r_1}$ on the $n$-cube is disjoint from the trace of the hyperplane $\hat{\mathfrak{h}}_{r_2}$. By Lemma [Lemma 16](#constraint_on_the_projection){reference-type="ref" reference="constraint_on_the_projection"}, if the halfspaces $\mathfrak{h}_{r_1}$ and $\mathfrak{h}_{r_2}$ intersect, then projection of their intersection into the $n$-cube $C$ lies inside $\hat{\mathfrak{h}}_{r_1}\cap \hat{\mathfrak{h}}_{r_2}$. Hence, for any such $r_1$ and $r_2$, the halfspaces $\mathfrak{h}_{r_1}$ and $\mathfrak{h}_{r_2}$ are disjoint. Therefore, the set $(H_r)_{r\in I}$ give us the desired infinite family of pairwise disjoint halfspaces of depth bigger than some uniform $\epsilon >0$. ◻ ## A comment on a weaker assumption in Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"} {#a-comment-on-a-weaker-assumption-in-theorem-thm_principal} One may weaken the assumption in Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"} and assume the action of $Isom(X)$ to be topologically transitive instead of transitive, that is, it admits a dense orbit. The same strategy works, we divide the theorem into condition on the existence of a facing triple in the neighbourhood of a points. Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"} adapts into the following: **Theorem 49**. *Let $X$ be a complete connected median space of rank $n$ which admits a topologically transitive action. If there exist $x\in X$ and $\epsilon>0$ such that the set of halfspaces $\mathcal{H}(B(x,\epsilon))$ contains no facing triple then the space $X$ is isomorphic to $(\mathbb{R}^n,\ell^1)$.* The second part reformulates into the following: **Proposition 50**. *Let $X$ be a complete connected median space of rank $n$ which admits a topologically transitive action. If there exists $x\in X$ such that for any $\epsilon>0$, the set of halfspaces $\mathcal{H}(B(x,\epsilon))$ contains a facing triple, then the space $X$ is not locally compact.* The proof of Proposition [Proposition 50](#prop1_bis){reference-type="ref" reference="prop1_bis"} follows exactly the path of the proof of the second part of Theorem [Theorem 4](#thm_principal){reference-type="ref" reference="thm_principal"}. Regarding the proof of Theorem [Theorem 49](#thm_principal_bis){reference-type="ref" reference="thm_principal_bis"}. One shows that under its assumption, we obtain the same result of Corollary [Corollary 42](#no_facing_triple){reference-type="ref" reference="no_facing_triple"}, that is, the space $X$ is locally isometrically modelled on $(\mathbb{R}^n,\ell^1)$. The rest of the arguments follow more or less the same path. # Actions with discrete orbit The aim of this section is to show that an isometric action which is Roller non elementary and Roller minimal on a complete locally compact median space of finite rank has discrete orbits. Let us first recall in the following subsection the nomenclature concerning isometric actions on the Roller compactification of a median space and some properties that we will be needing for the proof of Theorem [Theorem 1](#Theorem_discrete_orbit){reference-type="ref" reference="Theorem_discrete_orbit"}. ## Roller Boundary {#Roller_boundary} In [@Fior_median_property], the author extended the notion of Roller boundary of a CAT$(0)$ cube complex to the case of median spaces. When the median space $X$ is complete, locally convex and has compact interval, the latter boundary compactifies $X$. We call it ***the Roller compactification*** and we denote it by $\overline{X}$. We denote the ***Roller Boundary*** by $\partial X=\overline{X}\backslash X$. The metric on $X$ induces a stratifications of $\overline{X}$ into components, one component corresponds to the metric completion of $X$ and the remaining ones are contained in the Roller Boundary. Each component is endowed with a median metric given by the measure on the set of halfspaces of $X$. Any action of a group $G$ on $X$ extends to an action on $\overline{X}$. We say that the action is : - ***Roller Elementary*** if it has a finite orbit in $\overline{X}$ - ***Roller minimal*** if it does not fix a component of $\partial X$. We say that a median space is ***irreducible*** if it does not split as the $\ell^1$-product of two median space. We say that two halfspaces are strongly separated if they are disjoint and there is no halfspace which is transverse to both. The following lemma is borrowed from [@Fior_superrigidity] (Proposition 2.10, Proposition 2.12 and Lemma 4.1 therein) **Lemma 51**. *Let $X$ be an irreducible median space of finite rank. Let us assume that the action of $Isom(X)$ is Roller non elementary and Roller minimal. Then there exists a strongly separated facing triple $\mathfrak{h}_1,\mathfrak{h}_2,\mathfrak{h}_3\in \mathcal{H}(X)$ of positive depth, and a point $c\in X$ such that for any $x_i\in \mathfrak{h}_i$, we have $m(x_1,x_2,x_3)=c$. Note that the point $c$ lies in $\mathfrak{h}_1^c\cap\mathfrak{h}_2^c\cap\mathfrak{h}_3^c$.* ## Proof of Theorem [Theorem 1](#Theorem_discrete_orbit){reference-type="ref" reference="Theorem_discrete_orbit"} {#proof-of-theorem-theorem_discrete_orbit} Any complete connected median space $X$ is geodesic (see Lemma 13.3.2 [@Bowd_median-algebras]), hence by Hopf-Rinow Theorem, showing that a median space $X$ is not locally compact is equivalent to finding a closed ball which is not compact. **Lemma 52**. *Let $X$ be a median algebra of finite rank and let $\mathcal{H}\subseteq \mathcal{H}(X)$ be an infinite subset of halfspaces such that any $\mathfrak{h}_1,\mathfrak{h}_2\in \mathcal{H}$ are either transverse or disjoint. Then there exists an infinite subset $\mathcal{H}' \subseteq \mathcal{H}$ of pairwise disjoint halfspaces.* *Proof.* Let us consider the dual graph $\Gamma$ of $\mathcal{H}$, that is, the non oriented graph whose vertices are the halfspaces of $\mathcal{H}$ and two vertices are joined by and edge if the halfspaces labelling the vertices are transverse, i.e. $\Gamma:=(V,E)$ such that $V=\mathcal{H}$ and $(\mathfrak{h}_1,\mathfrak{h}_2)\in E$ if and only if $\mathfrak{h}_1$ and $\mathfrak{h}_2$ are transverse. Thus, finding an infinite family of pairwise disjoint halfspaces in $\mathcal{H}$ translates into finding an infinite subset $A$ of the graph $\Gamma$ consisting of vertices which are pairwise non adjacent. As the rank of the space $X$ is finite, the graph $\Gamma$ corresponds to the $1$-skeleton of finite dimensional simplicial complex. The set of vertices being infinite, the graph is unbounded with regard to its combinatoric metric. Therefore such subset $A$ exists. ◻ Let $G$ be a group acting by isometries on a median space $X$. We denote by $Stab_G(x)$ the subgroup of $G$ consisting of isometries which stabilize the point $x$. If $G=Isom(X)$ we simply write $Stab(x)$. We have the following proposition: **Proposition 53**. *Let $X$ be a complete connected locally compact median space of finite rank. Let us assume that $Isom(X)$ acts minimally on $X$. Then for any $x_0,x\in X$ the orbit $Stab(x_0).x$ is finite.* Before proving the proposition, we will be needing some results. We have the following lemma which states that any point $x_0\in X$ is determined by the couple $x\in X$ and $\mathcal{H}_{x}\cap \mathcal{H}(x_0,x)$: **Lemma 54**. *Let $X$ be a complete connected median space of finite rank. Let us consider $x,x_0\in X$ and set $C:=\displaystyle{\bigcap_{\mathfrak{h}\in\mathcal{H}_{x}\cap \mathcal{H}(x_0,x)}\mathfrak{h}}$. We have then $\pi_C(x_0):=x$.* *Note that the convex subset $C$ is closed by Remark [Remark 11](#closed_branched_halfspaces){reference-type="ref" reference="closed_branched_halfspaces"}, hence the nearest point projection onto $C$ exists.* *Proof.* By Proposition [Proposition 10](#halfspace_gateconvex){reference-type="ref" reference="halfspace_gateconvex"} and Proposition [Lemma 20](#strenghtening_separation_theorem){reference-type="ref" reference="strenghtening_separation_theorem"}, we have : $$C\cap [x_0,x]=(\bigcap_{\mathfrak{h}\in\mathcal{H}_{x}\cap \mathcal{H}(x_0,x)}\mathfrak{h})\cap [x_0,x]= \{x\}$$ We conclude by Lemma [Lemma 16](#constraint_on_the_projection){reference-type="ref" reference="constraint_on_the_projection"} that $\pi_C(x_0)=x$. ◻ **Lemma 55**. *Let $X$ be a complete connected median space of finite rank and let $x\in X$. Then for any isometry $g\in Stab(x_0)$ and a closed halfspace $\mathfrak{h}\in\mathcal{H}(X)$ such that $x_0\in\mathfrak{h}^c$, the halfspaces $\mathfrak{h}$ and $g.\mathfrak{h}$ are either transverse or disjoint.* *Proof.* Let us consider $g\in Stab(x_0)$ and a closed halfspace $\mathfrak{h}\in \mathcal{H}'$. As both $\mathfrak{h}^c$ and $(g.\mathfrak{h})^c$ contains $x_0$, it is enough to show that we have $g.\mathfrak{h}\subseteq \mathfrak{h}$ if and only if $g.\mathfrak{h}=\mathfrak{h}$. Note that the same conclusion will yield with regards to the case when $\mathfrak{h}\subseteq g.\mathfrak{h}$ as we have $g.\mathfrak{h}\subseteq \mathfrak{h}$ if and only if $\mathfrak{h}\subseteq g^{-1}.\mathfrak{h}$. Let us assume then that $g.\mathfrak{h}\subseteq \mathfrak{h}$. We set $\tilde{x}_0:=\pi_\mathfrak{h}(x_0)$ and first show that $\pi_{g.\mathfrak{h}}(x_0)=\tilde{x}_0$. In one hand, We have $\tilde{x}_0\in[\pi_{g.\mathfrak{h}}(x_0),x_0]$, which implies that : $$d(\pi_{g.\mathfrak{h}}(x_0),x_0)=d(\pi_{g.\mathfrak{h}}(x_0),\tilde{x}_0)+d(\tilde{x}_0,x)$$ In the other hand, we have $$d(x_0,\tilde{x}_0)=d(g.x_0,g.\tilde{x}_0)=d(x_0,\pi_{g.\mathfrak{h}}(g(x_0)))=d(x_0,\pi_{g.\mathfrak{h}}(x_0))$$ Hence, we have $d(\tilde{x}_0,\pi_{g.\mathfrak{h}}(x_0))=0$. Finally, we deduce that for any point $a\in\mathfrak{h}$ we have $\tilde{x}_0\in[a,x_0]$. As $x_0\in(g.\mathfrak{h})^c$ and $\tilde{x}_0\in(g.\mathfrak{h})$, the point $a$ cannot lie outside $g.\mathfrak{h}$. Therefore, we do have $g.\mathfrak{h}=\mathfrak{h}$. ◻ *Proof of Proposition [Proposition 53](#finite_orbit_elleptic_subgroup){reference-type="ref" reference="finite_orbit_elleptic_subgroup"}.* Let us consider $x_0,x\in X$ and $g\in Stab(x_0)$. We denote by $\mathcal{H}'_x$ the set of minimal halfspaces in $\mathcal{H}_x\cap \mathcal{H}(x_0,x)$. By Lemma [Lemma 54](#uniquely_determiner_by_branched_halfspace){reference-type="ref" reference="uniquely_determiner_by_branched_halfspace"}, any point $x\in X$ is determined by the point $x_0$ and the set $\mathcal{H}'_x$. Hence, it is enough to show that the orbit of any halfspace by $Stab(x_0)$ is finite. By Lemma [Lemma 55](#stabilizer_of_points_does_not_nest_halfspaces){reference-type="ref" reference="stabilizer_of_points_does_not_nest_halfspaces"}, the union of the orbit of each halfspace in $\mathcal{H}'_x$ under $Stab(x_0)$ constitutes a family of halfspaces which are either transverse or disjoint. The space $X$ being assumed to be locally compact, the finiteness of the latter family is ensured by Lemma [Lemma 52](#infinite_halfpsace){reference-type="ref" reference="infinite_halfpsace"} and Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"}. ◻ *Proof of Theorem [Theorem 1](#Theorem_discrete_orbit){reference-type="ref" reference="Theorem_discrete_orbit"}.* Let us show that if a median space $X$ admits an action which is Roller minimal and Roller non elementary with a non discrete orbit, then it is not locally compact. Let us set $G:=Isom(X)$ and let $x_0\in X$ such that $G.x_0$ is non discrete. By Lemma [Lemma 51](#Strongly_separated_facing_triple){reference-type="ref" reference="Strongly_separated_facing_triple"}, there exists a facing triple $\mathfrak{h}_1,\mathfrak{h}_2,\mathfrak{h}_3\in\mathcal{H}(X)$ which are uniquely determined by a point $x\in \mathfrak{h}_1^c\cap\mathfrak{h}_2^c\cap\mathfrak{h}_3^c$ in the sense that for any $x_i\in\mathfrak{h}_i$, we have $m(x_1,x_2,x_3)=x$. Let us fix $R>0$ and let $K\subseteq Isom(X)$ such that $d(x_0,g_i.x_0)\leq R$ for any $g\in K$. As the orbit of $x_0$ under $G$ is not discrete, the subset $K$ is infinite. If $K.x$ is finite, this implies that the orbit of $x_0$ under $Stab(x)$ is infinite. Hence, by Proposition [Proposition 53](#finite_orbit_elleptic_subgroup){reference-type="ref" reference="finite_orbit_elleptic_subgroup"} the space $X$ would not be locally compact. Let us assume then that $K.x$ is infinite. Combining Theorem [Theorem 2](#local_compactness){reference-type="ref" reference="local_compactness"} and Lemma [Lemma 52](#infinite_halfpsace){reference-type="ref" reference="infinite_halfpsace"}, it is enough to find an infinite subset in $\mathcal{H}:=K.\mathfrak{h}_1\cup K.\mathfrak{h}_2\cup K.\mathfrak{h}_3$ which consists of halfspaces which are either disjoint or transverse. If $\mathcal{H}$ does not contain an infinite chain then we are done. Let us assume then that there exists an infinite countable chain $\mathcal{H}_1\subseteq \mathcal{H}$. As $\mathfrak{h}_1$, $\mathfrak{h}_2$ and $\mathfrak{h}_3$ are disjoints, the chain $\mathcal{H}_1$ is given by $(g_{1,n}.\mathfrak{h}_{i_n})_{n\in\mathbb{N}}$ where $g_{1,n}\in K$ and $i_n\in\{1,2,3\}$. Let us set $\mathcal{H}':=\displaystyle{\bigcup_{i\in\mathbb{N}}(g_{1,n}.\mathfrak{h}_1\cup g_{1,n}.\mathfrak{h}_2\cup g_{1,n}.\mathfrak{h}_3 )}$ and note that $\mathcal{H}'\backslash \mathcal{H}_1$ is infinite. Again, If the subset $\mathcal{H}'\backslash \mathcal{H}_1$ does not contain an infinite chain then we are done. Let us assume then that $\mathcal{H}'\backslash \mathcal{H}_1$ contains an infinite chain $\mathcal{H}_2$. Such chain is given by $(g_{2,n}.\mathfrak{h}_{j_n})_{n\in\mathbb{N}}$ where $g_{2,n}\in K$ and $j_n\in\{1,2,3\}$. Note that as the halfspaces $\mathfrak{h}_1$, $\mathfrak{h}_2$ and $\mathfrak{h}_3$ are pairwise strongly separated, for any isometry $g\in G$, the halfspace $g.\mathfrak{h}_i$ cannot intersect two halfspaces in $\{\mathfrak{h}_1,\mathfrak{h}_2,\mathfrak{h}_3\}$. Hence, if for each $i\in\{1,2,3\}$ such that $g.\mathfrak{h}_i$ intersects a halfspace in $\{\mathfrak{h}_1,\mathfrak{h}_2,\mathfrak{h}_3\}$, then there exists a permutation $\sigma\in S_3$ such that $g.\mathfrak{h}_i$ intersects only the halfspace $\mathfrak{h}_{\sigma (i)}$. In the latter case, we necessarily have $g(x)=x$ as for any $x_i\in\mathfrak{h}_i$ $m(x_1,x_2,x_3)=x$. As we are considering the isometries $g\in K$ such that $g(x)\neq x$, we conclude that the infinite subset $\displaystyle{\bigcup_{i\in\mathbb{N}}(g_{2,n}.\mathfrak{h}_1\cup g_{2,n}.\mathfrak{h}_2\cup g_{2,n}.\mathfrak{h}_3 ) \backslash (\mathcal{H}_1\cup\mathcal{H}_2)}$ consists of pairwise disjoint halfspaces, which completes the proof. ◻ [^1]: *Add a reference maybe*

arxiv_math

--- abstract: | We revisit [@JK Theorem 6.3]. Following the main ideas used to prove this theorem, we establish a quantitative version of the strong unique continuation property for the Schödinger operator with unbounded potential. We also show that a combination of this result with a global quantitative unique continuation property from an arbitrary interior data yields a global quantitative strong unique continuation. address: Université de Lorraine, Nancy, France author: - Mourad Choulli title: Quantitative strong unique continuation property for the Schrödinger operator with unbounded potential --- # Introduction Let $n\ge 3$ be an integer, $p:=2n/(n+2)$ and $p':=2n/(n-2)$ its conjugate ($1/p+1/p'=1$). For sake of simplicity, the open ball of center $0$ and radius $r>0$ will denoted $B_r$. Let $B:=B_1$. We say that $w\in L^2(B)$ vanishes of infinite order at $0$ if there exists $0<\gamma < 1$ such that for every integer $m>0$ we have $$r^{-m}\int_{B_r}|w|^2\le c_m,\quad 0< r<\gamma,$$ where $c_m>0$ is a constant. Let $q\in L^{n/2}(B)$ and $P\in L^n(B)$. We say that the Schrödinger operator $\Delta +q$ (resp. the differential inequality $|\Delta u|\le |qu|+|P\cdot \nabla u|$) admits the strong unique continuation property (abbreviated in the sequel as SUCP) in the class $W^{2,p}(B)$ if every $u\in W^{2,p}(B)$ satisfying $(\Delta +q)u=0$ (resp. $|\Delta u|\le |qu|+|P\cdot \nabla u|$) in $B$ and $u$ vanishes of infinite order at $0$ must vanishes near $0$. To our best knowledge, the first result establishing that the differential inequality $|\Delta u|\le |qu|$ has the SUCP in the class $W^{2,p}(B)$ with potential $q\in L^{n/2}(B)$ is due to Jerison and Kenig [@JK]. The SUCP for the differential inequality $|\Delta u|\le |P\cdot \nabla u|$ when $P\in L^{(7n-2)/6}$ was obtained by Regbaoui [@Re]. While Wolff [@Wo2] proved that the SUCP holds for the differential inequality $|\Delta u|\le |qu|+|P\cdot \nabla u|$ with $q\in L^{n/2}$ and $P\in L^{\max(n,3n/2-2)}$. Sogge [@So] considered the case where $\Delta$ is replaced by a second order elliptic operator with smooth coefficients. He showed that the SUCP holds for the differential inequality $|\Delta u|\le |qu|+|P\cdot \nabla u|$ with $q\in L^{n/2}$ and $P\in L^\infty$. The result of Sogge [@So] was extended by Koch and Tataru [@KT] with the "best possible conditions", that is $q\in L^{n/2}$ and $P\in L^n$ (in fact they considered an operator with an additional term of the form $\mathrm{div}(Ru)$ with $R\in L^n$). By a counterexample, Jerison and Kenig showed in [@JK Remark 6.5] that $L^{n/2}(B)$ is the best possible space for the potential $q$. The reference [@KT] contains a brief history of the SUCP. Other references dealing with the SUCP for elliptic operators can found in [@KT; @So]. We aim in this article to quantify the SUCP for both the Schödinger operator $\Delta +q$ and the differential inequality $|\Delta u|\le |qu|+|P\cdot \nabla u|$. For sake of clarity, we limited ourselves on the case of the Schödinger operator $\Delta +q$ acting on the unit ball $B$. We briefly indicate in the last section the changes to be maid in order to obtain a variant of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"} and a quantitative SUCP for the differential inequality $|\Delta u|\le |qu|+|P\cdot \nabla u|$. We also give in this section a quantitative SUCP and a global quantitative SUCP for the operator $\Delta +q$ acting on an arbitrary domain of $\mathbb{R}^n$. Before precisely stating our main result, we introduce some definitions and notations. As usual, the set of positive integers is denoted in the sequel by $\mathbb{N}$. For each $m\in \mathbb{N}$, define $O_m$ as the set of functions $w\in L^2(B)$ satisfying $$\varrho_m (w):=\sup \{\eta^{-m}\|w\|_{L^2(B_\eta)};\; 0<\eta < 1\}<\infty.$$ It is clear that $\varrho_m (w)<\infty$ if and only if there exists $0<\gamma <1$ so that $$\varrho_m^\gamma (w):=\sup \{\eta^{-m}\|w\|_{L^2(B_\eta)};\; 0<\eta < \gamma\}<\infty.$$ Therefore, $u$ vanishes of infinite order at $0$ if and only if $\displaystyle u\in \bigcap_{m\in \mathbb{N}}O_m$. We use in this text that $W^{2,p}(B)$ is continuously embedded in $H^1(B)$, and that $H^1(B)$ is continuously embedded in $L^{p'}(B)$. Let $$\begin{aligned} &\sigma:=\sup \{ \|w\|_{L^{p'}(B)};\; w\in H_0^1(B),\; \|\nabla w\|_{L^2(B)}=1\}, \\ &Q:=\{ q\in L^{n/2}(B );\; \|q\|_{L^{n/2}(B)}\le \vartheta\},\end{aligned}$$ for some fixed $\vartheta>0$ satisfying $2\vartheta\sigma^2<1$, and $$\Lambda:=\{ \lambda >0;\; \mathrm{dist}(\lambda ,\mathbb{N}+2/(n-2))\ge 1/2\}.$$ Throughout, $0<\rho <1/2$ and $\phi\in C_0^\infty (B)$ satisfying $\phi =1$ in $B_{1/2}$ and $0\le \phi \le 1$ are fixed arbitrarily. **Theorem 1**. *There exists a constant $C=C(n,\vartheta)$ such that for any $m\in \mathbb{N}$, $\lambda \in \Lambda$, $q\in Q$, $u\in W^{2,p}(B)\cap O_m$ and $0<\eta < \rho/4$, we have $$\begin{aligned} &C\|u\|_{L^2(B_{\rho/2})}\le \eta^{-\lambda}\|(\Delta+q) u\|_{L^p(B)}\label{mat1.0} \\ &\hskip 3cm + 2^{-(\lambda+1)}\|(\Delta+q) (\phi u)\|_{L^{p}(B)}+3^m\eta^{m-(\lambda+3)}\varrho_m(u).\nonumber\end{aligned}$$* Let's explain why Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"} quantifies the SUCP for the operator $\Delta +q$, $q\in Q$, in the class $W^{2,p}(B)$. Pick $u\in W^{2,p}(B)$ satisfying $(\Delta+q)u=0$ in $B$ and $u$ vanishes of infinite order at $0$. For every $\lambda \in \Lambda$, $0<\eta < \rho/4$ and $m=[\lambda]+4$, where $[\lambda]$ is the entire part of $\lambda$, [\[mat1.0\]](#mat1.0){reference-type="eqref" reference="mat1.0"} yields $$C\|u\|_{L^2(B_{\rho/2})}\le 2^{-(\lambda+1)}\|(\Delta+q) (\phi u)\|_{L^{p}(B)}+3^{[\lambda]+4}\eta^{[\lambda]+1-\lambda}\varrho_{[\lambda]+4}(u).$$ Taking the limit when $\eta$ tends to zero in this inequality, we find $$C\|u\|_{L^2(B_{\rho/2})}\le 2^{-(\lambda+1)}\|(\Delta+q) (\phi u)\|_{L^{p}(B)}.$$ Choose a sequence $(\lambda_j)$ of $\Lambda$ tending to $\infty$ when $j$ goes to $\infty$. From the preceding inequality, we have $$C\|u\|_{L^2(B_{\rho/2})}\le 2^{-(\lambda_j+1)}\|(\Delta+q) (\phi u)\|_{L^{p}(B)}\quad \mbox{for each}\; j,$$ from which we derive in a straightforward manner that $u=0$ in $B_{\rho/2}$. Hence $u=0$ in $\displaystyle \bigcup_{0<\rho<1/2}B_{\rho/2}=B_{1/4}$. **Corollary 1**. *Let $q\in Q$. If $u\in W^{2,p}(B)$ satisfies $(\Delta +q)u=0$ in $B$ and $u$ vanishes of infinite order at $0$ then $u=0$ in $B_{1/4}$.* The main ingredient in the proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"} is the Carleman inequality [\[1\]](#1){reference-type="eqref" reference="1"} due to Jerison and Kenig [@JK]. In the case of elliptic operators with variable leading coefficients we still have a Carleman inequality [@KT]. However, this Carleman estimate is less flexible to use than [\[1\]](#1){reference-type="eqref" reference="1"} and we do not think it is the right tool to establish quantitative SUCP for operators with variable leading coefficients. We mention the paper by Malinnikova and Vessella [@MV] where, based on the Carleman inequality of Koch and Tataru [@KT], the authors proved a quantitative unique continuation result from a set of positive measure. When $q\in L^n(B)$ we derive from [\[mat1.0\]](#mat1.0){reference-type="eqref" reference="mat1.0"} and [@Ch Theorem 1.2 and comments in Section 1.3] a global quantitative SUCP. Precisely, we have the following result, where $$\mathbf{Q}:=\{q\in L^{n}(B);\; \|q\|_{L^n(B)}\le |B|^{-1/n}\vartheta \}\; (\subset Q).$$ **Theorem 2**. *Let $0<s,t<1/2$. For each $0<r <1$, $0<\eta<\rho/4$, $m\in \mathbb{N}$, $\lambda \in \Lambda$, $q\in \mathbf{Q}$, $u\in H^2(B)\cap O_m$, we have $$\begin{aligned} &C\|u\|_{H^{3/2+t}(B)}\le r^{\zeta}\|u\|_{H^2(B)} \\ &\hskip 1.5cm +e^{\mathfrak{b}r^{-\nu}}\Big(\eta^{-\lambda}\|(\Delta+q) u\|_{L^2(B)} \\ &\hskip 4cm + 2^{-(\lambda+1)}\|(\Delta+q) (\phi u)\|_{L^{p}(B)}+3^m\eta^{m-(\lambda+3)}\varrho_m(u)\Big),\end{aligned}$$ where $C=C(n,s, t, \vartheta)>0$, $\mathfrak{b}=\mathfrak{b}(n, s, t, \vartheta)>0$, $\nu=\nu(n)>0$ are constants and $\zeta =\min ((1/2-t)/(1+t),s/4)$.* By proceeding as above, we deduce from Theorem [Theorem 2](#mt2){reference-type="ref" reference="mt2"} the following global SUCP. **Corollary 2**. *Assume that $q\in \mathbf{Q}$. If $u\in H^2(B)$ satisfies $(\Delta +q) u=0$ in $B$ and $u$ vanishes of infinite order at $0$ then $u=0$ in $B$.* Let $E\subset B$ be a set of positive measure. Arguing as in [@Ch Section 3], we deduce from Corollary [Corollary 2](#corollary2){reference-type="ref" reference="corollary2"} the following consequence. **Corollary 3**. *Let $q\in \mathbf{Q}$ and $u\in H^2(B)$ satisfying $(\Delta +q) u=0$ in $B$ and $u=0$ in $E$. Then $u=0$ in $B$.* # Proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"} {#proof-of-theorem-mt1.0} ## Caccioppoli's type inequality **Lemma 1**. *Let $0<r,h <1$ satisfying $0<r+h <1$. There exists a constant $C=C(n,\vartheta)$ such that for any $u\in W^{2,p}(B)$ and $q\in Q$ we have $$\label{ca1} C\|\nabla u\|_{L^2(B_r)}\le \|(\Delta+q) u\|_{L^p(B)}+h^{-1}\|u\|_{L^2(B_{r+h})}.$$* *Proof.* Pick $\chi \in C_0^\infty (B_{r+h})$ satisfying $0\le \chi \le 1$, $\chi =1$ in a neighborhood of $B_r$ and $|\partial^\alpha\chi |\le c_0h^{-1}$ for each $|\alpha|=1$, where $c_0>0$ is a universal constant. Let $u\in W^{2,p}(B)$ and $\epsilon>0$. In light of Hölder's inequality, we have $$\left| \int_B\chi^2 u(\Delta+q) udx\right| \le \|(\Delta+q) u\|_{L^p(B)}\|\chi u\|_{L^{p'}(B)}$$ and hence $$\begin{aligned} \left| \int_B\chi^2 u(\Delta+q) udx\right| &\le (2\epsilon)^{-1}\|(\Delta+q) u\|_{L^p(B)}^2+2^{-1}\epsilon\|\chi u\|_{L^{p'}(B)}^2 \\ &\le (2\epsilon)^{-1}\|(\Delta+q) u\|_{L^p(B)}^2+2^{-1}\epsilon\sigma^2\|\nabla (\chi u)\|_{L^2(B)}^2.\end{aligned}$$ That is we have $$\begin{aligned} \left| \int_B\chi^2 u(\Delta+q) udx\right| \le (2\epsilon)^{-1}&\|(\Delta+q) u\|_{L^p(B)}^2\label{ca2} \\ &+\epsilon\sigma^2\left(\|u\nabla \chi\|_{L^2(B)}^2+\|\chi \nabla u\|_{L^2(B)}^2\right).\nonumber\end{aligned}$$ On the other hand, we get by making an integration by parts $$\int_B\chi^2 u\Delta udx=-\int_B\chi^2|\nabla u|^2dx-\int_B u\nabla \chi^2 \cdot \nabla udx.$$ Therefore, we find $$\|\chi \nabla u\|_{L^2(B)}^2\le \left|\int_B\chi^2 u\Delta udx\right|+2\left|\int_B u\chi \nabla \chi \cdot \nabla udx\right|.$$ Applying Cauchy-Schwarz inequality and then a convexity inequality to the second term of the right hand side of inequality above, we get $$\|\chi \nabla u\|_{L^2(B)}^2\le \left|\int_B\chi^2 u\Delta udx\right|+\epsilon^{-1}\|u\nabla \chi\|_{L^2(B)}^2+\epsilon\|\chi \nabla u\|_{L^2(B)}^2.$$ Whence, we have $$(1-\epsilon)\|\chi \nabla u\|_{L^2(B)}^2\le \left|\int_B\chi^2 u\Delta udx\right|+\epsilon^{-1}\|u\nabla \chi\|_{L^2(B)}^2,$$ from which we derive in a straightforward manner $$\label{ca3} (1-\epsilon)\|\chi \nabla u\|_{L^2(B)}^2\le \left|\int_B\chi^2 u(\Delta+q) udx\right|+ \left|\int_Bq\chi^2 u^2dx\right|+\epsilon^{-1}\|u\nabla \chi\|_{L^2(B)}^2.$$ Applying twice Hölder's inequality, we obtain $$\left|\int_Bq\chi^2 u^2dx\right|\le \|q\chi u\|_{L^p(B)}\|\chi u\|_{L^{p'}(B)}\le \|q\|_{L^{n/2}(B)}\|\chi u\|_{L^{p'}(B)}^2$$ Hence, we have $$\begin{aligned} \left|\int_Bq\chi^2 u^2dx\right|&\le \|q\|_{L^{n/2}(B)}\sigma^2\|\nabla (\chi u)\|_{L^2(B)}^2 \\ & \le 2\|q\|_{L^{n/2}(B)}\sigma^2\left(\|u\nabla \chi \|_{L^2(B)}^2+\|\chi \nabla u\|_{L^2(B)}^2\right) \\ & \le 2\vartheta\sigma^2\left(\|u\nabla \chi \|_{L^2(B)}^2+\|\chi \nabla u\|_{L^2(B)}^2\right).\end{aligned}$$ This inequality in [\[ca3\]](#ca3){reference-type="eqref" reference="ca3"} gives $$\label{ca4} (1-\epsilon-2\vartheta\sigma^2)\|\chi \nabla u\|_{L^2(B)}^2\le \left|\int_B\chi^2 u(\Delta+q) udx\right|+(\epsilon^{-1}+2\vartheta\sigma^2)\|u\nabla \chi\|_{L^2(B)}^2.$$ Putting [\[ca2\]](#ca2){reference-type="eqref" reference="ca2"} in [\[ca4\]](#ca4){reference-type="eqref" reference="ca4"}, we end up getting $$\begin{aligned} (1-2\vartheta\sigma^2-\epsilon(1+\sigma^2))\|\chi \nabla u\|_{L^2(B)}^2 &\le (2\epsilon)^{-1}\|(\Delta+q) u\|_{L^p(B)}^2 \\ &\qquad +(\epsilon^{-1}+\epsilon\sigma^2+2\vartheta\sigma^2)\|u\nabla \chi\|_{L^2(B)}^2.\end{aligned}$$ The expected inequality follows by taking $\epsilon=(1-2\vartheta\sigma^2)/(2+2\sigma^2)$. ◻ ## Carleman inequality Let $\dot{B}:=B\setminus\{0\}$. It follows from [@JK (3.4)] that $$\label{1} \||x|^{-\lambda}u\|_{p'}\le \kappa \||x|^{-\lambda}\Delta u\|_p,\quad u\in C_0^\infty(\dot{B}) ,\; \lambda \in \Lambda.$$ Here and henceforth $\kappa=\kappa(n)$ is a constant. For any $N>0$, let $\mathbf{B}_N$ denotes the closed ball of $L^{n/2}(B)$ with centrer $0$ and radius $N$. **Lemma 2**. *For every $N>0$, there exists a constant $\kappa_N=\kappa_N(n,N)>0$ such that $$\label{4} \||x|^{-\lambda}u\|_{L^{p'}(B)}\le \kappa_N \||x|^{-\lambda}(\Delta +q)u\|_{L^p(B)},\quad u\in C_0^\infty(\dot{B}),\; q\in\mathbf{B}_N,\; \lambda \in \Lambda.$$* *Proof.* Let $u\in C_0^\infty(\dot{B})$, $\lambda \in \Lambda$ and $q\in L^{n/2}(B)$ satisfying $$\label{2} 2\kappa \|q\|_{L^{n/2}(B)}\le 1.$$ In light of [\[1\]](#1){reference-type="eqref" reference="1"}, we have $$\begin{aligned} \||x|^{-\lambda}u\|_{L^{p'}(B)}&\le \kappa \||x|^{-\lambda}(\Delta +q) u\|_p +\kappa\|q|x|^{-\lambda}u\|_{L^p(B)} \\ &\le \kappa \||x|^{-\lambda}(\Delta +q) u\|_{L^p(B)} +\kappa\|q\|_{n/2}\||x|^{-\lambda}u\|_{L^{p'}(B)}\end{aligned}$$ which combined with [\[2\]](#2){reference-type="eqref" reference="2"} gives $$\label{3} \||x|^{-\lambda}u\|_{L^{p'}(B)}\le 2\kappa \||x|^{-\lambda}(\Delta +q)u\|_{L^p(B)}.$$ Let the set $\mathbf{N}$ consists in the constants $N>0$ for which there exists $\kappa_N>0$ such that for every $u\in C_0^\infty(\dot{B})$, $\lambda \in \Lambda$ and $q\in \mathbf{B}_N$, we have $$\||x|^{-\lambda}u\|_{L^{p'}(B)}\le \kappa_N \||x|^{-\lambda}(\Delta +q)u\|_{L^p(B)}.$$ By [\[3\]](#3){reference-type="eqref" reference="3"}, $(0,(2\kappa)^{-1}]\subset \mathbf{N}$. Let $N\in \mathbf{N}$ and $N_1=N+ (2\kappa_N)^{-1}$. For all $u\in C_0^\infty(\dot{B})$, $\lambda \in \Lambda$ and $q\in \mathbf{B}_{N_1}$, by using $$q=\frac{N}{N_1}q+\frac{(2\kappa_N)^{-1}}{N_1}q,$$ and by proceeding as for [\[3\]](#3){reference-type="eqref" reference="3"} we obtain $$\||x|^{-\lambda}u\|_{L^{p'}(B)}\le 2\kappa_N \||x|^{-\lambda}(\Delta +q)u\|_{L^p(B)}.$$ That is $N_1\in \mathbf{N}$ with $\kappa_{N_1}=2\kappa_N$. This proves that $\sup \mathbf{N}=\infty$. ◻ ## Proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"} {#proof-of-theorem-mt1.0-1} Pick a family of functions $(\psi_\eta)_{0<\eta < \rho/4}$ of $C^\infty(\mathbb{R}^n)$ so that $0\le \psi_\eta\le 1$, $\psi_\eta(x)=0$ if $|x|\le \eta$, $\psi_\eta(x)=1$ if $|x|\ge 2\eta$ and $$|\partial^\alpha \psi_\eta|\le c\eta^{-|\alpha|},\quad |\alpha|\le 2,$$ for each $0<\eta <\rho/4$, where $c>0$ is a universal constant. Let $m\in \mathbb{N}$, $q\in Q$, $u\in W^{2,p}(B)\cap O_m$ and set $v=\phi u$. Fix $0<\eta< \rho/4$ arbitrarily. By density, we get from [\[4\]](#4){reference-type="eqref" reference="4"} the following inequality $$\label{5} \||x|^{-\lambda}\psi_\eta v\|_{L^{p'}(B)}\le \kappa \||x|^{-\lambda}(\Delta+q) (\psi_\eta v)\|_{L^{p}(B)},$$ where $\kappa=\kappa_\vartheta$ is the constant appearing in [\[4\]](#4){reference-type="eqref" reference="4"} when $N=\vartheta$. Therefore, we have $$\begin{aligned} \||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_{\rho/2})}\le \kappa \||x|^{-\lambda}&(\Delta+q) (\psi_\eta v)\|_{L^{p}(B_\rho)} \\ &+\kappa\||x|^{-\lambda}(\Delta+q) (\psi_\eta v)\|_{L^{p}(\{\rho<|x|<1\})},\end{aligned}$$ and hence $$\begin{aligned} \||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_{\rho/2})}\le \kappa \||x|^{-\lambda}(\Delta+q) &(\psi_\eta u)\|_{L^{p}(B_\rho)}\label{5.0} \\ &+\kappa \rho^{-\lambda}\|(\Delta+q) v\|_{L^{p}(B)}.\nonumber\end{aligned}$$ In consequence, we obtain $$\begin{aligned} \||x|^{-\lambda} u\|_{L^{p'}(\{2\eta <|x]<\rho/2\})}\le \kappa \||x|^{-\lambda}(\Delta+q) &(\psi_\eta u)\|_{L^{p}(B_{\rho})}\label{5.1} \\ &+\kappa \rho^{-\lambda}\|(\Delta+q) v\|_{L^{p}(B)}.\nonumber\end{aligned}$$ On the other hand, Hölder's inequality implies $$\|u\|_{L^2(\{2\eta <|x]<\rho/2\})}\le \||x|^\lambda\|_{L^n(\{2\eta <|x|<\rho/2\})}\||x|^{-\lambda} u\|_{L^{p'}(\{2\eta <|x]<\rho/2\})}.$$ Since $$\||x|^\lambda\|_{L^n(\{2\eta <|x]<\rho/2\})}\le \||x|^\lambda\|_{L^n(B_{\rho/2})}=\left[\frac{\omega_n}{n(\lambda+1)}\right]^{1/n}(\rho/2)^{\lambda+1},$$ where $\omega_n=|\mathbb{S}^{n-1}|$, we get $$\label{5.2} \|u\|_{L^2(\{2\eta <|x]<\rho/2\})}\le\left[\frac{\omega_n}{n(\lambda+1)}\right]^{1/n}(\rho/2)^{\lambda+1}\||x|^{-\lambda} u\|_{L^{p'}(\{2\eta <|x]<\rho/2\})}.$$ Putting together [\[5.1\]](#5.1){reference-type="eqref" reference="5.1"} and [\[5.2\]](#5.2){reference-type="eqref" reference="5.2"}, we obtain $$\begin{aligned} &\|u\|_{L^2(B_{\rho/2})}\le \kappa_1 (1+\lambda)^{-1/n}(\rho/2)^{\lambda+1} \||x|^{-\lambda}(\Delta+q)(\psi_\eta u)\|_{L^{p}(B_\rho)} \\ &\hskip 5cm +\kappa_1 \rho2^{-(\lambda+1)}\|(\Delta+q) v\|_{L^{p}(B)}+\|u\|_{L^2(B_{2\eta})},\end{aligned}$$ where $\kappa_1=\kappa(\omega_n/n)^{1/n}$. Hence, the following inequality holds $$\begin{aligned} &\|u\|_{L^2(B_{\rho/2})}\le \kappa_1 \||x|^{-\lambda}(\Delta+q)(\psi_\eta u)\|_{L^{p}(B_\rho)}\label{5.3} \\ &\hskip 3cm +\kappa_1 2^{-(\lambda+1)}\|(\Delta+q) v\|_{L^{p}(B)}+\|u\|_{L^2(B_{2\eta})}.\nonumber\end{aligned}$$ Next, using the formula $$\Delta (\psi_\eta u)=\psi_\eta \Delta u+ 2\nabla \psi_\eta \cdot \nabla u+\Delta \psi_\eta u$$ and the fact that $\mbox{supp}(2\nabla \psi_\eta\cdot \nabla u+\Delta \psi_\eta u)\subset \{\eta<|x|<2\eta\}$ in order to derive the following inequality $$\begin{aligned} \||x|^{-\lambda}(\Delta+q) (\psi_\eta u)\|_{L^{p}(B_\rho)}\le &\eta^{-\lambda}\|(\Delta+q) u\|_{L^p(B)} \label{7} \\ &+ c_0\eta ^{-(\lambda+2)}\left(\|\nabla u\|_{L^2(B_{2\eta})}+\|u\|_{L^2(B_{2\eta})}\right),\nonumber\end{aligned}$$ for some constant $c_0=c_0(n)$. From inequality [\[ca1\]](#ca1){reference-type="eqref" reference="ca1"}, we get $$C\|\nabla u\|_{L^2(B_{2\eta})}\le \|(\Delta+q) u\|_{L^p(B)}+ \eta^{-1}\|u\|_{L^2(B_{3\eta})}.$$ Here and until the end of this proof, $C=C(n,\vartheta)$ is a generic constant. This inequality in [\[7\]](#7){reference-type="eqref" reference="7"} yields $$\begin{aligned} C\||x|^{-\lambda}(\Delta+q) (\psi_\eta u)&\|_{L^{p}(B_\rho)} \label{8} \\ &\le \eta^{-\lambda}\|(\Delta+q) u\|_{L^p(B)}+\eta^{-(\lambda+3)}\|u\|_{L^2(B_{3\eta})}.\nonumber\end{aligned}$$ Putting together [\[5.3\]](#5.3){reference-type="eqref" reference="5.3"} and [\[8\]](#8){reference-type="eqref" reference="8"}, we find $$\begin{aligned} &C\|u\|_{L^2(B_{\rho/2})}\le \eta^{-\lambda}\|(\Delta+q) u\|_{L^p(B)}+ 2^{-(\lambda+1)}\|(\Delta+q) v\|_{L^{p}(B)} \\ &\hskip 8cm +\eta^{-(\lambda+3)}\|u\|_{L^2(B_{3\eta})}.\end{aligned}$$ We end up getting $$\begin{aligned} &C\|u\|_{L^2(B_{\rho/2})}\le \eta^{-\lambda}\|(\Delta+q) u\|_{L^p(B)}+ 2^{-(\lambda+1)}\|(\Delta+q) (\phi u)\|_{L^{p}(B)} \\ &\hskip 8.5cm +3^m\eta^{m-(\lambda+3)}\varrho_m(u),\end{aligned}$$ which is the expected inequality. # Other results ## Quantitative SUCP without smallness condition on the potential Pick $q_0\in L^{n/2}(B)$ and fix $0<\underline{\rho}<1/2$ so that $\sigma^2\|q\|_{L^{n/2}(B_{\underline{\rho}})}<1/4$. Let $$\underline{Q}:=\{q\in L^{n/2}(B);\; \sigma^2\|q-q_0\|_{L^{n/2}(B_{\underline{\rho}})}<1/4\}.$$ The following inequality will be useful in the sequel $$\label{c1} \sigma^2\|q\|_{L^{n/2}(B_{\underline{\rho}})}<1/2,\quad q\in \underline{Q}.$$ Under the definitions and the notations of the proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"} with $\rho=\underline{\rho}$, instead of [\[5.0\]](#5.0){reference-type="eqref" reference="5.0"} we have $$\||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_{\underline{\rho}})}\le \kappa \||x|^{-\lambda}\Delta (\psi_\eta u)\|_{L^{p}(B_{\underline{\rho}})} +\kappa \rho^{-\lambda}\|\Delta v\|_{L^{p}(B)}.$$ In light of [\[c1\]](#c1){reference-type="eqref" reference="c1"}, this inequality implies the following one $$\||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_{\underline{\rho}})}\le 2\kappa \||x|^{-\lambda}(\Delta+q) (\psi_\eta u)\|_{L^{p}(B_{\underline{\rho}})} +2\kappa \rho^{-\lambda}\|\Delta v\|_{L^{p}(B)},$$ for any $q\in \underline{Q}$. In particular, we have $$\||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_{\underline{\rho}/2})}\le 2\kappa \||x|^{-\lambda}(\Delta+q) (\psi_\eta u)\|_{L^{p}(B_{\underline{\rho}})} +2\kappa \rho^{-\lambda}\|\Delta v\|_{L^{p}(B)},$$ for any $q\in \underline{Q}$. Modifying slightly the proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"}, we get the following result (note that Caccioppoli's inequality [\[ca1\]](#ca1){reference-type="eqref" reference="ca1"} remains valid when substituting $q\in Q$ by $q\in \underline{Q}$ and taking $0<r+h<\underline{\rho}$). **Theorem 3**. *There exists a constant $C=C(n)$ such that for any $m\in \mathbb{N}$, $\lambda \in \Lambda$, $q\in \underline{Q}$, $u\in W^{2,p}(B)\cap O_m$ and $0<\eta < \underline{\rho}/4$, we have $$\begin{aligned} &C\|u\|_{L^2(B_{\underline{\rho}/2})}\le \eta^{-\lambda}\|(\Delta+q) u\|_{L^p(B)} \\ &\hskip 3cm+ 2^{-(\lambda+1)}\|\Delta (\phi u)\|_{L^{p}(B)} +3^m\eta^{m-(\lambda+3)}\varrho_m(u).\end{aligned}$$* An immediate consequence of this theorem is **Corollary 4**. *Let $q\in L^{n/2}(B)$. If $u\in W^{2,p}(B)$ satisfies $(\Delta +q)u=0$ in $B$ and $u$ vanishes of infinite order at $0$ then $u=0$ in a neighborhood $V$ of $0$.* Of course, $V$ in Corollary [Corollary 4](#corollaryc1){reference-type="ref" reference="corollaryc1"} depends on $q$. ## Quantitative SUCP for differential inequalities Here again the notations are those of the proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"}. As in the beginning of the proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"}, we have $$\||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_\rho)}\le \kappa \||x|^{-\lambda}\Delta (\psi_\eta u)\|_{L^{p}(B_{\rho})} +\kappa \rho^{-\lambda}\|\Delta v\|_{L^{p}(B)},$$ where $\kappa=\kappa(n)$ is the constant in [\[1\]](#1){reference-type="eqref" reference="1"}. Similarly to [\[7\]](#7){reference-type="eqref" reference="7"}, we prove $$\begin{aligned} \||x|^{-\lambda}\Delta (\psi_\eta u)\|_{L^{p}(B_\rho)}\le &\||x|^{-\lambda}\psi_\eta \Delta u\|_{L^p(B_\rho)} \\ &c_0\eta ^{-(\lambda+2)}\left(\|\nabla u\|_{L^2(B_{2\eta})}+\|u\|_{L^2(B_{2\eta})}\right),\end{aligned}$$ for some constant $c_0=c_0(n)$. This inequality in the preceding one yields $$\begin{aligned} \||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_\rho)}\le \kappa \||x|^{-\lambda}\psi_\eta &\Delta u\|_{L^p(B_\rho)}+\kappa \rho^{-\lambda}\|\Delta v\|_{L^{p}(B)} \\ &+\kappa c_0\eta ^{-(\lambda+2)}\left(\|\nabla u\|_{L^2(B_{2\eta})}+\|u\|_{L^2(B_{2\eta})}\right).\end{aligned}$$ Pick $P\in L^n(B)$. Then under the assumption $$|\Delta u|\le |qu|+|P\cdot \nabla u|\quad \mbox{in}\; B,$$ the last inequality gives $$\begin{aligned} &\||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_\rho)}\le \kappa\left( \||x|^{-\lambda}\psi_\eta q u\|_{L^p(B_\rho)}+\||x|^{-\lambda}\psi_\eta P.\nabla u\|_{L^p(B_\rho)}\right) \\ &\hskip 2cm +\kappa \rho^{-\lambda}\|\Delta v\|_{L^{p}(B)}+\kappa c_0\eta ^{-(\lambda+2)}\left(\|\nabla u\|_{L^2(B_{2\eta})}+\|u\|_{L^2(B_{2\eta})}\right).\end{aligned}$$ Since $$\||x|^{-\lambda}\psi_\eta q u\|_{L^p(B_\rho)}\le \|q\|_{L^{n/2}(B_\rho)}\||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_\rho)}$$ and $$\||x|^{-\lambda}\psi_\eta P.\nabla u\|_{L^p(B_\rho)}\le \|P\|_{L^n(B_\rho)}\||x|^{-\lambda}\psi_\eta \nabla u\|_{L^2(B_\rho)},$$ if $q\in \underline{Q}$ and $\|P\|_{L^n(B_{\underline{\rho}})}\le K$, for some constant $K>0$, then we obtain $$\begin{aligned} &\||x|^{-\lambda}\psi_\eta u\|_{L^{p'}(B_{\underline{\rho}})}\le 2\kappa (c_0+K)\eta ^{-(\lambda+2)}\|\nabla u\|_{L^2(B_{2\eta})} \\ &\hskip 3.5cm +2\kappa \rho^{-\lambda}\|\Delta v\|_{L^{p}(B)}+2\kappa c_0\eta ^{-(\lambda+2)}\|u\|_{L^2(B_{2\eta})}.\end{aligned}$$ This inequality in hand, we can mimic the proof of Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"} in order to obtain the following result. **Theorem 4**. *For any $K>0$, $m\in \mathbb{N}$, $\lambda \in \Lambda$, $q\in \underline{Q}$, $P\in L^n(B)$ such that $\|P\|_{L^n(B_{\underline{\rho})}}\le K$, $u\in W^{2,p}(B)\cap O_m$ satisfying $|\Delta u|\le |qu|+|P\cdot \nabla u|$ in $B$ and $0<\eta < \underline{\rho}/4$, we have $$C\|u\|_{L^2(B_{\underline{\rho}/2})}\le 2^{-(\lambda+1)}\|\Delta(\phi u)\|_{L^{p}(B)} +3^m\eta^{m-(\lambda+3)}\varrho_m(u),$$ where $C=C(n,K)>0$ is a constant.* Inequality $|\Delta u|\le |qu|+|P\cdot \nabla u|$ in $B$ of Theorem [Theorem 4](#mt1.2){reference-type="ref" reference="mt1.2"} can be replaced by the following one $$c|\Delta u|\le |x|^{-2+\alpha_0}|u|+|x|^{-1+\alpha_1}|\nabla u|\quad \mbox{in}\; B,$$ where $c>0$, $\alpha_j>0$, $j=0,1$, are constants. This follows easily by observing that $|x|^{-2+\alpha_0}\in L^{n/2}(B)$ and $|x|^{-1+\alpha_1}\in L^n(B)$. As a consequence of Theorem [Theorem 4](#mt1.2){reference-type="ref" reference="mt1.2"}, we have the following corollary. **Corollary 5**. *Let $q\in L^{n/2}(B)$ and $P\in L^n(B)$. If $u\in W^{2,p}(B)$ satisfies $|\Delta u|\le |qu|+|P\cdot \nabla u|$ in $B$ and $u$ vanishes of infinite order at $0$ then $u=0$ in a neighborhood $V$ of $0$.* Corollary [Corollary 5](#corollaryc2){reference-type="ref" reference="corollaryc2"} with $P=0$ corresponds to [@JK Theorem 6.3]. We also point out that Corollary [Corollary 5](#corollaryc2){reference-type="ref" reference="corollaryc2"} is not true in general when $n=2$ (see [@Wo1 Theorem 2]). ## Quantitative SUCP for an arbitrary bounded domain Let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Fix $x_0\in \Omega$, let $d:=\mathrm{dist}(x_0,\mathbb{R}^n\setminus\Omega)$ and denote by $B_r(x_0)$ the open ball of center $x_0$ and radius $r>0$. Let $$\begin{aligned} &\tilde{Q}:=\{\tilde{q}\in L^{n/2}(\Omega);\; \|q\|_{L^2(B_d(x_0))}\le \vartheta\}, \\ &\tilde{\varrho}_m(\tilde{u}):=\sup \{ \eta^{-m}\|\tilde{u}\|_{L^2(B_\eta (x_0))};\; 0<\eta <d\},\quad \tilde{u}\in L^2(\Omega),\; m\in \mathbb{N}, \\ & \tilde{O}_m:=\{\tilde{u}\in L^2(\Omega);\; \tilde{\varrho}_m(\tilde{u})<\infty\}.\end{aligned}$$ Pick $m\in \mathbb{N}$, $\tilde{u}\in W^{2,p}(\Omega)\cap \tilde{O}_m$, $\tilde{q}\in \tilde{Q}$ and define $$u(x)=\tilde{u}(x_0+dx),\quad q(x)=d^2\tilde{q}(x_0+dx),\quad x\in B.$$ We check that $u\in W^{2,p}(B)\cap O_m$, $q\in Q$, $$\|q\|_{L^{n/2}(B)}=\|\tilde{q}\|_{L^{n/2}(B_d(x_0))},\quad \varrho_m(u)=d^{m-n/2}\tilde{\varrho}_m(\tilde{u})$$ and $$(\Delta +q)u=d^2(\Delta +\tilde{q}(x_0+d\, \cdot ))\tilde{u}(x_0+d\, \cdot )\quad \mbox{in}\; B.$$ Pick $0<\eta <\rho/4$. Applying Theorem [Theorem 1](#mt1.0){reference-type="ref" reference="mt1.0"}, we get $$\begin{aligned} &C\|u\|_{L^2(B_{\rho/2})}\le \eta^{-\lambda}\|(\Delta+q) u\|_{L^p(B)} \\ &\hskip 3cm + 2^{-(\lambda+1)}\|(\Delta+q) (\phi u)\|_{L^{p}(B)} +3^m\eta^{m-(\lambda+3)}\varrho_m(u).\end{aligned}$$ Here and henceforth $C=C(n,\vartheta)$ is a generic constant. Therefore, we obtain $$\begin{aligned} &Cd^{-n/2}\|\tilde{u}\|_{L^2(B_{d\rho/2})}\le d^{2-n/p}\eta^{-\lambda}\|(\Delta+q) \tilde{u}\|_{L^p(B_d(x_0))} \\ &\hskip 1.5cm + 2^{-(\lambda+1)}d^{2-n/p}\|(\Delta+\tilde{q}) (\tilde{\phi} \tilde{u})\|_{L^{p}(B_d(x_0))} +3^md^{m-n/2}\eta^{m-(\lambda+3)}\tilde{\varrho}_m(\tilde{u}),\end{aligned}$$ where $\tilde{\phi}(y)=\phi ((y-x_0)/d)$, $y\in B_d(x_0)$. In other words, we proved the following theorem. **Theorem 5**. *There exists a constant $C=C(n,\vartheta,d)$ such that for any $m\in \mathbb{N}$, $\lambda \in \Lambda$, $\tilde{q}\in \tilde{Q}$, $\tilde{u}\in W^{2,p}(\Omega)\cap \tilde{O}_m$ and $0<\eta < \rho/4$, we have $$\begin{aligned} &C\|\tilde{u}\|_{L^2(B_{d\rho/2}(x_0))}\le \eta^{-\lambda}\|(\Delta+\tilde{q}) u\|_{L^p(B_d(x_0))} \\ &\hskip 3cm + 2^{-(\lambda+1)}\|(\Delta+\tilde{q}) (\tilde{\phi} \tilde{u})\|_{L^{p}(B_d(x_0))}+(3d)^m\eta^{m-(\lambda+3)}\tilde{\varrho}_m(\tilde{u}).\end{aligned}$$* Theorem [Theorem 5](#Mt1){reference-type="ref" reference="Mt1"} together with [@Ch Theorem 1.2 and comments in Section 1.3] yield the following global quantitative SUCP, where $$\mathbf{\tilde{Q}}:=\{\tilde{q}\in L^{n}(\Omega);\; \|\tilde{q}\|_{L^n(B_d(x_0))}\le |B_d(x_0)|^{-1/n}\vartheta \}\; (\subset \tilde{Q}).$$ **Theorem 6**. *Assume that $\Omega$ is $C^{1,1}$. Let $0<s,t<1/2$. For each $0<r <1$, $0<\eta<\rho/4$, $m\in \mathbb{N}$, $\lambda \in \Lambda$, $\tilde{q}\in \mathbf{\tilde{Q}}$, $u\in H^2(\Omega)\cap \tilde{O}_m$, we have $$\begin{aligned} &C\|\tilde{u}\|_{H^{3/2+t}(\Omega)}\le r^{\zeta}\|\tilde{u}\|_{H^2(\Omega)} \\ &\hskip 1.5cm +e^{\mathfrak{b}r^{-\nu}}\Big(\eta^{-\lambda}\|(\Delta+\tilde{q}) \tilde{u}\|_{L^2(\Omega)} \\ &\hskip 4cm + 2^{-(\lambda+1)}\|(\Delta+q) (\tilde{\phi} \tilde{u})\|_{L^{p}(\Omega)}+(3d)^m\eta^{m-(\lambda+3)}\tilde{\varrho}_m(\tilde{u})\Big),\end{aligned}$$ where $C=C(n,s, t, \vartheta,\Omega, x_0,d)>0$, $\mathfrak{b}=\mathfrak{b}(n, s, t, \vartheta,\Omega,x_0,d)>0$, $\nu=\nu(n,\Omega)>0$ are constants and $\zeta =\min ((1/2-t)/(1+t),s/4)$.* 99 Choulli M. : Uniqueness of continuation for semilinear elliptic equations. arXiv:2208.08378. Jerison, D. ; Kenig, C. E. : Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. of Math. (2) 121 (1985), no. 3, 463-494. Koch H. ; Tataru, D. : Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Commun. Pure Appl. Math. 54 (3) (2001), 339-360. Malinnikova E. ; Vessella S. : Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann. 353 (4) (2012), 1157-1181. Regbaoui, R. : Unique continuation for differential equations of Schrödinger's type. Comm. Anal. Geom. 7 (2) (1999), 303-323. Sogge, C. D. : Strong uniqueness theorems for second order elliptic differential equations. Amer. J. Math. 112 (6) (1990), 943-984. Wolff, T. H : Note on counter examples in strong unique continuation problems. Proc. Amer. Math. Soc. 114 (2) (1992), 351-356. Wolff, T. H. : Recent work on sharp estimates in second order elliptic unique continuation problems. Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992), 99-128. Studies in Advanced Mathematics. CRC, Boca Raton, Fla., 1995.

arxiv_math

--- abstract: | In this article, we discuss some geometric infinitely divisible (gid) random variables using the Laplace exponents which are Bernstein functions and study their properties. The distributional properties and limiting behavior of the probability densities of these gid random variables at $0^+$ are studied. The autoregressive (AR) models with gid marginals are introduced. Further, the first order AR process is generalised to $k^{th}$ order AR process. We also provide the parameter estimation method based on conditional least square and method of moments for the introduced AR($1$) processes. address: Department of Mathematics, Indian Institute of Technology Ropar, Punjab-140001, India. author: - Monika S. Dhull - Arun Kumar bibliography: - bibfile.bib title: Geometric Infinitely Divisible Autoregressive Models --- Bernstein function ,autoregressive model ,geometric infinite divisibility ,non-Gaussian time series modeling 60E07,60G10 # Introduction {#intro} The classical autoregressive (AR) processes are the most popular time series models, since these models are easy to understand, interpretable, and have applications in different fields. In AR processes the next term is a linear combination of previous terms and an innovation term. The distribution of innovation terms plays an important role to capture the extreme events. The classical AR model assumes that the innovation terms has Gaussian distribution, which further leads to the marginals to be Gaussian. However, many real life time series marginals exhibit heavy-tailed behaviour or semi-heavy tailed behaviour. In literature various AR models are introduced with non-Gaussian innovations and marginals. The data with non-negative observations, binary outcomes, series of counts, proportions are some examples of non-Gaussian real life time series data (see e.g. [@GAG1993]). The first order AR models with different marginal distributions are very well studied in literature. [@Gav1980] considered the AR(1) model with marginals from gamma distribution. The authors discussed that the innovation terms have distribution same as the distribution of a non-negative random variable which is exponential for positive values and has a point mass at $0$. A second order autoregressive model is considered by [@Dew1985], where Laplace distributed marginals are assumed. [@Abba1999] considered inverse Gaussian autoregressive model where the marginals are inverse Gaussian distributed. The authors shows that after fixing a parameter of the inverse Gaussian to $0$ the innovation term is also inverse Gaussian distributed. Recently, [@niha2022] studied the AR($1$) model with one-sided tempered stable marginals and innovations and demonstrated that the model fits very well on real world data. The first order AR processes are also studied by [@Jose2004; @Jose2006] with geometric $\alpha$-Laplace and geometric Pakes generalised Linnik marginals. For a literature survey on unified view of AR(1) models see [@Grunwald1996].\ In this article, we use the concept of geometric infinitely divisible distributions to define a new class of generalised AR($1$) processes. We first define a class of gid random variables with Laplace transform of the form $\frac{1}{1+g(s)}$, where $g(s)$ is a Bernstein function which is the Laplace exponent of general subordinators. A large class of distributions e.g. the geometric $\alpha$-stable (or Mittag Leffler), geometric tempered stable, geometric gamma, geometric inverse Gaussian, one-sided geometric Laplace distributions and others come under this umbrella. We mainly focus on geometric tempered stable, geometric inverse Gaussian and geometric gamma distributions and corresponding AR models in this article. As per our knowledge these models and corresponding AR processes are not explored in literature. First the properties of gid marginals are studied in section [2](#section2){reference-type="ref" reference="section2"}. Section [3](#GBAR_model){reference-type="ref" reference="GBAR_model"} describes the AR$(1)$ model with gid marginals. We also obtain the distribution of innovation terms which are from the class of gid distributions except scale change. We then evaluate the pdf of innovation terms in integral form for the geometric tempered stable, geometric gamma, and geometric inverse Gaussian. Further, we generalise the result obtained in this section to the $k^{th}$ order autoregressive processes. In section [4](#section4){reference-type="ref" reference="section4"} we use the conditional least square (CLS) and method of moments (MOM) to estimate the parameters of the AR($1$) model defined in [Proposition 5](#ARmodel){reference-type="ref" reference="ARmodel"} and provide the simulation study for the same. Section [5](#conclusion){reference-type="ref" reference="conclusion"} concludes the work. # Geometric infinite divisible distributions {#section2} In this section, we introduce gid distributions. Infinitely divisible random variables (or distributions) with positive support play a crucial role in the study of subordinators [@applebaum]. The subordinators are real-valued non-decreasing stochastic processes having independent and stationary increments. The properties of subordinators are well studied in literature (for e.g. see, [@applebaum; @arun2018; @kumar_vellai]). Bernstein functions play a crucial role in the theory of non-decreasing Lévy processes or subordinators. A function $\phi: (0,\infty) \rightarrow (0, \infty)$ is called a Bernstein function if it admits the following representation $$\phi(s) = a + bs + \int_{0}^{\infty}(1-e^{-sx})\nu(dx),$$ where $a\geq 0$, $b\geq 0$ and $\nu$ is called the Lévy measure which is defined on $(0, \infty)$. Alternatively, the function $\phi: (0,\infty) \rightarrow (0, \infty)$ is a Bernstein function if $\phi$ is of class $\mathbb{C^{\infty}}$ and $\phi(s) \geq 0$, for all $s\geq 0$ and $(-1)^{n-1}\phi^{(n)}(s) \geq 0,$ for all $n\in \mathbb{N}$ and $s>0$ ([@book1]). For a subordinator $S(t)$, it follows that $\mathbb{E}[e^{-sS(t)}] = e^{-t\phi(s)}$, where $\phi(s)$ is also called the Laplace exponent. Next we define the gid' marginals.\ Consider a positive random variable $Y$ with Laplace transform $f(s)$ that is $\mathbb{E}[e^{-sY}] = f(s).$ Let $X$ be a positive infinitely divisible random variable with Laplace transform $\mathbb{E}[e^{-sX}] = e^{-g(s)}$, where $g(s)$ is a Bernstein function ([@book1]). Now we can write the Laplace transform of $X$ as $$\label{gid} \phi_X(s) = e^{-g(s)} = e^{1-g(s)-1} = e^{1-\frac{1}{(1+g(s))^{-1}}} = e^{1- \frac{1}{f(s)}},\; s>0.$$ It is known from [@Klebanov1985] that a distribution with Laplace transform $f(s)$ is geometrically infinitely divisible if and only if the distribution with Laplace transform $e^{1-\frac{1}{f(s)}}$ is infinitely divisible. One can define a class of geometrically infinitely divisible random variables by choosing $g(s)$ as the Laplace exponent of general subordinators and we call these random variables as the gid based random variables. We introduce a class of random variables $Y$ on ($0,\infty$) with Laplace transform of the form $f(s) = \frac{1}{1+g(s)}$, where $g(s)$ is the Laplace exponent of positive infinitely divisible random variables and study the relevant properties for time series autoregressive models. We extend the concept of geometric $\alpha-$Laplace ([@Jose2004]), geometric Pakes generalised Linnik marginals ([@Jose2006]) based autoregressive models to general class of autoregressive models with gid marginals. **Definition 1**. The random variable $Y$ on ($0,\infty$) is said to have gid marginals if its Laplace transform is given by $f(s) = \frac{1}{1+g(s)}$, where $g(s)$ is a Bernstein function and the Laplace exponent of positive infinitely divisible random variable. In particular, we consider the different functions for $g(s)$ and define the Laplace transform of corresponding gid marginals: (a) Geometric tempered stable: The Laplace transform of tempered stable random variable is $e^{-g(s)}$ with $g(s) = (s+\lambda)^{\beta}-\lambda^{\beta}$, then $f(s) = \frac{1}{1+(s+\lambda)^{\beta}-\lambda^{\beta}}, \, \lambda>0 \text{ and }\beta\in(0,1).$ (b) Geometric gamma: The Laplace transform of gamma random variable is $e^{-g(s)}$ with $g(s) = \alpha\log(1+\frac{s}{\beta}),$ then $f(s) = \frac{1}{1+\alpha \log\left(\frac{s+\beta}{\beta}\right)}, \, \alpha>0 \text{ and } \beta>0.$ (c) Geometric inverse Gaussian: The Laplace transform of inverse Gaussian random variable is $e^{-g(s)}$ with $g(s) = \delta\gamma\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}$, then $f(s) = \frac{1}{1+\delta\gamma\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}}, \, \delta>0 \text{ and } \gamma>0.$ **Proposition 1**. *Consider the identically and independently distributed random variables $X_1, X_2, \hdots,$ with gid marginals defined in Def. [Definition 1](#def2){reference-type="ref" reference="def2"}. Also consider $N(\theta)$ be a geometric random variable with mean $\frac{1}{\theta}$ and $P[N(\theta)=r] = \theta(1-\theta)^{r-1}$, $r = 1,2,\cdots, \, 0<\theta<1.$ Then $Y = \sum_{i=1}^{N(\theta)} X_{i}$ also have gid marginals with a scale change.* *Proof.* Let the Laplace transform of each $X_i$ for $i=1,2, \hdots$ is denoted by $\phi_X(s)$. Then, the Laplace transform of random variable $Y$ can be written as, $$\begin{aligned} \phi_{Y}(s) &= \sum_{r=1}^{\infty}[\phi_{X}(s)]^{r} \theta(1-\theta)^{r-1}\, =\sum_{r=1}^{\infty}\Big[\frac{1}{1+g(s)}\Big]^{r} \theta(1-\theta)^{r-1}\\ &=\frac{\frac{\theta}{1+g(s)}}{1-\big(\frac{1-\theta}{1+g(s)}\big)}\, =\frac{\theta}{\theta + g(s)}\,=\frac{1}{1+\frac{g(s)}{\theta}}. \end{aligned}$$ Hence, $Y$ is also a gid random variable. ◻ Next, we study the limiting behavior of the densities of gid random variables near $0^+$ using Laplace transform. Since the pdf of gid don't have closed form i.e. they have integral representations, therefore we apply the Tauberian theorem defined in [@Feller1971] (pp. 446, Theorem 4). In the following theorem we apply the Tauberian theorem on the geometric tempered stable and geometric inverse Gaussian cases. **Theorem 1**. *Consider the following Laplace transform $W(s)$ for their corresponding density function $f(x)$, then we obtain the asymptotic behavior of $f(x)$ at $0^{+}$.* (a) *For geometric tempered stable, $W(s)=\frac{1}{1+\theta((s+\lambda)^\beta-\lambda^\beta)}$, then as $x\to 0^+$ density $f(x)\sim \frac{x^{\beta-1}}{\theta\Gamma(\beta)}$, where $\lambda>0$ and $0 < \beta < 1$.* (b) *For geometric inverse Gaussian, $W(s)=\frac{1}{1+\theta\delta\{\sqrt{\gamma^2+2s}-\gamma\}}$, then as $x\to 0^{+}$ density $f(x)\sim \frac{x^{-1/2}}{\theta\delta\sqrt{2\pi}}$, where $\delta>0$, and $\gamma>0$.* *Proof.* (a) The Laplace transform of geometric tempered stable is $W(s)=\frac{1}{1+\theta((s+\lambda)^\beta-\lambda^\beta)}$, where $\lambda>0,\,0 < \beta < 1.$ For large $s$, $W(s)\sim \frac{1}{\theta s^{\beta}}.$ We apply Tauberian theorem with $\rho = \beta$ and the slowly varying function $L(1/s)=\frac{1}{\theta}$. Thus, the density $$f(x)\sim \frac{x^{\beta-1}}{\theta\Gamma(\beta)}, \text{ as } x\to 0^{+}.$$ (b) The Laplace transform of geometric inverse Gaussian is $W(s)=\frac{1}{1+\theta\delta\{\sqrt{\gamma^2+2s}-\gamma\}}$, where $\delta>0, \gamma>0$. Again for large $s$, $W(s)\sim \frac{1}{\theta \delta \sqrt{2s}}.$ The slowly varying function $L(1/s)=\frac{1}{\theta\delta \sqrt{2}}$ and $\rho=1/2.$ By Tauberian theorem, $f(x)\sim \frac{x^{-1/2}}{\theta\delta\sqrt{2\pi}},\text{ as } x\to 0^{+}$.  ◻ The density plots of the gid distributions specially, geometric tempered stable and geometric inverse Gaussian are also in accordance with the results obtained in above theorem. We use the method of Laplace transform applied in [@Ridout2009] to generate the data of length $1000$ in $R$ programming language. The density plots of gid with different parameter values as shown in Fig. [\[fig2\]](#fig2){reference-type="ref" reference="fig2"}. ## Mixture of gid random variables We introduce a new random variable $M$ which is a mixture of two independent gid random variables say $Y_1$ and $Y_2$. We define $g(s) = cg_1(s) + (1-c)g_2(s)$, where $0<c<1$, $g_1$ and $g_2$ are Laplace exponent functions for $Y_1$ and $Y_2$ respectively. We compute $\phi_M(s)$ the Laplace for each case as follows: $$\begin{aligned} e^{-g(s)} &= e^{-(c g_1(s)+(1-c)g_2(s))}= e^{1-1-(c g_1(s)+(1-c)g_2(s))} \\ &= e^{1-\frac{1}{(1+cg_1(s) +(1-c)g_2(s))^{-1}}} \end{aligned}$$ The Laplace of gid mixture $M$ is, $$\phi_M(s) = \frac{1}{1+cg_1(s) +(1-c)g_2(s)}\\ =\frac{1}{1 + g_2(s) + c(g_1(s)-g_2(s))},$$ for $0<c<1$. We substitute the values of functions $g_1(s)$ and $g_2(s)$ to obtain the Laplace transform of the following: (a) Geometric mixture tempered stable: For $0<c<1,\, 0<\beta_1, \beta_2<1$ and $\lambda_1, \lambda_2>0,$ $$\phi_M(s) = \frac{1}{1 + (s+\lambda_2)^{\beta_2}-\lambda_2^{\beta_2} + c((s+\lambda_1)^{\beta_1}-\lambda_1^{\beta_1}-(s+\lambda_2)^{\beta_2}+\lambda_2^{\beta_2})}.$$ (b) Geometric mixture gamma: For $0<c<1,\, \beta_1, \beta_2>0$ and $\alpha_1, \alpha_2>0,$ $$\phi_M(s) = \frac{1}{1 + \alpha_2\log(\frac{s+\beta_2}{\beta_2})+ c\{\alpha_1\log(\frac{s+\beta_1}{\beta_1})-\alpha_2\log(\frac{s+\beta_2}{\beta_2})\}}.$$ (c) Geometric mixture inverse Gaussian: For $0<c<1,\, \delta_1, \delta_2>0$ and $\gamma_1, \gamma_2>0,$ $$\phi_M(s) = \frac{1}{1 + \delta_2\Bigl(\sqrt{2s+\gamma_2^2}-\gamma_2\Bigr)+ c\Bigl(\delta_1\Bigl(\sqrt{2s+\gamma_1^2}-\gamma_1\Bigr) -\delta_2\Bigl(\sqrt{2s+\gamma_2^2}-\gamma_2\Bigr)\Bigr)}.$$ *Remark 1*. One can study the asymptotic behavior of gid mixture as done in [Theorem 1](#Theorem2.1){reference-type="ref" reference="Theorem2.1"}. Also, further these gid mixtures can be used to define autoregressive models as discussed in next section. # Autoregressive model {#GBAR_model} In this section, we develop the AR$(1)$ process with $\{Y_{n}\}$ as gid based marginals. Consider the following AR$(1)$ process: $$\label{GBmodel} Y_{n} = \begin{cases} \epsilon_n, & \text{ with probability } \theta,\\ Y_{n-1} + \epsilon_n, & \text{ with probability } 1-\theta, \end{cases}$$ where $0<\theta<1$. **Theorem 2**. *Consider a stationary AR$(1)$ process $\{Y_{n}\}$ as defined in Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"}. Then $\{Y_{n}\}$ and $\{\epsilon_{n}\}$ are identically distributed except for a scale change if and only if $\{Y_{n}\}$ has gid marginals.* *Proof.* First assume that the marginal distribution of $\{Y_n\}$ is are gid. Then the Laplace transform of $\{Y_{n}\}$ is $\phi_{Y_n}(s)= \frac{1}{1+g(s)}$ and innovation terms $\{\epsilon_n\}$ is denoted by $\phi_{\epsilon_n}(s)$. We can write Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"} in terms of Laplace transform as, $$\phi_{Y_n}(s) = \theta \phi_{\epsilon_n}(s) + (1-\theta)\phi_{Y_{n-1}}(s)\phi_{\epsilon_n}(s).$$ Using the stationarity condition and the Laplace transform $\phi_{Y}(s) =\frac{1}{1+g(s)}$, we rewrite above equation as, $$\begin{aligned} \phi_{Y}(s) &= \theta \phi_{\epsilon}(s) + (1-\theta)\phi_{Y}(s)\phi_{\epsilon}(s)\, = \phi_{\epsilon}(s)\{\theta + (1-\theta)\phi_{Y}(s)\}\\ \phi_{\epsilon}(s)&= \frac{\phi_{Y}(s)}{\theta + (1-\theta)\phi_{Y}(s)}\, = \frac{1}{1 + \theta g(s)}.\label{main_eq2} \end{aligned}$$ Hence, we conclude that innovation terms {$\epsilon_n$} are gid with $\theta$ scale change.\ Conversely, assume that $\{Y_{n}\}$ and $\{\epsilon_{n}\}$ are identically distributed except for scale change say $\phi_{\epsilon}(s) = \phi_{Y}(as)$, where $a$ is a constant, $\phi_{Y}(s)$ and $\phi_{\epsilon}(s)$ is Laplace transform of $\{Y_{n}\}$ and innovation terms $\{\epsilon_n\}$ respectively. Using $\phi_{\epsilon}(s) = \frac{1}{1+g(s)}$ and $\phi_{Y}(s)$, we get, $$\begin{aligned} \phi_{Y_n}(s) &= \theta \phi_{\epsilon_n}(s) + (1-\theta)\phi_{Y_n}(s)\phi_{\epsilon_n}(s)\, = \phi_{\epsilon}(s)\{\theta + (1-\theta)\phi_{Y}(s)\}\\ \phi_{Y}(s) &= \frac{\theta \phi_{\epsilon}(s)}{1-\{1-\theta \phi_{\epsilon}(s)\}}\,= \frac{\theta}{1+g(s)-(1-\theta)}\\ &=\frac{\theta}{\theta + g(s)}\,= \frac{1}{1+\frac{1}{\theta}g(s)}. \end{aligned}$$ Take $a = \frac{1}{\theta}$, we get that $\{Y_n\}$ are also gid with scale change of $\frac{1}{\theta}$.\ Hence, we have the Laplace transform of $\{\epsilon_n\}$ as gid with scale change of $\theta$ and the marginals $\{{Y_n}\}$ will also be same as innovation terms, that is gid. ◻ We find the joint Laplace transform of $Y_n$ and $Y_{n-1}$ and check for time reversibility of the process in the following result. **Proposition 2**. *Consider the stationary AR$(1)$ process as defined in Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"}. Then we can rewrite $Y_n$ as $Y_n = I_nY_{n-1} + \epsilon_n$, where $P[I_n=0] = \theta = 1-P[I_n=1], 0<\theta<1$. Then the $AR(1)$ process is not time reversible.* *Proof.* The Laplace transform of joint variables $(Y_n, Y_{n-1})$ is calculated as follows: $$\begin{aligned} \phi_{Y_{n-1}, Y_n}(s_1, s_2) &= \mathbb{E}[\exp(-s_1 Y_{n-1} - s_2Y_n)] \,= \mathbb{E}[\exp(-s_1 Y_{n-1} - s_2[I_nY_{n-1} + \epsilon_n]] \\ &= \mathbb{E}[\exp(-(s_1 + s_2I_n) Y_{n-1} - s_2\epsilon_n)] \, = \mathbb{E}[\exp(-(s_1 + s_2I_n) Y_{n-1})] \phi_{\epsilon_n}(s_2)\\ &=\frac{1}{1+\theta g(s_2)} \Big[\frac{\theta}{1+g(s_1)} + \frac{1-\theta}{1+g(s_1 + s_2)}\Big]. \end{aligned}$$ The joint Laplace transform of the process implies that it is not time reversible. ◻ First we generate the i.i.d. innovation terms $\{\epsilon_t\}$ using the Laplace transform as discussed in [@Ridout2009], then generate $\{Y_t\}$ from AR($1$) model defined in Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"} with $\theta=0.3$. The plot of innovation terms $\{\epsilon_t\}$ for geometric tempered stable marginals, geometric gamma marginals and geometric inverse Gaussian are shown in Fig. [\[fig4\]](#fig4){reference-type="ref" reference="fig4"}. The time series $\{Y_t\}$ from AR($1$) model with $\theta=0.3$ for all the three cases are shown in Fig. [\[fig3\]](#fig3){reference-type="ref" reference="fig3"}. In the next results, we discuss the form of the pdf of innovation terms using Laplace transform and complex inversion formula. For theory of complex inversion formula one can refer Chap. $4$ of [@Schiff1999]. **Theorem 3**. *Consider a stationary AR$(1)$ process $\{Y_{n}\}$ as defined in Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"}. If $\{Y_{n}\}$ is marginally distributed as geometric tempered stable with Laplace transform $\phi_{Y}(s)=\frac{1}{1+(s+\lambda)^\beta - \lambda^\beta}$, for $\beta=\frac{1}{m}, m=2,3,\cdots$, then the innovation terms $\{\epsilon_n\}$ also follow geometric tempered stable distribution with Laplace transform $\phi_{\epsilon}(s)=\frac{1}{1+\theta((s+\lambda)^\beta - \lambda^\beta)}$. Moreover, the pdf of innovation terms has the following integral form: $$f_\epsilon(x)=\frac{e^{(s_0-\lambda)x}}{\theta \beta s_0^{\beta-1}}+\frac{1}{\pi} \int_{0}^{\infty} \frac{\theta e^{-x(y+\lambda)}y^{\beta}\sin(\pi\beta)}{1+2\theta(y^{\beta}\cos(\pi\beta)-\lambda^\beta)+\theta^2(y^{2\beta}-2\lambda^\beta y^\beta \cos(\pi\beta)+\lambda^{2\beta})} \,dy.\label{main_eq1}$$ where $s_0=(\lambda^{\beta}-1/\theta)^{\beta-1}, \beta\in(0,1), \beta=\frac{1}{m}, m=2,3,\cdots \text{ and } \lambda>0.$* *Proof.* We substitute $g(s)=\frac{1}{1+(s+\lambda)^\beta - \lambda^\beta}$ in Eq. [\[main_eq2\]](#main_eq2){reference-type="eqref" reference="main_eq2"} to obtain the Laplace transform of innovation terms $\phi_{\epsilon}(s)=\frac{1}{1+\theta((s+\lambda)^\beta - \lambda^\beta)}, \,\beta=\frac{1}{m},\, m=2,3,4,\hdots$. Consider the function $G(s)= \frac{1}{1+\theta(s^\beta-\lambda^\beta)}.$ For the function $G(s)$, with $\beta=\frac{1}{m},\, m=2,3,\hdots$, it follows that $s_0 = (\lambda^\beta-1/\theta)^{1/\beta}$ is a simple pole and $s_1=(0,0)$ is the branch point. We use complex inversion formula to compute the Laplace inverse of $G(s)$ which in turn gives the pdf $f_\epsilon(x)$ of innovation terms $$f_{\epsilon}(x) = \frac{1}{2\pi\iota}\int_{x_0-i\infty}^{x_0+i\infty}e^{sx}\phi_\epsilon(s)\,ds,$$ where $x_0>a$ is chosen such that the integrand is analytic for $Re(s)>a.$ Consider the contour $ABCDEF$ in Fig. [1](#fig_1){reference-type="ref" reference="fig_1"} with branch point $s_1=(0,0)$, circular arcs $AB$ and $EF$ of radius $R$, arc $CD$ of radius $r$, line segments $BC$ and $DE$ parallel to $x-$axis and $AF$ is line segment from $x_0-iy$ to $x_0+iy$. For closed curve $C$ and poles $s_i$ inside $C$, Cauchy residue theorem states that, $$\frac{1}{2\pi\iota}\oint_C e^{sx}G(s)\,ds = \sum_{i=1}^{n}Res(e^{sx}G(s), s_i).$$ In limiting case, for contour $ABCDEF$, the integral on circular arcs $AB$ and $EF$ tend to $0$ as $R\to \infty.$ The integral over $CD$ also tends to $0$ as $r\to 0.$ We need to compute the following integral: $$\frac{1}{2\pi\iota}\int_{x_0-\iota \infty}^{x_0+\iota\infty} e^{sx}G(s)\,ds =Res(e^{sx}G(s), s_0) - \frac{1}{2\pi\iota}\int_{BC}e^{sx}G(s)\,ds - \frac{1}{2\pi\iota}\int_{DE}e^{sx}G(s)\,ds.\label{eq_2}$$ {#fig_1 width="70%"} Now, along $BC$, let $s=ye^{\iota\pi}$, then $ds=-dy$ and, $$\begin{aligned} \frac{1}{2\pi\iota}\int_{-R}^{-r}\frac{e^{sx}}{1+\theta(s^\beta-\lambda^\beta)}\,ds &= \frac{1}{2\pi\iota}\int_{r}^{R}\frac{e^{-xy}}{1+\theta(y^{\beta}e^{\iota\pi\beta}-\lambda^{\beta})}\,dy \\ &= \frac{1}{2\pi\iota}\int_{0}^{\infty}\frac{e^{-xy}}{1+\theta(y^{\beta}e^{\iota\pi\beta}-\lambda^{\beta})}\,dy.\label{eq_3} \end{aligned}$$ Along $DE$, let $s=ye^{-\iota\pi}$, then $ds=-dy$ and, $$\begin{aligned} -\frac{1}{2\pi\iota}\int_{-r}^{-R}\frac{e^{sx}}{1+\theta(s^\beta-\lambda^\beta)}\,ds &= -\frac{1}{2\pi\iota}\int_{r}^{R}\frac{e^{-xy}}{1+\theta(y^{\beta}e^{-\iota\pi\beta}-\lambda^{\beta})}\,dy \\ &= -\frac{1}{2\pi\iota}\int_{0}^{\infty}\frac{e^{-xy}}{1+\theta(y^{\beta}e^{-\iota\pi\beta}-\lambda^{\beta})}\,dy.\label{eq_4} \end{aligned}$$ Now substitute Eq. [\[eq_3\]](#eq_3){reference-type="eqref" reference="eq_3"} and [\[eq_4\]](#eq_4){reference-type="eqref" reference="eq_4"} in Eq. [\[eq_2\]](#eq_2){reference-type="eqref" reference="eq_2"}. $$\begin{aligned} \frac{1}{2\pi\iota}&\int_{x_0-\iota \infty}^{x_0+\iota\infty} e^{sx}G(s)\,ds =Res(e^{sx}G(s), s_0) - \frac{1}{2\pi\iota}\Big[\int_{0}^{\infty}\frac{e^{-xy}}{1+\theta(y^{\beta}e^{\iota\pi\beta}-\lambda^{\beta})}\,dy\\ &-\frac{1}{2\pi\iota}\int_{0}^{\infty}\frac{e^{-xy}}{1+\theta(y^{\beta}e^{-\iota\pi\beta}-\lambda^{\beta})}\,dy\Big]\\ &=Res(e^{sx}G(s), s_0) - \frac{1}{2\pi\iota}\int_{0}^{\infty}\frac{e^{-xy}(\theta y^\beta(e^{-\iota\pi\beta}-e^{\iota\pi\beta}))}{(1+\theta(y^{\beta}e^{\iota\pi\beta}-\lambda^{\beta}))(1+\theta(y^{\beta}e^{-\iota\pi\beta}-\lambda^{\beta}))}\,dy\\ %&=Res(e^{sx}G(s), s_0) - \frac{1}{2\pi\iota}\int_{0}^{\infty}\frac{-e^{-xy}(\theta y^\beta\sin(\pi\beta))}{1+\theta(y^{\beta}(\cos\pi\beta+\iota\sin\pi\beta)-2\lambda^{\beta}+y^{\beta}(\cos\pi\beta-\iota\sin\pi\beta))+\theta^2(y^{2\beta}-\lambda^{\beta}y^{\beta}(2\cos\pi\beta)+\lambda^{2\beta}))}\,dy\\ &=Res(e^{sx}G(s), s_0) + \frac{1}{\pi}\int_{0}^{\infty}\frac{e^{-xy}\theta y^\beta\sin(\pi\beta)}{1+2\theta(y^{\beta}\cos\pi\beta-\lambda^{\beta})+\theta^2(y^{2\beta}-2\lambda^{\beta}y^{\beta}\cos\pi\beta+\lambda^{2\beta})}\,dy \end{aligned}$$ Now we find the $Res(e^{sx}G(s))$ for poles at $s_0=(\lambda^\beta-1/\theta)^{m}, \, \beta =1/m,\; m=2,3,4,\cdots$. Also, note that all the poles $s_0<0$ are outside the analytic region, therefore we will calculate the residue at all those $s_0$ where $(\lambda^\beta-1/\theta)^m>0$ i.e. $\lambda>\frac{1}{\theta^m}$. We evaluate the residue as, $$\lim_{s \to s_0}\frac{(s-s_0)e^{sx}}{1+\theta(s^\beta-\lambda^\beta)}= \frac{e^{s_0x}}{\theta\beta(s_0)^{\beta-1}},$$ where $s_0=(\lambda^\beta-1/\theta)^{m}.$ The inverse Laplace of $G(s)$ is $$\mathcal{L}^{-1}\{G(s)\}=\frac{e^{s_0x}}{\theta\beta(s_0)^{\beta-1}}+\frac{1}{\pi}\int_{0}^{\infty}\frac{e^{-xy}\theta y^\beta\sin(\pi\beta)}{1+2\theta(y^{\beta}\cos\pi\beta-\lambda^{\beta})+\theta^2(y^{2\beta}-2\lambda^{\beta}y^{\beta}\cos\pi\beta+\lambda^{2\beta})}\,dy.$$ We obtain the pdf by using the first translational property of inverse Laplace. $$\begin{aligned} f_\epsilon(x) &= \mathcal{L}^{-1}\{G(s+\lambda)\}\, = e^{-\lambda x}\mathcal{L}^{-1}{G(s)}\\ &= e^{-\lambda x}\Big[\frac{e^{s_0x}}{\theta\beta(s_0)^{\beta-1}}+\frac{1}{\pi}\int_{0}^{\infty}\frac{e^{-xy}\theta y^\beta\sin(\pi\beta)}{1+2\theta(y^{\beta}\cos\pi\beta-\lambda^{\beta})+\theta^2(y^{2\beta}-2\lambda^{\beta}y^{\beta}\cos\pi\beta+\lambda^{2\beta})}\,dy\Big]. \end{aligned}$$ Hence, we obtain the form of pdf as mentioned in [\[main_eq1\]](#main_eq1){reference-type="eqref" reference="main_eq1"}. ◻ *Remark 2*. For $\lambda=0, \phi_Y(s)=\frac{1}{1+s^{\beta}}$ and $\phi_{\epsilon}(s)=\frac{1}{1+\theta s^{\beta}}$, which is the Laplace transform of one side geometric stable distribution also known as Mittag-Leffler distribution. For $\theta=1$, poles will be $s_0=(-1)^m,\, m=2,3,4\cdots$. The residue for the pole corresponding to $s_0=1$ is $e^x.$ Then, the pdf becomes, $$\begin{aligned} f_{\epsilon}(x) = e^{x}+\frac{1}{\pi}\int_{0}^{\infty}\frac{e^{-xy}y^\beta\sin(\pi\beta)}{1+2 y^{\beta}\cos(\pi\beta) + y^{2\beta}}\,dy, \text{ where }s_0=1. \label{dist1} \end{aligned}$$ *Remark 3*. In [\[dist1\]](#dist1){reference-type="eqref" reference="dist1"}, we take $\beta=1/2$ or equivalently $m=2$ and obtain the density using the results from [@Abram](pp. 303-304), $$\begin{aligned} f_{\epsilon}(x) &= e^{x}+\frac{1}{\pi}\int_{0}^{\infty}\frac{e^{-xy}\sqrt{y}}{1+y}\,dy = e^{x} +\frac{1}{\sqrt{\pi x}}-e^x+e^x Erf(\sqrt{x})\\ & =\frac{1}{\sqrt{\pi x}}+e^x Erf(\sqrt{x}). \end{aligned}$$ As $x\to 0,\, Erf(\sqrt{x}) \to 0 \text{ and }f_{\epsilon}(x) \sim \frac{1}{\sqrt{\pi x}},$ which match with the asymptotic behaviour discussed in Theorem [Theorem 1](#Theorem2.1){reference-type="ref" reference="Theorem2.1"}. **Proposition 3**. *For a stationary AR$(1)$ process with $\{Y_{n}\}$ defined as in Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"} with marginals distributed as geometric gamma with Laplace transform $\phi_{Y}(s)=\frac{1}{1+\alpha\log(\frac{s+\beta}{\beta})}$, then the innovation terms $\{\epsilon_n\}$ also follow geometric gamma with Laplace transform $\phi_{\epsilon}(s)=\frac{1}{1+\alpha\theta\log(\frac{s+\beta}{\beta})}$. Moreover, the pdf of innovation terms has the following integral form: $$f_{\epsilon}(x)=\frac{e^{s_0 x}(s_0+\beta)^{\alpha\theta}}{\beta^{\alpha\theta}} -e^{-\beta x} \int_{0}^{\infty}\frac{\alpha\theta e^{-xy}}{1+2\alpha\theta \log(y/\beta)+\pi^2\alpha^2\theta^2 +\alpha^2\theta^2(\log(y/\beta))^2}\,dy,$$ where $s_0=\beta(e^{-1/\alpha\theta}-1),\, \alpha>0 \text{ and } \beta>0.$* *Proof.* Again we substitute $g(s)=\frac{1}{1+\alpha\log(\frac{s+\beta}{\beta})}$ in Eq. [\[main_eq2\]](#main_eq2){reference-type="eqref" reference="main_eq2"} and get the Laplace transform of innovation terms $\phi_{\epsilon}(s)=\frac{1}{1+\alpha\theta\log(\frac{s+\beta}{\beta})}$. Again we use the complex inversion formula to obtain the pdf of innovation terms. We consider the function $G(s)=\frac{1}{1+\theta\alpha\log(\frac{s+\beta}{\beta})}$ and the pole for $G(s)$ at $s_0=\beta(e^{-1/\alpha\theta}-1)$ and branch point at $s = -\beta$. The residue corresponding to $s_0$ is $\frac{e^{s_0x}(s_0+\beta)^{\alpha\theta}}{\beta^{\theta\alpha}}.$ Using a similar contour as given in Fig. [1](#fig_1){reference-type="ref" reference="fig_1"} and the same steps to obtain the pdf as $$f_{\epsilon}(x)=\frac{e^{s_0 x}(s_0+\beta)^{\alpha\theta}}{\beta^{\alpha\theta}} -e^{-\beta x} \int_{0}^{\infty}\frac{\alpha\theta e^{-xy}}{1+2\alpha\theta \log(y/\beta)+\pi^2\alpha^2\theta^2 +\alpha^2\theta^2(\log(y/\beta))^2}\,dy.$$ ◻ **Proposition 4**. *For a stationary AR$(1)$ process with $\{Y_{n}\}$ defined as in Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"} with marginals distributed as geometric inverse Gaussian with Laplace transform $\phi_{Y}(s)=\frac{1}{1+\gamma\delta\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}}$, then the innovation terms $\{\epsilon_n\}$ also follows geometric inverse Gaussian with Laplace transform $\phi_{\epsilon}(s)=\frac{1}{1+\theta\gamma\delta\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}}$. Moreover, the pdf of innovation terms has the following integral form: $$f_{\epsilon}(x)=\frac{e^{s_0 x}\sqrt{2s_0+\gamma^2}}{\theta\delta} +\frac{\gamma^2e^{-x}}{2\pi}\int_{0}^{\infty}\frac{\theta\delta\sqrt{y}e^{-\frac{2xy}{\gamma^2}}}{1-2\theta\delta\gamma+\theta^2\delta^2(y+\gamma^2-2\gamma\sqrt{y}\cos{\theta})}\,dy,$$ where $s_0=\frac{\gamma^2}{2}(1-\frac{1}{\theta\gamma\delta})^2-\frac{\gamma^2}{2}, \,\delta>0 \text{ and } \gamma>0.$* *Proof.* The Laplace transform of innovation terms $\phi_{\epsilon}(s)=\frac{1}{1+\theta\gamma\delta\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}}$ is straight forward from Eq. [\[main_eq2\]](#main_eq2){reference-type="eqref" reference="main_eq2"}. Now we use the complex inversion formula to obtain the pdf of innovation terms as done in previous theorem. We consider the function $G(s)=\frac{1}{1+\theta\delta\gamma\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}}$ and the pole for $G(s)$ is $s_0=\frac{\gamma^2}{2}\Bigl\{1-\frac{1}{\theta\delta\gamma}\Bigr\} - \frac{\gamma^2}{2}.$ The residue corresponding to $s_0$ is $\frac{e^{s_0x}\sqrt{2s_0+\gamma^2}}{\theta\delta}.$ We use the same steps to obtain the pdf as $$f_{\epsilon}(x)=\frac{e^{s_0 x}\sqrt{2s_0+\gamma^2}}{\theta\delta} +\frac{\gamma^2e^{-x}}{2\pi}\int_{0}^{\infty}\frac{\theta\delta\sqrt{y}e^{-\frac{2xy}{\gamma^2}}}{1-2\theta\delta\gamma+\theta^2\delta^2(y+\gamma^2-2\gamma\sqrt{y}\cos{\theta})}\,dy.$$ ◻ We know that for a random process, the moments (if exists) uniquely describe the properties of the random variable $X$. We provide the $k^{th}$ order moments using the Laplace transform in the following result. The $k^{th}$ order moment of random variable $X$ is evaluated as: $\mathbb{E}(X^{k}) = (-1)^{k}\phi^{(k)}(s)$ for $s=0$ and $k\in \mathbb{N}$. (a) Geometric tempered stable innovations: $\phi_\epsilon(s) = \dfrac{1}{1+\theta\{(s+\lambda)^\beta-\lambda^\beta\}}$. Then,\ $\mathbb{E}(\epsilon) = \theta\beta\lambda^{\beta-1},\quad \mathbb{E}(\epsilon^2) = \theta\beta(\beta-1)\lambda^{\beta-2}-2\theta^2\beta^2(\lambda)^{2\beta-2},\lambda >0.$ (b) Geometric gamma innovations: $\phi_{\epsilon}(s)=\dfrac{1}{1+\alpha\theta\log\left(\dfrac{s+\beta}{\beta}\right)}$. Then,\ $\mathbb{E}(\epsilon) = \dfrac{\theta\alpha}{\beta}, \quad \mathbb{E}(\epsilon^2) = \dfrac{\theta\alpha + 2\theta^{2}\alpha^2}{\beta^2}.$ (c) Geometric inverse Gaussian innovations: $\phi_\epsilon(s) = \dfrac{1}{1+\theta\gamma\delta\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}}$. Then,\ $\mathbb{E}(\epsilon) = \dfrac{\theta\delta}{\gamma}, \quad \mathbb{E}(\epsilon^2) = \dfrac{2\theta^{2}\delta^{2}}{\gamma^2} + \dfrac{\theta\delta}{\gamma^3}.$ ## Generalisation to $k^{th}$ order autoregressive processes We define the generalised form of the process developed in previous section by Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"} as follows: $$Y_{n} = \begin{cases} \epsilon_n, & \text{ with probability } \theta,\\ Y_{n-1} + \epsilon_n, & \text{ with probability } 1-\theta_1,\\ Y_{n-2} + \epsilon_n, & \text{ with probability } 1-\theta_2,\\ \vdots\\ Y_{n-k} + \epsilon_n, & \text{ with probability } 1-\theta_k, \end{cases} \label{GGBmodel},$$ where $\sum_{i=1}^{k}\theta_i = 1-\theta, \,$ $0\leq\theta \leq1,\, i=1,2, \hdots, k.$ Also, $\{\epsilon_n\}$ are independent of $\{Y_{n-1}, Y_{n-2}, \cdots, \}$. We write the Laplace transform $\phi_{Y_n}(s)$ for the model defined in Eq. [\[GGBmodel\]](#GGBmodel){reference-type="eqref" reference="GGBmodel"}. $$\begin{aligned} \phi_{Y_n}(s) &= \theta\phi_{\epsilon_n}(s) + \theta_1\phi_{Y_{n-1}}(s)\phi_{\epsilon_n}(s) + \cdots + \theta_k\phi_{Y_{n-k}}(s)\phi_{\epsilon_n}(s)\\ &= \phi_{\epsilon_n}(s)\{\theta + \theta_1\phi_{Y_{n-1}(s)} + \cdots + \theta_k\phi_{Y_{n-k}(s)}\}. \end{aligned}$$ Since series is stationary, we get, $$\begin{aligned} \phi_{Y}(s)&=\phi_{\epsilon}(s){\theta + \sum_{i=1}^{k}\theta_i\phi_{Y}(s)} =\phi_{\epsilon}(s)\{\theta + {(1-\theta)}\phi_{Y}(s)\};\\ \phi_{\epsilon}(s)&=\frac{\phi_Y(s)}{\theta+(1-\theta)\phi_Y(s)}. \end{aligned}$$ Hence, we obtain the similar form for innovation terms $\{\epsilon_{n}\}$ as obtained in previous section. Next we define AR$(1)$ process with $|\theta|<1$ that is, stationary process and find the Laplace transform of the innovation terms. **Proposition 5**. *Consider the AR$(1)$ process $Y_n=\theta Y_{n-1}+ \epsilon_n, \, |\theta|<1$ is strictly stationary with Laplace transform of marginals as $\phi_{Y}(s)=\frac{1}{1+g(s)}$ then the Laplace transform of innovation terms $\{\epsilon_n\}$ is $\phi_{\epsilon}(s)=\frac{1+g(\theta s)}{1+g(s)}$.* *Proof.* We have stationary AR$(1)$ process with $|\theta|<1$ defined as $$\begin{aligned} Y_n &= \theta Y_{n-1} +\epsilon_n, \\ \phi_{Y_n}(s) &= \phi_{Y_{n-1}}(\theta s)\phi_{\epsilon}(s)\\ \phi_{\epsilon}(s) &= \frac{\phi_{Y}(s)}{\phi_{Y}(\theta s)}\, =\frac{1+g(\theta s)}{1+g(s)}. \end{aligned}$$ ◻ (a) Geometric tempered stable: For $g(s) = (s+\lambda)^{\beta}-\lambda^{\beta},$ then $\phi_\epsilon(s) = \frac{1+(\theta s+\lambda)^\beta-\lambda^\beta}{1 +(s+\lambda)^{\beta}-\lambda^{\beta}}, \, \lambda>0, \text{ and }\beta\in(0,1).$ (b) Geometric gamma: For $g(s) = \alpha\log(1+\frac{s}{\beta}),$ then $\phi_\epsilon(s) = \frac{1+\alpha\log\Bigl(\frac{\theta s+\beta}{\beta}\Bigr)}{1+\alpha\log\Bigl(\frac{s+\beta}{\beta}\Bigr)}, \, \alpha>0 \text{ and } \beta>0.$ (c) Geometric inverse Gaussian: For $g(s) = \delta\gamma\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}$, then $\phi_\epsilon(s) = \frac{1+\delta\gamma\Bigl\{\sqrt{1+\frac{2\theta s}{\gamma^2}}-1\Bigr\}}{1+\delta\gamma\Bigl\{\sqrt{1+\frac{2s}{\gamma^2}}-1\Bigr\}}, \, \delta>0 \text{ and } \gamma>0.$ # Parameter estimation and simulation {#section4} In this section, we estimate the parameters of the model defined in [Proposition 5](#ARmodel){reference-type="ref" reference="ARmodel"} using conditional least square (CLS) and method of moments (MOM). We first apply the CLS method to estimate the parameter $\theta$ and then use the MOM for the parameters of marginals in next subsection. ## Estimation by conditional least square and method of moments The conditional likelihood function is given by, $$L(\theta, \lambda,\beta)=\sum_{t=1}^{n}(Y_t-\mathbb{E}(Y_t|Y_{t-1}))^2,\text{ where } \mathbb{E}[Y_t|Y_{t-1}] = \theta Y_{t-1} + \mathbb{E}(\epsilon_t).$$ (a) Geometric tempered stable: For the model defined in [Proposition 5](#ARmodel){reference-type="ref" reference="ARmodel"}, first we assume that innovations are from distribution with Laplace transform $\phi_\epsilon(s) = \frac{1+(\theta s+\lambda)^\beta-\lambda^\beta}{1 +(s+\lambda)^{\beta}-\lambda^{\beta}}, \, \lambda>0 \text{ and }\beta\in(0,1).$ The first order and second order theoretical moments of innovation terms {$\epsilon_t\}$ are approximated by the empirical moments, which are given by $$\begin{aligned} \label{MOM1} \hat{m}_1 &= \sum_{t=1}^{n}\dfrac{\epsilon_t}{n}=(1-\theta)\beta\lambda^{\beta-1}, \\ \hat{m}_2 &= \sum_{t=1}^{n}\dfrac{\epsilon_t^2}{n} = \beta(\beta-1)(\theta^2-1)\lambda^{\beta-2} - 2\beta^2\lambda^{2\beta-2}(\theta-1). \end{aligned}$$ We substitute $\hat{m}_1$ from [\[MOM1\]](#MOM1){reference-type="eqref" reference="MOM1"} to likelihood function $L(\theta, \lambda, \beta)$ and obtain, $$L = \sum_{t=1}^{n}(Y_t-\theta Y_{t-1} - (1-\theta)\beta\lambda^{\beta-1})^2.$$ Take derivative with respect to unknown parameters $\theta, \lambda$ and $\beta$ and obtain the following relation, $$\begin{aligned} \dfrac{\partial{L}}{\partial{\lambda}} &= \sum_{t=1}^{n}2(Y_t-\theta Y_{t-1} - (1-\theta)\beta\lambda^{\beta-1})((1-\theta)\beta\lambda^{\beta-1}),\\ \dfrac{\partial{L}}{\partial{\beta}} &=\sum_{t=1}^{n}2(Y_t-\theta Y_{t-1} - (1-\theta)\beta\lambda^{\beta-1})((1-\theta)\lambda^{\beta-1}+(1-\theta)\beta\lambda^{\beta-1}\log{(\beta-1)}),\\ \dfrac{\partial{L}}{\partial{\theta}} &= \sum_{t=1}^{n}2(Y_t-\theta Y_{t-1} - (1-\theta)\beta\lambda^{\beta-1})(\beta\lambda^{\beta-1}- Y_{t-1}). \end{aligned}$$ Solving the above equations, we get the estimate for $\theta$, $$\hat{\theta} = \dfrac{\sum_{t=1}^{n}Y_tY_{t-1}-n\bar{Y}_{t-1}^2}{\sum_{t=1}^{n}(Y_{t-1}-\bar{Y}_{t-1})^2},$$ where $\bar{Y}_{t-1} = \dfrac{\sum_{t=1}^{n}Y_{t-1}}{n}.$ Now we use first and second order moments to estimate the remaining parameters $\lambda$ and $\beta$. $$\begin{aligned} \hat{m}_1 &= (1-\hat{\theta})\beta\lambda^{\beta-1}, \\ \hat{m}_2 &= \beta(\beta-1)(\hat{\theta}^2-1)\lambda^{\beta-2} - 2\beta^2\lambda^{2\beta-2}(\hat{\theta}-1). \end{aligned}$$ After rearranging the terms we get the following non-linear relation between $\lambda$ and $\beta$, $$\beta = 1 - \lambda\dfrac{\hat{m}_2(1-\hat{\theta})-2\hat{m}_1^2}{\hat{m}_1(1-\hat{\theta}^2)}.$$ To solve it further we use *fsolve()* function available in python scipy package. Also, note that the estimate for $\theta$ using CLS will be same for all the cases. (b) Geometric gamma: For this case the Laplace transform of innovation terms are\ $\phi_\epsilon(s) = \dfrac{1+(\log(\theta s+\beta/\beta))^\alpha}{1 +(\log(s+\beta/\beta))^{\alpha}}, \, \alpha>0 \text{ and }\beta>0.$ The first and second order moments of innovation terms will be, $$\begin{aligned} \label{MOM2} \hat{m}_1 &= (1-\hat{\theta})\dfrac{\alpha}{\beta}, \\ \hat{m}_2 &= \dfrac{\alpha(1-\hat{\theta}^2)}{\beta^2} + \dfrac{2\alpha^2(1-\hat{\theta})}{\beta^2}. \end{aligned}$$ Using these moments we get the the estimates as $\hat{\beta} = \dfrac{\hat{m}_1(1-\hat{\theta}^2)}{\hat{m}_2(1-\hat{\theta}-2\hat{m}_1^2)}$ and $\hat{\alpha} = \dfrac{\hat{m}_1\hat{\beta}}{1-\hat{\theta}}.$ (c) Geometric inverse Gaussian: The Laplace transform of innovation terms are\ $\phi_\epsilon(s) = \dfrac{1+\delta(\sqrt{2\theta s+\gamma^2}-\gamma)}{1+\delta(\sqrt{2s+\gamma^2}-\gamma)}, \, \gamma>0\text{ and }\delta>0.$ The first and second order moments of innovation terms will be, $$\begin{aligned} \hat{m}_1 &= (1-\hat{\theta})\dfrac{\delta}{\gamma}, \\ \hat{m}_2 &= 2(1-\hat{\theta})\dfrac{\delta^2}{\gamma^2} + (1-\hat{\theta}^2)\dfrac{\delta}{\gamma^3}. \end{aligned}$$ Using these moments we get the the estimates as $\hat{\gamma} = \sqrt{\dfrac{\hat{m}_1(1-\hat{\theta}^2)}{\hat{m}_2(1-\hat{\theta})-2\hat{m}_1^2}}$ and $\hat{\delta} = \dfrac{\hat{m}_1\hat{\gamma}}{1-\hat{\theta}}.$ ## Simulation We use simulation to further assess the performance of estimation method. We simulate the $500$ trajectories each of length $1000$ for geometric tempered and geometric gamma case. We use the method of Laplace transform to simulate the innovation terms $\{\epsilon_n\}$ as described in [@Ridout2009] and then generate the time series $\{Y_n\}$ from AR($1$) model. The true model parameter is $\theta = 0.3$ and distribution parameters for geometric tempered stable, geometric gamma and geometric inverse Gaussian are tabulated in [\[tab1\]](#tab1){reference-type="ref" reference="tab1"}. The boxplots for two cases are shown in Fig. [\[fig5\]](#fig5){reference-type="ref" reference="fig5"}. P2.0cmP2cmP2cmP2cm &&&\ & parameter $1$ & parameter $2$ & parameter $3$\ &&&\ True values & $\beta = 0.6$ & $\lambda = 1$ & $\rho = 0.3$\ Est. values & $\hat{\beta} = 0.206$ & $\hat{\lambda} = 0.917$ & $\hat{\rho} = 0.297$\ &&&\ True values &$\beta = 1$ & $\alpha = 2$ & $\rho = 0.3$\ Est. values & $\hat{\beta} = 1.189$ & $\hat{\alpha} = 2.422$ & $\hat{\rho} = 0.298$\ &&&\ True values &$\gamma = 1$ & $\delta = 2$ & $\rho = 0.3$\ Est. values & $\hat{\gamma} = 1.028$ & $\hat{\delta} = 2.054$ & $\hat{\rho} = 0.296$\ From the data in table and boxplots we observe that the CLS method gives good estimate for parameter $\rho$, whereas the estimation of other parameters by MOM have variance. The estimated value of $\beta$ from geometric tempered stable is not good. The relation of $\beta$ and $\lambda$ is non-linear therefore we solved it by using *fsolve* function defined in python's scipy library which is based on numerical methods. # Conclusion In this paper we use the Bernstein function $g(s)$ which is the Laplace exponent of a positive infinitely divisible random variable to define gid random variables with Laplace transform of the form $\dfrac{1}{1+g(s)}$. We also find the Laplace transform of mixtures of some particular gid random variables which is a new class of marginals to study. A new autoregressive process of order 1 with gid distribution is considered. We deduce that if marginals of AR$(1)$ defined in Eq. [\[GBmodel\]](#GBmodel){reference-type="eqref" reference="GBmodel"} are gid then the innovation terms are also gid with scale change of $\theta$. We find the integral form of the pdf of innovation terms using the Laplace transform and complex inversion method for three cases namely, geometric tempered stable, geometric gamma and geometric inverse Gaussian subordinators. Further, moments play an important role to study the characteristics of pdf. We have calculated the first and second order moments for these three gid random varaiables. Next we generalised the AR process to $k^{th}$ order and also proposed AR($1$) model defined in [Proposition 5](#ARmodel){reference-type="ref" reference="ARmodel"} with marginals having Laplace transform of the form $\dfrac{1}{1+g(s)}$. At last, we have estimated the parameters of the model defined in [Proposition 5](#ARmodel){reference-type="ref" reference="ARmodel"} using CLS and MOM and simulation study implies that the estimates are satisfactory.\ Monika S. Dhull would like to thank the Ministry of Education (MoE), India for supporting her PhD research. Further, Arun Kumar would like to express his gratitude to Science and Engineering Research Board (SERB), India for financial support under the MATRICS research grant MTR/2019/000286.\ No real world data is used for research described in this article.

arxiv_math

--- abstract: | This is the first of a series of papers in which we investigate the decay estimates for dispersive equations with Aharonov-Bohm solenoids in a uniform magnetic field. In this first starting paper, we prove the local-in-time dispersive estimates and Strichartz estimates for Schrödinger equation with one Aharonov-Bohm solenoid in a uniform magnetic field. The key ingredient is the construction of Schrödinger propagator, we provide two methods to construct the propagator. The first one is combined the strategies of [@FFFP1] and [@GYZZ22; @FZZ22], and the second one is based on the Schulman-Sunada formula in sprit of [@stov; @stov1] in which the heat kernel has been studied. In future papers, we will continue investigating this quantum model for wave with one or multiple Aharonov-Bohm solenoids in a uniform magnetic field. address: - Department of Mathematics, Beijing Institute of Technology, Beijing 100081; - Department of Mathematics, Beijing Institute of Technology, Beijing 100081; - Department of Mathematics, Beijing Institute of Technology, Beijing 100081; author: - Haoran Wang - Fang Zhang - Junyong Zhang title: "Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field I: Schrödinger equation" --- # Introduction Let us consider the electromagnetic Hamiltonian $$H_{A, V}=-(\nabla+iA(x))^2+V(x),$$ where the electric scalar potential $V: \mathbb{R}^n\to \mathbb{R}$ and the magnetic vector potential $$A(x)=(A^1(x),\ldots, A^n(x)): \, \mathbb{R}^n\to \mathbb{R}^n$$satisfies the Coulomb gauge condition $$\label{div0} \mathrm{div}\, A=0.$$ In three dimensions, the magnetic vector potential $A$ produces a magnetic field $B$, which is given by $$B=\mathrm{curl} (A)=\nabla\times A.$$ In general dimensions $n\geq2$, $B$ should be viewed as the matrix-valued field $B:\mathbb{R}^n\to \mathcal{M}_ {n\times n}(\mathbb{R})$ given by $$\label{B-n} B:=DA-DA^t,\quad B_{ij}=\frac{\partial A^i}{\partial x_j}-\frac{\partial A^j}{\partial x_i}.$$ The Schrödinger operators with electromagnetic potentials have been extensively studied from the aspects of spectral and scattering theory, we refer to Avron-Herbst-Simon [@AHS1; @AHS2; @AHS3] and Reed-Simon [@RS], in which many important physical potentials (e.g. the constant magnetic field and the Coulomb electric potential) are discussed. The purpose of our program here is to study how the electric or magnetic potentials affect the short-time or long-time behavior of the solutions for dispersive equations (e.g. the Schrödinger, wave and Klein-Gordon). Indeed, the program on the study of the decay estimates and Strichartz estimates of the dispersive equations has a long history due to their significance in analysis and PDEs fields. We refer to [@BPSS; @BPST; @CS; @DF; @DFVV; @EGS1; @EGS2; @S] and the references therein for the the classical and important Schrödinger and wave equations with electromagnetic potentials in mathematical and physical fields. However, since the different potentials has different effects, it is hard to provide a uniform argument to treat all of potentials, and the picture of the program is too far to completed, especially for some critical physic potentials. For example, the dispersive equations with the Aharonov-Bohm potential, as a diffraction physic model and scaling critical purely magnetic potential, has attracted more and more people to study from mathematic viewpoint. In [@FFFP; @FFFP1], Fanelli, Felli, Fontelos, and Primo studied the validity of the time-decay estimate for the Schrödinger equation associated with the operator included the Aharonov-Bohm potential. However, the argument in [@FFFP; @FFFP1] breaks down for wave equation due to the lack of pseudoconformal invariance (which was used for Schrödinger equation). Very recently, Fanelli, Zheng and the last author [@FZZ22] established the Strichartz estimate for wave equation by constructing the odd sin propagator. To solve open problems, raised in the survey [@Fanelli], about the the dispersive estimate for other equations (e.g. Klein-Gordon), Gao, Yin, Zheng and the last author [@GYZZ22] constructed the spectral measure and further proved the time decay and Strichartz estimates of Klein-Gordon equation. The potential models in [@FFFP; @FFFP1; @FZZ22; @GYZZ22] are scaling invariant and with no unbounded (at infinity) perturbations, which is a special cases of [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"} below with $B_0\equiv 0$. In this paper, as a sequence of recent paper [@GYZZ22; @FZZ22], we study the decay and Strichartz estimates for the Schrödinger equation in a mixed magnetic fields on the plane pierced by one infinitesimally thin Aharonov-Bohm solenoid and subjected to a perpendicular uniform magnetic field of constant magnitude $B_0$. More precisely, we focus on the typical 2D purely magnetic model Hamiltonians $$\label{H-A} H_{\alpha,B_0}=-(\nabla+i(A_B(x)+A_{\mathrm{hmf}}(x)))^2,$$ where $A_B(x)$ is the Aharonov-Bohm potential in [@AB59] $$\label{AB-potential} A_B(x)=\alpha\Big(-\frac{x_2}{|x|^2},\frac{x_1}{|x|^2}\Big),\quad x=(x_1,x_2)\in\mathbb{R}^2\setminus\{0\},$$ with $\alpha\in\mathbb{R}$ representing the circulation of $A_B$ around the solenoid and $A_{\mathrm{hmf}}(x)$ is the potential $$\label{A-hmf} A_{\mathrm{hmf}}(x)=\frac{B_0}{2}(-x_2,x_1),\quad B_0>0.$$ From [\[B-n\]](#B-n){reference-type="eqref" reference="B-n"}, the total magnetic filed is given by $B_0+\alpha \delta(x)$ where $\delta$ is the Dirac delta function. From mathematics viewpoint, it is worth emphasizing three key features that about the perturbations of the potentials here. The first one is that the Aharonov-Bohm potential producing the singular magnetic field has the same scaling of $\nabla$ which is homogenous degree of $-1$, so that the perturbation of Aharonov-Bohm potential [\[AB-potential\]](#AB-potential){reference-type="eqref" reference="AB-potential"} is non-trivial; the second one is that the magnetic poetential [\[A-hmf\]](#A-hmf){reference-type="eqref" reference="A-hmf"} is homogenous degree of $1$ so that the Schrödinger operator [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"} is not scaling invariant so is the Schrödinger equation. The potential $A_{\mathrm{hmf}}(x)$ is unbounded and the magnetic filed $B(x)=B_0$ from [\[B-n\]](#B-n){reference-type="eqref" reference="B-n"} is a uniform magnetic field caused a trapped well. On one hand, due to the perturbations of magnetic fields [\[A-hmf\]](#A-hmf){reference-type="eqref" reference="A-hmf"} are involved, the operator $H_{\alpha,B_0}$ has pure point spectrum, the dispersive behavior of Schrödinger equation associated with $H_{\alpha,B_0}$ becomes quite different from the model in [@FZZ22; @GYZZ22]. On the other hand, because of the Aharonov-Bohm effect, a feature of the Mehler kernel which is related to the Schrödinger kernel associated with pure uniform magnetic field breaks down. Indeed, if $\alpha=0$, the Schrödinger kernel can be written as $$e^{itH_{0,B_0}}(x,y)=\frac{B_0}{4\pi\sin(B_0t)}\exp\Big\{\frac{B_0}{4i}\big(\cot(B_0t)|x-y|^2-2x\wedge y\big)\Big\},$$ furthermore, one can write $$\begin{split} e^{itH_{0,B_0}}(x,y)=\frac{B_0}{4\pi\sin(B_0t)}&\exp\Big\{\frac{B_0}{4i}\cot(B_0t)\big(|x|^2+|y|^2\big)\Big\}\\ &\times\exp\Big\{i\frac{B_0y\cdot R(B_0t)x}{2\sin(B_0t)}\Big\}, \end{split}$$ where $R(\theta)$ is the usual $2\times 2$ rotation matrix given by $$R(\theta)=\left(\begin{array}{cc}\cos\theta &-\sin\theta\\ \sin\theta & \cos\theta \end{array}\right).$$ Therefore, see [@KL Theorem 2], one can prove the Strichartz estimates $$\|e^{itH_{0,B_0}}f\|_{L^q_t((0,\frac{\pi}{B_0});L^r_x(\mathbb{R}^2)}=(4\pi)^{1-\frac4q}\|e^{it\Delta}f\|_{L^q_t(\mathbb{R};L^r_x(\mathbb{R}^2)}\leq C\|f\|_{L^2(\mathbb{R}^2)},$$ where $(q,r)\in\Lambda_S$ defined in [\[adm:S\]](#adm:S){reference-type="eqref" reference="adm:S"} below. More precisely, we aim to study the dispersive behavior of the magnetic Schrödinger equation $$\label{equ:S} \begin{cases} i\partial_t u(t,x)- H_{\alpha,B_0} u(t,x)=0,\\ u(0,x)=u_0(x), \end{cases}$$ where $H_{\alpha,B_0}$ is given in [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"} with the perturbation of potentials $A_B(x)$ and $A_{\mathrm{hmf}}(x)$. Now we state our main results. **Theorem 1**. *Let $H_{\alpha,B_0}$ be in [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"} with the potentials being [\[AB-potential\]](#AB-potential){reference-type="eqref" reference="AB-potential"}-[\[A-hmf\]](#A-hmf){reference-type="eqref" reference="A-hmf"} and let $u(t,x)$ be the solution of [\[equ:S\]](#equ:S){reference-type="eqref" reference="equ:S"}. Then there exists a constant $C>0$ such that the dispersive estimate $$\label{dis-S} \|u(t,x)\|_{L^\infty(\mathbb{R}^2)}\leq C |\sin(tB_0)|^{-1}\|u_0\|_{L^1(\mathbb{R}^2)},\quad t\neq \frac{k\pi}{B_0},\, k\in\mathbb{Z},$$ and the Strichartz estimate holds for $$\label{str-S} \|u(t,x)\|_{L^q([0,T];L^p(\mathbb{R}^2))}\leq C \|u_0\|_{L^2(\mathbb{R}^2)},$$ where $T\in (0, \frac{\pi}{2B_0})$ and $(q,p)\in \Lambda_S$ with $$\label{adm:S} \Lambda_S:=\Bigg\{(q,p)\in[2,+\infty]\times [2,+\infty):\frac2q=2\Big(\frac12-\frac1p\Big)\Bigg\}.$$* **Remark 2**. The decay estimate [\[dis-S\]](#dis-S){reference-type="eqref" reference="dis-S"} is the periodic with period $\pi/B_0$. The endpoint of the time interval $T$ in the Strichartz estimates [\[str-S\]](#str-S){reference-type="eqref" reference="str-S"} depends on the coefficient $B_0$ of unbounded potential. Actually, if the unbounded potentials disappear, i.e. $B_0=0$, one can take $T$ to be $+\infty$ safely, that is, the global-in-time Strichartz estimates, which is corresponding to the Laplacian with the Aharonov-Bohm potential (see Theorem 1.3 of [@FFFP1]). However, in the present paper, as mentioned above, we only obtain local-in-time Strichartz estimates associated with the operator [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"} due to the unbounded potentials caused trapped well. **Remark 3**. Let $$A(x)=A_B(x)+A_{\mathrm{hmf}}(x),$$ one can verify that $\mathrm{div}\, A=0$ satisfies [\[div0\]](#div0){reference-type="eqref" reference="div0"}. Then, one observe that $$\begin{split} H_{\alpha,B_0}&=-(\nabla+i(A_B(x)+A_{\mathrm{hmf}}(x)))^2 \\&=-\Delta+\big(\frac{B_0}2\big)^2 |x|^2+\frac{\alpha^2}{|x|^2} +i B_0(-x_2,x_1)\cdot \nabla +i \frac{2\alpha}{|x|^2}(-x_2,x_1)\cdot \nabla+\alpha B_0. \end{split}$$ One will see that the operator is perturbed by the inverse-square potential and harmonic oscillator. This phenomenon is natural because the unbounded potential cause a trapped well, the energy cannot be dispersive for long time. This is closely relate to the models with pure harmonic oscillators, i.e. $H_{0,V}=-\Delta+|x|^2$, in which Koch and Tataru [@KT05] proved the decay is the periodic with period $\pi$ $$\|e^{it H_{0,V}}\|_{L^1(\mathbb{R}^n)\to L^\infty(\mathbb{R}^n)}\leq C|\sin t|^{-\frac n2}.$$ The paper is organized as follows. In Section [2](#sec:pre){reference-type="ref" reference="sec:pre"}, in a preliminary step, we recall the the self-adjoint extension of the operator $H_{\alpha,B_0}$, and study the spectrum of $H^{F}_{\alpha, B_0}$ (which is the Friedrichs of $H_{\alpha,B_0}$). In Section [3](#sec:const1){reference-type="ref" reference="sec:const1"}, we construct the Schrödinger propagator by combing the strategies of [@FFFP1] and [@GYZZ22; @FZZ22]. In Section [4](#sec:const2){reference-type="ref" reference="sec:const2"}, we construct the Schrödinger propagator by using another method based on the Schulman-Sunada formula. Finally, in Section [5](#sec:proof){reference-type="ref" reference="sec:proof"}, we prove the Theorem [Theorem 1](#thm:S){reference-type="ref" reference="thm:S"} by using the representation of the Schrödinger propagator constructed in previous sections. **Acknowledgments:**The authors thank L. Fanelli and P. Št'ovı́ček for helpful discussions. This work is supported by National Natural Science Foundation of China (12171031, 11901041, 11831004). # preliminaries {#sec:pre} In this section, we first recall the self-adjoint extension of the operator $H_{\alpha,B_0}$, the Friedrichs extension, and then we study the spectrum of $H^{F}_{\alpha,B_0}$ (which is the Friedrichs of $H_{\alpha,B_0}$). ## Quadratic form and the self-adjoint extension Define the space $\mathcal{H}_{\alpha,B_0}^1(\mathbb{R}^2)$ as the completion of $\mathcal{C}_c^\infty(\mathbb{R}^2\setminus\{0\};\mathbb{C})$ with respect to the norm $$\|f\|_{\mathcal{H}_{\alpha,B_0}^1(\mathbb{R}^2)}=\Big(\int_{\mathbb{R}^2}|\nabla_{\alpha,B_0} f(x)|^2 dx\Big)^{\frac12}$$ where $$\nabla_{\alpha,B_0} f(x)=\nabla f+i(A_B+A_{\mathrm{hmf}})f.$$ The quadratic form $Q_{\alpha,B_0}$ associated with $H_{\alpha,B_0}$ is defined by $$\begin{split} Q_{\alpha,B_0}: & \quad \quad \mathcal{H}_{\alpha,B_0}^1\to \mathbb{R}\\ Q_{\alpha,B_0}(f)&=\int_{\mathbb{R}^2}|\nabla_{\alpha,B_0} f(x)|^2dx. \end{split}$$ Then the quadratic form $Q_{\alpha,B_0}$ is positive definite, thus it implies that the operator $H_{\alpha,B_0}$ is symmetric semi bounded from below which admits a self-adjoint extension (Friedrichs extension) $H^{F}_{\alpha,B_0}$ with the natural form domain $$\mathcal{D}=\Big\{f\in \mathcal{H}_{\alpha,B_0}^1(\mathbb{R}^2): H^{F}_{\alpha,B_0}f\in L^{2}(\mathbb{R}^2)\Big\}$$ Even though the operator $H_{\alpha,B_0}$ has many other self-adjoint extensions (see [@ESV]) by the von Neumann extension theory, in this whole paper, we use the simplest Friedrichs extension and briefly write $H_{\alpha,B_0}$ as its Friedrichs extension $H^{F}_{A}$. ## The spectrum of the operator $H_{\alpha,B_0}$ In this subsection, we modify the argument of [@FFFP1] to obtain the eigenvalue and eigenfunction of the Schrödinger operator $$H_{\alpha,B_0}=-(\nabla+i(A_B(x)+A_{\mathrm{hmf}}(x)))^2,$$ where the magnetic vector potentials are in [\[AB-potential\]](#AB-potential){reference-type="eqref" reference="AB-potential"} and [\[A-hmf\]](#A-hmf){reference-type="eqref" reference="A-hmf"}. More precisely, we will prove that **Proposition 4** (The spectrum for $H_{\alpha,B_0}$). *Let $H_{\alpha,B_0}$ be the self-adjoint Schrödinger operator in [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"}. Then the eigenvalues of $H_{\alpha,B_0}$ are discrete and are given by $$\label{eigen-v} \lambda_{k,m}=(2m+1+|k+\alpha|)B_0+(k+\alpha)B_0,\quad m,\,k\in\mathbb{Z},\, m\geq0,$$ and the (finite) multiplicity of $\lambda_{k,m}$ is $$\#\Bigg\{j\in\mathbb{Z}:\frac{\lambda_{k,m}-(j+\alpha)B_0}{2B_0}-\frac{|j+\alpha|+1}{2}\in\mathbb{N}\Bigg\}.$$ Furthermore, let $\theta=\frac{x}{|x|}$, the corresponding eigenfunction is given by $$\label{eigen-f} V_{k,m}(x)=|x|^{|k+\alpha|}e^{-\frac{B_0 |x|^2}{4}}\, P_{k,m}\Bigg(\frac{B_0|x|^2}{2}\Bigg)e^{ik\theta}$$ where $P_{k,m}$ is the polynomial of degree $m$ given by $$P_{k,m}(r)=\sum_{n=0}^m\frac{(-m)_n}{(1+|k+\alpha|)_n}\frac{r^n}{n!}.$$ with $(a)_n$ ($a\in\mathbb{R}$) denoting the Pochhammer's symbol $$\begin{aligned} (a)_n= \begin{cases} 1,&n=0;\\ a(a+1)\cdots(a+n-1),&n=1,2,\cdots \end{cases}\end{aligned}$$* **Remark 5**. One can verify that the orthogonality holds $$\int_{\mathbb{R}^2}V_{k_1,m_1}(x) V_{k_2,m_2}(x) \,dx=0,\quad \text{if}\quad (k_1, m_1)\neq (k_2, m_2).$$ **Remark 6**. Let $L^\alpha_m(t)$ be the generalized Laguerre polynomials $$L^\alpha_m(t)=\sum_{n=0}^m (-1)^n \Bigg( \begin{array}{c} m+\alpha \\ m-n \\ \end{array} \Bigg)\frac{t^n}{n!},$$ and the well known orthogonality relation $$\int_0^\infty x^{\alpha} e^{-x}L^\alpha_m(x) L^\alpha_n (x)\, dx=\frac{\Gamma(n+\alpha+1)}{n!} \delta_{n,m},$$ where $\delta_{n,m}$ is the Kronecker delta. Let $\tilde{r}=\frac{B_0|x|^2}{2}$ and $\alpha_k=|k+\alpha|$, then $$\label{P-L} P_{k,m}(\tilde{r})=\sum_{n=0}^m\frac{(-1)^n m(m-1)\cdots(m-(n-1))}{(\alpha_k +1)(\alpha_k+2)\cdots(\alpha_k+n)}\frac{\tilde{r}^n}{n!}=\Bigg( \begin{array}{c} m+\alpha_k \\ m \\ \end{array} \Bigg)^{-1}L^{\alpha_k}_m(\tilde{r}).$$ Therefore, $$\label{V-km-2} \|V_{k,m}(x)\|^2_{L^2(\mathbb{R}^2)} =\pi \Big(\frac{2}{B_0}\Big)^{\alpha_k+1}\Gamma(1+\alpha_k) \Bigg( \begin{array}{c} m+\alpha_k \\ m \\ \end{array} \Bigg)^{-1}.$$ **Remark 7**. Recall the Poisson kernel formula for Laguerre polynomials [@AAR01 (6.2.25)]: for $a, b, c, \alpha>0$ $$\label{La-po} \begin{split} &\sum_{m=0}^\infty e^{-cm}\frac{m !}{\Gamma(m+\alpha+1)} L_m^{\alpha} (a) L_m^{\alpha} (b)\\ &=\frac{e^{\frac{\alpha c}2}}{(ab)^{\frac{\alpha}2}(1-e^{-c})} \exp\left(-\frac{(a+b)e^{-c}}{1-e^{-c}}\right) I_{\alpha}\left(\frac{2\sqrt{ab}e^{-\frac{c}2}}{1-e^{-c}}\right) \end{split}$$ then this together with [\[P-L\]](#P-L){reference-type="eqref" reference="P-L"} gives $$\label{Po-L} \begin{split} &\sum_{m=0}^\infty e^{-cm}\frac{m !}{\Gamma(m+\alpha_k+1)} \Bigg(\begin{array}{c} m+\alpha_k \\ m \\ \end{array} \Bigg)^2 P_{k,m} (a) P_{k,m} (b)\\ &=\frac{e^{\frac{\alpha_k c}2}}{(ab)^{\frac{\alpha_k}2}(1-e^{-c})} \exp\left(-\frac{(a+b)e^{-c}}{1-e^{-c}}\right) I_{\alpha_k}\left(\frac{2\sqrt{ab}e^{-\frac{c}2}}{1-e^{-c}}\right). \end{split}$$ *Proof.* Notice that the operator [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"}, in the polar coordinates $(r,\theta)$, has a nice representation $$H_{\alpha, B_0}=-\partial_r^2-\frac{1}{r}\partial_r+\frac{1}{r^2}\Big(-i\partial_\theta+\alpha+\frac{B_0r^2}{2}\Big)^2.$$ We want to solve the eigenfunction equation $$\label{eigen-p} H_{\alpha, B_0} g(x)=\lambda g(x)$$ in the domain [^1] of $H_{A, V}$. Define the projectors $P_k$ onto the eigenspaces of the angular momentum as $$P_kf(r,\theta)=\frac{1}{2\pi}\int_0^{2\pi}e^{ik(\theta-\theta')}f(r,\theta')d\theta':=f_k(r) e^{ik\theta},\quad k\in\mathbb{Z}$$ then it clear that the operator $H_{\alpha, B_0}$ commutes with the projectors $P_k$. In the polar coordinates $(r,\theta)$, [\[eigen-p\]](#eigen-p){reference-type="eqref" reference="eigen-p"} implies that for every $k\in\mathbb{Z}$, $$\label{eq:gk} g''_k(r)+\frac{1}{r}g'_k(r)-\frac{1}{r^2}\Big(k+\alpha+\frac{B_0 r^2}{2}\Big)^2g_k(r)=-\lambda g_k(r),$$ where $$g_k(r)= \frac{1}{2\pi}\int_0^{2\pi}e^{-ik\theta}g(r,\theta)d\theta.$$ Let $$\phi_k(s)=\Big(\frac{2s}{B_0}\Big)^{-\frac{|k+\alpha|}{2}}e^{\frac{s}{2}}g_k\Big(\sqrt{\frac{2s}{B_0}}\Big),$$ then $\phi_k$ satisfies $$\label{eq:phik} s\phi''_k(s)+(1+|k+\alpha|-s)\phi'_k(s)-\frac{1}{2}\Big(1+|k+\alpha|+k+\alpha-\frac{\lambda}{B_0}\Big)\phi_k(s)=0.$$ **Lemma 8**. *The Kummer Confluent Hypergeometric equation $$s\phi''(s)+(b-s)\phi'(s)-a\phi(s)=0,\quad s>0,$$ has two linearly independent solutions given by the Kummer function (also called confluent hypergeometric function) $$M(a,b,s)=\sum_{n=0}^\infty\frac{(a)_n}{(b)_n}\frac{s^n}{n!},\quad b\neq0,-1,-2,\cdots$$ where the Pochhammer's symbol $$\begin{aligned} (a)_n= \begin{cases} 1,&n=0;\\ a(a+1)\cdots(a+n-1),&n=1,2,\cdots \end{cases}\end{aligned}$$ and the Tricomi function (also called confluent hypergeometric function of the second kind) $$U(a,b,s)=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,s)+\frac{\Gamma(b-1)}{\Gamma(a)}s^{1-b}M(a-b+1,2-b,s).$$* By using Lemma [Lemma 8](#lem:KCH){reference-type="ref" reference="lem:KCH"}, we have two linearly independent solutions of [\[eq:phik\]](#eq:phik){reference-type="eqref" reference="eq:phik"}, hence two linearly independent solutions of [\[eq:gk\]](#eq:gk){reference-type="eqref" reference="eq:gk"} are given by $$\begin{aligned} g_k^1(\lambda;r)&=r^{|\alpha+k|}M\Big(\beta(k,\lambda),\gamma(k),\frac{B_0r^2}{2}\Big)e^{-\frac{B_0r^2}{4}}\\ g_k^2(\lambda;r)&=r^{|\alpha+k|}U\Big(\beta(k,\lambda),\gamma(k),\frac{B_0r^2}{2}\Big)e^{-\frac{B_0r^2}{4}}\end{aligned}$$ with $$\begin{aligned} \beta(k,\lambda)&=\frac{1}{2}\Big(1+k+\alpha+|k+\alpha|-\frac{\lambda}{B_0}\Big),\\ \gamma(k)&=1+|k+\alpha|.\end{aligned}$$ Therefore, the general solution of [\[eq:phik\]](#eq:phik){reference-type="eqref" reference="eq:phik"} is given by $$\begin{aligned} \label{sol:gk} g_k(r)=&A_kg_k^1(\lambda;r)+B_kg_k^2(\lambda;r)\\ &=r^{|\alpha+k|}e^{-\frac{B_0r^2}{4}} \Bigg(A_kM\Big(\beta(k,\lambda),\gamma(k),\frac{B_0r^2}{2}\Big)+B_kU\Big(\beta(k,\lambda),\gamma(k),\frac{B_0r^2}{2}\Big)\Bigg),\nonumber\end{aligned}$$ where $A_k,B_k$ are two constants which are dependent on $k$. **Lemma 9** ([@AS65],Chap.13). *The following properties hold:* - *The two functions $M(a,b,z)$ and $U(a,b,z)$ are linearly dependent if and only if $a\in-\mathbb{Z}_+$.* - *$M(a,b,z)$ is an entire function of $z$ and it is regular at $z=0$. However, $U(a,b,z)$ is singular at the origin provided that $b>1$ and $a\notin-\mathbb{Z}_+$ and it holds true that $$\label{U-0} \lim_{z\rightarrow0^+}z^{b-1}U(a,b,z)=\frac{\Gamma(b-1)}{\Gamma(a)}.$$ If $b\in(1,2)$, the asymptotic behavior of $U(a,b,z)$ as $z\rightarrow0^+$ is $$U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}+O(z^{2-b}).$$* - *If $-a\notin\mathbb{N}$, the asymptotic behavior as $z\rightarrow+\infty$ holds true: $$\label{M-inf} M(a,b,z)=\frac{\Gamma(b)}{\Gamma(b-a)}(-z)^{-a}(1+O(z^{-1}))+\frac{\Gamma(b)}{\Gamma(a)}e^z z^{a-b}(1+O(z^{-1})),$$ and $$U(a,b,z)=z^{-a}(1+O(z^{-1})).$$* Now we use this asymptotic lemma to conclude more detail information about the eigenvalues and eigenfunctions. Let $$m:=-\beta(k,\lambda)=-\frac{1}{2}\Big(1+k+\alpha+|k+\alpha|-\frac{\lambda}{B_0}\Big).$$ On one hand, from [\[M-inf\]](#M-inf){reference-type="eqref" reference="M-inf"}, if $m\notin \mathbb{N}$, the function $$M\big(-m,\gamma(k),\tilde{r}\big)\sim e^{ \tilde{r}} \tilde{r}^{-m-\gamma(k)},\quad \tilde{r}\rightarrow+\infty,$$ is singular at $+\infty$; while if $m\in\mathbb{N}=\{0,1,2,\cdots\}$, then $M(m,\gamma(k),\tilde{r})$ is in fact a polynomial of degree $m$ in $\tilde{r}$, which we shall denote as $P_{k,m}$, i.e., $$P_{k,m}(\tilde{r})=M(-m,1+|k+\alpha|,\tilde{r})=\sum_{n=0}^m\frac{(-m)_n}{(1+|k+\alpha|)_n}\frac{\tilde{r}^n}{n!}.$$ On the other hand, note that $\gamma(k)=1+|k+\alpha|\geq1$, from [\[U-0\]](#U-0){reference-type="eqref" reference="U-0"}, it follows that $$U\big(\beta(k,\lambda),\gamma(k),\tilde{r}\big)\sim \tilde{r}^{-|k+\alpha|},\quad \tilde{r}\rightarrow0+,$$ with the implicit constant depending only on $\lambda$ and $k$. Hence, by letting $\tilde{r}=\frac{B_0r^2}{2}$ and using [\[sol:gk\]](#sol:gk){reference-type="eqref" reference="sol:gk"} and Lemma [Lemma 9](#lem:asy){reference-type="ref" reference="lem:asy"}, we have for fixed $k\in\mathbb{Z}$ that $$g_k(r)\sim B_k r^{|\alpha+k|}e^{-\frac{B_0r^2}{4}} U\Big(\beta(k,\lambda),\gamma(k),\frac{B_0r^2}{2}\Big)\sim B_k r^{-|\alpha+k|},\quad\text{as}\quad r\rightarrow0+.$$ and $$g_k(r)\sim A_k r^{|\alpha+k|}e^{-\frac{B_0 r^2}{4}} M\Big(\beta(k,\lambda),\gamma(k),\frac{B_0 r^2}{2}\Big)\sim A_k e^{\frac{B_0r^2}{4}}r^{-1+k+\alpha-\frac{\lambda}{B_0}},\quad\text{as}\quad r\rightarrow \infty.$$ We now conclude that one must have $B_k\equiv0$. Indeed, otherwise, from the fact that the eigenfunction $g\in D(H_{\alpha,B_0})$, thus we have that $$\int_0^\infty g_k^2(r)\frac{dr}{r}\leq\int_{\mathbb{R}^2}\frac{g^2(x)}{|x|^2}dx<\infty,$$ which is obviously contradict with the fact that the integral $\int_0^\infty r^{-2|k+\alpha|-1} dr$ is divergent for all $k\in\mathbb{Z}$ at $0$. We next conclude that one must have $A_k\equiv0$ if $m\notin\mathbb{N}$. Actually, from the fact that the eigenfunction $g\in D(H_{\alpha,B_0})$ again, thus we have that $$\int_0^\infty g_k^2(r)\, rdr\leq\int_{\mathbb{R}^2} g^2(x)dx<\infty.$$ However, we note that $B_0>0$ and $$\int_0^\infty g_k^2(r)\, rdr\geq \int_1^\infty e^{\frac{B_0r^2}{4}}\, rdr$$ which is divergent. This is a contradiction. Therefore, we must have $$\mathbb{N} \ni m=-\frac{1}{2}\Big(1+k+\alpha+|k+\alpha|-\frac{\lambda}{B_0}\Big),$$ and $$g_k(r)=r^{|k+\alpha|}e^{-\frac{B_0r^2}{4}}\, P_{k,m}\Big(\frac{B_0r^2}{2}\Big).$$ Therefore, we prove the function $$V_{k,m}(x)=|x|^{|k+\alpha|}e^{-\frac{B_0|x|^2}{4}}\, P_{k,m}\Big(\frac{B_0|x|^2}{2}\Big)e^{ik\theta},$$ belongs to $D(H_{\alpha,B_0})$, thus providing an eigenfunction of the operator $H_{\alpha,B_0}$. Thus from $-m=\frac{1}{2}\Big(1+k+\alpha+|k+\alpha|-\frac{\lambda}{B_0}\Big)$, we solve [\[eigen-p\]](#eigen-p){reference-type="eqref" reference="eigen-p"} to obtain the eigenvalues $\lambda$ of $H_{\alpha,B_0}$ $$\lambda_{k,m}=(2m+1+|k+\alpha|)B_0+(k+\alpha)B_0,\quad k\in\mathbb{Z},\, m\in\mathbb{N}.$$ ◻ # The Schrödinger propagator {#sec:const1} In this section, we construct the Schrödinger propagator by using the spectrum property of Proposition [Proposition 4](#prop:spect){reference-type="ref" reference="prop:spect"}. The combined strategies are from [@FFFP1] and [@GYZZ22; @FZZ22]. More precisely, we will prove the following result. **Proposition 10**. *Let $H_{\alpha,B_0}$ be the operator in [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"} and suppose $x=r_1(\cos\theta_1,\sin\theta_1)$ and $y=r_2(\cos\theta_2,\sin\theta_2)$. Let $u(t,x)$ be the solution of the Schrödinger equation $$\begin{cases} \big(i\partial_t-H_{\alpha,B_0}\big)u(t,x)=0,\\ u(0,x)=f(x). \end{cases}$$ Then $$u(t,x)=e^{-itH_{\alpha,B_0}}f=\int_{\mathbb{R}^2} K_S(x,y) f(y)\, dy,$$ where $t\neq \frac{k\pi}{B_0},\, k\in\mathbb{Z}$. Let $\theta=\theta_1-\theta_2-tB_0\in\mathbb{R},$ there exists an integer $j_0$ satisfying $\theta+2j_0\pi\in [-\pi,\pi]$. Define $$\chi(\theta, j_0)= \begin{cases} \begin{array}{ll} 1, & \hbox{if\, $|\theta+2j_0\pi|<\pi$ ;} \\ e^{-i2\pi \alpha}+1 , & \hbox{if\, $\theta+2j_0\pi=-\pi;$}\\ e^{i2\pi\alpha}+1, & \hbox{if \,$\theta+2j_0\pi=\pi$.} \end{array} \end{cases}$$ Then the kernel of Schrödinger propagator $e^{-itH_{\alpha,B_0}}$ has the representation $$\label{S:express} \begin{split} K_S(x,y)&=\frac{B_0e^{-itB\alpha}}{8\pi^2 i\sin (tB_0)}e^{\frac{iB_0(r_1^2+r_2^2)}{4\tan (tB_0)}}\\ \times&\Big[e^{\frac{B_0r_1r_2}{2i\sin (tB_0)}\cos(\theta_1-\theta_2-tB_0)} e^{-i\alpha(\theta_1-\theta_2-tB_0+2j_0\pi)}\chi(\theta, j_0)\\ &-\frac{\sin(\pi\alpha)}{\pi}\int_{\mathbb{R}} e^{-\frac{B_0r_1r_2}{2i\sin (tB_0)}\cosh s}\frac{e^{-\alpha s}}{1+e^{-s+i(\theta_1-\theta_2-tB_0)}}\,ds\Big]. \end{split}$$* *Proof.* We construct the representation formula for the kernel of the Schrödinger flow $e^{-itH_{\alpha,B_0}}$ by combining the argument of [@FFFP1] and [@FZZ22; @GYZZ22]. Our starting point is the Proposition [Proposition 4](#prop:spect){reference-type="ref" reference="prop:spect"}. Let $\tilde{V}_{k,m}$ be the $L^2$-normalization of $V_{k,m}$ in [\[eigen-f\]](#eigen-f){reference-type="eqref" reference="eigen-f"}, then the eigenfunctions $\Big\{\tilde{V}_{k,m}\Big\}_{k\in\mathbb{Z}, m\in\mathbb{N}}$ form an orthonormal basis of $L^2(\mathbb{R}^2)$ corresponding to the eigenfunctions of $H_{\alpha,B_0}$. We expand the initial data $f(x)\in L^2$ as $$f(x)=\sum_{k\in\mathbb{Z}, \atop m\in\mathbb{N}} c_{k,m}\tilde{V}_{k,m}(x)$$ where $$\label{cmk1} c_{k,m}=\int_{\mathbb{R}^2}f(x)\overline{\tilde{V}_{k,m}(x)}\, dx.$$ The solution $u(t,x)$ of [\[equ:S\]](#equ:S){reference-type="eqref" reference="equ:S"} can be written as $$\label{sol:expan} u(t,x)=\sum_{k\in\mathbb{Z}, \atop m\in\mathbb{N}} u_{k,m}(t)\tilde{V}_{k,m}(x),$$ where $u_{k,m}(t)$ satisfies the ODE $$\begin{cases} &iu'_{k,m}(t)=\lambda_{k,m}u_{k,m}(t),\\ & u_{k,m}(0)=c_{k,m},\quad k\in\mathbb{Z}, \, m\in\mathbb{N}. \end{cases}$$ Thus we obtain $u_{k,m}(t)=c_{k,m}e^{-it\lambda_{k,m}}$. Therefore the solution [\[sol:expan\]](#sol:expan){reference-type="eqref" reference="sol:expan"} becomes $$u(t,x)=\sum_{k\in\mathbb{Z}, \atop m\in\mathbb{N}} c_{k,m}e^{-it\lambda_{k,m}}\tilde{V}_{k,m}(x).$$ Plugging [\[cmk1\]](#cmk1){reference-type="eqref" reference="cmk1"} into the above expression yields $$u(t,x)=\sum_{k\in\mathbb{Z}, \atop m\in\mathbb{N}} e^{-it\lambda_{k,m}}\left(\int_{\mathbb{R}^2}f(y)\overline{\tilde{V}_{k,m}(y)} dy\right)\tilde{V}_{k,m}(x).$$ We write $f$ in a harmonic spherical expansion $$f(y)=\sum_{k\in\mathbb{Z}} f_k(r_2) e^{ik\theta_2},$$ where $$\label{f-k} f_k(r_2)=\frac{1}{2\pi}\int_0^{2\pi} f(r_2, \theta_2) e^{-ik\theta_2}\, d\theta_2,\quad r_2=|y|,$$ we thus have $$\begin{aligned} u(t,x)&=\sum_{k\in\mathbb{Z}, \atop m\in\mathbb{N}} e^{-it\lambda_{k,m}}\frac{V_{k,m}(x)}{\|V_{k,m}\|^2_{L^2}}\Bigg(\int_0^\infty f_k(r_2)e^{-\frac{B_0r_2^2}{4}}\, P_{k,m}\Big(\frac{B_0r_2^2}{2}\Big) r_2^{1+\alpha_k}\mathrm{d}r_2\Bigg)\\ &=\Big(\frac{B_0}{2\pi}\Big)\sum_{k\in\mathbb{Z}}e^{ik\theta_1}\frac{B_0^{\alpha_k}e^{-it\beta_k}}{2^{\alpha_k}\Gamma(1+\alpha_k)}\Bigg[\sum_{m=0}^\infty \Bigg( \begin{array}{c} m+\alpha_k \\ m \\ \end{array} \Bigg)e^{-2itmB_0}\\ &\times\Bigg(\int_0^\infty f_k(r_2)(r_1r_2)^{\alpha_k}e^{-\frac{B_0(r_1^2+r_2^2)}{4}}P_{k,m}\left(\frac{B_0r_2^2}{2}\right)P_{k,m}\left(\frac{B_0r_1^2}{2}\right)r_2 \mathrm{d}r_2\Bigg)\Bigg],\end{aligned}$$ where $\alpha_k=|k+\alpha|$ and we use [\[V-km-2\]](#V-km-2){reference-type="eqref" reference="V-km-2"}, [\[eigen-v\]](#eigen-v){reference-type="eqref" reference="eigen-v"}, [\[eigen-f\]](#eigen-f){reference-type="eqref" reference="eigen-f"} and $$\begin{split} \lambda_{k,m}&=(2m+1+|k+\alpha|)B_0+(k+\alpha)B_0\\ &:=2mB_0+\beta_k \end{split}$$ with $\beta_k=(1+|k+\alpha|)B_0+(k+\alpha)B_0>0$. Notice that $P_{k,m}$ can be expressed as (see e.g. [@AAR01 (6.2.15)]) $$P_{k,m}\left(\frac{r^2}{2}\right)=\frac{\Gamma(1+\alpha_k)}{\Gamma(1+\alpha_k+m)}e^{\frac{r^2}{2}}r^{-\alpha_k}2^{\frac{\alpha_k}{2}}\int_0^\infty e^{-s}s^{m+\frac{\alpha_k}{2}}J_{\alpha_k}(\sqrt{2s}r)ds,$$ in terms of Bessel functions $J_{\alpha_k}$ of order $\alpha_k$. Hence, we have $$\begin{aligned} u(t,x)=&\frac{B_0}{2\pi}\sum_{k\in\mathbb{Z}}e^{ik\theta_1}\frac{B_0^{\alpha_k}e^{-it\beta_k}}{2^{\alpha_k}\Gamma(1+\alpha_k)}\Bigg[\sum_{m\in\mathbb{N}} \left( \begin{array}{c} m+\alpha_k \\ m \\ \end{array} \right)e^{-2itmB_0}\\ &\times\Big(\frac{\Gamma(1+\alpha_k)}{\Gamma(1+\alpha_k+m)}\Big)^2\Bigg(\int_0^\infty f_k(r_2)(r_1r_2)^{\alpha_k} e^{-\frac{B_0(r_1^2+r_2^2)}{4}}e^{\frac{B_0(r_1^2+r_2^2)}{2}}\Big(\frac{2}{B_0r_2^2}\Big)^{\frac{\alpha_k}{2}}\\ &\times\Big(\frac{2}{B_0r_1^2}\Big)^{\frac{\alpha_k}{2}}\Big(\int_0^\infty\int_0^\infty e^{-s_1-s_2}(s_1s_2)^{m+\frac{\alpha_k}{2}} J_{\alpha_k}(\sqrt{2B_0s_1}r_1)J_{\alpha_k}(\sqrt{2B_0s_2}r_2)ds_1ds_2\Big)r_2dr_2\Bigg)\Bigg]\\ =&\frac{B_0}{2\pi}\sum_{k\in\mathbb{Z}}e^{ik\theta_1}e^{-it\beta_k}\Gamma(1+\alpha_k)\Bigg[\sum_{m\in\mathbb{N}} \left( \begin{array}{c} m+\alpha_k \\ m \\ \end{array} \right)\frac{e^{-2itmB_0}}{(\Gamma(1+\alpha_k+m))^2}\\ &\times\Bigg(\int_0^\infty f_k(r_2)e^{\frac{B_0(r_1^2+r_2^2)}{4}}\Big(\int_0^\infty\int_0^\infty e^{-s_1-s_2}(s_1s_2)^{m+\frac{\alpha_k}{2}}\\ &\times J_{\alpha_k}(\sqrt{2B_0s_1}r_1)J_{\alpha_k}(\sqrt{2B_0s_2}r_2)ds_1ds_2\Big)r_2dr_2\Bigg)\Bigg],\quad \big(\text{variable changes:}\, s_1\to s_1^2,\, s_2\to s_2^2 \big)\\ =&\frac{2B_0}{\pi}\sum_{k\in\mathbb{Z}}e^{ik\theta_1}e^{-it\beta_k}\Bigg[\int_0^\infty f_k(r_2)e^{\frac{B_0(r_1^2+r_2^2)}{4}}e^{i\alpha_k(tB_0+\frac{\pi}{2})}\\ &\times\Bigg(\int_0^\infty\int_0^\infty\frac{s_1s_2}{e^{s_1^2+s_2^2}}\Bigg(\sum_{m\in\mathbb{N}} \frac{(-1)^me^{-i(tB_0+\frac{\pi}{2})(2m+\alpha_k)}}{\Gamma(1+m)\Gamma(1+\alpha_k+m)}(s_1s_2)^{2m+\alpha_k}\Bigg)\\ &\times J_{\alpha_k}\big(\sqrt{2B_0}s_1r_1\big)J_{\alpha_k}\big(\sqrt{2B_0}s_2 r_2\big)ds_1ds_2\Bigg)r_2dr_2\Bigg].\end{aligned}$$ Since (see e.g. [@AAR01 (4.5.2)]) $$\sum_{m=0}^\infty\frac{(-1)^me^{-i(tB_0+\frac{\pi}{2})(2m+\alpha_k)}}{\Gamma(1+\alpha_k+m)\Gamma(1+m)}(s_1s_2)^{2m+\alpha_k} =J_{\alpha_k}(2s_1s_2e^{-i(tB_0+\frac{\pi}{2})}),$$ then we have $$\begin{aligned} u(t,x)=\frac{2B_0}{\pi}\sum_{k\in\mathbb{Z}}e^{ik\theta_1}e^{it(\alpha_kB_0-\beta_k)+i\alpha_k\frac{\pi}{2}} \int_0^\infty e^{\frac{B_0(r_1^2+r_2^2)}{4}}f_k(r_2)G_{k,t}(r_1,r_2)r_2dr_2\end{aligned}$$ with $$\begin{split} G_{k,t}(r_1,r_2)=\int_0^\infty\int_0^\infty&\frac{s_1s_2}{e^{s_1^2+s_2^2}}J_{\alpha_k}(2s_1s_2e^{-i(tB_0+\frac{\pi}{2})}) \\&\times J_{\alpha_k}(\sqrt{2B_0}r_1s_1)J_{\alpha_k}(\sqrt{2B_0}r_2s_2)ds_1ds_2. \end{split}$$ Using formula (see [@Wat44 formula (1), P.395]) $$\begin{aligned} \label{formula} \int_0^\infty&e^{-p^2t^2}J_\nu(at)J_\nu(bt)tdt=\frac{1}{2p^2}e^{-\frac{a^2+b^2}{4p^2}}I_\nu\Big(\frac{ab}{2p^2}\Big),\nonumber\\ &\qquad \mathop{\mathrm{Re}}\nu>-1,\quad |\arg p|<\frac{\pi}{4},\quad I_\nu(r)=e^{-\frac{1}{2}\nu\pi i}J_\nu(re^{i\frac{\pi}{2}}),\end{aligned}$$ with $t=s_2,p=1,a=\sqrt{2B_0}r_2,b=2s_1e^{-i(tB_0+\frac{\pi}{2})},\nu=\alpha_k$, we get $$\begin{aligned} \int_0^\infty e^{-s_2^2}J_{\alpha_k}\big(\sqrt{2B_0}r_2s_2\big)&J_{\alpha_k}\Big(2s_1s_2e^{-i(tB_0+\frac{\pi}{2})}\Big)s_2ds_2\\ &=\frac{1}{2}e^{-\frac{B_0r_2^2+2s_1^2e^{-i(2tB_0+\pi)}}{2}}I_{\alpha_k}\big(\sqrt{2B_0}r_2s_1e^{-i(tB_0+\frac{\pi}{2})}\big),\end{aligned}$$ where $I_{\alpha_k}$ denotes the modified Bessel function of order $\alpha_k$. Hence $$\begin{aligned} G_{k,t}(r_1,r_2)&=\frac{1}{2}\int_0^\infty e^{-s_1^2}J_{\alpha_k}(\sqrt{2B_0}r_1s_1)e^{-\frac{B_0r_2^2+2s_1^2e^{-i(2tB_0+\pi)}}{2}} I_{\alpha_k}\big(\sqrt{2B_0}r_2s_1e^{-i(tB_0+\frac{\pi}{2})}\big)s_1ds_1\\ &=\frac{1}{4B_0}\int_0^\infty e^{-\frac{s_1^2}{2B_0}}J_{\alpha_k}(r_1s_1)e^{-\frac{B_0r_2^2+\frac{s_1^2}{B_0}e^{-i(2tB_0+\pi)}}{2}} I_{\alpha_k}(r_2s_1e^{-i(tB_0+\frac{\pi}{2})})s_1ds_1\\ &=\frac{1}{4B_0}e^{-i\alpha_k\frac{\pi}{2}}e^{-\frac{B_0r_2^2}{2}}\int_0^\infty e^{-\frac{s_1^2}{2B_0}(1+e^{-i(2tB_0+\pi)})}J_{\alpha_k}(r_1s_1)J_{\alpha_k}(r_2s_1e^{-itB_0})s_1ds_1\\ &=\frac{1}{4B_0}e^{-i\alpha_k\frac{\pi}{2}}e^{-\frac{B_0r_2^2}{2}}\frac{B_0}{1+e^{-i(2tB_0+\pi)}} e^{-\frac{B_0r_1^2+B_0r_2^2e^{-2itB_0}}{2(1+e^{-i(2tB_0+\pi)})}}I_{\alpha_k}(\frac{B_0r_1 r_2e^{-itB_0}}{1+e^{-i(2tB_0+\pi)}})\\ &=\frac{1}{4(1+e^{-i(2tB_0+\pi)})}e^{-i\alpha_k\frac{\pi}{2}} e^{-\frac{B_0(r_1^2+r_2^2)}{2(1+e^{-i(2tB_0+\pi)})}}I_{\alpha_k}\Big(\frac{B_0r_1 r_2e^{-itB_0}}{1+e^{-i(2tB_0+\pi)}}\Big)\end{aligned}$$ where we have used the fact $I_\nu(r)=e^{-i\frac{\pi}{2}\nu}J_\nu(re^{i\frac{\pi}{2}})$ in the third equality and [\[formula\]](#formula){reference-type="eqref" reference="formula"} with $t=s_1,p^2=\frac{1+e^{-i(2tB_0+\pi)}}{2B_0},a=r_1,b=r_2e^{-itB_0},\nu=\alpha_k$ in the last equality. Finally, we use the simple fact (for $tB_0\neq k\pi, k\in\mathbb{Z}$) $$\frac{1}{1+e^{-i(2tB_0+\pi)}}=\frac{e^{itB_0}}{2i\sin tB_0}$$ to obtain $$\begin{aligned} u(t,x)=&\frac{2B_0}{\pi}\frac{e^{itB_0}}{8i\sin tB_0}\sum_{k\in\mathbb{Z}}e^{ik\theta_1}e^{it(\alpha_kB_0-\beta_k)}\\ &\times\Bigg[\int_0^\infty e^{\frac{B_0(r_1^2+r_2^2)}{4}}f_k(r_2)e^{-\frac{B_0(r_2^2+r_1^2)}{4}\cdot\frac{e^{itB_0}}{i\sin tB_0}}I_{\alpha_k}\Bigg(\frac{B_0r_1r_2}{2i\sin (tB_0)}\Bigg)r_2dr_2\Bigg]\\ =&\frac{B_0e^{-itB_0\alpha}}{4\pi i\sin (tB_0)}\sum_{k\in\mathbb{Z}}e^{ik\theta_1}e^{-itB_0k}\Bigg(\int_0^\infty f_k(r_2)e^{-\frac{B_0(r_1^2+r_2^2)\cos (tB_0)}{4i\sin (tB_0)}}I_{\alpha_k}\Bigg(\frac{B_0r_1r_2}{2i\sin (tB_0)}\Bigg)r_2dr_2\Bigg),\end{aligned}$$ where we use $\beta_k=(k+\alpha)B_0+(1+\alpha_k)B_0$ in the last line. Recalling $f_k$ in [\[f-k\]](#f-k){reference-type="eqref" reference="f-k"}, we finally have $$\label{S:sum} \begin{split} u(t,x) =&\frac{B_0e^{-itB_0\alpha}}{8\pi^2 i\sin (tB_0)}\int_0^\infty \int_0^{2\pi}e^{-\frac{B_0(r_1^2+r_2^2)}{4i\tan (tB_0)}} \\&\times\sum_{k\in\mathbb{Z}} \Bigg( e^{ik(\theta_1-\theta_2-tB_0)} I_{\alpha_k}\bigg(\frac{B_0r_1r_2}{2i\sin (tB_0)}\bigg)\Bigg) f(r_2,\theta_2)r_2dr_2 d\theta_2. \end{split}$$ Now we consider the summation in $k$. Let $z=\frac{B_0r_1r_2}{2i\sin (tB_0)}$ and $\theta=\theta_1-\theta_2-tB_0$ and recall the following integral representation [@Wat44] of the modified Bessel function $I_\nu$ $$\label{mBessel:integral} I_\nu(z)=\frac{1}{\pi}\int_0^\pi e^{z\cos s}\cos(\nu s)d s-\frac{\sin(\pi\nu)}{\pi}\int_0^\infty e^{-z\cosh s}e^{-s\nu}ds.$$ Recall $\alpha_k=|\alpha+k|$ and $\alpha\in(0,1)$, we need to consider $$\label{equ:term1} \frac1\pi \sum_{k\in\mathbb{Z}} e^{ik\theta} \int_0^\pi e^{z\cos s} \cos(\alpha_k s) ds,$$ and $$\label{equ:term2} \frac1\pi \sum_{k\in\mathbb{Z}} e^{ik\theta } \sin(\pi\alpha_k)\int_0^\infty e^{-z\cosh s} e^{-s\alpha_k} ds.$$ Similarly as in [@FZZ22 Proposition 3.1], we use the Poisson summation formula $$\sum_{j\in\mathbb{Z}} \delta(x-2\pi j)=\sum_{k\in\mathbb{Z}} \frac1{2\pi} e^{i kx},$$ to obtain $$\begin{split} \frac1{\pi} \sum_{k\in\mathbb{Z}} e^{ik\theta} \cos(\alpha_k s) = \sum_{j\in\mathbb{Z}}\big[e^{is\alpha}\delta(\theta+s+2j\pi) +e^{-is\alpha}\delta(\theta-s +2j\pi)\big]. \end{split}$$ Therefore we have $$\begin{aligned} & \eqref{equ:term1}\nonumber\\ &=\sum_{j\in\mathbb{Z}}\int_0^\pi e^{z\cos s}\Bigg(e^{is\alpha}\delta(\theta+s+2\pi j)+e^{-is\alpha}\delta(\theta-s+2\pi j)\Bigg)d s\nonumber\\ &=e^{z\cos(\theta)}e^{-i\alpha\theta}\sum_{j\in\mathbb{Z}} \chi_{[-\pi,\pi]}(\theta+2\pi j)e^{-i2\pi j\alpha} \label{term1+2}.\end{aligned}$$ Recall $\theta=\theta_1-\theta_2-tB_0\in \mathbb{R}$ and observe that $$\theta+2\pi j_0\in (-\pi, \pi)\implies \theta+2\pi (j_0\pm m) \notin (-\pi, \pi), |m|\ge 1,$$ the summation in $j$ is only one term (except for $\theta+2j_0\pi=\pm\pi$). It's easily to verify that the identity [\[term1+2\]](#term1+2){reference-type="eqref" reference="term1+2"} are the same at $\theta+2j_0\pi=\pm\pi.$ For all $\theta\in\mathbb{R},$ there exists $j_0\in\mathbb{Z}$ such that $\theta+2j_0\pi\in[-\pi,\pi].$ Therefore, we get $$\begin{aligned} \eqref{equ:term1}=e^{z\cos(\theta)}e^{-i\alpha\theta}e^{-i2\pi j_0\alpha}\left\{ \begin{array}{ll} 1, & \hbox{$|\theta+2j_0\pi|<\pi$ ;} \\ e^{-i2\pi \alpha}+1 , & \hbox{$\theta+2j_0\pi=-\pi;$}\\ e^{i2\pi\alpha}+1, & \hbox{$\theta+2j_0\pi=\pi$.} \end{array} \right.\end{aligned}$$ Recall again that $\alpha_k=|\alpha+k|,k\in\mathbb{Z}$ and $\alpha\in(0,1)$, we have $$\begin{aligned} \alpha_k=|\alpha+k|= \begin{cases} k+\alpha,&k\geq0;\\ -\alpha-k,&k\leq-1, \end{cases}\end{aligned}$$ and hence $$\begin{aligned} \sin(\pi\alpha_k)=\sin(\pi|\alpha+k|)= \begin{cases} \cos k\pi\sin(\pi\alpha)=e^{ik\pi}\sin(\pi\alpha),&k\geq0;\\ -\cos k\pi\sin(\pi\alpha)=-e^{ik\pi}\sin(\pi\alpha),&k\leq-1. \end{cases}\end{aligned}$$ We can compute the summation as follows $$\begin{aligned} &\sum_{k\in\mathbb{Z}}\sin(\pi|\alpha+k|)e^{-s|\alpha+k|}e^{ik\theta}\\ &\quad=\sin(\pi\alpha)\sum_{k\geq0}e^{ik\pi}e^{-s(k+\alpha)}e^{ik\theta}-\sin(\pi\alpha)\sum_{k\leq-1}e^{ik\pi}e^{s(k+\alpha)}e^{ik\theta}\\ &\quad=\sin(\pi\alpha)\Big(e^{-\alpha s}\sum_{k\geq0}e^{ik(is+\pi+\theta)}-e^{\alpha s}\sum_{k\geq 1} e^{ik(is-\pi-\theta)}\Big)\\ &\quad=\sin(\pi\alpha)\Big(\frac{e^{-\alpha s}}{1-e^{i(is+\pi+\theta)}}-\frac{e^{\alpha s}e^{i(is-\pi-\theta)}}{1-e^{i(is-\pi-\theta)}}\Big)\\ &\quad=\sin(\pi\alpha)\Big(\frac{e^{-\alpha s}}{1+e^{-s+i\theta}}+\frac{e^{\alpha s}}{1+e^{s+i\theta)}}\Big).\end{aligned}$$ Therefore, we see $$\begin{aligned} & \eqref{equ:term2}\\ = &\frac{\sin(\alpha\pi)}{\pi}\int_0^\infty e^{-z\cosh s} \Big(\frac{e^{-\alpha s}}{1+e^{-s+i\theta}}+\frac{e^{\alpha s}}{1+e^{s+i\theta)}}\Big)\,ds\\ = &\frac{\sin(\alpha\pi)}{\pi}\int_{-\infty}^\infty e^{-z\cosh s} \frac{e^{-\alpha s}}{1+e^{-s+i\theta}}\,ds.\end{aligned}$$ So we put the two terms together and recall $\theta=\theta_1-\theta_2-tB_0$ to obtain $$\begin{split} K_S(x,y)&=\frac{B_0e^{-itB\alpha}}{8\pi^2 i\sin (tB_0)}e^{\frac{iB_0(r_1^2+r_2^2)}{4\tan (tB_0)}}\\ \times&\Big[e^{\frac{B_0r_1r_2}{2i\sin (tB_0)}\cos(\theta_1-\theta_2-tB_0)} e^{-i\alpha(\theta_1-\theta_2-tB_0+2j_0\pi)}\chi(\theta, j_0)\\ &-\frac{\sin(\pi\alpha)}{\pi}\int_{\mathbb{R}} e^{-\frac{B_0r_1r_2}{2i\sin (tB_0)}\cosh s}\frac{e^{-\alpha s}}{1+e^{-s+i(\theta_1-\theta_2-tB_0)}}\,ds\Big], \end{split}$$ which gives [\[S:express\]](#S:express){reference-type="eqref" reference="S:express"}. # The Construction of Schrödinger propagator based on the Schulman-Sunada formula {#sec:const2} In this section, we will use the Schulman-Sunada formula to reconstruct the Schrödinger propagator. We refer the reads to [@stov; @stov1]. Let $M=\mathbb{R}^2\setminus\{\vec{0}\}=(0,+\infty)\times \mathbb{S}^1$ where $\mathbb{S}^1$ is the unit circle. The universal covering space of $M$ is $\tilde{M}=(0,+\infty)\times \mathbb{R}$, then $M=\tilde{M}/\Gamma$ where the structure group $\Gamma=2\pi\mathbb{Z}$ acts in the second factor of the Cartesian product. Then Schulman's ansatz (see [@stov; @stov1]) enables us to compute the Schrödinger propagator $e^{-it H_{\alpha, B_0}}$ on $M$ by using the Schrödinger propagator $e^{-it \tilde{H}_{\alpha, B_0}}$ (see the operator $\tilde{H}_{\alpha, B_0}$ in [\[t-H\]](#t-H){reference-type="eqref" reference="t-H"} below) on $\tilde{M}$. More precisely, see [@stov1 (1)], we have $$\label{SS} e^{-it H_{\alpha, B_0}}(r_1,\theta_1; r_2, \theta_2)=\sum_{j\in\mathbb{Z}} e^{-it \tilde{H}_{\alpha, B_0}} (r_1,\theta_1+2j\pi; r_2, \theta_2).$$ This is similar to the construction of wave propagator on $\mathbb{T}^n$, see [@sogge (3.5.12)]. In the following subsections, we will construct the Schrödinger propagator $e^{-it \tilde{H}_{\alpha, B_0}}$. ## The eigenfunctions and eigenvalues Before we construct the propagator $e^{-it \tilde{H}_{\alpha, B_0}}$, let us give some remarks about the difference and advantages between $\tilde{H}_{\alpha, B_0}$ and $H_{\alpha, B_0}$. First, we recall the proof of Proposition [Proposition 4](#prop:spect){reference-type="ref" reference="prop:spect"}. From [\[H-A\]](#H-A){reference-type="eqref" reference="H-A"}, in the polar coordinates $(r,\theta)\in M$, then $$H_{\alpha, B_0}=-\partial_r^2-\frac{1}{r}\partial_r+\frac{1}{r^2}\Big(-i\partial_\theta+\alpha+\frac{B_0r^2}{2}\Big)^2,$$ which acts on $L^2(M, rdr\, d\theta)$. For $V_{k,m}(x)$ in [\[eigen-f\]](#eigen-f){reference-type="eqref" reference="eigen-f"} and $\lambda_{k,m}$ in [\[eigen-v\]](#eigen-v){reference-type="eqref" reference="eigen-v"}, we have show that $$H_{\alpha, B_0} V_{k,m}(x)=\lambda_{k,m} V_{k,m}(x).$$ We remark here that we choose $e^{ik\theta}$ as an eigenfunction of the operator $\Big(-i\partial_\theta+\alpha+\frac{B_0r^2}{2}\Big)^2$ on $L^2_\theta([0,2\pi))$ which satisfies that $$\begin{cases} \Big(-i\partial_\theta+\alpha+\frac{B_0r^2}{2}\Big)^2 \varphi(\theta)=\Big(k+\alpha+\frac{B_0r^2}{2}\Big)^2 \varphi(\theta)\\ \varphi(0)=\varphi(2\pi). \end{cases}$$ Instead, we consider the operator $$\label{t-H} \tilde{H}_{\alpha, B_0}=-\partial_r^2-\frac{1}{r}\partial_r+\frac{1}{r^2}\Big(-i\partial_\theta+\alpha+\frac{B_0r^2}{2}\Big)^2,$$ which acts on $L^2(\tilde{M}, rdr\, d\theta)$. We emphasize that the variable $\theta\in\mathbb{R}$ while not compact manifold $\mathbb{S}^1$. Then we choose $e^{i(\tilde{k}-\alpha)\theta}$ as an eigenfunction of the operator $\Big(-i\partial_\theta+\alpha+\frac{B_0r^2}{2}\Big)^2$ on $L^2_\theta(\mathbb{R})$ which satisfies that $$\label{eq:ef} \Big(-i\partial_\theta+\alpha+\frac{B_0r^2}{2}\Big)^2 \varphi(\theta)=\Big(\tilde{k}+\frac{B_0r^2}{2}\Big)^2 \varphi(\theta).$$ It worths to point out that $\tilde{k}\in\mathbb{R}$ is a real number while $k\in\mathbb{Z}$. More important, we inform move the $\alpha$ in right hand side of [\[eq:ef\]](#eq:ef){reference-type="eqref" reference="eq:ef"} to the $e^{i(\tilde{k}-\alpha)\theta}$ which will simplify the eigenfunctions. Now we modify the argument of Proposition [Proposition 4](#prop:spect){reference-type="ref" reference="prop:spect"}to solve $$\label{eigen-p'} \tilde{H}_{\alpha, B_0} g(x)=\lambda g(x)$$ to obtain the eigenfunctions of $\tilde{H}_{\alpha, B_0}$. Define the Fourier transform $F_{\theta\to\tilde{k}}$ with respect to the variable $\theta$ $$\label{Fourier'} F_{\theta\to\tilde{k}} f(r,\tilde{k})=\frac{1}{2\pi}\int_{\mathbb{R}}e^{i\tilde{k}\theta}f(r,\theta)\,d\theta:= \hat{f}(r,\tilde{k}),\quad \tilde{k}\in\mathbb{R}.$$ By taking the Fourier transform of [\[eigen-p\'\]](#eigen-p'){reference-type="eqref" reference="eigen-p'"}, in contrast to [\[eq:gk\]](#eq:gk){reference-type="eqref" reference="eq:gk"} we obtain $$\label{eq:gk'} \hat{g}''(r,\tilde{k})+\frac{1}{r}\hat{g}'(r,\tilde{k})-\frac{1}{r^2}\Big(\tilde{k}+\frac{B_0 r^2}{2}\Big)^2 \hat{g}(r,\tilde{k})=-\lambda \hat{g}(r,\tilde{k}).$$ Let $$\psi_{\tilde{k}}(s)=\Big(\frac{2s}{B_0}\Big)^{-\frac{|\tilde{k}|}{2}}e^{\frac{s}{2}} \hat{g}(\Big(\sqrt{\frac{2s}{B_0}}, \tilde{k}\Big),$$ then $\psi_{\tilde{k}}(s)$ satisfies $$\label{eq:psik} s\psi''_{\tilde{k}}(s)+(1+|\tilde{k}|-s)\psi'_{\tilde{k}}(s)-\frac{1}{2}\Big(1+|\tilde{k}|+\tilde{k}-\frac{\lambda}{B_0}\Big)\psi_{\tilde{k}}(s)=0.$$ which is same to [\[eq:phik\]](#eq:phik){reference-type="eqref" reference="eq:phik"} by replacing $|k+\alpha|$ by $\tilde{k}$. By using Lemma [Lemma 8](#lem:KCH){reference-type="ref" reference="lem:KCH"} again, we have two linearly independent solutions of [\[eq:psik\]](#eq:psik){reference-type="eqref" reference="eq:psik"}, hence two linearly independent solutions of [\[eq:gk\'\]](#eq:gk'){reference-type="eqref" reference="eq:gk'"} are given by $$\begin{aligned} \hat{g}^1(\lambda;r,\tilde{k})&=r^{|\tilde{k}|}M\Big(\tilde{\beta}(\tilde{k},\lambda), \tilde{\gamma}(\tilde{k}),\frac{B_0r^2}{2}\Big)e^{-\frac{B_0r^2}{4}}\\ \hat{g}^2(\lambda;r,\tilde{k})&=r^{|\tilde{k}|}U\Big(\tilde{\beta}(\tilde{k},\lambda),\tilde{\gamma}(\tilde{k}),\frac{B_0r^2}{2}\Big)e^{-\frac{B_0r^2}{4}}\end{aligned}$$ with $$\begin{aligned} \tilde{\beta}(\tilde{k},\lambda)&=\frac{1}{2}\Big(1+\tilde{k}+|\tilde{k}|-\frac{\lambda}{B_0}\Big),\\ \tilde{\gamma}(\tilde{k})&=1+|\tilde{k}|.\end{aligned}$$ Therefore, the general solution of [\[eq:psik\]](#eq:psik){reference-type="eqref" reference="eq:psik"} is given by $$\hat{g}(r,\tilde{k})=A_{\tilde{k}}\hat{g}^1(\lambda;r,\tilde{k})+B_{\tilde{k}}\hat{g}^2(\lambda;r,\tilde{k}).$$ where $A_{\tilde{k}}, B_{\tilde{k}}$ are two constants which depend on $\tilde{k}\in\mathbb{R}$. Let $$m:=-\tilde{\beta}(\tilde{k},\lambda)=-\frac{1}{2}\Big(1+\tilde{k}+|\tilde{k}|-\frac{\lambda}{B_0}\Big).$$ Similar as above, we use this asymptotic Lemma [Lemma 9](#lem:asy){reference-type="ref" reference="lem:asy"} to conclude that $m\in\mathbb{N}$ again, we omit the details. Therefore, we must have $$\mathbb{N} \ni m=-\frac{1}{2}\Big(1+\tilde{k}+|\tilde{k}|-\frac{\lambda}{B_0}\Big),$$ and $$\hat{g}(r,\tilde{k})=r^{|\tilde{k}|}e^{-\frac{B_0r^2}{4}}\, P_{\tilde{k}-\alpha,m}\Big(\frac{B_0r^2}{2}\Big).$$ Therefore, we obtain a complete set of generalized eigenfunctions of $\tilde{H}_{\alpha, B_0}$ $$\Big\{U_{m}(x, \tilde{k}): m\in \mathbb{N}, \tilde{k}\in\mathbb{R}\Big\}$$ where $$\label{eigen-f'} U_{m}(x, \tilde{k})=|x|^{|\tilde{k}|}e^{-\frac{B_0 |x|^2}{4}}\, P_{\tilde{k}-\alpha,m}\Bigg(\frac{B_0|x|^2}{2}\Bigg)e^{i(\tilde{k}-\alpha)\theta}$$ which belongs to $L^2(\tilde{M}, rdr\, d\theta)$. Thus from $-m=\frac{1}{2}\Big(1+\tilde{k}+|\tilde{k}|-\frac{\lambda}{B_0}\Big)$, we solve [\[eigen-p\'\]](#eigen-p'){reference-type="eqref" reference="eigen-p'"} to obtain the eigenvalues $\lambda$ of $\tilde{H}_{\alpha,B_0}$ $$\lambda_{\tilde{k},m}=(2m+1+|\tilde{k}|+\tilde{k})B_0,\quad \tilde{k}\in\mathbb{R},\, m\in\mathbb{N}.$$ We obtain analogue of [\[Po-L\]](#Po-L){reference-type="eqref" reference="Po-L"} by using [\[La-po\]](#La-po){reference-type="eqref" reference="La-po"} and [\[P-L\]](#P-L){reference-type="eqref" reference="P-L"} $$\begin{split} &\sum_{m=0}^\infty e^{-cm}\frac{m !}{\Gamma(m+|\tilde{k}|+1)} \Bigg(\begin{array}{c} m+|\tilde{k}| \\ m \\ \end{array} \Bigg)^2 P_{\tilde{k}-\alpha,m} (a) P_{\tilde{k}-\alpha,m} (b)\\ &=\frac{e^{\frac{|\tilde{k}| c}2}}{(ab)^{\frac{|\tilde{k}|}2}(1-e^{-c})} \exp\left(-\frac{(a+b)e^{-c}}{1-e^{-c}}\right) I_{|\tilde{k}|}\left(\frac{2\sqrt{ab}e^{-\frac{c}2}}{1-e^{-c}}\right). \end{split}$$ ## Construction of $e^{-it \tilde{H}_{\alpha, B_0}}$ Now we construct the propagator $e^{-it \tilde{H}_{\alpha, B_0}}$ by using the above eigenfunctions. Let $\tilde{U}_{m}(x, \tilde{k})$ be the $L^2$-normalization of $U_{m}(x, \tilde{k})$ in [\[eigen-f\'\]](#eigen-f'){reference-type="eqref" reference="eigen-f'"}. We wirte initial data $f(x)\in L^2$ as $$f(x)=\sum_{m\in\mathbb{N}}\int_{\mathbb{R}} c_{m}(\tilde{k}) \tilde{U}_{m}(x, \tilde{k}) \, d\tilde{k}$$ where $$c_{m}(\tilde{k})=\int_{\mathbb{R}^2}f(x)\overline{\tilde{U}_{m}(x, \tilde{k})}\, dx.$$ Using the Fourier transform [\[Fourier\'\]](#Fourier'){reference-type="eqref" reference="Fourier'"}, we solve $$\begin{cases} \big(i\partial_t-\tilde{H}_{\alpha,B_0}\big)u(t,x)=0,\quad (t, x)\in \mathbb{R}\times \tilde{M}\\ u(0,x)=f(x). \end{cases}$$ to obtain $$u(t,x)=\sum_{ m\in\mathbb{N}} \int_{\mathbb{R}} e^{-it\lambda_{\tilde{k},m}}\left(\int_{\mathbb{R}^2}f(y)\overline{\tilde{U}_{m}(y, \tilde{k})}\, dy\right)\tilde{U}_{m}(x, \tilde{k}) \, d\tilde{k}.$$ By repeating the the proof of Proposition [Proposition 10](#prop:S){reference-type="ref" reference="prop:S"}, we similarly show that $$\begin{split} u(t,x) =&\frac{B_0}{8\pi^2 i\sin (tB_0)}\int_0^\infty \int_{\mathbb{R}}e^{-\frac{B_0(r_1^2+r_2^2)}{4i\tan (tB_0)}} \\&\times \int_{\mathbb{R}} \Bigg( e^{i(\tilde{k}-\alpha)(\theta_1-\theta_2-tB_0)} I_{|\tilde{k}|}\bigg(\frac{B_0r_1r_2}{2i\sin (tB_0)}\bigg)\Bigg)\, d\tilde{k} f(r_2,\theta_2)r_2dr_2 d\theta_2. \end{split}$$ In contrast to [\[S:sum\]](#S:sum){reference-type="eqref" reference="S:sum"}, the difference here is that we replace the summation in $k\in\mathbb{Z}$ by integration on $\tilde{k}\in\mathbb{R}$. Hence the kernel of $e^{-it \tilde{H}_{\alpha, B_0}}$ is $$\begin{split} \tilde{K}_S(x,y) =&\frac{B_0}{4\pi i\sin (tB_0)}e^{-\frac{B_0(r_1^2+r_2^2)}{4i\tan (tB_0)}} \\&\times \int_{\mathbb{R}} \Bigg( e^{i(\tilde{k}-\alpha)(\theta_1-\theta_2-tB_0)} I_{|\tilde{k}|}\bigg(\frac{B_0r_1r_2}{2i\sin (tB_0)}\bigg)\Bigg)\, d\tilde{k}, \end{split}$$ where $x=(r_1,\theta_1)\in \tilde{M}$ and $y=(r_2,\theta_2)\in \tilde{M}$. Now, instead of summing in $k$ as before, we consider the integration in $\tilde{k}$. Again by letting $z=\frac{B_0r_1r_2}{2i\sin (tB_0)}$ and $\theta=\theta_1-\theta_2-tB_0$ and using [\[mBessel:integral\]](#mBessel:integral){reference-type="eqref" reference="mBessel:integral"}, we compute that $$\begin{split} & \frac1\pi \int_{\mathbb{R}} e^{i\tilde{k}\theta} \int_0^\pi e^{z\cos s} \cos(|\tilde{k}| s) dsd\tilde{k}\\ &=\frac1{2\pi} e^{z\cos \theta} \Big(\chi_{[0,\pi]}(\theta)+\chi_{[0,\pi]}(-\theta)\Big)= e^{z\cos \theta} \chi_{[-\pi,\pi]}(\theta), \end{split}$$ and $$\begin{split} & \frac1\pi \int_{\mathbb{R}} e^{i\tilde{k}\theta} \sin(\pi |\tilde{k}|)\int_0^\infty e^{-z\cosh s} e^{-s|\tilde{k}|} ds \,d\tilde{k}\\ &= \frac1\pi\int_0^\infty e^{-z\cosh s} \Big( \int_{0}^\infty e^{i\tilde{k}\theta}\frac{e^{i\pi \tilde{k}}-e^{-i\pi \tilde{k}}}{2i} e^{-s\tilde{k}} d\tilde{k} \\ & \qquad+ \int_{-\infty}^0 e^{i\tilde{k}\theta} \frac{e^{-i\pi \tilde{k}}-e^{i\pi \tilde{k}}}{2i} e^{s\tilde{k}} d\tilde{k}\Big) \,ds\\ &= \frac1{2\pi}\int_{-\infty}^\infty e^{-z\cosh s} \big(\frac{1}{\theta+\pi+is}-\frac{1}{\theta-\pi+is}\big) ds \end{split}$$ Therefore, we obtain $$\begin{split} &\tilde{K}_S(x,y) =\frac{B_0}{8\pi^2 i\sin (tB_0)}e^{-\frac{B_0(r_1^2+r_2^2)}{4i\tan (tB_0)}} e^{-i\alpha\theta}\\&\times \Big( e^{z\cos \theta} \chi_{[-\pi,\pi]}(\theta)-\frac1{2\pi}\int_{-\infty}^\infty e^{-z\cosh s} \big(\frac{1}{\theta+\pi+is}-\frac{1}{\theta-\pi+is}\big) ds \Big). \end{split}$$ Finally, by using [\[SS\]](#SS){reference-type="eqref" reference="SS"}, we have that $$\begin{split} &e^{-it H_{\alpha, B_0}}(r_1,\theta_1; r_2, \theta_2)\\ &=\frac{B_0}{8\pi^2 i\sin (tB_0)}e^{-\frac{B_0(r_1^2+r_2^2)}{4i\tan (tB_0)}} \sum_{j\in\mathbb{Z}} e^{-i\alpha(\theta+2j\pi)} \Big( e^{z\cos (\theta+2j\pi)} \chi_{[-\pi,\pi]}(\theta+2j\pi)\\&\qquad-\frac1{2\pi}\int_{-\infty}^\infty e^{-z\cosh s} \big(\frac{1}{(\theta+2j\pi)+\pi+is}-\frac{1}{(\theta+2j\pi)-\pi+is}\big) ds \Big). \end{split}$$ Due to the period property of $\cos$ function and for all $\theta\in\mathbb{R},$ there exists $j_0\in\mathbb{Z}$ such that $\theta+2j_0\pi\in [-\pi,\pi),$ the first term in the big bracket becomes $$\begin{aligned} &e^{z\cos \theta} e^{-i\alpha\theta} \sum_{j\in\mathbb{Z}} e^{-i\alpha 2j\pi} \chi_{[-\pi,\pi]}(\theta+2j\pi)\\ &=e^{z\cos \theta} e^{-i\alpha\theta}e^{-i2\pi j_0\alpha}\times\left\{ \begin{array}{ll} 1, & \hbox{$|\theta+2j_0\pi|<\pi$;} \\ e^{-i2\pi\alpha}+1, & \hbox{$\theta+2j_0\pi=-\pi$;}\\ e^{i2\pi\alpha}+1, & \hbox{$\theta+2j_0\pi=\pi$.} \end{array} \right.\end{aligned}$$ which is the same to [\[term1+2\]](#term1+2){reference-type="eqref" reference="term1+2"}. Hence it is the same to the first term of [\[S:express\]](#S:express){reference-type="eqref" reference="S:express"}. For the second term in the big bracket, we use the formula $$\sum_{j\in\mathbb{Z}} \frac{e^{-2\pi i\alpha j}}{\sigma+2\pi j}=\frac{i e^{i\alpha\sigma}}{e^{i\sigma}-1},\quad \alpha\in(0,1),\quad \sigma\in\mathbb{C}\setminus 2\pi\mathbb{Z},$$ to obtain $$\begin{split} \sum_{j\in\mathbb{Z}} e^{-2\pi i\alpha j}&\big(\frac{1}{(\theta+2j\pi)+\pi-is}-\frac{1}{(\theta+2j\pi)-\pi-is}\big)\\& =2\sin(\pi \alpha)\frac{e^{\alpha(s+i\theta)}}{1+e^{s+i\theta}}. \end{split}$$ Now we consider the second term $$\begin{split} &-e^{-i\theta\alpha}\frac{2\sin(\pi\alpha)}{2\pi}e^{i\theta\alpha} \int_{-\infty}^\infty e^{-z\cosh s} \frac{e^{\alpha s}}{1+e^{s+i\theta}}\,ds\\ =&-\frac{\sin(\pi\alpha)}{\pi}\int_{-\infty}^\infty e^{-z\cosh s} \frac{e^{-\alpha s}}{1+e^{-s+i\theta}}\,ds \end{split}$$ Recall $\theta=\theta_1-\theta_2-tB_0$ and $z=\frac{B_0r_1r_2}{2i\sin (tB_0)}$, we obtain $$\begin{split} K_S(x,y)&=\frac{B_0e^{-itB\alpha}}{8\pi^2 i\sin (tB_0)}e^{\frac{iB_0(r_1^2+r_2^2)}{4\tan (tB_0)}}\\ \times&\Big[e^{\frac{B_0r_1r_2}{2i\sin (tB_0)}\cos(\theta_1-\theta_2-tB_0)} e^{-i\alpha(\theta_1-\theta_2-tB_0+2j_0\pi)}\chi(\theta, j_0)\\ &-\frac{\sin(\pi\alpha)}{\pi}\int_{\mathbb{R}} e^{-\frac{B_0r_1r_2}{2i\sin (tB_0)}\cosh s}\frac{e^{-\alpha s}}{1+e^{-s+i(\theta_1-\theta_2-tB_0)}}\,ds\Big], \end{split}$$ which is exact same to [\[S:express\]](#S:express){reference-type="eqref" reference="S:express"}. ◻ # proof of Theorem [Theorem 1](#thm:S){reference-type="ref" reference="thm:S"} {#sec:proof} In this section, we prove the main Theorem [Theorem 1](#thm:S){reference-type="ref" reference="thm:S"} by using [\[S:express\]](#S:express){reference-type="eqref" reference="S:express"}. We first prove the dispersive estimate [\[dis-S\]](#dis-S){reference-type="eqref" reference="dis-S"}. To this end, it is enough to prove $$\begin{aligned} \Big|\int_{\mathbb{R}} e^{-\frac{B_0r_1r_2}{2i\sin (tB_0)}\cosh s}\frac{e^{-\alpha s}}{1+e^{-s+i(\theta_1-\theta_2-tB_0)}}\,ds\Big|\leq C,\end{aligned}$$ where $C$ is a constant independent of $t$, $r_1, r_2$ and $\theta_1, \theta_2$. We notice that $$\begin{aligned} &\int_{\mathbb{R}} e^{-\frac{B_0r_1r_2}{2i\sin (tB_0)}\cosh s}\frac{e^{-\alpha s}}{1+e^{-s+i(\theta_1-\theta_2-tB_0)}}\,ds\\ &=\int_{0}^\infty e^{-\frac{B_0r_1r_2}{2i\sin (tB_0)}\cosh s}\Big(\frac{e^{-\alpha s}}{1+e^{-s+i(\theta_1-\theta_2-tB_0)}}+\frac{e^{\alpha s}}{1+e^{s+i(\theta_1-\theta_2-tB_0)}}\Big)\,ds\end{aligned}$$ then we just need to verify that, for $\theta=\theta_1-\theta_2-tB_0,$ $$\begin{aligned} \int_{0}^\infty\Big|\frac{e^{-\alpha s}}{1+e^{-s+i\theta}}+\frac{e^{\alpha s}}{1+e^{s+i\theta}}\Big| \,ds\lesssim 1,\end{aligned}$$ where the implicit constant is independent of $\theta$. In fact, $$\begin{aligned} &\frac{e^{-\alpha s}}{1+e^{-s+i\theta}}+\frac{e^{\alpha s}}{1+e^{s+i\theta}}\\ &=\frac{\cosh(\alpha s)e^{-i\theta}+\cosh((1-\alpha)s)}{\cos\theta+\cosh s}\\ &=\frac{\cosh(\alpha s)\cos\theta+\cosh((1-\alpha)s)-i\sin\theta\cosh(\alpha s)}{2(\cos^2(\frac{\theta}{2})+\sinh^2(\frac s 2))}\\ &=\frac{2\cos^2(\frac{\theta}2)\cosh(\alpha s)+(\cosh((1-\alpha)s)-\cosh(\alpha s))-2i\sin(\frac\theta 2)\cos(\frac\theta 2)\cosh(\alpha s)}{2(\cos^2(\frac{\theta}{2})+\sinh^2(\frac s 2))}.\end{aligned}$$ Since that $\cosh x-1\sim\frac{x^2}2,\sinh x\sim x$, as $x\to 0;$ $\cosh x\sim e^x,\sinh x\sim e^{x}$, as $x\to \infty,$ we have $$\begin{aligned} \int_{0}^\infty\Big|\frac{\cos^2(\frac{\theta}2)\cosh(\alpha s)}{\cos^2(\frac{\theta}{2})+\sinh^2(\frac s 2)}\Big| \,ds\lesssim\int_0^1\frac{2|\cos(\frac\theta 2)|}{s^2+(2|\cos(\frac\theta 2)|)^2}ds+\int_1^\infty\ e^{(\alpha-1)s}ds\lesssim 1.\end{aligned}$$ Similarly, we obtain $$\begin{aligned} \int_{0}^\infty\Big|\frac{\sin(\frac\theta 2)\cos(\frac\theta 2)\cosh(\alpha s)}{\cos^2(\frac{\theta}{2})+\sinh^2(\frac s 2)}\Big| \,ds\lesssim 1.\end{aligned}$$ Finally, we verify that $$\begin{aligned} &\int_{0}^\infty\Big|\frac{\cosh((1-\alpha)s)-\cosh(\alpha s)}{\cos^2(\frac{\theta}{2})+\sinh^2(\frac s 2)}\Big| \,ds\\ &\lesssim\int_0^1\frac{|\frac{(1-\alpha)^2}2-\frac{\alpha^2}2|s^2}{s^2}ds+\int_1^\infty \big( e^{-\alpha s}+e^{(\alpha-1)s} \big)ds\lesssim 1.\end{aligned}$$ Therefore, we obtain the bound [\[dis-S\]](#dis-S){reference-type="eqref" reference="dis-S"} as desired. Next we prove [\[str-S\]](#str-S){reference-type="eqref" reference="str-S"}. For $T\in (0,\frac{\pi}{2B_0})$ and $t\in (0,T)$, we have $$\frac{2}{\pi}\leq\frac{\sin (tB_0)}{tB_0}\leq 1,$$ then the dispersive estimate [\[dis-S\]](#dis-S){reference-type="eqref" reference="dis-S"} gives $$\label{dis-S-0} \|e^{-itH_{\alpha,B_0}}\|_{L^1(\mathbb{R}^2)\rightarrow L^\infty(\mathbb{R}^2)}\lesssim\frac{1}{t},\quad \forall t\in(0,T],$$ when $T\in (0,\frac{\pi}{2 B_0}]$. On the other hand, by the spectral theory, we have the energy estimate $$\label{energy} \|e^{-itH_{\alpha,B_0}}\|_{L^2(\mathbb{R}^2)\rightarrow L^2(\mathbb{R}^2)}\lesssim 1.$$ To prove the Strichartz estimates [\[str-S\]](#str-S){reference-type="eqref" reference="str-S"}, we recall the abstract mechanism of Keel-Tao [@KT] **Theorem 11**. *(Keel-Tao [@KT]) Let $(X,d\mu)$ be a measure space and $H$ a Hilbert space. Suppose that for each time $t\in\mathbb{R}$, $U(t): H \rightarrow L^2(X)$ which satisfies the energy estimate $$\|U(t)\|_{H\rightarrow L^2}\leq C,\quad t\in \mathbb{R}$$ and that for some $\sigma>0$ either $$\|U(t)U(s)^*f\|_{L^\infty(X)}\leq C|t-s|^{-\sigma}\|f\|_{L^1(X)},\quad t\neq s.$$ Then the estimates hold $$\|U(t)f\|_{L^q_tL^r(X)}\lesssim \|f\|_{L^2(X)}.$$ where $$(q,r)\in\Lambda:=\Big\{(q,r)\in[2,+\infty]\times[2,+\infty): \frac2q=n(\frac12-\frac1r)\Big\}.$$* By making use of this theorem to $U(t)=e^{-itH_{\alpha,B_0}}\chi_{[0,T]}(t)$, since [\[dis-S-0\]](#dis-S-0){reference-type="eqref" reference="dis-S-0"} and [\[energy\]](#energy){reference-type="eqref" reference="energy"}, we obtain [\[str-S\]](#str-S){reference-type="eqref" reference="str-S"}. Therefore we finish the proof of Theorem [Theorem 1](#thm:S){reference-type="ref" reference="thm:S"}. 99 G. E. Andrews, R. Askey and R. Roy, *Special Functions* (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2001. Y. Aharonov and D. Bohm, *Significance of electromagnetic potentials in the quantum theory*, Phys. Rev. 115 (1959), no. 2, 485-491. J. E. Avron, I. W. Herbst and B. Simon, *Schrödinger operators with magnetic fields. I. General interactions*, Duke Math. J. 45 (1978), no. 4, 847-883. J. E. Avron, I. W. Herbst and B. Simon, *Separation of center of mass in homogeneous magnetic fields*, Ann. Phys. 114 (1978), no. 1-2, 431-451. J. E. Avron, I. W. Herbst and B. Simon, *Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field*, Comm. Math. Phys. 79 (1981), no. 4, 529-572. M. Abramowitz and I. A. Stegun, *Handbook of mathematical functions with formulas, graphs and mathematical tables*, U. S. Government Printing Office, Washington, DC, 1965. N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, *Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential*, J. Funct. Anal. 203 (2003), no. 2, 519-549. N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, *Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay*, Indiana Univ. Math. J. 53 (2004), no. 6, 1665-1680. S. Cuccagna and P. P. Schirmer, *On the wave equation with a magnetic potential*, Comm. Pure Appl. Math. 54 (2001), no. 2, 135-152. P. D'Ancona and L. Fanelli, *Decay estimates for the wave and Dirac equations with a magnetic potential*, Comm. Pure Appl. Math. 60 (2007), no. 3, 357-392. P. D'Ancona, L. Fanelli, L. Vega and N. Visciglia, *Endpoint Strichartz estimates for the magnetic Schrödinger equation*, J. Funct. Anal. 258 (2010), no. 10, 3227-3240. M. B. Erdoğan, M. Goldberg and W. Schlag, *Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions*, Forum Math. 21 (2009), no. 4, 687-722. M. B. Erdoğan, M. Goldberg and W. Schlag, *Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb{R}^3$*, J. Eur. Math. Soc. 10 (2008), no. 2, 507-531. P. Exner, P. Št'ovı́ček and P. Vytřas, *Generalised boundary conditions for the Aharonov-Bohm effect combined with a homogeneous magnetic field*, J. Math. Phys. 43 (2002), no. 5, 2151-2168. L. Fanelli, *Spherical Schrödinger Hamiltonians: spectral analysis and time decay*, A. Michelangeli, G. DellAntonio (eds.), Advances in Quantum Mechanics, Springer INdAM Ser., 18, Springer, Cham, 2017. L. Fanelli, V. Felli, M. A. Fontelos and A. Primo, *Time decay of scaling invariant electromagnetic Schrödinger equations on the plane*, Comm. Math. Phys. 337 (2015), no. 3, 1515-1533. L. Fanelli, V. Felli, M. A. Fontelos and A. Primo, *Time decay of scaling critical electromagnetic Schrödinger flows*, Comm. Math. Phys. 324 (2013), no. 3, 1033-1067. L. Fanelli, J. Zhang and J. Zheng, *Dispersive estimates for 2D-wave equations with critical potentials*, Adv. Math. 400 (2000), Paper No. 108333, 46 pp. X. Gao, Z. Yin, J. Zhang and J. Zheng, *Decay and Strichartz estimates in critical electromagnetic fields*, J. Funct. Anal. 282 (2022), no. 5, Paper No. 109350, 51 pp. T. F. Kieffer and M. Loss, *Non-linear Schrödinger equation in a uniform magnetic field*, Partial differential equations, spectral theory, and mathematical physics - the Ari Laptev anniversary volume, 247-265, EMS Ser. Congr. Rep. EMS Press, Berlin, 2021. M. Keel and T. Tao, *Endpoint Strichartz estimates*, Amer. J. Math. 120 (1998), no. 5, 955-980. H. Koch and D. Tataru, *$L^p$ eigenfunction bounds for the Hermite operator*, Duke Math. J. 128 (2005), no. 2, 369-392. M. Reed and B. Simon, *Methods of modern mathematical physics. II. Fourier analysis, self-adjointness*, Academic Press, New York-London, 1975. W. Schlag, *Dispersive estimates for Schrödinger operators: a survey*, Mathematical aspects of nonlinear dispersive equations, 255-285, Ann. of Math. Stud., 163, Princeton University Press, Princeton, NJ, 2007. C. D. Sogge, *Hangzhou lectures on eigenfunctions of the Laplacian*, Ann. of Math. Stud., 188, Princeton University Press, Princeton, NJ, 2014. P. Št'ovı́ček, *The heat kernel for two Aharonov-Bohm solenoids in a uniform magnetic field*, Ann. Phys. 376 (2017), 254-282. P. Št'ovı́ček, *The green's function for the two-solenoid Aharonov-Bohm effect*, Phys. Lett. A 142 (1989), no. 1, 5-10. G. N. Watson, *A Treatise on the Theory of Bessel Functions*, Cambridge University Press, CambridgeThe Macmillan Company, New York, 1944. [^1]: Here we use the Friedrichs self-adjoint extension.

arxiv_math

--- abstract: | In a recent research on degenerate points of steady axisymmetric gravity flows with general vorticity, it has been shown that the possible asymptotics near any stagnation point must be the "Stokes corner", the "horizontal cusp", or the "horizontal flatness"(Theorem 1.1, Du, Huang, Pu, *Commun. Math. Phys.*, 400, 2137-2179, 2023). In this paper, we focus on the horizontally flat singularity and show that it is *not possible*, and therefore the "Stokes corner" and the "cusp" are the only possible asymptotics at the stagnation points. The basic idea of our proof relies on a perturbation of the *frequency formula* for the two-dimensional problem (Vǎrvǎrucǎ, Weiss, *Acta Math.*, 206, 363-403, 2011). Our analysis also suggests that, for steady axisymmetric rotational gravity flows, the singular asymptotic profiles at stagnation points are similar to the scenario observed in two-dimensional waves with vorticity. (Vǎrvǎrucǎ, Weiss, *Ann. I. H. Poincaré-AN*, 29, 861-885, 2012). address: - School of Mathematical Sciences, Shenzhen University, Shenzhen, P. R. China - Department of Mathematics, Sichuan University, Chengdu, P. R. China author: - Lili Du - Chunlei Yang title: The non-existence of horizontally flat singularity for steady axisymmetric free surface flows near stagnation points --- [^1] 0.01 # Introduction The model problem we have in mind is the following semilinear Bernoulli-type free boundary problem, which describes an incompressible axisymmetric rotational flow acted on by gravity and with a free surface, $$\label{Formula: model problem} \begin{cases} \displaystyle\mathop{\mathrm{div}}\br{\frac{1}{x}\nabla\psi} = -xf(\psi) & \text{ in } \Omega\cap\{\psi> 0\}, \\ \displaystyle\frac{1}{x^{2}}|\nabla\psi|^{2} = -y & \text{ on } \Omega\cap\partial\{\psi> 0\}. \end{cases}$$ Here $\psi$ is the Stokes stream function and $\Omega$ is a connected open subset relative to the right half-plane ${\mathbb R}_{+}^{2}=\{(x,y)\in{\mathbb R}^{2}:x\geqslant 0\}$ (This implies that $(\Omega\cap\{x=0\})\cap\partial\Omega=\varnothing$). The given function $f$ in the first equation of [\[Formula: model problem\]](#Formula: model problem){reference-type="eqref" reference="Formula: model problem"} represents the strength of the vorticity, and $\Omega\cap\partial\{\psi>0\}$ denotes the free surface of the flow. Physically the equation [\[Formula: model problem\]](#Formula: model problem){reference-type="eqref" reference="Formula: model problem"} arises for example as the "axially symmetric rising jet" or "a bubble within a semi-infinitely long cylindrical tube" and we kindly refer readers to [@DHP2022 Section 2] for a comprehensive description. Regarding [\[Formula: model problem\]](#Formula: model problem){reference-type="eqref" reference="Formula: model problem"} as a free boundary problem, our primary interest lies in the geometry profiles of the free surface at which the relative fluid velocity $(\frac{1}{x}\psi_{y},-\frac{1}{x}\psi_{x})$ is the zero vector, and the regularity of the free surface when it is not the zero vector. In the formal case, a glance of the second equation in [\[Formula: model problem\]](#Formula: model problem){reference-type="eqref" reference="Formula: model problem"} indicates that such points are distributed along the axis of symmetry $\{x=0\}$ or the set $\{y=0\}$. As for the latter case, it is evident that at those free boundary points that are away from both the axis of symmetry and the set $\{y=0\}$, the gradient of $\psi$ can never reduce to zero. Due to the degeneracy or the non-degeneracy of the free boundary condition, we classify them as the set of stagnation points $S_{\psi}^{s}$ (Type I points, which includes those free boundary points $(x_{0},0)$, $x_{0}>0$), the set of symmetric axis points $S_{\psi}^{a}$ (Type II points, which includes free boundary points $(0,y_{0})$, $y_{0}<0$), the origin $O=(0,0)$ (which can be regarded as the intersection of Type I and Type II points), and the set of non-degenerate points $N_{\psi}$ (which consists of points on the free boundary points $(x_{0},y_{0})$, $x_{0}>0$, $y_{0}<0$). We now briefly review some results known to us with respect to the singularity and regularity of the free surface near the degenerate points (Type I, Type II and the origin $O$) and non-degenerate points ($N_{\psi}$). In [@DHP2022], the first author and et al. demonstrated that the possible singular asymptotics for any point in $S_{\psi}^{s}$ are the "Stokes corner" wherein the free surface exhibits a corner of $\frac{2}{3}\pi$, a "horizontal cusp", or a "horizontal flatness" (Figure [\[Fig: asymptotics for typeI point\]](#Fig: asymptotics for typeI point){reference-type="ref" reference="Fig: asymptotics for typeI point"}). Furthermore, at any point in axis $S_{\psi}^{a}$, the singular profiles of the free surface must be a "cusp". At the origin $O$, they concluded that the possibilities are the "Garabedian pointed bubble", a "horizontal cusp", or a "horizontal flatness" (we refer readers to [@DHP2022 Table 1] for these singular asymptotics). These findings suggest that the asymptotic profiles in proximity to various degenerate points for axisymmetric inviscid gravity flows with vorticity are akin to those observed in the irrotational case [@VW2014]. Ultimately, it is not beyond our expectations that the free boundary near all non-degenerate points can be expressed as a graph of a $C^{1,\alpha}$ $(0<\alpha<1)$ smooth function, and we would like to refer readers to our recent work [@DY2023] on this subject. In this paper, we focus on *horizontally flat singularities* near stagnation points (The third graph in Figure [\[Fig: asymptotics for typeI point\]](#Fig: asymptotics for typeI point){reference-type="ref" reference="Fig: asymptotics for typeI point"}). Originally, they were observed and excluded during the investigation of Stokes conjecture for water waves with zero vorticity [@VW2011 Theorem B], and also for water waves with non-zero vorticity [@VW2012 Theorem 4.6]. Furthermore, horizontally flat singularities do not exist at stagnation points of axisymmetric gravity irrotational flows [@VW2014 Theorem 3.8]. Physical intuitions that relate to this kind of singularities can be interpreted as there being an infinite number of connected components of the air region. Mathematically, these singularities correspond to a specific density, which we shall discuss in the subsequent section (cf. Proposition [Proposition 5](#Proposition: density estimates){reference-type="ref" reference="Proposition: density estimates"}), at the stagnation points. This density, which arises as the limits of rescaled variational solutions of the Weiss-boundary adjusted energy, plays an important role in classifying asymptotics at the stagnation points. One of the primary contributions of the work [@DHP2022 Theorem 1.1] was to compute all the possible blow-up limits and densities at the stagnation points when the vorticity is considered, and the objective of this article is to prove that the set of stagnation point that corresponds to the density of horizontal flat singularities is an empty set. The appearance of the horizontally flat singularities and the cusp singularities is caused by the lack of compactness for variational solutions of the problem [\[Formula: model problem\]](#Formula: model problem){reference-type="eqref" reference="Formula: model problem"}. It should be worth noting that the cusp singularities near the stagnation points were proved to be not possible when assuming the strong Bernstein estimates for water waves without vorticity [@VW2011 Lemma 4.4], and were conjectured to be impossible for water waves with vorticity with the additional Rayleigh-Taylor condition [@VW2012 Remark 4.7]. As for the horizontally flat singularities, Vǎrvǎrucǎ and Weiss considered for the first time the so-called *frequency formula* [@VW2011 Section 7], and successfully applied this formula to exclude them. The frequency of the harmonic functions was initially observed by Almgren in [@A2000] for $Q$-valued harmonic functions and later developed by Garofalo and Lin to more general elliptic equations in [@GL1986; @GL1987]. The aim of this paper is to develop ingenious tools introduced in [@VW2012; @VW2014]: a frequency formula (presented in Theorem [Theorem 9](#Theorem: freq){reference-type="ref" reference="Theorem: freq"}) in the context of axisymmetric rotational waves. The new frequency formula helps us to investigate the blow-up limits of the frequency function (presented in Proposition [Proposition 18](#Proposition: vm){reference-type="ref" reference="Proposition: vm"}). The situation is complicated here by the fact that the axisymmetric problem is *not* as the same as the true two-dimensional problem, and it is unavoidable to deal with the integrability of those additional terms caused by the axis of symmetry. A major part of this work, which is originally inspired by [@VW2014 Remark 3.11], is to prove that these terms are indeed behave like small *perturbations* of the two-dimensional problem (Presented in Proposition [Lemma 17](#Lemma: V(r)){reference-type="ref" reference="Lemma: V(r)"}). The flow of the paper is the following: after clarifying some notations and some preliminary results in Section [2](#Section: notation){reference-type="ref" reference="Section: notation"}, we follow in Section [3](#Section: frequency formula){reference-type="ref" reference="Section: frequency formula"} the approach in [@VW2012] in order to compute the frequency formula in our settings. Then, we apply in Section [4](#Section: blow-up limits){reference-type="ref" reference="Section: blow-up limits"} the ideas of [@VW2012; @VW2014] to study blow-up limits of the frequency sequence. Finally, we prove that the set of horizontally flat singularities is indeed empty and we prove this result in Section [5](#Section: conclusions){reference-type="ref" reference="Section: conclusions"}. # Notations and technical tools {#Section: notation} For the abuse of confusion, we adhere some notation utilized in the previous works. In what follows, we denote by $\chi_{A}$ the characteristic function of a set $A$, by $X=(x,y)$ a point in ${\mathbb R}{+}^{2}$, by $B_r(X_{0}):=\{X\in{\mathbb R}^{2}:|X-X_{0}|<r\}$ the ball of center $X_{0}$ and radius $r$ and by $B_r$ the ball $B_{r}(0)$. Also, ${\mathcal H}^{s}$ shall denote the $s$-dimensional Hausdorff measure and $\nu$ will always refer to the outer normal on a given surface. We will say $O(1)$ and $O(r)$, etc. if $|O(1)|\leqslant C$, $|O(r)|\leqslant Cr$ for some positive constant $C$. Moreover, $dX:=dxdy$ shall stand for the volume element and we further introduce weighted Lebesgue $L_{\mathrm{w}}^{2}(E)$ and Sobolev space $W_{\mathrm{w}}^{1,2}(E)$, which are defined for any open subset $E$ of ${\mathbb R}_{+}^{2}$ by $$\begin{aligned} L_{\mathrm{w}}^{2}(E):=\left\{\psi\colon E\to{\mathbb R}\text{ is measurable and }\int_{E}\frac{1}{x}|\psi|^{2}\:dX<+\infty\right\},\end{aligned}$$ and $$\begin{aligned} W_{\mathrm{w}}^{1,2}(E):=\left\{\psi\in L_{\mathrm{w}}^{2}(E):\pd{\psi}{x}\in L_{\mathrm{w}}^{2}(E)\text{ and }\pd{\psi}{y}\in L_{\mathrm{w}}^{2}(E)\right\},\end{aligned}$$ respectively. For the sake of convenience we are going to reflect the problem at the line $\{y=0\}$, namely, we study the solutions of the problem $$\label{Formula: model problem1} \begin{cases} \displaystyle\mathop{\mathrm{div}}\br{\frac{1}{x}\nabla\psi} = -xf(\psi) & \text{ in } \Omega\cap \{\psi> 0\}, \\ \displaystyle\frac{1}{x^{2}}|\nabla\psi|^{2} = y & \text{ on } \Omega\cap\partial\{\psi> 0\}. \end{cases}$$ The nonlinearity $f$ in the first equation of [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} is assumed at present a continuous function with primitive $F(z):=\int_{0}^{z}f(s)ds$. We define the set of stagnation points $S_{\psi}^{s}$ by $$\begin{aligned} S_{\psi}^{s}:=\{X_{0}=(x_{0},0)\in\Omega\cap\partial\{\psi>0\}\colon x_{0}>0\}. \end{aligned}$$ Since our problem is completely local, we do not impose any boundary condition on $\partial\Omega$. Finally, for any $X_{0}\in S_{\psi}^{s}$, we define the number $$\begin{aligned} \label{Formula: d} \delta:=\frac{\min\{x_{0},\mathop{\mathrm{dist}}(X_{0},\partial\Omega)\}}{2}.\end{aligned}$$ We now introduce our notion of a *variational solution* of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}. **Definition 1** (Variational solutions). *The function $\psi\in W_{\mathrm{w},\mathrm{loc}}^{1,2}(\Omega)$ such that $\psi\geqslant 0$ in $\Omega$, $\psi=0$ in $\Omega\cap\{x=0\}$ is called a *variational solution* of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}, provided that $\psi\in C^{0}(\Omega)\cap C^{2}(\Omega\cap\{ \psi>0 \})$, $\psi\equiv0$ in $\{y\leqslant 0\}$, $$\begin{aligned} \lim_{\substack{X\to X_{0},\\ X\in\Omega\cap\{\psi>0\}}}\frac{1}{x}\pd{\psi}{y}=0,\qquad\lim_{\substack{X\to X_{0},\\ X\in\Omega\cap\{\psi>0\}}}\frac{1}{x}\pd{\psi}{x}\text{ exists, } \end{aligned}$$ for any $X_{0}\in\Omega\cap\{x=0\}$, and $$\begin{aligned} \label{Formula: Variational solution} \begin{split} &\int_{\Omega}\left(\frac{1}{x}|\nabla\psi|^{2}-2xF(\psi)+xy\chi_{\{\psi>0\}}\right)\mathop{\mathrm{div}}\phi-\frac{2}{x}\nabla\psi D\phi\nabla\psi dX\\ &\qquad\qquad+\int_{\Omega}\left(-\frac{1}{x^{2}}|\nabla\psi|^{2}-2F(\psi)+y\chi_{\{\psi>0\}}\right)\phi_{1}dX+\int_{\Omega}x\chi_{\{\psi>0\}}\phi_{2}dX=0, \end{split} \end{aligned}$$ for any $\phi=(\phi_{1},\phi_{2})\in C_{0}^{\infty}(\Omega;{\mathbb R}^{2})$ such that $\phi_{1}=0$ on $\{x=0\}$.* **Remark 1**. *The equation [\[Formula: Variational solution\]](#Formula: Variational solution){reference-type="eqref" reference="Formula: Variational solution"} is nothing but the Euler-Lagrange equation of the energy $$\begin{aligned} E(\phi):=\int_{\Omega}\left(\frac{1}{x}|\nabla\phi|^{2}-2xF(\phi)+xy\chi_{\{\phi>0\}}\right)dX \end{aligned}$$ with respect to domain variation. Assume that the free boundary $\Gamma:=\Omega\cap\partial\{\psi>0\}$ and $\psi$ are $C^{2}$ smooth, then an integration by parts both in the water phase and on the free surface of the formula [\[Formula: Variational solution\]](#Formula: Variational solution){reference-type="eqref" reference="Formula: Variational solution"} shows that $\psi$ solves the equation [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} in the classical sense.* **Remark 2**. *In this remark, we introduce an energy identity (cf. [@DHP2022 (3.3)]) which is helpful for our future discussion. For any $X_{0}\in S_{\psi}^{s}$ and any $r\in(0,\delta)$, the following identity holds. $$\begin{aligned} \label{Formula: Energy identity} \int_{B_r(X_{0})}\left(\frac{1}{x}|\nabla\psi|^{2}-x\psi f(\psi)\right)dX=\int_{\partial B_r(X_{0})}\frac{1}{x}\psi\nabla\psi\cdot\nu\,d{\mathcal H}^1. \end{aligned}$$* The technical tools at our disposition comprise a monotonicity formula and a density estimate introduced in [@DHP2022]. For the sake of completeness, let us state them respectively. **Proposition 3** (Weiss's monotonicity formula, [@DHP2022 Lemma 3.1]). *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} and let $X_{0}\in S_{\psi}^{s}$. For any $r\in(0,\delta)$, define* *$$\begin{aligned} &I_{X_{0},\psi}(r)=I(r)=r^{-3}\int_{B_r(X_0)}\left(\frac{1}{x}|\nabla\psi|^{2}-x\psi f(\psi)+xy\chi_{\{\psi>0\}}\right)dX, \label{Formula: I(r)}\\ &J_{X_{0},\psi}(r)=J(r)=r^{-4}\int_{\partial B_r(X_0)}\frac{1}{x}\psi^{2}\,d{\mathcal H}^1, \label{Formula: J(r)}\\ &K_{X_{0},\psi}(r)=K(r)=r\int_{\partial B_r{X_0}}(2xF(\psi)-x\psi f(\psi))d\mathcal{H}^{d-1}+\int_{B_r(X_0)}(2x_{0}F(\psi)-6xF(\tau))dX,\label{Formula: K(r)}\\ &I_{1,X_{0},\psi}(r)=I_{1}(r)=\int_{B_r(X_0)}-\frac{x-x_{0}}{x^{2}}|\nabla\psi|^{2}dX,\label{Formula: I1(r)}\\ &I_{2,X_{0},\psi}(r)=I_{2}(r)=\int_{B_r(X_0)}(x-x_{0})y\chi_{\{\psi>0\}}dX,\label{Formula: I2(r)}, \end{aligned}$$ and* *[\[Formula: J1(r)\]]{#Formula: J1(r) label="Formula: J1(r)"} &J\_1,X\_0,(r)=J\_1(r)=\_B_r(X_0)\^2 dH\^1.&* *Then the Weiss-boundary adjusted energy $$\begin{aligned} \label{Formula: Weiss-boundary adjusted energy} \Phi_{X_{0},\psi}(r)=\Phi(r)=I(r)-\frac{3}{2}J(r) \end{aligned}$$ satisfies for a.e. $r\in(0,\delta)$, $$\begin{aligned} \label{Formula: derivatives of Weiss-boundary adjusted energy} \Phi'(r)=2r^{-3}\int\limits_{\partial B_r(X_0)}\frac{1}{x}\left(\nabla\psi\cdot\nu-\frac{3}{2}\frac{\psi}{r}\right)^{2}\,d{\mathcal H}^1+r^{-4}\sum_{i=1}^{2}I_{i}(r)+\frac{3}{2}r^{-5}J_{1}(r)+r^{-4}K(r). \end{aligned}$$* *Proof.* After a straightforward calculation (cf. [@DHP2022 proof of Lemma 3.1]), we obtain $$\begin{aligned} \label{Formula: I'(r)} I'(r)=r^{-4}\left(2r\int_{\partial B_{r}(X_{0})}\frac{1}{x}(\nabla\psi\cdot\nu)^{2}d\mathcal{H}^{1}-3\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi\nabla\psi\cdot\nu d\mathcal{H}^{1}+K(r)\right)+r^{-4}\sum_{i=1}^{2}I_{i}(r), \end{aligned}$$ and $$\begin{aligned} \label{Formula: J'(r)} J'(r)=r^{-5}\left(2r\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi\nabla\psi\cdot\nu d\mathcal{H}^{1}-3\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}-J_{1}(r)\right). \end{aligned}$$ ◻ We now recall the definition of weak solutions of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}. **Definition 2** (Weak solutions). *The function $\psi\in W_{\mathrm{w}}^{1,2}(\Omega)$ is called a *weak solution* of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}, provided that $\psi$ is a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} and the topological free boundary $\partial\{\psi>0\}\cap\Omega\cap\{x>0\}\cap\{y\neq0\}$ is locally a $C^{2,\alpha}$ curve.* **Remark 4**. *The smoothness of the topological free boundary of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} was recently proved in [@DY2023 Theorem 1.1].* For weak solutions of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}, the first author and his collaborators prove the following density estimates with respect to the stagnation points (cf. [@DHP2022 Proposition 3.3]). **Proposition 5** (Density estimates, [@DHP2022 Proposition 3.3]). *Let $\psi$ be a weak solution with the following growth assumption $$\begin{aligned} \label{Formula: growth assumption} \frac{|\nabla\psi|^{2}}{x^{2}}\leqslant C(|y|+|x-x_{0}|)\quad\text{ in }\quad B_{r}(X_{0}),\quad\text{ for any }\quad X_{0}\in\Omega\cap\partial\{\psi>0\}, \end{aligned}$$ and let $X_{0}\in S_{\psi}^{s}$. Then the following statements hold:* 1. *$\lim_{r\to0+}\Phi(r)$ exists and $$\begin{aligned} \Phi(0+)\in\left\lbrace 0,x_{0}\int\limits_{B_{1}}y^{+}\chi_{ \{x\colon\pi/6<\theta<5\pi/6\} }dX,x_{0}\int\limits_{B_{1}}y^{+}dX\right\rbrace. \end{aligned}$$* 2. *In the case $\Phi(0+)=x_{0}\int_{B_{1}}y^{+}\chi_{ \{x\colon\pi/6<\theta<5\pi/6\} }dX$, we have $$\begin{aligned} \frac{\psi((x_{0},0)+r(x,y))}{r^{3/2}}\to\frac{\sqrt{2}x_{0}}{3}\rho^{3/2}\cos\left(\frac{3}{2}\left(\min\left(\max\left(\theta,\frac{\pi}{6}\right),\frac{5\pi}{6}\right)-\frac{\pi}{2}\right)\right)\quad\text{ as }\quad r\to0+, \end{aligned}$$ strongly in $W_{\mathrm{loc}}^{1,2}(\mathbb{R}^{2})$ and locally uniformly in $\mathbb{R}^{2}$, where $X=(\rho\cos\theta,\rho\sin\theta)$.* 3. *In the case $\Phi(0+)\in\{0,x_{0}\int_{B_{1}}y^{+}dX\}$, we have $$\begin{aligned} \frac{\psi((x_{0},0)+r(x,y))}{r^{3/2}}\to0\quad\text{ as }\quad r\to0+, \end{aligned}$$ strongly in $W_{\mathrm{loc}}^{1,2}(\mathbb{R}^{2})$ and locally uniformly in $\mathbb{R}^{2}$.* In order to better describe the asymptotics of different singularities, authors in [@DHP2022] imposed an assumption (cf. [@DHP2022 Assumption 1.2]) on the free boundary of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}, which we states here. **Assumption 6** (Curve assumption). *Suppose that in a small neighborhood $B_{r_{0}}(X_{0})$ of $X_{0}\in S_{\psi}^{s}$, the free boundary $\partial\{\psi>0\}\cap B_{r_{0}}(X_{0})$ is a continuous injective curve $\gamma(t):=(\gamma_{1}(t),\gamma_{2}(t)):I\to{\mathbb R}$ with $\gamma(0)=X_{0}$ and for some $t$ contained in an interval $I$ of ${\mathbb R}$.* **Remark 7**. *It should be worth noted that the asymptotic profiles related to different densities can be graphed explicitly when assuming the Assumption [Assumption 6](#Assumption: curve assumption){reference-type="ref" reference="Assumption: curve assumption"}. In particular, we obtain in case (3) of Proposition [Proposition 5](#Proposition: density estimates){reference-type="ref" reference="Proposition: density estimates"} that if $\Phi(0+)=x_{0}\int_{B_{1}}y^{+}dX$, then (cf. Figure [\[Fig: horizontal flatness asymptotics\]](#Fig: horizontal flatness asymptotics){reference-type="ref" reference="Fig: horizontal flatness asymptotics"}) $\gamma_{1}(t)\neq x_{0}$ in $(-t_{1},t_{1})\setminus\{0\}$, $\sigma_{1}-x_{0}$ changes sign at $t=0$, and $$\begin{aligned} \lim_{t\to0}\frac{\gamma_{2}(t)}{\gamma_{1}(t)-x_{0}}=0. \end{aligned}$$* *If $\Phi(0+)=0$, then (cf. Figure [\[Fig: cusp asymptotics\]](#Fig: cusp asymptotics){reference-type="ref" reference="Fig: cusp asymptotics"}) $\gamma_{1}(t)\neq x_{0}$ in $(-t_{1},t_{1})\setminus\{0\}$, $\gamma_{1}-x_{0}$ does not change its sign at $t=0$, and $$\begin{aligned} \lim_{t\to0}\frac{\gamma(t)}{\gamma_{1}(t)-x_{0}}=0. \end{aligned}$$* In the forthcoming three sections, we will focus our analysis on stagnation points at which the density takes the value $x_{0}\int_{B_{1}}y^{+}dX$ (Figure [\[Fig: horizontal flatness asymptotics\]](#Fig: horizontal flatness asymptotics){reference-type="ref" reference="Fig: horizontal flatness asymptotics"}). To this end, we will first extend the frequency formula introduced by Vǎrvǎrucǎ and Weiss [@VW2012 Theorem 6.7] to the axisymmetric rotational case. Then we are able to apply a blow-up analysis to exclude the horizontal flat singularities. # Degenerate points and a frequency formula {#Section: frequency formula} This section aims to investigate qualitative properties of degenerate points and introduces a frequency formula. We begin with the definition of the set of degenerate points. **Definition 3** (Degenerate points). *Let $\psi$ be a variational solution of [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}. We define the set of degenerate points to be the set of free boundary points whose density takes the value $x_{0}\int_{B_{1}}y^{+}dX$. Namely, $$\begin{aligned} \Sigma_{\psi}:=\left\lbrace X_{0}\in S_{\psi}^{s}\colon\Phi(0+)=x_{0}\int_{B_{1}}y^{+}dX\right\rbrace. \end{aligned}$$* **Remark 8**. *The set $\Sigma_{\psi}$ is closed, as a consequence of the upper semicontinuity of the monotonicity formula $X\mapsto\Phi_{\psi,X}(0+)$ in $\{y=0\}$ and the characterization of the set of values of ${\mathcal D}(0+)$. (cf. [@DHP2022 Proposition 3.3]).* We now exhibit our first main result of our paper, namely, a frequency formula. **Theorem 9** (Frequency formula). *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}. Let $X_{0}\in S_{\psi}^{s}$ and assume that $\psi$ satisfies the growth assumption [\[Formula: growth assumption\]](#Formula: growth assumption){reference-type="eqref" reference="Formula: growth assumption"}. Define, for a.e. $r\in(0,\delta)$ $$\begin{aligned} \label{Formula: D(r)} D_{X_{0},\psi}(r)=D(r)=\frac{\displaystyle r\int_{B_r(X_0)}\left(\frac{1}{x}|\nabla\psi|^{2}-x\psi f(\psi)\right)dX}{\displaystyle\int_{\partial B_r(X_0)}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}}, \end{aligned}$$ and $$\begin{aligned} \label{Formula: V(r)} V_{X_{0},\psi}(r)=V(r)=\frac{\displaystyle r\int_{B_{r}(X_{0})}y^{+}(x_{0}-x\chi_{\{\psi>0\}})dX+\sum_{i=1}^{2}r^{4}\int_{0}^{r}t^{-4}I_{i}(t)dt+\frac{3}{2}r^{4}\int_{0}^{r}t^{-5}J_{1}(t)dt}{\displaystyle\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}}, \end{aligned}$$ where $I_{1}$, $I_{2}$ and $J_{1}$ are given in [\[Formula: I1(r)\]](#Formula: I1(r)){reference-type="eqref" reference="Formula: I1(r)"}, [\[Formula: I2(r)\]](#Formula: I2(r)){reference-type="eqref" reference="Formula: I2(r)"} and [\[Formula: J1(r)\]](#Formula: J1(r)){reference-type="eqref" reference="Formula: J1(r)"}, respectively. Moreover, set $$\begin{aligned} \label{Formula: Z(r)} Z_{X_{0},\psi}(r)=Z(r)=\frac{\displaystyle\int_{\partial B_{r}(X_{0})}\frac{x-x_{0}}{x^{2}}\psi^{2}d\mathcal{H}^{1}}{\displaystyle\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}}. \end{aligned}$$ Then the "frequency" $$\begin{aligned} \label{Formula: H(r)} H_{X_{0},\psi}(r)=H(r)=D(r)-V(r) \end{aligned}$$ satisfies for a.e. $r\in(0,\delta)$ the identities $$\begin{aligned} \label{Formula: H'(r)} \begin{split} H'(r)&=\frac{2}{r}\int\limits_{\partial B_{r}(X_{0})}\frac{1}{x}\left[\frac{r(\nabla\psi\cdot\nu)}{(\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}-D(r)\frac{\psi}{(\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}\right]^{2}d\mathcal{H}^{1}\\ &\quad+\frac{2}{r}V^{2}(r)+\frac{2}{r}V(r)\left(H(r)-\frac{3}{2}\right)+\frac{1}{r}Z(r)\left(H(r)-\frac{3}{2}\right)+\frac{K(r)}{\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}} \end{split}\\ \begin{split} &=\frac{2}{r}\int\limits_{\partial B_{r}(X_{0})}\frac{1}{x}\left[\frac{r(\nabla\psi\cdot\nu)}{(\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}-H(r)\frac{\psi}{(\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}\right]^{2}d\mathcal{H}^{1}\\ &\quad+\frac{2}{r}V(r)\left(H(r)-\frac{3}{2}\right)+\frac{1}{r}Z(r)\left(H(r)-\frac{3}{2}\right)+\frac{K(r)}{\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}}.\label{Formula: H'(r)(1)} \end{split} \end{aligned}$$ Here $K(r)$ is given in [\[Formula: K(r)\]](#Formula: K(r)){reference-type="eqref" reference="Formula: K(r)"}.* *Proof.* Define the quantity $$\begin{aligned} \label{Formula: M(r)} M(r):=I(r)-\sum_{i=1}^{2}\int_{0}^{r}t^{-4}I_{i}(t)dt-\frac{3}{2}\int_{0}^{r}t^{-5}J_{1}(t)dt, \end{aligned}$$ where $I(r)$, $I_{1}(r)$, $I_{2}(r)$ and $J_{1}(r)$ are defined in [\[Formula: I(r)\]](#Formula: I(r)){reference-type="eqref" reference="Formula: I(r)"}, [\[Formula: I1(r)\]](#Formula: I1(r)){reference-type="eqref" reference="Formula: I1(r)"}, [\[Formula: I2(r)\]](#Formula: I2(r)){reference-type="eqref" reference="Formula: I2(r)"} and [\[Formula: J1(r)\]](#Formula: J1(r)){reference-type="eqref" reference="Formula: J1(r)"} respectively. Then it follows from [\[Formula: I\'(r)\]](#Formula: I'(r)){reference-type="eqref" reference="Formula: I'(r)"} that for a.e. $r\in(0,\delta)$, $$\begin{aligned} \label{Formula: M'(r)} M'(r)=r^{-4}\left(2r\int_{\partial B_{r}(X_{0})}\frac{1}{x}(\nabla\psi\cdot\nu)^{2}d\mathcal{H}^{1}-3\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi\nabla\psi\cdot\nu d\mathcal{H}^{1}+K(r)\right)-\frac{3}{2}r^{-5}J_{1}(r). \end{aligned}$$ Observe now that by the definition [\[Formula: H(r)\]](#Formula: H(r)){reference-type="eqref" reference="Formula: H(r)"}, $$\begin{aligned} H(r)=D(r)-V(r)=\frac{M(r)-x_{0}\int_{B_{1}}y^{+}dX}{J(r)}. \end{aligned}$$ Thus, it follows from [\[Formula: J\'(r)\]](#Formula: J'(r)){reference-type="eqref" reference="Formula: J'(r)"}, [\[Formula: M\'(r)\]](#Formula: M'(r)){reference-type="eqref" reference="Formula: M'(r)"} and a direct computation that $$\begin{aligned} H'(r)&=\frac{M'(r)}{J(r)}-\frac{(M(r)-x_{0}\int_{B_{1}}y^{+}dX)}{J(r)}\frac{J'(r)}{J(r)}\\ &=\frac{\left(2r\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}(\nabla\psi\cdot\nu)^{2}d\mathcal{H}^{1}-3\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi\nabla\psi\cdot\nu d\mathcal{H}^{1}+K(r)\right)}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}-\frac{3}{2}\frac{Z(r)}{r}\\ &\quad-(D(r)-V(r))\frac{1}{r}\frac{\left(2r\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi\nabla\psi\cdot\nu d\mathcal{H}^{1}-3\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}-J_{1}(r)\right)}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}, \end{aligned}$$ where we have used the fact that $Z(r)=\frac{r^{-4}J_{1}(r)}{J(r)}$ (recalling [\[Formula: J(r)\]](#Formula: J(r)){reference-type="eqref" reference="Formula: J(r)"} and [\[Formula: J1(r)\]](#Formula: J1(r)){reference-type="eqref" reference="Formula: J1(r)"}). With the aid of the energy identity [\[Formula: Energy identity\]](#Formula: Energy identity){reference-type="eqref" reference="Formula: Energy identity"} mentioned in the Remark [Remark 2](#Remark: Energy identity){reference-type="ref" reference="Remark: Energy identity"}, it is easy to obtain the identity $$\begin{aligned} \label{Formula: D(r)(1)} D(r)=\frac{r\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi\nabla\psi\cdot\nu d\mathcal{H}^{1}}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}. \end{aligned}$$ Therefore, $$\begin{aligned} H'(r)&=\frac{2}{r}\left[\frac{r^{2}\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}(\nabla\psi\cdot\nu)^{2}d\mathcal{H}^{1}}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}-\frac{3}{2}D(r)\right]\\ &\quad-\frac{3}{2}\frac{Z(r)}{r}-\frac{2}{r}(D(r)-V(r))\left(D(r)-\frac{3}{2}-\frac{Z(r)}{2}\right)+\frac{K(r)}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}\\ &=\frac{2}{r}\left[\frac{r^{2}\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}(\nabla\psi\cdot\nu)^{2}d\mathcal{H}^{1}}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}-D^{2}(r)\right]\\ &\quad+\frac{2}{r}V(r)\left(V(r)+H(r)-\frac{3}{2}\right)+\frac{1}{r}Z(r)\left(H(r)-\frac{3}{2}\right)+\frac{K(r)}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}, \end{aligned}$$ where we have used the definition of $H(r)$ [\[Formula: H(r)\]](#Formula: H(r)){reference-type="eqref" reference="Formula: H(r)"} in the second identity. Using the identity [\[Formula: D(r)(1)\]](#Formula: D(r)(1)){reference-type="eqref" reference="Formula: D(r)(1)"} once again, we obtain $$\begin{aligned} H'(r)&=\frac{2}{r}\int_{\partial B_{r}(X_{0})}\frac{1}{x}\left[\frac{r\nabla\psi\cdot\nu}{(\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}-D(r)\frac{\psi}{(\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}\right]^{2}d\mathcal{H}^{1}\\ &\quad+\frac{2}{r}V^{2}(r)+\frac{2}{r}V(r)\left(H(r)-\frac{3}{2}\right)+\frac{1}{r}Z(r)\left(H(r)-\frac{3}{2}\right)+\frac{K(r)}{\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}}. \end{aligned}$$ This is exactly [\[Formula: H\'(r)\]](#Formula: H'(r)){reference-type="eqref" reference="Formula: H'(r)"}. As for [\[Formula: H\'(r)(1)\]](#Formula: H'(r)(1)){reference-type="eqref" reference="Formula: H'(r)(1)"}, we point out that it is a direct consequence of [\[Formula: H\'(r)\]](#Formula: H'(r)){reference-type="eqref" reference="Formula: H'(r)"}, with the help of the following identity $$\begin{aligned} &\int_{\partial B_{r}(X_{0})}\frac{1}{x}[r(\nabla\psi\cdot\nu)-D(r)\psi]^{2}d\mathcal{H}^{1}\\ &=\int_{\partial B_{r}(X_{0})}\frac{1}{x}[r(\nabla\psi\cdot\nu)-H(r)\psi]^{2}d\mathcal{H}^{1}-V^{2}(r)\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}. \end{aligned}$$ ◻ **Remark 10**. *In this remark, we make a comparison between the frequency formula [\[Formula: H\'(r)\]](#Formula: H'(r)){reference-type="eqref" reference="Formula: H'(r)"}, [\[Formula: H\'(r)(1)\]](#Formula: H'(r)(1)){reference-type="eqref" reference="Formula: H'(r)(1)"} in our context and that for the two-dimensional problem (cf. [@VW2012 (6.1), (6.2) in Theorem 6.7]), and explain the perturbation idea. Recalling first the frequency formulas for the two-dimensional problem in [@VW2012], which have the form $$\begin{aligned} H'(r)&=\frac{2}{r}\int_{\partial B_{r}(X_{0})}\left[\frac{r(\nabla\psi\cdot\nu)}{\int_{\partial B_{r}(X_{0})}\psi^{2}d\mathcal{H}^{1}}-D(r)\frac{\psi}{\int_{\partial B_{r}(X_{0})}\psi^{2}d\mathcal{H}^{1}}\right]^{2}d\mathcal{H}^{1}\\ &\quad+\frac{2}{r}V^{2}(r)+\frac{2}{r}V(r)\left(H(r)-\frac{3}{2}\right)+\frac{K(r)}{\int_{\partial B_{r}(X_{0})}\psi^{2}d\mathcal{H}^{1}}\\ &=\frac{2}{r}\int_{\partial B_{r}(X_{0})}\left[\frac{r(\nabla\psi\cdot\nu)}{\int_{\partial B_{r}(X_{0})}\psi^{2}d\mathcal{H}^{1}}-H(r)\frac{\psi}{\int_{\partial B_{r}(X_{0})}\psi^{2}d\mathcal{H}^{1}}\right]^{2}d\mathcal{H}^{1}\\ &\quad+\frac{2}{r}V(r)\left(H(r)-\frac{3}{2}\right)+\frac{K(r)}{\int_{\partial B_{r}(X_{0})}\psi^{2}d\mathcal{H}^{1}}. \end{aligned}$$ It is common that both two formulas consist of a non-negative part, which is the first integral on the right-hand side of $H'(r)$. We also note that in both of the two problems, $D(r)$ is the true "frequency" and is different because the solutions of the two-dimensional problem and our problem satisfy different equations. Here the major differences arise in $V(r)$ for two problems. In the two-dimensional problem it is defined to be the non-negative quantity $$\begin{aligned} V(r):=\frac{r\int_{B_{r}(X_{0})}y^{+}(1-\chi_{ \{ \psi>0\} })dX}{\int_{\partial B_{r}(X_{0})}\psi^{2}d{\mathcal H}^{1}}, \end{aligned}$$ and we point out that the existence of the limits $H(0+)$ and $V(0+)=0$ for the two-dimensional problem was highly dependent on the non-negativity of $V(r)$. However, recalling [\[Formula: V(r)\]](#Formula: V(r)){reference-type="eqref" reference="Formula: V(r)"}, we see that it has no sign and we cannot directly follow the same approach of [@VW2012]. The components of $V(r)$ defined by [\[Formula: V(r)\]](#Formula: V(r)){reference-type="eqref" reference="Formula: V(r)"} actually exhibits the feature of axisymmetric problems. In fact, it consists of a term $\int_{B_{r}(X_{0})}y^{+}(x_{0}-x\chi_{ \{ \psi>0\} })$ which can be viewed as the difference of the density with respect to the first variable, and two terms $\sum_{i=1}^{2}\int_{0}^{r}t^{-4}I_{i}(t)+\frac{3}{2}\int_{0}^{r}t^{-5}J_{1}(t)$ caused by the axis of symmetry. We will eventually prove that near each degenerate points, $V(r)=\widetilde{V}(r)+O(r)$ where $\widetilde{V}(r)\geqslant0$ for every $r\in(0,r_{0})$ with $r_{0}>0$ sufficiently small. Therefore, $V(r)$ can be regarded as a perturbation of a non-negative term $\widetilde{V}(r)$, so that we can borrow the idea from the celebrated paper [@VW2012]. Incidentally, although the term $K(r)$ in two formulas are also not the same, we still hope to deal with this term similarly as in [@VW2012], based on a simple fact that the weight $\frac{1}{x}$ is uniformly bounded in $B_{r}(X_{0})$ for every $r\in(0,\delta)$.* In what follows, we first consider the term in the frequency formula associates with the vorticity strength $f$. First note that when the vorticity is negative, the horizontally flat singularities are possible even for the two-dimensional problem. What is of particular interest to us, as suggested by Vǎrvǎrucǎ and Weiss in [@VW2012 Page 863], is the case when the vorticity is $0$ at the free surface, and may have infinitely many sign changes accumulating there. Thus, we impose the following growth assumption on $f$. **Assumption 11**. *There exists a constant $C<+\infty$ such that $$\begin{aligned} \label{Formula: f(z)} |f(z)|\leqslant Cz\quad\text{ for all }\quad z\in(0,z_{0}). \end{aligned}$$ Note that [\[Formula: f(z)\]](#Formula: f(z)){reference-type="eqref" reference="Formula: f(z)"} also implies $$\begin{aligned} \label{Formula: F(z)} |F(z)|\leqslant Cz^{2}/2\quad\text{ for all }\quad z\in(0,z_{0}). \end{aligned}$$* The following Lemma was inspired by [@GL1986 (4.11)] and [@VW2012 Lemma 6.9]. **Lemma 12**. *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}. Then for each $X_{0}=(x_{0},y_{0})\in\Omega$ and for all $r>0$ sufficiently small, $$\begin{aligned} \label{Formula: vort(1)} r\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}=\int_{B_{r}(X_{0})}\left[2\frac{\psi^{2}}{x}-\frac{x-x_{0}}{x^{2}}\psi^{2}+\left(\frac{|\nabla\psi|^{2}}{x}-x\psi f(\psi)\right)(r^{2}-|X-X_{0}|^{2})\right]dX. \end{aligned}$$* *Proof.* An integration by parts yields $$\begin{aligned} \label{Formula: vort(1-1)} \begin{split} &\int_{B_{r}(X_{0})}\mathop{\mathrm{div}}\br{\frac{\nabla(\psi^{2})}{x}}\cdot(r^{2}-|X-X_{0}|^{2})dX\\ &=-\int_{B_{r}(X_{0})}\frac{\nabla(\psi^{2})}{x}\cdot\nabla\br{r^{2}-|X-X_{0}|^{2}}dX+\underbrace{\int_{\partial B_{r}(X_{0})}\frac{\nabla(\psi^{2})}{x}\cdot\nu\cdot\br{r^{2}-|X-X_{0}|^{2}}d{\mathcal H}^{1}}_{=0}\\ &=\int_{B_{r}(X_{0})}\psi^{2}\mathop{\mathrm{div}}\br{\frac{\nabla(r^{2}-|X-X_{0}|^{2}}{x}}dX-\int_{\partial B_{r}(X_{0})}\frac{\psi^{2}}{x}\nabla\br{r^{2}-|X-X_{0}|^{2}}\cdot\frac{X-X_{0}}{|X-X_{0}|}d{\mathcal H}^{1}\\ &=\int_{B_{r}(X_{0})}\psi^{2}\frac{\Delta\br{r^{2}-|X-X_{0}|^{2}}}{x}dX-\int_{B_{r}(X_{0})}\frac{\psi^{2}}{x^{2}}\pd{(r^{2}-|X-X_{0}|^{2})}{x}dX+2r\int_{\partial B_{r}(X_{0})}\frac{\psi^{2}}{x}d{\mathcal H}^{1}\\ &=-4\int_{B_{r}(X_{0})}\frac{\psi^{2}}{x}dX+2\int_{B_{r}(X_{0})}\frac{x-x_{0}}{x^{2}}\psi^{2}dX+2r\int_{\partial B_{r}(X_{0})}\frac{\psi^{2}}{x}d{\mathcal H}^{1}. \end{split} \end{aligned}$$ On the other hand, noticing that $$\begin{aligned} \label{Formula: vort(1-2)} \mathop{\mathrm{div}}\br{\frac{\nabla(\psi^{2})}{x}}=2\mathop{\mathrm{div}}\br{\psi\frac{\nabla\psi}{x}}=2\nabla\psi\cdot\frac{\nabla\psi}{x}+2\psi\mathop{\mathrm{div}}\br{\frac{\nabla\psi}{x}}=2\frac{|\nabla\psi|^{2}}{x}-2x\psi f(\psi). \end{aligned}$$ A combination of [\[Formula: vort(1-1)\]](#Formula: vort(1-1)){reference-type="eqref" reference="Formula: vort(1-1)"} and [\[Formula: vort(1-2)\]](#Formula: vort(1-2)){reference-type="eqref" reference="Formula: vort(1-2)"} gives the desired result. ◻ As a direct corollary of Lemma [Lemma 12](#Lemma: vort){reference-type="ref" reference="Lemma: vort"} and the Assumption [Assumption 11](#Assumption: f){reference-type="ref" reference="Assumption: f"}, we obtain the following estimates for $K(r)$. **Corollary 13**. *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} with the growth assumption [\[Formula: growth assumption\]](#Formula: growth assumption){reference-type="eqref" reference="Formula: growth assumption"}. Assume that the nonlinearity $f$ satisfies [\[Formula: f(z)\]](#Formula: f(z)){reference-type="eqref" reference="Formula: f(z)"}. Then there exists $r_{0}>0$ sufficiently small such that $$\begin{aligned} \label{Formula: vort(2)} r\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}\geqslant\int_{B_{r}(X_{0})}\frac{1}{x}\psi^{2}dX\quad\text{ for all }\quad r\in(0,r_{0}), \end{aligned}$$ and $$\begin{aligned} \label{Formula: vort(3)} |K(r)|\leqslant C_{0}r\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ Here $C_{0}$ depends only on $x_{0}$.* *Proof.* It follows from [\[Formula: vort(1)\]](#Formula: vort(1)){reference-type="eqref" reference="Formula: vort(1)"} and the Assumption [Assumption 11](#Assumption: f){reference-type="ref" reference="Assumption: f"} that $$\begin{aligned} r\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}\geqslant\int_{B_{r}(X_{0})}\left(2-\frac{x-x_{0}}{x}-Cx^{2}(r^{2}-|X-X_{0}|^{2})\right)\frac{\psi^{2}}{x}dX. \end{aligned}$$ Consequently, there exists a $r_{0}\in(0,r)$ sufficiently small so that $$\begin{aligned} 2-\frac{x-x_{0}}{x}-Cx^{2}(r^{2}-|X-X_{0}|^{2})\geqslant1\quad\text{ for all }\quad r\in(0,r_{0})\quad\text{ and }\quad X\in B_{r}(X_{0}), \end{aligned}$$ and this proves [\[Formula: vort(2)\]](#Formula: vort(2)){reference-type="eqref" reference="Formula: vort(2)"}. As for [\[Formula: vort(3)\]](#Formula: vort(3)){reference-type="eqref" reference="Formula: vort(3)"}, recalling the definition of $K(r)$ [\[Formula: K(r)\]](#Formula: K(r)){reference-type="eqref" reference="Formula: K(r)"} and applying [\[Formula: F(z)\]](#Formula: F(z)){reference-type="eqref" reference="Formula: F(z)"}, one has $$\begin{aligned} |K(r)|\leqslant\int_{B_{r}(X_{0})}2x_{0}x\cdot\frac{F(\psi)}{x}+6x^{2}\cdot\frac{F(\psi)}{x}dX+r\int_{\partial B_{r}(X_{0})}2x^{2}\cdot\frac{F(\psi)}{x}+x^{2}\cdot\frac{\psi f(\psi)}{x}d\mathcal{H}^{1}. \end{aligned}$$ With the help of [\[Formula: vort(2)\]](#Formula: vort(2)){reference-type="eqref" reference="Formula: vort(2)"}, we obtain the desired result. ◻ **Proposition 14**. *Let $\psi$ be a variational solution of [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} with the growth assumption [\[Formula: growth assumption\]](#Formula: growth assumption){reference-type="eqref" reference="Formula: growth assumption"}. Assume that the nonlinearity $f$ satisfies [\[Formula: f(z)\]](#Formula: f(z)){reference-type="eqref" reference="Formula: f(z)"}. Let $X_{0}\in\Sigma_{\psi}$. Then the following holds, for some $r_{0}\in(0,\delta)$ sufficiently small.* 1. *There exists a positive constant $C_{1}$ such that $$\begin{aligned} H(r)-\frac{3}{2}\geqslant-C_{1}r^{2}\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$* 2. *The function $r\mapsto e^{\beta r^{2}}J(r)$ is nondecreasing on $(0,r_{0})$.* 3. *$r\mapsto\frac{1}{r}V^{2}(r)\in L^{1}(0,r_{0})$.* 4. *The function $r\mapsto H(r)$ has a right limit $H(0+)\geqslant\frac{3}{2}$.* 5. *The function $$\begin{aligned} H'(r)-\frac{2}{r}\int\limits_{\partial B_{r}(X_{0})}\frac{1}{x}\left[\frac{r(\nabla\psi\cdot\nu)}{(\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1})^{1/2}}-H(r)\frac{\psi}{(\int_{\partial B_{r}(x^{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1})^{1/2}}\right]^{2}\:d\mathcal{H}^{1} \end{aligned}$$ is bounded below by a function in $L^{1}(0,r_{0})$.* *Proof.* (1). Let $M(r)$ be defined in [\[Formula: M(r)\]](#Formula: M(r)){reference-type="eqref" reference="Formula: M(r)"}, then for $r$ sufficiently small, $$\begin{aligned} \label{Formula: derivation of Y(r)} \begin{alignedat}{5} &M(r)-\frac{3}{2}J(r)-\int_{B_{1}}x_{0}y^{+}dX\\ &=I(r)-\frac{3}{2}J(r)-\int_{0}^{r}\sum_{i=1}^{2}t^{-4}I_{i}(t)dt-\frac{3}{2}\int_{0}^{r}t^{-5}J_{1}(t)dt-\int_{B_{1}}x_{0}y^{+}dX\\ &=\Phi(r)-\Phi(0+)-\int_{0}^{r}\sum_{i=1}^{2}t^{-4}I_{i}(t)dt-\frac{3}{2}\int_{0}^{r}t^{-5}J_{1}(t)dt\\ &=\int_{0}^{r}\left[\Phi'(t)-\sum_{i=1}^{2}t^{-4}I_{i}(t)-\frac{3}{2}t^{-5}J_{1}(t)\right]dt\\ &=\int_{0}^{r}2t^{-3}\int_{\partial B_{t}(X_{0})}\frac{1}{x}\left(\nabla\psi\cdot\nu-\frac{3}{2}\frac{\psi}{r}\right)^{2}d\mathcal{H}^{1}dt+\int_{0}^{r}t^{-4}K(t)dt\\ &\geqslant-C_{0}\int_{0}^{r}t^{-3}\int_{\partial B_{t}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}dt, \end{alignedat} \end{aligned}$$ where we have used the definition of $\Phi(r)$ [\[Formula: Weiss-boundary adjusted energy\]](#Formula: Weiss-boundary adjusted energy){reference-type="eqref" reference="Formula: Weiss-boundary adjusted energy"} in the second identity, the fact on $\Phi'(r)$ [\[Formula: derivatives of Weiss-boundary adjusted energy\]](#Formula: derivatives of Weiss-boundary adjusted energy){reference-type="eqref" reference="Formula: derivatives of Weiss-boundary adjusted energy"} in the fourth identity and the estimate [\[Formula: vort(3)\]](#Formula: vort(3)){reference-type="eqref" reference="Formula: vort(3)"} in the last inequality. Let $Y(r):(0,r_{0})\to\mathbb{R}$ be defined by $$\begin{aligned} Y(r)=\int_{0}^{r}t^{-3}\int_{\partial B_{t}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}dt. \end{aligned}$$ It follows from [\[Formula: J(r)\]](#Formula: J(r)){reference-type="eqref" reference="Formula: J(r)"}, [\[Formula: Energy identity\]](#Formula: Energy identity){reference-type="eqref" reference="Formula: Energy identity"} and [\[Formula: J\'(r)\]](#Formula: J'(r)){reference-type="eqref" reference="Formula: J'(r)"} that $$\begin{aligned} \label{Formula: Y(r)(0)} \begin{split} &\frac{d}{dr}\left(\frac{Y'(r)}{r}\right)\\ &=\frac{2}{r}\left(r^{-3}\int_{B_{r}(X_{0})}(\frac{1}{x}|\nabla\psi|^{2}-x\psi f(\psi))dX-\frac{3}{2}r^{-4}\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}-\frac{1}{2}r^{-4}J_{2}(r)\right)\\ &=\frac{2}{r}\left[\left(M(r)-\frac{3}{2}J(r)-r^{-3}\int_{B_{r}(X_{0})}xy^{+}\chi_{\{\psi>0\}}dX\right)\right.\\ &\quad+\left.\left(\int_{0}^{r}\sum_{i=1}^{2}t^{-4}I_{i}(t)dt+\int_{0}^{r}t^{-5}J_{1}(t)dt\right)+\left(\frac{1}{2}\int_{0}^{r}t^{-5}J_{1}(t)dt-\frac{1}{2}r^{-4}J_{1}(r)\right)\right], \end{split} \end{aligned}$$ where we have used the definition of $M(r)$ in the second equality [\[Formula: M(r)\]](#Formula: M(r)){reference-type="eqref" reference="Formula: M(r)"}. Observe now that $$\begin{aligned} &r^{-3}\int_{B_{r}(X_{0})}xy^{+}\chi_{\{\psi>0\}}dX\\ &=r^{-3}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}dX+r^{-3}\int_{B_{r}(X_{0})}x_{0}y^{+}\chi_{\{\psi>0\}}dX\\ &\leqslant r^{-3}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}dX+r^{-3}\int_{B_{r}(X_{0})}x_{0}y^{+}dX\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ Thus, [\[Formula: Y(r)(0)\]](#Formula: Y(r)(0)){reference-type="eqref" reference="Formula: Y(r)(0)"} can be rewritten as $$\begin{aligned} \label{Formula: Y'(r)/r} \begin{split} \frac{d}{dr}\left(\frac{Y'(r)}{r}\right)&\geqslant\frac{2}{r}\left[\left(M(r)-\frac{3}{2}J(r)-r^{-3}\int_{B_{r}(X_{0})}x_{0}y^{+}dX\right)\right.\\ &\quad+\left(\int_{0}^{r}t^{-4}I_{1}(t)dt+\int_{0}^{r}t^{-5}J_{1}(t)dt\right)\\ &\quad+\left(\int_{0}^{r}t^{-4}I_{2}(t)dt-r^{-3}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}dX\right)\\ &\quad+\left.\left(\frac{1}{2}\int_{0}^{r}t^{-5}J_{1}(t)dt-\frac{1}{2}r^{-4}J_{1}(r)\right)\right]\quad\text{ for all }\quad r\in(0,r_{0}). \end{split} \end{aligned}$$ Let now $\varepsilon\in(0,t)$ be sufficiently small such that $\varepsilon+t\in(0,r)$. Define $t_{\varepsilon,s}:=\varepsilon+st\in(\varepsilon,\varepsilon+t)$ for some $s\in(0,1)$ and let $\varphi(t):=\int_{B_{t}}\frac{x-x_{0}}{x^{2}}\psi^{2}$. Then $\varphi(t)$ is an absolute continuous function on $(0,1)$, and we obtain $$\begin{aligned} t^{-5}J_{1}(t)&=t^{-5}\int_{\partial B_{t}(X_{0})}\frac{x-x_{0}}{x^{2}}\psi^{2}\,d{\mathcal H}^1\\ &=t^{-5}\int_{0}^{1}\int_{\partial B_{t}(X_{0})}\frac{x-x_{0}}{x^{2}}\psi^{2}\,d{\mathcal H}^1ds\\ &=t^{-5}\int_{0}^{1}\int_{\partial B_{t_{\varepsilon,s}}(X_{0})}\frac{x-x_{0}}{x^{2}}\psi^{2}\,d{\mathcal H}^1ds+O(\varepsilon)\\ &=t^{-6}(\varphi(t+\varepsilon)-\varphi(\varepsilon))+O(\varepsilon), \end{aligned}$$ where we have used the fact that $t^{-1}(\varphi(t+\varepsilon)-\varphi(\varepsilon))=\int_{0}^{1}\varphi'(t_{\varepsilon,s})ds$. Passing to the limit as $\varepsilon\to0+$ and writing $X=X_{0}+t\xi\in\partial B_{t}(X_{0})$ for $\xi\in\partial B_1$ yields $$\begin{aligned} t^{-5}J_{1}(t)&=t^{-6}\int_{B_{t}(X_{0})}\frac{x-x_{0}}{x^{2}}\psi^{2}dX\\ &=t^{-4}\int_{B_1}\frac{t\xi}{(x_{0}+t\xi)^{2}}(\psi(X_{0}+t\xi)-\psi(X_{0}))^{2}dX(\xi)\\ &=t^{-3}\int_{B_1}\frac{\xi}{(x_{0}+t\xi)^{2}}\left(\int_{0}^{t}\nabla\psi(X_{0}+s\xi)\cdot\xi ds\right)^{2}dX(\xi), \end{aligned}$$ where we have also used the fact that $\psi(X_{0})=0$ in the second identity since $X_{0}\in S_{\psi}^{s}$. On the other hand, notice that $$\begin{aligned} t^{-4}I_{1}(t)=t^{-3}\int_{B_{1}}-\frac{\xi}{(x_{0}+t\xi)^{2}}\left(\int_{0}^{t}\nabla\psi(X_{0}+s_{0}\xi)\cdot\xi ds\right)^{2}dX(\xi), \end{aligned}$$ for some $s_{0}\in(0,t)$. Since $t\in(0,r)$ and $r\in(0,r_{0})$ is sufficiently small, we obtain $$\begin{aligned} \label{Formula: r1} t^{-4}I_{1}(t)+t^{-5}J_{1}(t)=0\quad\text{ for all }\quad t\in(0,r_{0}). \end{aligned}$$ Moreover, recalling [\[Formula: I2(r)\]](#Formula: I2(r)){reference-type="eqref" reference="Formula: I2(r)"}, $$\begin{aligned} \label{Formula: calculation of r2} \begin{split} &r^{-1}\int_{0}^{r}t^{-4}I_{2}(t)dt-r^{-4}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}dX\\ &=r^{-1}\int_{0}^{r}t^{-4}\int_{B_{t}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}\:dX-r^{-4}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}\:dX\\ &=r^{-1}\left(\int_{0}^{2r}t^{-4}\int_{B_{t}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}\:dX-\int_{0}^{r}t^{-4}\int_{B_{t}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}\:dX\right)\\ &\quad-r^{-4}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}\:dX\\ &=\int_{0}^{1}r_{\varepsilon}^{-4}\int_{B_{r_{\varepsilon}}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}\:dX\:d\epsilon-r^{-4}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}\:dX, \end{split} \end{aligned}$$ where $r_{\varepsilon}:=(1+\varepsilon)r$. We have by passing to the limit as $\varepsilon\to0+$ the identity $$\begin{aligned} \label{Formula: r2} r^{-1}\int_{0}^{r}t^{-4}I_{2}(t)dt-r^{-4}\int_{B_{r}(X_{0})}(x-x_{0})y^{+}\chi_{\{\psi>0\}}dX=0,\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ Finally, it follows from the definition of $J_{1}(r)$ [\[Formula: J1(r)\]](#Formula: J1(r)){reference-type="eqref" reference="Formula: J1(r)"}, and a similar argument as in [\[Formula: calculation of r2\]](#Formula: calculation of r2){reference-type="eqref" reference="Formula: calculation of r2"}-[\[Formula: r2\]](#Formula: r2){reference-type="eqref" reference="Formula: r2"} that $$\begin{aligned} \label{Formula: r3} r^{-1}\int_{0}^{r}t^{-5}J_{1}(t)dt-r^{-5}J_{1}(r)=0\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ At this stage, we collect [\[Formula: r1\]](#Formula: r1){reference-type="eqref" reference="Formula: r1"}, [\[Formula: r2\]](#Formula: r2){reference-type="eqref" reference="Formula: r2"} and [\[Formula: r3\]](#Formula: r3){reference-type="eqref" reference="Formula: r3"} together and introduce them into the inequality [\[Formula: Y\'(r)/r\]](#Formula: Y'(r)/r){reference-type="eqref" reference="Formula: Y'(r)/r"}, and we obtain $$\begin{aligned} \label{Formula: Y'(r)/r(1)} \frac{d}{dr}\left(\frac{Y'(r)}{r}\right)&\geqslant2r^{-1}\left(M(r)-\frac{3}{2}J(r)-r^{-3}\int_{B_{r}(X_{0})}x_{0}y^{+}dX\right)\\ &\geqslant-\alpha\frac{Y(r)}{r}\nonumber, \end{aligned}$$ here we have used [\[Formula: derivation of Y(r)\]](#Formula: derivation of Y(r)){reference-type="eqref" reference="Formula: derivation of Y(r)"} in the second inequality. Here $\alpha<+\infty$ is a positive constant. As an application of the *Bessel type differential inequality* [@VW2012 (6.12)], we have that the function $r\mapsto Y(r)/r^{1/2}$ is a convex function on $(0,r_{0})$. A similar argument as in the proof of [@VW2012 Theorem 6.12] yields that $$\begin{aligned} \frac{3}{2}\frac{Y(r)}{r}\leqslant Y'(r)\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ This together with [\[Formula: derivation of Y(r)\]](#Formula: derivation of Y(r)){reference-type="eqref" reference="Formula: derivation of Y(r)"} gives that $$\begin{aligned} \label{Formula: Y(r)(1)} M(r)-\frac{3}{2}J(r)-\int_{B_{1}}x_{0}y^{+}dX\geqslant-\frac{2}{3}C_{0}r^{-2}\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}, \end{aligned}$$ which proves (1). (2). Recalling [\[Formula: J\'(r)\]](#Formula: J'(r)){reference-type="eqref" reference="Formula: J'(r)"}, [\[Formula: Y\'(r)/r(1)\]](#Formula: Y'(r)/r(1)){reference-type="eqref" reference="Formula: Y'(r)/r(1)"}, and [\[Formula: Y(r)(1)\]](#Formula: Y(r)(1)){reference-type="eqref" reference="Formula: Y(r)(1)"}, we have $$\begin{aligned} J'(r)=\frac{d}{dr}\left(\frac{Y'(r)}{r}\right)\geqslant2r^{-1}\cdot\left(-\frac{2}{3}C_{0}r^{-2}\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}\right)\geqslant-2\beta rJ(r), \end{aligned}$$ for some $\beta>0$. (3). Using [\[Formula: vort(3)\]](#Formula: vort(3)){reference-type="eqref" reference="Formula: vort(3)"} and the frequency formula [\[Formula: H\'(r)\]](#Formula: H'(r)){reference-type="eqref" reference="Formula: H'(r)"} obtained previously, we get that for a.e. $r\in(0,r_{0})$. $$\begin{aligned} H'(r)\geqslant\frac{2}{r}V^{2}(r)-2C_{1}r|V(r)|-C_{1}r|Z(r)|-C_{0}r. \end{aligned}$$ Since $X_{0}\in S_{\psi}^{s}$, we have $x-x_{0}=O(r)$ and by the definition of $Z(r)$ [\[Formula: Z(r)\]](#Formula: Z(r)){reference-type="eqref" reference="Formula: Z(r)"} that $$\begin{aligned} \label{Formula: order of Z(r)} |Z(r)|\leqslant C_{2}r\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ Moreover, $$\begin{aligned} 2C_{1}r|V(r)|\leqslant\frac{1}{r}V^{2}(r)+C_{1}^{2}r^{3}, \end{aligned}$$ we get for a.e. $r\in(0,r_{0})$ $$\begin{aligned} \label{Formula: lower bound for H'(r)} H'(r)\geqslant\frac{1}{r}V^{2}(r)-C_{1}^{2}r^{3}-C_{1}C_{2}r^{2}-C_{0}r. \end{aligned}$$ Since $r\mapsto H(r)$ is bounded below as $r\to0+$, $r\mapsto H(r)$ must be bounded below as $r\to0+$, this implies that $r\mapsto\tfrac{1}{r}V^{2}(r)\in L^{1}(0,r_{0})$. (4). The existence of the limit $\lim_{r\to0+}H(r)$ follows directly from [\[Formula: lower bound for H\'(r)\]](#Formula: lower bound for H'(r)){reference-type="eqref" reference="Formula: lower bound for H'(r)"} and then $H(0+)\geqslant\frac{3}{2}$. (5). Consider now [\[Formula: H\'(r)(1)\]](#Formula: H'(r)(1)){reference-type="eqref" reference="Formula: H'(r)(1)"}, a similar argument as in the proof of part (3) gives $$\begin{aligned} \label{Formula: H'(r)(2)} \begin{split} &H'(r)-\frac{2}{r}\int\limits_{\partial B_{r}(X_{0})}\frac{1}{x}\left[\frac{r(\nabla\psi\cdot\nu)}{(\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}-D(r)\frac{\psi}{(\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1})^{1/2}}\right]^{2}d\mathcal{H}^{1}\\ &\geqslant-2C_{1}r|V(r)|-C_{1}r|Z(r)|-C_{0}r\geqslant-\frac{1}{r}V^{2}(r)-C_{1}^{2}r^{3}-C_{1}C_{2}r^{2}-C_{0}r, \end{split} \end{aligned}$$ which, together with (3), gives (5). ◻ # Blow-up limits {#Section: blow-up limits} In this section, we consider the blow-up sequence $\{\phi_{m}\}$ defined in [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"}. With the aid of the frequency formula derived in the previous section, we are able to passing to the limit as $m\rightarrow\infty$. One particular difficulty arises here is that it is not obvious whether limits of $\phi_{m}$ are again variational solutions. To overcome this, we would like to employ a *concentration compactness* result for axisymmetric waves introduced in [@VW2014 Theorem 6.1], making it possible to pass to the limit in the domain variation formula for $\phi_{m}$ (presented in Proposition [Proposition 19](#Proposition: strong convergence){reference-type="ref" reference="Proposition: strong convergence"}). We begin with the following Lemma, which indicates the homogeneity of $\phi_{0}$. **Lemma 15**. *Let $\psi$ be a variational solution and let $X_{0}\in\Sigma_{\psi}$, and let $\phi_{m}$ be the sequence defined by $$\begin{aligned} \label{Formula: vm} \phi_{m}(X):=\frac{\psi(X_{0}+r_{m}X)}{\sqrt{r_{m}^{-1}\int\limits_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}}, \end{aligned}$$ for any $r_{m}\to0+$ as $m\to\infty$, then for every $0<\varrho<\sigma<1$, $$\begin{aligned} \label{Formula: vm(1)} \int\limits_{B_{\sigma}\setminus B_{\varrho}}\frac{1}{x_{0}}|x|^{-5}[\nabla\phi_{m}\cdot x-H_{X_{0},\psi}(0+)\phi_{m}]^{2}\:dX\to0\quad\text{ as }\quad m\to\infty. \end{aligned}$$* *Proof.* Recalling the inequality [\[Formula: H\'(r)(2)\]](#Formula: H'(r)(2)){reference-type="eqref" reference="Formula: H'(r)(2)"} and integrating it from $(r_{m}\varrho,r_{m}\sigma)$ with respect to $r$ for every $0<\varrho<\sigma$ gives $$\begin{aligned} \label{Formula: vm(2)} &\int\limits_{r_{m}\varrho}^{r_{m}\sigma}\frac{2}{r}\frac{\int\limits_{\partial B_{r}(X_{0})}\tfrac{1}{x}[r(\nabla\psi\cdot\nu)-H(r)\psi]^{2}\:d\mathcal{H}^{1}}{\int\limits_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}dr\\ &\leqslant\int_{r_{m}\varrho}^{r_{m}\sigma}H'(r)\:dr+\int_{r_{m}\varrho}^{r_{m}\sigma}\left(\frac{1}{r}V^{2}(r)+C_{1}^{2}r^{3}+C_{1}C_{2}r^{2}+C_{0}r\right)\:dr\nonumber. \end{aligned}$$ Changing variables $\tilde{X}:=\frac{X-X_{0}}{r_{m}}$ and set $\tilde{r}:=\frac{r}{r_{m}}$ for simplicity, we rewrite [\[Formula: vm(2)\]](#Formula: vm(2)){reference-type="eqref" reference="Formula: vm(2)"} as $$\begin{aligned} \int\limits_{\varrho}^{\sigma}\frac{2}{\tilde{r}r_{m}}\frac{\int\limits_{\partial B_{r/r_{m}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}[\tilde{r}r_{m}\nabla\psi(X_{0}+r_{m}\tilde{X})\cdot\nu-H(\tilde{r}r_{m})\psi(X_{0}+r_{m}\tilde{X})]^{2}r_{m}\:d\mathcal{H}^{1}}{\int\limits_{\partial B_{r/r_{m}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\psi^{2}(X_{0}+r_{m}\tilde{X})r_{m}\:d\mathcal{H}^{1}}r_{m}\:d\tilde{r} \end{aligned}$$ It follows from the definition of $\phi_{m}$ [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"} that $$\begin{aligned} &\int\limits_{\varrho}^{\sigma}\frac{2}{\tilde{r}}\frac{\int\limits_{\partial B_{\tilde{r}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\left[\tilde{r}\sqrt{r_{m}^{-1}\int\limits_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}\nabla \phi_{m}\cdot\nu-H(\tilde{r}r_{m})\psi(X_{0}+r_{m}\tilde{X})\right]^{2}\:d\mathcal{H}^{1}}{\int\limits_{\partial B_{\tilde{r}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\psi^{2}(X_{0}+r_{m}\tilde{X})\:d\mathcal{H}^{1}}\:d\tilde{r}\\ &=\int\limits_{\varrho}^{\sigma}\frac{2}{\tilde{r}}\frac{r_{m}^{-1}\int\limits_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}\int\limits_{\partial B_{\tilde{r}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\left[r\nabla\phi_{m}\cdot\nu-H(r_{m}\tilde{r})\tfrac{\psi(X_{0}+r_{m}\tilde{X})}{\sqrt{r_{m}^{-1}\int\limits_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}}\right]^{2}\:d\mathcal{H}^{1}}{\int\limits_{\partial B_{\tilde{r}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\psi^{2}(X_{0}+r_{m}\tilde{X})\:d\mathcal{H}^{1}}\:d\tilde{r}\\ &=\int\limits_{\varrho}^{\sigma}\frac{2}{\tilde{r}}\int\limits_{\partial B_{\tilde{r}}}\frac{1}{x_{0}+r_{m}\tilde{x}}\left[\frac{\tilde{r}(\nabla\phi_{m}\cdot\nu)}{\left(\int\limits_{\partial B_{\tilde{r}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\phi_{m}^{2}\:d\mathcal{H}^{1}\right)^{1/2}}-H(r_{m}\tilde{r})\tfrac{\phi_{m}}{\left(\int\limits_{\partial B_{\tilde{r}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\phi_{m}^{2}\:d\mathcal{H}^{1}\right)^{1/2}}\right]^{2}\:d\mathcal{H}^{1}d\tilde{r}\\ &\leqslant H(r_{m}\sigma)-H(r_{m}\varrho)+\int_{r_{m}\varrho}^{r_{m}\sigma}\left(\frac{1}{r}V^{2}(r)+C_{1}^{2}r^{3}+C_{1}C_{2}r^{2}+C_{0}r\right)\:dr\to0, \end{aligned}$$ as $m\to\infty$. Now note that for every $\tilde{r}\in(\varrho,\sigma)\subset(0,1)$ and all $m$ as above, it follows that $$\begin{aligned} \int\limits_{\partial B_{\tilde{r}}}\frac{1}{x_{0}+r_{m}\tilde{x}}\phi_{m}^{2}\:d\mathcal{H}^{1}&=\frac{\int\limits_{\partial B_{\tilde{r}}}\tfrac{1}{x_{0}+r_{m}\tilde{x}}\psi^{2}(X_{0}+r_{m}\tilde{X})\:d\mathcal{H}^{1}}{r_{m}^{-1}\int\limits_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}\\ &=\frac{r_{m}^{-1}\int\limits_{\partial B_{\tilde{r}r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}{r_{m}^{-1}\int\limits_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}\\ &=\frac{(r_{m}\tilde{r})^{4}J(r_{m}\tilde{r})}{r_{m}^{4}J(r_{m})}\leqslant\tilde{r}^{4}e^{\beta r_{m}^{2}(1-r^{2})}\to\tilde{r}^{4}, \end{aligned}$$ as $m\to\infty$, where we have used Proposition [Proposition 14](#Proposition: fre(2)){reference-type="ref" reference="Proposition: fre(2)"} (ii) and $r_{m}\tilde{r}\leqslant r_{m}$. Therefore, $$\begin{aligned} \int\limits_{\varrho}^{\sigma}\frac{2}{\tilde{r}^{5}}\int\limits_{\partial B_{\tilde{r}}}\frac{1}{x_{0}}\left[\tilde{r}\nabla\phi_{m}\cdot\nu-H(0+)\phi_{m}\right]^{2}\:d\mathcal{H}^{1}d\tilde{r}\to0\quad\text{ as }\quad m\to\infty. \end{aligned}$$ Thanks to the identity $$\begin{aligned} \tilde{r}^{-5}\int\limits_{\partial B_{\tilde{r}}}\frac{1}{x_{0}}[\tilde{r}\nabla\phi_{m}\cdot\nu-H(0+)\phi_{m}]^{2}\:d\mathcal{H}^{1}=\frac{d}{d\tilde{r}}\int_{B_{\tilde{r}}}\frac{|\tilde{x}|^{-5}}{x_{0}}\left[\nabla\phi_{m}\cdot\frac{\tilde{x}}{|\tilde{x}|}-H(0+)\phi_{m}\right]^{2}\:d\tilde{X}. \end{aligned}$$ We obtain for every $0<\varrho<\sigma<1$, $$\begin{aligned} \int_{B_{\sigma}\setminus B_{\varrho}}\frac{1}{x_{0}}|\tilde{x}|^{-5}[\nabla\phi_{m}-H(0+)\phi_{m}]^{2}\:d\tilde{X}\to0\quad\text{ as }\quad m\to\infty, \end{aligned}$$ as desired. ◻ **Remark 16**. *With the aid of Corollary [Corollary 13](#Corollary: vort){reference-type="ref" reference="Corollary: vort"}, for every $r$ sufficiently small, we have $$\begin{aligned} \left|D(r)-\frac{r\int_{B_{r}(X_{0})}\tfrac{1}{x}|\nabla\psi|^{2}dX}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}\right|\leqslant Cr^{2}, \end{aligned}$$ which implies that for any sequence $r_{m}\to0+$ as $m\to\infty$, the sequence $\phi_{m}$ defined in [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"} $$\begin{aligned} \label{Formula: D(rm)} \left|D(r_{m})-\int_{B_{1}}\frac{1}{x_{0}+r_{m}x}|\nabla\phi_{m}|^{2}dX\right|\leqslant Cr_{m}^{2}. \end{aligned}$$* In the following Lemma, we will show the fact we mentioned in Remark [Remark 10](#Remark: comparison of two pbm){reference-type="ref" reference="Remark: comparison of two pbm"}, that $V(r)=\widetilde{V}(r)+O(r)$ for some $\widetilde{V}(r)\geqslant0$. **Lemma 17**. *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} and let $V(r)$ be given as in [\[Formula: V(r)\]](#Formula: V(r)){reference-type="eqref" reference="Formula: V(r)"}. Then for every $X_{0}\in\Sigma_{\psi}$ there exists some sufficiently small $r_{0}\in(0,\delta)$ so that $$\begin{aligned} \label{Formula: tildeV(1)} V(r)=\widetilde{V}(r)+\frac{1}{2}Z(r)\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ Here, $$\begin{aligned} \label{Formula: tildeV(2)} \widetilde{V}(r):=\frac{\displaystyle r\int_{B_{r}(X_{0})}x_{0}y^{+}(1-\chi_{\{\psi>0\}})\:dX}{\displaystyle\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}\geqslant0. \end{aligned}$$* *Proof.* A direct computation gives that $$\begin{aligned} &r\int_{B_{r}(X_{0})}y^{+}(x_{0}-x\chi_{\{\psi>0\}})\:dX\\ &=r\int_{B_{r}(X_{0})}x_{0}y^{+}(1-\chi_{\{\psi>0\}})\:dX+r\int_{B_{r}(X_{0})}(x_{0}-x)y^{+}\chi_{\{\psi>0\}}\:dX. \end{aligned}$$ Let $\widetilde{V}(r)$ be defined as in [\[Formula: tildeV(2)\]](#Formula: tildeV(2)){reference-type="eqref" reference="Formula: tildeV(2)"}, then $$\begin{aligned} &\left(V(r)-\widetilde{V}(r)\right)\int_{\partial B_{r}(X_{0})}\frac{1}{x}\psi^{2}d\mathcal{H}^{1}\\ &=\left(r^{4}\int_{0}^{r}t^{-4}I_{1}(t)dt+r^{4}\int_{0}^{r}t^{-5}J_{1}(t)dt\right)+\left(r\int_{B_{r}(X)}(x_{0}-x)y^{+}\chi_{\{\psi>0\}}dX+r^{4}\int_{0}^{r}t^{-4}I_{2}(t)dt\right)\\ &\quad+\frac{1}{2}r^{4}\int_{0}^{r}t^{-5}J_{1}(t)dt. \end{aligned}$$ It follows from [\[Formula: r1\]](#Formula: r1){reference-type="eqref" reference="Formula: r1"} and [\[Formula: r2\]](#Formula: r2){reference-type="eqref" reference="Formula: r2"} that $$\begin{aligned} V(r)-\widetilde{V}(r)=\frac{1}{2}\frac{r^{4}\int_{0}^{r}t^{-5}J_{1}(t)dt}{\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}\quad\text{ for all }\quad r\in(0,r_{0}). \end{aligned}$$ Then the desired result follows from [\[Formula: r3\]](#Formula: r3){reference-type="eqref" reference="Formula: r3"} and the definition of $Z(r)$ (recalling [\[Formula: Z(r)\]](#Formula: Z(r)){reference-type="eqref" reference="Formula: Z(r)"}). ◻ With the aid of Lemma [Lemma 17](#Lemma: V(r)){reference-type="ref" reference="Lemma: V(r)"}, we are now able to prove the following result. **Proposition 18**. *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"}, and let $X_{0}\in\Sigma_{\psi}$. Then* 1. *There exist $\lim_{r\to0+}V(r)=0$ and $\lim_{r\to0+}D(r)=H(0+)$.* 2. *Let $\phi_{m}$ be defined in [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"} for any $r_{m}\to0+$ as $m\to\infty$, then the sequence is bounded in $W^{1,2}(B_{1})$.* 3. *For any sequence $r_{m}\to0+$ as $m\to\infty$ such that the sequence $\phi_{m}$ converges in $W^{1,2}(B_{1})$ to a blow-up limit $\phi_{0}$, then the function $\phi_{0}$ is a homogeneous function of degree $H_{X_{0},\psi}(0+)$ in $B_{1}$, and satisfies $$\begin{aligned} \phi_{0}\geqslant0\quad\text{ in }\quad B_{1},\qquad\phi_{0}\equiv0\quad\text{ in }\quad B_{1}\cap\{y\leqslant0\},\qquad\int_{\partial B_{1}}\frac{1}{x_{0}}\phi_{0}^{2}\:d\mathcal{H}^{1}=1. \end{aligned}$$* *Proof.* (1). Assume for the sake of contradiction this is not the case. Let $s_{m}\to0$ be such that the sequence $V(s_{m})$ is bounded away from $0$. The integrability of $r\mapsto\frac{1}{r}V^{2}\in L^{1}(0,r_{0})$ (Proposition [Proposition 14](#Proposition: fre(2)){reference-type="ref" reference="Proposition: fre(2)"} (3)) implies that $$\begin{aligned} \min_{r\in[s_{m},2s_{m}]}V(r)\to0\quad\text{ as }\quad m\to\infty. \end{aligned}$$ It follows from [\[Formula: tildeV(1)\]](#Formula: tildeV(1)){reference-type="eqref" reference="Formula: tildeV(1)"} that $$\begin{aligned} \min_{r\in[s_{m},2s_{m}]}\left(\widetilde{V}(r)+\frac{1}{2}Z(r)\right)\to0\quad\text{ as }\quad m\to\infty. \end{aligned}$$ Furthermore, it follows from [\[Formula: order of Z(r)\]](#Formula: order of Z(r)){reference-type="eqref" reference="Formula: order of Z(r)"} that $$\begin{aligned} \min_{r\in[s_{m},2s_{m}]}\widetilde{V}(r)\to0\quad\text{ as }\quad m\to\infty. \end{aligned}$$ Let $t_{m}\in[s_{m},s_{2m}]$ be such that $\widetilde{V}(t_{m})\to0$ as $m\to\infty$. For the choice $r_{m}:=t_{m}$ for every $m$, the sequence $\phi_{m}$ given by [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"} satisfies [\[Formula: vm(1)\]](#Formula: vm(1)){reference-type="eqref" reference="Formula: vm(1)"}. The fact that $V(r_{m})\to0$ implies that $D(r_{m})$ is bounded. It follows from [\[Formula: D(rm)\]](#Formula: D(rm)){reference-type="eqref" reference="Formula: D(rm)"} that $\phi_{m}$ is bounded in $W^{1,2}(B_{1})$. Let $\phi_{0}$ be any weak limit of $\phi_{m}$ along a subsequence. Note that by the compact embedding $W^{1,2}(B_{1})\hookrightarrow L^{2}(\partial B_{1})$, $\phi_{0}$ has norm $1$ on $L^{2}(\partial B_{1})$, since this is true for all $m$. It follows from [\[Formula: vm(1)\]](#Formula: vm(1)){reference-type="eqref" reference="Formula: vm(1)"} that $\phi_{0}$ is a homogeneous function of degree $H(0+)$. Thanks to Proposition [Proposition 14](#Proposition: fre(2)){reference-type="ref" reference="Proposition: fre(2)"} (2), one has $$\begin{aligned} \label{Formula: tildeVsm} \widetilde{V}(s_{m})&=\frac{s_{m}^{-3}\int_{B_{s_{m}}(X_{0})}x_{0}y^{+}(1-\chi_{\{\psi>0\}})\:dX}{s_{m}^{-4}\int_{\partial B_{s_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}\nonumber\\ &\leqslant\frac{s_{m}^{-3}\int_{B_{r_{m}}(X_{0})}x_{0}y^{+}(1-\chi_{\{\psi>0\}})\:dX}{e^{\beta[(r_{m}^{2}/4)-s_{m}^{2}]}\int_{\partial B_{r_{m}/2}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}\nonumber\\ &\leqslant\frac{e^{3\beta r_{m}^{2}/4}}{2}\frac{\int_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}{\int_{\partial B_{r_{m}/2}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}\widetilde{V}(r_{m})\nonumber\\ &=\frac{e^{3\beta r_{m}^{2}/4}}{2\int_{\partial B_{1/2}}\tfrac{1}{x_{0}+r_{m}x}\phi_{m}^{2}\:d\mathcal{H}^{1}}\widetilde{V}(r_{m}). \end{aligned}$$ Since, at least along a subsequence $$\begin{aligned} \int_{\partial B_{1/2}}\frac{1}{x_{0}+r_{m}x}\phi_{m}^{2}\:d\mathcal{H}^{1}\to\int_{\partial B_{1/2}}\frac{1}{x_{0}}\phi_{0}^{2}\:d\mathcal{H}^{1}>0, \end{aligned}$$ we have from [\[Formula: tildeVsm\]](#Formula: tildeVsm){reference-type="eqref" reference="Formula: tildeVsm"} that $\widetilde{V}(r_{m})\to0$, which implies that $\widetilde{V}(s_{m})\to0$. This together with $V(s_{m})=\widetilde{V}(s_{m})+\frac{1}{2}Z(s_{m})$ imply that $V(s_{m})\to0$ as $m\to\infty$. This contradicts to the choice of $V(s_{m})$. It follows indeed that $V(r)\to0$ as $r\to0+$, and therefore $D(r)\to H(0+)$. (2). It follows from [\[Formula: D(rm)\]](#Formula: D(rm)){reference-type="eqref" reference="Formula: D(rm)"} that the boundedness of the sequence $\phi_{m}$ in $W^{1,2}(B_{1})$ is equivalent to the boundedness of $D(r_{m})$, which is true by the statement (1). (3). Let $r_{m}\to0+$ be an arbitrary sequence such that $\phi_{m}$ converges weakly to $\phi_{0}$. The fact that $\phi_{0}$ is a homogeneous function of degree $H(0+)$ follows directly from [\[Formula: vm(1)\]](#Formula: vm(1)){reference-type="eqref" reference="Formula: vm(1)"}. Since $\phi_{0}$ belongs to $W^{1,2}(B_{1})$, its homogeneity implies that $\phi_{0}$ is continuous in $B_{1}$. The fact that $\int_{\partial B_{1}}\frac{1}{x_{0}}\phi_{0}^{2}\:d\mathcal{H}^{1}=1$ is a consequence of $\int_{\partial B_{1}}\tfrac{1}{x_{0}+r_{m}x}\phi_{m}^{2}\:d\mathcal{H}^{1}=1$ for all $m$. This concludes the proof. ◻ **Proposition 19**. *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} with the growth assumption [\[Formula: growth assumption\]](#Formula: growth assumption){reference-type="eqref" reference="Formula: growth assumption"}. Assume that the nonlinearity $f$ satisfies [\[Formula: f(z)\]](#Formula: f(z)){reference-type="eqref" reference="Formula: f(z)"}. Let $X_{0}\in\Sigma_{\psi}$ and let $r_{m}\to0+$ be such that the sequence $\phi_{m}$ given by [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"} converges weakly to $\phi_{0}$ in $W^{1,2}(B_{1})$. Then $\phi_{m}$ converges to $\phi_{0}$ strongly in $W_{\mathrm{loc}}^{1,2}(B_{1}\setminus\{0\})$, $\phi_{0}$ is continuous in $B_{1}$ and $\Delta \phi_{0}$ is a nonnegative Radon measure satisfying $\phi_{0}\Delta \phi_{0}=0$ in the sense of Radon measures in $B_{1}$.* *Proof.* The proof is similar to [@VW2014 Theorem 6.1] and we only exhibit the major differences in the following. Note first that $$\begin{aligned} \label{Formula: vm(2)(1)} \begin{split} \operatorname{div}\left(\frac{1}{x_{0}+r_{m}x}\nabla\phi_{m}(x)\right)&=\frac{r_{m}}{\sqrt{r_{m}^{-1}\int_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}}\operatorname{div}\left(\tfrac{\nabla \psi(X_{0}+r_{m}X)}{x_{0}+r_{m}x}\right)\\ &=\frac{-r_{m}(x_{0}+r_{m}x)f(\psi(X_{0}+r_{m}X))}{\sqrt{r_{m}^{-1}\int_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}}\\ &\geqslant\frac{-Cr_{m}(x_{0}+r_{m}x)\psi(X_{0}+r_{m}X)}{\sqrt{r_{m}^{-1}\int_{\partial B_{r_{m}}(X_{0})}\tfrac{1}{x}\psi^{2}\:d\mathcal{H}^{1}}}\\ &\geqslant-C_{1}r_{m}\phi_{m}\quad\text{ for }\quad\phi_{m}>0, \end{split} \end{aligned}$$ in the sense of distributions. We infer from [@GT1983 Theorem 8.17] that $$\begin{aligned} \operatorname{div}\left(\frac{1}{x_{0}+r_{m}x}\nabla\phi_{m}\right)\geqslant-C_{3}(\sigma)r_{m}\quad\text{ in }\quad B_{\sigma} \end{aligned}$$ in the sense of measures. Here we use the fact that $\phi_{m}$ is bounded in $L^{1}(B_{1})$. Set $\mu_{m}:=\operatorname{div}\left(\tfrac{1}{x_{0}+r_{m}x}\nabla\phi_{m}\right)$ for all $m$, and it follows that for each non-negative $\eta\in C_{0}^{\infty}(B_{1})$ such that $\eta=1$ in $B_{(\sigma+1)/2}$ $$\begin{aligned} \int_{B_{(\sigma+1)/2}}\eta\:d\mu_{m}\leqslant C_{4}\quad\text{ for all }\quad m\in\mathbb{N}, \end{aligned}$$ where we have used [\[Formula: vm(2)(1)\]](#Formula: vm(2)(1)){reference-type="eqref" reference="Formula: vm(2)(1)"} in the third inequality and the fact that $\phi_{m}$ is bounded in $L^{1}(B_{1})$ in the last inequality. Since $x_{0}>0$, we deduce from [\[Formula: vm(2)(1)\]](#Formula: vm(2)(1)){reference-type="eqref" reference="Formula: vm(2)(1)"} that $\Delta\phi_{0}$ is a non-negative Radon measure in $B_{1}$. The continuity of $\phi_{0}$ implies therefore that $\phi_{0}\Delta\phi_{0}$ is well defined as a non-negative Radon measure on $B_{1}$. In order to apply the concentration compactness by Delort in [@D1992], we modify each $\phi_{m}$ to $$\begin{aligned} \tilde{\phi}_{m}:=(\phi_{m}+C_{3}(\sigma)r_{m}(\tfrac{x_{0}x^{2}}{2}+\tfrac{r_{m}x^{3}}{3}))\star\varphi_{m}\in C^{\infty}(B_{1}), \end{aligned}$$ where $\star$ denotes the convolution and $\varphi_{m}$ is a standard mollifier such that $$\begin{aligned} \label{Formula: vm(3)} \operatorname{div}\left(\frac{1}{x_{0}+r_{m}x}\nabla\tilde{\phi}_{m}\right)\geqslant0\quad\text{ in }\quad B_{(\sigma+1)/2},\qquad\int_{B_{(\sigma+1)/2}}\eta d\tilde{\mu}_{m}\leqslant C<+\infty,\quad\text{ for all }m, \end{aligned}$$ and $$\begin{aligned} \label{Formula: vm(4)} \|\phi_{m}-\tilde{\phi}_{m}\|_{W^{1,2}(B_{\sigma})}\to0\quad\text{ as }\quad m\to\infty, \end{aligned}$$ where $\tilde{\mu}_{m}:=\operatorname{div}\left(\frac{1}{x_{0}+r_{m}x}\nabla\tilde{\phi}_{m}\right)$. Introduce a new set of coordinates $x'=x_{0}+r_{m}x$ and $y'=r_{m}y$ and consider the velocity vector field associated with $\phi_{m}(x',y')$: $$\begin{aligned} V^{m}(X,Y,Z):=\left(-\frac{1}{x'}\partial_{y'}\phi_{m}\cos\theta,-\frac{1}{x'}\partial_{y'}\phi_{m}\sin\theta,\frac{1}{x'}\partial_{x'}\phi_{m}\right), \end{aligned}$$ where $(X,Y,Z)=(x'\cos\varphi,x'\sin\varphi,y')$. We use the notation $\nabla'=(\partial_{x'},\partial_{y'})$ and $\operatorname{div}'=\partial_{x'}+\partial_{y'}$. We define the velocity vector field associated with the function $\tilde{\phi}_{m}(x',y')$, $$\begin{aligned} \widetilde{V}^{m}(X,Y,Z):=\left(-\frac{1}{x'}\partial_{y'}\tilde{\phi}_{m}\cos\theta,-\frac{1}{x'}\partial_{y'}\tilde{\phi}_{m}\sin\theta,\frac{1}{x'}\partial_{x'}\tilde{\phi}_{m}\right). \end{aligned}$$ We have that $\widetilde{V}^{m}$ is smooth and satisfies $\operatorname{div}'\widetilde{V}^{m}(X,Y,Z)=0$ and $$\begin{aligned} \operatorname{curl}\widetilde{V}^{m}=\operatorname{div}'\left(\frac{1}{x'}\nabla'\tilde{\phi}_{m}\right)(-\sin\theta,\cos\theta,0)\geqslant0, \end{aligned}$$ where we have used [\[Formula: vm(3)\]](#Formula: vm(3)){reference-type="eqref" reference="Formula: vm(3)"} in the last inequality. We consider also $$\begin{aligned} V^{0}(X,Y,Z):=\left(-\frac{1}{x_{0}}\partial_{y'}\phi_{0}\cos\theta,-\frac{1}{x_{0}}\partial_{y'}\phi_{0}\sin\theta,\frac{1}{x_{0}}\partial_{x'}\phi_{0}\right). \end{aligned}$$ Taking into account [\[Formula: vm(3)\]](#Formula: vm(3)){reference-type="eqref" reference="Formula: vm(3)"} we are in a position to proceed as a similar argument as in [@VW2014 Theorem 6.1] to conclude the following convergence $$\begin{aligned} \label{Formula: vm(5)} \frac{1}{x_{0}+r_{m}x}\partial_{x}\phi_{m}\partial_{y}\phi_{m}\to\frac{1}{x_{0}}\partial_{x}\phi_{0}\partial_{y}\phi_{0} \end{aligned}$$ and $$\begin{aligned} \label{Formula: vm(6)} \frac{1}{x_{0}+r_{m}x}((\partial_{x}\phi_{m})^{2}-(\partial_{y}\phi_{m})^{2})\to\frac{1}{x_{0}}((\partial_{x}\phi_{0})^{2}-(\partial_{y}\phi_{0})^{2}) \end{aligned}$$ in the sense of distributions in $B_{\sigma}$ as $m\to\infty$. Observe now that [\[Formula: vm(1)\]](#Formula: vm(1)){reference-type="eqref" reference="Formula: vm(1)"} shows that $$\begin{aligned} \label{Formula: vm(7)} \frac{1}{x_{0}+r_{m}x}\left(\nabla\phi_{m}\cdot X-H(0+)\phi_{m}\right)\to0 \end{aligned}$$ strongly in $L^{2}(B_{\sigma}\setminus B_{\tau})$ as $m\to\infty$. A combination of [\[Formula: vm(5)\]](#Formula: vm(5)){reference-type="eqref" reference="Formula: vm(5)"}- [\[Formula: vm(7)\]](#Formula: vm(7)){reference-type="eqref" reference="Formula: vm(7)"} and a similar argument as in the proof of [@VW2014 Theorem 6.1] gives that $$\begin{aligned} \int_{B_{\sigma}\setminus B_{\tau}}(\partial_{x}\phi_{m})^{2}\eta dX\to\int_{B_{\sigma}\setminus B_{\tau}}(\partial_{x}\phi_{0})^{2}\eta dX \end{aligned}$$ for every $0\leqslant\eta\in C_{0}^{0}(B_{\sigma}\setminus B_{\tau})$. Using [\[Formula: vm(5)\]](#Formula: vm(5)){reference-type="eqref" reference="Formula: vm(5)"} once more again yields that $\nabla\phi_{m}\to\nabla\phi_{0}$ strongly in $L_{\mathrm{loc}}^{2}(B_{\sigma}\setminus B_{\tau})$ with $0<\tau<\sigma<1$ arbitrarily. It follows that $\nabla\phi_{m}$ converges to $\nabla\phi_{0}$ strongly in $L_{\mathrm{loc}}^{2}(B_{1}\setminus\{0\})$. As a direct application of the strong convergence, $$\begin{aligned} \int_{B_{1}}\frac{1}{x_{0}}\nabla(\eta x_{0})\cdot\nabla\phi_{0}dX=0\quad\text{ for all }\quad\eta\in C_{0}^{1}(B_{1}\setminus\{0\}). \end{aligned}$$ Since $\phi_{0}=0$ in $B_{1}\cap\{y\leqslant0\}$, we have that $\phi_{0}\Delta\phi_{0}=0$ in the sense of distributions in $B_{1}$. ◻ With the aid of the strong convergence of $\phi_{m}$ to $\phi_{0}$, let $r_{m}\to0+$ be an arbitrary sequence so that the sequence $\phi_{m}$ defined in [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"} converges weakly in $W^{1,2}(B_{1})$ to a limit $\phi_{0}$. It follows from Proposition [Proposition 18](#Proposition: vm){reference-type="ref" reference="Proposition: vm"} (3) that $\phi_{0}\geqslant0$, $\phi_{0}$ is continuous and is a homogeneous function of degree $H(0+)\geqslant3/2$. Moreover, $\phi_{0}\Delta\phi_{0}$ is a non-negative Radon measure satisfying $\phi_{0}\Delta\phi_{0}=0$ in $B_{1}$. The strong convergence of $\phi_{m}$ to $\phi_{0}$ in $W^{1,2}(B_{1}\setminus\{0\})$ and $V(r_{m})\to0$ as $m\to\infty$ imply that $$\begin{aligned} 0=\int_{B_1}|\nabla\phi_{0}|^{2}\operatorname{div}\xi-2\nabla\phi_{0}D\xi\nabla\phi_{0}dX,\end{aligned}$$ for each $\xi\in C_{0}^{1}(B_{1}\cap\{y>0\};\mathbb{R}^{2})$. Then a similar argument as in the proof of [@VW2012 Theorem 9.1] gives the following result. **Proposition 20**. *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} with the growth assumption [\[Formula: growth assumption\]](#Formula: growth assumption){reference-type="eqref" reference="Formula: growth assumption"}. Assume that the nonlinearity $f$ satisfies [\[Formula: f(z)\]](#Formula: f(z)){reference-type="eqref" reference="Formula: f(z)"}. Then at each point $X_{0}$ of the set $\Sigma_{\psi}$ there exists an integer $N(X_{0})\geqslant2$ such that $$H_{X_{0},\psi}(0+)=N(X_{0})$$ and $$\begin{aligned} \label{Formula: freq} \frac{\psi(X_{0}+rX)}{\sqrt{r^{-1}\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}}\to\frac{\sqrt{x_{0}}\rho^{N(X_{0})}|\sin(N(X_{0})\min(\max(\theta,0),\pi))|}{\sqrt{\int_{0}^{\pi}\sin^{2}(N(X_{0})\theta)d\theta}}\quad\text{ as }\quad r\to0+, \end{aligned}$$ strongly in $W_{\mathrm{loc}}^{1,2}(B_{1}\setminus\{0\})$ and weakly in $W^{1,2}(B_{1})$, where $X=(\rho\cos\theta,\rho\sin\theta)$.* # Conclusions {#Section: conclusions} In this section, we collect two main results, the first states that $\Sigma_{\psi}$ is locally finite in $\Omega$, while the second one demonstrates that $\Sigma_{\psi}=\varnothing$. Thus, we proved that the set of horizontally flat singularities is an empty set. **Theorem 21**. *Let $\psi$ be a variational solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} with the growth assumption [\[Formula: growth assumption\]](#Formula: growth assumption){reference-type="eqref" reference="Formula: growth assumption"}. Assume that the nonlinearity $f$ satisfies [\[Formula: f(z)\]](#Formula: f(z)){reference-type="eqref" reference="Formula: f(z)"}. Then the set $\Sigma_{\psi}$ is a finite set locally in $\Omega$.* *Proof.* The proof is similar to [@VW2012 Theorem 9.2], we sketch the proof here. Assume for the sake of contradiction that there is a sequence of points $X_{m}\in\Sigma_{\psi}$ converging to $X_{0}\in\Omega$ with $X_{m}\neq X_{0}$ for all $m$. It follows from the upper semicontinuity (recalling Remark [Remark 8](#Remark: uppersemicontinuity){reference-type="ref" reference="Remark: uppersemicontinuity"}) that $X_{0}\in\Sigma_{\psi}$. Choose $r_{m}:=2|X_{m}-X_{0}|$, and assume without loss of generality that the sequence $(X_{m}-X_{0})/r_{m}$ is constant, with value $P\in\{(1/2,0),(-1/2,0)\}$. Let $\phi_{m}$ be the sequence defined in [\[Formula: vm\]](#Formula: vm){reference-type="eqref" reference="Formula: vm"}, and consider also the sequence $$\begin{aligned} \psi_{m}(X):=\frac{\psi(X_{0}+r_{m}X)}{r_{m}^{3/2}}.\end{aligned}$$ Note that each $\psi_{m}$ is a variational solution of the problem $$\begin{aligned} \begin{cases} \mathop{\mathrm{div}}\left(\frac{1}{x_{0}+r_{m}x}\nabla\psi_{m}\right)=-(x_{0}+r_{m}x)r_{m}^{1/2}\psi_{m}f(r_{m}^{3/2}\psi_{m})\quad&\text{ in }\quad B_{\delta/r_{m}}\cap\{\psi_{m}>0\},\\ |\nabla\psi_{m}|^{2}=(x_{0}+r_{m}x)^{2}y\quad&\text{ on }\quad B_{\delta/r_{m}}\cap\partial\{\psi_{m}>0\}. \end{cases} \end{aligned}$$ It is easy check that $\phi_{m}$ is a scalar multiple of $\psi_{m}$, that is, $$\begin{aligned} \label{Formula: vm(8)} \phi_m(X)=\frac{\psi_m(X)}{\sqrt{\int_{\partial B_1}\tfrac{1}{x_0+r_mx}\psi_m^2\,d{\mathcal H}^1}}. \end{aligned}$$ Since $X_{m}\in\Sigma_{\psi}$, it follows that $P\in\Sigma_{\psi_{m}}$. Therefore, Proposition [Proposition 14](#Proposition: fre(2)){reference-type="ref" reference="Proposition: fre(2)"} (1) shows that $$\begin{aligned} D_{P,\psi_{m}}(r)\geqslant V_{P,\psi_{m}}(r)+\frac{3}{2}-C_{1}r^{2}\quad\text{ for all }r\in(0,r_{0}), \end{aligned}$$ with $r_{0}\in(0,1/2)$. where $r_{0}\in(0,1/2)$ is sufficiently small. It then follows from Lemma [Lemma 17](#Lemma: V(r)){reference-type="ref" reference="Lemma: V(r)"} that $V(r)=\widetilde{V}(r)+\frac{1}{2}Z(r)$ for all $r\in(0,r_{0})$ and therefore, $$\begin{aligned} D_{P,\psi_{m}}(r)&\geqslant\widetilde{V}_{P,\psi_{m}}(r)+\frac{1}{2}Z_{P,\psi_{m}}(r)+\frac{3}{2}-C_{1}r^{2}\\ &\geqslant\frac{3}{2}+\frac{1}{2}Z_{P,\psi_{m}}(r)-C_{1}r^{2}, \end{aligned}$$ where we have used the fact that $\widetilde{V}_{P,\psi_{m}}(r)\geqslant 0$. Also observe that $$\begin{aligned} r\int_{B_r(P)}(x_{0}+r_{m}x)r_{m}^{1/2}\psi_{m}f(r_{m}^{3/2}\psi_{m})dX&=r\int_{B_r(P)}(x_{0}+r_{m}x)^{2}r_{m}^{1/2}\psi_{m}\frac{f(r_{m}^{3/2}\psi_{m})}{(x_{0}+r_{m}x)}\,d{\mathcal H}^1\\ &\geqslant-Cr^{2}r_{m}^{2}\int_{\partial B_r(P)}\tfrac{1}{x_{0}+r_{m}x}\psi_{m}^{2}\,d{\mathcal H}^1. \end{aligned}$$ where we have used the Assumption [Assumption 11](#Assumption: f){reference-type="ref" reference="Assumption: f"} in the last inequality. Therefore, $$\begin{aligned} r\int_{B_r(P)}\frac{1}{x_{0}+r_{m}x}|\nabla\psi_{m}|^{2}dX\geqslant\left(\frac{3}{2}+\frac{1}{2}Z(r)-C_{1}r^{2}-Cr^{2}r_{m}^{2}\right)\int_{\partial B_r(P)}\frac{1}{x_{0}+r_{m}x}\psi_{m}^{2}\,d{\mathcal H}^1. \end{aligned}$$ Recalling [\[Formula: vm(8)\]](#Formula: vm(8)){reference-type="eqref" reference="Formula: vm(8)"}, we obtain $$\begin{aligned} r\int_{B_r(P)}\frac{1}{x_{0}+r_{m}x}|\nabla\phi_{m}|^{2}dX\geqslant\left(\frac{3}{2}+\frac{1}{2}Z(r)-C_{1}r^{2}-Cr^{2}r_{m}^{2}\right)\int_{\partial B_r(P)}\frac{1}{x_{0}+r_{m}x}\phi_{m}^{2}\,d{\mathcal H}^1. \end{aligned}$$ It is a consequence of Proposition [Proposition 19](#Proposition: strong convergence){reference-type="ref" reference="Proposition: strong convergence"} that $\phi_{m}$ converges strongly in $W^{1,2}(B_{1/4}(P))$ to $\phi_{0}$ given by [\[Formula: freq\]](#Formula: freq){reference-type="eqref" reference="Formula: freq"}, hence $$\begin{aligned} r\int_{B_r(P)}\frac{1}{x_{0}}|\nabla\phi_{0}|^{2}dX\geqslant\left(\frac{3}{2}+\frac{1}{2}Z(r)-C_{1}r^{2}\right)\int_{\partial B_r(P)}\frac{1}{x_{0}}\phi_{0}^{2}\,d{\mathcal H}^1. \end{aligned}$$ However, this contradicts with the fact that $$\begin{aligned} \lim_{r\to0+}\frac{r\int_{B_r(P)}|\nabla\phi_{0}|^{2}dX}{\int_{\partial B_r(P)}\phi_{0}^{2}\,d{\mathcal H}^1}=1, \end{aligned}$$ where we have used the fact that $\lim_{r\to0+}Z(r)=0$. ◻ The next theorem states that the horizontally flat singularities do not exist. **Theorem 22**. *Let $\psi$ be a weak solution of the problem [\[Formula: model problem1\]](#Formula: model problem1){reference-type="eqref" reference="Formula: model problem1"} with the growth assumption [\[Formula: growth assumption\]](#Formula: growth assumption){reference-type="eqref" reference="Formula: growth assumption"}. Assume that the nonlinearity $f$ satisfies [\[Formula: f(z)\]](#Formula: f(z)){reference-type="eqref" reference="Formula: f(z)"} and that the free boundary satisfies the Assumption [Assumption 6](#Assumption: curve assumption){reference-type="ref" reference="Assumption: curve assumption"}. Then $\Sigma_{\psi}=\varnothing$.* *Proof.* The idea of the proof mainly borrows from [@VW2012 Theorem 10.1] Suppose towards a contradiction that there exists a point $X_{0}\in\Sigma_{\psi}$. We infer from Proposition [Proposition 20](#Proposition: v0){reference-type="ref" reference="Proposition: v0"} that there exists an integer $N(X_{0})\geqslant 2$ such that $$\begin{aligned} \label{Formula: vr} \begin{split} \phi_{r}(X)&:=\frac{\psi(X_{0}+rX)}{\sqrt{r^{-1}\int_{\partial B_{r}(X_{0})}\tfrac{1}{x}\psi^{2}d\mathcal{H}^{1}}}\to\frac{\sqrt{x_{0}}\rho^{N(X_{0})}|\sin(N(X_{0})\min(\max(\theta,0),\pi))|}{\sqrt{\int_{0}^{\pi}\sin^{2}(N(X_{0})\theta)d\theta}}\quad\text{ as }\quad r\to0+, \end{split} \end{aligned}$$ strongly in $W_{\mathrm{loc}}^{1,2}(B_{1}\setminus\{0\})$ and weakly in $W^{1,2}(B_{1})$, where $X=(\rho\cos\theta,\rho\sin\theta)$. On the other hand, it follows from that for any $\tilde{B}\subset B_{1}\cap\{y>0\}$, $\phi_{r}>0$ in $\tilde{B}$ for sufficiently small $r$. Consequently, recalling [\[Formula: vm(2)(1)\]](#Formula: vm(2)(1)){reference-type="eqref" reference="Formula: vm(2)(1)"}, $$\begin{aligned} \left|\operatorname{div}\left(\frac{1}{x_{0}+rx}\nabla\phi_{r}\right)\right|\leqslant C_{1}r\phi_{r}\quad\text{ in }\quad\tilde{B} \end{aligned}$$ for sufficiently small $r$. It follows that $\Delta\phi_{0}=0$ in $\tilde{B}$. This yields a contradiction to [\[Formula: vr\]](#Formula: vr){reference-type="eqref" reference="Formula: vr"} since $N(X_{0})\geqslant2$. Hence $\Sigma_{\psi}$ is indeed empty. ◻ 60 F. J. Almgren, Almgren's Big Regular Paper, World Scientific Monograph Series in Mathematcis, 1. World Scientific, River Edge, NJ, (2000). H. W. Alt, L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, *J. Reine Angew. Math.*, **325**, 105-144, (1981). Delort, J.-M., Une remarque sur le probleme des nappes de tourbillon axisymetriques sur $\mathbb{R}^{3}$, *J. Funct. Anal.*, **108**, (1992). L. L. Du, J. L. Huang, Y. Pu, The free boundary of steady axisymmetric inviscid flow with vorticity I: near the degenerate point, *Commun. Math. Phys.*, **400**, 2137-2179, (2023). L. L. Du, C. L. Yang, The free boundary of steady axisymmetric inviscid flow with vorticity II: near the non-degenerate points, *preprint*. N. Garofalo, F. H. Lin, Monotonicity properties of variational integrals: $A_{p}$ weights and unique continuation, *Indiana Univ. Math. J*, **35**, no. 2, 245-268, (1986). N. Garofalo, F. H. Lin, Unique continuation for elliptic operators: a geometric-variational approach, *Comm. Pure Appl, Math*, **XL**, 347-366, (1987). D. Gilbarg, N S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, *Grundlehren der Mathematischen Wissenschaften*, **224**, Springer-Verlag, Berlin, (1983). Vărvărucă, G. Weiss, A geometric approach to generalized Stokes conjectures, *Acta Math.*, **206**(2), 363-403, (2011). E. Vărvărucă, G. Weiss, The stokes conjecture for waves with vorticity, *Ann. Inst. H. Poincaré.*, **29**, 861-885, (2012). E. Vărvărucă, G. Weiss, Singularities of steady axisymmetric free surface flows with gravity, *Comm. Pure Appl. Math*, **67**(8), 1263-1306, (2014). [^1]: The first author is supported by National Nature Science Foundation of China Grant 11971331, 12125102, and Sichuan Youth Science and Technology Foundation 2021JDTD0024.

arxiv_math

--- abstract: | Let $f:X\to X$ be a non-isomorphic (i.e., $\deg f>1$) surjective endomorphism of a smooth projective threefold $X$. We prove that any birational minimal model program becomes $f$-equivariant after iteration, provided that $f$ is $\delta$-primitive. Here $\delta$-primitive means that there is no $f$-equivariant (after iteration) dominant rational map $\pi:X\dashrightarrow Y$ to a positive lower-dimensional projective variety $Y$ such that the first dynamical degree remains unchanged. This way, we further determine the building blocks of $f$. As the first application, we prove the Kawaguchi-Silverman conjecture for every non-isomorphic surjective endomorphism of a smooth projective threefold. As the second application, we reduce the Zariski dense orbit conjecture for $f$ to a terminal threefold with only $f$-equivariant Fano contractions. address: - " [School of Mathematical Sciences, Ministry of Education Key Laboratory of Mathematics and Engineering Applications & Shanghai Key Laboratory of PMMP]{.smallcaps} [East China Normal University, Shanghai 200241, China]{.smallcaps} " - " [Department of Mathematics]{.smallcaps} [National University of Singapore, Singapore 119076, Republic of Singapore]{.smallcaps} " author: - Sheng Meng, De-Qi Zhang title: Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds --- # Introduction Unless otherwise specified, we always assume that the base field $K$ is an algebraically closed field of characteristic $0$. Let $f:X\to X$ be a non-isomorphic (i.e., $\deg f>1$) surjective endomorphism of a normal projective variety $X$. It is fundamental to characterize $f$ and $X$. A natural reduction idea is to "decompose" $f$ and $X$ into lower-dimensional objects. A classical approach is to make use of various fibrations with universal properties such as the Iitaka fibration, special maximal rationally connected fibration, and the Albanese map. A more delicate surgery is to perform the Equivariant Minimal Model Program (EMMP) on a mildly singular $X$ as illustrated by the following commutative diagram $$\xymatrix{ X=X_1\ar@{-->}[r]^{\pi_1}\ar[d]_{f_1:=f^s} & X_2\ar@{-->}[r]^{\pi_2}\ar[d]^{f_2} &\cdots \ar@{-->}[r]^{\pi_{n-1}}&X_n\ar[d]^{f_n}\\ X=X_1\ar@{-->}[r]^{\pi_1} & X_2\ar@{-->}[r]^{\pi_2} &\cdots \ar@{-->}[r]^{\pi_{n-1}}&X_n }$$ for some positive integer $s>0$. Here $\pi_i$ is either a divisorial contraction, a flip, or a Fano contraction of a $K_{X_i}$-negative extremal ray. After iteration of $f_i$, a Fano contraction is always equivariant, while a divisorial contraction (resp. flip) is equivariant if we can show the exceptional locus (resp. flipping locus) is $f_i$-periodic, see [@MZ18 Lemma 6.2]. When $X$ is a normal projective surface, we can run EMMP due to the finiteness of negative curves by Nakayama [@Nak02 Section 2] (for the smooth case) and the authors [@MZ22 Section 5] (for the general case). The detailed characterization is further provided in [@JXZ23 Theorem 1.1]. In higher dimensions, one can also run EMMP when $X$ admits a polarized or an int-amplified endomorphism due to the finiteness of contractible extremal rays, in increased generality, proved in [@MZ18; @CMZ20; @Men20; @MZ20a; @MZ20b]. Of course, we do not expect every MMP to be EMMP in general and indeed there is an easy counter-example in dimension $3$ that breaks this illusion, see Example [Example 64](#example-emmp){reference-type="ref" reference="example-emmp"}. So it is important to explore the phenomenon when EMMP fails. *In the paper, we shall mainly focus on smooth projective threefolds.* Surprisingly, a new $f$-equivariant fibration, called *dynamical Iitaka fibration*, will emerge whenever the birational MMP starting from $X$ fails to be $f$-equivariant even after iteration. Further study on this fibration reveals that it preserves the *first dynamical degree* $$\delta_f:=\lim\limits_{m\to+\infty}((f^m)^*H\cdot H^{\dim X-1})^{1/m},$$ where $H$ is an ample divisor of $X$ (cf. Definition [Definition 18](#def-d1){reference-type="ref" reference="def-d1"}). Precisely, we show the following: **Theorem 1**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a smooth projective threefold $X$. Then, after iteration of $f$, either one of the following holds:* 1. *any birational MMP of $X$ is $f$-equivariant, or* 2. *there is an $f$-equivariant dominant rational map $\phi:X\dashrightarrow Y$ to a normal projective variety $Y$ with $0<\dim Y<3$ such that the first dynamical degree is preserved: $\delta_f=\delta_{f|_Y}$ (equvalently, $f$ is $\delta$-imprimitive as in Definition [Definition 2](#def_endo_notion){reference-type="ref" reference="def_endo_notion"}).* Now we introduce various notions of endomorphisms for the better exposition. **Definition 2**. Let $f:X\to X$ be a surjective endomorphism of a normal projective variety $X$. 1. $f$ is *non-isomorphic* if $\deg f>1$. 2. $f$ is *imprimitive* if there exists an $f^s$-equivariant dominant rational map $\pi:X\dashrightarrow Y$ for some $s>0$, with $0<\dim Y<\dim X$. 3. $f$ is *primitive* if it is not imprimitive. 4. $f$ is *$\delta$-imprimitive* if there exists an $f^s$-equivariant dominant rational map $\pi:X\dashrightarrow Y$ for some $s>0$, with $\delta_{f^s|_Y}=\delta_{f^s}$ and $0<\dim Y<\dim X$. 5. $f$ is *$\delta$-primitive* if it is not $\delta$-imprimitive. 6. $f$ is *weakly $\delta$-imprimitive* if there exists a finite surjective morphism $\pi:\widetilde{X}\to X$ such that $f$ lifts to a surjective endomorphism $\widetilde{f}:\widetilde{X}\to \widetilde{X}$ and $\widetilde{f}$ is $\delta$-imprimitive. 7. $f$ is *strongly $\delta$-primitive* if it is not weakly $\delta$-imprimitive. 8. $f$ is *strongly imprimitive* if there is an $f^s$-equivariant dominant rational map $\pi:X\dashrightarrow Y$ for some $s>0$, with $\dim Y=1$ (equivalently $\dim Y>0$ or $Y=\mathbb{P}^1$) and $f^s|_Y=\operatorname{id}_Y$. 9. $f$ is *weakly primitive* if it is not strongly imprimitive. 10. $f$ is *quasi-abelian* if it is a surjective endomorphism of a quasi-abelian variety, i.e., there exists a finite surjective morphism $\pi:A\to X$ étale in codimension $1$ such that $f^s$ lifts to a surjective endomorphism $\widetilde{f}:A\to A$ of an abelian variety $A$ for some $s>0$. 11. $f$ is *$q$-polarized* (or simply polarized) if $f^*H\sim qH$ for some $q>1$ and ample Cartier divisor $H$. With the above notion, Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"} says that: starting from a smooth projective threefold, we can run any birational EMMP if $f$ is non-isomorphic and $\delta$-primitive. A subsequent issue is whether a suitable choice of EMMP, especially the Fano contraction, preserves the first dynamical degree. So we further explore how the EMMP behaves. When the canonical divisor $K_X$ is pseudo-effective (equivalently the Kodaira dimension $\kappa(X)\ge 0$ by the three-dimensional minimal model theory, see [@KM98 §3.13]), Fujimoto and Nakayama [@FN07] proved that any EMMP involves only smooth $X_i$ and ends up with a smooth minimal model $X_n$. Furthermore, a full classification is given there and we can conclude that $f$ belongs to one of the three cases: weakly $\delta$-imprimitive, quasi-abelian, and strongly imprimitive. The situation is much more complicated when $K_X$ is not pseudo-effective. After finitely many steps of birational EMMP, we arrive at a Fano contraction $\tau:X_r\to X_{r+1}=:Y$. If $\dim Y=0$, then $f$ is polarized and one can explore more based on previous results. Here, we do not go into details on the literature review but only mention one recent result by the first author [@Men22] that such $X$ is *log Calabi-Yau*, i.e., $(X,\Delta)$ is log canonical for some effective $\mathbb{Q}$-divisor $\Delta$ such that $K_X+\Delta\sim_{\mathbb{Q}} 0$. Interestingly, the log Calabi-Yau and the log canonical pairs play an essential role in our treatment of the case $0<\dim Y<\dim X$. In this way, we can show that $f$ is either weakly primitive, $\delta$-primitive, or polarized after iteration. Now we state the structure theorem. **Theorem 3**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a smooth projective threefold $X$. Then the following hold in terms of the Kodaira dimension $\kappa(X)$.* 1. *If $\kappa(X)<0$, then $f$ is either $\delta$-imprimitive, strongly imprimitive, or polarized after iteration.* 2. *If $\kappa(X)=0$, then $f$ is either weakly $\delta$-imprimitive or quasi-abelian.* 3. *If $\kappa(X)>0$, then $f$ is strongly imprimitive.* Based on Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"}, we propose the following conjecture. **Conjecture 4** (Dream Building Blocks). *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a normal projective variety $X$. Then $f$ is either weakly $\delta$-imprimitive, strongly imprimitive, quasi-abelian, or polarized after iteration.* The Dream Building Blocks conjecture asserts: $$\textbf{``Problems on endomorphisms should be reduced to tractable cases.''}$$ Conjecture [Conjecture 4](#mainconj){reference-type="ref" reference="mainconj"} has been verified for surfaces (even with the possibility "strongly imprimitive" removed) by Theorem [Theorem 27](#thm-dbbc-surf){reference-type="ref" reference="thm-dbbc-surf"}, and smooth projective threefolds by Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"}. In the rest of this section, we discuss several applications. Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$ over $\overline{\mathbb{Q}}$. The *arithmetic degree* is defined as a function $$\alpha_f(x) := \lim\limits_{m\to+\infty} \max\{1, h_H (f^m(x))\}^{1/m},$$ where $h_H$ is a Weil height function associated with an ample divisor $H$ of $X$; see Definition [Definition 21](#def-a-deg){reference-type="ref" reference="def-a-deg"}. The following *Kawaguchi - Silverman Conjecture* [Conjecture 5](#conj-KSC){reference-type="ref" reference="conj-KSC"} (*KSC* for short, see [@KS16b]) asserts that $\alpha_f$ is well-defined, i.e., the limit exists, and the dynamical degree $\delta_f$ equals the arithmetic degree $\alpha_f(x)$ at any point $x$ with Zariski dense $f$-orbit. **Conjecture 5**. *(**Kawaguchi-Silverman Conjecture = KSC**) Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$ over $\overline{\mathbb{Q}}$. Then the following hold.* - *The limit defining arithmetic degree $\alpha_f(x)$ exists for any $x \in X(\overline{\mathbb{Q}})$.* - *If the (forward) orbit $O_f(x) = \{f^n(x)\, |\, n\ge 0\}$ is Zariski dense in $X$, then the arithmetic degree at $x$ is equal to the dynamical degree of $f$, i.e., $\alpha_f(x) = \delta_f$.* The original conjecture is formulated for dominant rational self-maps of smooth projective varieties. In our setting, Conjecture [Conjecture 5](#conj-KSC){reference-type="ref" reference="conj-KSC"} (1) has been proved by Kawaguchi and Silverman themselves (cf. [@KS16a]); more precisely, $\alpha_f(x)$ is either $1$ or the absolute value of an eigenvalue of $f^*|_{\operatorname{NS}(X)}$ for any $x\in X(\overline{\mathbb{Q}})$, see also [@Mat20b]. In particular, $\alpha_f(x)\le \delta_f$. We list the known results, to the best of our knowledge, towards KSC. **Remark 6**. The KSC holds for a surjective endomorphism $f$ on a projective variety $X$ which fits one of the following cases. 1. $f$ is polarized (cf. [@KS14]). 2. $f$ is quasi-abelian (cf. [@Sil17] and [@MZ22 Theorem 2.8]). 3. $X$ is a Mori dream space (cf. [@Mat20a]). 4. $\dim X\le 2$ (for smooth case, see [@MSS18]; for singular case, see [@MZ22]). 5. $X$ is a Hyperkähler manifold (cf. [@LS21]). 6. $X$ is a smooth rationally connected projective variety admitting an int-amplified endomorphism (see [@MZ22] and [@MY22] for respectively threefolds and $n$-folds). 7. $X$ is a projective threefold which admits an int-amplified endomorphism and has at worst $\mathbb{Q}$-factorial terminal singularities (cf. [@MMSZ23 Corollary 6.19]). 8. $X$ is a smooth projective threefold with irregularity $q(X)>0$ and $f$ is an automorphism (cf. [@CLO22]). We also refer to [@CLO22; @JSXZ21; @KS14; @MW22] for KSC in other settings. We briefly explain how Conjecture [Conjecture 4](#mainconj){reference-type="ref" reference="mainconj"} and the automorphic KSC would imply KSC. According to Remark [Remark 6](#rmk-ksc){reference-type="ref" reference="rmk-ksc"}, KSC holds for polarized and quasi-abelian cases. Since KSC itself requires a Zariski dense orbit, we may assume $f$ is weakly primitive. We may also assume $f$ is non-isomorphic if automorphic KSC holds. Conjecture [Conjecture 4](#mainconj){reference-type="ref" reference="mainconj"} then asserts that $f$ is $\delta$-imprimitive. Thus KSC holds by the induction on $\dim X$ and Lemma [Lemma 22](#lem-ksc-iff){reference-type="ref" reference="lem-ksc-iff"}. In particular, applying the structure Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"}, we have the following progress on KSC. **Theorem 7**. *Kawaguchi-Silverman conjecture holds for every non-isomorphic surjective endomorphism of a smooth projective threefold.* A stronger version of KSC, called the small Arithmetic Non-Density ($=$sAND) conjecture, also takes into account the points with no Zariski dense orbit, see [@MMSZ23 Conjecture 1.3] or Conjecture [Conjecture 75](#conj-sand){reference-type="ref" reference="conj-sand"}. The only troubled case is when $f$ is strongly imprimitive. Excluding this, the same strategy of the proof of Theorem [Theorem 7](#mainthm-C){reference-type="ref" reference="mainthm-C"} works: sAND holds for every weakly primitive and non-isomorphic surjective endomorphism of a smooth projective threefold, see Theorem [Theorem 76](#thm-sand){reference-type="ref" reference="thm-sand"}. Another related hard conjecture: Zariski dense orbit (ZDO for short) conjecture asserts that either $f$ is strongly imprimitive or $f$ has a Zariski dense orbit, see Conjecture [Conjecture 78](#conj-zdo){reference-type="ref" reference="conj-zdo"}. We refer to [@JXZ23 Remark 1.13] for a survey of known results. The following theorem gives a different version of Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"} from the aspect of ZDO. In other words, for a non-isomorphic surjective endomorphism of a smooth projective threefold, we can reduce Conjecture ZDO to the case of a $\mathbb{Q}$-factorial terminal projective threefold with no birational contractions, but only (automatically $f$-equivariant) Fano contractions. **Theorem 8**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a smooth projective threefold $X$. Suppose $f$ is weakly primitive and has no Zariski dense orbit. Then $X$ is uniruled and any birational MMP of $X$ is $f$-equivariant after iteration.* **Acknowledgement.** The authors would like to thank Fei Hu, Jia Jia, Chen Jiang, Yohsuke Matsuzawa, Joseph Silverman, Junyi Xie and Guolei Zhong for their valuable discussions and comments. The authors are supported respectively by Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014) and a National Natural Science Fund; and ARF: A-8000020-00-00 of NUS. Many thanks to IMS at NUS, NJU at Nanjing, and ECNU at Shanghai for the warm hospitality and kind support. # Preliminaries **Notation and Terminology 1**. Let $X$ and $Y$ be projective varieties of dimension $n$. Let $f:X\to X$ be a surjective endomorphism and $\pi:X\to Y$ a finite surjective morphism. Two $\mathbb{R}$-Cartier divisors $D_i$ of $X$ are *numerically equivalent*, denoted by $D_1\equiv D_2$, if $(D_1 - D_2)\cdot C=0$ for any curve $C$ on $X$. Two $r$-cycles $C_i$ of $X$ are *weakly numerically equivalent*, denoted by $C_1 \equiv_w C_2$, if $(C_1 - C_2)\cdot L_1 \cdots L_{n - r} = 0$ for all Cartier divisors $L_i$. For $\mathbb{R}$-Cartier divisors, the numerical equivalence coincides with the weak numerical equivalence; see [@MZ18 Section 2]. We use the following notation throughout the paper unless otherwise stated. ---------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------- $\operatorname{Pic}(X)$ the group of Cartier divisors of $X$ modulo linear equivalence $\sim$ $\operatorname{Pic}_{\mathbb{K}}(X)$ $\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{K}$ with $\mathbb{K}=\mathbb{Q}, \mathbb{R},\mathbb{C}$ $\operatorname{Pic}^0(X)$ the neutral connected component of $\operatorname{Pic}(X)$ $\operatorname{Pic}^0_{\mathbb{K}}(X)$ $\operatorname{Pic}^0(X)\otimes_{\mathbb{Z}}\mathbb{K}$ with $\mathbb{K}=\mathbb{Q}, \mathbb{R},\mathbb{C}$ $\operatorname{NS}(X)$ $\operatorname{Pic}(X)/\operatorname{Pic}^0(X)$, the Néron-Severi group $\operatorname{N}^1(X)$ $\operatorname{NS}(X)\otimes_{\mathbb{Z}}\mathbb{R}$, the space of $\mathbb{R}$-Cartier divisors modulo numerical equivalence $\equiv$ $\operatorname{NS}_{\mathbb{K}}(X)$ $\operatorname{NS}(X)\otimes_{\mathbb{Z}}\mathbb{K}$ with $\mathbb{K}=\mathbb{Q}, \mathbb{R},\mathbb{C}$ $\operatorname{N}_r(X)$ the space of $r$-cycles modulo weak numerical equivalence $\equiv_w$ $f^*|_V$ the pullback action on $V$, which is any group or space above $f_*|_V$ the pushforward action on $V$, which is any group or space above $\operatorname{Nef}(X)$ the cone of nef classes in $\operatorname{N}^1(X)$ $\operatorname{Amp}(X)$ the cone of ample classes in $\operatorname{N}^1(X)$ $\overline{\operatorname{NE}}(X)$ the cone of pseudo-effective classes in $\operatorname{N}_1(X)$ $\operatorname{PE}^1(X)$ the cone of pseudo-effective classes in $\operatorname{N}^1(X)$ $\operatorname{Supp}D$ the support of effective divisor $D=\sum a_i D_i$ which is $\bigcup_{a_i >0} D_i$, where the $D_i$ are prime divisors $R_{\pi}$ the ramification divisor of $\pi$ assuming that $X$ and $Y$ are normal $B_{\pi}$ $\pi(\operatorname{Supp}R_f)$ the (reduced) branch divisor of $\pi$ assuming that $X$ and $Y$ are normal $\kappa(X,D)$ Iitaka dimension of a $\mathbb{Q}$-Cartier divisor $D$ $\rho(X)$ Picard number of $X$ which is $\dim_{\mathbb{R}} \operatorname{N}^1(X)$ ---------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------- **Convention 1**. We refer to [@KM98; @BCHM10] for the standard notation and facts for singularities (*terminal, klt, lc*) and divisors involved in the minimal model program (MMP for short). Throughout this article, our minimal model program of $X$ involves no boundary part. By a divisorial contraction, a flipping contraction, a flip, or a Fano contraction, we always mean an operation of some $K_X$-negative extremal ray. We recall the following type of Hodge index theorem which is known to experts, see also [@MZg22 Lemma 4.1]. **Lemma 9**. *(cf. [@DS04 Corollarie 3.2] or [@Zha16 Lemma 3.2]) [\[lem-hodge\]]{#lem-hodge label="lem-hodge"} Let $X$ be a normal projective variety. Let $D_1\not\equiv0$ and $D_2\not\equiv 0$ be two nef $\mathbb{R}$-Cartier divisors such that $D_1\cdot D_2\equiv_w 0$. Then $D_1\equiv aD_2$ for some $a>0$.* The following type of projection formula (cf. [@MZ22 Proposition 3.7]) will be frequently used throughout this paper. **Proposition 10**. *Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$. Then $f_*f^*=f^*f_*=(\deg f)\operatorname{id}$ on $\operatorname{Pic}_{\mathbb{K}}(X)$. In particular, $f^*P\sim_{\mathbb{Q}} aP$ holds for some rational number $a>0$ when $P$ is an $f$-invariant $\mathbb{Q}$-Cartier prime divisor; and Iitaka dimension satisfies $\kappa(X, Q) = \kappa(X, f^*Q) = \kappa(X, f_*Q)$ if $Q$ and $f_*Q$ are $\mathbb{Q}$-Cartier divisors (see [@Uen75 Theorem 5.13], or Lemma [Lemma 31](#lem-pushforward-fiitka){reference-type="ref" reference="lem-pushforward-fiitka"} below).* **Definition 11**. Let $f:X\to X$ be a surjective endomorphism of a variety $X$. A subset $Z\subseteq X$ is said to be *$f$-invariant* (resp. *$f^{-1}$-invariant*) if $f(Z)=Z$ (resp. $f^{-1}(Z)=Z$), and $Z$ is said to be *$f$-periodic* (resp. *$f^{-1}$-periodic*) if $f^s(Z)=Z$ (resp. $f^{-s}(Z)=Z$) for some $s>0$. **Definition 12**. Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$. A prime divisor $E$ is said *$f$-prime* if $f^i(f^j(E))$ is irreducible for any $i, j\in \mathbb{Z}$. Denote by $$E_n:=f^n(E)$$ where $n\in \mathbb{Z}$. Note that an $f$-prime divisor $E$ is $f$-periodic if and only if it is $f^{-1}$-periodic. **Lemma 13**. *Let $f:X\to X$ be a surjective endomorphism of a $\mathbb{Q}$-factorial normal projective variety $X$. Let $E$ be a prime divisor with $\kappa(X,E)=0$. Then $E$ is $f$-prime.* *Proof.* By Proposition [Proposition 10](#prop-f*f*){reference-type="ref" reference="prop-f*f*"}, we have $$f^*f_*=f_*f^*=(\deg f) \operatorname{id}$$ on $\operatorname{Pic}_{\mathbb{Q}}(X)$. Suppose $f(E)=f(E')$ for some other prime divisor $E'$. Then $E\sim_{\mathbb{Q}} aE'$ for some rational number $a>0$ and hence $\kappa(X,E)>0$, a contradiction. Let $E_i:=f^i(E)$. We have $f^{-1}(E_{i+1})=E_i$ and $$\kappa(X,E_{i+1})=\kappa(X, f^*E_{i+1})=\kappa(X, E_i)=0$$ when $i>0$ by [@Uen75 Theorem 5.13]. Similarly, $E_i$ is irreducible when $i<0$. ◻ **Definition 14**. Let $X$ be a projective variety. The *numerical Cartier index* of $X$ is the minimal positive integer $r$ such that $rD\equiv D'$ with $D'$ Cartier for any $\mathbb{Q}$-Cartier Weil divisor $D$. The numerical Cartier index is always finite by the following lemma. The following lemma is suggested by Chen Jiang by using a similar proof of [@MZ22 Lemma 4.1]. **Lemma 15**. *Let $X$ be a projective variety. Then there is a positive integer $r$ such that $rD\equiv D'$ with $D'$ Cartier for any $\mathbb{Q}$-Cartier Weil divisor $D$.* *Proof.* Let $n:=\dim X$. Choose a basis $\{[H_i]\}_{i=1}^m$ of $\operatorname{N}^1(X)$ where $H_i$ is ample and Cartier. Consider the following space $$\operatorname{N}^{n-1}_{\mathbb{K}}(X):=\{\sum_{\text{\rm finite sum}} %^{<\infty} a x_1\cdots x_{n-1}\,|\, a\in\mathbb{K}, x_1,\cdots, x_{n-1} \text{ are Cartier divisors}\}/\equiv_w,$$ see also [@Men20 Section 3]. Note that $\operatorname{N}^{n-1}_{\mathbb{R}}(X)$ can be identified with $\operatorname{N}_1(X)$ by [@Zha16 Lemma 3.2]. Choose a basis $\{[C_j]\}_{j=1}^m$ of $\operatorname{N}^{n-1}_{\mathbb{Z}}(X)$. Then the intersection of $C_j$ with any Weil divisor is an integer. Let $A$ be the intersection matrix $(H_i\cdot C_j)_{i,j}$ which is invertible and integral. Let $D$ be a $\mathbb{Q}$-Cartier Weil divisor. Then there exists $a_j\in \mathbb{Q}$ such that $$D\cdot C_j=\sum_{i=1}^m a_i H_i\cdot C_j\in \mathbb{Z}$$ holds for each $i$ and hence $$(a_1,\cdots, a_m)\in A^{-1}(\mathbb{Z}^m)\subseteq \mathbb{Z}^m/\det(A).$$ We are done by letting $r =\det(A)$. ◻ **Definition 16**. Let $\pi:X\dashrightarrow Y$ be a dominant rational map of projective varieties. $\pi$ is *almost well-defined* (or *almost holomorphic* when $K = \mathbb{C}$) if it is well defined outside a Zariski closed proper subset of $Y$. **Definition 17**. Let $\pi:X\dashrightarrow Y$ be a dominant rational map of normal projective varieties. Let $D=\sum\limits_{i=1}^n a_iD_i$ be a Weil $\mathbb{R}$-divisor of $X$ where $D_i$ are prime divisors. Assume that $D_i$ dominates $Y$ when $i\le r$ and $D_i$ does not dominate $Y$ when $i>r$. Denote by $D^h:=\sum\limits_{i=1}^r a_i D_i$ the *$\pi$-horizontal part* of $D$ and $D^v:=\sum\limits_{i=r+1}^n a_i D_i$ the *$\pi$-verticle part* of $D$. **Definition 18**. (Dynamical degree; $\delta_f$, $\iota_f$) Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$. The (first) *dynamical degree* of $f$ is defined by $$\delta_f=\lim\limits_{m\to+\infty}((f^m)^*H\cdot H^{\dim X-1})^{1/m},$$ where $H$ is any nef and big Cartier divisor of $X$. This limit exists and is independent of the choice of $H$. Since our $f$ is well-defined, $\delta_f$ is the spectral radius of $f^*|_{\operatorname{N}^1(X)}$. Denote by $\iota_f$ the *minimal modulus of eigenvalues* of $f^*|_{\operatorname{N}^1(X)}$. Consider the equivariant dynamical systems of projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X \ar@{-->}[r]^{\pi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\pi$ is a dominant rational map. Define the *relative (first) dynamical degree* of $f$ as $$\delta_{f|_{\pi}}= \lim_{n \to \infty} ((f^n)^*H_X \cdot (\pi^*H_Y)^{\dim Y}\cdot H_X^{\dim X-\dim Y -1})^{1/n}$$ where $H_X$ and $H_Y$ are respectively nef and big Cartier divisors of $X$ and $Y$. The definition is independent of choices of $H_X$ and $H_Y$. Note that $\delta_{f|_\pi}$ is just the degree of $f$ on the general fibre of $\pi$ when $\dim X=\dim Y+1$. We recall the following facts. 1. $\delta_f=\deg f$ if $\dim X=1$. 2. $\delta_{f|_Z}\le \delta_f$ for any $f$-invariant irreducible closed subvariety $Z\subseteq X$. 3. Product formula: $\delta_f=\max\{\delta_{f|_{\pi}}, \delta_g\}$. In particular, $\delta_f=\delta_g$ if $\pi$ is generically finite and dominant. 4. $\delta_{f_1\times f_2}=\max\{\delta_{f_1},\delta_{f_2}\}$. 5. $\delta_{f^n}=\delta_f^n$ for any $n>0$. We refer to [@MMS+22 Notation 2.1] for a more general setting and references. **Lemma 19**. *Consider the equivariant dynamical systems of projective varieties: $$\xymatrix{ \save[]+<3.1pc,0.05pc>*{f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ } \restore &X\ar@{-->}[ld]_\phi\ar@{-->}[rd]^\psi\\ g \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ Y&&Z \ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ h }$$ where $\phi$ and $\psi$ are dominant rational maps. Suppose the induced map $\sigma:X\dashrightarrow Y\times Z$ is generically finite. Then either $\delta_f=\delta_g$ or $\delta_f=\delta_h$.* *Proof.* Note that $\delta_{g\times h}=\max\{\delta_g, \delta_h\}\le \delta_f$ by the dynamical product formula. Let $\overline{X}$ be the closure of the image $\sigma(X)$ in $Y\times Z$ and $\overline{f}:=(g\times h)|_{\overline{X}}$. Then $\delta_{\overline{f}}\le \delta_{g\times h}$. Since $\sigma$ is generically finite, we have $\delta_f=\delta_{\overline{f}}\le \max\{\delta_g, \delta_h\}\le \delta_f$. ◻ **Lemma 20**. *Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$ of dimension $n$. Suppose $f^*D\equiv aD$ for some $D\in \operatorname{N}_{n-1}(X)\backslash \{0\}$. Then $|a| \le \delta_f$.* *Proof.* Let $b$ be the spectral radius of $f^*|_{\operatorname{N}_{n-1}(X)}$, so $b\ge |a|$. Let $V$ be the cone of pseudo-effective cycles in $\operatorname{N}_{n-1}(X)$. Applying the Perron-Frobenius theorem to $f^*|_V$, $f^*D'\equiv bD'$ for some $D'\in V\backslash\{0\}$. Let $H$ be an ample Cartier divisor such that and $H-D'\in V$. Note that $D'\cdot H^{n-1}>0$. Then the lemma follows from the upper bound of $b$ below: $$\delta_f=\lim\limits_{m\to+\infty}((f^m)^*H\cdot H^{n-1})^{1/m}\ge \lim\limits_{m\to+\infty}((f^m)^*D'\cdot H^{n-1})^{1/m}=b.$$ ◻ **Definition 21**. (Weil height function and arithmetic degree) Let $X$ be a normal projective variety defined over $\overline{\mathbb{Q}}$. We refer to [@HS00 Part B], [@KS16b] or [@Mat20a §2.2] for the detailed definition of the *Weil height function* $h_D:X(\overline{\mathbb{Q}})\to \mathbb{R}$ associated with some $\mathbb{R}$-Cartier divisor $D$ on $X$. Here, we simply list some fundamental properties of the height function. - $h_D$ is determined by the divisor $D$ only up to a bounded function. - $h_{\sum a_i D_i}=\sum a_i h_{D_i}+O(1)$ where $O(1)$ means some bounded function. - $h_E$ is bounded below outside $\operatorname{Supp}E$ for any effective Cartier divisor $E$. - Let $\pi:X\to Y$ be a surjective morphism of normal projective varieties and $B$ some $\mathbb{R}$-Cartier divisor of $Y$. Then $h_B(\pi(x))=h_{\pi^*B}(x)+O(1)$ for any $x\in X(\overline{\mathbb{Q}})$. The *arithmetic degree* $\alpha_f(x)$ of $f$ at $x \in X(\overline{\mathbb{Q}})$ is defined as $$\alpha_f(x) = \lim\limits_{m\to+\infty} \max\{1, h_H (f^m(x))\}^{1/m},$$ where $H$ is an ample Cartier divisor. This limit exists and is independent of the choice of $H$ (cf. [@KS16a Theorem 2], [@KS16b Proposition 12]). Moreover, $\alpha_f(x)$ is either $1$ or the absolute value of an eigenvalue of $f^*|_{\operatorname{N}^1(X)}$ (cf. [@KS16a Remark 23]). Note that $\alpha_f(x)\le \delta_f$ and $\alpha_{f^s}(x)=\alpha_f(x)^s$. This allows us to replace $f$ by any positive power whenever needed. We recall the following lemma for the induction purpose of KSC, see [@Mat20a Lemma 5.6] and [@MZ22 Lemma 2.5]. **Lemma 22**. *Consider the equivariant dynamical systems of projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X \ar@{-->}[r]^{\pi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\pi$ is a dominant rational map. Then the following hold.* - *Suppose $\pi$ is generically finite. Then KSC holds for $f$ if and only if it holds for $g$.* - *Suppose $\delta_f=\delta_g$ and KSC holds for $g$. Then KSC holds for $f$.* # Dynamics on surfaces The following Lemma [Lemma 24](#lem-surf-toric){reference-type="ref" reference="lem-surf-toric"} is used in the proof of Lemma [Lemma 68](#lem-fano-toric){reference-type="ref" reference="lem-fano-toric"}. **Definition 23**. Let $f:X\to Y$ be a non-isomorphic finite surjective morphism of projective varieties. We say $f$ is *totally ramified* if $f^{-1}Q$ is prime for any primed divisor $Q\subseteq B_f$. Let $y\in Y$ be a closed point. We say $y$ is an *$f$-unsplitting point* if $f^{-1}(y)$ is a single point. **Lemma 24**. *Let $f:X\to Y$ be a totally ramified, non-isomorphic, finite surjective morphism of del Pezzo surfaces with $\rho(X)=\rho(Y)$. Suppose $(Y, B_f)$ is log Calabi-Yau. Then $(X,\operatorname{Supp}R_f){\stackrel{\mathclap{\sigma}}{\cong}} (Y, B_f)$ are toric pairs and $\operatorname{Sing}(B_f)$ is the set of $f$-unsplitting points.* *Proof.* There are exactly $5$ isomorphic classes of toric del Pezzo surfaces $$\mathbb{P}^1\times \mathbb{P}^1, \mathbb{P}^2, S_j \, (j \le 3)$$ where $S_j$ is the blowup of $\mathbb{P}^2$ at $j$ distinct points (not on the same line when $j = 3$). We remark (for late use) that there is no finite surjective morphism from $\mathbb{P}^1\times \mathbb{P}^1$ to $S_1$, or the other way round, since the latter has a negative curve while the former has none. Suppose we have proved that $(X,\operatorname{Supp}R_f)$ and $(Y, B_f)$ are toric pairs. Then there is an isomorphism $\sigma:X\to Y$ such that $\sigma^{-1}(B_f)=\operatorname{Supp}R_f$ since $\rho(X)=\rho(Y)$. Under the identification of $\sigma$, we can regard $f$ as a surjective endomorphism such that $f^{-1}(B_f)=B_f$ since $f$ is totally ramified. Since $(Y,B_f)$ is log Calabi-Yau and $Y$ is Fano, $B_f$ is ample and $R_f$ is big. Now $f^*B_f-B_f=R_f$ implies that $f$ is int-amplified (cf. [@Men20 Theorem 1.1]). Let $Q$ be an irreducible component $Q$ of $B_f$. After iteration, we may assume $f^{-1}(Q)=Q$. So $f|_Q$ is int-amplified for any irreducible component $Q$ of $B_f$. Note that $B_f$ is a simple normal crossing loop of smooth rational curves. Then $B_{f|_Q}=(B_f-Q)\cap Q$ and hence the set of $f$-unsplitting points is exactly $\operatorname{Sing}(B_f)=\operatorname{Sing}(\operatorname{Supp}R_f)$. Thus it suffices to show $(X,\operatorname{Supp}R_f)$ and $(Y, B_f)$ are toric pairs. We first consider two relatively minimal cases of $Y$. **Case $Y\cong \mathbb{P}^2$.** Note that $X\cong \mathbb{P}^2$ since $\rho(X)=\rho(Y)=1$. Since $f$ is totally ramified, $B_f$ is a union of lines by [@Gur03 Theorem]. Since $(Y, B_f)$ is log Calabi-Yau and hence LC, $B_f$ is a union of three lines that do not intersect at one point. Therefore, $(Y, B_f)$ is a toric pair. Since $f$ is totally ramified, $\operatorname{Supp}R_f$ has three irreducible components. Note that $K_X+\operatorname{Supp}R_f=f^*(K_Y+B_f)$. So $(X, \operatorname{Supp}R_f)$ is log Calabi-Yau (cf. [@KM98 Proposition 5.20]). Hence clearly it is a toric pair. **Case $Y\cong \mathbb{P}^1\times \mathbb{P}^1$.** Since $\rho(X)=\rho(Y)=2$, $X$ is either $\mathbb{P}^1\times \mathbb{P}^1$ or $S_1$. The existence of $f: X \to Y$ implies that $X = \mathbb{P}^1\times \mathbb{P}^1$ as remarked above. Then $f$ splits as two surjective endomorphisms $g, h$ of $\mathbb{P}^1$. Since $(Y, B_f)$ is log Calabi-Yau and hence LC, $B_f= \sum_{j=1}^2 (\mathbb{P}^1\times \{a_j\}+\{b_j\}\times \mathbb{P}^1)$ with $a_1\neq a_2$ and $b_1\neq b_2$. Since $f$ is totally ramified, $g$ and $h$ are totally ramified. Therefore, $(Y, B_f)$ and $(X, \operatorname{Supp}R_f)$ are toric pairs. Next, we consider the case when $Y$ is not relatively minimal and show by induction on $\rho(Y)$. Let $C_X$ be a $(-1)$-curve on $X$ and $C_Y:=f(C_X)$. Note that $f^*C_Y=qC_X$ for some integer $q>0$; so $C_Y^2<0$ and hence $C_Y$ is a $(-1)$-curve on $Y$ since $K_Y\cdot C_Y<0$ (cf. Proposition [Proposition 10](#prop-f*f*){reference-type="ref" reference="prop-f*f*"}). Note that $\deg f>1$ since $B_f$ is non-empty. So $q^2=\deg f>1$. Then $\deg f|_{C_X}=q>1$ and $C_Y\subseteq B_f$. Consider the following commutative diagram $$\xymatrix{ X\ar[r]^f\ar[d]_{\pi_{C_X}} &Y\ar[d]^{\pi_{C_Y}}\\ X'\ar[r]^{f'} &Y' }$$ where $\pi_{C_X}$ and $\pi_{C_Y}$ are the blowdowns of $C_X$ and $C_Y$ respectively. By the rigidity lemma (cf. [@Deb01 Lemma 1.15]), $f'$ is well-defined. Note that $\rho(X')=\rho(Y')=\rho(X)-1$ and $f'$ is finite surjective. Moreover, $f'$ is totally ramified and $B_{f'}=\pi_{C_Y}(B_f)$. Then $K_{Y'}+B_{f'}={\pi_{C_Y}}_*(K_Y+B_f)\equiv 0$ and hence $K_Y+B_f=\pi_{C_Y}^*(K_{Y'}+B_{f'})$. So $(Y', B_{f'})$ is log Calabi-Yau and hence a toric pair by induction. Note that $\operatorname{Sing}(B_{f'})$ is the set of $f'$-unsplitting points which contains $\pi_{C_Y}(C_Y)$. Then $B_{f'}$ has multiplicity $2$ at $\pi_{C_Y}(C_Y)$. So $\pi_{C_Y}(C_Y)$ is a closed toric orbit and $\pi_{C_Y}$ is a toric blowup. In particular, $(Y, B_f)$ is a toric pair. Similarly, $(X, \operatorname{Supp}R_f)$ is a toric pair. ◻ For convenience, we give a proof of the lemma below, which should be well-known. **Lemma 25**. *Let $f:X\to X$ be a surjective endomorphism of a projective surface $X$. Then $\deg f=\delta_f\cdot \iota_f$.* *Proof.* Consider the Jordan canonical form of $f^*|_{\operatorname{NS}_{\mathbb{C}}(X)}$. Let $x_1,\cdots, x_r$ be a basis of $\operatorname{NS}_{\mathbb{C}}(X)$ such that either $f^*x_i=\lambda_ix_i$ or $f^*x_i=\lambda_ix_i+x_{i+1}$. Note that $x_1\cdot x_j\neq 0$ and $x_1\cdot x_{j+1}=0$ for some $j$ (write $x_{r+1}=0$). By the projection formula, $\deg f=\lambda_1\cdot \lambda_j$. If we take $\lambda_1=\delta_f$, then $\deg f\ge \delta_f\cdot \iota_f$. If we take $\lambda_1=\iota_f$, then $\deg f\le \iota_f\cdot \delta_f$. ◻ We shall use the following lemma in the proof of Theorem [Theorem 41](#thm-fiitaka2){reference-type="ref" reference="thm-fiitaka2"}. **Lemma 26**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a normal projective surface $X$. Let $C$ be an $f$-prime curve which is not $f^{-1}$-periodic. Then $X$ is $\mathbb{Q}$-factorial and $\kappa(X,C)>0$.* *Proof.* If $f$ is int-amplified, then any $f$-prime curve $C$ is $f^{-1}$-periodic by [@Men20 Lemma 8.1]. So we may assume $f$ is not int-amplified. By [@JXZ23 Theorem 1.1], $X$ is further $\mathbb{Q}$-factorial. Indeed, if $K_X$ is pseudo-effective, then $X$ has quotient singularities and hence is $\mathbb{Q}$-factorial. If $K_X$ is not pseudo-effective, then $\rho(X)=2$ and $X$ admits a Fano contraction to a curve by [@MZ22 Theorem 5.4]. By [@Nak17 Proposition 2.33], $X$ is $\mathbb{Q}$-factorial. Let $C_i:=f^i(C)$ with $i\in \mathbb{Z}$. Since $C$ is not $f^{-1}$-periodic, $f^*C_{i+1}=C_i$ when $i\gg 1$. By the projection formula, $$C_i\cdot C_j=f^*C_{i+1}\cdot f^*C_{j+1}=(\deg f) C_{i+1}\cdot C_{j+1}.$$ Note that $\deg f>1$. Then we have $$\lim_{i,j\to+\infty} C_i\cdot C_j=0.$$ By Lemma [Lemma 15](#lem-index){reference-type="ref" reference="lem-index"}, we have $$r^2C_i\cdot C_j\in\mathbb{Z}$$ where $r$ is the numerical Cartier index of $X$. So we have $C_i\cdot C_j=0$ for $i,j\gg 1$. By the projection formula again, this holds for any $i,j\in \mathbb{Z}$. Thus, $C_i\cap C_j=\emptyset$ for any $i\neq j$. By [@BPS16 Theorem 1.1], each $C_i$ is semi-ample and hence $\kappa(X,C)>0$. ◻ **Theorem 27**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a normal projective surface $X$. Then $f$ is either $\delta$-imprimitive, quasi-abelian, or polarized after iteration. In particular, Conjecture [Conjecture 4](#mainconj){reference-type="ref" reference="mainconj"} holds for surfaces.* *Proof.* By [@MZ22 Theorems 5.1 and 5.4], we only need to consider the case where $f$ is quasi-étale, and $X$ is a quasi-étale quotient of $E\times T$ with $E$ an elliptic curve and $T$ a curve of genus $\ge 2$. Then $\kappa(X,K_X)=1$ and $K_X$ is semi-ample. By [@NZ09 Theorem A], after iteration, we have the following equivariant dynamical systems: $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X \ar[r]^{\varphi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g=\operatorname{id} }$$ where $\varphi$ is the Iitaka fibration and $\dim Y=1$. Note that $\delta_f=\deg f|_{X_y}=\deg f>1$ and $X$ is non-uniruled. So the general fibre $X_y$ of $\varphi$ is an elliptic curve. Note that $f|_{X_y}$ is not a translation. So the set of fixed points $\text{Fix}(f|_{X_y})$ is non-empty for general $y\in Y$. In particular, $\text{Fix}(f)$ contains an irreducible curve $C$ such that $\varphi|_{C}:C\to Y$ is finite surjective. Note that $f|_C=\operatorname{id}$ and $f^*C\sim_{\mathbb{Q}} (\deg f)C$ by Proposition [Proposition 10](#prop-f*f*){reference-type="ref" reference="prop-f*f*"}, so $C^2 = 0$. If $f^{-1}(C)=C$, then $f^*C=(\deg f)C$ and hence $C\subseteq B_f$, a contradiction since $f$ is quasi-étale. So there exists some irreducible curve $C'\neq C$ such that $f(C')=C$. Now $f^*C\sim_{\mathbb{Q}} (\deg f)C$ and Proposition [Proposition 10](#prop-f*f*){reference-type="ref" reference="prop-f*f*"} imply $f^*C'\sim_{\mathbb{Q}} (\deg f) C'$. Since $f|_{C'} = \operatorname{id}$ too and $$(f|_C)^*(C'|_C)\sim_{\mathbb{Q}} (\deg f)C'|_C ,$$ $C'\cap C=\emptyset$. Hence $C$ is semi-ample, so $\kappa(X,C)=1$ since $C^2 = 0$. Let $\psi:X\to Z$ be the Iitaka fibration of $C$. Then $\psi$ is $f$-equivariant and $\delta_{f|_Z}=\delta_f$ by the dynamical product formula. Thus $f$ is $\delta$-imprimitive. ◻ **Remark 28**. By the proof of Theorem [Theorem 27](#thm-dbbc-surf){reference-type="ref" reference="thm-dbbc-surf"}, we have indeed proved that "strongly $f$-imprimitive" is "$\delta$-imprimitive" for surfaces. # Dynamical Iitaka fibration In this section, we introduce the new concepts: *dynamical Iitaka dimension* and *dynamical Iitaka fibration*. **Definition 29** (Dynamical Iitaka dimension). Let $X$ be a normal projective variety and $D$ a $\mathbb{Q}$-Cartier divisor. Denote by $V_f(D)$ the subspace of $\operatorname{Pic}_{\mathbb{Q}}(X)$ spanned by $D_i:=(f^*)^i(D)$ with $i\in\mathbb{Z}$. By [@MZ22 Proposition 3.7] and noting that $f^*|_{\operatorname{Pic}_{\mathbb{Q}}(X)}$ is invertible, $V_f(D)$ is finite dimensional. We define the *dynamical $f$-Iitaka dimension* of $D$ as $$\kappa_f(X,D):=\max\{\kappa(X,D')\,|\,D'\in V_f(D)\}.$$ If $D_0,\cdots, D_n$ are effective and span $V_f(D)$, then $\kappa_f(X,D)=\kappa(X, \sum\limits_{i=0}^n D_i)$. **Lemma 30**. *Let $f:X\to X$ be a surjective endomorphism of a normal projective variety $X$. Let $D$ be a $\mathbb{Q}$-Cartier effective divisor of $X$ such that $\operatorname{Supp}D$ is not $f^{-1}$-periodic. Then $\kappa_f(X,D)>0$.* *Proof.* Let $D_i:=(f^*)^i(D)$. By [@KS16a Lemma 19], there is a monic integral polynomial $$P_f(t)=\sum_{i=0}^n a_i t^i\in \mathbb{Z}[t]$$ such that $P_f(f^*)$ annihilates $\operatorname{Pic}_{\mathbb{Q}}(X)$. So we have $$A:=\sum_{a_i>0} a_i D_i\sim_{\mathbb{Q}} \sum_{a_i<0} -a_i D_i=:B$$ where $\operatorname{Supp}D_n\subseteq \operatorname{Supp}A$. Since $\operatorname{Supp}D$ is not $f^{-1}$-periodic, we have $\operatorname{Supp}D_n\not\subseteq \bigcup\limits_{i=0}^{n-1} \operatorname{Supp}D_i$. In particular, $A\neq B$ and hence $\kappa_f(X,D)\ge \kappa(X,A)>0$. ◻ **Lemma 31**. *Consider the equivariant dynamical systems of normal projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X \ar@{-->}[r]^{\pi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ Then the following hold.* 1. *If $\pi$ is a surjective morphism, then $\kappa_f(X,\pi^*D)= \kappa_g(Y,D)$ for any $\mathbb{Q}$-Cartier divisor $D$ of $Y$.* 2. *If $\pi$ is a birational map and $Y$ is $\mathbb{Q}$-factorial, then $\kappa_f(X,D)\le \kappa_g(Y,\pi_*D)$ for any $\mathbb{Q}$-Cartier divisor $D$ of $X$.* *Proof.* (1) Note that $V_f(\pi^*D)=\pi^*(V_g(D))$ and $\kappa(X,\pi^*D)= \kappa(Y, D)$ for any $\mathbb{Q}$-Cartier divisor $D$ of $Y$ (cf. [@Uen75 Theorem 5.13]). Then $\kappa_f(X,\pi^*D)=\kappa_g(Y,D)$. \(2\) Let $W$ be the normalization of the graph of $\pi$ and $h:W\to W$ the lifting. Let $p_X:W\to X$ and $p_Y:W\to Y$ be the two projections. Then $\kappa_f(X,D)=\kappa_h(W,p_X^*D)$ by (1). Note that $\pi_*D={p_Y}_*p_X^*D$. So we may assume that $\pi$ is well-defined. We may also assume that $D$ is effective such that $\kappa_f(X,D)=\kappa(X,D)$. Then $n\pi^*\pi_*D=D+E$ with $E$ effective when $n\gg 1$. Then $\kappa_g(Y,\pi_*D)\ge \kappa(X,\pi^*\pi_*D)\ge \kappa(X,D)=\kappa_f(X,D)$. ◻ We recall the Chow reduction as introduced in [@Nak10 Proposition 4.14, Definition 4.15 and Theorem 4.19]; see also [@MZ22 Theorem 7.2] and the statement before it. **Theorem 32**. *Let $\pi:X\dashrightarrow Y$ be a dominant rational map of normal projective varieties defined over an algebraically closed field. Suppose $\pi$ has connected fibres. Then there exist a normal projective variety $Z$ and a birational map $\mu:Y\dashrightarrow Z$ satisfying the following conditions:* 1. *Let $\Gamma_Z\subseteq X\times Z$ be the graph of the composite $\mu\circ \pi: X\dashrightarrow Y\dashrightarrow Z$. Then $\Gamma_Z\to Z$ is equi-dimensional.* 2. *Let $\mu':Y\dashrightarrow Z'$ be a birational map to another normal projective variety $Z'$ such that the induced projection $\Gamma_{Z'}\to Z'$ is equi-dimensional. Then there exists a birational morphism $\nu:Z'\to Z$ with $\mu=\nu\circ\mu'$.* *In particular, for any surjective endomorphism $f:X\to X$ and a dominant self-map $g:Z\dashrightarrow Z$ with $g\circ (\mu\circ \pi)=(\mu\circ \pi)\circ f$, the map $g$ is a surjective endomorphism.* **Definition 33**. We call the composition $\mu\circ \pi:X\dashrightarrow Z$ in Theorem [Theorem 32](#thm-chow){reference-type="ref" reference="thm-chow"} the *Chow reduction* of $\pi$ which is unique by the condition (2). **Theorem 34**. *Let $f:X\to X$ be a surjective endomorphism of a normal projective variety $X$. Let $D$ be a $\mathbb{Q}$-Cartier divisor with $\kappa_f(X,D)\ge 0$. Then there is an $f$-equivariant dominant rational map $\phi_{f,D}:X\dashrightarrow Y$ to a normal projective variety $Y$ (with $f|_Y$ a surjective endomorphism too) satisfying the following conditions.* 1. *$\dim Y=\kappa_f(X,D)$.* 2. *Let $\Gamma$ be the graph of $\phi_{f,D}$. Then the induced projection $\Gamma\to Y$ is equi-dimensional.* 3. *$\phi_{f,D}$ is birational to the Iitaka fibration of any $D'\in V_f(D)$ with $\kappa(X,D')=\kappa_f(X,D)$.* 4. *Suppose further $D$ is an $f$-prime divisor dominating $Y$. Then $D$ is $f^{-1}$-periodic.* *Proof.* Let $D_i\in V_f(D)$ with $\kappa(X,D_i)=\kappa_f(X,D)$ and $i=1,2$. Let $\phi_{D_i}: X\dashrightarrow Y_{D_i}$ be the Iitaka fibration of $D_i$. Note that $$\kappa_f(X, D)\ge \kappa(X, D_1+D_2)\ge \kappa(X, D_1)=\kappa_f(X,D).$$ So we have $$\kappa(X, D_1+D_2)=\kappa(X, D_1)=\kappa(X, D_2).$$ Therefore, $\phi_{D_i}=\sigma_i\circ \phi_{D_1+D_2}$ for some birational maps $\sigma_i:Y_{D_1+D_2}\dashrightarrow Y_{D_i}$ (cf. [@MZ22 Lemma 7.3]). Then we simply take $\phi_{f,D}$ as the Chow reduction of $\phi_{D_1}$ in Theorem [Theorem 32](#thm-chow){reference-type="ref" reference="thm-chow"}. This proves (1), (2), and (3). For (4), suppose the contrary that $D$ is $f$-prime and dominating $Y$ and yet it is not $f^{-1}$-periodic. Let $P_i:=(f^*)^i(D)$. Then $\operatorname{Supp}P_i$ is irreducible and $\operatorname{Supp}P_i\neq \operatorname{Supp}P_j$. Let $n\ge 0$ be the minimal integer such that $P_0,\cdots, P_n$ span $V_f(D)$. Then we have $$D':=P_{n+1}+\sum_{i\in I} a_i P_i\sim_{\mathbb{Q}} \sum_{j\in J} b_j P_j$$ where $a_i,b_j$ are positive rational numbers, $J\neq\emptyset$, and $I \cap J=\emptyset$. Then $D'$ is $\mathbb{Q}$-movable. Note that $\phi_{f,D'}:X\dashrightarrow Y_{D'}$ factors through $\phi_{f,D}$ and $D'$ does not dominate $Y_{D'}$. So we get a contradiction. ◻ **Definition 35**. We call $\phi_{f,D}$ in Theorem [Theorem 34](#thm-fibration){reference-type="ref" reference="thm-fibration"} the *dynamical $f$-Iitaka fibration* ($f$-Iitaka fibration for short) of $D$. It is unique by the uniqueness of the Chow reduction and Theorem [Theorem 34](#thm-fibration){reference-type="ref" reference="thm-fibration"} (3). A special case of the $f$-Iitaka fibration has been studied in [@MZ22 Theorem 7.8]. **Theorem 36**. *Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$. Let $D$ be a $\mathbb{Q}$-Cartier divisor such that $f^*D\equiv qD$ for some $q>1$ and $\kappa(X,D)>0$. Let $\phi:X\dashrightarrow Y$ be the $f$-Iitaka fibration of $D$. Then $f|_Y$ is $q$-polarized.* *Proof.* The same proof of [@MZ22 Theorem 7.8] works after replacing $-K_X$ by $D$. ◻ # Dynamical Iitaka fibration of an $f$-prime divisor In this section, we further characterize the dynamical Iitaka fibration of the exceptional divisor. Theorem [Theorem 41](#thm-fiitaka2){reference-type="ref" reference="thm-fiitaka2"} is the main result. Our first result of the section is on $f$-prime divisors. **Theorem 37**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a $\mathbb{Q}$-factorial normal projective threefold $X$. Let $E$ be an $f$-prime divisor of $X$. Then either $E$ is $f^{-1}$-periodic or $0<\kappa_f(X,E)<3$.* We do some preparation for the proof of Theorem [Theorem 37](#thm-fiitaka1){reference-type="ref" reference="thm-fiitaka1"}. Let $E_i:=f^i(E)$ with $i\in\mathbb{Z}$. We may assume $E$ not $f^{-1}$-periodic. Then $E_i\neq E_j$ for any $i\neq j$. Moreover, $E_n\not\subseteq \operatorname{Supp}R_f$ when $|n|\gg 1$. Here $R_f$ is the ramification divisor of $f$. So we have $$f^*E_{n+1}=E_n$$ when $|n|\gg 1$. Now the story of the non-isomorphic $f$ starts with the following easy but useful lemma. **Lemma 38**. *$E_i\cdot E_j\cdot E_k=0$ for any $i,j,k\in \mathbb{Z}$.* *Proof.* Take $i,j,k\gg 1$, we have $$E_i\cdot E_j\cdot E_k=f^*E_{i+1}\cdot f^*E_{j+1}\cdot f^*E_{k+1}=(\deg f) E_{i+1}\cdot E_{j+1}\cdot E_{k+1}$$ by the projection formula. Note that $\deg f>1$. Then we have $$\lim_{i,j,k\to+\infty}E_i\cdot E_j\cdot E_k=0.$$ By Lemma [Lemma 15](#lem-index){reference-type="ref" reference="lem-index"}, we have $$r^3E_i\cdot E_j\cdot E_k\in \mathbb{Z}$$ where $r$ is the numerical Cartier index of $X$. So we have $$E_i\cdot E_j\cdot E_k=0$$ for $i,j,k\gg 1$. By the projection formula again, this holds for any $i,j,k\in \mathbb{Z}$. ◻ **Lemma 39**. *$E_s|_E$ is a nef $\mathbb{Q}$-Cartier divisor on $E$ when $s>0$.* *Proof.* Let $s>0$. We may assume $E_s\cap E\neq \emptyset$. Write $$E_{n+s}\cdot E_n=\sum a_{n,i} C_{n,i}$$ where $C_{n,i}$ is an irreducible curve and $a_{n,i}$ is a positive rational number. By Lemma [Lemma 15](#lem-index){reference-type="ref" reference="lem-index"}, $r^2a_{n,i}$ is an integer where $r$ is the numerical Cartier index of $X$. Note that $$f^*(E_{n+s}\cdot E_n)=E_{n+s-1}\cdot E_{n-1}$$ when $n\gg 1$. So we may assume that the number of irreducible components of $E_{n+s}\cdot E_n$ is stable when $n\gg 1$. Therefore, we may assume $$f^*C_{n,i}=t_{n,i}C_{n-1,i}$$ for some integer $t_{n,i}>0$ when $n\gg 1$. Then for $n\gg 1$ and any $m>0$, we have $$r^2a_{n,i}C_{n,i}=(f^m)^*r^2a_{n+m,i}C_{n+m,i}=(t_{n+m,i} t_{n+m-1,i}\cdots t_{n+1,i})\cdot (r^2a_{n+m,i}) C_{n,i}$$ and hence $t_{n+m,i}=1$ for $m\gg 1$. In particular, $t_{n,i}=1$ when $n\gg 1$. Then we have $$E_{n+s}\cdot C_{n,i}=(f^m)^*E_{n+m+s}\cdot (f^m)^*C_{n+m,i}=(\deg f)^m E_{n+m+s}\cdot C_{n+m,i}$$ when $n\gg 1$. So we have $$\lim_{m\to+\infty}E_{n+m+s}\cdot C_{n+m,i}=0$$ when $n\gg 1$. Note that $rE_{n+m+s}\cdot C_{n+m,i}$ is an integer for any $n,m$. Then $$E_{n+m+s}\cdot C_{n+m,i}=0$$ for $n\gg 1$ and $m\gg 1$. By the projection formula again, we have $$E_{n+s}\cdot C_{n,i}=0$$ for any $i$ when $n\gg 1$. So $E_{n+s}|_{E_n}$ is nef when $n\gg 1$. Finally, we see that $$E_s|_E=(f^n|_E)^*E_{n+s}|_{E_n}$$ is nef as desired. ◻ *Proof of Theorem [Theorem 37](#thm-fiitaka1){reference-type="ref" reference="thm-fiitaka1"}.* By Lemma [Lemma 30](#lem-fkappa>0){reference-type="ref" reference="lem-fkappa>0"}, $\kappa_f(X,E)>0$. We show in the following that $\kappa_f(X,E)<3$. Suppose the contrary. Then $\sum\limits_{i=1}^k E_i$ is big for some $k>0$. So $$\sum\limits_{i=1}^k E_i\sim_{\mathbb{Q}} A+F$$ where $A$ is an ample $\mathbb{Q}$-divisor and $F$ is an effective $\mathbb{Q}$-divisor. Note that there exists $n>0$ such that $E_{-n}\not\subseteq \operatorname{Supp}F$. Write $$(f^n)_*\sum\limits_{i=1}^k E_i=\sum\limits_{i=n+1}^{n+k} a_iE_i$$ where $a_i$ are positive integers. Then we have $$\sum\limits_{i=n+1}^{n+k} a_iE_i\sim_{\mathbb{Q}} (f^n)_*A+(f^n)_*F$$ where $(f^n)_*A$ is still ample. Note that $E\not\subseteq \operatorname{Supp}(f^n)_*F$. Indeed, if $E\subseteq \operatorname{Supp}(f^n)_*F$, then there is a prime divisor $F_1\subseteq \operatorname{Supp}F$ such that $f^n(F_1)=E$ and hence $F_1\subseteq f^{-n}(E)=E_{-n}$. So $E_{-n} = F_1 \subseteq \operatorname{Supp}F$, a contradiction. In particular, $$(\sum\limits_{i=n+1}^{n+k} a_iE_i)|_E\sim_{\mathbb{Q}} ((f^n)_*A)|_E+((f^n)_*F)|_E$$ is big. By Lemma [Lemma 39](#lem-nef){reference-type="ref" reference="lem-nef"}, $(\sum\limits_{i=n+1}^{n+k} a_iE_i)|_E$ is further nef. So $(\sum\limits_{i=n+1}^{n+k} a_iE_i)^2\cdot E>0$, contradicting Lemma [Lemma 38](#lem-ijk){reference-type="ref" reference="lem-ijk"}. ◻ **Remark 40**. We believe Theorem [Theorem 37](#thm-fiitaka1){reference-type="ref" reference="thm-fiitaka1"} can be generalized to higher dimensions provided that Lemma [Lemma 39](#lem-nef){reference-type="ref" reference="lem-nef"} can be so generalized too. Moreover, with Lemmas [Lemma 38](#lem-ijk){reference-type="ref" reference="lem-ijk"} and [Lemma 39](#lem-nef){reference-type="ref" reference="lem-nef"} holding true, our proof of Theorem [Theorem 37](#thm-fiitaka1){reference-type="ref" reference="thm-fiitaka1"} still works even when $f$ is an automorphism of positive entropy. This provides a new approach to Lesieutre's [@Les18 Theorem 1.7]. **Theorem 41**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a $\mathbb{Q}$-factorial normal projective threefold $X$. Let $E$ be a prime divisor of $X$ which is not $f^{-1}$-periodic and has $\kappa(X,E)=0$. Then the following hold.* 1. *$\kappa_f(X,E)=2$.* 2. *Let $\phi_{f,E}:X\dashrightarrow Y$ be the $f$-Iitaka fibration of $E$. Then $f|_Y$ is an automorphism.* 3. *$\phi_{f,E}$ is almost well-defined (cf. Definition [Definition 16](#def-awd){reference-type="ref" reference="def-awd"}).* *Proof.* By Lemma [Lemma 13](#lem-fprime){reference-type="ref" reference="lem-fprime"}, $E$ is $f$-prime. Let $\phi_{f,E}:X\dashrightarrow Y$ be an $f$-Iitaka fibration of $E$. Then $0<\dim Y=\kappa_f(X,E)<3$ by Theorem [Theorem 37](#thm-fiitaka1){reference-type="ref" reference="thm-fiitaka1"}. Let $E_i:=f^i(E)$. Consider the following equivariant dynamical systems $$\xymatrix{ \save[]+<2pc,0pc>*{h \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ } \restore &W \ar[r]^{p_X}\ar[rd]_{p_Y} & X \ar@{-->}[d]^{\phi_{f,E}} &\save[]+<-2pc,0pc>*{\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ f} \restore\\ && Y&\save[]+<-2pc,0pc>*{\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g} \restore }$$ where $W$ is the normalization of the graph of $\phi_{f,E}$. Let $E_i'$ be the strict transform of $E_i$ in $W$ and $C_i:=p_Y(E_i')$. By Theorem [Theorem 34](#thm-fibration){reference-type="ref" reference="thm-fibration"}, $E_i$ does not dominate $Y$. So $C_i\subsetneq Y$. Suppose the contrary $\dim Y=1$. Note that $p_Y$ has irreducible general fibres. Then $E_i'$ is some full general fibre of $p_Y$ when $i\gg 1$. By [@Uen75 Theorem 5.13], we have $$0=\kappa(X,E)=\kappa(X,(f^i)^*E_i)=\kappa(X,E_i)=\kappa(W, p_X^*E_i)\ge \kappa(W,E_i')=1$$ when $i\gg 1$, a contradiction. So $\dim Y=2$ and (1) is proved. Note that $p_Y$ has connected fibres. So $C_i=g^{-1}(C_{i+1})$ by [@CMZ20 Lemma 7.3]. In particular, $C_i$ is $g$-prime. Note that $p_Y$ is equi-dimensional (cf. Theorem [Theorem 34](#thm-fibration){reference-type="ref" reference="thm-fibration"}) and $p_Y$ has irreducible and reduced general fibres. So $E_i'=p_Y^{-1}(C_i)$ when $i\gg 1$. If $\deg g>1$, then $Y$ is $\mathbb{Q}$-factorial and $\kappa(Y, C_0)>0$ by Lemma [Lemma 26](#lem-surf-deg=1){reference-type="ref" reference="lem-surf-deg=1"}. However, by [@Uen75 Theorem 5.13], $$\kappa(X,E)=\kappa(X, E_i)=\kappa(W,p_X^*E_i)\ge \kappa(W, p_Y^*C_i)=\kappa(Y,C_i)>0$$ when $i\gg 1$, a contradiction. So (2) is proved. **Claim 42**. *Let $P$ be a $p_X$-exceptional prime divisor. Then $\dim p_X(P)=\dim p_Y(P)=1$.* After iteration, we may assume each $p_X$-exceptional prime divisor is $h^{-1}$-invariant. There is an effective exceptional Cartier divisor $F$ such that $-F$ is $p_X$-ample (cf. [@KM98 Lemma 2.62]). Write $$F=\sum_{q=1}^s F_q \text{ and } F_q=\sum_{i=1}^{n_q}a_{q,i} F_{q,i}$$ where $F_{q,i}$ are $p_X$-exceptional prime divisors with $h^*F_{q,i}=q F_{q,i}$ and $a_{q,i}>0$. Denote by $\text{Div}_{\mathbb{Q}}(W)$ the space of Weil $\mathbb{Q}$-divisors without modulo any equivalence. Denote by $\text{CDiv}_{\mathbb{Q}}(W)$ the space of $\mathbb{Q}$-Cartier divisors without modulo any equivalence. We can naturally regard $\text{CDiv}_{\mathbb{Q}}(W)$ as a subspace of $\text{Div}_{\mathbb{Q}}(W)$. Let $V$ be the finite dimensional subspace of $\text{Div}_{\mathbb{Q}}(W)$ spanned by the basis $F_{q,i}$. Then $h^*|_V$ is a diagonal action. Let $U=V\cap \text{CDiv}_{\mathbb{Q}}(W)$. Note that $(h^m)^*F/s^m\in U$ for each $m$ and $$\lim\limits_{m\to+\infty}(h^m)^*F/s^m=F_s.$$ So $F_s\in U$. Since $F\in U$, we have $\sum\limits_{q=1}^{s-1} F_q\in U$. In a similar way, we see $F_q\in U$ for each $q$. There exists some $q$ such that $P\subseteq \operatorname{Supp}F_q$. Suppose $p_X(P)$ is a point. Then $-F|_P$ is ample and $$-F_q|_P=-F|_P + (F-F_q)|_P$$ is big. Moreover, $$\tag{$*$} (h|_P)^*(-F_q|_P)\equiv q(-F_q|_P).$$ By [@MZ18 Proposition 1.1], $(h|_P)^*|_{\operatorname{N}^1(P)}$ admits an ample eigenvector with eigenvalue $q$. Then $\deg h|_P=q^2$ and hence $$1<\deg f=\deg h=q(\deg h|_P)=q^3.$$ So we have $q>1$. Then $h|_P$ and hence $g|_{p_Y(P)}$ are $q$-polarized by [@MZ18 Theorem 3.11]. Since $\dim p_Y(P)>0$, we get the following contradiction to (2): $$\deg g\ge \deg g|_{p_Y(P)}>1.$$ Suppose $p_Y(P)=Y$. Then $\deg h|_P=1$ by (2) and hence $$\deg f=\deg h=q(\deg h|_P)=q>1.$$ We remark that for any eigenvalue $\lambda$ of $(h|_P)^*|_{\operatorname{NS}(P)}$, $1/\lambda$ is an algebriac integer, so $\lambda$ can not be a rational number $>1$. Note that $\dim p_X(P)=1$. Then $-F|_P$ is $p_X|_P$-ample and $-F_q|_P$ is $p_X|_P$-big. In particular, $-F_q|_P\not\equiv 0$ and hence $q>1$ is an integral eigenvalue of $(h|_P)^*|_{\operatorname{N}^1(P)}$ by observing the same ($*$) above. This is absurd as remarked above. Thus the claim and hence (3) are proved since $\dim Y=2$ by (1). ◻ # Dynamical Iitaka fibration of $R_f$ In this section, we characterize the $f$-Iitaka fibration of the ramification divisor $R_f$, where $f:X\to X$ is a surjective endomorphism of a $\mathbb{Q}$-factorial normal projective variety $X$. We shall frequently use the following simple trick. Observe that $$R_{f^s}=\sum\limits_{i=0}^{s-1}(f^i)^*R_f$$ holds for any $s>0$. Then we have $\kappa_f(X,R_f)=\kappa(X, R_{f^s})$ for some $s\gg 1$. **Lemma 43**. *Consider the equivariant dynamical systems $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X \ar@{-->}[r]^{\pi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\pi$ is a dominant rational map of normal projective varieties with connected fibres. Suppose $\deg g=1$. Then $R_f^v=0$ where $R_f^v$ is the $\pi$-vertical part of $R_f$.* *Proof.* We may assume $\pi$ is a morphism after taking the normalization of the graph of $\pi$. For an irreducible closed subset $S\in Y$, we denote by $X_S$ the fibre product $X\times_Y S$. Note that $X$ is integral. Then the general fibre of $\pi$ is reduced since our base field has characteristic $0$. Let $Z$ be the closed subset of $Y$ such that $X_y$ is non-reduced exactly when $y\in Z$. Note that $\dim Z<\dim Y$. For any $y\in Z$, $f^*X_y=X_{g^{-1}(y)}$ is non-reduced and hence $g^{-1}(y)\in Z$. So $g^{-1}(Z)=Z$ and $\pi^{-1}(Z)$ is $f^{-1}$-invariant. Iterating $f$, we may assume that irreducible components of $Z$ and $\pi^{-1}(Z)$ are respectively $g^{-1}$- and $f^{-1}$-invariant. Let $Q\subseteq X$ be a prime divisor such that $f^*Q=rP +$(others) for some prime divisor $P$ and $r>1$. Suppose the contrary that $\pi(Q) \ne Y$. Then $Q \subseteq X_{\pi(Q)}$ implies that $f^*X_{\pi(Q)}=X_{g^{-1}(\pi(Q))}$ is non-reduced, hence $g^{-1}(\pi(Q))\subseteq Z$ and $\pi(Q))\subseteq g(Z) = Z$. Then $\deg g=1$ implies $f^*X_{\pi(Q)}=X_{\pi(Q)}$ and $f^*Q=Q$, a contradiction. ◻ **Lemma 44**. *Let $f:X\to X$ be a surjective endomorphism of a normal projective variety $X$. Suppose $R_f$ is $\mathbb{Q}$-Cartier. Let $\phi_{f,R_f}:X\dashrightarrow Y$ be the $f$-Iitaka fibration of $R_f$. Then either $\dim Y=0$ or $\deg f|_Y>1$.* *Proof.* Suppose $\deg f|_Y=1$. By Lemma [Lemma 43](#lem-reduced){reference-type="ref" reference="lem-reduced"}, even iterating $f$, every irreducible component of $\operatorname{Supp}R_f$ dominates $Y$. By the construction of $f$-Iitaka fibraiton, $\dim Y=0$. ◻ **Corollary 45**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a $\mathbb{Q}$-factorial normal projective threefold $X$. Let $E$ be a prime divisor of $X$ which is not $f^{-1}$-periodic and has $\kappa(X,E)=0$. Suppose $0<\kappa_f(X,R_f)<3$. Then $f$ is $\delta$-imprimitive.* *Proof.* Let $\phi_{f,E}:X\dashrightarrow Y$ be the $f$-Iitaka fibraiton of $E$ and $\phi_{f,R_f}:X\dashrightarrow Z$ the $f$-Iitaka fibraiton of $R_f$. By Theorem [Theorem 41](#thm-fiitaka2){reference-type="ref" reference="thm-fiitaka2"}, $\dim Y=2$ and $\deg f|_Y=1$. By Lemma [Lemma 44](#lem-fiitaka-rf){reference-type="ref" reference="lem-fiitaka-rf"}, $\deg f|_Z>1$. In particular, $\phi_{f,R_f}$ does not factor through $\phi_{f,E}$. Then the induced map $X\dashrightarrow Y\times Z$ is generically finite. By Lemma [Lemma 19](#lem-imprimitive){reference-type="ref" reference="lem-imprimitive"}, $f$ is $\delta$-imprimitive. ◻ Instead of simply considering $\kappa_f(X,R_f)=0$, we deal with a more general situation for the purpose of later treatment on the Fano contractions. **Lemma 46**. *Consider the equivariant dynamical systems $$\xymatrix{ \widetilde{f} \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ \widetilde{X} \ar[r]^{\pi} &X\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ f }$$ with the following settings:* 1. *$\pi$ is a finite surjective morphism of normal projective varieties with $\deg \pi=2$.* 2. *$X$ is smooth and $\delta_f>1$.* 3. *$P$ is a prime divisor with $f^*P=\delta_f P$.* 4. *$\pi^*P=P_1+P_2$ has two irreducible components.* *Then either $\delta_{f|_P}= \delta_f$, or $\kappa(X, P)>0$ and $f$ is $\delta$-imprimitive.* *Proof.* We may assume $\delta_{f|_P}\neq \delta_f=:\delta$. Then $\delta_{f|_P}< \delta_f$. Note that $\delta=\delta_f=\delta_{\widetilde{f}}>1$ is an integer by (3). After iteration, we may assume $\widetilde{f}^*P_i=\delta P_i$ for $i=1,2$. Note that $\delta_{\widetilde{f}|_{P_1}}=\delta_{f|_P}$. Then $\delta$ is not an eigenvalue of $(\widetilde{f}|_{P_1})^*|_{\operatorname{N}^1(P_1)}$. **Claim 47**. *$P_1$ and $P_2$ are $\mathbb{Q}$-Cartier in codimension $2$.* Note that $R_{\pi}=\pi^*B_{\pi}/2$ is a $\mathbb{Q}$-Cartier reduced divisor. Then $K_{\widetilde{X}}=\pi^*K_X+R_{\pi}$ is $\mathbb{Q}$-Cartier. Let $Z$ be the non-lc locus of $\widetilde{X}$ which has codimension at least $2$. Note that $\deg f\ge 2$ since $P\subseteq B_f$. Then $\widetilde{f}^{-1}(Z)=Z$ and each irreducible component of $Z$ is not contained in $P_1\cup P_2$ by [@BH14 Theorem 1.4]. In particular, $Z\cap P_i$ has codimension at least $3$ in $\widetilde{X}$. Let $U:=\widetilde{X}\backslash Z$. Then $U$ is log canonical and further $U$ is klt arround $(P_1\cup P_2)\cap U$ by [@Zha14 Proposition 2.2]. It is known that dlt and hence klt varieties are $\mathbb{Q}$-factorial in codimension $2$ (cf. [@GKKP11 Proposition 9.1]). So the claim is proved. **Claim 48**. *$R_{\pi}\cap P_1=R_{\pi}\cap P_2=\pi^{-1}(B_{\pi}\cap P)\subseteq P_1\cap P_2$.* It is clear that $R_{\pi}\cap P_1\subseteq \pi^{-1}(B_{\pi}\cap P)$. Conversely, for any $x\in B_{\pi}\cap P$, we have $\pi^{-1}(x)\cap P_i\neq \emptyset$ since $\pi|_{P_i}:P_i\to P$ is surjective. Note that $x\in B_{\pi}$ and $\deg \pi=2$. Then $\pi^{-1}(x)$ is a single point in $R_{\pi}$. So we have $\pi^{-1}(x)\subseteq R_{\pi}\cap P_i$ and the claim is proved. We continue the proof of Lemma [Lemma 46](#lem-kappa0){reference-type="ref" reference="lem-kappa0"}. Suppose $B_{\pi}\cap P=\emptyset$. By the purity of branch locus, $\pi$ is étale near $P$. Then $\widetilde{X}$ is smooth near $P_1\cup P_2$. In particular, $P_1$ and $P_2$ are Cartier. Note that $(\widetilde{f}|_{P_i})^*(P_i|_{P_j})=\delta P_i|_{P_j}$. So $P_i|_{P_j}\equiv 0$. Then $P_1$ and $P_2$ (and hence $P$) are nef and hence $P_1\equiv aP_2$ for some rational number $a>0$ by Lemma [\[lem-hodge\]](#lem-hodge){reference-type="ref" reference="lem-hodge"}. Note that $r(P_1-aP_2)\in \operatorname{Pic}^0(X)$ for some integer $r>0$ and $\widetilde{f}^*r(P_1-aP_2)=\delta\cdot r(P_1-aP_2)$. So $P_1-aP_2\sim_{\mathbb{Q}} 0$ by [@MMS+22 Proposition 6.3]. In particular, $\kappa(X, P)=\kappa(\widetilde{X}, P_1+P_2)>0$. Note that $P|_P\equiv 0$. So $0 < \kappa_f(X, P)=\kappa(X, P)<\dim X$. Let $\phi:X\dashrightarrow Y$ be the $f$-Iitaka fibraiton of $P$. Then $f|_Y$ is $\delta$-polarized by Theorem [Theorem 36](#thm-fiitaka-polarized){reference-type="ref" reference="thm-fiitaka-polarized"}. In particular, $f$ is $\delta$-imprimitive. Suppose $B_{\pi}\cap P\neq \emptyset$. By Claim [Claim 48](#claim-p1p2){reference-type="ref" reference="claim-p1p2"} and since $R_{\pi}$ is $\mathbb{Q}$-Cartier, $R_{\pi}\cap P_1$ is a non-empty pure codimension $2$ closed subset in $\widetilde{X}$. Then $P_1\cap P_2$ contains an irreducible component $S$ of codimension $2$ in $\widetilde{X}$. After iteration, we may assume $\widetilde{f}^{-1}(S)=S$. By Lemma [Lemma 20](#lem-d1-weil){reference-type="ref" reference="lem-d1-weil"}, $(\widetilde{f}|_{P_1})^*S=aS$ for some integer $a\le\delta_{\widetilde{f}|_{P_1}}<\delta$. By Claim [Claim 47](#claim-qfactorial){reference-type="ref" reference="claim-qfactorial"}, we can find some open subset $V\subseteq X$ with complement of codimension $3$, such that $P_2$ is $\mathbb{Q}$-Cartier on $V$ and $f^{-1}(V)$. Then $D:=P_2|_{P_1\cap V}$ (resp. $D':=P_2|_{P_1\cap f^{-1}(V)}$) is an effective $\mathbb{Q}$-Cartier divisor with $S\cap V$ (resp. $S\cap f^{-1}(V)$) being a component of $\operatorname{Supp}D$ (resp. $\operatorname{Supp}D'$). Let $g:=\widetilde{f}|_{P_1}$ and $h:=g|_{P_1\cap f^{-1}(V)}:P_1\cap f^{-1}(V)\to P_1\cap V$. We have $h^*D=(f^*P_2)|_{P_1\cap f^{-1}(V)}=\delta D'$. Then $h^*(S\cap V)=\delta (S\cap f^{-1}(V))$ and hence $g^*S=\delta S$, a contradiction. ◻ **Lemma 49**. *Consider the equivariant dynamical systems of normal projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X \ar@{-->}[r]^{\phi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ with the following settings:* 1. *$X$ is smooth with $\dim X=n$ and $\dim Y=n-1>0$.* 2. *$\phi$ is an almost well-defined dominant rational map.* 3. *$f$ is $\delta$-primitive.* 4. *$\kappa_f(X,R_f^h)=0$ where $R_f^h$ is the $\phi$-horizontal part of $R_f$.* *Then $R_f^h=0$ and the general fibre of $\phi$ is an elliptic curve.* *Proof.* Let $X_y$ be a general fibre of $\phi$. Since $\phi$ is almost well-defined, we may assume $X_y$ and $X_{g(y)}$ are smooth projective curves with the same genus. Since $f$ is $\delta$-primitive, $\delta_f>\delta_g \ge 1$. Then, $\dim X_y=1$ and the product formula imply $\deg f|_{X_y} = \delta_f >1$. By the adjunction formula and the ramification divisor formula, we have $$R_{f|_{X_y}}=R_f^h|_{X_y}.$$ If $R_f^h=0$, then $f|_{X_y}$ is étale and hence $X_y$ is an elliptic curve. Suppose the contrary that $R_f^h\neq 0$. Since $\kappa_f(X,R_f^h)=0$, each irreducible component $P_i$ of $P:=\operatorname{Supp}R_f^h$ is $f^{-1}$-invariant after iteration by Lemma [Lemma 30](#lem-fkappa>0){reference-type="ref" reference="lem-fkappa>0"}. Suppose further that $P$ has at least $2$ irreducible components $P_1$ and $P_2$. Write $f^*P_i=a_i P_i$. Then we have $$a_i=\deg f/\deg f|_{P_i}=\deg f/ \deg g=\deg f|_{X_y}=\delta_{f|_{\phi}}=\delta_f,$$ where the second equality is due to $P_i$ dominating $Y$ and the last equality is by the dynamical product formula and $\delta_f>\delta_g$ according to (3). Since $\phi|_{P_i}:P_i\dashrightarrow Y$ is generically finite, $\delta_{f|_{P_i}} = \delta_g<\delta_f$. So $(f|_{P_i})^*P_j|_{P_i}=\delta_f P_j|_{P_i}$ implies $P_j|_{P_i}\equiv 0$. Hence $P_i$ is nef and $P_1\equiv tP_2$ for some rational number $t>0$ by Lemma [\[lem-hodge\]](#lem-hodge){reference-type="ref" reference="lem-hodge"}. Note that $s(P_1-tP_2)\in \operatorname{Pic}^0(X)$ and $f^*s(P_1-tP_2)=\delta_f\cdot s(P_1-tP_2)$. Since $\delta_f>1$, we have $P_1-tP_2\sim_{\mathbb{Q}} 0$ by [@MMS+22 Proposition 6.3], contradicting (4). Thus $P$ is irreducible. Note that $R_{f|_{X_y}}=R_f^h|_{X_y}$. So $f|_{X_y}$ is totally ramified. Hence $X_y\cong \mathbb{P}^1$ and $\deg \phi|_P=2$. We construct an $f$-equivariant double cover of $X$ by the following commutative diagram: $$\xymatrix{ \widetilde{P}\ar[d]_{\widetilde{\phi|_P}}&\widetilde{W}\ar[r]^{p_{\widetilde{X}}}\ar[d]^{p_W}\ar[l]_{p_{\widetilde{P}}}&\widetilde{X}\ar[d]^{\pi}\\ Y&W\ar[l]^{\phi_Y}\ar[r]_{\phi_X}&X }$$ where $W$ is the normalization of the graph of $\phi$ with two projections $\phi_X$ and $\phi_Y$, $\widetilde{\phi|_P}:\widetilde{P}\to Y$ is the normalization of $Y$ in the function field $k(P)$, $\widetilde{W}$ is the main component of the normalization of $W\times_Y \widetilde{P}$, and $\pi:\widetilde{X}\to X$ is obtained by taking the Stein factorization of $\widetilde{W}\to X$. Note that this diagram is $f$-equivariant by [@CMZ20 Lemma 5.2]. Equivalently, $\widetilde{X}$ is the normalization of $X$ in $k(X)\otimes_{k(Y)} k(P)$. Note that $\deg \pi=\deg \widetilde{\phi|_P}=2$. The natural embedding of $k(P)\hookrightarrow k(P)\otimes_{k(Y)}k(P)$ implies that $\pi^*P=\pi^{-1}(P)=P_1+P_2$ has two irreducible components. By Lemma [Lemma 46](#lem-kappa0){reference-type="ref" reference="lem-kappa0"} and since $f$ is $\delta$-primitive, $\delta_f=\delta_{f|_P}=\delta_g$, a contradiction. ◻ **Corollary 50**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a smooth projective threefold $X$. Let $E$ be a prime divisor of $X$ which is not $f^{-1}$-periodic and has $\kappa(X,E)=0$. Suppose $\kappa_f(X,R_f)=0$. Then $f$ is either $\delta$-imprimitive or étale.* *Proof.* Let $\phi:X\dashrightarrow Y$ be the $f$-Iitaka fibration of $E$. By Theorem [Theorem 41](#thm-fiitaka2){reference-type="ref" reference="thm-fiitaka2"}, $\phi$ is almost well-defined, $\dim Y=2$ and $\deg f|_Y=1$. Suppose $f$ is $\delta$-primitive. Then $R_f = 0$ by Lemmas [Lemma 43](#lem-reduced){reference-type="ref" reference="lem-reduced"} and [Lemma 49](#lem-Rf=0){reference-type="ref" reference="lem-Rf=0"}. Hence $f$ is étale by the purity of branch locus. ◻ When $\kappa_f(X,R_f)=\dim X$, or equivalently when $R_f$ is big (after iterating $f$), we propose the following general question. Note that if Question [Question 51](#que-big){reference-type="ref" reference="que-big"} is answered affirmatively, we can apply the int-amplified EMMP theory when $R_f$ is big. **Question 51**. *Let $f:X\to X$ be a surjective endomorphism of a normal projective variety $X$. Suppose $R_f$ is a big Weil divisor. Is $f$ int-amplified?* For the purpose of this paper, it suffices for us to treat the following special cases. **Theorem 52**. *Let $f:X\to X$ be a non-isomorphic surjective endomorphism of a $\mathbb{Q}$-factorial lc projective threefold $X$. Suppose $X$ admits a divisorial contraction $\pi_0:X\to X_0$ of some $K_X$-negative extremal ray with the exceptional (prime) divisor $E$ not being $f^{-1}$-periodic. Then $\kappa_f(X,R_f)<3$.* *Proof.* Suppose the contrary that $\kappa_f(X,R_f)=3$. Then $R_f$ and hence $B_f=f(\operatorname{Supp}R_f)$ are big after iteration. Let $\phi:X\dashrightarrow Y$ be the $f$-Iitaka fibraiton of $E$ and $g:=f|_Y$. Note that $\kappa(X,E)=0$. By Theorem [Theorem 41](#thm-fiitaka2){reference-type="ref" reference="thm-fiitaka2"}, $\dim Y=2$, $\deg g=1$, and $\phi$ is almost well-defined. Let $E_i:=f^i(E)$ and $C_i$ the image of $E_i$ in $Y$ with $i\ge 0$. Note that $C_i$ is an irreducible curve by Theorem [Theorem 34](#thm-fibration){reference-type="ref" reference="thm-fibration"}, and $\phi|_{E_i}:E_i\dashrightarrow C_i$ is almost well-defined when $i\gg 1$. Let $\ell_0$ be an irreducible curve contracted by $\pi_0$. Let $\ell_i:=f^i(\ell_0)$ with $i\ge 0$. Then $E_i$ is covered by the curves in the extremal ray $\mathbb{R}_{\ge 0} [\ell_i]$. Since $R_f|_{E_i}$ is big when $i\gg 1$, there exists some $s>0$ such that $R_f\cdot \ell_i>0$ when $i\ge s$. Let $r$ be the numerical Cartier index of $X$. Then $r(f^j)^*R_f\cdot \ell_s=rR_f\cdot (f^j)_*\ell_s$ is a positive integer for any $j>0$. Therefore, $$\lim\limits_{n\to +\infty} R_{f^n}\cdot \ell_s=\sum_{i=0}^{+\infty}(f^j)^*R_f\cdot \ell_s=+\infty.$$ Take $n\gg 1$. By the ramification divisor formula, we may assume $$(f^n)^*K_X\cdot \ell_s=K_X\cdot \ell_s-R_{f^n}\cdot \ell_s<0.$$ By the projection formula, we have $K_X\cdot \ell_{s+n}<0$. By the cone theorem (cf. [@KM98 Theorem 3.7]), there exists a divisorial contraction $\pi_i:X\to X_i$ of $\ell_i$ with $E_i$ the excetpional divisor when $i\gg 1$. We may replace $E$ by $E_{n}$ with $n\gg 1$ and assume: *$K_X\cdot \ell_i<0$, $f^*E_{i+1} = E_i$ (and hence $\deg f|_{E_i} = \deg f$), $B_f|_{E_i}$ is a big divisor, $E_i\not\subseteq B_f\cup \operatorname{Sing}(X)$, and $\phi|_{E_i}:E_i\dashrightarrow C_i$ is almost well-defined for any $i\ge 0$.* Let $B_i:=\pi_i(E_i)$. If $B_i$ is a point, then $E_i^3 < 0$ by [@KM98 Lemma 2.62], contradicting Lemma [Lemma 38](#lem-ijk){reference-type="ref" reference="lem-ijk"}. So $B_i$ is a curve. By the rigidity lemma (cf. [@Deb01 Lemma 1.15]), we have the following commutative diagram $$\xymatrix{ X\ar[r]^{\pi_i}\ar[d]_f &X_i\ar[d]^{f_i}\\ X\ar[r]^{\pi_{i+1}} &X_{i+1} }$$ where $f_i$ is finite surjective with $f_i^{-1}(B_{i+1})=B_i$. Note that $-E_i$ is $\pi_i$-ample. So $E_i\cdot \ell_i<0$. On the other hand, $E_i$ is disjoint with the general fibre of $\phi$ and $\phi|_{E_i}:E_i\dashrightarrow C_i$ is almost well-defined when $i\gg 1$. Then the induced map $\sigma_i:E_i\dashrightarrow B_i\times C_i$ is generically finite. Consider the following commutative diagram $$\xymatrix{ E_i\ar@{-->}[rr]^{\sigma_i}\ar[d]_{f|_{E_i}} && B_i\times C_i\ar[d]^{f_i|_{B_i}\times g|_{C_i}}\\ E_{i+1}\ar@{-->}[rr]_{\sigma_{i+1}} && B_{i+1}\times C_{i+1} }$$ where $\deg f|_{E_i}=\deg f$ and $\deg (f_i|_{B_i}\times g|_{C_i})=\deg f_i|_{B_i}\le \deg f$. So $\deg \sigma_i\ge \deg \sigma_{i+1}$. We may further assume $\deg \sigma_i=\sigma_{i+1}$ for any $i\ge 0$ after replacing $E$ by $E_n$ with $n\gg 1$. Now $\deg g|_{C_0}=1$ implies $\deg f_0|_{B_0}=\deg f|_{E_0}=\deg f$ by the above diagram. Let $v\in B_1$ be a general point such that $\sharp f_0^{-1}(v)=\deg f$. This is possible because the base field has characteristic $0$. Note that $$f^{-1}(\pi_1^{-1}(v))=\bigcup_{u\in f_0^{-1}(v)}\pi_0^{-1}(u)$$ is a disjoint union. Then $\sharp f^{-1}(x)=\deg f$ for any $x\in \pi_1^{-1}(v)$. So $\pi_1^{-1}(v)\cap B_f=\emptyset$ and hence $B_f\cdot \ell_1=0$, contradicting $B_f|_{E_1}$ being a big divisor. ◻ **Theorem 53**. *Consider the following equivariant dynamical systems of $\mathbb{Q}$-factorial lc projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar[r]^{\tau} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\dim Y=1$, and $\tau$ is a Fano contraction of some $K_X$-negative extremal ray. Suppose $R_f$ is big. Then $f$ is int-amplified and $\deg g>1$.* *Proof.* Let $F$ be a general fibre of $\tau$ and $D$ another pseudo-effective divisor extremal in $\operatorname{PE}^1(X)$. After iteration, we may write $f^*F\equiv aF$ and $f^*D\equiv bD$ where $a$ and $b$ are positive integers. Write $K_X\equiv sF+tD$. Then the bigness of $$R_f=K_X-f^*K_X\equiv s(1-a)F+t(1-b)D$$ implies $a \ne 1 \ne b$, i.e., $a>1$ and $b>1$. By [@Men20 Theorem 1.1], $f$ is int-amplified. ◻ **Theorem 54**. *Consider the equivariant dynamical systems of $\mathbb{Q}$-factorial lc projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar[r]^{\tau} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ satisfying the following* 1. *$\dim X=n$ and $\dim Y=m>0$,* 2. *$\tau$ is a Fano contraction of some $K_X$-negative extremal ray,* 3. *$\tau$ is the only $K_X$-negative extremal contraction, and* 4. *$\delta_f>\delta_g$.* *Then $R_f$ is not big.* *Proof.* By Perron-Frobenius theorem, $f^*D \equiv \delta_f D$ for some nonzero nef $\mathbb{R}$-Cartier divisor $D$. If $D$ is $\tau$-trivial, then the cone theorem [@KM98 Theorem 3.7] implies that $D$ is pulled back from $Y$, contradicting that $\delta_g < \delta_f$. Thus $D$ is $\tau$-ample. Hence $D^{n-m}\not\equiv_w 0$. Since $\operatorname{N}^1(X)/\pi^*\operatorname{N}^1(Y)$ is 1-dimensional and $f^*|_{\operatorname{N}^1(X)/\pi^*\operatorname{N}^1(Y)} = q \operatorname{id}$ for some integer $q>0$, $\delta_f=q$. We may then choose $D$ to be Cartier, see also the second paragraph of the proof of [@MZ22 Proposition 9.2]. Since $f$ is not polarized (or else $\delta_f = \delta_g$, absurd), $D$ is not big (cf. [@MZ18 Proposition 1.1]). So $D^n=0$. Let $t$ be the integer such that $D^t\not\equiv_w 0$ and $D^{t+1}\equiv_w 0$. Fix an ample Cartier divisor $H$ of $X$. Let $a$ be a positive rational number such that $K_X+aD$ is $\tau$-trivial. Since $D$ is nef, any $(K_X+aD)$-negative extremal ray is also $K_X$-negative. Then $K_X+aD$ is nef by the assumption (3). Note that $$0\le (f^s)^*(K_X+aD)\cdot D^t\cdot H^{n-t-1}=(K_X-R_{f^s})\cdot D^t\cdot H^{n-t-1}$$ for any $s>0$. Suppose the contrary that $R_f$ is big. Then $R_f=(1/m)H+E$ for some effective $\mathbb{Q}$-Cartier divisor $E$ and $m>0$. Note that $$R_{f^s}\cdot D^t\cdot H^{n-t-1}\ge (1/m)\sum_{i=0}^{s-1} (f^i)^*H\cdot D^t\cdot H^{n-t-1}\ge s/m.$$ Then $(K_X-R_{f^s})\cdot D^t\cdot H^{n-t-1}<0$ when $s\gg 1$, contradicting the early inequality. ◻ **Corollary 55**. *Consider the equivariant dynamical systems of normal projective varieties $$\xymatrix{ \save[]+<2pc,0pc>*{f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ } \restore &X \ar@{-->}[r]^{\sigma} & X' \ar[d]^{\tau} &\save[]+<-1.93pc,0pc>*{\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ f'} \restore\\ && Y&\save[]+<-2pc,0pc>*{\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g} \restore }$$ satisfying the following* 1. *$X$ is smooth of dimension $n$, $\sigma$ is birational and $\sigma^{-1}$ contracts no divisors,* 2. *$X'$ is $\mathbb{Q}$-factorial lc and $\dim Y=n-1>0$,* 3. *$\tau$ is a Fano contraction, and* 4. *$\tau$ is the only $K_X$-negative extremal contraction.* *Then $f$ is $\delta$-imprimitive.* *Proof.* Suppose the contrary that $f$ is $\delta$-primitive. Then $f'$ is $\delta$-primitive too, and $\delta_{f'}>\delta_g$. Let $R_{f'}^h$ be the $\tau$-horizontal part of $R_f$. If $\kappa_{f'}(X, R_{f'}^h)=n$, then $R_{f'}^h$ and hence $R_{f'}$ are big after iteration, a contradiction to Theorem [Theorem 54](#thm-rf-big){reference-type="ref" reference="thm-rf-big"}. Let $\phi:X'\dashrightarrow Z$ be the $f'$-Iitaka fibration of $R_{f'}^h$. If $\dim Z=\kappa_{f'}(X, R_{f'}^h)>0$, then $\phi$ does not factor through $\tau$ and hence the induced map $X'\dashrightarrow Y\times Z$ is generically finite by noting that $\dim X'=\dim Y+1$. By Lemma [Lemma 19](#lem-imprimitive){reference-type="ref" reference="lem-imprimitive"}, $f'$ is $\delta$-imprimitive, a contradiction. So $\kappa_{f'}(X, R_{f'}^h)=0$. Let $R_f^h$ be the $\tau\circ\sigma$-horizontal part of $R_f$. Since $\sigma^{-1}$ contracts no divisors, we have $R_{f'}=\sigma_*R_f$. So $\kappa_f(X,R_f^h)=0$ by Lemma [Lemma 31](#lem-pushforward-fiitka){reference-type="ref" reference="lem-pushforward-fiitka"}. Since $\dim X=\dim Y+1$ and $\sigma^{-1}$ contracts no divisors, $\tau\circ\sigma$ is almost well-defined. By Lemma [Lemma 49](#lem-Rf=0){reference-type="ref" reference="lem-Rf=0"}, the general fibre of $\tau\circ\sigma$ (isomorphic to the general fibre of $\tau$) is an elliptic curve, contradicting that $\tau$ is a Fano contraction. ◻ # Equivariant flips In this section, we prove the equivariancy for flips in the birational MMP starting from a smooth projective threefold $X$. Theorem [Theorem 63](#thm-emmp-flip){reference-type="ref" reference="thm-emmp-flip"} is our main result. As usual, our MMP of $X$ has no boundary part and involves only $K_X$-negative extremal rays. We recall the following useful result from the proof of Claim [Claim 42](#claim-pxp){reference-type="ref" reference="claim-pxp"}. **Lemma 56**. *Consider the equivariant dynamical systems of normal projective varieties $$\xymatrix{ h \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ W \ar[r]^{\sigma} &X\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ f }$$ where $Y$ is $\mathbb{Q}$-factorial and $\sigma$ is birational. Suppose each exceptional prime divisor of $\sigma$ is $h^{-1}$-invariant. Then there is an effective Cartier $\sigma$-exceptional divisor $E=\sum\limits_q E_q$ with $\operatorname{Supp}E=\operatorname{Exc}(\sigma)$ such that $-E$ is $\sigma$-ample, $E_q$ is Cartier, and $h^*E_q=qE_q$. Moreover, $-E_q|_P$ is $\sigma|_P$-big for any prime divisor $P\subseteq \operatorname{Supp}E_q$. In particular, $h|_P$ is $q$-polarized when $\sigma(P)$ is a point.* **Definition 57**. Let $f:X\to X$ be a surjective endomorphism of a projective variety. We say $f$ is *$q$-amplified* if all the eigenvalues of $f^*|_{\operatorname{N}^1(X)}$ have the same modulos $q>1$. If $f$ is $q$-amplified (resp. $q$-polarized), then $f$ is int-amplified (resp. $q$-amplified). Like $q$-polarized and int-amplified endomorphisms, being $q$-amplified is preserved under generically finite lifting and dominant descending. **Lemma 58**. *Let $f:S\to S$ be a surjective endomorphism of a projective surface $S$ with $\deg f=q^2$ for some $q>1$ Let $D\in \operatorname{N}^1(S)\backslash\{0\}$ such that $D$ is pseudo-effective and $f^*D\equiv qD$. Then $f$ is $q$-amplified.* *Proof.* We may assume $S$ is normal. Suppose the contrary that $f$ is not $q$-amplified. Then $\delta_f>\iota_f$ (cf. Definition [Definition 18](#def-d1){reference-type="ref" reference="def-d1"}). By Lemma [Lemma 25](#lem-surf-deg){reference-type="ref" reference="lem-surf-deg"}, $q^2=\deg f=\delta_f\cdot \iota_f$. So $\delta_f > q > \iota_f$. Suppose $K_{S}$ is not pseudo-effective. By [@MZ22 Theorem 5.4], we may assume $\rho(S)=2$. After iteration, $f^*|_{\operatorname{N}^1(S)}$ is a diagonal action with at most two different eigenvalues, a contradiction. Suppose $K_{S}$ is pseudo-effective. Then $f$ is quasi-étale, and $S$ is a quasi-étale quotient of either an abelian surface or a product of an elliptic curve $E$ and a smooth projective curve $T$ of genus $\ge 2$, see [@MZ22 Theorem 5.1]. Suppose $S$ admits a negative curve $C$. Then the latter case occurs: $S\cong E\times T$. Note that $f^{-1}(C)=C$ after iteration (cf. [@MZ22 Lemma 4.3]) and hence $f^*C=qC$ with $q=\sqrt{\deg f}>1$ by the projection formula. However, $C\subseteq B_f$ and $f$ is not quasi-étale, a contradiction. In particular, $\operatorname{Nef}(S)=\operatorname{PE}^1(S)$. Then $D$ is further nef. Write $f^*D_+\equiv \delta_f D_+$ for some $D_+\in \operatorname{Nef}(S)\backslash\{0\}$. We have $D\cdot D_+\neq 0$ by [@MZg22 Lemma 4.1]. By the projection formula, $\deg f=\delta_f\cdot q$, a contradiction. ◻ **Lemma 59** (Connecting Lemma). *Consider the equivariant dynamical systems of normal projective threefolds $$\xymatrix{ h \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ W \ar[r]^{\sigma} &X\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ f }$$ where $X$ is klt and $\sigma$ is birational. Let $Q$ be an $h^{-1}$-invariant prime divisor of $W$ such that $h|_Q$ is $q$-amplified with $q^3=\deg h\ge 2$. Then $h|_P$ is $q$-amplified and $h^*P=qP$ for any $h^{-1}$-invarint prime divisor $P$ with $P\cap Q\neq\emptyset$.* *Proof.* After iteration, we may assume each exceptional prime divisor of $\sigma$ is $h^{-1}$-invariant. Set $q:=(\deg h)^{1/3}$. By the assumption and Lemma [Lemma 25](#lem-surf-deg){reference-type="ref" reference="lem-surf-deg"}, $\deg h|_Q=q^2$ and $h^*Q=qQ$. Write $h^*P=rP$ for some integer $r>0$. Note that $\deg h|_P=q^2$ if $r=q$. **Case $\dim \sigma(Q)=0$ and $\dim \sigma(P)=2$**. Note that $$\deg f|_{\sigma(P)}=\deg h|_P=\frac{\deg h}{r}.$$ So we have $$f^*\sigma(P)=r\sigma(P)$$ and hence $$h^*(\sigma^*\sigma(P))= r\sigma^*\sigma(P).$$ Note that $Q\subseteq \operatorname{Supp}\sigma^*\sigma(P)$. So $r=q$ and $\deg h|_P=q^2$. Since $Q$ is $\sigma$-exceptional, $Q\subseteq \operatorname{Supp}E_q$ by Lemma [Lemma 56](#lem-eq-cartier){reference-type="ref" reference="lem-eq-cartier"}. Then $E_q|_P$ is a non-zero effective Cartier divisor and $$(h|_P)^*E_q|_P\sim qE_q|_P.$$ By Lemma [Lemma 58](#lem-qamplified){reference-type="ref" reference="lem-qamplified"}, $h|_P$ is $q$-amplified. **Case $\dim \sigma(Q)=1$ and $\dim \sigma(P)=2$**. If $\sigma(Q)\subseteq\sigma(P)$, then we are done for the same reason as in the previous case. Suppose $\sigma(Q)\not\subseteq\sigma(P)$. Then $\sigma(P)|_{\sigma(Q)}$ is a non-zero effective $\mathbb{Q}$-Cartier divisor and $$(f|_{\sigma(Q)})^*\sigma(P)|_{\sigma(Q)}=r\sigma(P)|_{\sigma(Q)}.$$ Note that $h|_Q$ and hence $f|_{\sigma(Q)}$ are $q$-amplified. So $r=q$. Since $Q\subseteq \operatorname{Supp}E_q$, we have $$(h|_P)^*E_q|_P\sim qE_q|_P$$ where $E_q|_P$ is a non-zero effective Cartier divisor. By Lemma [Lemma 58](#lem-qamplified){reference-type="ref" reference="lem-qamplified"}, $h|_P$ is $q$-amplified . **Case $\dim \sigma(Q)=2$ and $\dim \sigma(P)=2$**. If $\sigma(Q)=\sigma(P)$, then $P=Q$ and we are done. Suppose $\sigma(Q)\neq \sigma(P)$. By looking at $\sigma(P)|_{\sigma(Q)}$, we have $r=q$ by a similar reason. Note that $\sigma(Q)|_{\sigma(P)}$ is a non-zero effective $\mathbb{Q}$-Catier divisor and $$(f|_{\sigma(P)})^*\sigma(Q)|_{\sigma(P)}=q\sigma(Q)|_{\sigma(P)}.$$ By Lemma [Lemma 58](#lem-qamplified){reference-type="ref" reference="lem-qamplified"}, $f|_{\sigma(P)}$ and hence $h|_P$ are $q$-amplified. **Case $\dim \sigma(P)\le 1$**. By Lemma [Lemma 56](#lem-eq-cartier){reference-type="ref" reference="lem-eq-cartier"}, $P\subseteq \operatorname{Supp}E_r$. If $Q\subseteq \operatorname{Supp}E_r$, then $r=q$. If $Q\not\subseteq \operatorname{Supp}E_r$, then $E_r|_Q$ is a non-zero effective Cartier divisor and $$(h|_Q)^*E_r|_Q\sim rE_r|_Q.$$ So $r=q$ all the time. Note that $$(h|_P)^*E_q|_P\sim qE_q|_P$$ where $E_q|_P$ is $\sigma|_P$-big by Lemma [Lemma 56](#lem-eq-cartier){reference-type="ref" reference="lem-eq-cartier"}. If $\dim \sigma(P)=0$, then $-E_q|_P$ is big. So $h|_P$ is $q$-polarized (cf. [@MZ18 Proposition 1.1]) and hence $q$-amplified. If $\dim \sigma(P)=1$, then $-E_q|_P\cdot \ell>0$ where $\ell$ is a general fibre of $\sigma|_P$. Note that $f|_{\sigma(P)}$ is $a$-polarized with $a=\deg f|_{\sigma(P)}$. Then $(f|_{\sigma(P)})^*\ell\equiv a\ell$ and $\deg h|_P=q\cdot a=q^2$ by the projection formula. So $a=q$. Note that $\ell$ is effective on $P$. By Lemma [Lemma 58](#lem-qamplified){reference-type="ref" reference="lem-qamplified"}, $h|_P$ is $q$-amplified. ◻ Lemma [Lemma 62](#lem-connected){reference-type="ref" reference="lem-connected"} tells the influence of earlier exceptional divisors on the later singular locus during the birational MMP starting from a smooth projective threefold. We begin with: **Lemma 60**. *Let $X$ be a $\mathbb{Q}$-Gorenstein normal projective threefold with canonical singularities. Let $\pi:X\to Y$ be the flipping contraction of some $K_X$-negative curve $C$. Then $C\cap \operatorname{Sing}(X)\neq \emptyset$.* *Proof.* If $C\cap \operatorname{Sing}(X)= \emptyset$, then $K_X\cdot C\le -1$. However, this contradicts [@Ben85 Theorem 0]. ◻ The lemma below is essentially due to [@Mor82]. We give a proof for the convenience. **Lemma 61**. *Let $X$ be a $\mathbb{Q}$-Gorenstein normal projective threefold and $\pi:X\to Y$ a divisorial contraction. Suppose the $\pi$-exceptional prime divisor $D$ is contracted to a curve $S$ in $Y$. Then $\pi^{-1}(y)\cong \mathbb{P}^1$ and $y\not\in \operatorname{Sing}(Y)$ for any $y\in S$ with $\pi^{-1}(y)\cap \operatorname{Sing}(X)=\emptyset$.* *Proof.* Let $C=\pi^{-1}(y)$ with $y\in S$ and $C\cap \operatorname{Sing}(X)=\emptyset$. Denote by $\mathcal{I}$ the ideal sheaf of $C$ in $X$. Let $C_n$ be the closed subscheme of $X$ defined by $\mathcal{I}^n$. By the same argument of [@Mor82 Lemmas 3.20], we have $\chi(\mathcal{O}_{C'})\ge 0$ for any closed subscheme $C'$ of $X$ with $(C')_{\mathrm{red}}=C$. In this step, we do not need $C\cap \operatorname{Sing}(X)=\emptyset$. Applying the same argument of [@Mor82 (3.38)], we further have $C\cong \mathbb{P}^1$ by noticing that $$\chi(\Omega_X^1\otimes\mathcal{O}_C)=K_X\cdot C<0$$ still holds. By the same discussion as in [@Mor82 (3.39)], we have $$\mathcal{I}/\mathcal{I}^2\cong \mathcal{O}_C\oplus\mathcal{O}_C(1).$$ It is easy to compute that $$\mathcal{I}^n/\mathcal{I}^{n+1}\cong \operatorname{Sym}^n(\mathcal{I}/\mathcal{I}^2)\cong \bigoplus_{i=0}^{n}\mathcal{O}_C(i)$$ and hence $$h^0(C_n,\mathcal{I}^n/\mathcal{I}^{n+1})=\frac{(n+1)(n+2)}{2}$$ Using the exact sequence $$0\to \mathcal{I}^n/\mathcal{I}^{n+1}\to \mathcal{O}_{C_n}\to \mathcal{O}_{C_{n+1}}\to 0,$$ we have $$H^0(C_n,\mathcal{O}_{C_n})\cong K[x,y,z]/(x,y,z)^n$$ by induction, where $K$ is the base field. Let $X_n$ be the closed subscheme of $X$ defined by $m_y^n\cdot\mathcal{O}_X$ where $m_y$ is the maximal ideal of $y$. Let $\hat{\mathcal{O}}_y$ be the completion of the stalk $\mathcal{O}_y$. By the theorem of formal functions, $$\hat{\mathcal{O}}_y\cong\lim_{\longleftarrow} H^0(X_n,\mathcal{O}_{X_n}).$$ Note that the sequence of ideals $m_y^n\cdot\mathcal{O}_X$ is cofinal with the sequence of ideals $\mathcal{I}^n$. So $$\hat{\mathcal{O}}_y\cong K[[x,y,z]].$$ Therefore, $Y$ is smooth at $y$. ◻ **Lemma 62**. *Consider the commutative diagram $$\xymatrix{ W_1\ar@{=}[d]^{p_1}& W_2\ar[l]_{\sigma_2}\ar[d]^{p_2} &\cdots\ar[l]_{\sigma_3} &W_{n-1}\ar[l]_{\sigma_{n-1}}\ar[d]^{p_{n-1}} &W_n\ar[l]_{\sigma_n}\ar[d]^{p_n}\\ X_1\ar[r]_{\pi_1}& X_2\ar@{-->}[r]_{\pi_2} &\cdots\ar@{-->}[r]_{\pi_{n-2}} &X_{n-1}\ar@{-->}[r]_{\pi_{n-1}} &X_n }$$ where $X_1$ is a smooth projective threefold, $\pi_i$ is either a divisorial contraction or a flip, $W_i$ is the normalization of the graph of $\pi_{i-1}\circ p_{i-1}:W_{i-1}\dashrightarrow X_i$. Let $x_n\in \operatorname{Sing}(X_n)$. Then there exists a connected reduced divisor $E\subseteq \operatorname{Exc}(p_n)$ such that $p_n^{-1}(x_n)\subseteq E$ and $E_1$ is contracted to a point in some $X_i$ for some prime divisor $E_1\subseteq E$.* *Proof.* We show by induction on $n$. Note that $\pi_1$ is a divisorial contraction by [@Mor82 Theorem 3.3]. When $n=2$, if $X_2$ is singular, we simply take $E=\operatorname{Exc}(\pi_1)$ by noting that $\operatorname{Sing}(X_2)=\pi_1(\operatorname{Exc}(\pi_1))$ is a point. In general, if we find some connected reduced divisor $F\subseteq\operatorname{Exc}(p_{n-1})$, then $(p_{n-1}\circ\sigma_n)^{-1}(p_{n-1}(F))$ is connected since $p_{n-1}\circ\sigma_n$ has connected fibres. Note that $\pi_{n-1}^{-1}$ contracts no divisors. So $(p_{n-1}\circ\sigma_n)^{-1}(p_{n-1}(F))\subseteq \operatorname{Exc}(p_n)$. Then we can take $E_F$ as the connected component of $\operatorname{Exc}(p_n)$ containing $(p_{n-1}\circ\sigma_n)^{-1}(p_{n-1}(F))$. Note that $p_n^{-1}(p_n(E_F))$ is connected. So $E_F= p_n^{-1}(p_n(E_F))$ and hence $p_n^{-1}(x_n)\subseteq E_F$ if and only if $x_n\in p_n(E_F)$. **Case: $\pi_{n-1}$ is isomorphic at $x_n$.** Let $x_{n-1}=\pi_{n-1}^{-1}(x_n)$. Then $x_{n-1}\in\operatorname{Sing}(X_{n-1})$. By induction, there exists a connected reduced divisor $F\subseteq \operatorname{Exc}(p_{n-1})$ such that $p_{n-1}^{-1}(x_{n-1})\subseteq F$ and $F_1$ is contracted to a point in some $X_i$ with $i\le n-1$ for some prime divisor $F_1\subseteq F$. Then we can take $E=E_F$. **Case: $\pi_{n-1}$ is not isomorphic at $x_n$ and $\pi_{n-1}$ is a divisorial contraction.** Let $P$ be the exceptional divisor of $\pi_{n-1}$. If $\pi_{n-1}(P)$ is a point, then $x_n=\pi_{n-1}(P)$ and we are done by simply taking $E$ as the connected component of $\operatorname{Exc}(p_n)$ containing $P$. So we may assume $\pi_{n-1}(P)=C$ is a curve. Note that $x_n\in C\cap\operatorname{Sing}(X_n)$. By Lemma [Lemma 61](#lem-mori){reference-type="ref" reference="lem-mori"}, we can find some $x_{n-1}\in \pi_{n-1}^{-1}(x_n)\cap \operatorname{Sing}(X_{n-1})$. By the same argument as in the first case, we can find the desired $F$. Note that $x_{n-1}\in P\cap p_{n-1}(F)$ and $(p_{n-1}\circ\sigma_n)^{-1}(P)\subseteq \operatorname{Exc}(p_n)$. By the construction, $(p_{n-1}\circ\sigma_n)^{-1}(P)\subseteq E_F$ and $x_n\in p_n(E_F)$. So we can take $E=E_F$. **Case: $\pi_{n-1}$ is not isomorphic at $x_n$ and $\pi_{n-1}$ is a flip.** Let $C\subseteq X_{n-1}$ and $C^+\subseteq X_n$ be the flipping loci which are connected by [@Mor88 Corollary 1.3]. Then there exists a $p_n$-exceptional connected reduced divisor $P$ of $W_n$ such that $p_{n-1}\circ\sigma_n(P)=C$ and $p_n(P)=C^+$. By Lemma [Lemma 60](#lem-ben){reference-type="ref" reference="lem-ben"}, there exists some $x_{n-1}\in C\cap \operatorname{Sing}(X_{n-1})$. By the same argument as in the previous cases, we can find the desired $F$ with $P\subseteq E_F$. Note that $\pi_{n-1}$ is not isomorphic at $x_n$. Then $x_n\in C^+\subseteq p_n(E_F)$. So we can take $E=E_F$. ◻ **Theorem 63**. *Consider the equivariant dynamical systems of a minimal model program $$f_1\ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X_1\to X_2\dashrightarrow\cdots\dashrightarrow X_n\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ f_n$$ where $f_1$ is a non-isomorphic surjective endomorphism of a smooth projective threefold $X_1$. Let $\pi_n:X_n\dashrightarrow X_{n+1}$ be a flip. Then $\pi_n$ is $f_n$-equivariant after iteration of $f$.* *Proof.* Let $C$ be the flipping locus of $\pi_n$. By Lemma [Lemma 60](#lem-ben){reference-type="ref" reference="lem-ben"}, there exists some $x_n\in \operatorname{Sing}(X_n)\cap C$. Let $\ell$ be an irreducible curve in $C$ containing $x_n$. Denote by $\ell_m=(f_n)^m(\ell)$ when $m\ge 0$. Note that $(f_n)^m(C)$ has the same number of irreducible components when $m\gg 1$. Then $f_n^{-1}(\ell_{m+1})=\ell_m$ when $m\gg 1$ (cf. [@MZ20a Lemma 4.2]). Write $(f_n)^*\ell_{m+1}= a_m \ell_m$ with $a_m$ being a positive integer. Suppose the contrary that the theorem is false, i.e., the $\ell_m$ are all different. We apply Lemma [Lemma 62](#lem-connected){reference-type="ref" reference="lem-connected"} and use the same notation there. There exists a connected $p_n$-exceptional divisor $E$ such that $p_n^{-1}(x_n)\subseteq E$ and some prime divisor $E_1\subseteq E$ is contracted to a point in some $X_i$ with $i\le n$. We may write $E=\sum\limits_{t=1}^s E_t$ such that the $E_t$ are prime divisors with $E_t\cap E_{t+1}\neq \emptyset$. Note that $f_n$ lifts to $h_n:W_n\to W_n$. After iteration, we may assume all $E_t$ are $h_n^{-1}$-invariant. Note that $\dim p_n(\operatorname{Exc}(p_n))\le 1$ and $p_n(\operatorname{Exc}(p_n))$ is $f_n^{-1}$-invariant. So $\ell_m\not\subseteq p_n(\operatorname{Exc}(p_n))$. Let $\gamma_m$ be the strict transform of $\ell_m$ in $W_n$. Note that $x_n\in \ell_0$ and $p_n^{-1}(x_n)\subseteq E$. So $\gamma_0\cap E\neq\emptyset$ and hence $\gamma_m\cap E\neq\emptyset$ for any $m\ge 0$. By Lemma [Lemma 56](#lem-eq-cartier){reference-type="ref" reference="lem-eq-cartier"}, $h_n|_{E_1}$ is $q$-polarized with $q=(\deg h)^{1/3}$. By the Connecting Lemma [Lemma 59](#lem-connection){reference-type="ref" reference="lem-connection"}, each $h_n|_{E_t}$ is $q$-amplified (by induction on $t \ge 1$). In particular, we have $h_n^*E=qE$. By Lemma [Lemma 56](#lem-eq-cartier){reference-type="ref" reference="lem-eq-cartier"}, there is a $p_n$-exceptional effective Cartier divisor $D$ such that $E\subseteq \operatorname{Supp}D$ and $h_n^*D=qD$. Since $\gamma_m\not\subseteq \operatorname{Exc}(p_n)$ and $\gamma_m\cap E\neq\emptyset$ for any $m\ge 0$, we have $D\cdot \gamma_m\in \mathbb{Z}_{>0}$ and $$1\le D\cdot \gamma_{m+1}=\frac{1}{q^3}h_n^*D\cdot h_n^*\gamma_{m+1}=\frac{a_m}{q^2}D\cdot \gamma_m.$$ So $a_m\ge q^2>1$ for $m\gg 1$. Note that $X_n$ has only finitely many singular points. By the purity of branch locus, $\ell_m\subseteq \operatorname{Supp}R_{f_n}$ when $m\gg 1$. After iteration, we may assume $P:=\overline{\bigcup_{m\gg 1} \ell_m}\subseteq \operatorname{Supp}R_{f_n}$ is a prime divisor of $X_n$. Note that for any prime divisor $P'$ with $f_n(P')=P$, we have $P'\subseteq \overline{\bigcup_{m-1\gg 1} \ell_m}$. So $f_n^{-1}(P)=P$. Let $Q$ be the strict transform of $P$ in $W_n$. Note that $\gamma_m\cap E\neq \emptyset$ implies $Q\cap E\neq \emptyset$. By the Connection Lemma [Lemma 59](#lem-connection){reference-type="ref" reference="lem-connection"}, $h_n^*Q=qQ$ and hence $f_n^*P=qP$. Then we have $$(f_n|_P)^*\ell_{m+1}=\frac{a_m}{q} \ell_m,$$ where $\frac{a_m}{q}\ge q>1$ when $m\gg 1$. So $f|_P$ is ramified along infinitely many curves, a contradiction as our base field has characteristic $0$. ◻ # Proof of Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"} {#proof-of-theorem-mainthm-a} *Proof of Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"}.* Suppose the contrary $f$ is $\delta$ primitive, and some birational MMP $$\xymatrix{ X=X_1\ar[r]^{\pi_1}& X_2\ar@{-->}[r]^{\pi_2} &\cdots\ar@{-->}[r]^{\pi_{n-1}} &X_{n}\ar@{-->}[r]^{\pi_{n}} &X_{n+1} }$$ has $\pi_i$ being $f_i:=f|_{X_i}$-equivariant after iteration for $i<n$ and $\pi_n$ not being $f_n$-equivariant even after iteration. By Theorem [Theorem 63](#thm-emmp-flip){reference-type="ref" reference="thm-emmp-flip"}, $\pi_n$ is a divisorial contraction. By [@MZ18 Lemma 6.2], the exceptional divisor of $\pi_n$ is not $f_n^{-1}$-periodic. By Theorem [Theorem 52](#thm-E-Rf-notbig){reference-type="ref" reference="thm-E-Rf-notbig"}, $\kappa_{f_n}(X,R_{f_n})<3$. By Corollary [Corollary 45](#cor-kappa-rf12){reference-type="ref" reference="cor-kappa-rf12"}, $\kappa_{f_n}(X,R_{f_n})=0$ since ${f_n}$ is $\delta$-primitive. Note that $$R_{f_n}=(\pi_{n-1}\circ\pi_{n-2}\circ\cdots\circ \pi_1)_*R_f.$$ So $\kappa_f(X, R_f)=0$ by Lemma [Lemma 31](#lem-pushforward-fiitka){reference-type="ref" reference="lem-pushforward-fiitka"}. Hence the general fibre of $\phi$ is an elliptic curve, by Lemma [Lemma 49](#lem-Rf=0){reference-type="ref" reference="lem-Rf=0"}, Let $E_{X_n}$ be the exceptional divisor of $\pi_n$ and $E \subseteq X$ its strict transform. By Lemma [Lemma 13](#lem-fprime){reference-type="ref" reference="lem-fprime"}, $E_{X_n}$ is $f_n$-prime, so $E$ is $f$-prime. Let $E_i:=f^i(E)$ with $i\ge 0$. Note that $E_i$ is covered by rational curves and $\kappa(X, E_i)=0$. Let $\phi:X\dashrightarrow Y$ be the $f$-Iitaka fibration of $E$. By Theorem [Theorem 41](#thm-fiitaka2){reference-type="ref" reference="thm-fiitaka2"}, $\dim Y=2$, $\deg f|_Y=1$ and $\phi$ is almost well-defined. By Corollary [Corollary 50](#cor-kappa-rf0){reference-type="ref" reference="cor-kappa-rf0"}, $f$ is quasi-étale and hence étale since $X$ is smooth. In particular, $X$ admits a non-trivial étale cover. So $X$ is not rationally connected (cf. [@Deb01 Corollary 4.18]). If $X$ is non-uniruled, then we can always run smooth birational EMMP for non-uniruled $X$ (cf. [@Fuj02 Section 4]), a contradiction. So, by [@Nak10 Theorem 4.19], there exists an $f$-equivariant special MRC fibration $$\psi:X\dashrightarrow Z$$ where $Z$ is non-uniruled normal projective variety with $0<\dim Z<3$ and $\psi$ has rationally connected general fibres. If $\phi \times \psi: X\dashrightarrow Y\times Z$ is generically finite, then $f$ is $\delta$-imprimitive by Lemma [Lemma 19](#lem-imprimitive){reference-type="ref" reference="lem-imprimitive"}, a contradiction. Note that the induced map $X\dashrightarrow Y\times Z$ is generically finite when $\dim Z=2$. So $\dim Z=1$ and $\psi$ factors through $\phi$. Write $\psi=\sigma\circ \phi$ where $\sigma:Y\dashrightarrow Z$ is a dominant rational map. Let $C$ be the image of $E_i$ in $Y$ with $i\gg 1$. Then $C$ is a rational curve since the general fibre of $\phi$ is elliptic and $E_i$ is covered by rational curves. Note that $Z$ is not a rational curve. So $C$ and hence $E_i$ do not dominate $Z$. Since $E_i$ is not $f^{-1}$-periodic, $Z$ is an elliptic curve and $\psi$ is a well-defined fibration. So $E_i=\psi^{-1}(\psi(E_i))$ for $i\gg 1$ since $\psi$ has irreducible general fibre. In particular, $E_i$ and hence $E=f^{-i}(E_i)$ are semi-ample, a contradiction. ◻ The following example is our first test on Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"} before it is proved. **Example 64**. There exists a smooth rational surface $S$ admitting an automorphism $g$ of positive entropy with no $g$-periodic curves (cf. [@Les21]). There are infinitely many choices of MMPs from $S$, but none of them is $g$-equivariant even after iteration. Let $E$ be an elliptic curve and $n_E:E\to E$ the multiplication map by $n$ with $n^2 > \delta_g$. Let $X:=S\times E$ and $f=g\times n_E$. We claim that $X$ admits no $f^{-1}$-periodic prime divisor. Indeed, let $P$ be an $f^{-1}$-invariant prime divisor and $e\in E$ the identity element. Then $P\cap S\times \{e\}$ is $(f|_{S\times\{e\}})^{-1}$-invariant and hence $P=S\times \{a\}$ for some $a$. However, $E$ admits no $(n_E)^{-1}$-periodic point. So the claim is true. Hence there are infinitely many choices of MMPs from $X$, but none of them is $f$-equivariant even after iteration. Nevertheless, $f$ is still $\delta$-imprimitive, via the projection $X\to E$. # Fano contraction to a curve In this section, we focus on the case of a Fano contraction $X \to Y$ to a curve (hence $\rho(X) = 2$). Theorem [Theorem 71](#thm-fano31){reference-type="ref" reference="thm-fano31"} is our main result. By a Fano contraction of $X$, we always mean a Fano contraction of a $K_X$-negative extremal ray. We begin with some lemmas. **Lemma 65**. *Let $f:X\to X$ be a $\delta$-primitive endomorphism of a normal projective variety $X$. Then $X$ admits at most one Fano contraction.* *Proof.* Suppose the contrary that there are two different Fano contractions $\pi:X\to Y$ and $\tau:X\to Z$. After iteration, we may assume $\pi$ and $\tau$ are $f$-equivariant (cf. [@MZ18 Remark 6.3]). Since $\rho(X)=\rho(Y)+1=\rho(Z)+1$, we have $\operatorname{N}^1(X)=\pi^*\operatorname{N}^1(Y)+\tau^*\operatorname{N}^1(Z)$. Therefore, $\delta_f=\max\{\delta_{f|_Y},\delta_{f|_Z}\}$. Note that $Y$ and $Z$ must be positive dimensional. So $f$ is $\delta$-imprimitive. ◻ **Lemma 66**. *Consider the following equivariant dynamical systems of smooth projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar[r]^{\tau} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\dim X=3$, $\dim Y=1$, and $\tau$ is a Fano contraction. Suppose $\deg g>1$. Then $f$ is $\delta$-imprimitive.* *Proof.* Suppose the contrary $f$ is $\delta$-primitive. Then $\delta_f>\delta_g>1$ and $f$ is int-amplified (cf. [@Men20 Theorem 1.1]). Then $-K_X$ is $\mathbb{Q}$-effective (cf. [@MZ22 Theorem 1.5]). We apply the proof of [@MZ22 Proposition 9.2]. First, we can find some nef $\tau$-ample Cartier divisor $D$ such that $f^*D\equiv \delta_fD$, see the second paragraph of the proof of [@MZ22 Proposition 9.2]. Let $a>0$ be a rational number such that $B:=D+aK_X$ is $\tau$-trivial. In the Situation 1 of the proof of [@MZ22 Proposition 9.2], i.e., when $B$ is pseudo-effective, we have $\kappa(X,-K_X)=0$ and $-K_X\sim_{\mathbb{Q}} D$ where $D=\operatorname{Supp}R_f$ is an $f^{-1}$-invariant prime divisor. This is also Case TIR (cf. [@MMSZ23 Section 6]) and it does not occur by [@MMSZ23 Theorem 6.6]. So we are left with Situation 2 in the proof of [@MZ22 Proposition 9.2]: $B$ is not pseudo-effective. Note that $\rho(X)=2$, $B$ is not nef, and $B$ is $\tau$-trivial. Then the extremal ray of $\overline{\operatorname{NE}}(X)$ not contracted by $\tau$ is $B$-negative and hence $K_X$-negative since $D$ is nef. Let $\phi:X\to Z$ be the $f$-equivariant contraction of the $B$-negative extremal ray. If $\dim Z=\dim X$, then $f|_Z$ and hence $f$ and $g$ are $\delta_f$-polarized (cf. [@MZ18 Section 3]), a contradiction. If $\dim Z<\dim X$, then $f$ is $\delta$-imprimitive by Lemma [Lemma 65](#lem-2fano){reference-type="ref" reference="lem-2fano"}. ◻ The following two lemmas collect the common part that we shall use in the proof of Lemmas [Lemma 69](#lem-elliptic){reference-type="ref" reference="lem-elliptic"} and [Lemma 70](#lem-rational){reference-type="ref" reference="lem-rational"}. **Lemma 67**. *Let $\pi:D\to Y$ be a surjective morphism where $D$ is a projective surface and $Y$ is a smooth projective curve. Suppose the general fibre of $\pi$ is a simple normal crossing loop of $m$ smooth rational curves. Let $Z$ be the union of curves in $\operatorname{Sing}(D)$ dominating $Y$. Then $Z$ has at most $2$ irreducible components.* *Proof.* Let $\nu:\overline{D}\to D$ be the normalization. Let $y\in Y$ be a general point. Then the general fibre $\overline{D}_y$ of $\pi\circ \nu$ has $m$ connected (and irreducible) components. Let $Q$ be an irreducible component of $\overline{D}_y:=(\pi\circ \nu)^{-1}(y)$. Note that $\nu|_{Q}$ is isomorphic. So we have $$\nu^{-1}(Z)\cap \overline{D}_y=\bigsqcup_{Q\subseteq \overline{D}_y} (\nu^{-1}(Z)\cap Q)=\bigsqcup_{Q\subseteq \overline{D}_y} (\nu^{-1}(Z\cap \nu(Q))\cap Q).$$ Note that $\sharp Z\cap \nu(Q)=2$. Then $\sharp \nu^{-1}(Z)\cap \overline{D}_y= 2m$ and hence $\deg (\pi|_D\circ \nu)|_{\nu^{-1}(Z)}=2m$. Let $p_1:\overline{D}\to \overline{Y}$ and $p_2:\overline{Y}\to Y$ form the Stein factorization of $\pi\circ \nu$. Note that $\deg p_2=m$. Then $\deg (p_1)|_{\nu^{-1}(Z)}=2m/m=2$. Since $\overline{Y}$ is irreducible, $\nu^{-1}(Z)$ and hence $Z$ have at most two irreducible components. ◻ **Lemma 68**. *Consider the following equivariant dynamical systems of smooth projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar[r]^{\tau} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\dim X=3$, $\dim Y=1$, and $\tau$ is a Fano contraction. Suppose $f$ is $\delta$-primitive and non-isomorphic. Then $\deg g=1$ and the following hold (after iterating $f$).* 1. *$B_f=\operatorname{Supp}R_f$ is an $f^{-1}$-invariant, nef and $\tau$-ample prime divisor such that $\kappa(X, B_f)=0$, $(X, B_f)$ is lc and $f^*B_f = \delta_f B_f$.* 2. *Let $Z$ be the union of $f^{-1}$-periodic curves dominating $Y$. Then $Z$ is the union of curves $C\subseteq \operatorname{Sing}(B_f)$ with $C$ dominating $Y$.* 3. *$B_f\cap X_y$ is a simple normal crossing loop of $m$ smooth rational curves with $m\ge 3$ for general $y\in Y$. In particular, $\deg \tau|_Z=m\ge 3$.* 4. *$Z$ has at most $2$ irreducible components.* *Proof.* Since $f$ is $\delta$-primitive, $\delta_f>\delta_g$. By Lemma [Lemma 66](#lem-degg>1){reference-type="ref" reference="lem-degg>1"}, $\deg g=1$. Let $V$ be the $1$-dimensional $\delta_f$-eigenspace of $f^*|_{\operatorname{N}^1(X)}$ (generated by a nef divisor). The other eigenvalue of $f^*|_{\operatorname{N}^1(X)}$ is $\delta_g=1$. Then $V=\text{Im} (f^*-\operatorname{id})|_{\operatorname{N}^1(X)}$. By the ramification divisor formula, $$R_f=f^*(-K_X)-(-K_X)\in V$$ and $f^*R_f\equiv \delta_f R_f$. Note that $f$ is not polarized (or else, $\delta_f = \delta_g$, absurd). So $R_f$ is not big even after any iteration (cf. [@MZ18 Proposition 1.1]). Then $\kappa_f(X, R_f)<3$. Let $y, y'\in Y$ be general points such that $g(y)=y'$. We may assume that the fibres $X_{y'}$ and $X_y$ are smooth del pezzo surfaces with the same Picard number. Note that $\deg f|_{X_y}=\deg f>1$. \(1\) By [@Deb01 Corollary 4.18], $X_{y'}$ has no non-trivial étale cover. By the purity of branch locus, $R_{f|_{X_y}}\neq 0$. By the adjunction formula and the ramification divisor formula, $R_f|_{X_y}=R_{f|_{X_y}}\neq 0$. So $R_f$ is $\tau$-ample and also an extremal ray of $\operatorname{PE}^1(X) = \operatorname{Nef}(X)$. Suppose the contrary that $\kappa_f(X, R_f)>0$. Then $\kappa(X, R_f)>0$ after iteration. Let $\phi:X\dashrightarrow Z$ be the $f$-Iitaka fibration of $R_f$. By Theorem [Theorem 36](#thm-fiitaka-polarized){reference-type="ref" reference="thm-fiitaka-polarized"}, $f|_Z$ is $\delta_f$-polarized. In particular, $f$ is $\delta$-imprimitive, a contradiction. So $\kappa_f(X, R_f)=\kappa(X,B_f)=0$ (cf. Proposition [Proposition 10](#prop-f*f*){reference-type="ref" reference="prop-f*f*"}). By Lemma [Lemma 30](#lem-fkappa>0){reference-type="ref" reference="lem-fkappa>0"}, each component of $B_f=\operatorname{Supp}R_f$ is $f^{-1}$-invariant after iteration. By [@Zha14 Proposition 2.1], the pair $(X,B_f)$ is log canonical. Suppose the contrary that $B_f$ contains at least two irreducible components $P_1$ and $P_2$. Since $R_f$ is extremal in $\operatorname{PE}^1(X)$, $P_1\equiv tP_2$ for some rational number $t>0$. Then $P_1-tP_2=\tau^*Q$ for some $\mathbb{Q}$-Cartier divisor $Q$ on $Y$ by the cone theorem (cf. [@KM98 Theorem 3.7]). Note that $g^*Q=\delta_f Q$ and $\delta_f>\delta_g$. So $Q\equiv 0$ and hence $rQ\in \operatorname{Pic}^0(Y)$ for some integer $r>0$. Note that $\delta_f^2>\delta_g$. By [@MMS+22 Proposition 6.3], $rQ\sim_{\mathbb{Q}} 0$ and hence $P_1\sim_{\mathbb{Q}} tP_2$. Then $\kappa(X,R_f)\ge \kappa(X, P_1)> 0$, a contradiction. So $B_f$ is irreducible. \(2\) By choosing $y$ general, we may assume $$(\operatorname{Supp}R_f)|_{X_y}=B_f|_{X_y}=B_f\cap X_y=\operatorname{Supp}R_{f|_{X_y}}.$$ Note that $$K_X+B_f=f^*(K_X+B_f)$$ and $\delta_f>1$. So $K_X+B_f\in \tau^*\operatorname{N}^1(Y)$ is $\tau$-trivial. By the adjunction formula, $$K_{X_y}+B_f\cap X_y=(K_X+B_f)|_{X_y}=(f|_{X_y})^*(K_{X_{y'}}+B_f\cap X_{y'})\equiv 0.$$ Note that $B_f\cap X_y$ has the same number of irreducible components for general $y$. So $f|_{X_y}$ is totally ramified. By [@KM98 Theorem 5.50], $(X_y, R_{f|_{X_y}})$ is log Calabi-Yau. By Lemma [Lemma 24](#lem-surf-toric){reference-type="ref" reference="lem-surf-toric"}, $(X_y, R_{f|_{X_y}})$ is further a toric pair and the unsplitting points of $f|_{X_y}$ are $\operatorname{Sing}(B_f\cap X_{y'})=\operatorname{Sing}(B_f)\cap X_{y'}$. Let $C$ be an $f^{-1}$-periodic curve dominating $Y$. Choosing $y, y'$ general, we may assume $\sharp C\cap X_y=\sharp C\cap X_{y'}$. Then the points in $C\cap X_{y'}$ are $f|_{X_y}$-unsplitting and hence $C\cap X_{y'}\subseteq \operatorname{Sing}(B_f)\cap X_{y'}$. So $C\subseteq \operatorname{Sing}(B_f)$. On the other hand, each irreducible curve $C'$ (not necessarily dominating $Y$) of $\operatorname{Sing}(B_f)$ is an lc centre of $(X,B_f)$ (cf. [@KM98 Lemma 2.29]). So $f^{-1}(C')=C'$ after iteration by [@BH14 Lemma 2.10]. Therefore, $Z$ is the union of of curves $C\subseteq \operatorname{Sing}(B_f)$ with $C$ dominating $Y$. \(3\) Let $y\in Y$ be a general point. By the classification of smooth toric Fano surfaces, the boundary $B_f\cap X_y$ is a simple normal crossing loop of $m$ smooth rational curves with $m\ge 3$. So $\sharp Z\cap X_y=\sharp \operatorname{Sing}(B_f\cap X_y)=m\ge 3$. Therefore, $\deg \tau|_Z\ge 3$. \(4\) This follows from Lemma [Lemma 67](#lem-2comp){reference-type="ref" reference="lem-2comp"}. ◻ **Lemma 69**. *Consider the following equivariant dynamical systems of smooth projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar[r]^{\tau} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\dim X=3$, $Y$ is an elliptic curve, $\deg g=1$, and $\tau$ is a Fano contraction. Suppose $f$ is $\delta$-primitive and non-isomorphic. Then $g=\operatorname{id}$ after iteration.* *Proof.* Suppose the contrary that $g$ is not an automorphism of finite order. Since $Y$ is an elliptic curve, $g$ is a non-torsion translation. Therefore, $$\textbf{``any $f^{-1}$-periodic irreducible closed subvariety of $X$ will dominate $Y$''}$$ by [@CMZ20 Lemma 7.5]. In particular, any $f^{-1}$-periodic irreducible curve is an elliptic curve étale over $Y$. Let $D:=B_f$ and $Z$ the union of $f^{-1}$-periodic irreducible curves. By Lemma [Lemma 68](#lem-fano-toric){reference-type="ref" reference="lem-fano-toric"}, $D$ is a nef $\tau$-ample prime divisor, $Z$ is the union of curves dominating $Y$ in $B_f$, and $\deg \tau|_Z\ge 3$. In particular, $D$ contains an $f^{-1}$-periodic irreducible curve. Note that $f^*D=\delta_f D$ and $D^2\cdot X_y>0$ for any general fibre $X_y$ of $\tau$. By the projection formula, $\deg f=\delta_f^2$. *We do replacements of $\tau:X\to Y$ to make every $f^{-1}$-periodic curve a section over $Y$.* Let $C$ be an $f^{-1}$-periodic curve such that $\deg \tau|_C>1$. Then the natural embedding $C\hookrightarrow C\times_Y C$ implies that the preimage of $C$ in $X\times_Y C$ splits into at least two irreducible components. Each time, we replace $\tau:X\to Y$ by $X\times_Y C\to C$, $D$ by $D\times_Y C$, and $f,g$ by the natural liftings. So after a suitable étale base change, we may assume $\deg \tau|_C=1$ for every $f^{-1}$-periodic curve $C$. However, $\tau$ may no longer be a Fano contraction and $D$ may no longer be irreducible. Note that $g$ is still a non-torsion translation. So any $f^{-1}$-periodic irreducible closed subvariety of $X$ still dominates $Y$. Then $D$ is the union $f^{-1}$-periodic irreducible closed subvarieties with dimension $\le 2$. Moreover, each irreducible component of $D$ contains an $f^{-1}$-periodic irreducible curve. $K_X$ is still not pseudo-effective since the general fibres over $Y$ are still toric surfaces. By Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"}, we can further run an $f$-equivariant (after iteration) birational MMP $$X\dashrightarrow X_2\dashrightarrow\cdots \dashrightarrow X_r$$ and a Fano contraction $\tau_r:X_r\to Y'$. Since $Y$ is an elliptic curve, the MMP is over $Y$. Note that $X$ is still smooth and there is no $g^{-1}$-periodic point on $Y$. So each $X_i\to X_{i+1}$ has to be a divisorial contraction with the exceptional divisor contracted to an elliptic curve étale over $Y$ (cf. [@CMZ20 Lemma 7.5]). In particular, $X_r$ is still smooth. Since $f$ is $\delta$-primitive, $\delta_{f|_{Y'}}<\delta_f$. Note that $\deg f=\delta_f^2$. If $\dim Y'=2$, then $\delta_f = \deg f|_{X_{y'}} = (\deg f)/(\deg f|_{Y'})$, $\deg f|_{Y'}=\delta_f$ and $\iota_{f|_{Y'}}=\delta_f/\delta_{f|_{Y'}}>1$ by Lemma [Lemma 25](#lem-surf-deg){reference-type="ref" reference="lem-surf-deg"}. In particular, $f|_{Y'}$ and hence $g$ are int-amplified, a contradiction. So $Y'\to Y$ is isomorphic by noting that $Y'\to Y$ has connected fibres. Let $C_r$ be an $f_r^{-1}$-periodic irreducible curve of $X_r$. Let $V$ be the preimage of $C_r$ in $X$. Since $V$ is $f^{-1}$-periodic, $V\subseteq D$. Now either $V$ is a curve or $V$ is an irreducible component of $D$ containing an $f^{-1}$-periodic curve. So $C_r$ is always dominated by some $f^{-1}$-periodic curve in $X$. In particular, $\deg \tau_r|_{C_r}=1$. Now, replacing $\tau$ by $\tau_r$, we finally have that $\tau$ is a Fano contraction as in Lemma [Lemma 68](#lem-fano-toric){reference-type="ref" reference="lem-fano-toric"}; meanwhile, $\deg \tau|_C=1$ holds for each $f^{-1}$-periodic curve $C$. We now apply Lemma [Lemma 68](#lem-fano-toric){reference-type="ref" reference="lem-fano-toric"} to our new setting again. So $\deg \tau|_Z\ge 3$ and $Z$ has at most $2$ irreducible components. However, this is impossible since each irreducible component of $Z$ is only a section over $Y$. ◻ **Lemma 70**. *Consider the following equivariant dynamical systems of smooth projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar[r]^{\tau} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\dim X=3$, $Y$ is a rational curve, $\deg g=1$, and $\tau$ is a Fano contraction. Suppose $f$ is $\delta$-primitive and non-isomorphic. Then $g=\operatorname{id}$ after iteration.* *Proof.* Suppose the contrary that $g$ is an automorphism of infinite order. Then $g$ has at most $2$ periodic points. By Lemma [Lemma 68](#lem-fano-toric){reference-type="ref" reference="lem-fano-toric"}, $D:=B_f\neq 0$ is an $f^{-1}$-invariant prime divisor and $\kappa(X, D)=0$. Let $Z$ be the union of $f^{-1}$-periodic curves dominating $Y$. Then $\deg \tau|_Z\ge 3$ and $Z$ has at most $2$ irreducible components. So there is an irreducible component $C$ of $Z$ such that $q:=\deg \tau|_C>1$. Let $\widetilde{C}$ be the normalization of $C$ and $p:\widetilde{C}\to Y$ the induced finite surjective morphism with $\deg p=q>1$. Let $\widetilde{X}$ the normalization of $X\times_Y \widetilde{C}$. Note that we can not guarantee the smoothness or even $\mathbb{Q}$-factorial singularities for $\widetilde{X}$. Now we have the following equivariant dynamical system: $$\xymatrix{ \save[]+<2pc,0.22pc>*{\widetilde{f} \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ } \restore &\widetilde{X} \ar[r]^{\widetilde{p}}\ar[d]_{\widetilde{\tau}} & X \ar[d]^{\tau} &\save[]+<-2pc,0.1pc>*{\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ f} \restore\\ \save[]+<2pc,0.1pc>*{\widetilde{g} \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ } \restore&\widetilde{C}\ar[r]_p& Y&\save[]+<-2pc,0.05pc>*{\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g} \restore }$$ where $\widetilde{g}$ is an automorphism of infinite order and hence $\widetilde{C}$ has genus $\le 1$. Note that $g$ has at least one fixed point by the Lefschetz trace formula. So $\widetilde{g}$ has a periodic point. Recall that an automorphism of an elliptic curve has finite order if it has a periodic point. Therefore, we have $\widetilde{C}\cong \mathbb{P}^1$. Since $\mathbb{A}^1$ and $\mathbb{P}^1$ are simply connected, the branch divisor $B_p$ contains at least $2$ points. On the other hand, $B_p$ is $g$-invariant and $g$ has at most $2$ periodic points. So after choosing a suitable coordinate of $Y$, we may assume $B_p=y_0+y_{\infty}$ where $y_0=[0:1]$ and $y_{\infty}=[1:0]$. After iteration, we may write $g([x_0:x_1])=[ax_0:x_1]$ for some non-zero $a$ which is not a root of unity. By choosing a suitable coordinate of $\widetilde{C}\cong Y$, we may write $p([x_0:x_1])=[x_0^q:x_1^q]$ and $\widetilde{g}([x_0:x_1])=[a^{1/q}x_0:x_1]$ satisfying $g\circ p=p\circ\widetilde{g}$. Note that $$\widetilde{C}\times_Y \widetilde{C}=\{([a_0:a_1],[b_0:b_1])\in \widetilde{C}\times \widetilde{C}\,|\, [a_0^q:a_1^q]=[b_0^q:b_1^q]\}$$ and $a_0^qb_1^q-b_0^qa_1^q=\prod_{i=1}^q(a_0b_1-\xi_q^ib_0a_1)$ where $\xi_q$ is the primitive $q$-th root of unity. So $\widetilde{C}\times_Y \widetilde{C}$ has exactly $q$ irreducible components. Recall that $\deg \tau|_Z\ge 3$. Then $\widetilde{p}^{-1}(Z)$ has at least $3$ irreducible components. Let $y\in Y$ and $P$ an irreducible component of $\tau^{-1}(y)$. Since $X$ is smooth, P is Cartier. By the cone theorem (cf. [@KM98 Theorem 3.7]), $P=\tau^*y$. So each fibre of $\tau$ is irreducible and reduced. Note that $\widetilde{p}^*X_{y_0}=\widetilde{\tau}^*p^*y_0=q\widetilde{\tau}^*y_0$. Then $\widetilde{\tau}^*y_0$ is irreducible and reduced by noting that $\deg \widetilde{p}=q$. In particular, $\deg \widetilde{p}|_{\widetilde{X}_{y_0}}=1$. The same argument holds for $y_{\infty}$. Note that $\widetilde{p}$ is étale outside $X_{y_0}$ and $X_{y_\infty}$. So $\widetilde{\tau}$ has irreducible and reduced fibres. Let $V$ be an irreducible component of the non-lc locus of the pair $(X, D+X_{y_0}+X_{y_\infty})$. By Lemma [Lemma 68](#lem-fano-toric){reference-type="ref" reference="lem-fano-toric"}, $(X,D)$ is log canonical. Then $V\subseteq X_{y_0}\cup X_{y_\infty}$ and $\dim V\le 1$. By [@BH14 Theorem 1.2], after iteration, $f^{-1}(V)=V$ and $\deg f|_V=\deg f\ge \deg f|_{X_{y_0}}$. Note that $D|_{X_{y_0}}$ is ample and $(f|_{X_{y_0}})^*D|_{X_{y_0}}=\delta_f D|_{X_{y_0}}$. So $f|_{X_{y_0}}$ and hence $f|_V$ are $\delta_f$-polarized. Then $\deg f|_{X_{y_0}}>\deg f|_V$, a contradiction. Therefore, $(X, D+X_{y_0}+X_{y_\infty})$ is log canonical. In particular, $X_{y_0}$ is smooth at every ($1$-dimensional) generic point $\eta$ of $D\cap X_{y_0}$ by [@KM98 Lemma 2.29]. Then $\widetilde{p}|_{\widetilde{X}_{y_0}}$ is isomorphic over $\eta$ by the Zariski main theorem. In particular, $\widetilde{p}^{-1}(x)$ consists of only $1$ point when $x$ is a general point of $D\cap X_{y_0}$, and we shall derive a contradiction based on this. Let $\widetilde{D}:=\widetilde{p}^*D=\widetilde{p}^{-1}(D)$. Note that $K_{\widetilde{X}}+\widetilde{D}+\widetilde{X}_{y_0}+\widetilde{X}_{y_\infty}=\widetilde{p}^*(K_X+D+X_{y_0}+X_{y_\infty})$. So the pair $(\widetilde{X}, \widetilde{D}+\widetilde{X}_{y_0}+\widetilde{X}_{y_\infty})$ is log canonical by [@KM98 Proposition 5.20]. Note that $\widetilde{p}^{-1}(Z)\subseteq \operatorname{Sing}(\widetilde{D})$ since $\widetilde{p}$ is étale outside $\widetilde{X}_{y_0}\cup \widetilde{X}_{y_\infty}$. Applying Lemma [Lemma 67](#lem-2comp){reference-type="ref" reference="lem-2comp"} to $\widetilde{D}\to \widetilde{C}$, we have $\widetilde{D}$ reducible since $\widetilde{p}^{-1}(Z)$ has at least $3$ irreducible components. Write $\widetilde{D}=\sum\limits_{i=1}^n \widetilde{D}_i$ where $\widetilde{D}_i$ is irreducible and $n\ge 2$. Let $\ell_i:=\widetilde{D}_i\cap \widetilde{X}_{y_0}$. Note that $\widetilde{p}(\ell_i)=D\cap X_{y_0}$. Then $\widetilde{X}$ is smooth at every generic point of $\ell_i$ (cf. [@KM98 Lemma 2.51]). By [@KM98 Lemma 2.29] again, $\ell_i$ and $\ell_j$ have no common irreducible component for $i\neq j$. Let $x\in D\cap X_{y_0}$ be a general point. Then $\widetilde{p}^{-1}(x)$ consists of at least $2$ points, a contradiction. ◻ **Theorem 71**. *Consider the following equivariant dynamical systems of smooth projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar[r]^{\tau} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\dim X=3$, $\dim Y=1$, and $\tau$ is a Fano contraction of some $K_X$-negative extremal ray. Suppose $f$ is $\delta$-primitive and non-isomorphic. Then $g=\operatorname{id}$ after iteration.* *Proof.* Since $f$ is $\delta$-primitive, $\deg g=1$ by Lemma [Lemma 66](#lem-degg>1){reference-type="ref" reference="lem-degg>1"}. Suppose $g$ is not an automorphism of finite order. Then $Y$ is either rational or elliptic. The theorem follows from Lemmas [Lemma 69](#lem-elliptic){reference-type="ref" reference="lem-elliptic"} and [Lemma 70](#lem-rational){reference-type="ref" reference="lem-rational"}. ◻ **Remark 72**. The proofs of Lemmas [Lemma 69](#lem-elliptic){reference-type="ref" reference="lem-elliptic"} and [Lemma 70](#lem-rational){reference-type="ref" reference="lem-rational"} adopt quite different strategies. The proof of Lemma [Claim 42](#claim-pxp){reference-type="ref" reference="claim-pxp"} is similar to the proof of [@MMSZ23 Theorem 6.6], which essentially makes use of "$f^{-1}$-periodic subvarieties must dominate $Y$" to keep the smoothness (or at least $\mathbb{Q}$-factorial singularities) of $X$ after base change and EMMP. This phenomenon no longer holds in Lemma [Lemma 70](#lem-rational){reference-type="ref" reference="lem-rational"} and we make a big effort to control the singularities of $X$ after base change. # Proof of Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"} {#proof-of-theorem-mainthm-b} **Theorem 73**. *Let $f:X\to X$ be a $\delta$-primitive non-isomorphic surjective endomorphism of a smooth projective threefold $X$ with Kodaira dimension $\kappa(X)<0$. Then, after iteration, $f$ is either polarized or strongly imprimitive.* *Proof.* Note that $K_X$ is not pseudo-effective. So we can run the following MMP $$\xymatrix{ X_1\ar[r]^{\pi_1}& X_2\ar@{-->}[r]^{\pi_2} &\cdots\ar@{-->}[r]^{\pi_{n-2}} &X_{n-1}\ar@{-->}[r]^{\pi_{n-1}} &X_n\ar[r]^\tau&Y }$$ where $\pi_i$ is either a divisorial contraction or a flip and $\tau$ is a Fano contraction. By the termination of flips (cf. [@Sho96]), we may assume that $X_n$ admits no $K_X$-negative extremal birational contraction. Since $f$ is $\delta$-primitive, the MMP is $f$-equivariant after iteration by Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"}. Denote by $f_i:=f|_{X_i}$ and $g:=f|_Y$. If $\dim Y=0$, then $f_n$ and hence $f$ are polarized (cf. [@MZ18 Corollary 3.12]). So we may assume $0<\dim Y<3$. Since $f$ is $\delta$-primitive, $\delta_f=\delta_{f_i}>\delta_g$. By Lemma [Lemma 65](#lem-2fano){reference-type="ref" reference="lem-2fano"}, $X_n$ has only one Fano contraction which is $\tau$. Then there is only one $K_{X_n}$-negative extremal ray. If $\dim Y=2$, then $f$ is $\delta$-imprimitive by Corollary [Corollary 55](#cor-fano-n-1){reference-type="ref" reference="cor-fano-n-1"} since $\pi_i^{-1}$ does not contract divisors. This is a contradiction. So $\dim Y=1$ and the Picard number $\rho(X_n)=2$. Write $f_n^*D_+\equiv \delta_f D_+$ for some nef $\tau$-ample $D_+\in \operatorname{N}^1(X_n)$ and $f_n^*D_-\equiv \delta_g D_-$ for some $D_-\in \tau^*(\operatorname{Amp}(Y))$. Then $D_+^2\cdot D_->0$ and hence $\deg f_n=\delta_f^2\cdot\delta_g$ by the projection formula. We claim that $X_n$ is smooth. Note that $\pi_1$ is a divisorial contraction by [@Mor82 Theorem 3.3]. Let $E$ be the exceptional divisor of $\pi_1$. Write $f^*E=aE$. If $\pi_1(E)$ is a point, then $E^3<0$ (cf. [@KM98 Lemma 2.62]) and $\deg f=a^3=\deg f_n=\delta_f^2\cdot \delta_g$. In particular, $\delta_g<a<\delta_f$. Moreover, $f|_E$ is $a$-polarized since $-E|_E$ is ample. Let $W$ be the normalization of the graph of $X\dashrightarrow X_n$ and $h:W\to W$ the lifting of $f$. Let $p_X:W\to X$ and $p_Y:W\to Y$ be the projections. Let $E'$ be the strict transform of $E$ in $W$. Then $h|_{E'}$ is $a$-polarized. If $p_Y(E')=Y$, then $g$ is also $a$-polarized (cf. [@MZ18 Theorem 1.3]) and hence $a=\delta_g$, absurd. So $y:=p_Y(E')$ is a point and $g^{-1}(y)=y$ by [@CMZ20 Lemma 7.5]. Let $D:=p_Y^*y$. Then $E'\subseteq \operatorname{Supp}D$. Note that $g^*y=\delta_g y$. Then $h^*D=\delta_g D$, a contradiction because $h^*E'=aE'$. So $\pi_1(E)$ is a curve. By [@Mor82 Theorem 3.3], $X_2$ is smooth and $\pi_2$ is also a divisorial contraction. Repeating the above argument for the subsequent steps, the claim is proved. The theorem then follows from Theorem [Theorem 71](#thm-fano31){reference-type="ref" reference="thm-fano31"}. ◻ *Proof of Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"}.* (1) is just Theorem [Theorem 73](#thm-kappa<0){reference-type="ref" reference="thm-kappa<0"} and (3) follows from [@NZ09 Theorem A]. For (2), by [@Fuj02 Main Theorem (A), Case 3, Page 61], an étale cover $\widetilde{X}$ of $X$ is either an abelian veriety or $\widetilde{X}\cong S\times C$ where $S$ is a K3 surface and $C$ is an elliptic curve. For the first case, $f$ is quasi-abelian. Indeed, we can choose the cover suitably, called the Albanese closure (cf. [@NZ10 Lemmas 2.12]), such that $f$ lifts to a surjective endomorphism $\widetilde{f}:\widetilde{X}\to \widetilde{X}$; further, in the second case, $\widetilde{f}=g\times h$ splits as an automorphism $g:S\to S$ and a non-isomorphic surjective endomorphism $h:C\to C$ (cf. [@NZ10 Proposition 3.5] and [@Men23 Proposition 7.4]). By Lemma [Lemma 19](#lem-imprimitive){reference-type="ref" reference="lem-imprimitive"}, $\widetilde{f}$ is $\delta$-imprimitive. ◻ # Applications: Proofs of Theorems [Theorem 7](#mainthm-C){reference-type="ref" reference="mainthm-C"}, [Theorem 76](#thm-sand){reference-type="ref" reference="thm-sand"} and [Theorem 8](#mainthm-D){reference-type="ref" reference="mainthm-D"} {#applications-proofs-of-theorems-mainthm-c-thm-sand-and-mainthm-d} As applications of the EMMP Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"} and the structure Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"}, we prove Theorem [Theorem 7](#mainthm-C){reference-type="ref" reference="mainthm-C"} for the Kawaguchi-Silverman conjecture, Theorem [Theorem 76](#thm-sand){reference-type="ref" reference="thm-sand"} for the sAND conjecture, and Theorem [Theorem 8](#mainthm-D){reference-type="ref" reference="mainthm-D"} for the Zariski dense orbit conjecture. *Proof of Theorem [Theorem 7](#mainthm-C){reference-type="ref" reference="mainthm-C"}.* Note that KSC holds for projective curves and surfaces and we may assume $f$ is weakly primitive and not quasi-abelian (cf. Remark [Remark 6](#rmk-ksc){reference-type="ref" reference="rmk-ksc"}). By Lemma [Lemma 22](#lem-ksc-iff){reference-type="ref" reference="lem-ksc-iff"}, we may assume $f$ is strongly $\delta$-primitive. By Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"}, $f$ is polarized after iteration. Note that KSC holds for polarized endomorphisms (cf. Remark [Remark 6](#rmk-ksc){reference-type="ref" reference="rmk-ksc"}). ◻ We generalize the KSC Conjecture [Conjecture 5](#conj-KSC){reference-type="ref" reference="conj-KSC"} to sAND Conjecture [Conjecture 75](#conj-sand){reference-type="ref" reference="conj-sand"} (cf. [@MMSZ23 Conjecture 1.3]). **Definition 74**. Let $X$ be a projective variety and $f$ a surjective endomorphism on $X$ over a number field $K$ with a fixed algebraic closure $\overline{K}=\overline{\mathbb{Q}}$. Let $d > 0$. Set $$X(d)=X(K,d)=\{ x \in X(L) \mid K \subseteq L \subseteq \overline K,\ [L:K] \leq d\},$$ $$Z_{f}=Z_f( \overline{K})=\{ x \in X(\overline{K}) \mid \alpha_f(x) < \delta_f \},$$ $$Z_{f}(d)=Z_f(K, d)=Z_f( \overline{K})\cap X(K,d).$$ **Conjecture 75** (**small Arithmetic Non-Density = sAND**). *Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$ defined over a number field $K$. Then the set $Z_f(K, d)$ is not Zariski dense in $X_{\overline{K}} = X \times_K \overline{K}$ for any positive constant $d>0$.* **Theorem 76**. *Let $f:X\to X$ be a weakly primitive non-isomorphic surjective endomorphism of a smooth projective threefold $X$. Then Conjecture sAND holds for $f$.* *Proof.* Note that Conjecture sAND holds for projective curves and surfaces by [@MMSZ23 Theorem 4.4]. Moreover, Conjecture sAND holds for quasi-abelian $f$ by [@MMSZ23 Corollary 5.9]. Then the proof of Theorem [Theorem 7](#mainthm-C){reference-type="ref" reference="mainthm-C"} works after replacing Lemma [Lemma 22](#lem-ksc-iff){reference-type="ref" reference="lem-ksc-iff"} by [@MMSZ23 Lemma 2.9]. ◻ **Remark 77**. We cannot fully deal with the strongly imprimitive $f$ for Conjecture sAND, which is also highly related to another very hard "Uniform Boundedness Conjecture" (cf. [@MMSZ23 Conjecture 1.9]), see [@MMSZ23 Section 7] for some special cases. In the rest of this section, we treat the following conjecture and prove Theorem [Theorem 8](#mainthm-D){reference-type="ref" reference="mainthm-D"}. **Conjecture 78** (**Zariski Dense Orbit = ZDO**). *Let $f: X \to X$ be a surjective endomorphism of a projective variety. Then either $f$ is strongly imprimitive or $f$ has a Zariski dense orbit, i.e., there is a closed point $x\in X$ such that the orbit $$O_f(x):=\{f^i(x)\,|\, i\ge 0\}$$ is Zariski dense in $X$.* Under a generically finite dominant map, the property of having a Zariski dense orbit is stable. The following lemma says that the same is true for the weakly primitive property, see [@Xie22 Lemma 2.1]. **Lemma 79**. *Consider the following equivariant dynamical systems of projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar@{-->}[r]^{\phi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ where $\phi$ is dominant and generically finite. Then $f$ is weakly primitive (resp. has a Zariski dense orbit) if and only if so is $g$.* In general, the product case for Conjecture ZDO is hard. We show a special case below. **Lemma 80**. *Let $g:Y\to Y$ and $h:Z\to Z$ be surjective endomorphisms of projective varieties. Suppose $g$ and $h$ have Zariski dense orbits, $\deg g<\deg h$ and $\dim Z=1$. Then $g\times h$ has a Zariski dense orbit.* *Proof.* Let $X:=Y\times Z$ and $f:=g\times h$. Let $y\in Y$ and $z\in Z$ such that $\overline{O_g(y)}=Y$ and $\overline{O_h(z)}=Z$. Then $\overline{O_{g\times h}(y,z)}$ dominates both $Y$ and $Z$. By [@MMSZ23 Lemma 2.7], we may assume $P:=\overline{O_{g\times h}(y,z)}$ is irreducible and $f$-invariant after suitable iteration and replacing $(y,z)$ by $(g(y), h(z))$. If $P\neq X$, then $P\to Y$ is generically finite and hence $$\deg g<\deg h\le \deg f|_P=\deg g,$$ a contradiction. So $P=X$ and the lemma is proved. ◻ **Lemma 81**. *Consider the following equivariant dynamical systems of normal projective varieties $$\xymatrix{ f \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ X\ar@{-->}[r]^{\phi} &Y\ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ g }$$ satisfying the following:* 1. *$\phi$ is dominant,* 2. *$X$ is $\mathbb{Q}$-factorial and $\dim X=\dim Y+1$,* 3. *$\deg f>\deg g=1$, and* 4. *$g$ has a Zariski dense orbit.* *Then $f$ has a Zariski dense orbit.* *Proof.* Note that Conjecture ZDO holds for curves (cf. [@Ame11 Corollary 9]). So we may assume $\dim Y>0$. Suppose the contrary that $f$ has no Zariski dense orbit. We claim that any $f$-periodic prime divisor $P$ of $X$ dominating $Y$ is $f^{-1}$-periodic. After iteration, we may assume $f(P)=P$. Suppose the contrary that $P$ is not $f^{-1}$-periodic. Then we can find an infinite sequence of prime divisors $P_i$ with $i\ge 0$ such that $f(P_{i+1})=P_i$, $P_i\neq P_j$ for any $i\neq j$, and $P_0=P$. Note that $\deg f|_P=\deg g=1$ by the assumption (3). By Proposition [Proposition 10](#prop-f*f*){reference-type="ref" reference="prop-f*f*"}, we have $f^*P\sim_{\mathbb{Q}} (\deg f) P$, and hence $f^*P_i\sim_{\mathbb{Q}} (\deg f) P_i$ for all $i$. Suppose $P_i\cap P_j\neq\emptyset$ for some $i>j\ge 0$. Then $P_{i-j}\cap P\neq \emptyset$ and hence $P_{i-j}|_P$ is a non-zero effective divisor on $P$. Since $f|_P$ is invertible, for any eigenvalue $\lambda$ of $(f|_P)^*|_{\operatorname{NS}(P)}$, $1/\lambda$ is an algebriac integer. In particular, $(f|_P)^*|_{\operatorname{NS}(P)}$ has no rational eigenvalue $>1$. However, $(f|_P)^*(P_{i-j}|_P)=(\deg f) P_{i-j}|_P$, a contradiction. So $P_i\cap P_j=\emptyset$ for any $i\neq j$. By [@BPS16 Theorem 1.1], $\kappa(X, P)=1$. Let $\psi:X\dashrightarrow Z$ be the $f$-Iitaka fibration of $P$. Then $\dim Z=1$ and the induced map $X\dashrightarrow Y\times Z$ is generically finite and dominant. By Theorem [Theorem 36](#thm-fiitaka-polarized){reference-type="ref" reference="thm-fiitaka-polarized"}, $f|_Z$ is $(\deg f)$-polarized and hence $f|_Z$ has a Zariski dense orbit (cf. [@Ame11 Corollary 9]). By Lemmas [Lemma 79](#lem-zdo-gf){reference-type="ref" reference="lem-zdo-gf"}, [Lemma 80](#lem-zdo-prod){reference-type="ref" reference="lem-zdo-prod"} and the assumption (4), $f$ has a Zariski dense orbit, a contradiction. The claim is proved. Let $y\in Y$ such that $O_g(y)$ is Zariski dense in $Y$. Note that $\dim X=\dim Y+1$ and $X$ is normal. Then $\phi^{-1}(y)$, the set of well-defined points of $\phi$ mapping to $y$, is non-empty and consists of infinitely many points. Let $x\in \phi^{-1}(y)$. Then $\overline{O_f(x)}$ dominates $Y$, is not equal to $X$, and contains an $f$-periodic prime divisor $Q_1$ dominating $Y$ (cf. [@MMSZ23 Lemma 2.7]). By the previous claim, $Q_1$ is $f^{-1}$-periodic. Then $\phi^{-1}(y)\backslash \bigcup\limits_{i=0}^{+\infty} f^{-i}(Q_1)\neq \emptyset$ after replacing $y$ by $g^s(y)$ for some $s>0$. Let $x_2\in \phi^{-1}(y)\backslash \bigcup\limits_{i=0}^{+\infty} f^{-i}(Q_1)$. We have $O_f(x_2)\cap Q_1=\emptyset$. So there is another $f$-periodic prime divisor $Q_2$ dominating $Y$. Repeating this process, we obtain an infinite sequence of different $f^{-1}$-periodic prime divisors $Q_i$ dominating $Y$. Let $\Sigma_0$ be the union of $f^{-1}$-periodic prime divisors in $\operatorname{Supp}R_f$ and $\Sigma:=\bigcup\limits_{j=0}^{+\infty} f^{-j}(\Sigma_0)$. Then $\Sigma$ is still a divisor. If $Q_i\subseteq \operatorname{Supp}R_{f^s}$ for some $s>0$, then $Q_i\subseteq \Sigma$. So when $i\gg 1$, we have $Q_i\not\subseteq \operatorname{Supp}R_{f^s}$ for any $s>0$. In particular, we have $f^*Q_i=Q_i$ for some $i$ after iteration. Then $\deg g=\deg f|_{Q_i}=\deg f>1$, a contradiction. ◻ Now we prove the birational EMMP: the ZDO version (compare with Theorem [Theorem 1](#mainthm-A){reference-type="ref" reference="mainthm-A"}). *Proof of Theorem [Theorem 8](#mainthm-D){reference-type="ref" reference="mainthm-D"}.* Since $f$ is weakly primitive, $\kappa(X)\le 0$. Note that Conjecture ZDO holds when $f$ is quasi-abelian (cf. [@GS17 Theorem 1.2]). Suppose $\kappa(X)=0$. We recall the proof of Theorem [Theorem 3](#mainthm-B){reference-type="ref" reference="mainthm-B"}. After an $f$-equivariant étale cover, $X\cong S\times C$ where $S$ is a K3 surface and $C$ is an elliptic curve. Further, $f=g\times h$ with $g:S\to S$ an automorphism and $h:C\to C$ of $\deg h>1$. Since $f$ is weakly primitive, so are $g$ and $h$. Since Conjecture ZDO holds for projective curves and surfaces (cf. [@Ame11 Corollary 9] and [@JXZ23 Theorem 1.9]), $g$ and $h$ have Zariski dense orbits. By Lemma [Lemma 80](#lem-zdo-prod){reference-type="ref" reference="lem-zdo-prod"}, $f$ has a Zariski dense orbit, a contradiction. So $X$ is uniruled. Suppose the contrary that there is a birational MMP $$\xymatrix{ X=X_1\ar[r]^{\pi_1}& X_2\ar@{-->}[r]^{\pi_2} &\cdots\ar@{-->}[r]^{\pi_{n-1}} &X_{n}\ar@{-->}[r]^{\pi_{n}} &X_{n+1} }$$ such that $\pi_i$ is $f_i:=f|_{X_i}$-equivariant after iteration for $i<n$ and $\pi_n$ is not $f_n$-equivariant even after any iteration. By Theorem [Theorem 63](#thm-emmp-flip){reference-type="ref" reference="thm-emmp-flip"}, $\pi_n$ is a divisorial contraction. Let $E$ be the strict transform of the exceptional divisor of $\pi_n$ in $X$. Note that $E$ is not $f^{-1}$-periodic by [@MZ18 Lemma 6.2] and $\kappa(X,E)=0$. Let $\phi:X\dashrightarrow Y$ be the $f$-Iitaka fibration of $E$. By Theorem [Theorem 41](#thm-fiitaka2){reference-type="ref" reference="thm-fiitaka2"}, $\dim Y=2$ and $\deg f|_Y=1$. Since $f$ is weakly primitive, so is $g$. Note that ZDO holds for projective surfaces by [@JXZ23 Theorem 1.9]. So $g$ has a Zariski dense orbit. Then we get a contradiction to Lemma [Lemma 81](#lem-zdo-reldim1){reference-type="ref" reference="lem-zdo-reldim1"}. ◻ MMS+22 E. Amerik, Existence of non-preperiodic algebraic points for a rational self-map of infinite order, Math. Res. Lett., **18**(2):251-256, 2011. X. Benveniste, Sur le cone des 1-cycles effectifs en dimension 3 Math. Ann. **272** (1985), 257-265. C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., **23**(2):405-468, 2010. F. Bogomolov, A. Pirutka and A. Silberstein, Families of disjoint divisors on varieties, Eur. J. Math. **2** (2016), no. 4, 917--928. A. Broustet and A. Höring, Singularities of varieties admitting an endomorphism, Math. Ann. **360** (2014), no. 1-2, 439-456. P. Cascini, S. Meng and D.-Q. Zhang, Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. **378** (2020), no. 1-2, 637-665. J. A. Chen, H. Lin and K. Oguiso, On the Kawaguchi--Silverman Conjecture for birational automorphisms of irregular varieties, [arxiv:2204.09845](https://arxiv.org/abs/2204.09845v2) O. Debarre, Higher-dimensional algebraic geometry, Universitext. Springer-Verlag, New York, 2001. T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne, Duke Math. J. **123**(2004), no. 2, 311--328. N. Fakhruddin, Questions on self-maps of algebraic varieties, J. Ramanujan Math. Soc., **18**(2):109-122, 2003. Y. Fujimoto, Endomorphisms of smooth projective $3$-folds with nonnegative Kodaira dimension, Publ. RIMS. Kyoto Univ. **38** (2002), 33-92. Y. Fujimoto and N. Nakayama, Endomorphisms of smooth projective 3-folds with nonnegative Kodaira dimension. II, J. Math. Kyoto Univ. **47** (2007), no. 1, 79-114. D. Ghioca and T. Scanlon, Density of orbits of endomorphisms of abelian varieties, Trans. Amer. Math. Soc. **369** (2017), no. 1, 447-466. D. Greb, S. Kebekus, S. J. Kovács and T. Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci.(2011), no.114, 87-169. R. V. Gurjar, On ramification of self-maps of $\bold P^2$, J. Algebra, **259** (2003), no. 1, 191--200. M. Hindry and J. Silverman, Diophantine geometry: An introduction, Graduate Texts in Mathematics, **201**. Springer-Verlag, New York, 2000. J. Jia, T. Shibata, J. Xie and D.-Q. Zhang, Endomorphisms of quasi-projective varieties: towards Zariski dense orbit and Kawaguchi-Silverman conjectures, Math. Res. Lett. (to appear), [arXiv:2104.05339](https://arxiv.org/abs/2104.05339) J. Jia, J. Xie and D.-Q. Zhang, Surjective endomorphisms of projective surfaces: the existence of infinitely many dense orbits, Math. Z. **303** (2023), no. 2, Paper No. 39, 23 pp. S. Kawaguchi and J. Silverman, Examples of dynamical degree equals arithmetic degree, Michigan Math. J. **63**(2014), no. 1, 41-63. S. Kawaguchi and J. Silverman, Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties, Trans. Amer. Math. Soc. **368** (2016), no. 7, 5009-5035. S. Kawaguchi and J. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. **713** (2016), 21-48. J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., **134** Cambridge Univ. Press, 1998. J. Lesieutre, Some constraints of positive entropy automorphisms of smooth threefolds, Ann. Sci. Éc. Norm. Supér. **51** (2018) 1507-1547. J. Lesieutre, Tri-Coble surfaces and their automorphisms, J. Mod. Dyn. **17** (2021), 267-284. J. Lesieutre and M. Santriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi--Silverman Conjecture, Int. Math. Res. Not. **2021**, no. 10, 7677--7714. Y. Matsuzawa, Kawaguchi--Silverman Conjecture for endomorphisms on several classes of varieties, Adv. Math. **366** (2020), 107086, 26 pp. Y. Matsuzawa, On upper bounds of arithmetic degrees, Amer. J. Math. **142** (2020), no. 6, 1797--1821. Y. Matsuzawa, S. Meng, T. Shibata and D.-Q. Zhang, Non-density of points of small arithmetic degrees, J. Geom. Anal. **33** (2023), no. 4, Paper No. 112, 41 pp. Y. Matsuzawa, S. Meng, T. Shibata, D.-Q. Zhang and G. Zhong, Invariant subvarieties with small dynamical degree, Int. Math. Res. Not., 2022, **2022**(15): 11448-11483. Y. Matsuzawa, K. Sano and T. Shibata, Arithmetic degrees and dynamical degrees of endomorphisms on surfaces, Algebra Number Theory **12** (2018), no. 7, 1635--1657. Y. Matsuzawa and L. Wang, Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps, [arxiv:2212.05804](https://arxiv.org/abs/2212.05804v1) Y. Matsuzawa and S. Yoshikawa, Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism, Math. Ann. **382** (2022), no. 3-4, 1681-1704. S. Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. **294** (2020), no. 3, 1727--1747. S. Meng, Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms, Int. Math. Res. Not. IMRN, (to appear), [doi:10.1093/imrn/rnac308](https://doi.org/10.1093/imrn/rnac308) S. Meng, On endomorphisms of projective varieties with numerically trivial canonical divisors, Internat. J. Math. **34** (2023), no. 1, Paper No. 2250093, 27 pp. S. Meng and D. -Q. Zhang, Building blocks of polarized endomorphisms of normal projective varieties, Adv. Math. **325** (2018), 243-273. S. Meng and D. -Q. Zhang, Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett. **27** (2020), no. 2, 523-550. S. Meng and D. -Q. Zhang, Normal projective varieties admitting polarized or int-amplified endomorphisms, Acta Math. Vietnam. **45** (2020), no. 1, 11-26. S. Meng and D. -Q. Zhang, Kawaguchi-Silverman conjecture for surjective endomorphisms, Doc. Math. **27**(2022), 1605-1642. S. Meng and G. Zhong, Rigidity of rationally connected smooth projective varieties from dynamical viewpoints, Math. Res. Lett. (to appear) [arxiv:2005.03983](https://arxiv.org/abs/2005.03983). S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) **116** (1982), no. 1, 133-176. S. Mori, Flip theorem and the existence of minimal models for 3-folds. J. Amer. Math. Soc. **1**(1988), no.1, 117-253. N. Nakayama, Ruled surfaces with non-trivial surjective endomorphisms, Kyushu J. Math. **56** (2002), no. 2, 433-446. N. Nakayama, Intersection sheaves over normal schemes, J. Math. Soc. Japan **62** (2010), no. 2, 487-595; also [RIMS1614](http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1614.pdf). N. Nakayama, A variant of Shokurov's criterion of toric surface, Algebraic varieties and automorphism groups, 287-392, Adv. Stud. Pure Math., **75**. Math. Soc. Japan, Tokyo, 2017. N. Nakayama, On the structure of normal projective surfaces admitting non-isomorphic surjective endomorphisms, RIMS Preprints **1934** (2020). N. Nakayama and D.-Q. Zhang, Building blocks of étale endomorphisms of complex projective manifolds, Proc. Lond. Math. Soc. (3) **99** (2009), no. 3, 725-756. N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. **346** (2010), no. 4, 991-1018. V. Shokurov, 3-fold log models, Algebraic geometry, 4. J. Math. Sci. **81** (1996), no. 3, 2667-2699. J. Silverman, Arithmetic and dynamical degrees on abelian varieties, J. Théor. Nombres Bordeaux **29** (2017), no. 1, 151-167. K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. **439**, Springer-Verlag, Berlin, 1975, Notes written in collaboration with P. Cherenack. J. Xie, The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker), J. Amer. Math. Soc. (published online) (2022). D.-Q. Zhang, Invariant hypersurfaces of endomorphisms of projective varieties, Advances in Mathematics, **252** (2014) 185-203. D.-Q. Zhang, $n$-dimensional projective varieties with the action of an abelian group of rank $n - 1$, Trans. Amer. Math. Soc. **368** (2016), no. 12, 8849-8872.

arxiv_math

--- abstract: | Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator $(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha<2$, converge to a solution of the classical case, with $-\Delta$, when $\alpha$ goes to $2$. Precisely, in the setting of mild solutions, we prove uniform convergence in both the time and spatial variables and derive a precise convergence rate, revealing some phenomenological effects. Finally, our results are also generalized to the coupled setting of the Magnetic-hydrodynamic system.\ **Keywords**: Navier-Stokes equations; Magnetic-hydrodynamic system; Fractional Laplacian operator; Mild solutions; Non-local to local convergence.\ **AMS Classification:** 35B40, 35B30. author: - "Oscar Jarrín[^1]" - "Geremy Loachamín[^2]" title: "**From non-local to local Navier-Stokes equations**" --- # Introduction For a velocity field $\vec{u}: [0,+\infty)\times \mathbb{R}^3\to \mathbb{R}^3$, and for a pressure term $p:[0,+\infty)\times \mathbb{R}^3\to \mathbb{R}$, we deal with the three-dimensional, incompressible and generalized Navier-Stokes equations in the whole space $\mathbb{R}^3$: $$\label{NS-Intro} \partial_t \vec{u}= -\nu (-\Delta)^{\alpha/2} \vec{u}- (\vec{u}\cdot \vec{\nabla})\vec{u}-\vec{\nabla} p, \quad \text{div}(\vec{u})=0, \qquad 1<\alpha \leq 2.$$ When $1<\alpha<2$, the diffusion term is given by the fractional Laplacian operator, which is easily defined in the Fourier variable by the symbol $|\xi|^\alpha$. Moreover, in the spatial variable, we have $$(-\Delta)^{\alpha/2} \vec{u}(t,x)= C_{\alpha}\, {\bf p.v.} \int_{\mathbb{R}^3} \frac{\vec{u}(t,x)- \vec{u}(t,y)}{|x-y|^{3+\alpha}} dy,$$ where $C_\alpha>0$ is a constant depending on $\alpha$, and ${\bf p.v.}$ denotes the principal value. The non-local behavior of this operator allows us to call the equations ([\[NS-Intro\]](#NS-Intro){reference-type="ref" reference="NS-Intro"}) the non-local Navier-Stokes equations. By contrast, when $\alpha=2$, the diffusion term is given by the classical Laplacian operator, and we shall refer to the classical (or local) Navier-Stokes equations. With a minor loss of generality, we shall set the viscosity constant $\nu$ equal to one. Numerical solutions to the classical Navier-Stokes equations (when $\alpha=2$) for engineering problems, turbulent fluid flows, and geophysical phenomena are not completely possible at present, see for example [@Dubois; @Pope]. In addition, the mathematical theory of global existence and regularity of solutions to these equations remains one of the most challenging open questions in mathematical analysis, as discussed in [@PGLR; @Temam]. In this context, the fractional Navier-Stokes equations (when $1<\alpha<2$) have been employed as a relevant modification of the classical equations to gain a better understanding of these mathematical and computational difficulties [@Holst; @Meerschaert; @Nan; @Olson-Titi]. In equation ([\[NS-Intro\]](#NS-Intro){reference-type="ref" reference="NS-Intro"}), for each $1<\alpha<2$ fixed, we have an associated fractional Navier-Stokes equation, and we will denote its solution by $(\vec{u}_{\alpha}, p_\alpha)$. The main objective of this note is to sharply study the *dynamics* of the family of solutions $(\vec{u}_{\alpha}, p_\alpha)_{1<\alpha<2}$ when the parameter $\alpha$ tends to $2$. This question is not only interesting from the theoretical point of view, but has also been pointed out in some experimental works involving fractional Burgers equations [@Funaki] and a fractional transport-type equation [@Zaslavshy]. More precisely, these *numerical* studies show that solutions to fractional equations behave as solutions to the classical ones (involving the Laplace operator) when $\alpha$ is sufficiently close to $2$. Inspired by these previous works, we aim to develop a *rigorous framework* to study the convergence $$\label{Convergence-Intro} (\vec{u}_{\alpha}, p_\alpha) \to (\vec{u}_2, p_2), \quad \mbox{when} \quad \alpha \to 2^{-},$$ where $(\vec{u}_2, p_2)$ denotes a solution to the classical Navier-Stokes equations. It is also worth mentioning that this question has been studied for some *elliptic* equations, including the nonlinear Schrödinger equation [@Bieganowski] and the fractional $p$-Laplacian problem [@Fernandez-Salort]. In these works, the authors mainly used variational methods and concentration-compactness principles to prove the convergence of *weak solutions* of the fractional problem to the classical problem. Specifically, in [@Bieganowski], this convergence was proven in the strong topology of the space $L^{2}_{loc}(\mathbb{R}^n)$ (with $n\geq 3$), whereas in [@Fernandez-Salort], the authors used the (more technical) notion of $\Gamma$-convergence. For the *parabolic* setting of equation ([\[NS-Intro\]](#NS-Intro){reference-type="ref" reference="NS-Intro"}), a first interesting study of the convergence ([\[Convergence-Intro\]](#Convergence-Intro){reference-type="ref" reference="Convergence-Intro"}) was done in [@Dlotko] for both the two-dimensional case and the three-dimensional case, in the setting of a bounded and smooth domain $\Omega \subset \mathbb{R}^n$ (with $n=2,3$), and for the sub-critical case when $\alpha >2$. Specifically, in the two-dimensional case, it is proven that a family of solutions $(\vec{u}_\alpha)_{2<\alpha \leq 5/2}$ to equations ([\[NS-Intro\]](#NS-Intro){reference-type="ref" reference="NS-Intro"}) convergences (when $\alpha \to 2^{+}$) in the *weak topology* of the space $L^2(\Omega)$ to a *weak Leray's solution* $\vec{u}_2$ of the classical Navier-Stokes equations. This result does not fulfill for the three-dimensional case, due to the lost of some key tools only available in $2D$ to handle the nonlinear term. Thus, in this case it is proven that solutions to the regularized equations $$\partial_t \vec{u}= \big(\Delta - \varepsilon (-\Delta)^{\alpha/2}\big) \vec{u}- (\vec{u}\cdot \vec{\nabla})\vec{u}-\vec{\nabla} p, \quad \text{div}(\vec{u})=0, \qquad \alpha>5/2,$$ converge (when $\varepsilon \to 0^{+}$) to a *weak Leray's solution* $\vec{u}_2$ of the classical Navier-Stokes equations in the *weak topology* of the energy space. The ideas to proof these results are mainly based on sharp *a priori* energy estimates, the weak formulation of solutions and concentration-compactness arguments. In this paper, we will employ a completely different approach (following some of the ideas presented in our previous work [@JarrinLoachamin]), which is principally based on the explicit structure of *mild solutions*. This approach and some sharp computations in the Fourier level (since we consider here the whole space $\mathbb{R}^3$) allow us to prove a *uniform convergence* ([\[Convergence-Intro\]](#Convergence-Intro){reference-type="ref" reference="Convergence-Intro"}) in the *strong topology* of the $L^{\infty}_{tx}$-space, and in the super-critical case when $\alpha <2$. Of course, under minor technical adaptations, our results hold for the sub-critical case when $\alpha>2$. Moreover, our approach also allows us to derive an *explicit convergence rate* which highlights some interesting phenomena, that we shall expose in detail below. **The main result.** We focus on the initial value problem for both the non-local ($1<\alpha<2$) and local ($\alpha=2$) Navier-Stokes equations: $$\label{NS} \begin{cases}\vspace{2mm} \partial_t \vec{u}_\alpha =- (-\Delta)^{\alpha/2} \vec{u}_\alpha - (\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha -\vec{\nabla} p_\alpha, \quad \text{div}(\vec{u}_\alpha)=0, \qquad 1<\alpha \leq 2, \\ \vec{u}_\alpha(0,\cdot)=\vec{u}_{0,\alpha}, \end{cases}$$ where $\vec{u}_{0,\alpha}:\mathbb{R}^3\to \mathbb{R}^3$ denotes the (divergence-free) initial datum. Recall that *mild solutions* to equations ([\[NS\]](#NS){reference-type="ref" reference="NS"}) are obtained using Banach's contraction principle by solving the following integral equation (due to Duhamel's formula) $$\label{Mild} \vec{u}_\alpha(t,\cdot)=e^{-(-\Delta)^{\alpha/2} t} \, \vec{u}_{0,\alpha} - \int_{0}^{t} e^{-(-\Delta)^{\alpha/2} (t-\tau)} \, \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha\right)(\tau, \cdot) d \tau, \qquad 1<\alpha \leq 2.$$ In this expression, for $1<\alpha<2$, we have $e^{-(-\Delta)^{\alpha/2} t} f= h_\alpha(t,\cdot)\ast f$, where the kernel $h_\alpha(t,x)$ is the fundamental solution to the fractional heat equation $\partial_t h_\alpha + (-\Delta)^{\frac{\alpha}{2}} h_\alpha=0$ when $t>0$, and $h_\alpha(0,\cdot)=\delta_0$ where $\delta_0$ is the Dirac mass at the origin. For $\alpha=2$, we have $e^{\Delta t} f= h(t,\cdot)\ast f$, where $h(t,x)$ is the well-known heat kernel. The operator $\mathbb{P}$ stands for Leray's projector, and it is well-known that the pressure $p_\alpha$ can be easily deduced from the velocity $\vec{u}_{\alpha}=(u_{\alpha,1},u_{\alpha,2},u_{\alpha,3})$ due to the divergence-free property of this latter. Then, we have $$\label{Pressure} p_\alpha= \frac{1}{-\Delta} \text{div}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha\right)= \sum_{i,j=1}^{3} \mathcal{R}_{i}\mathcal{R}_{j} (u_{\alpha,i} \, u_{\alpha,j}),$$ where $\mathcal{R}_{i}=\frac{\partial_i}{\sqrt{-\Delta}}$ denotes the Riesz transform. In the setting of non-homogeneous Sobolev spaces $H^s(\mathbb{R}^3)$, with $s>1/2$, the local well-posedness theory for mild solutions to the classical Navier-Stokes equations is a well-known issue [@Chemin]. In our next proposition, for the sake of completeness of this article, we revisit this result for the generalized case of equation ([\[NS\]](#NS){reference-type="ref" reference="NS"}) in the space $H^s(\mathbb{R}^3)$ with $s>3/2$. The (technical) constraint $s>3/2$ will be useful later to prove our *key tool*, given in Proposition [Proposition 2](#Key-Lemma){reference-type="ref" reference="Key-Lemma"} below, for the study of the convergence ([\[Convergence-Intro\]](#Convergence-Intro){reference-type="ref" reference="Convergence-Intro"}). Precisely, our approach is based on a sharp study of the convergence $h_\alpha(t,\cdot) \to h(t,\cdot)$ when $\alpha \to 2^{-}$, for the kernels in the mild formulation ([\[Mild\]](#Mild){reference-type="ref" reference="Mild"}). We emphasize that the proof of the proposition below is classical, but we also aim to determine how the existence time of the mild solution $\vec{u}_{\alpha}$, denoted by $T_\alpha$, explicitly depends on the parameter $\alpha$. **Proposition 1**. *Let $1<\alpha \leq 2$ be fixed. Let $s>3/2$, and let $\vec{u}_{0,\alpha} \in H^s(\mathbb{R}^3)$ be a divergence-free initial datum. There exists a time $$\label{Talpha} 0<T_\alpha = \frac{1}{2} \left( \frac{1- \frac{1}{\alpha}}{4C \| \vec{u}_{0,\alpha} \|_{H^s}} \right)^{\frac{\alpha}{\alpha-1}},$$ where $C>0$ is a generic constant, and there exists a unique mild solution $\vec{u}_\alpha$ to equation ([\[NS\]](#NS){reference-type="ref" reference="NS"}) such that $$\vec{u}_\alpha \in \mathcal{C}\big([0,T_\alpha], H^s(\mathbb{R}^3)\big) \quad \mbox{and} \quad p_\alpha \in \mathcal{C}\big([0,T_\alpha], H^s(\mathbb{R}^3)\big).$$* **Remark 1**. *Note that $0<T_\alpha$ as long as $1<\alpha$.* Once we have stated this proposition, we rigorously studied the convergence presented in ([\[Convergence-Intro\]](#Convergence-Intro){reference-type="ref" reference="Convergence-Intro"}). For the non-local case (when $1<\alpha<2$), we consider a family of initial data $(\vec{u}_{0,\alpha})_{1<\alpha <2} \subset H^s(\mathbb{R}^3)$ and the resulting family of solutions $(\vec{u}_{\alpha},p_\alpha)_{1<\alpha<2}$ obtained in Proposition [Proposition 1](#Prop:LWP){reference-type="ref" reference="Prop:LWP"}. Similarly, for the local case (when $\alpha=2$), we consider the initial datum $\vec{u}_{0,2}\in H^s(\mathbb{R}^3)$ and its associated solution $(\vec{u}_2,p_2)\in \mathcal{C}([0,T_2], H^s(\mathbb{R}^3))$. Our starting point is to assume the following convergence of initial data in the strong topology of the space $H^s(\mathbb{R}^3)$: $$\label{Conv-Data} \vec{u}_{0,\alpha} \to \vec{u}_{0,2}, \quad \alpha \to 2^{-}.$$ This assumption yields the following important facts. On one hand, this convergence will allow us to find a quantity $0<\varepsilon<\ll 1$ and a time $T_0$, depending only on $\varepsilon$, such that $$\label{Lower-bound-time} T_0\leq T_\alpha, \quad \mbox{for all} \quad 1+\varepsilon < \alpha \leq 2.$$ See Appendix [5](#AppendixA){reference-type="ref" reference="AppendixA"} for a rigorous justification of this fact. Consequently, for $1+\varepsilon<\alpha \leq 2$, each solution $(\vec{u}_{\alpha},p_\alpha)$ is defined at least on the time interval $[0,T_0]$, and this fact will be used when studying ([\[Convergence-Intro\]](#Convergence-Intro){reference-type="ref" reference="Convergence-Intro"}). On the other hand, since $s>3/2$, the space $H^s(\mathbb{R}^3)$ is continuously embedded in the space $L^\infty(\mathbb{R}^3)$, and the convergence ([\[Conv-Data\]](#Conv-Data){reference-type="ref" reference="Conv-Data"}) also holds in $L^\infty(\mathbb{R}^3)$. Thus, for the family of velocities $\vec{u}_{\alpha}$, we shall prove the following *uniform convergence*: $$\label{Convergence-velocities} \vec{u}_{\alpha}\to \vec{u}_2, \quad \alpha \to 2^{-}, \quad \mbox{in}\quad L^{\infty}([0,T_0]\times \mathbb{R}^3).$$ Recall that the pressures $p_\alpha$ are defined through Riesz transforms and the velocities $\vec{u}_{\alpha}$ in the expression ([\[Pressure\]](#Pressure){reference-type="ref" reference="Pressure"}). Nevertheless, since Riesz transforms are not bounded in the $L^{\infty}-$space, we need to consider the larger space $BMO(\mathbb{R}^3)$. See [@Grafakos Chapter $3$] for a definition and some properties of this space. In this setting, convergence ([\[Convergence-velocities\]](#Convergence-velocities){reference-type="ref" reference="Convergence-velocities"}) yields to prove: $$\label{Convergence-pressures} p_{\alpha}\to p_2, \quad \alpha \to 2^{-}, \quad \mbox{in}\quad L^{\infty}([0,T_0], BMO(\mathbb{R}^3)).$$ Furthermore, our main contribution is to quantify how fast the convergences ([\[Convergence-velocities\]](#Convergence-velocities){reference-type="ref" reference="Convergence-velocities"}) and ([\[Convergence-pressures\]](#Convergence-pressures){reference-type="ref" reference="Convergence-pressures"}) hold. To this end, for a parameter $\kappa>0$ fixed, we shall assume a convergence rate of initial data (see ([\[Conv-Rate-Data\]](#Conv-Rate-Data){reference-type="ref" reference="Conv-Rate-Data"}) below) which is measured in terms of $\kappa$. We thus aim to study when this convergence rate persists for solutions. In this context, we present our main result: **Theorem 1**. *Let $\displaystyle{(\vec{u}_{0,\alpha})_{1+\varepsilon<\alpha \leq 2}}$ be an initial data family, where $\vec{u}_{0,\alpha} \in H^s(\mathbb{R}^3)$ with $s>3/2$. Then, let $$(\vec{u}_\alpha, p_\alpha)_{1+\varepsilon<\alpha\leq 2} \subset \mathcal{C}\big([0,T_0], H^s(\mathbb{R}^3)\big),$$ be the associated family of solutions to equation ([\[NS\]](#NS){reference-type="ref" reference="NS"}), obtained in Proposition [Proposition 1](#Prop:LWP){reference-type="ref" reference="Prop:LWP"}.* *We assume the convergence given in ([\[Conv-Data\]](#Conv-Data){reference-type="ref" reference="Conv-Data"}). Moreover, for a parameter $\kappa>0$, we assume the convergence rate of initial data $$\label{Conv-Rate-Data} \| \vec{u}_{0,\alpha} - \vec{u}_{0,2} \|_{L^\infty} \leq {\bf c} (2-\alpha)^\kappa,$$ where ${\bf c}>0$ is a generic constant. There exists a constant $0<{\bf C}(T_0) \sim 1+T_0 +T^{2}_{0}$, depending on the initial datum $\vec{u}_{0,2}$, the quantity $\varepsilon$, the constant ${\bf c}$ and the time $T_0$, such that for all $1+\varepsilon < \alpha < 2$ the following estimate holds: $$\label{Conv-Rate-Sol} \sup_{0\leq t \leq T_0} \Big( \| \vec{u}_\alpha(t,\cdot) - \vec{u}_2(t,\cdot)\|_{L^\infty} + \| p_\alpha(t,\cdot) - p_2(t,\cdot)\|_{BMO}\Big) \leq {\bf C}(T_0) \Big( (2-\alpha)+(2-\alpha)^\kappa \Big).$$* Some remarks have been provided in order here. The uniform convergence ([\[Conv-Rate-Sol\]](#Conv-Rate-Sol){reference-type="ref" reference="Conv-Rate-Sol"}) is stronger than the ones obtained in the aforementioned works [@Bieganowski; @Dlotko; @Fernandez-Salort]. Moreover, in contrast to these works, we also derive a convergence rate (when $\alpha \to 2^{-}$) of the order $(2-\alpha)+(2-\alpha)^\kappa$. **Remark 2**. *In this last expression, we observe that increasing values of $\kappa$ makes our hypothesis ([\[Conv-Rate-Data\]](#Conv-Rate-Data){reference-type="ref" reference="Conv-Rate-Data"}) strong, but *this fact does not persist for solutions*. Precisely, for $\alpha$ sufficiently close to $2$, we obtain $$(2-\alpha)+(2-\alpha)^\kappa \sim (2-\alpha), \quad \mbox{when} \quad 1<\kappa,$$ and consequently, the convergence rate of solutions is *slower* than the one of initial data.* To understand this *unexpected phenomenon*, let us recall that mild solution to equation ([\[NS\]](#NS){reference-type="ref" reference="NS"}) are given in expression ([\[Mild\]](#Mild){reference-type="ref" reference="Mild"}), where the main difference between the fractional case and the classical one are the kernels $h_\alpha(t,\cdot)$ and $h(t,\cdot)$ respectively. In Proposition [Proposition 2](#Key-Lemma){reference-type="ref" reference="Key-Lemma"} below, we rigorously prove the convergence $h_\alpha(t,\cdot)\to h(t,\cdot)$ (when $\alpha \to 2^{-}$) with a *optimal* convergence rate of the order $(2-\alpha)$. Therefore, the convergence rate of solutions is given by a *competition* between the *assumed* convergence rate of initial date and the *phenomenological* convergence rate of the kernels in the mild formulation. **Remark 3**. *In the particular case of the same initial data for whole the family of equations ([\[NS\]](#NS){reference-type="ref" reference="NS"}): $\vec{u}_{0,\alpha}=\vec{u}_{0,2}$ for all $1+\varepsilon< \alpha <2$, the estimate ([\[Conv-Rate-Sol\]](#Conv-Rate-Sol){reference-type="ref" reference="Conv-Rate-Sol"}) becomes $$\sup_{0\leq t \leq T_0} \big( \| \vec{u}_\alpha(t,\cdot) - \vec{u}_2(t,\cdot)\|_{L^\infty} + \| p_\alpha(t,\cdot) - p_2(t,\cdot)\|_{BMO}\big) \leq {\bf C}(T_0) (2-\alpha),$$ where the convergence rate is purely determined by the convergence of the kernels $h_\alpha(t,x)\to h(t,x)$.* In the case of small initial data, it is well known that mild solutions to equation ([\[NS\]](#NS){reference-type="ref" reference="NS"}) are global in time, see [@PGLR Theorem $7.3$]. In this setting, we have **Corollary 1**. *Under the same hypothesis as in Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"}, assume that $$\label{Small-data} \sup_{1+\varepsilon<\alpha \leq 2}\| \vec{u}_{0,\alpha} \|_{H^s} \ll 1.$$ Then, for all $1+\varepsilon < \alpha < 2$, our main estimate ([\[Conv-Rate-Sol\]](#Conv-Rate-Sol){reference-type="ref" reference="Conv-Rate-Sol"}) writes down as $$\label{Conv-Rate-Sol-global} \sup_{0\leq t \leq T} \Big( \| \vec{u}_\alpha(t,\cdot) - \vec{u}_2(t,\cdot) \|_{L^\infty} + \| p_\alpha(t,\cdot) - p_2(t,\cdot) \|_{BMO}\Big) \\ \leq \, {\bf C}(T) \Big( (2-\alpha)+ (2-\alpha)^\kappa\Big),$$ where, for all time $0<T<+\infty$ we have ${\bf C}(T) \sim 1+T+T^2$. Moreover, the limit $\vec{u}_2$ is also a Leray's solution to the classical Navier-Stokes equations.* The convergence result presented in Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"} also allows us to study the convergence from the non-local to the local Navier-Stokes equation in the space $L^{p}((0,T_0), L^q(\mathbb{R}^3))$. In the above estimate, it is interesting to observe that the convergence rate depends on the parameters $\kappa$ and $q$ but not on the parameter $p$. **Corollary 2**. *Under the same hypothesis as in Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"}, for $1\leq p \leq +\infty$ and $2<q<+\infty$ the estimate holds: $$\label{Conv-Rate-Sol-Lp} \left\| \vec{u}_\alpha - \vec{u}_2 \right\|_{L^{p}_{t}L^{q}_{x}} + \left\| p_\alpha - p_2 \right\|_{L^{p}_{t}L^{q}_{x}} \\ \leq \, {\bf C}_{p,q}(T_0) \Big((2-\alpha) + (2-\alpha)^\kappa \Big)^{1-1/q},$$ with ${\bf C}_{p,q}(T_0) \sim (1+T_0+T^{2}_{0})$.* Finally, it is worth mentioning that Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"} also holds for to the two-dimensional case, where the regularity constraint $s>3/2$ is relaxed to $s>1$. In this sense, we complete the previous work [@Dlotko] with a non-local to local convergence result for $2D$ Navier-Stokes mild solutions. On the other hand, Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"} can be generalized to some relevant coupled systems in fluid dynamics, such as the Magneto-hydrodynamic (MHD) equations. See Appendix [6](#AppendixB){reference-type="ref" reference="AppendixB"} for all details. To conclude this section, we would like to make some final comments: as mentioned, the strategy developed to prove Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"} is strongly based on mild solutions of the equations ([\[NS\]](#NS){reference-type="ref" reference="NS"}). In future research, we aim to develop a different approach to study the convergence ([\[Convergence-Intro\]](#Convergence-Intro){reference-type="ref" reference="Convergence-Intro"}) in the setting of Leray's solutions. Moreover, by following some of the ideas in [@Bieganowski; @Fernandez-Salort], we think it would be interesting to study this convergence in the elliptic case of *stationary* (time-independent) solutions. **Organization of the article.** Section [2](#Sec:Prelimonaries){reference-type="ref" reference="Sec:Prelimonaries"} is essentially devoted to the proof of the key Proposition [Proposition 2](#Key-Lemma){reference-type="ref" reference="Key-Lemma"}. In Section [3](#Sec:LWP){reference-type="ref" reference="Sec:LWP"}, for the sake of completeness, we provide a brief proof of Proposition [Proposition 1](#Prop:LWP){reference-type="ref" reference="Prop:LWP"}. Finally, in Section [4](#Sec:NL-Ll){reference-type="ref" reference="Sec:NL-Ll"}, we prove our main results: Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"} and its Corollary [Corollary 2](#Corollary:LpLq){reference-type="ref" reference="Corollary:LpLq"}. # Preliminaries: Non-local to local heat equation {#Sec:Prelimonaries} Recall that the fractional kernel $h_\alpha(t,x)$ is easily defined in the Fourier level by the expression $\displaystyle{\widehat{h_\alpha}(t,\xi)= e^{-t\, \vert \xi \vert^\alpha}}$, while in the classical case, we have $\displaystyle{\widehat{h}(t,\xi)= e^{-t\, \vert \xi \vert^2}}$. In the following proposition, we study the strong convergence of the kernel $h_\alpha(t,x)$ to the heat kernel $h(t,x)$, when $\alpha \to 2^{-}$. This result will be our key tool in the sequel. **Proposition 2**. *Let $s>3/2$. There exists a constant $C=C_s>0$ such that, for all $1<\alpha < 2$ and for all time $0<T<+\infty$, the following estimate from above holds: $$\label{Above} \sup_{0\leq t \leq T} \left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert_{H^{-s}}\leq C \, T ( 2- \alpha ).$$ Moreover, there exists a constant $c=c_s \leq C$, and there exists quantity $0<\varepsilon_1 \ll 1$ such that for all $1+\varepsilon_1 < \alpha < 2$ the estimate from below holds: $$\label{Below} c \, \frac{T}{2} (2-\alpha) \leq \sup_{0\leq t \leq T} \left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert_{H^{-s}}.$$* **Proof.** We begin by verifying that the expression $\displaystyle{ \left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert^{2}_{H^{-s}}}$ is a continuous function of $t$. For $0\leq t_0, t\leq T$, we have $$\left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert^{2}_{H^{-s}} - \left\Vert h_\alpha(t_0,\cdot)-h(t_0,\cdot) \right\Vert^{2}_{H^{-s}} = \int_{\mathbb{R}^3} \left( \left\vert e^{-\vert \xi \vert^{\alpha} t} -e^{-\vert \xi \vert^2 t}\right\vert^2- \left\vert e^{-\vert \xi \vert^{\alpha} t_0} -e^{-\vert \xi \vert^2 t_0}\right\vert^2 \right)\frac{d \xi}{(1+\vert \xi \vert^2)^{s}}.$$ As $s>3/2$, we have $\displaystyle{\int_{\mathbb{R}^3}\frac{d \xi}{(1+\vert \xi \vert^2)^{s}} <+\infty}$, and a direct application of the well-known dominated convergence theorem yields $$\lim_{t \to t_0} \left( \left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert^{2}_{H^{-s}} - \left\Vert h_\alpha(t_0,\cdot)-h(t_0,\cdot) \right\Vert^{2}_{H^{-s}} \right)=0.$$ Once we have established this continuity property, there exists a time $0 < t_1 \leq T$ such that $$\sup_{0\leq t \leq T} \left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert_{H^{-s}}=\left\Vert h_\alpha(t_1,\cdot)-h(t_1,\cdot) \right\Vert_{H^{-s}}.$$ Now, we prove the estimate ([\[Above\]](#Above){reference-type="ref" reference="Above"}). We write $$\label{Id} \left\Vert h_\alpha(t_1,\cdot)-h(t_1,\cdot) \right\Vert^{2}_{H^{-s}} = \int_{\mathbb{R}^3} \vert e^{-\vert \xi \vert^{\alpha} t_1} -e^{-\vert \xi \vert^2 t_1}\vert^2 \frac{d \xi}{(1+\vert \xi \vert^2)^{s}}.$$ For $\xi \in \mathbb{R}^n\setminus \{0\}$ fixed, and for $1<\alpha < 2 +\delta$ (with $\delta>0$), we define the function $$\label{function} f_\xi (\alpha)= e^{- t_1 \vert \xi \vert^\alpha},$$ and by computing its derivative with respect to the variable $\alpha$, we get $$\label{function-der} f^{'}_{\xi}(\alpha)=- t_1\, e^{- t_1 \vert \xi \vert^\alpha} \, \vert \xi \vert^\alpha \ln (\vert \xi \vert)$$ Then, by the mean value theorem (in the variable $\alpha$), we can write $$\vert f_\xi(\alpha)-f_\xi(2) \vert \leq \Vert f^{'}_{\xi} \Vert_{L^{\infty}([1,2+\delta])}\, \vert 2-\alpha \vert.$$ Furthermore, we have the following uniform estimate with respect to the variable $\xi$: $$\label{Estim-Tech} \sup_{\xi \in \mathbb{R}^3} \Vert f^{'}_{\xi} \Vert_{L^{\infty}([1,2+\delta])} \leq C\, T.$$ Indeed, we write $$\left\Vert \Vert f^{'}_{\xi} \Vert_{L^{\infty}([1,2+\delta])} \right\Vert_{L^{\infty}(\mathbb{R}^3)}\leq \left\Vert \Vert f^{'}_{\xi} \Vert_{L^{\infty}([1,2+\delta])} \right\Vert_{L^{\infty}(\vert \xi \vert \leq 1)}+ \left\Vert \Vert f^{'}_{\xi} \Vert_{L^{\infty}([1,2+\delta])} \right\Vert_{L^{\infty}(\vert \xi \vert >1)} =A+B,$$ where we estimate the terms $A$ and $B$, separately. For the term $A$, as we have $\vert \xi \vert \leq 1$, $1<\alpha<2+\delta$, and moreover, since $\displaystyle{\lim_{\vert \xi \vert\to 0^{+}} \vert \xi \vert \ln(\vert \xi \vert)=0}$, we deduce the following control: $$A \leq T \, \left( \sup_{\xi \in \mathbb{R}^3} e^{-t_1 \vert \xi \vert^{2+\delta}} \vert \xi \vert \ln (\vert \xi \vert)\right)\leq C\, T.$$ For the term $B$, since $\vert \xi \vert >1$, we obtain $$B \leq T\, \left( \sup_{\xi \in \mathbb{R}^3} e^{-t_1 \vert \xi \vert} \vert \xi \vert^{2+\delta} \ln (\vert \xi \vert)\right)\leq C\, T.$$ Once we have the estimate ([\[Estim-Tech\]](#Estim-Tech){reference-type="ref" reference="Estim-Tech"}), we can write $$\vert f_\xi(\alpha) - f_\xi(2) \vert \leq C\, T (2-\alpha).$$ Finally, we get back to the identity ([\[Id\]](#Id){reference-type="ref" reference="Id"}) to get $$\begin{split} \Vert h_\alpha(t_1, \cdot) - h(t_1,\cdot) \Vert^{2}_{H^{-s}} = &\, \int_{\mathbb{R}^n} \vert f_\xi (\alpha) - f_\xi(2) \vert^2 \frac{d \xi}{(1+\vert \xi \vert^2)^s} \\ \leq &\, C\, T^2\, (2-\alpha)^2\, \int_{\mathbb{R}^3} \frac{d \xi}{(1+\vert \xi \vert^2)^s} \leq \, C_s \, T^2\, ( 2-\alpha )^2. \end{split}$$ We prove now the estimate ([\[Below\]](#Below){reference-type="ref" reference="Below"}). We write $$\sup_{0\leq t \leq T} \left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert^{2}_{H^{-s}} \geq \int_{2<|\xi |<4} \left\vert e^{-\vert \xi \vert^{\alpha} \frac{T}{2}} -e^{-\vert \xi \vert^2 \frac{T}{2}}\right\vert^2 \frac{d \xi}{(1+\vert \xi \vert^2)^{s}},$$ where we must study the expression at the right-hand side. By the function $f_{\xi}(\alpha)$ defined in ([\[function\]](#function){reference-type="ref" reference="function"}) (with $\frac{T}{2}$ instead of $t_1$) we have $$\int_{2<|\xi |<4} \left\vert e^{-\vert \xi \vert^{\alpha} \frac{T}{2}} -e^{-\vert \xi \vert^2 \frac{T}{2}}\right\vert^2 \frac{d \xi}{(1+\vert \xi \vert^2)^{s}}= \int_{2<|\xi |<4} \vert f_\xi(\alpha) - f_\xi(2)\vert^2 \frac{d \xi}{(1+\vert \xi \vert^2)^{s}}.$$ Moreover, by the expression ([\[function-der\]](#function-der){reference-type="ref" reference="function-der"}) (always with $\frac{T}{2}$ instead of $t_1$) and the Taylor formula we get $$\begin{split} | f_\xi(\alpha) - f_\xi(2)| =&\, | f'_{\xi}(2) (2-\alpha) + o (2-\alpha)|\\ \geq &\, | f'_{\xi}(2) (2-\alpha) - (- o (2-\alpha))|\\ \geq &\, |f'_{\xi}(2)| (2-\alpha) - |o (2-\alpha)|. \end{split}$$ Then, for $\frac{|f'_{\xi}(2)|}{2}>0$ there exists $0<\varepsilon_1\ll 1$ such that for $1+\varepsilon_1<\alpha <2$ we have $|o (2-\alpha)| \leq \frac{|f'_{\xi}(2)|}{2}(2-\alpha)$, hence, we get back to the previous estimate to obtain $$| f_\xi(\alpha) - f_\xi(2)| \geq \frac{|f'_{\xi}(2)|}{2}(2-\alpha).$$ Once we have this estimate at our disposal, we get back to the last integral to write $$\int_{2<|\xi |<4} \vert f_\xi(\alpha) - f_\xi(2)\vert^2 \frac{d \xi}{(1+\vert \xi \vert^2)^{s}} \geq \left(\frac{T}{2}\right)^2 \left( \min_{2<|\xi|<4} \frac{|f'_{\xi}(2)|}{2} \right)^2 \int_{2<|\xi|<2} \frac{d\xi}{(1+|\xi|^2)^s} =c^{2}_{s} \left(\frac{T}{2}\right)^2 (2-\alpha)^2,$$ which yields the wished estimate ([\[Below\]](#Below){reference-type="ref" reference="Below"}). Proposition [Proposition 2](#Key-Lemma){reference-type="ref" reference="Key-Lemma"} is proven. $\blacksquare$ # Proof of Proposition [Proposition 1](#Prop:LWP){reference-type="ref" reference="Prop:LWP"} {#Sec:LWP} The proof is rather standard, so we will only detail the main estimates. For a time $0<T<+\infty$, we consider the Banach space $\mathcal{C}\big([0,T], H^s(\mathbb{R}^3)\big)$, endowed with its natural norm $\| \cdot \|_{L^{\infty}_{t} H^{s}_{x}}$. On the right-hand side of equation ([\[Mild\]](#Mild){reference-type="ref" reference="Mild"}), the linear term is straightforward to estimate, and we have $\displaystyle{\left\| e^{-(-\Delta)^{\alpha/2} t} \, \vec{u}_{0,\alpha} \right\|_{L^{\infty}_{t} H^{s}_{x}} \leq \| \vec{u}_{0,\alpha} \|_{H^s}}$. Thereafter, for $0<t<T$ fixed, the bilinear term is estimated as follows $$\left\| \int_{0}^{t} e^{-(-\Delta)^{\alpha/2} (t-\tau)} \, \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha\right)(\tau, \cdot) d \tau \right\|_{H^s} \leq C \int_{0}^{t} \| \vec{\nabla} h_\alpha(t-\tau,\cdot) \|_{L^1} \, \| \vec{u}_\alpha \otimes \vec{u}_\alpha (\tau, \cdot) \|_{H^s} d \tau.$$ From [@Yu-Zhai Lemma $2.2$], we have $\displaystyle{ \| \vec{\nabla} h_\alpha(t-\tau,\cdot) \|_{L^1} \leq C (t-\tau)^{-\frac{1}{\alpha}}}$. On the other hand, since $s>3/2$, using the product laws in Sobolev spaces, we can write $\displaystyle{\| \vec{u}_\alpha \otimes \vec{u}_\alpha (\tau, \cdot) \|_{H^s} \leq C \| \vec{u}_\alpha (\tau,\cdot) \|^{2}_{H^s}}$. We thus obtain $$C \int_{0}^{t} \| \vec{\nabla} h_\alpha(t-\tau,\cdot) \|_{L^1} \, \| \vec{u}_\alpha \otimes \vec{u}_\alpha (\tau, \cdot) \|_{H^s} d \tau \leq C \left( \int_{0}^{t} (t-\tau)^{-\frac{1}{\alpha}}\right) \| \vec{u}_\alpha \|^{2}_{L^{\infty}_{t} H^{s}_{x}} \leq C\, \frac{T^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}}\, \| \vec{u}_\alpha \|^{2}_{L^{\infty}_{t} H^{s}_{x}}.$$ The existence and uniqueness of a mild solution $\vec{u}_\alpha$ follows from Picard's iterative schema, as long as $\displaystyle{4 C \| \vec{u}_{0,\alpha} \|_{H^s} \frac{T^{1-\frac{1}{\alpha}}}{1-\frac{1}{\alpha}} <1}$, which defines the time $T_\alpha$ as in ([\[Talpha\]](#Talpha){reference-type="ref" reference="Talpha"}). Proposition [Proposition 1](#Prop:LWP){reference-type="ref" reference="Prop:LWP"} is proven. $\blacksquare$ # From non-local to local Navier-Stokes equations {#Sec:NL-Ll} In the following, $C>0$ denotes a generic constant that may change in each line, but it does not depend on the parameter $\alpha$. ## Proof of Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"} {#proof-of-theorem-th-main} For a time $0<T \leq T_0$ fixed, we write $$\label{estim-base} \begin{split} &\, \sup_{0 \leq t \leq T} \Vert \vec{u}_{\alpha}(t,\cdot)-\vec{u}_2 (t,\cdot)\Vert_{L^{\infty}} \\ \leq &\, \sup_{0\leq t \leq T} \left\Vert e^{-(-\Delta)^{\alpha/2} t} \vec{u}_{0,\alpha} - e^{\Delta t} \vec{u}_{0,2} \right\Vert_{L^{\infty}} \\ &\, + \sup_{0\leq t \leq T}\left\Vert \int_{0}^{t} e^{-(-\Delta)^{\alpha/2} (t-\tau)} \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha\right)(\tau, \cdot) d \tau - \int_{0}^{t} e^{\Delta (t-\tau)} \mathbb{P}\left((\vec{u}_2 \cdot \vec{\nabla})\vec{u}_2 \right)(\tau, \cdot) d \tau \right\Vert_{L^{\infty}}\\ =& \, I_{\alpha}+J_{\alpha}. \end{split}$$ We begin by estimating each term on the right. For the term $I_\alpha$, we get $$\label{estim-lin-01} \begin{split} I_\alpha \leq &\, \sup_{0\leq t \leq T} \left\Vert \left( e^{-(-\Delta)^{\alpha/2} t} - e^{\Delta t} \right) \vec{u}_{0,\alpha} \right\Vert_{L^\infty} + \sup_{0\leq t \leq T} \left\Vert e^{\Delta t} \left( \vec{u}_{0,\alpha} - \vec{u}_{0,2}\right) \right\Vert_{L^\infty} \\ = &\, \sup_{0\leq t \leq T} \left\Vert \Big( h_\alpha(t,\cdot)-h(t,\cdot) \Big) \ast \vec{u}_{0,\alpha} \right\Vert_{L^\infty} + \sup_{0\leq t \leq T} \left\Vert h(t,\cdot) \ast \left( \vec{u}_{0,\alpha} - \vec{u}_{0,2}\right) \right\Vert_{L^\infty} \\ = &\, I_{\alpha,1}+I_{\alpha,2}. \end{split}$$ Afterwards, to estimate the term $I_{\alpha,1}$, one can apply the Bessel potential operators $(1-\Delta)^{-s/2}$ and $(1-\Delta)^{s/2}$ to deduce $$I_{\alpha,1} = \sup_{0\leq t \leq T}\left\Vert (1-\Delta)^{-s/2}\Big( h_\alpha(t,\cdot)-h(t,\cdot) \Big) \ast (1-\Delta)^{s/2} \vec{u}_{0,\alpha} \right\Vert_{L^{\infty}}.$$ Thus, thanks to Young's inequalities (with $1+1/\infty=1/2+1/2$), we can write $$\label{estim01} \begin{split} I_{\alpha,1} &\leq C\, \sup_{0\leq t \leq T} \left(\left\Vert (1-\Delta)^{-s/2} \Big( h_\alpha(t,\cdot)-h(t,\cdot) \Big) \right\Vert_{L^{2}}\, \left\Vert (1-\Delta)^{s/2} \vec{u}_{0,\alpha} \right\Vert_{L^{2}}\right) \\ & \leq C\left( \sup_{0\leq t \leq T} \left\Vert h_\alpha(t,\cdot)-h(t,\cdot) \right\Vert_{H^{-s}} \right)\, \left( \sup_{1+\varepsilon < \alpha < 2 } \Vert \vec{u}_{0,\alpha} \Vert_{H^{s}} \right), \end{split}$$ where each of the terms above must be estimated separately. Note that, for the first term on the right-hand side, it is natural to apply the estimate ([\[Above\]](#Above){reference-type="ref" reference="Above"}) proven in Proposition [Proposition 2](#Key-Lemma){reference-type="ref" reference="Key-Lemma"}, whereas the second term on the right-hand side can be controlled by the fact that the family $\displaystyle{(u_{0,\alpha})_{1+\varepsilon<\alpha <2}}$ is bounded in $H^s(\mathbb{R}^3)$. Therefore, the term $I_{\alpha,1}$ given in ([\[estim-lin-01\]](#estim-lin-01){reference-type="ref" reference="estim-lin-01"}) can be estimated as follows $$\label{Lim1-01} I_{\alpha,1} \leq C\, T\, (2-\alpha).$$ It is now time to study the term $I_{\alpha,2}$ in ([\[estim-lin-01\]](#estim-lin-01){reference-type="ref" reference="estim-lin-01"}). By Young's inequalities (with $1+ 1/\infty= 1 + 1/ \infty)$, the well-known properties of the heat kernel, and the assumption in ([\[Conv-Rate-Data\]](#Conv-Rate-Data){reference-type="ref" reference="Conv-Rate-Data"}), we have $$\label{Lim-02} I_{\alpha,2} \leq {\bf c} (2-\alpha)^\kappa.$$ Consequently, we set the constant ${\bf C}_1=\max(C, {\bf c})$, and by using equations ([\[Lim1-01\]](#Lim1-01){reference-type="ref" reference="Lim1-01"}) and ([\[Lim-02\]](#Lim-02){reference-type="ref" reference="Lim-02"}), we can derive the following estimate $$\label{Lin} I_\alpha \leq {\bf C}_1\,(1+T)\,\Big( (2+\alpha)+ (2-\alpha)^\kappa\Big).$$ Similarly, the term $J_{\alpha}$ in ([\[estim-base\]](#estim-base){reference-type="ref" reference="estim-base"}) can also be studied separately. $$\label{Estim-J} \begin{split} J_{\alpha} \leq & \sup_{0\leq t \leq T}\left\Vert \int_{0}^{t} h_\alpha(t-\tau,\cdot)\ast \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha\right)(\tau, \cdot) d \tau - \int_{0}^{t} h(t-\tau,\cdot) \ast \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha \right)(\tau, \cdot) d \tau \right\Vert_{L^{\infty}}\\ &+ \sup_{0\leq t \leq T}\left\Vert \int_{0}^{t} h(t-\tau,\cdot) \ast \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha\right)(\tau, \cdot) d \tau - \int_{0}^{t} h(t-\tau,\cdot) \ast \mathbb{P}\left((\vec{u}_2 \cdot \vec{\nabla})\vec{u}_2 \right)(\tau, \cdot) d \tau \right\Vert_{L^{\infty}}\\ \leq & \sup_{0\leq t \leq T} \left\Vert \int_{0}^{t}\Big(h_\alpha(t-\tau,\cdot)-h(t-\tau,\cdot)\Big)\ast \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha\right)(\tau, \cdot) d \tau\right\Vert_{L^{\infty}} \\ &+ \sup_{0\leq t \leq T} \left\Vert \int_{0}^{t} h(t-\tau,\cdot)\ast \mathbb{P}\left((\vec{u}_\alpha \cdot \vec{\nabla})\vec{u}_\alpha - (\vec{u}_2 \cdot \vec{\nabla})\vec{u}_2 \right)(\tau, \cdot) d \tau \right\Vert_{L^{\infty}}=J_{\alpha,1}+J_{\alpha,2}. \end{split}$$ For the term $J_{\alpha,1}$, we can leverage the properties of Leray's projector $\mathbb{P}$, and once again, we apply the operators $(1-\Delta)^{-s/2}$ and $(1-\Delta)^{s/2}$, along with Young's inequalities (with $1+1/\infty= 1/2 + 1/2$), to get the following estimates $$\label{estim-J1} \begin{split} J_{\alpha,1} & \leq \sup_{0\leq t \leq T} \left( \int_{0}^{t} \left\Vert \Big(h_\alpha(t-\tau,\cdot)-h(t-\tau,\cdot)\Big)\ast \mathbb{P}\left(\text{div}(\vec{u}_\alpha \otimes \vec{u}_\alpha)\right)(\tau, \cdot) \right\Vert_{L^{\infty}} d\tau \right) \\ & \leq \sup_{0\leq t \leq T} \left(\int_{0}^{t} \left\Vert \mathbb{P}\Big(\vec{\nabla}h_\alpha(t-\tau,\cdot) - \vec{\nabla}h(t-\tau,\cdot)\Big)\right\Vert_{H^{-s}}\, \left\Vert\left(\vec{u}_\alpha \otimes \vec{u}_\alpha\right)(\tau, \cdot) \right\Vert_{H^s} d\tau \right) \\ &\leq \sup_{0\leq t \leq T} \left( \int_{0}^{t} \left\Vert \vec{\nabla}h_\alpha(t-\tau,\cdot) - \vec{\nabla}h(t-\tau,\cdot)\right\Vert_{H^{-s}}\, \left\Vert \left(\vec{u}_\alpha \otimes \vec{u}_\alpha\right)(\tau, \cdot) \right\Vert_{H^s} d\tau \right) \\ &\leq T \left( \sup_{0\leq t \leq T} \left\Vert \vec{\nabla}h_\alpha(t,\cdot) - \vec{\nabla}h(t,\cdot)\right\Vert_{H^{-s}}\right) \left( \sup_{0\leq t \leq T} \Vert \left(\vec{u}_\alpha \otimes \vec{u}_\alpha\right)(t, \cdot) \Vert_{H^s}\right). \end{split}$$ To control the first term on the right-hand side, we can adapt Lemma [Proposition 2](#Key-Lemma){reference-type="ref" reference="Key-Lemma"} to the function $f_{\xi}(\alpha) = i\xi_j e^{-t|\xi|\alpha}$, with $j = 1, 2, 3$, this manner, we obtain $$\begin{aligned} \sup_{0\leq t \leq T} \left\Vert \vec{\nabla}h_\alpha(t,\cdot) - \vec{\nabla}h(t,\cdot)\right\Vert_{H^{-s}} \leq C T ( 2 - \alpha).\end{aligned}$$ For the remaining term on the right-hand side, we prove that there exists a constant ${\bf C}_2={\bf C}_2(\vec{u}_{2,0}, \varepsilon)$ that is sufficiently large and depends only on $\vec{u}_{0,2}$ and $\varepsilon$, such that the following uniform estimate holds: $$\sup_{1+\varepsilon < \alpha <2} \sup_{0\leq t \leq T} \| (\vec{u}_{\alpha}\otimes \vec{u}_{\alpha})(t,\cdot) \|_{H^s} \leq {\bf C}_2.$$ Indeed, recall that the solution $\vec{u}_\alpha \in \mathcal{C}\big([0,T], H^s(\mathbb{R}^3)\big)$ obtained in Proposition [Proposition 1](#Prop:LWP){reference-type="ref" reference="Prop:LWP"} by the Picard's iterative argument verifies $$\sup_{0\leq t \leq T} \Vert \vec{u}_\alpha(t,\cdot) \Vert_{H^s} \leq \sup_{0\leq t \leq T_\alpha} \Vert \vec{u}_\alpha(t,\cdot) \Vert_{H^s} \leq C\, \Vert \vec{u}_{0,\alpha} \Vert_{H^s}, \quad \text{where } T\leq T_0 \leq T_\alpha.$$ Moreover, based on the assumption ([\[Conv-Data\]](#Conv-Data){reference-type="ref" reference="Conv-Data"}), we have $\displaystyle{\sup_{1+\varepsilon < \alpha < 2} \| \vec{u}_{0,\alpha} \|_{H^s}}\leq {\bf C}_2$. Then, we obtain $$\label{Estim-unif-1} \sup_{1+\varepsilon < \alpha <2} \, \sup_{0\leq t \leq T} \| \vec{u}_{\alpha}(t,\cdot) \|_{H^s} \leq {\bf C}_2.$$ Thus, the desired estimate follows from the fact that $s>3/2$ and, using the product laws in Sobolev spaces, we can write $$\sup_{0 \leq t \leq T} \Vert \left(\vec{u}_\alpha \otimes \vec{u}_\alpha\right)(t,\cdot)\Vert_{H^s} \leq C\, \sup_{0 \leq t \leq T} \Vert \vec{u}_\alpha(t,\cdot)\Vert^{2}_{H^s} \leq \, C \left( \sup_{0 \leq t \leq T} \Vert \vec{u}_\alpha(t,\cdot)\Vert_{H^s} \right)^{2} \leq {\bf C}_2.$$ Returning to estimate ([\[estim-J1\]](#estim-J1){reference-type="ref" reference="estim-J1"}), the above inequality allows us to write $$\label{Lim2} J_{\alpha,1} \leq {\bf C}_2\, T^2 \, \vert 2-\alpha \vert \leq {\bf C}_2\, T^2\, \Big((2-\alpha)+ (2-\alpha)^\kappa\Big).$$ Subsequently, we study the term $J_{\alpha,2}$ given in ([\[Estim-J\]](#Estim-J){reference-type="ref" reference="Estim-J"}). For this propose we combine Leray's projector $\mathbb{P}$ properties and Young's inequalities (with $1+ 1/\infty= 1 + 1/ \infty$) as follows $$\label{eq:30} \begin{split} J_{\alpha,2} &\leq \sup_{0\leq t \leq T} \int_{0}^{t} \left\Vert h(t-\tau,\cdot)\ast \mathbb{P}\big(\text{div}(\vec{u}_\alpha \otimes \vec{u}_\alpha) - \text{div}(\vec{u}_2 \otimes \vec{u}_2)\big)(\tau, \cdot) \right\Vert_{L^{\infty}} d \tau\\ &\leq C\, \sup_{0\leq t \leq T} \int_{0}^{t} \Vert \nabla h(t-\tau,\cdot)\Vert_{L^1} \Vert \mathbb{P}\big((\vec{u}_\alpha \otimes \vec{u}_\alpha) - (\vec{u}_2 \otimes \vec{u}_2)\big)(\tau,\cdot) \Vert_{L^\infty} d\tau. \end{split}$$ Due to the well-known properties of the heat kernel $h(t,\cdot)$, we have $\Vert \nabla h(t-\tau,\cdot)\Vert_{L^1} \leq C (t-\tau)^{-1/2}$. Meanwhile, to estimate the term $\Vert \mathbb{P}\big((\vec{u}_\alpha \otimes \vec{u}_\alpha) - (\vec{u}_2 \otimes \vec{u}_2)\big)(\tau,\cdot) \Vert_{L^\infty}$, we make use of Leray's projector $\mathbb{P}$ properties, the uniform estimate inequality ([\[Estim-unif-1\]](#Estim-unif-1){reference-type="ref" reference="Estim-unif-1"}) and the fact that $s>3/2$. Thus, $$\begin{split} & \, \Vert \mathbb{P}\big((\vec{u}_\alpha \otimes \vec{u}_\alpha) - (\vec{u}_2 \otimes \vec{u}_2)\big)(\tau,\cdot) \Vert_{L^\infty} \\%= \left\Vert \P \big( (\vu_{\alpha}-\vu_{2}) \otimes (\vu_{\alpha}+\vu_{2}) \big) (\tau,\cdot) \right\Vert_{L^\infty} \\ = & \, \left\Vert \big(\vec{u}_{\alpha}(\tau,\cdot)-\vec{u}_{2}(\tau,\cdot)\big) \otimes \mathbb{P}(\vec{u}_{\alpha}+\vec{u}_{2}) (\tau,\cdot) \right\Vert_{L^\infty} \\ \leq &\, \left\Vert \vec{u}_{\alpha}(\tau,\cdot)-\vec{u}_{2}(\tau,\cdot)\right\Vert_{L^\infty} \big( \left\Vert \mathbb{P}(\vec{u}_{\alpha})(\tau,\cdot)\right\Vert_{L^\infty}+ \left\Vert \mathbb{P}(\vec{u}_{2})(\tau,\cdot) \right\Vert_{L^\infty}\big) \\ \leq & \, \left\Vert \vec{u}_{\alpha}(\tau,\cdot)-\vec{u}_{2}(\tau,\cdot)\right\Vert_{L^\infty} \big( \left\Vert\vec{u}_{\alpha}(\tau,\cdot)\right\Vert_{H^s}+ \left\Vert\vec{u}_{2}(\tau,\cdot) \right\Vert_{H^s}\big) \\ \leq & \, {\bf C}_2\, \Vert \vec{u}_{\alpha}(\tau,\cdot)-\vec{u}_{2}(\tau,\cdot) \Vert_{L^\infty}. \end{split}$$ These last two estimations allow us to control [\[eq:30\]](#eq:30){reference-type="eqref" reference="eq:30"} as follows $$\label{Lim3} J_{\alpha,2} \leq \, {\bf C}_2\, \sup_{0\leq t \leq T} \, \int_{0}^{t}(t-\tau)^{-1/2} \, \Vert \vec{u}_{\alpha}(\tau,\cdot)-\vec{u}_{2}(\tau,\cdot) \Vert_{L^\infty} d\tau \leq \, {\bf C}_2\, T^{1/2}\, \left( \sup_{0\leq t \leq T} \Vert \vec{u}_{\alpha}(t,\cdot)-\vec{u}_{2}(t,\cdot) \Vert_{L^\infty} \right).$$ Thus far, we have controlled the terms $I_\alpha$, $J_{\alpha,1}$ and $J_{\alpha,2}$ in ([\[Lin\]](#Lin){reference-type="ref" reference="Lin"}), ([\[Lim2\]](#Lim2){reference-type="ref" reference="Lim2"}), and ([\[Lim3\]](#Lim3){reference-type="ref" reference="Lim3"}), respectively. We set the constant ${\bf C}= \max({\bf C}_1, {\bf C}_2)$, and we get back to ([\[estim-base\]](#estim-base){reference-type="ref" reference="estim-base"}) to write $$\begin{split} \sup_{0\leq t \leq T} \Vert \vec{u}_\alpha(t,\cdot)- \vec{u}_2(t,\cdot)\Vert_{L^\infty} \leq &\, I_\alpha + J_{\alpha,1}+J_{\alpha, 2} \leq I_\alpha + J_{\alpha,1} + {\bf C}\, T^{1/2}\, \left( \sup_{0\leq t \leq T} \Vert \vec{u}_{\alpha}(t,\cdot)-\vec{u}_{2}(t,\cdot) \Vert_{L^\infty} \right). \end{split}$$ In the above estimate, we set a time $0<T_1\leq T$ such that $\displaystyle{ {\bf C}\, T^{1/2}_{1} \leq \frac{1}{2}}$. This way, we derive the following control: $$\sup_{0\leq t \leq T_1} \Vert \vec{u}_\alpha(t,\cdot)- \vec{u}_2(t,\cdot)\Vert_{L^\infty} \leq I_\alpha + J_{\alpha,1}+ \frac{1}{2}\, \left( \sup_{0\leq t \leq T_1} \Vert \vec{u}_{\alpha}(t,\cdot)-\vec{u}_{2}(t,\cdot) \Vert_{L^\infty} \right),$$ and we can write $$\frac{1}{2} \sup_{0\leq t \leq T_1} \Vert \vec{u}_\alpha(t,\cdot)- \vec{u}_2(t,\cdot)\Vert_{L^\infty} \leq I_\alpha + J_{\alpha,1}.$$ Then, by ([\[Lin\]](#Lin){reference-type="ref" reference="Lin"}) and ([\[Lim2\]](#Lim2){reference-type="ref" reference="Lim2"}), we obtain $$\label{Iteration-time} % \begin{split} \sup_{0\leq t \leq T_1} \Vert \vec{u}_\alpha(t,\cdot)- \vec{u}_2(t,\cdot)\Vert_{L^\infty} % & \, {\bf C}(1+T_1+T^{2}_{1}) \, \max\left( (2-\alpha)^\gamma, (2-\alpha)\right)\\ \leq \, {\bf C}(1+T_1+T^{2}_{1}) \, \Big( (2-\alpha)+(2-\alpha)^\kappa \Big). % \end{split}$$ By iterative application of this argument up to time $T_0 > 0$, we have $$\label{Key-estimate} \sup_{0\leq t \leq T_0} \Vert \vec{u}_\alpha(t,\cdot)- \vec{u}_2(t,\cdot)\Vert_{L^\infty} \leq \, {\bf C}(1+T_0+T^{2}_{0}) \, \Big((2-\alpha)+ (2-\alpha)^\kappa\Big).$$ To finish the proof of Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"}, we shall prove that estimate ([\[Key-estimate\]](#Key-estimate){reference-type="ref" reference="Key-estimate"}) yields $$\label{Pressures} \sup_{0\leq t \leq T_0} \Vert p_\alpha(t,\cdot)- p_2(t,\cdot)\Vert_{BMO} \leq \, {\bf C}(1+T_0+T^{2}_{0}) \, \Big((2-\alpha)+ (2-\alpha)^\kappa\Big).$$ Indeed, using expression ([\[Pressure\]](#Pressure){reference-type="ref" reference="Pressure"}), the estimate $\| \mathcal{R}_i f \|_{BMO} \leq C \| f \|_{L^\infty}$ (see, for instance, [@PGLR0 Theorem $6.2$]), and the uniform estimate ([\[Estim-unif-1\]](#Estim-unif-1){reference-type="ref" reference="Estim-unif-1"}), for $0<t\leq T_0$, we write $$\begin{split} \| p_\alpha(t,\cdot)-p_2(t,\cdot)\|_{BMO} \leq &\, C \| \vec{u}_{\alpha} \otimes \vec{u}_{\alpha} (t,\cdot) - \vec{u}_{2} \otimes \vec{u}_{2} (t,\cdot) \|_{L^\infty} \\ \leq &\, C \| \vec{u}_{\alpha}(t,\cdot) - \vec{u}_2(t,\cdot) \|_{L^\infty} \big( \| \vec{u}_{\alpha}(t,\cdot)\|_{L^\infty}+ \| \vec{u}_{2}(t,\cdot)\|_{L^\infty} \big) \\ \leq & \, {\bf C} \, \| \vec{u}_{\alpha}(t,\cdot) - \vec{u}_2(t,\cdot) \|_{L^\infty}, \end{split}$$ which yields ([\[Pressures\]](#Pressures){reference-type="ref" reference="Pressures"}). Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"} is now proven. $\blacksquare$ ## Proof of Corollary [Corollary 1](#Corollary:global){reference-type="ref" reference="Corollary:global"} {#proof-of-corollary-corollaryglobal} First, observe that in the case of global in time mild solutions (under the assumption ([\[Small-data\]](#Small-data){reference-type="ref" reference="Small-data"})) we can iterate ([\[Iteration-time\]](#Iteration-time){reference-type="ref" reference="Iteration-time"}) to obtain ([\[Conv-Rate-Sol-global\]](#Conv-Rate-Sol-global){reference-type="ref" reference="Conv-Rate-Sol-global"}) for any time $0<T<+\infty$. On the other hand, by the uniform estimate ([\[Estim-unif-1\]](#Estim-unif-1){reference-type="ref" reference="Estim-unif-1"}), the limit solution $\vec{u}_2$ verifies $\vec{u}\in L^{\infty}_{loc}([0,+\infty), H^s(\mathbb{R}^3))$, which yields that $\vec{u}_2$ belongs to the energy space $L^{\infty}_{t}L^{2}_{x}\cap (L^{2}_{loc})_t \dot{H}^{1}_{x}$ and it also verifies an energy equality. Corollary [Corollary 1](#Corollary:global){reference-type="ref" reference="Corollary:global"} is proven. $\blacksquare$ ## Proof of Corollary [Corollary 2](#Corollary:LpLq){reference-type="ref" reference="Corollary:LpLq"} {#proof-of-corollary-corollarylplq} Remark that the family of initial data also belongs to the space $L^2(\mathbb{R}^3)$, and by well-known arguments, for $1+\varepsilon<\alpha <1$, we have $$\| \vec{u}_{\alpha}(t,\cdot) \|^{2}_{L^2} \leq \| \vec{u}_{0,\alpha} \|^{2}_{L^2} \leq C \| \vec{u}_{0,\alpha}\|^{2}_{H^s}\leq {\bf C}_2.$$ Estimate ([\[Conv-Rate-Sol-Lp\]](#Conv-Rate-Sol-Lp){reference-type="ref" reference="Conv-Rate-Sol-Lp"}) follows from a standard interpolation argument (in Lebesgue spaces) between the estimate above and ([\[Conv-Rate-Sol\]](#Conv-Rate-Sol){reference-type="ref" reference="Conv-Rate-Sol"}). Then, Corollary [Corollary 2](#Corollary:LpLq){reference-type="ref" reference="Corollary:LpLq"} is proven. $\blacksquare$ # Appendix {#AppendixA} We now prove the lower bound ([\[Lower-bound-time\]](#Lower-bound-time){reference-type="ref" reference="Lower-bound-time"}). Using ([\[Conv-Data\]](#Conv-Data){reference-type="ref" reference="Conv-Data"}), we can set $0<\varepsilon \ll 1$ such that for all $1+\varepsilon < \alpha <2$, we have $\displaystyle{\left| \| \vec{u}_{0,\alpha} \|_{H^s}- \| \vec{u}_{0,2} \|_{H^s} \right| \leq \frac{1}{2} \| \vec{u}_{0,2} \|_{H^s}}$. Thus, we obtain $\displaystyle{\| \vec{u}_{0,\alpha} \|_{H^s} \leq \frac{3}{2} \| \vec{u}_{0,2} \|_{H^s}}$, and we can write $$\frac{1}{2} \left( \frac{1-\frac{1}{\alpha}}{4C \| \vec{u}_{0,2} \|_{H^s}} \right)^{\frac{\alpha}{\alpha-1}} \leq T_\alpha, \quad 1+\varepsilon < \alpha < 2.$$ Furthermore, the expression on the left-hand side is estimated from below by the quantity $$T_0= \frac{1}{2} \max\left[ \left( \frac{1-\frac{1}{1+\varepsilon}}{4C \| \vec{u}_{0,2} \|_{H^s}} \right)^{\frac{2}{\varepsilon}}, \left( \frac{1-\frac{1}{1+\varepsilon}}{4C \| \vec{u}_{0,2} \|_{H^s}} \right)^{1+\varepsilon}\right].$$ Indeed, as we have $1+\varepsilon < \alpha < 2$, then we get $1-\frac{1}{1+\varepsilon}< 1-\frac{1}{\alpha}$, and write $$\frac{1}{2} \left( \frac{1-\frac{1}{1+\varepsilon}}{4 C \Vert \vec{u}_{0,2} \Vert_{H^s}}\right)^{\frac{\alpha}{\alpha-1}} \leq \frac{1}{2}\left( \frac{1-\frac{1}{\alpha}}{4 C\Vert \vec{u}_{0,2} \Vert_{H^s}} \right)^{\frac{\alpha}{\alpha-1}}.$$ Thereafter, for the sake of simplicity, we denote $\displaystyle{A=\frac{1-\frac{1}{1+\varepsilon}}{4 C \Vert \vec{u}_{0,2} \Vert_{H^s}}}$, and we have $$\frac{1}{2} A^{\frac{\alpha}{\alpha-1}} \leq \frac{1}{2}\left[ \frac{1-\frac{1}{\alpha}}{4C \Vert \vec{u}_{0,2} \Vert_{H^s}} \right]^{\frac{\alpha}{\alpha-1}}.$$ Now, let us study the expression $\frac{\alpha}{\alpha-1}$. Since $1+\varepsilon < \alpha <2$, then we get $1+\varepsilon < \frac{\alpha}{\alpha-1} < \frac{2}{\varepsilon}$. Thus, on one hand, if the quantity $A$ above verifies $A<1$, then we have $\displaystyle{\frac{1}{2}A^{\frac{2}{\varepsilon}} \leq \frac{1}{2}A^{\frac{\alpha}{\alpha-1}}}$. On the other hand, if the quantity $A$ satisfies $1 \leq A$, then we obtain $\displaystyle{\frac{1}{2}A^{1+\varepsilon} \leq \frac{1}{2}A^{\frac{\alpha}{\alpha-1}}}$. # Appendix {#AppendixB} For $1<\alpha,\beta \leq 2$, we consider the initial value problem for the MHD equations $$\label{MHD} \begin{cases}\vspace{2mm} \partial_t \vec{u}= -(-\Delta)^{\alpha/2} \vec{u}- (\vec{u}\cdot \vec{\nabla})\vec{u}+ (\vec{b}\cdot \vec{\nabla} )\vec{b}-\vec{\nabla} p, \quad \text{div}(\vec{u})=0, \\ \vspace{2mm} \partial_t \vec{b}= -(-\Delta)^{\beta/2} \vec{b}- (\vec{u}\cdot \vec{\nabla})\vec{b}+ (\vec{b}\cdot \vec{\nabla} )\vec{u}, \quad \text{div}(\vec{b})=0, \\ \vec{u}_0(t,\cdot)=\vec{u}_0, \quad \vec{b}(0,\cdot)=\vec{b}_0, \end{cases}$$ where $\vec{u}: [0,+\infty) \times \mathbb{R}^3\to \mathbb{R}^3$ and $p:\vec{b}: [0,+\infty) \times \mathbb{R}^3\to \mathbb{R}$ always denote the velocity and the pressure of the fluid, $\vec{b}: [0,+\infty) \times \mathbb{R}^3\to \mathbb{R}^3$ is the magnetic field, and $\vec{u}_0, \vec{b}_0 : \mathbb{R}^3\to \mathbb{R}^3$ are the divergence-free initial data. Recall that mild solutions to the system ([\[MHD\]](#MHD){reference-type="ref" reference="MHD"}) write down as $$\label{Mild1} \begin{split} \vec{u}(t,\cdot)=e^{-(-\Delta)^{\alpha/2} t} \, \vec{u}_{0} - \underbrace{\int_{0}^{t} e^{-(-\Delta)^{\alpha/2} (t-\tau)} \, \mathbb{P}\left((\vec{u}\cdot \vec{\nabla})\vec{u}\right)(\tau, \cdot) d \tau}_{B_1(\vec{u},\vec{u})} + \underbrace{ \int_{0}^{t} e^{-(-\Delta)^{\alpha/2} (t-\tau)} \, \mathbb{P}\left( (\vec{b}\cdot \vec{\nabla})\vec{b}\right)(\tau, \cdot) d \tau}_{B_{2}(\vec{b},\vec{b})}, \end{split}$$ $$\label{Mild2} \begin{split} \vec{b}(t,\cdot)= e^{-(-\Delta)^{\beta/2} t} \, \vec{b}_{0} - \underbrace{\int_{0}^{t} e^{-(-\Delta)^{\beta/2} (t-\tau)} \, \mathbb{P}\left((\vec{u}\cdot \vec{\nabla})\vec{b}\right)(\tau, \cdot) d \tau}_{B_3(\vec{u},\vec{b})}+ \underbrace{\int_{0}^{t} e^{-(-\Delta)^{\beta/2} (t-\tau)} \, \mathbb{P}\left((\vec{b}\cdot \vec{\nabla})\vec{u}\right)(\tau, \cdot) d \tau}_{B_4(\vec{b},\vec{u})}, \end{split}$$ and the pressure term is related to the velocity $\vec{u}$ and the magnetic field $\vec{b}$ by the well-known expression $$\label{PressureMHD} p=\sum_{i,j=1}^{3} \mathcal{R}_{i}\mathcal{R}_{j} (u_{i} \, u_{j}+b_{i} \, b_{j}).$$ As in Proposition [Proposition 1](#Prop:LWP){reference-type="ref" reference="Prop:LWP"}, the existence of local-in-time mild $H^{s}$-solutions (with $s>3/2$) is rather classical, and we can state the following result adapted to the coupled system ([\[MHD\]](#MHD){reference-type="ref" reference="MHD"}): **Proposition 3**. *Let $1<\alpha,\beta \leq 2$ be fixed. Let $s>3/2$ and let $\vec{u}_{0,\alpha}, \vec{b}_{0,\beta} \in H^s(\mathbb{R}^3)$ be a divergence-free initial datum. There exists a time $$\label{Talpha-beta} 0<T_{\alpha,\beta} = \frac{1}{2} \min\left[\left( \frac{1- \frac{1}{\alpha}}{4C \| \vec{u}_{0,\alpha} \|_{H^s}} \right)^{\frac{\alpha}{\alpha-1}},\left( \frac{1- \frac{1}{\beta}}{4C \| \vec{b}_{0,\beta} \|_{H^s}} \right)^{\frac{\beta}{\alpha-1}}\right],$$ where $C>0$ is a generic constant, and there exists a unique mild solution $\vec{u}_{\alpha,\beta}, \vec{b}_{\alpha,\beta}$ to the system ([\[MHD\]](#MHD){reference-type="ref" reference="MHD"}), such that $$\vec{u}_{\alpha,\beta}, \ \vec{b}_{\alpha,\beta} \in \mathcal{C}\big([0,T_{\alpha,\beta}], H^s(\mathbb{R}^3)\big) \quad \mbox{and} \quad p_{\alpha,\beta} \in \mathcal{C}\big([0,T_{\alpha,\beta}], H^s(\mathbb{R}^3)\big).$$* As before, we shall assume the following strong convergence in the space $H^s(\mathbb{R}^3)$: $$\label{Conv-Data-MHD} \vec{u}_{0,\alpha} \to \vec{u}_{0,2} \quad \alpha \to 2, \quad \vec{b}_{0,\beta} \to \vec{b}_{0,2} \quad \beta \to 2,$$ which yields the existence of a parameter $0<\varepsilon\ll 1$ and a time $T_0$ (depending on $\varepsilon$), such that the entire family of solutions $(\vec{u}_{\alpha,\beta}, \vec{b}_{\alpha,\beta})_{1+\varepsilon<\alpha,\beta\leq 2}$ is at least well-defined in the interval $[0,T_0]$. See always Appendix [5](#AppendixA){reference-type="ref" reference="AppendixA"} for a rigorous explanation. Then, the non-local to local convergence adapted to the MHD system reads as follows: **Theorem 2**. *Let $(\vec{u}_{0,\alpha}, \vec{b}_{0,\beta})_{1+\varepsilon<\alpha, \beta \leq 2}$ be an initial data family, where $\vec{u}_{0,\alpha}, \vec{b}_{0,\beta} \in H^s(\mathbb{R}^3)$ with $s>3/2$. Let $\displaystyle{(\vec{u}_{\alpha,\beta}, \vec{b}_{\alpha,\beta}, p_{\alpha,\beta})_{1+\varepsilon<\alpha,\beta\leq 2} \subset \mathcal{C}\big([0,T_0], H^s(\mathbb{R}^3)\big)}$ be the corresponding family of solutions to the system ([\[MHD\]](#MHD){reference-type="ref" reference="MHD"}), given by Proposition [Proposition 3](#Prop:LWP-MHD){reference-type="ref" reference="Prop:LWP-MHD"}.* *We assume the convergence given in ([\[Conv-Data-MHD\]](#Conv-Data-MHD){reference-type="ref" reference="Conv-Data-MHD"}), and we assume the estimates $$\label{Conv-Rate-Data-MHD} \| \vec{u}_{0,\alpha} - \vec{u}_{0,2} \|_{L^\infty} \leq {\bf c_1} (2-\alpha)^{\kappa_1}, \quad \| \vec{b}_{0,\alpha} - \vec{b}_{0,2} \|_{L^\infty} \leq {\bf c_2} (2-\beta)^{\kappa_2}, \quad \mbox{with} \ \ 0<\kappa_1,\kappa_2,$$ and where ${\bf c_1}, {\bf c_2}>0$ are generic constants.* *Then, there exists a constant $0<{\bf C}(T_0)\sim 1+T_0 +T^{2}_{0}$, depending on $\vec{u}_{0,2}, \varepsilon$, ${\bf c}_1$, ${\bf c}_2$, and the time $T_0$, such that for all $1+\varepsilon < \alpha < 2$ the following estimate holds: $$\label{Conv-Rate-Sol-MHD} \begin{split} \sup_{0\leq t \leq T_0} & \big( \| \vec{u}_\alpha(t,\cdot) - \vec{u}_2(t,\cdot)\|_{L^\infty} + \| \vec{b}_\alpha(t,\cdot) - \vec{b}_2(t,\cdot)\|_{L^\infty} + \| p_\alpha(t,\cdot) - p_2(t,\cdot)\|_{BMO}\big) \\ \leq & {\bf C}(T_0) \max\Big( (2-\alpha)+(2-\alpha)^{\kappa_1}, (2-\beta)+(2-\beta)^{\kappa_2}\Big). \end{split}$$* As in Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"}, we observe that the convergence rate assumed for initial data in ([\[Conv-Rate-Data-MHD\]](#Conv-Rate-Data-MHD){reference-type="ref" reference="Conv-Rate-Data-MHD"}) does not always propagate to solutions due to the prescribed convergence rate of the kernels $h_\alpha(t,\cdot)\to h(t,\cdot)$ and $h_\beta(t,\cdot)\to h(t,\cdot)$, when $\alpha, \beta \to 2^{-}$. **Proof.** We essentially follow the same lines in the proof of Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"}, so it is enough to provide a brief sketch. For a time $0<T\leq T_0$, we start by writing $$\begin{split} &\, \sup_{0 \leq t \leq T} \left( \Vert \vec{u}_{\alpha}(t,\cdot)-\vec{u}_2 (t,\cdot)\Vert_{L^{\infty}} + \Vert \vec{b}_{\beta}(t,\cdot)-\vec{b}_2 (t,\cdot)\Vert_{L^{\infty}} \right)\\ \leq &\, \sup_{0\leq t \leq T} \left( \left\Vert e^{-(-\Delta)^{\alpha/2} t} \vec{u}_{0,\alpha} - e^{\Delta t} \vec{u}_{0,2} \right\Vert_{L^{\infty}} + \left\Vert e^{-(-\Delta)^{\beta/2} t} \vec{u}_{0,\beta} - e^{\Delta t} \vec{u}_{0,2} \right\Vert_{L^{\infty}} \right) \\ &\, + \sup_{0\leq t \leq T}\left\Vert B_1(\vec{u}_{\alpha,\beta}, \vec{u}_{\alpha,\beta}) - B_1(\vec{u}_2,\vec{u}_2) \right\Vert_{L^{\infty}} + \sup_{0\leq t \leq T}\left\Vert B_2(\vec{b}_{\alpha,\beta}, \vec{b}_{\alpha,\beta}) - B_2(\vec{b}_2,\vec{b}_2) \right\Vert_{L^{\infty}} \\ &\, + \sup_{0\leq t \leq T}\left\Vert B_3(\vec{u}_{\alpha,\beta}, \vec{b}_{\alpha,\beta}) - B_3(\vec{u}_2,\vec{b}_2) \right\Vert_{L^{\infty}} + \sup_{0\leq t \leq T}\left\Vert B_4(\vec{b}_{\alpha,\beta}, \vec{u}_{\alpha,\beta}) - B_4(\vec{b}_2,\vec{u}_2) \right\Vert_{L^{\infty}} \\ =& \, I_{\alpha}+ I_{\beta} + \sum_{i=1}^{4} J_{\alpha,\beta,i}. \end{split}$$ Terms $I_{\alpha}$ and $I_{\beta}$ are estimated as in ([\[Lin\]](#Lin){reference-type="ref" reference="Lin"}), and we have $$I_{\alpha}+ I_{\beta} \leq {\bf C}_1 (1+T) \max\Big( (2-\alpha)+(2-\alpha)^{\kappa_1}, (2-\beta)+(2-\beta)^{\kappa_2}\Big).$$ Thereafter, for $i=1,\cdots,4$, the terms $J_{\alpha,\beta,i}$ are estimated as in ([\[Lim2\]](#Lim2){reference-type="ref" reference="Lim2"}) and ([\[Lim3\]](#Lim3){reference-type="ref" reference="Lim3"}) to obtain $$\begin{split} \sum_{i=1}^{4} J_{\alpha,\beta,i} \leq &\, {\bf C}_2 T^2 \max\Big( (2-\alpha)+(2-\alpha)^{\kappa_1}, (2-\beta)+(2-\beta)^{\kappa_2}\Big) \\ &\, + {\bf C}_2 T^{1/2} \sup_{0 \leq t \leq T} \left( \Vert \vec{u}_{\alpha}(t,\cdot)-\vec{u}_2 (t,\cdot)\Vert_{L^{\infty}} + \Vert \vec{b}_{\beta}(t,\cdot)-\vec{b}_2 (t,\cdot)\Vert_{L^{\infty}} \right), \end{split}$$ and we conclude with the proof as in the end of the proof of Theorem [Theorem 1](#Th-Main){reference-type="ref" reference="Th-Main"}. $\blacksquare$ 40 B. Bieganowski, S. Secchi. *Non-local to local transition for ground states of fractional Schrödinger equations on $\mathbb{R}^N$*. J. Fixed Point Theory Appl. 22, 76 (2020). J.Y. Chemin. *Remarques sur l'existence globale pour le système de Navier-Stokes incompressible*, SIAM J. Math. Ann., 23:20-28, (1992). J.W. Cholewa and T. Dlotko. *Fractional Navier-Stokes Equations*, Discrete and Continuous Dynamical Systems, Series B, Volume 23, Number 8 (2018). T. Dlotko. *Navier--Stokes Equation and its Fractional Approximations*. Applied Mathematics & Optimization 77(1) (2018). T. Dubois, F. Jauberteau and R. Temam. *Dynamic Multilevel Methods and the Numerical Simulation of Turbulence*, Cambridge University Press, Cambridge, (1999). J. Fernández Bonder and A.M. Salort. *Stability of solutions for nonlocal problems*, Nonlinear Analysis, Volume 200, 112080, (2020). T. Funaki, D. Surgailis and W. A. Woyczynski. *Gibbs-Cox random fields and Burgers turbulence*, Ann. Appl. Prob. 5, 701-735 (1995). L. Grafakos. *Modern Fourier Analysis, Third Ed.*, Graduate Text in Mathematics 250, Springer (2014). J.M. Holst, E.M. Lunasin and G. Tsogtgerel. *Analysis of a General Family of Regularized Navier-Stokes and MHD Models*, Journal of Nonlinear Science 20(5) (2009). O. Jarrín and G. Loachamín. *From anomalous to classical diffusion in a nonlinear heat equation*. Journal of Elliptic and Parabolic Equations volume 9, pages 1071--1099 (2023). P.G. Lemarié-Rieusset. *Recent developments in the Navier-Stokes problem*, Chapman & Hall/CRC, (2002). P.G. Lemarié-Rieusset. *The Navier-Stokes Problem in the 21st Century*, Chapman & Hall/CRC, (2016). M.M. Meerschaert, D.A. Benson, B. Baeumer. *Multidimensional advection and fractional dispersion*, Phys. Rev. E 59 5026--5028 (1999). Z. Nan and Z. Zheng. *Existence and uniqueness of solutions for Navier--Stokes equations with hyper-dissipation in a large space*, J. Differential Equations 261: 3670--3703 (2016). E. Olson and E.S. Titi. *Viscosity versus vorticity stretching: Global well-posedness for a family of Navier--Stokes-alpha-like models*, Nonlinear Analysis 66: 2427--2458, (2007). S.B. Pope. *Turbulent Flows*, Cambridge University Press (2003). R. Temam. *Navier--Stokes Equations: Theory and Numerical Analysis, revised edition*, AMS Chelsea Publishing, (2001). X. Yu and Z. Zhai. *Well-posedness for fractional Navier--Stokes equations in the largest critical spaces $\dot{B}^{-2(\beta-1)}_{\infty,\infty}(\mathbb{R}^n)$*. Mathematical Methods in Applied Science, 35, pp. 676--683 (2012). G. M. Zaslavsky and S. S. Abdullaev. *Scaling properties and anomalous transport of particles inside the stochastic layer*, Phys. Rev. E 51, No. 5 3901-3910 (1995). # Statements and Declarations {#statements-and-declarations .unnumbered} Data sharing does not apply to this article as no datasets were generated or analyzed during the current study. This work has not received any financial support. In addition, the authors declare that they have no conflicts of interest, and all of them have equally contributed to this paper. [^1]: corresponding author: oscar.jarrin\@udla.edu.ec [^2]: geremy.loachamin\@uni.lu

arxiv_math

--- abstract: | Subgroup stability is a strong notion of quasiconvexity that generalizes convex cocompactness in a variety of settings. In this paper, we characterize stability of a subgroup by properties of its limit set on the Morse boundary. Given $H<G$, both finitely generated, $H$ is stable exactly when all the limit points of $H$ are conical, or equivalently when all the limit points of $H$ are horospherical, as long as the limit set of $H$ is a compact subset of the Morse boundary for $G$. author: - Jacob Garcia bibliography: - references.bib date: September 2023 title: Characterizations of Stability via Morse Limit Sets --- # Introduction An important example of Kleinian groups are called convex cocompact groups. These are exactly the subgroups $H$ whose orbit in $\mathbb{H}^3$ is convex cocompact. Additionally these groups admit compact Kleinian manifolds, and every infinite order element of a convex cocompact group is loxodromic. We highlight some of the other interesting properties of convex cocompact groups in the following theorem. **Theorem 1**. *([@marden_1974; @sullivan_1985]) A Kleinian group $H<\text{Iso}^+(\mathbb{H}^3)\cong PSL_2(\mathbb{C})$ is called convex cocompact if one of the following equivalent conditions hold:* 1. *$H$ acts cocompactly on the convex hull of its limit set $\Lambda H$.* 2. *Any $H$-orbit in $\mathbb{H}^3$ is quasiconvex.* 3. *Every limit point of $H$ is conical.* 4. *$H$ acts cocompactly on $\mathbb{H}^3\cup \Omega$, where $\Omega=\partial\mathbb{H}^3\setminus \Lambda H$. $\square$* However other more recent versions of this relationship have been shown. Swenson showed a generalization of this theorem for Gromov hyperbolic groups equipped with their visual boundaries [@swenson2001], and there has been recent interest in generalizing these relationships beyond the setting of word-hyperbolic groups. For example, convex cocompact subgroups of mapping class groups acting on Teichmüller space, equipped with the Thurston compactification, have been characterized by Farb-Mosher [@Farb2001ConvexCS] and Hamenstädt [@Hamenstadt2005] as exactly the subgroups which determine Gromov hyperbolic surface group extensions. There has also been recent work done in this direction for subgroups of $Out(F_n)$, relating convex cocompact subgroups to hyperbolic extensions of free groups [@Dowdall2017; @Hamenstdt2018; @MR_and_Stab2017]. There has also been interest in creating generalizations which are applicable for any finitely generated group. An important generalization comes from [@Durham_Taylor_ConvexCC_and_Stab_2015], where Durham and Taylor introduced *stability* (see Definition [Definition 28](#stabledefinition){reference-type="ref" reference="stabledefinition"}) to characterize convex cocompact subgroups of a mapping class group in a way which is intrinsic to the geometry of the mapping class group, and in fact, generalizes the notions of convex cocompactness to any finitely generated group. The concept of stability was later generalized to *strongly quasiconvex subgroup*, introduced in [@Tran2019]. We note that a subgroup is stable when it is undistorted and strongly quasiconvex. In the Kleinian, hyperbolic, and mapping class group settings, convex cocompactness is characterized by properties of the limit set on an appropriate boundary [@kent_leininger_2008]. For an arbitrary finitely generated group, it is possible to construct a (quasi-isometric invariant) boundary called the Morse boundary, which was introduced by Cordes in [@cordes_boundry2016] and expanded by Cordes and Hume in [@Cordes_Hume_2016_Stability]. A generalization of convex cocompactness developed by Cordes and Durham, called *boundary convex cocompactness* (see Definition [Definition 29](#BoundaryConvexCocompactDefinition){reference-type="ref" reference="BoundaryConvexCocompactDefinition"}), uses both the Morse boundary and stability to generalize item (1) of Theorem [Theorem 1](#MainKleinianTheorem){reference-type="ref" reference="MainKleinianTheorem"}, see [@boundarycc2016]. The purpose of this paper is fully generalize item (3) of Theorem [Theorem 1](#MainKleinianTheorem){reference-type="ref" reference="MainKleinianTheorem"} to the setting of finitely generated groups, thereby answering [@boundarycc2016 Question 1.15]. In fact, we additionally generalize some other characterizations from the hyperbolic setting found in [@swenson2001]. We summarize the results of this paper in the following theorem: **Theorem 2**. *Let $H$ be a finitely generated group acting by isometries on a proper geodesic metric space $X$. The following are equivalent:* 1. *Any $H$-orbit in $X$ is a stable embedding of $H\rightarrow X$.* 2. *$H$ acts boundary convex cocompactly on $X$.* 3. *Every point in $\Lambda H$ is a conical limit point of $H$, $\Lambda H\neq\emptyset$, and $\Lambda H$ is a compact subset of the Morse boundary of $H$.* 4. *Every point in $\Lambda H$ is a horospherical limit point of $H$, $\Lambda H\neq\emptyset$, and $\Lambda H$ is a compact subset of the Morse boundary of $H$.* **Remark 3**. The result $(1)\Leftrightarrow (2)$ is found in the main theorem of [@boundarycc2016]. We show $(3)\Rightarrow (4)$ in a combination of Propostion [Proposition 33](#MorseConicalImpliesMorseFunnelled){reference-type="ref" reference="MorseConicalImpliesMorseFunnelled"} and Theorem [Theorem 37](#MorseHoroLimitsAreMorseFunnelLimits){reference-type="ref" reference="MorseHoroLimitsAreMorseFunnelLimits"}, using methods similar to [@swenson2001]. We show $(4)\Rightarrow (2)$ in Theorem [Theorem 39](#AllMorseHoroImpliesCoboundedAction){reference-type="ref" reference="AllMorseHoroImpliesCoboundedAction"}, by first showing that non-cobounded actions on the weak convex hull of $\Lambda H$ admit a sequence of points $p_n$ which diverge quickly from the orbit (see Lemma [Lemma 38](#noncoboundedactionsarewellspreadinweakconvexhull){reference-type="ref" reference="noncoboundedactionsarewellspreadinweakconvexhull"}), but then showing that the $p_n$ converge to an element of $\Lambda H$, which ultimately contradicts the conical assumption. We give an alternate proof to $(2)\Rightarrow (3)$ in Proposition [Proposition 41](#BoundaryCCImpliesConical){reference-type="ref" reference="BoundaryCCImpliesConical"} which does not use the main theorem from [@boundarycc2016]. A limit point in $\Lambda H$ is conical if the limit point is accumulated by the orbit in a strong way: every geodesic ray representing the limit point gets boundedly close to the orbit. See Definitions [Definition 6](#hyperbolicdef){reference-type="ref" reference="hyperbolicdef"} and [Definition 31](#conical limit point def){reference-type="ref" reference="conical limit point def"}. In general, a geodesic ray which is constructed from geodesic segments $[x,hx]$ need not stay close to the orbit of $H$ even when $H$ is stably embedded in the hyperbolic setting. For an example, see [@swenson2001 Lemma 3]. A limit point in $\Lambda H$ is horospherical if it is accumulated by the orbit in a similar way: every horoball around a geodesic ray representing the limit point intersects the orbit, see Definition [Definition 31](#conical limit point def){reference-type="ref" reference="conical limit point def"}. We take a moment to provide a broad overview of stability in the recent liturature. In addition to results for the mapping class group from above in [@Farb2001ConvexCS; @kent_leininger_2008; @Hamenstadt2005; @Durham_Taylor_ConvexCC_and_Stab_2015], it is also known that infinite index Morse subgroups of the mapping class group exactly coincide with stable subgroups [@Kim2019], and stable subgroups of mapping class groups (and more generally, stable subgroups of Morse local-to-global groups) have interesting combination theorems [@Russell2021]. Stability has also been studied in the context of Morse local-to-global groups [@CordesRussellSprianoZalloum_RegOfMorseGeoAndGrowth], relatively hyperbolic groups [@MR_and_Stab2017], and hierarchically hyperbolic groups [@AbbottBehrstockDurham_AcylindricalStabilityInHHG_2021; @Russell2023]. It is also known that stable subgroups admit finite height [@Antolin2019]. Comparing Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"} to Theorem [Theorem 1](#MainKleinianTheorem){reference-type="ref" reference="MainKleinianTheorem"}, we see a cocompact action involving a domain of discontinuity in Theorem [Theorem 1](#MainKleinianTheorem){reference-type="ref" reference="MainKleinianTheorem"} which does not appear in Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"}. This is because the standard methods used for showing this property rely on the fact that the (Gromov-)hyperbolic boundary for a word hyperbolic group is a compactification, and thus finding the requisite compact set needed for a cocompact action boils down to finding an appropriate closed subset. In contrast, the Morse boundary usually does not compactify the underlying group, in fact the Morse boundary compactifies a finitely generated group $H$ if and only if $H$ is word hyperbolic, see [@cordes_boundry2016 Theorem 3.10] and [@boundarycc2016 Lemma 4.1]. This leads to an open question: **Open Question 1**. Does there exist an appropriate classification of boundary convex cocompactness via an appropriate action on a domain of discontinuity analog? For other properties in Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"}, we are able to address the need for some compactness in the Morse Boundary by assuming that the limit set of the group, $\Lambda H$, is compact. See Definition [Definition 21](#LimitSetDefinition){reference-type="ref" reference="LimitSetDefinition"} and Corollary [Corollary 23](#LimitSetsInstratumAreCompact){reference-type="ref" reference="LimitSetsInstratumAreCompact"}. It is not possible to remove the compactness condition in either point (3) or (4) of Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"}. For example, consider the group $G=\mathbb{Z}^2*\mathbb{Z}*\mathbb{Z}=\langle a,b\rangle * \langle c\rangle * \langle d\rangle$ with subgroup $H=\langle a,b,c\rangle$. As discussed in [@boundarycc2016 Remark 1.8], $H$ is isometrically embedded and convex in $G$, and so every point of $\Lambda H$ is conical with respect to $H$. In fact all rays representing a point in $\Lambda H$ travel through $H$ infinitely often. However $H$ is not hyperbolic, so $H$ is not stable. See [@boundarycc2016 Section 1.2] for a complete discussion. ## Applications to Mapping Class Groups Convex cocompact subgroups of mapping class groups have been well studied, see [@Farb2001ConvexCS; @Hamenstadt2005], but in particular conical limit point characterizations have been analyzed before. Let $S$ be a finite type surface, $\text{Mod}(S)$ its associated mapping class group, and let $\mathcal{T}(S)$ be its associated Teichmüller space. In [@kent_leininger_2008 Theorem 1.2], it is shown that a subgroup $H$ of $\text{Mod}(S)$ is convex cocompact if and only if all the limit points of $H$ in the Thurston compactification of $\mathcal{T}(S)$ are conical. A straightforward combination of Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"} and [@boundarycc2016 Thorem 1.18] gives the following direct comparison, which uses the intrinsic geometry of $\textup{Mod}(S)$ instead of the geometry of $\mathcal{T}(S)$. **Theorem 4**. *Let $S$ be a finite type surface, let $H<\textup{Mod}(S)$ be finitely generated. Then $H$ is a convex-cocompact subgroup of $\textup{Mod}(S)$ if, and only if, every point in $\Lambda H$ is a conical limit point of $H\curvearrowright\textup{Mod}(S)$, $\Lambda H\neq\emptyset$, and $\Lambda H$ is compact in the Morse boundary of $\textup{Mod}(S)$.* This theorem, combined with the above result of [@kent_leininger_2008], gives the following immediate corollary, which shows that conicality is a strong condition in the setting of mapping class groups: **Corollary 5**. *Let $S$ be a finite type surface, and let $H <\textup{Mod}(S)$ be finitely generated. The following are equivalent:* 1. *Every limit point of $H$ in the Morse boundary of $\textup{Mod}(S)$ is a conical limit point of $H\curvearrowright\textup{Mod}(S)$ and $\Lambda H$ is compact.* 2. *Every limit point of $H$ in the Thurston compactification of $\mathcal{T}(S)$ is a conical limit point of $H\curvearrowright\mathcal{T}(S)$.* We also show that there exists a natural $\textup{Mod}(S)$-equivariant map from $\textup{Mod}(S)$ to $\mathcal{T}(S)$ which sends conical limit points of $H<\textup{Mod}(S)$ in the Morse boundary of $\text{Mod}(S)$ to conical limit point of $H$ in the the Thurston compactification of $\mathcal{T}(S)$. This directly proves the implication $(1)\Rightarrow(2)$ in Corollary [Corollary 5](#TeichApplicationAllConical){reference-type="ref" reference="TeichApplicationAllConical"} without requiring results of [@kent_leininger_2008], and in fact, does not require $H$ to be a convex cocompact subgroup. See Theorem [Theorem 43](#TeichApplicationOneConical){reference-type="ref" reference="TeichApplicationOneConical"} for details. ## Acknowledgements I would like to thank my advisor Matthew Gentry Durham for his guidance and support during this project. Thanks to Elliot Vest for many conversations and for his comments on an earlier draft of this paper. # Background Here we collect some necessary material for the rest of the paper. We refer the reader to [@bridsonhaeflingertextbook] for many definitions and for needed background material, in particular for the construction of the visual boundary in a $\delta$-hyperbolic space. We begin by recalling some relevant definitions and setting some notation. **Definition 6**. Let $(X,d)$ be a proper, geodesic metric space, and let $\alpha:[a,\infty)\rightarrow X$ and $\beta:[b,\infty)\rightarrow X$ be two geodesic rays. - Given a closed set $S\subseteq X$, we define the *closest point projection to $S$* as $\pi_S(x)=\{s\in S:d(s,x)=d(S,x)\}$. - We say $\alpha$ and $\beta$ *$N$-asymptotically fellow-travel*, denoted by $\alpha \sim_N \beta$, if there exists $T\in\mathbb{R}$ so that whenever $t\geq T$, we have $d(\alpha(t),\beta(t))\leq N$. - Given $A\subset X$ and $K\geq0$, we denote the *$K$-neighborhood* of $A$ by $\mathcal{N}_K(A)=\{x\in X:d(x,A)\leq K\}$. In addition, if $X$ is $\delta$-hyperbolic we have the following definitions from [@swenson2001]: - We denote the *horoball about $\alpha$* by $H(\alpha)$ and define it as $H(\alpha)=\bigcup\{\beta([b,\infty)):\beta\sim_{6\delta} \alpha,~b\geq a\}$. - We denote the *funnel about $\alpha$* by $F(\alpha)$ and define it as $F(\alpha)=\{x\in X: d(x,\alpha)\leq d(\pi_\alpha(x),\alpha(a))\}$. - Given a point $x$ in the visual boundary of $X$ and a subset $A\subseteq X$, we say $x$ is a *horopherical limit point of $A$* if, for every geodesic ray $\alpha$ with $\alpha(\infty)=x$, we have $H(\alpha)\cap A\not=\emptyset$. - Given a point $x$ in the visual boundary of $X$ and a subset $A\subseteq X$, we say $x$ is a *funnelled limit point of $A$* if, for every geodesic ray $\alpha$ with $\alpha(\infty)=x$, we have $F(\alpha)\cap A\not=\emptyset$. - Given a point $x$ in the visual boundary of $X$ and a subset $A\subseteq X$, we say $x$ is a *conical limit point of $A$* if there exists $K>0$ such that, for every geodesic ray $\alpha$ with $\alpha(\infty)=x$, we have $\mathcal{N}_K(\alpha)\cap A\not=\emptyset$. We present here for completeness a relaxed version of a claim in [@swenson2001 pg 125] which shows that every horoball of a geodesic ray contains a funnel of an equivalent geodesic ray in a $\delta$-hyperbolic space. **Lemma 7**. *Let $(X,d)$ be a proper, geodesic, $\delta$-hyperbolic metric space, and let $\alpha:[0,\infty)\rightarrow X$ be a geodesic ray. Define $\alpha':[0,\infty)\rightarrow X$ by $\alpha'(t)=\alpha(t+6\delta)$. Then $F(\alpha')\subseteq H(\alpha)$.* *Proof.* See Figure [\[Figure1\]](#Figure1){reference-type="ref" reference="Figure1"}. Let $p\in F(\alpha')$. Construct a geodesic ray $\beta:[b,\infty)\rightarrow X$ such that $\beta\sim_{6\delta}\alpha$ and $\beta(b)=p$ (for details on the existence of such a geodesic ray, we refer to [@bridsonhaeflingertextbook pg 427-428]). Let $q\in \pi_{\alpha'}(p)$ such that $d(\alpha(0),q)=\min\{d(\alpha(0),x):x\in\pi_{\alpha'}(p)\}$, i.e., so that $q$ is the point in $\pi_{\alpha'}(p)$ closest to $\alpha(0)$. Notice that since $p\in F(\alpha')$ we have that $d(p,q)\leq d(q,\alpha'(0))$. Choose $T\geq 6\delta$ so that $q\in [\alpha'(0),\alpha'(T)]$ and so that for all $t\geq T$, we have $d(\alpha(t),\beta(t))<6\delta$. Then $$T-b=d(\beta(T),p)\leq d(\beta(T),\alpha(T))+d(\alpha(T),q)+d(q,p) \leq 6\delta + d(\alpha'(T-6\delta),q) + d(q,\alpha'(0))=6\delta+(T-6\delta).$$ This shows that $b\geq0$, and so $p=\beta(b)\in H(\alpha)$. ◻ \ We also include the complementary statement that every funnel of a geodesic ray contains a horoball of an equivalent geodesic ray. **Lemma 8**. **([@swenson2001 Lemma 5])* Let $(X,d)$ be a proper, geodesic, $\delta$-hyperbolic metric space, and let $\alpha:[0,\infty)\rightarrow X$ be a geodesic ray. Define $\alpha':[0,\infty)\rightarrow X$ by $\alpha'(t)=\alpha(t+12\delta)$. Then $H(\alpha')\subseteq F(\alpha)$. $\square$* The combination of Lemmas [Lemma 7](#funnel_in_horoball_hyperbolic){reference-type="ref" reference="funnel_in_horoball_hyperbolic"} and [Lemma 8](#horoball_in_funnel_hyperbolic){reference-type="ref" reference="horoball_in_funnel_hyperbolic"} give the following relationship, which was originally stated as a corollary in [@swenson2001]. **Corollary 9**. *In a proper, geodesic, $\delta$-hyperbolic metric space, the funneled limit points are exactly the horospherical limit points. $\square$* In a proper, geodesic, $\delta$-hyperbolic metric space, any two geodesic rays $\alpha$ and $\beta$ with $d_{Haus}(\alpha,\beta)<\infty$ can be re-parameterized so that they asymptotically fellow-travel, for a fellow-travelling constant depending only on $\delta$ [@swenson2001 Lemma 4]. This is an important part of the definition of a horoball in Definition [Definition 6](#hyperbolicdef){reference-type="ref" reference="hyperbolicdef"}. In order to define a horoball in a non-hyperbolic space, it will be necessary to develop an analogue of this fact for Morse rays. We begin by recalling the definition. **Definition 10**. *([@cordes_boundry2016 Definition 1.3])* A (quasi)-geodesic $\gamma$ in a metric space is called *$N$-Morse*, where $N$ is a function $[1,\infty)\times[0,\infty)\rightarrow[0,\infty)$, if for any $(K,C)$-quasi-geodesic $\varphi$ with endpoints on $\gamma$, we have $\varphi\subset \mathcal{N}_{N(K,C)}(\gamma)$. We call the function $N$ a *Morse gauge*. We say $\gamma$ is *Morse* if there exists a Morse gauge $N$ so that $\gamma$ is $N$-Morse. **Definition 11**. *([@cordes_boundry2016; @Cordes_Hume_2016_Stability])* Given a Morse gauge $N$ and a basepoint ${\mathfrak{o}}\in X$, the *$N$-Morse stratum*, denoted $X_{{\mathfrak{o}}}^N$, is defined as the set of all points $x$ such that $[{\mathfrak{o}},x]$ is an $N$-Morse geodesic. Each such stratum is $\delta$-hyperbolic for $\delta$ depending only on $N$ [@Cordes_Hume_2016_Stability Proposition 3.2], and thus has a well defined visual boundary, which we denote as $\partial X^N_{{\mathfrak{o}}}$. If $\mathcal{M}$ is the set of all Morse gauges, then there is a natural partial order on $\mathcal{M}$: $N\leq N'$ if $N(K,C)\leq N'(K,C)$ for all $K$ and $C$. Note the natural inclusion $\partial X_{{\mathfrak{o}}}^N \hookrightarrow \partial X_{{\mathfrak{o}}}^{N'}$ is continuous whenever $N\leq N'$ by [@cordes_boundry2016 Corollary 3.2]. We define the *Morse boundary based at ${\mathfrak{o}}$* as $$\partial X_{{\mathfrak{o}}} = \varinjlim_\mathcal{M} \partial X_{{\mathfrak{o}}}^N$$ with the induced direct limit topology. Given a Morse geodesic ray $\alpha$, we denote the associated point in $\partial X_{{\mathfrak{o}}}$ as $\alpha(\infty)$. **Remark 12**. Often when studying the Morse boundary, the basepoint is suppressed from the notation, as the Morse boundary is basepoint independent [@cordes_boundry2016 Proposition 2.5]. However, we will often make use of the basepoint explicitly in the arguments to come, thus we keep it in the notation. The following fact states that subrays of Morse rays are also Morse. This will be especially useful in Section 3, as many of the arguments which describe the relationships between horoballs, funnels, and cones require restriction to a subray, as illustrated in the proof of Lemma [Lemma 7](#funnel_in_horoball_hyperbolic){reference-type="ref" reference="funnel_in_horoball_hyperbolic"}. **Lemma 13**. *([@liu_2021 Lemma 3.1]) Let $X$ be a geodesic metric space. Let $\alpha:I\rightarrow X$ be an $N$-Morse $(\lambda,\epsilon)$-quasi-geodesic where $I$ is an interval of $\mathbb{R}$. Then for any interval $I'\subseteq I$, the $(\lambda,\epsilon)$-quasi-geodesic $\alpha'=\alpha|_{I'}$ is $N'$-Morse where $N'$ depends only on $\lambda,\epsilon$, and $N$. $\square$* We now present a combination of statements which will show that, given one Morse ray and another ray which fellow-travels with the first, then eventually the fellow-travelling constant is determined only by the Morse gauge of the first ray. **Proposition 14**. **([@cordes_boundry2016 Proposition 2.4])* Let $X$ be a geodesic metric space. Let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic ray. Let $\beta:[0,\infty)\rightarrow$ be a geodesic ray such that $d(\alpha(t),\beta(t))<K$ for $t\in[A,A+D]$ for some $A\in[0\infty)$ and $D\geq 6K$. Then for all $t\in[A+2K,A+D-2K]$, $d(\alpha(t),\beta(t))<4N(1,2N(5,0))+2N(5,0)+d(\alpha(0),\beta(0))$.$\square$* The proof of Proposition [Proposition 14](#thatonecordesprop){reference-type="ref" reference="thatonecordesprop"}, as presented in [@cordes_boundry2016], shows the following additional facts: **Corollary 15**. *Let $X$ be a geodesic metric space. Let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic ray. Let $\beta:[0,\infty)\rightarrow$ be a geodesic ray such that $d(\alpha(t),\beta(t))<K$ for $t\in[A,A+D]$ for some $A\in[0,\infty)$ and $D\geq 6K$. Then there exists $x,y\in[0,A+2K]$ such that $d(\alpha(x),\beta(y))<N(5,0)$.$\square$* **Corollary 16**. **([@cordes_boundry2016 Corollary 2.6])* Let $X$ be a geodesic metric space. Let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic ray. Let $\beta:[0,\infty)\rightarrow$ be a geodesic ray such that $d(\alpha(t),\beta(t))<K$ for all $t\in[0,\infty)$ (i.e. $\beta(\infty)=\alpha(\infty)$). Then for all $t\in[2K,\infty)$, $d(\alpha(t),\beta(t))<\max\{4N(1,2N(5,0))+2N(5,0),8N(3,0)\}+d(\alpha(0),\beta(0))$. $\square$* Combining Corollaries [Corollary 15](#cordescornumber1){reference-type="ref" reference="cordescornumber1"} and [Corollary 16](#cordescornumber2){reference-type="ref" reference="cordescornumber2"}, we get the following generalization of [@bridsonhaeflingertextbook Chapter 3, Lemma 3.3]. **Proposition 17**. *Let $X$ be a geodesic metric space. Let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic ray. Let $\beta:[0,\infty)\rightarrow X$ be a geodesic ray such that $d(\alpha(t),\beta(t))<K$ for all $t\in[0,\infty)$ (i.e. $\beta(\infty)=\alpha(\infty)$). Then there exists $T_1,T_2>0$ such that for all $t\in[0,\infty)$, $d(\alpha(T_1+t),\beta(T_2+t))<\max\{4N(1,2N(5,0))+2N(5,0),8N(3,0)\}+N(5,0)$.* *Proof.* By Corollary [Corollary 15](#cordescornumber1){reference-type="ref" reference="cordescornumber1"}, there exists $x,y\geq0$ so that $d(\alpha(x),\beta(y))<N(5,0)$. Define $\alpha'(t)=\alpha(x+t)$ and $\beta'(t)=\beta(y+t)$, and note in particular that $\alpha'(0)=\alpha(x)$ and $\beta'(0)=\beta(y)$. Applying Corollary [Corollary 16](#cordescornumber2){reference-type="ref" reference="cordescornumber2"} to $\alpha'$ and $\beta'$ produces the desired result. ◻ For convenience, we will denote $\delta_N=\max\{4N(1,2N(5,0))+2N(5,0),8N(3,0)\}+N(5,0)$. Using this notation, Proposition [Proposition 17](#betterprop2.4){reference-type="ref" reference="betterprop2.4"} leads to the following generalization of [@swenson2001 Lemma 4]. **Corollary 18**. *Let $X$ be a geodesic metric space. Let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic ray. Let $\beta:[0,\infty)\rightarrow X$ be a geodesic ray such that $\beta(\infty)=\alpha(\infty)$. Then there exists $a\in\mathbb{R}$ and an isometry $\rho:[a,\infty)\rightarrow[0,\infty)$ so that $\alpha\sim_{\delta_N}\beta\circ\rho$.* *Proof.* Apply Proposition [Proposition 17](#betterprop2.4){reference-type="ref" reference="betterprop2.4"} to find $T_1,T_2>0$ so that for all $t\in[0,\infty)$, $d(\alpha(T_1+t),\beta(T_2+t))<\delta_N$. Then let $\rho:[a,\infty)\rightarrow[0,\infty)$ be the unique isometry such that $\rho(T_1)=T_2$. ◻ **Proposition 19**. *Suppose $\alpha:[a,\infty)\rightarrow X$ is an $N$-Morse geodesic ray and $\beta:[b,\infty)\rightarrow X$ is a geodesic ray such that $\beta \sim_{\delta_N}\alpha$ and $\alpha(a)=\beta(b)$. Then $\beta$ is $M$-Morse where $M$ depends only on $N$.* *Proof.* It suffices to show that $d_{Haus}(\alpha,\beta)\leq K$ where $K\geq0$ depends only on $N$. Choose $T>0$ so that $d(\alpha(t),\beta(t))\leq \delta_N$ for all $t\geq T$. Note that $[\beta(b),\beta(t)]*[\beta(t),\alpha(t)]$ is a $(1,2\delta_N)$ quasi-geodesic, so by [@cordes_boundry2016 Lemma 2.1], $d_{Haus}([\alpha(a),\alpha(t)],[\beta(b),\beta(t)]*[\beta(t),\alpha(t)])\leq L$ for some $L$ depending only on $N$. But since $l([\alpha(t),\beta(t)])\leq\delta_N$, we have $d_{Haus}([\alpha(a),\alpha(t)],[\beta(b),\beta(t)])\leq L+\delta_N$. ◻ The above statement leads to the following generalization, which is very similar to [@cordes_boundry2016 Lemma 2.8]. This statement will be useful for showing a generalization of Corollary [Corollary 9](#limit_points_hyperbolic){reference-type="ref" reference="limit_points_hyperbolic"}, since our horoballs and funnels will be restricted to a single Morse stratum. See Theorem [Theorem 37](#MorseHoroLimitsAreMorseFunnelLimits){reference-type="ref" reference="MorseHoroLimitsAreMorseFunnelLimits"}. **Proposition 20**. *Suppose $x\in\partial X_{{\mathfrak{o}}}^N$ for a Morse gauge $N$. Then any geodesic ray $\alpha:[a,\infty)\rightarrow X$ with $\alpha(\infty)=x$ is $M$-Morse, where $M$ depends only on $N$ and the Morse gauge of $[\alpha(a),{\mathfrak{o}}]$.* *Proof.* See Figure [\[Figure2\]](#Figure2){reference-type="ref" reference="Figure2"}. Let $\beta:[b,\infty)\rightarrow X$ be $N$-Morse with $\beta(b)={\mathfrak{o}}, \beta(\infty)=\alpha(\infty)$, and let $N'$ be the Morse gauge of $[\alpha(a),\beta(b)]$. For each $n\in\mathbb{N}$, let $\gamma_n=[\alpha(a),\beta(b+n)]$. Note that $\beta|_{[b,b+n]}$ is Morse for some Morse gauge depending only on $N$ by Lemma [Lemma 13](#MorseSubraysAreMorse){reference-type="ref" reference="MorseSubraysAreMorse"}, and so by [@cordes_boundry2016 Lemma 2.3], $\gamma_n$ in $N''$-Morse for $N''$ depending only on $\max\{N,N'\}$. Then via a straightforward generalization of [@cordes_boundry2016 Lemma 2.10], there exists an $N''$-Morse geodesic ray $\gamma$ with $\gamma_n\rightarrow \gamma$ (uniformly on compact sets) and $\gamma(\infty)=\beta(\infty)$. Then Proposition [Proposition 19](#ParallelRaysWithSameStartareMorse){reference-type="ref" reference="ParallelRaysWithSameStartareMorse"} shows that $\alpha$ is Morse for an appropriate Morse gauge. ◻ \ ## Limit Sets and Weak Convex Hulls We now introduce limit sets and weak convex hulls, and give some useful properties that these sets have. We use these constructions to turn subsets of $X$ into subsets of the Morse boundary, and vice versa. **Definition 21**. ([@boundarycc2016 Definition 3.2]) Let $X$ be a proper, geodesic metric space and let $A\subseteq X$. The *limit set of* $A$, denoted as $\Lambda A$, is the set of points in $\partial X_{{\mathfrak{o}}}$ such that, for some Morse gauge $N$, there exists a sequence of points $(a_k)\subset A\cap X_{{\mathfrak{o}}}^N$ such that $[{\mathfrak{o}},a_k]$ converges (uniformly on compact sets) to a geodesic ray $\alpha$ with $\alpha(\infty)=x$. (Note $\alpha$ is $N$-Morse by [@cordes_boundry2016 Lemma 2.10].) In the case where $H$ acts properly by isometries on $X$, we use $\Lambda H$ to denote the limit set of $H{\mathfrak{o}}$. **Remark 22**. By [@boundarycc2016 Lemma 3.3], $\Lambda H$ can be defined as the limit set of *any* orbit of $H$, we merely choose the orbit $H{\mathfrak{o}}$ for convenience and simplicity in future arguments. We also prove a small fact about limit sets, which is similar to [@boundarycc2016 Lemma 4.1, Proposition 4.2]. **Corollary 23**. *Let $X$ be a proper, geodesic metric space and suppose $A\subseteq X$. If $\Lambda A\subseteq \partial X^N_{{\mathfrak{o}}}$ for some Morse gauge $N$, then $\Lambda A$ is compact.* *Proof.* By [@cordes_boundry2016 Proposition 3.12], this is equivalent to the condition that $\Lambda H$ is closed. ◻ **Remark 24**. By Corollary [Corollary 23](#LimitSetsInstratumAreCompact){reference-type="ref" reference="LimitSetsInstratumAreCompact"} and by [@boundarycc2016 Lemma 4.1], the requirement that $\Lambda H$ is compact is equivalent to the requirement that $\Lambda H$ is contained in the boundary of a single Morse stratum. **Definition 25**. ([@swenson2001; @boundarycc2016]) Let $X$ be a proper, geodesic metric space, and let $A\subseteq X\cup \partial X_{{\mathfrak{o}}}$. Then the *weak convex hull* of $A$, denoted $WCH(A)$, is the union of all geodesic (segments, rays, or lines) of $X$ which have both endpoints in $A$. We take a moment to highlight some nice interactions between the weak convex hull of a compact limit set with the Morse boundary. **Lemma 26**. *([@boundarycc2016 Proposition 4.2]) Let $X$ be a proper geodesic metric space and let $A\subseteq X$ such that $\Lambda A\subseteq \partial X_{{\mathfrak{o}}}^N$ for some Morse gauge $N$. Then there exists a Morse gauge $N'$, depending only on $N$, such that $WCH(\Lambda A)\subset X_{{\mathfrak{o}}}^{N'}$. $\square$* **Lemma 27**. *Let $X$ be a proper geodesic metric space and let $A\subseteq X$ such that $\Lambda A\subseteq \partial X_{{\mathfrak{o}}}^N$ for some Morse gauge $N$. Then $\Lambda(WCH(\Lambda A))\subseteq \Lambda A$.* *Proof.* We may assume $|\Lambda A|>1$. Let $x\in \Lambda(WCH(\Lambda A))$. By Definition [Definition 21](#LimitSetDefinition){reference-type="ref" reference="LimitSetDefinition"}, there exists $x_n\in WCH(\Lambda A)$ such that $[{\mathfrak{o}},x_n]$ converges to a geodesic ray $\gamma$ with $\gamma(\infty)=x$. We show that there exists $K>0$ so that for all $n$ there exists $a_n$ with $[{\mathfrak{o}},x_n]\subseteq \mathcal{N}_K([{\mathfrak{o}},a_n])$. Thus, (a subsequence of) the geodesics $[{\mathfrak{o}},a_n]$ converge to a geodesic ray $\alpha:[0,\infty)\rightarrow X$ with $\alpha(0)={\mathfrak{o}}$, and $\alpha(\infty)=\gamma(\infty)=x$ and so $x\in\Lambda A$. It remains to find $K$ so that $[{\mathfrak{o}},x_n]\subseteq \mathcal{N}_K([{\mathfrak{o}},a_n])$. Fix $n$. Since $x_n\in WCH(\Lambda A)$, $x\in\eta$ where $\eta:(-\infty,\infty):\rightarrow X$ is a geodesic with $\eta(\pm\infty)\in\Lambda A$. So, by Definition [Definition 21](#LimitSetDefinition){reference-type="ref" reference="LimitSetDefinition"}, there exists $a_k^+,a_k^-\in A\cap X^N_{{\mathfrak{o}}}$ so that $[{\mathfrak{o}},a_k^+]$ and $[{\mathfrak{o}},a_k^-]$ converge to geodesics $\beta^+$ and $\beta^-$, respectively, with $\beta^+(\infty)=\eta(\infty)$ and $\beta^-(-\infty)=\eta(-\infty)$. Since $\Lambda A\subseteq \partial X_{{\mathfrak{o}}}^N$, the triangle $\eta\cup\beta^+\cup\beta^-$ is $L$-slim for $L$ depending only on $N$ by [@boundarycc2016 Proposition 3.6], and as $x\in\eta$, there exists $y\in\beta^+\cup\beta^-$ so that $d(x_n,y)\leq L$. Without loss of generality, assume $y\in\beta^+$. Since $[{\mathfrak{o}},a_k^+]$ converges to $\beta^+$ uniformly on compact sets, choose $m$ large enough so that $d(y,[{\mathfrak{o}},a_m^+])\leq 1$. Let $z\in[{\mathfrak{o}},a_m^+]$ so that $d(y,z)\leq 1$. See Figure [\[FigureLimitSetAndWCH\]](#FigureLimitSetAndWCH){reference-type="ref" reference="FigureLimitSetAndWCH"}. Note that the concatenation $[{\mathfrak{o}},x_n]*[x_n,z]$ is a $(1,L+1)$-quasi-geodesic with endpoints on $[{\mathfrak{o}},a_m^+]$. Since $[{\mathfrak{o}},a_m^+]$ is $N$-Morse, we have that $[{\mathfrak{o}},x_n]\subseteq[{\mathfrak{o}},x_n]*[x_n,z]\subseteq \mathcal{N}_{N(1,L+1)}([{\mathfrak{o}},a_m^+])$. Since $K:=N(1,L+1)$ did not depend on the choice of $n$, this completes the proof. ◻ \ Finally, we finish this section by stating the definitions of stability and boundary convex cocompactness here for reference. **Definition 28**. *([@Durham_Taylor_ConvexCC_and_Stab_2015] [@boundarycc2016 Definition 1.3])* If $f:X\rightarrow Y$ is a quasi-isometric embedding between geodesic metric spaces, we say $X$ is a *stable* subspace of $Y$ if there exists a Morse Gauge $N$ such that every pair of points in $X$ can be connected by an $N$-Morse quasigeodesic in $Y$; we call $f$ a *stable embedding*. If $H< G$ are finitely generated groups, we say $H$ is *stable in* $G$ if the inclusion map $i:H\hookrightarrow G$ is a stable embedding. **Definition 29**. *([@boundarycc2016 Definition 1.4])* We say that $H$ acts *boundary convex cocompactly* on $X$ if the following conditions hold: 1. $H$ acts properly on $X$, 2. $\Lambda H$ is nonempty and compact, 3. The action of $H$ on $WCH(\Lambda H)$ is cobounded. # Limit point characterizations in the Morse Boundary The goal of this section is show that, given a set $A\subseteq X$, if $x\in\partial X_{{\mathfrak{o}}}$ is a Morse conical limit point of $A$, then $x$ is a Morse horospherical limit point of $A$. This was first shown in the hyperbolic case in [@swenson2001], here we generalize this fact into the setting of proper geodesic metric spaces. We begin by introducing definitions which generalize horospheres and funnels for Morse rays. **Definition 30** (Horoballs, Funnels). Let $X$ be a proper, geodesic metric space and let ${\mathfrak{o}}\in X$ be some designated point. Let $\alpha:[a,\infty)\rightarrow X$ be an $N'$-Morse geodesic ray, and let $N$ be some, potentially different, Morse gauge. We define the *$N$-Morse horoball around $\alpha$ based at ${\mathfrak{o}}$* as $$H^N_{{\mathfrak{o}}}(\alpha)=\{x\in X^N_{{\mathfrak{o}}}~|~\exists \beta:[b,\infty)\rightarrow X \text{ with }\beta\sim_{\delta_{N'}}\alpha\text{ and }b\geq a\text{ and }\beta(b)=x\}.$$ We define the *$N$-Morse funnel around $\alpha$ based at ${\mathfrak{o}}$* as $$F^N_{{\mathfrak{o}}}(\alpha)=\{x\in X^N_{{\mathfrak{o}}}~|~d(x,\pi_\alpha(x))\leq d(\alpha(a),\pi_\alpha(x))\}.$$ Comparing these definitions to Definition [Definition 6](#hyperbolicdef){reference-type="ref" reference="hyperbolicdef"} shows that a Morse horoball is a horoball about a Morse geodesic intersected with an appropriate Morse stratum, and similarly, a Morse funnel is a funnel about a Morse geodesic intersected with an appropriate Morse stratum. The following three definitions classify points on the Morse boundary by asking if every horoball, funnel, or cone intersects a given subset of $X$. **Definition 31**. Let $X$ be a proper, geodesic metric space and let ${\mathfrak{o}}\in X$ be some designated point. Let $A\subset X$. - We say that $x\in\partial X_{{\mathfrak{o}}}$ is a *Morse horospherical limit point of $A$* if for every Morse geodesic $\alpha$ with $\alpha(\infty)=x$, there exists a Morse gauge $N$ such that $H^{N}_{{\mathfrak{o}}}(\alpha)\cap A\neq\emptyset$. - We say that $x\in\partial X_{{\mathfrak{o}}}$ is a *Morse funnelled limit point of $A$* if for every Morse geodesic $\alpha$ with $\alpha(\infty)=x$, there exists a Morse gauge $N$ such that $F^{N}_{{\mathfrak{o}}}(\alpha)\cap A\neq\emptyset$. - We say that $x\in\partial X_{{\mathfrak{o}}}$ is a *Morse conical limit point of $A$* if there exists $K>0$ such that, for every Morse geodesic $\alpha$ with $\alpha(\infty)=x$, we have that $\mathcal{N}_K(\alpha)\cap A\neq\emptyset$. **Remark 32**. Notice that, in the case where $X$ is a $\delta$-hyperbolic space, these definitions agree with the definitions given in Definition [Definition 6](#hyperbolicdef){reference-type="ref" reference="hyperbolicdef"}, as every geodesic in a $\delta$-hyperbolic space is $N$-Morse for $N$ depending only on $\delta$. In light of this, we will use "conical limit point\" instead of "Morse conical limit point\" for the rest of this paper, except in cases where the difference between these definitions causes confusion. We similarly reduce "Morse horospherical limit point\" and "Morse funneled limit point\" to "horospherical limit point\" and "funnelled limit point,\" respectively. We now begin proving the new implications found in Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"}. We will first show that every conical limit point of $A$ is a funnelled limit point of $A$, and then we will show that the funnelled limit points of $A$ exactly coincide with the horospherical limit points of $A$. These arguments generalize the arguments found in [@swenson2001]. **Proposition 33**. *Let $X$ be a proper, geodesic metric space and let ${\mathfrak{o}}\in X$. Let $A\subseteq X$. If $x\in\partial X_{{\mathfrak{o}}}$ is a conical limit point of $A$, then $x$ is a funneled limit point of $A$.* *Proof.* See Figure [\[Figure4\]](#Figure4){reference-type="ref" reference="Figure4"}. Let $x\in\partial X_{{\mathfrak{o}}}$ be a conical limit point of $A\subseteq X$. Let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic with $\alpha(\infty)=x$. By Lemma [Lemma 13](#MorseSubraysAreMorse){reference-type="ref" reference="MorseSubraysAreMorse"}, there exists a Morse gauge $M$ so that every geodesic sub-ray of $\alpha$ is $M$-Morse. Thus by Definition [Definition 31](#conical limit point def){reference-type="ref" reference="conical limit point def"}, there exists $K\geq 0$ so that every subray of $\alpha$ gets at least $K$ close to $A$. Now define $\alpha'=\alpha|_{[3K,\infty)}$, and let $a\in A$ such that $a\in \mathcal{N}_K(\alpha')$. Then note that $d(a,\pi_\alpha(a))\leq d(a,\pi_{\alpha'}(a))\leq K$, and so $d(\pi_\alpha(a),\pi_{\alpha'}(a))\leq 2K$. By the triangle inequality, $\pi_\alpha(a)\subseteq \alpha|_{[K,\infty)}$. Therefore, $d(\pi_{\alpha}(a),a)\leq K=d(\alpha(0),\alpha(K)) \leq d(\alpha(0),\pi_{\alpha'}(a))$. It remains to show that $a\in X_{{\mathfrak{o}}}^{N'}$ for a Morse gauge $N'$ which is independent of the choice of $a\in A$. Let $L=d({\mathfrak{o}},\alpha(0))$, and let $p\in\pi_{\alpha'}(a)$, and note that $d(p,a)\leq K$. Thus, $[{\mathfrak{o}},\alpha(0)]$ and $[p,a]$ are both $N''$-Morse depending only on $\max\{K,L\}$, and $[{\mathfrak{o}},p]$ is $N'''$-Morse depending only on $N$ by Lemma [Lemma 13](#MorseSubraysAreMorse){reference-type="ref" reference="MorseSubraysAreMorse"}. Since $[{\mathfrak{o}},a]$ is one side of a quadrilateral whose other three sides are $\max\{N'',N'''\}$-Morse, $[{\mathfrak{o}},a]$ is $N'$-Morse where $N'$ does not depend on choice of $a\in A$ by [@cordes_boundry2016 Lemma 2.3]. ◻ \ Our next goal is to show that the funnelled limit points of $A$ coincide with the horospherical limit points of $A$. Towards this end, we show that, given a point $x$ in a horoball of a subray, the projection of $x$ to the subray is coarsely the same as the projection to the base ray. **Lemma 34**. *Suppose $\alpha$ is an $N$-Morse geodesic ray and let $\alpha'$ be a subray. Suppose $x\in H_{{\mathfrak{o}}}^{N'}(\alpha')$. If $\alpha(\infty)\in\partial X_{{\mathfrak{o}}}^{N''}$, then $d_{Haus}(\pi_\alpha(x),\pi_{\alpha'}(x))\leq K$, where $K\geq0$ depends only on $N$, $N'$, and $N''$.* *Proof.* See Figure [\[Figure5\]](#Figure5){reference-type="ref" reference="Figure5"}. Let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic ray and let $\alpha'=\alpha|_{[a,\infty)}$ for some $a\geq0$. By Lemma [Lemma 13](#MorseSubraysAreMorse){reference-type="ref" reference="MorseSubraysAreMorse"}, $\alpha'$ is $M$-Morse for $M$ depending only on $N$. Let $x\in H_{{\mathfrak{o}}}^{N''}(\alpha')$, thus there exists $\beta:[b,\infty)\rightarrow X$ a geodesic ray with $b\geq a$, $\beta(b)=x$, and $\beta\sim_{\delta_M}\alpha'$. By Proposition [Proposition 20](#ParallelRaysAreMorse){reference-type="ref" reference="ParallelRaysAreMorse"}, $\beta$ is $M'$-Morse for $M'$ depending only on $N$, $N'$, and $N''$. We note that if $\pi_\alpha(x)\subseteq \alpha'$, then $\pi_\alpha(x)=\pi_{\alpha'}(x)$. So, we assume that $\pi_\alpha(x)\not\subseteq\alpha'$. We shall show that in this case, $d(x,\pi_\alpha(x))$ and $d(x,\pi_{\alpha'}(x))$ are both bounded above by an appropriate constant, and this gives the desired result. Let $p\in\pi_\alpha(x)\setminus\alpha'$, and let $q\in\pi_{\alpha'}(x)$. Without loss of generality, let $T$ be large enough so that $q\in[\alpha'(a),\alpha'(T)]$ and $d(\alpha'(T),\beta(T))\leq \delta_N$. Put $\gamma=[\beta(b),\alpha'(a)]*[\alpha'(a),\alpha'(T)]*[\alpha'(T),\beta(T)]$, and note that $\gamma$ is a $(3,4\delta_N)$ quasi-geodesic. Thus there exists $w\in[\beta(b),\beta(T)]$ and $L\geq0$ such that $d(\alpha'(a),w)\leq L$, where $L$ depends only on $M'$ by [@cordes_boundry2016 Lemma 2.1]. Notice now that $|d(\alpha'(a),\alpha'(T))-d(w,\beta(T))|\leq\delta_N+L$. However, since $b\geq a$ and $w\in[\beta(b),\beta(T)]$, we know $|d(\alpha'(a),\alpha'(T))-d(w,\beta(T))|=d(\alpha'(a),\alpha'(T))-d(w,\beta(T))$. But then by the definition of the nearest point projection and the triangle inequality, we have $$d(x,p)\leq d(x,q)\leq d(x,\alpha'(a))\leq d(x,w)+d(w,\alpha'(a)) =d(x,\beta(T))-d(w,\beta(T))+d(w,\alpha'(a))$$ $$=d(\alpha'(b),\alpha'(T))-d(w,\beta(T))+d(w,\alpha'(a))\leq d(\alpha'(a),\alpha'(T))-d(w,\beta(T))+L\leq \delta_N+L+L.$$ Therefore, $d(\pi_\alpha(x),x)$ and $d(\pi_{\alpha'}(x),x)$ are both bounded above by $L$, which is a constant depending only on $N$, $N'$, and $N''$, as desired. ◻ \ We're now ready to show that Morse funnelled limit points are exactly Morse horospherical limit points. We proceed using the same overall strategy as the one found in [@swenson2001], by showing direct generalizations of Lemma [Lemma 7](#funnel_in_horoball_hyperbolic){reference-type="ref" reference="funnel_in_horoball_hyperbolic"} and Lemma [Lemma 8](#horoball_in_funnel_hyperbolic){reference-type="ref" reference="horoball_in_funnel_hyperbolic"} for the Morse case. **Proposition 35**. *Let $x\in\partial X_{{\mathfrak{o}}}^N$. Let $\alpha:[0,\infty)\rightarrow X$ be an $N'$-Morse geodesic with $\alpha(\infty)=x$. Then for every Morse gauge $N''$, there exists $T\geq0$ such that, for any subray $\alpha'$ of $\alpha$ with $d(\alpha(0),\alpha')\geq T$, we have $H_{{\mathfrak{o}}}^{N''}(\alpha')\subseteq F_{{\mathfrak{o}}}^{N''}(\alpha)$.* *Proof.* See Figure [\[Figure6\]](#Figure6){reference-type="ref" reference="Figure6"}. Let $\alpha'=\alpha|_{[a,\infty)}$ be a subray of $\alpha$. By Lemma [Lemma 13](#MorseSubraysAreMorse){reference-type="ref" reference="MorseSubraysAreMorse"}, $\alpha'$ is $M$-Morse where $M$ depends only on $N'$. Let $y\in H^{N''}_{{\mathfrak{o}}}(\alpha')$. Thus there exists $\beta:[b,\infty)\rightarrow X$ be a geodesic ray such that $b\geq a$, $\beta(b)=y$, and $\beta\sim_{\delta_{M}}\alpha'$. Note that $\beta$ is $M'$-Morse where $M'$ depends only on $N$, $N'$, and $N''$ by Proposition [Proposition 20](#ParallelRaysAreMorse){reference-type="ref" reference="ParallelRaysAreMorse"}. Choose $z\in\pi_\alpha(y)$ such that $d(\alpha(0),z)=d(\alpha(0),\pi_\alpha(y))$, i.e., so that $z$ is closest to $\alpha(0)$. By Lemma [Lemma 34](#HoroballProjectionToSubrayCloseToParentProjection){reference-type="ref" reference="HoroballProjectionToSubrayCloseToParentProjection"}, there exists $p\in\pi_{\alpha'}(x)$ so that $d(z,p)\leq L$ for some $L$ depending only on $N$, $N'$, and $N''$. Choose $t$ large enough so that $d(\alpha'(t),\beta(t))=d(\alpha(t),\beta(t))\leq\delta_M$ and $p,\alpha(b)\in[\alpha(a),\alpha(t)]$. Note that $[y,p]*[p,\alpha(t)]*[\alpha(t),\beta(t)]$ is a $(3,4\delta_M)$-quasi-geodesic, thus there exists $q\in[\beta(b),\beta(t)]$ and $\lambda\geq0$ such that $d(p,q)\leq\lambda$ where $\lambda$ depends only on $M'$ by [@cordes_boundry2016 Lemma 2.1]. It suffices to show that $d(y,z)\leq d(\alpha(0),z)$. Using the triangle inequality and the definition of $\pi_{\alpha}$, we find $$d(y,z)\leq d(y,p)\leq d(y,q)+d(q,p)\leq d(y,q)+\lambda =d(y,\beta(t))-d(q,\beta(t))+\lambda=d(\alpha(b),\alpha(t))-d(q,\beta(t))+\lambda$$ $$\leq d(\alpha(a),\alpha(t))-d(p,\alpha(t))+\lambda+\delta_M+\lambda=d(\alpha(a),p)+2\lambda+\delta_M\leq d(\alpha(a),z)+L+2\lambda+\delta_M.$$ So, if $a\geq L+2\lambda+\delta_M$, we have $d(y,z)\leq d(\alpha(a),z)+L+2\lambda+\delta_M\leq d(\alpha(a),z)+d(\alpha(0),\alpha(a))=d(\alpha(0),z)$. ◻ \ **Proposition 36**. *Let $x\in\partial X^N_{{\mathfrak{o}}}$. Let $\alpha:[0,\infty)\rightarrow X$ be an $N'$-Morse geodesic with $\alpha(\infty)=x$. Suppose $S=\delta_{N'}$. Define $\alpha'=\alpha|_{[S,\infty)}$. Then $F^{N''}_{{\mathfrak{o}}}(\alpha')\subseteq H^{N''}_{{\mathfrak{o}}}(\alpha)$ for any Morse gauge $N''$.* *Proof.* Let $y\in F_{{\mathfrak{o}}}^{N''}(\alpha')$. By definition, $d(y,\pi_{\alpha'}(y))\leq d(\alpha(S),\pi_{\alpha'}(y))$. Let $p\in\pi_{\alpha'}(y)$ such that $d(\alpha(S),p)=d(\alpha(S),\pi_{\alpha'}(y))$, i.e., let $p$ be the element of $\pi_{\alpha'}(y)$ which is closest to $\alpha(S)$. Then $d(y,p)\leq d(\alpha(S),p)$. Construct $\beta:[b,\infty)\rightarrow X$ such that $\beta(b)=y$ and $\beta\sim_{\delta_{N'}}\alpha$. We want to show that $b\geq0$. Choose $T\geq0$ so that $d(\beta(T),\alpha(T))\leq\delta_{N'}$. Then $T-b=d(y,\beta(T))\leq d(y,p)+d(p,\alpha(T))+d(\alpha(T),\beta(T))\leq d(\alpha(S),p)+d(p,\alpha(T))+\delta_{N'} = d(\alpha(S),\alpha(T))+\delta_{N'} = d(\alpha(0),\alpha(T))-d(\alpha(0),\alpha(S))+\delta_{N'} =T-S+\delta_{N'}=T$. In summary, $T-b\leq T$, but this immediately shows that $0\leq b$, as desired. ◻ \ **Theorem 37**. *Let $x\in\partial X_{{\mathfrak{o}}}$. Then $x$ is a Morse horospherical limit point of $A\subseteq X$ if and only if $x$ is a Morse funnelled limit point of $A$.* *Proof.* This is a direct consequence of Propositions [Proposition 35](#funnelscontainhoroballs){reference-type="ref" reference="funnelscontainhoroballs"} and [Proposition 36](#horoballscontainfunnels){reference-type="ref" reference="horoballscontainfunnels"}. ◻ # Limit Set Conditions For Stability In this section, we show that the horospherical limit point condition, combined with the limit set being compact, is enough for to show that the group action on the weak convex hull is cobounded. The main idea behind this argument is to show the contrapositive: when the group action is not cobounded, then geodesic rays in the space eventually end up very far from the orbit of the group. We begin by showing the following helpful fact, which states that if a group acts non-coboundedly on the weak convex hull of its compact limit set, then there exists a sequence of points $p_n$ in the weak convex hull that "maximally avoids\" the orbit. **Lemma 38** (Sliding Spheres). *Suppose $X$ is a proper geodesic metric space and suppose that $H$ acts properly on $X$ by isometries. Assume that $\Lambda H\neq\emptyset$, every point of $\Lambda H$ is a conical limit point of $H{\mathfrak{o}}$, and that $\Lambda H\subseteq \partial X_{{\mathfrak{o}}}^N$ for some Morse gauge $N$. If the action $H\curvearrowright WCH(\Lambda H)$ is not cobounded, there exists a sequence of points $p_n\in WCH(\Lambda H)$ such that, for sufficiently large $n$, $B_n(p_n)\cap H{\mathfrak{o}}=\emptyset$ and ${\mathfrak{o}}\in B_{n+1}(p_n)$.* *Proof.* Let $K>0$ be the conical limit point constant. Let $n\in\mathbb{N}$ with $n>K+1$. By [@liu_2021 Corollary 5.8], we may assume that $\Lambda H$ has at least two distinct points. Since $H\curvearrowright WCH(\Lambda H)$ is not cobounded, there exists $p\in WCH(\Lambda H)$ with $d(p,H{\mathfrak{o}})>n$. By definition, $p\in\gamma$ for some bi-infinite geodesic $\gamma$ with $\gamma(\pm \infty)\in\Lambda H$. Since $\Lambda H\subseteq \partial X_{{\mathfrak{o}}}^N$, we have by [@boundarycc2016 Proposition 4.2] that $\gamma$ is is Morse for some Morse gauge depending only on $N$. Since every point in $\Lambda H$ is a conical limit point of $H{\mathfrak{o}}$, there exists $h'\in H$ such that $d(h'{\mathfrak{o}},\gamma)<K$. Put $q\in\pi_\gamma(h'{\mathfrak{o}})$. We may assume that $\gamma(s)=q$ and $\gamma(s')=p$ with $s<s'$. Let $A=\{r\in[s,s']~:~n<d(\gamma(r),H{\mathfrak{o}})\}.$ (Equivalently, one may define $A=\{r\in[s,s']~:~B_n(\gamma(r))\cap H{\mathfrak{o}}=\emptyset\}$.) Note that $s'\in A$. Put $t=\inf A$. By the definition of $t$, we have $n\leq d(\gamma(t),H{\mathfrak{o}})$. See Figure [\[FigureBallsAvoidingNonCoboundedActions\]](#FigureBallsAvoidingNonCoboundedActions){reference-type="ref" reference="FigureBallsAvoidingNonCoboundedActions"}. We now claim that $d(\gamma(t),H{\mathfrak{o}})<n+1$. Suppose for contradiction that $n+1\leq d(\gamma(t),H{\mathfrak{o}})$. By the triangle inequality, $n\leq d(\gamma(t-1),H{\mathfrak{o}})$. So if $t-1\in[s,s']$, then $t-1\in A$, however $t=\inf A$. Thus $t-1\not\in [s,s']$. Therefore, $t\in[s,s+1]$, and so $n+1\leq d(\gamma(t),h{\mathfrak{o}})\leq d(\gamma(t),h'{\mathfrak{o}})\leq d(\gamma(t),q)+d(q,h'{\mathfrak{o}}) = d(\gamma(t),\gamma(s))+d(q,h'{\mathfrak{o}})\leq 1+K\leq n$, a contradiction. Therefore, there exists $h\in H$ such that $h{\mathfrak{o}}\in B_{n+1}(\gamma(t))$, but $B_n(\gamma(t))\cap H{\mathfrak{o}}=\emptyset$. Put $p_n=h^{-1}(\gamma(t))$. By [@boundarycc2016 Lemma 3.3], $h\gamma$ is a bi-infinite Morse geodesic with endpoints in $\Lambda H$, and since the action of $H$ on $X$ is by isometries, $B_n(p_n)\cap H{\mathfrak{o}}=\emptyset$, and ${\mathfrak{o}}\in B_{n+1}(p_n)$, as desired. ◻ \ We now prove that (4) implies (2) in the language of Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"}. We show that, if the action is not cobounded on the weak convex hull, then using Lemma [Lemma 38](#noncoboundedactionsarewellspreadinweakconvexhull){reference-type="ref" reference="noncoboundedactionsarewellspreadinweakconvexhull"} we can find a sequence of points which maximally avoid the orbit of $H$, but by Arzelà-Ascoli, geodesics connecting these points to the basepoint eventually fellow-travel a ray which has orbit points that at most linearly diverge from the ray. **Theorem 39**. *Suppose $X$ is a proper geodesic metric space and suppose $H$ acts properly on $X$ by isometries. Assume that $\Lambda H\neq \emptyset$, every point of $\Lambda H$ is a horospherical limit point of $H{\mathfrak{o}}$, and that there exists a Morse gauge $N$ such that $\Lambda H\subset \partial X^N_{{\mathfrak{o}}}$. Then the action of $H\curvearrowright WCH(\Lambda H)$ is cobounded.* *Proof.* For contradiction, assume that the action $H\curvearrowright WCH(\Lambda H)$ is not cobounded. By Lemma [Lemma 38](#noncoboundedactionsarewellspreadinweakconvexhull){reference-type="ref" reference="noncoboundedactionsarewellspreadinweakconvexhull"}, there exists a sequence of points $p_n\in WCH(\Lambda H)$ such that, for all $n>K$, $B_n(p_n)\cap H{\mathfrak{o}}=\emptyset$, and ${\mathfrak{o}}\in B_{n+1}(p_n)$. Let $\gamma_n:[0,d(0,p_n)]\rightarrow X$ be a geodesic with $\gamma_n(0)={\mathfrak{o}}$ and $\gamma_n(d(0,p_n))=p_n$. Notice that since $\Lambda H\subset\partial X^N_{{\mathfrak{o}}}$, we have that $\gamma_n$ is $N'$-Morse for some $N'$ depending only on $N$. Construct a subsequence $\gamma_{n_i}$ such that $\gamma_{n_i}$ converges, uniformly on compact subsets, to an $N'$-Morse geodesic ray $\gamma$ with $\gamma(0)={\mathfrak{o}}$. By construction and by Lemma [Lemma 27](#WeakConvexHullAndLimitSetContainments){reference-type="ref" reference="WeakConvexHullAndLimitSetContainments"}, $\gamma(\infty)\in \Lambda(WCH(\Lambda H))\subseteq\Lambda H$. So, by [@cordes_boundry2016 Corollary 2.6], there exists an $N$-Morse geodesic ray $\alpha$ with $\alpha(0)={\mathfrak{o}}$ and $d(\alpha(t),\gamma(t))<D$ for all $t\geq 0$, where $D\geq0$ is a constant that depends only on $N$. Let $T=2D+4$, and put $\alpha'=\alpha_{[T,\infty)}$. Since $\alpha'(\infty)\in \Lambda H$, and so by Theorem [Theorem 37](#MorseHoroLimitsAreMorseFunnelLimits){reference-type="ref" reference="MorseHoroLimitsAreMorseFunnelLimits"}, $\alpha'(\infty)$ is a funneled limit point of $H$. Thus there exists $h\in H$ so that $h{\mathfrak{o}}\in F_{{\mathfrak{o}}}^N(\alpha')$. Let $t_0=\min\{s:\alpha'(s)\in\pi_{\alpha'}(h{\mathfrak{o}})\}$. Since the sequence $\gamma_{n_i}$ converges uniformly on compact sets to $\gamma$, we may choose $n$ large enough so that $d(\gamma_n(t_0),\gamma(t_0))\leq 1$. See Figure [\[FigureHoroLimitsImplieCobouned\]](#FigureHoroLimitsImplieCobouned){reference-type="ref" reference="FigureHoroLimitsImplieCobouned"}. By the triangle inequality we have that $d(\gamma_n(t_0),\alpha'(t_0))\leq D+1$, and thus $|d(0,\gamma_n(t_0))-d(0,\alpha(t_0))|\leq D+1$. Also, by construction we have that $d(h{\mathfrak{o}},\alpha'(t_0))\leq d(\alpha(T),\alpha(t_0))$. Therefore we have $$d(p_n,h{\mathfrak{o}})\leq d(p_n,\gamma_n(t_0))+d(\gamma_n(t_0),\alpha'(t_0))+d(\alpha('t_0),h_0)\leq d(0,\gamma_n(t_0))+(D+1)+ d(\alpha(T),\alpha(t_0))$$ $$=d({\mathfrak{o}},p_n)-d({\mathfrak{o}},\gamma_n(t_0))+(D+1)+d(0,\alpha(t_0))-d(0,\alpha(T))\leq (n+1)+(D+1)+(D+1)-(2D+4)\leq n-1.$$ However, this contradicts the assumption that $B_n(p_n)\cap H{\mathfrak{o}}=\emptyset$. ◻ \ We now present an alternate definition of a conical limit point which agrees with Definition [Definition 31](#conical limit point def){reference-type="ref" reference="conical limit point def"} in the case where $\Lambda A$ is compact, and requires us to only consider of the geodesic rays which emanate from the given basepoint. By Corollary [Corollary 23](#LimitSetsInstratumAreCompact){reference-type="ref" reference="LimitSetsInstratumAreCompact"} and by [@boundarycc2016 Lemma 4.1], the requirement that $\Lambda H$ is compact is equivalent to the requirement that $\Lambda H$ is contained in the boundary of a single Morse stratum. **Proposition 40**. *Let $X$ be a proper, geodesic metric space. Let $Y\subseteq X$. Suppose $\Lambda Y\not=\emptyset$. Then the following are equivalent:* 1. *$x\in\partial X_{{\mathfrak{o}}}$ is a conical limit point of $Y$* 2. *There exists $K>0$ such that, for every $N$-Morse geodesic ray $\alpha:[0,\infty)\rightarrow X$ with $\alpha(0)={\mathfrak{o}}$ and $\alpha(\infty)=x$, and for every $T>0$, there exists $y\in Y$ such that $y\in \mathcal{N}_K(\alpha')$, where $\alpha':[0,\infty)\rightarrow X$ is defined by $\alpha'(t)=\alpha(t+T)$.* *Proof.* Showing that (1) implies (2) is a direct consequence of Lemma [Lemma 13](#MorseSubraysAreMorse){reference-type="ref" reference="MorseSubraysAreMorse"} and Definition [Definition 31](#conical limit point def){reference-type="ref" reference="conical limit point def"}. Instead assume (2). Let $\beta:[b,\infty)\rightarrow X$ be an $N'$-Morse ray with $\beta(\infty)=x$. Let $\alpha:[0,\infty)\rightarrow X$ an $N$-Morse geodesic ray with $\alpha(0)={\mathfrak{o}}$ and $\alpha(\infty)=x$. Without loss of generality, by Cor [Corollary 18](#betterswensonlemma4){reference-type="ref" reference="betterswensonlemma4"} there exists $T>0$ such that $d(\alpha(t),\beta(t))<\delta_N$ for all $t>T$. Put $\alpha':[0,\infty)\rightarrow X$ via $\alpha'(t)=\alpha(t+T)$. By hypothesis, there exists $y\in Y$ such that $y\in \mathcal{N}_K(\alpha')$. Say $s\in[0,\infty)$ such that $d(\alpha'(s),y)<K$, so via the triangle inequality we have $d(\beta(s+T),y)\leq d(\beta(s+t),\alpha(s+t))+d(\alpha'(s),y)\leq \delta_N+K$. Thus, $y\in \mathcal{N}_{K+\delta_N}(\beta)$, which shows (1). ◻ We conclude this section by showing that $(3) \Rightarrow (2)$ for Theorem [Theorem 2](#MyMainTheorem){reference-type="ref" reference="MyMainTheorem"}, which was first shown in [@boundarycc2016 Corollary 1.14], however here we present a direct proof that does not rely on [@boundarycc2016 Theorem 1.1]. **Proposition 41**. *Let $X$ be a proper geodesic metric space and let $H$ be a finitely generated group of isometries of $X$ such that the orbit map $H\rightarrow X$ via $h\mapsto h{\mathfrak{o}}$ is a stable mapping. If there exists a Morse gauge $N$ so that $\Lambda H\subseteq \partial X_{{\mathfrak{o}}}^N$, then every $x\in \Lambda H$ is a conical limit point of $H{\mathfrak{o}}$.* *Proof.* Let $x\in\Lambda H$, and let $\alpha:[0,\infty)\rightarrow X$ be an $N$-Morse geodesic ray with $\alpha(\infty)=x$, $\alpha(0)={\mathfrak{o}}$. Let $\alpha'=\alpha|_{[a,\infty)}$ be a subray of $\alpha$. Notice that $\alpha'$ is $N'$-Morse where $N'$ depends only on $N$ by Lemma [Lemma 13](#MorseSubraysAreMorse){reference-type="ref" reference="MorseSubraysAreMorse"}. By Proposition [Proposition 40](#ConicalLimitPointsOnlyNeedTailsOfRays){reference-type="ref" reference="ConicalLimitPointsOnlyNeedTailsOfRays"}, it suffices to show that there exists some $K\geq 0$, depending only on $N'$ and $H$, so that $H{\mathfrak{o}}\cap \mathcal{N}_K(\alpha')\not=\emptyset$. Since $H$ is a stable subgroup of isometries on $X$, we have that for any $h\in H$, there exists a $(\lambda,\lambda)$-quasi-geodesic $\gamma$ from ${\mathfrak{o}}$ to $h{\mathfrak{o}}$ such that, for any $p\in\gamma$, $B_{2\lambda}(p)\cap H{\mathfrak{o}}\not=\emptyset$. (To find such a path $\gamma$, take a geodesic in a Cayley graph for $H$ and embed it into $X$ by extending the orbit map along appropriate geodesic segments.) Now, since $x\in\Lambda H$, there exists a sequence $h_n\in H$ such that the sequence of geodesic segments, $\beta_n=[{\mathfrak{o}},h_n{\mathfrak{o}}]$, converges (uniformly on compact subsets) to a geodesic ray $\beta:[b,\infty)\rightarrow X$ with $\beta(\infty)=x$ and $\beta(b)={\mathfrak{o}}$. Since $H$ is a stable group of isometries, $\beta_n$ is $N''$-Morse by Definition [Definition 28](#stabledefinition){reference-type="ref" reference="stabledefinition"}. Up to potentially re-parameterizing $\beta$, there exists $T>a$ so that $d(\beta(T),\alpha(T))<\delta_N$ by Corollary [Corollary 18](#betterswensonlemma4){reference-type="ref" reference="betterswensonlemma4"}. Since $\beta_n$ converges to $\beta$ uniformly on $\overline{B_{T+1}({\mathfrak{o}})}$, the ball of radius $T+1$ centered at ${\mathfrak{o}}$, there exists $n\in\mathbb{N}$ and $p\in \beta_n$ so that $d(\beta(T),p)<1$. Since $\gamma_n$ is an $(\lambda,\lambda)$-quasi-geodesic with endpoints on $\beta_n$, there exists $q\in\gamma_n$ so that $d(p,q)\leq N''(\lambda,\lambda)$. Finally, there exists $h\in H$ so that $d(h{\mathfrak{o}},q)\leq\lambda$. Therefore by the triangle inequality, $d(\alpha(T),h{\mathfrak{o}})\leq d(\alpha(T),\beta(T))+d(\beta(T),p)+d(p,q)+d(q,h{\mathfrak{o}})\leq \delta_N+1+N''(\lambda,\lambda)+2\lambda$. As $\alpha(T)\in\alpha'$, this completes the proof. ◻ # Applictations to Teichmüller Space We conclude by illustrating applications to the above work in the setting of Teichmüller space for a finite type surface $S$. We begin by setting some notation. Let $\text{Mod}(S)$ denote the mapping class group of $S$ and let $\mathcal{T}(S)$ denote the associated Teichmüller space. We will denote the set of projective measured foliations on $S$ by $\text{PMF}(S)$. The Thurston compactification of Teichmüller space is $\overline{\mathcal{T}(S)} = \mathcal{T}(S)\cup\text{PMF}(S)$. We take a moment to restate Corollary [Corollary 5](#TeichApplicationAllConical){reference-type="ref" reference="TeichApplicationAllConical"} using the above notation: **Corollary 42**. *(Restatement of Corollary [Corollary 5](#TeichApplicationAllConical){reference-type="ref" reference="TeichApplicationAllConical"}.) Let $H$ be a finitely generated subgroup of $\textup{Mod}(S)$. The following are equivalent:* 1. *Every element of $\Lambda H\subset \partial \textup{Mod}(S)$ is a conical limit point of $H\curvearrowright\textup{Mod}(S)$ and $\Lambda H$ is compact (in the Morse boundary of $\textup{Mod}(S)$).* 2. *Every element of $\Lambda H\subset \textup{PMF}(S)$ is a conical limit point of $H\curvearrowright\mathcal{T}(S)$.* By work of Cordes, $\partial \text{Mod}(S)$ is homeomorphic to $\partial \mathcal{T}(S)$ (where $\partial$ refers to the Morse boundary) [@cordes_boundry2016 Theorem 4.12], and there exists a natural continuous injective map $h_\infty: \partial\mathcal{T}(S)\hookrightarrow \text{PMF}(S)$ [@cordes_boundry2016 Proposition 4.14]. Keeping this in mind, we denote the continuous inclusion $f_\infty: \partial\textup{Mod}(S)\hookrightarrow \text{PMF}(S)$. The purpose of this section is to prove the following theorem. **Theorem 43**. *Let $H$ be a subgroup of $\textup{Mod}(S)$, and let $x_\infty\in\Lambda H\subseteq \partial\textup{Mod}(S)$ be a conical limit point of $H\curvearrowright\textup{Mod}(S)$. Then $f_\infty(x_\infty)\in\textup{PMF}(S)$ is a conical limit point of $H\curvearrowright\mathcal{T}(S)$.* **Remark 44**. This theorem directly proves $(1)\Rightarrow(2)$ of Corollary [Corollary 5](#TeichApplicationAllConical){reference-type="ref" reference="TeichApplicationAllConical"}. Our proof of Theorem [Theorem 43](#TeichApplicationOneConical){reference-type="ref" reference="TeichApplicationOneConical"} uses several of the tools developed in [@cordes_boundry2016], so we take a moment to recall the construction and definitions presented therein and from [@MasurMinsky2000]. The *curve graph*, denoted $\mathcal{C}(S)$, is a locally infinite simplicial graph whose vertices are isotopy classes of simple closed curves on $S$. We join two vertices with an edge it there exists representative from each class that are disjoint. A set of (pairs of) curves $\mu=\{(\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_m,\beta_m)\}$ is called a complete clean *marking* of $S$ if the $\{\alpha_1,\dots,\alpha_m\}$ forms a pants decomposition of $S$, if each $\alpha_i$ is disjoint from $\beta_j$ whenever $i\not=j$, and if each $\alpha_i$ intersects $\beta_i$ once if the surface filled by $\alpha_i$ and $\beta_i$ is a one-punctured torus. (Otherwise, $\alpha_i$ and $\beta_i$ will intersect twice, and the filling surface is a four-punctured sphere.) We call $\{\alpha_1,\dots,\alpha_m\}$ the *base* of $\mu$ and we call $\beta_i$ the *transverse curve to $\alpha_i$ in $\mu$*. For the sake of completeness, we also define the marking graph, $\mathcal{M}(S)$, although the definition is not needed in this paper. $\mathcal{M}(S)$ is the simplicial graph whose vertices are markings as defined above, and two markings are joined by an edge if they differ by an elementary move. The marking graph $\mathcal{M}(S)$ is quasi-isometric to the mapping class group $\textup{Mod}(S)$, see [@MasurMinsky2000]. For each $\sigma\in\mathcal{T}(S)$ there is a *short marking*, which is constructed inductively by picking the shortest curves in $\sigma$ for the base and repeating for the transverse curves. Now define a map $\Upsilon:\mathcal{M}(S)\rightarrow\mathcal{T}(S)$ by taking a marking $\mu$ to the region in the $\epsilon$-thick part of $\mathcal{T}(S)$, denoted $\mathcal{T}_\epsilon(S)$, where $\mu$ is a short marking in that region. As stated in [@cordes_boundry2016], it is a well known fact that $\Upsilon$ is a coarsely well defined map which is coarsely Lipschitz. We take a moment to prove that this map is coarsely equivariant. **Lemma 45**. *Let $\Upsilon:\mathcal{M}(S)\rightarrow\mathcal{T}(S)$ be as above, and let $H<\textup{Mod}(S)$ be finitely generated. Then there exists a constant $K\geq0$ such that, for any marking $\mu\in\mathcal{M}$ and for any $h\in H$, $$d_{\mathcal{T}(S)}(h\Upsilon(\mu),\Upsilon(h\mu))\leq K.$$* *Proof.* Let $\mu=\{(\alpha_1,\beta_1),\dots,(\alpha_m,\beta_m)\}\in\mathcal{M}(S)$ and $h\in H$ be arbitrary. Let $\sigma\in\mathcal{T}(S)$ so that $\mu$ is a short marking on $\sigma$. (Equivalently, let $\sigma=\Upsilon(\mu)$.) Since the action of $H$ on $\mathcal{T}(S)$ permutes the lengths of curves, the length of each pair $(\alpha_i,\beta_i)$ with respect to $\sigma$ is the same as the length of the pair $(h\alpha_i,h\beta_i)$ with respect to $h\sigma$. Therefore as $\mu$ was a short marking for $\sigma$, this shows that $h \mu$ is a short marking for $h\sigma=h\Upsilon(\mu)$. However, by definition of $\Upsilon$, $h\mu$ is also a short marking for $\Upsilon(h\mu)$. As $\Upsilon$ was a coarsely well defined function, this shows that $d_{\mathcal{T}(S)}(h\Upsilon(\mu),\Upsilon(h\mu))\leq K$ for some $K\geq 0$, as desired. ◻ We now prove Theorem [Theorem 43](#TeichApplicationOneConical){reference-type="ref" reference="TeichApplicationOneConical"}, using the above lemma and several tools from [@cordes_boundry2016] to show that points in conical neighborhoods in $\mathcal{M}(S)$ end up in conical neighborhoods of $\mathcal{T}(S)$. *Proof.* Fix $\mu_0\in\mathcal{M}(S)$. Let $x\in\partial\mathcal{M}(S)_{\mu_0}$ be a conical limit point of $H\mu_0$. Put $\sigma_0=\Upsilon(\mu_0)$. We shall show that $f_\infty(x)$ is a conical limit point of $H\sigma_0$ by verifying the condition in Proposition [Proposition 40](#ConicalLimitPointsOnlyNeedTailsOfRays){reference-type="ref" reference="ConicalLimitPointsOnlyNeedTailsOfRays"}. Let $T\geq 0$ be arbitrary, and let $\lambda:[0,\infty)\rightarrow\mathcal{T}(S)$ be an arbitrary Morse geodesic ray with $\lambda(0)=\sigma_0$ and $\lambda(\infty)=f_\infty(x)$. Let $\alpha:\mathbb{N}\rightarrow\mathcal{M}(S)$ be an $N$-Morse geodesic with $\alpha(0)=\mu_0$ and $\alpha(\infty)=x$. By [@cordes_boundry2016 Lemma 4.9], $\Upsilon(\alpha)$ is an $N'$-Morse $(A,B)$-quasi-geodesic, for some $A$, $B$, and $N'$ depending only on $N$. Put $\beta=\Upsilon(\alpha)$. Notice that $\beta(0)=\sigma_0$ and, by the construction of $f_\infty$, we have $\beta(\infty)=f_\infty(x)$. (For details on the construction of $f_\infty$, we refer to [@cordes_boundry2016], specifically Proposition 4.11, Theorem 4.12, and Proposition 4.14.) Now let $\gamma_n = [\sigma_0, \beta(n)]$. Then each $\gamma_n$ is $N''$-Morse for $N''$ depending on $N$, and by Arzelá-Ascoli and [@cordes_boundry2016 Lemma 2.10], a subsequence of the $\gamma_n$ converges to a geodesic ray $\beta$ which is $N''$-Morse, and by [@cordes_boundry2016 Lemma 4.9], $\beta$ is bounded Hausdorff distance from $\gamma$, where the bound only depends on $N$. Say that $d_{Haus}(\beta,\gamma)\leq K_1$ for $K_1\geq0$. By [@cordes_boundry2016 Corollary 2.6], $d_{Haus}(\gamma,\lambda)\leq K_2$ where $K_2\geq0$ depends only on $N$. Choose $S\geq0$ so that, for all $s\geq S$, $d_{\mathcal{T}(S)}(\beta(s),\lambda_{[T,\infty)})\leq K_1+K_2$. By Proposition [Proposition 40](#ConicalLimitPointsOnlyNeedTailsOfRays){reference-type="ref" reference="ConicalLimitPointsOnlyNeedTailsOfRays"}, there exists $L\geq 0$ where, for all $r\geq0$, $d_{\mathcal{M}(S)}(h\mu_0,\alpha|_{[r,\infty)})\leq L$ for some $h\in H$. Since $\beta=\Upsilon(\alpha)$ and $\Upsilon$ is coarse Lipschitz, there exits $K_3\geq0$ and $h\in H$ so that $d_{\mathcal{T}(S)}(\Upsilon(h\mu_0),\beta|_{[S,\infty)})\leq K_3$. Let $s_0\in[S,\infty)$ so that $d_{\mathcal{T}(S)}(\Upsilon(h\mu_0),\beta(s_0))\leq K_3$. By Lemma [Lemma 45](#UpsilonIsCoarseEquivariant){reference-type="ref" reference="UpsilonIsCoarseEquivariant"}, there exists $K_4\geq0$ such that $d_{\mathcal{T}(S)}(\Upsilon(h\mu_0),h\Upsilon(\mu_0))\leq K_4$. By the triangle inequality, we have $$d_{\mathcal{T}(S)}(h\sigma_0,\lambda|_{[T,\infty)}) \leq d(h\Upsilon(\mu_0),\Upsilon(h\mu_0)) +d(\Upsilon(h\mu_0),\beta(s_0)) +d(\beta(s_0),\lambda|_{[T,\infty)}) \leq K_4+K_3+K_2+K_1$$ By Proposition [Proposition 40](#ConicalLimitPointsOnlyNeedTailsOfRays){reference-type="ref" reference="ConicalLimitPointsOnlyNeedTailsOfRays"}, $\lambda(\infty)=f_\infty(x)$ is a conical limit point of $H\sigma_0$. ◻

arxiv_math

--- abstract: | Given a two-sided noetherian ring $A$ with a dualizing complex, we show that the big finitistic dimension of $A$ is finite if and only if every bounded below Gorenstein-projective-acyclic cochain complex of Gorenstein-projective $A$-modules is contractible. If $A$ is further assumed to be an Artin algebra, we also prove a Gorenstein variant of a theorem of Rickard, showing its finitistic dimension is finite in case its Gorenstein-injective derived category is generated by the Gorenstein-injective modules. address: Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75 Praha, Czech Republic author: - Liran Shaul bibliography: - main.bib title: Gorenstein acyclic complexes and finitistic dimensions --- # Introduction Gorenstein homological algebra is a rapidly developing relative version of homological algebra in which the projective and injective modules are replaced by the more general Gorenstein-projective and Gorenstein-injective modules. The subject was initiated by Auslander, who proved a Gorenstein analogue of the fact that a commutative noetherian local ring is regular if and only if every finitely generated module has finite projective dimension. He showed that a noetherian local ring is Gorenstein if and only if every finitely generated module has a finite resolution made of Gorenstein-projective modules. This paper is dedicated to studying the big finitistic dimension of a ring $A$, which is defined to be $$\operatorname{FPD}(A) = \sup_{M \in \operatorname{Mod}(A), \operatorname{proj\,dim}_A(M) <\infty} (\operatorname{proj\,dim}_A(M)).$$ A major open problem in homological algebra, known as the big finitistic dimension conjecture states that if $A$ is an Artin algebra then $\operatorname{FPD}(A) < \infty$. In this paper we give Gorenstein homological versions of recent results from [@Rickard; @ShFinDim; @ShGor; @Zhang] concerning the question when $\operatorname{FPD}(A) < \infty$. It is often the case that results in ordinary homological algebra have analogues in Gorenstein homological algebra. Finding such analogues in the case of finitistic dimension is particularly attractive, because a result of Holm [@Holm Theorem 2.28] states that for any ring $A$, the Gorenstein version of the finitistic dimension coincides with the usual definition of $\operatorname{FPD}(A)$. To state our results, recall that the notion of an acyclic complex has Gorenstein versions in Gorenstein homological algebra. A cochain complex $X$ is called a $\mathcal{GP}$-acyclic complex if for any Gorenstein-projective $A$-module $G$, it holds that $\operatorname{Hom}_A(G,X)$ is acyclic. Every $\mathcal{GP}$-acyclic complex is acyclic, but the converse fails. This notion is recalled and studied in more detail in [2](#sec:prem){reference-type="ref" reference="sec:prem"} below. Our main result, repeated as [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} in the body of the paper states: **Theorem 1**. *Let $A$ be a ring, and consider the following statements:* 1. *$\operatorname{FPD}(A) < \infty$.* 2. *Every bounded below $\mathcal{GP}$-acyclic complex of Gorenstein-projective $A$-modules is contractible.* 3. *Every bounded below acyclic complex of projective $A$-modules is contractible.* *Then it always holds that (1) $\implies$ (2) $\implies$ (3). If $A$ is two-sided noetherian, and has a dualizing complex, then (3) $\implies$ (1), so that all three conditions are equivalent.* The fact that (1) $\implies$ (3), and the fact that (3) $\implies$ (1) when the ring is noetherian and admits a dualizing complexes are not new, and were proved in [@ShGor]. The major new contribution of the result above is the introduction of condition (2), and the fact that for two-sided noetherian rings with a dualizing complex (so in particular, for all Artin algebras), this condition is equivalent to $\operatorname{FPD}(A) < \infty$. The condition (3) implies that the finitistic dimension conjecture is reflected in the category $\operatorname{\mathsf{K}}_{\operatorname{ac}}(\operatorname{Proj}(A))$, the homotopy category of acyclic complexes of projective modules, which is a model for the singularity category of $A^{\operatorname{op}}$. Condition (2) of the above result suggests that the data whether $\operatorname{FPD}(A) < \infty$ is also encoded in the category $\operatorname{\mathsf{K}}_{\mathrm{gpac}}(\mathcal{GP}(A))$, the homotopy category of $\mathcal{GP}$-acyclic complexes of Gorenstein-projective $A$-modules, which is a kind of a Gorenstein singularity category. Our second main result is a Gorenstein version of a theorem of Rickard from [@Rickard]. In that paper, answering a question of Keller, Rickard showed that for a finite dimensional algebra $A$ over a field, if the localizing subcategory generated by the injective $A$-modules is equal to $\operatorname{\mathsf{D}}(A)$, then $\operatorname{FPD}(A) < \infty$. Our generation result concerns generation by the Gorenstein-injective modules, but we must replace the ordinary derived category by the Gorenstein-injective derived category $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$, which was introduced in [@GaoZhang], and in recalled in [2](#sec:prem){reference-type="ref" reference="sec:prem"} below. Using this notion, in [Corollary 1](#cor:gener){reference-type="ref" reference="cor:gener"} we show: **Theorem 2**. *Let $A$ be an Artin algebra, and suppose that the localizing subcategory of the Gorenstein-injective derived category $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$ generated by the Gorenstein-injective $A$-modules is equal to $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$. Then $\operatorname{FPD}(A) < \infty$.* One other result of interest is obtained as a byproduct of our methods. To explain it, recall that an acyclic complex $P$ of projective $A$-modules is contractible if and only if all its cocycles $\ker(d^n_P)$ are projective $A$-modules. In [@NeemanHo Theorem 8.6], Neeman showed that this is the case if and only if all its cocycles $\ker(d^n_P)$ are flat $A$-modules. In [Corollary 1](#cor:cont-criteria){reference-type="ref" reference="cor:cont-criteria"} we obtain the following new contractiblity criteria for acyclic complexes of projectives, provided that they are bounded below: **Theorem 3**. *Let $A$ be a ring, and let $P$ be an acyclic bounded below complex of projective $A$-modules. Then $P$ is contractible if and only if all its cocycles $\ker(d^n_P)$ are Gorenstein-projective $A$-modules.* # Preliminaries {#sec:prem} In this section we recall some basics about Gorenstein homological algebra and Gorenstein derived categories, and prove some new foundational results about the latter. For more background about this subject the reader may consult [@Christensen; @EnJen; @GaoZhang; @Holm2; @Holm; @Iacob] and their references. Rings are assumed to be associative and unital, modules are by default left modules, and we grade complexes cohomologically. A complex $M$ is called acyclic if it is exact, that is, if $\mathrm{H}^n(M) = 0$ for all $n \in \mathbb{Z}$. ## Gorenstein special modules Let $A$ be a ring. Following [@EJDef], an $A$-module $G$ is called a Gorenstein-projective $A$-module if there exist an acyclic complex of projective $A$-modules $P$, such that $G \cong \ker(d^0_P)$, and such that $\operatorname{Hom}_A(P,Q)$ is acyclic for any projective $A$-module $Q$. An $A$-module $J$ is called a Gorenstein-injective $A$-module if there is an acyclic complex of injective $A$-modules $I$, such that $J \cong \ker(d^0_I)$, and such that $\operatorname{Hom}_A(E,I)$ is an acyclic complex for all injective $A$-modules $E$. Finally, following [@EJFlat], an $A$-module $H$ is called a Gorenstein-flat $A$-module if there is an acyclic complex of flat $A$-modules $F$ such that $H \cong \ker(d^0_F)$, and such that $I\otimes_A F$ is an acyclic complex for all injective right $A$-modules $I$. It is immediate from the definitions that any projective module is Gorenstein-projective, any injective module is Gorenstein-injective, and any flat module is Gorenstein-flat. We will denote by $\mathcal{GP}(A)$ the category of Gorenstein-projective $A$-modules, and by $\mathcal{GI}(A)$ the category of Gorenstein-injective $A$-modules. ## Relative acyclic complexes Next, we recall some relative versions of acyclicity. Let $\mathcal{P}$ be any non-empty class of $A$-modules which is closed under isomorphisms. Recall that a complex $X$ of $A$-modules is called right $\mathcal{P}$-acyclic if for any $P \in \mathcal{P}$, the complex $\operatorname{Hom}_A(P,X)$ is acyclic. For example, if $\mathcal{P}$ is the class of all projective $A$-modules, then being right $\mathcal{P}$-acyclic is equivalent to being acyclic. It follows that if $\mathcal{P}$ contains the class of all projective $A$-modules, then any right $\mathcal{P}$-acyclic complex is acyclic. When $\mathcal{P} = \mathcal{GP}$, the class of Gorenstein-projective modules, such a complex is usually called a $\mathcal{GP}$-acyclic complex. Given a ring $A$, the collection of all $\mathcal{GP}$-acyclic complexes over $A$ is a triangulated subcategory of the homotopy category $\operatorname{\mathsf{K}}(A)$, which we will denote by $\operatorname{\mathsf{K}}_{\mathrm{gpac}}(A)$. A map of complexes $X\to Y$ is called a $\mathcal{GP}$-quasi-isomorphism if its cone is $\mathcal{GP}$-acyclic. Dually, for a non-empty class of objects $\mathcal{I}$ closed under isomorphisms, a complex $X$ will be called left $\mathcal{I}$-acyclic if for any $I \in \mathcal{I}$, the complex $\operatorname{Hom}_A(X,I)$ is acyclic. In the case where $\mathcal{I} = \mathcal{GI}$, the class of Gorenstein-injective modules, we will say that $X$ is a $\mathcal{GI}$-acyclic complex. The collection of all $\mathcal{GI}$-acyclic complexes over a ring $A$ will be denoted by $\operatorname{\mathsf{K}}_{\mathrm{giac}}(A)$. Again, this is a triangulated subcategory. ## Relative derived categories Given a ring $A$, following [@GaoZhang], we define the Gorenstein-projective derived category of $A$, denoted by $\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)$, to be the Verdier quotient $$\operatorname{\mathsf{D}}_{\mathcal{GP}}(A) := \operatorname{\mathsf{K}}(A) / \operatorname{\mathsf{K}}_{\mathrm{gpac}}(A).$$ This is a triangulated category, studied in detail in [@GaoZhang]. Dually, the Gorenstein-injective derived category of $A$ will be denoted by $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$. It is given as the Verdier quotient $$\operatorname{\mathsf{D}}_{\mathcal{GI}}(A) := \operatorname{\mathsf{K}}(A) / \operatorname{\mathsf{K}}_{\mathrm{giac}}(A).$$ Of special importance in this paper is the problem whether these categories admit products and coproducts. We shall now discuss this problem. We note that this is far from trivial, see [@NeeExa Example 4.1] for a similar situation in which products fail to exist. **Proposition 1**. *Let $A$ be a ring. Then the following hold:* 1. *A product of any collection of $\mathcal{GP}$-acyclic complexes over $A$ is $\mathcal{GP}$-acyclic.* 2. *A coproduct of any collection of $\mathcal{GI}$-acyclic complexes over $A$ is $\mathcal{GI}$-acyclic.* *Proof.* Let $\{X_i\}_{i \in I}$ be a collection of $\mathcal{GP}$-acyclic complexes over $A$, and let $P$ be a Gorenstein-projective $A$-module. We have that $$\operatorname{Hom}_A(P,\prod_{i \in I} X_i) \cong \prod_{i \in I} \operatorname{Hom}_A(P,X_i).$$ For each $i \in I$, since $X_i$ is an $\mathcal{GP}$-acyclic complex, by definition $\operatorname{Hom}_A(P,X_i)$ is an acyclic complex. Since the product of acyclic complexes is acyclic, the above isomorphism shows that $\operatorname{Hom}_A(P,\prod_{i \in I} X_i)$ is acyclic, which implies that $\prod_{i \in I} X_i$ is $\mathcal{GP}$-acyclic. The second statement is proved similarly. ◻ **Corollary 1**. *Let $A$ be a ring. Then the following hold:* 1. *The Gorenstein-projective derived category $\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)$ has arbitrary products.* 2. *The Gorenstein-injective derived category $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$ has arbitrary coproducts.* *Proof.* The first statement follows from [Proposition 1](#prop:hasprod){reference-type="ref" reference="prop:hasprod"}(1) and the dual of [@BokNee Lemma 1.5]. The second statement follows from [Proposition 1](#prop:hasprod){reference-type="ref" reference="prop:hasprod"}(2) and [@BokNee Lemma 1.5]. ◻ We do not know if the dual statements hold. The next result given a necessary and sufficient condition for them to hold: **Proposition 1**. *Let $A$ be a ring.* 1. *The Gorenstein-projective derived category $\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)$ has arbitrary coproducts which coincide with the coproducts in $\operatorname{\mathsf{K}}(A)$ if and only if any coproduct of a collection of $\mathcal{GP}$-acyclic complexes over $A$ is $\mathcal{GP}$-acyclic.* 2. *The Gorenstein-injective derived category $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$ has arbitrary products which coincide with the products in $\operatorname{\mathsf{K}}(A)$ if and only if any product of a collection of $\mathcal{GI}$-acyclic complexes over $A$ is $\mathcal{GI}$-acyclic.* *Proof.* If a coproduct of $\mathcal{GP}$-acyclic complexes is $\mathcal{GP}$-acyclic, then it follows from [@BokNee Lemma 1.5] that $\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)$ has arbitrary coproducts. Conversely, if $\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)$ has arbitrary coproducts, and $\{X_i\}_{i \in I}$ is a collection of $\mathcal{GP}$-acyclic complexes over $A$, then in $\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)$ it holds that each $X_i \cong 0$. Set $T:= \oplus_{i \in I} X_i$. Then $T \cong \oplus_{i \in I} 0 \cong 0$ in $\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)$. Let $P$ be a Gorenstein-projective $A$-module. Then for all $n$ we have that: $$0 = \operatorname{Hom}_{\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)}(P,0[n]) = \operatorname{Hom}_{\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)}(P,T[n]).$$ According to [@GaoZhang Proposition 2.8], we have that $$0 = \operatorname{Hom}_{\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)}(P,T[n]) \cong \operatorname{Hom}_{\operatorname{\mathsf{K}}(A)}(P,T[n]),$$ and as the latter by definition is equal to $\mathrm{H}^n(\operatorname{Hom}_A(P,T))$, we deduce that $\operatorname{Hom}_A(P,T)$ is acyclic, so that $T$ is $\mathcal{GP}$-acyclic as claimed. The second statement is proven similarly. ◻ ## Dualizing complexes The class of rings for which the main results of this paper hold are the two-sided noetherian rings which admit a dualizing complex. These were first introduced in noncommutative algebra by Yekutieli in [@Yek]. In this paper, following [@InKr], we need a slight variant of Yekutieli's definition, and define a dualizing complex $R$ over a ring $A$ to be a bounded complex of $A$-$A$-bimodules, which are injective both as $A$-modules and as $A^{\operatorname{op}}$-modules, have finitely generated cohomology both on the left and on the right, and such that the natural maps $A\to \operatorname{Hom}_A(R,R)$ and $A\to \operatorname{Hom}_{A^{\operatorname{op}}}(R,R)$ are quasi-isomorphisms. In particular, any Artin algebra $A$ admits a dualizing complex. In fact, in that case it is concentrated in a single degree, so we refer to it as the dualizing module of $A$. A foundational result in Gorenstein homological algebra [@JorRes], due to Jørgensen, states that over a two-sided noetherian ring $A$ with a dualizing complex, for any $A$-module $M$, there exist a $\mathcal{GP}$-quasi-isomorphism $P \to M$, where $P$ is a bounded above complex of Gorenstein-projective $A$-modules. Such a $P$ is called a proper Gorenstein-projective resolution of $M$. # Contractible complexes of Gorenstein-projectives In this section we show that certain relative acyclic complexes must be contractible over rings of finite finitistic dimension, establishing a relative version of the results of [@ShGor Appendix A]. It is well known that an acyclic complex of projective modules is contractible if and only if all its cocycles are projective. The next result is a relative version of this fact. **Proposition 1**. *Let $A$ be a ring, and let $\mathcal{P}$ be any non-empty class of $A$-modules which is closed under isomorphisms and direct summands, and contains all projective $A$-modules. Let $X$ be a right $\mathcal{P}$-acyclic complex, such that $X^n \in \mathcal{P}$ for all $n \in \mathbb{Z}$. Then $X$ is contractible if and only if all its cocycles $\ker(d^n_X)$ belong to $\mathcal{P}$.* *Proof.* Suppose first that $X$ is contractible. This implies that for any $n$, the short exact sequence $$0 \to \ker(d^n_X) \to X^n \to \ker(d^{n+1}_X) \to 0$$ is split, so in particular $\ker(d^n_X)$ is a direct summand of $X^n$, which implies that $\ker(d^n_X) \in \mathcal{P}$. Conversely, suppose that for all $n$, we have that $\ker(d^n_X) \in \mathcal{P}$. Since by assumption $X$ is acyclic, to show it is contractible, it is enough to show that the sequence the short exact sequence $$\label{eqn:to-split} 0 \to \ker(d^n_X) \to X^n \to \ker(d^{n+1}_X) \to 0$$ splits. To see this, we apply the functor $\operatorname{Hom}_A(\ker(d^{n+1}_X),-)$ to this sequence, and we claim that the resulting sequence $$\label{eqn:Hom-exact} 0 \to \operatorname{Hom}_A(\ker(d^{n+1}_X),\ker(d^n_X)) \to \operatorname{Hom}_A(\ker(d^{n+1}_X),X^n) \to \operatorname{Hom}_A(\ker(d^{n+1}_X),\ker(d^{n+1}_X)) \to 0$$ is still exact. To see this, by left exactness of the covariant hom functor, it is enough for us to show that the map $$\operatorname{Hom}_A(\ker(d^{n+1}_X),X^n) \to \operatorname{Hom}_A(\ker(d^{n+1}_X),\ker(d^{n+1}_X))$$ is surjective. Since by assumption $X$ is right $\mathcal{P}$-acyclic, and $\ker(d^{n+1}_X) \in \mathcal{P}$, it follows that the cochain complex $\operatorname{Hom}_A(\ker(d^{n+1}_X),X)$ is acyclic. In particular, the sequence $$\label{eqn:exact-X} \operatorname{Hom}_A(\ker(d^{n+1}_X),X^n) \to \operatorname{Hom}_A(\ker(d^{n+1}_X),X^{n+1}) \to \operatorname{Hom}_A(\ker(d^{n+1}_X),X^{n+2})$$ is exact. Given a map $\alpha \in \operatorname{Hom}_A(\ker(d^{n+1}_X),\ker(d^{n+1}_X))$, since $\ker(d^{n+1}_X) = \operatorname{Im}(d^n_X) \subseteq X^{n+1}$, we may consider $\alpha$ as an element of $\operatorname{Hom}_A(\ker(d^{n+1}_X),X^{n+1})$. The map $$\operatorname{Hom}_A(\ker(d^{n+1}_X),X^{n+1}) \xrightarrow{(d^{n+1}_X)_*} \operatorname{Hom}_A(\ker(d^{n+1}_X),X^{n+2})$$ is given by composition with $d^{n+1}_X$, which shows that $(d^{n+1}_X)_*(\alpha) = d^{n+1}_X\circ \alpha = 0$. Thus, exactness of [\[eqn:exact-X\]](#eqn:exact-X){reference-type="ref" reference="eqn:exact-X"} shows that $\alpha$ is in the image of $(d^n_X)_*$, and hence [\[eqn:Hom-exact\]](#eqn:Hom-exact){reference-type="ref" reference="eqn:Hom-exact"} is exact. In particular, there exist $\beta:\ker(d^{n+1}_X) \to X^n$, such that $d^n_X \circ \beta = 1_{\ker(d^{n+1}_X)}$, which shows that [\[eqn:to-split\]](#eqn:to-split){reference-type="ref" reference="eqn:to-split"} splits. ◻ Applying [Proposition 1](#prop:contr){reference-type="ref" reference="prop:contr"} to the class of Gorenstein-projective modules, we obtain: **Corollary 1**. *Let $A$ be a ring, and let $X$ be a $\mathcal{GP}$-acyclic complex of Gorenstein-projective modules. Then $X$ is contractible if and only if all its cocycles $\ker(d^n_X)$ are Gorenstein-projective.* Here is the main result of this section. **Theorem 1**. *Let $A$ be a ring such that $\operatorname{FPD}(A) < \infty$. Then any bounded below $\mathcal{GP}$-acyclic complex of Gorenstein-projective $A$-modules is contractible.* *Proof.* Suppose $\operatorname{FPD}(A) = s < \infty$. According to [@Holm Theorem 2.28], there is an equality $$\operatorname{FPD}(A) = \operatorname{FGPD}(A),$$ where $\operatorname{FGPD}(A)$ is the finitistic Gorenstein projective dimension of $A$. Let $X$ be a bounded below $\mathcal{GP}$-acyclic complex of Gorenstein-projective modules. Since any projective module is Gorenstein-projective, in particular $X$ is acyclic. The fact that $X$ is bounded below implies that $$0 \to \dots \to X^{n+s-1} \to X^{n+s} \xrightarrow{d^{n+s}_X} \operatorname{Im}(d^{n+s}_X)$$ is a Gorenstein-projective resolution of the $A$-module $\operatorname{Im}(d^{n+s}_X)$, which shows that $\operatorname{Im}(d^{n+s}_X)$ has finite Gorenstein-projective dimension. Hence, it has Gorenstein-projective dimension at most $s$, so by [@Holm Theorem 2.20], its $s$-syzygy, which is simply $\ker(d^n_X)$ must be Gorenstein-projective. [Corollary 1](#cor:GP-cont){reference-type="ref" reference="cor:GP-cont"} then implies that $X$ is contractible. ◻ **Remark 1**. One may easily obtain results dual to the results of this section, and show, for instance, that if a ring has finite finitistic injective dimension, then any bounded above $\mathcal{GI}$-acyclic complex of Gorenstein-injective $A$-modules is contractible. # Bounded below acyclic complexes of Gorenstein-projectives The aim of this section is to obtain a converse of [Theorem 1](#thm:FPDimplyCont){reference-type="ref" reference="thm:FPDimplyCont"}, showing that contractibility of certain complexes of Gorenstein-projectives implies finite finitistic dimension. The next result is similar to [@Holm2 Remark 3.3]. Because of its centrality to the argument below, we include full details. **Proposition 1**. *Let $A$ be a ring, and let $X$ be a bounded acyclic complex of $A$-modules. Suppose that for all $n \in \mathbb{Z}$, it holds that the $A$-module $X^n$ has finite projective dimension over $A$. Then $X$ is $\mathcal{GP}$-acyclic.* *Proof.* Let $G$ be some Gorenstein-projective $A$-module, and assume $X$ is of the form $$\label{eqn:proj-res} 0 \to P^{-n} \to \dots P^{-1} \to P^0 \to 0.$$ Applying the functor $\operatorname{Hom}_A(G,-)$ to the short exact sequence $$0 \to \ker(d^{-1}) \to P^{-1} \xrightarrow{d^{-1}} P^0 \to 0$$ yields $$0 \to \operatorname{Hom}_A(G,\ker(d^{-1})) \to \operatorname{Hom}_A(G,P^{-1}) \to \operatorname{Hom}_A(G,P^0) \to \operatorname{Ext}^1_A(G,\ker(d^{-1})).$$ The resolution ([\[eqn:proj-res\]](#eqn:proj-res){reference-type="ref" reference="eqn:proj-res"}) shows that $\ker(d^{-1})$ is isomorphic to a bounded complex of modules of finite projective dimension, and hence that $\operatorname{proj\,dim}_A(\ker(d^{-1})) < \infty$, so by [@Holm Proposition 2.3] we have that $\operatorname{Ext}^1_A(G,\ker(d^{-1})) = 0$, showing that the map $\operatorname{Hom}_A(G,P^{-1}) \to \operatorname{Hom}_A(G,P^0)$ is surjective. Suppose we know that the sequence $$\operatorname{Hom}_A(G,P^{-i}) \to \operatorname{Hom}_A(G,P^{-(i-1)}) \to \dots \to \operatorname{Hom}_A(G,P^0) \to 0$$ is exact, and let $K = \operatorname{Im}(d^{-(i+1)}:P^{-(i+1)} \to P^{-i})$. There is a short exact sequence $$0 \to N \to P^{-(i+1)} \to K \to 0,$$ and as $N$ is a syzygy of ([\[eqn:proj-res\]](#eqn:proj-res){reference-type="ref" reference="eqn:proj-res"}), by the same argument as above, it holds that $\operatorname{proj\,dim}_A(N) < \infty$. Thus, using [@Holm Proposition 2.3] again, the sequence $$0 \to \operatorname{Hom}_A(G,N) \to \operatorname{Hom}_A(G,P^{-(i+1)}) \to \operatorname{Hom}_A(G,K) \to 0$$ is exact, and as the map $d^{-(i+1)}:P^{-(i+1)} \to P^{-i}$ factors as $P^{-(i+1)} \twoheadrightarrow K \hookrightarrow P^{-i}$, it follows that $\operatorname{Hom}_A(G,P^{-(i+1)}) \to \operatorname{Hom}_A(G,P^{-i}) \to \operatorname{Hom}_A(G,P^{-(i-1)})$ is exact. Induction on the amplitude of $X$ now shows that $X$ is $\mathcal{GP}$-acyclic. ◻ Next, we extend the above proposition to bounded below complexes. **Theorem 1**. *Let $A$ be a ring, and let $X$ be a bounded below acyclic complex, with the property that $\operatorname{proj\,dim}_A(X^n) < \infty$ for all $n \in \mathbb{Z}$. Then $X$ is $\mathcal{GP}$-acyclic.* *Proof.* As $X$ is bounded below, there exist some $r \in \mathbb{Z}$ such that $X^n = 0$ for all $n<r$. Let $G$ be a Gorenstein-projective $A$-module. We must show that the complex $\operatorname{Hom}_A(G,X)$ is acyclic. For any $n \in \mathbb{Z}$, the fact that $X$ is bounded below implies the existence of an exact sequence of $A$-modules $$0 \to X^r \to X^{r+1} \to \dots \to X^{n-1} \to X^n \to X^{n+1} \xrightarrow{d^{n+1}_X} \operatorname{Im}(d^{n+1}_X) \to 0$$ As by assumption each $X^i$ has finite projective dimension, and as this sequence is exact, it follows that $\operatorname{proj\,dim}_A(\operatorname{Im}(d^{n+1}_X) < \infty$. Thus, we are in the situation of [Proposition 1](#prop:flat-is-gp){reference-type="ref" reference="prop:flat-is-gp"}, and it follows from it that the sequence $$\begin{gathered} 0 \to \operatorname{Hom}_A(G,X^r) \to \operatorname{Hom}_A(G,X^{r+1}) \to \dots \to \operatorname{Hom}_A(G,X^{n-1}) \to\\ \operatorname{Hom}_A(G,X^n) \to \operatorname{Hom}_A(G,X^{n+1}) \to \operatorname{Hom}_A(G,\operatorname{Im}(d^{n+1}_X)) \to 0\end{gathered}$$ is exact. In particular, for any $n \in \mathbb{Z}$, we see that the sequence $$\operatorname{Hom}_A(G,X^{n-1}) \to \operatorname{Hom}_A(G,X^n) \to \operatorname{Hom}_A(G,X^{n+1})$$ is exact, which shows that $\operatorname{Hom}_A(G,X)$ is acyclic, and hence that $X$ is $\mathcal{GP}$-acyclic. ◻ **Corollary 1**. *Over any ring, any acyclic bounded below complex of projective $A$-modules is $\mathcal{GP}$-acyclic.* Since projective modules are Gorenstein-projectives, combining this with [Corollary 1](#cor:GP-cont){reference-type="ref" reference="cor:GP-cont"} we obtain: **Corollary 1**. *Let $A$ be a ring, and let $P$ be an acyclic bounded below complex of projective $A$-modules. Then $P$ is contractible if and only if all its cocycles $\ker(d^n_P)$ are Gorenstein-projective $A$-modules.* Here is the main result of this section and of this paper: **Theorem 1**. *Let $A$ be a ring, and consider the following statements:* 1. *$\operatorname{FPD}(A) < \infty$.* 2. *Every bounded below $\mathcal{GP}$-acyclic complex of Gorenstein-projective $A$-modules is contractible.* 3. *Every bounded below acyclic complex of projective $A$-modules is contractible.* *Then it always holds that (1) $\implies$ (2) $\implies$ (3). If $A$ is two-sided noetherian, and has a dualizing complex, then (3) $\implies$ (1), so that all three conditions are equivalent.* *Proof.* The fact that (1) $\implies$ (2) is [Theorem 1](#thm:FPDimplyCont){reference-type="ref" reference="thm:FPDimplyCont"}, and that (2) $\implies$ (3) follows from [Corollary 1](#cor:proj-is-gp-acyclic){reference-type="ref" reference="cor:proj-is-gp-acyclic"}. Finally, if $A$ is two-sided noetherian and has a dualizing complex, then by [@ShGor Theorem 5.1], we have that (3) $\implies$ (1). ◻ **Remark 1**. In general, it could happen that (3) holds but (1) fails. See [@ShGor Theorem 5.1(d)] for details. We do not know if for an arbitrary ring (3) $\implies$ (2). See [@Posi] for some examples of (noncommutative non-noetherian) rings over which (3) fails. # Gorenstein-injective generation and acyclic complexes In this section we give a sufficient condition for the finitistic dimension conjecture to hold over Artin algebras. Over an Artin algebra, the flat and projective modules coincide, so it might be interesting to note (although this result is not used in the sequel) that the same is true for Gorenstein-projectives and Gorenstein flat modules: **Proposition 1**. *Over an Artin algebra $A$, the classes of Gorenstein-projective and Gorenstein-flat modules coincide.* *Proof.* Since over an Artin algebra $A$, flat modules are projective, given an acyclic complex $P$ of projective $A$-modules, it is thus enough to show that $\operatorname{Hom}_A(P,Q)$ is exact for all left projective $A$-modules $Q$ if and only if $I\otimes_A P$ is exact for all right injective $A$-modules $I$. To see this, note that if $R$ is the dualizing module of $A$, then any such $I$ is a direct summand of direct sums of copies of $R$, so it is enough to show that $\operatorname{Hom}_A(P,Q)$ is exact for all left projective $A$-modules $Q$ if and only if $R\otimes_A P$ is exact, and this is exactly [@JorRes Lemma 1.7]. ◻ In [@InKr Theorem 4.8], the authors constructed the covariant Grothendieck duality, which is an equivalence $\operatorname{\mathsf{K}}(\operatorname{Proj}(A)) \to \operatorname{\mathsf{K}}(\operatorname{Inj}(A))$ over any two-sided noetherian ring with a dualizing complex. The next result is a Gorenstein version of this result for Artin algebras. **Proposition 1**. *Let $A$ be an Artin algebra with a dualizing module $R$.* 1. *The functor $$R\otimes_A -: \mathcal{GP}(A) \to \mathcal{GI}(A)$$ is an equivalence of categories with a quasi-inverse $$\operatorname{Hom}_A(R,-):\mathcal{GI}(A) \to \mathcal{GP}(A).$$'* 2. *The functor $$R\otimes_A -: \operatorname{\mathsf{K}}(\mathcal{GP}(A)) \to \operatorname{\mathsf{K}}(\mathcal{GI}(A))$$ is an equivalence of categories with a quasi-inverse $$\operatorname{Hom}_A(R,-):\operatorname{\mathsf{K}}(\mathcal{GI}(A)) \to \operatorname{\mathsf{K}}(\mathcal{GP}(A)).$$* *Proof.* Since $R\otimes_A -$ and $\operatorname{Hom}_A(R,-)$ are additive functors, clearly the second statement follows from the first one. The first statement is contained in [@BelReit Proposition X.1.4]. ◻ Given a triangulated category $\mathcal{T}$ which has infinite coproducts, and given a non-empty set of objects $\mathbf{S}$ in $\mathcal{T}$, recall that the localizing subcategory generated by $\mathbf{S}$, denoted by $\operatorname{Loc}_{\mathcal{T}}(\mathbf{S})$ is the smallest triangulated subcategory of $\mathcal{T}$ which contains $\mathbf{S}$ and is closed under infinite coproducts. In case $\operatorname{Loc}_{\mathcal{T}}(\mathbf{S}) = \mathcal{T}$, we say that $\mathbf{S}$ generates $\mathcal{T}$. We have seen in [Corollary 1](#cor:has-prod){reference-type="ref" reference="cor:has-prod"} that over any ring $A$, the Gorenstein-injective derived category $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$ has arbitrary coproducts, so it makes sense to talk about localizing subcategories of it. In [@Rickard Theorem 4.3], it was shown that for a finite dimensional algebra $A$ over a field, if $\operatorname{Loc}_{\operatorname{\mathsf{D}}(A)}(\operatorname{Inj}(A)) = \operatorname{\mathsf{D}}(A)$, then $\operatorname{FPD}(A)<\infty$. This result was further generalized to larger classes of rings in [@Zhang Theorem 1.1] and [@ShFinDim Corollary 5.3]. Our next result (or rather a corollary of it) is a Gorenstein version of this fact. **Theorem 1**. *Let $A$ be an Artin algebra, and suppose that $$\operatorname{Loc}_{\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)}(\mathcal{GI}(A)) = \operatorname{\mathsf{D}}_{\mathcal{GI}}(A).$$ Then any bounded below $\mathcal{GP}$-acyclic complex of Gorenstein-projective modules is contractible.* *Proof.* Let $P$ be a bounded below $\mathcal{GP}$-acyclic complex of Gorenstein-projective modules. Let $n \in \mathbb{Z}$, and let $I$ be some Gorenstein-injective $A$-module. By [Proposition 1](#prop:GrothEquiv){reference-type="ref" reference="prop:GrothEquiv"}, $F = \operatorname{Hom}_A(R,I)$ is a Gorenstein-Projective $A$-module. The fact that $P$ is $\mathcal{GP}$-acyclic implies that $$\operatorname{Hom}_{\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)}(F[n],P) = 0$$ By [@GaoZhang Proposition 2.8], there is an equality $$\operatorname{Hom}_{\operatorname{\mathsf{D}}_{\mathcal{GP}}(A)}(F[n],P) = \operatorname{Hom}_{\operatorname{\mathsf{K}}(A)}(F[n],P),$$ and by [Proposition 1](#prop:GrothEquiv){reference-type="ref" reference="prop:GrothEquiv"}, we have that $$\operatorname{Hom}_{\operatorname{\mathsf{K}}(A)}(F[n],P) = \operatorname{Hom}_{\operatorname{\mathsf{K}}(A)}(R\otimes_A F[n],R\otimes_A P) = \operatorname{Hom}_{\operatorname{\mathsf{K}}(A)}(I[n],R\otimes_A P).$$ Since $P$ is a bounded-below complex of Gorenstein-projectives, it follows from [Proposition 1](#prop:GrothEquiv){reference-type="ref" reference="prop:GrothEquiv"} that $R\otimes_A P$ is a bounded-below complex of Gorenstein-injectives. By the dual result of [@GaoZhang Proposition 2.8], we obtain that $$0 = \operatorname{Hom}_{\operatorname{\mathsf{K}}(A)}(I[n],R\otimes_A P) = \operatorname{Hom}_{\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)}(I[n],R\otimes_A P).$$ Consider the following subcategory of $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$: $$\mathcal{T} = \{X \in \operatorname{\mathsf{D}}_{\mathcal{GI}}(A) \mid \forall n \in \mathbb{Z}, \operatorname{Hom}_{\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)}(X[n],R\otimes_A P) = 0 \}.$$ It is clear that $\mathcal{T}$ is a localizing subcategory of $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$, and we have seen that it contains all Gorenstein-injective $A$-modules. Hence, by the generation assumption, it follows that $\mathcal{T} = \operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$, so in particular $R\otimes_A P \in \mathcal{T}$, which shows that $R\otimes_A P \cong 0$ in $\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)$. Since it is a bounded-below complex of Gorenstein-injectives, again, the dual of [@GaoZhang Proposition 2.8] implies that $R\otimes_A P \cong 0$ in $\operatorname{\mathsf{K}}(A)$, and hence, [Proposition 1](#prop:GrothEquiv){reference-type="ref" reference="prop:GrothEquiv"} shows that $$P \cong \operatorname{Hom}_A(R,R\otimes_A P) \cong 0$$ in $\operatorname{\mathsf{K}}(A)$, as claimed. ◻ **Corollary 1**. *Let $A$ be an Artin algebra, and suppose that $$\operatorname{Loc}_{\operatorname{\mathsf{D}}_{\mathcal{GI}}(A)}(\mathcal{GI}(A)) = \operatorname{\mathsf{D}}_{\mathcal{GI}}(A).$$ Then $\operatorname{FPD}(A) < \infty$.* *Proof.* This follows from [Theorem 1](#thm:generation){reference-type="ref" reference="thm:generation"} and [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}. ◻

arxiv_math

--- abstract: | The analysis of a delayed generalized Burgers-Huxley equation (a non-linear advection-diffusion-reaction problem) with weakly singular kernels is carried out in this work. Moreover, numerical approximations are performed using the conforming finite element method (CFEM). The existence, uniqueness and regularity results for the continuous problem have been discussed in detail using the Faedo-Galerkin approximation technique. For the numerical studies, we first propose a semi-discrete conforming finite element scheme for space discretization and discuss its error estimates under minimal regularity assumptions. We then employ a backward Euler discretization in time and CFEM in space to obtain a fully-discrete approximation. Additionally, we derive a prior error estimates for the fully-discrete approximated solution. Finally, we present computational results that support the derived theoretical results. Burgers-Huxley equation, Galerkin approximation, weak solution, A priori error analysis, Conforming finite element method. author: - | Sumit Mahajan[^1] Arbaz Khan [^2], and Manil T. Mohan[^3]\ Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India bibliography: - IMANUM-refs.bib title: "Finite element approximation for a delayed generalized Burgers-Huxley equation with weakly singular kernels: Part I Well-posedness, Regularity and Conforming approximation" --- \ # Introduction The *generalized Burgers-Huxley equation (GBHE) with weakly singular kernels* describes a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transports with delays. The GBHE has various applications in different fields such as traffic flows, nuclear waste disposal, the mobility of fish populations, motion of the domain wall of a ferroelectric material in an electric field, etc. In this work, we consider a delayed generalized Burgers-Huxley equation with weakly singular kernels defined on $\Omega \times (0,T)$, where $\Omega\subset \mathbb{R}^d \ (d = 2, 3)$ be an open bounded and simply connected convex domain with Lipschitz boundary $\partial\Omega$. The delayed GBHE under our consideration is given by $$\label{GBHE} \left\{\begin{aligned} \frac{\partial u(x,t)}{\partial t}&+\alpha u(x,t)^{\delta}\sum_{i=1}^d\frac{\partial u(x,t)}{\partial x_i}-\nu\Delta u(x,t)-\eta\int_{0}^{t} K(t-s)\Delta u(x,s) \mathrm{~d}s\\&=\beta u(x,t)(1-u(x,t)^{\delta})(u(x,t)^{\delta}-\gamma)+f(x,t), \ (x,t)\in\Omega\times(0,T),\\ u(x,t)&=0, \ x \in \partial\Omega ,\ t\in[0,T),\\ u(x,0)&=u_0(x), \ x\in{\Omega}, \end{aligned}\right.$$ where $K(t)$ is a weakly singular kernel, $f(\cdot,\cdot)$ is an external forcing, $\alpha,\beta,\nu,\delta,\eta$ are parameters such that $\alpha>0$ is the advection coefficient, $\beta>0$, $\delta\geq 1$ is the retardation time, $\eta\geq 0$ is the relaxation time, $\gamma\in(0,1)$ and $d=2,3$. The integration term is used to study the delayed effect. For $\eta=0$, the equation ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) becomes $$\begin{aligned} \frac{\partial u}{\partial t}+\alpha u^{\delta}\sum_{i=1}^d\frac{\partial u}{\partial x_{i}}-\nu\Delta u = \beta u(1-u^{\delta})(u^{\delta}-\gamma)+f,\end{aligned}$$ which is the generalized Burgers-Huxley equation (GBHE). A wide literature is available on the GBHE and related models. Solitary and travelling wave solutions of the generalized Burgers-Huxley equation using a relevant non-linear transformation is obtained (@XYW [@JS; @TSE; @AMW1], etc). The global solvability results for the stochastic generalized Burgers-Huxley (SGBH) equation have also been proved in [@KMo]. Many researchers have established numerical solutions of the GBHE by using various numerical techniques such as [@HNS] used an Adomian decomposition method, @KHA provided a numerical solution of GBHE based on the collocation method using mesh-less radial basis functions, [@SGZ] performed numerical studies using a fourth-order Runge-Kutta (RK4) scheme in time combined with up to tenth-order finite difference scheme in space, [@KSM] used a three-step Taylor--Galerkin finite element scheme to approximate the solution of the singularly perturbed GBHE. For $\delta=1$, $\eta=0$, $\alpha\neq 0$ and $\beta\neq 0$, the equation ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) is known as the *Burgers-Huxley equation* (see [@XYW; @AMW3], etc.). For $\alpha=0$, $\eta=0$ and $\delta=1$, the equation ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) takes the form which is known as the *Huxley equation* which describes the nerve pulse propagation in nerve fibres and wall motion in liquid crystals [see @XYW1]. For $\beta = 0$, $\delta=1$, and $\alpha = 1$, the equation ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) reduces to the well-known viscous Burgers' equation which represents the mathematical modelling of the turbulence phenomena [see @JMB], far field of wave propagation in non-linear dissipative systems, shock wave theory, vorticity transportation, heat conduction, wave processes in thermoelastic medium, dispersion in porous media, hydrodynamic turbulence, elasticity, gas dynamics, etc. The kernel function $K(\cdot)$ is *weakly singular* such that $K\in\mathrm{L}^1(0,T)$. Moreover, we assume that the function $K(\cdot)$ is a *positive kernel*, that is, if for any $T> 0$, we have $$\begin{aligned} \label{pk} \int_0^T \int_0^t K(t-s)u(s)u(t)ds \mathrm{~d}t\geq 0,\ \ \text{ for all }\ u\in \mathrm{L}^2(0, T).\end{aligned}$$ One primary example of the weakly singular kernel is given by $$K(t) = \frac{1}{\Gamma(\alpha)}\frac{1}{t^{1-\alpha}}, \quad 0< \alpha < 1,$$ where $\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}e^{-t}\mathrm{d}t$ is the *Euler Gamma function*. The convolution integral can be interpreted as a fractional integral of order $\alpha$ [see, e.g., @AEL; @OS]. By varying the fractional integral exponent, we obtain a link between parabolic behavior (in the limit $\alpha = 0,$ the kernel $K$ approaches to the delta distribution) and hyperbolic behavior (specializing $\alpha =1$). The global solvability and numerical studies, for the one dimensional GBHE and the two-dimensional stationary generalized Burgers- Huxley equation, have been recently established in [@MKH] and [@KMR], respectively. The literature relevant to the finite element analysis of integro-differential equations of parabolic type is vast. The time discretization for integro-differential equations with a smooth kernel was introduced in [@STo]. A prior error estimates for linear parabolic equations can be found in [@CLY] and [@AKP], and for both parabolic and hyperbolic equations in [@YFa]. Error estimates for the semi-discrete finite element method (FEM) have been discussed in [@TZh]. Time discretization and error estimation using the Crank-Nicolson scheme and Padé-type scheme are discussed in [@CJL] and [@ZNy], respectively. Fully-discrete FEM for parabolic integro-differential equations with a weakly singular kernel using positive quadrature rule has been discussed in [@CTW] and [@MTh] . A second-order accurate numerical method for a semi-linear integro-differential equation using the Crank-Nicolson scheme with a weakly singular kernel is discussed in [@MMh]. Recently, [@ZXD] used the weak Galerkin finite element scheme for the parabolic integro-differential equation with a weakly singular kernel to obtain the optimal convergence order in $\mathrm{L}^2$. To the best of the authors' knowledge, this work appears to be the first in the field of theoretical and numerical partial differential equations to investigate the solvability of the model generalized Burgers-Huxley equation with memory [\[GBHE\]](#GBHE){reference-type="eqref" reference="GBHE"}. This research article makes significant contributions from both theoretical and numerical perspectives: - **Well-posedness results**: From theoretical point of view, the well-posedness of the generalized Burgers-Huxley equation with memory (Kernel) is proven for the first time in the literature (Section [2.3](#sec2.3){reference-type="ref" reference="sec2.3"}). The analysis is carried out in an open bounded and simply connected convex domain, $\Omega\subset \mathbb{R}^d (d = 2, 3)$, with a Lipschitz boundary $\partial\Omega$. - **Regularity results**: The novel regularity results for the equation [\[GBHE\]](#GBHE){reference-type="eqref" reference="GBHE"} in a convex domain or a domain with a $\mathrm{C}^2$-boundary under smoothness assumptions on the initial data and external forcing have been obtained in Section [2.4](#sec2.4){reference-type="ref" reference="sec2.4"}. These results are significant from a numerical analysis standpoint as well. - **Semi-discrete error estimates**: This work provides, for the first time, semi-discrete error estimates for non-linear integro-differential equations with weakly singular kernels under minimal regularity assumptions. - **Fully-discrete FEM error estimates**: The major contribution in the numerical aspect lies in the error estimates for the fully-discrete finite element method (FEM). Unlike the previous literature that necessitates $u_{tt}\in \mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$ ([@VTh]) to handle the time derivative term, which requires the boundary to be at least $\mathrm{C}^4$, our results do not impose such higher regularity requirements. Notably, the minimum regularity result has been established for the linear parabolic equation [@LSW], but our work is the first to extend it for a non-linear equation with weakly singular kernels. We show the existence and uniqueness of a global weak solution to the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}), using a Faedo-Galerkin method. Regularity results of the weak solution are obtained under smoothness assumptions on the initial data and external forcing. Here, we propose the conforming finite elements (CFEM) scheme to obtain the numerical approximation of the exact solution under minimal regularity and the required regularity is also proved for GBHE with weakly singular kernels. We rigorously derive a priori error estimates indicating first-order convergence in both space and time of the CFEM. We also include a set of computational tests that confirm the theoretical error bounds and show some model equation properties. The finite element approximation using non-conforming and discontinuous Galerkin (DG) finite element method has been discussed in [@GBHE]. The paper is organized as follows: In section [2](#sec1){reference-type="ref" reference="sec1"}, we will discuss the notational conventions and provide the abstract formulation of our problem with the necessary functional spaces. The global solvability results for the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) are established in Theorem [\[EWS\]](#EWS){reference-type="ref" reference="EWS"}; that is, the existence and uniqueness of a global weak solution is proved by taking the initial data $u_0\in \mathrm{L}^2(\Omega)$ and external forcing $f \in \mathrm{L}^2(0, T; \mathrm{H}^{-1}(\Omega))$. The uniqueness result of weak solutions under different choices on the initial data is discussed in Theorem [\[t3\]](#t3){reference-type="ref" reference="t3"}. The regularity results are obtained in Theorem [\[2M.thm3.2\]](#2M.thm3.2){reference-type="ref" reference="2M.thm3.2"} and Theorem [\[2M.thm33\]](#2M.thm33){reference-type="ref" reference="2M.thm33"} under smoothness assumptions on the initial data and the external forcing. In the subsequent section, the semi-discrete and the fully-discrete scheme are introduced, and a priori error estimates are derived for the CFEM in section [3](#sec3){reference-type="ref" reference="sec3"}. Finally, numerical results are presented in section [4](#sec4){reference-type="ref" reference="sec4"} for 2D and 3D to support our theoretical results. # Mathematical Setting and Solvability {#sec1} This section lists necessary function spaces and notations used throughout the paper. Also, we define the operators used to obtain the global solvability results of the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}), discuss its properties and establish the existence of a unique weak solution for our system and its regularity. ## Functional setting We use the standard notation for function spaces. In particular, the Lebesgue spaces are denoted by $\mathrm{L}^p(\Omega)$ for $p\in[1,\infty]$ and the corresponding norm is denoted by $\|\cdot\|_{\mathrm{L}^p}$ (omitting the domain specification whenever clear from the context). For $p=2$, $\mathrm{L}^2(\Omega)$ is a Hilbert space with the inner product denoted by $(\cdot,\cdot)$. The Hilbertian Sobolev spaces are denoted by $\mathrm{H}^{k}(\Omega)$. Let $\mathrm{C}_0^{\infty}(\Omega)$ denote the space of all infinitely differentiable functions with compact support in $\Omega$ and $\mathrm{H}_0^1(\Omega)$ be the closure of $\mathrm{C}_0^{\infty}(\Omega)$ in $\mathrm{H}^1$-norm. As $\Omega$ is a bounded domain, $\|\nabla\cdot\|_{\mathrm{L}^2}$ defines an equivalent norm on $\mathrm{H}^1_0(\Omega)$ by using the Poincaré inequality. Moreover, we have the Gelfand triplet $\mathrm{H}_0^1(\Omega)\hookrightarrow\mathrm{L}^2(\Omega)\hookrightarrow\mathrm{H}^{-1}(\Omega)$, where $\mathrm{H}^{-1}(\Omega)$ is the dual of $\mathrm{H}_0^1(\Omega)$, and the embedding $\mathrm{H}_0^1(\Omega)\hookrightarrow\mathrm{L}^2(\Omega)$ is compact. The duality paring between $\mathrm{H}_0^1(\Omega)$ and $\mathrm{H}^{-1}(\Omega)$ as well as ${\mathrm{L}}^p(\Omega)$ and its dual ${\mathrm{L}}^{p'}(\Omega)$, where $\frac{1}{p}+\frac{1}{p'}=1$ is denoted by $\langle\cdot,\cdot\rangle$. The sum space $\mathrm{H}^{-1}(\Omega)+{\mathrm{L}}^{p'}(\Omega)$ is well defined and $$\begin{aligned} (\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{p'}(\Omega))'= \mathrm{H}_0^1(\Omega)\cap\mathrm{L}^p(\Omega) \ \text{and} \ (\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^p(\Omega))'=\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{p'}(\Omega),\end{aligned}$$ where $\|y\|_{\mathrm{H}_0^1\cap\mathrm{L}^{p}}=\max\{\|y\|_{\mathrm{H}_0^1},\|y\|_{\mathrm{L}^p}\},$ which is equivalent to the norms $\|y\|_{\mathrm{H}_0^1}+\|y\|_{\mathrm{L}^{p}}$ and $\sqrt{\|y\|_{\mathrm{H}_0^1}^2+\|y\|_{\mathrm{L}^p}^2}$, and $$\begin{aligned} \|y\|_{\mathrm{H}^{-1}+\mathrm{L}^{p'}}&=\inf\{\|y_1\|_{\mathrm{H}^{-1}}+\|y_2\|_{\mathrm{L}^{p'}}:y=y_1+y_2, y_1\in\mathrm{H}^{-1}(\Omega) \ \text{and} \ y_2\in\mathrm{L}^{p'}(\Omega)\}\nonumber\\&= \sup\left\{\frac{|\langle y_1+y_2,f\rangle|}{\|f\|_{\mathrm{H}_0^1\cap\mathrm{L}^p}}:0\neq f\in\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^p(\Omega)\right\}.\end{aligned}$$ Note that $\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^p(\Omega)$ and $\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{p'}(\Omega)$ are Banach spaces. Moreover, we have the continuous embedding $\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^p(\Omega)\hookrightarrow\mathrm{H}_0^1(\Omega)\hookrightarrow\mathrm{L}^2(\Omega)\hookrightarrow\mathrm{H}^{-1}(\Omega)\hookrightarrow\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{p'}(\Omega)$. The following operators will be used in further analysis: ### Linear operator Let $A$ be the unbounded and self-adjoint linear operator on $\mathrm{L}^2(\Omega)$ defined by: $$\begin{aligned} A u:=-\Delta u, \ D(A)= \mathrm{H}^2(\Omega)\cap\mathrm{H}_0^1(\Omega),\end{aligned}$$ whenever $\Omega$ is convex, bounded open subset of $\mathbb{R}^n$ with a $\mathrm{C}^2$ boundary $\partial\Omega$ (Theorem 3.1.2.1, [@GPi]). Note that $D(A)\subset \mathrm{L}^p$, for $p\in[1,\infty]$ and $d=2,3$, using the Sobolev Embedding Theorem [see @MZ]. As the embedding $\mathrm{H}_0^1(\Omega)\hookrightarrow \mathrm{L}^2(\Omega)$ is compact, $A^{-1}$ exists and is a symmetric compact operator on $\mathrm{L}^2(\Omega)$. Then, by using spectral theorem [see @RDJL], there exists a sequence of eigenvalues of $A$, $0<\lambda_1\leq\lambda_2\leq\ldots\to\infty$ and an orthonormal basis $\{w_k\}_{k=1}^{\infty}$ of $\mathrm{L}^2(\Omega)$ consisting of eigenfunctions of $A$ [see @RDJL p. 504]. An integration by parts yields $$\langle Au,v\rangle=(\nabla u,\nabla v), \ \text{ for all } \ u,v\in\mathrm{H}_0^1(\Omega),$$ so that $A:\mathrm{H}_0^1(\Omega)\to\mathrm{H}^{-1}(\Omega)$. ### Non-linear operators Let us define the non-linear operator $b:(\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^{2(\delta+1)}(\Omega))\times\mathrm{H}_0^1(\Omega)\times (\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^{2(\delta+1)}(\Omega))\to\mathbb{R}$ by $$b(u,v,w)=\int_{\Omega}(u(x))^{\delta}\sum_{i=1}^{d}\frac{\partial v(x)}{\partial x_i}w(x)\mathrm{~d}x.$$ The map is well-defined since $|b(u,v,w)|\leq\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}\|v\|_{\mathrm{H}_0^1}\|w\|_{\mathrm{L}^{2(\delta+1)}},$ so that we can define a mapping $B(\cdot,\cdot):(\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^{2(\delta+1)}(\Omega))\times\mathrm{H}_0^1(\Omega)\to \mathrm{H}^{-1}(\Omega)+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega)$ by $\langle\mathrm{B}(u,v),w\rangle=b(u,v,w)$. For simplicity, we use the notation $B(u)=B(u,u)$. For all $p\geq 2$ and $u\in\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^{2(\delta+1)}(\Omega)$, using an integration by parts and boundary conditions, we have $$\begin{aligned} b(u,u,|u|^{p-2}u)&=\left(u^{\delta}\sum_{i=1}^{d}\frac{\partial u}{\partial x_i},|u|^{p-2}u\right)=\frac{1}{\delta+2}\sum_{i=1}^{d}\int_{\Omega}\frac{\partial}{\partial {x_i}}(u(x))^{\delta+2}|u(x)|^{p-2}\mathrm{~d} x\nonumber\\&=-\frac{1}{\delta+2}\sum_{i=1}^{d}\int_{\Omega}(u(x))^{\delta+2}\frac{\partial}{\partial {x_i}}|u(x)|^{p-2}\mathrm{~d} x=-\frac{p-2}{\delta+2}\sum_{i=1}^{d}\int_{\Omega}(u(x))^{\delta}|u(x)|^{p-2}\frac{\partial u(x)}{\partial{x_i}}\mathrm{~d}x\nonumber\\&=-\frac{p-2}{\delta+2}\left(u^{\delta}\sum_{i=1}^{d}\frac{\partial u}{\partial x_i},|u|^{p-2}u\right).\end{aligned}$$ The above expression implies $$\begin{aligned} \label{7a} b(u,u,|u|^{p-2}u)=\left(u^{\delta}\sum_{i=1}^{d}\frac{\partial u}{\partial x_i},|u|^{p-2}u\right)=0.\end{aligned}$$ Let us now show that the operator $B(\cdot)$ is locally Lipschitz. For $u,v,z \in \mathrm{H}_0^1(\Omega)\cap \mathrm{L}^{2(\delta+1)}(\Omega),$ and $w=u-v$, using integration by parts and Taylor's formula for $0<\theta_1<1$, we have $$\begin{aligned} & \langle B(u)-B(v),z\rangle \nonumber\\&= \left(u^{\delta}\sum_{i=1}^{d}\frac{\partial u}{\partial x_i} - v^{\delta}\sum_{i=1}^{d}\frac{\partial v}{\partial x_i},z\right) = \left(u^{\delta}\sum_{i=1}^{d}\frac{\partial w}{\partial x_i},z\right) + \left((u^{\delta}-v^{\delta})\sum_{i=1}^{d}\frac{\partial v}{\partial x_i},z\right)\nonumber \\& = -\delta\left(u^{\delta-1}\sum_{i=1}^{d}\frac{\partial u}{\partial x_i}w,z\right) -\left(u^{\delta}w,\sum_{i=1}^d\frac{\partial z}{\partial x_i}\right)+ \delta\left((\theta_1 u+(1-\theta_1)v)^{\delta-1}\sum_{i=1}^{d}\frac{\partial v}{\partial x_i}w,z\right)\nonumber\\&\leq\left(\delta\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta-1}\|\nabla u\|_{\mathrm{L}^2}\|z\|_{\mathrm{L}^{2(\delta+1)}}+\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}\|\nabla z\|_{\mathrm{L}^2}+\delta2^{\delta}(\|u\|^{\delta-1}_{\mathrm{L}^{2(\delta+1)}}+\|v\|^{\delta-1}_{\mathrm{L}^{2(\delta+1)}})\|\nabla v\|_{\mathrm{L}^2}\|z\|_{\mathrm{L}^{2(\delta+1)}}\right)\nonumber\\&\quad\times\|w\|_{\mathrm{L}^{2(\delta+1)}}\nonumber\\&\leq C\left[\left(\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta-1}+\|v\|_{\mathrm{L}^{2(\delta+1)}}^{\delta-1}\right)\left(\|u\|_{\mathrm{H}_0^1}+\|v\|_{\mathrm{H}_0^1}\right)+\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}\right]\|w\|_{\mathrm{L}^{2(\delta+1)}}\|z\|_{\mathrm{H}_0^1\cap\mathrm{L}^{2(\delta+1)}},\end{aligned}$$ so that $$\begin{aligned} \label{bllc} \|B(u)-B(v)\|_{\mathrm{H}^{-1}+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}} &\leq C\left[\left(\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta-1}+\|v\|_{\mathrm{L}^{2(\delta+1)}}^{\delta-1}\right)\left(\|u\|_{\mathrm{H}_0^1}+\|v\|_{\mathrm{H}_0^1}\right)+\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}\right]\|w\|_{\mathrm{L}^{2(\delta+1)}}\nonumber\\&\leq C r^{\delta}\|u-v\|_{\mathrm{L}^{2(\delta+1)}},\end{aligned}$$ for all $\|u\|_{\mathrm{H}_0^1\cap\mathrm{L}^{2(\delta+1)}}, \|v\|_{\mathrm{H}_0^1\cap\mathrm{L}^{2(\delta+1)}}\leq r$. Therefore, the operator $B : \mathrm{H}_0^1(\Omega)\cap \mathrm{L}^{2(\delta+1)}(\Omega) \to\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega)$ is locally Lipschitz. Let us now consider the non-linear operator $c(u)=u(1-u^{\delta})(u^{\delta}-\gamma)$. It should be noted that $$\begin{aligned} \label{7} (c(u),u)&=(u(1-u^{\delta})(u^{\delta}-\gamma),u)=((1+\gamma)u^{\delta+1}-\gamma u-u^{2\delta+1},u)\nonumber\\&=(1+\gamma)(u^{\delta+1},u)-\gamma\|u\|_{\mathrm{L}^2}^2-\|u\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\nonumber\\&\leq (1+\gamma)\|u\|^{\delta+1}_{\mathrm{L}^{2(\delta+1)}}\|u\|_{\mathrm{L}^2}-\gamma\|u\|_{\mathrm{L}^2}^2-\|u\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)},\end{aligned}$$ for all $u\in\mathrm{L}^{2(\delta+1)}(\Omega)$. Using Taylor's formula and Hölder's inequality, for $u,v,z\in\mathrm{L}^{2(\delta+1)}(\Omega)$ with $w=u-v$, $0<\theta_2<1$ and $0<\theta_3<1$, we get $$\begin{aligned} \label{cllc} \langle c(u)-c(v),z\rangle &=\langle (1+\gamma)(u^{\delta+1}-v^{\delta+1})-\gamma(u-v)-(u^{2\delta+1}-v^{2\delta+1}),z\rangle\\&= (1+\gamma)(\delta+1)( w(\theta_2u+(1-\theta_2)v)^{\delta},z)-\gamma(w,z)\nonumber\\&\quad-(2\delta+1)\langle w(\theta_3u+(1-\theta_3)v)^{2\delta},z\rangle\nonumber\\&\leq (1+\gamma)(\delta+1)2^{\delta-1}\left(\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}+\|v\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}\right)\|w\|_{\mathrm{L}^{2(\delta+1)}}\|z\|_{\mathrm{L}^2}+\gamma\|w\|_{\mathrm{L}^2}\|z\|_{\mathrm{L}^2}\nonumber\\&\quad+(2\delta+1)2^{2\delta-1}\left(\|u\|_{\mathrm{L}^{2(\delta+1)}}^{2\delta}+\|v\|_{\mathrm{L}^{2(\delta+1)}}^{2\delta}\right)\|w\|_{\mathrm{L}^{2(\delta+1)}}\|z\|_{\mathrm{L}^{2(\delta+1)}}\nonumber\\&\nonumber\leq \left((1+\gamma)(\delta+1)2^{\delta}|\Omega|^{\frac{\delta}{2(\delta+1)}}r^{\delta}+\gamma|\Omega|^{\frac{\delta}{\delta+1}}+(2\delta+1)2^{2\delta}r^{2\delta}\right)\|w\|_{\mathrm{L}^{2(\delta+1)}}\|z\|_{\mathrm{L}^{2(\delta+1)}},\end{aligned}$$ for $\|u\|_{\mathrm{L}^{2(\delta+1)}},\|v\|_{\mathrm{L}^{2(\delta+1)}}\leq r$, where $|\Omega|$ is the Lebesgue measure of $\Omega$. Thus, the operator $c(\cdot):\mathrm{L}^{2(\delta+1)}(\Omega)\to\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega)$ is locally Lipschitz. The following lemmas are useful for the further analysis: [@MTM][\[l1\]]{#l1 label="l1"} Let $K \in \mathrm{L}^1(0,T)$ and $\phi\in \mathrm{L}^2(0,T)$ for some $T>0$. Then $$\left(\int_0^{T}\left(\int_0^sK(s-\tau)\phi(\tau)\mathrm{~d}\tau\right)^2\mathrm{~d}s\right)^{\frac{1}{2}}\leq \left(\int_0^{T}|K(s)|\mathrm{~d}s\right)\left(\int_0^{T}\phi^2(s)\mathrm{~d}s\right)^\frac{1}{2}.$$ [\[l2\]]{#l2 label="l2"} If the kernel $K(\cdot)$ satisfies the conditions $$\left\{ \begin{aligned} &\nonumber K\in\mathrm{C}^2(0,\infty)\cap\mathrm{C}[0,\infty),\\ &\nonumber(-1)^kK^{(k)}(t)\geq 0,\textrm{ for all }t>0,k=0,1,2, \end{aligned} \right.$$ then for any right continuous function $f:[0,T]\to[0,\infty)$ with left limits and for $g:[0,T]\to\mathbb{R}$ such that $(K*g)(\cdot)$ is well defined, we have $$\begin{aligned} \label{211}\int_0^Tf(t)(K*g)(t)g(t)\/\mathrm{d}\/t\geq 0.\end{aligned}$$ [\[r1\]]{#r1 label="r1"} By approximating a weakly singular kernel with a smooth kernel having the properties obtained in Lemma [\[l2\]](#l2){reference-type="ref" reference="l2"}, one can obtain the result [\[211\]](#211){reference-type="eqref" reference="211"} for weakly singular kernels also. For instance, one can approximate $K(t) = \frac{1}{\Gamma(\alpha)}\frac{1}{t^{1-\alpha}}, \ 0< \alpha < 1,$ by $K_{\epsilon}(t) = \frac{1}{\Gamma(\alpha)}\frac{1}{(t+\epsilon)^{1-\alpha}},$ for some $\epsilon>0$. ## Abstract formulation By using the above functional setting, the abstract formulation of the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) is given by $$\label{AFGBHEM} \left\{ \begin{aligned} \frac{\mathrm{~d}u(t)}{\mathrm{~d}t}+\nu Au(t)+\eta ( K*Au)(t)&=-\alpha B(u(t))+\beta c(u(t))+f(t), \ \text{ in }\ \mathrm{H}^{-1}(\Omega) + \mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega),\\ u(0)&=u_0\in\mathrm{L}^2(\Omega), \end{aligned} \right.$$ for a.e. $t\in[0,T]$, where $(K*A u)(t)=\int_0^tK(t-s)Au(s)ds.$ Let us now provide the definition of weak solution of the system ([\[AFGBHEM\]](#AFGBHEM){reference-type="ref" reference="AFGBHEM"}). [\[weaksolution\]]{#weaksolution label="weaksolution"} For the initial data $u_0\in\mathrm{L}^{2}(\Omega)$ and external forcing $f\in\mathrm{L}^{2}(0,T;\mathrm{H}^{-1}(\Omega))$, we say that a function $$u\in \mathrm{L}^{\infty}(0,T; \mathrm{L}^{2}(\Omega))\cap\mathrm{L}^{2}(0,T; \mathrm{H}_0^1(\Omega))\cap\mathrm{L}^{2(\delta+1)}(0,T; \mathrm{L}^{2(\delta+1)}(\Omega)),$$ with $$\partial_tu\in\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(0,T;\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega)),$$ is called *a weak solution* to the system ([\[AFGBHEM\]](#AFGBHEM){reference-type="ref" reference="AFGBHEM"}), if, for a.e. $t\in(0,T)$, $u(\cdot)$ satisfies: $$\left\{ \begin{aligned} \nonumber\langle\partial_tu(t),v\rangle+\nu (\nabla u(t),\nabla v)+\alpha b(u(t),u(t),v)+\eta(( K*\nabla u)(s),\nabla v)&=\beta\langle c(u(t)),v\rangle+\langle f(t),v\rangle, \\ (u(0),v)&=(u_0,v), \end{aligned} \right.$$ for any $v\in\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^{2(\delta+1)}(\Omega)$. ## Existence and uniqueness of weak solution {#sec2.3} Let us now discuss the existence and uniqueness of the weak solution for the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}). [\[EWS\]]{#EWS label="EWS"} For $u_0\in\mathrm{L}^{2}(\Omega)$ and $f\in\mathrm{L}^{2}(0,T;\mathrm{H}^{-1}(\Omega)),$ there exists a weak solution to the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) in the sense of Definition [\[weaksolution\]](#weaksolution){reference-type="ref" reference="weaksolution"}. *Proof.* We establish the existence of a weak solution to the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) in the following steps: **Step 1:** *Faedo-Galerkin approximatiiron.* Let the set $\{w_k(x)\}_{k=1}^{\infty}$ be an orthogonal basis of $\mathrm{H}_0^1(\Omega)$ and orthonormal basis of $\mathrm{L}^2(\Omega)$ (page 504, [@RDJL]). We look for a function $u_m:[0,T]\to\mathrm{H}_0^1(\Omega)$ of the form $u_m(t)=\sum\limits_{k=1}^md_m^k(t)w_k$, where $m$ is a fixed positive integer and the coefficients $d_m^k(t)$, for $0\leq t\leq T$ and $k=1,\ldots,m$ are selected so that $d_m^k(0)=(u_0,w_k)$, for $k=1,2,\ldots,m$ and $$\begin{aligned} \label{8} &(u_m'(t),w_k)+\nu(\nabla u_m(t),\nabla w_k)+\alpha\left(u_m(t)^{\delta}\sum_{i=1}^{d}\frac{\partial u_m(t)}{\partial x_i},w_k\right)+\eta(( K*\nabla u_m)(t),\nabla w_k)\nonumber\\&=\beta(u_m(t)(1-(u_m(t))^{\delta})((u_m(t))^{\delta}-\gamma),w_k)+\langle f(t),w_k\rangle, \end{aligned}$$ for a.e. $0\leq t\leq T$ and $k=1,\ldots,m$. As the non-linear operators are locally Lipschitz (see ([\[bllc\]](#bllc){reference-type="ref" reference="bllc"}) and ([\[cllc\]](#cllc){reference-type="ref" reference="cllc"})) using Carathéodory's existence theorem, the finite dimensional problem ([\[8\]](#8){reference-type="ref" reference="8"}) has a local solution $u_m\in C([0,T];\mathrm{H}_m)$, where $\mathrm{H}_m=\mathrm{span}\{w_1,\ldots,w_m\}$. **Step 2:** *$\mathrm{L}^2$-energy estimate.* Multiplying ([\[8\]](#8){reference-type="ref" reference="8"}) by $d_m^k(t)$, and then summing for $k=1,\ldots,m$, we have $$\begin{aligned} &\frac{1}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|u_m(t)\|_{\mathrm{L}^2}^2+\nu\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\alpha\left(u_m(t)^{\delta}\sum_{i=1}^d\frac{\partial u_m(t)}{\partial x_i},u_m(t)\right)+\eta(( K*\nabla u_m)(t),\nabla u_m(t))\nonumber\\&=\beta(u_m(t)(1-(u_m(t))^{\delta})((u_m(t))^{\delta}-\gamma),u_m(t))+\langle f(t),u_m(t)\rangle, \end{aligned}$$ for a.e. $t\in[0,T]$. Using the property of the non-linear operator ([\[7a\]](#7a){reference-type="ref" reference="7a"}) and the estimate ([\[7\]](#7){reference-type="ref" reference="7"}) , we get $$\begin{aligned} &\frac{1}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|u_m(t)\|_{\mathrm{L}^2}^2+\nu\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\beta\gamma\|u_m(t)\|_{\mathrm{L}^2}^2+\beta\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\eta((K*\nabla u_m)(t),\nabla u_m(t))\nonumber\\&=\beta(1+\gamma)((u_m(t))^{\delta+1},u_m(t))+\langle f(t),u_m(t)\rangle\nonumber\\&\leq\beta(1+\gamma)\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{\delta+1}\|u_m(t)\|_{\mathrm{L}^2}+\|f(t)\|_{\mathrm{H}^{-1}}\|\nabla u_m(t)\|_{\mathrm{L}^2}\nonumber\\&\leq \frac{\beta}{2}\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\frac{\beta(1+\gamma)^2}{2}\|u_m(t)\|_{\mathrm{L}^2}^2+\frac{\nu}{2}\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\frac{1}{2\nu}\|f(t)\|_{\mathrm{H}^{-1}}^2. \end{aligned}$$ Integrating the above inequality from $0$ to $t$, we have $$\begin{aligned} \label{11} &\|u_m(t)\|_{\mathrm{L}^2}^2+\nu\int_0^t\|\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d} s+\beta\int_0^t\|u_m(s)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\mathrm{~d}s+2\eta\int_0^t((K*\nabla u_m)(s),\nabla u_m(s))\mathrm{~d} s\nonumber\\&\leq\|u_0\|_{\mathrm{L}^2}^2+\frac{1}{\nu}\int_0^t\|f(s)\|_{\mathrm{H}^{-1}}^2\mathrm{~d}s+\beta(1+\gamma^2)\int_0^t\|u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s, \end{aligned}$$ for all $t\in[0,T]$, where we have used the fact that $\|u_m(0)\|_{\mathrm{L}^2}\leq\|u_0\|_{\mathrm{L}^2}$. Using [\[pk\]](#pk){reference-type="eqref" reference="pk"} and an application of Gronwall's inequality in [\[11\]](#11){reference-type="eqref" reference="11"} yields $$\begin{aligned} &\|u_m(t)\|_{\mathrm{L}^2}^2+\nu\int_0^t\|\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d} s+\beta\int_0^t\|u_m(s)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\mathrm{~d}s\leq\left(\|u_0\|_{\mathrm{L}^2}^2+\frac{1}{\nu}\int_0^T\|f(t)\|_{\mathrm{H}^{-1}}^2\mathrm{~d}t\right)e^{\beta(1+\gamma^2)T}, \end{aligned}$$ for all $t\in[0,T]$. Taking supremum over time $0\leq t\leq T$, we have $$\begin{aligned} \label{13} &\sup_{0\leq t\leq T}\|u_m(t)\|_{\mathrm{L}^2}^2+\nu\int_0^T\|\nabla u_m(t)\|_{\mathrm{L}^2}^2\mathrm{~d} t+\beta\int_0^T\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\mathrm{~d}t\nonumber\\&\leq\left(\|u_0\|_{\mathrm{L}^2}^2+\frac{1}{\nu}\int_0^T\|f(t)\|_{\mathrm{H}^{-1}}^2\mathrm{~d}t\right)e^{\beta(1+\gamma^2)T}, \end{aligned}$$ and the right hand side of the inequality [\[13\]](#13){reference-type="eqref" reference="13"} is independent of $m$. **Step 3:** *Estimate for the time derivative.* For any $v\in\mathrm{H}_0^1(\Omega)\cap \mathrm{L}^{2(\delta+1)}(\Omega)$, with $\|v\|_{\mathrm{H}_0^1\cap \mathrm{L}^{2(\delta+1)}}\leq 1$, we have $$\begin{aligned} \langle u_m',v\rangle= (u_m',v)&=-\nu(\nabla u_m,\nabla v)-\alpha b(u_m,u_m,v)+\eta(K*\nabla u_m,\nabla v)+\beta \langle c(u_m),v\rangle+\langle f,v\rangle. \end{aligned}$$ An application of the Cauchy-Schwarz inequality and Hölder's inequality yield $$\begin{aligned} |\langle u_m',v\rangle|&\leq \nu|(\nabla u_m,\nabla v)|+\alpha|(b(u_m,u_m,v^1))|+\eta|(K*\nabla u_m,\nabla v)|+\beta|(c(u_m),v)|+|\langle f,v\rangle|\nonumber\\&\leq \left(\nu\|\nabla u_m\|_{\mathrm{L}^2}+\frac{\alpha}{\delta+1}\|u_m\|^{\delta+1}_{\mathrm{L}^{2(\delta+1)}}+\|f\|_{\mathrm{H}^{-1}}+\eta\|K*\nabla u_m\|_{\mathrm{L}^2}\right)\|\nabla v\|_{\mathrm{L}^2}\nonumber\\&\quad+\beta\left((1+\gamma)\|u_m\|^{\delta+1}_{\mathrm{L}^{2(\delta+1)}}+\gamma\|u_m\|_{\mathrm{L}^2}\right)\|v\|_{\mathrm{L}^2}+\beta\|u_m\|^{2\delta+1}_{\mathrm{L}^{2(\delta+1)}}\|v\|_{\mathrm{L}^{2(\delta+1)}}. \end{aligned}$$ Taking supremum over all $v \in \mathrm{H}_0^1(\Omega) \cap \mathrm{L}^{2(\delta+1)}(\Omega)$, we have $$\begin{aligned} \|u_m'\|_{\mathrm{H}^{-1}+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}} &\leq \bigg( \nu\|\nabla u_m\|_{\mathrm{L}^2}+\frac{\alpha}{\delta+1}\|u_m\|^{\delta+1}_{\mathrm{L}^{2(\delta+1)}}+\|f\|_{\mathrm{H}^{-1}}+\eta\|K*\nabla u_m\|_{\mathrm{L}^2}\nonumber\\&\qquad+C\beta(1+\gamma)\|u_m\|^{\delta+1}_{\mathrm{L}^{2(\delta+1)}}+C\gamma\|u_m\|_{\mathrm{L}^2}+\beta\|u_m\|^{2\delta+1}_{\mathrm{L}^{2(\delta+1)}} \bigg) . \end{aligned}$$ Integrating from 0 to T and using Young's inequality, we arrive at $$\begin{aligned} \label{19} \nonumber\int_{0}^{T} \|u_m'(t)\|_{\mathrm{H}^{-1}+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}}^{\frac{2(\delta+1)}{2\delta+1}}\mathrm{~d}t \leq& C\bigg\{ \nu^{\frac{2(\delta+1)}{2\delta+1}} \int_{0}^{T}\|\nabla u_m(t)\|_{\mathrm{L}^2}^{\frac{2(\delta+1)}{2\delta+1}}\mathrm{~d}t+\left(\frac{\alpha}{\delta+1}\right)^{\frac{2(\delta+1)}{2\delta+1}} \int_{0}^{T}\|u_m(t)\|^{\frac{2(\delta+1)^2}{2\delta+1}}_{\mathrm{L}^{2(\delta+1)}}\mathrm{~d}t\\\nonumber &+ \int_{0}^{T}\|f(t)\|_{\mathrm{H}^{-1}}^{\frac{2(\delta+1)}{2\delta+1}}\mathrm{~d}t+\eta^{\frac{2(\delta+1)}{2\delta+1}}\int_{0}^{T}\|K*\nabla u_m(t)\|_{\mathrm{L}^2}^{\frac{2(\delta+1)}{2\delta+1}}\mathrm{~d}t\nonumber\\&+(\beta(1+\gamma))^{\frac{2(\delta+1)}{2\delta+1}} \int_{0}^{T}\|u_m(t)\|^{\frac{2(\delta+1)^2}{2\delta+1}}_{\mathrm{L}^{2(\delta+1)}}\mathrm{~d}t\nonumber\\&+\gamma^{\frac{2\delta}{2\delta+1}} \int_{0}^{T}\|u_m(t)\|^{\frac{2(\delta+1)}{2\delta+1}}_{\mathrm{L}^2}\mathrm{~d}t+\beta^{\frac{2(\delta+1)}{2\delta+1}} \int_{0}^{T}\|u_m(t)\|^{2(\delta+1)}_{\mathrm{L}^{2(\delta+1)}} \mathrm{~d}t\bigg\}, \end{aligned}$$ where $\int_0^T\|K*\nabla u_m(t)\|_{\mathrm{L}^2}^{\frac{2(\delta+1)}{2\delta+1}} \mathrm{~d}t$ is bounded since (cf. Lemma [\[l1\]](#l1){reference-type="ref" reference="l1"}) $$\begin{aligned} \int_0^T\|K*\nabla u_m(t)\|_{\mathrm{L}^2}^{\frac{2(\delta+1)}{2\delta+1}} \mathrm{~d}t &\leq T^{\frac{\delta}{2\delta+1}}\left(\int_0^T\|K*\nabla u_m(t)\|_{\mathrm{L}^2}^2 \mathrm{~d}t\right)^{\frac{\delta+1}{2\delta+1}} \nonumber\\&\leq T^{\frac{\delta}{2\delta+1}} \left(\int_{0}^{T}|K(t)|\mathrm{~d}t\right)^{\frac{2(\delta+1)}{2\delta+1}} \left(\int_0^T\|\nabla u_m(t)\|_{\mathrm{L}^2}^2 \mathrm{~d}t\right)^{\frac{\delta+1}{2\delta+1}} < \infty, \end{aligned}$$ and $$\begin{aligned} \int_{0}^{T}\|f(t)\|_{\mathrm{H}^{-1}}^{\frac{2(\delta+1)}{2\delta+1}}\mathrm{~d}t \leq T^{\frac{\delta}{2\delta+1}} \left(\int_{0}^{T}\|f(t)\|_{\mathrm{H}^{-1}}^2\mathrm{~d}t\right)^{\frac{\delta+1}{2\delta+1}}. \end{aligned}$$ **Step 4:** *Passing to limit.* As the energy estimates ([\[13\]](#13){reference-type="ref" reference="13"}) and ([\[19\]](#19){reference-type="ref" reference="19"}) are uniformly bounded and independent of $m$, using the Banach-Alaoglu theorem, we can extract a subsequence $\{u_{m_j}\}_{j=1}^{\infty}$ of $\{u_m\}_{m=1}^{\infty}$ such that $$\label{2M.25} \left\{ \begin{aligned} &u_{m_j}\xrightarrow{w^*} u\ \text{ in }\ \mathrm{L}^{\infty}(0,T;\mathrm{L}^{2}(\Omega)),\\ &u_{m_j}\xrightarrow{w} u\ \text{ in }\ \mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega)), \\ &u_{m_j}\xrightarrow{w} u\ \text{ in }\ \mathrm{L}^{2(\delta+1)}(0,T;\mathrm{L}^{2(\delta+1)}(\Omega)),\\ &\partial_tu_{m_j}\xrightarrow{w} \partial_tu\ \text{ in }\ \partial_tu\in\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(0,T;\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega)), \end{aligned} \right.$$ as $j\to\infty$, for all $\delta\geq 1$. Since we have compact embedding of $\mathrm{H}_0^1(\Omega)\subset\mathrm{L}^2(\Omega)$ (page 284, [@LCE]), an application of the Aubin-Lions compactness lemma yields the following strong convergence: $$\begin{aligned} \label{2M.26} u_{m_j}\rightarrow u\ \text{ in }\ \mathrm{L}^{2}(0,T;\mathrm{L}^2(\Omega)), \end{aligned}$$ as $j\to\infty$. The above strong convergence also implies $u_{m_{k_j}}\rightarrow u$, for a.e. $(t,x)\in[0,T]\times\Omega$ as $j\to\infty$ (that is, along a subsequence). Let us now take limit in ([\[8\]](#8){reference-type="ref" reference="8"}) along the subsequence . Choose a function $v\in \mathrm{C}^1([0,T];\mathrm{H}_0^1(\Omega))$ having the form $v(t)=\sum\limits_{k=1}^Nd_m^k(t)w_k,$ where $\{d^k_m(t)\}_{k=1}^N$ are given smooth functions and $N$ is a fixed integer. Let us choose $m\geq N$, multiply ([\[8\]](#8){reference-type="ref" reference="8"}) by $d_m^k(t)$, sum from $k=1$ to $N$ and then integrate it from $0$ to $T$ to find $$\begin{aligned} \label{2M.27} &\int_0^T\left[\langle \partial_t u_{m}(t),v(t)\rangle +\nu(\nabla u_m(t),\nabla v(t))+\eta\langle(K*\nabla u_m(t)),\nabla v(t)\rangle+\alpha\left((u_m(t))^{\delta}\sum_{i=1}^d\frac{\partial u_m}{\partial x_i}(t),v(t)\right)\right]\mathrm{~d}t\nonumber\\&=\int_0^T\left[\beta(u_m(t)(1-(u_m(t))^{\delta})((u_m(t))^{\delta}-\gamma),v(t))+\langle f(t),v(t)\rangle\right]\mathrm{~d}t. \end{aligned}$$ Setting $m=m_{j}$ (more specifically $m_{k_j}$) and using the convergences given in ([\[2M.25\]](#2M.25){reference-type="ref" reference="2M.25"}) and ([\[2M.26\]](#2M.26){reference-type="ref" reference="2M.26"}), we can take weak limit in ([\[2M.27\]](#2M.27){reference-type="ref" reference="2M.27"}) to obtain $$\begin{aligned} \label{2M.28} &\nonumber\int_0^T\left[\langle \partial_t u(t),v(t)\rangle +\nu(\nabla u(t),\nabla v(t))+\eta\langle(K*\nabla u(t)),\nabla v(t)\rangle+\alpha\left((u(t))^{\delta}\sum_{i=1}^d\frac{\partial u_m}{\partial x_i}(t),v(t)\right)\right]\mathrm{~d}t\\&=\int_0^T\left[\beta(u(t)(1-(u(t))^{\delta})((u(t))^{\delta}-\gamma),v(t))+\langle f(t),v(t)\rangle \right]\mathrm{~d}t, \end{aligned}$$ provided if we show that $$\begin{aligned} B(u_{m_{k_j}})\xrightarrow{w} B(u) , \ \text{ and }\ c(u_{m_{k_j}})\xrightarrow{w} c(u )\ \text{ in }\ \mathrm{H}^{-1}(\Omega) + \mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega), \ \text{ as } \ j\to\infty. \end{aligned}$$ Using density argument $C_0^{\infty}(\Omega)$ is dense in $\mathrm{H}_0^1(\Omega)$, we will show that $b(u_{m_{k_j}},u_{m_{k_j}},v)\to b(u,u,v),$ for all $v \in C_0^{\infty}(\Omega)$ to obtain that $B(u_{m_{k_j}})\xrightarrow{w} B(u) \ \text{ in }\ H^{-1}(\Omega) + \mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega)$, as $j\to\infty$. By an application of Taylor's formula, integration by parts, Hölder's inequality, and convergence, we obtain $$\begin{aligned} & |b(u_{m_{k_j}},u_{m_{k_j}},v)- b(u,u,v)|\nonumber\\&= \left|\frac{1}{\delta+1}\sum_{i=1}^d\int_{\Omega}(u_{m_{k_j}}^{\delta+1}(x)-u^{\delta+1}(x))\frac{\partial v(x)}{\partial x_i}\mathrm{~d} x\right|\nonumber\\&=\left|\sum_{i=1}^d\int_{\Omega}(\theta u_{m_{k_j}}(x)+(1-\theta)u(x))^{\delta}(u_{m_{k_j}}(x)-u(x))\frac{\partial v(x)}{\partial x_i}\mathrm{~d} x\right|\nonumber\\&\leq\|u_{m_{k_j}}-u\|_{\mathrm{L}^2}\left(\|u_{m_{k_j}}\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}+\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}\right)\|\nabla v\|_{\mathrm{L}^{2(\delta+1)}}\nonumber\\&\to 0\ \text{ as } \ j\to\infty, \ \text{ for all } \ v\in C_0^{\infty}(\Omega). \end{aligned}$$ Again using Taylor's formula, interpolation and Hölder's inequalities, , we find $$\begin{aligned} & |(c(u_{m_{k_j}})-c(u),v)|\nonumber\\&\leq(1+\gamma)\left|\int_{\Omega}(u_{m_{k_j}}^{\delta+1}(x)-u^{\delta+1}(x))v(x)\mathrm{~d} x\right|+\left|\int_{\Omega}(u_{m_{k_j}}(x)-u(x))v(x)\mathrm{~d} x\right| \nonumber\\ &\quad+\left|\int_{\Omega}(u_{m_{k_j}}^{2\delta+1}(x)-u^{2\delta+1}(x))v(x)\mathrm{~d} x\right|\nonumber\\ &\leq \left((1+\gamma)(\delta+1) \left(\|u_{m_{k_j}}\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}+\|u\|_{\mathrm{L}^{2(\delta+1)}}^{\delta}\right)\|v\|_{\mathrm{L}^{2(\delta+1)}}+\|v\|_{\mathrm{L}^2}\right)\|u_{m_{k_j}}-u\|_{\mathrm{L}^2}\nonumber\\ &\quad+(1+2\delta)\|u_{m_{k_j}}-u\|_{\mathrm{L}^2}^{\frac{1}{\delta}}\left(\|u_{m_{k_j}}\|_{\mathrm{L}^{2(\delta+1)}}^{1-\frac{1}{\delta}}+\|u\|_{\mathrm{L}^{2(\delta+1)}}^{1-\frac{1}{\delta}}\right)\left(\|u_{m_{k_j}}\|_{\mathrm{L}^{2(\delta+1)}}^{2\delta}+\|u\|_{\mathrm{L}^{2(\delta+1)}}^{2\delta}\right)\|v\|_{\mathrm{L}^{\infty}}\nonumber\\&\to 0\ \text{ as } \ j\to\infty, \ \text{ for all } \ v\in C_0^{\infty}(\Omega). \end{aligned}$$ which completes the convergence proof. **Step 5:** *Initial data.* Let us now show that $u(0)=u_0$. From ([\[2M.28\]](#2M.28){reference-type="ref" reference="2M.28"}), we note that $$\begin{aligned} \label{2M.31} &\int_0^T\left[-\langle u(t), v'(t)\rangle +\nu(\nabla u(t),\nabla v(t))+\eta\langle(K*\nabla u(t)),\nabla v(t)\rangle+\alpha\left((u(t))^{\delta}\sum_{i=1}^2\frac{\partial u}{\partial x_i}(t),v(t)\right)\right]\mathrm{~d}t\nonumber\\&=\int_0^T\left[\beta(u(t)(1-(u(t))^{\delta})((u(t))^{\delta}-\gamma),v(t))+(f(t),v(t))\right]\mathrm{~d}t+(u(0),v(0)), \end{aligned}$$ for each $v\in \mathrm{C}^1([0,T];\mathrm{H}_0^1(\Omega))$ with $v(T)=0$. In a similar fashion, we deduce from ([\[2M.27\]](#2M.27){reference-type="ref" reference="2M.27"}) that $$\begin{aligned} &\int_0^T\left[-\langle u_{m}(t),v'(t)\rangle +\nu(\nabla u_m(t),\nabla v(t))+\eta\langle(K*\nabla u_m(t)),\nabla v(t)\rangle+\alpha\left((u_m(t))^{\delta}\sum_{i=1}^2\frac{\partial u_m}{\partial x_i}(t),v(t)\right)\right]\mathrm{~d}t\nonumber\\&=\int_0^T\left[\beta(u_m(t)(1-(u_m(t))^{\delta})((u_m(t))^{\delta}-\gamma),v(t))+(f(t),v(t))\right]\mathrm{~d}t+(u_m(0),v(0)). \end{aligned}$$ Taking $m=m_j$ in the above expression, using the above convergences (([\[2M.25\]](#2M.25){reference-type="ref" reference="2M.25"}) and ([\[2M.26\]](#2M.26){reference-type="ref" reference="2M.26"})) and passing then to limit, we find $$\begin{aligned} \label{2M.33} &\int_0^T\left[-\langle u(t), v'(t)\rangle +\nu(\nabla u(t),\nabla v(t))+\eta\langle(K*\nabla u(t)),\nabla v(t)\rangle+\alpha\left((u(t))^{\delta}\sum_{i=1}^2\frac{\partial u}{\partial x_i}(t),v(t)\right)\right]\mathrm{~d}t\nonumber\\&=\int_0^T\left[\beta(u(t)(1-(u(t))^{\delta})((u(t))^{\delta}-\gamma),v(t))+(f(t),v(t))\right]\mathrm{~d}t+(u_0,v(0)), \end{aligned}$$ since $u_m(0)\to u_0$ as $m\to\infty$ in $\mathrm{L}^2(\Omega)$. As $v(0)$ is arbitrary, comparing ([\[2M.31\]](#2M.31){reference-type="ref" reference="2M.31"}) and ([\[2M.33\]](#2M.33){reference-type="ref" reference="2M.33"}), one can easily conclude that $u(0)=u_0$ in $\mathrm{L}^2(\Omega)$. **Step 6:** *Energy equality.* As $u\in \mathrm{L}^{2}(0,T;\mathrm{H}_0^1(\Omega))\cap\mathrm{L}^{2(\delta+1)}(0,T;\mathrm{L}^{2(\delta+1)}(\Omega))$ with $\partial_tu\in\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(0,T;\newline\mathrm{H}^{-1}(\Omega)+\mathrm{L}^{\frac{2(\delta+1)}{2\delta+1}}(\Omega))$ using an argument similar to Theorem 7.2 ([@JCR] (Exercise 8.2)) we have $u\in \mathrm{C}^0([0,T];\mathrm{L}^2(\Omega))$ and the following energy equality holds $$\begin{aligned} &\|u(t)\|_{\mathrm{L}^2}^2+2\nu\int_0^t\|\nabla u(s)\|_{\mathrm{L}^2}^2\mathrm{~d} s + 2\eta\int_0^t(K*\nabla u(s), \nabla u(s)) \mathrm{~d} s+ 2\beta\gamma \int_{0}^{t}\|u(s)\|^2_{\mathrm{L}^2}\mathrm{~d} s \nonumber\\& +2\beta\int_0^T\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\mathrm{~d}s = \|u_0\|_{\mathrm{L}^2}^2 + 2\beta\int_{0}^t(u^{\delta+1}(s),u(s)) \mathrm{~d} s+\int_0^t\langle f(s), u(s)\rangle \mathrm{~d}s, \end{aligned}$$ for all $t\in[0,T].$ This proves the existence of weak solution. ◻ [\[r2\]]{#r2 label="r2"} If we take the initial data $u_0\in\mathrm{L}^{2(\delta+1)}(\Omega)$, and the external forcing, $f\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$, then we can obtain the $\mathrm{L}^{2(\delta+1)}$-energy estimate as follows: Multiplying ([\[8\]](#8){reference-type="ref" reference="8"}) by $d_m^k(t)|u_m(t)|^{2\delta}$, and then summing for $k=1,\ldots,m$, we obtain for a.e. $t\in[0,T]$ $$\begin{aligned} &(\partial_t u_m(t),|u_m(t)|^{2\delta}u_m(t))+\nu(2\delta+1)(\nabla u_m(t),|u_m(t)|^{2\delta}\nabla u_m(t))\nonumber\\&\quad +\eta(2\delta+1)((K*\nabla u_m)(t),|u_m(t)|^{2\delta}\nabla u_m(t))+\alpha\left((u_m(t))^{\delta}\sum_{i=1}^{d}\frac{\partial u_m}{\partial x_i}(t),|u_m(t)|^{2\delta}u_m(t)\right)\nonumber\\&= \beta(u_m(t)(1-(u_m(t))^{\delta})((u_m(t))^{\delta}-\gamma,|u_m(t)|^{2\delta}u_m(t))+(f,|u_m(t)|^{2\delta}u_m(t)). \end{aligned}$$ Now, using ([\[7a\]](#7a){reference-type="ref" reference="7a"}), the Cauchy-Schwarz, Hölder's and Young's inequalities, we arrive at $$\begin{aligned} &\frac{1}{2(\delta+1)}\frac{\mathrm{~d}}{\mathrm{~d}t}\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\nu(2\delta+1)\||u_m(t)|^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2\nonumber\\&\quad+\eta(2\delta+1)((K*\nabla u_m)(t),|u_m(t)|^{2\delta}\nabla u_m(t))+\beta\|u_m(t)\|_{\mathrm{L}^{4\delta+2}}^{4\delta+2}+\beta\gamma\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\nonumber\\&\leq\frac{\beta}{2}\|u_m(t)\|_{\mathrm{L}^{4\delta+2}}^{4\delta+2}+\beta(1+\gamma)^2\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\frac{1}{\beta}\|f(t)\|_{\mathrm{L}^2}^2, \end{aligned}$$ for a.e. $t\in[0,T]$. Integrating the above inequality from $0$ to $t$ and using Lemma [\[l2\]](#l2){reference-type="ref" reference="l2"} and Remark [\[r1\]](#r1){reference-type="ref" reference="r1"}, we have $$\begin{aligned} &\frac{1}{2(\delta+1)}\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\nu(2\delta+1)\int_0^t\||u_m(s)|^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+\frac{\beta}{2}\int_0^t\|u_m(s)\|_{\mathrm{L}^{4\delta+2}}^{4\delta+2}\mathrm{~d}s\nonumber\\&\leq\frac{1}{2(\delta+1)}\|u_0\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\frac{1}{\beta}\int_0^t\|f(s)\|_{\mathrm{L}^{2}}^2\mathrm{~d}s+\beta(1+\gamma+\gamma^2)\int_0^t\|u_m(s)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\mathrm{~d}s, \end{aligned}$$ for all $t\in[0,T]$. Applying Gronwall's inequality, we get $$\begin{aligned} &\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\leq \left(\|u_0\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\frac{2(\delta+1)}{\beta}\int_0^T\|f(t)\|_{\mathrm{L}^{2}}^2\mathrm{~d}t\right)e^{\beta(1+\gamma+\gamma^2)T}, \end{aligned}$$ for all $t\in[0,T]$. Thus, it is immediate that $$\begin{aligned} \label{2M.20} \sup_{0\leq t\leq T}\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}&+\nu(2\delta+1)(2(\delta+1))\int_0^T\||u_m(t)|^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t+\frac{\beta (2(\delta+1))}{2}\int_0^T\|u_m(t)\|_{\mathrm{L}^{4\delta+2}}^{4\delta+2}\mathrm{~d}t\nonumber\\&\leq \left(\|u_0\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+\frac{2(\delta+1)}{\beta}\int_0^T\|f(t)\|_{\mathrm{L}^{2}}^2\mathrm{~d}t\right)e^{2\beta(1+\gamma+\gamma^2)T}, \end{aligned}$$ and the right hand side of the above inequality is independent of $m$. Since the right hand side of [\[2M.20\]](#2M.20){reference-type="eqref" reference="2M.20"} is independent of $m$, an application of the Banach-Alaoglu theorem yields $u\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^{2(\delta+1)}(\Omega))\cap\mathrm{L}^{2(2\delta+1)}(0,T;\mathrm{L}^{2(2\delta+1)}(\Omega))$. We will now discuss a result on the uniqueness of the weak solution that we have obtained in Theorem [\[EWS\]](#EWS){reference-type="ref" reference="EWS"}. [\[t3\]]{#t3 label="t3"} For $u_0\in\mathrm{L}^{d\delta}(\Omega)$, for $d=2,3$, the weak solution to the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) obtained in Theorem [\[EWS\]](#EWS){reference-type="ref" reference="EWS"} is unique. *Proof.* Let $u_1(\cdot)$ and $u_2(\cdot)$ be two weak solutions of the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}) with the same external forcing $f(\cdot,\cdot)$ and initial data $u_0$ with the regularity given in the statement. Then, $w=u_1-u_2$ satisfies: $$\label{34} \left\{\begin{aligned} &\nonumber\frac{\partial w(x,t)}{\partial t}+\alpha \left[u_1(x,t)^{\delta}\sum_{i=1}^d\frac{\partial u_1(x,t)}{\partial x_i}-u_2(x,t)^{\delta}\sum_{i=1}^d\frac{\partial u_2(x,t)}{\partial x_i}\right]-\nu\Delta w(x,t)-\eta\Big(K*\Delta w\Big)(x,t)\\&=\beta \left[u_1(x,t)(1-u_1(x,t)^{\delta})(u_1(x,t)^{\delta}-\gamma)-u_2(x,t)(1-u_2(x,t)^{\delta})(u_2(x,t)^{\delta}-\gamma)\right] , \\ &w(x,t)=0, \ x \in \partial\Omega ,\ t\in(0,T), \\ &w(x,0)=0, \end{aligned}\right.$$ for $(x,t)\in\Omega\times(0,T)$ in the weak sense. One can write the above system in the weak sense as $$\begin{aligned} \label{235} \langle w' ,v\rangle+\alpha (B(u_1)-B(u_2),v)+\nu(\nabla w,\nabla v)+\eta((K*\nabla w),\nabla v)=\beta\langle c(u_1)-c(u_2),v\rangle, \end{aligned}$$ for all $v\in \mathrm{H}_0^1(\Omega)\cap \mathrm{L}^{2(\delta+1)}(\Omega)$ and a.e. $t\in[0,T]$. Using $v=w$ in [\[235\]](#235){reference-type="eqref" reference="235"}, we find $$\begin{aligned} \label{23} \frac{1}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|w\|_{\mathrm{L}^2}^2+\nu\|\nabla w\|_{\mathrm{L}^2}^2+\eta( (K*\nabla w),\nabla w)= -\alpha(B(u_1)-B(u_2),w)+\beta\langle c(u_1)-c(u_2),w\rangle. \end{aligned}$$ It can be easily seen that $$\begin{aligned} \label{24} \beta(c(u_1)-c(u_2),w)&=\beta\left[(u_1(1-u_1^{\delta})(u_1^{\delta}-\gamma)-u_2(1-u_2^{\delta})(u_2^{\delta}-\gamma),w)\right]\nonumber\\&=-\beta\gamma\|w\|_{\mathrm{L}^2}^2-\beta(u_1^{2\delta+1}-u_2^{2\delta+1},w)+\beta(1+\gamma)(u_1^{\delta+1}-u_2^{\delta+1},w). \end{aligned}$$ Estimating the term $-\beta(u_1^{2\delta+1}-u_2^{2\delta+1},w)$ from ([\[24\]](#24){reference-type="ref" reference="24"}) as $$\begin{aligned} \label{25} -\beta(u_1^{2\delta+1}-u_2^{2\delta+1},w)&= -\beta(u_1^{2\delta},w^2) -\beta(u_2^{2\delta},w^2)-\beta(u_1^{2\delta}u_2-u_2^{2\delta}u_1,w)\nonumber\\&=-\beta\|u_1^{\delta}w\|_{\mathrm{L}^2}^2-\beta\|u_2^{\delta}w\|_{\mathrm{L}^2}^2-\beta(u_1u_2,u_1^{2\delta}+u_2^{2\delta})+\beta(u_1^2,u_2^{2\delta})+\beta(u_2^2,u_1^{2\delta})\nonumber\\&=-\frac{\beta}{2}\|u_1^{\delta}w\|_{\mathrm{L}^2}^2-\frac{\beta}{2}\|u_2^{\delta}w\|_{\mathrm{L}^2}^2-\frac{\beta}{2}(u_1^{2\delta}-u_2^{2\delta},u_1^2-u_2^2)\nonumber\\&\leq -\frac{\beta}{2}\|u_1^{\delta}w\|_{\mathrm{L}^2}^2-\frac{\beta}{2}\|u_2^{\delta}w\|_{\mathrm{L}^2}^2. \end{aligned}$$ We estimate the term $\beta(1+\gamma)(u_1^{\delta+1}-u_2^{\delta+1},w)$ from ([\[24\]](#24){reference-type="ref" reference="24"}), using Taylor's formula, Hölder's and Young's inequalities as $$\begin{aligned} \label{26} \beta(1+\gamma)(u_1^{\delta+1}-u_2^{\delta+1},w)&=\beta(1+\gamma)(\delta+1)((\theta_1u_1+(1-\theta)u_2)^{\delta}w,w)\nonumber\\&\leq \beta(1+\gamma)(\delta+1)2^{\delta-1}(\|u_1^{\delta}w\|_{\mathrm{L}^2}+\|u_2^{\delta}w\|_{\mathrm{L}^2})\|w\|_{\mathrm{L}^2}\nonumber\\&\leq\frac{\beta}{4}\|u_1^{\delta}w\|_{\mathrm{L}^2}^2+\frac{\beta}{4}\|u_2^{\delta}w\|_{\mathrm{L}^2}^2+\frac{\beta}{2}2^{2\delta}(1+\gamma)^2(\delta+1)^2\|w\|_{\mathrm{L}^2}^2. \end{aligned}$$ Combining ([\[25\]](#25){reference-type="ref" reference="25"})-([\[26\]](#26){reference-type="ref" reference="26"}) and substituting it in ([\[24\]](#24){reference-type="ref" reference="24"}), we obtain $$\begin{aligned} \label{27} &\beta\left[(u_1(1-u_1^{\delta})(u_1^{\delta}-\gamma)-u_2(1-u_2^{\delta})(u_2^{\delta}-\gamma),w)\right]\nonumber\\&\leq -\beta\gamma\|w\|_{\mathrm{L}^2}^2-\frac{\beta}{4}\|u_1^{\delta}w\|_{\mathrm{L}^2}^2-\frac{\beta}{4}\|u_2^{\delta}w\|_{\mathrm{L}^2}^2+\frac{\beta}{2}2^{2\delta}(1+\gamma)^2(\delta+1)^2\|w\|_{\mathrm{L}^2}^2. \end{aligned}$$ On the other hand, we derive a bound for $-\alpha\langle B(u_1)-B(u_2),w\rangle$, using Taylor's formula, Hölder's, Ladyzhenskaya's and Young's inequalities as $$\begin{aligned} \label{30} -\alpha\langle B(u_1)-B(u_2),w\rangle&=\frac{\alpha}{\delta+1} \left((u_1^{\delta+1}-u_2^{\delta+1})\left(\begin{array}{c}1\\\vdots\\1\end{array}\right),\nabla w\right)\nonumber\\&=\alpha\left((u_1-u_2)(\theta u_1+(1-\theta) u_2)^{\delta}\left(\begin{array}{c}1\\\vdots\\1\end{array}\right),\nabla w\right)\nonumber\\&\leq 2^{\delta-1} \alpha(\|u_1\|^{\delta}_{\mathrm{L}^{4\delta}}+\|u_2\|^{\delta}_{\mathrm{L}^{4\delta}})\|w\|_{\mathrm{L}^{4}}\|\nabla w\|_{\mathrm{L}^2}\nonumber\\&\leq2^{\delta-1} \alpha(\|u_1\|^{\delta}_{\mathrm{L}^{4\delta}}+\|u_2\|^{\delta}_{\mathrm{L}^{4\delta}})\|w\|_{\mathrm{L}^{2}}^{\frac{4-d}{4}}\|\nabla w\|_{\mathrm{L}^2}^{\frac{4+d}{4}}\nonumber\\&\leq \frac{\nu}{2}\|\nabla w\|_{\mathrm{L}^2}^2+C(\alpha,\nu)\left(\|u_1\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u_2\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\|w\|_{\mathrm{L}^2}^2. \end{aligned}$$ where $C(\alpha, \nu) = \left(\frac{4+d}{4\nu}\right)^{\frac{4+d}{4-d}}\left(\frac{4-d}{8}\right)(2^{\delta-1}\alpha)^{\frac{8}{4-d}}$. Combining ([\[27\]](#27){reference-type="ref" reference="27"}) and ([\[30\]](#30){reference-type="ref" reference="30"}), we have $$\begin{aligned} \label{332} &\frac{\mathrm{~d}}{\mathrm{~d}t}\|w(t)\|_{\mathrm{L}^2}^2+\nu\|\nabla w(t)\|_{\mathrm{L}^2}^2+\frac{\beta}{2}\|u_1^{\delta}(t)w(t)\|_{\mathrm{L}^2}^2+\frac{\beta}{2}\|u_2^{\delta}(t)w(t)\|_{\mathrm{L}^2}^2+\beta\gamma\|w(t)\|_{\mathrm{L}^2}^2\nonumber\\&\leq \beta 2^{2\delta+1}(1+\gamma)^2(\delta+1)^2\|w(t)\|_{\mathrm{L}^2}^2+C(\alpha,\nu)\left(\|u_1(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u_2(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\|w(t)\|_{\mathrm{L}^2}^2, \end{aligned}$$ for a.e. $t\in[0,T]$. An application of Gronwall's inequality in ([\[332\]](#332){reference-type="ref" reference="332"}) gives for all $t\in[0,T]$ $$\begin{aligned} \label{333} \|w(t)\|_{\mathrm{L}^2}^2\leq\|w(0)\|_{\mathrm{L}^2}^2e^{ \beta 2^{2\delta}(1+\gamma)^2(\delta+1)^2T}\exp\left\{C(\alpha,\nu)\int_0^T\left(\|u_1(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u_2(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\mathrm{~d} t\right\}. \end{aligned}$$ For $d=2$ and $u_0\in\mathrm{L}^{2\delta}(\Omega)$, from Remark [\[r2\]](#r2){reference-type="ref" reference="r2"}, it is clear that $u_1,u_2\in\mathrm{L}^{4\delta}(0,T;\mathrm{L}^{4\delta}(\Omega))$, and hence the uniqueness follows. For $d=3,$ $u_0\in\mathrm{L}^{3\delta}(\Omega)$, the integral appearing in the exponential is finite by using interpolation inequality as $$\begin{aligned} \int_0^T\|u_1(t)\|_{\mathrm{L}^{4\delta}}^{8\delta} \mathrm{~d} t \leq\sup_{t\in[0,T]}\|u_1(t)\|_{\mathrm{L}^{3\delta}}^{3\delta} \int_{0}^{T}\|u_1(t)\|_{\mathrm{L}^{5\delta}}^{5\delta}\mathrm{~d} t< \infty. \end{aligned}$$ Hence, the uniqueness follows if we take the initial data $u_0 \in \mathrm{L}^{d\delta}(\Omega)$. ◻ [\[rk2.4\]]{#rk2.4 label="rk2.4"} For $d=3$, using Gagliardo-Nirenberg's inequality in [\[30\]](#30){reference-type="eqref" reference="30"}, one can find a sufficient criterion for the uniqueness of weak solution for $u_0\in\mathrm{L}^{3\delta}(\Omega)$ as $$\int_0^T\|u(t)\|_{\mathrm{L}^{r\delta}}^{s\delta}\/\mathrm{d}\/t<\infty,\ \text{ where }\ \frac{3}{r}+\frac{2}{s}=1, \ r\in(3,\infty).$$ [\[rem2.4\]]{#rem2.4 label="rem2.4"} If we consider our initial data $u_0 \in \mathrm{L}^{2}(\Omega)$ and $\beta\nu>(2^{\delta}\alpha)^2,$ then there exists a unique weak solution to the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}). One can estimate $-\alpha\langle B(u_1)-B(u_2),w\rangle$ in [\[30\]](#30){reference-type="eqref" reference="30"} using integrating by parts, Taylor's formula, Hölder's and Young's inequalities as: $$\begin{aligned} \label{28} -\alpha\langle B(u_1)-B(u_2),w\rangle&=\frac{\alpha}{\delta+1} \left((u_1^{\delta+1}-u_2^{\delta+1})\left(\begin{array}{c}1\\\vdots\\1\end{array}\right),\nabla w\right)\nonumber\\&=\alpha\left((u_1-u_2)(\theta u_1+(1-\theta) u_2)^{\delta}\left(\begin{array}{c}1\\\vdots\\1\end{array}\right),\nabla w\right)\nonumber\\&\leq 2^{\delta-1}\alpha\|\nabla w\|_{\mathrm{L}^2}\left(\|u_1^{\delta}w\|_{\mathrm{L}^2}+\|u_2^{\delta}w\|_{\mathrm{L}^2}\right)\nonumber\\&\leq\frac{\nu}{2}\|\nabla w\|_{\mathrm{L}^2}^2+\frac{2^{2\delta}\alpha^2}{4\nu}\|u_1^{\delta} w\|_{\mathrm{L}^2}^2+\frac{2^{2\delta}\alpha^2}{4\nu}\|u_2^{\delta} w\|_{\mathrm{L}^2}^2. \end{aligned}$$ Combining ([\[27\]](#27){reference-type="ref" reference="27"}) and ([\[28\]](#28){reference-type="ref" reference="28"}) in ([\[23\]](#23){reference-type="ref" reference="23"}), we find $$\begin{aligned} & \frac{1}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|w(t)\|_{\mathrm{L}^2}^2+\frac{\nu}{2}\|\nabla w(t)\|_{\mathrm{L}^2}^2+\frac{\beta}{4}\|u_1^{\delta}(t)w(t)\|_{\mathrm{L}^2}^2+\frac{\beta}{4}\|u_2(t)^{\delta}w(t)\|_{\mathrm{L}^2}^2+\beta\gamma\|w(t)\|_{\mathrm{L}^2}^2+\eta( (K*\nabla w)(t),\nabla w(t))\nonumber\\&\leq \frac{\beta}{2}2^{2\delta}(1+\gamma)^2(\delta+1)^2\|w(t)\|_{\mathrm{L}^2}^2+\frac{\alpha^2}{4\nu}2^{2\delta}\|u_1^{\delta}w(t)\|_{\mathrm{L}^2}^2+\frac{\alpha^2}{4\nu}2^{2\delta}\|u_2(t)^{\delta}w(t)\|_{\mathrm{L}^2}^2. \end{aligned}$$ From the above expression, it is immediate that $$\begin{aligned} &\frac{\mathrm{~d}}{\mathrm{~d}t}\|w(t)\|_{\mathrm{L}^2}^2+\nu\|\nabla w(t)\|_{\mathrm{L}^2}^2+\left(\frac{\beta}{2}-\frac{\alpha^2}{2\nu}2^{2\delta}\right)\|u_1(t)^{\delta}w(t)\|_{\mathrm{L}^2}^2\nonumber\\&\quad+\left(\frac{\beta}{2}-\frac{\alpha^2}{2\nu}2^{2\delta}\right)\|u_2(t)^{\delta}w(t)\|_{\mathrm{L}^2}^2+\eta( (K*\nabla w)(t),\nabla w(t))\nonumber\\&\leq \beta 2^{2\delta}(1+\gamma)^2(\delta+1)^2\|w(t)\|_{\mathrm{L}^2}^2. \end{aligned}$$ For $\beta\nu>(2^{\delta}\alpha)^2$, integrating the above inequality from $0$ to $t$, ([\[pk\]](#pk){reference-type="ref" reference="pk"}) and then applying of Gronwall's inequality results to $$\begin{aligned} & \|w(t)\|_{\mathrm{L}^2}^2\leq\|w(0)\|_{\mathrm{L}^2}^2e^{ \beta 2^{2\delta}(1+\gamma)^2(\delta+1)^2T}, \end{aligned}$$ for all $t\in[0,T]$. Since $w_0=0$ and $u_1$ and $u_2$ are weak solutions of the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}), uniqueness follows easily. ## Regularity results {#sec2.4} If we take our initial data $u_0\in\mathrm{H}_0^1(\Omega)\cap \mathrm{L}^{2(\delta+1)}(\Omega)$, and the external forcing, $f\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$, where $\Omega\subset\mathbb{R}^d, d=2,3,$ is either convex, or a domain with $\mathrm{C}^2$-boundary, then we can obtain the $\mathrm{H}_0^1(\Omega)$-energy estimate. Under smoothness assumptions on the domain, initial data and external forcing, let us now establish the regularity of the weak solution obtained in Theorem [\[EWS\]](#EWS){reference-type="ref" reference="EWS"}. [\[2M.thm3.2\]]{#2M.thm3.2 label="2M.thm3.2"} Let $u$ be a weak solution to the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}). If $\Omega\subset\mathbb{R}^d,d=2,3,$ is either convex, or a domain with $\mathrm{C}^2$-boundary and let $u_0\in\mathrm{H}_0^1(\Omega)\cap \mathrm{L}^{2(\delta+1)}(\Omega)$ and $f\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$ be given. Then, $u$ is a strong solution with $$\begin{aligned} u\in\mathrm{C}([0,T];\mathrm{H}_0^1(\Omega))\cap\mathrm{L}^2(0,T;\mathrm{H}^2(\Omega))\cap\mathrm{L}^{2(\delta+1)}(0,T;\mathrm{L}^{6(\delta+1)}(\Omega)), \end{aligned}$$ and $\partial_tu\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$ and the following system is satisfied: $$\label{2M.336} \left\{ \begin{aligned} \nonumber\frac{\mathrm{~d}u(t)}{\mathrm{~d}t}+\nu Au(t)+\eta ( K*Au)(s)&=-\alpha B(u(t))+\beta c(u(t))+f(t), \ \text{ in }\ \mathrm{L}^2(\Omega),\\ u(0)&=u_0\in\mathrm{L}^2(\Omega), \end{aligned} \right.$$ for a.e. $t\in[0,T]$. *Proof.* **Step 1:** *$\mathrm{H}_0^1$-energy estimate.* Let $\lambda_k$ denote the $k^{\mathrm{th}}$ eigenvalue of $-\Delta$ in $\mathrm{H}_0^1(\Omega)$. Multiplying the identity ([\[8\]](#8){reference-type="ref" reference="8"}) by $\lambda_kd_k^m(t)$ and summing it from $k=1,2,\ldots,m$, we have $$\begin{aligned} \label{2m.14} (\partial_t u_m,Au_m)-\nu(\Delta u_m, Au_m)-\eta(K*\Delta u_m, Au_m)=-\alpha (B(u_m ),Au_m )+\beta(c(u_m),Au_m)+(f,Au_m). \end{aligned}$$ Now, using integrating by parts and applying Hölder's and Young's inequalities, we find $$\begin{aligned} \alpha |(B(u_m ),Au_m )|\leq\alpha \|B(u_m)\|_{\mathrm{L}^2}\|A u_m\|_{\mathrm{L}^2}\leq\alpha\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}\|Au_m\|_{\mathrm{L}^2}\leq\frac{\nu}{4}\|Au_m\|_{\mathrm{L}^2}^2+\frac{\alpha^2}{\nu}\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2. \end{aligned}$$ Estimating $\beta(c(u_m),Au_m)$ using Hölder's and Young's inequalities as follows: $$\begin{aligned} &\beta(c(u_m),Au_m)\nonumber\\&=\beta((1+\gamma)u_m^{\delta+1}-\gamma u_m-u_m^{2\delta+1},Au_m)\nonumber\\ &=\beta((1+\gamma)(\delta+1)u_m^{\delta}\nabla u_m-\gamma\nabla u_m-(2\delta+1)u_m^{2\delta}\nabla u_m,\nabla u_m)\nonumber\\ &=\beta(1+\gamma)(\delta+1)(u_m^{\delta}\nabla u_m,\nabla u_m)-\beta\gamma(\nabla u_m,\nabla u_m)-\beta(2\delta+1)(u_m^{2\delta}\nabla u_m,\nabla u_m)\nonumber\\ &\leq \beta(1+\gamma)(\delta+1)\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}\|\nabla u_m\|_{\mathrm{L}^2}-\beta\gamma\|\nabla u_m\|_{\mathrm{L}^2}^2-\beta(2\delta+1)\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2\nonumber\\ &\leq\frac{\beta(1+\gamma)^2(\delta+1)^2}{2(2\delta+1)}\|\nabla u_m\|_{\mathrm{L}^2}^2+\frac{\beta(2\delta+1)}{2}\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2 -\beta\gamma\|\nabla u_m\|_{\mathrm{L}^2}^2-\beta(2\delta+1)\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2\nonumber\\ &\leq \frac{\beta(1+\gamma)^2(\delta+1)^2}{2(2\delta+1)}\|\nabla u_m\|_{\mathrm{L}^2}^2-\beta\gamma\|\nabla u_m\|_{\mathrm{L}^2}^2-\frac{\beta(2\delta+1)}{2}\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2. \end{aligned}$$ Combining the above estimates and substituting in ([\[2m.14\]](#2m.14){reference-type="ref" reference="2m.14"}), we get $$\begin{aligned} & \frac{1}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\frac{\nu}{2}\|\Delta u_m(t) \|_{\mathrm{L}^2}^2+\frac{\beta(2\delta+1)}{2}\|u_m(t)^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2+\eta((K*\Delta u_m)(t),\Delta u_m(t))\nonumber\\&\leq \frac{\alpha^2}{\nu}\|u_m(t)^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2+\frac{\beta((1+\gamma^2)(\delta+1)^2+2\gamma\delta^2)}{2(2\delta+1)}\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\frac{1}{\nu}\|f(t)\|_{\mathrm{L}^2}^2. \end{aligned}$$ Integrating from $0$ to $t$ and using ([\[pk\]](#pk){reference-type="ref" reference="pk"}), we find $$\begin{aligned} \label{2M.53} &\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\nu\int_{0}^{t}\|\Delta u_m(s) \|_{\mathrm{L}^2}^2\mathrm{~d}s+\beta(2\delta+1)\int_{0}^{t}\|u_m(s)^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\leq\|u_0\|_{\mathrm{H}_0^1}^2+ \frac{2\alpha^2}{\nu}\int_{0}^{t}\|u_m(s)^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+\frac{\beta((1+\gamma^2)(\delta+1)^2+2\gamma\delta^2)}{(2\delta+1)}\int_{0}^{t}\|\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\quad+\frac{2}{\nu}\int_{0}^{t}\|f(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\leq C\left(\|u_0\|_{\mathrm{H}_0^1},\|f\|_{\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))},\alpha,\beta,\gamma,\delta,\nu,T\right), \end{aligned}$$ since $u_m(\cdot)$ satisfies the energy estimates ([\[13\]](#13){reference-type="ref" reference="13"}) and ([\[2M.20\]](#2M.20){reference-type="ref" reference="2M.20"}). Whenever $u_0\in\mathrm{H}_0^1(\Omega)$ and $f\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$, from ([\[2M.53\]](#2M.53){reference-type="ref" reference="2M.53"}), we have $$\begin{aligned} u_m\in\mathrm{L}^{\infty}(0,T;\mathrm{H}_0^1(\Omega))\cap\mathrm{L}^2(0,T;\mathrm{H}^2(\Omega)), \end{aligned}$$ by using the elliptic regularity (Theorem 3.1.2.1, [@GPi]). Using Sobolev's inequality, $\|u\|_{\mathrm{L}^6} \leq C\|\nabla u||_{\mathrm{L}^2}$, we have $\|u_m^{\delta+1}\|_{\mathrm{L}^6}^2 \leq C\|\nabla u_m^{\delta+1}||_{\mathrm{L}^2}^2\leq C(\delta+1)\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2},$ so that $$\begin{aligned} \int_{0}^t\|u_m(s)\|^{2(\delta+1)}_{\mathrm{L}^{6(\delta+1)}}\mathrm{~d}s= \int_{0}^t\|u_m(s)^{\delta+1}\|_{\mathrm{L}^6}^2 \mathrm{~d}s\leq C(\delta+1)^2\int_{0}^t\|u_m(s)^{\delta}\nabla u_m(s)||_{\mathrm{L}^2}^2\mathrm{~d}s< \infty. \end{aligned}$$ Thus, we infer $$\begin{aligned} u_m\in\mathrm{L}^{2(\delta+1)}(0,T;\mathrm{L}^{6(\delta+1)}(\Omega)). \end{aligned}$$ **Step 2:** *Estimate for time derivative.* For $m\geq 1$, multiplying ([\[8\]](#8){reference-type="ref" reference="8"}) by $d{_m^k}'(t)$ and sum it over $k=1,2,\ldots,m,$ we have $$\begin{aligned} \label{2M.54} &\|\partial_t u_m(t)\|_{\mathrm{L}^2}^2+\frac{\nu}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|\nabla u_m(t)\|_{\mathrm{L}^2}^2\nonumber\\&=\eta((K*\Delta u_{m})(t),\partial_{t}u_{m}(t))-\alpha\left((u_m(t))^{\delta}\sum_{i=1}^{d}\frac{\partial u_m}{\partial x_i}(t),\partial_{t}u_{m}(t)\right)\nonumber\\&\quad+\beta(u_m(t)(1-(u_m(t))^{\delta})((u_m(t))^{\delta}-\gamma),\partial_tu_m(t))+(f(t),\partial_tu_m(t))\nonumber\\&=\eta((K*\Delta u_{m})(t),\partial_{t}u_{m}(t)) -\alpha\left((u_m(t))^{\delta}\sum_{i=1}^{d}\frac{\partial u_m}{\partial x_i}(t),\partial_{t}u_{m}(t)\right)+\beta(1+\gamma)(u_m(t)^{\delta+1},\partial_tu_m(t))\nonumber\\&\quad-\beta\gamma\frac{\mathrm{~d}}{\mathrm{~d}t}\|u_m(t)\|_{\mathrm{L}^2}^2-\frac{\beta}{2(\delta+1)}\frac{\mathrm{~d}}{\mathrm{~d}t}\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+(f(t),\partial_tu_m(t)). \end{aligned}$$ Using Hölder's and Young's inequalities, we estimate the first term from the right hand side of the equality ([\[2M.54\]](#2M.54){reference-type="ref" reference="2M.54"}) as $$\begin{aligned} \label{2M.55} \alpha\left((u_m)^{\delta}\sum_{i=1}^{d}\frac{\partial u_m}{\partial x_i},\partial_{t}u_{m}\right)\leq\frac{1}{4}\|\partial_tu_m\|_{\mathrm{L}^2}^2+\alpha^2\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2. \end{aligned}$$ Similarly, we estimate $\beta(1+\gamma)(u_m^{\delta+1},\partial_tu_m)$ and $\eta(K*\Delta u_{m},\partial_{t}u_{m})$ as $$\begin{aligned} \nonumber\beta(1+\gamma)|(u_m^{\delta+1},\partial_tu_m)|&\leq\frac{1}{4}\|\partial_tu_m\|_{\mathrm{L}^2}^2+\beta^2(1+\gamma)^2\|u_m\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)},\\ \eta(K*\Delta u_{m},\partial_{t}u_{m})&\leq\frac{\eta^2}{2}\|K*\Delta u_{m}\|_{\mathrm{L}^2}^2+ \frac{1}{2}\|\partial_t u_m\|_{\mathrm{L}^2}^2, \end{aligned}$$ The final term from the right hand side of the equality ([\[2M.54\]](#2M.54){reference-type="ref" reference="2M.54"}) can be estimated as $$\begin{aligned} \label{2M.5.7} |(f,\partial_tu_m)|\leq\|f\|_{\mathrm{L}^2}\|\partial_tu_m\|_{\mathrm{L}^2}\leq\frac{1}{4}\|\partial_tu_m\|_{\mathrm{L}^2}^2+\|f\|_{\mathrm{L}^2}^2. \end{aligned}$$ Combining ([\[2M.55\]](#2M.55){reference-type="ref" reference="2M.55"})-([\[2M.5.7\]](#2M.5.7){reference-type="ref" reference="2M.5.7"}), substituting it in ([\[2M.54\]](#2M.54){reference-type="ref" reference="2M.54"}) and then integrating from $0$ to $t$, we get $$\begin{aligned} &2\nu\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\frac{2\beta}{\delta+1}\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+4\beta\gamma\|u_m(t)\|_{\mathrm{L}^2}^2+\frac{1}{2}\int_0^t\|\partial_tu_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\leq 2\nu\|u_0\|_{\mathrm{H}_0^1}^2+\frac{2\beta}{\delta+1}\|u_0\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+4\beta\gamma\|u_0\|_{\mathrm{L}^2}^2+4\int_0^t\|f(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+4\alpha^2\int_0^t\|u_m^{\delta}(s)\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\quad+\frac{\eta^2C}{2}\int_0^t\|\Delta u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+4\beta^2(1+\gamma)^2\int_0^t\|u_m(s)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\mathrm{~d}s. \end{aligned}$$ Applying Gronwall's inequality, we arrive at $$\begin{aligned} \label{2M.5.9} &2\nu\|\nabla u_m(t)\|_{\mathrm{L}^2}^2+\frac{2\beta}{\delta+1}\|u_m(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+4\beta\gamma\|u_m(t)\|_{\mathrm{L}^2}^2+\frac{1}{2}\int_0^t\|\partial_tu_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\leq\left(2\nu\|u_0\|_{\mathrm{H}_0^1}^2+\frac{2\beta}{\delta+1}\|u_0\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}+4\beta\gamma\|u_0\|_{\mathrm{L}^2}^2+4\int_0^T\|f(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t+4\alpha^2\int_0^T\|u_m^{\delta}(t)\nabla u_m(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t\right)\nonumber\\&\quad\times\exp\left({4\beta(1+\gamma)^2(1+\delta)T}\right)e^{\frac{\eta^2C}{2\nu}T}, \end{aligned}$$ for all $t\in[0,T]$. Using the estimate given in ([\[2M.53\]](#2M.53){reference-type="ref" reference="2M.53"}), we obtain that the right hand side of ([\[2M.5.9\]](#2M.5.9){reference-type="ref" reference="2M.5.9"}) is independent of $m$. Thus, it is immediate that $\partial_tu_m\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$. Using arguments similar to Steps 4 and 5 in Theorem [\[EWS\]](#EWS){reference-type="ref" reference="EWS"}, we obtain the existence of a $\tilde{u}(\cdot)$ such that $$\begin{aligned} \tilde{u}&\in\mathrm{L}^{\infty}(0,T;\mathrm{H}_0^1(\Omega)\cap\mathrm{L}^{2(\delta+1)}(\Omega))\cap\mathrm{L}^2(0,T;\mathrm{H}^2(\Omega))\cap\mathrm{L}^{2(\delta+1)}(0,T;\mathrm{L}^{6(\delta+1)}(\Omega)) \ \text{ and }\\ \partial_t\tilde{u}&\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega)), \end{aligned}$$ so that we get $\tilde{u}\in\mathrm{C}([0,T];\mathrm{H}_0^1(\Omega))$ and is a strong solution to the system ([\[GBHE\]](#GBHE){reference-type="ref" reference="GBHE"}). The uniqueness follows from the estimate ([\[333\]](#333){reference-type="ref" reference="333"}) and we have $\tilde{u}=u$. ◻ [\[2M.thm33\]]{#2M.thm33 label="2M.thm33"} Let $1\leq \delta < \infty$ for $d=2$ and, $1\leq \delta \leq 2$ for $d=3$ and assume that $u_0\in\mathrm{H}^2(\Omega)\cap\mathrm{H}_0^1(\Omega)$ 1. If $f\in\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))$, then, we have $\partial_tu\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))\cap\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega)).$ 2. If $f\in\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))\cap \mathrm{L}^2(0,T;\mathrm{H}^1(\Omega)),$ then $u \in \mathrm{L}^{\infty}(0,T;\mathrm{H}^2(\Omega))$ . *Proof.* We divide the proof into the following steps: **Step 1.** $\partial_tu\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega)$, and $\partial_t\nabla u\in\mathrm{L}^{2}(0,T;\mathrm{L}^2(\Omega)$. Let $1\leq \delta < \infty$ for $d=2$ and, $1\leq \delta \leq 2$ for $d=3$. Differentiating [\[8\]](#8){reference-type="eqref" reference="8"} with respect to $t$, we find $$\begin{aligned} \label{2M.62} &(\partial_{tt}u_m(t),w_k)+\nu(\partial_{t}\nabla u_m(t),\nabla w_k)\nonumber\\&+\alpha\sum_{i=1}^{d}\left((u_m(t))^{\delta}\partial_{tx_i} u_m(t)+\delta u_m(t)^{\delta-1}\partial_t u_m(t)\partial{x_i} u_m(t),w_k\right)-\eta((K*\partial_t \Delta u_m)(t)+K(t)\Delta u_m(0), w_k)\nonumber\\&=\beta((\gamma+1)(\delta+1)u_m(t)^{\delta}\partial_tu_m(t)-\gamma\partial_tu_m(t)-(2\delta+1)u_m(t)^{2\delta}\partial_tu_m(t),w_k)+(\partial_tf(t),w_k). \end{aligned}$$ Multiplying [\[2M.62\]](#2M.62){reference-type="eqref" reference="2M.62"} with $d{_m^k}'(\cdot)$ and summing it over $k=1,2,\ldots,m$, we obtain $$\begin{aligned} \label{2M.63} \nonumber&\frac{1}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|\partial_tu_m(t)\|_{\mathrm{L}^2}^2+\nu\|\partial_{t}\nabla u_m(t)\|_{\mathrm{L}^2}^2+\beta\gamma\|\partial_tu_m(t)\|_{\mathrm{L}^2}^2+\beta(2\delta+1)\|u_m(t)^{\delta}\partial_tu_m(t)\|_{\mathrm{L}^2}^2\\&\nonumber\quad+\eta((K*\partial_t \nabla u_m)(t),\partial_t\nabla u_m(t))\nonumber\\&=-\alpha\sum_{i=1}^{d}\left((u_m(t))^{\delta}\partial_{tx_i}u_m(t),\partial_tu_m\right)+\beta(\gamma+1)(\delta+1)(u_m(t)^{\delta}\partial_tu_m(t),\partial_tu_m(t))\nonumber\\&\quad+(\partial_tf(t),\partial_tu_m(t)) + \eta(K(t)\Delta u_m(0),\partial_t u_m(t)). \end{aligned}$$ We estimate the term $-\alpha\sum_{i=1}^{d}((u_m)^{\delta}\partial_{tx_i}u_m,\partial_tu_m)$ using integration by parts, Hölder's, Ladyzhenskaya's inequality and Young's inequalities as $$\begin{aligned} \label{2M.64} \nonumber-&\sum_{i=1}^{d}\alpha\left((u_m)^{\delta}\partial_{tx_i}u_m,\partial_tu_m\right)\\\nonumber&= -\frac{1}{2}\sum_{i=1}^{d}\alpha\left(u_m^{\delta}, \frac{\partial }{\partial x_i}\left(\partial_tu_m\right)^2\right) = \frac{\alpha\delta}{2}\sum_{i=1}^{d}\left(u_m^{\delta-1}\frac{\partial u_m }{\partial x_i}, \left(\partial_tu_m\right)^2\right)\\\nonumber& =\frac{\alpha\delta}{2}\sum_{i=1}^{d} \int_{\Omega} u_m^{\delta-1} \left(\frac{\partial u_m }{\partial x_i}\right)^{\frac{\delta-1}{\delta}}\left(\frac{\partial u_m }{\partial x_i}\right)^{\frac{1}{\delta}}\left(\partial_tu_m\right)^2 \mathrm{~d}x \\\nonumber&\leq \frac{\alpha\delta}{2}\sum_{i=1}^{d} \left(\int_{\Omega} u_m^{2\delta} \left(\frac{\partial u_m }{\partial x_i}\right)^2\mathrm{~d}x\right)^{\frac{\delta-1}{2\delta}} \left(\int_{\Omega}\left(\frac{\partial u_m }{\partial x_i}\right)^{\frac{2}{\delta+1}}\left(\partial_tu_m\right)^{\frac{4\delta}{\delta+1}} \mathrm{~d}x\right)^{\frac{\delta+1}{2\delta}}\\\nonumber& \leq \frac{\alpha\delta}{2}\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^{\frac{\delta-1}{\delta}} \|(\nabla u_m)^{\frac{1}{\delta+1}}\left(\partial_tu_m\right)^{\frac{2\delta}{\delta+1}}\|_{\mathrm{L}^2} ^{\frac{\delta+1}{\delta}}\\\nonumber& \leq \frac{\alpha\delta}{2}\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^{\frac{\delta-1}{\delta}} \|\nabla u_m\|_{\mathrm{L}^2}^{\frac{1}{\delta}}\|\partial_tu_m\|_{\mathrm{L}^4} ^{2} \\\nonumber &\leq \frac{\alpha\delta}{\sqrt{2}} \|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^{\frac{\delta-1}{\delta}} \|\nabla u_m\|_{\mathrm{L}^2}^{\frac{1}{\delta}}\|\partial_tu_m\|_{\mathrm{L}^2} ^{\frac{4-d}{2}}\|\partial_t\nabla u_m\|_{\mathrm{L}^2} ^{\frac{d}{2}}\\& \leq \frac{\nu}{2} \|\partial_t\nabla u_m\|_{\mathrm{L}^2} ^{2} + \left(\frac{4-d}{4}\right)\left(\frac{d}{2\nu}\right)^{\frac{d}{4-d}}\left(\frac{\alpha\delta}{\sqrt{2}}\right)^{\frac{4}{4-d}} \|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^{\frac{4(\delta-1)}{(4-d)\delta}} \|\nabla u_m\|_{\mathrm{L}^2}^{\frac{4}{\delta(4-d)}}\|\partial_tu_m\|_{\mathrm{L}^2} ^{2}. \end{aligned}$$ We estimate $\eta\left|\left(K(t)\Delta u_m(0),\partial_t u_m\right)\right|$ as $$\begin{aligned} \nonumber\eta\left|\left(K(t)\Delta u_m(0),\partial_t u_m(t)\right)\right|\leq \eta |K(t)|\|\Delta u_m(0)\|_{\mathrm{L}^2}\|\partial_t u_m(t)\|_{\mathrm{L}^2}\leq \frac{\eta}{2}|K(t)|(\|\Delta u_m(0)\|_{\mathrm{L}^2}^2+\|\partial_t u_m(t)\|^2_{\mathrm{L}^2}). \end{aligned}$$ Similarly, we estimate the terms $\beta(\gamma+1)(\delta+1)(u_m^{\delta}\partial_tu_m,\partial_tu_m)$ and $(\partial_tf,\partial_tu_m)$ as $$\begin{aligned} \label{2M.65} \nonumber \beta(\gamma+1)(\delta+1)|(u_m^{\delta}\partial_tu_m,\partial_tu_m)|&\leq \beta(\gamma+1)(\delta+1)\|u_m^{\delta}\partial_tu_m\|_{\mathrm{L}^{2}}\|\partial_tu_m\|_{\mathrm{L}^2}\\\nonumber&\leq \frac{\beta(2\delta+1)}{2}\|u_m^{\delta}\partial_tu_m\|_{\mathrm{L}^{2}}^2+\frac{\beta(\gamma+1)^2(\delta+1)^2}{2(2\delta+1)}\|\partial_tu_m\|_{\mathrm{L}^2}^2 ,\\ |(\partial_tf,\partial_tu_m)|\leq \|\partial_tf\|_{\mathrm{L}^2}\|\partial_tu_m\|_{\mathrm{L}^2}&\leq\frac{1}{2\beta\gamma}\|\partial_tf\|_{\mathrm{L}^2}^2+\frac{\beta\gamma}{2}\|\partial_tu_m\|_{\mathrm{L}^2}^2. \end{aligned}$$ Substituting the estimates [\[2M.64\]](#2M.64){reference-type="eqref" reference="2M.64"}-[\[2M.65\]](#2M.65){reference-type="eqref" reference="2M.65"} in [\[2M.63\]](#2M.63){reference-type="eqref" reference="2M.63"}, we have $$\begin{aligned} \nonumber&\frac{\mathrm{~d}}{\mathrm{~d}t}\|\partial_tu_m(t)\|_{\mathrm{L}^2}^2+\nu\|\partial_{t}\nabla u_m(t)\|_{\mathrm{L}^2}^2+\beta\gamma\|\partial_tu_m(t)\|_{\mathrm{L}^2}^2+\beta(2\delta+1)\|u_m(t)^{\delta}\partial_tu_m(t)\|_{\mathrm{L}^2}^2\\&\nonumber\quad+2\eta((K*\partial_t \nabla u_m)(t),\partial_t\nabla u_m(t))\\&\nonumber\leq \eta|K(t)|\|\Delta u_m(0)\|_{\mathrm{L}^2}^2 + \frac{1}{\beta\gamma}\|\partial_tf(t)\|_{\mathrm{L}^2}^2 \\& \quad+ \left(C(\alpha,\nu,\delta) \|u_m(t)^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^{\frac{4(\delta-1)}{(4-d)\delta}} \|\nabla u_m(t)\|_{\mathrm{L}^2}^{\frac{4}{\delta(4-d)}}+ \eta|K(t)| +C(\beta,\gamma,\delta)\right)\|\partial_tu_m(t)\|_{\mathrm{L}^2}^2, \end{aligned}$$ where $C(\alpha,\nu,\delta)=\left(\frac{4-d}{2}\right)\left(\frac{d}{2\nu}\right)^{\frac{d}{4-d}}\left(\frac{\alpha\delta}{\sqrt{2}}\right)^{\frac{4}{4-d}}$ and $C(\beta,\gamma,\delta)= \frac{\beta(\gamma+1)^2(\delta+1)^2}{(2\delta+1)}$. $$\begin{aligned} &\|\nonumber\partial_tu_m(t)\|_{\mathrm{L}^2}^2+\nu\int_0^t\|\partial_{t}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+\beta\gamma\int_0^t\|\partial_tu_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s +\beta(2\delta+1)\int_0^t\|u_m(s)^{\delta}\partial_tu_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\\\nonumber&\leq \|\partial_tu_m(0)\|_{\mathrm{L}^2}^2 +\eta\|\Delta u_m(0)\|_{\mathrm{L}^2}^2 \int_0^t|K(s)|\mathrm{~d}s + \frac{1}{\beta\gamma}\int_0^t\|\partial_tf(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s \\& \quad + \int_0^t\left(C(\alpha,\nu,\delta) \|u_m(s)^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^{\frac{4(\delta-1)}{(4-d)\delta}} \|\nabla u_m(s)\|_{\mathrm{L}^2}^{\frac{4}{\delta(4-d)}}+ \eta |K(s)| +C(\beta,\gamma,\delta)\right)\|\partial_tu_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s. \end{aligned}$$ As $K(t) \in \mathrm{L}^1(0,T)$ and note that $\|u_m(0)\|_{\mathrm{H}^2}\leq C\|u_0\|_{\mathrm{H}^2}$ (see page 385, [@LCE]) and an application of Gronwall's inequality yields $$\begin{aligned} \label{2M.68} &\|\partial_tu_m(t)\|_{\mathrm{L}^2}^2+\nu\int_0^t\|\partial_{t}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+\beta\gamma\int_0^t\|\partial_tu_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\qquad +\beta(2\delta+1)\int_0^t\|u_m(s)^{\delta}\partial_tu_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\leq\left( \|\partial_tu_m(0)\|_{\mathrm{L}^2}^2 + \eta\| u_0\|_{\mathrm{H}^2}\int_0^t|K(s)|\mathrm{~d}s + \frac{1}{\beta\gamma}\int_0^t\|\partial_tf(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\right)\nonumber\\&\quad\times\exp\left(\int_{0}^{t}\left[C(\alpha,\nu,\delta) \|u_m(s)^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^{\frac{4(\delta-1)}{(4-d)\delta}} \|\nabla u_m(s)\|_{\mathrm{L}^2}^{\frac{4}{\delta(4-d)}}+ \eta\|u_0\|_{\mathrm{H}^2} |K(s)| +C(\beta,\gamma,\delta)\right]\right)\nonumber\\&\leq C\left(\|u_0 \|_{\mathrm{H}^2},\|f\|_{\mathrm{H}^{1}(0,T;\mathrm{L}^2(\Omega))},\alpha,\nu,\beta,\gamma,\delta,T\right), \end{aligned}$$ for all $t\in[0,T]$, where the first term in exponential is bounded , provided $1\leq \delta <\infty$, for $d=2$ and, $1\leq \delta \leq 2,$ for $d=3$. Taking supremum over time $0\leq t\leq T$, we have $$\begin{aligned} \sup_{0\leq t\leq T}\|\partial_tu_m(t)\|_{\mathrm{L}^2}^2+\nu\int_0^T\|\partial_{t}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\leq C\left(\|u_0\|_{\mathrm{H}^2},\|f\|_{\mathrm{H}^{1}(0,T;\mathrm{L}^2(\Omega))},\alpha,\beta,\gamma,\delta,\nu,T\right). \end{aligned}$$ **Step 2.** $\nabla(u^{\delta+1})\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))$. In ([\[8\]](#8){reference-type="ref" reference="8"}), if we take $w_k = -u_m^{2\delta}\Delta u_m$, then we have for a.e. $t\in[0,T]$ $$\begin{aligned} \label{2r.14} \nonumber &(\partial_t u_m(t),-u_m(t)^{2\delta}\Delta u_m(t))+\nu(\Delta u_m(t),u_m(t)^{2\delta}\Delta u_m(t))+\eta((K*\Delta u_m)(t),u_m(t)^{2\delta}\Delta u_m(t))\\&=\alpha (B(u_m(t) ),u_m(t)^{2\delta}\Delta u_m(t) )-\beta(c(u_m(t)),u_m(t)^{2\delta}\Delta u_m(t))-(f(t),u_m(t)^{2\delta}\Delta u_m(t)). \end{aligned}$$ Using integration by parts, one can estimate the time derivative term as $$\begin{aligned} \nonumber(\partial_t u_m,-u_m^{2\delta}\Delta u_m) &= -\frac{1}{2\delta+1}\left(\partial_t u_m^{2\delta+1},\Delta u_m\right) =\frac{1}{2\delta+1}\left(\nabla\left(\partial_t u_m^{2\delta+1}\right),\nabla u_m\right)\\&\nonumber =\left(\partial_t\left( u_m^{2\delta}\nabla u_m\right),\nabla u_m\right)= \frac{\mathrm{d}}{\mathrm{dt}}\left( u_m^{2\delta}\nabla u_m,\nabla u_m\right) - \left( u_m^{2\delta}\nabla u_m,\partial t\nabla u_m\right)\\&= \frac{\mathrm{d}}{\mathrm{dt}}\|u^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2 - \left( u_m^{2\delta}\nabla u_m,\partial_t\nabla u_m\right). \end{aligned}$$ Using the above estimate, [\[2r.14\]](#2r.14){reference-type="eqref" reference="2r.14"} can be rewritten as $$\begin{aligned} \label{2r.15} \nonumber &\frac{\mathrm{d}}{\mathrm{dt}}\|u_m(t)^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2 +\nu\|u_m(t)^{\delta}\Delta u\|_{\mathrm{L}^2}^2+\eta((K*\Delta) u_m(t),u_m(t)^{2\delta}\Delta u_m(t))\\&= ( u_m(t)^{2\delta}\nabla u_m(t),\partial_t\nabla u_m(t))+\alpha (B(u_m(t) ),u^{2\delta}\Delta u_m(t) )\nonumber\\&\quad-\beta(c(u_m(t)),u_m(t)^{2\delta}\Delta u_m(t))-(f(t),u_m(t)^{2\delta}\Delta u_m(t)), \end{aligned}$$ for a.e. $t\in[0,T]$. For the term $-\beta(c(u_m),u_m^{2\delta}\Delta u_m)$, using integration by parts and Young's inequality, we get $$\begin{aligned} \nonumber-&\beta(c(u_m),u_m^{2\delta}\Delta u_m) =- \beta(u_m(1-u_m^{\delta})(u_m^{\delta}-\gamma),u_m^{2\delta}\Delta u_m) \\&\nonumber= -\beta(1+\gamma)(u_m^{\delta+1},u_m^{2\delta}\Delta u_m) + \beta\gamma(u_m,u_m^{2\delta}\Delta u_m)+ \beta(u^{2\delta+1},u_m^{2\delta}\Delta u_m)\\&\nonumber= -\beta(1+\gamma)(u_m^{3\delta+1},\Delta u_m) + \beta\gamma(u_m^{2\delta+1},\Delta u_m)+ \beta(u_m^{4\delta+1},\Delta u_m)\\&\nonumber= \beta(1+\gamma)(3\delta+1)(u_m^{3\delta}\nabla u_m,\nabla u_m) -\beta\gamma(2\delta+1)(u_m^{2\delta}\nabla u_m, \nabla u_m)- \beta(4\delta+1)(u_m^{4\delta}\nabla u_m,\nabla u_m)\\&\nonumber\leq \frac{\beta}{4}(4\delta+1)\|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^2 + \frac{\beta(1+\gamma)^2(3\delta+1)^2}{(4\delta+1)}\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2\nonumber\\&\quad-\beta\gamma(2\delta+1)\|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2- \beta(4\delta+1)\|u^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^2. \end{aligned}$$ An application of Cauchy-Schwarz and Young's inequalities yields $$\begin{aligned} \alpha (B(u_m ),u_m^{2\delta}\Delta u_m ) &= \sum_{i=1}^d\left(u_m^{\delta}\frac{\partial u_m}{\partial x_i},u_m^{2\delta}\Delta u_m\right) \leq \|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}\|u_m^{\delta}\Delta u_m \|_{\mathrm{L}^2} \nonumber\\&\leq \frac{1}{\nu}\|u_m\|_{\mathrm{L}^{\infty}}^{2\delta} \|u_m^{\delta}\nabla u_m\|_{\mathrm{L}^2}^2 + \frac{\nu}{4}\|u_m^{\delta}\Delta u_m \|_{\mathrm{L}^2} ^2 ,\\ (f,u_m^{2\delta}\Delta u_m) & = (u_m^{\delta}f ,u_m^{\delta}\Delta u_m)\leq \frac{\nu}{4}\|u_m^{\delta}\Delta u_m \|_{\mathrm{L}^2} ^2 + \frac{1}{\nu}\|u_m^{\delta}f\|_{\mathrm{L}^2}^2 \leq \frac{\nu}{4}\|u_m^{\delta}\Delta u_m \|_{\mathrm{L}^2} ^2 + \frac{1}{\nu}\|u_m\|_{\mathrm{L}^{\infty}}^{2\delta}\|f\|_{\mathrm{L}^2}^2,\\ ( u_m^{2\delta}\nabla u_m,\partial_t\nabla u_m) &\leq \|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^2\|\partial_t \nabla u_m\|_{\mathrm{L}^2}^2\leq \frac{\beta}{4}(4\delta+1)\|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^2 + \frac{1}{\beta(4\delta+1)}\|\partial_t\nabla u_m\|_{\mathrm{L}^2}^2. \end{aligned}$$ Combining the above estimates and integrating from $0$ to $t$, and using Lemma [\[l2\]](#l2){reference-type="ref" reference="l2"} and Remark [\[r1\]](#r1){reference-type="ref" reference="r1"}, we have $$\begin{aligned} \label{2r.17} \nonumber &\|u_m(t)^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2 + \frac{\nu}{2}\int_0^t\|u_m(s)^{\delta}\Delta u_m\|_{\mathrm{L}^2}^2\mathrm{~d}s % +\int_0^t\eta((K*\Delta u_m)(s),u_m(s)^{2\delta}\Delta u_m(s))\mathrm{~d}s + \frac{\beta}{2}(4\delta+1)\int_0^t\|u_m(s)^{2\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s \\&\nonumber\quad+\beta\gamma(2\delta+1)\int_0^t\|u_m(s)^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\\& \leq \|u_m(0)^{\delta}\nabla u_m(0)\|_{\mathrm{L}^2}^2 + \frac{1}{\beta(4\delta+1)}\int_0^t\|\partial_t\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+ \frac{\beta(1+\gamma)^2(3\delta+1)^2}{(4\delta+1)}\int_0^t\|u_m(s)^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s \nonumber\\&\quad+\frac{1}{\nu}\int_0^t\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta} \|u_m(s)^{\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^{2\delta}\mathrm{~d}s+ \frac{1}{\nu}\int_0^t\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta}\|f(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s. \end{aligned}$$ As $f\in \mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))$ , we have $f\in \mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$ and $\partial_tf\in L^2(0,T;\mathrm{L}^2(\Omega))$, which implies $f\in \mathrm{C}([0,T], \mathrm{L}^2(\Omega))$, so we have $$\int_0^t\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta}\|f(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s \leq \sup_{0\leq t\leq T}\|f(t)\|_{\mathrm{L}^2}^2 \int_0^t\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta}\mathrm{~d}s.$$ Using the last estimate and an application of Gronwall's inequality in [\[2r.17\]](#2r.17){reference-type="eqref" reference="2r.17"} yield $$\begin{aligned} \nonumber &\sup_{0\leq t\leq T}\|u_m(t)^{\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2 + \frac{\nu}{2}\int_0^t\|u_m(s)^{\delta}\Delta u_m\|_{\mathrm{L}^2}^2\mathrm{~d}s+ \frac{\beta}{2}(4\delta+1)\int_0^t\|u_m(s)^{2\delta}\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s % +\int_0^t\eta((K*\Delta u_m)(s),u_m(s)^{2\delta}\Delta u_m(s))\mathrm{~d}s \\& \leq \bigg(\|u_m(0)^{\delta}\nabla u_m(0)\|_{\mathrm{L}^2}^2 + \frac{1}{\beta(4\delta+1)}\int_0^T\|\partial_t\nabla u_m(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s+ \sup_{0\leq t\leq T}\|f(t)\|_{\mathrm{L}^2}^2 \int_0^T\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta}\mathrm{~d}s\bigg)\nonumber\\& \quad\times e^{\left( \frac{\beta(1+\gamma)^2(3\delta+1)^2}{(4\delta+1)}\right)} \exp\bigg\{\frac{1}{\nu}\int_0^T\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta}\mathrm{~d}s\bigg\}. \end{aligned}$$ For the right hand side to be bounded, we need to show that $\int_0^T\|u_m(t)\|_{\mathrm{L}^{\infty}}^{2\delta}\mathrm{~d}s<\infty$. As $u \in \mathrm{L}^{\infty}(0,T;\mathrm{H}_0^1(\Omega))\cap \mathrm{L}^2(0,T;\mathrm{H}^2(\Omega))$, using Agmon's inequality we have for $3D$ $(d=3)$ $$\begin{aligned} \int_0^T\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta}\mathrm{~d}s \leq C \int_0^T\|u_m(s)\|_{\mathrm{H}^1}^{\delta}\|u_m(s)\|_{\mathrm{H}^2}^{\delta}\mathrm{~d}s \leq C \sup_{0\leq t\leq T}\|u_m(s)\|_{\mathrm{H}^1}^{\delta}\int_0^T\|u_m(s)\|_{\mathrm{H}^2}^{\delta}\mathrm{~d}s < \infty, \end{aligned}$$ for $1\leq \delta\leq 2$. For the $2D$ $(d=2)$, for $\epsilon >0$, using Agmon's inequality $\|u\|_{\mathrm{L}^{\infty}}\leq C \|u\|_{\mathrm{H}^{1-\epsilon}}^{\frac{1}{1+\epsilon}}\|u\|_{\mathrm{H}^{2}}^{\frac{\epsilon}{1+\epsilon}}$, so $$\begin{aligned} \int_0^T\|u_m(s)\|_{\mathrm{L}^{\infty}}^{2\delta}\mathrm{~d}s \leq C \int_0^T \|u_m(s)\|_{\mathrm{H}^{1-\epsilon}}^{\frac{2\delta}{1+\epsilon}}\|u_m(s)\|_{\mathrm{H}^{2}}^{\frac{2\epsilon\delta}{1+\epsilon}}\mathrm{~d}s\leq C \sup_{0\leq t\leq T}\|u_m(s)\|_{\mathrm{H}^{1-\epsilon}}^{\frac{2\delta}{1+\epsilon}}\int_0^t\|u_m(s)\|_{\mathrm{H}^2}^{\frac{2\epsilon\delta}{1+\epsilon}}\mathrm{~d}s < \infty, \end{aligned}$$ provided $\frac{2\epsilon\delta}{1+\epsilon} \leq 2 \implies \delta \leq \frac{1}{\epsilon}+1$. Since $\epsilon>0$ is arbitrary, the above estimate is finite for $1\leq \delta < \infty$. Hence $\nabla u_m^{\delta+1} \in \mathrm{L}^{\infty}(0,T;\mathrm{L}^{2}(\Omega))$. **Step 3.** $u\in\mathrm{L}^{\infty}(0,T;\mathrm{H}^2(\Omega))$. Let $\lambda_k$ denote the $k^{\mathrm{th}}$ eigenvalue of $\Delta$ in $\mathrm{H}_0^1(\Omega)$. Multiplying the identity ([\[8\]](#8){reference-type="ref" reference="8"}) by $-\lambda_k^2d_k^m(t)$ and summing it from $k=1,2,\ldots,m$, we have $$\begin{aligned} \nonumber &(\partial_t u_m(t),A^2u_m(t))-\nu(Au_m(t), A^2u_m(t))-\eta(K*A u_m(t), A^2u_m(t))\\&=-\alpha (B(u_m(t) ),A^2u_m(t) )+\beta(c(u_m(t)),A^2u_m(t))+(f(t),A^2u_m(t)). \end{aligned}$$ An application of integration by parts yields $$\begin{aligned} \nonumber &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|Au_m(t)\|_{\mathrm{L}^2}^2+\nu\|A^{\frac{3}{2}}u_m(t)\|_{\mathrm{L}^2}^2+\eta(K*A^{\frac{3}{2}} u_m(t),A^{\frac{3}{2}}u_m(t))\\&=-\alpha (B(u_m(t)),A^2u_m(t))+\beta(c(u_m(t)),A^2u_m(t))+(f(t),A^2u_m(t)). \end{aligned}$$ Note that $\mathrm{D}(A^{\alpha})=\left\{\begin{array}{cl}\mathrm{H}^{2\alpha}(\Omega)&\text{ if }0<\alpha<1/4,\\ \mathrm{H}_0^{2\alpha}(\Omega)&\text{ if }1/4<\alpha<1.\end{array}\right.$ Since $u_m\big|_{\partial\Omega}=0$, using integration by parts, Cauchy-Schwarz and Young's inequalities, we have $$\begin{aligned} \alpha |(B(u_m ),A^2u_m )|&\leq\alpha \|A^{\frac{1}{2}}B(u_m)\|_{\mathrm{L}^2}\|A^{\frac{3}{2}} u_m\|_{\mathrm{L}^2}\leq\alpha\|A^{\frac{1}{2}}(u_m^{\delta}\nabla u_m)\|_{\mathrm{L}^2}\|A^{\frac{3}{2}}u_m\|_{\mathrm{L}^2}\nonumber\\ &\nonumber\leq\frac{\nu}{8}\|A^{\frac{3}{2}}u_m\|_{\mathrm{L}^2}^2+\frac{2\alpha^2}{\nu(\delta+1)}\|A^{\frac{1}{2}}\nabla u_m^{\delta+1}\|_{\mathrm{L}^2}^2\\&\leq\frac{\nu}{8}\|A^{\frac{3}{2}}u_m\|_{\mathrm{L}^2}^2+\frac{C\alpha^2}{\nu}\left(\|u_m^{\delta} Au_m\|_{\mathrm{L}^2}^2 + \delta\| u_m^{\delta-1}(\nabla u_m)^2\|_{\mathrm{L}^2}^2\right). \end{aligned}$$ Now we estimate $\| u_m^{\delta-1}(\nabla u_m)^2\|_{\mathrm{L}^2}^2$ using Sobolev embedding (for $d=2$, $1\leq \delta<\infty$ and for $d=3$, $1\leq\delta\leq 2$), Hölder's and Young's inequalities as $$\begin{aligned} \|u_m^{\delta-1}(\nabla u_m)^2\|_{\mathrm{L}^2}^2 &= \int_{\Omega} |u_m|^{2(\delta-1)}|\nabla u_m|^4 \mathrm{d}x = \int_{\Omega} |u_m|^{2(\delta-1)}|\nabla u_m|^{\frac{\delta-1}{\delta}}|\nabla u_m|^{\frac{3\delta+1}{\delta}} \mathrm{d}x\\& \leq \left(\int_{\Omega} |u_m|^{4\delta}|\nabla u_m|^2 \mathrm{d}x\right)^{\frac{\delta-1}{2\delta}}\left(\int_{\Omega}|\nabla u_m|^{\frac{2(3\delta+1)}{\delta+1}} \mathrm{d}x\right)^{\frac{\delta+1}{2\delta}}\\& = \|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^{\frac{\delta-1}{\delta}}\|\nabla u_m\|_{\mathrm{L}^{\frac{2(3\delta+1)}{\delta+1}}}^{\frac{3\delta+1}{\delta}}\\& \leq C\|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^{\frac{\delta-1}{\delta}}\|A u_m\|_{\mathrm{L}^2}^{\frac{3\delta+1}{\delta}}= C\||u_m|^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^{\frac{\delta-1}{\delta}}\|A u_m\|_{\mathrm{L}^2}^{\frac{\delta-1}{\delta}}\|A u_m\|_{\mathrm{L}^2}^{\frac{2(\delta+1)}{\delta}}\\&\leq \frac{1}{2}\|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^2\|Au_m\|_{\mathrm{L}^2}^2 +C \left(\frac{2(\delta-1)}{2\delta}\right)^{\frac{\delta-1}{\delta+1}}\left(\frac{\delta+1}{2\delta}\right)\|Au_m\|_{\mathrm{L}^2}^4. \end{aligned}$$ Combining, we have $$\begin{aligned} \alpha |(B(u_m ),A^2u_m )|&\leq \frac{\nu}{8}\|A^{\frac{3}{2}}u_m\|_{\mathrm{L}^2}^2+\frac{C\alpha^2}{\nu}\|u_m^{\delta} Au_m\|_{\mathrm{L}^2}^2 + \frac{\delta}{2}\|u_m^{2\delta}\nabla u_m\|_{\mathrm{L}^2}^2 \|Au_m\|_{\mathrm{L}^2}^2+ C(\alpha,\nu,\delta)\|Au_m\|_{\mathrm{L}^2}^4. \end{aligned}$$ Again using Cauchy-Schwarz and Young's inequalities, we get $$\begin{aligned} &|\beta(c(u_m),A^2u_m)|\nonumber=|\beta((1+\gamma)u_m^{\delta+1}-\gamma u_m-u_m^{2\delta+1},A^2u_m)|\nonumber\\ &=\beta(1+\gamma)(A^{\frac{1}{2}}u_m^{\delta+1},A^{\frac{3}{2}}u_m)+\beta\gamma (A u_m, A u_m) + \beta(A^{\frac{1}{2}} u_m^{2\delta+1},A^{\frac{3}{2}} u_m)\nonumber\\ &\leq \beta(1+\gamma)(\delta+1)\|u_m^{\delta}A^{\frac{1}{2}}u_m\|_{\mathrm{L}^2}\|A^{\frac{3}{2}}u_m\|_{\mathrm{L}^2}+\beta\gamma \|A u_m\|_{\mathrm{L}^2}^2+\beta(2\delta+1)\|u_m^{2\delta}A^{\frac{1}{2}} u_m\|_{\mathrm{L}^2}\|A^{\frac{3}{2}}u_m\|_{\mathrm{L}^2}\nonumber\\ &\leq \frac{\beta^2(1+\gamma)^2(\delta+1)^2}{\nu}\|u_m^{\delta}A^{\frac{1}{2}}u_m\|^2_{\mathrm{L}^2} +\beta\gamma \|A u_m\|^2_{\mathrm{L}^2}+\frac{\beta^2(2\delta+1)^2}{\nu}\|u_m^{2\delta}A^{\frac{1}{2}} u_m\|^2_{\mathrm{L}^2} + \frac{\nu}{4}\|A^{\frac{3}{2}}u_m\|_{\mathrm{L}^2}^2. \end{aligned}$$ Finally, for the forcing term using Cauchy-Schwarz and Young's inequalities, we obtain $$\begin{aligned} |(f, A^2u_m)| \leq |(A^{\frac{1}{2}}f, A^{\frac{3}{2}} u_m)| \leq \frac{1}{\nu}\|A^{\frac{1}{2}}f\|_{\mathrm{L}^2}^2+\frac{\nu}{8}\|A^{\frac{3}{2}}u_m\|^2_{\mathrm{L}^2}. \end{aligned}$$ Combing the above estimates we have $$\begin{aligned} \nonumber &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|Au_m(t)\|_{\mathrm{L}^2}^2+\frac{\nu}{2}\|A^{\frac{3}{2}}u_m(t)\|_{\mathrm{L}^2}^2+\eta(K*A^{\frac{3}{2}} u_m(t),A^{\frac{3}{2}}u_m(t))\\&\leq \frac{\beta^2(1+\gamma)^2(\delta+1)^2}{\nu}\|u_m(t)^{\delta}A^{\frac{1}{2}}u_m(t)\|^2_{\mathrm{L}^2} +\frac{\beta^2(2\delta+1)^2}{\nu}\|u_m(t)^{2\delta}A^{\frac{1}{2}} u_m(t)\|^2_{\mathrm{L}^2}+ \frac{1}{\nu}\|A^{\frac{1}{2}}f(t)\|_{\mathrm{L}^2}^2 \nonumber\\& \quad +\frac{C\alpha^2}{\nu}\|u_m(t)^{\delta} Au_m(t)\|_{\mathrm{L}^2}^2+\left( \frac{\delta}{2}\|u_m(t)^{2\delta}\nabla u_m(t)\|_{\mathrm{L}^2}^2 +\beta\gamma + C(\alpha,\nu,\delta)\|Au_m(t)\|_{\mathrm{L}^2}^2\right)\|Au_m(t)\|_{\mathrm{L}^2}^2. \end{aligned}$$ Integrating from $0$ to $t$, using positivity of the kernel $K$ and Gronwall's inequality, and taking supremum over time, we have $$\begin{aligned} \nonumber &\sup_{0\leq t\leq T}\|Au_m(t)\|_{\mathrm{L}^2}^2+\nu\int_0^t\|A^{\frac{3}{2}}u_m(s)\|_{\mathrm{L}^2}^2 \mathrm{d}s \\&\leq\bigg( \frac{2\beta^2(1+\gamma)^2(\delta+1)^2}{\nu}\int_0^T\|u_m(s)^{\delta}A^{\frac{1}{2}}u_m(s)\|^2_{\mathrm{L}^2}\mathrm{d}s +\frac{2\beta^2(2\delta+1)^2}{\nu}\int_0^T\|u_m(s)^{2\delta}A^{\frac{1}{2}} u_m(s)\|^2_{\mathrm{L}^2}\mathrm{d}s\nonumber\\&\qquad+ \frac{2}{\nu}\int_0^T\|A^{\frac{1}{2}}f(s)\|_{\mathrm{L}^2}^2\mathrm{d}s + \|u_0\|_{\mathrm{H}^2}\bigg)\nonumber\\&\quad\times \exp\left( \delta\int_0^T\|u_m(s)^{2\delta}\nabla u_m(s)|_{\mathrm{L}^2}^2 \mathrm{d}s +2\beta\gamma T+ C(\alpha,\nu,\delta)\int_0^T\|Au_m(s)\|_{\mathrm{L}^2}^2\mathrm{d}s \right). \end{aligned}$$ Using the estimate [\[2M.53\]](#2M.53){reference-type="eqref" reference="2M.53"} and [\[2r.17\]](#2r.17){reference-type="eqref" reference="2r.17"} the right hand side is bounded. Hence we have $u_m\in\mathrm{L}^{\infty}(0,T;\mathrm{H}^2(\Omega))$, $\ \partial_tu_m\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))\cap\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))$. Passing limit along a subsequence $m_j\to\infty$, we obtain the required bound for $u$. ◻ In Theorem [\[2M.thm33\]](#2M.thm33){reference-type="ref" reference="2M.thm33"} $(ii)$, we need $f \in \mathrm{L}^2(0,T;\mathrm{H}^1(\Omega))$ for the memory term. However, if we take $\eta = 0$, that is, GBHE without memory using elliptic regularity we can prove that for $u_0\in\mathrm{H}^2(\Omega)\cap\mathrm{H}_0^1(\Omega)$ and $f\in\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))$, we have $u \in \mathrm{L}^{\infty}(0,T;\mathrm{H}^2(\Omega))$ . # Finite Element Method {#sec3} ## Semidiscrete Galerkin approximation and error estimates {#2M.se7} This section is devoted to the error estimates for the semidiscrete Galerkin approximation. We partition the domain $\Omega$ into shape-regular meshes (triangular or rectangles) denoted by $\mathcal{T}_h$ and consider a finite dimensional space $V_h$, defined as $$\begin{aligned} \label{31} V_{h}=\left\{v_{h}: v_{h} \in C^0(\bar{\Omega})\cap\mathrm{H}_0^1(\Omega),\left.v_{h}\right|_{T} \in\mathbb{P}_1 (T) \ \text{ for all }\ T \in \mathcal{T}_{h}\right\},\end{aligned}$$ where $\mathbb{P}_1$ is the space of polynomials which have degree at most $1$. Note that, $V_h$ is a finite dimensional subspace of $\mathrm{H}_0^1(\Omega)$ and $0<h<1$ is the associated small mesh parameter. The aim is to prove the estimate of the type [see @VTh] $$\begin{aligned} \inf_{\chi\in V_h}\left\{\|u-\chi\|_{\mathrm{L}^2}+h\|\nabla(u-\chi)\|_{\mathrm{L}^2}\right\}\leq Ch^2\|u\|_{\mathrm{H}^2}, \end{aligned}$$ for all $u\in\mathrm{H}^2(\Omega)\cap\mathrm{H}_0^1(\Omega)$. The continuous time Galerkin approximation of the problem [\[AFGBHEM\]](#AFGBHEM){reference-type="eqref" reference="AFGBHEM"} is defined in the following way: Find $u_h(\cdot,t)\in V_h,$ such that for $t\in(0,T)$, $$\begin{aligned} \label{7p1} \nonumber \langle\partial_tu_h(t),\chi\rangle+\nu (\nabla u_h(t),\nabla \chi)+\alpha b(u_h(t),u_h(t),\chi)+\eta((K*\nabla u_h)(t),\nabla \chi)&=\beta(c(u_h(t)),\chi)+\langle f(t),\chi\rangle, \\ (u_h(0),\chi)&=(u_0^h,\chi), \ \text{ for }\ \chi\in V_h, \end{aligned}$$ where $u_0^h$ approximates $u_0$ in $V_h$. Central to the analysis of finite element methods, we define an elliptic projection associated to our model onto $V_h$, the Ritz projection ([@VTh]) of $u$ defined by $$\begin{aligned} (\nabla R_hu ,\nabla \chi) = (\nabla u,\nabla \chi), \ \text{ for all } \ \chi\in V_h. \end{aligned}$$ By setting $\chi=R_hu$ in the above equality, we obtain that the Ritz projection is stable, that is, $\|\nabla R_hu\|_{\mathrm{L}^2}\leq\|\nabla u\|_{\mathrm{L}^2}$, for all $u\in\mathrm{H}_0^1(\Omega)$. If we take $\Lambda = R_hu-u$, then, we have the following estimate for $\Lambda$ ([@VTh]), $$\begin{aligned} \label{7a1} \|\Lambda(t)\|_{\mathrm{L}^2}+h\|\nabla\Lambda(t)\|_{\mathrm{L}^2}\leq Ch^s\|u(t)\|_{\mathrm{H}^s}, \text{ for }\ s=1,2. \end{aligned}$$ The discrete equation [\[7p1\]](#7p1){reference-type="eqref" reference="7p1"} admits at least one solution $u_h\in V_h$. *Proof.* It follows as a direct consequence of Theorem [\[EWS\]](#EWS){reference-type="ref" reference="EWS"}. ◻ \ For all $t\in[0,T]$, the error estimates in this semi-discretization are given in the following theorem, [\[thm7.1\]]{#thm7.1 label="thm7.1"} Let $V_h$ be the finite dimensional subspace of $\mathrm{H}_0^1(\Omega)$ with parameter $h$ defined in [\[31\]](#31){reference-type="eqref" reference="31"}. Assume that $u_0\in \mathrm{L}^{d\delta}(\Omega)\cap\mathrm{H}_0^1(\Omega)$ and $f\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$, and $u(\cdot)$, the solution of [\[weaksolution\]](#weaksolution){reference-type="eqref" reference="weaksolution"} has the following regularity: $$\begin{aligned} u\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^{2(\delta+1)}(\Omega))\cap\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))\cap\mathrm{L}^{(d+2)\delta}(0,T;\mathrm{L}^{(d+2)\delta}(\Omega)) , \ \partial_tu\in\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega)). \end{aligned}$$ Then the error in the semi-discretization satisfies the following estimate: $$\begin{aligned} \label{7a3} & \|u_h-u\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 + \|u_h-u\|_{\mathrm{L}^{2}(0,T;\mathrm{H}_0^1(\Omega))}^2 \nonumber\\&\leq C\bigg\{\|u^h_0-u_0\|_{\mathrm{L}^2}^2+h^2\int_0^T\|\partial_tu(t)\|^2_{\mathrm{H}_0^1}\mathrm{~d} t + h^2\int_0^T\|u(t)\|_{\mathrm{H}_0^1}^2\mathrm{~d} t + h^2\int_0^T\|u(t)\|_{\mathrm{H}^2}^2\mathrm{~d} t\bigg \}, \end{aligned}$$ where the constant $C$ is independent of $h$ and depends only on $\|u_0\|_{\mathrm{H}_0^1},\nu,\alpha,\beta,\gamma,\eta,\delta,T,\|f\|_{\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))}$. **Note:** If $u_0\in \mathrm{L}^{d\delta}(\Omega)\cap\mathrm{H}_0^1(\Omega)$ and $f\in\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))$, then using Theorem [\[2M.thm3.2\]](#2M.thm3.2){reference-type="ref" reference="2M.thm3.2"} we have $u\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^{2(\delta+1)}(\Omega))\cap\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))\cap\mathrm{L}^{(d+2)\delta}(0,T;\mathrm{L}^{(d+2)\delta}(\Omega))$.\ *Proof.* Using [\[weaksolution\]](#weaksolution){reference-type="eqref" reference="weaksolution"} and [\[7p1\]](#7p1){reference-type="eqref" reference="7p1"}, we know that $(u_h-u)(\cdot)$ satisfies $$\begin{aligned} \label{7p2} & \langle\partial_t(u_h(t)-u(t)),\chi\rangle+\nu (\nabla (u_h(t)-u(t)),\nabla \chi)+\eta( (K*\nabla(u_h-u))(t),\nabla\chi)\nonumber\\&= -\alpha[b(u_h(t),u_h(t),\chi)-b(u(t),u(t),\chi)]+\beta[c(u_h(t),\chi)-c(u(t),\chi)], \end{aligned}$$ for all $\chi\in V_h$ and a.e. $t\in[0,T]$. Let us choose $\chi=u_h-W\in V_h$ in [\[7p2\]](#7p2){reference-type="eqref" reference="7p2"} to obtain for a.e. $t\in[0,T]$ $$\begin{aligned} \label{7p3} & \langle\partial_t(u_h(t)-u(t)),u_h(t)-W(t)\rangle+\nu (\nabla(u_h(t)-u(t)),\nabla(u_h(t)-W(t)))\nonumber\\&\quad +\eta\langle (K*\nabla(u_h-u))(t),\nabla(u_h(t)-W(t))\rangle\nonumber\\&= -\alpha[b(u_h(t),u_h(t),u_h(t)-W(t))-b(u(t),u(t),u_h(t)-W(t))]\nonumber\\&\quad+\beta[c(u_h(t),u_h(t)-W(t))-c(u(t),u_h(t)-W(t))]. \end{aligned}$$ Next, we write $u_h-W$ as $u_h-u+u-W$ in [\[7p3\]](#7p3){reference-type="eqref" reference="7p3"} to find $$\begin{aligned} \label{minreg} & \frac{1}{2}\frac{\mathrm{~d}}{\mathrm{~d}t}\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2+\nu\|\nabla(u_h(t)-u(t))\|_{\mathrm{L}^2}^2+\eta( (K*\nabla(u_h-u))(t),\nabla(u_h(t)-u(t)))\nonumber\\&=-\langle\partial_t(u_h(t)-u(t)),u(t)-W(t)\rangle-(\nabla(u_h(t)-u(t)),\nabla(u(t)-W(t)))\nonumber\\&\quad -\eta\langle (K*\nabla(u_h-u))(t),\nabla(u(t)-W(t))\rangle\nonumber\\&\quad-\alpha\left({u_h(t)}^{\delta}\sum_{i=1}^d\frac{\partial u_h(t)}{\partial x_i}-u(t)^{\delta}\sum_{i=1}^d\frac{\partial u(t)}{\partial x_i},u_h(t)-u(t)\right) \nonumber\\&\quad-\alpha\left({u_h(t)}^{\delta}\sum_{i=1}^d\frac{\partial u_h(t)}{\partial x_i}-u(t)^{\delta}\sum_{i=1}^d\frac{\partial u(t)}{\partial x_i}, u(t)-W(t)\right)\nonumber\\&\quad+\beta\left(u_h(t)(1-u_h(t)^{\delta})(u_h(t)^{\delta}-\gamma)-u(t)(1-u(t)^{\delta})(u(t)^{\delta}-\gamma),u_h(t)-u(t)\right)\nonumber\\&\quad+\beta\left(u_h(t)(1-u_h(t)^{\delta})(u_h(t)^{\delta}-\gamma)-u(t)(1-u(t)^{\delta})(u(t)^{\delta}-\gamma),u(t)-W(t)\right), \end{aligned}$$ for a.e. $t\in[0,T]$. Using calculations similar to [\[332\]](#332){reference-type="eqref" reference="332"} yields $$\begin{aligned} \label{7p5} & \frac{\mathrm{~d}}{\mathrm{~d} t}\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2+\nu\|\nabla(u_h(t)-u(t))\|_{\mathrm{L}^2}^2+2\eta ((K*\nabla(u_h-u))(t),\nabla(u_h(t)-u(t)))\nonumber\\&\quad+\frac{\beta}{2}\|u_h(t)^{\delta}(u_h(t)-u(t))\|_{\mathrm{L}^2}^2+\frac{\beta}{2}\|u(t)^{\delta}(u_h(t)-u(t))\|_{\mathrm{L}^2}^2 + \beta\gamma\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2\nonumber\\&\leq \left[C(\beta,\gamma,\delta) +C(\alpha,\nu)\left(\|u_h(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\right]\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2 \nonumber\\&\quad -2\partial_t(u_h(t)-u(t),u(t)-W(t))+2(u_h(t)-u(t),\partial_t(u(t)-W(t)))\nonumber\\&\quad-2(\nabla(u_h(t)-u(t)),\nabla(u(t)-W(t)))-2\eta\langle (K*\nabla(u_h-u))(t),\nabla(u(t)-W(t))\rangle\nonumber\\&\quad-2\alpha\left({u_h(t)}^{\delta}\sum_{i=1}^d\frac{\partial u_h(t)}{\partial x_i}-u(t)^{\delta}\sum_{i=1}^d\frac{\partial u(t)}{\partial x_i},u(t)-W(t)\right)\nonumber\\&\quad+2\beta\left(u_h(t)(1-u_h(t)^{\delta})(u_h(t)^{\delta}-\gamma)-u(t)(1-u(t)^{\delta})(u(t)^{\delta}-\gamma),u(t)-W(t)\right)\nonumber\\&= \left(C(\beta,\gamma,\delta)+C(\alpha,\nu)\left(\|u_h(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\right)\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2 \nonumber\\&\quad -2\partial_t(u_h(t)-u(t),u(t)-W(t)) -2\eta\langle (K*\nabla(u_h-u))(t),\nabla(u(t)-W(t))\rangle+\sum_{i=1}^4J_i, \end{aligned}$$ where the positive constants $C(\beta,\gamma,\delta)= \beta 2^{2\delta}(1+\gamma)^2(\delta+1)^2$ and $C(\alpha, \nu) = \left(\frac{4+d}{4\nu}\right)^{\frac{4+d}{4-d}}\left(\frac{4-d}{8}\right)(2^{\delta-1}\alpha)^{\frac{4-d}{8}}$. We estimate $J_1$ and $J_2$ using the Cauchy-Schwarz and Young's inequalities as $$\begin{aligned} \label{7p6} |J_1|&\leq 2\|u_h-u\|_{\mathrm{L}^2}\|\partial_t(u-W)\|_{\mathrm{L}^2}\leq\|u_h-u\|_{\mathrm{L}^2}^2+\|\partial_t(u-W)\|_{\mathrm{L}^2}^2,\\|J_2|&\leq 2\|\nabla(u_h-u)\|_{\mathrm{L}^2}\|\nabla(u-W)\|_{\mathrm{L}^2}\leq\frac{\nu}{4}\|\nabla(u_h-u)\|_{\mathrm{L}^2}^2+\frac{4}{\nu}\|\nabla(u-W)\|_{\mathrm{L}^2}^2\label{1.M}. \end{aligned}$$ Using an integration by parts, Taylor's formula, Hölder's and Young's inequalities, we rewrite $J_3$ as $$\begin{aligned} \label{7p7} J_3&=-\frac{2\alpha}{\delta+1}\sum_{i=1}^d\left(\frac{\partial}{\partial{x_i}}({u_h}^{\delta+1}-u^{\delta+1}),u-W\right)=\frac{2\alpha}{\delta+1}\sum_{i=1}^d\left(\left(u_h^{\delta+1}-u^{\delta+1}\right),\frac{\partial}{\partial x_i}(u-W)\right)\nonumber\\&=2\alpha\sum_{i=1}^d\left((\theta u_h+(1-\theta)u)^{\delta}(u_h-u),\frac{\partial}{\partial{x_i}}(u-W)\right)\nonumber\\&\leq 2\alpha\sum_{i=1}^d\|(\theta u_h+(1-\theta)u)^{\delta}(u_h-u)\|_{\mathrm{L}^2}\left\|\frac{\partial}{\partial x_i}{(u-W)}\right\|_{\mathrm{L}^2}\nonumber\\&\leq 2^{\delta}\alpha\left(\|{u_h}^{\delta}(u_h-u)\|_{\mathrm{L}^2}+\|{u}^{\delta}(u_h-u)\|_{\mathrm{L}^2}\right)\|\nabla(u-W)\|_{\mathrm{L}^2}\nonumber\\&\leq\frac{\beta}{8}\|{u_h}^{\delta}(u_h-u)\|_{\mathrm{L}^2}^2+\frac{\beta}{8}\|{u}^{\delta}(u_h-u)\|_{\mathrm{L}^2}^2+\frac{2^{2(\delta+1)}\alpha^2}{\beta}\|\nabla(u-W)\|_{\mathrm{L}^2}^2. \end{aligned}$$ Let us rewrite $J_4$ as $$\begin{aligned} \label{2.M} J_4&=2\beta(1+\gamma)({u_h}^{\delta+1}-u^{\delta+1},u-W)-2\beta\gamma(u_h-u,u-W)\nonumber\\&\quad-2\beta({u_h}^{2\delta+1}-u^{2\delta+1},u-W):=\sum_{i=5}^7J_i. \end{aligned}$$ We estimate $J_5$ using Taylor's formula, Hölder's and Young's inequalities as $$\begin{aligned} \label{3.M} J_5&=2\beta(1+\gamma)(\delta+1)((\theta u_h+(1-\theta)u)^{\delta}(u_h-u),u-W) \nonumber\\&\leq 2^{\delta}\beta(1+\gamma)(\delta+1)\left(\|{u_h}^{\delta}(u_h-u)\|_{\mathrm{L}^2}+\|{u}^{\delta}(u_h-u)\|_{\mathrm{L}^2}\right)\|u-W\|_{\mathrm{L}^2}\nonumber\\&\leq\frac{\beta}{8}\|{u_h}^{\delta}(u_h-u)\|_{\mathrm{L}^2}^2+\frac{\beta}{8}\|{u}^{\delta}(u_h-u)\|_{\mathrm{L}^2}^2+2^{2(\delta+1)}\beta(1+\gamma)^2(\delta+1)^2\|u-W\|_{\mathrm{L}^2}^2. \end{aligned}$$ Using the Cauchy-Schwarz and Young's inequalities, we estimate $J_6$ as $$\begin{aligned} \label{4.M} J_6&\leq 2\beta\gamma\|u_h-u\|_{\mathrm{L}^2}\|u-W\|_{\mathrm{L}^2}\leq \beta\gamma\|u_h-u\|_{\mathrm{L}^2}^2+\beta\gamma\|u-W\|_{\mathrm{L}^2}^2. \end{aligned}$$ Making use of Taylor's formula, Hölder's, Young's and Gagliardo-Nirenberg inequalities, we estimate $J_7$ as $$\begin{aligned} \label{7p11} J_7&=-2(2\delta+1)\beta\left((\theta u_h+(1-\theta)u)^{2\delta}(u_h-u),u-W\right)\nonumber\\&\leq 2^{2\delta}(2\delta+1)\beta\left(( |u_h|^{2\delta}+|u|^{2\delta})(u_h-u),u-W\right)\nonumber\\&\leq 2^{2\delta}(2\delta+1)\beta\left(\|u_h\|_{\mathrm{L}^{4\delta}}^{2\delta}+\|u\|_{\mathrm{L}^{4\delta}}^{2\delta}\right)\|u_h-u\|_{\mathrm{L}^{\frac{2d}{d-1}}}\|u-W\|_{\mathrm{L}^{2d}}\nonumber\\&\leq 2^{2\delta}(2\delta+1)\beta\left(\|u_h\|_{\mathrm{L}^{4\delta}}^{2\delta}+\|u\|_{\mathrm{L}^{4\delta}}^{2\delta}\right)\|u_h-u\|_{\mathrm{L}^2}^{\frac{1}{2}}\|\nabla(u_h-u)\|_{\mathrm{L}^2}^{\frac{1}{2}}\|u-W\|_{\mathrm{L}^{2d}}\nonumber\\&\leq 2^{2\delta-1}(2\delta+1)\beta\|\nabla(u-W)\|_{\mathrm{L}^2}^2 + 2^{2\delta-1}(2\delta+1)\beta\left(\|u_h\|_{\mathrm{L}^{4\delta}}^{4\delta}+\|u\|_{\mathrm{L}^{4\delta}}^{4\delta}\right)\|u_h-u\|_{\mathrm{L}^2}\|\nabla(u_h-u)\|_{\mathrm{L}^2} \nonumber\\&\leq 2^{2\delta-1}(2\delta+1)\beta\|\nabla(u-W)\|_{\mathrm{L}^2}^2 + \frac{\nu}{4}\|\nabla(u_h-u)\|_{\mathrm{L}^2}^2 \nonumber\\&\quad + \frac{2^{4\delta-2}(2\delta+1)^2\beta^2}{\nu}\left(\|u_h\|_{\mathrm{L}^{4\delta}}^{8\delta}+\|u\|_{\mathrm{L}^{4\delta}}^{8\delta}\right)\|u_h-u\|_{\mathrm{L}^2}^2, \end{aligned}$$ where we used the fact that $\mathrm{H}_0^1(\Omega)\subset\mathrm{L}^{6}(\Omega)$. Note also that $$\begin{aligned} \label{7p12} \|u_h-u\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}&=\int_{\Omega}|u_h(x)-u(x)|^{2\delta}|u_h(x)-u(x)|^{2}\mathrm{~d} x\nonumber\\&\leq 2^{2\delta-1}\int_{\Omega}\left(|u_h(x)|^{2\delta}+|u(x)|^{2\delta}\right)|u_h(x)-u(x)|^{2}\mathrm{~d}x\nonumber\\&=2^{2\delta-1}\|{u_h}^{\delta}(u_h-u)\|_{\mathrm{L}^2}^2+2^{2\delta-1}\|{u}^{\delta}(u_h-u)\|_{\mathrm{L}^2}^2. \end{aligned}$$Combining [\[7p6\]](#7p6){reference-type="eqref" reference="7p6"}-[\[7p12\]](#7p12){reference-type="eqref" reference="7p12"} and then substituting it in [\[7p5\]](#7p5){reference-type="eqref" reference="7p5"}, we deduce $$\begin{aligned} & \frac{\mathrm{~d}}{\mathrm{~d}t}\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2+\frac{\nu}{2}\|\nabla(u_h(t)-u(t))\|_{\mathrm{L}^2}^2+\eta\langle (K*\nabla(u_h-u))(t),\nabla(u_h(t)-u(t))\rangle\nonumber\\&+\frac{\beta}{2^{2\delta+1}}\|u_h(t)-u(t))\|_{\mathrm{L}^{2\delta+2}}^{2\delta+2}\nonumber\\&\leq -2\partial_t(u_h(t)-u(t),u(t)-W(t))+\|\partial_t(u(t)-W(t))\|_{\mathrm{L}^2}^2\nonumber\\&\quad-\eta\langle (K*\nabla(u_h-u))(t),\nabla(u(t)-W(t))\rangle+\left(2^{2\delta-1}(2\delta+1)\beta+\frac{4}{\nu}+\frac{2^{2(\delta+1)}\alpha^2}{\beta}\right)\|\nabla (u(t)-W(t))\|_{\mathrm{L}^2}^2\nonumber\\&\quad+\left(2^{2(\delta+1)}\beta(1+\gamma)^2(\delta+1)^2+\frac{\beta\gamma}{2}\right)\|u(t)-W(t)\|_{\mathrm{L}^2}^2+\bigg(1+2\beta\gamma+C(\beta,\gamma,\delta)\bigg)\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2\nonumber\\&\quad + \left(\frac{2^{4\delta-2}(2\delta+1)^2\beta^2}{\nu}\left(\|u_h(t)\|_{\mathrm{L}^{4\delta}}^{8\delta}+\|u(t)\|_{\mathrm{L}^{4\delta}}^{8\delta}\right)+C(\alpha,\nu)\left(\|u_h(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\right)\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2. \end{aligned}$$ Integrating the above inequality from $0$ to $t$ and using positivity of kernel $K(\cdot)$, we have $$\begin{aligned} \label{715} &\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2+\frac{\nu}{2}\int_0^t\|\nabla(u_h(s)-u(s))\|_{\mathrm{L}^2}^2\mathrm{~d}s+\frac{\beta}{2^{2\delta+1}}\int_0^t\|u_h(s)-u(s))\|_{\mathrm{L}^{2\delta+2}}^{2\delta+2}\mathrm{~d}s\nonumber\\&\leq \|u_h^0-u_0\|_{\mathrm{L}^2}^2-2(u_h(t)-u(t),u(t)-W(t))+2(u_h^0-u_0,u_0-W(0))\nonumber\\&+\int_0^t\|\partial_t(u(s)-W(s))\|_{\mathrm{L}^2}^2\mathrm{~d}s + \left(2^{2\delta-1}(2\delta+1)\beta+\frac{4}{\nu}+\frac{2^{2(\delta+1)}\alpha^2}{\beta} + \frac{\eta}{4C_K\nu}\right)\int_0^t\|\nabla (u(s)-W(s))\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&+\left(2^{2(\delta+1)}\beta(1+\gamma)^2(\delta+1)^2+\frac{\beta\gamma}{2}\right)\int_0^t\|u(s)-W(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s +\bigg(1+2\beta\gamma+C(\beta,\alpha,\delta)\bigg)\int_0^t\|u_h(s)-u(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s\nonumber\\& +\int_0^t\left(\frac{2^{4\delta-2}(2\delta+1)^2\beta^2}{\nu}\left(\|u_h(s)\|_{\mathrm{L}^{4\delta}}^{8\delta}+\|u(s)\|_{\mathrm{L}^{4\delta}}^{8\delta}\right)+C(\alpha,\nu)\left(\|u_h(s)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u(s)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\right)\|u_h(s)-u(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s, \end{aligned}$$ for all $t\in[0,T]$, where we estimate the memory term using Lemma [\[l1\]](#l1){reference-type="ref" reference="l1"} as $$\begin{aligned} &\eta\int_0^t\langle K*\nabla(u_h(s)-u(s)),\nabla(u(s)-W(s))\rangle\mathrm{~d}s\\&\leq\eta \int_0^t\| K*\nabla(u_h(s)-u(s))\|_{\mathrm{L}^2}\|\nabla(u(s)-W(s))\|_{\mathrm{L}^2}\mathrm{~d}s\nonumber\\&\leq\frac{\nu}{4C_K}\int_0^t \| K*\nabla(u_h(s)-u(s))\|_{\mathrm{L}^2}^2\mathrm{~d}s+\frac{C_K\eta^2}{\nu}\int_0^t\|\nabla(u(s)-W(s))\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber\\&\leq \frac{\nu}{4}\int_0^t\|\nabla(u_h(s)-u(s))\|_{\mathrm{L}^2}^2\mathrm{~d}s+\frac{C_K\eta^2}{\nu}\int_0^t\|\nabla(u(s)-W(s))\|_{\mathrm{L}^2}^2\mathrm{~d}s, \end{aligned}$$ for the constant $C_K = \left(\int_0^T|K(t)|\mathrm{d}t\right)^2$. Using the Cauchy-Schwarz inequality and Young's inequality, we estimate $-2(u_h-u,u-W)$ and $2(u_h^0-u_0,u_0-W(0))$ as $$\begin{aligned} -2(u_h-u,u-W)\leq 2\|u_h-u\|_{\mathrm{L}^2}\|u-W\|_{\mathrm{L}^2}\leq\frac{1}{2}\|u_h-u\|_{\mathrm{L}^2}^2+2\|u-W\|_{\mathrm{L}^2}^2,\\ 2(u_h^0-u_0,u_0-W(0))\leq 2\|u_h^0-u_0\|_{\mathrm{L}^2}\|u_0-W(0)\|_{\mathrm{L}^2}\leq\|u_h^0-u_0\|_{\mathrm{L}^2}^2+\|u_0-W(0)\|_{\mathrm{L}^2}^2. \end{aligned}$$ Using the above estimates in [\[715\]](#715){reference-type="eqref" reference="715"} and then applying Gronwall's inequality, we get $$\begin{aligned} \label{716} &\sup_{t\in[0,T]}\|u_h(t)-u(t)\|_{\mathrm{L}^2}^2+\nu\int_0^T\|\nabla(u_h(t)-u(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t+\left(\frac{\beta}{2^{2\delta}}\right)\int_0^T\|u_h(t)-u(t)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\mathrm{~d}t\nonumber\\&\leq 2\bigg\{2\|u_h^0-u_0\|_{\mathrm{L}^2}^2+\|u_0-W(0)\|_{\mathrm{L}^2}^2+\int_0^T\|\partial_t(u(t)-W(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t\nonumber\\&\quad+\left(2^{2\delta-1}(2\delta+1)\beta+\frac{4}{\nu}+\frac{2^{2(\delta+1)}\alpha^2}{\beta} + \frac{\eta}{4C\nu}\right)\int_0^T\|\nabla(u(t)-W(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t\nonumber\\&\quad+\bigg(2^{2(\delta+1)}\beta(1+\gamma)^2(\delta+1)^2+\frac{\beta\gamma}{2}\bigg)\int_0^T\|u(t)-W(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t\bigg\}\times e^{\left(1+2\beta\gamma+C(\beta,\gamma,\delta)\right)T}\nonumber\\&\quad\times\exp\bigg\{\int_0^T\left(\frac{2^{4\delta-2}(2\delta+1)^2\beta^2}{\nu}\left(\|u_h(t)\|_{\mathrm{L}^{4\delta}}^{8\delta}+\|u(t)\|_{\mathrm{L}^{4\delta}}^{8\delta}\right)+C(\alpha,\nu)\left(\|u_h(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}+\|u(t)\|^{\frac{8\delta}{4-d}}_{\mathrm{L}^{4\delta}}\right)\right)\mathrm{~d}t\bigg\}. \end{aligned}$$ The term appearing in the exponential is uniformly bounded and is independent of $h$ for $u_0\in \mathrm{L}^{d\delta}$ (see [\[333\]](#333){reference-type="ref" reference="333"}). Using [\[7a1\]](#7a1){reference-type="eqref" reference="7a1"}, we get $$\begin{aligned} \|u_0-W(0)\|_{\mathrm{L}^2}^2&=\|u_0-R^hu_0\|^2_{\mathrm{L}^2}\leq Ch^2\|u_0\|_{\mathrm{H}_0^1}, \\ \int_0^T\|u(t)-W(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t&=\int_0^T\|u(t)-R^hu(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t\leq Ch^4\int_0^T \|u(t)\|_{\mathrm{H}^2}^2\mathrm{~d} t,\\ \int_0^T\|\nabla(u(t)-W(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t&=\int_0^T\|\nabla(u(t)-R^hu(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t\leq Ch^2\int_0^T\|u(t)\|_{\mathrm{H}^2}^2 \mathrm{~d} t,\\ \int_0^T\|\partial_t(u(t)-W(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t&=\int_0^T\|\partial_t(u(t)-R^hu(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t\leq Ch^2\int_0^T \|\partial_tu(t)\|_{\mathrm{H}_0^1}^2\mathrm{~d} t. \end{aligned}$$ Thus, [\[716\]](#716){reference-type="eqref" reference="716"} implies that $$\begin{aligned} & \nu\int_0^T\|\nabla(u^h(t)-u(t))\|_{\mathrm{L}^2}^2\mathrm{~d}t\leq C(\|u_0\|_{\mathrm{H}_0^1},\nu,\alpha,\beta,\gamma,\eta,\delta,T,\|f\|_{\mathrm{L}^2(0,T;\mathrm{L}^2(\Omega))})\nonumber\\&\times\bigg\{\|u^h_0-u_0\|_{\mathrm{L}^2}^2+h^2\int_0^T\|\partial_tu(t)\|^2_{\mathrm{H}_0^1}\mathrm{~d} t + h^2\int_0^T\|u(t)\|_{\mathrm{H}_0^1}^2\mathrm{~d} t + h^2\int_0^T\|u(t)\|_{\mathrm{H}^2}^2\mathrm{~d} t\bigg \}, \end{aligned}$$ and hence the estimate [\[7a3\]](#7a3){reference-type="eqref" reference="7a3"} follows. ◻ 1\. For $d=2$ with $\delta\in[1,\infty)$ and $d=3$ with $\delta\in[1,2]$, we infer from Theorem [\[2M.thm33\]](#2M.thm33){reference-type="ref" reference="2M.thm33"} $\partial_tu\in\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))$. For large values of $\alpha,\beta,\gamma$ (cf. Remark [\[rem2.4\]](#rem2.4){reference-type="ref" reference="rem2.4"}), one can get the same regularity for the case $d=3$ with $\delta\in(2,\infty)$. 2\. In the above Theorem we need $\partial_tu\in \mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))$. However, if we assume $u_h \in \mathrm{H}^1(0,T;V_h)$ and $P_h$ denotes the $\mathrm{L}^2(\Omega)$ projection from $\mathrm{L}^2(\Omega)$ onto $V_h$ as in [@LSW] and [@KLS], namely, for each $v \in \mathrm{L}^2(\Omega)$ $$\begin{aligned} \label{L2proj} \left(P_h v-v, w^h\right)=0 \quad\text{for all}\quad w^h \in V_h,\text{a.e. } t. \end{aligned}$$ Also, there exists a constant $C$ independent of $v$ and $h$ such that $$\begin{aligned} \|u-P_hv\|_{\mathrm{H}^r} &\leq C \|u-R_hv\|_{\mathrm{H}^r},\quad \text{for all}\quad u\in \mathrm{H}_0^1(\Omega), \ r \in[0,1], \end{aligned}$$ We can estimate the first term on the right hand side of [\[minreg\]](#minreg){reference-type="eqref" reference="minreg"} using the $\mathrm{L}^2$ projection [\[L2proj\]](#L2proj){reference-type="eqref" reference="L2proj"} repeatedly as $$\begin{aligned} \nonumber -\langle\partial_t&(u_h(t)-u(t)),u(t)-P_hu(t)\rangle = \langle\partial_t u(t), u(t) - P_hu(t)\rangle\\& = \langle\partial_t (u(t)-P_hu(t)), u(t) - P_hu(t)\rangle = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|u(t)-P_hu(t)\|_{\mathrm{L}^2}^2 \end{aligned}$$ for a.e. $t\in[0,T]$, which upon integration can be estimated as $$\begin{aligned} \int_0^T\|u(t)-P_hu(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t&\leq\int_0^T\|u(t)-R^hu(t)\|_{\mathrm{L}^2}^2\mathrm{~d}t\leq Ch^4\int_0^T \|u(t)\|_{\mathrm{H}^2}^2\mathrm{~d} t. \end{aligned}$$ So, we need only our initial data $u_0\in \mathrm{L}^{d\delta}(\Omega)\cap \mathrm{H}_0^1(\Omega)$ and $\partial_tu\in \mathrm{L}^2(0,T;\mathrm{L}^2(\Omega)).$ ## Fully-discrete Galerkin approximation and error estimates In this section, we introduce a fully-discrete conforming finite element scheme. We partition the time interval $[0,T]$ into $0 =t_0<t_1<t_2\cdots < t_N=T$ with uniform time stepping $\Delta t$, that is, $t_k = k\Delta t$. We replace the time derivative with a backward Euler discretization and approximating the memory term in the following way: We shall approximate $\psi$ in $J(\psi) = \int_0^{t} K(t-s)\psi(s) \mathrm{d}s$ by the piecewise constant function taking value $\psi^k=\psi(t_k)$ in $(t_{k-1},t_{k})$ and the quadrature rule as $$\begin{aligned} J(\psi) = \int_0^{t} K(t-s)\psi(s) \mathrm{d}s \approx \frac{1}{(\Delta t)^2} \int_{t_{k-1}}^{t_k}\int_0^t K(t-s)\Delta t \psi(s)\mathrm{~d}s\mathrm{d}t = \sum_{j=1}^k \omega_{kj}\Delta t \psi^j,\end{aligned}$$ where $\omega_{k j}=\frac{1}{(\Delta t)^2 } \int_{t_{k-1}}^{t_k} \int_{t_{j-1}}^{\min \left(t, t_j\right)} K(t-s) \mathrm{~d}s \mathrm{~d}t$, for $1\leq k\leq N.$ We denote $C$ as a generic constant which is independent of $h$ and $\Delta t$. We then define a generic finite element approximate solution $u_{kh}$ by $$\begin{aligned} \label{gfds} u_{kh} |_{[t_{k-1},t_k]} = u_h^{k-1}+\left(\frac{t-t_{k-1}}{\Delta t}\right)(u_h^{k}-u_h^{k-1}), \qquad 1\leq k\leq N, \qquad \qquad \text{for } t\in [t_{k-1},t_k]. \end{aligned}$$ Note here that $\partial_tu_{kh} = \frac{u_h^k - u_h^{k-1}}{\Delta t}$. The continuous time Galerkin approximation of the problem [\[AFGBHEM\]](#AFGBHEM){reference-type="eqref" reference="AFGBHEM"} is defined in the following way: Find $u_h^k\in V_h,$ for $k=1,2,\ldots,N$ such that for $\chi\in V_h$: $$\label{fd1} \left\{ \begin{aligned} \left(\frac{u_h^k-u_h^{k-1}}{\Delta t},\chi\right)+\nu a(u_h^k,\chi)&+ \alpha b(u_h^k,u_h^k,\chi)+\Big(\sum_{j=1}^k \omega_{kj}(\Delta t) \nabla u_{h}^j,\nabla \chi\Big) \\&=\beta(u_h^k(1-(u_h^k)^{\delta})((u_h^k)^{\delta}-\gamma),\chi)+\langle f^k,\chi\rangle,\\ (u_h^0(x_i),\chi)&=(u_0(x_i),\chi), \ \text{ for }\ i=1,2,\ldots,N, \end{aligned} \right.$$ $f^k = (\Delta t)^{-1}\int_{t_{k-1}}^{t_k} f(s) \mathrm{d}s$ for $f\in \mathrm{L}^2(0,T;\mathrm{H}^{-1}(\Omega))$, $u^0_h$ approximates $u_0$ in $V_h$ and $x_i,$ $i=1,\ldots,n,$ represent the nodes of the triangulation $\mathcal{T}^h$ of $\Omega$. The existence of the discrete solution can be proved using an application of Brouwer's fixed point theorem \[Lemma 1.4, [@Te]\] as done in Theorem 3.1 [@KMR]. To show the error estimates, we need a stability result. It is natural to see that for the quadratic form analogous to the double integral in [\[pk\]](#pk){reference-type="eqref" reference="pk"}, we have $$\begin{aligned} \label{pkd} \sum_{k=1}^N \left(\sum_{j=1}^{k}\omega_{kj}\Delta t \nabla u_h^{j} ,\nabla u_h^k\right) \geq 0,\end{aligned}$$ as discussed in [@MMW] and Lemma 4.7 [@MTh]. The following lemma is useful for the analysis: [\[lemma3.1\]]{#lemma3.1 label="lemma3.1"} Assume $f\in \mathrm{L}^2(0,T;\mathrm{H}^{-1}(\Omega)).$ Then the set $\{f^k\}_{k=1}^N$ defined by $f^k = (\Delta t)^{-1}\int_{t_{k-1}}^{t_k} f(s) \mathrm{d}s$ satisfies $$\begin{aligned} \Delta t \sum_{k=1}^N\|f^k\|_{\mathrm{H}^{-1}}^2 \leq C \|f\|^2_{\mathrm{L}^2(0,T;\mathrm{H}^{-1}(\Omega))} \end{aligned}$$ and $$\begin{aligned} \sum_{k=1}^N\int_{t_{k-1}}^{t_k}\|f^k-f(t)\|_{\mathrm{H}^{-1}}^2 \mathrm{d}t \rightarrow 0 \quad \Delta t \rightarrow 0. \end{aligned}$$ If $f\in H^{\gamma}(0,T;\mathrm{H}^{-1}(\Omega))$ for some $\gamma\in [0,1]$, then $$\begin{aligned} % \label{3.25} \sum_{k=1}^N\int_{t_{k-1}}^{t_k}\|f^k-f(t)\|_{\mathrm{H}^{-1}}^2\mathrm{d}t \leq C (\Delta t)^{2\gamma}\|f\|^2_{\mathrm{H}^{\gamma}(0,T;\mathrm{H}^{-1}(\Omega))}. \end{aligned}$$ *Proof.* For a proof, we refer to Lemma 3.2, [@LSW]. ◻ Let us now discuss the stability of the fully-discrete approximation [\[fd1\]](#fd1){reference-type="eqref" reference="fd1"}. [\[4.1\]]{#4.1 label="4.1"} Assume $f\in \mathrm{L}^2(0,T;\mathrm{H}^{-1}(\Omega))$ and $u_h^0 \in \mathrm{L}^2(\Omega)$. Let $\{u_h^k\}_{k=1}^N \subset V_h$ be defined by [\[fd1\]](#fd1){reference-type="eqref" reference="fd1"}. Then $$\begin{aligned} \label{3p33} \sum_{k=1}^{N} \Delta t\|\nabla u_h^k\|_{\mathrm{L}^2}^2 \leq \left(\frac{1}{\nu }\|f\|^2_{\mathrm{L}^2(0,T;H^{-1}(\Omega))} +\| u_h^0\|_{\mathrm{L}^2}^2\right) \times e^{\beta T(1+\gamma)^2}.\end{aligned}$$ *Proof.* Suppose $f\in \mathrm{L}^2(0,T;\mathrm{H}^{-1}(\Omega))$, taking $\chi = u_h^k$ in [\[fd1\]](#fd1){reference-type="eqref" reference="fd1"}, we obtain $$\begin{aligned} &\frac{1}{2\Delta t}\|u_h^k\|_{\mathrm{L}^2}^2 - \frac{1}{2\Delta t}\|u_h^{k-1}\|_{\mathrm{L}^2}^2 + \frac{1}{2\Delta t}\|u_h^k-u_h^{k-1}\|_{\mathrm{L}^2}^2+\nu a(u_h^k,u_h^k)+ \alpha b(u_h^k,u_h^k,u_h^k) \\&+\eta \left(\sum_{j=1}^{k}\Delta t \omega_{kj}\nabla u_h^{j} ,\nabla u_h^k\right)=\beta(u_h^k(1-(u_h^k)^{\delta})((u_h^k)^{\delta}-\gamma),u_h^k)+\langle f^k,u_h^k\rangle. \end{aligned}$$ Using [\[7a\]](#7a){reference-type="eqref" reference="7a"}, [\[7\]](#7){reference-type="eqref" reference="7"}, Cauchy-Schwarz and Young's inequalities, we have $$\begin{aligned} &\frac{1}{2\Delta t}\|u_h^k\|_{\mathrm{L}^2}^2 - \frac{1}{2\Delta t}\|u_h^{k-1}\|_{\mathrm{L}^2}^2 + \frac{1}{2\Delta t}\|u_h^k-u_h^{k-1}\|_{\mathrm{L}^2}^2+\nu \|\nabla u_h^k\|_{\mathrm{L}^2}^2+ \beta\gamma\|u_h^k\|_{\mathrm{L}^2}^2+\beta\|u_h^k\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\\&+\eta \left(\sum_{j=1}^{k}\Delta t \omega_{kj}\nabla u_h^{j} ,\nabla u_h^k\right)\leq \frac{\beta}{2}\| u_h^k\|^{2(\delta+1)}_{\mathrm{L}^{2(\delta+1)}}+\frac{\beta(1+\gamma)^2}{2}\|u_h^k\|_{\mathrm{L}^2}^2+ \frac{1}{2\nu}\|f^k\|_{\mathrm{H}^{-1}}^2 + \frac{\nu}{2} \|\nabla u_h^k\|_{\mathrm{L}^2}^2. \end{aligned}$$ This may be simplified as $$\begin{aligned} &\frac{1}{2\Delta t}\|u_h^k\|_{\mathrm{L}^2}^2 - \frac{1}{2\Delta t}\|u_h^{k-1}\|_{\mathrm{L}^2}^2+\frac{\nu}{2} \|\nabla u_h^k\|_{\mathrm{L}^2}^2+\eta \left(\sum_{j=1}^{k}\omega_{kj}\Delta t \nabla u_h^{j} ,\nabla u_h^k\right)\leq \frac{\beta(1+\gamma)^2}{2}\|u_h^k\|_{\mathrm{L}^2}^2+ \frac{1}{2\nu}\|f^k\|_{\mathrm{H}^{-1}}^2. \end{aligned}$$ Summing over $k$ for $k = 1, 2, \ldots, N$ and using positivity of the Kernel $K(\cdot)$ [\[pkd\]](#pkd){reference-type="eqref" reference="pkd"} and Lemma [\[lemma3.1\]](#lemma3.1){reference-type="ref" reference="lemma3.1"}, we have $$\begin{aligned} &\frac{1}{\Delta t}\|u_h^N\|_{\mathrm{L}^2}^2 - \frac{1}{\Delta t}\|u_h^0\|_{\mathrm{L}^2}^2+\nu \sum_{k=1}^N\|\nabla u_h^k\|_{\mathrm{L}^2}^2\leq \beta(1+\gamma)^2\sum_{k=1}^N\|u_h^k\|_{\mathrm{L}^2}^2 + \frac{1}{\nu\Delta t }\Delta t\sum_{k=1}^N\|f^k\|_{\mathrm{H}^{-1}}^2. \end{aligned}$$ Finally, an application of the discrete Gronwall inequality (Lemma 9 [@AKP]) yields the required bound [\[3p33\]](#3p33){reference-type="eqref" reference="3p33"}. ◻ [\[thm7.2\]]{#thm7.2 label="thm7.2"} Let $1\leq \delta < \infty$ for $d=2$ and, $1\leq \delta \leq 2$ for $d=3$ and assume that $u_0\in \mathrm{H}^2(\Omega)\cap \mathrm{H}_0^1(\Omega)$ and $f\in \mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))$. Let $u_h$ be the semi-discrete solution [\[7p1\]](#7p1){reference-type="eqref" reference="7p1"} and $u_{kh}$ be the generic fully-discrete solution defined in [\[gfds\]](#gfds){reference-type="eqref" reference="gfds"}, then $$\begin{aligned} \nonumber\|u_h - u_{kh}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 + \|u_h - u_{kh}\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2&\leq C(\Delta t)^2(\|f\|^2_{\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))} + \|u_0\|^2_{\mathrm{H}^2})\\&\quad+ \eta^2(\Delta t)^2\sup_{k,j} \omega_{kj}^2(\|f\|^2_{\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))} + \|u_0\|^2_{\mathrm{H}^2}). \end{aligned}$$ *Proof.* We divide our proof into two steps: In Step 1, we prove the error estimates on the nodal values and in Step 2, we prove the final estimate. **Step 1:** *Estimates at $t_k :$* Integrating the semi-discrete scheme [\[7p1\]](#7p1){reference-type="eqref" reference="7p1"} from $t_{k-1}$ to $t_k$, we obtain $$\begin{aligned} \label{sdi} &\nonumber (u_h({t_{k}})-u_h(t_{k-1}),\chi)+\nu \left(\int_{t_{k-1}}^{t_k}\nabla u_h(t) \mathrm{~d} t,\nabla \chi\right)+\alpha \left(\int_{t_{k-1}}^{t_k} B(u_h(t)) \mathrm{~d} t,\chi \right)\\&\qquad+\eta\left(\int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t,\nabla \chi\right) = \beta\left(\int_{t_{k-1}}^{t_k}c(u_h(t))\mathrm{~d} t,\chi\right)+\left( \int_{t_{k-1}}^{t_k}f(t)\mathrm{~d} t,\chi\right). \end{aligned}$$ The fully-discrete scheme [\[fd1\]](#fd1){reference-type="eqref" reference="fd1"} is given as $$\begin{aligned} \label{fd3} &\nonumber \left(\frac{u_h^k-u_h^{k-1}}{\Delta t},\chi\right)+\nu (\nabla u_h^k,\nabla \chi)+ \alpha b(u_h^k,u_h^k,\chi)+\Big(\sum_{j=1}^k \omega_{kj} \Delta t \nabla u_{h}^k,\nabla \chi\Big) \\&\qquad=\beta(u_h^k(1-(u_h^k)^{\delta})((u_h^k)^{\delta}-\gamma),\chi)+( f^k,\chi). \end{aligned}$$ Subtracting [\[fd3\]](#fd3){reference-type="eqref" reference="fd3"} from [\[sdi\]](#sdi){reference-type="eqref" reference="sdi"}, we have $$\begin{aligned} &(u_h(t_k)-u_h^k,\chi)-(u_h(t_{k-1})-u_h^{k-1},\chi)+\nu \left(\int_{t_{k-1}}^{t_k}\nabla u_h(t) \mathrm{~d}t-\Delta t\nabla u_h^k,\nabla \chi\right) +\eta\left(\int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t,\nabla \chi\right) \\&- \Delta t \eta \left(\sum_{j=1}^{k}\Delta t \omega_{kj}\nabla u_h^{j} ,\nabla\chi\right) +\alpha \left(\left(\int_{t_{k-1}}^{t_k} B(u_h(t)) \mathrm{~d} t\right) - \Delta t B(u_h^k),\chi \right)=\beta\left(\int_{t_{k-1}}^{t_k}c(u_h(t))\mathrm{~d} t - \Delta tc(u_h^k),\chi\right). \end{aligned}$$ Rewriting the above equation and taking $\chi = u_h(t_k)-u_h^k$ , we deduce $$\begin{aligned} &\|u_h(t_k)-u_h^k\|_{\mathrm{L}^2}^2+\nu \Delta t \|\nabla (u_h(t_k) -u_h^k)\|_{\mathrm{L}^2}^2 + \Delta t \eta \left(\sum_{j=1}^{k}\omega_{kj}\Delta t\nabla (u_h(t_j)-u_h^{j}) ,\nabla(u_h(t_k) -u_h^k)\right) \\&= (u_h(t_{k-1})-u_h^{k-1},u_h(t_k) -u_h^k) - \nu \left(\int_{t_{k-1}}^{t_k}\nabla u_h(t) \mathrm{~d}t-\Delta t\nabla u_h(t_k),\nabla (u_h(t_k)-u_h^k)\right)\\&\quad + \eta \left(\Delta t \sum_{j=1}^{k}\omega_{kj}\Delta t\nabla u_h(t_j) - \int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t,\nabla (u_h(t_k)-u_h^k)\right) \\&\quad-\alpha \left(\int_{t_{k-1}}^{t_k} B(u_h(t)) \mathrm{~d} t- \Delta t B(u_h(t_k)),u_h(t_k)-u_h^k \right)-\alpha \Delta t\left(B(u_h(t_k) ) - B(u_h^k),u_h(t_k)-u_h^k \right)\\&\quad+\beta \left(\int_{t_{k-1}}^{t_k} c(u_h(t)) \mathrm{~d} t - \Delta t c(u_h(t_k)),u_h(t_k)-u_h^k \right)+\beta \Delta t\left(c(u_h(t_k) ) - c(u_h^k),u_h(t_k)-u_h^k \right). \end{aligned}$$ Using Cauchy-Schwarz and Young's inequalities, we can estimate the first term on right hand side as $$\begin{aligned} (u_h(t_{k-1})-u_h^{k-1},u_h(t_k) -u_h^k) \leq \frac{1}{2}\|u_h(t_{k-1})-u_h^{k-1}\|_{\mathrm{L}^2}^2 + \frac{1}{2}\|u_h(t_k) -u_h^k\|_{\mathrm{L}^2}^2\label{fde1}, \end{aligned}$$ Again an application of the Cauchy-Schwarz and Young's inequalities yield $$\begin{aligned} &\nonumber\nu \left(\int_{t_{k-1}}^{t_k}\nabla u_h(t) \mathrm{~d}t-\Delta t\nabla u_h(t_k),\nabla (u_h(t_k)-u_h^k)\right)\\&\nonumber\leq \nu\Big\|\int_{t_{k-1}}^{t_k}\nabla u_h(t)\mathrm{~d}t-\Delta t\nabla u_h(t_k) \Big\|_{\mathrm{L}^2}\big\|\nabla (u_h(t_k)-u_h^k)\big\|_{\mathrm{L}^2}\\&\leq \frac{2}{\nu\Delta t}\Big\|\int_{t_{k-1}}^{t_k}\nabla u_h(t)\mathrm{~d}t -\Delta t\nabla u_h(t_k) \Big\|_{\mathrm{L}^2}^2+\frac{\nu\Delta t}{8}\big\|\nabla (u_h(t_k)-u_h^k)\big\|_{\mathrm{L}^2}^2, \end{aligned}$$ where the first term on the right hand side can be estimated as $$\begin{aligned} \label{int1} \nonumber&\frac{1}{\Delta t}\big\|\int_{t_{k-1}}^{t_k}\nabla u_h(t)\mathrm{~d}t-\Delta t\nabla u_h(t_k) \big\|_{\mathrm{L}^2}^2 = \frac{1}{\Delta t}\big\|\int_{t_{k-1}}^{t_k}(\nabla u_h(t)-\nabla u_h(t_k)) \mathrm{~d}t\big\|_{\mathrm{L}^2}^2 \\&\nonumber= \frac{1}{\Delta t}\big\|\int_{t_{k-1}}^{t_k}\int_{t_k}^{t}\nabla \partial_tu_h(s) \mathrm{~d}s\mathrm{~d}t\big\|_{\mathrm{L}^2}^2\leq \frac{1}{\Delta t}\left(\int_{t_{k-1}}^{t_k}\int_{t_{k-1}}^{t_k}\| \partial_t \nabla u_h(s)\|_{\mathrm{L}^2} \mathrm{~d}s\mathrm{~d}t\right)^2\\&\leq\Delta t\left(\int_{t_{k-1}}^{t_k}\| \partial_t \nabla u_h(s)\|_{\mathrm{L}^2} \mathrm{~d}s\right)^2\leq (\Delta t)^2\int_{t_{k-1}}^{t_k} \| \partial_t \nabla u_h(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s. \end{aligned}$$ The memory term can be estimated as $$\begin{aligned} & \eta \left(\Delta t \sum_{j=1}^{k}\Delta t\omega_{kj}\nabla u_h(t_j) - \int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t,\nabla (u_h(t_k)-u_h^k)\right) \\&\leq \eta \Big\|\Delta t \sum_{j=1}^{k}\Delta t\omega_{kj}\nabla u_h(t_j) - \int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t\Big\|_{\mathrm{L}^2}\|\nabla (u_h(t_k)-u_h^k)\|_{\mathrm{L}^2} \\& \leq \frac{2\eta^2}{\nu\Delta t} \Big\|(\Delta t)^2 \sum_{j=1}^{k}\omega_{kj}\nabla u_h(t_j) - \int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t\Big\|^2_{\mathrm{L}^2}+\frac{\nu\Delta t}{8}\|\nabla (u_h(t_k)-u_h^k)\|^2_{\mathrm{L}^2}, \end{aligned}$$ where the term $\frac{2\eta^2}{\nu\Delta t} \big\|(\Delta t)^2 \sum_{j=1}^{k}\omega_{kj}\nabla u_h(t_j) - \int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t\big\|^2_{\mathrm{L}^2}$ can be estimated using [\[int1\]](#int1){reference-type="eqref" reference="int1"} as $$\begin{aligned} &\frac{1}{\Delta t}\Big\|(\Delta t)^2 \sum_{j=1}^{k}\omega_{kj}\nabla u_h(t_j) - \int_{t_{k-1}}^{t_k}(K*\nabla u_h)(t)\mathrm{~d} t\Big\|_{\mathrm{L}^2}^2 \\&= \frac{1}{\Delta t}\Big\|\int_{t_{k-1}}^{t_k}\sum_{j=1}^k\left(\int_{t_{j-1}}^{\min(t,t_j)}K(t-s)(\nabla u_h(t_j)- \nabla u_h(s))\mathrm{~d}s \right)\mathrm{~d} t\Big\|^2_{\mathrm{L}^2} \\&\leq\frac{1}{\Delta t}\Big\|\sum_{j=1}^k\int_{t_{k-1}}^{t_k}\left(\int_{t_{j-1}}^{\min(t,t_j)}|K(t-s)|\int_{t_{j-1}}^{t_j}|\partial_t\nabla u_h(\tau)|\mathrm{~d}\tau \right)\mathrm{~d}s\mathrm{~d} t\Big\|^2_{\mathrm{L}^2}\\&\leq\frac{1}{\Delta t}\Big\|\sum_{j=1}^k(\Delta t)^2\left(\frac{1}{(\Delta t)^2}\int_{t_{k-1}}^{t_k}\int_{t_{j-1}}^{\min(t,t_j)}|K(t-s)| \mathrm{~d}s\mathrm{~d} t\int_{t_{j-1}}^{t_j}|\partial_t\nabla u_h(\tau)|\mathrm{~d}\tau \right)\Big\|^2_{\mathrm{L}^2}\\&\leq (\Delta t)^3\Big\|\sum_{j=1}^k \overline{K}_{kj}\int_{t_{j-1}}^{t_j}|\partial_t\nabla u_h(\tau)|\mathrm{~d}\tau\Big\|^2_{\mathrm{L}^2} \\&\leq k(\Delta t)^4\sum_{j=1}^k \overline{K}_{kj}^2\int_{t_{j-1}}^{t_j}\|\partial_t\nabla u_h(\tau)\|_{\mathrm{L}^2}^2\mathrm{~d}\tau\\&\leq T(\Delta t)^3\sum_{j=1}^k \overline{K}_{kj}^2\int_{t_{j-1}}^{t_j}\|\partial_t\nabla u_h(\tau)\|_{\mathrm{L}^2}^2\mathrm{~d}\tau, \end{aligned}$$ where $\overline{K}_{kj} = \frac{1}{(\Delta t)^2}\int_{t_{k-1}}^{t_k}\int_{t_{j-1}}^{\min(t,t_j)}|K(t-s)| \mathrm{~d}s\mathrm{~d} t$. An application of integration by parts and Cauchy-Schwarz inequality gives $$\begin{aligned} \label{nl1} \nonumber &\alpha \left(\int_{t_{k-1}}^{t_k} B(u_h(t)) \mathrm{~d} t - \Delta t B(u_h(t_k)),u_h(t_k)-u_h^k \right) \\&\nonumber= \frac{\alpha}{\delta+1}\sum_{i=1}^{d}\left(\int_{t_{k-1}}^{t_k} u_h^{\delta+1}(t) \mathrm{d}t - \Delta tu_h^{\delta+1}(t_k), \frac{\partial}{\partial x_i}(u_h(t_k)-u_h^k)\right)\\&\nonumber \leq \frac{\alpha}{\delta+1}\Big\|\int_{t_{k-1}}^{t_k} u_h^{\delta+1}(t) \mathrm{d}t - \Delta t u_h^{\delta+1}(t_k)\Big\|_{\mathrm{L}^2} \big\|\nabla(u_h(t_k)-u_h^k)\big\|_{\mathrm{L}^2}\\&\leq \frac{2\alpha^2}{\nu(\delta+1)^2\Delta t}\Big\|\int_{t_{k-1}}^{t_k} u_h^{\delta+1}(t) \mathrm{d}t - \Delta t u_h^{\delta+1}(t_k)\Big\|_{\mathrm{L}^2}^2+\frac{\nu\Delta t}{8}\big\|\nabla(u_h(t_k)-u_h^k)\big\|^{2}_{\mathrm{L}^2}. \end{aligned}$$ Now, we estimate $\frac{1}{\Delta t}\big\|\int_{t_{k-1}}^{t_k}u_h^{\delta+1}(t)\mathrm{~d}t-\Delta t u_h^{\delta+1}(t_k) \big\|_{\mathrm{L}^2}^2$ similar to [\[int1\]](#int1){reference-type="eqref" reference="int1"} as $$\begin{aligned} &\frac{1}{\Delta t}\Big\|\int_{t_{k-1}}^{t_k}u_h^{\delta+1}(t)\mathrm{~d}t-\Delta t u_h^{\delta+1}(t_k)\Big\|_{\mathrm{L}^2}^2 = \frac{1}{\Delta t}\Big\|\int_{t_{k-1}}^{t_k}( u_h^{\delta+1}(t)-u_h^{\delta+1}(t_k)) \mathrm{~d}t\Big\|_{\mathrm{L}^2}^2\\&= \frac{1}{\Delta t}\Big\|\int_{t_{k-1}}^{t_k}\int_{t_k}^{t} \partial_tu_h^{\delta+1}(s) \mathrm{~d}s\mathrm{~d}t\Big\|_{\mathrm{L}^2}^2\leq \frac{1}{\Delta t}\left(\int_{t_{k-1}}^{t_k}\int_{t_{k-1}}^{t_k}\| \partial_tu_h^{\delta+1}(s)\|_{\mathrm{L}^2} \mathrm{~d}s\mathrm{~d}t\right)^2\\&\leq\Delta t\left(\int_{t_{k-1}}^{t_k}\| \partial_tu_h^{\delta+1}(s)\|_{\mathrm{L}^2} \mathrm{~d}s\right)^2\leq (\Delta t)^2\int_{t_{k-1}}^{t_k} \| \partial_tu_h^{\delta+1}(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s \\&= (\Delta t)^2(\delta+1)^2\int_{t_{k-1}}^{t_k} \|u_h^{\delta}(s)\partial_tu_h(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s. \end{aligned}$$ Finally the reaction term can be simplified as $$\begin{aligned} \left(\int_{t_{k-1}}^{t_k} c(u_h(t)) \mathrm{~d} t\right)& - \Delta t c(u_h(t_k))= \beta (1+\gamma)\left(\int_{t_{k-1}}^{t_k} u_h^{\delta+1}(t) \mathrm{~d} t-\Delta t u_h^{\delta+1}(t_k)\right)\\&-\beta\gamma\left(\left(\int_{t_{k-1}}^{t_k} u_h(t) \mathrm{~d} t\right)-\Delta t u_h(t_k)\right)-\beta\left(\int_{t_{k-1}}^{t_k} u_h^{2\delta+1}(t) \mathrm{~d} t-\Delta t u_h^{2\delta+1}(t_k)\right). \end{aligned}$$ Firstly, we estimate the term with coefficient $\gamma$, using Cauchy-Schwarz inequality and the estimate similar to [\[int1\]](#int1){reference-type="eqref" reference="int1"} as $$\begin{aligned} \nonumber\beta\gamma\left(\int_{t_{k-1}}^{t_k} u_h(t) \mathrm{~d} t-\Delta t u_h(t_k),u_h(t_k)-u_h^k \right) &\leq \frac{(\beta\gamma)^2}{\nu\Delta t}\Big\|\int_{t_{k-1}}^{t_k} u_h(t) \mathrm{~d} t-\Delta t u_h(t_k)\Big\|_{\mathrm{L}^2}^2 +\frac{\nu\Delta t}{8}\|u_h(t_k)-u_h^k\|^{2}_{\mathrm{L}^2}\\&\leq \frac{(\beta\gamma)^2(\Delta t)^2}{\nu}\int_{t_{k-1}}^{t_k} \| \partial_tu_h(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s +\frac{\nu\Delta t}{8}\|(u_h(t_k)-u_h^k)\|^{2}_{\mathrm{L}^2}. \end{aligned}$$ Using an estimate similar to [\[nl1\]](#nl1){reference-type="eqref" reference="nl1"}, we have $$\begin{aligned} &\beta(1+\gamma)\left(\int_{t_{k-1}}^{t_k} u_h^{\delta+1}(t) \mathrm{~d} t-\Delta t u_h^{\delta+1}(t_k), u_h(t_k)-u_h^k \right)\\&\leq \frac{(\Delta t)^2(1+\gamma)^2\beta^2(\delta+1)^2}{\nu}\int_{t_{k-1}}^{t_k} \|u_h^{\delta}(s)\partial_tu_h(s)\|_{\mathrm{L}^2}^2 + \frac{\nu\Delta t}{8}\|u_h(t_k)-u_h^k\|^{2}_{\mathrm{L}^2}. \end{aligned}$$ Also for the final term, we obtain $$\begin{aligned} &\Big|\beta\left(\int_{t_{k-1}}^{t_k} u_h^{2\delta+1}(t) \mathrm{~d} t-\Delta t u_h^{2\delta+1}(t_k), u_h(t_k)-u_h^k \right)\Big|= \Big|\beta\int_{t_{k-1}}^{t_k} \left(u_h^{2\delta+1} (t)-u_h^{2\delta+1} (t_k), u_h(t_k)-u_h^k \right)\mathrm{~d} t\Big|\\& = \Big|\beta\int_{t_{k-1}}^{t_k}\left(\int_{t_{k}}^{t}\partial_tu_h^{2\delta+1}(s) \mathrm{~d}s,u_h(t_k)-u_h^k \right)\mathrm{~d} t\Big| \leq \beta(2\delta+1) \int_{t_{k-1}}^{t_k}\int_{t_{k-1}}^{t_k}\left|\left(u_h^{2\delta}(s)\partial_tu_h(s),u_h(t_k)-u_h^k \right)\right| \mathrm{~d}s\mathrm{~d} t\\&\leq \beta(2\delta+1)\Delta t\int_{t_{k-1}}^{t_k}\left|\left(u_h^{\delta}(t)(u_h^{\delta}(t)\partial_tu_h(t)) ,u_h(t_k)-u_h^k \right)\right| \mathrm{~d} t\\&\leq \beta(2\delta+1)\Delta t \int_{t_{k-1}}^{t_k}\|u_h^{\delta}(t)\|_{\mathrm{L}^{\frac{2(\delta+1)}{\delta}}}\|u_h^{\delta}(t)\partial_tu_h(t)\|_{\mathrm{L}^2}\|u_h(t_k)-u_h^k\|_{\mathrm{L}^{2(\delta+1)}} \mathrm{~d} t \\&\leq \beta(2\delta+1)\Delta t\|u_h\|_{\mathrm{L}^{\infty}((t_{k-1},t_k);\mathrm{L}^{2(\delta+1)}(\Omega))}^{\delta}\int_{t_{k-1}}^{t_k}\|u_h^{\delta}(t)\partial_tu_h(t)\|_{\mathrm{L}^2}\|\nabla(u_h(t_k)-u_h^k)\|_{\mathrm{L}^{2}} \mathrm{~d} t \\&\leq\frac{2\beta^2(2(\delta+1))^2(\Delta t)^2}{\nu}\|u_h\|_{\mathrm{L}^{\infty}(0,T; \mathrm{L}^{2(\delta+1)}(\Omega))}^{2\delta}\int_{t_{k-1}}^{t_k}\|u_h^{\delta}(t)\partial_tu_h(t)\|^2_{\mathrm{L}^2} \mathrm{~d} t +\frac{\nu\Delta t}{8} \|\nabla(u_h(t_k)-u_h^k)\|^2_{\mathrm{L}^{2}}, \end{aligned}$$ for $1\leq\delta<\infty$ ($d=2$) and $1\leq \delta\leq 2$ ($d=3$). For, $w= u_h(t_k)-u_h^k$, we derive a bound for $-\alpha\Delta t(B(u_h(t_k))-B(u_h^k),w)$, using Taylor's formula, Hölder's, Gagliardo-Nirenberg interpolation and Young's inequalities as $$\begin{aligned} -\alpha \Delta t\left(B(u_h(t_k) ) - B(u_h^k),u_h(t_k)-u_h^k \right)&=\frac{\alpha}{\delta+1} \left((u_h(t_k)^{\delta+1}-(u_h^k)^{\delta+1})\left(\begin{array}{c}1\\\vdots\\1\end{array}\right),\nabla w\right)\nonumber\\&=\alpha\left((u_h(t_k)-u_h^k)(\theta u_h(t_k)+(1-\theta) u_h^k)^{\delta}\left(\begin{array}{c}1\\\vdots\\1\end{array}\right),\nabla w\right)\nonumber\\&\leq 2^{\delta-1} \alpha(\|u_h(t_k)\|^{\delta}_{\mathrm{L}^{2(\delta+1)}}+\|u_h^k\|^{\delta}_{\mathrm{L}^{2(\delta+1)}})\|w\|_{\mathrm{L}^{2(\delta+1)}}\|\nabla w\|_{\mathrm{L}^2} \nonumber\\&\leq 2^{\delta-1}C \alpha(\|u_h(t_k)\|^{\delta}_{\mathrm{L}^{2(\delta+1)}}+\|u_h^k\|^{\delta}_{\mathrm{L}^{2(\delta+1)}})\|\nabla w\|_{\mathrm{L}^{2}}^{\frac{(2+d)\delta+2}{2(\delta+1)}}\| w\|_{\mathrm{L}^2}^{\frac{(2-d)\delta+2}{2(\delta+1)}}\nonumber\\&\leq \frac{\nu}{8}\|\nabla w\|_{\mathrm{L}^2}^2+C(\alpha,\nu)\left(\|u_h(t_k)\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}+\|u_h^k\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}\right)\|w\|_{\mathrm{L}^2}^2.\end{aligned}$$ where $C(\alpha, \nu) = \left(\frac{2((2+d)\delta+2)}{\nu(\delta+1)}\right)^{\frac{(2-d)\delta+2}{(2+d)\delta+2}}\times\left(\frac{(2-d)\delta+2}{4(\delta+1)}\right)(2^{\delta-1}\alpha)^{\frac{4(\delta+1)}{(2-d)\delta+2}}$. Combining the above estimates and using the calculations similar to [\[26\]](#26){reference-type="eqref" reference="26"}, we deduce $$\begin{aligned} &\frac{1}{2}\|u_h(t_k)-u_h^k\|_{\mathrm{L}^2}^2-\frac{1}{2}\|u_h(t_{k-1})-u_h^{k-1}\|_{\mathrm{L}^2}^2+\frac{\nu}{2} \Delta t \|\nabla (u_h(t_k) -u_h^k)\|_{\mathrm{L}^2}^2\\&\quad+ \Delta t \eta \left(\sum_{j=1}^{k}\omega_{kj}\Delta t \nabla (u_h(t_j)-u_h^{j}) ,\nabla(u_h(t_k) -u_h^k)\right) +\frac{\beta\Delta t}{2}\|u_h(t_k)^{\delta}(u_h(t_k)-u_h^k)\|_{\mathrm{L}^2}^2\\&\quad+\frac{\beta\Delta t}{2}\|(u_h^k)^{\delta}(u_h(t_k)-u_h^k)\|_{\mathrm{L}^2}^2+\beta\gamma\Delta t\|(u_h(t_k)-u_h^k)\|_{\mathrm{L}^2}^2\\& \leq\frac{2\alpha^2(\Delta t)^2}{\nu}\int_{t_{k-1}}^{t_k} \| \partial_t \nabla u_h(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s + \frac{(\Delta t)^2\beta^2\gamma^2}{\nu} \int_{t_{k-1}}^{t_k} \|\partial_tu_h(s)\|_{\mathrm{L}^2}^2 \mathrm{d}s \\&\quad+\Delta t\left(C(\alpha,\nu)\left(\|u_h(t_k)\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}+\|u_h^k\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}\right)+\frac{\beta}{2}2^{2\delta}(1+\gamma)^2(\delta+1)^2 + \frac{3\nu}{8}\right)\|u_h(t_k)-u_h^k\|_{\mathrm{L}^2}^2\\&\quad +\frac{(\Delta t)^2}{\nu} \left(2\alpha^2 + \beta^2(1+\gamma)^2(\delta+1)^2\right)\int_{t_{k-1}}^{t_k} \|u_h^{\delta}(s)\partial_tu_h(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s + \sum_{j=1}^k \frac{2\eta^2T(\Delta t)^3}{\nu}\overline{K}_{kj}^2\int_{t_{j-1}}^{t_j}\|\partial_t\nabla u_h(\tau)\|_{\mathrm{L}^2}^2\mathrm{~d}\tau. \end{aligned}$$ Summing over all $k = 1,2,\ldots, N$ , and using the positivity of the kernel (cf. [\[pkd\]](#pkd){reference-type="eqref" reference="pkd"}), we attain $$\begin{aligned} &\|u_h(t_N)-u_h^N\|_{\mathrm{L}^2}^2+\sum_{k=1}^N\nu \Delta t \|\nabla (u_h(t_k) -u_h^k)\|_{\mathrm{L}^2}^2\nonumber\\& \leq\frac{4\alpha^2(\Delta t)^2}{\nu}\int_{0}^{T} \| \partial_t \nabla u_h(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s +\frac{2(\Delta t)^2\beta^2\gamma^2}{\nu} \int_{0}^{T} \|\partial_tu_h(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s \nonumber \\&\quad+2\Delta t \sum_{k=1}^N\left(C(\alpha,\nu)\left(\|u_h(t_k)\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}+\|u_h^k\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}\right)+\frac{\beta}{2}2^{2\delta}(1+\gamma)^2(\delta+1)^2 + \frac{3\nu}{8}\right)\|u_h(t_k)-u_h^k\|_{\mathrm{L}^2}^2\nonumber\\&\quad+\frac{2(\Delta t)^2}{\nu} \left(2\alpha^2 + C \beta^2(1+\gamma)^2(\delta+1)^2\right)\int_{0}^{T} \|u_h(s)^{\delta}\partial_tu_h(s)\|_{\mathrm{L}^2}^2\mathrm{~d}s\nonumber \\&\quad + \frac{4\eta^2T(\Delta t)^3}{\nu}\sum_{k=1}^N\sum_{j=1}^k \overline{K}_{kj}^2\int_{t_{j-1}}^{t_j}\|\partial_t\nabla u_h(\tau)\|_{\mathrm{L}^2}^2\mathrm{~d}\tau.\label{329} \end{aligned}$$ For the memory term, we get $$\begin{aligned} T(\Delta t)^3\sum_{k=1}^N\sum_{j=1}^k \overline{K}_{kj}^2\int_{t_{j-1}}^{t_j}\|\partial_t\nabla u_h(\tau)\|_{\mathrm{L}^2}^2\mathrm{~d}\tau & \leq \sup_{k,j} \overline{K}_{kj}^2 T(\Delta t)^3\sum_{k=1}^N\sum_{j=1}^k \int_{t_{j-1}}^{t_j}\|\partial_t\nabla u_h(\tau)\|_{\mathrm{L}^2}^2\mathrm{~d}\tau\\& \leq \sup_{k,j}\overline{K}_{kj}^2 T^2(\Delta t)^2 \int_{0}^{T}\|\partial_t\nabla u_h(\tau)\|_{\mathrm{L}^2}^2\mathrm{~d}\tau. \end{aligned}$$ Using the above estimate and an application of Gronwall inequality in [\[329\]](#329){reference-type="eqref" reference="329"} yields $$\begin{aligned} \label{fdest} &\nonumber\|u_h(t_N)-u_h^N\|_{\mathrm{L}^2}^2+\sum_{k=1}^N\nu \Delta t \|\nabla (u_h(t_k) -u_h^k)\|_{\mathrm{L}^2}^2\\&\nonumber \leq C(\Delta t)^2\left(\int_{0}^{T} \| \partial_t \nabla u_h(s)\|_{\mathrm{L}^2}^2 \mathrm{~d}s +\int_{0}^{T} \|u_h(s)^{\delta}\partial_tu_h(s)\|_{\mathrm{L}^2}^2+\frac{4\eta^2T^2}{\nu}\sup_{k,j} \overline{K}_{kj}^2 \int_{0}^{T} \|\partial_tu_h(s)\|_{\mathrm{L}^2}^2 \mathrm{d}s \right)\\&\quad \times \exp\left\{2\Delta t\left[C(\alpha,\nu)\left(\|u_h(t_k)\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}+\|u_h^k\|_{\mathrm{L}^{2(\delta+1)}}^{\frac{4\delta(\delta+1)}{(2-d)\delta+2}}\right)+\frac{\beta}{2}2^{2\delta}(1+\gamma)^2(\delta+1)^2 + \frac{3\nu}{8}\right]\right\}. \end{aligned}$$ One can prove the estimates obtained in [\[2M.68\]](#2M.68){reference-type="eqref" reference="2M.68"} for the semi-discrete solution $u_h$ also, so that the right hand side of [\[fdest\]](#fdest){reference-type="eqref" reference="fdest"} is bounded (see Remark [\[rem3.2\]](#rem3.2){reference-type="ref" reference="rem3.2"} below). **Step 2:** *Estimate for any $t\in [t_{k-1},t_k]$.* Let us now introduce a linear interpolation $\mathcal{I} u_h$ for the semi-discrete solution $u_h$ \[[@YGU] (Section 3.1)\] by: $$\mathcal{I} u_h(t) = u_h(t_{k-1})+\left(\frac{t-t_{k-1}}{\Delta t}\right)(u_h(t_{k})-u_h(t_{k-1})) \qquad \text{for } t\in [t_{k-1},t_k].$$ Let us write $u_h-u_{kh} = u_h -\mathcal{I}u_h + \mathcal{I}u_h -u_{kh}$, where $\mathcal{I}u_h$ is the linear interpolation as defined above. Using the triangle inequality, we have $$\begin{aligned} \|u_h - u_{kh}\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2 \leq 2 \|u_h-\mathcal{I}u_h\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2 +2 \|\mathcal{I}u_h -u_{kh}\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2. \end{aligned}$$ The first term can be estimated using (Lemma 3.2 \[[@YGU]\]) and the final term using [\[fdest\]](#fdest){reference-type="eqref" reference="fdest"} as $$\begin{aligned} \label{sim} &\nonumber\|u_h-\mathcal{I} u_h\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2 = \sum_{i=1}^N\int_{t_{i-1}}^{t_i}\|u_h(t)-\mathcal{I} u_h(t)\|_{\mathrm{H}_0^1(\Omega)}^2 \mathrm{d}t\leq C(\Delta t )^2 \int_{0}^{T}\left\|\partial_tu_h(t)\right\|_{\mathrm{H}_0^1}^2 \mathrm{d}t,\\& \|\mathcal{I}u_h -u_{kh}\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2 = \sum_{i=1}^N\int_{t_{i-1}}^{t_i} \|\mathcal{I}u_h(t) -u_{kh}(t)\|_{\mathrm{H}_0^1(\Omega)}^2\mathrm{d}t \leq C \sum_{i=1}^N\Delta t \|u_h(t_i) -u_h^i\|_{\mathrm{H}_0^1}^2. \end{aligned}$$ Again using triangle inequality, we have $$\begin{aligned} \|u - u_{kh}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 \leq 2 \|u_h-\mathcal{I}u_h\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 + 2\|\mathcal{I}u_h -u_{kh}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2. \end{aligned}$$ Similarly as in [\[sim\]](#sim){reference-type="eqref" reference="sim"}, by using (Corollary 3.1 \[[@YGU]\]) and the estimate [\[fdest\]](#fdest){reference-type="eqref" reference="fdest"}, we deduce $$\begin{aligned} &\|u_h-\mathcal{I} u_h\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 \leq\sup_{1\leq i \leq N}\Big(\sup_{t_{i-1}\leq t\leq t_{i}}\|u_h-\mathcal{I} u_h\|^2_{\mathrm{L}^2(\Omega)}\Big) \leq C(\Delta t )^2 \|u_h\|_{W^{1,\infty}(0,T ; \mathrm{L}^2(\Omega))}^2, \\& \|\mathcal{I}u_h -u_{kh}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 \leq\sup_{1\leq i \leq N}\Big(\sup_{t_{i-1}\leq t\leq t_{i}}\|\mathcal{I}u_h -u_{kh}\|^2_{\mathrm{L}^2(\Omega)}\Big) \leq C\sup_{1\leq i \leq N}\Big(\|u_h(t_i) -u_h^i\|^2_{\mathrm{L}^2(\Omega)}\Big), \end{aligned}$$ and the required result follows. ◻ The main result of this section is [\[thm7.3\]]{#thm7.3 label="thm7.3"} Assume that $u _0\in \mathrm{H}^2(\Omega)\cap \mathrm{H}_0^1(\Omega)$ and $f\in\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))$ such that $u\in\mathrm{L}^{\infty}(0,T;\mathrm{H}^2(\Omega)),$ $u_t\in \mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))\cap \mathrm{L}^{2}(0,T;\mathrm{H}_0^1(\Omega))$ be the solution of [\[weaksolution\]](#weaksolution){reference-type="eqref" reference="weaksolution"} on the interval $(0,T]$. Then, the finite element approximation $u_{kh}$ converges to $u$ as $\Delta t, h\rightarrow 0$. In addition, there exists a constant $C\geq0$ such that the approximation $u_{kh}$ satisfies the following error estimate: $$\begin{aligned} % \label{fdes} & \|u- u_{kh}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 + \|u - u_{kh}\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2\nonumber\\&\leq \eta^2(\Delta t)^2\sup_{k,j} \overline{K}_{kj}^2(\|f\|^2_{\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))} + \|u_0\|^2_{\mathrm{H}^2})+ C((\Delta t)^2 + h^2)(\|f\|^2_{\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))} + \|u_0\|^2_{\mathrm{H}^2}). \end{aligned}$$ *Proof.* The proof is an application of triangle inequality, Theorem [\[thm7.1\]](#thm7.1){reference-type="ref" reference="thm7.1"} and Lemma [\[thm7.2\]](#thm7.2){reference-type="ref" reference="thm7.2"}. ◻ [\[rem3.2\]]{#rem3.2 label="rem3.2"} 1. In the literature most of the estimates needs higher regularity $u_{tt}\in \mathrm{L}^1(0,T;\mathrm{L}^2(\Omega))$ but we have proved estimates under the minimal regularity assumptions. 2. For the generalised Burgers- Huxley equation without memory (that is, $\eta =0$), we have obtained the optimal rate of convergence. Also, if we assume that $K(\cdot) \in \mathrm{L}^{\infty}(0,T),$ then $\overline{K}_{kj}^2 \leq C$, so that we obtain the optimal convergence again. 3. As proved in [@MTW] and [@MMu], if we assume $|K(t-s) |\leq |t-s|^{-\alpha}$, for $0<\alpha<1,$ then we have $\overline{K}_{kj}^2 \leq O((\Delta t)^{-2\alpha})$ and in that case the error estimates are given as $$\begin{aligned} \|u- u_{kh}\|_{\mathrm{L}^{\infty}(0,T;\mathrm{L}^2(\Omega))}^2 + \|u - u_{kh}\|_{\mathrm{L}^2(0,T;\mathrm{H}_0^1(\Omega))}^2\leq C((\Delta t)^{2(1-\alpha)}+h^2)(\|f\|^2_{\mathrm{H}^1(0,T;\mathrm{L}^2(\Omega))} + \|u_0\|^2_{\mathrm{H}^2}). \end{aligned}$$ For $\alpha = \frac{1}{2}$, we have sub-optimal convergence rate and as $\alpha\rightarrow0,$ the convergence tends to an optimal one. 4. If we assume $K(\cdot)\in \mathrm{L}^{p}(0,T)$, then we have $\overline{K}_{kj}^2\leq (\Delta t)^{\frac{2-2q}{q}}$, where $\frac{1}{p}+ \frac{1}{q} =1$. Therefore, the error estimates are of order $((\Delta t)^{\frac{2}{q}} + h^2).$ For $p = q =2$, that is, $K(\cdot)\in \mathrm{L}^2(0,T),$ then we have the error estimates are of order $(\Delta t + h^2)$. 5. If we use the time discretization with non uniform refinement as done in [@MTW] and [@MMu], we can get the optimal convergence rate. # Numerical studies {#sec4} In this section, we perform numerical computations to verify the theoretical results obtained in Theorem [\[thm7.3\]](#thm7.3){reference-type="ref" reference="thm7.3"}. The results are obtained in the open-source finite element library FEniCS [@ABJ; @LHL]. The error tables shown below present the error estimates and order of convergence for the generalized Burgers-Huxley equation (GBHE) with kernel $(\eta \neq 0)$ and without Kernel $(\eta=0)$ for the equation $$\begin{aligned} \label{1} \frac{\partial u}{\partial t}+\alpha u^{\delta}\sum_{i=1}^d\frac{\partial u}{\partial x_{i}}-\nu\Delta u-\eta \int_{0}^{t} K(t-s)\Delta u(s)\mathrm{~d}s=\beta u(1-u^{\delta})(u^{\delta}-\gamma)+f, \end{aligned}$$ under the following norm $$|\!|\!|u^k- u^k_h|\!|\!| = \|u_h(t_N)-u_h^N\|_{\mathrm{L}^2}^2+\sum_{k=1}^N\nu \Delta t \|\nabla (u_h(t_k) -u_h^k)\|_{\mathrm{L}^2}^2.$$ In all the experiments, we are using a backward Euler discretization in time and FEM in space. Consider the temporal discretization $t_k=k\Delta t$, for a given $M$, $\Delta t=\frac{T}{M}$, $t_k=k\Delta t$ and $h$ is the spatial discretization parameter. We choose $\Delta t \propto h$ and the rate of convergence is given by $r = \log(\|u- u^1_h\|/\|u - u_h^2\|) /\log(h^1/h^2)$. ## Accuracy verification for the smooth kernel Consider the problem [\[1\]](#1){reference-type="eqref" reference="1"} defined on the domain $(0,1)^d$. Table [1](#table1){reference-type="ref" reference="table1"} represents the errors and convergence rates for the numerical solution $u_h$ for the smooth kernel $K(t) = e^{-\delta t},$ $t\in[0,T]$ with the exact solution $u = \Pi_{i=1}^{d}e^{-t}\sin(\pi x_i),$ where we have used positive quadrature rule for the memory term. We choose the value of the parameters as follows: $\alpha = \beta = \delta = \nu=1,$ and $\gamma= 0.5$, the forcing term has been calculated using the closed-form solution. We are using first-order polynomials to approximate the numerical solution for all the stimulations. Table [1](#table1){reference-type="ref" reference="table1"} contains the error history for a sequence of refined uniform meshes for the numerical solution constructed using CFEM. The convergence results for both the GBHE with and without memory are represented in the table for both 2D and 3D. In the table, we can observe that the error in the energy norm decreases with the mesh size at the rate $O(h)$. We require at most three Newton iterations and three linear solver iterations to reach the prescribed tolerance of $10^{-10}$. Error history in 2D --------------------- ------------------------- ------------------------------------------- -------- -------------------------------------------- -------- -- -- CGFEM Mesh $|\!|\!|\cdot|\!|\!|$-error for $\eta =0$ $O(h)$ $|\!|\!|\cdot|\!|\!|$-error for $\eta = 1$ $O(h)$ $4\times 4$ $7.21(-01)$ $-$ $6.77(-01)$ $-$ $8\times 8$ $3.54(-01)$ $1.03$ $3.28(-01)$ $1.04$ $16\times 16$ $1.74(-01)$ $1.02$ $1.60(-01)$ $1.04$ $32\times 32$ $8.62(-02)$ $1.01$ $7.90(-02)$ $1.02$ $64\times 64$ $4.29(-02)$ $1.01$ $3.92(-02)$ $1.01$ $128\times 128$ $2.14(-02)$ $1.00$ $1.95(-02)$ $1.00$ CGFEM Mesh $|\!|\!|\cdot|\!|\!|$-error for $\eta =0$ $O(h)$ $|\!|\!|\cdot|\!|\!|$-error for $\eta = 1$ $O(h)$ $2\times 2 \times 2$ $1.35(00)$ $-$ $1.33(00)$ $-$ $4\times 4 \times4$ $7.40(-01)$ $0.86$ $7.13(-01)$ $0.90$ $8\times 8 \times 8$ $3.71(-01)$ $1.00$ $3.53(-01)$ $1.01$ $16\times 16 \times16$ $1.83(-01)$ $1.02$ $1.73(-01)$ $1.03$ $32\times 32 \times 32$ $9.06(-02)$ $1.01$ $8.55(-02)$ $1.02$ $64\times 64 \times 64$ $4.50(-02)$ $1.01$ $4.24(-02)$ $1.01$ : Errors and convergence rates for the numerical solutions $u_h^k$ for the smooth function $K(t)= e^{-\delta t}$ and the given solution $u = \Pi_{i=1}^{d}e^{-t}\sin(\pi x_i).$ ## Weakly singular kernel Consider the problem ([\[1\]](#1){reference-type="ref" reference="1"}) defined on the domain $(0,1)^d$. We shall approximate $\psi$ in $J_k(\psi) = \int_0^{t_n} K(t_n-s)\psi(s) \mathrm{d}s$ by the piecewise constant function taking value $\psi(t_j)$ in $(t_{j-1},t_{j})$. We have obtained the errors and convergence rates for the numerical solution $u^k_h$ for the weakly singular kernel $K(t) =\frac{1}{\sqrt{t}}$ with two different expressions of the exact solution $$\begin{aligned} \text{Case 1 : }u = (t^3-t^2+1)\prod\limits_{i=1}^{d}\sin(\pi x_i), \qquad \text{Case 2 : }u = t\sqrt{t}\prod\limits_{i=1}^{d}\sin(\pi x_i). \end{aligned}$$ Error history in 2D --------------------- ------------------------- ------------------------------------------- -------- -------------------------------------------- -------- -- -- CGFEM Mesh $|\!|\!|\cdot|\!|\!|$-error for $\eta =0$ $O(h)$ $|\!|\!|\cdot|\!|\!|$-error for $\eta = 1$ $O(h)$ $4\times 4$ $8.04(-01)$ $-$ $8.05(-01)$ $-$ $8\times 8$ $4.14(-01)$ $0.96$ $4.14(-01)$ $0.96$ $16\times 16$ $2.09(-01)$ $0.99$ $2.09(-01)$ $0.99$ $32\times 32$ $1.05(-01)$ $0.99$ $1.05(-01)$ $0.99$ $64\times 64$ $5.26(-02)$ $1.00$ $5.25(-02)$ $1.00$ $128\times 128$ $2.63(-02)$ $1.00$ $2.62(-02)$ $1.00$ CGFEM Mesh $|\!|\!|\cdot|\!|\!|$-error for $\eta =0$ $O(h)$ $|\!|\!|\cdot|\!|\!|$-error for $\eta = 1$ $O(h)$ $2\times 2 \times 2$ $1.48(00)$ $-$ $1.50(00)$ $-$ $4\times 4 \times 4$ $8.62(-01)$ $0.78$ $8.64(-01)$ $0.80$ $8\times 8 \times 8$ $4.51(-01)$ $0.93$ $4.51(-01)$ $0.94$ $16\times 16 \times 16$ $2.23(-01)$ $1.01$ $2.28(-01)$ $0.98$ $32\times 32 \times 32$ $1.15(-01)$ $0.96$ $1.15(-01)$ $0.99$ $64\times 64 \times 64$ $5.74(-02)$ $1.00$ $5.74(-02)$ $1.00$ : Errors and convergence rates for the numerical solutions $u_h^k$ for the weakly singular kernel $K(t)= \frac{1}{\sqrt{t}}$ and the given solution $u = \prod\limits_{i=1}^{d}(t^3-t^2+1)\sin(\pi x_i).$ The parameters are chosen as in [@KMR], that is, $\alpha = \beta = \delta = \nu=1,$ and $\gamma= 0.5$. Table [2](#table2){reference-type="ref" reference="table2"} represents the convergence results related to the Case 1 and Table [3](#table3){reference-type="ref" reference="table3"} corresponds to the Case 2. In both cases, we can observe that the error in the energy norm decreases with the mesh size at the rate $O(h)$. We require at most three Newton iterations and three linear solver iterations to reach the prescribed tolerance of $10^{-10}$. Error history in 2D with weakly singular kernel, $u=t\sqrt{t}\sin(2\pi x)\sin(2\pi y)$ ---------------------------------------------------------------------------------------- ----------------- ------------------------------------------- -------- -------------------------------------------- -------- CGFEM Mesh $|\!|\!|\cdot|\!|\!|$-error for $\eta =0$ $O(h)$ $|\!|\!|\cdot|\!|\!|$-error for $\eta = 1$ $O(h)$ $4\times 4$ $1.44(00)$ $-$ $1.47(00)$ $-$ $8\times 8$ $9.24(-01)$ $0.64$ $9.35(-01)$ $0.65$ $16\times 16$ $5.02(-01)$ $0.88$ $5.06(-01)$ $0.89$ $32\times 32$ $2.59(-01)$ $0.95$ $2.60(-01)$ $0.96$ $64\times 64$ $1.31(-01)$ $0.98$ $1.32(-01)$ $0.99$ $128\times 128$ $6.59(-02)$ $0.99$ $6.61(-02)$ $1.00$ CGFEM Mesh $|\!|\!|\cdot|\!|\!|$-error for $\eta =0$ $O(h)$ $|\!|\!|\cdot|\!|\!|$-error for $\eta = 1$ $O(h)$ $4\times 4$ $1.31(00)$ $-$ $1.32(00)$ $-$ $8\times 8$ $9.20(-01)$ $0.51$ $9.26(-01)$ $0.51$ $16\times 16$ $5.15(-01)$ $0.84$ $5.18(-01)$ $0.84$ $32\times 32$ $2.68(-01)$ $0.94$ $2.61(-01)$ $0.99$ $64\times 64$ $1.36(-01)$ $0.98$ $1.37(-01)$ $0.93$ : Errors and convergence rates for the numerical solutions $u^k_h$ for the weakly singular kernel $K(t)= \frac{1}{\sqrt{t}}$ and the given solution $u = t\sqrt{t}\sin(2\pi x)\sin(2\pi y).$ ## Application to Nerve Pulse Propagation We conclude this section by showing a difference of qualitative illustration between a classical bistable equation with a weakly singular kernel (without advection, $\alpha = 0$ and with a simplified cubic nonlinearity induced by $\delta = 1$) and the generalized Burgers-Huxley equation with a weakly singular kernel. In this study, we perform a straightforward simulation to a transient problem which incorporates an additional ordinary differential equation (ODE). This inclusion allows for the emergence of self-sustained patterns, as discussed in [@DCS]. The system reads $$\begin{aligned} \label{TBH} \nonumber\partial_t u + \alpha u^\delta \sum_{i=1}^d\partial_i u - \nu \Delta u -\eta\int_{0}^{t} K(t-s)\Delta u(s) \mathrm{~d}s &= \beta u (1-u^\delta)(u^\delta-\gamma) -v ,\\\partial_t v &= \varepsilon(u-\rho v), \end{aligned}$$ where $\varepsilon$ and $\rho$ are parameters that can change the rest state and dynamics. Setting $\delta = 1$, $\eta =0$ and $\alpha = 0$, one recovers the well-known FitzHugh-Nagumo equations $$\partial_t u - \nu \Delta u - \beta u (1-u)(u-\gamma) + v = 0, \qquad \partial_t v = \varepsilon(u-\rho v).$$ To obtain the weak formulation similar to [\[fd1\]](#fd1){reference-type="eqref" reference="fd1"} for the CFEM, we apply a simple backward Euler time discretization with constant time step $\Delta t = 0.2$ and other parameters similar to [@KMR]. For this example, we prescribe Neumann boundary conditions for $u$ on $\partial\Omega$. Figure [9](#fig:ex3){reference-type="ref" reference="fig:ex3"} depicts three snapshots of the evolution of $u$ (representing the action potential propagation in a piece of nerve tissue, cardiac muscle, or any excitable media) for the classical FitzHugh-Nagumo system with weakly singular kernel vs. the modified delayed generalized Burgers-Huxley system with weakly singular kernel [\[TBH\]](#TBH){reference-type="eqref" reference="TBH"}. All numerical solutions computed using the CFEM setting. The difference in the spiral dynamics appears to be more responsive to the degree of non-linearity (governed by $\delta$) rather than the additional advection term (influenced by $\alpha$). {#fig:ex3 width="32.5%"} {#fig:ex3 width="32.5%"} {#fig:ex3 width="32.5%"}\ {#fig:ex3 width="32.5%"} {#fig:ex3 width="32.5%"} {#fig:ex3 width="32.5%"}\ {#fig:ex3 width="32.5%"} {#fig:ex3 width="32.5%"} {#fig:ex3 width="32.5%"} # Acknowledgement The first author would like to thank Ministry of Education, Government of India (Prime Minister Research Fellowship, PMRF ID : 2801816), for financial support to carry out his research work. [^1]: Email: sumit\_m\@ma.iitr.ac.in [^2]: Corresponding author. Email: arbaz\@ma.iitr.ac.in [^3]: Email: maniltmohan\@ma.iitr.ac.in, maniltmohan\@gmail.com.

arxiv_math

--- abstract: | We discuss in this note the $\mathbb{C}$-algebra $H^0(X,\mathsf{S}^{\scriptscriptstyle\bullet}T_X)$ for a smooth complex projective variety $X$. We compute it in some simple examples and survey a few cases where it is trivial. address: | Université Côte d'Azur\ CNRS -- Laboratoire J.-A. Dieudonné\ Parc Valrose\ F-06108 Nice cedex 2, France author: - Arnaud Beauville title: The algebra of symmetric tensors on smooth projective varieties --- [^1] # Introduction Let $X$ be a smooth complex projective variety. We will denote by $T^*X$ its cotangent bundle, and by $\mathbb{P}T^*X$ its projectivization. In this note we are interested in the graded $\mathbb{C}$-algebra $$S(X):=\bigoplus_{p\geq 0}H^0(X,\mathsf{S}^pT_X)= \mathscr{O}(T^*X)=\bigoplus_{p\geq 0} H^0(\mathbb{P}T^*X,\mathscr{O}_{\mathbb{P}T^*X}(p))\,.$$ ($\mathscr{O}(T^*X)$ is the algebra of regular functions on $T^*X$, with the grading defined by the linear action of $\mathbb{C}^*$.) Despite its simple definition, this is an intriguing object, which is usually quite complicated, even for a variety as simple as the quadric (see Proposition [Proposition 3](#quadric){reference-type="ref" reference="quadric"} below). We describe it in some particular cases in §2. Then we survey the (frequent) cases where it is known to be trivial (§3), and we discuss briefly its Krull dimension (§4). **Notations**: We work over the complex numbers. If $X$ is a variety endowed with an action of $\mathbb{C}^*$, we denote by $\mathscr{O}(X)$ the $\mathbb{C}$-algebra of regular functions on $X$, with the grading defined by the $\mathbb{C}^*$-action. By a vector space we mean a complex, finite-dimensional vector space. # Some examples ## Abelian varieties We start with a trivial case: if $X$ is an abelian variety of dimension $n$, we have $T_X\cong \mathscr{O}_X^n$, hence $S(X)$ is a polynomial algebra in $n$ variables. ## Projective space Let $V$ be a vector space. We let $I\in V\otimes V^*$ be the image of the identity by the isomorphism $\operatorname{End}(V)\vbox{\hbox to .8cm{\hfill{$\scriptstyle\sim$}\hfill} \nointerlineskip\hbox to .8cm{{\hfill$\longrightarrow $\hfill}} }V\otimes V^*$. **Proposition 1**. *The graded algebra $S(\mathbb{P}(V))$ is isomorphic to the quotient of $\displaystyle\bigoplus_{d\geq 0}\bigl(\mathsf{S}^dV\otimes \mathsf{S}^dV^*\bigr)$ by the ideal generated by $I$.* *Proof* : The projective cotangent bundle $\mathbb{P}T^*_{\mathbb{P}(V)}$ can be identified with the incidence hypersurface $Z\subset \mathbb{P}(V)\times \mathbb{P}(V^*)$ consisting of pairs $(x,H)$ with $x\in H$; the tautological line bundle $\mathscr{O}_{Z}(1)$ is induced by $\mathscr{O}_{\mathbb{P}(V)}(1)\boxtimes \mathscr{O}_{\mathbb{P}(V^*)}(1)$. The Proposition follows from the exact sequence $$0\rightarrow \mathscr{O}_{\mathbb{P}(V)}(d-1)\boxtimes \mathscr{O}_{\mathbb{P}(V^*)}(d-1)\xrightarrow{\ \times I\ }\mathscr{O}_{\mathbb{P}(V)}(d)\boxtimes \mathscr{O}_{\mathbb{P}(V^*)}(d)\rightarrow \mathscr{O}_Z(d)\rightarrow 0\,.\qed$$ ## Rational homogeneous manifolds In this section we will use some general facts about nilpotent orbits, which can be found for example in [@Fu]. Let $X=G/P$, where $G$ is a reductive algebraic group and $P$ a parabolic subgroup. We denote by $\mathfrak{g}$ and $\mathfrak{p}$ their Lie algebras, and by $\mathfrak{n}$ the nilradical of $\mathfrak{p}$. The Killing form of $\mathfrak{g}$ provides an isomorphism of $G$-modules $\mathfrak{n}\vbox{\hbox to .8cm{\hfill{$\scriptstyle\sim$}\hfill} \nointerlineskip\hbox to .8cm{{\hfill$\longrightarrow $\hfill}} }(\mathfrak{g}/\mathfrak{p})^*$; using this we identify the cotangent bundle $T^*(G/P)$ to the homogeneous bundle $G\times ^P\mathfrak{n}$. Associating to a pair $(g,N)$ in $G\times \mathfrak{n}$ the element $\operatorname{Ad}(g)\cdot N$ of $\mathfrak{g}$ defines a generically finite, $\mathbb{C}^*$-equivariant map $\pi :T^*(G/P)\rightarrow \mathfrak{g}$, whose image $\mathscr{N}$ is the closure of a nilpotent orbit. We will consider the case where the induced map $\bar{\pi }:T^*(G/P)\rightarrow \mathscr{N}$ is birational. In this case $\bar{\pi }$ is a resolution of the normalization $\tilde{\mathscr{N}}$ of $\mathscr{N}$, and we have $S(X)=\mathscr{O}(\tilde{\mathscr{N} })$. For $G=\operatorname{GL}(n)$ all parabolic subgroups have this property, and $\mathscr{N}$ is normal, so $S(X)=\mathscr{O}( \mathscr{N} )$. In the other classical cases there is a precise description of the parabolic subgroups for which $\bar{\pi}$ is birational [@Fu 3.3]; we will content ourselves with the example of quadrics. ## Flag varieties, Grassmannians Let $V$ be a vector space, and let $(0)=V_0\subset V_1\subset\allowbreak\ldots \subset V_{s+1}=V$ be a (partial) flag in $V$. The stabilizer $P$ of this flag is a parabolic subgroup of $\operatorname{GL}(V)$, and all parabolics are obtained in this way. The variety $G/P$ is the variety of flags $(0)=F_0\subset F_1\subset\ldots \subset F_{s+1}=V$ with $\dim F_i=\dim V_{i}$. The Lie algebra $\mathfrak{p}$ is the stabilizer of $(V_i)$ in $\operatorname{End}(V)$, and its nilradical $\mathfrak{n}$ is the subspace of $u\in \operatorname{End}(V)$ satisfying $u(V_{i+1})\subset V_i$ for $0\leq i\leq s$. Therefore $\mathscr{N}$ is the subvariety of endomorphisms $u\in\operatorname{End}(V)$ for which there exists a flag $(F_i)$ in $G/P$ with $u(F_{i+1})\subset F_i$ for $0\leq i\leq s$. Let us spell out this in the case of the Grassmannian $\mathbb{G}:=\mathbb{G}(r,V)$ of $r$-dimensional subspaces of $V$. We put $n:=\dim V$. **Proposition 2**. *$S(\mathbb{G}(r,V))=\mathscr{O}(\mathscr{N})$, where $\mathscr{N}\subset \operatorname{End}(V)$ is the subvariety of endomorphisms $u$ satisfying $u^2=0$ and $\operatorname{rk} u\leq \min\{r,n-r\}$.* *Proof* : Since $\mathbb{G}(r,V)\cong \mathbb{G}(n-r,V)$, we can assume $r\leq n/2$. By the previous discussion, $\mathscr{N}$ consists of endomorphisms $u$ for which there exists a $r$-dimensional subspace $W\subset V$ with $u(V)\subset W$ and $u(W)=0$, that is, $\operatorname{Im}u\subset W\subset\operatorname{Ker}u$. This implies $u^2=0$ and $\operatorname{rk}u\leq r$; conversely, if this is satisfied, we have $\operatorname{Im}u\subset \operatorname{Ker}u$ and $\dim\operatorname{Ker}u=n-\operatorname{rk}u\geq n-r\geq r$, so any $r$-dimensional subspace $W$ with $\operatorname{Im}u\subset W\subset\operatorname{Ker}u$ does the job.0◻ *Remarks*$.-$ 1) Taking $r=1$ we recover Proposition [Proposition 1](#proj){reference-type="ref" reference="proj"}. ) If $r=\lfloor\frac{n}{2} \rfloor$ the condition $u^2=0$ implies $\operatorname{rk}u \leq r$, so $\mathscr{N}$ is simply the variety of square zero endomorphisms of $V$. ## Quadrics Let $V$ be a vector space, and let $q$ be a non-degenerate quadratic form on $V$, defining a quadric $Q:=V(q)$ in $\mathbb{P}(V)$. **Proposition 3**. *$S(Q)$ is isomorphic to the quotient of the homogeneous coordinate ring of $\mathbb{G}(2,V)\subset \mathbb{P}(\bigwedge^2V)$ by the ideal generated by $\wedge^2q$.* *Proof* : Let $\ell$ be an isotropic line in $V$ and let $P$ be the stabilizer of $\ell$, so that $Q= \operatorname{O}(V)/P$. The Lie algebra $\mathfrak{o}(V)$ consists of endomorphisms of $V$ which are skew-symmetric (with respect to $q$), and $\mathfrak{p}$ is the stabilizer of $\ell$ in $\mathfrak{o}(V)$. The nilradical $\mathfrak{n}$ of $\mathfrak{p}$ consists of skew-symmetric endomorphisms $u$ such that $u(\ell^{\perp})\subset \ell$ and $u(\ell)=0$. Such a map is of the form $$\label{map} x\,\mapsto\, q(w,x)v-q(v,x)w, \mbox{ where } v\in\ell \mbox{ and } w\in \ell^{\perp} \,.$$ Varying $\ell$, we see that $\mathscr{N}$ consists of the maps of the form ([\[map\]](#map){reference-type="ref" reference="map"}) such that the restriction of $q$ to $\langle {v,w} \rangle$ has rank $\leq 1$. Such maps correspond bijectively to decomposable bivectors $v\wedge w\in \bigwedge^2V$, and the condition on $q$ can be written $\wedge^2q(v\wedge w)=0$. This implies the Proposition.0◻ ## Intersection of two quadrics The following result is proved in [@BEHLV]: **Proposition 4**. *Let $X\subset \mathbb{P}^{n+2}$ be a smooth complete intersection of two quadrics. Then $S(X)$ is a polynomial algebra in $n$ variables of degree $2$.* It is somewhat surprising that the answer is much simpler in this case that for a single quadric. ## Completely integrable systems Let $V$ be a graded vector space, endowed with the associated $\mathbb{C}^*$-action. Suppose that we have a $\mathbb{C}^*$-equivariant morphism $\Phi :T^*X\rightarrow V$ whose general fiber is of the form $Y\smallsetminus Z$, where $Y$ is a complete variety and $Z$ a closed subvariety of codimension $\geq 2$. Then the functions on $T^*X$ are constant on the fibers of $\Phi$, hence the homomorphism $\Phi ^*: \mathscr{O}(V)=\mathsf{S}^{\scriptscriptstyle\bullet}V^*\rightarrow \mathscr{O}(T^*X)=S(X)$ is an isomorphism of graded algebras. A famous example of this situation is given by the *Hitchin fibration* [@Hi]. Let $C$ be a curve of genus $g\geq 2$. We fix coprime integers $r,d\geq 1$, and consider the moduli space $\mathscr{M}$ of stable vector bundles on $C$ of rank $r$ and degree $d$. It is a smooth projective variety. By deformation theory the tangent space $T_E(\mathscr{M})$ at a point $E$ of $\mathscr{M}$ identifies with $H^1(C,\mathscr{E}nd(E))$; by Serre duality, its dual $T^*_E\mathscr{M}$ identifies with $\operatorname{Hom}(E,E\otimes K_C)$. Let $V$ be the graded vector space $\bigoplus_{i=1}^rH^0(C, K_C^{i})$ (with $\deg H^0(C, K_C^{i})=i$). For $u\in \operatorname{Hom}(E,E\otimes K_C)$, we have $\operatorname{Tr}\wedge^i u\in H^0(C,K^i_C)$. Associating to $u$ the vector $\operatorname{Tr}u+\ldots +\operatorname{Tr}\wedge^r$ gives a $\mathbb{C}^*$-equivariant map $\Phi :T^*\mathscr{M}\rightarrow V$. **Proposition 5**. *The homomorphism $\Phi^*:\mathscr{O}(V)=\mathsf{S}^{\scriptscriptstyle\bullet}V^*\rightarrow \mathscr{O}(T^*\mathscr{M})=S(\mathscr{M})$ is an isomorphism.* *Proof* : $T^*\mathscr{M}$ admits an open embedding into the moduli space $\mathscr{H}$ of stable Higgs bundles (of rank $r$ and degree $d$), and $\Phi$ extends to a proper map $\bar{\Phi }:\mathscr{H}\rightarrow V$ [@Hi]. The codimension of $\mathscr{H}\smallsetminus T^*X$ is $\geq 2$ [@Fa Theorem II.6], hence $\operatorname{codim}(\bar{\Phi }^{-1}(v)\smallsetminus\Phi^{-1}(v))\geq 2$ for $v$ general in $V$. By the previous remarks this implies the result.0◻ There are a number of variations on this theme. First of all, one can fix a line bundle $L$ of degree $d$ on $X$ and consider the subspace $\mathscr{M}_L$ of $\mathscr{M}$ parameterizing the vector bundles $E$ with $\det E=L$; then $\Phi$ maps $\mathscr{M}_L$ onto the graded subspace $V_0:= \bigoplus_{i=2}^rH^0(C, K_C^{i})$ of $V$, and we get as before an isomorphism of $S(\mathscr{M}_L)$ with $\mathsf{S}^{\scriptscriptstyle\bullet}V_0^*$. Note that in the case $g=r=2$ $\mathscr{M}_L$ is a complete intersection of two quadrics in $\mathbb{P}^5$, so we recover the case $n=3$ of Proposition [Proposition 4](#2Q){reference-type="ref" reference="2Q"}. We can also consider the moduli space $\mathscr{M}_{\operatorname{par} }$ of stable parabolic vector bundles on $C$ of rank $r$, degree $d$ and weights $\alpha$, with a parabolic structure along a divisor $D=p_1+\ldots +p_s$ --- we refer for instance to [@BGL] for the precise definitions. For generic weights $\mathscr{M}_{\operatorname{par} }$ is smooth and projective; the Hitchin map $\Phi : T^*\mathscr{M}_{\operatorname{par} }\rightarrow V_{\operatorname{par} }$ takes its values in the vector space $V_{\operatorname{par} }:=\bigoplus_{i=1}^r H^0(C,K_C((i-1)D))$. It extends to a proper map from the moduli space $\mathscr{H}_{\operatorname{par} }$ of parabolic Higgs bundle to $V_{\operatorname{par} }$, and $\mathscr{M}_{\operatorname{par} }\smallsetminus T^*\mathscr{M}_{\operatorname{par} }$ has codimension $\geq 2$ provided $g\geq 4$, or $g=3$ and $r\geq 3$, or $g=2$ and $r\geq 5$ [@BGL Proposition 5.10]. If this holds, we get as before an isomorphism $\mathsf{S}^{\scriptscriptstyle\bullet}V_{\operatorname{par} }^*\vbox{\hbox to .8cm{\hfill{$\scriptstyle\sim$}\hfill} \nointerlineskip\hbox to .8cm{{\hfill$\longrightarrow $\hfill}} }S(\mathscr{M}_{\operatorname{par} })$. ## An example: ruled surfaces Contrary to what the previous examples might suggest, $S(X)$ is *not* invariant under deformation of $X$; a typical example is provided by ruled surfaces. Let $C$ be a curve of genus $\geq 2$, and $E$ a stable rank 2 vector bundle on $C$ with trivial determinant[^2]. We put $X=\mathbb{P}_C(E)$. **Proposition 6**. *For general $E$ we have $S(X)=\mathbb{C}$.* *Proof* : Denote by $p:X\rightarrow C$ the structure map and by $\mathscr{O}_X(1)$ the tautological line bundle. The exact sequence $$0\rightarrow \mathscr{O}_X(2)\rightarrow T_X\rightarrow p^*T_C\rightarrow 0\,.$$ gives rise to exact sequences $$\label{ex} 0\rightarrow \mathscr{O}_X(2p)\rightarrow \mathsf{S}^pT_X\rightarrow \mathsf{S}^{p-1}T_X\otimes p^*T_C \rightarrow 0\,.$$ We claim that $H^0(X,\mathsf{S}^{p-1}T_X\otimes p^*T_C)=0$. Indeed we get from ([\[ex\]](#ex){reference-type="ref" reference="ex"}) exact sequences $$0\rightarrow \mathscr{O}_X(2q)\otimes p^*T_C^{r}\rightarrow \mathsf{S}^qT_X \otimes p^*T_C^{r}\rightarrow \mathsf{S}^{q-1}T_X\otimes p^*T_C^{r+1} \rightarrow 0\,.$$ We have $H^0(X,\mathscr{O}_X(2q)\otimes p^*T_C^{r})=H^0(C, \mathsf{S}^{2q}E\otimes T_C^{r}))=0$ for $r\geq 1$, because $\mathsf{S}^{2q}E$ is semi-stable [@Ha ch. I, Theorem 10.5] and $\deg T_C<0$. Since $H^0(C, T_C^{r+1}))=0$, we get by induction $H^0(X,\mathsf{S}^{q}T_X\otimes p^*T_C)=0$, hence ([\[ex\]](#ex){reference-type="ref" reference="ex"}) gives isomorphisms $$\label{iso} H^0(X,\mathsf{S}^p T_X)\cong H^0(X,\mathscr{O}_X(2p))\cong H^0(C,\mathsf{S}^{2p}E)\,.$$ Now for general $E$ the bundles $\mathsf{S}^qE$ are stable [@Ha *loc. cit.*], so $H^0(X, \mathsf{S}^pT_X)=0$ for $p>0$.0◻ For special bundles $E$ the algebra $S(X)$ can be quite nontrivial. If $E$ is *unstable* the tangent bundle $T_{X}$ is big [@Ki], hence $S(X)$ has Krull dimension $3$. This does not hold if $E$ is stable, but one can get interesting algebras of dimension 2. Let $V$ be a 2-dimensional Hermitian space, and let $G$ be a finite subgroup of $\operatorname{SU}(V)$, acting irreducibly on $V$. Recall that $G$ is the pull-back by the covering map $\operatorname{SU}(2)\rightarrow \operatorname{SO}(3)$ of a group $\bar{G}$ isomorphic to the dihedral group $D_n$ or to $\mathfrak{A}_4,\mathfrak{S}_4$ or $\mathfrak{A}_5$. Given a Galois covering $\pi :\tilde{C}\rightarrow C$ with group $G$, the vector bundle $E_{\pi }:= \tilde{C}\times ^{G}V$ on $C$ is stable, of rank 2, with trivial determinant. The space $H^0(C,\mathsf{S}^pE_{\pi })$ is canonically isomorphic to the $G$-invariant subspace of $\mathsf{S}^pV$. Note that this is zero if $p$ is odd, since $G$ contains the element $-1_V$. Therefore it follows from ([\[iso\]](#iso){reference-type="ref" reference="iso"}) that $S(X)$ *is isomorphic to the graded algebra of invariants* $(\mathsf{S}^{\scriptscriptstyle\bullet}V)^{G}$, the algebra of regular functions on the quotient variety $V/G$. The determination of $(\mathsf{S}^{\scriptscriptstyle\bullet}V)^{G}$ goes back to Klein [@Kl Ch. II]. It is generated by 3 homogeneous elements $x,y,z$, subject to one weighted homogeneous relation $F(x,y,z)=0$. Putting $\mathbf{d}=(\deg x, \deg y, \allowbreak\deg z)$, we have: $\bullet$ For $\bar{G}=D_n$, $\mathbf{d}=(2n+2,2n,4)$, $F=x^2+y^2z+z^{n+1}$. $\bullet$ For $\bar{G}=\mathfrak{A}_4$, $\mathbf{d}=(4,4,6)$, $F=x^2+y^3+z^3$. $\bullet$ For $\bar{G}=\mathfrak{S}_4$, $\mathbf{d}=(12,8,6)$, $F= x^2+y^3+z^4$. $\bullet$ For $\bar{G}=\mathfrak{A}_5$, $\mathbf{d}=(30,20,12)$, $F= x^2+y^3+z^5$. # Cases with $S(X)=\mathbb{C}$ ## Varieties with $c_1(X)=0$ The following result, proved in [@Ko], is a direct consequence of Yau's theorem: **Proposition 7**. *Let $X$ be a compact Kähler variety with $c_1(X)=0$ in $H^2(X,\mathbb{Q})$, and $\pi _1(X)$ finite. Then $S(X)=\mathbb{C}$.* With no assumption on $\pi _1(X)$, we know that $X$ is the quotient of a product $A\times Y$, where $A$ is a complex torus and $Y$ is simply connected, by a finite group $G$ acting freely [@B]. It follows that $S(X)$ *is isomorphic to the invariant subring* $(\mathsf{S}^{\scriptscriptstyle\bullet}T_0(A))^G$. ## Varieties of general type **Proposition 8**. *Let $X$ be a variety of general type. Then $S(X)=\mathbb{C}$.* This is a consequence of the stronger result that $T_X$ is not pseudo-effective [@HP Proposition 4.11]. ## Hypersurfaces The following result is proved in [@HLS]: **Proposition 9**. *Let $X$ be a smooth hypersurface of degree $d\geq 3$ and dimension $\geq 2$. Then $S(X)=0$.* In fact the authors prove the stronger result $H^0(X,\mathsf{S}^p T_X(d-3))=0$, and also that $T_X$ is not pseudoeffective. # The Krull dimension of S(X) A complete description of the ring $S(X)$ is in general intractable, but we can still ask for some of its properties: for instance its Krull dimension. It is equal to $1+\kappa (\mathscr{O}_{\mathbb{P}T^*X}(1))$, where $\kappa$ denotes the *Iitaka dimension* (see for instance [@La Ch. 2]). We have $0\leq \dim S(X)\allowbreak \leq 2\dim X$, and all cases can occur. In particular, $$\dim S(X)=2\dim X\ \Longleftrightarrow \ \mathscr{O}_{\mathbb{P}T^*X}(1)\mbox{ big }\Longleftrightarrow\ T_X\mbox{ big}\,.$$ This property holds for toric varieties [@Hs] and for all rational homogeneous varieties [@GW Corollary 4.4]. The paper [@Li1] contains a number of other examples of varieties with a group action whose tangent bundle is big. Though the most interesting cases occur when the Kodaira dimension $\kappa (X)$ is $-\infty$, one may ask what can be said when $\kappa (X)\geq 0$. The condition $S(X)\neq \mathbb{C}$, or the weaker condition that $T_X$ is pseudo-effective, imposes strong restrictions on $X$ --- see [@HP Proposition 4.11]. In fact, the following result, which I had proposed as a conjecture in the first version of this note, has been proved by J. Liu [@Li2]: **Proposition 10**. *$\dim S(X)\leq \dim X-\kappa (X)$. If equality holds, a finite étale covering of $X$ is isomorphic to $A\times Y$, where $A$ is an abelian variety and $Y$ a variety of general type.* It follows in particular that $\dim S(X)>\dim X$ implies $\kappa (X)=-\infty$. BEHLV A. Beauville : *Some remarks on Kähler manifolds with* $c_1=0$. Classification of algebraic and analytic manifolds, 1-26, Progr. Math. **39**, Birkhäuser Boston, 1983. A. Beauville, A. Etesse, A. Höring, J. Liu, C. Voisin : *Symmetric tensors on the intersection of two quadrics and Lagrangian fibration*. Preprint `arxiv:2304.10929`. I. Biswas, T. Gómez, M. Logares : *Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles*. Canad. J. Math. **68** (2016), no. 3, 504-520. G. Faltings : *Stable $G$-bundles and projective connections*. J. Algebraic Geom. 2 (1993), no. 3, 507-568. B. Fu : *Symplectic resolutions for nilpotent orbits*. Invent. Math. **151** (2003), no. 1, 167-186. D. Greb, M. Wong : *Canonical complex extensions of Kähler manifolds*. J. Lond. Math. Soc. (2) **101** (2020), no. 2, 786-827. R. Hartshorne : *Ample subvarieties of algebraic varieties*. Lecture Notes in Math. **156**. Springer-Verlag, Berlin-New York, 1970. N. Hitchin : *Stable bundles and integrable systems*. Duke Math. J. **54** (1987), no. 1, 91-114. J-C. Hsiao : *A remark on bigness of the tangent bundle of a smooth projective variety and D-simplicity of its section rings*. J. Algebra Appl. **14** (2015), no. 7, 1550098. A. Höring, J. Liu, F. Shao : *Examples of Fano manifolds with non-pseudoeffective tangent bundle*. J. Lond. Math. Soc. (2) **106** (2022), no. 1, 27-59. A. Höring, T. Peternell : *Stein complements in compact Kähler manifolds*. Preprint `arxiv:2111.03303`. J-S. Kim : *Bigness of the tangent bundles of projective bundles over curves*. Preprint `arxiv:2206.15413`. F. Klein : *Lectures on the icosahedron and the solution of equations of the fifth degree*. Dover Publications, Inc., New York, 1956. S. Kobayashi : *Chern class and holomorphic tensor fields*. Nagoya Math. J. 77 (1980), 5-11. R. Lazarsfeld : *Positivity in algebraic geometry I*. Ergebnisse der Math. **48**. Springer-Verlag, Berlin, 2004. J. Liu : *On moment map and bigness of tangent bundles of $G$-varieties*. Alg. Number Th. **17** (2023) 1501-1532. J. Liu : *Krull dimension of symmetric algebras and Beauville's conjecture*. Preprint. [^1]: I am indebted to Feng Shao for several useful comments and references. [^2]: Such a bundle is isomorphic to its dual, so we will not bother to distinguish them.

arxiv_math

--- author: - Matt DeVos Kathryn Nurse Robert Šámal title: Nowhere-zero 8-flows in cyclically 5-edge-connected, flow-admissible signed graphs --- # Introduction

arxiv_math

--- abstract: | Let $\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also, $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this paper, we show that if $\Gamma\in \mathfrak{C}$, then all strongly outer actions of $\Gamma$ on the Razak-Jacelon algebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szabó's result for strongly self-absorbing C$^*$-algebras. address: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Yamadaoka 1-5, Suita, Osaka 565-0871, Japan author: - Norio Nawata title: Strongly outer actions of certain torsion-free amenable groups on the Razak-Jacelon algebra --- [^1] # Introduction Let $\mathcal{W}$ be the Razak-Jacelon algebra studied in [@J] (see also [@Raz]). By classification results in [@CE] and [@EGLN] (see also [@Na4]), $\mathcal{W}$ is the unique simple separable nuclear monotracial $\mathcal{Z}$-stable C$^*$-algebra that is $KK$-equivalent to $\{0\}$. Also, $\mathcal{W}$ is regarded as a stably finite analog of the Cuntz algebra $\mathcal{O}_2$. More generally, we can consider that $\mathcal{W}$ is a non-unital analog of strongly self-absorbing C$^*$-algebras. (Note that every strongly self-absorbing C$^*$-algebra is unital by definition.) In this paper, we study group actions on $\mathcal{W}$ and show an analogous result of Szabó's result in [@Sza7] for group actions on strongly self-absorbing C$^*$-algebras (see also [@IM1], [@IM2], [@IM3], [@M1], [@M2], [@MS1], [@MS2] and [@Sza3] for pioneering works). We refer the reader to [@I] for the importance and some difficulties of studying group actions on C$^*$-algebras. Gabe and Szabó classified outer actions of countable discrete amenable groups on Kirchberg algebras up to cocycle conjugacy in [@GS]. In their classification, $\mathcal{O}_2$ and $\mathcal{O}_{\infty}$ play central roles. Hence it is natural to expect that $\mathcal{W}$ plays a central role in the classification theory of group actions on "classifiable" stably finite (at least stably projectionless) C$^*$-algebras. Let $\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ is the same class as in [@Sza7 Definition B]. It is easy to see that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups, and $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. Szabó showed that if $\Gamma\in \mathfrak{C}$ and $\mathcal{D}$ is a strongly self-absorbing C$^*$-algebra, then there exists a unique strongly outer action of $\Gamma$ on $\mathcal{D}$ up to cocycle conjugacy ([@Sza7 Corollary 3.4]). In this paper, we show an analogous result of this result. Indeed, the main theorem in this paper is the following theorem. **Theorem A 1**. (Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"}.)  \ Let $\Gamma$ be a countable discrete group in $\mathfrak{C}$, and let $\alpha$ be a strongly outer action of $\Gamma$ on $\mathcal{W}$. Then $\alpha$ is cocycle conjugate to $\mu^{\Gamma}\otimes \mathrm{id}_{\mathcal{W}}$ on $M_{2^{\infty}}\otimes\mathcal{W}$ where $\mu^{\Gamma}$ is the Bernoulli shift action of $\Gamma$ on $\bigotimes_{g\in \Gamma}M_{2^{\infty}}\cong M_{2^{\infty}}$. We say that an action $\alpha$ on $\mathcal{W}$ is *$\mathcal{W}$-absorbing* if there exists a simple separable nuclear monotracial C$^*$-algebra $A$ and an action $\beta$ on $A$ such that $\alpha$ is cocycle conjugate to $\beta\otimes\mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$. The proof of the main theorem above is based on a characterization in [@Na5] of strongly outer $\mathcal{W}$-absorbing actions of countable discrete amenable groups. Actually, we use the following theorem that is a slight variant of [@Na5 Theorem 8.1]. Note that $F(\mathcal{W})$ is Kirchberg's central sequence C$^*$-algebra of $\mathcal{W}$. **Theorem B 1**. (Theorem [Theorem 4](#thm:8.1){reference-type="ref" reference="thm:8.1"}.)  \ Let $\alpha$ be a strongly outer action of a countable discrete amenable group $\Gamma$ on $\mathcal{W}$. Then $\alpha$ is cocycle conjugate to $\mu^{\Gamma}\otimes\mathrm{id}_{\mathcal{W}}$ on $M_{2^{\infty}}\otimes \mathcal{W}$ if and only if $\alpha$ satisfies the following properties:  \ (i) there exists a unital homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$,  \ (ii) if $x$ and $y$ are normal elements in $F(\mathcal{W})^{\alpha}$ such that $\mathrm{Sp}(x)=\mathrm{Sp}(y)$ and $0<\tau_{\mathcal{W}, \omega} (f(x))=\tau_{\mathcal{W}, \omega}(f(y))$ for any $f\in C(\mathrm{Sp}(x))_{+}\setminus\{0\}$, then $x$ and $y$ are unitary equivalent in $F(\mathcal{W})^{\alpha}$,  \ (iii) there exists an injective homomorphism from $\mathcal{W}\rtimes_{\alpha}\Gamma$ to $\mathcal{W}$. We use a first cohomology vanishing type theorem (Corollary [Corollary 8](#cor:vanishing){reference-type="ref" reference="cor:vanishing"}) for showing that if $\Gamma\in \mathfrak{C}$ and $\alpha$ is a strongly outer action of $\Gamma$ on $\mathcal{W}$, then $F(\mathcal{W})^{\alpha}$ satisfies the properties (i) and (ii) in the theorem above. Kishimoto's techniques for Rohlin type theorems in [@Kis0] and [@Kis00], Hermann-Ocneanu's argument in [@HO] and homotopy type arguments in [@Na2] enable us to show this first cohomology vanishing type theorem. Also, note that our arguments for $F(\mathcal{W})^{\alpha}$ are based on results that are shown by techniques around (equivariant) property (SI) in [@MS], [@MS2], [@MS3], [@Sa0], [@Sa], [@Sa2] and [@Sza6]. # Preliminaries {#sec:pre} ## Notations and basic definitions Let $\alpha$ and $\beta$ be actions of a countable discrete group $\Gamma$ on C$^*$-algebras $A$ and $B$, respectively. We say that $\alpha$ is *conjugate to* $\beta$ if there exists a isomorphism $\varphi$ from $A$ onto $B$ such that $\varphi\circ \alpha_g=\beta_g\circ \varphi$ for any $g\in \Gamma$. An $\alpha$-cocycle on $A$ is a map from $\Gamma$ to the unitary group of $M(A)$ such that $u_{gh}=u_{g}\alpha_g(u_h)$ for any $g,h\in \Gamma$, where $M(A)$ is the multiplier algebra of $A$. We say that $\alpha$ is *cocycle conjugate to* $\beta$ if there exist an isomorphism $\varphi$ from $A$ onto $B$ and a $\beta$-cocycle $u$ such that $\varphi\circ \alpha_g=\mathrm{Ad}(u_g)\circ \beta_g \circ \varphi$ for any $g\in\Gamma$. An action $\alpha$ of $\Gamma$ on $A$ is said to be *outer* if $\alpha_g$ is not an inner automorphism of $A$ for any $g\in \Gamma\setminus \{\iota\}$ where $\iota$ is the identity of $\Gamma$. We denote by $A^{\alpha}$ and $A\rtimes_{\alpha}\Gamma$ the fixed point algebra and the reduced crossed product C$^*$-algebra, respectively. Assume that $A$ has a unique tracial state $\tau_{A}$. Let $(\pi_{\tau_{A}}, H_{\tau_{A}})$ be the Gelfand-Naimark-Segal (GNS) representation of $\tau_{A}$. Then $\pi_{\tau_{A}}(A)^{''}$ is a finite factor and $\alpha$ induces an action $\tilde{\alpha}$ on $\pi_{\tau_{A}}(A)^{''}$. We say that $\alpha$ is *strongly outer* if $\tilde{\alpha}$ is an outer action on $\pi_{\tau_{A}}(A)^{''}$. (We refer the reader to [@GHV] and [@MS2] for the definition of strongly outerness for more general settings.) We denote by $\mathcal{R}_0$ and $M_{2^{\infty}}$ the injective II$_1$ factor and the CAR algebra, respectively. ## Fixed point algebras of Kirchberg's central sequence C$^*$-algebras {#sec:fixed} Let $\omega$ be a free ultrafilter on $\mathbb{N}$, and put $$A^{\omega}:= \ell^{\infty}(\mathbb{N}, A)/\{\{x_n\}_{n\in\mathbb{N}}\in \ell^{\infty}(\mathbb{N}, A)\; | \; \lim_{n\to\omega} \|x_n \|=0 \}.$$ We denote by $(x_n)_n$ a representative of an element in $A^{\omega}$. We identify $A$ with the C$^*$-subalgebra of $A^{\omega}$ consisting of equivalence classes of constant sequences. Set $$\mathrm{Ann}(A, A^{\omega}):= \{(x_n)\in A^{\omega}\cap A^{\prime}\; |\; \lim_{n\to \omega}\| x_na\|=0 \text{ for any } a\in A\}.$$ Then $\mathrm{Ann}(A, A^{\omega})$ is an closed ideal of $A^{\omega}\cap A^{\prime}$, and define $$F(A):= A^{\omega}\cap A^{\prime}/\mathrm{Ann}(A, A^{\omega}).$$ See [@Kir2] for basic properties of $F(A)$. For a tracial state $\tau_A$ on $A$, define a map $\tau_{A, \omega}$ from $F(A)$ to $\mathbb{C}$ by $\tau_{A, \omega}([(x_n)_n])=\lim_{n\to\omega}\tau_A(x_n)$ for any $[(x_n)_n]\in F(A)$. Then $\tau_{A, \omega}$ is a well defined tracial state on $F(A)$ by [@Na2 Proposition 2.1]. Put $J_{\tau_{A}}:=\{x\in F(A)\; |\; \tau_{A, \omega}(x^*x)=0\}$. If $A$ is separable and $\tau_{A}$ is faithful, then $\pi_{\tau_{A}}$ induces an isomorphism from $F(A)/J_{\tau_{A}}$ onto the von Neumann algebraic central sequence algebra (or the asymptotic centralizer) $\pi_{\tau_{A}}(A)^{''}_{\omega}$ by essentially the same argument as in the proof of [@KR Theorem 3.3]. In this paper, the reindexing argument and the diagonal argument (or Kirchberg's $\varepsilon$-test [@Kir2 Lemma A.1]) are frequently used. We refer the reader to [@BBSTWW Section 1.3] and [@Oc Chapter 5] for details of these arguments. Every action $\alpha$ of a countable discrete group on $A$ induces an action on $F(A)$. We denote it by the same symbol $\alpha$ for simplicity. Note that if $\alpha$ on $A$ are cocycle conjugate to $\beta$ on $B$, then $\alpha$ on $F(A)$ are conjugate to $\beta$ on $F(B)$. If $A$ is simple, separable and monotracial, then $\tilde{\alpha}$ also induces an action on $\pi_{\tau_{A}}(A)^{''}_{\omega}$. We also denote it by the same symbol $\tilde{\alpha}$. By [@Na5 Proposition 3.9], we see that $\pi_{\tau_{A}}$ induces an isomorphism from $F(A)^{\alpha}/J_{\tau_{A}}^{\alpha}$ onto $(\pi_{\tau_{A}}(A)^{''})_{\omega}^{\tilde{\alpha}}$. The following proposition is an immediate consequence of [@Na5 Theorem 3.6], [@Na5 Proposition 3.11] and [@Na5 Proposition 3.12]. Note that these propositions are based on results in [@MS], [@MS2], [@MS3], [@Sa0], [@Sa], [@Sa2] and [@Sza6]. **Proposition 1**. Let $\alpha$ be an outer action of a countable discrete amenable group on $\mathcal{W}$.  \ (1) The Razak-Jacelon algebra $\mathcal{W}$ has property (SI) relative to $\alpha$, that is, if $a$ and $b$ are positive contractions satisfying $\tau_{\mathcal{W}, \omega}(a)=0$ and $\inf_{m\in\mathbb{N}}\tau_{\mathcal{W}, \omega}(b^m)>0$, then there exists an element $s$ in $F(\mathcal{W})^{\alpha}$ such that $bs=s$ and $s^*s=a$.  \ (2) The fixed point algebra $F(\mathcal{W})^{\alpha}$ is monotracial.  \ (3) If $a$ and $b$ are positive elements satisfying $d_{\tau_{\mathcal{W}, \omega}}(a)< d_{\tau_{\mathcal{W}, \omega}}(b)$, then there exists an element $r$ in $F(\mathcal{W})^{\alpha}$ such that $r^*br=a$. **Definition 2**. Let $\alpha$ be an action of a countable discrete group $\Gamma$ on $\mathcal{W}$. We say that $\alpha$ has *property W* if $\alpha$ satisfies the following properties:  \ (i) there exists a unital homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$,  \ (ii) if $x$ and $y$ are normal elements in $F(\mathcal{W})^{\alpha}$ such that $\mathrm{Sp}(x)=\mathrm{Sp}(y)$ and $0<\tau_{\mathcal{W}, \omega} (f(x))=\tau_{\mathcal{W}, \omega}(f(y))$ for any $f\in C(\mathrm{Sp}(x))_{+}\setminus\{0\}$, then $x$ and $y$ are unitary equivalent in $F(\mathcal{W})^{\alpha}$. Note that if there exists a unital homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$, then $\alpha$ on $\mathcal{W}$ is cocycle conjugate to $\alpha\otimes\mathrm{id}_{M_{2^{\infty}}}$ on $\mathcal{W}\otimes M_{2^{\infty}}$. Indeed, there exists a unital homomorphism from $M_{2^{\infty}}$ to $F(\mathcal{W})^{\alpha}$ by a similar argument as [@Kir2 Corollary 1.13]. Hence [@Sza1 Corollary 3.8] (see also [@Sza1c]) implies this cocycle conjugacy. Using this observation, Proposition [Proposition 1](#pro:MS-pro){reference-type="ref" reference="pro:MS-pro"} and Definition [Definition 2](#def:property-w){reference-type="ref" reference="def:property-w"} instead of $M_{2^{\infty}}$-stability of $\mathcal{W}$, [@Na2 Proposition 4.1], [@Na2 Theorem 5.3] and [@Na2 Theorem 5.8], we obtain the following theorem by essentially the same arguments as in the proofs of [@Na2 Proposition 4.2], [@Na2 Theorem 5.7] and [@Na2 Corollary 5.11](or [@Na3 Corollary 5.5]). **Theorem 3**. Let $\alpha$ be an outer action of a countable discrete amenable group on $\mathcal{W}$. Assume that $\alpha$ has property W.  \ (1) For any $\theta\in [0,1]$, there exists a projection $p$ in $F(\mathcal{W})^{\alpha}$ such that $\tau_{\mathcal{W}, \omega}(p)=\theta$.  \ (2) For any unitary element $u$ in $F(\mathcal{W})^{\alpha}$, there exists a continuous path of unitaries $U: [0,1]\to F(\mathcal{W})^{\alpha}$ such that $$U(0)=1,\quad U(1)=u \quad \text{and} \quad \mathrm{Lip}(U)\leq 2\pi$$ where $\mathrm{Lip}(U)$ is the Lipschitz constant of $U$.  \ (3) If $p$ and $q$ are projections in $F(\mathcal{W})^{\alpha}$ such that $0<\tau_{\mathcal{W}, \omega}(p)=\tau_{\mathcal{W}, \omega}(q)$, then $p$ and $q$ are Murray-von Neumann equivalent. For any countable discrete group $\Gamma$, let $\mu^{\Gamma}$ be the Bernoulli shift action of $\Gamma$ on $\bigotimes_{g\in \Gamma}M_{2^{\infty}}\cong M_{2^{\infty}}$. The following theorem is a slight variant of [@Na5 Theorem 8.1]. **Theorem 4**. Let $\alpha$ be a strongly outer action of a countable discrete amenable group $\Gamma$ on $\mathcal{W}$. Then $\alpha$ is cocycle conjugate to $\mu^{\Gamma}\otimes\mathrm{id}_{\mathcal{W}}$ on $M_{2^{\infty}}\otimes \mathcal{W}$ if and only if $\alpha$ has property W and there exists an injective homomorphism from $\mathcal{W}\rtimes_{\alpha}\Gamma$ to $\mathcal{W}$. *Proof.* [@Na5 Proposition 4.2], [@Na5 Theorem 4.5] and [@Na5 Theorem 8.1] imply the only if part. The if part is an immediate consequence of [@Na5 Theorem 8.1] and Theorem [Theorem 3](#thm:homotopy){reference-type="ref" reference="thm:homotopy"}. ◻ # First cohomology vanishing type theorem In this section, we shall show a first cohomology vanishing type theorem (Corollary [Corollary 8](#cor:vanishing){reference-type="ref" reference="cor:vanishing"}). This is a corollary of a Rohlin type theorem (Theorem [Theorem 7](#thm:Rohlin){reference-type="ref" reference="thm:Rohlin"}). The following lemma is well known among experts. See, for example, [@Ka Theorem 4.8] for a similar (but not the same) result. For reader's convenience, we shall give a proof based on Ocneanu's classification theorem [@Oc Corollary 1.4]. **Lemma 5**. Let $\Gamma$ be a countable discrete amenable group, and let $N$ be a normal subgroup of $\Gamma$. If $\gamma$ is an outer action of $\Gamma$ on the injective II$_1$ factor $\mathcal{R}_0$ and $g_0\notin N$, then $\gamma_{g_0}$ induces a properly outer automorphism of $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. *Proof.* Since $N$ is a normal subgroup, it is clear that $\gamma_{g_0}$ induces an automorphism of $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. First, we shall show that $\gamma_{g_0}$ is not trivial as an automorphism of $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. Let $\pi$ be the quotient map from $\Gamma$ to $\Gamma/ N$, and let $\beta$ be the Bernoulli shift action of $\Gamma /N$ on $\mathcal{R}_0\cong \bigotimes_{\pi (g)\in \Gamma /N}\mathcal{R}_0$. Define an action $\delta$ of $\Gamma$ on $\mathcal{R}_0\cong \mathcal{R}_0 \bar{\otimes}\mathcal{R}_{0}$ by $\delta_g:= \gamma_g\otimes \beta_{\pi (g)}$ for any $g\in\Gamma$. By Ocneanu's classification theorem [@Oc Corollary 1.4], $\gamma$ on $\mathcal{R}_0$ and $\delta$ on $\mathcal{R}_0 \bar{\otimes}\mathcal{R}_{0}$ are cocycle conjugate. Hence there exists an isomorphism $\Phi$ from $(\mathcal{R}_0)_{\omega}$ onto $(\mathcal{R}_0 \bar{\otimes}\mathcal{R}_{0})_{\omega}$ such that $\Phi\circ \gamma_g=\delta_g\circ \Phi$ for any $g\in \Gamma$. Since $\beta_{\pi (g_0)}$ is an outer automorphism of $\mathcal{R}_0$, there exists an element $(x_n)_n$ in $(\mathcal{R}_0)_{\omega}$ such that $(\beta_{\pi(g_0)}(x_n))_n\neq (x_n)_n$ by [@C3 Theorem 3.2]. Put $(y_n)_n:=\Phi^{-1}((1\otimes y_n)_n)\in (\mathcal{R}_0)_{\omega}$. Then it is easy to see that we have $(y_n)_n\in (\mathcal{R}_0)_{\omega}^{\gamma|_N}$ and $(\gamma_{g_0}(y_n))_n\neq (y_n)_n$. Finally, we shall show that $\gamma_{g_0}$ is properly outer as an automorphism of $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. Since $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$ is a factor (see, for example, [@MS2 Lemma 4.1]), it is enough to show that $\gamma_{g_0}$ is outer as an automorphism of $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. In particular, we shall show that for any element $(u_n)_n$ in $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$, there exists an element $(z_n)_n$ in $(\mathcal{R}_0)_{\omega}^{\gamma|_N}$ such that $(u_nz_n)_n=(z_nu_n)_n$ and $(\gamma_{g_0}(z_n))_n\neq (z_n)_n$. Taking a suitable subsequence of $(y_n)_n$ (or the reindexing argument), we obtain the desired element $(z_n)_n$. Consequently, the proof is complete. ◻ The following lemma is an analogous lemma of [@Na2 Lemma 6.2]. See also [@MS1 Theorem 3.4]. **Lemma 6**. Let $\Gamma$ be a semidirect product $N\rtimes \mathbb{Z}$ where $N$ is a countable discrete amenable group, and let $\alpha$ be a strongly outer action of $\Gamma$ on $\mathcal{W}$. Then for any $k\in\mathbb{N}$, there exists a positive contraction $f$ in $F(\mathcal{W})^{\alpha|_{N}}$ such that $$\tau_{\mathcal{W}, \omega}(f)=\frac{1}{k}\quad \text{and} \quad f\alpha_{(\iota,j)}(f)=0$$ for any $1\leq j \leq k-1$. *Proof.* Since $\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{''}$ is isomorphic to the injective II$_1$ factor, Lemma [Lemma 5](#lem:outer){reference-type="ref" reference="lem:outer"} implies that $\tilde{\alpha}_{(\iota, 1)}$ is an aperiodic automorphism of $(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{''})_{\omega}^{\tilde{\alpha}|_N}$. Hence it follows from [@C2 Theorem 1.2.5] that there exists a partition of unity $\{P_{j}\}_{j=1}^k$ consisting of projections in $(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{''})_{\omega}^{\tilde{\alpha}|_N}$ such that $\tilde{\alpha}_{(\iota, 1)}(P_{j})=P_{j+1}$ for any $1\leq j \leq k-1$. Since $\pi_{\tau_{\mathcal{W}}}$ induces an isomorphism from $F(\mathcal{W})^{\alpha|_N}/J_{\tau_{\mathcal{W}}}^{\alpha|_N}$ onto $(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{''})_{\omega}^{\tilde{\alpha}|_N}$ (see Section [2.2](#sec:fixed){reference-type="ref" reference="sec:fixed"}), there exists a positive contraction $[(e_n)_n]$ in $F(\mathcal{W})^{\alpha|_N}$ such that $(\pi_{\tau_{\mathcal{W}}}(e_n))_n=P_1$ in $(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{''})_{\omega}^{\tilde{\alpha}|_N}$. Then we have $$\lim_{n\to\omega} \| \pi_{\tau_{\mathcal{W}}}(e_n \alpha_{(\iota ,j)}(e_n))\|_2=0 \quad \text{and} \quad \tau_{\mathcal{W}, \omega}([(e_n)_n])=\tilde{\tau}_{\mathcal{W}, \omega}(P_1)=\dfrac{1}{k}$$ for any $1\leq j \leq k-1$, where $\tilde{\tau}_{\mathcal{W}, \omega}$ is the induced tracial state on $(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{''})_{\omega}^{\tilde{\alpha}|_N}$ by $\tau_{\mathcal{W}}$. The rest of the proof is the same as [@Na2 Lemma 6.2]. (See also [@MS1 Proposition 3.3].) ◻ Using Proposition [Proposition 1](#pro:MS-pro){reference-type="ref" reference="pro:MS-pro"}, Theorem [Theorem 3](#thm:homotopy){reference-type="ref" reference="thm:homotopy"} (we need to assume that $\alpha|_N$ has property W) and Lemma [Lemma 6](#lem:6.2){reference-type="ref" reference="lem:6.2"} instead of [@Na2 Proposition 4.1], [@Na2 Proposition 4.2], [@Na2 Theorem 5.8] and [@Na2 Lemma 6.2], we obtain the following Rohlin type theorem by essentially the same arguments in the proofs of [@Na2 Lemma 6.3] and [@Na2 Theorem 6.4]. Note that these arguments are based on [@Kis0] and [@Kis00]. **Theorem 7**. Let $\Gamma$ be a semidirect product $N\rtimes \mathbb{Z}$ where $N$ is a countable discrete amenable group, and let $\alpha$ be a strongly outer action of $\Gamma$ on $\mathcal{W}$. Assume that $\alpha|_{N}$ has property W. Then for any $k\in\mathbb{N}$, there exists a partition on unity $\{p_{1,i}\}_{i=0}^{k-1}\cup \{p_{2,j}\}_{j=0}^{k}$ consisting of projections in $F(\mathcal{W})^{\alpha|_N}$ such that $$\alpha_{(\iota, 1)}(p_{1,i})=p_{1, i+1}\quad \text{and} \quad \alpha_{(\iota, 1)}(p_{2,j})=p_{2,j+1}$$ for any $0\leq i\leq k-2$ and $0\leq j\leq k-1$. Theorem [Theorem 3](#thm:homotopy){reference-type="ref" reference="thm:homotopy"}, Theorem [Theorem 7](#thm:Rohlin){reference-type="ref" reference="thm:Rohlin"} and Hermann-Ocneanu's argument [@HO Theorem 1] (see also remarks after [@HO Lemma 1], [@I0] and [@Kis3]) imply the following corollary. **Corollary 8**. Let $\Gamma$ be a semidirect product $N\rtimes \mathbb{Z}$ where $N$ is a countable discrete amenable group, and let $\alpha$ be a strongly outer action of $\Gamma$ on $\mathcal{W}$. Assume that $\alpha|_{N}$ has property W and $S$ is a countable subset in $F(\mathcal{W})^{\alpha|_N}$. For any unitary element $u$ in $F(\mathcal{W})^{\alpha|_N}\cap S^{\prime}$, there exists a unitary element $v$ in $F(\mathcal{W})^{\alpha|_N}\cap S^{\prime}$ such that $u=v\alpha_{(\iota, 1)}(v)^{*}$. # Main theorem In this section we shall show the main theorem. Recall that $\mathfrak{C}$ is the smallest class of countable discrete groups with the following properties:(i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that if $\Gamma$ is an extension of $N$ by $\mathbb{Z}$, then $\Gamma$ is isomorphic to a semidirect product $N\rtimes\mathbb{Z}$. The following lemma is an easy consequence of the definition of property W and the diagonal argument. **Lemma 9**. Let $\Gamma$ be an increasing union $\bigcup_{m\in\mathbb{N}}\Gamma_m$ of discrete countable groups $\Gamma_m$, and let $\alpha$ be an action of $\Gamma$ on $\mathcal{W}$. If $\alpha|_{\Gamma_m}$ has property W for any $m\in\mathbb{N}$, then $\alpha$ has property W. The following lemma is an application of Corollary [Corollary 8](#cor:vanishing){reference-type="ref" reference="cor:vanishing"}. **Lemma 10**. Let $\Gamma$ be a semidirect product $N\rtimes \mathbb{Z}$ where $N$ is a countable discrete amenable group, and let $\alpha$ be a strongly outer action of $\Gamma$ on $\mathcal{W}$. If $\alpha|_{N}$ has property W, then $\alpha$ has property W. *Proof.* (i) There exists a unital homomorphism $\varphi$ from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha|_N}$ by the assumption. Let $\{e_{ij}\}_{i,j=1}^{2}$ be the standard matrix units of $M_{2}(\mathbb{C})$. Since we have $0<\tau_{\mathcal{W}, \omega}(\varphi(e_{11})) =\tau_{\mathcal{W}, \omega}(\alpha_{(\iota,1)}(\varphi(e_{11})))$, there exists an element $w$ in $F(\mathcal{W})^{\alpha|_N}$ such that $w^*w =\alpha_{(\iota, 1)}(\varphi(e_{11}))$ and $ww^*=\varphi(e_{11})$ by Theorem [Theorem 3](#thm:homotopy){reference-type="ref" reference="thm:homotopy"}. Put $u:= \sum_{i=1}^{2}\varphi(e_{i1})w\alpha_{(\iota ,1)}(\varphi(e_{1i}))$. Then $u$ is a unitary element in $F(\mathcal{W})^{\alpha|_N}$ such that $\alpha_{(\iota, 1)}(\varphi(x))=u^*\varphi(x)u$ for any $x\in M_{2}(\mathbb{C})$. By Corollary [Corollary 8](#cor:vanishing){reference-type="ref" reference="cor:vanishing"}, there exists a unitary element $v$ in $F(\mathcal{W})^{\alpha|_N}$ such that $u=v\alpha_{(\iota ,1)}(v)^*$. We have $\alpha_{(\iota, 1)}(v^*\varphi(x)v)= v^*\varphi(x)v$ for any $x\in M_{2}(\mathbb{C})$. Hence the map $\psi$ defined by $\psi(x):=v^*\varphi(x)v$ for any $x\in M_{2}(\mathbb{C})$ is a unital homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$. (ii) Let $x$ and $y$ be normal elements in $F(\mathcal{W})^{\alpha}$ such that $\mathrm{Sp}(x)=\mathrm{Sp}(y)$ and $0<\tau_{\mathcal{W}, \omega} (f(x))=\tau_{\mathcal{W}, \omega}(f(y))$ for any $f\in C(\mathrm{Sp}(x))_{+}\setminus\{0\}$. Since $x$ and $y$ are also elements in $F(\mathcal{W})^{\alpha|_N}$, there exists a unitary element $u$ in $F(\mathcal{W})^{\alpha|_N}$ such that $uxu^*=y$ by the assumption. Note that $u\alpha_{(\iota, 1)}(u)^*$ is a unitary element in $F(\mathcal{W})^{\alpha|_N}\cap \{ y\}^{\prime}$. Hence Corollary [Corollary 8](#cor:vanishing){reference-type="ref" reference="cor:vanishing"} implies that there exists a unitary element $v$ in $F(\mathcal{W})^{\alpha|_N}\cap \{ y\}^{\prime}$ such that $u\alpha_{(\iota, 1)}(u)^*=v\alpha_{(\iota, 1)}(v)^*$. We have $\alpha_{(\iota ,1)}(v^*u)=v^*u$ and $v^*uxu^*v= v^*yv=y$. Therefore $x$ and $y$ are unitary equivalent in $F(\mathcal{W})^{\alpha}$. By (i) and (ii), $\alpha$ has property W. ◻ The following theorem is the main theorem in this paper. **Theorem 11**. Let $\Gamma$ be a countable discrete group in $\mathfrak{C}$, and let $\alpha$ be a strongly outer action of $\Gamma$ on $\mathcal{W}$. Then $\alpha$ is cocycle conjugate to $\mu^{\Gamma}\otimes \mathrm{id}_{\mathcal{W}}$ on $M_{2^{\infty}}\otimes\mathcal{W}$. *Proof.* Every action of the trivial group on $\mathcal{W}$ has property W by results in [@Na2] (or [@Na4 Theorem 3.8]). By Lemma [Lemma 9](#lem:unions){reference-type="ref" reference="lem:unions"} and Lemma [Lemma 10](#lem:extension-z){reference-type="ref" reference="lem:extension-z"}, we see that $\alpha$ has property W. Note that this implies that $\mathcal{W}\rtimes_{\alpha}\Gamma$ is $M_{2^{\infty}}$-stable because $\alpha$ is cocycle conjugate to $\alpha\otimes\mathrm{id}_{M_{2^{\infty}}}$ on $\mathcal{W}\otimes M_{2^{\infty}}$. Since the class of separable C$^*$-algebras that are $KK$-equivalent to $\{0\}$ is closed under countable inductive limits and crossed products by $\mathbb{Z}$, [@Na4 Theorem 6.1] implies that $\mathcal{W}\rtimes_{\alpha}\Gamma$ is isomorphic to $\mathcal{W}$. Therefore we obtain the conclusion by Theorem [Theorem 4](#thm:8.1){reference-type="ref" reference="thm:8.1"}. ◻ The following corollary is an immediate consequence of the theorem above. **Corollary 12**. Let $\Gamma$ be a countable discrete group in $\mathfrak{C}$. Then there exists a unique strongly outer action of $\Gamma$ on $\mathcal{W}$ up to cocycle conjugacy. 99 J. Bosa, N. P. Brown, Y. Sato, A. Tikuisis, S. White and W. Winter, *Covering dimension of C$^*$-algebras and 2-coloured classification*, Mem. Amer. Math. Soc. **257** (2016), no. 1233. J. Castillejos and S. Evington, *Nuclear dimension of simple stably projectionless C$^*$-algebras*, Anal. PDE 13 (2020), no. 7, 2205--2240. A. Connes, *Outer conjugacy classes of automorphisms of factors*, Ann. Sci. Ecole Norm. Sup. (4) **8** (1975), no. 3, 383--419. A. Connes, *Periodic automorphisms of the hyperfinite factor of type II$_1$*, Acta Sci. Math. (Szeged) **39** (1977), no. 1-2, 39--66. G. A. Elliott, G. Gong, H. Lin and Z. Niu, *The classification of simple separable KK-contractible C$^*$-algebras with finite nuclear dimension*, J. Geom. Phys. **158** (2020), 103861, 51 pp. E. Gardella, I. Hirshberg and A. Vaccaro, *Strongly outer actions of amenable groups on $\mathcal{Z}$-stable nuclear C$^*$-algebras*, J. Math. Pures Appl. (9)162(2022), 76--123. J. Gabe and G. Szabó, *The dynamical Kirchberg-Phillips theorem*, preprint, arXiv:2205.04933 \[math.OA\], to appear in Acta Math. R. H. Herman and A. Ocneanu, *Stability for integer actions on UHF C$^*$-algebras*, J. Funct. Anal. **59** (1984), 132--144. M. Izumi, *The Rohlin property for automorphisms of $C^*$-algebras*, Mathematical physics in mathematics and physics (Siena, 2000) Fields Inst. Commun., vol. **30**, Amer. Math. Soc., Providence, RI, 2001, pp. 191--206. M. Izumi, *Group actions on operator algebras*, in: Proceedings of the International Congress of Mathematicians, vol. III, Hyderabad, India, 2010, Hindustan Book Agency, 2010, pp. 1528--1548. M. Izumi and H. Matui, *$\mathbb{Z}^2$-actions on Kirchberg algebras* Adv. Math. **224** (2010), no.2, 355---400. M. Izumi and H. Matui, *Poly-$\mathbb{Z}$ group actions on Kirchberg algebras I*, Int. Math. Res. Not. IMRN (2021), no.16, 12077---12154. M. Izumi and H. Matui, *Poly-$\mathbb{Z}$ group actions on Kirchberg algebras II*, Invent. Math. **224** (2021), no.3, 699---766. B. Jacelon, *A simple, monotracial, stably projectionless C$^*$-algebra*, J. Lond. Math. Soc. (2) **87** (2013), no. 2, 365--383. Y. Katayama, *A duality for an action of a countable amenable group on a hyperfinite II$_1$-factor*, J. Operator Theory **21** (1989), no.2, 297---314. E. Kirchberg, *Central sequences in C$^*$-algebras and strongly purely infinite algebras*, Operator Algebras: The Abel Symposium 2004, 175--231, Abel Symp., **1**, Springer, Berlin, 2006. E. Kirchberg and M. Rørdam, *Central sequence $C^*$-algebras and tensorial absorption of the Jiang-Su algebra*, J. Reine Angew. Math. **695** (2014), 175--214. A. Kishimoto, *The Rohlin property for automorphisms of UHF algebras*, J. Reine Angew. Math. **465** (1995), 183--196. A.Kishimoto. *The Rohlin property for shifts on UHF algebras and automorphisms of Cuntz algebras*, J. Funct. Anal. **140** (1996), no. 1, 100--123. A. Kishimoto, *The Rohlin property for shift automorphisms*, Rev. Math. Phys. **12** (2000), no. 7, 965--980 H. Matui, *Classification of outer actions of $\mathbb{Z}^N$ on $\mathcal{O}_2$*, Adv. Math. **217** (2008), no.6, 2872---2896. H. Matui, *$\mathbb{Z}^N$-actions on UHF algebras of infinite type*, J. Reine Angew. Math. **657** (2011), 225-244. H. Matui and Y. Sato, *$\mathcal{Z}$-stability of crossed products by strongly outer actions*, Comm. Math. Phys. **314** (2012), no. 1, 193--228. H. Matui and Y. Sato, *Strict comparison and $\mathcal{Z}$-absorption of nuclear C$^*$-algebras*, Acta Math. **209** (2012), no. 1, 179--196. H. Matui and Y. Sato, *$\mathcal{Z}$-stability of crossed products by strongly outer actions II* , Amer. J. Math. **136** (2014), no. 6, 1441--1496. H. Matui and Y. Sato, *Decomposition rank of UHF-absorbing C$^*$-algebras*, Duke Math. J. **163** (2014), no. 14, 2687--2708. N. Nawata, *Trace scaling automorphisms of the stabilized Razak-Jacelon algebra*, Proc. Lond. Math. Soc. (3) **118** (2019), no. 3, 545--576. N. Nawata, *Rohlin actions of finite groups on the Razak-Jacelon algebra*, Int. Math. Res. Not. IMRN 2021, no. 4, 2991--3020. N. Nawata, *A characterization of the Razak-Jacelon algebra*, to appear in Anal. PDE, arXiv:2008.10235 \[math.OA\]. N, Nawata, *Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra*, J. Funct. Anal.285(2023), no.8, Paper No. 110088, 44 pp. A. Ocneanu, *Actions of discrete amenable groups on von Neumann algebras*, Lecture Notes in Mathematics, vol. **1138**, Springer-Verlag, Berlin, 1985. S. Razak, *On the classification of simple stably projectionless C$^*$-algebras*, Canad. J. Math. **54** (2002), no. 1, 138--224. Y. Sato, *Certain aperiodic automorphisms of unital simple projectionless C$^*$-algebras* Internat. J. Math. **20** (2009), no.10 , 1233--1261. Y. Sato, *The Rohlin property for automorphisms of the Jiang-Su algebra*, J. Funct. Anal. **259** (2010), no. 2, 453--476. Y. Sato, *Actions of amenable groups and crossed products of $\mathcal{Z}$-absorbing C$^*$-algebras*, Operator algebras and mathematical physics, 189--210, Adv. Stud. Pure Math. **80** Math. Soc. Japan, Tokyo, 2019. G. Szabó, *Strongly self-absorbing C\*-dynamical systems III*, Adv. Math. **316** (2017), no. 20, 356--380. G. Szabó, *Strongly self-absorbing C\*-dynamical systems*, Trans. Amer. Math. Soc. **370** (2018), no. 1, 99--130. G. Szabó, *Corrigendum to \"Strongly self-absorbing C$^*$-dynamical systems"*, Trans. Amer. Math. Soc. **373** (2020), no. 10, 7527--7531. G. Szabó, *Equivariant Kirchberg-Phillips-type absorption for amenable group actions*, Comm. Math. Phys. **361** (2018), no. 3, 193--228. G. Szabó, *Actions of certain torsion-free elementary amenable groups on strongly self-absorbing C$^*$-algebras*, Comm. Math. Phys. **371** (2019), no.1, 267---284. G. Szabó, *Equivariant property (SI) revisited*, Anal. PDE **14** (2021), no. 4, 1199--1232. [^1]: This work was supported by JSPS KAKENHI Grant Number 20K03630

arxiv_math

--- abstract: | An important problem in biological modeling is choosing the right model. Given experimental data, one is supposed to find the best mathematical representation to describe the real-world phenomena. However, there may not be a unique model representing that real-world phenomena. Two distinct models could yield the same exact dynamics. In this case, these models are called *indistinguishable*. In this work, we consider the indistinguishability problem for linear compartmental models, which are used in many areas, such as pharmacokinetics, physiology, cell biology, toxicology, and ecology. We exhibit sufficient conditions for indistinguishability for models with a certain graph structure: paths from input to output with "detours". The benefit of applying our results is that indistinguishability can be proven using only the graph structure of the models, without the use of any symbolic computation. This can be very helpful for medium-to-large sized linear compartmental models. These are the first sufficient conditions for indistinguishability of linear compartmental models based on graph structure alone, as previously only necessary conditions for indistinguishability of linear compartmental models existed based on graph structure alone. We prove our results by showing that the indistinguishable models are the same up to a renaming of parameters, which we call *permutation indistinguishability*. address: - California State University, Stanislaus - Santa Clara University author: - Cashous Bortner - Nicolette Meshkat bibliography: - AIM.bib title: Graph-based sufficient conditions for indistinguishability of linear compartmental models --- # Introduction {#sec:intro} Two state space models are *indistinguishable* if for any choice of parameters in the first model, there is a choice of parameters in the second model that will yield the same dynamics in both models, and vice versa. The indistinguishability problem can be related to the problem of structural identifiability, which concerns determining if all of the unknown parameters in a model can be recovered from input/output data, although it is not necessary to test identifiability in order to test indistinguishability. Previous work has focused on testing indistinguishability of given models and also on finding the set of all indistinguishable models from a given model. Vajda [@vajda1982] presents the problem in terms of structural equivalence and gives a method for generating a whole set of indistinguishable models. Godfrey and Chapman [@godfrey-chapman] give necessary conditions for indistinguishability based on the graph structure of the model, which they call geometric rules. Zhang et al [@zhang1991] present a method applying the Godfrey and Chapman geometric rules to generate all identifiable indistinguishable models given a starting model. Davidson et al [@davidson2017] present the program DISTING to compute all indistinguishable models from an identifiable subgraph of a given linear compartmental model. There has also been some work on indistinguishability of nonlinear compartmental models [@godfrey-chapman-vajda-1994; @chapman1994; @chapman1996]. Yates et al [@yates] review the problems of identifiability and indistinguishability for compartmental models. While there are necessary conditions for indistinguishability via the graph structure (see the Godfrey-Chapman geometric rules in Section [3.1](#subsection:gfgeorules){reference-type="ref" reference="subsection:gfgeorules"}) that can be applied without the use of symbolic computation, to the best our knowledge, there are no sufficient conditions for indistinguishability based on the graph structure. Our goal in this work is to find sufficient conditions for indistinguishability based on graph structure for a certain class of models. The starting point is a simple *path model* from input to output and we examine the effect of creating and moving a "detour" to the path. We prove our results by showing that the models we consider are indistinguishable via a permutation, or renaming, of parameters. This renaming of parameters allows us to easily prove indistinguishability. Our main results are the following. Given a path from compartment $1$ to compartment $n$, with input in compartment $1$ and output in compartment $n$, the class of models with a leak from compartment $i$, where $i=1,\ldots,n-1$, are all indistinguishable (see Theorem [Theorem 24](#thm:leak){reference-type="ref" reference="thm:leak"}). Replacing the leak with an edge from compartment $n$ to compartment $n-1$ also gives an indistinguishable model (see Theorem [Theorem 26](#thm:terminalcycle){reference-type="ref" reference="thm:terminalcycle"}). We also generalize Theorem [Theorem 24](#thm:leak){reference-type="ref" reference="thm:leak"} to apply to moving any "detour" down the path (see Theorem [Theorem 27](#thm:detour){reference-type="ref" reference="thm:detour"}). Finally, we show how to generalize even further with multiple source-sink paths (see Corollary [Corollary 29](#cor:sink){reference-type="ref" reference="cor:sink"} and Corollary [Corollary 32](#cor:source){reference-type="ref" reference="cor:source"}). To the best of our knowledge, our results provide the first sufficient conditions for indistinguishability of linear compartmental models based on graph structure alone. Thus, with our results, indistinguishability can be proven without the use of symbolic computation for a certain class of path models. The outline of our paper is the following. We first give some background on linear compartmental models, input-output equations, and identifiability in Section [2](#section:background){reference-type="ref" reference="section:background"}. We then define indistinguishability and provide many motivating examples in Section [3](#section:indist){reference-type="ref" reference="section:indist"}. We then provide sufficient conditions for indistinguishability for paths with leaks (Section [4](#section:leak){reference-type="ref" reference="section:leak"}), paths with detours (Section [5](#section:detour){reference-type="ref" reference="section:detour"}), and multiple sinks or mutliple sources (Section [6](#section:sink){reference-type="ref" reference="section:sink"} and Section [7](#section:source){reference-type="ref" reference="section:source"}). Finally, we discuss our results and future work in Section [8](#section:discussion){reference-type="ref" reference="section:discussion"}. # Background {#section:background} ## Linear Compartmental Models Let $G$ be a directed graph with vertex set $V$ and set of directed edges $E$. Each vertex $i \in V$ corresponds to a compartment in our model and an edge $j \rightarrow i$ denotes a direct flow of material from compartment $j$ to compartment $i$. Also introduce three subsets of the vertices $In, Out, Leak \subseteq V$ corresponding to the set of input compartments, output compartments, and leak compartments, respectively. To each edge $j \rightarrow i$ we associate an independent parameter $a_{ij}$, the rate of flow from compartment $j$ to compartment $i$. To each leak node $i \in Leak$, we associate an independent parameter $a_{0i}$, the rate of flow from compartment $i$ leaving the system. We associate a matrix $A(G)$, called the *compartmental matrix* to the graph and the set $Leak$ in the following way: $$A(G)_{ij} = \left\{ \begin{array}{l l l} -a_{0i}-\sum_{k: i \rightarrow k \in E}{a_{ki}} & \quad \text{if $i=j$ and } i \in Leak\\ -\sum_{k: i \rightarrow k \in E}{a_{ki}} & \quad \text{if $i=j$ and } i \notin Leak\\ a_{ij} & \quad \text{if $j\rightarrow{i}$ is an edge of $G$}\\ 0 & \quad \text{otherwise}\\ \end{array} \right.$$ For brevity, we will often use $A$ to denote $A(G)$. Then we construct a system of linear ODEs with inputs and outputs associated to the quadruple $(G, In, Out, Leak)$ as follows: $$\label{eq:main} \dot{x}(t)=Ax(t)+u(t) \quad \quad y_i(t)=x_i(t) \mbox{ for } i \in Out$$ where $u_{i}(t) \equiv 0$ for $i \notin In$. The coordinate functions $x_{i}(t)$ are the state variables, the functions $y_{i}(t)$ are the output variables, and the nonzero functions $u_{i}(t)$ are the inputs. The resulting model is called a *linear compartmental model*. We will indicate output compartments by this symbol: . Input compartments are labeled by "in", and leaks are indicated by edges which go to no vertex. **Example 1**. The model $\mathcal{M}=(G,\{1\},\{2\},\{1\})$ with $G$ given in Figure [\[fig:nonpermute\]](#fig:nonpermute){reference-type="ref" reference="fig:nonpermute"} is a linear compartmental model with equations given by: $$\begin{aligned} \label{eq:A-ex} \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} &~=~ \begin{pmatrix} - a_{01} - a_{21} & a_{12} \\ a_{21} & -a_{12} \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} u_1 \\ 0 \end{pmatrix}~,\end{aligned}$$ with output equation $y_2=x_2$. **Definition 2**. A *path* from vertex $i_{0}$ to vertex $i_{k}$ in a directed graph $G$ is a sequence of vertices $i_{0},i_{1}, i_{2}, \ldots, i_{k}$ such that $i_{j} \to i_{j+1}$ is an edge for all $j = 0, \ldots, k-1$. Now we give some definitions from [@linear-i-o] regarding an important subgraph to this work: **Definition 3**. For a linear compartmental model $\mathcal{M}=(G,In,Out,Leak)$, let $i \in Out$. The *output-reachable subgraph to $i$* (or *to $y_i$*) is the induced subgraph of $G$ containing all vertices $j$ for which there is a directed path in $G$ from $j$ to $i$. A linear compartmental model is *output connectable* if every compartment has a directed path leading from it to an output compartment. Likewise, a linear compartmental model is *input connectable* if every compartment has a directed path leading to it from an input compartment. ## Input-output equations We will be using the so-called *differential algebra approach* to find *input-output equations* to analyze both structural identifiability [@Ljung; @Ollivier] and indistinguishability. In the differential algebra approach, we view the model equations as differential polynomials in a differential polynomial ring $R(p)[u,y,x]$, i.e., the ring of polynomials in state variable vector $x$, output vector $y$, input vector $u$, and their derivatives, with coefficients in $R(p)$ for parameter vector $p$. Since the unmeasured state variables $x_i$ cannot be determined, the goal in this approach is to use differential elimination to eliminate all unknown state variables and their derivatives. The resulting equations are only in terms of input variables, output variables, their derivatives, and parameters, so these equations have the following form: $$\begin{aligned} \label{eq:general-i-o} \sum_i{c_i(p)\Psi_i(u,y)} =0~.\end{aligned}$$ An equation of the form [\[eq:general-i-o\]](#eq:general-i-o){reference-type="eqref" reference="eq:general-i-o"} is called an *input-output equation* for $\mathcal M$. For nonlinear models, one standard "reduced" generating set for these input-output equations is formed by those equations in a *characteristic set* (defined precisely in [@glad]) that do not involve the $x_i$'s or their derivatives. In a characteristic set, which can be computed using the software `DAISY` [@daisy], each $\Psi_i(u,y)$ in each input-output equation [\[eq:general-i-o\]](#eq:general-i-o){reference-type="eqref" reference="eq:general-i-o"} is a differential monomial, i.e., a monomial purely in terms of input variables, output variables, and their derivatives. The terms $c_i(p)$ are called the coefficients of the input-output equations. These coefficients can be fixed uniquely by normalizing the input-output equations to make them monic [@daisy]. However, for linear models, it has been shown that these input-output equations can be found much more easily using the Transfer Function approach [@Bellman] or even a trick with Cramer's Rule [@MeshkatSullivant]. We will be taking the latter approach to get an explicit formula for the input-output equations. We now state Theorem 3.8 from [@linear-i-o] with input $i$ and output $j$, which gives the input-output equation in $y_j$ in terms of the output-reachable subgraph to $y_j$. **Theorem 4**. *Let $\mathcal{M}=(G, In, Out, Leak)$ be a linear compartmental model with at least one input. Let $j \in Out$, and assume that there exists a directed path from some input compartment to compartment-$j$. Let $H$ denote the output-reachable subgraph to $y_j$, let $A_H$ denote the compartmental matrix for the restriction $\mathcal{M}_H$, and let $\partial I$ be the the product of the differential operator $d/dt$ and the $|V_G| \times |V_G|$ identity matrix. Then the following is an input-output equation for $\mathcal M$ involving $y_j$: $$\begin{aligned} \label{eq:io-det} \det (\partial I -{A}_H) y_j ~=~ \sum_{i \in In \cap V_H} (-1)^{i+j} \det \left( \partial I-{A}_H \right)^{i,j} u_i ~, \end{aligned}$$ where $\left( \partial I-{A}_H \right)^{i,j}$ denotes the matrix obtained from $\left( \partial I-{A}_H \right)$ by removing the row corresponding to compartment-$i$ and the column corresponding to compartment-$j$.* **Example 5** (Continuation of Example [Example 1](#ex:continuing){reference-type="ref" reference="ex:continuing"}). The model $\mathcal{M}=(G,\{1\},\{2\},\{1\})$ with $G$ given in Figure [\[fig:nonpermute\]](#fig:nonpermute){reference-type="ref" reference="fig:nonpermute"} has the following input-output equation: $$\begin{aligned} \ddot{y}_2+(a_{01}+a_{21}+a_{12})\dot{y}_2+(a_{01}a_{12})y_2 = a_{21}u_1\end{aligned}$$ **Remark 6**. We note that the input-output equation has $y_j$ and derivatives of $y_j$ on the left-hand side and $u_i$ and derivatives of $u_i$ on the right-hand side. For brevity we collectively refer to them as derivatives of $y_j$ and $u_i$. We also note there are polynomial functions of parameters as coefficients of these derivatives, appearing on both sides. Therefore, in the rest of the paper, we will refer to the *right-hand side coefficients* and the *left-hand side coefficients* as the polynomial functions of parameters that appear in front of the derivatives of $u_i$ and $y_j$, respectively. ## Identifiability We will see shortly that sometimes the question of identifiability can help answer questions about indistinguishability. We give some definitions of identifiability from [@MeshkatSullivantEisenberg]. **Definition 7**. Let $(G, In, Out, Leak)$ be a linear compartment model and let $c$ denote the vector of all nonzero and nonmonic coefficient functions of all the linear input-output equations derived in Theorem [Theorem 4](#thm:ioscc){reference-type="ref" reference="thm:ioscc"} for each $i \in Out$. The function $c$ defines a map $c: \mathbb{R}^{|E| + |Leak|} \rightarrow \mathbb{R}^{k}$, where $k$ is the total number of coefficients which we call the *coefficient map*. The linear compartment model $(G, In, Out, Leak)$ is: - *globally identifiable* if $c$ is a one-to-one function, and is *generically globally identifiable* if global identifiability holds everywhere in $\mathbb{R}^{|E| + |Leak|}$, except possibly on a set of measure zero. - *locally identifiable* if around any neighborhood of a point in $\mathbb{R}^{|E| + |Leak|}$, $c$ is a one-to-one function, and is *generically locally identifiable* if local identifiability holds everywhere in $\mathbb{R}^{|E| + |Leak|}$, except possibly on a set of measure zero. - *unidentifiable* if $c$ is infinite-to-one. **Example 8** (Continuation of Example [Example 1](#ex:continuing){reference-type="ref" reference="ex:continuing"}). The model $\mathcal{M}=(G,\{1\},\{2\},\{1\})$ with $G$ given in Figure [\[fig:nonpermute\]](#fig:nonpermute){reference-type="ref" reference="fig:nonpermute"} has the following coefficient map: $$\begin{aligned} c(a_{12}, a_{21}, a_{01})= (a_{01}+a_{21}+a_{12}, a_{01}a_{12}, a_{21})\end{aligned}$$ We can test injectivity of this map by solving the system of polynomial equations: $$\begin{aligned} a_{01}+a_{21}+a_{12} = a^*_{01}+a^*_{21}+a^*_{12} \\ a_{01}a_{12} = a^*_{01}a^*_{12} \\ a_{21}=a^*_{21}\end{aligned}$$ We get two solutions: $$\begin{aligned} (a_{12}, a_{21}, a_{01})=(a^*_{12}, a^*_{21}, a^*_{01}) \text{ or} \\ (a_{12}, a_{21}, a_{01})=(a^*_{01}, a^*_{21}, a^*_{12})\end{aligned}$$ Thus the model is locally identifiable. # Indistinguishability {#section:indist} Recall two state space models are *indistinguishable* if for any choice of parameters in the first model, there is a choice of parameters in the second model that will yield the same dynamics in both models, and vice versa. There have been several definitions and approaches to solve this problem in the literature [@godfrey-chapman; @raksanyi; @walter1984; @zhang1991]. To start with, to be indistinguishable, two models must have the same input and output variables. Since indistinguishable models give the same dynamics, the structures of their input-output equations should be the same. In the case that there is one output variable in both models, there is a single input-output equation. To say the input-output equations have the same structure means that exactly the same differential monomials (i.e. $y_i$, $\dot{y}_i$, \...,$u_i$, $\dot{u}_i$, \..., etc) appear in both input-output equations. **Example 9**. \[**Distinguishable via input-output equation structure**\] Consider the models given in Figure [\[fig:small2\]](#fig:small2){reference-type="ref" reference="fig:small2"}. The model $\mathcal{M}$ has the form: $$\begin{aligned} \dot{x}_1 = -a_{01} x_1 + a_{12} x_2 + u_1\\ \dot{x}_2 = -a_{12}x_2 \\ y_1 = x_1 \end{aligned}$$ while the model $\mathcal{M}'$ has the form: $$\begin{aligned} \dot{x}_1 = + a_{12} x_2 + u_1\\ \dot{x}_2 = -(a_{02} + a_{12})x_2 \\ y_1 = x_1 \end{aligned}$$ The input-output equations for $\mathcal{M}$ and $\mathcal{M}'$ are, respectively: $$\begin{aligned} \ddot{y}_1 + (a_{01} + a_{12}) \dot{y}_1 + a_{01}a_{12} y_1 = \dot{u}_1 + a_{12} u_1 \\ \ddot{y}_1 + (a_{02} + a_{12}) \dot{y}_1 = \dot{u}_1 + (a_{02} + a_{12}) u_1 \end{aligned}$$ Since these two models do not have the same structure of the input-output equations, e.g. $\mathcal{M}$ has a term in $y_1$ while $\mathcal{M}'$ does not, then these two models are *distinguishable*. Supposing that the two models have the same structures as described above, we can let $c(p)$ and $c'(p')$ denote the corresponding coefficient maps of the two models, respectively. Here $c : \Theta \rightarrow \mathbb{R}^m$ and $c': \Theta' \rightarrow \mathbb{R}^m$, and the components are ordered so that the components correspond to each other as coming from the same differential monomial. Note that the dimensions of the parameter spaces $\Theta$ and $\Theta'$ might be different. We further assume that both coefficient maps are monic on the same coefficient. Indistinguishability is characterized in terms of the coefficient maps $c$ and $c'$. **Definition 10**. Suppose that Model 1 and Model 2 have the same input-output equation structure. Let $c: \Theta \rightarrow \mathbb{R}^{m}$ and $c': \Theta^{'} \rightarrow \mathbb{R}^{m}$ be the coefficient maps for Model 1 and Model 2, respectively. We say that: - Model 1 and Model 2 are *indistinguishable* if for all $p' \in \Theta'$, there exists at least one $p\in \Theta$ such that $c(p)=c'(p')$, and vice versa; - Model 1 and Model 2 are *distinguishable* if they are not indistinguishable. **Remark 11**. The definition of indistinguishability is equivalent to saying that $c(\Theta) = c'(\Theta')$, i.e. the images of the coefficient maps are the same. A simple observation on distinguishability is that indistinguishable models must have the same vanishing ideal on the image of the parametrization [@meshkat-rosen-sullivant]. This means that the coefficients must satisfy the same algebraic dependency relationships. This is usually easy to check in small to medium-sized examples. **Example 12**. \[**Distinguishable via coefficient dependency relationships**\] Consider the following models in Figure [\[fig:twoinputdist\]](#fig:twoinputdist){reference-type="ref" reference="fig:twoinputdist"}. The model $\mathcal{M}$ has the form: $$\begin{aligned} \dot{x}_1 = -(a_{01}+a_{21})x_1 + u_1 \\ \dot{x}_2 = a_{21}x_1 -a_{32}x_2 + u_2 \\ \dot{x}_3 = a_{32}x_2 \\ y_3 = x_3 \end{aligned}$$ while the model $\mathcal{M}'$ has the form: $$\begin{aligned} \dot{x}_1 = -a_{21}x_1 + u_1 \\ \dot{x}_2 = a_{21}x_1 -(a_{02} + a_{32})x_2 + u_2 \\ \dot{x}_3 = a_{32}x_2 \\ y_3 = x_3 \end{aligned}$$ The input-output equations for $\mathcal{M}$ and $\mathcal{M}'$ are, respectively: $$\begin{aligned} \dddot{y}_3 + (a_{32} + a_{01} + a_{21})\ddot{y}_3 + (a_{01}a_{32} + a_{21}a_{32})\dot{y}_3 = a_{21}a_{32}u_1 + a_{32}\dot{u}_2 + (a_{01}a_{32} + a_{21}a_{32}) u_2 \\ \dddot{y}_3 + (a_{32} + a_{02} + a_{21})\ddot{y}_3 + (a_{02}a_{21} + a_{21}a_{32})\dot{y}_3 = a_{21}a_{32}u_1 + a_{32}\dot{u}_2 + a_{21}a_{32} u_2\end{aligned}$$ If we label the coefficients $c_1,...,c_5$ for $\mathcal{M}$ and $c'_1,...,c'_5$ for $\mathcal{M}'$, then the algebraic dependency relationships among the coefficients for $\mathcal{M}$ and $\mathcal{M}'$ are the following, respectively: $$\begin{aligned} c_2 - c_5 = 0, c_1c_4 - c_4^2 -c_5 = 0 \\ c'_3 - c'_5 = 0, c'_2{c'_4}^2 -c'_1c'_4c'_5 + {c'_5}^2 =0\end{aligned}$$ Since the algebraic dependency relationships among the coefficients are different, the models are distinguishable. Once the same vanishing ideal has been established, an approach for checking indistinguishability is to construct the equation system $c(p) = c'(p')$ and attempt to "solve" for one set of parameters in terms of the other, and vice versa, using Gröbner basis calculations [@meshkat-rosen-sullivant]. Once this has been done, one must check the resulting solutions to determine if they satisfy the necessary inequality constraints of the parameter spaces $\Theta$ and $\Theta'$. We note that identifiable models with coefficient maps satisfying the same algebraic dependence relationships can always be solved for one set of parameters in terms of the other, and vice versa, but the parameter constraints (e.g. positive parameters) must still be checked for indistinguishability to hold [@meshkat-rosen-sullivant]. While the method above gives an approach to test indistinguishability, we note that this approach involves symbolic computation, which can be computationally expensive for medium to large-sized examples. Thus, in this work, we seek out a different approach to test indistinguishability. The motivation comes from the next example, where we consider indistinguishability of linear compartmental models via a permutation of parameters. Thus, rather than testing that the images of the coefficient maps are the same using Gröbner basis calculations, we can simply show that the coefficient maps are identical up to a renaming of parameters: **Example 13** (**Indistinguishable via permutation of parameters**). Consider the models in Figure [\[fig:permute\]](#fig:permute){reference-type="ref" reference="fig:permute"}. The model $\mathcal{M}$ has the form: $$\begin{aligned} \dot{x}_1 = -(a_{01}+a_{21})x_1 + u_1 \\ \dot{x}_2 = a_{21}x_1 -a_{32}x_2 \\ \dot{x}_3 = a_{32}x_2 \\ y_3 = x_3 \end{aligned}$$ while the model $\mathcal{M}'$ has the form: $$\begin{aligned} \dot{x}_1 = -a_{21}x_1 + u_1 \\ \dot{x}_2 = a_{21}x_1 -(a_{02} + a_{32})x_2 \\ \dot{x}_3 = a_{32}x_2 \\ y_3 = x_3 \end{aligned}$$ and the model $\mathcal{M}''$ has the form: $$\begin{aligned} \dot{x}_1 = -a_{21}x_1 + u_1 \\ \dot{x}_2 = a_{21}x_1 -a_{32}x_2 + a_{23}x_3 \\ \dot{x}_3 = a_{32}x_2 -a_{23}x_3 \\ y_3 = x_3 \end{aligned}$$ The input-output equations are of the form: $$\begin{aligned} \mathcal{M}: &\quad \dddot{y}_3+(a_{32}+a_{01}+a_{21})\ddot{y}_3+(a_{01}a_{32}+a_{21}a_{32})\dot{y}_3 = (a_{21}a_{32})u_1\\ \mathcal{M}': & \quad \dddot{y}_3+(a_{21}+a_{02}+a_{32})\ddot{y}_3+(a_{02}a_{21}+a_{32}a_{21})\dot{y}_3 = (a_{32}a_{21})u_1\\ \mathcal{M}'': & \quad \dddot{y}_3+(a_{21}+a_{23}+a_{32})\ddot{y}_3+(a_{23}a_{21}+a_{32}a_{21})\dot{y}_3 = (a_{32}a_{21})u_1\end{aligned}$$ The three coefficient maps are exactly the same up to a renaming of parameters, i.e., we can rename the parameters of $\mathcal{M}$ (first column), $\mathcal{M}'$ (second column), $\mathcal{M}''$ (third column), respectively: $$\begin{aligned} a_{32} \equiv a_{21} \equiv a_{21}\\ a_{01} \equiv a_{02} \equiv a_{23}\\ a_{21} \equiv a_{32} \equiv a_{32}\end{aligned}$$ Thus we can see these models are indistinguishable, via this renaming of parameters. From Example [Example 13](#ex:permute){reference-type="ref" reference="ex:permute"}, it appears that one way we can go from model to model is by moving the leak. However, the situation is a little bit more nuanced as the next example shows: **Example 14**. \[**Moving leak does not always result in indistinguishability**\] Consider now the model $\mathcal{M}'''$ in Figure [\[fig:movingleaknonindist\]](#fig:movingleaknonindist){reference-type="ref" reference="fig:movingleaknonindist"}, where we move a leak from compartment 2 in $\mathcal{M}'$ in Figure [\[fig:permute\]](#fig:permute){reference-type="ref" reference="fig:permute"} to compartment 3 in $\mathcal{M}'''$. The model $\mathcal{M}'''$ has the form: $$\begin{aligned} \dot{x}_1 = -a_{21}x_1 + u_1 \\ \dot{x}_2 = a_{21}x_1 -a_{32}x_2 \\ \dot{x}_3 = a_{32}x_2 -a_{03}x_3 \\ y_3 = x_3 \end{aligned}$$ Here the structure of the input-output equation is different from the models $\mathcal{M}, \mathcal{M}', \mathcal{M}''$, as the input-output equation has a constant term in $y_3$: $$\begin{aligned} \mathcal{M}''': & \quad \dddot{y}_3+(a_{21}+a_{32}+a_{03})\ddot{y}_3+(a_{03}a_{32}+a_{32}a_{21})\dot{y}_3 + (a_{21}a_{32}a_{03})y_3 = (a_{32}a_{21})u_1\\\end{aligned}$$ Thus, $\mathcal{M}'''$ is distinguishable from models $\mathcal{M}, \mathcal{M}', \mathcal{M}''$. While indistinguishability from a permutation of parameters is easy to check, we note that it is possible to have indistinguishable models that do not result from a permutation of parameters. Consider the following example: **Example 15**. \[**Indistinguishable, but not from permutation of parameters**\] Consider the models in Figure [\[fig:nonpermute\]](#fig:nonpermute){reference-type="ref" reference="fig:nonpermute"}. The model $\mathcal{M}$ has the form: $$\begin{aligned} \dot{x}_1 = -(a_{01}+a_{21})x_1 + a_{12}x_2 + u_1 \\ \dot{x}_2 = a_{21}x_1 \\ y_2 = x_2 \end{aligned}$$ while the model $\mathcal{M}'$ has the form: $$\begin{aligned} \dot{x}_1 = a_{21}x_1 + a_{12}x_2 + u_1 \\ \dot{x}_2 = a_{21}x_1 - (a_{02} + a_{12})x_2 \\ y_2 = x_2 \end{aligned}$$ and the model $\mathcal{M}''$ has the form: $$\begin{aligned} \dot{x}_1 = -(a_{01}+a_{21})x_1 + u_1 \\ \dot{x}_2 = a_{21}x_1 -a_{02}x_2 \\ y_2 = x_2 \end{aligned}$$ The input-output equations are of the form: $$\begin{aligned} \mathcal{M}: &\quad \ddot{y}_2+(a_{01}+a_{21}+a_{12})\dot{y}_2+(a_{01}a_{12})y_2 = a_{21}u_1\\ \mathcal{M}': &\quad \ddot{y}_2+(a_{02}+a_{21}+a_{12})\dot{y}_2+(a_{02}a_{21})y_2 = a_{21}u_1\\ \mathcal{M}'': &\quad \ddot{y}_2+(a_{01}+a_{21}+a_{02})\dot{y}_2+(a_{01}a_{02}+a_{21}a_{02})y_2 = a_{21}u_1\\\end{aligned}$$ We see that $a_{21}$ must play the same role in all three models. However, there is no renaming of parameters that will map one coefficient map to the next. Nevertheless, these three models are indistinguishable. We showed in Example [Example 8](#ex:ident){reference-type="ref" reference="ex:ident"} that model $\mathcal{M}$ is locally identifiable. Likewise, it can easily be shown that $\mathcal{M}'$ and $\mathcal{M}''$ are also (at least) locally identifiable. Since the coefficient maps are (at most) finite-to-one from 3 parameters to 3 coefficients, this means they are all surjective maps, thus their images are the same as they are all equal to $\mathbb{R}^3$. One must still check any constraints on the parameters, e.g. we can assume the parameters must be positive. Since the coefficients are all positive assuming positive parameters, then their images are the same as they are equal to $\mathbb{R}_{>0}^3$, and the models are indistinguishable, but *not* via a renaming of the parameters. Alternatively, instead of checking (local) injectivity and surjectivity, one could construct the equation system $c(p) = c'(p')$ and attempt to "solve" for one set of parameters in terms of the other, and vice versa, using Gröbner basis calculations. For example, letting $c(p)$ correspond to model $\mathcal{M}'$ where $p=(a_{02},a_{21},a_{12})$and letting $c'(p')$ correspond to model $\mathcal{M}''$ where $p'=(b_{01},b_{21},b_{02})$, we obtain $p$ in terms of $p'$: $$a_{21}=b_{21}, a_{02}=\frac{b_{02}b_{01}+b_{02}b_{21}}{b_{21}}, a_{12}=\frac{-b_{01}b_{02}+b_{01}b_{21}}{b_{21}}$$ and likewise we can solve for $p'$ in terms of $p$: $$\begin{gathered} b_{01} = \frac{1}{2}(a_{02}+a_{12}-a_{21} \pm \sqrt{a_{02}^2+2a_{02}a_{12}+a_{12}^2-2a_{02}a_{21}+2a_{12}a_{21}+a_{21}^2}), \\ b_{21}=a_{21}, \\ b_{02} = \frac{1}{2}(a_{02}+a_{12}+a_{21} \mp \sqrt{(-a_{02}-a_{12}-a_{21})^2-4a_{02}a_{21}})\end{gathered}$$ See [@meshkat-rosen-sullivant] for more examples of this technique. ## Godfrey-Chapman geometric rules {#subsection:gfgeorules} Our goal in this paper is to find sufficient conditions for indistinguishability based on graph structure. Below, we outline previous work that gives necessary conditions for indistinguishability based on graph structure, which we refer to as the Godfrey-Chapman geometric rules [@godfrey-chapman]. ### *Rule 1.* The length of the shortest path from any perturbed (input) compartment to any observed (output) compartment is preserved. ### *Rule 2.* The number of compartments with a path to any given observed compartment is preserved. Note that this is satisfied if we assume our model is output connectable (see Definition [Definition 3](#defn:outputreachable){reference-type="ref" reference="defn:outputreachable"}). ### Rule 3. The number of compartments that can be reached from a given perturbed compartment is preserved. Note that this is satisfied if we assume our model is input connectable (see Definition [Definition 3](#defn:outputreachable){reference-type="ref" reference="defn:outputreachable"}). ### Rule 4. The number of traps is preserved. A trap is a compartment or a strongly connected set of compartments from which no path exists to any compartment outside the trap, including the environment (i.e. leaks). Note that if we again assume output connectability, then the only possible traps must include an output. We will see shortly that the class of graphs we consider automatically satisfy these conditions. **Remark 16**. Rule 1 directly connects to the number of coefficients on the right-hand side of the input-output equation. In the case when there is a single input and single output, the number of nontrivial coefficients on the right-hand side of the corresponding input-output equation according the Corollary 3.4 of [@treemodels] is: $$\# \text{ on RHS} = \begin{cases} n-1 & \text{ if } In=Out \\ n-\text{dist}(In,Out) & \text{ if } In \neq Out \end{cases}.$$ Here, $\text{dist}(In,Out)$ corresponds to the length of the shortest directed path from $In$ to $Out$. In the case of multiple inputs or multiple outputs, we simply apply this formula for each distinct input/output pair. ## Permutation indistinguishability and class of graphs The previous examples demonstrate the need to find the input-output equations first before testing indistinguishability. However, finding the input-output equations and then testing the sufficient conditions requires symbolic computation, which can be tedious for large models. Our goal in this paper is to circumvent that step of finding the input-output equations and instead determine indistinguishability of certain models from the graph itself. We will restrict our efforts to a certain class of models and thus find sufficient conditions for indistinguishability based on the graph structure. We narrow down our class of models to *paths* from input to output with various detours. The basic starting point is a *path model*: **Definition 17**. An n-compartment model $\mathcal{M}= (G, In, Out, Leak)$ is a *path model* if $G$ is given by a path from compartment $1$ to compartment $n$, $In=\{1\}$, and $Out=\{n\}$. We denote the graph $G$ as $P_n$. The graph in Figure [\[fig:path\]](#fig:path){reference-type="ref" reference="fig:path"} gives a path model without any leaks. Notice that in a path model, the output-reachable subgraph is the entire graph, thus the model is output connectable. It is also input connectable. That is, every compartment has a path to the output and every compartment can be reached from the input. Thus, these models automatically satisfy Rules 2 and 3 of the Godfrey-Chapman necessary conditions. Rule 1 is automatically satisfied as each model contains a simple path from input to output. To satisfy Rule 4, we require that compartment $n$ does not contain a leak, which explains why the model $\mathcal{M}'''$ in Figure [\[fig:movingleaknonindist\]](#fig:movingleaknonindist){reference-type="ref" reference="fig:movingleaknonindist"} is not indistinguishable with models $\mathcal{M}, \mathcal{M}'$, and $\mathcal{M}''$ in Example [Example 14](#ex:movingleaknonindist){reference-type="ref" reference="ex:movingleaknonindist"}. We will also be considering variations on path models that have extra vertices, edges, or leaks. One such example is a path model with a cycle along the last two vertices, which we call a *terminal cycle*, as in model $\mathcal{M}'$ in Figure [\[fig:p3leak2ex\]](#fig:p3leak2ex){reference-type="ref" reference="fig:p3leak2ex"}. We call models such as these *skeletal path models*, as the graph contains a "backbone" of a path along with other vertices, edges, or leaks: **Definition 18**. A model $\mathcal{M}= (G, In, Out, Leak)$ is a *skeletal path model* if $G$ contains a path from compartment $1$ to $n$, $In={1}$, and $Out={n}$. It is clear that Rules 1-4 of the Godfrey-Chapman necessary conditions are satisfied for skeletal path models as long as the shortest path from input to output is the "skeletal path" and there is no leak from compartment $n$. Now we will prove indistinguishability via a simple permutation of parameters: **Definition 19**. We say two models $\mathcal{M}$ and $\mathcal{M}'$ are *indistinguishable via a permutation of parameters* or *permutation indistinguishable* if models $\mathcal{M}$ and $\mathcal{M}'$ have the same input-output equations up to a renaming of the parameters. This means there is a bijection $\Phi$ from the parameters in $\mathcal{M}$ to the parameters in $\mathcal{M}'$ such that the coefficients of $\mathcal{M}'$ are exactly the coefficients of $\mathcal{M}$ under $\Phi$. **Remark 20**. Under the map $\Phi$, we have $p \mapsto p'$. We will abuse notation and apply $\Phi$ to polynomial functions $f(p)$ as well, where $\Phi(f(p)) = f(\Phi(p))$. We now show models that are permutation indistinguishable are, in fact, indistinguishable. **Proposition 21**. *Assume $\mathcal{M}$ and $\mathcal{M}'$ are permutation indistinguishable. Then they are also indistinguishable.* *Proof.* By assumption, $\mathcal{M}$ and $\mathcal{M}'$ have the same input-output equations up to a renaming of the parameters. Thus they have the same input-output equation structure and satisfy the same coefficient dependency relationships. Let $c(p)$ be the coefficient map for $\mathcal{M}$ and $c(p')$ be the coefficient map for $\mathcal{M}'$, where $p'$ is a renaming of the parameters $p$. Then clearly the image of $c(p)$ equals the image of $c(p')$, and thus the models are indistinguishable. ◻ We end this section by noting that permutation indistinguishability is an equivalence relation. **Proposition 22**. *Permutation indistinguishability is an equivalence relation.* *Proof.* - Reflexive: A model $\mathcal{M}$ is permutation indistinguishable from itself via the identity map. - Symmetric: Assume a model $\mathcal{M}$ is indistinguishable from a model $\mathcal{M}'$. This means there is a bijection $\Phi$ from the coefficients of $\mathcal{M}$ to the coefficients of $\mathcal{M}'$. The inverse map $\Phi^{-1}$ proves that $\mathcal{M}'$ is indistinguishable from $\mathcal{M}$. - Transitive: Assume $\mathcal{M}$ is indistinguishable from $\mathcal{M}'$ and $\mathcal{M}'$ is indistinguishable from $\mathcal{M}''$. This means there is a bijection $\Phi$ from the coefficients $\mathcal{M}$ to the coefficients of $\mathcal{M}'$ and also a bijection $\Psi$ from the coefficients $\mathcal{M}'$ to the coefficients of $\mathcal{M}''$. Composing these bijections, we get that there is a bijection $\Psi (\Phi)$ from the coefficients $\mathcal{M}$ to the coefficients of $\mathcal{M}''$, thus $\mathcal{M}$ is indistinguishable from $\mathcal{M}''$.  ◻ # Paths with leaks and terminal cycles {#section:leak} Recall Figure [\[fig:permute\]](#fig:permute){reference-type="ref" reference="fig:permute"} and Example [Example 13](#ex:permute){reference-type="ref" reference="ex:permute"}, which we revisit now: **Example 23**. Consider the two models $\mathcal{M}= (P_3,\{1\},\{3\},\{1\})$ and $\mathcal{M}'=(P_3,\{1\},\{3\},\{2\})$ shown in Figure [\[fig:permute\]](#fig:permute){reference-type="ref" reference="fig:permute"}. The input-output equations of $\mathcal{M}$ and $\mathcal{M}'$ respectively are $$\begin{aligned} \mathcal{M}: &\quad \dddot{y}_3+(a_{32}+a_{01}+a_{21})\ddot{y}_3+(a_{01}a_{32}+a_{21}a_{32})\dot{y}_3 = (a_{21}a_{32})u_1\\ \mathcal{M}': & \quad \dddot{y}_3+(a_{21}+a_{02}+a_{32})\ddot{y}_3+(a_{02}a_{21}+a_{32}a_{21})\dot{y}_3 = (a_{32}a_{21})u_1\\\end{aligned}$$ Recall these models are indistinguishable via a renaming of parameters. We describe this renaming using the map $\Phi$ we defined in Definition [Definition 19](#defn:permutation){reference-type="ref" reference="defn:permutation"}. The models $\mathcal{M}$ and $\mathcal{M}'$ are indistinguishable under the map $$\Phi \colon \begin{pmatrix} a_{01} \\ a_{21} \\ a_{32} \end{pmatrix} \mapsto \begin{pmatrix} a_{02} \\ a_{32} \\ a_{21} \end{pmatrix}$$ We would now like to generalize this example by considering a path model with $n$ vertices. The basic path model with no leaks has a compartmental matrix of the form: $$\label{eq:comp-mat} A = \begin{bmatrix} -a_{21} & 0 & \cdots & \cdots & 0 & 0 \\ a_{21} & -a_{32} & \ddots & \ddots & \vdots & \vdots \\ 0 & a_{32} & -a_{43} & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \ddots &0 & 0 \\ 0 & \cdots & 0 & \ddots & -a_{n(n-1)} & 0 \\ 0 & \cdots & 0 & 0& a_{n(n-1)} & 0 \end{bmatrix}$$ The compartmental matrix of a basic path model with the first row and last column removed has the form: $$\label{eq:comp-mat1n} A^{1,n} = \begin{bmatrix} a_{21} & -a_{32} & 0 & \cdots & 0 \\ 0 & a_{32} & -a_{43} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots &0 \\ 0 & \cdots & 0 & \ddots & -a_{n(n-1)} \\ 0 & \cdots & 0 & 0& a_{n(n-1)} \end{bmatrix}$$ **Theorem 24**. *The path models $\mathcal{M}_i=(P_n,\{1\},\{n\},\{i\})$ and $\mathcal{M}_j = (P_n,\{1\},\{n\},\{j\})$ are indistinguishable for all $i,j <n$.* *Proof.* Let $A$ be the compartmental matrix of $\mathcal{M}_i$, and $B$ be the compartmental matrix of $\mathcal{M}_j$. Both $A$ and $B$ have form similar to Equation [\[eq:comp-mat\]](#eq:comp-mat){reference-type="ref" reference="eq:comp-mat"} with an additional $a_{0i}$ and $a_{0j}$ subtracted from the $i^\text{th}$ and $j^\text{th}$ diagonal elements respectively. The same is true for the matrices $\partial I -A$ and $\partial I -B$. From Theorem [Theorem 4](#thm:ioscc){reference-type="ref" reference="thm:ioscc"}, and specifically Equation [\[eq:io-det\]](#eq:io-det){reference-type="ref" reference="eq:io-det"}, the left-hand sides of the input-output equations of $\mathcal{M}$ and $\mathcal{M}'$ are generated by the determinant of the $\partial I -A$ and $\partial I -B$ matrices respectively. Since $\partial I -A$ is a lower triangular matrix, the differential operator of the left-hand side of the input-output equation of $\mathcal{M}$ is: $$\label{eq:lhs-leakmove} \left( \frac{d}{dt} + a_{0i} + a_{(i+1)i} \right) \left( \frac{d}{dt} \right) \prod_{\substack{k \in [n],\\ k \neq i}} \left( \frac{d}{dt} + a_{(k+1)k} \right)$$ Similarly, the differential operator of the left-hand side of the input-output equation of $\mathcal{M}'$ is: $$\left( \frac{d}{dt} + a_{0j} + a_{(j+1)j} \right) \left( \frac{d}{dt} \right) \prod_{\substack{k \in [n],\\ k \neq j}} \left( \frac{d}{dt} + a_{(k+1)k} \right)$$ According to Theorem [Theorem 4](#thm:ioscc){reference-type="ref" reference="thm:ioscc"}, and again Equation [\[eq:io-det\]](#eq:io-det){reference-type="ref" reference="eq:io-det"}, the right-hand side of the input-output equations of $\mathcal{M}$ and $\mathcal{M}'$ are the determinants of $(\partial I - A)^{1,n}$ and $(\partial I - B)^{1,n}$ respectively. Note that in either case, since $\mathcal{M}$ and $\mathcal{M}'$ are identical outside of the leak element, and both $(\partial I - A)^{1,n}$ and $(\partial I - B)^{1,n}$ are upper-triangular, as seen in Equation [\[eq:comp-mat1n\]](#eq:comp-mat1n){reference-type="ref" reference="eq:comp-mat1n"}, with no leak terms on the main diagonal. Therefore, the right-hand side differential operators of both $\mathcal{M}$ and $\mathcal{M}'$ are $$\label{eq:rhs-leakmove} (-1)^{1+n}\prod_{k\in [n]} \left(- a_{(k+1)k} \right) = (-1)^{1+n} \cdot (-1)^{n} \prod_{k\in [n]} \left(a_{(k+1)k} \right) = - \prod_{k\in [n]} \left(a_{(k+1)k} \right).$$ Thus, under the \[bijective\] map $\Phi$ from the parameters of $\mathcal{M}$ to the parameters of $\mathcal{M}'$ defined as: $$\Phi( a_{uv} ) = \begin{cases} a_{0j} & \text{ for }u=0\ \& \ v=i \\ a_{(j+1)j} & \text{ for }u=(i+1)\ \&\ v=i \\ a_{(i+1)i} & \text{ for }u=(j+1) \ \&\ v=j \\ a_{uv} & \text{ otherwise} \end{cases}$$ $\mathcal{M}$ and $\mathcal{M}'$ are permutation indistinguishable. ◻ We now again consider Figure [\[fig:permute\]](#fig:permute){reference-type="ref" reference="fig:permute"} and Example [Example 13](#ex:permute){reference-type="ref" reference="ex:permute"}, which we revisit and rename the models to those in Figure [\[fig:p3leak2ex\]](#fig:p3leak2ex){reference-type="ref" reference="fig:p3leak2ex"}. **Example 25**. Consider the two models $\mathcal{M}= (P_3,\{1\},\{3\},\{2\})$ and $\mathcal{M}'=(P_3\cup \{3 \to 2\},\{1\},\{3\},\emptyset)$ shown in Figure [\[fig:p3leak2ex\]](#fig:p3leak2ex){reference-type="ref" reference="fig:p3leak2ex"}. The input-output equations of $\mathcal{M}$ and $\mathcal{M}'$ respectively are $$\begin{aligned} \mathcal{M}: & \quad \dddot{y}_3+(a_{21}+a_{02}+a_{32})\ddot{y}_3+(a_{02}a_{21}+a_{32}a_{21})\dot{y}_3 = (a_{32}a_{21})u_1\\ \mathcal{M}': & \quad \dddot{y}_3+(a_{21}+a_{23}+a_{32})\ddot{y}_3+(a_{23}a_{21}+a_{32}a_{21})\dot{y}_3 = (a_{32}a_{21})u_1 \end{aligned}$$ Recall these models are indistinguishable via a renaming of parameters. We describe this renaming using the map $\Phi$ we defined in Definition [Definition 19](#defn:permutation){reference-type="ref" reference="defn:permutation"}. The models $\mathcal{M}$ and $\mathcal{M}'$ are indistinguishable under the map $$\Phi \colon \begin{pmatrix} a_{02} \\ a_{21} \\ a_{32} \end{pmatrix} \mapsto \begin{pmatrix} a_{23} \\ a_{21} \\ a_{32} \end{pmatrix}$$ We would now like to generalize this example. **Theorem 26**. *The path models $\mathcal{M}=(P_n,\{1\},\{n\},\{n-1\})$ and $\mathcal{M}' = (P_n \cup \{n \to n-1\},\{1\},\{n\},\emptyset)$ are indistinguishable.* *Proof.* The model $\mathcal{M}'$ has compartmental matrix of the form: $$\label{eq:comp-mat2} A' = \begin{bmatrix} -a_{21} & 0 & \cdots & \cdots & 0 & 0 \\ a_{21} & -a_{32} & \ddots & \ddots & \vdots & \vdots \\ 0 & a_{32} & -a_{43} & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \ddots &0 & 0 \\ 0 & \cdots & 0 & \ddots & -a_{n(n-1)} & a_{(n-1)n} \\ 0 & \cdots & 0 & 0& a_{n(n-1)} & -a_{(n-1)n} \end{bmatrix}$$ Again, via Theorem [Theorem 4](#thm:ioscc){reference-type="ref" reference="thm:ioscc"} the left-hand side of the input-output equation corresponding to $\mathcal{M}'$ has form: $$\left[\underbrace{\left( \frac{d}{dt}+a_{(n-1)n}\right)\left(\frac{d}{dt}+a_{n(n-1)} \right)-a_{n(n-1)}a_{(n-1)n}}_{\text{bottom right $2\times 2$ block}} \right] \underbrace{\prod_{i\in [n-1]} \left( \frac{d}{dt} + a_{(k+1)k} \right)}_{\text{first $n-2$ diagonal elements}}$$ equivalently $$\label{eq:leak2exio} \left(\frac{d}{dt}+a_{(n-1)n}+a_{n(n-1)}\right) \left(\frac{d}{dt}\right) \prod_{i\in [n-1]} \left( \frac{d}{dt} + a_{(k+1)k} \right)$$ Removing the first row and last column of the $A'$ compartmental matrix yields a matrix of the same form as that in Equation [\[eq:comp-mat1n\]](#eq:comp-mat1n){reference-type="ref" reference="eq:comp-mat1n"}. Therefore, the right-hand side of the input-output equation of $\mathcal{M}'$ has the same form as Equation [\[eq:rhs-leakmove\]](#eq:rhs-leakmove){reference-type="ref" reference="eq:rhs-leakmove"}. Thus, under the \[bijective\] map $\Phi$ from the parameters of $\mathcal{M}$ to the parameters of $\mathcal{M}'$ defined as: $$\Phi( a_{uv} ) = \begin{cases} a_{(n-1)n} & \text{ for }u=0\ \& \ v=(n-1)\\ a_{uv} & \text{ otherwise} \end{cases}$$ $\mathcal{M}$ and $\mathcal{M}'$ are permutation indistinguishable. ◻ # Detour Indistinguishability {#section:detour} We now consider a scenario as in Figure [\[fig:detour1\]](#fig:detour1){reference-type="ref" reference="fig:detour1"}, where a basic path model has a "detour". These models still satisfy our conditions of skeletal path models. **Theorem 27**. *Let $P_n$ be the directed path graph from $1$ to $n$, and let $D$ be some connected directed graph. Then, for some $1 \leq i < j \leq n-1$ and $s,t \in V(D)$, the *detour model* $$\mathcal{M}= (P_n \cup D \cup \{i \to s\} \cup \{t \to j\} , \{1\},\{n\},Leak)$$ is indistinguishable to the model $$\mathcal{M}' = (P_n \cup D \cup \{(i+1) \to s\} \cup \{t \to (j+1)\} , \{1\},\{n\},Leak)$$* *Proof.* We claim the following map between the coefficients of $\mathcal{M}$ and $\mathcal{M}'$ yields indistinguishability between these two models: $$\Phi(a_{uv}) = \begin{cases} a_{(u+1)(v+1)}, \text{ if } u=v+1 \text{ and } u < n & (\text{edges along $P_n$ not including last edge})\\ a_{21}, \text{ if } u=n \text{ and } v=n-1& (\text{last edge on $P_n$}) \\ %a_{u(v+1)} a_{s(i+1)}, \text{ if } u = s \text{ and } v = i & (\text{off-ramp off of path}) \\ %a_{(u+1)v} a_{(j+1)t}, \text{ if } u = j \text{ and } v = t & (\text{on-ramp back to path}) \\ a_{uv}, \text{ otherwise } & (\text{on detour}) \end{cases}$$ First, we construct the compartmental matrix of the detour model, which we call $A$, using the compartment matrices of the elements of the detour model. Let $A_D$ be the compartmental matrix of the underlying detour with an additional $a_{jt}$ in the $j^{\text{th}}$ diagonal sum. Then, the compartmental matrix of the detour model $\mathcal{M}$ can be generated as a block matrix as follows: $$\partial I - A = \left[\begin{array}{cccccc|c} \partial + a_{21} &0 &\cdots &\cdots & 0 & 0& \mathbf{0} \\ -a_{21} & \ddots&\ddots &&\vdots&\vdots &\mathbf{0} \\ 0 & \ddots & \partial + a_{si}+a_{(i+1)i} & \ddots &\vdots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & 0& -a_{jt}\\ \vdots & & \ddots & \ddots & \partial + a_{n(n-1)} & 0 & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{n(n-1)} & \partial &\mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} & -a_{si} & \mathbf{0} & \mathbf{0} &\mathbf{0} & \partial I - A_D \end{array} \right]$$ where the top-right block only contains the element $a_{jt}$ (in row $j$ and column $t$) corresponding to the edge from the detour back onto the path, and similarly the bottom-left block only contains the element $a_{si}$ (in row $s$ and column $i$) corresponding to the edge from the path to the detour, with all other entries being zero. The second detour model $\mathcal{M}'$ has compartmental matrix of the form: $$\partial I - A' = \left[\begin{array}{cccccc|c} \partial + a_{21} &0 &\cdots &\cdots & 0 & 0& \mathbf{0} \\ -a_{21} & \ddots&\ddots &&\vdots&\vdots &\mathbf{0} \\ 0 & \ddots & \partial + a_{s(i+1)}+a_{(i+2)(i+1)} & \ddots &\vdots &\vdots & \mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & 0& -a_{(j+1)t}\\ \vdots & & \ddots & \ddots & \partial + a_{n(n-1)} & 0 & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{n(n-1)} & \partial &\mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} & -a_{s(i+1)} & \mathbf{0} & \mathbf{0} &\mathbf{0} & \partial I - A_D' \end{array} \right]$$ According to Equation [\[eq:io-det\]](#eq:io-det){reference-type="ref" reference="eq:io-det"}, the coefficients of the left-hand side of the input-output equation of $\mathcal{M}$ and $\mathcal{M}'$ come from the determinants $\det(\partial I - A)$ and $\det(\partial I - A')$ respectively. Note that in both determinants, we can expand along the $n^{\text{th}}$ (the last column in the upper-left block), which only contains a $\partial$ yielding: $$\label{eq:lhsA} \det(\partial I - A) =\partial \det\left( \begin{array}{ccccc|c} \partial + a_{21} &0 &\cdots &\cdots & 0 & \mathbf{0} \\ -a_{21} & \ddots&\ddots &&\vdots &\mathbf{0} \\ 0 & \ddots & \partial + a_{si}+a_{(i+1)i} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{jt}\\ 0 &\cdots & 0 & -a_{(n-1)(n-2)} & \partial + a_{n(n-1)} & \mathbf{0} \\ \hline \mathbf{0} & \mathbf{0} & -a_{si} & \mathbf{0} & \mathbf{0} & \partial I - A_D \end{array} \right)$$ and $$\label{eq:lhsAprime} \det(\partial I - A') =\partial \det\left( \begin{array}{ccccc|c} \partial + a_{21} &0 &\cdots &\cdots & 0 & \mathbf{0} \\ -a_{21} & \ddots&\ddots &&\vdots &\mathbf{0} \\ 0 & \ddots & \partial + a_{s(i+1)}+a_{(i+2)(i+1)} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{(j+1)t}\\ 0 &\cdots & 0 & -a_{(n-1)(n-2)} & \partial + a_{n(n-1)} & \mathbf{0} \\ \hline \mathbf{0} & \mathbf{0} & -a_{s(i+1)} & \mathbf{0} & \mathbf{0} & \partial I - A_D' \end{array} \right).$$ Expanding down the $n-1^{\text{st}}$ column of the determinant in Equation [\[eq:lhsA\]](#eq:lhsA){reference-type="ref" reference="eq:lhsA"} and the first row in Equation [\[eq:lhsAprime\]](#eq:lhsAprime){reference-type="ref" reference="eq:lhsAprime"}, we get $$\label{eq:lhsAshifted2} \det(\partial I - A) =\partial(\partial + a_{n(n-1)}) \det\left( \begin{array}{ccccc|c} \partial +a_{21} &0 &\cdots & \cdots & 0 & \mathbf{0} \\ -a_{21} & \ddots&\ddots & & \vdots &\mathbf{0} \\ 0 & \ddots & \partial + a_{si}+a_{(i+1)i} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{jt}\\ 0 &\cdots & 0 & -a_{(n-2)(n-3)} & \partial + a_{(n-1)(n-2)} & \mathbf{0} \\ \hline \mathbf{0} & \mathbf{0} & -a_{si} & \mathbf{0} & \mathbf{0} & \partial I - A_D \end{array} \right)$$ and $$\label{eq:lhsAprime2} \det(\partial I - A') =\partial(\partial + a_{21}) \det\left( \begin{array}{ccccc|c} \partial +a_{32} &0 &\cdots &\cdots & 0 & \mathbf{0} \\ -a_{32} & \ddots&\ddots &&\vdots &\mathbf{0} \\ 0 & \ddots & \partial + a_{s(i+1)}+a_{(i+2)(i+1)} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{(j+1)t}\\ 0 &\cdots & 0 & -a_{(n-1)(n-2)} & \partial + a_{n(n-1)} & \mathbf{0} \\ \hline \mathbf{0} & \mathbf{0} & -a_{s(i+1)} & \mathbf{0} & \mathbf{0} & \partial I - A_D' \end{array} \right).$$ Note that under the map $\Phi$, the entries of the matrix in the determinant in Equation [\[eq:lhsAshifted2\]](#eq:lhsAshifted2){reference-type="ref" reference="eq:lhsAshifted2"} are exactly the entries of matrix in the determinant in Equation [\[eq:lhsAprime2\]](#eq:lhsAprime2){reference-type="ref" reference="eq:lhsAprime2"}. Also, since $\Phi(a_{n(n-1)})=a_{21}$, then $\Phi$ applied to the coefficients of $\det(\partial I -A))$ gives $\det(\partial I - A')$, meaning the coefficients of the derivatives of $y_n$ are indistinguishable under $\Phi$. Now we consider the coefficients of the derivatives of $u_1$ which, again according to Equation [\[eq:io-det\]](#eq:io-det){reference-type="ref" reference="eq:io-det"}, come from the determinants of $(\partial I - A)^{(1,n)}$ and $(\partial I - A')^{(1,n)}$, respectively. We split our argument into three cases: $i=1$ or $i =2$ or $i \geq 3$. For the case where $i \geq 3$ we have: $$(\partial I - A)^{1,n} = \left[\begin{array}{ccccc|c} -a_{21} & \partial + a_{32} &0 &\cdots &0 &\mathbf{0} \\ 0 & \ddots & \partial + a_{si}+a_{(i+1)i} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{jt}\\ \vdots & & \ddots & \ddots & \partial + a_{n(n-1)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{n(n-1)} &\mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} & -a_{si} & \mathbf{0} & \mathbf{0} & \partial I - A_D \end{array} \right]$$ and $$(\partial I - A')^{1,n} = \left[\begin{array}{ccccc|c} -a_{21} & \partial + a_{32} &0 & \cdots & 0 &\mathbf{0} \\ 0 & \ddots & \partial + a_{s(i+1)}+a_{(i+2)(i+1)} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots & \ddots & \ddots & 0 & -a_{(j+1)t}\\ \vdots & & \ddots & \ddots & \partial + a_{n(n-1)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{n(n-1)} &\mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} & -a_{s(i+1)} & \mathbf{0} & \mathbf{0} & \partial I - A_D' \end{array} \right]$$ Expanding along the $n-1^{\text{st}}$ row (last row in the top-left block) in $(\partial I - A)^{1,n}$ and the first column in $(\partial I - A')^{1,n}$ we get $$\det(\partial I - A)^{1,n} = -a_{n(n-1)} \det\left(\begin{array}{ccccc|c} -a_{21} & \partial + a_{32} &0 &\cdots &0 &\mathbf{0} \\ 0 & \ddots & \partial + a_{si}+a_{(i+1)i} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots & \ddots & \ddots & 0 & -a_{jt}\\ \vdots & & \ddots & \ddots & \partial + a_{(n-1)(n-2)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{(n-1)(n-2)} &\mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} & -a_{si} & \mathbf{0} & \mathbf{0} & \partial I - A_D \end{array} \right)$$ and $$\det(\partial I - A')^{1,n} = -a_{21}\det\left(\begin{array}{ccccc|c} -a_{32} & \partial + a_{43} &0 & \cdots & 0 &\mathbf{0} \\ 0 & \ddots & \partial + a_{s(i+1)}+a_{(i+2)(i+1)} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{(j+1)t}\\ \vdots & & \ddots & \ddots & \partial + a_{n(n-1)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{n(n-1)} &\mathbf{0}\\ \hline \mathbf{0} & \mathbf{0} & -a_{s(i+1)} & \mathbf{0} & \mathbf{0} & \partial I - A_D' \end{array} \right).$$ Thus, by the same argument as for the coefficients of the derivatives of $y_n$, the coefficients of $u_1$ are the same up to parameter permutation, as described by $\Phi$. For the case $i=1$, we can still expand along the $n-1^{st}$ row (last row in the top-left block) in $(\partial I - A)^{1,n}$ and the first column in $(\partial I - A')^{1,n}$, but our matrices look like: $$\det(\partial I - A)^{1,n} = -a_{n(n-1)} \det\left(\begin{array}{ccccc|c} -a_{21} & \partial + a_{32} &0 &\cdots &0 &\mathbf{0} \\ 0 & \ddots & \partial + a_{43} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots & \ddots & \ddots & 0 & -a_{jt}\\ \vdots & & \ddots & \ddots & \partial + a_{(n-1)(n-2)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{(n-1)(n-2)} &\mathbf{0}\\ \hline -a_{s1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \partial I - A_D \end{array} \right)$$ and $$\det(\partial I - A')^{1,n} = -a_{21}\det\left(\begin{array}{ccccc|c} -a_{32} & \partial + a_{43} &0 & \cdots & 0 &\mathbf{0} \\ 0 & \ddots & \partial + a_{54} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{(j+1)t}\\ \vdots & & \ddots & \ddots & \partial + a_{n(n-1)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{n(n-1)} &\mathbf{0}\\ \hline -a_{s2} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \partial I - A_D' \end{array} \right).$$ Thus, the coefficients of $u_1$ are the same up to parameter permutation, as described by $\Phi$. Likewise, for the case $i=2$, we can still expand along the $n-1^{st}$ row (last row in the top-left block) in $(\partial I - A)^{1,n}$ and the first column in $(\partial I - A')^{1,n}$, but our matrices look like: $$\det(\partial I - A)^{1,n} = -a_{n(n-1)} \det\left(\begin{array}{ccccc|c} -a_{21} & \partial + a_{32} + a_{s2} &0 &\cdots &0 &\mathbf{0} \\ 0 & -a_{32} & \partial + a_{43} &\vdots &\mathbf{0} \\ \vdots &\ddots & \ddots & \ddots & 0 & -a_{jt}\\ \vdots & & \ddots & \ddots & \partial + a_{(n-1)(n-2)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{(n-1)(n-2)} &\mathbf{0}\\ \hline \mathbf{0} & -a_{s2} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \partial I - A_D \end{array} \right)$$ and $$\det(\partial I - A')^{1,n} = -a_{21}\det\left(\begin{array}{ccccc|c} -a_{32} & \partial + a_{43} + a_{s3} &0 & \cdots & 0 &\mathbf{0} \\ 0 & -a_{43} & \partial + a_{54} & \ddots &\vdots &\mathbf{0} \\ \vdots &\ddots &\ddots & \ddots & 0 & -a_{(j+1)t}\\ \vdots & & \ddots & \ddots & \partial + a_{n(n-1)} & \mathbf{0} \\ 0 & \cdots & \cdots & 0 & -a_{n(n-1)} &\mathbf{0}\\ \hline \mathbf{0} & -a_{s3} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \partial I - A_D' \end{array} \right).$$ Thus, the coefficients of $u_1$ are the same up to parameter permutation, as described by $\Phi$. ◻ **Example 28**. Consider the linear compartmental models from Figure [\[fig:detour2\]](#fig:detour2){reference-type="ref" reference="fig:detour2"} with compartmental matrices: $$A = \begin{bmatrix} -a_{21} & 0 & 0 & 0 & 0 \\ a_{21} & -a_{32}-a_{52} & 0 & 0 & 0 \\ 0 & a_{32} & -a_{43} & 0 & a_{35} \\ 0 & 0 & a_{43} & 0 & 0 \\ 0 & a_{52} & 0 & 0 & -a_{35} \end{bmatrix} \text{ and } B =\begin{bmatrix} -a_{21} & 0 & 0 & 0 & 0 \\ a_{21} & -a_{32} & 0 & 0 & 0 \\ 0 & a_{32} & -a_{43}-a_{53} & 0 & 0 \\ 0 & 0 & a_{43} & 0 & a_{45} \\ 0 & 0 & a_{53} & 0 & -a_{45} \end{bmatrix}$$ The input-output equations are given by: $$(\partial + a_{21}) (\partial + a_{32}+a_{52})(\partial + a_{43}) \partial (\partial + a_{35}) y_4 =(-a_{21}a_{32}a_{43}(\partial + a_{35}) - a_{21}a_{35}a_{43}a_{52}) u_1$$ and $$(\partial + a_{21}) (\partial + a_{32})(\partial + a_{43}+a_{53}) \partial (\partial + a_{45})y_4 = (-a_{21}a_{32}a_{43}(\partial + a_{45}) - a_{32}a_{45}a_{21}a_{53}) u_1$$ These two models are indistinguishable via the following map: $$\begin{pmatrix} a_{21} \\ a_{32} \\ a_{43} \\ a_{35} \\ a_{52} \end{pmatrix} \mapsto \begin{pmatrix} a_{32} \\ a_{43} \\ a_{21} \\ a_{45} \\ a_{53} \end{pmatrix}$$ # Sink Models {#section:sink} We now extend the idea of indistinguishability of path models to a family of models called *sink models*. A sink model is defined as a combination of skeletal path models which all of have same output, as depicted in Figure [\[fig:basicsink\]](#fig:basicsink){reference-type="ref" reference="fig:basicsink"}. **Corollary 29**. *Suppose two sink models $\mathcal{M}$ and $\mathcal{M}'$ are identical outside of one path on each model which on their own are permutation indistinguishable, then $\mathcal{M}$ and $\mathcal{M}'$ are permutation indistinguishable.* *Proof.* Let $A$ and $A'$ be the compartmental matrices corresponding to $\mathcal{M}$ and $\mathcal{M}'$ respectively. Let $A_1$ be the compartmental matrix of the path model in $\mathcal{M}$ which differs from the corresponding path in $\mathcal{M}'$, but is indistinguishable. Let $A_1'$ be the corresponding compartmental matrix in $\mathcal{M}'$. $$A = \left[ \begin{array}{ccccc} A_1 & * & * & * & * \\ 0 & A_2 & 0 & \cdots & 0 \\ 0 & 0 & A_3 & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0\\ 0 & \cdots & \cdots & 0& A_k \end{array} \right]$$ Note that each $*$ in the matrix above corresponds to a single $a_{ni}$ in the $n^{\text{th}}$ row of $A$ corresponding to the edge which connects each of the paths corresponding to $A_2,A_3,\ldots , A_k$ to the output, $n$. Similarly, the compartmental matrix corresponding to $\mathcal{M}'$ is $$A' = \left[ \begin{array}{ccccc} A_1' & * & * & * & * \\ 0 & A_2' & 0 & \cdots & 0 \\ 0 & 0 & A_3' & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0\\ 0 & \cdots & \cdots & 0& A_k' \end{array} \right] = \left[ \begin{array}{ccccc} A_1 '& * & * & * & * \\ 0 & A_2 & 0 & \cdots & 0 \\ 0 & 0 & A_3 & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0\\ 0 & \cdots & \cdots & 0& A_k \end{array} \right]$$ Note that each of the paths not corresponding to $A_1$ or $A_1'$ in $\mathcal{M}$ and $\mathcal{M}'$ respectively are exactly the same, hence $A_i=A_i'$ for each of the $i\geq 2$. Thus, the coefficients of the left-hand side of the all $k$ of the input-output equations of $\mathcal{M}$ and $\mathcal{M}'$ respectively are given by $$\det(\partial I - A) = \det(\partial I - A_1) \det(\partial I - A_2) \cdots \det(\partial I - A_k)$$ and $$\begin{aligned} \det(\partial I - A') &= \det(\partial I - A_1') \det(\partial I - A_2') \cdots \det(\partial I - A_k') \\ &=\det(\partial I - A_1') \det(\partial I - A_2) \cdots \det(\partial I - A_k) \end{aligned}$$ Since the models corresponding to $A_1$ and $A_1'$ are indistinguishable, then $\det(\partial I - A_1)$ and $\det(\partial I - A_1')$ are identical up to permutation of parameters. Thus, $\det(\partial I - A)$ and $\det(\partial I - A')$ are identical up to permutation of parameters. Similarly, the coefficients for each of the derivatives of $u_{j}$ on the right-hand side of each of the input-output equations of $\mathcal{M}$ and $\mathcal{M}'$ is generated by $$\begin{aligned} \det(\partial I - A)^{j,n}.\end{aligned}$$ For $j=1$ corresponding to the input in the path model in $\mathcal{M}$ which is different but indistinguishable from the path model in $\mathcal{M}'$, we get $$\begin{aligned} \det(\partial I - A)^{1,n} &= \det(\partial I - A_1)^{1,n} \det(\partial I - A_2) \cdots \det(\partial I - A_k) \end{aligned}$$ since the first row and $n^{\text{th}}$ column are both in $A_1$. Again, for the $j=1$ coefficient on the right-hand side of $\mathcal{M}'$ we get $$\begin{aligned} \det(\partial I - A')^{(1,n)} &= \det(\partial I - A_1')^{1,n} \det(\partial I - A_2) \cdots \det(\partial I - A_k) \end{aligned}$$ Similar to the argument for the left-hand side coefficients, since the models corresponding to $A_1$ and $A_1'$ are indistinguishable, then $\det(\partial I - A_1)^{(1,n)}$ and $\det(\partial I - A_1')$ are identical up to permutation of parameters. Thus, $\det(\partial I - A)$ and $\det(\partial I - A')$ are identical up to permutation of parameters. ◻ **Example 30**. Sink model examples from Figure [\[fig:sink\]](#fig:sink){reference-type="ref" reference="fig:sink"}. Consider the linear compartmental models with compartmental matrices: $$A = \begin{bmatrix} -a_{21}-a_{41} & 0 & 0 & 0 & 0 \\ a_{21} & -a_{32} & 0 & a_{24} & 0 \\ 0 & a_{32} & 0 & 0 & a_{35} \\ a_{41} & 0 & 0 & -a_{24} & 0 \\ 0 & 0 & 0 & 0 & -a_{35} \end{bmatrix} \text{ and } A' =\begin{bmatrix} -a_{21} & 0 & 0 & 0 & 0 \\ a_{21} & -a_{32}-a_{42} & 0 & 0 & 0 \\ 0 & a_{32} & 0 & a_{34} & a_{35} \\ 0 & a_{42} & 0 & -a_{34} & 0 \\ 0 & 0 & 0 & 0 & -a_{35} \end{bmatrix}$$ The input-output equations are given by: $$\begin{gathered} (\partial + a_{21} + a_{41}) (\partial + a_{32})\partial(\partial + a_{24}) (\partial + a_{35}) y_3 = \\ (-a_{21}a_{32}(\partial + a_{24})(\partial + a_{35})+a_{41}a_{35}a_{32}a_{24})u_1 + (\partial + a_{21} + a_{41})(\partial + a_{32})a_{35}a_{24}u_5\end{gathered}$$ and $$\begin{gathered} (\partial + a_{21}) (\partial + a_{32} + a_{42})\partial(\partial + a_{34}) (\partial + a_{35}) y_3 = \\ (-a_{21}a_{32}(\partial + a_{34})(\partial + a_{35})+a_{42}a_{35}a_{21}a_{34})u_1 + (\partial + a_{21})(\partial + a_{32} + a_{42})a_{35}a_{34}u_5\end{gathered}$$ $\quad$ These two models are indistinguishable via the following map: $$\begin{pmatrix} a_{21} \\ a_{24} \\ a_{32} \\ a_{41} \\ a_{35} \end{pmatrix} \mapsto \begin{pmatrix} a_{32} \\ a_{34} \\ a_{21} \\ a_{42} \\ a_{35} \end{pmatrix}$$ # Source Models {#section:source} We now extend the idea of indistinguishability of path models to a family of models called *source models*. A source model is defined as a combination of skeletal path models which all of have same input, as depicted in Figure [\[fig:basicsource\]](#fig:basicsource){reference-type="ref" reference="fig:basicsource"}. **Lemma 31**. *The models $\mathcal{M}=(G,S,\{i\},Leak)$ and $\mathcal{M}'=(\tilde{G},\{i\},S,Leak)$, where the graph $\tilde{G}$ is the graph $G$ with the arrows reversed and $S$ is a subset of the vertex set $V$, have the same coefficient maps.* *Proof.* Consider the input-output equation for the model $\mathcal{M}$. The left hand side coefficients of the input-output equation are given by $\det(\partial I -A)$ and the right-hand side coefficients are given by $\det(\partial I - A)^{(j,i)}$ for each $j \in S$. Now consider the model $\mathcal{M}'$. Since the graph $\tilde{G}$ is the graph $G$ but with the arrows reversed, this means the matrix $A$ is now $A^T$ for the model $\mathcal{M}'$. Since there are multiple outputs, we get one input-output equation for each output corresponding to $j \in S$. However, the left-hand side for each of these input-output equations are given by $\det(\partial I - A^T) y_j$, thus the coefficients are identical on the left-hand side for each output. The right-hand side coefficients are given by $\det(\partial I - A^T)^{(i,j)}$ for each $j \in S$. This is precisely the transpose of $\det(\partial I - A)^{(j,i)}$, thus the coefficients are identical. ◻ **Corollary 32**. *Suppose two source models $\mathcal{M}$ and $\mathcal{M}'$ are identical outside of one path on each model which on their own are permutation indistinguishable, then $\mathcal{M}$ and $\mathcal{M}'$ are permutation indistinguishable.* *Proof.* This following from Corollary [Corollary 29](#cor:sink){reference-type="ref" reference="cor:sink"} and Lemma [Lemma 31](#lemma:swapinandout){reference-type="ref" reference="lemma:swapinandout"}. ◻ # Discussion {#section:discussion} In this work, we have considered a class of models containing a path from input to output and have found sufficient conditions based on the structure of the graph for the models to be indistinguishable. To the best of our knowledge, these are the first sufficient conditions for indistinguishability of linear compartmental models based on graph structure alone. We first showed that, given a path from compartment $1$ to $n$, with input in $1$ and output in $n$, the class of models with a leak from compartment $i$, where $i=1,..,n-1$, are all indistinguishable (see Theorem [Theorem 24](#thm:leak){reference-type="ref" reference="thm:leak"}), as well as the model with an edge from $n$ to $n-1$ instead of a leak (see Theorem [Theorem 26](#thm:terminalcycle){reference-type="ref" reference="thm:terminalcycle"}). We then generalized this result to apply to moving any "detour" down the path (see Theorem [Theorem 27](#thm:detour){reference-type="ref" reference="thm:detour"}). Finally, we showed how to generalize further with multiple source-sink paths (see Corollary [Corollary 29](#cor:sink){reference-type="ref" reference="cor:sink"} and Corollary [Corollary 32](#cor:source){reference-type="ref" reference="cor:source"}). An obvious next step would be to consider models that do not contain this "skeletal path" structure. For example, models that consist of cycles. Another direction for future work is to consider indistinguishability not arising in the permutation indistinguishability framework that we have examined, but as in Example [Example 15](#ex:nonpermute){reference-type="ref" reference="ex:nonpermute"}. The original inspiration for this study of indistinguishability was the recent work of the authors of this text and collaborators on a novel graphical characterization of the coefficients of linear compartmental model input-output equations [@aim]. From this graphical understanding of the input-output equations, as discussed in Section [3.1](#subsection:gfgeorules){reference-type="ref" reference="subsection:gfgeorules"}, we had an immediate understanding of the necessary conditions for indistinguishability. Several of the proofs in this work started as graph theoretic arguments relating the coefficients of two indistinguishable models, and we suspect that this result, though not necessarily heavily utilized in this work, will be a strong tool in future works related to indistinguishability.

arxiv_math

--- abstract: | The Gromov--Hausdorff distance between two metric spaces measures how far the spaces are from being isometric. It has played an important and longstanding role in geometry and shape comparison. More recently, it has been discovered that the Gromov--Hausdorff distance between unit spheres equipped with the geodesic metric has important connections to Borsuk--Ulam theorems and Vietoris--Rips complexes. We develop a discrete framework for obtaining upper bounds on the Gromov--Hausdorff distance between spheres, and provide the first quantitative bounds that apply to spheres of all possible pairs of dimensions. As a special case, we determine the exact Gromov--Hausdorff distance between the circle and any even-dimensional sphere, and determine the asymptotic behavior of the distance from the 2-sphere to the $k$-sphere up to constants. author: - "Michael Harrison[^1]" - "R. Amzi Jeffs[^2]" bibliography: - dgh.bib date: September 2023 title: | Quantitative upper bounds on the\ Gromov--Hausdorff distance between spheres --- # Introduction {#sec:intro} The Gromov--Hausdorff distance $d_\mathrm{GH}(X,Y)$ between two compact metric spaces $X$ and $Y$ captures how closely the spaces can be aligned with one another---in particular, $d_\mathrm{GH}(X,Y) = 0$ if and only if $X$ and $Y$ are isometric [@BBI2001; @petersen2006]. The Gromov--Hausdorff distance was first defined in 1975 by Edwards [@Edwards75], and then rediscovered in 1981 by Gromov [@Gromov81], and for decades it has played an important role in metric geometry (see e.g. [@cheeger1997; @colding1996]). In more recent years, it has proved to be a natural tool in the context of data analysis and shape matching [@BBK2008; @memoli2007; @MS05; @CM2010]. Precisely determining the Gromov--Hausdorff distance between given metric spaces is computationally challenging [@schmiedl2017; @memoli2012; @AFNSW2018]. In particular, computing the Gromov--Hausdorff distance between finite metric spaces is NP-hard [@memoli2007]. Exact values of the Gromov--Hausdorff distance in special cases have been obtained only very recently---for example, between certain discrete metric spaces [@IT19], between an interval and a circle [@JT22], between vertex sets of regular polygons [@Talipov22]. The first nontrivial bounds on the Gromov--Hausdorff distance between unit spheres equipped with the geodesic metric were obtained in recent work of Lim, Mémoli, and Smith [@LMS22], who provided lower bounds using novel connections to Borsuk--Ulam type theorems, and upper bounds via explicit constructive techniques. They computed the exact Gromov--Hausdorff distance between $S^n$ and $S^k$ for $n=0$, $k=\infty$, and $(n,k)\in \{(1,2), (1,3), (2,3)\}$. They also provided quantitative upper bounds in the case $k = n+1$, and for general $n$ and $k$ they showed that $d_\mathrm{GH}(S^n, S^k)$ is strictly less than the trivial upper bound of $\tfrac{\pi}{2}$, which follows from the fact that $S^n$ and $S^k$ have diameter $\pi$. The aforementioned lower bounds were subsequently improved in a large polymath project [@dghpolymath], which included the present authors. This project built on insights of Adams, Bush, and Frick [@ABF21] and Lim, Mémoli, and Smith [@LMS22] to demonstrate quantitative relationships between the Gromov--Hausdorff distance $d_\mathrm{GH}(S^n, S^k)$, Borsuk--Ulam type theorems concerning functions $S^k\to S^n$, and the topology of Vietoris--Rips complexes over $S^n$ and $S^k$. These quantitative connections further motivate the question of determining the exact value of the Gromov--Hausdorff distance between spheres. Explicit upper bounds on $d_\mathrm{GH}(S^n, S^k)$ for general $n$ and $k$ have remained elusive, and our work addresses this scarcity by computing the exact value of $d_\mathrm{GH}(S^1,S^k)$ for all even $k$ and giving quantitative upper bounds on $d_\mathrm{GH}(S^n,S^k)$ for all $n$ and $k$. #### Quantitative results. For reasons of notational convenience, we will work with the quantity $2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k)$. The statement of below contains two equivalent formulations: first an upper bound on $2\hspace{0.1em}d_{\mathrm{GH}}(S^n,S^k)$, and then a lower bound on the gap between $2\hspace{0.1em}d_\mathrm{GH}(S^n,S^k)$ and the elementary upper bound of $\pi$. When $n=1$ and $k$ is even, is strong enough to meet the existing lower bounds. Forthcoming work will establish a tight upper bound on $2\hspace{0.1em}d_\mathrm{GH}(S^1, S^k)$ in the case that $k$ is odd, expanding on an approach developed by Facundo Mémoli and Zane Smith. The lower bounds for $2\hspace{0.1em}d_\mathrm{GH}(S^1, S^{k})$ were achieved in [@dghpolymath], where it is shown that, for all $n$ and $k$, strong bounds for $2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k)$ can be obtained in terms of the equivariant topology of Vietoris--Rips complexes. shows that these bounds are sharp in infinitely many cases, providing affirmative evidence for [@dghpolymath Question 8.1]. It could be the case that these topological lower bounds for $2\hspace{0.1em}d_\mathrm{GH}(S^n,S^k)$ are sharp for *every* $n$ and $k$, and provides the strongest evidence to date in this direction. theoremntok[\[thm:ntok\]]{#thm:ntok label="thm:ntok"} For every $1 \le n < k <\infty$, we have $$2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k) \le \frac{\pi k}{k+1}\quad\quad \text{or equivalently} \quad \quad \pi - 2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k) \ge \frac{\pi}{k+1}.$$ When $n=1$ and $k$ is even, we have equality. We will establish the $n=1$ case of in . The general case is treated in . #### Packings, coverings, and asymptotic results. The bound of has the advantage that it is concrete and simple to state. However, we are able to prove a more granular set of results which gives an asymptotic improvement on for fixed $n\ge 2$, and gives a sharp (up to constants) asymptotic description of $2\hspace{0.1em}d_\mathrm{GH}(S^2,S^k)$. To discuss these results, we first introduce some definitions and notation. We will consider projective space $\mathbb{R}\mathrm{P}^n$, endowed with the quotient metric induced by the geodesic metric on $S^n$. We denote this metric by $d_{\mathbb{R}\mathrm{P}^n}$. Below, we will make repeated use of the following two parameters: $$\begin{aligned} p_m(\mathbb{R}\mathrm{P}^n) &\stackrel{\mbox{\normalfont\tiny def}}{=}\sup \{\varepsilon > 0 \mid \text{$\exists x_1,\ldots, x_m$ in $\mathbb{R}\mathrm{P}^n$ so that $d_{\mathbb{R}\mathrm{P}^n}(x_i,x_j) \ge \varepsilon$ for all $i\neq j$ }\},\\ c_m(\mathbb{R}\mathrm{P}^n) &\stackrel{\mbox{\normalfont\tiny def}}{=}\, \inf \{\varepsilon > 0 \mid \text{$\exists x_1,\ldots, x_m$ in $\mathbb{R}\mathrm{P}^n$ so that for all $x\in \mathbb{R}\mathrm{P}^n$ there is $i$ with $d_{\mathbb{R}\mathrm{P}^n}(x, x_i)\le \varepsilon$ }\}.\end{aligned}$$ The parameter $p_m(\mathbb{R}\mathrm{P}^n)$ is the minimum distance between distinct points in an optimal packing of $m$ points in $\mathbb{R}\mathrm{P}^n$ (i.e. a projective code). The parameter $c_m(\mathbb{R}\mathrm{P}^n)$ is the smallest radius needed to cover $\mathbb{R}\mathrm{P}^n$ by $m$-many metric balls. By considering an optimal packing with the additional property that as few points as possible have pairwise distance exactly $p_m(\mathbb{R}\mathrm{P}^n)$, one can see that $c_m(\mathbb{R}\mathrm{P}^n) \le p_m(\mathbb{R}\mathrm{P}^n)$. A standard argument using the fact that the volume of a ball in $\mathbb{R}\mathrm{P}^n$ is proportional to the $n$-th power of its radius shows that, for fixed $n$, $p_m(\mathbb{R}\mathrm{P}^n)$ and $c_m(\mathbb{R}\mathrm{P}^n)$ are bounded above and below by constant multiples of $\frac{1}{\sqrt[n]{m}}$. With these parameters in hand, we proceed to our results. theorempacking[\[thm:packing\]]{#thm:packing label="thm:packing"} For all $2\le n < k < \infty$, we have $$2\hspace{0.1em}d_{\mathrm{GH}}(S^n, S^k) \le \max \left \{\arccos\left(\frac{-(k-1)}{k+1}\right) ,\,\, \pi - p_{k+1}(\mathbb{R}\mathrm{P}^n),\,\, 2p_{k+1}(\mathbb{R}\mathrm{P}^n) \right\}.$$ Using , we obtain an asymptotic result describing the gap between $2\hspace{0.1em}d_{\mathrm{GH}}(S^n, S^k)$ and $\pi$ for fixed $n$. Below, $\Omega\left(\frac{1}{\sqrt{k}}\right)$ denotes a function that is bounded below by a positive multiple of $\frac{1}{\sqrt{k}}$. corollaryasymptoticcor[\[cor:asymptotic\]]{#cor:asymptotic label="cor:asymptotic"} For fixed $n\ge 2$, we have $$\pi - 2\hspace{0.1em}d_{\mathrm{GH}}(S^n, S^k) = \Omega\left(\frac{1}{\sqrt{k}}\right).$$ Previous work (see [@dghpolymath Theorem 5.3]) has shown that $2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k) \ge \pi - 2c_{k}(\mathbb{R}\mathrm{P}^n)$. In particular $\pi - 2\hspace{0.1em}d_\mathrm{GH}(S^n,S^k) \le \Theta\left(\tfrac{1}{\sqrt[n]{k}}\right)$, where $\Theta\left(\tfrac{1}{\sqrt[n]{k}}\right)$ denotes a positive function that is bounded above and below by constant multiples of $\tfrac{1}{\sqrt[n]{k}}$. For $n=2$, this matches (up to constants) the upper bound of , allowing us to exactly determine the asymptotic behavior of $\pi - 2\hspace{0.1em}d_\mathrm{GH}(S^2, S^k)$. **Corollary 1**. *The Gromov--Hausdorff distance between $S^2$ and $S^k$ satisfies $$\pi - 2\hspace{0.1em}d_{\mathrm{GH}}(S^2, S^k) = \Theta\left(\frac{1}{\sqrt{k}}\right).$$* We conjecture that in general the $n$-th root bound is the correct answer for the asymptotic behavior of $\pi - 2\hspace{0.1em}d_\mathrm{GH}(S^n,S^k)$ when $n$ is fixed. Note that and prove this conjecture for $n=1$ and $n=2$ respectively. conjectureasymptoticconj[\[conj:asymptotic\]]{#conj:asymptotic label="conj:asymptotic"} For fixed $n\ge 1$, we have $$\pi - 2\hspace{0.1em}d_{\mathrm{GH}}(S^n, S^k) = \Theta\left(\frac{1}{\sqrt[n]{k}}\right).$$ #### Our methods. To prove upper bounds on $2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k)$, it suffices to construct (possibly discontinuous) correspondences between $S^n$ and $S^k$ which do not distort the metric too much (see ). Lim, Mémoli, and Smith [@LMS22] constructed optimal correspondences between low-dimensional spheres by decomposing the larger sphere into discrete chunks, which were then collapsed to points in the lower-dimensional sphere. They used a similar approach to build correspondences between $S^{n+1}$ and $S^n$ for all $n$, a construction which was further developed in [@dghpolymath Section 6]. Continuing in a similar vein, our approach provides a clean framework for constructing correspondences between $S^n$ and $S^k$ for any $n$ and $k$, and it bounds the resulting distortion (see ). Despite the thematic through-line in the development of these methods, the correspondences that we construct are wholly new---existing correspondences in the literature are not a special case of them. More specifically, our approach uses finite centrally symmetric point sets in $S^n$ and $S^k$ to break the spheres into discrete chunks, which are then collapsed into the opposite sphere in such a way that the distortion is quantifiable. It is possible that our bounds could be improved in some cases with different choices of centrally symmetric point sets, but we suspect that determining the exact value of $d_{\mathrm{GH}}(S^n,S^k)$ for all $n$ and $k$ will require constructions that go beyond straightforward applications of our discrete method. # Background {#sec:background} The Gromov--Hausdorff distance $d_\mathrm{GH}(X, Y)$ between metric spaces $X$ and $Y$ is the infimum over all isometric embeddings of $X$ and $Y$ into a larger metric space, of the Hausdorff distance between their images. In other words, $d_\mathrm{GH}(X,Y)$ captures how closely $X$ and $Y$ can be "aligned\" in some ambient space. Below we give an equivalent definition of the Gromov--Hausdorff distance due to Kalton and Ostrovskii [@KO99], which will prove convenient for our purposes. We first require some additional terminology. A *correspondence* between two sets $X$ and $Y$ is a relation $R\subseteq X\times Y$ with the property that $\pi_X(R) = X$ and $\pi_Y(R) = Y$; that is, the coordinate projections of $R$ to $X$ and $Y$ are surjective. Equivalently, a correspondence relates every point of $X$ to a point of $Y$, and vice versa. If $X$ and $Y$ are metric spaces, the *distortion* of a correspondence $R$ is $$\mathop{\mathrm{dis}}(R) \stackrel{\mbox{\normalfont\tiny def}}{=}\sup_{(x,y), (x',y')\in R} \big|d_X(x,x') - d_Y(y,y')\big|.$$ Above, $d_X$ and $d_Y$ denote the respective metrics on $X$ and $Y$. Informally, $\mathop{\mathrm{dis}}(R)$ is small if $R$ does not "pull apart\" or "push together\" points too much. Finally, the *Gromov--Hausdorff distance* between compact metric spaces $X$ and $Y$ is given by the relationship $$\label{eq:dgh} 2\hspace{0.1em}d_\mathrm{GH}(X,Y) \stackrel{\mbox{\normalfont\tiny def}}{=}\inf_{R} \mathop{\mathrm{dis}}(R)$$ where the infimum is taken over all correspondences $R\subseteq X\times Y$. Our metric spaces of interest are unit spheres $S^n \stackrel{\mbox{\normalfont\tiny def}}{=}\{x\in \mathbb{R}^{n+1} \mid \langle x,x\rangle = 1\}$, equipped with the *geodesic metric*$$d_{S^n}(x,x') \stackrel{\mbox{\normalfont\tiny def}}{=}\arccos(\langle x, x'\rangle).$$ We will always consider spheres of positive finite dimension, as the cases of $S^0$ and $S^\infty$ have been fully treated by previous work [@LMS22]. With the geodesic metric, $S^n$ has diameter equal to $\pi$, and a quick consequence is that $2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k)$ is at most $\pi$. As previously mentioned, Lim, Mémoli, and Smith [@LMS22] showed that in fact $2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k)$ is strictly less than $\pi$ when $1\le n < k < \infty$. We say that a set $P\subseteq S^n$ is *antipodal* or *centrally symmetric* if $x\in P$ implies $-x\in P$. We often write finite antipodal sets as $P = \{\pm p_1, \ldots, \pm p_m\}$. For a metric space $(X, d_X)$, the *separation* of a finite set $P\subseteq X$ is the minimum distance between distinct points in the set, that is $$\mathop{\mathrm{sep}}_X(P) \stackrel{\mbox{\normalfont\tiny def}}{=}\min_{p\neq p'}\{d_{X}(p,p')\}$$ where the minimum above is over all $p,p'\in P$ with $p\neq p'$. Note that if $P\subseteq S^n$ is an antipodal set containing more than two points, then $\mathop{\mathrm{sep}}_{S^n}(P)\le\frac{\pi}{2}$. Each finite subset $P = \{p_1,\ldots, p_m\}$ of a metric space $(X, d_X)$ can be associated to its *Voronoi diagram* $\{X_1, \ldots, X_m\}$ where $X_i\stackrel{\mbox{\normalfont\tiny def}}{=}\{x\in X\mid d_X(x,p_i)\le d_X(x, p_j) \text{ for all $j\neq i$}\}$. We call $X_i$ the *Voronoi cell* associated to $p_i$. If $P = \{p_1,\ldots, p_m\}$ is a finite subset of a metric space $(X, d_X)$ with Voronoi diagram $\{X_1, \ldots, X_m\}$, then the *Voronoi diameter* of $P$ is $$\mathop{\mathrm{vdiam}}_X(P) \stackrel{\mbox{\normalfont\tiny def}}{=}\max_{i\in[m]}\big\{\mathop{\mathrm{diam}}_{X}(X_i)\big \}.$$ Note that if $P\subseteq S^n$ is antipodal, then the Voronoi diagram of $P$ is centrally symmetric. Also note that $\mathop{\mathrm{vdiam}}_X(P) \le 2\hspace{0.1em}d_\mathrm{H}(X, P)$ where $d_\mathrm{H}$ denotes Hausdorff distance. # Correspondences induced by antipodal sets First, we informally describe our construction. Choose finite antipodal sets in $S^n$ and $S^k$ with equal size. Each of these sets decomposes $S^n$ and $S^k$ into Voronoi cells. Since there are equally many cells and points in each sphere, we may collapse cells in $S^n$ to points in $S^k$, and vice versa, obtaining a correspondence (see below). Provided that the cells are not too large and the points are sufficiently separated, such correspondences will have distortion bounded away from $\pi$. The following theorem formalizes and quantifies this technique. **Theorem 2**. *Let $P = \{\pm p_1, \ldots, \pm p_m\}\subseteq S^n$ and $Q=\{\pm q_1,\ldots, \pm q_m\}\subseteq S^k$ be finite antipodal sets of equal size. Let $\{\pm F_1, \ldots, \pm F_m\}$ and $\{\pm G_1,\ldots, \pm G_m\}$ be the Voronoi diagrams in $S^n$ and $S^k$ induced by $P$ and $Q$ respectively. Define the correspondence $$R_{P, Q} \stackrel{\mbox{\normalfont\tiny def}}{=}\big\{\pm (p_i, q) \,\big\vert\, i\in[m]\text{ and }q\in G_i\big\} \cup \big\{\pm (p, q_i)\,\big\vert\, i\in[m]\text{ and }p\in F_i\big\}.$$ The distortion of $R_{P, Q}$ is at most the maximum of the following four quantities: $$\begin{aligned} \mathop{\mathrm{vdiam}}_{S^n}(P) &\qquad\pi-\mathop{\mathrm{sep}}_{S^n}(P)\\ \mathop{\mathrm{vdiam}}_{S^k}(Q) &\qquad \pi-\mathop{\mathrm{sep}}_{S^k}(Q).\end{aligned}$$* $$\includegraphics{induced.eps}$$ *Proof.* First note that $R_{P,Q}$ is indeed a correspondence, as every point in $S^n$ appears in some Voronoi cell $\pm F_i$, and every point in $S^k$ appears in some Voronoi cell $\pm G_i$. It remains to bound the quantity$$\tag{$\dagger$}\label{eq:distortion} \sup_{(x,y), (x',y')\in R_{P,Q}} \left|d_{S^n}(x,x') - d_{S^k}(y,y')\right|.$$ We achieve this by considering a variety of cases, based on the definition of $R_{P,Q}$. Many of these cases are symmetric, thanks to the symmetry in the definition of $R_{P,Q}$, but we enumerate all of them for completeness. To see that these nine cases cover all possibilities, note that if $(x,y)\in R_{P,Q}$ then $x\in P$ or $y\in Q$ (and possibly both are true). **Case 1:** $x = x' = \pm p_i$.\ We have $y,y'\in G_i$ or $y,y'\in -G_i$. Thus ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is equal to $d_{S^k}(y,y')$, which is at most $\mathop{\mathrm{vdiam}}_{S^k}(Q)$. **Case 2:** $x = -x' = \pm p_i$.\ In this case $d_{S^n}(x,x') = \pi$, so ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is equal to $\pi-d_{S^k}(y,y')$. We have $y\in G_i$ and $y'\in -G_i$. By central symmetry of the Voronoi diagram in $S^k$, we have $-y'\in G_i$, and so $$d_{S^k}(y,y') = \pi-d_{S^k}(y,-y')\ge \pi - \mathop{\mathrm{diam}}_{S^k}(G_i) \ge \pi-\mathop{\mathrm{vdiam}}_{S^k}(Q).$$ Plugging in and simplifying, we see ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is again at most $\mathop{\mathrm{vdiam}}_{S^k}(Q)$. **Case 3:** $x = \pm p_i$ and $x'= \pm p_j$ for $i\neq j$.\ Here we have $\mathop{\mathrm{sep}}_{S^n}(P) \le d_{S^n}(x,x') \le \pi-\mathop{\mathrm{sep}}_{S^n}(P)$ since $x$ and $x'$ are distinct, non-antipodal points in $P$. From this, we see that ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is at most $\pi-\mathop{\mathrm{sep}}_{S^n}(P)$. **Case 4:** $x = p_i$ and $y' = q_i$, or $x = -p_i$ and $y' = -q_i$.\ By central symmetry, it suffices to consider the case $x = p_i$ and $y' = q_i$. We have $y\in G_i$ and $x'\in F_i$. Since $P$ and $Q$ are antipodal, we have $d_{S^n}(x,x') \le \pi/2$ and $d_{S^k}(y,y')\le \pi/2$. Thus ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is at most $\pi/2$, which is in turn at most $\pi-\mathop{\mathrm{sep}}_{S^n}(P)$ since $\mathop{\mathrm{sep}}_{S^n}(P)\le \pi/2$. **Case 5:** $x = p_i$ and $y' = -q_i$, or $x=-p_i$ and $y' = q_i$.\ By central symmetry, it suffices to consider the case $x = p_i$ and $y' = -q_i$. Here we have $x'\in -F_i$ and $y\in G_i$. Using central symmetry of $P$ and $Q$, the former implies $d_{S^n}(p_i,x') \ge \pi/2$, and the latter implies $d_{S^k}(y, -q_i) \ge \pi/2$. These inequalities imply that ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is at most $\pi/2$, which is in turn at most $\pi-\mathop{\mathrm{sep}}_{S^n}(P)$ since $\mathop{\mathrm{sep}}_{S^n}(P)\le \pi/2$. **Case 6:** $x = \pm p_i$ and $y' = \pm q_j$ for $i\neq j$.\ Here $x'$ lies in $\pm F_j$, and $y$ lies in $\pm G_i$. We claim that$$\mathop{\mathrm{sep}}_{S^n}(P)/2 \le d_{S^n}(x,x') \le \pi-\mathop{\mathrm{sep}}_{S^n}(P)/2 \text{ and } \mathop{\mathrm{sep}}_{S^k}(Q)/2 \le d_{S^k}(y,y') \le \pi-\mathop{\mathrm{sep}}_{S^k}(Q)/2.$$ To see this, consider the case $x'\in F_j$ and $x=p_i$. Note by triangle inequality that $$\mathop{\mathrm{sep}}_{S^n}(P) \le d_{S^n}(p_i, p_j) \le d_{S^n}(p_i, x') + d_{S^n}(x', p_j) \le 2d_{S^n}(p_i, x')$$ where the last inequality follows from the fact that $x'\in F_j$. This gives $\mathop{\mathrm{sep}}_{S^n}(P)/2 \le d_{S^n}(x,x')$, and the remaining cases follow by an analogous triangle inequality argument. Now, using the inequalities above we find that ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is at most $\pi-(\mathop{\mathrm{sep}}_{S^n}(P) + \mathop{\mathrm{sep}}_{S^k}(Q))/2$, which is in turn bounded above by the maximum of $\pi - \mathop{\mathrm{sep}}_{S^n}(P)$ and $\pi-\mathop{\mathrm{sep}}_{S^k}(Q)$. **Case 7:** $y = \pm q_i$ and $y' = \pm q_j$ for $i \neq j$.\ Symmetric to Case 3. We have $\mathop{\mathrm{sep}}_{S^k}(Q) \le d_{S^n}(y,y') \le \pi-\mathop{\mathrm{sep}}_{S^k}(Q)$ since $y$ and $y'$ are distinct, non-antipodal points in $Q$. From this, we see that ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is at most $\pi-\mathop{\mathrm{sep}}_{S^k}(Q)$. **Case 8:** $y = -y' = \pm q_i$.\ Symmetric to Case 2. Since $d_{S^k}(y,y') = \pi$, ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is equal to $\pi-d_{S^n}(x,x')$. However, $x\in F_i$ and $x'\in -F_i$, which implies $d_{S^n}(x,x') \ge \pi-\mathop{\mathrm{vdiam}}_{S^n}(P)$. Thus we find ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is at most $\mathop{\mathrm{vdiam}}_{S^n}(P)$. **Case 9:** $y = y' = \pm q_i$.\ Symmetric to Case 1. Here we have $x,x'\in F_i$ or $x,x'\in -F_i$, and so ([\[eq:distortion\]](#eq:distortion){reference-type="ref" reference="eq:distortion"}) is equal to $d_{S^n}(x,x')$, which is at most $\mathop{\mathrm{diam}}_{S^n}(F_i) \le \mathop{\mathrm{vdiam}}_{S^n}(P)$. ◻ *Remark 3*. One may also wish to construct correspondences $R_{P,Q}$ using Voronoi diagrams of non-antipodal point sets $P$ and $Q$. This can certainly be done, and a theorem similar to could be obtained, but the casework would be slightly different and more laborious. The advantage of the antipodal symmetry in our theorem is that $R_{P,Q}$ will generally send antipodes to antipodes, and we need not worry that points which lie distance $\mathop{\mathrm{diam}}(S^n) = \mathop{\mathrm{diam}}(S^k) = \pi$ from another are "pulled together" by the correspondence. Without the antipodal assumption, several further quantities---such as the diameter of $P$ and $Q$---would need to be considered. We have chosen the antipodal formulation since it simplifies our statements and constructions, and we are not aware of any cases where a non-antipodal argument would improve any of our bounds. This is consistent with the "helmet trick\" established by Lim, Mémoli, and Smith [@LMS22], which shows that one may reduce to the case of antipode-preserving (or "odd\") functions between $S^n$ and $S^k$ when bounding distortion. *Remark 4*. The requirement that $P$ and $Q$ have the same size is one of the main technical obstacles in applying . There are many natural antipodal sets in $S^n$, such as the vertices of the cross-polytope, the positive and negative copies of the vertices of a geodesic simplex, or the (normalized) vertices of a hypercube. However, it is usually not so straightforward to find a nice antipodal set of matching size in $S^k$. # From the 1-sphere to the $k$-sphere {#sec:1tok-initial} To use , we must construct antipodal point sets in $S^n$ and $S^k$ with small Voronoi diameter and large separation, to the extent that this is possible. Doing so in $S^1$ is straightforward: the best choice is simply to space points evenly around the circle. In $S^k$ there are several natural choices, and one that will prove useful for us is the vertex set of the cross-polytope, i.e. the standard basis vectors and their negatives. We start by characterizing the Voronoi diameter and separation of this set. **Lemma 5**. *Let $Q = \{\pm e_1, \ldots, \pm e_{k+1}\}\subseteq S^k$ be the standard basis vectors and their negatives. Then for $k\ge 1$ we have$$\mathop{\mathrm{vdiam}}_{S^k}(Q) = \arccos\left(\frac{-(k-1)}{k+1}\right) \qquad \text{ and } \qquad \mathop{\mathrm{sep}}_{S^k}(Q) = \frac{\pi}{2}.$$* *Proof.* The equality $\mathop{\mathrm{sep}}_{S^k}(Q) = \frac{\pi}{2}$ is immediate since every pair of non-antipodal points in $Q$ are distance $\frac{\pi}{2}$ apart. To obtain the Voronoi diameter, it suffices by symmetry to compute the diameter of the cell associated to $e_1$. This cell consists of all points in $S^k$ whose first coordinate is positive and maximum in absolute value among all the coordinates. In particular, every point in this cell has first coordinate at least as large as $\frac{1}{\sqrt{k+1}}$. Thus if $x$ and $x'$ lie in this cell, then $$\begin{aligned} \langle x, x'\rangle \ = \ \sum_{i=1}^{k+1} x_ix_i' \ \ge \ \frac{1}{k+1} + \sum_{i=2}^{k+1} x_ix_i' \ \ge \ \frac{1}{k+1} - \frac{k}{k+1} \ = \ \frac{-(k-1)}{k+1}.\end{aligned}$$ Above, the last inequality follows by applying the Cauchy--Schwarz inequality to the truncated vectors obtained by deleting the first coordinates of $x$ and $x'$, noting that each truncated vector has norm at most $\sqrt{\tfrac{k}{k+1}}$. Taking the arccosine of both sides flips the direction of the inequality, and we obtain $d_{S^k}(x,x') \le \arccos\left(\tfrac{-(k-1)}{k+1}\right)$. Moreover, equality is achieved when $x$ has all coordinates equal to $\tfrac{1}{\sqrt{k+1}}$ and $x'$ has first coordinate equal to $\tfrac{1}{\sqrt{k+1}}$ and all other coordinates equal to $\tfrac{-1}{\sqrt{k+1}}$. ◻ To analyze correspondences from $S^1$ to $S^k$, we first require a small technical lemma. **Lemma 6**. *For $k\ge 3$, we have $\arccos\left(\frac{-(k-1)}{k+1}\right) \le \frac{(k-1)\pi}{k}$.* *Proof.* For $k=3$, we have equality. For $k\ge 4$, we use the estimate $\arccos(x)\le \pi - \sqrt{2x+2}$, and immediately see that $\arccos\left(\frac{-(k-1)}{k+1}\right) \le \pi - \frac{2}{\sqrt{k+1}}$. As $k\ge 4$, we have $4k^2 \ge \pi^2(k+1)$, which implies that $\frac{2}{\sqrt{k+1}} \ge \frac{\pi}{k}$. Hence $\arccos\left(\frac{-(k-1)}{k+1}\right) \le \pi - \frac{\pi}{k} = \frac{(k-1)\pi}{k}$ as desired. ◻ **Theorem 7**. *For $k\ge 2$, we have $2\hspace{0.1em}d_\mathrm{GH}(S^1, S^k) \le \frac{k\pi}{k+1}$.* *Proof.* Let $P\subseteq S^1$ consist of $2(k+1)$ evenly spaced points, and let $Q\subseteq S^k$ be the standard basis vectors and their negatives. The correspondence $R_{P,Q}$ of (shown in ) will have distortion at most the maximum of $\mathop{\mathrm{vdiam}}(P)$, $\pi-\mathop{\mathrm{sep}}(P)$, $\mathop{\mathrm{vdiam}}(Q)$, and $\pi-\mathop{\mathrm{sep}}(Q)$. We may immediately note that $\mathop{\mathrm{vdiam}}(P) = \mathop{\mathrm{sep}}(P) = \frac{\pi}{k+1}$, and $\mathop{\mathrm{sep}}(Q) = \frac{\pi}{2}$. tells us that $\mathop{\mathrm{vdiam}}(Q) = \arccos\left(\frac{-(k-1)}{k+1}\right)$. When $k=2$, this is $\arccos\left(-\frac{1}{3}\right) < \arccos\left(-\frac{1}{2}\right) = \frac{2\pi}{3} = \frac{k\pi}{(k+1)}$, and for $k\ge 3$ provides a slightly stronger upper bound of $\frac{(k-1)\pi}{k}$. Plugging all of this in, we see that $R_{P,Q}$ has distortion at most $\pi - \mathop{\mathrm{sep}}(P) = \frac{k\pi}{k+1}$ as desired. ◻ *Remark 8*. The $k=2$ case of shows that $2\hspace{0.1em}d_{\mathrm{GH}}(S^1,S^2) = \frac{2\pi}{3}$. Lim, Mémoli, and Smith [@LMS22] previously constructed a correspondence between $S^1$ and $S^2$ with this optimal distortion, and their construction likewise involved decomposing $S^2$ into chunks---six isometric triangular regions, three in the upper hemisphere and three in the lower hemisphere---that were then collapsed to points in $S^1$. Interestingly, our correspondence $R_{P,Q}$ uses a different decomposition of $S^2$ consisting of the six centrally projected facets of the cube. Another difference is that Lim, Mémoli, and Smith relate points in $S^1$ to the equator in $S^2$ in an isometric way, whereas we collapse intervals in $S^1$ to points in $S^2$. illustrates these decompositions of $S^1$ and $S^2$ that are used to build $R_{P,Q}$. $$\includegraphics{1to2.eps}$$ *Remark 9*. In [@dghpolymath] the lower bound of $2\hspace{0.1em}d_\mathrm{GH}(S^1, S^k) \ge \frac{k\pi}{k+1}$ was proved for $k$ even. Together with , this gives the $n=1$ case of . # From the $n$-sphere to the $k$-sphere {#sec:general} To prove , we will construct point sets in $S^n$ (with $n\ge 2$) with Voronoi diameter at most $\frac{k\pi}{k+1}$, and separation at least $\frac{\pi}{k+1}$. These sets will consist of $2(k+1)$ points, matching the number of vertices of the cross-polytope in $S^k$, so that we may apply . It turns out that there is a relatively simple construction: start with the vertices of a cross-polytope in $S^n$, then add the remaining points along the projected edges of the cross-polytope, spacing them as evenly as possible. The following lemma explains this construction. **Lemma 10**. *For each $2\le n < k < \infty$, there exists an antipodal set $P\subseteq S^n$ consisting of $2(k+1)$ points, with $$\mathop{\mathrm{vdiam}}(P)\le \frac{\pi n}{n+1}\quad \text{ and } \quad \mathop{\mathrm{sep}}(P) \ge \frac{\pi}{2\left( \left\lceil \frac{k-n}{n(n+1)}\right\rceil + 1\right)}\ge \frac{\pi}{k-n+3}.$$* *Proof.* To start, set $P = \{\pm e_1,\ldots, \pm e_{n+1}\}\subseteq S^n$. By we have $\mathop{\mathrm{vdiam}}(P)\le \frac{\pi n}{n+1}$, and adding further points can only improve this inequality. We must add $2(k+1)-2(n+1) = 2(k-n)$ points to $P$. Consider the $2n(n+1)$-many geodesic arcs in $S^n$ between pairs of points $\{\pm e_i, \pm e_j\}$, where $i\neq j$. Split these arcs into antipodal pairs, choosing one element of each pair as the "positive\" copy of the arc. Set $N\stackrel{\mbox{\normalfont\tiny def}}{=}\left\lceil \frac{k-n}{n(n+1)}\right\rceil$. Into the interior of each of the $n(n+1)$-many positive arcs, place up to $N$-many points, evenly spaced with distances between consecutive points (including the endpoints of the arc) at least $\frac{\pi}{2(N+1)}$. Copy the points from the positive arcs antipodally to the negative arcs. Collecting everything, we have up to $2(k+1)$ points in $P$, as shown in . $$\includegraphics{arcs.eps}$$ Now, we claim that $\mathop{\mathrm{sep}}(P) \ge \frac{\pi}{2(N+1)}$. By construction, this bound holds when we restrict our attention to any copy of $S^1$ containing $\{\pm e_i, \pm e_j\}$. The only other case to worry about is the distance between $p,p'\in P$, where $p$ and $p'$ lie in the interiors of arcs that are not coplanar. In particular, the nonzero coordinates in $p$ and $p'$ do not occur in the same indices. As each point in $P$ has at most two nonzero coordinates, we either have $\langle p, p'\rangle = 0$ or $\langle p, p'\rangle \le c^2$ where $c$ is the maximum absolute value among the coordinates of all points in $P$ that lie on the interiors of arcs. This implies that $d_{S^n}(p,p')$ is at least as large as the distance from $p$ (respectively, $p'$) to the nearest $\pm e_i$. This proves that $\mathop{\mathrm{sep}}(P) \ge \frac{\pi}{2(N+1)}$ as desired. The final inequality follows from the fact that $n\ge 2$, so $2\left\lceil \frac{k-n}{n(n+1)}\right\rceil \le k-n+1$. ◻ With this construction, we can now recall and prove . *Proof.* The case $n=1$ follows from and the lower bounds of [@dghpolymath]. For $n\ge 2$, let $P\subseteq S^n$ be the antipodal set of $2(k+1)$ points of , and let $Q\subseteq S^k$ be the standard basis vectors and their negatives. The correspondence $R_{P,Q}$ of will have distortion at most the maximum of $\mathop{\mathrm{vdiam}}(P)$, $\pi-\mathop{\mathrm{sep}}(P)$, $\mathop{\mathrm{vdiam}}(Q)$, and $\pi-\mathop{\mathrm{sep}}(Q)$. guarantees that $\mathop{\mathrm{vdiam}}_{S^n}(P)\le \frac{\pi n}{n+1} < \frac{\pi k}{k+1}$, and $\pi-\mathop{\mathrm{sep}}_{S^n}(P)$ is bounded above by $\pi-\frac{\pi}{k-n+3} = \frac{\pi(k-n+2)}{k-n+3}\le \frac{k\pi}{k+1}$. guarantees that $\mathop{\mathrm{vdiam}}_{S^k}(Q)$ and $\pi-\mathop{\mathrm{sep}}_{S^k}(Q)$ are bounded above by $\frac{\pi k}{k+1}$. The result follows. ◻ The proof of likewise proceeds by choosing appropriate antipodal sets and applying . As before, in the higher dimensional sphere we use the vertices of a cross-polytope. In the lower dimensional sphere, we simply use an optimal packing of antipodal points. *Proof.* Let $P\subseteq S^n$ be an antipodal set of $2(k+1)$ points whose image $\overline P$ in $\mathbb{R}\mathrm{P}^n$ is an optimal packing, with as few points as possible at pairwise distance exactly $p_{k+1}(\mathbb{R}\mathrm{P}^n)$. Let $Q\subseteq S^k$ be the vertices of a cross-polytope. Applying , it will suffice to give appropriate upper bounds on $\mathop{\mathrm{vdiam}}(P)$, $\pi-\mathop{\mathrm{sep}}(P)$, $\mathop{\mathrm{vdiam}}(Q)$, and $\pi-\mathop{\mathrm{sep}}(Q)$. The latter two quantities are bounded above by $\arccos\left(\frac{-(k-1)}{k+1}\right)$ by , and so it remains to consider the first two quantities. The separation of $P$ in $S^n$ is exactly equal to $p_{k+1}(\mathbb{R}\mathrm{P}^n)$. Indeed, the quotient map $S^n \to \mathbb{R}\mathrm{P}^n$ exactly preserves the distance between any two points that are within a distance of $\pi/2$ from one another. As $S^n$ is antipodal, the closest pair of points in $S^n$ has distance at most $\pi/2$ between them. The Voronoi diameter of $P$ is at most $2d_\mathrm{H}(P, S^n)$, and we further claim that $d_\mathrm{H}(P, S^n) \le p_{k+1}(\mathbb{R}\mathrm{P}^n)$. First note that because $P$ is antipodal, $d_\mathrm{H}(P, S^n)$ is equal to $d_\mathrm{H}(\overline P, \mathbb{R}\mathrm{P}^n)$. Since $\overline P$ has as few pairs of points as possible at distance exactly $p_{k+1}(\mathbb{R}\mathrm{P}^n)$, we see that $d_\mathrm{H}(\overline P, \mathbb{R}\mathrm{P}^n)\le p_{k+1}(\mathbb{R}\mathrm{P}^n)$; otherwise we could delete a point in $\overline P$ with distance exactly $p_{k+1}(\mathbb{R}\mathrm{P}^n)$ to another point in $\overline P$ and replace it by a point at a strictly larger distance to all other points in $\overline P$. This proves the result. ◻ *Proof.* By , we see that $$\begin{aligned} \pi - 2\hspace{0.1em}d_{\mathrm{GH}}(S^n,S^k) &\ge \pi - \max \left \{\arccos\left(\frac{-(k-1)}{k+1}\right) ,\,\, \pi - p_{k+1}(\mathbb{R}\mathrm{P}^n),\,\, 2p_{k+1}(\mathbb{R}\mathrm{P}^n) \right\}\\ & = \min \left\{\arccos\left(\frac{k-1}{k+1}\right), p_{k+1}(\mathbb{R}\mathrm{P}^n), \pi - 2p_{k+1}(\mathbb{R}\mathrm{P}^n)\right\}.\end{aligned}$$ It thus suffices to bound the three quantities above. Using the bound $\arccos(x) \ge \sqrt{2-2x}$, we see that $\arccos\left(\frac{k-1}{k+1}\right)$ is at least $\frac{2}{\sqrt{k+1}}$. The quantity $\pi - 2p_{k+1}(\mathbb{R}\mathrm{P}^n)$ tends to $\pi$ as $k\to\infty$, and so can be safely ignored. Lastly, the fact that $p_{k+1}(\mathbb{R}\mathrm{P}^n)$ is bounded below by a multiple of $\frac{1}{\sqrt[n]{k}} \ge \frac{1}{\sqrt{k}}$ completes the proof. ◻ *Remark 11*. For any $m$, one can chose optimal packings of $m$ points in $\mathbb{R}\mathrm{P}^n$ and $\mathbb{R}\mathrm{P}^k$, and a similar argument to the one above shows that $$2\hspace{0.1em}d_\mathrm{GH}(S^n, S^k) \le \max \{\pi-p_m(\mathbb{R}\mathrm{P}^n),2p_m(\mathbb{R}\mathrm{P}^k)\}.$$ Choosing $m = k+1$, we note that $2p_{k+1}(\mathbb{R}\mathrm{P}^k) = \pi$, and so this formulation does not generalize or improve on . We are also not aware of examples where choosing some $m > k+1$ improves on the bounds in our previously stated theorems via this formulation. # Acknowledgements {#acknowledgements .unnumbered} We are grateful to Henry Adams, Boris Bukh, Florian Frick, Sunhyuk Lim, and Facundo Mémoli for helpful discussions and feedback. We are also grateful to all our collaborators in the Gromov--Hausdorff, Borsuk--Ulam, Vietoris--Rips polymath project [@dghpolymath] which initiated our work on this project. [^1]: Institute for Advanced Study, Princeton, NJ 08540, USA. Supported by the National Science Foundation through Award No. 1926686. [^2]: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Supported by the National Science Foundation through Award No. 2103206.

arxiv_math

--- abstract: | Predicting the endemic/epidemic transition during the temporal evolution of a contagious disease.\ *Methods:* Defining indicators for detecting the transition endemic/epidemic, with four scalars to be compared, calculated from the daily reported news cases: coefficient of variation, skewness, kurtosis, and entropy. The indicators selected are related to the shape of the empirical distribution of the new cases observed over 14 days. This duration has been chosen to smooth out the effect of weekends when fewer new cases are registered. For finding a forecasting variable, we have used the PCA (principal component analysis), whose first principal component (a linear combination of the selected indicators) explains a large part of the observed variance and can then be used as a predictor of the phenomenon studied (here the occurrence of an epidemic wave).\ *Results:* A score has been built from the four proposed indicators using a Principal Component Analysis (PCA), which allows an acceptable level of forecasting performance by giving a realistic retro-predicted date for the rupture of the stationary endemic model corresponding to the entrance in the epidemic exponential growth phase. This score is applied to the retro-prediction of the limits of the different phases of the COVID-19 outbreak in successive endemic/epidemic transitions and three countries, France, India, and Japan.\ *Conclusion:* We provided a new forecasting method for predicting an epidemic wave occurring after an endemic phase for a contagious disease. author: - | [Jacques Demongeot $^{1}$, Pierre Magal$^{2}$[^1], and Kayode Oshnubi $^{1}$]{.smallcaps}\ *$^{1}$ AGEIS laboratory, UGA, 38700 La Tronche, France*\ *$^{2}$ Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France.*\ *CNRS, IMB, UMR 5251, F-33400 Talence, France.*\ title: "**Forecasting the changes between endemic and epidemic phases of a contagious disease, with the example of COVID-19**" --- **Keywords:** *Contagious disease; Endemic phase; Epidemic wave; Endemic/epidemic transition forecasting; COVID-19 epidemic wave prediction* *The paper is dedicated to James D. Murray, whose pioneering work in mathematical biology we admire.* # Introduction Finding a reliable prediction method of the frontiers between different stationary and non-stationary periods of a time series is a challenging problem. Since the seminal work by Deshayes and Picard on the stationarity rupture in time series [@Deshayes-Picard1979; @Deshayes-Picard1984; @Picard1985], many works have dealt with the break in stationarity [@4; @5; @6; @7; @8; @9], the most recent using the concepts of functional statistics [@10; @11; @12; @13; @14; @15; @16]. Indeed, stationarity is crucial as many forecasting models of time series rely on stationarity for easy modeling and obtaining reliable results. A stationary time series presents statistical properties which do not change over time, as the empirical distribution of the random variable observed in the series, with its main characteristic parameters, mean, coefficient of variation, moments, and entropy. In the event of a break in stationarity, there may be a sudden transition with a sudden change in the values of these parameters and the appearance of a non-constant trend. The problem of the existence of this transition arises with particular acuity in the case of contagious diseases, which alternate stationary endemic periods and epidemic peaks having an initial exponential trend, which must be predicted to prevent the spread of the disease does not give rise to a pandemic. The term endemic phase is understood to mean a period in which there is an equilibrium in a model whose parameters have changed in value following an epidemic phase, due to mitigation measures or a change in the virulence of the infectious agent. In the case of chronic diseases observed outside epidemic phases (bacterial meningitis, rabies, smallpox before its eradication, etc.), the definition of endemicity corresponds more to the sporadic appearance (by birth or human displacement) of individuals much more susceptible than the general population to the infectious agent [@Bernoulli1760; @Blower; @Murray]. We will propose in this article a method to estimate the breakdown of endemic stationarity based on four parameters linked to the empirical distribution of the number of daily reported new cases of COVID-19 in several countries, parameters whose isolated or joint predictive power will be analyzed. These parameters are the coefficient of variation , the skewness, the kurtosis, and the entropy of the stationary empirical measure calculated in a moving window. The pair represented by the succession of an endemic phase and an epidemic wave in COVID-19 outbreak can be considered as a functional unit, the break between the two phases having to be found [@16_1; @16_2; @16_3; @16_4; @16_5; @16_6]. The endemic phase is characterized by a low average level of new cases, with low variance. At the start of the epidemic phase, the average number of cases will grow exponentially, and the standard deviation will grow proportionally at the beginning, then saturate, like that of an additive noise in part independent of growth, which explains the increase then decrease in the coefficient of variation and kurtosis, therefore those of the first principal component at the endemic/epidemic boundary. # Materials and Methods We use in the following a moving window of length 14 days for calculating the empirical distribution of the random variable equal to the number of daily reported new cases. The empirical distribution $N_t$ on day $t$ is obtained from the daily number of reported new cases considered as a random variable $N_t= \left(N(t-13), N(t-12), \ldots, N(t)\right)$. The length of the window has been chosen for eliminating the effect of the week periodicity observed in data due to the lack of reporting each weekend. Indeed, daily new infection cases are highly affected by weekends, such that new case numbers are lowest at the start of the week, and increase afterwards [@16A]. ## Empirical first four moments In the following we will use the terminology *endemic period*, to describe a period during which the daily new cases occurs randomly around some mean value. An *epidemic wave* is a period during which the daily new cases occurs by contact between susceptible and infected. Our goal in this paper is to explore the transition between the endemic period and the following epidemic wave which will be studied by calculating several parameters in a moving window around the frontier of 14 days on which we suspect this transition occurred. We consider the first four moments of $N_t$. We start with the *mean* $$\mu=E(N_t) = \dfrac{ \displaystyle \sum_{i=0}^{13} N(t-i)}{14},$$ where $E$ is the expectation operator, with the *standard deviation* $$\sigma=E\left(\left(N_t-\mu\right)^2\right)^{1/2} = \sqrt{ \dfrac{ \displaystyle \sum_{i=0}^{13} \left(N(t-i)-\mu\right)^2 }{14}}.$$ From these two first parameters, we can compute the *coefficient of variation* $$CV(N_t)=\dfrac{\sigma}{\mu}.$$ The *skewness* of the random variable $N_t$ is the third standardized moment, defined as $$Skew(N_t)=E\left(\left(\dfrac{N_t-\mu}{\sigma} \right)^3\right).$$ Recall that the skewness verifies $$Skew(N_t)= \dfrac{E(N_t^3)-3 \mu \sigma^2 - \mu^3}{\sigma^3}= \dfrac{E(N_t^3)}{\sigma^3}- \left(3\dfrac{1}{CV}+\dfrac{1}{CV^3} \right).$$ The *kurtosis* is the fourth standardized moment, defined as $$Kurt(N_t)=E\left(\left(\dfrac{N_t-\mu}{\sigma} \right)^4\right) .$$ The *empirical entropy* $\mathcal{E}$ of the empirical distribution is defined as follows: $$\mathcal{E} (N_t) = - \sum_{i=1:d \text{ with } p_i >0} p_i \log p_i,$$ where the $p_i$ are the weights of a histogram on d value intervals of $N_t$. In the Results' section, we use the *approximate entropy*. We refer to [@Pincus] for more details. ## Phenomenological model used for multiple epidemic waves To represent the data, we used a phenomenological model to fit the curve of cumulative reported cases. Such an idea is not new since it was already proposed by Bernoulli [@Bernoulli1760] in 1760 in the context of the smallpox epidemic. Here we used the so-called Bernoulli--Verhulst [@Verhulst1838] model to describe the epidemic phase. Bernoulli [@Bernoulli1760] investigated an epidemic phase followed by an endemic phase. This appears clearly in Figures 9 and 10 of the paper by Dietz, and Heesterbeek [@Dietz-Heesterbeek] who revisited the original article of Bernoulli. We also refer to Blower [@Blower] for another article revisiting the original work of Bernoulli. Several works comparing cumulative reported cases data and the Bernoulli--Verhulst model appear in the literature (see [@Hsieh; @Wang-Wu-Yang; @Zhou-Yan]). The Bernoulli--Verhulst model is sometimes called Richard's model, although Richard's work came much later in 1959. The phenomenological model deals with data series of new infectious cases decomposed into two successive phases: 1) endemic phases followed by 2) epidemic phases. **Endemic phase:** During the endemic phase, the dynamics of new cases appears to fluctuate around an average value independently of the number of cases. Therefore the average cumulative number of cases is given by $$\label{15.11} {\rm CR}(t)=N_0+ (t-t_{0}) \times a, \text{ for } t \in [t_0 , t_1],$$ where $t_0$ denotes the beginning of the endemic phase, $N_0$ is the number of new cases at time $t_0$, and $a$ is the average value of the daily number of new cases. We assume that the average daily number of new cases is constant. Therefore the daily number of new cases is given by $$\label{15.12} {\rm CR}'(t)= a.$$ **Epidemic phase:** In the epidemic phase, the new cases are contributing to produce secondary cases. Therefore the daily number of new cases is no longer constant, but varies with time as follows $$\label{15.13} {\rm CR}(t)= N_{\rm{base}}+ \dfrac{\mathrm{e}^{\chi (t-t_0)} N_0 }{\left[ 1+ \dfrac{ N_0^\theta}{N_\infty^\theta} \left(\mathrm{e}^{\chi \theta \left(t- t_0 \right) } -1 \right) \right]^{ 1/\theta} }, \text{ for } t \in [t_0 , t_1].$$ In other words, the daily number of new cases follows the Bernoulli--Verhulst [@Bernoulli1760; @Verhulst1838] equation. Namely, by setting $$\label{15.14} N(t)={\rm CR}(t)-N_{\rm{base}},$$ we obtain $$\label{15.15} N'(t)= \chi \, N(t) \, \left[ 1 - \left( \dfrac{N(t) }{N_\infty}\right)^\theta \right],$$ completed with the initial value $$N(t_0)=N_0.$$ In the model, $N_{\rm{base}}+N_0$ corresponds to the value ${\rm CR}(t_0)$ of the cumulative number of cases at time $t=t_0$. The parameter $N_\infty+N_{\rm{base}}$ is the maximal value of the cumulative reported cases after the time $t=t_0$. $\chi>0$ is a Malthusian growth parameter, and $\theta$ regulates the speed at which ${\rm CR}(t)$ increases to $N_\infty+N_{\rm{base}}$. # Results Here we use cumulative numbers of reported new cases for COVID-19 in France, India, and Japan taken from WHO [@Data]. Data shows a succession of endemic periods (yellow background color regions) followed by epidemic waves (blue background color regions). ## Data for France {#Fig1} In Figure [2](#Fig2){reference-type="ref" reference="Fig2"}, each colored segment corresponds to a single endemic or epidemic phenomenological model. This change of color may occur due to a change of dynamic inside an epidemic phase when a second wave comes before the end of the previous one. {#Fig2} {#Fig3} By making a principal component analysis (i.e. the matlab function PCA) between the standardized variables $CV_s(N_t), Skew_s(N_t), Kurt_s(N_t), \mathcal{E}_s(N_t)$, we obtain the percentage of the variance explained by each principal component $$Explain= \left( \begin{array}{cccc} 70.71\\ 21.92\\ 4.94\\ 2.43\\ \end{array} \right)$$ and the matrix giving the projection coefficients of the principal components $$coeff =\left( \begin{array}{cccc} 0.5527 & -0.1480 & 0.7163 & -0.3995\\ 0.5631 & -0.2162 & -0.0348 & 0.7968\\ 0.5577 & -0.0795 & -0.6955 & -0.4461\\ 0.2577 & 0.9618 & 0.0449 & 0.0808 \end{array} \right).$$ By using the first column of the above matrix, we deduce the first principal component $$\label{8} 0.55 CV_s(N_t)+ 0.56 Skew_s(N_t)+ 0.56 Kurt_s(N_t)+ 0.26 \mathcal{E}_s(N_t)$$ which explains $70.71 \%$ of the variability. We deduce that Kurtosis, Skewness, and the coefficient of variation (in decreasing order of importance) best explain the variability. ![*In this figure we plot the first principal component for France (see formula [\[8\]](#8){reference-type="eqref" reference="8"}). The horizontal green lines correspond to $\pm 1$.* ](Figure/Figure11FR.pdf){#Fig4} ## Data for India In this subsection, we consider the data for India. We present the same curves as for France, describing successively raw data and simulated results by the phenomenological models. {#Fig5} In Figure [6](#Fig6){reference-type="ref" reference="Fig6"}, each colored segment corresponds to a single endemic or epidemic phenomenological model. This change of color may occur due to a change of dynamic inside an epidemic phase when a second wave comes before the end of the previous one. {#Fig6} {#Fig7} By making a principal component analysis (i.e. the matlab function PCA) between the standardized variables $CV_s(N_t), Skew_s(N_t), Kurt_s(N_t)$, and $\mathcal{E}_s(N_t)$, we obtain the percentage of the variance explained by each principal component $$Explain= \left( \begin{array}{cccc} 53.13\\ 22.55\\ 14.22\\ 10.09 \\ \end{array} \right)$$ and the matrix giving the projection coefficients of the principal components $$coeff =\left( \begin{array}{cccc} 0.5161 & -0.2296 & 0.8212 & -0.0810\\ 0.5714 & -0.0587 & -0.4434 & -0.6881\\ 0.5660 &-0.2235 &-0.3478 & 0.7133\\ 0.2946 & 0.9455 & 0.0896 & 0.1062 \end{array} \right).$$ By using the first column of the above matrix, we deduce the first principal component $$\label{9} 0.52 CV_s(N_t)+ 0.57 Skew_s(N_t)+ 0.57 Kurt_s(N_t)+ 0.29 \mathcal{E}_s(N_t)$$ which explains $53 \%$ of the variability. ![*In this figure we plot the first principal component for India (see formula [\[9\]](#9){reference-type="eqref" reference="9"}). The horizontal green lines correspond to the values $\pm 1$.* ](Figure/Figure11IN.pdf){#Fig8} ## Data for Japan In this subsection, we consider the data for Japan. We present the same curves as for France and India, describing successively raw data and simulated results by the phenomenological models. {#Fig9} {#Fig10} {#Fig11} By making a principal component analysis (i.e. the matlab function PCA) between the standardized variables $CV_s(N_t), Skew_s(N_t), Kurt_s(N_t)$, and $\mathcal{E}_s(N_t)$, we obtain the percentage of the variance explained by each principal component $$Explain= \left( \begin{array}{cccc} 71.62\\ 17.54 \\ 6.87\\ 3.97 \end{array} \right)$$ and the matrix giving the projection coefficients of the principal components $$coeff =\left( \begin{array}{cccc} 0.5234 & -0.2243 & 0.7969 & -0.2018\\ 0.5452 &-0.2406 & -0.2310 & 0.7691\\ 0.5401 & -0.1760 & -0.5575 & -0.6054\\ 0.3703 & 0.9278 & 0.0270 & 0.0358 \end{array} \right).$$ By using the first column of the above matrix, we deduce the first principal component $$\label{10} 0.52 CV_s(N_t)+ 0.55 Skew_s(N_t)+ 0.54 Kurt_s(N_t)+ 0.37 \mathcal{E}_s(N_t)$$ which explains $71 \%$ of the variability. We deduce that Kurtosis, Skewness, and the coefficient of variation (in decreasing order of importance) best explain the variability. ![*In this figure we plot the first principal component for Japan (see formula [\[10\]](#10){reference-type="eqref" reference="10"}). The horizontal green lines correspond to the values $\pm 1$.* ](Figure/Figure11JP.pdf){#Fig12} # Discussion The forecasting of the epidemic waves of the COVID-19 outbreak is based on a change in the nature of the time series dynamics related to the number of daily new reported cases of this contagious disease. This change can concern the moments or the entropy of the empirical distribution of the stationary component at the end of the endemic phase, which disrupts when a not constant trend occurs, marking the start of an epidemic wave. From a careful examination of Figures [4](#Fig4){reference-type="ref" reference="Fig4"}, [8](#Fig8){reference-type="ref" reference="Fig8"} and [12](#Fig12){reference-type="ref" reference="Fig12"}, we can conclude that there are not constant but frequent patterns for $C_1(t)$ identifiable in the three studied countries and for a majority of their endemic/epidemic transitions. The predictive power of the first principal component $C_1(t)$ can be quantified by its performance ratio, that is, by the percentage of correct retro-predictions obtained by fixing variation thresholds to forecast the occurrence of an epidemic wave. For France, if we fix the threshold to the value $1$, $C_1(t)$ predicts correctly an epidemic outbreak a weak after a decrease from this threshold value not reached elsewhere in the endemic phase. This prediction is correct at $53\%$ only for India. For Japan, the performance of the retro-prediction is $71\%$ and $70\%$ for France. It is clear that the level of prediction is not very high (71% in the best case), but, in the absence of a currently reliable predictor, we can consider that it is sufficient to trigger mitigation measures at level of a population. A more systematic study of the changing shape of the empirical distribution is needed, looking at many epidemic waves in many countries. A parameter measuring the deviation from classical laws (such as those linked to the Kolmogorov-Smirnov, chi-square or Shapiro-Wilk tests in the case of the normal law) could thus be added in subsequent works. As previously noticed in [@Weitz], a classical epidemic peak with a near-symmetric growth and decline may be preceded or followed by a shoulder-like behavior corresponding to a prematurely stopped wave followed by another. Shoulder-like behavior for epidemic waves can be explained by using multiple sub-group epidemic models. This idea was first explored for SARS-CoV-1 in [@Magal] and reconsidered for SARS-CoV-2 by [@Chowell]. But the changes between endemic and epidemic are still challenging to model. # Conclusions We have studied in this article the evolution of four parameters related to the dynamics of the number of the daily reported new cases $N_t$ at day $t$ of a contagious disease, which can serve as early indicators of the appearance of epidemic waves from a previous endemic state. By applying this parameter calculation to COVID-19, we showed that a score obtained by PCA based on the linear combination of the four chosen parameters with specific coefficients for each one could forecast the variations of the empirical distribution of the daily reported number of new cases $N_t$, then could be considered often as a good predictor (for the countries and the epidemic waves in these countries) of the endemic-epidemic transition. A systematic study of contagious diseases other than COVID-19 is necessary to confirm this forecasting property's existence. Still, we can already propose this score as a realistic indicator of the next occurrence of an epidemic outbreak from a change in the dynamics of the observed daily new cases during the endemic periods. 999 Deshayes, J, & Picard, D. Application aux tests de rupture de régression. *Astérisque* 1979, 68, 73-98. Deshayes, J., & Picard, D. Lois asymptotiques des tests et estimateurs de rupture dans un modèle statistique classique. *Annales de l'I.H.P. Probabilités et statistiques* 1984, 20, 309-327. Picard, D. Testing and estimating change-points in time series. *Advances in applied probability*, 1985, 17(4), 841-867. Vogt, M., & Dette, H. Detecting gradual changes in locally stationary processes. *The Annals of Statistics* 2015, 43, 713-740. van Delft, A., & Eichler, M. Locally stationary functional time series. *Electronic Journal of Statistics* 2018, 12, 107-170. Palachy, S. Detecting stationarity in time series data. *Medium Towards Data Sci.* (2019), 9, 53. Ting, K.M., Liu, Z., Zhang, H., & Zhu, Y. A new distributional treatment for time series and an anomaly detection investigation. *Proceedings of the VLDB Endowment* (2022), 15, 2321-2333. Hauber, A.L., Sigloch, C., & Timmer, J. Detecting frequency modulation in stochastic time-series data. *Physical Review E* (2022), 106, 024204. Bawdekar, A.A., Prusty, B.R., & Bingi, K. Sensitivity Analysis of Stationarity Tests' Outcome to Time Series Facets and Test Parameters. *Mathematical Problems in Engineering* 2022, 2022, 2402989. Demongeot, J., Laksaci, A., Madani, F., & Rachdi, M. Functional data: local linear estimation of the conditional density and its application. *Statistics* (2013), 47, 26-44. Rachdi, M., Laksaci, A., Demongeot, J., Abdali, A., & Madani, F. Theoretical and practical aspects on the quadratic error in the local linear estimation of the conditional density for functional data. *Comp. Stat. Data Anal.* (2014), 73, 53-68. Demongeot, J., Laksaci, A., Rachdi, M., & Rahmani, S. On the local linear modelization of the conditional distribution for functional data. *Sankhya A* (2014), 76, 328-355. Demongeot, J., Hamie, A., Laksaci, A., & Rachdi, M. Relative-Error Prediction in Nonparametric Functional Statistics: Theory and Practice. *Journal of Multivariate Analysis* (2016), 146, 261-268. Belkis, A., Demongeot, J., Laksaci, A., & Rachdi, M. Functional data analysis: Estimation of the relative error in functional re-gression under random left-truncation. *J. Nonparametric Statistics* (2018), 30, 472-490. Oshinubi, K., Ibrahim, F., Rachdi, M., & Demongeot, J. Functional Data Analysis: Application to Daily Observation of COVID-19 Prevalence in France. *AIMS Mathematics* (2022), 7, 5347-5385. Jelassi, M., Oshinubi, K., Rachdi, M., & Demongeot, J. Epidemic Dynamics on Social Interaction Networks. AIMS Bioengineering (2022), 9, 348-361. Bernoulli, D. Essai d'une nouvelle analyse de la petite Vérole, & des avantages de l'Inoculation pour la prévenir, *Mémoire Académie Royale des Sciences, Paris*, 1760. Blower S, & Bernoulli D. An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it. (1766). *Rev Med Virol.* (2004) Sep-Oct;14(5):275-88. Murray J.D., Stanley E.A., Brown D.L. On the spatial spread of rabies among foxes. *Proc R Soc Lond B Biol Sci.* (1986) Nov 22;229 (1255):111-50. Cinaglia, P., & Cannataro, M. Forecasting COVID-19 epidemic trends by combining a neural network with $R_t$ estimation. *Entropy* (2022), 24(7), 929. Srivastava, S. R., Meena, Y. K., & Singh, G. Forecasting on Covid-19 infection waves using a rough set filter driven moving average models. *Appl. Soft. Comput.* (2022) Dec;131:109750. Lynch, C. J., & Gore, R. Short-range forecasting of COVID-19 during early onset at county, health district, and state geographic levels using seven methods: comparative forecasting study. *Journal of medical Internet research*, 23(3) (2021), e24925. Lynch C.J., & Gore R. Application of one-, three-, and seven-day forecasts during early onset on the COVID-19 epidemic dataset using moving average, autoregressive, autoregressive moving average, autoregressive integrated moving average, and naïve forecasting methods. *Data Brief.* (2021) Apr;35:106759. Yu CS, Chang SS, Chang TH, Wu JL, Lin YJ, Chien HF, & Chen RJ. A COVID-19 Pandemic Artificial Intelligence-Based System With Deep Learning Forecasting and Automatic Statistical Data Acquisition: Development and Implementation Study. *J Med Internet Res.* (2021) May 20;23(5):e27806. ArunKumar KE, Kalaga DV, Sai Kumar CM, Chilkoor G, Kawaji M, & Brenza TM. Forecasting the dynamics of cumulative COVID-19 cases (confirmed, recovered and deaths) for top-16 countries using statistical machine learning models: Auto-Regressive Integrated Moving Average (ARIMA) and Seasonal Auto-Regressive Integrated Moving Average (SARIMA). *Appl Soft Comput.* 2021 May;103:107161. Demongeot, J., Oshinubi, K., Rachdi, M., Seligmann, H., Thuderoz, F., & Waku, J. Estimation of daily reproduction rates in COVID-19 outbreak, *Computation*, 9, 109 (2021). Demongeot, J., & Magal, P. Spectral method in epidemic time series. *Biology* (2022), 11, 1825. Worldometer. Available online: https://www.worldometers.info/coronavirus (Accessed on 6 January 2023). Caswell, H. *Matrix population models: construction, analysis and interpretation*, 2nd ed. Sinauer: Sunderland, (2001). Demongeot, J., & Demetrius, L. Complexity and Stability in Biological Systems. *Int. J. Bifurcation and Chaos* (2015), 25, 40013. Nicol-Harper, A., Dooley, C., Packman, D., Mueller, M., Bijak, J., Hodgson, D., Townley, S., & Ezard, T. Inferring transient dynamics of human populations from matrix non-normality. *Popul Ecol.* (2018), 60, 185-196. Xu, Z., Yang, D., Wang, J., & Demongeot, J. Statistical Analysis Supports UTR (untranslated region) Deletion Theory in SARS-CoV-2. *Virulence* (2022), 13, 1772-1789. Demongeot, J., Griette, Q., Maday, Y., & Magal, P. A Kermack-McKendrick model with age of infection starting from a single or multiple cohorts of infected patients. *Proc. Royal Society A* (2023), 479, 2022.0381. Kammegne, B., Oshinubi, K., Babasola, T., Peter, O.J., Longe, O.B., Ogunrinde, R.B., Titiloye, E.O., & Demongeot, J. Mathematical modelling of spatial distribution of COVID-19 outbreak using diffusion equation. *Pathogens* (2023), 12, 88. Xu, Z., Wei, D., Zeng, Q., Zhang, H., Sun, Y., & Demongeot, J. More or less deadly? A mathematical model that predicts 1 SARS-CoV-2 evolutionary direction. *Computers in Biology & Medicine* 2023, 153, 106510. Pincus, S. M. Approximate entropy as a measure of system complexity. *Proceedings of the National Academy of Sciences*, 88(6) (1991), 2297-2301. Verhulst, P. F. Notice sur la loi que la population poursuit dans son accroissement, *Corresp. Mathématique Phys.*, (1938),113-121. Dietz, K., & Heesterbeek, J. A. P. Daniel Bernoulli's epidemiological model revisited, *Math. Biosci.*, 180 (2002), 1-21. Hsieh, Y. H. Richards model: a simple procedure for real-time prediction of outbreak severity, in *Modeling and dynamics of infectious diseases*, (2009), 216--236. Wang, X.S., Wu, J., & Yang, Y. Richards model revisited: validation by and application to infection dynamics, *J. Theoret. Biol.*, 313 (2012), 12-19. Zhou, G., & Yan, G. Severe acute respiratory syndrome epidemic in Asia, *Emerg. Infect. Dis.*, 9 (2003), 1608-1610. Data from WHO. Accessed: July 20, 2022. <https://COVID19.who.int/WHO-COVID-19-global-data.csv> Weitz, J. S., Park, S. W., Eksin, C., & Dushoff, J. Awareness-driven behavior changes can shift the shape of epidemics away from peaks and toward plateaus, shoulders, and oscillations. *Proceedings of the National Academy of Sciences*, 117(51) (2020), 32764-32771. Magal, P., Seydi, O., & Webb, G. Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission. *Mathematical biosciences*, 301 (2018), 59-67. Chowell, G., Dahal, S., Tariq, A., Roosa, K., Hyman, J. M., & Luo, R. An ensemble n-sub-epidemic modeling framework for short-term forecasting epidemic trajectories: Application to the COVID-19 pandemic in the USA. *PLoS Computational Biology*, 18(10) (2022), e1010602. [^1]: Corresponding author. e-mail: <pierre.magal@u-bordeaux.fr>

arxiv_math

--- abstract: | We find all K-polystable limits of smooth Fano threefolds in family . address: - "*Ivan Cheltsov* [University of Edinburgh, Edinburgh, Scotland]{.nodecor} [`i.cheltsov@ed.ac.uk`]{.nodecor}" - "*Alan Thompson* [Loughborough University, Loughborough, England]{.nodecor} [`a.m.thompson@lboro.ac.uk`]{.nodecor}" author: - Ivan Cheltsov and Alan Thompson title: K-moduli of Fano threefolds in family --- Throughout this paper, all varieties are assumed to be projective and defined over $\mathbb{C}$. # Introduction {#section:intro} Let $Q$ be a smooth quadric threefold in $\mathbb{P}^4$, let $C_1$ and $C_2$ be disjoint smooth irreducible conics in the quadric $Q$. Then we can choose coordinates $x$, $y$, $z$, $t$, $w$ on $\mathbb{P}^4$ such that $$\begin{aligned} C_1&=\{x=0,y=0,w^2+zt=0\},\\ C_2&=\{z=0,t=0,w^2+xy=0\},\end{aligned}$$ and $Q$ is one of the following quadrics: $$\begin{aligned} Q&=\big\{w^2+xy+zt=a(xt+yz)+b(xz+yt)\big\},\ \text{$(a,b)\in\mathbb{C}^2$ such that $a\pm b\ne \pm 1$},\tag{$\beth$}\\ Q&=\big\{w^2+xy+zt=a(xt+yz)+xz\big\},\ \text{$a\in\mathbb{C}$ such that $a\ne\pm 1$},\tag{$\gimel$}\\ Q&=\big\{w^2+xy+zt=xt+xz\big\}.\tag{$\daleth$}\end{aligned}$$ Let $\pi\colon X\to Q$ be the blow up of the quadric threefold $Q$ along the conics $C_1$ and $C_2$. Then $X$ is a smooth Fano threefold in the deformation family , and all smooth members of this deformation family can be obtained in this way. Alternatively, we can describe $X$ as a complete intersection in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$ of three smooth divisors of degree $(1,0,1)$, $(0,1,1)$, $(0,0,2)$. Namely, if $Q$ is the quadric $(\beth)$, then $$\label{equation:beth}\tag{$\beth$} X=\big\{u_1x=v_1y,u_2z=v_2t,w^2+xy+zt=a(xt+yz)+b(xz+yt)\big\}$$ where $([u_1:v_1],[u_2:v_2])$ are coordinates on $\mathbb{P}^1\times\mathbb{P}^1$. If $Q$ is the quadric $(\gimel)$, then $$\label{equation:gimel}\tag{$\gimel$} X=\big\{u_1x=v_1y,u_2z=v_2t,w^2+xy+zt=a(xt+yz)+xz\big\}$$ Finally, if $Q$ is the quadric $(\daleth)$, then $$\label{equation:daleth}\tag{$\daleth$} X=\big\{u_1x=v_1y,u_2z=v_2t,w^2+xy+zt=xt+xz\big\}$$ We know all K-polystable smooth Fano threefolds in the deformation family . Namely, it has been proved in [@Book] that $X$ is K-polystable $\iff$ $Q$ is the quadric $(\beth)$. The goal of this paper is prove the following result: **Main Theorem 1**. *All K-polystable smoothable Fano threefolds in the family are complete intersections in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$ which can be described as follows: $$\label{equation:K-beth}\tag{$\beth$} \big\{u_1x=v_1y,u_2z=v_2t,w^2+xy+zt=a(xt+yz)+b(xz+yt)\big\},$$ $$\label{equation:V-aleph}\tag{$\aleph$} \big\{u_1x=v_1y,u_2z=v_2t,w^2+r(x+y)^2+r(z+t)^2=(2s+2)(xt+yz)+(2s-2)(xz+yt)\big\},$$ where $(a,b)\in\mathbb{C}^2$ and $[r:s]\in\mathbb{P}^1$.* K-polystable Fano threefolds in the Main Theorem form an irreducible two-dimensional component of the K-moduli of smoothable Fano threefolds, see Section [6](#section:K-moduli){reference-type="ref" reference="section:K-moduli"} for the global description of this component. Another two-dimensional component of this moduli space has been described in [@Gokova], and its one-dimensional components are described in [@London]. Let us say few words about the threefolds in the Main Theorem. Let $\mathscr{X}$ be one of them. Then, using the natural projection $\mathscr{X}\to\mathbb{P}^4$, we get a birational morphism $\varpi\colon\mathscr{X}\to\mathscr{Q}$, where $\mathscr{Q}$ is a (possibly singular) irreducible quadric in $\mathbb{P}^4$ such that either $$\label{equation:K-beth-quadric}\tag{$\beth$} \mathscr{Q}=\big\{w^2+xy+zt=a(xt+yz)+b(xz+yt)\big\}$$ or $$\label{equation:V-aleph-quadric}\tag{$\aleph$} \mathscr{Q}=\big\{w^2+r(x+y)^2+r(z+t)^2=(2s+2)(xt+yz)+(2s-2)(xz+yt)\big\},$$ where $(a,b)\in\mathbb{C}^2$ and $[r:s]\in\mathbb{P}^1$. The morphism $\varpi$ is a blow up of two conics: $$\begin{aligned} \mathscr{C}_1&=\mathscr{Q}\cap \{x=0,y=0\},\\ \mathscr{C}_2&=\mathscr{Q}\cap\{z=0,t=0\}.\end{aligned}$$ These conics are contained in the smooth locus of the quadric $\mathscr{Q}$, and - $\mathscr{C}_1$ and $\mathscr{C}_2$ are smooth if $\mathscr{Q}$ is the quadric ($\beth$), - $\mathscr{C}_1$ and $\mathscr{C}_2$ are reduced and singular if $\mathscr{Q}$ is the quadric ($\aleph$) with $[r:s]\ne [0:1]$, - $\mathscr{C}_1$ and $\mathscr{C}_2$ are non-reduced if $\mathscr{Q}$ is the quadric ($\aleph$) with $[r:s]=[0:1]$. If $\mathscr{Q}$ is the quadric ($\beth$), then $\mathscr{X}$ is singular $\iff$ $\mathscr{Q}$ is singular $\iff$ $a\pm b\pm 1=0$. In this case, the singularities of the quadric $\mathscr{Q}$ can be described as follows: - if $a\pm b\pm 1=0$ and $ab\ne 0$, then $\mathscr{Q}$ has one singular point, - if $a\pm b\pm 1=0$ and $ab=0$, then $\mathscr{Q}$ is singular along a line. Similarly, if $\mathscr{Q}$ is the quadric ($\aleph$), then $\mathscr{X}$ is singular, because both $\mathscr{C}_1$ and $\mathscr{C}_2$ are singular. In this case, $\mathscr{Q}$ is smooth $\iff$ $[r:s]\ne[\pm 1:1]$, and $|\mathrm{Sing}(\mathscr{Q})|=1$ when $[r:s]=[\pm 1:1]$. **Acknowledgements.** We started this research project in New York back in October 2022 during the conference *MMP and Moduli*, and we completed it in Banff in February 2023 during the workshop *Explicit Moduli Problems in Higher Dimensions*. We would like to thank the Simons Foundation and Banff International Research Station for very good working conditions. We would like to thank Hamid Abban and Kento Fujita for comments, and Yuchen Liu for his help with arguments in Section [6](#section:K-moduli){reference-type="ref" reference="section:K-moduli"}. Ivan Cheltsov is supported by EPSRC grant EP/V054597/1. Alan Thompson is supported by EPSRC grant EP/V005545/1. # First GIT quotient (bad moduli space) {#section:GIT-1} In this section we describe a compact moduli space for Fano threefolds in family , by applying GIT techniques to the description of such threefolds as blow-ups of quadrics along two smooth disjoint conics. As in Section [1](#section:intro){reference-type="ref" reference="section:intro"}, we fix two smooth disjoint conics $$\begin{aligned} C_1&=\{x=0,y=0,w^2+zt=0\}\subset\mathbb{P}^4,\\ C_2&=\{z=0,t=0,w^2+xy=0\}\subset\mathbb{P}^4,\end{aligned}$$ where $[x:y:z:t:w]\in\mathbb{P}^4$. Let $Q$ be a quadric threefold in $\mathbb{P}^4$ that contains $C_1$ and $C_2$. Then $Q$ is given by an equation of the form $$\label{equation:quadric} \alpha(xy+zt+w^2)+\beta xz+\gamma xt+\delta yz+\epsilon yt=0,$$ where $[\alpha:\beta:\gamma:\delta:\epsilon] \in \mathbb{P}^4$. Let $\pi\colon X\to Q$ be the blow up of of the conics $C_1$ and $C_2$. If $Q$ is smooth, then $X$ is a smooth Fano threefold in the deformation family . Note that equations of the form [\[equation:quadric\]](#equation:quadric){reference-type="eqref" reference="equation:quadric"} are preserved by the action of $(\mathbb{C}^\ast)^2$ given by $$[x:y:z:t:w]\mapsto\Big[\lambda x: \tfrac{1}{\lambda}y: \mu z: \tfrac{1}{\mu} t: w\Big]$$ for $(\lambda,\mu)\in(\mathbb{C}^\ast)^2$. This induces a $(\mathbb{C}^\ast)^2$-action on the parameter space $\mathbb{P}^4$ given by $$[\alpha:\beta:\gamma:\delta:\epsilon] \mapsto \Big[\alpha:\lambda\mu \beta: \tfrac{\lambda}{\mu} \gamma: \tfrac{\mu}{\lambda} \delta: \tfrac{1}{\lambda\mu} \epsilon\Big]$$ for $(\lambda,\mu)\in(\mathbb{C}^\ast)^2$. The GIT quotient by this action is isomorphic to $\mathbb{P}^2$ with coordinates $$[\mathbf{x}_0:\mathbf{x}_1:\mathbf{x}_2] =[\alpha^2: \beta\epsilon: \gamma\delta].$$ We now classify the unstable, semistable, and stable points, and corresponding quadrics. ## Unstable points. The unstable points $[\alpha:\beta:\gamma:\delta:\epsilon] \in \mathbb{P}^4$ and the corresponding quadric threefolds can be described as follows: $$\begin{aligned} \alpha = \beta =\gamma = 0&\ \text{and}\ Q=\{y(\delta z+\epsilon t)=0\},\\ \alpha = \beta = \delta = 0&\ \text{and}\ Q=\{t(\gamma x+\epsilon y)=0\}, \\ \alpha = \gamma = \epsilon = 0&\ \text{and}\ Q=\{z(\beta x+\delta y)=0\}, \\ \alpha = \delta = \epsilon = 0&\ \text{and}\ Q=\{x( \beta z+\gamma t)=0\}.\end{aligned}$$ In these cases, the quadric $Q$ degenerates to a union of two hyperplanes. ## Stable points. A point $[\alpha:\beta:\gamma:\delta:\epsilon] \in \mathbb{P}^4$ is stable if and only if its orbit is closed and its stabiliser is finite. This occurs if and only if all of $\beta$, $\gamma$, $\delta$ and $\epsilon$ are nonzero, corresponding to points in the GIT quotient $\mathbb{P}^2$ away from the lines $\{\mathbf{x}_1=0\}$ and $\{\mathbf{x}_2=0\}$. We separate the corresponding quadric threefolds into three cases. 1. Stable orbits with $\alpha \neq 0$ correspond to points in $\mathbb{P}^2$ with all of $\mathbf{x}_0, \mathbf{x}_1, \mathbf{x}_2$ nonzero. Taking the affine chart $\mathbf{x}_0 = 1$, each such orbit contains quadrics of the form $$w^2+xy+zt+a(xt+yz)+b(xz+yt)=0,$$ for $(a,b)\in(\mathbb{C}^*)^2$ given by $a = \pm \sqrt{\mathbf{x}_2}$ and $b = \pm \sqrt{\mathbf{x}_1}$. The quadric is singular if and only if $a \pm b \pm 1 = 0$, which corresponds to the affine curve $$1-2\mathbf{x}_1-2\mathbf{x}_2+\mathbf{x}_1^2-2\mathbf{x}_1\mathbf{x}_2+\mathbf{x}_2^2=0.$$ 2. Orbits with $\alpha=0$ and $\gamma \delta \neq \beta \epsilon$ correspond to points in the quotient $\mathbb{P}^2$ that satisfy the following conditions: $\mathbf{x}_0 = 0$ and $\mathbf{x}_1 \neq \mathbf{x}_2$. So, taking the affine chart $\mathbf{x}_1=1$, we see that each such orbit contains a quadric of the form $$a(xt+yz)+xz+yt= 0,$$ for $a \in \mathbb{C}^* \setminus \{\pm 1\}$ given by $a = \pm\sqrt{\mathbf{x}_2}$. Such quadrics have one singular point. 3. There is a unique stable orbit with $\alpha=0$ and $\gamma \delta = \beta \epsilon \neq 0$, corresponding to the point $[\mathbf{x}_0:\mathbf{x}_1:\mathbf{x}_2] = [0:1:1]$ in the quotient. It contains the threefold $$(x+y)(z+t)=0,$$ which is a union of two hyperplanes. ## Strictly semistable points. A point $[\alpha:\beta:\gamma:\delta:\epsilon]$ is strictly semistable if and only if it is not unstable, and at least one of $\beta,\gamma,\delta,\epsilon$ is zero. Such points have non-closed orbits or infinite stabilisers. **Example 1**. If $\beta\ne 0$, $\gamma\ne 0$ and $\delta\ne 0$, then the orbit of $[\alpha:\beta:\gamma:\delta:0]$ contains the point $[\alpha:0:\gamma:\delta:0]$ in its closure, and the orbit of $[\alpha:0:\gamma:\delta:0]$ is closed but has infinite stabiliser given by $(\lambda,\lambda)\in (\mathbb{C}^\ast)^2$. The strictly semistable points belonging to minimal orbits (strictly polystable points) are described below, along with their corresponding quadric threefolds. 1. All points of the form $[\alpha:0:\gamma:\delta:0]$ or $[\alpha:\beta:0:0:\epsilon]$, where $\alpha\beta\gamma\delta\epsilon\ne 0$, and either $\alpha^2 \neq \gamma\delta$ or $\alpha^2 \neq \beta\epsilon$, respectively. Such orbits lie over points in the quotient $$\begin{aligned} \Big\{[\mathbf{x}_0:0:\mathbf{x}_2]\ \big|\ \mathbf{x}_0, \mathbf{x}_2 \in \mathbb{C}^*,\ \mathbf{x}_0 \neq \mathbf{x}_2\Big\},\\ \Big\{[\mathbf{x}_0:\mathbf{x}_1:0]\ \big|\ \mathbf{x}_0, \mathbf{x}_1 \in \mathbb{C}^*,\ \mathbf{x}_0 \neq \mathbf{x}_1\Big\}.\end{aligned}$$ The corresponding quadrics are $$\begin{aligned} \big\{w^2+xy+zt+a(xt+yz)&=0\big\},\\ \big\{w^2+xy+zt+b(xz+yt)&=0\big\},\end{aligned}$$ where $a,b \in \mathbb{C}^* \setminus \{\pm 1\}$. They are smooth. 2. The orbit of the point $[1:0:0:0:0]$. It gives the point $[1:0:0]$ in the quotient. The corresponding quadric threefold is $\{w^2+xy+zt=0\}$ --- it is smooth. 3. Two orbits of all points $[0:0:\gamma:\delta:0]$ and $[0:\beta:0:0:\epsilon]$ such that $\beta\gamma\delta\epsilon\ne 0$, which lie over the points $[0:0:1]$ and $[0:1:0]$ in the quotient $\mathbb{P}^2$, respectively. This gives two singular quadrics: $\{xt+yz=0\}$ and $\{xz+yt=0\}$. 4. Two orbits consisting of points of the form $[\alpha:0:\gamma:\delta:0]$ or $[\alpha:\beta:0:0:\epsilon]$, where $\alpha\beta\gamma\delta\epsilon\ne 0$, and either $\alpha^2 = \gamma\delta$ or $\alpha^2=\beta\epsilon$, respectively. These orbits lie over the points $[1:0:1]$ and $[1:1:0]$ in the quotient, and the quadrics are $$\begin{aligned} w^2+xy+zt+xt+yz&=0,\\ w^2+xy+zt+xz+yt&=0.\end{aligned}$$ Both of them are irreducible and singular along the line $\{w=y-t=x-z=0\}$. ## Quotient space Note that the space of quadrics of the form [\[equation:quadric\]](#equation:quadric){reference-type="eqref" reference="equation:quadric"} admits a discrete action of the group $\boldsymbol{\mu}_2^3$ given by $$\begin{aligned} \sigma_1\colon [x:y:z:t:w] &\mapsto [y:x:z:t:w],\\ \sigma_2\colon [x:y:z:t:w] &\mapsto [z:t:x:y:w],\\ \sigma_3\colon [x:y:z:t:w] &\mapsto [x:y:z:t:-w].\end{aligned}$$ These involutions act on the parameter space $\mathbb{P}^4$ by $$\begin{aligned} \sigma_1\colon [\alpha:\beta:\gamma:\delta:\epsilon]&\mapsto [\alpha:\delta:\epsilon:\beta:\gamma],\\ \sigma_2\colon [\alpha:\beta:\gamma:\delta:\epsilon]&\mapsto [\alpha:\beta:\delta:\gamma:\epsilon],\\ \sigma_3\colon [\alpha:\beta:\gamma:\delta:\epsilon]&\mapsto [\alpha:\beta:\gamma:\delta:\epsilon].\end{aligned}$$ The involutions $\sigma_2$ and $\sigma_3$ act trivially on the GIT quotient, whilst $\sigma_1$ acts by $$\sigma_1\colon [\mathbf{x}_0:\mathbf{x}_1:\mathbf{x}_2] \mapsto [\mathbf{x}_0:\mathbf{x}_2:\mathbf{x}_1].$$ Therefore, to get a moduli space from our GIT quotient, we must further quotient it by the involution $\sigma_1$ to get a copy of the weighted projective space $\mathbb{P}(1,1,2)$ with coordinates $$[\xi:\eta:\zeta]=\big[\mathbf{x}_0:\mathbf{x}_1+\mathbf{x}_2:(\mathbf{x}_1 - \mathbf{x}_2)^2\big]=\big[\alpha^2:\beta\epsilon+\gamma\delta:(\beta\epsilon-\gamma\delta)^2\big].$$ Over this quotient it is easy to describe where the stable and semistable threefolds occur. Namely, the stable threefolds occur where $\eta^2 \neq \zeta$ as follows: 1. the locus with $\xi \neq 0$ and $\eta^2 \neq \zeta$, and singular quadrics lie over $\xi^2-\xi\eta+\zeta=0$, 2. the locus $\{[0:\eta:\zeta]\mid \eta^2 \neq \zeta \text{ and } \zeta \neq 0\}$, 3. the point $[\xi:\eta:\zeta] = [0:1:0]$. Likewise, the strictly semistable quadrics occur along the curve $\eta^2 = \zeta$ as follows: 1. the locus where $\eta^2 = \zeta$ and $[\xi:\eta:\zeta] \notin \{[1:0:0], [0:1:1], [1:1:1]\}$, 2. the point $[\xi:\eta:\zeta] =[1:0:0]$, 3. the point $[\xi:\eta:\zeta] = [0:1:1]$, 4. the point $[\xi:\eta:\zeta] = [1:1:1]$. Unfortunately, the constructed moduli space is not the moduli space of K-polystable threefolds we are looking for, because the threefold $X$ obtained by blowing up the quadric in the class (S3) is reducible. Moreover, all threefolds obtained by blowing up quadrics in the class (S2) are isomorphic, and the quadrics in the class $\mathrm{(S2)}$ give K-unstable threefolds: **Lemma 1**. *Let $Q$ be the quadric $\{a(xt+yz)+xz+yt=0\}\subset\mathbb{P}^4$ for $a\in\mathbb{C}\setminus\{0,\pm 1\}$, and let $\pi\colon X\to Q$ be the blow up of the quadric $Q$ along $C_1$ and $C_2$. Then $X$ is K-unstable.* *Proof.* Observe that both $C_1$ and $C_2$ do not contain the singular point of the quadric $Q$, the quadric threefold $Q$ is a cone over $\mathbb{P}^1\times\mathbb{P}^1$ with vertex at the point $[0:0:0:0:1]$, and there exists the following commutative diagram: $$\xymatrix{ &\widetilde{X}\ar@{->}[dl]_{\varphi}\ar@{->}[dr]^{\varpi}\ar@/^2.5pc/@{->}[ddrr]^{\nu}&\\% X\ar@{->}[dr]_{\pi}&&\widetilde{Q}\ar@{->}[dl]^{\phi}\ar@{->}[dr]^{\upsilon}&\\% &{Q}&&\mathbb{P}^1\times\mathbb{P}^1}$$ where $\phi$ and $\varphi$ are blow ups of the singular points $\mathrm{Sing}(Q)$ and $\mathrm{Sing}(X)$, $\upsilon$ is a $\mathbb{P}^1$-bundle, $\varpi$ is the blow up of the preimages of the conics $C_1$ and $C_2$, and $\nu$ is a conic bundle. Let $E_1$ and $E_2$ be proper transforms on the threefold $\widetilde{X}$ of the $\pi$-exceptional surfaces such that $\pi\circ\phi(E_1)=C_1$ and $\pi\circ\phi(E_2)=C_2$. Then $\upsilon\circ\varpi(E_1)$ and $\upsilon\circ\varpi(E_2)$ are disjoint rulings of the surface $\mathbb{P}^1\times\mathbb{P}^1$, because planes spanned by $C_1$ and $C_2$ are contained in $Q$. Therefore, we may assume that both $\upsilon\circ\varpi(E_1)$ and $\upsilon\circ\varpi(E_2)$ are divisors of degree $(1,0)$. Let $H_1$ and $H_2$ be the pull back on $\widetilde{X}$ of the divisors on $\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,0)$ and $(0,1)$, let $S_1$ and $S_2$ be proper transforms on $\widetilde{X}$ of the planes in $Q$ that contain $C_1$ and $C_2$, and let $F$ be the $\varphi$-exceptional surface. Then $H_1\sim S_1+E_1$ and $H_1\sim S_2+E_2$, so that $$\varphi^*(-K_X)\sim 3(H_1+H_2)+3F-E_1-E_2\sim 3(S_1+S_2)+3F+2(E_1+E_2).$$ Now, take $u\in\mathbb{R}_{\geqslant 0}$. Then $$\varphi^*(-K_X)-uF\sim_{\mathbb{R}} 3(H_1+H_2)+(3-u)F-E_1-E_2\sim_{\mathbb{R}} 3(S_1+S_2)+(3-u)F+2(E_1+E_2),$$ so this divisor is pseudoeffective $\iff$ $u\leqslant 3$. Moreover, this divisor is nef $\iff$ $u\leqslant 1$. Furthermore, if $u\in(1,3)$, then its Zariski decomposition can be described as follows: $$\varphi^*(-K_X)-uF\sim_{\mathbb{R}}\underbrace{\varphi^*(-K_X)-uF-\frac{u-1}{2}\big(S_1+S_2\big)}_{\text{positive part}}+\underbrace{\frac{u-1}{2}\big(S_1+S_2\big)}_{\text{negative part}}.$$ Thus, we see that $$\begin{gathered} \quad\quad\quad\beta(F)=2-\frac{1}{26}\int\limits_{0}^{1}\big(3(H_1+H_2)+(3-u)F-E_1-E_2\big)^3du-\\ -\frac{1}{26}\int\limits_{2}^{3}\Big(3(H_1+H_2)+(3-u)F-E_1-E_2-\frac{u-1}{2}\big(2H_1-E_1-E_2\big)\Big)^3du.\end{gathered}$$ To compute these integrals, observe that $$\begin{aligned} H_1^2&=0, & H_2^2&=0, & E_1\cdot E_2&=0, & E_1\cdot F&=0, & E_2\cdot F&=0,\\ F^3&=2, & H_1\cdot F^2&=-1, & H_2\cdot F^2&=-1, & E_1^3&=-4, & E_2^3&=-4, \\ F\cdot H_1\cdot H_2&=1, & H_1\cdot E_1&=0, & H_1\cdot E_2&=0, & H_2\cdot E_1^2&=-2, & H_2\cdot E_2^2&=-2.\end{aligned}$$ This gives $\beta(F)=-\frac{3}{52}<0$, which implies that $X$ is K-unstable [@Fujita2019; @Li]. ◻ # Second GIT quotient (good moduli space) {#section:GIT-2} In this section, we construct another compact GIT moduli space for Fano threefolds in the deformation family  3.10. Recall that all K-polystable smooth Fano threefolds in this family are contained in class $(\beth)$, and the quadric $Q$ in this case is invariant under the action of $\boldsymbol{\mu}_2^3$ generated by the following three involutions: $$\begin{aligned} \tau_1\colon [x:y:z:t:w] &\mapsto [y:x:t:z:w],\\ \tau_2\colon [x:y:z:t:w] &\mapsto [z:t:x:y:w],\\ \tau_3\colon [x:y:z:t:w] &\mapsto [x:y:z:t:-w].\end{aligned}$$ In light of this, we take $Q$ to be a quadric invariant under this $\boldsymbol{\mu}_2^3$-action: $$\label{equation:newquadric} \alpha w^2 + \tfrac{\beta}{2}(x^2+y^2+z^2+t^2) + \gamma(xt+yz) + \delta(xz+yt) + \epsilon(xy+zt) = 0,$$ where $[\alpha:\beta:\gamma:\delta:\epsilon] \in\mathbb{P}^4$. As before, we take $C_1$ and $C_2$ to be the two conics given by the intersection with the  planes $\{x=y=0\}$ and $\{z=t=0\}$, namely $$\begin{aligned} C_1 &= \{x=0,\, y=0,\, \alpha w^2 + \tfrac{\beta}{2}(z^2+t^2) + \epsilon zt = 0\},\\ C_2 &= \{z=0,\, t=0,\, \alpha w^2 + \tfrac{\beta}{2}(x^2+y^2) + \epsilon xy = 0\}.\end{aligned}$$ Unlike the construction from Section [2](#section:GIT-1){reference-type="ref" reference="section:GIT-1"}, this general form allows $C_1$ and $C_2$ to degenerate. A simple computer calculation shows that equations of the form [\[equation:newquadric\]](#equation:newquadric){reference-type="eqref" reference="equation:newquadric"} are preserved by a subgroup of $\mathrm{PGL}(5,\mathbb{C})$ isomorphic to $(\mathbb{C}^\ast)^2$, given explicitly by matrices of the form $$\begin{pmatrix} \lambda + \mu & \lambda - \mu & 0 & 0 & 0 \\ \lambda - \mu & \lambda + \mu & 0 & 0 & 0 \\ 0 & 0 & \lambda - \mu & \lambda + \mu & 0 \\ 0 & 0 & \lambda + \mu & \lambda - \mu & 0 \\ 0 & 0 & 0 & 0 & -\tfrac{2}{\lambda^2\mu^2} \end{pmatrix}$$ acting on the coordinates $[x:y:z:t:w]$, where $(\lambda,\mu) \in (\mathbb{C}^\ast)^2$ and we have chosen our matrix representative to have determinant $2^5 = 32$ (to avoid having lots of factors of $\frac{1}{2}$). This induces a $(\mathbb{C}^\ast)^2$-action on the parameter space $\mathbb{P}^4$ given by $$\begin{pmatrix} \tfrac{4}{\lambda^4\mu^4} & 0 & 0 & 0 \\ 0 & 2(\lambda^2+ \mu^2) & 0 & 0 & 2(\lambda^2-\mu^2)\\ 0 & 0 & 2(\lambda^2-\mu^2) & 2(\lambda^2 + \mu^2) & 0 \\ 0 & 0 & 2(\lambda^2+\mu^2) & 2(\lambda^2 - \mu^2) & 0 \\ 0 & 2(\lambda^2- \mu^2) & 0 & 0 & 2(\lambda^2+\mu^2) \end{pmatrix}$$ acting on the coordinates $[\alpha:\beta:\gamma:\delta:\epsilon]$, where $(\lambda,\mu) \in (\mathbb{C}^\ast)^2$. The ring of invariants for this action is generated by polynomials of the form $$\alpha f_1f_2g_1g_2$$ with $f_i \in \{(\beta+\epsilon),\,(\gamma+\delta)\}$ and $g_i \in \{(\beta-\epsilon),\,(\gamma-\delta)\}$. There are nine such polynomials, corresponding to coordinates on $\mathbb{P}^8$. One can show that the GIT quotient of the parameter space is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1 \subset \mathbb{P}^8$, embedded by the Segre embedding of bidegree $(2,2)$. The isomorphism with $\mathbb{P}^1 \times \mathbb{P}^1$ is given explicitly by $$\big([\mathbf{x}_0:\mathbf{x}_1],[\mathbf{y}_0:\mathbf{y}_1]\big) = \big([\beta+\epsilon:\gamma + \delta], [\beta - \epsilon:\gamma - \delta]\big).$$ We now classify the unstable and stable points, and corresponding quadrics. There are no strictly semistable points under this action. ## Unstable points. The unstable points $[\alpha:\beta:\gamma:\delta:\epsilon] \in \mathbb{P}^4$ come in two types. The first are the points with $\alpha = 0$, which have corresponding quadrics $$Q=\Big\{\tfrac{\beta}{2}(x^2+y^2+z^2+t^2) + \gamma(xt+yz) + \delta(xz+yt) + \epsilon(xy+tz) = 0\Big\}.$$ In this case, the quadric $Q$ degenerates to a cone over a quadric surface. The second type are those with either $$\begin{aligned} \beta - \epsilon = \gamma - \delta = 0&\ \text{and}\ Q=\Big\{\alpha w^2 + \tfrac{\beta}{2}((x+y)^2+(z+t)^2) + \gamma(x+y)(z+t)=0\Big\}, \\ \beta + \epsilon = \gamma + \delta = 0&\ \text{and}\ Q=\Big\{\alpha w^2 + \tfrac{\beta}{2}((x-y)^2+(z-t)^2) + \gamma(x-y)(z-t) =0\Big\}.\end{aligned}$$ In these cases $Q$ is singular along a line and both of the conics $C_1$ and $C_2$ become reducible. ## Stable points. The stable points are those that are not unstable --- there are no strictly semistable points. We separate the  corresponding quadric threefolds into 2 cases. 1. (Class $(\beth)$) Stable orbits with $\alpha \neq 0$ and $\beta \neq \pm \epsilon$ correspond to points with $$\mathbf{x}_0\mathbf{y}_0 \neq 0$$ in the quotient space $\mathbb{P}^1 \times \mathbb{P}^1$. Taking the affine chart with $\mathbf{x}_0=1$ and $\mathbf{y}_0=-1$, we see that each such orbit contains quadrics of the form $$w^2+xy+zt+a(xt+yz)+b(xz+yt)=0,$$ for $a,b\in \mathbb{C}^*$ given by $a = \frac{1}{2}(\mathbf{x}_1 + \mathbf{y}_1)$ and $b = \frac{1}{2}(\mathbf{x}_1 - \mathbf{y}_1)$. This quadric is singular if and only if $a \pm b \pm 1 = 0$, corresponding to $\mathbf{x}_1 = \pm 1$ or $\mathbf{y}_1 = \pm 1$. 2. (Class $(\aleph)$) Stable orbits with $\alpha\neq 0$ and $\beta = -\epsilon$ or $\beta = \epsilon$, which correspond to points in the quotient $\mathbb{P}^1 \times \mathbb{P}^1$ with $\mathbf{x}_0 = 0$ or $\mathbf{y}_0 = 0$, respectively. Taking the affine chart $\mathbf{x}_1=1$ or $\mathbf{y}_1=1$, respectively, we see that such orbits contain the quadric $$w^2 + r(x-y)^2+r(z-t)^2 + (2s+2)(xt+yz)+(2-2s)(xz+yt)= 0$$ or $$w^2 + r(x+y)^2+r(z+t)^2 + (2s+2)(xt+yz)+(2s-2)(xz+yt)= 0$$ for $[r:s] \in \mathbb{P}^1$ given by $[r:s] = [2\mathbf{y}_0:2\mathbf{y}_1]$ or $[r:s] = [2\mathbf{x}_0:2\mathbf{x}_1]$, respectively. Generically this defines a smooth quadric where the conics $C_1$ and $C_2$ are reduced and singular (pairs of lines). The exceptions are - $[r:s] = [\pm 1 :1]$, when the quadric acquires an isolated singularity, - $[r:s] = [0:1]$, when $C_1$ and $C_2$ become non-reduced (double lines). In Sections [4](#section:beth){reference-type="ref" reference="section:beth"} and [5](#section:aleph){reference-type="ref" reference="section:aleph"}, we will show that all Fano threefolds $X$ obtained as blow-ups of quadrics $Q$ from classes $(\beth)$ and $(\aleph)$ above are K-polystable. ## Quotient space. {#section:quotient-space} Observe that the space of quadric threefolds of the form [\[equation:newquadric\]](#equation:newquadric){reference-type="eqref" reference="equation:newquadric"} admits an additional $\boldsymbol{\mu}_2$-action given by $$[x:y:z:t:w]\mapsto [x:-y:-z:t:w],$$ which acts on the parameter space $\mathbb{P}^4$ as $$[\alpha:\beta:\gamma:\delta:\epsilon]\mapsto [\alpha:\beta:\gamma:-\delta:-\epsilon].$$ This involution induces an involution on the GIT quotient given by $$\big([\mathbf{x}_0:\mathbf{x}_1],[\mathbf{y}_0:\mathbf{y}_1]\big) \mapsto \big([\mathbf{y}_0:\mathbf{y}_1],[\mathbf{x}_0:\mathbf{x}_1]\big).$$ So, to get a moduli space from our GIT quotient, we must further quotient it by this involution. The result is a copy of $\mathbb{P}^2$ with coordinates $$[\xi:\eta:\zeta]=[\mathbf{x}_0\mathbf{y}_0:\mathbf{x}_1\mathbf{y}_1:\mathbf{x}_0\mathbf{y}_1+\mathbf{x}_1\mathbf{y}_0] = [\beta^2-\epsilon^2:\gamma^2-\delta^2: 2\beta\gamma-2\epsilon\delta].$$ Over this quotient it is easy to describe where each of the types of stable threefolds occur: 1. the locus with $\xi \neq 0$, singular quadrics lie over the lines $\xi+\eta = \pm \zeta$. 2. the locus with $\xi=0$, singular quadrics lie over the points $[0:1:1]$ and $[0:-1:1]$, and the conics $C_1$ and $C_2$ become non-reduced at the point $[0:1:0]$. Note that this quotient identifies the two normal forms in class $(\aleph)$. # Threefolds in the class $(\beth)$ {#section:beth} Let $X$ be the complete intersection $$\big\{u_1x=v_1y, u_2z=v_2t, w^2+xy+zt=a(xt+yz)+b(xz+yt)\big\}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4,$$ where $(a,b)\in\mathbb{C}^2$, and $([u_1:v_1],[u_2:v_2],[x:y:z:t:w])$ are coordinates on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$. Then $X$ is singular $\iff$ $a\pm b\pm 1=0$. If $a\pm b\pm 1=0$ and $ab\ne 0$, then $$\mathrm{Sing}(X)=\left\{\aligned &\big([1:1],[1:1],[1:1:-1:-1:0]\big)\ \text{if $a+b+1=0$}, \\ &\big([1:1],[1:1],[1:1:1:1:0]\big)\ \text{if $a+b-1=0$}, \\ &\big([-1:1],[-1:1],[1:-1:1:-1:0]\big)\ \text{if $a-b+1=0$}, \\ &\big([-1:1],[-1:1],[1:-1:-1:1:0]\big)\ \text{if $a-b-1=0$}. \endaligned \right.$$ If $a\pm b\pm 1=0$ and $ab=0$, then $X$ is singular along the curve $$\big\{w=0, x=z, y=t, u_1z=v_1t, u_2z=v_2t, v_1u_2=u_1v_2\big\}.$$ Let $Q$ be the quadric threefold in $\mathbb{P}^4$ given by $w^2+xy+zt=a(xt+yz)+b(xz+yt)$, and let $\pi\colon{X}\to{Q}$ be the morphism that is induced by the natural projection ${X}\to\mathbb{P}^4$. Then $\pi$ is a blow up of the following two smooth conics: $$\begin{aligned} {C}_1&=\{w^2+zt=0,x=0,y=0\},\\ {C}_2&=\{w^2+xy=0,z=0,t=0\}.\end{aligned}$$ The quadric $Q$ is smooth $\iff$ $a\pm b\pm 1=0$. If $a\pm b\pm 1=0$ and $ab\ne 0$, then $$\mathrm{Sing}(Q)=\left\{\aligned &[1:1:-1:-1:0]\ \text{if $a+b+1=0$}, \\ &[1:1:1:1:0]\ \text{if $a+b-1=0$}, \\ &[1:-1:1:-1:0]\ \text{if $a-b+1=0$}, \\ &[1:-1:-1:1:0]\ \text{if $a-b-1=0$}. \endaligned \right.$$ If $a\pm b\pm 1=0$ and $ab\ne 0$, then $Q$ is singular along the line $\{w=0, x=z, y=t\}$. The threefold $X$ admits an action of the group $\boldsymbol{\mu}_2^3$ given by $$\begin{aligned} \tau_1\colon \big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big) &\mapsto \big([v_1:u_1],[v_2:u_2],[y:x:t:z:w]\big),\\ \tau_2\colon \big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big) &\mapsto \big([u_2:v_2],[u_1:v_1],[z:t:x:y:w]\big),\\ \tau_3\colon \big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big) &\mapsto \big([u_1:v_1],[u_2:v_2],[x:y:z:t:-w]\big).\end{aligned}$$ Let $G$ be the subgroup in $\mathrm{Aut}(X)$ generated by the involutions $\tau_1$, $\tau_2$, $\tau_3$. Then $$G\cong\boldsymbol{\mu}_2^3,$$ the blow up $\pi\colon X\to Q$ is $G$-equivariant, and the $G$-action on $Q$ induces a $G$-action on the projective space $\mathbb{P}^4$. Therefore, we can also consider $G$ as a subgroup in $\mathrm{PGL}_4(\mathbb{C})$, where $\tau_1$, $\tau_2$, $\tau_3$ act on $\mathbb{P}^4$ as in Section [3](#section:GIT-2){reference-type="ref" reference="section:GIT-2"}: $$\begin{aligned} \tau_1\colon [x:y:z:t:w] &\mapsto [y:x:t:z:w],\\ \tau_2\colon [x:y:z:t:w] &\mapsto [z:t:x:y:w],\\ \tau_3\colon [x:y:z:t:w] &\mapsto [x:y:z:t:-w].\end{aligned}$$ Observe that the only $G$-fixed points in $\mathbb{P}^4$ are $$[1:1:-1:-1:0],[1:1:1:1:0],[1:-1:1:-1:0],[1:-1:-1:1:0].$$ **Lemma 1**. *The following two assertions hold:* - *if $X$ is smooth, then $X$ does not have $G$-fixed points;* - *if $X$ has one singular point, then $\mathrm{Sing}(X)$ is the only $G$-fixed point in $X$.* *Proof.* If $Q$ is smooth, it does not contain any $G$-fixed points. Vice versa, if $Q$ is singular, then its singular point is the only $G$-fixed point in $Q$. The claim follows. ◻ We will also need the following technical lemma: **Lemma 1**. *Suppose that $a\pm b\pm 1=0$ and $ab\ne 0$. Let $O$ be the singular point $\mathrm{Sing}(X)$, let $\varphi\colon\widetilde{X}\to X$ be the blow up of the point $O$, and let $F$ be the $\varphi$-exceptional surface. Then* - *$\varphi$ is $G$-equivariant,* - *the group $G$ acts faithfully on $F$,* - *$\mathrm{rk}\,\mathrm{Pic}^G(F)=1$,* - *$F$ has no $G$-fixed points.* *Proof.* We only consider the case when $b=1-a$ and $a\not\in\{0,1\}$; the other cases are similar. Then $$Q=\big\{w^2+xy+zt-a(xt+yz)-(1-a)(xz+yt)=0\big\},$$ and $\pi(O)=[1:1:1:1:0]$. Let us introduce new coordinates on $\mathbb{P}^4$ as follows: $$\left\{\aligned &\mathbf{x}=x-y, \\ &\mathbf{y}=x+y-z-t, \\ &\mathbf{z}=z-t, \\ &\mathbf{t}=x+y+z+t, \\ &\mathbf{w}=w. \endaligned \right.$$ In these coordinates, $Q=\{\mathbf{x}^2-\mathbf{y}^2+\mathbf{z}^2=(4a-2)\mathbf{x}\mathbf{z}+4\mathbf{w}^2\}$ and $\pi(O)=[0:0:0:1:0]$. Moreover, the involutions $\tau_1$, $\tau_2$, $\tau_3$ acts as follows: $$\begin{aligned} \tau_1\colon \big[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:\mathbf{w}\big]&\mapsto\big[-\mathbf{x}:\mathbf{y}:-\mathbf{z}:\mathbf{t}:\mathbf{w}\big],\\ \tau_2\colon \big[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:\mathbf{w}\big]&\mapsto\big[\mathbf{z}:-\mathbf{y}:\mathbf{x}:\mathbf{t}:\mathbf{w}\big],\\ \tau_3\colon \big[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:\mathbf{w}\big]&\mapsto\big[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:-\mathbf{w}\big].\end{aligned}$$ Furthermore, we can $G$-equivariantly identify $F$ with the projectivization of the tangent cone to $Q$ at the point $\pi(O)$. So, we can consider $F$ as the quadric surface in $\mathbb{P}^3$ given by $$\mathbf{x}^2-\mathbf{y}^2+\mathbf{z}^2=(4a-2)\mathbf{x}\mathbf{z}+4\mathbf{w}^2,$$ where now we consider $[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{w}]$ as coordinates on $\mathbb{P}^3$. Now, all assertions are easy to check looking on how the group $G$ acts on the surface $F$. ◻ We have the following $G$-equivariant commutative diagram $$\xymatrix{ &X\ar@{->}[dl]_{\pi}\ar@{->}[dr]^{\eta}&\\% {Q}\ar@{-->}[rr]_{\rho}&&\mathbb{P}^1\times\mathbb{P}^1}$$ where $\eta$ is a conic bundle given by the natural projection $X\to\mathbb{P}^1\times\mathbb{P}^1$, and $\rho$ is a rational map that is given by $[x:y:z:t:w]\mapsto([x:y],[z:t])$. *Remark 1*. The only $G$-fixed points in $\mathbb{P}^1\times\mathbb{P}^1$ are $([1:1],[1:1])$ and $([-1:1],[-1:1])$. Let $\Delta$ be the discriminant curve of the conic bundle $\eta$. Then $\Delta$ is given by $$\begin{gathered} \quad\quad\quad a^2u_1^2v_2^2+2abu_1^2u_2v_2+b^2u_1^2u_2^2+(2a^2+2b^2-4)v_1v_2u_1u_2+\\ +2abu_1v_1u_2^2+2abu_1v_1v_2^2+a^2u_2^2v_1^2+2abu_2v_1^2v_2+b^2v_1^2v_2^2=0,\quad\quad\quad\end{gathered}$$ where we consider $([u_1:v_1],[u_2:v_2])$ as coordinates on $\mathbb{P}^1\times\mathbb{P}^1$. *Remark 1*. If $a\pm b\pm 1\ne 0$, then $\Delta$ is smooth. If $a\pm b\pm 1=0$ and $ab\ne 0$, then $$\mathrm{Sing}(\Delta)=\left\{\aligned &\big([1:1],[1:1]\big)\ \text{if $a+b\pm 1=0$}, \\ &\big([-1:1],[-1:1]\big)\ \text{if $a-b\pm 1=0$}. \\ \endaligned \right.$$ Let $H$ be a proper transform on $X$ of a general hyperplane section of the quadric $Q$, and let $E_1$ and $E_2$ be the $\pi$-exceptional divisors such that $\pi(E_1)=C_1$ and $\pi(E_2)=C_2$. **Lemma 1**. *Suppose that $X$ has at most isolated singularities. Then $$\mathrm{Pic}^G(X)=\mathrm{Cl}^G(X)=\mathbb{Z}[H]\oplus\mathbb{Z}[E_1+E_2].$$* *Proof.* If the threefold $X$ is smooth, the assertion is obvious, since $G$ swaps $E_1$ and $E_2$. If $X$ has one singular point, the assertion follows from Lemma [Lemma 1](#lemma:ODP-G-action){reference-type="ref" reference="lemma:ODP-G-action"}. ◻ Now, let us describe the cone of $G$-invariant effective divisors on $X$. **Lemma 1**. *Suppose that $X$ has at most isolated singularities. Let $S$ be an $G$-invariant surface in $X$. Then $S\sim n_1(E_1+E_2)+n_2(2H-E_1-E_2)+n_3H$ for some $n_1,n_2,n_3\in\mathbb{Z}_{\geqslant 0}$.* *Proof.* By Lemma [Lemma 1](#lemma:class-groups){reference-type="ref" reference="lemma:class-groups"}, we see that $S\sim k_1H+k_2(E_1+E_2)$ for some integers $k_1$ and $k_2$. On the other hand, the conic bundle $\eta\colon X\to\mathbb{P}^1\times\mathbb{P}^1$ is given by $|2H-E_1-E_2|$. Then $$S\sim_{\mathbb{Q}} m_1(E_1+E_2)+m_2(2H-E_1-E_2)$$ for some non-negative rational numbers $m_1$ and $m_2$. We have $k_1=2m_2$ and $m_1-m_2=k_2$. If $k_1$ is even, then $m_1$ and $m_2$ are integers and we are done, since $\mathrm{Pic}(X)$ has no torsion. So, we may assume that $k_1=2n+1$, where $n\in\mathbb{Z}$. Then $m_2=n+\frac{1}{2}$ and $m_1=k_2+n+\frac{1}{2}$, which gives $n\geqslant 0$ and $k_2+n\geqslant 0$, since $m_1\geqslant 0$ and $m_2\geqslant 0$. Hence, we have $$S\sim_{\mathbb{Q}} (k_2+n)(E_1+E_2)+n(2H-E_1-E_2)+H,$$ which gives $S\sim (k_2+n)(E_1+E_2)+n(2H-E_1-E_2)+H$, since $\mathrm{Pic}(X)$ has no torsion. ◻ **Corollary 1**. *Suppose that $X$ has at most isolated singularities. Let $S$ be a $G$-invariant irreducible surface in $X$. Then $\beta(S)>0$.* *Proof.* We have $\beta(S)=1-S_X(S)$, and it follows from Lemma [Lemma 1](#lemma:Eff-cone){reference-type="ref" reference="lemma:Eff-cone"} that $$S_X(S)=\frac{1}{26}\int\limits_{0}^{\infty}\mathrm{vol}\big(-K_X-uS\big)du\leqslant\frac{1}{26}\int\limits_{0}^{\infty}\mathrm{vol}\big(-K_X-uF\big)du,$$ where $F$ is one divisor among $H$, $E_1+E_2$, $2H-E_1-E_2$. On the other hand, we have $$\frac{1}{26}\int\limits_{0}^{\infty}\mathrm{vol}\big(-K_X-uH\big)du=\frac{1}{26}\int\limits_{0}^{1}\big(-K_X-uH\big)^3du=\frac{1}{26}\int\limits_{0}^{1}2(1-u)(u^2-8u+13)du=\frac{21}{52}.$$ Similarly, we have $$\frac{1}{26}\int\limits_{0}^{\infty}\mathrm{vol}\big(-K_X-u(E_1+E_2)\big)du=\frac{1}{26}\int\limits_{0}^{\frac{1}{2}}2(2u-1)(2u^2-2u-13)du=\frac{53}{208}.$$ Finally, we compute $$\begin{gathered} \frac{1}{26}\int\limits_{0}^{\infty}\mathrm{vol}\big(-K_X-u(2H-E_1-E_2)\big)du= \frac{1}{26}\int\limits_{0}^{1}\big((3-2u)H+(u-1)E_1+(u-1)E_2\big)^3du+\\ +\frac{1}{26}\int\limits_{1}^{\frac{3}{2}}\big((3-2u)H\big)^3du=\frac{1}{26}\int\limits_{0}^{1}12u^2-36u+26du+\frac{1}{26}\int\limits_{1}^{\frac{3}{2}}2(3-2u)^3du=\frac{49}{104}.\end{gathered}$$ Thus, we conclude that $\beta(S)>0$, as claimed. ◻ We conclude this section with the following technical result: **Proposition 1**. *Suppose that $X$ has at most isolated singularities, and let $S$ be a smooth surface in one of the linear systems $|H-E_1|$ or $|H-E_2|$. Then $$\delta_P(X)\geqslant\frac{104}{99}$$ for every point $P\in S$ such that $P\not\in E_1\cup E_2$.* *Proof.* We may assume that $S\in |H-E_1|$. Let $u$ be a non-negative real number. Then $$-K_X-uS\sim_{\mathbb{R}}(2-u)S+(H-E_2)+E_1\sim_{\mathbb{R}} (3-u)H-(1-u)E_1-E_2.$$ Then $-K_X-uS$ is nef $\iff$ $u\in[0,1]$, and $-K_X-uS$ is pseudo-effective $\iff$ $u\in[0,2]$. Moreover, if $u\in[1,2]$, then its Zariski decomposition can be described as follows: $$-K_X-uS\sim_{\mathbb{R}}\underbrace{(3-u)H-E_2}_{\text{positive part}}+\underbrace{(u-1)E_1}_{\text{negative part}}.$$ Thus, for transparency, we let $$P(u)=\left\{\aligned &(3-u)H-(1-u)E_1-E_2\ \text{if $0\leqslant u\leqslant 1$}, \\ &(3-u)H-E_2\ \text{if $1\leqslant u\leqslant 2$}, \endaligned \right.$$ and $$N(u)=\left\{\aligned &0\ \text{if $0\leqslant u\leqslant 1$}, \\ &(u-1)E_1\ \text{if $1\leqslant u\leqslant 2$}. \endaligned \right.$$ Then $$\big(P(u)\big)^3=\left\{\aligned &26-18u\ \text{if $0\leqslant u\leqslant 1$}, \\ &40-2u^3+18u^2-48u\ \text{if $1\leqslant u\leqslant 2$}. \endaligned \right.$$ Then $$S_X(S)=\frac{1}{(-K_{X})^3}\int\limits_{0}^{2}\mathrm{vol}\big(-K_X-uS\big)du=\frac{1}{26}\int\limits_{0}^{2}\big(P(u)\big)^3du=\frac{3}{4},$$ so that $\beta(S)=1-S_X(S)=\frac{1}{3}$. Let $P$ be a point in $S$, and let $C$ be an irreducible smooth curve in $S$ such that $P\in C$. Write $N(u)\vert_{S}=N^\prime(u)+\mathrm{ord}_{C}(N(u)\vert_{S})C$. For every $u\in[0,2]$, set $$t(u)=\mathrm{sup}\Big\{v\in\mathbb{R}_{\geqslant 0}\ \big\vert\ \text{the~divisor $P(u)\big\vert_{S}-vC$ is pseudo-effective}\Big\}.$$ Let $v$ be a real number in $[0,t(u)]$, let $P(u,v)$ be the positive part of the Zariski decomposition of the $\mathbb{R}$-divisor $P(u)\vert_{S}-vC$, and let $N(u,v)$ be its negative part. Set $$S\big(W^S_{\bullet,\bullet};C\big)=\frac{3}{(-K_X)^3}\int\limits_0^2\big(P(u)\big\vert_{S}\big)^2\cdot\mathrm{ord}_{C}\big(N(u)\big\vert_{S}\big)du+\frac{3}{(-K_X)^3}\int\limits_0^2\int\limits_0^{t(u)}\big(P(u,v)\big)^2dvdu$$ and $$\begin{gathered} S\big(W_{\bullet, \bullet,\bullet}^{S,C};P\big)=\frac{3}{(-K_X)^3}\int\limits_0^2\int\limits_0^{t(u)}\big(P(u,v)\cdot C\big)^2dvdu+\\ +\frac{6}{(-K_X)^3} \int\limits_0^2\int\limits_0^{t(u)}\big(P(u,v)\cdot C\big)\cdot \mathrm{ord}_P\big(N^\prime(u)\big|_C+N(u,v)\big|_C\big)dvdu.\end{gathered}$$ Then it follows from [@AbbanZhuang; @Book] that $$\delta_P(X)\geqslant\min\Bigg\{\frac{1}{S_X(S)},\frac{1}{S\big(W^S_{\bullet,\bullet};C\big)},\frac{1}{S\big(W_{\bullet, \bullet,\bullet}^{S,C};P\big)}\Bigg\}.$$ But $S_X(S)=\frac{3}{4}$. Hence, to complete the proof, it is enough to find an irreducible smooth curve $C\subset S$ such that $P\in C$, and $S(W^S_{\bullet,\bullet};C)\leqslant\frac{99}{104}\geqslant S(W_{\bullet, \bullet,\bullet}^{S,C};P)$ if $P\not\in E_1\cup E_2$. Now, suppose that $P\not\in E_1\cup E_2$. In particular, this assumption implies $P\not\in\mathrm{Supp}(N(u))$, so that the formulas for $S(W^S_{\bullet,\bullet};C)$ and $S(W_{\bullet, \bullet,\bullet}^{S,C};P)$ simplify as follows: $$S\big(W^S_{\bullet,\bullet};C\big)=\frac{3}{(-K_X)^3}\int\limits_0^2\int\limits_0^{t(u)}\big(P(u,v)\big)^2dvdu$$ and $$S\big(W_{\bullet, \bullet,\bullet}^{S,C};P\big)=\frac{3}{(-K_X)^3}\int\limits_0^2\int\limits_0^{t(u)}\big(P(u,v)\cdot C\big)\Big (\big(P(u,v)\cdot C\big)+2\mathrm{ord}_P\big(N(u,v)\big|_C\big)\Big)dvdu.$$ To find the required curve $C$, recall that $S$ is a del Pezzo surface of degree $6$, and $$E_2\big\vert_{S}=\mathbf{e}_1+\mathbf{e}_2,$$ where $\mathbf{e}_1$ and $\mathbf{e}_2$ are disjoint $(-1)$-curves in the surface $S$. Let $Z$ be the fiber of the conic bundle $S\to\mathbb{P}^1$ given by $([u_1:v_1],[u_2:v_2],[x:y:z:t:w])\mapsto[u_2:v_2]$ such that $P\in Z$. Then $Z\cdot \mathbf{e}_1=Z\cdot\mathbf{e}_2=1$ and $$E_1\big\vert_{S}\sim H\big\vert_{S}\sim Z+\mathbf{e}_1+\mathbf{e}_2.$$ Let us choose $C$ to be an irreducible component of the fiber $Z$ that contains the point $P$. A priori, we may have the following two possibilities: 1. $Z$ is an irreducible smooth rational curve and $Z^2=0$; 2. $Z=\ell_1+\ell_2$ for two $(-1)$-curves $\ell_1$ and $\ell_2$ such that $\ell_1\cdot\ell_2=\ell_1\cdot \mathbf{e}_1=\ell_2\cdot\mathbf{e}_2=1$. In the first case, we have $C=Z$. In the second case, we may assume $P\in\ell_1$, so $C=\ell_1$. Let us compute $S(W^S_{\bullet,\bullet};C)$ and $S(W_{\bullet, \bullet,\bullet}^{S,C};P)$ in each of these cases. First, we note that $$P(u)\big\vert_{S}=\left\{\aligned &2Z+\mathbf{e}_1+\mathbf{e}_2\ \text{if $0\leqslant u\leqslant 1$}, \\ &(3-u)Z+(2-u)(\mathbf{e}_1+\mathbf{e}_2)\ \text{if $1\leqslant u\leqslant 2$}. \endaligned \right.$$ Let $v$ be a non-negative real number. If $C=Z$, then $$P(u)\big\vert_{S}-vC=\left\{\aligned &(2-v)C+\mathbf{e}_1+\mathbf{e}_2\ \text{if $0\leqslant u\leqslant 1$}, \\ &(3-u-v)C+(2-u)(\mathbf{e}_1+\mathbf{e}_2)\ \text{if $1\leqslant u\leqslant 2$}, \endaligned \right.$$ which implies that $$t(u)=\left\{\aligned &2\ \text{if $0\leqslant u\leqslant 1$}, \\ &3-u\ \text{if $1\leqslant u\leqslant 2$}. \endaligned \right.$$ Furthermore, if $C=Z$ and $u\in[0,1]$, then $$P(u,v)=\left\{\aligned &(2-v)C+\mathbf{e}_1+\mathbf{e}_2\ \text{if $0\leqslant v\leqslant 1$}, \\ &(2-v)\big(C+\mathbf{e}_1+\mathbf{e}_2\big)\ \text{if $1\leqslant v\leqslant 2$}, \endaligned \right.$$ and $$N(u,v)=\left\{\aligned &0\ \text{if $0\leqslant v\leqslant 1$}, \\ &(v-1)\big(\mathbf{e}_1+\mathbf{e}_2\big)\ \text{if $1\leqslant v\leqslant 2$}, \endaligned \right.$$ which gives $$\big(P(u,v)\big)^2=\left\{\aligned &6-4v\ \text{if $0\leqslant v\leqslant 1$}, \\ &2(v-2)^2\ \text{if $1\leqslant v\leqslant 2$}, \endaligned \right.$$ and $$P(u,v)\cdot C=\left\{\aligned &2\ \text{if $0\leqslant v\leqslant 1$}, \\ &4-2v\ \text{if $1\leqslant v\leqslant 2$}. \endaligned \right.$$ Similarly, if $C=Z$ and $u\in[1,2]$, then $$P(u,v)=\left\{\aligned &(3-u-v)C+(2-u)(\mathbf{e}_1+\mathbf{e}_2)\ \text{if $0\leqslant v\leqslant 1$}, \\ &(3-u-v)\big(C+\mathbf{e}_1+\mathbf{e}_2\big)\ \text{if $1\leqslant v\leqslant 3-u$}, \endaligned \right.$$ and $$N(u,v)=\left\{\aligned &0\ \text{if $0\leqslant v\leqslant 1$}, \\ &(v-1)\big(\mathbf{e}_1+\mathbf{e}_2\big)\ \text{if $1\leqslant v\leqslant 3-u$}, \endaligned \right.$$ so that $$\big(P(u,v)\big)^2=\left\{\aligned &2(2-u)(4-u-2v)\ \text{if $0\leqslant v\leqslant 1$}, \\ &2(3-u-v)^2\ \text{if $1\leqslant v\leqslant 3-u$}, \endaligned \right.$$ and $$P(u,v)\cdot C=\left\{\aligned &4-2u\ \text{if $0\leqslant v\leqslant 1$}, \\ &6-2u-2v\ \text{if $1\leqslant v\leqslant 3-u$}. \endaligned \right.$$ In particular, if $C=Z$, then $P\not\in\mathrm{Supp}(N(u,v))$, because $P\not\in E_2$ by our assumption. Now, integrating, we get $S(W^S_{\bullet,\bullet};C)=\frac{3}{4}<\frac{99}{104}$ and $S(W_{\bullet, \bullet,\bullet}^{S,C};P)=\frac{21}{26}<\frac{99}{104}$. Hence, to complete the proof, we may assume that $Z=\ell_1+\ell_2$ and $C=\ell_1$. Then $$P(u)\big\vert_{S}-vC=\left\{\aligned &(2-v)C+2\ell_2+\mathbf{e}_1+\mathbf{e}_2\ \text{if $0\leqslant u\leqslant 1$}, \\ &(3-u-v)C+(3-u)\ell_2+(2-u)(\mathbf{e}_1+\mathbf{e}_2)\ \text{if $1\leqslant u\leqslant 2$}, \endaligned \right.$$ which implies that $$t(u)=\left\{\aligned &2\ \text{if $0\leqslant u\leqslant 1$}, \\ &3-u\ \text{if $1\leqslant u\leqslant 2$}. \endaligned \right.$$ Further, if $u\in[0,1]$, then $$P(u,v)=\left\{\aligned &(2-v)C+2\ell_2+\mathbf{e}_1+\mathbf{e}_2\ \text{if $0\leqslant v\leqslant 1$}, \\ &(2-v)\big(C+\mathbf{e}_1\big)+(3-v)\ell_2+\mathbf{e}_2\ \text{if $1\leqslant v\leqslant 2$}, \endaligned \right.$$ and $$N(u,v)=\left\{\aligned &0\ \text{if $0\leqslant v\leqslant 1$}, \\ &(v-1)\big(\mathbf{e}_1+\ell_2\big)\ \text{if $1\leqslant v\leqslant 2$}, \endaligned \right.$$ which gives $$\big(P(u,v)\big)^2=\left\{\aligned &6-v^2-2v\ \text{if $0\leqslant v\leqslant 1$}, \\ &(2-v)(4-v)\ \text{if $1\leqslant v\leqslant 2$}, \endaligned \right.$$ and $$P(u,v)\cdot C=\left\{\aligned &1+v\ \text{if $0\leqslant v\leqslant 1$}, \\ &3-v\ \text{if $1\leqslant v\leqslant 2$}. \endaligned \right.$$ Likewise, if $u\in[1,2]$, then $$P(u,v)=\left\{\aligned &(3-u-v)C+(3-u)\ell_2+(2-u)(\mathbf{e}_1+\mathbf{e}_2)\ \text{if $0\leqslant v\leqslant 2-u$}, \\ &(3-u-v)C+(5-2u-v)\ell_2+(2-u)(\mathbf{e}_1+\mathbf{e}_2)\ \text{if $2-u\leqslant v\leqslant 1$}, \\ &(3-u-v)(C+\mathbf{e}_1)+(5-2u-v)\ell_2+(2-u)\mathbf{e}_2\ \text{if $1\leqslant v\leqslant 3-u$}, \endaligned \right.$$ and $$N(u,v)=\left\{\aligned &0\ \text{if $0\leqslant v\leqslant 2-u$}, \\ &(v-2+u)\ell_2\ \text{if $2-u\leqslant v\leqslant 1$}, \\ &(v-2+u)\ell_2+(v-1)\mathbf{e}_1\ \text{if $1\leqslant v\leqslant 3-u$}, \endaligned \right.$$ so that $$\big(P(u,v)\big)^2=\left\{\aligned &2u^2+2uv-v^2-12u-4v+16\ \text{if $0\leqslant v\leqslant 2-u$}, \\ &(u-2)(3u+4v-10)\ \text{if $2-u\leqslant v\leqslant 1$}, \\ &(3-u-v)(7-3u-v)\ \text{if $1\leqslant v\leqslant 3-u$}, \endaligned \right.$$ and $$P(u,v)\cdot C=\left\{\aligned &2-u+v\ \text{if $0\leqslant v\leqslant 2-u$}, \\ &4-2u\ \text{if $2-u\leqslant v\leqslant 1$}, \\ &5-2u-v\ \text{if $1\leqslant v\leqslant 3-u$}. \endaligned \right.$$ Now, integrating, we get $S(W^S_{\bullet,\bullet};C)=\frac{99}{104}$. Note that $P\not\in\mathbf{e}_1$, since $P\not\in E_2$ by assumption. Therefore, if $P\not\in\ell_2$, then $P\not\in\mathrm{Supp}(N(u,v))$, which implies that $S(W_{\bullet, \bullet,\bullet}^{S,C};P)=\frac{37}{52}<\frac{99}{104}$. Similarly, if $P\in\ell_2$, then $$\begin{gathered} S\big(W_{\bullet, \bullet,\bullet}^{S,C};P\big)=\frac{37}{52}+\frac{6}{26} \int\limits_0^2\int\limits_0^{t(u)}\big(P(u,v)\cdot C\big)\cdot \mathrm{ord}_P\big(N(u,v)\big|_C\big)dvdu=\\ =\frac{37}{52}+\frac{6}{26}\int\limits_0^1\int\limits_1^{2}\big(P(u,v)\cdot C\big)(v-1)dvdu+\frac{6}{26}\int\limits_1^2\int\limits_{2-u}^{3-u}\big(P(u,v)\cdot C\big)(v-2+u)dvdu=\\ =\frac{37}{52}+\frac{6}{26}\int\limits_0^1\int\limits_1^{2}(3-v)(v-1)dvdu+\frac{6}{26}\int\limits_1^2\int\limits_{2-u}^{1}(4-2u)(v-2+u)dvdu+\\ +\frac{6}{26}\int\limits_1^2\int\limits_1^{3-u}(5-2u-v)(v-2+u)dvdu=\frac{99}{104}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{gathered}$$ This completes the proof of Proposition [Proposition 1](#proposition:delta-dP4){reference-type="ref" reference="proposition:delta-dP4"}. ◻ ## Smooth threefolds. {#section:beth-smooth} We continue to use the notation introduced earlier in this section. The goal of this subsection is to give a new proof of the following result. **Theorem 1** ([@Book Proposition 5.79]). *Let $X$ be a smooth Fano threefold from class $(\beth)$. Then $X$ is K-polystable.* We begin with a technical lemma that will also be useful in the singular case. **Lemma 1**. *Suppose that $X$ has at worst isolated singularities. Let $\mathbf{F}$ be a $G$-invariant prime divisor $\mathbf{F}$ over the threefold $X$ with $$\beta(\mathbf{F})=A_X(\mathbf{F})-S_X(\mathbf{F})\leqslant 0,$$ and let $Z$ be its center on $X$. Suppose that $Z$ is not a singular point of the threefold $X$. Then $Z$ is an irreducible curve with $\eta(Z)=([1:1],[1:1])$ or $\eta(Z)=([1:-1],[1:-1])$.* *Proof.* Note first that $Z$ is a $G$-invariant curve, by Lemma [Lemma 1](#lemma:G-action){reference-type="ref" reference="lemma:G-action"} and Corollary [Corollary 1](#corollary:divisorial-stability){reference-type="ref" reference="corollary:divisorial-stability"}, and $$\delta_P(X)\leqslant 1$$ for every point $P\in Z$. Observe also that $Z\not\subset E_1\cup E_2$, because the surfaces $E_1$ and $E_2$ are disjoint and swapped by the action of the group $G$. Let $S$ be any surface in $|H-E_1|$. If $S\cdot Z\ne 0$, then choosing $S$ to be general enough, we see that $S$ is a smooth del Pezzo surface and $S\cap Z$ contains a point $P\not\in E_1\cup E_2$. This would give $\delta_P(X)>1$ by Proposition [Proposition 1](#proposition:delta-dP4){reference-type="ref" reference="proposition:delta-dP4"}, which is a contradiction. So we have $$\big(H-E_1\big)\cdot Z=0.$$ Similarly, we get $(H-E_2)\cdot Z=0$. Therefore, we conclude that $\eta(Z)$ is a point in $\mathbb{P}^1\times\mathbb{P}^1$. Finally, using Remark [Remark 1](#remark:G-action-P1-P1){reference-type="ref" reference="remark:G-action-P1-P1"} we get $\eta(Z)=([1:1],[1:1])$ or $\eta(Z)=([1:-1],[1:-1])$. ◻ *Proof of Theorem [Theorem 1](#theorem:K-stability-smooth){reference-type="ref" reference="theorem:K-stability-smooth"}.* Suppose for a contradiction that the Fano threefold $X$ is smooth, but $X$ is not K-polystable. It follows from [@Fujita2019; @Li; @Zhuang] that there exists a $G$-invariant prime divisor $\mathbf{F}$ over $X$ such that $\beta(\mathbf{F})\leqslant 0$. Let $Z$ be the center of $\mathbf{F}$ on $X$. Then by Lemma [Lemma 1](#lemma:G-invariant-center){reference-type="ref" reference="lemma:G-invariant-center"}, the center $Z$ is a curve with $\eta(Z)=([1:1],[1:1])$ or $\eta(Z)=([1:-1],[1:-1])$. Let $S$ be the unique surface in the pencil $|H-E_1|$ that contains $Z$, and let $\ell=\eta(S)$. If $\eta(Z)=([1:\pm 1],[1:\pm 1])$, then $\ell=\{u_1\pm v_1=0\}$. Moreover, one can check that - $\{u_1-v_1=0\}$ intersects $\Delta$ in a single point $\iff$ $b+a\pm 1=0$, - $\{u_1+v_1=0\}$ intersects $\Delta$ in a single point $\iff$ $b-a\pm 1=0$. Since $Q$ is smooth by assumption, we have $b\pm a\pm 1\ne 0$, and $\ell$ intersects the discriminant curve $\Delta$ in two distinct points. This implies that $S$ is a smooth del Pezzo surface. Now, applying Proposition [Proposition 1](#proposition:delta-dP4){reference-type="ref" reference="proposition:delta-dP4"}, we get $\delta_P(X)>1$ for a general point $P$ in the curve $Z$, which contradicts $\beta(\mathbf{F})\leqslant 0$. This completes the proof of Theorem [Theorem 1](#theorem:K-stability-smooth){reference-type="ref" reference="theorem:K-stability-smooth"}. ◻ ## Threefolds with one singular point. {#section:beth-singular} We continue to use the notation introduced at the beginning of this section. In this subsection, we will prove the following result. **Theorem 1**. *Suppose that $X$ is a threefold from class $(\beth)$ with one singular point. Then $X$ is K-polystable.* Suppose that the threefold $X$ has one singular point. Then $a\pm b\pm 1=0$ and $ab\ne 0$. Up to a change of coordinates, we may assume that $b=1-a$ and $a\not\in\{0,1\}$. Let $O=\mathrm{Sing}(X)$. Recall that $$X=\big\{u_1x=v_1y,u_2z=v_2t,w^2+xy+zt=a(xt+yz)+(1-a)(xz+yt)\big\}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4,$$ and $O=([1:1],[1:1],[1:1:1:1:0])$. Similarly, we have $$Q=\big\{w^2+xy+zt=a(xt+yz)+(1-a)(xz+yt)\big\}\subset\mathbb{P}^4,$$ and $\mathrm{Sing}(Q)=\pi(O)=[1:1:1:1:0]$. Note that $\eta(O)=([1:1],[1:1])$. The fiber of the conic bundle $\eta$ over the point $([1:1],[1:1])$ is a reducible reduced conic $L_1+L_2$, where $L_1$ and $L_2$ are two smooth rational curves such that $L_1\cap L_2=O$, and $\pi(L_1)$ and $\pi(L_2)$ are the only lines in $Q$ that intersect $C_1$ and $C_2$. We have $$\begin{aligned} \pi(L_1)&=\{x=y,z=t,w+x-z=0\},\\ \pi(L_2)&=\{x=y,z=t,w-x+z=0\}.\end{aligned}$$ We will use the curves $L_1$ and $L_2$ in the proofs of the next two lemmas. **Lemma 1**. *Let $\mathbf{F}$ be a $G$-invariant prime divisor $\mathbf{F}$ over the threefold $X$ with $\beta(\mathbf{F})\leqslant 0$, and let $Z$ be its center on the threefold $X$. Then $Z=O$.* *Proof.* Suppose that $Z\ne O$. Applying Lemma [Lemma 1](#lemma:G-invariant-center){reference-type="ref" reference="lemma:G-invariant-center"}, we find that $Z$ is an irreducible curve such that $\eta(Z)=([1:1],[1:1])$ or $\eta(Z)=([1:-1],[1:-1])$. But $Z\ne L_1$ and $Z\ne L_2$, since neither of the curves $L_1$ nor $L_2$ is $G$-invariant, so $\eta(Z)=([1:-1],[1:-1])$. Let $S$ be the surface in $|H-E_1|$ that is cut out on $X$ by the equation $u_1+v_1=0$. Then $S$ is smooth and $Z\subset S$, so $\delta_P(X)>1$ for a general point $P\in Z$ by Proposition [Proposition 1](#proposition:delta-dP4){reference-type="ref" reference="proposition:delta-dP4"}. This contradicts $\beta(\mathbf{F})\leqslant 0$. ◻ **Lemma 1**. *Let $\mathbf{F}$ be any $G$-invariant prime divisor $\mathbf{F}$ over $X$ such that $C_X(\mathbf{F})=O$. Then $\beta(\mathbf{F})>0$.* *Proof.* Let $\varphi\colon\widetilde{X}\to X$ and $\phi\colon\widetilde{Q}\to Q$ be blow ups of the points $O$ and $\pi(O)$, respectively. As in the proof of Lemma [Lemma 1](#lemma:K-unstable){reference-type="ref" reference="lemma:K-unstable"}, we have the following $G$-equivariant commutative diagram: $$\xymatrix{ &\widetilde{X}\ar@{->}[dl]_{\varphi}\ar@{->}[dr]^{\varpi}\ar@/^2.5pc/@{->}[ddrr]^{\nu}&\\% X\ar@{->}[dr]_{\pi}&&\widetilde{Q}\ar@{->}[dl]^{\phi}\ar@{->}[dr]^{\upsilon}&\\% &{Q}&&\mathbb{P}^1\times\mathbb{P}^1}$$ where $\varpi$ is the blow up of the proper transforms of the conics $C_1$ and $C_2$, $\upsilon$ is a $\mathbb{P}^1$-bundle, and $\nu$ is a conic bundle. Let $F$ be the $\varphi$-exceptional surface. Then $\beta(F)=2-S_X(F)$. Let $\widetilde{H}=\varphi^*(H)$, let $\widetilde{E}_1$ and $\widetilde{E}_2$ be the proper transforms on $\widetilde{X}$ of the $\pi$-exceptional surfaces $E_1$ and $E_2$, respectively, let $\widetilde{S}_1$ and $\widetilde{S}_2$ be the proper transforms of the quadric cones in $Q$ that contain $C_1$ and $C_2$, respectively. We have $\varpi^*(-K_{X})\sim 3\widetilde{H}-\widetilde{E}_1-\widetilde{E}_2$. We also have $\widetilde{S}_1\sim \widetilde{H}-\widetilde{E}_1-F$ and $\widetilde{S}_2\sim \widetilde{H}-\widetilde{E}_2-F$. Take $u\in\mathbb{R}_{\geqslant 0}$. Then $$\label{equation:ODP-F} \varpi^*(-K_{X})-uF\sim_{\mathbb{R}} \frac{3}{2}\big(\widetilde{S}_1+\widetilde{S}_2\big)+\frac{1}{2}\big(\widetilde{E}_1+\widetilde{E}_2\big)+(3-u)F.$$ Hence, intersecting $\varpi^*(-K_{X})-uF$ with a sufficiently general fiber of the morphism $\nu$, we see that $\varpi^*(-K_{X})-uF$ is pseudoeffective $\iff$ $u\in[0,3]$. Moreover, we have $$\begin{aligned} \widetilde{H}^3&=2, & \widetilde{E}_1^3&=-4, & \widetilde{E}_2^3&=-4, & F^3&=2, & \widetilde{E}_1\cdot \widetilde{H}^2&=0,\\ \widetilde{E}_2\cdot \widetilde{H}^2&=0, & \widetilde{E}_2\cdot \widetilde{E}_1^2&=0, & \widetilde{E}_1\cdot \widetilde{E}_2^2&=0, & F\cdot \widetilde{E}_1^2&=0, & F\cdot \widetilde{E}_2^2&=0,\\ \widetilde{E}_1\cdot F^2&=0, & \widetilde{E}_2\cdot F^2&=0, & \widetilde{H}\cdot F^2&=0, & F\cdot \widetilde{H}^2&=0, & \widetilde{H}\cdot \widetilde{E}_1^2&=-2,\\ \widetilde{H}\cdot \widetilde{E}_2^2&=-2, & \widetilde{E}_1\cdot \widetilde{E}_2\cdot \widetilde{H}&=0, & \widetilde{E}_1\cdot \widetilde{E}_2\cdot F&=0, & \widetilde{E}_1\cdot F\cdot \widetilde{H}&=0, & \widetilde{E}_2\cdot F\cdot \widetilde{H}&=0.\end{aligned}$$ Furthermore, the divisor $\varpi^*(-K_{X})-uF$ is nef for $u\in[0,1]$. Thus, we have $$\begin{gathered} S_X(F)=\frac{1}{26}\int\limits_{0}^{1}\big(3\widetilde{H}-\widetilde{E}_1-\widetilde{E}_2-uF\big)^3du+\frac{1}{26}\int\limits_{1}^{3}\mathrm{vol}\big(\varpi^*(-K_{X})-uF\big)du=\\ =\frac{1}{26}\int\limits_{0}^{1}26-2u^3du+\frac{1}{26}\int\limits_{1}^{3}\mathrm{vol}\big(\varpi^*(-K_{X})-uF\big)du=\frac{51}{2}+\frac{1}{26}\int\limits_{1}^{3}\mathrm{vol}\big(\varpi^*(-K_{X})-uF\big)du.\end{gathered}$$ If $\ell$ is the preimage on $\widetilde{X}$ of a general ruling of one of the cones $\pi\circ\varphi(\widetilde{S}_1)$ and $\pi\circ\varphi(\widetilde{S}_2)$, then $(\varpi^*(-K_{X})-uF)\cdot \ell=2-u$, which shows that $\varpi^*(-K_{X})-uF$ is not nef for $u>2$. Let $\widetilde{L}_1$ and $\widetilde{L}_2$ be the proper transforms on $\widetilde{X}$ of the curves $L_1$ and $L_2$, respectively. Then $\varpi^*(-K_{X})-uF$ is not nef for $u>1$, because $$\big(\varpi^*(-K_{X})-uF\big)\cdot \widetilde{L}_1=\big(\varpi^*(-K_{X})-uF\big)\cdot \widetilde{L}_2=1-u.$$ Using [\[equation:ODP-F\]](#equation:ODP-F){reference-type="eqref" reference="equation:ODP-F"}, one can show that $\widetilde{L}_1$ and $\widetilde{L}_2$ are the only curves in $\widetilde{X}$ that have negative intersection with $\varpi^*(-K_{X})-uF$ for $u\in(1,2]$. Note that $\widetilde{L}_1\cong\widetilde{L}_2\cong\mathbb{P}^1$ and their normal bundles are isomorphic to $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)$. Let $\psi\colon\widehat{X}\to\widetilde{X}$ be the blow up of $\widetilde{L}_1$ and $\widetilde{L}_2$, let $G_1$ and $G_2$ be the $\psi$-exceptional surfaces such that $\psi(G_1)=\widetilde{L}_1$ and $\psi(G_2)=\widetilde{L}_2$, let $\widehat{H}$, $\widehat{E}_1$, $\widehat{E}_2$, $\widehat{S}_1$, $\widehat{S}_2$, $\widehat{F}$, be the proper transforms on the threefold $\widehat{X}$ of the surfaces $\widetilde{H}$, $\widetilde{E}_1$, $\widetilde{E}_2$, $\widetilde{S}_1$, $\widetilde{S}_2$, $F$, respectively. Then $$(\varpi\circ\psi)^*(-K_{X})\sim 3\widehat{H}-\widehat{E}_1-\widehat{E}_2.$$ We have $\widehat{S}_1\sim \widehat{H}-\widehat{E}_1-\widehat{F}-G_1$ and $\widehat{S}_2\sim \widehat{H}-\widehat{E}_2-\widehat{F}-G_2$. This gives $$\label{equation:ODP-F-blow-up} (\varpi\circ\psi)^*(-K_{X})-u\widehat{F} \sim_{\mathbb{R}} \frac{3}{2}\big(\widehat{S}_1+\widehat{S}_2\big)+\frac{1}{2}\big(\widehat{E}_1+\widehat{E}_2\big)+\frac{3}{2}\big(G_1+G_2\big)+(3-u)\widehat{F}.$$ For $u\in[0,3]$, the Zariski decomposition of this divisor can be computed on $\widehat{X}$ as follows: - if $u\in[0,1]$, then the divisor $(\varpi\circ\psi)^*(-K_{X})-u\widehat{F}$ is nef, - if $u\in[1,2]$, then the positive (nef) part of the Zariski decomposition is $$\begin{gathered} (\varpi\circ\psi)^*(-K_{X})-u\widehat{F}-(u-1)(G_1+G_2)\sim_{\mathbb{R}}\\ \sim_{\mathbb{R}}3\widehat{H}-\widehat{E}_1-\widehat{E}_2-(u-1)(G_1+G_2)\sim_{\mathbb{R}}\\ \sim_{\mathbb{R}}\frac{3}{2}\big(\widehat{S}_1+\widehat{S}_2\big)+\frac{1}{2}\big(\widehat{E}_1+\widehat{E}_2\big)+\frac{5-2u}{2}\big(G_1+G_2\big)+(3-u)\widehat{F},\end{gathered}$$ and the negative part of the Zariski decomposition is $(u-1)(G_1+G_2)$, - if $u\in(2,3]$, then the positive part of the Zariski decomposition is $$\begin{gathered} (\varpi\circ\psi)^*(-K_{X})-u\widehat{F}-(u-1)(G_1+G_2)-(u-2)(\widehat{S}_1+\widehat{S}_2)\sim_{\mathbb{R}}\\ \sim_{\mathbb{R}}3\widehat{H}-\widehat{E}_1-\widehat{E}_2-(u-1)(G_1+G_2)-(u-2)(\widehat{S}_1+\widehat{S}_2)\sim_{\mathbb{R}}\\ \sim_{\mathbb{R}}(7-2u)\widehat{H}-(3-u)\big(\widehat{E}_1+\widehat{E}_2+G_1+G_2\big)+(u-4)F\sim_{\mathbb{R}}\\ \sim_{\mathbb{R}}\frac{7-2u}{2}\big(\widehat{S}_1+\widehat{S}_2\big)+\frac{1}{2}\big(\widehat{E}_1+\widehat{E}_2\big)+\frac{5-2u}{2}\big(G_1+G_2\big)+(3-u)\widehat{F},\end{gathered}$$ and the negative part is $(u-1)(G_1+G_2)+(u-2)(\widehat{S}_1+\widehat{S}_2)$. The intersections of the divisors $\widehat{H}$, $\widehat{E}_1$, $\widehat{E}_2$, $\widehat{F}$, $G_1$, $G_2$ on $\widehat{X}$ can be computed as follows: $$\begin{aligned} \widehat{H}^3&=2, & \widehat{E}_1^3&=-4, & \widehat{E}_2^3&=-4, & G_1^3&=2,\\ G_2^3&=2, & \widehat{F}^3&=2, & \widehat{H}\cdot \widehat{E}_1^2&=-2, & \widehat{H}\cdot \widehat{E}_2^2&=-2,\\ \widehat{F}\cdot G_1^2&=-1, & \widehat{F}\cdot G_2^2&=-1, & \widehat{E}_1\cdot G_1^2&=-1, & \widehat{E}_2\cdot G_1^2&=-1,\\ \widehat{E}_1\cdot G_2^2&=-1, & \widehat{E}_2\cdot G_2^2&=-1, & \widehat{H}\cdot G_1^2&=-1, & \widehat{H}\cdot G_2^2&=-1,\end{aligned}$$ and other triple intersections are zero. Now we are ready to compute $S_X(F)$. We have $$\begin{gathered} S_X(F)=\frac{1}{26}\int\limits_{0}^{1}\Big((\pi\circ\varpi)^*(-K_{X})-u\widehat{F}\Big)^3du+\frac{1}{26}\int\limits_{1}^{2}\Big(3\widehat{H}-\widehat{E}_1-\widehat{E}_2-(u-1)(G_1+G_2)\Big)^3du+\\ +\frac{1}{26}\int\limits_{2}^{3}\Big((7-2u)\widehat{H}-(3-u)\big(\widehat{E}_1+\widehat{E}_2+G_1+G_2\big)+(u-4)F\Big)^3du=\\ =\frac{1}{26}\int\limits_{0}^{1}26-2u^3du+\frac{1}{26}\int\limits_{1}^{2}24+6u-6u^2du+\frac{1}{26}\int\limits_{2}^{3}6(2-u)(4-u)du=\frac{99}{52}.\end{gathered}$$ Thus, we see that $\beta(F)=\frac{5}{52}>0$. Now, suppose that $\beta(\mathbf{F})\leqslant 0$. Let $\widetilde{Z}$ be the center of the divisor $\mathbf{F}$ on the threefold $\widetilde{X}$. Then $\widetilde{Z}$ is a $G$-invariant irreducible proper subvariety of the surface $F$, since $\beta(F)>0$. Recall from Lemma [Lemma 1](#lemma:ODP-G-action){reference-type="ref" reference="lemma:ODP-G-action"} that $G$ acts faithfully on $F$, and $F$ does not have $G$-fixed points. Hence, we conclude that $\widetilde{Z}$ is a curve. Let $\widehat{Z}$ be the proper transform of this curve on $\widehat{X}$. Then $\widehat{Z}=C_{\widehat{X}}(\mathbf{F})$ and $\widehat{Z}\subset\widehat{F}$. Let us apply Abban--Zhuang theory [@AbbanZhuang] to the flag $\widehat{Z}\subset\widehat{F}$. For every $u\in[0,3]$, let $P(u)$ be the positive (nef) part of the Zariski decomposition of the divisor $(\varpi\circ\psi)^*(-K_{X})-u\widehat{F}$ (described above), and let $N(u)$ be its negative part. Observe that $\widehat{Z}\not\subset\mathrm{Supp}(N(u))$, because $\widehat{Z}$ is irreducible and $G$-invariant. Set $$t(u)=\mathrm{sup}\Big\{v\in\mathbb{R}_{\geqslant 0}\ \big\vert\ \text{the~divisor $P(u)\big\vert_{\widehat{F}}-v\widehat{Z}$ is pseudo-effective}\Big\}.$$ For $v\in[0,t(u)]$, let $P(u,v)$ be the nef part of the Zariski decomposition of $P(u)\vert_{\widehat{F}}-v\widehat{Z}$, and let $N(u,v)$ be the negative part of this Zariski decomposition. Set $$S\big(W^{\widehat{F}}_{\bullet,\bullet};\widehat{Z}\big)=\frac{3}{26}\int\limits_0^3\int\limits_0^{t(u)}\big(P(u,v)\big)^2dvdu.$$ Then it follows from [@AbbanZhuang Theorem 3.3] and [@Book Corollary 1.109] that $$1\geqslant\frac{A_X(\mathbf{F})}{S_X(\mathbf{F})}\geqslant\min\Bigg\{\frac{A_X(\widehat{F})}{S_X(\widehat{F})},\frac{1}{S\big(W^{\widehat{F}}_{\bullet,\bullet};\widehat{Z}\big)}\Bigg\}=\min\Bigg\{\frac{104}{99},\frac{1}{S\big(W^{\widehat{F}}_{\bullet,\bullet};\widehat{Z}\big)}\Bigg\}.$$ Therefore, we conclude that $S(W^{\widehat{F}}_{\bullet,\bullet};\widehat{Z})\geqslant 1$. Let us show that $S(W^{\widehat{F}}_{\bullet,\bullet};\widehat{Z})<1$. The morphism $\psi$ induces a $G$-equivariant birational morphism $\sigma\colon \widehat{F}\to F$ that blows up two points $\widetilde{L}_1\cap F$ and $\widetilde{L}_2\cap F$. By Lemma [Lemma 1](#lemma:ODP-G-action){reference-type="ref" reference="lemma:ODP-G-action"}, $\mathrm{rk}\,\mathrm{Pic}^G(F)=1$, where $F\cong\mathbb{P}^1\times\mathbb{P}^1$. Thus, we see that $\widehat{F}$ is a smooth del Pezzo surface of degree $6$, $\mathrm{rk}\,\mathrm{Pic}^G(\widehat{F})=2$, and there exists the following $G$-equivariant diagram: $$\xymatrix{ &\widehat{F}\ar@{->}[dl]_{\sigma}\ar@{->}[dr]^{\xi}&\\% F &&\mathbb{P}^1}$$ where $\xi$ is a $G$-minimal conic bundle with $2$ singular fibers. Set $\mathbf{g}_1=G_1\vert_{\widehat{F}}$ and $\mathbf{g}_2=G_2\vert_{\widehat{F}}$. Let $\mathbf{h}$ be a curve on $F$ of degree $(1,1)$, and let $\mathbf{f}$ be a smooth fiber of the morphism $\xi$. Then $\mathbf{f}\sim\sigma^*\big(\mathbf{h}\big)-\mathbf{g}_1-\mathbf{g}_2$. Since $\widehat{Z}$ is $G$-invariant, we have $\widehat{Z}\sim n\mathbf{f}+m(\mathbf{g}_1+\mathbf{g}_2)$ for some non-negative integers $n$ and $m$. Since $\widehat{Z}$ is irreducible, we have $n>0$, so that $$S\big(W^{\widehat{F}}_{\bullet,\bullet};\widehat{Z}\big)=\frac{3}{26}\int\limits_0^3\int\limits_0^{\infty}\mathrm{vol}\big(P(u)\vert_{\widehat{F}}-v\widehat{Z}\big)dvdu\leqslant\frac{3}{26}\int\limits_0^3\int\limits_0^{\infty}\mathrm{vol}\big(P(u)\vert_{\widehat{F}}-v\mathbf{f}\big)dvdu.$$ Hence, to show that $S(W^{\widehat{F}}_{\bullet,\bullet};\widehat{Z})<1$, we may assume that $\widehat{Z}=\mathbf{f}$. We have $G_1\vert_{\widehat{F}}=\mathbf{g}_1$, $G_2\vert_{\widehat{F}}=\mathbf{g}_2$, $\widehat{F}\vert_{\widehat{F}}\sim\sigma^*\big(\mathbf{h}\big)\sim\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2$ and $\widehat{H}\vert_{\widehat{F}}\sim \widehat{E}_1\vert_{\widehat{F}}\sim \widehat{E}_2\vert_{\widehat{F}}\sim 0$. This gives $$P(u)\big\vert_{\widehat{F}}-v\mathbf{f}\sim_{\mathbb{R}}\left\{\aligned &(u-v)\mathbf{f}+u\big(\mathbf{g}_1+\mathbf{g}_2\big)\ \text{if $0\leqslant u\leqslant 1$}, \\ &(u-v)\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2\ \text{if $1\leqslant u\leqslant 2$}, \\ &(4-u-v)\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2\ \text{if $2\leqslant u\leqslant 3$}, \endaligned \right.$$ which implies that $$t(u)=\left\{\aligned &u\ \text{if $0\leqslant u\leqslant 2$}, \\ &4-u\ \text{if $2\leqslant u\leqslant 3$}. \endaligned \right.$$ If $u\in[0,1]$, then $P(u,v)=(u-v)(\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2)$ and $N(u,v)=v(\mathbf{g}_1+\mathbf{g}_2)$ for $v\in[0,u]$, which gives $(P(u,v))^2=2(u-v)^2$ for every $(u,v)\in[0,1]\times[0,u]$. If $u\in[1,2]$, then $$P(u,v)=\left\{\aligned &(u-v)\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2\ \text{if $0\leqslant v\leqslant u-1$}, \\ &(u-v)\big(\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2\big)\ \text{if $u-1\leqslant v\leqslant u$}, \endaligned \right.$$ and $$N(u,v)=\left\{\aligned &0\ \text{if $0\leqslant v\leqslant u-1$}, \\ &(v-u+1)\big(\mathbf{g}_1+\mathbf{g}_2\big)\ \text{if $u-1\leqslant u\leqslant u$}, \endaligned \right.$$ which gives $$\big(P(u,v)\big)^2=\left\{\aligned &4u-4v-2\ \text{if $0\leqslant v\leqslant u-1$}, \\ &2(u-v)^2\ \text{if $u-1\leqslant v\leqslant u$}. \endaligned \right.$$ Finally, if $u\in[2,3]$, then $$P(u,v)=\left\{\aligned &(4-u-v)\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2\ \text{if $0\leqslant v\leqslant 3-u$}, \\ &(4-u-v)\big(\mathbf{f}+\mathbf{g}_1+\mathbf{g}_2\big)\ \text{if $3-u\leqslant v\leqslant 4-u$}. \endaligned \right.$$ and $$N(u,v)=\left\{\aligned &0\ \text{if $0\leqslant v\leqslant 3-u$}, \\ &(v+u-3)\big(\mathbf{g}_1+\mathbf{g}_2\big)\ \text{if $3-u\leqslant u\leqslant 4-u$}, \endaligned \right.$$ which gives $$\big(P(u,v)\big)^2=\left\{\aligned &14-4u-4v\ \text{if $0\leqslant v\leqslant 3-u$}, \\ &2(4-u-v)^2\ \text{if $3-u\leqslant v\leqslant 4-u$}. \endaligned \right.$$ Now, integrating, we get $$\begin{gathered} S(W^{\widehat{F}}_{\bullet,\bullet};\mathbf{f})=\frac{3}{26}\int\limits_0^1\int\limits_0^{u}2(u-v)^2dvdu+\frac{3}{26}\int\limits_1^2\int\limits_0^{u-1}4u-4v-2dvdu+\frac{3}{26}\int\limits_1^2\int\limits_{u-1}^u2(u-v)^2dvdu+\\ +\frac{3}{26}\int\limits_2^3\int\limits_0^{3-u}14-4u-4vdvdu+\frac{3}{26}\int\limits_2^3\int\limits_{3-u}^{4-u}2(4-u-v)^2dvdu=\frac{29}{52}<1,\end{gathered}$$ which is a contradiction. Lemma [Lemma 1](#lemma:K-stability-isolated-2){reference-type="ref" reference="lemma:K-stability-isolated-2"} is proved. ◻ Now, using the $G$-equivariant valuative criterion for K-polystability [@Zhuang Corollary 4.14], we see that Theorem [Theorem 1](#theorem:K-stability-isolated){reference-type="ref" reference="theorem:K-stability-isolated"} follows from Lemmas [Lemma 1](#lemma:K-stability-isolated-1){reference-type="ref" reference="lemma:K-stability-isolated-1"} and [Lemma 1](#lemma:K-stability-isolated-2){reference-type="ref" reference="lemma:K-stability-isolated-2"}. ## Threefolds with non-isolated singularities. {#section:beth-special} We continue to use the notation introduced at the beginning of this section. Suppose, in addition, that $a\pm b\pm 1=ab=0$. Up to a change of coordinates, we may assume that $$X=\big\{u_1x=v_1y,u_2z=v_2t,w^2+(x+z)(y+t)=0\big\}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4,$$ and $Q=\{w^2+(x+z)(y+t)=0\}\subset\mathbb{P}^4$. Then $X$ is singular along the curve $$C=\big\{w=0,x+z=0,y+t=0,u_1z=v_1t,u_2z=v_2t,v_1u_2=u_1v_2\big\},$$ and $Q$ is a quadric in $\mathbb{P}^4$ that is singular along the line $L=\{w=0,x+z=0,y+t=0\}$. Recall that $\pi\colon X\to Q$ is a blow up of the following two smooth conics: $$\begin{aligned} {C}_1&=\{w^2+zt=0,x=0,y=0\},\\ {C}_2&=\{w^2+xy=0,z=0,t=0\},\end{aligned}$$ which both lie in the smooth locus of the quadric $Q$. We have $L=\pi(C)$. In this subsection we will prove the following result. **Theorem 1**. *The threefold $X$ from class $(\beth)$ described above, which has non-isolated singularities, is K-polystable.* Let $\Pi_1$ and $\Pi_2$ be the planes $\{x+z=0,w=0\}$ and $\{y+t=0,w=0\}$ in $\mathbb{P}^4$, respectively. Then $\Pi_1$ and $\Pi_2$ are contained in the quadric $Q$. In fact, we have $\Pi_1+\Pi_1=Q\cap\{w=0\}$. Now, we let $S_1$ and $S_2$ be the proper transforms on $X$ of the planes $\Pi_1$ and $\Pi_2$, respectively. Then $S_1\cap S_2=X\cap\{w=0\}$, $\Pi_1\cap\Pi_2=\mathrm{Sing}(Q)=L$ and $S_1\cap S_2=\mathrm{Sing}(X)=C$. Recall that $\tau_3$ is the involution in $\mathrm{Aut}(X)$ given by $$\big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big)\mapsto \big([u_1:v_1],[u_2:v_2],[x:y:z:t:-w]\big).$$ We set $W=X/\tau_3$, and let $\theta\colon X\to W$ be the quotient map. Then $\theta$ is ramified in $S_1+S_2$, and $W$ is the smooth Fano threefold in the family , which can be identified with $$\big\{u_1x=v_1y,u_2z=v_2t\big\}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^3,$$ where we consider $([u_1:v_1],[u_2:v_2],[x:y:z:t])$ as coordinates on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^3$. Moreover, we have the following commutative diagram: $$\label{equation:beth-double-cover} \xymatrix{ {Q}\ar@{-->}[drr]\ar@{->}[rrrr]^{\vartheta}&&&&\mathbb{P}^3\ar@{-->}[dll]\\ &&\mathbb{P}^1\times\mathbb{P}^1&&\\ {X}\ar@{->}[rru]_{\eta}\ar@{->}[uu]^{\pi}\ar@{->}[rrrr]_{\theta}&&&&{W}\ar@{->}[uu]_{\varpi}\ar@{->}[ull]}$$ where - $\vartheta$ is the double cover given by $[x:y:z:t:w]\mapsto[x:y:z:t]$, - $\varpi$ is a blow up of the lines $\vartheta(C_1)=\{x=0,y=0\}$ and $\vartheta(C_2)=\{z=0,t=0\}$, - the map $Q\dasharrow \mathbb{P}^1\times\mathbb{P}^1$ is given by $[x:y:z:t:w]\mapsto([x:y],[z:t])$, - the map $\mathbb{P}^3\dasharrow\mathbb{P}^1\times\mathbb{P}^1$ is given by $[x:y:z:t]\mapsto([x:y],[z:t])$, - $\eta$ and $W\to\mathbb{P}^1\times\mathbb{P}^1$ are natural projections. Observe that the double cover $\vartheta$ is ramified in $\Pi_1+\Pi_2$, the morphism $\eta$ is a conic bundle, and the morphism $W\to\mathbb{P}^1\times\mathbb{P}^1$ is a $\mathbb{P}^1$-bundle. In addition to the finite subgroup $G\cong\boldsymbol{\mu}_2^3$ described in the beginning of this section, the group $\mathrm{Aut}(X)$ contains a subgroup $\Gamma\cong\mathbb{C}^\ast$ that consists of the automorphisms $$\big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big)\mapsto \Big(\Big[\frac{u_1}{\lambda}:\lambda v_1\Big],\Big[\frac{u_2}{\lambda}:\lambda v_2\Big],\Big[\lambda x:\frac{y}{\lambda}:\lambda z:\frac{t}{\lambda}:w\Big]\Big),$$ where $\lambda\in\mathbb{C}^\ast$. Let $\mathbf{G}$ be the subgroup in $\mathrm{Aut}(X)$ generated by the subgroup $\Gamma$ together with the involutions $\tau_1$, $\tau_2$, $\tau_3$ described earlier in this section. Note that $\tau_3\in\Gamma$, so that $$\mathbf{G}\cong\big(\mathbb{C}^\ast\rtimes\boldsymbol{\mu}_2\big)\times\boldsymbol{\mu}_2,$$ because $\tau_1\circ\gamma=\gamma\circ\tau_1$ and $\tau_2\circ\gamma\circ\tau_2=\gamma^{-1}$ for every $\gamma\in\Gamma$. Note also that the commutative diagram [\[equation:beth-double-cover\]](#equation:beth-double-cover){reference-type="eqref" reference="equation:beth-double-cover"} is $\mathbf{G}$-equivariant, and the curve $C$ is $\mathbf{G}$-invariant. Set $$C^\prime=\big\{u_1x=v_1y,u_2z=v_2t,w^2+(x+z)(y+t)=0,x=z,t=y\big\}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4.$$ Then $C^\prime$ is a $\mathbf{G}$-invariant irreducible smooth curve such that $\pi(C^\prime)$ is a smooth conic. **Lemma 1**. *The following assertions holds:* 1. *$X$ does not have $\mathbf{G}$-fixed points;* 2. *the only $\mathbf{G}$-invariant irreducible curves in $X$ are $C$ and $C^\prime$.* *Proof.* Since the diagram [\[equation:beth-double-cover\]](#equation:beth-double-cover){reference-type="eqref" reference="equation:beth-double-cover"} is $\mathbf{G}$-equivariant and $\tau_2$ swaps the $\pi$-exceptional surfaces, it is enough to prove the following assertions for the induced $\mathbf{G}$-action on $\mathbb{P}^3$: 1. $\mathbb{P}^3$ does not have $\mathbf{G}$-fixed points; 2. the only $\mathbf{G}$-invariant irreducible curves in $\mathbb{P}^3$ are the lines $\vartheta(L)$ and $\vartheta\circ\pi(C^\prime)$. Both these assertions are easy to check, since $\tau_1$ and $\tau_2$ act on $\mathbb{P}^3$ as $$\begin{aligned} \tau_1\colon [x:y:z:t] &\mapsto [y:x:t:z],\\ \tau_2\colon [x:y:z:t] &\mapsto [z:t:x:y],\end{aligned}$$ and the subgroup $\Gamma$ acts on $\mathbb{P}^3$ as $[x:y:z:t:w]\mapsto[\lambda^2 x:y:\lambda^2 z:t]$, where $\lambda\in\mathbb{C}^\ast$. ◻ Let $Q\dasharrow\mathbb{P}^2$ be the rational map given by $[x:y:z:t:w]\mapsto[x+z:y+t:w]$. This map is undefined along the line $\pi(C)=\mathrm{Sing}(Q)=\{x+z=0,y+t=0,w=0\}$, and the closure of its image is a smooth conic $\mathscr{C}\subset\mathbb{P}^2$. Therefore, we have a $\mathbf{G}$-equivariant dominant map $\chi\colon Q\dasharrow \mathscr{C}$, which fits the following $\mathbf{G}$-equivariant commutative diagram: $$\label{equation:beth-big} \xymatrix{ &\mathbb{P}^1\times\mathbb{P}^1&&\\ &\widetilde{X}\ar@{->}[dl]_{\phi}\ar@{->}[dr]\ar@{->}[rr]^{\rho}&&\mathscr{C}\times\mathbb{P}^1\times\mathbb{P}^1\ar@{->}[dd]^{\mathrm{pr}_1}\ar@/_2pc/@{->}[ull]_{\mathrm{pr}_{2,3}}\\% X\ar@/^1pc/@{->}[uur]^{\eta}\ar@{->}[dr]_{\pi}&&\widetilde{Q}\ar@{->}[dl]\ar@{->}[rd]&&\\% &{Q}\ar@{-->}[rr]_{\chi}&&\mathscr{C}}$$ where both $\widetilde{X}$ and $\widetilde{Q}$ are smooth threefolds, and - $\phi$ is the blow of the curve $C$, - $\widetilde{Q}\to Q$ is the blow up of the line $L$, - $\widetilde{X}\to\widetilde{Q}$ is the blow up of the preimages of the conics $C_1$ and $C_2$, - $\mathrm{pr}_{1}$ is the projection to the first factor, - $\mathrm{pr}_{2,3}$ is the projection to the product of the second and the third factors, - $\widetilde{Q}\to\mathscr{C}$ is a $\mathbb{P}^2$-bundle, - $\rho$ is a birational morphism. To describe $\rho$ more explicitly, let $\mathscr{S}$ be the surface in $Q$ that is cut out by $xt=yz$, and let $\widetilde{\mathscr{S}}$ be its proper transform on $\widetilde{X}$. Then $\rho$ contracts the surface $\widetilde{\mathscr{S}}$ to a smooth curve. Note that the birational map $Q\dasharrow\mathscr{C}\times\mathbb{P}^1\times\mathbb{P}^1$ in [\[equation:beth-big\]](#equation:beth-big){reference-type="eqref" reference="equation:beth-big"} is given by $$[x:y:z:t:w]\mapsto\big([x+z:y+t:w],[x:y],[z:t]\big).$$ Furthermore, there exists a $\mathbf{G}$-equivariant ramified double cover $\upsilon\colon\mathscr{C}\to\mathbb{P}^1$ that fits into the following $\mathbf{G}$-equivariant commutative diagram: $$\label{equation:beth-quotient-long} \xymatrix{ Q\ar@{->}[d]_{\vartheta}&&X\ar@{->}[ll]_{\pi}\ar@{->}[d]_{\theta} &&\widetilde{X}\ar@{->}[ll]_{\phi}\ar@{->}[d]_{\tilde{\theta}}\ar@{->}[rr]^(.4){\rho}&&\mathscr{C}\times\mathbb{P}^1\times\mathbb{P}^1\ar@{->}[d]^{\upsilon\times\mathrm{Id}_{\mathbb{P}^1}\times\mathrm{Id}_{\mathbb{P}^1}}\\% \mathbb{P}^3&& W\ar@{->}[ll]_{\varpi}&& \widetilde{W}\ar@{->}[ll]_{\varphi}\ar@{->}[rr]^(.4){\varrho}&&\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1}$$ where $\tilde{\theta}$ is the quotient by the involution $\tau_3$, the map $\varphi$ is the blow up of the curve $\theta(C)$, and $\varrho$ is the blow up of the tridiagonal. Note that $\widetilde{W}$ is the unique smooth Fano threefold in the deformation family , and $\varpi\circ\varphi$ blows up the lines $\vartheta(L)$, $\vartheta(C_1)$, $\vartheta(C_2)$. Let $H$ be the proper transform on $X$ of a general hyperplane section of the quadric $Q$, let $E_1$ and $E_2$ be the $\pi$-exceptional surfaces such that $\pi(E_1)=C_1$ and $\pi(E_2)=C_2$. Then $$-K_X\sim 3H-E_1-E_2.$$ Let $\widetilde{H}$, $\widetilde{E}_1$, $\widetilde{E}_2$ be the proper transforms on $\widetilde{X}$ of the surfaces $H$, $E_1$, $E_2$, respectively, and let $F$ be the $\phi$-exceptional surface. Then $-K_{\widetilde{X}}\sim\phi^*(-K_X)\sim 3\widetilde{H}-\widetilde{E}_1-\widetilde{E}_2$ and $$\widetilde{\mathscr{S}}\sim 2\widetilde{H}-\widetilde{E}_1-\widetilde{E}_2-F.$$ Note that $F$ and $\widetilde{\mathscr{S}}$ are $\mathbf{G}$-invariant prime divisors in $\widetilde{X}$, which intersect transversally along a smooth irreducible $\mathbf{G}$-invariant curve. Furthermore, using [\[equation:beth-quotient-long\]](#equation:beth-quotient-long){reference-type="eqref" reference="equation:beth-quotient-long"}, we also see that the surface $\widetilde{\theta}(F)$ is $\varphi$-exceptional, and the surface $\widetilde{\theta}(\widetilde{\mathscr{S}})$ is $\varrho$-exceptional. **Lemma 1**. *The following assertions hold:* (a) *One has $F\cong\mathbb{P}^1\times\mathbb{P}^1$, and $F$ contains exactly two $\mathbf{G}$-invariant irreducible curves, which are both smooth and contained in the linear system $|\widetilde{\mathscr{S}}\vert_{F}|$.* (b) *One has $\widetilde{\mathscr{S}}\cong\mathbb{P}^1\times\mathbb{P}^1$, and $\widetilde{\mathscr{S}}$ contains exactly two $\mathbf{G}$-invariant irreducible curves, the intersection $F\cap \widetilde{\mathscr{S}}$ and the proper transform of the curve $C^\prime$, which are both smooth and contained in the linear system $|F\vert_{\widetilde{\mathscr{S}}}|$.* *Proof.* An easy explicit calculation using [\[equation:beth-quotient-long\]](#equation:beth-quotient-long){reference-type="eqref" reference="equation:beth-quotient-long"} and the formulas for the $\mathbf{G}$-action. ◻ On $\widetilde{X}$, intersections of the divisors $\widetilde{H}$, $\widetilde{E}_1$, $\widetilde{E}_2$, $F$ can be computed as follows: $$\widetilde{H}^3=2, \widetilde{E}_1^3=\widetilde{E}_2^3=-4, F^3=-4, \widetilde{H}\cdot F^2=-2, \widetilde{H}\cdot \widetilde{E}_1^2=\widetilde{H}\cdot \widetilde{E}_2^2=-2,$$ and all other triple intersections are zero. **Lemma 1**. *Let $S$ be an $\mathbf{G}$-invariant prime divisor in $\widetilde{X}$. Then $\beta(S)>0$.* *Proof.* Let $\widetilde{S}_1$ and $\widetilde{S}_2$ be proper transforms on $\widetilde{X}$ of the surfaces $S_1$ and $S_2$, respectively. Using [\[equation:beth-big\]](#equation:beth-big){reference-type="eqref" reference="equation:beth-big"}, one can show that $S\sim a\widetilde{H}+b\big(\widetilde{E}_1+\widetilde{E}_2\big)+c\big(\widetilde{S}_1+\widetilde{S}_2\big)+d\widetilde{\mathscr{S}}+eF$ for some non-negative integers $a$, $b$, $c$, $d$, $e$. Thus, to prove the lemma, it is enough to show that $$\frac{1}{26}\int\limits_{0}^{\infty}\mathrm{vol}\big(-K_{\widetilde{X}}-uD\big)du<1$$ for each $D\in\{\widetilde{H}, \widetilde{E}_1+\widetilde{E}_2, \widetilde{S}_1+\widetilde{S}_2, \widetilde{\mathscr{S}}, F\}$. The computation in the first three cases is very similar to the proof of Corollary [Corollary 1](#corollary:divisorial-stability){reference-type="ref" reference="corollary:divisorial-stability"}; we'll illustrate the cases $D=F$ and $D=\widetilde{\mathscr{S}}$. First, we compute $\beta(F)$. Let $u$ be a non-negative real number. Then $$-K_{\widetilde{X}}-uF\sim_{\mathbb{R}}3\widetilde{H}-\widetilde{E}_1-\widetilde{E}_2-uF\sim_{\mathbb{R}}\widetilde{\mathscr{S}}+\widetilde{S}_1+\widetilde{S}_2+(2-u)F,$$ because $2\widetilde{S}_1\sim 2\widetilde{S}_2\sim \widetilde{H}-F$. This shows that $-K_{\widetilde{X}}-uF$ is pseudo-effective $\iff$ $u\leqslant 2$. Similarly, we see that $-K_{\widetilde{X}}-uF$ is nef $\iff$ $u\in[0,1]$. Furthermore, if $u\in[1,2]$, then the Zariski decomposition of this divisor can be described as follows: $$-K_{\widetilde{X}}-uF\sim_{\mathbb{R}}\underbrace{-K_{\widetilde{X}}-uF-(u-1)\widetilde{\mathscr{S}}}_{\text{positive part}}+\underbrace{(u-1)\widetilde{\mathscr{S}}}_{\text{negative part}}.$$ Hence, we have $$\begin{gathered} \beta(F)=1-\frac{1}{26}\int\limits_{0}^{1}\big(-K_{\widetilde{X}}-uF\big)^2du-\frac{1}{26}\int\limits_{1}^{2}\big(-K_{\widetilde{X}}-uF-(u-1)\widetilde{\mathscr{S}}\big)^2du=\\ =1-\frac{1}{26}\int\limits_{0}^{1}\big(3\widetilde{H}-\widetilde{E}_1-\widetilde{E}_2-uF\big)^2du-\frac{1}{26}\int\limits_{1}^{2}\big((5-2u)\widetilde{H}-(2-u)\widetilde{E}_1-(2-u)\widetilde{E}_2-F\big)^2du=\\ =1-\frac{1}{26}\int\limits_{0}^{1}4u^3-18u^2+26du-\frac{1}{26}\int\limits_{1}^{2}12(u-2)^2du=\frac{1}{26}>0.\end{gathered}$$ Now, we compute $\beta(\widetilde{\mathscr{S}})$. As above, we observe that $$-K_{\widetilde{X}}-u\widetilde{\mathscr{S}}\sim_{\mathbb{R}} \frac{3-2u}{2}\widetilde{\mathscr{S}}+\frac{1}{2}\big(\widetilde{E}_1+\widetilde{E}_2\big)+\frac{3}{2}F.$$ This shows that the divisor $-K_{\widetilde{X}}-u\widetilde{\mathscr{S}}$ is pseudo-effective if and only if $u\leqslant\frac{3}{2}$. If $u\in[0,1]$, then the positive part of the Zariski decomposition of this divisor is $$\frac{3-2u}{2}\big(\widetilde{\mathscr{S}}+F\big)+\frac{1}{2}\big(\widetilde{E}_1+\widetilde{E}_2\big),$$ and the negative part is $uF$. If $1\leqslant u\leqslant\frac{3}{2}$, the Zariski decomposition is the following: $$-K_{\widetilde{X}}-u\widetilde{\mathscr{S}} \sim_{\mathbb{R}}\underbrace{ \frac{3-2u}{2}\big(\widetilde{\mathscr{S}}+F+\widetilde{E}_1+\widetilde{E}_2\big)}_{\text{positive part}}+\underbrace{uF+(u-1)\big(\widetilde{E}_1+\widetilde{E}_2\big)}_{\text{negative part}}.$$ Note that $\widetilde{\mathscr{S}}+F+\widetilde{E}_1+\widetilde{E}_2\sim 2\widetilde{H}$. Integrating, we get $\frac{1}{26}\int\limits_{0}^{\infty}\mathrm{vol}\big(-K_{\widetilde{X}}-u\widetilde{\mathscr{S}}\big)du=\frac{49}{104}$. ◻ Now we are ready to prove that $X$ is K-polystable. Suppose that $X$ is not K-polystable. By [@Zhuang Corollary 4.14], there exists a $\mathbf{G}$-invariant prime divisor $\mathbf{F}$ over $X$ with $\beta(\mathbf{F})\leqslant 0$. Let $Z$ be its center on $\widetilde{X}$. Then $Z$ is a $\mathbf{G}$-invariant curve by Lemmas [Lemma 1](#lemma:beth-special-curves){reference-type="ref" reference="lemma:beth-special-curves"} and [Lemma 1](#lemma:beth-special-Eff-cone){reference-type="ref" reference="lemma:beth-special-Eff-cone"}. **Lemma 1**. *One has $Z\not\subset F$.* *Proof.* Let $\mathbf{s}$ be a section of the projection $F\to C$ such that $\mathbf{s}^2=0$, and let $\mathbf{f}$ be its fiber. Then $\widetilde{E}_1\cap F=\varnothing$ and $\widetilde{E}_2\cap F=\varnothing$. Moreover, we compute $F\vert_{F}\sim -2\mathbf{s}+\mathbf{f}$ and $\widetilde{H}\vert_{F}\sim\mathbf{f}$, which gives $\widetilde{\mathscr{S}}\vert_{F}\sim 2\mathbf{s}+\mathbf{f}$. Suppose that $Z\subset F$. Then $Z\sim 2\mathbf{s}+\mathbf{f}$, by Lemma [Lemma 1](#lemma:beth-G-action){reference-type="ref" reference="lemma:beth-G-action"}(a). For every number $u\in[0,2]$, let $P(u)$ be the positive part of the Zariski decomposition of $-K_{\widetilde{X}}-uF$ described in the proof of Lemma [Lemma 1](#lemma:beth-special-Eff-cone){reference-type="ref" reference="lemma:beth-special-Eff-cone"}, and let $N(u)$ be its negative part. Take $v\in\mathbb{R}_{\geqslant 0}$. Then $$P(u)\big\vert_{F}-vZ=\left\{\aligned &(2u-2v)\mathbf{s}+(3-u-v)\mathbf{f}\ \text{if $0\leqslant u\leqslant 1$}, \\ &(2-2v)\mathbf{s}+(4-2u-v)\mathbf{f}\ \text{if $1\leqslant u\leqslant 2$}, \endaligned \right.$$ and $$N(u)\big\vert_{F}=\left\{\aligned &0\ \text{if $0\leqslant u\leqslant 1$}, \\ &(u-1)\widetilde{\mathscr{S}}\big\vert_{F}\ \text{if $1\leqslant u\leqslant 2$}. \endaligned \right.$$ For $u\in[0,2]$, set $t(u)=\mathrm{sup}\{v\in\mathbb{R}_{\geqslant 0}\ \vert\ \text{$P(u)\vert_{F}-vZ$ is pseudo-effective}\}$. Then $$t(u)=\left\{\aligned &u\ \text{if $0\leqslant u\leqslant 1$}, \\ &1\ \text{if $1\leqslant u\leqslant \frac{3}{2}$}, \\ &4-2u\ \text{if $\frac{3}{2}\leqslant u\leqslant 2$}. \endaligned \right.$$ Moreover, it follows from [@AbbanZhuang Theorem 3.3] and [@Book Corollary 1.109] that $$1\geqslant\frac{A_X(\mathbf{F})}{S_X(\mathbf{F})}\geqslant\min\Bigg\{\frac{1}{S_X(F)},\frac{1}{S\big(W^{F}_{\bullet,\bullet};Z\big)}\Bigg\},$$ where $$S\big(W^{F}_{\bullet,\bullet};Z\big)= \frac{3}{26}\int\limits_0^2\big(P(u)\big\vert_{F}\big)^2\mathrm{ord}_{Z}\big(N(u)\big\vert_{F}\big)du+\frac{3}{26}\int\limits_0^2\int\limits_0^{t(u)}\big(P(u)\vert_{F}-vZ\big)^2dvdu.$$ In the proof of Lemma [Lemma 1](#lemma:beth-special-Eff-cone){reference-type="ref" reference="lemma:beth-special-Eff-cone"}, we computed $S_X(F)=\frac{25}{26}$, so $S(W^{F}_{\bullet,\bullet};Z)\geqslant 1$. But $$\begin{gathered} S\big(W^{F}_{\bullet,\bullet};Z\big)= \frac{3}{26}\int\limits_1^2\big(16-8u\big)^2(u-1)\mathrm{ord}_{Z}\big(\widetilde{\mathscr{S}}\big\vert_{F}\big)du+\\ +\frac{3}{26}\int\limits_0^1\int\limits_0^{u}4(3-u-v)(u-v)dvdu+\frac{3}{26}\int\limits_1^\frac{3}{2}\int\limits_0^{1}4(1-v)(4-2u-v)dvdu+\\ +\frac{3}{26}\int\limits_\frac{3}{2}^2\int\limits_0^{4-2u}4(1-v)(4-2u-v)dvdu= \frac{2}{13}\mathrm{ord}_{Z}\big(\widetilde{\mathscr{S}}\big\vert_{F}\big)+\frac{33}{104}\leqslant \frac{49}{104}<1,\end{gathered}$$ which is a contradiction ◻ **Lemma 1**. *One has $Z\not\subset \widetilde{\mathscr{S}}$.* *Proof.* Set $\mathbf{s}=F\vert_{\widetilde{\mathscr{S}}}$. Then $\mathbf{s}$ is a ruling of $\widetilde{\mathscr{S}}\cong\mathbb{P}^1\times\mathbb{P}^1$. Let $\mathbf{f}$ be a ruling of this surface such that $\mathbf{s}\cdot \mathbf{f}=1$ and $\mathbf{f}^2=0$. Then $\widetilde{E}_1\vert_{\widetilde{\mathscr{S}}}\sim \widetilde{E}_2\vert_{\widetilde{\mathscr{S}}}\sim \mathbf{s}$, $\widetilde{H}\vert_{\widetilde{\mathscr{S}}}\sim\mathbf{s}+2\mathbf{f}$, $\widetilde{\mathscr{S}}\vert_{\widetilde{\mathscr{S}}}\sim -\mathbf{s}+4\mathbf{f}$. Suppose $Z\subset \widetilde{\mathscr{S}}$. Then $Z\sim \mathbf{s}$ by Lemma [Lemma 1](#lemma:beth-G-action){reference-type="ref" reference="lemma:beth-G-action"}(b). For $u\in[0,\frac{3}{2}]$, let $P(u)$ be the positive part of the Zariski decomposition of $-K_{\widetilde{X}}-u\widetilde{\mathscr{S}}$ described in the proof of Lemma [Lemma 1](#lemma:beth-special-Eff-cone){reference-type="ref" reference="lemma:beth-special-Eff-cone"}, and let $N(u)$ be its negative part. Take $v\in\mathbb{R}_{\geqslant 0}$. Then $$P(u)\big\vert_{\widetilde{\mathscr{S}}}-vZ=\left\{\aligned &(1-v)\mathbf{s}+(6-4u)\mathbf{f} \ \text{if $0\leqslant u\leqslant 1$}, \\ &(3-2u-v)\mathbf{s}+(6-4u)\mathbf{f}\ \text{if $1\leqslant u\leqslant \frac{3}{2}$}, \endaligned \right.$$ and $$N(u)\big\vert_{\widetilde{\mathscr{S}}}=\left\{\aligned &u\mathbf{s}\ \text{if $0\leqslant u\leqslant 1$}, \\ &u\mathbf{s}+(u-1)\widetilde{E}_1\big\vert_{\widetilde{\mathscr{S}}}+(u-1)\widetilde{E}_1\big\vert_{\widetilde{\mathscr{S}}}\ \text{if $1\leqslant u\leqslant \frac{3}{2}$}. \endaligned \right.$$ Note that $Z\not\subset \widetilde{E}_1\cup \widetilde{E}_2$, because $Z$ is $\mathbf{G}$-invariant. For $u\in[0,\frac{3}{2}]$, set $t(u)=\mathrm{sup}\{v\in\mathbb{R}_{\geqslant 0}\ \vert\ \text{$P(u)\vert_{\widetilde{\mathscr{S}}}-vZ$ is pseudo-effective}\}$. Then $$t(u)=\left\{\aligned &1\ \text{if $0\leqslant u\leqslant 1$}, \\ &3-2u\ \text{if $1\leqslant u\leqslant \frac{3}{2}$}. \endaligned \right.$$ Moreover, it follows from [@AbbanZhuang Theorem 3.3] and [@Book Corollary 1.109] that $$1\geqslant\frac{A_X(\mathbf{F})}{S_X(\mathbf{F})}\geqslant\min\Bigg\{\frac{1}{S_X(\widetilde{\mathscr{S}})},\frac{1}{S\big(W^{\widetilde{\mathscr{S}}}_{\bullet,\bullet};Z\big)}\Bigg\},$$ where $$S\big(W^{\widetilde{\mathscr{S}}}_{\bullet,\bullet};Z\big)= \frac{3}{26}\int\limits_0^\frac{3}{2}\big(P(u)\big\vert_{\widetilde{\mathscr{S}}}\big)^2\mathrm{ord}_{Z}\big(N(u)\big\vert_{\widetilde{\mathscr{S}}}\big)du+\frac{3}{26}\int\limits_0^\frac{3}{2}\int\limits_0^{t(u)}\big(P(u)\vert_{\widetilde{\mathscr{S}}}-vZ\big)^2dvdu.$$ We know from the proof of Lemma [Lemma 1](#lemma:beth-special-Eff-cone){reference-type="ref" reference="lemma:beth-special-Eff-cone"} that $S_X(\widetilde{\mathscr{S}})=\frac{49}{54}$. Then $S(W^{\widetilde{\mathscr{S}}}_{\bullet,\bullet};Z)\geqslant 1$. But $$\begin{gathered} S\big(W^{\widetilde{\mathscr{S}}}_{\bullet,\bullet};Z\big)=\frac{3}{26}\int\limits_0^14u(3-2u)\mathrm{ord}_{Z}\big(\mathbf{s}\big)du+\frac{3}{26}\int\limits_1^\frac{3}{2}4u(3-2u)^2\mathrm{ord}_{Z}\big(\mathbf{s}\big)du+\\ +\frac{3}{26}\int\limits_0^1\int\limits_0^{1}4(1-v)(3-2u)dvdu+\frac{3}{26}\int\limits_1^\frac{3}{2}\int\limits_0^{3-2u}4(3-2u)(3-2u-v)dvdu=\frac{49\mathrm{ord}_{Z}\big(\mathbf{s}\big)+51}{104},\end{gathered}$$ which implies that $S(W^{\widetilde{\mathscr{S}}}_{\bullet,\bullet};Z)\leqslant \frac{25}{26}$. This is a contradiction. ◻ By Lemmas [Lemma 1](#lemma:beth-Z-not-F){reference-type="ref" reference="lemma:beth-Z-not-F"} and [Lemma 1](#lemma:beth-Z-not-S){reference-type="ref" reference="lemma:beth-Z-not-S"}, we have $Z\not\subset F\cup\widetilde{\mathscr{S}}$. But $Z\subset F\cup\widetilde{\mathscr{S}}$ by Lemma [Lemma 1](#lemma:beth-special-curves){reference-type="ref" reference="lemma:beth-special-curves"}. The obtained contradiction implies that the Fano threefold $X$ is K-polystable, completing the proof of Theorem [Theorem 1](#theorem:K-stability-non-isolated){reference-type="ref" reference="theorem:K-stability-non-isolated"} # Threefolds in the class $(\aleph)$ {#section:aleph} Let $X$ be the following complete intersection in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$: $$\big\{u_1x=v_1y, u_2z=v_2t, w^2+r(x+y)^2+r(z+t)^2=(2s+2)(xt+yz)+(2s-2)(xz+yt)\big\},$$ where $[r:s]\in\mathbb{P}^1$, and $([u_1:v_1],[u_2:v_2],[x:y:z:t:w])$ are coordinates on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$. For every $[r:s]$, the threefold $X$ is singular at the following two points $$\begin{aligned} \big([1:1],[-1:1],[0:0:-1:1:0]\big),\\ \big([-1:1],[1:1],[-1:1:0:0:0]\big),\end{aligned}$$ which are both isolated ordinary double singularities of the threefold $X$ if $[r:s]\ne [0:1]$. Furthermore, if $[r:s]=[\pm 1:1]$, then the threefold $X$ has an additional isolated double singularity at the point $$\big([1:1],[1:1],[1:1:\pm 1:\pm 1:0]\big).$$ Finally, if $[r:s]=[0:1]$, then $\mathrm{Sing}(X)$ consists of the following two curves: $$\begin{aligned} \big\{x=0&,y=0,w=0,u_1z+v_1t=0, u_2z-v_2t=0, v_1u_2+u_1v_2=0\big\},\\ \big\{z=0&,t=0,w=0,u_1x-v_1y=0, u_2x+v_2y=0, v_1u_2+u_1v_2=0\big\}.\end{aligned}$$ This is a very special case (see Section [5.3](#section:aleph-0){reference-type="ref" reference="section:aleph-0"} for more details). Let $Q$ be the quadric $\{w^2+r(x+y)^2+r(z+t)^2=(2s+2)(xt+yz)+(2s-2)(xz+yt)\}\subset\mathbb{P}^4$. Then $Q$ is singular $\iff$ $[r:s]=[\pm 1:1]$. If $[r:s]=[\pm 1:1]$, then $$\mathrm{Sing}(Q)=\big[1:1:\pm 1:\pm 1:0\big].$$ Let $\pi\colon{X}\to{Q}$ be the birational morphism that is induced by the projection ${X}\to\mathbb{P}^4$. Then $\pi$ is a blow up of the following two singular conics: $$\begin{aligned} {C}_1&=\{x=0,y=0,w^2+r(z+t)^2=0\},\\ {C}_2&=\{z=0,t=0,w^2+r(x+y)^2=0\}.\end{aligned}$$ These conics are contained in the smooth locus of the quadric $Q$, and they are reduced unless $[r:s]=[0:1]$. If $[r:s]=[0:1]$, both conics are double lines. As in Section [4](#section:beth){reference-type="ref" reference="section:beth"}, we see that $\mathrm{Aut}(X)$ contains the following commuting involutions: $$\begin{aligned} \tau_1\colon \big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big) &\mapsto \big([v_1:u_1],[v_2:u_2],[y:x:t:z:w]\big),\\ \tau_2\colon \big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big) &\mapsto \big([u_2:v_2],[u_1:v_1],[z:t:x:y:w]\big),\\ \tau_3\colon \big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big) &\mapsto \big([u_1:v_1],[u_2:v_2],[x:y:z:t:-w]\big).\end{aligned}$$ Note that $\pi$ is $\mathrm{Aut}(X)$-equivariant, and these involutions act on $\mathbb{P}^4$ as in Section [3](#section:GIT-2){reference-type="ref" reference="section:GIT-2"}: $$\begin{aligned} \tau_1\colon [x:y:z:t:w] &\mapsto [y:x:t:z:w],\\ \tau_2\colon [x:y:z:t:w] &\mapsto [z:t:x:y:w],\\ \tau_3\colon [x:y:z:t:w] &\mapsto [x:y:z:t:-w].\end{aligned}$$ These involutions generate a subgroup in $\mathrm{Aut}(X)$ isomorphic to $\boldsymbol{\mu}_2^3$. But $\mathrm{Aut}(X)$ is larger. In fact, it is infinite. Indeed, consider the monomorphism $\mathbb{C}^\ast\hookrightarrow\mathrm{PGL}_4(\mathbb{C})$ given by $$\lambda\mapsto\begin{pmatrix} \frac{1+\lambda}{2} & \frac{1-\lambda}{2} & 0 & 0 & 0\\ \frac{1-\lambda}{2} & \frac{1+\lambda}{2} & 0 & 0 & 0\\ 0 & 0 & \frac{\lambda +1}{2 \lambda} & \frac{\lambda -1}{2 \lambda} & 0\\ 0 & 0 & \frac{\lambda -1}{2 \lambda} & \frac{\lambda +1}{2 \lambda} & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}.$$ Note that this $\mathbb{C}^\ast$-action leaves $Q$ invariant. This gives a monomorphism $\mathbb{C}^\ast\hookrightarrow\mathrm{Aut}(Q)$. Since this $\mathbb{C}^\ast$-action also leaves both planes $\{x=0,y=0\}$ and $\{z=0,t=0\}$ invariant, it leaves both conics $C_1$ and $C_2$ invariant, because $$\begin{aligned} C_1&=Q\cap\{x=0,y=0\},\\ C_2&=Q\cap\{z=0,t=0\}.\end{aligned}$$ Thus the $\mathbb{C}^\ast$-action lifts to $X$. Let $\Gamma\cong\mathbb{C}^\ast$ be the corresponding subgroup in $\mathrm{Aut}(X)$. Let $\mathbf{G}$ be the subgroup in $\mathrm{Aut}(X)$ generated by $\Gamma$ together with $\tau_1$, $\tau_2$, $\tau_3$. Then $$\mathbf{G}\cong\big(\mathbb{C}^\ast\rtimes\boldsymbol{\mu}_2\big)\times\boldsymbol{\mu}_2,$$ because $\tau_1\in\Gamma$, and $\tau_3\circ\lambda=\lambda\circ\tau_3$ and $\tau_2\circ\lambda\circ\tau_2=\lambda^{-1}$ for every $\lambda\in\Gamma$. **Lemma 1**. *If $[r:s]\ne [\pm 1:1]$, then the Fano threefold $X$ does not have $\mathbf{G}$-fixed points. If $[r:s]=[\pm 1:1]$, then the only $\mathbf{G}$-fixed point in $X$ is the singular point $$\big([1:1],[1:1],[1:1:\pm 1:\pm 1:0]\big).$$* *Proof.* To simplify the $\mathbf{G}$-action, let us introduce new coordinates on $\mathbb{P}^4$ as follows: $$\left\{\aligned &\mathbf{x}=x-y, \\ &\mathbf{y}=x+y-z-t, \\ &\mathbf{z}=z-t, \\ &\mathbf{t}=x+y+z+t, \\ &\mathbf{w}=w. \endaligned \right.$$ In new coordinates, the defining equation of the quadric $Q$ simplifies to $$2\mathbf{x}\mathbf{z}+\frac{r+s}{2}\mathbf{y}^2+\frac{r-s}{2}\mathbf{t}^2+\mathbf{w}^2=0.$$ Moreover, we have ${C}_1=\{\mathbf{x}=0,\mathbf{y}+\mathbf{t}=0\}\cap Q$ and ${C}_2=\{\mathbf{z}=0,\mathbf{y}-\mathbf{t}=0\}\cap Q$. The involutions $\tau_2$ and $\tau_3$ act as $$\begin{aligned} \tau_2\colon \big[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:\mathbf{w}\big]&\mapsto\big[\mathbf{z}:-\mathbf{y}:\mathbf{x}:\mathbf{t}:\mathbf{w}\big],\\ \tau_3\colon \big[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:\mathbf{w}\big]&\mapsto\big[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:-\mathbf{w}\big],\end{aligned}$$ and the action of the group $\Gamma\cong\mathbb{C}^\ast$ simplifies to $$[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{t}:\mathbf{w}]\mapsto\Big[\lambda\mathbf{x}:\mathbf{y}:\frac{\mathbf{z}}{\lambda}:\mathbf{t}:\mathbf{w}\Big],$$ where $\lambda\in\mathbb{C}^\ast$. In new coordinates, the $\mathbf{G}$-fixed points in $\mathbb{P}^4$ are $$\begin{aligned} ,\\ [0:0:0:1:0],\\ [0:0:0:0:1].\end{aligned}$$ Note that $[0:1:0:0:0]\in Q\iff r+s=0$, and $[0:0:0:1:0]\in Q\iff r-s=0$. Moreover, we have $[0:0:0:0:1]\not\in Q$ for every $[r:s]\in\mathbb{P}^1$, so the assertion follows. ◻ Arguing as in the proof of Lemma [Lemma 1](#lemma:aleph-G-gixed-points){reference-type="ref" reference="lemma:aleph-G-gixed-points"}, we obtain the following result. **Lemma 1**. *Suppose that $[r:s]=[\pm 1:1]$. Let $O=([1:1],[1:1],[1:1:\pm 1:\pm 1:0])$, let $\varphi\colon\widetilde{X}\to X$ be the blow up of the point $O$, and let $F$ be the $\varphi$-exceptional surface. Then* - *$\varphi$ is $\mathbf{G}$-equivariant,* - *the group $\mathbf{G}$ acts faithfully on $F$,* - *$\mathrm{rk}\,\mathrm{Pic}^{\mathbf{G}}(F)=1$,* - *$F$ does not contain $\mathbf{G}$-fixed points,* - *$F\cong\mathbb{P}^1\times\mathbb{P}^1$ contains exactly two $\mathbf{G}$-invariant irreducible curves,\ which have bidegree $(1,1)$.* *Proof.* Let us only consider the case when $[r:s]=[1:1]$; the other case is analogous. We use the setup and notation introduced in the proof of Lemma [Lemma 1](#lemma:aleph-G-gixed-points){reference-type="ref" reference="lemma:aleph-G-gixed-points"}. It follows from this proof that we can $\mathbf{G}$-equivariantly identify the surface $F$ with the quadric surface $$\big\{\mathbf{w}^2+2\mathbf{x}\mathbf{z}+\mathbf{y}^2=0\big\}\subset\mathbb{P}^3,$$ where we consider $[\mathbf{x}:\mathbf{y}:\mathbf{z}:\mathbf{w}]$ as coordinates on $\mathbb{P}^3$. Now our claims are easy to check. For instance, if $Z$ is a $\mathbf{G}$-invariant irreducible curve in $F$, then $\Gamma$ acts faithfully on $Z$, which implies that $\tau_2$ fixes a point on $Z$. Then, analyzing $\langle\tau_2\rangle$-fixed points in $F$, we see that either $Z=F\cap\{\mathbf{y}=0\}$ or $Z=F\cap\{\mathbf{w}=0\}$ as claimed. ◻ We have the following $G$-equivariant diagram: $$\xymatrix{ &X\ar@{->}[dl]_{\pi}\ar@{->}[dr]^{\eta}&\\% {Q}&&\mathbb{P}^1\times\mathbb{P}^1}$$ where $\eta$ is the conic bundle that is given by the natural projection $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4\to \mathbb{P}^1\times\mathbb{P}^1$. Let $\Delta$ be its discriminant curve in $\mathbb{P}^1\times\mathbb{P}^1$. Then $\Delta=\Delta_1+\Delta_2$, where $$\begin{aligned} \Delta_1&=\big\{(r-s+1)u_1u_2+(r-s-1)u_1v_2+(r-s-1)u_2v_1+(r-s+1)v_1v_2=0\big\},\\ \Delta_2&=\big\{(r+s-1)u_1u_2+(r+s+1)u_1v_2+(r+s+1)u_2v_1+(r+s-1)v_1v_2=0\big\}.\end{aligned}$$ If $[r:s]\ne[0:1]$, then they meet transversally at $([1:-1],[1:1])$ and $([1:1],[1:-1])$. The curves $\Delta_1$ and $\Delta_2$ are smooth for $[r:s]\ne [\pm 1:1]$. If $[r:s]=[1:1]$, then $$\begin{aligned} \Delta_1&=\big\{(u_1-v_1)(u_2-v_2)=0\big\},\\ \Delta_2&=\big\{u_1u_2+3u_1v_2+3u_2v_1+v_1v_2=0\big\}.\end{aligned}$$ Similarly, if $[r:s]=[1:-1]$, then $$\begin{aligned} \Delta_1&=\big\{3u_1u_2+u_1v_2+u_2v_1+3v_1v_2=0\big\},\\ \Delta_2&=\big\{(u_1-v_1)(u_2-v_2)=0\big\}.\end{aligned}$$ Finally, if $[r:s]=[0:1]$, then $\Delta_1=\Delta_2$. We will deal with this case in Section [5.3](#section:aleph-0){reference-type="ref" reference="section:aleph-0"}. *Remark 1*. The only $\mathbf{G}$-fixed points in $\mathbb{P}^1\times\mathbb{P}^1$ are $([1:1],[1:1])$ and $([1:-1],[1:-1])$. Suppose $[r:s]\ne[0:1]$. Let $E_1$ and $E_2$ be the $\langle\tau_3\rangle$-irreducible reducible $\pi$-exceptional surfaces such that $\pi(E_1)=C_1$ and $\pi(E_2)=C_2$. Set $H=\pi^*(\mathcal{O}_Q(1))$. Then $$\mathrm{Pic}^{\mathbf{G}}(X)=\mathrm{Cl}^{\mathbf{G}}(X)=\mathbb{Z}[H]\oplus\mathbb{Z}[E_1+E_2].$$ This easily follows from Lemma [Lemma 1](#lemma:ODP-G-action-aleph){reference-type="ref" reference="lemma:ODP-G-action-aleph"}, since $C_1+C_2$ is $\mathbf{G}$-irreducible. **Lemma 1**. *Let $S$ be any $\mathbf{G}$-invariant irreducible surface in $X$. Then $\beta(S)>0$.* *Proof.* The conic bundle $\eta\colon X\to\mathbb{P}^1\times\mathbb{P}^1$ is given by the linear system $|2H+E_1+E_2|$. Therefore, arguing as in the proof of Lemma [Lemma 1](#lemma:Eff-cone){reference-type="ref" reference="lemma:Eff-cone"}, we see that $$S\sim a(E_1+E_2)+b(2H-E_1-E_2)+cH$$ for some non-negative integers $a$, $b$, $c$. So, arguing exactly as in the proof of Corollary [Corollary 1](#corollary:divisorial-stability){reference-type="ref" reference="corollary:divisorial-stability"}, we obtain the required assertion. ◻ We conclude this section with the following technical lemma: **Proposition 1**. *Let $S$ be a smooth surface in $|H-E_1|$ or $|H-E_2|$. Then $$\delta_P(X)\geqslant\frac{104}{99}$$ for every point $P\in S$ such that $P\not\in E_1\cup E_2$.* *Proof.* The proof is identical to the proof of Proposition [Proposition 1](#proposition:delta-dP4){reference-type="ref" reference="proposition:delta-dP4"}. ◻ ## Threefolds with two singular points. {#section:aleph-two-points} As before, we continue to use the notation and assumptions introduced earlier in this section. Suppose that $$[r:s]\ne [\pm 1:1]$$ and $[r:s]\ne [0:1]$. The goal of this subsection is to prove the following. **Theorem 1**. *Let $X$ be a singular Fano threefold from class $(\aleph)$ with two singular points. Then $X$ is K-polystable.* The proof of this theorem follows the method used to prove Theorem [Theorem 1](#theorem:K-stability-smooth){reference-type="ref" reference="theorem:K-stability-smooth"}. We begin with a technical lemma. **Lemma 1**. *Suppose that $X$ is a Fano threefold from class $(\aleph)$ with isolated singularities. Let $\mathbf{F}$ be a $\mathbf{G}$-invariant prime divisor over $X$ with $\beta(\mathbf{F}) \leqslant 0$, and let $Z$ be its center on $X$. Suppose that $Z$ is not a $\mathbf{G}$-fixed singular point of the threefold $X$. Then $Z$ is an irreducible curve such that $\eta(Z)=([1:1],[1:1])$ or $\eta(Z)=([1:-1],[1:-1])$.* *Proof.* The proof is identical to the proof of Lemma [Lemma 1](#lemma:G-invariant-center){reference-type="ref" reference="lemma:G-invariant-center"}, using Lemmas [Lemma 1](#lemma:aleph-G-gixed-points){reference-type="ref" reference="lemma:aleph-G-gixed-points"} and [Lemma 1](#lemma:divisorial-stability-aleph){reference-type="ref" reference="lemma:divisorial-stability-aleph"}, and also Proposition [Proposition 1](#proposition:delta-dP4-aleph){reference-type="ref" reference="proposition:delta-dP4-aleph"}. ◻ Now we prove Theorem [Theorem 1](#theorem:K-stability-aleph-two-singularities){reference-type="ref" reference="theorem:K-stability-aleph-two-singularities"}. Suppose that the singular threefold $X$ is not K-polystable. By [@Zhuang Corollary 4.14], there is a $\mathbf{G}$-invariant prime divisor $\mathbf{F}$ over $X$ such that $\beta(\mathbf{F})\leqslant 0$. Let $Z$ be the center of $\mathbf{F}$ on $X$. We seek a contradiction. By Lemma [Lemma 1](#lemma:aleph-G-gixed-points){reference-type="ref" reference="lemma:aleph-G-gixed-points"}, the threefold $X$ does not contain $\mathbf{G}$-fixed singular points. Furthermore, the fibers of the conic bundle $\eta$ over the points $([1:1],[1:1])$ and $([1:-1],[1:-1])$ are smooth. So, applying Lemma [Lemma 1](#lemma:G-invariant-center){reference-type="ref" reference="lemma:G-invariant-center"}, we see that $Z$ is the fiber of the conic bundle $\eta$ over one of the points $([1:1],[1:1])$ or $([1:-1],[1:-1])$. Let $S$ be the unique surface in $|H-E_1|$ that contains $Z$. Observe the following: - if $\eta(Z)=([1:1],[1:1])$, then $S$ is cut out on $X$ by $u_1-v_1=0$, and $$\mathrm{Sing}(S)=\big([1:1],[-1:1],[0:0:-1:1:0]\big)\in\mathrm{Sing}(X),$$ the surface $\pi(S)$ is a quadric cone with vertex $[0:0:-1:1:0]=\mathrm{Sing}(C_1)$, the intersection $\pi(S)\cap C_2$ consists of two distinct smooth points of the conic $C_2$, and $\pi$ induces a birational morphism $S\to\pi(S)$ that blows up these two points; - if $\eta(Z)=([1:-1],[1:-1])$, then $S$ is cut out on $X$ by $u_1+v_1=0$, and $$\mathrm{Sing}(S)=\big([1:-1],[1,1],[-1:1:0:0:0]\big)\in\mathrm{Sing}(X),$$ the quadric surface $\pi(S)$ is smooth, $\mathrm{Sing}(C_2)\in\pi(S)$, and $\pi$ induces a birational morphism $S\to\pi(S)$, which is a weighted blow up of the point $\pi(S)\cap C_2$. In both cases, the surface $S$ is a singular sextic del Pezzo surface with one ordinary double point, and the curve $Z$ is contained in the smooth locus of the surface $S$. Let us apply Abban--Zhuang theory [@AbbanZhuang; @Book] to the flag $Z\subset S$. Take $u\in\mathbb{R}_{\geqslant 0}$. Set $$P(u)=\left\{\aligned &(3-u)H-(1-u)E_1-E_2\ \text{if $0\leqslant u\leqslant 1$}, \\ &(3-u)H-E_2\ \text{if $1\leqslant u\leqslant 2$}, \endaligned \right.$$ and $$N(u)=\left\{\aligned &0\ \text{if $0\leqslant u\leqslant 1$}, \\ &(u-1)E_1\ \text{if $1\leqslant u\leqslant 2$}. \endaligned \right.$$ Then it follows from the proof of Proposition [Proposition 1](#proposition:delta-dP4){reference-type="ref" reference="proposition:delta-dP4"} that $P(u)$ and $N(u)$ are the positive and the negative parts of the Zariski decomposition of $-K_X-uS$ for $u\in[0,2]$, respectively. Moreover, if $u>2$, then $-K_X-uS$ is not pseudo-effective. This gives $S_X(S)=\frac{3}{4}$. Now, we take $v\in\mathbb{R}_{\geqslant 0}$ and consider the divisor $P(u)\vert_{S}-vZ$. For $u\in[0,2]$, set $$t(u)=\mathrm{sup}\Big\{v\in\mathbb{R}_{\geqslant 0}\ \big\vert\ \text{the~divisor $P(u)\big\vert_{S}-vZ$ is pseudo-effective}\Big\}.$$ Let $P(u,v)$ be the positive part of the Zariski decomposition of $P(u)\vert_{S}-vZ$ for $v\leqslant t(u)$. Note that $Z\not\subset\mathrm{Supp}(N(u))$ for every $u\in[0,2]$. Thus, it follows from [@AbbanZhuang; @Book] that $$\frac{A_X(\mathbf{F})}{S_X(\mathbf{F})}\geqslant\min\Bigg\{\frac{1}{S_X(S)},\frac{1}{S\big(W^S_{\bullet,\bullet};Z\big)}\Bigg\},$$ where $$S\big(W^S_{\bullet,\bullet};Z\big)=\frac{3}{(-K_X)^3}\int\limits_0^2\int\limits_0^{t(u)}\big(P(u,v)\big)^2dvdu.$$ Since $S_X(S)=\frac{3}{4}$ and $\beta(\mathbf{F})=A_X(\mathbf{F})-S_X(\mathbf{F})\leqslant 0$, we conclude that $S(W^S_{\bullet,\bullet};Z)\geqslant 1$. Let us compute $S(W^S_{\bullet,\bullet};Z)$. Set $\mathbf{e}=E_2\vert_{S}$. Then $\mathbf{e}^2=-2$ and $Z\cdot \mathbf{e}=2$. We have $$P(u)\big\vert_{S}-vZ=\left\{\aligned &(2-v)Z+\mathbf{e}\ \text{if $0\leqslant u\leqslant 1$}, \\ &(3-u-v)Z+(2-u)\mathbf{e}\ \text{if $1\leqslant u\leqslant 2$}, \endaligned \right.$$ which gives $$t(u)=\left\{\aligned &2\ \text{if $0\leqslant u\leqslant 1$}, \\ &3-u\ \text{if $1\leqslant u\leqslant 2$}, \endaligned \right.$$ because $\mathrm{Supp}(\mathbf{e})$ is contractible. Furthermore, if $u\in[0,1]$, then $$P(u,v)=\left\{\aligned &(2-v)Z+\mathbf{e}\ \text{if $0\leqslant v\leqslant 1$}, \\ &(2-v)\big(Z+\mathbf{e}\big)\ \text{if $1\leqslant v\leqslant 2$}, \endaligned \right.$$ which gives $$\big(P(u,v)\big)^2=\left\{\aligned &6-4v\ \text{if $0\leqslant v\leqslant 1$}, \\ &2(v-2)^2\ \text{if $1\leqslant v\leqslant 2$}. \endaligned \right.$$ Similarly, if $u\in[1,2]$, then $$P(u,v)=\left\{\aligned &(3-u-v)Z+(2-u)\mathbf{e}\ \text{if $0\leqslant v\leqslant 1$}, \\ &(3-u-v)\big(Z+\mathbf{e}\big)\ \text{if $1\leqslant v\leqslant 3-u$}, \endaligned \right.$$ which gives $$\big(P(u,v)\big)^2=\left\{\aligned &2(2-u)(4-u-2v)\ \text{if $0\leqslant v\leqslant 1$}, \\ &2(3-u-v)^2\ \text{if $1\leqslant v\leqslant 3-u$}. \endaligned \right.$$ Integrating, we obtain $S(W^S_{\bullet,\bullet};Z)=\frac{3}{4}$. But we already proved earlier that $S(W^S_{\bullet,\bullet};Z)\geqslant 1$. This is a contradiction. Hence, we see that $X$ is K-polystable. ## Threefolds with three singular points. {#section:aleph-node} We continue to use the notation and assumptions introduced at the beginning of this section. Suppose that $[r:s]=[\pm 1:1]$. The goal of this subsection is to prove the following. **Theorem 1**. *Let $X$ be a singular threefold from class $(\aleph)$ with three singular points. Then $X$ is K-polystable.* The proof follows the method used to prove Theorem [Theorem 1](#theorem:K-stability-isolated){reference-type="ref" reference="theorem:K-stability-isolated"}. After changing coordinates, we may assume that $[r:s]=[1:1]$. Then $$X=\big\{u_1x=v_1y,u_2z=v_2t,w^2+(x+y)^2+(z+t)^2=4(xt+yz)\big\}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4,$$ and the singular locus of the threefold $X$ consists of the following three points: $$\begin{aligned} \big([1:1],[-1:1]&,[0:0:-1:1:0]\big),\\ \big([-1:1],[1:1]&,[-1:1:0:0:0]\big),\\ \big([1:1],[1:1]&,[1:1:1:1:0]\big),\end{aligned}$$ which are isolated ordinary double singularities of the threefold $X$. We have $$Q=\big\{w^2+(x+y)^2+(z+t)^2=4(xt+yz)\big\}\subset\mathbb{P}^1\times\mathbb{P}^4,$$ the surface $Q$ is a cone with vertex $[1:1:1:1:0]$, and $\pi$ is a blow up of the conics $$\begin{aligned} {C}_1&=\{x=0,y=0,w^2+(z+t)^2=0\},\\ {C}_2&=\{z=0,t=0,w^2+(x+y)^2=0\}.\end{aligned}$$ Note that $C_1$ and $C_2$ are reduced singular conics that do not contain $\mathrm{Sing}(Q)$. The fiber of the conic bundle $\eta$ over the point $([1:1],[1:1])$ is a reducible conic $L_1+L_2$, where $L_1$ and $L_2$ are smooth irreducible curves such that $$\begin{aligned} \pi(L_1)&=\{x=y,z=t,w+2i(x-z)=0\},\\ \pi(L_2)&=\{x=y,z=t,w-2i(x-z)=0\}.\end{aligned}$$ Note that $\pi(L_1)$ and $\pi(L_2)$ are the only lines in $Q$ that intersect both conics $C_1$ and $C_2$, and each of them intersects exactly one irreducible component of the conics $C_1$ and $C_2$. Observe that the intersection $L_1\cap L_2$ is the singular point $([1:1],[1:1],[1:1:1:1:0])$. We denote this singular point by $O$. **Lemma 1**. *Let $\mathbf{F}$ be a $\mathbf{G}$-invariant prime divisor $\mathbf{F}$ over the threefold $X$ with $\beta(\mathbf{F})\leqslant 0$, and let $Z$ be its center on the threefold $X$. Then $Z=O$.* *Proof.* By Lemma [Lemma 1](#lemma:aleph-G-gixed-points){reference-type="ref" reference="lemma:aleph-G-gixed-points"}, $O$ is the unique $\mathbf{G}$-fixed singular point in $X$. Thus, by Lemma [Lemma 1](#lemma:G-invariant-center-aleph){reference-type="ref" reference="lemma:G-invariant-center-aleph"}, we see that one of the following three cases holds: - either $Z=O$, - or $Z$ is an irreducible component of the fiber of $\eta$ over $([1:1],[1:1])$, - or $Z$ is an irreducible component of the fiber of $\eta$ over $([1:-1],[1:-1])$. In the second case, either $Z=L_1$ or $Z=L_2$, which is impossible, because neither of these curves is $\mathbf{G}$-invariant. Moreover, the fiber of $\eta$ over $([1:-1],[1:-1])$ is irreducible and smooth. So, either $Z=O$ or $Z$ is the fiber of the conic bundle $\eta$ over $([1:-1],[1:-1])$. Suppose that $Z$ is the fiber of $\eta$ over $([1:-1],[1:-1])$. Set $S=X\cap\{u_1+v_1=0\}$. Then $S$ is a singular sextic del Pezzo surface that contains $Z$. Moreover, we have $$\mathrm{Sing}(S)=\big([1:-1],[1,1],[-1:1:0:0:0]\big)\in\mathrm{Sing}(X),$$ this point is an ordinary double point of the surface $S$, the quadric surface $\pi(S)$ is smooth, and $\pi$ induces a weighted blow up $S\to\pi(S)$ of the single intersection point $\pi(S)\cap C_2$. Now, arguing as in the end of Section [5.1](#section:aleph-two-points){reference-type="ref" reference="section:aleph-two-points"}, we obtain a contradiction. ◻ Now we are ready to prove that $X$ is K-polystable. Suppose for a contradiction that the singular Fano threefold $X$ is not K-polystable. Then there is a $\mathbf{G}$-invariant prime divisor $\mathbf{F}$ over $X$ such that $\beta(\mathbf{F})\leqslant 0$. By Lemma [Lemma 1](#lemma:K-stability-3-points-1){reference-type="ref" reference="lemma:K-stability-3-points-1"}, its center on $X$ is the point $O$. Let $\varphi\colon\widetilde{X}\to X$ be the blow up of the point $O$, and let $F$ be the $\varphi$-exceptional divisor, and let $\widetilde{Z}$ be the center on $\widetilde{X}$ of the prime divisor $\mathbf{F}$. Then $\varphi$ is $\mathbf{G}$-equivariant, and - either $\widetilde{Z}=F$ and $\beta(F)=\beta(\mathbf{F})$, or - $\widetilde{Z}$ is a $\mathbf{G}$-invariant irreducible curve in $F$, or - $\widetilde{Z}$ is a $\mathbf{G}$-fixed point in $F$. Case $(\mathrm{3})$ is impossible by Lemma [Lemma 1](#lemma:ODP-G-action-aleph){reference-type="ref" reference="lemma:ODP-G-action-aleph"}. Moreover, in case $(\mathrm{2})$ we have two choices for $Z$, as described in the proof of Lemma [Lemma 1](#lemma:ODP-G-action-aleph){reference-type="ref" reference="lemma:ODP-G-action-aleph"}. Now, arguing exactly as in the proof of Lemma [Lemma 1](#lemma:K-stability-isolated-2){reference-type="ref" reference="lemma:K-stability-isolated-2"}, we obtain a contradiction. This proves that $X$ is K-polystable. ## Threefolds with non-isolated singularities. {#section:aleph-0} It remains to study the singular threefold $X$ from class $(\aleph)$ with $[r:s]=[0:1]$. Recall that $X$ is the complete intersection $$\big\{u_1x=v_1y,u_2z=v_2t,w^2=4(xt+yz)\big\}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4,$$ where $([u_1:v_1],[u_2:v_2],[x:y:z:t:w])$ are coordinates on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$. The goal of this subsection is to prove the following. **Theorem 1**. *The threefold $X$ from class $(\aleph)$ described above, which has non-isolated singularities, is K-polystable.* Let $$Q=\big\{w^2=4(xt+yz)\big\}\subset\mathbb{P}^4,$$ and let $\pi\colon{X}\to{Q}$ be the birational morphism that is induced by the projection ${X}\to\mathbb{P}^4$. Then $Q$ is a smooth quadric threefold, and the morphism $\pi$ is a weighted blow up with weights $(1,2)$ of the lines ${L}_1=\{x=0,y=0,w=0\}$ and ${L}_2=\{z=0,t=0,w=0\}$. Note that the singular locus of the threefold ${X}$ consists of two disjoint smooth curves $$\begin{aligned} \{v_1u_2+u_1v_2=0,u_1z+v_1t=0,u_2z=v_2t,x=0,y=0,w=0\},\\ \{v_1u_2+u_1v_2=0,u_2x+v_2y=0,u_1x=v_1y,z=0,t=0,w=0\}.\end{aligned}$$ Let ${S}$ be the strict transform on ${X}$ of the surface in ${Q}$ that is cut out on $Q$ by $w=0$, let ${E}_1$ and ${E}_2$ be the two $\pi$-exceptional surfaces such that $\pi({E}_1)={L}_1$ and $\pi({E}_2)={L}_2$, and let $\iota$ be the involution in $\mathrm{Aut}(X)$ given by $$\iota \colon \big([u_1:v_1],[u_2:v_2],[x:y:z:t:w]\big)\mapsto \big([u_1:v_1],[u_2:v_2],[x:y:z:t:-w]\big).$$ We set $W=X/\iota$ and let $\theta\colon X\to W$ be the quotient map. Then $W$ is a smooth threefold and $\theta$ is ramified in $S\cup E_1\cup E_2$. Moreover, we have the following commutative diagram: $$\xymatrix{ {X}\ar@{->}[d]_{\pi}\ar@{->}[rr]^{\theta}&&{W}\ar@{->}[d]^{\varpi}\\% {Q}\ar@{->}[rr]_{\vartheta}&&\mathbb{P}^3}$$ where $\vartheta$ is the double cover ramified in $\pi(S)$ given by $[x:y:z:t:w]\mapsto[x:y:z:t]$ and $\varpi$ is the blow up of the disjoint lines: $\{x=y=0\}$ and $\{z=t=0\}$. Note that $W$ is the unique smooth Fano threefold in the family . The threefold ${X}$ can be also obtained via the following commutative diagram: $$\xymatrix{ {V}\ar@{->}[d]_{\alpha}&&&{Y}\ar@{->}[lll]_{\beta}\\% {Q}&&&{X}\ar@{->}[lll]_{\pi}\ar@{->}[u]_{\gamma}}$$ where $\alpha$ blows up the lines ${L}_1$ and ${L}_2$, $\beta$ blows up the $(-1)$-curves in the $\alpha$-exceptional surfaces (which are both isomorphic to $\mathbb{F}_1$), and $\gamma$ contracts the $\alpha$-exceptional surfaces. Note that ${V}$ is the smooth Fano threefold in the family . One can check that $$\mathrm{Aut}\big({X}\big)\cong\mathrm{Aut}\big({V}\big)\cong\Big(\mathbb{C}^\ast\rtimes\boldsymbol{\mu}_2\Big)\times\mathrm{PGL}_2\big(\mathbb{C}\big),$$ and ${S}$ is the only $\mathrm{Aut}({X})$-invariant prime divisor over ${X}$. Set ${H}=\pi^*(\mathcal{O}_{Q}(1))$, and observe that $-K_{{X}}\sim 3{H}-2{E}_1-2{E}_2$ and $S\sim {H}-{E}_1-{E}_2$. Then, arguing as in the proof of Lemma [Lemma 1](#lemma:beth-special-Eff-cone){reference-type="ref" reference="lemma:beth-special-Eff-cone"}, we get $$\begin{gathered} \beta({S})=1-\frac{1}{26}\int\limits_{0}^{2}\big(-K_{X}-u{S}\big)^3du-\frac{1}{26}\int\limits_{2}^{3}\big(-K_{X}-u{S}-(u-2)({E}_1+{E}_2)\big)^3du=\\ =1-\frac{1}{26}\int\limits_{0}^{2}\big((3-u){H}+(u-2)({E}_1+{E}_2)\big)^3du-\frac{1}{26}\int\limits_{2}^{3}\big((3-u){H}\big)^3du=\\ =1-\frac{1}{26}\int\limits_{0}^{2}3u^2-18u+26du-\frac{1}{26}\int\limits_{2}^{3}2(3-u)^3du=1-\frac{49}{52}=\frac{3}{52}>0.\end{gathered}$$ Thus it follows from [@Zhuang Corollary 4.14] that ${X}$ is K-polystable. # Proof of Main Theorem {#section:K-moduli} In Section [3](#section:GIT-2){reference-type="ref" reference="section:GIT-2"} we presented the parameter space $T=\mathbb{P}^4$ whose open part parametrizes all K-polystable smooth Fano threefolds in the family  3.10, and described a $(\mathbb{C}^\ast)^2$-action on $T$ such that the GIT quotient $T/\!/(\mathbb{C}^\ast)^2\cong\mathbb{P}^1\times\mathbb{P}^1$ parametrizes the Fano threefolds from the Main Theorem, which include all K-polystable smooth Fano threefolds in this family. Now, arguing as in [@Gokova Section 6], we obtain the assertion of the Main Theorem. Namely, our construction implies that there is a $\mathbb{Q}$-Gorenstein family of Fano varieties over the GIT moduli stack $[\mathring{T}/(\mathbb{C}^\ast)^2]$ that contains all K-polystable smooth members of the family , where $\mathring{T}$ is the GIT semistable open subset of the parameter space $T$. In Sections [4](#section:beth){reference-type="ref" reference="section:beth"} and [5](#section:aleph){reference-type="ref" reference="section:aleph"}, we proved that all GIT-polystable objects in our family are in fact K-polystable Fano threefolds. So, we obtain a morphism of stacks $$\big[\mathring{T}/(\mathbb{C}^\ast)^2\big]\longrightarrow\mathcal{M}^{\mathrm{Kss}}_{3,26},$$ where $\mathcal{M}^{\mathrm{Kss}}_{3,26}$ is the K-moduli stack parametrizing K-semistable Fano 3-folds of degree $26$. By [@Alper Theorem 6.6], this morphism descends to good moduli spaces $$\mathring{T}/(\mathbb{C}^\ast)^2\longrightarrow M^{\mathrm{Kps}}_{3,26},$$ where points of $M^{\mathrm{Kps}}_{3,26}$ parameterize K-polystable Fano threefolds of degree $26$. Since both good moduli spaces are known to be proper, we see that the image is an irreducible component of the K-moduli space. This implies the Main Theorem. We further note that the morphism $\mathring{T}/(\mathbb{C}^\ast)^2\to M^{\mathrm{Kps}}_{3,26}$ factors as $$\xymatrix{ \mathring{T}/(\mathbb{C}^\ast)^2\ar@{->}[rrrr]^(0.55){\text{quotient by an involution}}&&&&\mathbb{P}^2\ar@{->}[rr]^{\varsigma}&&M^{\mathrm{Kps}}_{3,26}},$$ where the map $\mathring{T}/(\mathbb{C}^\ast)^2\longrightarrow\mathbb{P}^2$ is described in Section [3.3](#section:quotient-space){reference-type="ref" reference="section:quotient-space"}. Then $\varsigma(\mathbb{P}^2)$ is the irreducible component of the moduli space $M^{\mathrm{Kps}}_{3,26}$ that contains K-polystable smooth Fano threefolds in the deformation family . 3 H. Abban, I. Cheltsov, E. Denisova, E. Etxabarri-Alberdi,\ A.-S. Kaloghiros, D. Jiao, J. Martinez-Garcia, T. Papazachariou, *One-dimensional components in\ the K-moduli of smooth Fano 3-folds*, to appear. H. Abban, Z. Zhuang, *K-stability of Fano varieties via admissible flags*,\ Forum of Mathematics Pi **10** (2022), 1--43. J. Alper, *Good moduli spaces for Artin stacks*,\ Annales de l'Institut Fourier **63** (2013), 2349--2402. C. Araujo, A.-M. Castravet, I. Cheltsov, K. Fujita, A.-S. Kaloghiros, J. Martinez-Garcia,\ C. Shramov, H. Süß, N. Viswanathan, *The Calabi problem for Fano threefolds*,\ Lecture Notes in Mathematics, Cambridge University Press, 2023. I. Cheltsov, T. Duarte Guerreiro, K. Fujita,\ I. Krylov, J. Martinez-Garcia, *K-stability of Casagrande--Druel varieties*, to appear. K. Fujita, *A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties*,\ Journal für die Reine und Angewandte Mathematik **751** (2019), 309--338. C. Li, *K-semistability is equivariant volume minimization*,\ Duke Mathematical Journal **166** (2017), 3147--3218. Z. Zhuang, *Optimal destabilizing centers and equivariant K-stability*,\ Inventiones Mathematicae **226** (2021), 195--223.

arxiv_math

--- abstract: | We study the behavior of Frobenius operators on smooth proper pushforwards of prismatic $F$-crystals. In particular we show that the $i$-th pushforward has its Frobenius height increased by at most $i$. Our proof crucially uses the notion of prismatic $F$-gauges introduced by Drinfeld and Bhatt--Lurie and its relative version, and we give a self-contained treatment without using the stacky formulation. address: - "H. Guo: Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany" - "S. Li: Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China" author: - Haoyang Guo - Shizhang Li bibliography: - template.bib title: Frobenius height of prismatic cohomology with coefficients --- # Introduction ## Background and main result {#background-and-main-result .unnumbered} Fix a prime number $p$, throughout this paper we let $K$ be a $p$-adic discretely valued field with perfect residue field $k$, and let $\mathcal{O}_K$ be the ring of integers. For smooth proper $p$-adic formal schemes over $\mathcal{O}_K$, the prismatic cohomology introduced by Bhatt--Scholze in [@BS19] (see also [@BMS18] and [@BMS19]) can be regarded as the universal cohomology theory. Recently there has been many works concerning "general coefficients" for this cohomology theory, known as *prismatic $F$-crystals*, to name a few: [@MT20], [@Wu21], [@BS21], [@DuLiu21], [@MW21], [@DLMS] and [@GR22]. In this paper, we study the behavior of these general coefficients under smooth proper pushforwards. More precisely, we are interested in estimating the height of the induced Frobenius operator. Our main result is the following (combining and ): **Theorem 1** (Main Theorem). *Let $f \colon X\to Y$ be a smooth proper morphism of smooth $p$-adic formal schemes over $\mathcal{O}_K$ of relative equidimension $n$, and let $(\mathcal{E},\varphi_{\mathcal{E}})$ be an $\mathcal{I}$-torsionfree coherent prismatic $F$-crystal over $X$ with Frobenius height in $[a,b]$.* *Then for each integer $i$, the $i$-th relative prismatic cohomology $R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}$ induces a coherent prismatic $F$-crystal over $Y_{\mathlarger{\mathbbl{\Delta}}}$, whose $\mathcal{I}$-torsionfree quotient has Frobenius height in $[a+\max\{0 ,i-n\}, b+\min\{i,n\}]$.* *Moreover there exists a Verschiebung operator $$\psi_i \colon \mathcal{I}^{\otimes (i+b)}\otimes_{\mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}} R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E} \longrightarrow \varphi_{{\mathlarger{\mathbbl{\Delta}}}_Y}^* R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E},$$ which is inverse to the Frobenius operator of $R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}$ up to multiplying $\mathcal{I}^{\otimes (i+b)}$.* According to [@DLMS Theorem 1.3] or [@GR22 Theorem A], any $\mathbb{Z}_p$-crystalline local system comes from an $I$-torsionfree coherent prismatic $F$-crystal. Moreover, in [@GR22 §8] the first named author and Reinecke have described alternative approaches to showing that the induced Frobenius operator on $Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}$ is an isogeny. The approach in this paper has the advantage that we can allow torsions in coefficients and cohomology, and we give precise bounds on Frobenius height. Notice that prismatic $F$-crystals can give rise to usual crystalline $F$-crystals in characteristic $p$ as well as local systems in characteristic $0$. Our estimate of Frobenius height is compatible with the work of Kedlaya [@Ked06 Thm. 6.7.1] bounding Frobenius slopes of rigid cohomology with coefficients as well as the work of Shimizu [@Shimizu18 Theorem 5.10] controlling generalized Hodge--Tate weights of cohomology of local systems. In the case of constant coefficient $\mathcal{E} = \mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}, X}$, this result has been established by Bhatt--Scholze in [@BS19 §15]. In fact, our proof is in spirit a generalization of theirs: We found a canonical way to promote the $F$-crystal to a "Nygaard-filtered $F$-crystal", a.k.a. *prismatic $F$-gauge*, whose graded pieces are controlled and can be estimated by coherent cohomologies. The framework of prismatic $F$-gauges had been laid down by Drinfeld [@Drin20] and Bhatt--Lurie [@BL22a], [@BL22b]. Many detailed discussions can be found in lecture notes by Bhatt [@Bha23], and our paper is inspired by results therein. However we choose to not follow their stacky approach, and instead just discuss prismatic ($F$-)gauges and relative ($F$-)gauges using slightly more concrete terms. In the remainder of this introduction, let us briefly indicate the proof idea, which will also allow us to summarize various sections of this article. ## Constructions {#constructions .unnumbered} As some preliminary discussions, in we explain the existence of a standard $t$-structure on the category of prismatic ($F$-)crystals on $X$ (see ), so that we have well-defined functors $R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *}$ sending coherent crystals on $X$ to those on $Y$, and expressions like "coherent" and "$I$-torsionfree" appearing in the main theorem above make sense. Here comes the main construction. In , we give a definition of *(absolute) ($F$-)gauges* on the formal spectra of qrsp algebras $S$ and show that the assignment $S \mapsto {\mathrm{(F\text{-})Gauge}}({\mathrm{Spf}}(S))$ is a quasi-syntomic sheaf (see ). By the unfolding process (c.f. [@BMS19 Proposition 4.31]), this gives rise to the notion of ($F$-)gauges on our $X$. Then we show the following: **Theorem 1** (= ). *There is a functor $$\Pi_X \colon \{I\text{-torsionfree coherent } F\text{-crystals on } X\} \longrightarrow \{\text{coherent } F\text{-gauges on } X\},$$ given by equipping an $F$-crystal with the "saturated Nygaardian filtration".* This is inspired by the special case of $X={\mathrm{Spf}}(\mathcal{O}_K)$ in [@Bha23 Theorem 6.6.13] and extends the result in loc. cit. to arbitrary smooth $p$-adic formal schemes over ${\mathrm{Spf}}(\mathcal{O}_K)$. Next in , we study the analogous notion of *relative ($F$-)gauges* on formal schemes $Z$ that are quasi-syntomic over the reduction of some base prism $(A, I)$ (see also [@Tang22 §2]), which again satisfies the quasi-syntomic sheaf property (see ). By a natural base change process, we have: **Theorem 1** (= ). *Let $(A, I)$ be a bounded prism in $Y_{{\mathlarger{\mathbbl{\Delta}}}}$. There is a natural functor $$BC \colon \{F\text{-gauges on } X\} \longrightarrow \{\text{relative } F\text{-gauges on } (X_{\bar{A}}/A)\}.$$* Moreover, we can associate a relative $F$-gauge with a *filtered Higgs field* and a *filtered flat connection*. This is analogous to Hodge--Tate comparison and de Rham comparison of prismatic cohomology in [@BS19], except that we work with more general coefficients. Below for a quasi-syntomic formal scheme $Z$ over $\bar{A}$, we let ${\mathrm{FilHiggs}}(Z/A)$ and ${\mathrm{FilConn}}(Z/\bar{A})$ be the categories of filtered Higgs fields over $Z/A$ and filtered flat connections over $Z/\bar{A}$ separately. These are filtered derived enhancements of the usual notion of Higgs fields and flat connections (see and for details). Then we have the following constructions: **Theorem 1** (see §[3.3](#sec filtered Higgs){reference-type="ref" reference="sec filtered Higgs"} and §[3.4](#Completeness of Nygaard filtration via filtered de Rham realization){reference-type="ref" reference="Completeness of Nygaard filtration via filtered de Rham realization"} for details). *Let $(A,I)$ be a bounded prism, and let $Z$ be a quasi-syntomic formal scheme over $\bar{A}$.* 1. **Hodge--Tate realization:* There is a natural functor $$\{\text{relative } F\text{-gauges on } (Z/A)\} \longrightarrow {\mathrm{FilHiggs}}(Z/A)$$* 2. **de Rham realization:* There is a natural functor $$\{\text{relative } F\text{-gauges on } (Z/A)\} \longrightarrow {\mathrm{FilConn}}(Z/\bar{A}).$$* To summarize: at this point we have functorially attached filtrations to our prismatic $F$-crystal restricted to $(X_{\bar{A}}/A)$. It remains to study the induced filtration. ## Properties {#properties .unnumbered} In we realize that the graded pieces of an absolute gauge $E$ is naturally equipped with a finite increasing exhaustive filtration, which we call the *weight filtration* (). The key feature for this weight filtration is that its graded pieces come from a graded coherent complexes ${\mathrm{Red}}_{\bullet}(E)$ on $X$ via base change along $\mathcal{O}_X \to {\mathrm{Gr}}^{\bullet}_N \mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}$. Moreover, when an $F$-gauge comes from an $I$-torsionfree coherent $F$-crystal with Frobenius height in $[a, b]$ as in , we can relate the weight filtrations on the graded piece ${\mathrm{Gr}}^\bullet E$ with the knowledge of the original Frobenius action; see . In particular, the index of the weight filtration, which we call the *weights*, also lies in $[a, b]$. In fact, completely analogous statements hold as well in the relative setting and for filtered Higgs fields. Moreover the weight filtrations in absolute and relative settings are compatible under the realization functor in , see . We can thus understand cohomology of the filtered Higgs field $M$ using coherent cohomology of ${\mathrm{Red}}_{\bullet}(M)$ and that of sheaves of differentials $\Omega^i_{Z/\bar{A}}$: **Theorem 1** (= ). *Assume $Z$ is smooth of relative dimension $n$ over $\bar{A}$, and let $M$ be a filtered Higgs field of weight $[a,b]$. For each $i\in \mathbb{Z}$, the complex $R\Gamma(Z/A, {\mathrm{Gr}}_i M)$ admits a finite increasing exhaustive filtration of range $[a,b]$, such that each graded piece is isomorphic to $R\Gamma(Z, {\mathrm{Red}}_u(M)\otimes_{\mathcal{O}_Z} \Omega^v_{Z/\bar{A}})$ for some $u,v\in \mathbb{Z}$.* Combining these facts, we also get the following byproduct: the Higgs field coming from a coherent $I$-torsionfree $F$-crystal has nilpotent Higgs structure, and the order of nilpotence is bounded above by the length of the range of Frobenius heights. On the other hand, recall in [@BS19 §15] the authors showed that the graded pieces of Nygaard filtration of prismatic cohomology are conjugate filtrations of Hodge--Tate cohomology. Analogous results hold true in the setting of relative $F$-gauges: Namely for a relative $F$-gauge $E$ over $Z/A$, its Frobenius structure induces isomorphisms between the graded pieces of $E^{(1)}\colonequals \varphi_A^* E$ and the filtrations of the associated filtered Higgs field (). Combine this with the analysis in , we can thus understand the cohomology of graded pieces of $F$-gauges using coherent cohomology. To understand the filtration of $F$-gauges, another ingredient is the fact that we work in the world of perfect filtered/graded complexes for the entire construction process. In we utilize this fact, coupled with the completeness of the relative Nygaard filtration on $\mathrm{R\Gamma}(X_{\bar{A}}/A, {\mathlarger{\mathbbl{\Delta}}}^{(1)})$ and some descendability results, to show that the induced "Nygaard filtration" on the cohomology of $E^{(1)}$ is also complete. Thanks to the completeness, to bound the cohomology of the filtration of $E^{(1)}$, we are allowed to do so on its graded pieces, and hence reduce the calculation to aforementioned Nygaard-conjugate isomorphism. Finally in we harvest the fruits of the above discussion and deduce several results, including those described in the above main theorem. ## Byproducts and comments {#byproducts-and-comments .unnumbered} At the end of this introduction, we mention several byproducts of our work and give some comments on related topics. **Remark 1** (Torsion of étale cohomology with coefficients). Let $X$ be a smooth $p$-adic formal scheme. As showed in [@GR22 Theorem 6.1], taking derived direct image along proper smooth morphism commutes with étale realization functor on the category of prismatic $F$-crystal in perfect complexes (see also ). Moreover, we observe () that étale realization functor is $t$-exact. Combine the above observations with , we get the following refinement of "weak étale comparison" (c.f. [@GR22 Thm. 6.1]): **Corollary 1** (= + ). *Let $f \colon X \to Y$ be a smooth proper morphism between smooth formal schemes over ${\mathrm{Spf}}(\mathcal{O}_K)$, then derived pushforward of $F$-crystals in perfect complexes on $X$ are $F$-crystals in perfect complexes on $Y$, and the following diagram commutes functorially: $$\xymatrix{ {\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(X) \ar[d]_{Rf_{{\mathlarger{\mathbbl{\Delta}}}, *}} \ar[r]^{T(-)} & D^{(b)}_{lisse}(X_{\eta}, \mathbb{Z}_p) \ar[d]^{Rf_{\eta, *}} \\ {\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(Y) \ar[r]^{T(-)} & D^{(b)}_{lisse}(Y_{\eta}, \mathbb{Z}_p). }$$ Moreover we have $T(R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}) = R^i f_{\eta, *}(T(\mathcal{E}))$.* This allows us to study torsions of individual étale cohomology groups of a given lisse étale complex over $\mathbb{Z}_p$ that is *crystalline* [^1], using the Frobenius structure and its bound on prismatic cohomology. **Remark 1** ($p$-divisible group and $F$-gauges). Given a quasi-syntomic $p$-adic formal scheme $X$, it is shown in [@ALB23 Thm 1.16] that there is a natural equivalence for the following categories: $$\{\text{admissible prismatic } F\text{-crystals in vector bundles of height } [0,1] \text{ on }X\} \simeq \{p\text{-divisible groups on }X\}.$$ We show in the aforementioned categories are further equivalent to the category of $F$-gauges in vector bundles of weight $[0,1]$. We also mention two natural variants of our results. **Remark 1** (Coherent $F$-gauge). As shown in , an $F$-gauge that arises from an $I$-torsionfree coherent $F$-crystal satisfies some extra condition on the associated graded pieces, and we call such a condition *weakly reflexive*. This is related to the notion of *reflexiveness* introduced in [@Bha23 Def. 6.6.4], but is slightly more genereal (as we do not assume the underlying $F$-crystal to be locally free). We expect that our method can be used to obtain an analogous result of concerning these general $F$-gauges. **Remark 1** (Positive characteristic case). Although we only work with smooth $p$-adic formal schemes $X$ over $\mathcal{O}_K$ and $F$-crystals/gauges on $X$, the entire article can be carried to the characteristic $p$ setting: Namely we may instead assume $X$ to be a smooth variety over a perfect field $k$, and consider cohomology of crystalline $F$-crystals/gauges on $X$. In this case, $p$-torsionfree coherent $F$-crystals naturally give rise to coherent $F$-gauges, and the Frobenius operator of their cohomology satisfies the same bound as in . Finally, we mention an ongoing extension of our work. **Remark 1** (Torsion Fontaine--Laffaille modules). The relationship between Fontaine--Laffaille modules in geometric setting and analytic prismatic $F$-crystals with Frobenius height in a restrictive range has been discussed by [@Wue23] (see also the interesting discussion in the forthcoming [@IKY §3] as well as the upcoming work of Christian Hokaj). In an ongoing joint work with Hui Gao and Daxin Xu, we attempt to understand the more subtle relation between coherent prismatic $F$-gauges and torsion Fontaine--Laffaille modules, as well as comparing their cohomologies. ## Notation and conventions {#notation-and-conventions .unnumbered} We work in infinity categorical language throughout the article, and assume the basics of prisms and quasi-syntomic rings, which we refer the reader to [@Lur09], [@Lur17], [@BS19] and [@BMS19] for details. Moreover, we assume all the constructions, including tensor products, pullback functors, Hodge complexes and de Rham complexes, to be derived $p$-complete or $(p,I)$-complete whenever applicable (unless otherwise mentioned), where $I$ is some ideal in a given ring. A (decreasingly) filtered complex ${\mathrm{Fil}}^\bullet C$ is defined as an object in the filtered $\infty$-category $\mathrm{Fun}(\mathbb{Z}^\mathrm{op}, D(\mathbb{Z}))$, where the latter is equivalent to the $\infty$-category of quasi-coherent complexes over the algebraic stack $\mathbb{A}^1/\mathbb{G}_m$ and is naturally equipped with a symmetric monoidal structure. A filtered algebra is then defined as an $\mathbb{E}_\infty$-algebra over the filtered $\infty$-category. Similarly, we can define the notion of a graded complex and a graded algebra in the derived content. Here for a filtered complex ${\mathrm{Fil}}^\bullet C$, we use ${\mathrm{Gr}}^\bullet C=\bigoplus {\mathrm{Gr}}^i C$ to denote its associated graded complex, and use ${\mathrm{Fil}}^\bullet C \langle i\rangle$ to denote its $i$-th shift of filtered degree, whose value at $n\in \mathbb{Z}$ is ${\mathrm{Fil}}^{i+n}C$. For our applications, we also use ${\mathrm{Fil}}_\bullet C$ and ${\mathrm{Gr}}_\bullet C$ to denote an increasingly filtered complex and its associated graded complex. For more details, we refer to [@BMS19 §5.1] and [@Bha23 §2.2.1] for brief introductions on filtered and graded objects. We always fix $K$ to be a complete $p$-adic discretely valued field with perfect residue field $k$, and let $\mathcal{O}_K$ be the ring of integers. To differentiate the notions, we use $\mathcal{E}$ to denote an ($F$-)crystal, and use $E$ to denote an ($F$-)gauge. ## Acknowledgements {#acknowledgements .unnumbered} The intellectual debt of this paper to Bhatt's notes [@Bha23] should be obvious to readers. We have benefited from helpful conversations with the following mathematicians: Johannes Anschütz, Bhargav Bhatt, Heng Du, Hui Gao, Arthur-César Le Bras, Tong Liu, Emanuel Reinecke, Peter Scholze, Yupeng Wang, and Daxin Xu. We would like to express special thanks to Bhargav Bhatt for discussions on various parts of this paper. We are grateful to Tong Liu for mentioning this problem to us. During the writing of this paper, HG is supported by Max Planck Institut für Mathematik, and SL is supported by the National Natural Science Foundation of China (No. 12288201). # Absolute prismatic $F$-crystals and prismatic $F$-gauges In this section, we recall the notion of absolute prismatic $F$-crystals and absolute prismatic $F$-gauges, and consider their relation. ## The coherent prismatic ($F$-)crystals {#The coherent prismatic ($F$-)crystals} Let $X$ be a quasi-syntomic $p$-adic formal scheme over $\mathcal{O}_K$. Recall from [@BS19] and [@BS21] that we can attach to $X$ the following ringed sites: **Definition 1**. The *(absolute) prismatic site of $X$* is defined as the opposite category $X_{\mathlarger{\mathbbl{\Delta}}}$ of bounded prisms $(A,I)$ over $X$ with the $(p,I)$-complete flat topology, and is equipped with a sheaf of rings and a sheaf of ideals $$\mathcal{O}_{\mathlarger{\mathbbl{\Delta}}}\colonequals (A,I) \longmapsto A;~\mathcal{I}_{\mathlarger{\mathbbl{\Delta}}}\colonequals (A,I) \longmapsto I.$$ The prismatic structure sheaf $\mathcal{O}_{\mathlarger{\mathbbl{\Delta}}}$ admits a natural Frobenius action which lifts the usual Frobenius on its reduction mod $p$. **Definition 1**. The *quasiregular semiperfectoid site of $X$* is the opposite category $X_{\mathrm{qrsp}}$ of quasiregular semiperfectoid algebras which are quasi-syntomic over $X$, equipped with the quasi-syntomic topology. It comes with a sheaf of prisms: $$({\mathlarger{\mathbbl{\Delta}}}_\bullet, I_\bullet) \colonequals S \longmapsto ({\mathlarger{\mathbbl{\Delta}}}_S, I_{{\mathlarger{\mathbbl{\Delta}}}_S})$$ sending $S\in X_{\mathrm{qrsp}}$ to the initial prism of ${\mathrm{Spf}}(S)_{\mathlarger{\mathbbl{\Delta}}}$ as in [@BS19 Prop. 7.2]. The fact that they are quasi-syntomic sheaves follows from the theory of Nygaard filtration and quasi-syntomic descent for conjugate filtrations, see [@BS19 §12]. Notice that for every $S$, the ideal $I_{{\mathlarger{\mathbbl{\Delta}}}_S}$ is non-canonically principal. The Frobenius endomorphism of prisms induces a natural endomorphism $\varphi$ on ${\mathlarger{\mathbbl{\Delta}}}_\bullet$. The above allows us to define various notions of coefficients associated to $X$. Below we let $*\in \{{\mathrm{vect}}, {\mathrm{perf}},\emptyset\}$ to denote the corresponding category with coefficients in either vector bundles, perfect complexes or general complexes. **Definition 1**. Denote by ${\mathrm{(F\text{-})Crys}}^*(X)$ the category of *prismatic* ($F$-)*crystals* (in either vector bundles or perfect complexes or general complexes) over $X$, in the sense of [@BS21 Definition 4.1]. Concretely, an $F$-crystal consists of a vector bundle/perfect complex/general complex $\mathcal{E}$ over the prismatic site $X_{\mathlarger{\mathbbl{\Delta}}}$ together with a Frobenius-isogeny, namely an $\mathcal{O}_{\mathlarger{\mathbbl{\Delta}}}$-linear isomorphism $$\varphi_\mathcal{E}: (\varphi_{\mathcal{O}_{\mathlarger{\mathbbl{\Delta}}}}^*\mathcal{E})[1/\mathcal{I}] \simeq \mathcal{E}[1/\mathcal{I}].$$ By [@BS21 Proposition 2.13 and Proposition 2.14], one also has the following descriptions when $*\in \{{\mathrm{vect}}, {\mathrm{perf}}\}$ $${\mathrm{(F\text{-})Crys}}^*(X) \simeq \lim_{S\in X_{\mathrm{qrsp}}} {\mathrm{(F\text{-})Crys}}^*({\mathrm{Spf}}(S)).$$ Among objects in $X_{\mathlarger{\mathbbl{\Delta}}}$, the following type of prisms plays a very useful role. **Definition 1**. Assume $X$ is smooth over $\mathcal{O}_K$. A *Breuil--Kisin* prism $(A,I)$ in $X_{{\mathlarger{\mathbbl{\Delta}}}}$ is a prism such that ${\mathrm{Spf}}(\bar{A})\to X$ is an open immersion (c.f. [@DLMS Example 3.4] and [@GR22 proof of Theorem 5.10]). Below we show that such $A$ is always with a faithfully flat Frobenius endomorphism $\phi_A:A\to A$. We have the following simple description of Breuil--Kisin prisms. **Lemma 1**. *Let $(A,I)$ be a Breuil--Kisin prism over $X$ as above, and let $\widetilde{R}$ be a smooth lift of $\bar{A}/\pi$ over $W$.* 1. *If $e > 1$, then there is an isomorphism of pairs $(A, I) \simeq (\widetilde{R}[\![u]\!], E(u))$, where $E(u)$ is the Eisenstein polynomial of a uniformizer $\pi$.* 2. *If $e = 1$, assuming ${\mathrm{Spf}}(A)$ is oriented, then we have an isomorphism of pairs $(A, I) \simeq (\widetilde{R}[\![u]\!], u - p)$.* *Proof.* By deformation theory, we can lift the isomorphism $\widetilde{R}/p \simeq \bar{A}/\pi$ to an isomorphism $\widetilde{R} \otimes_{W} \mathcal{O}_K \xrightarrow{\simeq} \bar{A}$. Again by deformation theory, we can lift the morphism $\widetilde{R} \to \bar{A}$ further to $\widetilde{R} \to A$. Now we can start the proof. Assume $e > 1$, pick a uniformizer $\pi \in \mathcal{O}_K \subset \bar{A}$, lift it to an element $\widetilde{\pi} \in A$. Noticing that $\widetilde{\pi}$ is topologically nilpotent in $A$, we get a map $\widetilde{R}[\![u]\!] \to A$, sending $u$ to $\widetilde{\pi}$. Now we observe that the Eisenstein polynomial $E(u)$ is sent to zero in $\bar{A}$, hence $E(u)$ necessarily lands in $I \subset A$, and the source is $E(u)$-adically complete whereas the target is $I$-adically complete. Therefore to finish our proof, we just need to check that $E(\widetilde{\pi})$ generates $I$ in $A$. Writing $E(\widetilde{\pi}) = \widetilde{\pi}^n + p \cdot f$, where $f$ is a unit, and let us compute $$p \delta(E(\widetilde{\pi})) = \big(\widetilde{\pi}^p + p \delta(\widetilde{\pi})\big)^n + p \cdot \varphi(f) - E(\widetilde{\pi})^p.$$ Using the fact that $n > 1$, one can check that the right hand side is $p \cdot \varphi(f) + p \cdot g$ where $g \in (p, \widetilde{\pi})$. Since $A$ is $p$-torsion free, this implies that $\delta(E(\widetilde{\pi}))$ is a unit. Therefore [@BS19 Lemma 2.24] gives $E(\widetilde{\pi}) = I$. If instead $e = 1$ and $(A, I)$ is oriented. Let us choose a generator $d$ of $I$. Then we get an isomorphism $\widetilde{R}[\![\widetilde{u}]\!] \simeq A$ by sending $\widetilde{u}$ to $d$. Now we just let $u = \widetilde{u} + p$. ◻ **Corollary 1**. *Let $(A,I)$ be a Breuil--Kisin prism over $X$ as above, then $A$ is a $p$-torsionfree regular ring and the Frobenius endomorphism $\phi_A:A\to A$ is quasi-syntomic, finite, and faithfully flat.* *Proof.* The fact that $A$ is $p$-torsionfree and regular follows immediately from the concrete description of $A$ in . Also from the description, we see that the Frobenius on $A/p$ is finite and lci, therefore its lift is also finite and quasi-syntomic thanks to $p$-completeness of $A$. Using regularity, finiteness, and miracle flatness [@stacks-project [Tag 00R4](https://stacks.math.columbia.edu/tag/00R4)], we get that $\phi_A$ is flat. Faithfulness follows from $p$-completeness: the induced map on spectrum is a homeomorphism on the locus $V(p)$ which contains all closed points by $p$-completeness, noticing that flat map is open we see that the induced map on spectrum has to be surjective. ◻ If $X={\mathrm{Spf}}(R)$ is further assumed to be affine, then we can find such an $A$ so that $(A,I)$ covers the final object in $Shv(X_{\mathlarger{\mathbbl{\Delta}}})$. It is because with mild assumptions, we can construct absolute product of any prism with Breuil--Kisin prisms in $X_{{\mathlarger{\mathbbl{\Delta}}}}$, and they enjoy good complete flatness properties. **Proposition 1**. *Assume $X$ is separated and smooth over ${\mathrm{Spf}}(\mathcal{O}_K)$, let $(A,I)$ be a Breuil--Kisin prism in $X_{{\mathlarger{\mathbbl{\Delta}}}}$ and let $(A', I')$ be a prism in $X_{{\mathlarger{\mathbbl{\Delta}}}}$. Then their product in $Shv(X_{{\mathlarger{\mathbbl{\Delta}}}})$ exists and is given by a prism $(C, K)$. It has the following properties:* 1. *The map $(A', I') \to (C, K = I' C)$ is $(p, I')$-completely flat;* 2. *If $\mathrm{Im}({\mathrm{Spf}}(\bar{A'}) \to X) \subset {\mathrm{Spf}}(\bar{A})$, then the map above is $(p, I')$-completely faithfully flat;* 3. *If the map ${\mathrm{Spf}}(\bar{A'}) \to X$ is $p$-completely flat, then the map $(A, I) \to (C, K = I C)$ is $(p,I)$-completely flat.* 4. *If in (3) we assume furthermore that ${\mathrm{Spf}}(\bar{A}) \subset \mathrm{Im}({\mathrm{Spf}}(\bar{A'}) \to X)$, then the map above is $(p, I)$-completely faithfully flat.* *Consequently, if $X={\mathrm{Spf}}(R)$ is affine, then we can find a Breuil--Kisin prism $(A, I)$ which covers the final object $* \in Shv(X_{{\mathlarger{\mathbbl{\Delta}}}})$.* This Proposition is certainly well-known to experts, for the convenience of the reader we spell out the proof in detail. Here we temporarily abuse the notation of $K$. *Proof.* Let us construct the prism $(C,K)$ first: it is supposed to be the absolute pushout of $(A, I)$ and $(A', I')$ in the category of prisms in $X_{{\mathlarger{\mathbbl{\Delta}}}}$. Unwinding definition, we see that $(C, K)$ is the initial object of prisms $(D, L)$ receiving map from both $(A, I)$ and $(A', I')$ with the property that the induced map $\bar{A} \widehat{\otimes}_{W(k)} \bar{A'} \to D/L$ factors through $\bar{A} \widehat{\otimes}_{\mathcal{O}_X} \bar{A'}$. Here we make two remarks: - For any prism $(D, L) \in X_{{\mathlarger{\mathbbl{\Delta}}}}$, using the fact that $D/L^n$ are $p$-complete, we see that the map $W(k) \to \mathcal{O}_X \to D/L$ lifts uniquely to a map $W(k) \to D$ of $\delta$-rings. This explains why the first tensor product above is over $W(k)$. - Since $X$ was assumed to be separated, the (completed) tensor product $\bar{A} \widehat{\otimes}_{\mathcal{O}_X} \bar{A'}$ is given by a (derived) $p$-complete ring corresponding to the formal scheme ${\mathrm{Spf}}(\bar{A}) \times_X {\mathrm{Spf}}(\bar{A'})$. Back to the construction of $(C, K)$, from the above description we see that it is necessarily given by the delta envelope of $(B, J)$, whose meaning we shall explain below: Here $B \coloneqq A \widehat{\otimes^L}_{W(k)} A'$ denotes the derived $(p, 1 \otimes I', I \otimes 1)$-completed tensor product, and one checks that it is concentrated in degree $0$ and is a $(p, I')$-completely flat $\delta$-$A'$-algebra. Indeed if we derived mod $B$ by $1 \otimes I'$, it becomes the derived $(p, I)$-completed tensor product $A \widehat{\otimes^L}_{W(k)} \bar{A'}$ which is seen to be a $p$-completely flat $\bar{A'}$-algebra due to . Let $J$ denote the kernel ideal of the map $B \to \bar{A} \widehat{\otimes}_{\mathcal{O}_X} \bar{A'}$. Below we shall check that our ideal $J$ meets the regularity condition in [@BS19 Proposition 3.13]. Granting this the cited proposition constructs the prismatic envelope and moreover checks that it satisfies the property (1). Now let us check the regularity condition. Due to the Zariski locality of the regularity condition, we may assume that $A$ is given by the concrete description in . Now we see that the ideal $J/(1 \otimes I')$ is Zariski locally on ${\mathrm{Spf}}(B/(1 \otimes I'))$ given by $u - 1 \otimes \pi$ and (the lift of) a regular sequence corresponding to the kernel of $\widetilde{R} \widehat{\otimes}_{W(k)} \bar{A'} \twoheadrightarrow \bar{A} \widehat{\otimes}_{\mathcal{O}_X} \bar{A'}$, where both completed tensor product are $p$-completed tensor product. The said kernel is Zariski locally a regular sequence because $\widetilde{R}$ is $p$-completely smooth $W(k)$-algebra and $\bar{A}$ defines an open of $X$, so the surjection is from a $p$-completely smooth $\bar{A'}$-algebra to a $p$-completely Zariski open $\bar{A'}$-algebra, which is lci mod $p$. By the above paragraph, we have constructed $(C, K)$ and it satisfies the properties listed in [@BS19 Proposition 3.13]. In particular, it satisfies (1) of our proposition. Let us check property (2): we need to utilize the perfection $(A_{\infty}, I_{\infty}) \coloneqq (A, I)_{\mathrm{perf}}$ of $(A, I)$ introduced in [@BS19 Lemma 3.9]. Now says that the Frobenius $\phi_A$ is quasi-syntomic and faithfully flat, by construction spelled out in loc. cit. we see that $(A, I) \to (A_{\infty}, I_{\infty})$ is also quasi-syntomic and $(p, I)$-completely faithfully flat. Let us stare at the following diagram: $$\xymatrix{ (A', I') \ar[r] & (C, K) \ar[r] & (E, M) \\ & (A, I) \ar[u] \ar[r] & (A_{\infty}, I_{\infty}) \ar[u], }$$ where the square is a $(p, I)$-completely pushing out defining the prism $(E, M)$. By the above discussion, we know the rightward arrows in this square are $(p, I)$-completely faithfully flat. Therefore we reduce our problem to checking the composite $(A', I') \to (E, M)$ is $(p, I')$-completely faithfully flat. To that end, let us study $(E, M)$ from a slightly different perspective: By how it is defined, we see that $(E, M)$ is the absolute pushout of $(A_{\infty}, I_{\infty})$ and $(A', I')$ in the category of prisms in $X_{{\mathlarger{\mathbbl{\Delta}}}}$. Since perfect prisms are initial prisms of their reductions ([@BS19 Theorem 3.10]), we see that $(E, M)$ is the initial object in $\big(S/A'\big)_{{\mathlarger{\mathbbl{\Delta}}}}$, where $S \coloneqq \bar{A'} \widehat{\otimes}_{\bar{A}} \bar{A_{\infty}}$. This is where we use the assumption that $\mathrm{Im}({\mathrm{Spf}}(\bar{A'}) \to X) \subset {\mathrm{Spf}}(\bar{A})$. Now we check that $S$ is a large quasi-syntomic $p$-completely faithfully flat $\bar{A'}$-algebra (see [@BS19 Definition 15.1] for the meaning of being "large"): It is the $p$-complete base change of a quasi-syntomic $p$-completely faithfully flat morphism; as for largeness, just notice that any perfectoid algebra is large (being a quotient of the Witt vector of a perfect ring), hence so are their base changes. By [@BS19 Theorem 15.2], we learn that $E$ is given by the derived prismatic cohomology ${\mathlarger{\mathbbl{\Delta}}}_{S/A'}$ explained in [@BS19 Construction 7.6]. Recall that we wanted to show $\bar{A'} \to \bar{E}$ is $p$-completely faithfully flat, the map factors through the $p$-completely faithfully flat map $\bar{A'} \to S$, hence it suffices to show $S \to \bar{E}$ is $p$-completely faithfully flat. To that end, one just notices that this map is the $0$-th conjugate filtration on the derived Hodge--Tate cohomology $\bar{E} = \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A'}$, and the higher graded pieces of the conjugate filtration are $p$-completely flat $S$-modules: indeed they are given by $\Gamma^{\bullet}_S(\mathbb{L}_{S/A'}[-1])^{\wedge}$ and the shifted cotangent complex $\mathbb{L}_{S/A'}[-1]$ is a $p$-completely flat $S$-module (c.f. the discussion right after [@BS19 Definition 15.1]). The proof of (3) is fairly easy: it is equivalent to asking the map $\bar{A} \to \bar{C}$ to be $p$-completely flat. We stare at the following commutative diagram: $$\xymatrix{ {\mathrm{Spf}}(\bar{C}) \ar[r] \ar[d] & {\mathrm{Spf}}(\bar{A}) \ar[d] \\ {\mathrm{Spf}}(\bar{A'}) \ar[r] & X. }$$ Our assumption, together with (1) above, implies that the map ${\mathrm{Spf}}(\bar{C}) \to X$ is $p$-completely flat. Since the map ${\mathrm{Spf}}(\bar{A}) \to X$ is an open immersion, we get what we want. As for (4): since $X$ is separated, the preimage of ${\mathrm{Spf}}(\bar{A}) \subset X$ defines an affine open in ${\mathrm{Spf}}(\bar{A'})$ which is also an affine open $U \subset {\mathrm{Spf}}(A')$, we may replace $A'$ by $\mathcal{O}_{{\mathrm{Spf}}(A')}(U)$ and replace $X$ by ${\mathrm{Spf}}(\bar{A})$ without changing $(C, K)$. Now we are reduced to the case where $X = {\mathrm{Spf}}(\bar{A}) = \mathrm{Im}({\mathrm{Spf}}(A'))$. Looking again at the above diagram, we see that ${\mathrm{Spf}}(\bar{C}) \to X = {\mathrm{Spf}}(\bar{A})$ is $p$-completely faithfully flat due to (2) and (3). This finishes the proof of properties (1)-(4) listed in this Proposition. For the last sentence, simply notice that when $X$ is affine, one can find a Breuil--Kisin prism $(A, I)$ with ${\mathrm{Spf}}(A/I) \xrightarrow{\simeq} X$, see for instance [@DLMS Example 3.4] or [@GR22 proof of Theorem 5.10]. The statement now follows from property (2) of product with any other prism in $X_{{\mathlarger{\mathbbl{\Delta}}}}$. ◻ This allows us to put a $t$-structure on prismatic ($F$-)crystals in perfect complexes on $X$ smooth over $\mathcal{O}_K$, as follows. **Construction 1**. Let us assume first that $X$ is separated and smooth over $\mathcal{O}_K$. By , we see that one can find a family of Breuil--Kisin prisms $(A_{\lambda}, I_{\lambda})$ indexed by ${\lambda \in \Lambda}$ which jointly covers $X_{{\mathlarger{\mathbbl{\Delta}}}}$. Now we may form the Cech nerve of the cover $\bigsqcup_{\lambda \in \Lambda} (A_{\lambda}, I_{\lambda}) \twoheadrightarrow *$, whose $n$-th spot is given by $(A_{\lambda_1}, I_{\lambda_1}) \times \cdots \times (A_{\lambda_{n+1}}, I_{\lambda_{n+1}})$ indexed by $\Lambda^{n+1}$. Below we will simply denote this Cech nerve by ${\mathrm{Spf}}(A^{[\bullet]})$ with the corresponding family of rings $A^{[\bullet]}$, so each $A^{[n]}$ really stands for the above family of rings indexed by $\Lambda^{n+1}$. By $(p, I)$-completely faithfully flat descent (c.f. [@Mat22 Theorem 5.8] and [@BS21 Theorem 2.2]), the $\infty$-category ${\mathrm{(F\text{-})Crys}}^{{\mathrm{perf}}}(X)$ is given by the cosimplicial limit of the derived $\infty$-categories of ($F$-)perfect complexes on ${\mathrm{Spf}}(A^{[\bullet]})$, we define a $t$-structure by requiring an ($F$-)crystals in perfect complexes lives in $\leq 0$ part (resp. $\geq 0$ part) if the underlying complexes on ${\mathrm{Spf}}(A^{[\bullet]})$ is concentrated in cohomological degrees $\leq 0$ (resp. $\geq 0$). By (2), the maps from $A^{[0]}$ to any of $A^{[\bullet]}$ are all $(p, I)$-completely flat, so are all the maps composed with the Frobenius on $A^{[\bullet]}$. Hence all these maps are already flat as $A^{[0]}$ consists of $(p, I)$-complete Noetherian rings. Therefore it suffices to check the cohomological degrees for the underlying complexes on ${\mathrm{Spf}}(A_{\lambda})$'s. We call this the *standard $t$-structure*, below we shall check that it is indeed a $t$-structure and is independent of the choice of the family $(A_{\lambda}, I_{\lambda})$. We refer readers to [@BBDG82 Chapter 1] and [@Lur17 Section 1.2.1] for a general discussion on $t$-structures. **Lemma 1**. *defines a $t$-structure on ${\mathrm{(F\text{-})Crys}}^{{\mathrm{perf}}}(X)$. When $X$ is quasi-compact, this $t$-structure is exhaustive and separated.* *Proof.* Given $\mathcal{F}$ (resp. $\mathcal{G}$) in ${\mathrm{(F\text{-})Crys}}^{{\mathrm{perf}}}(X)$ with its values on ${\mathrm{Spf}}(A_{\lambda})$'s concentrated in degrees $\leq 0$ (resp.$\geq 1$), we see immediately that $\mathrm{Map}_{{\mathrm{(F\text{-})Crys}}^{{\mathrm{perf}}}(X)}(\mathcal{F}, \mathcal{G})$ is the cosimplicial limit of contractible spaces: This is simply because their values on each ${\mathrm{Spf}}(A^{\bullet})$'s have contractible mapping spaces for degree reason. Therefore we get that the mapping space above is itself a contractible space. To see the existence of $\tau^{\leq 0}$ truncation, one just notices that term-wise $\tau^{\leq 0}$ truncation still satisfies the crystal condition thanks to the discussion prior to this Lemma. Moreover, the resulting object is a perfect complex: By crystal condition it suffices to check this on the family of rings $A^{[0]} \coloneqq \{A_{\lambda}\}_{\lambda \in \Lambda}$, but this is automatic as Breuil--Kisin prisms are regular by . Due to flatness of $A^{[0]} \to A^{[i]} \xrightarrow{\varphi_{A^{[i]}}} A^{[i]}$, we see that the $\leq 0$ truncation of $\varphi_{A^{[i]}}^* \mathcal{F}(A^{[i]})$ is the same as the Frobenius twist of the $\leq 0$ truncation of $\mathcal{F}(A^{[i]})$. Hence the linearized Frobenius extends canonically to the truncations, proving the existence of truncations for objects in ${\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(X)$. When $X$ is quasi-compact, one can find a disjoint union of finitely many Breuil--Kisin prisms to cover $X_{{\mathlarger{\mathbbl{\Delta}}}}$, and the exhaustiveness and separatedness follows from the same properties of the standard $t$-structure on ${\mathrm{Perf}}(R)$ for any regular ring $R$. ◻ Granting the claim that in the separated case the standard $t$-structure is independent of the choice of covering families of Breuil--Kisin prisms, we make the following definition. **Definition 1**. Let $X$ be smooth over $\mathcal{O}_K$, we define the *standard $t$-structure* on ${\mathrm{(F\text{-})Crys}}^{{\mathrm{perf}}}(X)$ by: An object lives in $\leq 0$ part (resp. $\geq 0$ part) if its restriction to ${\mathrm{(F\text{-})Crys}}^{{\mathrm{perf}}}(U)$ is so for any separated open $U \subset X$. We say an object in ${\mathrm{(F\text{-})Crys}}^{{\mathrm{perf}}}(X)$ is *coherent* if it belongs to the heart of the standard $t$-structure, we denote this heart by ${\mathrm{(F\text{-})Crys}}^{{\mathrm{coh}}}(X)$. By taking its associated homotopy category, we may regard ${\mathrm{(F\text{-})Crys}}^{{\mathrm{coh}}}(X)$ as an abelian category. Applying the usual Zariski descent to Breuil--Kisin prisms, one sees that it suffices to check the above condition for a family of separated opens $U \subset X$ which jointly covers $X$. What is left to show is the claim about independence of the choice of covering family of Breuil--Kisin prisms. In fact, we prove the following stronger statement. **Proposition 1**. *Let $X$ be separated and smooth over $\mathcal{O}_K$. Fix a covering family of Breuil--Kisin prisms $\bigsqcup_{\lambda \in \Lambda} (A_{\lambda}, I_{\lambda}) \twoheadrightarrow *$ which gives rise to a $t$-structure as in . Let $(A', I') \in X_{{\mathlarger{\mathbbl{\Delta}}}}$ and suppose that ${\mathrm{Spf}}(\bar{A'}) \to X$ is $p$-completely flat, then the evaluation functor $\mathrm{Crys}^{{\mathrm{perf}}}(X) \to {\mathrm{Perf}}(A')$ is $t$-exact with respect to the standard $t$-structures on both sides.* In particular, being in the heart of the $t$-structure with respect to one covering family of Breuil--Kisin prisms forces its value on any other Breuil--Kisin prism to concentrate in degree $0$. This way, we see that the $t$-structure we get is indeed independent of the choice of covering families of Breuil--Kisin prisms. *Proof.* Let $(C_{\lambda}, K_{\lambda})$ be the product of $(A_{\lambda}, I_{\lambda})$ and $(A', I')$ as in . Then property (2) in said Proposition implies that ${\mathrm{Spf}}(C_{\lambda})$ gives rise to a $(p, I')$-completely flat cover of ${\mathrm{Spf}}(A')$. By definition, the product $(C^{\bullet}, K^{\bullet})$ of $(A^{\bullet}, I^{\bullet})$ with $(A', I')$ exists and is given by the Cech nerve of the cover ${\mathrm{Spf}}(C^{[0]}) \coloneqq \bigsqcup_{\lambda \in \Lambda} {\mathrm{Spf}}(C_{\lambda}) \twoheadrightarrow {\mathrm{Spf}}(A')$. Now let $\mathcal{F} \in \mathrm{Crys}^{{\mathrm{coh}}}(X)$ be in the heart of the $t$-structure defined by $(A_{\lambda}, I_{\lambda})$. Then its values on $C^{[\bullet]}$ defines a descent datum giving rise to the value on $A'$ (here we suppresses the ideal for simplicity). By (3), we know that all of $C^{[\bullet]}$ are $(p, I)$-completely flat over $A^{[0]}$. Since $A^{[0]}$ consists of derived $(p, I)$-complete rings which are Noetherian, we see that all of $C^{[\bullet]}$ are flat over $A^{[0]}$. The complexes in descent datum are (non-canonically) given by $\mathcal{F}(C^{[\bullet]}) \simeq \mathcal{F}(A^{[0]}) \otimes_{A^{[0]}} C^{[\bullet]}$, we learn that they consist of modules sitting in degree $0$. Now below ensures that the descent $\mathcal{F}(A')$ also lives in degree $0$, hence proving the assertion. ◻ **Lemma 1**. *Let $A' \to C^{[0]}$ be a $J$-completely faithfully flat map of derived $J$-complete rings, where $J = (f_1, \cdots, f_r) \subset A'$ is a finitely generated ideal. Denote the Cech nerve by $C^{[\bullet]}$. Then in the equivalence $D_{J-\text{comp}}(A') \simeq \lim_{\Delta} D_{J-\text{comp}}(C^{[\bullet]})$ given by $J$-complete faithfully flat descent, a descent datum of the right hand side consisting of complexes $M^{[\bullet]}$ concentrated in cohomological degree $0$ has its descent $M \in D_{J-\text{comp}}(A')$ concentrated in cohomological degree $0$ as well.* *Proof.* Since the descent $M$ is calculated by $\lim_{\Delta} M^{[\bullet]}$, it is concentrated in cohomological degrees $\geq 0$. Below we show the reverse cohomological degree estimate. We look at the following base change formula for any natural number $m$: $$M \otimes^{L}_{A'} \mathrm{Kos}(A'; f_i^m) \otimes^{L}_{\mathrm{Kos}(A'; f_i^m)} \mathrm{Kos}(C^{[0]}; f_i^m) \cong M^{[0]} \otimes^{L}_{C^{[0]}} \mathrm{Kos}(C^{[0]}; f_i^m).$$ The right hand side lives in degrees $\leq 0$ by our condition that $M^{[0]}$ lives in degree $0$. Therefore we see $M \otimes^{L}_{A'} \mathrm{Kos}(A'; f_i^m)$ also lives in degrees $\leq 0$, as $A' \to C^{[0]}$ is $J$-completely faithfully flat. By the same argument, the transition maps $\pi_0(M \otimes^{L}_{A'} \mathrm{Kos}(A'; f_i^{m+1})) \to \pi_0(M \otimes^{L}_{A'} \mathrm{Kos}(A'; f_i^m))$ are all surjective. Using the fact that $M$ is derived $J$-complete, we get that $M = \mathrm{R}\lim_m \big( M \otimes^{L}_{A'} \mathrm{Kos}(A'; f_i^m) \big)$, which lives in degrees $\leq 0$ due to the above analysis. ◻ Let us point out that we do not know if the converse to the above holds true: Namely given a $J$-complete $A$-module $M$, base change $M$ along a $J$-completely flat ring map $A \to B$ might not be concentrated in degree $0$ anymore. Now we can show the aforementioned independence of the choice of Breuil--Kisin prisms. **Corollary 1**. *Let $X$ be a smooth formal scheme over $\mathcal{O}_K$. An object $\mathcal{F}\in {\mathrm{(F\text{-})Crys}}^{\mathrm{perf}}(X)$ lives in $\leq 0$ part (resp. $\geq 0$ part) if and only if any of the following equivalent condition holds:* 1. *There exists a covering family of Breuil--Kisin prisms $(A_\lambda, I_\lambda)$ in $X_{\mathlarger{\mathbbl{\Delta}}}$, such that $\mathcal{F}(A_\lambda, I_\lambda)$ lives in $D^{\leq 0}(A_\lambda)$ (resp. $D^{\geq 0}(A_\lambda)$).* 2. *For any Breuil--Kisin prism $(A,I)\in X_{\mathlarger{\mathbbl{\Delta}}}$, we have $\mathcal{F}(A,I)\in D^{\leq 0}(A)$ (resp. $D^{\geq 0}(A)$).* 3. *There exists a covering family of prisms $(A,I)\in X_{\mathlarger{\mathbbl{\Delta}}}$ with ${\mathrm{Spf}}(\bar{A})\to X$ being $p$-completely flat, such that $\mathcal{F}(A,I)\in D^{\leq 0}(A)$ (resp. $D^{\geq 0}(A)$).* 4. *For any prism $(A,I)\in X_{\mathlarger{\mathbbl{\Delta}}}$ such that ${\mathrm{Spf}}(\bar{A})\to X$ is $p$-completely flat, we have $\mathcal{F}(A,I)\in D^{\leq 0}(A)$ (resp. $D^{\geq 0}(A)$).* *Proof.* We shall prove the "living in $\leq 0$ part" statement, as the other part can be proved similarly. It is easy to see that (4) implies (3) and (2), (2) implies (1), and (1) implies that $\mathcal{F}$ lives in $\leq 0$ part. The last condition implies (4): This is exactly . What is left to show is that (3) also implies that $\mathcal{F}$ lives in $\leq 0$ part. Now we look at the truncation $\mathcal{G} \coloneqq \tau^{>0}(\mathcal{F})$. Equivalently, we need to show $\mathcal{G}$ is zero. By and condition (3), we see that $\mathcal{G}$ has value $0$ on a covering family of prisms, hence it is zero. ◻ **Definition 1** ($I$-torsionfree crystal). We say an $\mathcal{E} \in {\mathrm{(F\text{-})Crys}}^{{\mathrm{coh}}}(X)$ is $I$-torsionfree if the map of coherent crystals: $\mathcal{I}_{{\mathlarger{\mathbbl{\Delta}}}} \otimes_{\mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}} \mathcal{E} \to \mathcal{E}$ is injective, i.e. has no kernel with respect to the standard $t$-structure. Equivalently, this is asking $\mathcal{E}(A)$ to be $I$-torsionfree for all (or a covering family of) Breuil--Kisin prisms $(A, I) \in X_{{\mathlarger{\mathbbl{\Delta}}}}$. **Remark 1**. We have the following fully faithful inclusion of categories: $${\mathrm{(F\text{-})Crys}}^{\mathrm{vect}}(X) \subset {\mathrm{(F\text{-})Crys}}^\mathrm{an}(X) \subset {\mathrm{(F\text{-})Crys}}^{I\text{-tf}} \subset {\mathrm{(F\text{-})Crys}}^{\mathrm{coh}}(X),$$ for the last inclusion, where the target is regarded as an abelian category (since it is the heart of the standard $t$-structure), see [@GR22 Theorem 5.10]. By definition, we have a natural fully faithful inclusion of $\infty$-categories $${\mathrm{(F\text{-})Crys}}^{\mathrm{coh}}(X) \subset {\mathrm{(F\text{-})Crys}}^{\mathrm{perf}}(X).$$ Recall in [@BS21 §3] the authors introduced the notion of Laurent $F$-crystals (see Definition 3.2 in loc. cit.). According to the Corollary 3.7 in loc. cit., Laurent $F$-crystals on a bounded $p$-adic formal scheme $X$ are functorially identified with the locally constant derived category of the adic generic fiber of $X$: $$D_{\mathrm{perf}}(X_{{\mathlarger{\mathbbl{\Delta}}}}, \mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}[1/I_{{\mathlarger{\mathbbl{\Delta}}}}]^{\wedge}_p)^{\varphi = 1} \simeq D^{(b)}_{lisse}(X_{\eta}, \mathbb{Z}_p).$$ Notice that the right hand side has its own standard $t$-structure. **Lemma 1**. *Let $X$ be a smooth formal scheme over ${\mathrm{Spf}}(\mathcal{O}_K)$, then the base change functor compose with the above étale realization functor $\big(- \otimes_{\mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}} \mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}[1/I_{{\mathlarger{\mathbbl{\Delta}}}}]^{\wedge}_p\big)^{\varphi = 1} \colon {\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(X) \to D^{(b)}_{lisse}(X_{\eta}, \mathbb{Z}_p)$ is $t$-exact.* *Proof.* It suffices to prove the statement locally on $X$, so we may assume $X = {\mathrm{Spf}}(R)$ is affine which has a Breuil--Kisin prism $(A, I)$ with $A/I = R$ and the Frobenius $\varphi_A \colon A/I[1/p] \to A/(\varphi(I))[1/p]$ is finite étale: Indeed, we may assume that $X$ admits a toric chart, then the Breuil--Kisin prism constructed in [@DLMS Example 3.4] has these properties. Suppose $\mathcal{E} \in {\mathrm{F\text{-}Crys}}^{{\mathrm{coh}}}(X)$, we need to show its étale realization $T(\mathcal{E}) \in D^{(b)}_{lisse}(X_{\eta}, \mathbb{Z}_p)$ sits in cohomological degree $0$. To that end, let $R_{\infty} \coloneqq A_{{\mathrm{perf}}}/I$ be the reduction of the perfection of $(A, I)$, whose adic generic fiber is an affinoid perfectoid pro-finite-étale cover of $X$. It suffices to show that the restriction of $T(\mathcal{E})$ to ${\mathrm{Spf}}(R_{\infty})_{\eta}$ sits in cohomological degree $0$. Tracing through the proof of [@BS21 Corollary 3.7], we see that the restriction of $T(\mathcal{E})$ is computed (via the tilting equivalence) by the $\varphi$-invariants (as a pro-étale sheaf) of $\mathcal{E}(A_{{\mathrm{perf}}})[1/I]^{\wedge}_p = \mathcal{E}(A) \otimes_A A_{{\mathrm{perf}}}[1/I]^{\wedge}_p$. By definition, the perfect complex $\mathcal{E}(A)$ is given by a finitely presented $A$-module in degree $0$. The ring map $A \to A_{{\mathrm{perf}}}[1/I]^{\wedge}_p$ is $p$-completely flat, hence is flat thanks to Noetherianity of $A$. Therefore we see that $\mathcal{E}(A_{{\mathrm{perf}}})[1/I]^{\wedge}_p$ is also concentrated in degree $0$. Now our claim is equivalent to the statement that $\varphi - 1$ on $\mathcal{E}(A_{{\mathrm{perf}}})[1/I]^{\wedge}_p$ is surjective (as pro-étale $\mathbb{Z}_p$-sheaves on $A_{{\mathrm{perf}}}[1/I]^{\wedge}_p$). This is well-known, but let us still sketch a proof: Since $\mathcal{E}(A_{{\mathrm{perf}}})[1/I]^{\wedge}_p$ is derived $p$-complete, the cokernel is also derived $p$-complete, therefore it suffices to show its mod $p$ reduction is $0$. Equivalently it suffices to show the (derived) $\varphi$-invariants of the derived mod $p$ reduction $\mathcal{E}(A_{{\mathrm{perf}}})[1/I]/p$ lives in degrees $\leq 0$, let us denote it by $T(\mathcal{E}/^L p)$. Finally we may use [@BS21 Proposition 3.6]: The authors show that after trivializing $T(\mathcal{E} /^L p)$ on an étale neighborhood $U$ of $\mathrm{Spec}(A_{{\mathrm{perf}}}[1/I]/p)$, so that $T(\mathcal{E} /^L p)(U)$ is just an $\mathbb{F}_p$ complex (which we need to show to be concentrated in degrees $\leq 0$), we have an isomorphism $T(\mathcal{E} /^L p)(U) \otimes_{\mathbb{F}_p} \mathcal{O}_U \simeq \mathcal{E}(A_{{\mathrm{perf}}})[1/I]/^L p \otimes_{A_{{\mathrm{perf}}}[1/I]/p} \mathcal{O}_U$, as pro-étale sheaves. Now we claim victory as the right hand side lives in degrees $\leq 0$. ◻ Another notion of central interest to our article is the *height* of an $I$-torsionfree prismatic $F$-crystal. **Definition 1**. Let $X$ be a smooth formal scheme over ${\mathrm{Spf}}(\mathcal{O}_K)$, and let $(\mathcal{E},\varphi_\mathcal{E}) \in {\mathrm{F\text{-}Crys}}^{I\text{-tf}}(X)$. We say it has *height* in $[a, b]$ if the linearized Frobenius $$\varphi_\mathcal{E} \colon \varphi_{\mathcal{O}_{\mathlarger{\mathbbl{\Delta}}}}^* \mathcal{E} \longrightarrow \mathcal{E}[1/I]$$ has its image squeezed in the following manner: $I^b \mathcal{E} \subset \mathrm{Im}(\varphi_\mathcal{E}) \subset I^a \mathcal{E}$, viewed as sub-$\mathcal{O}_{\mathlarger{\mathbbl{\Delta}}}$-modules in $\mathcal{E}[1/I]$. When $[a,b] = [0,h]$, we simply say it is effective of height $\leq h$, this terminology is compatible with [@DLMS Definition 3.14]. **Remark 1**. One can check height of $(\mathcal{E}, \varphi_\mathcal{E})$ on a covering family of prisms whose reduction is flat over $X$. **Example 1**. Fix notation as above, the prismatic structure sheaf with its Frobenius is effective and has height $\leq 0$. Assuming that $X$ is smooth proper of relative equidimension $d$ over ${\mathrm{Spf}}(\mathcal{O}_K)$, then the $i$-th derived pushforward of $\mathcal{O}_{{\mathlarger{\mathbbl{\Delta}}}}$ has its $I$-torsionfree quotient being effective of height $\leq \min\{i, d\}$: [@BS19 Corollary 15.5] shows the $\leq i$ inequality, and the $\leq d$ inequality can be directly proved using [@LL20 Lemma 7.8]. In fact the bound in terms of dimension $d$ holds more generally, one can translate an argument due to Bhatt (see [@GR22 Remark 8.11]) to the following concrete estimate: if $X \to Y$ is an equi-$d$-dimensional smooth (without properness!) map of smooth formal schemes over ${\mathrm{Spf}}(\mathcal{O}_K)$ and let $(\mathcal{E}, \varphi_\mathcal{E})$ be an $I$-torsionfree effective $F$-crystal on $X$ with height $\leq h$, then for any integer $i$ the $i$-th derived pushforward $(R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *}\mathcal{E}, \varphi_\mathcal{E})$ has its $I$-torsionfree quotient of height $\leq (d + h)$. Our paper aims at generalizing these height estimates, see . ## Prismatic $F$-gauges {#Prismatic $F$-gauges} In this subsection we shall recall the notion of prismatic $F$-gauges, introduced by Drinfeld and Bhatt--Lurie. Let us start with $F$-gauges over a quasiregular semiperfectoid ring. **Definition 1** ([@Bha23 Definition 5.5.17 and Example 6.1.7]). Let $S$ be a quasiregular semiperfectoid ring. A *prismatic gauge* on ${\mathrm{Spf}}(S)$ is a $(p,I)$-complete filtered complex ${\mathrm{Fil}}^\bullet E$ over ${\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S$. A *prismatic $F$-gauge* $E=(E, {\mathrm{Fil}}^\bullet E, \widetilde{\varphi}_{E})$ on ${\mathrm{Spf}}(S)$ consists of the following data: - a prismatic gauge $(E, {\mathrm{Fil}}^\bullet E)$, called *the underlying gauge* of $E$; - a map of filtered complexes $$\widetilde{\varphi}_{E}: {\mathrm{Fil}}^\bullet E \longrightarrow I^\mathbb{Z}{\mathlarger{\mathbbl{\Delta}}}_S\otimes_{{\mathlarger{\mathbbl{\Delta}}}_S} E,$$ which is linear over the filtered map $\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S} \colon {\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S \to I^\bullet{\mathlarger{\mathbbl{\Delta}}}_S$; - such that the filtered linearization $$\varphi_{E}: {\mathrm{Fil}}^\bullet E\otimes_{{\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S} I^\mathbb{Z} {\mathlarger{\mathbbl{\Delta}}}_S \longrightarrow I^\mathbb{Z} E$$ is a filtered isomorphism in ${\mathcal{DF}}_{(p,I)\text{-comp}}(I^\mathbb{Z}{\mathlarger{\mathbbl{\Delta}}}_S) \simeq \mathrm{D}_{(p,I)\text{-comp}}({\mathlarger{\mathbbl{\Delta}}}_S)$. Following the convention of [@Bha23], we call ${\mathrm{Fil}}^\bullet E$ a *Nygaardian filtration* on $E$. So a prismatic gauge is nothing but a prismatic crystal equipped with a Nygaardian filtration. **Definition 1**. Let $S$ be a quasiregular semiperfectoid ring. 1. We use ${\mathrm{(F\text{-})Gauge}}({\mathrm{Spf}}(S))$ to denote the category of prismatic ($F$-)gauges over ${\mathrm{Spf}}(S)$. 2. We use ${\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S))$ with $*\in\{{\mathrm{vect}}, {\mathrm{perf}}\}$ to denote the subcategory of ${\mathrm{(F\text{-})Gauge}}({\mathrm{Spf}}(S))$ consisting of those $E\in {\mathrm{(F\text{-})Gauge}}({\mathrm{Spf}}(S))$ such that the underlying filtered ${\mathrm{Fil}}^\bullet_N{\mathlarger{\mathbbl{\Delta}}}_S$-complex ${\mathrm{Fil}}^\bullet E$ is in ${\mathrm{Vect}}({\mathrm{Fil}}_N^\bullet{\mathlarger{\mathbbl{\Delta}}}_S)\colonequals {\mathrm{Vect}}({\mathrm{Spf}}({\mathrm{Rees}}({\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S))/\mathbb{G}_m)$ (resp. ${\mathrm{Perf}}({\mathrm{Spf}}({\mathrm{Rees}}({\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S))/\mathbb{G}_m)$). 3. We use ${\mathrm{(F\text{-})Gauge}}^{\mathrm{coh}}({\mathrm{Spf}}(S))$ to denote the full subcategory of ${\mathrm{(F\text{-})Gauge}}^{\mathrm{perf}}({\mathrm{Spf}}(S))$ such that the underlying gauge is a module over ${\mathlarger{\mathbbl{\Delta}}}_S$ filtered by submodules. **Remark 1**. The mapping space between two coherent ($F$-)gauges on ${\mathrm{Spf}}(S)$ is discrete. **Remark 1**. For a map of quasiregular semiperfectoid rings $S_1\to S_2$ in $X_{\mathrm{qrsp}}$ and $*\in\{\emptyset, {\mathrm{vect}}, {\mathrm{perf}}\}$, the completed filtered base change induces a natural functor $$\begin{aligned} \Phi_{(S_1,S_2)}:{\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S_1)) &\longrightarrow {\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S_2)),\\ (E, {\mathrm{Fil}}^\bullet E, (\widetilde{\varphi}_{E})) &\longmapsto (E\bigotimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} {\mathlarger{\mathbbl{\Delta}}}_{S_2}, {\mathrm{Fil}}^\bullet E\bigotimes_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_1}} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_2}, (\widetilde{\varphi}_{E}\otimes \varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S_2}})). \end{aligned}$$ From the construction, the induced functor on underlying ($F$-)crystals is the natural pullback functor. By taking limit of the above categories, we obtain the definition of ($F$-)gauges for more general schemes. **Definition 1**. Let $X$ be a quasi-syntomic formal scheme. The category of *prismatic ($F$-)gauges* over $X$ is defined as the limit of $\infty$-categories $${\mathrm{(F\text{-})Gauge}}^*(X)\coloneqq\lim_{S\in X_{\mathrm{qrsp}}} {\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S)),$$ where $*\in \{\emptyset, {\mathrm{vect}}, {\mathrm{perf}}, {\mathrm{coh}}\}$. **Remark 1**. There is a natural functor from ($F$-)gauges to ($F$-)crystals by forgetting the filtration, and it is compatible with the limit formula in [@BS21 Proposition 2.14]. **Remark 1**. In the stacky formalism of [@Bha23], the categories ${\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S))$ for $*\in\{\emptyset, {\mathrm{vect}}, {\mathrm{perf}}\}$ are equivalent to the categories of complexes (resp. vector bundles, resp. perfect complexes) over the filtered prismatization $X^{\mathcal{N}}$ (resp. syntomification $X^\mathrm{syn}$) of $X$. However the analogous statement is probably not true when $* = {\mathrm{coh}}$. A key fact is that the category of ($F$-)gauges form a sheaf under quasi-syntomic topology. Before proving this, let us record some useful flatness criteria in commutative algebra. **Lemma 1**. *Let $A \to B$ be a ring map, let $f \in A$ be a nonzerodivisor in both $A$ and $B$. Assume that $A/f \to B/f$ and $A[1/f] \to B[1/f]$ are (faithfully) flat, then $A \to B$ is (faithfully) flat. If $J \subset A$ is a finitely generated ideal, then the analogous statement with all "(faithfully) flat" replaced by "$J$-completely (faithfully) flat" also holds.* *Proof.* Let us prove the "flat" statement first, the faithful part is easy. Let $M$ be an $A$ module, denote by $N$ the submodule consisting of $f$-power torsion elements, then $M/N$ is an $f$-torsion free $A$ module. Since $\mathrm{Tor}_i^A(M, B)$ is squeezed between $\mathrm{Tor}_i^A(N, B)$ and $\mathrm{Tor}_i^A(M/N, B)$, it is equivalent to showing the later two vanish for $i > 0$. Let us treat $N$ first. Since $N$ is the filtered colimit of its finitely generated submodules $N'$ each of which is annihilated by a power (depending on the finitely generated submodule) of $f$, we are further reduced to showing: If $f^c$ annihilates $N'$, then $N' \otimes^L_A B$ lives in degree $0$. Now we write $N' \otimes^L_A B = N' \otimes^L_{A/f^c} A/f^c \otimes^L_A B = N' \otimes^L_{A/f^c} B/f^c$: For the second equality we use the fact that $f$ is a nonzerodivisor in $B$. We may appeal to [@stacks-project [Tag 051C](https://stacks.math.columbia.edu/tag/051C)] and see that $A/f^c \to B/f^c$ is flat. Now we treat $M/N$. First we observe $M/N \otimes^L_A B \otimes^L_B B/f = (M/N)/^L f \otimes^L_{A/f} B/f$ which lives in degree $0$ by $f$-torsion free assumption on $M/N$ and flatness of $A/f \to B/f$. Therefore we see that $\mathrm{Tor}_i^A(M/N \otimes^L_A B)$ are $f$-invertible for all $i > 0$: Indeed one simply observes that $\mathrm{Tor}_i^A(M/N \otimes^L_A B)/f$ and $\mathrm{Tor}_{i-1}^A(M/N \otimes^L_A B)[f]$ are subquotients of $\mathrm{H}_i((M/N \otimes^L_A B)\otimes^L_B B/f)$. Hence we may invert $f$ and get $\mathrm{Tor}_i^A(M/N, B) = \mathrm{Tor}_i^{A[1/f]}(M/N[1/f], B[1/f]) = 0$ for all $i > 0$, here the first equality follows from previous sentence and the second equality follows from the flatness assumption on $A[1/f] \to B[1/f]$. For the $J$-complete analog, one simply runs the above argument with the starting assumption that $M$ is an $A/J$ module. ◻ **Lemma 1**. *Let $(A, {\mathrm{Fil}}_{\bullet}(A)) \to (B, {\mathrm{Fil}}_{\bullet}(B))$ be a filtered map of commutative unital rings equipped with increasing exhaustive multiplicative $\mathbb{N}$-indexed filtrations. If the graded ring map ${\mathrm{Gr}}_{\bullet}(A) \to {\mathrm{Gr}}_{\bullet}(A)$ is (faithfully) flat, then $A \to B$ is (faithfully) flat. If $J \subset {\mathrm{Fil}}_0(A)$ is a finitely generated ideal and if the increasing filtrations on $A$ and $B$ are $J$-completely exhaustive, then the analogous statement with all "(faithfully) flat" replaced by "$J$-completely (faithfully) flat" also holds.* *Proof.* It is equivalent to showing the following: let $K \subset A$ be an ideal, then $K \otimes_A B \to B$ is injective (for the faithfully flat statement we need to further show that this map is not surjective unless $K = A$ is the unit ideal). Now we equip $K$ with the induced filtration ${\mathrm{Fil}}_i(K) = K \cap {\mathrm{Fil}}_i(A)$, and consider the following filtered map ${\mathrm{Fil}}_{\bullet}(K \otimes_A B) \coloneqq {\mathrm{Fil}}_{\bullet}(K) \otimes^L_{{\mathrm{Fil}}_{\bullet}(A)} {\mathrm{Fil}}_{\bullet}(B) \to {\mathrm{Fil}}_{\bullet}(B)$. The source is equipped with exhaustive increasing $\mathbb{N}$-indexed filtration and the above map has its graded pieces given by ${\mathrm{Gr}}_{\bullet}(K \otimes_A B) = {\mathrm{Gr}}_{\bullet}(K) \otimes^L_{{\mathrm{Gr}}_{\bullet}(A)} {\mathrm{Gr}}_{\bullet}(B) \to {\mathrm{Gr}}_{\bullet}(B)$. As ${\mathrm{Gr}}_{\bullet}K$ is a graded ideal inside of ${\mathrm{Gr}}_{\bullet}A$, the (faithfully) flatness of ${\mathrm{Gr}}_{\bullet}(A) \to {\mathrm{Gr}}_{\bullet}(B)$ implies that the above has its source living in degree $0$ and the induced map is injective (and not surjective unless ${\mathrm{Gr}}_{\bullet}(K) = {\mathrm{Gr}}_{\bullet}(A)$). This together with the snake lemma implies that $K \to A = {\mathrm{colim}}_{i \to \infty} ({\mathrm{Fil}}_i(K) \to {\mathrm{Fil}}_i(A))$ is injective (and not surjective unless $K = A$). For the completely (faithfully) flat statement, one just runs the above argument with the assumption that $K$ contains $J$. ◻ The quasi-syntomic descent of ($F$-)gauges follows from [@Bha23 Theorem 5.5.10 and Remark 5.5.18], and since our definition (which does not involve the stacky approach) a priori does not agree with the one given in Bhatt's notes, we give a direct proof. **Proposition 1** (c.f. [@Bha23 Theorem 5.5.10 and Remark 5.5.18]). *Let $S\to S^{(0)}$ be a quasi-syntomic cover of quasiregular semiperfectoid rings, and let $S^{(\bullet)}$ be the $p$-completed Čech nerve. Then for $*\in \{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$, we have a natural equivalence $${\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S)) \simeq \lim_{[n]\in \Delta} {\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S^{(n)})).$$* *Proof.* Let us treat the descent of gauges first, we fix a map $R \to S$ where $R$ is perfectoid with its associated perfect prism $(A, I = (d))$. Via Rees's construction (see [@Bha23 §2.2]), we can regard gauges as graded $(p, I)$-complete modules over the graded ring $\mathrm{Rees}({\mathrm{Fil}}^{\bullet}_N) \coloneqq \bigoplus_{i \in \mathbb{Z}} {\mathrm{Fil}}^i_N t^{-i}$. Therefore the descent of gauges follows from the following two statements: 1. The map $\mathrm{Rees}({\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}_S)) \to \mathrm{Rees}({\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}_{S^{(0)}}))$ is $(p, I)$-completely faithfully flat. 2. For any map of qrsp algebras $S \to \widetilde{S}$ with $\widetilde{S}^{(0)} \coloneqq \widetilde{S} \widehat{\otimes}_S S^{(0)}$, we have a filtered isomorphism: $${\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}_{\widetilde{S}}) \widehat{\otimes}_{{\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}_S)} {\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}_{S^{(0)}}) \xrightarrow{\cong} {\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}_{\widetilde{S}^{(0)}}).$$ Let us prove (1) first. Using (with the $f$ there being $t$ here), we see that it suffices to show that both the underlying ring map ${\mathlarger{\mathbbl{\Delta}}}_S \to {\mathlarger{\mathbbl{\Delta}}}_{S^{(0)}}$ and the graded algebra map $$\bigoplus_{i \in \mathbb{N}} {\mathrm{Gr}}^i_N({\mathlarger{\mathbbl{\Delta}}}_S) t^{-i} \to \bigoplus_{i \in \mathbb{N}} {\mathrm{Gr}}^i_N({\mathlarger{\mathbbl{\Delta}}}_{S^{(0)}}) t^{-i}$$ are completely faithfully flat. By [@BS19 Theorem 12.2], the graded algebra is identified with Rees's construction of the (twisted) conjugate filtered Hodge--Tate algebra (in particular the filtration satisfies the assumption in ), namely $$\bigoplus_{i \in \mathbb{N}} {\mathrm{Gr}}^i_N({\mathlarger{\mathbbl{\Delta}}}_S) t^{-i} \cong \bigoplus_{i \in \mathbb{N}}{\mathrm{Fil}}_{i}^{\mathrm{conj}}\overline{{\mathlarger{\mathbbl{\Delta}}}}_S\{i\} t^{-i}$$ (similarly with $S$ replaced by $S^{(0)}$). Applying again (with the $f$ there being $d/t$ here), we see that it suffices to show $\overline{{\mathlarger{\mathbbl{\Delta}}}}_S \to \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S^{(0)}}$ and $\bigoplus_{i \in \mathbb{N}} {\mathrm{Gr}}_i^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_S) t^{-i} \to \bigoplus_{i \in \mathbb{N}} {\mathrm{Gr}}_i^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S^{(0)}}) t^{-i}$ are completely faithfully flat. Here we have also used the fact that ${\mathlarger{\mathbbl{\Delta}}}_S \to {\mathlarger{\mathbbl{\Delta}}}_{S^{(0)}}$ is $(p, I)$-completely faithfully flat if and only if $\overline{{\mathlarger{\mathbbl{\Delta}}}}_S \to \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S^{(0)}}$ is $p$-completely faithfully flat. Now we apply to the conjugate filtered Hodge--Tate rings, we see that all we need to show is the $p$-completely faithfully flatness of the graded algebra map (of conjugate filtration). Since the graded algebra of conjugate filtration is the (derived) Hodge algebra, the map ($p$-completed) is identified with $\Gamma^*_{S}(\mathbb{L}_{S/R}[-1])^{\wedge} \to \Gamma^*_{S^{(0)}}(\mathbb{L}_{S^{(0)}/R}[-1])^{\wedge}$. Since the target is $S^{(0)}$-algebra, we see this map can be factored through the base change along $S \to S^{(0)}$ which is $p$-completely faithfully flat. Finally, we are reduced to showing the map $\Gamma^*_{S^{(0)}}(\mathbb{L}_{S/R}[-1] \widehat{\otimes}_S S^{(0)})^{\wedge} \to \Gamma^*_{S^{(0)}}(\mathbb{L}_{S^{(0)}/R}[-1])^{\wedge}$ (induced by the natural map of cotangent complexes) is $p$-completely faithfully flat. This follows from the fact that $$\mathrm{Cone}(\mathbb{L}_{S/R}[-1] \widehat{\otimes}_S S^{(0)} \to \mathbb{L}_{S^{(0)}/R}[-1]) = \mathbb{L}_{S^{(0)}/S}[-1]$$ is a $p$-completely flat $S^{(0)}$-module thanks to the quasi-syntomicity of $S \to S^{(0)}$ and the assumption that $S^{(0)}$ is semiperfectoid. Let us now show the statement (2) above. Such a filtered map is a filtered isomorphism if and only if its underlying and graded maps are both isomorphisms. By [@BS19 Theorem 12.2], the graded map is identified with: $$\big(\bigoplus_{i \in \mathbb{N}} {\mathrm{Fil}}_i^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}}) t^{-i}\big) \widehat{\otimes}_{\big(\bigoplus_{i \in \mathbb{N}} {\mathrm{Fil}}_i^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_S) t^{-i}\big)} \big(\bigoplus_{i \in \mathbb{N}} {\mathrm{Fil}}_i^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S^{(0)}}) t^{-i}\big) \to \big(\bigoplus_{i \in \mathbb{N}} {\mathrm{Fil}}_i^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}^{(0)}}) t^{-i}\big).$$ Via the Rees's construction, the above being an isomorphism is equivalent to the following filtered map being an isomorphism: $${\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}}) \widehat{\otimes}_{{\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_S)} {\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S^{(0)}}) \xrightarrow{\cong} {\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}^{(0)}}).$$ Now for a map of increasingly $\mathbb{N}$-indexed $p$-completely exhaustively filtered complexes, being a filtered isomorphism is equivalent to its graded map being an isomorphism. After taking graded algebras, we are reduced to showing the following graded map being an isomorphism: $$\Gamma^*_{\widetilde{S}}(\mathbb{L}_{\widetilde{S}/R}[-1])^{\wedge} \widehat{\otimes}_{\Gamma^*_{S}(\mathbb{L}_{S/R}[-1])^{\wedge}} \Gamma^*_{S^{(0)}}(\mathbb{L}_{S^{(0)}/R}[-1])^{\wedge} \xrightarrow{\cong} \Gamma^*_{\widetilde{S}^{(0)}}(\mathbb{L}_{\widetilde{S}^{(0)}/R}[-1])^{\wedge}.$$ The left hand side is identified with $\Gamma^*_{\widetilde{S}^{(0)}}\big(\mathrm{Cone}(\mathbb{L}_{S/R}[-1] \widehat{\otimes}_S \widetilde{S}^{(0)} \to \mathbb{L}_{S^{(0)}/R}[-1] \widehat{\otimes}_{S^{(0)}} \widetilde{S}^{(0)} \oplus \mathbb{L}_{\widetilde{S}/R}[-1] \widehat{\otimes}_{\widetilde{S}} \widetilde{S}^{(0)})\big)^{\wedge}$, and we are further reduced to showing the following map of $\widetilde{S}^{(0)}$-modules is an isomorphism: $$\mathrm{Cone}(\mathbb{L}_{S/R}[-1] \widehat{\otimes}_S \widetilde{S}^{(0)} \to \mathbb{L}_{S^{(0)}/R}[-1] \widehat{\otimes}_{S^{(0)}} \widetilde{S}^{(0)} \oplus \mathbb{L}_{\widetilde{S}/R}[-1] \widehat{\otimes}_{\widetilde{S}} \widetilde{S}^{(0)}) \to \mathbb{L}_{\widetilde{S}^{(0)}/R}[-1]^{\wedge},$$ which follows from the fact that $\widetilde{S}^{(0)}$ is given by $p$-completely (derived) tensor of $S^{(0)}$ and $\widetilde{S}$ over $S$ and functoriality of cotangent complexes. This finishes the proof of the graded part of the Nygaard-filtered base change formula, and it also simultaneously proves the part about underlying complexes: Since in the process we have showed the following filtered map being an isomorphism: $${\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}}) \widehat{\otimes}_{{\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_S)} {\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S^{(0)}}) \xrightarrow{\cong} {\mathrm{Fil}}_{\bullet}^{\mathrm{conj}}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}^{(0)}}).$$ Taking its underlying map, we get the following isomorphism: $\overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}} \widehat{\otimes}_{\overline{{\mathlarger{\mathbbl{\Delta}}}}_S} \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S^{(0)}} \xrightarrow{\cong} \overline{{\mathlarger{\mathbbl{\Delta}}}}_{\widetilde{S}^{(0)}}$, which in turn shows that its $I$-completely deformed map is also an isomorphism: ${\mathlarger{\mathbbl{\Delta}}}_{\widetilde{S}} \widehat{\otimes}_{{\mathlarger{\mathbbl{\Delta}}}_S} {\mathlarger{\mathbbl{\Delta}}}_{S^{(0)}} \xrightarrow{\cong} {\mathlarger{\mathbbl{\Delta}}}_{\widetilde{S}^{(0)}}$ (by derived Nakayama's Lemma). As for the descent of $F$-gauges, since we have established the descent of gauges we are reduced to showing descent of the Frobenius morphism which follows from flat descent. ◻ **Lemma 1**. *In , the descent of coherent ($F$-)gauges is again coherent.* *Proof.* Perfectness can be checked flat locally, see [@stacks-project [Tag 068T](https://stacks.math.columbia.edu/tag/068T)], so all we need to check is the cohomological concentration property. This follows immediately from . ◻ The following is our main theorem in this subsection, which says that any coherent $I$-torsionfree prismatic $F$-crystal can be canonically extended to a coherent prismatic $F$-gauge equipped with "the saturated Nygaardian filtration". This extends the special case of $X={\mathrm{Spf}}(\mathcal{O}_K)$ in [@Bha23 Theorem 6.6.13] to arbitrary smooth $p$-adic formal schemes over ${\mathrm{Spf}}(\mathcal{O}_K)$. **Theorem 1**. *Let $X$ be a smooth formal scheme over $\mathcal{O}_K$.* 1. *There is a functor $$\Pi_X: {\mathrm{F\text{-}Crys}}^{I\text{-}\mathrm{tf}}(X) \longrightarrow {\mathrm{F\text{-}Gauge}}^{\mathrm{coh}}(X),$$ characterized by the requirement that for any $S \in X_{\mathrm{qrsp}}$, where ${\mathrm{Spf}}(S) \to X$ is $p$-completely flat, the restriction of $\Pi_X(\mathcal{E})$ on ${\mathrm{Spf}}(S)_{\mathlarger{\mathbbl{\Delta}}}$ satisfies ${\mathrm{Fil}}^\bullet(\Pi_X(\mathcal{E})({\mathlarger{\mathbbl{\Delta}}}_S)) = \widetilde{\varphi}_\mathcal{E}^{-1}(I^\bullet \mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S))$. Here we are regarding both the source and target as additive categories.* 2. *The functor $\Pi_X$ is right adjoint to the forgetful functor from $I$-torsionfree $F$-gauges to $I$-torsionfree $F$-crystals. In fact, for any pair of $(E_1, {\mathrm{Fil}}^{\bullet} E_1, \widetilde{\varphi}_{E_1}) \in {\mathrm{F\text{-}Gauge}}^{{\mathrm{coh}}}(X)$ and $(\mathcal{E}_2, \widetilde{\varphi}_{\mathcal{E}_2}) \in {\mathrm{F\text{-}Crys}}^{I\text{-}\mathrm{tf}}(X)$, we have an identification of homomorphisms: $$\mathrm{Hom}_{{\mathrm{F\text{-}Crys}}^{{\mathrm{coh}}}(X)}(E_1, \mathcal{E}_2) = \mathrm{Hom}_{{\mathrm{F\text{-}Gauge}}^{{\mathrm{coh}}}(X)}(E_1, \Pi_X(\mathcal{E}_2)).$$* 3. *The functor $\Pi_X$ is compatible with étale pullback in $X$.* We need some preparatory discussion on local situations first. We fix the following assumption for the rest of the subsection. **Situation 1**. Let $U = {\mathrm{Spf}}(R)$ be an affine open of $X$, and let $(A, I = (d))$ be an oriented Breuil--Kisin prism of $U$ whose existence is again guaranteed by deformation theory (see for example [@DLMS Example 3.4]). Denote the perfection $(A_{{\mathrm{perf}}}, I A_{{\mathrm{perf}}})$ by $(A^{(0)}, I A^{(0)})$. We denote the value of $\mathcal{E}$ on $(A, I)$ and $(A^{(0)}, I A^{(0)})$ by $M$ and $M^{(0)}$, they are equipped with linearized Frobenii $\varphi_M$ and $\varphi_{M^{(0)}}$ respectively. **Construction 1**. Recall that the $F$-crystal structure provides us with a linearized Frobenii $\varphi^*_A M \xrightarrow{\varphi_M} M[1/I]$ and $\varphi^*_{A^{(0)}} M^{(0)} \xrightarrow{\varphi_{M^{(0)}}} M^{(0)}[1/I]$. We define the *twisted filtration* on $\varphi^*_A M$ (resp. $\varphi^*_{A^{(0)}} M^{(0)}$) by the following formula $${\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_A M \coloneqq \varphi_M^{-1}\big(I^{\bullet} M\big) \text{ (resp.~}{\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_{A^{(0)}} M^{(0)} \coloneqq \varphi_{M^{(0)}}^{-1}\big(I^{\bullet} M^{(0)}\big)\text{).}$$ **Lemma 1**. *Let $E$ be an $I$-torsionfree $F$-crystal having height in $[a, b]$, and denote the associated graded of the twisted filtration on $\varphi^*_A M = \varphi^*_A E(A)$ above by $N$, viewed as a graded $R[u] = {\mathrm{Gr}}_{I^{\mathrm{N}}}(A)$-module. Then $N$ is $u$-torsionfree, lives in degrees $\geq a$, and $N^{\deg = i} \xrightarrow{\cdot u} N^{\deg = i+1}$ is an isomorphism provided $i \geq b$. In particular, both $N$ and $N[1/u]/N$ have bounded $p$-power torsion.* *Proof.* Since $E$ is $I$-torsionfree, we know that the linearized Frobenius is injective. Therefore we have an identification: ${\mathrm{Fil}}^i_{\mathrm{tw}}(M) = \mathrm{Im}(\varphi_M) \cap I^i M$. Consequently, we have $$N^{\deg = i} = \mathrm{Im}(\mathrm{Im}(\varphi_M) \cap I^i M \to I^{i}M/I^{i+1}M),$$ which immediately implies $u$-torsionfreeness. The rest of the second sentence follows from the height condition. It follows that the $R[u]$-module $N$ is generated by its degree $[a,b]$ pieces, and since each graded piece is finitely generated over $R$, we see $N$ is finitely generated over $R[u]$. Since $R[u]$ is Noetherian, we see that $N$ has bounded $p$-power torsion. The $R$-module $N[1/u]/N$ is given by the direct sum of infinite copies of $M/I$ together with $\mathrm{Coker}(N^{\deg = i} \xrightarrow{\cdot u^{b-i}} N^{\deg = b})$ where $i \in [a,b-1]$. This explicit description shows the boundedness claim of $N[1/u]/N$. ◻ We get the following consequence concerning perfectness of the twisted filtration on $\varphi^*_A M$. **Lemma 1**. *The ${\mathrm{Rees}}({\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_A M)$ is a perfect module of ${\mathrm{Rees}}(I^{\mathrm{N}}) = A[t, u]/(ut - d)$.* *Proof.* Since perfectness can be tested point-wise in the sense of [@stacks-project [Tag 068V](https://stacks.math.columbia.edu/tag/068V)], it suffices to show perfectness after $(p, I)$-completely inverting or quotient by $t$. Invert $t$, the ring becomes $A[t^{\pm 1}]^{\wedge}$ which is regular, and the module becomes $\varphi^*_A M[t^{\pm 1}]^{\wedge}$ which is finitely generated over the aforesaid ring. The perfectness in this case follows from [@stacks-project [Tag 066Z](https://stacks.math.columbia.edu/tag/066Z)]. Modulo $t$, our ring becomes $R[u]$ and our module becomes $N$ as in the previous lemma. During the proof of the said lemma, we have showed that $N$ is finitely generated over $R[u]$, hence the same argument in the invert $t$ case applies again. ◻ We need the following auxiliary lemma which says that the saturated Nygaardian filtration on $\Pi_U(E) \mid_{A^{(0)}}$ has a model over the filtered ring $(A, I^{\mathbb{N}})$. **Lemma 1**. *We have the following filtered base change formula between twisted filtrations constructed in : $${\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_A M \widehat{\otimes}_{(A, I^{\mathbb{N}})} (A^{(0)}, I^{\mathbb{N}} A^{(0)}) \xrightarrow{\cong} {\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_{A^{(0)}} M^{(0)},$$ as well as the following filtered base change formula between the twisted filtration and the saturated Nygaardian filtration: $${\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_{A^{(0)}} M^{(0)} \widehat{\otimes}_{(A^{(0)}, I^{\mathbb{N}} A^{(0)}), \varphi^{-1}} (A^{(0)}, \varphi^{-1}(I^{\mathbb{N}} A^{(0)}) = {\mathrm{Fil}}^{\bullet}_N(A^{(0)})) \xrightarrow{\cong} {\mathrm{Fil}}^{\bullet} M^{(0)}.$$* Here we are using the fact that $A^{(0)}$ is a perfect prism, hence its Frobenius is invertible. *Proof.* Note that we have the following relation between Rees algebras associated with various filtered rings showing up in the statement: $$\mathrm{Rees}(I^{\mathbb{N}}A) \widehat{\otimes}_A A^{(0)} \cong \mathrm{Rees}(I^{\mathbb{N}} A^{(0)})^{\wedge} \text{ and }$$ $$\mathrm{Rees}(I^{\mathbb{N}} A^{(0)}) \widehat{\otimes}_{A^{(0)}, \varphi^{-1}} A^{(0)} \cong \mathrm{Rees}(\varphi^{-1}(I^{\mathbb{N}} A^{(0)}))^{\wedge} = \mathrm{Rees}({\mathrm{Fil}}^{\bullet}_{N}{\mathlarger{\mathbbl{\Delta}}}_{A^{(0)}/I})^{\wedge}.$$ Via Rees's construction (see [@Bha23 §2.2]), our first base change formula is equivalent to the following formula: $${\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_A M \widehat{\otimes}_A A^{(0)} \xrightarrow{\cong} {\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_{A^{(0)}} M^{(0)}$$ which follows from the fact that $A \to A^{(0)}$ is flat and under this flat map the $\varphi_M$ is base changed to $\varphi_{M^{(0)}}$. Similarly, our second base change formula is equivalent to: $${\mathrm{Fil}}^{\bullet}_{\mathrm{tw}} \varphi^*_{A^{(0)}} M^{(0)} \widehat{\otimes}_{A^{(0)}, \varphi^{-1}} A^{(0)} \xrightarrow{\cong} {\mathrm{Fil}}^{\bullet} M^{(0)}.$$ Recall that the filtration ${\mathrm{Fil}}^{\bullet} M^{(0)}$ is defined by the preimage of $I$-adic filtration under the semi-linear Frobenius $\widetilde{\varphi}_{M^{(0)}}$. Therefore our second base change formula follows from the fact that $\varphi^{-1}_{A^{(0)}}$ is an isomorphism and upon base changing the source via $\varphi^{-1}_{A^{(0)}}$ the linearized Frobenius $\varphi_{M^{(0)}}$ becomes the semi-linear Frobenius $\widetilde{\varphi}_{M^{(0)}}$. ◻ The following crucial proposition, which is inspired by the proof of [@Bha23 Lemma 6.6.10], shows that the saturated Nygaardian filtration satisfies filtered base change formula in a particular situation. **Proposition 1**. *Let $R^{(0)} \coloneqq A^{(0)}/I$ and let $S$ be a $p$-completely flat quasi-syntomic $R^{(0)}$-algebra, then the natural filtered base change is a filtered isomorphism: $$\left(M^{(0)}, \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} M^{(0)})\right) \widehat{\otimes}_{(A^{(0)}, {\mathrm{Fil}}^{\bullet}_N)} ({\mathlarger{\mathbbl{\Delta}}}_S, {\mathrm{Fil}}^{\bullet}_N) \xrightarrow{\cong} \left(\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S), \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} \mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S))\right).$$ Consequently, if $S_1 \to S_2$ is a morphism in $X_{{\mathrm{qrsp}}}$ and suppose they are $p$-completely flat over some $R^{(0)}$ constructed out of an oriented Breuil--Kisin prism $(A, I = (d))$, then the natural filtered base change is a filtered isomorphism: $$\left(\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{S_1}), \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} \mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{S_1}))\right) \widehat{\otimes}_{({\mathlarger{\mathbbl{\Delta}}}_{S_1}, {\mathrm{Fil}}^{\bullet}_N)} ({\mathlarger{\mathbbl{\Delta}}}_{S_2}, {\mathrm{Fil}}^{\bullet}_N) \xrightarrow{\cong} \left(\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{S_2}), \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} \mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{S_2}))\right).$$* *Proof.* Since the map always induces an isomorphism of the underlying complex, thanks to $\mathcal{E}$ being a crystal, all we need to check is that the tensor product filtration (defined by the left hand side of the above equation) and the saturated Nygaardian filtration (defined by the right hand side of the above equation) agree. Now we recall in the proof of [@Bha23 Lemma 6.6.10], Bhatt observes the following characterization of the saturated Nygaardian filtration: **Porism 1** (Follows from the proof of [@Bha23 Lemma 6.6.10]). *Let $A$ be a ring and let $(M_1, {\mathrm{Fil}}^{\bullet} M_1) \xrightarrow{\widetilde{\varphi}} (M_2, {\mathrm{Fil}}^{\bullet} M_2)$ be a map of two decreasing filtered objects in ${\mathcal{DF}}(A)$ such that* 1. *${\mathrm{Fil}}^{\ll 0} M_1 \to M_1$ is an isomorphism, all ${\mathrm{Fil}}^{\bullet} M_1$'s are connective;* 2. *The graded complexes ${\mathrm{Gr}}^{\bullet}(M_1)$ are coconnective;* 3. *$({\mathrm{Fil}}^{\bullet} M_2)$ are $A$-modules with injective transitions; and* 4. *The graded map ${\mathrm{Gr}}(\widetilde{\varphi}) \colon {\mathrm{Gr}}^{\bullet}(M_1) \to {\mathrm{Gr}}^{\bullet}(M_2)$ have coconnective cone.* *Then $(M_1, {\mathrm{Fil}}^{\bullet} M_1)$ is also an honest decreasingly filtered $A$-module with its filtration given by ${\mathrm{Fil}}^{\bullet} M_1 = \widetilde{\varphi}^{-1} ({\mathrm{Fil}}^{\bullet} M_2)$.* *Proof.* This is an exercise in homological algebra, below we give some hints. The conditions (i) and (ii) imply that $(M_1, {\mathrm{Fil}}^{\bullet} M_1)$ is an honest decreasingly filtered $A$-module. Then condition (iii) and (iv) implies that ${\mathrm{Fil}}^{i+1} M_1 = {\mathrm{Fil}}^i M_1 \cap \widetilde{\varphi}^{-1} ({\mathrm{Fil}}^{i+1} M_2)$, finishing the proof. ◻ We shall verify that the filtered map $$\left(M^{(0)}, \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} M^{(0)})\right) \widehat{\otimes}_{(A^{(0)}, {\mathrm{Fil}}^{\bullet}_N)} ({\mathlarger{\mathbbl{\Delta}}}_S, {\mathrm{Fil}}^{\bullet}_N) \xrightarrow{\widetilde{\varphi}_\mathcal{E}} \left(\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S)[1/I], I^{\bullet} \mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S)\right).$$ satisfies the conditions of , which will show that the tensor product filtration agrees with the saturated Nygaardian filtration. The condition (i) is stable under filtered base change between honestly filtered algebras, therefore the tensor product filtration satisfies the condition (i) as $(M^{(0)}, \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} M^{(0)}))$ clearly satisfies it. The condition (iii) is easily seen to be satisfied. Finally we need to verify (ii) and (iv) in the Porism above concerning the behavior of graded pieces. For (ii): The graded pieces of the tensor product filtration is given by ${\mathrm{Gr}}(M^{(0)}) \widehat{\otimes}_{{\mathrm{Gr}}_N(A^{(0)})} {\mathrm{Gr}}_N({\mathlarger{\mathbbl{\Delta}}}_S)$. By , we get the following description of ${\mathrm{Gr}}(M^{(0)}) = {\mathrm{Gr}}_{\mathrm{tw}}(\varphi^*_A M) \widehat{\otimes}_{R[u]} {\mathrm{Gr}}_N(A^{(0)})$. Here the tensor is $p$-completed and the base change map is given by taking graded map of composition of the following filtered maps (where $u$ is the image of $d$ in $I/I^2$): $$(A, I^{\mathbb{N}} A) \to (A^{(0)}, I^{\mathbb{N}} A^{(0)}) \xrightarrow{\varphi^{-1}_{A^{(0)}}} (A^{(0)}, {\mathrm{Fil}}^{\bullet}_N).$$ In the proof of , we have seen that the map ${\mathrm{Gr}}_N(A^{(0)}) \to {\mathrm{Gr}}_N({\mathlarger{\mathbbl{\Delta}}}_S)$ is $p$-completely flat. Precomposing with the $p$-completely flat map $R[u] \to {\mathrm{Gr}}_N(A^{(0)})$, we see that the induced map $R[u] \to {\mathrm{Gr}}_N({\mathlarger{\mathbbl{\Delta}}}_S)$ is $p$-completely flat. Therefore we see that ${\mathrm{Gr}}_{\mathrm{tensor}}(E({\mathlarger{\mathbbl{\Delta}}}_S)) = {\mathrm{Gr}}_{\mathrm{tw}}(\varphi^*_A M) \widehat{\otimes}_{R[u]} {\mathrm{Gr}}_N({\mathlarger{\mathbbl{\Delta}}}_S)$ is concentrated in degree $0$ because ${\mathrm{Gr}}_{\mathrm{tw}}(\varphi^*_A M)$ has bounded $p$-power torsion (). As for (iv): just like the proof of [@Bha23 Lemma 6.6.10], we may identify the map $${\mathrm{Gr}}_{\mathrm{tensor}}(E({\mathlarger{\mathbbl{\Delta}}}_S)) \xrightarrow{\widetilde{\varphi}_\mathcal{E}} {\mathrm{Gr}}_{I\text{-adic}}(E({\mathlarger{\mathbbl{\Delta}}}_S)[1/I])$$ as the $p$-completely base change of ${\mathrm{Gr}}_N(M^{(0)}) \to {\mathrm{Gr}}_N(M^{(0)})[1/u]$ along the map ${\mathrm{Gr}}_N(A^{(0)}) \to {\mathrm{Gr}}_N({\mathlarger{\mathbbl{\Delta}}}_S)$. Using again, we may identify the above map further as the following $p$-completely base changed map $$\left({\mathrm{Gr}}_{\mathrm{tw}}(\varphi^*_A M) \to {\mathrm{Gr}}_{\mathrm{tw}}(\varphi^*_A M)[1/u]\right) \widehat{\otimes}_{R[u]} {\mathrm{Gr}}_N({\mathlarger{\mathbbl{\Delta}}}_S).$$ Therefore it suffices to notice that $\left({\mathrm{Gr}}_{\mathrm{tw}}(\varphi^*_A M)[1/u]/{\mathrm{Gr}}_{\mathrm{tw}}(\varphi^*_A M)\right)$ has bounded $p$-power torsion (). ◻ Now we are ready to prove , let us stress again that it is inspired by [@Bha23 Lemma 6.6.10]. *Proof of .* (1): we adopt the notation in the discussion right after the statement. Let $U = {\mathrm{Spf}}(R)$ be an affine open of $X$. Let $R^{(0)} \coloneqq A^{(0)}/I$ and let $R^{(\bullet)}$ be the Cech nerve of the quasi-syntomic cover $R \to R^{(0)}$ with their absolute prismatic cohomology $A^{(\bullet)} \coloneqq {\mathlarger{\mathbbl{\Delta}}}_{R^{(\bullet)}}$. We denote $E(A^{(\bullet)})$ by $M^{(\bullet)}$, and by abuse of notation we denote the semi-linear Frobenii on these $M^{(\bullet)}$ by the same symbol $\widetilde{\varphi}_\mathcal{E}$. Since $F$-gauges form a quasi-syntomic sheaf (), we need to first check that the "saturated Nygaardian filtrations" on $\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{R^{(i)}})$ satisfies filtered base change with respect to the various maps induced by the simplicial maps between the $R^{(i)}$'s. To that end, let $[i] \to [j]$ be an arrow in $\Delta$, we need to verify the natural map $$\left(M^{(i)}, \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} M^{(i)})\right) \widehat{\otimes}_{(A^{(i)}, {\mathrm{Fil}}^{\bullet}_N)} (A^{(j)}, {\mathrm{Fil}}^{\bullet}_N) \to \left(M^{(j)}, \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} M^{(j)})\right)$$ is a filtered isomorphism. This immediately follows from . This shows that the saturated Nygaardian filtration defines an $F$-gauge $\Pi_U(\mathcal{E}|_U)$ on $U$, and the construction is easily seen to be functorial in $\mathcal{E}$ and $U$. The perfectness follows from combining and . Next we verify that these $\Pi_U(\mathcal{E}|_U)$'s glue to an $F$-gauge on $X$. To that end, it suffices to check the following **Claim 1**. *For any ${\mathrm{Spf}}(S) \in U_{{\mathrm{qrsp}}}$ with $p$-completely flat structural map to $U$, the value $\Pi_U(\mathcal{E}|_U)({\mathrm{Spf}}(S))$ has its filtration given by the saturated Nygaardian filtration.* Granting the above claim, then for any two affine opens $U$ and $V$, the restriction of $\Pi_U(\mathcal{E}|_U)|_{U \cap V}$ and $\Pi_V(\mathcal{E}|_V)|_{U \cap V}$ will be canonically identified: their values on any ${\mathrm{Spf}}(S)$ which are flat and quasi-syntomic over $U \cap V$ are canonically identified (the filtrations are given by the saturated Nygaardian filtration), finally we just notice that these ${\mathrm{Spf}}(S)$'s form a basis of $(U \cap V)_{{\mathrm{qrsp}}}$, and $F$-gauges form a quasi-syntomic sheaf (). *Proof of the above claim:.* Let us base change the Cech nerve $R^{(\bullet)}$ along the map $R \to S$, and denote the resulting Cech nerve $S^{(\bullet)}$. The underlying gauge of $\Pi_U(\mathcal{E}|_U)({\mathrm{Spf}}(S))$ is the descent of the gauges on ${\mathrm{Spf}}(S^{(\bullet)})$, and the latter gauges are given by the filtered base change $\left(M^{(i)}, \widetilde{\varphi}_\mathcal{E}^{-1}(I^{\bullet} M^{(i)})\right) \widehat{\otimes}_{(A^{(i)}, {\mathrm{Fil}}^{\bullet}_N)} ({\mathlarger{\mathbbl{\Delta}}}_{S^{(i)}}, {\mathrm{Fil}}^{\bullet}_N)$. Using , we see that the latter gauges are nothing but the saturated Nygaardian filtered module $\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{S^{(i)}})$. By , we see the descent gauge $(\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S), {\mathrm{Fil}}_{\mathrm{descent}})$ is coherent, namely it is an honest decreasingly filtered ${\mathlarger{\mathbbl{\Delta}}}_S$-module. Finally we apply for the Frobenius filtered map $(\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S), {\mathrm{Fil}}_{\mathrm{descent}}^{\bullet}) \xrightarrow{\widetilde{\varphi}_\mathcal{E}} (\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S)[1/I], I^{\bullet}\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S))$: we have verified conditions (i) and (ii); the condition (iii) follows from coherence of $\mathcal{E}$ and ; finally the condition (iv) is stable under descent as taking limit preserves co-connectivity, hence it is satisfied for our map (as this map is the descent of maps which satisfy condition (iv)). Therefore we may conclude that the descent filtration is again the saturated Nygaardian filtration, which finishes our proof of the claim. ◻ To finish the proof of (1), we need to see that for any ${\mathrm{Spf}}(S) \in X_{{\mathrm{qrsp}}}$ with $p$-completely flat structural map to $X$, the value $\Pi_X(\mathcal{E})({\mathrm{Spf}}(S))$ has its filtration given by the saturated Nygaardian filtration. We may choose a Zariski cover of ${\mathrm{Spf}}(S)$ by affine opens such that each affine open maps to an affine open in $X$. Then the filtration on $\Pi_X(\mathcal{E})({\mathrm{Spf}}(S))$ is given by descent of saturated Nygaardian filtration (by the above claim), therefore we can use the same argument as above again to conclude that the filtration we get is the saturated Nygaardian filtration on $\mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_S)$. Next we show (2): Let $E_1$ and $\mathcal{E}_2$ be as in the statement. Notice that mapping spaces $\mathrm{Map}_{{\mathrm{F\text{-}Crys}}^{{\mathrm{coh}}}(X)}(E_1, \mathcal{E}_2)$ and $\mathrm{Map}_{{\mathrm{F\text{-}Gauge}}^{{\mathrm{coh}}}(X)}(E_1, \Pi_X(\mathcal{E}_2))$ are discrete, we see that the Hom groups are glued from the Hom groups on affine opens of $U \subset X$, so we have reduced ourselves to the case of $X$ being an affine. In this case, the Hom groups are computed by the corresponding Hom groups of values of $E_1$, $\mathcal{E}_2$, and $\Pi_X(\mathcal{E}_2)$ at those ${\mathrm{Spf}}(S) \in X_{{\mathrm{qrsp}}}$ which are flat over $X$. So we are finally reduced to checking the analogous statement for $X$ replaced by these qrsp ${\mathrm{Spf}}(S)$. Since any Frobenius-equivariant map $E_1({\mathlarger{\mathbbl{\Delta}}}_S) \to \mathcal{E}_2({\mathlarger{\mathbbl{\Delta}}}_S)$ uniquely extends to a filtered map $E_1({\mathlarger{\mathbbl{\Delta}}}_S) \to \Pi_X(\mathcal{E}_2)({\mathlarger{\mathbbl{\Delta}}}_S)$, as the latter is equipped with the saturated Nygaardian filtration, the two Hom groups are canonically identified as desired. Lastly we deal with (3): Let $V \to U \to X$ be two affines in $X_{\acute{e}t}$, we just need to show the natural functor ${\mathrm{F\text{-}Gauge}}^{\mathrm{perf}}(U) \to {\mathrm{F\text{-}Gauge}}^{\mathrm{perf}}(V)$ sends $\Pi_U(\mathcal{E} \mid_U) \mapsto \Pi_V(\mathcal{E} \mid_V)$. Choose a Breuil--Kisin prism $(A, I)$ for $U$, the étale map $V \to U$ canonically lifts $(A, I)$ to a Breuil--Kisin prism $(A', IA')$ over $(A, I)$ in $U_{{\mathlarger{\mathbbl{\Delta}}}}$. Notice that $A \to A'$ is $(p, I)$-completely flat, hence flat by Noetherianity of $A$. Tracing through the argument of proving (1), we see that it suffices to show the twisted filtrations on $\varphi_A^* \mathcal{E}(A)$ and $\varphi_{A'}^* \mathcal{E}(A')$ are related via $${\mathrm{Fil}}_{\mathrm{tw}} \varphi_A^* \mathcal{E}(A) \widehat{\otimes}_A A' \xrightarrow{\cong} {\mathrm{Fil}}_{\mathrm{tw}} \varphi_{A'}^* \mathcal{E}(A'),$$ which follows directly from the flatness of $A \to A'$ and how the twisted filtrations are defined. ◻ **Remark 1**. In fact we may define the functor on all of coherent $F$-crystals, by pulling back filtrations from their $I$-torsionfree quotients. The resulting graded modules will agree with the graded modules of the $F$-gauges of their $I$-torsionfree quotients. Since we do not need this greater generality, we restrict ourselves to working with $I$-torsionfree coherent $F$-crystals only. ## Weight filtration on graded pieces of gauges {#absolute weight filtration subsection} In this subsection, we introduce a so-called weight filtration on the associated graded of a gauge. To start, we first define the notion of *weight* for an ($F$-)gauge, using a reduction functor that sends the graded piece of a gauge over $X$ to a graded complex over $\mathcal{O}_X$. **Construction 1**. Let $X$ be a quasi-syntomic $p$-adic formal scheme, and let $\ast\in \{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$. We define the *reduction functor* on the category ${\mathcal{DG}}^\ast_{p\text{-comp}}(X_{\mathrm{qrsp}}, {\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}) \simeq \lim_{S\in X_{\mathrm{qrsp}}} {\mathcal{DG}}^\ast_{p\text{-comp}} ({\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S)$ as the following composition of functors $$\begin{tikzcd} {\mathrm{Red}}_X \colonequals {\mathcal{DG}}^\ast_{p\text{-comp}}({\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}) \arrow[rr, "-\otimes_{{\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}} \mathcal{O}_{\mathrm{qrsp}}"] && {\mathcal{DG}}^\ast_{p\text{-comp}}(\mathcal{O}_X);\\ M^\bullet \arrow[rr, mapsto] && M^\bullet\bigotimes_{{\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}} \mathcal{O}_{\mathrm{qrsp}}, \end{tikzcd}$$ where $\mathcal{O}_{\mathrm{qrsp}}= {\mathrm{Gr}}_N^0 {\mathlarger{\mathbbl{\Delta}}}$ is regarded as a graded ${\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}$-algebra via projection onto degree $0$ piece. The $i$-th graded piece of ${\mathrm{Red}}_X(-)$ is denoted as ${\mathrm{Red}}_{i,X}(-)$. Here we implicitly use the $p$-completely flat descent of $p$-complete complexes (resp. perfect complexes, resp. vector bundles) for the target category, namely $${\mathcal{DG}}^\ast_{p\text{-comp}}(\mathcal{O}_X) \simeq \lim_{S\in X_{\mathrm{qrsp}}} {\mathcal{DG}}^\ast_{p\text{-comp}}(S).$$ In particular, we have the natural limit formula for the reduction functor $${\mathrm{Red}}_X \simeq \lim_{S\in X_{\mathrm{qrsp}}} {\mathrm{Red}}_S.$$ When the choice of the formal scheme $X$ is clear, we omit $X$ in the subscripts and use ${\mathrm{Red}}$ and ${\mathrm{Red}}_i$ to abbreviate ${\mathrm{Red}}_X$ and ${\mathrm{Red}}_{i,X}$. By a slight abuse of notation, for a gauge $(E,{\mathrm{Fil}}^\bullet E)$ over $X$, we also abbreviate the notation ${\mathrm{Red}}_X({\mathrm{Gr}}^\bullet E)$ as ${\mathrm{Red}}_X(E)$ when there is no confusion. We then define the notion of weights for a gauge. **Definition 1**. Let $X$ be a quasi-syntomic formal scheme, and let $[a, b]$ be an interval in $\mathbb{R}\cup \{-\infty, \infty\}$. For a graded complex $M^\bullet\in {\mathcal{DG}}^\ast_{p\text{-comp}}(X_{\mathrm{qrsp}}, {\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}})$, we say it has *weights* in $[a,b]$ if $M^n = 0$ for $n \ll 0$ and $${\mathrm{Red}}_i(M^\bullet)=0,~\forall i\notin [a,b].$$ For a gauge $E=(E,{\mathrm{Fil}}^\bullet E)$ on $X$, we say it has *weights* in $[a,b]$ if its associated graded $M^\bullet={\mathrm{Gr}}^\bullet E$ is so. **Remark 1**. In the stacky language, the reduction functor on the category of gauges can be translated in terms of the pullback functor along the closed immersion $X\times B\mathbb{G}_m \to X^N$ as in [@Bha23 Rmk. 5.3.14]. Similarly the "weights" defined above corresponds to the "Hodge--Tate weights" defined in loc. cit. Using the reduction functor, we now introduce the weight filtration on the associated graded of a gauge. **Theorem 1**. *Let $X$ be a quasi-syntomic $p$-adic formal scheme, let $\ast\in \{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$, and let $a\leq b$ be two integers. Assume $M^\bullet \in {\mathcal{DG}}^\ast_{p\text{-}\mathrm{comp}}(X_{\mathrm{qrsp}}, {\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}})$ has weights in $[a,b]$. There exists a unique finite increasing and exhaustive filtration ${\mathrm{Fil}}^{\mathrm{wt}}_i (M^\bullet)$ on $M^\bullet$ indexed by $i \in [a,b]$, such that ${\mathrm{Gr}}^{\mathrm{wt}}_i(M^\bullet)$ is canonically isomorphic to ${\mathrm{Red}}_i(M^\bullet)\otimes_{\mathcal{O}_X} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}$.* Note that in the special case when $M^\bullet={\mathrm{Gr}}^\bullet E$ where $E=(E,{\mathrm{Fil}}^\bullet E)\in \mathrm{Gauge}^\ast(X)$, we get a weight filtration ${\mathrm{Fil}}^{\mathrm{wt}}_i({\mathrm{Gr}}^\bullet E)$ on ${\mathrm{Gr}}^\bullet E$ with ${\mathrm{Gr}}^{\mathrm{wt}}_i({\mathrm{Gr}}^\bullet E) \simeq {\mathrm{Red}}_i (E)\otimes_{\mathcal{O}_X} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}$. As a preparation, we first prove the following graded Nakayama lemma. **Lemma 1**. *Let $R^\bullet$ be a commutative ring which is $\mathbb{N}$-graded, and let $M^\bullet\in {\mathcal{DG}}(R^\bullet)$ be a $\mathbb{Z}$-graded complex over $R^\bullet$ such that $M^n=0$ for all $n \ll 0$.* 1. *Assume $M^\bullet \not= 0$ and let $c$ be the smallest integer $n$ such that $M^n\neq 0$. Then the graded $R^0$-complex $M^\bullet \otimes_{R^\bullet} R^0$ has no $\deg < c$ piece, and its $\deg = c$ piece is naturally isomorphic to $M^c$.* 2. *Consequently, if the base change $M^\bullet \otimes_{R^\bullet} R^0 \simeq 0$, then we have $M^\bullet \simeq 0$.* *Proof.* (2) follows from (1), below we prove (1). We take the graded tensor product of $M^\bullet$ with the following fiber sequence of $R^\bullet$-graded complexes $$R^{\geq 1} \longrightarrow R^\bullet \longrightarrow R^0,$$ resulting a fiber sequence of graded $R^0$-complexes: $$M^\bullet \otimes_{R^\bullet} R^{\geq 1} \to M^\bullet \to M^\bullet \otimes_{R^\bullet} R^0.$$ We now claim that $(M^\bullet \otimes_{R^\bullet} R^{\geq 1})^{\deg \leq c} = 0$, which implies the first statement. To show the claim, we use the bar resolution of $M^\bullet \otimes_{R^\bullet} R^{\geq 1}$, which says that it can be computed by the colimit of the simplicial diagram of $R^0$-graded complexes $$M^\bullet \otimes_{R^\bullet} R^{\geq 1} \simeq {\mathrm{colim}}_{n\in \Delta} (M^\bullet\otimes_{R^0} R^\bullet \otimes_{R^0} \cdots \otimes_{R^0} R^\bullet \otimes_{R^0} R^{\geq 1}),$$ where the $n$-th term in the right hand side has $n$ copies of $R^\bullet$ in the tensor product. Notice that for a given $n\in \Delta$, the degree $i$ piece of the $n$-th term is isomorphic to the direct sum $$\bigoplus_{u+w_1+\cdots+w_n+v=i} M^u \otimes_{R^0} R^{w_1} \otimes_{R^0} \cdots \otimes_{R^0} R^{w_n} \otimes_{R^0} R^v,$$ where each direct summand is zero unless $u\geq c$, $w_j\geq 0$ and $v\geq 1$. As a consequence, the $\deg \leq c$ piece of the entire simplicial diagram vanishes. ◻ *Proof of .* We construct the weight filtration by induction on the length of the interval $[a,b]$. If the length is negative, which means the interval is empty, then by assumption we have ${\mathrm{Red}}_X(M^{\bullet}) = 0$. By graded Nakayama  (2) we see that $M^{\bullet} = 0$ and we put trivial filtration in this case. Now we move to the induction step. We first notice that the assumption on $M^\bullet$ implies that $M^i = 0$ for $i<a$. To see this, let $S$ be any algebra in $X_{\mathrm{qrsp}}$, and let $c$ be the least integer such that $M^c \neq 0$, which exists by assumption. Using , we see that $(M^\bullet(S)) \otimes_{{\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S} S$ is a graded $S$-complex whose $\deg < c$ pieces vanish, and whose $\deg = c$ piece is exactly $M^c \not = 0$. Since $M^\bullet$ has weights in $[a,b]$, by we have $c\geq a$. The map of graded $\mathcal{O}_{\mathrm{qrsp}}$-complexes $M^a \to M^\bullet$ induces a natural map of graded ${\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}$-complexes $$M^a \bigotimes_{\mathcal{O}_{\mathrm{qrsp}}} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}\longrightarrow M^\bullet, \tag{$\ast$}$$ which is an isomorphism on the $\deg = a$ piece. We let $C^{\bullet}$ be the cofiber of the map $(\ast)$. We claim that it has weights in $[a+1,b]$ such that $C^n = 0$ for $n \ll 0$, and the natural map $M^{\bullet} \to C^{\bullet}$ induces a natural isomorphism between their $a+1 \leq \deg \leq b$ reductions. Granting this claim, then by induction we have exhibited the weight filtration on $C^{\bullet}$, which together with ${\mathrm{Fil}}^{{\mathrm{wt}}}_a \coloneqq M^a \bigotimes_{\mathcal{O}_{\mathrm{qrsp}}} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}\to M^\bullet$ forms a weight filtration on $M^{\bullet}$ as desired. Finally we verify the above claim. To see $C^n = 0$ for $n \ll 0$, one just observes that both terms in the map $(\ast)$ have no $\deg < a$ term: The latter was showed two paragraphs above, and for the former we just observe that ${\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}$ has no negative degree piece. As for the rest of the claim, we apply the reduction functor to the map $(\ast)$ and get $M^a \to {\mathrm{Red}}_X(M^{\bullet})$. By the first statement in , we see that this identifies the source as the $\deg = a$ piece of the target, hence its cofiber (which is nothing but the reduction of $C^{\bullet}$) is precisely the $a+1 \leq \deg \leq b$ reductions of $M^{\bullet}$. As for the uniqueness: Suppose there are two such filtrations denoted by ${\mathrm{Fil}}_i$ and ${\mathrm{Fil}}'_i$ ($i \in [a, b]$) respectively. We first claim that the map ${\mathrm{Fil}}_a(M^{\bullet}) \to M^{\bullet}$ canonically factors through ${\mathrm{Fil}}'_a(M^{\bullet})$. Equivalently, we need to show the induced map $${\mathrm{Fil}}_a(M^{\bullet}) \cong {\mathrm{Red}}_a(M^\bullet) \otimes_{\mathcal{O}_X} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}} \to M^{\bullet}/{\mathrm{Fil}}'_a(M^{\bullet})$$ is canonically $0$. Now we observe that such a map of graded-${\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}$-modules is induced by a map of graded-$\mathcal{O}_X$-module: $${\mathrm{Red}}_a(M^\bullet) \to M^{\bullet}/{\mathrm{Fil}}'_a(M^{\bullet}),$$ the source has degree $a$ whereas the target has degrees strictly larger than $a$ by assumption of ${\mathrm{Fil}}'$ being a weight filtration. The claim implies that we get a natural map between ${\mathrm{Fil}}_a$ and ${\mathrm{Fil}}'_a$, both by assumption are isomorphic to ${\mathrm{Red}}_a(M^\bullet) \otimes_{\mathcal{O}_X} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}$, and the induced map on reductions is an isomorphism. Hence we see that the two filtrations will have the initial piece (the $a$-th filtration) being the same, and an induction on the length of $[a,b]$ implies that the induced filtrations after quotient out the common initial piece are also the same. Therefore the two filtrations are identified. This proves the case of $* = \emptyset$. The cases of $* = {\mathrm{perf}}\text{ or } {\mathrm{vect}}$ follows from observing that the reductions are either graded perfect complexes or graded vector bundles respectively. ◻ **Remark 1**. For an integer $a$ and some $b\in \mathbb{Z}_{\geq a}\cup \{\infty\}$, we let ${\mathcal{DG}}^\ast_{p\text{-comp}}(X, {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}})_{[a,b]}^+$ be the full subcategory of ${\mathcal{DG}}^\ast_{p\text{-comp}}(X, {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}})$ whose objects have weights in $[a,b]$. Then the natural full embedding ${\mathcal{DG}}^\ast_{p\text{-comp}}(X, {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}})_{[a+1,b]}^+ \to {\mathcal{DG}}^\ast_{p\text{-comp}}(X, {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}})_{[a,b]}^+$, admits a left adjoint, sending an object $M^\bullet$ to the grade complex $C$ as in the proof of . We have the following concrete understanding of the reduction of graded pieces of $F$-gauges which come from $I$-torsionfree $F$-crystals as in . **Theorem 1**. *Let $X$ be a smooth formal scheme over $\mathcal{O}_K$, and let $(\mathcal{E}, \varphi_\mathcal{E}) \in {\mathrm{F\text{-}Crys}}^{I\text{-}\mathrm{tf}}(X)$ having height in $[a, b]$.* 1. *Then the gauge $\Pi_X(\mathcal{E})$ constructed in has the reduction of its graded pieces given by a graded coherent sheaf having gradings in $[a,b]$.* 2. *In the setting of , we have the following concrete description of $${\mathrm{Red}}_U({\mathrm{Gr}}^{\bullet}(\Pi_U(\mathcal{E}|_U))) \cong {\mathrm{Gr}}^{\bullet}_{\mathrm{tw}}(\varphi_A^*M) /^L u \otimes_R \mathcal{O}_{{\mathrm{qrsp}}}.$$* We remind readers that the twisted filtration on $\varphi_A^*M$ was discussed around , and the symbol $u$ stands for the image of $d \in I$ in ${\mathrm{Gr}}^{\bullet}(I^{\mathbb{N}}A)$ (so it has degree $1$). *Proof.* Let us show (2) first. Tracing through the proof of , especially and (notice that the $S$ there forms a basis of the qrsp site of $U$), we get the following relation between twisted filtration on $\varphi_A^*M$ and saturated filtration on $\Pi_U(\mathcal{E})$: $${\mathrm{Fil}}^{\bullet}_{\mathrm{tw}}(\varphi_A^*M) \otimes_{(A, I^{\mathbb{N}})} {\mathrm{Fil}}^{\bullet}_N {\mathlarger{\mathbbl{\Delta}}}\cong {\mathrm{Fil}}^{\bullet}_N \Pi_U(\mathcal{E}).$$ Passing to graded pieces and applying the reduction functor, we get: $${\mathrm{Red}}_U({\mathrm{Gr}}^{\bullet}(\Pi_U(\mathcal{E}|_U))) \cong {\mathrm{Gr}}^{\bullet}_{\mathrm{tw}}(\varphi_A^*M) \otimes_{R[u]} {\mathrm{Gr}}^{\bullet}_N{\mathlarger{\mathbbl{\Delta}}} \otimes_{{\mathrm{Gr}}^{\bullet}_N{\mathlarger{\mathbbl{\Delta}}}} \mathcal{O}_{{\mathrm{qrsp}}} \cong {\mathrm{Gr}}^{\bullet}_{\mathrm{tw}}(\varphi_A^*M) \otimes_{R[u]} R (=R[u]/u) \otimes_{R} \mathcal{O}_{{\mathrm{qrsp}}}.$$ To prove (1), we may work Zariski locally on $X$. Now the statement follows from (2) and the explicit knowledge of ${\mathrm{Gr}}^{\bullet}_{\mathrm{tw}}(\varphi_A^*M)$ as a graded-$R[u]$-module, see . ◻ In the remainder of this subsection, we discuss the relation between weights of an $F$-gauge and heights of its underlying $F$-crystal. First we observe that the Frobenius twist of an $F$-gauge is eventually $I$-adically filtered. **Proposition 1**. *Let $S$ be a quasiregular semiperfectoid ring, let $E=(E,{\mathrm{Fil}}^\bullet E, \widetilde{\varphi}_E)\in {\mathrm{F\text{-}Gauge}}^{\mathrm{perf}}(X)$ be of weight $[a,b]$. Denote by ${\mathrm{Fil}}^\bullet (\varphi^*E)$ the filtered base change of ${\mathrm{Fil}}^\bullet E$ along $\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}:{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S\to I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S$.* (i) *The Frobenius structure $\widetilde{\varphi}_E$ induces a filtered map $v:{\mathrm{Fil}}^\bullet (\varphi^*E) \to I^\bullet\otimes E$, which is an isomorphism on ${\mathrm{Fil}}^{\geq b}(-)$. In particular, the filtered complex ${\mathrm{Fil}}^{\geq b} (\varphi^*E)$ is isomorphic to the $I$-adic filtration $I^{\geq b}\otimes E$.* (ii) *For $i\in \mathbb{Z}$, the weight filtration induces a finite increasing exhaustive filtration of range $[a,b]$ on the map ${\mathrm{Gr}}^i (v): {\mathrm{Gr}}^i(\varphi^*E) \to \bar{I}^{i}E$, such that its $j$-th graded piece of the map is $$\begin{cases} {\mathrm{Red}}_j(E)\otimes_S \bar{I}^{i-j} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S \xrightarrow{\simeq} {\mathrm{Red}}_j(E)\otimes_S \bar{I}^{i-j} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S,~i\geq j;\\ 0 \longrightarrow {\mathrm{Red}}_j(E)\otimes_S \bar{I}^{i-j} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S,~i<j. \end{cases}$$* *Proof.* We first notice that as ${\mathrm{Fil}}^\bullet E$ is a filtered perfect complex over ${\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$, its filtered base change $${\mathrm{Fil}}^\bullet E \underset{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}}{\bigotimes} I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S$$ is also a filtered perfect complex over $I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S$. Moreover, as the ring ${\mathlarger{\mathbbl{\Delta}}}_S$ is $(p,I)$-complete, the filtered ring $I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S$ is thus filtered complete. In particular, by [^2] and the perfectness, the filtered complex ${\mathrm{Fil}}^\bullet (\varphi^* E)$ and hence ${\mathrm{Fil}}^{\geq b} (\varphi^* E)$ is also filtered complete and $(p,I)$-complete automatically. Notice that by the perfectness again, the ${\mathlarger{\mathbbl{\Delta}}}_S$-complex $E$ is $I$-adic complete. So to show the filtered map ${\mathrm{Fil}}^{\geq b}(v)$ is a filtered isomorphism, it suffices to check the associated map of the graded pieces. We then note by construction that the map $v$ can be factored as the composition $$\begin{tikzcd} {\mathrm{Fil}}^\bullet E \underset{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}}{\widehat{\bigotimes}} I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S \arrow[r,"u"]& {\mathrm{Fil}}^\bullet E \underset{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}}{\widehat{\bigotimes}} I^\mathbb{Z} {\mathlarger{\mathbbl{\Delta}}}_S \arrow[r, "\varphi_E", "\simeq"'] & I^\mathbb{Z} E, \end{tikzcd}$$ where the map $\varphi_E$ is a filtered isomorphism, and to prove (i) it suffices to show that ${\mathrm{Gr}}^{\geq b} (u)$ is a graded isomorphism. On the other hand, the weight filtration of ${\mathrm{Gr}}^\bullet E$ induces a finite increasing exhaustive filtration on the graded map ${\mathrm{Gr}}^\bullet (u)$ above, whose $j$-th associated graded is $$V_j \otimes_S \bar{I}^\mathbb{N} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S \longrightarrow V_j \otimes_S \bar{I}^\mathbb{Z} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S, \tag{$\ast$}$$ where $V_j\simeq {\mathrm{Red}}_j(E)$ is a perfect $S$-complex of graded degrees $j\in [a,b]$. This finishes the proof of (i), since $j$ is assumed to be $\leq b$. By applying ${\mathrm{Gr}}^i(-)$ at the graded map $(\ast)$, we further obtain (ii). ◻ **Remark 1**. It is clear from the proof that can be extended to more general $F$-gauges $E=(E,{\mathrm{Fil}}^\bullet E, \widetilde{\varphi}_E)$: we are only using their weights are in $[a,b]$ and ${\mathrm{Fil}}^\bullet E$ is filtered complete. In the following, we let ${\mathrm{F\text{-}Gauge}}^{I\text{-}\mathrm{tf}}(X)$ be the category of coherent $F$-gauges with $I$-torsionfree underlying $F$-crystals. **Corollary 1**. *Assume $X$ is smooth over $\mathcal{O}_K$, and $E=(E,{\mathrm{Fil}}^\bullet E, \widetilde{\varphi}_E)\in {\mathrm{F\text{-}Gauge}}^{I\text{-}\mathrm{tf}}(X)$ is of height $[a,a+1]$. Then $E = \Pi_X(\mathcal{E})$ where $\mathcal{E}$ is the underlying $F$-crystal of $E$ and $\Pi_X$ is the functor obtained in . In particular, $E$ is saturated.* *Proof.* We adopt the notation in . In particular, we let $U = {\mathrm{Spf}}(R)$ be an affine open of $X$, and let $(A, I)$ be a Breuil--Kisin prism of $U$ with perfection $(A^{(0)}, I A^{(0)})$ and $S \coloneqq \overline{A^{(0)}}$. Next we show that the value of $E$ at ${\mathrm{Spf}}(S)$ is equipped with the saturated Nygaardian filtration. As $E$ is of height $\geq a$, we have ${\mathrm{Fil}}^a E(S) = E(S)$. Moreover, by .(i), ${\mathrm{Fil}}^{\geq a+1} (\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}^* E(S))$ is isomorphic to the $I$-adic filtration on $I^{a+1}E(S)$ and thus equals to $\varphi_E^{-1}(I^{\geq a+1} E(S))$. Notice that since $\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}:{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S \to I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S$ is a filtered isomorphism, we can untwist $\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}$ to get ${\mathrm{Fil}}^{\geq a+1} E(S) \simeq \widetilde{\varphi}_E^{-1} (I^{\geq a+1} E(S))$. Hence the value $E(S)$ is saturated. Now the same proof of (1) shows that the value of $E$ at ${\mathrm{Spf}}(\overline{A^{(i)}/I})$ are equipped with saturated Nygaardian filtrations for all $i$. Lastly the same proof of shows that for any ${\mathrm{Spf}}(T) \in X_{{\mathrm{qrsp}}}$ with $p$-completely flat structural map to $U$, the value of $E$ at ${\mathrm{Spf}}(T)$ has saturated Nygaardian filtration, this finishes the proof. ◻ We also give a relation on the weight of an $F$-gauge and the height of the underlying $F$-crystal. **Proposition 1**. *Let $X$ be a quasi-syntomic $p$-adic formal scheme, and let $E=(E,{\mathrm{Fil}}^\bullet E, \widetilde{\varphi}_E)\in {\mathrm{F\text{-}Gauge}}^{I\text{-}\mathrm{tf}}(X)$.* (i) *If $E$ is of weight $[a,b]$, then its underlying $F$-crystal $(\mathcal{E},\varphi_\mathcal{E})$ has height within $[a,b]$ as well.* (ii) *Conversely, assume $(\mathcal{E},\varphi_\mathcal{E})$ is of height $[a',b']$ with $E$ being saturated, and there is a quasi-syntomic cover of $X$ by perfectoids. Then $E$ is of weight $[a',b']$.* Note that ${\mathrm{Fil}}^\bullet E$ is saturated when $E$ is an $F$-gauge in vector bundle (). As a consequence, in this case the notion of weight of $E$ and the height of the underlying $F$-crystal coincide. *Proof.* As both weight and height can be checked locally with respect to quasi-syntomic topology, it suffices to assume $X={\mathrm{Spf}}(S)$ is quasiregular semiperfectoid. We first assume that $E$ is of weight $[a,b]$. By construction (cf. ), we have ${\mathrm{Fil}}^i E = E$ for $i\leq a$. In particular, since ${\mathrm{Fil}}^i E\subseteq \widetilde{\varphi}_E^{-1}(I^iE)$, the image $\widetilde{\varphi}_E(E)$ and hence the image of the linearlized map $\varphi_E(E)$ are contained in $I^a E$; namely the $F$-crystal $(\mathcal{E},\varphi_\mathcal{E})$ has height $\geq a$. Moreover, by , since the filtered map $v$ is an isomorphism on ${\mathrm{Fil}}^{\geq b}(-)$, by forgetting the filtration and looking at the underlying map of modules, we get the image of the linearized map $\varphi_E(E)$ contains the submodule $I^bE$, and thus the associated $F$-crystal is of height $\leq b$. Here we also note that if ${\mathrm{Red}}_b(E)$ is nonzero, then by looking at the $b$-th graded factor of the map $(\ast)$, for any $j<b$ we have $${\mathrm{Gr}}^b(V_j\otimes_S \bar{I}^\mathbb{N} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S )=0 \longrightarrow {\mathrm{Gr}}^b(V_j \otimes_S \bar{I}^\mathbb{Z} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S) = V_j\otimes_S \overline{{\mathlarger{\mathbbl{\Delta}}}}_S \neq 0.$$ In particular, for any given integer $l<b$, the map ${\mathrm{Gr}}^l u$ and hence ${\mathrm{Fil}}^l u$ is not surjective and hence not an isomorphism. For (ii), we assume the underlying $F$-crystal of $E$ is of height $[a',b']$, and $S$ is perfectoid. Then the image $\varphi_\mathcal{E}(\mathcal{E})$ satisfies the inclusions $$I^{b'}\mathcal{E} \subseteq \varphi_\mathcal{E}(\mathcal{E}) \subseteq I^{a'} \mathcal{E}.$$ So by taking the preimage along the ${\mathlarger{\mathbbl{\Delta}}}_S$-linear map $\varphi_\mathcal{E}$, we see $\varphi^{-1}_\mathcal{E}(I^{a'} \mathcal{E}) = \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}^* \mathcal{E}$, and the map $\varphi_\mathcal{E}$ induces an isomorphism $\varphi^{-1}_\mathcal{E}(I^{\geq b'} \mathcal{E}) \simeq I^{\geq b'} \mathcal{E}$. On the other hand, since $\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}$ is a filtered isomorphism, by the assumption that $E$ has saturated filtration, we get $${\mathrm{Fil}}^\bullet (\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}^* E) = \varphi^{-1}_\mathcal{E}(I^\bullet \mathcal{E}).$$ This in particular implies that ${\mathrm{Fil}}^{a'} E=E$, and $E$ is of weight $\geq a'$. Moreover, by .(ii), the map $\varphi^{-1}_E(I^i E) \to I^i E$ is not an isomorphism unless $i\geq \max\{l~|~{\mathrm{Red}}_l(E)\neq 0\}$, where the number $\max\{l~|~{\mathrm{Red}}_l(E)\neq 0\}$ is by definition the largest possible weight of $E$ (). Thus $E$ is of weight $\leq b'$. ◻ ## $F$-gauges in vector bundles and $p$-divisible groups Recall that in [@ALB23] the authors established, for every quasi-syntomic $p$-adic formal scheme $X$, an anti-equivalence between $p$-divisible groups over $X$ and admissible prismatic Dieudonné crystals over $X$ (see [@ALB23 Theorem 1.4.4]). Here a prismatic Dieudonné crystal over $X$ is the same as an $F$-crystal in vector bundles over ${\mathrm{Spf}}(R)_{{\mathlarger{\mathbbl{\Delta}}}}$ which is effective of height $\leq 1$, and admissibility is a technical condition introduced in [@ALB23 Def. 4.5], and is automatic if $X$ is regular and admits a quasi-syntomic cover by perfectoids, (see [@ALB23 Definition 1.3.1 and Remark 1.4.9][^3]). Below, for quasi-syntomic $X$, we let ${\mathrm{F\text{-}Crys}}^{\mathrm{adm},{\mathrm{vect}}}_{[0,1]}(X)$ be the subcategory of ${\mathrm{F\text{-}Crys}}^{\mathrm{vect}}_{[0,1]}(X)$ that consists of admissible prismatic $F$-crystals of height $[0,1]$. We then establish an equivalence of the above category with yet another category. As a preparation, we show that the filtration on an $F$-gauge in vector bundles is always saturated. **Proposition 1**. *Let $S$ be a quasiregular semiperfectoid ring, and let $E=(E,{\mathrm{Fil}}^\bullet E, \widetilde{\varphi}_E)$ be an $F$-gauge in vector bundles over $S$. Then $E$ is saturated, or more explicitly we have ${\mathrm{Fil}}^i E = \widetilde{\varphi}_E^{-1}(I^i E)$ for $i\in \mathbb{Z}$.* **Corollary 1**. *Let $X$ be a quasi-syntomic $p$-adic formal scheme. Then the forgetful functor below from $F$-gauges in vector bundles to its underlying $F$-crystals is fully faithful $${\mathrm{F\text{-}Gauge}}^{\mathrm{vect}}(X) \longrightarrow {\mathrm{F\text{-}Crys}}^{\mathrm{vect}}(X).$$* *Proof.* By assumption, as ${\mathrm{Fil}}^\bullet E$ is a filtered vector bundle over the filtered ordinary ring ${\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$, each ${\mathrm{Fil}}^i E$ is a submodule inside $E$ such that $\widetilde{\varphi}_E({\mathrm{Fil}}^i E) \subset I^iE$. Thus ${\mathrm{Fil}}^i E \subseteq \widetilde{\varphi}_E^{-1}(I^i E)$, and we want to show that each inclusion is an equality. Here we note that by the assumption of ${\mathrm{Fil}}^\bullet E$, we know the inclusion is an equality when $i<<0$. For each $i\in \mathbb{Z}$, the Frobenius structure $\widetilde{\varphi}_E$ induces the following commutative diagram of short exact sequences $$\begin{tikzcd} 0 \ar[r] & {\mathrm{Fil}}^{i+1} E \arrow[d, "\widetilde{\varphi}_E^{i+1}"] \ar[r] & {\mathrm{Fil}}^i E \arrow[d, "\widetilde{\varphi}_E^i"] \ar[r] & {\mathrm{Gr}}^i E \arrow[d, "{\mathrm{Gr}}^i \widetilde{\varphi}"] \ar[r] & 0\\ 0 \ar[r] & I^{i+1} E \ar[r] & I^i E \ar[r] & I^iE/I^{i+1}E \ar[r] &0, \end{tikzcd}$$ where $\widetilde{\varphi}_E^i$ is the restriction of $\widetilde{\varphi}_E$ on ${\mathrm{Fil}}^i E$. We then claim that the map ${\mathrm{Gr}}^i E \to I^iE/I^{i+1}E$ is injective for each $i\in \mathbb{Z}$. Granting the claim, a simple diagram chasing implies that ${\mathrm{Fil}}^{i+1} E= (\widetilde{\varphi}_E^i)^{-1}(I^{i+1} E) = \widetilde{\varphi}_E^{-1} ( I^{i+1} E)$. Hence we get the saturatedness of ${\mathrm{Fil}}^\bullet E$ by induction and the observation that ${\mathrm{Fil}}^i E=E$ for $i<<0$. Now we check the above claim. Recall that the graded map ${\mathrm{Gr}}^\bullet \widetilde{\varphi}:{\mathrm{Gr}}^\bullet E \to \bar{I}^\bullet E$ factors through the linearlization $$\begin{tikzcd} {\mathrm{Gr}}^\bullet E \ar[r] & {\mathrm{Gr}}^\bullet E \underset{{\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, {\mathrm{Gr}}^\bullet \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}}{\widehat{\bigotimes}} \bar{I}^\mathbb{N} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S \arrow[rr, "{\mathrm{Gr}}^\bullet \varphi_E"] && \bar{I}^\bullet E, \end{tikzcd}$$ such that a further base change to $\bar{I}^\mathbb{Z} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S$ is a graded isomorphism (cf. ). In particular, to show the injection of ${\mathrm{Gr}}^i E \to \bar{I}^i E$, it suffices to show that the following map is a graded injection $${\mathrm{Gr}}^\bullet E \longrightarrow {\mathrm{Gr}}^\bullet E\underset{{\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, {\mathrm{Gr}}^\bullet \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}}{\widehat{\bigotimes}} \bar{I}^\mathbb{Z} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S.$$ We then notice that since ${\mathrm{Gr}}^\bullet E$ is a graded vector bundle over ${\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$, its reduction ${\mathrm{Red}}_S (E)$, which is the graded base change of ${\mathrm{Gr}}^\bullet E$ from ${\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$ to $S$, is a graded vector bundle over $S$. Moreover, by , the complex ${\mathrm{Gr}}^\bullet E$ is filtered by a finite exhaustive weight filtration with graded pieces being ${\mathrm{Red}}_{j}(E) \otimes_S {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$. Thus by a finite induction, it suffices to assume ${\mathrm{Gr}}^\bullet E$ is of the form $V\otimes_S {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$ for a finite projective $S$-module $V$. Finally, as $S$ is quasiregular semiperfectoid, by [@BS19 Thm. 12.2] the graded map ${\mathrm{Gr}}^\bullet \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}: {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S \to \bar{I}^\mathbb{N} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S$ and thus ${\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S \to \bar{I}^\mathbb{Z} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S$ is injective. So by the flatness of $V$ over $S$, we get the injection of $$V\otimes_S {\mathrm{Gr}}^i_N {\mathlarger{\mathbbl{\Delta}}}_S \longrightarrow {\mathrm{Gr}}^i\left( (V\otimes_S {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S) \underset{{\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, {\mathrm{Gr}}^\bullet \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}}{\widehat{\bigotimes}} \bar{I}^\mathbb{Z} \overline{{\mathlarger{\mathbbl{\Delta}}}}_S \right) \simeq V\otimes_S \bar{I}^i \overline{{\mathlarger{\mathbbl{\Delta}}}}_S,$$ hence the claim. ◻ **Theorem 1**. *Let $X$ be a quasi-syntomic $p$-adic formal scheme. There is a natural equivalence $${\mathrm{F\text{-}Gauge}}^{\mathrm{vect}}_{[0,1]}(X) \simeq {\mathrm{F\text{-}Crys}}^{\mathrm{adm},{\mathrm{vect}}}_{[0,1]}(X).$$ It is compatible with when $X$ is smooth over ${\mathrm{Spf}}(\mathcal{O}_K)$.* The subscript $[0,1]$ denotes objects which are effective of height $\leq 1$. *Proof.* Let $(\mathcal{E},\varphi_\mathcal{E})$ be a given object in ${\mathrm{F\text{-}Crys}}^{\mathrm{adm},{\mathrm{vect}}}_{[0,1]}(X)$. We use $(M_S, {\mathrm{Fil}}^\bullet M_S, \widetilde{\varphi}_{M_S})$ to denote the value of $\Pi_{{\mathrm{Spf}}(S)}(\mathcal{E}|_{{\mathrm{Spf}}(S)_{\mathrm{qrsp}}})$ at $S\in X_{\mathrm{qrsp}}$, which by (1) satisfies $${\mathrm{Fil}}^n M_S = \widetilde{\varphi}_{M_S}^{-1} (I^nM_S) \subset M_S.$$ We then claim that (1) [\[claim p div 1\]]{#claim p div 1 label="claim p div 1"} the filtered module ${\mathrm{Fil}}^\bullet M_S$ is a filtered vector bundle over ${\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$, and (2) [\[claim p div 2\]]{#claim p div 2 label="claim p div 2"} for a map of rings $S_1\to S_2$ in $X_{\mathrm{qrsp}}$, the natural filtered linearlization map below is a filtered isomorphism $${\mathrm{Fil}}^\bullet M_1 \otimes_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_1}} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_2} \longrightarrow {\mathrm{Fil}}^\bullet M_2.$$ Here we use $M_i$ to simplify the notation $M_{S_i}$. Granting the above two claims, we see every admissible prismatic $F$-crystal in vector bundles of height $[0,1]$ naturally underlies an $F$-gauge in vector bundles of weight $[0,1]$. Conversely, given an $F$-gauge in vector bundles of weight $[0,1]$, its underlying $F$-crystal is an $F$-crystal in vector bundle of height $[0,1]$ by .(i). We let $f\in {\mathrm{Fil}}^1_N {\mathlarger{\mathbbl{\Delta}}}_S$ be any nonzero divisor such that $d=\varphi(f)$ generates the ideal $I$ in ${\mathlarger{\mathbbl{\Delta}}}_S$. By [@ALB23 Prop. 4.29, Lem. 4.23, Prop. 4.22] (applied at $(A,{\mathrm{Fil}}A,\varphi,\varphi_1)=({\mathlarger{\mathbbl{\Delta}}}_S,{\mathrm{Fil}}^1_N {\mathlarger{\mathbbl{\Delta}}}_S,\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}, \frac{1}{d}\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S})$, which satisfies the assumption of [@ALB23 Prop. 4.22] by loc. cit. Lem 4.27, Lem. 4.28), there are finite projective ${\mathlarger{\mathbbl{\Delta}}}_S$-modules $L_0$ and $L_1$, such that $$M_S \simeq L_0\oplus L_1, \quad {\mathrm{Fil}}^1 M_S \simeq {\mathrm{Fil}}^1_N {\mathlarger{\mathbbl{\Delta}}}_S \otimes_{{\mathlarger{\mathbbl{\Delta}}}_S} L_0 \oplus L_1,$$ Moreover, as $(\mathcal{E},\varphi_\mathcal{E})$ is of height $[0,1]$, by we have $$\begin{aligned} &{\mathrm{Fil}}^i M=M,~i\leq 0;\\ &{\mathrm{Fil}}^i M=f^{i-1}\cdot {\mathrm{Fil}}^1 M,~i\geq 1. \end{aligned}$$ This in particular implies that as a filtered module over ${\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$, we have $${\mathrm{Fil}}^\bullet M = L_0\otimes_{{\mathlarger{\mathbbl{\Delta}}}_S} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S \oplus L_1\otimes_{{\mathlarger{\mathbbl{\Delta}}}_S} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S,$$ where $L_i$ is the constant filtered object siting in the filtered degrees $\leq i$. This finishes the proof of [\[claim p div 1\]](#claim p div 1){reference-type="ref" reference="claim p div 1"}. For [\[claim p div 2\]](#claim p div 2){reference-type="ref" reference="claim p div 2"}, we apply [\[claim p div 1\]](#claim p div 1){reference-type="ref" reference="claim p div 1"} at $S=S_1$, to get the aforementioned decomposition. Then by [@ALB23 Lem. 4.14] we have the following equality of submodules in $M_2$ $${\mathrm{Fil}}^1 M_2 = {\mathrm{Fil}}^1_N {\mathlarger{\mathbbl{\Delta}}}_{S_2}\cdot M_2 + \mathrm{Im}({\mathlarger{\mathbbl{\Delta}}}_{S_2} \otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} {\mathrm{Fil}}^1 M_1 \to M_{S_2})$$ which by the decomposition formula is $$\begin{aligned} &= {\mathrm{Fil}}^1_N {\mathlarger{\mathbbl{\Delta}}}_{S_2} \otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} (L_0\oplus L_1) + \mathrm{Im}\bigl( {\mathlarger{\mathbbl{\Delta}}}_{S_2} \otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} ({\mathrm{Fil}}^1_N {\mathlarger{\mathbbl{\Delta}}}_{S_1} \otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} L_0 \oplus L_1) \to {\mathlarger{\mathbbl{\Delta}}}_{S_2}\otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} (L_0\oplus L_1) \bigr)\\ &= {\mathrm{Fil}}^1_N {\mathlarger{\mathbbl{\Delta}}}_{S_2} \otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} L_0\oplus {\mathlarger{\mathbbl{\Delta}}}_{S_2}\otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} L_1. \end{aligned}$$ As a consequence, we see ${\mathrm{Fil}}^\bullet M_2 \simeq {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_2} \otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} L_0 \oplus {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_2} \otimes_{{\mathlarger{\mathbbl{\Delta}}}_{S_1}} L_1$, and the natural filtered linearization ${\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_2} \otimes_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_1}} {\mathrm{Fil}}^\bullet M_1 \to {\mathrm{Fil}}^\bullet M_2$ is a filtered derived isomorphism. Finally we show the essential surjectivity. By , an $F$-gauge in vector bundle has the saturated Nygaardian filtration and is uniquely determined by the underlying $F$-crystal. So it is left to check that the underlying $F$-crystal is admissible in the sense of [@ALB23 Def. 4.5, Def. 4.10]. Namely for $S\in X_{\mathrm{qrsp}}$, we want to show that the image of $M_S \xrightarrow{\widetilde{\varphi}_{M_S}} M_S \to M_S/IM_S$ is a vector bundle $F_S$ over $S$, such that the linearization $F_S \otimes_S \overline{{\mathlarger{\mathbbl{\Delta}}}}_S \to M_S/IM_S$ is injective. By , ${\mathrm{Fil}}^1 M_S$ is equal to $\widetilde{\varphi}^{-1}(IM_S)$, and thus the image $F_S$ is equal to $M_S/{\mathrm{Fil}}^1 M_S={\mathrm{Gr}}^0 M_S$. The zero-th graded piece ${\mathrm{Gr}}^0 M_S=F_S$ is a vector bundle over ${\mathrm{Gr}}^0_N {\mathlarger{\mathbbl{\Delta}}}_S=S$, thanks to the assumption that ${\mathrm{Fil}}^\bullet M_S$ is a filtered vector bundle over ${\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$. Moreover, by taking the zero-th graded piece of the filtered isomorphism $\varphi_{M_S}:{\mathrm{Fil}}^\bullet M_S \otimes_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}} I^\mathbb{Z} {\mathlarger{\mathbbl{\Delta}}}_S \simeq I^\mathbb{Z} M_S$, the required injectivity of the linearlization above is equivalent to the injectivity of the map $${\mathrm{Gr}}^0 M_S \otimes_S \overline{{\mathlarger{\mathbbl{\Delta}}}}_S \longrightarrow {\mathrm{Gr}}^0 \left({\mathrm{Fil}}^\bullet M_S \otimes_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S, \varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}} I^\mathbb{Z} {\mathlarger{\mathbbl{\Delta}}}_S \right).$$ Here by an induction on the weight filtration, we may assume ${\mathrm{Gr}}^\bullet M_S =V \otimes_S {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S$ for a finite projective $S$-module $V$ of graded degree $0$ or $1$. In both cases, the injection follows from that of the structure sheaf, which finishes the proof. ◻ # Relative prismatic $F$-gauges and realizations We now turn our focus to a relative version of prismatic $F$-gauges, and its relation with the absolute version studied in the previous section. Throughout this section, let us fix $(A, I)$ a bounded prism with $\bar{A} \coloneqq A/I$. ## Quasi-syntomic sheaves in relative setting Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme. Consider $(Z/\bar{A})_{{\mathrm{qrsp}}} \subset Z_{{\mathrm{qSyn}}}$, where $(Z/\bar{A}){{\mathrm{qrsp}}}$ denote the category of large quasi-syntomic $\bar{A}$-algebras (in the sense of [@BS19 Definition 15.1]) together with a quasi-syntomic map to $Z$ equipped with quasi-syntomic topology. Then we have the following analog of the "*unfolding process*" in [@BMS19 Proposition 4.31]: **Proposition 1**. *Restriction to qrsp objects induces an equivalence $$\mathrm{Shv}_{\mathcal{C}}(Z_{{\mathrm{qSyn}}}) \cong \mathrm{Shv}_{\mathcal{C}}((Z/\bar{A})_{{\mathrm{qrsp}}})$$ for any presentable $\infty$-category $\mathcal{C}$.* *Proof.* The proof is identical to that in loc. cit. ◻ **Remark 1**. Concretely, suppose we are given a quasi-syntomic sheaf $\mathcal{F}$ on $(Z/\bar{A})_{{\mathrm{qrsp}}}$. Let $R$ be an object in $Z_{{\mathrm{qSyn}}}$, and suppose we are given a large quasi-syntomic cover $R \to S^{(0)}$ with its Cech nerve being $S^{(\bullet)}$, then the unfolding evaluated at $R$ can be computed as the cosimplicial limit of $\mathcal{F}(S^{(\bullet)})$. **Example 1** (see [@BMS19 §1.3 and §5.2] and [@BS19 §15]). Here are some examples of quasi-syntomic sheaves on ${\mathrm{Spf}}(\bar{A})_{{\mathrm{qSyn}}}$: the relative prismatic cohomology ${\mathlarger{\mathbbl{\Delta}}}_{-/A}$ and its $\varphi_A^*$-variant ${\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A} \coloneqq {\mathlarger{\mathbbl{\Delta}}}_{-/A} \widehat{\otimes}_{A, \varphi_A} A$, the $i$-th Nygaard filtration on ${\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}$ for any $i$, the relative Hodge--Tate cohomology $\overline{{\mathlarger{\mathbbl{\Delta}}}}_{-/A} \coloneqq {\mathlarger{\mathbbl{\Delta}}}_{-/A}/I$ and its $i$-th conjugate filtration for any $i$, the $p$-adic derived de Rham cohomology $\mathrm{dR}_{-/\bar{A}}$ and its $i$-th Hodge filtration for any $i$. By restriction, we get quasi-syntomic sheaves on $Z_{{\mathrm{qrsp}}}$. The following relation between the various filtrations appearing above will be constantly used (see [@BS19 Theorem 15.2 and 15.3]): we have fiber sequences $${\mathrm{Fil}}^{i+1}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A} \to {\mathrm{Fil}}^{i}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A} \to {\mathrm{Gr}}^{i}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A} \cong {\mathrm{Fil}}_i^{{\mathrm{conj}}} \overline{{\mathlarger{\mathbbl{\Delta}}}}_{-/A} \{i\}; \text{ and }$$ $${\mathrm{Fil}}^{i-1}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A} \otimes_A I \to {\mathrm{Fil}}^{i}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A} \to {\mathrm{Fil}}^i_H \mathrm{dR}_{-/\bar{A}}.$$ In terms of Rees's construction, we can formulate them uniformly: **Proposition 1**. *For simplicity, let us assume $(A, I = (d))$ is a bounded oriented prism. Now we view ${\mathrm{Rees}}({\mathrm{Fil}}^{\bullet}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}) \coloneqq \bigoplus_{i \in \mathbb{Z}} {\mathrm{Fil}}^i_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A} \cdot t^{-i}$ as a graded ${\mathrm{Rees}}(I^{\mathbb{N}}) \coloneqq A[t, \frac{d}{t}]$-algebra. Then we have $${\mathrm{Rees}}({\mathrm{Fil}}^{\bullet}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}) /^L (t) \cong \bigoplus_{i \in \mathbb{N}} {\mathrm{Fil}}_i^{{\mathrm{conj}}} \overline{{\mathlarger{\mathbbl{\Delta}}}}_{-/A} \cdot (\frac{d}{t})^i; \text{ and }$$ $${\mathrm{Rees}}({\mathrm{Fil}}^{\bullet}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}) /^L (\frac{d}{t}) \cong {\mathrm{Rees}}({\mathrm{Fil}}^{\bullet}_H \mathrm{dR}_{-/\bar{A}}) \coloneqq \bigoplus_{i \in \mathbb{Z}} {\mathrm{Fil}}^i_H \mathrm{dR}_{-/\bar{A}} \cdot t^{-i}.$$* The following flatness result is the analog of (proof of) . **Proposition 1**. *Let $S_1 \to S_2$ be a quasi-syntomic flat (resp. faithfully flat) map of two quasi-syntomic $\bar{A}$-algebras and let $S_1 \to S_3$ be an arbitrary map of quasi-syntomic $\bar{A}$-algebras with $S_4 \coloneqq S_2 \widehat{\otimes}_{S_1} S_3$, then:* 1. *The map $\mathrm{Hdg}(S_1/\bar{A}) \to \mathrm{Hdg}(S_2/\bar{A})$ is $p$-completely flat (resp. faithfully flat), and the graded map $$\mathrm{Hdg}(S_2/\bar{A}) \widehat{\otimes}_{\mathrm{Hdg}(S_1/\bar{A})} \mathrm{Hdg}(S_3/\bar{A}) \to \mathrm{Hdg}(S_4/\bar{A})$$ is a graded isomorphism;* 2. *The filtered map ${\mathrm{Fil}}^{{\mathrm{conj}}}_{\bullet}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S_1/A}) \to {\mathrm{Fil}}^{{\mathrm{conj}}}_{\bullet}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S_2/A})$ has its corresponding Rees's construction map being $p$-completely flat (resp. faithfully flat), and the filtered map $${\mathrm{Fil}}^{{\mathrm{conj}}}_{\bullet}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S_2/\bar{A}}) \widehat{\otimes}_{{\mathrm{Fil}}^{{\mathrm{conj}}}_{\bullet}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S_1/\bar{A}})} {\mathrm{Fil}}^{{\mathrm{conj}}}_{\bullet}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S_3/\bar{A}}) \to {\mathrm{Fil}}^{{\mathrm{conj}}}_{\bullet}(\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S_4/\bar{A}})$$ is a filtered isomorphism;* 3. *The filtered map ${\mathrm{Fil}}^{\bullet}_H(\mathrm{dR}_{S_1/A}) \to {\mathrm{Fil}}^{\bullet}_H(\mathrm{dR}_{S_2/A})$ has its corresponding Rees's construction map being $p$-completely flat (resp. faithfully flat), and the filtered map $${\mathrm{Fil}}^{\bullet}_H(\mathrm{dR}_{S_2/\bar{A}}) \widehat{\otimes}_{{\mathrm{Fil}}^{\bullet}_H(\mathrm{dR}_{S_1/\bar{A}})} {\mathrm{Fil}}^{\bullet}_H(\mathrm{dR}_{S_3/\bar{A}}) \to {\mathrm{Fil}}^{\bullet}_H(\mathrm{dR}_{S_4/\bar{A}})$$ is a filtered isomorphism;* 4. *The filtered map ${\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}^{(1)}_{S_1/A}) \to {\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}^{(1)}_{S_2/A})$ has its corresponding Rees's construction map being $(p, I)$-completely flat (resp. faithfully flat), and the filtered map $${\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}^{(1)}_{S_2/A}) \widehat{\otimes}_{{\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}^{(1)}_{S_1/A})} {\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}^{(1)}_{S_3/A}) \to {\mathrm{Fil}}^{\bullet}_N({\mathlarger{\mathbbl{\Delta}}}^{(1)}_{S_4/A})$$ is a filtered isomorphism.* *Proof.* The proof follows from similar argument appearing in the proof of : the proof of (1) has appeared there. Then similar to there, with in mind, we can use (1) together with and to prove (2), and use (2) and to prove both of (3) and (4). ◻ ## Relative prismatic $F$-gauges {#Relative prismatic $F$-gauges} Let $(A,I)$ be a fixed bounded prism. For a quasi-syntomic $\bar{A}$-formal scheme $Z$, we denote by $(Z/\bar{A})_{\mathrm{qrsp}}$ the category of large quasi-syntomic $\bar{A}$-algebra over $Z$ together with quasi-syntomic topology, in the sense of [@BS19 Definition 15.1]. **Definition 1** (c.f. [@Tang22 Definition 2.14]). Let $S$ be a large quasi-syntomic $\bar{A}$-algebra. A *prismatic gauge* $E=(E, {\mathrm{Fil}}^\bullet E^{(1)})$ over ${\mathrm{Spf}}(S)/A$ consist of the following data: - a derived $(p,I)$-complete complex $E$ over ${\mathlarger{\mathbbl{\Delta}}}_{S/A}$; - a derived $(p,I)$-complete filtration ${\mathrm{Fil}}^\bullet E^{(1)}$ on $E^{(1)} \coloneqq \varphi_A^* E$, linear over the Nygaard filtered relative prismatic cohomology ${\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S/A}^{(1)}$. A *prismatic $F$-gauge* $E=(E, {\mathrm{Fil}}^\bullet E^{(1)}, \widetilde{\varphi}_{E})$ over ${\mathrm{Spf}}(S)/A$ consist of the following data: - a prismatic gauge $E=(E, {\mathrm{Fil}}^\bullet E^{(1)})$ over ${\mathlarger{\mathbbl{\Delta}}}_{S/A}$ called the underlying gauge of $E$; - a map of filtered complexes $$\widetilde{\varphi}_{E} \colon {\mathrm{Fil}}^\bullet E^{(1)} \longrightarrow I^\mathbb{Z} E = I^\mathbb{Z}{\mathlarger{\mathbbl{\Delta}}}_{S/A} \widehat{\otimes}_{{\mathlarger{\mathbbl{\Delta}}}_{S/A}} E,$$ such that the filtered linearization $$\varphi_{E} \colon {\mathrm{Fil}}^\bullet E^{(1)} \widehat{\otimes}_{{\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S/A}^{(1)}, \varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S}}} I^\mathbb{Z} {\mathlarger{\mathbbl{\Delta}}}_S \longrightarrow I^\mathbb{Z} E$$ is a filtered isomorphism in ${\mathcal{DF}}_{(p,I)\text{-comp}}(I^\mathbb{Z}{\mathlarger{\mathbbl{\Delta}}}_{S/A})\simeq \mathcal{D}_{(p,I)\text{-comp}}({\mathlarger{\mathbbl{\Delta}}}_{S/A})$. We use ${\mathrm{(F\text{-})Gauge}}({\mathrm{Spf}}(S)/A)$ to denote the natural $\infty$-category whose objects are as above. **Remark 1** (c.f. [@Tang22 Proposition 2.1 and Remark 2.5]). Here we notice that by [@BS19 Thm. 15.2], the filtered morphism of $A$-algebras $$\varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S}} \colon {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S/A}^{(1)} \to I^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S/A}$$ identifies the target as $(p, I)$-completely inverting $I \subset {\mathrm{Fil}}^1_N$ of the source as filtered $A$-algebras. In particular, the above linarization condition can be rewritten as a filtered isomorphism $$\left( {\mathrm{Fil}}^\bullet E^{(1)}[1/I] \right)^\wedge_{(p,I)} \xlongrightarrow[\cong]{\varphi_{E}} I^\mathbb{Z} E,$$ or equivalently $$\left( {\mathrm{colim}}_{i\in \mathbb{Z}} (\cdots \to {\mathrm{Fil}}^i E^{(1)} \otimes I^{-i} \to {\mathrm{Fil}}^{i+1} E^{(1)} \otimes I^{-i-1}\to \cdots) \right)^\wedge_{(p,I)} \xrightarrow[\cong]{\varphi_E} E.$$ As in , we can also consider the analogous subcategories consisting of vector bundles (resp. perfect complexes, resp. coherent objects: coherence here means the underlying complex is a perfect complex concentrated in cohomological degree $0$ with filtrations given by submodules). **Remark 1**. Similar to , for a map of large quasi-syntomic $\bar{A}$-algebras $S_1\to S_2$, the completed filtered base change induces a natural functor $$\begin{aligned} \Phi_{(S_1,S_2)}:{\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S_1)) &\longrightarrow {\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S_2)),\\ (E, ~{\mathrm{Fil}}^\bullet E^{(1)}, (\varphi_{E})) & \longmapsto (E \widehat{\bigotimes}_{{\mathlarger{\mathbbl{\Delta}}}_{S_1/A}} {\mathlarger{\mathbbl{\Delta}}}_{S_2/A}, {\mathrm{Fil}}^\bullet E^{(1)} \widehat{\bigotimes}_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_1/A}^{(1)}} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S_2/A}^{(1)}, (\widetilde{\varphi}_{E} \otimes_{\varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S_1/A}}} \varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S_2/A}})).\end{aligned}$$ **Definition 1**. Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme. The category of *prismatic ($F$-)gauges* over $Z/A$ is defined as the limit of $\infty$-categories $${\mathrm{(F\text{-})Gauge}}^*(Z/A)\colonequals \lim_{S\in (Z/\bar{A})_{\mathrm{qrsp}}} {\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S)/A),$$ where $*\in \{\emptyset, {\mathrm{vect}}, {\mathrm{perf}}, {\mathrm{coh}}\}$. As in the absolute case (), we have the following sheaf property. **Proposition 1**. *Let $S\to S^{(0)}$ be a quasi-syntomic cover of large quasi-syntomic $\bar{A}$-algebras, and let $S^{(\bullet)}$ be the $p$-completed Čech nerve. Then for $*\in \{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$, we have a natural equivalence $${\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S)/A) \simeq \lim_{[n]\in \Delta} {\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S^{(n)})/A).$$* *Proof.* The proof is very similar to that of : via Rees's dictionary, the descent property follows from (4). ◻ Our main construction in this section is the following. **Theorem 1**. *Let $X$ be a quasi-syntomic formal scheme, and let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme, together with a map $Z\to X$. Then for $*\in\{\emptyset, {\mathrm{perf}}, {\mathrm{vect}}\}$, there is a natural functor $$BC_{X, Z/A} \colon {\mathrm{(F\text{-})Gauge}}^*(X) \longrightarrow {\mathrm{(F\text{-})Gauge}}^*(Z/A),$$ compatible with the natural base change functor of prismatic $F$-crystals ${\mathrm{(F\text{-})Crys}}^*(X)\to {\mathrm{(F\text{-})Crys}}^*(Z/A)$.* As a preparation, we first consider the special case of quasiregular semiperfectoid rings. **Lemma 1**. *Let $S$ be a quasi-regular semiperfectoid ring, and let $S'$ a large quasi-syntomic $\bar{A}$-algebra, together with a map ${\mathrm{Spf}}(S') \to {\mathrm{Spf}}(S)$. Then there is a functorial commutative diagram of filtered rings $$\begin{tikzcd} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S \arrow[d, "\varphi_{{\mathlarger{\mathbbl{\Delta}}}_S}"] \ar[r]& {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)} \arrow[d, "\varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}}"] \\ I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S \ar[r] & I^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A}, \end{tikzcd}$$ such that the top row underlies the composition of homomorphisms of rings ${\mathlarger{\mathbbl{\Delta}}}_S \to {\mathlarger{\mathbbl{\Delta}}}_{S'/A} \to {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}$.* *Proof.* There is a natural commutative diagram: $$\begin{tikzcd} {\mathlarger{\mathbbl{\Delta}}}_{S} \ar[r] \arrow[dd, "\varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S}}"] & {\mathlarger{\mathbbl{\Delta}}}_{S'/A} \arrow[d, "\mathrm{id} \otimes 1"]\\ & {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)} \coloneqq {\mathlarger{\mathbbl{\Delta}}}_{S'/A} \widehat{\otimes}_{A, \varphi_A} A \arrow[d, "\varphi_{rel}"] \\ {\mathlarger{\mathbbl{\Delta}}}_{S} \ar[r] & {\mathlarger{\mathbbl{\Delta}}}_{S'/A}, \end{tikzcd}$$ where the horizontal maps follow from the initial property of ${\mathlarger{\mathbbl{\Delta}}}_{S}$ in the category ${\mathrm{Spf}}(S)_{\mathlarger{\mathbbl{\Delta}}}$, and the right vertical map is the natural factorization of $\varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S'/A}}$, with $\varphi_{rel}$ being the relative Frobenius map for prismatic cohomology of $S'$ over $A$. Moreover, if an element $x\in {\mathlarger{\mathbbl{\Delta}}}_{S}$ is sent into $I^n{\mathlarger{\mathbbl{\Delta}}}_{S}$ under $\varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S}}$, then by the commutativity, the image of $x$ in ${\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}$ is sent into $I^n{\mathlarger{\mathbbl{\Delta}}}_{S'/A}$ under $\varphi_{rel}$. By definition of Nygaard filtration for quasiregular semiperfectoid rings [@BS19 Theorem 15.2.(1)], this in particular means that the above diagram carries ${\mathrm{Fil}}^n_N {\mathlarger{\mathbbl{\Delta}}}_{S}$ into ${\mathrm{Fil}}^n_N {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}$ under the top right composition: $$\begin{tikzcd} {\mathlarger{\mathbbl{\Delta}}}_{S} \ar[r] & {\mathlarger{\mathbbl{\Delta}}}_{S'/A} \arrow[r, "\mathrm{id} \otimes 1"] & {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}. \end{tikzcd}$$ Thus we get the filtered map as needed. ◻ *Proof of .* We prove the version with $F$, the one without $F$ is similar and easier. Assume $S\in X_{\mathrm{qrsp}}$ is any quasiregular semiperfectoid ring over $X$. Then since ${\mathrm{Spf}}(S)\times_X Z$ is a quasi-syntomic formal scheme over $Z$, and $Z$ is quasi-syntomic over ${\mathrm{Spf}}(\bar{A})$, we see ${\mathrm{Spf}}(S)\times_X Z$ admits quasi-syntomic covers by formal spectra of large quasi-syntomic $\bar{A}$-algebras. Let $S'$ be any large quasi-syntomic $\bar{A}$-algebra that is quasi-syntomic over ${\mathrm{Spf}}(S)\times_X Z$. Since both $F$-gauges and relative $F$-gauges satisfy quasi-syntomic descent ( and ), it suffices to construct functors $BC_{(S,S')} \colon {\mathrm{F\text{-}Gauge}}^*({\mathrm{Spf}}(S)) \longrightarrow {\mathrm{F\text{-}Gauge}}^*({\mathrm{Spf}}(S')/A)$ compatible with base change in both $S$ and $S'$ (defined in and respectively). Lastly we define the functor $BC_{(S,S')}$, for any pair $(S, S')$ as above, by taking the filtered base change: $$\begin{aligned} BC_{(S,S')} \colon {\mathrm{F\text{-}Gauge}}^*({\mathrm{Spf}}(S)) &\longrightarrow {\mathrm{F\text{-}Gauge}}^*({\mathrm{Spf}}(S')/A),\\ E=(E, {\mathrm{Fil}}^\bullet E, \varphi_{E}) &\longmapsto \left(E \widehat{\bigotimes}_{{\mathlarger{\mathbbl{\Delta}}}_{S}} {\mathlarger{\mathbbl{\Delta}}}_{S'/A}, {\mathrm{Fil}}^\bullet E \widehat{\bigotimes}_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S}} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}, \widetilde{\varphi}_{E} \otimes \varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}} \right).\end{aligned}$$ Here, unraveling definitions, the filtered map $\widetilde{\varphi}_{E} \otimes \varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}}$ is given by $$\begin{tikzcd} {\mathrm{Fil}}^\bullet E \widehat{\bigotimes}_{{\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S} {\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)} \arrow[rrr, "\widetilde{\varphi}_{E} \otimes_{\varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S}}} \varphi_{{\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}}"] & & & I^\bullet E \widehat{\bigotimes}_{I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S} I^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A} \simeq I^\bullet (E \widehat{\otimes}_{{\mathlarger{\mathbbl{\Delta}}}_S} {\mathlarger{\mathbbl{\Delta}}}_{S'/A}), \end{tikzcd}$$ and by the commutative diagram in , its linearization is $$\begin{aligned} \left( {\mathrm{Fil}}^\bullet E \widehat{\bigotimes}_{{\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_S} {\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)} \right) \widehat{\bigotimes}_{{\mathrm{Fil}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}} I^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A} & \cong \left({\mathrm{Fil}}^\bullet E \widehat{\bigotimes}_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S}} I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S \right) \widehat{\bigotimes}_{I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S} I^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A} \\ & \xlongrightarrow{\varphi_E \otimes \mathrm{id}} I^\bullet E \widehat{\bigotimes}_{I^\bullet {\mathlarger{\mathbbl{\Delta}}}_S} I^\bullet {\mathlarger{\mathbbl{\Delta}}}_{S'/A},\end{aligned}$$ which is a filtered isomorphism by assumption of $E=(E, {\mathrm{Fil}}^\bullet E, \varphi_{E}) \in {\mathrm{F\text{-}Gauge}}({\mathrm{Spf}}(S))$. ◻ **Remark 1**. Let us note that the functor in satisfies base change with respect to the base prism: in the setting of the theorem, suppose we are given a map of prisms $(A, I) \to (\widetilde{A}, I\widetilde{A})$ and let $\widetilde{Z} \coloneqq Z \times_{\bar{A}} \overline{\widetilde{A}}$. Then we have a commutative diagram: $$\xymatrix{ {\mathrm{(F\text{-})Gauge}}^*(X) \ar[rr] \ar[rd] & & {\mathrm{(F\text{-})Gauge}}^*(Z/A) \ar[ld]^{- \widehat{\otimes}_A \widetilde{A}} \\ & {\mathrm{(F\text{-})Gauge}}^*(\widetilde{Z}/\widetilde{A}), & }$$ where the two unlabelled arrows are what constructed in . Indeed, tracing through the construction in the proof of , we are reduced to a similar statement with $X$ and $Z$ replaced by $S$ and $S'$ (and $\widetilde{Z}$ replaced by $\widetilde{S'} \coloneqq S' \widehat{\otimes}_{\bar{A}} \overline{\widetilde{A}}$), which immediately follows from the base change formula of prismatic cohomology $${\mathlarger{\mathbbl{\Delta}}}_{S'/A} \widehat{\otimes}_A \widetilde{A} \cong {\mathlarger{\mathbbl{\Delta}}}_{\widetilde{S'}/\widetilde{A}},$$ as well as similar base change formulas for Nygaard filtrations ([@BS19 Theorem 15.2.(3)]). Similar to , there is a natural notion of weight and weight filtration on the associated graded of a relative prismatic $F$-gauge. **Definition 1**. Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme, and let $[a, b]$ be an interval in $\mathbb{R}\cup \{-\infty, \infty\}$. For a graded complex $M^\bullet \in {\mathcal{DG}}^\ast_{p\text{-comp}}((Z/\bar{A})_{\mathrm{qrsp}}, {\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{-/A}^{(1)})$. 1. The reduction of $M^\bullet$, as a graded complex over $((Z/\bar{A})_{\mathrm{qrsp}}, \mathcal{O}_{\mathrm{qrsp}})$, is defined as the graded base change $${\mathrm{Red}}_{Z/A}(M^\bullet) \colonequals M^\bullet \widehat{\bigotimes}_{{\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{-/A}^{(1)}} \mathcal{O}_{\mathrm{qrsp}}.$$ The $i$-th graded piece of ${\mathrm{Red}}_{Z/A}(M^\bullet)$ is denoted as ${\mathrm{Red}}_{i,Z/A}(M^\bullet)$. 2. The graded complex $M^\bullet$ is said to *have weights in* $[a,b]$ if $M^i=0$ for $i\ll 0$ and $${\mathrm{Red}}_{i,Z/A}(M^\bullet)=0, \forall i\notin [a,b].$$ For a relative ($F$-)gauge $E=(E,{\mathrm{Fil}}^\bullet E^{(1)}, \varphi_E)$ over $Z/A$ we define its reduction and weights by that of its associated graded $M^\bullet={\mathrm{Gr}}^\bullet E^{(1)}$. **Proposition 1**. *Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme, let $\ast\in \{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$, and let $a\leq b$ be two integers.* 1. *Assume $M^\bullet\in {\mathcal{DG}}^\ast_{p\text{-}\mathrm{comp}}(Z_{\mathrm{qrsp}}, {\mathrm{Gr}}_N^\bullet {\mathlarger{\mathbbl{\Delta}}}_{-/A}^{(1)})$ has weights in $[a,b]$. There exists a unique finite increasing and exhaustive filtration ${\mathrm{Fil}}^{\mathrm{wt}}_i (M^\bullet)$ on $M^\bullet$ indexed by $i \in [a,b]$, such that ${\mathrm{Gr}}^{\mathrm{wt}}_i(M^\bullet)$ is canonically isomorphic to ${\mathrm{Red}}_{i,Z/A}(M^\bullet) \widehat{\otimes}_{\mathcal{O}_Z} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{-/A}^{(1)}$.* 2. *Assume there is a map of $p$-adic formal schemes $Z \to X$ such that $X$ is quasi-syntomic. Let $E=(E,{\mathrm{Fil}}^\bullet E) \in \mathrm{Gauge}^\ast(X)$ having weights in $[a,b]$, and let $E'$ be the associated relative gauge over $Z/A$ as in . Then $E'$ also has weights in $[a,b]$. In fact we have a natural graded tensor product formula: $${\mathrm{Red}}_{Z/A}(E') \cong {\mathrm{Red}}_X(E)\widehat{\otimes}_{\mathcal{O}_X} \mathcal{O}_Z.$$ Moreover the weight filtrations on $M^{\bullet} \coloneqq {\mathrm{Gr}}^{\bullet}(E)$ and $N^{\bullet} \coloneqq {\mathrm{Gr}}^{\bullet}(E')$ are related via a natural filtered tensor product formula: $${\mathrm{Fil}}^{{\mathrm{wt}}}_i(N^{\bullet}) \cong {\mathrm{Fil}}^{{\mathrm{wt}}}_i(M^{\bullet}) \widehat{\otimes}_{{\mathrm{Gr}}^\bullet_N} {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{-/A}^{(1)}.$$* *Proof.* Part (i) is identical with the proof of and we do not repeat here. For part (ii), it suffices to check the claim quasi-syntomic locally, and we assume there is a quasiregular semiperfectoid algebra $S\in X_{\mathrm{qrsp}}$, a large quasi-syntomic algebra $S'\in (Z/\bar{A})_{\mathrm{qrsp}}$, together with a map $S\to S'$ that is compatible with $X\to Z$. Then by proof of , we have ${\mathrm{Fil}}^\bullet E'(S')^{(1)}= {\mathrm{Fil}}^\bullet E(S)\widehat{\otimes}_{{\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S} {\mathrm{Fil}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)}$. Notice that we also have a commutative diagram of graded rings $$\begin{tikzcd} S \arrow[r, "\iota"] \ar[d] & {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_S \ar[d] \arrow[r, "{\mathrm{Gr}}^0"] & S \ar[d] \\ S' \arrow[r, "\iota"] & {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S'/A}^{(1)} \arrow[r, "{\mathrm{Gr}}^0"] & S'. \end{tikzcd}$$ As a consequence, by the local formula of the reduction functors (, ), we get the tensor product formula $${\mathrm{Red}}_{S'/A}(E') \simeq {\mathrm{Red}}_S(E) \widehat{\otimes}_S S',$$ and the last formula follows from uniqueness of the weight filtration on $N^{\bullet}$. ◻ ## Filtered Higgs complex and Hodge--Tate realization {#sec filtered Higgs} In eye of , we study the mod $t$ specialization of a relative prismatic $F$-gauge in this subsection. We start by considering the notion of filtered Higgs complex and the induced conjugate filtration on Hodge--Tate cohomology. Let $(A,I)$ be a bounded prism. Recall that for a large quasi-syntomic formal scheme $S$ over $\bar{A}$, the relative Hodge--Tate cohomology ring $\overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}$ admits an increasing filtration called *conjugate filtration*, whose $i$-th graded factor is $\wedge^i\mathbb{L}_{S/\bar{A}}\{-i\}[-i]$ (see [@BS19 Construction 7.6]). We use ${\mathrm{Gr}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A} = \bigoplus_{i \in \mathbb{N}} \wedge^i\mathbb{L}_{S/\bar{A}}\{-i\}[-i]$ to denote the associated graded algebra over $S$. **Definition 1**. Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme. 1. The category of *filtered Higgs fields* over $Z/A$ is defined as the limit of $\infty$-categories $${\mathrm{FilHiggs}}(Z/A)\colonequals \lim_{S \in (Z/\bar{A})_{\mathrm{qrsp}}} {\mathrm{FilHiggs}}({\mathrm{Spf}}(S)/A),$$ where ${\mathrm{FilHiggs}}({\mathrm{Spf}}(S)/A)\colonequals {\mathcal{DF}}_{p\text{-comp}}({\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A})$ is the category of $p$-complete filtered complexes over the filtered ring ${\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}$. 2. The category of *graded Higgs fields* over $Z/A$ is defined as the limit of $\infty$-categories $${\mathrm{GrHiggs}}(Z/A) \colonequals \lim_{S \in (Z/\bar{A})_{\mathrm{qrsp}}} {\mathrm{GrHiggs}}({\mathrm{Spf}}(S)/A),$$ where ${\mathrm{GrHiggs}}({\mathrm{Spf}}(S)/A)$ is the category of $p$-complete graded complexes over the graded ring ${\mathrm{Gr}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}$. Here, similar to , for a map of large quasi-syntomic $\bar{A}$-algebras $S_1\to S_2$, the completed filtered/graded base change induces natural functors between ${\mathrm{FilHiggs}}({\mathrm{Spf}}(S_i)/A)$ and ${\mathrm{GrHiggs}}({\mathrm{Spf}}(S_i)/A)$. Similar to , for $*\in \{{\mathrm{vect}}, {\mathrm{perf}}\}$, we use ${\mathrm{FilHiggs}}^*(Z/A)$ (resp. ${\mathrm{GrHiggs}}^*(Z/A)$) to denote the subcategory of filtered (resp. graded) Higgs bundles/perfect Higgs complexes. **Construction 1** (Associated graded functor). There is a natural functor by taking associated graded: $${\mathrm{Gr}}: {\mathrm{FilHiggs}}^*(Z/A) \longrightarrow {\mathrm{GrHiggs}}^*(Z/A),$$ for $*=\{\emptyset, {\mathrm{vect}}, {\mathrm{perf}}\}$. When $Z = {\mathrm{Spf}}(S)$ is large quasi-syntomic, it sends a filtered ${\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}$-complex ${\mathrm{Fil}}_\bullet M$ to the graded ${\mathrm{Gr}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}$-complex ${\mathrm{Gr}}_\bullet M = \bigoplus_{i \in \mathbb{N}} {\mathrm{Fil}}_i M/ {\mathrm{Fil}}_{i-1} M$. We also have the quasi-syntomic sheaf property for categories of filtered and graded Higgs complexes. **Proposition 1**. *Let $S \to S^{(0)}$ be a quasi-syntomic cover of large quasi-syntomic rings over $\bar{A}$, and let $S^{(\bullet)}$ be the $p$-completed Čech nerve. Then for $*\in \{{\mathrm{perf}},{\mathrm{vect}}\}$, we have a natural equivalence $$\begin{aligned} {\mathrm{FilHiggs}}^*({\mathrm{Spf}}(S)/A) & \simeq \lim_{[n]\in \Delta} {\mathrm{FilHiggs}}^*({\mathrm{Spf}}(S^{(n)})/A),\\ {\mathrm{GrHiggs}}^*({\mathrm{Spf}}(S)/A) & \simeq \lim_{[n]\in \Delta} {\mathrm{GrHiggs}}^*({\mathrm{Spf}}(S^{(n)})/A). \end{aligned}$$* *Proof.* One just mimics the proof of , and uses (2) as the main ingredient. ◻ **Remark 1**. Let us justify the appearance of "Higgs" in the above discussion. By construction, there is a forgetful functor from ${\mathrm{FilHiggs}}(Z/A)$ to the category of Hodge--Tate crystals over $(Z/A)_{\mathlarger{\mathbbl{\Delta}}}$, by forgetting the filtration structure. Moreover, when $Z$ is affine and smooth over $\bar{A}$, showing in [@BL22b Corollary 6.6] (independently by [@Tian21 Thm. 4.12], and for affine smooth $Z$ that is small by [@MW22 Thm. 1.1]) Hodge--Tate crystals over $Z/\bar{A}$ are equivalent to quasi-nilpotent Higgs fields in complexes of $Z/A$. So we can think of objects in ${\mathrm{FilHiggs}}(Z/A)$ as derived filtered generalizations of the usual notion of Higgs fields, over $p$-adic formal schemes that may not be smooth. To relate the relative prismatic ($F$-)gauge with the filtered Higgs field, we first recall the following observation on Nygaard graded pieces of relative prismatic cohomology from [@BS19 §15]. Assume the ideal $I$ is generated by an element $d$, which we fix for the rest of this subsection. Then one can define a twisted version of the conjugate filtered Hodge--Tate cohomology, by $$S\longmapsto \left( {\mathrm{colim}}~{\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A} \{\bullet\} \right)^\wedge_p,$$ where the transition map is given by the tensor product of the canonical map ${\mathrm{Fil}}^{\mathrm{conj}}_i \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A} \to {\mathrm{Fil}}^{\mathrm{conj}}_{i+1} \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}$ with the twisting map $d:I^i/I^{i+1} \simeq I^{i+1}/I^{i+2}$. **Fact 1**. *Let $d$ be a generator of the ideal $I$. For a large quasi-syntomic $\bar{A}$-algebra over $S$, the relative Frobenius $\varphi_{rel} \colon {\mathlarger{\mathbbl{\Delta}}}_{S/A}^{(1)} \to {\mathlarger{\mathbbl{\Delta}}}_{S/A}$ identifies the graded ring ${\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S/A}^{(1)}$ with the Rees construction of the twisted conjugate filtered Hodge--Tate cohomology ${\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}\{\bullet\}$. As a consequence, for $*\in\{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$, we have an equivalence $${\mathcal{DG}}_{p\text{-}\mathrm{comp}}^*({\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S/A}^{(1)}) \simeq {\mathcal{DF}}_{p\text{-}\mathrm{comp}}^*( {\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}).$$* *Proof.* The first identification of rings is in [@BS19 Thm. 15.2.(2)] and also recorded in . To get the equivalence of categories, we simply notice that there is an equivalence of filtered derived categories $$\begin{aligned} {\mathcal{DF}}({\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}) & \longrightarrow {\mathcal{DF}}({\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}\{\bullet\}),\\ {\mathrm{Fil}}_\bullet M &\longmapsto {\mathrm{Fil}}_\bullet M\otimes_{\bar{A}} \bar{I}^\bullet/\bar{I}^{\bullet+1}, \end{aligned}$$ which is compatible with the twisting of the conjugate filtered Hodge--Tate cohomology. ◻ By taking limit ranging over all $S \in (Z/\bar{A})_{\mathrm{qrsp}}$, the above naturally extends to general quasi-syntomic $\bar{A}$-formal schemes. **Corollary 1**. *Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme, and let $*\in\{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$. There is an equivalence of categories $${\mathcal{DG}}_{p\text{-}\mathrm{comp}}^*((Z/\bar{A}))_{\mathrm{qrsp}}, {\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{-/A}^{(1)}) \simeq {\mathrm{FilHiggs}}^*(Z/A).$$* **Definition 1**. Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme, and let $*\in\{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$. We define the *Hodge--Tate realization* functor $${\mathrm{(F\text{-})Gauge}}^*(Z/A) \longrightarrow {\mathrm{FilHiggs}}^*(Z/A)$$ to be the composition of the associated graded functor with the equivalence in . Namely it is the limit of compositions $${\mathrm{(F\text{-})Gauge}}^*({\mathrm{Spf}}(S)/A) \longrightarrow {\mathcal{DG}}_{p\text{-comp}}^*({\mathrm{Gr}}^\bullet_N {\mathlarger{\mathbbl{\Delta}}}_{S/A}^{(1)}) \simeq {\mathcal{DF}}_{p\text{-comp}}^*( {\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{S/A}) = {\mathrm{FilHiggs}}^*({\mathrm{Spf}}(S)/\bar{A}),$$ where $S$ ranges over all $(Z/\bar{A})_{\mathrm{qrsp}}$. The following relates Hodge--Tate cohomology and Higgs cohomology. **Proposition 1**. *Let $Z$ be a quasi-syntomic formal scheme over $\bar{A}$, and let $*\in \{\emptyset, {\mathrm{perf}}, {\mathrm{vect}}\}$. Let $E=(E,{\mathrm{Fil}}^\bullet E^{(1)},\varphi_{E})\in {\mathrm{F\text{-}Gauge}}^*(Z/A)$ and let $(M, {\mathrm{Fil}}_\bullet M)$ be the associated filtered Higgs field. Then $\varphi_{E}$ induces the following isomorphisms* 1. *twisted graded pieces ${\mathrm{Gr}}^i E^{(1)}\{-i\}$ of ${\mathrm{Fil}}^\bullet E^{(1)}$ and filtrations ${\mathrm{Fil}}_i M$ of $M$;* 2. *Hodge--Tate cohomology of $E$ and Higgs cohomology of $M$.* As a consequence, for a relative $F$-gauge $E$ over $X/A$, we get a natural filtration on $R\Gamma((X/A)_{\mathlarger{\mathbbl{\Delta}}}, \overline{E})$, which we call the *conjugate filtration* on Hodge--Tate cohomology of $E$. *Proof of .* Let $E=(E, {\mathrm{Fil}}^\bullet E^{(1)}, \varphi_{E})$ be an object in ${\mathrm{F\text{-}Gauge}}^*({\mathrm{Spf}}(S)/A)$ for a given $S\in (Z/\bar{A})_{\mathrm{qrsp}}$, and let $(M,{\mathrm{Fil}}_\bullet M)$ be the associated filtered Higgs field. On the one hand, by the map $\varphi_{E}$ induces an isomorphism of complexes $$\left( {\mathrm{colim}}_{i\in \mathbb{Z}} (\cdots \to {\mathrm{Fil}}^i E^{(1)} \otimes I^{-i} \to {\mathrm{Fil}}^{i+1} E^{(1)} \otimes I^{-i-1}\to \cdots) \right)^\wedge_{(p,I)} \simeq E,$$ whose reduction mod $I$ is the Hodge--Tate cohomology, namely $$\left( A/I \otimes_A {\mathrm{colim}}_{i\in \mathbb{Z}} (\cdots \to {\mathrm{Fil}}^i E^{(1)} \otimes I^{-i} \to {\mathrm{Fil}}^{i+1} E^{(1)} \otimes I^{-i-1}\to \cdots) \right)^\wedge_p \simeq \overline{E}.$$ Moreover, by taking the mod $I$ (where $I$ is having filtration degree $1$) reduction within the colimit, we can rewrite the left hand side as $$\left( {\mathrm{colim}}_{i\in \mathbb{Z}} (\cdots \to {\mathrm{Gr}}^i E^{(1)} \otimes \bar{I}^{-i} \to {\mathrm{Gr}}^{i+1} E^{(1)} \otimes \bar{I}^{-i-1}\to \cdots) \right)^\wedge_p \simeq \overline{E}.$$ On the other hand, the twisted graded factor ${\mathrm{Gr}}^\bullet E^{(1)}\{-\bullet\}$ is the Rees construction for the associated filtered Higgs field ${\mathrm{Fil}}_\bullet M$. In particular, the Higgs cohomology, which is the underlying complex $M$ of the increasingly filtered objects ${\mathrm{Fil}}_i M$, is equal to $$\left( {\mathrm{colim}}_{i\in \mathbb{Z}} (\cdots \to {\mathrm{Fil}}_i M \to {\mathrm{Fil}}_{i+1} M \to \cdots ) \right)^\wedge_p \simeq \left( {\mathrm{colim}}_{i\in \mathbb{Z}} (\cdots \to {\mathrm{Gr}}^i E^{(1)}\{-i\} \to {\mathrm{Gr}}^{i+1} E^{(1)}\{-i-1\} \to \cdots ) \right)^\wedge_p.$$ Hence the relative Frobenius identifies the Hodge--Tate cohomology of the $F$-gauge $E$ with the Higgs cohomology of the associated filtered Higgs field $M$, and sends ${\mathrm{Gr}}^i E^{(1)}\{-i\}$ isomorphically onto ${\mathrm{Fil}}_i M$. For arbitrary quasi-syntomic $\bar{A}$-formal scheme $Z$ and a relative $F$-gauge $E \in {\mathrm{F\text{-}Gauge}}^*(Z/A)$, since the above isomorphisms for $E(S)$ is functorial in $S \in (Z/\bar{A})_{\mathrm{qrsp}}$, by passing to limit we may conclude the isomorphism of cohomology in general. ◻ The identification in in particular allows us to study the notion of weights on filtered Higgs complexes. Following , for $\ast\in \{\emptyset, {\mathrm{perf}}, {\mathrm{vect}}\}$, we let ${\mathrm{Red}}_{Z/A}$ be the base change functor $$\begin{tikzcd} {\mathrm{FilHiggs}}^\ast(Z/\bar{A}) \arrow[rrr, "-\otimes_{{\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{-/A}} \mathcal{O}_{\mathrm{qrsp}}"] &&& {\mathcal{DG}}^\ast_{p\text{-comp}}(\mathcal{O}_Z), \end{tikzcd}$$ which by construction factors through the quotient functor ${\mathrm{Gr}}\colon {\mathrm{FilHiggs}}^\ast(Z/\bar{A}) \to {\mathrm{GrHiggs}}^\ast(Z/\bar{A})$. A filtered Higgs complex $M$ is of *weight* $[a,b]$ if ${\mathrm{Red}}_{Z/A}(M)$ lives within degree $[a,b]$. Translating using , we get the weight filtration on filtered Higgs fields. **Corollary 1**. *Fix a generator $d$ of the ideal $I$, let $Z$ be a quasi-syntomic $p$-adic formal scheme over $\bar{A}$, and let $*\in\{\emptyset, {\mathrm{perf}},{\mathrm{vect}}\}$. For a filtered Higgs complex $M \in {\mathrm{FilHiggs}}^\ast(Z/\bar{A})$ of weight $[a,b]$, there is a finite increasing and exhaustive filtration ${\mathrm{wt}}_\bullet(M)$ of range $[a,b]$ on $M$ such that the $l$-th graded piece ${\mathrm{wt}}_l(M)$ is naturally isomorphic to ${\mathrm{Red}}_{l,Z/A}(M)\otimes_{\mathcal{O}_Z} {\mathrm{Fil}}^{\mathrm{conj}}_\bullet \overline{{\mathlarger{\mathbbl{\Delta}}}}_{-/A}$.* By taking the direct image to the Zariski site of $Z$, we can analyze the graded factors of the conjugate filtration on Hodge--Tate cohomology. **Construction 1**. Denote by $\lambda \colon Z_{\mathrm{qrsp}}\to Z_\mathrm{Zar}$ the natural map of ringed sites. Then there is a natural graded derived pushforward functor between two graded derived category of presheaves $$R\lambda_* \colon {\mathcal{DG}}((Z/\bar{A})_{\mathrm{qrsp}},\mathcal{O}_{\mathrm{qrsp}}^\mathrm{PSh}) \to {\mathcal{DG}}(Z_\mathrm{Zar}, \mathcal{O}_{Z}^\mathrm{PSh}).$$ **Remark 1**. For any open $U \subset Z$, the evaluation of $R\lambda_*$ at $U$ of quasi-syntomic sheaves is given by the unfolding process, see and . We now use weight filtration to subdivide the Hodge--Tate cohomology and Higgs cohomology into smaller pieces, where each of them can be understood using the reduction functor and differential forms. **Corollary 1**. *Let $Z$ be a smooth $\bar{A}$-formal scheme of relative dimension $n$, let $*\in \{{\mathrm{vect}},{\mathrm{perf}}\}$, and let $M\in {\mathrm{FilHiggs}}^*(Z/\bar{A})$ be of weight $[a,b]$. The $j$-th graded factor ${\mathrm{Gr}}_j R\lambda_* M$ is zero unless $j\in [a,b+n]$. In the latter case it admits a finite increasing filtration such that the $i$-th graded factor of this filtration is $$({\mathrm{Red}}_{a+i, Z/A}(M)) \otimes_{\mathcal{O}_X} \Omega^{j-a-i}_{Z/\bar{A}}\{a+i-j\}[a+i-j],\quad \max\{0,j-a-n\}\leq i \leq \min\{j,b\}-a.$$ In particular, if $\bigoplus_l {\mathrm{Red}}_{l,Z/A}(M)$ is given by a coherent sheaf over $\mathcal{O}_Z$, then we have $${\mathrm{Gr}}_j R\lambda_* M \in D^{[\max\{j-b,0\}, \min\{j-a,n\}]}_\mathrm{coh}(\mathcal{O}_Z).$$* Here in the calculation, we implicitly use the fact that for a smooth $\bar{A}$-formal scheme of relative dimension $n$, the relative sheaf of differential $\Omega^l_{Z/\bar{A}}$ is zero unless $l\in [0,n]$. For the future usage, we also record a special case when an $F$-gauge comes from an $I$-torsionfree coherent $F$-crystal, where we can show that the reduction of the associated filtered Higgs field (which is the same as the reduction of the gauge by ) is coherent. This is summarized in the next result. **Corollary 1**. *Let $f:X\to Y$ be a smooth morphism of smooth $p$-adic formal schemes over $\mathcal{O}_K$, and let $(\mathcal{E},\varphi_{\mathcal{E}})$ be an $I$-torsionfree coherent $F$-crystal over $X$. For a bounded prism $(A,I)\in Y_{\mathlarger{\mathbbl{\Delta}}}$ such that ${\mathrm{Spf}}(\bar{A})\to Y$ is $p$-completely flat, let $E' = BC_{X, X_{\bar{A}}/A}(\Pi_X(\mathcal{E}))$ be the associated relative $F$-gauge over $X_{\overline{A}}/A$ via and . Then the reduction ${\mathrm{Red}}_{X_{\overline{A}}/A}(E')$ is a graded coherent sheaf over $\mathcal{O}_X$.* *Proof.* Using the identification in , this follows from the complete base change formula ${\mathrm{Red}}_{X_{\overline{A}}/A}(E')\simeq {\mathrm{Red}}_X(E)\widehat{\otimes}_{\mathcal{O}_X} \mathcal{O}_{X_{\overline{A}}}$ in .(ii): since ${\mathrm{Red}}_X(E)$ is coherent over $\mathcal{O}_X$ () and $X_{\overline{A}}\to X$ is $p$-completely flat, the complete base change is also coherent. ◻ ## Completeness of Nygaard filtration via filtered de Rham realization {#Completeness of Nygaard filtration via filtered de Rham realization} In this subsection, in eye of , we study the mod $\frac{d}{t}$ specialization of a relative prismatic $F$-gauge in this subsection. The following definition and sheaf-property proposition is completely analogous to previous subsections. **Definition 1**. Let $Z$ be a quasi-syntomic $\bar{A}$-formal scheme. The category of *filtered connections* over $Z/\bar{A}$ is defined as the limit of $\infty$-categories $${\mathrm{FilConn}}(Z/\bar{A})\colonequals \lim_{S \in (Z/\bar{A})_{\mathrm{qrsp}}} {\mathrm{FilConn}}({\mathrm{Spf}}(S)/\bar{A}),$$ where ${\mathrm{FilConn}}({\mathrm{Spf}}(S)/\bar{A}) \colonequals {\mathcal{DF}}_{p\text{-comp}}({\mathrm{Fil}}^\bullet_H \mathrm{dR}(S/\bar{A}))$ is the category of $p$-complete filtered complexes over the filtered ring ${\mathrm{Fil}}^\bullet_H \mathrm{dR}(S/\bar{A})$. Here, similar to , for a map of large quasi-syntomic $\bar{A}$-algebras $S_1\to S_2$, the completed filtered base change induces natural functors between ${\mathrm{FilConn}}({\mathrm{Spf}}(S_i)/\bar{A})$. Similar to , for $*\in \{{\mathrm{vect}}, {\mathrm{perf}}\}$, we use ${\mathrm{FilConn}}^*(Z/\bar{A})$ to denote the subcategory of filtered bundles/perfect complexes with flat connection. Just like before, all these form quasi-syntomic sheaves. **Proposition 1**. *Let $S \to S^{(0)}$ be a quasi-syntomic cover of large quasi-syntomic rings over $\bar{A}$, and let $S^{(\bullet)}$ be the $p$-completed Čech nerve. Then for $*\in \{{\mathrm{perf}},{\mathrm{vect}}\}$, we have a natural equivalence $$\begin{aligned} {\mathrm{FilConn}}^*({\mathrm{Spf}}(S)/\bar{A}) & \simeq \lim_{[n]\in \Delta} {\mathrm{FilConn}}^*({\mathrm{Spf}}(S^{(n)})/\bar{A}). \end{aligned}$$* *Proof.* One just mimics the proof of , and uses (3) as the main ingredient. ◻ Before we proceed to the next result, we need to discuss the notion of "perfect complex" in great generality. We collect the discussion in the following: **Digression 1** ([@Lur17 §7.2.4]). We begin by recalling [@Lur17 Definition 7.2.4.1]: Let $R$ be an $\mathbb{E}_1$-ring, then the subcategory of left perfect $R$-complexes in the $\infty$-category of left $R$-complexes is defined as the smallest stable subcategory containing $R$ and closed under retracts, objects of this subcategory are called left perfect $R$-complexes. Immediately after this definition, Lurie proved [@Lur17 Proposition 7.2.4.2]: In the above setting, left perfect $R$-complexes are the same as compact objects among left $R$-complexes, and they generate the $\infty$-category of left $R$-complexes. In fact there is a third characterization of perfectness [@Lur17 Proposition 7.2.4.4]: it is equivalent to being dualizable. For our purpose, we need analogous results in the filtered setting. Via Rees's construction [@Bha23 §2.2], we are led to consider analogs of the above in the graded setting. **Definition 1**. Let $R^{\bullet}$ be a graded $\mathbb{E}_1$-ring, then the subcategory of graded left perfect $R^{\bullet}$-complexes in the $\infty$-category of graded left $R^{\bullet}$-complexes is defined as the smallest stable subcategory containing $R^{\bullet}\langle i \rangle$ for all $i$ and closed under retracts. Here $(-)\langle i \rangle$ denotes shift of grading by $i$. We denote this subcategory by $\mathcal{G}\mathrm{LMod}^{{\mathrm{perf}}}_{R^{\bullet}}$, objects of which are called graded left perfect $R^{\bullet}$-complexes. Recall that given a graded $\mathbb{E}_1$-ring $R^{\bullet}$, we may form its *direct sum totalization* (abbreviated as totalization below) $\bigoplus_{i \in \mathbb{Z}} R^i$, similarly for a graded left $R^{\bullet}$-complex, we may form $\bigoplus_{i \in \mathbb{Z}} M^i$, these are $\mathbb{E}_1$-ring and left modules over it respectively. Here is an analog of [@Lur17 Proposition 7.2.4.2]. **Proposition 1**. *Let $R^{\bullet}$ be a graded $\mathbb{E}_1$-ring. Then:* 1. *The $\infty$-category $\mathcal{G}\mathrm{LMod}_{R^{\bullet}}$ is compactly generated.* 2. *An object of $\mathcal{G}\mathrm{LMod}^{{\mathrm{perf}}}_{R^{\bullet}}$ is compact if and only if it is perfect.* *Moreover the property of being perfect/compact is equivalent to the totalization being perfect/compact as a complex over the totalization of $R^{\bullet}$.* *Proof.* The proof of (1) and (2) are exactly the same as that in loc. cit: Tracing through the proof there, we just need to observe that if $\mathrm{Map}_{R^{\bullet}}(R^{\bullet}\langle i \rangle[j], M^{\bullet}) = 0$ for all $i, j$, then $M^{\bullet} = 0$. As for the last statement: it simply follows from the fact that taking totalization has a right adjoint preserving filtered colimit: $$\mathrm{LMod}_{\bigoplus_{i} R^i} \to \mathcal{G}\mathrm{LMod}_{R^{\bullet}}, M \mapsto M \otimes_{(\bigoplus_{i} R^i)} \left(\bigoplus_{j \in \mathbb{Z}} R^{\bullet}\langle j \rangle\right),$$ where the map $\bigoplus_{i} R^i \to \bigoplus_{j \in \mathbb{Z}} R^{\bullet}\langle j \rangle$ is the canonical map realizing the source as the $\deg = 0$ piece of the target. ◻ **Remark 1**. If $R^{\bullet}$ is a graded discrete commutative ring, then graded $R^{\bullet}$-complexes are the same as quasi-coherent complexes on the stack $[\mathrm{Spec}(\bigoplus_i R^i)/\mathbb{G}_m]$, whose perfectness is defined by that of its pullback to $\mathrm{Spec}(\bigoplus_i R^i)$. This is a well-defined notion due to flat descent of perfectness (see [@stacks-project [Tag 066X](https://stacks.math.columbia.edu/tag/066X)] and [@stacks-project [Tag 068T](https://stacks.math.columbia.edu/tag/068T)]). Our proposition above shows that in this special case, the perfectness in classical sense agrees with our definition here. Now let us translate our discussion of graded rings/complexes to filtered setting via the Rees construction. **Definition 1**. Let ${\mathrm{Fil}}^{\bullet} S$ be a filtered $\mathbb{E}_1$-ring with its associated Rees construction $R^{\bullet}$, viewed as a graded $\mathbb{E}_1$-ring. Then the Rees construction (c.f. [@Bha23 Proposition 2.6]) induces an equivalence of $\infty$-categories $$\mathcal{F}\mathrm{LMod}_{{\mathrm{Fil}}^{\bullet} S} \cong \mathcal{G}\mathrm{LMod}_{R^{\bullet}}.$$ We then define a filtered left complex ${\mathrm{Fil}}^{\bullet} M$ to be perfect if its image under the above equivalence is perfect. Since the above is an equivalence, we immediately see that the subcategory of filtered left perfect ${\mathrm{Fil}}^{\bullet} S$-complexes can be alternatively defined as the smallest stable subcategory containing $({\mathrm{Fil}}^{\bullet} S)\langle i \rangle$ for all $i$ and closed under retracts. Here once again, we use $(-)\langle i \rangle$ to denote shift of filtration by $i$. The following is a simple translation of via the Rees equivalence. **Proposition 1**. *Let ${\mathrm{Fil}}^{\bullet} S$ be a filtered $\mathbb{E}_1$-ring. Then:* 1. *The $\infty$-category $\mathcal{F}\mathrm{LMod}_{{\mathrm{Fil}}^{\bullet} S}$ is compactly generated.* 2. *An object of $\mathcal{F}\mathrm{LMod}_{{\mathrm{Fil}}^{\bullet} S}$ is compact if and only if it is perfect.* *Moreover the property of being perfect/compact is equivalent to its underlying Rees construction being perfect/compact as a ${\mathrm{Rees}}({\mathrm{Fil}}^{\bullet} S)$-complex.* We also get the following useful property of filtered left perfect complexes over a complete filtered ring. **Proposition 1**. *Let ${\mathrm{Fil}}^{\bullet} S$ be a filtered $\mathbb{E}_1$-ring which is complete with respect to its filtration. Any filtered left perfect ${\mathrm{Fil}}^{\bullet} S$-complex is complete with respect to its filtration.* *Proof.* We simply observe that such filtered left complexes form a stable subcategory containing $({\mathrm{Fil}}^{\bullet} S)\langle i \rangle$ for all $i$ and closed under retracts. Therefore it must contain the subcategory of perfect complexes. ◻ Analogs of flat locality of perfectness (see [@stacks-project [Tag 068T](https://stacks.math.columbia.edu/tag/068T)]) in derived algebraic setting has been considered and worked out by various people, we shall use the notion of "descendability" by Mathew, see [@Mat16 Definition 3.18]. Below let us collect some useful facts about descendable maps between $\mathbb{E}_{\infty}$-rings. **Digression 1** ([@Mat16 §3], see also [@BS17 §11.2]). Descendability is preserved under base change, see [@Mat16 Corollary 3.21]. If $R \to S$ is a descendable map between $\mathbb{E}_{\infty}$-rings, and let $S^{(\bullet)}$ be the Cech nerve, then [@Mat16 Proposition 3.22] gives the "usual descent" statement: the $\infty$-category of $R$-complexes is equivalent to that of "descent data of $S^{(\bullet)}$-complexes". Combining with [@Mat16 Proposition 3.28], we see that an $R$-complex $M$ defined by an $S$-complex $N$ together with a descent data is in $\mathrm{Mod}_R^{{\mathrm{perf}}}$ if $N \in \mathrm{Mod}_S^{{\mathrm{perf}}}$. Since we shall be working in the derived complete world, let us remark that all above statements work in the derived complete setting. In fact Mathew works with general "*stable homotopy theory*", examples of which include $p$-complete spectra, and $p$-complete (or $(p, I)$-complete) $A$-complexes for some bounded prism $(A, I)$. In [@BS19 Lemma 8.6] one finds an interesting example of descendable map between two $(p, I)$-complete $\mathbb{E}_\infty$-rings, what we need is the following slight generalization. **Proposition 1**. *Let $(A, I = (d))$ be a bounded prism with $\bar{A} \coloneqq A/I$, let $\bar{A}\langle \underline{X} \rangle \to R$ be a $p$-completely étale map, and let $R_{\infty} \coloneqq R \widehat{\otimes}_{\bar{A}\langle \underline{X} \rangle} \bar{A}\langle \underline{X}^{1/p^\infty} \rangle$. Then the filtered maps $${\mathrm{Fil}}^{\bullet}_N \mathrm{R\Gamma}({\mathrm{Spf}}(R)_{{\mathrm{qSyn}}}, {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}) \to {\mathrm{Fil}}^{\bullet}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{R_{\infty}/A}$$ and $${\mathrm{Fil}}^{\bullet}_H \mathrm{dR}(R/\bar{A}) \to {\mathrm{Fil}}^{\bullet}_H \mathrm{dR}(R_{\infty}/\bar{A})$$ induce descendable maps between their underlying Rees algebras.* *Proof.* Since descendability is preserved under base change, we are immediately reduced to the case where $\bar{A}\langle \underline{X} \rangle = R$. It suffices to prove the statement when there is only one variable, as the general case follows from tensoring up one variable case. Just like the proof of [@BS19 Lemma 8.6], we use $F$ to denote the fiber in both Rees algebra maps, and we just need to show $F^{\otimes 2}$ has only nullhomotopic maps to the Rees algebra for the filtered sheaves evaluated at ${\mathrm{Spf}}(R)$. For the case of Nygaard filtered ${\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}$, by completeness, it suffices to check the above claim after modulo $\frac{d}{t}$ (using notation from ); by the we are reduced to the case of Hodge filtered $\mathrm{dR}(-/\bar{A})$. At this point, we simply proceed as in the proof of [@BS19 Lemma 8.6]: By completeness we may further reduce mod $p$, then our map has a model defined over $\mathbb{F}_p$, and the proof as in loc. cit. applies to show the relevant mapping space has only one component. ◻ **Theorem 1**. *Let $Z$ be a smooth $\bar{A}$-formal scheme. Let $E=(E, {\mathrm{Fil}}^\bullet E^{(1)})$ be a prismatic gauge in perfect complexes over $Z/A$, let $\mathcal{E}$ be a prismatic crystal in perfect complexes over $Z/A$, and let $(\mathcal{F}, {\mathrm{Fil}}^\bullet \mathcal{F})$ be a filtered perfect complex with connection over $Z/\bar{A}$.* 1. *For any affine open $U = {\mathrm{Spf}}(R) \subset Z$ which admits a formal étale map to $\widehat{\mathbb{A}^n}_{\bar{A}}$, the filtered complexes ${\mathrm{Fil}}^{\bullet} E^{(1)}(U)$ and ${\mathrm{Fil}}^\bullet \mathcal{F}(U)$ are perfect as filtered ${\mathrm{Fil}}^{\bullet}_N {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{U/A}$-complexes and ${\mathrm{Fil}}^{\bullet}_H \mathrm{dR}(U/\bar{A})$-complexes respectively.* 2. *The assignments $S \mapsto \mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{S/A})$ and $S \mapsto \mathcal{E}^{(1)}({\mathlarger{\mathbbl{\Delta}}}_{S/A}) \coloneqq \mathcal{E}({\mathlarger{\mathbbl{\Delta}}}_{S/A}) \widehat{\otimes}_{A, \varphi_A} A$ defines quasi-syntomic sheaves on $(Z/A)_{{\mathrm{qrsp}}}$. With notation as in (1), the complex $\mathcal{E}(U)$ is perfect over ${\mathlarger{\mathbbl{\Delta}}}_{U/A}$. Moreover the natural map $$\mathcal{E}(U) \widehat{\otimes}_{A, \varphi_A} A \to \mathcal{E}^{(1)}(U)$$ is an isomorphism.* 3. *The filtrations ${\mathrm{Fil}}^{\bullet} E^{(1)}(Z)$ and ${\mathrm{Fil}}^\bullet \mathcal{F}(Z)$ are complete.* Here the evaluation at $U$ of quasi-syntomic sheaves is defined by the unfolding process, see and . *Proof.* For (1): Fix a $p$-completely étale map $\bar{A}\langle \underline{X} \rangle \to R$. Let $R^{(\bullet)}$ be the Cech nerve of $R \to R_{\infty}$ where $R_{\infty} \coloneqq R \widehat{\otimes}_{\bar{A}\langle \underline{X} \rangle} \bar{A}\langle \underline{X}^{1/p^\infty} \rangle$. By , the values ${\mathrm{Fil}}^{\bullet} E^{(1)}(U)$ and ${\mathrm{Fil}}^\bullet \mathcal{F}(U)$ are given by the descent of their values on $R^{(\bullet)}$. By , we need to check their associated Rees's construction is perfect. Notice that the values on each of the $R^{\bullet}$ are perfect by assumption, so we only need to descend perfectness down. Finally we are reduced to , by descent of perfectness along descendable map (). For (2): the first statement follows from the fact that ${\mathlarger{\mathbbl{\Delta}}}_{-/A}$ and ${\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}$ are quasi-syntomic sheaves (by Hodge--Tate and de Rham comparison respectively), and the perfectness assumption on $\mathcal{E}$. Then the perfectness of $\mathcal{E}(U)$ over ${\mathlarger{\mathbbl{\Delta}}}_{U/A}$ can be proved exactly as in (1). Lastly, with notations as in the proof of (1), we need to show the following natural arrow: $$\mathcal{E}(U) \widehat{\otimes}_{A, \varphi_A} A \to \lim_{\bullet \in \Delta} \mathcal{E}^{(1)}(R^{(\bullet)})$$ is an isomorphism. We first observe base change formulas: $\mathcal{E}(U) \widehat{\otimes}_{A, \varphi_A} A \cong \mathcal{E}(U) \widehat{\otimes}_{{\mathlarger{\mathbbl{\Delta}}}_{U/A}} {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{U/A}$ as well as $\mathcal{E}^{(1)}(R^{(\bullet)}) \cong \mathcal{E}(U) \widehat{\otimes}_{{\mathlarger{\mathbbl{\Delta}}}_{U/A}} {\mathlarger{\mathbbl{\Delta}}}^{(1)}_{R^{(\bullet)}/A}$. By perfectness of $\mathcal{E}(U)$ over ${\mathlarger{\mathbbl{\Delta}}}_{U/A}$, using the above base change formulas, we are reduced to showing that the unfolding of ${\mathlarger{\mathbbl{\Delta}}}^{(1)}_{-/A}$ at $U$ is ${\mathlarger{\mathbbl{\Delta}}}_{U/A} \widehat{\otimes}_{A, \varphi_A} A$: This follows from the de Rham comparison as in [@BS19 Theorem 15.3]. As for (3): quasi-syntomic sheaves are in particular Zariski sheaves, so it suffices to check completeness for the values on a basis of opens in $Z_{\mathrm{Zar}}$. To that end, we simply use (1) and to reduce ourselves to the completeness of the Nygaard and Hodge filtration on smooth algebras. The Hodge filtration for smooth formal schemes is a finite filtration, hence complete. The claim about Nygaard filtration can be found in [@LL20 Lemma 7.8.(1)], the proof is quite easy: by $\frac{d}{t}$-completeness, we may check the completeness of filtration after modulo $\frac{d}{t}$, but then reduces us to checking completeness of Hodge filtration for smooth formal schemes. ◻ In a private communication, Bhatt told us that [@BL22b Theorem 7.17] is proved by a similar argument. # Height of prismatic cohomology {#Height of prismatic cohomology} In this section, we study Verschiebung operator and the Frobenius height of prismatic cohomology, with coefficients in coherent prismatic $F$-crystals. Recall that in we defined a natural graded derived pushforward functor between two graded derived category of presheaves $$R\lambda_*: {\mathcal{DG}}((X/\bar{A})_{\mathrm{qrsp}},\mathcal{O}_{\mathrm{qrsp}}^\mathrm{PSh}) \to {\mathcal{DG}}(X_\mathrm{Zar}, \mathcal{O}_{X}^\mathrm{PSh}).$$ Also recall that in we have attached an $F$-gauge $\Pi_X(\mathcal{E})$ on $X$ to an $F$-crystal $(\mathcal{E}, \varphi_{\mathcal{E}})$ on $X$, then according to , we obtain a relative $F$-gauge $BC_{X, X_{\bar{A}}/A}(\Pi_X(\mathcal{E}))$ over $X_{\bar{A}}/A$. Using these constructions, our first main theorem is the following. **Theorem 1**. *Let $f \colon X\to Y$ be a smooth morphism of smooth $p$-adic formal schemes over $\mathcal{O}_K$ that is of relative dimension $n$, and let $(A,I)$ be a bounded prism over $Y$ such that ${\mathrm{Spf}}(\bar{A})\to Y$ is $p$-completely flat. Assume $(\mathcal{E},\varphi_{\mathcal{E}})\in {\mathrm{F\text{-}Crys}}^{\mathrm{coh}}(X)$ is $I$-torsionfree of height $[a,b]$, with $(E,{\mathrm{Fil}}^\bullet E^{(1)},\widetilde{\varphi}_E)$ the associated $F$-gauge.* 1. *[\[thm height of coh local graded piece\]]{#thm height of coh local graded piece label="thm height of coh local graded piece"} For each $j\in \mathbb{Z}$, the complex $R\lambda_* {\mathrm{Gr}}^jE^{(1)}$ lives in $D^{[0,n]}_{{\mathrm{coh}}}(\mathcal{O}_{X_{\bar{A}}})$.* 2. *[\[thm height of coh local with I-adic filtration\]]{#thm height of coh local with I-adic filtration label="thm height of coh local with I-adic filtration"} The filtered complex $R\lambda_* {\mathrm{Fil}}^{\geq n+b}E^{(1)}$ is equivalent to the $I$-adic filtration $R\lambda_* I^{\geq n+b} E^{(1)}=I^{\geq n+b}\otimes R\lambda_* E^{(1)}$.* 3. *[\[thm height of coh local bound\]]{#thm height of coh local bound label="thm height of coh local bound"} The map $\widetilde{\varphi}_E$ induces a natural isomorphism of the truncations $$\varphi_{R^{\leq i} \lambda_*}: R^{\leq i} \lambda_* ({\mathrm{Fil}}^{i+b} E^{(1)}) \simeq R^{\leq i} \lambda_* (I^{i+b} E)=I^{i+b}\otimes R^{\leq i}\lambda_* E, ~\forall i\in \mathbb{N}.$$* Combine (i) and (ii) above, we see that $R\lambda_* \mathcal{E}$ and $R\lambda_* {\mathrm{Fil}}^\bullet E^{(1)}$ live in $D^{[0,n]}(X_{\overline{A}})$. **Remark 1**. Note that in the special case when $\mathcal{E}$ is the prismatic structure sheaf $\mathcal{O}_{\mathlarger{\mathbbl{\Delta}}}$, this is shown in [@LL20 Lem. 7.8], which essentially follows from the calculation of graded pieces of Nygaard filtration as in [@BS19 Thm. 15.2]. *Proof.* Denote by $(\overline{\mathcal{E}}, {\mathrm{Fil}}_\bullet \overline{\mathcal{E}})$ the associated filtered Higgs field as in . By and , for each $l\in \mathbb{Z}$, we have $$R\lambda_* {\mathrm{Gr}}_l \overline{\mathcal{E}} \in D_{\mathrm{coh}}^{[0,n]}(\mathcal{O}_X).$$ Since $R\lambda_* {\mathrm{Fil}}_j \overline{\mathcal{E}}$ admits a finite increasing and exhaustive filtration with graded pieces being $R\lambda_* {\mathrm{Gr}}_l \overline{\mathcal{E}}$, we see the cohomological bound and coherence in (i) hold for $R\lambda_* {\mathrm{Fil}}_j \overline{\mathcal{E}}$. To compare the two filtrations, by the filtered completeness of ${\mathrm{Fil}}^{\bullet}E^{(1)}$ in and that of the $I$-adic filtration, it suffices to compare the map of associated graded pieces. We let $C^j$ be the cone of the Frobenius map $\varphi_{\mathcal{E}}:{\mathrm{Fil}}^j E^{(1)} \to I^j E$, regarded as a descending filtered complex via the filtration $C^{\geq j}$. Then for each $j\in \mathbb{Z}$, the graded piece ${\mathrm{Gr}}^j C^\bullet$ by definition is $$\mathrm{Cone}(\varphi_\mathcal{E}:{\mathrm{Gr}}^j E^{(1)} \longrightarrow \overline{\mathcal{E}}\{j\}),$$ with the left hand side identified with ${\mathrm{Fil}}_j \overline{\mathcal{E}}\{j\}$ by . (i). So we get ${\mathrm{Gr}}^j C^\bullet \simeq {\mathrm{Fil}}_{\geq j+1} \overline{\mathcal{E}}\{j\}$. By and , the filtered complex $R\lambda_* ( {\mathrm{Fil}}_{\geq j+1} \overline{\mathcal{E}}\{j\})$ has a finite increasing and exhaustive filtration ranging from the filtered degree $j+1-b$ to $n$ (and the entire term vanishes when $j+1-b>n$.), such that the $l$-th graded piece lives in $D_{\mathrm{coh}}^{[\max\{l,0\},n]}(\mathcal{O}_{X_{\overline{A}}})$. So by induction and the aforementioned facts in again, we have $$R\lambda_* {\mathrm{Gr}}^j C^\bullet \in D_{\mathrm{coh}}^{[\max\{j+1-b,0\}, n]}(\mathcal{O}_{X_{\overline{A}}}), \tag{$\ast$}$$ which vanishes when $j+1-b>n$. This means that the Frobenius map induces an isomorphism from $R\lambda_* {\mathrm{Gr}}^j E^{(1)}$ to $R\lambda_* \overline{\mathcal{E}}\{j\}$ for $j+1-b>n$, and $R\lambda_* ({\mathrm{Fil}}^\bullet E^{(1)})$ is eventually the $I$-adic filtration. Finally by the completeness of the filtration, item (iii) follows from a reformulation of $(\ast)$ that for each $j\geq i+b$, we have $R\lambda_* {\mathrm{Gr}}^j C^\bullet$ lives in $D_{\mathrm{coh}}^{[i+1, n]}(\mathcal{O}_{X_{\overline{A}}})$. ◻ By taking the higher direct image along a proper smooth morphism, we can estimate the height of each individual prismatic cohomology. This is analogous to the work of Kedlaya bounding slopes of rigid cohomology as in [@Ked06 Thm. 6.7.1]. **Theorem 1**. *Let $f \colon X\to Y$ be a smooth morphism of smooth $p$-adic formal schemes over $\mathcal{O}_K$ that is of relative dimension $n$, and let $(A,I)\in Y_{\mathlarger{\mathbbl{\Delta}}}$ such that ${\mathrm{Spf}}(\bar{A})\to Y$ is $p$-completely flat. Assume $(\mathcal{E},\varphi_{\mathcal{E}})\in {\mathrm{F\text{-}Crys}}^{\mathrm{coh}}(X)$ is $I$-torsionfree of height $[a,b]$, and we denote $E=(E,{\mathrm{Fil}}^\bullet E^{(1)},\widetilde{\varphi}_E)$ the associated relative $F$-gauge over $X_{\bar{A}}/A$. For an integer $j\geq a$ and a non-negative integer $i$, the Frobenius structure on the $i$-th relative prismatic cohomology induces a natural commutative diagram $$\begin{tikzcd} \mathrm{H}^i((X/\bar{A})_{\mathrm{qrsp}}, {\mathrm{Fil}}^j E^{(1)}) \arrow[r, "u^j"] \arrow[d, "\varphi^j"'] & \mathrm{H}^i((X/\bar{A})_{\mathrm{qrsp}},E^{(1)}) \arrow[d, "\varphi"] \\ I^j\otimes \mathrm{H}^i((X/\bar{A})_{\mathrm{qrsp}}, E) \arrow[r]& I^a\otimes \mathrm{H}^i((X/\bar{A})_{\mathrm{qrsp}}, E), \end{tikzcd}$$ where the horizontal arrows are defined by the canonical maps and the vertical arrows are given by Frobenius morphisms on $E$. Then we have:* 1. *[\[thm height of coh global upper\]]{#thm height of coh global upper label="thm height of coh global upper"} the map $\varphi^j$ is an isomorphism when $j \geq b+\min\{i,n\}$, and* 2. *[\[thm height of coh global lower\]]{#thm height of coh global lower label="thm height of coh global lower"} the map $u^j$ is an isomorphism when $j \leq a+\max\{0,i-n\}$.* *Proof.* We first notice that as the $F$-crystal $(\mathcal{E},\varphi_\mathcal{E})$ is $I$-torsionfree and of height $[a,b]$, the Frobenius map sends $\mathcal{E}^{(1)}$ into $I^a\mathcal{E}\simeq I^a\otimes \mathcal{E}\subset \mathcal{E}[1/I]$, and sends ${\mathrm{Fil}}^\bullet E^{(1)}$ into $I^a \otimes E$. So the diagram in the statement can be obtained by applying the functor $\mathrm{H}^i(X,-)$ at the following diagram of complexes over $(X_{\bar{A}})_\mathrm{Zar}$: $$\begin{tikzcd} R\lambda_* {\mathrm{Fil}}^j E^{(1)} \arrow[r, "v^j"] \arrow[d, "R\lambda_*\widetilde{\varphi}^j_E"'] & R\lambda_* E^{(1)} \arrow[d, "R\lambda_*\widetilde{\varphi}_E"] \\ I^j\otimes R\lambda_* E \arrow[r] & R\lambda_* (I^a\otimes E) \simeq I^a \otimes R\lambda_* E. \end{tikzcd}$$ To show item [\[thm height of coh global upper\]](#thm height of coh global upper){reference-type="ref" reference="thm height of coh global upper"}, we apply $\mathrm{H}^i(X,-)$ at the isomorphism of sheaves from .[\[thm height of coh local bound\]](#thm height of coh local bound){reference-type="ref" reference="thm height of coh local bound"}, which immediately implies that the map $\varphi^j$ is an isomorphism when $j \geq b+i$. It is then left to show that the map $\varphi^j$ is an isomorphism when $j \geq b+n$, and we assume now that $i \geq n$. Since both $R\lambda_* {\mathrm{Fil}}^\bullet E^{(1)}$ and $R\lambda_* E$ live in cohomological degree $[0,n]$ (.[\[thm height of coh local graded piece\]](#thm height of coh local graded piece){reference-type="ref" reference="thm height of coh local graded piece"} and [\[thm height of coh local with I-adic filtration\]](#thm height of coh local with I-adic filtration){reference-type="ref" reference="thm height of coh local with I-adic filtration"}), the Leray spectral sequence for the map $\varphi^j=\mathrm{H}^i(X, R\lambda_*\widetilde{\varphi}_E^j)$ is computed by the following $n$-terms $$\mathrm{H}^{i-n}(X, R^n\lambda_* \widetilde{\varphi}_E^j), \ldots, \mathrm{H}^i(X, R^0\lambda_* \widetilde{\varphi}_E^j).$$ Notice that by .[\[thm height of coh local bound\]](#thm height of coh local bound){reference-type="ref" reference="thm height of coh local bound"}, the map $R^l\lambda_*\widetilde{\varphi}_E^j$ is an isomorphism when $j\geq b+l$. Thus the map $\varphi^j$ is an isomorphism when $j \geq b+n$, as the number $l$ ranges from $0$ to $n$. For [\[thm height of coh global lower\]](#thm height of coh global lower){reference-type="ref" reference="thm height of coh global lower"}, the sheaf of complexes $\mathrm{Cone}(v^j)$ admits a decreasing filtration, which by satisfies $${\mathrm{Gr}}^l(\mathrm{Cone}(v^j))= \begin{cases} {\mathrm{Fil}}_l \bar{\mathcal{E}}\{l\},~ l \leq j-1,\\ 0,~ l \geq j. \end{cases}$$ Moreover, by and , each ${\mathrm{Fil}}_l \bar{\mathcal{E}}\{l\}$ lives in $D^{[0, \min\{l-a,n\}]}_\mathrm{coh}(\mathcal{O}_X)$. So by the filtered completeness of $\mathrm{Cone}(v^j)$, the complex $\mathrm{Cone}(v^j)$ itself lives in $D^{[0, \min\{ j-1-a,n\}]}_\mathrm{coh}(\mathcal{O}_X)$, which is zero when $j-1-a <0$, or equivalently $j \leq a$. This in particular implies that when $j\leq a$, the map $v^j$ and hence $u^j$ are always isomorphisms for all the integers $i$. To finish the proof of [\[thm height of coh global lower\]](#thm height of coh global lower){reference-type="ref" reference="thm height of coh global lower"}, we are left to consider the case when $i \geq n$. Using Leray spectral sequence again, the cone of the map $u^j=\mathrm{H}^i(v^j)$ is computed by the following terms $$\mathrm{H}^{i-n}(X, \mathcal{H}^n(\mathrm{Cone}(v^j)), \ldots, \mathrm{H}^n(X, \mathcal{H}^{i-n}(\mathrm{Cone}(v^j)),$$ where we implicitly use the fact that $\mathrm{H}^l(X,\mathcal{H}^{i-l}(\mathrm{Cone}(v^j))=0$ for $l >n$. In particular, when $j \leq a+i-n$, as the complex $\mathrm{Cone}(v^j)$ lives in $D^{[0, \min\{ j-1-a,n\}]}_\mathrm{coh}(\mathcal{O}_X) \subset D^{[0, \min\{ i-n-1,n\}]}_\mathrm{coh}(\mathcal{O}_X)$, the terms computing $\mathrm{Cone}(u^j)$ above are all equal to zero. Hence the map $u^j$ is an isomorphism when $j\leq a+i-n$, under the assumption that $i \geq n$. ◻ Analogous to [@BS19 Cor. 15.5], we can extend the Verschiebung operator on individual prismatic cohomology sheaf to general coefficients. **Corollary 1**. *Keep the same assumptions and notations as in . Then for $i\in \mathbb{N}$, there is a natural map $$\psi: I^{i+b}\otimes_A R^{\leq i} \lambda_* E \longrightarrow R^{\leq i} \lambda_* E^{(1)},$$ such that its compositions with Frobenius morphism $\varphi: R^{\leq i} \lambda_* E^{(1)} \to R^{\leq i} \lambda_* E$ are canonical maps induced from inclusions $I^{i+b}\subset A$, namely $$\begin{aligned} &\varphi\circ \psi \simeq \mathrm{can}:I^{i+b}\otimes R^{\leq i} \lambda_* E\longrightarrow R^{\leq i} \lambda_* E;\\ &\psi\circ\varphi \simeq \mathrm{can} :I^{i+b}\otimes R^{\leq i} \lambda_* E^{(1)} \longrightarrow R^{\leq i} \lambda_* E^{(1)}. \end{aligned}$$* *Proof.* Let $C$ be the sheaf of complexes $R^{\leq i} \lambda_* E$, and let $C^{(1)}$ be its $\varphi_A$-linear twist, which by flatness is $R^{\leq i} \lambda_* E^{(1)}$. Similarly we let ${\mathrm{Fil}}^j C^{(1)}$ be the complex $R^{\leq i} \lambda_* ({\mathrm{Fil}}^j E^{(1)})$. Define the map $\psi$ by the following composition $$\begin{tikzcd} C\otimes I^{i+b} \arrow[rr, "\varphi^{-1}_{R^{\leq i} \lambda_*}", "\simeq"'] && {\mathrm{Fil}}^{i+b} C^{(1)} \arrow[r, "\mathrm{can}"] & C^{(1)}. \end{tikzcd}$$ The first composition formula then follows from the following enlarged commutative diagram $$\begin{tikzcd} C\otimes I^{i+b} \arrow[rr, "\varphi^{-1}_{R^{\leq i} \lambda_*}", "\simeq"'] \arrow[rrd, "\psi"'] && {\mathrm{Fil}}^{i+b} C^{(1)} \arrow[d, "\mathrm{can}"] \arrow[r, "\varphi", "\simeq"'] & I^{i+b}\otimes C\arrow[d, "\mathrm{can}"] \\ && C^{(1)} \arrow[r, "\varphi"] & C. \end{tikzcd}$$ For the second formula, it follows from the observation that the composition below is the canonical map $$\begin{tikzcd} I^{i+b}\otimes C^{(1)} \arrow[r, "\varphi"] & I^{i+b}\otimes C \arrow[rr, "\varphi^{-1}_{R^{\leq i} \lambda_*}", "\simeq"'] && {\mathrm{Fil}}^{i+b} C^{(1)}. \end{tikzcd}$$ ◻ Recall that evaluating at prisms whose reduction is $p$-completely flat is $t$-exact, see . By taking all the bounded prisms $(A,I)\in Y_{\mathlarger{\mathbbl{\Delta}}}$ such that ${\mathrm{Spf}}(\bar{A})$ is $p$-completely flat over $Y$, the above implies the following result on individual relative prismatic cohomology crystal. **Corollary 1**. *Let $f:X\to Y$ be a smooth proper morphism of smooth $p$-adic formal schemes over $\mathcal{O}_K$, and assume $(\mathcal{E},\varphi_{\mathcal{E}})\in {\mathrm{F\text{-}Crys}}^{\mathrm{coh}}(X)$ is $I$-torsionfree of height $[a,b]$. For $i\in \mathbb{N}$, the $R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}$ is a coherent prismatic $F$-crystal over $Y_{\mathlarger{\mathbbl{\Delta}}}$, such that the image of its Frobenius morphism $\varphi$ within $\mathcal{I}_{\mathlarger{\mathbbl{\Delta}}}^a\otimes R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}$ satisfies the inclusions: $$\mathrm{Im}(\mathcal{I}_{\mathlarger{\mathbbl{\Delta}}}^{b+\min\{i,n\}}\otimes R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}) \subseteq \mathrm{Im}(\varphi) \subseteq \mathrm{Im}(\mathcal{I}_{\mathlarger{\mathbbl{\Delta}}}^{a+\max\{0 ,i-n\}}\otimes R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}).$$ In particular, the $I$-torsionfree quotient of $R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}$ has Frobenius height in $[a+\max\{0 ,i-n\}, b+\min\{i,n\}]$.* Here we implicitly use .(2) to identify $R^if_{{\mathlarger{\mathbbl{\Delta}}},*} \mathcal{E}^{(1)}$ with $\varphi_{Y_{\mathlarger{\mathbbl{\Delta}}}}^* R^if_{{\mathlarger{\mathbbl{\Delta}}},*} \mathcal{E}$ for the Frobenius morphism of $R^if_{{\mathlarger{\mathbbl{\Delta}}},*} \mathcal{E}$. The following lemma says that passing to $I$-power torsions or $I$-torsionfree quotient preserves $F$-crystals. **Lemma 1**. *Let $X$ be a smooth $p$-adic formal schemes over $\mathcal{O}_K$, and let $(\mathcal{E},\varphi_{\mathcal{E}}) \in {\mathrm{F\text{-}Crys}}^{\mathrm{coh}}(X)$ be a coherent $F$-crystal. Then $\mathcal{F} \coloneqq \mathcal{E}[I^{\infty}]$ is $(I, p)$-power-torsion, hence $\varphi_{X_{\mathlarger{\mathbbl{\Delta}}}}^* \mathcal{F}[1/I] = 0 = \mathcal{F}[1/I]$. In particular, we have a coherent $F$-crystal $(\mathcal{F}, 0)$, and the $\varphi_{\mathcal{E}}$ induces a Frobenius isogeny on $\mathcal{E}/\mathcal{F}$ making it an $I$-torsionfree coherent $F$-crystal.* *Proof.* On Breuil--Kisin prisms $(A, I) \in X_{{\mathlarger{\mathbbl{\Delta}}}}$, we have $\mathcal{F}(A) = \mathcal{E}(A)[I^{\infty}]$. It suffices to know that $\mathcal{F}(A)$ is $p$-power torsion. This follows from [@DLMS Proposition 4.13]: the authors showed that $\mathcal{E}(A)[p^{-1}]$ is a finite projective $A[p^{-1}]$-module, hence $\mathcal{F}(A)[p^{-1}] = 0$. ◻ Next we show the derived pushforward of $I$-power torsion prismatic $F$-crystals will have isogenous Frobenius, for trivial reasons. **Proposition 1**. *Let $f \colon X\to Y$ be a qcqs smooth morphism of smooth $p$-adic formal schemes over $\mathcal{O}_K$, and let $(\mathcal{E},\varphi_{\mathcal{E}}) \in {\mathrm{F\text{-}Crys}}^{\mathrm{coh}}(X)$ be an $I^{\infty}$-torsion $F$-crystal on $X_{{\mathlarger{\mathbbl{\Delta}}}}$. Then both $Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}[1/I] = 0$ and $\varphi_{Y_{{\mathlarger{\mathbbl{\Delta}}}}}^*Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}[1/I] = 0$. In particular, we have $(Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}, Rf_*(\varphi_{\mathcal{E}})) \in {\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(Y)$.* *Proof.* The claim about $Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}[1/I] = 0$ follows immediately from the assumption that $E[1/I] = 0$ and the map being qcqs. Below we show the other claim. By standard argument, we are immediately reduced to the case of $Y = {\mathrm{Spf}}(R_0)$ is an affine and $X = {\mathrm{Spf}}(R)$ is also an affine which moreover admits an étale chart: Namely we may assume that there is an étale map $R_0\langle T_i^{\pm 1} \rangle \to R$. Mimicking [@DLMS Example 3.4], we see that one can find a Breuil--Kisin prism $(A, I)$ covering $Y_{{\mathlarger{\mathbbl{\Delta}}}}$ as well as a Breuil--Kisin prism $(B, J)$ covering the relative prismatic site $(X/A)_{{\mathlarger{\mathbbl{\Delta}}}}$, and moreover the relative Frobenius $\varphi_{B/A} \colon B \widehat{\otimes}_{A, \varphi_A} A \to B$ is faithfully flat. Similar to , the object $(B, J = IB)$ is weakly final in $(X/A)_{{\mathlarger{\mathbbl{\Delta}}}}$, and its $(i+1)$-fold self product in $(X/A)_{{\mathlarger{\mathbbl{\Delta}}}}$ exist for all $i \in \mathbb{N}$ and are all given by affine objects denoted as $(B^{(i)}, IB^{(i)})$. With the above notation, the vanishing we are after translates[^4] to: $$\lim_{\bullet \in \Delta} \mathcal{E}(B^{(\bullet)}, IB^{(\bullet)}) \widehat{\otimes}_{A, \varphi_A} A[1/I] = 0.$$ To that end, it suffices to show $\mathcal{E}(B^{(\bullet)}, IB^{(\bullet)}) \widehat{\otimes}_{A, \varphi_A} A[1/I] = 0$, and in fact it suffices to show this when $\bullet = 0$ (as the latter ones are base changed from this case). Finally, since the relative Frobenius on $B = B^{(0)}$ is faithfully flat, we are reduced to knowing $\mathcal{E}(B, IB) \widehat{\otimes}_{B, \varphi_B} B[1/I] = 0$. But this follows from the Frobenius isogeny property of $\mathcal{E}$: $$\mathcal{E}(B, IB) \widehat{\otimes}_{B, \varphi_B} B[1/I] \xrightarrow[\varphi_{\mathcal{E}}]{\cong} \mathcal{E}(B, IB) [1/I]$$ and the assumption that $\mathcal{E}$ is $I^{\infty}$-torsion (so the latter module in the above equation is $0$). As for the last sentence, we just note that (see for instance [@GR22 Proposition 5.11]) $Rf_{{\mathlarger{\mathbbl{\Delta}}}, *}\mathcal{E}$ is a prismatic crystal in perfect complex over $Y$. ◻ Now we are ready to generalize the "Frobenius isogeny property" and "weak étale comparison" in [@GR22]. **Theorem 1**. *Let $f \colon X \to Y$ be a smooth proper morphism between smooth formal schemes over ${\mathrm{Spf}}(\mathcal{O}_K)$, then derived pushforward of $F$-crystals in perfect complexes on $X$ are $F$-crystals in perfect complexes on $Y$, moreover the following diagram commutes functorially: $$\xymatrix{ {\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(X) \ar[d]_{Rf_{{\mathlarger{\mathbbl{\Delta}}}, *}} \ar[r]^{T(-)} & D^{(b)}_{lisse}(X_{\eta}, \mathbb{Z}_p) \ar[d]^{Rf_{\eta, *}} \\ {\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(Y) \ar[r]^{T(-)} & D^{(b)}_{lisse}(Y_{\eta}, \mathbb{Z}_p). }$$* *Proof.* Let $(\mathcal{E},\varphi_\mathcal{E})$ be in ${\mathrm{F\text{-}Crys}}^{\mathrm{perf}}(X)$. It is known (see for instance [@GR22 Proposition 5.11]) that $Rf_{{\mathlarger{\mathbbl{\Delta}}}, *}\mathcal{E}$ is a prismatic crystal in perfect complexes over $Y$. To see the induced Frobenius on $Rf_{{\mathlarger{\mathbbl{\Delta}}}, *}\mathcal{E}$ must be an isogeny, we first reduce ourselves to the case where $(\mathcal{E},\varphi_{\mathcal{E}}) \in {\mathrm{F\text{-}Crys}}^{\mathrm{coh}}(X)$. By considering the $I^{\infty}$-torsions of $\mathcal{E}$ and its $I$-torsionfree quotient, which are coherent $F$-crystals by , we are done thanks to and respectively. To see the commutative diagram, we recall by construction in [@BS21 Cor. 3.7] that the étale realization $T(Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E})$ is isomorphic to the sheaf of complexes over $Y_{\eta,\mathrm{pro\acute{e}t}}$, sending an affinoid perfectoid Huber pair $(S[1/p],S)$ over $Y_\eta$ onto the following complexes $$\mathrm{fib}\left(\begin{tikzcd} (Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}[1/I]^\wedge_p)(\mathrm{A}_{\inf}(S),I) \arrow[rr, "\varphi-\mathrm{id}"] && (Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}[1/I]^\wedge_p)(\mathrm{A}_{\inf}(S),I), \end{tikzcd}\right)$$ namely $$T(Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}): (S[1/p],S) \longmapsto \left( (Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}[1/I]^\wedge_p) (\mathrm{A}_{\inf}(S),I)\right)^{\varphi=1}.$$ On the other hand, by applying derived global section at weak étale comparison in [@GR22 Thm. 6.1], for any perfect prism $(A,I)$ in $Y_{\mathlarger{\mathbbl{\Delta}}}$, there is a natural isomorphism of $\mathbb{Z}_p$-complexes $$R\Gamma((X_{\overline{A}})_{\eta, \mathrm{pro\acute{e}t}},T(\mathcal{E})) \simeq \left( (Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}[1/I]^\wedge_p)(A,I)\right)^{\varphi=1},$$ where $X_{\overline{A}}$ is the complete base change $X\times_Y {\mathrm{Spf}}(\bar{A})$. As a consequence, by taking the inverse system with respect to perfect prisms associated to affinoid perfectoid Huber pairs $(S[1/p], S)$ over $Y_\eta$, we get a natural isomorphism of $\mathbb{Z}_p$-complete complexes over $Y_\eta$ $$Rf_{\eta,*}T(\mathcal{E}) \simeq T(Rf_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}).$$ ◻ Combining the above with , we get the following slight refinement of the "weak étale comparison" [@GR22 Theorem 6.1]. **Corollary 1**. *Let $f \colon X \to Y$ be a smooth proper morphism between smooth formal schemes over ${\mathrm{Spf}}(\mathcal{O}_K)$, and let $(\mathcal{E},\varphi_\mathcal{E}) \in {\mathrm{F\text{-}Crys}}^{{\mathrm{perf}}}(X)$. Then $T(R^i f_{{\mathlarger{\mathbbl{\Delta}}}, *} \mathcal{E}) = R^i f_{\eta, *}(T(\mathcal{E}))$.* [^1]: We refer the reader to [@Bha23 §1.2] for the notion and the related discussions. [^2]: This Proposition and in fact the whole , is independent of contents before it. [^3]: Although the authors only stated that admissibility is automatic for complete regular local rings with a perfect characteristic $p$ residue field, their proof works for any $p$-complete ring $R$ admitting a quasi-syntomic cover by a perfectoid. [^4]: To see this translation, we refer readers to the discussion around [@BS19 4.17-4.18 and their footnote 10.]

arxiv_math

--- abstract: | A map of a set to itself admits a representation by a directed graph with vertices being the elements of the set and directed edges from vertexes to their images. Disregarding directionality, one defines communities as the maximal connected components and notices that each community is uni-cyclic. The distributions of the sizes of communities and lengths of cycles in graphs representing unconstrained random maps is a classical subject. We define experts as images, followers as vertexes that are not experts, and further notions including prophets, founders, egocentrics, introverts, etc. We introduce and analyze classes of random maps with sociological flavor. author: - P. L. Krapivsky bibliography: - references-nets.bib title: Random Maps with Sociological Flavor --- # Introduction {#sec:intro} A map of a set $S$ to itself, $f: S\to S$, can be represented by a directed graph in which elements of $S$ are vertices and a directed edge goes from each $x\in S$ to $f(x)$. This graph decomposes into communities defined as the maximal connected components. We denote communities by $C_c$ with $c=1,\ldots,C$, where $C$ is the total number of communities. Each community has exactly one cycle. The length $\ell_c$ of the cycle in community $C_c$ satisfies the obvious bounds: $1\leq \ell_c\leq |C_c|$. The number of elements in the set $S$ is $M = |S| = \sum_{1\leq c\leq C} |C_c|$. Let us interpret elements of set $S$, equipped with map $f: S\to S$, as individuals. An individual $y\in S$ is an expert if there is at least one individual $x\in S$ such that $f(x)=y$; followers are individuals who are not experts. Denote by $E_c$ and $F_c$ the numbers of experts and followers in the community $C_c$. The total numbers of experts and followers are $E =\sum_{1\leq c\leq C} E_c$ and $F =\sum_{1\leq c\leq C} F_c$. Every community decomposes into experts and followers, $E_c+ F_c=|C_c|$, and similarly $E+F = |S| = M$. We define egocentrics as loops in the graph: $f(x)=x$ for an egocentric $x\in S$. Introverts are isolated egocentrics, i.e., egocentrics without followers. In an example in Fig. [1](#Fig:random-map-ill){reference-type="ref" reference="Fig:random-map-ill"}, there are four communities with $|C_1|=1, |C_2|=|C_3|=5, |C_4|=17$ individuals and $F_1=0, F_2=1, F_3=2, F_4=7$ followers; overall there are 10 followers and 18 experts. The lengths of the cycles are $\ell_1=1, \ell_2=\ell_3=3, \ell_4=6$. There is one egocentric who happens to be an introvert. {#Fig:random-map-ill width="7.89cm"} For a random map $f: S\to S$, the image $f(x)$ of each $x\in S$ is chosen randomly. For random maps without any constraint (classical random maps), all $M^M$ maps are considered equally likely. See [@Kruskal; @Katz; @Stepanov69; @Stepanov71] for earlier work on classical random maps and [@Harris; @Kolchin] for reviews of earlier research. Random maps have applications in modeling epidemic processes, analyses of cryptographic systems, and random number generation, etc. (see [@Gertsbakh; @Pittel83; @Flajolet90; @Quisquater90; @Jaworski08; @Aldous; @Jaworski14] and references therein). In physics, random maps have appeared in the context of spin glasses and disordered materials [@Witten-RM]. Special types of random maps arise in the context of synchronous dynamics of random Boolean networks such as Kauffman's NK model [@BN-Kauffman69; @BN-Kauffman89; @Kauffman; @BN-Hilhorst87; @BN-Derrida87; @BN-Flyvbjerg; @BN-Parisi; @BN-Sam; @BN-Barbara-1; @BN-Barbara-2; @BN-Mori]. Random maps can be used to mimic belief dynamics [@Mirta]. Other applications are anticipated in social and communication networks [@Wasserman; @Castellano09; @Tomasello]. Even for classical random maps, Sec. [2](#sec:maps){reference-type="ref" reference="sec:maps"}, results related to followers, experts, egocentrics, and introverts could be challenging to find in the existing literature [@Harris; @Kolchin; @Flajolet90]. Some results exist but appear under different names; e.g., experts are known as terminal nodes [@Flajolet90]. Sociological terminology suggests novel classes of random maps, and our chief goal is to propose and analyze such random maps. Specifically, we introduce (Sec. [3](#sec:FM){reference-type="ref" reference="sec:FM"}) random maps compatible with fitness and ranking (Sec. [4](#sec:MRM){reference-type="ref" reference="sec:MRM"}). In Table [1](#Tab:loops){reference-type="ref" reference="Tab:loops"}, we collect a few results for three random map models compatible with ranking and for a one-parametric class of fitness models in which the image $j=f(i)$ for each $i\in S$ is chosen with probability proportional to $j^a$, the fitness of the image; see [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"} for the precise definition. When $a>-1$, the fraction of experts $e(a)$ and the average number of introverts $A(a)$ are given by $$\begin{aligned} \label{e-a} &e(a) = 1 - \int_0^1 dx\,e^{-(a+1)x^a}\\ \label{A-a} & A(a) = (a+1)\int_0^1 dx\,x^a e^{-(a+1)x^a}\end{aligned}$$ as shown in Sec. [3](#sec:FM){reference-type="ref" reference="sec:FM"}. Maps Fitness Uniform Linear $\text{(Linear)}^{-1}$ ------------- --------- --------------- ------------------- ------------------------ -- Experts $e(a)$ $\frac{1}{2}$ $\frac{9-e^2}{4}$ $1-(\ln M)^{-1}$ Egocentrics 1 $\ln M$ $2$ $M/\ln M$ Introverts $A(a)$ 1 $(e^2-1)/M$ $e^{-1} M/\ln M$ : The fraction of experts and the average numbers of egocentrics and introverts. We present results for the class of fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"} with $a>-1$, and for random map models \[[\[map:more\]](#map:more){reference-type="eqref" reference="map:more"}, [\[map:linear-rank\]](#map:linear-rank){reference-type="eqref" reference="map:linear-rank"}, [\[map-linear\]](#map-linear){reference-type="eqref" reference="map-linear"} from left to right\] compatible with ranking. For the general class of fitness models [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"}, there is on average one egocentric individual independently of $M$. Random maps compatible with ranking (Sec. [4](#sec:MRM){reference-type="ref" reference="sec:MRM"}) are defined as follows. We label elements of $S$ from $1$ to $M$, interpret labels as ranks, and postulate that a map $f: S\to S$ is compatible with ranking if $f(i)\geq i$ for all $i=1,\ldots,M$. All $M!$ of maps compatible with ranking are considered equiprobable for recursive random map (RRM). For the RRM, the probability distribution of the random variable $F$ is $$\label{PFM:Euler} \Pi(F,M)=\frac{1}{M!}\,\eulerian{M}{F}$$ where $\eulerian{M}{F}$ are Eulerian numbers [@Euler36; @Euler55; @Knuth]. We also derive an exact formula for the probability distribution of the number of communities $$\label{PCM:Stirling} P(C,M)=\frac{1}{M!}\,{M\brack C}$$ where ${M\brack C}$ are the Stirling numbers of the first kind [@Knuth]. The degree $k$ of the individual with highest rank, the founder, has identical probability distribution $$\label{PkM:Stirling} \Phi(k,M)=\frac{1}{M!}\,{M\brack k}$$ We focus on static random maps, yet some static maps admit a dynamical reformulation. The prime example is the static RRM. This model is isomorphic to a growing random map where individuals arrive one by one, and each new individual maps to an individual randomly chosen amongst those already present (including itself). Stopping the process after $M$ steps gives the RRM of size $M$. In Appendix [8](#ap:RRM){reference-type="ref" reference="ap:RRM"}, we explain this isomorphism and derive [\[PFM:Euler\]](#PFM:Euler){reference-type="eqref" reference="PFM:Euler"} and [\[PCM:Stirling\]](#PCM:Stirling){reference-type="eqref" reference="PCM:Stirling"}. Other evolving mechanisms are possible, and in Sec. [5](#sec:rewire){reference-type="ref" reference="sec:rewire"}, we briefly discuss random maps of the same set evolving through rewiring. In Sec. [6](#sec:prophet){reference-type="ref" reference="sec:prophet"}, we define prophets and founders and briefly analyze their properties in random map models. In particular, we derive the degree distribution [\[PkM:Stirling\]](#PkM:Stirling){reference-type="eqref" reference="PkM:Stirling"} of the founder. We conclude with a discussion (Sec. [7](#sec:disc){reference-type="ref" reference="sec:disc"}). # Classical Random Maps {#sec:maps} In this section, we consider classical random maps: all $M^M$ maps of a set of size $|S|=M$ to itself are equiprobable. Many results in this and the following sections are valid for arbitrary $M$, but we are primarily interested in the behavior in the $M\to\infty$ limit. We begin by outlining a few basic statistical properties of classical random maps. Denote by $P(C,M)$ the probability that there are $C$ communities. The average number of communities, $\langle C\rangle = \sum_{C\geq 1}C P(C,M)$, grows logarithmically with the total number $M$ of individuals $$\label{c-av} \langle C\rangle = \tfrac{1}{2}\ln M + O(1)$$ when $M\gg 1$. The full probability distribution $P(C,M)$ is also known [@Harris]. The probability to have exactly one community is [@Katz]: $$\label{P1-M} P(1,M) = \frac{(M-1)!}{M^M}\sum_{m=0}^{M-1} \frac{M^m}{m!}$$ For large $M$ the probability $P(1,M)$ simplifies to $$\label{P1} P(1,M)\simeq \sqrt{\frac{\pi}{2M}}$$ The probability $\Pi_1(\ell,M)$ to end up with a single community that has a cycle of length $\ell$ is also known [@Katz]: $$\Pi_1(\ell,M) = \frac{M!}{(M-\ell)! M^{\ell + 1}}$$ This probability distribution simplifies to $$\Pi_1(\ell,M) = M^{-1}\,\exp\!\left[-\frac{x^2}{2}\right]$$ in the scaling limit $$\label{scaling} M\to\infty, \quad \ell\to\infty , \quad x = \frac{\ell}{\sqrt{M}} = \text{finite}$$ A randomly chosen individual is a follower with probability $(1-1/M)^M$. Thus $$\label{Fav:random} \langle F\rangle = M(1-1/M)^M$$ For classical random maps, followers and experts were called terminal nodes and image points in [@Flajolet90] where the asymptotic behaviors $\langle F\rangle = Me^{-1}$ and $\langle E\rangle = M(1-e^{-1})$ were derived. The probability distribution $Q(\mathcal{L},M)$ to have $\mathcal{L}$ loops is the binomial distribution $$\label{Q:binom} Q(\mathcal{L},M) = \binom{M}{\mathcal{L}}\left(1-\frac{1}{M}\right)^{M-\mathcal{L}} \left(\frac{1}{M}\right)^\mathcal{L}$$ The average number of loops is independent on $M$ $$\label{loops-av} \langle \mathcal{L}\rangle = \sum_{\mathcal{L}=0}^M \mathcal{L} Q(\mathcal{L},M) = 1$$ while higher moments depend on $M$: $$\label{loops-higher} \langle \mathcal{L}^2\rangle = 2 - M^{-1}, \quad \langle \mathcal{L}^3\rangle = 5 - 6M^{-1}+ 2 M^{-2}$$ etc. Specializing [\[Q:binom\]](#Q:binom){reference-type="eqref" reference="Q:binom"} to $\mathcal{L}=0$ and $\mathcal{L}=M$ we otain the probabilities to observe extremal numbers of loops: $$\label{Q:min-max} Q(0,M) = \big(1-M^{-1}\big)^M, \quad Q(M,M) = M^{-M}$$ The probability to have maximal number of communities, $C=M$, is of course identical to the probability to have the maximal number of loops: $$\label{QMM:classical} P(M,M) = Q(M,M) = M^{-M}$$ In the $M\to \infty$ limit, the binomial distribution [\[Q:binom\]](#Q:binom){reference-type="eqref" reference="Q:binom"} simplifies to the Poisson distribution $$\label{Q:Poisson} Q(\mathcal{L},\infty) = \frac{e^{-1}}{\mathcal{L}!}$$ A loop $f(i)=i$ implies that individual $i$ is egocentric. Thus $Q(\mathcal{L},M)$ is the probability that there are $\mathcal{L}$ egocentrics. An introvert is an isolated egocentric, i.e., someone who is not an expert for anyone else. The probability $R(I,M)$ to have $I$ introverts can be found recurrently. The probability $R(1,M)$ satisfies $$\label{R1} R(1,M) = M^{-1} \binom{M}{1} \left(1-\frac{1}{M}\right)^{M-1}R(0,M-1)$$ Here $M^{-1}$ is the probability to form a loop, the binomial factor accounts for the label of an introvert, $(1-M^{-1})^{M-1}$ assures that remaining individuals map into themselves and with probability $R(0,M-1)$ there are no introverts among them. When $M\to\infty$, Eq. [\[R1\]](#R1){reference-type="eqref" reference="R1"} simplifies to $$\label{R1:inf} R(1,\infty) = e^{-1}R(0,\infty)$$ Similarly to [\[R1\]](#R1){reference-type="eqref" reference="R1"}, the probability $R(2,M)$ to have two introverts satisfies $$\label{R2} R(2,M) = M^{-2}\binom{M}{2}\left(1-\frac{2}{M}\right)^{M-2}R(0,M-2)$$ which leads to $$\label{R2:inf} R(2,\infty) = \frac{e^{-2}}{2!}\,R(0,\infty)$$ Generally for arbitrary $I\geq 1$ we obtain $$\label{RI:inf} R(I,\infty) = \frac{e^{-I}}{I!}\,R(0,\infty)$$ The normalization requirement $\sum_{I\geq 0}R(I,\infty) = 1$ fixes the probability to have no introverts $$\label{R0} R(0,\infty) = \exp\!\big[-e^{-1}\big] = 0.6922006275553\ldots$$ Thus $R(I,\infty)$ is the Poisson distribution $$\label{RI:dist} R(I,\infty) = \frac{1}{I!}\,\exp\!\big[-I -e^{-1}\big]$$ Using [\[RI:dist\]](#RI:dist){reference-type="eqref" reference="RI:dist"} we find the average number of introverts $$\label{introverts-av} \langle I\rangle = e^{-1}=0.3678794411714\ldots$$ We emphasize that [\[loops-av\]](#loops-av){reference-type="eqref" reference="loops-av"} and [\[introverts-av\]](#introverts-av){reference-type="eqref" reference="introverts-av"} give average numbers, not fractions---in classical random maps, the number of egocentrics, let alone introverts, remains finite in the $M\to\infty$ limit. The degree of the vertex is the number of its neighbors. Followers have degree 1, while experts have degrees larger than one. It makes sense to stratify experts according to their degree $k$. Denote by $N_k(M)$ the number of individuals of degree $k$. We have $$\label{Nk:EF} F = N_1, \quad E = \sum_{k\geq 2} N_k(M)$$ The notion of degree is easy to appreciate in undirected graphs. In a directed graph, each vertex has two natural degrees: an in-degree $k_\text{in}$ and an out-degree $k_\text{out}$. The full degree of the vertex is the sum $$k_\text{in} + k_\text{out} = k$$ For directed graphs generated by maps all out-degrees are equal to unity, $k_\text{out}(x)=1$ for each $x\in S$. Therefore it suffices to consider the full degree. If $k(x)=1$, vertex $x$ is a follower; if $k(x)\geq 2$, vertex $x$ is an expert. Each edge contributes a unity to the total in and out degrees, $$\sum_{x\in S}k_\text{in}(x) = \sum_{x\in S}k_\text{out}(x) = M,$$ and therefore $$\label{sum:deg} \sum_{x\in S}k(x) = 2M$$ for an arbitrary map. The sum rules $$\begin{aligned} \label{norm} & \sum_{k\geq 1} N_k(M) = M\\ \label{map:def} & \sum_{k\geq 1} kN_k(M) = 2M\end{aligned}$$ are valid for an arbitrary map: [\[norm\]](#norm){reference-type="eqref" reference="norm"} follows from [\[Nk:EF\]](#Nk:EF){reference-type="eqref" reference="Nk:EF"} and $E+F=M$, while [\[map:def\]](#map:def){reference-type="eqref" reference="map:def"} follows from [\[sum:deg\]](#sum:deg){reference-type="eqref" reference="sum:deg"}. For classical random maps, and for most models studied below, $N_k(M)$ are random self-averaging quantities with fractions $$n_k(M) = M^{-1} \langle N_k(M) \rangle$$ saturating in the $M\to\infty$ limit and providing the major information about the asymptotic behavior of $N_k(M)$. For classical random maps the fractions $n_k$ read $$\label{nk:binom} n_k(M) = \binom{M}{k-1}\left(1-\frac{1}{M}\right)^{M-k+1} \left(\frac{1}{M}\right)^{k-1}$$ In the $M\to\infty$ limit, the binomial distribution [\[nk:binom\]](#nk:binom){reference-type="eqref" reference="nk:binom"} becomes the Poisson distribution $$\label{nk:Poisson} n_k(\infty) = \frac{e^{-1}}{(k-1)!}$$ This result appears in [@Flajolet90] and perhaps in earlier work. The knowledge of the degree distribution allows one to probe the behavior of the maximal degree $D(M)$. Using the criterion $$\label{D:criterion} M\sum_{k\geq D(M)}n_k\sim 1$$ and [\[nk:Poisson\]](#nk:Poisson){reference-type="eqref" reference="nk:Poisson"} we find $$\label{DM:nu} D(M) \simeq \frac{\ln M}{\ln(\ln M)}$$ # Fitness Models {#sec:FM} We assign a positive number, fitness, to each individual and investigate random map models compatible with fitness. We label individuals from 1 to $M$, denote by $\phi(i)$ the fitness of $i\in S$, and postulate that the probability of a map $i\to j$ is proportional to the fitness $\phi(j)$ of the image. Thus $$\label{map:fitness} \text{Prob}[f(i)=j] = \frac{\phi(j)}{Z}\,, \quad Z = \sum_{k=1}^M \phi(k)$$ Choosing $\phi(i)=1$ for all $i\in S$ we recover classical random maps. For the model with linear fitness, $\phi(i)=i$, the rule [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"} becomes $$\label{linear:fitness} \text{Prob}[f(i)=j] = \frac{2j}{M(M+1)}$$ More generally, one can postulate that the probability of a map $i\to j$ is proportional to $\phi_i(j)$, the fitness of $j$ in eye of $i$. Thus $$\label{gen:fitness} \text{Prob}[f(i)=j] = \frac{\phi_i(j)}{Z_i}, \quad Z_i = \sum_{k=1}^M \phi_i(k)$$ Each model [\[gen:fitness\]](#gen:fitness){reference-type="eqref" reference="gen:fitness"} is specified by $M^2$ fitness parameters. In the following we limit ourselves to the class of models [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"} if not stated otherwise. Vertex $i$ is a follower with probability $[1-Z^{-1}\phi(i)]^M$, so the average number of followers is $$\label{Lav:fitness} \langle F\rangle = \sum_{i=1}^M \left[1-\frac{\phi(i)}{Z}\right]^M$$ To deduce concrete results from the general solution [\[Lav:fitness\]](#Lav:fitness){reference-type="eqref" reference="Lav:fitness"} we consider special fitness models from the class of models [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"}. For the linear fitness model, $\phi(i)=i$, the large $M$ behavior of the sum in [\[Lav:fitness\]](#Lav:fitness){reference-type="eqref" reference="Lav:fitness"} reads $$\begin{split} &\left[1-\frac{\phi(i)}{Z}\right]^M = \left[1-\frac{2i}{M(M+1)}\right]^M \simeq e^{-2i/M}\\ & \sum_{i=1}^M e^{-2i/M} = \frac{1-e^{-2}}{e^{2/M}-1} \simeq \frac{1-e^{-2}}{2}\,M \end{split}$$ Thus $$\label{L-av-fitness} \lim_{M\to\infty}\frac{\langle E\rangle}{M} = \frac{1+e^{-2}}{2}\,, \quad \lim_{M\to\infty}\frac{\langle F\rangle}{M} = \frac{1-e^{-2}}{2}$$ More generally, consider a one-parameter family of fitness models with algebraic fitness $\phi(i)=i^a$, that is $$\label{fitness-a} \text{Prob}[f(i)=j] = \frac{j^a}{Z}\,, \quad Z = \sum_{k=1}^M k^a$$ Interesting behaviors occur when $a\geq -1$. We shall tacitly assume that $a>-1$ if not stated otherwise; occasionally, we comment on subtle behaviors in the marginal $a=-1$ case. For the one-parameter family of fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"} the average number of followers [\[Lav:fitness\]](#Lav:fitness){reference-type="eqref" reference="Lav:fitness"} becomes $$\langle F\rangle = \sum_{i=1}^M \left[1-\frac{i^a}{Z}\right]^M, \quad Z=\sum_{i=1}^M i^a$$ When $a>-1$, the 'partition function' $Z$ diverges as $M^{a+1}/(a+1)$ when $M\gg 1$. This simplifies asymptotic analysis allowing us to replace summation by integration. The asymptotic fraction of the followers reads $$\label{Fav} \lim_{M\to\infty}\frac{\langle F\rangle}{M} = \int_0^1 dx\, e^{-(a+1)x^a} = \frac{\gamma(a^{-1}, a+1)}{a(a+1)^{1/a}}$$ Here $$\gamma(\beta,B)=\int_0^B dy\,y^{\beta-1} e^{-y}$$ is an incomplete gamma function. The fraction of experts is complimentary since $E+F=M$; this fraction is given by [\[e-a\]](#e-a){reference-type="eqref" reference="e-a"}. The dependence of these fractions on the parameter $a$ are presented in Fig. [2](#Fig:followers_experts){reference-type="ref" reference="Fig:followers_experts"}. At $a=0$, i.e., for classical random maps, the number of experts is maximal among all models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"}. We also note that when $a\to\infty$, the fraction of experts approaches zero as $$\label{E-av:asymp} \lim_{M\to\infty}\frac{\langle E\rangle}{M} = \frac{\ln a + \gamma}{a} + \ldots$$ where $\gamma \approx 0.5772156649$ is the Euler constant [@Knuth]. The asymptotic approach is slow due to the presence of a logarithm and therefore in [\[E-av:asymp\]](#E-av:asymp){reference-type="eqref" reference="E-av:asymp"} we also kept the sub-leading term $\gamma/a$ involving Euler's constant $\gamma$. At two special values of $a$ which are roots of $$\frac{1}{2} = \int_0^1 dx\, e^{-(a+1)x^a}$$ the number of followers is equal to the number of experts; this occurs when $a\approx 1.9329$ and $a\approx -0.6289$. ![The fractions of experts and followers for the family of fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"}. The fraction of experts reaches maximum $M^{-1}\langle E\rangle = 1-e^{-1}$ at $a=0$ corresponding to classical random maps, and it vanishes when $a\to -1$ or $a\to\infty$. The fraction of followers exhibits a complementary behavior. ](FE.pdf){#Fig:followers_experts width="7.77cm"} The behavior is particularly subtle when $a=-1$. We need to compute $$\label{Eav:1} \langle E\rangle = \sum_{k=1}^M \left(1-\left[1-\frac{1}{k H_M}\right]^M\right)$$ where $H_M= \sum_{1\leq m\leq M} m^{-1}$. The harmonic number $H_M$ grows logarithmically with $M$. More precisely $$\label{Harmonic} H_M = \ln M + \gamma + \frac{1}{2M} - \frac{1}{12M^2} + \frac{1}{120M^3} + \ldots$$ Replacing summation over $1\leq k\leq M$ in [\[Eav:1\]](#Eav:1){reference-type="eqref" reference="Eav:1"} by integration over $\kappa=k H_M/M$ we reduce [\[Eav:1\]](#Eav:1){reference-type="eqref" reference="Eav:1"} to $$\label{Eav:int} \frac{\langle E\rangle}{M} \simeq \frac{1}{H_M}\int_{M^{-1}H_M}^{H_M}d\kappa \,\big(1-e^{-1/\kappa}\big)$$ The dominant contribution to the integral is gathered when $\kappa$ is large and equals to $\ln H_M$ in the leading order. Using the asymptotic [\[Harmonic\]](#Harmonic){reference-type="eqref" reference="Harmonic"} we arrive at $$\label{Eav:asymp} \frac{\langle E\rangle}{M} \simeq \nu, \qquad \nu = \frac{\ln(\ln M)}{\ln M}$$ Thus when $a=-1$, the fraction of experts vanishes, albeit very slowly, when $M\to\infty$. Consider egocentrics, i.e., loops. For the class of models [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"}, vertex $i$ is the loop with probability $Z^{-1}\phi(i)$. Thus on average, there is one egocentric individual, no dependence on fitness and the population size $M$: $$\langle \mathcal{L}\rangle = \sum_{i=1}^M \frac{\phi(i)}{Z} = 1$$ To probe the probability distribution $Q(\mathcal{L},M)$, we first notice that there are no loops with probability $$\label{Q0M} Q(0,M) = \prod_{i=1}^M \left[1-\frac{\phi(i)}{Z}\right]$$ The probability [\[Q0M\]](#Q0M){reference-type="eqref" reference="Q0M"} tends to $Q(0,\infty) = e^{-1}$ for all fitness models [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"} in which $Z$ diverges when $M\to\infty$. For instance, for the class of models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"} with $a\geq -1$, the asymptotic behavior of $Q(0,M)$ is $$\label{Q0M:asymp} \frac{Q(0,M)}{Q(0,\infty)} - 1 \simeq - \begin{cases} \frac{(a+1)^2}{2(2a+1)}\,M^{-1} & a>-1\\ \frac{\pi^2}{12}\,(\ln M)^{-2} & a=1 \end{cases}$$ The probability to have one loop is $$Q(1,M) = \prod_{i=1}^M \left[1-\frac{\phi(i)}{Z}\right] \sum_{j=1}^M \frac{\phi(j)}{Z-\phi(j)}$$ If $Z$ diverges in the $M\to\infty$ limit, $Q(1,M)$ approaches to the same universal value $Q(1,\infty) = e^{-1}$ as the probability to have no loops. Generally fixing $\mathcal{L}$ and taking the $M\to\infty$ limit, one finds that $Q(\mathcal{L},\infty)$ is the Poisson distribution [\[Q:Poisson\]](#Q:Poisson){reference-type="eqref" reference="Q:Poisson"} as in the case of random maps. It is remarkable that $Q(\mathcal{L},\infty)$ is the same for all models [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"} modulo just one property: $Z$ must diverge in the $M\to\infty$ limit. The dependence of $Q(\mathcal{L},M)$ on fitnesses becomes appreciable when $\mathcal{L}$ is large. For instance, the number of loops is maximal with probability $$\label{QMM:fitness} Q(M,M) = \prod_{i=1}^M \frac{\phi(i)}{Z}$$ For models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"} with small integer values of the exponent $a$ the number of loops is maximal with probability $$\label{QMM:0123} Q(M,M) = \begin{cases} M^{-M} & a=0\\ \frac{2^M \cdot M!}{[M(M+1)]^M} & a=1\\ \frac{6^M \cdot (M!)^2}{[M(M+1)(2M+1)]^M} & a=2\\ \frac{4^M \cdot (M!)^3}{[M(M+1)]^{2M}} & a=3 \end{cases}$$ Let us now look at the distribution of introverts. Extending the argument that led to [\[R1\]](#R1){reference-type="eqref" reference="R1"} we obtain $$\label{R1:fitness} \frac{R(1,M)}{R(0,M-1)} = \sum_{i=1}^M \frac{\phi(i)}{Z} \left[1-\frac{\phi(i)}{Z}\right]^{M-1}$$ Similarly $$\label{R2:fitness} \frac{R(2,M)}{R(0,M-2)} = \sum_{1\leq i<j\leq M} \varphi(i) \, \varphi(j)$$ where we have used the shorthand notation $$\varphi(i) = \frac{\phi(i)}{Z} \left[1-\frac{\phi(i)}{Z}\right]^{M-1}$$ Generally $$\label{RI:fitness} \frac{R(I,M)}{R(0,M-I)} = \sum_{1\leq i_1<\ldots<i_I\leq M} \prod_{p=1}^I\varphi(i_p)$$ To deduce more concrete results from [\[RI:fitness\]](#RI:fitness){reference-type="eqref" reference="RI:fitness"} we consider again the one-parameter family of fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"}. Taking the $M\to\infty$ limit and replacing summation in Eq. [\[R1:fitness\]](#R1:fitness){reference-type="eqref" reference="R1:fitness"} by integration we obtain $$\label{R10} \frac{R(1,\infty)}{R(0,\infty)} = A(a) \equiv (a+1) \int_0^1 dx\, x^a e^{-(a+1)x^a}$$ Similarly Eq. [\[R2:fitness\]](#R2:fitness){reference-type="eqref" reference="R2:fitness"} simplifies to $$\label{R20} \frac{R(2,\infty)}{R(0,\infty)} = \frac{A^2}{2!}$$ Continuing we arrive at the Poisson distribution $$R(I,\infty) = \frac{A^I}{I!}\,e^{-A}$$ The average number of introverts $$\label{introverts-av-a} \langle I\rangle = A(a)$$ and the probability $R(0,\infty) = e^{-A(a)}$ that there are no introverts are plotted in Fig. [3](#Fig:introverts){reference-type="ref" reference="Fig:introverts"}. The average number of introverts attains the maximal value at $a=0$, i.e. for the classical random map; $A(a)$ vanishes when $a\to -1$ and $a\to\infty$, e.g. $A(a)\simeq a^{-1}$ when $a\gg 1$. A few concrete values of $\langle I\rangle = A(a)$: $$\begin{split} &A(0)=e^{-1}=0.3678794411714\ldots\\ &A(1)=\frac{1}{2}-\frac{3}{2e^2}=0.296997075145\ldots\\ &A(2)=\sqrt{\frac{\pi}{48}}\,\,\text{Erf}\!\left[\sqrt{3}\right]-\frac{1}{2e^3} =0.22727824593\ldots \end{split}$$ ![Bottom: The average number of introverts, $A(a)$, see [\[R10\]](#R10){reference-type="eqref" reference="R10"} and [\[introverts-av-a\]](#introverts-av-a){reference-type="eqref" reference="introverts-av-a"}. Top: The probability $e^{-A(a)}$ that there are no introverts. Shown is the dependence on $a$ parametrizing fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"}. The average number $A(a)$ of introverts coincides with the fraction of vertices of degree 2, see [\[n2AI\]](#n2AI){reference-type="eqref" reference="n2AI"}. ](int-av-0.pdf){#Fig:introverts width="7.77cm"} The above results [\[R10\]](#R10){reference-type="eqref" reference="R10"}--[\[introverts-av-a\]](#introverts-av-a){reference-type="eqref" reference="introverts-av-a"} are valid when $a>-1$. In the marginal $a=-1$ case the behavior is again more subtle. Strictly speaking all $R(I,\infty)$ vanish for all $I\geq 1$, but the large $M$ behavior is still essentially described by the Poisson distribution $$R(I,M) = \frac{\nu^I}{I!}\,e^{-\nu}\,, \qquad \nu = \frac{\ln(\ln M)}{\ln M}$$ We now turn to the degree distribution. For the class of models [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"}, vertex $i$ has degree $k$ with probability $$\binom{M}{k-1}\left[1-\frac{\phi(i)}{Z}\right]^{M-k+1}\left[\frac{\phi(i)}{Z}\right]^{k-1}$$ Therefore the fraction of vertices with degree $k$ is $$\label{nk:sum-fitness} n_k = \sum_{i=1}^M \frac{1}{M}\binom{M}{k-1}\left[1-\frac{\phi(i)}{Z}\right]^{M-k+1}\left[\frac{\phi(i)}{Z}\right]^{k-1}$$ Comparing [\[R1:fitness\]](#R1:fitness){reference-type="eqref" reference="R1:fitness"} and [\[nk:sum-fitness\]](#nk:sum-fitness){reference-type="eqref" reference="nk:sum-fitness"} we deduce a curious result that the fraction of vertices of degree 2 is related to the distribution of introverts: $$\label{R1n2} n_2 = \frac{R(1,M)}{R(0,M-1)}$$ For the one-parameter family of fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"}, we use [\[R10\]](#R10){reference-type="eqref" reference="R10"} and [\[introverts-av-a\]](#introverts-av-a){reference-type="eqref" reference="introverts-av-a"} to obtain $$\label{n2AI} n_2 = A(a) = \langle I\rangle$$ when $M\to\infty$. Specializing [\[nk:sum-fitness\]](#nk:sum-fitness){reference-type="eqref" reference="nk:sum-fitness"} to fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"} and taking the $M\to\infty$ limit we obtain $$\label{nk:fitness} n_k = \frac{1}{(k-1)!}\int_0^1 dx\, \left[(a+1)x^a\right]^{k-1} e^{-(a+1)x^a}$$ Setting $a=0$, we recover the Poisson degree distribution [\[nk:Poisson\]](#nk:Poisson){reference-type="eqref" reference="nk:Poisson"} describing classical random maps. The dependence of $n_1=M^{-1}\langle F\rangle$ on $a$ is shown in Fig. [2](#Fig:followers_experts){reference-type="ref" reference="Fig:followers_experts"}. The dependence of $n_2=A(a)=\langle I\rangle$ on $a$ is shown in Fig. [3](#Fig:introverts){reference-type="ref" reference="Fig:introverts"} (bottom curve). The fractions of vertices with degrees $k=3,4,5$ are shown in Fig. [4](#Fig:n345){reference-type="ref" reference="Fig:n345"}. ![The fitness model [\[map:fitness\]](#map:fitness){reference-type="eqref" reference="map:fitness"} with $\phi(i)=i^a$. The fractions $n_k$ with degrees $k=3,4,5$ (top to bottom) ](n345.pdf){#Fig:n345 width="7.77cm"} The tail of the distribution [\[nk:fitness\]](#nk:fitness){reference-type="eqref" reference="nk:fitness"} changes from factorial to algebraic when $a$ passes through $a=0$ corresponding to random maps: $$\label{nk:tail} n_k \simeq \begin{cases} a^{-1}e^{-a-1}\,\frac{(a+1)^{k-1}}{k!} & a>0\\ - a^{-1}(1+a)^{-1/a}\,k^{-1+1/a} &-1<a<0 \end{cases}$$ when $k\gg 1$. Equations [\[nk:fitness\]](#nk:fitness){reference-type="eqref" reference="nk:fitness"}--[\[nk:tail\]](#nk:tail){reference-type="eqref" reference="nk:tail"} apply when $a>-1$. In the marginal $a=-1$ case the behavior is more subtle: $$n_k \simeq \begin{cases} 1 & k=1\\ \nu & k=2\\ (k-1)^{-1}(\ln M)^{-1} & 3\leq k<\ln M\\ 0 & k\gg \ln M \end{cases}$$ with $\nu$ defined in Eq. [\[Eav:asymp\]](#Eav:asymp){reference-type="eqref" reference="Eav:asymp"}. Finally, we give an estimate of the maximal degree: $$\label{DM:nu-a} D(M) \simeq \begin{cases} \nu^{-1} & a\geq 0\\ \frac{a+1}{M^a} & -1<a<0 \\ \frac{M}{\ln M} & a= -1 \end{cases}$$ The remarkably universal growth law of $D(M)$ in the $a\geq 0$ range is established using the criterion [\[D:criterion\]](#D:criterion){reference-type="eqref" reference="D:criterion"} and the asymptotic [\[nk:tail\]](#nk:tail){reference-type="eqref" reference="nk:tail"}. A more precise estimate coming from [\[nk:tail\]](#nk:tail){reference-type="eqref" reference="nk:tail"} is implicitly determined by $$D(M)[\ln D(M) - 1 - \ln(a+1)]\simeq \ln M$$ When $-1<a<0$, the maximal degree can be determined by first noting that the average degree of the vertex with label $i$ is $d_i=(a+1)i^a/M^a$. These average degrees are significantly separated (a gap between adjacent degrees greatly exceeds fluctuations of those degrees). Hence the maximal degree is $D(M)=d_1=(a+1)/M^a$ as stated in [\[DM:nu-a\]](#DM:nu-a){reference-type="eqref" reference="DM:nu-a"}. Similarly, when $a=-1$, the maximal degree is very close to the average degree $d_1=M/\ln M$ of the vertex with label 1. # Random Maps Compatible with Rank {#sec:MRM} Every finite set can be linearly ordered. Let us identify the ordering with labeling and interpret labels as ranks: $i<j$ asserts that individual $i$ has a lower rank than individual $j$. It is natural to consider maps compatible with ranking, namely maps $f:S\to S$ satisfying $f(i)\geq i$ for all elements $i\in S$. A subclass of the general fitness models [\[gen:fitness\]](#gen:fitness){reference-type="eqref" reference="gen:fitness"} satisfying $\phi_i(j) =\psi(j-i)$ and $\psi(n) = 0$ when $n<0$ is compatible with ranking. The simplest uniform choice is $\psi(n) = 1$ when $n\geq 0$, so for each $i$, an expert is chosen uniformly at random among all $j\geq i$. Thus $$\label{map:more} \text{Prob}[f(i)=j] = \begin{cases} 0 & 1\leq j<i\\ (M-i+1)^{-1} & i\leq j \leq M \end{cases}$$ If we want to mimic the desire to map to the higher rank individuals with higher probabilities, we may choose $\psi(n) = n+1$. In this model $$\label{map:linear-rank} \text{Prob}[f(i)=j] = \begin{cases} 0 & 1\leq j<i\\ \frac{2(j-i+1)}{(M-i+1)(M-i+2)} & i\leq j \leq M \end{cases}$$ Choosing $\psi(n)$ decreasing with $n$ mimics peer affinity relevant in many social settings. For the inverse linear function $\psi(n) = 1/(n+1)$ $$\label{map-linear} \text{Prob}[f(i)=j] = \begin{cases} 0 & 1\leq j<i\\ \frac{1}{(j-i+1) H_{M-i+1}} & i\leq j \leq M \end{cases}$$ A one-parametric family of ranking models interpolating between models [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"}--[\[map-linear\]](#map-linear){reference-type="eqref" reference="map-linear"} is defined by the rule $$\label{map-a} \text{Prob}[f(i)=j] = \begin{cases} 0 & 1\leq j<i\\ \frac{(j-i+1)^a}{S_{M-i+1}(a)} & i\leq j \leq M \end{cases}$$ with $S_k(a)=\sum_{1\leq p\leq k} p^a$. The interesting range is again $a\geq -1$. In this section we analyze these models. ## Uniform choice: Model [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"} {#sec:rank1} The average number of egocentric individuals, i.e., the average number of loops, is the harmonic number $$\langle \mathcal{L}(M)\rangle = H_M = \sum_{m=1}^M \frac{1}{m}$$ Indeed, the top rank individual maps to itself, the second highest rank individual maps to itself with probability 1/2, the third highest rank individual maps to itself with probability 1/3, etc. There is always at least one loop as the top rank individual maps to itself, $M\to M$. The probability to have exactly one loop is $$Q(1,M) = \prod_{i=2}^{M} \left[1-\frac{1}{i}\right] = \frac{1}{M}$$ The probability to have exactly two loops is $$Q(2,M) = \prod_{i=2}^{M} \left[1-\frac{1}{i}\right] \sum_{j=2}^{M} \frac{1}{j-1}= M^{-1}\, H_{M-1}$$ The probability to have exactly three loops is $$Q(3,M) = \frac{1}{M} \sum_{2\leq j<k\leq M} \frac{1}{(j-1)(k-1)}$$ Computing the sum one gets $$Q(3,M) = \frac{1}{2M} \left[(H_{M-1})^2-H_{M-1}^{(2)}\right]$$ where $H_n^{(2)}=\sum_{1\leq k\leq n}k^{-2}$ is the generalized harmonic number. Thus $$Q(3,M) \simeq \frac{1}{2M} \left[(\ln M + \gamma)^2-\frac{\pi^2}{6}\right]$$ when $M\gg 1$. Generally the probability to have exactly $\mathcal{L}$ loops is $$\label{QLM} Q(\mathcal{L},M) = \frac{1}{M} \sum_{2\leq j_1<\ldots<j_\mathcal{L}\leq M} \prod_{\nu=1}^\mathcal{L} \frac{1}{j_\nu-1}$$ When $\mathcal{L}$ is finite and $M\gg 1$, Eq. [\[QLM\]](#QLM){reference-type="eqref" reference="QLM"} simplifies to $$Q(\mathcal{L},M) \simeq M^{-1}\,\frac{(\ln M)^{\mathcal{L}-1}}{(\mathcal{L}-1)!}$$ We also note that $\mathcal{L}=M$ occurs with probability $$\label{QMM:RRM} Q(M,M) = \frac{1}{M!}$$ To determine the average number of followers we note that vertex $1$ is a follower with probability $1-1/M$ as it can be an expert only to itself; vertex $2$ is a follower with probability $[1-\frac{1}{M}][1-\frac{1}{M-1}]=1-2/M$; generally vertex $i$ is a follower with probability $1-i/M$. Thus $$\label{F:MRM} \langle F\rangle = \sum_{i=1}^M \left[1-\frac{i}{M}\right] = \frac{M-1}{2}$$ In Appendix [8](#ap:RRM){reference-type="ref" reference="ap:RRM"}, we show that the model [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"} is isomorphic to recursive random maps and derive the announced formula [\[PCM:Stirling\]](#PCM:Stirling){reference-type="eqref" reference="PCM:Stirling"} for the probability distribution $$\label{PCM:def} P(C,M) = \text{Prob}[C(M)=C]$$ of the number of communities via the Stirling numbers of the first kind [@Knuth]. Using [\[PCM:Stirling\]](#PCM:Stirling){reference-type="eqref" reference="PCM:Stirling"} we find the average number of communities $$\label{C-av-RRM} \langle C\rangle = \sum_{C\geq 1}C P(C,M) = H_M$$ Thus the average number of communities grows logarithmically with $M$ as for classical random maps, Eq. [\[c-av\]](#c-av){reference-type="eqref" reference="c-av"}, but with twice larger amplitude \[cf. [\[Harmonic\]](#Harmonic){reference-type="eqref" reference="Harmonic"}\]. In Appendix [8](#ap:RRM){reference-type="ref" reference="ap:RRM"}, we also establish a neat formula [\[PFM:Euler\]](#PFM:Euler){reference-type="eqref" reference="PFM:Euler"} for the probability distribution of the number of followers $$\label{PFM:def} \Pi(F,M) = \text{Prob}[F(M)=F]$$ The distributions $P(C,M)$ and $\Pi(F,M)$ do not become stationary (i.e., independent on $M$) in the $M\to\infty$ limit. This is obvious from explicit formulae for $P(C,M)$ at small $C$ such as $$\label{P12} P(1,M)=M^{-1}, \qquad P(2,M)=M^{-1}H_{M-1}$$ and from explicit formulae for $\Pi(F,M)$ at small $F$ $$\label{P01} \Pi(0,M)=\frac{1}{M!}, \qquad \Pi(1,M)=\frac{2^M-M-1}{M!}$$ The distribution of introverts $R(I,M)$ is easier to compute than the distributions $P(C,M)$ and $\Pi(F,M)$. It turns out that each individual is introvert with probability $1/M$. This is obvious for the first individual, and requires a little calculation $$\frac{1}{M-1}\left(1-\frac{1}{M}\right) = \frac{1}{M}$$ for the second individual. Generally the $k^\text{th}$ individual is introvert with probability $$\label{Ik} \frac{1}{M-k+1}\prod_{i=1}^{k-1}\left(1-\frac{1}{M-i+1}\right) = \frac{1}{M}$$ Therefore the distribution of introverts is binomial $$\label{R:binom} R(I,M) = \binom{M}{I}\left(1-\frac{1}{M}\right)^{M-I} \left(\frac{1}{M}\right)^I$$ from which we conclude that $\langle I\rangle=1$ for any $M$. In contrast to the distributions $P(C,M)$ and $\Pi(F,M)$, the distribution of introverts is asymptotically stationary $$\label{R:Poisson} R(I,\infty) = \frac{e^{-1}}{I!}$$ Equation [\[F:MRM\]](#F:MRM){reference-type="eqref" reference="F:MRM"} yields $n_1(\infty)=\frac{1}{2}$. In Appendix [8](#ap:RRM){reference-type="ref" reference="ap:RRM"}, we show that the degree distribution is asymptotically pure exponential $$\label{nk:RRM} \lim_{M\to\infty}n_k(M) = 2^{-k}$$ When $M$ is large but finite, [\[nk:RRM\]](#nk:RRM){reference-type="eqref" reference="nk:RRM"} is applicable as long $\langle N_k(M)\rangle\gg 1$. Alternatively, $k$ should be much smaller than the maximal degree $D(M)$. The maximal degree is found from the criterion [\[D:criterion\]](#D:criterion){reference-type="eqref" reference="D:criterion"} together with [\[nk:RRM\]](#nk:RRM){reference-type="eqref" reference="nk:RRM"} to yield $$\label{D-max:RRM} D(M) \simeq \frac{\ln M}{\ln 2}$$ When $k\gtrsim \log_2(M)$, fluctuations of $N_k(M)$ become too large and these quantities are no longer self-averaging. ## Linear choice: Model [\[map:linear-rank\]](#map:linear-rank){reference-type="eqref" reference="map:linear-rank"} The average number of egocentric individuals, i.e., the average number of loops is $$\langle \mathcal{L}(M)\rangle = \sum_{i=1}^M \frac{2}{(M-i+1)(M-i+2)}=\frac{2M}{M+1}$$ Thus $\langle \mathcal{L}\rangle \to 2$ when $M\to\infty$. There is again at least one loop as the top rank individual maps to itself, $M\to M$. The probability to have one loop $$\label{Q1:linear-rank} Q(1,M) = \prod_{k=2}^{M} \left[1-\frac{2}{k(k+1)}\right] = \frac{M+2}{3M}$$ approaches to 1/3 when $M\to\infty$. The probability to have two loops $$Q(2,M) = \frac{M+2}{3M} \sum_{j=2}^{M} \frac{2}{(j-1)(j+2)}$$ reduces to $$\label{Q2:linear-rank} Q(2,M) = \frac{(M-1)(11M^2+26M+12)}{27M^2(M+1)}$$ and approaches to 11/27 when $M\to\infty$. The probability to have three loops is $$Q(3,M) = \frac{M+2}{3M} \sum_{2\leq j<k\leq M} \frac{4}{(j-1)(j+2)(k-1)(k+2)}$$ An exact expression of the sum through special functions is rather cumbersome, so we just give $$\label{Q3:linear-rank} Q(3,\infty)= \frac{107 - 6 \pi^2}{243} = 0.196\,635\,282\ldots$$ Generalizing above results for small number of loops we arrive at the general formula $$Q(\mathcal{L}+1,M) = \frac{M+2}{3M} \sum_{2\leq j_1<\ldots<j_\mathcal{L}\leq M} \prod_{\nu=1}^\mathcal{L} \frac{2}{(j_\nu-1)(j_\nu+2)}$$ Taking the $M\to\infty$ limit and keeping $\mathcal{L}$ finite we obtain $$\label{QL:linear-rank} Q(\mathcal{L}+1,\infty) = \frac{2^\mathcal{L}}{3} \sum_{1\leq j_1<\ldots<j_\mathcal{L}} \prod_{\nu=1}^\mathcal{L} \frac{1}{j_\nu(j_\nu+3)}$$ The asymptotic probability to have four loops admits a simple expression resembling [\[Q3:linear-rank\]](#Q3:linear-rank){reference-type="eqref" reference="Q3:linear-rank"}: $$\label{Q4:linear-rank} Q(4,\infty)= \frac{1003 - 90 \pi^2}{2187} = 0.052\,462\,553\ldots$$ We have not found compact formulas for asymptotic probabilities $Q(\mathcal{L},\infty)$ with $\mathcal{L}>4$. Returning to finite $M$ we also note that the maximal number of loops, i.e., the probability that all individuals are introverts, is $$\label{QMM:linear} Q(M,M) = \frac{2^M}{M!(M+1)!}$$ Using the definition of the map, Eq. [\[map:linear-rank\]](#map:linear-rank){reference-type="eqref" reference="map:linear-rank"}, one finds the average number of followers is given by the sum $$\label{Fav:sum} \langle F\rangle = \langle N_1\rangle =\sum_{j=1}^M \Psi_j(M)$$ where we have used shorthand notation $$\label{Psi:def} \Psi_j(M) = \prod_{i=1}^j \left[1-\frac{2(j-i+1)}{(M-i+1)(M-i+2)}\right]$$ A straightforward analysis (see Appendix [9](#ap:derive){reference-type="ref" reference="ap:derive"} for details) yields the large $M$ asymptotic: $$\label{Lav:linear-asymp} \lim_{M\to\infty}\frac{\langle F\rangle}{M} = n_1 = \frac{e^2-5}{4}= 0.59726402473\ldots$$ In Appendix [9](#ap:derive){reference-type="ref" reference="ap:derive"} we also determine the degree distribution for the model [\[map:linear-rank\]](#map:linear-rank){reference-type="eqref" reference="map:linear-rank"}: $$\label{nk:linear} n_{k} = \int_0^1 dx\, (1-x)^2 e^{2x}\, \frac{[J(x)]^{k-1}}{(k-1)!}$$ where we shortly write $$\label{J:def} J(x) = \int_0^x dy\,\frac{2(x-y)}{(1-y)^2} = -2[x + \ln(1-x)]$$ An asymptotic analysis of [\[nk:linear\]](#nk:linear){reference-type="eqref" reference="nk:linear"}--[\[J:def\]](#J:def){reference-type="eqref" reference="J:def"} allows one to deduce the large $k$ tail of the degree distribution $$\label{nk:tail-linear} n_k\simeq \frac{1}{2 e}\left(\frac{2}{3}\right)^k$$ Using criterion [\[D:criterion\]](#D:criterion){reference-type="eqref" reference="D:criterion"} together with [\[nk:tail-linear\]](#nk:tail-linear){reference-type="eqref" reference="nk:tail-linear"} we find that the maximal degree exhibits a logarithmic growth $$\label{DM:linear} D(M) \simeq \frac{\ln M}{\ln(3/2)}$$ Similarly to [\[Ik\]](#Ik){reference-type="eqref" reference="Ik"} we find that the $k^\text{th}$ individual is introvert with probability $$\label{Ik:linear} I_k(M)=\frac{2\Psi_k(M)}{(M-k+1)(M-k+2)-2}$$ Using the asymptotic behavior of $\Psi_k(M)$ which is derived by employing a continuum approach, see Appendix [9](#ap:derive){reference-type="ref" reference="ap:derive"}, we deduce $$\label{Ik:linear-asymp} I_k(M)\simeq \frac{2}{M^2}\,e^{2k/M}$$ The average number of introverts decays as $$\langle I(M)\rangle=\sum_{k=1}^M I_k(M)\simeq \frac{e^2-1}{M}$$ ## Inverse linear choice: Model [\[map-linear\]](#map-linear){reference-type="eqref" reference="map-linear"} The average number of egocentric individuals is $$\langle \mathcal{L}(M)\rangle = \sum_{k=1}^M \frac{1}{H_k}$$ In previous models the average number of egocentric individuals remained finite when $M\to\infty$, the only exception is the model [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"} where the number of egocentric individuals diverges, albeit very slowly (logarithmically). In the present model the divergence is much more fast: $$\label{Lav-linear} \langle \mathcal{L}(M)\rangle \simeq \frac{M}{\ln M}$$ The probability to have exactly one egocentric $$\label{Q1-linear} Q(1,M) = \prod_{k=2}^{M} \left[1-\frac{1}{H_k}\right]$$ is very small when $M$ is large $$\ln Q(1,M) = -\frac{M}{\ln M} + \mathcal{O}\!\left[\frac{M}{(\ln M)^2}\right]$$ The maximal number of egocentrics occurs with probability $$\label{QMM:harmonic} Q(M,M) = \prod_{k=1}^{M} \frac{1}{H_k}$$ which decays faster than exponentially but slower than factorially with $M$: $$\label{QMM:harm-decay} Q(M,M) \asymp e^{-M\ln(\ln M)}$$ (Here $A \asymp B$ when $M\to\infty$ means the asymptotic equality of the logarithms: $\lim_{M\to\infty} \frac{\ln A}{\ln B}=1$.) The average number of followers is given by $$\label{Fav-linear-exact} \langle F\rangle = \sum_{j=1}^M \prod_{i=1}^j \left[1-\frac{1}{(j-i+1) H_{M-i+1}}\right]$$ The asymptotic behavior is slightly sub-linear, namely $$\label{Fav-linear} \langle F\rangle \simeq \frac{M}{\ln M}$$ as we show in Appendix [9](#ap:derive){reference-type="ref" reference="ap:derive"}. Hence in the model [\[map-linear\]](#map-linear){reference-type="eqref" reference="map-linear"} the fraction of followers asymptotically vanishes, although slowly as inverse logarithm: $n_1\simeq (\ln M)^{-1}$. Moreover $$\label{nk-linear} n_k \simeq \frac{1}{\ln M}\,\frac{1}{(k-1)!}$$ as we show in Appendix [9](#ap:derive){reference-type="ref" reference="ap:derive"}. As before we find that the $k^\text{th}$ individual is introvert with probability $$\label{Ik:linear-} I_k(M)=\frac{1}{H_{M-k+1}}\prod_{j=2}^k \left[1-\frac{1}{j H_{M-k+j}}\right]$$ Using this exact formula one finds that the average number of introverts diverges as $$\label{Iav-linear} \langle I(M)\rangle \simeq e^{-1}\,\frac{M}{\ln M}$$ Thus in the realm of model [\[map-linear\]](#map-linear){reference-type="eqref" reference="map-linear"} the number of egocentrics is asymptotically the same as the number of followers, cf. [\[Lav-linear\]](#Lav-linear){reference-type="eqref" reference="Lav-linear"} and [\[Fav-linear\]](#Fav-linear){reference-type="eqref" reference="Fav-linear"}, and about $37\%$ of egocentrics are introverts. ## The algebraic family of models [\[map-a\]](#map-a){reference-type="eqref" reference="map-a"} The average number of egocentric individuals $$\langle \mathcal{L}(M)\rangle = \sum_{k=1}^M \frac{1}{S_k(a)}\,, \qquad S_k(a)=\sum_{p=1}^k p^a$$ The average number of egocentric individuals diverges as $$\langle \mathcal{L}(M)\rangle \simeq -\frac{a+1}{aM^a}$$ when $-1<a<0$ and saturates to $$\label{sigma-a} \lim_{M\to\infty}\langle \mathcal{L}(M)\rangle = \sigma(a) = \sum_{k=1}^\infty \frac{1}{S_k(a)}$$ when $a>0$. This $\sigma(a)$ is a monotonically decaying function of $a$, see Fig. [5](#Fig:sigma){reference-type="ref" reference="Fig:sigma"}. Here are a few special values and asymptotic behaviors: $$\begin{split} &\sigma(1) = 2\\ &\sigma(2) = 6(3-4\ln 2) = 1.364\,467\ldots\\ &\sigma(3) = \tfrac{4}{3}\pi^2-12 = 1.159\,472\ldots \\ &\sigma(a) \simeq a^{-1} ~~\qquad 0<a\ll 1\\ &\sigma(a) - 1 \simeq 2^{-a} \quad a\gg 1 \end{split}$$ ![Top: The average number of egocentrics $\sigma(a)$ given by [\[sigma-a\]](#sigma-a){reference-type="eqref" reference="sigma-a"}. Bottom: The probability $Q_1(a)$ to have exactly one egocentric individual given by [\[Q1-a\]](#Q1-a){reference-type="eqref" reference="Q1-a"}. Shown are results for the one-parameter family of models [\[map-a\]](#map-a){reference-type="eqref" reference="map-a"}. ](sigma_Q1.pdf){#Fig:sigma width="7.77cm"} The probability to have exactly one egocentric is $$\label{Q1-a} Q(1,M) = \prod_{k=2}^{M} \left[1-\frac{1}{S_k(a)}\right]$$ from which $$\label{Q1-a-asymp} Q(1,M) = \begin{cases} \asymp \exp\!\left[\frac{a+1}{aM^a}\right] & -1<a<0\\ M^{-1} & a=0\\ \simeq Q_1(a) & a>0 \end{cases}$$ with $$\label{Q1-a-prod} Q_1(a) = \prod_{k=2}^{\infty} \left[1-\frac{1}{S_k(a)}\right]$$ Here are a few special values and asymptotic behaviors: $$\begin{split} &Q_1(1) = \frac{1}{3}\\ &Q_1(2) = \frac{3\sqrt{\pi}}{2\Gamma\!\left[\frac{13-\sqrt{-23}}{4}\right] \Gamma\!\left[\frac{13+\sqrt{-23}}{4}\right]} = 0.676\,285\ldots\\ &Q_1(3) = (12\pi)^{-1}\cosh\!\left[\tfrac{\pi\sqrt{7}}{2}\right] = 0.846\,536\ldots \\ &Q_1(a) \asymp e^{-1/a} \quad\qquad 0<a\ll 1\\ &1-Q_1(a) \simeq 2^{-a} \qquad a\gg 1 \end{split}$$ The probability $Q_1(a)$ to have one egocentric individual is an increasing function of $a$, see Fig. [5](#Fig:sigma){reference-type="ref" reference="Fig:sigma"}. The probability to have two egocentric individuals is given by $$\label{Q2-a} Q(2,M) = Q(1,M)\sum_{k=2}^{M} \frac{1}{S_k(a)-1}$$ Generally $Q(\mathcal{L},M)\to Q(\mathcal{L},\infty)\equiv Q_\mathcal{L}(a)$ when $a>0$ with $$\label{QL-a} Q_\mathcal{L}(a) = Q_1(a) \sum_{2\leq j_1<\ldots<j_\mathcal{L}} \prod_{n=1}^\mathcal{L} \frac{1}{S_{j_n}(a)-1}\\$$ The probability that all individuals are egocentrics (they are all introverts in this extreme situation) is $$\label{QMM:Sa} Q(M,M) = \prod_{k=1}^{M} \frac{1}{S_k(a)}$$ # Rewiring {#sec:rewire} So far, we have analyzed static random maps. Evolving random maps constitute a large and unexplored class of random maps. Let us briefly look at random maps evolving through rewiring: The population size remains fixed while the map varies. For instance, individuals may want to rewire to more connected individuals (i.e., those with higher degrees). We assume that individuals attempting rewiring rely only on local information. Here is a specific example of the rewiring procedure. Starting with a random initial map, we implement rewiring as follows: 1. At each time step a randomly chosen individual, say $x$, inspects its expert $f(x)$ and the secondary expert $f[f(x)]$. 2. If the degree of the secondary expert exceeds the degree of the expert, $k(f[f(x)])>k(f(x))$, the switch from $x\to f(x)$ to $x\to f[f(x)]$ occurs; otherwise, the rewiring attempt fails. Thus the final map obeys the constraints: $$k(f[f(x)]) \leq k(f(x)) \quad\text{for all} \quad x\in S$$ {#Fig:graphs width="7.77cm"} After a certain number of rewiring steps, the process comes to a halt. One would like to understand how the average number of rewiring steps scales with $M$. For instance, applying the rewiring procedure to the map shown in Fig. [1](#Fig:random-map-ill){reference-type="ref" reference="Fig:random-map-ill"} reduces each community to the directed star graph. An example of a directed star graph is shown in Fig. [6](#Fig:graphs){reference-type="ref" reference="Fig:graphs"} (bottom). Generally, during the rewiring procedure, a follower remains a follower, while an expert may become a follower. Thus the number of followers increases, and the number of experts decreases. If there is a single individual with a maximal degree in a community, the degree of this individual may only increase. Only cycles and directed star graphs are stable under rewiring (see also Fig. [6](#Fig:graphs){reference-type="ref" reference="Fig:graphs"}). Thus communities of a final map are either cycles and directed star graphs. We know the average number of communities in classical random graphs, Eq. [\[c-av\]](#c-av){reference-type="eqref" reference="c-av"}. If the community is a cycle, it remains a cycle; otherwise, it turns into a directed star graph. Each directed star graph has one expert; for a cycle $S_\ell$ of length $\ell$, the number of experts is $\ell$. Thus we have a lower bound $E\geq C$ for the total number of experts after rewiring, leading to $$\label{E-av-low} \langle E\rangle \geq \langle C\rangle = \tfrac{1}{2}\ln M + O(1)$$ This lower bound seems qualitatively correct, namely we anticipate a logarithmic growth law $$\label{E-av} \langle E\rangle = B\ln M$$ An obvious challenge is to determine the amplitude $B$. # Experts, Prophets and founders {#sec:prophet} We shortly call a prophet an expert of the highest degree. For random maps compatible with ranking, the founder is the highest-rank individual. The founder maps to itself, so the founder is an expert, moreover, an egocentric. One would like to compute the probability $F_M$ that the founder is a prophet. The total number of experts varies within the bounds $1\leq E \leq M$. There is one map with $E=1$, a directed star with a loop from the center to itself; for $M=7$, such map is shown in Fig. [6](#Fig:graphs){reference-type="ref" reference="Fig:graphs"} (bottom). Maps with $E=M$ are collections of cycles. The total number of prophets also varies within the bounds $$1\leq P \leq M$$ Maps with the maximal number of prophets, $P=M$, are collections of $M$ loops. The probabilities of such extreme outcome in various random map models are given by [\[QMM:classical\]](#QMM:classical){reference-type="eqref" reference="QMM:classical"}, [\[QMM:fitness\]](#QMM:fitness){reference-type="eqref" reference="QMM:fitness"}, [\[QMM:0123\]](#QMM:0123){reference-type="eqref" reference="QMM:0123"}, [\[QMM:RRM\]](#QMM:RRM){reference-type="eqref" reference="QMM:RRM"}, [\[QMM:linear\]](#QMM:linear){reference-type="eqref" reference="QMM:linear"}, [\[QMM:harmonic\]](#QMM:harmonic){reference-type="eqref" reference="QMM:harmonic"}, [\[QMM:Sa\]](#QMM:Sa){reference-type="eqref" reference="QMM:Sa"}. The opposite extreme is more challenging for analyses. The numbers $S(M)$ of maps with a single prophet when all $M^M$ maps are possible have been recently computed via complicated recurrences [@Sloane]. Thus, the fraction of maps $S_M/M^M$ with a single prophet is the probability that a classical random map has a single prophet. In scale-free networks, there is a single prophet with probability approaching unity in the $M\to\infty$ limit [@KR13]. This is not so for classical random maps. For instance, according to Monte-Carlo simulations, a classical random map has a single prophet with probability close to $0.35$ for $M=3.2 \cdot 10^6$. {#Fig:SM width="7.89cm"} Exact results for $M$ up to a few hundreds exhibit intriguing non-monotonic dependence on $M$, see Fig. [7](#Fig:SM){reference-type="ref" reference="Fig:SM"}. Numerical results obtained by Monte-Carlo simulations for larger range of $M$ indicate never-ending oscillations, see Fig. [8](#Fig:SM-log){reference-type="ref" reference="Fig:SM-log"} suggesting that the period of oscillations increases logarithmically with $M$. Heuristic arguments relying on the Poisson form [\[nk:Poisson\]](#nk:Poisson){reference-type="eqref" reference="nk:Poisson"} of the degree distribution of classical random maps suggest a slightly slower growth of period, namely $S(M)/M^M$ is a periodic function of $\ln(M)/\ln[\ln(M)]$. More precisely, the local maxima of $S(M)/M^M$ apparently approach a positive constant, while the local minima slowly decay to zero. {#Fig:SM-log width="7.89cm"} In some random map models, the number of experts is proportional to the population size. The linear scaling $\langle E\rangle\sim M$ is not necessarily a drawback, e.g., if expertise in everyday matters, a more active spouse in a family may be essentially an expert. Still, one anticipates that the number of experts can grow slower than linearly with population size. Perhaps a logarithmic growth is realized for the model with rewiring, cf. [\[E-av\]](#E-av){reference-type="eqref" reference="E-av"}. Strictly speaking, we have only established the lower bound [\[E-av-low\]](#E-av-low){reference-type="eqref" reference="E-av-low"}, so the logarithmic growth [\[E-av\]](#E-av){reference-type="eqref" reference="E-av"} has not been proven. For classical random maps, the maximal degree is given by [\[DM:nu\]](#DM:nu){reference-type="eqref" reference="DM:nu"}, i.e., it grows even slightly slower than logarithmically with $M$. Recall that for the family of fitness models [\[fitness-a\]](#fitness-a){reference-type="eqref" reference="fitness-a"}, the maximal degree grows according to [\[DM:nu-a\]](#DM:nu-a){reference-type="eqref" reference="DM:nu-a"}: if $a\geq 0$, the growth law identical to the growth law [\[DM:nu\]](#DM:nu){reference-type="eqref" reference="DM:nu"} of the classical random maps; if $-1<a<0$, the growth is algebraic, $D\sim M^{-a}$; if $a=-1$, a prophet has an almost macroscopic fraction of individuals among the followers, $D\simeq M/\ln M$. Characteristics of prophets are sensitive to the type of random map. For models compatible with ranking, the most natural candidate for being a prophet is the founder, i.e., the highest rank individual. For the model [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"}, the average degree $\delta$ of the founder and the prophet \[cf. Eq. [\[D-max:RRM\]](#D-max:RRM){reference-type="eqref" reference="D-max:RRM"}\] grow logarithmically with $M$ but with different amplitudes: $$\label{deg-rank:RRM} \delta = H_M\simeq \ln M, \quad D(M)\simeq \frac{\ln M}{\ln 2}$$ Hence for large $M$, it is unlikely that the prophet is the founder (the first individual in the dynamical interpretation of the RRM). For the RRM, one can derive the entire degree distribution $\Phi(k,M)$ of the founder. This distribution satisfies $$M \Phi(k,M) = \Phi(k-1,M-1) + (M-1) \Phi(k,M-1)$$ Solving this recurrence starting with $\Phi(k,1)=\delta_{k,1}$ gives the announced result [\[PkM:Stirling\]](#PkM:Stirling){reference-type="eqref" reference="PkM:Stirling"}. Using the identity [@Knuth] $$\label{Stirling:GF} \sum_{k=1}^M {M\brack k} x^k = x(x+1)\cdots(x+M-1)$$ one can deduce exact formulas for the average $\delta = \langle k\rangle$ and higher moments. In the RRM of the large size $M$, the founder is the prophet with probability $$\label{Prophet:RRM} F_M\sim \sum_{k>D(M)}\Phi(k,M)$$ The degree of the prophet is a random variable with small fluctuation when $M\gg 1$. This feature justifies using the average degree $D(M)$ of the prophet in the sum in [\[Prophet:RRM\]](#Prophet:RRM){reference-type="eqref" reference="Prophet:RRM"}. Using [\[PkM:Stirling\]](#PkM:Stirling){reference-type="eqref" reference="PkM:Stirling"}, [\[Stirling:GF\]](#Stirling:GF){reference-type="eqref" reference="Stirling:GF"} and the Cauchy theorem, we express $\Phi(k,M)$ as the contour integral $$\Phi(k,M) = \frac{1}{2\pi\sqrt{-1}}\oint \frac{dx}{x^{k}}\,\frac{(x+1)\cdots(x+M-1)}{M!}$$ We then compute the integral in the $M\to\infty$ limit by applying the saddle point technique. In the interesting situation when $k=\kappa \ln M$ with $\kappa>1$ we get $$\Phi(k,M) \sim M^{\kappa-1-\kappa\ln\kappa}(\ln M)^{-1/2}$$ Combining this with [\[Prophet:RRM\]](#Prophet:RRM){reference-type="eqref" reference="Prophet:RRM"} and noticing that $k=D(M)$ when $\kappa=1/\ln 2$, we deduce the probability $$\label{first} F_M\sim M^{-\epsilon} (\ln M)^{-1/2}, \quad \epsilon = 1-\frac{1+\ln \ln 2}{\ln 2}$$ that the founder is the prophet. The exponent $\epsilon$ arises in a surprisingly large number of similarly unrelated problems related to divisors [@Divisors; @Ford08; @Anatomy], random permutations [@Peres16; @Ford16a; @Ford16b], averaging processes [@BK-21], etc. The RRMs resemble recursive random trees (RRTs) [@RRT-Tapia; @RRT-Moon; @RRT-Pittel], a null model of growing networks [@Newman; @book; @Frieze] where individuals arrive one by one and map upon arriving to already present individuals. There are no loops in the RRT, but the distinction between the models vanishes as $M\to\infty$, and the chief properties of the maximal degree are expected to be similar. For RRTs, the evolution of the maximal degree was studied in [@KR02] where [\[first\]](#first){reference-type="eqref" reference="first"} was derived. Another interesting result of [@KR02] is that the expected difference between the maximal rank $M$ and the rank of the prophet, equivalently, the population size $\mathcal{P}$ when the prophet was born, scales sub-linearly $$\label{born} \mathcal{P} \sim M^\psi, \quad \psi=2-\log_2 3$$ For the model [\[map:linear-rank\]](#map:linear-rank){reference-type="eqref" reference="map:linear-rank"}, the average degree of the founder $$\delta = \sum_{i=1}^M \frac{2}{M-i+2}= 2[H_{M+1}-1] \simeq 2\ln M$$ grows logarithmically with $M$ as the degree of the prophet, Eq. [\[DM:linear\]](#DM:linear){reference-type="eqref" reference="DM:linear"}, but the latter grows with larger amplitude $[\ln(3/2)]^{-1}\approx 2.466$. For the model [\[map-linear\]](#map-linear){reference-type="eqref" reference="map-linear"}, the average degree of the founder grows much slower than the degree of the prophet $$\label{delta-D} \delta = \sum_{k=1}^M \frac{1}{k H_k} \simeq \ln(\ln M), \quad D(M)\simeq \frac{\ln M}{\ln(\ln M)}$$ The degree of the prophet is found from the criterion $$1 \sim M\sum_{k>D(M)}n_k\simeq \frac{M/\ln M}{[D(M)]!}$$ where we have used the degree distribution [\[nk-linear\]](#nk-linear){reference-type="eqref" reference="nk-linear"} and then extracted the asymptotic using the Stirling formula. # Discussion {#sec:disc} We devised random map models with sociological flavor, and explored their basic properties, e.g., the numbers of experts, followers, prophets, egocentrics, and introverts. Specifically, we analyzed random maps compatible with fitness or ranking. Some of our results for the average number of egocentrics and introverts in various random map models are collected in Table [1](#Tab:loops){reference-type="ref" reference="Tab:loops"}. The number of egocentrics may increase with $M$. This phenomenon occurs in models [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"} and [\[map-linear\]](#map-linear){reference-type="eqref" reference="map-linear"}. In the former model, the number of egocentrics diverges logarithmically with the total number of individuals $M$; in the latter model, it diverges almost linearly, cf. Eq. [\[Lav-linear\]](#Lav-linear){reference-type="eqref" reference="Lav-linear"}. In Sect. [6](#sec:prophet){reference-type="ref" reference="sec:prophet"}, we analyzed prophets and founders. Random maps in which the number of individuals increases provide an attractive setup for studying prophets. One can ask about the size of the population when a prophet was born, the probability that an earlier prophet remains the prophet throughout the evolution, the total number of distinct prophets that appeared throughout history. Equations [\[born\]](#born){reference-type="eqref" reference="born"}--[\[first\]](#first){reference-type="eqref" reference="first"} give answers to some of these questions for RRMs. The average number of lead changes increases as $\ln M$ for RRMs, see [@KR02]. The number of distinct prophets is not affected by leap-frogging, so it is smaller than the number of lead change. The average number of distinct prophets that appeared during the RRM evolution up to size $M$ is unknown. Random maps compatible with ranking naturally arise in the dynamical setting. Individuals arrive one by one, and each new individual $x$ immediately maps to $f(x)$, one of already present individuals (including itself). The RRM corresponds to the uniform choice. We now define another simple model. The *preliminary* choice is uniform. We postulate that $f(x)=x$ is accepted, but if $f(x)\ne x$, we redirect to the image of the preliminary choice, i.e., the final choice is $f[f(x)]$. We thus obtain recursive random maps with redirection (RRMR). In the RRMR, the community of the founder is a directed star graph as in Fig. [6](#Fig:graphs){reference-type="ref" reference="Fig:graphs"} (bottom). As the RRM model, the RRMR model is parameter-free and may be tractable. Some characteristics of these two models are identical, e.g., the probability distribution of the number of communities is given by the same formula [\[PCM:Stirling\]](#PCM:Stirling){reference-type="eqref" reference="PCM:Stirling"}. Other characteristics differ and are sometimes easier to determine for the RRMR. For instance, the degree distribution $\Phi(k,M)$ of the founder of the RRMR satisfies $$M \Phi(k,M) = (k-1)\Phi(k-1,M-1) + (M-k) \Phi(k,M-1)$$ Solving this recurrence starting with $\Phi(k,1)=\delta_{k,1}$ we find that the degree distribution of the founder is flat $$\label{Phi:RRMR} \Phi(k,M) = M^{-1}, \qquad k = 1, \ldots, M$$ For the RRMR, the degree of the founder tends to be higher than in the RRM. Indeed, using [\[Phi:RRMR\]](#Phi:RRMR){reference-type="eqref" reference="Phi:RRMR"} we find the average degree of the founder $$\label{Phi:RRMR} \delta = \frac{M+1}{2}$$ significantly exceeding the average degree of the founder $\delta=H_M$ for the RRM. The dynamical setting may help to appropriate the techniques developed for studying similar types of growing random trees [@KR02; @JML08; @JML10] to exploring more subtle questions about prophets and the founder for the RRMR and other dynamical random maps compatible with ranking. Vertices of anomalously high degrees are widespread in modern communication and social networks. Referring to such vertices as prophets is questionable. Intuitively, a prophet is more like a central vertex in the largest community, and the size of the community appears to be a more interesting quantity than the degree of the prophet. Thus, rather than studying the degrees of prophets and founder (in random maps compatible with ranking), one could investigate the largest community and the primordial community (containing the founder). The size distribution of the largest community is difficult to probe analytically. In most random map models, the distribution $\Psi(s,M)$ of the size $s$ of the largest community acquires the scaling form $$\Psi(s,M) = M^{-1}\psi(\sigma)$$ in the scaling limit $$s\to\infty, \quad M\to\infty, \quad \sigma = \frac{s}{M} = \text{finite}$$ The rescaled size distribution $\psi(\sigma)$ has singularities at $\sigma=1/m$ with $m=2,3,\ldots$ that become weaker and weaker as $m$ increases. The origin of these singularities is easy to appreciate: One community may have size exceeding $M/2$, i.e., in the $\frac{1}{2}<\sigma<1$ range; two communities may have size in the $\frac{1}{3}<\sigma<\frac{1}{2}$ range; etc. This infinite set of singularities prevents an analytical determination of $\psi(\sigma)$. The rescaled size distribution for classical random maps was probed numerically in [@BN-Derrida87]. The average size of the largest community is close to $0.75782\,M$, see [@Flajolet90]. For the RRM, the rescaled size distribution and the average size of the largest community are unknown. The primordial community keeps identity throughout the evolution, and its properties are easy to access. The size distribution $\mathcal{P}(s,M)$ of the primordial community is flat $$\label{P:RRM} \mathcal{P}(s,M) = M^{-1}, \qquad s = 1, \ldots, M$$ Therefore, the average size of the primordial community is $\langle s \rangle = \frac{M+1}{2}$. The average size of the largest community exceeds the average size of the primordial community and also scales linearly with $M$, namely, as $\lambda M$ with $\lambda>\frac{1}{2}$. Computing the amplitude $\lambda$ might be feasible; we leave it for future work. I benefitted from discussions with Mirta Galesic and Sid Redner. I am very grateful to Aaron Schweiger, the author of the sequence [@Sloane], for preparing Figs. [7](#Fig:SM){reference-type="ref" reference="Fig:SM"}--[8](#Fig:SM-log){reference-type="ref" reference="Fig:SM-log"}, reading the manuscript, and providing helpful suggestions. # Recursive random maps (RRMs) {#ap:RRM} Here we consider an evolving random map with a growing number of individuals. Namely, we postulate that individuals arrive one by one, and each new individual $x$ immediately maps to an individual that we denote $f(x)$. As in classical random maps, we postulate that the image $f(x)$ is chosen uniformly and randomly among all present individuals, including $x$. The outcome is a recursive random map (RRM) model. If the choice is uniform but excludes the new individual, $f(x)\ne x$, the outcome is a recursive random tree [@Newman; @book; @Frieze]. As a random map, it contains a single community and no egocentrics and is less interesting for our purposes than RRMs. A model of growing random hypergraphs known as random recursive hypergraphs [@PK-hyper] also overlaps with RRMs. The number of communities $C(M)$ is a random variable that grows according to $$\label{CM} C(M) = \begin{cases} C(M-1) & \text{prob} ~~ 1-M^{-1}\\ C(M-1) + 1 & \text{prob} ~~ M^{-1} \end{cases}$$ In particular, the average satisfies the recurrence $$\label{CM-av} \langle C(M) \rangle = \langle C(M-1) \rangle + M^{-1}$$ which is solved to yield $$\label{CM-av-sol} \langle C(M) \rangle = H_M$$ More generally, using [\[CM\]](#CM){reference-type="eqref" reference="CM"} we find that the probability distribution defined by Eq. [\[PCM:def\]](#PCM:def){reference-type="eqref" reference="PCM:def"} satisfies $$\begin{aligned} \label{PCM} P(C,M) &=& (1-M^{-1})P(C,M-1) \nonumber \\ &+& M^{-1}P(C-1,M-1)\end{aligned}$$ The initial condition is $$\label{P:IC} P(C,1)=\delta_{C,1}$$ When $C=1$, Eq. [\[PCM\]](#PCM){reference-type="eqref" reference="PCM"} reduces to $$P(1,M) = (1-M^{-1})P(1,M-1)$$ Solving this recurrence subject to $P(1,1)=1$ yields $$\label{P1M-sol} P(1,M) = M^{-1}$$ Using [\[P1M-sol\]](#P1M-sol){reference-type="eqref" reference="P1M-sol"} we recast [\[PCM\]](#PCM){reference-type="eqref" reference="PCM"} for $C=2$ into $$\label{P2M} P(2,M) = (1-M^{-1})P(2,M-1)+\frac{1}{M(M-1)}$$ Solving this recurrence subject to $P(2,1)=0$ yields $$\label{P2M-sol} P(2,M) = M^{-1}H_{M-1}$$ for $M\geq 2$. Collecting [\[P1M-sol\]](#P1M-sol){reference-type="eqref" reference="P1M-sol"} and [\[P2M-sol\]](#P2M-sol){reference-type="eqref" reference="P2M-sol"} give [\[P12\]](#P12){reference-type="eqref" reference="P12"}. We now derive the announced general solution [\[PCM:Stirling\]](#PCM:Stirling){reference-type="eqref" reference="PCM:Stirling"}. We multiply [\[PCM\]](#PCM){reference-type="eqref" reference="PCM"} by $M!$ and notice that $M! P(C,M)$ satisfies an addition formula for the Stirling numbers ${M\brack C}$ of the first kind [@Knuth]. The initial condition [\[P:IC\]](#P:IC){reference-type="eqref" reference="P:IC"} agrees with the standard initial condition for the Stirling numbers of the first kind, and we arrive at [\[PCM:Stirling\]](#PCM:Stirling){reference-type="eqref" reference="PCM:Stirling"}. Using [\[PCM:Stirling\]](#PCM:Stirling){reference-type="eqref" reference="PCM:Stirling"} and identities for the relevant Stirling numbers of the first kind, we extract the probabilities that the number of communities is close to maximal: $$\label{PMM3} \begin{split} P(M,M) &= \frac{1}{M!} \\ P(M-1,M) & = \frac{1}{M!}\binom{M}{2} \\ P(M-2,M) &= \frac{1}{M!}\,\frac{3M-1}{4}\binom{M}{3} \\ P(M-3,M) &= \frac{1}{M!}\binom{M}{2}\binom{M}{4} \end{split}$$ etc. In the complimentary limit of $C=O(1)$ and $M\gg 1$ $$\label{PCM:log} P(C,M) \simeq M^{-1}\,\frac{(\ln M)^{C-1}}{(C-1)!}$$ Every new community starts as a loop, so $C=\mathcal{L}$. This is not applicable to all maps, e.g., for the map shown in Fig. [1](#Fig:random-map-ill){reference-type="ref" reference="Fig:random-map-ill"} we have $C=3$ and $\mathcal{L}=1$. For RRMs, however, the equality $C=\mathcal{L}$ always holds. Thus [\[P12\]](#P12){reference-type="eqref" reference="P12"} imply $$Q(1,M) = M^{-1}, \quad Q(2,M) = M^{-1}H_{M-1}$$ and one can similarly re-express [\[CM-av-sol\]](#CM-av-sol){reference-type="eqref" reference="CM-av-sol"} and [\[PMM3\]](#PMM3){reference-type="eqref" reference="PMM3"}--[\[PCM:log\]](#PCM:log){reference-type="eqref" reference="PCM:log"}. These results are identical to the results for *static* random maps compatible with ranking and satisfying [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"}. To establish an isomorphism between the dynamic RRM model and the static model [\[map:more\]](#map:more){reference-type="eqref" reference="map:more"} we label individuals in the dynamic RRM model in reverse order. Indeed, if we grow the random map to size $M$, we assign rank $M$ to the first individual, rank $M-1$ to the second individual, etc.; the equivalence of the rules is then easy to appreciate. Let us now compute the distribution of the number of followers. The definition of the RRM implies that $F(M)$ is a random variable evolving according to $$\label{FM:eq} F(M+1) = \begin{cases} F(M) & \text{prob} ~~\frac{F(M)+1}{M+1}\\ F(M) + 1 & \text{prob} ~~\frac{M-F(M)}{M+1} \end{cases}$$ Indeed, the number of followers does not change if the newly-introduced individual maps to a follower or itself. This happens with probability $\frac{F(M)+1}{M+1}$. Otherwise, the number of followers increases by one. The stochastic rule [\[FM:eq\]](#FM:eq){reference-type="eqref" reference="FM:eq"} implies that the probability distribution defined by Eq. [\[PFM:def\]](#PFM:def){reference-type="eqref" reference="PFM:def"} satisfies $$\begin{aligned} \label{PFM} M\Pi(F,M) &=& (F+1)\Pi(F,M-1) \nonumber \\ &+& (M-F)\Pi(F-1,M-1)\end{aligned}$$ Multiplying Eq. [\[PFM\]](#PFM){reference-type="eqref" reference="PFM"} by $(M-1)!$ and noticing that $M! \Pi(F,M)$ satisfies recurrence identical to the addition formula for Eulerian numbers [@Euler36; @Euler55; @Knuth] we arrive at the announced result [\[PFM:Euler\]](#PFM:Euler){reference-type="eqref" reference="PFM:Euler"}. Explicit expressions [@Knuth] $$\begin{split} \eulerian{M}{0} &=1 \\ \eulerian{M}{1} &=2^M-M-1 \\ \eulerian{M}{2} &=3^M-(M+1)2^M+\binom{M+1}{2} \\ \eulerian{M}{3} &=4^M-(M+1)3^M+2^M\binom{M+1}{2}-\binom{M+1}{3} \end{split}$$ for Eulerian numbers give $\Pi(F,M)$ for $F=0,1,2,3$. While $\Pi(0,M)=1/M!$ is obvious from the definition of the RRM, a straightforward derivation of $\Pi(F,M)$ for $F=1,2,3$ is laborious. Explicit results for $\Pi(F,M)$ with $F=M-1,M-2,M-3, M-4$ then follow from the mirror symmetry between Eulerian numbers [@Knuth] $$\label{mirror:E} \eulerian{M}{F}=\eulerian{M}{M-1-F}$$ We now derive the announced asymptotic behavior [\[nk:RRM\]](#nk:RRM){reference-type="eqref" reference="nk:RRM"} of the degree distribution. The number of vertices of degree $k\geq 2$ evolves according to stochastic rules $$\label{NkM} N_k(M+1)= \begin{cases} N_k(M)-1 & \mathrm{prob}~~ \frac{N_k(M)}{M+1}\\[2mm] N_k(M)+1 & \mathrm{prob}~~ \frac{N_{k-1}(M)}{M+1}\\[2mm] N_k(M) & \mathrm{prob}~~ 1-\frac{N_{k-1}(M)+N_k(M)}{M+1} \end{cases}$$ Averaging above equation yields the recurrence $$\label{Nkav} (M+1)\langle N_k(M+1)\rangle =M\langle N_k(M)\rangle +\langle N_{k-1}(M)\rangle$$ When $k=2$, this recurrence gives $$\label{N2av} (M+1)\langle N_2(M+1)\rangle =M\langle N_2(M)\rangle + \frac{M-1}{2}$$ where we have used $$\left\langle N_1(M)\right\rangle = \langle F(M)\rangle = \frac{M-1}{2}$$ The initial graph is a loop. Solving [\[N2av\]](#N2av){reference-type="eqref" reference="N2av"} subject to the initial condition $N_2(1)=1$ yields $$\label{n2M} n_2(M) = \frac{1}{4}-\frac{3}{4}M^{-1}+\frac{3}{2}M^{-2}$$ Specializing [\[Nkav\]](#Nkav){reference-type="eqref" reference="Nkav"} to $k=3$ and using [\[n2M\]](#n2M){reference-type="eqref" reference="n2M"} we obtain $$\label{N3av} (M+1)\langle N_3(M+1)\rangle = M\langle N_3(M)\rangle + \frac{M}{4} -\frac{3}{4} +\frac{3}{2}M^{-1}$$ which we solve subject to $N_3(1)=0$ to yield $$\label{n3M} n_3(M) = \frac{1}{8}-\frac{7}{8}M^{-1}+\frac{3+6H_{M-1}}{4M^2}$$ Continuing one deduces exact results for $n_k(M)$ that become more and more cumbersome as $k$ increases. The most important leading and sub-leading terms are simple. In the $M\to\infty$ limit, the recurrence [\[Nkav\]](#Nkav){reference-type="eqref" reference="Nkav"} simplifies to $2n_k=n_{k-1}$ from which we deduce [\[nk:RRM\]](#nk:RRM){reference-type="eqref" reference="nk:RRM"}. A more careful asymptotic analysis of [\[Nkav\]](#Nkav){reference-type="eqref" reference="Nkav"} allows one to extract the sub-leading term: $$\label{nkM} n_k(M) = 2^{-k}-\big(1-2^{-k}\big)M^{-1}+\ldots$$ # Derivation of [\[Lav:linear-asymp\]](#Lav:linear-asymp){reference-type="eqref" reference="Lav:linear-asymp"}--[\[nk:linear\]](#nk:linear){reference-type="eqref" reference="nk:linear"} and [\[Fav-linear\]](#Fav-linear){reference-type="eqref" reference="Fav-linear"}--[\[nk-linear\]](#nk-linear){reference-type="eqref" reference="nk-linear"} {#ap:derive} To compute the sum in Eq. [\[Fav:sum\]](#Fav:sum){reference-type="eqref" reference="Fav:sum"} we first simplify $\Psi_j(M)$ defined by [\[Psi:def\]](#Psi:def){reference-type="eqref" reference="Psi:def"}. Taking the logarithm of the product in [\[Psi:def\]](#Psi:def){reference-type="eqref" reference="Psi:def"} we obtain the sum $$\ln \Psi_j(M) = \sum_{k=1}^j \ln\left[1-\frac{2k}{(M-j+k)(M-j+k+1)}\right]$$ where $k=j-i+1$. Replacing summation on the right-hand side by integration (this is justifiable since we are interested in the $M\to\infty$ behavior) we arrive at $$\ln \Psi_j(M) \simeq \int_0^j dk\,\ln\!\left[1-\frac{2k}{(M-j+k)^2}\right]$$ in the leading order. Expanding the logarithm and keeping the leading term we obtain $$\frac{\ln \Psi_j(M)}{2} \simeq -\int_0^j \frac{k\,dk}{(M-j+k)^2} = \frac{j}{M} + \ln\!\left(1-\frac{j}{M}\right)$$ Plugging this into [\[Fav:sum\]](#Fav:sum){reference-type="eqref" reference="Fav:sum"} and replacing summation over $j$ by integration over $x=j/M$ we obtain $$\label{Lav:int} \langle F\rangle \simeq M\int_0^1 dx\, (1-x)^2 e^{2x}$$ Similarly to $\langle F\rangle = \langle N_1\rangle$ one finds $$\langle N_2\rangle = \sum_{j=1}^M \sum_{i=1}^j \frac{2(j-i+1)\,\Psi_j(M)}{(M-i+1)(M-i+2)-2(j-i+1)}$$ Replacing summations by integrations one gets $$\label{n2:linear} n_2 = \int_0^1 dx\, (1-x)^2 e^{2x} J(x)$$ with $J(x)$ given by [\[J:def\]](#J:def){reference-type="eqref" reference="J:def"}. Similarly one derives a general formula for $\langle N_k\rangle$ involving $k$ sums. Summations can be replaced by integrations when $M\to\infty$. This leads to the announced Eq. [\[nk:linear\]](#nk:linear){reference-type="eqref" reference="nk:linear"}. One can explicitly compute $n_k$ for small $k$. The integral in [\[Lav:int\]](#Lav:int){reference-type="eqref" reference="Lav:int"} is $(e^2-5)/4$ leading to the announced result [\[Lav:linear-asymp\]](#Lav:linear-asymp){reference-type="eqref" reference="Lav:linear-asymp"}. Computing the integral in [\[n2:linear\]](#n2:linear){reference-type="eqref" reference="n2:linear"} we obtain $$\label{n2:linear-sol} n_2 = -1+ \tfrac{1}{2}e^2 C, \quad C = \gamma + \ln 2 - 1 + \int_{2}^\infty \frac{dt}{t}\,e^{-t}$$ where we have introduced the quantity $C$ that appears in $n_k$ probably for all $k\geq 2$. Indeed, computing the integrals in Eq. [\[nk:linear\]](#nk:linear){reference-type="eqref" reference="nk:linear"} one gets $$\label{n345} \begin{split} &n_3 = 2e^2S_3 -\tfrac{5e^2+1}{4}-e^2 C\\ &n_4 = 4e^2(S_4-S_3) - \tfrac{e^2+9}{12} - \tfrac{1}{2}\,e^2 C \\ &n_5 = 2e^2(4S_5 - 4S_4 - S_3) + \tfrac{13e^2-11}{12} + \tfrac{5}{6}\,e^2 C \end{split}$$ etc. where we shortly write $$S_p \equiv \sum_{j\geq 0} \frac{1}{(j+1)^p}\,\frac{(-2)^j}{j!}$$ Here are a few numerical values: $$\begin{split} &n_1 = 0.59726402473266\ldots \\ &n_2 = 0.17952742450736\ldots \\ &n_3 = 0.08878616080733\ldots \\ &n_4 = 0.05031763625502\ldots \\ &n_5 = 0.03034754225441\ldots \end{split}$$ To establish [\[Fav-linear\]](#Fav-linear){reference-type="eqref" reference="Fav-linear"} we use [\[Fav:sum\]](#Fav:sum){reference-type="eqref" reference="Fav:sum"} with $$\label{Psi:2} \Psi_j(M) = \prod_{i=1}^j \left[1-\frac{1}{(j-i+1)H_{M-i+1}}\right]$$ Taking the logarithm we obtain the sum $$\ln \Psi_j(M) = \sum_{k=1}^j \ln\left[1-\frac{1}{kH_{M-j+k}}\right]$$ where $k=j-i+1$. The asymptotic behavior is $$\label{P-HH} \ln \Psi_j(M) \simeq -\frac{H_j}{H_M}$$ Combining [\[Fav:sum\]](#Fav:sum){reference-type="eqref" reference="Fav:sum"} and [\[P-HH\]](#P-HH){reference-type="eqref" reference="P-HH"} we obtain $$\begin{aligned} \langle N_1\rangle &=& \sum_{j=1}^M \Psi_j(M) \\ & \simeq & \int_1^M dj\,\exp\!\left[-\frac{\ln j}{\ln M}\right] \simeq \int_1^M dj\,\left[1-\frac{\ln j}{\ln M}\right] \end{aligned}$$ leading to [\[Fav-linear\]](#Fav-linear){reference-type="eqref" reference="Fav-linear"}. Similarly we find $$\begin{aligned} \langle N_2\rangle &=& \sum_{j=1}^M \sum_{i=1}^j \frac{\Psi_j(M)}{(j-i+1)H_{M-i+1}-1} \\ &\simeq& \sum_{j=1}^M \frac{\Psi_j(M) H_j}{H_M} \simeq \int_1^M dj\,\left[1-\frac{\ln j}{\ln M}\right] \frac{\ln j}{\ln M}\end{aligned}$$ leading to $n_2\simeq (\ln M)^{-1}$. A general formula for $\langle N_k\rangle$ involves $k$ sums. Simplifying it as above and replacing summations by integrations one establishes [\[nk-linear\]](#nk-linear){reference-type="eqref" reference="nk-linear"}.

arxiv_math

--- abstract: | We develop a new approach to the study of spectral asymmetry. Working with the operator $\operatorname{curl}:=*\mathrm{d}$ on a connected oriented closed Riemannian 3-manifold, we construct, by means of microlocal analysis, the asymmetry operator --- a scalar pseudodifferential operator of order $-3$. The latter is completely determined by the Riemannian manifold and its orientation, and encodes information about spectral asymmetry. We argue that the asymmetry operator generalises and contains the classical eta invariant traditionally associated with the asymmetry of the spectrum, which can be recovered by computing its regularised operator trace. Remarkably, the whole construction is direct and explicit, and does not involve analytic continuation or algebraic topology.   **Keywords:** curl, Maxwell's equations, spectral asymmetry, eta invariant, pseudodifferential projections.   **2020 MSC classes:** primary 58J50; secondary 35P20, 35Q61, 47B93, 47F99; 58J28, 58J40. author: - "Matteo Capoferri[^1]" - "Dmitri Vassiliev[^2]" date: 5 September 2023 title: | Beyond the Hodge Theorem:\ curl and asymmetric pseudodifferential projections --- # Main results {#Main results} Let $(M,g)$ be a connected closed Riemannian manifold of dimension $d\ge2$ and let $\Omega^k(M)$ be the Hilbert space of real-valued $k$-forms, $1\le k\le d-1$. Hodge's Theorem [@jost Corollary 3.4.2] tells us that $\Omega^k(M)$ decomposes into a direct sum of three orthogonal closed subspaces $$\label{Hodge decompostion} \Omega^k(M) = \mathrm{d}\Omega^{k-1}(M) \oplus \delta\Omega^{k+1}(M) \oplus \mathcal{H}^k(M),$$ where $\mathrm{d}\Omega^{k-1}(M)$, $\delta\Omega^{k+1}(M)$ and $\mathcal{H}^k(M)$ are the Hilbert subspaces of exact, coexact and harmonic $k$-forms, respectively. The overarching idea of our paper is that in the special case when $d=3$, $k=1$ and $M$ is oriented the space $\delta\Omega^{k+1}(M)$ admits a further decomposition into a distinguished pair of orthogonal Hilbert subspaces $\delta\Omega_\pm^{k+1}(M)$. Moreover, this further decomposition is effectively described in terms of pseudodifferential projections for which the symbols can be written down explicitly in terms of curvature and its covariant derivatives. Remarkably, this leads to a new approach to the study of *spectral asymmetry*. The study of spectral asymmetry, that is, the difference in the distribution of positive and negative eigenvalues of (pseudo)differential operators, has a long and noble history, initiated by the seminal series of papers by Atiyah, Patodi and Singer [@asymm1; @asymm2; @asymm3; @asymm4]. In a nutshell, the classical approach goes as follows. One considers the *eta function* of the operator at hand --- say, curl or Dirac --- defined as $$\label{eta function intro} \eta(s):=\sum_{\lambda_k\ne 0}\operatorname{sgn}(\lambda_k) |\lambda_k|^{-s}, \qquad s\in \mathbb{C},$$ where $\lambda_k$ are the nonzero eigenvalues of the operator. The complex function $\eta(s)$ can be easily shown to be holomorphic for $\operatorname{Re} s>d$; one then tries to give a meaning to the quantity $\eta(0)$, called *eta invariant*, by examining the meromorphic extension of [\[eta function intro\]](#eta function intro){reference-type="eqref" reference="eta function intro"} for $\operatorname{Re}s<d$ and showing that there is no pole at $s=0$. The motivation for considering $\eta(0)$ as a measure of spectral asymmetry is that, when the operator in question is simply a Hermitian matrix, $\eta(0)$ is precisely the number of positive eigenvalues minus the number of negative eigenvalues. Classically, it has been shown that $\eta(0)$ is a geometric invariant which can be computed resorting to complex analysis and algebraic topology.   Perhaps surprisingly, there is almost no literature on the spectrum of curl on a 3-manifold with or without boundary. Indeed, if one looks carefully, the overwhelming majority of existing papers deal, effectively, with $\operatorname{curl}^2$ as opposed to $\operatorname{curl}$ itself. Many authors [@safarov_curl; @birman_curl; @birman_curl2; @gureev; @filonov_curl1; @filonov_curl2] have studied Maxwell's equations on domains in $\mathbb{R}^3$ subject to appropriate boundary conditions, which leads to the analysis of spectral problems of the form $$\label{Maxwell resonator} \begin{pmatrix} 0&i\operatorname{curl}\\ -i\operatorname{curl}&0 \end{pmatrix} \begin{pmatrix} E\\ B \end{pmatrix} = \lambda \begin{pmatrix} E\\ B \end{pmatrix}.$$ One would think that the spectral problem [\[Maxwell resonator\]](#Maxwell resonator){reference-type="eqref" reference="Maxwell resonator"} reduces to the spectral problem for $\operatorname{curl}$ by means of a unitary transformation $$\label{unitary transformation} \begin{pmatrix} E\\ B \end{pmatrix} \mapsto \begin{pmatrix} v_+\\ v_- \end{pmatrix} := \frac{1}{\sqrt{2}} \begin{pmatrix} -i&1\\ i&1 \end{pmatrix} \begin{pmatrix} E\\ B \end{pmatrix}$$ but this argument fails because physically meaningful boundary conditions do not agree with [\[unitary transformation\]](#unitary transformation){reference-type="eqref" reference="unitary transformation"}. The presence or absence of a boundary is a major issue in the subject. To the best of our knowledge, the only paper dedicated to the study of the spectrum of $\operatorname{curl}$ on a closed oriented Riemannian manifold is [@baer_curl]. In [@baer_curl] the author examined the qualitative properties of the spectrum of $\operatorname{curl}$ depending on the dimension $d$ and established one-term asymptotic formulae (Weyl law) for the (global) eigenvalue counting functions, alongside a number of explicit examples. Although not very close in spirit to the aims of the current paper, we also refer the reader to [@peralta1; @peralta2] examining the first eigenvalue of $\operatorname{curl}$ on domains in $\mathbb{R}^3$.   Let us now formulate our problem and state our main results, postponing proofs and detailed arguments until later sections. In the rest of this paper, unless otherwise stated, the dimension $d$ is 3 and the manifold is orientable and oriented. Let $*$ be the Hodge dual (see Appendix [10](#Exterior calculus){reference-type="ref" reference="Exterior calculus"} for our sign convention), and define $\operatorname{curl}$ to be the differential expression $$\label{curl differential experssion} \operatorname{curl}:=\ast \mathrm{d}$$ acting in $\Omega^1(M)$. We will show in Section [2](#The operator curl){reference-type="ref" reference="The operator curl"} --- more precisely, in Theorem [Theorem 9](#theorem properties of curl){reference-type="ref" reference="theorem properties of curl"} --- that the differential expression [\[curl differential experssion\]](#curl differential experssion){reference-type="eqref" reference="curl differential experssion"} gives rise to a self-adjoint operator (denoted by the same symbol) in the space of coexact 1-forms $\delta\Omega^2(M)$. The latter has discrete spectrum, accumulating to both $+\infty$ and $-\infty$, and not necessarily symmetric about zero. As a way of illustrating that spectral asymmetry actually occurs, in Appendices [11](#The spectrum of the Laplacian on a Berger sphere){reference-type="ref" reference="The spectrum of the Laplacian on a Berger sphere"} and [12](#The spectrum of curl on a Berger sphere){reference-type="ref" reference="The spectrum of curl on a Berger sphere"} we write down explicitly the spectrum of $\operatorname{curl}$ on a Berger sphere and compute the corresponding (classical) eta invariant.   Note that the definition of $\operatorname{curl}$ does not involve the concept of connection. In this respect, $\operatorname{curl}$ is one of the most fundamental operators of mathematical physics, like the Laplace--Beltrami operator. Furthermore, $\operatorname{curl}$ lies at the heart of (homogeneous) Maxwell's equations. Namely, seeking solutions harmonic in time reduces Maxwell's equations on $M\times\mathbb{R}$ to the spectral problem for $\operatorname{curl}$. And the solution of the Cauchy problem for Maxwell's equations can be expressed in terms of eigenvalues and eigenforms of $\operatorname{curl}$, see formula [\[solution to Maxwell as a series\]](#solution to Maxwell as a series){reference-type="eqref" reference="solution to Maxwell as a series"}. Finally, it is worth emphasising that the sign of the eigenvalues of curl has a physical meaning: it coincides with the sign of electromagnetic chirality of time-harmonic polarised solutions of Maxwell's equations generated by the corresponding $\operatorname{curl}$ eigenpair (see [\[harmonic solution 1\]](#harmonic solution 1){reference-type="eqref" reference="harmonic solution 1"}, [\[harmonic solution 2\]](#harmonic solution 2){reference-type="eqref" reference="harmonic solution 2"}). We refer the reader to Appendix [13](#Maxwell's equations and electromagnetic chirality){reference-type="ref" reference="Maxwell's equations and electromagnetic chirality"} for further details and a self-contained exposition of this fact. These are but few of the reasons why a more detailed examination of the spectrum of curl --- and in particular of its spectral asymmetry --- is very timely, if not overdue.   As a first step, we introduce the following definitions. **Definition 1**. The operator $P_0$ is defined as the extension to $\Omega^1(M)$ of the identity operator on $\mathrm{d}\Omega^0(M)$. Here the extension is specified by the requirement that the orthogonal complement of $\mathrm{d}\Omega^0(M)$ in $\Omega^1(M)$ maps to zero. **Definition 2**. Let Let $\theta:\mathbb{R}\to\mathbb{R}$, $$\label{Heaviside function} \theta(z):= \begin{cases} 0&\text{if}\quad z\le0, \\ 1&\text{if}\quad z>0 \end{cases}$$ be the Heaviside function. The operators $P_\pm$ are defined as the extensions to $\Omega^1(M)$ of the operators $\theta(\pm\operatorname{curl})$ on $\delta\Omega^2(M)$. Here the extensions are specified by the requirement that the orthogonal complement of $\delta\Omega^2(M)$ in $\Omega^1(M)$ maps to zero. The operator $P_0$ is the orthogonal projection onto the kernel of $\operatorname{curl}$ with harmonic 1-forms removed, whereas the operators $P_\pm$ are the positive (+) and negative (-) spectral projections of $\operatorname{curl}$. Further on we use the notation $$\label{norm of xi} \|\xi\| %:=(g^{\mu\nu}(x)\,\xi_\mu\xi_\nu)^{1/2} :=\sqrt{g^{\mu\nu}(x)\,\xi_\mu\xi_\nu}$$ and we denote by $|\,\cdot\,|$ the Euclidean norm of vectors. Throughout this paper we use Greek letters for tensor indices. **Theorem 3**. (a) *The operators $P_0$, $P_+$ and $P_-$ are pseudodifferential of order zero.* (b) *Their principal symbols read $$\label{main theorem 1 equation 1} [(P_0)_\mathrm{prin}]_\alpha{}^\beta(x,\xi) = \|\xi\|^{-2}\,\xi_\alpha \,g^{\beta\gamma}(x)\,\xi_\gamma\,,$$ $$\label{main theorem 1 equation 2} [(P_\pm)_\mathrm{prin}]_\alpha{}^\beta(x,\xi) = \frac12 \left[ \delta_\alpha{}^\beta - [(P_0)_\mathrm{prin}]_\alpha{}^\beta(x,\xi) \pm i\, \|\xi\|^{-1}\, E_\alpha{}^{\gamma\beta}(x)\, \xi_\gamma \right],$$ where $$\label{main theorem 1 equation 3} %\label{Levi-Civita tensor} E_{\alpha\beta\gamma}(x):=\rho(x)\,\varepsilon_{\alpha\beta\gamma}\,,$$ $\rho$ is the Riemannian density and $\varepsilon$ is the totally antisymmetric symbol, $\varepsilon_{123}:=+1$.* (c) *Their subprincipal symbols, defined in accordance with [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"}, are zero: $$\label{main theorem 1 equation 4} (P_0)_\mathrm{sub}=0, \qquad (P_\pm)_\mathrm{sub}=0.$$* Euclidean versions of formulae [\[main theorem 1 equation 1\]](#main theorem 1 equation 1){reference-type="eqref" reference="main theorem 1 equation 1"} and [\[main theorem 1 equation 2\]](#main theorem 1 equation 2){reference-type="eqref" reference="main theorem 1 equation 2"} appeared in [@lerner]. On the other hand, the notion of subprincipal symbol for operators acting on 1-forms was never previously defined in full generality. We address this matter in detail, including a discussion of known results, in Section [3](#Pseudodifferential operators acting on 1-forms){reference-type="ref" reference="Pseudodifferential operators acting on 1-forms"}.   Remarkably, the pseudodifferential projections $P_0$ and $P_\pm$ can be constructed explicitly. In Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"} we will provide an algorithm for the calculation of their full symbols. The latter proposition and Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"} give a detailed description of the structure of the projection operators $P_0$ and $P_\pm$, which is our first main result.   The projections $P_+$ and $P_-$ are the key ingredients of our new approach to spectral asymmetry. One starts by observing that, at a formal level, we have $$\label{at a formal level} \#\{\text{positive eigenvalues}\} \,-\, \#\{\text{negative eigenvalues}\} =\operatorname{Tr}(P_+-P_-).$$ Here and further on $\operatorname{Tr}$ stands for operator trace, whereas $\operatorname{tr}$ stands for matrix trace. Unfortunately, one is immediately presented with two major issues: (i) the LHS of [\[at a formal level\]](#at a formal level){reference-type="eqref" reference="at a formal level"} is a difference of two infinities and (ii) the (matrix) operator $P_+-P_-$ in the RHS of [\[at a formal level\]](#at a formal level){reference-type="eqref" reference="at a formal level"} is not of trace class. Our strategy is to assign a meaning to the RHS of [\[at a formal level\]](#at a formal level){reference-type="eqref" reference="at a formal level"} by decomposing the procedure of taking operator trace into two steps: 1) take the pointwise matrix trace of the matrix pseudodifferential operator $P_+-P_-\,$, 2) compute the operator trace of the resulting scalar pseudodifferential operator. Step 1 above defines, up to some additional technical subtleties, a scalar pseudodifferential operator which we call the *asymmetry operator* and denote by $A$ --- see Definition [Definition 40](#definition asymmetry operator){reference-type="ref" reference="definition asymmetry operator"}. The operator $A$ is *prima facie* a pseudodifferential operator of order 0. However, taking the matrix trace brings about unexpected cancellations, so that the operator $A$ turns out, in fact, to be of order $-3$. This is our second main result. **Theorem 4**. *The asymmetry operator $A$ is a pseudodifferential operator of order $-3$ and its principal symbol reads $$\label{main theorem 2 equation 1} A_\mathrm{prin}(x,\xi)=-\frac1{2\|\xi\|^5}E^{\alpha\beta \gamma}(x)\, \nabla_\alpha \operatorname{Ric}_{\beta}{}^\rho(x)\, \xi_\gamma\xi_\rho\,,$$ where $\operatorname{Ric}$ is the Ricci tensor[^3].* *Remark 5*. Note that formula [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"} can be equivalently rewritten as $$\label{main theorem 2 equation 1 trace-free} A_\mathrm{prin}(x,\xi)=-\frac1{2\|\xi\|^5}E^{\alpha\beta \gamma}(x)\, \nabla_\alpha \left( \operatorname{Ric}_{\beta}{}^\rho(x) - \frac{1}{3}\, \delta_{\beta}{}^\rho\,\operatorname{Sc}(x) \right) \xi_\gamma\xi_\rho\,,$$ i.e. one can replace the Ricci tensor with its trace-free part. In other words, the principal symbol of the asymmetry operator does not feel scalar curvature. Now, it is well known [@shubin Corollary 27.1 and §12.1] that a sufficient condition for an operator to be of trace class and to have continuous integral kernel is for its order to be strictly less than $-d$, where $d$ is the dimension of the manifold. In the case at hand, the operator $A$ has order $-3$ and the dimension of the manifold is 3, so we are looking at a borderline situation. One would hope that with a mild regularisation one could cross the finish line and be able to define a notion of regularised trace. Indeed, we will establish in Theorem [Theorem 49](#singularity theorem){reference-type="ref" reference="singularity theorem"} that the integral kernel $\mathfrak{a}(x,y)$ of $A$ decomposes as $\mathfrak{a}(x,y)=\mathfrak{a}_d(x,y)+\mathfrak{a}_c(x,y)$, where $\mathfrak{a}_c$ is continuous, whereas $\mathfrak{a}_d$ is possibly discontinuous but bounded, and integrates to zero over small spheres centred at either $x$ or $y$. This leads us to our third main result. **Theorem 6**. *The integral kernel $\mathfrak{a}(x,y)$ of $A$ is a bounded function, smooth outside the diagonal. Furthermore, for any $x\in M$ the limit $$\psi_{\operatorname{curl}}^\mathrm{loc}(x) := \lim_{r\to0^+} \frac{1}{4\pi r^2} \int_{\mathbb{S}_r(x)} \mathfrak{a}(x,y)\,\mathrm{d}S_y$$ exists and defines a continuous scalar function $\,\psi_{\operatorname{curl}}^\mathrm{loc}:M\to\mathbb{R}\,$. Here $\mathbb{S}_r(x)=\{y\in M|\operatorname{dist}(x,y)=r\}$ is the sphere of radius $r$ centred at $x$ and $\mathrm{d}S_y$ is the surface area element on this sphere.* We call the function $\psi_{\operatorname{curl}}^\mathrm{loc}(x)$ and the number $\,\psi_{\operatorname{curl}}:=\int_M\psi_{\operatorname{curl}}^\mathrm{loc}(x) \,\rho(x)\,\mathrm{d}x\,$ the local and global regularised trace of $A$, see Definitions [Definition 55](#local regularised trace of the asymmetry operator){reference-type="ref" reference="local regularised trace of the asymmetry operator"} and [Definition 56](#global regularised trace of the asymmetry operator){reference-type="ref" reference="global regularised trace of the asymmetry operator"}. These two quantities are geometric invariants, in that they are determined by the Riemannian 3-manifold and its orientation. In particular, the global regularised trace is a real number measuring the asymmetry of the spectrum of $\operatorname{curl}$. Although this matter will be comprehensively addressed elsewhere, we conjecture that the latter is precisely the classical eta invariant for $\operatorname{curl}$, see Section [8](#A conjecture){reference-type="ref" reference="A conjecture"} for further details. *Remark 7*. Pseudodifferential projections are very natural tools in the study of the spectral properties of systems. At first glance, one may be tempted to think they are unnecessarily complicated, especially for operators acting on trivial bundles, in that one could try and diagonalise the operator along the lines of [@diagonalisation] instead, thus reducing the problem at hand to the examination a collection of scalar operators. However, after examining the matter more carefully, diagonalisation turns out to be topologically obstructed in numerous examples of physical relevance. In particular, it was shown in [@obstructions Section 3.3] that for the case of the operator $\operatorname{curl}$ one cannot even diagonalise the principal symbol $\operatorname{curl}_\mathrm{prin}$ [\[principal symbol curl\]](#principal symbol curl){reference-type="eqref" reference="principal symbol curl"} in the whole cotangent fibre at given point of $M$, let alone the operator itself.   All in all, our results offer a new approach to the subject of spectral asymmetry, one that possesses the following elements of novelty. 1. We characterise asymmetry in terms of a pseudodifferential operator of negative order, as opposed to a single number. Here the fact that the order is negative is key as it opens the way to defining the regularised trace. 2. Our approach is not operator-specific. It is versatile and can be deployed in a variety of different scenarios. For example, our strategy can be applied to the Dirac operator. 3. Our technique is explicit and based on direct computations. Moreover, it rigorously implements the intuitive notion of spectral asymmetry as difference between numbers of positive and negative eigenvalues, without relying on "black boxes" such as analytic continuation. *Remark 8*. Throughout this paper, we focus our analysis on the space of real-valued 1-forms $\Omega^1(M)$. One could perform a completely analogous analysis working in the space of real-valued 2-forms $\Omega^2(M)$; indeed, on a 3-manifold $\Omega^1(M)$ and $\Omega^2(M)$ are isomorphic by Hodge duality. ## Structure of the paper {#structure-of-the-paper .unnumbered} Our paper is structured as follows. In Section [2](#The operator curl){reference-type="ref" reference="The operator curl"} we put $\operatorname{curl}$ on a rigorous operator-theoretic footing, and state and prove its basic properties. In doing so, we rely on the auxiliary operator $\operatorname{curl}_E$, called *extended curl*, which can be viewed as an elliptic "extension\" of $\operatorname{curl}$. Although most of the results from this section are known at least in some form, detailed proofs, often hard to find in the literature, are provided here in full and in a self-contained fashion, for the convenience of the reader and future reference. In Section [3](#Pseudodifferential operators acting on 1-forms){reference-type="ref" reference="Pseudodifferential operators acting on 1-forms"} we temporarily move away from dimension 3 and develop an invariant calculus for pseudodifferential operators acting on 1-forms over a Riemannian closed $d$-manifold. In particular we define a notion of subprincipal symbol for these operators, and write down the composition formula. In Section [4](#Trace of pseudodifferential operators acting on 1-forms){reference-type="ref" reference="Trace of pseudodifferential operators acting on 1-forms"} we prepare the ground for the formulation of our main results and examine the issue of taking the trace of pseudodifferential operators on 1-forms. The highlight of this section is the notion of matrix trace $\operatorname{tr}_\epsilon\,$, defined by [\[Q acting on 1-forms 6\]](#Q acting on 1-forms 6){reference-type="eqref" reference="Q acting on 1-forms 6"} and [\[Q acting on 1-forms 7\]](#Q acting on 1-forms 7){reference-type="eqref" reference="Q acting on 1-forms 7"}, which addresses the issue of taking the pointwise matrix trace of integral operators whose integral kernel is a two-point tensor. In Section [5](#The operators P0 and Ppm){reference-type="ref" reference="The operators P0 and Ppm"} we study the pseudodifferential projections $P_0$, $P_+$ and $P_-$, and provide an explicit algorithm for the construction of their full symbols. This section contains the proof of Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"}. In Section [6](#The asymmetry operator){reference-type="ref" reference="The asymmetry operator"} we define the asymmetry operator $A$ (subsection [6.1](#Definition of the asymmetry operator subsection){reference-type="ref" reference="Definition of the asymmetry operator subsection"}), prove that it is a pseudodifferential operator of order $-3$ (subsection [6.2](#The order of the asymmetry operator subsection){reference-type="ref" reference="The order of the asymmetry operator subsection"}), and compute its principal symbol $A_\mathrm{prin}$ (subsection [6.3](#The principal symbol of the asymmetry operator){reference-type="ref" reference="The principal symbol of the asymmetry operator"}). This section establishes Theorem [Theorem 4](#main theorem 2){reference-type="ref" reference="main theorem 2"}. Section [7](#The regularised trace of $A$){reference-type="ref" reference="The regularised trace of $A$"} is the centrepiece of our paper. After examining the singular structure of the integral kernel $\mathfrak{a}(x,y)$ of $A$ near the diagonal (subsection [7.1](#Structure of Aprin near the diagonal){reference-type="ref" reference="Structure of Aprin near the diagonal"}), we devise a regularisation procedure which allows one to define the local and global trace of the asymmetry operator, geometric invariants measuring spectral asymmetry (subsection [7.2](#Local and global trace){reference-type="ref" reference="Local and global trace"}). This yields Theorem [Theorem 6](#main theorem 3){reference-type="ref" reference="main theorem 3"}. In Section [8](#A conjecture){reference-type="ref" reference="A conjecture"} we formulate conjectures relating our geometric invariants to known invariants (local and global eta invariants), and provide some heuristic justification. A rigorous proof of these conjectures is a serious endeavour, beyond the scope of this paper, and will be addressed in full elsewhere. In Section [9](#Challenges in higher dimensions){reference-type="ref" reference="Challenges in higher dimensions"} we briefly elaborate on the challenges involved in extending our results to dimensions higher than 3, and why doing so could be interesting. The paper is complemented by a list of notation given at the end of this section, and by six appendices. In Appendix [10](#Exterior calculus){reference-type="ref" reference="Exterior calculus"} we summarise our notation and conventions on exterior calculus. Appendices [11](#The spectrum of the Laplacian on a Berger sphere){reference-type="ref" reference="The spectrum of the Laplacian on a Berger sphere"} and [12](#The spectrum of curl on a Berger sphere){reference-type="ref" reference="The spectrum of curl on a Berger sphere"} are concerned with spectral theory on Berger 3-spheres: in Appendix [11](#The spectrum of the Laplacian on a Berger sphere){reference-type="ref" reference="The spectrum of the Laplacian on a Berger sphere"} we recall the definition of a Berger sphere and list the eigenvalues of the Laplace--Beltrami operator, whereas in Appendix [12](#The spectrum of curl on a Berger sphere){reference-type="ref" reference="The spectrum of curl on a Berger sphere"} we list the eigenvalues of $\operatorname{curl}$ on a Berger sphere as an illustration of spectral asymmetry, and compute the eta invariant by explicitly examining the eta function for curl. In Appendix [13](#Maxwell's equations and electromagnetic chirality){reference-type="ref" reference="Maxwell's equations and electromagnetic chirality"} we examine the relation between the spectral problem for curl and solutions of Maxwell's equations, and discuss the physical meaning of the sign of the eigenvalues of $\operatorname{curl}$. In Appendix [14](#appendix parallel transport){reference-type="ref" reference="appendix parallel transport"} we derive series expansions for the parallel transport maps. Finally, in Appendix [15](#An alternative derivation of formula){reference-type="ref" reference="An alternative derivation of formula"} we independently verify formula [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"}. ## Notation {#notation .unnumbered} --------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- **Symbol** **Description** $\sim$ Asymptotic expansion $\ast$ Hodge dual [\[definition of Hodge star\]](#definition of Hodge star){reference-type="eqref" reference="definition of Hodge star"} $\|\,\cdot\,\|$ Riemannian norm [\[norm of xi\]](#norm of xi){reference-type="eqref" reference="norm of xi"} $|\,\cdot\,|$ Euclidean norm $\langle\,\cdot\,\rangle$ Japanese bracket [\[Japanese bracket\]](#Japanese bracket){reference-type="eqref" reference="Japanese bracket"} $A$ Asymmetry operator, Definition [Definition 40](#definition asymmetry operator){reference-type="ref" reference="definition asymmetry operator"} $A_\mathrm{diag}$, $A_\mathrm{pt}$ Local decomposition of $A$ as per [\[A diag\]](#A diag){reference-type="eqref" reference="A diag"} and [\[A pt\]](#A pt){reference-type="eqref" reference="A pt"} $\mathfrak{a}(x,y)$ Integral kernel of the asymmetry operator $A$ $\operatorname{curl}$ Curl, as a differential expression [\[curl differential experssion\]](#curl differential experssion){reference-type="eqref" reference="curl differential experssion"} and as an operator [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"} $\operatorname{curl}_E$ Extended curl, Definition [Definition 11](#definition extended curl){reference-type="ref" reference="definition extended curl"} $\operatorname{curl}_{E,\mathrm{d}},\operatorname{curl}_{E,\delta},\operatorname{curl}_{E,\mathcal{H}}$ Orthogonal summands of $\operatorname{curl}_E$, Lemma [Proposition 14](#lemma invariant subspaces extended curl){reference-type="ref" reference="lemma invariant subspaces extended curl"} $d$ Dimension of the manifold $M$, $d\ge2$ $\mathrm{d}$ Exterior derivative, Appendix [10](#Exterior calculus){reference-type="ref" reference="Exterior calculus"} $\delta$ Codifferential, Appendix [10](#Exterior calculus){reference-type="ref" reference="Exterior calculus"} $\Delta:=-\delta\mathrm{d}$ (Nonpositive) Laplace--Beltrami operator $\boldsymbol{\Delta}:=-(\mathrm{d}\delta+\delta\mathrm{d})$ (Nonpositive) Hodge Laplacian $\operatorname{dist}$ Geodesic distance $e_j{}^\alpha(x)$, $e^k{}_\beta(x)$ Framing and dual framing [\[dual framing\]](#dual framing){reference-type="eqref" reference="dual framing"} $\widetilde e_j{}^\alpha(x)$, $\widetilde e^k{}_\beta(x)$ Levi-Civita framing and dual Levi-Civita framing, Definition [Definition 22](#definition Levi-Civita framing){reference-type="ref" reference="definition Levi-Civita framing"} $\varepsilon_{\alpha\beta\gamma}$ Totally antisymmetric symbol, $\varepsilon_{123}=+1$ $E_{\alpha\beta\gamma}$ Totally antisymmetric tensor [\[main theorem 1 equation 3\]](#main theorem 1 equation 3){reference-type="eqref" reference="main theorem 1 equation 3"} $f_{x^\alpha}$ Partial derivative of $f$ with respect to $x^\alpha$ $g$ Riemannian metric $G$ Einstein tensor $\gamma(x,y;\tau)$ Geodesic connecting $x$ to $y$, with $\gamma(x,y;0)=x$ and $\gamma(x,y;1)=y$ $\Gamma^\alpha{}_{\beta\gamma}$ Christoffel symbols $\eta_Q^\mathrm{loc}(x;s)$ Local eta function of the operator $Q$ (Definition [Definition 57](#definition of local eta function){reference-type="ref" reference="definition of local eta function"} for $Q=\operatorname{curl}$) $\eta_Q(s)$ Eta function of the operator $Q$ $H^s(M)$ Generalisation of the usual Sobolev spaces $H^s$ to differential forms $\mathcal{H}^k(M)$ Harmonic $k$-forms over $M$ $(\lambda_j, u_j)$, $j=\pm1, \pm2, \ldots$ Eigensystem for $\operatorname{curl}$ $\theta$ Heaviside theta function [\[Heaviside function\]](#Heaviside function){reference-type="eqref" reference="Heaviside function"} $\operatorname{I}$ Identity matrix $\operatorname{Id}$ Identity operator $(\mu_j, f_j)$, $j=0,1,2,\ldots$ Eigensystem for $-\Delta$ $M$ Connected oriented closed manifold $\operatorname{mod} \ \Psi^{-\infty}$ Modulo an integral operator with infinitely smooth kernel $P_0$, $P_\pm$ Definitions [Definition 1](#definition of the operator P0){reference-type="ref" reference="definition of the operator P0"} and [Definition 2](#definition of the operators Ppm){reference-type="ref" reference="definition of the operators Ppm"} $p_\pm(x,y)$ Full symbol of $P_\pm$ $Q_\mathrm{prin}$ Principal symbol of the pseudodifferential operator $Q$ $Q_{\mathrm{prin},s}$ Principal symbol of $Q$, a pseudodifferential operator of order $-s$ $Q_{\mathrm{sub}}$ Subprincipal symbol of $Q$, for operators on 1-forms see Definition [Definition 19](#definition subprincipal symbol operators 1 forms){reference-type="ref" reference="definition subprincipal symbol operators 1 forms"} $\operatorname{Riem}$, $\operatorname{Ric}$, $\operatorname{Sc}$ Riemann curvature tensor, Ricci tensor and scalar curvature $\rho(x)$ Riemannian density $\mathbb{S}_r(x)$ Geodesic sphere of radius $r$ centred at $x\in M$ $\operatorname{tr}$ Matrix trace [\[Q acting on column of scalars 3\]](#Q acting on column of scalars 3){reference-type="eqref" reference="Q acting on column of scalars 3"} $\operatorname{tr}_\epsilon$ Matrix trace with $\epsilon$-cut-off, Definition [Definition 30](#definition of matrix trace of a pdo acting on 1-forms){reference-type="ref" reference="definition of matrix trace of a pdo acting on 1-forms"} $\operatorname{Tr}$ Operator trace [\[Q acting on column of scalars 2\]](#Q acting on column of scalars 2){reference-type="eqref" reference="Q acting on column of scalars 2"} $TM$, $T^*M$ Tangent and cotangent bundle $\Omega^k(M)$ Differential $k$-forms over $M$ $\psi_{\operatorname{curl}}^\mathrm{loc}(x)$ Regularised local trace of $A$, Definition [Definition 55](#local regularised trace of the asymmetry operator){reference-type="ref" reference="local regularised trace of the asymmetry operator"} $\psi_{\operatorname{curl}}$ Regularised global trace of $A$, Definition [Definition 56](#global regularised trace of the asymmetry operator){reference-type="ref" reference="global regularised trace of the asymmetry operator"} $\Psi^s$ Classical pseudodifferential operators of order $s$ $\Psi^{-\infty}$ Infinitely smoothing operators $\eqref{Psi minus infinity}$ $\zeta_Q(s)$ Zeta function of the operator $Q$ $Z$ Parallel transport map [\[Q acting on 1-forms 3\]](#Q acting on 1-forms 3){reference-type="eqref" reference="Q acting on 1-forms 3"}, see also Appendix [14](#appendix parallel transport){reference-type="ref" reference="appendix parallel transport"} --------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- # The operator curl {#The operator curl} In this section we put $\operatorname{curl}$ on a rigorous mathematical foundation, defining it within the framework of the theory of unbounded self-adjoint operators in Hilbert spaces. This would provide the reader with an account of the fundamental properties of $\operatorname{curl}$, in a self-contained fashion. The main challenge that one faces when dealing with $\operatorname{curl}$ is that it is not elliptic[^4]. Indeed, the formula for the principal symbol of $\operatorname{curl}$ (which happens to coincide with its full symbol) reads $$\label{principal symbol curl} [(\operatorname{curl})_\mathrm{prin}]_\alpha{}^\beta(x,\xi)= -i\,E_\alpha{}^{\beta\gamma}(x)\,\xi_\gamma\,,$$ where the tensor $E$ is defined in accordance with [\[main theorem 1 equation 3\]](#main theorem 1 equation 3){reference-type="eqref" reference="main theorem 1 equation 3"}. Although $\xi$ is not a 1-form in its own right, at a formal level the RHS of [\[principal symbol curl\]](#principal symbol curl){reference-type="eqref" reference="principal symbol curl"} can be rewritten, concisely, as $-i*\xi$. A straightforward calculation shows that $$\label{determinant principal symbol curl} \det (\operatorname{curl})_\mathrm{prin}=0.$$ The eigenvalues of $(\operatorname{curl})_\mathrm{prin}$ are simple and read $$\label{eigenvalues principal symbol curl} h^{(0)}(x,\xi)=0, \qquad h^{(\pm)}(x,\xi)=\pm\|\xi\|\,,$$ for all $(x,\xi)\in T^*M\setminus \{0\}$. Consequently, standard elliptic theory does not apply and care is required in defining the appropriate Hilbert space, domain of the operator and establishing the mapping properties of $\operatorname{curl}$. ## Operator theoretic definition and main properties Let $H^s(M)$, $s>0$, be the space of differential forms that are square integrable together with their partial derivatives up to order $s$. We do not carry in our notation for Sobolev spaces the degree of differential forms: this will be clear from the context. Henceforth, to further simplify notation we drop the $M$ and write $\Omega^k$ for $\Omega^k(M)$ and $H^s$ for $H^s(M)$.   To begin with, let us observe that $\operatorname{curl}$ maps smooth coexact 1-forms to smooth coexact 1-forms. Therefore, it is natural to define $\operatorname{curl}$ as an operator $$\label{definition curl} \operatorname{curl}=*\mathrm{d}: \delta\Omega^2 \cap H^1\to \delta\Omega^2,$$ where $\delta\Omega^2$ is the Hilbert space of real-valued coexact 1-forms with inner product $$\label{inner product} \langle u,v \rangle:=\int_M *u \wedge v=\int_M u \wedge * v.$$ The following theorem, the main result of this section, establishes the basic properties of the operator [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"}. It essentially tells us that, despite not being elliptic, $\operatorname{curl}$ enjoys numerous properties characteristic of elliptic operators. **Theorem 9**. *Let $(M,g)$ be a connected oriented closed Riemannian 3-manifold.* (a) *The operator $\operatorname{curl}$ [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"} is self-adjoint.* (b) *The spectrum of $\operatorname{curl}$ is discrete and accumulates to $+\infty$ and $-\infty$.* (c) *Zero is not an eigenvalue of $\operatorname{curl}$.* (d) *The operator $\operatorname{curl}^{-1}$ is a bounded operator from $\delta\Omega^2 \cap H^s$ to $\delta\Omega^2 \cap H^{s+1}$ for all $s\ge0$.* *Remark 10*. Let us point out, once again, that throughout this paper $\operatorname{curl}$ comes in two guises: as a differential expression [\[curl differential experssion\]](#curl differential experssion){reference-type="eqref" reference="curl differential experssion"} and as a self-adjoint operator [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"}. Both will be denoted by "$\operatorname{curl}$"; which is which will be clear from the context. We would like to emphasise the significance of property (c) in the above theorem. If we start deforming the metric the eigenvalues of $\operatorname{curl}$, understood as those of a self-adjoint operator [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"}, will never turn to zero or cross zero (change sign). This is in stark contrast with the Dirac operator where zero may be an eigenvalue, depending on the choice of metric [@hitchin]. In fact, the study of zero modes of the Dirac operator (harmonic spinors) is an active area of research. In order to prove the above theorem, let us introduce an auxiliary elliptic operator: *extended curl*. ## Extended curl It is known that one can extend curl to a $4\times 4$ elliptic operator acting in $\Omega^1\oplus \Omega^0$. **Definition 11**. We define *extended curl* to be the operator $$\label{definition extended curl equation 1} \operatorname{curl}_E:=\begin{pmatrix} \operatorname{curl}&\mathrm{d}\\ \delta&0 \end{pmatrix}: (\Omega^1 \cap H^1) \oplus (\Omega^0\cap H^1) \to \Omega^1\oplus \Omega^0\,.$$ The operator [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} appeared in [@millson], see also [@asymm1 p. 229], [@asymm2 pp. 44--45 and p. 66], [@asymm3 p. 405] and [@singerICM pp. 190--191]. *Remark 12*. Note that $\operatorname{curl}_E$ is an operator of Dirac type, namely, its square is $\delta \mathrm{d}+\mathrm{d}\delta$ on $\Omega^1\oplus\Omega^0$ (minus the nonpostive Hodge Laplacian). Furthermore, [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} is the restriction of $d+\delta$ to differential forms of even degree, in the sense that $$\begin{pmatrix} *&0\\ 0&1 \end{pmatrix} \operatorname{curl}_E \begin{pmatrix} *&0\\ 0&1 \end{pmatrix} =(\mathrm{d}+\delta)*: \Omega^2\oplus\Omega^0\to\Omega^2\oplus\Omega^0.$$ A straightforward calculation shows that the principal symbol of extended curl reads $$\label{principal symbol extended curl} (\operatorname{curl}_E)_\mathrm{prin}(x,\xi) = \begin{pmatrix} [(\operatorname{curl})_\mathrm{prin}]_\alpha{}^\beta(x,\xi) & i\xi_\alpha\\ -i g^{\beta\gamma}(x)\,\xi_\gamma & 0 \end{pmatrix}.$$ The eigenvalues of [\[principal symbol extended curl\]](#principal symbol extended curl){reference-type="eqref" reference="principal symbol extended curl"} are $$\label{eigenvalues principal symbol extended curl} \pm\|\xi\|\, ,$$ each with multiplicity two, so that we have $$\label{determinant principal symbol extended curl} \det (\operatorname{curl}_E)_\mathrm{prin}=\|\xi\|^4\,,$$ Compare formulae [\[eigenvalues principal symbol extended curl\]](#eigenvalues principal symbol extended curl){reference-type="eqref" reference="eigenvalues principal symbol extended curl"} and [\[determinant principal symbol extended curl\]](#determinant principal symbol extended curl){reference-type="eqref" reference="determinant principal symbol extended curl"} with [\[eigenvalues principal symbol curl\]](#eigenvalues principal symbol curl){reference-type="eqref" reference="eigenvalues principal symbol curl"} and [\[determinant principal symbol curl\]](#determinant principal symbol curl){reference-type="eqref" reference="determinant principal symbol curl"} respectively. Formulae [\[determinant principal symbol extended curl\]](#determinant principal symbol extended curl){reference-type="eqref" reference="determinant principal symbol extended curl"} and [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} imply that $\operatorname{curl}_E$ is elliptic and self-adjoint. **Definition 13** ([@schmudgen Definition 1.7]). Let $L:\mathcal{D}(L)\to H$ be a self-adjoint (possibly unbounded) linear operator in the Hilbert space $H$. We say that a closed vector subspace $V\subseteq H$ is an *invariant subspace* of the operator $L$ if $$\label{definition invariant subspace equation 1} L(\mathcal{D}(L) \cap V) \subseteq V.$$ **Proposition 14**. *The Hilbert space $\Omega^1\oplus \Omega^0$ decomposes into a direct sum of three orthogonal Hilbert subspaces $$\label{lemma invariant subspaces extended curl equation 1} \Omega^1\oplus \Omega^0=[\mathrm{d}\Omega^0 \oplus \delta\Omega^1] \oplus[\delta\Omega^2 \oplus 0] \oplus [ \mathcal{H}^1 \oplus \mathcal{H}^0]$$ which are invariant subspaces of the operator $\operatorname{curl}_E$ in the sense of Definition [Definition 13](#definition invariant subspace){reference-type="ref" reference="definition invariant subspace"}. Accordingly, the operator $\operatorname{curl}_E$ decomposes into a direct sum of three operators $$\label{lemma invariant subspaces extended curl equation 2} \operatorname{curl}_E=\operatorname{curl}_{E,\mathrm{d}} \oplus \operatorname{curl}_{E,\delta} \oplus\operatorname{curl}_{E,\mathcal{H}}\,,$$ where $$\label{lemma invariant subspaces extended curl equation 3} \operatorname{curl}_{E,\mathrm{d}}:(\mathrm{d}\Omega^0 \cap H^1) \oplus (\delta\Omega^1\cap H^1)\to \mathrm{d}\Omega^0 \oplus \delta\Omega^1,$$ $$\label{lemma invariant subspaces extended curl equation 4} \operatorname{curl}_{E,\delta}:(\delta\Omega^2 \cap H^1) \oplus 0 \to \delta\Omega^2 \oplus 0,$$ $$\label{lemma invariant subspaces extended curl equation 5} \operatorname{curl}_{E,\mathcal{H}}: \mathcal{H}^1 \oplus \mathcal{H}^0 \to \mathcal{H}^1 \oplus \mathcal{H}^0.$$* *Proof.* That the three subspaces in the RHS of [\[lemma invariant subspaces extended curl equation 1\]](#lemma invariant subspaces extended curl equation 1){reference-type="eqref" reference="lemma invariant subspaces extended curl equation 1"} are mutually orthogonal and their direct sum gives the whole space $\Omega^1\oplus\Omega^0$ follows immediately from the Hodge decomposition theorem. Let us show that they are invariant subspaces of $\operatorname{curl}_E\,$. We will do this in several steps. **Step 1**. Observe that $\operatorname{curl}_E$ maps $\mathcal{H}^1\oplus\mathcal{H}^0$ to zero. Hence, $\mathcal{H}^1\oplus\mathcal{H}^0$ is an invariant subspace of $\operatorname{curl}_E\,$. **Step 2**. The facts that - $\mathcal{H}^1\oplus\mathcal{H}^0$ is an invariant subspace of $\operatorname{curl}_E$ and - $[\mathrm{d}\Omega^0\oplus\delta\Omega^1]\oplus[\delta\Omega^2\oplus 0]$ is the orthogonal complement of $\mathcal{H}^1\oplus\mathcal{H}^0$ imply that $[\mathrm{d}\Omega^0\oplus\delta\Omega^1]\oplus[\delta\Omega^2\oplus 0]$ is an invariant subspace of $\operatorname{curl}_E\,$. **Step 3**. Choose a $\lambda$ in the resolvent set $\rho(\operatorname{curl}_E)$ of the operator $\operatorname{curl}_E$. It will be convenient for us to choose a real $\lambda$, $$\label{lemma invariant subspaces extended curl equation Dima 1} \lambda\in\mathbb{R}.$$ This can always be achieved because the operator $\operatorname{curl}_E$ is elliptic and self-adjoint, hence its spectrum is real and discrete. Note that zero is an eigenvalue of $\operatorname{curl}_E$ (see Step 1 above), so $$\label{lemma invariant subspaces extended curl equation Dima 2} \lambda\ne0.$$ Let $\,\Delta:=-\delta\mathrm{d}\,$ be the (nonpositive) Laplace--Beltrami operator. Let $$\label{eigenvalues of Laplace--Beltrami operator} 0=\mu_0<\mu_1\le\mu_2\le\dots$$ be the eigenvalues of $\,-\Delta\,$ enumerated in increasing order with account of multiplicities. It will be convenient for us to further restrict the choice of $\lambda$ by imposing the condition $$\label{lemma invariant subspaces extended curl equation Dima extra} \lambda\not\in\{\pm\sqrt{\mu_1}\,,\,\pm\sqrt{\mu_2}\,,\dots\}.$$ **Step 4**. Observe that in order to prove that the two subspaces $\mathrm{d}\Omega^0\oplus\delta\Omega^1$ and $\delta\Omega^2\oplus 0$ are invariant subspaces of $\operatorname{curl}_E$ it is sufficient to prove that these are invariant subspaces of the resolvent $\mathcal{R}(\lambda):=(\operatorname{curl}_E- \lambda\operatorname{Id})^{-1}\,$. **Step 5**. In view of [\[lemma invariant subspaces extended curl equation Dima 1\]](#lemma invariant subspaces extended curl equation Dima 1){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 1"} the operator $\mathcal{R}(\lambda)$ is self-adjoint. The facts that - $[\mathrm{d}\Omega^0\oplus\delta\Omega^1]\oplus[\delta\Omega^2\oplus 0]$ is an invariant subspace of $\mathcal{R}(\lambda)$ and - the subspaces $\mathrm{d}\Omega^0\oplus\delta\Omega^1$ and $\delta\Omega^2\oplus 0$ are mutually orthogonal imply that it suffices to prove that only one of the subspaces $\mathrm{d}\Omega^0\oplus\delta\Omega^1$ and $\delta\Omega^2\oplus 0$ is an invariant subspace of $\mathcal{R}(\lambda)$. In what follows we prove that $\delta\Omega^2\oplus 0$ is an invariant subspace of $\mathcal{R}(\lambda)$. **Step 6**. Let $\Omega^k_\infty$ denote the vector space of infinitely smooth real-valued $k$-forms and let $$\label{lemma invariant subspaces extended curl equation Dima 3} \mathrm{d}\Omega^{k-1}_\infty := \{ \mathrm{d}\omega\ |\ \omega\in\Omega^{k-1}_\infty \},$$ $$\label{lemma invariant subspaces extended curl equation Dima 4} \delta\Omega^{k+1}_\infty := \{ \delta\omega\ |\ \omega\in\Omega^{k+1}_\infty \}.$$ The Hilbert spaces $\Omega^k$, $\mathrm{d}\Omega^{k-1}$ and $\delta\Omega^{k+1}$ are defined as $L^2$ closures of the vector spaces $\Omega^k_\infty$, $\mathrm{d}\Omega^{k-1}_\infty$ and $\delta\Omega^{k+1}_\infty$ respectively, see [@jost Corollary 3.4.2]. This implies that the vector space $\delta\Omega^2_\infty\oplus 0$ is dense in the Hilbert spaces $\delta\Omega^2\oplus 0$. As $\mathcal{R}(\lambda)$ is a bounded operator, in order to prove that $\delta\Omega^2\oplus 0$ is an invariant subspace of $\mathcal{R}(\lambda)$ it is sufficient to prove that $$\label{lemma invariant subspaces extended curl equation Dima 5} [\mathcal{R}(\lambda)](\delta\Omega^2_\infty\oplus 0) \subseteq \delta\Omega^2_\infty\oplus 0\,.$$ **Step 7**. Let $z = \begin{pmatrix} z_1 \\ 0 \end{pmatrix} \in \delta\Omega^2_\infty\oplus 0$. Put $$\label{lemma invariant subspaces extended curl equation Dima 6} v:=\mathcal{R}(\lambda)z = \begin{pmatrix} v_1 \\ v_0 \end{pmatrix} \in [\mathrm{d}\Omega^0_\infty\oplus\delta\Omega^1_\infty]\oplus[\delta\Omega^2_\infty\oplus 0].$$ Here $z_1$, $v_1$ are 1-forms and $v_0$ is a 0-form (scalar). The $v_1$ and $v_0$ are infinitely smooth because the operator $\operatorname{curl}_E$ is elliptic. Formula [\[lemma invariant subspaces extended curl equation Dima 6\]](#lemma invariant subspaces extended curl equation Dima 6){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 6"} can be rewritten more explicitly as $$\begin{aligned} \label{lemma invariant subspaces extended curl equation Dima 7} \operatorname{curl}v_1+\mathrm{d}v_0-\lambda v_1 &= z_1\,, \\ \label{lemma invariant subspaces extended curl equation Dima 8} \delta v_1-\lambda v_0 &= 0\,.\end{aligned}$$ We have [@warner Section 6.8, p. 223] $$\label{lemma invariant subspaces extended curl equation Dima 9} \Omega^k_\infty = \mathrm{d}\Omega^{k-1}_\infty \oplus \delta\Omega^{k+1}_\infty \oplus \mathcal{H}^k,$$ compare with [\[Hodge decompostion\]](#Hodge decompostion){reference-type="eqref" reference="Hodge decompostion"}. We already know (see Step 2) that the 1-form $v_1$ does not contain a contribution from harmonic 1-forms, so formulae [\[lemma invariant subspaces extended curl equation Dima 9\]](#lemma invariant subspaces extended curl equation Dima 9){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 9"}, [\[lemma invariant subspaces extended curl equation Dima 3\]](#lemma invariant subspaces extended curl equation Dima 3){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 3"} and [\[lemma invariant subspaces extended curl equation Dima 4\]](#lemma invariant subspaces extended curl equation Dima 4){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 4"} give us $$\label{lemma invariant subspaces extended curl equation Dima 10} v_1=v_{1\mathrm{d}}+v_{1\delta}\,,$$ where $$\label{lemma invariant subspaces extended curl equation Dima 11} v_{1\mathrm{d}}=\mathrm{d}w_0\,,$$ $$\label{lemma invariant subspaces extended curl equation Dima 12} v_{1\delta}=\delta w_2$$ for some infinitely smooth 0-form (scalar) $w_0$ and some infinitely smooth 2-form $w_2\,$. **Step 8**. Applying the operator $\delta$ to [\[lemma invariant subspaces extended curl equation Dima 7\]](#lemma invariant subspaces extended curl equation Dima 7){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 7"} and using formulae [\[lemma invariant subspaces extended curl equation Dima 10\]](#lemma invariant subspaces extended curl equation Dima 10){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 10"}--[\[lemma invariant subspaces extended curl equation Dima 12\]](#lemma invariant subspaces extended curl equation Dima 12){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 12"} as well as the fact that $\delta z_1=0$, we get $\Delta(\lambda w_0-v_0)=0$, which implies $$\label{lemma invariant subspaces extended curl equation Dima 13} v_0=\lambda w_0+C,$$ where $C\in\mathbb{R}$ is a constant. Substituting [\[lemma invariant subspaces extended curl equation Dima 10\]](#lemma invariant subspaces extended curl equation Dima 10){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 10"} and [\[lemma invariant subspaces extended curl equation Dima 13\]](#lemma invariant subspaces extended curl equation Dima 13){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 13"} into [\[lemma invariant subspaces extended curl equation Dima 8\]](#lemma invariant subspaces extended curl equation Dima 8){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 8"} and using formulae [\[lemma invariant subspaces extended curl equation Dima 11\]](#lemma invariant subspaces extended curl equation Dima 11){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 11"}, [\[lemma invariant subspaces extended curl equation Dima 12\]](#lemma invariant subspaces extended curl equation Dima 12){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 12"}, we get $$\label{lemma invariant subspaces extended curl equation Dima 14} -\Delta w_0-\lambda^2 w_0=\lambda C.$$ In view of [\[lemma invariant subspaces extended curl equation Dima 2\]](#lemma invariant subspaces extended curl equation Dima 2){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 2"} and [\[lemma invariant subspaces extended curl equation Dima extra\]](#lemma invariant subspaces extended curl equation Dima extra){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima extra"} equation [\[lemma invariant subspaces extended curl equation Dima 14\]](#lemma invariant subspaces extended curl equation Dima 14){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 14"} has the unique solution $w_0=-\lambda^{-1}C$, which immediately implies $$\label{lemma invariant subspaces extended curl equation Dima 15} v_0=0,\qquad v_{1\mathrm{d}}=0.$$ It only remains to observe that formulae [\[lemma invariant subspaces extended curl equation Dima 6\]](#lemma invariant subspaces extended curl equation Dima 6){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 6"}, [\[lemma invariant subspaces extended curl equation Dima 10\]](#lemma invariant subspaces extended curl equation Dima 10){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 10"}, [\[lemma invariant subspaces extended curl equation Dima 12\]](#lemma invariant subspaces extended curl equation Dima 12){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 12"} and [\[lemma invariant subspaces extended curl equation Dima 15\]](#lemma invariant subspaces extended curl equation Dima 15){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 15"} imply $v\in\delta\Omega^2_\infty\oplus 0$, thus proving the inclusion [\[lemma invariant subspaces extended curl equation Dima 5\]](#lemma invariant subspaces extended curl equation Dima 5){reference-type="eqref" reference="lemma invariant subspaces extended curl equation Dima 5"}. ◻ ## Decomposition of the spectrum of extended curl {#Decomposition of the spectrum of extended curl} Recall that extended curl [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} is a self-adjoint elliptic operator, hence its spectrum is discrete and eigenfunctions infinitely smooth. **Proposition 15**. *The spectrum of $\operatorname{curl}_E$ decomposes into three parts as $$\label{decomposition of spectrum of extended curl} \sigma(\operatorname{curl}_E) = \sigma(\operatorname{curl}) \cup \sigma(\operatorname{curl}_{E,\mathrm{d}}) \cup \sigma(\operatorname{curl}_{E,\mathcal{H}}),$$ where $\operatorname{curl}$, $\operatorname{curl}_{E,\mathrm{d}}$ and $\operatorname{curl}_{E,\mathcal{H}}$ are the operators [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"}, [\[lemma invariant subspaces extended curl equation 3\]](#lemma invariant subspaces extended curl equation 3){reference-type="eqref" reference="lemma invariant subspaces extended curl equation 3"} and [\[lemma invariant subspaces extended curl equation 5\]](#lemma invariant subspaces extended curl equation 5){reference-type="eqref" reference="lemma invariant subspaces extended curl equation 5"} respectively. Formula [\[decomposition of spectrum of extended curl\]](#decomposition of spectrum of extended curl){reference-type="eqref" reference="decomposition of spectrum of extended curl"} is understood as the (disjoint) union of spectra with account of multiplicities.* *Proof.* Proposition [Proposition 15](#corollary decomposition of spectrum of extended curl){reference-type="ref" reference="corollary decomposition of spectrum of extended curl"} is an immediate consequence of Proposition [Proposition 14](#lemma invariant subspaces extended curl){reference-type="ref" reference="lemma invariant subspaces extended curl"} and the observation that the operator $\,\operatorname{curl}_{E,\delta}\,$ defined by formulae [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} and [\[lemma invariant subspaces extended curl equation 4\]](#lemma invariant subspaces extended curl equation 4){reference-type="eqref" reference="lemma invariant subspaces extended curl equation 4"} is precisely our original operator $\operatorname{curl}$ defined by formula [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"}. ◻ Proposition [Proposition 15](#corollary decomposition of spectrum of extended curl){reference-type="ref" reference="corollary decomposition of spectrum of extended curl"} tells us that the spectrum of curl sits inside the spectrum of extended curl and the two spectra differ by $\sigma(\operatorname{curl}_{E,\mathrm{d}}) \cup \sigma(\operatorname{curl}_{E,\mathcal{H}})$. So the task at hand is working out the spectra of $\operatorname{curl}_{E,\mathrm{d}}$ and $\operatorname{curl}_{E,\mathcal{H}}\,$. The following two lemmata address this issue. **Lemma 16**. *We have $\sigma(\operatorname{curl}_{E,\mathcal{H}})=\{0\}$, zero being an eigenvalue of multiplicity $\operatorname{dim}\mathcal{H}^1+1$.* *Proof.* Lemma [Lemma 16](#spectrum zero){reference-type="ref" reference="spectrum zero"} is an immediate consequence of formulae [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} and [\[lemma invariant subspaces extended curl equation 5\]](#lemma invariant subspaces extended curl equation 5){reference-type="eqref" reference="lemma invariant subspaces extended curl equation 5"}. ◻ **Lemma 17**. *We have $$\label{spectrum plus minus square root of Laplacian Dima 0} \sigma(\operatorname{curl}_{E,\mathrm{d}})=\{\pm\sqrt{\mu_1}\,,\,\pm\sqrt{\mu_2}\,,\dots\},$$ where the $\mu_j$ are the eigenvalues [\[eigenvalues of Laplace\--Beltrami operator\]](#eigenvalues of Laplace--Beltrami operator){reference-type="eqref" reference="eigenvalues of Laplace--Beltrami operator"} of $\,-\Delta\,$.* *Proof.* Let $v= \begin{pmatrix} \mathrm{d}w_0\\ v_0 \end{pmatrix}$ be an eigenfunction of $\operatorname{curl}_{E,\mathrm{d}}$ corresponding to an eigenvalue $\lambda$. Here $w_0$ and $v_0$ are some infinitely smooth 0-forms (scalars). We have $$\begin{aligned} \label{spectrum plus minus square root of Laplacian Dima 1} \mathrm{d}v_0 &= \lambda\mathrm{d}w_0\,, \\ \label{spectrum plus minus square root of Laplacian Dima 2} -\Delta w_0 &= \lambda v_0\,.\end{aligned}$$ Elementary analysis of [\[spectrum plus minus square root of Laplacian Dima 1\]](#spectrum plus minus square root of Laplacian Dima 1){reference-type="eqref" reference="spectrum plus minus square root of Laplacian Dima 1"} [\[spectrum plus minus square root of Laplacian Dima 2\]](#spectrum plus minus square root of Laplacian Dima 2){reference-type="eqref" reference="spectrum plus minus square root of Laplacian Dima 2"} yields [\[spectrum plus minus square root of Laplacian Dima 0\]](#spectrum plus minus square root of Laplacian Dima 0){reference-type="eqref" reference="spectrum plus minus square root of Laplacian Dima 0"}. ◻ Proposition [Proposition 15](#corollary decomposition of spectrum of extended curl){reference-type="ref" reference="corollary decomposition of spectrum of extended curl"} and Lemmata [Lemma 16](#spectrum zero){reference-type="ref" reference="spectrum zero"}, [Lemma 17](#spectrum plus minus square root of Laplacian){reference-type="ref" reference="spectrum plus minus square root of Laplacian"} tell us that the spectrum of $\operatorname{curl}$ can be recovered from the spectrum of $\operatorname{curl}_E\,$, provided one knows the spectrum of the Laplace--Beltrami operator $\Delta$. Moreover, the eta functions of $\operatorname{curl}$ and $\operatorname{curl}_E\,$ are the same, because contributions to $\sigma(\operatorname{curl}_E)$ coming from the Laplace--Beltrami operator are symmetric about zero. ## Proof of Theorem [Theorem 9](#theorem properties of curl){reference-type="ref" reference="theorem properties of curl"} {#proof-of-theorem-theorem-properties-of-curl} *Proof of Theorem [Theorem 9](#theorem properties of curl){reference-type="ref" reference="theorem properties of curl"}.*   \(a\) Standard elliptic theory tells us that the operator $\operatorname{curl}_E$ defined by formula [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} is self-adjoint in the operator theoretic sense. Proposition [Proposition 14](#lemma invariant subspaces extended curl){reference-type="ref" reference="lemma invariant subspaces extended curl"} and formulae [\[definition extended curl equation 1\]](#definition extended curl equation 1){reference-type="eqref" reference="definition extended curl equation 1"} and [\[lemma invariant subspaces extended curl equation 4\]](#lemma invariant subspaces extended curl equation 4){reference-type="eqref" reference="lemma invariant subspaces extended curl equation 4"} tell us that our original operator $\operatorname{curl}$ defined by formula [\[definition curl\]](#definition curl){reference-type="eqref" reference="definition curl"} is part of the operator $\operatorname{curl}_E\,$; here the concept of 'part of an operator' is understood in accordance with [@schmudgen Definition 1.7]. Self-adjointness of $\operatorname{curl}$ now follows from [@schmudgen Proposition 3.11]. \(b\) Discreteness of the spectrum of $\operatorname{curl}$ follows from Proposition [Proposition 15](#corollary decomposition of spectrum of extended curl){reference-type="ref" reference="corollary decomposition of spectrum of extended curl"}. As to the statement that the spectrum of $\operatorname{curl}$ is unbounded both from above and from below, we prove it by writing down spectral asymptotics. Let $\lambda_k$, $k\in\mathbb{Z}\setminus\{0\}$, be the eigenvalues of $\operatorname{curl}$. The choice of particular enumeration is irrelevant for our purposes, but what is important is that eigenvalues are enumerated with account of their multiplicity. We define the positive ($+$) and negative ($-$) counting functions for $\operatorname{curl}$ as $$\label{global counting functions for curl} N^\pm_{\operatorname{curl}}(\lambda) := \begin{cases} 0\quad&\text{for}\quad\lambda\le0, \\ \sum_{0<\pm\lambda_k<\lambda} \,1 \quad&\text{for}\quad\lambda>0. \end{cases}$$ We define, in a similar fashion, counting functions $N^\pm_{\operatorname{curl}_E}(\lambda)$ and $N^\pm_{\operatorname{curl}_{E,d}}(\lambda)$ for the operators $\operatorname{curl}_E$ and $\operatorname{curl}_{E,d}$ respectively. Proposition [Proposition 15](#corollary decomposition of spectrum of extended curl){reference-type="ref" reference="corollary decomposition of spectrum of extended curl"} and Lemma [Lemma 16](#spectrum zero){reference-type="ref" reference="spectrum zero"} tell us that $$\label{difference of two couunting functions} N^\pm_{\operatorname{curl}}(\lambda) = N^\pm_{\operatorname{curl}_E}(\lambda) - N^\pm_{\operatorname{curl}_{E,d}}(\lambda).$$ Lemma [Lemma 17](#spectrum plus minus square root of Laplacian){reference-type="ref" reference="spectrum plus minus square root of Laplacian"} tells us that the evaluation of $N^\pm_{\operatorname{curl}_{E,d}}(\lambda)$ reduces to the evaluation of eigenvalues of the Laplace--Beltrami operator. The latter is a subject which has been extensively studied over the last 100 years. Application of [@Hor; @1968 Theorem 1.1] gives the following asymptotic formula for $N^\pm_{\operatorname{curl}_{E,d}}(\lambda)$ with sharp remainder term estimate: $$\label{asymptotic formula 1} N^\pm_{\operatorname{curl}_{E,d}}(\lambda) = \frac{V}{6\pi^2}\lambda^3+O(\lambda^2) \qquad \text{as} \qquad \lambda\to+\infty,$$ where $V$ is the Riemannian volume of the manifold. Application of [@Ivr82 Theorem 0.1] gives the following asymptotic formula for $N^\pm_{\operatorname{curl}_E}(\lambda)$: $$\label{asymptotic formula 2} N^\pm_{\operatorname{curl}_E}(\lambda) = \frac{V}{3\pi^2}\lambda^3+O(\lambda^2) = 2N^\pm_{\operatorname{curl}_{E,d}}(\lambda)+O(\lambda^2) \qquad \text{as} \qquad \lambda\to+\infty.$$ The factor 2 appears in the RHS of [\[asymptotic formula 2\]](#asymptotic formula 2){reference-type="eqref" reference="asymptotic formula 2"} because the eigenvalues of the principal symbol of the operator $\operatorname{curl}_E\,$, given by formula [\[eigenvalues principal symbol extended curl\]](#eigenvalues principal symbol extended curl){reference-type="eqref" reference="eigenvalues principal symbol extended curl"}, have multiplicity two. Formulae [\[difference of two couunting functions\]](#difference of two couunting functions){reference-type="eqref" reference="difference of two couunting functions"}--[\[asymptotic formula 2\]](#asymptotic formula 2){reference-type="eqref" reference="asymptotic formula 2"} imply Bär's [@baer_curl Theorem 3.6] asymptotic formula $$\label{asymptotic formula 3} N^\pm_{\operatorname{curl}}(\lambda) = \frac{V}{6\pi^2}\lambda^3+O(\lambda^2) \qquad \text{as} \qquad \lambda\to+\infty.$$ Formula [\[asymptotic formula 3\]](#asymptotic formula 3){reference-type="eqref" reference="asymptotic formula 3"} shows that the spectrum of $\operatorname{curl}$ accumulates to $+\infty$ and $-\infty$. \(c\) Suppose that zero is an eigenvalue of $\operatorname{curl}$. Let $$\label{zero mode 1} u\in\delta\Omega^2_\infty$$ be the corresponding eigenform (zero mode). Then $\operatorname{curl}u=0$, which implies $$\label{zero mode 2} \mathrm{d}u=0.$$ But [\[zero mode 1\]](#zero mode 1){reference-type="eqref" reference="zero mode 1"} implies $$\label{zero mode 3} \delta u=0.$$ Formulae [\[zero mode 2\]](#zero mode 2){reference-type="eqref" reference="zero mode 2"} and [\[zero mode 3\]](#zero mode 3){reference-type="eqref" reference="zero mode 3"} tell us that the 1-form $u$ is harmonic. As $\delta\Omega^2\cap\mathcal{H}^1=\{0\}$, we conclude that $u=0$, which is a contradiction. \(d\) Proposition [Proposition 15](#corollary decomposition of spectrum of extended curl){reference-type="ref" reference="corollary decomposition of spectrum of extended curl"}, Lemmata [Lemma 16](#spectrum zero){reference-type="ref" reference="spectrum zero"} and [Lemma 17](#spectrum plus minus square root of Laplacian){reference-type="ref" reference="spectrum plus minus square root of Laplacian"} and part (c) of Theorem [Theorem 9](#theorem properties of curl){reference-type="ref" reference="theorem properties of curl"} tell us that zero is an eigenvalue of $\operatorname{curl}_E$ of multiplicity $\operatorname{dim}\mathcal{H}^1+1$. Let $P_{\mathcal{H}^1}$ and $P_{\mathcal{H}^0}$ be orthogonal projections onto the spaces of harmonic 1-forms and 0-forms respectively. Put $$\label{definition extended curl equation 1 invertible} \widetilde\operatorname{curl}_E := \operatorname{curl}_E + \begin{pmatrix} P_{\mathcal{H}^1}&0\\ 0&P_{\mathcal{H}^0} \end{pmatrix}.$$ The operator $\widetilde\operatorname{curl}_E$ is invertible. By standard elliptic theory $\widetilde\operatorname{curl}{}_E^{-1}$ is a bounded operator acting from the Sobolev space $H^s$ to the Sobolev space $H^{s+1}$. The operator $\operatorname{curl}^{-1}$ is part of the operator $\widetilde\operatorname{curl}{}_E^{-1}\,$, hence it possesses the same mapping properties. ◻ # Pseudodifferential operators acting on 1-forms {#Pseudodifferential operators acting on 1-forms} In this section we develop an invariant calculus for pseudodifferential operators acting on 1-forms over a closed Riemannian manifold $(M,g)$ of arbitrary dimension $d$. Let $Q$ be a classical polyhomogeneous pseudodifferential operator of order $s$ acting on 1-forms, and let $$\label{11 June 2021 equation 1} Q: u_\alpha(x) \mapsto v_\alpha(x) =(2\pi)^{-d} \int e^{i(x-y)^\gamma\xi_\gamma}\,q_\alpha{}^\beta(x,\xi)\, u_\beta(y)\,\mathrm{d}y\,\mathrm{d}\xi\,,$$ $$\label{11 June 2021 equation 2} q_\alpha{}^\beta(x,\xi) \sim [q_s]_\alpha{}^\beta(x,\xi) + [q_{s-1}]_\alpha{}^\beta(x,\xi) +\dots$$ be its representation in local coordinates (the same for $x$ and $y$). Here $\sim$ stands for asymptotic expansion [@shubin § 3.3], and 'polyhomogeneous' means that $[q_{s-k}]_\alpha{}^\beta(x, \lambda \,\xi)=\lambda^{s-k}\,[q_{s-k}]_\alpha{}^\beta(x, \xi)$ for any positive $\lambda$ and $k=0,1,2,\dots$. Throughout the paper we denote pseudodifferential operators with upper case letters, and their symbols and homogeneous components of symbols --- with lower case. For the principal and subprincipal symbol we use upper case letters and denote them by $Q_\mathrm{prin}$ and $Q_\mathrm{sub}$ respectively, with the subscript indicating that we are looking at invariant (or covariant) objects 'living' on the cotangent bundle as opposed to the operator $Q$ itself which 'lives' (acts) on the base manifold. Later on we will sometimes write a pseudodifferential operator as an integral operator with distributional integral kernel (Schwartz kernel), see, for example, formula [\[Q acting on 1-forms 1\]](#Q acting on 1-forms 1){reference-type="eqref" reference="Q acting on 1-forms 1"}. We will use lower case Fraktur font for the Schwartz kernel. Returning to the pseudodifferential operator [\[11 June 2021 equation 1\]](#11 June 2021 equation 1){reference-type="eqref" reference="11 June 2021 equation 1"}, [\[11 June 2021 equation 2\]](#11 June 2021 equation 2){reference-type="eqref" reference="11 June 2021 equation 2"}, it is easy to see that $$\label{formal definition of principal symbol} [Q_\mathrm{prin}]_\alpha{}^\beta(x,\xi):=[q_s]_\alpha{}^\beta(x,\xi)$$ provides an invariant definition for the principal symbol $Q_\mathrm{prin}$ of $Q$. However, the subleading term, $[q_{s-1}]_\alpha{}^\beta\,$, is *not* invariant under change of coordinates. The task at hand is to define and provide a formula for the subprincipal symbol of $Q$, by determining appropriate correction terms to $[q_{s-1}]_\alpha{}^\beta$. The original definition of subprincipal symbol goes back to J.J. Duistermaat and L. Hörmander [@DuHo Eqn. (5.2.8)]. They defined the subprincipal symbol for operators acting on half-densities. Duistermaat and Hörmander's definition of subprincipal symbol extends to operators acting on scalar fields because scalars can be identified with half-densities: it is just a matter of multiplying or dividing by $\sqrt\rho\,$. This leads to the appearance of an additional correction term. Namely, given a pseudodifferential operator $Q$ of order $s$ $$\label{11 June 2021 equation 1 scalar} Q: f(x) \mapsto (2\pi)^{-d} \int e^{i(x-y)^\gamma\xi_\gamma}\,q(x,\xi)\, f(y)\,\mathrm{d}y\,\mathrm{d}\xi\,,$$ $$\label{11 June 2021 equation 2 scalar} q(x,\xi) \sim q_s(x,\xi) + q_{s-1}(x,\xi) +\dots$$ acting on scalar fields, its subprincipal symbol reads $$\label{subprincipal symbol of operator acting on a scalar field} Q_\mathrm{sub} := q_{s-1} + \frac{i}2\dfrac{\partial^2 q_s}{\partial x^\gamma\partial\xi_\gamma} + \frac{i}2\, \frac{\partial\ln\rho}{\partial x^\gamma}\, \frac{\partial q_s}{\partial\xi_\gamma}\,,$$ where $\rho$ is the Riemannian density. In formula [\[subprincipal symbol of operator acting on a scalar field\]](#subprincipal symbol of operator acting on a scalar field){reference-type="eqref" reference="subprincipal symbol of operator acting on a scalar field"} one can, in principle, instead of the Riemannian density use any other prescribed positive reference density. However, in this paper we always stick with the Riemannian density. In particular, this implies the identity $$\label{relation between Riemannian density and Christoffel symbols} \frac{\partial\ln\rho}{\partial x^\gamma} = \Gamma^{\alpha}{}_{\gamma\alpha}\,.$$ The definition of subprincipal symbol [\[subprincipal symbol of operator acting on a scalar field\]](#subprincipal symbol of operator acting on a scalar field){reference-type="eqref" reference="subprincipal symbol of operator acting on a scalar field"} extends further, without change, to operators acting on $m$-columns of scalar fields (sections of the trivial $\mathbb{R}^m$- or $\mathbb{C}^m$-bundle over $M$). A notion of subprincipal symbol for operators acting on 1-forms is available in the literature in the special case when the principal symbol is of the form $fI$, where $f$ is a scalar function on $T^*M\setminus\{0\}$ and $I$ is the $d\times d$ identity matrix, see, e.g., [@hintz; @JS]. **Theorem 18**. *Let $Q$ be a pseudodifferential operator of order $s$ acting on 1-forms with symbol [\[11 June 2021 equation 2\]](#11 June 2021 equation 2){reference-type="eqref" reference="11 June 2021 equation 2"}. Then the quantity $$\label{subprincipal symbol operators 1-forms} [Q_\mathrm{sub}]_\mu{}^\nu:= \left([q_{s-1}]_\mu{}^\nu+\frac{i}2\dfrac{\partial^2 [q_s]_\mu{}^\nu}{\partial x^\gamma\partial\xi_\gamma} \right) %\\ +\frac{i}2 \left( \Gamma^{\alpha}{}_{\gamma\alpha} \dfrac{\partial [q_s]_\mu{}^\nu}{\partial\xi_\gamma}\ - \Gamma^\alpha{}_{\gamma \mu} \dfrac{\partial [q_s]_\alpha{}^\nu}{\partial\xi_\gamma} - \Gamma^\nu{}_{\gamma\alpha} \dfrac{\partial [q_s]_\mu{}^\alpha}{\partial\xi_\gamma} \right)$$ is covariant under change of local coordinates.* **Definition 19**. We call [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} the *subprincipal symbol* of $Q$. *Remark 20*. 'Covariance' in Theorem [Theorem 18](#theorem subprincipal symbol operators 1 forms){reference-type="ref" reference="theorem subprincipal symbol operators 1 forms"} means that one doesn't get derivatives of Jacobians under change of local coordinates. More precisely, let $x$ and $\widetilde x$ be two local coordinate systems, and let $Q_\mathrm{sub}$ and $\widetilde Q_\mathrm{sub}$ be the quantity [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} evaluated in coordinates $x$ and $\widetilde x$ respectively. Put $J^\alpha{}_\beta:=\partial\widetilde x^\alpha/\partial x^\beta$, $[J^{-1}]^\mu{}_\nu:=\partial x^\mu/\partial\widetilde x^\nu$, $\widetilde\xi_\alpha:=[J^{-1}]^\beta{}_\alpha\,\xi_\beta\,$. Then the statement of the theorem is that $$\label{covariance explained in detail} [\widetilde Q_\mathrm{sub}]_\alpha{}^\beta(\widetilde x,\widetilde\xi)= [J^{-1}]^\mu{}_\alpha(\widetilde x) \ [Q_\mathrm{sub}]_\mu{}^\nu(x(\widetilde x),\xi(\widetilde x,\widetilde\xi)) \ J^\beta{}_\nu(x(\widetilde x))\ .$$ The above transformation law also applies to the principal symbol [\[formal definition of principal symbol\]](#formal definition of principal symbol){reference-type="eqref" reference="formal definition of principal symbol"}. *Remark 21*. Note that the RHS of formula [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} does not reduce to $$[q_{s-1}]_\mu{}^\nu+\frac{i}2 \nabla_\gamma \dfrac{\partial[q_s]_\mu{}^\nu}{\partial\xi_\gamma}$$ by formally treating $\frac{\partial[q_s]_\mu{}^\nu}{\partial\xi_\gamma}$ as a $(2,1)$-tensor on $M$. The issue here is that $\frac{\partial[q_s]_\mu{}^\nu}{\partial\xi_\gamma}$ is not a true tensor, and even if it were, the sign in front of the last Christoffel symbol is not what one would expect. We now proceed to the proof of Theorem [Theorem 18](#theorem subprincipal symbol operators 1 forms){reference-type="ref" reference="theorem subprincipal symbol operators 1 forms"}. One way of proving the theorem is by a lengthy explicit calculation establishing [\[covariance explained in detail\]](#covariance explained in detail){reference-type="eqref" reference="covariance explained in detail"}. However, we will present an alternative proof which provides insight into the origins of formula [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"}. In order to do so we need to introduce first some geometric notions that will be useful here and later on in the paper. By a *frame* at a point $x\in M$ we mean a basis of orthonormal vectors in $T_xM$. By a *framing* in an open set $\mathcal{U}\subset M$ we mean the choice of a frame at each point $x$ of $\mathcal{U}$, depending smoothly on the base point $x$. Given a framing $\{e_j{}^\alpha\}_{j=1}^d$, the corresponding *dual framing* $\{e^k{}_\beta\}_{k=1}^d$ is defined as $$\label{dual framing} e^k{}_\beta(x):=\delta^{jk}\,g_{\alpha\beta}(x)\,e_j{}^\alpha(x),$$ $\delta$ being the Kronecker symbol. Here and further on, $e_j{}^\alpha$ denotes the $\alpha$-th component of the $j$-th vector field and $e^k{}_\beta$ denotes the $\beta$-th component of the $k$-th covector field. **Definition 22**. Let $z\in M$ be given and let $\{V_j\}_{j=1}^d$ be a frame at $z$. Let $\mathcal{U}_z$ be a sufficiently small neighbourhood of $z$. We define the *Levi-Civita framing* $\{\widetilde e_j\}_{j=1}^d$ to be the framing in $\mathcal{U}_z$ obtained by parallel-transporting $\{V_j\}_{j=1}^d$ along geodesics emanating from $z$. Let us emphasise that the Levi-Civita framing is uniquely determined in the neighbourhood of every point $z\in M$ up to a rigid (constant) orthogonal transformation, which reflects the freedom in choosing the frame $\{V_j\}_{j=1}^d$ at the point $z$. By definition, the Levi-Civita framing and its dual satisfy $$\label{14 June 2021 equation 1} \left. \left[ \partial_\beta \widetilde e_j{}^\alpha+\Gamma^\alpha{}_{\beta\gamma}\,\widetilde e_j{}^\gamma \right] \right|_{x=z} =0\,,$$ $$\label{14 June 2021 equation 2} \left. \left[ \partial_\beta \widetilde e\,{}^j{}_\alpha-\Gamma^\gamma{}_{\beta\alpha}\,\widetilde e\,{}^j{}_\gamma \right] \right|_{x=z} =0,$$ see also [@dirac Lemma 7.2]. *Proof of Theorem [Theorem 18](#theorem subprincipal symbol operators 1 forms){reference-type="ref" reference="theorem subprincipal symbol operators 1 forms"}.* Let us fix a point $z\in M$. Let $\mathcal{U}_z$ be a small open neighbourhood of $z$ and let $\{\widetilde{e}_j\}_{j=1}^d$ be a Levi-Civita framing centred at $z$. Consider the operator $S$ mapping, locally, 1-forms on $\mathcal{U}_z$ to sections of the trivial $\mathbb{R}^d$-bundle over $\mathcal{U}_z$, defined in accordance with $$\label{proof subprincipal symbol operators 1 forms equation 1} S: u_\alpha\mapsto \widetilde e_j{}^\beta\, u_\beta.$$ We claim that [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} is the map $$\label{proof subprincipal symbol operators 1 forms equation 2} (z ,\xi)\mapsto \left[ S^{-1}(\rho^{1/2}SQS^{-1} \rho^{-1/2})_\mathrm{sub} S\right]_\alpha{}^\beta(z,\xi),$$ where $$\label{proof subprincipal symbol operators 1 forms equation 3} (\rho^{1/2}SQS^{-1} \rho^{-1/2})_\mathrm{sub}$$ denotes the usual subprincipal symbol [@DuHo Eqn. (5.2.8)] of the operator $Q_{1/2}:=\rho^{1/2}SQS^{-1} \rho^{-1/2}$ acting on $d$-columns of half-densities. The task at hand is to show, by means of an explicit computation, that [\[proof subprincipal symbol operators 1 forms equation 2\]](#proof subprincipal symbol operators 1 forms equation 2){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 2"} and [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} coincide. As a by-product of the upcoming calculation and [\[proof subprincipal symbol operators 1 forms equation 2\]](#proof subprincipal symbol operators 1 forms equation 2){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 2"} it will follow that [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} is covariant under changes of local coordinates and independent of the choice of the Levi-Civita framing $\{\widetilde e_j\}_{j=1}^3$. The operator $Q_{1/2}$ acts as $$\label{proof subprincipal symbol operators 1 forms equation 4} (Q_{1/2}f)_j(x)=\frac{1}{(2\pi)^d}\int e^{i(x-y)^\gamma\xi_\gamma}\,[\widetilde q_{1/2}]_j{}^k(x,y,\xi)\, f_k(y)\,\mathrm{d}y\,\mathrm{d}\xi\,,$$ where $$\label{proof subprincipal symbol operators 1 forms equation 5} [\widetilde q_{1/2}]_j{}^k(x,y,\xi):=\left[\frac{\rho(x)}{\rho(y)} \right]^{1/2} \widetilde e_j{}^\alpha(x)\, q_{\alpha}{}^\beta(x,\xi) \,\widetilde e\,{}^k{}_\beta(y)\,.$$ The left (polyhomogeneous) symbol $q_{1/2}\sim \sum_{k=0}^{+\infty} \,[q_{1/2}]_{s-k}$ of $Q_{1/2}$ is obtained by excluding the $y$-dependence from $\widetilde q_{1/2}$ via the amplitude-to-symbol operator $$\label{proof subprincipal symbol operators 1 forms equation 6} \mathcal{S}_\mathrm{left}\sim \sum_{k=0}^{+\infty} \mathcal{S}_\mathrm{-k}, \qquad \mathcal{S}_0:=\left.(\cdot)\right|_{y=x}, \qquad \mathcal{S}_{-k}:=\frac{1}{k!}\left. \left[ \left(-i\dfrac{\partial^2}{\partial y^\gamma\partial\xi_\gamma} \right)^k(\cdot)\right] \right|_{y=x}.$$ The terms positively homogeneous in $\xi$ of degree $s$ and $s-1$ in $$[q_{1/2}]_j{}^k(x,\xi):=\mathcal{S}_\mathrm{left}([\widetilde q_{1/2}]_j{}^k(x,y,\xi))$$ read $$\label{proof subprincipal symbol operators 1 forms equation 7} [(q_{1/2})_s]_j{}^k(x,\xi)=\widetilde e_j{}^\alpha(x)\, [q_s]_{\alpha}{}^\beta(x,\xi) \,\widetilde e\,{}^k{}_\beta(x)$$ and $$\label{proof subprincipal symbol operators 1 forms equation 8} [(q_{1/2})_{s-1}]_j{}^k(x,\xi)=\widetilde e_j{}^\alpha\,[q_{s-1}]_\alpha{}^\beta \widetilde e\,{}^{k}{}_\beta-i \widetilde e_j{}^\alpha\dfrac{\partial [q_s]_\alpha{}^\beta}{\partial\xi_\gamma}\dfrac{\partial \widetilde e\,{}^{k}{}_\beta}{\partial x^\gamma} +\frac{i}2 (\ln \rho)_{x^\gamma} \,\widetilde e_j{}^\alpha\dfrac{\partial [q_s]_\alpha{}^\beta}{\partial\xi_\gamma}\widetilde e\,{}^{k}{}_\beta\,,$$ respectively. In the RHS of [\[proof subprincipal symbol operators 1 forms equation 8\]](#proof subprincipal symbol operators 1 forms equation 8){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 8"} framings are evaluated at $x$, and homogeneous components of $q$ and their derivatives at $(x,\xi)$. On account of [\[proof subprincipal symbol operators 1 forms equation 7\]](#proof subprincipal symbol operators 1 forms equation 7){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 7"} and [\[proof subprincipal symbol operators 1 forms equation 8\]](#proof subprincipal symbol operators 1 forms equation 8){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 8"} one obtains $$\begin{gathered} \label{proof subprincipal symbol operators 1 forms equation 9} [(Q_{1/2})_\mathrm{sub}]_j{}^k(z,\xi)= \left.\left([(q_{1/2})_{s-1}]_j{}^k(x,\xi)+\frac{i}{2}\dfrac{\partial^2 [(q_{1/2})_s]_\alpha{}^\beta}{\partial x^\gamma\partial\xi_\gamma}(x,\xi) \right) \right|_{x=z} \\ = \widetilde e_j{}^\alpha\,\left([(q_{1/2})_{s-1}]_\alpha{}^\beta+\frac{i}2\dfrac{\partial^2 [(q_{1/2})_s]_\alpha{}^\beta}{\partial x^\gamma\partial\xi_\gamma} \right)\widetilde e\,{}^{k}{}_\beta +\frac{i}2 \dfrac{\partial \widetilde e_j{}^\alpha}{\partial x^\gamma}\dfrac{\partial [(q_{1/2})_s]_\alpha{}^\beta}{\partial\xi_\gamma}\widetilde e\,{}^{k}{}_\beta \\ - \frac{i}2 \widetilde e_j{}^\alpha \dfrac{\partial [(q_{1/2})_s]_\alpha{}^\beta}{\partial\xi_\gamma} \dfrac{\partial \widetilde e\,{}^{k}{}_\beta}{\partial x^\gamma} +\frac{i}2 (\ln \rho)_{x^\gamma} \,\widetilde e_j{}^\alpha\dfrac{\partial [(q_{1/2})_s]_\alpha{}^\beta}{\partial\xi_\gamma}\widetilde e\,{}^{k}{}_\beta\,.\end{gathered}$$ In the RHS of [\[proof subprincipal symbol operators 1 forms equation 9\]](#proof subprincipal symbol operators 1 forms equation 9){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 9"} framings are evaluated at $z$, and homogeneous components of $q$ and their derivatives at $(z,\xi)$. In view of the identities $$\label{proof subprincipal symbol operators 1 forms equation 10} \left. \widetilde e\,{}^j{}_\mu\dfrac{\partial \widetilde e_j{}^\alpha}{\partial x^\gamma}\right|_{x=z}=-\Gamma^\alpha{}_{\gamma \mu}(z), \qquad \left.\dfrac{\partial \widetilde e\,{}^{k}{}_\beta}{\partial x^\gamma}\,\widetilde e_k{}^\nu\right|_{x=z}=\Gamma^\nu{}_{\gamma\beta}(z),$$ which follow from [\[14 June 2021 equation 1\]](#14 June 2021 equation 1){reference-type="eqref" reference="14 June 2021 equation 1"}, [\[14 June 2021 equation 2\]](#14 June 2021 equation 2){reference-type="eqref" reference="14 June 2021 equation 2"}, and the identity [\[relation between Riemannian density and Christoffel symbols\]](#relation between Riemannian density and Christoffel symbols){reference-type="eqref" reference="relation between Riemannian density and Christoffel symbols"}, a straightforward calculation tells us that the quantity $$\label{proof subprincipal symbol operators 1 forms equation 12} \widetilde e\,{}^j{}_\mu(z)\,[(Q_{1/2})_\mathrm{sub}]_j{}^k(z,\xi)\, \widetilde e_k{}^\nu(z)$$ coincides with [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"}. The quantity [\[proof subprincipal symbol operators 1 forms equation 12\]](#proof subprincipal symbol operators 1 forms equation 12){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 12"} is invariant under rigid orthogonal transformations of the Levi-Civita framing, hence, this quantity is a covariant object. As [\[proof subprincipal symbol operators 1 forms equation 12\]](#proof subprincipal symbol operators 1 forms equation 12){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 12"} coincides with [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"}, this implies that [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} is covariant. ◻ Let us consider the special case when $M$ is an oriented 3-manifold. In this case we have the following result. **Lemma 23**. *The subprincipal symbol of $\operatorname{curl}$ is zero, $$\label{lemma subprincipal symbol curl equation} (\operatorname{curl})_\mathrm{sub}=0.$$* *Proof.* It follows from [\[principal symbol curl\]](#principal symbol curl){reference-type="eqref" reference="principal symbol curl"} that $$\label{proof subprincipal curl equation 1} \frac{\partial [\operatorname{curl}_\mathrm{prin}]_\alpha{}^\beta}{\partial \xi_\gamma}=-i\,E_\alpha{}^{\beta\gamma}(x) = \frac{i}{\rho(x)}\varepsilon^{\gamma\beta \mu} \,g_{\mu\alpha}(x),$$ where the second equality is a straightforward consequence of [\[main theorem 1 equation 3\]](#main theorem 1 equation 3){reference-type="eqref" reference="main theorem 1 equation 3"} and the properties of the totally antisymmetric symbol. In light of [\[relation between Riemannian density and Christoffel symbols\]](#relation between Riemannian density and Christoffel symbols){reference-type="eqref" reference="relation between Riemannian density and Christoffel symbols"}, the substitution of [\[proof subprincipal curl equation 1\]](#proof subprincipal curl equation 1){reference-type="eqref" reference="proof subprincipal curl equation 1"} and [\[principal symbol curl\]](#principal symbol curl){reference-type="eqref" reference="principal symbol curl"} into [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} gives us $$\label{proof subprincipal curl equation 2} \begin{split} 2\rho\, [\operatorname{curl}_\mathrm{sub}]_\alpha{}^\beta & = -\rho\frac{ \partial [\rho^{-1}\,\varepsilon^{\nu\beta\mu}\,g_{\mu\alpha}] } {\partial x^\nu} - \Gamma^{\lambda}{}_{\nu\lambda} [\varepsilon^{\nu\beta\mu}\,g_{\mu\alpha}] + \Gamma^\lambda{}_{\nu \alpha} [\varepsilon^{\nu\beta\mu}\,g_{\mu\lambda}] + \Gamma^\beta{}_{\nu\lambda} [\varepsilon^{\nu\lambda\mu}\,g_{\mu\alpha}] \\ & = - \varepsilon^{\nu\beta\mu} \frac{\partial g_{\mu\alpha}}{\partial x^\nu} + \varepsilon^{\nu\beta\mu}\,\Gamma^\lambda{}_{\nu \alpha}\,g_{\mu\lambda} \\ & = - \varepsilon^{\nu\beta\mu} \left(\Gamma^\lambda{}_{\nu \alpha}g_{\mu\lambda}+\Gamma^\lambda{}_{\nu \mu}g_{\alpha\lambda} \right) + \varepsilon^{\nu\beta\mu}\,\Gamma^\lambda{}_{\nu \alpha}\,g_{\mu\lambda} \\ & = 0. \end{split}$$ Alternatively, one can prove [\[lemma subprincipal symbol curl equation\]](#lemma subprincipal symbol curl equation){reference-type="eqref" reference="lemma subprincipal symbol curl equation"} by fixing an arbitrary point on $M$ and carrying out the above calculations in normal coordinates, using the fact that the subprincipal symbol is covariant by Theorem [Theorem 18](#theorem subprincipal symbol operators 1 forms){reference-type="ref" reference="theorem subprincipal symbol operators 1 forms"}. ◻ Returning to the analysis of the general case of a Riemannian manifold of arbitrary dimension, let us examine the second order differential operators $\delta d$ and $d\delta$ acting on 1-forms. Similarly to [\[lemma subprincipal symbol curl equation\]](#lemma subprincipal symbol curl equation){reference-type="eqref" reference="lemma subprincipal symbol curl equation"}, one can show that $$\label{subprincipal symbol of delta d} (\delta \mathrm{d})_\mathrm{sub}=0,$$ $$\label{subprincipal symbol of d delta} (\mathrm{d}\delta)_\mathrm{sub}=0.$$ Of course, formulae [\[subprincipal symbol of delta d\]](#subprincipal symbol of delta d){reference-type="eqref" reference="subprincipal symbol of delta d"} and [\[subprincipal symbol of d delta\]](#subprincipal symbol of d delta){reference-type="eqref" reference="subprincipal symbol of d delta"} imply that the Hodge Laplacian $\,\boldsymbol{\Delta}:=-(\delta \mathrm{d}+\mathrm{d}\delta)\,$ acting on 1-forms has zero subprincipal symbol. The inner product $$\label{general inner product on 1-forms} \langle u,v \rangle:=\int_M g^{\alpha\beta}(x)\,\overline{u_\alpha(x)}\,v_\beta(x)\,\rho(x)\,\mathrm{d}x$$ allows us to define the formal adjoint $Q^*$ of a pseudodifferential operator $Q$ acting on 1-forms. **Lemma 24**. *We have $$\begin{aligned} \label{lemma about the formal adjoint of a pseudodifferential operator equation principal} [(Q^*)_\mathrm{prin}]_\mu{}^\nu \!\!\! &= g_{\mu\beta} \, \overline{ [Q_\mathrm{prin}]_\alpha{}^\beta } \, g^{\alpha\nu}\,, \\ \label{lemma about the formal adjoint of a pseudodifferential operator equation subprincipal} [(Q^*)_\mathrm{sub}]_\mu{}^\nu \!\!\! &= g_{\mu\beta} \, \overline{ [Q_\mathrm{sub}]_\alpha{}^\beta } \, g^{\alpha\nu}\,.\end{aligned}$$* *Proof.* Formula [\[lemma about the formal adjoint of a pseudodifferential operator equation principal\]](#lemma about the formal adjoint of a pseudodifferential operator equation principal){reference-type="eqref" reference="lemma about the formal adjoint of a pseudodifferential operator equation principal"} is obvious. As to formula [\[lemma about the formal adjoint of a pseudodifferential operator equation subprincipal\]](#lemma about the formal adjoint of a pseudodifferential operator equation subprincipal){reference-type="eqref" reference="lemma about the formal adjoint of a pseudodifferential operator equation subprincipal"}, it can be established by a lengthy explicit calculation, which shows that all additional terms involving derivatives of the metric and the Riemannian density cancel out. However, a shorter way of proving it is to argue as in the alternative version of the proof of Lemma [Lemma 23](#lemma subprincipal symbol curl){reference-type="ref" reference="lemma subprincipal symbol curl"}: fix an arbitrary point on $M$ and carry out calculations in normal coordinates, using the fact that the subprincipal symbol is covariant by Theorem [Theorem 18](#theorem subprincipal symbol operators 1 forms){reference-type="ref" reference="theorem subprincipal symbol operators 1 forms"}. ◻ **Theorem 25**. *Let $Q$ and $R$ pseudodifferential operators acting on 1-forms. Then $$\label{theorem about subprincipal symbol of composition equation 1} (QR)_\mathrm{sub} = Q_\mathrm{prin} R_\mathrm{sub} + Q_\mathrm{sub} R_\mathrm{prin} +\frac i2 \{\{Q_\mathrm{prin},R_\mathrm{prin}\}\},$$ where $$\begin{gathered} \label{theorem about subprincipal symbol of composition equation 2} \{\{Q_\mathrm{prin},R_\mathrm{prin}\}\}_\alpha{}^\beta := \left( \frac{\partial [Q_\mathrm{prin}]_\alpha{}^\kappa}{\partial x^\gamma} - \Gamma^{\alpha'}{}_{\gamma\alpha}[Q_\mathrm{prin}]_{\alpha'}{}^\kappa + \Gamma^\kappa{}_{\gamma\kappa'}[Q_\mathrm{prin}]_\alpha{}^{\kappa'} \right) \frac{\partial[R_\mathrm{prin}]_\kappa{}^\beta}{\partial\xi_\gamma} \\ - \frac{\partial[Q_\mathrm{prin}]_\alpha{}^\kappa}{\partial\xi_\gamma} \left( \frac{\partial [R_\mathrm{prin}]_\kappa{}^\beta}{\partial x^\gamma} - \Gamma^{\kappa'}{}_{\gamma\kappa}[R_\mathrm{prin}]_{\kappa'}{}^\beta + \Gamma^\beta{}_{\gamma\beta'}[R_\mathrm{prin}]_\kappa{}^{\beta'} \right)\end{gathered}$$ is the generalised Poisson bracket.* *Remark 26*. Note that if we *formally* treat principal symbols as $(1,1)$-tensors on $M$ and put $$\label{6 July 2021 equation 3} \widetilde\nabla_\alpha[Q_\mathrm{prin}]_\beta{}^\gamma := \frac{\partial [Q_\mathrm{prin}]_\beta{}^\gamma}{\partial x^\alpha} - \Gamma^{\beta'}{}_{\alpha\beta}[Q_\mathrm{prin}]_{\beta'}{}^\gamma + \Gamma^\gamma{}_{\alpha\gamma'}[Q_\mathrm{prin}]_\beta{}^{\gamma'},$$ then formula [\[theorem about subprincipal symbol of composition equation 2\]](#theorem about subprincipal symbol of composition equation 2){reference-type="eqref" reference="theorem about subprincipal symbol of composition equation 2"} simplifies and takes the form $$\label{6 July 2021 equation 4} \{\{Q_\mathrm{prin},R_\mathrm{prin}\}\}_\alpha{}^\beta = \widetilde\nabla_\gamma[Q_\mathrm{prin}]_\alpha{}^\kappa \, \frac{\partial[R_\mathrm{prin}]_\kappa{}^\beta}{\partial\xi_\gamma} - \frac{\partial[Q_\mathrm{prin}]_\alpha{}^\kappa}{\partial\xi_\gamma} \, \widetilde\nabla_\gamma[R_\mathrm{prin}]_\kappa{}^\beta\,.$$ Compare with Remark [Remark 21](#remark minus sign){reference-type="ref" reference="remark minus sign"}. It is possible to introduce genuine covariant derivatives on the cotangent bundle and reformulate formulae [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} and [\[theorem about subprincipal symbol of composition equation 2\]](#theorem about subprincipal symbol of composition equation 2){reference-type="eqref" reference="theorem about subprincipal symbol of composition equation 2"} in terms of the latter; see, for example, [@dirac Remark 4.5]. We refrain from doing this because it would take us away from the core subject of our paper. *Proof of Theorem [Theorem 25](#theorem about subprincipal symbol of composition){reference-type="ref" reference="theorem about subprincipal symbol of composition"}.* Let us fix a point $z\in M$. Let $\mathcal{U}_z$ be a small open neighbourhood of $z$ and let $\{\widetilde{e}_j\}_{j=1}^3$ be a Levi-Civita framing centred at $z$. Arguing as in the proof of Theorem [Theorem 18](#theorem subprincipal symbol operators 1 forms){reference-type="ref" reference="theorem subprincipal symbol operators 1 forms"}, we denote by $Q_{1/2}$ and $R_{1/2}$ the pseudodifferential operators acting on half-densities obtained from the operators $Q$ and $R$ (which act on 1-forms), see formula [\[proof subprincipal symbol operators 1 forms equation 1\]](#proof subprincipal symbol operators 1 forms equation 1){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 1"} and following text. The task at hand is to show that, when $x=z$, the RHS of [\[theorem about subprincipal symbol of composition equation 1\]](#theorem about subprincipal symbol of composition equation 1){reference-type="eqref" reference="theorem about subprincipal symbol of composition equation 1"} coincides with the quantity $$\label{proof subprincipal symbol composition equation 1} \left.\left(\widetilde e\,{}^j{}_\alpha\,[(Q_{1/2}R_{1/2})_\mathrm{sub}]_j{}^k \,\widetilde e_k{}^\beta\right)\right|_{x=z}\,,$$ where $(Q_{1/2}R_{1/2})_\mathrm{sub}$ denotes the usual subprincipal symbol for operators acting on half-densities and for which the composition formula is known $$\label{proof subprincipal symbol composition equation 2} (Q_{1/2}R_{1/2})_\mathrm{sub}=(Q_{1/2})_\mathrm{prin}(R_{1/2})_\mathrm{sub}+(Q_{1/2})_\mathrm{sub}(R_{1/2})_\mathrm{prin}+\frac{i}2\{(Q_{1/2})_\mathrm{prin},(R_{1/2})_\mathrm{prin}\}\,.$$ Here $$\label{Poisson bracket on matrix-functions} \{(Q_{1/2})_\mathrm{prin},(R_{1/2})_\mathrm{prin}\} := \frac{\partial(Q_{1/2})_\mathrm{prin}}{\partial x^\gamma} \, \frac{\partial(R_{1/2})_\mathrm{prin}}{\partial\xi_\gamma} \,-\, \frac{\partial(Q_{1/2})_\mathrm{prin}}{\partial\xi_\gamma} \, \frac{\partial(R_{1/2})_\mathrm{prin}}{\partial x^\gamma}$$ is the Poisson bracket on matrix-functions. We have $$\begin{gathered} \label{dima composition proof equation 1} \left. \left( \widetilde e\,{}^j{}_\alpha\,[(Q_{1/2})_\mathrm{prin}(R_{1/2})_\mathrm{sub}]_j{}^k \,\widetilde e_k{}^\beta \right) \right|_{x=z} = \left. \left( \widetilde e\,{}^j{}_\alpha \,[(Q_{1/2})_\mathrm{prin}]_j{}^l \,[(R_{1/2})_\mathrm{sub}]_l{}^k \,\widetilde e_k{}^\beta \right) \right|_{x=z} \\ = \left. \left( \widetilde e\,{}^j{}_\alpha \,[(Q_{1/2})_\mathrm{prin}]_j{}^l \,\widetilde e_l{}^\kappa \,\widetilde e\,{}^m{}_\kappa \,[(R_{1/2})_\mathrm{sub}]_m{}^k \,\widetilde e_k{}^\beta \right) \right|_{x=z} \\ = \left. \left( \widetilde e\,{}^j{}_\alpha \,[(Q_{1/2})_\mathrm{prin}]_j{}^l \,\widetilde e_l{}^\kappa \right) \right|_{x=z} \left. \left( \widetilde e\,{}^m{}_\kappa \,[(R_{1/2})_\mathrm{sub}]_m{}^k \,\widetilde e_k{}^\beta \right) \right|_{x=z} \\ = \left. \left( [Q_\mathrm{prin}]_\alpha{}^\kappa \right) \right|_{x=z} \left. \left( [R_\mathrm{sub}]_\kappa{}^\beta \right) \right|_{x=z} = \left. \left( [Q_\mathrm{prin}\,R_\mathrm{sub}]_\alpha{}^\beta \right) \right|_{x=z}\,.\end{gathered}$$ Here we used the fact that, according to [\[proof subprincipal symbol operators 1 forms equation 2\]](#proof subprincipal symbol operators 1 forms equation 2){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 2"}, the expression $\left. \left( \widetilde e\,{}^m{}_\kappa \,[(R_{1/2})_\mathrm{sub}]_m{}^k \,\widetilde e_k{}^\beta \right) \right|_{x=z}$ is precisely $\left. \left( [R_\mathrm{sub}]_\kappa{}^\beta \right) \right|_{x=z}\,$. Similarly, we have $$\label{dima composition proof equation 2} \left. \left( \widetilde e\,{}^j{}_\alpha\,[(Q_{1/2})_\mathrm{sub}(R_{1/2})_\mathrm{prin}]_j{}^k \,\widetilde e_k{}^\beta \right) \right|_{x=z} = \left. \left( [Q_\mathrm{sub}\,R_\mathrm{prin}]_\alpha{}^\beta \right) \right|_{x=z}\,.$$ In view of [\[dima composition proof equation 1\]](#dima composition proof equation 1){reference-type="eqref" reference="dima composition proof equation 1"} and [\[dima composition proof equation 2\]](#dima composition proof equation 2){reference-type="eqref" reference="dima composition proof equation 2"} the proof of the theorem reduces to establishing the identity $$\label{dima composition proof equation 3} \left. \left( \{\{Q_\mathrm{prin},R_\mathrm{prin}\}\}_\alpha{}^\beta \right) \right|_{x=z} = \left. \left( \widetilde e\,{}^j{}_\alpha\,\{(Q_{1/2})_\mathrm{prin},(R_{1/2})_\mathrm{prin}\}_j{}^k \,\widetilde e_k{}^\beta \right) \right|_{x=z}\,.$$ Recalling that $$\label{proof subprincipal symbol composition equation 3} [(Q_{1/2})_\mathrm{prin}]_j{}^l= \widetilde e_j{}^\mu\,[Q_\mathrm{prin}]_\mu{}^\nu\,\widetilde e\,{}^l{}_\nu\,, \qquad [(R_{1/2})_\mathrm{prin}]_l{}^k= \widetilde e_l{}^\rho\,[R_\mathrm{prin}]_\rho{}^\sigma\,\widetilde e\,{}^k{}_\sigma\,,$$ a straightforward calculation gives us $$\begin{gathered} %\label{proof subprincipal symbol composition equation 4} %\begin{split} \left.\{(Q_{1/2})_\mathrm{prin},(R_{1/2})_\mathrm{prin}\}_j{}^k\right|_{x=z} %& \\ = \Bigl[ \widetilde e_j{}^\mu [Q_\mathrm{prin}]_\mu{}^\nu \dfrac{\partial \widetilde e\,{}^l{}_\nu}{\partial x^\gamma} \widetilde e_l{}^\rho \frac{\partial[R_\mathrm{prin}]_\rho{}^\sigma}{\partial\xi_\gamma} \widetilde e\,{}^k{}_\sigma - \widetilde e_j{}^\mu \frac{\partial[Q_\mathrm{prin}]_\mu{}^\nu}{\partial\xi_\gamma} \widetilde e\,{}^l{}_\nu \dfrac{\partial \widetilde e_l{}^\rho}{\partial x^\gamma}[R_\mathrm{prin}]_\rho{}^\sigma \widetilde e\,{}^k{}_\sigma \\ %& + \frac{\partial \widetilde e_j{}^\mu}{\partial x^\gamma}[Q_\mathrm{prin}]_\mu{}^\nu \frac{\partial[R_\mathrm{prin}]_\nu{}^\sigma}{\partial\xi_\gamma} \widetilde e\,{}^k{}_\sigma - \widetilde e_j{}^\mu \frac{\partial[Q_\mathrm{prin}]_\mu{}^\nu}{\partial\xi_\gamma} [R_\mathrm{prin}]_\nu{}^\sigma \dfrac{\partial \widetilde e\,{}^k{}_\sigma}{\partial x^\gamma} \\ %& \left. + \widetilde e_j{}^\mu\,\{Q_\mathrm{prin},R_\mathrm{prin}\}_\mu{}^\sigma\,\widetilde e\,{}^k{}_\sigma\Bigr]\right|_{x=z}\,, %\end{split}\end{gathered}$$ so that $$\begin{gathered} \label{proof subprincipal symbol composition equation 5} \left. \left( \widetilde e\,{}^j{}_\alpha\,\{(Q_{1/2})_\mathrm{prin},(R_{1/2})_\mathrm{prin}\}_j{}^k\,\widetilde e_k{}^\beta \right) \right|_{x=z} %& \\ = \Bigl[ [Q_\mathrm{prin}]_\alpha{}^\nu \dfrac{\partial \widetilde e\,{}^l{}_\nu}{\partial x^\gamma} \widetilde e_l{}^\rho \frac{\partial[R_\mathrm{prin}]_\rho{}^\beta}{\partial\xi_\gamma} - \frac{\partial[Q_\mathrm{prin}]_\alpha{}^\nu}{\partial\xi_\gamma} \widetilde e\,{}^l{}_\nu \dfrac{\partial \widetilde e_l{}^\rho}{\partial x^\gamma}[R_\mathrm{prin}]_\rho{}^\beta \\ %& + \widetilde e\,{}^j{}_\alpha \frac{\partial \widetilde e_j{}^\mu}{\partial x^\gamma}[Q_\mathrm{prin}]_\mu{}^\nu \frac{\partial[R_\mathrm{prin}]_\nu{}^\beta}{\partial\xi_\gamma} - \frac{\partial[Q_\mathrm{prin}]_\alpha{}^\nu}{\partial\xi_\gamma} [R_\mathrm{prin}]_\nu{}^\sigma \dfrac{\partial \widetilde e\,{}^k{}_\sigma}{\partial x^\gamma}\widetilde e_k{}^\beta \\ %& \left. + \{Q_\mathrm{prin},R_\mathrm{prin}\}_\alpha{}^\beta\Bigr]\right|_{x=z}\,.\end{gathered}$$ On account of [\[proof subprincipal symbol operators 1 forms equation 10\]](#proof subprincipal symbol operators 1 forms equation 10){reference-type="eqref" reference="proof subprincipal symbol operators 1 forms equation 10"} and [\[theorem about subprincipal symbol of composition equation 2\]](#theorem about subprincipal symbol of composition equation 2){reference-type="eqref" reference="theorem about subprincipal symbol of composition equation 2"}, formula [\[proof subprincipal symbol composition equation 5\]](#proof subprincipal symbol composition equation 5){reference-type="eqref" reference="proof subprincipal symbol composition equation 5"} implies [\[dima composition proof equation 3\]](#dima composition proof equation 3){reference-type="eqref" reference="dima composition proof equation 3"}. ◻ **Example 27**. Let $M$ be an oriented 3-manifold. Then Theorem [Theorem 25](#theorem about subprincipal symbol of composition){reference-type="ref" reference="theorem about subprincipal symbol of composition"} and Lemma [Lemma 23](#lemma subprincipal symbol curl){reference-type="ref" reference="lemma subprincipal symbol curl"} imply $$\label{subprincipal symbol curl squared} (\operatorname{curl}^2)_\mathrm{sub}=0.$$ Alternatively, formula [\[subprincipal symbol curl squared\]](#subprincipal symbol curl squared){reference-type="eqref" reference="subprincipal symbol curl squared"} follows from [\[subprincipal symbol of delta d\]](#subprincipal symbol of delta d){reference-type="eqref" reference="subprincipal symbol of delta d"} and the fact that $\operatorname{curl}^2=\delta \mathrm{d}$. # Trace of pseudodifferential operators acting on 1-forms {#Trace of pseudodifferential operators acting on 1-forms} The aim of this section is to extend, in a natural way, the notion of trace of a pseudodifferential operator acting on 1-forms. Given a pseudodifferential operator $Q$ acting on 1-forms, we will define a scalar operator $\operatorname{tr}_\epsilon Q$ is such a way that, if $Q$ is of trace class, then the operator-theoretic trace of $Q$ coincides with the operator-theoretic trace of $\operatorname{tr}_\epsilon Q$. Later in the paper we will use this definition without assuming that $Q$ is of trace class: it may happen that $Q$ itself is not of trace class, but $\operatorname{tr}_\epsilon Q$ is --- which is the case, up to an additional minor regularisation, for $Q=\theta(\operatorname{curl})-\theta(-\operatorname{curl})$. In this section, as in the previous one, we work on a closed Riemannian manifold $(M,g)$ of arbitrary dimension $d$. Let us start with the simpler case when $Q$ is a pseudodifferential operator of order $s$ acting on $m$-columns of scalar fields. Suppose that $$\label{s less than d} s<d.$$ Then $Q$ is of trace class [@shubin Corollary 27.1]. Furthermore [@shubin §12.1], in this case $Q$ is an integral operator with continuous integral kernel, $$\label{Q acting on column of scalars 1} Q: u_j(x)\mapsto\int_M\mathfrak{q}_j{}^k(x,y)\,u_k(y)\,\rho(y)\,\mathrm{d}y\,.$$ We introduced the factor $\rho(y)$ in the above integral in order to make the integral kernel $\mathfrak{q}_j{}^k(x,y)$ a (matrix-valued) scalar function on $M\times M$. We have $$\label{Q acting on column of scalars 2} \operatorname{Tr}Q=\int_M(\operatorname{tr}\mathfrak{q})(x,x)\,\rho(x)\,\mathrm{d}x\,,$$ where $(\operatorname{tr}\mathfrak{q})(x,y):=\mathfrak{q}_j{}^j(x,y)$ is the trace of the matrix-function $\mathfrak{q}_j{}^k(x,y)$. We define the scalar operator $$\label{Q acting on column of scalars 3} \operatorname{tr}Q: f(x)\mapsto\int_M (\operatorname{tr}\mathfrak{q})(x,y)\,f(y)\,\rho(y)\,\mathrm{d}y$$ and call it *the matrix trace of the operator $Q$*. It is easy to see that $$\label{Q acting on column of scalars 4} \operatorname{Tr}Q=\operatorname{Tr}(\operatorname{tr}Q).$$ Note that [\[Q acting on column of scalars 3\]](#Q acting on column of scalars 3){reference-type="eqref" reference="Q acting on column of scalars 3"} is well defined for any $s\in\mathbb{R}$, not necessarily satisfying condition [\[s less than d\]](#s less than d){reference-type="eqref" reference="s less than d"}. More precisely, if $Q$ is of order $s$, it is easy to see that (i) $\operatorname{tr}Q$ is also an operator of order $s$, (ii) $(\operatorname{tr}Q)^*=\operatorname{tr}(Q^*)$, where the star refers to formal adjoints with repsect to the natural inner products, (iii) $(\operatorname{tr}Q)_\mathrm{prin}=\operatorname{tr}(Q_\mathrm{prin})$, (iv) $(\operatorname{tr}Q)_\mathrm{sub}=\operatorname{tr}(Q_\mathrm{sub})$. Of course, when condition [\[s less than d\]](#s less than d){reference-type="eqref" reference="s less than d"} is not satisfied the integral kernel $\mathfrak{q}_j{}^k(x,y)$ appearing in formula [\[Q acting on column of scalars 1\]](#Q acting on column of scalars 1){reference-type="eqref" reference="Q acting on column of scalars 1"} should be understood in the distributional sense (Schwartz kernel), but this does not prevent us from taking its matrix trace. The distribution $(\operatorname{tr}\mathfrak{q})(x,y)$ will be the Schwartz kernel of the scalar operator [\[Q acting on column of scalars 3\]](#Q acting on column of scalars 3){reference-type="eqref" reference="Q acting on column of scalars 3"}. We will now adapt the above construction to the case of a pseudodifferential operator $$\label{Q acting on 1-forms 1} Q: u_\alpha(x)\mapsto\int_M \mathfrak{q}_\alpha{}^\beta(x,y)\,u_\beta(y)\,\rho(y)\,\mathrm{d}y$$ acting on 1-forms. Here we encounter a problem: the quantity $\mathfrak{q}_\alpha{}^\alpha(x,y)$ is not a scalar on $M\times M$ because the tensor indices $\alpha$ and $\beta$ in $q_\alpha{}^\beta(x,y)$ 'live' at different points, $x$ and $y$ respectively. More precisely, the problem here is that the Schwartz kernel $\mathfrak{q}_\alpha{}^\beta(x,y)$ of the operator [\[Q acting on 1-forms 1\]](#Q acting on 1-forms 1){reference-type="eqref" reference="Q acting on 1-forms 1"} is a two-point tensor: it is a covector at $x$ in the tensor index $\alpha$ and a vector at $y$ in the tensor index $\beta$. We overcome this impediment as follows. Let $x$ and $y$ be two points on the manifold $M$ which are sufficiently close. Then there is a unique shortest geodesic $\gamma(x,y)$ connecting $x$ and $y$. Given a point $z\in\gamma(x,y)$, put $$\label{Q acting on 1-forms 2} t=\frac{\operatorname{dist}(z,x)}{\operatorname{dist}(x,y)}\,,$$ which is the variable arc length of the geodesic normalised by its total length. The variable $t$ provides a convenient parameterisation of our geodesic $\gamma(x,y;\,\cdot\,):[0,1]\to M$, so that $\gamma(x,y;0)=x$ and $\gamma(x,y;1)=y$. We denote by $$\label{Q acting on 1-forms 3} Z: T_xM\ni u^\alpha\mapsto u^\alpha\,Z_\alpha{}^\beta(x,y)\in T_yM$$ the linear map realising the parallel transport of vectors from $x$ to $y$ along the unique shortest geodesic connecting $x$ and $y$. Note that the result of parallel transport is independent of the choice of the particular parameterisation of the curve along which it takes place. In what follows we will be raising and lowering indices in the 2-point tensor $Z_\alpha{}^\beta(x,y)$ using the Riemannian metric $g(x)$ in the first index and $g(y)$ in the second. Of course, $$\label{Q acting on 1-forms 3a extra} Z_\alpha{}^\kappa(x,y)\,Z_\kappa{}^\beta(y,x) \,=\, \delta_\alpha{}^\beta\,, \qquad Z^\alpha{}_\kappa(x,y)\,Z^\kappa{}_\beta(y,x) \,=\, \delta^\alpha{}_\beta\,,$$ $$\label{Q acting on 1-forms 3a} Z_\alpha{}^\kappa(x,y)\,Z^\beta{}_\kappa(x,y) \,=\, \delta_\alpha{}^\beta\,, \qquad Z_\kappa{}^\alpha(x,y)\,Z^\kappa{}_\beta(x,y) \,=\, \delta^\alpha{}_\beta\,.$$ Formula [\[Q acting on 1-forms 3a extra\]](#Q acting on 1-forms 3a extra){reference-type="eqref" reference="Q acting on 1-forms 3a extra"} expresses the fact that if we parallel transport a vector/covector from $x$ to $y$ along the shortest geodesic connecting these two points, and then parallel transport it back from $y$ to $x$ along the same geodesic, we get the original vector/covector. Formula [\[Q acting on 1-forms 3a\]](#Q acting on 1-forms 3a){reference-type="eqref" reference="Q acting on 1-forms 3a"} is a consequence of the fact that the Levi-Civita connection is metric compatible. *Remark 28*. For later use, let us observe that formulae [\[Q acting on 1-forms 3a extra\]](#Q acting on 1-forms 3a extra){reference-type="eqref" reference="Q acting on 1-forms 3a extra"} and [\[Q acting on 1-forms 3a\]](#Q acting on 1-forms 3a){reference-type="eqref" reference="Q acting on 1-forms 3a"} imply $$\label{aux4} Z_\beta{}^\alpha(x,y) = Z^\alpha{}_\beta(y,x).$$ Indeed, take the first identity [\[Q acting on 1-forms 3a extra\]](#Q acting on 1-forms 3a extra){reference-type="eqref" reference="Q acting on 1-forms 3a extra"}, multiply it by $Z^\alpha{}_\gamma(x,y)$ and then use the second identity [\[Q acting on 1-forms 3a\]](#Q acting on 1-forms 3a){reference-type="eqref" reference="Q acting on 1-forms 3a"}. This gives us $\,\delta^\kappa{}_\gamma\,Z_\kappa{}^\beta(y,x) \,=\, \delta_\alpha{}^\beta \,Z^\alpha{}_\gamma(x,y) \,$, which is equivalent to [\[aux4\]](#aux4){reference-type="eqref" reference="aux4"}. Let us introduce a one-parameter family of scalar distributions defined by $$\label{Q acting on 1-forms 5} (\operatorname{tr}_\tau \mathfrak{q})(x,y) := Z^\alpha{}_\kappa(x,\gamma(x,y;\tau)) \, \mathfrak{q}_{\alpha}{}^\beta(x,y) \, Z_\beta{}^\kappa(y,\gamma(x,y;\tau))\,,$$ where $\mathfrak{q}_{\alpha}{}^\beta(x,y)$ is the Schwartz kernel from [\[Q acting on 1-forms 1\]](#Q acting on 1-forms 1){reference-type="eqref" reference="Q acting on 1-forms 1"} and $\tau\in[0,1]$. What happens here is that tensor indices are parallel transported to the common point $z=\gamma(x,y;\tau)$ on the geodesic $\gamma(x,y)$ connecting $x$ and $y$. Thus, the quantity $(\operatorname{tr}_\tau\mathfrak{q})(x,y)$ is a genuine scalar. As we are dealing with distributions we need to clarify the precise meaning of formula [\[Q acting on 1-forms 5\]](#Q acting on 1-forms 5){reference-type="eqref" reference="Q acting on 1-forms 5"}. Consider a map from smooth scalar functions of two variables to smooth $(1,1)$ two-point tensors defined as $$C^\infty(M\times M)\ni f(x,y) \mapsto Z^\alpha{}_\kappa(x,\gamma(x,y;\tau)) \, Z_\beta{}^\kappa(y,\gamma(x,y;\tau)) \, f(x,y)$$ The quantity $\mathfrak{q}_{\alpha}{}^\beta(x,y)$ in [\[Q acting on 1-forms 5\]](#Q acting on 1-forms 5){reference-type="eqref" reference="Q acting on 1-forms 5"} is a distribution (continuous complex-valued linear functional) acting on smooth $(1,1)$ two-point tensors. Formula [\[Q acting on 1-forms 5\]](#Q acting on 1-forms 5){reference-type="eqref" reference="Q acting on 1-forms 5"} should be understood as a composition of the latter with the former. **Lemma 29**. *The quantity [\[Q acting on 1-forms 5\]](#Q acting on 1-forms 5){reference-type="eqref" reference="Q acting on 1-forms 5"} does not depend on $\tau$.* *Proof.* The statement of the lemma is an immediate consequence of the fact that the Levi-Civita connection is metric compatible. ◻ Since the definition [\[Q acting on 1-forms 5\]](#Q acting on 1-forms 5){reference-type="eqref" reference="Q acting on 1-forms 5"} is independent of the choice of $\tau$, henceforth we will set, for convenience, $\tau=0$, in which case formula [\[Q acting on 1-forms 5\]](#Q acting on 1-forms 5){reference-type="eqref" reference="Q acting on 1-forms 5"} simplifies and reads $$\label{Q acting on 1-forms 5 with tau zero} (\operatorname{tr}\mathfrak{q})(x,y) := \mathfrak{q}_{\alpha}{}^\beta(x,y) \, Z_\beta{}^\alpha(y,x)\,.$$ Furthermore, we drop the subscript $\tau$. There is still one problem with formula [\[Q acting on 1-forms 5 with tau zero\]](#Q acting on 1-forms 5 with tau zero){reference-type="eqref" reference="Q acting on 1-forms 5 with tau zero"}: the linear operator $Z$ appearing in this formula is defined only for $x$ and $y$ sufficiently close. In order to view [\[Q acting on 1-forms 5 with tau zero\]](#Q acting on 1-forms 5 with tau zero){reference-type="eqref" reference="Q acting on 1-forms 5 with tau zero"} as a well-defined distribution we need a cut-off. Let $\chi:[0,+\infty)\to \mathbb{R}$ be a compactly supported infinitely smooth scalar function such that $\chi=1$ in a neighbourhood of zero. We modify formula [\[Q acting on 1-forms 5\]](#Q acting on 1-forms 5){reference-type="eqref" reference="Q acting on 1-forms 5"} to read $$\label{Q acting on 1-forms 6} (\operatorname{tr}_\epsilon\mathfrak{q})(x,y) := \mathfrak{q}_{\alpha}{}^\beta(x,y) \, Z_\beta{}^\alpha(y,x) \,\chi(\operatorname{dist}(x,y)/\epsilon)\,,$$ where $\epsilon>0$ is a small parameter which ensures that $(\operatorname{tr}_\epsilon\mathfrak{q})(x,y)$ vanishes when $x$ and $y$ are not sufficiently close. **Definition 30**. The scalar operator $$\label{Q acting on 1-forms 7} \operatorname{tr}_\epsilon Q: f(x)\mapsto\int_M (\operatorname{tr}_\epsilon\mathfrak{q})(x,y)\,f(y)\,\rho(y)\,\mathrm{d}y$$ is called *the matrix trace* of the operator [\[Q acting on 1-forms 1\]](#Q acting on 1-forms 1){reference-type="eqref" reference="Q acting on 1-forms 1"}. Thus, given a pseudodifferential operator $Q$ acting on 1-forms we defined the scalar pseudodifferential operator $\operatorname{tr}_\epsilon Q\,$. The latter depends on the small parameter $\epsilon>0$, but, as we will see, the choice of this parameter does not affect the main results of our paper. **Proposition 31**. (a) *Under the condition [\[s less than d\]](#s less than d){reference-type="eqref" reference="s less than d"} we have $$\label{properties of matrix trace of a pdo acting on 1-forms lemma equation 3} \operatorname{Tr}(\operatorname{tr}_\epsilon Q)=\operatorname{Tr}Q\,.$$* (b) *We have $$\label{properties of matrix trace of a pdo acting on 1-forms lemma equation 4} (\operatorname{tr}_\epsilon Q)^*=\operatorname{tr}_\epsilon(Q^*)\,,$$ where the star refers to formal adjoints with respect to the natural inner products on scalar functions and 1-forms respectively.* (c) *We have $$\label{properties of matrix trace of a pdo acting on 1-forms lemma equation 1} (\operatorname{tr}_\epsilon Q)_\mathrm{prin}=\operatorname{tr}Q_\mathrm{prin}\,,$$ $$\label{properties of matrix trace of a pdo acting on 1-forms lemma equation 2} (\operatorname{tr}_\epsilon Q)_\mathrm{sub}=\operatorname{tr}Q_\mathrm{sub}\,.$$* *Remark 32*. Let us emphasise that, crucially, the right-hand sides of [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 3\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 3){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 3"}, [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 1\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 1){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 1"} and [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 2\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 2){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 2"} do not depend on $\epsilon$. *Proof of Proposition [Proposition 31](#properties of matrix trace of a pdo acting on 1-forms lemma){reference-type="ref" reference="properties of matrix trace of a pdo acting on 1-forms lemma"}.* (a) Formula [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 3\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 3){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 3"} follows from the fact that $\,Z_\alpha{}^\beta(x,x)=\delta_\alpha{}^\beta\,$ and $\chi(0)=1$.   \(b\) We have $$\begin{gathered} \label{aux1} (\operatorname{tr}_\epsilon Q)^*: f(x)\mapsto\int_M \overline { (\operatorname{tr}_\epsilon\mathfrak{q})(y,x) } \,f(y)\,\rho(y)\,\mathrm{d}y \\ = \int_M \overline { \mathfrak{q}_{\alpha}{}^\beta(y,x) } \, Z_\beta{}^\alpha(x,y) \,\chi(\operatorname{dist}(x,y)/\epsilon) \,f(y)\,\rho(y)\,\mathrm{d}y\,,\end{gathered}$$ $$\label{aux2} Q^*: u_\alpha(x)\mapsto\int_M \overline { \mathfrak{q}^\beta{}_\alpha(y,x) } \,u_\beta(y)\,\rho(y)\,\mathrm{d}y\,,$$ $$\begin{gathered} \label{aux3} \operatorname{tr}_\epsilon(Q^*): f(x)\mapsto \int_M \overline { \mathfrak{q}^\beta{}_\alpha(y,x) } \, Z_\beta{}^\alpha(y,x) \,\chi(\operatorname{dist}(x,y)/\epsilon) \,f(y)\,\rho(y)\,\mathrm{d}y \\ = \int_M \overline { \mathfrak{q}_\alpha{}^\beta(y,x) } \, Z^\alpha{}_\beta(y,x) \,\chi(\operatorname{dist}(x,y)/\epsilon) \,f(y)\,\rho(y)\,\mathrm{d}y\,.\end{gathered}$$ Formulae [\[aux1\]](#aux1){reference-type="eqref" reference="aux1"}, [\[aux3\]](#aux3){reference-type="eqref" reference="aux3"} and [\[aux4\]](#aux4){reference-type="eqref" reference="aux4"} imply [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 4\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 4){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 4"}.   \(c\) Formula [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 1\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 1){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 1"} follows from the fact that $\,Z_\alpha{}^\beta(x,x)=\delta_\alpha{}^\beta\,$ and $\chi(0)=1$. The proof of [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 2\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 2){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 2"} is more subtle. Formula [\[11 June 2021 equation 1\]](#11 June 2021 equation 1){reference-type="eqref" reference="11 June 2021 equation 1"} tells us that, modulo an infinitely smooth contribution, the Schwartz kernel $\mathfrak{q}_\alpha{}^\beta(x,y)$ of the operator [\[Q acting on 1-forms 1\]](#Q acting on 1-forms 1){reference-type="eqref" reference="Q acting on 1-forms 1"} reads $$\label{properties of matrix trace of a pdo acting on 1-forms lemma proof equation 1} \mathfrak{q}_\alpha{}^\beta(x,y) = \frac{1}{(2\pi)^d\,\rho(y)} \int e^{i(x-y)^\gamma\xi_\gamma}\,q_\alpha{}^\beta(x,\xi) \,\mathrm{d}\xi\,,$$ where the symbol $q_\alpha{}^\beta(x,\xi)$ admits the asymptotic expansion [\[11 June 2021 equation 2\]](#11 June 2021 equation 2){reference-type="eqref" reference="11 June 2021 equation 2"}. We also have $$\label{properties of matrix trace of a pdo acting on 1-forms lemma proof equation 2} Z_\alpha{}^\beta(x,y)=\delta_\alpha{}^\beta -\Gamma^\beta{}_{\lambda\alpha}(x)\,(y-x)^\lambda +O(|y-x|^2)\,.$$ Substituting [\[properties of matrix trace of a pdo acting on 1-forms lemma proof equation 1\]](#properties of matrix trace of a pdo acting on 1-forms lemma proof equation 1){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma proof equation 1"} and [\[properties of matrix trace of a pdo acting on 1-forms lemma proof equation 2\]](#properties of matrix trace of a pdo acting on 1-forms lemma proof equation 2){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma proof equation 2"} into [\[Q acting on 1-forms 6\]](#Q acting on 1-forms 6){reference-type="eqref" reference="Q acting on 1-forms 6"} we arrive at a representation of the scalar pseudodifferential operator [\[Q acting on 1-forms 7\]](#Q acting on 1-forms 7){reference-type="eqref" reference="Q acting on 1-forms 7"} with amplitude depending on $x$, $y$ and $\xi$. In order to write down the (left) symbol of the operator [\[Q acting on 1-forms 7\]](#Q acting on 1-forms 7){reference-type="eqref" reference="Q acting on 1-forms 7"} we need to exclude the dependence on $y$, which is achieved in the standard manner, by means of integration by parts on the basis of the formula $$\label{properties of matrix trace of a pdo acting on 1-forms lemma proof equation 4} (x-y)^\lambda \,e^{i(x-y)^\gamma\xi_\gamma} =-i\,\frac{\partial e^{i(x-y)^\gamma\xi_\gamma}}{\partial\xi_\lambda}\,.$$ This produces the following expressions for the leading and subleading terms of the symbol $(\operatorname{tr}_\epsilon q)(x,\xi)$ of the scalar pseudodifferential operator [\[Q acting on 1-forms 7\]](#Q acting on 1-forms 7){reference-type="eqref" reference="Q acting on 1-forms 7"}: $$\label{properties of matrix trace of a pdo acting on 1-forms lemma proof equation 5} (\operatorname{tr}_\epsilon q)_s = [q_s]_\alpha{}^\alpha\,,$$ $$\label{properties of matrix trace of a pdo acting on 1-forms lemma proof equation 6} (\operatorname{tr}_\epsilon q)_{s-1} = [q_{s-1}]_\alpha{}^\alpha \,-\, i \,\Gamma^\alpha{}_{\gamma\beta} \,\frac{\partial[q_s]_\alpha{}^\beta}{\partial\xi_\gamma}\,.$$ Note that the right-hand sides of [\[properties of matrix trace of a pdo acting on 1-forms lemma proof equation 5\]](#properties of matrix trace of a pdo acting on 1-forms lemma proof equation 5){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma proof equation 5"} and [\[properties of matrix trace of a pdo acting on 1-forms lemma proof equation 6\]](#properties of matrix trace of a pdo acting on 1-forms lemma proof equation 6){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma proof equation 6"} do not depend on the parameter $\epsilon$. Substituting [\[properties of matrix trace of a pdo acting on 1-forms lemma proof equation 5\]](#properties of matrix trace of a pdo acting on 1-forms lemma proof equation 5){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma proof equation 5"} and [\[properties of matrix trace of a pdo acting on 1-forms lemma proof equation 6\]](#properties of matrix trace of a pdo acting on 1-forms lemma proof equation 6){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma proof equation 6"} into [\[subprincipal symbol of operator acting on a scalar field\]](#subprincipal symbol of operator acting on a scalar field){reference-type="eqref" reference="subprincipal symbol of operator acting on a scalar field"} and using [\[relation between Riemannian density and Christoffel symbols\]](#relation between Riemannian density and Christoffel symbols){reference-type="eqref" reference="relation between Riemannian density and Christoffel symbols"}, we get $$\label{properties of matrix trace of a pdo acting on 1-forms lemma proof equation 7} (\operatorname{tr}_\epsilon Q)_\mathrm{sub} = [q_{s-1}]_\alpha{}^\alpha \,+\, \frac{i}2\dfrac{\partial^2 [q_s]_\alpha{}^\alpha}{\partial x^\gamma\partial\xi_\gamma} \,+\, \frac{i}2 \,\Gamma^\beta{}_{\gamma\beta} \,\frac{\partial [q_s]_\alpha{}^\alpha}{\partial\xi_\gamma} \,-\, i \,\Gamma^\alpha{}_{\gamma\beta} \,\frac{\partial[q_s]_\alpha{}^\beta}{\partial\xi_\gamma}\,.$$ But taking the trace of [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} gives the same expression as in the right-hand side of [\[properties of matrix trace of a pdo acting on 1-forms lemma proof equation 7\]](#properties of matrix trace of a pdo acting on 1-forms lemma proof equation 7){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma proof equation 7"}. This completes the proof of formula [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 2\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 2){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 2"}. Alternatively, one can prove formula [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 2\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 2){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 2"} by fixing an arbitrary point on $M$ and carrying out calculations in normal coordinates, using the covariance of subprincipal symbols defined in accordance with formulae [\[subprincipal symbol of operator acting on a scalar field\]](#subprincipal symbol of operator acting on a scalar field){reference-type="eqref" reference="subprincipal symbol of operator acting on a scalar field"} and [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"}. With this approach all the Christoffel symbols disappear. ◻ Proposition [Proposition 31](#properties of matrix trace of a pdo acting on 1-forms lemma){reference-type="ref" reference="properties of matrix trace of a pdo acting on 1-forms lemma"} part (b) immediately implies **Corollary 33**. *If $Q$ is formally self-adjoint then so is $\operatorname{tr}_\epsilon Q\,$.* # The projection operators $P_0$ and $P_\pm$ {#The operators P0 and Ppm} This section is devoted to the study of the operators $P_0$ and $P_\pm$. We will show that they are pseudodifferential operators whose full symbols can be constructed via an explicit algorithm. Furthermore, we will compute their principal and subprincipal symbols, thus proving Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"}.   By $\Psi^s$ we denote the space of classical pseudodifferential operators of order $s$ with polyhomogeneous symbols acting on 1-forms, recall [\[11 June 2021 equation 1\]](#11 June 2021 equation 1){reference-type="eqref" reference="11 June 2021 equation 1"}, [\[11 June 2021 equation 2\]](#11 June 2021 equation 2){reference-type="eqref" reference="11 June 2021 equation 2"}. We define $$\label{Psi minus infinity} \Psi^{-\infty}:=\bigcap_s \Psi^s$$ and we write $Q=R \mod \Psi^{-\infty}$ if $Q-R$ is an integral operator with (infinitely) smooth integral kernel.   Let us begin by proving parts (a) and (b) of Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"}. The matter of subprincipal symbols (part (c) of Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"}) and the construction of our operators will be addressed afterwards. *Proof of Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"}, (a) and (b).* Let $f_j$ be the orthonormalised eigenfunctions corresponding to the eigenvalues [\[eigenvalues of Laplace\--Beltrami operator\]](#eigenvalues of Laplace--Beltrami operator){reference-type="eqref" reference="eigenvalues of Laplace--Beltrami operator"} of $\,-\Delta\,$. Put $$\label{8 November 2022 equation 2} v_j:=\mu_j^{-1/2}\mathrm{d}f_j,\qquad j=1,2,\dots.$$ The $v_j$ form an orthonormal basis in the Hilbert space $d\Omega^0(M)$ and the operator $P_0$ can be written as $$\label{8 November 2022 equation 3} P_0:=\sum_{j=1}^{+\infty}v_j\langle v_j\,,\,\cdot\,\rangle.$$ Let $$\label{8 November 2022 equation 4} \Delta^{-1} := -\sum_{j=1}^{+\infty}\mu_j^{-1}f_j\langle f_j\,,\,\cdot\,\rangle$$ be the pseudoinverse of the Laplace--Beltrami operator $\Delta$. Combining formulae [\[8 November 2022 equation 2\]](#8 November 2022 equation 2){reference-type="eqref" reference="8 November 2022 equation 2"}--[\[8 November 2022 equation 4\]](#8 November 2022 equation 4){reference-type="eqref" reference="8 November 2022 equation 4"} we get an explicit formula for the projection operator $P_0$ $$\label{8 November 2022 equation 5} P_0=-\mathrm{d}\,\Delta^{-1}\delta\,.$$ In particular, formula [\[8 November 2022 equation 5\]](#8 November 2022 equation 5){reference-type="eqref" reference="8 November 2022 equation 5"} tells us that $P_0$ is a pseudodifferential operator of order zero and its principal symbol is given by formula [\[main theorem 1 equation 1\]](#main theorem 1 equation 1){reference-type="eqref" reference="main theorem 1 equation 1"}. Consider the operator $\operatorname{curl}$ in the Hilbert space $\delta\Omega^2(M)$. Let $\lambda_j$ be its eigenvalues and $u_j$ its orthonormalised eigenforms. Here we enumerate using positive integers $j$ for positive eigenvalues and negative integers $j$ for negative eigenvalues, so that $$\label{6 November 2022 equation 1} \ldots\le\lambda_{-2}\le\lambda_{-1}<0<\lambda_1\le\lambda_2\le\ldots,$$ with account of multiplicities. The $u_j$ form an orthonormal basis in the Hilbert space $\delta\Omega^2(M)$ and the operators $P_\pm$ can be written as $$\label{7 November 2022 equation 1} P_+=\sum_{j=1}^{+\infty}u_j\langle u_j\,,\,\cdot\,\rangle,$$ $$\label{7 November 2022 equation 2} P_-=\sum_{j=-1}^{-\infty}u_j\langle u_j\,,\,\cdot\,\rangle.$$ We have $$\label{formula for Pplus plus Pminus} P_++P_-=\operatorname{Id}-P_0-P_{\mathcal{H}^1}\,,$$ where $\operatorname{Id}$ is the identity operator on 1-forms and $P_{\mathcal{H}^1}$ is the orthogonal projection onto the finite-dimensional space of harmonic 1-forms. Of course, $P_{\mathcal{H}^1}$ is an integral operator with infinitely smooth integral kernel, $$\label{PH is smooth} P_{\mathcal{H}^1}\in\Psi^{-\infty}.$$ Consider the Hodge Laplacian $$\label{Hodge Laplacian} \boldsymbol{\Delta}:=-(\mathrm{d}\delta+\delta \mathrm{d}) = - \sum_{j=1}^{+\infty}\mu_j\,v_j\langle v_j\,,\,\cdot\,\rangle - \sum_{j\in\mathbb{Z}\setminus\{0\}}\lambda_j^2\,u_j\langle u_j\,,\,\cdot\,\rangle$$ acting on 1-forms. This is an elliptic self-adjoint nonpositive differential operator which commutes with the differential operator $\operatorname{curl}$. Given an $s\in\mathbb{R}$, the operator $$\label{Hodge Laplacian to the power s} (-\boldsymbol{\Delta})^{s} := \sum_{j=1}^{+\infty}\mu_j^{s}\,v_j\langle v_j\,,\,\cdot\,\rangle + \sum_{j\in\mathbb{Z}\setminus\{0\}}|\lambda_j|^{2s}\,u_j\langle u_j\,,\,\cdot\,\rangle$$ is a pseudodifferential operator of order $2s$. It is easy to see that $$\label{formula for Pplus minus Pminus} P_+-P_- = (-\boldsymbol{\Delta})^{-1/2} \, \operatorname{curl} = (-\boldsymbol{\Delta})^{-1/4} \, \operatorname{curl} \, (-\boldsymbol{\Delta})^{-1/4} = \operatorname{curl} \, (-\boldsymbol{\Delta})^{-1/2} \,.$$ Equations [\[formula for Pplus plus Pminus\]](#formula for Pplus plus Pminus){reference-type="eqref" reference="formula for Pplus plus Pminus"} and [\[formula for Pplus minus Pminus\]](#formula for Pplus minus Pminus){reference-type="eqref" reference="formula for Pplus minus Pminus"} are a system of two linear algebraic equations for the two unknowns $P_+$ and $P_-\,$, whose unique solution is $$\label{whose unique solution is} P_\pm = \frac{1}{2} \left[ \operatorname{Id}-P_0-P_{\mathcal{H}^1} \pm (-\boldsymbol{\Delta})^{-1/4} \, \operatorname{curl} \, (-\boldsymbol{\Delta})^{-1/4} \right].$$ We have $$\label{principal symbol of the identity operator} [\operatorname{Id}_\mathrm{prin}]_\alpha{}^\beta = \delta_\alpha{}^\beta\,,$$ $$\label{principal symbol of the Hodge Laplacian} [-\boldsymbol{\Delta}_\mathrm{prin}]_\alpha{}^\beta = \|\xi\|^{2}\, \delta_\alpha{}^\beta\,.$$ Formulae [\[whose unique solution is\]](#whose unique solution is){reference-type="eqref" reference="whose unique solution is"}--[\[principal symbol of the Hodge Laplacian\]](#principal symbol of the Hodge Laplacian){reference-type="eqref" reference="principal symbol of the Hodge Laplacian"}, [\[main theorem 1 equation 1\]](#main theorem 1 equation 1){reference-type="eqref" reference="main theorem 1 equation 1"}, [\[PH is smooth\]](#PH is smooth){reference-type="eqref" reference="PH is smooth"} and [\[principal symbol curl\]](#principal symbol curl){reference-type="eqref" reference="principal symbol curl"} imply [\[main theorem 1 equation 2\]](#main theorem 1 equation 2){reference-type="eqref" reference="main theorem 1 equation 2"}. ◻ Recall that, as explained in Section [2](#The operator curl){reference-type="ref" reference="The operator curl"}, $\operatorname{curl}_\mathrm{prin}$ has three simple eigenvalues $h^{(\aleph)}$, $\aleph\in\{0,+,-\}$, given by [\[eigenvalues principal symbol curl\]](#eigenvalues principal symbol curl){reference-type="eqref" reference="eigenvalues principal symbol curl"}. Let us denote by $P^{(\aleph)}$, $\aleph\in\{0,+,-\}$, the corresponding eigenprojections. It is easy to see that these are exactly the principal symbols [\[main theorem 1 equation 1\]](#main theorem 1 equation 1){reference-type="eqref" reference="main theorem 1 equation 1"}, [\[main theorem 1 equation 2\]](#main theorem 1 equation 2){reference-type="eqref" reference="main theorem 1 equation 2"} of our pseudodifferential operators $P_\aleph$, $\aleph \in\{0,+,-\}$. Observe that the pseudodifferential operators $P_\aleph$, $\aleph \in\{0,+,-\}$, satisfy $$\label{theorem about projections equation 2} P_\aleph^*= P_\aleph, \qquad P_\aleph P_\beth= \delta_{\aleph\beth} P_\aleph\,, \qquad \sum_\aleph \ P_\aleph=\mathrm{Id}\,, \qquad [\operatorname{curl}, P_\aleph]=0\,,$$ where $\delta$ is the Kronecker delta. Next, let us examine what happens if we relax [\[theorem about projections equation 2\]](#theorem about projections equation 2){reference-type="eqref" reference="theorem about projections equation 2"}, asking that they are satisfied not exactly, but only modulo $\Psi^{-\infty}$, cf. [\[formula for Pplus plus Pminus\]](#formula for Pplus plus Pminus){reference-type="eqref" reference="formula for Pplus plus Pminus"} and [\[PH is smooth\]](#PH is smooth){reference-type="eqref" reference="PH is smooth"}. Namely, let us seek operators $\widetilde P_\aleph\in\Psi^0$, $\aleph\in\{0,+,-\}$, satisfying [\[theorem about projections tilde equation all\]]{#theorem about projections tilde equation all label="theorem about projections tilde equation all"} $$\label{theorem about projections tilde equation 2} (\widetilde P_\aleph)^*=\widetilde P_\aleph \mod \Psi^{-\infty},$$ $$\label{theorem about projections tilde equation 3} \widetilde P_\aleph \widetilde P_\beth= \delta_{\aleph\beth} \widetilde P_\aleph\mod \Psi^{-\infty}\,,$$ $$\label{theorem about projections tilde equation 4} \sum_\aleph \widetilde P_\aleph=\mathrm{Id}\mod \Psi^{-\infty}\,,$$ $$\label{theorem about projections tilde equation 5} [\operatorname{curl}, \widetilde P_\aleph]=0 \mod \Psi^{-\infty}\,.$$ **Theorem 34**. *Let $\widetilde P_0$, $\widetilde P_+$ and $\widetilde P_-$ be pseudodifferential operators of order zero, acting on 1-forms, satisfying $$\label{theorem about projections tilde equation 1} (\widetilde P_\aleph)_\mathrm{prin}=P^{(\aleph)}\,, \qquad \aleph\in\{0,+,-\},$$ and [\[theorem about projections tilde equation all\]](#theorem about projections tilde equation all){reference-type="eqref" reference="theorem about projections tilde equation all"}. Then $$\label{theorem about Pj and tilde Pj equation 1} \widetilde P_\aleph=P_\aleph \mod \Psi^{-\infty}\,,\qquad\aleph\in\{0,+,-\}\,.$$* *Proof.* The claim follows from a straightforward adaptation of [@part1 Theorem 2.2] to the setting of the current paper. The original result from [@part1] was formulated under the assumption of ellipticity but ellipticity was never used in establishing the existence of projections. What was used is simplicity of eigenvalues of the principal symbol, which holds for $\operatorname{curl}$. Furthermore, the original result from [@part1] was formulated under the assumption that the operator acts on columns of half-densities, but this assumption was never used in an essential way; what was used is the presence of an inner product. ◻ In fact, we have a stronger version of Theorem [Theorem 34](#theorem about projections tilde){reference-type="ref" reference="theorem about projections tilde"}. **Theorem 35**. *Let $\widetilde P_0$, $\widetilde P_+$ and $\widetilde P_-$ be pseudodifferential operators of order zero, acting on 1-forms, satisfying conditions [\[theorem about projections tilde equation 5\]](#theorem about projections tilde equation 5){reference-type="eqref" reference="theorem about projections tilde equation 5"}, [\[theorem about projections tilde equation 1\]](#theorem about projections tilde equation 1){reference-type="eqref" reference="theorem about projections tilde equation 1"} and $$\label{theorem about projections tilde stronger version equation 1} (\widetilde P_\aleph)^2=\widetilde P_\aleph\mod \Psi^{-\infty} \,, \qquad \aleph\in\{0,+,-\}.$$ Then we have [\[theorem about Pj and tilde Pj equation 1\]](#theorem about Pj and tilde Pj equation 1){reference-type="eqref" reference="theorem about Pj and tilde Pj equation 1"}.* What happens here is that in Theorem [Theorem 34](#theorem about projections tilde){reference-type="ref" reference="theorem about projections tilde"} assumptions [\[theorem about projections tilde equation 2\]](#theorem about projections tilde equation 2){reference-type="eqref" reference="theorem about projections tilde equation 2"} and [\[theorem about projections tilde equation 4\]](#theorem about projections tilde equation 4){reference-type="eqref" reference="theorem about projections tilde equation 4"} are redundant, and [\[theorem about projections tilde equation 3\]](#theorem about projections tilde equation 3){reference-type="eqref" reference="theorem about projections tilde equation 3"} can be relaxed to [\[theorem about projections tilde stronger version equation 1\]](#theorem about projections tilde stronger version equation 1){reference-type="eqref" reference="theorem about projections tilde stronger version equation 1"}. *Proof of Theorem [Theorem 35](#theorem about projections tilde stronger version){reference-type="ref" reference="theorem about projections tilde stronger version"}.* The proof is analogous to that of Theorem [Theorem 34](#theorem about projections tilde){reference-type="ref" reference="theorem about projections tilde"}, only additionally taking into account [@part1 Theorem 4.1]. ◻ The overall strategy of our paper hinges on the key observation [\[theorem about Pj and tilde Pj equation 1\]](#theorem about Pj and tilde Pj equation 1){reference-type="eqref" reference="theorem about Pj and tilde Pj equation 1"}. Indeed, this gives us an effective way of constructing the full symbols of our projections $P_\aleph$, $\aleph \in\{0,+,-\}$, without resorting explicitly to the representations [\[8 November 2022 equation 5\]](#8 November 2022 equation 5){reference-type="eqref" reference="8 November 2022 equation 5"} and [\[whose unique solution is\]](#whose unique solution is){reference-type="eqref" reference="whose unique solution is"}. Let introduce refined notation for the principal symbol. Namely, we denote by $(\,\cdot\,)_{\mathrm{prin},s}$ the principal symbol of the expression within brackets, regarded as an operator in $\Psi^{-s}$. To appreciate the need for such notation, consider the following example. Let $Q$ and $R$ be pseudodifferential operators in $\Psi^{-s}$ with the same principal symbol. Then, as an operator in $\Psi^{-s}$, $Q-R$ has vanishing principal symbol: $(Q-R)_{\mathrm{prin},s}=0$. But this tells us that $Q-R$ is, effectively, an operator in $\Psi^{-s-1}$ and, as such, it may have nonvanishing principal symbol $(Q-R)_{\mathrm{prin},s+1}$. This refined notation will be used whenever there is risk of confusion. **Proposition 36**. *The operators $P_0$, $P_\pm$ can be explicitly constructed, modulo $\Psi^{-\infty}$, as follows.* - ***Step 1**. Choose three arbitrary pseudodifferential operators $P_{\aleph,0}\in \Psi^0$, $\aleph\in\{0,+,-\}$, such that $(P_{\aleph,0})_\mathrm{prin}=P^{(\aleph)}$.* - ***Step 2**. For $k=1,2,\ldots$ define $$\label{algorithm equation 1} P_{\aleph,k}:=P_{\aleph,0}+\sum_{n=1}^k X_{\aleph,n}, \qquad X_{\aleph,n}\in \Psi^{-n}.$$* *Assuming we have determined the pseudodifferential operator $P_{\aleph,k-1}$, compute, one after the other, the following quantities:* (a) *$R_{\aleph,k}:=-((P_{\aleph,k-1})^2-P_{\aleph,k-1})_\mathrm{prin,k}\,,$* (b) *$S_{\aleph,k}:=-R_{\aleph,k}+P^{(\aleph)} R_{\aleph,k}+R_{\aleph,k}P^{(\aleph)}\,,$* (c) *$T_{\aleph,k}:=[P_{\aleph,k-1},\operatorname{curl}]_{\mathrm{prin},k-1}+[S_{\aleph,k},\operatorname{curl}_\mathrm{prin}].$* - ***Step 3**. Choose a pseudodifferential operator $X_{\aleph,k}\in \Psi^{-k}$ satisfying $$\displaystyle (X_{\aleph,k})_\mathrm{prin}=S_{\aleph,k}+\sum_{\beth\ne \aleph}\frac{P^{(\aleph)} T_{\aleph,k}P^{(\beth)}-P^{(\beth)} T_{\aleph,k}P^{(\aleph)}}{h^{(\aleph)}-h^{(\beth)}}\,.$$* - ***Step 4**. Then $$\label{proposition algorighm Pj equation 1} P_\aleph \sim P_{\aleph,0}+\sum_{n=1}^{+\infty}X_{\aleph,n}\,.$$ Here $\sim$ stands for an asymptotic expansion in 'smoothness': truncated sums give an approximation modulo pseudodifferential operators of lower order.* *Proof.* The algorithm presented here is a straightforward adaptation of that from [@part1 subsection 4.3], which can be applied to the operator $\operatorname{curl}$ in view of Theorem [Theorem 34](#theorem about projections tilde){reference-type="ref" reference="theorem about projections tilde"}. ◻ *Remark 37*. The paper [@part1] offers two different algorithms for the construction of an orthonormal basis of pseudodifferential projections. The first algorithm, using the full set of conditions from Theorem [Theorem 34](#theorem about projections tilde){reference-type="ref" reference="theorem about projections tilde"}, is given in [@part1 subsection 3.4]. The second algorithm, using the minimal set of conditions from Theorem [Theorem 35](#theorem about projections tilde stronger version){reference-type="ref" reference="theorem about projections tilde stronger version"}, is given in [@part1 subsection 4.3]. Of course, the end result is the same, modulo $\Psi^{-\infty}$, but the latter is much simpler. *Remark 38*. Let us stress once again that the algorithm from Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"} does not require ellipticity of the original operator, but it *does* require the eigenvalues of the principal symbol to be simple --- see Step 3. We are now in a position to prove part (c) of Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"}, which will be quite useful in the analysis of the asymmetry operator. *Proof of Theorem [Theorem 3](#main theorem 1){reference-type="ref" reference="main theorem 1"}(c).* For starters, let us observe that the claim [\[main theorem 1 equation 4\]](#main theorem 1 equation 4){reference-type="eqref" reference="main theorem 1 equation 4"} is *invariant* under changes of local coordinates, because the $(P_\aleph)_\mathrm{sub}$ are covariant under such transformations. Therefore, it suffices to fix a point $z\in M$, choose normal coordinates at $z$, and check that [\[main theorem 1 equation 4\]](#main theorem 1 equation 4){reference-type="eqref" reference="main theorem 1 equation 4"} is satisfied at $(z,\zeta)\in T^*M\setminus\{0\}$ for all $\zeta$, in the chosen coordinate system. Indeed, let us fix a point $z$ and work in normal coordinates centred at $z$. Let us choose arbitrary pseudodifferential operators $P_{\aleph,0}\in \Psi^0$, $\aleph\in\{0,+,-\}$, satisfying $(P_{\aleph,0})_\mathrm{prin}=(P_\aleph)_\mathrm{prin}$ (recall that the latter are given by [\[main theorem 1 equation 1\]](#main theorem 1 equation 1){reference-type="eqref" reference="main theorem 1 equation 1"} and [\[main theorem 1 equation 2\]](#main theorem 1 equation 2){reference-type="eqref" reference="main theorem 1 equation 2"}) and $$\label{Proof of main theorem 1 equation 1} (P_{\aleph,0})_\mathrm{sub}=0.$$ Then it is easy to see from the algorithm in Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"} --- in particular, from formula [\[proposition algorighm Pj equation 1\]](#proposition algorighm Pj equation 1){reference-type="eqref" reference="proposition algorighm Pj equation 1"} --- that $$\label{Proof of main theorem 1 equation 2} (P_\aleph)_\mathrm{sub}=(X_{\aleph,1})_\mathrm{prin}.$$ So proving [\[main theorem 1 equation 4\]](#main theorem 1 equation 4){reference-type="eqref" reference="main theorem 1 equation 4"} reduces to proving that $$\label{Proof of main theorem 1 equation 3} [(X_{\aleph,1})_\mathrm{prin}(z,\zeta)]_\alpha{}^\beta=0$$ in the chosen coordinate system. Let us compute, one after the other, the quantities from Step 2 in Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"}. Using [\[theorem about subprincipal symbol of composition equation 1\]](#theorem about subprincipal symbol of composition equation 1){reference-type="eqref" reference="theorem about subprincipal symbol of composition equation 1"} and [\[Proof of main theorem 1 equation 1\]](#Proof of main theorem 1 equation 1){reference-type="eqref" reference="Proof of main theorem 1 equation 1"} we get $$\label{Proof of main theorem 1 equation 4} [R_{\aleph,1}]_\alpha{}^\beta(z,\zeta)=-[(P_\aleph^2)_\mathrm{sub}]_\alpha{}^\beta(z,\zeta) = -\frac{i}2\{\{(P_\aleph)_\mathrm{prin}\,, (P_\aleph)_\mathrm{prin}\}\}_\alpha{}^\beta(z,\zeta)=0\,.$$ In the last equality we used the fact that first spatial derivatives of $(P_\aleph)_\mathrm{prin}$ and Christoffel symbols vanish at $(z,\zeta)$ in normal coordinates. Formula [\[Proof of main theorem 1 equation 4\]](#Proof of main theorem 1 equation 4){reference-type="eqref" reference="Proof of main theorem 1 equation 4"} immediately implies $[S_{\aleph,k}]_\alpha{}^\beta(z,\zeta)=0$. Hence, on account of Lemma [Lemma 23](#lemma subprincipal symbol curl){reference-type="ref" reference="lemma subprincipal symbol curl"}, we have $$\label{Proof of main theorem 1 equation 5} [T_{\aleph,1}]_\alpha{}^\beta(z,\zeta)=\frac{i}2 \{\{(P_\aleph)_\mathrm{prin}\,,\operatorname{curl}_\mathrm{prin}\}\}(z,\zeta)- \frac{i}2\{\{\operatorname{curl}_\mathrm{prin}\,,(P_\aleph)_\mathrm{prin}\}\}(z,\zeta)\,.$$ But the latter also vanishes in normal coordinates, so one arrives at [\[Proof of main theorem 1 equation 3\]](#Proof of main theorem 1 equation 3){reference-type="eqref" reference="Proof of main theorem 1 equation 3"}. ◻ *Remark 39*. Observe that [\[main theorem 1 equation 1\]](#main theorem 1 equation 1){reference-type="eqref" reference="main theorem 1 equation 1"}--[\[main theorem 1 equation 4\]](#main theorem 1 equation 4){reference-type="eqref" reference="main theorem 1 equation 4"} and [\[definition subprincipal symbol operators 1 forms\]](#definition subprincipal symbol operators 1 forms){reference-type="eqref" reference="definition subprincipal symbol operators 1 forms"} give us explicit formulae for the homogeneous components of degree $-1$ of the symbols of the operators $P_0$ and $P_\pm$ in arbitrary local coordinates. # The asymmetry operator {#The asymmetry operator} ## Definition of the asymmetry operator {#Definition of the asymmetry operator subsection} The remainder of this paper is devoted to the study of the pseudodifferential operator $P_+-P_-$ which encodes information about the asymmetry of the Riemannian manifold $(M,g)$ under change of orientation. In particular, one expects this information to be carried by the trace of $P_+-P_-\,$, either in its pointwise or global (operator-theoretic) versions. Unfortunately, one encounters a problem: the pseudodifferential operator $P_+-P_-$ is of order zero and its spectrum is the set of three points $\{-1,0,+1\}$, all three being eigenvalues of infinite multiplicity. Hence, it is not trace class. In order to circumvent this problem, acting in the spirit of Section [4](#Trace of pseudodifferential operators acting on 1-forms){reference-type="ref" reference="Trace of pseudodifferential operators acting on 1-forms"}, we introduce the following definition. **Definition 40**. We define the *asymmetry operator* to be the (scalar) pseudodifferential operator $$\label{definition asymmetry operator formula} A:=\operatorname{tr}_\epsilon (P_+-P_-)\,,$$ where $\operatorname{tr}_\epsilon$ is defined in accordance with Definition [Definition 30](#definition of matrix trace of a pdo acting on 1-forms){reference-type="ref" reference="definition of matrix trace of a pdo acting on 1-forms"}. The above definition warrants a number of remarks. *Remark 41*. [.]{style="color: white"} (a) The asymmetry operator depends, *a priori*, on the choice of the parameter $\epsilon$ and the cut-off $\chi$. However, we suppress the dependence on $\epsilon$ and $\chi$ in our notation for the asymmetry operator because, as we shall see later on, our main results do not depend on these choices. The dependence on $\epsilon$ and $\chi$ is, in a sense, trivial: two operators $A$ corresponding to different choices of $\epsilon$ and $\chi$ differ by an integral operator with infinitely smooth integral kernel which vanishes in a neighbourhood of the diagonal in $M\times M$. The trace of the latter is clearly zero. (b) The Schwartz kernel of $P_+-P_-$ is real, hence the Schwartz kernel of $A$ is also real. (c) The operator $P_+-P_-$ is self-adjoint, hence, by Corollary [Corollary 33](#self-adjointness of matrix trace){reference-type="ref" reference="self-adjointness of matrix trace"} the asymmetry operator $A$ is also self-adjoint. ## The order of the asymmetry operator {#The order of the asymmetry operator subsection} Definition [Definition 40](#definition asymmetry operator){reference-type="ref" reference="definition asymmetry operator"} implies that, *a priori*, our asymmetry operator $A$ is a pseudodifferential operator of order 0. The goal of this subsection is to show that, as a result of cancellations in the symbol, the order of $A$ is in fact $-3$. This will be done in three steps. 1. First, we will show that $A$ is of order $-2$. This will follow easily from the facts already established earlier in the paper. 2. Next, we will show that, in normal coordinates, parallel transport does not contribute to homogeneous components of the symbol of $A$ of degree $-2$ and $-3$. 3. Finally, we will show that $A$ is of order $-3$. This is the subject of Theorem [Theorem 45](#Theorem A order -3){reference-type="ref" reference="Theorem A order -3"} which will come in the end of the subsection and is one of the main results of this paper. **Lemma 42**. *The asymmetry operator $A$ is a pseudodifferential operator of order $-2$.* *Proof.* The claim of the lemma is equivalent to the following two statements: $$\label{Aprin is zero} A_\mathrm{prin}=0\,,$$ $$\label{Asub is zero} A_\mathrm{sub}=0\,.$$ Formula [\[Aprin is zero\]](#Aprin is zero){reference-type="eqref" reference="Aprin is zero"} follows from [\[definition asymmetry operator formula\]](#definition asymmetry operator formula){reference-type="eqref" reference="definition asymmetry operator formula"}, [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 1\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 1){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 1"} and [\[main theorem 1 equation 2\]](#main theorem 1 equation 2){reference-type="eqref" reference="main theorem 1 equation 2"}, whereas formula [\[Asub is zero\]](#Asub is zero){reference-type="eqref" reference="Asub is zero"} follows from [\[definition asymmetry operator formula\]](#definition asymmetry operator formula){reference-type="eqref" reference="definition asymmetry operator formula"}, [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 2\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 2){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 2"} and [\[main theorem 1 equation 4\]](#main theorem 1 equation 4){reference-type="eqref" reference="main theorem 1 equation 4"}. ◻ Let us now fix an arbitrary point $z\in M$ and work in normal coordinates centred at $z$. In our chosen coordinate system the operator $A$ (modulo $\Psi^{-\infty}$) reads $$\label{A in normal coordinates} A: f(x) \mapsto \frac{1}{(2\pi)^3} \int e^{i(x-y)^\mu\xi_\mu}\, [p_+-p_-]_{\alpha}{}^\beta(x,\xi) \, Z_\beta{}^\alpha(y,x) \, f(y) \,\mathrm{d}y\,\mathrm{d}\xi\,,$$ where $$_\alpha{}^\beta(x,\xi)\sim \sum_{j=0}^{+\infty} [(p_\pm)_{-j}]_\alpha{}^\beta(x,\xi), \qquad [(p_\pm)_{-k}]_\alpha{}^\beta(x,\lambda\xi)=\lambda^{-k}[(p_\pm)_{-k}]_\alpha{}^\beta(x,\xi) \quad \forall\lambda>0,$$ is the full symbol of $P_\pm$ in the chosen coordinate system. Next, we observe that [\[A in normal coordinates\]](#A in normal coordinates){reference-type="eqref" reference="A in normal coordinates"} can be recast as $$\label{A in normal coordinates v2} A=A_\mathrm{diag}+A_\mathrm{pt}\,,$$ where $$\label{A diag} A_\mathrm{diag}:=\frac{1}{(2\pi)^3} \int e^{i(x-y)^\mu\xi_\mu}\, [p_+-p_-]_{\alpha}{}^\alpha(x,\xi) \, f(y) \,\mathrm{d}y\,\mathrm{d}\xi$$ and $$\label{A pt} A_\mathrm{pt}:=A-A_\mathrm{diag}.$$ Here the subscript "pt" stands for "parallel transport". The operator $A_\mathrm{pt}$ describes the contribution to the asymmetry operator arising from the fact that our original operator $\operatorname{curl}$ acts on 1-forms and not on 3-columns of scalar fields, The decomposition [\[A in normal coordinates v2\]](#A in normal coordinates v2){reference-type="eqref" reference="A in normal coordinates v2"} is not invariant, it relies on our particular choice of local coordinates but it is convenient for our purposes. Namely, in what follows we will show that $A_\mathrm{pt}$ will not contribute to the principal symbol of order $-3$ of the asymmetry operator $A$ at the point $z$. Let us denote by $a_\mathrm{pt}(x,\xi)\sim \sum_{j=0}^{+\infty} (a_\mathrm{pt})_{-j}(x,\xi)$ the full symbol of $A_\mathrm{pt}\,$. Our task is to show that $$\label{components of apt are zero} (a_\mathrm{pt})_{-j}(z,\xi)=0 \qquad \text{for} \ j=0,1,2,3.$$ In what follows, we will be using the expansion [\[prop: expansion of parallel transoport equation 3 tau zero\]](#prop: expansion of parallel transoport equation 3 tau zero){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3 tau zero"} for $Z_{\alpha}{}^\beta$. The claim [\[components of apt are zero\]](#components of apt are zero){reference-type="eqref" reference="components of apt are zero"} for $j=0$ and $j=1$ follows immediately from [\[prop: expansion of parallel transoport equation 3 tau zero\]](#prop: expansion of parallel transoport equation 3 tau zero){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3 tau zero"}. **Lemma 43**. *We have $$\label{lemma apt -2 is zero equation} (a_\mathrm{pt})_{-2}(z,\xi)=0.$$* *Proof.* Upon integration by parts, we get from [\[prop: expansion of parallel transoport equation 3 tau zero\]](#prop: expansion of parallel transoport equation 3 tau zero){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3 tau zero"} $$\label{apt -2 equation 1} (a_\mathrm{pt})_{-2}(z,\xi)=\frac16\, \operatorname{Riem}^{\alpha}{}_{\mu\kappa\nu}(z)\,\dfrac{\partial^2[(p_+-p_-)_{0}]_{\alpha}{}^\kappa(z,\xi)}{\partial \xi_\mu \partial \xi_\nu} =\frac16 \,\operatorname{Riem}_{\alpha\mu\kappa\nu}(z)\,\dfrac{\partial^2[i\, |\xi|^{-1}\, \varepsilon^{\alpha\gamma\kappa}\, \xi_\gamma]}{\partial \xi_\mu \partial \xi_\nu}\,,$$ where $|\cdot|$ is the Euclidean norm. By the symmetries of the Riemann tensor we have $$\label{riemann1} \operatorname{Riem}_{\alpha\mu\kappa\nu}=\frac12(R_{\alpha\mu\kappa\nu}+\operatorname{Riem}_{\kappa\nu\alpha\mu}).$$ Since $$\label{riemann1a} \frac{\partial^2[i\, |\xi|^{-1}\, \varepsilon^{\alpha\gamma\kappa}\, \xi_\gamma]}{\partial \xi_\mu \partial \xi_\nu}$$ is symmetric in the pair of indices $\mu$ and $\nu$, we can replace [\[riemann1\]](#riemann1){reference-type="eqref" reference="riemann1"} in [\[apt -2 equation 1\]](#apt -2 equation 1){reference-type="eqref" reference="apt -2 equation 1"} with its symmetrised version $$\label{riemann2} \operatorname{Riem}_{\alpha\mu\kappa\nu}=\frac12(\operatorname{Riem}_{\alpha\mu\kappa\nu}+\operatorname{Riem}_{\kappa\nu\alpha\mu}+\operatorname{Riem}_{\alpha\nu\kappa\mu}+\operatorname{Riem}_{\kappa\mu\alpha\nu}).$$ But the quantity [\[riemann2\]](#riemann2){reference-type="eqref" reference="riemann2"} is symmetric in the pair of indices $\alpha$ and $\kappa$, whereas [\[riemann1a\]](#riemann1a){reference-type="eqref" reference="riemann1a"} is antisymmetric in the same pair of indices. Therefore [\[apt -2 equation 1\]](#apt -2 equation 1){reference-type="eqref" reference="apt -2 equation 1"} vanishes. ◻ Next, we observe that formulae [\[main theorem 1 equation 4\]](#main theorem 1 equation 4){reference-type="eqref" reference="main theorem 1 equation 4"}, [\[subprincipal symbol operators 1-forms\]](#subprincipal symbol operators 1-forms){reference-type="eqref" reference="subprincipal symbol operators 1-forms"} and [\[main theorem 1 equation 2\]](#main theorem 1 equation 2){reference-type="eqref" reference="main theorem 1 equation 2"} imply $$\label{ppm minus 1 is zero} [(p_+-p_-)_{-1}]_\alpha{}^\beta(z,\xi)=0.$$ This fact is needed in establishing the following lemma. **Lemma 44**. *We have $$\label{lemma apt -3 is zero equation} (a_\mathrm{pt})_{-3}(z,\xi)=0.$$* *Proof.* The expansion [\[prop: expansion of parallel transoport equation 3 tau zero\]](#prop: expansion of parallel transoport equation 3 tau zero){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3 tau zero"} and formula [\[ppm minus 1 is zero\]](#ppm minus 1 is zero){reference-type="eqref" reference="ppm minus 1 is zero"} imply, upon integration by parts, $$\begin{gathered} \label{apt -3 equation 1} (a_\mathrm{pt})_{-3}(z,\xi)=-\frac{i}{6}\, \dfrac{\partial^2\Gamma^\alpha{}_{\sigma\kappa}}{\partial x^\mu \partial x^\nu}(z)\,\dfrac{\partial^3[(p_+-p_-)_{0}]_{\alpha}{}^\kappa(z,\xi)}{\partial \xi_\sigma\partial \xi_\mu \partial \xi_\nu}\\ = -\frac{i}{6} \,\delta_{\alpha\beta}\,\dfrac{\partial^2\Gamma^\beta{}_{\sigma\kappa}}{\partial x^\mu \partial x^\nu}(z)\,\dfrac{\partial^3[i\, |\xi|^{-1}\, \varepsilon^{\alpha\gamma\kappa}\, \xi_\gamma]}{\partial \xi_\sigma\partial \xi_\mu \partial \xi_\nu}\,.\end{gathered}$$ We observe that in formula [\[apt -3 equation 1\]](#apt -3 equation 1){reference-type="eqref" reference="apt -3 equation 1"} one can replace the quantity $\delta_{\alpha\beta}\,\dfrac{\partial^2\Gamma^\beta{}_{\sigma\kappa}}{\partial x^\mu \partial x^\nu}(z)$ with its symmetrised version in the three indices $\sigma$, $\mu$ and $\nu$. It is known [@expansions Eqn. (11.9)] that the latter is equal to $$\label{symmetrised christoffel xx} \frac12 \nabla_\nu \operatorname{Riem}_{\alpha \sigma \mu \kappa}(z)\,.$$ Using the symmetries of the Riemann tensor, [\[symmetrised christoffel xx\]](#symmetrised christoffel xx){reference-type="eqref" reference="symmetrised christoffel xx"} can be equivalently rewritten as $$\label{symmetrised christoffel yy} \frac12 \nabla_\nu \operatorname{Riem}_{\alpha \sigma \mu \kappa} = \frac14 [\nabla_\nu \operatorname{Riem}_{\alpha \sigma \mu \kappa}+\nabla_\nu \operatorname{Riem}_{\mu \kappa \alpha \sigma}].$$ Now, when contracted with $$\label{apt -3 equation 2} \dfrac{\partial^3[i\, |\xi|^{-1}\, \varepsilon^{\alpha\gamma\kappa}\, \xi_\gamma]}{\partial \xi_\sigma\partial \xi_\mu \partial \xi_\nu},$$ the quantity [\[symmetrised christoffel yy\]](#symmetrised christoffel yy){reference-type="eqref" reference="symmetrised christoffel yy"} can be replaced with its symmetrised version in the pair of indices $\mu$ and $\sigma$: $$\label{apt -3 equation 3} \frac18 [\nabla_\nu \operatorname{Riem}_{\alpha \sigma \mu \kappa}+\nabla_\nu \operatorname{Riem}_{\mu \kappa \alpha \sigma} + \nabla_\nu \operatorname{Riem}_{\alpha \mu \sigma \kappa}+\nabla_\nu \operatorname{Riem}_{\sigma \kappa \alpha \mu}]\,.$$ But the quantity [\[apt -3 equation 3\]](#apt -3 equation 3){reference-type="eqref" reference="apt -3 equation 3"} is now symmetric in the pair of indices $\alpha$ and $\kappa$, whereas [\[apt -3 equation 2\]](#apt -3 equation 2){reference-type="eqref" reference="apt -3 equation 2"} is antisymmetric in the same pair of indices. Therefore [\[apt -3 equation 1\]](#apt -3 equation 1){reference-type="eqref" reference="apt -3 equation 1"} vanishes. ◻ Lemmata [Lemma 43](#lemma apt -2 is zero){reference-type="ref" reference="lemma apt -2 is zero"} and [Lemma 44](#lemma apt -3 is zero){reference-type="ref" reference="lemma apt -3 is zero"} give us [\[components of apt are zero\]](#components of apt are zero){reference-type="eqref" reference="components of apt are zero"}.   We are now in a position to prove the main result of this subsection. **Theorem 45**. *The asymmetry operator $A$ is a pseudodifferential operator of order $-3$.* *Remark 46*. Before rigorously proving Theorem [Theorem 45](#Theorem A order -3){reference-type="ref" reference="Theorem A order -3"}, let us give an intuitive explanation of why the claim should be true. The principal symbol $A_{\mathrm{prin},2}$ (that is, the principal symbol of the asymmetry operator $A$ viewed as an operator of order $-2$) should be proportional to - Ricci curvature --- because $a_{-2}$ involves two derivatives of the metric and because in dimension three the Riemann curvature tensor can be expressed in terms of the Ricci tensor; - the totally antisymmetric tensor $E$ --- because the asymmetry operator $A$ is expected to feel orientation. If one attempts to construct a scalar quantity out of the above two objects, the metric, and momentum $\xi$, one realises that the only possible outcome is the the scalar zero. *Proof of Theorem [Theorem 45](#Theorem A order -3){reference-type="ref" reference="Theorem A order -3"}.* Let us denote by $a_\mathrm{diag}$ the full symbol of $A_\mathrm{diag}\,$. Formulae [\[main theorem 1 equation 2\]](#main theorem 1 equation 2){reference-type="eqref" reference="main theorem 1 equation 2"} and [\[ppm minus 1 is zero\]](#ppm minus 1 is zero){reference-type="eqref" reference="ppm minus 1 is zero"} immediately imply $$(z,\xi)=0$$ and $$(z,\xi)=0,$$ respectively. Hence, in view of [\[A in normal coordinates v2\]](#A in normal coordinates v2){reference-type="eqref" reference="A in normal coordinates v2"} and [\[components of apt are zero\]](#components of apt are zero){reference-type="eqref" reference="components of apt are zero"}, proving the claim reduces to showing that $$\label{proof Theorem A order -3 equation 1} [(a_\mathrm{diag})_{-2}](z,\xi)=[(p_+-p_-)_{-2}]_\alpha{}^\alpha(z,\xi)=0.$$ Our strategy for proving [\[proof Theorem A order -3 equation 1\]](#proof Theorem A order -3 equation 1){reference-type="eqref" reference="proof Theorem A order -3 equation 1"} is to check it explicitly by implementing the algorithm from Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"} in chosen normal coordinates. As the latter is computationally challenging, we describe below how to do so in a cunning way. In what follows $z=0$ (the centre of our normal coordinates).   **Simplification 1**. It is well known that $$\label{14 September 2021 equation 1} g_{\alpha\beta}(x) = \delta_{\alpha\beta} - \frac13 \operatorname{Riem}_{\alpha\mu\beta\nu}(0)\,x^\mu x^\nu +O(|x|^3).$$ The approximation [\[14 September 2021 equation 1\]](#14 September 2021 equation 1){reference-type="eqref" reference="14 September 2021 equation 1"} is sufficiently accurate for the computation of [\[proof Theorem A order -3 equation 1\]](#proof Theorem A order -3 equation 1){reference-type="eqref" reference="proof Theorem A order -3 equation 1"}.   **Simplification 2**. Express the Riemann tensor $\operatorname{Riem}$ in terms of the Ricci tensor $\operatorname{Ric}$ via the identity $$\begin{gathered} \label{14 September 2021 equation 5} \operatorname{Riem}_{\alpha\beta\gamma\delta}(x) = \operatorname{Ric}_{\alpha\gamma}(x)\,g_{\beta\delta}(x) - \operatorname{Ric}_{\alpha\delta}(x)\,g_{\beta\gamma}(x) + \operatorname{Ric}_{\beta\delta}(x)\,g_{\alpha\gamma}(x) - \operatorname{Ric}_{\beta\gamma}(x)\,g_{\alpha\delta}(x) \\ + \frac{\operatorname{Sc}(x)}{2} ( g_{\alpha\delta}(x)g_{\beta\gamma}(x) - g_{\alpha\gamma}(x)g_{\beta\delta}(x) )\,,\end{gathered}$$ which holds in dimension three, where $\operatorname{Sc}(x):=g^{\alpha\beta}(x)\operatorname{Ric}_{\alpha\beta}(x)$ is scalar curvature. Then substitute [\[14 September 2021 equation 5\]](#14 September 2021 equation 5){reference-type="eqref" reference="14 September 2021 equation 5"} into [\[14 September 2021 equation 1\]](#14 September 2021 equation 1){reference-type="eqref" reference="14 September 2021 equation 1"} and discard cubic and higher order terms.   **Simplification 3**. It is clear that $\operatorname{Ric}$ will appear in $[(a_\mathrm{diag})_{-2}](z,\xi)=[(p_+-p_-)_{-2}]_\alpha{}^\alpha(0,\xi)$ in a linear fashion. This means that we can set $$\label{Ric at 0 simplification 3} \operatorname{Ric}(0)= \begin{pmatrix} c_1 & c_2 & c_3\\ c_2 & c_4 & c_5\\ c_3 & c_5 & c_6 \end{pmatrix}$$ and do calculations assuming that only one of the constants $c_j$, $j=1,\dots,6$, is nonzero. In particular, one can choose the nonzero $c_j$ to be equal to 1. To establish [\[proof Theorem A order -3 equation 1\]](#proof Theorem A order -3 equation 1){reference-type="eqref" reference="proof Theorem A order -3 equation 1"} one just needs to run the algorithm six times.   **Simplification 4**. Without loss of generality, evaluate the final quantities (as soon as there are no more differentiations with respect to momentum) at $$\label{xi0} \xi_0=\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.$$ This is sufficient because - one can rotate the normal coordinate system so that $\xi$ aligns with the third spatial coordinate and - one can scale $\xi$ so as to normalise it, using the homogeneity of $[(a_\mathrm{diag})_{-2}](0,\xi)$. Moreover, one can write $\xi=\xi_0+\eta$ and replace $\|\xi\|$ with its Taylor expansion in $\eta$. The latter simplifies differentiations in $\xi$.   **Simplification 5**. In implementing the algorithm, expand all quantities at the $k$-th step in powers of $x$ and $\eta$ retaining only terms up to order $2-k$, jointly in $x$ and $\eta$. Here we are assuming that $x$ and $\eta$ are of the same order.   Simplifications 1--5 reduce our algorithm to the manipulation of polynomials. Note that the vector spaces of polynomials in the variables $x^\alpha$ and $\eta_\alpha\,$, $\alpha=1,2,3$, of degree 0, 1 and 2 have dimension 1, 7 and 28 respectively[^5].   We implement below the algorithm in the special case $$\label{case c1=1} c_j=\delta_{1j}.$$ The other cases are analogous. Formulae [\[14 September 2021 equation 1\]](#14 September 2021 equation 1){reference-type="eqref" reference="14 September 2021 equation 1"}--[\[Ric at 0 simplification 3\]](#Ric at 0 simplification 3){reference-type="eqref" reference="Ric at 0 simplification 3"} and [\[case c1=1\]](#case c1=1){reference-type="eqref" reference="case c1=1"} imply that in the chosen coordinate system the metric tensor reads $$\label{proof Theorem A order -3 equation 2} g_{\alpha\beta}(x) = \begin{pmatrix} 1-\frac16(x^2)^2-\frac16(x^3)^2&\frac16x^1x^2&\frac16x^1x^3\\ \frac16x^1x^2&1-\frac16(x^1)^2+\frac16(x^3)^2&-\frac16x^2x^3\\ \frac16x^1x^3&-\frac16x^2x^3&1-\frac16(x^1)^2+\frac16(x^2)^2 \end{pmatrix} +O(|x|^3)$$ and $$\label{proof Theorem A order -3 equation 3} \rho(x)=1-\frac16(x^1)^2+O(|x|^3).$$ Let us now retrace, step by step, the algorithm from Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"}. In view of Lemma [Lemma 42](#Lemma A order -2){reference-type="ref" reference="Lemma A order -2"}, let us assume that we have carried out the first iteration of the algorithm and have determined pseudodifferential operators $P_{\pm,1}\in \Psi^0$ satisfying $$\label{proof Theorem A order -3 equation 4} P_{\pm,1}^2=P_{\pm,1} \mod \Psi^{-2}, \qquad P_{+,1}P_{-,1}=0\mod \Psi^{-2},\qquad [P_{\pm,1},\operatorname{curl}]=0 \mod \Psi^{-1}\,.$$ This gives us $$\label{proof Theorem A order -3 equation 8} (p_{\pm,1})_{-1}(x,\xi_0+\eta)=-\frac1{12}\left( \begin{array}{ccc} i x^3 &\pm x^3 & 0 \\ \pm x^3& -i x^3 & 0 \\ \pm x^2-3 i x^1 & \pm x^1+i x^2& 0 \\ \end{array} \right) +O(|x|^2+|\eta|^2)\,.$$ Without loss of generality, we can assume that $$\label{proof Theorem A order -3 equation 5} (p_{\pm,1})_{-2}=0.$$ The latter can be achieved by choosing $X_{\pm,1}$ appropriately. Therefore, proving the theorem reduces to showing that $$\label{proof Theorem A order -3 equation 6} [\operatorname{tr}(X_{+,2}-X_{-,2})_\mathrm{prin}](0,\xi_0)=0.$$ The first quantity we need to compute is $R_{\pm,2}$. Due to [\[proof Theorem A order -3 equation 5\]](#proof Theorem A order -3 equation 5){reference-type="eqref" reference="proof Theorem A order -3 equation 5"} we have $$\label{proof Theorem A order -3 equation 7} R_{\pm,2}=-(P_{\pm,1}^2)_{-2}$$ where $(P_{\pm,1}^2)_{-2}$ stands for the homogeneous component of degree $-2$ of the symbol of the operator $P_{\pm,1}^2\,$. To compute the latter, one needs to use the formula for the symbol of the composition of pseudodifferential operators: given two operators $A$ and $B$ with symbols $a$ and $b$, the symbol $\sigma_{BA}$ of their composition $BA$ is given by the formula $$\label{5 November 2021 equation 1} \sigma_{BA}\sim\sum_{k=0}^{\infty} \frac{1}{i^k k!} \dfrac{\partial^k b}{\partial \xi_{\alpha_1}\dots \partial \xi_{\alpha_k}}\dfrac{\partial^k a}{\partial x^{\alpha_1}\dots \partial x^{\alpha_k}}\,,$$ see [@shubin Theorem 3.4]. By performing the relevant calculations with the simplifications described above, one obtains $$\label{proof Theorem A order -3 equation 9} R_{\pm,2}(0,\xi_0)= -\frac1{12} \begin{pmatrix} 1 & \mp 2i &0\\ \pm 2i& 1 & 0\\ 0 & 0 & 0 \end{pmatrix}\,,$$ $$\label{proof Theorem A order -3 equation 10} S_{\pm,2}(0,\xi_0)= -\frac1{12} \begin{pmatrix} 2 & \mp i &0\\ \pm i& 2 & 0\\ 0 & 0 & 0 \end{pmatrix}\,,$$ $$\label{proof Theorem A order -3 equation 11} T_{\pm,2}(0,\xi_0)= \pm\frac1{12} \begin{pmatrix} 1 & 0 &0\\ 0 & -1 & 0\\ 0 & 0 & 0 \end{pmatrix} \,$$ and, finally, $$\label{proof Theorem A order -3 equation 12} (X_{\pm,2})_\mathrm{prin}(0,\xi_0)= -\frac1{24} \begin{pmatrix} 4 & \mp 3i &0\\ \pm i & 4 & 0\\ 0 & 0 & 0 \end{pmatrix} \,.$$ Formula [\[proof Theorem A order -3 equation 12\]](#proof Theorem A order -3 equation 12){reference-type="eqref" reference="proof Theorem A order -3 equation 12"} implies [\[proof Theorem A order -3 equation 6\]](#proof Theorem A order -3 equation 6){reference-type="eqref" reference="proof Theorem A order -3 equation 6"}. ◻ ## The principal symbol of the asymmetry operator {#The principal symbol of the asymmetry operator} This subsection is concerned with the proof of Theorem [Theorem 4](#main theorem 2){reference-type="ref" reference="main theorem 2"}. As in the previous subsection, in what follows we also assume to have chosen normal coordinates centred at $z$. *Remark 47*. Before moving on to the proof, let us try to reproduce the intuitive argument from Remark [Remark 46](#remark common sense 1){reference-type="ref" reference="remark common sense 1"} at the next order of homogeneity. The principal symbol $A_{\mathrm{prin},3}$ (that is, the principal symbol of the asymmetry operator $A$ viewed as an operator of order $-3$) should be proportional to - the covariant derivative of Ricci curvature --- because $a_{-3}$ involves three derivatives of the metric and because in dimension three the Riemann curvature tensor can be expressed in terms of the Ricci tensor; - the totally antisymmetric tensor $E$ --- because the operator $A$ is expected to feel orientation. If one attempts to construct a scalar quantity out of the above two objects, the metric, and momentum $\xi$, one realises that the only possible nonzero outcome, up to a constant scaling, is $$E^{\alpha\beta \gamma}(x)\, \nabla_\alpha \operatorname{Ric}_{\beta}{}^\rho(x)\, \frac{\xi_\gamma\xi_\rho}{\|\xi\|^5}\,.$$ *Proof of Theorem [Theorem 4](#main theorem 2){reference-type="ref" reference="main theorem 2"}.* In view of Lemma [Lemma 44](#lemma apt -3 is zero){reference-type="ref" reference="lemma apt -3 is zero"}, proving [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"} reduces to showing that, in chosen normal coordinates, we have $$\label{Proof of Aprin equation 1} [(p_+-p_-)_{-3}]_{\alpha}{}^\alpha(z,\xi)=-\frac1{2|\xi|^5}\varepsilon^{\alpha\beta \gamma}\, \nabla_\alpha \operatorname{Ric}_{\beta}{}^\rho(z)\, \xi_\gamma\xi_\rho\,.$$ The strategy for proving [\[Proof of Aprin equation 1\]](#Proof of Aprin equation 1){reference-type="eqref" reference="Proof of Aprin equation 1"} consists in implementing the algorithm from Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"} in chosen normal coordinates up to order $-3$. Even with the help of modern computer algebra, doing this to the required accuracy is a highly nontrivial task. Therefore, as in the proof of Theorem [Theorem 45](#Theorem A order -3){reference-type="ref" reference="Theorem A order -3"}, we describe below some helpful simplifications, which once again reduce all calculations to the manipulation of polynomials. In what follows $z=0$ (the centre of our normal coordinates).   **Simplification 1**. It is known, see [@Schoen; @and; @Yau p. 211 formula (3.4)] or [@Uwe; @Muller formula (6)], that $$\label{14 September 2021 equation 1bis} g_{\alpha\beta}(x) = \delta_{\alpha\beta} - \frac13 \operatorname{Riem}_{\alpha\mu\beta\nu}(0)\,x^\mu\,x^\nu - \frac16 (\nabla_\sigma \operatorname{Riem}_{\alpha\mu\beta\nu})(0)\,x^\sigma\,x^\mu\,x^\nu +O(|x|^4).$$ The approximation [\[14 September 2021 equation 1bis\]](#14 September 2021 equation 1bis){reference-type="eqref" reference="14 September 2021 equation 1bis"} is sufficiently accurate for the computation of [\[Proof of Aprin equation 1\]](#Proof of Aprin equation 1){reference-type="eqref" reference="Proof of Aprin equation 1"}.   **Simplification 2**. Express the Riemann tensor $\operatorname{Riem}$ in terms of the Ricci tensor $\operatorname{Ric}$ via the identity [\[14 September 2021 equation 5\]](#14 September 2021 equation 5){reference-type="eqref" reference="14 September 2021 equation 5"}. Then substitute [\[14 September 2021 equation 5\]](#14 September 2021 equation 5){reference-type="eqref" reference="14 September 2021 equation 5"} into [\[14 September 2021 equation 1bis\]](#14 September 2021 equation 1bis){reference-type="eqref" reference="14 September 2021 equation 1bis"} and discard quartic and higher order terms.   **Simplification 3**. It is clear that $\operatorname{Ric}$ and $\nabla\operatorname{Ric}$ will appear in $(p_{\pm})_{-3}$ in a linear fashion. We can choose [\[Ric at 0 simplification 3\]](#Ric at 0 simplification 3){reference-type="eqref" reference="Ric at 0 simplification 3"} and $$\begin{gathered} \label{nablaG at 0 simplification 3} \nabla_1\operatorname{Ric}_{\alpha\beta}(0) = \begin{pmatrix} c_7&c_8&c_9\\ c_8&c_{10}&c_{11}\\ c_9&c_{11}&c_{12} \end{pmatrix}, \qquad \nabla_2\operatorname{Ric}_{\alpha\beta}(0) = \begin{pmatrix} c_{13}&c_{14}&c_{15}\\ c_{14}&c_{16}&c_{17}\\ c_{15}&c_{17}&c_{18} \end{pmatrix}, \\ \nabla_3\operatorname{Ric}_{\alpha\beta}(0) = \begin{pmatrix} c_{19}&c_{20}&c_{21}\\ c_{20}&c_{22}&c_{23}\\ c_{21}&c_{23}&c_{24} \end{pmatrix},\end{gathered}$$ and do calculations assuming that only one of the constants $c_j$, $j=1,\dots,24$, is nonzero. In particular, one can choose the nonzero $c_j$ to be equal to 1. To establish [\[Proof of Aprin equation 1\]](#Proof of Aprin equation 1){reference-type="eqref" reference="Proof of Aprin equation 1"} one just needs to run the algorithm 24 times.   **Simplification 4**. Without loss of generality, evaluate the final quantities (as soon as there are no more differentiations with respect to momentum) at the particular $\xi_0$ given by [\[xi0\]](#xi0){reference-type="eqref" reference="xi0"}. This is sufficient due to rotational symmetry and homogeneity. Moreover, one can write $\xi=\xi_0+\eta$ and replace $\|\xi\|$ with its Taylor expansion in $\eta$. The latter simplifies differentiations in $\xi$.   **Simplification 5**. In implementing the algorithm, expand all quantities at the $k$-th step in powers of $x$ and $\eta$ retaining only terms up order $3-k$, jointly in $x$ and $\eta$. Here we are assuming that $x$ and $\eta$ are of the same order.   Simplifications 1--5 reduce our algorithm to the manipulation of polynomials.   We implement below the algorithm in the special case $$\label{case c5=1} c_{11}=1, \qquad c_j=0, \ \text{for }j\ne 11.$$ The other cases are analogous. Formulae [\[Ric at 0 simplification 3\]](#Ric at 0 simplification 3){reference-type="eqref" reference="Ric at 0 simplification 3"} and [\[14 September 2021 equation 1bis\]](#14 September 2021 equation 1bis){reference-type="eqref" reference="14 September 2021 equation 1bis"}--[\[case c5=1\]](#case c5=1){reference-type="eqref" reference="case c5=1"} imply that in the chosen coordinate system the metric tensor reads $$\label{Proof of Aprin equation 2} g_{\alpha\beta}(x) = \begin{pmatrix} 1-\frac{x^1x^2x^3}3&\frac{(x^1)^2x^3}6&\frac{(x^1)^2x^2}6\\ \frac{(x^1)^2x^3}6&1&-\frac{(x^1)^3}6\\ \frac{(x^1)^2x^2}6&-\frac{(x^1)^3}6&1 \end{pmatrix} +O(|x|^4)$$ and $$\label{Proof of Aprin equation 3} \rho(x)=1-\frac{x^1x^2x^3}6+O(|x|^4).$$ Let us now retrace, step by step, the algorithm from Proposition [Proposition 36](#proposition algorighm Pj){reference-type="ref" reference="proposition algorighm Pj"}. In view of Theorem [Theorem 45](#Theorem A order -3){reference-type="ref" reference="Theorem A order -3"}, let us assume that we have carried out the first two iterations of the algorithm and have determined pseudodifferential operators $P_{\pm,2}\in \Psi^0$ satisfying $$\label{Proof of Aprin equation 4} P_{\pm,2}^2=P_{\pm,2} \mod \Psi^{-3}, \qquad P_{+,2}P_{-,2}=0\mod \Psi^{-3},\qquad [P_{\pm,2},\operatorname{curl}]=0 \mod \Psi^{-2}\,.$$ This gives us $$\label{Proof of Aprin equation 5} (p_{\pm,2})_{-1}(x,\xi_0+\eta)= \frac1{24} \begin{pmatrix} \mp5\,(x^1)^2-4ix^1 x^2 & 6i\,(x^1)^2\mp3x^1 x^2 &0\\ -2i\,(x^1)^2\pm x^1 x^2& \mp3\, (x^1)^2 & 0\\ 2i\,x^2 x^3 & -6i\,x^1 x^3 & 0 \end{pmatrix}+ O(|x|^3+|\eta|^3)$$ and $$\label{Proof of Aprin equation 6} (p_{\pm,2})_{-2}(x,\xi_0+\eta)= -\frac1{24} \begin{pmatrix} \pm ix^3& 6 x^3 &0\\ -2x^3& \pm3ix^3 & 0\\ \pm 9i\,x^1 -2x^2 & 6x^1\pm3i x^2& 0 \end{pmatrix}+ O(|x|^2+|\eta|^2)\,.$$ Without loss of generality, we can assume that $$\label{Proof of Aprin equation 7} (p_{\pm,1})_{-3}=0.$$ The latter can be achieved by choosing $X_{\pm,2}$ appropriately. Therefore, proving [\[Proof of Aprin equation 1\]](#Proof of Aprin equation 1){reference-type="eqref" reference="Proof of Aprin equation 1"} reduces to showing that $$\label{Proof of Aprin equation 8} [\operatorname{tr}(X_{+,3}-X_{-,3})_\mathrm{prin}](0,\xi_0)=-\frac12.$$ By performing the relevant calculations with the simplifications described above and with account of [\[5 November 2021 equation 1\]](#5 November 2021 equation 1){reference-type="eqref" reference="5 November 2021 equation 1"}, one obtains $$\label{Proof of Aprin equation 9} R_{\pm,3}(0,\xi_0)= \frac{i}8 \begin{pmatrix} 0 & 1 &0\\ -1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix},$$ $$\label{Proof of Aprin equation 10} S_{\pm,3}(0,\xi_0)=\mp\frac18 \begin{pmatrix} 1 & 0 &0\\ 0& 1 & 0\\ 0 & 0 & 0 \end{pmatrix},$$ $$\label{Proof of Aprin equation 11} T_{\pm,3}(0,\xi_0)= \mp\frac{i}4 \begin{pmatrix} 0 & 1 &0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix},$$ and, finally, $$\label{Proof of Aprin equation 12} (X_{\pm,3})_\mathrm{prin}(0,\xi_0)= \mp\frac14 \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}.$$ Formula [\[Proof of Aprin equation 12\]](#Proof of Aprin equation 12){reference-type="eqref" reference="Proof of Aprin equation 12"} implies [\[Proof of Aprin equation 8\]](#Proof of Aprin equation 8){reference-type="eqref" reference="Proof of Aprin equation 8"}. ◻ *Remark 48*. Observe that neither in this section nor elsewhere in the paper our arguments make use of the second (differential) Bianchi identity. Of course, the second Bianchi identity implies that the number of independent components in formula [\[nablaG at 0 simplification 3\]](#nablaG at 0 simplification 3){reference-type="eqref" reference="nablaG at 0 simplification 3"} is 15, not 18. Namely, $c_{21}$, $c_{23}$ and $c_{24}$ can be expressed in terms of the remaining components. The second Bianchi identity may be needed for the calculation of the subprincipal symbol of the asymmetry operator. This matter is outside the scope of the current paper. Since formula [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"} is one of the key results of this paper, we provide in Appendix [15](#An alternative derivation of formula){reference-type="ref" reference="An alternative derivation of formula"} an independent verification relying on formula [\[formula for Pplus minus Pminus\]](#formula for Pplus minus Pminus){reference-type="eqref" reference="formula for Pplus minus Pminus"}. # The regularised trace of the asymmetry operator {#The regularised trace of $A$} ## Structure of the integral kernel near the diagonal {#Structure of Aprin near the diagonal} The asymmetry operator $A$ can be written as an integral operator $$\label{structure frac a equation 1} (Af)(x)=\int_M \mathfrak{a}(x,y)\,f(y)\,\rho(y)\,\mathrm{d}y \,.$$ Now, the operator $A$ is not guaranteed to be of trace class, so one cannot apply formula [\[properties of matrix trace of a pdo acting on 1-forms lemma equation 3\]](#properties of matrix trace of a pdo acting on 1-forms lemma equation 3){reference-type="eqref" reference="properties of matrix trace of a pdo acting on 1-forms lemma equation 3"} to it. The issue here is that in dimension 3 a sufficient condition for a pseudodifferential operator to be of trace class is that its order be strictly less than $-3$, whereas according to Theorem [Theorem 4](#main theorem 2){reference-type="ref" reference="main theorem 2"} for a generic Riemannian 3-manifold the order of $A$ is exactly $-3$. This means that we need to perform some sort of very basic regularisation in order to get over the finish line and define, in an invariant and geometrically meaningful manner, the trace of $A$ . In order to define an appropriate regularisation, we need to understand the structure of the integral kernel $\mathfrak{a}(x,y)$. More precisely, we need to characterise explicitly the way in which the latter fails to be continuous. **Theorem 49**. *In a neighbourhood of the diagonal $\{(x,x)\,|\, x\in M\}\subset M\times M$ the integral kernel of $A$ can be written in local coordinates as $$\label{singularity theorem equation 1} \mathfrak{a}(x,y)=\mathfrak{a}_d(x,y)+\mathfrak{a}_c(x,y)\,,$$ where $$\label{singularity theorem equation 2} \mathfrak{a}_d(x,y):=\frac{1}{12\pi^2}\, E^{\alpha\beta}{}_\gamma(x)\, \nabla_\alpha \operatorname{Ric}_{\beta\rho}(x)\, \frac{(x-y)^\gamma (x-y)^\rho}{\operatorname{dist}^2(x,y)}$$ and $\mathfrak{a}_c$ is continuous as a function of two variables. Recall that the tensor $E$ is defined in accordance with [\[main theorem 1 equation 3\]](#main theorem 1 equation 3){reference-type="eqref" reference="main theorem 1 equation 3"}.* *Remark 50*. The tensor $\,E^{\alpha\beta}{}_\gamma\,\nabla_\alpha \operatorname{Ric}_{\beta\rho}\,$ appearing in formula [\[singularity theorem equation 2\]](#singularity theorem equation 2){reference-type="eqref" reference="singularity theorem equation 2"} is trace-free, $$\label{singularity theorem tensor is trace-free equation} E^{\alpha\beta}{}_\gamma\,\nabla_\alpha \operatorname{Ric}_{\beta\rho} \,g^{\gamma\rho} =0.$$ This is an important property which will be used on two occasions in the subsequent arguments in this section. We will see that property [\[singularity theorem tensor is trace-free equation\]](#singularity theorem tensor is trace-free equation){reference-type="eqref" reference="singularity theorem tensor is trace-free equation"} implies that the singularity of the integral kernel of our asymmetry operator $A$ is weaker than one would expect of a generic pseudodifferential operator of order $-3$. For a generic pseudodifferential operator of order $-3$ one would have a logarithm in the leading term of the singularity of the integral kernel, but there is no logarithm in [\[singularity theorem equation 2\]](#singularity theorem equation 2){reference-type="eqref" reference="singularity theorem equation 2"}. In order to prove Theorem [Theorem 49](#singularity theorem){reference-type="ref" reference="singularity theorem"} we need to establish first a number of preparatory lemmata. Until further notice we will be working in Euclidean space $\mathbb{R}^3$ equipped with Cartesian coordinates. In what follows we use the *Japanese bracket* notation $$\label{Japanese bracket} \langle\xi\rangle := (1+|\xi|^2)^{1/2}.$$ **Lemma 51**. *We have $$\label{lemma Bessel equation} \frac1{(2\pi)^3}\int_{\mathbb{R}^3}\frac{1}{\langle\xi\rangle^5} \,e^{-i y^\mu\,\xi_\mu}\, \, \mathrm{d}\xi= \frac{1}{6\pi^2} |y|\,K_1(|y|)\,,$$ where $K_1$ is the modified Bessel function of the second kind [@AS § 9.6, p. 374].* *Proof.* Let us observe that the function $$\label{proof lemma Bessel equation 1} y \mapsto \frac1{(2\pi)^3}\int_{\mathbb{R}^3}\frac{1}{\langle\xi\rangle^5} \,e^{-i y^\mu\,\xi_\mu}\, \, \mathrm{d}\xi$$ is real and spherically symmetric. Therefore, we have $$\label{proof lemma Bessel equation 2} \begin{split} \frac1{(2\pi)^3}\int_{\mathbb{R}^3}\frac{1}{\langle\xi\rangle^5} \,e^{-i y^\mu\,\xi_\mu}\, \, \mathrm{d}\xi & = \frac1{(2\pi)^3}\int_{\mathbb{R}^3}\frac{1}{\langle\xi\rangle^5} \,\cos \left( |y|\,\xi_1\right)\, \, \mathrm{d}\xi\, \\ & = \frac{2\pi}{(2\pi)^3}\int_{-\infty}^{+\infty}\cos \left( |y|\,\xi_1\right) \left(\int_{0}^{+\infty}\frac{r}{(1+(\xi_1)^2+r^2)^{5/2}} \, dr\right) \, \mathrm{d}\xi_1\, \\ & = \frac{1}{12\pi^2}\int_{-\infty}^{+\infty}\frac{\cos \left(|y|\,\xi_1\right)}{\langle \xi_1\rangle^3} \, \mathrm{d}\xi_1\,, \end{split}$$ where at the second step we performed the change of variable $(\xi_2,\xi_3)=(r \cos \theta, r \sin \theta)$. Formula [\[lemma Bessel equation\]](#lemma Bessel equation){reference-type="eqref" reference="lemma Bessel equation"} now follows by explicitly evaluating the integral in the single variable $\xi_1$ appearing in the RHS of [\[proof lemma Bessel equation 2\]](#proof lemma Bessel equation 2){reference-type="eqref" reference="proof lemma Bessel equation 2"}: $$\label{proof lemma Bessel equation 3} \int_{-\infty}^{+\infty}\frac{\cos \left(|y|\,\xi_1\right)}{\langle \xi_1\rangle^3} \, d\xi_1=2 |y|\, K_1(|y|)\,.$$ The latter is known as Basset's Integral [@DLMF [(10.32.11)](https://dlmf.nist.gov/10.32.E11)]. ◻ **Lemma 52**. *We have $$\label{lemma Bessel singularity equation} \frac1{(2\pi)^3}\int_{\mathbb{R}^3}\frac{1}{\langle\xi\rangle^5} \,e^{-i y^\mu\,\xi_\mu}\, \, \mathrm{d}\xi= \frac{1}{12\pi^2} \,|y|^2\,\ln |y| + h(y), \quad\text{where}\quad h\in C^3(\mathbb{R}^3).$$* *Proof.* It is known that the modified Bessel function of the second kind $K_1$ admits the asymptotic expansion $$\label{proof lemma Bessel singularity equation 1} K_1(t)=\frac1t+\frac{t}4 \left(2 \ln t +2\gamma -1-\ln 4 \right) + O(t^3\ln t) \quad \text{as}\quad t\to 0^+,$$ where $\gamma$ is the Euler--Mascheroni constant --- see, e.g., [@DLMF [(10.31.11)](https://dlmf.nist.gov/10.31.E1) and [(10.25.2)](https://dlmf.nist.gov/10.25.E2)]. By combining [\[proof lemma Bessel singularity equation 1\]](#proof lemma Bessel singularity equation 1){reference-type="eqref" reference="proof lemma Bessel singularity equation 1"} and [\[lemma Bessel equation\]](#lemma Bessel equation){reference-type="eqref" reference="lemma Bessel equation"} one obtains [\[lemma Bessel singularity equation\]](#lemma Bessel singularity equation){reference-type="eqref" reference="lemma Bessel singularity equation"}. ◻ **Lemma 53**. *We have $$\label{lemma Bessel singularity differentiated equation} \frac1{(2\pi)^3}\int_{\mathbb{R}^3}\frac{\xi_\gamma\xi_\rho}{\langle\xi\rangle^5} \,e^{-i y^\mu\,\xi_\mu}\, \, \mathrm{d}\xi = -\frac{1}{12\pi^2} \left[ 2\,\frac{y^\gamma y^\rho}{|y|^2} + (1+2\ln|y|) \,\delta_{\gamma\rho} \right] + h_{\gamma\rho}(y), \quad\text{where}\quad h_{\gamma\rho}\in C^1(\mathbb{R}^3).$$* *Proof.* Formula [\[lemma Bessel singularity differentiated equation\]](#lemma Bessel singularity differentiated equation){reference-type="eqref" reference="lemma Bessel singularity differentiated equation"} is a straightforward consequence of [\[lemma Bessel singularity equation\]](#lemma Bessel singularity equation){reference-type="eqref" reference="lemma Bessel singularity equation"}, once one observes that $$\frac{\xi_\gamma\xi_\rho}{\langle\xi\rangle^5} \,e^{-i y^\mu\,\xi_\mu}=-\frac{\partial^2}{\partial y^\gamma \partial y^\rho} \left( \frac{1}{\langle\xi\rangle^5} \,e^{-i y^\mu\,\xi_\mu}\right)\,.$$ ◻ *Proof of Theorem [Theorem 49](#singularity theorem){reference-type="ref" reference="singularity theorem"}.* In the previous section we have established that the asymmetry operator $A$ is a pseudodifferential operator of order $-3$ with principal symbol [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"}. Locally, the symbol $a(x,\xi)$ of $A$ and its integral kernel $\mathfrak{a}(x,y)$ are related via the (distributional) identity $$\label{structure frac a equation 2} \mathfrak{a}(x,y)=\frac{1}{(2\pi)^3\,\rho(y)}\int_{\mathbb{R}^3} e^{i (x-y)^\mu\,\xi_\mu}\, a(x,\xi)\, \mathrm{d}\xi\,.$$ Since a pseudodifferential operator of order $-4$ on a 3-manifold has a continuous integral kernel, formulae [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"}, [\[main theorem 1 equation 3\]](#main theorem 1 equation 3){reference-type="eqref" reference="main theorem 1 equation 3"} and [\[structure frac a equation 2\]](#structure frac a equation 2){reference-type="eqref" reference="structure frac a equation 2"} imply that problem at hand reduces, in normal coordinates centred at $x$, to examining the quantity $$\label{structure frac a equation 3} -\frac{1}{16 \pi^3} \varepsilon^{\alpha\beta\gamma}\, \nabla_\alpha \operatorname{Ric}_{\beta}{}^\rho(0)\,\int_{\mathbb{R}^3} e^{-iy^\mu\,\xi_\mu}\, \frac{\xi_\gamma\xi_\rho}{|\xi|^5} \, (1-\chi(|\xi|))\,\mathrm{d}\xi\,,$$ where $\chi$ is a cut-off as in [\[Q acting on 1-forms 6\]](#Q acting on 1-forms 6){reference-type="eqref" reference="Q acting on 1-forms 6"}. But $$\label{structure frac a equation 4} \frac{\xi_\gamma\xi_\rho}{|\xi|^5} = \frac{\xi_\gamma\xi_\rho}{\langle\xi\rangle^5} + O \left( \frac{1}{|\xi|^5} \right),$$ so the required result follows from Lemma [Lemma 53](#lemma Bessel singularity differentiated){reference-type="ref" reference="lemma Bessel singularity differentiated"}. Note that the term with $\delta_{\gamma\rho}$ from [\[lemma Bessel singularity differentiated equation\]](#lemma Bessel singularity differentiated equation){reference-type="eqref" reference="lemma Bessel singularity differentiated equation"} vanishes upon contraction with $\,\varepsilon^{\alpha\beta\gamma}\, \nabla_\alpha \operatorname{Ric}_{\beta}{}^\rho(0)\,$ due to formula [\[singularity theorem tensor is trace-free equation\]](#singularity theorem tensor is trace-free equation){reference-type="eqref" reference="singularity theorem tensor is trace-free equation"}. ◻   Theorem [Theorem 49](#singularity theorem){reference-type="ref" reference="singularity theorem"} tells us that the singularity of the integral kernel $\mathfrak{a}(x,y)$ of our asymmetry operator $A$ is very weak. We are looking at a bounded function of two variables with a discontinuity on the diagonal. The discontinuity of $\mathfrak{a}(x,y)$ on the diagonal manifests itself in the fact that limit of $\mathfrak{a}(x,y)$ as $y$ tends to $x$ depends on the direction along which $y$ tends to $x$. The bottom line is that working with the asymmetry operator $A$ does not require the use of the theory of distributions or microlocal analysis. It is an integral operator whose integral kernel has a mild discontinuity on the diagonal. ## Local and global trace {#Local and global trace} The decomposition [\[singularity theorem equation 1\]](#singularity theorem equation 1){reference-type="eqref" reference="singularity theorem equation 1"}, [\[singularity theorem equation 2\]](#singularity theorem equation 2){reference-type="eqref" reference="singularity theorem equation 2"} depends on the choice of local coordinates. However, the quantity $$\label{local trace definitiion 1} \psi_{\operatorname{curl}}^\mathrm{loc}(x):=\mathfrak{a}_c(x,x)$$ does not depend on the choice of local coordinates, i.e. is a true scalar function $\psi_{\operatorname{curl}}:M\to\mathbb{R}$. Let us elaborate. Denote $$\label{elaborate 1} f(x,y) := 12\pi^2 \,\mathfrak{a}_d(x,y) \,\operatorname{dist}^2(x,y) =E^{\alpha\beta}{}_\gamma(x)\, \nabla_\alpha \operatorname{Ric}_{\beta\rho}(x)\,(x-y)^\gamma (x-y)^\rho\,.$$ The issue at hand is that the expression [\[elaborate 1\]](#elaborate 1){reference-type="eqref" reference="elaborate 1"} is not a scalar because the quantities $(x-y)^\gamma$ and $(x-y)^\rho$ are not vectors based at the point $x$. Let us switch from local coordinates $x$ to local coordinates $\widetilde x$ and let $\widetilde f(\widetilde x,\widetilde y)$ be the representation of the quantity [\[elaborate 1\]](#elaborate 1){reference-type="eqref" reference="elaborate 1"} in these new coordinates. It is easy to see, by writing down the appropriate Jacobians, that $$\label{elaborate 2} \widetilde f(\widetilde x(x),\widetilde y(y))-f(x,y) = O(\operatorname{dist}^3(x,y)).$$ Now, let $\widetilde{\mathfrak{a}}_c(\widetilde x,\widetilde y)$ be the representation of the quantity $\mathfrak{a}_c(x,y)$ in the new coordinates. Formulae [\[singularity theorem equation 1\]](#singularity theorem equation 1){reference-type="eqref" reference="singularity theorem equation 1"}, [\[singularity theorem equation 2\]](#singularity theorem equation 2){reference-type="eqref" reference="singularity theorem equation 2"}, [\[elaborate 1\]](#elaborate 1){reference-type="eqref" reference="elaborate 1"}, [\[elaborate 2\]](#elaborate 2){reference-type="eqref" reference="elaborate 2"} and the fact that $\mathfrak{a}(x,y)$ is a true scalar in two variables imply $$\label{elaborate 3} \widetilde{\mathfrak{a}}_c(\widetilde x(x),\widetilde y(y))-\mathfrak{a}_c(x,y) = O(\operatorname{dist}(x,y)),$$ which, in turn,implies $\widetilde{\mathfrak{a}}_c(\widetilde x(x),\widetilde x(x))=\mathfrak{a}_c(x,x)$. Using formulae [\[singularity theorem equation 1\]](#singularity theorem equation 1){reference-type="eqref" reference="singularity theorem equation 1"}, [\[singularity theorem equation 2\]](#singularity theorem equation 2){reference-type="eqref" reference="singularity theorem equation 2"} and [\[local trace definitiion 1\]](#local trace definitiion 1){reference-type="eqref" reference="local trace definitiion 1"} for the calculation of the function $\psi_{\operatorname{curl}}$ is impractical. The following lemma provides an alternative equivalent representation. **Lemma 54**. *We have $$\label{lemma averaging over sphere equation} \psi_{\operatorname{curl}}^\mathrm{loc}(x) = \lim_{r\to0^+} \frac{1}{4\pi r^2} \int_{\mathbb{S}_r(x)} \mathfrak{a}(x,y)\,\mathrm{d}S_y\,,$$ where $\mathfrak{a}(x,y)$ is the integral kernel of the asymmetry operator [\[definition asymmetry operator formula\]](#definition asymmetry operator formula){reference-type="eqref" reference="definition asymmetry operator formula"} defined in accordance with [\[structure frac a equation 1\]](#structure frac a equation 1){reference-type="eqref" reference="structure frac a equation 1"}, $\mathbb{S}_r(x)=\{y\in M|\operatorname{dist}(x,y)=r\}$ is the sphere of radius $r$ centred at $x$ and $\mathrm{d}S_y$ is the surface area element on this sphere.* *Proof.* Let us fix an $x\in M$ and choose normal coordinates centred at $x$. Theorem [Theorem 49](#singularity theorem){reference-type="ref" reference="singularity theorem"} tells us that $$\label{lemma averaging over sphere proof equation 1} \mathfrak{a}(x,y) = \mathfrak{a}_c(x,y) + c_{\gamma\rho}(x)\,\frac{(x-y)^\gamma(x-y)^\rho}{|x-y|^2}\,,$$ where $$\label{lemma averaging over sphere proof equation 2} c_{\gamma\rho}(x) = \frac{1}{12\pi^2}\, E^{\alpha\beta}{}_\gamma(x)\, \nabla_\alpha \operatorname{Ric}_{\beta\rho}(x)\,.$$ We have $$\label{lemma averaging over sphere proof equation 3} \int_{\mathbb{S}_r(x)} (x-y)^\gamma(x-y)^\rho \,\mathrm{d}S_y = \begin{cases} \frac{4\pi r^2}{3}\quad&\textbf{if}\quad\gamma=\rho, \\ \ \ 0\quad&\textbf{if}\quad\gamma\ne\rho, \end{cases}$$ hence, $$\label{lemma averaging over sphere proof equation 4} \frac{1}{4\pi r^2} \int_{\mathbb{S}_r(x)} \mathfrak{a}(x,y)\,dS_y = \frac{1}{4\pi r^2} \int_{\mathbb{S}_r(x)} \mathfrak{a}_c(x,y)\,\mathrm{d}S_y + \frac{c_{11}(x)+c_{22}(x)+c_{33}(x)}{3}\,.$$ But formula [\[singularity theorem tensor is trace-free equation\]](#singularity theorem tensor is trace-free equation){reference-type="eqref" reference="singularity theorem tensor is trace-free equation"} implies $c_{11}(x)+c_{22}(x)+c_{33}(x)=0$, so the second term in the right-hand side of [\[lemma averaging over sphere proof equation 4\]](#lemma averaging over sphere proof equation 4){reference-type="eqref" reference="lemma averaging over sphere proof equation 4"} vanishes. Formulae [\[lemma averaging over sphere proof equation 4\]](#lemma averaging over sphere proof equation 4){reference-type="eqref" reference="lemma averaging over sphere proof equation 4"} and [\[local trace definitiion 1\]](#local trace definitiion 1){reference-type="eqref" reference="local trace definitiion 1"} now imply [\[lemma averaging over sphere equation\]](#lemma averaging over sphere equation){reference-type="eqref" reference="lemma averaging over sphere equation"} by the continuity of $\mathfrak{a}_c(x,y)$. ◻ The advantage of formula [\[lemma averaging over sphere equation\]](#lemma averaging over sphere equation){reference-type="eqref" reference="lemma averaging over sphere equation"} is that it does not involve the use of local coordinates. Hence, it can be viewed as a more convenient and natural definition of the scalar function $\psi_{\operatorname{curl}}^\mathrm{loc}\,$. **Definition 55**. We call the scalar continuous function $\psi_{\operatorname{curl}}^\mathrm{loc}(x)$, $\psi_{\operatorname{curl}}^\mathrm{loc}:M\to\mathbb{R}$, defined in accordance with formulae [\[lemma averaging over sphere equation\]](#lemma averaging over sphere equation){reference-type="eqref" reference="lemma averaging over sphere equation"} and [\[structure frac a equation 1\]](#structure frac a equation 1){reference-type="eqref" reference="structure frac a equation 1"} *the regularised local trace of the asymmetry operator*. **Definition 56**. We call the number $$\label{global regularised trace of the asymmetry operator equation} \psi_{\operatorname{curl}} := \int_M \psi_{\operatorname{curl}}^\mathrm{loc}(x) \,\rho(x) \,\mathrm{d}x$$ *the regularised global trace of the asymmetry operator*. The function $\psi_{\operatorname{curl}}^\mathrm{loc}(x)$ and the number $\psi_{\operatorname{curl}}$ are determined by the Riemannian 3-manifold and its orientation. This means that we are looking at geometric invariants. # A conjecture {#A conjecture} The natural question is how are our geometric invariants $\psi_{\operatorname{curl}}(x)$ and $\psi_{\operatorname{curl}}$ related to classical eta invariants. Recall that we denote by $\lambda_k$ the eigenvalues of $\operatorname{curl}$ written with account of multiplicity and by $u_k$ the corresponding orthonormal eigenforms. Recall also that zero is not in the spectrum of $\operatorname{curl}$, see Theorem [Theorem 9](#theorem properties of curl){reference-type="ref" reference="theorem properties of curl"}, so that $\lambda_k\ne0$ for all $k$. The (global) eta function for the operator $\operatorname{curl}$ $$\label{global eta function for curl} \eta_{\operatorname{curl}}(s) := \sum_{k\in\mathbb{Z}\setminus\{0\}}\frac{\operatorname{sgn}\lambda_k}{|\lambda_k|^s}$$ was introduced in [@asymm1 p. 229], see also [@asymm1 p. 229], [@asymm2 pp. 44--45], [@asymm3 p. 405] and [@singerICM pp. 190--191]. It was shown in [@asymm1; @asymm2; @asymm3; @asymm4] that the series [\[global eta function for curl\]](#global eta function for curl){reference-type="eqref" reference="global eta function for curl"} converges absolutely for $\operatorname{Re}s>3$, that it defines a meromorphic function in $\mathbb{C}$ by analytic continuation, with possible first order poles at $s=3,2,1,0,-1,-2,\ldots$, and that $\eta_{\operatorname{curl}}(s)$ is holomorphic at $s=0$. The (real) number $\eta_{\operatorname{curl}}(0)$ is called the (global) eta invariant for the operator $\operatorname{curl}$. **Definition 57**. We define the *local eta function* for the operator $\operatorname{curl}$ as $$\label{local eta function for curl} \eta_{\operatorname{curl}}^\mathrm{loc}(x;s) := \sum_{k\in\mathbb{Z}\setminus\{0\}}\frac{\operatorname{sgn}\lambda_k}{|\lambda_k|^s}\,*(u_k\wedge *u_k)(x)\,,$$ where $*(u_k\wedge *u_k)(x)=g^{\alpha\beta}(x)\,[u_k]_\alpha(x)\,[u_k]_\beta(x)$ is the pointwise norm of the eigenform $u_k$. We believe that for any fixed $x\in M$ the local eta function has the same properties as the global one: the series [\[local eta function for curl\]](#local eta function for curl){reference-type="eqref" reference="local eta function for curl"} converges absolutely for $\operatorname{Re}s>3$, that it defines a meromorphic function in $\mathbb{C}$ by analytic continuation, with possible first order poles at $s=3,2,1,0,-1,-2,\ldots$, and that $\eta_{\operatorname{curl}}^\mathrm{loc}(x;s)$ is holomorphic at $s=0$. Of course, the local and global eta functions are related by the identity $$\label{relation between local and global eta functions} \eta_{\operatorname{curl}}(s) = \int_M \eta_{\operatorname{curl}}^\mathrm{loc}(x;s)\,\rho(x)\,\mathrm{d}x\,.$$ **Definition 58**. We call the (real-valued) function $\eta_{\operatorname{curl}}^\mathrm{loc}(x;0)$ the *local eta invariant* for the operator $\operatorname{curl}$. **Conjecture 59**. (a) *The function $\eta_{\operatorname{curl}}(s)$ is holomorphic in the half-plane $\operatorname{Re}s>-2$.* (b) *For every $x\in M$ the function $s\mapsto\eta_{\operatorname{curl}}^\mathrm{loc}(x;s)$ is holomorphic in the half-plane $\operatorname{Re}s>-2$.* *Remark 60*. It is known [@asymm2 Theorem 4.14(iii)] that the function $\eta_{\operatorname{curl}}(s)$ is holomorphic in the half-plane $\operatorname{Re}s>-1/2$. **Conjecture 61**. (a) *We have $$\label{conjecture relating invariants equation a} \psi_{\operatorname{curl}}=\eta_{\operatorname{curl}}(0),$$ where $\psi_{\operatorname{curl}}$ is our geometric invariant introduced in Definition [Definition 56](#global regularised trace of the asymmetry operator){reference-type="ref" reference="global regularised trace of the asymmetry operator"}.* (b) *Moreover, we have $$\label{conjecture relating invariants equation b} \psi_{\operatorname{curl}}^\mathrm{loc}(x)=\eta_{\operatorname{curl}}^\mathrm{loc}(x;0).$$* We will examine the validity of the above conjectures in a separate paper because doing this requires very extensive and delicate analysis. However, we give below an indication of why we believe Conjecture [Conjecture 59](#conjecture holomorphic for real s greater than -2){reference-type="ref" reference="conjecture holomorphic for real s greater than -2"} to be true.   There are two ways of approaching Conjecture [Conjecture 59](#conjecture holomorphic for real s greater than -2){reference-type="ref" reference="conjecture holomorphic for real s greater than -2"}.   **First argument**. Arguing in the spirit of Seeley [@seeley] (with appropriate modifications) one expects the residues of the poles of the eta function, global or local, to be expressed via local geometric invariants: curvature, its covariant derivatives and the totally antisymmetric tensor. More precisely, the residues in question should be scalar quantities formed out of products of these invariants containing exactly one appearance of the totally antisymmetric tensor. Let us assign to such scalar invariants degrees according to the following rules: - the degrees of factors add up, - curvature carries degree $-2$, - each covariant differentiation contributes $-1$ to the degree. After analysing possible scalar invariants of this type one concludes that the nonvanishing invariant of highest degree is the invariant $$(\nabla_\alpha\operatorname{Ric}_{\beta\gamma})\,\operatorname{Ric}^\beta{}_\nu\,E^{\alpha\gamma\nu}$$ which has degree $-5$. An invariant of degree $n$ corresponds to a pole at $s=3+n$. Hence, we expect the first pole to be at $s=3-5=-2$. This would agree with similar results obtained for the massless Dirac operator [@bismut; @branson].   **Second argument**. Consider the global counting function for $\operatorname{curl}$ [\[global counting functions for curl\]](#global counting functions for curl){reference-type="eqref" reference="global counting functions for curl"} and its local analogue $$\label{local counting functions for curl} N^\pm_{\operatorname{curl}}(x;\lambda) := \begin{cases} 0\quad&\text{for}\quad\lambda\le0, \\ \sum_{0<\pm\lambda_k<\lambda} \,*(u_k\wedge *u_k)(x) \quad&\text{for}\quad\lambda>0. \end{cases}$$ Of course, the local and global counting functions are related by the identity $$\label{relation between local and global counting functions} N^\pm_{\operatorname{curl}}(\lambda) = \int_M N^\pm_{\operatorname{curl}}(x;\lambda)\,\rho(x)\,dx\,.$$ Let $\widehat{\mu}:\mathbb{R}\to \mathbb{C}$ be a smooth function such that $\widehat{\mu}= 1$ in some neighbourhood of the origin and $\operatorname{supp} \widehat{\mu}\subset(-T_0,T_0)$, where $T_0$ is the infimum of lengths of all the geodesic loops originating from all the points of the manifold. Let $\mu$ be the inverse Fourier transform of $\widehat{\mu}$. Although the operator $\operatorname{curl}$ is not elliptic, arguing in the spirit of [@wave Appendix B] and references therein, it is natural to expect that the mollified derivative of the local and global counting functions admits complete asymptotic expansions of the form $$\label{local counting function mollified} ((N^\pm_{\operatorname{curl}})'*\mu)(x;\lambda) = c^\pm_2(x)\,\lambda^2 + c^\pm_1(x)\,\lambda + c^\pm_0(x) + c^\pm_{-1}(x)\,\lambda^{-1} +\ldots,$$ $$\label{global counting function mollified} ((N^\pm_{\operatorname{curl}})'*\mu)(\lambda) = c^\pm_2\,\lambda^2 + c^\pm_1\,\lambda + c^\pm_0 + c^\pm_{-1}\,\lambda^{-1} +\ldots,$$ as $\lambda\to+\infty$. Here the star stands for convolution in the variable $\lambda$. We call the coefficients $c^\pm_k$ appearing in [\[local counting function mollified\]](#local counting function mollified){reference-type="eqref" reference="local counting function mollified"} and [\[global counting function mollified\]](#global counting function mollified){reference-type="eqref" reference="global counting function mollified"} *Weyl coefficients*, local and global respectively. Of course, the local and global Weyl coefficients are related by the identity $c^\pm_k=\int_M c^\pm_k(x)\,\rho(x)\,\mathrm{d}x\,$. It is known [@baer_curl] that $c^\pm_2(x)=\frac{1}{2\pi^2}\,$. In the remainder of the argument we, for the sake of brevity, deal only with local functions. The local eta function is expressed via the local counting functions as $$\label{eta function is expressed via the counting functions} \eta_{\operatorname{curl}}^\mathrm{loc}(x;s) = \int_0^{+\infty}\lambda^{-s} \left[ (N^+_{\operatorname{curl}})'(x;\lambda)-(N^-_{\operatorname{curl}})'(x;\lambda) \right] \mathrm{d}\lambda\,.$$ One can show [@fang_PhD Theorem 3.2.2] that the residues of the local eta function are given by $$\mathrm{Res}(\eta_{\operatorname{curl}}^\mathrm{loc}(x;\,\cdot\,),n) =c_{n-1}^+(x)-c_{n-1}^-(x)\,, \qquad n=3,2,1,0,-1,-2,\ldots.$$ We are pursuing a separate project aimed at calculating the Weyl coefficients $c^\pm_k(x)$, $k=1,0,-1,\ldots$, for the operator $\operatorname{curl}$, and we are doing this along the lines of [@dirac] where similar analysis was initiated for the massless Dirac operator. We expect the first five Weyl coefficients for $N^+_{\operatorname{curl}}$ and $N^-_{\operatorname{curl}}$ to coincide, $$c_k^+(x)=c_k^-(x),\qquad k=2,1,0,-1,-2,$$ which would imply that the local eta function is holomorphic in the half-plane $\operatorname{Re}s>-2$. # Challenges in higher dimensions {#Challenges in higher dimensions} Let $(M,g)$ be a connected closed Riemannian manifold of dimension $$\label{higher dimensions} d=4k+3,\qquad k=1,2,\ldots.$$ In this case one can define $\operatorname{curl}$ in the usual way [\[curl differential experssion\]](#curl differential experssion){reference-type="eqref" reference="curl differential experssion"} as an operator acting in $\Omega^{2k+1}(M)$. It is easy to see that $\operatorname{curl}$ remains formally self-adjoint and at a formal level everything appears to be similar to what happens in dimension 3. *Remark 62*. One could have also considered the case $d=4k+1$, $k=1,2,\ldots$, with $\operatorname{curl}$ acting in $\Omega^{2k}(M)$ in accordance with [\[curl differential experssion\]](#curl differential experssion){reference-type="eqref" reference="curl differential experssion"}. However, in this case $\operatorname{curl}$ is formally anti-self-adjoint, the spectrum is purely imaginary and there is no spectral asymmetry [@baer_curl Theorem 3.2]. As our interest is in spectral asymmetry, we do not discuss the case $d=4k+1$. Note also that the spectral problem for $\operatorname{curl}$ in dimension $d=4k+1$ cannot be formulated in terms of the spectral theory of self-adjoint operators in *real* Hilbert spaces: one can introduce the factor $\,i\,$ in the right-hand side of [\[curl differential experssion\]](#curl differential experssion){reference-type="eqref" reference="curl differential experssion"} to make the operator formally self-adjoint (as was done in [@baer_curl Definition 2.2]), but the resulting operator will no longer be an operator acting in the vector space of *real* differential forms. The bottom line is that the two cases, $d=4k+3$ and $d=4k+1$, are fundamentally different. We stick with [\[higher dimensions\]](#higher dimensions){reference-type="eqref" reference="higher dimensions"} because this gives the only genuine generalisation of the operator $\operatorname{curl}$ to higher dimensions.   Analysis of higher dimensions [\[higher dimensions\]](#higher dimensions){reference-type="eqref" reference="higher dimensions"} presents two major challenges.   **First challenge**. We expect that in higher dimensions the asymmetry operator will still be a pseudodifferential operator of order $-3$, as it is in the 3-dimensional case. However, a sufficient condition for a pseudodifferential operator to be of trace class is that its order be less than $-d$. This means that defining the regularised trace of the asymmetry operator becomes challenging: one would need to analyse singularities of the integral kernel and perform regularisation reducing the order of the singularity by $4k+1$. In other words, one would need to perform $4k+1$ steps of regularisation. Doing this in a geometrically invariant fashion will not be an easy task.   **Second challenge**. In the current paper we used the results of [@part1] which apply to operators whose principal symbols have simple eigenvalues (see also [@part2]). In dimension 3 the operator $\operatorname{curl}$ does possess this property --- its principal symbol has three simple eigenvalues [\[eigenvalues principal symbol curl\]](#eigenvalues principal symbol curl){reference-type="eqref" reference="eigenvalues principal symbol curl"}. In higher dimensions the eigenvalues of the principal symbol of the operator $\operatorname{curl}$ are still given by formulae [\[eigenvalues principal symbol curl\]](#eigenvalues principal symbol curl){reference-type="eqref" reference="eigenvalues principal symbol curl"}, only now they have multiplicity: zero is an eigenvalue of multiplicity $\binom{4k+2}{2k}$, whereas each of the two nonzero eigenvalues has multiplicity $\frac{1}{2}\binom{4k+2}{2k+1}$. This means that addressing the issue of higher dimensions would require extending the results of [@part1] to operators whose principal symbols have multiple eigenvalues. This will be a difficult task as the construction presented in [@part1] relied heavily on the assumption that the eigenvalues of the principal symbol of the operator are simple. Overall, partial differential operators whose principal symbols have multiple eigenvalues are known to present a major challenge in micrololcal analysis.   Despite the challenges involved, examining the asymmetry operator in higher dimension could be an piece of research worth pursuing in the future. For example, since the seminal works by Milnor and Kervaire [@milnor1; @milnor2] the study of exotic smooth structures on 7-spheres has been an active area of research in differential geometry. Our asymmetry operator, being very sensitive to the underlying geometry, might help to make some progress in this area. # Acknowledgements {#acknowledgements .unnumbered} MC was partially supported by a Leverhulme Trust Research Project Grant RPG-2019-240, by a Research Grant (Scheme 4) of the London Mathematical Society, and by a grant of the Heilbronn Institute for Mathematical Research (HIMR) via the UKRI/EPSRC Additional Funding Programme for Mathematical Sciences. We are grateful to Christian Bär, Bruno Colbois, Michal Kwasigroch, Jason Lotay and Nikolai Saveliev for useful comments and suggestions. We also thank Michael Levitin for kindly sharing his Mathematica scripts on the Berger sphere. # Exterior calculus {#Exterior calculus} In this appendix we set out out conventions on exterior calculus. Henceforth, we identify differential forms with covariant antisymmetric tensors, and $M$ is an oriented 3-manifold equipped with Riemannian metric $g$ and Levi-Civita connection $\nabla$. It is well known that the metric $g$ induces a canonical isomorphism between the tangent bundle $TM$ and the contangent bundle $T^*M$, the so-called *musical isomorphism*. We denote it by $\flat:TM\to T^*M$ (lowering of indices) and its inverse by $\sharp:T^*M\to TM$ (raising of indices). Given a scalar field $f\in C^\infty(M)$, its exterior derivative $\mathrm{d}f$ is defined as the gradient. Given a 1-form $u\in\Omega^1(M)$, its exterior derivative $\mathrm{d}u\in\Omega^2(M)$ is defined, componentwise, as $$(\mathrm{d}u)_{\alpha\beta} = \partial_{x^\alpha}u_\beta-\partial_{x^\beta}u_\alpha\,.$$ We define the action of the Hodge star $*:\Omega^k(M)\to \Omega^{3-k}(M)$ on a rank $k$ antisymmetric tensor as $$\label{definition of Hodge star} (*v)_{\mu_{k+1}\ldots\mu_3}\!:=\frac1{k!}\, \sqrt{\det g_{\alpha\beta}} \ v^{\mu_1\ldots\mu_k}\,\varepsilon_{\mu_1\mu_2\mu_3}\,,$$ where $\varepsilon$ is the totally antisymmetric symbol, $\varepsilon_{123}:=+1$. With regards to the totally antisymmetric symbol, we adopt the convention of raising indices using the Kronecker symbol, so that, for example, $$\varepsilon^{\alpha_1\alpha_2\alpha_3}:= \varepsilon_{\alpha_1\alpha_2\alpha_3}.$$ Let $\wedge$ denote the exterior product. Given a pair of real-valued rank $k$ covariant antisymmetric tensors $u$ and $v$ we define their $L^2$ inner product as $$%\label{inner product of antisymmetric tensors} \begin{split} \langle u,v\rangle :&= \int_M \frac{1}{k!}\, u_{\alpha_1\dots\alpha_k} \, v_{\beta_1\dots\beta_k} \, g^{\alpha_1\beta_1}\cdots g^{\alpha_k\beta_k} \,\sqrt{\det g_{\mu\nu}} \ \mathrm{d}x \\ & = \int_M u\wedge *v =\int_M *u\wedge v\,. \end{split}$$ For the sake of clarity, let us mention that the exterior product of 1-forms reads $$(u\wedge v)_{\alpha\beta}=u_\alpha v_\beta-u_\beta v_\alpha\,.$$ Given $w\in\Omega^k(M)$ and $u\in\Omega^{k-1}(M)$ we define the action of the codifferential $\delta:\Omega^k(M)\to\Omega^{k-1}(M)$ in accordance with $$\langle w,\mathrm{d}u\rangle=\langle\delta w,u\rangle.$$ In particular, when $u\in\Omega^1(M)$ and $w\in\Omega^2(M)$, we get in local coordinates $$\delta u=-\nabla^\alpha u_\alpha,$$ $$(\delta w)_\alpha=\nabla^\beta w_{\alpha\beta}.$$ # The spectrum of the Laplacian on a Berger sphere {#The spectrum of the Laplacian on a Berger sphere} In preparation for the calculations that will appear in Appendix [12](#The spectrum of curl on a Berger sphere){reference-type="ref" reference="The spectrum of curl on a Berger sphere"}, we list here, in the form of a theorem, the eigenvalues of the Laplacian on a Berger sphere. These were first obtained in [@tanno Lemma 4.1] and reproduced in [@lotay Theorem 5.5] and [@lauret Proposition 3.9]. ## Definitions and notation {#subsec:Definitions and notation berger} In Euclidean space $\mathbb{R}^4$ equipped with Cartesian coordinates $\mathbf{x}^\alpha$, $\alpha=1,2,3,4$, let $$\mathbb{S}^3:=\{\mathbf{x}\in \mathbb{R}^4 \ | \ \|\mathbf{x}\|=1\}$$ be the $3$-sphere. We use bold script for 4-dimensional objects and normal script for 3-dimensional objects. We prescribe orientation of $\mathbb{S}^3$ as follows. We define local coordinates $y=(y^1,y^2,y^3)$ on $\mathbb{S}^3$ to be positively oriented if $$\label{definition of orientatioin via embedding} \det \begin{pmatrix} \frac{\partial\mathbf{x}^1}{\partial y^1}(y)& \frac{\partial\mathbf{x}^1}{\partial y^2}(y)& \frac{\partial\mathbf{x}^1}{\partial y^3}(y)& \mathbf{x}^1(y) \\ \frac{\partial\mathbf{x}^2}{\partial y^1}(y)& \frac{\partial\mathbf{x}^2}{\partial y^2}(y)& \frac{\partial\mathbf{x}^2}{\partial y^3}(y)& \mathbf{x}^2(y) \\ \frac{\partial\mathbf{x}^3}{\partial y^1}(y)& \frac{\partial\mathbf{x}^3}{\partial y^2}(y)& \frac{\partial\mathbf{x}^3}{\partial y^3}(y)& \mathbf{x}^3(y) \\ \frac{\partial\mathbf{x}^4}{\partial y^1}(y)& \frac{\partial\mathbf{x}^4}{\partial y^2}(y)& \frac{\partial\mathbf{x}^4}{\partial y^3}(y)& \mathbf{x}^4(y) \end{pmatrix} <0\,.$$ The above definition of orientation agrees with that from [@sphere Appendix A]. Spherical coordinates $$\label{spherical coordinates for 3-sphere} \begin{pmatrix} \mathbf{x}^1\\ \mathbf{x}^2\\ \mathbf{x}^3\\ \mathbf{x}^4 \end{pmatrix} = \begin{pmatrix} \cos y^1\\ \sin y^1\cos y^2\\ \sin y^1\sin y^2\cos y^3\\ \sin y^1\sin y^2\sin y^3\\ \end{pmatrix}, \quad y^1,y^2\in(0,\pi), \quad y^3\in[0,2\pi),$$ are an example of positively oriented local coordinates on $\mathbb{S}^3$. Of course, if one is only interested in the study of the Laplace--Beltrami operator, orientation is irrelevant. However, with $\operatorname{curl}$ or the Dirac operator [@dirac subsection 9.1] prescribing orientation of the manifold is essential. Consider the vector fields $$\label{berger equation 1} \begin{aligned} &\mathbf{V}_{1}:= - \textbf{x}^4\dfrac{\partial}{\partial \textbf{x}^1}- \textbf{x}^3\frac{\partial}{\partial \textbf{x}^2} +\textbf{x}^2\dfrac{\partial}{\partial \textbf{x}^3}+\textbf{x}^1\dfrac{\partial}{\partial \textbf{x}^4}\,,\\ &\mathbf{V}_{2}:= \phantom{-} \textbf{x}^3\dfrac{\partial}{\partial \textbf{x}^1}-\textbf{x}^4\frac{\partial}{\partial \textbf{x}^2} -\textbf{x}^1 \dfrac{\partial}{\partial \textbf{x}^3}+\textbf{x}^2\dfrac{\partial}{\partial \textbf{x}^4}\,,\\ &\mathbf{V}_{3}:= - \textbf{x}^2\dfrac{\partial}{\partial \textbf{x}^1}+ \textbf{x}^1\frac{\partial}{\partial \textbf{x}^2} -\textbf{x}^4\dfrac{\partial}{\partial \textbf{x}^3}+\textbf{x}^3\dfrac{\partial}{\partial \textbf{x}^4}\,.\\ \end{aligned}$$ The vector fields [\[berger equation 1\]](#berger equation 1){reference-type="eqref" reference="berger equation 1"} are tangent to $\mathbb{S}^3$, i.e., they are orthogonal to the gradient of $\|\textbf{x}\|$, or, equivalently, $\mathbf{V}_{j} (\|\textbf{x}\|^2)=0$, $j=1,2,3$. For $j=1,2,3$ let $V_j$ be the restriction of $\mathbf{V}_{j}$ to $\mathbb{S}^3$. Let $a\in \mathbb{R}$ be a positive parameter. Consider the differential operator $$\label{berger equation 2} B:=V_1^2+V_2^2+\frac{1}{a^2}V_3^2\,.$$ The *Berger metric* $g$ on $\mathbb{S}^3$ is defined via the identity $$\label{berger equation 3} B_\mathrm{prin}(y,\eta)=-g^{\alpha\beta}(y)\eta_\alpha\eta_\beta$$ or, equivalently, $$\label{berger equation 4} g^{\alpha\beta}(y):=-\frac{\partial^2 B_\mathrm{prin}}{\partial \eta_\alpha\partial\eta_\beta}\,.$$ We call $(\mathbb{S}^3,g)$ the *Berger sphere*. The notion of a Berger sphere is well established in differential geometry, nevertheless, for the benefit of a wider audience, we provide here an explicit description in the simplest possible coordinate system. Working in the southern hemisphere $\mathbf{x}^4<0$, consider the positively oriented local coordinates $$\label{berger equation 5} y^\alpha=\mathbf{x}^\alpha,\qquad \alpha=1,2,3,$$ so that $$\label{berger equation 6} \mathbf{x}^4 = -\sqrt{1-(\mathbf{x}^1)^2-(\mathbf{x}^2)^2-(\mathbf{x}^3)^2} = -\sqrt{1-|y|^2}.$$ In these coordinates the vector fields $V_j$ read $$\label{berger equation 9} \begin{split} V_1 & = \sqrt{1-|y|^2} \dfrac{\partial}{\partial y^1} - y^3 \dfrac{\partial}{\partial y^2} + y^2 \dfrac{\partial}{\partial y^3}, \\ V_2 & = y^3 \dfrac{\partial}{\partial y^1} + \sqrt{1-|y|^2} \dfrac{\partial}{\partial y^2} - y^1 \dfrac{\partial}{\partial y^3}, \\ V_3 & = -y^2 \dfrac{\partial}{\partial y^1} + y^1 \dfrac{\partial}{\partial y^2} + \sqrt{1-|y|^2} \dfrac{\partial}{\partial y^3}. \end{split}$$ Formulae [\[berger equation 2\]](#berger equation 2){reference-type="eqref" reference="berger equation 2"}, [\[berger equation 3\]](#berger equation 3){reference-type="eqref" reference="berger equation 3"} and [\[berger equation 9\]](#berger equation 9){reference-type="eqref" reference="berger equation 9"} imply $$\label{berger equation 10} \footnotesize g^{\alpha\beta}(y)= \begin{pmatrix} 1-(y^1)^2-\frac{a^2-1}{a^2}(y^2)^2& - \frac{y^1 y^2}{a^2} &- y^1 y^3+\frac{a^2-1}{a^2}y^2\sqrt{1-|y|^2} \\ - \frac{y^1 y^2}{a^2} &1-(y^2)^2-\frac{a^2-1}{a^2}(y^1)^2& - y^2 y^3-\frac{a^2-1}{a^2}y^1\sqrt{1-|y|^2} \\ -y^1 y^3+\frac{a^2-1}{a^2}y^2\sqrt{1-|y|^2} & - y^2 y^3-\frac{a^2-1}{a^2}y^1\sqrt{1-|y|^2} &\frac{a^2-1}{a^2}(y^2)^2+\frac{a^2-1}{a^2}(y^2)^2-\frac{1-(y^3)^2}{a^2}\\ \end{pmatrix}.$$ We summarise the properties of the vector fields $V_j$, $j=1,2,3$, and the Berger metric $g$ in the following Lemma. The proof amounts to straightforward calculations and is omitted. **Lemma 63**. *We have the following properties.* (a) *$[V_j,V_k]=-2 \varepsilon_{jkl}\,V_l\,$, $j,k=1,2,3$.* (b) *The vector fields $V_j$ are positively oriented, i.e. $\det V_j{}^\alpha>0$, $j,\alpha=1,2,3$. Here the $V_j{}^\alpha$ are the components of the vector fields $V_j$ with respect to the basis $\frac{\partial}{\partial y^\alpha}\,$.* (c) *The vector fields $$\label{orthonormalised vector fields} \widetilde V_1:=V_1\,, \qquad \widetilde V_2:=V_2\,, \qquad \widetilde V_3:=\frac{1}{a}V_3$$ are orthonormal with respect to the metric $g$ defined by [\[berger equation 10\]](#berger equation 10){reference-type="eqref" reference="berger equation 10"}.* (d) *The Riemannian density $\rho(y)$ generated by the Berger metric $g$ reads $$\rho(y)=a\,\rho_0(y),$$ where $\rho_0(y)$ is the standard Riemannian density of the round $\mathbb{S}^3$.* (e) *The vector fields $V_j$, $j=1,2,3$, viewed as differential operators are skew-Hermitian.* (f) *The vector fields $V_j$, $j=1,2,3$, are Killing vector fields for the standard round metric $g_0$ on $\mathbb{S}^3$.* (g) *The vector field $V_3$ is a Killing vector field for the Berger metric $g$.* (h) *The operator $B$ defined in accordance with [\[berger equation 2\]](#berger equation 2){reference-type="eqref" reference="berger equation 2"} is the Laplace--Beltrami operator (henceforth denoted by $\Delta$) on the Berger sphere.* (i) *The operator $\operatorname{curl}$ on the Berger sphere can be represented as $$\label{berger equation 11} \operatorname{curl}_s = \begin{pmatrix} \frac{2}{a} &-\frac1{a}V_3 & V_2\\ \frac1{a}V_3& \frac{2}{a} & - V_1\\ -V_2 & V_1& 2a \\ \end{pmatrix}.$$ Here the subscript $\,s\,$ indicates that the operator appearing in [\[berger equation 11\]](#berger equation 11){reference-type="eqref" reference="berger equation 11"} acts on 3-columns of scalar functions. The latter can be obtained by projecting 1-forms onto the orthonormal framing [\[orthonormalised vector fields\]](#orthonormalised vector fields){reference-type="eqref" reference="orthonormalised vector fields"}. The operators $\operatorname{curl}$ and $\operatorname{curl}_s$ are related as $$\operatorname{curl}_\alpha{}^\beta = \widetilde V^j{}_\alpha\,[\operatorname{curl}_s]_j{}^k\,\widetilde V_k{}^\beta, \qquad \widetilde V^j{}_\alpha:=\delta^{jl}\,\widetilde V_l{}^\gamma\,g_{\alpha\gamma}\,.$$ See also the projection operator $\mathbf{S}$ in [@obstructions Section 5.1].* ## The spectrum **Theorem 64**. *The eigenvalues of the operator $-\Delta$ on the Berger 3-sphere are $$\label{eigenvalue laplacian berger formula 1} n(n+2) + \left( a^{-2}-1 \right) \left( n-2l \right)^2 \,, \quad n=0,1,2,3,\dots, \quad 0 \le l \le \left\lfloor \frac{n}{2} \right\rfloor$$ with* (i) *multiplicity $$1$$ if $n=0$,* (ii) *multiplicity $$2n+2$$ if $n$ is odd,* (iii) *multiplicity $$2n+2$$ if $n$ is even and $0\le l< \frac{n}{2}$, and* (iv) *multiplicity $$n+1$$ if $n$ is even and $l =\frac{n}{2}$.* For a proof of the above theorem we refer the reader to [@lotay Theorem 5.5]. Somewhat more involuted proofs may also be found in [@tanno Lemma 4.1] and [@lauret Proposition 3.9]. *Remark 65*. When $a=1$ Theorem [Theorem 64](#theorem spectrum berger laplacian){reference-type="ref" reference="theorem spectrum berger laplacian"} reduces to the usual spectrum of $\,-\Delta\,$ on the round 3-sphere: eigenvalues $n(n+2)$, $n=0,1,2,\dots$, with multiplicity $(n+1)^2\,$. # The spectrum of curl on a Berger sphere {#The spectrum of curl on a Berger sphere} In this appendix we will write down explicitly the eigenvalues and the $\eta$-invariant for the operator curl on a Berger sphere. This will be done using classical (as opposed to pseudodifferential) techniques and will serve as an illustration of the spectral asymmetry of curl.   Throughout this appendix we use the notation from Appendix [11.1](#subsec:Definitions and notation berger){reference-type="ref" reference="subsec:Definitions and notation berger"}. ## The spectrum The positive spectrum of curl [\[berger equation 11\]](#berger equation 11){reference-type="eqref" reference="berger equation 11"} was first computed by Lotay in [@lotay Proposition 5.6]. Explicit formulae for the negative spectrum do not appear to be available in the literature. We give below, for future reference and in the form of a theorem, explicit formulae for the *full* spectrum of curl: Lotay's result (slightly reformulated) as well as negative eigenvalues. The proof is omitted. **Theorem 66**. *The spectrum of the operator $\operatorname{curl}$ on the Berger 3-sphere is the (disjoint) union of the following four series of eigenvalues:* I. *Eigenvalues $$\label{20 August 2021 equation 1} \frac{n}{a}\,,\qquad n=2,3,\dots,$$ with multiplicity $$\label{20 August 2021 equation 2} 2n-2.$$* II. *Eigenvalues $$\label{20 August 2021 equation 3} \frac{n+2(a^2-1)}{a}\,,\qquad n=2,3,\dots.$$ Here the multiplicity is as follows.* (a) *If $n=2$ the multiplicity is $$\label{20 August 2021 equation 4} 1.$$* (b) *If $n=3,4,\dots$ the multiplicity is $$\label{20 August 2021 equation 5} 2n-2.$$* III. *Eigenvalues $$\label{20 August 2021 equation 6} a + \sqrt{ a^2 + n(n+2) + \left( a^{-2}-1 \right) \left( n-2l \right)^2 }\ , \quad n=2,3,\dots, \quad 1 \le l \le \left\lfloor \frac{n}{2} \right\rfloor.$$ Here the multiplicity is as follows.* (a) *If $n$ is odd the multiplicity is $$\label{20 August 2021 equation 7} 2n+2.$$* (b) *If $n$ is even and $l<\frac{n}{2}$ the multiplicity is $$\label{20 August 2021 equation 8} 2n+2.$$* (c) *If $n$ is even and $l=\frac{n}{2}$ the multiplicity is $$\label{20 August 2021 equation 9} n+1.$$* IV. *Eigenvalues $$\label{20 August 2021 equation 10} a - \sqrt{ a^2 + n(n+2) + \left( a^{-2}-1 \right) \left( n-2l \right)^2 }\ , \quad n=2,3,\dots, \quad 1 \le l \le \left\lfloor \frac{n}{2} \right\rfloor,$$ with multiplicities as in [\[20 August 2021 equation 7\]](#20 August 2021 equation 7){reference-type="eqref" reference="20 August 2021 equation 7"}--[\[20 August 2021 equation 9\]](#20 August 2021 equation 9){reference-type="eqref" reference="20 August 2021 equation 9"}.* *Remark 67*. (i) Series I, II and III form the positive spectrum, whereas series IV is the negative spectrum. (ii) For $a=1$ the Berger sphere reduces to the standard 3-sphere. It is not hard to check that for $a=1$ Theorem [Theorem 66](#theorem spectrum of curl){reference-type="ref" reference="theorem spectrum of curl"} gives us eigenvalues $$\label{spectrum curl 3 sphere eigenvalues} \pm n, \qquad n=2,3,\dots,$$ with multiplicity $$\label{spectrum curl 3 sphere multiplicity} n^2-1.$$ In particular, the spectrum is symmetric about $0$. (iii) Formulae [\[spectrum curl 3 sphere eigenvalues\]](#spectrum curl 3 sphere eigenvalues){reference-type="eqref" reference="spectrum curl 3 sphere eigenvalues"} and [\[spectrum curl 3 sphere multiplicity\]](#spectrum curl 3 sphere multiplicity){reference-type="eqref" reference="spectrum curl 3 sphere multiplicity"} agree with [@baer_curl Theorem 5.2]. (iv) Note that in [\[20 August 2021 equation 6\]](#20 August 2021 equation 6){reference-type="eqref" reference="20 August 2021 equation 6"} the lowest value the integer $l$ can take is $1$. The case $l=0$ is special: in this case the square root becomes superfluous and the expression [\[20 August 2021 equation 6\]](#20 August 2021 equation 6){reference-type="eqref" reference="20 August 2021 equation 6"} turns into [\[20 August 2021 equation 3\]](#20 August 2021 equation 3){reference-type="eqref" reference="20 August 2021 equation 3"}. We have highlighted the special status of the case $l=0$ by separating out the series of eigenvalues [\[20 August 2021 equation 3\]](#20 August 2021 equation 3){reference-type="eqref" reference="20 August 2021 equation 3"}. (v) Observe that the expression [\[eigenvalue laplacian berger formula 1\]](#eigenvalue laplacian berger formula 1){reference-type="eqref" reference="eigenvalue laplacian berger formula 1"} appears under the square root in formulae [\[20 August 2021 equation 6\]](#20 August 2021 equation 6){reference-type="eqref" reference="20 August 2021 equation 6"} and [\[20 August 2021 equation 10\]](#20 August 2021 equation 10){reference-type="eqref" reference="20 August 2021 equation 10"}. ## The eta invariant In this appendix we will compute the eta invariant for the operator $\operatorname{curl}$ on a Berger 3-sphere directly, using the explicit formulae for the eigenvalues established above in Theorem [Theorem 66](#theorem spectrum of curl){reference-type="ref" reference="theorem spectrum of curl"}. Recall that the eta-function is defined, in general, in accordance with [\[eta function intro\]](#eta function intro){reference-type="eqref" reference="eta function intro"}. **Theorem 68**. *Consider the operator $\operatorname{curl}$ on a Berger 3-sphere, defined as in Appendix [11.1](#subsec:Definitions and notation berger){reference-type="ref" reference="subsec:Definitions and notation berger"}. We have $$\label{theorem eta invariant berger equation 1} \eta_{\operatorname{curl}}(0)=\frac23 (a^2-1)^2.$$* *Remark 69*. The above theorem warrants a number of remarks. (i) The eta invariant for the Dirac operator on a Berger 3-sphere was computed by Hitchin [@hitchin p. 34] and reads $$\label{eta invariant Dirac} \eta_{\mathrm{Dirac}}(0)=-\frac{1}{6}(a^2-1)^2\,.$$ Note that the parameter $\lambda$ in [@hitchin] corresponds to our parameter $a$, and that Hitchin adopts the same (or rather, equivalent) conventions as we do when defining the Dirac operator. The latter can be seen if one compares the spectra of the Dirac operator in [@hitchin] and [@sphere]: Hitchin's [@hitchin Proposition 3.2] with $p=q=1$ and minus sign in front of the square root agrees with [@sphere formulae (6.5) and (6.8)]. The result [\[eta invariant Dirac\]](#eta invariant Dirac){reference-type="eqref" reference="eta invariant Dirac"} was obtained under the assumption $a<4$. We do not need to impose any restriction on $a$ because, unlike for Dirac operator, where the dimension of the space of harmonic spinors is highly dependent on the Riemannian metric, zero is never an eigenvalue of curl (see Theorem [Theorem 9](#theorem properties of curl){reference-type="ref" reference="theorem properties of curl"}). (ii) For a general connected oriented closed Riemannian 3-manifold the invariants $\eta_{\operatorname{curl}}(0)$ and $\eta_{\mathrm{Dirac}}(0)$ are related as $$\label{relation between eta curl and eta Dirac} \eta_{\operatorname{curl}}(0)= -4\eta_{\mathrm{Dirac}}(0)+ k$$ for some $k\in \mathbb{Z}$. This follows from abstract algebraic topology arguments involving Hirzebruch polynomials, see [@asymm1 Section 1] and [@asymm2 Theorems 4.2 and 4.14]. Theorem [Theorem 68](#theorem eta invariant berger){reference-type="ref" reference="theorem eta invariant berger"} establishes that, in the case of a Berger 3-sphere, we have [\[relation between eta curl and eta Dirac\]](#relation between eta curl and eta Dirac){reference-type="eqref" reference="relation between eta curl and eta Dirac"} with $k=0$ for $a<4$. The significance of the value $a=4$ is that for this particular value of the parameter $a$ the Dirac operator has a zero mode (harmonic spinor). (iii) The properties of the Dirac operator on a Berger 3-sphere, including an alternative (to Hitchin's [@hitchin]) derivation of the spectrum, were further studied by Bär in [@baer92]. Note that Bär adopts the opposite convention when defining the Dirac operator, so that the eigenvalues of the Dirac operator in [@baer92] differ from those in [@hitchin; @sphere] by sign . In preparation for the proof of Theorem [Theorem 68](#theorem eta invariant berger){reference-type="ref" reference="theorem eta invariant berger"}, let us make the following observation. **Lemma 70**. *The function $\eta_{\operatorname{curl}}$ admits the representation $$\label{eta curl dima 1} \eta_{\operatorname{curl}}(s) = \theta(s) + (2a)^{-s}+4a^s \zeta(s-1)\,,$$ where $$\label{eta curl dima 2} \theta(s) := \sum_{j=1}^\infty \left[ \left( \sqrt{a^2+\mu_j} \,+\,a \right)^{-s} - \left( \sqrt{a^2+\mu_j} \,-\,a \right)^{-s} \right],$$ the $\mu_j$ are the eigenvalues [\[eigenvalues of Laplace\--Beltrami operator\]](#eigenvalues of Laplace--Beltrami operator){reference-type="eqref" reference="eigenvalues of Laplace--Beltrami operator"} of $\,-\Delta\,$ enumerated in increasing order with account of multiplicities, and $\,\zeta(s):=\sum_{n=1}^\infty n^{-s}\,$ is the Riemann zeta function.* *Proof.* Formulae [\[eta curl dima 1\]](#eta curl dima 1){reference-type="eqref" reference="eta curl dima 1"} and [\[eta curl dima 2\]](#eta curl dima 2){reference-type="eqref" reference="eta curl dima 2"} are a straightforward consequence of Theorems [Theorem 64](#theorem spectrum berger laplacian){reference-type="ref" reference="theorem spectrum berger laplacian"} and [Theorem 66](#theorem spectrum of curl){reference-type="ref" reference="theorem spectrum of curl"}. ◻ **Definition 71**. We define $$\label{Minakshisundaram--Pleijel zeta function definitiion equation 1} \zeta_{-\Delta}(s) := \sum_{j=1}^\infty\mu_j^{-s}$$ to be the zeta function[^6] associated with (minus) the Laplace--Beltrami operator $-\Delta$. Similarly, we define $$\label{Minakshisundaram--Pleijel zeta function definitiion equation 2} \zeta_{\sqrt{-\Delta}}(s) := \sum_{j=1}^\infty\mu_j^{-s/2} = \zeta_{-\Delta}(s/2)\,.$$ It is known [@minak p. 243] that the function $\zeta_{\sqrt{-\Delta}}(s)$ is meromorphic in $\mathbb{C}$ with simple poles at $s=3,1,-1,\dots$. **Lemma 72**. *For a closed Riemannian 3-manifold $(M,g)$ we have $$\label{new lemma zeta function laplacian equation 1} \zeta_{\sqrt{-\Delta}}(s)=\frac{1}{2\pi^2}\left[\frac{\operatorname{Vol}(M)}{s-3}+\frac{\int_M\operatorname{Sc}(x)\,\rho(x)\,\mathrm{d}x}{12(s-1)}+\ldots \right].$$* *Proof.* Let $N_{\sqrt{-\Delta}}(\lambda)$ be the counting function for the positive eigenvalues of $\sqrt{-\Delta}$, see, e.g., [@wave Appendix B]. The functions $\zeta_{\sqrt{-\Delta}}$ and $N_{\sqrt{-\Delta}}$ are related as $$\label{proof new lemma zeta function laplacian equation 1} \zeta_{\sqrt{-\Delta}}(s)=\int_{0}^{+\infty} \lambda^{-s}\, N'_{\sqrt{-\Delta}}(\lambda)\, \mathrm{d}\lambda\,,$$ where the prime stands for differentiation with respect to $\lambda$. By combining [@wave equation (B.1) and Theorem B.2] with formula [\[proof new lemma zeta function laplacian equation 1\]](#proof new lemma zeta function laplacian equation 1){reference-type="eqref" reference="proof new lemma zeta function laplacian equation 1"}, we obtain [\[new lemma zeta function laplacian equation 1\]](#new lemma zeta function laplacian equation 1){reference-type="eqref" reference="new lemma zeta function laplacian equation 1"}. ◻ Since for the Berger 3-sphere $\operatorname{Vol}(M)=2\pi^2a$ and $\operatorname{Sc}=8-2a^2$, we have the following. **Corollary 73**. *For the Berger 3-sphere $$\label{corollary zeta function laplacian equation 1} \zeta_{\sqrt{-\Delta}}(s)=a\left[\frac{1}{s-3}+\frac{4-a^2}{6(s-1)}+\ldots \right].$$* We are now in a position to prove Theorem [Theorem 68](#theorem eta invariant berger){reference-type="ref" reference="theorem eta invariant berger"}. *Proof of Theorem [Theorem 68](#theorem eta invariant berger){reference-type="ref" reference="theorem eta invariant berger"}.* Since $\,\zeta(-1)=-\frac1{12}\,$, formula [\[eta curl dima 1\]](#eta curl dima 1){reference-type="eqref" reference="eta curl dima 1"} implies that the proof of [\[theorem eta invariant berger equation 1\]](#theorem eta invariant berger equation 1){reference-type="eqref" reference="theorem eta invariant berger equation 1"} reduces to the proof of $$\label{proof of eta invariant berger curl dima 1} \theta(0) = \frac23\, a^2(a^2-2)\,,$$ where $\theta$ is the function [\[eta curl dima 2\]](#eta curl dima 2){reference-type="eqref" reference="eta curl dima 2"}. Formulae [\[proof of eta invariant berger curl dima 1\]](#proof of eta invariant berger curl dima 1){reference-type="eqref" reference="proof of eta invariant berger curl dima 1"} and [\[eta curl dima 2\]](#eta curl dima 2){reference-type="eqref" reference="eta curl dima 2"} reduce the analysis of the operator $\,\operatorname{curl}\,$ to the analysis of the Laplace--Beltrami operator. To establish [\[proof of eta invariant berger curl dima 1\]](#proof of eta invariant berger curl dima 1){reference-type="eqref" reference="proof of eta invariant berger curl dima 1"} we use a trick due to Hitchin [@hitchin p. 34]. We expand the terms in the RHS of [\[eta curl dima 2\]](#eta curl dima 2){reference-type="eqref" reference="eta curl dima 2"} in inverse powers of $\mu_j$ as $$\begin{gathered} \label{proof of eta invariant berger curl dima 2} %\left( \Bigl( \sqrt{a^2+\mu_j} \,\pm\,a \Bigr)^{-s} %\right)^{-s} = \mu_j^{-s/2} %\left( \Bigl( \sqrt{1+a^2\mu_j^{-1}} \,\pm\,a\mu_j^{-1/2} \Bigr)^{-s} %\right)^{-s} \\ = \mu_j^{-s/2} %\left( \Bigl( 1 \pm a\mu_j^{-1/2} + \frac{1}{2}a^2\mu_j^{-1} + O\left(\mu_j^{-2}\right) \Bigr)^{-s} %\right)^{-s} \\ = \mu_j^{-s/2} %\left( \Bigl( 1 - s \Bigl[ \pm a\mu_j^{-1/2} + \frac{1}{2}a^2\mu_j^{-1} \Bigr] + \frac{s(s+1)}{2} \Bigl[ a^2\mu_j^{-1} \pm a^3\mu_j^{-3/2} \Bigr] \mp \frac{s(s+1)(s+2)}{6} a^3\mu_j^{-3/2} + O\left(\mu_j^{-2}\right) \Bigr) %\right) \\ = \mu_j^{-s/2} %\left( \Bigl( 1 \mp sa\mu_j^{-1/2} + \frac{s^2}{2} a^2\mu_j^{-1} \pm \frac{s(1-s^2)}{6} a^3\mu_j^{-3/2} + O\left(\mu_j^{-2}\right) \Bigr). %\right).\end{gathered}$$ Substituting [\[proof of eta invariant berger curl dima 2\]](#proof of eta invariant berger curl dima 2){reference-type="eqref" reference="proof of eta invariant berger curl dima 2"} into [\[eta curl dima 2\]](#eta curl dima 2){reference-type="eqref" reference="eta curl dima 2"} we get $$\label{proof of eta invariant berger curl dima 3} \theta(s) = -\,2sa\,\zeta_{\sqrt{-\Delta}}(s+1) \,+\,\frac{s(1-s^2)}{3}\,a^3\,\zeta_{\sqrt{-\Delta}}(s+3) \,+\,h(s)\,,$$ where $h$ is analytic at $s=0$ and $$\label{proof of eta invariant berger curl dima 4} h(0)=0\,.$$ Formulae [\[proof of eta invariant berger curl dima 3\]](#proof of eta invariant berger curl dima 3){reference-type="eqref" reference="proof of eta invariant berger curl dima 3"}, [\[corollary zeta function laplacian equation 1\]](#corollary zeta function laplacian equation 1){reference-type="eqref" reference="corollary zeta function laplacian equation 1"} and [\[proof of eta invariant berger curl dima 4\]](#proof of eta invariant berger curl dima 4){reference-type="eqref" reference="proof of eta invariant berger curl dima 4"} imply $$\label{proof of eta invariant berger curl dima 5} \lim_{s\to0}\theta(s) = -\,2a^2\,\frac{4-a^2}{6} \,+\,\frac{1}{3}\,a^4 = \frac23\, a^2(a^2-2)\,,$$ which proves [\[proof of eta invariant berger curl dima 1\]](#proof of eta invariant berger curl dima 1){reference-type="eqref" reference="proof of eta invariant berger curl dima 1"}. ◻ # Maxwell's equations and electromagnetic chirality {#Maxwell's equations and electromagnetic chirality} The focus of our paper is the study of asymmetry between positive and negative eigenvalues of the operator $\operatorname{curl}$. It is natural to ask the question: does the sign of an eigenvalue of $\operatorname{curl}$ have a physical meaning? We show in this appendix that it does. The key here is the notion of *electromagnetic chirality*, see Definition [Definition 76](#definition of chirality){reference-type="ref" reference="definition of chirality"}. Let $(M,g)$ be a connected oriented closed Riemannian 3-manifold. Consider homogeneous vacuum Maxwell equations on $M\times\mathbb{R}$ $$\label{Maxwell 1} \begin{pmatrix} 0&\operatorname{curl}\\ -\operatorname{curl}&0 \end{pmatrix} \begin{pmatrix} E\\ B \end{pmatrix} = \frac{\partial}{\partial t} \begin{pmatrix} E\\ B \end{pmatrix},$$ $$\label{Maxwell 2} \delta E=\delta B=0,$$ where $E$ (electric field) and $B$ (magnetic field) are the unknown quantities, time-dependent real-valued 1-forms. We list below two basic symmetries of Maxwell's equations [\[Maxwell 1\]](#Maxwell 1){reference-type="eqref" reference="Maxwell 1"}, [\[Maxwell 2\]](#Maxwell 2){reference-type="eqref" reference="Maxwell 2"}. All the results in this appendix are given without proof because the corresponding proofs are elementary. **Definition 74**. Given a pair of real-valued 1-forms $E$ and $B$, we define the *duality transform* as the linear map $$\label{definition of duality transform equation} \operatorname{dual}: \begin{pmatrix} E\\ B \end{pmatrix} \mapsto \begin{pmatrix} -B\\ E \end{pmatrix}.$$ The duality transform can be interpreted as the action of the 4-dimensional Lorentzian Hodge star on the electromagnetic tensor $$F_{\alpha\beta} = \begin{pmatrix} 0&E_1&E_2&E_3\\ -E_1&0&-B_3&B_2\\ -E_2&B_3&0&-B_1\\ -E_3&-B_2&B_1&0 \end{pmatrix}.$$ Observe that the duality transform is a skew-involution, i.e. applying it twice gives minus identity. **Lemma 75**. *Maxwell's equations [\[Maxwell 1\]](#Maxwell 1){reference-type="eqref" reference="Maxwell 1"}, [\[Maxwell 2\]](#Maxwell 2){reference-type="eqref" reference="Maxwell 2"} are invariant under the action of the duality transform [\[definition of duality transform equation\]](#definition of duality transform equation){reference-type="eqref" reference="definition of duality transform equation"}, i.e. the duality transform maps solutions to solutions.* **Definition 76** (). Given a pair of real-valued 1-forms $E$ and $B$, we define *electromagnetic chirality* as the quadratic functional $$\label{definition of chirality equation} \operatorname{chir}(E,B) := \int_M(E\wedge\mathrm{d}E+B\wedge\mathrm{d}B).$$ **Lemma 77**. *Chirality [\[definition of chirality equation\]](#definition of chirality equation){reference-type="eqref" reference="definition of chirality equation"} is a conserved quantity for Maxwell's equations [\[Maxwell 1\]](#Maxwell 1){reference-type="eqref" reference="Maxwell 1"}, [\[Maxwell 2\]](#Maxwell 2){reference-type="eqref" reference="Maxwell 2"}, i.e. if $(E,B)$ is a solution then $\operatorname{chir}(E,B)$ does not depend on time $t$.* We will be seeking harmonic (in the time variable $t$) solutions of Maxwell's equations. **Definition 78**. A harmonic solution $(E,B)$ of Maxwell's equations [\[Maxwell 1\]](#Maxwell 1){reference-type="eqref" reference="Maxwell 1"}, [\[Maxwell 2\]](#Maxwell 2){reference-type="eqref" reference="Maxwell 2"} is said to be *polarised* if $\,(\dot E,\dot B)=\pm\lambda\operatorname{dual}(E,B)\,$. Here the dot stands for derivative in $t$ and $\lambda$ is the angular frequency. Recall that $(\lambda_j, u_j)$, $j=\pm1, \pm2, \ldots$, is the eigensystem for $\operatorname{curl}$. It is easy to see that $$\label{harmonic solution 1} \begin{pmatrix} E\\ B \end{pmatrix} = \begin{pmatrix} u_j\cos(\lambda_jt)\\ -u_j\sin(\lambda_jt) \end{pmatrix}$$ and $$\label{harmonic solution 2} \begin{pmatrix} E\\ B \end{pmatrix} = \begin{pmatrix} u_j\sin(\lambda_jt)\\ u_j\cos(\lambda_jt) \end{pmatrix}$$ are harmonic polarised solutions of Maxwell's equations [\[Maxwell 1\]](#Maxwell 1){reference-type="eqref" reference="Maxwell 1"}, [\[Maxwell 2\]](#Maxwell 2){reference-type="eqref" reference="Maxwell 2"}. The two solutions [\[harmonic solution 1\]](#harmonic solution 1){reference-type="eqref" reference="harmonic solution 1"} and [\[harmonic solution 2\]](#harmonic solution 2){reference-type="eqref" reference="harmonic solution 2"} are related via the duality transform [\[definition of duality transform equation\]](#definition of duality transform equation){reference-type="eqref" reference="definition of duality transform equation"} and describe waves with opposite polarisations. We see that the correspondence between eigenvalues and eigenforms of $\operatorname{curl}$ on the one hand and harmonic polarised solutions of Maxwell's equations on the other is one-to-two. Let $E_0$ and $B_0$ be a pair of infinitely smooth real-valued 1-forms which do not depend on $t$ and satisfy the condition $\delta E_0=\delta B_0=0$. Consider the Cauchy problem $$\label{Cauchy problem} \left.E\right|_{t=0}=E_0, \qquad \left.B\right|_{t=0}=B_0$$ for Maxwell's equations [\[Maxwell 1\]](#Maxwell 1){reference-type="eqref" reference="Maxwell 1"}, [\[Maxwell 2\]](#Maxwell 2){reference-type="eqref" reference="Maxwell 2"}. It is easy to see that the solution can be written as a series $$\label{solution to Maxwell as a series} \begin{pmatrix} E\\ B \end{pmatrix} = \sum_{j\in\mathbb{Z}\setminus\{0\}} \left[ \begin{pmatrix} u_j\cos(\lambda_jt)\\ -u_j\sin(\lambda_jt) \end{pmatrix} \langle E_0\,,u_j\rangle + \begin{pmatrix} u_j\sin(\lambda_jt)\\ u_j\cos(\lambda_jt) \end{pmatrix} \langle B_0\,,u_j\rangle \right] + \begin{pmatrix} P_{\mathcal{H}^1}E_0\\ P_{\mathcal{H}^1}B_0 \end{pmatrix},$$ where $P_{\mathcal{H}^1}$ is the orthogonal projection onto the space of harmonic 1-forms and $\langle\,\cdot\ ,\,\cdot\,\rangle$ is the inner product [\[inner product\]](#inner product){reference-type="eqref" reference="inner product"}. We see that our harmonic solutions [\[harmonic solution 1\]](#harmonic solution 1){reference-type="eqref" reference="harmonic solution 1"} and [\[harmonic solution 2\]](#harmonic solution 2){reference-type="eqref" reference="harmonic solution 2"} provide a basis for the Cauchy problem [\[Cauchy problem\]](#Cauchy problem){reference-type="eqref" reference="Cauchy problem"} for Maxwell's equations [\[Maxwell 1\]](#Maxwell 1){reference-type="eqref" reference="Maxwell 1"}, [\[Maxwell 2\]](#Maxwell 2){reference-type="eqref" reference="Maxwell 2"}. The following proposition is the main result of this appendix. **Proposition 79**. *For both solutions [\[harmonic solution 1\]](#harmonic solution 1){reference-type="eqref" reference="harmonic solution 1"} and [\[harmonic solution 2\]](#harmonic solution 2){reference-type="eqref" reference="harmonic solution 2"} we have $$\label{proposition about chirality equation} \operatorname{chir}(E,B)=\lambda_j\,.$$* Formula [\[proposition about chirality equation\]](#proposition about chirality equation){reference-type="eqref" reference="proposition about chirality equation"} tells us that the sign of an eigenvalue of $\operatorname{curl}$ has the physical meaning of the sign of chirality of the corresponding harmonic polarised solution of Maxwell's equations. And here it does not matter which polarisation one chooses when writing down the harmonic solution of Maxwell's equations. # Taylor expansions for the operator of parallel transport {#appendix parallel transport} This appendix is concerned with local expansions of parallel transport maps in normal coordinates. These can, in principle, be found in many guises and with varied degrees of accuracy in the literature. Nevertheless, for the convenience of the reader and for future reference, we provide here a concise self-contained derivation, one which agrees with the definitions and sign conventions set out in this paper.   Let us fix a point $x\in M$ and let us choose normal coordinates centred at $x$. Let $y\in M$ be a point in a (small) neighbourhood of $x$. Recall that $Z_\alpha{}^\beta(x,y)$ is the map realising the parallel transport of vectors along the unique geodesic connecting $x$ to $y$, whereas $Z^\alpha{}_\beta(x,y)$ is the map realising the parallel transport of covectors along the same geodesic, see [\[Q acting on 1-forms 3\]](#Q acting on 1-forms 3){reference-type="eqref" reference="Q acting on 1-forms 3"}--[\[Q acting on 1-forms 3a\]](#Q acting on 1-forms 3a){reference-type="eqref" reference="Q acting on 1-forms 3a"}. **Proposition 80**. *We have $$\label{prop: expansion of parallel transoport equation 1} Z_\alpha{}^\beta(0,y)=\delta_\alpha{}^\beta+\frac16 \operatorname{Riem}^\beta{}_{\mu\alpha\nu}(0)\,y^\mu y^\nu -\frac16 \frac{\partial^2\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu \partial y^\rho}(0) \,y^\mu y^\nu y^\rho+O(|y|^4)\,$$ and $$\label{prop: expansion of parallel transoport equation 2} Z^\alpha{}_\beta(0,y)=\delta^\alpha{}_\beta-\frac16 \operatorname{Riem}^\alpha{}_{\mu\beta\nu}(0)\, y^\mu y^\nu +\frac16 \frac{\partial^2\Gamma^\alpha{}_{\mu\beta}}{\partial y^\nu \partial y^\rho}(0)\,y^\mu y^\nu y^\rho+O(|y|^4)\,.$$ Furthermore, for all $\tau\in[0,1]$ we have $$\label{prop: expansion of parallel transoport equation 3} Z_\alpha{}^\beta(y,\tau y)=\delta_\alpha{}^\beta+\frac{\tau^2-1}6 \operatorname{Riem}^\beta{}_{\mu\alpha\nu}(0)\,y^\mu y^\nu -\frac{\tau^3-1}6 \frac{\partial^2\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu \partial y^\rho}(0) \,y^\mu y^\nu y^\rho+O(|y|^4)\,$$ and $$\label{prop: expansion of parallel transoport equation 4} Z^\alpha{}_\beta(y,\tau y)=\delta^\alpha{}_\beta-\frac{\tau^2-1}6 \operatorname{Riem}^\alpha{}_{\mu\beta\nu}(0)\, y^\mu y^\nu +\frac{\tau^3-1}6 \frac{\partial^2\Gamma^\alpha{}_{\mu\beta}}{\partial y^\nu \partial y^\rho}(0) \,y^\mu y^\nu y^\rho+O(|y|^4)\,.$$* Note that formula [\[prop: expansion of parallel transoport equation 1\]](#prop: expansion of parallel transoport equation 1){reference-type="eqref" reference="prop: expansion of parallel transoport equation 1"} agrees with [@dirac formula (7.1)]. *Proof of Proposition [Proposition 80](#prop: expansion of parallel transoport){reference-type="ref" reference="prop: expansion of parallel transoport"}.* Let us start with a vector $v^\alpha$ at the origin and let us parallel transport it along a straight line to $y$ --- the unique shortest geodesic in the chosen coordinate system. This means imposing the condition $$\label{proof expansion of parallel transoport equation 1} y^\mu\nabla_\mu v^\beta= y^\mu \left( \partial_\mu v^\beta+\Gamma^\beta{}_{\mu\alpha}v^\alpha \right)=0$$ along the straight line. Let us now parameterise our straight line with parameter $t\in[0,1]$ as $z(t)=ty$. Then $$\label{proof expansion of parallel transoport equation 2} y^\mu \partial_\mu v^\beta =\dot v^\beta.$$ Formulae [\[proof expansion of parallel transoport equation 1\]](#proof expansion of parallel transoport equation 1){reference-type="eqref" reference="proof expansion of parallel transoport equation 1"} and [\[proof expansion of parallel transoport equation 2\]](#proof expansion of parallel transoport equation 2){reference-type="eqref" reference="proof expansion of parallel transoport equation 2"} imply $$\begin{gathered} \label{proof expansion of parallel transoport equation 3} \left.v^\beta\right|_{t=1} = \left.v^\beta\right|_{t=0} - \int_0^1 y^\mu\Gamma^\beta{}_{\mu\gamma}(ty)\,v^\gamma(ty) \,dt \\ = \left.v^\beta\right|_{t=0} - \int_0^1 y^\mu \left[ \frac{\partial\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu}(0)\,y^\nu\,t + \frac12\frac{\partial^2\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu\partial y^\rho}(0)\,y^\nu y^\rho\,t^2 + O(|y|^3) \right] \left[ \left.v^\alpha\right|_{t=0} + O(|y|^2) \right] \,dt\,.\end{gathered}$$ In the above equation we used the fact that $$\label{proof expansion of parallel transoport equation 4} \left.\dot{v}^\alpha\right|_{t=0}=0\,,$$ which follows from [\[proof expansion of parallel transoport equation 1\]](#proof expansion of parallel transoport equation 1){reference-type="eqref" reference="proof expansion of parallel transoport equation 1"}, [\[proof expansion of parallel transoport equation 2\]](#proof expansion of parallel transoport equation 2){reference-type="eqref" reference="proof expansion of parallel transoport equation 2"} and the fact that Christoffel symbols vanish at the origin. Using the elementary differential-geometric identity $$\label{proof expansion of parallel transoport equation 5} \frac{\partial\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu}(0)=-\frac13\left( \operatorname{Riem}^\beta{}_{\mu\alpha\nu}(0) + \operatorname{Riem}^\beta{}_{\alpha\mu\nu}(0) \right)$$ and performing integration in [\[proof expansion of parallel transoport equation 3\]](#proof expansion of parallel transoport equation 3){reference-type="eqref" reference="proof expansion of parallel transoport equation 3"}, we obtain $$\label{proof expansion of parallel transoport equation 6} \left.v^\beta\right|_{t=1} = \left.v^\beta\right|_{t=0} + \frac{1}{6} \operatorname{Riem}^\beta{}_{\mu\alpha\nu}(0) \, y^\mu y^\nu \left.v^\alpha\right|_{t=0} - \frac16 \frac{\partial^2\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu\partial y^\rho}(0)\,y^\mu y^\nu y^\rho\left.v^\alpha\right|_{t=0} +O(|y|^4)\,.$$ The latter implies [\[prop: expansion of parallel transoport equation 1\]](#prop: expansion of parallel transoport equation 1){reference-type="eqref" reference="prop: expansion of parallel transoport equation 1"}. Combining [\[prop: expansion of parallel transoport equation 1\]](#prop: expansion of parallel transoport equation 1){reference-type="eqref" reference="prop: expansion of parallel transoport equation 1"} with [\[Q acting on 1-forms 3a\]](#Q acting on 1-forms 3a){reference-type="eqref" reference="Q acting on 1-forms 3a"} one immediately obtains [\[prop: expansion of parallel transoport equation 2\]](#prop: expansion of parallel transoport equation 2){reference-type="eqref" reference="prop: expansion of parallel transoport equation 2"}. In order to obtain [\[prop: expansion of parallel transoport equation 3\]](#prop: expansion of parallel transoport equation 3){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3"} let us parallel transport along our straight line from $y$ to $\tau y$. Arguing as above, we get $$\begin{gathered} \label{proof expansion of parallel transoport equation 7} \left.v^\beta\right|_{t=\tau} = \left.v^\beta\right|_{t=1} - \int_1^{\tau} y^\mu\,\Gamma^\beta{}_{\mu\gamma}(ty)\,v^\gamma(ty) \,ds \\ = \left.v^\beta\right|_{t=1} - \int_1^{\tau} y^\mu \left[ -\frac{1}{3} \operatorname{Riem}^\beta{}_{\mu\alpha\nu}(0)\,y^\nu\,t + \frac12\frac{\partial^2\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu\partial y^\rho}(0)\,y^\nu y^\rho\,t^2 + O(|y|^3) \right] \left[ \left.v^\alpha\right|_{t=1} + O(|y|^2) \right] \,ds \\ = \left.v^\beta\right|_{t=1} + \frac{\tau^2-1}{6} \operatorname{Riem}^\beta{}_{\mu\alpha\nu}(0) \, y^\mu y^\nu \left.v^\alpha\right|_{t=1} - \frac{\tau^3-1}6 \frac{\partial^2\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu\partial y^\rho}(0)\,y^\mu y^\nu y^\rho\left.v^\alpha\right|_{t=1} +O(|y|^4)\,.\end{gathered}$$ The latter implies [\[prop: expansion of parallel transoport equation 3\]](#prop: expansion of parallel transoport equation 3){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3"} which, in turn, combined with [\[Q acting on 1-forms 3a\]](#Q acting on 1-forms 3a){reference-type="eqref" reference="Q acting on 1-forms 3a"} immediately gives us [\[prop: expansion of parallel transoport equation 4\]](#prop: expansion of parallel transoport equation 4){reference-type="eqref" reference="prop: expansion of parallel transoport equation 4"}. ◻ *Remark 81*. For $\tau=0$ expansions [\[prop: expansion of parallel transoport equation 3\]](#prop: expansion of parallel transoport equation 3){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3"} and [\[prop: expansion of parallel transoport equation 4\]](#prop: expansion of parallel transoport equation 4){reference-type="eqref" reference="prop: expansion of parallel transoport equation 4"} simplify to read $$\label{prop: expansion of parallel transoport equation 3 tau zero} Z_\alpha{}^\beta(y,0)=\delta_\alpha{}^\beta-\frac16 \operatorname{Riem}^\beta{}_{\mu\alpha\nu}(0)\,y^\mu y^\nu +\frac16 \frac{\partial^2\Gamma^\beta{}_{\mu\alpha}}{\partial y^\nu \partial y^\rho}(0) \,y^\mu y^\nu y^\rho+O(|y|^4)\,$$ and $$\label{prop: expansion of parallel transoport equation 4 tau zero} Z^\alpha{}_\beta(y,0)=\delta^\alpha{}_\beta+\frac16 \operatorname{Riem}^\alpha{}_{\mu\beta\nu}(0)\, y^\mu y^\nu -\frac16 \frac{\partial^2\Gamma^\alpha{}_{\mu\beta}}{\partial y^\nu \partial y^\rho}(0) \,y^\mu y^\nu y^\rho+O(|y|^4)\,.$$ Formulae [\[prop: expansion of parallel transoport equation 1\]](#prop: expansion of parallel transoport equation 1){reference-type="eqref" reference="prop: expansion of parallel transoport equation 1"}, [\[prop: expansion of parallel transoport equation 2\]](#prop: expansion of parallel transoport equation 2){reference-type="eqref" reference="prop: expansion of parallel transoport equation 2"}, [\[prop: expansion of parallel transoport equation 3 tau zero\]](#prop: expansion of parallel transoport equation 3 tau zero){reference-type="eqref" reference="prop: expansion of parallel transoport equation 3 tau zero"} and [\[prop: expansion of parallel transoport equation 4 tau zero\]](#prop: expansion of parallel transoport equation 4 tau zero){reference-type="eqref" reference="prop: expansion of parallel transoport equation 4 tau zero"} agree with [\[Q acting on 1-forms 3a extra\]](#Q acting on 1-forms 3a extra){reference-type="eqref" reference="Q acting on 1-forms 3a extra"} and [\[Q acting on 1-forms 3a\]](#Q acting on 1-forms 3a){reference-type="eqref" reference="Q acting on 1-forms 3a"}. # An alternative derivation of formula [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"} {#An alternative derivation of formula} In this appendix we verify, using an alternative method, formula [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"}, one of the main results of our paper. In what follows we work in normal coordinates and we assume that curvature (but not its covariant derivative) vanishes at the origin, so that $$\label{14 September 2021 equation 1bis simplified} g_{\alpha\beta}(x) = \delta_{\alpha\beta} - \frac16 (\nabla_\sigma \operatorname{Riem}_{\alpha\mu\beta\nu})(0)\,x^\sigma\,x^\mu\,x^\nu +O(|x|^4).$$ We continue using the notation $\boldsymbol{\Delta}=-(\mathrm{d}\delta+\delta \mathrm{d})$ for the Hodge Laplacian on 1-forms [\[Hodge Laplacian\]](#Hodge Laplacian){reference-type="eqref" reference="Hodge Laplacian"}; $|\xi|$ stands for the Euclidean norm of $\xi$, whereas $\|\xi\|$ stands for the Riemannian norm thereof. **Proposition 82**. *Let $$_\alpha{}^\beta(x,\xi) \sim \|\xi\|^{-1}\delta_\alpha{}^\beta + [s_{-2}]_\alpha{}^\beta(x,\xi) + [s_{-3}]_\alpha{}^\beta(x,\xi) + [s_{-4}]_\alpha{}^\beta(x,\xi) + \dots$$ be the (left) symbol of the operator $(-\boldsymbol{\Delta})^{-1/2}$ (recall [\[Hodge Laplacian to the power s\]](#Hodge Laplacian to the power s){reference-type="eqref" reference="Hodge Laplacian to the power s"}). Then $$\label{29 July 2023 equation 6} A_\mathrm{prin}(x,\xi) = - \varepsilon_\beta{}^{\alpha\gamma} \left( \,([s_{-3}]_\alpha{}^\beta)_{x^\gamma} +i \,\xi_\gamma \,[s_{-4}]_\alpha{}^\beta \right) + O(|\xi|^{-3}|x|)\,.$$* *Proof.* The claim follows from [\[formula for Pplus minus Pminus\]](#formula for Pplus minus Pminus){reference-type="eqref" reference="formula for Pplus minus Pminus"}, [\[components of apt are zero\]](#components of apt are zero){reference-type="eqref" reference="components of apt are zero"}, [\[5 November 2021 equation 1\]](#5 November 2021 equation 1){reference-type="eqref" reference="5 November 2021 equation 1"} and the fact that the full symbol of $\operatorname{curl}$ is [\[principal symbol curl\]](#principal symbol curl){reference-type="eqref" reference="principal symbol curl"}. ◻ In the remainder of this appendix, we will derive explicit formulae for $[s_{-3}]_\alpha{}^\beta$ and $[s_{-3}]_\alpha{}^\beta$. The proofs are straightforward, hence omitted. **Lemma 83**. *Let $$_\alpha{}^\beta(x,\xi) \sim \|\xi\|^{2}\delta_\alpha{}^\beta + [q_1]_\alpha{}^\beta(x,\xi) + [q_0]_\alpha{}^\beta(x,\xi)$$ be the (left) symbol of the operator $-\boldsymbol{\Delta}$. We have $$\label{10 August 2023 equation 16} [q_1]_\alpha{}^\beta(x,\xi) = i\,a_\alpha{}^{\beta\gamma}{}_{\mu\nu}\,\xi_\gamma\,x^\mu\,x^\nu + O(|\xi|\,|x|^3)\,,$$ $$\label{10 August 2023 equation 17} [q_0]_\alpha{}^\beta(x) = b_\alpha{}^\beta{}_\nu\,x^\nu + O(|x|^2)\,,$$ where $$\begin{gathered} \label{10 August 2023 equation 18} a_\alpha{}^{\beta\gamma}{}_{\mu\nu} := \left[ \frac12\nabla_\mu\operatorname{Ric}^\gamma{}_\nu - \frac1{12}\nabla^\gamma\operatorname{Ric}_{\mu\nu} \right] \delta_\alpha{}^\beta \\ - \frac16 \left[ \nabla_\alpha\operatorname{Riem}^\gamma{}_\mu{}^\beta{}_\nu -3 \nabla_\mu\operatorname{Riem}^\gamma{}_\alpha{}^\beta{}_\nu +5 \nabla_\nu\operatorname{Riem}^\gamma{}_\mu{}^\beta{}_\alpha \right],\end{gathered}$$ $$\label{10 August 2023 equation 19} b_\alpha{}^\beta{}_\nu := - \frac{1}{6} \nabla^\beta\operatorname{Ric}_{\alpha\nu} + \frac{1}{2} \nabla_\alpha\operatorname{Ric}^\beta{}_\nu + \frac{1}{2} \nabla_\nu\operatorname{Ric}_\alpha{}^\beta\,.$$* **Lemma 84**. *The symbols $$_\alpha{}^\beta(x,\xi) \sim \|\xi\|\delta_\alpha{}^\beta + [r_0]_\alpha{}^\beta(x,\xi) + [r_{-1}]_\alpha{}^\beta(x,\xi) + [r_{-2}]_\alpha{}^\beta(x,\xi) + \dots$$ and $s_\alpha{}^\beta$ of $(-\boldsymbol{\Delta})^{1/2}$ and $(-\boldsymbol{\Delta})^{-1/2}$ are expressed in terms of [\[10 August 2023 equation 16\]](#10 August 2023 equation 16){reference-type="eqref" reference="10 August 2023 equation 16"} and [\[10 August 2023 equation 17\]](#10 August 2023 equation 17){reference-type="eqref" reference="10 August 2023 equation 17"} via the following hierarchy of identities: $$\label{10 August 2023 equation 20} [r_0]_\alpha{}^\beta=\frac1{2|\xi|}[q_1]_\alpha{}^\beta-\frac{1}{2i|\xi|^2} \xi^\mu(\|\xi\|)_{x^\mu}\,\delta_\alpha{}^\beta + O(|x|^3)\,,$$ $$\label{10 August 2023 equation 21} [r_{-1}]_\alpha{}^\beta \\ = \frac1{2|\xi|}[q_0]_\alpha{}^\beta -\frac1{2i|\xi|^2} \xi^\mu ([r_0]_\alpha{}^\beta)_{x^\mu} \\ +\frac1{4|\xi|}(|\xi|)_{\xi_\mu \xi_\nu}(\|\xi\|)_{x^\mu x^\nu}\,\delta_\alpha{}^\beta \\ + O(|\xi|^{-1}|x|^2)\,,$$ $$\begin{gathered} \label{10 August 2023 equation 22} [r_{-2}]_\alpha{}^\beta = -\frac1{2i|\xi|^2} \xi^\mu([r_{-1}]_\alpha{}^\beta)_{x^\mu} \\ +\frac1{4|\xi|} (|\xi|)_{\xi_\mu\xi_\nu}([r_0]_\alpha{}^\beta)_{x^\alpha x^\beta} +\frac1{12i|\xi|}(|\xi|)_{\xi_\mu \xi_\nu\xi_\rho}(\|\xi\|)_{x^\mu x^\nu x^\rho}\,\delta_\alpha{}^\beta \\ + O(|\xi|^{-2}|x|)\,,\end{gathered}$$ $$\label{10 August 2023 equation 5} [s_{-2}]_{\alpha}{}^\beta= -|\xi|^{-2}[r_{0}]_{\alpha}{}^\beta- \frac{1}{i|\xi|^2}\,\xi^\mu\,(\|\xi\|^{-1})_{x^\mu} \,\delta_\alpha{}^\beta + O(|\xi|^{-2}|x|^3)\,,$$ $$\begin{gathered} \label{10 August 2023 equation 6} [s_{-3}]_\alpha{}^\beta = - |\xi|^{-2}[r_{-1}]_\alpha{}^\beta -\frac1{i|\xi|^2} \xi^\mu\,([s_{-2}]_\alpha{}^\beta)_{x^\mu} \\ +\frac1{2|\xi|} (|\xi|)_{\xi_\mu\xi_\nu}(\|\xi\|^{-1})_{x^\mu x^\nu}\, \delta_\alpha{}^\beta + O(|\xi|^{-3}|x|^2)\,,\end{gathered}$$ $$\begin{gathered} \label{10 August 2023 equation 7} [s_{-4}]_\alpha{}^\beta = - |\xi|^{-2}[r_{-2}]_\alpha{}^\beta -\frac1{i|\xi|^2}\, \xi^\mu\,([s_{-3}]_\alpha{}^\beta)_{x^\mu} \\ +\frac1{2|\xi|} (|\xi|)_{\xi_\mu\xi_\nu}([s_{-2}]_\alpha{}^\beta)_{x^\mu x^\nu} +\frac{1}{6i|\xi|}(|\xi|)_{\xi_\mu\xi_\nu\xi_\rho}(\|\xi\|^{-1})_{x^\mu x^\nu x^\rho}\, \delta_\alpha{}^\beta \\ + O(|\xi|^{-4}|x|)\,.\end{gathered}$$* *Remark 85*. Lemmata [Lemma 83](#alternative lemma 1){reference-type="ref" reference="alternative lemma 1"} and [Lemma 84](#alternative lemma 2){reference-type="ref" reference="alternative lemma 2"} are true in any dimension $d$, not only in dimension $d=3$. In order to verify our formula [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"} for $A_\mathrm{prin}\,$, it only remains to compute (for example, with the help of computer algebra) the right-hand sides of [\[10 August 2023 equation 6\]](#10 August 2023 equation 6){reference-type="eqref" reference="10 August 2023 equation 6"} and [\[10 August 2023 equation 7\]](#10 August 2023 equation 7){reference-type="eqref" reference="10 August 2023 equation 7"}, and substitute the resulting expressions into [\[29 July 2023 equation 6\]](#29 July 2023 equation 6){reference-type="eqref" reference="29 July 2023 equation 6"}. For instance, for the particular choice of metric determined by $$\operatorname{Ric}(x) = \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & x^1\\ 0 & x^1 & 0\\ \end{pmatrix} + O(|x|^2)\$$ (recall [\[14 September 2021 equation 5\]](#14 September 2021 equation 5){reference-type="eqref" reference="14 September 2021 equation 5"} and [\[14 September 2021 equation 1bis simplified\]](#14 September 2021 equation 1bis simplified){reference-type="eqref" reference="14 September 2021 equation 1bis simplified"}) we get $$\label{s-3 and s-4} \left.s_{-3}\right|_{(x,\xi_0)}=-\frac1{12} \begin{pmatrix} 0 & 3x^3 & 3x^2\\ -x^3 & 0 & 3x^1\\ -x^2 & 3x^1 & 0 \end{pmatrix} + O(|x|^2), \quad \left.s_{-4}\right|_{(x,\xi_0)}=-\frac{i}{2} \begin{pmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix} + O(|x|)\,,$$ where $\xi_0$ is given by [\[xi0\]](#xi0){reference-type="eqref" reference="xi0"}. Substituting [\[s-3 and s-4\]](#s-3 and s-4){reference-type="eqref" reference="s-3 and s-4"} into [\[29 July 2023 equation 6\]](#29 July 2023 equation 6){reference-type="eqref" reference="29 July 2023 equation 6"} we obtain $$A_\mathrm{prin}(0,\xi_0)=-\frac12\,,$$ which agrees with [\[main theorem 2 equation 1\]](#main theorem 2 equation 1){reference-type="eqref" reference="main theorem 2 equation 1"} --- see also [\[Proof of Aprin equation 8\]](#Proof of Aprin equation 8){reference-type="eqref" reference="Proof of Aprin equation 8"}. 42 M. Abramowitz and I.A. Stegun, *Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables*. Dover Books on Advanced Mathematics, Dover Publications, 1965. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry, *Bull. London Math. Soc.* **5** (1973) 229--234. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry I, *Math. Proc. Camb. Phil. Soc.* **77** (1975) 43--69. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry II, *Math. Proc. Camb. Phil. Soc.* **78** (1975) 405--432. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry III, *Math. Proc. Camb. Phil. Soc.* **79** (1976) 71--99. C. Bär, The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces, *Arch. Math.* **59** (1992) 65--79 (1992). C. Bär, The curl operator on odd-dimensional manifolds, *J. Math. Phys.* **60** (2019) 031501. M. Sh. Birman and M. Z. Solomyak, The Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, *Vestn. Leningr. Univ. Math.* **20** no. 3 (1987) 15--21. M. Sh. Birman and M. Z. Solomyak, $L^2$-Theory of the Maxwell operator in arbitrary domains, *Uspekhi Mat. Nauk* **42** no. 6 (1987) 61--76; *Russian Math. Surveys* **42** no. 6 (1987) 75--96. J.-M. Bismut and D. S. Freed, The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem, *Comm. Math. Phys.* **107** (1986) 103--163. T. P. Branson and P. B. Gilkey, Residues of the eta function for an operator of Dirac type, *J. Funct. Anal.* **108** (1992) 47--87. L. Brewin, Riemann normal coordinate expansions using Cadabra. Preprint arXiv:0903.2087v3. M. Capoferri, Diagonalization of elliptic systems via pseudodifferential projections, *J. Differential Equations* **313** (2022) 157--187. M. Capoferri, M. Levitin and D. Vassiliev, Geometric wave propagator on Riemannian manifolds, *Comm. Anal. Geom.* **30** no. 8 (2022) 1713--1777. M. Capoferri, G. Rozenblum, N. Saveliev and D. Vassiliev, Topological obstructions to the diagonalisation of pseudodifferential systems *Proc. Amer. Math. Soc. (Ser. B)* **9** (2022) 472--486. M. Capoferri and D. Vassiliev, Global propagator for the massless Dirac operator and spectral asymptotics. *Integral Equations Operator Theory* **94** (2022) 30. M. Capoferri and D. Vassiliev, Invariant subspaces of elliptic systems I: pseudodifferential projections. *J. Funct. Anal.* **282** no. 8 (2022) 109402. M. Capoferri and D. Vassiliev, Invariant subspaces of elliptic systems II: spectral theory, *J. Spectr. Theory* **12** no. 1 (2022) 301--338. M.N. Demchenko and N. Filonov, Spectral asymptotics of the Maxwell operator on Lipschitz manifolds with boundary, in: *Spectral Theory of Differential Operators: M.Sh. Birman 80th Anniversary Collection* (Amer. Math. Soc., Providence, RI, 2008), 73--90. *NIST Digital Library of Mathematical Functions*. <https://dlmf.nist.gov/>, Release 1.1.9 of 2023-03-15. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl, and M.A. McClain, eds. J. J. Duistermaat and L. Hörmander, Fourier integral operators. II., *Acta Math.* **128** no. 3--4 (1972) 183--269. A. Enciso, W. Gerner and D. Peralta-Salas, Optimal convex domains for the first curl eigenvalue, *T*rans. Amer. Math. Soc., in press. A. Enciso and D. Peralta-Salas, Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue, *Ann. Sc. Norm. Super. Pisa Cl. Sci.* **XXIV** (2023) 311-327. Y.-L. Fang, M. Levitin and D. Vassiliev, Spectral analysis of the Dirac operator on a 3-sphere, *Operators and Matrices* **12** (2018) 501--527. Y.-L. Fang, *Analysis of first order systems on manifolds without boundary: A spectral theoretic approach*. PhD Thesis, University College London, 2017. URL: <https://discovery.ucl.ac.uk/id/eprint/1560979>. N. Filonov, Weyl asymptotics of the spectrum of the Maxwell operator in Lipschitz domains of arbitrary dimension, *Algebra i Anal.* **25** no. 1 (2013) 170--215. T.E. Gureev, Exact asymptotics of the spectrum of the Maxwell operator in a solid resonator, *Funct. Anal. Appl.* **24** no. 3 (1990) 235--237. P. Hintz, Resonance expansions for tensor-valued waves on asymptotically Kerr--de Sitter spaces, *J. Spectr. Theory* **7** no. 2 (2017) 519--557. N. Hitchin, Harmonic spinors, *Adv. Math.* **14** no. 1 (1974) 1--55. L. Hörmander, The spectral function of an elliptic operator, *Acta Math.* **121** (1968) 193--218. V. Ivrii, Accurate spectral asymptotics for elliptic operators that act in vector bundles *Funct. Anal. Appl.* **16** (1982) 101--108. D. Jakobson and A. Strohmaier, High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows, *Comm. Math. Phys.* **270** (2007) 813--833. J. Jost, *Riemannian Geometry and Geometric Analysis*, Springer-Verlag, 2011. M.A. Kervaire and J.W. Milnor, Groups of homotopy spheres: I, *Ann. of Math.* **77** no. 3 (1963) 504--537. E.A. Lauret, The smallest Laplace eigenvalue of homogeneous 3-spheres, *Bull. LMS* **51** no. 1 (2019) 49--69. N. Lerner and F. Vigneron, On some properties of the curl operator and their consequences for the Navier-Stokes system. ArXiv preprint arXiv:2203.07950 (2022). D.M. Lipkin, Existence of a new conservation law in electromagnetic theory, *J. Math. Phys.* **5** (1964) 696--700. J. Lotay, Stability of coassociative conical singularities, *Comm. Anal. Geom.* **20** no. 4 (2012) 803--867. J.J. Millson, *Chern-Simons invariants of constant curvature manifolds*. Ph.D. thesis, University of California, Berkeley, 1973. J.W. Milnor, On manifolds homeomorphic to the 7-sphere, *Ann. of Math.* **64** no. 2 (1956) 399--405. S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, *Canadian J. Math.* **1** (1949) 242--256. U. Muller, C. Schubert and A.E.M. van de Ven, A closed formula for the Riemann normal coordinate expansion, *Gen. Relativ. Gravit.* **31** (1999) 1759--1768. Yu. Safarov, Asymptotic behavior of the spectrum of the Maxwell operator, *J. Sov. Math.* **27** (1984) 2655--2661. K. Schmüdgen, *Unbounded self-adjoint operators on Hilbert space*, Graduate Texts in Mathematics **265**, Springer Netherlands, 2012. R. Schoen and S.-T. Yau, *Lectures on differential geometry*, International Press, 2010. R.T. Seeley, Complex powers of an elliptic operator, In: *Proc. Symp. Pure Math.* **10**, Amer. Math. Soc., Providence (RI), 1967, 288--307. M.A. Shubin, *Pseudodifferential operators and spectral theory*, Springer, 2001. I.M. Singer, Eigenvalues of the Laplacian and invariants of manifolds, *Proceedings of the International Congress of Mathematicians* **1** (1974) 187--200. Y. Tang and A.E. Cohen, Optical chirality and its interaction with matter, *Phys. Rev. Lett.* **104** (2010) 163901. S. Tanno, The first eigenvalue of the Laplacian on spheres, *Tohoku Math. J.* **31** (1979) 179--185. F.W. Warner, *Foundations of differentiable manifolds and Lie groups*, Springer, 1983. [^1]: MC: Maxwell Institute for Mathematical Sciences, Edinburgh *and* Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK; m.capoferri\@hw.ac.uk, <https://mcapoferri.com>. [^2]: DV: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK; D.Vassiliev\@ucl.ac.uk, <http://www.ucl.ac.uk/~ucahdva/>. [^3]: *The Riemann curvature tensor $\operatorname{Riem}$ has components ${\operatorname{Riem}^\kappa}_{\lambda\mu\nu}$ defined in accordance with $${\operatorname{Riem}^\kappa}_{\lambda\mu\nu}:= \mathrm{d}x^\kappa(\operatorname{Riem}(\partial_\mu\,,\partial_\nu)\,\partial_\lambda) = \partial_\mu{\Gamma^\kappa}_{\nu\lambda} -\partial_\nu{\Gamma^\kappa}_{\mu\lambda} +{\Gamma^\kappa}_{\mu\eta}{\Gamma^\eta}_{\nu\lambda} -{\Gamma^\kappa}_{\nu\eta}{\Gamma^\eta}_{\mu\lambda}\,,$$ the $\Gamma$'s being Christoffel symbols. The Ricci tensor is defined as $\operatorname{Ric}_{\mu\nu}:=R^\alpha{}_{\mu\alpha\nu}$ and $\operatorname{Sc}:=g^{\mu\nu}\operatorname{Ric}_{\mu\nu}$ is scalar curvature.* [^4]: Recall that, by definition, a matrix (pseudo)differential operator is elliptic if the determinant of its principal symbol is nowhere vanishing on $T^*M\setminus\{0\}$. [^5]: The dimension of the vector space of polynomials of degree $k$ in $n$ variables is ${{k+n}\choose{k}}\,.$ [^6]: The function $s\mapsto \zeta_{-\Delta}(s)$ is often called the *Minakshisundaram--Pleijel zeta function*, see [@minak equation (6)].

arxiv_math

--- abstract: | This article proves, in the case of split groups over arbitrary fields, that all fibers of convolution morphisms attached to parahoric affine flag varieties are paved by products of affine lines and affine lines minus a point. This applies in particular to the affine Grassmannian and to the convolution morphisms in the context of the geometric Satake correspondence. The second part of the article extends these results over $\mathbb Z$. Those in turn relate to the recent work of Cass-van den Hove-Scholbach on the geometric Satake equivalence for integral motives, and provide some alternative proofs for some of their results. address: Department of Mathematics, University of Maryland, College Park, MD 20742-4015, DC, USA author: - by Thomas J. Haines title: Cellular pavings of fibers of convolution morphisms --- [^1] # Introduction and Main Results Let $G$ be a split connected reductive group over any field $k$. Let $W$ be the Iwahori-Weyl group of $LG(k) = G(k(\!(t)\!))$, and for each $r$-tuple $w_\bullet = (w_1, \dots, w_r) \in W^r$ and choice of standard parahoric subgroup ${\mathcal P}\subset LG(k)$ consider the convolution morphism $$m_{w_\bullet, {\mathcal P}}: X_{\mathcal P}(w_\bullet) := X_{\mathcal P}(w_1) \widetilde{\times} \cdots \widetilde{\times} X_{\mathcal P}(w_r) \rightarrow X_{\mathcal P}(w_*)$$ defined on the twisted product of Schubert varieties $X_{\mathcal P}(w_i) \subset {\rm Fl}_{\mathcal P}$ (see $\S$[2](#Notation_sec){reference-type="ref" reference="Notation_sec"} and $\S$[3](#Review_sec){reference-type="ref" reference="Review_sec"}). Such morphisms have long played an important role in the geometric Langlands program and in the study of the geometry of Schubert varieties. For example, if $w_\bullet = (s_1,\dots, s_r)$ is a sequence of simple affine reflections, $w = s_1\cdots s_r$ is a reduced word, and ${\mathcal P}$ is the standard Iwahori subgroup ${\mathcal B}$, then $X_{\mathcal B}(s_\bullet) \rightarrow X_{\mathcal B}(w)$ is the Demazure resolution (of singularities) of $X_{\mathcal B}(w)$. If ${\mathcal P}= L^+G$ is the positive loop group and $w_\bullet = \mu_\bullet = (\mu_1, \dots, \mu_r)$ is a tuple of cocharacters in $G$, the corresponding convolution morphism is used to define the convolution of $L^+G$-equivariant perverse sheaves on the affine Grassmannian ${\rm Gr}_G = LG/L^+G$, and hence it plays a key role in the geometric Satake correspondence. Numerous applications stem from the study of the fibers of convolution morphisms, their dimensions and irreducible components, and possible pavings of them by affine spaces or related spaces. This article will focus on pavings of fibers by affine spaces, or by closely related spaces. We recall that a variety $X$ is *paved by varieties in a class $\mathcal C$* provided that there exists a finite exhaustion by closed subvarieties $\emptyset = X_0 \subset X_1 \subset \cdots \subset X_l = X$ such that each locally closed difference $X_i - X_{i-1}$ for $1 \leq i \leq r$ isomorphic to a member of the class $\mathcal C$. The fact that the fibers of Demazure resolutions admit pavings by affine spaces was for a long time a folklore result, until a proof appeared in [@H05] and later more generally in [@dHL18] (see [@dHL18 Thm. 2.5.2]). This has been used in proving various parity vanishing and purity results in Kazhdan-Lusztig theory [@H05; @dHL18] and in the geometric Satake correspondence [@Ga01; @MV07; @Ri14]. The paving of certain fibers related to the affine Grassmannian for ${\rm GL}_n$ gives a different approach to paving by affines of some Springer-Spaltenstein varieties, which are certain partial Springer resolutions of the nilpotent cone for ${\rm GL}_n$; see [@Hai06 Prop. 8.2, **ff.**], and [@Sp]. One could conjecture that all fibers of general convolution morphisms $X_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*)$ are paved by affine spaces. In the special case of a sequence of minuscule cocharacters $w_\bullet = \mu_\bullet$ and the associated convolution morphism $X_{L^+G}(\mu_\bullet) \to X_{L^+G}(|\mu_\bullet|)$ for the affine Grassmannian $Gr_G = LG/L^+G$, this was proved in [@Hai06 Cor. 1.2]. In general for the affine Grassmannian, it is not known which fibers are paved by affine spaces (see [@Hai06 Question 3.9]). The existence of an affine space paving of fibers of $m_{w_\bullet, {\mathcal P}}$ in the general case seems to be an interesting open question -- and the author is not aware of any counterexamples. One can consider the analogous question of when fibers of *uncompactified* convolution morphisms $Y_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*)$ are paved by affine spaces. This turns out to usually fail (for examples, see Remarks [Remark 16](#s,s_eg){reference-type="ref" reference="s,s_eg"} and [Remark 17](#KLM_eg){reference-type="ref" reference="KLM_eg"} below). However, a weaker result does always hold. **Theorem 1**. *Every fiber of a convolution morphism $Y_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*)$ is paved by finite products of copies of $\mathbb A^1$ and $\mathbb A^1 - \mathbb A^0$.* As a corollary we obtain the following result on fibers of the usual convolution morphisms. **Corollary 2**. *Every fiber of any convolution morphism $X_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*)$ is paved by finite products of copies of $\mathbb A^1$ and $\mathbb A^1 - \mathbb A^0$.* The previous two results show that the fibers in question are *cellular $k$-schemes*, in the sense of [@RS20 Def. 3.1.5]. We adopt a similar terminology and declare that they admit *cellular pavings*. A weaker version of Corollary [Corollary 2](#corB){reference-type="ref" reference="corB"} was stated without proof in [@dHL18 Rem. 2.5.4]. The proof is given here in $\S$[6](#P=B_sec){reference-type="ref" reference="P=B_sec"}, $\S$[7](#thmA_sec){reference-type="ref" reference="thmA_sec"}, and $\S$[8](#corB_sec){reference-type="ref" reference="corB_sec"}. The special case of the affine Grassmannian ${\rm Gr}_G = LG/L^+G$ over $k = \mathbb C$ is related to but is not implied by [@GL05 Thm. B] (for one thing, their Demazure-Hansen-Bott-Samelson desingularization is not the same as our convolution morphism). One situation where paving by affine spaces is known is given by the following result. **Theorem 3**. *Suppose $w_\bullet = s_\bullet = (s_1, s_2, \dots, s_r)$ is a sequence of simple reflections with Demazure product $s_* = s_1 * s_2 * \cdots * s_r$. Then the fibers of $X_{\mathcal B}(s_\bullet) \rightarrow X_{{\mathcal B}}(s_*)$ are paved by affine spaces.* This theorem was proved in [@dHL18 Thm. 2.5.2]. However, here we give a different proof, which has the advantage that it can be easily adapted to prove the special case of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} where ${\mathcal P}= {\mathcal B}$ and every $w_i$ is a simple reflection. This in turn is used to prove the general case of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"}. The results above should all have analogues at least for connected reductive groups $G$ which are defined and tamely ramified over a field $k(\!(t)\!)$ with $k$ perfect (see Remark [Remark 6](#k_perf_rem){reference-type="ref" reference="k_perf_rem"}). The proofs will necessarily be more involved and technical, and the author expects them to appear in a separate work. In section [11](#Z_sec){reference-type="ref" reference="Z_sec"}, we extend all the preceding results over ${\mathbb Z}$. We prove in that section the following result (Theorem [Theorem 38](#CvdHS_gen){reference-type="ref" reference="CvdHS_gen"}), the second part of which recovers [@CvdHS+ Thm. 1.2]. **Theorem 4**. *Assume $G_{\mathbb Z}\supset B_{\mathbb Z}\supset T_{\mathbb Z}$ is a connected reductive group over ${\mathbb Z}$ with Borel pair defined over ${\mathbb Z}$. Consider a parahoric subgroup ${\mathcal P}_{\mathbb Z}$ and the associated Schubert schemes $X_{{\mathcal P}, {\mathbb Z}}(w) \subset {\rm Fl}_{{\mathcal P},{\mathbb Z}}$. The convolution morphisms attached to $w_\bullet = (w_1, \dots, w_r) \in W^r$ may be constructed over ${\mathbb Z}$ $$m_{w_\bullet, {\mathcal P}_{\mathbb Z}}: X_{{\mathcal P}, {\mathbb Z}}(w_\bullet) \rightarrow X_{{\mathcal P}, {\mathbb Z}}(w_*).$$ and for any $v \leq w_*$, the fiber $m_{w_\bullet, {\mathcal P}_{\mathbb Z}}^{-1}(v\,e_{{\mathcal P}_{\mathcal Z}})$ has a cellular paving over ${\mathbb Z}$. Furthermore for any standard parabolic subgroup $P_{\mathbb Z}= M_{\mathbb Z}N_{\mathbb Z}$ and any pair $(\mu, \lambda) \in X_*(T)^+ \times X_*(T)^{+_M}$, the subscheme $L^+M_{\mathbb Z}LN_{\mathbb Z}x_\lambda \cap L^+G_{\mathbb Z}x_\mu$ of the affine Grassmannian ${\rm Gr}_{G, {\mathbb Z}}$ has a cellular paving over ${\mathbb Z}$.* Here is an outline of the contents of this article. In $\S$[2](#Notation_sec){reference-type="ref" reference="Notation_sec"} and $\S$[3](#Review_sec){reference-type="ref" reference="Review_sec"} we give our notation and recall the basic definitions related to convolution morphisms. The main idea of the proof of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} is to prove it by induction on $r$: one projects from the fiber onto the $r-1$ term in the twisted product; then one needs to show that the image is paved by locally closed subvarieties, each of which has a $\mathcal C$-paving, and over which the aformentioned projection morphism is trivial. The strategy of proof is given in more detail in $\S$[6.2](#thmC_proof){reference-type="ref" reference="thmC_proof"}. The required triviality statements are proved in $\S$[4](#Fact_sec){reference-type="ref" reference="Fact_sec"} and $\S$[5](#Strat_triv_sec){reference-type="ref" reference="Strat_triv_sec"}. The core of the article is found in $\S$[6](#P=B_sec){reference-type="ref" reference="P=B_sec"} -- $\S$[8](#corB_sec){reference-type="ref" reference="corB_sec"}. First, Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} is proved in $\S$[6.2](#thmC_proof){reference-type="ref" reference="thmC_proof"}, and this proof is then adapted to prove the special case of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} for ${\mathcal P}= {\mathcal B}$ and all $w_i$ simple reflections, in $\S$[6.3](#ThmA_spec_case_subsec){reference-type="ref" reference="ThmA_spec_case_subsec"}. This is used to deduce the special case of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} with ${\mathcal P}= {\mathcal B}$ in $\S$[7.2](#ThmA_P=B_proof){reference-type="ref" reference="ThmA_P=B_proof"}. Finally, the general case of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} is proved in $\S$[7.3](#ThmA_general){reference-type="ref" reference="ThmA_general"}, using the previous special cases as stepping-stones. In $\S$[8](#corB_sec){reference-type="ref" reference="corB_sec"} we quickly deduce Corollary [Corollary 2](#corB){reference-type="ref" reference="corB"} from Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"}. In $\S$[9](#HA_app){reference-type="ref" reference="HA_app"} we give an application to structure constants for parahoric Hecke algebras. In $\S$[11](#Z_sec){reference-type="ref" reference="Z_sec"} we develop all the needed machinery to extend the above results over ${\mathbb Z}$. The paper ends with Errata for [@dHL18] in $\S$[12](#Errata){reference-type="ref" reference="Errata"}. I express my thanks to Thibaud van den Hove, whose questions about [@dHL18 Rem. 2.5.4] prompted me to write up these results. I also thank him for helpful comments on an early version of this paper, and for giving me access to an advance copy of the revised version of [@CvdHS+]. # Notation {#Notation_sec} Generally speaking, we follow the same notation and conventions as [@dHL18]. Let $G$ be a split connected reductive group over a field $k$ with algebraic closure $\bar{k}$ and separable closure $k^{\rm sep}$. Fix a Borel pair $G \supset B \supset T$, also split and defined over $k$. This gives rise to the based absolute root system $(X^*(T) \supset \Phi, X_*(T) \supset \Phi^\vee, \Delta),$ the real vector space $V = X_*(T) \otimes \mathbb R$, and the canonical perfect pairing $\langle \cdot, \cdot \rangle: X^*(T) \times X_*(T) \rightarrow \mathbb Z$. The affine roots $\Phi_{\rm aff} = \{ a = \alpha + n \, | \, \alpha \in \Phi, ~~ n \in \mathbb Z\}$ are affine-linear functionals on $V$. We denote the origin by ${\bf 0} \in V$ and the $B$-dominant Weyl chamber $\mathfrak C = \{ v \in V \, | \, \langle \alpha, v \rangle > 0, \,\forall \alpha \in \Delta\}$ with apex at ${\bf 0}$. We denote the set of dominant cocharacters by $X_*(T)^+ := X_*(T) \cap \mathfrak C$. We also fix the base alcove ${\bf a} \subset \mathfrak C$ whose closure contains ${\bf 0}$. The positive simple affine roots $\Delta_{\rm aff}$ are the minimal affine roots $a = \alpha + n$ taking positive values on ${\bf a}$. We use the convention that $\lambda \in X_*(T)$ acts on $V$ by translation by $-\lambda$. The finite Weyl group is the Coxeter group $(W_0, S)$ generated by the simple reflection $\{ s_\alpha \in S\}$ on $V$, for $\alpha \in \Delta$; the group $W_0$ fixes the origin ${\bf 0}$. The extended affine Weyl group $W = X_*(T) \rtimes W_0$ acts on $V$ and hence on the set $\Phi_{\rm aff}$ by precomposition. Let $(W_{\rm aff}, S_{\rm aff})$ denote the Coxeter group generated by $S_{\rm aff}$, the simple affine reflections $s_a$ for $a \in \Delta_{\rm aff}$. It has a Bruhat order $\leq$ and a length function $\ell: W_{\rm aff} \rightarrow \mathbb Z_{\geq 0}$. Let $\Omega \subset W$ be the subgroup stabilizing ${\bf a} \subset V$. The group decomposition $W = W_{\rm aff} \rtimes \Omega$ allows us to extend $\leq$ and $\ell$ from $W_{\rm aff}$ to $W$, by declaring $\Omega$ to be the set of length zero elements in $W$. Fix the field $F = k(\!(t)\!)$ and ring of integers $\mathcal O = k[\![t]\!]$. The Iwahori-Weyl group $N_G(T)(F)/T(\mathcal O)$ may be naturally identified with $W$. We choose once and for all lifts of $w \in W_0$ in $N_GT(\mathcal O)$ and we lift $\lambda \in X_*(T)$ to the element $t^\lambda := \lambda(t) \in T(F)$. Altering these lifts by any elements in $T(\mathcal O)$ does not affect anything in what follows. We define the loop group $LG$ (resp. positive loop group $L^+G$) to be the group ind-scheme (resp., group scheme) over $k$ representing the group functor on $k$-algebras $LG(R) = G(R(\!(t)\!))$ (resp., $L^+G(R) = G(k[\![t]\!])$). For a facet ${\bf f}$ contained in the closure of ${\bf a}$, we obtain the 'standard'' parahoric group scheme $P_{\bf f}$ (see [@BT84; @HR08]). We often write ${\mathcal P}:= L^+P_{\bf f}$, and regard this as a (standard) parahoric group in $LG$. Note that $L^+G = L^+P_{\bf 0}$. The (standard) Iwahori subgroup will be denoted ${\mathcal B}:= L^+P_{\bf a}$. Let $W_{\bf f} =: W_{{\mathcal P}} \subset W_{\rm aff}$ be the subgroup which fixes ${\bf f}$ pointwise; it is a Coxeter group generated by the simple affine reflections which fix ${\bf f}$. The Bruhat order $\leq$ on $W$ descends to a Bruhat order $\leq$ on coset spaces such as $W_{\mathcal P}\backslash W / W_{\mathcal P}$ and $W/W_{\mathcal P}$. Let $^{\bf f} W ^{\bf f}$ denote the elements $w \in W$ which are the unique $\leq$-maximal elements in their double cosets $W_{\mathcal P}w W_{\mathcal P}$. The partial affine flag variety is by definition the étale sheafification of the presheaf on the category ${\rm Aff}_k$ of affine schemes ${\rm Spec}(R)$ over $k$ given by $R \mapsto LG(R)/L^+P_{\bf f}(R)$. It is represented by an ind-projective ind-scheme denoted simply by ${\rm Fl}_{\mathcal P}= LG/L^+P_{\bf f}$, and it carries a left action by ${\mathcal P}= L^+P_{\bf f}$. Denote by $e_{\mathcal P}$ its natural base point. It is well-known (see e.g. [@HR08]) that for any two standard parahoric subgroups ${\mathcal Q}$ and ${\mathcal P}$, we have a natural bijection on the level of $k$-points and $k^{\rm sep}$-points $$\label{BT_decomp} {\mathcal Q}(k) \backslash {\rm Fl}_{\mathcal P}(k) = W_{\mathcal Q} \backslash W/W_{\mathcal P}= {\mathcal Q}(k^{\rm sep}) \backslash {\rm Fl}_{\mathcal P}(k^{\rm sep}).$$ The elements of Bruhat-Tits theory used in [@HR08 Prop. 8, Rem. 9] work for split groups without any assumption that the residue field $k$ is perfect (cf. also Remark [Remark 6](#k_perf_rem){reference-type="ref" reference="k_perf_rem"}). Alternatively, for split $G$, ([\[BT_decomp\]](#BT_decomp){reference-type="ref" reference="BT_decomp"}) can be proved directly for any residue field $k$ (including $\bar{k}$), using BN-pair relations. For $w \in W$, let $Y_{\mathcal P}(w)$ (resp. $Y_{{\mathcal B}{\mathcal P}}(w)$) denote the ${\mathcal P}$-orbit (resp. ${\mathcal B}$-orbit) of $we_{\mathcal P}$ in ${\rm Fl}_{\mathcal P}$. When ${\mathcal P}= {\mathcal B}$ we will often omit the subscripts. Define the *Schubert variety* $X_{\mathcal P}(w)$ to be the Zariski closure of $Y_{\mathcal P}(w) \subset {\rm Fl}_{\mathcal P}$, endowed with reduced structure. Similarly, define $X(w) = X_{\mathcal B}(w)$ and $X_{{\mathcal B}{\mathcal P}}(w)$. In the part of this paper where we work over a field $k$, the schemes which arise are finite-type separated schemes over $k$ (not necessarily irreducible). We will always give them reduced structure, and we will call them "varieties". The morphisms of varieties we consider will always be defined over $k$, and will usually be described on the level of points in an unspecified algebraic closure of $k$. # Review of convolution morphisms {#Review_sec} For $w \in W$, define $\overline{{\mathcal P}w {\mathcal P}} = \coprod_{v \leq w} {\mathcal P}v {\mathcal P}$, where $v$ ranges over elements $v \in W_{\mathcal P}\backslash W /W_{\mathcal P}$. For any $r$-tuple $w_\bullet = (w_1, \dots, w_r) \in W^r$, we define $X_{\mathcal P}(w_\bullet)$ to be the quotient of ${\mathcal P}^r = (L^+P_{\bf f})^r$ acting on $$\overline{{\mathcal P}w_1 {\mathcal P}} \times \overline{{\mathcal P}w_2 {\mathcal P}} \times ~ \cdots ~ \times \overline{{\mathcal P}w_r {\mathcal P}}$$ by the right action $$\label{r-fold_action} (g_1, g_2, \dots, g_r) \cdot (p_1, p_2, \dots, p_r) := (g_1 p_1, p^{-1}_1g_2 p_2, \dots, p^{-1}_{r-1} g_r p_r).$$ We define $Y_{\mathcal P}(w_\bullet)$ similarly, with each $\overline{{\mathcal P}w_i {\mathcal P}}$ replaced by ${\mathcal P}w_i {\mathcal P}$. The quotients should be understood as étale sheafifications of presheaf quotients on the category ${\rm Aff}_k$. It is well-known that $X_{\mathcal P}(w_\bullet)$ (resp., $Y_{\mathcal P}(w_\bullet)$) is represented by an irreducible projective (resp., quasi-projective) $k$-variety. We regard the above objects as "twisted products\": $X_{\mathcal P}(w_\bullet) = X_{\mathcal P}(w_1) \widetilde{\times} X_{\mathcal P}(w_2) \widetilde{\times} \cdots \widetilde{\times} X_{\mathcal P}(w_r)$ (resp., $Y_{\mathcal P}(w_\bullet) = Y_{\mathcal P}(w_1) \widetilde{\times} Y_{\mathcal P}(w_2) \widetilde{\times} \cdots \widetilde{\times} Y_{\mathcal P}(w_r)$), consisting of tuples $(g_1{\mathcal P}, g_2{\mathcal P}, \dots, g_r{\mathcal P})$ such that $g_{i-1}^{-1}g_i \in \overline{{\mathcal P}w_i {\mathcal P}}$ (resp.  ${\mathcal P}w_i {\mathcal P}$) for all $1 \leq i \leq r$ (here $g_0 = 1$ by convention). Recall that the Demazure product $W^r \rightarrow W$, $(w_1, w_2, \dots, w_r) \mapsto w_1 * w_2 * \cdots * w_r$ is an associative operation. It induces an associative product $(\,^{\bf f}W^{\bf f})^r \rightarrow \, ^{\bf f}W^{\bf f}$, see e.g. [@dHL18 $\S$`<!-- -->`{=html}4]. Given any $w \in W$, let $^{\bf f}w^{\bf f}$ denote the unique $\leq$-maximal element in $W_{\mathcal P}w W_{\mathcal P}$. Given $w_\bullet = (w_1, w_2, \dots, w_r) \in W^r$, define $$w_* := \,^{\bf f}w_1^{\bf f} * \,^{\bf f}w_2^{\bf f} * \cdots * \,^{\bf f}w_r^{\bf f}.$$ Then the multiplication map $(LG)^r \rightarrow LG$ descends to the quotient and defines the *convolution morphisms* $$\begin{aligned} m_{w_\bullet} &= m_{w_\bullet, {\mathcal P}} ~ : ~ X_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*) \\ p_{w_\bullet} &= p_{w_\bullet, {\mathcal P}} ~ : ~ Y_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*), \end{aligned}$$ see [@dHL18 $\S$`<!-- -->`{=html}4]. We might describe those of the the second kind as *uncompactified convolution morphisms*. # Consequences of a factorization of the pro-unipotent Iwahori subgroup {#Fact_sec} Recall that $\bf f$ is a facet in the closure of ${\bf a}$ and the facets ${\bf f}$ and ${\bf a}$ give rise to the Iwahori and parahoric subgroups ${\mathcal B}= L^+P_{\bf a}$ and ${\mathcal P}= L^+P_{\bf f}$. The group $\mathcal U$ is the pro-unipotent radical of the Iwahori subgroup ${\mathcal B}$; it is the preimage of the unipotent radical $U \subset B$ under the natural homomorphism ${\mathcal B}\rightarrow B$ induced by $t \mapsto 0$; see [@dHL18 $\S$`<!-- -->`{=html}3.7]. Recall also that $\overline{\mathcal U}_{{\mathcal P}} = L^{--}P_{\bf f}$ is the ind-affine group ind-scheme defined in [@dHL18 Def. 3.6.1], called the negative parahoric loop group. The definition is given over $\mathbb Z$ in ([\[neg_parahoric_def\]](#neg_parahoric_def){reference-type="ref" reference="neg_parahoric_def"}). For an affine root $a$, the notation $a \overset{\bf f}{>} 0$ means that $a$ takes positive values on the facet ${\bf f}$. When ${\bf f} = {\bf a}$, we usually simply write $a > 0$. The notations $a \overset{\bf f}{\geq} 0$, $a \overset{\bf f}{<} 0$, etc, have the obvious meaning. **Proposition 5**. *Let ${\mathcal P}\supset {\mathcal B}$ be any fixed parahoric subgroup as above, and let $v \in W$ be an arbitrary element. Let ${\bf f}$ be the facet in the closure of the base alcove ${\bf a}$ which corresponds to ${\mathcal P}$.* 1. *We have a factorization of group functors $$\mathcal U = (\mathcal U \cap \,^v\overline{\mathcal U}_{\mathcal P}) \, \cdot \, (\mathcal U \cap \,^v {\mathcal P}).$$* 2. *There is an isomorphism of schemes $\mathcal U \cap \,^v\overline{\mathcal U}_{\mathcal P}\cong \prod_a U_a$ where $U_a$ ranges over the affine root groups corresponding to affine roots with $a > 0$ and $v^{-1}a \overset{\bf f}{<} 0$, and the product is taken in any order.* *Proof.* This is [@dHL18 Prop. 3.7.4]. The proof over $\mathbb Z$ given in Proposition [Proposition 27](#dHL.3.7.4_Z){reference-type="ref" reference="dHL.3.7.4_Z"} works here as well. ◻ **Remark 6**. Usually the hypothesis that $k$ is perfect is implicit in Bruhat-Tits theory and the theory of parahoric subgroups: all residue fields of the complete discretely-valued fields $F$ one works over should be assumed to be perfect, so that Steinberg's theorem applies to show that every reductive group over the completion $\breve{F}$ of a maximal unramified extension of $F$ is quasi-split. (This assumption on residue fields is missing from [@HR08], and should be added. I am grateful to Gopal Prasad for pointing out this oversight.) Since we are assuming our group $G$ is already split over $k$, it is automatically quasi-split over $\breve{k(\!(t)\!)} = k^{\rm sep}(\!(t)\!)$. Therefore we do not need to assume $k$ is perfect when invoking Bruhat-Tits theory for $G$. Note that the hypothesis that $k$ is perfect appears to be used in the proof of [@dHL18 Prop. 3.7.4], since that proof relies on [@dHL18 Rem. 3.1.1]. However, the latter actually holds for all $k$: we see the key point that $B$ is *$k$-triangularizable* in the sense of [@Spr $\S14.1$] by invoking [@Spr 16.1.1, 14.1.2] applied to $B$. **Lemma 7**. *Assume that $v$ is right-$\bf f$-minimal, ie., it is the unique minimal element in its coset $v W_{\langle {\bf f}\rangle}$, where $W_{\langle \bf f \rangle}$ is the Coxeter subgroup of $W_{\rm aff}$ which fixes $\bf f$ pointwise. Then for any positive affine root $a >0$, we have $$v^{-1}a \overset{\bf f}{<} 0 \, \Leftrightarrow \, v^{-1}a < 0.$$* *Proof.* The implication $(\Rightarrow)$ uses only that ${\bf f}$ belongs to the closure of ${\bf a}$, and holds for any $v \in W$. Next we prove $(\Leftarrow)$: Assuming $v^{-1}a < 0$, we wish to prove $v^{-1}a \overset{\bf f}{<} 0$. Suppose on the contrary that $v^{-1}a \overset{\bf f}{\geq} 0$. Combined with $v^{-1}a < 0$, we deduce $v^{-1}a \overset{\bf f}{=} 0$, that is, $v^{-1}s_a v \in W_{\langle \bf f \rangle}$. Since $v$ is right-$\bf f$-minimal and $s_a v \in vW_{\langle \bf f \rangle}$, we deduce that $s_a v > v$. On the other hand, since $a$ is positive on ${\bf a}$ and $a$ is negative on $v\bf a$, we see that $v{\bf a}$ and ${\bf a}$ are on opposite sides of the affine root hyperplane $H_a$, which means $s_a v < v$, a contradiction. ◻ **Proposition 8**. *If $v \in W$ is right-$\bf f$-minimal, then we have isomorphisms $$Y_{{\mathcal B}{\mathcal P}}(v)~\cong~\mathcal U \cap \,^v\overline{\mathcal U}_{\mathcal P}~\cong~\mathcal U \cap \,^v\overline{\mathcal U}~\cong~Y_{{\mathcal B}{\mathcal B}}(v).$$* *Proof.* Since $v$ normalizes $T(\mathcal O)$, we have $Y_{{\mathcal B}{\mathcal P}}(v) = \mathcal U v {\mathcal P}/{\mathcal P}$, which identifies with $\mathcal U \cap \,^v\overline{\mathcal U}_{\mathcal P}$ by Proposition [Proposition 5](#dHL.3.7.4){reference-type="ref" reference="dHL.3.7.4"}(a) and by the fact that $\overline{\mathcal U}_{{\mathcal P}} \rightarrow {\rm Fl}_{\mathcal P}$ is an (open) immersion, see e.g. [@dHL18 Thm. 2.3.1]. This is identified with $\mathcal U \cap \,^v\overline{\mathcal U}$ by Proposition [Proposition 5](#dHL.3.7.4){reference-type="ref" reference="dHL.3.7.4"}(b) and Lemma [Lemma 7](#root_prod){reference-type="ref" reference="root_prod"}. ◻ **Remark 9**. The proof of Proposition [Proposition 8](#BP_comp){reference-type="ref" reference="BP_comp"} given here ultimately relies on the negative parahoric loop group introduced in [@dHL18]. Another proof which is more general and which avoids this reliance, is given in [@HaRi2 Lem. 3.3]. We give the proof above because it is an almost immediate consequence of Proposition [Proposition 5](#dHL.3.7.4){reference-type="ref" reference="dHL.3.7.4"}, which we need anyway to establish Proposition [Proposition 10](#strat_triv){reference-type="ref" reference="strat_triv"} below. # Stratified triviality of convolution morphisms {#Strat_triv_sec} **Proposition 10**. *The morphism $m_{w_\bullet}: X_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*)$ is trivial over every ${\mathcal B}$-orbit in its image.* *Proof.* Writing $m := m_{w_\bullet}$, we prove the triviality of the map $m$ over $\mathcal B$-orbits contained in its image. Assume $Y_{{\mathcal B}{\mathcal P}}(v) \subset X_{\mathcal P}(w_*)$. By Proposition [Proposition 8](#BP_comp){reference-type="ref" reference="BP_comp"}, an element $\mathcal P' \in Y_{{\mathcal B}{\mathcal P}}(v)$ can be written in the form $$\mathcal P' = {uv}{\mathcal P}$$ for a unique element $u \in \mathcal U \cap \, ^v\overline{\mathcal U}_{\mathcal P}$. We can then define an isomorphism $$m^{-1}(Y_{{\mathcal B}{\mathcal P}}(v)) ~ \overset{\sim}{\longrightarrow} ~ \, m^{-1}(v\mathcal P) \times Y_{{\mathcal B}{\mathcal P}}(v)$$ by sending $(\mathcal P_1, \dots, \mathcal P_{r-1}, {uv}\mathcal P)$ to $(\, {u^{-1}}\mathcal P_1, \cdots, {u^{-1}}\mathcal P_{r-1}, {v}\mathcal P) \times {uv}\mathcal P$. Obviously the first factor belongs to $m^{-1}(v\mathcal P)$. ◻ # Paving results for the case ${\mathcal P}= {\mathcal B}$ {#P=B_sec} ## BN-pair relations and lemmas on retractions {#bnpair} $\;$ The following statements can be interpreted at the level of $k$ or $\bar{k}$-points, but we will suppress this from the notation. Recall that given $\mathcal B_1 = {g_1}\mathcal B$, $\mathcal B_2 = {g_2}\mathcal B$ and $w \in W$, we say the pair $(\mathcal B_1, \mathcal B_2)$ is in relative position $w$ (and we write $\mathcal B_1 \, \overset{w}{\textendash \textendash} \,\mathcal B_2$) if and only if $g_1^{-1}g_2 \in \mathcal B w \mathcal B$. We write $$\mathcal B_1 \, \overset{\leq w}{\textendash \textendash} \,\mathcal B_2 \,\,\,\,\,\,\,\, \mbox{if and only if} \,\,\,\,\,\,\,\, \mathcal B_1 \, \overset{v}{\textendash \textendash} \, \mathcal B_2 \,\,\,\,\mbox{for some $v \leq w$}.$$ We have $$Y_\mathcal B(w) = \{ \mathcal B' ~ | ~ \mathcal B \, \overset{w}{\textendash \textendash}\, \mathcal B' \} \hspace{.25in} \mbox{and} \hspace{.25in} X_\mathcal B(w) = \{ \mathcal B' ~ | ~ \mathcal B \, \overset{\leq w}{\textendash \textendash} \, \mathcal B' \}.$$ The BN-pair relations hold for $v \in W$ and $s \in S_{\rm aff}$: $$\begin{aligned} \label{BN-pair_eq} \mathcal B v \mathcal B s \mathcal B &= \begin{cases} \mathcal B vs \mathcal B, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \mbox{if $v < vs$,} \\ \mathcal B vs \mathcal B \cup \mathcal B v \mathcal B, \,\,\,\,\, \mbox{if $vs < v$.} \end{cases} \\ %s\mathcal B s &\nsubseteq \mathcal B. \notag\end{aligned}$$ Note that for every $v \in W$ and $s \in S_{\rm aff}$, there is an isomorphism $\{ \mathcal B' ~ | ~ v\mathcal B \,\, \overset{\leq s}{\textendash \textendash} \,\mathcal B' \} \cong \mathbb P^1$ and $\{ \mathcal B' ~ | ~ v\mathcal B \, \,\overset{\leq s}{\textendash \textendash} \,\mathcal B' \} \subset Y_\mathcal B(v) \cup Y_\mathcal B(vs)$. **Lemma 11**. *Suppose $s \in S_{\rm aff}$ and $v \in W$.* 1. *If $v < vs$, then $\{ \mathcal B' ~ | ~ v\mathcal B \,\, \overset{\leq s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(v) = \{ v\mathcal B \} \cong \mathbb A^0$.* 2. *If $v < vs$, then $\{ \mathcal B' ~ | ~ v \mathcal B \,\, \overset{\leq s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(vs) \cong \mathbb A^1$.* 3. *If $vs < v$, then $\{ \mathcal B' ~ | ~ v\mathcal B \,\, \overset{\leq s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(v) \cong \mathbb A^1$.* 4. *If $vs < v$, Then $\{ \mathcal B' ~ | ~ v\mathcal B \,\, \overset{\leq s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(vs) = \{ {vs}\mathcal B\} \cong \mathbb A^0$.* *Proof.* This is obvious from properties of the retraction map from the building associated to $G$ onto the apartment corresponding to $T$, with respect to an alcove in that apartment. A reference for how such retractions "work" is [@HKM $\S6$]. ◻ In a similar way, we get an analogous lemma. **Lemma 12**. *Suppose $s \in S_{\rm aff}$ and $v \in W$.* 1. *If $v < vs$, then $\{ \mathcal B' ~ | ~ v\mathcal B \,\, \overset{s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(v) = \emptyset$.* 2. *If $v < vs$, then $\{ \mathcal B' ~ | ~ v \mathcal B \,\, \overset{s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(vs) \cong \mathbb A^1$.* 3. *If $vs < v$, then $\{ \mathcal B' ~ | ~ v\mathcal B \,\, \overset{s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(v) \cong \mathbb A^1- \mathbb A^0$.* 4. *If $vs < v$, Then $\{ \mathcal B' ~ | ~ v\mathcal B \,\, \overset{s}{\textendash \textendash} \, \mathcal B' \} \cap Y_\mathcal B(vs) = \{ {vs}\mathcal B\} \cong \mathbb A^0$.* ## Proof of Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} {#thmC_proof} Since $\mathcal B$ is understood, we will write $Y(w_\bullet)$ for $Y_\mathcal B(w_\bullet)$, and $X(w_\bullet)$ for $X_\mathcal B(w_\bullet)$ in what follows. Let $s_\bullet = (s_1, \dots, s_r) \in \mathcal S^r$. There is no requirement here that $s_1 \cdots s_r$ be reduced. Recall the subvariety $X(s_\bullet) \subset (\mathcal G/\mathcal B)^r$ which consists of the $r$-tuples $(\mathcal B_1, \dots, \mathcal B_r)$ such that $\mathcal B_{i-1} \overset{\leq s_i}{\textendash \textendash} \mathcal B_i$ for all $i = 1, \dots, r$ (with the convention that $\mathcal B_0 = \mathcal B$). We are going to prove the paving by affine spaces of the fibers of the morphism $$m~ :~ X(s_\bullet) \longrightarrow X(s_*) \subset \mathcal G/\mathcal B, \,\qquad (\mathcal B_1, \dots, \mathcal B_r) \longmapsto \mathcal B_r.$$ We proceed by induction on $r$. The case $r = 1$ is trivial, so we assume $r > 1$ and that the theorem holds for $r-1$. Let $s'_* := s_1 * \cdots * s_{r-1}$. Let $s'_\bullet = (s_1, \dots, s_{r-1})$. By our induction hypothesis, the theorem holds for $$m': X(s'_\bullet) \longrightarrow X(s'_*), \qquad (\mathcal B_1, \dots, \mathcal B_{r-1}) \longmapsto \mathcal B_{r-1}.$$ Now suppose $v \leq s_*$, so that $v\mathcal B \in {\rm Im}(m)$. For an element $(\mathcal B_1, \dots, \mathcal B_{r-1}, v\mathcal B) \in m^{-1}(v\mathcal B)$, we have $$\mathcal B \, \overset{v}{\textendash \textendash} \, v\mathcal B \, \,\overset{\leq s_r}{\textendash \textendash} \, \mathcal B_{r-1}.$$ It follows from the BN-pair relations that $\mathcal B_{r-1} \in Y(v) \cup Y(vs_r)$. We consider the map $$\begin{aligned} \xi: m^{-1}(v\mathcal B) &\longrightarrow Y(v) \cup Y(vs_r) \\ (\mathcal B_1, \dots, \mathcal B_{r-1}, v\mathcal B) &\longmapsto \mathcal B_{r-1}.\end{aligned}$$ We will examine the subsets ${\rm Im}(\xi) \cap Y(v)$ and ${\rm Im}(\xi) \cap Y(vs_r)$. We will show that 1. these subsets are affine spaces (either empty, a point, or $\mathbb A^1$); one of them, denoted $\mathbb A_1$, is closed in ${\rm Im}(\xi)$, and the other, denoted $\mathbb A_2$, is nonempty, open, and dense in ${\rm Im}(\xi)$; 2. if $\mathbb A_i \neq \emptyset$, then $\mathbb A_i$ belongs to ${\rm Im}(m')$; furthermore $\xi^{-1}(\mathbb A_i) \cong m'^{-1}(\mathbb A_i)$ under the obvious identification, and $\xi: \xi^{-1}(\mathbb A_i) \rightarrow \mathbb A_i$ corresponds to the morphism $m': m'^{-1}(\mathbb A_i) \rightarrow \mathbb A_i$. These facts are enough to prove Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"}. Indeed, applying $\xi^{-1}$ to the decomposition $${\rm Im}(\xi) = \mathbb A_1 \cup \mathbb A_2$$ and using (ii) gives us a decomposition $$m^{-1}(v\mathcal B) = \xi^{-1}(\mathbb A_1) \cup \xi^{-1}(\mathbb A_2) = m'^{-1}(\mathbb A_1) \cup m'^{-1}(\mathbb A_2)$$ where the first is closed and the second is nonempty and open. By the induction hypothesis, the fibers of $m'$ are paved by affine spaces. Since $\mathbb A_i$ is contained in a $\mathcal B$-orbit, we see $m'$ is trivial over each $\mathbb A_i$ by Proposition [Proposition 10](#strat_triv){reference-type="ref" reference="strat_triv"}, and hence each $m'^{-1}(\mathbb A_i)$ is paved by affine spaces. Thus $m^{-1}(v\mathcal B)$ is paved by affine spaces. To verify the properties (i,ii), we need to consider various cases. We start with two cases which arise from the following standard lemma about the Bruhat order (see e.g. [@Hum Prop. 5.9]). **Lemma 13**. *Let $(W,S)$ be a Coxeter group and $x,y \in W$ and $s \in S$. Then $x \leq y$ implies $x \leq ys$ or $xs \leq ys$ [(]{.upright}or both[)]{.upright}.* Recall $v \leq s_*$ by assumption. The two cases we need to consider are Case I: $s'_* < s'_* s_r$, so that $s_* = s'_* s_r$. Thus by Lemma [Lemma 13](#trick){reference-type="ref" reference="trick"}, $v \leq s'_*$ or $v s_r \leq s'_*$. Case II: $s'_* s_r < s'_*$, so that $s_* = s'_*$. Thus $v \leq s'_*$. We will break each of these into subcases, depending on whether $v < vs_r$ or $vs_r < v$. We then consider further subcases depending on which of $v$ or $vs_r$ precedes $s'_*$ in the Bruhat order. Case I.1: $v < vs_r$. So $v \leq s'_*$ is automatic. There are two subcases: I.1a: $v < vs_r \leq s'_*$; I.1b: $v \leq s'_*$ but $vs_r \nleq s'_*$. Case I.2: $vs_r < v$. So $vs_r \leq s'_*$ is automatic. There are two subcases: I.2a: $vs_r < v \leq s'_*$; I.2b: $vs_r \leq s'_*$ but $v \nleq s'_*$. Case II.1: $v < vs_r$. As $v \leq s'_*$ is automatic, there are two subcases: II.1a: $v < vs_r \leq s'_*$. II.1b: $v \leq s'_*$ but $vs_r \nleq s'_*$. Case II.2: $vs_r < v$. Here $vs_r \leq s'_*$ and $v \leq s'_*$, so there are no further subcases. Consider any element $(\mathcal B_1, \dots, \mathcal B_{r-1}, v\mathcal B)$ in $m^{-1}(v\mathcal B)$. As noted already above, we have $\mathcal B \, \overset{v}{\textendash \textendash} \, v\mathcal B \, \overset{\leq s_r}{\textendash \textendash} \, \mathcal B_{r-1}$. Then Lemma [Lemma 11](#BN_pair_lem){reference-type="ref" reference="BN_pair_lem"} tells us the shape of ${\rm Im}(\xi) \cap Y(v)$ and ${\rm Im}(\xi) \cap Y(vs_r)$ in all the cases enumerated above. We record the results in the following table. Case ${\rm Im}(\xi) \cap Y(v)$ ${\rm Im}(\xi) \cap Y(vs_r)$ ------- --------------------------- ------------------------------ I.1a $\mathbb A^0$ $\mathbb A^1$ I.1b $\mathbb A^0$ $\emptyset$ I.2a $\mathbb A^1$ $\mathbb A^0$ I.2b $\emptyset$ $\mathbb A^0$ II.1a $\mathbb A^0$ $\mathbb A^1$ II.1b $\mathbb A^0$ $\emptyset$ II.2 $\mathbb A^1$ $\mathbb A^0$ .3cm In each case it is clear which piece should be labelled $\mathbb A_1$ or $\mathbb A_2$. This proves the main part of (i,ii); the other assertions are clear. This completes the proof of Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"}. 0◻ **Remark 14**. Each $\mathbb A^1$ appearing in the table may be identified with a suitable affine root group $U_{\alpha +n}$, the $k$-group with $k$-points $$U_{\alpha + n}(k) = \{ u_\alpha(x t^n) \, | \, x \in k\},$$ where $u_\alpha : \mathbb G_a \rightarrow G$ is the root homomorphism corresponding to the root $\alpha$. For example, consider Case I.1a. Then ${\rm Im}(\xi) \cap Y(vs_r)$ is $\{{\mathcal B}_{r-1} \, | \, v{\mathcal B}\overset{s_r}{\textendash \textendash} {\mathcal B}_{r-1}\}$. Each such ${\mathcal B}_{r-1}$ can be expressed as ${\mathcal B}_{r-1} = {v u s_r}{\mathcal B}$ for a unique $u \in \mathcal U \cap \, ^{s_r}\overline{\mathcal U}_{\mathcal B}$. Now use Proposition [Proposition 5](#dHL.3.7.4){reference-type="ref" reference="dHL.3.7.4"}(b). ## Proof Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} in a special case {#ThmA_spec_case_subsec} We will now prove Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} in the case where ${\mathcal P}={\mathcal B}$ and $w_i = s_i$ is a simple reflection for all $1 \leq i \leq r$. The argument is by induction on $r$, as in the previous subsection. We consider the analogues $p$ and $p'$ of the morphisms $m$ and $m'$ $$\begin{aligned} p: Y(s_\bullet) &\rightarrow X(s_*) \hspace{.25in} , \hspace{.25in} ({\mathcal B}_1, \dots, {\mathcal B}_{r-1}, {\mathcal B}_r) \mapsto {\mathcal B}_{r} \\ p': Y(s'_\bullet) &\rightarrow X(s'_*) \hspace{.25in} , \hspace{.25in} ({\mathcal B}_1, \dots, {\mathcal B}_{r-2}, {\mathcal B}_{r-1}) \mapsto {\mathcal B}_{r-1}\end{aligned}$$ and for $v{\mathcal B}$ in the image of $p$, we consider the map $$\begin{aligned} \xi^\circ: p^{-1}(v\mathcal B) &\longrightarrow Y(v) \cup Y(vs_r) \\ (\mathcal B_1, \dots, \mathcal B_{r-1}, v\mathcal B) &\longmapsto \mathcal B_{r-1}.\end{aligned}$$ The locally triviality of $p$ over ${\mathcal B}$-orbits in its image still holds, and similarly for $p'$ (see the proof of Proposition [Proposition 10](#strat_triv){reference-type="ref" reference="strat_triv"}), and it suffices to establish the analogues of (i, ii) above. We consider the same cases as above, and we list the possibilities for ${\rm Im}(\xi^\circ) \cap Y(v)$ and ${\rm Im}(\xi^\circ) \cap Y(vs_r)$ in the table below, determined in each case with the help of Lemma [Lemma 12](#strict-BN_pair_lem){reference-type="ref" reference="strict-BN_pair_lem"}. Case ${\rm Im}(\xi^\circ) \cap Y(v)$ ${\rm Im}(\xi^\circ) \cap Y(vs_r)$ ------- ---------------------------------------------- ------------------------------------ I.1a $\emptyset$ $\mathbb A^1$ or $\emptyset$ I.1b $\emptyset$ $\emptyset$ I.2a $\mathbb A^1 - \mathbb A^0$  or  $\emptyset$ $\mathbb A^0$ or $\emptyset$ I.2b $\emptyset$ $\mathbb A^0$ or $\emptyset$ II.1a $\emptyset$ $\mathbb A^1$ or $\emptyset$ II.1b $\emptyset$ $\emptyset$ II.2 $\mathbb A^1 - \mathbb A^0$  or  $\emptyset$ $\mathbb A^0$ or $\emptyset$ .3cm Let us explain the meaning of entries such as "$\mbox{$\mathbb A^1 - \mathbb A^0$ ~or~ $\emptyset$}$", for example in the entry in case I.2a for ${\rm Im}(\xi^\circ) \cap Y(v)$. Note that $v \leq s'_*$ implies that $Y(v) \subset {\rm Im}(m')$, but $Y(v) \subset {\rm Im}(p')$ is not automatic. However, since $p'$ is ${\mathcal B}$-equivariant we either have $Y(v) \cap {\rm Im}(p') = \emptyset$, or $Y(v) \subset {\rm Im}(p')$. If $Y(v) \cap {\rm Im}(p') = \emptyset$, the table entry is $\emptyset$. If $Y(v) \subset {\rm Im}(p')$, the intersection ${\rm Im}(\xi^\circ) \cap Y(v)$ is precisely the part of $Y(v)$ which is exactly of relative position $s_r$ from $v{\mathcal B}$, and this identifies with $\mathbb A^1 - \mathbb A^0$ in the case where $vs_r < v$. The analogues of (i,ii) above hold, except that here, both $\mathbb A_1$ and $\mathbb A_2$ can be empty, and when nonempty the larger subset can be either $\mathbb A^0$, $\mathbb A^1 - \mathbb A^0$, or $\mathbb A^1$. The morphism $p'$ is trivial over every ${\mathcal B}$-orbit in its image (comp. Proposition [Proposition 10](#strat_triv){reference-type="ref" reference="strat_triv"}), and by induction the nonempty fibers of $p'$ are paved by finite products of copies of $\mathbb A^1$ and $\mathbb A^1- \mathbb A^0$. Therefore the fibers of $p$ also have the desired property. This proves Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} in the special case where ${\mathcal P}= {\mathcal B}$ and each $w_i$ is a simple reflection $s_i$. 0◻ **Remark 15**. As in Remark [Remark 14](#A1_rem){reference-type="ref" reference="A1_rem"}, each $\mathbb A^1$ in the table may be identified with an affine root group $U_{\alpha + n}$, and each $\mathbb A^ 1- \mathbb A^0$ may be identified with a suitable variety of non-identity elements $U^*_{\alpha + n}$. For example, consider Case I.2a. Then ${\rm Im}(\xi^\circ) \cap Y(v)$ is $\{{\mathcal B}_{r-1} \, | \, v{\mathcal B}\overset{s_r}{\textendash \textendash} {\mathcal B}_{r-1}\} \cap Y(v)$. We may write such ${\mathcal B}_{r-1}$ as $${\mathcal B}_{r-1} = vus_r{\mathcal B}$$ for a unique $u \in \mathcal U \cap \,^{s_r}\overline{\mathcal U}_{\mathcal B}$ such that $u \neq e$. Now use Proposition [Proposition 5](#dHL.3.7.4){reference-type="ref" reference="dHL.3.7.4"}(b). **Remark 16**. In the cases I.2a and II.2, the $\mathbb A^0$ piece is in the closure of the $\mathbb A^1 - \mathbb A^0$ piece, and it is tempting to consider the union of these as $\mathbb A^1$. Indeed, if one ignores the possibiliy of $\emptyset$ in cases I.2a and II.2, the table seems to show that in every case ${\rm Im}(\xi^\circ)$ is an affine space ($\emptyset$, $\mathbb A^0$, or $\mathbb A^1$), and one could ask whether the argument does not in fact prove (by induction again) that every fiber of $p$ is paved by affine spaces. However, one cannot ignore the empty set, and in fact in Case II.2 it is possible to have ${\rm Im}(\xi^\circ) \cap Y(v) = \mathbb A^1 - \mathbb A^0$, while ${\rm Im}(\xi^\circ) \cap Y(vs_r) = \emptyset$. Letting $s \in S_{\rm aff}$, this happens for $s_\bullet = (s_1, s_2) = (s,s)$ and $v = s_2= s$. This situation is reflected by the quadratic relation in the Iwahori-Hecke algebra $T_s * T_s = (q-1)T_s + qT_1$. In addition, even in a special situation where ${\rm Im}(\xi^\circ)$ is always an affine space, the affine space paving would remain elusive, as it is not clear that $p'$ would be trivial over all of ${\rm Im}(\xi^\circ)$ whenever it is not contained in single ${\mathcal B}$-orbit. **Remark 17**. The above remark "explains" why we cannot hope to improve Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} to assert that all fibers of $Y_{\mathcal P}(w_\bullet) \rightarrow Y_{\mathcal P}(w_*)$ are paved by *affine spaces*. For a concrete example related to the affine Grassmannian ${\rm Gr}_G = LG/L^+P_{\bf 0}$ over a finite field $k = \mathbb F_q$, take $G = {\rm SO}(5)$, and let $$\mu_1 = \mu_2 = \mu_3 = \alpha^\vee_1 + \alpha^\vee_2 = (1,1),$$ where $\alpha^\vee_i$ are the two simple coroots of $G$. Here we use notation following the conventions of [@Bou]. In [@KLM $\S 8.5$] it is shown that the Hecke algebra structure constant $c^0_{\mu_\bullet}(q)$ (the coefficient of the unit element in the product $1_{K t^{\mu_1}K} * 1_{K t^{\mu_2}K} * 1_{K t^{\mu_3}K}$ for $K = L^+P_{\bf 0}({\mathbb F}_q)$) satisfies $c^0_{\mu_\bullet}(q) = q^5 - q$. This shows that the fiber over the base point $e_0$ of $Y_{L^+P_{\bf 0}}(\mu_\bullet) \rightarrow X_{L^+P_{\bf 0}}(|\mu_\bullet|)$ cannot be paved by affine spaces over $\mathbb F_q$. # Proof of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} {#thmA_sec} ## Schubert cells in ${\rm Fl}_{\mathcal B}$ as convolution spaces If $\tau s_1 \cdots s_r = w$ is a reduced expression, we sometimes write $Y(\tau s_1 \cdots s_r)$ for $Y(w)$. This Schubert cell has the following well-known moduli description of its $k$-points. **Lemma 18**. *Fix the reduced expression $w = \tau s_1 \cdots s_r$ as above.* 1. *Giving a $k$-point of $Y(w)$ is equivalent to giving a $k$-point of $\tau^{-1}Y(w)$, which is equivalent to giving a sequence of Iwahori subgroups $({\mathcal B}_0, {\mathcal B}_1, \dots, {\mathcal B}_r)$ such that $${\mathcal B}=: {\mathcal B}_0\, \overset{s_1}{\textendash \textendash} \, {\mathcal B}_1 \, \overset{s_2}{\textendash \textendash} \, {\mathcal B}_2 \, \overset{s_3}{\textendash \textendash} \,\cdots \, \overset{s_r}{\textendash \textendash} \,{\mathcal B}_r.$$* 2. *For any element $y \in LG(k)$, giving a $k$-point of $y^{-1}Y(w)$ is equivalent to giving a sequence of Iwahori subgroups $({\mathcal B}_0, {\mathcal B}_1, \cdots, {\mathcal B}_r)$ such that $$^{y^{-1}}{\mathcal B}=: {\mathcal B}_0\, \overset{s_1}{\textendash \textendash} \, {\mathcal B}_1 \, \overset{s_2}{\textendash \textendash} \, {\mathcal B}_2 \, \overset{s_3}{\textendash \textendash} \,\cdots \, \overset{s_r}{\textendash \textendash} \,{\mathcal B}_r.$$* *Proof.* In both cases, note that $\tau$ normalizes the Iwahori ${\mathcal B}$. ◻ ## Proof of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} for ${\mathcal P}= {\mathcal B}$ {#ThmA_P=B_proof} Consider the morphism $p_{w_\bullet, {\mathcal B}}: Y_{\mathcal B}(w_\bullet) \rightarrow X_{\mathcal B}(w_*)$. For each $1 \leq i \leq r$, we choose a reduced expression $$w_i = \tau_i s_{i1} \cdots s_{i n_i}$$ for $s_{ij} \in S_{\rm aff}$ and $\tau_i \in \Omega$. Since conjugation by $\tau_i$ normalizes ${\mathcal B}$, permutes $S_{\rm aff}$, and preserves the Demazure product, we may reduce the study of fibers to the case where each $\tau_i = 1$. Then we have $$w_* = s_{11} * \cdots * s_{1 n_1} * s_{21} * \cdots * s_{2 n_2}* \cdots \cdots * s_{r1} * \cdots * s_{r n_r} =: s_{**}.$$ By Lemma [Lemma 18](#unwind){reference-type="ref" reference="unwind"}, the morphism $p_{w_\bullet, {\mathcal B}}$ is identified with the morphism $p_{s_{\bullet \bullet}, {\mathcal B}} : Y(s_{\bullet \bullet}) \rightarrow Y(s_{**})$. By $\S\ref{ThmA_spec_case_subsec}$, its fibers possess the required pavings. 0◻ ## Proof of Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} in general {#ThmA_general} Let $\mathcal C$ be the class of varieties which are finite products of copies of $\mathbb A^1$ and $\mathbb A^1 - \mathbb A^0$. We consider the morphism $p = p_{w_\bullet, {\mathcal P}} : Y_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*)$, and suppose $v{\mathcal P}$ lies in the image. We prove that the fiber $p^{-1}(v{\mathcal P})$ has a $\mathcal C$-paving by induction on $r$. As before, consider the morphism $p' : Y_{\mathcal P}(w_1, \dots, w_{r-1}) \rightarrow {\rm Fl}_{\mathcal P}$ given by $({\mathcal P}_1,{\mathcal P}_2, \dots, {\mathcal P}_{r-1}) \mapsto {\mathcal P}_{r-1}$, and, by a slight abuse, its restriction $\xi^\circ = p'|_{p^{-1}(v{\mathcal P})} : p^{-1}(v{\mathcal P}) \rightarrow {\rm Fl}_{\mathcal P}$, defined by $({\mathcal P}_1, \dots, {\mathcal P}_{r-1}, v{\mathcal P}) \mapsto {\mathcal P}_{r-1}$. We have $${\rm Im}(\xi^\circ) = {\rm Im}(p') \cap vY_{\mathcal P}(w_{r}^{-1}).$$ We claim that for any $y \in W$ with corresponding ${\mathcal B}$-orbit $Y_{{\mathcal B}{\mathcal P}}(y)$, the intersection ${\rm Im}(\xi^\circ) \cap Y_{{\mathcal B}{\mathcal P}}(y)$ is either empty, or has a $\mathcal C$-paving. Then since such locally closed subsets cover ${\rm Im}(\xi^\circ)$ and since $p'$ is trivial over each such subset, the $\mathcal C$-paving of $p^{-1}(v{\mathcal P})$ will follow by our induction hypothesis applied to $p'$. Note that if ${\rm Im}(p') \cap Y_{{\mathcal B}{\mathcal P}}(y)$ is nonempty, then $Y_{{\mathcal B}{\mathcal P}}(y) \subset {\rm Im}(p')$, and we are trying to produce a $\mathcal C$-paving of $$Y_{{\mathcal B}{\mathcal P}}(y) \cap vY_{{\mathcal P}}(w_r^{-1}).$$ We can pass to ${\mathcal B}$-orbits by writing $W_{\mathcal P}w_r^{-1} W_{\mathcal P}= \coprod_{\eta_m} \eta_m W_{\mathcal P}$, for $\eta_m \in W$ a finite collection of right-$\bf f$-minimal elements. We then have a locally closed decomposition $Y_{{\mathcal P}}(w_r^{-1}) = \coprod_{\eta_m} Y_{{\mathcal B}{\mathcal P}}(\eta_m)$. Thus we need to show that each $$Y_{{\mathcal B}{\mathcal P}}(y) \cap vY_{{\mathcal B}{\mathcal P}}(\eta_m)$$ has a $\mathcal C$-paving. We may assume $y$ is also right-$\bf f$-minimal. Then by Lemma [Proposition 8](#BP_comp){reference-type="ref" reference="BP_comp"}, this is isomorphic to $$Y_{{\mathcal B}}(y) \cap vY_{{\mathcal B}}(\eta_m).$$ This is turn is equal to the fiber over $v{\mathcal B}$ of the morphism $Y_{\mathcal B}(y) \widetilde{\times} Y_{\mathcal B}(\eta_m^{-1}) \rightarrow X_{\mathcal B}(y * \eta_m^{-1})$. But each fiber of this morphism has a $\mathcal C$-paving by $\S\ref{ThmA_P=B_proof}$. 0◻ # Proof of Corollary [Corollary 2](#corB){reference-type="ref" reference="corB"} {#corB_sec} This follows immediately from Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"}, as we have a decomposition into locally closed subvarieties $$\label{conv_cl_rel} X_{\mathcal P}(w_\bullet) = \coprod_{v_\bullet} Y_{\mathcal P}(v_\bullet)$$ where $v_\bullet$ ranges over all tuples $(v_1, v_2, \dots, v_r) \in W_{\mathcal P}\backslash W/W_{\mathcal P}$ such that $v_i \leq w_i$ in the Bruhat order on $W_{\mathcal P}\backslash W/W_{\mathcal P}$ for all $i$. Thus the fiber has a corresponding decomposition, and the result follows from Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"}. 0◻ # Application to structure constants for parahoric Hecke Algebras {#HA_app} Fix a nonarchimedean field $F$ with ring of integers $\mathcal O_F$ and residue field $k_F = \mathbb F_q$. Let us suppose $G$ is a split group over $\mathbb Z$, and fix a Borel pair $B \supset T$ in $G$, also split and defined over $\mathbb Z$. This gives rise to the extended affine Weyl group $W$ defined using $G \supset B \supset T$ (it agrees with the extended affine Weyl group attached to $G_F \supset B_F \supset T_F$). For any parahoric subgroup ${\mathcal P}\subset G(F)$, consider the parahoric Hecke algebra $\mathcal H(G(F)/\!/{\mathcal P}) = C_c({\mathcal P}\backslash G(F) /{\mathcal P}, \mathbb C)$, give the structure of a unital associative $\mathbb C$-algebra with convolution $*$ defined using the Haar measure on $G(F)$ giving ${\mathcal P}$ volume $1$. Consider the $\mathbb C$-basis of characteristic functions $f_w := 1_{\mathcal P w \mathcal P}$ indexed by elements $w \in W_{\mathcal P}\backslash W/W_{\mathcal P}$. We can represent such cosets by maximal length elements $w \in \,^{\bf f} W ^{\bf f}$. **Proposition 19**. *For any $w_1, w_2 \in \,^{\bf f} W ^{\bf f}$, we have $$f_{w_1} * f_{w_2} = \sum_{v \in \,^{\bf f} W ^{\bf f}} c^v_{w_1, w_2}(q) \, f_v$$ where the structure constant is a non-negative integer of the form $$c^v_{w_1, w_2}(q) = \sum_{a,b \in \mathbb Z_{\geq 0}} m_{a,b} ~q^a (q-1)^b$$ for certain non-negative integers $m_{a,b}$ which vanish for all but finitely many pairs $(a,b)$.* *Proof.* The combinatorics of parahoric Hecke algebras over characteristic zero local fields $F$ are the same as those for $F=\mathbb F_q(\!(t)\!)$ (the parahoric subgroups in each setting chosen to correspond to each other in the obvious way, suitably identifying apartments for $G_F \supset T_F$ and $G_{\mathbb F_q(\!(t)\!)} \supset T_{\mathbb F_q(\!(t)\!)}$ and facets therein -- for a much more general statement, see [@PZ13 4.1.2]). Therefore we can assume $F$ is of the later form. Then note that $c^v_{w_1, w_2}(q)$ is the number of $\mathbb F_q$-rational points in the fiber over $v{\mathcal P}$ of the corresponding convolution morphism $Y_{\mathcal P}(w_1) \widetilde{\times} Y_{\mathcal P}(w_2) \rightarrow X_{\mathcal P}(w_*)$. Thus the result follows from Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"}. ◻ This gives rise to general parahoric variants (in the equal parameter case) of combinatorial results on structure constants for spherical affine Hecke algebras due to Parkinson [@Par06 Thm. 7.2] and Schwer [@Schw06]. By virtue of the Macdonald formula (see e.g. [@HKP Thm. 5.6.1]), the function $P_\lambda$ considered (albeit with differing normalizations) by Parkinson and Schwer agrees up to an explicit normalizing factor with the Satake tranform $f^\vee_\lambda$ of the basis elements $f_{\lambda} = 1_{G(\mathbb F_q[\![t]\!]) t^\lambda G(\mathbb F_q[\![t]\!])}$ above, for any dominant $\lambda \in X_*(T)$. In particular, Proposition [Proposition 19](#str_const_prop){reference-type="ref" reference="str_const_prop"} shows that suitably renormalized versions of the functions $C^\nu_{\lambda \mu}$ appearing in [@Schw06 Thm. 1.3] lie in $\mathbb Z_{\geq 0}[q-1]$. # Cellular paving of certain subvarieties in the affine Grassmannian {#KU_GrG_sec} In this section we will restrict our attention to certain generalizations of the intersections containing the Mirkovic-Vilonen cycles in the affine Grassmannian. Let ${\mathcal P}= {\mathcal P}_{\bf 0} = L^+G$, and consider the affine Grassmannian ${\rm Gr}_G = {\rm Fl}_{\mathcal P}$. We fix any standard parabolic subgroup $P \supset B$ with Levi factorization $P = MN$, for a Levi subgroup $M \supset T$ and unipotent radical $N \subset U$. Here $B = TU$ is the Levi decomposition of the fixed Borel subgroup $B$. We abbreviate $K = L^+G$ and note that the intersection $K_M := K \cap M$ in $LG$ can be identified with $L^+M$. We define $K_P := K_M \cdot LN$. This is a semidirect group ind-scheme over $k$, since $K_M$ normalizes $LN$. For $\lambda \in X_*(T)$, denote the corresponding point by $x_\lambda := \lambda(t) e_{\mathcal P}\in {\rm Gr}_G(k)$. Fix $\mu \in X_*(T)^+$. Recall [@HKM Def. 3.1], in which we declare $\nu \in X_*(T)$ satisfies $\nu \geq^P \mu$ provided that - $\langle \alpha, \nu \rangle = 0$ for all $T$-roots $\alpha$ appearing in ${\rm Lie}(M)$; - $\langle \alpha, \nu + \lambda \rangle > 0$ for all $T$-roots $\alpha$ appearing in ${\rm Lie}(N)$ and for all $\lambda \in \Omega(\mu)$. Here $\Omega(\mu) = \{ \lambda \in X_*(T) \, | \, \mu - w\lambda ~\mbox{is a sum of positive coroots, for all $w \in W_0$}\}$. Also, let $X_*(T)^{+_M}$ be the cocharacters which are dominant for the roots appearing in ${\rm Lie}(B \cap M)$. **Proposition 20**. *If $\nu \geq^P \mu$ for $\mu \in X_*(T)^+$, and if $\lambda \in \Omega(\mu) \cap X_*(T)^{+_M}$, then there is an equality of $k$-subvarieties in ${\rm Gr}_G$ $$\label{HKM_eq} (t^{-\nu}Kt^\nu)x_\lambda \,\cap\, K x_\mu = K_Px_\lambda \,\cap\, K x_\mu.$$* *Proof.* The equality $(t^{-\nu}Kt^\nu)x_\lambda \,\cap\, \overline{K x_\mu} = K_Px_\lambda \,\cap\, \overline{K x_\mu}$ follows on combining [@HKM Prop. 7.1] and [@HKM Lem. 7.3]. The desired equality without the closures follows formally from this one. ◻ The left hand side of ([\[HKM_eq\]](#HKM_eq){reference-type="ref" reference="HKM_eq"}) admits a cellular paving by Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"}. Indeed, we have $$(t^{-\nu}Kt^\nu)x_\lambda \,\cap\, K x_\mu = p_{w_\bullet, L^+G}^{-1} (t^{-\nu} e_{L^+G}),$$ for $w_\bullet = (t_\mu, t_{-\nu-\lambda})$. Hence we deduce the following result. **Corollary 21**. *For $\mu, \lambda$ as above, the variety $L^+M\,LN\,x_\lambda \cap L^+G\,x_\mu$ in ${\rm Gr}_G$ admits a cellular paving. In particular, for $P = B$, the Mirkovic-Vilonen variety $LUx_\lambda \cap L^+G x_\mu$ admits a cellular paving.* Note that this applies to all pairs $(\mu, \lambda) \in X_*(T)^+ \times X_*(T)^{+_M}$: if the intersection is non-empty, then $\lambda \in \Omega(\mu)$ is automatic, by [@HKM Lem. 7.2(b)]. # Paving results over $\mathbb Z$ {#Z_sec} The goal of what follows is to extend the constructions and results above to work over $\mathbb Z$. Because there is no building attached to a group over ${\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}]$, the main challenge is to give purely group theoretic arguments for certain results which are usually proved with the aid of buildings. ## Basic constructions over $\mathbb Z$ We shall recall the basic notions attached to groups over $\mathbb Z$. One useful reference is [@RS20 $\S4$], but in places we have chosen a slightly different way to justify the foundational results (for example, we do not assume the existence of the Demazure resolutions over $\mathbb Z$ -- a result stated without proof in [@Fal03] -- and instead we construct them as a special case of the convolution morphisms over ${\mathbb Z}$). We assume $G$ is a reductive group over $\mathbb Z$, more precisely, a smooth affine group scheme over $\mathbb Z$ whose geometric fibers are connected reductive groups, and which admits a maximal torus $T$ over $\mathbb Z$, which is automatically split (see [@Co14 $\S6.4$, Ex. 5.1.4]). We fix a Borel pair over $\mathbb Z$, given by $G \supset B \supset T$ (Borel subgroups $B \supset T$ exist, by e.g. [@Co14 pf. of Thm. 5.1.13]). Following [@RS20 $\S4$], we have the usual objects: the standard apartment endowed with its Coxeter complex structure given by the affine roots, the base alcove ${\bf a}$ and other facets ${\bf f}$ therein, the Weyl group $W_0$, the Iwahori-Weyl group $W$, the affine Weyl group $W_{\rm aff}$, and the stabilizer subgroups $W_{\bf f} \subset W_{\rm aff}$. The Iwahori-Weyl group $W := N_G(T)(\mathbb Z(\!(t)\!))/T(\mathbb Z[\![t]\!])$ can be identified with the extended affine Weyl group $X_*(T) \rtimes W_0$ where using [@Co14 Prop. 5.1.6] we may identify $W_0 = N_G(T)(\mathbb Z[\![t]\!])/T(\mathbb Z[\![t]\!])$. As $X_*(T) \rtimes W_0$ remains unchanged upon base changing along $\mathbb Z \rightarrow k$ for any field $k$, it inherits a Bruhat order $\leq$ as in the classical theory over a field. Similarly, the apartment is canonically identified with the apartments attached to $(G_{{\mathbb Q} (\hspace{-0,7mm}( {t} )\hspace{-0,7mm}) }, T_{{\mathbb Q} (\hspace{-0,7mm}( {t} )\hspace{-0,7mm}) })$ or $(G_{{\mathbb F}_p (\hspace{-0,7mm}( {t} )\hspace{-0,7mm}) }, T_{{\mathbb F}_p (\hspace{-0,7mm}( {t} )\hspace{-0,7mm}) })$ for any prime number $p$. We define in the obvious way the positive loop group $L^+G_\mathbb Z$ (a pro-smooth affine group scheme over $\mathbb Z$) and the loop group $LG_\mathbb Z$ (an ind-affine group ind-scheme over $\mathbb Z$). For representability, see e.g. [@HR20 Lem. 3.2]. The following result is essentially due to Pappas and Zhu, and this precise form was checked jointly with Timo Richarz. **Lemma 22**. *Let ${\bf f}$ be any facet of the apartment corresponding to $T$ in the Bruhat-Tits building of $G({\mathbb Q} (\hspace{-0,7mm}( {t} )\hspace{-0,7mm}) )$, and let ${\mathcal G}_{\bf f_{\mathbb Q}}$ be the associated parahoric ${\mathbb Q} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}]$-group scheme with connected fibers and with generic fiber $G\otimes_{\mathbb Z}{\mathbb Q} (\hspace{-0,7mm}( {t} )\hspace{-0,7mm})$. Then there exists a unique smooth affine fiberwise connected ${\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}]$-group scheme ${\mathcal G}_{\bf f}$ of finite type extending ${\mathcal G}_{\bf f_{\mathbb Q}}$ with the following properties:\ i[)]{.upright} There is an identification of ${\mathbb Z} (\hspace{-0,7mm}( {t} )\hspace{-0,7mm})$-groups ${\mathcal G}_{\bf f} \otimes_{{\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}] }{\mathbb Z} (\hspace{-0,7mm}( {t} )\hspace{-0,7mm}) =G\otimes_{\mathbb Z}{\mathbb Z} (\hspace{-0,7mm}( {t} )\hspace{-0,7mm})$.\ ii[)]{.upright} For every prime number $p$, the group scheme ${\mathcal G}_{\bf f}\otimes_{{\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}] }{\mathbb F}_p [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}]$ is the Bruhat-Tits group scheme with connected fibers for $G\otimes_{\mathbb Z}{\mathbb F}_p (\hspace{-0,7mm}( {t} )\hspace{-0,7mm})$ associated with ${\bf f}$.* *Proof.* This is proven in [@PZ13 4.2.2]. Note that the base ring in *loc. cit.* is the polynomial ring ${\mathcal O}[t]$ where ${\mathcal O}$ is discretely valued. The same proof remains valid over the base ring ${\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}]$ using [@BT84 3.9.4]. ◻ For each ${\bf f}$, we define the "parahoric" subgroup $L^+{\mathcal G}_{\bf f} \subset LG_{\mathbb Z}$, and we often abbreviate by writing ${\mathcal P}_{\mathbb Z}:= L^+{\mathcal G}_{\bf f}$. This has the property that for each homomorphism $\mathbb Z \rightarrow k$ for $k$ a field, we have ${\mathcal P}_{{\mathbb Z}} \otimes_{{\mathbb Z}} k \cong {\mathcal P}_k$ where the latter is the object defined earlier when working over the field $k$. We define the (partial) affine flag variety $${\rm Fl}_{{\mathcal P},{\mathbb Z}} = (LG_{\mathbb Z}/ {\mathcal P}_{\mathbb Z})^{\mbox{\rm \'{e}t}},$$ the étale sheafification of the quotient presheaf on ${\rm Aff}_\mathbb Z$. This is represented by an ind-projective ind-scheme over $\mathbb Z$; see [@HR20 Cor. 3.11], where the proof is given for objects defined over $\mathcal O[t]$ for any Noetherian ring $\mathcal O$ -- a similar proof works in our setting over ${\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}]$. We denote the base point in ${\rm Fl}_{{\mathcal P},{\mathbb Z}}$ by $e_{{\mathcal P}, {\mathbb Z}}$ We have a notion of a negative parahoric loop group and a correponding open cell in ${\rm Fl}_{{\mathcal P}, {\mathbb Z}}$. We define $L^{--}G_{\mathbb Z}:= {\rm ker}(L^-G_{\mathbb Z}\rightarrow G_{\mathbb Z})$, $t^{-1} \mapsto 0$, where $L^-G_{\mathbb Z}(R) = G(R[t^{-1}])$. Following [@dHL18], we define $L^{--}{\mathcal G}_{\bf a, {\mathbb Z}} = L^{--}G_{{\mathbb Z}} \rtimes \overline{U}_{\mathbb Z}$. Then for any facet ${\bf f}$ in the closure of ${\bf a}$, we define the negative parahoric loop group $$\label{neg_parahoric_def} L^{--}{\mathcal G}_{\bf f,{\mathbb Z}} := \bigcap_{w \in W_{\bf f}} \,^w(L^{--}{\mathcal G}_{\bf a, {\mathbb Z}}),$$ the intersection being taken in $LG_{\mathbb Z}$. **Lemma 23**. *The multiplication map $L^{--}{\mathcal G}_{\bf f, {\mathbb Z}} \times L^+{\mathcal G}_{\bf f, {\mathbb Z}} ~ \rightarrow ~ LG_{\mathbb Z}$ is representable by a quasi-compact open immersion.* *Proof.* This is proved in the same way as [@HLR Lem. 3.6], which proves the analogous result when the base ring is a ring of Witt vectors ${\mathbb W}$ instead of ${\mathbb Z}$; the same argument works for our group schemes ${\mathcal G}_{\bf f, {\mathbb Z}}$ over $D_{\mathbb Z}:= {\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}]$. We omit the details. ◻ From now on, we often write ${\mathcal G}$ for ${\mathcal G}_{\bf f}$ and ${\mathcal P}_{\mathbb Z}$ for $L^+{\mathcal G}_{\bf f}$. We recall the interpretation of partial affine flag varieties in terms of suitable spaces of torsors. For any ring $R$, denote $D_R = {\rm Spec}(R [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}] )$ and $D^*_R = {\rm Spec}(R (\hspace{-0,7mm}( {t} )\hspace{-0,7mm}) )$. Recall that we define the sheaf ${\rm Gr}_{{\mathcal G}}$ on ${\rm Aff}_{\mathbb Z}$ to be the functor sending $R$ to the set ${\rm Gr}_{\mathcal G}(R)$ of isomorphism classes of pairs $({\mathcal E}, \alpha)$ where ${\mathcal E}$ is a right étale torsor for ${\mathcal G}_{D_R} = {\mathcal G}\times_{D_{\mathbb Z}} D_R$ over $D_R$, and where $\alpha \in {\mathcal E}(D^*_R)$, that is, an isomorphism of ${\mathcal G}_{D^*_R}$-torsors ${\mathcal E}_0|_{D^*_R} \overset{\sim}{\rightarrow} {\mathcal E}|_{D^*_R}$, where ${\mathcal E}_0$ is the trivial ${\mathcal G}_{D_R}$-torsor. The left action of $g \in LG(R)$ on ${\rm Gr}_{\mathcal G}(R)$ sends $({\mathcal E}, \alpha)$ to $({\mathcal E}, \alpha \circ g^{-1})$. Then ${\rm Gr}_{\mathcal G}(R) \cong {\rm Fl}_{{\mathcal P}, {\mathbb Z}}(R)$, functorially in $R$ (see e.g. [@HR20 Lem. 3.4]). **Remark 24**. For the groups $G$ over ${\mathbb Z}$ we consider, one can show using negative parahoric loop groups that the morphism $LG_{\mathbb Z}\rightarrow {\rm Fl}_{{\mathcal P}, {\mathbb Z}}$ has sections locally in the Zariski-topology, and hence for any semi-local ring $R$ we have ${\rm Fl}_{{\mathcal P}, {\mathbb Z}}(R) = LG_{\mathbb Z}(R)/{\mathcal P}_{\mathbb Z}(R)$. This can be seen by reducing to the case of fields, as in [@RS20 $\S4.3$]. One can also deduce it from a recent result of $\check{\rm C}$esnavi$\check{\rm c}$ius [@Ces22 Thm. 1.7] that the affine Grassmannian ${\rm Gr}_{G, {\mathbb Z}}$ agrees with the Zariski sheafification of the presheaf quotient $LG_{{\mathbb Z}}/L^+G_{\mathbb Z}$. To use this to prove the corresponding result for a general parahoric ${\mathcal P}_{\mathbb Z}$, one first deduces the result for ${\mathcal P}_{\mathbb Z}= {\mathcal B}_{\mathbb Z}$, using the lifting for ${\mathcal P}_{\mathbb Z}= L^+G_{\mathbb Z}$ and the fact that the fiber of ${\rm Fl}_{{\mathcal B}, {\mathbb Z}} \rightarrow {\rm Gr}_{G, {\mathbb Z}}$ over the base point is $(G/B)_{\mathbb Z}$ and $G \rightarrow (G/B)_{\mathbb Z}$ is Zariski locally trivial. Then finally one uses the topological surjectivity of ${\rm Fl}_{{\mathcal B}, {\mathbb Z}} \rightarrow {\rm Fl}_{{\mathcal P}, {\mathbb Z}}$ to prove that a cover given by translates of the big cell in the source maps to a cover of translates of the big cell in the target. In fact one can use translates $wL^{--}{\mathcal G}_{\bf a} e_{{\mathcal B}, {\mathbb Z}}$ for $w \in W$ to cover ${\rm Fl}_{{\mathcal B}, {\mathbb Z}}$, thanks to the Birkhoff decomposition of $LG$ over fields (see [@Fal03 Lem. 4]). I am grateful to Thibaud van den Hove for a clarifying discussion about this remark, which we shall not need in the rest of this article. **Lemma 25**. *Fix a ring $R$ and $({\mathcal E}, \alpha) \in {\rm Gr}_{{\mathcal G}}(R)$. Then the presheaf ${\rm Gr}_{{\mathcal G}, {\mathcal E}, \alpha}$ sending ${\rm Spec}(R') \rightarrow {\rm Spec}(R)$ to the set of isomorphism classes of pairs $({\mathcal E}', \alpha')$ consisting of a ${\mathcal G}_{D_{R'}}$-torsor ${\mathcal E}' \rightarrow D_{R'}$ and an isomorphism of ${\mathcal G}_{D^*_{R'}}$-torsors $\alpha' : {\mathcal E}_{D^*_{R'}} \overset{\sim}{\rightarrow} {\mathcal E}'_{ D^*_{R'}}$ is representable by an ind-projective ind-flat ind-scheme over $R$.* *Proof.* If we fix a representative $({\mathcal E}, \alpha)$ within its isomorphism class, then the map $$({\mathcal E}', \alpha') \mapsto ({\mathcal E}', \alpha' \circ \alpha)$$ is a well-defined isomorphism of presheaves ${\rm Gr}_{{\mathcal G}, {\mathcal E}, \alpha} \overset{\sim}{\rightarrow} {\rm Gr}_{{\mathcal G}} \times {\rm Spec}(R)$. Now recall that ${\rm Gr}_{{\mathcal G}}$ is ind-flat over $\mathbb Z$ by adapting the proof of [@HLR Prop. 8.9], or by reducing to the case ${\mathcal P}_{\mathbb Z}= L^+G_{{\mathbb Z}}$ and then invoking [@HLR Prop, 8.8]. ◻ ## Ingredients needed for paving over $\mathbb Z$ ### Iwahori decompositions of ${\mathcal B}_{\mathbb Z}$ and $\mathcal U_{\mathbb Z}$ Our choice of base Iwahori subgroup ${\mathcal B}_{\mathbb Z}$ is compatible with our choice of Borel subgroup $B = TU$ over ${\mathbb Z}$ in the following sense: for any algebra $R$, we have $${\mathcal B}_{\mathbb Z}(R) = \{ g \in L^+G_{\mathbb Z}(R) \, | \, \bar{g} \in B(R) \}$$ where $\bar{g}$ is the image of $g$ under the canonical projection $L^+G_{\mathbb Z}(R)\rightarrow G(R)$. We define the pro-unipotent radical ${\mathcal U}_{\mathbb Z}\subset {\mathcal B}_{\mathbb Z}$ by requiring ${\mathcal U}_{\mathbb Z}(R)$ to be the preimage of $U(R)$ under the projection $g \mapsto \bar{g}$. Let ${\mathcal T}_{\mathbb Z}$ denote the group scheme ${\mathcal T}_{\mathbb Z}= L^+T_{\mathbb Z}$. Let $\overline{B} = T\overline{U}$ be the Borel subgroup such that $B \cap \overline{B} = T$. For any integer $m \geq 1$, let $L^{(m)}G_{\mathbb Z}(R)$ denote the kernel of the natural homomorphism $L^+G_{\mathbb Z}(R) \rightarrow G(R/t^mR)$. Write ${\mathcal T}^{(1)}_{\mathbb Z}:= L^{(1)}T_{\mathbb Z}$. **Proposition 26**. *The group schemes ${\mathcal B}_{\mathbb Z}$ and ${\mathcal U}_{\mathbb Z}$ possess Iwahori decompositions with respect to $B = TU$, that is, there are unique factorizations of functors $$\begin{aligned} \label{Iwahori_decomp_Z_eq} {\mathcal B}_{\mathbb Z}&= ({\mathcal B}_{\mathbb Z}\cap L\overline{U}_{\mathbb Z}) \cdot {\mathcal T}_{\mathbb Z}\cdot ({\mathcal B}_{\mathbb Z}\cap LU_{\mathbb Z}) \\ {\mathcal U}_{\mathbb Z}&= ({\mathcal U}_{\mathbb Z}\cap L\overline{U}_{\mathbb Z}) \cdot {\mathcal T}^{(1)}_{\mathbb Z}\cdot ({\mathcal B}_{\mathbb Z}\cap LU_{\mathbb Z}).\end{aligned}$$* *Proof.* First we note that the uniqueness in the decomposition follows from the uniqueness of the decomposition in the big cell in $\overline{U} \cdot T \cdot U$ in $G$. We shall prove only the first decomposition (the second is completely similar). Consider $g \in {\mathcal B}_{\mathbb Z}(R)$, with reduction modulo $t$ given by $\bar{g} = \bar{b}$ for some $b \in B(R) \subset L^+B_{\mathbb Z}(R)$. Then $g^{(1)} := g b^{-1} \in L^{(1)}G_{\mathbb Z}(R)$, and it suffices to show this element lies in $$\label{level_1_decomp} ({\mathcal B}_{\mathbb Z}\cap L^{(1)}\overline{U}_{\mathbb Z}) \cdot {\mathcal T}^{(1)}_{\mathbb Z}\cdot ({\mathcal B}_{\mathbb Z}\cap L^{(1)}U_{\mathbb Z}).$$ The filtration $\cdots \subset L^{(m+1)}G_{\mathbb Z}\subset L^{(m)}G_{\mathbb Z}\subset \cdots \subset L^+G_{\mathbb Z}$ has abelian quotients isomorphic to ${\rm Lie}(G)_{\mathbb Z}= {\rm Lie}(\overline{U})_{\mathbb Z}\oplus {\rm Lie}(T)_{\mathbb Z}\oplus {\rm Lie}(U)_{\mathbb Z}$. We claim that we can write $$g^{(1)} = \underset{m \rightarrow \infty}{\rm lim} \bar{u}_m \cdot t_m \cdot u_m$$ with $\bar{u}_m, t_m, u_m$ lying in the $R$-points of the appropriate factors of ([\[level_1\_decomp\]](#level_1_decomp){reference-type="ref" reference="level_1_decomp"}), and such that the limit converges in the $t$-adic topology. Indeed, decomposing the image modulo $t^2$ of $g^{(1)}$ in terms of the Lie algebra and lifting, we can write $$g^{(1)} ~ = ~ \bar{u}^{(1,2)} \cdot g^{(2)} \cdot t^{(1,2)} \cdot u^{(1,2)}$$ where $\bar{u}^{(1,2)} \in L^{(1)}\overline{U}_{\mathbb Z}$, $t^{(1,2)} \in L^{(1)}T_{\mathbb Z}$, and $u^{(1,2)} \in L^{(1)}U_{\mathbb Z}$ and where $g^{(2)} \in L^{(2)}G_{\mathbb Z}$. Here we have used that $\bar{u}^{(1,2)}$ normalizes $L^{(2)}G_{\mathbb Z}$. We then repeat this process with $g^{(2)}$, and get an expression $$g^{(1)} = (\bar{u}^{(1,2)}\bar{u}^{(2,3)}) \cdot g^{(3)} \cdot (t^{(1,2)}t^{(2,3)}) \cdot (u^{(2,3)} u^{(1,2)})$$ where $g^{(3)} \in L^{(3)}G_{\mathbb Z}$ and where $?^{(2,3)}$ refers to a component of $g^{(2)}$ lying in an appropriate $L^{(2)}?$ group, with $g^{(3)}$ viewed as an "error term". Here we have used again the normality of $L^{(m)}G_{\mathbb Z}$ in $L^+G_{\mathbb Z}$, the commutativity of $L^+T_{\mathbb Z}$, and the fact that $L^+T_{\mathbb Z}$ normalizes each $L^{(m)}U_{\mathbb Z}$. Continuing this, we define the sequences $\bar{u}^{(m-1,m)}$, $t^{(m-1,m)}$, $u^{(m-1, m)}$, and $g^{(m)}$ and then set $$\begin{aligned} \bar{u}_m &= \bar{u}^{(1,2)} \cdots \bar{u}^{(m-1,m)} \\ t_m &= t^{(1,2)} \cdots t^{(m-1,m)} \\ u_m &= u^{(m-1,m)} \cdots u^{(1,2)}.\end{aligned}$$ In the $t$-adic topology, these products converge and the terms $g^{(m)}$ approach the identity element $e \in G$. Hence this proves the claim, and thus the Proposition. ◻ ### Proposition [Proposition 5](#dHL.3.7.4){reference-type="ref" reference="dHL.3.7.4"} over ${\mathbb Z}$ {#proposition-dhl.3.7.4-over-mathbb-z} **Proposition 27**. *The analogue over $\mathbb Z$ of Proposition [Proposition 5](#dHL.3.7.4){reference-type="ref" reference="dHL.3.7.4"} holds.* *Proof.* There are only finitely many affine roots $a$ such that $U_{a,{\mathbb Z}}$ is contained in ${\mathcal U}_{\mathbb Z}\cap \,^v\overline{{\mathcal U}}_{{\mathcal P}, {\mathbb Z}}$, namely the finitely many $a$ with $a > 0$ and $v^{-1}a \overset{\bf f}{<} 0$. By the Iwahori decomposition Proposition [Proposition 26](#Iwahori_decomp_Z){reference-type="ref" reference="Iwahori_decomp_Z"} and the root group filtrations in $U_{\mathbb Z}$ and $\overline{U}_{\mathbb Z}$, we easily see that there exist finitely many positive affine roots $a_1, \dots, a_N$ such that $$\label{product} {\mathcal U}_{\mathbb Z}= U_{a_1, {\mathbb Z}} \cdots U_{a_N, {\mathbb Z}} \,\, ({\mathcal U}_{\mathbb Z}\cap\, ^v{\mathcal P}_{\mathbb Z}).$$ In what follows, we suppress the subscript $\mathbb Z$. Give a total order $\preceq$ to the set of positive affine roots $a_i$ in this list, by letting $a \prec b$ if and only if $a(x_0) < b(x_0)$ for a suitably general point $x_0 \in {\bf a}$. Let $r_1 \prec r_2 \prec \cdots \prec r_M$ be the totally order subset of the $a_i$'s with the property that $v^{-1}r_i \overset{\bf f}{<} 0$. The root group $U_{r_1}$ appears finitely many times in ([\[product\]](#product){reference-type="ref" reference="product"}). Starting from the left, we commute the first $U_{r_1}$ to the left past any preceding $U_b$'s. By the commutator relations (e.g. [@dHL18 (3.6)]), in moving all the $U_{r_1}$ groups all the way to the left, we introduce finitely many additional affine root groups $U_c$ with $r_1 \prec c$. Then we consider the part of the product which now involves only root groups in of the form $U_{r_2}, \dots, U_{r_M}$ and certain $U_c$ with $v^{-1}c \overset{\bf f}{\geq} 0$. Then we repeat the above process with $r_2$ in place of $r_1$. Continuing, we eventually move all the $U_r$ factors with $v^{-1}r \overset{\bf f}{<} 0$ all the way to the left. We have proved that $$\label{first_decomp} \mathcal U = \prod_{r} U_r \, ({\mathcal U}\cap \,^v{\mathcal P})$$ where $r$ ranges over the affine roots with $r > 0$ and $v^{-1}r \overset{\bf f}{<} 0$. We claim that the obvious inclusion $\prod_r U_r \subset {\mathcal U}\cap \, ^v\overline{{\mathcal U}}_{{\mathcal P}}$ is an equality, and the resulting product is a decomposition. Both statements follow easily using the theory of the big cell, Lemma [Lemma 23](#big_cell_Z){reference-type="ref" reference="big_cell_Z"}. ◻ **Corollary 28**. *The analogues over ${\mathbb Z}$ of Propositions [Proposition 8](#BP_comp){reference-type="ref" reference="BP_comp"} and [Proposition 10](#strat_triv){reference-type="ref" reference="strat_triv"} hold.* ### Schubert cells and Schubert schemes over ${\mathbb Z}$ Fix $w \in W$ and fix a lift $\dot{w} \in N_GT({\mathbb Z} [\hspace{-0,5mm}[ {t} ]\hspace{-0,5mm}] )$ of $w$. We usually suppress the dot from now on, since no construction depends on this choice. The group ${\mathcal P}_{\mathbb Z}$ acts on the left on ${\rm Fl}_{{\mathcal P}, {\mathbb Z}}$, we define the *Schubert scheme* $X_{{\mathcal P}, {\mathbb Z}}(w) \subset {\rm Fl}_{{\mathcal P}, {\mathbb Z}}$ to be the scheme-theoretic image of the morphism $${\mathcal P}_{\mathbb Z}\rightarrow {\rm Fl}_{{\mathcal P}, {\mathbb Z}}, \hspace{.3in} p \mapsto p \dot{w} e_{{\mathcal P}, {\mathbb Z}}.$$ Similarly we can define $X_{{\mathcal Q}{\mathcal P}, {\mathbb Z}}(w)$ for any parahoric subgroup ${\mathcal Q}_{\mathbb Z}$, in particular we have $X_{{\mathcal B}{\mathcal P}, {\mathbb Z}}$. We define $Y_{{\mathcal P}, {\mathbb Z}}(w) \subset {\rm Fl}_{{\mathcal P}, {\mathbb Z}}$ to be the étale sheaf-theoretic image of the morphism of sheaves ${\mathcal P}_{{\mathbb Z}} \rightarrow {\rm Fl}_{{\mathcal P}, {\mathbb Z}}$, $p \mapsto p \dot{w}e_{\mathcal P}$, and as before we define similarly $Y_{{\mathcal Q}{\mathcal P}, {\mathbb Z}}$ for any parahoric subgroup ${\mathcal Q}_{\mathbb Z}\subset LG_{\mathbb Z}$. **Lemma 29**. *Let ${\mathcal P}_{\mathbb Z}\subset LG$ be the parahoric subgroup fixed above [(]{.upright}similar statements apply to any ${\mathcal Q}$-orbits in ${\rm Fl}_{{\mathcal P}, {\mathbb Z}}$[)]{.upright}.* 1. *The scheme $X_{{\mathcal P}, {\mathbb Z}}(w)$ is an integral scheme which is projective and faithfully flat over ${\rm Spec}({\mathbb Z})$, and $X_{{\mathcal P}, {\mathbb Z}}(w) \otimes {\mathbb Q}= X_{{\mathcal P}, {\mathbb Q}}(w)$.* 2. *The morphism $Y_{{\mathcal P},{\mathbb Z}}(w) \rightarrow {\rm Fl}_{{\mathcal P}, {\mathbb Z}}$ of étale sheaves factors canonically as $$Y_{{\mathcal P}, {\mathbb Z}}(w) \rightarrow X_{{\mathcal P}, {\mathbb Z}}(w) \rightarrow {\rm Fl}_{{\mathcal P}, {\mathbb Z}},$$ and the first morphism is represented by a quasi-compact open immersion of schemes.* 3. *The scheme $Y_{{\mathcal P}, {\mathbb Z}}(w)$ is smooth over ${\rm Spec}({\mathbb Z})$, and its formation commutes with base change along an arbitrary homomorphism $\mathbb Z \rightarrow R$.* *Proof.* The projectivity in (a) is proved in [@RS20 Def 4.3.4, ff]. Part (b) can be proved by adapting the argument of [@Ri16b Cor. 3.14]. Part (c) holds since $Y_{{\mathcal P}, {\mathbb Z}}(w)$ is the orbit under a smooth group scheme over $\mathbb Z$. Hence $Y_{{\mathcal P}, {\mathbb Z}}(w) \otimes {\mathbb Q}= Y_{{\mathcal P}, {\mathbb Q}}(w)$. The formation of scheme-theoretic image of a quasi-compact morphism commutes with flat base change (see [@StaPro Lem. 29.25.16]). So the generic fiber of $X_{{\mathcal P}, {\mathbb Z}}(w)$ is the schematic-closure of $Y_{{\mathcal P}, {\mathbb Q}}(w)$ in ${\rm Fl}_{{\mathcal P}, {\mathbb Q}}$, that is, $X_{{\mathcal P}, {\mathbb Z}}(w) \otimes {\mathbb Q}= X_{{\mathcal P}, {\mathbb Q}}(w)$. Now the flat closure of the latter in $X_{{\mathcal P}, {\mathbb Z}}$ contains the scheme-theoretic closure of $Y_{{\mathcal P}, {\mathbb Z}}(w)$, which is all of $X_{{\mathcal P}, {\mathbb Z}}(w)$. This shows that the latter is faithfully flat over $\mathbb Z$. Clearly $X_{{\mathcal P}, {\mathbb Z}}(w)$ is irreducible since it is the scheme-theoretic image of a morphism with irreducible source. Moreover $X_{{\mathcal P}, {\mathbb Z}}(w)$ is reduced, since is it the flat-closure of a $\mathbb Q$-variety. ◻ ### Reduction to neutral element of $\Omega$ In the theory over fields $k$, it is easy to see that $\tau \in \Omega$ has the property that $\tau$ normalizes the standard Iwahori subgroup ${\mathcal B}_k \subset LG_k$ corresponding to the base alcove ${\bf a}$. We need to know that this remains true over ${\mathbb Z}$. **Lemma 30**. *If $\tau \in \Omega$, then $^\tau{\mathcal B}_{\mathbb Z}= {\mathcal B}_{\mathbb Z}$ as subgroups of $LG_{\mathbb Z}$.* *Proof.* The identification follows by the uniqueness characterization of the group scheme ${\mathcal G}_{\bf a, {\mathbb Z}}$ in Lemma [Lemma 22](#PZ_ext_lem){reference-type="ref" reference="PZ_ext_lem"} and the fact that it holds after base change to every field $k$. ◻ ### Twisted products over ${\mathbb Z}$ As above we fix ${\mathcal P}_{\mathbb Z}= L^+{\mathcal G}_{\bf f, {\mathbb Z}}$. Again abbreviate ${\mathcal G}:= {\mathcal G}_{\bf f, {\mathbb Z}}$. Fix $r \in {\mathbb N}$ and consider the right action of ${\mathcal P}_{\mathbb Z}^r$ on $LG_{\mathbb Z}^r$ given by the same formula as ([\[r-fold_action\]](#r-fold_action){reference-type="ref" reference="r-fold_action"}). **Definition 31**. We define the $r$-fold twisted product $$\widetilde{\rm Gr}_{{\mathcal G}} := LG_{\mathbb Z}\times^{{\mathcal P}_{\mathbb Z}} LG_{\mathbb Z}\times^{{\mathcal P}_{\mathbb Z}} \cdots \times^{{\mathcal P}_{\mathbb Z}} LG_{\mathbb Z}/{\mathcal P}_{\mathbb Z}=: {\rm Gr}_{{\mathcal G}} \tilde{\times} \cdots \tilde{\times} {\rm Gr}_{{\mathcal G}}$$ to be the étale quotient sheaf for the presheaf $(LG_{\mathbb Z})^r/({\mathcal P}_{\mathbb Z})^r$ defined above. It is clear from the fact that every ${\mathcal G}$-bundle over $D_R$ is trivializable over $D_{R'}$ for some étale ring extension $R \rightarrow R'$, that we can identify $\widetilde{\rm Gr}_{{\mathcal G}}(R)$ with the set of equivalence classes of tuples $$({\mathcal E}_\bullet, \alpha_\bullet) = ({\mathcal E}_1, \dots, {\mathcal E}_r; \alpha_1, \dots, \alpha_r)$$ such that each ${\mathcal E}_i$ is a $\mathcal G_{D_R}$-torsor over $D_R$, and the $\alpha_i : {\mathcal E}_{i-1}|_{D^*_R} \overset{\sim}{\rightarrow} {\mathcal E}_i|_{D^*_R}$ are isomorphisms of $\mathcal G_{D^*_R}$-torsors over $D^*_R$ for all $i = 1, \dots, r$ (with the convention that ${\mathcal E}_0$ is the trivial torsor). **Lemma 32**. *The sheaf $LG_{\mathbb Z}\times^{{\mathcal P}_{\mathbb Z}} LG_{\mathbb Z}\times^{{\mathcal P}_{\mathbb Z}} \cdots \times^{{\mathcal P}_{\mathbb Z}} LG_{\mathbb Z}/{\mathcal P}_{\mathbb Z}$ is represented by an ind-proper ind-scheme which is faithfully flat over ${\mathbb Z}$.* *Proof.* We proceed by induction on $r$. The case $r =1$ is clear: the ind-flatness of ${\rm Gr}_{{\mathcal G}} \rightarrow {\rm Spec}({\mathbb Z})$ is proved by an easy reduction to the case ${\mathcal P}_{\mathbb Z}= L^+G_{\mathbb Z}$, which is then handled by [@HLR Prop. 8.8]. Now assume $r > 1$ and that the result holds for $r-1$-fold quotients. The projection onto the first factor gives a morphism $$p: LG_{\mathbb Z}\times^{{\mathcal P}_{\mathbb Z}} LG_{\mathbb Z}\times^{{\mathcal P}_{\mathbb Z}} \cdots \times^{{\mathcal P}_{\mathbb Z}} LG_{\mathbb Z}/{\mathcal P}_{\mathbb Z}\rightarrow LG_{\mathbb Z}/{\mathcal P}_{\mathbb Z}.$$ Now the induction hypothesis and the proof of Lemma [Lemma 25](#new_base_pt){reference-type="ref" reference="new_base_pt"} shows that this morphism is representable by an ind-proper ind-flat ind-scheme, and hence the total space is represented by an ind-scheme. Locally in the étale topology on the target, $p$ is locally trivial with flat fiber, hence is flat. It follows that the source of $p$ is flat over ${\mathbb Z}$. It remains to prove the source of $p$ is proper over ${\mathbb Z}$. If we knew that $p$ were Zariski-locally trivial, then by induction we could conclude that $p$ would be ind-proper and hence its source would be ind-proper over ${\mathbb Z}$. However, we do not know that there are enough translates of the open subset $L^{--}{\mathcal G}e_{{\mathcal P}_{\mathbb Z}}$ to cover the topological space of ${\rm Gr}_{\mathcal G}$, and so Zariski-local triviality is not clear. We know that ind-locally in the étale topology on the target, $LG \rightarrow {\rm Gr}_{\mathcal G}$ has sections, and hence after passing to an étale cover $p$ becomes Zariski-locally trivial with ind-proper over ${\mathbb Z}$ fibers, by using translates of the big cell (see Lemma [Lemma 23](#big_cell_Z){reference-type="ref" reference="big_cell_Z"}). Since properness descends along étale covers, we conclude that $p$ is ind-proper, as desired. Note that this argument becomes even simpler if we use the Zariski local triviality result of $\check{\rm C}$esnavi$\check{\rm c}$ius (see Remark [Remark 24](#Ces_rem){reference-type="ref" reference="Ces_rem"}) but we do not need this more sophisticated result. ◻ Let $w \in W$. Denoting the quotient morphism by $q: LG_{\mathbb Z}\rightarrow {\rm Fl}_{{\mathcal P}, {\mathbb Z}}$, note that ${\mathcal P}_{\mathbb Z}w {\mathcal P}_{\mathbb Z}= q^{-1}(Y_{{\mathcal P}, {\mathbb Z}}(w))$ (an equality of étale subsheaves of $LG_{\mathbb Z}$), where by definition ${\mathcal P}_{\mathbb Z}w {\mathcal P}_{\mathbb Z}$ denotes the étale sheaf quotient ${\mathcal P}_{\mathbb Z}\times^{w, {\mathcal P}_{\mathbb Z}} {\mathcal P}_{\mathbb Z}$ of ${\mathcal P}_{\mathbb Z}\times {\mathcal P}_{\mathbb Z}$ by the right action of ${\mathcal P}_{\mathbb Z}\cap w {\mathcal P}_{\mathbb Z}w^{-1}$ given by $(p,p')\cdot \delta = (p\delta, w^{-1}\delta^{-1}w p)$. In this vein, we *define* $\widetilde{{\mathcal P}_{\mathbb Z}w {\mathcal P}_{\mathbb Z}} := q^{-1}(X_{{\mathcal P}, {\mathbb Z}}(w))$. The quotient is locally trivial in the Zariski topology thanks to the isomorphism ${\mathcal P}_{\mathbb Z}\cap \,^w\overline{{\mathcal U}}_{{\mathcal P},{\mathbb Z}} \overset{\sim}{\rightarrow} {\mathcal P}_{\mathbb Z}/ ({\mathcal P}_{\mathbb Z}\cap\,^w{\mathcal P}_{\mathbb Z})$ (see Proposition [Proposition 27](#dHL.3.7.4_Z){reference-type="ref" reference="dHL.3.7.4_Z"} ---FIX!!!!!), and hence for fields $k$ we have the expected relation $({\mathcal P}_{\mathbb Z}w {\mathcal P}_{\mathbb Z})(k) = {\mathcal P}_{\mathbb Z}(k) w {\mathcal P}_{\mathbb Z}(k)$. **Definition 33**. Let $w_\bullet = (w_1, w_2, \dots, w_r) \in W^r$. We define $$\begin{aligned} Y_{{\mathcal P}, {\mathbb Z}}(w_\bullet) &:= {\mathcal P}_{{\mathbb Z}} w_1 {\mathcal P}_{{\mathbb Z}} \times^{{\mathcal P}_{\mathbb Z}} {\mathcal P}_{{\mathbb Z}} w_2 {\mathcal P}_{\mathbb Z}\times^{{\mathcal P}_{\mathbb Z}} \cdots \times^{{\mathcal P}_{\mathbb Z}} {\mathcal P}_{\mathbb Z}w_r {\mathcal P}_{{\mathbb Z}} /{\mathcal P}_{\mathbb Z}= Y_{{\mathcal P}, {\mathbb Z}}(w_1) \tilde{\times} Y_{{\mathcal P}, {\mathbb Z}}(w_2) \tilde{\times} \cdots \tilde{\times} Y_{\mathcal P}(w_r) \\ X_{{\mathcal P}, {\mathbb Z}}(w_\bullet) &:= \widetilde{{\mathcal P}_{{\mathbb Z}} w_1 {\mathcal P}_{{\mathbb Z}}} \times^{{\mathcal P}_{\mathbb Z}} \widetilde{{\mathcal P}_{{\mathbb Z}} w_2 {\mathcal P}_{\mathbb Z}} \times^{{\mathcal P}_{\mathbb Z}} \cdots \times^{{\mathcal P}_{\mathbb Z}} \widetilde{{\mathcal P}_{\mathbb Z}w_r {\mathcal P}_{{\mathbb Z}}} /{\mathcal P}_{\mathbb Z}= X_{{\mathcal P}, {\mathbb Z}}(w_1) \tilde{\times} X_{{\mathcal P}, {\mathbb Z}}(w_2) \tilde{\times} \cdots \tilde{\times} X_{\mathcal P}(w_r) \end{aligned}$$ to be the étale quotient sheaves as in Definition [Definition 31](#twisted_Gr){reference-type="ref" reference="twisted_Gr"}. **Lemma 34**. *The sheaves $X_{{\mathcal P},{\mathbb Z}}(w_\bullet)$ and $Y_{{\mathcal P}, {\mathbb Z}}(w_\bullet)$ are represented by integral schemes which are finite type and flat over ${\mathbb Z}$. Moreover, $X_{{\mathcal P}, {\mathbb Z}}(w_\bullet)$ is proper over ${\mathbb Z}$.* *Proof.* The proof goes by induction on $r$, in the same manner as Lemma [Lemma 32](#conv_repble){reference-type="ref" reference="conv_repble"}. ◻ ### Demazure morphisms and closure relations over ${\mathbb Z}$ We need to construct the Demaure resolutions over $\mathbb Z$. This is stated without proof in [@Fal03] and is implicit in some literature (e.g. [@PR08; @RS20]) but we think some extra discussion is needed. For $s \in S_{\rm aff}$, let ${\mathcal G}_{s, {\mathbb Z}} : = {\mathcal G}_{{\bf f}, {\mathbb Z}}$, where ${\bf f}$ is the facet fixed by $s$. Let ${\mathcal P}_{s, {\mathbb Z}} = L^+{\mathcal G}_{s, {\mathbb Z}}$. We have ${\mathcal P}_{s, {\mathbb Z}} = {\mathcal B}_{{\mathbb Z}} \cup {\mathcal B}_{{\mathbb Z}} s {\mathcal B}_{\mathbb Z}$ as schemes (to show this we use Lemma [Lemma 29](#rep_lem_Z){reference-type="ref" reference="rep_lem_Z"}(c), and the fact that the inclusion ${\mathcal B}_{{\mathbb Z}}(R) \cup {\mathcal B}_{{\mathbb Z}} s {\mathcal B}_{\mathbb Z}(R) \hookrightarrow {\mathcal P}_{s,{\mathbb Z}}(R)$ is surjective when $R$ is any field, but we warn that this equality fails for general $R$, in particular for $R = {\mathbb Z}$). We have an identification ${\mathbb P}^1_{\mathbb Z}= {\mathcal P}_{s, {\mathbb Z}}/{\mathcal B}_{\mathbb Z}$. Furthermore, the foregoing shows we have an open immersion $\mathbb A^1_{\mathbb Z}= {\mathcal B}_{\mathbb Z}s {\mathcal B}_{\mathbb Z}/{\mathcal B}_{\mathbb Z}\hookrightarrow {\mathbb P}^1_{\mathbb Z}$ with closed complement $\mathbb A^0_{\mathbb Z}= {\mathcal B}_{\mathbb Z}/{\mathcal B}_{{\mathbb Z}} \hookrightarrow {\mathbb P}^1_{{\mathbb Z}}$. The BN-pair relations hold: **Lemma 35**. *For any $w \in W$ and $s \in S_{\rm af}$, we have equalities of sub-ind-schemes in $LG_{\mathbb Z}$ $${\mathcal B}_{\mathbb Z}w {\mathcal B}_{\mathbb Z}s {\mathcal B}_{\mathbb Z}= \begin{cases} {\mathcal B}_{\mathbb Z}ws {\mathcal B}_{\mathbb Z}, \hspace{.65in} \mbox{if $w < ws$} \\ {\mathcal B}_{\mathbb Z}w {\mathcal B}_{\mathbb Z}\cup {\mathcal B}_{\mathbb Z}ws {\mathcal B}_{\mathbb Z}, \hspace{.1in} \mbox{if $ws < w$.} \end{cases}$$* *Proof.* Both cases are proved by induction on $\ell(w)$. The first case follows from the case of fields and Lemma [Lemma 29](#rep_lem_Z){reference-type="ref" reference="rep_lem_Z"}(c). For the second case, it is enough to prove the result for $w =s$. But ${\mathcal B}_{\mathbb Z}s {\mathcal B}_{\mathbb Z}s {\mathcal B}_{\mathbb Z}= {\mathcal B}_{\mathbb Z}s {\mathcal B}_{\mathbb Z}\cup {\mathcal B}_{\mathbb Z}= {\mathcal P}_{s, {\mathbb Z}}$ follows because ${\mathcal P}_{s, {\mathbb Z}}$ is a group subscheme of $LG_{\mathbb Z}$ and $s{\mathcal B}_{\mathbb Z}s \not\subset {\mathcal B}_{\mathbb Z}$. ◻ Let $w = s_1 \dots s_r$ be a reduced word in $W$. Consider the Demazure morphism given by projecting to the final coordinate $$m_{s_\bullet, {\mathbb Z}}: D(s_\bullet)_{\mathbb Z}:= {\mathcal P}_{s_1, {\mathbb Z}} \times^{{\mathcal B}_{\mathbb Z}} {\mathcal P}_{s_2, {\mathbb Z}} \times^{{\mathcal B}_{\mathbb Z}} \cdots \times^{{\mathcal B}_{\mathbb Z}} {\mathcal P}_{s_r, {\mathbb Z}}/ {\mathcal B}_{\mathbb Z}\rightarrow X_{{\mathcal B}, {\mathbb Z}}(w).$$ The image lies in $X_{{\mathcal B}, {\mathbb Z}}(w)$ by flatness and properness, and by the fact that this holds over ${\mathbb Q}$. By the BN-pair relations, it gives an isomorphism over $Y_{{\mathcal B}, {\mathbb Z}}(w)$. Furthermore, it implies the closure relations $$\label{cl_rel_Z} X_{{\mathcal P}, {\mathbb Z}}(w) = \coprod_{v} Y_{{\mathcal P}, {\mathbb Z}}(v)$$ where $v \in W_{\mathcal P}\backslash W/W_{\mathcal P}$ is such that $v \leq w$ in the Bruhat order on $W_{\mathcal P}\backslash W/W_{\mathcal P}$. In particular, we see $\widetilde{{\mathcal P}_{\mathbb Z}w {\mathcal P}_{\mathbb Z}} = \coprod_v {\mathcal P}_{\mathbb Z}v {\mathcal P}_{\mathbb Z}$. Here and in ([\[cl_rel_Z\]](#cl_rel_Z){reference-type="ref" reference="cl_rel_Z"}) the union indicates a union of locally closed subschemes, and every subscheme appearIng is reduced by construction. With the existence of Demazure resolutions over ${\mathbb Z}$ in hand, one can prove the following result by copying the argument of [@HLR Prop. 3.4] (Demazure resolutions over ${\mathbb Z}$ are used to prove that Schubert varieties attached to simply-connected groups over a field $k$ are normal, following the argument in [@PR08 $\S9$]). **Corollary 36**. *For any field $k$, $(X_{{\mathcal P}, {\mathbb Z}}(w) \otimes_{\mathbb Z}k)_{\rm red} = X_{{\mathcal P}, k}(w)$. Further, $X_{{\mathcal P}, {\mathbb Z}}(w) \otimes_{{\mathbb Z}} k$ is reduced if and only if $X_{{\mathcal P}, k}(w)$ is normal.* ### Convolution morphisms over ${\mathbb Z}$ Given the BN-pair relations involving subschemes of $LG_{\mathbb Z}$, we have the following. **Lemma 37**. *For any $w_\bullet = (w_1, w_2, \dots, w_r) \in W^r$ with Demazure product $$w_* := \,^{\bf f}w^{\bf f}_1 * \,^{\bf f}w^{\bf f}_2 * \cdots *\,^{\bf f}w^{\bf f}_r,$$ we have the convolution morphisms and uncompactified convolution morphisms over ${\mathbb Z}$: $$\begin{aligned} m_{w_\bullet, {\mathcal P}_{\mathbb Z}}: X_{{\mathcal P}, {\mathbb Z}}(w_\bullet) &\rightarrow X_{{\mathcal P}, {\mathbb Z}}(w_*) \\ p_{w_\bullet, {\mathcal P}_{\mathbb Z}}: Y_{{\mathcal P}, {\mathbb Z}}(w_\bullet) &\rightarrow X_{{\mathcal P}, {\mathbb Z}}(w_*).\end{aligned}$$* ## Main results over $\mathbb Z$ The following theorem gives the ${\mathbb Z}$-versions of Corollaries [Corollary 2](#corB){reference-type="ref" reference="corB"} and [Corollary 21](#MN_cor){reference-type="ref" reference="MN_cor"}. **Theorem 38**. *In the notation above, for any $v \in W$ the fiber $m_{w_\bullet, {\mathcal P}_{\mathbb Z}}^{-1}(v\,e_{{\mathcal P}_{\mathbb Z}})$ has a cellular paving over ${\mathbb Z}$, that is, it is paved by finite products of ${\mathbb A}^1_{\mathbb Z}$ and ${\mathbb A}^1_{\mathbb Z}- {\mathbb A}^0_{\mathbb Z}$. Further, for every standard parabolic subgroup $P_{\mathbb Z}= M_{\mathbb Z}N_{\mathbb Z}\subset G_{\mathbb Z}$, and every pair of cocharacters $(\mu, \lambda) \in X_*(T)^+ \times X_*(T)^{+_M}$, the scheme $L^+M_{\mathbb Z}\,LN_{\mathbb Z}\,x_\lambda \cap L^+G_{\mathbb Z}\,x_\mu$ in ${\rm Gr}_{G_{\mathbb Z}}$ has a cellular paving over ${\mathbb Z}$.* Note that the second statement gives an alternate proof of a recent result of Cass-van den Hove-Scholbach, namely [@CvdHS+ Thm. 1.2]. *Proof.* The proofs of the results over fields can be directly imported to the context over ${\mathbb Z}$, using in particular Corollary [Corollary 28](#BP_comp_Z){reference-type="ref" reference="BP_comp_Z"}, Lemma [Lemma 30](#Omega_Z){reference-type="ref" reference="Omega_Z"}, Lemma [Lemma 35](#BN_pair_lem_Z){reference-type="ref" reference="BN_pair_lem_Z"}, and equation ([\[cl_rel_Z\]](#cl_rel_Z){reference-type="ref" reference="cl_rel_Z"}). With these tools in hand, the proof over fields works over ${\mathbb Z}$ with no changes. Note that we do not really need the language of retractions at any point in the proof: every fact justified using retractions is equivalent to a purely group-theoretic statement. See for example Remarks [Remark 14](#A1_rem){reference-type="ref" reference="A1_rem"} and [Remark 15](#Gm_rem){reference-type="ref" reference="Gm_rem"}. ◻ # Errata for [@dHL18] {#Errata} We take this opportunity to point out a few minor mistakes in [@dHL18]. In [@dHL18 Prop. 3.10.2], we stated that all Schubert varieties $X_{\mathcal P}(w)$ in partial affine flag varieties ${\rm Fl}_{\mathcal P}$ are normal. This is true for classical Schubert varieties (those contained in $G/P$ for a parabolic subgroup $P$ in $G$) but is false in general. Pappas and Rapoport proved in [@PR08] that normality does hold for all affine Schubert varieties attached to $G$ over a field $k$, as long as the characteristic of $k$ is coprime to the order of the Borovoi fundamental group $\pi_1(G_{\rm der})$ (see [@Bo98]). However, when ${\rm char}(k)$ divides $|\pi_1(G_{\rm der})|$, it is proved in [@HLR Thm. 2.5] that *most* Schubert varieties in ${\rm Fl}_{\mathcal P}$ are *not normal*. The normality of Schubert varieties is invoked in [@dHL18 Cor. 4.1.4] to prove that the convolution space $X_{\mathcal P}(w_\bullet)$ is normal. This also fails in general, but is true when ${\rm char}(k) \nmid |\pi_1(G_{\rm der})|$. In addition, normality of Schubert varieties is used in one of the proofs in [@dHL18] that the fibers of convolution morphisms $X_{\mathcal P}(w_\bullet) \rightarrow X_{\mathcal P}(w_*)$ are geometrically connected. More precisely, a normality hypothesis plays a role in [@dHL18 Prop. 4.4.4], which in turn is used to prove the geometric connectedness of the fibers in [@dHL18 Cor. 4.4.5]. This proof is not valid in general, but is valid, again, under the hypothesis ${\rm char}(k) \nmid |\pi_1(G_{\rm der})|$. Fortunately, in [@dHL18 Thm. 2.2.2], another proof of the geometric connectedness of the fibers is given, which does not rely on any normality of Schubert varieties. Futhermore, the polynomials $F_{p,v}(q)$ appearing in [@dHL18 Eqn. (2.1)] were defined incorrectly as the Poincaré polynomials of the fibers $p^{-1}(v{\mathcal B})$. They are rather the functions $$F_{p,v}(q) = {\rm tr}\big({\rm Frob}_q ~,~ \sum_i (-1)^i \, {\rm H}^i(p^{-1}(v{\mathcal B}) \,,\, {\mathcal IC}_{X_{{\mathcal B}}(w_\bullet)})\big).$$ The fact that $F_{p,v}(q) \in \mathbb Z_{\geq 0}[q]$ is not *a priori* obvious, but it follows from [@dHL18 Eqn. (2.1)]. 99999999 N. Bourbaki: *Éléments de Mathématique. Fasc. XXXIV. Groupes et algèbre de Lie. Chapitre: IV: Groupes de Coxeter et système de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines*, in: Actualités Scientifiques et Industrielles, vol. 1337, Hermann, Paris, 1968, 288 pp. M.  Borovoi: *Abelian Galois cohomology of reductive algebraic groups*, Mem. Amer. Soc. Math. **626** (1998). F.  Bruhat, J.  Tits: *Groupes réductifs sur un corps local II. Schémas en groupes. Existence d'une donnée radicielle valuée*, Inst. Hautes Études Sci. Publ. Math. **60** (1984), 197-376. R. Cass, T. van den Hove, J. Scholbach: *The geometric Satake equivalence for integral motives*, arXiv:2211.04832. K. $\check{\rm C}$esnavi$\check{\rm c}$ius: *Torsors on the complement of a smooth divisor*, arXiv:2204.08233. B. Conrad: *Reductive group schemes*, in autour des schémas en groupes, Group Schemes, A celebration of SGA3 , Volume I, Panoramas et synthèses **42-43** (2014). Mark Andrea de Cataldo, Thomas J Haines, and Li Li, *Frobenius semisimplicity for convolution morphisms*, Math.  Z. **289**(1):119-169, (2018). G. Faltings: *Algebraic loop groups and moduli spaces of bundles*, J. Eur. Math. Soc. (JEMS) **5** (2003), no. 1, 41-68. D. Gaitsgory: *Construction of central elements in the affine Hecke algebra via nearby cycles*, Invent. Math. **144** (2001), no. 2, 253-280. T. Haines, *A proof of the Kazhdan-Lusztig purity theorem via the decomposition theorem of BBD*, note, around 2005. Available at www.math.umd.edu/$\sim$tjh. T. Haines, *Equidimensionality of convolution morphisms and applications to saturation problems*, Advances in Math. **207**, no. 1 (2006), 297-327. T. Haines, M. Kapovich, J. J. Millson, *Ideal triangles in Euclidean buildings and branching to Levi subgroups*, J. Algebra **361** (2012), 41-78. T. Haines, R. Kottwitz, A. Prasad, *Iwahori-Hecke algebras*, J. Ramanujan Math. Soc. **25**, No.2 (2010), 113-145. T. Haines, J. Lourenço, T. Richarz, *On normality of Schubert varieties: remaining cases in positive characteristic*, 51 pages, to appear, Ann. Sci. École Norm. Sup. arXiv:1806.11001. T. Haines, M. Rapoport: *On parahoric subgroups*, Adv. Math. **219** (2008), 188-198. T. Haines, T. Richarz: *The test function conjecture for local models of Weil-restricted groups*, Compos. ,Math. **156**(7), (2020), 1348-1404. T. Haines, T. Richarz: *Normality and Cohen-Macaulayness of parahoric local models*, to appear, J. Eur. Math. Soc. DOI 10.4171/JEMS/1192. J.E. Humphreys, *Reflection groups and Coxeter groups*, Cambridge Studies in Adv. Math. **29**, Cambridge Univ. Press. 1990. 204 pp. +xii. M. Kapovich, B. Leeb, J. Millson, *The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra*, Memoirs of the AMS **192**, (2008). I. Mirković, K. Vilonen: *Geometric Langlands duality and representations of algebraic groups over commutative rings*, Ann. of Math. (2) **166** (2007), no. 1, 95-143. G. Pappas, M. Rapoport: *Twisted loop groups and their affine flag varieties*, Adv. Math. **219** (2008), 118-198. G. Pappas, X. Zhu: *Local models of Shimura varieties and a conjecture of Kottwitz*, Invent. Math. **194** (2013), 147-254. J. Parkinson: *Buildings and Hecke algebras*, J. Alg. **297** (2006), 1-49. T. Richarz: *A new approach to the geometric Satake equivalence*, Documenta Mathematica **19** (2014) 209-246. T. Richarz, J. Scholbach: *The intersection motive of the moduli stack of shtukas*, Forum of Math. Sigma **8**, (2020), paper e8, 99 pp. https://umd.zoom.us/j/7912376276. T. Richarz: *Affine Grassmannians and Geometric Satake equivalences*, Int. Math. Res. Not. (2016), Issue **12**, 3717-3767. C. Schwer: *Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra*, IMRN, (2006), 1-31. N. Spaltenstein, *The fixed point set of a unipotent transformation on the flag manifold*, Nederl. Akad. Wetensch. Proc. Ser. A **79**=Indag. Math **38** (1976), no. 5, 452-456. T. A. Springer, *Linear Algebraic Groups*, 2nd ed. Progress in Math. vol. **9**, Birkhäuser, 1998. Stacks Project, Authors of the stacks project, available at http://stacks.math.columbia.edu/. [^1]: Research partially supported by NSF DMS-2200873.

arxiv_math

--- abstract: | The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution. This is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify the lump type solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller, we show that the lump type solutions are rational and their tau function has to be a polynomial of degree $k(k+1)$. In particular, this implies that the lump solution is the unique ground state of the KP-I equation (as conjectured by Klein and Saut in [@Klein0]). Our result generalizes a theorem by Airault-McKean-Moser on the classification of rational solutions for the KdV equation. address: - Department of Mathematics, University of Science and Technology of China, Hefei, China. - Department of Mathematics, University of British Columbia, Vancouver, B.C. Canada, V6T 1Z2. - Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan, China. author: - Yong Liu - Juncheng Wei - Wen Yang title: Uniqueness of lump solutions of KP-I equation --- # Introduction The KP equation first appeared in the 1970 paper [@KP] by Kadomtsev and Petviashvili, where they studied the transverse stability of the line solitons of KdV equation. It can be written as $$\partial_{x}\left( \partial_{t}U+\partial_{x}^{3}U+3\partial_{x}\left( U^{2}\right) \right) -\sigma\partial_{y}^{2}U=0.$$ If $\sigma=1,$ then it is called KP-I equation, while the case of $\sigma=-1$ is of negative dispersion and called KP-II. KP equation is an integrable system and can be regarded as a two dimensional generalization of the classical KdV equation. It is an important PDE both in mathematics and physics. Up to now, there are vast literature on KP equation. In the sequel, we shall briefly mention some results which are most closely related to our objective. There are various different ways to study the KP equation. One of them is to use the inverse scattering transform. Manakov[@Manakov] studied the inverse scattering transform of KP equation on a formal level. Segur in [@Segur] analyzed the direct scattering and rigorously obtained the solution for the direct problem under a small norm assumption, but lump solutions are not investigated in these works. Then Fokas-Ablowitz [@Fokas] obtained the lump solution in their inverse scattering setting. Their results are later extended to higher order rational solutions in [@Vi]. Zhou[@Zhou; @Zhou1; @Zhou2] then studied the KP-I equation and related problems in a rigorous way. There the lump solutions correspond to poles of the associated eigenfunctions. Since the pole structure is actually still not well understood in the general case, therefore the lump solutions are not treated. Observe that if $U$ is a traveling wave of the form $u\left( x-t,y\right) ,$ then the KP-I equation reduces to the following Boussinesq equation: $$\partial_{x}^{2}\left( \partial_{x}^{2}u+3u^{2}-u\right) -\partial_{y}% ^{2}u=0. \label{Bous}$$ Due to the above mentioned difficulties, we would like to study the traveling wave solutions of the KP-I equation using the inverse scattering transform of the Boussinesq equation. This IST is first carried out in [@Deift]. The first equation of the Lax pair turns out to be a third order ODE, in contrast with the second order ODE of the KdV equation. In this direction, the IST for first order ODE systems with generic potential(the poles are all simple) is studied in [@Beals1], [@Beals2] and the case of higher order ODE is treated in [@Beals]. The case of general potentials have been studied in [@DeiftZh; @Zhou1] using the augmented contour approach. Let us write the solution $u$ of ([\[Bous\]](#Bous){reference-type="ref" reference="Bous"}) in terms of the $\tau$ function: $u:=2\partial_{x}^{2}\ln \tau.$ Then it is well known that the Boussinesq equation in bilinear form is $$\left( D_{x}^{4}-D_{x}^{2}-D_{y}^{2}\right) \tau\cdot\tau=0, \label{k}%$$ where $D$ is the bilinear derivative operator. One can check that the function $\tau\left( x,y\right) =x^{2}+y^{2}+3$ is a solution of this bilinear equation. Note that this function is even in both $x$ and $y$ variables and corresponding to the lump solution. We point out that actually the lump solution is first obtained in [@Manakovz; @Sat] using other methods. We have proved in [@Liu] using the Backlund transformation that the lump solution is nondegenerated, in the sense that the linearized KP-I operator at this solution does not have nontrivial kernels. It is also known that KP equation is closely related to many other PDEs. For instance, in [@Be](see also the references therein), it is shown that KP equation is related to the GP equation. The nondegeneracy result for the lump can then be used to construct traveling wave solutions of the GP equation with subsonic speed, with a perturbation argument, see [@LWW]. More general rational solutions of $\left( \ref{k}\right)$ with degree $k\left( k+1\right)$ have been found in [@Pelinovskii95], [@Pelinovskii]. Then in [@Pelinovskii93] it is proved that around the higher energy lump type solutions, the KP-I equation has anomalous scattering with infinite phase shift. This indicates that the dynamics of the KP-I equation will be more complicated than the KdV equation. Hence it is important to understand the structure of lump type solutions for the Boussinesq, as well as the KP-I equation. We also point out that KP-I is globally well-posed in the natural energy space, see, for instance [@Kenig],[@Moli],[@Moli2] and the references therein for related results in this direction. A very fascinating and in-depth description of KP equation and related dynamical, variational, and other properties of its solutions, including lump, can be found in the book of Klein-Saut[@Klein]. We refer to it and also its references for a detailed introduction on this subject. It is worth mentioning that $\left( \ref{Bous}\right)$ is a special case of Boussinesq-type equation(its original form described by Boussinesq in 1870s) $$\partial_{x}^{2}\left( pu+u^{2}+\partial_{x}^{2}u\right) +\sigma^{2}% \partial_{y}^{2}u=0,$$ where $\sigma^{2}=\pm1$ and $p$ is a constant. Rational solutions of this equation has been studied in many papers, such as [@Airault; @Moser; @Bog; @Clarkson; @Pelinovskii95; @H]. For instance, the special case of $\left( \sigma,p\right) =\left( 1,0\right)$ is considered in [@Clarkson], using the theory of Painlevé equations. In view of all these developments, it is desirable to have some classification on the solutions of the Boussinesq equation. In this paper, we would like to classify all the "lump type" solutions. The main result of this paper is the following **Theorem 1**. *Suppose $u$ is a real valued smooth solution of the equation $$\label{KP1} \partial_{x}^{2}\left( \partial_{x}^{2}u+3u^{2}-u\right) -\partial_{y}% ^{2}u=0\text{ in }\mathbb{R}^{2}.$$ Assume $$u\left( x,y\right) \rightarrow0\text{, as }x^{2}+y^{2}\rightarrow+\infty.$$ Then $u=2\partial_{x}^{2}\ln\tau_{k},$ where $\tau_{k}$ is a polynomial in $x,y$ of degree $k\left( k+1\right)$ for some $k\in\mathbb{N}$.* As in [@DS1997], problem ([\[KP1\]](#KP1){reference-type="ref" reference="KP1"}) admits a variational structure which defines a ground state solution in suitable Sobolev space. As a corollary of this theorem and the algebraic decaying property of ground states (see Theorem 3.1 of [@DS1997]), we see that the classical lump solution is the unique ground state of the KP-I equation, due to the fact that the energy is determined by the degree of the tau function. This answers affirmatively the uniqueness question raised in Remark 18 and Remark 19 of [@Klein0]. As already pointed out there, while the uniqueness of ground state of the Schrodinger equation can be proved using ODE shooting method, the uniqueness of lump is more complicated since it is not radially symmetric. To our knowledge, our result seems to be the first classification result for solutions of semilinear elliptic equations without symmetry(also without any other assumptions like stable or finite Morse index). Solutions satisfying the assumption of this theorem will be called lump type solutions. We remark that for each fixed $k,$ there is a family of lump type solutions. We expect that all lump type solutions should be included in the family found in [@Pelinovskii95]. Those solutions will be recalled in the next section. However, a full classification of this type would need further detailed analysis, which will not be pursued in this paper. Such a full classification would presumably yields some information of the lump type solutions of the generalized KP equation. Let us sketch the main ideas of our proof. We first use the robust inverse scattering transform developed by Bilman-Miller [@Bilman] to show that lump type solutions has to be rational. Then we use the technique of [@Moser], appealing to the Boussinesq hierarchy, to show that the degree of the $\tau$ function has to be $k\left( k+1\right) .$ This technique is used in [@Moser] to prove that the $\tau$ function of the rational solution of the KdV equation necessarily is a polynomial of degree $k\left( k+1\right) /2.$ This paper is organized in the following way. In Section 2, we recall the construction of lump type solution appeared in various papers of Pelinovskii and his collaborators. We emphasize that actually there are many other constructions, using different methods, which we will not recall here. In Section 3, we use the robust inverse scattering transform to show that the solution is rational. In Section 4, we investigated the degree of the $\tau$ function, the main theorem then follows immediately. Some numerical computation for the even solution is carried out in the last section. **Acknowledgment.** Y. Liu is supported by the National Key R&D Program of China 2022YFA1005400 and NSFC 11971026, NSFC 12141105. J. Wei is partially supported by NSERC of Canada. W. Yang is partially supported by National Key R&D Program of China 2022YFA1006800, NSFC No. 12171456 and NSFC No. 12271369. # Family of lump type solutions For each $n\in\mathbb{N},$ real valued rational solutions of the Boussinesq equation are obtained in [@Pelinovskii] by a limiting procedure. These solutions are even with respect to $x$ and $y$ variable. Then in [@Pelinovskii95], families of rational solutions are derived using Wronskian representation of the solutions to the KP equation. For reader's convenience, Let us recall these results in this section. We adopt the notations used in [@Pelinovskii95]. Consider the KP-I equation $$\left( -4u_{t_{3}}+\left( 3u^{2}\right) t_{1}+u_{t_{1}t_{1}t_{1}}\right) _{t_{1}}-3u_{t_{2}}u_{t_{2}}=0.$$ This equation has solutions expressed in terms of the $\tau$ function: $$u\left( t_{1},t_{2},t_{3}\right) =2\partial_{t_{1}}^{2}\ln\tau\left( t_{1},t_{2},t_{3}\right) .$$ There are different forms for the $\tau$ functions. Let us explain it now. Let $\Psi_{n}^{\pm}$ be solutions of the system of differential equations $$\left\{ \begin{array} [c]{c}% \pm i\partial_{t_{2}}\Psi_{n}^{\pm}=\partial_{t_{1}}^{2}\Psi_{n}^{\pm},\\ \partial_{t_{3}}\Psi_{n}^{\pm}=\partial_{t_{1}}^{3}\Psi_{n}^{\pm}. \end{array} \right.$$ We fix an integer $N$ and define $$\tau=\det M_{N}, \label{tau1}%$$ where $M_{N}$ is the $N\times N$ matrix whose entries are given by $c_{nk}+I_{nk},1\leq n,k\leq N.$ Here $c_{nk}$ are arbitrary complex parameters and $$I_{nk}=\int_{-\infty}^{t_{1}}\Psi_{n}^{+}\left( s,t_{2},t_{3}\right) \Psi_{k}^{-}\left( s,t_{2},t_{3}\right) ds.$$ The KP-I equation has another family of solutions, for which the $\tau$ function has the Wronskian form: $$\tau=W\left( \Psi_{1}^{\pm},...,\Psi_{N}^{\pm}\right) =\det\left( J_{nk}^{\pm}\right) , \label{tau2}%$$ where $J_{nk}^{\pm}=\partial_{t_{1}}^{k-1}\Psi_{n}^{\pm}.$ The two forms $\left( \ref{tau1}\right)$ and $\left( \ref{tau2}\right)$ are related to each other. If we choose in $\left( \ref{tau1}\right)$ the function $$\Psi_{k}^{-}=\exp\left( p_{k}t_{1}-p_{k}^{2}t_{2}+p_{k}^{3}t_{3}\right) =\exp\left( \Phi_{k}^{-}\right) ,$$ then integration by parts yields $$I_{nk}=\left( \frac{\Psi_{n}^{+}}{p_{k}}-\frac{\partial_{t_{1}}\Psi_{n}^{+}% }{p_{k}^{2}}+\frac{\partial_{t_{1}}^{2}\Psi_{n}^{+}}{p_{k}^{3}}+...\right) \exp\left( \Phi_{k}^{-}\right) .$$ Assuming $p_{k}>>1,$ the leading terms of $\tau$ can be written as the product of a Vandermont determinant and the Wronskian $W\left( \Psi_{1}^{+}% ,...,\Psi_{N}^{+}\right) .$ Hence $$\tau=\left[ \frac{% %TCIMACRO{\dprod \limits_{1\leq m<k\leq N}}% %BeginExpansion {\displaystyle\prod\limits_{1\leq m<k\leq N}} %EndExpansion \left( p_{k}-p_{m}\right) }{% %TCIMACRO{\dprod \limits_{k=1}^{N}}% %BeginExpansion {\displaystyle\prod\limits_{k=1}^{N}} %EndExpansion p_{k}^{N}}W\left( \Psi_{1}^{+},...,\Psi_{N}^{+}\right) +O\left( p^{-\left( \frac{N\left( N+1\right) }{2}+1\right) }\right) \right] \exp\left( %TCIMACRO{\dsum \limits_{k=1}^{N}}% %BeginExpansion {\displaystyle\sum\limits_{k=1}^{N}} %EndExpansion \Phi_{k}^{-}\right) .$$ Dividing the right hand side by $\exp\left( %TCIMACRO{\dsum \limits_{k=1}^{N}}% %BeginExpansion {\displaystyle\sum\limits_{k=1}^{N}} %EndExpansion \Phi_{k}^{-}\right)$ and the constant before the Wronskian $W,$ letting $p\rightarrow0,$ we get $\left( \ref{tau2}\right) .$ Let $K\leq N$ be a fixed integer. If the above limiting procedure is only carried out for $\Phi_{k}^{-},k=K+1,...,N,$ then we obtain $$\tau=\det\left( S_{nk}\right) , \label{tau3}%$$ where $$S_{nk}=\left\{ \begin{array} [c]{l}% I_{nk},\text{ for }k=1,...,K,\\ J_{n,k-K}^{+},\text{ for }k=K+1,...,N. \end{array} \right.$$ Let us now consider the function $\phi_{m}:=\partial_{p}^{m}\exp\left( \Phi ^{+}\left( t_{1},t_{2},t_{3},p\right) \right) ,$ where $$\Phi^{+}\left( t_{1},t_{2},t_{3},p\right) =% %TCIMACRO{\dsum \limits_{j=1}^{\infty}}% %BeginExpansion {\displaystyle\sum\limits_{j=1}^{\infty}} %EndExpansion \left( p^{j}t_{j}\right) .$$ We have $$\phi_{m}=P_{m}\exp\left( \Phi^{+}\left( t_{1},t_{2},t_{3},p\right) \right) .$$ Here $P_{m}$ is a polynomial of the variables $\theta_{1},...,\theta_{m},$ given by $\theta_{j}=\frac{1}{m!}\partial_{p}^{j}\Phi^{+}.$ In particular, $$\begin{aligned} \theta_{1} & =t_{1}+2pt_{2}+3p^{2}t_{3}+...,\\ \theta_{2} & =t_{2}+3pt_{3}+...\\ \theta_{3} & =t_{3}+..., \end{aligned}$$ and $\theta_{j}$ only depends on $t_{j},t_{j+1},....$ We have $$P_{1}=\theta_{1},P_{2}=2\theta_{2}+\theta_{1}^{2}.$$ To obtain a solution of the Boussinesq equation, let $v=-\frac{1}{3p}$ and define the vertex operator $$\mathcal{S}\left( v\right) =\exp\left( -% %TCIMACRO{\dsum \limits_{m=1}^{+\infty}}% %BeginExpansion {\displaystyle\sum\limits_{m=1}^{+\infty}} %EndExpansion \frac{v^{m}}{m}\partial_{\theta_{m}}\right) .$$ Then $$\mathcal{S}\left( v\right) P_{n}=\left( 1-v\partial_{t_{1}}\right) P_{n}.$$ The tau function will be a solution of the Boussinesq equation if it depends on the variable $x=t_{1}+3p^{2}t_{3}$ and $t_{2}.$ This requires $$\partial_{\theta_{2}}\tau=v\partial_{\theta_{3}}\tau.$$ The next step to construct solutions of the Boussinesq equation is choose $p$ to be $1$ and define $$\begin{aligned} \Psi_{n}^{+} & =S^{N-n}\left( v\right) P_{2n-1}^{+}\exp\left( \Phi ^{+}\left( t_{1},t_{2},t_{3},p\right) \right) ,\text{ for }1\leq n\leq N,\\ \Psi_{k}^{-} & =S^{K-k}P_{2k-1}^{-}\exp\left( \Phi^{-}\left( t_{1}% ,t_{2},t_{3},p\right) \right) ,1\leq k\leq K. \end{aligned}$$ Then the Tau function defined by $\left( \ref{tau3}\right)$ will correspond to a solution of the Boussinesq equation. In general, this solution is complex valued. But in the particular case of $K=N,$ if we choose $P_{k}^{+},P_{k}% ^{-}$ such that $P_{k}^{+}=\bar{P}_{k}^{-},$ then $$\tau=\det\left( w^{+}\left( w^{-}\right) ^{T}\right) ,$$ where $$\begin{aligned} \left( w^{+}\right) _{nk} & =\left( -\mu\right) ^{k-1}\partial_{t_{1}% }^{k-1}\left[ S^{-k}\left( \mu\right) S^{N-n}\left( v\right) P_{2n-1}% ^{+}\right] ,1\leq n\leq N,1\leq k\leq2N-1,\\ w_{nk}^{-} & =\bar{w}_{nk}^{+}. \end{aligned}$$ Hence this determinant can be written as the sum of positive terms. Note that the condition $P_{k}^{+}=\bar{P}_{k}^{-}$ requires $t_{2k}$ is imaginary and $t_{2k+1}$ is real. Hence there are in total $2N$ free (real)parameters, or $N$ complex parameters. We point out that in [@Yang], an explicit family of rational solutions is obtained with different methods. It is not clear whether these two family of solutions are same. It is also worth mentioning that there are quite a vast literature on the construction and analysis of solutions to the KP equation, for instances, [@Ablowitz; @Cha0; @Cha1; @Prada; @W; @Ito; @Kac; @Pelnovsky94; @Pelnovsky98], just to list a few. # Inverse scattering of the Boussinesq equation and the rationality of lump type solutions In this section, we would like to prove that lump type solutions are indeed rational. We consider the Boussinesq equation in the following form $$q_{yy}=3q_{xxxx}-12\left( q^{2}\right) _{xx}-24q_{xx}. \label{B}%$$ This equation can be obtained from the original Boussinesq equation $\left( \ref{Bous}\right)$ by a rescaling, namely, by setting $q\left( x,y\right) =-6u\left( 2\sqrt{2}x,8\sqrt{3}y\right) .$ We are interested in the solutions of equation ([\[B\]](#B){reference-type="ref" reference="B"}) which decay to zero at infinity. The usual inverse scattering of the Boussinesq equation is developed in [@Deift], with different boundary condition. It can be seen later on that in our case, the situation is more complicated. In this section, we will adopt the powerful method of robust inverse scattering transform, first developed in [@Bilman] for the Schrodinger equation, to show that solution of the Boussinesq equation has to be rational. Introducing a new function $p$, the equation $\left( \ref{B}\right)$ can also be written into the following system of ODEs: $$\left\{ \begin{array} [c]{l}% q_{y}=-3p_{x},\\ p_{y}=-q_{xxx}+8qq_{x}+8q_{x}. \end{array} \right.$$ This system has the following Lax pair(see [@Deift; @Zah]): $$\frac{dL}{dy}=QL-LQ=[Q,L],$$ where $$\left\{ \begin{array} [c]{l}% L=i\frac{d^{3}}{dx^{3}}-i\left[ \left( 2\left( q+1\right) \frac{d}% {dx}+q_{x}\right) \right] +p,\\ Q=i\left( 3\frac{d^{2}}{dx^{2}}-4\left( q+1\right) \right) . \end{array} \right.$$ Let $k\in\mathbb{C}\,\ $be a complex parameter. We consider the equation $$Lf=\left( k^{3}+2k\right) f,$$ which can be recasted into the matrix form: $$\frac{d}{dx}\left( \begin{array} [c]{c}% f_{1}\\ f_{2}\\ f_{3}% \end{array} \right) =\left( \begin{array} [c]{ccc}% 0 & 1 & 0\\ 0 & 0 & 1\\ q_{x}+pi-i\left( k^{3}+2k\right) & 2\left( q+1\right) & 0 \end{array} \right) \left( \begin{array} [c]{c}% f_{1}\\ f_{2}\\ f_{3}% \end{array} \right) . \label{firstorder}%$$ The coefficient matrix of this system, denoted by $A,$ will depend on the potential $q.$ As $x\rightarrow\pm\infty,$ $A$ will tend to the constant matrix $$T:=\left( \begin{array} [c]{ccc}% 0 & 1 & 0\\ 0 & 0 & 1\\ -i\left( k^{3}+2k\right) & 2 & 0 \end{array} \right) .$$ The eigenvalues of $T$ can be explicitly computed. They depend on the parameter $k$ and are given by $$\lambda_{1}=ik,\quad\lambda_{2}=\frac{-ik+\sqrt{3k^{2}+8}}{2},\quad\lambda _{3}=\frac{-ik-\sqrt{3k^{2}+8}}{2}.$$ It follows that $T$ can be written as $PMP^{-1}$, where $$P\left( k\right) =\left( \begin{matrix} 1 & 1 & 1\\ ik & \frac{-ik+\sqrt{3k^{2}+8}}{2} & \frac{-ik-\sqrt{3k^{2}+8}}{2}\\ -k^{2} & \frac{k^{2}+4-ik\sqrt{3k^{2}+8}}{2} & \frac{k^{2}+4+ik\sqrt{3k^{2}% +8}}{2}% \end{matrix} \right) ,M(k)=\left( \begin{matrix} \lambda_{1} & 0 & 0\\ 0 & \lambda_{2} & 0\\ 0 & 0 & \lambda_{3}% \end{matrix} \right) .$$ Recall that for any constant matrix $B,$ the matrix $e^{Tx}B$ is a solution of the equation $U^{\prime}=TU.$ We choose $B=P$ and get the following matrix solution $$U_{bg}(k,x):=n(k)P\left( k\right) e^{M(k)x}:=E\left( k\right) e^{M\left( k\right) x}.$$ Here $n\left( k\right)$ is chosen such that $\det\left( U_{bg}\right) =1$. Explicitly, $$n\left( k\right) =\left( \left( 3k^{2}+2\right) \sqrt{3k^{2}+8}\right) ^{-1}.$$ One can see that as $k$ tends to $\sqrt{8/3}i$ or $-\sqrt{8/3}i$, $n$ blows up. Let $k=s+ti$, with $x,y\in \mathbb{R}.$ Direct computation tells us that the condition $\operatorname{Re}\left( \lambda_{2}\right) =\operatorname{Re}\left( \lambda_{3}\right)$ implies $\operatorname{Re}\sqrt{3k^{2}+8}=0.$ That is, $$s=0,t^{2}>\frac{8}{3}.$$ On the other hand, $\operatorname{Re}\left( \lambda_{1}\right) =\operatorname{Re}\left( \lambda_{2}\right)$ or $\operatorname{Re}\left( \lambda_{3}\right)$ requires $$s^{2}-3t^{2}+2=0.$$ Let $r$ be a fixed large constant and $B_{r}$ be the ball of radius $r$ centered at the origin. In the region $B_{r}^{c}:=\mathbb{R}^{2}\backslash B_{r},$ we consider the curve $$\Sigma_{1}:=\left\{ \left( s,t\right) \in B_{r}^{c}:s^{2}-3t^{2}% +2=0\right\} \cup\left\{ \left( s,t\right) \in B_{r}^{c}:s=0\text{ and }t^{2}>\frac{8}{3}\right\} .$$ Let us define $$\Omega_{1}=B_{r}^{c}\backslash\Sigma_{1}.$$ Note that $\Omega_{1}$ has six connected components, which we will denote them by $\Omega_{1,1},...,\Omega_{1,6}.$ In the ball $B_{r},$ we consider the curve $$\Sigma_{2}:=\left\{ \left( s,t\right) :s=0,t^{2}\leq\frac{8}{3}\right\} .$$ We also define $\Omega_{2}:=B_{r}\backslash\Sigma_{2}.$ Next we define the Beals-Coifman solution for $\left( \ref{firstorder}% \right) .$ Note that if the matrix $\phi$ satisfies $\phi^{\prime}=A\phi,$ then $g:=\phi e^{-Mx}$ will satisfy $$\begin{aligned} g^{\prime} & =\phi^{\prime}e^{-Mx}+\phi\left( -M\right) e^{-Mx}\\ & =Ag-gM. \end{aligned}$$ For $k\in \mathbb{R}^{2}\backslash B_{r},$ we choose to be the matrix solution such that $\left\Vert g\left( x\right) \right\Vert _{L^{\infty}\left( \mathbb{R}\right) }<+\infty,$ and $$g\left( x\right) \rightarrow E\left( k\right) ,\text{ as }x\rightarrow -\infty.$$ We then define $U^{ou}=ge^{Mx}.$ This solution is meromorphic in each $\Omega_{1,j},j=1,...,6.$ The restriction of $U^{ou}$ to $\Omega_{1,j}$ will be denoted by $U_{j}^{ou}.$ On the common boundaries of $\Omega_{1,j}$ and $\Omega_{1,j+1},$ $U_{j+1}^{ou}$ and $U_{j}^{ou}$ are related by the transfer matrix $V_{j}.$ That is, for $j=1,...,6,$ $$U_{j+1}^{ou}=U_{j}^{ou}V_{j}.$$ Here we set $U_{7}^{ou}=U_{1}^{ou}.$ **Lemma 2**. *The transfer matrix $V_{j}$ is equal to the identity matrix $I.$* *Proof.* The matrix $V_{j}$ is independent of $x,y$. We would like to analyze the limit of $V_{j}$ as $y$ tends to infinity. We only like to estimate the difference between $U_{j}^{ou}$ and $U_{bg}=E\left( k\right) e^{-M\left( k\right) x}.$ Note that $$e^{-Mx}=% \begin{bmatrix} e^{-\lambda_{1}x} & 0 & 0\\ 0 & e^{-\lambda_{2}x} & 0\\ 0 & 0 & e^{-\lambda_{3}x}% \end{bmatrix} .$$ Let us denote the $j$-th column of $E\left( k\right)$ by $\xi_{j}.$ For each $j=1,2,3$, we would like to use an iteration scheme to construct a vector valued solution $\eta$ satisfies the following condition: If $\operatorname{Re}\left( \lambda_{j}\right) \geq0,$ then $$\eta_{j}e^{\lambda_{j}x}\rightarrow\xi_{j}\text{ as }x\rightarrow+\infty,$$ If $\operatorname{Re}\left( \lambda_{j}\right) \leq0,$ then $$\eta_{j}e^{\lambda_{j}x}\rightarrow\xi_{j}\text{ as }x\rightarrow-\infty.$$ Using the same arguments as that of the proof Lemma 5.2 in [@LiuWei], we find that $$\left\vert \eta_{j}e^{\lambda_{j}x}-\xi_{j}\right\vert \leq\int_{x}^{\pm \infty}\frac{C}{s^{2}+y^{2}}ds=\frac{C}{y}\arctan\frac{x}{y}.$$ Hence as $y\rightarrow\infty,$ we find that uniformly $$\eta_{j}e^{\lambda_{j}x}\rightarrow\xi_{j}.$$ This implies that the transfer matrix is identity. ◻ In $\Omega_{2},$ we define $U^{in}$, matrix solution of $\left( \ref{firstorder}\right)$, such that $$U^{in}\left( 0\right) =I.$$ A key property is that $U^{in}$ is holomorphic in $\Omega_{2}.$ In general, assuming the jump matrix from the interior to the outer solutions at the boundary circle $\partial B_r$ has the form $$G\left( k\right) E\left( k\right) .$$ Then we have the following relation: $$U^{ou}=U^{in}G\left( k\right) E\left( k\right) . \label{holom}%$$ Taking $\left( x,y\right) =\left( 0,0\right) ,$ we get $$U^{ou}\left( 0,0\right) =G\left( k\right) E\left( k\right) .$$ It turns our that the Beals-Coifman fundamental solution $U^{ou}\left( k;x,y\right)$ has the form $$U^{ou}\left( k;x,y\right) =\left[ I+% %TCIMACRO{\dsum \limits_{\lambda_{j}}}% %BeginExpansion {\displaystyle\sum\limits_{\lambda_{j}}} %EndExpansion% %TCIMACRO{\dsum \limits_{s=1}^{n_{j}}}% %BeginExpansion {\displaystyle\sum\limits_{s=1}^{n_{j}}} %EndExpansion \left( \frac{A_{j,s}\left( x,y\right) }{\left( k-\lambda_{j}\right) ^{k}% }\right) \right] E\left( k\right) e^{M\left( k\right) x}.$$ Here $A_{j,s}$ are $3\times3$ matrices. We then obtain $$\left[ I+% %TCIMACRO{\dsum \limits_{\lambda_{j}}}% %BeginExpansion {\displaystyle\sum\limits_{\lambda_{j}}} %EndExpansion% %TCIMACRO{\dsum \limits_{s=1}^{n_{j}}}% %BeginExpansion {\displaystyle\sum\limits_{s=1}^{n_{j}}} %EndExpansion \left( \frac{A_{j,s}\left( x,y\right) }{\left( k-\lambda_{j}\right) ^{k}% }\right) \right] E\left( k\right) e^{Mx}E\left( k\right) ^{-1}G\left( k\right) ^{-1}=U^{in}. \label{holo}%$$ As we already mentioned, $U^{in}$ is a holomorphic function in the radius $r$ disk. This will yield a system of equations for the entries of $A_{j,s}.$ Next we would like to show that the system has a unique solution. **Lemma 3**. *For fixed $G,$ the system $\left( \ref{holom}\right)$ has a unique solution.* *Proof.* We have $$U^{ou}\left( k;x,y\right) =U^{in}\left( k;x,y\right) G\left( k\right) .$$ Suppose there is another pair $\left( \tilde{U}^{ou},\tilde{U}^{in}\right)$ such that $$\tilde{U}^{ou}\left( k;x,y\right) =\tilde{U}^{in}\left( k;x,y\right) G\left( k\right) .$$ We claim that $\tilde{U}^{ou}=U^{ou}.$ Indeed, since $\tilde{U}^{ou}$ is invertible, the matrix $U^{ou}\left( \tilde{U}^{ou}\right) ^{-1}$ is holomorphic outside $B_{r}$, while $U^{in}\left( \tilde{U}^{in}\right) ^{-1}$ is holomorphic inside $B_{r}.$ Moreover, they are equal to each other on $\partial B_{r}.$ Hence they patch up to an entire holomorphic function which is also bounded. Hence in view of their asymptotics at infinity, we obtain $$U^{ou}=\tilde{U}^{ou},\text{ and }U^{in}=\tilde{U}^{in}.$$ This finishes the proof. ◻ With this result at hand, next we show that the solution $q$ has to be rational. **Theorem 4**. *Suppose $q$ is a solution of the Boussinesq equation satisfying the assumption of Theorem, then $q$ is rational.* *Proof.* The solution $q,$ which appears as a potential in the Lax pair equation, is determined by the Beals-Coifman function $U^{ou}$ and $U^{in}.$ Hence we need to determine the matrices $A_{j,s}$ in $\left( \ref{holo}\right) .$ Using the arguments of Lemma [Lemma 2](#transfer){reference-type="ref" reference="transfer"}, one can show that the possible poles $\lambda_{j}$ in $\left( \ref{holo}\right)$ has to be $\lambda_{\pm }^{\ast}:=\pm\sqrt{\frac{8}{3}}i.$ The matrix $E\left( k\right) e^{Mx}E\left( k\right) ^{-1}$ is holomorphic(with removable singularities) in $k.$ As $k\rightarrow \lambda_{\pm}^{\ast},$ direct computation shows that it(as well as its derivatives in $k$) tends to a matrix whose entries are polynomial functions of $x$(multiplied by some exponential functions in $x$)$.$ Since the right hand side of $\left( \ref{holo}\right)$ is holomorphic in $k,$ we see that the matrices $A_{j,s}$ satisfy a system of linear equation whose entries are polynomial in $x.$ Now by Lemma [Lemma 3](#unique){reference-type="ref" reference="unique"}, the solution has to be unique. Hence $q$ is rational in $x.$ It then follows from the Krichever theorem(see [@Krichever78; @Krichever83]) that the solution is also rational in $y.$ ◻ # The Boussinesq hierarchy and the structure of the rational solutions In this section, we will use the technique developed in [@Moser] for the KdV equation, to classify the degree of the tau function of the rational solution of the Boussinesq equation. Indeed, this problem is mentioned in P. 123-P. 124 of [@Moser], for the hyperbolic case of Boussinesq equation. In [@Mckean], Mckean found the Boussinesq hierarchy associated to the Boussinesq equation. The Boussinesq equation he studied has the following form $$\partial_{y}^{2}q=3\partial_{x}^{2}\left( \partial_{x}^{2}q+4q^{2}\right) . \label{eq:Bou}%$$ Related works on Boussinesq hierarchy can be found in [@G]. We use $D$ to denote the differentiation with respect to the $x$ variable(not the bilinear derivative operator). Define the operator $$\mathcal{D}=% \begin{bmatrix} 0 & D\\ D & 0 \end{bmatrix} .$$ Let $$\begin{aligned} L_{0} & :=D^{5}+5\left( qD^{3}+D^{3}q\right) \\ & -3\left( q^{\prime\prime}D+Dq^{\prime\prime}\right) +16qDq, \end{aligned}$$ and define $$\mathcal{K}_{0}=% \begin{bmatrix} D^{3}+qD+Dq & 3pD+2p^{\prime}\\ 3pD+p^{\prime} & \frac{1}{3}L_{0}% \end{bmatrix} .$$ He then uses $\mathcal{K}_{0}$ to define recursively a sequence of vector fields, which generate the Boussinesq hierarchy. In our case, we are actually considering those solutions of the Boussinesq equation $\left( \ref{eq:Bou}\right)$ with nonzero boundary condition, say $\tilde{q}\rightarrow-\frac{1}{8}$ as $x^{2}+y^{2}\rightarrow+\infty.$ Indeed, introducing new variable $\tilde{q}$ by $q=\tilde{q}-\frac{1}{8}$ in $\left( \ref{eq:Bou}\right) ,$ we obtain $$\partial_{y}^{2}\tilde{q}=3\partial_{x}^{2}\left( \partial_{x}^{2}\tilde {q}+4\tilde{q}^{2}-\tilde{q}\right) . \label{eq:B}%$$ If we set $\tilde{q}\left( x,y\right) =\frac{3}{4}u\left( x,\sqrt {3}y\right) ,$ then $u$ satisfies the version of Boussinesq equation appeared in the first section, that is: $$\partial_{x}^{2}\left( \partial_{x}^{2}u+3u^{2}-u\right) -\partial_{y}% ^{2}u=0.$$ We are thus lead to consider the shifted operator $L$ defined by $$\begin{aligned} L & :=D^{5}+5\left[ \left( q-\frac{1}{8}\right) D^{3}+D^{3}\left( q-\frac{1}{8}\right) \right] \\ & -3\left( q^{\prime\prime}D+Dq^{\prime\prime}\right) +16\left( q-\frac {1}{8}\right) D\left( q-\frac{1}{8}\right) . \end{aligned}$$ Note that $$L=L_{0}-\frac{5}{4}D^{3}-2\left( qD+Dq\right) +\frac{1}{4}D$$ We then define $$\mathcal{K}_{i}=% \begin{bmatrix} D^{3}+\left( q-\frac{1}{8}\right) D+D\left( q-\frac{1}{8}\right) & 3\left( p+\left( -1\right) ^{i}a\right) D+2p^{\prime}\\ 3\left( p+\left( -1\right) ^{i}a\right) D+p^{\prime} & \frac{1}{3}L \end{bmatrix} .$$ Here the constant $a$ is chosen such that $$\left( 3a\right) ^{2}+\frac{1}{48}=0.$$ We will explain later on why $a$ should be chosen in this way. Let $H_{0}=\int\frac{3p}{2}.$ Then a series of vector fields can be defined recursively by $$X_{n+1}=\mathcal{K}_{n}\mathcal{\nabla}H_{n},\text{ and }\mathcal{D}\nabla H_{n}=X_{n}.$$ More precisely, once we obtained $X_{j},$ we can find $\nabla H_{j}$ by using the relation $\mathcal{D}\nabla H_{j}=X_{j}.$ Then we can find $X_{j+1}$ by $X_{j+1}=\mathcal{K\nabla}H_{j}.$ In particular, $$X_{0}=\mathcal{D}\nabla H_{0}=0,X_{1}=\mathcal{K}\nabla H_{0}=\left( 3p^{\prime},q^{\prime\prime\prime}+8qq^{\prime}-q^{\prime}\right) .$$ Hence $X_{2}$ is the Boussinesq flow. Here $^{\prime}$ represents the derivative with respect to the $x$ variable. For real valued solutions, as we will see, the main order term of the $\tau$ function is $\left( x^{2}+3y^{2}\right) ^{n}.$ Suppose $u$ is a rational solution of the KP-I equation. Then from [@Krichever78; @Krichever83], we know that $u$ can be written in the form $$u=-\frac{3}{2}\sum_{i=1}^{N}\frac{1}{\left( x-\xi_{i}\left( y,t\right) \right) ^{2}}.$$ In this case, $u=\frac{3}{2}\partial_{x}^{2}\ln\tau,$ where $\tau$ is a polynomial in the $x$ variable. For rational solutions $q$ of the Boussinesq equation, we have $$q=-\frac{3}{2}\sum_{i=1}^{N}\frac{1}{\left( x-\eta_{i}\left( y\right) \right) ^{2}}. \label{qBou}%$$ inserting this into the equation $\left( \ref{eq:B}\right) ,$ we find that for each fixed index $i=1,...,n,$ there holds $$\partial_{y}^{2}\eta_{i}-\frac{72}{\left( \eta_{i}-\eta_{j}\right) ^{3}}=0, \label{Cal}%$$ $$\eta_{i}^{\prime2}+36\Sigma_{j\neq i}\left( \eta_{j}-\eta_{i}\right) ^{-2}+3=0.$$ Note that $\left( \ref{Cal}\right)$ is the Caloger-Moser system. More precisely, let $\partial_{y}\eta_{i}=\beta_{i}.$ Then the CM flow is $$\left\{ \begin{array} [c]{l}% \partial_{y}\eta_{i}=\beta_{i},\\ \partial_{y}\beta_{i}=\frac{72}{\left( \eta_{i}-\eta_{j}\right) ^{3}}. \end{array} \right. \label{cm}%$$ Indeed, one can show that the function $\left( \ref{qBou}\right)$ solves the Boussinesq equation if and only $\left( \eta,\beta\right)$ satisfies the CM system $\left( \ref{cm}\right)$ restricted to the set $$M:=\left\{ \left( \eta,\beta\right) \in\mathbb{C}^{2n}:\nabla\left( F_{3}+F_{1}\right) =0\right\} ,$$ where $$\begin{aligned} F_{1} & =3% %TCIMACRO{\dsum \limits_{j=1}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{j=1}^{n}} %EndExpansion \beta_{j},\text{ }\\ F_{3} & =\frac{1}{3}% %TCIMACRO{\dsum \limits_{j=1}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{j=1}^{n}} %EndExpansion \beta_{j}^{3}+36% %TCIMACRO{\dsum \limits_{j=1}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{j=1}^{n}} %EndExpansion% %TCIMACRO{\dsum \limits_{k\neq j}}% %BeginExpansion {\displaystyle\sum\limits_{k\neq j}} %EndExpansion \frac{\beta_{j}}{\left( \eta_{j}-\eta_{k}\right) ^{2}}. \end{aligned}$$ The proof follows from similar lines as that of [@Moser], although the case of hyperbolic Boussinesq equation is treated there, instead of the elliptic case we are studying now. Therefore we omit the details. **Proposition 5**. *The $k$-th Boussinesq flow $e\left( tX_{k}\right)$ induces a flow on $M.$* *Proof.* The Boussinesq equation reads as $$\left\{ \begin{array} [c]{l}% q_{y}=3p^{\prime},\\ p_{y}=q^{\prime\prime\prime}+8qq^{\prime}-q^{\prime}. \end{array} \right.$$ The rational solution $q$ of the Boussinesq equation can be written as $$q=-\frac{3}{2}% %TCIMACRO{\dsum _{j=1}^{n}}% %BeginExpansion {\displaystyle\sum_{j=1}^{n}} %EndExpansion \frac{1}{\left( x-\eta_{j}\right) ^{2}}.$$ Therefore, $$p=\frac{1}{2}% %TCIMACRO{\dsum _{j=1}^{n}}% %BeginExpansion {\displaystyle\sum_{j=1}^{n}} %EndExpansion \frac{\beta_{j}}{\left( x-\eta_{j}\right) ^{2}}.$$ where $\beta_{j}=\partial_{y}\eta_{j}.$ We would like to write down the explicit form of each vector field $X_{k}$ acting on $\left( q,p\right)$ in an inductive way. In view of the form of $\left( q,p\right) ,$ it is natural to expect that $$X_{k}\left( q,p\right) =\left( 6% %TCIMACRO{\dsum _{j=1}^{n}}% %BeginExpansion {\displaystyle\sum_{j=1}^{n}} %EndExpansion \frac{a_{j}}{\left( x-\eta_{j}\right) ^{3}},-% %TCIMACRO{\dsum _{j=1}^{n}}% %BeginExpansion {\displaystyle\sum_{j=1}^{n}} %EndExpansion \left( \frac{2\beta_{j}a_{j}}{\left( x-\eta_{j}\right) ^{3}}+\frac{b_{j}% }{\left( x-\eta_{j}\right) ^{2}}\right) \right) . \label{Bouvec}%$$ Here $a_{j}=\frac{d\eta_{j}}{dy},b_{j}=\frac{d\beta_{j}}{dy}$ lie in the tangent space of $M.$ Once this is proved, we then deduce that $X_{k}$ induces a flow on the locus. We use the recursive formula of $X_{i}$ defined through the operator $\mathcal{K}.$ Let us set $$m_{j}\left( x\right) :=\frac{1}{\left( x-\eta_{j}\right) ^{2}}% ,n_{j}\left( x\right) =\frac{1}{x-\eta_{j}}.$$ We also use $\Sigma_{j}=% %TCIMACRO{\dsum _{j=1}^{n}}% %BeginExpansion {\displaystyle\sum_{j=1}^{n}} %EndExpansion .$ Under these notations, we have $$q=-\frac{3}{2}\Sigma_{j}m_{j},p=\frac{1}{2}\Sigma_{j}\left( \beta_{j}% m_{j}\right) .$$ For this vector, we have $$\begin{aligned} \nabla H_{i} & =% \begin{bmatrix} 0 & D^{-1}\\ D^{-1} & 0 \end{bmatrix} \left( 6\Sigma_{j}\frac{a_{j}}{\left( x-\eta_{j}\right) ^{3}},-\Sigma _{j}\left( \frac{2\beta_{j}a_{j}}{\left( x-\eta_{j}\right) ^{3}}% +\frac{b_{j}}{\left( x-\eta_{j}\right) ^{2}}\right) \right) ^{T}\\ & =\left( \left( \Sigma_{j}\left( \beta_{j}a_{j}m_{j}+b_{j}n_{j}\right) \right) ,-3\Sigma_{j}\left( a_{j}m_{j}\right) \right) . \end{aligned}$$ Let us denote the first component of $\mathcal{K}X_{k}$ by $\left( \mathcal{K}X_{k}\right) ^{\left( 1\right) }.$ Then assuming $\mathcal{K}% X_{k-1}$ has the form $\left( \ref{Bouvec}\right)$, we find that $\left( \mathcal{K}X_{k}\right) ^{\left( 1\right) }$ equals $$\begin{aligned} & \left( D^{3}+\left( q-\frac{1}{8}\right) D+D\left( q-\frac{1}% {8}\right) \right) \left( \Sigma_{j}\left( \beta_{j}a_{j}m_{j}+b_{j}% n_{j}\right) \right) \\ & +\left( 3\left( p+\alpha\right) D+2p^{\prime}\right) \left( -3\Sigma_{j}\left( a_{j}m_{j}\right) \right) . \end{aligned}$$ The points $\eta_{j},j=1,...,$ are possible poles. To analyze this function, we would like to expand it around each pole $\eta_{j}$. Let us fix an index $j.$ The coefficient before $\frac{1}{\left( x-\eta _{j}\right) ^{5}}$ is $$\begin{aligned} & \left( -24+2\left( -\frac{3}{2}\right) \left( -2\right) +\left( -\frac{3}{2}\right) \left( -2\right) \right) \beta_{j}a_{j}\\ & +\left( -3\right) \left( 3\left( \frac{1}{2}\right) \left( -2\right) +2\left( \frac{1}{2}\right) \left( -2\right) \right) \beta_{j}a_{j}\\ & =0. \end{aligned}$$ Therefore, $\left( \mathcal{K\nabla}H_{k}\right) ^{\left( 1\right) }$ does not have pole of order $5.$ Next we consider the term $\frac{1}{\left( x-\eta_{j}\right) ^{4}}.$ We see that it only comes from $$\left( D^{3}+qD+Dq\right) \left( b_{j}n_{j}\right) .$$ The coefficient vanishes, due to the fact that $$b_{j}\left( \left( -1\right) \left( -2\right) \left( -3\right) +\left( -\frac{3}{2}\right) \left( -1\right) +\left( -\frac{3}{2}\right) \left( -3\right) \right) =0.$$ For the term $\frac{1}{\left( x-\eta_{j}\right) ^{3}},$ it comes from $$\begin{aligned} & \left( 2q-\frac{1}{4}\right) %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( \beta_{j}a_{j}m_{j}^{\prime}\right) +q^{\prime}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( \beta_{j}a_{j}m_{j}\right) \\ & +q^{\prime}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( b_{j}n_{j}\right) -9\left( p+\alpha\right) %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( a_{j}m_{j}^{\prime}\right) -6p^{\prime}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( a_{j}m_{j}\right) . \end{aligned}$$ The coefficient $I_{3}$ equals $$\begin{aligned} & 2\left( -\frac{3}{2}\right) \Sigma_{k}^{\prime}\left( \left( -2\right) \beta_{j}a_{j}m_{k}\left( x_{j}\right) \right) -\frac{1}{4}\left( -2\right) \beta_{j}a_{j}\\ & +\left( -\frac{3}{2}\right) \left( -2\right) \Sigma_{k}^{\prime}\left( \beta_{k}a_{k}m_{k}\left( x_{j}\right) +b_{k}n_{k}\left( x_{j}\right) \right) \\ & -9\left( \frac{1}{2}\right) \Sigma_{k}^{\prime}\left( -2\beta_{k}% a_{j}m_{k}\left( x_{j}\right) \right) -9\alpha\left( -2\right) a_{j}-6\left( \frac{1}{2}\right) \left( -2\right) \Sigma_{k}^{\prime }\left( \beta_{j}a_{k}m_{k}\left( x_{j}\right) \right) . \end{aligned}$$ That is $$\begin{aligned} I_{3} & =6\Sigma_{k}^{\prime}\left( \beta_{j}a_{j}m_{k}\left( x_{j}\right) \right) +\frac{1}{2}\beta_{j}a_{j}\\ & +3\Sigma_{k}^{\prime}\beta_{k}a_{k}m_{k}\left( x_{j}\right) +3\Sigma _{k}^{\prime}\left( b_{k}n_{k}\left( x_{j}\right) \right) \\ & +9\Sigma_{k}^{\prime}\left( \beta_{k}a_{j}m_{k}\left( x_{j}\right) \right) +18\alpha a_{j}+6\Sigma_{k}^{\prime}\left( \beta_{j}a_{k}% m_{k}\left( x_{j}\right) \right) . \end{aligned}$$ For the $\frac{1}{\left( x-\eta_{j}\right) ^{2}}$ term, it is related to $$\begin{aligned} & 2q% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( \beta_{j}a_{j}m_{j}^{\prime}+b_{j}n_{j}^{\prime}\right) +q^{\prime}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( \beta_{j}a_{j}m_{j}+b_{j}n_{j}\right) -\frac{1}{4}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( b_{j}n_{j}^{\prime}\right) \\ & +3p% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( -3a_{j}m_{j}^{\prime}\right) +2p^{\prime}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( -3a_{j}m_{j}\right) . \end{aligned}$$ Let us use $\Sigma_{k}^{\prime}$ to denote the summation over the index $k$ which is not equal to $j.$ Using the formula of $q$ and $p,$ we can compute its coefficient $I_{2}:$ $$\begin{aligned} & 2\left( -\frac{3}{2}\right) \left[ \left( -2\right) \Sigma_{k}% ^{\prime}\left[ \beta_{j}a_{j}m_{k}^{\prime}\left( x_{j}\right) \right] +\Sigma_{k}^{\prime}\left[ \beta_{k}a_{k}m_{k}^{\prime}\left( x_{j}\right) \right] \right] \\ & +\left( -\frac{3}{2}\right) \left[ \Sigma_{k}^{\prime}\left[ \beta _{j}a_{j}m_{k}^{\prime}\left( x_{j}\right) \right] +\left( -2\right) \Sigma_{k}\left[ \beta_{k}a_{k}m_{k}^{\prime}\left( x_{j}\right) \right] \right] \\ & +2\left( -\frac{3}{2}\right) \left( -1\right) b_{j}\Sigma_{k}^{\prime }\left( m_{k}\left( x_{j}\right) \right) +2\left( -\frac{3}{2}\right) \Sigma_{k}^{\prime}\left[ b_{k}n_{k}^{\prime}\left( x_{j}\right) \right] +\left( -\frac{3}{2}\right) \left( -2\right) \Sigma_{k}^{\prime}\left[ b_{k}n_{k}^{\prime}\left( x_{j}\right) \right] -\frac{1}{4}\left( -1\right) b_{j}\\ & -9\left( \frac{1}{2}\right) \left[ \left( -2\right) \Sigma_{k}% ^{\prime}\left[ \beta_{k}a_{j}m_{k}^{\prime}\left( x_{j}\right) \right] +\Sigma_{k}^{\prime}\left[ \beta_{j}a_{k}m_{k}^{\prime}\left( x_{j}\right) \right] \right] \\ & -6\left( \frac{1}{2}\right) \left[ \Sigma_{k}^{\prime}\left[ \beta _{k}a_{j}m_{k}^{\prime}\left( x_{j}\right) \right] +\left( -2\right) \Sigma_{k}^{\prime}\left[ \beta_{j}a_{k}m_{k}^{\prime}\left( x_{j}\right) \right] \right] . \end{aligned}$$ Then $I_{2}$ equals $$\begin{aligned} & \frac{9}{2}\Sigma_{k}^{\prime}\left[ \beta_{j}a_{j}m_{k}^{\prime}\left( x_{j}\right) \right] +6\Sigma_{k}^{\prime}\left[ \beta_{k}a_{j}% m_{k}^{\prime}\left( x_{j}\right) \right] +\frac{3}{2}\Sigma_{k}^{\prime }\left[ \beta_{j}a_{k}m_{k}^{\prime}\left( x_{j}\right) \right] \\ & +3b_{j}\Sigma_{k}^{\prime}\left( m_{k}\left( x_{j}\right) \right) +\frac{1}{4}b_{j}. \end{aligned}$$ Note that $$12\Sigma_{k}^{\prime}\left( m_{k}\left( x_{j}\right) \right) +1=\frac {1}{3}\beta_{j}^{2},$$ $$\Sigma_{k}^{\prime}\left[ \beta_{k}m_{k}^{\prime}\left( x_{j}\right) \right] =-\Sigma_{k}^{\prime}\left[ \beta_{j}m_{k}^{\prime}\left( x_{j}\right) \right] .$$ It follows that $$I_{2}=\frac{3}{2}\Sigma_{k}^{\prime}\left[ \beta_{j}\left( a_{k}% -a_{j}\right) m_{k}^{\prime}\left( x_{j}\right) \right] +\frac{1}{12}% b_{j}\beta_{j}^{2}.$$ Since $\left( a_{1},...,a_{N},b_{1},...,b_{N}\right)$ is in the tangent space of $M,$ we obtain $I_{2}=0.$ Next, we compute the $\frac{1}{\left( x-\eta_{j}\right) }$ term. It comes from $$2q% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( \beta_{j}a_{j}m_{j}^{\prime}+b_{j}n_{j}^{\prime}\right) +q^{\prime}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( \beta_{j}a_{j}m_{j}+b_{j}n_{j}\right) -9p% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( a_{j}m_{j}^{\prime}\right) -6p^{\prime}% %TCIMACRO{\dsum _{j=1}^{N}}% %BeginExpansion {\displaystyle\sum_{j=1}^{N}} %EndExpansion \left( a_{j}m_{j}\right) .$$ The corresponding coefficient $I_{1}$ is $$\begin{aligned} & 2\left( -\frac{3}{2}\right) \Sigma_{k}^{\prime}\left( \beta_{k}% a_{k}m_{k}^{\prime\prime}\left( x_{j}\right) +b_{k}n_{k}^{\prime\prime }\left( x_{j}\right) \right) +2\left( -\frac{3}{2}\right) \Sigma _{k}^{\prime}\left( \left( -2\right) \frac{1}{2}\beta_{j}a_{j}m_{k}% ^{\prime\prime}\left( x_{j}\right) \right) +2\left( -\frac{3}{2}\right) \Sigma_{k}^{\prime}\left( \left( -1\right) b_{j}m_{k}^{\prime}\left( x_{j}\right) \right) \\ & +\left( -\frac{3}{2}\right) \left( -2\right) \Sigma_{k}^{\prime}\left( \frac{1}{2}\beta_{k}a_{k}m_{k}^{\prime\prime}\left( x_{j}\right) +\frac {1}{2}b_{k}n_{k}^{\prime\prime}\left( x_{j}\right) \right) +\left( -\frac{3}{2}\right) \Sigma_{k}^{\prime}\left( \beta_{j}a_{j}m_{k}% ^{\prime\prime}\left( x_{j}\right) \right) +\left( -\frac{3}{2}\right) \Sigma_{k}^{\prime}\left( b_{j}m_{k}^{\prime}\right) \\ & -9\left( \frac{1}{2}\right) \left( \Sigma_{k}^{\prime}\left( \beta _{j}a_{k}m_{k}^{\prime\prime}\left( x_{j}\right) \right) +\Sigma _{k}^{\prime}\left( \left( -2\right) \frac{1}{2}\beta_{k}a_{j}m_{k}% ^{\prime\prime}\right) \right) \\ & -6\left( \frac{1}{2}\right) \left( \left( -2\right) \Sigma_{k}% ^{\prime}\left( \frac{1}{2}\beta_{j}a_{k}m_{k}^{\prime\prime}\left( x_{j}\right) \right) +\Sigma_{k}^{\prime}\beta_{k}a_{j}m_{k}^{\prime\prime }\right) . \end{aligned}$$ It follows that $$\begin{aligned} I_{1} & =\frac{3}{2}\Sigma_{k}^{\prime}\left( \beta_{j}a_{j}m_{k}% ^{\prime\prime}\left( x_{j}\right) \right) +\frac{3}{2}\Sigma_{k}^{\prime }\left( \beta_{k}a_{j}m_{k}^{\prime\prime}\left( x_{j}\right) \right) -\frac{3}{2}\Sigma_{k}^{\prime}\left( \beta_{j}a_{k}m_{k}^{\prime\prime }\left( x_{j}\right) \right) -\frac{3}{2}\Sigma_{k}^{\prime}\left( \beta_{k}a_{k}m_{k}^{\prime\prime}\left( x_{j}\right) \right) \\ & +\frac{3}{2}\Sigma_{k}^{\prime}\left( b_{j}m_{k}^{\prime}\left( x_{j}\right) \right) +\frac{3}{2}\Sigma_{k}^{\prime}\left( b_{k}% m_{k}^{\prime}\left( x_{j}\right) \right) . \end{aligned}$$ Note that on $M,$ we have, for each fixed index $j,$ $$\Sigma_{k}^{\prime}\left( \left( \beta_{k}+\beta_{j}\right) m_{k}^{\prime }\left( x_{j}\right) \right) =0.$$ As a consequence, $$\Sigma_{k}^{\prime}\left( \left( b_{k}+b_{j}\right) m_{k}^{\prime}\left( x_{j}\right) \right) +\Sigma_{k}^{\prime}\left( \left( \beta_{k}+\beta _{j}\right) \left( a_{j}-a_{k}\right) m_{k}^{\prime\prime}\left( x_{j}\right) \right) =0$$ This implies that $I_{1}=0.$ Now we consider the second component $\left( \mathcal{K\nabla}H_{j}\right) ^{\left( 2\right) }$ of the vector field $\mathcal{K\nabla}H_{j}.$ We have $$\left( \mathcal{K\nabla}H_{j}\right) ^{\left( 2\right) }=-L\Sigma _{j}\left( a_{j}m_{j}\right) +\left( 3\left( p-\alpha\right) D+p^{\prime }\right) \Sigma_{j}\left( \beta_{j}a_{j}m_{j}+b_{j}n_{j}\right) .$$ Similar(but more tedious, the most complicated term is $16qDq$) computation as above shows that the term $\frac{1}{\left( x-\eta_{j}\right) ^{k}}$ vanishes for $k=1,4,5,6,7.$ Let us now compute the coefficient $J_{3}$ of $\frac {1}{\left( x-\eta_{j}\right) ^{3}}.$ Recall that $$L=L_{0}-\frac{5}{4}D^{3}-2\left( qD+Dq\right) +\frac{1}{4}D$$ $$L_{0}:=D^{5}+5\left( qD^{3}+D^{3}q\right) -3\left( q^{\prime\prime }D+Dq^{\prime\prime}\right) +16qDq,$$ Observe that $D^{3}\left( q\Sigma_{j}\left( a_{j}m_{j}\right) \right)$ does not contain $\frac{1}{\left( x-\eta_{j}\right) ^{3}}$ term. Hence from the operator $L_{0},$ the contribution to the coefficient is: $$\begin{aligned} & 5\left( -\frac{3}{2}\right) \left( -24\right) \Sigma_{k}^{\prime }\left( \frac{1}{2}a_{j}m_{k}^{\prime\prime}\left( x_{j}\right) \right) \\ & -3\left( -\frac{3}{2}\right) \left( 2\left( 6\right) \Sigma _{k}^{\prime}a_{k}m_{k}^{\prime\prime}\left( x_{j}\right) +2\left( -2\right) \Sigma_{k}^{\prime}\left( a_{j}m_{k}^{\prime\prime}\left( x_{j}\right) \right) +\left( -24\right) \Sigma_{k}^{\prime}\left( \frac{1}{2}a_{k}m_{k}^{\prime\prime}\left( x_{j}\right) \right) \right) \\ & +16\left( \frac{9}{4}\right) \left( \Sigma_{k}^{\prime}\left( \left( -2\right) a_{j}m_{k}^{\prime\prime}\left( x_{j}\right) \right) +\Sigma _{k}^{\prime}\left( m_{k}\left( x_{j}\right) \right) \left( -2\right) \Sigma_{k}^{\prime}\left( \left( a_{k}+a_{j}\right) m_{k}\left( x_{j}\right) \right) \right) . \end{aligned}$$ From the operator $-\frac{5}{4}D^{3}-2\left( qD+Dq\right) +\frac{1}{4}D,$ we get $$-2\left( -\frac{3}{2}\right) \left( 2\left( -2\right) \Sigma_{k}^{\prime }\left( a_{j}m_{k}\left( x_{j}\right) \right) +\left( -2\right) \Sigma_{k}^{\prime}\left( a_{k}m_{k}\left( x_{j}\right) \right) \right) +\frac{1}{4}\left( -2a_{j}\right)$$ Finally, from $$\left( 3\left( p+\alpha\right) D+p^{\prime}\right) \Sigma_{j}\left( \beta_{j}a_{j}m_{j}+b_{j}n_{j}\right) ,$$ we obtain $$\begin{aligned} & 3\left( \frac{1}{2}\right) \Sigma_{k}^{\prime}\left( \left( -2\right) \beta_{k}\beta_{j}a_{j}m_{k}\left( x_{j}\right) \right) +3\alpha\left( -2\right) \beta_{j}a_{j}\\ & +\left( \frac{1}{2}\right) \left( -2\right) \Sigma_{k}^{\prime}\left( \beta_{j}\beta_{k}a_{k}m_{k}\left( x_{j}\right) \right) +\frac{1}{2}\left( -2\right) \Sigma_{k}^{\prime}\beta_{j}b_{k}n_{k}\left( x_{j}\right) . \end{aligned}$$ Combining these, we obtain $$\begin{aligned} J_{3} & =72\Sigma_{k}^{\prime}\left( m_{k}\left( x_{j}\right) \right) \Sigma_{k}^{\prime}\left( \left( a_{k}+a_{j}\right) m_{k}\left( x_{j}\right) \right) \\ & +12\Sigma_{k}^{\prime}\left( a_{j}m_{k}\left( x_{j}\right) \right) +6\Sigma_{k}^{\prime}\left( a_{k}m_{k}\left( x_{j}\right) \right) +\frac{1}{2}a_{j}\\ & -3\Sigma_{k}^{\prime}\left( \beta_{k}\beta_{j}a_{j}m_{k}\left( x_{j}\right) \right) -\Sigma_{k}^{\prime}\left( \beta_{j}\beta_{k}% a_{k}m_{k}\left( x_{j}\right) \right) \\ & -\Sigma_{k}^{\prime}\left( \beta_{j}b_{k}n_{k}\left( x_{j}\right) \right) -6\alpha\beta_{j}a_{j}. \end{aligned}$$ Now using the identity $$\beta_{j}^{2}+36\Sigma_{k}^{\prime}m_{k}\left( x_{j}\right) +3=0,$$ we then see that $J_{3}=\beta_{j}I_{3}.$ The coefficient of $\frac{1}{\left( x-\eta_{j}\right) ^{2}}$ is also nonzero, and can be computed in a similar way. However, to prove the assertion of the proposition, it is not necessary to know its explicit formula. In the sequel, for $j=1,...,n$, let us use $I_{j}$ to denote the coefficient of degree $-3$ term for the pole $\eta_{j}$. We have now proved that $\mathcal{K}\nabla H_{k}$ has the form $$\left( 6% %TCIMACRO{\dsum _{j=1}^{n}}% %BeginExpansion {\displaystyle\sum_{j=1}^{n}} %EndExpansion \frac{I_{j}}{\left( x-\eta_{j}\right) ^{3}},-% %TCIMACRO{\dsum _{j=1}^{n}}% %BeginExpansion {\displaystyle\sum_{j=1}^{n}} %EndExpansion \left( \frac{2\beta_{j}I_{j}}{\left( x-\eta_{j}\right) ^{3}}+\frac{B_{j}% }{\left( x-\eta_{j}\right) ^{2}}\right) \right) =\mathcal{D}\nabla H_{k+1},$$ Our next aim is to show that the vector $$\left( I_{1},...,I_{n},B_{1},...,B_{n}\right)$$ lies in the tangent space of $M$ at the point $\left( \eta_{1},...,\eta _{n},\beta_{1},...,\beta_{n}\right) .$ To see this, it will be suffice to show that $\mathcal{K}\nabla H_{k+1}$ is residue free at each pole, because due to our previous computation, this means exactly the it is in the tangent space of the locus $M$. Let us write the operator $\mathcal{K}$ as $$\mathcal{K}=% \begin{bmatrix} K_{11} & K_{12}\\ K_{21} & K_{22}% \end{bmatrix} .$$ We also write $\nabla H_{k+1}=\left( \phi_{1},\phi_{2}\right) ^{T}.$ That is, $$\left( \phi_{1},\phi_{2}\right) =\left( \Sigma_{j}\left( I_{j}\beta _{j}m_{j}+B_{j}n_{j}\right) ,-3\Sigma_{j}\left( I_{j}m_{j}\right) \right) .$$ Introducing $$\begin{aligned} \sigma & =\Sigma_{j}\left( a_{j}\beta_{j}m_{j}+b_{j}n_{j}\right) ,\\ \tau & =-3\Sigma_{j}\left( a_{j}m_{j}\right) , \end{aligned}$$ we get $$\begin{aligned} \phi_{1}^{\prime} & =K_{21}\sigma+K_{22}\tau,\label{fi}\\ \phi_{2}^{\prime} & =K_{11}\sigma+K_{12}\tau.\nonumber \end{aligned}$$ Let $l$ be a closed path around the pole $\eta_{j}$ in the complex $x$ plane. To see that the residue is zero(that is, does not have $\frac{1}{x-\eta_{j}}$ term in the Laurent expansion around $\eta_{j}$), we compute the integral $$\begin{aligned} Q & :=\int_{l}\left( \mathcal{K}\nabla H_{k+1}\right) ^{T}dx\\ & =\int_{l}\left[ K_{11}\phi_{1}+K_{12}\phi_{2},K_{21}\phi_{1}+K_{22}% \phi_{2}\right] dx. \end{aligned}$$ It is important to observe that each operator $K_{11},K_{22}$ is skew-symmetric, and moreover the adjoint of $K_{12}$ is $-K_{21},$ that is, $$\int\left( gK_{12}h\right) =-\int\left( hK_{21}g\right) .$$ This is to say that the matrix operator $\mathcal{K}$ is skew-symmetric. Integrating by parts tells us that $Q$ equals $$-\int_{l}\left[ \phi_{1}K_{11}\left( 1\right) +\phi_{2}K_{21}\left( 1\right) ,\phi_{1}K_{12}\left( 1\right) +\phi_{2}K_{22}\left( 1\right) \right] dx.$$ Let us define $\ $ $$\mu=% \begin{bmatrix} \mu_{1}\\ \mu_{2}% \end{bmatrix} :=\mathcal{K}% \begin{bmatrix} 1\\ 0 \end{bmatrix} \text{ and }v=% \begin{bmatrix} v_{1}\\ v_{2}% \end{bmatrix} :=\mathcal{K}% \begin{bmatrix} 0\\ 1 \end{bmatrix} .$$ Then for some functions $w,s,$ we have $$\mu=\mathcal{D}\left( w_{1},w_{2}\right) ^{T}\text{ and }v=\mathcal{D}% \left( s_{1},s_{2}\right) ^{T}.$$ Explicitly, $$\mu=\left( q^{\prime},p^{\prime}\right) ^{T},\text{ }v=\left( 2p^{\prime },\frac{1}{3}\left( 2q^{\prime\prime\prime}+16qq^{\prime}-2q^{\prime}\right) \right) ^{T}.$$ With these notations, $$\begin{aligned} Q & =-\int_{l}\left[ \phi_{1}\mu_{1}+\phi_{2}\mu_{2},\phi_{1}v_{1}+\phi _{2}v_{2}\right] dx\\ & =-\int_{l}\left[ \phi_{1}w_{2}^{\prime}+\phi_{2}w_{1}^{\prime},\phi _{1}s_{2}^{\prime}+\phi_{2}s_{1}^{\prime}\right] dx\\ & =\int_{l}\left[ \phi_{1}^{\prime}w_{2}+\phi_{2}^{\prime}w_{1},\phi _{1}^{\prime}s_{2}+\phi_{2}^{\prime}s_{1}\right] dx. \end{aligned}$$ Using $\left( \ref{fi}\right) ,$ we find that $Q$ is equal to $$\begin{aligned} & \int_{l}\left[ \left( K_{21}\sigma+K_{22}\tau\right) w_{2}+\left( K_{11}\sigma+K_{12}\tau\right) w_{1},\left( K_{21}\sigma+K_{22}\tau\right) s_{2}+\left( K_{11}\sigma+K_{12}\tau\right) s_{1}\right] dx\\ & =-\int_{l}\left[ \left( K_{11}w_{1}+K_{12}w_{2}\right) \sigma+\left( K_{21}w_{1}+K_{22}w_{2}\right) \tau,\left( K_{11}s_{1}+K_{12}s_{2}\right) \sigma+\left( K_{21}s_{1}+K_{22}s_{2}\right) \tau\right] dx. \end{aligned}$$ Direct computation shows that $$\begin{aligned} \mathcal{K}% \begin{bmatrix} w_{1}\\ w_{2}% \end{bmatrix} & =\left( -3\Sigma_{j}\left( s_{j}m_{j}^{\prime}\right) ,\Sigma_{j}\left( s_{j}\beta_{j}m_{j}^{\prime}+t_{j}n_{j}^{\prime}\right) \right) ^{T},\\ \mathcal{K}% \begin{bmatrix} s_{1}\\ s_{2}% \end{bmatrix} & =\left( -3\Sigma_{j}\left( \bar{s}_{j}m_{j}^{\prime}\right) ,\Sigma _{j}\left( \bar{s}_{j}\beta_{j}m_{j}^{\prime}+\bar{t}_{j}n_{j}^{\prime }\right) \right) ^{T}, \end{aligned}$$ where $\left( s_{1},...,s_{n},t_{1},...,t_{n}\right)$ and $\left( \bar {s}_{1},...,\bar{s}_{n},\bar{t}_{1},...,\bar{t}_{n}\right)$ are in the tangent space of $M.$ Then $$\begin{aligned} & \int_{l}\left( \left( K_{11}w_{1}+K_{12}w_{2}\right) \sigma+\left( K_{21}w_{1}+K_{22}w_{2}\right) \tau\right) dx\\ & =-3\int_{l}\left[ \Sigma_{k}\left( s_{k}m_{k}^{\prime}\right) \Sigma _{j}\left( a_{j}\beta_{j}m_{j}+b_{j}n_{j}\right) +\Sigma_{j}\left( s_{j}\beta_{j}m_{j}^{\prime}+t_{j}n_{j}^{\prime}\right) \Sigma_{k}\left( a_{k}m_{k}\right) \right] dx. \end{aligned}$$ Residue computation shows that this integral is zero. This finishes the proof.   ◻ We remark that this result is consistent with the results in [@Shiota], where the relation between CM hierarchy and the KP hierarchy is studied. Next we show that the $k$ the flow is trivial, if $k$ is large. **Lemma 6**. *Let $n$ be fixed. Then for $k$ large, at $y=0,$ $X_{k}^{\left( 1\right) }=0.$* *Proof.* By our choice of the parameter $a,$ if the index $k$ is an odd number, then the main order term of $X_{k}$ is $O\left( \frac{1}{x^{k+2}}\right) .$ We define $$\pi_{k}=% %TCIMACRO{\dsum \limits_{j=0}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{j=0}^{n}} %EndExpansion \eta_{j}^{k},\Pi_{k}=% %TCIMACRO{\dsum \limits_{j=0}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{j=0}^{n}} %EndExpansion \left( \beta_{j}\eta_{j}^{k}\right) .$$ Since $$\frac{1}{\left( 1-t\right) ^{2}}=\sum\limits_{k}\left( k+1\right) t^{k},$$ we can write $$\begin{aligned} q & =-\frac{3}{2}% %TCIMACRO{\dsum \limits_{k=0}^{\infty}}% %BeginExpansion {\displaystyle\sum\limits_{k=0}^{\infty}} %EndExpansion \left( x^{-k-2}\left( %TCIMACRO{\dsum \limits_{j}}% %BeginExpansion {\displaystyle\sum\limits_{j}} %EndExpansion \eta_{j}^{k}\right) \right) ,\\ p & =\frac{1}{2}% %TCIMACRO{\dsum \limits_{k=0}^{\infty}}% %BeginExpansion {\displaystyle\sum\limits_{k=0}^{\infty}} %EndExpansion \left( x^{-k-2}\left( %TCIMACRO{\dsum \limits_{j}}% %BeginExpansion {\displaystyle\sum\limits_{j}} %EndExpansion \beta_{j}\eta_{j}^{k}\right) \right) . \end{aligned}$$ It follows that $$X_{k}\left( q,p\right) =% %TCIMACRO{\dsum \limits_{j=0}^{\infty}}% %BeginExpansion {\displaystyle\sum\limits_{j=0}^{\infty}} %EndExpansion \left( -\frac{3\left( j+1\right) X_{k}^{\left( 1\right) }\pi_{j}% }{2x^{j+2}},\frac{\left( j+1\right) X_{k}^{\left( 2\right) }\Pi_{j}% }{2x^{j+2}}\right) .$$ Since the main order of $X_{k}$ is $x^{-k-2},$ we see that if $j<k,$ then $$X_{k}^{\left( 1\right) }\pi_{j}=0\text{ and }X_{k}^{\left( 2\right) }% \Pi_{j}=0.$$ However, if $k_{0}\geq n,$ then $\pi_{1},...,\pi_{k_{0}}$ form a basis of the locus. From this we deduce that if $$n<k,$$ then the first component of the flow $X_{k}$ is trivial. ◻ **Lemma 7**. *Suppose $\eta$ is a complex-valued homogeneous polynomial in $x,y$ of degree $m\ $and $$\left( D_{x}^{2}+D_{y}^{2}\right) \eta\cdot\eta=0.$$ Then $$\eta\left( x,y\right) =a\left( x^{2}+y^{2}\right) ^{j}\left( x+yi\right) ^{k},$$ where $a$ is a constant and $2j+k=m.$ In particular, if $\eta$ is real-valued, then $\eta=a\left( x^{2}+y^{2}\right) ^{m}$ for some real number $a.$* *Proof.* In the polar coordinate $\left( r,\theta\right) ,$ where $r=\sqrt {x^{2}+y^{2}},$ we can write $\eta=r^{m}g\left( \theta\right) .$ Then $$\begin{aligned} \left( D_{x}^{2}+D_{y}^{2}\right) \eta\cdot\eta & =2\left( \eta\Delta \eta-\left\vert \nabla\eta\right\vert ^{2}\right) \\ & =2r^{m}g\left( m^{2}r^{m-2}g+r^{m-2}g^{\prime\prime}\right) -2\left( m^{2}r^{2m-2}g^{2}+r^{2m-2}g^{\prime2}\right) . \end{aligned}$$ From this we obtain $$gg^{\prime\prime}-g^{\prime2}=0,$$ which implies $g\left( \theta\right) =ae^{b\theta}$ for some constants $a$ and $b.$ Since $g$ has to be $2\pi$-periodic in $\theta,$ we have $b=ki$ for some integer $k.$ It follows that $$\eta=ar^{m}\left( e^{i\theta}\right) ^{k}=ar^{m-k}\left( x+yi\right) ^{k}.$$ Setting $j=\frac{m-k}{2},$ we arrive at the desired result. ◻ Let $\tau$ be a polynomial solution of the bilinear equation $$\left( D_{x}^{4}-D_{x}^{2}-D_{y}^{2}\right) \tau\cdot\tau=0, \label{bilinear}%$$ with $\deg\left( \tau\right) =m.$ By Lemma [Lemma 7](#l1){reference-type="ref" reference="l1"}, we can assume without loss of generality that the highest degree terms of $\tau$ are of the form $$\left( x^{2}+y^{2}\right) ^{j}\left( x+yi\right) ^{k}=z^{j+k}\bar{z}% ^{j}:=\tau_{m},$$ where $z=x+yi$ and $\bar{z}=x-yi$ are complex variables. Let us denote those terms of $\tau$ with degree $m-1$ by $\tau_{m-1}.$ The previous lemma can also be proved using the $\left( z,\bar{z}\right)$ coordinate. Indeed, we have the following **Lemma 8**. *$\tau_{m-1}=a_{1}z^{j+k-1}\bar{z}^{j}+a_{2}z^{j+k}\bar{z}^{j-1}$ for some constants $a_{1},a_{2}.$ In particular, if $\tau$ is real-valued, then $$\tau_{m-1}=az^{j-1}\bar{z}^{j}+\bar{a}z^{j}\bar{z}^{j-1}.$$* *Proof.* The terms of degree $2m-3$ in the left hand side of $\left( \ref{bilinear}% \right)$ are of the form $$-\left( D_{x}^{2}+D_{y}^{2}\right) \tau_{m}\cdot\tau_{m-1}.$$ Suppose $z^{r}\bar{z}^{s}$ is a term appearing in $\tau_{m-1}$, then there holds $$D_{z}D_{\bar{z}}\left( z^{j+k}\bar{z}^{j}\right) \cdot\left( z^{r}\bar {z}^{s}\right) =0.$$ Direct computation tells us that $$\left[ \left( j+k\right) j-\left( j+k\right) s-jr+rs\right] z^{j+k+r-1}\bar{z}^{j+s-1}=0.$$ In the case of $k=0,$ we have $$j^{2}-js-jr+rs=0.$$ That is, $r=j$ or $s=j.$ If the solution is real valued, then the degree $j+s$ term has to be $$\begin{aligned} g\left( x,y\right) & :=cz^{j}\bar{z}^{s}+\bar{c}z^{s}\bar{z}^{j}\\ & =\left( a+bi\right) \left( x+yi\right) ^{j}\left( x-yi\right) ^{s}+\left( a-bi\right) \left( x+yi\right) ^{s}\left( x-yi\right) ^{j}. \end{aligned}$$ Using $r+s=m-1$ and $2j+k=m,$ we obtain $$\left( j+k\right) j-\left( j+k\right) s-j\left( m-1-s\right) +\left( m-1-s\right) s=0.$$ That is, $$-s^{2}+\left( 2j-1\right) s-j\left( j-1\right) =0.$$ Therefore, $s=j$ or $j-1.$ If in addition $\tau$ is real-valued, then $k=0$ and $\tau_{m}=z^{j}\bar {z}^{j}.$ Hence $$\tau_{m-1}=az^{j-1}\bar{z}^{j}+\bar{a}z^{j}\bar{z}^{j-1}.$$ ◻ By this lemma, in the real-valued case, if we introduce new variables $Z=z+\frac{a}{j}$ and $\bar{Z}=\bar{z}+\frac{\bar{a}}{j},$ then we see that $$z^{j}\bar{z}^{j}+az^{j-1}\bar{z}^{j}+\bar{a}z^{j}\bar{z}^{j-1}=Z^{j}\bar {Z}^{j}+P,$$ where $P$ is a polynomial of $Z,\bar{Z}$ with degree less than $j-1.$ This means that we can find real numbers $b_{1},b_{2}$ such that in the new variables $\tilde{x}=x+b_{1},\tilde{y}=y+b_{2},$ the highest degree term of $\tau$ is $\left( \tilde{x}^{2}+\tilde{y}^{2}\right) ^{j}$ and $\tau$ does not have terms with degree $2j-1.$ **Lemma 9**. *Suppose $q=\frac{3}{2}\partial_{x}^{2}\ln\tau$ is a real valued rational solution of the Boussinesq equation $\left( \ref{eq:B}\right)$, where $\tau$ is a polynomial of degree $2n.$ Let $p=\int_{-\infty}^{x}\partial _{y}qdx.$ Then for $x$ large, at $y=0,$ $$q=-\frac{3n}{x^{2}}+O\left( x^{-3}\right) ,p=O\left( \frac{1}{x^{5}% }\right) .$$* *Proof.* Since $\tau$ is real valued, after a possible translation of the coordinate(and a scaling of the $y$ variable), it has the form $$\tau\left( x,y\right) =\left( x^{2}+y^{2}\right) ^{n}+% %TCIMACRO{\dsum \limits_{j+k\leq2n-2}}% %BeginExpansion {\displaystyle\sum\limits_{j+k\leq2n-2}} %EndExpansion \left( a_{jk}x^{j}y^{k}\right) .$$ Therefore, $q=\frac{3n}{x^{2}}+O\left( x^{-3}\right) .$ We also have $$q=\frac{3n}{2}\partial_{x}^{2}\ln\left( x^{2}+y^{2}\right) +2\partial _{x}^{2}\ln\left( \frac{1}{\left( x^{2}+y^{2}\right) ^{n}}% %TCIMACRO{\dsum \limits_{j+k\leq2n-2}}% %BeginExpansion {\displaystyle\sum\limits_{j+k\leq2n-2}} %EndExpansion \left( a_{jk}x^{j}y^{k}\right) \right) .$$ Hence $$\partial_{y}q=2n\left( \frac{i}{\left( x+yi\right) ^{3}}-\frac{i}{\left( x-yi\right) ^{3}}\right) +O\left( x^{-5}\right) .$$ The result then readily follows. ◻ Now we are at a position to prove the following **Theorem 10**. *Assume that $\tilde{q}$ is a real valued rational solution of the Boussinesq equation $\left( \ref{eq:B}\right)$ with $$\tilde{q}\left( x,y\right) \rightarrow0,\text{ as }x^{2}+y^{2}% \rightarrow+\infty.$$ Then $\tilde{q}=\frac{3}{2}\partial_{x}^{2}\ln\tau,$ where $\tau$ is a polynomial in $x,y$ with degree $k\left( k+1\right) ,$ $k\in\mathbb{N}.$* *Proof.* Since $\left( 3a\right) ^{2}+\frac{1}{48}=0,$ we compute $$% \begin{bmatrix} -\frac{D}{4} & -3aD\\ -3aD & \frac{D}{12}% \end{bmatrix}% \begin{bmatrix} 0 & D^{-1}\\ D^{-1} & 0 \end{bmatrix}% \begin{bmatrix} -\frac{D}{4} & 3aD\\ 3aD & \frac{D}{12}% \end{bmatrix} =0.$$ This identity guarantees that if the main order term of the $X_{j}$ is $x^{-k},$ then the main order term of $X_{j+2}$ will be at the order $x^{-k-2}.$ For $x$ large, the main order term of $\tilde{q}$ is $\frac{m}{x^{2}}.$ Since the degree of the polynomial is expected to be $k\left( k+1\right) ,$ we expect $m$ to be $-\frac{3}{2}k\left( k+1\right) .$ We compute $$\begin{aligned} & \left( D^{3}+qD+Dq\right) \left( \frac{1}{x^{j}}\right) \\ & =\left[ -j\left( j+1\right) \left( j+2\right) -m\left( j+j+2\right) \right] \frac{1}{x^{j+3}}\\ & =-\left( j+1\right) \left( j\left( j+2\right) +2m\right) \frac {1}{x^{j+3}}:=b_{m}\left( j\right) \frac{1}{x^{j+3}}. \end{aligned}$$ Similarly, $$\begin{aligned} & \left( -\frac{5}{4}D^{3}-2\left( qD+Dq\right) \right) \left( \frac {1}{x^{j}}\right) \\ & =\frac{5}{4}j\left( j+1\right) \left( j+2\right) +2m\left( j+j+2\right) \frac{1}{x^{j+3}}\\ & =\frac{1}{8}\left( j+1\right) \left( 10j\left( j+2\right) +32m\right) \frac{1}{x^{j+3}}=:B_{m}\left( j\right) \frac{1}{x^{j+3}}. \end{aligned}$$ Vanishing of terms requires $$\frac{1}{4}b_{m}\left( j\right) -\frac{1}{4}B_{m}\left( j\right) =0.$$ That is, $$m=-\frac{3}{8}j\left( j+2\right) .$$ Let $j=2k,$ we find that $$m=-\frac{3}{2}k\left( k+1\right) .$$ This completes the proof. ◻ # Even solutions from numerical point of view In this section, we would like to carry out some numerical computation to verify that the degree of the corresponding $\tau$ function of the even solution is $k\left( k+1\right) .$ Suppose $q$ is an even solution, that is, $q\left( x,y\right) =q\left( x,-y\right) =q\left( -x,y\right) .$ We assume that the degree of its tau function $f$ is $2n.$ From Lemma [Lemma 7](#l1){reference-type="ref" reference="l1"}, we can assume that the sum of the degree $2n$ terms of $f$ is $T_{n}=\left( x^{2}+y^{2}\right) ^{n}.$ We also denote the sum of the degree $2n-2j$ terms of $f$ by $T_{n,j}.$ Let us define functions $$g_{j}:=\left( x^{2}+y^{2}\right) ^{n-3j}x^{2j}y^{2j}\text{, \ }\xi _{i,j}:=\left( D_{x}^{2}+D_{y}^{2}\right) g_{i}\cdot g_{j}.$$ Observe that actually $\xi_{i,j}$ can be divided by $\left( x^{2}% +y^{2}\right) ^{2n-3i-3j-1}.$ We introduce the constants $$d_{i,j}:=\frac{\left( D_{x}^{2}+D_{y}^{2}\right) g_{i}\cdot g_{j}}{\left( x^{2}+y^{2}\right) ^{2n-3i-3j-1}}\bigg|_{\left( x^{2}=-1,y^{2}=1\right) }.$$ **Lemma 11**. *$d_{i,j}=-12\left( i-j\right) ^{2}\left( -1\right) ^{i+j}.$* *Proof.* We compute $$\left( D_{x}^{2}+D_{y}^{2}\right) g_{i}\cdot g_{j}=\Delta g_{i}g_{j}% +g_{i}\Delta g_{j}-2\nabla g_{i}\cdot\nabla g_{j}.$$ Note that $$\begin{aligned} \Delta g_{j} & =\left( 2n-6j\right) ^{2}r^{2n-6j-2}x^{2j}y^{2j}+8j\left( 2n-6j\right) r^{2n-6j-2}x^{2j}y^{2j}+r^{2n-6j}\Delta\left( x^{2j}% y^{2j}\right) \\ & =\left( 2n+2j\right) \left( 2n-6j\right) r^{2n-6j-2}x^{2j}% y^{2j}+r^{2n-6j}\Delta\left( x^{2j}y^{2j}\right) . \end{aligned}$$ $$\begin{aligned} & \nabla g_{i}\cdot\nabla g_{j}\\ & =\left( x^{2}+y^{2}\right) ^{2n-3i-3j-2}x^{2i+2j-2}y^{2i+2j}\left( \left( 2n-6i\right) x^{2}+2i\left( x^{2}+y^{2}\right) \right) \left( \left( 2n-6j\right) x^{2}+2j\left( x^{2}+y^{2}\right) \right) \\ & +\left( x^{2}+y^{2}\right) ^{2n-3i-3j-2}x^{2i+2j}y^{2i+2j-2}\left( \left( 2n-6i\right) y^{2}+2i\left( x^{2}+y^{2}\right) \right) \left( \left( 2n-6j\right) y^{2}+2j\left( x^{2}+y^{2}\right) \right) . \end{aligned}$$ Therefore, $$\begin{aligned} \frac{\left( D_{x}^{2}+D_{y}^{2}\right) g_{i}\cdot g_{j}}{\left( x^{2}+y^{2}\right) ^{2n-3i-3j-1}}|_{x^{2}=-1,y^{2}=1} & =\left( 2n+2i\right) \left( 2n-6i\right) \left( -1\right) ^{i+j}\\ & +\left( 2n+2j\right) \left( 2n-6j\right) \left( -1\right) ^{i+j}\\ & -2\left( \left( 2n-6i\right) \left( 2n-6j\right) +2\left( 2n-6i\right) 2j+2\left( 2n-6j\right) 2i\right) \left( -1\right) ^{i+j}\\ & =-12\left( i-j\right) ^{2}\left( -1\right) ^{i+j}. \end{aligned}$$ ◻ Let us now consider the function $D_{x}^{4}\left( x^{2}+y^{2}\right) ^{n-3i}\cdot\left( x^{2}+y^{2}\right) ^{n-3j}.$ Since we have taken the fourth order derivative, this function is dividable by $\left( x^{2}% +y^{2}\right) ^{2n-3i-3j-4}.$ We also define $$p_{i,j}=\frac{D_{x}^{4}g_{i}\cdot g_{j}}{\left( x^{2}+y^{2}\right) ^{2n-3i-3j-4}}\bigg|_{\left( x^{2}=-1,y^{2}=1\right) .}%$$ These constants depending on $n,i,j.$ Explicitly, $\left( -1\right) ^{i+j}p_{i,j}$ is equal to $$\begin{aligned} & 1296i^{4}-5184i^{3}j+7776i^{2}j^{2}-5184ij^{3}+1296j^{4}+2592i^{3}% -2592i^{2}j\\ & -1728i^{2}n-2592ij^{2}+3456ijn+2592j^{3}-1728j^{2}n+1584i^{2}-1440ij\\ & -576in+1584j^{2}-576jn+192n^{2}+288i+288j-192n. \end{aligned}$$ In the special case $i=j,$ $$p_{i,j}=\left( -1\right) ^{i+j}192\left( 3j-n+1\right) \left( 3j-n\right) .$$ When $i=j-1,$ we have $p_{i,j}=\left( -1\right) ^{i+j}192\left( 3j-n+7\right) \left( 3j-n\right) .$ If $i=j-2,$ then $$p_{i,j}=\left( -1\right) ^{i+j}192\left( 3j-n+30\right) \left( 3j-n+1\right) .$$ Moreover, if $i=0,$ then $$p_{i,j}=1296j^{4}+2592j^{3}-1728j^{2}n+1584j^{2}-576jn+192n^{2}+288j-192n.$$ Now we would like to define a sequence of numbers $a_{m},m=0,1,...,$ depending on $n,$ in the following way. Take $a_{0}=1.$ Then $a_{m}$ is determined by $a_{1},...,a_{m-1}$ through the following recursive relation: $$\sum\limits_{i,j\leq m,i+j=m}a_{i}a_{j}d_{i,j}=\sum\limits_{i,j\leq m,i+j=m-1}a_{i}a_{j}p_{i,j}.$$ We regard $a_{j}$ as a polynomial of the variable $n.$ Now let us define the constant $$J_{n}:=\sum\limits_{i,j\leq\lbrack n/3],i+j=[n/3]+1}a_{i}a_{j}d_{i,j}% -\sum\limits_{i,j\leq\lbrack n/3],i+j=[n/3]}a_{i}a_{j}p_{i,j}.$$ **Proposition 12**. *Let $n$ be a fixed integer. If $J_{n}\neq0,$ then the Boussinesq equation has no rational even solution with degree $2n.$* *Proof.* First of all, we claim that $T_{n,j}$ has the form $$a_{j}\left( x^{2}+y^{2}\right) ^{n-3j}x^{2j}y^{2j}+\left( x^{2}% +y^{2}\right) ^{n-3j+1}\Gamma\left( x,y\right) ,$$ where $\Gamma$ is a homogeneous polynomial in $x,y$ with degree $4j-2.$ Indeed, Let us denote the function $\left( D_{x}^{4}-D_{x}^{2}-D_{y}^{2}\right) f\cdot f$ by $K_{f}.$ Since we have chosen $a_{0}$ to be $1,$ $K_{f}$ is a polynomial of degree at most $4n-4.$ The terms with degree $4n-4$ are given by $$D_{x}^{4}T_{n,0}\cdot T_{n,0}-\left( D_{x}^{2}+D_{y}^{2}\right) T_{n,0}\cdot T_{n,1}.$$ This function is dividable by $\left( x^{2}+y^{2}\right) ^{2n-4}.$ We write is as $$b_{1}\left( x^{2}+y^{2}\right) ^{2n-4}x^{2}y^{2}+\left( x^{2}+y^{2}\right) ^{2n-3}M\left( x,y\right) .$$ Inserting $x^{2}=-1,y^{2}=1$ into this function, we find that necessary $b_{1}=0.$ Therefore, we get $$a_{0}^{2}p_{0,0}-a_{0}a_{1}d_{0,1}=0.$$ Similarly, consider the terms with degree $4n-6,$ we get $$D_{x}^{4}T_{n,0}\cdot T_{n,1}-\left( D_{x}^{2}+D_{y}^{2}\right) T_{n,1}\cdot T_{n,1}-\left( D_{x}^{2}+D_{y}^{2}\right) T_{n,0}\cdot T_{n,2}=0.$$ Then $$a_{0}a_{1}p_{0,1}-a_{1}^{2}d_{1,1}-a_{0}a_{2}d_{0,2}=0.$$ Similarly, for $m\leq\left[ n/3\right] ,$ $$\sum\limits_{i,j\leq m,i+j=m}a_{i}a_{j}d_{i,j}=\sum\limits_{i,j\leq m,i+j=m-1}a_{i}a_{j}p_{i,j}.$$ Since we require that the solution is a polynomial, the function $$\sum\limits_{i,j\leq\lbrack n/3],i+j=[n/3]+1}a_{i}a_{j}\left( D_{x}^{2}% +D_{y}^{2}\right) T_{n,i}\cdot T_{n,j}-\sum\limits_{i,j\leq\left[ n/3\right] ,i+j=\left[ n/3\right] }a_{i}a_{j}D_{x}^{4}T_{n,i}\cdot T_{n,j}%$$ should dividable by $\left( x^{2}+y^{2}\right) ^{n-1},$ this implies that  $J_{n}=0.$ ◻ We have computed the constants $a_{j}$ and $J_{n},$ using mathematical software. It turns out that at least for $n\leq300,$ $J_{n}$ is equal to zero if and only if $n=\frac{k\left( k+1\right) }{2}$ for some integer $k.$ 99 M.J. Ablowitz, S. Chakravarty, A.D. Trubatch, J. Villarroel, A novel class of solutions of the non-stationary Schrödinger and the Kadomtsev-Petviashvili I equations, *Phys. Lett.* A 267 (2000), no. 2-3, 132-146. H. Airault, Solutions of the Boussinesq equation. *Phys. D* 21 (1986), no. 1, 171-176. H. Airault, H.P. McKean, J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. *Comm. Pure Appl. Math.* 30 (1977), no. 1, 95-148. R. Beals, P. Deift, C. Tomei, Direct and inverse scattering on the line. Mathematical Surveys and Monographs, 28. American Mathematical Society, Providence, RI, 1988. R. Beals, R.R. Coifman, Scattering and inverse scattering for first order systems. *Comm. Pure Appl. Math.* 37 (1984), no. 1, 39-90. R. Beals, R.R. Coifman, Scattering and inverse scattering for first-order systems. II. *Inverse Problems*, 3 (1987), no. 4, 577-593. F. Bethuel, P. Gravejat, J.C. Saut, On the KP-I transonic limit of two-dimensional Gross-Pitaevskii travelling waves, *Dynamics of Partial Differential Equations.* 5 (2008) 241-280. D. Bilman, P.D. Miller, A robust inverse scattering transform for the focusing nonlinear Schrödinger equation. *Comm. Pure Appl. Math.* 72 (2019), no. 8, 1722-1805. L.V. Bogdanov, V.E. Zakharov, The Boussinesq equation revisited. *Phys. D* 165 (2002), no. 3-4, 137-162. S. Chakravarty, M. Zowada, Classification of KP-I lumps, *J. Phys. A: Math. Theor.* 55 (2022) 215701. S. Chakravarty, M. Zowada, Multi-lump wave patterns of KP-I via integer partitions, *Physica D*. 446 (2023) 133644. P.A. Clarkson, Rational solutions of the Boussinesq equation. *Anal. Appl.* 6 (2008), no. 4, 349-369. P. Deift, C. Tomoi, E. Trubowitz, Inverse scattering and the Boussinesq equation, *Communications on Pure and Applied Mathematics*, 35, (1982), 567-628. P. Deift, X. Zhou, Direct and inverse scattering on the line with arbitrary singularities. *Comm. Pure Appl. Math.* 44 (1991), no.5, 485-533. A. de Bouard, J.C. Saut, Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves. *SIAM J. Math. Anal.* 28 (1997), no. 5, 1064--1085. R. Dickson, F. Gesztesy, K. Unterkofler, A new approach to the Boussinesq hierarchy. *Math. Nachr.* 198 (1999), 51-108. P.G. Estevez, J. Prada, Lump solutions for PDE's: Algorithmic construction and Classification, *Journal of nonlinear Mathematical Physics*, 15 (2008), suppl. 3, 166-175. A.S. Fokas, M.J. Ablowitz, On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili equation. *Stud. Appl. Math.* 69 (1983), no. 3, 211-228. K.A. Gorshkov, D.E. Pelinovskii, Y.A. Stepanyants, Normal and anomalous scattering, formation and decay of bound states of two-dimensional solitons described by the Kadomtsev-Petviashvili equation, *Soviet Journal of Experimental and Theoretical Physics Letters*, 77(2), 1993, 237-245. V.M. Galkin, D.E. Pelinovsky; Y.A. Stepanyants, The structure of the rational solutions to the Boussinesq equation. *Phys. D* 80 (1995), no. 3, 246-255. K. Hikami, M. Wadati, On additional symmetry: The many body problem related to the KP hierarchy, *Physics Letter A*, 191(1994), 87--90. M. Ito, Symmetries and conservation laws of the classical Boussinesq equation, *Physics Letters A*, 104 (1984) 248--250. V.G. Kac, J.W. van de Leur, Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions, *Japan. J. Math.* 13, (2018), 235-271. B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media. *Soviet Physics-Doklady*, 15, (1970), 539--541. C.E. Kenig, On the local and global well-posedness theory for the KP-I equation, *Ann. Inst. H. Poincare Anal. Non-Linire*. 21 (2004), 827--838. C. Klein, J.C. Saut, IST versus PDE: a comparative study, Fields Inst. Commun., 75 Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2015, 383-449. C. Klein, J.C. Saut, Nonlinear Dispersive Equations: Inverse Scattering and PDE Methods, Springer International Publishing, Cham, 2021. I.M. Krichever, Rational solutions of the Kadomtsev--Petviashvili equation and integrable systems of $N$ particles on a line, *Functional Analysis and Its Applications*, 1978, 12(1), 59-61. I.M. Krichever, Rational solutions of the Zakharov-Shabat equations and completely integrable systems of $N$ particles on a line, *Journal of Soviet Mathematics*, 21, 1983, 335--345. Q.M. Liu, X.B. Hu, Y. Li, Rational solutions of the classical Boussinesq hierarchy, *J. Phy. A: Math. Gen.* 23 (1990), 585-591. Y. Liu, J. Wei, Nondegeneracy, Morse Index and Orbital Stability of the KP-I Lump Solution, *Arch Rational Mech Anal.* 234 (2019) 1335-1389. Y. Liu, Z.P. Wang, J.C. Wei, W. Yang, From KP-I lump solution to travelling waves of Gross-Pitaevskii equation, arXiv:2110.15472, preprint. Y. Liu, J. Wei, Classification of finite Morse index solutions to the elliptic sine-Gordon equation in the plane, *Rev. Mat. Iberoam.* 38 (2021) 355-432. H.P. McKean, Boussinesq's equation as a Hamiltonian system. Topics in functional analysis (essays dedicated to M. G. Krein on the occasion of his 70th birthday), pp. 217--226, Adv. Math. Suppl. Stud., 3, Academic Press, New York-London, 1978. S.V. Manakov, The inverse scattering transform for the time-dependent Schrodinger equation and Kadomtsev-Petviashvili equation, *Physica D*, 3(1981), 420--427. S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its, V.B. Matveev, Two dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. *Physics letter A*, 63, 1977, 205-206. L. Molinet, J.C. Saut, N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, *Duke Math. J.* 115 (2002). L. Molinet, J.C. Saut, N. Tzvetkov, Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution, *Commun. Math. Phys.* 272 (2007), 775--810. D.E. Pelinovskii, Y.A. Stepanyants, New multisoliton solutions of the Kadomtsev-Petviashvili equation, *Soviet Journal of Experimental and Theoretical Physics Letters*, 57(1), 1993, 24-28. D.E. Pelinovsky, Rational solutions of the Kadomtsev-Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rational solution. *J. Math. Phys.* 35 (1994), no. 11, 5820-5830. D.E. Pelinovsky, Rational solutions of the KP hierarchy and the dynamics of their poles. II. Construction of the degenerate polynomial solutions. *J. Math. Phys.* 39 (1998), no. 10, 5377-5395. J. Satsuma, M.J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems. *J. Math. Phys.* 20(7), 1496-1503, 1979. H. Segur, Comments on inverse scattering for the Kadomtsev-Petviashvili equation. Mathematical methods in hydrodynamics and integrability in dynamical systems (La Jolla, Calif., 1981), 211--228, AIP Conf. Proc., 88, Amer. Inst. Physics, New York, 1982. T. Shiota, Calogero-Moser hierarchy and KP hierarchy, *Journal of Mathematical Physics*, 35, 5844 (1994). J. Villarroel, M.J. Ablowitz, On the discrete spectrum of the nonstationary Schrodinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation. *Comm. Math. Phys*. 207 (1999), no. 1, 1-42. B. Yang, J. Yang, General Rogue Waves in the Boussinesq Equation, *J. Phys. Soc. Jpn.* 89 (2020) 024003. V.E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, *Soviet Physics JETP*, Vol. 38, p.108, 1974. X. Zhou, Inverse scattering transform for the time dependent Schrodinger equation with applications to the KPI equation, *Comm. Math. Phys.* 128 (1990), no. 3, 551--564. X. Zhou, Direct and inverse scattering transforms with arbitrary spectral singularities. *Comm. Pure Appl. Math.* 42 (1989), no. 7, 895--938. X. Zhou, The Riemann--Hilbert Problem and Inverse Scattering, *SIAM J. Math. Anal.* 20 (1989) 966-986.

arxiv_math

--- abstract: | In this article, we provide some necessary and sufficient coefficients conditions for a harmonic mapping to be hereditarily spirallike. Also, we give growth estimate for certain harmonic hereditarily spirallike mappings. Moreover, we connect the concept of harmonic hereditarily spirallike mapping to analytic spirallike mapping and provide some examples in support of our results. address: - Md Firoz Ali, Department of Mathematics, National Institute of Technology Durgapur, 713209, West Bengal, India. - Sushil Pandit, Department of Mathema tics, National Institute of Technology Durgapur, 713209, West Bengal, India. author: - Md Firoz Ali\* - Sushil Pandit title: On Harmonic Univalent Spirallike Mappings --- [^1] # Introduction Let $\mathcal{A}$ denote the class of all analytic functions $h$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ of the form $$\begin{aligned} \label{p3-001} h(z)=z+\sum\limits_{n=2}^\infty a_nz^n,\end{aligned}$$ and $\mathcal{S}$ be the class of univalent functions in $\mathcal{A}.$ An analytic function $h\in\mathcal{S}$ is called starlike if it maps the unit disk $\mathbb{D}$ onto a starlike domain with respect to origin i.e., if it satisfies ${\rm Re\,}\left(zh'(z)/h(z)\right)>0$ for $z\in\mathbb{D}\setminus\{0\}.$ Similarly, an analytic function $h\in\mathcal{S}$ is called convex if it maps the unit disk $\mathbb{D}$ onto a convex domain i.e., if it satisfies ${\rm Re\,}\left(1+zh''(z)/h'(z)\right)>0$ for $z\in\mathbb{D}\setminus\{0\}.$ A domain $\Omega$ containing the origin is called $\lambda-$ spirallike if for each point $w_0$ in $\Omega,$ the arc of $\lambda-$ spiral from origin to $w_0$ lies entirely in $\Omega.$ For $\lambda\in(-\pi/2,\pi/2),$ a function $h\in\mathcal{S}$ is called $\lambda-$ spirallike if $h(\mathbb{D})$ is a $\lambda-$ spirallike domain i.e., if ${\rm Re\,}\left(e^{-i\lambda}zh'(z)/h(z)\right)>0$ for $z\in\mathbb{D}\setminus\{0\}.$ For more information about these classes, we refer to [@Duren-1983].\ The differential operator $D$ for a continuously differentiable function $f\in C^1(\mathbb{D})$ is defined by $$Df(z)=zf_z(z)-\overline{z}f_{\overline{z}}(z),$$ where $f_z=(f_x-if_y)/2$ and $f_{\overline{z}}=(f_x+if_y)/2.$ Here $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to $x={\rm Re\,}z$ and $y={\rm Im\,}z,$ respectively. A continuous twice differentiable complex valued function $f=u+iv$ is called harmonic in a complex domain $\Omega$ if both $u$ and $v$ are real harmonic in $\Omega.$ In any simply connected domain $\Omega,$ every harmonic mapping $f$ can be represented as $f=h+\overline{g}$ where $h$ and $g$ are analytic in $\Omega.$ This representation is known as canonical representation and $h$ is called analytic part whereas $g$ is called co-analytic part of $f.$ A harmonic mapping $f=h+\overline{g}$ is sense-preserving if the Jacobian $J_f=|h'|^2-|g'|^2$ is positive and sense-reversing if $J_f$ is negative. Lewy [@Lewy-1936] showed that $f$ is locally univalent if $J_f$ is non vanishing. Let $\mathcal{H}$ be class of all sense-preserving harmonic functions $f=h+\overline{g}$ in $\mathbb{D}$ with $$\begin{aligned} \label{p3-010} h(z)=z+\sum\limits_{n=2}^\infty a_nz^n,~g(z)=\sum\limits_{n=1}^\infty b_nz^n\end{aligned}$$ and $\mathcal{H}'$ be class of harmonic functions $f=h+\overline{g}$ in $\mathcal{H}$ with the representation $$\begin{aligned} \label{p3-020} h(z)=z-\sum\limits_{n=2}^\infty |a_n|z^n, ~g(z)=\sum\limits_{n=1}^\infty |b_n|z^n.\end{aligned}$$ Let $\mathcal{S}_H$ denote the family of functions in $\mathcal{H}$ that are univalent. In 1984, Clunie and Sheil-Small [@Clunie-Small-1984] studied several geometric properties of functions in $\mathcal{S}_H$ alongwith functions in its subclasses of starlike functions, convex functions, close-to-convex functions, etc.\ It is well known that the properties convexity and starlikeness of an analytic and univalent mapping are hereditary i.e., if an analytic and univalent function $f$ is convex or starlike then $f(\mathbb{D}_r)$ is a convex or starlike domain, respectively for each $0<r<1,$ where $\mathbb{D}_r=\{z\in\mathbb{C}:|z|<r\}.$ But the properties convexity and starlikeness are not hereditary for a harmonic univalent map $f$. For example, the half-plane harmonic mapping $l(z)=h(z)+\overline{g(z)}$ with $$\begin{aligned} h(z)=\frac{z-\frac{1}{2}z^2}{(1-z)^2},\quad g(z)=\frac{-\frac{1}{2}z^2}{(1-z)^2}\end{aligned}$$ maps $\mathbb{D}$ univalently onto ${\rm Re\,}w >-1/2$, whereas $l(\mathbb{D}_r)$ is not convex for $\sqrt{2}-1<r<1$ (see [@Ma-Ponnusamy-Sugawa-2022]). On the other hand, the harmonic Koebe function $k(z)=h(z)+\overline{g(z)}$ with $$\begin{aligned} h(z)=\frac{z-\frac{1}{2}z^2+\frac{1}{6}z^3}{(1-z)^3},\quad g(z)=\frac{\frac{1}{2}z^2+\frac{1}{6}z^3}{(1-z)^3}\end{aligned}$$ map $\mathbb{D}$ onto a slit domain $\mathbb{C}\setminus(-\infty,-1/6)$ in one-to-one manner (see [@Clunie-Small-1984]) but, recently, Ma et al. [@Ma-Ponnusamy-Sugawa-2022] showed that $k(\mathbb{D}_r)$ is not starlike for $r=\sqrt{5}/3.$ Due to this phenomena, in 2004, Chuaqui et al. [@Chuaqui-Duren-Osgood-2004] introduced the concept of fully convexity and fully starlikeness for harmonic mappings. A harmonic mapping $f=h+\overline{g}\in\mathcal{H}$ is called fully convex if it maps every circle $C_r=\{z\in\mathbb{C}:|z|=r\},~0<r<1,$ in an one-to-one manner onto a convex curve. It is well known that, a sense-preserving harmonic mapping $f=h+\overline{g}\in\mathcal{H}$ is fully convex if ${\rm Re\,}\left(\frac{D^2f(z)}{Df(z)}\right)>0$ for every $z=re^{i\theta}\in\mathbb{D}\setminus\{0\}.$ Similarly, a harmonic mapping $f=h+\overline{g}\in\mathcal{H}$ is said to be fully starlike if it maps every circle $C_r,~0<r<1,$ in an one-to-one manner onto a curve that bounds a domain starlike with respect to the origin. It is well known that, a sense-preserving harmonic mapping $f=h+\overline{g}\in\mathcal{H}$ with $f(0)=0$ if and only if $z=0$ is fully starlike if ${\rm Re\,}\left(\frac{Df(z)}{f(z)}\right)>0$ for every $z=re^{i\theta}\in\mathbb{D}\setminus\{0\}.$ In the same paper [@Chuaqui-Duren-Osgood-2004], authors showed that a harmonic fully convex function $f=h+\overline{g}\in\mathcal{H}$ is also univalent but a harmonic fully starlike function $f=h+\overline{g}\in\mathcal{H}$ may not be even locally univalent. As in [@Ma-Ponnusamy-Sugawa-2022], we call a harmonic fully starlike and univalent mapping $f=h+\overline{g}\in\mathcal{H}$ as harmonic hereditarily starlike function. Let $\mathcal{ST}_H$ denotes the class of harmonic mappings $f\in\mathcal{H}$ which are hereditarily starlike and $\mathcal{ST}_{H'}$ be the class of harmonic mappings $f\in\mathcal{ST}_H$ of the form [\[p3-020\]](#p3-020){reference-type="eqref" reference="p3-020"}. Different geometric aspects of starlikeness and hereditarily starlikeness for harmonic mappings in the class $\mathcal{H}$ and $\mathcal{H}'$ have been studied in [@Chuaqui-Duren-Osgood-2004; @Jahangiri-1999; @Mocanu-1980; @Nagpal-Ravichandran-2013; @Ponnusamy-Kaliraj-2014].\ Clunie and Sheil-Small [@Clunie-Small-1984] successfully extended the theory of starlikeness, convexity and close-to-convexity from analytic functions to harmonic functions. So it is natural to think whether spirallkeness can be extended from analytic functions to harmonic functions. Affirmatively, Ma et al. [@Ma-Ponnusamy-Sugawa-2022] introduced the concept of hereditarily spirallikeness for harmonic mappings in the unit disk $\mathbb{D}$ and investigated its geometric properties. For a real number $\lambda$ with $|\lambda|<\frac{\pi}{2}$, a harmonic function $f=h+\overline{g}$ in $\mathcal{H}$ is called hereditarily $\lambda-$spirallike if $f$ is univalent on $\mathbb{D}$ and $f(\mathbb{D}_r)$ is a $\lambda-$spirallike domain for each $r<1.$ The following analytic characterization of hereditarily spirallike functions has been obtained in [@Ma-Ponnusamy-Sugawa-2022]. **Lemma 1**. *[@Ma-Ponnusamy-Sugawa-2022] [\[p3-030\]]{#p3-030 label="p3-030"} Let $\lambda\in (-\pi/2,\pi/2)$. Suppose that a function $f\in C^1(\mathbb{D})$ satisfies the condition that $f(z)=0$ if and only if $z=0,$ and that $J_f=|f_z|^2-|f_{\overline{z}}|^2>0$ on $\mathbb{D}.$ Then $f$ is one-one in $\mathbb{D}$ and $f(\mathbb{D}_r)$ is $\lambda-$spirallike for each $0<r<1$ if and only if $$\begin{aligned} {\rm Re\,}\left(e^{-i\lambda}\frac{Df(z)}{f(z)}\right)>0,\quad z\in\mathbb{D}\setminus\{0\}.\end{aligned}$$* Let $\mathcal{SP}_H(\lambda)$ and $\mathcal{SP}_{H'}(\lambda)$ denote the classes of harmonic hereditarily $\lambda-$spirallike functions in $\mathcal{H}$ and $\mathcal{H}',$ respectively.\ In this article, we provide some necessary and sufficient conditions for a harmonic mapping to be hereditarily spirallike function. Also, we give growth estimate for certain harmonic hereditarily spirallike mappings. Moreover, we connect this concept of harmonic hereditarily spirallike mapping with analytic spirallike function and develop some examples in support of our results. In Section [2](#Background And Main Results){reference-type="ref" reference="Background And Main Results"}, we present our results and examples. In Section [3](#Proof of Main Results){reference-type="ref" reference="Proof of Main Results"}, we provide detail proof of the results. # Main Results {#Background And Main Results} In 2004, Chuaqui et al. [@Chuaqui-Duren-Osgood-2004] obtained a necessary and sufficient coefficient condition for harmonic fully starlike and harmonic fully convex mappings. In this context, we obtain a similar result for harmonic hereditarily spirallike mappings. **Theorem 1**. *Let $f=h+\overline{g}$ be a sense-preserving harmonic mapping on $\mathbb{D}$ such that $f(0)=0$ only for $z=0.$ Then $f\in\mathcal{SP}_H(\lambda)$ if and only if $$\begin{aligned} \label{p3-050} |h(z)|^2{\rm Re\,}\left(e^{-i\lambda}\frac{zh'(z)}{h(z)}\right) &>|g(z)|^2{\rm Re\,}\left(e^{i\lambda}\frac{zg'(z)}{g(z)}\right)\\ &\qquad\quad +{\rm Re\,}\left(ze^{i\lambda}(h(z)g'(z)-e^{-2i\lambda}g(z)h'(z))\right)\nonumber.\end{aligned}$$* Theorem [Theorem 1](#p3-040){reference-type="ref" reference="p3-040"} can be used to identify whether a harmonic mapping is hereditarily spirallike or not. In this context, we present the following example. **Example 1**. For $\alpha\in\mathbb{D},$ the harmonic mapping $f_1(z)=z+\alpha\overline{z}$ is not hereditarily $\pi/4-$spirallike. Because for $\alpha=-1/2$ and $z=1/2(1+i)$ the inequality [\[p3-050\]](#p3-050){reference-type="eqref" reference="p3-050"} becomes $3/4>1.$ Here we note that, for $\alpha\in\mathbb{D}$ and an analytic univalent mapping $h\in\mathcal{S},$ the harmonic mapping $f=h+\alpha\overline{h}$ is sense-preserving fully starlike in $\mathbb{D}$ if and only if $h$ is starlike in $\mathbb{D},$ because $${\rm Re\,}\left(\frac{Df(z)}{f(z)}\right)=\frac{(1-|\alpha|^2)|h(z)|^2}{|h(z)+\alpha\overline{h(z)}|^2}{\rm Re\,}\left(\frac{zh'(z)}{h(z)}\right).$$ But the Example [Example 1](#p3-060){reference-type="ref" reference="p3-060"} shows that this does not happen in the case of spirallike mappings, and so the class $\mathcal{SP}_H(\lambda)$ is not affine invariant.\ In 1998, Silverman [@Silverman-1998] considered the classes $\mathcal{H}$ and $\mathcal{H}'$ of harmonic mappings and obtained sufficient conditions for functions in $\mathcal{H}$ and $\mathcal{H}'$ to be a member of the class $\mathcal{ST}_H$ and $\mathcal{ST}_{H'},$ respectively, in term of coefficients. Furthermore, the author proved that the obtained condition is also necessary for the the later case. For convenient of the reader, we state the result here. **Lemma 2**. *[@Silverman-1998] Let $f=h+\overline{g}\in\mathcal{H}$ be of the form [\[p3-010\]](#p3-010){reference-type="eqref" reference="p3-010"} such that $$\begin{aligned} \label{p3-080} \sum\limits_{n=1}^\infty n\left(|a_n|+|b_n|\right)\leq 2.\end{aligned}$$ Then $f\in\mathcal{ST}_H.$ Moreover, the condition [\[p3-080\]](#p3-080){reference-type="eqref" reference="p3-080"} is necessary and sufficient for a harmonic mapping $f=h+\overline{g}\in\mathcal{H}'$ of the form [\[p3-020\]](#p3-020){reference-type="eqref" reference="p3-020"} to be a member of the class $\mathcal{ST}_H'.$* We present a few similar results for harmonic hereditarily spirallike mappings in the unit disk. Before that we note down a few notations which we will use throughout the paper. For $\lambda\in(-\pi/2,\pi/2)$ and $n\geq 1,$ let $$\begin{aligned} \label{p3-090} A_n=\left|1+ne^{-i\lambda}\right|+\left|1-ne^{-i\lambda}\right|\quad\text{and}\quad B=\left|1+e^{-i\lambda}\right|-\left|1-e^{-i\lambda}\right|.\end{aligned}$$ **Theorem 2**. *Let $\lambda\in(-\pi/2,\pi/2)$ and $f=h+\overline{g}$ be of the form [\[p3-010\]](#p3-010){reference-type="eqref" reference="p3-010"} such that $$\begin{aligned} \label{p3-110} \sum\limits_{n=2}^\infty\frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty\frac{A_n}{B}|b_n|\leq 1\end{aligned}$$ where $A_n$ and $B$ are given by [\[p3-090\]](#p3-090){reference-type="eqref" reference="p3-090"}. Then $f$ is harmonic hereditarily $\lambda-$spirallike function.* For $\lambda=0,$ Theorem [Theorem 2](#p3-100){reference-type="ref" reference="p3-100"} reduces to Lemma [Lemma 2](#p3-070){reference-type="ref" reference="p3-070"}, which is expected because $\mathcal{SP}_H(0)=\mathcal{ST}_H.$\ There are harmonic hereditarily $\lambda-$spirallike mappings for which the value of the left hand side of [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"} is $1.$ For example, the harmonic hereditarily $\lambda-$spirallike mappings $$\begin{aligned} f(z)=h(z)+\overline{g(z)}=z+\sum\limits_{n=2}^\infty\frac{B}{A_n}x_nz^n+\sum\limits_{n=1}^\infty\frac{B}{A_n}\overline{y_nz^n}\end{aligned}$$ where $\sum\limits_{n=2}|x_n|+\sum\limits_{n=1}|y_n|=1$ have this property.\ The inequality [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"} is useful to construct harmonic univalent spirallike functions in the unit disk. Using the inequality [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"}, one can easily check that $f_2(z)=z+\alpha B/A_2\overline{z^2}$ and $f_3(z)=z+B\alpha/A_1\overline{z}+B/A_3(1-|\alpha|)\overline{z^3}$ where $\alpha\in\mathbb{D}$ are sense-preserving harmonic hereditarily $\lambda-$spirallike mappings for every $\lambda\in(-\pi/2,\pi/2)$. The images of $\mathbb{D}$ under $f_2$ for certain values of $\alpha$ and $\lambda$ are shown in Figure [\[fig-1\]](#fig-1){reference-type="ref" reference="fig-1"}. \ The condition [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"} in the Theorem [Theorem 2](#p3-100){reference-type="ref" reference="p3-100"} is not a necessary condition. This can be seen by an example of a hereditarily $\lambda-$spirallike function which does not satisfy the condition [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"}. We consider the function $f_4(z)=z(1-z)^{i-1}=z+\sum\limits_{n=2}^\infty a_nz^n\in\mathcal{H}.$ It is known that $f_4$ is hereditarily $\pi/4-$spirallike function (see [@Robertson-1969]). Clearly, $$a_2=\frac{f_3''(0)}{2}=1-i.$$ But for $\lambda=\pi/4,$ we have $$\sum\limits_{n=2}^\infty\frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty\frac{A_n}{B}|b_n|\geq \frac{A_2}{B}|a_2|=\sqrt{2}\frac{A_2}{B}>2.$$ Our next result gives necessary and sufficient condition for harmonic mappings in $\mathcal{H}'$ to be a member of the class $\mathcal{SP}_{H'}(\lambda).$ **Theorem 3**. *Let $\lambda\in(-\pi/2,\pi/2)$ and $f=h+\overline{g}\in\mathcal{H}'$ be of the form [\[p3-020\]](#p3-020){reference-type="eqref" reference="p3-020"}. If $f$ satisfies [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"} then $f\in\mathcal{SP}_{H'}(\lambda).$ Conversely, every harmonic function $f\in\mathcal{SP}_{H'}(\lambda)$ satisfies the coefficient inequality $$\begin{aligned} \label{p3-130} \sum\limits_{n=2}^\infty\frac{B}{A_n}|a_n|+\sum\limits_{n=1}^\infty\frac{B}{A_n}|b_n|\leq 1\end{aligned}$$ where $A_n$ and $B$ are given by [\[p3-090\]](#p3-090){reference-type="eqref" reference="p3-090"}.* The proof of the converse part of the Theorem [Theorem 3](#p3-120){reference-type="ref" reference="p3-120"} provides the following result. **Corollary 1**. *Let $\lambda\in(-\pi/2,\pi/2)$ and $f=h+\overline{g}\in\mathcal{SP}_{H'}(\lambda)$ be of the form [\[p3-020\]](#p3-020){reference-type="eqref" reference="p3-020"}. Then $\sum\limits_{n=1}^\infty(n|a_n|+n|b_n|)\leq 2,$ and $f\in\mathcal{ST}_H'.$* By Theorem [Theorem 2](#p3-100){reference-type="ref" reference="p3-100"} and Theorem [Theorem 3](#p3-120){reference-type="ref" reference="p3-120"}, we have constructed harmonic hereditarily $\lambda-$spirallike mapping in infinite series form. **Example 2**. Let $h_1(z)=z,~ h_n(z)=z+B/A_nz^n,~n\geq2$ and $g_n(z)=z+B/A_n\overline{z^n},~n\geq1$ and $X_n,Y_n\geq0$ are such that $\sum\limits_{n=1}^\infty(X_n+Y_n)=1.$ Then $f(z)=\sum\limits_{n=1}^\infty(X_nh_n(z)+Y_ng_n(z))$ is hereditarily $\lambda-$spirallike.\ Clearly, $f$ can be written as $$\begin{aligned} f(z)&=\sum\limits_{n=1}^\infty(X_nh_n(z)+Y_ng_n(z))\\ &=\sum\limits_{n=1}^\infty(X_n+Y_n)z+\sum\limits_{n=2}^\infty\left(\frac{X_nB}{A_n}z^n\right)+\sum\limits_{n=1}^\infty\left(\frac{Y_nB}{A_n}\overline{z^n}\right)\\ &=z+\sum\limits_{n=2}^\infty\left(\frac{X_nB}{A_n}z^n\right)+\sum\limits_{n=1}^\infty\left(\frac{Y_nB}{A_n}\overline{z^n}\right).\end{aligned}$$ Thus the function $f$ is of the form [\[p3-010\]](#p3-010){reference-type="eqref" reference="p3-010"} and satisfies $$\begin{aligned} \sum\limits_{n=2}^\infty\frac{A_n}{B}\left(\frac{X_nB}{A_n}\right)+\sum\limits_{n=1}^\infty\frac{A_n}{B}\left(\frac{Y_nB}{A_n}\right)=\sum\limits_{n=2}^\infty X_n+\sum\limits_{n=1}^\infty Y_n=1-X_1\leq1.\end{aligned}$$ Hence by the Theorem [Theorem 2](#p3-100){reference-type="ref" reference="p3-100"}, $f$ is hereditarily $\lambda-$spirallike. **Remark 1**. If we take $h_n(z)=z-B/A_nz^n,~n\geq2$ in the Example [Example 2](#p3-150){reference-type="ref" reference="p3-150"}, then the infinite series $\sum\limits_{n=1}^\infty(X_nh_n(z)+Y_ng_n(z))$ ultimately belong to the class $\mathcal{SP}_{H'}(\lambda).$ **Example 3**. Let $f=h+\overline{g}\in\mathcal{SP}_{H'}(\lambda).$ Then $f$ can be written as $f(z)=\sum\limits_{n=1}^\infty(X_nh_n(z)+Y_ng_n(z)),$ where $h_1(z)=z,~ h_n(z)=z-B/A_nz^n,~n\geq2$ and $g_n(z)=z+B/A_n\overline{z^n},~n\geq1$ with some $X_n,Y_n\geq0$ are such that $\sum\limits_{n=1}^\infty(X_n+Y_n)=1.$ Since $f=h+\overline{g}\in\mathcal{SP}_{H'}(\lambda)$ is of the form [\[p3-020\]](#p3-020){reference-type="eqref" reference="p3-020"}, by Theorem [Theorem 3](#p3-120){reference-type="ref" reference="p3-120"} the condition [\[p3-130\]](#p3-130){reference-type="eqref" reference="p3-130"} holds. Let $$\begin{aligned} h_1(z)=z,~h_n(z)=z-\frac{A_n}{B}z^n,~n\geq2\quad\text{and}\quad g_n(z)=z+\frac{A_n}{B}\overline{z^n},~n\geq1\end{aligned}$$ and set $$\begin{aligned} X_n=\frac{B|a_n|}{A_n},~n\geq2\quad\text{and}\quad Y_n=\frac{B|b_n|}{A_n},~n\geq1\end{aligned}$$ with $X_1=1-\sum\limits_{n=2}^\infty X_n-\sum\limits_{n=1}^\infty Y_n.$ Then from the condition [\[p3-130\]](#p3-130){reference-type="eqref" reference="p3-130"}, it follows that $0\leq X_n,~Y_n\leq1$ for $n\geq1.$ Therefore, $$\begin{aligned} \sum\limits_{n=1}^\infty(X_nh_n(z)+Y_ng_n(z))&=\sum\limits_{n=1}^\infty (X_n+Y_n)z-\sum\limits_{n=2}^\infty X_n\frac{A_n}{B}z^n+\sum\limits_{n=1}^\infty Y_n\frac{A_n}{B}\overline{z^n}\\ &=z-\sum\limits_{n=2}^\infty |a_n|z^n+\sum\limits_{n=1}^\infty |b_n|\overline{z^n}=f(z).\end{aligned}$$ The growth estimate for sense-preserving harmonic $\lambda-$spirallike functions is given below. **Theorem 4**. *Let $f=h+\overline{g}\in\mathcal{H}$ be a harmonic mapping of the form [\[p3-010\]](#p3-010){reference-type="eqref" reference="p3-010"} which satisfies the condition [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"}. Then the sharp inequality $$\begin{aligned} \left(1-\frac{B}{A_1}\right)r \leq |f(z)| \leq \left(1+\frac{B}{A_1}\right)r \quad\text{for}~|z|=r\end{aligned}$$* holds and $$\begin{aligned} \left\{w\in\mathbb{C}:|w|<1-\frac{B}{A_1}\right\}\subset f(\mathbb{D}).\end{aligned}$$ There is an interesting relation between analytic starlike and spirallike functions which was obtained by Başgöze and Keogh [@Basgoze-Keogh-1970]. For convenient, we state the result below. **Lemma 3**. *[@Basgoze-Keogh-1970] Let $\lambda\in(-\pi/2,\pi/2).$ For each analytic $\lambda-$spirallike function $h\in\mathcal{A}$ there exist an unique analytic starlike function $g\in\mathcal{A}$ such that $$\begin{aligned} \label{p3-190} \frac{h(z)}{z}=\left(\frac{g(z)}{z}\right)^{e^{-i\lambda}\cos\lambda}.\end{aligned}$$* We extend the Lemma [Lemma 3](#p3-180){reference-type="ref" reference="p3-180"} for functions in the class $C^1(\mathbb{D}).$ **Theorem 5**. *Let $\lambda$ be a real number with $|\lambda|<\pi/2.$ For a hereditarily $\lambda-$spirallike function $h\in C^1(\mathbb{D})$ satisfying $h(z)=0$ only for $z=0$ and $J_h=|h_z|^2-|h_{\overline{z}}|^2>0$ for $z\in\mathbb{D}$ there exists a fully starlike function $g\in C^1(\mathbb{D})$ satisfying $g(z)=0$ only for $z=0$ and $J_g=|g_z|^2-|g_{\overline{z}}|^2>0$ for $z\in\mathbb{D}$ such that relation [\[p3-190\]](#p3-190){reference-type="eqref" reference="p3-190"} is satisfied.* It is important to note that, the relation [\[p3-190\]](#p3-190){reference-type="eqref" reference="p3-190"} hold for analytic functions, but not for complex-valued harmonic functions as the exponential power of harmonic function may not be harmonic. For example, for the harmonic function $g(z)=z+\alpha\overline{z},~\alpha\in\mathbb{D},$ the corresponding mapping $h$ define by the relation [\[p3-190\]](#p3-190){reference-type="eqref" reference="p3-190"}, is not harmonic. Now, we give a relation between harmonic spirallike and starlike functions in some different way. **Theorem 6**. *Let $F(z)=z-\sum\limits_{n=2}^\infty |a_n|z^n+\overline{\sum\limits_{n=1}^\infty |b_n|z^n}$ be harmonic hereditarily starlike mapping. Then $f(z)=z+\sum\limits_{n=2}^\infty d_na_nz^n+\overline{\sum\limits_{n=1}^\infty d_nb_nz^n},$ where $\{d_n\}$ is a sequence such that $|d_n|\leq nB/A_n$ for $n\geq 1,$ is harmonic hereditarily $\lambda-$spirallike mapping. Conversely, if $f(z)=z-\sum\limits_{n=2}^\infty |a_n|z^n+\overline{\sum\limits_{n=1}^\infty |b_n|z^n}$ is harmonic hereditarily $\lambda-$spirallike mapping then $F(z)=z+\sum\limits_{n=2}^\infty a_nz^n+\overline{\sum\limits_{n=1}^\infty b_nz^n}$ is hereditarily starlike mapping.* For $0<\alpha<1,$ the harmonic mapping $f_5(z)=z+B\alpha/A_1\overline{z}+B/A_2(1-\alpha)\overline{z^2}$ is hereditarily $\lambda-$spirallike as from Lemma [\[p3-080\]](#p3-080){reference-type="ref" reference="p3-080"}, the function $F(z)=z+\alpha\overline{z}+1/2(1-\alpha)\overline{z^2}$ is hereditarily starlike. Here, we note that $d_n=nB/A_n.$ The images of $\mathbb{D}$ under $f_5$ for certain values of $\alpha$ and $\lambda$ are shown in Figure [\[fig-3\]](#fig-3){reference-type="ref" reference="fig-3"}. \ Thus we see that one can construct harmonic spirallike functions from given starlike mapping, and vice-versa.\ As many geometric properties of analytic univalent functions are already known, it provide more geometric exposures. So, it is quite interesting to relate harmonic function theory to analytic function theory. In this regards, Nagpal and Ravichandran [@Nagpal-Ravichandran-2013] established a relation between harmonic fully starlike function and analytic starlike function. **Lemma 4**. *[@Nagpal-Ravichandran-2013] A sense-preserving harmonic function $f=h+\overline{g}$ is fully starlike in $\mathbb{D}$ if the analytic functions $h+\epsilon g$ are starlike in $\mathbb{D}$ for each $|\epsilon|=1.$* Our next result for spirallike function is based on similar concept. **Theorem 7**. *Let $\lambda\in(-\pi/2,\pi/2)$ and $\{d_n\}$ be sequence of complex numbers with $|d_n|\leq nB/A_n$ for $n\geq 1.$ Let $H(z)=z-\sum\limits_{n=2}^\infty |a_n|z^n$ and $G(z)=\sum\limits_{n=1}^\infty |b_n|z^n$ be two analytic functions. If $$F_\epsilon(z)=z\left(\frac{H(z)+\epsilon G(z)}{z}\right)^{e^{i\lambda}\cos\lambda}$$ is analytic and $\lambda-$spirallike for each $|\epsilon|=1$ then the harmonic mapping $$f(z)=h(z)+\overline{g(z)}=z+\sum\limits_{n=2}^\infty d_na_nz^n+\overline{\sum\limits_{n=1}^\infty d_nb_nz^n}$$ is hereditarily $\lambda-$spiralike.* # Proof of Main Results {#Proof of Main Results} In this section, we will proof all the results sequentially. ***Proof of Theorem [Theorem 1](#p3-040){reference-type="ref" reference="p3-040"}**.* Since $f\in\mathcal{SP}_H(\lambda),$ it follows that $$\begin{aligned} {\rm Re\,}\left(e^{-i\lambda}\frac{Df(z)}{f(z)}\right) &= {\rm Re\,}\left(e^{-i\lambda}\frac{zh'(z)-\overline{zg'(z)}}{h(z)+g(z)}\right)\\ & = {\rm Re\,}\left(e^{-i\lambda}\frac{\left(zh'(z)-\overline{zg'(z)}\right)(\overline{h(z)}+g(z))}{|h(z)+\overline{g(z)}|^2}\right)\\ &= \frac{1}{|h(z)+\overline{g(z)}|^2}{\rm Re\,}\left[ e^{-i\lambda}\frac{zh'(z)}{h(z)}|h(z)|^2-\overline{e^{i\lambda}\frac{zg'(z)}{g(z)}}|g(z)|^2 \right.\\ &\hspace{40mm}\left. + e^{-i\lambda}zh'(z)g(z)-\overline{e^{i\lambda}zg'(z)h(z)}\right]\\ & >0.\end{aligned}$$ Upon using the fact ${\rm Re\,}w={\rm Re\,}(\overline{w})$ for $w\in\mathbb{C}$ in the last inequality, we get [\[p3-050\]](#p3-050){reference-type="eqref" reference="p3-050"}.\ Conversely, a simple calculation leads the condition [\[p3-050\]](#p3-050){reference-type="eqref" reference="p3-050"} to ${\rm Re\,}\left(e^{-i\lambda}\frac{Df(z)}{f(z)}\right)>0.$ And from the hypothesis, $f$ is sense-preserving and $f(z)=0$ only for $z=0.$ Therefore Lemma [\[p3-030\]](#p3-030){reference-type="ref" reference="p3-030"} implies that the function $f$ is harmonic hereditarily $\lambda-$spirallike function. ◻ ***Proof of Theorem [Theorem 2](#p3-100){reference-type="ref" reference="p3-100"}**.* We first note that $$\begin{aligned} \label{p3-240} \frac{A_n}{B} =\frac{\left|1+ne^{-i\lambda}\right|+\left|1-ne^{-i\lambda}\right|}{\left|1+e^{-i\lambda}\right|-\left|1-e^{-i\lambda}\right|}\geq \frac{\left|(1+ne^{-i\lambda})-(1-ne^{-i\lambda})\right|}{|1+e^{-i\lambda}|}\geq \frac{2n}{2}=n\end{aligned}$$ and so by [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"} we have $$\begin{aligned} |h'(z)| & \geq 1-\sum\limits_{n=2}^\infty n|a_n|r^{n-1}> 1-\sum\limits_{n=2}^\infty n|a_n|\geq 1-\sum\limits_{n=2}^\infty \frac{A_n}{B}|a_n|\\ & \geq \sum\limits_{n=1}^\infty \frac{A_n}{B}|b_n|\geq \sum\limits_{n=1}^\infty n|b_n|>\sum\limits_{n=1}^\infty n|b_n|r^{n-1}=|g'(z)|.\end{aligned}$$ Thus the function $f$ is sense-preserving and locally univalent in $\mathbb{D}.$ Next, we see that $$\sum\limits_{n=2}^\infty n|a_n|+\sum\limits_{n=1}^\infty n|b_n|\leq \sum\limits_{n=2}^\infty \frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty \frac{A_n}{B}|b_n|\leq1$$ and hence, from Lemma [Lemma 2](#p3-070){reference-type="ref" reference="p3-070"}, it follows that $f$ is univalent. Now, to show that $f$ is hereditarily $\lambda-$spirallike, it is sufficient to show $$\begin{aligned} {\rm Re\,}\left(e^{-i\lambda}\frac{Df(z)}{f(z)}\right)>0,\quad z\in\mathbb{D}\setminus\{0\},\end{aligned}$$ which is equivalent to, $$\begin{aligned} \left|1+e^{-i\lambda}\frac{Df(z)}{f(z)}\right|>\left|1-e^{-i\lambda}\frac{Df(z)}{f(z)}\right|,\end{aligned}$$ or, $$\begin{aligned} \label{p3-250} M:=\left|f(z)+e^{-i\lambda}Df(z)\right|-\left|f(z)-e^{-i\lambda}Df(z)\right|>0.\end{aligned}$$ Now $$\begin{aligned} \label{p3-260} M &=\left|f(z)+e^{-i\lambda}Df(z)\right|-\left|f(z)-e^{-i\lambda}Df(z)\right| \\\nonumber &=\left|(h(z)+e^{-i\lambda}zh'(z))+\overline{(g(z)-e^{i\lambda}zg'(z))}\right|\\\nonumber &\qquad-\left|(h(z)-e^{-i\lambda}zh'(z))+\overline{(g(z)+e^{i\lambda}zg'(z))}\right|\\\nonumber &=\left|(1+e^{-i\lambda})z+\sum\limits_{n=2}\left(1+ne^{-i\lambda}\right)a_nz^n+\sum\limits_{n=1}\left(1-ne^{-i\lambda}\right)\overline{b_nz^n}\right|\\\nonumber &\qquad\quad\quad-\left|(1-e^{-i\lambda})z+\sum\limits_{n=2}\left(1-ne^{-i\lambda}\right)a_nz^n+\sum\limits_{n=1}\left(1+ne^{-i\lambda}\right)\overline{b_nz^n}\right|\\\nonumber & \geq \left(\left|1+e^{-i\lambda}\right|-\left|1-e^{-i\lambda}\right|\right)|z|-\sum\limits_{n=2}^\infty\left(\left|1+ne^{-i\lambda}\right|+\left|1-ne^{-i\lambda}\right| \right)|a_n||z|^n\\\nonumber &\qquad\quad\quad-\sum\limits_{n=1}^\infty\left(\left|1-ne^{-i\lambda}\right|+\left|1+ne^{-i\lambda}\right| \right)|b_n||z|^n\\\nonumber &=|z|\left[1-\left(\sum\limits_{n=2}^\infty\frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty\frac{A_n}{B}|b_n|\right)|z|^{n-1} \right]\end{aligned}$$ where $$\begin{aligned} A_n &=\left|1+ne^{-i\lambda}\right|+\left|1-ne^{-i\lambda}\right|\quad\text{and}\quad B=\left|1+e^{-i\lambda}\right|-\left|1-e^{-i\lambda}\right|.\end{aligned}$$ Thus from [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"} and [\[p3-260\]](#p3-260){reference-type="eqref" reference="p3-260"}, for $z\in\mathbb{D}\setminus\{0\},$ we have $$\begin{aligned} M &\geq |z|\left[1-\left(\sum\limits_{n=2}^\infty\frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty\frac{A_n}{B}|b_n|\right)|z|^{n-1} \right]\\ &>|z|\left[1-\left(\sum\limits_{n=2}^\infty\frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty\frac{A_n}{B}|b_n|\right)\right]>0.\end{aligned}$$ ◻ ***Proof of Theorem [Theorem 3](#p3-120){reference-type="ref" reference="p3-120"}**.* First part of the theorem follows immediately from the proof the the Theorem [Theorem 2](#p3-100){reference-type="ref" reference="p3-100"}. So we will prove the converse part. Let $f=h+\overline{g}\in\mathcal{SP}_{H'}(\lambda).$ Then $f$ is univalent and satisfies $$\begin{aligned} \label{p3-270} {\rm Re\,}\left(e^{-i\lambda}\frac{Df(z)}{f(z)}\right)>0 ~~\text{for}~z\in\mathbb{D}\setminus\{0\}.\end{aligned}$$ For $0\leq r<1,$ we define $$\phi(r)=1-\left(\sum_{n=2}^{\infty} |a_n| -\sum_{n=1}^{\infty} |b_n|\right)r^{n-1}$$ and $$\psi(r)=\sum_{n=2}^{\infty} n|a_n|r^{n-1} +\sum_{n=1}^{\infty} n|b_n|r^{n-1}.$$ Then on the positive real axis, $f$ can be written as $f(r)=r\phi(r),~0\leq r<1.$ Since $f(z)\neq 0$ for $z\neq 0$ it follows that $\phi(r)\neq 0$ in $(0,1).$ Indeed, we show that $\phi(r)>0$ for each $0<r<1$. On the contrary, suppose that there exist a $r_0\in (0,1)$ such that $\phi(r_0)<0$. Since $\phi$ is continuous in $[0,r_0]$ and $\phi(0)=1>0,$ it follows that $\phi$ has a zero in $(0,r_0),$ which contradicts the fact that $f(z)\neq 0$ for $z\neq 0.$ Now, $$\begin{aligned} \frac{Df(z)}{f(z)} =\frac{zh'(z)-\overline{zg'(z)}}{h(z)+\overline{g(z)}}=\frac{z-\sum\limits_{n=2}^\infty n|a_n|z^n-\sum\limits_{n=1}^\infty n|b_n|z^n}{z-\sum\limits_{n=2}^\infty |a_n|z^n+\sum\limits_{n=1}^\infty|b_n|z^n}.\end{aligned}$$ Taking the values of $z$ on the positive real axis, i.e., $z=r,~0<r<1,$ we get $$\begin{aligned} {\rm Re\,}\left(e^{-i\lambda}\frac{Df(r)}{f(r)}\right)= \cos\lambda\frac{1-\psi(r)}{\phi(r)}.\end{aligned}$$ As $f=h+\overline{g}\in\mathcal{SP}_{H'}(\lambda)$ and $\phi(r)>0,$ it follows that $1-\psi(r)>0$ for $0<r<1.$ Thus $\psi(r)<1$ in $(0,1).$ Also, we see that $\psi$ is monotonic increasing function of $r\in(0,1)$ and hence $$\begin{aligned} \label{p3-280} \lim_{r\rightarrow 1^{-}}\psi(r)=\sum\limits_{n=2}^\infty n|a_n|+\sum\limits_{n=1}^\infty n|b_n|)\leq1.\end{aligned}$$ Since $B/A_n\leq n$ for $n\geq1$ (see [\[p3-240\]](#p3-240){reference-type="eqref" reference="p3-240"}), it follows that $$\begin{aligned} \sum\limits_{n=2}^\infty \frac{B}{A_n}|a_n|+\sum\limits_{n=1}^\infty \frac{B}{A_n}|b_n|\leq \sum\limits_{n=2}^\infty n|a_n|+\sum\limits_{n=1}^\infty n|b_n|\leq1.\end{aligned}$$ This completes the proof. ◻ ***Proof of Corollary [Corollary 1](#p3-140){reference-type="ref" reference="p3-140"}**.* If $f=h+\overline{g}\in\mathcal{SP}_{H'}(\lambda)$ then by the argument given in the proof of the Theorem [Theorem 3](#p3-120){reference-type="ref" reference="p3-120"}, the inequality [\[p3-280\]](#p3-280){reference-type="eqref" reference="p3-280"} hold. Now the proof follows from Lemma [Lemma 2](#p3-070){reference-type="ref" reference="p3-070"}. ◻ ***Proof of Theorem [Theorem 4](#p3-170){reference-type="ref" reference="p3-170"}**.* Let $|z|=r<1.$ As the harmonic function $f(z)=z+\sum\limits_{n=2}^\infty a_nz^n+\overline{\sum\limits_{n=1}^\infty b_nz^n}$ satisfies the condition [\[p3-110\]](#p3-110){reference-type="eqref" reference="p3-110"}, it follows that $$\begin{split} |f(z)| & \leq |z|+\sum\limits_{n=2}^\infty|a_n||z|^n+\sum\limits_{n=1}^\infty|b_n||z|^n\\ & \leq r+\sum\limits_{n=2}^\infty|a_n|r+\sum\limits_{n=1}^\infty|b_n|r\\ & =r+\frac{B}{A_1}\left(\sum\limits_{n=2}^\infty \frac{A_1}{B}|a_n|+\sum\limits_{n=1}^\infty \frac{A_1}{B}|b_n|\right)r\\ & \leq r+\frac{B}{A_1}\left(\sum\limits_{n=2}^\infty \frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty \frac{A_n}{B}|b_n|\right)r\\ & \leq r+\frac{B}{A_1}r\\ &= \left(1+\frac{B}{A_1}\right)r. \end{split}$$ On the other hand, $$\begin{aligned} \label{p3-290} |f(z)|\nonumber & \geq |z|-\sum\limits_{n=2}^\infty|a_n||z|^n-\sum\limits_{n=1}^\infty|b_n||z|^n\\\nonumber & \geq r-\sum\limits_{n=2}^\infty|a_n|r-\sum\limits_{n=1}^\infty|b_n|r\\\nonumber & =r-\frac{B}{A_1}\left(\sum\limits_{n=2}^\infty \frac{A_1}{B}|a_n|+\sum\limits_{n=1}^\infty \frac{A_1}{B}|b_n|\right)r\\\nonumber & \geq r-\frac{B}{A_1}\left(\sum\limits_{n=2}^\infty \frac{A_n}{B}|a_n|+\sum\limits_{n=1}^\infty \frac{A_n}{B}|b_n|\right)r\\\nonumber & \geq r-\frac{B}{A_1}r\\ &= \left(1-\frac{B}{A_1}\right)r.\end{aligned}$$ Left hand estimate is sharp for the hereditarily $\lambda-$spirallike function $f_6(z)=z-B/A_1\overline{z}$ whereas the right hand estimate is sharp for $f_7(z)=z+B/A_1\overline{z}.$\ On taking $r\rightarrow 1^{-}$ in [\[p3-290\]](#p3-290){reference-type="eqref" reference="p3-290"}, we get $$\begin{aligned} \left\{w\in\mathbb{C}:|w|<1-\frac{B}{A_1}\right\}\subset f(\mathbb{D}).\end{aligned}$$ ◻ ***Proof of Theorem [Theorem 5](#p3-200){reference-type="ref" reference="p3-200"}**.* As $h(z)/z$ and $g(z)/z$ are non zero in the unit disk $\mathbb{D},$ from the relation [\[p3-190\]](#p3-190){reference-type="eqref" reference="p3-190"}, we have $$e^{i\lambda}\log\frac{h(z)}{z}=\cos\lambda\log\frac{g(z)}{z}.$$ Upon taking the differential operator D, we get $$e^{i\lambda}\left(\frac{Dh(z)}{h(z)}-1\right)=\cos\lambda\left(\frac{Dg(z)}{g(z)}-1\right)$$ which is equivalent to $$e^{i\lambda}\frac{Dh(z)}{h(z)}=\cos\lambda\frac{Dg(z)}{g(z)}+i\sin\lambda.$$ Thus $${\rm Re\,}\left(e^{i\lambda}\frac{Dh(z)}{h(z)}\right)=\cos\lambda{\rm Re\,}\left(\frac{Dg(z)}{g(z)}\right)$$ from which the result follows. ◻ ***Proof of Theorem [Theorem 6](#p3-210){reference-type="ref" reference="p3-210"}**.* Since $F(z)=z-\sum\limits_{n=2}^\infty |a_n|z^n+\overline{\sum\limits_{n=1}^\infty |b_n|z^n}$ is harmonic hereditarily starlike mapping of the form [\[p3-020\]](#p3-020){reference-type="eqref" reference="p3-020"}, by Lemma [Lemma 2](#p3-070){reference-type="ref" reference="p3-070"}, the coefficient condition [\[p3-080\]](#p3-080){reference-type="eqref" reference="p3-080"} holds. Since $\{d_n\}$ is a sequence of complex numbers with $|d_n|\leq nB/A_n$ for all $n\geq 1,$ then $$\begin{aligned} \sum\limits_{n=2}^\infty\frac{A_n}{B}|d_na_n|+\sum\limits_{n=1}^\infty\frac{A_n}{B}|d_nb_n| & = \sum\limits_{n=2}^\infty\frac{A_n}{B}|d_n||a_n|+\sum\limits_{n=1}^\infty\frac{A_n}{B}|d_n||b_n|\\ & \leq \sum\limits_{n=2}^\infty n|a_n|+\sum\limits_{n=1}^\infty n|b_n|\\ & \leq 1.\end{aligned}$$ Hence by Theorem [Theorem 2](#p3-100){reference-type="ref" reference="p3-100"}, $f(z)=z+\sum\limits_{n=2}^\infty d_n a_nz^n+\overline{\sum\limits_{n=1}^\infty d_n b_nz^n}$ is hereditarily $\lambda-$spirallike. Conversely, if $f(z)=z-\sum\limits_{n=2}^\infty |a_n|z^n+\overline{\sum\limits_{n=1}^\infty |b_n|z^n}$ is harmonic hereditarily $\lambda-$spirallike mapping, then from the Corollary [Corollary 1](#p3-140){reference-type="ref" reference="p3-140"}, the coefficient condition [\[p3-080\]](#p3-080){reference-type="eqref" reference="p3-080"} holds. Hence by Lemma [Lemma 2](#p3-070){reference-type="ref" reference="p3-070"}, the function $F(z)=z+\sum\limits_{n=2}^\infty a_n z^n+\overline{\sum\limits_{n=1}^\infty b_n z^n}$ is hereditarily starlike. ◻ ***Proof of Theorem [Theorem 7](#p3-230){reference-type="ref" reference="p3-230"}**.* As $F_\epsilon(z)=z\left(\frac{H(z)+\epsilon G(z)}{z}\right)^{e^{i\lambda}\cos\lambda}$ is $\lambda-$spirallike for each $|\epsilon|=1$, from the Lemma [Lemma 3](#p3-180){reference-type="ref" reference="p3-180"}, it follows that $H(z)+\epsilon G(z)$ is analytic starlike for each $|\epsilon|=1.$ Consequently, by Lemma [Lemma 4](#p3-220){reference-type="ref" reference="p3-220"}, the associated harmonic function $F(z)=H(z)+\overline{G(z)}$ is fully starlike of the form [\[p3-020\]](#p3-020){reference-type="eqref" reference="p3-020"}. Thus, from Theorem [Theorem 6](#p3-210){reference-type="ref" reference="p3-210"}, $f=z+\sum\limits_{n=2}^\infty d_na_nz^n+\overline{\sum\limits_{n=1}^\infty d_nb_nz^n}$ is harmonic hereditarily $\lambda-$spirallike. ◻ **Declarations:\ ** **Data availability:** Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.\ **Authors Contributions:** All authors contributed equally to the investigation of the problem and the order of the authors is given alphabetically according to their surname. All authors read and approved the final manuscript.\ **Acknowledgement:** The second named author thanks the Department Of Science and Technology, Ministry Of Science and Technology, Government Of India for the financial support through DST-INSPIRE Fellowship (No. DST/INSPIRE Fellowship/2018/IF180967).\ **Conflict of interest:** The authors declare that they have no conflict of interest. 99 T. Başgöze, F. R. Keogh, The Hardy class of a spiral-like function and its derivative, *Proc. Amer. Math. Soc.*, **26**(2) (1970), 266-269. H. Chen, P.M. Gauthier,W. Hengartner, Bloch constants theorem for planar harmonic mappings, *Proc. Amer. Math. Soc.*, **128**(11) (2000), 3231-3240. M. Chuaqui, P. Duren, B. Osgood , Curvature properties of planar harmonic mappings, *Comput. Methods Funct. Theory*, **4**(1) (2004), 127-142. J. Clunie, T. Sheil-Small, Harmonic univalent functions, *Ann. Acad. Sci. Fenn.*, **9** (1984), 3-25. P. Duren, Univalent Functions, Springer-Verlag, New York (1983). J. M. Jahangiri, Harmonic functions starlike in the unit disk, *J. Math. Anal. Appl.*, **235**(2) (1999), 470-477. H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, *Bull. Amer. Math. Soc.*, **42**(10) (1936), 689-692. G. Liu, S. Ponnusamy, Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm, *Indag. Math.* **29**(2) (2018), 752-778. X. S. Ma, S. Ponnusamy, T. Sugawa, Harmonic spirallike functions and harmonic strongly starlike functions, *Monatsh. Math.*, **199**(2) (2022), 363-375. P. T. Mocanu, Starlikeness and convexity for nonanalytic functions in the unit disc, *Mathematica (Cluj)*, **22**(45) (1980), 77-83. S. Nagpal, V. Ravichandran, Fully starlike and fully convex harmonic mappings of order $\alpha$, *Ann. Polon. Math.*, **108**(1) (2013), 85-107. S. Ponnusamy, A. Sairam Kaliraj, Univalent harmonic mappings convex in one direction, *Anal. Math. Phys.* **4**(3) (2014), 221-236. M.S. Robertson, Univalent functions $f (z)$ for which $zf^{\prime}(z)$ is spirallike, *Michigan Math. J.* **16**(1) (1969), 97-101. H. Silverman, Harmonic univalent functions with negative coefficients, *J. Math. Anal. Appl.*, **220** (1998), 283-289. [^1]:

arxiv_math

--- abstract: | We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We then present a simple application, related to possible correlations between trace functions and dynamical sequences. address: D-MATH, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland author: - Emmanuel Kowalski title: Rational approximation with chosen numerators --- # Introduction The following statement is motivated by certain specific applications concerning possible correlations between "trace functions" and "dynamical" sequences (see Section [4](#sec-appli){reference-type="ref" reference="sec-appli"} for a concrete statement and the notes [@ergodic] for a more general perspective). **Theorem 1**. *Let $c>0$ be a real number with $c\leqslant 1/2$. There exist sequences $(a_p)$ indexed by prime numbers, with $a_p$ an integer such that $0\leqslant a_p<p$ for all $p$, such that for almost all $x\in [0,1]$, the set of primes $p$ with $$\Bigl|x-\frac{a_p}{p}\Bigr|\leqslant\frac{c}{p}$$ is infinite.* So, informally, we consider a problem of diophantine approximation where, for each denominator, only one numerator is allowed (and with the additional restriction, coming from the original motivation, that the denominators are primes). The approximation is of course then worse than what is allowed by varying the numerator (and of course not every choice of numerator can be successful). We will give two proofs in Sections [2](#sec-proof){reference-type="ref" reference="sec-proof"} and [3](#sec-proof2){reference-type="ref" reference="sec-proof2"}. The first is (probably unsurprisingly) very elementary, but has some nice aspects, especially an analogy with sieve. The second proof is more straightforward in principle, but involves more sophisticated ingredients, especially about the distribution of primes. This simple result suggests some questions: 1. Can one describe an explicit sequence $(a_p)$ which has the desired property? Note that the second proof will show that it is a generic property (in the sense of the natural probability measure on the space of sequences $(a_p)$, described below). 2. More specifically, if we define $(a_p)$ using the "greedy" algorithm (taking $a_p$ for successive primes so as to always maximize the measure of the union of the intervals up to that point), does it work? 3. It is easy to see that the "exceptional set" will always contain infinitely many rational numbers. Are there other elements in this exceptional set? If Yes, can we describe the elements that belong to it, or compute its Hausdorff dimension? 4. For suitable $(a_p)$ and $x$, can we estimate asymptotically as $X\to +\infty$ the number of primes $p\leqslant X$ such that $|x-a_p/p|\leqslant c/p$? Heuristically, one can hope to have something like $$2c\sum_{p\leqslant X}\frac{1}{p}\sim 2c\log\log X$$ such primes $\leqslant X$; can this be established for suitable choices of the $a_p$? The first proof provides some weaker quantitative information, with high probability (with respect to $x$). 5. What about multidimensional versions? Variants on manifolds? **Remark 2**. Although problems of this kind do not seem to be standard in diophantine approximation, one can interpret the question roughly as asking whether there exists a sequence $(a_p)$ such that the set of points $a_p/p$ is "eutaxic" with respect to the radii $c/p$ (see, e.g., the survey of Durand [@durand Ch. 8], and also Bugeaud's book [@bugeaud Ch. 6]). A related, but more delicate, question was solved by Shepp [@shepp] (after previous work of Dvoretzky and others), who found a sharp criterion which ensures that a non-increasing sequence $(\ell_n)_{n\geqslant 1}$ of positive real numbers has the property that, almost surely, the union of arcs of length $\ell_n$ with independent and uniform centers will cover *entirely* a circle of length $1$. Shepp proved that this holds if and only if $$\sum_{n\geqslant 1}\frac{1}{n^2}\exp(\ell_1+\cdots+\ell_n)=+\infty.$$ As observed by Dvoretzky [@dvoretzky], this is a different condition than asking that the union covers almost all points of the circle, which occurs if and only if the series $\sum \ell_n$ diverges, by an elementary argument as in Section [3](#sec-proof2){reference-type="ref" reference="sec-proof2"}. ## Notation {#notation .unnumbered} We use the Vinogradov notation $f\ll g$ (for complex-valued functions $f$ and $g$ defined on some set $X$): it means that there exists a real number $c\geqslant 0$ (the "implied constant") such that $|f(x)|\leqslant cg(x)$ for all $x\in X$. ## Acknowledgements {#acknowledgements .unnumbered} Thanks to Y. Bugeaud for interesting comments and references, in particular to the work of Shepp. # First proof {#sec-proof} We denote by $\lambda$ the Lebesgue measure. For a sequence $\text{\boldmath${a}$}=(a_p)$ with $0\leqslant a_p<p$ for all primes $p$, define $$A_{\text{\boldmath${a}$}}=\{x\in [0,1]\,\mid\, \Bigl|x-\frac{a_p}{p}\Bigr|\leqslant \frac{c}{p}\text{ for infinitely many~$p$}\}.$$ We thus want to find $\text{\boldmath${a}$}$ with $\lambda(A_{\text{\boldmath${a}$}})=1$. For real parameters $X$ and $Y$ with $1\leqslant X<Y$, we consider the set $$\Omega_{X,Y}(\text{\boldmath${a}$})=\Bigl\{ x\in [0,1]\,\mid\, \Bigl|x-\frac{a_p}{p}\Bigr|> \frac{c}{p}\text{ for $X< p\leqslant Y$} \Bigr\}.$$ We observe that the set $\mathscr{A}$ of all sequences $\text{\boldmath${a}$}$ is naturally a compact set, as a product of finite sets. In particular, there is a natural product probability measure on this set, where each $a_p$ is uniform over the integers from $0$ to $p-1$. We will use the notation $\mathbf{P}(\cdot)$ and $\mathbf{E}(\cdot)$ below to indicate probability and expectation according to this measure. The crucial lemma is the following. We view it as a kind of sieve statement, on average over $\mathscr{A}$. **Lemma 3**. *Let $$H_{X,Y}=\sum_{X< p\leqslant Y}\frac{1}{p}.$$* *We have $$\mathbf{E}(\lambda(\Omega_{X,Y})) \ll \frac{1}{H_{X,Y}},$$ where the implied constant depends only on $c$.* *Proof.* Let $\varphi_p\colon [0,1]\to \{0,1\}$ denote the characteristic function of the interval $I_p(a_p)=[a_p/p-c/p,a_p/p+c/p]$, each being viewed as random variables on $\mathscr{A}$ (the $\varphi_p$ are random functions, the $I_p$ are random intervals). Let $$N_{X,Y}=\sum_{X< p\leqslant Y}\varphi_p,$$ again a random variable on $\mathscr{A}$. We denote also $$\nu_{X,Y}=\int_0^1N_{X,Y}(x)dx$$ and note that $\nu_{X,Y}=2cH_{X,Y}$, independently of the value of $\text{\boldmath${a}$}$. Noting that $\Omega_{X,Y}$ is the set of those $x$ where $N_{X,Y}=0$, we deduce from Markov's inequality (on $[0,1]$ with the Lebesgue measure) the upper bound $$\lambda(\Omega_{X,Y}) \leqslant \lambda\Bigl(\Bigl\{ x\in[0,1]\,\mid\, |N_{X,Y}(x)-\nu_{X,Y}|\geqslant\nu_{X,Y} \Bigr\}\Bigr)\leqslant\frac{\alpha_{X,Y}}{\nu_{X,Y}^2}$$ where $$\alpha_{X,Y}=\int_0^1\Bigl(N_{X,Y}(x)-\nu_{X,Y}\Bigr)^2dx$$ (again a random variable on $\mathscr{A}$). We have $$\alpha_{X,Y}=\int_0^1\Bigl(\sum_{X\leqslant p\leqslant Y}\Bigl(\varphi_p(x)-\frac{2c}{p}\Bigr)\Bigr)^2dx= \sum_{X< p_1,p_2\leqslant Y} \int_0^1 \Bigl(\varphi_{p_1}(x)-\frac{2c}{p}\Bigr) \Bigl(\varphi_{p_2}(x)-\frac{2c}{p}\Bigr) dx.$$ For $p_1=p_2$, the integral is equal to $$\int_0^1 \Bigl(\varphi_{p_1}(x)-\frac{2c}{p_1}\Bigr)^2dx= \frac{2c}{p_1}\Bigl(1-\frac{2c}{p_1}\Bigr)\leqslant\frac{2c}{p_1}$$ (variance of a Bernoulli random variable with probability of success $2c/p_1$), again independently of $\text{\boldmath${a}$}$. Thus $$\sum_{X< p_1\leqslant Y}\mathbf{E}\Bigl(\int_0^1 \Bigl(\varphi_{p_1}(x)-\frac{2c}{p_1}\Bigr)^2dx\Bigr) \leqslant 2cH_{X,Y}.$$ We now suppose that $p_1\not=p_2$. We then have $$\label{eq-var1} \int_0^1 \Bigl(\varphi_{p_1}(x)-\frac{2c}{p_1}\Bigr) \Bigl(\varphi_{p_2}(x)-\frac{2c}{p_2}\Bigr)dx =\lambda(I_{p_1}\cap I_{p_2})-\frac{4c^2}{p_1p_2},$$ where the first term depends on $\text{\boldmath${a}$}$. We next estimate the expectation $$\mathbf{E}\Bigl( \lambda(I_{p_1}\cap I_{p_2}) \Bigr)$$ over $\text{\boldmath${a}$}$. For this purpose, we may (and do) assume that $p_1<p_2$. We then have the formula $$\mathbf{E}\Bigl( \lambda(I_{p_1}\cap I_{p_2}) \Bigr)= \frac{1}{p_1p_2} \sum_{0\leqslant a<p_1}\lambda\Bigl( I_{p_1}(a)\cap \bigcup_{0\leqslant b<p_2}\Bigl[ \frac{b}{p_2}-\frac{c}{p_2}, \frac{b}{p_2}+\frac{c}{p_2} \Bigr] \Bigr).$$ Drawing a picture if need be, we get $$\mathbf{E}\Bigl( \lambda(I_{p_1}\cap I_{p_2}) \Bigr)= \frac{1}{p_1p_2}\times p_1\times \Bigl( \frac{4c^2}{p_1}+O\Bigl( \frac{1}{p_2} \Bigr) \Bigr)=\frac{4c^2}{p_1p_2} +O\Bigl( \frac{1}{p_2^2} \Bigr).$$ Combined with ([\[eq-var1\]](#eq-var1){reference-type="ref" reference="eq-var1"}), this leads to $$\sum_{X\leqslant p_1<p_2\leqslant Y} \mathbf{E} \Bigl(\int_0^1 \Bigl(\varphi_{p_1}(x)-\frac{2c}{p_1}\Bigr) \Bigl(\varphi_{p_2}(x)-\frac{2c}{p_2}\Bigr)dx \Bigr)\ll H_{X,Y}.$$ Multiplying by two to account for the case $p_1>p_2$ and adding the contribution where $p_1=p_2$, we conclude that $\mathbf{E}( \alpha_{X,Y})\ll H_{X,Y}$, and hence $$\mathbf{E}(\lambda(\Omega_{X,Y}))\ll H_{X,Y}^{-1},$$ as claimed. ◻ **Remark 4**. The start of the argument is essentially of form of sieve inequality, especially similar to those used in certain geometric group theory works by Lubotzky and Meiri, see for instance the account in [@expanders Th. 5.3.1]. More generally, sieve methods in analytic number theory lead to bounds (also sometimes lower bounds) for the sizes of sets of the form $$\{n\leqslant N\,\mid\, n\,(\mathrm{mod}\,{p})\notin I_p\text{ for } p\leqslant X\}$$ for suitable choices of subsets $I_p\subset \mathbf{Z}/p\mathbf{Z}$ and of parameters $N$ and $X$. It was pointed out in [@sieve] that in fact some of the basic techniques (e.g., the so-called "large sieve") can be extended to much more general settings than the integers, and the lemma above provides another illustration. We now conclude the proof of the theorem. Since $$\lim_{Y\to +\infty}H_{X,Y}=\sum_{p>X} \frac{1}{p}=+\infty$$ for any $X\geqslant 1$ (one of the most elementary quantitative forms of the infinitude of primes, already known to Euler), Lemma [Lemma 3](#lm-sieve){reference-type="ref" reference="lm-sieve"} implies that for any $X\geqslant 2$ and any $\varepsilon>0$, we can find $(a_p)_{X< p\leqslant Y}$ (with $0\leqslant a_p<p$) such that $$\lambda\Bigl(\Bigl\{ x\in [0,1]\,\mid\, \Bigl|x-\frac{a_p}{p}\Bigr|\leqslant\frac{c}{p}\text{ for some prime $p$ with $X< p\leqslant Y$} \Bigr\}\Bigr)\geqslant 1-\varepsilon.$$ Let $X_1=1$. Apply the previous remark first with (say) $X=1$ and $\varepsilon=1/2$, and denote $X_2$ a suitable value of $Y$. Then apply the assumption with $X=X_2$ and $\varepsilon=1/4$, calling $X_3$ the value of $Y$; repeating, we obtain a strictly increasing sequence $(X_n)_{n\geqslant 1}$ of integers and a sequence $(a_p)\in\mathscr{A}$ such that the set $$B_n=\Bigl\{ x\in [0,1]\,\mid\, \Bigl|x-\frac{a_p}{p}\Bigr|\leqslant\frac{c}{p}\text{ for some prime $p$ with $X_n<p\leqslant X_{n+1}$} \Bigr\}$$ satisfies $\lambda(B_n)\geqslant 1-2^{-n}$ for any $n\geqslant 1$. If $x\in [0,1]$ belongs to infinitely sets $B_n$, then $x\in A_{\text{\boldmath${a}$}}$. On the other hand, since $$\sum_{n\geqslant 1}\lambda([0,1]\mathchoice {\mathbin{\vrule height .62ex width 1.61ex depth -.38ex}}% 12 {\mathbin{\vrule height .62ex width 1.61ex depth -.38ex}}% 12 {\mathbin{\vrule height .50ex width 0.85ex depth -.28ex}}% 9 {\mathbin{\vrule height .20ex width 0.570ex depth -.24ex}}% 7 B_n)<+\infty,$$ the easy Borel--Cantelli Lemma shows that almost every $x\in [0,1]$ belongs at most to finitely many sets $[0,1]\mathchoice {\mathbin{\vrule height .62ex width 1.61ex depth -.38ex}}% 12 {\mathbin{\vrule height .62ex width 1.61ex depth -.38ex}}% 12 {\mathbin{\vrule height .50ex width 0.85ex depth -.28ex}}% 9 {\mathbin{\vrule height .20ex width 0.570ex depth -.24ex}}% 7 B_n$. # Second proof {#sec-proof2} We now give the second proof. This is based on an application of Fubini's Theorem (which is a standard approach, as in the first few lines of Dvoretzky's paper [@dvoretzky]). We write again $I_p(\text{\boldmath${a}$})=[a_p/p-c/p,a_p/p+c/p]$, viewed as random intervals on the probability space $\mathscr{A}$ to which $\mathbf{P}(\cdot)$ and $\mathbf{E}(\cdot)$ refer. Let $x\in [0,1]$. We then have $$\mathbf{P}(x\in I_p)=\frac{1}{p}\sum_{\substack{0\leqslant a<p\\|x-a/p|<c/p}}1$$ and hence $\mathbf{P}(x\in I_p)$ is either $0$ or $1/p$, depending on whether there exists an integer $a$ such that the fractional part of $xp$ is $<c$, or not. It is a non-trivial fact from the distribution of primes that, if $x$ is irrational, then we have $$\label{eq-vino} \sum_{\{xp\}<c}\frac{1}{p}=+\infty$$ (precisely, this follows by summation by parts from the much more precise results of Vinogradov [@vinogradov Ch. XI] which give an asymptotic formula for the number of primes $p\leqslant X$ satisfying $\{xp\}<c$; we note in passing that this result has been improved since then, notably by Vaughan). Thus, since the events $\{x\in I_p\}$ are independent by construction, the non-trivial direction of the Borel--Cantelli Lemma implies $$\mathbf{P}(x\in I_p\text{ for infinitely many } p)=1$$ for any irrational $x$. Now by Fubini's Theorem, we obtain $$\begin{aligned} \mathbf{E}(\lambda(A_{\text{\boldmath${a}$}}))&= \mathbf{E}\Bigl( \int_{0}^1 1_{\{x\in I_p\text{ for infinitely many } p\}}\ dx \Bigr)\\ &=\int_0^1 \mathbf{P}(x\in I_p\text{ for infinitely many } p)dx =1,\end{aligned}$$ and since $\lambda(A_{\text{\boldmath${a}$}})\leqslant 1$, this means that $A_{\text{\boldmath${a}$}}$ has measure $1$ for almost all sequences $(a_p)$. **Remark 5**. We do not require the full force of Vinogradov's theorem, but in any case, the formula ([\[eq-vino\]](#eq-vino){reference-type="ref" reference="eq-vino"}) for an arbitrary irrational number $x$ seems to be comparable to the similar divergence of the sum of inverses of primes in an arithmetic progression. It would also be enough to know that the divergence of the series ([\[eq-vino\]](#eq-vino){reference-type="ref" reference="eq-vino"}) holds for almost all $x$ (instead of all irrationals), and it is quite likely that this can be proved more easily. # Application {#sec-appli} Let $X=(\mathbf{R}/\mathbf{Z})^2$ and $\mu$ the Lebesgue measure on $X$. Further, let $f\colon X\to X$ be the map defined by $f(x,y)=(x+y,y)$. We have $f_*\mu=\mu$. Define $\varphi\colon X\to \mathbf{C}$ by $\varphi(x,y)=e(x)$. We chose a sequence $(a_p)$ as in Theorem [Theorem 1](#th-approx){reference-type="ref" reference="th-approx"} with $c=1/2$. For $p$ prime and $n\in\mathbf{Z}$, we define $t_p(n)=e(-na_p/p)$. (This is a trace function modulo $p$, but this aspect is not important here.) **Proposition 6**. *For $p$ prime, define $s_p\colon X\to \mathbf{C}$ by $$s_p(x,y)=\frac{1}{p}\sum_{0\leqslant n<p}t_p(n)\varphi(f^n(x,y)).$$* *The following properties hold:* **(1)* The sequence $(s_p)$ does not converge almost everywhere as $p\to +\infty$.* **(2)* If $\mathsf{P}$ is an infinite set of primes such that $$\sum_{p\in\mathsf{P}}\frac{\log p}{p}<+\infty,$$ then the sequence $(s_p)_{p\in\mathsf{P}}$ converges almost everywhere to $0$.* *Proof.* Since $f^n(x,y)=(x+ny,y)$ for all integers $n\in\mathbf{Z}$, we can compute $s_n$ by summing a finite geometric progression, and we obtain $$s_p(x,y)=\frac{e(x)}{p}\frac{\sin(\pi p(y-a_p/p))}{\sin(\pi (y-a_p/p))} e\Bigl(\frac{(p-1)}{2}(y-a_p/p)\Bigr).$$ It follows that $s_p(x,y)\to 0$ along any infinite set $\mathsf{P}$ of primes such that $$\lim_{\substack{p\to +\infty\\p\in \mathsf{P}}} \ p\Bigl|y-\frac{a_p}{p}\Bigr|=+\infty.$$ Thus (2) follows because the assumption there implies that almost all $(x,y)$ satisfy $$\Bigl|y-\frac{a_p}{p}\Bigr|\geqslant\frac{\log p}{p}$$ for all but finitely many $p\in\mathsf{P}$, by the easy Borel--Cantelli lemma. We now prove (1). Note that if $(s_p)$ converges almost everywhere, the limit must be zero according to (2). But the formula for $s_p$ and the defining property of $(a_p)$ imply that for all $x$ and almost all $y\in\mathbf{R}/\mathbf{Z}$, we have $|s_p(x,y)|\gg 1$ for infinitely many primes $p$. Thus there is almost surely a subsequence which does not converge to $0$. ◻ **Remark 7**. The condition in (2) can be replaced by $$\sum_{p\in\mathsf{P}}\frac{\psi_p}{p}<+\infty,$$ where $(\psi_p)_p$ is an arbitrary sequence of non-negative real numbers such that $\psi_p\to +\infty$. CCC Y. Bugeaud: *Approximation by algebraic numbers*, Cambridge Tracts in Math. 160, Cambridge Univ. Press, 2004. A. Durand: *Describability via ubiquity and eutaxy in Diophantine approximation*, Ann. Math. Blaise Pascal 22 (2015), 1--149. A. Dvoretzky: *On covering a circle by randomly placed arcs*, Proc. Nat. Acad. Sci. 42 (1956), 199--203. H. Iwaniec and E. Kowalski: *Analytic Number Theory*, A.M.S Colloquium Publ. 53, 2004. E. Kowalski: *The large sieve and its applications*, Cambridge Tracts in Math. 175, Cambridge Univ. Press, 2008. E. Kowalski: *An introduction to expander graphs*, Cours Spécialisés 26, S.M.F, 2019. E. Kowalski: *Unmotivated ergodic averages*, preprint (2019--2023); available at [www.math.ethz.ch/\~kowalski/ergodic-trace.pdf](www.math.ethz.ch/~kowalski/ergodic-trace.pdf). L.A. Shepp: *Covering the circle with random arcs*, Israel J. Math. 11 (1972), 328--345. I.M. Vinogradov: *The method of trigonometrical sums in the theory of numbers*, Interscience Publishers, 1963.

arxiv_math

--- abstract: | We show that the boundary of a projectively compact Einstein manifold of dimension $n$ can be extended by a line bundle naturally constructed from the projective compactification. This extended boundary is such that its automorphisms can be identified with asymptotic symmetries of the compactification. The construction is motivated by the investigation of a new curved orbit decomposition for a $n+1$ dimensional manifold which we prove results in a line bundle over a projectively compact order one Einstein manifolds. author: - "Jack Borthwick[^1]" - "Yannick Herfray[^2]" bibliography: - ./../../bibPC2.bib - ./../../Carrollian-references.bib title: Asymptotic symmetries of projectively compact order one Einstein manifolds --- # Introduction In the study of objects living on non-compact manifolds it is often enlightening to try to make sense of their asymptotics. Curved orbit decompositions arising from holonomy reductions of (normal) Cartan geometries, as first studied in full generality in [@cap_holonomy_2014], provide a fascinating framework in which one can seemingly make sense of the asymptotics of geometric structures themselves. In many interesting cases, the orbit decomposition has a large open orbit $\mathcal{O}$ equipped with a certain geometric structure, e.g. an Einstein metric. Its complement can then be viewed as a boundary "at infinity" and splits into pieces endowed with other geometries that one may interpret as "limits" of the structure of $\mathcal{O}$. Defining and studying the asymptotics of pseudo-Riemannian manifolds $(M,g)$ is of particular interest due to their importance in physics. For instance, conformal compactification is key for the formulation of a notion of isolated system in General Relativity [@geroch_asymptotic_1977; @penrose_spinors_1984; @frauendiener_conformal_2004; @valiente_kroon_conformal_2016]. It is also a way to approach the study of the asymptotic behaviour to partial differential equations on non-compact pseudo-Riemannian manifolds, see for instance [@gover_poincare-einstein_2015; @curry_introduction_2018; @Mason:2004wg; @AIF_2016__66_3_1175_0; @borthwick2023peeling; @mason_nicolas_2009] in the conformal case or [@Borthwick_Proca_Projective] in the projective case, for examples of ideas and work in this direction. In [@cap_projective_2012; @cap_holonomy_2014; @Cap:2014aa; @Cap:2014ab; @cap_projective_2016-1; @Flood:2018aa], a specific type of holonomy reduction of *projective* geometries is studied that gives rise to an Einstein metric on the open orbit, suggesting that projective geometry could play an important role in understanding their asymptotics. This is intimately related to the notion of projectively compact metric of order $\alpha$ introduced by Čap--Gover in [@Cap:2014aa]. The order of a projectively compact metric is related to its volume asymptotics, and can a priori take any positive real value. For Einstein metrics, the cases $\alpha =1$ (vanishing scalar curvature) and $\alpha=2$ (non-vanishing scalar curvature) are the most relevant. Both cases were thoroughly investigated by Čap--Gover in [@Cap:2014ab]. In particular, it is shown in this reference that, for projectively compact Einstein metrics of order $1$, this notion coincides with asymptotic flatness at spatial infinity as introduced by Ashtekar and Romano in [@ashtekar_spatial_1992], however with the restriction that only spacetime with vanishing mass aspect can be obtained in this way. Asymptotic flatness at spatial infinity is by now a classical subject in general relativity, see e.g. [@beig_einsteins_1982; @compere_relaxing_2011; @mohamed_comparison_2021; @compere_asymptotic_2023]. A physically important feature of these spacetimes is their asymptotic symmetry group: the -- infinite dimensional -- Spi group of [@ashtekar_spatial_1992]. The identification of spatial infinity with projective compactifications of order 1 however presents us with a conundrum: at null infinity, the group of asymptotic symmetry (the BMS group of [@bondi_gravitational_1962; @sachs_gravitational_1962]) can be identified with the automorphisms of the boundary geometry [@geroch_asymptotic_1977; @ashtekar_geometry_2015; @duval_conformal_2014] and act on the induced Cartan geometry [@herfray_asymptotic_2020; @herfray_tractor_2022; @Herfray:2021qmp] encoding the radiative data. On the other hand the boundary geometry of projective compactification of order 1 of an Einstein metric is again a normal projective geometry with a holonomy reduction corresponding to projective compactification of order $2$ of an Einstein metric [@Cap:2014ab] and thus necessarily has a finite dimensional group of automorphisms. So how exactly does the Spi group of Ashtekar--Romano plays a role in the projective compactification picture? In this work we wish to resolve this tension by introducing an extended boundary at spatial infinity (closely related to spatial infinity in the sense of Ashtekar--Hansen [@AhstekarHansen78]). The properties of this boundary parallel exactly that of null infinity: its geometry is not rigid and the corresponding group of automorphisms 1) coincides with the Spi group, 2) acts on a space of induced Cartan geometry encoding some asymptotic gravitational data. We also obtain a similar structure at time like infinity. In order to introduce and motivate this step, we will study a closely related curved orbit decomposition of Cartan geometries [@cap_holonomy_2014] generalising the work of Cap--Gover [@Cap:2014ab]. The majority of this article will in fact be dedicated to the discussion of the curved version of a flat model discussed in great detail by Figueroa-O'Farrill and collaborators in [@Figueroa-OFarrill:2021sxz]. This homogeneous model generalises the projectively compactified 4D model $\mathbb{R}\text{P}^{4}$ to a 5D model $\mathbb{R}\text{P}^{5}\setminus \{pt\} \simeq \mathbb{R} \times \mathbb{R}\text{P}^{4}$ fibering over the usual projective compactification. $$\label{Introduction: Homogeneous model} \begin{array}{cccccc} \mathbb{R}\text{P}^{5}\setminus \{pt\} &\simeq& \left( \mathbb{R} \times M^4\right) &\sqcup& \left(\textrm{Ti}^4 \sqcup \mathscr{I}^3 \sqcup \textrm{Spi}^4\right)\\[1em] \downarrow && \downarrow && \downarrow\\[1em] \mathbb{R}\text{P}^{4} &\simeq& M^4 &\sqcup& \left( H^{3} \sqcup S^{2} \sqcup dS^{3}\right). \end{array}$$ This orbit decomposition will be discussed in more details (and generic dimension) in the core of this article, however we point here its essential features: while the action of the Poincaré group has a trivial action along the fibres[^3] $\mathbb{R} \times M^4 \to M^4$ over Minkowski space, it has a non-trivial action over the extended boundary $\textrm{Ti}^4 \sqcup \mathscr{I}^3 \sqcup \textrm{Spi}^4$: $$\label{Introduction: Boundary Homogeneous space} \begin{array}{ccccccc} \textrm{Ti}^4 &\simeq& \frac{\text{ISO}(1,3)}{\mathbb{R}^{3}\rtimes \text{SO}(3)} &\to& H^{3} &\simeq& \frac{\text{SO}(1,3)}{\text{SO}(3)} \\[1em] \mathscr{I}^3 &\simeq& \frac{\text{ISO}(1,3)}{\mathbb{R}^{3} \rtimes (\mathbb{R}^* \times \text{ISO}(2)) } &\to& S^{2} &\simeq& \frac{\text{SO}(1,3)}{\mathbb{R}^{2} \rtimes (\mathbb{R}^* \times \text{SO}(2) )} \\[1em] \textrm{Spi}^4 &\simeq& \frac{\text{ISO}(1,3)}{ \mathbb{R}^{3} \rtimes \text{SO}(1,2)} &\to& dS^{3} &\simeq& \frac{\text{SO}(1,3)}{\text{SO}(1,2)}. \end{array}$$ This points to the possibility of a non trivial geometrical extension of the boundary geometry of projectively compactified order 1 Einstein metrics. We shall prove in this article that this intuition is correct. The article is organised as follows. In section [2](#section: Projective tractors and the homogeneous model){reference-type="ref" reference="section: Projective tractors and the homogeneous model"} we recall the elements of projective tractor calculus that we will need and discuss in more details the homogeneous model [\[Introduction: Homogeneous model\]](#Introduction: Homogeneous model){reference-type="eqref" reference="Introduction: Homogeneous model"}. In section [3](#Section: curved orbit decompostion){reference-type="ref" reference="Section: curved orbit decompostion"} we detail the different curved orbits of the corresponding holonomy reduction, under suitable assumptions this results in a line bundle $M\to \mathcal{Q}$. In section [4](#section: Geometry of the open orbit O){reference-type="ref" reference="section: Geometry of the open orbit O"} we prove that the open curved orbits $\mathcal{O}$ is of the form $\mathbb{R}\times \mathcal{Q}_{\mathcal{O}}$ where $\mathcal{Q}_{\mathcal{O}}$ is an Einstein metric. We then prove, in section [5](#section: Geometry of Q){reference-type="ref" reference="section: Geometry of Q"}, that as a submanifold of $\mathcal{Q}$, this is projectively compact of order $1$ with boundary $\mathcal{H}$. In section [6](#section: The geometry of Sigma to H){reference-type="ref" reference="section: The geometry of Sigma to H"} we investigate the induced geometry of the extended boundary $\Sigma \to \mathcal{H}$, first as a submanifold of $M\to \mathcal{Q}$ and then a natural extension of $\mathcal{H} \subset \mathcal{Q}$. Finally, in section [7](#section: Asymptotic symmetries of projectively compact Einstein manifolds){reference-type="ref" reference="section: Asymptotic symmetries of projectively compact Einstein manifolds"}, we discuss asymptotic symmetries and their action on the extended boundary. # Acknowledgements {#acknowledgements .unnumbered} The first author gratefully acknowledges that this research is supported by NSERC Discovery Grant 105490-2018. # Projective tractors and the homogeneous model {#section: Projective tractors and the homogeneous model} It should be understood that all affine connections on the tangent bundle are torsion-free. We make extensive use of the Penrose abstract index notation [@Penrose:1984aa]. In index notation, tractor indices (see below) will be denoted by capital Latin letters, $A,B,C\dots,$. The signature of a bilinear form will be written $(q,p)$: "$q$ '$-$'s and $p$ '$+$'s". ## Projective tractor calculus {#Projective Tractor Calculus} We here review the elements of projective tractor calculus that we will need in the rest of the article. Let $M$ be a manifold of dimension $n+1$ endowed with an equivalence class of projectively equivalent torsion-free connections $[\nabla]$. It is a standard result[^4] that two torsion-free affine connections $\hat{\nabla}$ and $\nabla$ are projectively equivalent if and only if there is a 1-form $\Upsilon_a$ such that for any vector field $\xi^b$: $$\hat{\nabla}_a\xi^b=\nabla_a\xi^b + \Upsilon_a\xi^b +\Upsilon_c\xi^c\delta_a^b. $$ We will use $\hat{\nabla}=\nabla + \Upsilon$ as shorthand for this condition. We denote by $\mathcal{E}(w)$ the associated vector bundle to the frame bundle $P^1(M)$ determined by the $1$-dimensional representation of $GL_{n+1}(\mathbb{R})$: $$A \mapsto \lvert \det {A}\rvert ^{\frac{w}{n+2}}.$$ Sections of this bundle will be called *projective densities* of weight $w$. Their definition is so that for any $\sigma \in \Gamma(\mathcal{E}(w))$ under the projective change of connection $\hat{\nabla} = \nabla + \Upsilon$, $$\hat{\nabla}\sigma = \nabla \sigma + w\Upsilon \sigma.$$ For any vector bundle $\mathcal{B}$ with base $M$, we will write: $\mathcal{B}(w)=\mathcal{B}\otimes \mathcal{E}(w)$ and we will say that sections of $\mathcal{B}(w)$ are of weight $w$. Let $G=\textnormal{PGL}(n+1,\mathbb{R})$ and $H=(\mathbb{R}^{n+1})^*\rtimes GL(n+1,\mathbb{R})$ the isotropy subgroup of a given line. The class $[\nabla]$ determines a reduction of the second-order frame bundle $P^2(M)$ to a $H$-principle bundle $P$, on which there is a unique *normal* Cartan connection $\omega$ [@Cartan:1924aa; @Kobayashi:1995aa]; this is a normal projective geometry modeled on projective geometry as described in [@Sharpe:1997aa]. The *standard (projective) tractor bundle*, $\mathcal{T}$ can be defined as the associated vector bundle to $P$ determined by the restriction to $H$ of the $(n+2)$-dimensional representation of $G$ given by: $A\mapsto \lvert\det A\rvert^{-\frac{1}{n+2}}A$. The Cartan connection $\omega$ induces a linear connection $\nabla^{\mathcal{T}}$ on $\mathcal{T}$. The bundle $\mathcal{T}$ fits into a canonical short exact sequence of vector bundles: A choice of connection in the projective class $[\nabla]$ provides a non-canonical isomorphism $\mathcal{T}\simeq \mathcal{E}(-1)\oplus TM(-1)$, described above by the non-canonical maps $Y$ and $W$. When thinking of these maps as sections of $\mathcal{T}^*\otimes\mathcal{E}(-1)$ and $T^*M(1)\otimes\mathcal{T}$ respectively, they transform under change of connection $\hat{\nabla}=\nabla + \Upsilon$ according to: $$\hat{Y}_A=Y_A - \Upsilon_aZ^a_A, \quad \widehat{W}^A_a=W^A_a + \Upsilon_a X^A. $$ Fixing a choice of connection $\nabla$ in the projective class, sections of the standard tractor bundle will be written: $$T^A=\xi^aW_a^A + \rho X^A,$$ similar expressions will be used with sections of tensor powers of the tractor bundle. The tractor bundle should be thought of as a generalisation to the curved setting of the ambient $\mathbb{R}^{n+2}$ in the standard picture of projective space. The dual tractor bundle $\mathcal{T}^*$ is usually identified with the first jet prolongation $J^1\mathcal{E}(1)$ of the weight $1$ projective density bundle [@Bailey:1994aa]. This description has the advantage of being more direct than the principal bundle approach outlined above, and is independent of the projective structure. In addition, it allows for straightforward comparisons of tractors on different manifolds; in particular, tractors in the "bulk" and on distinguished hypersurfaces. We also recall from [@Bailey:1994aa] the standard decomposition of the curvature tensor $R_{ab\phantom{c}d}^{\phantom{ab}c}$: $$ R_{ab\phantom{c}d}^{\phantom{ab}c}=W_{ab\phantom{c}d}^{\phantom{ab}c} + 2\delta^c_{[a}P_{b]d} + \beta_{ab}\delta^c_d,$$ where the Weyl tensor $W$ is trace-free and $P_{ab}$ is the projective Schouten tensor. It is easily verified that: $$nP_{ab}= R_{ab}+\beta_{ab},\quad \beta_{ab}=-\frac{2}{n+2}R_{[ab]}=-P_{[ab]},$$ where $R_{ab}=R_{da\phantom{d}b}^{\phantom{da}d}$ is the Ricci tensor. The tensor $\beta_{ab}$ can be thought of as the curvature tensor of the density bundle $\mathcal{E}(1)$. A connection $\nabla$ is said to be *special* if it preserves a nowhere vanishing density $\sigma$. In this case $\beta_{ab}=0$ (in particular all density bundles are flat) and we say that $\nabla$ is the *scale* determined by $\sigma$. Even when it does not preserve a nowhere vanishing density, we will refer to a choice of connection in the class as a choice of scale. Having fixed a projective connection $\nabla$ in the class, and split the short-exact sequence, the action of the normal Cartan connection can be summarised conveniently by the relations: $$\label{TractorConnection} \nabla_a X^A= W^A_a, \quad \nabla_a W^B_b=-P_{ab}X^B,\quad \nabla_a Y_A=P_{ab}Z^b_A, \quad \nabla_a Z^b_B=-\delta_{a}^bY_B. $$ The tractor curvature, defined by, the identity $\Omega_{ab\phantom{C}D}^{\phantom{ab}C}T^D= 2 \nabla_{[a}\nabla_{b]}T^C$, is known to be given in an arbitrary scale by: $$\label{TractorCurvature} \Omega_{ab\phantom{C}D}^{\phantom{ab}C}=W_{ab\phantom{c}d}^{\phantom{ab}c}W^C_cZ^d_D -Y_{abd}X_CZ^d_D$$ Where: $Y_{abd}=2\nabla_{[a}P_{b]d}$ is the projective Cotton tensor. A projective class $[\nabla]$ is said to be *metric*, if it admits a solution $\zeta^{ab} \in \Gamma(S^2(T^*M)(-2))$ to the Eastwood-Matveev metrisability equation [@Eastwood:2008aa]: $$\textnormal{trace-free}\left( \nabla_c\zeta^{ab} \right)=0,$$ or equivalently $\nabla_c \zeta^{ab} = 2 \delta^{(a}_c \lambda^{b)}$ for some $\lambda^a$. Solutions of this equation have been shown [@Flood:2018aa] to be in one-to-one correspondence with tractors $H^{AB}$ that satisfy: $$\label{TractorMetrisability}\nabla_c H^{AB} + \frac{2}{n+1}X^{(A}W_{cE\phantom{B}F}^{\phantom{cE}B)}H^{EF}=0, $$ where $W_{cE\phantom{B}F}^{\phantom{cE}B}=Z^c_C\Omega_{ce\phantom{B}F}^{\phantom{ce}B}$ and $\Omega_{ce\phantom{B}F}^{\phantom{ce}B}$ is the tractor curvature. In this case $H^{AB}$ is of the form: $$H^{AB}= \zeta^{ab}W^A_aW^B_b + \lambda^aX^{(A}W^{B)}_b+ \tau X^AX^B.$$ with $$\begin{aligned} \label{H decomposition} \lambda^a &= - \frac{2}{n+2}\nabla_c\zeta^{ca},& \tau&= \frac{\nabla_a\nabla_b \zeta^{ab}}{(n+1)(n+2)}+\frac{1}{n+1}P_{ab}\zeta^{ab}.\end{aligned}$$ If $\nabla_c H^{AB}=0$, then it can be shown that Eq. [\[TractorMetrisability\]](#TractorMetrisability){reference-type="eqref" reference="TractorMetrisability"} is automatically satisfied; these solutions are said to be *normal*. This is equivalent to $$\begin{aligned} \label{DH=0} \nabla_c \zeta^{ab} - 2 \delta^{(a}_c \lambda^{b)} &=0, & \nabla_c \lambda^a + \delta^a_c \tau - P_{cb}\zeta^{ba} &=0.\end{aligned}$$ (and imply [\[H decomposition\]](#H decomposition){reference-type="eqref" reference="H decomposition"}). Normal solutions are known to correspond to Einstein metrics [@Cap:2014aa]. If one introduces[^5] $$\mathop{\mathrm{\textnormal{\textbf{det}}}}: \begin{array}{ccc} \mathcal{E}^{ab}(w) & \longrightarrow &\mathcal{E}[(n+1)(w+2) + 2 ] \\ h^{ab} &\longmapsto& \epsilon^2_{a_1 ... a_{n+1}b_1... b_{n+1}} h^{a_1b_1}... h^{a_{n+1}b_{n+1}} \end{array}$$ and[^6] $$\det : \begin{array}{ccl} \mathcal{E}^{AB} & \longrightarrow & C^{\infty}(M) \\ H^{AB} &\longmapsto& \epsilon^2_{A_0 A_1 ... A_{n+1}B_1 B_1... B_{n+1}} H^{A_0B_0}... H^{A_{n+1}B_{n+1}} \end{array}$$ then $\mathop{\mathrm{\textnormal{\textbf{det}}}}(\zeta)\zeta^{ab} \in \mathcal{E}^{ab}$ is Einstein with scalar curvature $R= n(n+1) \det(H)$. It will also be useful to keep in mind the following algebraic result (which we take from [@Flood:2018aa]): **Lemma 1**. *If $H^{AB}$ is invertible of inverse $\Phi_{AB}$ then $X^A X^B \Phi_{AB} =\det(H) \mathop{\mathrm{\textnormal{\textbf{det}}}}(\zeta)$ and if $\det(H)=0$ then the kernel is generated by $D_{A}\tau$ with $\tau^2 = \frac{\mathop{\mathrm{\textnormal{\textbf{det}}}}(\zeta)}{n+1}$.* ## The Model {#model} We now review from [@Figueroa-OFarrill:2021sxz] the homogeneous model on which is based the holonomy reduction of Cartan geometry which is considered in this article. We will write elements of the $n+1$-dimensional projective space $\mathbb{R}\text{P}^{n+1} \simeq \frac{\text{SL}(n+2)}{\mathbb{R}^{n+1} \rtimes \text{GL}(n+1)}$ in terms of homogeneous coordinates $[X]$ with $X\in \mathbb{R}^{n+2}$. Introducing a metric $\Phi$ of signature $(2,n)$ on $\mathbb{R}^{n+2}$ together with a null element $I \in \mathbb{R}^{n+2}$, $\Phi(I,I)=0$ picks a Poincaré subgroup $$\text{ISO}(1,n-1) \subset \text{SL}(n+2).$$ stabilising both $\Phi$ and $I$. Since by definition the action of the Poincaré group $\text{ISO}(1,n-1)$ stabilises $I$ this defines a point-like orbit $[I]$ in $\mathbb{R}\text{P}^{n+1}$. Introducing $$\mathcal{M}_{\lambda, \sigma} = \{ X\in \mathbb{R}^{n+2} \;s.t.\; \Phi(X,X)=\lambda, \Phi(X,I)=\sigma, X\not \propto I\}$$ the orbit decomposition of $\mathbb{R}\text{P}^{n+1}$ under $\text{ISO}(1,n-1)$ is given by $$\label{ModelOrbitDecomposition} \mathbb{R}\text{P}^{n+1} \simeq \left(\sqcup_{\lambda \in \mathbb{R}} \mathcal{M}_{\lambda, \sigma=1}\right) \sqcup \left(\mathcal{M}_{-1, 0} \sqcup \mathcal{M}_{0, 0} \sqcup \mathcal{M}_{1, 0}\right) \sqcup [I].$$ Each of the orbits in the first continuous set is isomorphic to $n$-dimensional Minkowski space: $$\begin{aligned} \forall \lambda &\in \mathbb{R}, \quad \mathcal{M}_{\lambda, \sigma=1} \simeq \frac{\text{ISO}(1,n-1)}{\text{SO}(1,n-1)}.\end{aligned}$$ The next three orbits are "extended boundaries", respectively denoted as $\text{Ti}^{n} = \mathcal{M}_{-1, 0}$, $\mathscr{I}^{n-1} = \mathcal{M}_{0, 0}$ and $\text{Spi}^{n} = \mathcal{M}_{1, 0}$ in [@Figueroa-OFarrill:2021sxz]. These are naturally real line bundles over hyperbolic space, the conformal sphere and de Sitter space respectively: $$\label{Boundary Homogeneous space} \begin{array}{rcccccc} \textrm{Ti}^n &\simeq& \frac{\text{ISO}(1,n-1)}{\mathbb{R}^{n-1}\rtimes \text{SO}(n-1)} &\to& H^{n-1} &\simeq& \frac{\text{SO}(1,n-1)}{\text{SO}(n-1)} \\[1em] \mathscr{I}^{n-1} &\simeq& \frac{\text{ISO}(1,n-1)}{\mathbb{R}^{n-1} \rtimes (\mathbb{R}^* \times \text{ISO}(n-2)) } &\to& S^{n-2} &\simeq& \frac{\text{SO}(1,n-1)}{\mathbb{R}^{n-2} \rtimes (\mathbb{R}^* \times \text{SO}(n-2) )} \\[1em] \textrm{Spi}^n &\simeq& \frac{\text{ISO}(1,n-1)}{ \mathbb{R}^{n-1} \rtimes \text{SO}(1,n-2)} &\to& dS^{n-1} &\simeq& \frac{\text{SO}(1,n-1)}{\text{SO}(1,n-2)}. \end{array}$$ We now briefly recall how this orbit decomposition relates both to conformal and projective compactifications of Minkowski space. The action of the Poincaré group on the Klein quadric $N = \{ [X]\in \mathbb{R}\text{P}^{n+1} \;s.t.\; \Phi(X,X)=0\}$ gives an orbit decomposition $$N \simeq \mathcal{M}_{\lambda=0, \sigma=1} \sqcup \mathcal{M}_{0, 0} \sqcup [I].$$ This realises the conformal compactification of Minkowski space and was taken as the model for the holonomy reduction studied in [@Herfray:2020rvq; @Herfray:2021xyp; @Herfray:2021qmp]. On the other hand, as depicted in Figure [\[Figure: homogeneous space\]](#Figure: homogeneous space){reference-type="ref" reference="Figure: homogeneous space"}, $\mathbb{R}\text{P}^{n+1}\setminus [I]$ is naturally a fibre bundle over $\mathbb{R}\text{P}^{n}$, where the projection is obtained by quotienting by $I$. This results in a fibration of the orbit decomposition (also depicted, in the dimensionally reduced $n=1$ case, in Figure [1](#Figure: 2D picture of the model){reference-type="ref" reference="Figure: 2D picture of the model"}): $$\label{Homogeneous model: diagram decomposition} \begin{array}{cccccc} \mathbb{R}\text{P}^{n+1}\setminus [I] &\simeq& \left(\sqcup_{\lambda \in \mathbb{R}} \mathcal{M}_{\lambda, \sigma=1}\right) &\sqcup& \left(\mathcal{M}_{-1, 0} \sqcup \mathcal{M}_{0, 0} \sqcup \mathcal{M}_{1, 0}\right)\\[1em] \downarrow && \downarrow && \downarrow\\[1em] \mathbb{R}\text{P}^{n} &\simeq& M^n &\sqcup& \left( H^{n-1} \sqcup S^{n-2} \sqcup dS^{n-1}\right) \end{array}$$ This quotient therefore realises the projective compactification of Minkowski space which was the starting point for the holonomy reduction studied in [@Cap:2014aa; @Cap:2014ab; @Flood:2018aa]. We therefore see that this extended model in a sense bridges between the conformal and projective compactification of Minkowski space. The investigation of the corresponding interplay between conformal and projective aspects in the associated curved orbit decomposition is the subject of this article. ![Dimensionally reduced depiction of $\mathbb{R}\textnormal{P}^{n+1}\setminus[I]$. The oriented lines indicate the fibres of the projection $\mathbb{R}\textnormal{P}^{n+1}\setminus[I]\to\mathbb{R}\textnormal{P}^{n}$; apart for those along the extended boundary at infinity -depicted here by a golden line- the fibres cross successive surfaces of constant growing $\frac{\Phi(X,X)}{(\Phi(I,X)^2}$ (ranging from $-\infty$ to $+\infty$).](2D_picture_of_the_model.pdf){#Figure: 2D picture of the model width="100%"} # Curved orbit decomposition {#Section: curved orbit decompostion} The object of this section is to study the curved picture of the previous homogenous model. Throughout, $\widetilde{M}$ is a compact manifold of dimension $n+1$ equipped with a projective structure $[\nabla]$. We assume that it is additionally endowed with a normal *non-degenerate* solution $H^{AB}$ of the metrisability equation of signature[^7] $(2,n)$, and a nowhere vanishing parallel null-tractor $I^A$: $$\begin{aligned} \label{Holonomy reduction conditions DH=0, DI=0} \nabla_a H^{AB} &=0, & \nabla_a I^A &=0,\end{aligned}$$ with $\det(H^{AB})\neq0$ and $I^AI^B\Phi_{AB}=0$, where $\Phi_{AB}$ is the pointwise inverse of $H^{AB}$. From Eq. [\[TractorConnection\]](#TractorConnection){reference-type="eqref" reference="TractorConnection"}, it is easily seen that in an arbitrary scale $\nabla$ the components of $$I^A= n^aW_a^A + \rho X^A,$$ satisfy the equations: $$\label{IParallel} \nabla_cI^A =\left(\nabla_c n^a + \delta_c^a \rho \right)W_a^A + (\nabla_c \rho - P_{ca}n^a)X^A=0 \Leftrightarrow \begin{cases} \nabla_c n^a= -\delta_c^a \rho \\ \nabla_c\rho = P_{ca}n^a \end{cases}. $$ We emphasise that $n^a \in \Gamma(T\widetilde{M}(-1))$ is a scale-independent quantity whilst $\rho$ is not. On top of this vector field, we also have at our disposal two distinguished densities (of weight 1 and 2 respectively) that will be central to our development: $$\begin{aligned} \sigma&=\Phi_{AB}I^AX^B,\label{DefSigma}\\ \lambda&=\Phi_{AB}X^AX^B.\label{DefLambda}\end{aligned}$$ These will allow to state our main result concerning the curved orbits. ## Resulting curved orbits **Theorem 1**. *[\[Thrm: Curved orbits decomposition\]]{#Thrm: Curved orbits decomposition label="Thrm: Curved orbits decomposition"}* *The nowhere-vanishing null parallel tractor $I^A$ determines a first decomposition of $\widetilde{M}$ into two parts: $$\widetilde{M} = M\sqcup \mathcal{Z}(n),$$ where $\mathcal{Z}(n) := \{x\in M, n^a=0 \}$ is composed of finitely many isolated points and $M$ is the complement. The additional data of the parallel tractor $H^{AB}$ then provides a further decomposition $$M= \mathcal{O} \sqcup \Sigma$$ Here $\mathcal{O}$ is the open submanifold such that $\sigma \neq 0$ and (if non-empty) $\Sigma := \mathcal{Z}(\sigma)\setminus \mathcal{Z}(n)$ is a smooth embedded hypersurface. Finally $\Sigma$ itself decomposes into further smooth submanifolds: $$\Sigma= \Sigma^- \sqcup \Sigma^0 \sqcup \Sigma^+$$ with $\Sigma^- :=\mathcal{Z}(\sigma)\cap\{\lambda <0\}$, $\Sigma^+:=\mathcal{Z}(\sigma)\cap\{\lambda > 0\}$ and $\Sigma^0:=\mathcal{Z}(\sigma)\cap\mathcal{Z}(\lambda)\setminus\mathcal{Z}(n).$* *Overall we have $$\label{Decomposition of the manifold} M=\mathcal{O} \sqcup \Sigma^- \sqcup \Sigma^0 \sqcup \Sigma^+ \sqcup \mathcal{Z}(n).$$* It should be noted that $\widetilde{M}$ also admits another curved orbit decomposition relative to the data $(\lambda, H^{AB})$: $$\label{Decomposition of O} \widetilde{M} = U^-\sqcup \Lambda\sqcup U^+,$$ where $U^-$, $\Lambda$ and $U^-$ are the sets $\{\lambda<0\}$, $\{\lambda=0\}$ and $\{\lambda>0\}$ respectively. In particular, it follows from [@Cap:2014ab; @Cap:2014aa; @Flood:2018aa] that $\Lambda$ is the projective boundary of $U^{\pm}$. This would additionally lead to a further decomposition of $\mathcal{O}$, however, this will not be our central focus because the decomposition [\[Decomposition of O\]](#Decomposition of O){reference-type="eqref" reference="Decomposition of O"} is *not* preserved by the flow of the vector field $n^a$. Rather we have: **Theorem 2**. *[\[Thrm: Curved orbits decomposition: quotient\]]{#Thrm: Curved orbits decomposition: quotient label="Thrm: Curved orbits decomposition: quotient"} The flow of $n^a$ preserves the decomposition [\[Decomposition of the manifold\]](#Decomposition of the manifold){reference-type="eqref" reference="Decomposition of the manifold"}. Furthermore, the quotient $\mathcal{Q}_\mathcal{O}$ of $\mathcal{O}$ by this flow is a $n$-dimensional manifold.* *If we assume that the quotient $\mathcal{Q}$ of $M=\widetilde{M}\setminus \mathcal{Z}(n)$ is a manifold we obtain the decomposition* *generalising [\[Homogeneous model: diagram decomposition\]](#Homogeneous model: diagram decomposition){reference-type="eqref" reference="Homogeneous model: diagram decomposition"} in our curved setting.* Our present goal is to identify the geometric structures of each part and the relationships between them when they are all (save possibly $\mathcal{Z}(n)$) non-empty. ## Derivation of Theorem [\[Thrm: Curved orbits decomposition\]](#Thrm: Curved orbits decomposition){reference-type="ref" reference="Thrm: Curved orbits decomposition"} We recall that the Thomas $D$-operator is defined by the following equation in an arbitrary scale: $$\label{ThomasD} D_A T^{\mathscr{B}} = \nabla_a T^{\mathscr{B}}Z^a_A +w T^{\mathscr{B}}Y_A.$$ Where $T^{\mathscr{B}}$ is a section of an arbitrary bundle with tractor indices, of weight $w$; note that $D_A$ reduces weight by $1$. The following lemma will be useful: **Lemma 2**. *$$\begin{aligned} \label{RelationISigma} I^A= H^{AB}D_B\sigma,\\ \label{RelationXLambda} \Phi_{AB}X^B=\frac{1}{2}D_A\lambda, \\ \label{InverseTractorMetric} \Phi_{AB}=\frac{1}{2}D_AD_B \lambda. \end{aligned}$$* *Proof.* In any scale, the identity tractor satisfies: $$ \delta^A_B=W^A_aZ_B^b\delta_b^a + X^AY_B=D_BX^A.$$ Since the Thomas $D$ operator satisfies the Leibniz rule, we have: $$D_A\sigma = D_A \Phi_{BC}I^BX^C + \Phi_{BC}D_AI^BX^C+\Phi_{BC}I^BD_AX^C$$ However, $\Phi$ and $I$ are weightless parallel tractors, so, by definition of the Thomas $D$ operator, $D_A\Phi_{BC}=0$ and $D_A I^B=0$. Thus: $$D_A\sigma = \Phi_{BA}I^B. $$ Multiplying by $H^{AB}$ leads to the first equation. In a similar fashion: $$D_A\lambda= 2\Phi_{BC}(D_AX^B)X^C=2\Phi_{AC}X^C$$ Applying $D_B$ once more to this equation gives the last identity. ◻ We use this to prove the first elementary decomposition of the manifold $M$. **Proposition 1**. *For any connection $\nabla$ in the projective class $\Sigma \cap \mathcal{Z}(\nabla_c\sigma) = \emptyset$. Consequently, when non-empty, $\Sigma$ is a smooth hypersurface of $M$.* *Proof.* From [\[RelationISigma\]](#RelationISigma){reference-type="eqref" reference="RelationISigma"}, we see that on $\mathcal{Z}(\sigma)\cap\mathcal{Z}(\nabla_c\sigma)$ we must have: $$0= D_A\sigma= \Phi_{AB}I^B.$$ Since by assumption, $\Phi_{AB}$ is non-degenerate this implies $I^B=0$ at such points. The result follows as we assume $I^B$ non-vanishing. ◻ The hypersurface $\Sigma$ will be interpreted as points at infinity of the region $\{\sigma \neq 0\}$; this is justified by: **Proposition 2**. *Let $\mathcal{O}_\pm=\{\pm \sigma > 0\}$, then the scale $\nabla^\sigma$ determined by $\sigma$ on $\mathcal{O}_\pm$ is projectively compact of order one and $\mathcal{O}_\pm \cup \Sigma$ is a projective compactification of $\mathcal{O}_\pm$.* *Proof.* This is a direct consequence of [@Flood:2018aa Proposition 2.4]. ◻ We recall from [@Cap:2014ab Proposition 2.3] that in this case the points are indeed "at infinity" with respect to the (parametrised) geodesics of $\nabla^\sigma$, they cannot be reached for a finite value of the affine parameter. Whilst the decomposition of $\mathcal{O}$ associated with $H^{AB}$ and the density $\lambda$ studied in [@Flood:2018aa] is not of direct interest to us, the set $\mathcal{Z}(\sigma) \cap \mathcal{Z}(\lambda)$ is an important region. Ideally, we would like it to be a codimension $2$ submanifold of $X$, the following proposition studies some possible singular points. **Proposition 3**. *Recall that $I^A$ defines a weight $-1$ vector field $n^a=Z^a_AI^A$. The set of points $\mathcal{Z}(n)$, or equivalently points at which $I\propto X$, is composed of isolated points that lie in $\mathcal{Z}(\lambda)\cap\mathcal{Z}(\sigma)$. Since $\widetilde{M}$ is compact $\mathcal{Z}(n)$ is a finite.* *Proof.* We introduce an arbitrary scale $\nabla$, and recall that $I^A=\rho X^A + n^aZ_a^A$. Thus $n$ vanishes exactly at points where $I\propto X$. Since $I$ is null, and by definition of $\sigma$, when $I\propto X$ it follows that we have both $\lambda=\Phi_{AB}X^AX^B \propto \Phi_{AB}I^AI^B=0$ and $\sigma=\Phi_{AB}X^AI^B \propto \Phi_{AB}X^AX^B=0.$ So $\mathcal{Z}(n)\subset \mathcal{Z}(\lambda)\cap\mathcal{Z}(\sigma)$. Let us prove that these are isolated points. Using Eq. [\[TractorConnection\]](#TractorConnection){reference-type="eqref" reference="TractorConnection"} we see that: $$\nabla_b n^a= \nabla_b(I^AZ_A^a)=-\delta_b^aY_AI^A=-\delta_b^a \rho.$$ Since $I$ is nowhere vanishing, $\rho$ does not vanish on $\mathcal{Z}(n)$ and for any vector field $v^b$ that does not vanish on $\mathcal{Z}(n)$, we must have: $v^b\nabla_b n^a=\rho v^b \neq 0$ therefore $\mathcal{Z}(n)$ consists of isolated points. ◻ The following justifies the introduction of $\mathcal{Z}(n)$: **Proposition 4**. *Let $\Sigma^0$ be the relative complement of $\mathcal{Z}(n)$ in $\mathcal{Z}(\sigma)\cap \mathcal{Z}(\lambda)$, then $\Sigma^0$ is a smooth submanifold of $M$ with codimension $2$.* *Proof.* From Eqs. [\[RelationISigma\]](#RelationISigma){reference-type="eqref" reference="RelationISigma"} and [\[RelationXLambda\]](#RelationXLambda){reference-type="eqref" reference="RelationXLambda"} one has $D_A\lambda= 2\Phi_{AB}X^B$ and $D_A\sigma= \Phi_{AB}I^B$. When evaluated at points of $\mathcal{Z}(\sigma)\cap\mathcal{Z}(\lambda)$ these give $$\begin{aligned} 2\Phi_{AB}X^B &= 0 \,Y_A + \nabla_a\lambda \,Z^a_{A} & \Phi_{AB}I^B &= 0 \,Y_A + \nabla_a\sigma \,Z^a_{A}. \end{aligned}$$ To conclude the proof we need to show that there are no points of $\Sigma^0$ where $\nabla_a\lambda \propto \nabla_a\sigma$. However, at such points we would have for some density $f$: $$2\Phi_{AB}X^B + f \Phi_{AB}I^B =0.$$ Since $\Phi_{AB}$ is invertible by hypothesis, this would mean $2X^B + f I^B =0$ which is excluded by definition of $\Sigma^0$. ◻ ## Derivation of Theorem [\[Thrm: Curved orbits decomposition: quotient\]](#Thrm: Curved orbits decomposition: quotient){reference-type="ref" reference="Thrm: Curved orbits decomposition: quotient"} We will first need the following **Lemma 3**. *In a generic scale $\nabla$, one has $$\begin{aligned} n^a \nabla_a \sigma &= - \rho \sigma \\ n^a \nabla_a \lambda &= 2\sigma - 2\lambda \rho \end{aligned}$$* *Proof.* Since $I$ is null, it follows from Eq. [\[RelationISigma\]](#RelationISigma){reference-type="eqref" reference="RelationISigma"} that: $I^AD_A\sigma=0$. In a generic scale $\nabla$ we have: $$I^A= n^aW_a^A+\rho X^A, \quad D_A\sigma=\sigma Y_A + \nabla_a\sigma Z^a_A,$$ and therefore $$I^AD_A\sigma = n^a\nabla_a\sigma + \rho\sigma =0.$$ Similarly from Eq. [\[RelationXLambda\]](#RelationXLambda){reference-type="eqref" reference="RelationXLambda"}, and by definition of $\sigma$, we have: $$2\sigma=I^BD_B\lambda= n^a\nabla_a\lambda + 2\rho\lambda.$$ ◻ The weight $-1$ vector field $n^a$ determines a distinguished family of unparametrised curves on $\widetilde{M}$. These are precisely the integral curves of any vector field $N^a$ obtained from $n^a$ by multiplying by a nowhere vanishing density $\nu$ of weight $1$ on $\widetilde{M}$; only the parametrisation depends on the choice of $\nu$. Since $\widetilde{M}$ is compact, any choice of $N^a$ leads to a complete vector field on $\widetilde{M}$ which restricts to a complete vector field on $M$. Lemma [Lemma 3](#CovariantDerivativesDirectionn){reference-type="ref" reference="CovariantDerivativesDirectionn"} shows that $n^a$ is tangent to the submanifolds $\Sigma$ and $\Sigma^0$ from which it follows immediately: **Corollary 1**. *For any choice of nowhere vanishing density $\nu$ of weight $1$ on $\tilde{M}$, the action of $(\mathbb{R},+)$ given by the global flow of the vector field $N=\nu n$ on $N$, preserves the decomposition [\[Decomposition of the manifold\]](#Decomposition of the manifold){reference-type="eqref" reference="Decomposition of the manifold"}.* **Definition 1**. *Define $\mathcal{Q}$ to be the quotient space $M/\mathbb{R}$, equipped with the quotient topology, where the action of $\mathbb{R}$ is provided by the flow of any $N=\nu n$ and $\nu$ is a nowhere vanishing scale on $M$ that has a smooth extension to $\widetilde{M}$.* *Let $\pi: M \rightarrow \mathcal{Q}$ be the natural projection, we define the distinguished open subset $\mathcal{Q}_\mathcal{O}=\pi(\mathcal{O})$ of $\mathcal{Q}$ ($\pi$ is an open map). Since $\mathcal{O}$ is stable under the action of $\mathbb{R}$ this can also be defined as the quotient space $\mathcal{O}/\mathbb{R}$ where the action of $\mathbb{R}$ is that of the flow of $N^\sigma=\sigma n$ equipped with the quotient topology.* We will now show that $\mathcal{Q}_\mathcal{O}$ has a natural manifold structure. **Theorem 3**. *Let $\pi: \mathcal{O} \rightarrow \mathcal{Q}_\mathcal{O}$ be the canonical projection. Then there is a unique smooth structure on $\mathcal{Q}_\mathcal{O}$ such that $\pi$ has the structure of a (right) $\mathbb{R}$-principal fibre bundle. The zero set of the function $l:= \sigma^{-2}\lambda : \mathcal{O} \to \mathbb{R}$ defines a global section and yields a preferred global diffeomorphism $$\begin{array}{ccc} \mathcal{O} & \simeq &\mathbb{R} \times \mathcal{Q}_\mathcal{O}\\ x & \mapsto & ( \frac{1}{2}l(x) , \pi(x) ) \end{array}.$$ In particular, $\omega_a=\frac{1}{2} \nabla_a l$ is a canonical principal connection.* *Proof of Theorem [Theorem 3](#TheoFibreBundleInside BIS){reference-type="ref" reference="TheoFibreBundleInside BIS"}.* We apply the characterisation of principle fibre bundles [@Sharpe:1997aa Theorem **4**.2.4]. It suffices to show that action is free and proper. We shall work in the distinguished scale determined by $\sigma$ on $\mathcal{O}$. We define $l=\sigma^{-2}\lambda$, and let $\gamma$ denote an arbitrary integral curve of $N=\sigma n$, then Eq. [\[sigma scale =\> N\^a nabla_a l= 2\]](#sigma scale => N^a nabla_a l= 2){reference-type="eqref" reference="sigma scale => N^a nabla_a l= 2"} leads to: $$\label{LambdaAlongCurve BIS}  \frac{\textnormal{d}}{\textnormal{d}t}\lambda(\gamma(t))= (N^a\nabla_a l)|_{\gamma(t)}=2.$$ It follows that $l=2t +l_0$ along the curve $N$, consequently $l$ is strictly monotonous and unbounded along such a curve. Thus the action is: - free: the isotropy subgroup of any point $x$ is either $\mathbb{R}$ (i.e. $\forall t \in \mathbb{R}, \phi^N_t(x)=x$) or $T\mathbb{Z}$ for some $T\in \mathbb{R}$. The first case is excluded as $n^a$ does not vanish on $\mathcal{O}$. If it was of the form $T\mathbb{Z}$ with $T>0$ then the curve $t\mapsto \phi^N_t(x)$ would be periodic, but this is excluded because $l$ is strictly monotonous along such a curve. Hence, $T=0$. - proper: Let $K$ be a compact subset of $M\setminus \Sigma$, then, $l$, being continuous, is bounded above and below on $K$ say: $\forall x \in K, m \leq l(x)\leq M$. Let $x \in K$, then: $$l(\phi^N_t(x)) = 2 t + l(x) \geq 2t + m.$$ So for any $x\in K$, when $t \geq (\frac{M-m +1}{2})$ $\phi^N_t(x)$ has definitively left $K$. This implies that $\{t\in \mathbb{R}, t K \cap K \neq \emptyset\}$ is compact. Finally, Eq. [\[LambdaAlongCurve BIS\]](#LambdaAlongCurve BIS){reference-type="eqref" reference="LambdaAlongCurve BIS"} implies that $\frac{1}{2}l(\phi^N_t(p))=\frac{1}{2}l(p) + t$, which shows that $\frac{1}{2}l$ provides a canonical trivialisation of $\mathcal{O}$. ◻ *Remark 1*. For future computations it will be useful to note that $N$ is by definition a fundamental vector field on $\mathcal{O}$. Recall that if $P$ is a principal $G$ bundle then the fundamental vector fields $X^*$ associated with $X\in \mathfrak{g}$ is defined at $p\in P$ by $$A^*_p=\left.\frac{\textnormal{d}}{\textnormal{d}t}(p \exp(Xt))\right|_{t=0},$$ in the simple case we consider of the abelian Lie algebra $\mathbb{R}$, the exponential map satisfies $\exp(1t)=t$ therefore: $$1^*_x = \left. \frac{\textnormal{d}}{\textnormal{d}t}(p\cdot t)\right|_{t=0}= \left. \frac{\textnormal{d}}{\textnormal{d}t}\phi^N_t(x)\right|_{t=0}=N_x.$$ It is not clear that $\mathcal{Q}$ itself can be equipped with a smooth structure, as the behaviour of $n$ near $\Sigma$ (to which it is tangent) is unclear. However, to simplify our discussion we shall make the assumption that $\mathcal{Q}$ can be equipped with a smooth structure such that $\pi : M\rightarrow \mathcal{Q}$ is a fibre bundle with fibre $\mathbb{R}$. We conclude this section by recording the following useful result. **Lemma 4**. *Let $\nu$ be any nowhere vanishing scale and $N^a = \nu n^a$, $g^{ab} = \nu^2 \zeta^{ab}$ then $$\mathcal{L}_{N} g^{ab} = 2 N^{(a} \lambda^{b)} + 2 \rho g^{ab},$$ where $\lambda^{a}$ and $\rho$ are respectively defined by equations [\[DH=0\]](#DH=0){reference-type="eqref" reference="DH=0"} and [\[IParallel\]](#IParallel){reference-type="eqref" reference="IParallel"}.* *Proof.* For any vector field $X$, we have the identity $$\mathcal{L}_{X}g^{ab} = \nabla_X g^{ab} - 2 \nabla_c X^{(a} g^{b)c}.$$ Taking $X^a = N^a$ and making use of [\[DH=0\]](#DH=0){reference-type="eqref" reference="DH=0"} and [\[IParallel\]](#IParallel){reference-type="eqref" reference="IParallel"} one obtains $$\mathcal{L}_{N} g^{ab} = 2 N^{(a} \lambda^{b)} + 2 \rho g^{ab}.$$ ◻ # Geometry of the open orbit $\mathcal{O}$ {#section: Geometry of the open orbit O} In this section we restrict ourselves to the curved orbit $\mathcal{O}$ on which $\sigma$ is nowhere vanishing. We can therefore form the undensitised bi-vector, vector field and function $$\begin{aligned} g^{ab} &= \sigma^2 \zeta^{ab}, & N^a &= \sigma n^a& l&=\sigma^{-2}\lambda.\end{aligned}$$ The scale $\nabla$ determined by $\sigma$ (i.e. such that $\nabla\sigma=0$) has the following important properties: **Lemma 5**. *In the scale $\nabla$ determined by $\nabla \sigma =0$, we have: $$\begin{aligned} \nabla_c n^a &= -\delta^a_c \rho =0, \label{sigma scale => rho=0} \\ \lambda n^a&=-\frac{1}{2}\sigma \zeta^{ab}\nabla_b\lambda, \label{sigma scale => n=grad(l)} \\ \nabla_c \zeta^{ab}&=-2\sigma^{-1}\delta^{(a}_{c}n^{b)} \label{sigma scale => lambda^a =n^a}. \end{aligned}$$* *Proof.* Equations [\[sigma scale =\> rho=0\]](#sigma scale => rho=0){reference-type="eqref" reference="sigma scale => rho=0"} follows from Eqs. [\[IParallel\]](#IParallel){reference-type="eqref" reference="IParallel"}, [\[RelationISigma\]](#RelationISigma){reference-type="eqref" reference="RelationISigma"}, $I^2=0$ and the definition of the scale: $$\rho = I^AY_A= I^A(\sigma^{-1}D_A\sigma) =\sigma^{-1}I^2 =0.$$ To prove the second relation $\eqref{sigma scale => n=grad(l)}$ we remark that from Eq. [\[RelationXLambda\]](#RelationXLambda){reference-type="eqref" reference="RelationXLambda"} one has: $X^A=\frac{1}{2}H^{AB}D_B\lambda$. Contracting with $Z_A^a$ then gives $$0=\frac{1}{2}Z_A^aH^{AB}D_B\lambda= \frac{1}{2}\zeta^{ab}\nabla_b\lambda + Z_A^a\underbrace{H^{AB}Y_B}_{\sigma^{-1}H^{AB}D_B\sigma}\lambda=\frac{1}{2}\zeta^{ab}\nabla_b\lambda +\sigma^{-1}n^a\lambda,$$ where we have used Eq. [\[RelationISigma\]](#RelationISigma){reference-type="eqref" reference="RelationISigma"}. Finally, the last relation $\eqref{sigma scale => lambda^a =n^a}$ comes from: $$\nabla_c \zeta^{ab}= \nabla_c Z_A^aH^{AB}Z_B^b=-\delta^{a}_{c}Y_AH^{AB}Z_B^b-\delta^{b}_{c}Z_B^bH^{AB}Y_B=-2\sigma^{-1}\delta^{(a}_{c}n^{b)}.$$ ◻ *Remark 2*. Since $\sigma$ is covariantly constant for the connection $\nabla$, Eqs. [\[sigma scale =\> rho=0\]](#sigma scale => rho=0){reference-type="eqref" reference="sigma scale => rho=0"},[\[sigma scale =\> n=grad(l)\]](#sigma scale => n=grad(l)){reference-type="eqref" reference="sigma scale => n=grad(l)"}, and [\[sigma scale =\> lambda\^a =n\^a\]](#sigma scale => lambda^a =n^a){reference-type="eqref" reference="sigma scale => lambda^a =n^a"} translate directly in terms of the quantities $g,N,f$: $$\nabla_c N^a =0, \quad N^a= -\frac{1}{2}l^{-1}g^{ab}\nabla_b l, \quad \nabla_c g^{ab}=-2\delta_{c}^{(a}N^{b)}.$$  Equation [\[sigma scale =\> rho=0\]](#sigma scale => rho=0){reference-type="eqref" reference="sigma scale => rho=0"} means that the integral lines of $N^a$ are geodesically parametrised for $\nabla$. In fact, combining this with Lemma [Lemma 3](#CovariantDerivativesDirectionn){reference-type="ref" reference="CovariantDerivativesDirectionn"}, one sees that $$\label{sigma scale => N^a nabla_a l= 2} N^a \nabla_a l= 2$$ in other terms $l$ coincides with this parametrisation. This however fails to be a gradient flow for $g^{ab}$: Eq. [\[sigma scale =\> n=grad(l)\]](#sigma scale => n=grad(l)){reference-type="eqref" reference="sigma scale => n=grad(l)"} recovers that $g^{ab}$ is degenerate along $l=0$, and one rather has $N^a = -\frac{1}{2} g^{ab}(l^{-1} \nabla_b l)$. The last identity [\[sigma scale =\> lambda\^a =n\^a\]](#sigma scale => lambda^a =n^a){reference-type="eqref" reference="sigma scale => lambda^a =n^a"} is crucial to derive the following canonical form: **Proposition 5**. *$$g^{ab} = -l N^a N^b + \tilde{h}^{ab}$$ where $\tilde{h}^{ab}\nabla_b l=0$ and $\mathcal{L}_{N}\tilde{h}^{ab}=0$.* *Proof.* We recall from Lemma [Lemma 4](#LieDerivativeofg){reference-type="ref" reference="LieDerivativeofg"} that in a generic scale $\nu$ $$\mathcal{L}_{N} g^{ab} = 2 N^{(a} \lambda^{b)} + 2 \rho g^{ab},$$ where $\lambda^a$ is defined by the relation $\nabla_c \zeta^{ab} - 2 \delta^{(a}_c \lambda^{b)} =0$. When evaluated in the scale $\sigma$ this becomes, by making use of [\[sigma scale =\> rho=0\]](#sigma scale => rho=0){reference-type="eqref" reference="sigma scale => rho=0"} and [\[sigma scale =\> lambda\^a =n\^a\]](#sigma scale => lambda^a =n^a){reference-type="eqref" reference="sigma scale => lambda^a =n^a"}, $$\mathcal{L}_{N} g^{ab} = -2N^{a} N^b.$$ Since $N^a \nabla_a l=2$, this is equivalent to $\mathcal{L}_{N}( g^{ab} + l N^{a} N^b)=0$ and therefore one can write $$g^{ab} = -l N^{a} N^b + \tilde{h}^{ab}$$ with $\mathcal{L}_{N} \tilde{h}^{ab}=0$. We conclude by computing $\tilde{h}^{ab} \nabla_b l$ $$\begin{aligned} \tilde{h}^{ab} \nabla_b l &= g^{ab}\nabla_b l +l N^{a} N^b\nabla_b l\\ &= -2 l N^a + 2 l N^a =0 \end{aligned}$$ where we made use of [\[sigma scale =\> n=grad(l)\]](#sigma scale => n=grad(l)){reference-type="eqref" reference="sigma scale => n=grad(l)"}. ◻ Using Lemma [Lemma 5](#Lemma: sigma scale => identities){reference-type="ref" reference="Lemma: sigma scale => identities"} we have: **Proposition 6**. *At points of $\mathcal{O}$ where $g^{ab}$ is invertible, the corresponding metric is Ricci-flat.* *Remark 3*. Note that the Ricci flat metric $\sigma^2\zeta^{ab}$ of this proposition differs from the Einstein metric with a non-vanishing scalar curvature $\lambda \zeta^{ab}$ that is given by the projective compactification of order 2 as in [@cap_einstein_2014]. *Proof.* Starting from $$\begin{aligned} 0= D_B I^A =D_B H^{AC}D_C\sigma=H^{AC}D_B (\sigma Y_C)&=\sigma H^{AC}Z_B^bP_{bc}Z^c_C \\&=\left(\zeta^{ac}W_a^A +n^c\sigma^{-1} X^A)P_{bc}Z^b_B \right)\\&=\zeta^{ac}P_{bc} W_a^AZ^b_B, \end{aligned}$$ where we have used Eq. [\[IParallel\]](#IParallel){reference-type="eqref" reference="IParallel"}. It follows that when $\zeta^{ab}$ is non-degenerate, $P_{bc}=0$ which extends by density to all of $\mathcal{O}$. Define, when $l\neq 0$, a torsion-free connection $\tilde{\nabla}$ by: $$\label{RelationScaleSigmaLeviCivita} \tilde{\nabla}_c \xi^b = \nabla_c \xi^b +N^bg_{cd}\xi^d,$$ for any vector field $\xi^b$. *Remark 4*. Note that this is *not* a projective change of connection. It is straightforward to check that $\tilde{\nabla}$ is the Levi-Civita connection of $g$. Let us calculate the Ricci tensor $\tilde{R}_{ab}$ of $\tilde{\nabla}$ with respect to that of $\nabla$ which we denote by $R_{ab}$. In fact, $R_{ab}$ vanishes since $\sigma$ is a special scale and $R_{ab}=(n-1)P_{ab}=0$. We have: $$\tilde{R}_{ab}= \underbrace{R_{ab}}_{=0} + \underbrace{N^c\nabla_c g_{ab}}_{\frac{1}{2}\Upsilon_a\Upsilon_b} + \frac{1}{2}\nabla_a\Upsilon_b -\frac{1}{4}\Upsilon_a\Upsilon_b-\frac{1}{2}\underbrace{(N^c\Upsilon_c)}_{2\sigma^2\lambda^{-1}}g_{ab}$$ Thus: $$\tilde{R}_{ab}= \frac{1}{2}\nabla_a\Upsilon_b +\frac{1}{4}\Upsilon_a\Upsilon_b-\sigma^2\lambda^{-1}g_{ab}$$ where we have introduced: $\Upsilon=\lambda^{-1}\nabla \lambda$. Using $H^{AB}\Phi_{BC}=\delta^A_C$ and Equation [\[InverseTractorMetric\]](#InverseTractorMetric){reference-type="eqref" reference="InverseTractorMetric"} one finds that: $$ \lambda^{-1}\sigma^2 g_{ab}= \frac{1}{2} \left(\lambda^{-1}\nabla_a\nabla_b\lambda - \frac{1}{2}\lambda^{-2}\nabla_a \lambda\nabla_b \lambda \right)=\frac{1}{2}\left(\nabla_a \Upsilon_b + \frac{1}{2}\Upsilon_a\Upsilon_b \right),$$ which shows that: $$ \tilde{R}_{ab}= 0.$$ ◻ Proposition [Proposition 5](#Proposition: canonical form on O){reference-type="ref" reference="Proposition: canonical form on O"} suggests that $\tilde{h}^{ab}$ might induce a smooth tensor on the quotient $\mathcal{Q}_{\mathcal{O}}$ of $\mathcal{O}$ by the integral lines of $N^a$. In turn, Eqs [\[sigma scale =\> lambda\^a =n\^a\]](#sigma scale => lambda^a =n^a){reference-type="eqref" reference="sigma scale => lambda^a =n^a"} and [\[RelationScaleSigmaLeviCivita\]](#RelationScaleSigmaLeviCivita){reference-type="eqref" reference="RelationScaleSigmaLeviCivita"} suggest that the connection $\nabla$ factors to the quotient and this factorisation will yield the Levi-Civita connection of such a tensor field; the following results confirm this intuition. **Proposition 7**. *$g^{ab}$ induces a bilinear form $h^{ab}$ on $\mathcal{Q}_{\mathcal{O}}$ which is everywhere non-degenerate.* *Proof.* Let $\alpha,\beta$ be $1$-form fields on $\mathcal{Q}_{\mathcal{O}}$ and consider the smooth function on $\mathcal{O}$ defined by $g(\pi^*\alpha,\pi^*\beta)$, Lemma [Proposition 5](#Proposition: canonical form on O){reference-type="ref" reference="Proposition: canonical form on O"} implies that $g(\pi^*\alpha,\pi^*\beta)= \tilde{h}(\pi^*\alpha,\pi^*\beta)$ is constant on the fibres of $\mathcal{O} \to \mathcal{Q}_{\mathcal{O}}$ and so factorises to a well-defined smooth map on $\mathcal{Q}_{\mathcal{O}}$, that we denote by $h(\alpha,\beta)$. It is clearly symmetric and tensorial so determines a bilinear form on $T^*\mathcal{Q}_{\mathcal{O}}$. Now $\zeta^{ab}$ is degenerate exactly at points where $\lambda =0$ and so $g^{ab} = \sigma^{2}\zeta^{ab}$ is invertible at any point where $l= \sigma^{-2}\lambda \neq 0$. However, at any point of $\mathcal{O}$, one has the decomposition $T^*\mathcal{O} = \text{Span}(\nabla_a l) \oplus \pi^* (T^*\mathcal{Q}_{\mathcal{O}})$ and by Lemma [Proposition 5](#Proposition: canonical form on O){reference-type="ref" reference="Proposition: canonical form on O"} the corresponding decomposition of $g^{ab}$ is $\text{diag}(-l, h^{ab})$. Therefore $g^{ab}$ can only be invertible at points where $l\neq0$ if $h^{ab}$ is. ◻ **Proposition 8**. *The connection $\nabla$ is invariant under the flow of $N$.* *Proof.* First we consider instead the principal connection $\alpha$ determined by $\nabla$ on the frame bundle $P^1(\mathcal{O})$. The $1$-parameter group of diffeomorphisms generated by $N$ induces a $1$-parameter group of diffeomorphisms on $P^1(\mathcal{O})$. Since $N$ is parallel, the infinitesimal generator $\tilde{N}$ of the induced $1$-parameter group of diffeomorphisms[^8] is precisely the *horizontal lift* of $N$ with respect to the principal connection $\alpha$. This can be seen, for instance, by a swift computation in a canonical coordinate system of $P^1(\mathcal{O})$. Since the Lie derivative satisfies the Leibniz rule we have: $$(\mathcal{L}_{\tilde{N}}\alpha)(\tilde{X})=\tilde{N}(\alpha(\tilde{X})) - \alpha([\tilde{N},\tilde{X}]).$$ If $\tilde{X}=X^*, X\in \mathfrak{gl}(n+1,\mathbb{R})$ is a fundamental vector field, this vanishes entirely; indeed: $\alpha(X^*)=X$ is constant and the Lie bracket between a fundamental field and a horizontal vector field vanishes. Hence, $\mathcal{L}_{\tilde{N}}\alpha$ is a horizontal and equivariant $\mathfrak{gl}(n+1,\mathbb{R})$ valued 1-form. In fact, restricting to the case where $\tilde{X}$ is horizontal ($\alpha(\tilde{X})=0$), we see that it reduces to $$(\mathcal{L}_{\tilde{N}}\alpha)(\tilde{X})=-\alpha([\tilde{N},\tilde{X}])=\Omega(\tilde{N},\tilde{X}), $$ where $\Omega$ is the curvature $2$-form. It follows that we should compute: $N^aR_{ab\phantom{c}d}^{\phantom{ab}c}.$ Using Lemma [Lemma 5](#Lemma: sigma scale => identities){reference-type="ref" reference="Lemma: sigma scale => identities"}, we see that although $\nabla$ is not the Levi-Civita connection of $g$, it does satisfy: $\nabla_d \nabla_c g^{ab}=0,$ which leads to the following symmetry of the curvature tensor: $$R^{abcd}=-R^{abdc},$$ where all indices are raised with $g^{ab}$. This implies that $R^{abcd}$ has all the symmetries of the curvature tensor of the Levi-Civita connection of a metric. In particular, at points where $l\neq 0$, $$N^aR_{a}^{\phantom{a}bcd}=-\frac{1}{2}l^{-1}\nabla_a l R^{abcd}=\frac{1}{2}l^{-1}\nabla_al R^{cdba}=-R^{cdb}_{\phantom{cdb}a}N^a.$$ However, since $N^a$ is parallel, we must have: $R_{ab\phantom{c}d}^{\phantom{ab}c}N^d=0$, thus at points where $l\neq 0$ (at which $g^{ab}$ is non-degenerate): $$ N^aR_{ab\phantom{c}d}^{\phantom{ab}c}=0,$$ but these points are dense in $\mathcal{O}$, so the above identity extends to $\mathcal{O}$ by continuity. Consequently: $$\mathcal{L}_{\tilde{N}}\alpha=0.$$ ◻ Before we state the immediate corollary, we first record a consequence of the final steps of our computation: **Lemma 6**. *At every point of $M$: $$n^aW_{ab\phantom{c}d}^{\phantom{ab}c}=0$$* *Proof.* Both the Weyl tensor $W_{ab\phantom{c}d}^{\phantom{ab}c}$ and $n^a$ are projectively invariant, hence this is a projectively invariant statement and it suffices to check in one particular scale. On $\mathcal{O}$, $\sigma$ is a scale and we have seen above that the Riemann tensor of the connection $\nabla^\sigma$ satisfies: $$n^aR_{ab\phantom{c}d}^{\phantom{ab}c}=0. $$ However, we saw in the proof of Proposition [Proposition 6](#Proposition: einstein eq sur g){reference-type="ref" reference="Proposition: einstein eq sur g"} that $P_{bc}=0$ in this *special* scale, i.e. $\beta_{ab}=0$ too. Hence: $R_{ab\phantom{c}d}^{\phantom{ab}c}=W_{ab\phantom{c}d}^{\phantom{ab}c}$ and thus the result holds on $\mathcal{O}$, since $\mathcal{O}$ is dense in $M$ it extends to $M$ by density. ◻ The immediate corollary of Proposition [Proposition 8](#Connection invariant under flow O){reference-type="ref" reference="Connection invariant under flow O"} is that : **Corollary 2**. *[\[Proposition: connection on Q\]]{#Proposition: connection on Q label="Proposition: connection on Q"} $\nabla$ induces a torsion-free connection $\bar{\nabla}$ on the quotient $\mathcal{Q}_{\mathcal{O}}$.* *Proof.* Let $\bar{X},\bar{Y}$ be vector fields on $\mathcal{Q}_{\mathcal{O}}$, and $X,Y$ their horizontal lifts to vector fields $\mathcal{O}$ using the principal connection $\omega$ in Theorem [Theorem 3](#TheoFibreBundleInside BIS){reference-type="ref" reference="TheoFibreBundleInside BIS"} and define: $$\bar{\nabla}_X Y = \pi_*(\nabla_{\overline{X}}\overline{Y})$$ By the previous proposition, this is independent of choice on the fibre and so is a well-defined vector field on $Q$. This is clearly linear in $X$ and satisfies the Leibniz rule in $Y$ and therefore is a connection. It is torsion-free as: $$ \bar{\nabla}_{\bar{Y}}{\bar{X}}-\bar{\nabla}_{\bar{X}}{\bar{Y}}=\pi_*(\nabla_X Y -\nabla_Y X)= \pi_*([X,Y])=[\pi_*(X),\pi_*(Y)]=[\bar{X},\bar{Y}],$$ by naturality of the pushforward. ◻ **Proposition 9**. *The induced connection $\bar{\nabla}$ on $\mathcal{Q}_{\mathcal{O}}$ is the Levi-Civita connection of the form $h$ defined in Proposition [Proposition 7](#Proposition: invertible metric on Q_O){reference-type="ref" reference="Proposition: invertible metric on Q_O"}; furthermore the Lorentzian geometry $(Q,h_{ab})$ is Ricci-flat.* *Proof.* If $\bar{X}$ is a vector field on $\mathcal{Q}_{\mathcal{O}}$, $X$ its horizontal lift to a vector field on $\mathcal{O}$, and $\alpha, \beta$ are 1-forms on $\mathcal{Q}_{\mathcal{O}}$, then at every point $x\in \mathcal{O}$ we have: $$ \begin{aligned} (\bar{\nabla}_{\bar{X}} h)(\alpha,\beta)(\pi(x))&= \nabla_{\bar{X}}g(\pi^*\alpha, \pi^*\beta)(x)\\&=-(\pi^*\alpha)(N)(\pi^*\beta)(X)(x)-(\pi^*\beta)(N)(\pi^*\alpha)(X)\\&=0. \end{aligned}$$ This shows that $\bar{\nabla}$ is the Levi-Civita connection for $h$. Now, if $\bar{X},\bar{Y}$ and $\bar{Z}$ are vector fields on $\mathcal{Q}_\mathcal{O}$ and $X,Y,Z$ their horizontal lifts to $\mathcal{O}$ then, noting that $\nabla_a l[X,Y]^a=0$, and using that $N$ is parallel for $\nabla$, we have: $$\bar{R}(\bar{X},\bar{Y})\bar{Z}= \pi_*(R(X,Y)Z). $$ If $(\bar{E}_i)$ is an arbitrary local frame of $T\mathcal{Q}_{\mathcal{O}}$ then lifting them horizontally to fields on $\mathcal{O}$, we obtain a local frame $(E_i,N)$ of $TM$, with dual frame: $(\omega^i, \frac{1}{2}\textnormal{d}l)$. Now: $$\textrm{Ric}(Y,Z)= \sum_i \omega^i(R(E_i, Y)Z) + \frac{1}{2}\textnormal{d}l(R(N,Y)Z)= \sum_i \omega^i(R(E_i, Y)Z).$$ By definition, $\omega^i(E_j)=\delta^i_j$, $\omega^i(N)=0$ therefore: $\mathcal{L}_N \omega^i(E_j) = -\omega^i([N,E_j])=0$. Thus, the $\omega^i$ are the pullbacks of a local frame $\bar{\omega}^i$ of $\mathcal{Q}_\mathcal{O}$ and it follows that: $$ 0=\textrm{Ric}(Y,Z)=\overline{\textrm{Ric}}(\tilde{Y},\tilde{Z}).$$ We can also check this in terms of the global trivialisation given in Theorem [Theorem 3](#TheoFibreBundleInside BIS){reference-type="ref" reference="TheoFibreBundleInside BIS"}: $$\begin{array}{ccc} T\mathcal{O} & \to & \text{Span}(N) \oplus T\mathcal{Q}_\mathcal{O}\\ v^a & \mapsto &( \frac{1}{2}(v^b dl_b) N^a , v^b \nabla_bx^{\mu}) \end{array}$$ the decomposition of the tangent bundle given by $l$. At any point where $\lambda \neq 0$ the inverse of $g^{ab}$ is, in this decomposition, $$g_{ab} = \begin{pmatrix}-\frac{1}{l} & 0 \\ 0 & h_{\mu\nu}\end{pmatrix}.$$ The local connection form of the Levi-Civita connection in coordinates $(x^0=\frac{1}{2} l, x^\mu)$ adapted the global trivialisation is then given by $$\label{Levi-civita in the scale sigma} (\omega^i_{\, j})_c= \nabla_c x^0\begin{pmatrix} -\frac{1}{2l} & 0 \\ 0 & 0 \end{pmatrix} + \nabla_c x^\rho \begin{pmatrix} 0 & 0 \\ 0 & \gamma^{\mu}_{\rho\nu} \end{pmatrix}$$ where $\gamma^{\mu}_{\rho\nu}$ are the Christoffel symbols of the Levi-Civita connection of $h$. The local curvature form is $$\label{Riemann in the scale sigma} (F^i{}_{j})_{cd}= \nabla_cx^{\rho}\wedge\nabla_d x^{\sigma} \begin{pmatrix} 0 & 0 \\ 0 & R^{\mu}{}_{\nu\rho\sigma} \end{pmatrix}$$ where $R^{\mu}{}_{\nu\rho\sigma}$ is the Riemann tensor of $h$. Since by Proposition [Proposition 6](#Proposition: einstein eq sur g){reference-type="ref" reference="Proposition: einstein eq sur g"} $g_{ab}$ is Ricci-flat it immediately follows that $h_{ab}$ is also. ◻ # Geometry of $\mathcal{Q}$ {#section: Geometry of Q} We recall that $M := \widetilde{M}\setminus \mathcal{Z}(n)$ and assume, as in Theorem [\[Thrm: Curved orbits decomposition: quotient\]](#Thrm: Curved orbits decomposition: quotient){reference-type="ref" reference="Thrm: Curved orbits decomposition: quotient"}, that its quotient $\mathcal{Q}$ by the flow of $n^a$ can be equipped with a smooth manifold structure. It then decomposes as $$\mathcal{Q}= \mathcal{Q}_\mathcal{O} \cup \mathcal{H}^- \cup \mathcal{H}^0 \cup \mathcal{H}^+ .$$ From the previous section, in particular Proposition [Proposition 7](#Proposition: invertible metric on Q_O){reference-type="ref" reference="Proposition: invertible metric on Q_O"} and Proposition [Proposition 9](#Proposition: einstein eq sur h){reference-type="ref" reference="Proposition: einstein eq sur h"}, we learned that $\mathcal{Q}_\mathcal{O}$ is equipped with a Ricci flat Lorentzian metric. We will now show that this metric is projectively compact of order 1 with $\mathcal{H}^- \cup \mathcal{H}^0 \cup \mathcal{H}^+$ forming the projective boundary. In order to prove this we will proceed in 3 steps. First, we will discuss how to identify densities and tractor bundles on $Q$ with subbundles of bundles on $M$ via pullback. This first step can be thought as an infinitesimal version of the relations between the homogeneous models [\[Figure: homogeneous space\]](#Figure: homogeneous space){reference-type="ref" reference="Figure: homogeneous space"}. In particular, since $H^{AB}$ and $I_A$ are covariantly constant on the whole of $M$ they will define corresponding tractor field $\bar{H}^{AB}$ and $\bar{I}_A$ on $\mathcal{Q}$. As a result of the quotient by $I^A$ which is necessary for the identification, $\bar{H}^{AB}$ then is degenerate of kernel $\bar{I}_A$. Secondly, we will show that the projective tractor connection $\nabla$ of $M$ induces a normal projective tractor connection $\bar{\nabla}$ on $\mathcal{Q}$ and that $\bar{\nabla}_c\bar{H}^{AB}=0$, $\bar{\nabla}_c\bar{I}_{A}=0$. This will imply by [@Cap:2014ab; @Flood:2018aa] that $\mathcal{Q}$ is equipped with a Ricci flat metric projectively compact of order 1. Finally we will show that on $\mathcal{Q}_\mathcal{O}$ this metric coincides with the Ricci flat metric resulting from the previous section. ## Densities and tractors of $\mathcal{Q}$ We will first show how to relate densities on $M$ with densities on $\mathcal{Q}$. Extending this at the level of $1$-jets will yield the corresponding identification between projective tractors $\mathcal{T}_M$ on $M$ with projective tractors $\mathcal{T}_\mathcal{Q}$ of $\mathcal{Q}$. ### Densities on $\mathcal{Q}$ In the sequel, we shall assume that the flow of $n^a \in \Gamma\left(TM(-1)\right)$ on $M := \widetilde{M} \setminus \mathcal{Z}(n)$ defines a fibre bundle $$\label{M -> Q} \pi :\begin{array}{ccc} M & \longrightarrow &\mathcal{Q} \end{array}$$ with fibres given by the integral lines, [and general fibre $\mathbb{R}$]{.roman}. Let $\mathcal{E}_M(w)$ and $\mathcal{E}_\mathcal{Q}(w)$ be projective densities of $M$ and $\mathcal{Q}$ respectively. **Proposition 10**. *There is a canonical isomorphism $$\label{Isomorphism between densities} \pi^*\mathcal{E}_{\mathcal{Q}}(w)\simeq \mathcal{E}_M(w).$$* *Proof.* By definition of projective densities (see Section [2.1](#Projective Tractor Calculus){reference-type="ref" reference="Projective Tractor Calculus"}), the bundle $\mathcal{E}_{M}(w)$ is canonically isomorphic to $\big(\textnormal{Vol}(M)\big)^{-\frac{w}{n+2}}$ where $\textnormal{Vol}(M) = \big(\Lambda^{n+1}T^*M\otimes\Lambda^{n+1} T^*M\big)^{\frac{1}{2}}$ is the bundle of volume densities of $M$. Therefore, there always exists a canonical nowhere vanishing section $\varepsilon^2_{a_1\dots a_{n+1}, b_1 \dots b_{n+1}}$ of $[(\Lambda^{n+1}T^*M)\otimes(\Lambda^{n+1} T^*M)](2(n+2))$ realising the canonical isomorphism $$\varepsilon^2_{a_1\dots a_{n+1}, b_1 \dots b_{n+1}} : \begin{array}{ccc} \mathcal{E}_M(-2(n+2)) &\simeq& \Lambda^{n+1}T^*M \otimes \Lambda^{n+1}T^*M \end{array}$$ We will note $TM/n$ the fibre bundle over $M=\widetilde{M}\setminus \mathcal{Z}(n)$ obtained by taking the point-wise quotient of $TM$ by the span of $s n^a$ where $s$ is any "true scale", i.e. nowhere vanishing density of weight $1$; the quotient is independent of the choice of $s$. By construction, the annihilator $\mathcal{N}^*$ of $n^a$ in $T^*M$ is identified with $(TM/n)^*$ as well as with the pull-back bundle $\pi^*(T^*Q)$. It follows that: $$\label{proof: nu volume definition} \nu_{a_1\dots a_n, b_1\dots b_{n}}:= \varepsilon^2_{aa_1\dots a_{n}, bb_1 \dots b_{n}}n^an^b,$$ is identified with a nowhere vanishing section of $\pi^*\big(\Lambda^n T^*Q \otimes \Lambda^n T^*Q \big)\otimes \mathcal{E}_M\big(2(n+1)\big)$ and thus realises an isomorphism $$\label{Proof: intermediate density isomorphism} \mathcal{E}_M\big(-2(n +1)\big) \simeq \pi^*\big(\Lambda^n T^*Q \otimes \Lambda^n T^*Q \big) \simeq \pi^*\mathcal{E}_{\mathcal{Q}}(-2(n+1)).$$ Taking roots yields the desired isomorphism for arbitrary weights $w$. ◻ This identification enables us to construct a natural differential operator on projective densities of $M$. Indeed, the action of the Lie derivative on $\pi^*\big(\Lambda^n T^*Q \otimes \Lambda^n T^*Q \big)$ along $n^a$ induces a differential operator $D_n$ on $\mathcal{E}_{M}(w)$ by closing the following commutative diagram: **Proposition 11**. *The maps appearing in the commutative diagram in Figure [\[Commutative diagram for Lie derivatives\]](#Commutative diagram for Lie derivatives){reference-type="ref" reference="Commutative diagram for Lie derivatives"} are well defined linear differential operator of order $1$. In a given projective scale, we have $$\label{Vertical derivative, definition} \begin{array}{cccc} D_n : &\mathcal{E}_{M}(w) &\longrightarrow & \mathcal{E}_{M}(w-1)\\ &\lambda &\longmapsto & (\nabla_{n} +w\rho)\lambda \end{array}$$ where $\rho := -\tfrac{1}{n+1} \nabla_a n^a=Y_AI^A.$* *Proof.* Let $s$ be an arbitrary non-vanishing density of weight $1$ on $M$, and $X^a=sn^a$. Since $\pi^*(\Lambda^nT^*Q)$ can be identified with $n$-forms on $M$ that vanish whenever at least one of its arguments is a vertical vector, we can apply the Lie derivative in the usual sense. Let $\nabla$ be the scale determined by $s$ in the projective class then for any $\alpha \in \pi^*(\Lambda^nT^*Q)$: $$\mathcal{L}_X\alpha= X \lrcorner \textnormal{d}\alpha = \nabla_X \alpha - n s\rho \alpha= s(n^a\nabla_a\alpha -n\rho \alpha),$$ where we recall $\nabla_b n^a = - \rho \delta_b^a$. Since this is proportional to the scale $s$, we see that this yields a well-defined linear operator $\mathcal{L}_{n} : \pi^*(\Lambda^n T^*Q) \to \pi^*(\Lambda^n T^*Q)(-1)$. According to the commutative diagram in Figure [\[Commutative diagram for Lie derivatives\]](#Commutative diagram for Lie derivatives){reference-type="ref" reference="Commutative diagram for Lie derivatives"}, to define $D_n$ on densities of weight $-2(n+1)$, we must set: $$\begin{aligned} D_n \lambda = \nu^{-1}\mathcal{L}_n (\nu \lambda)&= \nu^{-1} \left(\nabla_n (\nu \lambda) - 2n\rho \nu\lambda\right) \\ &=\nu^{-1} \left(\nu \nabla_n \lambda -2\rho \nu \lambda - 2n\rho \nu \lambda \right)\\&= \nabla_n \lambda - 2(n+1)\rho \lambda. \end{aligned}$$ where $\nu$ is a short-hand notation for the section [\[proof: nu volume definition\]](#proof: nu volume definition){reference-type="eqref" reference="proof: nu volume definition"}. Eq. [\[Vertical derivative, definition\]](#Vertical derivative, definition){reference-type="eqref" reference="Vertical derivative, definition"} follows for densities of arbitrary weight $w$. ◻ Proposition [Proposition 10](#Proposition: Isomorphism between densities){reference-type="ref" reference="Proposition: Isomorphism between densities"} allows us to identify projective densities $\mathcal{E}_{M}(w)$ on $M$ with pullbacks of densities on $\mathcal{Q}$. Since sections $\lambda$ of $\pi^*\mathcal{E}_{\mathcal{Q}}(w)$ which are the pullback $\lambda = \pi^* \bar{\lambda}$ of sections of $\mathcal{E}_{\mathcal{Q}}(w)$ are precisely those with vanishing Lie derivative in the direction $n^a$ we obtain, from the commutative diagram [\[Commutative diagram for Lie derivatives\]](#Commutative diagram for Lie derivatives){reference-type="ref" reference="Commutative diagram for Lie derivatives"}, the following characterisation of densities on $Q$: **Proposition 12**. *Densities $\tilde{\lambda}$ of weight $w$ on $\mathcal{Q}$ are canonically identified with densities (of weight $w$) $\lambda$ on $M$ satisfying $D_n\lambda=0$: $$\begin{aligned} \label{Equivalence of densities on Q and M} D_n \lambda &=0,\quad\lambda \in \Gamma(\mathcal{E}_{M}(w)) & \Leftrightarrow& & \lambda &= \pi^* \tilde\lambda, \qquad \tilde\lambda \in \Gamma(\mathcal{E}_{\mathcal{Q}}(w)).\end{aligned}$$* In the righthand side of the above equivalence, and throughout the following sections of this article, we abuse notation and use the isomorphism [\[Isomorphism between densities\]](#Isomorphism between densities){reference-type="eqref" reference="Isomorphism between densities"} to identify sections of $\mathcal{E}_M(w)$ and $\pi_*\left(\mathcal{E}_{\mathcal{Q}}(w)\right)$, i.e. we will think of $\pi^*\lambda$ as a section of $\mathcal{E}_M(w)$. *Remark 5*. This shows that when $\mathcal{Q}$ can be equipped with a smooth manifold structure, there is a nowhere vanishing density $s$ of weight $1$ on $M$ such that $\rho=Y_AI^A=0$ in the scale $\nabla^s$. Indeed, let $\tilde{s}$ be a nowhere vanishing scale on $\mathcal{Q}$; we have shown that this corresponds to a nowhere vanishing density $s = \pi^* \tilde{s}$ such that $I^AD_As =0$, writing this last equation in the scale determined by $s$ shows that $s\rho =0 \Rightarrow \rho=0.$ The above remark motivates the following definition: **Definition 2**. *A nowhere vanishing $1$-density on $M$ that satisfies $D_n \nu =0$ will be called an **adapted scale** on $M$.* These scales are useful as they guarantee the most compatibility between the structures on $M$ and $\mathcal{Q}$; we record below of their some fundamental properties. **Lemma 7**. *Let $s$ be an adapted scale on $M$, and $\nabla$ the scale it determines in the projective class. Then: $$\rho = Y_A I^A =0, \quad P_{ab}n^b = 0,\quad \nabla_a n^b =0.$$* Using their characterisation and Eq. [\[CovariantDerivativesDirectionn\]](#CovariantDerivativesDirectionn){reference-type="eqref" reference="CovariantDerivativesDirectionn"}, we can immediately identify some important adapted scales on $\mathcal{O}$ and $\Sigma\setminus \{\lambda= 0\}$: **Corollary 3**. - *The density $\sigma := H^{AB}X_A I_B$ on $M$ induces a weight 1 density $\bar{\sigma}$ on $$\mathcal{Q}= \mathcal{Q}_\mathcal{O} \cup \underbrace{\mathcal{H}^- \cup \mathcal{H}^0 \cup \mathcal{H}^+}_{\mathcal{H}}$$ such that $\bar{\sigma}(x) =0 \;\Leftrightarrow\; x \in \mathcal{H}=\Sigma/\mathbb{R}$; in particular it is a boundary defining function for $\mathcal{H}$.* - *The density $\lambda := H^{AB}X_A X_B$ on $M$ induces a weight 2 density $\bar{\lambda}$, on $$\mathcal{H} = \mathcal{H}^- \cup \mathcal{H}^0 \cup \mathcal{H}^+$$ such that $\bar{\lambda}(x) =0 \;\Leftrightarrow\; x \in \mathcal{H}^0$.* ### Tractors of $\mathcal{Q}$ {#sssection: Tractors of Q} In the previous subsection we established an equivalence [\[Equivalence of densities on Q and M\]](#Equivalence of densities on Q and M){reference-type="eqref" reference="Equivalence of densities on Q and M"} between densities on $\mathcal{Q}$ and densities on $M$ which are in the kernel of [\[Vertical derivative, definition\]](#Vertical derivative, definition){reference-type="eqref" reference="Vertical derivative, definition"}. In this section we will extend this equivalence to the corresponding tractor bundles $\mathcal{T}_{M} = \left(J^1 \mathcal{E}_{M}(1)\right)^* \rightarrow M$ and $\mathcal{T}_{\mathcal{Q}} = \left(J^1 \mathcal{E}_{\mathcal{Q}}(1)\right)^* \rightarrow \mathcal{Q}$. **Proposition 13**. *The isomorphism of Proposition [Proposition 10](#Proposition: Isomorphism between densities){reference-type="ref" reference="Proposition: Isomorphism between densities"} extends at the level of 1-jet and yields the canonical isomorphisms $$\begin{aligned} \pi^*(\mathcal{T}_{\mathcal{Q}})^* &\simeq I^\circ \subset \mathcal{T}^*_{M}, & \pi^*\mathcal{T}_{\mathcal{Q}} &\simeq \mathcal{T}_M /I. \end{aligned}$$* *Proof.* As we have seen, any section $\lambda$ of $\mathcal{E}_M(1)$ that satisfies $D_n\lambda=0$, arises as $\pi^*\bar{\lambda}$ for some section $\bar{\lambda}$ of $\mathcal{E}_{\mathcal{Q}}(1)$. Now $0=D_n\lambda= I^AD_A \lambda$ so that at each point $p\in M$, $D_A \lambda$ is in $I^\circ_p$. Consider the map $D_A \lambda \mapsto \pi^* D_A \bar{\lambda}$. By construction, $D_A \bar{\lambda}_{\pi(p)}$ only depends on the $1$-jet of $\bar{\lambda}$ at $\pi(p)$, and similarly for $D_A \lambda$ at $p$, therefore this defines a vector bundle morphism $I^\circ \to \pi^*\mathcal{T}_{\mathcal{Q}}^*$, which is injective in the fibres. Since they have the same dimension this is a vector bundle isomorphism. The other isomorphism is obtained by duality. ◻ Working in adapted scales (see Definition [Definition 2](#AdaptedScales){reference-type="ref" reference="AdaptedScales"}), we can determine an analogue of Proposition [Proposition 12](#Proposition: equivalence of densities){reference-type="ref" reference="Proposition: equivalence of densities"} that tells us when a section of $I^\circ \subset \mathcal{T}_M^*$ arises as a pullback of a section of $\mathcal{T}_{\mathcal{Q}}^*$. First, let us point out that the splitting sections $Y$ are $W$ determined by a connection $\nabla$ in a projective class in fact only depend on the connection $\nabla$ induces on the density bundle $\mathcal{E}(1)$. Therefore, one can define $Y$ and $W$ even *in the absence of a projective structure* by simply choosing a connection on $\mathcal{E}(1)$. In particular, for every nowhere vanishing density $\nu$, there is always a connection on $\mathcal{E}(1)$ for which it is parallel. **Corollary 4**. *A co-tractor field $V_A$ on $M$ can be written as the pullback $V_A =\pi^*\bar{V}_A$ of a tractor field $\bar{V}_A$ on $Q$ if and only if: $$\begin{aligned} n^a\nabla_a V_A &=0 &&\textrm{ and } &I^AV_A&=0.\end{aligned}$$ Let $\nu = \pi^*\bar{\nu}$ be an adapted scale on $M$, by the preceding remark, $\nu$ and $\bar{\nu}$ determine respectively connections on $\mathcal{E}_\mathcal{Q}(1)$, $\mathcal{E}_{M}(1)$ and splitting operators $\overline{Y},\overline{W}$ and $Y,W$ on[^9] $(\mathcal{T}_\mathcal{Q})^*$ and $(\mathcal{T}_{M})^*$; this is summarised by Figure [\[Figure: Compatible splittings on Q/M\]](#Figure: Compatible splittings on Q/M){reference-type="ref" reference="Figure: Compatible splittings on Q/M"}. Using these connections to simultaneously split $\mathcal{T}_{M}$ and $\mathcal{T}_{\mathcal{Q}}$, the sections $$\begin{aligned} V_A&=\xi Y_A + \mu_c Z_A^c, & \bar{V}_A &= \bar{\xi} \bar{Y}_A + \bar{\mu}_c\bar{Z}_A^c,\end{aligned}$$ of $(\mathcal{T}_M)^*$ and $(\mathcal{T}_{\mathcal{Q}})^*$ are identified via the relations $\pi^*\bar{\xi} = \xi$ and $\pi^*\bar{\mu}=\mu$.* *Proof.* Writing out the conditions on $V_A$ in terms of its components in a generic scale we have: $$\label{proof: explicit equivalence of tractors} \begin{aligned} I^A V_A = 0 \Leftrightarrow n^a\mu_a + \xi \rho =0, \quad \nabla_{n} V_A = 0 \Leftrightarrow \begin{cases} \nabla_{n} \xi = n^a\mu_a, \\ \nabla_{n}\mu_a +n^aP_{ab}\xi =0.\end{cases} \end{aligned}$$ Working in an adapted scale and making use of Lemma [Lemma 7](#Lemma: PropertiesAdaptedScales){reference-type="ref" reference="Lemma: PropertiesAdaptedScales"}, the above equations become equivalent to requiring that each components $\xi$ and $\mu$ are the pullback of corresponding fields $\bar{\xi}$ and $\bar{\mu}$ on $\mathcal{Q}$. Now, by construction, the isomorphism of Proposition [Proposition 13](#Proposition: Isomorphism of tractors on Q/M){reference-type="ref" reference="Proposition: Isomorphism of tractors on Q/M"} maps $D_A\nu=\nu Y_A$ to $D_A \bar{\nu}= \bar{\nu} \bar{Y}_A$. It follows that $\xi Y_A \mapsto \bar{\xi} \bar{Y}_A$ and then by linearity: $V_A -\xi Y_A \mapsto \bar{V}_A - \bar{\xi}\bar{Y}_A$. ◻ ## The projective structure and the tractor connection of $\mathcal{Q}$ ### The induced projective structure on $\mathcal{Q}$ We have already established in Proposition [Proposition 8](#Connection invariant under flow O){reference-type="ref" reference="Connection invariant under flow O"} that the projective structure of $M$ induces a projective structure on $\mathcal{Q}_\mathcal{O}$; we now extend this to all of $\mathcal{Q}$. **Proposition 14**. *Each adapted scale $s$ determines a connection $\bar{\nabla}^s$ on $\mathcal{Q}$. Furthermore, all such connections belong to the same projective structure.* *Proof.* Let $s$ be an adapted scale and let us consider the connection $\nabla$ it determines in the projective class. As in the proof of Proposition [Proposition 8](#Connection invariant under flow O){reference-type="ref" reference="Connection invariant under flow O"}, we can evaluate the Lie derivative of the connection. It follows from Lemma [Lemma 7](#Lemma: PropertiesAdaptedScales){reference-type="ref" reference="Lemma: PropertiesAdaptedScales"} and Lemma [Lemma 6](#ContractionNandWeyl){reference-type="ref" reference="ContractionNandWeyl"} that: $$ (\mathcal{L}_{N^s}\nabla^s)^a_{bc} = (N^s)^a P_{bc},$$ Let $\beta$ be a $1$-form on $\mathcal{Q}$, and $\bar{Z}$ a vector field on $\mathcal{Q}$ to vector fields on $M$. Locally lift $\bar{Z}$ to a vector field $Z$ on $M$ with the property that $\nabla_n Z=0$ and set: $$\pi^*\bar{\nabla}^s_{\bar{Z}} \alpha = \nabla_{Z}(\pi^*\beta).$$ To check that the definition make sense we must prove that right-hand side is independent of the lift and can be written as the pullback of a form on $\mathcal{Q}$. First note that $\nabla_{N^s} (\pi^*\beta)=0$, hence, two such lifts differ by a vector that is proportional to $N^s$, the formula, if well-defined, is independent of the choice of lift. Second: $$\mathcal{L}_{N^s} \nabla_{Z}(\pi^*\beta)=\nabla_{[N^s,Z]}(\pi^*\beta) + \pi^*\beta\left(\mathcal{L}_{N^s}\nabla^s(Z, \cdot)\right) +\nabla_{Z} \mathcal{L}_{N^s}\pi^*\beta=0,  $$ whilst $(\nabla_{\bar{Z}}\pi^*\beta)(N^s)=0$, showing that the connection is well-defined. If $s_1$ is another adapted scale then $$\nabla^{s_1} = \nabla^{s} - s_1^{-1}\nabla s_1= \nabla^{s}-f^{-1}\nabla f,$$ where $f=s_1s^{-1}$. As $D_n s_1= D_n s= 0$ it follows that $f=\pi^* \bar{f}$ and $\frac{df}{f}= \pi^*\frac{d\bar{f}}{\bar{f}}$, so that: $$\begin{aligned} \nabla_{Z}^{s_1}\pi^*\beta - \nabla_{Z}^{s}\pi^*\beta&= f^{-1}\textnormal{d}f(\pi^*\beta)(Z) + f^{-1}\textnormal{d}f(Z)\pi^*\beta \\&= (\pi^*\bar{f})^{-1}\textnormal{d}(\pi^*\bar{f})\pi^*(\beta(\bar{Z}))+ (\pi^*\bar{f})^{-1}\textnormal{d}(\pi^*\bar{f})(Z) \pi^*\beta \\&=\pi^*\left(\bar{f}^{-1}\textnormal{d}\bar{f}\beta(\bar{Z})\right)+ (\pi^*\bar{f})^{-1}\pi^*(\textnormal{d}\bar{f})(Z) \pi^*\beta \\&= \pi^*\left(\bar{f}^{-1}\textnormal{d}\bar{f}\beta(\bar{Z})+ \bar{f}^{-1} \bar{f}(\bar{Z})\beta\right). \end{aligned}$$ Therefore all connections constructed in this manner are projectively related. ◻ The following is an immediate consequence of the definition the connection: **Proposition 15**. *If $s=\pi^*\bar{s}$ is an adapted scale then the curvature tensors $R$ and $\bar{R}$ of $\nabla^s$ and $\bar{\nabla}^{s}$ satisfy: $$\pi^*(\alpha(\bar{R}(\bar{X},\bar{Y})(-))=(\pi^*\alpha)(R(X,Y)(-)),$$ for every $1$-form $\alpha$ and vector fields $\bar{X},\bar{Y}$, on $\mathcal{Q}$ and $X,Y$ are arbitrary lifts of $\bar{X},\bar{Y}$ respectively that satisfy $\nabla_nX = \nabla_n Y=0$.* **Corollary 5**. *Let $s=\pi^*\bar{s}$ be an adapted scale then for any vector fields $\bar{Y},\bar{Z}$ on $\mathcal{Q}$ we have: $$\pi^*(\bar{P}(\bar{Y},\bar{Z}))= P(Y,Z), $$ where $P$ and $\bar{P}$ are the Schouten tensors of $\nabla^s$ and $\bar{\nabla}^{s}$ respectively and $Y$ and $Z$ are lifts of $\bar{Y}, \bar{Z}$ such that $\nabla_n Y=\nabla_n Z$.* *Proof.* It follows immediately from the above that: $$ \pi^*(\bar{R}(\bar{Y},\bar{Z}))=\underbrace{R(Y,Z)}_{\substack{\rotatebox{90}{$\,=$}\\(n-1)P(Y,Z)}} - P(Y,Z)=(n-2)P(Y,Z).$$ Since the Schouten tensor $P$ is symmetric ($\nabla^s$ is special) it follows that the Ricci tensor $\bar{R}$ is symmetric (therefore we recover that $\bar{\nabla}^s$ is special), and so $\bar{R}(\bar{Y},\bar{Z})=(n-2)\bar{P}(\bar{Y},\bar{Z})$ ◻ **Corollary 6**. *For any adapted scale $s=\pi^*\bar{s}$: $\bar{\nabla}^s\bar{s}=0$.* *Proof.* Let $\nu = s^{-(2n+2)}=\pi^*\bar{\nu}$. By definition, we have $\bar{s}^{-(2n+2)}=\bar{\nu}$. These densities are identified with sections of $(\Lambda^nT^*\mathcal{Q})^{\otimes 2}$, we can apply the definition of $\bar{\nabla}$ and the Leibniz rule to see that: $$\pi^*\bar{\nabla}_{\bar{Y}}\bar{\nu}=\nabla_{Y} \pi^*\bar{\nu} = \nabla_{Y}\nu=0. $$ ◻ ### The induced tractor connection on $\mathcal{Q}$ The aim of this section is to show that the projective structures fit together nicely. **Proposition 16**. *The tractor connection on $\mathcal{T}^*_M$ induces a connection on $I^\circ$ and a connection $\bar{\nabla}$ on the bundle $\mathcal{T}^*_{\mathcal{Q}}$.* *Proof.* Since $I^A$ is parallel if $I^AV_A=0$ then $I^A\nabla_a V_A=0$, so $\nabla$ restricts to a connection on $I^\circ$. To define a connection on $\mathcal{T}^*_{\mathcal{Q}}$, let $\bar{Z}$ be a vector field on $Q$ and let $Z$ be any local lift of $\bar{Z}$ to $M$ that satisfies $\nabla_n Z=0$. Let $\phi: \pi^* \mathcal{T}_{\mathcal{Q}} \rightarrow I^\circ$ denote the bundle morphism established in the previous section and set: $$\pi^*\bar{\nabla}_{\bar{Z}} V= \phi^{-1}\nabla_{Z}\phi(\pi^* V).  $$ This is independent on the choice of lift $Z$ as any other lift $Z'$ is such that $(Z-Z')\propto n$ and $\nabla_n \phi(\pi^* V)=0$. This second identity can be seen using an adapted scale in which the components $\xi$ and $\mu_c$ of $\phi(\pi^*V)$ must satisfy $\nabla_n \xi =0$, $\nabla_n \mu_c=0$ and $\mu_cn^c=0$. Since in adapted scales $P_{ab}n^a=0$, this means precisely that the tractor is covariantly constant in the direction of $n$. It remains to check that $\nabla_n \nabla_Z \phi(\pi^*V) =0$ so that the right-hand side is indeed a pullback section. For any choice of adapted scale $\nu$, setting $N=\nu n$, we have the identity for cotractors $V_A$: $$\begin{aligned} \nabla_N (\nabla_Z V_A) - \nabla_Z (\nabla_N V_A) &= [Z,N]^a \nabla_a V_A -N^a Z^b\Omega_{ab}{}^B{}_A V_B \end{aligned}$$ Using this with $V=\phi(\pi^*V)$ and our assumption on the lift: $$\begin{aligned} \nabla_N (\nabla_Z V_A) &= -N^a Z^b\Omega_{ab}{}^B{}_A V_B. \end{aligned}$$ The vanishing of this last term, concluding the proof, will be given by the following Lemma: **Lemma 8**. *$$n^a \Omega_{ab}{}^A{}_B =0.$$* *Proof.* The tractor curvature is a projectively invariant tensor which, in any given scale, is, by Eq. [\[TractorCurvature\]](#TractorCurvature){reference-type="eqref" reference="TractorCurvature"}, completely parametrised by the (projective) Weyl and (projective) Cotton tensors. Evaluating their contraction with $n^a$ in the scale $\sigma$ one sees from [\[Riemann in the scale sigma\]](#Riemann in the scale sigma){reference-type="eqref" reference="Riemann in the scale sigma"}, [\[Levi-civita in the scale sigma\]](#Levi-civita in the scale sigma){reference-type="eqref" reference="Levi-civita in the scale sigma"} that their contraction with $n^a$ always vanishes. This extends to the points where $\sigma=0$ by continuity. ◻ It follows that $\nabla_N(\nabla_Z V_A)$ vanishes and $\bar{\nabla}$ is well-defined. ◻ **Proposition 17**. 1. *$H^{AB}$ induces a parallel section $\bar{H}^{AB}$ on $\mathcal{Q}$. It is a symmetric pairing of rank $n-1$ on $\mathcal{T}_{\mathcal{Q}}^*$.* 2. *$I_A=D_A\sigma$ induces a parallel section $\bar{I}_A$ of $\mathcal{T}_{\mathcal{Q}}^*$ and $\bar{I}_A$ is in the radical of $\bar{H}^{AB}$.* 3. *$\bar{I}_A=D_A\bar{\sigma}$.* *Proof.* 1. Let $\tilde{H}^{AB}$ denote the induced metric in $I^\circ$, let $\bar{V}_A,\bar{W}_B$ be sections of $\mathcal{T}_{\mathcal{Q}}^*$ that we can pullback an identify with sections $V_A,W_B$ of $I^\circ$ that satisfy, $\nabla_n V_A=\nabla_n W_B= 0.$ Since $H^{AB}$ is parallel it follows that $H^{AB}V_AW_B$ is constant along the fibres of $M \rightarrow \mathcal{Q}$; this can this be used as the definition of $\bar{H}^{AB}$. To show that $\bar{H}^{AB}$ is parallel observe that we have the scalar identity: $$\bar{\nabla}_a(\bar{H}^{AB}\bar{V}_A\bar{W}_B)=\bar{\nabla}_a(\bar{H}^{AB}\bar{V}_A\bar{W}_B) - \bar{H}^{AB}\bar{\nabla}_a\bar{V}_A\bar{W}_B - \bar{H}^{AB}\bar{V}_A\bar{\nabla}_a\bar{W}_B,$$ which, by definition of $\bar{\nabla}$ and $\bar{H}^{AB}$, must pull back to the identity: $$\nabla_a(H^{AB}V_AW_B)=\nabla_aH^{AB}V_AW_B - H^{AB}\nabla_aV_AW_B - H^{AB}V_A\nabla_aW_B=0. $$ In the above we have defined[^10]: $V_A=\pi^*\bar{V}_A$ and $W_A=\pi^*\bar{W}_B$. 2. This follows from the fact that $\nabla_c D_A \sigma =0$ and $I^AD_A \sigma = \Phi_{AB}I^AI^B=0$. It is immediate from the definition of $\bar{\nabla}$ that $\bar{I}_A$ is parallel for $\bar{\nabla}$.  ◻ **Proposition 18**. *The connection $\bar{\nabla}$ on $\mathcal{T}^*_{\mathcal{Q}}$ is the Cartan normal connection associated to the projective class of $[\bar{\nabla}^s]$ where $s$ is any adapted scale.* *Proof.* Choose a fixed adapted scale $s$ and introduce the splittings of $\mathcal{T}_M^*$ and $\mathcal{T}_Q^*$ as in Corollary [Corollary 4](#Proposition: explicit tractor isomorphism){reference-type="ref" reference="Proposition: explicit tractor isomorphism"}. Let $\bar{V}_A= \bar{\xi}\bar{Y}_A + \bar{\mu}_c\bar{Z}^c_A$ be a section of $\mathcal{T}_Q^*$, we have from the aforementioned Corollary that: $$V_A=\pi^*\bar{V}_A= \xi Y_A + \mu_c Z^c_A$$ where $\xi = \pi^*\xi$ and $\mu=\pi^*\mu$. Let $\bar{Z}$ be a vector field on $\mathcal{Q}$ and $Z$ a lift to $M$ such that $\nabla_n Z=0$, then: $$\nabla_Z V_A= (\nabla_Z\xi -\mu(Z) )Y_A + (\nabla_Z \mu_c +P(Z,-)_c\xi)Z^c_C.$$ By definition of the connection $\bar{\nabla}$, the Leibniz rule and Corollary [Corollary 5](#SchoutenTensorMagic){reference-type="ref" reference="SchoutenTensorMagic"} we have: $$\begin{cases} \nabla_Z \xi - \mu(Z)= \pi^*\left(\bar{\nabla}_{\bar{Z}}\bar{\xi}-\bar{\mu}(\bar{Z})\right)\\(\nabla_Z \mu +P(Z,-)\xi)= \pi^*\left(\nabla_{\bar{Z}}\bar{\mu} + \bar{P}(\bar{Z},-)\xi \right). \end{cases}$$ Hence, using again Corollary [Corollary 4](#Proposition: explicit tractor isomorphism){reference-type="ref" reference="Proposition: explicit tractor isomorphism"}, it follows that: $$\bar{\nabla}_c\bar{V}_A=(\bar{\nabla}_c \bar{\xi} - \bar{\mu}_c)\bar{Y}_A + (\bar{\nabla}_c\bar{\mu}_a+\bar{P}_{ca})Z^c_A.$$ ◻ ### Projective compactification of order 1 It now follows from Proposition [Proposition 17](#Proposition: TractorGeometryofQ){reference-type="ref" reference="Proposition: TractorGeometryofQ"} and Proposition [Proposition 18](#Proposition: normal tractor connection on Q){reference-type="ref" reference="Proposition: normal tractor connection on Q"} that $\mathcal{Q}$ is equipped with a normal solution to the metrisability equation of rank $n-1$ whose geometry is therefore described by [@Flood:2018aa Theorem 3.14]: **Theorem 4**. *Let $g^{ab}=\bar{\sigma}^{-2}\bar{H}^{AB}\bar{Z}_A^b\bar{Z}_B^b$ on $\mathcal{Q}\cap \{\bar{\sigma}\neq 0\}$. Then the restriction of $g_{ab}$ to $\mathcal{Q}\cap\{\pm \overline{\sigma} >0\}$ is projectively compact (Ricci-flat) Einstein metric of order $1$ and $\mathcal{Q}\cap\{\pm \overline{\sigma} \geq0\}$ is a projective compactification with projective boundary $\mathcal{H}=\mathcal{Q}\cap\{\bar{\sigma}=0\}$.* ## An additional induced tractor bundle on $\mathcal{Q}$ The isomorphisms in Proposition [Proposition 13](#Proposition: Isomorphism of tractors on Q/M){reference-type="ref" reference="Proposition: Isomorphism of tractors on Q/M"} are convenient for the classical description of the projective geometry on $\mathcal{Q}$. However, the infinitesimal geometric structure on $\mathcal{Q}$ is *richer* than that of the usual projective tractor geometry. We will first illustrate this by constructing from the ambient projective tractor bundle $\mathcal{T}_M \to M$ a vector bundle $\tilde{\mathcal{T}} \to \mathcal{Q}$ on $\mathcal{Q}$, whose fibres of dimension $n+2$ will be able to carry additional information compared to $\mathcal{T}_{\mathcal{Q}} \to \mathcal{Q}$ (which is a $n+1$ dimensional vector bundle). This might at first seem incidental, but it turns out to be important in elucidating the geometry at the projective boundary $\mathcal{H} \subset \mathcal{Q}$. The bundle $\widetilde{\mathcal{T}}$ inherits a connection and parallel sections $\tilde{I}^A$, $\tilde{H}^{AB}$, mimicking the original setup on $M$ seen here, however, as a structure on $\mathcal{Q}$. A natural question one might ask is: does this vector bundle correspond to the tractor bundle of a Cartan geometry on $\mathcal{Q}$ and what, if any, additional geometric structures may it induce? The answer we propose is that this is in fact an alternative form of *projective* geometry on $\mathcal{Q}$, however, based on a *non-effective* homogeneous model. Referring to the model of our holonomy reduction discussed in the Introduction and Section [2.2](#model){reference-type="ref" reference="model"}, one might observe there that the base $\mathbb{R}P^n$ arises equivalently as either the set of planes containing the line generated by $I$ or, the set of projective lines passing through the fixed point $[I]$. It can then be described as the Klein geometry $\tilde{G}/\tilde{P}$ where $\tilde{G}$ is the isotropy subgroup of $[I]$ in $PGL(n+1)$ and $\tilde{P}$ the stabiliser of some given plane containing $\mathbb{R}I$. It is clear that the action has a relatively large kernel conjugate to: $$\tilde{K}=\left\{ \begin{pmatrix} c & \upsilon \\ 0 & \lambda I_{n+1} \end{pmatrix} \!\!\!\mod Z(GL(n+2))\right\}.$$ Given any Cartan geometry[^11] $(C \to \mathcal{Q}, \omega)$ with this model, it is natural to construct a bundle $\mathscr{T}$ associated to the restriction to $\tilde{P}$ of the $(n+2)$-dimensional representation $u \mapsto \lvert\det u\rvert^{-\frac{1}{n+2}}u$ of $PGL(n+1,\mathbb{R})$. There is also a canonical line bundle $\mathscr{L} \rightarrow \mathcal{Q}$ associated to the 1-dimensional representation of $\tilde{P}$ given by: $$\begin{pmatrix} c& \upsilon_\nu & \upsilon_0 \\ 0 & A^\mu_\nu & 0 \\ 0 & \Upsilon_\mu & a \end{pmatrix} \!\!\!\mod Z(GL(n+2)) \mapsto \left\lvert\frac{c}{a}\right\rvert^\frac{1}{n+2}. $$ As with projective densities, let $\mathscr{L}(w)$ denote the $w$-th power of $\mathscr{L}$. A generic $\tilde{\mathfrak{g}}$-valued 1-form on $\mathcal{Q}$ can be parametrised as follows: $$\label{GeneralParametrisationConnection} \begin{pmatrix}-\frac{1}{n+2}\alpha^\sigma_\sigma + \frac{n+1}{n+2}\kappa & \upsilon_\nu& \upsilon_0\\ 0 & \alpha^\mu_\nu - \frac{1}{n+2}\alpha^\sigma_\sigma \delta^\mu_\nu -\frac{1}{n+2}\kappa\delta^\mu_\nu & \omega^\mu \\ 0 & P_\nu & -\frac{1}{n+2}\alpha^\sigma_\sigma -\frac{1}{n+2}\kappa \end{pmatrix}. $$ The 1-form $\kappa$ parametrises a linear connection on $\mathscr{L}$ with local connection form $\frac{1}{n+2}\kappa$. Modulo the kernel $\mathfrak{k}$, this leads to the usual parametrisation of a projective connection and a natural condition on $\omega$ is to require that $\tilde{\omega}=\omega \!\!\!\mod \mathfrak{k}$ is the normal Cartan connection of the corresponding projective class. We make this assumption in the rest of the paragraph, thus $P_\nu$ is completely determined in terms of the Projective Schouten tensor. Let us now make a few further general observations about these Cartan geometries. First, we see that there is always a distinguished direction in $\mathscr{T}$; this should be assimilated to the origin $[I]$ in the definition of the non-effective homogenous model. Secondly, we have a natural exact sequence realising $\mathscr{T}$ as an extension of a weighted version of the usual tractor bundle $\mathcal{T}_Q$: Now, if we assume that there is a distinguished *parallel* section $\tilde{I}$ of $\mathscr{T}$ in the privileged direction there are local gauges in which $\kappa= \frac{1}{n+1}\alpha^\mu_\mu$ and it is apparent that we should assimilate $\mathscr{L} \approx \mathcal{E}(\frac{1}{n+2})$. The short exact sequence then becomes: Under the above compatibility assumption we therefore see that parallel sections of $\mathscr{T}$ canonically project to parallel sections of $\mathcal{T}_\mathcal{Q}$; so a holonomy reduction of the Cartan geometry $(C,\omega)$ given by parallel sections of $\mathscr{T}$ leads to a holonomy reduction of the underlying normal projective geometry. For instance, if we assume that there is a non-degenerate parallel section $\tilde{H}^{\tilde{A}\tilde{B}}$ of signature $(2,n)$ such that $\tilde{\Phi}_{\tilde{A}\tilde{B}}\tilde{I}^{\tilde{A}}\tilde{I}^{\tilde{B}}=0$, where $\tilde{\Phi}$ is the inverse of $\tilde{H}$, then we get a parallel section: $H^{AB}=\Pi_{\tilde{A}}^A\Pi_{\tilde{B}}^B\tilde{H}^{\tilde{A}\tilde{B}}$. Additionally, the section: $\Pi^B_{\tilde{B}}\tilde{H}^{\tilde{A}\tilde{B}}$ induces an isomorphism $\mathcal{T}_\mathcal{Q}^* \simeq \tilde{I}^\perp$. Since $\tilde{I} \in \tilde{I}^\perp$, it induces a cotractor $I_A$ such that $H^{AB}I_A=0$, as expected. There are few unresolved issues in pushing the general discussion further. For instance, the general connection in Eq. [\[GeneralParametrisationConnection\]](#GeneralParametrisationConnection){reference-type="eqref" reference="GeneralParametrisationConnection"} has unspecified components $\upsilon_0,\upsilon_\nu$ that do not seem immediately linked to the underlying normal projective structure. However, one may expect that the existence of a parallel section $H^{\tilde{A}\tilde{B}}$ could exhaust these remaining degrees of freedom. Indeed, an argument in favour of this is that there will be an induced Ricci-flat Einstein metric $g$ on $\mathcal{Q}$ and, where $g$ is non-degenerate, we will be able to find local gauges of the form: $$\begin{pmatrix}0& -\omega^\mu\eta_{\mu\nu}& 0\\ 0 & \alpha^\mu_\nu & \omega^\mu \\ 0 & 0 &0\end{pmatrix},$$ where $\alpha$ is the Levi-Civita connection form expressed in an orthogonal basis and $\eta=\textrm{diag}(-1,1,\dots,1)$. Another related issue is that it is not clear how to split the exact sequence in Figure [\[Decomposition_Tscr\]](#Decomposition_Tscr){reference-type="ref" reference="Decomposition_Tscr"}. From the strict of point of view of the bundle $\mathscr{T}$, this is a choice of a point in the affine space $\mathfrak{a}=\{ T_{\tilde{A}} \in \mathscr{T}^* \, \lvert \, T_{\tilde{A}}\tilde{I}^{\tilde{A}}=1\}$, directed by sections of $\tilde{I}^{\circ} \simeq\mathcal{T}_Q^*$. However, it is unclear how one may interpret this in terms of a geometric structure on $\mathcal{Q}$. Supposing a choice can be made, if we denote $\rotatebox[origin=c]{180}{$\Pi$}$ the inverse induced splitting section, then we note that it follows directly from Eq. [\[GeneralParametrisationConnection\]](#GeneralParametrisationConnection){reference-type="eqref" reference="GeneralParametrisationConnection"} that the connection will act on $T^{\tilde{A}}= f \tilde{I}^{\tilde{A}}+ \rotatebox[origin=c]{180}{$\Pi$}^{\tilde{A}}_A t^A$ ($f\in C^{\infty}(\mathcal{Q}), t\in \Gamma(\mathcal{T}_{\mathcal{Q}})$) according to: $$\label{Desiredformofconnection} \nabla_c T^{\tilde{A}}= (\nabla_c f + \upsilon_{cB} t^B)\tilde{I}^{\tilde{A}} + \rotatebox[origin=c]{180}{$\Pi$}^{\tilde{A}}_A \nabla_c t^A, $$ where $\upsilon_{cA}$ is a co-tractor valued $1$-form determined by the connection and depending on the splitting. In this section we construct a vector bundle $\widetilde{\mathcal{T}}\rightarrow \mathcal{Q}$ that we conjecture carries the structure of $\mathscr{T}$. In the case we study, the distinguished section $\tilde{I}$ is parallel and we will show that $\widetilde{\mathcal{T}}$ has the same decomposition structure as in Figure [\[Decomposition_Tscr\]](#Decomposition_Tscr){reference-type="ref" reference="Decomposition_Tscr"}. The bundle $\widetilde{\mathcal{T}}\rightarrow \mathcal{Q}$ is constructed as follows: if $\nu$ is a density on $\widetilde{M}$ that restricts to a nowhere vanishing density on $M$, the flow of $N^a =\nu n^a$ defines a canonical action of $\mathbb{R}$ on the tractor bundle $\pi_{\mathcal{T}}: \mathcal{T}_{M} \rightarrow M$ that parallel transports tractors along the integral curve of $N$, $t\mapsto \phi^N_t(x)$. The quotient $\pi_{\mathcal{T}_{M}\to \tilde{\mathcal{T}}}: \mathcal{T}_{M} \longrightarrow \widetilde{\mathcal{T}}$ is a $(n+2)$-dimensional vector bundle $\widetilde{\mathcal{T}} \to \mathcal{Q}$, with projection defined by factorisation of the map $\pi_{\tilde{\mathcal{T}}}$ (see Figure [\[DefinitionTauTilde\]](#DefinitionTauTilde){reference-type="ref" reference="DefinitionTauTilde"}); this construction is independent of the choice of scale $\nu$. This leads to: **Definition 3**. *Define $\tilde{\mathcal{T}}$ to be the quotient space $\mathcal{T}/\mathbb{R}$ constructed as above. It naturally fits into the commutative square:* A point in the fibre $\pi_{\tilde{\mathcal{T}}}^{-1}(\{q\})$ can be thought of as a parallel vector field along the integral curve $q\in \mathcal{Q}$ of $N$ or, in other words, a covariantly constant section along $\pi^{-1}(\{q\})$. From Figure [\[DefinitionTauTilde\]](#DefinitionTauTilde){reference-type="ref" reference="DefinitionTauTilde"}, it can be seen that sections of $\mathcal{T}_M$ which are covariantly constant along the fibres of $M \to \mathcal{Q}$ map to sections of $\tilde{\mathcal{T}}$. Conversely, a section $\tilde{V}$ of $\tilde{\mathcal{T}}$ gives rise to such a section of $\mathcal{T}_M$. Indeed, we can define a map on $M=\coprod_{q\in \mathcal{Q}} \pi^{-1}(\{q\})$, that to each $x\in \pi^{-1}(\{q\})$ gives the value of the section $V(q)$ at $x$. Using the maps $q,t$ of a local trivialisation $\psi: \pi^{-1}(U_i) \rightarrow U_i\times \mathbb{R}$, locally this coincides with $V(x)=\bar{V}(q(x))(t(x))$ where for each $q\in U_i\subset \mathcal{Q}$, $\bar{V}(q)$ is the solution of the ordinary differential equation, $$\begin{cases} \nabla_{\dot{\gamma}} \bar{V}(t)=0, \\ \bar{V}(0)=\tilde{V}(q)(\psi^{-1}(q,0)), \\ \gamma(t)=\psi^{-1}(q,t)=\phi^{\tilde{N}}_t(x_0), \textrm{where $x_0=\phi^{-1}(q,0)$}. \end{cases}$$ Above, $\tilde{N}$ is the image of the vector field $\partial_t$ on $U_i \times \mathbb{R}_t$ under the diffeomorphism $\psi$; the section is smooth by smoothness with respect to initial conditions. We record this observation in the following lemma. **Lemma 9**. *Sections of $\widetilde{\mathcal{T}}$ are in one-to-one correspondence with sections of $\mathcal{T}$ that are covariantly constant along the fibres of $M\longrightarrow \mathcal{Q}$.* As claimed we see that the bundle comes equipped with a natural set of geometric data. **Proposition 19**. *$\widetilde{\mathcal{T}}$ inherits from $\mathcal{T}$ the follow data:* 1. *[\[connection_tilde\]]{#connection_tilde label="connection_tilde"} a natural linear connection defined by the relation: $$\widetilde{\nabla_{\bar{X}}T}_p= (\nabla_X \widetilde{T})_p,$$ where $p\in M$, $\bar{X}$ is a vector field on $Q$ near $q=\pi(p)$, $T\in \Gamma(\widetilde{\mathcal{T}})$, $X$ is lift of $\bar{X}$ near $p$ that satisfies $\nabla_n X=0$, and $~\widetilde{}$ denotes the correspondence of Lemma [Lemma 9](#correspondence_sections_T_tilde){reference-type="ref" reference="correspondence_sections_T_tilde"},* 2. *canonical sections $\tilde{H}^{AB}$ and $\tilde{I}^A$ of $\tilde{\mathcal{T}}\otimes\tilde{\mathcal{T}}$ and $\tilde{\mathcal{T}}$.* *Furthermore, the sections $\tilde{I}$ and $\tilde{H}$ are parallel for the connection defined in [\[connection_tilde\]](#connection_tilde){reference-type="ref" reference="connection_tilde"}.* *Proof.* 1. The construction is similar to that in the proof of Proposition [Proposition 16](#Proposition: InducedConnectionOnT*Q){reference-type="ref" reference="Proposition: InducedConnectionOnT*Q"}. 2. Since $\nabla_a H^{AB}=0$ and $\nabla_a I^A=0$ these both descend to corresponding sections $\tilde{H}^{AB}$ and $\tilde{I}^A$. That they are parallel follows immediately from the definition of the connection. ◻ To develop further the idea that this data should be thought of as an intrinsic Cartan geometry on $\mathcal{Q}$ modeled on $\tilde{G}/\tilde{P}$, we now confirm that we have the expected decomposition sequence of Figure [\[Decomposition_Tscr\]](#Decomposition_Tscr){reference-type="ref" reference="Decomposition_Tscr"} realising it as an extension of the standard projective tractor bundle. **Proposition 20**. *There are canonical isomorphisms: $$\begin{aligned} \tilde{I}^{\circ} &\simeq (\mathcal{T}_{\mathcal{Q}})^*, & \tilde{\mathcal{T}}/\tilde{I} &\simeq \mathcal{T}_{\mathcal{Q}}.\end{aligned}$$ In particular, we have the decomposition sequence:* *Proof.* We refer the reader to the proof of Proposition [Proposition 13](#Proposition: Isomorphism of tractors on Q/M){reference-type="ref" reference="Proposition: Isomorphism of tractors on Q/M"} as the arguments are similar. ◻ As in the discussion of the model, one can split the exact sequence with a choice of section of $\widetilde{\mathcal{T}}^*$ that does not annihilate the distinguished tractor $\tilde{I}$; in fact on $\mathcal{Q}_{\mathcal{O}}$, we have a canonical choice: **Lemma 10**. *The cotractor $L_C=\frac{1}{2}D_C(\sigma^{-1}\lambda)$ descends to a section $\tilde{L}_{\tilde{C}}$ of $\widetilde{\mathcal{T}}^*$ such that $\tilde{L}_{\tilde{C}}\tilde{I}^{\tilde{C}}=1$.* *Proof.* Using Lemma [Lemma 2](#Lemma: Relations ISigma, XLambda, etc){reference-type="ref" reference="Lemma: Relations ISigma, XLambda, etc"}: $$\begin{aligned} n^d\nabla_d L_C=\frac{1}{2}I^DD_DD_C(\sigma^{-1}\lambda) %&= \frac{1}{2}I^D D_D( -\sigma^{-2}\lambda D_C\sigma + \sigma^{-1}D_C\lambda) \\ &=\frac{1}{2}I^D\left(2\sigma^{-3}\lambda D_DD_C\sigma -2\sigma^{-2}D_{(D}\lambda D_{C)} \sigma +\sigma^{-1}D_DD_C\lambda \right) \\ &= - \sigma^{-2}\frac{1}{2}I^DD_D\lambda D_C\sigma + \sigma^{-1}I^D\Phi_{DC}\\ &= -\sigma^{-1}D_C\sigma + \sigma^{-1}D_C\sigma =0. \end{aligned}$$ The first claim then follows from Lemma [Lemma 9](#correspondence_sections_T_tilde){reference-type="ref" reference="correspondence_sections_T_tilde"} and a straightforward computation, using again Lemma [Lemma 2](#Lemma: Relations ISigma, XLambda, etc){reference-type="ref" reference="Lemma: Relations ISigma, XLambda, etc"}, shows the second. ◻ This provides us with the means to test the following statements on $\mathcal{Q}_\mathcal{O}$: **Corollary 7**. 1. *The bundle $\widetilde{\mathcal{T}}$ with its connection $\tilde{\nabla}$ and parallel tractor $\tilde{I}$, endows $\mathcal{Q}$ with a Cartan geometry modeled on the non-homogenous model of projective geometry $\tilde{G}/\tilde{P}$. [\[conj1\]]{#conj1 label="conj1"}* 2. *The linear connection on $\mathcal{T}_{\mathcal{Q}}^*$ induced by the isomorphism $\mathcal{T}_{\mathcal{Q}}^* \simeq \tilde{I}^{\circ}$ coincides with the normal projective connection $\bar{\nabla}$.* *Proof.* We only sketch the proof of [\[conj1\]](#conj1){reference-type="ref" reference="conj1"} on $\mathcal{Q}_{\mathcal{O}}$. Since $\tilde{I}$ is parallel it follows from its definition that the connection $\tilde{\nabla}$ will adopt the desired form given by Eq. [\[Desiredformofconnection\]](#Desiredformofconnection){reference-type="eqref" reference="Desiredformofconnection"}, where the connection on $\mathcal{\mathcal{T}}_Q$ is induced from $\tilde{\nabla}$ by the isomorphism of Proposition [Proposition 20](#Proposition: tractor isomorphism){reference-type="ref" reference="Proposition: tractor isomorphism"}. In particular, we may compute the connection cotractor 1-form $\upsilon$ from the $\tilde{\nabla}_b\tilde{L}_{\tilde{C}}$. Working in the scale determined by $\sigma$ on $\mathcal{O}$, it can be determined from: $\frac{1}{2}W^B_bD_BD_C(\sigma^{-1}\lambda)=\frac{1}{2}\sigma^{-1}W^B_b\nabla_a\nabla_b \lambda Z^a_A=\sigma g_{ab}Z^b_B$, in the notation of Section [4](#section: Geometry of the open orbit O){reference-type="ref" reference="section: Geometry of the open orbit O"}. Pushing down to $\mathcal{Q}_{\mathcal{O}}$ through the different identifications of this and the previous sections, we arrive at: $$\upsilon_{aB}= -\bar{\sigma} h_{ab}\bar{Z}^b_B,$$ where $h$ is the metric of Proposition [Proposition 7](#Proposition: invertible metric on Q_O){reference-type="ref" reference="Proposition: invertible metric on Q_O"}. ◻ In this situation, where the geometry arises as an inherited structure from an ambient space, there is an obvious interpretation of the additional degrees of freedom in the non-effective Cartan geometry as the remnants of the projective degree of freedom of the projective geometry on the ambient space; however it is still unclear how they should be interpreted in terms of $\mathcal{Q}$ alone. It should also be noted that Proposition [Proposition 20](#Proposition: tractor isomorphism){reference-type="ref" reference="Proposition: tractor isomorphism"} does not rely on the existence of the parallel tractor $\tilde{H}$ on $\widetilde{\mathcal{T}}$. Taking it into account adds a further interesting element to the geometric picture surrounding of $\widetilde{\mathcal{T}}$; relating it to conformal geometry. By Proposition [Proposition 7](#Proposition: invertible metric on Q_O){reference-type="ref" reference="Proposition: invertible metric on Q_O"}, $\mathcal{Q}_{\mathcal{O}} := \mathcal{Q}\cap \{\sigma \neq0 \}$ is equipped with a conformal class of metrics $[h^{ab}]$ and therefore defines a *conformal* tractor bundle. **Proposition 21**. *On $\mathcal{Q}_{\mathcal{O}}$ the bundle $\tilde{\mathcal{T}}$ is canonically isomorphic to the conformal tractor bundle. What is more the induced connection $\tilde{\nabla}$ coincides with the normal conformal tractor connection.* *Proof.* By Theorem [Theorem 3](#TheoFibreBundleInside BIS){reference-type="ref" reference="TheoFibreBundleInside BIS"}, the line bundle $\mathcal{O} \to \mathcal{Q}_{\mathcal{O}}$ has a preferred section $s : \mathcal{Q}_{\mathcal{O}} \to \mathcal{O}\cap\{\lambda=0\}$. This map thus canonically identifies $\tilde{\mathcal{T}}$ with the pull back of the projective tractor bundle $i^*\mathcal{T}_{\mathcal{O}}\to \mathcal{O}\cap\{\lambda=0\}$ as well as the corresponding induced connections. Now we recall that the submanifold $\mathcal{O}\cap\{\lambda=0\}$ can also be thought as part of the conformal boundary for a projective compactification of order $2$. What is more the corresponding conformal structure coincides with the conformal structure on $\mathcal{Q}_{\mathcal{O}}$ (this follows from their respective definitions). However, by the results from [@cap_projective_2016-1 Section 4], the pullback bundle $i^*\mathcal{T}_{\mathcal{O}}$ of projective tractors along the projective boundary is canonically identified with the conformal tractor bundle and the restriction of the projective tractor connection coincides with the normal conformal tractor connection. ◻ In other terms $\tilde{\mathcal{T}}$ (and its induced connection $\tilde{\nabla}$) are continuous extensions of the conformal tractor bundle (and the conformal tractor connection) from $\mathcal{Q}_{\mathcal{O}}$ to the whole of the projective compactification $\mathcal{Q}$. The points we develop in this section are the first steps towards the desirable picture in which this structure can be constructed from geometrical data on $\mathcal{Q}$ alone; they suggest an interesting picture and interplay between projective and conformal geometry that we hope to unravel in future work. # The geometry of $\Sigma \to \mathcal{H}$ {#section: The geometry of Sigma to H} ## The geometry of $\Sigma$ {#ssection: the geometry of Sigma} The hypersurface $\Sigma \subset M$ is the projective boundary of $\mathcal{O}$ given by a holonomy reduction from $\text{PGL}(n+1)$ to $\text{Aff}(n+1)$: this follows from $\nabla_c I_A=0$ and $I_{A} = D_A\sigma$ (cf Lemma [Lemma 2](#Lemma: Relations ISigma, XLambda, etc){reference-type="ref" reference="Lemma: Relations ISigma, XLambda, etc"}). We have the following short sequences, dual to that in [@Flood:2018aa Corollary 3.15], **Proposition 22**. *[\[Proposition: short sequences of tractors on Sigma\]]{#Proposition: short sequences of tractors on Sigma label="Proposition: short sequences of tractors on Sigma"}* *Proof.* Since $\sigma$ is boundary defining function for $\Sigma$ we have $J^1\mathcal{E}_{\Sigma}(1) = i^*(J^1\mathcal{E}_{M}(1))/ J^1\sigma$ or equivalently $(\mathcal{T}_{\Sigma})^* = i^*(\mathcal{T}_{M})^* / I_A$. Passing to the dual gives $\mathcal{T}_{\Sigma} = i^*\big(I_A\big)^{\circ} \subset i^*(\mathcal{T}_{M})$. ◻ Recall that we note $\Phi_{AB}$ the inverse of $H^{AB}$. Since we have $\nabla_c \Phi_{AB}=0$, $\nabla_c I^A=0$ and $I^2=0$ on $M$ it follows from the above isomorphisms that we have **Proposition 23**. *The tractor bundle $\mathcal{T}_{\Sigma}$ of $\Sigma\subset M$ inherits from $M$ a projective tractor connection $\nabla_c$, a covariantly constant section $I^A$ and a covariantly constant pairing $\Phi_{AB}$ with one dimensional kernel such that $\Phi_{AB} I^B=0$.* *Remark 6*. We point out that contrary to what we observed in Proposition [Proposition 18](#Proposition: normal tractor connection on Q){reference-type="ref" reference="Proposition: normal tractor connection on Q"} for the quotient $\mathcal{Q}$, the projective tractor connection $\nabla_c$ induced on the boundary, is generically *not* normal (as a projective connection). The geometry induced on $\Sigma$ is far richer than just a projective geometry. Let us here restrict to $\Sigma^{\pm} \subset M^{\pm}$ where $\lambda$ is nowhere vanishing and thus defines a preferred scale[^12]. In this case, the induced geometries are known as Carrollian geometries [@duval_carroll_2014]: **Theorem 5**. *The hypersurface $\Sigma^{\pm} \subset M$ is naturally equipped with* - *a vector field $N^a$* - *a degenerate metric $q_{ab}$ of rank $n-1$ satisfying $q_{ab}N^b=0$, $\mathcal{L}_{N}q_{ab}=0$,* - *a compatible affine connection $\nabla_c$ satisfying $\nabla_cN^a=0$, and $\nabla_c q_{ab}=0$.* *On $\Sigma^{\pm}$, the signature of $q_{ab}$ is $(0, \mp, + ... +)$.* *Proof.* Recall that, from Lemma [Lemma 2](#Lemma: Relations ISigma, XLambda, etc){reference-type="ref" reference="Lemma: Relations ISigma, XLambda, etc"}, $\Phi_{AB} = D_A D_B \lambda$ at every point of $M$. On $M^{\pm}$ it follows from [@cap_holonomy_2014; @Flood:2018aa] that, in the scale $\lambda$, the representative connection $\nabla_c$ in the projective class is the Levi-Civita connection of the metric $g_{ab}$ defined by $$\label{Proof: tractor metric on Sigma} \Phi_{AB} = \lambda( Y_A Y_B + g_{ab} Z^a_A Z^b_B).$$ What is more when $\Phi_{AB}X^AX^B = \pm|\lambda|\neq0$, since $\Phi_{AB}$ has signature $(--+...+)$, $g_{ab}$ has signature $(-\mp+...+)$. Now Lemma [Lemma 3](#CovariantDerivativesDirectionn){reference-type="ref" reference="CovariantDerivativesDirectionn"} implies, when evaluated in the scale $\lambda$ that $\rho=0$ on $\Sigma^{\pm}$ i.e. $i^*(I^A) = n^a W_a^A$. Since by Lemma [Lemma 2](#Lemma: Relations ISigma, XLambda, etc){reference-type="ref" reference="Lemma: Relations ISigma, XLambda, etc"} $i^*(D_A\sigma) = i^*(\Phi_{AB}I^B)$ we find, in the scale $\lambda$, $$\begin{aligned} \sqrt{\lambda}\nabla_a\sigma = g_{ab} N^b\end{aligned}$$ where $$N^a := i^*(\sqrt{\lambda} n^a)$$ is therefore the normal to $\Sigma$. Since $0= I^2= g_{ab} N^a N^b$ this normal is null and the pull back $$q_{ab} := i^*g_{ab}$$ is degenerate with one dimensional kernel spanned by $N^a$ and signature $(0\mp+...+)$. We already know that $\nabla_a$ is the Levi-Civita connection of $g_{ab}$. To conclude that it induces a connection on the tangent bundle of $\Sigma$ which satisfies, $$\begin{aligned} \nabla_c q_{ab}&=0, & \nabla_c N^a&=0.\end{aligned}$$ we need only to verify that $i^*(\nabla_a N^b)=0$. However this follows from evaluating [\[IParallel\]](#IParallel){reference-type="eqref" reference="IParallel"} in the scale $\lambda$. One concludes by noting that $\mathcal{L}_N q_{ab} = i^*(\mathcal{L}_N g_{ab}) = 2i^*(\nabla_{(a} N_{b)})=0$. ◻ Carrollian geometries have been extensively studied in e.g. [@duval_carroll_2014; @bergshoeff_carroll_2017; @bekaert_connections_2018; @morand_embedding_2020; @figueroa-ofarrill_gauging_2022; @figueroa-ofarrill_non-lorentzian_2022]. As was shown in [@hartong_gauging_2015; @Herfray:2021qmp] they define a unique normal Cartan geometry modelled on the homogeneous spaces $\mathrm{Ti}^n$ and $\mathrm{Spi}^n$ of [\[Boundary Homogeneous space\]](#Boundary Homogeneous space){reference-type="eqref" reference="Boundary Homogeneous space"}. **Proposition 24**. *([@Herfray:2021qmp])[\[Proposition: Carollian Cartan geometries\]]{#Proposition: Carollian Cartan geometries label="Proposition: Carollian Cartan geometries"} Carrollian geometries, as obtained on $\Sigma^{-}$ and $\Sigma^+$ by Theorem [Theorem 5](#Theorem: induced geometry on Sigma){reference-type="ref" reference="Theorem: induced geometry on Sigma"}, define unique normal Cartan connections respectively modelled on the homogeneous spaces $\mathrm{Ti}^n$ and $\mathrm{Spi}^n$ defined by Eq [\[Boundary Homogeneous space\]](#Boundary Homogeneous space){reference-type="eqref" reference="Boundary Homogeneous space"}).* Without much surprise these Cartan geometries coincide with the induced tractor geometries given by Proposition [Proposition 23](#Proposition: Induced tractor geometry on Sigma){reference-type="ref" reference="Proposition: Induced tractor geometry on Sigma"}. **Proposition 25**. *The tractor connections respectively induced on $\mathcal{T}_{\Sigma^-} \to \Sigma^-$ and $\mathcal{T}_{\Sigma^+} \to \Sigma^+$ coincides with the normal Cartan connections associated with the geometry of Theorem [Theorem 5](#Theorem: induced geometry on Sigma){reference-type="ref" reference="Theorem: induced geometry on Sigma"}.* *Proof.* In the proof of Theorem [Theorem 5](#Theorem: induced geometry on Sigma){reference-type="ref" reference="Theorem: induced geometry on Sigma"} we already saw that, in the scale given by $\lambda$, the induced tractor pairing $\Phi_{AB}$ is in the diagonal form [\[Proof: tractor metric on Sigma\]](#Proof: tractor metric on Sigma){reference-type="eqref" reference="Proof: tractor metric on Sigma"} while $I^A$ is simply proportional to $n^a$. Moreover, the linear connection $\nabla_a$ induced by the tractor connection is metric compatible. It follows that the induced tractor connections define Cartan geometries modelled on $\mathcal{M}_{\pm1,0}$. We thus only need to prove that the induced tractor connection is normal, which in this case just reduces to torsion-freeness. However it is straightforward to see that the induced tractor connection on $\Sigma$ is torsion-free as a result of the fact that the tractor connection on $M$ is. ◻ ## The geometry of $\mathcal{H}$ By definition $\mathcal{H}^{\pm}$ are obtained by taking the quotient of $\Sigma^{\pm}$ by the flow of $n^a$. What is more, by Corollary [Corollary 3](#Corollary: Densities induced on Q){reference-type="ref" reference="Corollary: Densities induced on Q"} the boundary defining function $\sigma$ of $\Sigma$ descends to a boundary defining $\bar{\sigma}$ function for $\mathcal{H}$. By taking the corresponding quotient by $n^a$ and $I^A$, the short sequences of Proposition [\[Proposition: short sequences of tractors on Sigma\]](#Proposition: short sequences of tractors on Sigma){reference-type="ref" reference="Proposition: short sequences of tractors on Sigma"} then descends to **Proposition 26**. *[\[Diagramm: tractor bundles along H\]]{#Diagramm: tractor bundles along H label="Diagramm: tractor bundles along H"}* *Proof.* This follows from Proposition [Proposition 20](#Proposition: tractor isomorphism){reference-type="ref" reference="Proposition: tractor isomorphism"}, Proposition [\[Proposition: short sequences of tractors on Sigma\]](#Proposition: short sequences of tractors on Sigma){reference-type="ref" reference="Proposition: short sequences of tractors on Sigma"} and the fact that $\tilde{I}_A = D_A \bar{\sigma}$. ◻ The relationship between the different tractor bundles that we discussed up to now are summarised by the following commutative diagram From Proposition [\[Proposition: connection on Q\]](#Proposition: connection on Q){reference-type="ref" reference="Proposition: connection on Q"} the tractor connection on $\Sigma$ induces a tractor connection on $\mathcal{H}$. Now, by Theorem [Theorem 4](#Theorem: Projective compactification order 1){reference-type="ref" reference="Theorem: Projective compactification order 1"}, $\mathcal{H}$ is the projective boundary of a projectively compact Einstein metric which, by [@Cap:2014ab], also induces a tractor connection on $\mathcal{H}$. From the above relationship between the tractor bundle and our previous results these connections coincide and, by the results of [@Cap:2014ab] one has: **Corollary 8**. *The projective tractor connections induced on $\mathcal{T}_{\mathcal{H}^-} \to \mathcal{H}^-$ and $\mathcal{T}_{\mathcal{H}^+} \to \mathcal{H}^+$ inherit a holonomy reduction to $SO(n-2,1)$ and respectively define normal Cartan connections modelled on $H^{n-1}$ and $dS^{n-1}$. From [@Cap:2014ab] these correspond to projectively compact Einstein manifolds with projective boundary $\mathcal{H}^0$.* ## The geometry of $\mathcal{I}\to \mathcal{H}$ According to Corollary [Corollary 3](#Corollary: Densities induced on Q){reference-type="ref" reference="Corollary: Densities induced on Q"}, $\tilde{\sigma}\in \Gamma(\mathcal{E}_{\mathcal{Q}}(1))$ is a boundary defining function for $\mathcal{H} = \mathcal{H}^{-}\cup\mathcal{H}^{0}\cup \mathcal{H}^{+}$. This implies the short sequence (dual to the second line of [\[Diagramm: tractor bundles along H\]](#Diagramm: tractor bundles along H){reference-type="eqref" reference="Diagramm: tractor bundles along H"}): What is more, from Corollary [Corollary 3](#Corollary: Densities induced on Q){reference-type="ref" reference="Corollary: Densities induced on Q"}, $\sqrt{\bar{\lambda}}\in \Gamma(\mathcal{E}_{\mathcal{H^{\pm}}}(1))$ is a nowhere vanishing smooth density on $\mathcal{H}^{\pm}$. This defines a section $$\label{Jlambda section} \phi: \mathcal{H}^{\pm} \xrightarrow{J^1 \sqrt{\bar{\lambda}}} J^1 \mathcal{E}_{\mathcal{H}}(1) \simeq \frac{i^*(J^1 \mathcal{E}_{\mathcal{Q}}(1))}{ i^* (J^1\bar{\sigma})}.$$ However, there is no canonical way to extend $\sqrt{\bar{\lambda}}$ as a density in a neighbourhood of $\mathcal{H}$ in $\mathcal{Q}$ and therefore no preferred way to extend $J^1 \sqrt{\bar{\lambda}}$ to a section of $i^*(J^1 \mathcal{E}_{\mathcal{Q}}(1))$, that we will view as a line bundle over $J^1\mathcal{E}_{\mathcal{H}}(1)$. This fact is the basis of the following definition. **Definition 4**. *The line bundle $\mathcal{I}^{\pm} \to \mathcal{H}^{\pm}$ is defined as the pull-back bundle $$\phi^* \Big( i^*(J^1 \mathcal{E}_{Q}(1)) \Big) \to \mathcal{H}^{\pm}.$$* Sections of $\mathcal{I}^{\pm}$ therefore correspond to choosing an extension of $J^1 \sqrt{\bar{\lambda}}$ as a section of $i^*(J^1 \mathcal{E}_{\mathcal{Q}}(1))$. This is naturally a $\mathbb{R}$-principal bundle: if one notes $J^1\sqrt{\Lambda}$ the 1-jet extension of $\sqrt{\bar{\lambda}}$ in $i^*(J^1 \mathcal{E}_{\mathcal{Q}}(1))$ corresponding to a point in $\mathcal{I}^{\pm}$, then the $\mathbb{R}$-action is given by $$\label{R action on I} \begin{array}{ccc} \mathbb{R}\,\times\, \mathcal{I}^{\pm} & \to & \mathcal{I}^{\pm}\\[0.4em] (k\,,\, J^1\sqrt{\Lambda}) & \mapsto & J^1\sqrt{\Lambda} + k \,J^1\sigma. \end{array}$$ This bundle is intrinsic to $\mathcal{H}^{\pm}$ as a submanifold of $\mathcal{Q}$ in the sense that it only depends on data coming from the projective compactification. Nevertheless it also nicely ties up with the geometry of $M$. **Proposition 27**. *$\mathcal{I}^{\pm}$ is canonically isomorphic to $\Sigma^{\pm}$.* *Furthermore, the action [\[R action on I\]](#R action on I){reference-type="eqref" reference="R action on I"} coincides, through this isomorphism, to the $\mathbb{R}$-action induced on $\Sigma^{\pm}$ by the flow of $\sqrt{\lambda}n^a$.* *Proof.* In the previous sections we found that points in the fibre $i^*(J^1\mathcal{E}_{\mathcal{Q}}(1))_x$ at $x\in \mathcal{H}^{\pm}$ correspond to covariantly constant sections of $(I^A)^{\circ}_{\pi^{-1}(x)} \subset (\mathcal{T}_{M})^*_{\pi^{-1}(x)}$ while points in the fibre $J^1\mathcal{E}_{\mathcal{H}}(1)_x$ correspond to covariantly constant sections of $\big((I^A)^{\circ}/I_A\big)_{\pi^{-1}(x)}$. In particular, since we have $$\nabla_{n} D_{A}\sqrt{\lambda} = I^B D_{B}\Big( \tfrac{1}{2} \lambda^{-\frac{1}{2}}D_{A}\lambda\Big)= -\tfrac{\lambda^{-\frac{3}{2}}}{4}I^B D_{B}\lambda \;D_{A}\lambda+\lambda^{-\frac{1}{2}}I^B D_{B}D_{A}\lambda$$ which implies (by making use of Lemma [Lemma 2](#Lemma: Relations ISigma, XLambda, etc){reference-type="ref" reference="Lemma: Relations ISigma, XLambda, etc"}) $$\Big(\nabla_{n} D_{A}\sqrt{\lambda}\Big)\Big|_{\Sigma} = 0+\lambda^{-\frac{1}{2}}I_A$$ it follows that $D_{A}\sqrt{\lambda}\big|_{\Sigma}$ defines a covariantly constant section of $i^*\big((I^{A})^{\circ}/I_A\big) \simeq (\mathcal{T}_{\mathcal{H}})^*$. One can easily prove that this section is identified with the section [\[Jlambda section\]](#Jlambda section){reference-type="eqref" reference="Jlambda section"}. From this remark and the definition of $\mathcal{I}^{\pm}$ we thus find that sections of $\mathcal{I}^{\pm}$ are identified with sections of $i^*(I^{A})^{\circ} \subset i^*(\mathcal{T}_{M})^*$ of the form $$\label{proof: scrI.Sigma equivalence chi def} D_{A}\sqrt{\lambda} + \chi I_A,$$ where $\chi$ is a function on $\Sigma$, and which are covariantly constant along the fibres of $\Sigma^{\pm} \to \mathcal{H}^{\pm}$. This last condition is equivalent to requiring that $\chi$ satisfies $$\label{proof: scrI.Sigma equivalence edp} \nabla_N \chi = -1$$ where $N^a = \lambda^{\frac{1}{2}}n^a$. In a coordinate system $(s,y)$ adapted to $\Sigma^{\pm} \to \mathcal{H}^{\pm}$, and such that $N^a= \partial_s$, solutions are of the form $$\label{proof: scrI.Sigma coordinate expression for chi} \chi = \xi(y) - s.$$ It follows that the set of points where $\chi$ vanishes defines a section $y \mapsto (\xi(y), y)$ of $\Sigma^{\pm} \to \mathcal{H}^{\pm}$. Such sections are thus in one-to-one correspondence with solutions of [\[proof: scrI.Sigma equivalence edp\]](#proof: scrI.Sigma equivalence edp){reference-type="eqref" reference="proof: scrI.Sigma equivalence edp"} and therefore in one-to-one correspondence with sections of $\mathcal{I}^{\pm}$. Finally comparing [\[R action on I\]](#R action on I){reference-type="eqref" reference="R action on I"} with [\[proof: scrI.Sigma equivalence chi def\]](#proof: scrI.Sigma equivalence chi def){reference-type="eqref" reference="proof: scrI.Sigma equivalence chi def"} one sees that the $\mathbb{R}$-action $\eqref{R action on I}$ amounts to adding $k\in \mathbb{R}$ to $\chi$. The coordinate expression [\[proof: scrI.Sigma coordinate expression for chi\]](#proof: scrI.Sigma coordinate expression for chi){reference-type="eqref" reference="proof: scrI.Sigma coordinate expression for chi"} then shows that this is equivalent to the action of the flow of $N^a= \partial_s$. ◻ **Definition 5**. *The $\mathbb{R}$-action allows to parallel transport 1-jets $J^1\mathcal{E}_{\mathcal{I}^{\pm}}(1)$ on $\mathcal{I}^{\pm}$, we note $$(\bm{T}_{\mathcal{H}})^* \to \mathcal{H}$$ the resulting bundle on $\mathcal{H}$.* **Proposition 28**. *$\bm{T}_{\mathcal{H}} \to \mathcal{H}^{\mp}$ coincides with the bundle $\tilde{\mathcal{T}}_{\mathcal{H}} \to \mathcal{H}^{\mp}$ obtained by taking the quotient of $\mathcal{T}_{\Sigma} \to \Sigma^{\mp}$ by the flow of $n^a$ (as in section [\[sssection: Tractors of Q\]](#sssection: Tractors of Q){reference-type="eqref" reference="sssection: Tractors of Q"}). In particular it is equipped with a normal tractor connection modelled on the homogeneous space $\mathrm{Ti}^n$, $\mathrm{Spi}^n$ (see [\[Boundary Homogeneous space\]](#Boundary Homogeneous space){reference-type="eqref" reference="Boundary Homogeneous space"}).* *Proof.* It follows from the definition and Proposition [Proposition 27](#Proposition: scrI.Sigma equivalence){reference-type="ref" reference="Proposition: scrI.Sigma equivalence"}. ◻ From the diagram in Figure [\[Figure: The tractor bundles along the boundary\]](#Figure: The tractor bundles along the boundary){reference-type="ref" reference="Figure: The tractor bundles along the boundary"} we then find: **Lemma 11**. *One has a canonical isomorphism $$\bm{T}_{\mathcal{H}} \simeq i^*(\tilde{I}_A)^{\circ} \subset i^* \tilde{\mathcal{T}}.$$* Since, on $\mathcal{Q}_{\mathcal{O}}$ and by Proposition [Proposition 21](#Proposition: ambient tractors extend conformal){reference-type="ref" reference="Proposition: ambient tractors extend conformal"}, $\tilde{\mathcal{T}}$ is isomorphic to the conformal tractor bundle, the preceding Lemma means that $\bm{T}_{\mathcal{H}}$ is in a certain sense the extension by continuity of this conformal tractor bundle along the projective boundary. Similarly the Cartan connection induced on these bundles by Proposition [Proposition 28](#Proposition: Cartan geometries on I){reference-type="ref" reference="Proposition: Cartan geometries on I"} must be an extension by continuity of the conformal tractor connection. The importance of these last results stems from the fact that normal tractor connections modelled on the homogeneous space $\mathrm{Ti}^n$ and $\mathrm{Spi}^n$ are not unique, see e.g. [@Herfray:2021qmp], not even after fixing a flat normal tractor connection on $\mathcal{H}^{\pm}$. The resulting freedom encodes asymptotic data at space/time-like infinity induced by the projective compactification. # Asymptotic symmetries of projectively compact Einstein manifolds of order 1 {#section: Asymptotic symmetries of projectively compact Einstein manifolds} Let $(\mathcal{Q}, \bar{\nabla}, \bar{H}^{AB}, \bar{I}_{A} )$ be a projectively compact order 1 Einstein manifold (i.e. as obtained in the previous sections). We will note $Sym(\mathcal{Q})$ the space of diffeomorphisms $\Phi : \mathcal{Q}\to \mathcal{Q}$ preserving the boundary $\Phi(\mathcal{H}^{\pm}) = \mathcal{H}^{\pm}$ and such that $$\begin{aligned} \left(\Phi^* \bar{H}^{AB}\right) \big|_{\mathcal{H}^{\pm}} & = \bar{H}^{AB}\big|_{\mathcal{H}^{\pm}}, & \left(\Phi^* \bar{I}_{A}\right) \big|_{\mathcal{H}^{\pm}} & = \bar{I}_{A}\big|_{\mathcal{H}^{\pm}}.\end{aligned}$$ In particular, these diffeomorphisms must induce an isometry $\phi : \mathcal{H}^{\pm} \to \mathcal{H}^{\pm}$ of $(\mathcal{H}^{\pm} , \lambda \zeta^{ab})$ and therefore preserve the induced Cartan geometry given by Proposition [Corollary 8](#Proposition: Cartan geometries on H){reference-type="ref" reference="Proposition: Cartan geometries on H"} (but will not in general preserve the Cartan geometry induced on $\mathcal{I}^{\pm}$ by Proposition [Proposition 28](#Proposition: Cartan geometries on I){reference-type="ref" reference="Proposition: Cartan geometries on I"}). We will note $Sym_0(\mathcal{Q})$ the space of diffeomorphisms $\Phi : \mathcal{Q}\to \mathcal{Q}$ which act trivially on the boundary up to first order $i^*(d\Phi) = Id$. **Definition 6**. *We will call asymptotic symmetries of a projectively compact Einstein manifold $(\mathcal{Q}, \bar{\nabla}, \bar{H}^{AB}, \bar{I}_{A})$ the quotient $Asym(\mathcal{Q}) = Sym(\mathcal{Q})/Sym_0(\mathcal{Q})$.* **Theorem 6**. *The action of the asymptotic symmetries on $\mathcal{I}^{\pm}$ coincides with the action of the principal bundle automorphisms of $\mathcal{I}^{\pm}$ preserving the Einstein metric on $\mathcal{H}^{\pm}$ and we thus have $$\begin{aligned} Sym(Q)/Sym_0(Q) &\simeq C^{\infty}(\mathcal{H}^{\pm}) \rtimes ISO(\lambda\zeta^{ab})\end{aligned}$$* *In particular, the group of asymptotic symmetries naturally acts on the normal Cartan connections induced on $\mathcal{I}^{\pm}$ by Proposition [Proposition 28](#Proposition: Cartan geometries on I){reference-type="ref" reference="Proposition: Cartan geometries on I"}.* *Proof.* Let $\Phi:\mathcal{Q}\to \mathcal{Q}$ be an asymptotic symmetry, we note $\phi: \mathcal{H} \to \mathcal{H}$ the induced action on $\mathcal{H}$. Since $\Phi$ preserves $i^*(\bar{H}^{AB})$ it follows that $\phi$ must be an isometry of $\lambda \zeta^{ab}$. We recall that along the boundary $\mathcal{H}$ the restriction of the tractor metric $\bar{H}^{AB}$ to $\mathcal{T}_\mathcal{H}$ is invertible of inverse $\tfrac{1}{2}D_AD_B\bar{\lambda}$ and that $D_A\bar{\lambda} = (D_AD_B\bar{\lambda}) X^B$. It follows that an asymptotic symmetry must preserve both $i^*I_A$ and $D_A\sqrt{\bar{\lambda}}$. Let $J^1_x\sqrt{\Lambda}$ be a point in $\mathcal{I}^{\pm}$ i.e. a point in $J^1_x \mathcal{E}_{\mathcal{Q}}(1)$ whose quotient by $J^1_x\bar{\sigma}$ coincides with $J^1_x\sqrt{\lambda}$. By the previous remark, the image of this point by an asymptotic symmetry must of the form $$\Phi^* (J^1\sqrt{\Lambda})_x = J^1_{x}\sqrt{\Lambda} + \chi J^1_{x} \sigma$$ where $\chi$ is some function that might at this stage depend on the points of the fibres of $J^1_x\sqrt{\Lambda} \to J^1_x\sqrt{\lambda}$. To conclude we need to prove that symmetry is compatible with the $\mathbb{R}$ action $J^1\sqrt{\Lambda} \mapsto J^1_x\sqrt{\Lambda} + k J^1_{x} \sigma$, $k\in \mathbb{R}$. However, $$\Phi^* (J^1_x\sqrt{\Lambda} + k J^1_{x} \sigma ) = J^1_{\phi(x)}\sqrt{\Lambda} + (\chi + k) J^1_{\phi(x)} \sigma$$ follows from the fact that a symmetry must preserve $i^*(I_A)$. Therefore, the asymptotic symmetry induces an automorphism of the bundle. One concludes the proof by observing that such automorphism completely fixes the first order jets of the symmetry and is thus equivalent to an asymptotic symmetry. ◻ [^1]: <jack.borthwick@mcgill.ca> [^2]: <yannick.herfray@univ-tours.fr> [^3]: Here $M^4 = \frac{ISO(1,3)}{SO(3,1)}$ is the usual Minkowski spacetime. [^4]: See, for instance, [@Kobayashi:1995aa Proposition 7.2] [^5]: $\epsilon^2$ is the section that realises the identification $(\Lambda^{n+1}TM)^{\otimes 2} \simeq \mathcal{E}(2(n+2))$; if $M$ is orientable, then it can be thought of as the square of the volume form. [^6]: We recall from [@Flood:2018aa] that the projective tractor volume form is defined by $$\epsilon^2_{A_0 A_1 ... A_{n+1}B_1 B_1... B_{n+1}} := \epsilon^2_{a_1 ... a_{n+1}b_1... b_{n+1}} Y_{[A_0} Z_{A_1}^{a_1} ... Z_{A_{n+1}]}^{a_{n+1}} Y_{[B_0} Z_{B_1}^{b_1} ... Z_{B_{n+1}]}^{b_{n+1}}$$ [^7]: For definiteness, we work in this article with the physically interesting case of Lorentz signature. However, most of our considerations are independent of this choice. [^8]: This is also referred to as the complete or natural lift of $N$, see for instance [@Cordero:1983aa]. [^9]: *One has $Y_AI^A=\nu^{-1}I^AD_A \nu=\nu^{-1}D_n \nu=0$ and these are thus splitting operators on $I^{\perp} \subset \mathcal{T}_{M}^*$.* [^10]: We omit, by abuse of notation, the bundle isomorphism [^11]: $C$ is $\tilde{P}$-principal bundle. [^12]: The Cartan geometry induced on $M^0$ is similar to that of null infinity, the conformal boundary of an asymptotically flat spacetime. This has been studied in details in [@herfray_asymptotic_2020; @herfray_tractor_2022] and can be thought as a conformal version of the geometries induced on $M^{\pm}$, see [@Herfray:2021qmp].

arxiv_math

--- abstract: | This note is an (exact) copy of the report of Jaak Peetre, \"Generalizing Ovchinnikov's Theorem\". Published as Technical Report, Lund (1981). Some more recent general references have been added, some references updated though (in *italics*) and some misprints corrected. address: - Lund Institute of Technology, Lund, Sweden - "Nilsson: Stockholm, Sweden" author: - Jaak Peetre - Per G. Nilsson (Typist) title: Generalizing Ovchinnikov's Theorem --- Generalizing Ovchinnikov's Theorem. Jaak Peetre # 0. Introduction {#introduction .unnumbered} Whereas in the past much work in interpolation spaces more or less consciously has been connected with direct applications to various questions in Analysis the last 4 - 5 years have witnessed a considerable progress on the theoretical side. I am in the first place thinking of the results of Ovchinnikov (including joint work with Dmitriev) and the important recent paper by Janson - see the bibliography [@23-Ovc], [@24-Ovc], [@25-Ovc],[@9-DmKrOv] [@10-DmOv] and [@16-Jan] respectively. Unfortunately most of Ovchinnikov's papers are hard to read and so is to some extent Janson's too. A common basis for all this work is a fundamental theorem by Aronszajn and Gagliardo [@1-AG65], published already in 1965, but which for quite a long time seemed to have fallen almost into oblivion [^1] .Perhaps we are finally beginning to gain a better understanding of the true inner meaning of the various interpolation methods which often have come up in a rather ad hoc fashion[^2]. $% \medskip$ [0.1]{.ul} A central result in Ovchinnikov's work is the following interpolation theorem [@23-Ovc]. Let $T$ be any bounded linear operator mapping the couple of weighted $l^{\infty }$ spaces $l^{\infty }\left( \overline{w}\right) =\left( l^{\infty }\left( w^{0}\right) ,l^{\infty }\left( w^{1}\right) \right)$ into the couple of weighted $l^{1}$ spaces $% l^{1}\left( \overline{w}\right) =\left( l^{1}\left( w^{0}\right) ,l^{1}\left( w^{1}\right) \right)$ in symbols: $T:l^{\infty }\left( \overline{w}\right) \rightarrow l^{1}\left( \overline{w}\right) .$ Then $T$ maps the space $l^{\infty }\left( w^{0}/\rho \left( w^{1}/w^{0}\right) \right)$ into $l^{1}\left( w^{0}/\rho \left( w^{1}/\rho \left( w^{1}\right) \right) \right) ,$ in symbols $T:l^{\infty }\left( w^{0}/\rho \left( w^{1}/w^{0}\right) \right) \rightarrow l^{1}\left( w^{0}/\rho \left( w^{1}/w^{0}\right) \right) .$ Here $\rho$ is any pseudo-concave function $% \left( \rho \in \mathcal{P}\right) .$ Ovchinnikov's original proof was based on Grothendieck's fundamental theorem [@14-Gr]. Subsequently Janson [@16-Jan] managed to give another more elementary proof which in particular does not involve the latter result (cf. sec. [5]{.ul} of this paper). (If $\rho \in \mathcal{P}^{+-}$ a proof based on Xincin's (= Khintchine's) inequality was found by Gustavsson [@15-Gu].) In connection with his interpolation theorem Ovchinnikov [@23-Ovc] had introduced three new interpolation methods, indexed by the letters $l,m$ and $u$ (l for \"lower\", m for \"middle\" and u for \"upper\"), which fall in the general scheme of Aronszajn and Gagliardo [@1-AG65]. Janson [@16-Jan] in the other hand clarifies the role of l and u. However m seems now to fall out of the picture. $\medskip$ [0.2]{.ul} The purpose of this paper is to extend Ovchinnikov's theorem in various directions. In our first rather trivial generalization (sec. [1]{.ul}) we try to replace the spaces $l^{\infty }\left( w^{0}/\rho \left( w^{1}/w^{0}\right) \right)$ and $l^{1}\left( w^{0}/\rho \left( w^{1}/w^{0}\right) \right)$ by general interpolation (or more generally only intermediate) spaces $A$ and $B$ with respect to $l^{\infty }\left( \overline{w}\right)$ and $l^{1}\left( \overline{w}\right)$ respectively. Our result is roughly speaking that $T:l^{\infty }\left( \overline{w}\right) \rightarrow l^{1}\left( \overline{w}\right) ,$ where $T$ is a bounded linear operator implies $T:A\rightarrow B$ if and only if this assertion is already true when $T$ is a multiplier transform. Next (sec. [2]{.ul}) we consider vector valued and the continuous case. As a corollary ([ 2.5]{.ul} theorem) we get the following Ovchinnikov type theorem for Besov couples: If $T:B_{p}^{\overline{s},\infty }\left( \mathbb{R}^{n}\right) \overset{\text{def}}{=}\left( B_{p}^{s_{0},\infty }\left( \mathbb{R}% ^{n}\right) ,B_{p}^{s_{1},\infty }\left( \mathbb{R}^{n}\right) \right) \rightarrow B_{q}^{\overline{t},1}\left( \mathbb{R}^{m}\right) \overset{% \text{def }}{=}\left( B_{q}^{t_{0},1}\left( \mathbb{R}^{m}\right) ,B_{q}^{t_{1},1}\left( \mathbb{R}^{m}\right) \right)$ then $% T:B_{p}^{s,\infty }\left( \mathbb{R}^{n}\right) \rightarrow B_{q}^{t,1}\left( \mathbb{R}^{m}\right)$ where $s=\left( 1-\theta \right) s_{0}+\theta s_{1},t=\left( 1-\theta \right) t_{0}+\theta t_{1}$ with $% \theta \in \left( 0,1\right) .$ In the following section (sec. [3]{.ul} ) we introduce an abstract setting for Ovchinnikov type theorems. If $% \overline{X}=\left( X_{0},X_{1}\right)$ and $\overline{Y}=\left( Y_{0},Y_{1}\right)$ are general Banach couples we say they satisfy condition $\left( O\right)$ if $T:\overline{X}\rightarrow \overline{Y}$ implies $T:K^{A}\left( \overline{X}\right) \rightarrow J^{B}\left( \overline{% Y}\right)$ where $A$ and $B$ are any two of the spaces of sec. [1]{.ul} and $K^{A}$ and $J^{B}$ stands for the corresponding $K$ and $J$ functors respectively. We put this in relation with the work of Janson [@16-Jan]. Janson has observed - in the special case $A=l^{\infty }\left( \rho \left( \overline{w}\right) \right)$ and $B=l^{1}\left( \rho \left( \overline{w}% \right) \right) ,$which is really the only case which matters - that [^3] $% K^{A}$ is the maximal functor with $K^{A}\left( l^{\infty }\left( \overline{w% }\right) \right) \subseteq A$ and $J^{B}$ the minimal functor with $% J^{B}\left( l^{1}\left( \overline{w}\right) \right) \supset B$ - maximal and minimal of course in the sense of the Aronszajn-Gagliardo theorem [@1-AG65] . Let now $G^{A}$ be the minimal functor with $G^{A}\left( l^{\infty }\left( \overline{w}\right) \right) \supset A$ and $H^{B}$ the maximal functor with $H^{B}\left( l^{1}\left( \overline{w}\right) \right) \subseteq B.$ We say that the couple $\overline{X}$ is of type $\left( l\right)$ if $G^{A}\left( \overline{X}\right) =K^{A}\left( \overline{X}% \right)$ for all $A$ and that the couple $\overline{Y}$ is of type $\left( u\right)$ if $H^{B}\left( \overline{Y}\right) =J^{B}\left( \overline{Y}% \right)$ for all $B.$ Then trivially, if $\overline{X}$ is of type $\left( l\right)$ and $\overline{Y}$ of type $\left( u\right) ,$ the couples $% \overline{X}$ and $\overline{Y}$ satisfies condition $\left( O\right) .$ For instance, we show that Marcinkiewicz couples are of type $\left( l\right)$ and Lorentz couples of type $\left( u\right) .$Thus we get an Ovchinnikov type theorem with Marcinkiewicz and Lorentz spaces instead of weighted $% l^{\infty }$ and $l^{1}.$ In Sec. [4]{.ul} we relate those notions with the idea of nuclearity. In this context we give also a direct proof of Ovchinnikov's theorem [@23-Ovc], along the lines of the original one, thus relying to Grothendieck's fundamental theorem [@14-Gr]. En revanche, in a attempt to make this paper as well-contained as possible, we include what is essentially Janson's proof [@16-Jan] but free of the context of Aronszajn-Gagliardo functors - whether this really is an advantage from the conceptual point of view, we leave to the reader to judge. Finally in the short sec. [6]{.ul} , pointing out some open problems, we outline a program for further research.$\medskip$ [0.3.]{.ul} Some of our terminology is put forward in Sec. $\frac{1}{2}$ - obtained thus by interpolation between the present sec. [0]{.ul} and sec. [1]{.ul}. As far as practicable we try to adhere to the one of Bergh-Löfström [@3-BeL0].$\medskip$ We make the following convention: $\frac{a}{b}<\infty$ is interpreted to mean $a=0$ if $b=0.\medskip$ The sign $\approx$ further stands as usual for equivalence. Thus $u\approx v$ where $u$ and $v$ are positive quantities, means that we have inequalities $u\leq C_{1}v,v\leq C_{2}u;$ what the constants $C_{1}$ and $% C_{1}$ depend on is usually clear from the context.$\medskip$ [0.4]{.ul} The principal results of this paper - in a somewhat premature form though - were already announced at the Scandinavian Mathematical Congress held at Aarhus, Denmark in aug. 1980 [@26-Pe]. In the meantime we have read Janson [@16-Jan], which has lead to considerable improvements. # $\frac{1}{2}$. Definitions. {#frac12.-definitions. .unnumbered} [$\frac{1}{2}.1$]{.ul}$.$ Let $\overline{X}=\left( X_{0},X_{1}\right)$ and $\overline{Y}=\left( Y_{0},Y_{1}\right)$ be two Banach pairs. As usual we put $\Delta \overline{X}=X_{0}\cap X_{1}$ and $\Sigma \overline{X}% =X_{0}+X_{1}$ (similarly for $\overline{Y}).$ Moreover $\Sigma _{0}\overline{% X}$ denote the closure of $\Delta X$ in $\Sigma \overline{X}.$ If $\Sigma \overline{X}$ is [regular]{.ul} (i.e. if $\Delta \overline{X}$ is dense in both $X_{0}$ and $X_{1}$) then $\Sigma _{0}\overline{X}=\Sigma \overline{X% }$.$\medskip$ We say that we have a [bounded linear]{.ul} operator from $\overline{X}$ into $\overline{Y}$ if there is given a linear map $T$ from $\Sigma \overline{X}$ into $\Sigma \overline{Y}$ such that $T\left( X_{i}\right) \subseteq Y_{i}$ ($i=0,1),$ the restriction $T_{i}=T\mid _{X_{i}}$of $T$ to $% X_{i}$ defining a continuous linear operator from $X_{i}$ into $Y_{i}$ in the ordinary sense. We then write $T:\overline{X}\rightarrow \overline{Y}$ or more completely $T:\overline{X}\overset{b}{\rightarrow }\overline{Y}.$ We define the [operator norm]{.ul} of $T$ by $\left\Vert T\right\Vert =\max \left( \left\Vert T_{0}\right\Vert ,\left\Vert T_{1}\right\Vert \right)$ where $\left\Vert T_{i}\right\Vert$ is the usual operator norm of $T_{i},$ i.e. $\left\Vert T_{i}\right\Vert =\sup_{0\neq x\in X_{i}}\left\Vert Tx\right\Vert _{Y_{i}}/\left\Vert x_{i}\right\Vert _{X_{i}}$ ($i=0,1).\medskip$ We say that $T$ is [nuclear]{.ul} operator from $\overline{X}$ into $% \overline{Y}$ and we use the notation $T:\overline{X}\overset{n}{\rightarrow }\overline{Y}$ , if $T$ is a linear map from $\Sigma \overline{X}$ into $% \Sigma \overline{Y}$ such that there exists a family $\left\{ b_{n}\right\}$ in $\Delta \overline{Y}$ and a family $\left\{ l_{n}\right\}$ in $\left( \Sigma \overline{X}\right) ^{^{\prime }}$ with[^4] $$\left( 1\right) :\dsum \max \left\{ \left\Vert l_{n}\right\Vert _{X_{0}^{^{\prime }}}\left\Vert b_{n}\right\Vert _{Y_{0}},\left\Vert l_{n}\right\Vert _{X_{1}^{^{\prime }}}\left\Vert b_{n}\right\Vert _{Y_{1}}\right\} <\infty \tag{1}$$ such that $Ta=\dsum l_{n}\left( a\right) b_{n}$ for any $a\in \Sigma \overline{X}.$ (Here $n$ rungs through some index set $I,$which we without loss of generality may take at most denumerable infinite.). Note that the series $\dsum l_{n}\left( a\right) b_{n}$ is always summable in $\Sigma \overline{Y}.$ Indeed writing $\alpha =a_{0}+a_{1}$ with $a_{i}\in X_{i}$ $% \left( i=0,1\right)$ we see that each of the series $\dsum l_{n}\left( a_{i}\right) b_{n}$ is summable in $Y_{i},$ so the sum $Ta$ is really an element of $\Sigma _{0}\overline{Y}.$ This also shows that $T:\overline{X}% \overset{n}{\rightarrow }\overline{Y}$ implies $T:\overline{X}\overset{b}{% \rightarrow }\overline{Y}.$ We define the [*nuclear norm*]{.ul} of $T,$ denoted by $\left\Vert T\right\Vert _{n},$ as the infimum of all expressions appearing in $\left( 1\right) .$ Clearly $\left\Vert T\right\Vert \leq \left\Vert T\right\Vert _{n}.\medskip$ [*Remark.*]{.ul}* *If $T:\overline{X}\overset{n}{\rightarrow }\overline{Y}$ then already each of the maps $T:X_{i}\rightarrow Y_{i}$ $% \left( i=0,1\right) \footnote{% See Notes \underline{$\frac{1}{2}$}:$\left\langle 2\right\rangle .$}$ is nuclear; we may say that $T$ is \"separately nuclear\". But the converse is not true. There are counter-examples showing that a separately nuclear operator need not to be nuclear (in our sense).$\medskip$ [$\frac{1}{2}.2$]{.ul}$.$ We introduce the following \"orderings\" between elements of $\Sigma \overline{Y}$ and $\Sigma \overline{X}$ (or $% \Sigma _{0}\overline{Y}$ and $\Sigma _{0}\overline{X}),$ $\lambda$ denoting a fixed number $>1.$ $\left( i\right) :y<_{b}x$ means that for any $\epsilon >0$ there exists an operator $T:\overline{X}\overset{b}{\rightarrow }\overline{Y}$ with $% \left\Vert T\right\Vert <1+\epsilon$ such that $y=Tx.$ $\left( ii\right) :y<_{n}x$ means that for any $\epsilon >0$ there exists an operator $T:\overline{X}\overset{n}{\rightarrow }\overline{Y}$ with $% \left\Vert T\right\Vert _{n}<1+\epsilon$ such that $y=Tx$ $\left( iii\right) :y<_{K\left( \lambda \right) }x$ means that $K\left( \lambda ^{k},y\right) \leq K\left( \lambda ^{k},x\right)$ for all $k\in \mathbb{Z}$. $\left( iv\right) :y<_{J\left( \lambda \right) }x$ means that for every [representation]{.ul}* *$\widehat{x}=\left\{ x_{k}\right\} _{k\in \mathbb{Z}}$ of $x\footnote{% See Notes \underline{$\frac{1}{2}$}:$\left\langle 3\right\rangle .$}$ and every $\epsilon >0$ there exists a representation $\widehat{y}=\left\{ y_{k}\right\} _{k\in \mathbb{Z}}$ of $y$ such that $J\left( \lambda ^{k},y_{k}\right) \leq \left( 1+\epsilon \right) J\left( \lambda ^{k},x_{k}\right)$ for all $k\in \mathbb{Z}$. $\left( v\right) :y<_{J/K\left( \lambda \right) }x$ means that for every $% \epsilon >0$ there exists a representation $\widehat{y}=\left\{ y_{k}\right\} _{k\in \mathbb{Z}}$ of $y$ such that $\dsum_{k\in \mathbb{Z}}% \frac{J\left( \lambda ^{k},y_{k}\right) }{K\left( \lambda ^{k},x\right) }% <1+\epsilon .$ Orderings seem to be destined to play a prominent role in interpolation theory. E.g. [@6-CwPe] is mainly devoted to a study of the orderings $% \left( iii\right)$ and $\left( iv\right) .$ The orderings $\left( i\right) ,\left( iii\right)$ and $\left( iv\right)$ are [transitive]{.ul} in the obvious sense. Let $\overline{Z}=\left( Z_{0},Z_{1}\right)$ be another Banach pair (besides $\overline{X}$ and $% \overline{Y}$). Then e.g. $z<_{b}y,y<_{b}x$ implies $z<_{b}x.$ The orderings $\left( ii\right)$ and $\left( v\right)$ have the [ideal property ]{.ul} with respect to the ordering $\left( i\right) .$That is $$\begin{aligned} z &<&_{b}y,y<_{n}x\Longrightarrow z<_{n}x, \\ z &<&_{n}y,y<_{b}x\Longrightarrow z<_{n}x.\end{aligned}$$ On the other hand the ordering $\left( v\right)$ has the [left ideal property]{.ul} with respect to ordering $\left( iv\right)$ and the right ideal property with respect to the ordering $\left( iii\right) .$That is; $$\begin{aligned} z &<&_{J\left( \lambda \right) }y,y<_{J/K\left( \lambda \right) }x\Longrightarrow z<_{J/K\left( \lambda \right) }x; \\ z &<&_{J/K\left( \lambda \right) }y,y<_{K\left( \lambda \right) }x\Longrightarrow z<_{J/K\left( \lambda \right) }x.\end{aligned}$$ This is also true for the ordering $\left( ii\right)$ but this not quite obvious. For the proof sec. [4.1]{.ul} cor. 2.$\medskip$ Let $<_{0}\left( o=b,n,...\right)$ be any of the orderings $\left( i\right) -\left( v\right) .$Then we use the symbol $<<_{0}$ in the following sense. $% y<<_{0}x$ means that $y<_{0}cx$ for some $c.$(Sometimes it is intended that the constant $c$ depends only on the pairs involved. This will in most cases be clear from the context.) In the case of the orderings, $\left( iii\right) -\left( v\right)$ the value of $\lambda$ is then immaterial so we write simply $y<<_{K}x$ in place of $y<<_{K\left( \lambda \right) }x$ etc.$% \medskip$ [$\frac{1}{2}.3$]{.ul}$.$We recall now the definition of the \"classical\" $K-$ and $J-$ spaces (see e.g. [@6-CwPe]; cf. [@3-BeL0], [@4-BrKr]), in fact in a slightly more general form as suggested by the work of Janson [@16-Jan]. Let $A$ be any intermediate space with respect to the weighted $l^{\infty }$ couple $l^{\infty }\left( \overline{w}\right) =\left( l^{\infty }\left( w^{0}\right) ,l^{\infty }\left( w^{1}\right) \right)$ and $B$ be one with respect to the weighted $l^{1}$ couple $% l^{1}\left( \overline{w}\right) =\left( l^{1}\left( w^{0}\right) ,l^{1}\left( w^{1}\right) \right) ,\overline{w}=\left( w^{0},w^{1}\right) =\left( \left\{ w_{n}^{0}\right\} ,\left\{ w_{n}^{1}\right\} \right)$ be any given couple of weight sequences.[^5] Let $% \overline{X}=\left( X_{0},X_{1}\right)$ be any Banach couple. Then we set $$x\in K^{A}\left( \overline{X}\right) \text{ iff }\left\{ w_{n}^{0}K\left( \frac{w_{n}^{0}}{w_{n}^{1}},x\right) \right\} \in A,$$ $$\begin{aligned} x &\in &J^{B}\left( \overline{Y}\right) \text{ iff for some representation }% \widehat{x}=\left\{ x_{n}\right\} ~\text{of }x\text{ holds} \\ \left\{ w_{n}^{0}J\left( \frac{w_{n}^{0}}{w_{n}^{1}},x\right) \right\} &\in &B.\end{aligned}$$ The properties of these spaces are well-known. If we want to emphasize the dependence on $\overline{w}$ we use the more complete notation $K_{\overline{% w}}^{A}$ and $J_{\overline{w}}^{B},$in place of $K^{A}$ and $J^{B}$ respectively. (From Sec. 3 on we shall again specialize to the traditional case $w_{n}^{0}=1,w_{n}^{1}=\lambda ^{-n}.)$. Sometimes we also put, following the historical tradition, $K^{A}\left( \overline{X}\right) =% \overline{X}_{A;K}$ and $J^{B}\left( \overline{Y}\right) =\overline{Y}% _{B;J}.$ In particular let $A$ and/or $B$ be the space $l^{p}\left( \rho \left( \overline{w}\right) \right)$ corresponding to the norm $$\left\Vert c\right\Vert =\left( \dsum_{n}\left( w_{n}^{0}\left\vert c_{n}\right\vert /\rho \left( \frac{w_{n}^{0}}{w_{n}^{1}}\right) \right) ^{p}\right) ^{1/p}\text{ (with }c=\left\{ c_{n}\right\} )$$ where $p\in \left[ 1,\infty \right]$ (with the usual interpretation for $% p=\infty$) and $\rho$ any positive function, usually pseudo-concave though ($\rho \in \mathcal{P)}$ . Then we write $\overline{X}_{\rho p;K}$ and $% \overline{X}_{\rho p;J}$ respectively for these spaces. If $\rho \left( t\right) =t^{\theta }$ $\left( \theta \in \left[ 0,1\right] \right)$ we get the space $\overline{X}_{\theta p;K}$ and $\overline{X}_{\theta p;J}.$ (If $% \theta \in \left( 0,1\right)$ we can, in view of the equivalence theorem, omit the additional subscripts $K$ and $J$ writing $\overline{X}_{\theta p}$ for both of them.)$\medskip$ [$\frac{1}{2}.4$]{.ul}$.$ We mention also some basic facts connected with the Aronszajn-Gargliardo theorem [@1-AG65] already referred to in the introduction.$\medskip$ Let $\overline{A}=\left( A_{0},A_{1}\right)$ and $\overline{B}=\left( B_{0},B_{1}\right)$ be any given two Banach couples and $A$ and $B$ intermediate spaces with respect to $\overline{A}$ and $\overline{B}$ respectively. Then there exists a minimal interpolation functor $Orb_{A}$ (or $Orb_{A,\overline{A}})$ such that $Orb_{A}\left( \overline{A}\right) \supseteq A$ and a maximal one $Corb_{B}$ (or $Corb_{B,\overline{B}})$ such that $Corb_{B}\left( \overline{B}\right) \subseteq B.$ If $\overline{X}% =\left( X_{0},X_{1}\right)$ is a generic Banach couple we say that $% Orb_{A}\left( \overline{X}\right)$ is the [orbit]{.ul} of $A$ in $% \overline{X}$ and that $Corb_{B}\left( \overline{X}\right)$ is the [coorbit]{.ul} of $B$ in $\overline{X}.$ It is easy to see that if $A$ and $B$ are [relative interpolation]{.ul}* *spaces with respect to $\overline{A}$ and $\overline{B}$ then $Orb_{A}\subseteq Corb_{B}.\medskip$ In particular if $\overline{A}=l^{\infty }\left( \overline{w}\right)$ or $% B=l^{1}\left( \overline{w}\right)$ we put $G^{A}=Orb_{A}$ and $% H^{B}=Corb_{B}$ respectively. In the special case $A=l^{\infty }\left( \rho \left( \overline{w}\right) \right)$ and $B=l^{1}\left( \rho \left( \overline{w}\right) \right)$ $\left( \rho \in \mathcal{P}\right)$ these functors were investigated by Janson [@16-Jan]. In the same situation Janson [@16-Jan] also observed that (in our present notation) $% K^{A}=Corb_{A}$ and $J^{B}=Orb_{B}.$ It is easy to see that this is also the case for general $A$ and $B.$ # 1. General intermediate spaces. {#general-intermediate-spaces. .unnumbered} In this sec. we wish to extend Ovchinnikov's theorem [@23-Ovc] (as stated in the Introduction) in the following direction. Instead of $% l^{\infty }\left( \rho \left( \overline{w}\right) \right)$ and $l^{1}\left( \rho \left( \overline{w}\right) \right)$ we consider quite general [intermediate]{.ul} spaces $A$ and $B$ with respect to $l^{\infty }\left( \overline{w}\right)$ and $l^{1}\left( \overline{w}\right)$ respectively. We ask under which conditions it is true that $$T:l^{\infty }\left( \overline{w}\right) \rightarrow l^{1}\left( \overline{w}% \right) \Longrightarrow T:A\rightarrow B \tag{1}$$ * *We first derive a necessary condition, that is rather straightforward. [Proposition 1]{.ul}. Let $A$ and $B$ be intermediate spaces, with respect to $l^{\infty }\left( \overline{w}\right)$ and $l^{1}\left( \overline{w}\right)$ respectively Then $\left( 1\right)$ implies that $$a=\left\{ a_{k}\right\} \in A,\dsum_{k}\frac{\left\vert b_{k}\right\vert }{% \left\vert a_{k}\right\vert }<\infty \Longrightarrow b=\left\{ b_{n}\right\} \in B \tag{2}$$ (According to the convention made in the introduction, we interpret $\frac{% \left\vert b_{k}\right\vert }{\left\vert a_{k}\right\vert }$ as $0$ if $% b_{k}=0.)$ [Proof:]{.ul}* *Assume $a\in A,\dsum \frac{\left\vert b_{k}\right\vert }{\left\vert a_{k}\right\vert }<\infty .$ Let $T$ be defined by $Tx=\left\{ \frac{b_{k}}{a_{k}}x_{k}\right\}$ if $x=\left\{ x_{k}\right\} .$($T$ is thus a multiplier transform.). Then clearly $% T:l^{\infty }\left( \overline{w}\right) \rightarrow l^{1}\left( \overline{w}% \right)$ and $Ta=b.$Thus $\left( 1\right)$ gives at once $b\in B.$\#$% \medskip \medskip$ [Example.]{.ul}* *If $A=l^{\infty }\left( \rho \left( \overline{w}% \right) \right) ,B=l^{1}\left( \rho \left( \overline{w}\right) \right)$ - Ovchinnikovs' case - then $\left( 2\right)$ is certainly is fulfilled. Indeed if $\sup \left\vert a_{k}\right\vert <\infty ,\dsum \frac{\left\vert b_{k}\right\vert }{\left\vert a_{k}\right\vert }<\infty$ then $$\dsum \left\vert b_{k}\right\vert =\dsum \frac{\left\vert b_{k}\right\vert }{% \left\vert a_{k}\right\vert }\left\vert a_{k}\right\vert \leq \dsum \frac{% \left\vert b_{k}\right\vert }{\left\vert a_{k}\right\vert }\sup \left\vert a_{k}\right\vert <\infty .$$ We now claim that the condition $\left( 2\right)$ is essentially also necessary for $\left( 1\right)$ to hold true. We must specify what \"essentially\" means.$\medskip$ To this end we make first the following general observation based on the Aronszajn-Gagliardo theorem [@1-AG65]. If $A$ and $B$ are relative interpolation spaces with respect to any Banach couples $\overline{A}$ and $% \overline{B}$ then by enlarging $A$ and diminishing $B$ we may as well assume that $A$ is an interpolation space with respect to $\overline{A}$ and $B$ an interpolation space with respect to $\overline{B}.$ Indeed we may, if necessary, replace $A$ by $Orb_{A}\left( \overline{A}\right) \supseteq A$ and $B$ by $Corb_{B}\left( \overline{B}\right) \subseteq B;Orb_{A}\left( \overline{A}\right)$ and $Corb_{B}\left( \overline{B}\right)$ are then again relative interpolation spaces with respect to $\overline{A}$ and $% \overline{B}.$ Conversly if $Orb_{A}\left( \overline{A}\right)$ and $% Corb_{B}\left( \overline{B}\right)$ are relative interpolation spaces with respect to $\overline{A}$ and $\overline{B}$ so are obviously $A$ and $% B.\medskip$ Returning to the case $\overline{A}=l^{\infty }\left( \overline{w}\right) ,% \overline{B}=l^{1}\left( \overline{w}\right)$ we may therefore from now assume that $A$ and $B$ are not only intermediate spaces but also [ interpolation]{.ul} spaces, with respect to $l^{\infty }\left( \overline{w}% \right)$ and $l^{1}\left( \overline{w}\right)$ respectively. But then we have a rather explicit representation of the norm in $A$ and $B.$ In fact $A$ can be renormed in such a fashion that $$\left\Vert a\right\Vert _{A}=\left\Vert \left\{ K\left( \tau _{n},a\right) \right\} \right\Vert _{A} \tag{3}$$ and $B$ in such a fashion that $$\left\Vert b\right\Vert _{B}=\inf_{b}\left\Vert \left\{ J\left( \tau _{n},b_{n}\right) \right\} \right\Vert \tag{4}$$ * * where the inf is extended over all representations $\widehat{b}=\left\{ b_{n}\right\}$ of $b.$We have further put $\tau _{n}=w_{n}^{0}/w_{n}^{1}.$ We also know that $$a\in l^{\infty }\left( \rho \left( \overline{w}\right) \right) \iff \sup \frac{K\left( \tau _{n},a\right) }{\rho \left( \tau _{n}\right) }<\infty \tag{5}$$ and that $$f\in l^{1}\left( \rho \left( \overline{w}\right) \right) \iff \dsum \frac{% J\left( \tau _{n},b_{n}\right) }{\rho \left( \tau _{n}\right) }<\infty \tag{6}$$ for some representation $\widehat{b}=\left\{ b_{n}\right\}$ on $b.$ For the proof of these facts see the Appendix.$\medskip$ It is now easy to prove [Proposition 2.]{.ul} Let $A$ and $B$ be interpolation spaces with respect to $l^{\infty }\left( \overline{w}\right)$ and $l^{1}\left( \overline{w}\right) .$ Then $\left( 2\right)$ implies $\left( 1\right) .$ [Proof]{.ul} :Let $T:l^{\infty }\left( \overline{w}\right) \rightarrow l^{1}\left( \overline{w}\right)$ and take any $a\in A.$ We want to prove under the hypothesis of $\left( 2\right)$ that $Ta\in B.$ We apply Ovcinnikov's theorem [@23-Ovc] with $\rho \left( t\right) =K\left( t,a\right) .$Then $a\in l^{\infty }\left( \rho \left( \overline{w}\right) \right)$* *by $\left( 5\right)$ so we get $Ta\in l^{1}\left( \rho \left( \overline{w}\right) \right) .$ Therefore by $\left( 6\right)$ $Ta$ has a representation * *$\widehat{b}$ with $\dsum \frac{J\left( \tau _{k},b_{k}\right) }{K\left( \tau _{k},a\right) }<\infty .$ But by $\left( 3\right)$ holds $\left\{ K\left( \tau _{k},a\right) \right\} \in A$ so $% \left( 2\right)$ gives $\left\{ J\left( \tau _{k},b_{k}\right) \right\} \in B.$ Then by $\left( 4\right)$ $Ta\in B.$\# # 2. The vector valued and the continuous cases. Retracts and partial retracts. {#the-vector-valued-and-the-continuous-cases.-retracts-and-partial-retracts. .unnumbered} We next wish to prove a vector valued analogue of Ovchinnikov's theorem [@23-Ovc]. At the same time we will find that in the proof of Ovchinnikov's theorem it suffices to consider geometric progressions only, that is, we can take $\tau _{k}=\lambda ^{k}$ (\"the $\lambda -adic$ case\") where $\lambda$ is a given number $>1.$(In most cases it would be sufficient to take $% \lambda =2;$ the only raison d'être for allowing a general $\lambda$ is that by choosing $\lambda$ close to $1,$ one gets the constants as close to $1$ as one wants to.[^6]) On the other hand in Janson's proof of this result [@16-Jan] an essential ingredient is precisely the use of general (\"sparse\") sequences of weights. (Cf. Sec. [$5$]{.ul} ).$\medskip$ We begin with some general considerations (\"abstract nonsense\") on retracts and partial retracts (cf. [@27-Pe], [@28-Pe]), that perhaps might be of use in other contexts too.$\medskip$ [$2.1$]{.ul}$.$ Let $\overline{A}$ and $\overline{X}$ be any two Banach couples. We say that $\overline{X}$ is a [partial retract]{.ul} of $% \overline{A}$ if for any $x\in \Sigma \overline{X}$ there exists linear operators $\pi :\overline{A}\rightarrow \overline{X}$ (projection) and $% \iota :\overline{X}\rightarrow \overline{A}$ (retraction, section of $\pi )$ such that $\pi \iota x=x.$ If we can choose $\pi$ and $\iota$ independently of $x$ we say that we have a [retract]{.ul} (omitting \"partial\"). That is $\pi \iota =id$ and we have the commutative diagram $$\begin{tabular}{ccc} $\overline{A}$ & & \\ $\downarrow \pi $ & $\nwarrow \iota $ & \\ $\overline{X}$ & $\overset{id}{\rightarrow }$ & $\overline{X}$% \end{tabular}%$$ In the general case we have the commutative diagram $$\begin{tabular}{ccc} $\overline{A}$ & & \\ $\downarrow \pi $ & $\nwarrow \iota $ & \\ $\overline{X}$ & & $\overline{X}$ \\ $\uparrow \widetilde{x}$ & $\nearrow \widetilde{x}$ & \\ $\overline{\mathbb{R}}$ & & \end{tabular}%$$ Here $\overline{\mathbb{R}}=\left( \mathbb{R},\mathbb{R}\right)$ stands for a 1-dimensional (scalar) couple and $\widetilde{x}:c\rightarrow cx$ denotes multiplication by $x.\medskip$ Let thus $\overline{X}$ be a partial retract of $\overline{A}$ and consider another two Banach couples $\overline{B}$ and $\overline{Y}$ such that $% \overline{Y}$ is partial retract of $\overline{B}.$ For $y\in \Sigma \overline{Y}$ denote by $\pi ^{^{\prime }}$ and $\iota ^{^{\prime }}$ the corresponding operators.$\medskip$ The following elementary lemma is fundamental in our discussion, although it will never be used very directly. [Lemma 1]{.ul}. $y<<_{b}x\iff \iota ^{^{\prime }}y<<_{b}\iota x.$ [Proof]{.ul}* :* $1):$ If $y<<_{b}x$ then $y=Tx$ for some $T:% \overline{X}\rightarrow \overline{Y}.$ But then $\iota ^{^{\prime }}y=\iota ^{^{\prime }}T\pi \iota x=S\iota x$ where $S=\iota ^{^{\prime }}T\pi :\overline{A}\rightarrow \overline{B}.$ This gives $\iota ^{^{\prime }}y<<_{b}\iota x.$* * $2):$ Conversely if $\iota ^{^{\prime }}y<<_{b}\iota x$ then $\iota ^{^{\prime }}y=S\iota x$ for some $S:\overline{A}\rightarrow \overline{B}.$ Now we get $y=\pi ^{^{\prime }}\iota ^{^{\prime }}y=\pi ^{^{\prime }}S\iota x=Tx$ where $T=\pi ^{^{\prime }}S\iota :\overline{X}\rightarrow \overline{Y}% .$ This proves $y<<_{b}x.$\# $\medskip$ [Remark]{.ul}* .*The meaning of lemma $1$ is of course the following. The fundamental question of interpolation theory is to decide for given couples $\overline{X}$ and $\overline{Y}$ when the relation $x<<_{b}y$ is true (whatever we mean by \"decide\" and by \"true\"). So the lemma simply says that in the case at hand this question can can be reduced to the analogous question for the couples $\overline{A}$ and $\overline{B}$.$% \medskip$ The following corollary illustrates the usefulness of this result. [Corollary]{.ul}.* *If $\overline{A}$ and $\overline{B}$ are relative Calderon so are $\overline{X}$ and $\overline{Y}$. [Proof]{.ul}[ :]{.roman}We have to prove $y<<_{K}x$ implies $y<<_{b}x.$ But trivially $\iota ^{^{\prime }}y<<_{K}y$ and moreover, since $\pi \iota x=x,x<<_{K}\iota x.$So $\iota ^{^{\prime }}y<<_{K}\iota x.$ Since $\overline{% A}$ and $\overline{B}$ are relative Calderon this gives $\iota ^{^{\prime }}y<<_{b}\iota x.$Lemma $1$ allows us to conclude that indeed $y<<_{b}x.$\#$% \medskip$ We go on assuming that $\overline{X}$ is a partial retract of $\overline{A}$ and retain the notation $\pi ,\iota$ for the operators corresponding to $x$ in $\Sigma \overline{X}.$ Then we have [Lemma 2]{.ul}*.* Let $X$ and $A$ be relative interpolation spaces with respect to $\overline{X}$ and $\overline{A}.$ Then $Orb_{X}\subseteq Orb_{A}$ [Proof]{.ul}: Consider any Banach couple $\overline{U}$ and an element $% u\in Orb_{X}\left( \overline{U}\right) .$We have to show that $u\in Orb_{A}\left( \overline{U}\right)$ It is sufficient to consider the case when $u$ is of the form $u=Tx$ with $x\in X$ and $T:\overline{X}\rightarrow \overline{U}.$ Then $u=T\pi \iota x=Sa$ with $S=T\pi :\overline{A}% \rightarrow \overline{U}$ and $a=\iota x\in A$ (since $X$ and $A$ are relative interpolation spaces with respect to $\overline{X}$ and $\overline{A% }$ and $x\in X)$. This gives $u\in Orb_{A}\left( \overline{U}\right) .$\#$% \medskip$ The result dual to lemma 2 is [Lemma 2$^{^{\prime }}$]{.ul}. In the same hypothesis $% Corb_{A}\subseteq Corb_{X}$ [Proof]{.ul}:* *Let again $\overline{U}$ be any Banach couple and let $u\in Corb_{A}\left( \overline{U}\right) .$We wish to show that $u\in Corb_{X}\left( \overline{U}\right) .$ Since $u\in Corb_{A}\left( \overline{U}% \right)$ we have $Su\in A$ for any $S:\overline{U}\rightarrow \overline{A}.$ Consider now an arbitrary operator $T:\overline{U}\rightarrow \overline{X.}$ We have to show that $x=Tu\in X.$ Let $\iota ,\pi$ correspond to this $x$ and set $S=\iota T.$ Then clearly $S:\overline{U}\rightarrow \overline{A}$ so $Su\in A.$ But then $\pi Su\in X$, since $A$ and $X$ are relative interpolation spaces with respect to $\overline{A}$ and $\overline{X}.$ Since $Tu=\pi Su$ this establishes our claim that $u\in Corb_{X}\left( \overline{U}\right) ,$\#$\medskip$ From lemma 2 and lemma $2^{^{\prime }}$we obtain the following [Corollary]{.ul}*.* Assume that $\overline{A}$ too is retract of $% \overline{X}$ and that $X$ and $A$ are relative interpolation spaces with respect to $\overline{X}$ and $\overline{A}.$ Then $% Orb_{X}=Orb_{A},Corb_{X}=Corb_{A}.$ Thus, informally speaking, couples in the same partial retract class give the same orbits and the same coorbits.$\medskip$ [$2.2$]{.ul}$.$We now return to the special case that really interest us. We consider the Banach couple[^7] $$l^{p}\left( \overline{w}D\right) =\left( l^{p}\left( w^{0}D\right) ,l^{p}\left( w^{1}D\right) \right)$$ where $$1\leq p\leq \infty ,$$ $$\overline{w}=\left( w^{0},w^{1}\right) =\left( \left\{ w_{n}^{0}\right\} ,\left\{ w_{n}^{1}\right\} \right)$$ is a couple of weighted sequences (the index n runs through some denumerable index set $I$), $$D=\left\{ D_{n}\right\} \text{ is a sequence of Banach spaces }D_{n}$$ $$l^{p}\left( w^{i}D\right) ,\left( i=0,1\right)$$ denotes the space of sequences $x=\left\{ x_{n}\right\}$ such that $% x_{n}\in D_{n}$ for each $n\in I$ and $$\left( \dsum_{n}\left( w_{n}^{i}\left\Vert x_{n}\right\Vert _{D_{n}}\right) ^{p}\right) ^{1/p}<\infty$$ (with the usual interpretation if $p=\infty ).$ If $D_{n}=\mathbb{R}$ for each $n$ we write just $l^{p}\left( \overline{w}% \right) .$ In particular we put $$\overline{l}_{\lambda }^{p}=\left( l^{p}\left( \left\{ 1\right\} \right) ,l^{p}\left( \left\{ \lambda ^{-k}\right\} \right) \right) =\left( l^{p},l^{p}\left( \lambda ^{-k}\right) \right)$$ where $\lambda$ is a given number $>1;$ in this case $k$ is assumed to run over $\mathbb{Z}$ , the set of integers. (The $\overline{}$ in the notation is to remind us that this is couple.)$\medskip$ [Proposition 1]{.ul}* .* $l^{p}\left( \overline{w}D\right)$ is a partial retract of $\overline{l}_{\lambda }^{p}.$ [Proof]{.ul}: For any $k\in \mathbb{Z}$ set $$e_{k}=\left\{ n:w_{n}^{0}\leq \lambda ^{k}w_{n}^{1}<\lambda w_{n}^{0}\right\} . \tag{1}$$ * *Clearly $e_{k}\cap e_{l}=\emptyset$ $\left( k\neq l\right)$ and $% \cup _{k}e_{k}=I$ so we have a partition of $I$ (our index set). Let $x\in \Sigma l^{p}\left( \overline{w}D\right) .$ We wish to exhibit the corresponding operators $\pi :\overline{l}_{\lambda }^{p}\rightarrow l^{p}\left( \overline{w}D\right)$ and $\iota :l^{p}\left( \overline{w}% D\right) \rightarrow \overline{l}_{\lambda }^{p}$ with $\pi \iota x=x.$ Begin with $\iota .$ By the Hahn-Banach theorem we can find linear functionals. $\alpha _{k}$ $\left( k\in \mathbb{Z}\right)$ on $\Sigma l^{p}\left( \overline{w}D\right)$ such that $$\alpha _{k}\left( x\right) =\left( \dsum_{n\in e_{k}}\left( w_{n}^{0}\left\Vert x_{n}\right\Vert _{D_{n}}\right) ^{p}\right) ^{1/p}=s_{k} \tag{2}$$ and $$\left\vert \alpha _{k}\left( y\right) \right\vert \leq \left( \dsum_{n\in e_{k}}\left( w_{n}^{0}\left\Vert y_{n}\right\Vert _{D_{n}}\right) ^{p}\right) ^{1/p} \tag{3}$$ for every $y=\left\{ y_{n}\right\} \in \Sigma l^{p}\left( \overline{w}% D\right) .$Then by the first inequality defining $e_{k}$ (see $\left( 1\right) )$ holds also $$\lambda ^{k}\left\vert \alpha _{k}\left( y\right) \right\vert \leq \left( \dsum_{n\in e_{k}}\left( w_{n}^{1}\left\Vert y_{n}\right\Vert _{D_{n}}\right) ^{p}\right) ^{1/p}. \tag{4}$$ (Note that the expression to the right in $\left( 3\right)$ and $\left( 4\right)$ is just $\left\Vert \chi _{e_{k}}y\right\Vert _{l^{p}\left( w^{i}D\right) }$ $\left( i=0,1\right)$ where $\chi _{e_{k}}$ denotes the characteristic function of the set $e_{k}.).$ We set now $\iota \left( y\right) =\left\{ \alpha _{k}\left( y\right) \right\} .$ Then $\iota :l^{p}\left( \overline{w}D\right) \rightarrow \overline{l}_{\lambda }^{p}$. Indeed adding up $\left( 3\right)$ and $\left( 4\right)$ we get $% \left\Vert {}\right\Vert \leq 1.$ Also $\left( 2\right)$ gives $\iota \left( x\right) =s=\left\{ s_{k}\right\} .$ Next let us construct $\pi .$ If $n\in e_{k}$ put $\phi _{n}=k.$ (Thus $% n\rightarrow \phi _{n}$ is the \"inverse\" of the set valued function $% k\rightarrow e_{k}.).$For $a=\left\{ a_{k}\right\} \in \Sigma \overline{l}% _{\lambda }^{p}$ set $\pi \left( a\right) =\left\{ a_{\phi _{n}}x_{n}/s_{\phi _{n}}\right\} .$ Then by the second inequality defining $% e_{k}$ (see $\left( 1\right) )$ we get $\pi :\overline{l}_{\lambda }^{p}\rightarrow l^{p}\left( \overline{w}D\right) ,\left\Vert \pi \right\Vert \leq \lambda .$ Also clearly $\pi \left( a\right) =s,$ so, since $\iota \left( x\right) =s,$we finally obtain $\pi \iota x=x.$\#$% \medskip$ In the other direction, we prove [Proposition 2*.*]{.ul}* *Assume that for each $k\in \mathbb{% Z}$ we can find $n\in e_{k}$ (where $e_{k}$ as in the proof of prop. 1, see $% \left( 2\right) )$ such that $D_{n}\neq 0.$Then $\overline{l}_{\lambda }^{p}$ is a retract (not only a partial one) of $l^{p}\left( \overline{w}D\right) .$ [Proof]{.ul}: Let $n=n\left( k\right)$ be the index corresponding to $% k$ in the hypothesis of our proposition. Pick up an element $\theta _{k}\in D_{n\left( k\right) }$ with $\left\Vert \theta _{k}\right\Vert =1.$ We have to construct operators $\iota :\overline{l}_{\lambda }^{p}\rightarrow l^{p}\left( \overline{w}D\right)$ and $\pi :l^{p}\left( \overline{w}% D\right) \rightarrow \overline{l}_{\lambda }^{p}$ such that $\pi \iota =id.$ Let $\widehat{\theta _{k}}=\left( 0,...,0,\theta _{k},0,,,\right)$ be the sequence whose $n\left( k\right)$ th entry is $\theta _{k}$, all other entries being zero. Then for $a=\left\{ a_{k}\right\} \in \Sigma \overline{l}% _{\lambda }^{p}$ we set $\iota \left( a\right) =\dsum a_{k}\widehat{\theta _{k}}/w_{n\left( k\right) }^{0}.$ The second inequality in $\left( 1\right)$ readily gives $\left\Vert \iota \right\Vert \leq \lambda .$ Conversly select by the Hahn-Banach theorem linear functionals. on $% D_{n\left( k\right) }$ such that $\eta _{k}\left( \theta _{k}\right) =1,\left\Vert \eta _{k}\right\Vert =1.$Set $\pi \left( x\right) =\left\{ w_{n\left( k\right) }^{0}\eta _{k}\left( x_{n\left( k\right) }\right) \right\}$ for $x=\left\{ x_{n}\right\} \in \Sigma l^{p}\left( \overline{w}% D\right) .$Now the first inequality in $\left( 1\right)$ gives $\left\Vert \pi \right\Vert \leq \lambda .$ Since clearly $\pi \iota \left( x\right) =x$ for any $x,$ the proof is complete. \#$\medskip$ [2.3]{.ul}* .* After these lengthy preparations we are now ready to prove the main result of this section.$\medskip$ We consider two couples $l^{\infty }\left( \overline{w}D\right)$ and $% l^{1}\left( \overline{z}E\right)$ where $\overline{z}$ and $E$ have a similar meaning as $\overline{w}$ and $D.$* * [Theorem]{.ul}.* *$T:l^{\infty }\left( \overline{w}D\right) \rightarrow l^{1}\left( \overline{z}E\right) \Longrightarrow T:l^{\infty }\left( \rho \left( \overline{w}\right) D\right) \rightarrow l^{1}\left( \rho \left( \overline{z}\right) E\right)$ for any $\rho \in \mathcal{P}$. [Proof]{.ul}* :* By [$2.2$]{.ul}$,$ prop. 1, $l^{\infty }\left( \overline{w}D\right)$ is a partial retract of $\overline{l}% _{\lambda }^{\infty }$ and $l^{1}\left( \overline{z}E\right)$ one of $% \overline{l}_{\lambda }^{1}.$Consider any element of $l^{\infty }\left( \rho \left( \overline{w}\right) D\right) .$We have to show that $y=Tx\in l^{1}\left( \rho \left( \overline{z}\right) E\right) .$Let $\pi ,\iota$ and $\pi ^{^{\prime }},\iota ^{^{\prime }}$be the operators corresponding to $x$ and $y$ respectively. (cf. the proof of [$2.1$]{.ul}$,$lemma $1.)$ As in that proof we set $S=\iota ^{^{\prime }}T\pi .$ Thus we have the following (only \"partially commutative diagram) $$\begin{tabular}{ccc} $l^{\infty }\left( \overline{w}D\right) $ & $\overset{T}{\rightarrow }$ & $% l^{1}\left( \overline{z}E\right) $ \\ $\iota \downarrow \uparrow \pi $ & & $\iota ^{^{\prime }}\downarrow \uparrow \pi ^{^{\prime }}$ \\ $\overline{l}_{\lambda }^{\infty }$ & $\overset{S}{\rightarrow }$ & $% \overline{l}_{\lambda }^{1}$% \end{tabular}%$$ We take for granted that $l^{\infty }\left( \frac{1}{\rho \left( \lambda ^{l}\right) }\right)$ and $l^{1}\left( \frac{1}{\rho \left( \lambda ^{k}\right) }\right)$ are relative interpolation spaces with respect to $% \overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1};$ this is of course just the $\lambda -$adic special case of Ovchinnikov's theorem [@23-Ovc]; see introduction. It is clear that $\iota x\in l^{\infty }\left( 1/\rho \left( \lambda ^{-k}\right) \right) .$ By the result just quoted thus $S\iota x\in l^{1}\left( 1/\rho \left( \lambda ^{k}\right) \right) .$It follows that $\pi ^{^{\prime }}S\iota x\in l^{1}\left( \rho \left( \overline{z}\right) E\right) .$But $\pi ^{^{\prime }}S\iota x=\pi ^{^{\prime }}\iota ^{^{\prime }}T\pi \iota x=Tx=y.$ This completes the proof. \#$\medskip$ [Remark]{.ul}* *For a direct proof of this theorem and the following generalization, by the method of Janson [@16-Jan], see sec. 5.$% \medskip$ [2.4]{.ul}.* *We now extend the previous results $\left( \underline{2.2}-\underline{2.3}\right)$ to the continuous (or rather the measurable) case.$\medskip$ Thus let $W$ be any measure space equipped with a measure $\mu .$ We are interested in couples $$L^{p}\left( \overline{w}D\right) =\left( L^{p}\left( w^{0}D\right) ,L^{p}\left( w^{1}D\right) \right)$$ As before $1\leq p\leq \infty .$Now $\overline{w}=\left( w^{0},w^{1}\right)$ is a couple of positive measurable functions on $W$ (\"weight functions\") and $D$ a continuous Banach bundle over $W.$ By the latter we intend a space $D$ together with a projection $\pi :D\rightarrow W$ such that each fiber $% D_{\omega }=\pi ^{-1}\left( \omega \right)$ ($\omega \in W)$ is a Banach space. In addition $D$ should be equipped with a \"continuity structure\", i.e. there is a given vector space $\Gamma$ of sections of $D$ (called \"principal sections\") such that $\left( i\right) :$ for each $x\in \Gamma$ the map $W\rightarrow \mathbb{R}% ,\omega \rightarrow \left\Vert x\left( \omega \right) \right\Vert _{D_{\omega }}$ is continuous, $\left( ii\right) :$ for each $\omega \in W$ the evaluation map $\Gamma \rightarrow D_{\omega },x\rightarrow x\left( \omega \right)$ has a dense image in $D_{\omega }.\medskip$ Given this a continuous Banach bundle one can develop Bourbaki style integration theory pretty much as in the special case of a trivial bundle for which case we get the integral. In particular one can define the space of sections $L^{p}\left( w^{i}D\right)$ $\left( i=0,1\right)$ by taking the completion of $\Gamma$ in the norm $$\left\Vert x\right\Vert =\left( \tint\nolimits_{W}\left( w^{i}\left( \omega \right) \left\Vert x\left( \omega \right) \right\Vert _{D_{\omega }}\right) ^{p}d\mu \left( \omega \right) \right) ^{1/p}.$$ [Remark]{.ul}.* *(historical) The notion of \"continuous Banach bundle\" (our word) seems to be due to Godement [@12-God], [@13-God], the French expression being \"champs continu d'espaces de Banach\". Godement develops also the corresponding integration theory.[^8] (See Dixmier [@7-Dix] where ample references can be found.). If all the fibers are Hilbert spaces and if $p=2$ one gets what is known as a \"direct integral of Hilbert spaces\". (see Dixmier [@8-Dix].). If $D$ is a trivial bundle, i.e. all spaces $D_{\omega }$ coincide we take for $\Gamma$ the \"constant\" sections and we are back in the classical case of vector valued integration.$\medskip$ Now it is easy to carry over [$2.2$]{.ul}$\emph{,}$ prop. 1. The definition of $e_{k}$ now takes the form (cf. [$\emph{2.2}$]{.ul}, $% \left( 1\right) )$ $$e_{k}=\left\{ \omega :w^{0}\left( \omega \right) \leq \lambda ^{k}w^{1}\left( \omega \right) <\lambda w^{0}\left( \omega \right) \right\}$$ In this way we get a partition of $W$ into measurable subsets. We define linear functionals. $\alpha _{k}$ such that $\alpha _{k}\left( x\right) =s$ with now $s_{k}=\left\Vert \chi _{e_{k}}x\right\Vert _{L^{p}\left( w^{0}D\right) }.$The rest of the proof is pretty much the same. At any rate [we find that]{.ul} $L^{p}\left( \overline{w}D\right)$ [always is a partial retract of]{.ul} $\overline{l}_{\lambda }^{p}.$ In the same we can generalize $\underleftarrow{2.2},$ prop. $2,$ provided one puts the proper restriction on $D$ generalizing the one of that proposition. We will not give any details, because we are not going to use this result anyhow.$\medskip$ Next let us consider pairs $L^{\infty }\left( \overline{w}D\right)$ and $% L^{1}\left( \overline{z}E\right)$ where $\overline{z}$ and $E$ has a similar meaning as $\overline{w}$ and $D;\overline{z}=\left( z_{0},z_{1}\right)$ is thus a pair of weight functions over a measure space $Z,$ with measure $\nu ,$say, and $E$ is continuous Banach bundle over that space. (Notice that since $p=\infty$ in the former case we get the same couple $L^{\infty }\left( \overline{w}D\right)$ if we replace the measure on $W,$ say, $\mu$ by an equivalent measure, not so in the latter case.) We put $\rho \left( \overline{w}\right) =w^{0}/\rho \left( w^{1}/w^{0}\right)$ $\left( \rho \in \mathcal{P}\right) ;$ this is again a weight function on $% W.$We define $\rho \left( \overline{z}\right)$ is a similar manner. Using the generalization just outline (*underlined*) of [$2.2$]{.ul}$,$ prop. 1 (the cases $p=1$ and $p=\infty$ respectively), we then get the following analogue of [$2.2$]{.ul}, theorem.$\medskip$ [Theorem]{.ul}.* * $T:L^{\infty }\left( \overline{w}D\right) \rightarrow L^{1}\left( \overline{z}E\right) \Longrightarrow T:L^{\infty }\left( \rho \left( \overline{w}\right) D\right) \rightarrow L^{1}\left( \rho \left( \overline{z}\right) E\right)$ for any $\rho \in \mathcal{P}$ .$% \medskip$ [$2.5$]{.ul}$.$ Let us indicate an application of the previous theorem . Consider the [Besov Couple]{.ul} $$B_{p}^{\overline{s}r}\left( \mathbb{R}^{n}\right) =\left( B_{p}^{s_{0}r}\left( \mathbb{R}^{n}\right) ,B_{p}^{s_{1}r}\left( \mathbb{R}% ^{n}\right) \right)$$ with $\overline{s}=\left( s_{0},s_{1}\right)$ and $r,p\in \left[ 1,\infty % \right] .$ It is well-known (see e.g. [@3-BeL0 chap. 6] or [@29-Pe chap. 5]) that $B_{p}^{\overline{s}r}\left( \mathbb{R}^{n}\right)$ is a retract of $l^{p}\left( 2^{k\overline{s}}L_{p}\right)$ where $l^{p}\left( 2^{k\overline{s}}\right)$ denotes the couple of weighted sequences $% l^{p}\left( \left\{ 2^{ks_{0}}\right\} ,\left\{ 2^{ks_{1}}\right\} \right)$ and $L_{p}=L_{p}\left[ 0,1\right] .$Using this fact [$2.4$]{.ul}$,$ theorem thus yields as a corollary e.g. the following result.$\medskip$ [Theorem]{.ul}.* *$T:B_{p}^{\overline{s}\infty }\left( \mathbb{R}% ^{n}\right) \rightarrow B_{q}^{\overline{t}1}\left( \mathbb{R}^{m}\right) \Longrightarrow T:B_{p}^{s\infty }\left( \mathbb{R}^{n}\right) \rightarrow B_{q}^{t1}\left( \mathbb{R}^{m}\right)$ provided $s=\left( 1-\theta \right) s_{0}+\theta s_{1},t=\left( 1-\theta \right) t_{0}+\theta t_{1},\theta \in \left( 0,1\right) .$ Here $\overline{s}=\left( s_{0},s_{1}\right) ,\overline{% t}=\left( t_{0},t_{1}\right) ,p,q\in \left[ 1,\infty \right]$ are arbitrary. $\medskip$ [Example]{.ul}.* *Taking $m=n$ this result is formally applicable to the convolution operator $Tf=a\ast f$ where $a\in B_{r}^{\gamma r}$ where $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1$ and $t_{i}=s_{i}+\gamma$ $\left( i=0,1\right) .$The conclusion in this case follows of course also directly, without any recourse to such a sophisticated device as Ovchinnikov's theorem [@23-Ovc]. [This is perhaps symptomatic]{.ul}*.* At any rate we - this is of course no proof - do not know of any non-trivial application of this result to the usual operators of Analysis. # 3. Condition $\left( O\right) .$Type $\left( l\right)$ and type $% \left( u\right) .$ {#condition-left-oright-.type-left-lright-and-type-left-uright-. .unnumbered} Now we wish to pass to a more abstract situation.$\medskip$ [3.1.]{.ul} Let us first return to the primitive situation of sec. [1]{.ul}. (We have not been able to extend the results of that sec. to the vector valued case of sec. [2]{.ul}.). In view of the results of sec. [2]{.ul}. it is no great case of generality to restrict oneself to the $\lambda -$adic case, i.e. we consider (in the notation of that sec.) the couples $\overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1}.$ If $A$ and $B$ are relative interpolation spaces with respect to $% \overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1}$ then we must have the inclusion $G^{A}\subseteq H^{B}.$ If in addition $A$ is an interpolation space with respect to $\overline{l}_{\lambda }^{\infty }$ and $% B$ one with respect to $\overline{l}_{\lambda }^{1}$ then by Janson [@16-Jan] we get $G^{A}\left( \overline{l}_{\lambda }^{\infty }\right) =K^{A}\left( \overline{l}_{\lambda }^{\infty }\right) =A,H^{B}\left( \overline{l}_{\lambda }^{1}\right) =J^{B}\left( \overline{l}_{\lambda }^{1}\right) =B.$ (For the definition of the functors $% G^{A},K^{A},H^{B},J^{B}$ sec. [$\frac{1}{2}$]{.ul}. ). This leads to the following question: Given any two Banach couples $\overline{X}$ and $% \overline{Y}$ under which conditions can we assert that $T:\overline{X}% \rightarrow \overline{Y}$ implies $T:K^{A}\left( \overline{X}\right) \rightarrow J^{B}\left( \overline{Y}\right)$ for any $A$ and $B,$i.e. that $% K^{A}\left( \overline{X}\right)$ and $J^{B}\left( \overline{Y}\right)$ are relative interpolation spaces with respect to $\overline{X}$ and $\overline{Y% }.$ (That $T:\overline{X}\rightarrow \overline{Y}$ implies $T:G^{A}\left( \overline{X}\right) \rightarrow H^{B}\left( \overline{Y}\right)$ is by what we just said obvious.). We then say that $\overline{X}$ [and]{.ul} $% \overline{Y}$ [satisfy condition ]{.ul} $\left( O\right) .$ $\medskip$ We have thus already met (see sec. [2]{.ul}) several instances of couples $\overline{X}$ and $\overline{Y}$ satisfying condition $\left( O\right) ,\overline{X}=\overline{l}^{\infty },\overline{Y}=\overline{l}^{1}$ being the primitive case, more generally $\overline{X}=l^{\infty }\left( \overline{w},D\right) ,\overline{Y}=l^{1}\left( \overline{z},E\right)$ or $% \overline{X}=L^{\infty }\left( \overline{w},D\right) ,\overline{Y}% =L^{1}\left( \overline{z},E\right) ,$ likewise $\overline{X}=B_{p}^{% \overline{s},\infty }\left( \mathbb{R}^{n}\right) ,\overline{Y}=B_{q}^{% \overline{t},1}\left( \mathbb{R}^{m}\right) .$ As is easy to convince oneself various combinations of these couples will also do.$\medskip$ Let us therefore coin the following additional terminology. We say that $% \overline{X}$ [is of type ]{.ul} $\left( l\right)$ if $K^{A}\left( \overline{X}\right) =G^{A}\left( \overline{X}\right)$ for any interpolation space $A$ with respect to $\overline{l}_{\lambda }^{\infty }$ and that $% \overline{Y}$ [is of type ]{.ul} $\left( u\right)$ if $H^{B}\left( \overline{Y}\right) =J^{B}\left( \overline{Y}\right)$ for any interpolation space $B$ with respect to $\overline{l}_{\lambda }^{1}.$ E.g., $\overline{l}% _{\lambda }^{r},l^{r}\left( \overline{w},D\right) ,L^{r}\left( \overline{w}% ,D\right) ,B_{p}^{\overline{s},q}\left( \mathbb{R}^{n}\right)$ are of type $% \left( l\right)$ if $r=\infty$ and of type $\left( u\right)$ if $r=1.$ (The letters l and u are of course chosen to honour Ovchinnikov.)$\medskip$ The following result is almost trivial. (It has of course been implicit the above discussion.)$\medskip$ [Proposition. ]{.ul} Let $\overline{X}$ be of type $\left( l\right)$ and $\overline{Y}$ of type $\left( u\right) .$Then $\overline{X}$ and $% \overline{Y}$ satisfy condition $\left( O\right) .$ [Proof ]{.ul}: Let $T:\overline{X}\rightarrow \overline{Y}.$Then $% T:G^{A}\left( \overline{X}\right) \rightarrow H^{B}\left( \overline{Y}% \right)$ since $G^{A}\subseteq H^{B}$ whenever $A$ and $B$ are relative interpolation spaces with respect to $\overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1}.$ But $K^{A}\left( \overline{X}\right) =G^{A}\left( \overline{X}\right) ,H^{B}\left( \overline{Y}\right) =J^{B}\left( \overline{Y}\right)$ by our assumption on $\overline{X}$ and $% \overline{Y}.$ Se we get indeed $T:K^{A}\left( \overline{X}\right) \rightarrow J^{B}\left( \overline{Y}\right) .$\#$\medskip$ [Remark]{.ul}. Indeed we know of no single non-trivial instance of two couples $\overline{X}$ and $\overline{Y}$ satisfying condition $\left( O\right)$ where $\overline{X}$ is not of type $\left( l\right)$ and $% \overline{Y}$ not of type $\left( u\right) .\medskip$ [3.2]{.ul}. In proving that two given Banach couples satisfy condition $% \left( O\right)$ it is not necessary to consider the most general interpolation spaces with respect to $\overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1}.$ Indeed the following holds true.$\medskip$ [Proposition]{.ul}. Let $\overline{X}$ and $\overline{Y}$ be any two Banach couples . The following conditions are equivalent. $\left( i\right) :$ $\overline{X}$ and $\overline{Y}$ satisfy condition $% \left( O\right) .$ $\left( ii\right) :T:\overline{X}\rightarrow \overline{Y}\Longrightarrow T:% \overline{X}_{\rho \infty :K}\rightarrow \overline{Y}_{\rho 1:J}$ for any $% \rho \in \mathcal{P}$. $\left( iii\right) :$ For any $y\in \Sigma _{0}\overline{Y}$ and $x\in \Sigma \overline{X}$ holds $y<<_{b}x\Longrightarrow y<<_{J/K}x.$ [Proof]{.ul}. The implications $\left( i\right) \Longrightarrow \left( ii\right) \Longrightarrow \left( iii\right)$ are trivial. There remains the implication $\left( iii\right) \Longrightarrow \left( i\right) ,$ which is proved just by adopting the proof of [1]{.ul}, prop. 2. Indeed assume that $x\in K^{A}\left( \overline{X}\right)$ i.e. $\left\{ K\left( \lambda ^{k},x\right) \right\} \in A$ and let there be given a bounded linear operator $T:\overline{X}\rightarrow \overline{Y}.$ We must show that $% y=Tx\in J^{B}\left( \overline{Y}\right) .$ (Here $A$ and $B$ are relative interpolation spaces with respect to $\overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1}.).$ Since $y<<_{b}x$ the condition $\left( iii\right)$ gives us a representation $\widehat{y}$ of $y$ such that $% \Sigma J\left( \lambda ^{k},y_{k}\right) /K\left( \lambda ^{k},x\right) <\infty .$ Now, by our assumption on $A$ and $B$ and by $\underline{1},$ prop. [1]{.ul}, condition 1, $\left( 2\right)$ must be true. Therefore we conclude $\left\{ J\left( \lambda ^{k},y_{k}\right) \right\} \in B.$ Hence $y\in J^{B}\left( \overline{Y}\right) .$\#$\medskip$ [Corollary]{.ul}. If $\overline{X}$ and $\overline{Y}$ satisfy condition $\left( O\right)$ then $\overline{Y}$ must be regular $\left( \Sigma _{0}\overline{Y}=\Sigma \overline{Y}\right) .\medskip$ [3.3]{.ul}. Next we give useful criteria for a given couple to be type $% \left( l\right)$ or of type $\left( u\right) .$ We begin with the following. $\medskip$ [Proposition 1]{.ul}. Let $\overline{X}$ and $\overline{X}^{\left( 1\right) }$ be any two Banach couples. Assume that for each $x\in \Sigma \overline{X}$ there exists an element $x^{\left( 1\right) }\in \Sigma \overline{X}^{\left( 1\right) }$ and a linear operator $\pi :\overline{X}% ^{\left( 1\right) }\rightarrow \overline{X}$ such that $x=\pi x^{\left( 1\right) }$ and $x^{\left( 1\right) }<<_{K}x.$ If $\overline{X}^{\left( 1\right) }$ is of type $\left( l\right)$ so is $\overline{X}.$ [Proof]{.ul}. It is convenient to picture the situation diagrammatically $$\begin{tabular}{cc} & $\overline{X}$ \\ $\pi $ & $\uparrow $ \\ & $\overline{X}^{\left( 1\right) }$% \end{tabular}% ,% \begin{tabular}{cc} $x$ & \\ $\uparrow $ & $<<_{K}$ \\ $x^{\left( 1\right) }$ & \end{tabular}%$$ Assuming that $x\in K^{A}\left( \overline{X}\right) ,A$ an interpolation space with respect to $\overline{l}_{\lambda }^{\infty },$ we have to show that $x\in G^{A}\left( \overline{X}\right) .$ Now $\left\{ K\left( \lambda ^{j},x\right) \right\} \in A,$ since $x\in K^{A}\left( \overline{X}\right) ,$ so $x^{\left( 1\right) }<<_{K}x$ gives $\left\{ K\left( \lambda ^{j},x^{\left( 1\right) }\right) \right\} \in A.$ Thus $x^{\left( 1\right) }$ is a linear combinations of elements of the form $Ta$ where $a\in A$ and $T:% \overline{l}_{\lambda }^{\infty }\rightarrow \overline{X}^{\left( 1\right) }.$ Then $x$ is linear combination of elements $T^{^{\prime }}a$ where $% a\in A$ and $T^{^{\prime }}=\pi T:\overline{l}_{\lambda }^{\infty }\rightarrow \overline{X}.$ This proves $x\in G^{A}\left( \overline{X}% \right) .$ Thus $K^{A}\left( \overline{X}\right) =G^{A}\left( \overline{X}% \right)$ (for any $A)$ and $\overline{X}$ is effectively of type $\left( l\right) .$\#$\medskip$ Next we prove the dual result. [Proposition 2]{.ul}. Let $\overline{Y}$ and $\overline{Y}^{\left( 1\right) }$ be any two Banach couples. Assume that for each $y\in \Sigma _{0}% \overline{Y}$ and a linear operator $\iota :\overline{Y}\rightarrow \overline{Y}^{\left( 1\right) }$ such that $y^{\left( 1\right) }=\iota y$ and $y<<_{J}y^{\left( 1\right) }.$ If $\overline{Y}^{\left( 1\right) }$ is of type $\left( u\right)$ then so is $\overline{Y}$. [Proof]{.ul}. The proof is parallel to the one of prop. 1, is possibly still simpler. The relevant diagram is now $$\begin{tabular}{cc} & $\overline{Y}$ \\ $\iota $ & $\downarrow $ \\ & $\overline{Y}^{\left( 1\right) }$% \end{tabular}% ,% \begin{tabular}{cc} $y$ & \\ $\downarrow $ & $<<_{J}$ \\ $y^{\left( 1\right) }$ & \end{tabular}%$$ Let $y\in H^{B}\left( \overline{Y}\right) ,$ $B$ an interpolation space with respect to $\overline{l}_{\lambda }^{1}.$ We must verify that $y\in J^{B}\left( \overline{Y}\right) .$ Since $y\in H^{B}\left( \overline{Y}% \right)$ for any $T:\overline{Y}\rightarrow \overline{l}_{\lambda }^{1}$ we have $Ty\in B.$But then in particular $T^{1}\iota y\in B$ for any $T^{1}:% \overline{Y}^{\left( 1\right) }\rightarrow \overline{l}_{\lambda }^{1},$ that is $T^{1}y^{\left( 1\right) }\in B.$ Thus, by the same token as above, we conclude $y^{\left( 1\right) }\in H^{B}\left( \overline{Y}^{\left( 1\right) }\right)$ or, since $H^{B}\left( \overline{Y}^{\left( 1\right) }\right) =J^{B}\left( \overline{Y}^{\left( 1\right) }\right)$ by hypothesis, $y^{\left( 1\right) }\in J^{B}\left( \overline{Y}^{\left( 1\right) }\right) .$ Since $y<<_{J}y^{\left( 1\right) }$ this gives $y\in J^{B}\left( \overline{Y}\right) .$ We have proved that $H^{B}\left( \overline{Y}\right) =J^{B}\left( \overline{Y}\right)$ (for any $B)$ and $% \overline{Y}$ is of type $\left( u\right) .$\#$\medskip$ [Remark]{.ul}. In the somewhat obscure terminology of [@27-Pe] the hypothesis of prop. 2 says that $\overline{Y}$ is a lb-pseudoretract of $% \overline{Y}^{\left( 1\right) }.$ Similarly the condition of prop. 1 means that $\overline{X}$ is a lb-pseudoretract of $\overline{X}^{\left( 1\right) }.$ In sec. [2]{.ul} of the present paper we further used \"partial retract\" in place for lb-pseudoretract. (b for \"bounded\", l for \"linear\".)$% \medskip$ In praxis the relation $<<_{J}$ is difficult to verify (except in trivial cases). Luckily we can in the special case of interest for us (cf. infra [3.4]{.ul}) substitute it for $<<_{K}.$Namely there holds the following. [Corollary]{.ul}. Assume that $\overline{Y}$ satisfies the \"strong form of the fundamental lemma\" (see [@6-CwPe], p. 33, condition $\left( 3\right)$). Then we can make the same conclusion as in prop. 2 also if we only assume $y<<_{K}y^{\left( 1\right) }$ (not necessarily $y<<_{J}y^{\left( 1\right) }).$ $\medskip$ Before proving this result we make some clarifications. [Remark]{.ul}. Let $\overline{A}$ be any Banach couple. The \"fundamental lemma\" (see e.g. [@3-BeL0], p. 45) says that any $a\in \Sigma _{0}\overline{A}$ has a representation $\widehat{a}=\left\{ a_{k}\right\} _{k\in \mathbb{Z}}$ such that $J\left( \lambda ^{k},a_{k}\right) \leq CK\left( \lambda ^{k},a\right)$ for all $k\in \mathbb{Z}$ where $C$ is a constant depending in $\lambda$ only. The \"strong form of the fundamental lemma\" (see loc. cit.) says that one can choose $\widehat{a}$ such that $\Sigma _{j}\min \left( 1,\frac{\lambda ^{k}}{% \lambda ^{j}}\right) J\left( \lambda ^{j},a_{j}\right) \leq C^{^{\prime }}K\left( \lambda ^{k},a\right)$ (or in brief $\Omega \left\{ J\left( \lambda ^{j},a_{j}\right) \right\} \leq C^{^{\prime }}K\left( \lambda ^{k},a\right) ;\Omega$ is the Calderon operator, see the appendix) for all $% k\in \mathbb{Z}$ where $C^{^{\prime }}$ now depends on $\overline{A}$ too. What couples $\overline{A}$ admit this strong form of the fundamental lemma is not quite clear yet (cf. [@3-BeL0]).$\medskip$ After this digression we proceed to the [Proof]{.ul} (of the corollary of prop. 2): Assume that $% y<<_{K}y^{\left( 1\right) },y^{\left( 1\right) }\in J^{B}\left( \overline{Y}% ^{\left( 1\right) }\right) .$We have to prove that $y\in J^{B}\left( \overline{Y}\right) .$ Let the norm in $B$ be given by $$\left\Vert b\right\Vert _{B}\approx \Phi \left( \left\{ K\left( \lambda ^{k},b\right) \right\} \right) \approx \inf_{\widehat{b}}\Phi \left( \Omega \left( \left\{ J\left( \lambda ^{k},b_{k}\right) \right\} \right) \right)$$ where $\Phi$ is a sequence norm and (in the last expression) $\Omega$ stands for the Calderon transformation, the inf being taken over all representations $\widehat{b}$ of $b$ (see the appendix). Then we must have $% \Phi \left( \Omega \left( \left\{ J\left( \lambda ^{k},y_{k}^{\left( 1\right) }\right) \right\} \right) \right) <\infty$ for some representation $\widehat{y}^{\left( 1\right) }=\left\{ y_{k}^{\left( 1\right) }\right\}$ of $y^{\left( 1\right) }.$ Since always $K\left( \lambda ^{k},y^{\left( 1\right) }\right) \leq \Omega \left\{ J\left( \lambda ^{k},y_{k}^{\left( 1\right) }\right) \right\}$ (see e.g. [@3-BeL0], p.44) this gives $\Phi \left( \left\{ K\left( \lambda ^{k},y^{\left( 1\right) }\right) \right\} \right) <\infty$ and thus a fortiori - remember that $y<<_{K}y^{\left( 1\right) }$- $\Phi \left( \left\{ K\left( \lambda ^{k},y\right) \right\} \right) <\infty .$ But if $\widehat{y}$ is the representation of $y$ provided by the strong form of the fundamental lemma then also $\Phi \left( \Omega \left( \left\{ J\left( \lambda ^{k},y_{k}\right) \right\} \right) \right) <\infty .$ This proves that $y\in J^{B}\left( \overline{Y}\right) .$ \#$\medskip$ [3.4]{.ul}. As an application of the proceeding considerations we now show that Marcinkiewicz and Lorentz couples are of type $\left( l\right)$ and of type $\left( u\right)$ respectively. $\medskip$ Let us first recall the definition of Marcinkiewicz $M\left( \psi \right)$ and Lorentz spaces $\Lambda \left( \phi \right) .$ (A good introduction to Marcinkiewicz and Lorentz spaces can be found in chap. 2 of the book by Krein-Petunin-Semenov [@17-KrPeSe].)$\medskip$ In what follows $\psi$ and $\phi$ denote positive concave functions on $% \left( 0,\infty \right)$ $\left( \psi ,\phi \in \mathcal{K}\right)$ such that $$\tint\nolimits_{0}^{t}\frac{du}{\psi \left( u\right) }\leq C_{1}\frac{t}{% \psi \left( t\right) } \tag{1}$$ and $$\tint\nolimits_{0}^{t}\phi \left( u\right) \frac{du}{u}\leq C_{2}\phi \left( t\right) \tag{2}$$ respectively, with suitable constants $C_{1}$ and $C_{2}$.$\medskip$ We consider a fixed otherwise unspecified measure space. (Below we shall specialize to the case of the interval $\left( 0,\infty \right)$ equipped with the measure $dt.$). We say that a measurable function $f$ on this space belongs to $M\left( \psi \right)$ if $$\sup_{t}\frac{\psi \left( t\right) }{t}\tint\nolimits_{0}^{t}f^{\ast }\left( u\right) du<\infty \tag{3}$$ or equivalently $\sup_{t}\psi \left( t\right) f^{\ast }\left( t\right) <\infty$ and to $\Lambda \left( \phi \right)$ if $$\tint\nolimits_{0}^{\infty }f^{\ast }\left( t\right) d\phi \left( t\right) <\infty \tag{4}$$ or equivalently $\tint\nolimits_{0}^{\infty }f^{\ast }\left( t\right) \phi \left( t\right) \frac{dt}{t}<\infty .$ Here $f^{\ast }$ stands for the non-increasing rearrangement of $f.$ The equivalence of the two conditions in $\left( 3\right)$ or $\left( 4\right)$ results from $\left( 1\right)$ and $\left( 2\right)$ respectively.[^9]$\medskip$ One can show that (see [@17-KrPeSe]) $$M\left( \psi _{0}\right) +M\left( \psi _{1}\right) =M\left( \min \left( \psi _{0},\psi _{1}\right) \right) \tag{5}$$ and that $$\Lambda \left( \phi _{0}\right) +\Lambda \left( \phi _{1}\right) =\Lambda \left( \min \left( \phi _{0},\phi _{1}\right) \right) \tag{6}$$ up to equivalence of norm. Replacing here $\psi _{1}$ and $\phi _{1}$ by $% t\psi _{1}$ and $t\phi _{1}$ we get estimates for the K-functional in the $% \underline{\text{Marcinkiewicz couple}}$ $M\left( \overline{\psi }\right) =\left( M\left( \psi _{0}\right) ,M\left( \psi _{1}\right) \right)$ and the [Lorentz couple]{.ul} $\Lambda \left( \overline{\phi }\right) =\left( \Lambda \left( \phi _{0}\right) ,\Lambda \left( \phi _{1}\right) \right) .$ (One has similar estimates for the J-functional (see [@17-KrPeSe]) but we do not need them here.).$\medskip$ We can now announce the following result [Theorem]{.ul}. $\left( i\right)$ The Marcinkiewicz couple $M\left( \overline{\psi }\right) =\left( M\left( \psi _{0}\right) ,M\left( \psi _{1}\right) \right)$, where each $\psi _{i}$ $\left( i=0,1\right)$ is subject to $\left( 1\right)$ is of type $\left( l\right) .$ $\left( ii\right)$ The Lorentz couple $\Lambda \left( \overline{\phi }\right) =\left( \Lambda \left( \phi _{0}\right) ,\Lambda \left( \phi _{1}\right) \right) ,$where each $\phi _{i}$ $\left( i=0,1\right)$ is subject to $% \left( 2\right) ,$ is of type $\left( u\right) .$ [Proof]{.ul}. For simplicity we give the proof only for the case when the underlying measure space is the interval $\left( 0,\infty \right)$ equipped with the Lebesque measure $dt.$ [Ad $\left( i\right)$]{.ul} . We wish to apply [3.3]{.ul} prop 1 with $\overline{X}=\left( M\left( \psi _{0}\right) ,M\left( \psi _{1}\right) \right)$ and $\overline{X}^{\left( 1\right) }=\left( L^{\infty }\left( \psi _{0}\right) ,L^{\infty }\left( \psi _{1}\right) \right) .$ To a given $f$ in $\Sigma X$ we have to produce a suitable \"projection\" $\pi :\overline{X}% ^{\left( 1\right) }\rightarrow \overline{X}.$ With no loss of generality we may assume that $f=f^{\ast }.$ We then let $\pi$ be simply the identity map. Its continuity is established as follows. Let $g$ be an element of $% L^{\infty }\left( \psi \right)$ where $\psi$ is any function (in $\mathcal{% K}$) subject to $\left( 1\right) .$Then $\left\vert g\left( t\right) \right\vert \leq \frac{C}{\psi \left( t\right) }$ a.e. on $\left( 0,\infty \right)$ for some constant $C.$ Now $\frac{1}{\psi }$ is a non-increasing function. Thus we get $g^{\ast }\left( t\right) \leq \frac{C}{\psi \left( t\right) }.$ This shows (cf. $\left( 1\right) )$ that $g\in M\left( \psi \right) .$ Thus the identity map from $L^{\infty }\left( \psi \right)$ into $M\left( \psi \right)$ is continuous. In particular taking $\psi =\psi _{i}$ $\left( i=0,1\right)$ this substantiates our previous claim. Since our $f$ has the same K-functional in both couples, up to equivalence $$K\left( t,f\right) \approx \sup \min \left( \psi _{0}\left( u\right) ,t\psi _{1}\left( u\right) \right) f^{\ast }\left( u\right)$$ - here we use $\left( 5\right)$- all the conditions for the application of [3.3]{.ul} prop. 1 are met. We conclude that, since $\overline{X}% ^{\left( 1\right) }=L^{\infty }\left( \overline{\psi }\right)$ is of type $% \left( l\right)$ (see [3.2]{.ul} ), so is $\overline{X}=M\left( \overline{\psi }\right) .$ [Ad $\left( ii\right)$]{.ul}. This time we wish to apply [3.2 ]{.ul} cor. of prop. 2 taking $\overline{Y}=\Lambda \left( \overline{\phi }% \right) ,\overline{Y}^{\left( 1\right) }=L^{1}\left( \frac{\overline{\phi }}{% t}\right)$ (where $\frac{\overline{\phi }}{t}$ is the couple of weight functions $\left( \frac{\phi _{0}\left( t\right) }{t},\frac{\phi _{1}\left( t\right) }{t}\right) ).$ Again we know that this $\overline{Y}^{\left( 1\right) }$ is of type $\left( u\right)$ (see [3.3]{.ul}). That $% \overline{Y}$ admits the strong form of the fundamental lemma is indicated in [@6-CwPe], p. 33. To a given $f$ in $\Sigma \overline{Y}$ we have to produce a suitable \"retraction\" $\iota :\overline{Y}\rightarrow \overline{Y}% ^{\left( 1\right) }$ and again we fix attention to the case $f=f^{\ast }$ in which case we can take $\iota$ to be simply the identity map. The proof of its continuity runs as follows. Consider quite generally $g$ in $\Lambda \left( \phi \right)$ when $\phi$ is any function (in $\mathcal{K)}$ subject to $\left( 2\right) .$Then by $\left( 2\right)$ $% \tint\nolimits_{0}^{\infty }g^{\ast }\left( t\right) \frac{\phi \left( t\right) }{t}<\infty .$But $\frac{\phi \left( t\right) }{t}$ is a non-increasing function. Thus by a basic property of rearrangements (see e.g. [@3-BeL0], p. 5) it follows that $\tint\nolimits_{0}^{\infty }\left\vert g\left( t\right) \right\vert \frac{\phi \left( t\right) }{t}% dt<\infty .$This proves $g\in L^{1}\left( \frac{\phi \left( t\right) }{t}% \right) .$ In particular taking $\phi =\phi _{i}$ $\left( i=0,1\right)$ we get the desired continuity of our map $\iota .$ For the K-functional of $f$ we have in both couples the estimate (by $\left( 6\right)$ [and ]{.ul} $\left( 2\right) )$ $$K\left( t,f\right) \approx \tint\nolimits_{0}^{\infty }f^{\ast }\left( u\right) \min \left( \phi _{0}\left( u\right) ,t\phi _{1}\left( u\right) \right) \frac{du}{u}$$ Thus [3.2]{.ul} cor. of prop. 2 is effectively applicable and we conclude that our $\Lambda \left( \overline{\phi }\right)$ indeed is of type $\left( u\right) .$\#$\medskip$ The following result is now immediate (formal application of [3.1]{.ul} prop.) [Corollary]{.ul}. Consider any Marcinkiewicz couple $M\left( \overline{% \psi }\right)$ and any Lorentz couple $\Lambda \left( \overline{\phi }% \right)$ (possibly over [different]{.ul} measure spaces). Then $% M\left( \overline{\psi }\right)$ and $\Lambda \left( \overline{\phi }% \right)$ satisfy condition $\left( O\right) .$ In particular thus if $% T:M\left( \overline{\psi }\right) \rightarrow \Lambda \left( \overline{\phi }% \right)$ then $T:M\left( \rho \left( \overline{\psi }\right) \right) \rightarrow \Lambda \left( \rho \left( \overline{\phi }\right) \right)$ for any $\rho \in \mathcal{P}$.$\medskip$ [Remark]{.ul}. As in the case of Besov couples (sec. [2.5]{.ul}, remark) we know of no interesting operators mapping a space $M\left( \psi \right)$ into space $\Lambda \left( \phi \right) .$ # 4. Nuclearity. {#nuclearity. .unnumbered} [4.1]{.ul}. The following result, which is fundamental for our discussion, appears in the literature in many guises It is often associated with the names of Bergh and Cwikel. See the following remark (historical). [Lemma]{.ul}. If $y<_{n}x$ then $y<_{J/K\left( \lambda \right) }\lambda x.$ Conversely if $y<_{J/K\left( \lambda \right) }x$ then $y<_{n}x$ [Proof]{.ul}: Assume $y<_{n}x.$ Then by definition $y=Tx$ for some $T:% \overline{X}\overset{n}{\rightarrow }\overline{Y}$ with $\left\Vert T\right\Vert _{n}<1+\epsilon ,\epsilon >0,$ i.e. $Tx=\dsum_{n}l_{n}\left( a\right) b_{n}$ with $\dsum_{n}\max_{i=0,1}\left\Vert l_{n}\right\Vert _{X_{i}^{^{\prime }}}\left\Vert b_{n}\right\Vert _{Y_{i}}<1+\epsilon .$ Let $% e_{k}$ $\left( k\in \mathbb{Z}\right)$ be the subset of the our (denumerable) index set $I,$ say defined by the condition $$\left\Vert b_{n}\right\Vert _{Y_{0}}\leq \lambda ^{k}\left\Vert b_{n}\right\Vert _{Y_{1}}<\lambda \left\Vert b_{n}\right\Vert _{Y_{0}}.$$ Then $$J\left( \lambda ^{-k},l_{n}\right) J\left( \lambda ^{k},b_{n}\right) \leq \lambda \max_{i=0,1}\left\Vert l_{n}\right\Vert _{X_{i}^{^{\prime }}}\left\Vert b_{n}\right\Vert _{Y_{i}}. \tag{1}$$ With no loss of generality we may assume $b_{n}\neq 0$ for all $n.$So each $% n$ belongs to exactly one $e_{k}$ and we have a partition of $I:I=\cup _{k\in \mathbb{Z}}e_{k},e_{k}\cap e_{l}=\emptyset$ if $k\neq l.$ If we set $% y_{k}=\dsum_{n\in e_{k}}l_{n}\left( x\right) b_{n}$ (the series is clearly summable in $\Sigma \overline{Y})$ we therefore have $y=\dsum_{k\in \mathbb{Z% }}y_{k}$, that is $\widehat{y}=\left\{ y_{k}\right\} _{k\in \mathbb{Z}}$ is a representation of $y.$Furthermore by $\left( 1\right)$ $$\begin{aligned} J\left( \lambda ^{k},y_{k}\right) &\leq &\dsum_{n\in e_{k}}J\left( \lambda ^{-k},l_{n}\right) K\left( \lambda ^{k},x\right) J\left( \lambda ^{k},b_{n}\right) \\ &\leq &\lambda \dsum_{n\in e_{k}}\max_{i=0,1}\left\Vert l_{n}\right\Vert _{X_{i}^{^{\prime }}}\left\Vert b_{n}\right\Vert _{Y_{i}}K\left( \lambda ^{k},x\right)\end{aligned}$$ which yields $$\dsum_{k}\frac{J\left( \lambda ^{k},y_{k}\right) }{K\left( \lambda ^{k},x\right) }\leq \lambda \dsum_{n}\max_{i=0,1}\left( \left\Vert l_{n}\right\Vert _{X_{i}^{^{\prime }}}\left\Vert b_{n}\right\Vert _{Y_{i}}\right) <\lambda \left( 1+\epsilon \right) .$$ Since $\epsilon >0$ is arbitrary, this shows that $y\leq _{J/K\left( \lambda \right) }\lambda x.$ Conversly let $y<_{J/K\left( \lambda \right) }x.$ Then, for any $\epsilon >0,$ $y$ has a representation $y=\left\{ y_{k}\right\} _{k\in \mathbb{Z}}$ such that $$\dsum_{k}\frac{J\left( \lambda ^{k},y_{k}\right) }{K\left( \lambda ^{k},x\right) }<1+\epsilon .$$ By Hahn-Banach's theorem choose $l_{k}\in \left( \Sigma \overline{X}\right) ^{^{\prime }}$such that $l_{k}\left( x\right) =1,\left\vert l_{k}\left( a\right) \right\vert \leq \frac{K\left( \lambda ^{k},a\right) }{K\left( \lambda ^{k},x\right) }$ for $a\in \Sigma \overline{X}.$ In particular we then have $\left\Vert l_{k}\right\Vert _{X_{i}^{^{\prime }}}\leq \frac{% \lambda ^{ki}}{K\left( \lambda ^{k},x\right) }$ $\left( i=0,1\right) .$ Define the operator $T$ by the formula $Ta=\dsum_{k}l_{k}\left( a\right) y_{k}.$ Then $$\begin{aligned} \dsum_{k}\left\Vert l_{k}\right\Vert _{X_{i}^{^{\prime }}}\left\Vert y_{k}\right\Vert _{Y_{i}} &\leq &\dsum_{k}\frac{\max \lambda ^{ki}\left\Vert y_{k}\right\Vert _{Y_{i}}}{K\left( \lambda ^{k},x\right) } \\ &=&\dsum_{k}\frac{J\left( \lambda ^{k},y_{k}\right) }{K\left( \lambda ^{k},x\right) }<1+\epsilon\end{aligned}$$ so $T:\overline{X}\overset{n}{\rightarrow }\overline{y}$ with $\left\Vert T\right\Vert _{n}<1+\epsilon .$ Since clearly $Tx=y$ (in view of $% l_{k}\left( x\right) =1$ and $y=\dsum_{k}y_{k}),$moreover $\epsilon >0$ being arbitrary, we conclude that indeed $y<_{n}x.$ The proof is complete. \# $\medskip$ In particular we get the following corollaries. [Corollary 1]{.ul}. $y<<_{n}x$ iff $y<<_{J/K\left( \lambda \right) }x.$ [Corollary 2]{.ul}. The analogue of [$\frac{1}{2}.2$]{.ul}$\left( 1\right)$ with $<_{n}$ in place of $<_{J/K\left( \lambda \right) }.\medskip$ [Remark]{.ul}. Bergh in his unfortunately unpublished master's thesis [@2-BE71] (cf. [@3-BeL0]) showed that if for some $t$ holds $J\left( t,y\right) \leq K\left( t,x\right)$ then $y<_{b}x.$He gave also a number of interesting concrete applications of this result. Bergh's result apparently is a special case of the above lemma. With $t=\lambda ^{k}$ apply it to the trivial representation $y$ such that $y_{j}=\left\{ \begin{array}{c} y:j=k \\ 0:j\neq k% \end{array}% \right\} .$ Cwikel in [@5-Cw] on the other hand proved that if we have $% K\left( \lambda ^{k},y\right) \leq c_{k}K\left( \lambda ^{k},x\right)$ with $\dsum_{k}c_{k}<\infty$ then $y<_{b}x.\footnote{% See Notes \underline{4}:$\left\langle 1\right\rangle $}$ This follow from our result by taking $\widehat{y}$ to be the representation provided by the fundamental lemma ([@3-BeL0], p.33). The results of Bergh and Cwikel now gain a new dimension in the light of our notion of nuclearity.$\medskip$ [4.2]{.ul}. We can now prove our principal result.$\medskip$ [Theorem]{.ul}. The couples $\overline{X}$ and $\overline{Y}$ satisfies condition $\left( O\right)$ iff $y<<_{b}x$ implies $y<<_{n}x.$ [Proof]{.ul}: Let the couples $\overline{X}$ and $\overline{Y}$ satisfy condition $\left( O\right) .$Suppose that $y<<_{b}x.$Then by [3.2]{.ul} , prop. we have $y<<_{J/K\left( \lambda \right) }x.$Therefore $y<<_{n}x$ by [4.1]{.ul}, cor. 1. Thus $y<<_{b}x$ implies $y<<_{n}x.$ Conversly assume that this is the case. Then reversing the previous reasoning we find that $\overline{X}$ and $\overline{Y}$ indeed satisfy condition $\left( O\right) .$\#$\medskip$ [4.3]{.ul}. We apply our previous result to give a proof of Ovchinnikov's theorem [@23-Ovc]; until now we have thus assumed this theorem to be known. Our proof is just a variations of Ovchinnikov's proof but perhaps slightly simpler; in contract to Janson [@16-Jan] it still involves Grothendieck's fundamental theorem [@14-Gr]. $\medskip$ In sec. [3]{.ul} we have seen that it suffices yo consider the case of $% \lambda$ -adic weights, i.e. the case $\overline{w}=\left( 1,\lambda ^{-k}\right) .$ We thus have to show that the couples $\overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1}$ satisfy condition $\left( O\right) .$ Let $y<<_{b}x$. i.e. for some operator $T:\overline{l}_{\lambda }^{\infty }\rightarrow \overline{l}_{\lambda }^{1}$ holds $y=Tx$ where $x$ and $y$ are given elements in $\Sigma \overline{l}_{\lambda }^{\infty }$ and $\Sigma \overline{l}_{\lambda }^{1}$ respectively. We are going to show that $y<_{J/K\left( \lambda \right) }x.$ By [3]{.ul}, prop. it will then follows that $\overline{l}_{\lambda }^{\infty }$ and $\overline{l}_{\lambda }^{1}$ indeed satisfy condition $\left( O\right) .$ To achieve this let $\overline{H}=\left( H_{0},H_{1}\right)$ be a suitable Hilbert couple (i.e. $H_{0}$ and $H_{1}$ are Hilbert spaces). Pick up an element $u\in \Sigma \overline{H}$ such that $x<<_{b}u$ and $u<<_{K}x.$ That such a $u$ exists follows at one from the \"universal property\" of the couple $\overline{l}_{\lambda }^{\infty }$ (see [@27-Pe], [@28-Pe], [@6-CwPe]; cf. Ovchinnikov's treatment [@23-Ovc]). In particular we have then also a bounded linear operator $S:\overline{H}\rightarrow \overline{l}% _{\lambda }^{\infty }$ with $x=Su.$ The composition thus has the factorization $TS:\overline{H}\rightarrow \overline{l}_{\lambda }^{\infty }\rightarrow \overline{l}_{\lambda }^{1}.$ Especially we have $% TS:H_{i}\rightarrow l^{\infty }\left( \lambda ^{-ik}\right) \rightarrow l^{1}\left( \lambda ^{-ik}\right)$ $\left( i=0,1\right) .$ Grothendieck's fundamental theorem has as consequence ([@14-Gr], p. 65[^10]) that $% TS:H_{i}\rightarrow l^{1}\left( \lambda ^{-ik}\right)$ is indeed nuclear. Thus $TS:\overline{H}\rightarrow \overline{l}_{\lambda }^{1}$ is \"separately nuclear\". In sec. [$\frac{1}{2}$]{.ul} we said that in general \"separately nuclear\" does not entail \"nuclear\". But in this special case it does.[^11]. So we have established $TS:\overline{H}\overset{n}{\rightarrow }\overline{l}_{\lambda }^{1}.$ It follows that $y<<_{n}u.$ Since  $u<<_{K\left( \lambda \right) }x$ the ideal property for the ordering $<<_{n}$ (sec. [4.1]{.ul} cor. 2) gives $y<<_{J/K\left( \lambda \right) }x.$ The proof is complete. # 5. Janson's proof of Ovchinnikov's theorem. {#jansons-proof-of-ovchinnikovs-theorem. .unnumbered} We have already given one proof of Ovchinnikov's theorem [@23-Ovc], essentially his own proof; see [4.3]{.ul}. For the benefit of mankind we give here another one, essentially the one of Janson [@16-Jan]. It is entirely self-contained in particular independent of the Aronszajn-Gagliardo theorem [@1-AG65]. In fact we can without any extra labor directly threat the continuous case of sec. [2.4]{.ul}. Thus $L^{\infty }\left( \overline{w}D\right)$ and $L^{1}\left( \overline{z}E\right)$ having the same meaning as there our goal is to establish the following result.$% \medskip$ [Theorem]{.ul}. If $T:L^{\infty }\left( \overline{w}D\right) \rightarrow L^{1}\left( \overline{z}E\right)$ then $T:L^{\infty }\left( \rho \left( \overline{w}\right) D\right) \rightarrow L^{1}\left( \rho \left( \overline{z}\right) E\right)$ for any $\rho \in \mathcal{P}$ . $\medskip$ The proof will be broken up into a series of lemmata.$\medskip$ The key lemma, due to Janson, replaces Grothendieck's fundamental theorem in Ovchinnikov's treatment. [Lemma 1]{.ul}. ([@16-Jan], lemma 8[^12]). Let $T:L^{\infty }\left( \overline{w}\right) \rightarrow l^{1}\left( \overline{z}\right) .$ Then $T:l^{\infty }\rightarrow l^{1}$ provided the following condition is fulfilled: $\left( 1\right) :\dsum_{m\in \mathbb{Z}}\epsilon _{m}<\infty$ for some (positive) sequence $\left\{ \epsilon _{m}\right\}$ such that $\min \left( w_{k}^{0}/z_{j}^{0},w_{k}^{1}/z_{j}^{1}\right) \leq \epsilon _{m}$ if $% j-k=m.\medskip$ [Remark]{.ul}. As will be clear from the proof the same conclusion holds if we just assume $T:c_{0}\left( \overline{w}\right) \rightarrow l^{1}\left( \overline{z}\right)$ ($c_{0}$ denotes of course the space of (doubly infinite) sequences tending to $0.).$We see also that $T:l^{\infty }\rightarrow l^{1}$ is in fact a [nuclear]{.ul} operator; this observation is perhaps new.$\medskip$ [Proof]{.ul} (after Janson [@16-Jan]): On the space $\Sigma c_{0}\left( \overline{w}\right) =c_{0}\left( \min \left( w_{0},w_{1}\right) \right)$ the operator $T$ is given by a (doubly infinite) matrix $\left( t_{jk}\right) ,$so that if $\left\{ x_{j}\right\} \in \Sigma \left( c_{0}\left( \overline{w}\right) \right)$ we have $Tx_{j}=% \dsum_{k}t_{jk}x_{k}$ which again entails that $$\dsum_{j}z_{j}^{i}\left\vert \dsum_{k}t_{jk}x_{k}\right\vert \leq C\sup_{k}w_{k}^{i}\left\vert x_{k}\right\vert ,\left( i=0,1\right) \tag{2}$$ for some constant $C.$Taking $j=0$ in $\left( 1\right)$ we see in particular $$\min \left( w_{k}^{0},w_{k}^{1}\right) \rightarrow 0\text{ as }\left\vert k\right\vert \rightarrow \infty . \tag{$1^{^{\prime }}$}$$ Therefore we have $l^{\infty }\subseteq \Sigma \left( c_{0}\left( \overline{w% }\right) \right) .$To complete the proof it thus suffices to show that $% \dsum_{j,k}\left\vert t_{jk}\right\vert <\infty ,$ which again - in view of $% \left( 1\right)$ - will follows if we can can show that $$\dsum_{j-k=m}\left\vert t_{jk}\right\vert \leq C\epsilon _{m} \tag{3}$$ To this end we set $a_{jk}=\max \left( z_{j}^{0}/w_{k}^{0},z_{j}^{1}/w_{k}^{1}\right) t_{jk}.$Then $\left( 2\right)$ can be written simply as $$\dsum_{j}\left\vert \dsum_{k}a_{jk}\zeta _{k}\right\vert \leq C\sup \left\vert \zeta _{k}\right\vert . \tag{$2^{^{\prime }}$}$$ That is, the matrix $\left( a_{jk}\right)$ defines a bounded linear operator $A:l^{\infty }\rightarrow l^{1}.$We see that in order to establish $% \left( 3\right)$ it suffices to show that $\left( 2^{^{\prime }}\right)$ implies $$\dsum_{j-k=m}\left\vert a_{jk}\right\vert \leq C \tag{$3^{^{\prime }}$}$$ for each $m.$ Now $\left( 2^{^{\prime }}\right)$ entails that for each real number $x$ holds $$\dsum_{j}\left\vert \dsum_{k}a_{jk}e^{i\left( j-k-m\right) x}\right\vert \leq C$$ Taking averages $\left( 3^{^{\prime }}\right)$ then follows at once. \#$% \medskip$ [Remark]{.ul}. As already mentioned (see Introduction) a different \"elementary\" method in connection with Ovchinnikov's theorem [@23-Ovc] based on Khinchine's (=Xincin's) inequality is used in Gustavsson [@15-Gu].$\medskip$ Now we restate lemma 1 in a form more useful for the application we have in mind. [Lemma $1^{^{\prime }}$]{.ul}. Let $T:\left( l^{\infty },l^{\infty }\left( \frac{1}{\tau }\right) \right) \rightarrow \left( l^{1},l^{1}\left( \frac{1}{\sigma }\right) \right) .$Then $T:l^{\infty }\left( \frac{1}{\rho \left( \tau \right) }\right) \rightarrow l^{1}\left( \frac{1}{\lambda \left( \sigma \right) }\right)$ provided $\dsum_{m\in \mathbb{Z}}\epsilon _{m}<\infty$ where $$\min \left( 1,\frac{\sigma _{j}}{\tau _{k}}\right) \frac{\rho \left( \tau _{k}\right) }{\lambda \left( \sigma _{j}\right) }\leq \epsilon _{m}\text{ if }k-j=m$$ Here $\tau =\left\{ \tau _{j}\right\}$ and $\sigma =\left\{ \sigma _{j}\right\}$ are any two given sequences and $\rho$ and $\lambda$ are given positive functions on $\left( 0,\infty \right) ;\rho \left( \tau \right)$ stands for the sequence $\left\{ \rho \left( \tau _{k}\right) \right\}$ with a similar meaning for $\lambda \left( \sigma \right) ;1$ denotes the sequence $\left\{ 1\right\} ,\frac{1}{\tau }$ the sequence $% \left\{ \frac{1}{\tau _{k}}\right\} ,$ similarly for $\frac{1}{\sigma },% \frac{1}{\rho \left( \lambda \right) },\frac{1}{\lambda \left( \sigma \right) }.$ [Proof]{.ul}: Indeed we at once reduce it to lemma $1$ with $w^{0}=\rho \left( \tau \right) ,w^{1}=\frac{\rho \left( \tau \right) }{\tau }% ,z^{0}=\lambda \left( \sigma \right) ,z^{1}=\frac{\lambda \left( \sigma \right) }{\sigma }.\#\medskip$ In particular if $\tau =\sigma ,\rho =\lambda$ condition $\left( 1\right)$ takes the form $$\dsum_{m}\epsilon _{m}<\infty \text{ where }\min \left( 1,\frac{\tau _{j}}{% \tau _{k}}\right) \frac{\rho \left( \tau _{k}\right) }{\rho \left( \tau _{j}\right) }\text{ if }k-j=m. \tag{$1^{^{\prime \prime }}$}$$ $\medskip$ [Example]{.ul}. Notice that $\left( 1^{^{\prime \prime }}\right)$ certainly is fulfilled if $\tau _{k}=\lambda ^{k},\rho \in \mathcal{P}^{+-}.$ Indeed denoting (for any $\rho )$ by $s_{\rho }$ the corresponding dilation function [^13] i.e. $$s_{\rho }\left( t\right) =\sup_{u}\frac{\rho \left( tu\right) }{\rho \left( u\right) },$$ we have $\min \left( 1,\lambda ^{j-k}\right) \frac{\rho \left( \lambda ^{k}\right) }{\rho \left( \lambda ^{j}\right) }\leq \min \left( 1,\lambda ^{-m}\right) s_{\rho }\left( \lambda ^{m}\right)$ for $k-j=m$ so $\left( 1^{^{\prime \prime }}\right)$ follows if $\rho \in \mathcal{P}^{+-}$ because then $s_{\rho }\left( t\right) =O\left( \min \left( t^{\alpha _{0}},t^{\alpha _{1}}\right) \right)$ for suitable $\alpha _{0},\alpha _{1}$ with $0<\alpha _{0}<\alpha _{1}<1.\medskip$ [Lemma 2]{.ul}. Let $x\in \Sigma L^{\infty }\left( \overline{w}D\right)$ and let $\tau =\left\{ \tau _{k}\right\}$ be any [increasing]{.ul} positive sequence such that $$K\left( t,x\right) \leq C\sup_{k}\min \left( 1,\frac{t}{\tau _{k}}\right) K\left( \tau _{k},x\right) \tag{4}$$ for some $C.$Then we can find an element $a\in \Sigma l^{\infty }\left( 1,% \frac{1}{\tau }\right)$ and a bounded linear map $\iota :l^{\infty }\left( 1,\frac{1}{\tau }\right) \rightarrow L^{\infty }\left( \overline{w}D\right)$ with $x=\iota a$ such that $a<<_{K}x.$ [Proof]{.ul}: We first assume that the underlying measure space is $% W=\left( 0,\infty \right)$ and that the measure is $\mu =dt$ and take $D$ to be the trivial bundle all of whose fibers are equal to $\mathbb{R}$. We thus have the couple $L^{\infty }\left( 1,\frac{1}{\tau }\right) .$We further take $w^{0}=1,w^{1}=\frac{1}{t}.$ Then we have the well-known estimates (see e.g. [@30-Pe]) $$K\left( t,x\right) \approx \sup_{s\in \left( 0,\infty \right) }\min \left( 1,% \frac{t}{s}\right) \left\vert x\left( s\right) \right\vert \text{ for }x\in \Sigma L^{\infty }\left( 1,\frac{1}{t}\right) . \tag{5}$$ We further assume that $x\in \mathcal{P}_{1},$ which in view of $\left( 5\right)$ implies that $K\left( t,x\right) \approx x\left( t\right)$ and also [linear]{.ul} in the intervals $\left( \tau _{k},\tau _{k+1}\right) .$(Then $\left( 4\right)$ is automatically fulfilled). We set $a=\left\{ x\left( \tau _{k}\right) \right\}$ and define the map $\iota$ as follows. If $b=\left\{ b_{k}\right\} \in \Sigma l^{\infty }\left( 1,\frac{% 1}{\tau }\right)$ we put $\iota b\left( \tau _{k}\right) =b_{k}$ and extend this by linearity to general $t,$ that is if $t$ is in the interval $\left( \tau _{k},\tau _{k+1}\right)$ we put $\iota b\left( t\right) =\left( 1-\theta \right) b_{k}+\theta b_{k+1}$ where we have written $t=\left( 1-\theta \right) \tau _{k}+\theta \tau _{k+1}$ with $0<\theta <1.$It is clear that $$\begin{aligned} \left\vert \iota b\left( t\right) \right\vert &\leq &\left( 1-\theta \right) \left\vert b_{k}\right\vert +\theta \left\vert b_{k}\right\vert \leq \sup_{k}\left\vert b_{k}\right\vert =\left\Vert b\right\Vert _{l^{\infty },} \\ \frac{1}{t}\left\vert \iota b\left( t\right) \right\vert &\leq &\left( 1-\theta \right) \frac{\tau _{k}}{t}\frac{\left\vert b_{k}\right\vert }{\tau _{k}}+\theta \frac{\tau _{k+1}}{t}\frac{\left\vert b_{k+1}\right\vert }{% t_{k+1}}\leq \\ &\leq &\sup_{k}\frac{\left\vert b_{k}\right\vert }{\tau _{k}}=\left\Vert b\right\Vert _{l^{\infty }\left( \frac{1}{\tau }\right) }\end{aligned}$$ which establishes the continuity of $\iota$, and that $\iota a=x.$ Also by the discrete analogue of $\left( 5\right)$ and by $\left( 4\right)$ $$K\left( t,a\right) \approx \sup_{k}\min \left( 1,\frac{t}{\tau _{k}}\right) x\left( \tau _{k}\right) \approx x\left( t\right) \approx K\left( t,x\right)$$ so in particular $a<<_{K}x.$ To treat the general case we shall make use of the fact that the couples $% L^{\infty }\left( 1,\frac{1}{t}\right) =\left( L^{\infty },L^{\infty }\left( \frac{1}{t}\right) \right)$ and $L^{\infty }\left( \overline{w}D\right)$ are relative Calderon. In the case of a trivial bundle with all fibers equal to $\mathbb{R}$ (\"scalar case\", this is well-known (see [@27-Pe], [@28-Pe], [@6-CwPe]). The general case can easily brought back to this case, since $L^{\infty }\left( \overline{w}D\right)$ is a partial retract of a scalar couple (cf. [2.4]{.ul}). Let thus $x$ be given in $\Sigma L^{\infty }\left( \overline{w}D\right) .$ Let $\widetilde{x}$ be the function in $\Sigma L^{\infty }\left( 1,\frac{1}{t% }\right)$ which is linear in each interval $\left( \tau _{k},\tau _{k+1}\right)$ and such that $\widetilde{x}\left( \tau _{k}\right) =K\left( \tau _{k},x\right) .$ It is clear from $\left( 4\right)$ that $x<<_{K}% \widetilde{x}$ so we get a map $T:L^{\infty }\left( 1,\frac{1}{t}\right) \rightarrow L^{\infty }\left( \overline{w}D\right)$ with $x=T\widetilde{x}.$ Also trivially $\widetilde{x}<<_{K}x.$Furthermore the previous proof on the special case $Z=\left( 0,\infty \right)$ yields a map $\widetilde{\iota }% :l^{\infty }\left( 1,\frac{1}{\tau }\right) \rightarrow L^{\infty }\left( 1,% \frac{1}{t}\right)$ with $\widetilde{x}=\widetilde{\iota }a$ for some $a\in \Sigma \left( l^{\infty },l^{\infty }\left( \frac{1}{\tau }\right) \right)$ with $a<<_{K}\widetilde{x}.$ We obtain the desired map $\iota :l^{\infty }\left( 1,\frac{1}{\tau }\right) \rightarrow L^{\infty }\left( \overline{w}% D\right)$ as the composite $T\widetilde{\iota }.$The proof is completed if we notice that by the transitivity of the ordering $<<_{K}$ (sec. $% \underline{\frac{1}{2}.2}$ ) follows $a<<_{K}x.$\#$\medskip$ The result dual to lemma 2 reads. [Lemma 3]{.ul}.Let $y\in \Sigma L^{1}\left( \overline{z}E\right)$ and let $\tau =\left\{ \tau _{k}\right\}$ be any [increasing]{.ul} positive sequence such that $$\inf_{k}\max \left( 1,\frac{t}{\tau _{k}}\right) \rho \left( \tau _{k}\right) \leq C\rho \left( t\right) \tag{6}$$ for some constant $C.$Then we can find an element $b\in \Sigma l^{1}\left( 1,% \frac{1}{\tau }\right)$ and a bounded linear map $\pi :L^{1}\left( \overline{z}E\right) \rightarrow l^{1}\left( 1,\frac{1}{\tau }\right)$ with $b=\pi y$ such that if $b\in l^{1}\left( \frac{1}{\rho \left( \tau \right) }% \right)$ then $y\in L^{1}\left( \rho \left( \overline{z}\right) E\right) .$ [Proof]{.ul}: Let $\left\{ H_{k}\right\}$ be a partition of the underlying measure space $Z$ into disjoint subsets such that $$\rho \left( \tau _{k}\right) \max \left( 1,\frac{z^{0}}{z^{1}\tau _{k}}% \right) \leq C\rho \left( \frac{z^{0}}{z^{1}}\right) \tag{7}$$ on $H_{k}.$ (It is for the existence of such a partition we need $\left( 6\right) ).$ By the Hahn-Banach theorem (cf. 2.2, proof of prop. 1) we get linear functionals. $\beta _{k}$ $\left( k\in \mathbb{Z}\right)$ on $\Sigma L^{1}\left( \overline{z}E\right)$ such that $$\beta _{k}\left( y\right) =\tint\nolimits_{H_{k}}z^{0}/\rho \left( z^{0}/z^{1}\right) \left\Vert y\right\Vert _{E_{\omega }}dv$$ and $$\left\vert \beta _{k}\left( v\right) \right\vert \leq \tint\nolimits_{H_{k}}z^{0}/\rho \left( z^{0}/z^{1}\right) \left\Vert v\right\Vert _{E_{\omega }}dv$$ for every $v\in \Sigma L^{1}\left( \overline{z}E\right) .$ Then by $\left( 7\right)$ we have also $$\begin{aligned} \left\vert \beta _{k}\left( v\right) \right\vert &\leq &\frac{C}{\rho \left( \tau _{k}\right) }\tint\nolimits_{H_{k}}z^{0}\left\Vert v\right\Vert _{E_{\omega }}dv \\ \left\vert \beta _{k}\left( v\right) \right\vert &\leq &\frac{C\tau _{k}}{% \rho \left( \tau _{k}\right) }\tint\nolimits_{H_{k}}z^{1}\left\Vert v\right\Vert _{E_{\omega }}dv\end{aligned}$$ We define $\pi$ by the formula $\pi v_{k}=\rho \left( \tau _{k}\right) \beta _{k}\left( v\right)$ for $v\in \Sigma L^{1}\left( \overline{z}% E\right) .$Then obviously $$\begin{aligned} \left\Vert \pi v\right\Vert _{l^{1}} &=&\dsum_{k}\left\vert \pi v_{k}\right\vert \leq C\dsum_{k}\tint\nolimits_{H_{k}}z^{0}\left\Vert v\right\Vert _{E_{\omega }}dv=C\left\Vert v\right\Vert _{L^{1}\left( z^{0}E\right) } \\ \left\Vert \pi v\right\Vert _{l^{1}\left( \frac{1}{\tau }\right) } &=&\dsum_{k}\frac{1}{\tau _{k}}\left\vert \pi v_{k}\right\vert \leq C\dsum_{k}\tint\nolimits_{H_{k}}z^{1}\left\Vert v\right\Vert _{E_{\omega }}dv=C\left\Vert v\right\Vert _{L^{1}\left( z^{1}E\right) }\end{aligned}$$ Moreover for $\pi y=b=\left\{ b_{k}\right\}$ we obtain $$\begin{aligned} \left\Vert b\right\Vert _{l^{1}\left( \frac{1}{\rho \left( \tau \right) }% \right) } &=&\dsum_{k}\frac{\left\vert b_{k}\right\vert }{\rho \left( \tau _{k}\right) }=\dsum_{k}\tint\nolimits_{H_{k}}z^{0}/\rho \left( z^{0}/z^{1}\right) \left\Vert y\right\Vert _{E_{\omega }}dv \\ &=&\tint\nolimits_{Z}z^{0}/\rho \left( z^{0}/z^{1}\right) \left\Vert y\right\Vert _{E_{\omega }}dv=\left\Vert y\right\Vert _{L^{1}\left( \rho \left( \overline{z}\right) E\right) }\end{aligned}$$ Thus $y\in L^{1}\left( z^{0}/\rho \left( z^{0}/z^{1}\right) E\right)$ iff $% b\in l^{1}\left( \frac{1}{\rho \left( \tau \right) }\right) .$\#$\medskip$ Finally we establish the following result also taken over in principle from Janson [@16-Jan]. [Lemma 4]{.ul}. Let $x$ be in $\Sigma L^{\infty }\left( \overline{w}% D\right)$ and set $\rho \left( t\right) =K\left( t,x\right)$ (so $\rho \in \mathcal{P)}$. Assume the range of the function $\rho \left( t\right)$ as well as that of $\rho ^{^{\prime }}\left( t\right) =\frac{t}{\rho \left( t\right) }$ is $\left( 0,\infty \right) .$Then there exists an [ increasing]{.ul} positive sequence $\tau =\left\{ \tau _{k}\right\} _{k\in \mathbb{Z}}$ such that $\left( 4\right) ,\left( 6\right)$ and $\left( 1^{^{\prime \prime }}\right)$ are fulfilled. [Proof]{.ul}: We begin by noticing that $\left( 4\right)$ says that $$\rho \left( t\right) \leq C\cdot \inf_{k}\min \left( 1,\frac{t}{\tau _{k}}% \right) \rho \left( \tau _{k}\right) \tag{8}$$ and that $\left( 6\right)$ is formally $\left( 8\right)$ for the function $% \rho ^{^{\prime }}.$ Following Janson [@16-Jan] we define $\tau _{k}$ inductively by $\tau _{0}=1$ and $\min \left( \frac{\rho \left( \tau _{k+1}\right) }{\rho \left( \tau _{k}\right) },\frac{\rho ^{^{\prime }}\left( \tau _{k+1}\right) }{\rho ^{^{\prime }}\left( \tau _{k}\right) }% \right) =2.$ (It is here the assumption about the range comes in.). Then $$\rho \left( \tau _{j}\right) \geq 2^{j-k}\rho \left( \tau _{k}\right) \text{ or }\frac{\rho \left( \tau _{k}\right) }{\rho \left( \tau _{j}\right) }\leq 2^{k-j}\text{ if }j\geq k$$ Similarly $$\rho ^{^{\prime }}\left( \tau _{k}\right) \geq 2^{k-j}\rho ^{^{\prime }}\left( \tau _{j}\right) \text{ or }\frac{\tau _{j}\rho \left( \tau _{k}\right) }{\tau _{k}\rho \left( \tau _{j}\right) }\leq 2^{j-k}\text{ if }% j\leq k$$ Thus at any rate $$\min \left( 1,\frac{\tau _{j}}{\tau _{k}}\right) \frac{\rho \left( \tau _{k}\right) }{\rho \left( \tau _{j}\right) }\leq 2^{-\left\vert j-k\right\vert }$$ so $\left( 1^{^{\prime \prime }}\right)$ follows indeed, with $\epsilon _{m}=2^{-\left\vert m\right\vert }.$To establish $\left( 6\right)$ let $% t\in \left( \tau _{k},\tau _{k+1}\right) .$Then there are two cases: [Case 1]{.ul}. $\rho \left( \tau _{k+1}\right) \leq 2\rho \left( \tau _{k}\right) .$Then, since $\rho \in \mathcal{P}_{1},$ we get $\rho \left( t\right) \leq \rho \left( \tau _{k+1}\right) \leq 2\rho \left( \tau _{k}\right) =2\min \left( 1,\frac{t}{\tau _{k}}\right) \rho \left( \tau _{k}\right) .$ [Case 2]{.ul}. $\frac{\rho \left( t_{k+1}\right) }{\tau _{k+1}}\leq 2\cdot \frac{\rho \left( \tau _{k}\right) }{\tau _{k}}.$We now have $\frac{% \rho \left( t\right) }{t}\leq 2\frac{\rho \left( \tau _{k+1}\right) }{\tau _{k+1}}$ or $\rho \left( t\right) \leq 2\frac{t}{\tau _{k+1}}\rho \left( \tau _{k+1}\right) =2\min \left( 1,\frac{t}{\tau _{k+1}}\right) \rho \left( \tau _{k+1}\right) .$ Therefore we have proven $\left( 8\right)$ (or $\left( 4\right) )$ with $% C=2.$Since $\left( 6\right) ,$ as we have just remarked, is just $\left( 8\right)$ with $\rho ^{^{\prime }}$ in place of $\rho$ and further $\rho$ and $\rho ^{^{\prime }}$ enter in a symmetric fashion into the definition of $\tau$ we need no particular proof of $\left( 6\right) .$ \#$\medskip$ These lemmas being established we can at last proceed to the [Proof]{.ul} (of theorem/completed/). Let thus $T:L^{\infty }\left( \overline{w}D\right) \rightarrow L^{1}\left( \overline{z}E\right)$ and $% x\in L^{\infty }\left( \rho \left( \overline{w}\right) D\right) .$ We have to show that $Tx\in L^{1}\left( \rho \left( \overline{z}\right) E\right) .$ Clearly it suffices to prove this if $\rho \left( t\right) =K\left( t,x\right) .$(cf. e.g. the argument in [1]{.ul}, proof of prop. 2). Make first the assumption about the ranges in lemma 4. Let then $\tau$ be the sequence provided by that lemma and select according to lemma 2, $b,\pi$ according to lemma $3,$ with $y=Tx.$ Put $S=\pi T$ so that we have the commutative diagram $$\begin{tabular}{ccc} $L^{\infty }\left( \overline{w}D\right) $ & $\overset{T}{\rightarrow }$ & $% L^{1}\left( \overline{z}E\right) $ \\ $\uparrow \iota $ & & $\downarrow \pi $ \\ $l^{\infty }\left( 1,\frac{1}{\tau }\right) $ & $\overset{S}{\rightarrow }$ & $l^{1}\left( 1,\frac{1}{\tau }\right) $% \end{tabular}%$$ By lemma $1$ $S:l^{\infty }\left( \frac{1}{\rho \left( \tau \right) }\right) \rightarrow l^{1}\left( \frac{1}{\rho \left( \tau \right) }\right) .$Now $% a\in l^{\infty }\left( \frac{1}{\rho \left( \tau \right) }\right)$ so $% Sa\in l^{1}\left( \frac{1}{\rho \left( \tau \right) }\right) .$But $Sa=\pi T\iota a=\pi Tx=\pi y.$ Therefore $y\in L^{1}\left( \rho \left( \overline{z}% \right) E\right) .$ This completes the proof in the above assumption about the ranges. If this assumptions is [not]{.ul} fulfilled one must take recourse to suitable one-sided version of the preceding lemmata. In order to avoid tedious repetition we omit the details.$\#$ # 6. Open questions. {#open-questions. .unnumbered} [6.1]{.ul}. [\"The philosophy of weights\"]{.ul}. We would like to give an at least heuristic explanation why weights play such a great role in the theory of interpolation spaces. We proceed in several steps.$\medskip$ Q: Which are the simplest pairs? A: The pairs $\overline{A}=\left( A_{0},A_{1}\right)$ for which $A_{0}$ and $A_{1}$ are similar. By the latter we mean that there exists a Banach space $% E$ and linear maps $\Lambda _{i}:A_{i}\rightarrow E$ $\left( i=0,1\right)$ such that $\left\Vert a\right\Vert _{A_{i}}=\left\Vert \Lambda _{i}a\right\Vert _{E},a\in A_{i},$ i.e. $A_{i}$ is isometric to a (closed) subspace of $E.\medskip$ [Example]{.ul}. If the $A_{i}$ are Hilbert spaces then this is always the case (with $E\ $Hilbert too); cf. [@11-FoLi]. In this context it is interesting to consider interpolation spaces $A$ with respect to $\overline{A}$ such that $\left\Vert a\right\Vert _{A}=\left\Vert \Lambda a\right\Vert _{E},a\in A,$ for a suitable linear map $\Lambda .$ Indeed $\Lambda$ may be conceived as a kind of function of $\Lambda _{0}$ and $\Lambda _{1}.$ We have then the [problem of interpolation functions]{.ul}, a line of development starting with the classical paper by Foias-Lions [@11-FoLi] already referred to.$\medskip$ Q. Which are the simplest Banach spaces? And which are the simplest operators? A. $E=l^{1},l^{\infty },l^{2},l^{p}$ and there continuous analogues $% L^{1},L^{\infty },L^{2},L^{p}$, more generally rearrangement invariant spaces (both sequence and function spaces), in particular thus Marcinkiewicz and Lorentz spaces, $BMO,H^{1},FL^{1},FL^{\infty }$ etc. As for $\Lambda$ a natural candidate in each of these cases is the multiplication operator $% \Lambda f=wf.\medskip$ We have thus a general program, albeit somewhat vague in its contours. [To generalize the ]{.ul} [theory of Ovchinnikov]{.ul} [ and Janson to the context of more general weighted pairs]{.ul} $E\left( \overline{% w}\right) .\medskip$ Let us consider a few somewhat more specialized questions. Is $E\left( \overline{w}\right)$ always Calderon?. And when is $E\left( \overline{w}\right)$ tame in Ovchinnikov's sense [@23-Ovc]. (The case $% FL^{1}$ treated by Janson [@16-Jan] shows that $E\left( \overline{w}% \right)$ is not always tame.) To relate the properties of $E\left( \overline{w}\right)$ with the dual pair $E^{\ast }\left( 1/\overline{w}\right) .$ To characterize all (relative) interpolation spaces with respect to the pairs $l^{p}\left( \overline{w}\right)$ and $l^{q}\left( \overline{z}% \right) .$ (Some results in this direction are given in Ovchinnikov [@24-Ovc]. His paper [@25-Ovc] on the other hand might be considered as a beginning of a study of the (Hilbert) pair $l^{2}\left( \overline{w}\right)$ in this spirit. As we have already noticed the case $E=FL^{1}$ as well as $% E=FL^{\infty }$ appears in Janson [@16-Jan].)$\medskip$ [Remark]{.ul}. The special role of $l^{\infty }\left( \overline{w}% \right)$ and $l^{1}\left( \overline{z}\right) ,$ further accentuated by the recent work of Brudnyi-Krugljak. [@4-BrKr], is apparent already in [@26-Pe] (cf. [@6-CwPe] too); from the category point of view advocated there they simply appear as [projective]{.ul} and [injective]{.ul} objects respectively.$\medskip$ [6.2]{.ul}. [Lion's problem]{.ul}. Lions [@20-Lio] long ago raised the question whether the space $\overline{X}_{\theta p}$ always depends [effectively]{.ul} on their parameters $\theta ,p.$ This is vaguely related to the topics of sec. [4]{.ul}. (Cf. [@27-Pe] for quite a different idea.). For the complex spaces $\overline{X}_{\left[ \theta \right] }$ the analogues question was solved by Stafney [@32-Sta] . His method is however quite general and in the case of the space $% \overline{X}_{\theta p}$ leads to the following partial result (cf. [@3-BeL0] p.82, exc. 21 and p.104): $\left( \ast \right)$ [Assume there exists]{.ul} $\theta _{0},p_{0}$ [and]{.ul} $\theta _{1},p_{1}$ [with]{.ul} $\theta _{0},\theta _{1}$ [in]{.ul} $\left( 0,1\right) ,\theta _{0}\neq \theta _{1}$ [such that]{.ul} $\overline{X}_{\theta _{0}p_{0}}=\overline{X}_{\theta _{1}p_{1}}.$[Assume also that]{.ul} $% \overline{X}$ is [regular]{.ul} . [Then]{.ul} $X_{0}=X_{1}.$ There remains thus the more difficult case $\theta _{0}=\theta _{1}$, that is (writing $\theta =\theta _{0}=\theta _{1})$ the case when $\overline{X}% _{\theta p_{0}}=\overline{X}_{\theta p_{1}}$for some $\theta$ in $\left( 0,1\right)$ with $p_{0}\neq p_{1}.$ Applying $\left( \ast \right)$ in conjunction with a suitable reiteration theorem we conclude that $\overline{X% }_{\theta p}$ in this case is independent of $p$ if $p<\infty ;$ regarding the limiting case $p=\infty$ we can only say that the norm of $\overline{X}% _{\theta \infty }$ restricted to $\overline{X}_{\theta 1}$ coincides up to equivalence with that of $\overline{X}_{\theta 1}.$ On the other hand by a theorem of Lions (see [@3-BeL0], p. 103) holds $$\begin{aligned} \left( \overline{X}_{\theta p},\overline{X}_{\theta q}\right) _{\left[ \tau % \right] } &=&\overline{X}_{\zeta r}\text{ if }\zeta =\left( 1-\tau \right) \theta +\tau \theta \\ \frac{1}{r} &=&\frac{1-\tau }{p}+\frac{\tau }{q},\left( \tau \in \left( 0,1\right) \right) .\end{aligned}$$ Thus we arrive at least to the following conclusion. If $\overline{X}% _{\theta p}$ is independent of $p$ for some $\theta$ in $\left( 0,1\right)$ then the same is true for any $\theta$ in $\left( 0,1\right) .\medskip$ If we now generalizing (replacing $\rho \left( t\right) =t^{\theta }$ by a general function), we have the problem to decide for any particular $\rho \in \mathcal{P}$ (or for all $\rho \in \mathcal{P)}$ whether one can have $% \overline{X}_{\rho \infty ;K}=\overline{X}_{\rho 1;J}.$ Let us remark that if there exists $x\in \Sigma X$ such that $K\left( t,x\right) \approx \rho \left( t\right)$ then this certainly cannot be the case. (This is because $% \Delta$ is dense in $\overline{X}_{\rho 1;J}.).$But then we have at least the modified question whether the norms of $\overline{X}_{\rho \infty ;K}$ and $\overline{X}_{\rho 1;J}$ coincide (up to equivalence) on $\Delta .$ Put in more concrete terms if $x\in \Sigma \overline{X}$ and $K\left( t,x\right) =o\left( \rho \left( t\right) \right)$ does there exists a representation $% \widehat{x}=\left\{ x_{k}\right\} _{k\in \mathbb{\mathbb{Z}}}$ of $x$ such that $$\dsum_{k}\frac{J\left( 2^{k},x_{k}\right) }{\rho \left( 2^{k}\right) }% <\infty .$$ This is about to say that the identity map $\overline{X}$ is in some sense nuclear. Let us remark that if we pass to the more general situation of $% \underline{\text{locally convex spaces }}$ (instead of just Banach spaces) this very well can be the case. This is essentially contained already in a classic example in [@18-LiPe], chap. $VIII.\medskip$ [6.3]{.ul}. We conclude by a few scattered remarks pertaining mainly to the topics on sec. [3]{.ul} . We can ask if $\overline{Y}$ of type $\left( u\right)$ implies $\overline{Y}% ^{^{\prime }}$ (the dual pair, assuming that $\overline{Y}$ is regular) then is of type $\left( l\right) .\medskip$ We have also a natural notion of \"type $\left( l_{0}\right) ^{"}$which is roughly speaking obtained if we replace $l^{\infty }$ by $c_{0}.$ (Janson [@16-Jan] makes a great show of the distinction between $% l^{\infty }$ and $c_{0},$ a distinction which we have here almost entirely neglected.). And corresponding to it a \"condition $\left( O_{0}\right) ^{".}$ The question is then if $\overline{X}$ is of type $\left( l_{0}\right)$ implies $\overline{X}^{^{\prime }}$ is type $\left( u\right) .$ And if from the facts that $\overline{X}$ and $\overline{Y}$ satisfies condition $\left( O_{0}\right)$ follows that $\overline{Y}^{^{\prime }}$ and $\overline{X}% ^{^{\prime }}$ satisfy condition $\left( O\right) .\medskip$ In this connection we also ask if $\overline{X}$ is Calderon is then $% \overline{X}^{^{\prime }}$ Calderon too. If $\overline{X}$ is tame in Ovchinnikov's sense [@23-Ovc] does it follows that $G^{A}\left( \overline{X}\right) =H^{B}\left( \overline{X}% \right)$ where $A$ and $B$ are in the same relation as $l^{\infty }\left( \rho \left( \overline{w}\right) \right)$ and $l^{1}\left( \rho \left( \overline{w}\right) \right)$ (cf. sec. 1). # Appendix. {#appendix. .unnumbered} Consider any Banach couple $\overline{A}=\left\{ A_{0},A_{1}\right\} .$It is well-known (see e.g. [@6-CwPe]) $\overline{A}$ is termed [ Calderon]{.ul} if $b<<_{K}a$ implies $b<<_{b}a.$ $\overline{A}$ will be called $% \sigma -$additive if for any (infinite) sequence $\left\{ a_{j}\right\} \subseteq \Sigma A$ such that for some (or any) $t$ holds $\dsum_{j}K\left( t,a_{j}\right) <\infty$ we can find a sequence $\left\{ b_{j}\right\} \subseteq \Sigma A$ such that for any $t$ holds $K\left( t,\dsum b_{j}\right) \approx \dsum_{j}K\left( t,b_{j}\right)$ and (for each $j)$ $% K\left( t,a_{j}\right) \approx K\left( t,b_{j}\right) .$[^14]. Assuming that $% \overline{A}$ is both Calderon and $\sigma$-additive one can then easily prove the following fact (see [@3-BeL0], p. 12 remark 2.2, where the additive case is considered). If $A$ is any interpolation space with respect to $\overline{A}$ then there exists a sequence norm $\Phi$ such that $$\left\Vert a\right\Vert _{A}\approx \Phi \left( \left\{ K\left( \lambda ^{j},a\right) \right\} \right) . \tag{1}$$ (Every interpolation is a classical $K$ space.). Suppose moreover that $% \overline{A}$ satisfies the [strong form of the fundamental lemma]{.ul} , i.e. every $a\in \Sigma _{0}\overline{A}$ has a representation $a=\left\{ a_{k}\right\}$ such that for any $t$ holds $$\dsum \min \left( 1,\frac{t}{\lambda ^{k}}\right) J\left( \lambda ^{k},a_{k}\right) \leq C\cdot K\left( t,a\right)$$ with $C$ depending only on $\overline{A}$ (see [@6-CwPe], p. 31, th. 4.4, \"property $\left( 2\right) ^{^{\prime \prime }}),$ or $\Omega \left( \left\{ J\left( \lambda ^{k},a_{k}\right) \right\} \right) \leq C\cdot K\left( \lambda ^{k},a\right)$ where we have introduced the discrete Calderon transform $\Omega ,$i.e. $$\Omega \left( c\right) =\left\{ \dsum_{k}\min \left( 1,\lambda ^{j-k}\right) c_{k}\right\} _{j}\text{ if }c=\left\{ c_{k}\right\} _{k}.$$ Then follows readily (see [@6-CwPe], p.35-36, th. 4.8 and th. 4.9) that we have $$\left\Vert a\right\Vert _{A}=\inf_{a}\Psi \left( \left\{ J\left( \lambda ^{k},a\right) \right\} \right) , \tag{2}$$ where the inf goes over all representation $a=\left\{ a_{k}\right\}$ of $% a,\Psi$ being the sequence norm defined as $\Psi =\Phi \circ \Omega$. (All interpolation spaces are classical J spaces.)$\medskip$ Consider now the special case when $1^{\circ }$ $\overline{A}=l^{\infty }\left( \overline{w}\right)$ or $2^{\circ }$ $\overline{A}=l^{1}\left( \overline{w}\right)$ fixing first attention to the $\lambda$ -adic case, i.e. $\overline{w}=\left( \left\{ 1\right\} ,\left\{ \lambda ^{-k}\right\} \right)$ (so that $\overline{A}=\overline{l}_{\lambda }^{\infty }$ or $% \overline{A}=\overline{l}_{\lambda }^{1}$ respectively in the notation of sec. 3). $\medskip$ It is easy to see that in both cases we have a $\sigma$-additive Calderon couple and that in the second case also the strong form of the fundamental lemma holds. [^15].$\medskip$ In case $1^{\circ },$i.e. $\overline{A}=\overline{l}_{\lambda }^{\infty },$ substituting $a$ for the sequence $a^{^{\prime }}=\left\{ K\left( \lambda ^{j},a\right) \right\}$ in $\left( 1\right) ,$we then get since $K\left( \lambda ^{j},a^{^{\prime }}\right) \approx K\left( \lambda ^{j},a\right) :$ $$\left\Vert \left\{ K\left( \lambda ^{j},a\right) \right\} \right\Vert _{A}\approx \Phi \left( \left\{ K\left( \lambda ^{j},a\right) \right\} \right) .$$ so using $\left( 1\right)$ once more we see that $$\left\Vert a\right\Vert _{A}\approx \left\Vert \left\{ K\left( \lambda ^{j},a\right) \right\} \right\Vert _{A}. \tag{3}$$ In the same way in case $2^{\circ },$i.e. $\overline{A}=\overline{l}% _{\lambda }^{1},$ we obtain using $\left( 2\right)$ $$\left\Vert a\right\Vert _{A}\approx \inf_{a}\left\Vert \left\{ J\left( \lambda ^{j},a_{j}\right) \right\} \right\Vert _{A}. \tag{4}$$ It is easy to see that upon introducing an equivalent norm in $A$ we can achieve equality in $\left( 3\right)$ or $\left( 4\right) .$ (That much is of course not needed for the aims we are attempting at.)$\medskip$ We have thus established [1]{.ul}, formulas $\left( 3\right)$ and $% \left( 4\right)$ in the special case of $\lambda$-adic sequences.$\medskip$ The general case can be handled similarly by observing first that the K-functional $K\left( t,a\right)$ is up to equivalence determined by its restriction to the sequence $\left\{ \tau _{n}\right\} .$(Recall that $\tau _{n}=w_{n}^{0}/w_{n}^{1}.).\medskip$ There remains to establish [1]{.ul}, formulas $\left( 5\right)$ and $% \left( 6\right) .$ With no loss of generality we may take $% w_{n}^{1}=1,w_{n}^{1}=1/\tau _{n}.$ As for [1]{.ul}, $\left( 5\right)$ we notice as is well-known (see e.g. [@30-Pe]), that $$K\left( t,a\right) \approx \sup_{n}\min \left( 1,\frac{t}{\tau _{n}}\right) \left\vert a_{n}\right\vert \text{ if }a=\left\{ a_{n}\right\} .$$ We then see that $$\begin{aligned} \sup_{n}\frac{K\left( \tau _{n},a\right) }{\rho \left( \tau _{n}\right) } &\approx &\sup_{m}\sup_{n}\frac{\min \left( 1,\frac{\tau _{m}}{\tau _{n}}% \right) }{\rho \left( \tau _{n}\right) }\left\vert a_{m}\right\vert \\ &\approx &\sup_{m}\frac{\left\vert a_{m}\right\vert }{\rho \left( \tau _{m}\right) }=\left\Vert a\right\Vert _{l^{\infty }\left( \rho \left( \overline{w}\right) \right) }.\end{aligned}$$ Here we have effectively made use of $\rho \in \mathcal{P}$. This proves [1]{.ul}, $\left( 5\right) .$ [1]{.ul},$\left( 6\right)$ can be proven along similar lines. we omit the details (cf. [@6-CwPe]). # Notes {#notes .unnumbered} [Introduction.]{.ul} $\left\langle 1\right\rangle :$ After most of this was completed there appeared the note by Brudnyi and Krugjlak [@4-BrKr] listing a number of even more spectacular results. In particular the \"strong form of the fundamental lemma\" (cf. e.g. the appendix of the present paper) seems to be applicable to all couples with practically no reserve. As a result large portions of this paper could have been presented in a neater form. See also the forthcoming paper by Nilsson [@22-Nil] where the results of Brudnyi and Krugjlak are further exploited. $\left\langle 2\right\rangle .$ The same result appears also in the note by Brudnyi and Krugjlak [@4-BrKr].$\medskip$ [$\frac{1}{2}$]{.ul} . $\left\langle 1\right\rangle :$ If $\overline{X}$ is regular then $\left( \Sigma \overline{X}\right) ^{^{\prime }}\approx \Delta \overline{X}% ^{^{\prime }}$ where $\overline{X}^{^{\prime }}=\left( X_{0}^{^{\prime }},X_{1}^{^{\prime }}\right)$ is the dual pair. If $\overline{X}$ is not regular then $\overline{X}^{^{\prime }}$ is not formally defined but we have at any rate always canonical maps $\left( \Sigma \overline{X}\right) ^{^{\prime }}\rightarrow X_{j}^{^{\prime }},\left( j=0,1\right)$ which however need not be injective; $\left\Vert l_{n}\right\Vert _{X_{j}^{^{\prime }}}$ in $\left( 1\right)$ is thus an abusive notation for the norm of the of the image of $l_{n}.$ $\left\langle 2\right\rangle .$ By abuse of notation we now write $T$ in place of $T_{i}=T\mid _{X_{i}}.$We intend to follows this praxis in what follows. $\left\langle 3\right\rangle .$ Following [@6-CwPe] we say that we have a representation of $x$ (where $x\in \Sigma _{0}\overline{X})$ if there is given a sequence $\widehat{x}=\left\{ x_{v}\right\} _{v\in \mathbb{\mathbb{Z}% }}$ in $\Delta \overline{X}$ such that $x=\dsum x_{v}$ (convergence in $% \Sigma \overline{X}).$ $\left\langle 4\right\rangle :$ For any sequence $w=\left\{ w_{v}\right\} _{v\in \mathbb{Z}}$ of positive real numbers and $p\in \left[ 1,\infty % \right]$ we let $l^{p}\left( w\right)$ be the set of real sequences $% a=\left\{ a_{k}\right\} _{k\in \mathbb{Z}}$such that $\left( \dsum_{k\in \mathbb{\mathbb{Z}}}\left( w_{k}\left\vert a_{k}\right\vert \right) ^{p}\right) ^{1/p}<\infty$, with the usual interpretation if $p=\infty .\medskip$ [2]{.ul}. $\left\langle 1\right\rangle .$ This really gives an indication that one should have worked with the continuous version, not the discrete one. $\left\langle 2\right\rangle .$ This notation is inspired by Ovchinnikov [@25-Ovc]. $\left\langle 3\right\rangle .$I am grateful to Lars Gårding for directing my attention to Godement's work.$\medskip$ [3]{.ul}. $\left\langle 1\right\rangle .$ If $\left( 1\right)$ and $\left( 2\right)$ are not stipulated one usual takes the former condition as a definition; cf. [@17-KrPeSe].$\medskip$ [4]{.ul}. $\left\langle 1\right\rangle :$ Cwikel [@5-Cw] begins also the investigation of those couples $\overline{X}$ and $\overline{Y}$ for which the same conclusion holds if we substitute $\dsum_{k\in \mathbb{\mathbb{Z}}% }c_{k}<\infty$ for $\left( \dsum_{k\in \mathbb{Z}}c_{k}^{r}\right) ^{1/r}<\infty ,r$ a fixed number $>1.$(If we can pass to the limit $r=\infty$ we have a Calderon pair.). The infimum of the numbers $r$ which are permissible here apparently is an important invariant for the couples $% \overline{X}$ and $\overline{Y}.$ $\left\langle 2\right\rangle .$ Ovchinnikov quotes also the paper by Lindenstrauss-Pelczynski [@18-LiPe] - the main [raison d'etre]{.ul} of the latter was indeed to make Grothendieck's work available for a general audience - but as far as we have been able to see this particular consequence is not, at least not very explicitly, treated there. $\left\langle 3\right\rangle .$ The nuclearity. of $R:B\rightarrow l^{1}\left( w\right) ,$ where $B$ is any Banach space implies indeed that $% \dsum_{k}\left\Vert ^{t}R\left( e_{k}\right) \right\Vert _{B^{^{\prime }}}w_{k}<\infty$ where $e_{k}$ is the k'th basis vector of $l^{1}\left( w\right) .\medskip$ [5]{.ul}. $\left\langle 1\right\rangle .$ Cf. [@19-LiTz], prop. 1.c.8 where a similar device is used; I owe this observation to Per Nilsson. $\left\langle 2\right\rangle .$We borrow this term from [@17-KrPeSe]. $% \medskip$ [Appendix]{.ul}. $\left\langle 1\right\rangle .$From the work of Brudnyi-Krugljak. [@4-BrKr] referred to already in the Introduction we infer that the assumption of $\sigma$ - additivity is essentially superfluous. $\left\langle 2\right\rangle .$That we have a Calderon couple results from [@27-Pe] (cf. [@28-Pe], [@6-CwPe] and [@31-SeSe] respectively). The rest is easy to verify.$\medskip$ 10 N. Aronszajn - E. Gagliardo, Interpolation spaces and interpolation methods. Ann. Math. Pura Appl. 68 (1965), 51-118. J. Bergh, On the interpolation of normed linear spaces. Technical report, Lund, 1971. J. Bergh - J. Löfström, Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften 223, Springer, Berlin-Heidelberg-New York, 1976. Ju. A. Brudnyi - H. Ja. Krugljak, Real Interpolation functors. Dokl. Akad. Nauk SSSR 256 (1980), 14-17. (Russian.) M. Cwikel, Monotonicity properties of interpolation spaces II. *Arkiv för Matematik 19, 123--136 (1981).* M. Cwikel-J. Peetre, Abstract K and J spaces. J. Math. Pures Appl. 59 (1980), 1-49. J. Dixmier, Les $C^{\ast }-$algebres et leurs representations. Deuxieme ed. , Gauthier-Villiars, Paris, 1969. J. Dixmier, Les algebres d'operateurs dans l'espace hilbertien (algebres de von Neumann). Gauthier-Villars, Paris, 1957. V.I. Dmitriev-S.G. Krein-V.I. Ovchinnikov, Fundamentals of the theory of interpolation of linear operators. Geometry of linear spaces and operator theory, pp. 31-74. Jaroslav Gos. Univ., Jaroslavl, 1977. V.I. Dmitriev-V.I. Ovchinnikov, Interpolation in real method spaces. Dokl. Akad. Nauk SSSR 246 (1979), 794-797. (Russian.) C. Foias-J.L. Lions, Sur certains theoremes d'interpolation. Acta Szeged 22 (1961), 269-282. R. Godement, Sur la theorie des representations unitaires. Ann. Math. 53 (1951), 68-124. R. Godement, Theorie generale des sommes continues d'espaces de Banach. C.R. Acad.Sci. Paris 228 (1949), 1321-1323. A. Grothendieck, Resume de la theorie metrique des produits tensoriels topologiquies. Bol. Soc. Mat. Sao Paulo 8 (1956), 1-79. J. Gustavsson, On Interpolation of weighted $L^{p}$ spaces and Ovchinnikov's theorem. *Studia Mathematica 72, 237-251 (1982)*. S. Janson, Minimal and maximal methods of interpolation. *J. Functional Analysis 44, 50-73 (1981).* S. Krein-Ju. Petunin-E.Semenov, Interpolation of linear operators. Nauka. Moscow, 1978 (Russian.) J. Lindenstrauss-A.Pelczynski, Absolutely summing operators in $\mathfrak{L}_{p}$ spaces and their applications. Studia. Math. 29 (1968), 275-326. J. Lindenstrauss-L.Tzafriri, Classical Banach spaces I. Erg. der Math. 92, Springer, Berlin-Heidelberg-New York, 1977. J.L.Lions, personal communication. J.L. Lions-J.Peetre, Sur une classe d'espaces d'interpolation. Inst. Hautes Etudes Sc. Publ. Math. 19 (1964), 5-68. P. Nilsson, Reiteration theorems for real interpolation and approximation spaces. *Ann. Mat. Pura App., IV, Ser. 132, 291-339 (1982).* V.I.Ovchinnikov, Interpolation theorems resulting from Grothendieck's inequality. Funkcional. Anal. i Prilozen. 10 (1976), 45-54. (Russian.) V.I. Ovchinnikov, On the description of interpolation orbits. Funkcional. Anal. i Prilozen. 13 (1979), 85-86. (Russian.) V.I.Ovchinnikov, Interpolation of operators of the class $\mathfrak{S}_{p}$ into Hilbert pairs. Mat. Zametki 27 (1980), 273-282. J. Peetre, Remarks on Ovchinnikov's theorem. Abstracts, 19th scand. congr. math., Aarhus, Aug. 18-22, 1980, Various Publications Series No. 33, Matematisk instititut, Aarhus universitet, 1980. J. Peetre, Banach couples, I. Technical report, Lund, 1971, *see also arXiv.org 2303.06622 math.FA (2023).* J. Peetre, Interpolation functors and Banach couples. Actes Congres Intern. Math. vol. 2, 1970, pp. 373-381. Gauthiers-Villars, Paris, 1971. J. Peetre, New thoughts on Besov spaces. Mathematics Department, Duke University, Durham, 1976. J. Peetre, On an interpolation theorem of Foias and Lions. Acta Szeged 25 (1964), 255-261. A. Sedaev-E.Semenov, On the possibility of describing interpolation spaces in terms of Peetre's K-method. Optimizaciya 4 (1971), 98-114. (Russian.) J. Stafney, Analytic interpolation of certain interpolation spaces. Pac. J. Math. 32 (1960), 241-248. *Y.A. Brudnyi and N.Y. Kruglyak, Interpolation Functors and Interpolation Spaces 1. Elsevier, North-Holland (1991).* *Y.A Brudnyi, S.G. Krein, and E.M. Semenov, Interpolation of linear operators. J. Soviet Math 42, 2009--2113 (1988)*. *S. Kaijser, J.W. Pelletier, Interpolation theory and duality. Lecture Notes in Math. 1070 (1984), 152-168.* *S. Kaijser, J.W. Wick-Pelletier, Interpolation Functors and Duality, Lecture Notes in Math 1208 (1986).* *P. Nilsson, Interpolation of Calderón and Ovčinnikov pairs. Ann. Mat. Pura Appl. 134, 201--232 (1983)*. *V. I. Ovchinnikov, The Method of Orbits in Interpolation Theory, Math. Reports 1 (1984), 349--516.* [^1]: For references, post 1981, see notably Brudnyi-Krugljak [@33-BK Chapter 2, Chapter 4], Brudnyi-Krein-Semenov [@34-BrKrSe], Kaijer-Pelletier [@35-KaPe],[@36-KaPe-II],, Ovchinnikov [@38-Ov84], Nilsson [@37-Ni94] and the references listed there (especially [@34-BrKrSe]). [^2]: See Notes: [Introductioṅ]{.ul}:$\left\langle 1\right\rangle .$ [^3]: See Notes: [Introductioṅ]{.ul}:$\left\langle 2\right\rangle .$ [^4]: See Notes [$\frac{1}{2}$]{.ul}:$\left\langle 1\right\rangle .$ [^5]: See Notes [$\frac{1}{2}$]{.ul}:$\left\langle 4\right\rangle .$ [^6]: See Notes [2]{.ul}:$\left\langle 1\right\rangle .$ [^7]: See Notes [2]{.ul}:$\left\langle 2\right\rangle .$ [^8]: See Notes [2]{.ul}:$\left\langle 3\right\rangle .$ [^9]: See Notes [3]{.ul}:$\left\langle 1\right\rangle$ [^10]: See Notes [5]{.ul}:$\left\langle 2\right\rangle$ [^11]: See Notes [4]{.ul}:$\left\langle 3\right\rangle$ [^12]: See Notes [5]{.ul}:$\left\langle 1\right\rangle$ [^13]: See Notes [5]{.ul}.$\left\langle 2\right\rangle$ [^14]: See Notes Appendix:$\left\langle 1\right\rangle .$ [^15]: See notes Appendix: $\left\langle 2\right\rangle .$

arxiv_math

--- abstract: | We establish $L^2$-exponential decay properties for linear dissipative kinetic equations, including the time-relaxation and Fokker-Planck models, in bounded spatial domains with general boundary conditions that may not conserve mass. Their diffusion asymptotics in $L^2$ is also derived under general Maxwell boundary conditions. The proofs are simply based on energy estimates together with previous ideas from $L^2$-hypocoercivity and relative entropy methods. address: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK author: - Yuzhe Zhu title: A note on asymptotics of linear dissipative kinetic equations in bounded domains --- # Introduction {#intro} Consider the time-relaxation operator $\mathcal{L}_1$ and the Fokker-Planck operator $\mathcal{L}_2$ defined by $$\begin{aligned} \mathcal{L}_1f(v) := \int_{\mathbb{R}^d}f(v)\mathop{}\!\mathrm{d}\mu-f(v) {\quad\rm and\quad} \mathcal{L}_2f(v) := (\nabla_v-v)\cdot\nabla_v f(v) {\quad\rm for\ } f(v)\in C_c^\infty(\mathbb{R}^d). \end{aligned}$$ Here we write $\mu(v):=(2\pi)^{-d/2}e^{-|v|^2/2}$ and $\mathop{}\!\mathrm{d}\mu:=\mu(v)\mathop{}\!\mathrm{d}v$ to denote the Gaussian function and the Gaussian measure, respectively. Let $\Omega$ be a bounded $C^{1,1}$-domain in $\mathbb{R}^d$, and $n_x$ be the unit outward normal vector at $x\in\partial\Omega$. We split the phase boundary $\Sigma:=\partial\Omega\times\mathbb{R}^d$ into the outgoing part $\Sigma_+$, incoming part $\Sigma_-$ and grazing part $\Sigma_0$, which are precisely defined as follows, $$\begin{aligned} \Sigma_\pm:= \left\{ (x,v)\in\Sigma :\, \pm\;\! n_x\cdot v >0 \right\} {\quad\rm and\quad} \Sigma_0:= \left\{ (x,v)\in\Sigma :\, n_x\cdot v =0 \right\}. \end{aligned}$$ Let $i=1$ or $2$. We are concerned with asymptotic behaviours of the solution $f_\varepsilon=f_\varepsilon(t,x,v)$, $(t,x,v)\in\mathbb{R}_+\times\Omega\times\mathbb{R}^d$, to the following dissipative kinetic equation with the parameter $\varepsilon\in(0,1]$, $$\label{rfp} \left\{ \begin{aligned} \ &\varepsilon\partial_t f_\varepsilon+v\cdot\nabla_xf_\varepsilon-\nabla_x\phi\cdot\nabla_vf_\varepsilon=\varepsilon^{-1}\mathcal{L}_i f_\varepsilon\quad {\rm in\ } \mathbb{R}_+\times\Omega\times\mathbb{R}^d, \\ \ &\,f_\varepsilon|_{t=0} =f_{\rm in}\quad {\rm in\ } \Omega\times\mathbb{R}^d, \\ \ &\,f_\varepsilon=\alpha\mathcal{D}f_\varepsilon+\beta\mathcal{R}f_\varepsilon{\quad\rm in\ }\mathbb{R}_+\times\Sigma_-. \\ \end{aligned} \right.$$ Here the external potential $\phi\in C^{0,1}(\Omega)$, and the coefficients $\alpha,\beta:\partial\Omega\rightarrow[0,1]$ are measurable and satisfy $\alpha+\beta\le1$ and $\alpha+\beta\in C^{0,1}(\partial\Omega)$. For $f(x,v)\in C_c^\infty(\Sigma)$, the diffuse reflection boundary operator $\mathcal{D}$ and the specular reflection operator $\mathcal{R}$ are defined by $$\begin{aligned} \mathcal{D}f(x):=c_w \int_{\mathbb{R}^d}f(x,v)\,(n_x\cdot v)_+\mathop{}\!\mathrm{d}\mu {\quad\rm and\quad} \mathcal{R}f(x,v):=f(x,v-2(n_x\cdot v)\;\!n_x), \end{aligned}$$ where the constant $c_w:=\sqrt{2\pi}$ so that $c_w\int_{\mathbb{R}^d}(n_x\cdot v)_\pm\mathop{}\!\mathrm{d}\mu=1$. For brevity, we also set $$\begin{aligned} \mathop{}\!\mathrm{d}m:=e^{-\phi(x)}\mu(v)\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}v {\quad\rm and\quad} \|\cdot\|:=\|\cdot\|_{L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)}. \end{aligned}$$ Throughout the article, we assume that the solutions mentioned, which exist in bounded spatial domains in the weak sense, have suitable traces and satisfy Green's (renormalization) formula. One may refer to [@Mischler; @DHHM] for details in this regard. We aim in this work at investigating the long-time asymptotics of the solution $f_\varepsilon$ to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} and its limit as $\varepsilon\rightarrow0$. When $\alpha+\beta=1$, [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} is associated with the so-called *Maxwell boundary condition* which conserves the mass. Generally, [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} is not conservative. Our first result gives hypocoercive estimates with uniform-in-$\varepsilon$ exponential decay rates for solutions to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} associated with general boundary conditions. **Theorem 1**. *Let $\varepsilon\in(0,1]$, $f_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$, $M_0:=\left(\int_\Omega e^{-\phi}\mathop{}\!\mathrm{d}x\right)^{\!-1} \int_{\Omega\times\mathbb{R}^d} f_{\rm in}\mathop{}\!\mathrm{d}m$, and the functions $\alpha,\beta:\partial\Omega\rightarrow[0,1]$ satisfy $\alpha+\beta\le1$ and $\alpha+\beta\in C^{0,1}(\partial\Omega)$, and $f_\varepsilon$ be the solution to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}. If $\alpha+\beta=1$, then there are some constants $C,\lambda>0$ depending only on $d,\Omega,\|\phi\|_{C^{0,1}(\Omega)}$ such that for any $t>0$, $$\|f_\varepsilon(t)-M_0\| \le C e^{-\lambda t}\|f_{\rm in}-M_0\|.$$ If $\alpha+\beta\in[0,\delta]$ for some constant $\delta\in[0,1)$, then there are some constants $C',\lambda'>0$ depending only on $d,\Omega,\|\phi\|_{C^{0,1}(\Omega)},\delta,\|\nabla(\alpha+\beta)\|_{L^\infty(\partial\Omega)}$ such that for any $t>0$, $$\|f_\varepsilon(t)\| \le C' e^{-\lambda't}\|f_{\rm in}\|.$$* The second result shows that the asymptotic dynamics of [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} (with $\beta=1-\alpha$) as $\varepsilon\rightarrow0$ is governed by the following parabolic equation associated with the Neumann boundary condition, $$\label{limiteq} \left\{ \begin{aligned} \ &\partial_t\rho=\Delta_x\rho -\nabla_x\phi\cdot\nabla_x\rho {\quad\rm in\ } \mathbb{R}_+\times\Omega, \\ \ &\, \rho|_{t=0} = \rho_{\rm in}{\quad\rm in\ } \Omega,\\ \ &\, n_x\cdot\nabla_x\rho=0 {\quad\rm in\ } \mathbb{R}_+\times\partial\Omega.\\ \end{aligned} \right.$$ **Theorem 2**. *Let $\varepsilon\in(0,1]$, $f_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$, and $\alpha:\partial\Omega\rightarrow[0,1]$. Suppose that $f_\varepsilon$ is the solution to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} (with $\beta=1-\alpha$) associated with the Maxwell boundary condition $f_\varepsilon=\alpha\mathcal{D}f_\varepsilon+(1-\alpha)\mathcal{R}f_\varepsilon$ on $\mathbb{R}_+\times\Sigma_-$, and $\rho$ is the solution to [\[limiteq\]](#limiteq){reference-type="eqref" reference="limiteq"} associated with the initial data $\rho_{\rm in}:=\int_{\mathbb{R}^d}f_{\rm in}\mathop{}\!\mathrm{d}\mu$. Then we have the strong convergence that for any $T>0$ and any positive function $\tau$ defined on $(0,1]$ such that $\tau(\varepsilon)/\varepsilon^2\rightarrow\infty$ as $\varepsilon\rightarrow0$, $$\begin{aligned} \|f_\varepsilon-\rho\|_{L^2([0,T]\times\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}t\mathop{}\!\mathrm{d}m)} \rightarrow 0,\quad \|f_\varepsilon-\rho\|_{L^\infty([\tau(\varepsilon),\infty);L^2(\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}m))} \rightarrow 0, {\quad\rm as\ }\varepsilon\rightarrow0.\end{aligned}$$ If $f_{\rm in}\in L_v^2H_x^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$ satisfies $|v|\nabla_xf_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$ and the compatibility condition that $f_{\rm in}$ depends only on the variable $x$ in $\partial\Omega\times\mathbb{R}^d$, then for any $T>0$ and for $T_\varepsilon:=\varepsilon^2|\log\sqrt{\varepsilon}|$, $$\begin{aligned} \|f_\varepsilon-\rho\|_{L^2([0,T]\times\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}t\mathop{}\!\mathrm{d}m)} \le C \sqrt{\varepsilon T} \left(\|f_{\rm in}\| +\|(1+|v|)\nabla_xf_{\rm in}\| +\|D_x^2f_{\rm in}\| \right),\\ \|f_\varepsilon-\rho\|_{L^\infty([T_\varepsilon,\infty);L^2(\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}m))} \le C \sqrt{\varepsilon} \left(\|f_{\rm in}\| +\|(1+|v|)\nabla_xf_{\rm in}\| +\|D_x^2f_{\rm in}\| \right). \end{aligned}$$ If $f_{\rm in}$ is well-prepared in the sense that $f_{\rm in}\in H_x^2(\Omega)$ depends only on the variable $x$ in $\Omega\times\mathbb{R}^d$, then $$\begin{aligned} \|f_\varepsilon-\rho\|_{L^\infty(\mathbb{R}_+;\;\!L^2(\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}m))} \le C\sqrt{\varepsilon} \|f_{\rm in}\|_{H_x^2(\Omega)}. \end{aligned}$$ Here the constant $C>0$ depends only on $d,\Omega,\|\phi\|_{C^{0,1}(\Omega)}$.* The model [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}, with $i=1$ or $2$, is a kinetic description of the probability distribution of a certain system of interacting particles, submitted to an external force derived from the potential $\phi$, at time $t$ located at the position $x$ in the physical space $\Omega\subset\mathbb{R}^d$ with the velocity $v\in\mathbb{R}^d$. It is complemented with the proper boundary conditions corresponding to the behaviour of particles at the boundary. The boundary condition in [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} is prescribed as zero influx when $\alpha=\beta=0$, and it is generally governed by the balance relations between the distribution of particles at the incoming and outgoing boundaries. The operator $\mathcal{L}_1$ serves as a prototype for describing collision processes of linear relaxation type, including the case of neutron transport. The operator $\mathcal{L}_2$ derived from the Ornstein-Uhlenbeck velocity process captures the impact of collisions with particles of a surrounding bath. The collision operators act only on the velocity variable and conserve local mass. They are both dissipative in the velocity variable and contribute to the relaxation process towards local equilibrium. The *hypocoercivity* theory seeks to understand how the interaction between transport and collisions leads to time-decay convergence for inhomogeneous collisional kinetic equations such as [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}. The (small) parameter $\varepsilon\in(0,1]$ representing the ratio of the collisional mean free path and the observation length is introduced to measure the balance between the transport part $v\cdot\nabla_x$ and the collision part $\mathcal{L}_i$, while diffusion phenomena can be observed on long time scales of order $\varepsilon^{-1}$. This regime with $\varepsilon\rightarrow0$ is often referred to as the *diffusion limit*. In the absence of boundary conditions, the systematic development of $H^1$-hypocoercivity for kinetic diffusion equations (such as [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} with $i=2$) can be found in [@Villanihypo], which typically involves high order regularity estimates for solutions. Nevertheless loss of regularity may occur near the grazing boundaries; see [@YZ Appendix A]. The $L^2$-hypocoercivity instead provides $L^2$-decay for a large class of dissipative kinetic equations without assuming regularity on the solution; see for instance [@Herau; @DMS1; @EGKM; @DMS2; @BDMMS], where the spatial domain is either the periodic box $\mathbb{T}^d$ or the whole space $\mathbb{R}^d$. The approaches given in [@BCMT] and [@DHHM] yield $L^2$-decay estimates when considering boundary conditions that conserve mass. In [@DD], $L^2$-decay estimates were established for a specific scenario involving exponentially decaying influx boundary conditions. We also point out that, for kinetic Fokker-Planck equations, constructive exponential convergence result was given in [@AM2021] by means of a Poincaré-type inequality under the zero influx boundary condition, and nonconstructive estimates in a general non-conservative setting were given in [@FGM] by revisiting the Krein-Rutman theory. The mathematical study of approximating linear kinetic equations through macroscopic diffusion equations has been investigated in previous works such as [@LK; @BLP; @BSS; @Degond]. It was noticed in [@WG] that the presence of grazing boundaries may cause a breakdown in the diffusive asymptotics within the $L^\infty$ framework. We refer to [@Wu3; @Ouyang] for recent development of the diffusion limit of the neutron transport equation associated with boundary conditions. In addition to the analysis through formal expansions as employed in the aforementioned works, the relative entropy method has also emerged as a valuable approach in addressing the hydrodynamic asymptotics; see for instance [@BGL; @LSR]. Our study draws inspiration from the $L^2$-hypocoercivity and relative entropy methods used in [@AZ] which focused on a nonlinear kinetic Fokker-Planck model with the spatial domain of either $\mathbb{T}^d$ or $\mathbb{R}^d$. By making simple observations on the boundary terms and employing basic energy estimates, we are able to derive quantitative $L^2$-hypocoercive estimates uniformly in $\varepsilon\in(0,1]$ and $L^2$-diffusion asymptotics as $\varepsilon\rightarrow0$ for solutions to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} in the presence of boundary conditions. The article is organized as follows. In Section [2](#pre){reference-type="ref" reference="pre"}, we introduce some notations and present preliminary estimates related to our results. We prove Theorem [Theorem 1](#longtime){reference-type="ref" reference="longtime"} in Section [3](#hypocoercive){reference-type="ref" reference="hypocoercive"} and Theorem [Theorem 2](#asympto){reference-type="ref" reference="asympto"} in Section [4](#diffusionlimit){reference-type="ref" reference="diffusionlimit"}. Some basic elliptic estimates are recalled in Appendix [5](#append){reference-type="ref" reference="append"}. # Preliminaries {#pre} ## Notations Let us introduce the notations that will be used. For any couple of (scalar, vector or $d\times d$-matrix valued) functions $\Psi_1,\Psi_2\in L^2\left(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m\right)$, we denote their inner product with respect to the measure $\mathop{}\!\mathrm{d}m=e^{-\phi}\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}\mu$ by $$\begin{aligned} \left(\Psi_1,\, \Psi_2\right) :=\int_{\Omega\times\mathbb{R}^d} \Psi_1\Psi_2\mathop{}\!\mathrm{d}m, \end{aligned}$$ where the multiplication between the couple in the integrand is replaced by scalar contraction product, if $\Psi_1,\Psi_2$ is a couple of vectors or matrices. Then $\|\cdot\|$ is its induced norm. Since $\Omega$ is a bounded domain and $\phi$ is Lipschitz, the norm $\|\cdot\|$ is equivalent to the norm $\|\cdot\|_{L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}\mu)}$. Similarly, with $\mathop{}\!\mathrm{d}\sigma_x$ denoting the surface measure on $\partial\Omega$, the inner product on the phase boundary $\partial\Omega\times\mathbb{R}^d$ for $\Psi_1,\Psi_2\in L^2\left(\partial\Omega\times\mathbb{R}^d,e^{-\phi}\mathop{}\!\mathrm{d}\sigma_x\mathop{}\!\mathrm{d}\mu\right)$ is defined by $$\begin{aligned} \left(\Psi_1,\, \Psi_2\right)_\partial:=\int_{\partial\Omega\times\mathbb{R}^d} \Psi_1\Psi_2\,e^{-\phi}\mathop{}\!\mathrm{d}\sigma_x\mathop{}\!\mathrm{d}\mu. \end{aligned}$$ Its induced norm is denoted by $\|\cdot\|_\partial$. In this setting, we have $$\begin{aligned} \left(\psi_1,\,(\nabla_v-v)\psi_2\right) = -\left(\nabla_v\psi_1,\,\psi_2\right), \end{aligned}$$ and the transport operator $v\cdot\nabla_x-\nabla_x\phi\cdot\nabla_v$ satisfies $$\begin{aligned} \left((v\cdot\nabla_x-\nabla_x\phi\cdot\nabla_v)\psi_1,\,\psi_2\right) +\left(\psi_1,\, (v\cdot\nabla_x-\nabla_x\phi\cdot\nabla_v)\psi_2\right) =\left(\psi_1\psi_2,\, n_x\cdot v\right)_\partial, \end{aligned}$$ for any scalar functions $\psi_1,\psi_2\in C_c^\infty(\overline{\Omega}\times\mathbb{R}^d)$. For $\Psi\in L^1(\mathbb{R}^d,\mathop{}\!\mathrm{d}\mu)$, we denote the projection onto Span$\{\mu\}$ and its orthogonal complement by $$\begin{aligned} \langle\Psi\rangle:=\int_{\mathbb{R}^d} \Psi\mathop{}\!\mathrm{d}\mu {\quad\rm and\quad} \Psi^\perp:=\Psi-\langle\Psi\rangle.\end{aligned}$$ Taking the bracket $\langle\cdot\rangle$ after multiplying the equation in [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} with $1$ and $v$ leads to the following macroscopic equations, $$\begin{aligned} &\varepsilon\partial_t\langle f_\varepsilon\rangle+\nabla_x\cdot\langle vf_\varepsilon\rangle-\nabla_x\phi\cdot\langle vf_\varepsilon\rangle= 0, \label{hydroFP1} \\ &\varepsilon\partial_t\langle vf_\varepsilon\rangle+ \nabla_x\cdot\langle v\otimes vf_\varepsilon\rangle-\nabla_x\phi\cdot\langle(v\otimes v-{\rm I})f_\varepsilon\rangle= -\varepsilon^{-1}\langle vf_\varepsilon\rangle. \label{hydroFP2}\end{aligned}$$ Throughout the rest of the article, the notation $X\lesssim Y$ means that $X\le CY$ for some constant $C>0$ depending only on $d,\Omega,\|\phi\|_{C^{0,1}(\Omega)},\delta,\|\nabla(\alpha+\beta)\|_{L^\infty(\partial\Omega)}$. ## Boundary estimates We start by presenting the following two identities, which are applicable to functions subject to general boundary conditions and will play a crucial role in handling the boundary terms in the later estimates. **Lemma 3**. *Let the functions $\alpha,\beta:\partial\Omega\rightarrow[0,1]$ satisfy $\alpha+\beta\le1$, and the vector $U:\partial\Omega\rightarrow\mathbb{R}^d$ lying in $L^2(\partial\Omega)$ satisfy $n_x\cdot U=0$. For any function $f:\Sigma\rightarrow\mathbb{R}$ such that $f=\alpha\mathcal{D}f+\beta\mathcal{R}f$, we have $$\begin{aligned} &\left(v\cdot U f,\,n_x\cdot v\right)_\partial= \left(v\cdot U [(1-\beta)f-\alpha\mathcal{D}f],\,(n_x\cdot v)_+\right)_\partial, \label{identity1} \\ &\left(f^2,\,n_x\cdot v\right)_\partial =\left((1-\beta^2)(f-\mathcal{D}f)^2,\,(n_x\cdot v)_+\right)_\partial+ \left((1-(\alpha+\beta)^2)(\mathcal{D}f)^2,\,(n_x\cdot v)_+\right)_\partial\label{identity2} \\ &\qquad\qquad\quad\;\,= \left((\alpha^2+2\alpha\beta)(f-\mathcal{D}f)^2,\,(n_x\cdot v)_+\right)_\partial+ \left((1-(\alpha+\beta)^2)f^2,\,(n_x\cdot v)_+\right)_\partial. \nonumber\end{aligned}$$* *Proof.* According to the definition, $$\begin{aligned} \left(v\cdot Uf,\,n_x\cdot v\right)_\partial =\left(v\cdot Uf,\,(n_x\cdot v)_+\right)_\partial-\left(v\cdot U \alpha\mathcal{D}f,\,(n_x\cdot v)_-\right)_\partial -\left(v\cdot U\beta\mathcal{R}f,\,(n_x\cdot v)_-\right)_\partial. \end{aligned}$$ On the one hand, since the vector field $U\mathcal{D}f$ is independent of $v$ and tangent to $n_x$, we have $$\begin{aligned} \left(v\cdot U \alpha\mathcal{D}f,\,(n_x\cdot v)_\pm\right)_\partial=0. \end{aligned}$$ On the other hand, by a change of variables, $$\begin{aligned} \left(v\cdot U\beta\mathcal{R}f,\,(n_x\cdot v)_-\right)_\partial =\left(v\cdot U\beta f,\,(n_x\cdot v)_+\right)_\partial -\left(2(n_x\cdot v)n_x\cdot U\beta f,\,(n_x\cdot v)_+\right)_\partial, \end{aligned}$$ where the second term on the right hand side vanishes due to the assumption on $U$. The above three identities thus shows [\[identity1\]](#identity1){reference-type="eqref" reference="identity1"}. As for [\[identity2\]](#identity2){reference-type="eqref" reference="identity2"}, we observe that for any bounded function $a_b$ defined on $\partial\Omega$ (we will take $a_b$ to be either $1-\beta^2$ or $\alpha^2+2\alpha\beta$), we have $$(a_bf\mathcal{D}f,\,(n_x\cdot v)_+)_\partial=(a_b(\mathcal{D}f)^2,\,(n_x\cdot v)_+)_\partial,$$ which implies that $$\begin{aligned} \left(a_b(f-\mathcal{D}f)^2,\,(n_x\cdot v)_+\right)_\partial =\left(a_bf^2,\,(n_x\cdot v)_+\right)_\partial- \left(a_b(\mathcal{D}f)^2,\,(n_x\cdot v)_+\right)_\partial. \end{aligned}$$ By definition and changes of variables, we have $$\begin{aligned} \left(f^2,\,n_x\cdot v\right)_\partial = \left(f^2,\,(n_x\cdot v)_+\right)_\partial-\left(\alpha^2(\mathcal{D}f)^2,\,(n_x\cdot v)_-\right)_\partial\\ -\left(2\alpha\beta\mathcal{D}f\mathcal{R}f,\,(n_x\cdot v)_-\right)_\partial-\left(\beta^2(\mathcal{R}f)^2,\,(n_x\cdot v)_-\right)_\partial\\ = \left(f^2,\,(n_x\cdot v)_+\right)_\partial-\left(\alpha^2(\mathcal{D}f)^2,\,(n_x\cdot v)_+\right)_\partial -\left(2\alpha\beta f\mathcal{D}f,\,(n_x\cdot v)_+\right)_\partial-\left(\beta^2 f^2,\,(n_x\cdot v)_+\right)_\partial\\ =\left((1-\beta^2)f^2,\,(n_x\cdot v)_+\right)_\partial- \left((\alpha^2+2\alpha\beta)(\mathcal{D}f)^2,\,(n_x\cdot v)_+\right)_\partial&. \end{aligned}$$ The proof of [\[identity2\]](#identity2){reference-type="eqref" reference="identity2"} is complete by gathering the above two identities. ◻ **Lemma 4**. *Let the functions $\alpha,\beta:\partial\Omega\rightarrow[0,1]$ satisfy $\alpha+\beta\le1$, and the vector $U:\partial\Omega\rightarrow\mathbb{R}^d$ lying in $L^2(\partial\Omega)$ satisfy $n_x\cdot U=0$. Suppose that the function $c_b:=(1-\alpha-\beta)/(\alpha+\beta)\in L^\infty(\partial\Omega)$. For any functions $\varrho\in L^2(\partial\Omega)$ and $f:\Sigma\rightarrow\mathbb{R}$ such that $f=\alpha\mathcal{D}f+\beta\mathcal{R}f$, we have $$\begin{aligned} \left|\left(v\cdot(U-c_b\varrho n_x) f,\,n_x\cdot v\right)_\partial\right| \lesssim \left(\|U-c_b\varrho n_x\|_\partial+ \|\sqrt{c_b}\varrho\|_\partial\right) \left(f^2,\, n_x\cdot v\right)_\partial^\frac{1}{2}. \end{aligned}$$* *Proof.* By a change of variables, $$\left(c_b\varrho f,\,(n_x\cdot v)^2\right)_\partial=\left(c_b\rho[(1+\beta)f+\alpha\mathcal{D}f],\,(n_x\cdot v)_+^2\right)_\partial.$$ In view of [\[identity1\]](#identity1){reference-type="eqref" reference="identity1"} of Lemma [Lemma 3](#identity){reference-type="ref" reference="identity"}, we obtain $$\begin{aligned} \left(v\cdot(U-c_b\varrho n_x) g,\,n_x\cdot v\right)_\partial =\left(v\cdot U[(1-\beta)f-\alpha\mathcal{D}f],\,(n_x\cdot v)_+\right)_\partial - \left(c_b\varrho f,\,(n_x\cdot v)^2\right)_\partial\\ =\left(v\cdot(U-c_b\varrho n_x)[(1-\beta)f-\alpha\mathcal{D}f],\,(n_x\cdot v)_+\right)_\partial - 2\left(c_b\varrho(\beta f+\alpha\mathcal{D}f),\,(n_x\cdot v)_+^2\right)_\partial&. \end{aligned}$$ By using Hölder's inequality, we obtain $$\begin{aligned} \left|\left(v\cdot(U-c_b\varrho n_x) g,\,n_x\cdot v\right)_\partial\right| \lesssim \|U-c_b\varrho n_x\|_\partial\left((1-\beta)^2(f-\mathcal{D}f)^2 +(1-\alpha-\beta)^2(\mathcal{D}f)^2,\, (n_x\cdot v)_+ \right)_\partial^\frac{1}{2} \\ + \|\sqrt{c_b}\varrho\|_\partial\left((1-\alpha-\beta)\big[f^2+(\mathcal{D}f)^2\big],\,(n_x\cdot v)_+\right)_\partial^\frac{1}{2} &. \end{aligned}$$ Applying [\[identity2\]](#identity2){reference-type="eqref" reference="identity2"} of Lemma [Lemma 3](#identity){reference-type="ref" reference="identity"}, as well as the elementary inequalities that $(1-\beta)^2\le1-\beta^2$ and $(1-\alpha-\beta)^2\le 1-\alpha-\beta \le1-(\alpha+\beta)^2$, we then arrive at the desired result. ◻ ## Entropy dissipation The dissipation property exhibited by the operator $\mathcal{L}_i$, which acts only on the velocity variable, is formulated in the following lemma. **Lemma 5**. *If $f_\varepsilon$ is a solution to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} associated with the initial data $f_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$ and the boundary condition $f_\varepsilon=\alpha\mathcal{D}f_\varepsilon+\beta\mathcal{R}f_\varepsilon$ on $\mathbb{R}_+\times\Sigma_-$, then we have $$\begin{aligned} \label{dissipation-h} \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t} \|f_\varepsilon\|^2 \le -\frac{2}{\varepsilon^2}\|f_\varepsilon^\perp\|^2- \frac{1}{\varepsilon}\left(f_\varepsilon^2,\, n_x\cdot v\right)_\partial, \end{aligned}$$ with $\left(f_\varepsilon^2,\, n_x\cdot v\right)_\partial\ge0$. In particular, for any $t>0$, $$\begin{aligned} \label{dissipationR} \|f_\varepsilon(t)\|^2 +\frac{2}{\varepsilon^2} \int_0^t \|f_\varepsilon^\perp\|^2 +\frac{1}{\varepsilon}\int_0^t\left(f_\varepsilon^2,\,n_x\cdot v\right)_\partial \le \|f_{\rm in}\|^2. \end{aligned}$$* *Proof.* A direct computation by integrating [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} against $\mu e^{-\phi}f_\varepsilon$ yields that $$\frac{1}{2}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|f_\varepsilon\|^2 =\left(f_\varepsilon,\,\partial_t f_\varepsilon\right) =\frac{1}{\varepsilon^2} \left(\mathcal{L}_if_\varepsilon,f_\varepsilon\right) -\frac{1}{2\varepsilon} \left(f_\varepsilon^2,\,n_x\cdot v\right)_\partial \le -\frac{1}{\varepsilon^2}\|f_\varepsilon^\perp\|^2 -\frac{1}{2\varepsilon} \left(f_\varepsilon^2,\,n_x\cdot v\right)_\partial.$$ Indeed, we used the definition of $\mathcal{L}_i$ and the Gaussian-Poincaré inequality with $\langle f_\varepsilon^\perp\rangle=0$ to see that $$\begin{aligned} \left(\mathcal{L}_1f_\varepsilon,f_\varepsilon\right) = -\|f_\varepsilon^\perp\|^2, \quad \left(\mathcal{L}_2f_\varepsilon,f_\varepsilon\right) = -\|\nabla_vf_\varepsilon^\perp\|^2 \le -\|f_\varepsilon^\perp\|^2.\end{aligned}$$ Here we point out that $\left(f_\varepsilon^2,\,n_x\cdot v\right)_\partial\ge0$ due to [\[identity2\]](#identity2){reference-type="eqref" reference="identity2"} of Lemma [Lemma 3](#identity){reference-type="ref" reference="identity"}. Integrating along time, we derive [\[dissipationR\]](#dissipationR){reference-type="eqref" reference="dissipationR"} as claimed. ◻ The following lemma gives a weighted dissipation estimate for solutions to the homogeneous counterpart of [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}, which will be used in Section [4](#diffusionlimit){reference-type="ref" reference="diffusionlimit"} to address the initial layer correction. It is worth noting that the variable $x$ involved in the lemma below serves as a parameter and therefore does not affect the relaxation process. **Lemma 6**. *There exists some constant $c_0>0$ such that for any solution $\psi_\varepsilon$ to $$\label{layereq} \left\{ \begin{aligned} \ &\partial_t\psi_\varepsilon=\varepsilon^{-2}\mathcal{L}_i\psi_\varepsilon\quad {\rm in\ } \mathbb{R}_+\times\Omega\times\mathbb{R}^d, \\ \ &\psi_\varepsilon|_{t=0}=\psi_{\rm in}\quad {\rm in\ } \Omega\times\mathbb{R}^d, \\ \end{aligned} \right.$$ with the initial data $\psi_{\rm in}$ satisfying $(1+|v|)\psi_{\rm in}\in L^2(\mathbb{R}^d,\mathop{}\!\mathrm{d}\mu)$ and $\langle\psi_{\rm in}\rangle=0$, we have $$\begin{aligned} \|\psi_\varepsilon(t)\|^2 +\frac{2}{\varepsilon^2} \int_0^t \|\psi_\varepsilon\|^2 \le \|\psi_{\rm in}\|^2, \label{layerdecay1} \\ \|(1+|v|)\psi_\varepsilon(t)\| \lesssim e^{-c_0t/\varepsilon^2}\|(1+|v|)\psi_{\rm in}\|. \label{layerdecay2}\end{aligned}$$* *Proof.* We first observe that $\langle\psi_\varepsilon(t)\rangle=0$ for any $t\ge0$. One is thus able to arrive at [\[layerdecay1\]](#layerdecay1){reference-type="eqref" reference="layerdecay1"} in the same way as the derivation of [\[dissipationR\]](#dissipationR){reference-type="eqref" reference="dissipationR"} in Lemma [Lemma 5](#dissipation){reference-type="ref" reference="dissipation"}. In order to show the weighted estimate [\[layerdecay2\]](#layerdecay2){reference-type="eqref" reference="layerdecay2"} for solutions to [\[layereq\]](#layereq){reference-type="eqref" reference="layereq"} with $i=1$ (the time-relaxation case), we integrate [\[layereq\]](#layereq){reference-type="eqref" reference="layereq"} against $\mu e^{-\phi}\psi_\varepsilon$ and $\mu e^{-\phi}|v|^2\psi_\varepsilon$ so that $$\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|\psi_\varepsilon\|^2= -\frac{2}{\varepsilon^2}\|\psi_\varepsilon\|^2,\quad \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|v\psi_\varepsilon\|^2 = -\frac{2}{\varepsilon^2}\|v\psi_\varepsilon\|^2,$$ which then implies [\[layerdecay2\]](#layerdecay2){reference-type="eqref" reference="layerdecay2"} from Grönwall's inequality directly. As for [\[layereq\]](#layereq){reference-type="eqref" reference="layereq"} with $i=2$ (the Fokker-Planck case), we apply integration by parts, Gaussian-Poincaré inequality and Cauchy-Schwarz inequality so that $$\begin{aligned} \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|\psi_\varepsilon\|^2 &=-\frac{2}{\varepsilon^2}\|\nabla_v\psi_\varepsilon\|^2 \le -\frac{1}{\varepsilon^2}\left(\|\psi_\varepsilon\|^2+\|\nabla_v\psi_\varepsilon\|^2\right),\\ \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|v\psi_\varepsilon\|^2 &=-\frac{2}{\varepsilon^2}\int_{\Omega\times\mathbb{R}^d}\left(|v|^2|\nabla_v\psi_\varepsilon|^2+2\psi_\varepsilon v\cdot\nabla_v\psi_\varepsilon\right)\mathop{}\!\mathrm{d}m \le \frac{2}{\varepsilon^2}\|\psi_\varepsilon\|^2. \end{aligned}$$ Combining these two estimates with the fact that $\|v\psi_\varepsilon\|\lesssim\|\nabla_v\psi_\varepsilon\|+\|\psi_\varepsilon\|$ (a weighted $L^2(\mathbb{R}^d,\mathop{}\!\mathrm{d}\mu)$ inequality via integration by parts) and Grönwall's inequality, we then derive [\[layerdecay2\]](#layerdecay2){reference-type="eqref" reference="layerdecay2"} as claimed. ◻ # Long-time asymptotics {#hypocoercive} This section is devoted to the proof of Theorem [Theorem 1](#longtime){reference-type="ref" reference="longtime"}. We consider dissipative kinetic equations in bounded spatial domains with general boundary conditions that may not conserve mass, and discuss its hypocoercive property concerning the (uniform-in-$\varepsilon$) long-time behaviour of the solutions. We recall the initial mass $M_0:=\left(\int_\Omega e^{-\phi}\mathop{}\!\mathrm{d}x\right)^{-1} \int_{\Omega\times\mathbb{R}^d} f_{\rm in}\mathop{}\!\mathrm{d}m$. It can be readily verified that the mass conservation of [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} holds when $\alpha+\beta=1$. Let us turn to the proof of Theorem [Theorem 1](#longtime){reference-type="ref" reference="longtime"}. *Proof of Theorem [Theorem 1](#longtime){reference-type="ref" reference="longtime"}.* The proof will proceed in three steps. *Step 1. Elliptic problem associated with macroscopic quantities.*\ We consider the solution $u$ to the following elliptic equation with the Robin boundary condition, $$\label{elliptic-u} \left\{ \begin{aligned} \ & u-\Delta_x u+ \nabla_x\phi\cdot\nabla_xu=\langle f_\varepsilon\rangle-M_c {\quad\rm in\ }\Omega, \\ \ & n_x\cdot\nabla_xu +c_bu =0 {\quad\rm on\ }\partial\Omega. \\ \end{aligned} \right.$$ Here we set $c_b:=(1-\alpha-\beta)/(\alpha+\beta)$ (we may always assume $\alpha+\beta\ge\iota>0$ so that $c_b$ is bounded, and eventually let $\iota\rightarrow0$); we take $M_c:=0$ if $c_b|_{\partial\Omega}>0$, and $M_c:=M_0$ if $c_b|_{\partial\Omega}=0$. In particular, when $c_b|_{\partial\Omega}=0$, that is, $\alpha+\beta=1$ on $\partial\Omega$, we deduce from [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} that the mass conservation $\left(\langle f_\varepsilon\rangle-M_c,\,1\right)=0$ is satisfied. Let us consider the quantity $$\begin{aligned} \mathcal{A}:=\left(\langle f_\varepsilon\rangle-M_c-u,\,\langle f_\varepsilon\rangle-M_c\right). \end{aligned}$$ By [\[elliptic-z\]](#elliptic-z){reference-type="eqref" reference="elliptic-z"} of Lemma [Lemma 8](#elliptic-est){reference-type="ref" reference="elliptic-est"} (see Appendix [5](#append){reference-type="ref" reference="append"} below), we obtain $$\begin{aligned} \label{AA} \|\langle f_\varepsilon\rangle-M_c\| + \|u\| + \|\nabla_xu\| + \|D_x^2u\| + \|\sqrt{c_b}u\|_\partial\lesssim \mathcal{A}^\frac{1}{2}. \end{aligned}$$ Since $\langle vf_\varepsilon\rangle$=$\langle vf_\varepsilon^\perp\rangle$, the local conservation law [\[hydroFP1\]](#hydroFP1){reference-type="eqref" reference="hydroFP1"} can be written as $$\begin{aligned} \varepsilon\partial_t\langle f_\varepsilon\rangle+\nabla_x\cdot\langle vf_\varepsilon^\perp\rangle-\nabla_x\phi\cdot\langle vf_\varepsilon^\perp\rangle=0. \end{aligned}$$ Combining this relation with [\[elliptic-u\]](#elliptic-u){reference-type="eqref" reference="elliptic-u"} and applying integration by parts, we have $$\label{elliptic1} \begin{split} \|\partial_tu\|^2 +\|\nabla_x\partial_tu\|^2 + \|\sqrt{c_b}\partial_tu\|_\partial^2 = \left( \partial_tu,\,\partial_t(u-\Delta_xu+\nabla_x\phi\cdot\nabla_xu)\right) = \left( \partial_tu,\,\partial_t\langle f_\varepsilon\rangle\right) \\ = \varepsilon^{-1}\left( \partial_tu,\,(\nabla_x\phi-\nabla_x)\cdot\langle vf_\varepsilon^\perp\rangle\right) = \varepsilon^{-1}\left( \nabla_x\partial_tu,\,\langle vf_\varepsilon^\perp\rangle\right) - \varepsilon^{-1}\left( \partial_tu,\,n_x\cdot\langle vf_\varepsilon^\perp\rangle\right)_\partial&. \end{split}$$ By Hölder's inequality, $$\begin{aligned} \label{elliptic2} \left|\left(\nabla_x\partial_tu,\,\langle vf_\varepsilon^\perp\rangle\right)\right| \le \|v\cdot\nabla_x\partial_tu\| \;\! \|f_\varepsilon^\perp\| = \|\nabla_x\partial_tu\| \;\! \|f_\varepsilon^\perp\|. \end{aligned}$$ Based on the boundary condition of $f_\varepsilon$, the boundary term above can be recast as $$\begin{aligned} \left(\partial_tu,\,n_x\cdot\langle vf_\varepsilon^\perp\rangle\right)_\partial= \left( \partial_tu\;\!(1-\alpha-\beta)(f_\varepsilon-M_c),\,(n_x\cdot v)_+\right)_\partial. \end{aligned}$$ By using Hölder's inequality, the trace inequality $\|\partial_tu\|_{L^2(\partial\Omega)}\lesssim \|\partial_tu\|_{H_x^1(\Omega)}$, and [\[identity2\]](#identity2){reference-type="eqref" reference="identity2"} of Lemma [Lemma 3](#identity){reference-type="ref" reference="identity"} with $f:=f_\varepsilon-M_c$, we have $$\label{elliptic3} \begin{split} \left|\left( \partial_tu,\,n_x\cdot\langle vf_\varepsilon^\perp\rangle\right)_\partial\right| &\lesssim \|\partial_tu\|_{L^2(\partial\Omega)} \left((1-\alpha-\beta)^2(f_\varepsilon-M_c)^2,\,(n_x\cdot v)_+\right)_\partial^\frac{1}{2}\\ &\lesssim \left(\|\partial_tu\| +\|\nabla_x\partial_tu\|\right) \left((f_\varepsilon-M_c)^2,\,n_x\cdot v\right)_\partial^\frac{1}{2}. \end{split}$$ It follows from [\[elliptic1\]](#elliptic1){reference-type="eqref" reference="elliptic1"}, [\[elliptic2\]](#elliptic2){reference-type="eqref" reference="elliptic2"}, [\[elliptic3\]](#elliptic3){reference-type="eqref" reference="elliptic3"} and Hölder's inequality that $$\begin{aligned} \label{Poisson2} \|\partial_tu\| +\|\nabla_x\partial_tu\| \lesssim \varepsilon^{-1}\|f_\varepsilon^\perp\| + \varepsilon^{-1}\left((f_\varepsilon-M_c)^2,\,n_x\cdot v\right)_\partial^\frac{1}{2}. \end{aligned}$$ *Step 2. Macro-micro decomposition.*\ Taking the inner product with $v\cdot\nabla_xu(t,x)$ on both sides of [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} yields that $$\begin{aligned} \varepsilon\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t} \left( v\cdot\nabla_xu,\, f_\varepsilon-M_c \right) =&\ \varepsilon\left (v\cdot\nabla_x\partial_tu,\, f_\varepsilon-M_c \right) +\varepsilon^{-1} \left( \mathcal{L}_i v\cdot\nabla_xu,\, f_\varepsilon-M_c \right) \\ &+\left( (v\cdot\nabla_x-\nabla_x\phi\cdot\nabla_v)(v\cdot\nabla_xu),\, f_\varepsilon-M_c \right) +\Sigma_\gamma, \end{aligned}$$ where the boundary term $\Sigma_\gamma:=-\left((v\cdot\nabla_xu)(f_\varepsilon-M_c),\, n_x\cdot v \right)_\partial$. Taking into account the macro-micro decomposition, given as $f_\varepsilon-M_c=\langle f_\varepsilon\rangle-M_c+f_\varepsilon^\perp$, we obtain $$\begin{aligned} \varepsilon\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t} \left( v\cdot\nabla_xu,\, f_\varepsilon^\perp \right) =&\ \varepsilon\left( v\cdot\nabla_x\partial_tu,\, f_\varepsilon^\perp \right) -\varepsilon^{-1}\left( v\cdot\nabla_xu,\, f_\varepsilon^\perp \right) \\ &+ \left( \Delta_x u- \nabla_x\phi\cdot\nabla_xu,\, \langle f_\varepsilon\rangle-M_c \right) + \left( (v\otimes v):D^2_xu-\nabla_x\phi\cdot\nabla_xu ,\, f_\varepsilon^\perp \right) +\Sigma_\gamma, \end{aligned}$$ where we used the facts that $\mathcal{L}_iv=-v$ for $i\in\{1,2\}$ and $\langle v_jv_k\rangle=\delta_{jk}$ for $j,k\in\{1,\ldots,d\}$. It then turns out from the definition of $u$ and Hölder's inequality that $$\begin{aligned} \mathcal{A}+ \varepsilon\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\left( v\cdot\nabla_xu,\, f_\varepsilon^\perp \right) \lesssim \left( \|D^2_xu\| + \varepsilon\|\nabla_x\partial_tu\| +\varepsilon^{-1}\|\nabla_xu\| \right)\|f_\varepsilon^\perp\| + |\Sigma_\gamma|. \end{aligned}$$ By applying Lemma [Lemma 4](#identity-e){reference-type="ref" reference="identity-e"} with $U:=\nabla_x u+c_bun_x$ and $f:=f_\varepsilon-M_c$, as well as the trace inequality $\|\nabla_xu\|_{L^2(\partial\Omega)}\lesssim\|u\|_{H_x^2(\Omega)}$, we have $$\begin{aligned} |\Sigma_\gamma| \lesssim&\, \left(\|u\|_{H_x^2(\Omega)} + \|\sqrt{c_b}u\|_{L^2(\partial\Omega)}\right) \left((f_\varepsilon-M_c)^2,\, n_x\cdot v \right)_\partial^\frac{1}{2}. \end{aligned}$$ Gathering the above two estimates with [\[AA\]](#AA){reference-type="eqref" reference="AA"} and [\[Poisson2\]](#Poisson2){reference-type="eqref" reference="Poisson2"}, we arrive at $$\begin{aligned} \mathcal{A}+ \varepsilon\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\left(v\cdot\nabla_xu,\, f_\varepsilon^\perp\right) \lesssim&\ \|f_\varepsilon^\perp\| \left(\varepsilon^{-1}\mathcal{A}^\frac{1}{2} + \|f_\varepsilon^\perp\| +\left((f_\varepsilon-M_c)^2,\,n_x\cdot v\right)_\partial^\frac{1}{2}\right)\\ &\ +\mathcal{A}^\frac{1}{2}\left((f_\varepsilon-M_c)^2,\, n_x\cdot v \right)_\partial^\frac{1}{2}. \end{aligned}$$ Applying the Cauchy-Schwarz inequality and utilizing [\[AA\]](#AA){reference-type="eqref" reference="AA"} again, we know that for some constant $c_*>0$, $$\begin{aligned} \label{necessario} c_*\|\langle f_\varepsilon\rangle-M_c\|^2 + \varepsilon\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t} \left(v\cdot\nabla_xu,\, f_\varepsilon^\perp\right) \lesssim \varepsilon^{-2}\|f_\varepsilon^\perp\|^2 + \left((f_\varepsilon-M_c)^2,\, n_x\cdot v \right)_\partial. \end{aligned}$$ *Step 3. Construction of the modified entropy.*\ Let $\kappa>0$ be a constant. We define the modified entropy as follows, $$\begin{aligned} E_\varepsilon:=\|f_\varepsilon-M_c\|^2 + \kappa \varepsilon\left(v\cdot\nabla_xu,\, f_\varepsilon^\perp\right) .\end{aligned}$$ If the constant $\kappa>0$ is sufficiently small, the modified entropy $E_\varepsilon$ is equivalent to $\|f_\varepsilon-M_c\|^2$ independent of $\varepsilon$; indeed, by Hölder's inequality and [\[elliptic-regularity\]](#elliptic-regularity){reference-type="eqref" reference="elliptic-regularity"} of Lemma [Lemma 8](#elliptic-est){reference-type="ref" reference="elliptic-est"}, we have $$\begin{aligned} \left|\left(v\cdot\nabla_xu,\, f_\varepsilon^\perp\right) \right| \lesssim \|\langle f_\varepsilon\rangle-M_c\| \;\! \|f_\varepsilon^\perp\| \le \|f_\varepsilon-M_c\|^2. \end{aligned}$$ In light of this fact, we conclude from [\[dissipation-h\]](#dissipation-h){reference-type="eqref" reference="dissipation-h"} of Lemma [Lemma 5](#dissipation){reference-type="ref" reference="dissipation"} (with $f_\varepsilon$ replaced by $f_\varepsilon-M_c$) and [\[necessario\]](#necessario){reference-type="eqref" reference="necessario"} that $$\begin{aligned} \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t} E_\varepsilon \lesssim -\varepsilon^{-2}\|f_\varepsilon^\perp\|^2-\|\langle f_\varepsilon\rangle-M_c\|^2 \le -\|f_\varepsilon-M_c\|^2 \lesssim - E_\varepsilon. \end{aligned}$$ By using Grönwall's inequality and leveraging the equivalence between $E_\varepsilon$ and $\|f_\varepsilon-M_c\|^2$, we arrive at the desired result. ◻ # Diffusion asymptotics {#diffusionlimit} This section is dedicated to proving Theorem [Theorem 2](#asympto){reference-type="ref" reference="asympto"}, which establishes the diffusion asymptotics of the kinetic model [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}. Throughout this section, we assume $\alpha+\beta=1$ when referring to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}, specifically associated with the Maxwell boundary condition $f_\varepsilon=\alpha\mathcal{D}f_\varepsilon+(1-\alpha)\mathcal{R}f_\varepsilon$ on $\mathbb{R}_+\times\Sigma_-$. **Proposition 7**. *Let $\varepsilon\in(0,1]$, $f_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$, and $f_\varepsilon$ be the solution to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"} associated with the initial data $f_{\rm in}$ and the Maxwell boundary condition. Suppose that $\rho$ is the solution to [\[limiteq\]](#limiteq){reference-type="eqref" reference="limiteq"} associated with the initial data $\rho_{\rm in}:=\langle f_{\rm in}\rangle$, and $\psi_\varepsilon$ is the solution to [\[layereq\]](#layereq){reference-type="eqref" reference="layereq"} associated with the initial data $\psi_{\rm in}:=f_{\rm in}-\langle f_{\rm in}\rangle$. Then we have the strong convergence that $$\label{asympto1} \|f_\varepsilon-\rho-\psi_\varepsilon\|_{L^\infty(\mathbb{R}_+;\;\!L^2(\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}m))} \rightarrow 0 {\quad\rm as\ }\varepsilon\rightarrow0.$$ If $f_{\rm in}\in L_v^2H_x^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$ satisfies $|v|\nabla_xf_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$ and the compatibility condition that $f_{\rm in}$ depends only on the variable $x$ in $\partial\Omega\times\mathbb{R}^d$, then we have $$\begin{aligned} \label{asympto2} \|f_\varepsilon-\rho-\psi_\varepsilon\|_{L^\infty(\mathbb{R}_+;\;\!L^2(\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}m))} \lesssim \sqrt{\varepsilon} \left(\|f_{\rm in}\| +\|(1+|v|)\nabla_xf_{\rm in}\| +\|D_x^2f_{\rm in}\| \right). \end{aligned}$$ If the initial data is well-prepared, that is, $f_{\rm in}=\langle f_{\rm in}\rangle\in H_x^2(\Omega)$, then we have $$\begin{aligned} \label{asympto3} \|f_\varepsilon-\rho\|_{L^\infty(\mathbb{R}_+;\;\!L^2(\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}m))} \lesssim \sqrt{\varepsilon} \|f_{\rm in}\|_{H_x^2(\Omega)}. \end{aligned}$$* *Proof.* The proof will proceed in three steps. *Step 1. Evolution of the entropy functional.*\ We abbreviate $g_\varepsilon:=f_\varepsilon-\psi_\varepsilon$. The distance between $g_\varepsilon$ and $\rho$ reads $$\begin{aligned} \|g_\varepsilon-\rho\|^2=\|g_\varepsilon\|^2+\|\rho\|^2-2(g_\varepsilon,\,\rho).\end{aligned}$$ Since $\{v_1,\ldots,v_d\}$ is a part of the orthonormal basis of $L^2(\mathbb{R}^d,\mathop{}\!\mathrm{d}\mu)$ provided by the Hermite polynomials, we have $$|\langle vf_\varepsilon\rangle|^2=|\langle vf_\varepsilon^\perp\rangle|^2\le \langle|f_\varepsilon^\perp|^2\rangle.$$ It then follows from [\[dissipation-h\]](#dissipation-h){reference-type="eqref" reference="dissipation-h"} of Lemma [Lemma 5](#dissipation){reference-type="ref" reference="dissipation"} that $$\begin{aligned} \frac{1}{2}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|f_\varepsilon\|^2 \le-\varepsilon^{-2}\|\langle vf_\varepsilon\rangle\|^2 -(2\varepsilon)^{-1}\left(f_\varepsilon^2,\,n_x\cdot v\right)_\partial. \end{aligned}$$ Taking the evolution of $\psi_\varepsilon$ into account, we similarly derive $$\label{entropy1} \begin{split} \frac{1}{2}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|g_\varepsilon\|^2 &\le-\varepsilon^{-2}\|\langle vg_\varepsilon\rangle\|^2 -\varepsilon^{-1}\left(g_\varepsilon,\,v\cdot\nabla_x\psi_\varepsilon\right) -(2\varepsilon)^{-1}\left(g_\varepsilon^2,\,n_x\cdot v\right)_\partial\\ &=-\varepsilon^{-2}\|\langle vg_\varepsilon\rangle\|^2 -\varepsilon^{-1}\left(f_\varepsilon,\,v\cdot\nabla_x\psi_\varepsilon\right) -(2\varepsilon)^{-1}\left(g_\varepsilon^2-\psi_\varepsilon^2,\,n_x\cdot v\right)_\partial. \end{split}$$ By the limiting equation [\[limiteq\]](#limiteq){reference-type="eqref" reference="limiteq"} and the macroscopic relation [\[hydroFP1\]](#hydroFP1){reference-type="eqref" reference="hydroFP1"}, we have $$\label{entropy2} \begin{split} \frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}(g_\varepsilon,\,\rho) =\left(\langle g_\varepsilon\rangle,\,\partial_t\rho\right)+\left(\partial_t\langle f_\varepsilon\rangle,\,\rho\right) =& \left(\varepsilon^{-1}\langle vg_\varepsilon\rangle-\nabla_x\langle g_\varepsilon\rangle,\,\nabla_x\rho\right) +\varepsilon^{-1}\left(\rho,\,v\cdot\nabla_x\psi_\varepsilon\right)\\ &+\left(g_\varepsilon,\,n_x\cdot\nabla_x\rho\right)_\partial-\varepsilon^{-1}\left(f_\varepsilon\rho,\,n_x\cdot v\right)_\partial, \end{split}$$ and $$\begin{aligned} \label{entropy3} \frac{1}{2}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|\rho\|^2 =\left(\rho,\,\partial_t\rho\right) =-\|\nabla_x\rho\|^2 +\left(\rho,\,n_x\cdot\nabla_x\rho\right)_\partial.\end{aligned}$$ We thus obtain from [\[entropy1\]](#entropy1){reference-type="eqref" reference="entropy1"}, [\[entropy2\]](#entropy2){reference-type="eqref" reference="entropy2"}, [\[entropy3\]](#entropy3){reference-type="eqref" reference="entropy3"} that $$\begin{aligned} \frac{1}{2}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|g_\varepsilon-\rho\|^2 \le& -\left\|\varepsilon^{-1}\langle vg_\varepsilon\rangle+\nabla_x\rho\right\|^2 + \left(\varepsilon^{-1}\langle vg_\varepsilon\rangle+\nabla_x\langle g_\varepsilon\rangle,\,\nabla_x\rho\right) -\left(\rho+f_\varepsilon,\,\varepsilon^{-1}v\cdot\nabla_x\psi_\varepsilon\right) \\ &+\left( \rho-g_\varepsilon,\,n_x\cdot\nabla_x\rho\right)_\partial- (2\varepsilon)^{-1}\left(g_\varepsilon^2-\psi_\varepsilon^2-2f_\varepsilon\rho,\, n_x\cdot v\right)_\partial. \end{aligned}$$ It implies that $$\label{distevolution} \begin{split} \frac{1}{2}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|f_\varepsilon-\rho-\psi_\varepsilon\|^2 \le \left(\varepsilon^{-1}\langle vf_\varepsilon\rangle+\nabla_x\langle f_\varepsilon\rangle,\,\nabla_x\rho\right) +\Sigma_1 +R_\psi, \end{split}$$ where we collected the terms $$\begin{aligned} \Sigma_1:=&\, \left( \rho-f_\varepsilon+\psi_\varepsilon,\,n_x\cdot\nabla_x\rho\right)_\partial - (2\varepsilon)^{-1}\left(f_\varepsilon^2-2f_\varepsilon\rho-2f_\varepsilon\psi_\varepsilon+2\rho\psi_\varepsilon,\, n_x\cdot v\right)_\partial,\\ R_\psi:=&\, -\left(f_\varepsilon,\,\varepsilon^{-1}v\cdot\nabla_x\psi_\varepsilon\right) - \left(\nabla_x\psi_\varepsilon,\,\nabla_x\rho\right). \end{aligned}$$ Since $\langle vf_\varepsilon\rangle=\langle vf_\varepsilon^\perp\rangle$ and $\langle(v\otimes v-{\rm I}) f_\varepsilon\rangle=\langle v\otimes v f_\varepsilon^\perp\rangle$, the macroscopic relation [\[hydroFP2\]](#hydroFP2){reference-type="eqref" reference="hydroFP2"} can be recast as $$\begin{aligned} \varepsilon^{-1}\langle vf_\varepsilon\rangle+\nabla_x\langle f_\varepsilon\rangle = -\varepsilon\partial_t\langle vf_\varepsilon^\perp\rangle-\nabla_x\cdot \langle v\otimes vf_\varepsilon^\perp\rangle +\nabla_x\phi\cdot\langle v\otimes vf_\varepsilon^\perp\rangle. \end{aligned}$$ Therefore, we have $$\begin{aligned} \int_0^t \left(\varepsilon^{-1}\langle vf_\varepsilon\rangle+\nabla_x\langle f_\varepsilon\rangle,\,\nabla_x\rho\right) =\varepsilon\int_0^t\left(\langle vf_\varepsilon^\perp\rangle,\,\nabla_x\partial_t\rho\right) +\int_0^t\left(\langle v\otimes v f_\varepsilon^\perp\rangle,\, \nabla_x\otimes\nabla_x\rho\right)&\\ -\varepsilon\left(\langle vf_\varepsilon^\perp\rangle,\,\nabla_x\rho\right)(t) +\varepsilon\left(\langle vf_\varepsilon^\perp\rangle,\,\nabla_x\rho\right)(0) -\int_0^t\left(\langle v\otimes v f_\varepsilon^\perp\rangle,\, n_x\otimes\nabla_x\rho\right)_\partial&. \end{aligned}$$ Combining this with [\[distevolution\]](#distevolution){reference-type="eqref" reference="distevolution"} yields that $$\label{entropy-relative} \begin{split} \frac{1}{2}\|f_\varepsilon-\rho-\psi_\varepsilon\|^2 - \int_0^t(\Sigma_2+R_\psi) \lesssim&\left( \int_0^t \left(\varepsilon^2\|\nabla_x\partial_t\rho\|^2 + \|D_x^2\rho\|^2\right) \right)^{\!\frac{1}{2}} \left(\int_0^t\|f_\varepsilon^\perp\|^2\right)^{\!\frac{1}{2}}\\ &\,+\varepsilon\sup\nolimits_{[0,t]}\|\nabla_x\rho\|\|f_\varepsilon\| , \end{split}$$ where the new collection $\Sigma_2$ of boundary terms is defined by $$\begin{aligned} \Sigma_2:=&\ \Sigma_1+\left(({\rm I}-v\otimes v)f_\varepsilon,\,n_x\otimes\nabla_x\rho\right)_\partial\\ =&\,-\left((v\cdot\nabla_x\rho)f_\varepsilon,\,n_x\cdot v\right)_\partial +\left(\rho+\psi_\varepsilon,\,n_x\cdot\nabla_x\rho\right)_\partial -(2\varepsilon)^{-1}\left(f_\varepsilon^2-2f_\varepsilon\rho-2f_\varepsilon\psi_\varepsilon+2\rho\psi_\varepsilon,\, n_x\cdot v\right)_\partial. \end{aligned}$$ *Step 2. Taking into account boundary conditions and initial layer correction.*\ We have to address the boundary term $\Sigma_2$ and the initial layer term $R_\psi$ appeared in [\[entropy-relative\]](#entropy-relative){reference-type="eqref" reference="entropy-relative"}. Let us assume the compatibility condition on the initial data that $f_{\rm in}$ depends only on the variable $x$ in $\partial\Omega\times\mathbb{R}^d$. By [\[layereq\]](#layereq){reference-type="eqref" reference="layereq"}, we know that $\psi_\varepsilon|_{\Sigma}$ depends only on the variable $x$. Consequently, according to the boundary condition of $f_\varepsilon$, we have $$\begin{aligned} \left(f_\varepsilon^2-2f_\varepsilon\rho-2f_\varepsilon\psi_\varepsilon+2\rho\psi_\varepsilon,\, n_x\cdot v\right)_\partial =\left(f_\varepsilon^2,\,n_x\cdot v\right)_\partial\ge0. \end{aligned}$$ It turns out that $$\begin{aligned} \Sigma_2 =-\left((v\cdot\nabla_x\rho)f_\varepsilon,\,n_x\cdot v\right)_\partial -(2\varepsilon)^{-1}\left(f_\varepsilon^2,\,n_x\cdot v\right)_\partial. \end{aligned}$$ By Lemma [Lemma 4](#identity-e){reference-type="ref" reference="identity-e"} with $U:=\nabla_x\rho$ and $\beta:=1-\alpha$, the trace inequality $\|\nabla_x\rho\|_{L^2(\partial\Omega)}\lesssim\|\rho\|_{H_x^2(\Omega)}$ and the Cauchy-Schwarz inequality, we have $$\begin{aligned} \left|\left((v\cdot\nabla_x\rho)f_\varepsilon,\,n_x\cdot v\right)_\partial\right| \lesssim \|\rho\|_{H_x^2(\Omega)} \left(f_\varepsilon^2,\, n_x\cdot v\right)_\partial^\frac{1}{2} \lesssim C_1\varepsilon\|\rho\|_{H_x^2(\Omega)}^2 + C_1^{-1}\varepsilon^{-1}\left(f_\varepsilon^2,\, n_x\cdot v\right)_\partial. \end{aligned}$$ Here the constant $C_1>0$ is chosen to be sufficiently large so that $$\begin{aligned} \label{entropy-boundary} \Sigma_2 \lesssim \varepsilon\|\rho\|_{H_x^2(\Omega)}^2.\end{aligned}$$ In order to estimate the term $R_\psi=-\left(\varepsilon^{-1}vf_\varepsilon+\nabla_x\rho,\,\nabla_x\psi_\varepsilon\right)$, we apply [\[layerdecay2\]](#layerdecay2){reference-type="eqref" reference="layerdecay2"} of Lemma [Lemma 6](#layer){reference-type="ref" reference="layer"} with $\psi:=\nabla_x\psi_\varepsilon$ and $\psi_{\rm in}:=\nabla_xf_{\rm in}-\nabla_x\langle f_{\rm in}\rangle$ to see that for some constant $c_0>0$, $$\begin{aligned} |R_\psi| \le \varepsilon^{-1}\left(\|f_\varepsilon\|+\|\nabla_x\rho\|\right)\|(1+|v|)\nabla_x\psi_\varepsilon\| \lesssim \varepsilon^{-1}e^{-c_0t/\varepsilon^2} \left(\|f_\varepsilon\|+\|\nabla_x\rho\|\right)\|(1+|v|)\nabla_xf_{\rm in}\|. \end{aligned}$$ Integrating along time and noticing the fact that $\int_0^\infty e^{-c_0t/\varepsilon^2}\mathop{}\!\mathrm{d}t=\varepsilon^2/c_0$, we obtain $$\begin{aligned} \label{entropy-remainder} \int_0^t|R_\psi| \lesssim \varepsilon\|(1+|v|)\nabla_xf_{\rm in}\| \sup\nolimits_{[0,t]}\left(\|f_\varepsilon\|+\|\nabla_x\rho\|\right). \end{aligned}$$ *Step 3. Conclusion and approximation.*\ We are in a position to leverage the entropy dissipation and the regularity of the limiting equation to draw our conclusion. Recall the parabolic estimate that for $\rho$ solving [\[limiteq\]](#limiteq){reference-type="eqref" reference="limiteq"} and for any $t>0$, $$\begin{aligned} \sup\nolimits_{[0,t]}\left(\|\rho\|_{H_x^1(\Omega)}^2+\|\partial_t\rho\|^2\right) + \int_0^t\left(\|D_x^2\rho\|^2 +\|\nabla_x\partial_t\rho\|^2\right) \lesssim \|\rho_{\rm in}\|_{H^2(\Omega)}^2; \end{aligned}$$ one may refer to [@Evans § 7.1.3] in this regard. It then follows from [\[entropy-relative\]](#entropy-relative){reference-type="eqref" reference="entropy-relative"}, [\[entropy-boundary\]](#entropy-boundary){reference-type="eqref" reference="entropy-boundary"}, [\[entropy-remainder\]](#entropy-remainder){reference-type="eqref" reference="entropy-remainder"} that $$\begin{aligned} \|f_\varepsilon-\rho-\psi_\varepsilon\|^2 \lesssim \|\rho_{\rm in}\|_{H^2(\Omega)} \left(\int_0^t\|f_\varepsilon^\perp\|^2\right)^{\!\frac{1}{2}} +\varepsilon\|\rho_{\rm in}\|_{H^2(\Omega)}\sup\nolimits_{[0,t]}\|f_\varepsilon\| +\varepsilon\|\rho_{\rm in}\|_{H^2(\Omega)}^2\\ +\varepsilon\|(1+|v|)\nabla_xf_{\rm in}\| \sup\nolimits_{[0,t]}\left(\|f_\varepsilon\|+\|\rho_{\rm in}\|_{H^2(\Omega)}\right)&. \end{aligned}$$ By the entropy dissipation [\[dissipationR\]](#dissipationR){reference-type="eqref" reference="dissipationR"} of Lemma [Lemma 5](#dissipation){reference-type="ref" reference="dissipation"} and the Cauchy-Schwarz inequality, we have $$\begin{aligned} \label{fepr} \|f_\varepsilon-\rho-\psi_\varepsilon\|^2 \lesssim \varepsilon\left(\|f_{\rm in}\|^2 +\|(1+|v|)\nabla_xf_{\rm in}\|^2 +\|D_x^2f_{\rm in}\|^2 \right). \end{aligned}$$ We thus obtain [\[asympto2\]](#asympto2){reference-type="eqref" reference="asympto2"} under the compatibility assumption on $f_{\rm in}$. We remark that [\[asympto3\]](#asympto3){reference-type="eqref" reference="asympto3"} can be derived in a similar manner, based on the observation that the well-preparedness of $f_{\rm in}$ implies $\psi_\varepsilon\equiv0$. As for general initial data $f_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d}m)$, we consider its approximation $$\begin{aligned} f_{\rm in}^N:=\varphi_Nf_{\rm in}*\varrho_N. \end{aligned}$$ Here $N\in\mathbb{N}_+$, $\varrho_N(x):=N^d\varrho_1(Nx)$ with $0\le\varrho_1(x)\in C^\infty_0(\mathbb{R}^d)$ such that $\int_{\mathbb{R}^d}\varrho_1(x)\mathop{}\!\mathrm{d}x=1$, and $\varphi_N(x,v)\in C^\infty_0(\Omega\times\mathbb{R}^d)$ satisfies $$\begin{aligned} 0\le\varphi_N(x,v)\le1,\quad |\nabla_x\varphi_N(x,v)|\lesssim N,\quad |D^2_x\varphi_N(x,v)|\lesssim N^2 {\quad\rm in\ } \Omega\times\mathbb{R}^d, \\ \varphi_N(x,v)=1 {\quad\rm in\ } \{(x,v)\in\Omega\times\mathbb{R}^d:\, {\rm dist}(x,\partial\Omega)\ge N^{-1} {\rm\ and\ } |v|\le N\},\\ \varphi_N(x,v)=0 {\quad\rm in\ }\{(x,v)\in\Omega\times\mathbb{R}^d:\, {\rm dist}(x,\partial\Omega)\le (2N)^{-1} {\rm\ or\ } |v|\ge 2N\}. \end{aligned}$$ In this setting, we know that $\|f_{\rm in}-f_{\rm in}^N\|\rightarrow0$ as $N\rightarrow\infty$, and by definition and Young's inequality, $$\begin{aligned} \label{fepr1} \|(1+|v|)\nabla_xf_{\rm in}^N\| + \|D_x^2f_{\rm in}^N\| \lesssim \|(|v||\nabla_x\varphi_N|+|D_x^2\varphi_N|)f_{\rm in}*\varrho_N\| \nonumber\\ + \|(|v|\varphi_N+|\nabla_x\varphi_N|)f_{\rm in}*(\nabla_x\varrho_N)\| + \|\varphi_Nf_{\rm in}*(D_x^2\varrho_N)\|\\ \lesssim\left(N^2\|\varrho_N\|_{L^1(\mathbb{R}^d)} +N\|\nabla_x\varrho_N\|_{L^1(\mathbb{R}^d)} +\|D_x^2\varrho_N\|_{L^1(\mathbb{R}^d)}\right) \|f_{\rm in}\| \lesssim N^2\|f_{\rm in}\|&. \nonumber\end{aligned}$$ Let $f_\varepsilon^N$, $\rho^N$, $\psi_\varepsilon^N$ be the solutions to [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}, [\[limiteq\]](#limiteq){reference-type="eqref" reference="limiteq"}, [\[layereq\]](#layereq){reference-type="eqref" reference="layereq"} associated with the initial conditions $f_\varepsilon^N|_{t=0}=f_{\rm in}^N$, $\rho^N|_{t=0}=\langle f_{\rm in}^N\rangle$, $\psi_\varepsilon^N|_{t=0}=f_{\rm in}^N-\langle f_{\rm in}^N\rangle$, respectively. In view of the dissipation estimates for [\[rfp\]](#rfp){reference-type="eqref" reference="rfp"}, [\[layerdecay1\]](#layerdecay1){reference-type="eqref" reference="layerdecay1"} provided by Lemma [Lemma 5](#dissipation){reference-type="ref" reference="dissipation"}, [Lemma 6](#layer){reference-type="ref" reference="layer"}, as well as the standard dissipation property for [\[limiteq\]](#limiteq){reference-type="eqref" reference="limiteq"}, we have $$\label{fepr2} \begin{split} \|f_\varepsilon-f_\varepsilon^N\| \le \|f_{\rm in}-f_{\rm in}^N\|, \quad \|\rho-\rho^N\| \le \|\langle f_{\rm in}\rangle-\langle f_{\rm in}^N\rangle\|\le \|f_{\rm in}-f_{\rm in}^N\|, \\ \|\psi_\varepsilon-\psi_\varepsilon^N\| \le \|f_{\rm in}-\langle f_{\rm in}\rangle-f_{\rm in}^N+\langle f_{\rm in}^N\rangle\| \le 2\|f_{\rm in}-f_{\rm in}^N\|. \end{split}$$ According to [\[fepr\]](#fepr){reference-type="eqref" reference="fepr"}, [\[fepr1\]](#fepr1){reference-type="eqref" reference="fepr1"}, [\[fepr2\]](#fepr2){reference-type="eqref" reference="fepr2"}, we obtain $$\begin{aligned} \|f_\varepsilon-\rho-\psi_\varepsilon\| &\le \|f_\varepsilon-f_\varepsilon^N\| + \|\rho-\rho^N\|+ \|\psi_\varepsilon-\psi_\varepsilon^N\| + \|f_\varepsilon^N-\rho^N-\psi_\varepsilon^N\| \\ &\lesssim \|f_{\rm in}-f_{\rm in}^N\| + \sqrt{\varepsilon} N^2\|f_{\rm in}\|. \end{aligned}$$ Sending $\varepsilon\rightarrow0$ and then $N\rightarrow\infty$ yields [\[asympto1\]](#asympto1){reference-type="eqref" reference="asympto1"}. This completes the proof. ◻ Theorem [Theorem 2](#asympto){reference-type="ref" reference="asympto"} is a direct consequence of Proposition [Proposition 7](#asympto0){reference-type="ref" reference="asympto0"}. *Proof of Theorem [Theorem 2](#asympto){reference-type="ref" reference="asympto"}.* By the dissipation estimate [\[layerdecay1\]](#layerdecay1){reference-type="eqref" reference="layerdecay1"} of Lemma [Lemma 6](#layer){reference-type="ref" reference="layer"} and Grönwall's inequality, we know that $$\|\psi_\varepsilon\| \le e^{-t/\varepsilon^2}\|\psi_{\rm in}\|\le e^{-t/\varepsilon^2}\|f_{\rm in}\|.$$ On the one hand, integrating along time, we obtain $$\|\psi_\varepsilon\|_{L^2(\mathbb{R}_+\times\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}t\mathop{}\!\mathrm{d}m)} \le \|e^{-t/\varepsilon^{2}}\|_{L^2(\mathbb{R}_+)} \|f_{\rm in}\| \le \varepsilon\|f_{\rm in}\|,$$ so that for any $T>0$, we have $$\label{psi1} \|f_\varepsilon-\rho\|_{L^2([0,T]\times\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}t\mathop{}\!\mathrm{d}m)} \le \sqrt{T}\|f_\varepsilon-\rho-\psi_\varepsilon\|_{L^\infty(\mathbb{R}_+;\;\!L^2(\Omega\times\mathbb{R}^d,\;\!\mathop{}\!\mathrm{d}m))} + \varepsilon\|f_{\rm in}\|.$$ On the other hand, for any $t\ge\tau(\varepsilon)$, we have $$\label{psi2} \|(f_\varepsilon-\rho)(t)\| \le \|(f_\varepsilon-\rho-\psi_\varepsilon)(t)\| + \|\psi_\varepsilon(t)\| \le \|(f_\varepsilon-\rho-\psi_\varepsilon)(t)\| + e^{-\tau(\varepsilon)/\varepsilon^2}\|f_{\rm in}\|.$$ In view of [\[psi1\]](#psi1){reference-type="eqref" reference="psi1"} and [\[psi2\]](#psi2){reference-type="eqref" reference="psi2"}, together with Proposition [Proposition 7](#asympto0){reference-type="ref" reference="asympto0"}, we conclude the proof. ◻ # Elliptic estimates under the Robin boundary condition {#append} This appendix is dedicated to clarifying the elliptic estimates used in Section [3](#hypocoercive){reference-type="ref" reference="hypocoercive"}. Though standard, we include their proofs below for the sake of completeness. **Lemma 8**. *Let $\Omega$ be a bounded $C^{1,1}$-domain, the function $\phi:\Omega\rightarrow\mathbb{R}$ be Lipschitz, the source term $S\in L^2(\Omega)$, and the coefficient $c_b:\partial\Omega\rightarrow[0,\infty)$. Then there exists a unique solution $u$ to $$\label{elliptic-S} \left\{ \begin{aligned} \ & u-\Delta_x u+ \nabla_x\phi\cdot\nabla_xu=S {\quad\rm in\ }\Omega, \\ \ & n_x\cdot\nabla_xu +c_bu =0 {\quad\rm on\ }\partial\Omega. \\ \end{aligned} \right.$$ Supposing that $c_b$ is Lipschitz and satisfies $|\nabla c_b|\le C_0 c_b$ on $\partial\Omega$ for some constant $C_0>0$, we then have the regularity estimate $$\begin{aligned} \label{elliptic-regularity} \|u\| + \|\nabla_x u\| + \|D^2_x u\| + \|\sqrt{c_b}u\|_\partial\le C \|S\| . \end{aligned}$$ If additionally either $c_b|_{\partial\Omega}=\left(S,\,1\right)=0$, or $c_b|_{\partial\Omega}\ge\delta_b$ for some constant $\delta_b>0$, then we have the Poincaré type inequality $$\begin{aligned} \label{elliptic-Poincare} \|u\| \le C\left( \|\nabla_x u\| + \|\sqrt{c_b}u\|_\partial\right). \end{aligned}$$ Furthermore, under the above assumptions, we have $$\begin{aligned} \label{elliptic-z} \|S\| + \|u\| + \|\nabla_xu\| + \|D_x^2u\| + \|\sqrt{c_b}u\|_\partial \le C \left(S-u,\,S\right)^\frac{1}{2} . \end{aligned}$$ Here the constant $C>0$ depends only on $d,\Omega,\|\phi\|_{C^{0,1}(\Omega)},C_0,\delta_b$.* **Remark 9**. *For any Lipschitz function $c_b:\partial\Omega\rightarrow[0,\infty)$ such that $|\nabla c_b|\le C_0c_b$ on $\partial\Omega$ and $c_b(x_0)=0$ for some constant $C_0>0$ and some point $x_0\in\partial\Omega$, it can be shown that $c_b\equiv0$ on $\partial\Omega$. In light of this observation, we assume that $c_b$ is strictly positive whenever it is not identically zero.* *Proof.* First of all, one may find the existence and uniqueness results in [@Lieberman Corollary 5.27]. One may also refer to [@Lieberman § 5.7] for the regularity issues. We sketch the proof of [\[elliptic-regularity\]](#elliptic-regularity){reference-type="eqref" reference="elliptic-regularity"} in our scenario. Indeed, taking the inner product with $u$ on both sides of [\[elliptic-S\]](#elliptic-S){reference-type="eqref" reference="elliptic-S"} yields that $$\begin{aligned} \|u\|^2 + \|\nabla_xu\|^2 = \left(S,\,u\right) + \left(n_x\cdot\nabla_xu,\,u\right)_\partial = \left(S,\,u\right) - \|\sqrt{c_b}u\|_\partial^2, \end{aligned}$$ which implies from the Cauchy-Schwarz inequality that $$\begin{aligned} \label{eu1} \|u\| + \|\nabla_xu\| + \|\sqrt{c_b}u\|_\partial\lesssim \|S\| . \end{aligned}$$ In order to derive higher regularity of $u$, we consider $R>0$ and $\overline{x}\in\partial\Omega$, and assume for simplicity that $\partial\Omega\cap B_R(\overline{x})=\{x\in B_R(\overline{x}):x_d=0\}$ and $\Omega\cap B_R(\overline{x})=\{x\in B_R(\overline{x}):x_d>0\}$; it can be verified that a similar computation as the one below remains applicable for general curved $C^{1,1}$-domains based on a boundary flattening argument. Pick a cut-off function $\eta\in C^\infty_0(B_R(\overline{x}))$ valued in $[0,1]$ such that $\eta|_{B_{R/2}(\overline{x})}=1$. Let us abbreviate $\partial_k=\partial_{x_k}$ for $k\in\{1,\ldots,d\}$. Taking the inner product with $\eta^2\partial_ku$ on both sides of the equation satisfies by $\partial_ku$ yields that for any $k\in\{1,\ldots,d-1\}$, $$\begin{aligned} \left(\eta^2\partial_ku,\,\partial_ku\right) + \left(\nabla_x(\eta^2\partial_ku),\,\nabla_x\partial_ku\right) =&\left(\partial_k(\eta^2\partial_ku),\,\nabla_x\phi\cdot\nabla_xu\right) +\left(\eta^2\partial_ku,\,\nabla_x\phi\cdot\nabla_x\partial_ku\right) \\ &-\left(\partial_k(\eta^2\partial_ku),\,S\right) -\left(\eta^2\partial_ku,\,\partial_k(c_bu)\right)_\partial. \end{aligned}$$ Here we notice that the boundary term $$\left(\eta^2\partial_ku,\,\partial_k(c_bu)\right)_\partial=\left(\eta^2\partial_ku,\, u\partial_kc_b \right)_\partial+ \|\sqrt{c_b}\eta\partial_ku\|_\partial^2,$$ and due to the assumption that $|\partial_kc_b|\le C_0|c_b|$, $$\left|\left(\eta^2\partial_ku,\, u\partial_kc_b \right)_\partial\right| \le C_0\|\sqrt{c_b}\eta u\|_\partial\|\sqrt{c_b}\eta\partial_ku\|_\partial.$$ Applying the Cauchy-Schwarz inequality, we deduce $$\begin{aligned} \|\eta\nabla_x\partial_ku\| + \|\sqrt{c_b}\eta\partial_ku\|_\partial \lesssim \|(\eta+|\nabla_x\eta|)\partial_ku\| + \|\eta S\| +\|\sqrt{c_b}\eta u\|_\partial. \end{aligned}$$ We observe from [\[elliptic-S\]](#elliptic-S){reference-type="eqref" reference="elliptic-S"} that $\partial_d^2u=-S+u-\sum_{k=1}^{d-1}\partial_k^2u+\nabla_x\phi\cdot\nabla_xu$ so that $$\label{eu2} \|D_x^2u\| \lesssim \|\nabla_xu\| + \|S\| +\|\sqrt{c_b}u\|_\partial.$$ Gathering [\[eu1\]](#eu1){reference-type="eqref" reference="eu1"} and [\[eu2\]](#eu2){reference-type="eqref" reference="eu2"}, we arrive at [\[elliptic-regularity\]](#elliptic-regularity){reference-type="eqref" reference="elliptic-regularity"}. As for [\[elliptic-Poincare\]](#elliptic-Poincare){reference-type="eqref" reference="elliptic-Poincare"}, we first assume that $c_b|_{\partial\Omega}=\left(S,\,1\right)=0$. Combining the Poincaré inequality that $\|u\|\lesssim\|\nabla u\|+|(u,\,1)|$ with the fact that $(u,\,1) = (n_x\cdot\nabla_xu,\,1)_\partial=0$, we obtain [\[elliptic-Poincare\]](#elliptic-Poincare){reference-type="eqref" reference="elliptic-Poincare"}. Otherwise, for $c_b|_{\partial\Omega}\ge\delta_b>0$, the Poincaré inequality that $\|u\| \lesssim \|\nabla_xu\| + \|u\|_\partial$ (see for instance [@Lieberman Lemma 5.22]) also implies [\[elliptic-Poincare\]](#elliptic-Poincare){reference-type="eqref" reference="elliptic-Poincare"}. Finally, in view of [\[elliptic-S\]](#elliptic-S){reference-type="eqref" reference="elliptic-S"}, we notice that $$\begin{aligned} \left(S-u,\,S\right) = \left(-\Delta_x u+ \nabla_x\phi\cdot\nabla_xu,\,u-\Delta_x u+ \nabla_x\phi\cdot\nabla_xu\right) \ge \left(-\Delta_x u+ \nabla_x\phi\cdot\nabla_xu,\,u\right)\\ = \|\nabla_xu\|^2 - \left(n_x\cdot\nabla_xu ,\,u\right)_\partial = \|\nabla_xu\|^2 + \|\sqrt{c_b}u\|_\partial^2&, \end{aligned}$$ and $$\begin{aligned} \left(S-u,\,S\right) = \|S\|^2 - \left(u,\,u-\Delta_x u+ \nabla_x\phi\cdot\nabla_xu\right) =\|S\|^2 - \|u\|^2 - \|\nabla_xu\|^2 - \|\sqrt{c_b}u\|_\partial^2. \end{aligned}$$ Combining these two facts with [\[elliptic-regularity\]](#elliptic-regularity){reference-type="eqref" reference="elliptic-regularity"} and [\[elliptic-Poincare\]](#elliptic-Poincare){reference-type="eqref" reference="elliptic-Poincare"}, we conclude [\[elliptic-z\]](#elliptic-z){reference-type="eqref" reference="elliptic-z"}. The proof is complete. ◻ DHHM22 Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat, and Matthew Novack. Variational methods for the kinetic Fokker--Planck equation. , 2019. Francesca Anceschi and Yuzhe Zhu. On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics. , 2021. Armand Bernou, Kleber Carrapatoso, Stéphane Mischler, and Isabelle Tristani. Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition. , 40(2):287--338, 2023. Emeric Bouin, Jean Dolbeault, Stéphane Mischler, Clément Mouhot, and Christian Schmeiser. Hypocoercivity without confinement. , 2(2):203--232, 2020. Claude Bardos, François Golse, and C. David Levermore. Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. , 46(5):667--753, 1993. Alain Bensoussan, Jacques-L. Lions, and George C. Papanicolaou. Boundary layers and homogenization of transport processes. , 15(1):53--157, 1979. Claude Bardos, Rafael Santos, and Rémi Sentis. Diffusion approximation and computation of the critical size. , 284(2):617--649, 1984. Dingqun Deng and Renjun Duan. Spectral gap formation to kinetic equations with soft potentials in bounded domain. , 397(3):1441--1489, 2023. Helge Dietert, Frédéric Hérau, Harsha Hutridurga, and Clément Mouhot. Quantitative geometric control in linear kinetic theory. , 2022. Pierre Degond and Sylvie Mas-Gallic. Existence of solutions and diffusion approximation for a model Fokker-Planck equation. In *Proceedings of the conference on mathematical methods applied to kinetic equations (Paris, 1985)*, volume 16, pages 589--636, 1987. Jean Dolbeault, Clément Mouhot, and Christian Schmeiser. Hypocoercivity for kinetic equations with linear relaxation terms. , 347(9-10):511--516, 2009. Jean Dolbeault, Clément Mouhot, and Christian Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. , 367(6):3807--3828, 2015. Raffaele Esposito, Yan Guo, Chanwoo Kim, and Rossana Marra. Non-isothermal boundary in the Boltzmann theory and Fourier law. , 323(1):177--239, 2013. Lawrence C. Evans. , volume 19 of *Graduate Studies in Mathematics*. American Mathematical Society, Providence, RI, second edition, 2010. Claudia Fonte Sanchez, Pierre Gabriel, and Stéphane Mischler. On the Krein-Rutman theorem and beyond. , 2023. Frédéric Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. , 46(3-4):349--359, 2006. Gary M. Lieberman. . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. Edward W. Larsen and Joseph B. Keller. Asymptotic solution of neutron transport problems for small mean free paths. , 15:75--81, 1974. Stéphane Mischler. Kinetic equations with Maxwell boundary conditions. , 43(5):719--760, 2010. Zhimeng Ouyang. Diffusive limit of the unsteady neutron transport equation in bounded domains. , 2023. Laure Saint-Raymond. , volume 1971 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin, 2009. Cédric Villani. Hypocoercivity. , 202(950):iv+141, 2009. Lei Wu and Yan Guo. Geometric correction for diffusive expansion of steady neutron transport equation. , 336(3):1473--1553, 2015. Lei Wu. Diffusive limit of transport equation in 3D convex domains. , 4(2):203--284, 2021. Yuzhe Zhu. Regularity of kinetic Fokker-Planck equations in bounded domains. , 2022.

arxiv_math

--- abstract: | In this paper we relate minimal left ideals on Clifford algebras with special geometric structures in dimensions $6,7,$ and $8$. address: Departament of Mathematics, UNITO, Torino, Italy author: - Ricardo Suárez title: A Minimal left ideal description of Geometric structures in dimensions $6$, $7$, and $8$ --- # Introduction Spinorial descriptions of special geometric structures in dimensions $6,7,$ and $8$ have been recently explored in Riemannian geometry. The spaces of spinors that are often used are $\Delta={\mathbb R}^{m}$, where $m=8,8,16$, for $SU(3)$, $G_2$, and $Spin(7)$ structures respectively. The goal of this paper is to canonically identify special geometric structures in dimensions $6,7,$ and $8$ with the left Clifford modules generated by a primitive idempotents, i.e. a minimal left ideals for the appropriate dimensions. For instance, in dimension $6$, the $SU(3)$ structure $\psi_+,\psi_{-},\omega_{0}$ defines the minimal left ideal ${\mathbb R}_{0,6}\cdot \frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$, where $\frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$ is the unit element in the induced minimal left ideal. In dimension $7$, the fundamental $3$-form $\phi$ that defines the $G_2$ structure induces the minimal left ideal ${\mathbb R}_{0,7}\cdot \frac{1}{112}(\star q^{*}(\phi\wedge \star \phi)+7 q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge \star \phi))$, where $\frac{1}{112}(\star q^{*}(\phi\wedge \star \phi)+7 q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge \star \phi))$ defines the unit element in this minimal left ideal. Lastly, in dimension $8$, the $Spin(7)$ structure induced by the $4$-form $\Omega_{0}$ defines the minimal left ideal ${\mathbb R}_{0,8}\cdot \frac{1}{128}(\star q(\Omega_0\wedge \Omega_{0})-8q(\Omega_0)+q(\Omega_0\wedge\Omega_{0}))$. # Geometric structures in dimension $6,7$, and $8$ We begin with the definition of a $G$-structure on a differential manifold $M$. **Definition 1**. *Let $G$ be a closed Lie subgroup of $GL(n,{\mathbb R})$. A **$G$-structure** on $M$ is a reduction of the structure group of the frame bundle $\mathcal{F}(M)$, $GL(n,{\mathbb R})$, to $G$. That is, a $G$-structure is a principal sub-bundle $Q$ of $\mathcal{F}(M)$ with fibre $G$. If $G$ is one of the groups that appear in the Berger classification theorem, then $G$ is called a **special geometric structure**.* Let $(M,g)$ be an oriented Riemannian spin manifold; that is, $(M,g)$ is a Riemannian manifold with vanishing first and second Stiefel Whitney classes. These manifolds admit at least two tensor fields, the Riemannian metric $g$ and the volume form $dV_{g}$, whose stabilizer group at any given point is the special orthogonal group $SO(n)$. As is well known for Riemannian manifolds the holonomy group $Hol(g)$ is strictly smaller than $SO(n)$ if and only if there exist additional nontrivial parallel tensor fields on $M$ with respect to the Levi-Civita connection $\nabla^{g}$. Thus $G$-structures in general can be viewed in terms of the additional tensor fields assigned to our oriented Riemannian manifolds. In 1955 Marcel Berger classified the possible Lie subgroups of $O(n)$ for $n$-dimensional, simply connected manifolds with an irreducible, non-symmetric Riemannian metric (see \[14\]). **Theorem 2**. *Suppose that $M$ is a simply connected manifold of dimension $n$ and $g$ is an irreducible, non-symmetric Riemannian metric on $M$. Then exactly one of the following seven cases holds.* 1. *$Hol(g)=SO(n)$.* 2. *$n=2m$ with $m\ge 2$, and $Hol(g)=U(m)$ in $SO(2m)$.* 3. *$n=2m$ with $m\ge 2$, and $Hol(g)=SU(m)$ in $SO(2m)$.* 4. *$n=4m$ with $m\ge 2$, and $Hol(g)=Sp(m)$ in $SO(4m)$.* 5. *$n=4m$ with $m\ge 2$, and $Hol(g)=Sp(m)Sp(1)$ in $SO(4m)$.* 6. *$n=7$ and $Hol(g)=G_2$ in $SO(7)$.* 7. *$n=8$ and $Hol(g)=Spin(7)$ in $SO(8)$.* In dimension $6$, an $SU(3)$ structure can be defined by a $2$-form and a stable $3$-form which can locally be expressed as $\omega=e^{12}+e^{34}+e^{56}$ and $\psi_+=e^{135}-e^{146}-e^{236}-e^{245}$. Moreover, on a $6$-dimensional oriented Riemannian manifold, we have two additional tensors, the almost complex structure $J$ and a complex volume form $\Psi$ which can be expressed entirely in terms of $J$ and $\psi_{+}$: $\Psi=\psi_{+}+i\psi_{-}$, where $\psi_{-}=J\psi_{+}$. Moreover, the Riemannian metric can be expressed in terms of $\omega$ and $J$, given by the tensor equation $g=\omega(\cdot,J\cdot)$. In dimension $7$, a $G_2$ structure on a $7$-dimensional Riemannain manifold is defined by a stable $3$-form $\phi$ which can be locally (point-wise) defined as $\phi=e^{123}+e^{145}-e^{257}-e^{347}+e^{167}-e^{356}+e^{246}$ with respect to the local frame $e^{1},\ldots,e^{7}$ of $T_{p}^{*}M$. For $G_2$ manifolds, the Riemannian metric and volume form are expressed in terms of the fundamental $3$-form via the equation $g(X,Y)dV=\frac{1}{6}(\iota_X\phi)\wedge (\iota_Y\phi)\wedge \phi$ for any pair of vector fields $X,Y$. On dimension $8$ Riemannian manifolds, a $Spin(7)$ structure is defined as an admissible $4$-form that is locally expressed as: $$\Omega=e^{1234}+e^{1256}+e^{1278}+e^{1357}-e^{1368}-e^{1458}$$ $$-e^{1467}-e^{2358}-e^{2367}-e^{2457}+e^{2468}+e^{3456}+e^{3478}+e^{5678}.$$ # Real Clifford algebras and minimal left ideals generated by primitive idempotents Let $V$ be a finite dimensional ${\mathbb R}$ vector space with quadratic form $q$, and $V^{\otimes}=\bigoplus_k V^{\otimes k}$ its tensor algebra. We define the **Clifford algebra** of the pair $(V,q)$ as the quotient of the tensor algebra with the two-sided ideal $I_q=\langle v\otimes v+q(x)1_{V^{\otimes}}\rangle$; that is, $C_{q}(V)=V^{\otimes}/I_q$. The Clifford algebra $C_{q}(V)$ carries a natural $\mathbb{Z}_2$ grading, where $C_{q}^{0}(V)$ denotes the elements of even degree, while $C_{q}^{1}(V)$ denotes the elements of odd degree. Choosing a basis for $(V,q)$, say $e_1,\ldots,e_n$, we get the following canonical basis for the Clifford algebra generated by the following $2^n$ monomials: $$\label{canonicalbasis} \{1_V\} \cup \{e_{i_1} \cdots e_{i_k} : i_1 < i_2 < \cdots < i_k, k = 1, \ldots , n\}.$$ As is well known, for any quadratic space over ${\mathbb R}$ we have an isomorphism with a quadratic space of the form ${\mathbb R}^{p,q}$, with a signature $(p,q)$, where $p$ is the number of positive definite generators and $q$ the number of negative definite generators. For the quadratic spaces ${\mathbb R}^{p,q}$ and ${\mathbb R}^{n}$, we denote the associated Clifford algebras by ${\mathbb R}_{p,q}$ and ${\mathbb R}_{0,n}$ respectively. ${\mathbb R}_{0,n}$ is canonically isomorphic to $\bigwedge {\mathbb R}^n$ as ${\mathbb R}$ vector spaces. This isomorphism is achieved via the assignment of the **quantization map** $q:\bigwedge {\mathbb R}^n\rightarrow {\mathbb R}_{0,n}$, where $e_{i_1}\wedge \cdots \wedge e_{i_k}\mapsto e_{i_1} \cdots e_{i_k}$. The natural grading of both algebras is preserved via this assignment. The inverse of this isomorphism is what is called the **symbol map**, which we denote $\sigma: {\mathbb R}_{0,n}\rightarrow \bigwedge {\mathbb R}^n$, given by $\sigma(e_{i_1} \cdots e_{i_k})=e_{i_1}\wedge \cdots \wedge e_{i_k}$. These maps have natural extensions onto the spaces of exterior forms on ${\mathbb R}^n$. We denote these extensions by $q^{*}:\bigwedge^{*}({\mathbb R}^n)^{*}\rightarrow {\mathbb R}_n$ where $q^{*}(e^{i_i}\wedge \cdots \wedge e^{i_k})=e_{i_1} \cdots e_{i_k}$, and $\sigma^{*}:{\mathbb R}_n\rightarrow \bigwedge^{*} ({\mathbb R}^n)^{*}$ where $\sigma^{*}(e_{i_1} \cdots e_{i_k})=e^{i_1}\wedge \cdots \wedge e^{i_k}$, where $e^{1}, \ldots ,e^{n}$ is the dual basis to the canonical basis $e_1, \ldots ,e_n$; that is, $e^{i}(e_j)=\delta_{ij}$. It is well known that Clifford algebras are isomorphic to matrix algebras over the division algebras ${\mathbb R},{\mathbb C},$ and $\mathbb{H}$. This classification is given in the following theorem (see \[7\]). **Theorem 3**. *The Clifford algebra ${\mathbb R}_{p,q}$, where $p+q=n$, has the following minimal representations over ${\mathbb R}$, ${\mathbb C}$, and $\mathbb{H}$:* 1. *${\mathbb R}_{p,q}\cong M_{2^{\frac{n}{2}}}({\mathbb R})$ if $q-p = 0,6 \mod 8$.* 2. *${\mathbb R}_{p,q}\cong M_{2^{\frac{n-1}{2}}}({\mathbb C})$ if $q-p = 1,5 \mod 8$.* 3. *${\mathbb R}_{p,q}\cong M_{2^{\frac{n-2}{2}}}(\mathbb{H})$ if $q-p = 2,4 \mod 8$.* 4. *${\mathbb R}_{p,q}\cong M_{2^{\frac{n-3}{2}}}(\mathbb{H})\oplus M_{2^{\frac{n-3}{2}}}(\mathbb{H})$ if $q-p = 3 \mod 8$.* 5. *${\mathbb R}_{p,q}\cong M_{2^{\frac{n-1}{2}}}({\mathbb R})\oplus M_{2^{\frac{n-1}{2}}}({\mathbb R})$ if $q-p = 7 \mod 8$.* For any semisimple algebra $A$, a minimal left ideal is of type $A\cdot e$, where $\cdot$ is multiplication in the algebra and $e\in A$ is a **primitive idempotent**. An element $e$ is primitive if it cannot be written as a sum of orthogonal idempotents ($e=f+g$; $f^{2}=f$, $g^{2}=g$, $f\cdot g=g\cdot f=0$ see \[17\]). An idempotent is called **minimal** if it is minimal with respect to the partial ordering $f\le e$, which happens if and only if $ef=f=fe$ (see \[17\]). A **minimal left ideal** $I\subset A$ is a left ideal that does not contain any other nonzero left ideals. It is known that if $I$ is a minimal left ideal of our algebra $A$ then either $I^2=0$ or $I=A\cdot e$ for some idempotent $e\in A$ (see \[17\]). For our Clifford algebras, any minimal left ideal is of the form ${\mathbb R}_{p,q}\cdot f$ where $f$ is a primitive idempotent. ${\mathbb R}_{p,q}\cdot f$ is clearly a left ${\mathbb R}_{p,q}$ module, where module multiplication is given by ${\mathbb R}_{p,q}\times {\mathbb R}_{p,q}\cdot f\rightarrow {\mathbb R}_{p,q} \cdot f$, via $(\phi,\psi\cdot f)\mapsto (\phi\cdot \psi)\cdot f$, for all $\phi\in{\mathbb R}_{p,q}$ and $\psi\cdot f\in{\mathbb R}_{p,q}\cdot f.$ **Definition 4**. *The minimal left ideals of the Clifford algebra ${\mathbb R}_{p,q}$ are called spinor spaces, and the elements of the minimal left ideals are called algebraic spinors.* As is consistent with spin geometry, these spinor spaces generate our spinor representations of the appropriate dimension. The following theorem gives us the construction and classification of minimal left ideals in ${\mathbb R}_{p,q}$ (see \[17\]). **Theorem 5**. *A minimal left ideal of ${\mathbb R}_{p,q}$ is of type ${\mathbb R}_{p,q}\cdot f$ where $f=\dfrac{1+e_{t_1}}{2}\cdot \cdots \cdot \dfrac{1+e_{t_k}}{2}$ is a primitive idempotent in ${\mathbb R}_{p,q}$ and $e_{t_1}, \ldots ,e_{t_k}$ is a set of commuting elements of the canonical basis such that $e_{t_i}^2=1$ for all $i=1, \ldots ,k=q-r_{q-p}$; moreover, the generators form a multiplicative group of order $2^{q-r_{q-p}}$. The numbers $r_i$ are called the Randon-Hurwitz numbers, given by the recurrence formula $r_{q-p}$ subject to the conditions: $r_0=0,\ r_1=1,\ r_2=2,\ r_3=2,\ r_j=3$ where $4\le j\le 7$, $r_{i+8}=r_i+4$ for $i\ge 0$, $r_{-1}=-1$, and $r_{-i}=1-i+r_{i-2}$ for $i\ge 2$.* The $k$ commuting elements generate $2^k$ different idempotents which yield the decomposition of the Clifford algebra given by: $${\mathbb R}_{p,q}=\bigoplus_{all\ \pm\ combinations} {\mathbb R}_{p,q}\cdot \prod_{\alpha} \dfrac{1\pm e_{\alpha}}{2},$$ where each minimal left ideal ${\mathbb R}_{p,q}\cdot \prod_{\alpha} \dfrac{1\pm e_{\alpha}}{2}$ is of real dimension $2^{p+q-k}$. The algebra of endomorphisms is isomorphic to the real matrix algebra of dimensions matching those of the above theorem; that is, ${\mathbb R}_{p,q}\cong End({\mathbb R}_{p,q}\cdot f)\cong M_{2^{p+q-k}}({\mathbb R})$. Restricting our representations to the spin sub-groups $\rho: Spin(p,q)\rightarrow Aut({\mathbb R}_{p,q}\cdot f)$ gives us the usual spinor representations. In the next three sections, we properly recover geometric structures in dimensions $6,7,$ and $8$ in terms of these minimal left ideals. # Recovering $SU(3)$ structures from Algebraic spinors in dimension $6$ In this section we make the association between the local description of the $SU(3)$ structure in dimension six and its associated minimal left ideal of ${\mathbb R}_{0,6}$. ## An $SU(3)$ structure recovered from a minimal left ideal in dimension $6$ An $SU(3)$ structure in dimension $6$ is given by tensors $\omega_0, \psi_{\pm},J_0$ such that their stabilizer in ${\mathbb R}^6$ is the group $SU(3)$. For the Clifford algebra ${\mathbb R}_{0,6}$ we define the primitive idempotent $f=\left(\dfrac{1+e_{135}}{2}\right)\left(\dfrac{1-e_{146}}{2}\right)\left(\dfrac{1-e_{236}}{2}\right)=\frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})$, where ${\mathbb R}_{0,6}\cdot f$ is the associated minimal left ideal. Now we normalize the idempotent to get unit coefficients which we denote $W=8f=1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256}$ in ${\mathbb R}_{0,6}$. Utilizing the quantization and symbol maps, $q^{*}:\bigwedge ({\mathbb R}^6)^{*}\rightarrow {\mathbb R}_{0,6}$ and $\sigma^{*}:{\mathbb R}_{0,6}\rightarrow \bigwedge ({\mathbb R}^6)^{*}$, as well as the grading projection map $\pi_{\alpha}: {\mathbb R}_{0,6}\rightarrow {\mathbb R}_{0,6}^{\alpha}$, where $\pi_{\alpha}(x)=\langle x \rangle_{\alpha}$ is the $\alpha$ graded component, we have $\langle W \rangle_0=1,\langle W \rangle_3=e_{135}-e_{146}-e_{236}-e_{245},$ and $\langle W \rangle_4=e_{3456}-e_{1234}-e_{1256}$. This gives us the graded decomposition $W=\langle W \rangle_0+\langle W \rangle_3+\langle W \rangle_4$. Using the Clifford Hodge dual, we have the relationship $\star \langle W \rangle_3=e_{246}-e_{235}-e_{145}-e_{136}$, and $\star \langle W \rangle_4=-(e_{12}+e_{56}+e_{34})$. With these relations, we recover the $SU(3)$ structure given by $\sigma^{*}(\langle W \rangle_3)=\psi_+\in \bigwedge^{3} ({\mathbb R}^{6})^{*}$, $-\sigma^{*}(\star \langle W \rangle_3)=\psi_{-}\in \bigwedge^{3} ({\mathbb R}^{6})^{*}$, $-\sigma^{*}(\star \langle W \rangle_4)=\omega_{0}\in \bigwedge^{2} ({\mathbb R}^{6})^{*}$. **Proposition 6**. *Fix the primitive idempotent $f=\frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})$ in ${\mathbb R}_{0,6}$ that defines the minimal left ideal ${\mathbb R}_{0,6}\cdot f$. Normalize the left ideal so that we have unit coefficients in the sense $W=8f$. From this we recover the tensor fields that define an $SU(3)$ structure in dimension $6$. That is, $\sigma^{*}( \langle W \rangle_3)=\psi_+, -\sigma^{*}(\star \langle W \rangle_3)=\psi_{-}$, and $-\sigma^{*}(\star \langle W \rangle_4)=\omega_{0}$.* ## Minimal left ideal induced from an $SU(3)$ structure For the converse, we fix an $SU(3)$ structure in ${\mathbb R}^{6}$ defined by the local tensors $\omega_0=e^{12}+e^{34}+e^{56}$ and $\psi_+=e^{135}-e^{146}-e^{236}-e^{245}$, and $\psi_{-}=J\cdot \psi_{+}$, where $J$ is the complex structure in ${\mathbb R}^{6}$. It is easy to see that $q^{*}(\psi_+)=e_{135}-e_{246}-e_{236}-e_{145}$, $q^{*}(\psi_{-})=e_{136}+e_{145}+e_{235}-e_{246}$, and $q^{*}(\omega_0)=e_{12}+e_{56}+e_{34}$. The volume form can be expressed in terms of $\psi_{+}$ and $\psi_{-}$ using the formula $\psi_{+}\wedge \psi_{-}=4e^{123456}$; hence the volume element of the Clifford algebra ${\mathbb R}_{0,6}$ can be expressed by the formula $\frac{1}{4} q^{*}(\psi_{+}\wedge \psi_{-})=e_{123456}$. Now via the Clifford Hodge product we have $\frac{1}{4}\star q^{*}(\psi_{+}\wedge \psi_{-})=1$. Hence we have the formula $\frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))= \frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})=\left(\dfrac{1+e_{135}}{2}\right)\left(\dfrac{1-e_{146}}{2}\right)\left(\dfrac{1-e_{236}}{2}\right)$, where $e_{135}, e_{146}, e_{236}$ are commuting positive definite elements in the canonical basis. Hence we have a primitive idempotent, establishing the following: **Proposition 7**. *The $SU(3)$ structure $\omega_0,\psi_{+},\psi_{-}$ in ${\mathbb R}^6$ determines the primitive idempotent $f=\frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$ in the Clifford algebra ${\mathbb R}_{0,6}$, and the resulting minimal left ideal is ${\mathbb R}_{0,6}\cdot \frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$.* Definitionally, elements of our minimal left ideal $\sigma\in {\mathbb R}_{0,6}\cdot \frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$ are algebraic spinors generated by $SU(3)$ structures. Although it is enough to have this canonical identification, from a well defined $SU(3)$ structure $\psi_{+},\psi_{-},\omega_{0}$ we can always generate a primitive idempotent and hence a minimal left ideal in this canonical manner. Now the primitive idempotent, $f=\frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$, represents the identity class in the canonical projection $\phi\mapsto \phi\cdot f$. Moreover, we have the ${\mathbb R}$ basis $f,e_2f,e_3f,e_5f,e_{23}f,e_{25}f,e_{35}f,e_{235}f$ in ${\mathbb R}_{0,6}\cdot f$, where each basis element is a unit spinor in the module. # Algebraic spinors in dimension $7$ In dimension $7$, a $G_2$ structure is a positive $3$-form whose local tensor in ${\mathbb R}^7$ is given by $\phi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}\in\bigwedge^{3}({\mathbb R}^{7})^{*}$, and whose fundamental $4$-form with respect to the Hodge star operator is given by $\star \phi_0=e^{4567}+e^{2367}+e^{2345}+e^{1357}-e^{1346}-e^{1256}-e^{1247}\in \bigwedge^{4}({\mathbb R}^7)^{*}$. ## $G_2$ structure recovered from a minimal left ideal. We begin by fixing the primitive idempotent $f=\frac{1}{16}(1+e_{123})(1+e_{145})(1-e_{257})(1+e_{167})$, which generates the minimal left ideal ${\mathbb R}_{0,7}\cdot f$. We normalize the idempotent to obtain unit scalars via $W=16f=1+e_{123}+e_{145}-e_{2345}-e_{257}-e_{1357}+e_{1247}-e_{347}+e_{167}-e_{2367}-e_{4567}-e_{1234567}+e_{1256}-e_{356}-e_{246}-e_{1346}$. Using the symbol and projection maps, we have $\langle W \rangle_3=e_{123}+e_{145}+e_{167}+e_{246}-e_{257}-e_{347}-e_{356}$, and $\langle W \rangle_4=e_{2367}-e_{4567}+e_{1346}+e_{1256}-e_{2345}-e_{1357}+e_{1247}$. In terms of the projection decomposition, we write $W=1+ \langle W \rangle_3+ \langle W \rangle_4-e_{1234567}$, where $e_{1234567}$ is the volume element in ${\mathbb R}_{0,7}$. Using the Clifford Hodge dual, we have $\star \langle W \rangle_3= \langle W \rangle_4$ and $\star e_{1234567}=1$, and thus we have $W=\star e_{1234567}+ \langle W \rangle_3+\star \langle W \rangle_3-e_{1234567}$. Hence, using the extended symbol map, we have $\sigma^{*}(\langle W \rangle_3)=\phi_{0}\in\bigwedge^{3} ({\mathbb R}^7)^{*}$, which is the desired $G_2$ structure in ${\mathbb R}^7$. The associated $4$-form that comes with a $G_2$ structure is given by $-\sigma^{*}(\langle W \rangle_4)=\star \phi_0$. From this we have the following. **Proposition 8**. *Fix the primitive idempotent $f=\frac{1}{16}(1+e_{123})(1+e_{145})(1-e_{257})(1+e_{167})$, with ${\mathbb R}_{0,7}\cdot f$ the associated minimal left ideal. We can then obtain a $G_2$ structure in ${\mathbb R}^7$ generated from the normalized tensor $W$ via $\sigma^{*}(\langle W \rangle_3)=\phi_0$, and the associated $4$-form is then given by $-\sigma^{*}(\langle W \rangle_4)=\star \phi_0$.* ## Algebraic spinors generated by $G_2$ structures in dimension $7$ Fix a $G_2$ structure $\phi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}\in\bigwedge^{3}({\mathbb R}^{7})^{*}$. Using the quantization map, we have $q^{*}(\phi)=e_{123}+e_{145}+e_{167}+e_{246}-e_{257}-e_{347}-e_{356}$ and $q^{*}(\star\phi)=-e_{2367}+e_{4567}-e_{1346}-e_{1256}+e_{2345}+e_{1357}-e_{1247}$. The volume form in ${\mathbb R}^7$ expressed in terms of the $G_2$ structure is given by the formula $\phi\wedge \star\phi=7 e^{1234567}$, and thus we have on ${\mathbb R}_{0,7}$ the equation $q^{*}(\phi\wedge \star \phi)=7e_{1234567}$. Using the Clifford Hodge star operator, we have $\star q^{*}(\phi\wedge \star\phi)=7$. Putting this all together, we get the following primitive idempotent element in ${\mathbb R}_{0,7}$ induced by the $G_2$ structure: $$f_{\phi}=\frac{1}{112}(\star q^{*}(\phi\wedge \star \phi)+7 q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge \star \phi))=\frac{1}{16}(1+e_{123})(1+e_{145})(1-e_{257})(1+e_{167}).$$ **Proposition 9**. *Fix the local $G_2$ structure $\phi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}$ in ${\mathbb R}^7$. The $3$-form $\phi$ then determines the primitive idempotent $f_{\phi}=\frac{1}{112}(\star q^{*}(\phi\wedge \star \phi)+7 q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge \star \phi))$ in the Clifford algebra ${\mathbb R}_{0,7}$, with the resulting minimal left ideal ${\mathbb R}_7\cdot \frac{1}{112}(\star q^{*}(\phi\wedge \star \phi)+7 q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge \star \phi))=:{\mathbb R}_7\cdot f_{\phi}$.* Elements of the minimal left ideal ${\mathbb R}_{0,7}\cdot f_{\phi}$ are algebraic spinors generated by the $G_2$ structure. Although it is enough to have this canonical identification, from a well defined $G_2$ structure $\phi$ we can always generate a primitive idempotent and hence a minimal left ideal in this canonical manner. As vector spaces we have ${\mathbb R}_{0,7}\cdot f_{\phi}\cong {\mathbb R}^8$, with the basis of equivalence classes given by $f_{\phi},e_1f_{\phi},e_2f_{\phi},e_3f_{\phi},e_{4}f_{\phi},e_{5}f_{\phi},e_{6}f_{\phi},e_{7}f_{\phi}$. # Algebraic spinors in dimension $8$ In dimension $8$, a $Spin(7)$ structure is defined by the model tensor $\Omega_{0}=e^{1234}+e^{1256}+e^{1278}+e^{1357}-e^{1368}-e^{1458}-e^{1467}-e^{2358}-e^{2367}-e^{2457}+e^{2468}+e^{3456}+e^{3478}+e^{5678}\in\wedge^{4} ({\mathbb R}^{8})^{*}.$ Using the quantization map, we have the following: $q^{*}(\Omega_0)=e_{1234}+e_{1256}+e_{1278}+e_{1357}-e_{1368}-e_{1458}-e_{1467}-e_{2358}-e_{2367}-e_{2457}+e_{2468}+e_{3456}+e_{3478}+e_{5678},$ $q^{*}(\Omega_0\wedge \star\Omega_0)=8e_{12345678}$, and $\star q^{*}(\Omega_0\wedge \star\Omega_0)=8$. Now the formula $$f_{\Omega}=\frac{1}{128}(\star q(\Omega_0\wedge \Omega)-8q(\Omega_0)+q(\Omega_0\wedge\Omega_{0}))$$ is a primitive idempotent in ${\mathbb R}_{0,8}$, as it factors out as $f_{\Omega}=\frac{1}{16}(1-e_{1234})(1-e_{1256})(1-e_{1278})(1-e_{1357})$. Thus ${\mathbb R}_{0,8}\cdot\frac{1}{128}(\star q(\Omega_0\wedge \Omega)-8q(\Omega_0)+q(\Omega_0\wedge\Omega_{0}))$ is the minimal left ideal induced by our $Spin(7)$ structure. Moreover, ${\mathbb R}_{0,8}\cdot \frac{1}{128}(\star q(\Omega_0\wedge \Omega)-8q(\Omega_0)+q(\Omega_0\wedge\Omega_{0}))\cong\Delta_{8}={\mathbb R}^{16}$. We summarize this with the following proposition. **Proposition 10**. *The $Spin(7)$ structure $\Omega_0$ in ${\mathbb R}^8$ determines the primitive idempotent $f_{\Omega}=\frac{1}{128}(\star q(\Omega_0\wedge \Omega)-8q(\Omega_0)+q(\Omega_0\wedge\Omega_{0}))$ in the Clifford algebra ${\mathbb R}_{0,8}$, and the resulting minimal left ideal is ${\mathbb R}_{0,8}\cdot \frac{1}{128}(\star q(\Omega_0\wedge \Omega)-8q(\Omega_0)+q(\Omega_0\wedge\Omega_{0}))$.* Conversely, for ${\mathbb R}_{0,8}$ we have four commuting generators from our canonical basis $e_{1234}, e_{1256}, e_{1278}, e_{1357}$, in which the idempotent $f=\frac{1}{16}(1-e_{1234})(1-e_{1256})(1-e_{1278})(1-e_{1357})$ defines the spinor space ${\mathbb R}_{0,8}\cdot f$ isomorphic to $\Delta_{8}={\mathbb R}^{16}$. The normalized tensor, $W=16f=1-(e_{1234}+e_{1256}+e_{1278}+e_{1357}-e_{1368}-e_{1458}-e_{1467}-e_{2358}-e_{2367}-e_{2457}+e_{2468}+e_{3456}+e_{3478}+e_{5678})+e_{12345678}$, decomposes as $W=\star e_{12345678}-\langle W \rangle_4+e_{12345678}$. Hence we define $\sigma^{*}(\langle W \rangle_4)=\Omega_0$, and $\sigma^{*}(\star\langle W \rangle_4)=\star\Omega_0=\Omega_0$. This results in the following proposition: **Proposition 11**. *Fix the primitive idempotent $f=\frac{1}{16}(1-e_{1234})(1-e_{1256})(1-e_{1278})(1-e_{1357})$ such that ${\mathbb R}_{0,8}\cdot f$ is the minimal left ideal. We can associate a $Spin(7)$ structure in ${\mathbb R}^8$ generated from the normalized tensor $W$ via $\sigma^{*}(\langle W \rangle_4)=\sigma^{*}(\star\langle W \rangle_4)=\Omega_0$.* # Primitive idempotents in dimension $6$ to $G_2$ structures in dimension $7$ We conclude this paper by relating the constructions in dimension $6$ and dimension $7$ by viewing ${\mathbb R}^{7}={\mathbb R}^{6}\oplus {\mathbb R}$, where the orthogonal dimension is given by the basis vector $e_7$. We define a generic $SU(3)$ structure given in the ${\mathbb R}^6$ component by $\omega_0=e^{12}+e^{34}+e^{56}$, $\psi_{+}=e^{135}-e^{146}-e^{236}-e^{245}$, and $\psi_{-}=e^{136}+e^{145}+e^{235}-e^{246}$. For the associated Clifford algebra ${\mathbb R}_{0,6}$ we have the associated primitive idempotent $f=\left(\dfrac{1+e_{135}}{2}\right)\left(\dfrac{1-e_{146}}{2}\right)\left(\dfrac{1-e_{236}}{2}\right)=\frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})$, with ${\mathbb R}_{0,6}\cdot f$ being the associated minimal left ideal. As we saw above from the normalized idempotent $W$ in dimension $6$, we recover the $SU(3)$ structure via $\sigma^{*}(\langle W \rangle_3)=\psi_{+},\sigma^{*}(\star\langle W \rangle_3)=\psi_{-}$, and $\sigma^{*}(\star \langle W \rangle_4)=\omega_0$. Now from the induced $SU(3)$ structure in dimension $6$, we can recover a $G_2$ structure in dimension $7$ via $\phi_0=\sigma^{*}(\star\langle W \rangle_4)\wedge e^7+\sigma^{*}(\langle W \rangle_3)$. Conversely, starting with the $SU(3)$ we have the primitive idempotent $\frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$, which gives us the minimal left ideal ${\mathbb R}_{0,6}\cdot \frac{1}{32}(\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0))$. Now the normalized idempotent with unit scalars $W$ in this formulation is given by $W=\star q^{*}(\psi_{+}\wedge \psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_0)$, and hence we define the primitive idempotent in dimension $7$ as follows: $f_{\phi}= \frac{1}{112}(\star q^{*}(\sigma^{*}(\star\langle W \rangle_4)\wedge e^7+\sigma^{*}(\langle W \rangle_3)\wedge \star \sigma^{*}(\star\langle W \rangle_4)\wedge e^7+\sigma^{*}(\langle W \rangle_3))+7 q^{*}(\sigma^{*}(\star\langle W \rangle_4)\wedge e^7+\sigma^{*}(\langle W \rangle_3))-7q^{*}(\star\sigma^{*}(\star\langle W \rangle_4)\wedge e^7+\sigma^{*}(\langle W \rangle_3))-q^{*}(\sigma^{*}(\star\langle W \rangle_4)\wedge e^7+\sigma^{*}(\langle W \rangle_3)\wedge \star \sigma^{*}(\star\langle W \rangle_4)\wedge e^7+\sigma^{*}(\langle W \rangle_3) )).$ Thus ${\mathbb R}_{0,7}\cdot f_{\phi}$ is the spinor module induced from the $SU(3)$ structures recovered from the primitive idempotents in dimension $6$. # Future research With the correspondences established above, we can construct algebraic spinor bundles from the induced spinor spaces and provide classification equations for $G_2$, $SU(3)$, and $Spin(7)$ manifolds in terms of algebraic spinors. # Acknowledgments I would like to thank my advisor, Dr. Ivona Grzegorczyk, and my supervisor at UNITO, Dr. Anna Fino, as well as UNITO for financing my research and providing me guidance in this topic. EMG M. Atiyah, R. Bott and A. Shapiro, *Clifford Modules*, Topology 3, 1964. A. Besse, *Einstein Manifolds*. Springer, 1987. A. Bilge, S. Koçak, and S. Uğuz, *Canonical Bases for Real Representations of Clifford Algebras*, Linear Algebra and Its Applications **2** (2008), 417-439. G. Calvaruso, M. Castrillon Lopez, *Pseudo Riemannian Homogeneous Structures*. Springer Developments in Mathematics, 2019. \[D\] G.M Dixon, *Division Algebras: Octonions, Quaternions, Complex numbers and the Algebraic Design of Physics*. Springer-Science+ Buisness Media B.V. 290, 1994. A. Dimakis, *A New Representation for Spinors in Real Clifford Algebras*, Clifford Algebras and Their Applications in Mathematical Physics, Chisholm, J. S. R. Springer, Netherlands, 49-60, 1986. G. Hile and P. Lounesto, *Matrix Representations of Clifford Algebras*, Linear Algebra and Its Applications 128 (1990), 51-63. D. Hussenmoller, *Fibre bundles*, GTM, Springer, New York 1975. D. Joyce, *Riemannian Holonomy Groups and Calibrated Geometry*. Oxford Graduate Texts In Mathematics, 2007. F. Harvey, *Spinors and Calibrations*. Academic Press, San Diego, 1990. M. Karoubi, *K-Theory, an Introduction*, Springer Verlag, Berlin-Heidelberg-New York 1978. S. Karigiannis, N. Leung, J. Lotay, *Lectures and Surveys on $G_2$ manifolds and Related Topics*, Springer, Fields Institute Communications, Volume 84, 2020. S. Kobayashi, *Transformation Groups in Differential Geometry*. Springer, 1995. S. Kobayashi, K. Nomizu, *Foundations of Differential Geometry Volume 1*. Wiley Classics Library, 1963. H. B. Lawson Jr. and M-L Michelsohn, *Spin Geometry*, Princeton University Press, Princeton, NJ 1989. P. Lounesto, *Clifford Algebras and Spinors, Second Edition*. Cambridge University Press, Cambridge, 2001. P. Lounesto and G. Wene, *Idempotent Structure of Clifford Algebras*, Acta Applicandae Mathematica (1987), 9. E. Meinrenken, *Clifford Algebras and Lie Theory*. Springer, Berlin, 2013. A. Mukherjee, *Atiyah-Singer Index Theorem - An Introduction*. Hindustan Book Agency, 2013. Ian R. Porteous, *Clifford Algebras and the Classical Groups*. Cambridge University Press, Cambridge, 1995. A. Raffero, *Non-integrable special geometric structures in dimensions six and seven*. Ph.D. Thesis, Università degli Studi di Torino, 2016. L. Tu, *An Introduction to Manifolds*. Springer Universitext, 2011. R. O. Wells Jr., *Differential Analysis on Complex Manifolds*, Springer Verlag, New York 1980.

arxiv_math

--- abstract: | We introduce a new, Kac-Moody-flavoured construction for Lie superalgebras, which seeks to incorporate phenomena of the queer Lie superalgebra. The idea of the generalization is to replace the maximal torus by a maximal quasitoral subalgebra, which has the representation theory of a family of (degenerate) Clifford superalgebras. Remarkably, we find that the theory is quite rigid, and a natural class of Lie superalgebras becomes apparent, which we call queer Kac-Moody algebras. We classify finite growth queer Kac-Moody algebras, which includes an $\mathfrak{so}(3)$-superconformal algebra, and give a new perspective on the distinctiveness of the queer Lie superalgebra. author: - Alexander Sherman, Lior Silberberg title: A Queer Kac-Moody Construction --- # Introduction In 1968, Kac [@K3] and Moody [@Moody] introduced a class of Lie algebras that are now known as Kac-Moody algebras. On the surface level, the theory of such algebras is a simple yet beautiful extension of the theory of semisimple Lie algebras as developed by Lie, Killing, and Cartan. Namely, from the data of an $n\times n$ complex matrix $A$ (often assumed to satisfy certain conditions), one produces a Lie algebra $\mathfrak{g}(A)$, and if $A$ is nice enough (i.e. symmetrizable, generalized Cartan) then $\mathfrak{g}(A)$ admits a neat presentation in terms of generators (the Chevalley generators) and relations (the Chevalley-Serre relations). Although the applications were not immediately apparent, in the 70s and 80s it dawned that Kac-Moody algebras, and in particular the class of affine Lie algebras, have rich connections to physics, number theory, modular forms, theta functions, differential equations, the Virasoro algebra, and more. We refer to [@K] for a comprehensive study of such algebras and their applications to other fields, as well as a wealth of references to other works. Somewhat in parallel, the theory of Lie superalgebras had a start, with a major milestone being Kac's seminal 1977 paper [@K2] in which, among other things, the simple finite-dimensional Lie superalgebras were classified. It was realized already in [@K2] that the theory of Kac-Moody Lie algebras admits an obvious extension to the super setting, where one starts with an $n\times n$ matrix $A$ along with a parity function $p:\{1,\dots,n\}\to\mathbb{Z}/2\mathbb{Z}$. One obtains (up to central quotients and derived subalgebras) all simple, basic (admitting an even invariant form) classical Lie superalgebras from this approach. Finite growth Kac-Moody superalgebras were classified later on in the works of Kac ([@K4]) (the case with no isotropic simple roots), Van-de Leur ([@L]) (the symmetrizable case), and finished by Hoyt and Serganova (see [@C] and [@CS]). The theory of affine Lie superalgebras has connections once again to physics and number theory, amongst other fields, and such connections have been explored in, e.g. [@KW1], [@KW2], and [@KW3]. See [@GHS] for a more recent approach to Kac-Moody superalgebras, which nicely clarifies the role of the Weyl groupoid. ## It's here and it's queer Amidst the progress in understanding Kac-Moody Lie superalgebras, a certain Lie superalgebra was left out: the queer Lie superalgebra. In the theory of finite-dimensional associative superalgebras over the complex numbers, there are two families of central simple superalgebras: namely, $\operatorname{End}(\mathbb{C}^{m|n})$ and $Q(n)$, where the latter is the queer associative superalgebra, and it can be viewed as the subalgebra of $\operatorname{End}(\mathbb{C}^{n|n})$ consisting of endomorphisms commuting with a particular odd automorphism. One may present $Q(n)$ as matrices of the form $$\begin{bmatrix} A & B\\ B & A \end{bmatrix}$$ where $A,B$ are arbitrary $n\times n$ matrices. Thus one arrives at the Lie superalgebra $\mathfrak{q}(n)$, which is the natural Lie superalgebra obtained from $Q(n)$ with the supercommutator operation. On the face of it, $\mathfrak{q}(n)$ is a nice Lie superalgebra, whose even part is $\mathfrak{g}\mathfrak{l}(n)$ and whose odd part is the adjoint representation of $\mathfrak{g}\mathfrak{l}(n)$. However, it is distinctive in that its Cartan subalgebra, $$\mathfrak{h}=\begin{bmatrix} D & D'\\ D' & D \end{bmatrix},$$ where $D,D'$ are diagonal, is *not* purely even or commutative. It is instead what we call *quasitoral*, i.e. it satisfies $[\mathfrak{h}_{\overline{0}},\mathfrak{h}]=0$. This property prevents it from ever being described in classical Kac-Moody terms, because in that setup one always begins with a self-normalizing torus. In fact the root system of $\mathfrak{q}(n)$ is nothing but the classical $A_{n-1}$ root system. Thus it is clear that to have a Kac-Moody type construction that gives rise to the queer Lie superalgebra, one needs to begin with a self-normalizing quasitoral subalgebra. ## The classical construction Without going into too many details, we explain the general construction we provide here. Let us start with the classical construction. Starting with an $n\times n$ matrix $A=(a_{ij})$ and parity function $p:\{1,\dots,n\}\to\mathbb{Z}/2\mathbb{Z}$, one begins by finding a finite-dimensional abelian Lie algebra $\mathfrak{h}$ along with linearly independent sets $$\Pi=\{\alpha_1,\dots,\alpha_n\}\subseteq\mathfrak{h}^*, \ \text{ and } \ \Pi^\vee=\{h_1,\dots,h_n\}\subseteq\mathfrak{h}, \text{ satisfying } \alpha_j(h_i)=a_{ij}.$$ Then one takes the algebra generated by $\mathfrak{h}$ and Chevalley generators $e_1,\dots,e_n,f_1,\dots,f_n$, where the parity of $e_i$ and $f_i$ is $p(i)$, and we have the relations $$[h,e_{i}]=\alpha_i(h)e_i, \ \ \ [h,f_i]=-\alpha_i(h)f_i, \ \ \ [e_i,f_j]=\delta_{ij}h_i.$$ One then defines $\mathfrak{g}(A)$ to be the quotient of this algebra by the maximal set of relations that don't kill any elements in $\mathfrak{h}$--if $A$ is generalized Cartan and symmetrizable, then this is just the Serre relations. For the Lie superalgebra case, see [@LS] for more on the relations. To understand our generalization, one should view the choices of $\alpha_i\in\mathfrak{h}^*$ and $p(i)\in\mathbb{Z}/2\mathbb{Z}$ as a choice of irreducible representations of $\mathfrak{h}$: indeed, $\alpha_i$ tells you the action, and $p(i)$ tells you the parity of the representation. Thus the Chevalley generators are just nonzero elements (i.e. bases) of corresponding irreducible representations of $\mathfrak{h}$. ## The queer construction To generalize to the queer setting, one should start with a quasi-toral Lie superalgebra $\mathfrak{h}$, where we set $\mathfrak{t}:=\mathfrak{h}_{\overline{0}}$. Then the irreducible representations of $\mathfrak{h}$ are in bijection (up to parity) with $\mathfrak{t}^*$; however even when $\mathfrak{t}$ acts semisimply, which we assume, the category of $\mathfrak{h}$-modules is not semisimple, which adds both a complication and richness to the theory. With $\mathfrak{h}$ in hand, one should take a linearly independent set $\Pi=\{\alpha_1,\dots,\alpha_n\}\subseteq\mathfrak{t}^*$. The natural generalization of the Chevalley generators is to choose irreducible representations $$\mathfrak{g}_{\alpha_1},\dots,\mathfrak{g}_{\alpha_n},\mathfrak{g}_{-\alpha_1},\dots,\mathfrak{g}_{-\alpha_n}$$ of $\mathfrak{h}$ with specified weights $\pm\alpha_1,\dots,\pm\alpha_n$. Finally, to have coroots, one needs (nontrivial) $\mathfrak{h}$-equivariant maps $[-,-]_{i}:\mathfrak{g}_{\alpha_{i}}\otimes\mathfrak{g}_{-\alpha_i}\to\mathfrak{h}$. We call the package of data specified above a Cartan datum, $\mathcal{A}$, and one may now construct a Lie superalgebra $\mathfrak{g}(\mathcal{A})$ in a completely analogous fashion as in the classical Kac-Moody setup. In particular $\mathfrak{g}(\mathcal{A})$ will contain $\mathfrak{h}$ as a self-normalizing subalgebra, a root decomposition, it will contain $\mathfrak{g}_{\pm\alpha_i}$ for all $i$, and it will admit a natural triangular decomposition $\mathfrak{g}(\mathcal{A})=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}$. ## Clifford Kac-Moody algebras The above setup is very general however (in fact one doesn't even need the $\mathfrak{h}$-modules $\mathfrak{g}_{\alpha_i}$ to be irreducible), and to get something with more structure it makes sense to impose the condition of integrability (see Definition [\[definition-integrable\]](#definition-integrable){reference-type="ref" reference="definition-integrable"}). This gives us the class of what we call Clifford Kac-Moody algebras. Here Clifford indicates that the simple root spaces are representations are Clifford algebras. It is natural to then classify Clifford Kac-Moody algebras with one simple root, and determine how the roots can interact with one another. We have done this in Sections 5 and 6, and the answer we obtain is depicted in the diagram below. To explain, the 'nodes' of the diagram represent possible Clifford Kac-Moody algebras with one simple root $\alpha$, and we have drawn an arrow $\alpha\to\beta$ if it is possible (under our integrability assumption) that $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$, where $\mathfrak{h}_{\alpha}=[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]$ are the $\alpha$-coroots. This is analogous to the condition that $\alpha_j(h_i)\neq0$ in a Kac-Moody algebra. In particular, a loop at a node means that it is possible for two simple roots of that type to interact. Here the node 'Super KM' contains the three possible root subalgebras $\mathfrak{s}\mathfrak{l}(2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$, and $\mathfrak{s}\mathfrak{l}(1|1)$. These subalgebras can interact with in each in every possible way, so to speak, which is what the self arrow represents. They give rise to classical Kac-Moody superalgebras. Similarly, the node 'Queer KM' represents the three possible root subalgebras which appear in our definition of a *queer Kac-Moody* algebra: they are (up to central quotient) of the form $\mathfrak{t}\mathfrak{s}:=\mathfrak{s}\otimes\mathbb{C}[\xi]\oplus\mathbb{C}\langle c\rangle$, where $\mathbb{C}[\xi]=\mathbb{C}\langle1,\xi\rangle$ is a supersymmetric polynomial algebra on one odd variable, $c$ is central, and $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2),$ or $\mathfrak{s}\mathfrak{l}(1|1)$. The formula for the bracket is given in Example [\[example-takiff\]](#example-takiff){reference-type="ref" reference="example-takiff"}, and is entirely analogous to the loop algebra construction. Such an 'odd' affinization is what we refer to as a 'Takiff' construction, following an established convention, and in recognition of Takiff's work [@T]. Thus we see that queer Kac-Moody nodes are obtained from super Kac-Moody nodes by applying the Takiff construction, so to speak. And further, the diagram shows that queer Kac-Moody nodes can interact with each other in interesting ways. $$\xymatrix{ &&&\text{Super KM} \ar@(ul,ur) \ar@/_3pc/[ddddlll] \ar@/^3pc/[ddddrrr]\ar[ddr]\ar[ddl] \ar@/^2pc/[ddddl]\ar@/_2pc/[ddddr]&&&\\ &&&&&&\\ &&\text{Queer KM}\ar[dd] \ar[ddrr] \ar@(ul,ur)\ar@/^2pc/@{-->}[rr] &&\ar[ll] \ar[dd]\ar[lldd]H_1 &&\\ &&&&&&\\ \mathfrak{h}\mathfrak{e}(0) & & \mathfrak{h}\mathfrak{e}(2)^{\Pi} && H_n,\ n\geq 3 & & \mathfrak{h}\mathfrak{e}(0)^{\Pi} }$$ The other nodes are all of 'Heisenberg' type, which means that $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\alpha}]=0$. The diagram illustrates that they cannot interact as nicely with the other nodes under our integrability assumption. (See Section [6](#section-connectivity){reference-type="ref" reference="section-connectivity"} for further explanation and for the meaning of the dashed arrow). ## Queer Kac-Moody algebras As has already been stated, queer Kac-Moody (qKM) algebras are those constructed from simple roots of type $\mathfrak{t}\mathfrak{s}$, where $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$, or $\mathfrak{s}\mathfrak{l}(1|1)$. Note also that $\mathfrak{t}\mathfrak{s}\mathfrak{l}(2)\cong\mathfrak{sq}(2)=[\mathfrak{q}(2),\mathfrak{q}(2)]$. Thus arises the question of classification of such algebras, and in particular the classification of the finite growth qKM algebras. One simple way to produce a qKM algebra is to start with a Kac-Moody superalgebra $\mathfrak{s}$ and apply the Takiff construction to obtain, $\mathfrak{s}\otimes\mathbb{C}[\xi]\oplus\langle c,\partial_{\xi},\dots\rangle$. The ellipsis indicates that there are certain central extensions and derivations that may be added, but we ignore this for now. Although this is a beautiful and important superalgebra (and in the finite-dimensional case a slight variant of it was studied by [@CC]), we view the Takiff construction as somewhat degenerate: in particular every simple root shares the central element $c$ as a coroot. We thus call such algebras 'completely coupled' (or $Y$-coupled to be more precise). There are in fact notions of being completely $X$-coupled, completely $Y$-coupled, and completely uncoupled, the latter being the most 'interesting' case (see Definition [\[definition_coupling\]](#definition_coupling){reference-type="ref" reference="definition_coupling"}). In Section [9](#section_coupled_v_uncoupled){reference-type="ref" reference="section_coupled_v_uncoupled"} we prove the following: An indecomposable, finite growth qKM algebra is either completely $X$-coupled, completely $Y$-coupled, or completely uncoupled. This allows us to separate our study into the three separate classes. For the following, we note that qKM algebras have Dynkin diagrams with vertices for $\mathfrak{t}\mathfrak{s}\mathfrak{l}(2)$ simple roots, for $\mathfrak{t}\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ simple roots, and $\diamondtimes$ for $\mathfrak{tsl}(1|1)$ simple roots. We put labels on edges to indicate values of pairings of roots with coroots (see Section [7.4](#section_dynkin){reference-type="ref" reference="section_dynkin"}). Let $\mathfrak{g}$ be an indecomposable, completely $X$-coupled algebra with more than one simple root, without any assumptions on growth conditions. Then $\mathfrak{g}$ is of finite growth with GK dimension 1, and has one of three possible Dynkin diagrams: For the next theorem, let $\Theta$ denote the map which takes a Dynkin diagram of a qKM algebra and turns the diamonds into circles, thus giving the Dynkin diagram of Kac-Moody superalgebra. [\[thm_intro_Y\_coupled\]]{#thm_intro_Y_coupled label="thm_intro_Y_coupled"} Suppose that $\mathfrak{g}$ is a finite growth qKM algebra with Dynkin diagram $D$. Let $\mathfrak{s}$ be the Kac-Moody superalgebra obtained from the Dynkin diagram $\Theta(D)$. If $\mathfrak{g}$ is completely $Y$-coupled, then $\mathfrak{g}$ is constructed from $\mathfrak{s}$ via the Takiff construction (see Theorem [\[theorem-Takiffs\]](#theorem-Takiffs){reference-type="ref" reference="theorem-Takiffs"} for a precise statement). ## Completely uncoupled qKM algebras Theorem [\[thm_intro_Y\_coupled\]](#thm_intro_Y_coupled){reference-type="ref" reference="thm_intro_Y_coupled"} tells us that any possible finite-growth Dynkin diagram can appear in our construction. However if we pass to the completely uncoupled situation, things become more rigid. Namely we prove the following: [\[theorem_intro_completely_uncoupled\]]{#theorem_intro_completely_uncoupled label="theorem_intro_completely_uncoupled"} The possible Dynkin diagrams of an indecomposable completely uncoupled qKM algebra of finite growth with more than one simple root are the following: 1. Type $A(n)$ for $n \in \mathbb{Z}_{\geq 2}$, giving rise to $\mathfrak{q}(n+1)$: with $n$ vertices. 2. Type $A(n)^{(1)}$ for $n \in \mathbb{Z}_{\geq 2}$, giving rise to $\mathfrak{q}(n+1)^{(1)}$, an affinization of $\mathfrak{q}(n)$: with $n+1$ vertices. 3. Type $A(1)^{(1)}$, giving rise to two superalgebras $\mathfrak{q}_{(2,2)}^{\pm}$: Note that $\mathfrak{q}(2)^{(1)}$ is actually $Y$-coupled, and comes from a Takiff construction. One outcome of Theorem [\[theorem_intro_completely_uncoupled\]](#theorem_intro_completely_uncoupled){reference-type="ref" reference="theorem_intro_completely_uncoupled"} is that the only finite-type, completely uncoupled qKM root system we can obtain is $A(n)$, and ultimately the only algebra we can get (up to central extensions/quotients and derivations/commutator subalgebras) is $\mathfrak{q}(n)$. Thus it shows that there cannot be a queer version of other simple root systems, which is clear from the lack of any simple algebra of this form, but difficult to perceive otherwise. Another outcome is the existence of two nontrivial completely uncoupled finite-growth Lie superalgebra $\mathfrak{q}^{\pm}_{(2,2)}$. These Lie superalgebras both have root system of type $A^{(1)}_{1}$, and contain $\widehat{\mathfrak{s}\mathfrak{l}}_{2}$ as a subalgebra. In a future work it will be shown that $\mathfrak{q}^{+}_{(2,2)}$ is the $\mathfrak{s}\mathfrak{o}(3)$-superconformal algebra $K(3,1,0,0)$ in the notation of [@KL]. ## Serre relations Establishing Serre relations is important in the study of Kac-Moody superalgebras, in particular for quantizations of superalgebra. It is natural to ask when qKM algebras admit nice presentations. We have the following, which is proven in [@Si]. Let $\mathfrak{g}$ be a finite growth, completely uncoupled Lie superalgebra with simple roots $\{\alpha_1,\dots,\alpha_n\}$, where $n>1$. Let $A=(a_{ij})$ be its Cartan matrix (see Section [7.4](#section_dynkin){reference-type="ref" reference="section_dynkin"}). Then $\mathfrak{g}$ is generated by $\mathfrak{h}$ and simple root spaces $\mathfrak{g}_{\pm\alpha_1},\dots,\mathfrak{g}_{\pm\alpha_i}$, subject to the following relations: 1. Chevalley-type relations (those defining $\mathfrak{g}(\mathcal{A})$, as listed in Section [3.1](#section_cartan_datum){reference-type="ref" reference="section_cartan_datum"}); and 2. Serre relations: for $i\neq j$ we have $$\operatorname{ad}(\mathfrak{g}_{\alpha_i})^{1-a_{ij}}(\mathfrak{g}_{\alpha_j})=0, \ \ \ \ \operatorname{ad}(\mathfrak{g}_{-\alpha_i})^{1-a_{ij}}(\mathfrak{g}_{-\alpha_j})=0.$$ ## Future work The importance of affine Lie (super)algebras to other areas of mathematics and physics cannot be understated, and thus an important problem, which will be addressed in a future work, is an explicit realization of the finite growth qKM algebras described above. So far, we have determined that $\mathfrak{q}^+(2,2)$ is the $\mathfrak{s}\mathfrak{o}(3)$-superconformal algebra $K(3,1,0,0)$, but there are 4 others that we do not yet have realizations for: $\mathfrak{q}^{-}(2,2)$, and the three $X$-coupled superalgebras in Theorem [\[theorem_X\_coupled\]](#theorem_X_coupled){reference-type="ref" reference="theorem_X_coupled"}. It is very important to understand if these are already known superalgebras or not, and if they are, whether there are further insights that the qKM structure can provide. Another important avenue is the study of representation theory of qKM algebras. One can ask for descriptions of integrable representations, if there are character formulas, denominator identities, if one can compute the Shapovalov determinant (to extend the work of [@G]), and more. In particular we hope that nice identities can be found in the Grothendieck ring of $\mathfrak{h}$, such as was begun for $\mathfrak{q}(n)$ in [@GSS]. ## Acknowledgements The authors would like to thank Maria Gorelik and Vera Serganova for many helpful discussions and suggestions. The first author was partially supported by ARC grant DP210100251. The second author was partially supported by the funding received from the MINERVA Stiftung with the funds from the BMBF of the Federal Republic of Germany. ## Table of contents ## List of notation - $\mathcal{C}\ell(V,B)$ the Clifford superalgebra on a vector space $V$ with symmetric bilinear form $B$; - $\mathcal{S}(V)$ the supersymmetric algebra on $V$; - $\mathfrak{h}$ a finite-dimensional quasi-toral Lie superalgebra (Sec. [2.2](#section-irreducible-h-modules){reference-type="ref" reference="section-irreducible-h-modules"}); - $\mathfrak{t}=\mathfrak{h}_{\overline{0}}$ the even part of a quasitoral Lie superalgebra; - $\mathcal{F}(\mathfrak{h})$ the category of finite-dimensional $\mathfrak{h}$-modules with semisimple action of $\mathfrak{t}$ (Sec. [2.2](#section-irreducible-h-modules){reference-type="ref" reference="section-irreducible-h-modules"}); - $B_{\lambda}$ the symmetric bilinear form on $\mathfrak{h}_{\overline{1}}$ given by $B_{\lambda}(x,y)=\lambda([x,y])$ (Sec. [2.2](#section-irreducible-h-modules){reference-type="ref" reference="section-irreducible-h-modules"}); - $C_{\lambda}$ an irreducible representation of $\mathfrak{h}$ of weight $\lambda$ (Sec. [2.2](#section-irreducible-h-modules){reference-type="ref" reference="section-irreducible-h-modules"}); - $V^\vee:=V^{\omega_{\mathfrak{h}}^{-1}}$ the twist of an $\mathfrak{h}$-module $V$ by $\omega_{\mathfrak{h}}^{-1}$ (Sec. [2.3](#section_dualities){reference-type="ref" reference="section_dualities"}); - $\mathcal{A}$ a Cartan datum (Def. [\[definition-datum\]](#definition-datum){reference-type="ref" reference="definition-datum"}); - $\Pi=\{\alpha_1,\dots,\alpha_n\}\subseteq\mathfrak{t}^*$ simple roots (Sec. [3.1](#section_cartan_datum){reference-type="ref" reference="section_cartan_datum"}); - $\mathfrak{h}_{\alpha}:=[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]$ the coroot space of $\alpha$ (Def. [\[definition_coroot\]](#definition_coroot){reference-type="ref" reference="definition_coroot"}); - $\tilde{\mathfrak{g}}(\mathcal{A})$ the Lie superalgebra constructed in Sec. [3.1](#section_cartan_datum){reference-type="ref" reference="section_cartan_datum"}; - $\mathfrak{g}(\mathcal{A}):=\tilde{\mathfrak{g}}(\mathcal{A})/\mathfrak{r}$, see Sec. [3.3](#section-g(A)){reference-type="ref" reference="section-g(A)"}; - $\omega$ the Chevalley automorphism on $\mathfrak{g}(\mathcal{A})$, see Cor. [\[corollary-Chevalley\]](#corollary-Chevalley){reference-type="ref" reference="corollary-Chevalley"}; - $\mathfrak{g}\langle\alpha\rangle$ the subalgebra generated by $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$ (Sec. [5](#section-one-root){reference-type="ref" reference="section-one-root"}); - $T\mathfrak{s}$, $\mathfrak{t}\mathfrak{s}$, $\mathfrak{pt}\mathfrak{s}$ are Takiff constructions, see Ex. [\[example-takiff\]](#example-takiff){reference-type="ref" reference="example-takiff"}; - $\mathfrak{h}\mathfrak{e}(n)^{(\Pi)}$ a Heisenberg superalgebra determined by a rank $n$ simple root, see Sec. [5.1](#section_heisenberg){reference-type="ref" reference="section_heisenberg"}; - $Tak(\mathfrak{s})$ a type for a simple root, see the table at the start of Sec. [6](#section-connectivity){reference-type="ref" reference="section-connectivity"}; - $e_i,E_i,f_i,F_i$ are 'Chevalley generators' in a qKM algebra (Sec. [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}); - $H_i,h_i,c_i$ are the pure coroots of $\alpha_i$ in qKM algebra (Sec. [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}); - $x_{ij},y_{ij}$ satisfy $[H_i,e_j+E_j]=x_{ij}E_j+y_{ij}e_j$ (Sec. [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}). # Quasi-toral Lie superalgebras {#section_quasi_toral} In this section we review some known results on the representation theory of quasi-toral Lie superalgebras (see Definition [\[definition-quasi-toral\]](#definition-quasi-toral){reference-type="ref" reference="definition-quasi-toral"}), which will be used frequently in our work. For more details, we refer the reader to [@G] and [@GSS]. ## Clifford superalgebras {#section_cliff_algs} Let $V$ be a finite-dimensional vector space and let $B$ be a symmetric (not necessarily non-degenerate) bilinear form on $V$. Let $\mathcal{C}\ell(V,B)$ denote the corresponding Clifford superalgebra, with the elements of $V$ being odd. Explicitly, $\mathcal{C}\ell(V,B)$ is the quotient of the tensor superalgebra $\mathcal{T}V:=\bigoplus\limits_{n\geq0}V^{\otimes n}$ by the ideal generated by expressions of the form $v\otimes w+w\otimes v-B(v,w)$, where $v,w\in V$. Because this ideal is homogeneous (in particular it is generated by even elements), the quotient inherits the structure of an associative superalgebra. The form $B$ induces a non-degenerate symmetric bilinear form on $V/\ker B$, which we shall also denote by $B$. We have a well-known, non-canonical isomorphism of superalgebras $$\begin{aligned} \label{eq_iso_cliff} \mathcal{C}\ell(V,B) \simeq \mathcal{C}\ell(V/\ker B,B) \otimes \mathcal{S} (\ker B), \end{aligned}$$ where $\mathcal{S}(\ker B)$ is the superalgebra of supercommutative polynomials in elements of $\ker B$, a purely odd vector space. In particular $\mathcal{S}(\ker B)$ is isomorphic as an algebra to the Grassmann algebra on $\ker B$ viewed as an even vector space, and thus is finite-dimensional. Let $m$ denote the dimension of $V$, and suppose that $B$ is non-degenerate. For the following facts we refer to [@CW], Exercise 3.11. - $\mathcal{C}\ell(V,B)$ is always a simple superalgebra; - all modules over $\mathcal{C}\ell(V,B)$ are completely reducible; - if $m$ is odd, $\mathcal{C}\ell(V,B)$ admits a unique, parity invariant, irreducible module; - if $m$ is even, $\mathcal{C}\ell(V,B)$ admits two irreducible modules which differ by parity. - if $m \neq 0$ and $E$ is an irreducible $\mathcal{C}\ell(V,B)$-module, then $\dim E_{\overline{0}} = \dim E_{\overline{1}} = 2^{\lfloor \frac{m-1}{2} \rfloor}$, where $\lfloor-\rfloor$ denotes the floor function. ## Irreducible $\mathfrak{h}$-modules {#section-irreducible-h-modules} For general background on Lie superalgebras, we refer to [@CW] or [@M]. [\[definition-quasi-toral\]]{#definition-quasi-toral label="definition-quasi-toral"} A Lie superalgebra $\mathfrak{h}$ is said to be quasi-toral if $[\mathfrak{h}_{\overline{0}},\mathfrak{h}] = 0$, i.e. $\mathfrak{h}_{\overline{0}}$ is central in $\mathfrak{h}$. [\[example of h_n\]]{#example of h_n label="example of h_n"} For $n\geq0$, let $\mathfrak{h}(n)$ be the Lie superalgebra with $\mathfrak{h}(n)_{\overline{0}}=\mathbb{C}\langle c\rangle$, and $\mathfrak{h}(n)_{\overline{1}}=\mathbb{C}^n$. We impose that $c$ is a nonzero, central element of $\mathfrak{h}(n)$, and for $x,y\in\mathfrak{h}(n)_{\overline{1}}$ we set $[x,y]=(x,y)c$, where $(-,-)$ is a fixed, nondegenerate symmetric bilinear form on $\mathbb{C}^n$. Then $\mathfrak{h}(n)$ is a quasitoral Lie superalgebra. For $\mathfrak{h}$ a finite-dimensional quasi-toral Lie superalgebra, we write $\mathfrak{t} \coloneq \mathfrak{h}_{\overline{0}}$. Throughout the paper, we will consider only $\mathfrak{h}$-modules with semisimple $\mathfrak{t}$-action. Write $\mathcal{F}(\mathfrak{h})$ for the category of finite-dimensional $\mathfrak{h}$-modules with semisimple action of $\mathfrak{t}$. To a weight $\lambda \in \mathfrak{t}^*$, we associate a symmetric bilinear form $B_{\lambda}$ on $\mathfrak{h}_{\overline{1}}$, given by $B_{\lambda}(H_1,H_2) = \lambda([H_1,H_2])$. [\[definition-rank\]]{#definition-rank label="definition-rank"} For a weight $\lambda \in \mathfrak{t}^*$, we define the rank of $\lambda$ to be $\operatorname{rk}\lambda:=\operatorname{rk}B_{\lambda} \in \mathbb{Z}_{\geq 0 }$, where $\mathrm{rk}B_{\lambda}$ is the rank of the bilinear form $B_{\lambda}$ on $\mathfrak{h}_{\overline{1}}$. Observe that $\operatorname{rk}c\lambda=\operatorname{rk}\lambda$ for $c\in\mathbb{C}^\times$. We consider the universal enveloping algebra $\mathcal{U}(\mathfrak{h})$ as an algebra over the central polynomial subalgebra $\mathcal{U}(\mathfrak{t})=\mathcal{S}(\mathfrak{t})$. Let $\lambda \in \mathfrak{t}^*$ and let $I(\lambda)$ be the kernel of the algebra homomorphism $\mathrm{ev}_{\lambda} : \mathcal{S}(\mathfrak{t})\rightarrow \mathbb{C}$, induced by evaluation at $\lambda$. We consider the Clifford superalgebra $$\begin{aligned} \mathcal{C}\ell(\lambda) \coloneq \mathcal{C}\ell(\mathfrak{h}_{\overline{1}},B_{\lambda}) = \mathcal{U}(\mathfrak{h})/I(\lambda)\mathcal{U}(\mathfrak{h}). \end{aligned}$$ Then we have a non-canonical isomorphism of superalgebras (see ([\[eq_iso_cliff\]](#eq_iso_cliff){reference-type="ref" reference="eq_iso_cliff"})) $$\begin{aligned} \mathcal{C}\ell(\lambda) \simeq \mathcal{C}\ell(\mathfrak{h}_{\overline{1}}/\ker B_\lambda,B_{\lambda}) \otimes \mathcal{S} (\ker B_\lambda), \end{aligned}$$ where $B_{\lambda}$, by abuse of notation, also denotes the induced form on $\mathfrak{h}_{\overline{1}}/\ker B_\lambda$. We have a decomposition of $\mathcal{F}(\mathfrak{h})$ according to central character given by $$\mathcal{F}(\mathfrak{h})=\bigoplus\limits_{\lambda\in\mathfrak{t}^*}\mathcal{F}_{\lambda},$$ where $\mathcal{F}_{\lambda}$ consists of those modules on which $\mathfrak{t}$ acts by $\lambda$. Further, $\mathcal{F}_{\lambda}$ is equivalent to the category of $\mathcal{C}\ell(\lambda)$-modules. Thus all irreducible $\mathfrak{h}$-modules arise as $\mathcal{C}\ell(\lambda)$-modules, for some $\lambda \in \mathfrak{t}^*$. We denote by $C_\lambda$ a choice of unique (up to parity) irreducible $\mathfrak{h}$-module on which $\mathfrak{t}$ acts via $\lambda$, where we assume that $C_0$ is the trivial module $\mathbb{C}$. Recall from Section [2.1](#section_cliff_algs){reference-type="ref" reference="section_cliff_algs"} that if $m=\operatorname{rk}\lambda$, then $\dim (C_{\lambda})_{\overline{0}}= \dim(C_{\lambda})_{\overline{1}}=2^{\lfloor \frac{m-1}{2} \rfloor}$. Note that the blocks of $\mathcal{F}(\mathfrak{h})$ are more finely parameterized in the following way. If $B_\lambda$ is degenerate, then $\mathcal{F}_{\lambda}$ is a block, and as stated above, this block is equivalent to the category of finite-dimensional modules over $\mathcal{C}\ell(\lambda)$. If $B_{\lambda}$ is non-degenerate, then $\mathcal{F}_{\lambda}$ is semisimple; if $\operatorname{rk}\lambda$ is odd there is exactly one simple module in $\mathcal{F}_{\lambda}$, and if $\operatorname{rk}\lambda$ is even there are two. ## Dualities on $\mathcal{F}(\mathfrak{h})$ {#section_dualities} The category $\mathcal{F}(\mathfrak{h})$ admits the usual duality $(-)^*$ which is induced by the anti-automorphism of $\mathcal{U}(\mathfrak{h})$ defined by $-\mathrm{id}_{\mathfrak{h}}$. We have another duality on $\mathcal{F}(\mathfrak{h})$ which we denote by $(-)^\#$. It is induced by the anti-automorphism of $\mathcal{U}(\mathfrak{h})$ defined by $\mathrm{id}_{\mathfrak{h}_{\overline{0}}}\oplus(\sqrt{-1}\cdot\mathrm{id}_{\mathfrak{h}_{\overline{1}}})$. We have the formula $$C_{\lambda}^\#\cong \begin{cases} C_{\lambda} & \text{ if }\operatorname{rk}\lambda \text{ is odd or }\operatorname{rk}\lambda\equiv0\ \pmod4,\\ \Pi C_{\lambda} & \text{ if }\operatorname{rk}\lambda\equiv 2\ \pmod4. \end{cases}$$ The quasi-toral Lie superalgebra $\mathfrak{h}$ admits an automorphism $\omega_{\mathfrak{h}} = (-\mathrm{id}_{\mathfrak{h}_{\overline{0}}})\oplus(\sqrt{-1}\cdot\mathrm{id}_{\mathfrak{h}_{\overline{1}}})$. Given an irreducible $\mathfrak{h}$-module $C_{\lambda}$, we define the twisted module $$C_{\lambda}^{\vee} \coloneq C_{\lambda}^{\omega_{\mathfrak{h}}^{-1}}$$ (note the inverse - it will simplify notation later). On $\mathcal{F}(\mathfrak{h})$, we have a natural isomorphism of functors $(-)^\vee \cong ((-)^*)^\#$. From Theorem 3.6.4 of [@GSS] we obtain the following isomorphisms of $\mathfrak{h}$-modules which we use later: if $\operatorname{rk}\lambda$ is odd, then $$\begin{aligned} \label{equation-2Z+1} C_{\lambda}\otimes C_{\lambda}^\vee \simeq \mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_{\lambda})\oplus \Pi \mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_{\lambda}), \end{aligned}$$ where $\mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_{\lambda})$ is the Grassmann algebra on $\mathfrak{h}_{\overline{1}}/\ker B_{\lambda}$, and has the natural action by $\mathcal{C}\ell(0)\cong\mathcal{S}(\mathfrak{h}_{\overline{1}})$ from left multiplication in the quotient. In particular $\mathfrak{t}=\mathfrak{h}_{\overline{0}}$ will act trivially. Note that $\mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_{\lambda})$ has a simple socle and top, and thus in particular is indecomposable. If $\operatorname{rk}\lambda \in 4\mathbb{Z}$, then we have $$\begin{aligned} \label{equation-4Z} C_{\lambda}\otimes C_{\lambda}^\vee \simeq \mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_{\lambda}), \end{aligned}$$ while if $\operatorname{rk}\lambda \in 4\mathbb{Z}+2$, then $$\begin{aligned} \label{equation-4Z+2} C_{\lambda}\otimes C_{\lambda}^\vee \simeq \Pi\mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_{\lambda}). \end{aligned}$$ We now obtain the following corollary: [\[corollary-tensor-of-irreducibles\]]{#corollary-tensor-of-irreducibles label="corollary-tensor-of-irreducibles"} Let $\lambda \in \mathfrak{t}^*$ and let $\phi : C_{\lambda}\otimes C_{\lambda}^\vee \rightarrow \mathfrak{h}$ be an $\mathfrak{h}$-module homomorphism. 1. If $\operatorname{rk}\lambda \in 4\mathbb{Z}$, the $\operatorname{Im}\phi$ is purely even and at most one-dimensional; and 2. if $\operatorname{rk}\lambda \in 4\mathbb{Z}+2$, then the odd part of the image of $\phi$ is at most one-dimensional, and it generates $\operatorname{Im}\phi$ as an $\mathfrak{h}$-module. *Proof.* This follows immediately from $[\mathfrak{h}_{\overline{0}},\mathfrak{h}]=0$ and the structure of $C_{\lambda}\otimes C_{\lambda}^\vee$ given in [\[equation-4Z\]](#equation-4Z){reference-type="eqref" reference="equation-4Z"} and [\[equation-4Z+2\]](#equation-4Z+2){reference-type="eqref" reference="equation-4Z+2"}. ◻ ## Nondegenerate morphisms {#section_nondegenerate} Let $\mathfrak{h}$ be a quasitoral Lie superalgebra, and let $U,V,$ and $W$ be $\mathfrak{h}$-modules. We say a morphism of $\mathfrak{h}$-modules $\phi:U\otimes V\to W$ is nondegenerate if for any submodules $U'\subseteq U$ and $V'\subseteq V$ we have both $\phi(U'\otimes V)\neq0$ and $\phi(U\otimes V')\neq0$. Observe that if $U$ and $V$ are simple $\mathfrak{h}$-modules, then a map $\phi:U\otimes V\to W$ is nondegenerate if and only if $\phi\neq0$. ## Realization of $C_{\lambda}$ {#subsection-realization-irreducible} We consider the polynomial superalgebra $\mathbb{C}[\xi_1,...,\xi_m]$, where $\xi_1,\dots,\xi_m$ are odd, supercommuting variables, i.e.  $\xi_i\xi_j = -\xi_j\xi_i$ for all $i,j$. In particular $\xi_i^2 = 0$. For a set $I = \{i_1,...,i_k\} \subseteq \{1,...,m\}$, we denote $\xi_I \coloneq \xi_{i_1}\cdots \xi_{i_k}$ for $i_1<...<i_k$ (in particular $\xi_\emptyset = 1$). We now explain how to concretely realize the irreducible $\mathfrak{h}$-modules $C_{\lambda}$ as polynomial superalgebras, depending on the rank of $\lambda$. If $\operatorname{rk}\lambda = 2m$ for some $m \in \mathbb{Z}_{\geq0}$, then there exist linearly independent vectors $H_1,...,H_m,\bar{H}_1,...,\bar{H}_m$ of $\mathfrak{h}_{\overline{1}}$ which satisfy $$\begin{aligned} \begin{array}{ccc} B_\lambda(H_i,\bar{H_j}) = \delta_{ij}, & B_\lambda(H_i,H_j) = 0, & B_\lambda(\bar{H}_i,\bar{H}_j)=0, \end{array} \end{aligned}$$ for $1\leq i,j \leq m$. We can identify $C_\lambda$ (up to parity) with $\mathbb{C}[\xi_1,...,\xi_m]$, subject to the following action of $\mathfrak{h}_{\overline{1}}$: $H_i$ acts by multiplication by $\xi_i$ and $\bar{H}_i$ acts by the odd derivation $\partial_{\xi_i}$. If, on the other hand, $\operatorname{rk}\lambda = 2m+1$ for some $m \in \mathbb{Z}_{\geq 0}$, then there exist linearly independent vectors $H_1,...,H_m,\bar{H}_1,...,\bar{H}_m,\tilde{H}$ of $\mathfrak{h}_{\overline{1}}$ which satisfy $$\begin{aligned} \begin{array}{ccc} B_\lambda(H_i,\bar{H_j}) = \delta_{ij}, & B_\lambda(H_i,H_j) = 0, & B_\lambda(\bar{H}_i,\bar{H}_j)=0, \\ B_\lambda(\tilde{H},\tilde{H}) = 2, & B_\lambda(H_i,\tilde{H}) = 0, & B_\lambda(\bar{H}_i,\tilde{H}) = 0, \end{array} \end{aligned}$$ for $1\leq i,j \leq m$. The $\mathfrak{h}$-module $C_\lambda$ can be realized as $\mathbb{C}[\xi_1,...,\xi_{m+1}]$ with the following action of $\mathfrak{h}_{\overline{1}}$: $H_i$ acts by multiplication by $\xi_i$, $\bar{H}_i$ acts by the odd derivation $\partial_{\xi_i}$, and $\tilde{H}$ acts by $\xi_{m+1}+\partial_{\xi_{m+1}}$. # General Construction of $\mathfrak{g}(\mathcal{A})$ In this section we give a construction, in the spirit of the Kac-Moody approach, for Lie superalgebras admitting a quasi-toral Cartan subalgebra. It is a generalization of the constructions given in [@K] and [@K3] for Lie algebras, and [@K2] for Lie superalgebras. ## Cartan datum and $\tilde{\mathfrak{g}}(\mathcal{A})$ {#section_cartan_datum} [\[definition-datum\]]{#definition-datum label="definition-datum"} A Cartan datum $\mathcal{A}$ consists of the following information: 1. A finite-dimensional quasi-toral Lie superalgebra $\mathfrak{h}$. We write $\mathfrak{t} \coloneq \mathfrak{h}_{\overline{0}}$ and $[-,-]_{\mathfrak{h}}$ for the Lie bracket on $\mathfrak{h}$; 2. A linearly independent subset $\Pi = \{\alpha_1,...,\alpha_n\}\subseteq\mathfrak{t}^*$; 3. For each $\alpha \in \pm \Pi$, an $\mathfrak{h}$-module $\mathfrak{g}_{\alpha}$ in $\mathcal{F}_{\alpha}$; i.e., as a $\mathfrak{t}$-module, $\mathfrak{g}_{\alpha}$ is a weight space of weight $\alpha$. We write $m_{\alpha} : \mathfrak{h} \times \mathfrak{g}_{\alpha} \rightarrow \mathfrak{g}_{\alpha}$ for the action of $\mathfrak{h}$ on $\mathfrak{g}_{\alpha}$; 4. [\[definition-datum-step-4\]]{#definition-datum-step-4 label="definition-datum-step-4"} For each $\alpha \in \Pi$, a nondegenerate morphism (see Section [2.4](#section_nondegenerate){reference-type="ref" reference="section_nondegenerate"}) of $\mathfrak{h}$-modules $[-,-]_{\alpha} : \mathfrak{g}_{\alpha}\otimes \mathfrak{g}_{-\alpha} \rightarrow \mathfrak{h}$. [\[remark on vanishing\]]{#remark on vanishing label="remark on vanishing"} Observe that for any $\alpha\in\Pi$, the morphism $[-,-]_{\alpha}$ must vanish on the submodule generated by odd elements in the radical (as an $\mathfrak{h}$-module) of $\mathfrak{g}_{\alpha}\otimes\mathfrak{g}_{-\alpha}$. To a Cartan datum $\mathcal{A}$ we associate a Lie superalgebra $\tilde{\mathfrak{g}}(\mathcal{A})$, which is generated by the super vector space $$\begin{aligned} \label{generating-set} \mathcal{C} = \bigoplus_{i=1}^n \mathfrak{g}_{-\alpha_i} \oplus \mathfrak{h} \oplus \bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i}, \end{aligned}$$ subject to the following relations: 1. For any $h_1,h_2 \in \mathfrak{h}$ we have $[h_1,h_2] = [h_1,h_2]_{\mathfrak{h}}$. 2. For any $h \in \mathfrak{h}$, $\alpha \in \pm\Pi$ and $x \in \mathfrak{g}_{\alpha}$, we have $[h,x] = m_{\alpha}(h,x)$. 3. For any $\alpha,\beta \in \Pi$ with $x \in \mathfrak{g}_{\alpha}$ and $y \in \mathfrak{g}_{-\beta}$, we have $[x,y] = [x,y]_{\alpha}$ if $\alpha = \beta$ and $[x,y] = 0$ otherwise. We denote by $\tilde{\mathfrak{n}}^+$ and $\tilde{\mathfrak{n}}^-$ the subalgebras of $\tilde{\mathfrak{g}}(\mathcal{A})$ generated by $\bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i}$ and $\bigoplus_{i=1}^n \mathfrak{g}_{-\alpha_i}$, respectively. We set $Q_+ \coloneq \mathbb{Z}_{\geq 0} \Pi$. We obtain the following theorem, whose statement and proof are direct generalizations of Theorem 1.2 in [@K]. [\[theorem-half-baked-structure\]]{#theorem-half-baked-structure label="theorem-half-baked-structure"} Let $\tilde{\mathfrak{g}} \coloneq \tilde{\mathfrak{g}}(\mathcal{A})$ and $\tilde{\mathfrak{n}}^\pm$ be as above. Then 1. [\[bullet1-half-baked\]]{#bullet1-half-baked label="bullet1-half-baked"} The subalgebras $\tilde{\mathfrak{n}}^+$ and $\tilde{\mathfrak{n}}^-$ are freely generated as Lie superalgebras by $\bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i}$ and $\bigoplus_{i=1}^n \mathfrak{g}_{-\alpha_i}$, respectively; 2. [\[bullet2-half-baked\]]{#bullet2-half-baked label="bullet2-half-baked"} $\tilde{\mathfrak{g}} = \tilde{\mathfrak{n}}^- \oplus \mathfrak{h} \oplus \tilde{\mathfrak{n}}^+$ as super vector spaces; 3. [\[bullet3-half-baked\]]{#bullet3-half-baked label="bullet3-half-baked"} As a $\mathfrak{t}$-module, $\tilde{\mathfrak{g}}$ admits a root space decomposition: $$\begin{aligned} \tilde{\mathfrak{g}} = \bigoplus_{\substack{\alpha \in Q_+ \\ \alpha \neq 0}} \tilde{\mathfrak{g}}_{-\alpha} \oplus \mathfrak{h} \oplus \bigoplus_{\substack{\alpha \in Q_+ \\ \alpha \neq 0}} \tilde{\mathfrak{g}}_{\alpha}. \end{aligned}$$ In particular, $\mathfrak{h}$ is the centralizer of $\mathfrak{t}$ in $\tilde{\mathfrak{g}}$, and is self-normalizing; 4. [\[bullet4-half-baked\]]{#bullet4-half-baked label="bullet4-half-baked"} $\tilde{\mathfrak{g}}$ contains a unique maximal ideal $\mathfrak{r}$ that intersect $\mathfrak{h}$ trivially; it satisfies $\mathfrak{r} = (\mathfrak{r}\cap \tilde{\mathfrak{n}}^+)\oplus (\mathfrak{r}\cap \tilde{\mathfrak{n}}^-)$. *Proof.* Let $W$ be any representation of $\mathfrak{h}$, and consider the super vector space $V = \mathcal{T}(\oplus_{i=1}^n \mathfrak{g}_{\alpha_i})\otimes W$. The space $V$ is naturally graded, with $V_s = (\oplus_{i=1}^n \mathfrak{g}_{\alpha_i})^{\otimes s}\otimes W$. We define a representation $\pi_W : \tilde{\mathfrak{g}} \rightarrow \operatorname{End}(V)$ by the following action of the generators of $\tilde{\mathfrak{g}}$. Let $e \in \oplus_{i=1}^n \mathfrak{g}_{\alpha_i}$, $f \in \oplus_{i=1}^n \mathfrak{g}_{-\alpha_i}$, and $h\in \mathfrak{h}$ be homogeneous elements. We let $f$ act on $V_0$ trivially and $h$ act on $V_0=W$ by the existing action of $\mathfrak{h}$ on $W$. For $a\in V$, we define inductively: $$\begin{aligned} & e(a) = e\otimes a, \\ & h(e\otimes a) = [h,e]\otimes a + (-1)^{\bar{h}\bar{e}} e\otimes h(a) \\ & f(e\otimes a) = [f,e]\otimes a + (-1)^{\bar{f}\bar{e}} e\otimes f(a). \end{aligned}$$ To show this defines an action of $\tilde{\mathfrak{g}}$ on $V$, we need to check it respects the defining relations imposed on $\tilde{\mathfrak{g}}$, which is an immediate yet technical generalization of the proof of Theorem 1.2 in [@K]. We demonstrate the general argument of the proof by showing, inductively, that $[h,f](a) = (hf-(-1)^{\bar{h}\bar{f}}fh)(a)$ for all $a \in V$. Assuming this equality holds for $a \in V_s$, then $$\begin{aligned} & (hf-(-1)^{\bar{h}\bar{f}}fh)(e\otimes a) \\ & = h[f,e](a) + (-1)^{\bar{f}\bar{e}}h(e\otimes f(a)) -(-1)^{\bar{h}\bar{f}}f([h,e]\otimes a) - (-1)^{\bar{h}(\bar{f}+\bar{e})}f(e\otimes h(a)) \\ & = h[f,e](a) + (-1)^{\bar{f}\bar{e}}[h,e]\otimes f(a) + (-1)^{(\bar{f}+\bar{h})\bar{e}}e\otimes hf(a) - (-1)^{\bar{h}\bar{f}} [f,[h,e]](a) \\ & - (-1)^{\bar{e}\bar{f}}[h,e]\otimes f(a) - (-1)^{\bar{h}(\bar{f}+\bar{e})}[f,e]h(a) - (-1)^{\bar{e}(\bar{f}+\bar{h})+\bar{h}\bar{f}}e\otimes fh(a) \\ & = [h,[f,e]](a) - (-1)^{\bar{h}\bar{f}}[f,[h,e]](a) + (-1)^{\bar{e}(\bar{f}+\bar{h})}e\otimes(hf-(-1)^{\bar{h}\bar{f}}fh)(a) \\ & = [[h,f],e](a) + (-1)^{\bar{e}(\bar{f}+\bar{h})}e\otimes[h,f](a) = [h,f](e\otimes a). \end{aligned}$$ The equality $[h,f](a) = (hf-(-1)^{\bar{h}\bar{f}}fh)(a)$ for $a \in V$ follows by induction. As the representation $\pi_W$ is defined for every $\mathfrak{h}$-module $W$, we obtain that $\mathfrak{h}$ injects into $\tilde{\mathfrak{g}}$. Moreover, the action of $\bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i}$ via $\pi_W$ shows that $\bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i}$ injects into $\tilde{\mathfrak{g}}$. Let $W_0$ denote the trivial $\mathfrak{h}$-module. The map $\phi:\tilde{\mathfrak{n}}^+ \rightarrow \mathcal{T}(\bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i})$ given by $n \mapsto \pi_{W_0}(n)(1)$ establishes $\mathcal{T}(\bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i})$ as an enveloping algebra of $\tilde{\mathfrak{n}}^+$, and is easily verified to be its universal enveloping algebra. In particular, $\tilde{\mathfrak{n}}^+$ is freely generated by $\bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i}$. The result for $\tilde{\mathfrak{n}}^-$ is analogous. This proves [\[bullet1-half-baked\]](#bullet1-half-baked){reference-type="ref" reference="bullet1-half-baked"}. We certainly have $\tilde{\mathfrak{g}} = \tilde{\mathfrak{n}}^- + \mathfrak{h} + \tilde{\mathfrak{n}}^+$. Suppose $n^- + h + n^+ = 0$ for some $n^\pm \in \tilde{\mathfrak{n}}^{\pm}$ and $h\in \mathfrak{h}$. Then $0 = \pi_W(n^-+h+n^+)(a)$ for $a \in V_0 = W$. For $W = W_0$ we obtain $\phi(n^+) = \pi_{W_0}(n^-+h+n^+)(1) = 0$, so $n^+ = 0$. Now $0 = \pi_W(n^-+h)(a) = \pi_W(h)(a)$ for all $a \in V_0 = W$, which implies $h = 0$. We conclude that $n^- = 0$, which establishes [\[bullet2-half-baked\]](#bullet2-half-baked){reference-type="ref" reference="bullet2-half-baked"}. Part [\[bullet3-half-baked\]](#bullet3-half-baked){reference-type="ref" reference="bullet3-half-baked"} is an immediate consequence of [\[bullet2-half-baked\]](#bullet2-half-baked){reference-type="ref" reference="bullet2-half-baked"}. Finally, a standard linear algebraic argument gives that any ideal $I$ of $\tilde{\mathfrak{g}}$ decomposes as a $\mathfrak{t}$-module in the following way: $$\begin{aligned} \label{equation-decomposition-of-ideals} I = \bigoplus_{\substack{\alpha \in Q_+ \\ \alpha \neq 0}}(I\cap\tilde{\mathfrak{g}}_{-\alpha}) \oplus (I\cap \mathfrak{h}) \oplus \bigoplus_{\substack{\alpha \in Q_+ \\ \alpha \neq 0}}(I\cap\tilde{\mathfrak{g}}_{\alpha}) = (I\cap \tilde{\mathfrak{n}}^-) \oplus (I\cap \mathfrak{h}) \oplus (I\cap \tilde{\mathfrak{n}}^+). \end{aligned}$$ Therefore, if $I$ intersects $\mathfrak{h}$ trivially then $I = (I\cap \tilde{\mathfrak{n}}^-) \oplus (I\cap \tilde{\mathfrak{n}}^+)$. Summing over all such ideals of $\tilde{\mathfrak{g}}$, we get a unique maximal $\mathfrak{r}$ intersecting $\mathfrak{h}$ trivially, which satisfies $\mathfrak{r} = (\mathfrak{r}\cap\tilde{\mathfrak{n}}^-)\oplus(\mathfrak{r}\cap\tilde{\mathfrak{n}}^+)$, according to [\[equation-decomposition-of-ideals\]](#equation-decomposition-of-ideals){reference-type="eqref" reference="equation-decomposition-of-ideals"}. This proves part [\[bullet4-half-baked\]](#bullet4-half-baked){reference-type="ref" reference="bullet4-half-baked"}. ◻ By the same argument, Theorem [\[theorem-half-baked-structure\]](#theorem-half-baked-structure){reference-type="ref" reference="theorem-half-baked-structure"} also holds in the case that $\mathfrak{g}_{\pm \alpha_i}$ are generalized weight spaces of weight $\pm \alpha_i$ as $\mathfrak{t}$-modules. The following lemma is a direct generalization of Lemma 1.5 in [@K], with the same proof. [\[lemma-maximal-ideal\]]{#lemma-maximal-ideal label="lemma-maximal-ideal"} If $x \in \tilde{\mathfrak{n}}^+$ is such that $[\mathfrak{g}_{-\alpha},x] \in \mathfrak{r}$ for all $\alpha \in \Pi$, then $x \in \mathfrak{r}$. Analogously, if $y \in \tilde{\mathfrak{n}}^-$ satisfies $[\mathfrak{g}_{\alpha},y] \in \mathfrak{r}$ for all $\alpha \in \Pi$, then $y \in \mathfrak{r}$. ## Chevalley automorphism of $\tilde{\mathfrak{g}}(\mathcal{A})$ {#subsection-Chevalley} Recall the automorphism $\omega_{\mathfrak{h}}$ of $\mathfrak{h}$ defined in Section [2.2](#section-irreducible-h-modules){reference-type="ref" reference="section-irreducible-h-modules"}. Let $\mathcal{A}$ be a Cartan datum; a Chevalley automorphism of $\tilde{\mathfrak{g}}(\mathcal{A})$ is an extension $\tilde{\omega}$ of $\omega_{\mathfrak{h}}$ to an automorphism of all of $\tilde{\mathfrak{g}}(\mathcal{A})$. In contrast with the usual Kac-Moody construction, the Lie superalgebra $\tilde{\mathfrak{g}}(\mathcal{A})$ is not guaranteed to admit a Chevalley automorphism; in particular, a necessary condition is that $\mathfrak{g}_{-\alpha}\cong\mathfrak{g}_{\alpha}^\vee$ for all roots $\alpha$. We now describe a condition which ensures the existence of such an automorphism. Suppose $\mathcal{A}$ is a Cartan datum for which we have identifications $\mathfrak{g}_{-\alpha}=\mathfrak{g}_{\alpha}^\vee$ for all $\alpha \in \Pi$, where $\mathfrak{g}_{\alpha}^\vee$ is the twist by the automorphism $\omega_{\mathfrak{h}}^{-1}$ as introduced in Section [2.2](#section-irreducible-h-modules){reference-type="ref" reference="section-irreducible-h-modules"}. For each $\alpha\in\Pi$, let $\omega_{\alpha}:\mathfrak{g}_{\alpha}\rightarrow \mathfrak{g}_{\alpha}^\vee$ be the identity map of the underlying super vector spaces. We will also write $x^\vee \coloneq \omega_{\alpha}(x)$ for $x \in \mathfrak{g}_{\alpha}$. We set $\omega_{-\alpha} \coloneq \delta_{\mathfrak{g}_{\alpha}}\circ\omega_{\alpha}^{-1}:\mathfrak{g}_{\alpha}^\vee \rightarrow \mathfrak{g}_{\alpha}$, where $\delta_{\mathfrak{g}_{\alpha}}(v)=(-1)^{\overline{v}}v$ is the grading operator on $\mathfrak{g}_{\alpha}$. It follows immediately that for $\alpha\in \Pi$, $x \in \mathfrak{g}_{\alpha}$, $y \in \mathfrak{g}_{-\alpha}$ and $h \in \mathfrak{h}$ we have $$\begin{aligned} \begin{array}{cc} \omega_{\alpha}([h,x]) = [\omega_{\mathfrak{h}}(h),\omega_{\alpha}(x)], & \omega_{-\alpha}([h,y]) = [\omega_{\mathfrak{h}}(h),\omega_{-\alpha}(y)]. \end{array} \end{aligned}$$ [\[theorem-half-baked-Chevalley\]]{#theorem-half-baked-Chevalley label="theorem-half-baked-Chevalley"} Let $\mathcal{A}$ be a Cartan datum satisfying $\mathfrak{g}_{-\alpha}=\mathfrak{g}_{\alpha}^\vee$ for all $\alpha \in \Pi$. Assume $\omega_{\mathfrak{h}}([x,y]) = [\omega_{\alpha}(x),\omega_{-\alpha}(y)]$ for all $x\in \mathfrak{g}_{\alpha}$ and $y \in \mathfrak{g}_{-\alpha}$, for all $\alpha \in \Pi$. Then there exists a Chevalley automorphism $\tilde{\omega}$ of $\tilde{\mathfrak{g}}(\mathcal{A})$ of order $4$, which preserves $\mathfrak{r}$ and satisfies $\tilde{\omega}|_{\mathfrak{h}} = \omega_{\mathfrak{h}}$ and $\tilde{\omega}|_{\mathfrak{g}_{\alpha}} = \omega_{\alpha}$ for $\alpha \in \pm \Pi$. *Proof.* Let $\omega_{\mathcal{C}} \coloneq \bigoplus_{i=1}^n \omega_{-\alpha_i} \oplus \omega_{\mathfrak{h}} \oplus \bigoplus_{i=1}^n \omega_{\alpha_i}$ be the endomorphism of the super vector space $\mathcal{C}$ defined in [\[generating-set\]](#generating-set){reference-type="eqref" reference="generating-set"}. It is immediate that $\omega_{\mathcal{C}}$ is an automorphism and that $\omega_{\mathcal{C}}^2 = \delta_{\mathcal{C}}$, where $\delta_{\mathcal{C}}$ is the grading operator of $\mathcal{C}$. We wish to extend $\omega_{\mathcal{C}}$ to a Lie superalgebra endomorphism $\tilde{\omega}$ of $\tilde{\mathfrak{g}}(\mathcal{A})$. By assumption we know that $\omega_{\mathcal{C}}([x,y]) = [\omega_{\mathcal{C}}(x),\omega_{\mathcal{C}}(y)]$ whenever $x,y,[x,y] \in \mathcal{C}$. Moreover, because $\omega_\mathcal{C}$ is an even linear map, we have $$\begin{aligned} + (-1)^{\bar{x}\bar{y}}[\omega_{\mathcal{C}}(y),\omega_{\mathcal{C}}(x)] = 0 \end{aligned}$$ and $$\begin{aligned} ,\omega_{\mathcal{C}}(z)] = [\omega_{\mathcal{C}}(x),[\omega_{\mathcal{C}}(y),\omega_{\mathcal{C}}(z)]] + (-1)^{\bar{x}\bar{y}} [\omega_{\mathcal{C}}(y),[\omega_{\mathcal{C}}(x),\omega_{\mathcal{C}}(z)]] \end{aligned}$$ for homogeneous $x,y,z \in \mathcal{C}$. It follows that we may extend $\tilde{\omega}$ to $\tilde{\mathfrak{g}}(\mathcal{A})$ by requiring that $\tilde{\omega}([x,y]) = [\tilde{\omega}(x),\tilde{\omega}(y)]$ for $x,y \in \tilde{\mathfrak{g}}(\mathcal{A})$ and $\tilde{\omega}|_{\mathcal{C}} \coloneq \omega_{\mathcal{C}}$. From the definition of $\omega_{\mathcal{C}}$ follows that $\tilde{\omega}^2 = \delta_{\tilde{\mathfrak{g}}(\mathcal{A})}$, where $\delta_{\tilde{\mathfrak{g}}(\mathcal{A})}$ is the grading operator of $\tilde{\mathfrak{g}}(\mathcal{A})$. In particular, we obtain that $\tilde{\omega}$ is an automorphism of order $4$. It is immediate that $\tilde{\omega}(\tilde{\mathfrak{g}}_{\alpha}) = \tilde{\mathfrak{g}}_{-\alpha}$ for any $\alpha \in \pm Q_+$. This implies $\tilde{\omega}(\mathfrak{r})\cap \mathfrak{h} = 0$, so $\tilde{\omega}(\mathfrak{r}) \subseteq \mathfrak{r}$. A similar argument gives $\tilde{\omega}^{-1}(\mathfrak{r}) \subseteq \mathfrak{r}$. Altogether, we obtain $\tilde{\omega}(\mathfrak{r}) = \mathfrak{r}$. ◻ [\[proposition-rank-2-chevalley\]]{#proposition-rank-2-chevalley label="proposition-rank-2-chevalley"} Let $\mathcal{A}$ be a Cartan datum such that $\mathfrak{g}_{-\alpha} = \mathfrak{g}_{\alpha}^\vee$ for all $\alpha \in \Pi$. Assume, moreover, that $\mathfrak{g}_{\alpha}$ is irreducible and that $\operatorname{rk}\alpha \leq 2$ for all $\alpha \in \Pi$. Then the conditions of Theorem [\[theorem-half-baked-Chevalley\]](#theorem-half-baked-Chevalley){reference-type="ref" reference="theorem-half-baked-Chevalley"} hold, and thus $\tilde{\mathfrak{g}}(\mathcal{A})$ admits a Chevalley automorphism. *Proof.* Fix a simple root $\alpha \in \Pi$. If $\operatorname{rk}\alpha = 0$, then $\mathfrak{g}_{\alpha}$ is $1$-dimensional with basis $\{v\}$. Then $\{v^\vee\}$ is a basis for $\mathfrak{g}_{-\alpha}$ (where $v^\vee = \omega_{\alpha}(v)$) and $[v,v^\vee] \in \mathfrak{t}$. Therefore $$\begin{aligned} \omega_{\mathfrak{h}}([v,v^\vee]) = -[v,v^\vee] = - [(-1)^{\bar{v}}\omega_{-\alpha}(v^\vee),\omega_{\alpha}(v)] = [\omega_{\alpha}(v),\omega_{-\alpha}(v^\vee)]. \end{aligned}$$ If $\operatorname{rk}\alpha=1$ or $2$, then $\mathfrak{g}_{\alpha}$ is $(1|1)$-dimensional. Let $\{v,w\}$ be a homogeneous basis for $\mathfrak{g}_{\alpha}$. From the irreducibility of $\mathfrak{g}_{\alpha}$, there exists $H \in \mathfrak{h}_{\overline{1}}$ such that $H\cdot v = w$. As $[v,v^\vee] \in \mathfrak{t}$, we have $[\mathfrak{h},[v,v^\vee]] = 0$. So $$\begin{aligned} 0 = [H,[v,v^\vee]] = [H\cdot v, v^\vee] + (-1)^{\bar{v}} [v,H\cdot v^\vee] = [w,v^\vee] - \sqrt{-1}\cdot (-1)^{\bar{v}}[v,w^\vee]. \end{aligned}$$ As $[w,v^\vee] \in \mathfrak{h}_{\overline{1}}$, we must have $$\begin{aligned} \omega_{\mathfrak{h}}([w,v^\vee]) = \sqrt{-1}[w,v^\vee] = -(-1)^{\bar{v}}[v,w^\vee] = - [\omega_{-\alpha}(v^\vee),\omega_{\alpha}(w)] = [\omega_{\alpha}(w),\omega_{-\alpha}(v^\vee)]. \end{aligned}$$ A similar computation gives $\omega_{\mathfrak{h}}([v,w^\vee]) = [\omega_{\alpha}(v),\omega_{-\alpha}(w^\vee)]$. Finally, the computation in the $\operatorname{rk}\alpha = 0$ case gives $\omega_{\mathfrak{h}}([v,v^\vee]) = [\omega_{\alpha}(v),\omega_{-\alpha}(v^\vee)]$ and $\omega_{\mathfrak{h}}([w,w^\vee]) = [\omega_{\alpha}(w),\omega_{-\alpha}(w^\vee)]$. The claim follows by linearity. ◻ ## The Lie superalgebra $\mathfrak{g}(\mathcal{A})$ {#section-g(A)} We define $\mathfrak{g}(\mathcal{A}) \coloneq \tilde{\mathfrak{g}}(\mathcal{A})/\mathfrak{r}$, the quotient of $\tilde{\mathfrak{g}}(\mathcal{A})$ by the maximal ideal intersecting $\mathfrak{h}$ trivially, as in Theorem [\[theorem-half-baked-structure\]](#theorem-half-baked-structure){reference-type="ref" reference="theorem-half-baked-structure"}. We say $\mathfrak{g}(\mathcal{A})$ is the Lie superalgebra associated to the Cartan datum $\mathcal{A}$. We also write $\mathfrak{g}$ instead of $\mathfrak{g}(\mathcal{A})$ when clear from context. The natural map $$\mathcal{C} = \bigoplus_{i=1}^n \mathfrak{g}_{-\alpha_i} \oplus \mathfrak{h} \oplus \bigoplus_{i=1}^n \mathfrak{g}_{\alpha_i}\to\mathfrak{g}(\mathcal{A})$$ is an embedding of super vector spaces. Thus we may identify $\mathfrak{h}$ and $\mathfrak{g}_{\alpha}$ for $\alpha\in\pm\Pi$ with their images in $\mathfrak{g}(\mathcal{A})$. *Proof.* This immediately follows from the definition of $\mathfrak{r}$ and the assumption of nondegeneracy on our maps $\mathfrak{g}_{\alpha}\otimes\mathfrak{g}_{-\alpha}\to\mathfrak{h}$. ◻ Since $\mathfrak{t}$ acts on $\mathfrak{g}$ semisimply, we have a weight space decomposition $\mathfrak{g}=\bigoplus\limits_{\alpha\in\mathfrak{t}^*}\mathfrak{g}_{\alpha}$. We call $\alpha \in \mathfrak{t}^*\setminus\{0\}$ a root of $\mathfrak{g}$ if $\mathfrak{g}_{\alpha} \neq 0$. We denote by $\Delta$ the set of roots of $\mathfrak{g}$, and set $\Delta_+ \coloneq \Delta \cap Q_+$. If $\alpha = \sum_{i=1}^{s} k_i\alpha_i$ is a root for some $k_i \in \mathbb{Z}_{\geq 0}$, we say $\alpha$ is of height $\sum_{i=1}^sk_i$. We denote by $\mathfrak{n}^+$ and $\mathfrak{n}^-$ the images of $\tilde{\mathfrak{n}}^+$ and $\tilde{\mathfrak{n}}^-$ in $\mathfrak{g}$, respectively. Theorem [\[theorem-half-baked-structure\]](#theorem-half-baked-structure){reference-type="ref" reference="theorem-half-baked-structure"} implies the triangular decomposition $\mathfrak{g} = \mathfrak{n}^-\oplus \mathfrak{h} \oplus \mathfrak{n}^+$ as a super vector space and the decomposition $$\begin{aligned} \mathfrak{g} = \bigoplus_{\alpha\in\Delta_+} \mathfrak{g}_{-\alpha} \oplus \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta_+} \mathfrak{g}_{\alpha} \end{aligned}$$ as a $\mathfrak{t}$-module. We notice that $\mathfrak{g}_{\alpha}$ is an $\mathfrak{h}$-module for any $\alpha \in Q_+$, so the decomposition also holds as an $\mathfrak{h}$-module. Further, $\mathfrak{h}$ is the centralizer of $\mathfrak{t}$ in $\mathfrak{g}(\mathcal{A})$, and is self-normalizing. [\[definition_coroot\]]{#definition_coroot label="definition_coroot"} For a root $\alpha \in \Delta$, we call $\mathfrak{h}_{\alpha}:=[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]$ the *coroot-space* corresponding to $\alpha$; it is an ideal of $\mathfrak{h}$. An element of $\mathfrak{h}_{\alpha}$ shall be called an *$\alpha$-coroot*, or a coroot corresponding to $\alpha$. We say an $\alpha$-coroot $h$ is *pure* if $h = [x,y]$ for homogeneous non-zero $x \in \mathfrak{g}_{\alpha}$ and $y \in \mathfrak{g}_{-\alpha}$. [\[proposition-roots-connectivity\]]{#proposition-roots-connectivity label="proposition-roots-connectivity"} Let $\Pi_1,\Pi_2 \subseteq \Pi$ be disjoint subsets such that $$\begin{aligned} = [\mathfrak{h}_{\beta},\mathfrak{g}_{\alpha}] = 0 \end{aligned}$$ for any $\alpha \in \Pi_1$, $\beta \in \Pi_2$. Let $Q^+_s = \mathbb{Z}_{\geq 0}\Pi_s$ for $s=1,2$. If $\theta \in \Delta$ is a root of $\mathfrak{g}$ that satisfies $\theta \in Q^+_1 + Q^+_2$, then necessarily $\theta \in Q^+_1$ or $\theta \in Q^+_2$. *Proof.* Let us show that $[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}] = 0$ for $\alpha \in \Pi_1$, $\beta \in \Pi_2$. The Jacobi identity implies $$\begin{aligned} ] \subseteq [[\mathfrak{g}_{-\alpha},\mathfrak{g}_{\alpha}],\mathfrak{g}_{\beta}] + [\mathfrak{g}_{\alpha},[\mathfrak{g}_{-\alpha},\mathfrak{g}_{\beta}]]. \end{aligned}$$ But $[[\mathfrak{g}_{-\alpha},\mathfrak{g}_{\alpha}],\mathfrak{g}_{\beta}] = 0$ by assumption and $[\mathfrak{g}_{-\alpha},\mathfrak{g}_{\beta}]=0$ by the defining relations of $\mathfrak{g}$. So $[\mathfrak{g}_{-\alpha},[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]] = 0$. A similar argument shows $[\mathfrak{g}_{-\beta},[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]] = 0$. We certainly have $[\mathfrak{g}_{-\gamma},[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]] = 0$ for any $\gamma \in \Pi \setminus \{\alpha,\beta\}$. As $[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}] \subseteq \mathfrak{n}^+$ and $[\mathfrak{g}_{-\gamma},[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]]=0$ for any $\gamma \in \Pi$, Corollary [\[corollary-maximal-ideal\]](#corollary-maximal-ideal){reference-type="ref" reference="corollary-maximal-ideal"} implies $[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}] = 0$. Let $\mathfrak{g}^{(s)}$ be the subalgebra of $\mathfrak{g}$ generated by $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$ for $\alpha \in \Pi_{s}$. Then what we have shown so far implies $[\mathfrak{g}^{(1)},\mathfrak{g}^{(2)}] = 0$. Because $\theta \in Q^+_1 + Q^+_2$ we must have that $\mathfrak{g}_{\theta}$ is contained in the algebra generated by $\mathfrak{g}^{(1)}$ and $\mathfrak{g}^{(2)}$. But then $\mathfrak{g}_{\theta}$ is contained in either $\mathfrak{g}^{(1)}$ or $\mathfrak{g}^{(2)}$, which means $\theta \in Q^+_1$ or $\theta \in Q^+_2$. ◻ We now state several obvious corollaries: [\[corollary-center\]]{#corollary-center label="corollary-center"} Let $\mathfrak{c}$ be the center of $\mathfrak{g}$. Then $\mathfrak{c} \subseteq \bigcap_{i=1}^n (\mathrm{Ann}_{\mathfrak{h}} \mathfrak{g}_{\alpha_i}\cap \mathrm{Ann}_{\mathfrak{h}} \mathfrak{g}_{-\alpha_i})$ and $\mathfrak{c}_{\overline{0}} = \bigcap_{i=1}^n \ker \alpha_i \subseteq \mathfrak{t}$. Lemma [\[lemma-maximal-ideal\]](#lemma-maximal-ideal){reference-type="ref" reference="lemma-maximal-ideal"} immediately implies the following corollary: [\[corollary-maximal-ideal\]]{#corollary-maximal-ideal label="corollary-maximal-ideal"} If $x \in \mathfrak{n}^+$ satisfies $[\mathfrak{g}_{-\alpha},x] = 0$ for all $\alpha \in \Pi$, then $x = 0$. Similarly, if $y \in \mathfrak{n}^-$ is such that $[\mathfrak{g}_{\alpha},y] = 0$ for all $\alpha \in \Pi$, then $y=0$. Theorem [\[theorem-half-baked-Chevalley\]](#theorem-half-baked-Chevalley){reference-type="ref" reference="theorem-half-baked-Chevalley"} and Proposition [\[proposition-rank-2-chevalley\]](#proposition-rank-2-chevalley){reference-type="ref" reference="proposition-rank-2-chevalley"} give the following corollary: [\[corollary-Chevalley\]]{#corollary-Chevalley label="corollary-Chevalley"} Let $\mathcal{A}$ be a Cartan datum satisfying $\mathfrak{g}_{-\alpha} = \mathfrak{g}_{\alpha}^\vee$ for all $\alpha \in \Pi$. Assume $$\begin{aligned} \label{chevalley-equation} \omega_{\mathfrak{h}}([x,y]) = [\omega_{\alpha}(x),\omega_{-\alpha}(y)] \end{aligned}$$ for all $x \in \mathfrak{g}_{\alpha}$ and $y \in \mathfrak{g}_{-\alpha}$, for all $\alpha \in \Pi$ (see Section [3.2](#subsection-Chevalley){reference-type="ref" reference="subsection-Chevalley"} for notation). Then $\mathfrak{g}(\mathcal{A})$ admits a Chevalley automorphism $\omega$ with $\omega^2=\delta$, where $\delta(v)=(-1)^{\overline{v}}v$. In particular, if the $\mathfrak{h}$-modules $\mathfrak{g}_{\alpha}$ are irreducible and $\operatorname{rk}\alpha \leq 2$ for all $\alpha \in \Pi$, then [\[chevalley-equation\]](#chevalley-equation){reference-type="eqref" reference="chevalley-equation"} holds. As a generalization of Lemma 1.6 in [@K] we have: ## Cartan subdata and morphisms {#section_subdatum} Let $\mathfrak{h}'\subseteq\mathfrak{h}$ be any Lie subalgebra of $\mathfrak{h}$, and for each $\alpha\in\pm\Pi$ choose $\mathfrak{h}'$-submodules $\mathfrak{g}_{\alpha}'\subseteq\mathfrak{g}_{\alpha}$ such that $[\mathfrak{g}_{\alpha}',\mathfrak{g}_{-\alpha}']\subseteq\mathfrak{h}'$. Then we obtain what we call a *Cartan subdatum* $\mathcal{A}'$ of $\mathcal{A}$, which by definition consists of the above information, and constitutes a Cartan datum of its own apart from possibly failing the condition that the map $\mathfrak{g}_{\alpha}'\otimes\mathfrak{g}_{-\alpha}'\to\mathfrak{h}'$ is nondegenerate. Nevertheless, we may define $\tilde{\mathfrak{g}}(\mathcal{A}')$ in the natural way, and we will have an obvious map $\tilde{\mathfrak{g}}(\mathcal{A}')\to\mathfrak{g}(\mathcal{A})$. There are two distinguished types of Cartan subdata. Let $\Pi'\subseteq\Pi$ be a subset of simple roots. Then we obtain a natural Cartan subdatum $\mathcal{A}':=\mathcal{A}_{\Pi'}$ of $\mathcal{A}$ with $\mathfrak{h}'=\mathfrak{h}$, and $\mathfrak{g}_{\alpha}'=\mathfrak{g}_{\alpha}$ if $\pm\alpha\in\Pi'$, and $\mathfrak{g}_{\alpha}'=0$ otherwise. In this case $\mathcal{A}'$ will be a Cartan datum in its own right. We call this a *full* Cartan subdatum. The other important case is when $\mathfrak{h}'=\mathfrak{t}$, and we choose vectors $e_1,\dots,e_n,f_1,\dots,f_n$ with $e_i\in\mathfrak{g}_{\alpha_i}$, $f_i\in\mathfrak{g}_{-\alpha_i}$, $e_i,f_i$ of the same parity. We call a Cartan subdatum of this form *classical*. Setting $h_i:=[e_i,f_i]$, we obtain in this way a Cartan matrix $A'=(\alpha_j(h_i))$, and with it a natural map $\tilde{\mathfrak{g}}'(A')\to\mathfrak{g}(\mathcal{A})$, where $\tilde{\mathfrak{g}}'(A')$ is the derived subalgebra of $\tilde{\mathfrak{g}}(A')$. The following result will be of use later on. If there exists a classical Cartan subdatum of $\mathcal{A}$ with Cartan matrix $A'$ for which $\mathfrak{g}(A')$ is not of finite growth, then $\mathfrak{g}(\mathcal{A})$ is also not of finite growth. *Proof.* By the setup described above, we have a map $\tilde{\mathfrak{g}}(A')\to\mathfrak{g}(\mathcal{A})$. The image of this map admits as a subquotient the Lie superalgebra $\mathfrak{g}'(A')$ quotient by its center, where $\mathfrak{g}'(A')$ is the derived subalgebra of $\mathfrak{g}(A')$. Since this is of finite growth if and only if $\mathfrak{g}(A')$ is, we are done. ◻ # Clifford Kac-Moody algebras [\[definition-integrable\]]{#definition-integrable label="definition-integrable"} We say that a Cartan datum $\mathcal{A}$ (and by extension, the Lie superalgebra $\mathfrak{g}(\mathcal{A})$) is integrable if for any $\alpha,\beta \in \pm \Pi$ there exists $n \in \mathbb{N}$ such that $(\mathrm{ad}\ \mathfrak{g}_{\alpha})^n \mathfrak{g}_{\beta} = 0$. We call $\mathfrak{g}(\mathcal{A})$ Clifford Kac-Moody if it satisfies the following properties: 1. The $\mathfrak{h}$-module $\mathfrak{g}_{\alpha}$ is irreducible for any $\alpha \in \pm\Pi$; and 2. $\mathcal{A}$ is integrable. In this situation, we will also say that the Cartan datum $\mathcal{A}$ is Clifford Kac-Moody. [\[remark-almostKM-subdatum\]]{#remark-almostKM-subdatum label="remark-almostKM-subdatum"} For any full Cartan subdatum $\mathcal{A}'$ of $\mathcal{A}$ (see Section [3.4](#section_subdatum){reference-type="ref" reference="section_subdatum"}), $\mathcal{A}'$ is Clifford Kac-Moody whenever $\mathcal{A}$ is. For symmetrizable Kac-Moody superalgebras admitting odd isotropic simple roots, one is led to consider non-conjugate positive systems obtained via odd reflections. Thus a 'better' behaved notion of integrability is to require that a root vector acts ad-locally integrably whenever it is a simple root vector for some base obtained from some number of odd reflections from a fixed starting one. We ignore this subtlety here, since it is more important when discussing representation theory, which for now we do not discuss. For any $\alpha \in \Pi$, choose a non-zero pure even coroot $h_{\alpha}$ and set $A \coloneq (\beta(h_{\alpha}))_{\alpha,\beta\in\Pi}$. Then the integrability of $\mathfrak{g}(\mathcal{A})$ implies that if $A$ is elemental as in Sec. 2 of [@CS], then it satisfies the condition of Lem. 3.1 in [@CS]. ## Examples of Clifford Kac-Moody algebras Let $\mathfrak{g}=\mathfrak{q}(n)$ denote the Lie subalgebra of $\mathfrak{g}\mathfrak{l}(n|n)$ consisting of matrices of the form $$\begin{bmatrix} A & B\\ B &A \end{bmatrix},$$ where $A$ and $B$ are arbitrary. Set $\mathfrak{h}\subseteq\mathfrak{q}(n)$ to be the subalgebra of matrices of the form $$\begin{bmatrix} D & D'\\ D' & D\end{bmatrix},$$ where $D,D'$ are diagonal. Then $\mathfrak{h}$ is quasi-toral and self-normalizing in $\mathfrak{g}$. Let $\mathfrak{t}=\mathfrak{h}_{\overline{0}}$, the subalgebra of block diagonal matrices, and let $\epsilon_1,\dots,\epsilon_n\in\mathfrak{t}^*$ be the coordinate projections. Then if we set $\alpha_i:=\epsilon_i-\epsilon_{i+1}$, we obtain a Cartan datum from $\mathfrak{h}$ and $\mathfrak{g}_{\pm\alpha_1},\dots,\mathfrak{g}_{\pm\alpha_{n-1}}$, and all roots are of rank 2. The root spaces $\mathfrak{g}_{\alpha_i}$ are irreducible, and since integrability is easily checked, we see that $\mathfrak{g}$ is Clifford Kac-Moody. Write $e_i:=e_{ii}+e_{n+i,n+i}$ and $E_{i}:=e_{i,n+i}+e_{n+i,i}$. Then the pure coroots in $\mathfrak{h}_{\alpha_i}$ are given by $h_i:=e_i-e_{i+1}$, $c_i:=e_i+e_{i+1},$ and $H_i:=E_{i}-E_{i+1}$. It is interesting to note that the pure coroots are distinct for different simple roots. One should view $\mathfrak{q}(n)$ as the primary, motivating example of a Clifford Kac-Moody algebra, and of our construction at large. [\[example_sqn\]]{#example_sqn label="example_sqn"} Let $\mathfrak{g}=\mathfrak{s}\mathfrak{q}(n)=[\mathfrak{q}(n),\mathfrak{q}(n)]$. Then $\mathfrak{g}$ has Cartan subalgebra given by $\mathfrak{h}$ the set of matrices of the form $$\begin{bmatrix} D & D'\\ D' & D \end{bmatrix},$$ where $\operatorname{tr}(D')=0$. The rest of the Cartan datum remains unchanged from that of $\mathfrak{q}(n)$, and $\mathfrak{s}\mathfrak{q}(n)$ is Clifford Kac-Moody whenever $n\geq 3$ (notice that $\mathfrak{g}_{\alpha_1}$ is not irreducible in $\mathfrak{s}\mathfrak{q}(2)$). [\[example_psq(n)\]]{#example_psq(n) label="example_psq(n)"} Let $\mathfrak{g}=\mathfrak{p}\mathfrak{s}\mathfrak{q}(n):=\mathfrak{s}\mathfrak{q}(n)/\mathbb{C}I_{n|n}$. Similarly to the previous example, one can verify that $\mathfrak{g}$ is Clifford Kac-Moody if $n\geq3$. [\[example-takiff\]]{#example-takiff label="example-takiff"} An interesting class of examples of Clifford Kac-Moody algebras is obtained via extensions of Takiff superalgebras, which can be described as follows (see also Example 5.1 in [@S2]). Let $\mathfrak{s}$ be a symmetrizable (see Example [\[example_takiff_non_symm\]](#example_takiff_non_symm){reference-type="ref" reference="example_takiff_non_symm"} for when $\mathfrak{s}$ is not symmetrizable) integrable (as in Sec. 3 of [@CS]) Kac-Moody superalgebra with invariant form $(-,-)$. We define $$T\mathfrak{s} \coloneq \mathfrak{s}\otimes\mathbb{C}[\xi]/(\xi^2)\oplus \mathbb{C}\langle\partial_\xi,c,z\rangle,$$ where $\xi$ and $\partial_\xi$ are odd, $c$ and $z$ even and central, and we have the following bracket (note that there are signs missing in the formula for the bracket in [@S2]): $$\begin{aligned} &[x\otimes p+ a\partial_\xi , y\otimes q+ b\partial_\xi] = \\ & (-1)^{\overline{p}\overline{y}}[x,y]\otimes pq+(-1)^{\overline{y}}ay\otimes q'+(-1)^{\overline{p}}bx\otimes p'+(-1)^{\overline{y}}(x,y)Res(p'q)c + abz \end{aligned}$$ for $x,y \in \mathfrak{s}$, $a,b \in \mathbb{C}$, and $p,q\in\mathbb{C}[\xi]$. Here we write e.g. $p':=\partial_{\xi}p$, and $Res(a+b\xi)=b$. To check that $\mathfrak{g}:=T\mathfrak{s}$ is Clifford Kac-Moody, let $\overline{\mathfrak{t}}\subseteq\mathfrak{s}$ be a maximal torus of $\mathfrak{s}$, and let $\mathfrak{h}=\overline{\mathfrak{t}}\otimes\mathbb{C}[\xi]\oplus\mathbb{C}\langle\partial_{\xi},c,z\rangle$. Then $\mathfrak{h}$ is quasi-toral and self-normalizing in $T\mathfrak{s}$. Write $\mathfrak{t}:=\mathfrak{h}_{\overline{0}}$. If $\overline{\Pi}=\{\overline{\alpha_1},\dots,\overline{\alpha_n}\}\subseteq(\overline{\mathfrak{t}})^{*}$ denotes the simple roots of $\mathfrak{s}$, let $\alpha_i\in\mathfrak{t}^*$ denote the weight satisfying $\alpha_i(t)=\overline{\alpha_i}(t)$ for $t\in\overline{\mathfrak{t}}$, and $\alpha_i(c)=\alpha_i(z)=0$. Then $\Pi=\{\alpha_1,\dots,\alpha_n\}\subseteq\mathfrak{t}^*$ is linearly independent, and $\mathfrak{h}$ along with the root spaces $\mathfrak{g}_{\pm\alpha_i}$ give rise to a Cartan datum. The root spaces $\mathfrak{g}_{\alpha_i}$ are spanned by $e_{i}\otimes1,e_i\otimes\xi$ where $e_i$ is the Chevalley generator of $\mathfrak{s}$ for the root $\overline{\alpha_i}$, and it is easy to see this forms an irreducible $\mathfrak{h}$-module (with the help of $\partial_{\xi}$). Finally, $\mathfrak{g}$ will be integrable because the same is true of $\mathfrak{s}$. Observe that every simple root of $T\mathfrak{s}$ is of rank 2. The $\alpha$ coroot-space $\mathfrak{h}_{\alpha}$ for a simple root $\alpha$ is given by $h_{\overline{\alpha}}\otimes \mathbb{C}[\xi]\oplus\mathbb{C}\langle c\rangle$, where $h_{\overline{\alpha}}$ denotes the coroot in $\mathfrak{s}$ of the simple root $\overline{\alpha}$. Thus the pure coroots are $h_{\overline{\alpha}}\otimes 1$, $h_{\overline{\alpha}}\otimes\xi$, and $c$; in particular every coroot space shares a pure coroot, and this coroot is central. As a special case of the above construction, we have that $$T\mathfrak{s}\mathfrak{l}(2)/(c-z)\cong\mathfrak{q}(2).$$ We will see that in some ways the property of two coroot spaces sharing a pure, central coroot is characteristic of the superalgebras $T\mathfrak{s}$. This construction represents the only known 'general' construction of nontrivial Clifford Kac-Moody algebras, i.e. ones with simple roots of rank bigger than 0. [\[example_takiff_non_symm\]]{#example_takiff_non_symm label="example_takiff_non_symm"} In the above example, we can drop the condition that $\mathfrak{s}$ be symmetrizable at the cost of removing the central extension $c$. We will still obtain a Clifford Kac-Moody algebra in this way, and we will still call this superalgebra $T\mathfrak{s}$. [\[example_ts_pts\]]{#example_ts_pts label="example_ts_pts"} For examples of superalgebras that are not Clifford Kac-Moody, let $\mathfrak{s}$ be a symmetrizable Kac-Moody Lie superalgebra, and write $\mathfrak{t}\mathfrak{s}$ for the subalgebra of $T\mathfrak{s}$ spanned by $\mathfrak{s}\otimes\mathbb{C}[\xi]$ and $c$. It $\overline{\mathfrak{t}}$ is a maximal torus of $\mathfrak{s}$, then we set $\mathfrak{h}=\overline{\mathfrak{t}}\otimes\mathbb{C}[\xi]\oplus\mathbb{C}\langle c\rangle$ to obtain a self-normalizing quasi-toral subalgebra. However in this case the root spaces of simple roots, spanned by $e_i\otimes 1$ and $e_i\otimes\xi$ for a Chevalley generator $e_i$ of $\mathfrak{s}$, will not be irreducible $\mathfrak{h}$ modules because we no longer have the derivation $\partial_{\xi}$. Thus $\mathfrak{t}\mathfrak{s}$ is not Clifford Kac-Moody (however it does arise naturally from our more general construction, as $\mathfrak{t}\mathfrak{s}$ contains no nontrivial ideals that intersect $\mathfrak{h}$ trivially). Another example is obtained by considering $\mathfrak{p}\mathfrak{t}\mathfrak{s}:=\mathfrak{t}\mathfrak{s}/(c)\cong\mathfrak{s}\otimes\mathbb{C}[\xi]$; and indeed if $\mathfrak{s}$ is not symmetrizable then this is the natural superalgebra to consider. This case is again clearly not Clifford Kac-Moody, but once again arises naturally from our construction. Finally, we note that $$\mathfrak{t}\mathfrak{s}\mathfrak{l}(2)\cong\mathfrak{s}\mathfrak{q}(2) \ \text{ and } \ \mathfrak{pt}\mathfrak{s}\mathfrak{l}(2)\cong\mathfrak{p}\mathfrak{s}\mathfrak{q}(2).$$ # Cartan datum with one simple root {#section-one-root} In light of Section [3.4](#section_subdatum){reference-type="ref" reference="section_subdatum"} and Remark [\[remark-almostKM-subdatum\]](#remark-almostKM-subdatum){reference-type="ref" reference="remark-almostKM-subdatum"}, it is wise to begin our study with Clifford Kac-Moody algebras $\mathfrak{g}(\mathcal{A})$ having one simple root. Thus let $\mathcal{A}$ be a Cartan datum with $\Pi = \{\alpha\}$ and irreducible $\mathfrak{h}$-modules $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$. The integrability condition on $\mathfrak{g}(\mathcal{A})$ means that $(\mathrm{ad}\ \mathfrak{g}_{\alpha})^n \mathfrak{g}_{\alpha} = 0$ and $(\mathrm{ad}\ \mathfrak{g}_{-\alpha})^n \mathfrak{g}_{-\alpha} = 0$ for some $n \in \mathbb{N}$. For a simple root $\alpha$ of any Cartan datum, write $$\mathfrak{g}\langle\alpha\rangle:=\mathfrak{h}_{\alpha}\oplus\bigoplus\limits_{n\in\mathbb{Z}_{\neq0}}\mathfrak{g}_{n\alpha}.$$ Said otherwise, $\mathfrak{g}\langle\alpha\rangle$ is the subalgebra generated by $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$. ## Roots of Heisenberg type {#section_heisenberg} We say that a simple root $\alpha$ is of Heisenberg type if any of the following equivalent conditions hold: 1. $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\alpha}]=0$; 2. $\mathfrak{h}_{\alpha}\subseteq\operatorname{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$; 3. $\mathfrak{h}_{\alpha}$ is central in $\mathfrak{g}\langle\alpha\rangle$. The one part of the above definition that may not be clear is why $\mathfrak{h}_{\alpha}$ need be abelian; however if there exists $H_1,H_2\in(\mathfrak{h}_{\alpha})_{\overline{1}}$, then $[H_1,H_2]\in[H_1,[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]]=0$. If $\alpha$ is of Heisenberg type, then $\Delta = \{\pm \alpha\}$. *Proof.* The Jacobi identity implies $$\begin{aligned} ] \subseteq [\mathfrak{g}_{\alpha},[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]]. \end{aligned}$$ Hence our assumption $[\mathfrak{g}_{\alpha},[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]] = 0$ yields $[\mathfrak{g}_{-\alpha},[\mathfrak{g}_{\alpha},\mathfrak{g}_{\alpha}]] = 0$. From Corollary [\[corollary-maximal-ideal\]](#corollary-maximal-ideal){reference-type="ref" reference="corollary-maximal-ideal"} we obtain $[\mathfrak{g}_{\alpha},\mathfrak{g}_{\alpha}] = 0$. A similar argument shows $[\mathfrak{g}_{-\alpha},\mathfrak{g}_{-\alpha}] = 0$. ◻ [\[definition he_n\]]{#definition he_n label="definition he_n"} For each $n\geq0$, let $\mathfrak{h}\mathfrak{e}(n)$ denote the following Lie superalgebra: let $\mathfrak{h}(n)$ be the Lie superalgebra constructed in Example [\[example of h_n\]](#example of h_n){reference-type="ref" reference="example of h_n"}, and let $C_1$ denote an irreducible representation of $\mathfrak{h}(n)$ in which $c$ acts as the identity; assume that $C_1$ is purely even if $n=0$. Now set $C_{-1}:=C_1^\vee$, and let $\mathfrak{h}$ denote the quotient of the $\mathfrak{h}(n)$-module $C_1\otimes C_{-1}$ by the submodule generated by odd elements lying in its radical (see Remark [\[remark on vanishing\]](#remark on vanishing){reference-type="ref" reference="remark on vanishing"}). Then $C_{-1}\oplus \mathfrak{h}\oplus\mathfrak{h}(n)\oplus C_1$ is Clifford Kac-Moody, where $\mathfrak{h}$ acts trivially on $C_{\pm1}$, and the bracket map $[-,-]:C_{1}\otimes C_{-1}\to\mathfrak{h}\oplus \mathfrak{h}(n)$ is the quotient map onto $\mathfrak{h}$. We set $\mathfrak{h}\mathfrak{e}(n):=C_{-1}\oplus\mathfrak{h}\oplus C_1$. Similarly let $\mathfrak{h}^{\Pi}$ denote the quotient of $C_{1}\otimes\Pi C_{-1}$ by the submodule generated by odd elements lying in the radical. Then $\Pi C_{-1}\oplus\mathfrak{h}^{\Pi}\oplus\mathfrak{h}(n)\oplus C_1$ will be Clifford Kac-Moody in a similar way, and we set $\mathfrak{h}\mathfrak{e}(n)^{\Pi}:=\Pi C_{-1}\oplus \mathfrak{h}\oplus C_1$. [\[example_he(n)\]]{#example_he(n) label="example_he(n)"} If $n=0$ in the above, we obtain the purely even algebra $\mathfrak{h}\mathfrak{e}(0)=\mathbb{C}\langle e,h,f\rangle$ with $h$ central and $[e,f]=h$. On the other hand, $\mathfrak{h}\mathfrak{e}(0)^{\Pi}=\mathbb{C}\langle e,h,f\rangle$ where $e,h$ are odd, $f$ is even, $h$ is central, and $[e,f]=h$. Finally we note that $\mathfrak{h}\mathfrak{e}(2)\cong\mathfrak{t}\mathfrak{s}\mathfrak{l}(1|1)\cong\mathfrak{t}\mathfrak{h}\mathfrak{e}(0)$. The following lemma is straightforward, and mostly given for purposes of clarity. If $\alpha$ is a simple root of rank $n$ and of Heisenberg type, then $\mathfrak{g}\langle\alpha\rangle$ is a quotient by a central ideal of $\mathfrak{h}$ of one of the following: 1. if $n$ is odd, then is it a quotient of $\mathfrak{h}\mathfrak{e}(n)$; 2. if $n\equiv 2$(mod 4), then it is a quotient of either $\mathfrak{h}\mathfrak{e}(n)$ or $\mathfrak{h}\mathfrak{e}(n)^{\Pi}$; 3. if $n\equiv 0$(mod 4), then it is a quotient of either $\mathfrak{h}\mathfrak{e}(n),$ $\mathfrak{h}\mathfrak{e}(n)^{\Pi}$, or $\Pi C_{-1}\oplus \mathfrak{h}\oplus\Pi C_1$ in the notation of Definition [\[definition he_n\]](#definition he_n){reference-type="ref" reference="definition he_n"}. ## Simple roots of non-Heisenberg type The goal for the rest of this section is to prove the following theorem: [\[theorem-one-simple-root\]]{#theorem-one-simple-root label="theorem-one-simple-root"} Suppose $\mathfrak{h}_{\alpha} \nsubseteq \mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$. Then $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$, and one of the following happens: 1. $\operatorname{rk}\alpha = 0$, $\Delta = \{\pm \alpha\}$, and $\mathfrak{g}\langle \alpha \rangle$ is isomorphic to $\mathfrak{s}\mathfrak{l}(2)$. 2. $\operatorname{rk}\alpha = 0$, $\Delta = \{\pm \alpha,\pm 2\alpha\}$, and $\mathfrak{g}\langle \alpha \rangle$ is isomorphic to $\mathfrak{osp}(1|2)$. 3. $\operatorname{rk}\alpha = 2$, $\Delta = \{\pm \alpha\}$, and $\mathfrak{g}\langle \alpha \rangle$ is isomorphic to either $\mathfrak{tsl}(2)\cong\mathfrak{s}\mathfrak{q}(2)$ or $\mathfrak{ptsl}(2)\cong\mathfrak{p}\mathfrak{s}\mathfrak{q}(2)$ (see Examples [\[example_sqn\]](#example_sqn){reference-type="ref" reference="example_sqn"}, [\[example_psq(n)\]](#example_psq(n)){reference-type="ref" reference="example_psq(n)"}, and [\[example_ts_pts\]](#example_ts_pts){reference-type="ref" reference="example_ts_pts"}). 4. $\operatorname{rk}\alpha = 2$, $\Delta = \{\pm \alpha,\pm 2\alpha\}$, and $\mathfrak{g}\langle \alpha \rangle$ is isomorphic to either $\mathfrak{t}\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ or $\mathfrak{p}\mathfrak{t}\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ (see Example [\[example_ts_pts\]](#example_ts_pts){reference-type="ref" reference="example_ts_pts"}). Stated more simply, the above theorem says that if $\alpha$ is not of Heisenberg type, then either it gives rise to a $\mathfrak{s}\mathfrak{l}(2)$, $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$, or a Takiff construction on one of these. We deduce Theorem [\[theorem-one-simple-root\]](#theorem-one-simple-root){reference-type="ref" reference="theorem-one-simple-root"} from the following results. [\[lemma-root-subalgebra\]]{#lemma-root-subalgebra label="lemma-root-subalgebra"} Suppose $\mathfrak{h}_{\alpha} \nsubseteq \mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$. There exist $e \in \mathfrak{g}_{\alpha}$ and $f \in \mathfrak{g}_{-\alpha}$ such that the subalgebra they generate is isomorphic to either $\mathfrak{s}\mathfrak{l}(2)$ or $\mathfrak{osp}(1|2)$. *Proof.* First let us show that we can find an even $\alpha$-coroot $h \in \mathfrak{h}_{\alpha}$ such that $\alpha(h) \neq 0$. By assumption, there exists a homogeneous $\alpha$-coroot that does not annihilate $\mathfrak{g}_{\alpha}$. If it is even, take $h$ to be this coroot. If it is odd denote it by $H$. It follows from $[H,\mathfrak{g}_{\alpha}]\neq 0$ and from the irreducibility of $\mathfrak{g}_{\alpha}$ that $H \notin \ker B_\alpha$. So there exists $K \in \mathfrak{h}_{\overline{1}}$ such that $\alpha([K,H]) \neq 0$. Then $h \coloneq [K,H] \in \mathfrak{h}_{\alpha}$ is the even coroot we are looking for. Now, by definition, there exists a finite index set $I$ and homogeneous elements $e_i \in \mathfrak{g}_{\alpha}$ and $f_i \in \mathfrak{g}_{-\alpha}$ of the same parity for each $i\in I$ such that $h = \sum_{i\in I} [e_i,f_i]$. Hence $\alpha(h) = \alpha(\sum_{i\in I} [e_i,f_i]) \neq 0$, which implies that $\alpha([e_j,f_j]) \neq 0$ for some $j \in I$. Now by our integrability assumption, it is straightforward to check that $e:=e_j$, $f:=f_j$ generate a subalgebra isomorphic to either $\mathfrak{s}\mathfrak{l}(2)$ or $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$, depending on whether $e,f$ are even or odd. ◻ We denote by $\mathfrak{s}$ the subalgebra isomorphic to either $\mathfrak{s}\mathfrak{l}(2)$ or $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ obtained in Lemma [\[lemma-root-subalgebra\]](#lemma-root-subalgebra){reference-type="ref" reference="lemma-root-subalgebra"}. [\[lemma-sl2-osp2-subalgebras\]]{#lemma-sl2-osp2-subalgebras label="lemma-sl2-osp2-subalgebras"} Suppose $\mathfrak{h}_{\alpha}\nsubseteq \mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$, and let $\mathfrak{s}$ be as above. 1. If $\mathfrak{s} \simeq \mathfrak{s}\mathfrak{l}(2)$, then $\Delta = \{\pm \alpha\}$. 2. If $\mathfrak{s}\simeq \mathfrak{osp}(1|2)$, then $\Delta = \{\pm \alpha,\pm 2\alpha\}$. Moreover, $\operatorname{rk}\alpha \leq 2$ and $\dim \mathfrak{g}_\beta = \dim \mathfrak{g}_{\alpha}$ for any $\beta \in \Delta$. *Proof.* The integrability of $\mathfrak{g}(\mathcal{A})$ implies that it is finite-dimensional, and in particular a finite dimensional $\mathfrak{s}$-module. If $e \in \mathfrak{s}\cap \mathfrak{g}_{\alpha}$ is non-zero, then by the finite-dimensional representation theory of $\mathfrak{s}\mathfrak{l}(2)$ and $\mathfrak{osp}(1|2)$, the map $$\begin{aligned} \label{equation-sl2-action} \mathrm{ad}\ e : \mathfrak{g}_{k\alpha} \rightarrow \mathfrak{g}_{(k+1)\alpha} \end{aligned}$$ is surjective for all $k \geq 0$. In particular $\dim \mathfrak{g}_{k\alpha} \leq \dim \mathfrak{g}_{\alpha}$ for all $k > 0$. Since $\mathfrak{g}_{\alpha}$ is irreducible and $\operatorname{rk}\alpha = \operatorname{rk}k\alpha$ for all $k\neq 0$, we have $\dim \mathfrak{g}_{k\alpha} \geq \dim \mathfrak{g}_{\alpha}$ whenever $\mathfrak{g}_{k\alpha} \neq 0$. As a consequence, $\mathrm{ad}\ e : \mathfrak{g}_{k\alpha} \rightarrow \mathfrak{g}_{(k+1)\alpha}$ is a linear isomorphism for $k > 0$ whenever $\mathfrak{g}_{(k+1)\alpha}\neq 0$. If $\mathfrak{s}\simeq \mathfrak{s}\mathfrak{l}(2)$, then $[e,e] = 0$, hence $\mathfrak{g}_{2\alpha} = 0$. If $\mathfrak{s}\simeq \mathfrak{osp}(1|2)$, then $[e,e]\neq 0$ and $[e,[e,e]] = 0$, so $\mathfrak{g}_{3\alpha} = 0$. The description of $\Delta$ follows immediately. Finally, from [\[equation-sl2-action\]](#equation-sl2-action){reference-type="eqref" reference="equation-sl2-action"}, we have $[e,\mathfrak{h}] = \mathfrak{g}_{\alpha}$. As $[e,\mathfrak{t}] = \mathbb{C}\langle e\rangle$, we obtain that either $\dim (\mathfrak{g}_\alpha)_{\overline{0}} = 1$ or $\dim (\mathfrak{g}_\alpha)_{\overline{1}} = 1$. Therefore $\operatorname{rk}\alpha \leq 2$ by our dimension formulas for irreducibles given in Section [2](#section_quasi_toral){reference-type="ref" reference="section_quasi_toral"}. ◻ [\[lemma-rk0-rk2\]]{#lemma-rk0-rk2 label="lemma-rk0-rk2"} If $\mathfrak{h}_{\alpha} \nsubseteq \mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$, then $\operatorname{rk}\alpha = 0$ or $\operatorname{rk}\alpha = 2$. *Proof.* Let us assume $\operatorname{rk}\alpha \neq 0$. Then according to Lemma [\[lemma-sl2-osp2-subalgebras\]](#lemma-sl2-osp2-subalgebras){reference-type="ref" reference="lemma-sl2-osp2-subalgebras"}, we have $1 \leq \operatorname{rk}\alpha \leq 2$. The irreducibility of $\mathfrak{g}_{\alpha}$ as an $\mathfrak{h}$-module implies $\dim \mathfrak{g}_{\alpha} = (1|1)$. Let $k \in \mathbb{Z}$ be the maximal integer such that $k\alpha \in \Delta$ and fix $\beta = k\alpha$. Then $\dim \mathfrak{g}_\beta = (1|1)$ from Lemma [\[lemma-sl2-osp2-subalgebras\]](#lemma-sl2-osp2-subalgebras){reference-type="ref" reference="lemma-sl2-osp2-subalgebras"}. Choose non-zero elements $x \in (\mathfrak{g}_{\beta})_{\overline{0}}$, $X \in (\mathfrak{g}_{\beta})_{\overline{1}}$ and $y \in (\mathfrak{g}_{-\beta})_{\overline{0}}$, and set $H \coloneq [X,y]$. We notice that $x$ and $y$ belong to an $\mathfrak{s}\mathfrak{l}(2)$-subalgebra, so $\beta([x,y]) \neq 0$. We have $$\begin{aligned} = [[X,y],x] = [X,[y,x]] = \beta([x,y])X \neq 0, \end{aligned}$$ and $$\begin{aligned} = [[X,y],X] = [X,[y,X]] = [[y,X],X] = -[[X,y],X] = -[H,X]. \end{aligned}$$ Thus $[H,X] = 0$, and we conclude that $[H,\mathfrak{g}_{\beta}] \neq 0$ and $\beta([H,H]) = 0$. This is impossible if $\operatorname{rk}\beta = 1$,and thus we must have $\operatorname{rk}\beta=\operatorname{rk}\alpha = 2$. ◻ [\[corollary-twist\]]{#corollary-twist label="corollary-twist"} If $\mathfrak{h}_{\alpha} \nsubseteq \mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$, then $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$. *Proof.* If $\operatorname{rk}\alpha = 0$, then $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$ because $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$ are of the same parity. If $\operatorname{rk}\alpha = 2$, then according to [\[equation-4Z+2\]](#equation-4Z+2){reference-type="eqref" reference="equation-4Z+2"} in Section [2.2](#section-irreducible-h-modules){reference-type="ref" reference="section-irreducible-h-modules"}, we have $\mathfrak{g}_{\alpha}\otimes \mathfrak{g}_{-\alpha} \simeq \Pi \mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_\alpha)$ if $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$ and $\mathfrak{g}_{\alpha}\otimes \mathfrak{g}_{-\alpha} \simeq \mathcal{S}(\mathfrak{h}_{\overline{1}}/\ker B_\alpha)$ otherwise. However in the latter situation, the top of $\mathcal{S}(\mathfrak{h}_{\overline{1}}/\operatorname{ker} B_\alpha)$ as an $\mathfrak{h}$-module is even, and thus $\mathfrak{h}_{\alpha}=\operatorname{Im}([-,-]:\mathfrak{g}_{\alpha}\otimes\mathfrak{g}_{-\alpha}\to \mathfrak{h})$ would be purely even. However the proof of Lemma [\[lemma-rk0-rk2\]](#lemma-rk0-rk2){reference-type="ref" reference="lemma-rk0-rk2"} has shown that $\mathfrak{h}$ contains a non-trivial odd element. Our result follows. ◻ ## Realization of $\mathfrak{g}\langle \alpha \rangle$ as in Theorem [\[theorem-one-simple-root\]](#theorem-one-simple-root){reference-type="ref" reference="theorem-one-simple-root"} {#section-realization} Again, suppose $\mathfrak{h}_{\alpha} \nsubseteq \mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$. We continue building on the results of Lemmas [\[lemma-sl2-osp2-subalgebras\]](#lemma-sl2-osp2-subalgebras){reference-type="ref" reference="lemma-sl2-osp2-subalgebras"} and [\[lemma-rk0-rk2\]](#lemma-rk0-rk2){reference-type="ref" reference="lemma-rk0-rk2"} along with Corollary [\[corollary-twist\]](#corollary-twist){reference-type="ref" reference="corollary-twist"}. If $\operatorname{rk}\alpha =0$, then it is clear that $\mathfrak{g}\langle \alpha \rangle$ is isomorphic to either $\mathfrak{s}\mathfrak{l}(2)$ or $\mathfrak{osp}(1|2)$. So assume $\operatorname{rk}\alpha = 2$. In this case $\mathfrak{g}_{\alpha}$ is $(1|1)$-dimensional. If $\Delta=\{\pm\alpha\}$, let $e$ and $E$ be even and odd basis vectors of $\mathfrak{g}_{\alpha}$, respectively. As $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$, under this identification we may write $f \coloneq e^\vee$ and $F \coloneq \sqrt{-1}E^\vee$ as in Proposition [\[proposition-rank-2-chevalley\]](#proposition-rank-2-chevalley){reference-type="ref" reference="proposition-rank-2-chevalley"}, to form a (unique up to scalar) homogeneous basis for $\mathfrak{g}_{-\alpha}$. If $\Delta=\{\pm\alpha,\pm2\alpha\}$, we instead let $E$ and $e$ be even and odd basis vectors of $\mathfrak{g}_{\alpha}$ (note the change of order), and set $f=\sqrt{-1}e^{\vee}$ and $F=E^\vee$. Now in either case, we define the following pure coroots: $$\begin{aligned} \begin{array}{ccc} h \coloneq [e,f], & c \coloneq [E,F], & H \coloneq [E,f]. \end{array} \end{aligned}$$ If $\Delta=\{\pm\alpha\}$, then we normalize $e$ in the such that $\alpha(h) = 2$, and if $\Delta=\{\pm\alpha,\pm2\alpha\}$ we normalize $e$ so that $\alpha(h)=1$. Then the proof of Proposition [\[proposition-rank-2-chevalley\]](#proposition-rank-2-chevalley){reference-type="ref" reference="proposition-rank-2-chevalley"} and a direct computation give the following relations: $$\begin{aligned} \begin{array}{ccccc} H = [e,F], & [H,e] =\alpha(h)E, & [H,E] = 0, & [H,H] = 2c, & \alpha(c)=0. \end{array} \end{aligned}$$ It is now straightforward to check that $\mathfrak{g}\langle\alpha\rangle$ is isomorphic to either $\mathfrak{t}\mathfrak{s}$ or $\mathfrak{pt}\mathfrak{s}$ where $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2)$ if $\Delta=\{\pm\alpha\}$ and $\mathfrak{s}=\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ if $\Delta=\{\pm\alpha,\pm2\alpha\}$. *Proof of Theorem [\[theorem-one-simple-root\]](#theorem-one-simple-root){reference-type="ref" reference="theorem-one-simple-root"}.* The theorem is an immediate consequence of Lemmas [\[lemma-root-subalgebra\]](#lemma-root-subalgebra){reference-type="ref" reference="lemma-root-subalgebra"} and [\[lemma-sl2-osp2-subalgebras\]](#lemma-sl2-osp2-subalgebras){reference-type="ref" reference="lemma-sl2-osp2-subalgebras"}, and [\[lemma-rk0-rk2\]](#lemma-rk0-rk2){reference-type="ref" reference="lemma-rk0-rk2"}, along with Corollary [\[corollary-twist\]](#corollary-twist){reference-type="ref" reference="corollary-twist"} and Section [5.3](#section-realization){reference-type="ref" reference="section-realization"}. ◻ # On Connectivity of simple roots {#section-connectivity} Let $\mathfrak{g}(\mathcal{A})$ be a Clifford Kac-Moody algebra. For a simple root $\alpha \in \Pi$, let $\mathfrak{g}\langle\alpha\rangle$ denote the subalgebra of $\mathfrak{g}(\mathcal{A})$ generated by $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$. Using the results of Section [5](#section-one-root){reference-type="ref" reference="section-one-root"}, we define the root type of a simple root $\alpha \in \Pi$ according to the following table: root type the root $\alpha$ satisfies -------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\mathfrak{s}\mathfrak{l}(2)$ $\mathfrak{g}\langle\alpha\rangle\cong\mathfrak{s}\mathfrak{l}(2)$ $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ $\mathfrak{g}\langle\alpha\rangle\cong\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ $\mathfrak{s}\mathfrak{l}(1|1)$ $\mathfrak{g}\langle\alpha\rangle\cong\mathfrak{s}\mathfrak{l}(1|1)$ $\mathfrak{h}\mathfrak{e}(0)$ $\mathfrak{g}\langle\alpha\rangle$ a purely even Heisenberg of rank 0 (see Example [\[example_he(n)\]](#example_he(n)){reference-type="ref" reference="example_he(n)"}) $\mathfrak{h}\mathfrak{e}(0)^{\Pi}$ a 'mixed' parity Heisenberg of rank 0 (see Example [\[example_he(n)\]](#example_he(n)){reference-type="ref" reference="example_he(n)"}) $Tak(\mathfrak{s}\mathfrak{l}(2))$ $\operatorname{rk}\alpha=2$ and $\mathfrak{g}\langle\alpha\rangle$ a central quotient of $\mathfrak{t}\mathfrak{s}\mathfrak{l}(2)$ (case (iii) of Theorem [\[theorem-one-simple-root\]](#theorem-one-simple-root){reference-type="ref" reference="theorem-one-simple-root"}) $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$ $\operatorname{rk}\alpha=2$ and $\mathfrak{g}\langle\alpha\rangle$ a central quotient of $\mathfrak{t}\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ (case (iv) of Theorem [\[theorem-one-simple-root\]](#theorem-one-simple-root){reference-type="ref" reference="theorem-one-simple-root"}) $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ $\operatorname{rk}\alpha=2$ and $\mathfrak{g}\langle\alpha\rangle$ a central quotient of $\mathfrak{t}\mathfrak{s}\mathfrak{l}(1|1)\cong\mathfrak{t}\mathfrak{h}\mathfrak{e}(0)$ (see Example [\[example_ts_pts\]](#example_ts_pts){reference-type="ref" reference="example_ts_pts"}) $H_n$ $\alpha$ is of Heisenberg type and $\operatorname{rk}\alpha=n$ (see Section [5.1](#section_heisenberg){reference-type="ref" reference="section_heisenberg"})   We emphasize that each simple root in a Clifford Kac-Moody algebra has a well-defined type. The above table hints that simple roots of rank $0$ and rank $2$ are distinguished among all simple roots. In this section, we indeed show that if we want 'interesting' interactions between simple roots, i.e. without any obvious ideals, we should ask that they are all of rank zero or rank 2. ## Connectivity {#section_connectivity} The next natural question to address is when two simple roots can interact, according to their type in the above table. To be precise, we are interested in the following question: **Question:** for which ordered pairs of simple roots $(\alpha,\beta)$ of given type in the above table do the following equivalent condition hold: 1. $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$; 2. $[(\mathfrak{h}_{\alpha})_{\overline{0}},\mathfrak{g}_{\beta}]\neq0$; 3. there exists a pure, even $\alpha$-coroot $h$ (see Section [3.3](#section-g(A)){reference-type="ref" reference="section-g(A)"}) for which $\beta(h)\neq0$. Notice that when the above question has a positive answer, we will have that $\mathfrak{g}_{\alpha+\beta}\neq0$, i.e. $\alpha+\beta\in\Delta$. Equivalently, the above question asks when it is a necessary condition that $\mathfrak{h}_{\alpha}\subseteq\operatorname{Ann}_{\mathfrak{h}}\mathfrak{g}_{\beta}$, according to the root type, as listed in the above table. As an example, if $\alpha$ is of $\mathfrak{h}\mathfrak{e}(0)$-type and $\beta$ is of any other type, then we always have $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]=0$. This is because even Heisenbergs have no finite-dimensional representations with nontrivial central character. This section seeks to answer the above question, and the answer is depicted in the picture below. Namely, in the below diagram we positioned all root types in separate places, and then we draw an arrow from root type $\alpha$ to root type $\beta$ if it is possible that $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$ in a Clifford Kac-Moody algebra.     $$\xymatrix{ &\mathfrak{s}\mathfrak{l}(2) \ar[rr] \ar@(ul,ur) \ar@{=>}[d] \ar@/^3pc/[rrrr] & & \ar[ll] \mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2) \ar@(ul,ur) \ar@{=>}[d]\ar[rr] & & \ar@/_3pc/[llll] \ar[ll] \mathfrak{s}\mathfrak{l}(1|1) \ar@(ul,ur) \ar@{=>}[d] & \\ &&&&&&\\ &Tak(\mathfrak{s}\mathfrak{l}(2)) \ar@/^3pc/@{-->}[rrrrr] \ar@(ul,ur) \ar[rr] \ar[rrdd] \ar[rrrrdd] \ar@/^2pc/[rrrr] & & \ar[ll] Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)) \ar@(ul,ur) \ar[dd] \ar@/^1pc/[ddrr]\ar[rr] & & \ar@/_2pc/[llll] \ar[ll] \ar[dd] \ar[ddll] Tak(\mathfrak{s}\mathfrak{l}(1|1)) \ar@(ul,ur) & \ar[l]\ar[ddl]\ar[ddlll]H_1 \\ &&&&&&\\ & \mathfrak{h}\mathfrak{e}(0) & \mathfrak{h}\mathfrak{e}(0)^{\Pi} & \mathfrak{h}\mathfrak{e}(2)^{\Pi} & & H_n, \ n\geq 3 & }$$ First of all, we note this is an expanded version of Figure [\[Fig1a\]](#Fig1a){reference-type="ref" reference="Fig1a"} from the introduction. In the above diagram, the bold-faced downward arrows $\Rightarrow$ from $\mathfrak{s}\mathfrak{l}(2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$, and $\mathfrak{s}\mathfrak{l}(1|1)$ signify that if $\alpha$ is one of any of these three root types and $\beta$ is another root type, then it is possible that $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$. We have used this notation to avoid an overwhelming thicket of arrows from these three root types. Further, the dashed arrow pointing to $H_1$ is meant to signify that if $\alpha$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$ and $\beta$ is of type $H_1$, and if we have $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$, then $[\mathfrak{h}_{\beta},\mathfrak{g}_{\gamma}]=0$ for any simple root $\gamma$, i.e. $\beta$ is forced to become a 'sink'. ## Sinks In light of condition (3) in our above question, an important question is when all pure, even coroots of $\mathfrak{g}\langle\alpha\rangle$ lie in an even Heisenberg triple, i.e. lie in a subalgebra isomorphic to $\mathfrak{h}\mathfrak{e}(0)$. In this case $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]=0$ for all simple roots $\beta$, i.e. $(\mathfrak{h}_{\alpha})_{\overline{0}}$ will be central in $\mathfrak{g}(\mathcal{A})$. One of the main results of this section is the following: [\[prop_sinks\]]{#prop_sinks label="prop_sinks"} All pure, even coroots of $\mathfrak{g}\langle\alpha\rangle$ lie in an even Heisenberg triple whenever $\alpha$ is one of the following types: 1. $\mathfrak{h}\mathfrak{e}(0)$; 2. $\mathfrak{h}\mathfrak{e}(0)^{\Pi}$; 3. $\mathfrak{h}\mathfrak{e}(2)^{\Pi}$; 4. $H_n$ for $n\geq 3$. In particular, in these cases, $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]=0$ for all simple roots $\beta$. The above proposition tells us that if $\alpha$ is a simple root of the above type, then $\mathfrak{g}\langle\alpha\rangle$ generates an ideal containing no other simple roots. Thus they are 'sinks' in the theory, which we see from the diagram above. *Proof of Proposition [\[prop_sinks\]](#prop_sinks){reference-type="ref" reference="prop_sinks"}.* The statement for $\mathfrak{h}\mathfrak{e}(0)$ is clear, and for $\mathfrak{h}\mathfrak{e}(0)^{\Pi}$ there are no even coroots, so that statement is vacuously true. For $\mathfrak{h}\mathfrak{e}(2)^{\Pi}$, let $e$ and $E$ be even and odd basis vectors of $\mathfrak{g}_{\alpha}$, and $f$ and $F$ even and odd basis vectors of $\mathfrak{g}_{-\alpha}$. Then our claim reduces to showing $[E,F] \in \mathbb{C}\langle[e,f]\rangle$. Choose $K\in\mathfrak{h}$ such that $\alpha(K^2)\neq0$, so that $K$ defines an odd automorphism of $\mathfrak{g}_{\pm\alpha}$. Then $[K,e] = aE$ and $[K,F] = bf$ for non-zero $a,b \in \mathbb{C}$. we know from Section [2.3](#section_dualities){reference-type="ref" reference="section_dualities"} that $\mathfrak{h}_{\alpha}$ is one-dimensional and purely even, so that $[e,F]=0$. Thus $$\begin{aligned} 0 = [K,[e,F]] = a[E,F] + b[e,f], \end{aligned}$$ which gives our result. Finally, we deal with case of $H_n$ for $n\geq 3$. In the proof, we will use the realization of a simple $\mathfrak{h}$-module given in Section [2.5](#subsection-realization-irreducible){reference-type="ref" reference="subsection-realization-irreducible"}. We remind here that $\mathbb{C}[\xi_1,...,\xi_m]$ is the superalgebra of polynomial in odd variables $\xi_1,...,\xi_m$ satisfying the relation $\xi_i\xi_j = -\xi_j\xi_i$ for all $i,j$. For a set $I=\{i_1,...,i_k\} \subseteq \{1,...,m\}$, we write $\xi_I \coloneq \xi_{i_1}\cdot...\cdot \xi_{i_k}$ for $i_1<...<i_k$. We first consider the case where $\operatorname{rk}\alpha$ is even, let $m \in \mathbb{Z}$ be such that $\operatorname{rk}\alpha = 2m$. Then our assumption on $\operatorname{rk}\alpha$ implies $m \geq 2$. According to Section [2.5](#subsection-realization-irreducible){reference-type="ref" reference="subsection-realization-irreducible"}, we can identify $\mathfrak{g}_{\alpha}$ as $\mathbb{C}[\xi_1,...,\xi_m]$ up to parity, and choose $H_1,...,H_m,\bar{H}_1,...,\bar{H}_m \in \mathfrak{h}_{\overline{1}}$ such that $H_i$ acts on $\mathfrak{g}_{\alpha}$ via multiplication by $\xi_i$, and $\bar{H}_i$ by the derivation $\partial_{\xi_i}$. Similarly, we can identify $\mathfrak{g}_{-\alpha}$ as $\mathbb{C}[\phi_1,...,\phi_m]$ up to parity, such that $H_i$ acts on $\mathfrak{g}_{-\alpha}$ via multiplication by $-\phi_i$, and $\bar{H}_i$ be the derivation $\partial_{\phi_i}$. Let $\xi_I$ and $\phi_J$ be odd elements; we will show that $[\xi_I,\phi_J]$ is either zero or a bracket of two even elements from $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$. We will use the fact that $\mathfrak{h}_{\alpha} = [\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}] \subseteq \mathfrak{h}$, and so $[\mathfrak{h}_{\overline{1}},[\mathfrak{h}_{\overline{1}},\mathfrak{h}_{\alpha}]] = 0$ because $\mathfrak{h}$ is quasi-toral. If $\lvert I\cap J\rvert\geq 2$, then we can find $i,j \in I\cap J$ such that $i\neq j$, so $$\begin{aligned} 0 = [H_i,[H_j,[\xi_{I\setminus\{i,j\}},\phi_J]]] = \pm [\xi_I,\phi_J]. \end{aligned}$$ If $I=J=\{i\}$, then because $m\geq2$, we can find $j\notin I$ so that $$\begin{aligned} 0=[H_i,[\bar{H}_j,[\xi_j,\phi_i]]] = [H_i,[\xi_\emptyset,\phi_i]] = [\xi_i,\phi_i]. \end{aligned}$$ If $I=J=\emptyset$, then because $m\geq 2$, we can find $i,j \in \{1,...,m\}$ such that $i\neq j$, so $$\begin{aligned} 0 = [\bar{H}_i,[\bar{H}_j,[\xi_i,\phi_j]]] = \pm [\bar{H}_i,[\xi_i,\phi_{\emptyset}]] = \pm [\xi_\emptyset,\phi_\emptyset]. \end{aligned}$$ Finally, if none of the above happens, then it must be that $I\neq J$. If $i \in I\setminus J$, then $$\begin{aligned} 0 = [\bar{H}_i,[H_i,[\xi_I,\phi_J]]] = [\bar{H}_i,[\xi_I,-\phi_i\phi_J]] = \pm[\xi_{I\setminus\{i\}},\phi_{J\cup\{i\}}] \pm [\xi_I,\phi_J]. \end{aligned}$$ If $i \in J\setminus I$, then $$\begin{aligned} 0 = [\bar{H}_i,[H_i,[\xi_I,\phi_J]]] = [\bar{H}_i,[\xi_i\xi_I,\phi_J]] = [\xi_I,\phi_J] \pm [\xi_{I\cup\{i\}},\phi_{J\setminus\{i\}}]. \end{aligned}$$ In the above two equations, we see that our term of interest, $[\xi_I,\phi_J]$, is equal to a bracket of two even elements, as desired. We now consider the case where $\operatorname{rk}\alpha$ is odd, let $m \in \mathbb{Z}$ be such that $\operatorname{rk}\alpha = 2m+1$. Again, from Section [2.5](#subsection-realization-irreducible){reference-type="ref" reference="subsection-realization-irreducible"}, we can find $H_1,...,H_m,\bar{H}_1,...,\bar{H}_m,\tilde{H} \in \mathfrak{h}_{\overline{1}}$ such that $\mathfrak{g}_{\alpha}$ can be realized as $\mathbb{C}[\xi_1,...,\xi_{m+1}]$, such that $H_i$ acts by $\xi_i$, $\bar{H}_i$ acts by $\partial_{\xi_i}$, and $\tilde{H}$ acts by $\xi_{m+1}+\partial_{\xi_{m+1}}$. Similarly, $\mathfrak{g}_{-\alpha}$ can be realized as $\mathbb{C}[\phi_1,...,\phi_{m+1}]$, such that $H_i$ acts by $-\phi_i$, $\bar{H}_i$ acts by $\partial_{\phi_i}$, and $\tilde{H}$ acts by $-\phi_{m+1}+\partial_{\phi_{m+1}}$. As in the previous case, let $\xi_I$ and $\phi_J$ be odd elements. We notice that in the current setting ($\operatorname{rk}\alpha = 2m+1$), it must be that $\lvert I \rvert$ and $\lvert J \rvert$ are odd integers. We will show that $[\xi_I,\phi_J]$ is either zero or a bracket of two even elements from $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$. The only computations left (which are not analogous to the case when $\operatorname{rk}\alpha$ is even) are for $I\setminus \{m+1\} = J\setminus \{m+1\}$ and $\lvert I\setminus \{m+1\} \rvert \leq 1$. If $I=J$ and $I\setminus\{m+1\} = \{i\}$, then $$\begin{aligned} 0 = [\tilde{H},[H_i,[\xi_{m+1},\phi_i]]] = [\tilde{H},[\xi_i\xi_{m+1},\phi_i]] = \pm [\xi_i,\phi_i] \pm [\xi_i\xi_{m+1},\phi_i\phi_{m+1}]. \end{aligned}$$ If $I = J = \{m+1\}$, then $$\begin{aligned} 0 = [\tilde{H},[\bar{H}_1,[\xi_1,\phi_{m+1}]]] = [\tilde{H},[\xi_\emptyset,\phi_{m+1}]] = [\xi_{m+1},\phi_{m+1}] \pm [\xi_{\emptyset},\phi_{\emptyset}]. \end{aligned}$$ This concludes our proof. ◻ ## Connectivity properties of $Tak(\mathfrak{s}\mathfrak{l}(2))$, $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, and $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ In this section we assume that $\mathfrak{g}\langle\alpha\rangle$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, or $Tak(\mathfrak{s}\mathfrak{l}(1|1))$; equivalently we assume that $\operatorname{rk}\alpha=2$ and $\mathfrak{g}_{\alpha}^\vee\cong\mathfrak{g}_{-\alpha}$. In this case, by Section [2.3](#section_dualities){reference-type="ref" reference="section_dualities"}, $\mathfrak{h}_{\alpha}$ is generated by an odd element which we call $H_{\alpha}$. If $\beta$ is a simple root with $\operatorname{rk}\beta=0$, we have $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]=0$. *Proof.* Since $[H_{\alpha},\mathfrak{g}_{\beta}]=0$ we are done. ◻ [\[lemma_conn_tak\]]{#lemma_conn_tak label="lemma_conn_tak"} Suppose that $\beta$ is of rank 1. Then 1. if $\alpha$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, then $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]=0$; 2. if $\alpha$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$ and $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$, then $[\mathfrak{h}_{\beta},\mathfrak{g}_{\gamma}]=0$ for all simple roots $\gamma$ (i.e. $H_1$ 'becomes' a sink, justifying the dashed arrow in our diagram). *Proof.* For $(1)$, we have $H_{\alpha}^2=0$ for $Tak(\mathfrak{s}\mathfrak{l}(1|1))$, while $H_{\alpha}^2$ is the center of an even Heisenberg triple for $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$. Thus we necessarily have $\beta(H_{\alpha}^2)=0$ for these two cases. Since $\beta$ is of rank one this implies that $[H_{\alpha},\mathfrak{g}_{\beta}]=0$, and we are done. For $(2)$, if $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$, then clearly $H_{\alpha}$ acts by an automorphism on $\mathfrak{g}_{\beta}$. Write $e,E$ and $f,F$ for homogeneous bases of $\mathfrak{g}_{\beta}$ and $\mathfrak{g}_{-\beta}$ respectively, and set $H_{\beta}=[E,f]$. Then we see that $$[H_{\alpha},H_{\beta}]=a[e,f]+b[E,F],$$ for some nonzero $a,b\in\mathbb{C}$. On the other hand we may write $\mathfrak{h}_{\alpha}\ni[H_{\alpha},H_{\beta}]=rh_{\alpha}+sc_{\alpha}$, where $h_{\alpha}$ is the pure $\alpha$ coroot lying in an $\mathfrak{s}\mathfrak{l}(2)$-triple, and $c_{\alpha}$ is central in $\mathfrak{g}\langle\alpha\rangle$. Further, because $H_{\beta}^2=0$, we see that either $r=0$ or $s=0$, and perhaps both are 0. Now since $$a[e,f]+b[E,F]=rh_{\alpha}+sc_{\alpha},$$ and it is clear that $\beta(a[e,f]+b[E,F])=0$, we also must have $r\beta(h_{\alpha})+s\beta(c_{\alpha})=0$. By $\mathfrak{s}\mathfrak{l}(2)$-representation theory we must have $\beta(h_{\alpha})\neq0$, so this forces $r=0$, which tells us that $$a[e,f]+b[E,F]=sc_{\alpha}.$$ However since $\alpha(c_{\alpha})=0$, this forces $a\alpha([e,f])+b\alpha([E,F])=0$. Since $[e,f]$ lies in a Heisenberg triple, we must have $\alpha([e,f])=0$, so necessarily $\alpha([E,F])=0$. Thus $\alpha((\mathfrak{h}_{\beta})_{\overline{0}})=0$, which implies $[H_{\alpha},H_{\beta}]=0$. It follows that $$a[e,f]+b[E,F]=0,$$ meaning that $[E,F]\in\mathbb{C}\langle[e,f]\rangle$, so that all pure, even coroots of $\mathfrak{g}\langle\beta\rangle$ lie in a Heisenberg triple. This completes the argument. ◻ ## Connectivity properties of $H_1$ [\[lemma-rank-1\]]{#lemma-rank-1 label="lemma-rank-1"} Let $\alpha \in \Pi$ be such that $\operatorname{rk}\alpha= 1$, and let $\beta \in \Pi$ be any simple root with either $\operatorname{rk}\beta<2$ or $\operatorname{rk}\beta=2$ of type $Tak(\mathfrak{s}\mathfrak{l}(2))$ or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$. Then $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]=0$. *Proof.* For the proof, let us first realize the subalgebra of $\mathfrak{g}(\mathcal{A})$ generated by $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$. Because $\operatorname{rk}\alpha = 1$, we have $\mathfrak{g}_{-\alpha} \cong\mathfrak{g}_{\alpha}^\vee$, and thus we identify $\mathfrak{g}_{-\alpha}$ and $\mathfrak{g}_{\alpha}^\vee$. Let $e$ and $E$ be even and odd basis vectors for $\mathfrak{g}_{\alpha}$, respectively. Then $f \coloneq e^\vee$ and $F \coloneq \sqrt{-1}E^\vee$ form a homogeneous basis for $\mathfrak{g}_{-\alpha}$. Let $h \coloneq [e,f]$, $c \coloneq [E,F]$ and $H \coloneq [e,F]$. We notice that $e,h,f$ form a Heisenberg subalgebra, so the integrability of $\mathfrak{g}(\mathcal{A})$ implies $\beta(h) = 0$. Suppose first that $[H,\mathfrak{g}_{\beta}]=0$. The irreducibility of $\mathfrak{g}_{\alpha}$ implies the existence of $K\in\mathfrak{h}_{\overline{1}}$ such that $[K,\mathfrak{g}_{\alpha}]=\mathfrak{g}_{\alpha}$. In particular, $[K,H] = ah+bc$ for some non-zero $a,b \in \mathbb{C}$. Therefore $$\begin{aligned} 0 = [[K,H],\mathfrak{g}_{\beta}] = [ah+bc,\mathfrak{g}_{\beta}]. \end{aligned}$$ As $\beta(h) = 0$, we must have $\beta(c) = 0$. Thus we obtain $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}] = 0$, as desired. We conclude: $$\begin{aligned} \label{equation-root-implication} [H,\mathfrak{g}_{\beta}]=0 \implies [\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]= 0. \end{aligned}$$ We now prove the lemma case by case, depending on the properties of $\beta$. If $\operatorname{rk}\beta = 0$, we must have $[H,\mathfrak{g}_{\beta}] = 0$, so we are done. Similarly if $\operatorname{rk}\beta=1$, then since $H^2=0$ we must have $[H,\mathfrak{g}_{\beta}]=0$, so we are done. Suppose now $\beta$ is of $Tak(\mathfrak{s}\mathfrak{l}(2))$-type or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$-type. In the latter case, $\mathfrak{g}\langle\alpha\rangle$ contains $(\mathfrak{p})\mathfrak{sq}(2)$ as a subalgebra with the same coroots, so it suffices to assume we are in this case. However by Lemma [\[lemma_conn_tak\]](#lemma_conn_tak){reference-type="ref" reference="lemma_conn_tak"}, if $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]\neq0$ then necessarily $[\mathfrak{h}_{\beta},\mathfrak{g}_{\alpha}]=0$. If we write $H_{\beta}$ for a nonzero odd coroot of $\mathfrak{h}_{\beta}$, then this implies that $[H_1,H_\beta]=0$. Thus $H,H_\beta$ have commuting actions on $\mathfrak{g}_{\beta}$, so since $H_1$ acts nontrivially it must act nontrivially on the $\mathfrak{s}\mathfrak{l}(2)$ triple inside $\mathfrak{g}\langle\beta\rangle$. However if $h_{\beta}$ denotes the even coroot of said $\mathfrak{s}\mathfrak{l}(2)$ triple, then this implies $\alpha(h_{\beta})\neq0$, a contradiction. ◻ # Queer Kac-Moody algebras {#section-QKM} ## Regularity and indecomposability [\[definition-regular\]]{#definition-regular label="definition-regular"} We say that a Cartan datum $\mathcal{A}$ (and by extension, the Lie superalgebra $\mathfrak{g}(\mathcal{A})$) is regular if $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}] = 0$ implies $[\mathfrak{h}_{\beta},\mathfrak{g}_{\alpha}] = 0$ for any $\alpha,\beta \in \Pi$. From Proposition[\[proposition-roots-connectivity\]](#proposition-roots-connectivity){reference-type="ref" reference="proposition-roots-connectivity"} we deduce: [\[corollary-regular\]]{#corollary-regular label="corollary-regular"} If $\mathfrak{g}$ is regular and $\alpha,\beta \in \Pi$ are such that $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}] = 0$, then $\alpha+\beta$ is not a root of $\mathfrak{g}$. Let $\mathcal{A}$ be a Cartan datum. We say that $\mathcal{A}$ is indecomposable if there does not exist a nontrivial partition $\Pi=\Pi_1\sqcup\Pi_2$ such that $[\mathfrak{h}_{\alpha},\mathfrak{g}_{\beta}]=[\mathfrak{h}_{\beta},\mathfrak{g}_{\alpha}]=0$ for all $\alpha\in\Pi_1$, $\beta\in\Pi_2$. Our work in the previous section immediately implies: [\[theorem-connectivity-aKM\]]{#theorem-connectivity-aKM label="theorem-connectivity-aKM"} Let $\mathfrak{g}(\mathcal{A})$ be a regular, indecomposable Clifford Kac-Moody algebra with $|\Pi|>1$. Then one of two possibilities can occur: 1. $\operatorname{rk}\alpha =0$ and $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$ for all $\alpha\in\Pi$; 2. $\operatorname{rk}\alpha =2$ and $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$ for all $\alpha\in\Pi$. ## Queer Kac-Moody superalgebras Following Theorem [\[theorem-connectivity-aKM\]](#theorem-connectivity-aKM){reference-type="ref" reference="theorem-connectivity-aKM"}, it makes sense to consider the class of Clifford Kac-Moody algebras in which any simple root $\alpha\in \Pi$ satisfies $\operatorname{rk}\alpha = 2$ and $\mathfrak{g}_{-\alpha} \simeq \mathfrak{g}_{\alpha}^\vee$. [\[definition-qkm\]]{#definition-qkm label="definition-qkm"} Let $\mathcal{A}$ be a Cartan datum. We say the Lie superalgebra $\mathfrak{g}(\mathcal{A})$ is queer Kac-Moody (qKM) if the following conditions are satisfied: 1. [\[qKM-integrable\]]{#qKM-integrable label="qKM-integrable"} $\mathcal{A}$ is integrable, see Definition [\[definition-integrable\]](#definition-integrable){reference-type="ref" reference="definition-integrable"}; 2. [\[qKM-regular\]]{#qKM-regular label="qKM-regular"} $\mathcal{A}$ is regular, see Definition [\[definition-regular\]](#definition-regular){reference-type="ref" reference="definition-regular"}; 3. [\[qKM-rank2\]]{#qKM-rank2 label="qKM-rank2"} every $\alpha \in \Pi$ is of rank $2$; 4. [\[qKM-irrtwist\]]{#qKM-irrtwist label="qKM-irrtwist"} for any $\alpha\in\Pi$, $\mathfrak{h}$-module $\mathfrak{g}_{\alpha}$ is irreducible, and $\mathfrak{g}_{-\alpha}\simeq \mathfrak{g}_{\alpha}^\vee$; 5. [\[qKM-linear-independence\]]{#qKM-linear-independence label="qKM-linear-independence"} for any $\alpha \in \Pi$ we have $\mathfrak{h}_{\alpha} \nsubseteq \sum_{\beta\in\Pi\setminus\{\alpha\}}\mathfrak{h}_{\beta}$. In this situation we will also say that $\mathcal{A}$ is a queer Kac-Moody Cartan datum. We notice that every qKM algebra is a Clifford Kac-Moody algebra, and the possible simple root types are $Tak(\mathfrak{s}\mathfrak{l}(2))$, $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, and $Tak(\mathfrak{s}\mathfrak{l}(1|1))$. [\[remark-Chevalley\]]{#remark-Chevalley label="remark-Chevalley"} From Corollary [\[corollary-Chevalley\]](#corollary-Chevalley){reference-type="ref" reference="corollary-Chevalley"} we deduce that any qKM algebra admits a Chevalley automorphism $\omega$ extending $\omega_{\mathfrak{h}}$ and satisfying $\omega^2=\delta$. Condition [\[qKM-regular\]](#qKM-regular){reference-type="ref" reference="qKM-regular"} should be viewed as a convenient assumption; we expect that if we would remove the regularity condition, we would obtain algebras with a nontrivial ideal inside generated by 'non-regular' simple roots. However for the sake of tightening the scope of this paper, we limit our considerations. Property [\[qKM-linear-independence\]](#qKM-linear-independence){reference-type="ref" reference="qKM-linear-independence"} is a condition on the linear independence of the simple coroots of $\mathfrak{g}(\mathcal{A})$. In a Clifford Kac-Moody algebra with all simple roots of rank $0$, this condition coincides with the usual linear independence of coroots condition as in [@K]. Let $\mathcal{A}$ be a Cartan datum with simple roots $\Pi=\{\alpha_1,\dots,\alpha_n\}$ satisfying (KM1)-(KM4) of Definition [\[definition-qkm\]](#definition-qkm){reference-type="ref" reference="definition-qkm"}. Then $\mathcal{A}$ satisfies (KM5) if and only if the odd coroots in $\mathfrak{h}_{\alpha_1},\dots,\mathfrak{h}_{\alpha_n}$ are linearly independent. *Proof.* The backward direction is clear. For the forward direction, suppose that $\mathcal{A}$ satisfies (KM5) but the odd coroots are not linearly independent. Since $\dim(\mathfrak{h}_{\alpha_i})_{\overline{1}}=1$ for all $i$, let $H_i\in(\mathfrak{h}_{\alpha_i})_{\overline{1}}$ be any nonzero element. Then by our assumption, for some $i$ we have $H_i\in\sum_{j\neq i}\mathfrak{h}_{\alpha_j}$. However $H_i$ generates $\mathfrak{h}_{\alpha_i}$ as an $\mathfrak{h}$-module; thus $$\mathfrak{h}_{\alpha_i}=\mathbb{C}\langle H_i\rangle\oplus[\mathfrak{h},H_i]\subseteq\sum\limits_{j\neq i}[\mathfrak{h},\mathfrak{h}_{\alpha_i}]\subseteq\sum_{j\neq i}\mathfrak{h}_{\alpha_j},$$ contradiction (KM5). ◻ It follows from our work that if $\mathcal{A}$ is qKM, then for a simple root $\alpha$, the subalgebra $\mathfrak{g}\langle\alpha\rangle$ is isomorphic to a central quotient of $\mathfrak{t}\mathfrak{s}$ for a Kac-Moody Lie superalgebra with one simple root (see Example [\[example_ts_pts\]](#example_ts_pts){reference-type="ref" reference="example_ts_pts"}). This illustrates a beauty in the theory of queer Kac-Moody superalgebras: they are built by 'glueing together' extensions of Takiffs of Kac-Moody superalgebras with one simple root. A trivial way to glue together such Takiffs is via the Takiff construction applied to an arbitrary Kac-Moody Lie superalgebra; however, in a sense, this is a less interesting type of glueing. We will find that if we avoid such constructions, we are led to more 'interesting' Lie superalgebras such as $\mathfrak{q}(n)$, and also that the theory becomes very rigid. ## Cartan datum for qKM algebras {#section-notation-qKM} Because qKM algebras only have simple roots of type $Tak(\mathfrak{s})$ for $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$, or $\mathfrak{s}\mathfrak{l}(1|1)$, we can more explicitly present their Cartan datum. We do that now, while also introducing notation that will be used henceforth throughout the paper. Let $\mathcal{A}$ be a qKM Cartan datum, and let $\Pi=\{\alpha_1,\dots,\alpha_n\}$ be the simple roots. For each $i$, we have the subalgebra $\mathfrak{g}\langle\alpha_i\rangle$ generated by $\mathfrak{g}_{\alpha_i}$ and $\mathfrak{g}_{-\alpha_i}$. We choose a homogeneous bases of $\mathfrak{g}_{\pm\alpha_i}$, using the identification $\mathfrak{g}_{-\alpha_i}=\mathfrak{g}_{\alpha_i}^\vee$, as follows: 1. if $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, set $e_i$ and $E_i$ to be nonzero even and odd elements of $\mathfrak{g}_{\alpha_i}$, and let $f_i:=e_i^\vee$ and $F_i:=\sqrt{-1}E_i^\vee$. Rescale the choice of $e_i$ so that $\alpha_i([e_i,f_i])=2$; 2. if $\alpha_i$ is of type $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$ or $Tak(\mathfrak{s}\mathfrak{l}(1|1))$, set $E_i$ and $e_i$ to be nonzero even and odd elements of $\mathfrak{g}_{\alpha_i}$ (notice the change in order), and let $f_i:=\sqrt{-1}e_i^\vee$ and $F_i:=E_i^\vee$. If $\alpha_i$ is of type $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, then rescale the choice of $e_i$ so that $\alpha_i([e_i,f_i])=1$. It follows from the above choices that: 1. if $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, then $e_i,[e_i,f_i],f_i$ form an $\mathfrak{s}\mathfrak{l}(2)$ triple; 2. if $\alpha_i$ is of type $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, then $[e_i,e_i],e_i,[e_i,f_i],f_i,[f_i,f_i]$ form an $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$-quintuple; 3. if $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$, then $e_i,[e_i,f_i],f_i$ form an $\mathfrak{s}\mathfrak{l}(1|1)$-triple. On should view the $e_i,E_i,f_i,$ and $F_i$ as the 'Chevalley generators' for a qKM algebra. ### Uniqueness of generators {#section_uniqueness_gens} Before going further, we note that the above choices are not unique: in particular we may simultaneously rescale $e_i$ and $f_i$ by $(-1)$, or simultaneously rescale $E_i$ and $F_i$ by $\lambda\in\mathbb{C}^\times$. For $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ we may also rescale $e_i,f_i$ by any $\lambda\in\mathbb{C}^\times$. ### Pure coroots Set $$H_i:=[E_i,f_i];$$ one may check that $H_i\neq0$, and we also always have $$H_i=[e_i,F_i].$$ Further: $$h_i:=[e_i,f_i] \ \ \ \ c_i:=[E_i,F_i].$$ The following table summarizes some of the important properties, which are easy to prove, of the pure coroots we have introduced. $Tak(\mathfrak{s}\mathfrak{l}(2))$ $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$ $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ ------------------- ---------------------------------------- ------------------------------------------------------- ---------------------------------------- $H_i^2=$ $c_i$ $c_i$ 0 $c_i$ lies in\... $\mathfrak{s}\mathfrak{l}(1|1)$-triple even Heisenberg triple even Heisenberg triple $\alpha_i(c_i)=0$ $\beta(c_i)=0$ for all roots $\beta$ $\beta(c_i)=0$ for all roots $\beta$ $h_i$ lies in\... $\mathfrak{s}\mathfrak{l}(2)$-triple $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$ quintuple $\mathfrak{s}\mathfrak{l}(1|1)$-triple $\alpha_i(h_i)=2$ $\alpha_i(h_i)=1$ $\alpha_i(h_i)=0$ [\[remark_root_subalg_really_takiff\]]{#remark_root_subalg_really_takiff label="remark_root_subalg_really_takiff"} A more concrete viewpoint on the elements described above is given using the fact that each $\mathfrak{g}\langle\alpha_i\rangle$ is a central quotient of $\mathfrak{t}\mathfrak{s}=\mathfrak{s}\otimes\mathbb{C}[\xi]\oplus\mathbb{C}\langle c\rangle$ for $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)$, or $\mathfrak{s}\mathfrak{l}(1|1)$. In these terms, $e_i,f_i$ are the Chevalley generators of $\mathfrak{s}$, $h_i$ is the coroot of $\mathfrak{s}$, $c_i=c$, and $H_i=h_i\otimes \xi$. ### $X$ and $Y$ matrices {#section_X_Y_matrices} Let $\mathcal{A}$ be a qKM Cartan datum. We now introduce two $n\times n$ matrices $X(\mathcal{A})=X=(x_{ij})$ and $Y(\mathcal{A})=Y=(y_{ij})$ of complex numbers, which encode much of the Cartan datum. The entries are defined as follows: $$[H_j,e_i]=x_{ji}E_i, \ \text{ and } \ [H_j,E_i]=y_{ji}e_i.$$ Then because $\mathfrak{g}_{-\alpha_i}\cong\mathfrak{g}_{\alpha_i}^\vee$, we have $$[H_i,f_j]=-(-1)^{\overline{e_i}}x_{ij}F_j, \ \text{ and } \ [H_i,F_j]=(-1)^{\overline{e_i}}y_{ij}f_j.$$ [\[remark_rescaling\]]{#remark_rescaling label="remark_rescaling"} The entries of the matrices $X(\mathcal{A})$ and $Y(\mathcal{A})$ are not unique; indeed, in addition to permuting indices, if we rescaled our choices of generators $e_i$ and $E_i$ (where permitted) we would scale certain entries in the matrices. Indeed, if we rescale $E_i$ by $\lambda$ the entries change as follows: $$y_{ij},y_{ji}\mapsto \lambda y_{ij},\lambda y_{ji}, \ \ \ \ x_{ij},x_{ji}\mapsto\lambda x_{ij},x_{ji}/\lambda.$$ ### Formulas [\[lemma_qKM_formulas\]]{#lemma_qKM_formulas label="lemma_qKM_formulas"} We have the following formulas: 1. for any $i,j$: $$[H_i,H_j]=x_{ij}c_j+y_{ij}h_j = x_{ji}c_i+y_{ji}h_i;$$ 2. for any $i,j$: $$\alpha_k([H_i,H_j])=x_{ik}y_{jk} + x_{jk}y_{ik},$$ in particular, $$\alpha_j(H_i^2)=x_{ij}y_{ij};$$ 3. $$y_{ii}=0 \ \text{ for all }i;$$ 4. $$x_{ii}=\alpha_i(h_i)=\begin{cases} 2 & \text{ if }\alpha_i\text{ of type }Tak(\mathfrak{s}\mathfrak{l}(2)),\\ 1 & \text{ if }\alpha_i\text{ of type }Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2)),\\ 0 & \text{ if }\alpha_i\text{ of type }Tak(\mathfrak{s}\mathfrak{l}(1|1)). \end{cases}$$ 5. $$\alpha_i([H_i,H_j])=y_{ij}\alpha_i(h_i).$$ *Proof.* Formula (1) follows from applying the Jacobi identity $[H_i,H_j]$ with either $H_i=[E_i,f_i]$ or $H_j=[E_j,f_j]$. Formula (2) holds because $\alpha_k([H_i,H_j])$ is given by the scalar action of $[H_i,H_j]$ on $\mathfrak{g}_{\alpha_k}$; however this is the action of the operator $H_iH_j+H_jH_i$, and one sees that it acts by the given scalar. Formulas (3) and (4) are straightforward, and formula (5) follows from (2), (3), and (4). ◻ [\[lemma_nontrivial_relation_on_ys\]]{#lemma_nontrivial_relation_on_ys label="lemma_nontrivial_relation_on_ys"} For two simple roots $\alpha_i,\alpha_j$ of type $Tak(\mathfrak{s}\mathfrak{l}(2))$ or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$ we have $$x_{ij}x_{ji}y_{ji}+y_{ij}\alpha_i(h_j)=y_{ij}\alpha_i(h_i),$$ and $$y_{ij}y_{ji}(\alpha_i(h_i)-\alpha_j(h_j))=y_{ij}^2\alpha_i(h_j)-y_{ji}^2\alpha_j(h_i).$$ In particular if $\alpha_i(h_i)=\alpha_j(h_j)$, then $$y_{ij}^2\alpha_i(h_j)=y_{ji}^2\alpha_j(h_i).$$ *Proof.* Indeed, if we apply $\alpha_i$ to formula (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} we obtain $$x_{ij}\alpha_i(c_j)+y_{ij}\alpha_i(h_j)=y_{ij}\alpha_i(h_i).$$ Now we use that for $Tak(\mathfrak{s}\mathfrak{l}(2))$ and $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$ type we have $H_i^2=c_i$, and apply (2) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"}, we obtain the first formula. To obtain the second formula, swap the indices on $i$ and $j$ in the first formula, and subtract one from the other. The third formula is immediate. ◻ ## Dynkin diagrams {#section_dynkin} Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra, with notation as explained in Section [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}. We call $A = (a_{ij})=(\alpha_j(h_i))$ the Cartan matrix of $\mathfrak{g}(\mathcal{A})$. Define $D=D(\mathcal{A})$ to be the Dynkin diagram associated with the Cartan datum $\mathcal{A}$ as follows: draw a vertex for each $i=1,\dots,n$ according to the following rule: 1. draw if $a_{ii}=2$, or equivalently $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$; 2. draw if $a_{ii}=1$, or equivalently $\alpha_i$ is of type $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$; 3. draw $\diamondtimes$ if $a_{ii}=0$, or equivalently $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$. Next we draw an edge from vertex $i$ to vertex $j$ if $a_{ij}a_{ji}\neq0$, and we label the edge by the pair $a_{ij}, a_{ji}$. We will sometimes draw an unlabeled edge between two vertices to indicate that we assume $a_{ij}a_{ji}\neq0$. If we disregard the type of a simple root, we draw $\square$ for its corresponding vertex. The Dynkin diagram of $\mathfrak{q}(n)$ looks as follows: This is also the Dynkin diagram of $T\mathfrak{s}\mathfrak{l}(n)$. From the results of Section [8](#section-two-simple-roots){reference-type="ref" reference="section-two-simple-roots"} we see that two simple roots $\alpha$ and $\beta$ of a qKM algebra $\mathfrak{g}(\mathcal{A})$ are connected in the associated Dynkin diagram $S(A)$ if and only if $\alpha+\beta$ is a root of $\mathfrak{g}(\mathcal{A})$, that is, $\mathfrak{g}_{\alpha+\beta}\neq 0$. We say two distinct simple roots $\alpha,\beta$ are connected if the same is true in the Dynkin diagram. Note that by the following lemma, this is equivalent to $\alpha_i(h_j)\alpha_j(h_i)=a_{ij}a_{ji}\neq0$. [\[lemma-qKM-notation\]]{#lemma-qKM-notation label="lemma-qKM-notation"} Suppose that $\mathcal{A}$ is a qKM Cartan datum, and $\alpha_i,\alpha_j$ are distinct simple roots. Then the following are equivalent: 1. $\alpha_i$ and $\alpha_j$ are connected; 2. $\mathfrak{g}_{\alpha_i+\alpha_j}\neq0$; 3. $[\mathfrak{h}_{\alpha_i},\mathfrak{g}_{\alpha_j}]\neq0$; 4. $[H_i,\mathfrak{g}_{\alpha_j}]\neq0$; 5. $x_{ij}$ and $y_{ij}$ are not both zero; 6. $\alpha_j((\mathfrak{h}_{\alpha_i})_{\overline{0}})\neq0$. 7. $\alpha_{j}(h_i)\neq0$. *Proof.* The implications between all but the last condition a straightforward application of regularity, the equivalence of the conditions in the question at the start of Section [6](#section-connectivity){reference-type="ref" reference="section-connectivity"}, and the fact that $H_i$ generates $\mathfrak{h}_{\alpha_i}$ as an ideal of $\mathfrak{h}$. The final condition is equivalent by $\mathfrak{s}\mathfrak{l}(2)$-representation theory if $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$ or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, and for $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ it is because $\alpha_j(c_i)=0$ for all roots $\alpha_j$ (see the table in Section [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}). ◻ [\[remark_dynkin_diagram\]]{#remark_dynkin_diagram label="remark_dynkin_diagram"} Consider the map $\Theta$ from Dynkin diagrams of qKM algebras to Dynkin diagrams of Kac-Moody superalgebras, obtained by turning diamonds into circles (see [@K2] for more on Dynkin diagrams of Kac-Moody superalgebras). We notice that if the Dynkin diagram $D$ of a qKM algebra $\mathfrak{g}(\mathcal{A})$ has that $\Theta(D)$ is of infinite growth, then $\mathfrak{g}(\mathcal{A})$ is also of infinite growth, by Section [3.4](#section_subdatum){reference-type="ref" reference="section_subdatum"}. # Two simple roots in a qKM algebra {#section-two-simple-roots} [\[lemma_coupled_defn_justif\]]{#lemma_coupled_defn_justif label="lemma_coupled_defn_justif"} Let $\alpha_i,\alpha_j$ be connected simple roots in a qKM algebra $\mathfrak{g}(\mathcal{A})$. Then either $x_{ij}x_{ji}\neq0$ or $y_{ij}y_{ji}\neq0$. *Proof.* Suppose that neither of the above two equations holds; because $\alpha_i$ and $\alpha_j$ are connected, we may assume WLOG that $x_{ij}=y_{ji}=0$ and $x_{ji}y_{ij}\neq0$. Then by (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"}, we learn that $$x_{ji}c_i=y_{ij}h_j.$$ However since $\alpha_i(c_i)=0$, this implies $\alpha_i(h_j)=0$, contradicting Lemma [\[lemma-qKM-notation\]](#lemma-qKM-notation){reference-type="ref" reference="lemma-qKM-notation"}. ◻ [\[definition_coupling\]]{#definition_coupling label="definition_coupling"} Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra. We say two distinct simple roots $\alpha_i,\alpha_j \in \Pi$ are coupled if they are connected and we have $\alpha_i(c_j)=\alpha_j(c_i)=0$. Equivalently, by Lemma [\[lemma_coupled_defn_justif\]](#lemma_coupled_defn_justif){reference-type="ref" reference="lemma_coupled_defn_justif"}, either $x_{ij}=x_{ji}=0$, in which case we say $\alpha_i$ and $\alpha_j$ are $X$-coupled, or $y_{ij}=y_{ji}=0$, in which case we say $\alpha_i$ and $\alpha_j$ are $Y$-coupled. [\[remark_coroot_coupling\]]{#remark_coroot_coupling label="remark_coroot_coupling"} By (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"}, we have for two simple roots $\alpha_i$ and $\alpha_j$ the relation: $$x_{ij}c_j+y_{ij}h_j = x_{ji}c_i+y_{ji}h_i.$$ Thus if $\alpha_i$ and $\alpha_j$ are coupled, the above formula gives a nontrivial relation between one pure even coroot on each side. In particular, if they are $X$-coupled we obtain $$y_{ij}h_j=y_{ji}h_i, \ \ \ \ y_{ij}y_{ji}\neq0,$$ and if they are $Y$-coupled we obtain $$x_{ij}c_j=x_{ji}c_i, \ \ \ \ x_{ij}x_{ji}\neq0.$$ The prototypical example of a qKM algebra with coupled simple roots is the extension of the Takiff superalgebra $T\mathfrak{s}$ for any Kac-Moody superalgebra $\mathfrak{s}$. Indeed, in this case we have $Y=0$, i.e. all connected simple roots are $Y$-coupled. For $\mathfrak{g}=\mathfrak{q}(n)$, no distinct simple roots are coupled with one another. The aim for this section is to prove the following theorem: [\[theorem_coupling\]]{#theorem_coupling label="theorem_coupling"} Suppose that $\mathcal{A}$ is qKM and $\alpha_i,\alpha_j\in\Pi$ are distinct simple roots. Then one of the following cases must occur: 1. $\alpha_i$ and $\alpha_j$ are not connected; 2. $\alpha_i$ and $\alpha_j$ are $Y$-coupled; 3. $\alpha_i$ and $\alpha_j$ are $X$-coupled, $\alpha_i(h_j)\alpha_j(h_i)=4$, and one of the following holds up to swapping the indices $i$ and $j$: 1. $\alpha_i$ and $\alpha_j$ are of type $Tak(\mathfrak{s}\mathfrak{l}(2))$; or, 2. $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, $\alpha_j$ is of type $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, and we have $\alpha_j(h_i)=-2$ and $\alpha_i(h_j)=-1$; 4. $\alpha_i$ and $\alpha_j$ are not coupled, both of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, and $y_{ij}y_{ji}\neq0$. Note that the following is false for the entries of the $X$-matrix (in particular it fails for $\mathfrak{q}^{-}_{(2,2)}$, see Section [11.3](#section_q_22){reference-type="ref" reference="section_q_22"}). [\[corollary_ys\]]{#corollary_ys label="corollary_ys"} If $\mathcal{A}$ is qKM, then $y_{ij}=0\Rightarrow y_{ji}=0$. *Proof.* This follows immediately from Theorem [\[theorem_coupling\]](#theorem_coupling){reference-type="ref" reference="theorem_coupling"}. ◻ The proof of Theorem [\[theorem_coupling\]](#theorem_coupling){reference-type="ref" reference="theorem_coupling"} will occupy the rest of the section, and we will also prove auxiliary results which will be important in future sections. We begin by dealing with the case of $Tak(\mathfrak{s}\mathfrak{l}(1|1))$: Suppose that $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ and is connected to $\alpha_j$. Then $\alpha_i$ and $\alpha_j$ are $Y$-coupled. *Proof.* Suppose that $\alpha_i$ and $\alpha_j$ are connected, so that $\alpha_i(h_j)\alpha_j(h_i)\neq0$. If $\alpha_i$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$, then $H_i^2=0$, implying that $x_{ij}y_{ij}=0$. Suppose that $x_{ij}=0$, so that (1) of [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} gives $$y_{ij}h_j=x_{ji}c_i+y_{ji}h_i.$$ Applying $\alpha_i$ to the RHS we obtain 0, implying that $\alpha_i(h_j)=0$, a contradiction. Thus we must have $y_{ij}=0$, implying that $$x_{ij}c_j=x_{ji}c_i+y_{ji}h_i.$$ If we apply $\alpha_j$, we know from the table in Section [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"} that $\alpha_j(c_j)=\alpha_j(c_i)=0$, so we obtain $y_{ij}\alpha_j(h_i)=0$; since $\alpha_j(h_i)\neq0$, this forces $y_{ji}=0$. Thus we have proven that $\alpha_i$ is $Y$-coupled to $\alpha_j$. ◻ [\[lemma-sq2-rep\]]{#lemma-sq2-rep label="lemma-sq2-rep"} Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra and let $\alpha \in \Pi$ be of type $Tak(\mathfrak{s}\mathfrak{l}(2))$. Let $e,E,f,F,h,c,H$ be the spanning set for $\mathfrak{g}\langle\alpha\rangle\cong(\mathfrak{p})\mathfrak{sq}(2)$ as defined in Section [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}. Let $\beta \in \Pi$ be a simple root different from $\alpha$. Then: 1. $\mathfrak{g}_{\beta-\beta(h)\alpha}\neq0$, and $\mathfrak{g}_{\beta+(1-\beta(h))\alpha}=0$; 2. if $\beta(h)=-1$ and $v \in \mathfrak{g}_{\beta}$, then $$[E,v] + [e,[H,v]] =0.$$ *Proof.* The first statement follows from the integrability of the action of $\mathfrak{s}\mathfrak{l}(2)$. Now suppose that $\beta(h) = -1$. Then $\mathfrak{g}_{\beta+2\alpha}=0$ so that $[e,[E,v]] = 0$ . Therefore $$\begin{aligned} 0 & = -[f,[e,[E,v]]] = -[[f,e],[E,v]] - [e,[[f,E],v]] \\ & = [h,[E,v]] + [e,[H,v]] = [E,v] + [e,[H,v]]. \end{aligned}$$ ◻ [\[propistion-two-sq2\]]{#propistion-two-sq2 label="propistion-two-sq2"} Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra and let $\alpha_i, \alpha_j \in \Pi$ be two simple roots type $Tak(\mathfrak{s}\mathfrak{l}(2))$. If $\alpha_i$ and $\alpha_j$ are connected, then exactly one of the following happens: 1. $\alpha_i$ and $\alpha_j$ are coupled, in which case either: 1. $\alpha_i$ and $\alpha_j$ are $Y$-coupled; or 2. $\alpha_i$ and $\alpha_j$ are $X$-coupled and $\alpha_i(h_j)\alpha_j(h_i)=4$. 2. $\alpha_i(h_j) = \alpha_j(h_i) = -1$ and $\alpha_i(c_j)\alpha_j(c_i) \neq 0$; 3. $\alpha_i(h_j)=\alpha_j(h_i) = -2$, and exactly one of $x_{ij}$, $x_{ji}$ is zero; 4. $\alpha_i(h_j),\alpha_j(h_i) \in \mathbb{Z}_{\leq -2}$ and $\alpha_i(c_i)\alpha_j(c_i) \neq 0$. *Proof.* Because we assume $\alpha_i,\alpha_j$ are connected, we have $\alpha_i(h_j)\alpha_j(h_i)\neq0$. Now because $\alpha_i(h_i)=\alpha_j(h_j)$, Lemma [\[lemma_nontrivial_relation_on_ys\]](#lemma_nontrivial_relation_on_ys){reference-type="ref" reference="lemma_nontrivial_relation_on_ys"} implies that: $$\begin{aligned} y_{ij}^2\alpha_i(h_j) = y_{ji}^2\alpha_j(h_i). \end{aligned}$$ So if $y_{ij}y_{ji} = 0$, it must be that both $y_{ij}=y_{ji} = 0$, implying $\alpha_i$ and $\alpha_j$ are $Y$-coupled. If, on the other hand, $x_{ij}x_{ji} = 0$, then Lemma [\[lemma_nontrivial_relation_on_ys\]](#lemma_nontrivial_relation_on_ys){reference-type="ref" reference="lemma_nontrivial_relation_on_ys"} gives: $$\begin{aligned} 2y_{ji} = y_{ij}\alpha_i(h_j), \\ 2y_{ij} = y_{ji}\alpha_j(h_i). \end{aligned}$$ Thus $y_{ij},y_{ji}$ is a non-trivial solution to the linear system defined by the matrix $$\begin{bmatrix} 2 & -\alpha_i(h_j)\\ -\alpha_j(h_i) & 2\end{bmatrix},$$ implying that the determinant is 0, which gives $$\alpha_i(h_j)\alpha_j(h_i) = 4.$$ In particular, this shows that if $\alpha_i$ and $\alpha_j$ are $X$-coupled, then $\alpha_i(h_j)\alpha_j(h_i)=4$, as desired. On the other hand, if $\alpha_i$ and $\alpha_j$ are not coupled but we still have $x_{ij}x_{ji}=0$, then exactly one of $x_{ij}$, $x_{ji}$ is zero, and we have $\alpha_i(h_j)\alpha_j(h_i)=4$. It remains to show then that $\alpha_i(h_j)=\alpha_j(h_i)=-2$, or equivalently that $\alpha_i(h_j)\neq-1$ and $\alpha_j(h_i)\neq-1$, which will follow from our final argument. Thus suppose that $\alpha_i(h_j) = -1$ and $\alpha_i$, $\alpha_j$ are not coupled. Then Lemma [\[lemma-sq2-rep\]](#lemma-sq2-rep){reference-type="ref" reference="lemma-sq2-rep"} implies $[E_j,e_i] + x_{ji}[e_j,E_i] = 0$. As a result $$\begin{aligned} 0 = [f_i,[E_j,e_i] + x_{ji}[e_j,E_i]] = [E_j,-h_i]+ x_{ji}[e_j,-H_i] = (\alpha_j(h_i)+x_{ji}x_{ij})E_j, \end{aligned}$$ and $$\begin{aligned} 0 = [F_i,[E_j,e_i] + x_{ji}[e_j,E_i]] = [E_j,H_i] + x_{ji}[e_j,c_i] = (y_{ij}-x_{ij}x_{ji}y_{ij})e_j. \end{aligned}$$ Because $\alpha_i$ and $\alpha_j$ are not coupled, we have $y_{ij} \neq 0$, so $x_{ij}x_{ji}=1$ and $\alpha_j(h_i) = -1$. This completes our proof. ◻ [\[prop_coupling_sl_osp\]]{#prop_coupling_sl_osp label="prop_coupling_sl_osp"} Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra. Let $\alpha_i \in \Pi$ be of $Tak(\mathfrak{s}\mathfrak{l}(2))$-type and $\alpha_j \in \Pi$ be of $Tak(\mathfrak{osp}(1|2))$-type. If $\alpha_i$ is connected to $\alpha_j$, then either they are Y-coupled, or they are $X$-coupled and we have $\alpha_j(h_i)=-2$ and $\alpha_i(h_j)=-1$. *Proof.* First of all, we have $\alpha_i(c_j) = 0$ by the table from Section [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}. In particular $x_{ji}y_{ji}=0$. We may consider the $\mathfrak{s}\mathfrak{q}(2)$-subquotient of $\mathfrak{g}\langle\alpha_j\rangle$ generated by $\mathfrak{g}_{2\alpha_j}$ and $\mathfrak{g}_{-2\alpha_j}$. Then Proposition [\[propistion-two-sq2\]](#propistion-two-sq2){reference-type="ref" reference="propistion-two-sq2"} tells us that $y_{ji}=0$ implies $y_{ij}=0$ as well, so that $\alpha_i$ and $\alpha_j$ are $Y$-coupled. Let us assume they are not $Y$-coupled, so that $x_{ji}=0$, $y_{ij}y_{ji}\neq0$. Then Proposition [\[propistion-two-sq2\]](#propistion-two-sq2){reference-type="ref" reference="propistion-two-sq2"} tells us that $\alpha_i(h_j)(2\alpha_j)(h_i)=4$. If $\alpha_j(h_i)=-1$, then Lemma [\[lemma-sq2-rep\]](#lemma-sq2-rep){reference-type="ref" reference="lemma-sq2-rep"} then gives $[E_i,e_j] + [e_i,[H_i,e_j]] = 0$, thus $$\begin{aligned} \label{equation-contradicting-tosp} 0 = [F_j,[E_i,e_j] + [e_i,[H_i,e_j]]] = -[E_i,H_j] + [e_i,[F_j,[H_i,e_j]]]. \end{aligned}$$ As $[H_i,e_j] \in \mathbb{C}\langle E_j\rangle$, we have $[F_j,[H_i,e_j]] \in \mathbb{C}\langle c_j\rangle$, so $[e_i,[F_j,[H_i,e_j]]] = 0$. Together with [\[equation-contradicting-tosp\]](#equation-contradicting-tosp){reference-type="eqref" reference="equation-contradicting-tosp"}, we obtain $$\begin{aligned} 0 = [E_i,H_j] = y_{ji}e_i \neq 0, \end{aligned}$$ a contradiction. Thus we must have $\alpha_j(h_i)=-2$ and $\alpha_i(h_j)=-1$. Now if $\alpha_i$ and $\alpha_j$ were not $X$-coupled, by then Proposition [\[propistion-two-sq2\]](#propistion-two-sq2){reference-type="ref" reference="propistion-two-sq2"} we would have $2\alpha_j(h_i)=\alpha_i(h_j)=-2$, which is not the case, meaning instead they must be $X$-coupled, and we are done. ◻ We now finish the proof of Theorem [\[theorem_coupling\]](#theorem_coupling){reference-type="ref" reference="theorem_coupling"} with the following proposition. If $\alpha_i$ and $\alpha_j$ are both of type $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$ and are connected, then they cannot be $X$-coupled (in particular they must be $Y$-coupled). *Proof.* Suppose that $\alpha_i$ and $\alpha_j$ are $X$-coupled. By considering $2\alpha_i$ and $\alpha_j$ as $X$-coupled simple roots, Proposition [\[prop_coupling_sl_osp\]](#prop_coupling_sl_osp){reference-type="ref" reference="prop_coupling_sl_osp"} tells us that we have $\alpha_j(h_i)=-2$ and $(2\alpha_i)(h_j)=-1$. However this implies that $\alpha_i(h_j)=-1/2$, which is impossible. ◻ # Completely coupled vs. completely uncoupled {#section_coupled_v_uncoupled} It is interesting to understand what are the possible configurations of simple roots of a qKM algebra. If we consider qKM algebras of finite growth, the coupling property is very well behaved. First, a definition. Let $\mathcal{A}$ a be qKM Cartan datum with matrices $X=X(\mathcal{A})$ and $Y=Y(\mathcal{A})$ as in Section [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}. We say that $\mathcal{A}$, and also $\mathfrak{g}(\mathcal{A})$, is 1. completely coupled if any two simple roots that are connected are coupled; 2. completely $X$-coupled if $x_{ij}=0$ for all $i\neq j$; 3. completely $Y$-coupled if $Y=0$, i.e. $y_{ij}=0$ for all $i,j$; 4. completely uncoupled if for all $i\neq j$, either $\alpha_i$ and $\alpha_j$ are not connected, or they are connected and uncoupled. The main theorem to be proven in this section is: Suppose that $\mathfrak{g}(\mathcal{A})$ is an indecomposable, finite growth qKM algebra. Then $\mathcal{A}$ is either completely $X$-coupled, completely $Y$-coupled, or completely uncoupled. We also have explicit descriptions of completely $X$-coupled and completely $Y$-coupled algebras. [\[theorem_X\_coupled\]]{#theorem_X_coupled label="theorem_X_coupled"} If $\mathfrak{g}(\mathcal{A})$ is an indecomposable, completely $X$-coupled qKM algebra, then its Cartan datum satisfies one of the following up to isomorphism, and $\mathfrak{g}(\mathcal{A})$ is finite growth: 1. $\Pi=\{\alpha_1,\alpha_2\}$ with $\alpha_1,\alpha_2$ both of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, and $h_1=-h_2$. We have $$X=\begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}, \ \ \ \ Y=\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.$$ The Dynkin diagram is given by: 2. $\Pi=\{\alpha_1,\alpha_2\}$ with $\alpha_1,\alpha_2$ both of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, and $h_1=-2h_2$. We have $$X=\begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}, \ \ \ \ Y=\begin{bmatrix} 0 & -2\\ 1 & 0 \end{bmatrix}.$$ The Dynkin diagram is given by: 3. $\Pi=\{\alpha_1,\alpha_2\}$ with $\alpha_1$ of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, $\alpha_2$ of type $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, and $h_1=-2h_2$. We have $$X=\begin{bmatrix} 2 & 0\\ 0 & 1 \end{bmatrix}, \ \ \ \ Y=\begin{bmatrix} 0 & -2\\ 1 & 0 \end{bmatrix}.$$ The Dynkin diagram is given by: For the completely $Y$-coupled case, see Theorem [\[theorem-Takiffs\]](#theorem-Takiffs){reference-type="ref" reference="theorem-Takiffs"}. ## $X$-couplings [\[lemma_X\_coupling_isolated\]]{#lemma_X_coupling_isolated label="lemma_X_coupling_isolated"} Suppose that $\mathcal{A}$ is an indecomposable qKM Cartan datum. Then if $\alpha_i,\alpha_j$ are distinct, connected, $X$-coupled simple roots, then $\Pi=\{\alpha_i,\alpha_j\}$, i.e. there are no other simple roots. *Proof.* By Theorem [\[theorem_coupling\]](#theorem_coupling){reference-type="ref" reference="theorem_coupling"}, our assumption implies that $\alpha_k(h_k)=1$ or $2$ for $k=i,j$. On the other hand, since $x_{ij}=x_{ji}=0$ by assumption, (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} implies that $h_i=\lambda h_j$ for $\lambda=y_{ij}/y_{ji}\neq0$. We see that $$0<\alpha_i(h_i)=\lambda\alpha_i(h_j).$$ Since $\alpha_i(h_j)\in\mathbb{Z}_{<0}$, this forces $\lambda\in\mathbb{R}_{<0}$. If $\gamma$ were any other simple root in $\mathcal{A}$, then we must have $\gamma(h_i),\gamma(h_j)\leq0$; on the other hand this means that $0\leq \lambda\gamma(h_j)=\gamma(h_i)$. Thus $\gamma(h_i)=0$, and similarly $\gamma(h_j)=0$. Thus $\gamma$ is not connected to either $\alpha_i$ or $\alpha_j$, which by indecomposability implies that $\Pi=\{\alpha_i,\alpha_j\}$, and we are done. ◻ *Proof of Theorem [\[theorem_X\_coupled\]](#theorem_X_coupled){reference-type="ref" reference="theorem_X_coupled"}.* It remains to check that the only possible Cartan data are the ones listed. However this follows from (3) of Theorem [\[theorem_coupling\]](#theorem_coupling){reference-type="ref" reference="theorem_coupling"}, which tells us the possible simple root types and the values of $\alpha_i(h_j)$, and we can rescale $y_{ij}$ and $y_{ji}$ to make it as above, by rescaling our choice of generators (see Remark [\[remark_rescaling\]](#remark_rescaling){reference-type="ref" reference="remark_rescaling"}). The relations on $h_1$ and $h_2$ come from (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"}. ◻ ## Complete coupling for finite growth qKM algebras Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra. Let $\alpha_1$, $\alpha_2$ and $\alpha_3$ be simple roots, such that their corresponding full subdiagram of the Dynkin diagram of $\mathfrak{g}(\mathcal{A})$ is: If $\alpha_1$ and $\alpha_2$ are coupled, then $\alpha_2$ and $\alpha_3$ are coupled. *Proof.* Since we may assume by Lemma [\[lemma_X\_coupling_isolated\]](#lemma_X_coupling_isolated){reference-type="ref" reference="lemma_X_coupling_isolated"} that $\alpha_1$ and $\alpha_2$ are $Y$-coupled, (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} implies that $$x_{12}c_2=x_{21}c_1,$$ where $x_{12},x_{21}\neq0$. Thus we obtain that $\alpha_1(c_2)=0$, since the same is true of $c_1$. Applying $\alpha_1$ to the formula $$x_{23}c_3+y_{23}h_3=x_{32}c_2+y_{32}h_2$$ gives $$0=x_{32}\alpha_1(c_2)+y_{32}\alpha_1(h_2)=y_{32}\alpha_1(h_2).$$ Since $\alpha_1(h_2)\neq0$, this forces $y_{32}=0$, so by Lemma [\[lemma_coupled_defn_justif\]](#lemma_coupled_defn_justif){reference-type="ref" reference="lemma_coupled_defn_justif"} we obtain $y_{23}=0$ also, and we are done. ◻ Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra of finite growth. Let $\alpha_1$, $\alpha_2$ and $\alpha_3$ be simple roots, such that their corresponding full subdiagram of the Dynkin diagram of $\mathfrak{g}(\mathcal{A})$ is: If $\alpha_1$ and $\alpha_2$ are coupled, then $\alpha_3$ is coupled with $\alpha_1$ and $\alpha_2$. *Proof.* We again may assume by Lemma [\[lemma_X\_coupling_isolated\]](#lemma_X_coupling_isolated){reference-type="ref" reference="lemma_X_coupling_isolated"} that any coupling that occurs here is $Y$-coupling. If $\alpha_3$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, then we are done by Theorem [\[theorem_coupling\]](#theorem_coupling){reference-type="ref" reference="theorem_coupling"}. If $\alpha_1$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$, then it must be $Y$-coupled to $\alpha_2$ and $\alpha_3$, so by Remark [\[remark_coroot_coupling\]](#remark_coroot_coupling){reference-type="ref" reference="remark_coroot_coupling"} we learn that $c_2$ and $c_3$ are proportional. This implies that $\alpha_2(c_3)=\alpha_3(c_2)=0$, so $\alpha_2$ and $\alpha_3$ are coupled. Similar considerations apply if $\alpha_2$ is of type $Tak(\mathfrak{s}\mathfrak{l}(1|1))$ or $Tak(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2))$. Thus we may assume that $\alpha_1$, $\alpha_2$ and $\alpha_3$ are all of type $Tak(\mathfrak{s}\mathfrak{l}(2))$, and we know that $y_{12}=y_{21}=0$, since they are $Y$-coupled. If either $\alpha_1$ or $\alpha_2$ were coupled to $\alpha_3$, then we would have that $c_1$ is a nonzero multiple of $c_2$ is a nonzero multiple of $c_3$, meaning that $\alpha_i(c_j)=0$ for all $i,j$, so all roots are coupled. Thus we assume $\alpha_3$ is not coupled to either $\alpha_1$ or $\alpha_2$; in particular $y_{13}y_{31}y_{23}y_{32}\neq0$. Now suppose that $x_{13} = 0$; then (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} gives $$\begin{aligned} y_{13}h_3 = x_{31}c_1 + y_{31}h_1. \end{aligned}$$ Since $\alpha_2(c_1)=\alpha_1(c_1) = 0$, so evaluating $\alpha_1$ and $\alpha_2$ on the above equation, we obtain $$\begin{aligned} &y_{13}a_{31} = 2y_{31}, \\ &y_{13}a_{32} = y_{31}a_{12}. \end{aligned}$$ However this implies $a_{31}/a_{32}=2/a_{12}$, which is impossible, as $a_{31},a_{32},a_{12} < 0$. Thus we must have $x_{13}\neq0$, and for similar reasons $x_{23}\neq0$. By (1) or Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} we have $[H_1,H_2] = x_{21}c_1 = x_{12}c_2$, and by (2) of the same lemma we obtain: $$\begin{aligned} x_{13}y_{23}+x_{23}y_{13} = x_{21}x_{13}y_{13} = x_{12}x_{23}y_{23}. \end{aligned}$$ We may rescale $E_1$ and $E_2$ such that $y_{13}=y_{23}= 1$, so from the above equation we obtain $$\begin{aligned} x_{13}(1-x_{21})+x_{23} = x_{13} + x_{23}(1-x_{12}) = 0. \end{aligned}$$ As $x_{13}x_{23} \neq 0$, the above equations imply that $(1-x_{21})(1-x_{12}) = 1$, so $$\begin{aligned} \label{equation-x-relations} x_{12}+x_{21} = x_{12}x_{21}. \end{aligned}$$ We again use equations (1) and (2) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} to deduce that $$\begin{aligned} & x_{12}y_{32} = \alpha_2([H_1,H_3]) = y_{31}a_{12}, \\ & x_{21}y_{31} = \alpha_1([H_2,H_3]) = y_{32}a_{21}. \end{aligned}$$ From these equations we obtain $x_{12}x_{21} = a_{12}a_{21}$ and $\frac{x_{12}}{x_{21}} = \frac{y_{31}^2a_{12}}{y_{32}^2a_{21}}$. As in the proof of Proposition [\[propistion-two-sq2\]](#propistion-two-sq2){reference-type="ref" reference="propistion-two-sq2"}, we must have $y_{31}^2a_{13} = y_{13}^2a_{31} = a_{31}$ and $y_{32}^2a_{23} = y_{23}^2a_{32} = a_{32}$, so $$\begin{aligned} \frac{x_{12}}{x_{21}} = \frac{a_{12}a_{23}a_{31}}{a_{21}a_{13}a_{32}} > 0. \end{aligned}$$ Squaring [\[equation-x-relations\]](#equation-x-relations){reference-type="eqref" reference="equation-x-relations"} and dividing by $x_{12}x_{21}$, we get $$\begin{aligned} \frac{x_{12}}{x_{21}} + \frac{x_{21}}{x_{12}} + 2 = x_{12}x_{21} \end{aligned}$$ But $\frac{x_{12}}{x_{21}} > 0$ and $x_{12}x_{21} = a_{12}a_{21}$, so $a_{12}a_{21} > 2$. It is now easy to see that $A$ is not a finite type Cartan matrix, meaning $\mathfrak{g}(\mathcal{A})$ is not of finite growth by Remark [\[remark_dynkin_diagram\]](#remark_dynkin_diagram){reference-type="ref" reference="remark_dynkin_diagram"}. ◻ We immediately obtain the following corollary: If $\mathfrak{g}(\mathcal{A})$ is an indecomposable qKM algebra of finite growth, then it is either completely coupled or completely uncoupled. ## $Y$-couplings: realization via the Takiff construction [\[theorem-Takiffs\]]{#theorem-Takiffs label="theorem-Takiffs"} Let $\mathfrak{g}(\mathcal{A})$ be an indecomposable, completely $Y$-coupled qKM algebra (i.e. $Y=0$). Let $A=(a_{ij})=(\alpha_j(h_i))$ be the Cartan matrix of $\mathfrak{g}(\mathcal{A})$, and set $\mathfrak{s}=\mathfrak{s}(A)$ to be the corresponding contragredient, Kac-Moody Lie superalgebra, with root lattice $Q_{\mathfrak{s}}$. Finally, let $T\mathfrak{s}$ be as in Example [\[example-takiff\]](#example-takiff){reference-type="ref" reference="example-takiff"}. Set $$J_{\mathcal{A}}:=\bigcap\limits_{i=1}^{n}\operatorname{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha_i}, \ \ \ \ J_{\mathfrak{s}}:=\bigcap\limits_{\alpha\in Q_{\mathfrak{s}}}\operatorname{Ann}_{T\mathfrak{s}}(\mathfrak{s}_{\alpha}\otimes\mathbb{C}[\xi]).$$ Then: 1. Up to rescaling of generators (see Remark [\[remark_rescaling\]](#remark_rescaling){reference-type="ref" reference="remark_rescaling"}), we have an equality of matrices $X=A$; 2. we have an embedding $$\mathfrak{g}(\mathcal{A})/J_\mathcal{A}\hookrightarrow T\mathfrak{s}(A)/J_{\mathfrak{s}},$$ which is an isomorphism on all root spaces. Thus Theorem [\[theorem-Takiffs\]](#theorem-Takiffs){reference-type="ref" reference="theorem-Takiffs"} says if $Y=0$, $\mathfrak{g}(\mathcal{A})$ is, up to extensions and derivations in $\mathfrak{h}$, a Takiff superalgebra. *Proof of Theorem [\[theorem-Takiffs\]](#theorem-Takiffs){reference-type="ref" reference="theorem-Takiffs"}.* Fix $\alpha_i$ a simple root of $\mathfrak{g}(\mathcal{A})$. We first show that after rescaling $E_1,\dots,E_n$, we obtain that $X=A$. The irreducibility of $\mathfrak{g}_{\alpha_i}$ as an $\mathfrak{h}$-module implies the existence of $I \in \mathfrak{h}_{\overline{1}}$ such that $[I,E_i] = e_i$. For each $j$, let $y_j \in \mathbb{C}$ be the scalar satisfying $[I,E_j] = y_je_j$. We compute: $$\begin{aligned} x_{ij}y_j = \alpha_j([I,H_i]) = \alpha_j(h_i) = a_{ij} \end{aligned}$$ Thus it suffices to show that we can rescale $E_1,\dots,E_n$ so that $y_j=1$ for all $j$; or equivalently that $y_j\neq0$ for all $j$. However the above formula clearly shows that if $\alpha_j$ is connected to $\alpha_i$, then $y_i\neq0\Rightarrow y_j\neq0$. Since the Dynkin diagram is connected, we are done, and we obtain, after rescaling, $x_{ij}=a_{ij}$ for all $i,j$. For any $X \in \mathfrak{h}_{\overline{1}}$ define the complex numbers $\langle X , e_i \rangle$ and $\langle X , E_i \rangle$ by the formula: $$[X,e_i+E_i] = \langle X , e_i \rangle E_i + \langle X , E_i \rangle e_i.$$ Repeating the argument given for $I$ and the fact that $X=A$, we have $\langle X , E_i \rangle = \langle X , E_j \rangle$ for any $i,j$. Write $\mathfrak{t}_{\mathfrak{s}}$ for the Cartan subalgebra of $\mathfrak{s}$, and let $t_1,\dots,t_n\in\mathfrak{t}_{\mathfrak{s}}$ be such that $\alpha_i(t_j)=\delta_{ij}$. Define a map $\tilde{\phi} : \tilde{\mathfrak{g}}(\mathcal{A})\rightarrow T\mathfrak{s}(A)/J_\mathfrak{s}$ as follows: $$\begin{aligned} e_i & \mapsto \tilde{e}_i \\ f_i & \mapsto \tilde{f}_i \\ E_i & \mapsto \tilde{e}_i\otimes \xi\\ F_i & \mapsto \tilde{f}_i\otimes \xi \\ \mathfrak{t}\ni h& \mapsto \sum\limits_{i}\alpha_i(h)t_i \\ \mathfrak{h}_{\overline{1}}\ni X & \mapsto \langle X,E_1 \rangle\partial_\xi + \sum_{i=1}^n \langle X,e_i \rangle t_i\otimes\xi \end{aligned}$$ It is an immediate verification that the above assignment determines a well defined homomorphism of Lie superalgebras. We further notice that $(\tilde{\phi}(\tilde{\mathfrak{g}}(\mathcal{A})))_{\tilde{\alpha}} = (T\mathfrak{s}(A)/Z_\mathfrak{s})_{\tilde{\alpha}}$ for any nonzero $\tilde{\alpha} \in \mathbb{Z}\langle \tilde{\alpha}_1,...,\tilde{\alpha}_n \rangle$. As $T\mathfrak{s}(A)/Z_\mathfrak{s}$ admits no ideals intersecting $(T\mathfrak{s}(A)/Z_\mathfrak{s})_{\overline{0}}$ (its Cartan subalgebra) trivially, we see that the map $\tilde{\phi}$ factors through a map $\phi : \mathfrak{g}(\mathcal{A}) \rightarrow T\mathfrak{s}(A)/J_\mathfrak{s}$, which is bijective on all root spaces. Finally, it is easy to check that $\phi(J_\mathfrak{g}) = 0$, so that $\phi$ factors through an map $\mathfrak{g}(\mathcal{A})/J_{\mathfrak{g}}\rightarrow T\mathfrak{s}(A)/J_\mathfrak{s}$, which is easily checked to be injective. ◻ # Completely uncoupled qKM algebras In this section, we consider the structure of completely uncoupled qKM algebras $\mathfrak{g}(\mathcal{A})$, by which we mean that $\alpha$ and $\beta$ are uncoupled for any distinct $\alpha,\beta\in \Pi$. In particular, we will be interested in the possible Dynkin diagrams, and a classification of those with finite growth. By Theorem [\[theorem_coupling\]](#theorem_coupling){reference-type="ref" reference="theorem_coupling"}, we may assume all simple roots are of $Tak(\mathfrak{s}\mathfrak{l}(2))$-type. One should view this as the more nontrivial setting, that is far away from Takiff superalgebras. We will see in fact that being completely uncoupled makes a qKM algebra very rigid. We begin with a lemma stating some of the properties we will use of completely uncoupled algebras. [\[lemma_uncoupled_facts\]]{#lemma_uncoupled_facts label="lemma_uncoupled_facts"} Let $\mathcal{A}$ be a completely uncoupled qKM Cartan datum. Then: 1. any full Cartan subdatum (see Section [3.4](#section_subdatum){reference-type="ref" reference="section_subdatum"}) $\mathcal{A}'$ of a completely uncoupled qKM Cartan datum $\mathcal{A}$ is also completely uncoupled; 2. two distinct simple roots $\alpha_i$ and $\alpha_j$ are connected if and only if $[H_i,H_j]\neq0$; 3. we have, for all $i,j$, $y_{ij}y_{ji}\neq0$, and: $$y_{ij}^2\alpha_i(h_j)=y_{ji}^2\alpha_j(h_i).$$ *Proof.* Part (1) is clear, and (3) is an immediate application of Lemma [\[lemma_nontrivial_relation_on_ys\]](#lemma_nontrivial_relation_on_ys){reference-type="ref" reference="lemma_nontrivial_relation_on_ys"} and Corollary [\[corollary_ys\]](#corollary_ys){reference-type="ref" reference="corollary_ys"}. For part (2), the backward direction is obvious. For the forward direction, (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} tells us that $$[H_i,H_j]=x_{ij}c_j+y_{ij}h_j.$$ However we must have $y_{ij}\neq0$, and since $h_j$ cannot be a multiple of $c_j$, we must have $[H_i,H_j]\neq0$. ◻ [\[theorem-root-system\]]{#theorem-root-system label="theorem-root-system"} Let $\mathfrak{g}(\mathcal{A})$ be a completely uncoupled qKM algebra and let $A$ be its Cartan matrix. Let $\mathfrak{s} = \mathfrak{s}(A)$ be the (central quotient of the) Kac-Moody Lie algebra corresponding to $A$, with Cartan subalgebra $\mathfrak{t}$ [^1], set of simple roots $\Pi$ and simple coroots $h_1,...,h_n$. Then the root system $\Delta$ of $\mathfrak{g}(\mathcal{A})$ and the root system $\Delta_{\mathfrak{s}}$ of $\mathfrak{s}$ coincide (as subsets of $\mathfrak{t}^*$). Moreover, $\mathfrak{g}(\mathcal{A})$ is finite-dimensional if and only if $\Delta$ is finite. *Proof.* It is immediate that $\Delta_\mathfrak{s} \subseteq \Delta$. Suppose $\Delta_\mathfrak{s} \neq \Delta$; then the Chevalley automorphism on $\mathfrak{g}$ (see Remark [\[remark-Chevalley\]](#remark-Chevalley){reference-type="ref" reference="remark-Chevalley"}) implies that $\Delta^+ \setminus \Delta_{\mathfrak{s}} \neq \emptyset$. Let $\beta \in \Delta^+ \setminus \Delta_{\mathfrak{s}}$ be of minimal height. Then $\beta$ is not proportional to any simple root. Since $\beta$ is a root of $\mathfrak{g}(\mathcal{A})$, there exists $\alpha \in \Pi$ such that $\beta - \alpha \in \Delta^+$, so the minimality of the height of $\beta$ implies $\beta - \alpha \in \Delta_\mathfrak{s}^+$. Let $\mathfrak{s_{\alpha}} \subseteq \mathfrak{s}$ be the $\mathfrak{s}\mathfrak{l}(2)$ subalgebra corresponding to $\alpha$. From the integrability of $\mathfrak{g}(\mathcal{A})$ we obtain that $\bigoplus_{k\in\mathbb{Z}} \mathfrak{g}_{\beta + k \alpha}$ is a finite-dimensional $\mathfrak{s}_{\alpha}$-module. But then $r_{\alpha}(\beta)$ is a positive root of $\mathfrak{g(\mathcal{A})}$, of smaller height than $\beta$, so $r_{\alpha}(\beta) \in \Delta_{\mathfrak{s}}^+$. This implies that $\beta \in \Delta_{s}$, a contradiction. Hence $\Delta = \Delta_{\mathfrak{s}}$. If $\Delta$ is finite then because $\dim \mathfrak{g}_{\beta}<\infty$ for all $\beta \in \Delta_\mathfrak{g}$, we have $\mathfrak{g}$ is finite-dimensional. ◻ [\[lemma-Dynkin-D\]]{#lemma-Dynkin-D label="lemma-Dynkin-D"} There does not exist a completely uncoupled qKM algebra $\mathfrak{g}(\mathcal{A})$ with the following Dynkin diagram: . *Proof.* We write $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ for the simple roots and assume $\alpha_1$ corresponds to the middle vertex of the Dynkin diagram. Lemma [\[lemma_uncoupled_facts\]](#lemma_uncoupled_facts){reference-type="ref" reference="lemma_uncoupled_facts"} implies that $[H_1,H_2] \neq 0$. As $[H_1,H_2] \in \mathfrak{h}_{\alpha_1}\cap\mathfrak{h}_{\alpha_2}$, we have $[H_1,H_2] \in \ker \alpha_4$. So $$\begin{aligned} \in \mathbb{C}\langle c_1,h_1\rangle\cap\ker\alpha_4. \end{aligned}$$ A symmetric argument implies $$\begin{aligned} \in\mathbb{C}\langle c_1,h_1 \rangle \cap \ker \alpha_4. \end{aligned}$$ However, since $\alpha_4(h_1)\neq0$, the above subspaces are all one-dimensional and thus equal. Hence: $$\begin{aligned} \mathbb{C}\langle[H_1,H_2]\rangle = \mathbb{C}\langle c_1,h_1 \rangle \cap \ker \alpha_4 = \mathbb{C}\langle[H_1,H_3]\rangle. \end{aligned}$$ Therefore $\mathbb{C}\langle [H_1,H_2] \rangle\in(\mathfrak{h}_{\alpha_3})_{\overline{0}}\subseteq\ker\alpha_2$. Thus by Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} we have $0=\alpha_2([H_1,H_2])=y_{12}\alpha_2(h_2)$, i.e. $y_{12}=0$, a contradiction of being uncoupled. ◻ [\[lemma-A3-diagrams\]]{#lemma-A3-diagrams label="lemma-A3-diagrams"} Suppose $\mathfrak{g}(\mathcal{A})$ is a completely uncoupled qKM algebra with the following Dynkin diagram: Then $\mathfrak{g}(\mathcal{A})$ is necessarily of type $A_3$, that is, its Cartan matrix is $A = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}$. *Proof.* Using the notation of Section [7.3](#section-notation-qKM){reference-type="ref" reference="section-notation-qKM"}, we have, by the proof of Proposition [\[propistion-two-sq2\]](#propistion-two-sq2){reference-type="ref" reference="propistion-two-sq2"}, $$\begin{aligned} &[H_i,H_j] = x_{ij}c_j + y_{ij}h_j = x_{ji}c_i + y_{ji}h_i, \label{equation-commutator-odd-coroots-general}\\ &y_{ij}^2\alpha_i(h_j) = y_{ji}^2 \alpha_j(h_i) \label{equation-y-dependence-general} \end{aligned}$$ As $\mathfrak{g}(\mathcal{A})$ is completely uncoupled, we have $y_{12}y_{21}y_{23}y_{32} \neq 0$. We also have $$\begin{aligned} \alpha_2([H_1,H_3]) e_2 = [[H_1,H_3],e_2] = (x_{32}y_{12} + x_{12}y_{32})e_2. \end{aligned}$$ By Lemma [\[lemma-qKM-notation\]](#lemma-qKM-notation){reference-type="ref" reference="lemma-qKM-notation"} we have $[H_1,H_3] = 0$, so $$\begin{aligned} \label{equation-non-touching-eval} 0=\alpha_2([H_1,H_3]) = x_{32}y_{12} + x_{12}y_{32}. \end{aligned}$$ For $i=2$, $j=3$, equation [\[equation-commutator-odd-coroots-general\]](#equation-commutator-odd-coroots-general){reference-type="eqref" reference="equation-commutator-odd-coroots-general"} evaluated on $\alpha_1$ gives $$\begin{aligned} \label{equation-touching-eval} 0 = x_{23}\alpha_1(c_3) + y_{23}\alpha_1(h_3) =x_{32}\alpha_1(c_2) + y_{32}\alpha_1(h_2). \end{aligned}$$ As $y_{32} \neq 0$, the linear system [\[equation-non-touching-eval\]](#equation-non-touching-eval){reference-type="eqref" reference="equation-non-touching-eval"} and [\[equation-touching-eval\]](#equation-touching-eval){reference-type="eqref" reference="equation-touching-eval"} in the variables $x_{32}$ and $y_{32}$ admits a non-trivial solution. Therefore $$\begin{aligned} \label{equation-determinant-relation} 0 = \begin{vmatrix} \alpha_1(c_2) & \alpha_1(h_2) \\ y_{12} & x_{12} \end{vmatrix} = x_{12}\alpha_1(c_2) - y_{12}\alpha_1(h_2). \end{aligned}$$ Using [\[equation-commutator-odd-coroots-general\]](#equation-commutator-odd-coroots-general){reference-type="eqref" reference="equation-commutator-odd-coroots-general"} again for $i=1$, $j=2$ and evaluating on $\alpha_1$ we obtain $$\begin{aligned} x_{12}\alpha_1(c_2) + y_{12}\alpha_1(h_2) = x_{21}\alpha_1(c_1) + y_{21}\alpha_1(h_1) = 2y_{21}. \end{aligned}$$ Together with [\[equation-determinant-relation\]](#equation-determinant-relation){reference-type="eqref" reference="equation-determinant-relation"}, we deduce $y_{12}\alpha_1(h_2) = y_{21}$. Finally, [\[equation-y-dependence-general\]](#equation-y-dependence-general){reference-type="eqref" reference="equation-y-dependence-general"} implies $$\begin{aligned} \alpha_1(h_2) (y_{21}^2\alpha_2(h_1)) = \alpha_1(h_2)(y_{12}^2\alpha_1(h_2)) = (y_{12}\alpha_1(h_2))^2 = y_{21}^2, \end{aligned}$$ hence $\alpha_1(h_2)\alpha_2(h_1) = 1$. A symmetric argument gives $\alpha_3(h_2)\alpha_2(h_3) = 1$, and our claim follows. ◻ Using Remark [\[remark_dynkin_diagram\]](#remark_dynkin_diagram){reference-type="ref" reference="remark_dynkin_diagram"}, we obtain the following as an immediate corollary to Lemmas [\[lemma-Dynkin-D\]](#lemma-Dynkin-D){reference-type="ref" reference="lemma-Dynkin-D"} and [\[lemma-A3-diagrams\]](#lemma-A3-diagrams){reference-type="ref" reference="lemma-A3-diagrams"}, and Proposition [\[propistion-two-sq2\]](#propistion-two-sq2){reference-type="ref" reference="propistion-two-sq2"}: [\[corollary-fg-cc-qKM\]]{#corollary-fg-cc-qKM label="corollary-fg-cc-qKM"} Let $\mathfrak{g}(\mathcal{A})$ be a completely uncoupled indecomposable qKM algebra (with at least two simple roots). If $\mathfrak{g}(\mathcal{A})$ is of finite growth, then its Dynkin diagram is one of the following types: 1. Type $A(n)$ for $n \in \mathbb{Z}_{\geq 2}$, which corresponds to with $n$ vertices. 2. Type $A(n)^{(1)}$ for $n \in \mathbb{Z}_{\geq 2}$, which corresponds to with $n+1$ vertices. 3. Type $A(1)^{(1)}$, which corresponds to In the next section we will construct finite growth qKM algebras associated to each of the above diagrams, and see they are of finite growth. ## Question on general Dynkin diagrams [\[lemma_triangle\]]{#lemma_triangle label="lemma_triangle"} Suppose that $\mathfrak{g}(\mathcal{A})$ is a completely uncoupled qKM algebra with Dynkin diagram: Then if $a_{12}a_{21}\neq1$, $a_{23}a_{32}a_{31}a_{13}\neq1$, then $a_{12}a_{23}a_{31}=a_{21}a_{32}a_{13}$. The upshot of Lemma [\[lemma_triangle\]](#lemma_triangle){reference-type="ref" reference="lemma_triangle"} is that a Dynkin diagram with underlying graph of a triangle generically cannot correspond to a completely uncoupled qKM algebra. This section in general shows that completely uncoupled qKM algebras are very rigid. We do not prove this lemma here since we do not use it, but we state it in order to motivate the following question. **Question** Do the Dynkin diagrams listed in Corollary [\[corollary-fg-cc-qKM\]](#corollary-fg-cc-qKM){reference-type="ref" reference="corollary-fg-cc-qKM"}, along with $\begin{tikzpicture} \node[diamond,aspect=1,scale=0.6,draw] (A) at (-0.75,0) {}; \node[diamond,aspect=1,scale=0.6,draw] (B) at (0.75,0) {}; \draw (A) edge node[above,font=\tiny] {\(-m,-n\)} (B) ; \end{tikzpicture}$ with $m,n\in\mathbb{Z}_{\geq2}$, exhaust all connected Dynkin diagrams of completely uncoupled qKM algebras? # Finite growth, completely uncoupled qKM algebras In this section we give explicit presentations for the superalgebras with the Dynkin diagrams of Corollary [\[corollary-fg-cc-qKM\]](#corollary-fg-cc-qKM){reference-type="ref" reference="corollary-fg-cc-qKM"}, i.e. the finite growth Dynkin diagrams. ## Type $A(n)$ {#subsection-An} The proof for the following theorem can be found in [@Si]. There, an explicit list of possibilities for $\mathfrak{g}(\mathcal{A})$ is given, under the assumption that $$\bigcap_{\alpha \in \Pi} \mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha} \subseteq \sum_{\alpha\in \Pi} \mathfrak{h}_{\alpha}.$$ [\[theorem-finite-dimensional-QKM\]]{#theorem-finite-dimensional-QKM label="theorem-finite-dimensional-QKM"} Let $\mathfrak{g}(\mathcal{A})$ be an uncoupled qKM algebra Dynkin diagram of type $A(n)$ for $n \in \mathbb{Z}_{\geq 2}$. Then the derived subalgebra of $\mathfrak{g}(\mathcal{A})$ is isomorphic to a trivial central extension of either $\mathfrak{s}\mathfrak{q}(n)$ or $\mathfrak{p}\mathfrak{s}\mathfrak{q}(n)$. ## Type ${A(n)}^{(1)}$ Let $\mathfrak{g}(\mathcal{A})$ be a completely coupled qKM algebra with Dynkin diagram ${A(n)}^{(1)}$. We aim to show that $\mathfrak{g}(\mathcal{A})$ is an affinization of a qKM algebra with a Dynkin diagram of type $A(n)$. By an odd affinization we shall mean the following: Let $\mathfrak{g}$ be a Lie superalgebra that admits an odd, invariant, super-symmetric bilinear form $(-,-)$. Then the Lie superalgebra $$\begin{aligned} \hat{\mathfrak{g}} \coloneq \mathfrak{g}\otimes \mathbb{C}[t,t^{-1}] \oplus \mathbb{C}\langle t\partial_t, K \rangle \end{aligned}$$ with $t$ even, $K$ odd and central, and bracket given by $$\begin{aligned} = [x,y]\otimes t^{m+n} + m \delta_{m+n,0} (x,y) K, \end{aligned}$$ is called the odd affinization of $\mathfrak{g}$ with respect to the bilinear form $(-,-)$. Now let $\mathcal{A}$ be a completely uncoupled Cartan datum with Dynkin diagram of type $A(n)^{(1)}$. Denote the simple roots of $\mathfrak{g}(\mathcal{A})$ by $\alpha_0,...,\alpha_n$; we will consider all indices mod $n+1$. A direct computation (see [@Si]) shows that we can rescale $E_i$ such that $y_{i+1,i} = - y_{i,i+1} = 1$ and $x_{i+1,i} = x_{i,i+1} = -1$. In particular $x_{ij} = a_{ij}$ for any $i,j$. Moreover, it is shown there that $\sum_{i=0}^n h_i = 0$. We notice that $\sum_{i=0}^n H_i \in \bigcap_{\alpha\in\Pi}\mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$, and that any other linear combination of $H_0,\dots,H_n$ that lies in $\bigcap_{\alpha\in\Pi}\mathrm{Ann}_{\mathfrak{h}}\mathfrak{g}_{\alpha}$ must be proportional to $\sum_{i=0}^n H_i$. We denote $K \coloneq \sum_{i=0}^n H_i$ and notice that $[K,K] = 0$. Part (1) of Lemma [\[lemma_qKM_formulas\]](#lemma_qKM_formulas){reference-type="ref" reference="lemma_qKM_formulas"} and the above values of $x_{ij}$ and $y_{ij}$ imply: $$\begin{aligned} &-c_0 - h_0 = -c_n + h_n, \\ &-c_0 + h_0 = -c_1 - h_1, \end{aligned}$$ so that $$\begin{aligned} \label{equation-psqn-realtions} 2c_0 = c_1 + h_1 + c_n - h_n. \end{aligned}$$ Finally, let $\alpha_0,...,\alpha_n,\beta_1,...,\beta_m$ be a basis for $\mathfrak{h}_{\overline{0}}^*$, we let $\alpha_0^*,...\alpha_n^*,\beta_1^*,...\beta_m^*$ be the dual basis for $\mathfrak{h}_{\overline{0}}$. Let $\mathfrak{g}(\mathcal{A})$ be a qKM algebra with Dynkin diagram of type ${A(n)}^{(1)}$ for $n \in \mathbb{Z}_{\geq 2}$. Then $\hat{\mathfrak{psq}_n}$ can be identified with a subquotient of $\mathfrak{g}(\mathcal{A})$, where $\hat{\mathfrak{psq}}_n$ is the odd affinization of $\mathfrak{psq}_n$ with respect to its non-degenerate form. Concretely, there exists a qKM algebra $\mathfrak{g}(\mathcal{A}')$ that can be identified with an ideal of $\mathfrak{g}(\mathcal{A})$ such that $\mathfrak{g}(\mathcal{A}) = \mathfrak{g}(\mathcal{A}') + \mathfrak{h}_{\overline{1}}$, and $\hat{\mathfrak{psq}_n} \simeq \mathfrak{g}(\mathcal{A}')/\mathfrak{c}'_{\overline{0}}$, where $\mathfrak{c}'_{\overline{0}}$ is the even part of the center of $\mathfrak{g}(\mathcal{A}')$. *Proof.* We use the notation of [@G] for a basis of $\mathfrak{psq}_n$. Let $\mathfrak{h}'$ be a subalgebra of $\mathfrak{h}$ given by $\mathfrak{t}' = \mathfrak{t}$ and $\mathfrak{h}' = \mathbb{C}\langle K,H_1,...,H_n \rangle$ (as in the notation above). Let $\mathcal{A}'$ be the Cartan datum obtain from $\mathcal{A}$ by replacing $\mathfrak{h}$ by $\mathfrak{h}'$ (and restricting the $\mathfrak{h}$-modules $\mathfrak{g}_{\alpha}$ to $\mathfrak{h}'$). Clearly $\mathfrak{g}(\mathcal{A}')$ is a qKM algebra and can be embedded naturally as an ideal of $\mathfrak{g}(\mathcal{A})$ satisfying $\mathfrak{g}(\mathcal{A}) = \mathfrak{g}(\mathcal{A}')+\mathfrak{h}_{\overline{1}}$. Now define a map from $\tilde{\mathfrak{g}}(\mathcal{A}')$ onto $\hat{\mathfrak{psq}}_n$ as follows: $$\begin{aligned} &e_i \mapsto X_{E_{i,i+1}, 0}\otimes 1 \text{ for } 1\leq i \leq n, \\ &E_i \mapsto X_{0, E_{i,i+1}}\otimes 1 \text{ for } 1\leq i \leq n, \\ &f_i \mapsto X_{E_{i+1,i}, 0}\otimes 1 \text{ for } 1\leq i \leq n, \\ &F_i \mapsto X_{0, E_{i+1,i}}\otimes 1 \text{ for } 1\leq i \leq n, \\ &e_0 \mapsto X_{E_{n,0},0}\otimes t, \\ &E_0 \mapsto X_{0,E_{n,0}}\otimes t, \\ &f_0 \mapsto X_{E_{0,n},0}\otimes t^{-1}, \\ &F_0 \mapsto X_{0,E_{0,n}}\otimes t^{-1}, \\ &\alpha_0^* \mapsto t\delta_t, \\ &h_i \mapsto X_{E_{i,i}-E_{i+1,i+1},0}\otimes 1 \text{ for } 1 \leq i \leq n, \\ &\beta_i^* \mapsto 0, \\ &K \mapsto K, \\ &H_i \mapsto X_{0,E_{i,i}-E_{i+1,i+1}}\otimes 1 \text{ for } 1 \leq i \leq n. \end{aligned}$$ It is straightforward yet tedious to verify that the above assignment determines a well defined homomorphism of Lie superalgebras, which surjects onto $\hat{\mathfrak{psq}}_n$. As $\hat{\mathfrak{psq}}_n$ has no non-trivial ideals intersecting $\mathfrak{h}$ trivially, this homomorphism factors through a map $\mathfrak{g}(\mathcal{A})\rightarrow \hat{\mathfrak{psq}}_n$. It is clear that the kernel of this lies entirely in the even part of the center. ◻ It should be noted that the odd part of the Cartan subalgebra of $\mathfrak{g}(\mathcal{A})$ cannot be much more complicated than that of $\hat{\mathfrak{psq}}_n$. Indeed let $Y \in \mathfrak{h}_{\overline{1}}$. Because $x_{ij} = a_{ij}$ and the submatrix of $A$ obtained by removing the first row and the first column is non-degenerate, there is a unique linear combination $\sum_{i=1}^n t_i H_i$ such that $[Y-\sum_{i=1}^n t_i H_i,e_j] = 0$ for $1\leq j \leq n$. So we assume $[Y,e_j] = 0$ for all $1\leq j \leq n$. Let $r \in \mathbb{C}$ be the scalar satisfying $[Y,E_1] = re_1$. Then $[Y,H_1] = rh_1$, so $$\begin{aligned} ra_{12}e_2 = [rh_1, e_2] = [[Y,H_1],e_2] = x_{12}[Y,E_2] = a_{12}[Y,E_2], \end{aligned}$$ implying $[Y,E_2]=re_2$. Repeating this argument, we obtain $[Y,E_j] = re_j$ for all $1\leq j \leq n$. For $j=0$, we obtain the following: $$\begin{aligned} -re_0 &= ra_{10}e_0 = [rh_1, e_0] = [[H_1,Y], e_0] = [H_1,[Y,e_0]] + a_{10}[Y,E_0] = [H_1,[Y,e_0]] - [Y,E_0],\\ -re_0 & = ra_{n0}e_0 = [rh_n, e_0] = [[H_n,Y], e_0] = [H_n,[Y,e_0]] + a_{n0}[Y,E_0] = [H_n,[Y,e_0]] - [Y,E_0]. \end{aligned}$$ These equations imply that $[H_1,[Y,e_0]]=[H_n,[Y,e_0]]$; however since $y_{n,0}=-y_{1,0}$, this implies that $[Y,e_0]=0$. Thus we obtain further that $[Y,E_0] = re_0$. It is immediate that $[Y,Y]$ is central. ## Type ${A(1)}^{(1)}$ {#section_q_22} Let $s \in \{\pm 1\}$ and $x_{12},x_{21},y_{12},y_{21} \in \mathbb{C}$ satisfy $\frac{y_{12}}{y_{21}} = s\sqrt{\frac{a_{12}}{a_{21}}}$ and $x_{12}x_{21} = 2 + s\sqrt{a_{12}a_{21}}=2(1+s)$. We assume without loss of generality that $x_{12}\neq 0$ and define a datum $\mathcal{A}^{s}_{(2,2)}$. Let $\mathfrak{h}$ be a $(3|2)$-dimensional quasi-toral superalgebra, defined as follows. Let $\{h_1,h_2,h_3\}$ be a basis for $\mathfrak{t}$ and $\{H_1,H_2\}$ a basis for $\mathfrak{h}_{\overline{1}}$. Let $\alpha_1,\alpha_2,\alpha_3$ be a basis of $\mathfrak{t}^*$, defined by $$\begin{aligned} \alpha_j(h_i) = \begin{pmatrix} 2 & -2 & 0 \\ -2 & 2 & 1 \\ 0 & 1 & 0 \end{pmatrix}. \end{aligned}$$ We also write $\delta = \alpha_1+\alpha_2$. We let $c_1 = x_{12}y_{12} h_3$ and $c_2 = \frac{1}{x_{12}}(x_{21}c_1+y_{21}h_1 - y_{12}h_2)$. We define the relations in $\mathfrak{h}$ by $$\begin{aligned} &[H_i,H_i] = 2c_i, \\ &[H_1,H_2] = x_{12}c_2+ y_{12}h_2 = x_{21}c_1 + y_{21}h_1. \end{aligned}$$ We set $\Pi = \{\alpha_1,\alpha_2\}$. It is clear that $\alpha_1$ and $\alpha_2$ are linearly independent in $\mathfrak{t}^*$. We let $\mathfrak{g}_{\alpha_i}$ be an irreducible $\mathfrak{h}$-module with $e_i$ and $E_i$ even and odd basis vectors, the action of $\mathfrak{h}$ given by $$\begin{aligned} & H_i\cdot(e_i+E_i) = 2E_i, \\ & H_j\cdot (e_i+E_i) = x_{ji}E_i + y_{ji}e_i. \end{aligned}$$ Our definition of $\mathfrak{h}$ ensures this is a well defined action. Now we let $\mathfrak{g}_{-\alpha_i} \coloneq \mathfrak{g}_{\alpha_i}^{\vee}$ and let $f_i \coloneq e_i^\vee$ and $F_i \coloneq \sqrt{-1} \cdot E_i^\vee$ be a basis for it. Finally, we let $[-,-]_{\alpha_i}:\mathfrak{g}_{\alpha_i}\otimes \mathfrak{g}_{-\alpha_i} \rightarrow \mathfrak{h}$ be the $\mathfrak{h}$-module homomorphism defined by $[e_i,F_i]_{\alpha_i} = H_i$. All this information defines a Cartan datum $\mathcal{A}_{(2,2)}^s$. It then follows that $\mathfrak{g}(\mathcal{A}_{(2,2)}^s)$ is a qKM algebra. We write $\mathfrak{q}^+_{(2,2)} \coloneq \mathfrak{g}(\mathcal{A}_{(2,2)}^{1})$ and $\mathfrak{q}^-_{(2,2)} \coloneq \mathfrak{g}(\mathcal{A}_{(2,2)}^{-1})$. $\mathfrak{q}^\pm_{(2,2)}$ are of finite growth, their real root spaces are of dimension $(1|1)$ and their imaginary root space is of dimension $(2|2)$. *Proof.* Let $\mathfrak{g}$ denote either $\mathfrak{q}^+_{(2,2)}$ or $\mathfrak{q}^-_{(2,2)}$. Let $\mathfrak{s}$ be the Kac-Moody Lie algebra corresponding to $\mathfrak{g}$ as in Theorem [\[theorem-root-system\]](#theorem-root-system){reference-type="ref" reference="theorem-root-system"}. Then $\mathfrak{s} \simeq \mathfrak{s}\mathfrak{l}(2)^{(1)}$ and is of finite growth. We can associate to $\mathfrak{g}$ the same Weyl group $W$ as of $\mathfrak{s}$, and every reflection of $W$ can be lifted to an automorphism of $\mathfrak{g}$. As $\dim \mathfrak{g}_{\alpha} = (1|1)$ for any $\alpha \in \Pi$, we have $\dim \mathfrak{g}_{\beta} = (1|1)$ for any real root $\beta\in\Delta$. If we write $\delta = \alpha_1+\alpha_2$ then the last result can be stated as $\dim \mathfrak{g}_{\alpha+k\delta} = (1|1)$ for any $\alpha \in \Pi$, $k \in \mathbb{Z}$. Finally, Proposition [\[propistion-two-sq2\]](#propistion-two-sq2){reference-type="ref" reference="propistion-two-sq2"} implies, without loss of generality, that $\alpha_2(c_1) \neq 0$. We consider the $\mathfrak{s}\mathfrak{q}(2)$ subalgebra of $\mathfrak{g}$ corresponding to $\alpha_1$. The representation theory of $\mathfrak{s}\mathfrak{q}(2)$ implies that $\dim \mathfrak{g}_{k\delta} \geq (2|2)$, for any $k \in \mathbb{Z}\setminus\{0\}$. We notice that $\delta(c_1) \neq 0$. If $\mathfrak{g}_{k\delta} > (2|2)$, then there exists $\theta \in \mathfrak{g}_{k\delta}$ such that $[\mathfrak{g}_{-\alpha_1},\theta] = 0$. This implies $0 = [c_1,\theta] = k\delta(c_1)\theta$, so $\delta(c_1) = 0$, a contradiction. We conclude that $\dim \mathfrak{g}_{k\delta}= (2|2)$ for any $k \in \mathbb{Z}\setminus\{0\}$. This concludes the proof. ◻ 99999999 S-J. Cheng and K. Coulembier. *Representation theory of a semisimple extension of the Takiff superalgebra*, International Mathematics Research Notices, Vol. 18 (2022): 14454-14495. S-J. Cheng and W. Wang. *Dualities and representations of Lie superalgebras*, American Mathematical Society (2012). M. Gorelik. *Shapovalov determinants of Q-type Lie superalgebras*, International Mathematical Research Papers (2006). M. Gorelik, V. Hinich, and V. Serganova. *Root groupoid and related superalgebras*, arXiv preprint 2209.06253. M. Gorelik, V. Serganova and A. Sherman. *On the Grothendieck ring of a quasireductive Lie superalgebra*, arXiv preprint 2206.07709. C. Hoyt. *Regular Kac--Moody superalgebras and integrable highest weight modules*, Journal of Algebra, Vol. 324, no. 12 (2010): 3308-3354. C. Hoyt and V. Serganova. *Classification of finite-growth general Kac--Moody superalgebras*, Communications in Algebra, Vol. 35, no. 3 (2007): 851-874. V.G. Kac. *Infinite-dimensional Lie algebras*, Cambridge University Press (1990). V.G. Kac. *Lie superalgebras*, Advances in Mathematics, Vol. 26 (1977): 8--96. V.G. Kac. *Simple irreducible graded Lie algebras of finite growth*, Math. USSR Izv., Vol. 2 (1968): 1271--1311. V.G. Kac. *Infinite-dimensional algebras, Dedekind's $\eta$-function, classical Möbius function and the very strange formula*, Advances in Mathematics, Vol. 30, no. 2 (1978): 85-136. V.G. Kac and J.W. Van de Leur. *On classification of superconformal algebras*, Rijksuniversiteit Utrecht. Mathematisch Instituut, 1988. V.G. Kac and M. Wakimoto. *Modular invariant representations of infinite-dimensional Lie algebras and superalgebras*, Proceedings of the National Academy of Sciences, Vol. 85, no. 14 (1988): 4956-4960. V.G. Kac and M. Wakimoto. *Integrable highest weight modules over affine superalgebras and number theory*, Lie Theory and Geometry: In Honor of Bertram Kostant, Birkhäuser Boston (1994): 415-456. V.G. Kac and M. Wakimoto. *Integrable highest weight modules over affine superalgebras and Appell's function*, Communications in Mathematical Physics, Vol. 215 (2001): 631-682. D. Leites and V. Serganova. *Defining relations for classical Lie superalgebras. I. superalgebras with Cartan matrix or Dynkin-type diagram* Proc. Topological and Geometrical Methods in Field Theory (1992): 194-201. J.W. Van De Leur. *A classification of contragredient Lie superalgebras of finite growth*, Communications in algebra, Vol. 17, no. 8 (1989): 1815-1841. R. Moody. *A new class of Lie algebras*, Journal of Algebra, Vol. 10, no. 2 (1968): 211-230. I.M. Musson. *Lie superalgebras and enveloping algebras*, American Mathematical Society, Vol. 131 (2012). V. Serganova. *Kac-Moody superalgebras and integrability*, Developments and Trends in Infinite-Dimensional Lie Theory, Birkhäuser Boston (2011): 169--218. V. Serganova. *Quasireductive supergroups*, New Developments in Lie Theory and Its Applications, American Mathematical Society, Vol. 544 (2011): 141--159. L. Silberberg. *A queer Kac-Moody construction*, Weizmann Institute of Science, Master's Thesis (2023). S. Takiff. *Rings of invariant polynomials for a class of Lie algebras*, Trans. Am. Math. Soc., Vol. 160 (1971): 249-262. [Alexander Sherman, Dept. of Mathematics, University of Sydney, Camperdown, Australia]{.smallcaps} *Email address:* `xandersherm@gmail.com` [Lior Silberberg, Dept. of Mathematics, Weizmann Institute of Science, Rehovot, Israel]{.smallcaps} *Email address:* `lior-david.silberberg@weizmann.ac.il ` [^1]: this is a Kac-Moody Lie algebra in the sense of [@GHS], with the condition on the dimension of the Cartan subalgebra $\mathfrak{t}$ more relaxed.

arxiv_math

--- abstract: | During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $31$ series identities involving harmonic numbers, three of which were previously conjectured by the second author. For example, we obtain that $$\sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}} \left( \frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2} \right) = \frac{\pi^4}{720}.$$ and $$\sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 \zeta(3)^2,$$ where $H_n^{(m)}$ denotes $\sum_{0<j\leqslant n}j^{-m}$. address: - School of Mathematics, Tianjin University, Tianjin 300072, People's Republic of China - Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China author: - Qing-Hu Hou - Zhi-Wei Sun title: Taylor coefficients and series involving harmonic numbers --- Preprint # Introduction For each $m\in\mathbb{Z}^+=\{1,2,3,\ldots\}$, the $m$th harmonic numbers are those rational numbers $$H_n^{(m)}=\sum_{0<k\le n}\frac 1{k^m}\ \ (n\in\mathbb{N}=\{0,1,2,\ldots\}).$$ The numbers $H_n=H_n^{(1)}\ (n\in\mathbb{N})$ are the usual harmonic numbers. Z.-W. Sun [@Sun22; @Sun23] formulated many conjectural series identities whose summands involve generalized harmonic numbers of small orders. Motivated by this, we confirmed some conjectures of Sun and evaluate more series involving harmonic numbers of small orders. Pilehroods [@PP12] used the WZ method to prove the following identity proposed by the second author in [@Sun11]: $$\label{27-a} \sum_{k=1}^\infty \frac{(-27)^{k-1}(15k-4)}{k^3 {2k \choose k}^2{3k \choose k}} = \sum_{k=1}^\infty \frac{\left( \frac{k}{3} \right)}{k^2},$$ where $\left(-\right)$ denotes the Legendre symbol. Motivated by their method, we confirm Conjecture 3.8(i) in [@Sun22]. Namely, we have the following theorem. **Theorem 1**. *We have the identity $$\label{eq-bi-27} \sum_{n=1}^\infty \frac{(-27)^{k}}{k^3 {2k \choose k}^2 {3k \choose k}} \left( (15k-4) (3 H_{3k-1} - H_{k-1}) - 9 \right) = -\frac{4\pi^3}{\sqrt3}.$$* Our next theorem confirms Conjectures 6.1(i) and 6.4(i) of Sun [@Sun23]. **Theorem 2**. *We have $$\label{eq-bi3} \sum_{k=1}^\infty \frac{(21k-8) H_{k-1}^{(2)} + 2/k}{k^3 {2k \choose k}^3} = \frac{\pi^4}{360}$$ and $$\label{eq-16bi3} \sum_{k=1}^\infty \frac{16^k}{k^3 {2k \choose k}^3} \left( (6k-2) H_{k-1}^{(2)} + \frac{1}{k} \right) = \frac{\pi^4}{24}.$$* Recently, Wei and Xu [@WX2308] confirmed the idenities $$\sum_{k=0}^\infty\frac{\binom{2k}k^5}{(-4096)^k}\left((20k^2+8k+1)H_k^{(3)}+\frac 8{2k+1}\right)=\frac{64\zeta(3)}{\pi^2}$$ and $$\sum_{k=0}^\infty\frac{\binom{2k}k^5}{(-2^{20})^k}\left((820k^2+180k+13)\left(9H_{2k}^{(3)}-H_k^{(3)}\right)+\frac{125}{2k+1}\right)=1024\frac{\zeta(3)}{\pi^2},$$ conjectured by Sun [@Sun23 Conjectures 6.20 and 6.22(i)]. Our third theorem gives similar results involving harmonic numbers of order $4$ conjectured by Sun [@Sun23 Conjectures 6.21(i) and the first identity of 6.23(i)]. **Theorem 3**. *We have $$\label{eq-bi5-12} \sum_{k=0}^\infty \frac{{2k \choose k}^5} {(-4096)^k} (20k^2+8k+1) (64 H_{2k}^{(4)} - 3H_k^{(4)}) = -\frac{56}{45} \pi^2$$ and $$\label{eq-bi5-20} \sum_{k=0}^\infty \frac{{2k \choose k}^5} {(-2^{20})^k} \left( (820k^2+180k+13) (49 H_{2k}^{(4)} - 3H_k^{(4)}) - \frac{195}{(2k+1)^2} \right) = -\frac{896}{45} \pi^2.$$* Chu and Zhang [@ChZh] presented several transformation formulas. By taking suitable linear combinations of the Taylor coefficients and using Au's package on multiple zeta functions [@Au], we are able to derive some identities involving harmonic numbers which are similar to some conjectures in Sections 4 and 5 of Sun [@Sun22]. [@ChZh Example 84] has the equivalent form: $$\label{Ex84} \sum_{k=1}^{\infty} \frac{16^k(6 k-1)}{(2 k-1) k^2{2k \choose k}{4k \choose 2k}} = 8G,$$ where $G=\sum_{k=0}^\infty(-1)^k/(2k+1)^2$ is the Catalan constant. In contrast, we obtain the following result. **Theorem 4**. *We have $$\label{eq-16-b2b4} \sum_{k=1}^{\infty} \frac{16^k(6 k-1)(4 H_{2 k-1}-H_{k-1})}{(2 k-1) k^2{2k \choose k}{4k \choose 2k}} =\pi^3.$$* [@ChZh Example 11] has the equivalent form: $$\label{Ex11} \sum_{k=1}^{\infty} \frac{30 k-11}{(2 k-1) k^3{2k \choose k}^2} = 4\zeta(3).$$ Wei [@Wei2303] deduced some variants of this identity involving harmonic numbers of order $1$ or $2$. In contrast, we obtain the following result. **Theorem 5**. *We have $$\label{eq-bi2} \sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 \zeta(3)^2.$$* [@ChZh Example 21] has the equivalent form: $$\label{Ex21} \sum_{k=1}^\infty\frac{(-1)^{k-1}(56k^2-32k+5)}{(2k-1)^2k^3\binom{2k}k\binom{3k}k}=4\zeta(3).$$ Motivated by this and Conjectures 4.5 and 4.6of Sun [@Sun23], we establish the following theorem. **Theorem 6**. *We have $$\label{eq-Chu21} \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^3 {2k \choose k} {3k \choose k}} \left( \frac{56k^2-32k+5}{(2k-1)^2} H_{2k-1} + \frac{1}{2k} \right) = \frac{\pi^4}{20}$$ and $$\label{eq-1.12} \sum_{k=1}^\infty \frac{(-1)^k \big( (56k^2-32k+5) (H_{3k-1}-\frac{46}{3} H_{2k-1} - H_{k-1}) -12k \big)}{(2k-1)^2 k^3 {2k \choose k} {3k \choose k}} =\frac{7}{10} \pi^4.$$* [@ChZh Example 24] has the equivalent form: $$\label{Ex24} \sum_{k=1}^\infty\frac{(-1)^k(7k-2)}{(2k-1)k^2\binom{2k}k\binom{3k}k}=-\frac{\pi^2}{12}.$$ Motivated by this and Conjecture 4.2 of Sun [@Sun23], we establish the following theorem. **Theorem 7**. *We have $$\label{eq-Chu24a} \sum_{k=1}^\infty \frac{(-1)^k}{k^2 {2k \choose k} {3k \choose k}} \left( \frac{7k-2}{2k-1} (2 H_{2k-1} - H_{k-1}) + \frac{1}{k} \right) = - \frac{3}{2} \zeta(3)$$ and $$\label{eq-Chu24b} \sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}}\left(\frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2}\right)=\frac{\pi^4}{720}.$$* [@ChZh Example 1] has the equivalent form: $$\label{Ex1} \sum_{k=1}^\infty \frac{(-16)^k (20k^2-8k+1) }{(2k-1)^2 k^3 {2k \choose k} {4k \choose 2k}} = - 14 \zeta(3).$$ Motivated by this and Conjecture 4.11 of Sun [@Sun23], we get the following result. **Theorem 8**. *We have $$\label{eq-1.13} \sum_{k=1}^\infty \frac{(-16)^k \big( (20k^2-8k+1) (2H_{4k-1}-9H_{2k-1}) -8k \big)}{(2k-1)^2 k^3 {2k \choose k} {4k \choose 2k}} = \pi^4.$$* [@ChZh Example 50] has the equivalent form: $$\label{Ex50} \sum_{k=1}^\infty\frac{256^k(22k-1)}{(2k-1)k^2\binom{3k}k\binom{6k}{3k}}=128G.$$ Motivated by this and Conjecture 4.15 of Sun [@Sun23], we obtain the following theorem. **Theorem 9**. *We have $$\label{eq-1.14} \sum_{k=1}^\infty \frac{256^k \big( (22k-1)\mathcal H(k)-50 \big)}{(2k-1) k^2 {3k \choose k} {6k \choose 3k}}= -16 \pi^3,$$ where $\mathcal H(k) = 10H_{6k-1}-5H_{3k-1}-16H_{2k-1}+5H_{k-1}$.* [@ChZh Example 41] has the equivalent form: $$\label{Ex41} \sum_{k=1}^\infty\frac{(-256)^k(86k^2-21k+2)}{(2k-1)^2k^3\binom{3k}k\binom{6k}{3k}}=-224\zeta(3).$$ Motivated by this, we establish the following theorem. **Theorem 10**. *We have $$\label{eq-4.33} \sum_{k=1}^\infty \frac{(-256)^k \big( (86k^2-21k+2) H_{2k-1} +25k(6k-1)/(2k-1) \big)}{(2k-1)^2 k^3 {3k \choose k} {6k \choose 3k}}= - 8 \pi^4$$ and $$\label{eq-4.34} \sum_{k=1}^\infty \frac{(-256)^k \big( (86k^2-21k+2) c_k + 2k(232k-45)/(2k-1) \big)}{(2k-1)^2 k^3 {3k \choose k} {6k \choose 3k}}= - 32 \pi^4,$$ where $c_k = 2H_{6k-1}-H_{3k-1}+H_{k-1}$. Also, $$\label{eq-4.35} \sum_{k=1}^\infty \frac{(-256)^k \big( (86k^2-21k+2) H_{2k-1}^{(2)} - 61k(6k-1)/(2k-1)^2 \big)}{(2k-1)^2 k^3 {3k \choose k} {6k \choose 3k}}= 992 \zeta(5)$$ and $$\begin{gathered} \label{eq-4.36} \sum_{k=1}^\infty \frac{(-256)^k \big( (86k^2-21k+2) (15 H_{2k-1}^{(3)} - 2 H_{k-1}^{(3)}) - 83k(6k-1)/(2k-1)^3 \big)}{(2k-1)^2 k^3 {3k \choose k} {6k \choose 3k}}\\ = -1568 \zeta(3)^2.\end{gathered}$$* [@ChZh Example 93] has the equivalent form: $$\label{Ex93} \sum_{k=1}^\infty\frac{(-1)^{k-1}(112k^3-8k^2-6k+1)\binom{2k}k^2}{(2k-1)^3k^2\binom{3k}k\binom{6k}{3k}}=\frac 23\pi^2.$$ Motivated by this and the spirit of Sun [@Sun23], we establish the following theorem. **Theorem 11**. *Let $P(k) = 112k^3-8k^2-6k+1$. Then $$\label{eq-1.11a} \sum_{k=1}^\infty \frac{(-1)^{k-1} {2k \choose k}^2 \big( P(k) H_{2k-1} + 20k^2(6k-1)/(2k-1) \big)}{(2k-1)^3 k^2 {3k \choose k} {6k \choose 3k}} = 11 \zeta(3)$$ and $$\label{eq-1.11b} \sum_{k=1}^\infty \frac{(-1)^{k-1} {2k \choose k}^2 \big( P(k) (2 H_{6k-1}-H_{3k-1}+H_{k-1}) + 8k^2 \frac{86k-11}{2k-1} \big)}{(2k-1)^3 k^2 {3k \choose k} {6k \choose 3k}} = 50 \zeta(3).$$* We are unable to prove the second author's following conjectural identities: $$\begin{aligned} \sum_{k=1}^\infty\frac{(-1)^k\binom{2k}k^2(P(k)(6H_{2k-1}^{(2)}-H_{k-1}^{(2)})-172k^2(6k-1)/(2k-1)^2)} {(2k-1)^3k^2\binom{3k}k\binom{6k}{3k}} =\frac{11}{60}\pi^4 \end{aligned}$$ and $$\begin{aligned}&\sum_{k=1}^\infty\frac{(-1)^k\binom{2k}k^2(P(k)(7H_{2k-1}^{(3)}-H_{k-1}^{(3)})-136k^2(6k-1)/(2k-1)^3)} {(2k-1)^3k^2\binom{3k}k\binom{6k}{3k}} \\&\qquad=\frac{165}{4}\zeta(5)-\frac{11}3\pi^2\zeta(3). \end{aligned}$$ [@ChZh Example 25] has the equivalent form: $$\label{Ex25} \sum_{k=1}^\infty\frac{(-16)^{k}(112k^3-116k^2+35k-3)}{k^2(2k-1)(4k-1)(4k-3) \binom{3k}k\binom{6k}{3k}}=- \frac{\pi^2}{4}.$$ Motivated by this and the spirit of Sun [@Sun23], we establish the following theorem. **Theorem 12**. *Let $P(k) = 112k^3-116k^2+35k-3$. Then $$\label{eq-1.12a} \sum_{k=1}^\infty \frac{(-16)^{k} \big( P(k) H_{2k-1} -\frac{ (4k-1)(6k-1)(8k-3) }{4(2k-1)} \big)}{k^2(2k-1)(4k-1)(4k-3) \binom{3k}k\binom{6k}{3k}} = \frac{\pi^2}{2} \log 2 - \frac{7}{2} \zeta(3)$$ and $$\begin{gathered} \label{eq-1.12b} \sum_{k=1}^\infty \frac{(-16)^{k} \big( P(k) (2H_{6k-1}-H_{3k-1}-3H_{k-1}) -64k^2+46k -17/2 \big)}{k^2(2k-1)(4k-1)(4k-3) \binom{3k}k\binom{6k}{3k}} \\ = 7 \zeta(3) -2\pi^2 \log 2. \end{gathered}$$* We are unable to prove the second author's following conjectural identity: $$\sum_{k=1}^\infty \frac{(-16)^{k} \big( P(k) (H_{2k-1}-2H_{k-1}) - \frac{(4k-1)(64k^2-2k-3)}{8k} \big)}{k^2(2k-1)(4k-1)(4k-3) \binom{3k}k\binom{6k}{3k}} \\ = \frac{21}{4} \zeta(3) - \pi^2 \log 2.$$ [@ChZh Example 72] has the equivalent form: $$\label{Ex72} \sum_{k=1}^\infty\frac{(3k-1)4^{k}}{(2k-1)k^3 {2k \choose k}^2}= \frac{7}{4} \zeta(3).$$ Motivated by this and conjecture 4.1(i) of [@Sun23], we establish the following theorem. **Theorem 13**. *We have $$\label{eq-1.13a} \sum_{k=1}^\infty \frac{4^k}{(2k-1)k^3{2k \choose k}^2} \left( (3k-1)(2H_{2k-1}-3H_{k-1}) + \frac{2k}{2k-1}\right) = \frac{\pi^4}{16},$$ $$\label{eq-1.13b} \sum_{k=1}^\infty \frac{(3k-1)4^k}{(2k-1)k^3{2k \choose k}^2} \left( 4H^{(2)}_{2k-1} - 5H_{k-1}^{(2)} \right) = \frac{31}{4} \zeta(5),$$ and $$\label{eq-1.13c} \sum_{k=1}^\infty \frac{(3k-1)4^k}{(2k-1)k^3{2k \choose k}^2} \left(8H^{(3)}_{2k-1} +7H_{k-1}^{(3)} \right) = \frac{49}{4} \zeta(3)^2.$$* [@ChZh Example 47] has the equivalent form: $$\label{Ex47} \sum_{k=1}^\infty\frac{(22k^2-17k+3) 16^k {4k \choose 2k}}{k (4k-1)(4k-3) {3k \choose k}{6k \choose 3k}}= 2 \pi.$$ Motivated by this and the spirit of Sun [@Sun23], we establish the following theorem. **Theorem 14**. *Let $Q(k) = (2k-1)(11k-3) = 22k^2-17k+3$. We have $$\label{eq-1.14a} \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} + (6k-1)(4k-3)/(2k-1) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} = 16G - 4 \pi \log 2,$$ $$\label{eq-1.14b} \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{2k-1} + (6k-1)(k-3)/(6k-3) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} = \frac{4}{3} (8G - \pi \log 2),$$ and $$\begin{gathered} \label{eq-1.14c} \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) (2H_{6k-1}-H_{3k-1})+ 4(18k^2-20k+3)/(6k-3) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ = \frac{4}{3} (16G + \pi \log 2). \end{gathered}$$* We are unable to prove the second author's following conjectural identity: $$\label{eq-1.14d} \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) (H_{2k-1}^{(2)}- \frac{3}{16} H_{k-1}^{(2)}) - 3(6k-1)/(4k-2) \big)}{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} = \frac{\pi^3}{12}.$$ [@ChZh Example 14] has the equivalent form: $$\label{Ex14} \sum_{k=1}^\infty\frac{(60k^2-43k+8) {4k \choose 2k}} {k^3(4k-1) {2k \choose k}^4}= \frac{\pi}{3}.$$ Motivated by this and the spirit of Sun [@Sun23], we establish the following theorem. **Theorem 15**. *Let $Q(k) = 60k^2-43k+8$. We have $$\label{eq-1.15a} \sum_{k=1}^\infty \frac{{4k \choose 2k} \big( Q(k) (3H_{2k-1}-2H_{k-1}) -34k+25/2 \big) }{(4k-1)k^3{2k \choose k}^4} = 6 \zeta(3),$$ $$\label{eq-1.15b} \sum_{k=1}^\infty \frac{{4k \choose 2k} \big( Q(k) H_{4k-1} -(304k^2-216k+39)/(4(4k-1)) \big) }{(4k-1)k^3{2k \choose k}^4} = 4 \zeta(3),$$ $$\label{eq-1.15c} \sum_{k=1}^\infty \frac{{4k \choose 2k} \big( Q(k) H_{k-1}^{(2)} +(44k-15)/(4k) \big) }{(4k-1)k^3{2k \choose k}^4} = \frac{\pi^4}{90},$$ and $$\label{eq-1.15d} \sum_{k=1}^\infty \frac{{4k \choose 2k} \big( Q(k) H_{2k-1}^{(2)} +(52k-21)/(4k) \big) }{(4k-1)k^3{2k \choose k}^4} = \frac{2\pi^4}{45}.$$* We are unable to prove the second author's following conjectural identities: $$\label{eq-1.15e} \sum_{k=1}^\infty \frac{{4k \choose 2k} \big( Q(k) H_{k-1}^{(3)} -(184k^2-114k+19)/(8k^2(2k-1)) \big) }{(4k-1)k^3{2k \choose k}^4} = \frac{\pi^2}{3} \zeta(3) - 5 \zeta(5),$$ $$\label{eq-1.15f} \sum_{k=1}^\infty \frac{{4k \choose 2k} \big( Q(k) H_{2k-1}^{(3)} -(584k^2-390k+65)/(8k^2(2k-1)) \big) }{(4k-1)k^3{2k \choose k}^4} =30\zeta(5)- 2\pi^2 \zeta(3),$$ and $$\label{eq-1.15g} \sum_{k=1}^\infty \frac{{4k \choose 2k} \big( Q(k) (2H_{2k-1}^{(4)} + H_{k-1}^{(4)}) - 25(252k-631)/(504k^2) \big) }{(4k-1)k^3{2k \choose k}^4} = \frac{11\pi^6}{1890}.$$ Our method to prove the identities in our theorems is to extract the multivariate Taylor coefficients of a hypergeometric identity. We will utilize two types of identities. In Section 2, we focus on identities constructed through the WZ method. Recall that a *WZ pair* $(F(n,k), G(n,k))$ refers to a pair of hypergeometric functions that satisfy $$F(n+1,k) - F(n,k) = F(n,k+1) - F(n,k), \quad \forall\, n,k \in \mathbb{N}.$$ It has been demonstrated in [@MZ96] that $$\label{WZ-o} \sum_{k=0}^\infty F(0,k) - \lim_{n \to \infty} \sum_{k=0}^n F(n,k) = \sum_{n=0}^\infty G(n,0) - \lim_{k \to \infty} \sum_{n=0}^k G(n,k).$$ In most cases, both limits vanish and $F(n,k)$ and $G(n,k)$ involve additional parameters. By comparing the Taylor expansions of these parameters, we obtain identities that involve harmonic numbers. Following this approach, Au [@Au] confirmed several conjectures made by the second author. We will prove Theorems [Theorem 1](#bi-27){reference-type="ref" reference="bi-27"} and [Theorem 2](#bi3){reference-type="ref" reference="bi3"} in this way. It is worth noting that if $(F(n,k), G(n,k))$ is a WZ pair, then so is $(F(n+a,k+b), G(n+a,k+b))$. Consequently, any WZ pair can be generalized by introducing two extra parameters $a$ and $b$. In Section 3, we focus on identities established from summation formulas or transformation formulas of hypergeometric series. By applying differential operations, one can derive identities that involve harmonic numbers. This method has been explored by Chu and his coauthor [@Chu97; @CL23] and Wei [@Wei2303; @Wei2306]. We emphasize that symbolic computation plays an important role in our work. We use the `Maple` command `coeftayl` to extract the Taylor coefficients, the `Maple` package `APCI` (which is available at the first author's homepage) to compute the WZ pair $(F(n,k), G(n,k))$ once $F(n,k)$ is given, and the `Mathematica` package `MultipleZetaValue` developed by Au [@Au] to evaluate series involving rational functions and harmonic numbers. We also search for proper linear combinations of the coefficients by solving linear equations. Let us first introduce some notations. We use the standard notation of raising factorial $$(a)_n = \begin{cases} a (a+1) \cdots (a+n-1), &\text{if}\ n>0, \\ 1, &\text{if}\ n=0. \end{cases}$$ We will use $[x^i y^j]f(x,y)$ to denote the Taylor coefficient of $x^i y^j$ in $f(x,y)$. If there is no confusion, we write notions as $a[x^2]+b[xy]$ to denote the linear combinations of these coefficients. # Identities from WZ pairs *Proof of Theorem [Theorem 1](#bi-27){reference-type="ref" reference="bi-27"}.* Let $(F(n,k),G(n,k))$ be the WZ pair given by Pilehroods in [@PP12]: $$F(n,k) = (-1)^n (n+2k+1) H(n,k),$$ $$G(n,k) = (-1)^n \frac{18k^2 + 54kn + 45n^2 + 36k + 63n + 22}{18(2n+1)} H(n,k),$$ where $$H(n,k) = \frac{(1/3)_k^2 (2/3)_k^2 (2/3)_n (4/3)_n n!^2}{(1/3)_{n+k+1}^2 (2/3)_{n+k+1}^2 {2n \choose n}}.$$ For the WZ-pair $(F(n,k+b), G(n,k+b))$, we have $$\sum_{k=0}^\infty F(0,k+b) = \sum_{n=0}^\infty G(n,b).$$ The Taylor coefficient $[b^1]$ at $b=0$ of the right hand side is $$2\sum_{n=1}^\infty \frac{(-27)^{n}}{n^3 {3n \choose n} {2n \choose n}^2} \left( (15n-4) (3 H_{3n-1} - H_{n-1}) - 9 \right).$$ While the Taylor coefficient $[b^1]$ at $b=0$ of the left hand side is $$162 \sum_{k=1}^\infty \left( \frac{1}{(3k-1)^3} - \frac{1}{(3k-2)^3} \right)=-162\sum_{k=1}^\infty\frac{(\frac k3)}{k^3}.$$ It is known that $$\sum_{k=1}^\infty\frac{(\frac k3)}{k^3}=\frac{4\pi^3}{81\sqrt3}.$$ Therefore we obtain the desired [\[bi-27\]](#bi-27){reference-type="eqref" reference="bi-27"}. ------------------------------------------------------------------------ *Proof of Theorem [Theorem 2](#bi3){reference-type="ref" reference="bi3"}.* Let $$\begin{aligned} & H(a,b,c,d,k) = (-1)^{a+b} \frac{\Gamma(a+k+1) \Gamma(b+k+1)}{\Gamma(-a-b+c+d+1))} \nonumber \\ & \quad \times \frac{\Gamma(-a+c+1) \Gamma(-a+d+1) \Gamma(-b+c+1) \Gamma(-b+d+1)}{\Gamma(c+k+2) \Gamma(d+k+2) } . \label{H}\end{aligned}$$ \(i\) Au [@Au] showed that for $$F(n,k) = H(a,b,c+n,d+n,k+n)$$ there exists a hypergeometric term $G(n,k)$ such that $(F(n,k), G(n,k))$ forms a WZ pair. One can compute the explicit $G(n,k)$ by the `Maple` package `APCI`. By the WZ pair $$\tilde{F}(n,k)=F(n,k)/F(0,0), \quad \tilde{G}(n,k)=G(n,k)/F(0,0),$$ we derive from [\[WZ-o\]](#WZ-o){reference-type="eqref" reference="WZ-o"} that $$\begin{gathered} \label{eq-1.2} \sum_{k=0}^\infty \frac{(a+1)_k (b+1)_k}{(c+2)_k (d+2)_k} = \sum_{n=0}^\infty p_1(n) \frac{(a+1)_n (b+1)_n}{(c+2)_{2n+1} (d+2)_{2n+1}} \\ \quad \times \frac{ (-a+c+1)_n (-a+d+1)_n (-b+c+1)_n (-b+d+1)_n}{ (-a-b+c+d+1)_{2n+2}},\end{gathered}$$ where $p_1(n)$ is a polynomial in $n$ with parameters $a,b,c,d$. Now consider the Taylor expansion in $a,b,c,d$ at $a=b=c=d=0$. For the left hand side of [\[eq-1.2\]](#eq-1.2){reference-type="eqref" reference="eq-1.2"}, we have $$[ab]-2[b^2] = \sum_{k=0}^\infty \frac{H_k^{(2)}}{(k+1)^2} = \sum_{k=1}^\infty \frac{H_k^{(2)}}{k^2} - \zeta(4) = \frac{\zeta(2)^2 + \zeta(4)}{2} - \zeta(4) = \frac{\pi^4}{120}.$$ For the right hand side of [\[eq-1.2\]](#eq-1.2){reference-type="eqref" reference="eq-1.2"}, we have $$[ab]-2[b^2] = 3 \sum_{n=1}^\infty \frac{(21n-8) H_{n-1}^{(2)} + 2/n}{n^3 {2n \choose n}^3}.$$ Therefore, [\[eq-bi3\]](#eq-bi3){reference-type="eqref" reference="eq-bi3"} follows immediately. \(ii\) For $$F(n,k) = H(a,b,c+n,d+n,k),$$ it is routine to compute $G(n,k)$ such that $(F(n,k), G(n,k))$ forms a WZ pair. By the WZ pair $$\tilde{F}(n,k)=F(n,k)/F(0,0), \quad \tilde{G}(n,k)=G(n,k)/F(0,0),$$ we derive from [\[WZ-o\]](#WZ-o){reference-type="eqref" reference="WZ-o"} that $$\begin{gathered} \label{eq-1.3} \sum_{k=0}^\infty \frac{(a+1)_k (b+1)_k}{(c+2)_k (d+2)_k} \\ = \sum_{n=0}^\infty p_2(n) \frac{ (-a+c+1)_n (-a+d+1)_n (-b+c+1)_n (-b+d+1)_n}{(c+2)_n (d+2)_n (-a-b+c+d+1)_{2n+2}},\end{gathered}$$ where $p_2(n)$ is a polynomial in $n$ with parameters $a,b,c,d$. Now consider the Taylor expansion in $a,b,c,d$ at $a=b=c=d=-1/2$. For the left hand side of [\[eq-1.3\]](#eq-1.3){reference-type="eqref" reference="eq-1.3"}, we have $$[ab]-2[b^2] = \sum_{k=0}^\infty \frac{4H_{2k}^{(2)} - H_k^{(2)}} {(2k+1)^2} .$$ By `MultipleZetaValues`, we see that this sum is equal to $\pi^4/96$. For the right hand side of [\[eq-1.3\]](#eq-1.3){reference-type="eqref" reference="eq-1.3"}, we have $$[ab]-2[b^2] = \frac{1}{4} \sum_{n=1}^\infty \frac{2(3n-1) H_{n-1}^{(2)} +1/n}{n^3 {2n \choose n}^3} 16^n.$$ Therefore, [\[eq-16bi3\]](#eq-16bi3){reference-type="eqref" reference="eq-16bi3"} follows immediately. In view of the above, we have completed the proof of Theorem [Theorem 2](#bi3){reference-type="ref" reference="bi3"}. ------------------------------------------------------------------------ # Hypergeometric identities In this section, we use some hypergeometric identities to show the rest theorems. \(i\) We quote Identity (3.3) from [@WX2308]: $$\begin{gathered} \label{eq-WX} \sum_{k=0}^{\infty}\left(\frac{-1}{4}\right)^k \frac{(c)_k(d-c)_k(e-d)_k(1-c)_k(1+c-d)_k(1+d-e)_k \beta_k(a, c, d, e)}{\left(\frac{1}{2}\right)_k(1)_k(1+a-c)_k(1+a+c-d)_k(1+a+d-e)_k(2+a-e)_k} \\ \quad=\frac{\Gamma(1+a-c) \Gamma(1+a+c-d) \Gamma(1+a+d-e) \Gamma(2+a-e)}{\Gamma(a) \Gamma(1+a-d) \Gamma(1+a+c-e) \Gamma(1+a-c+d-e)},\end{gathered}$$ where $$\begin{gathered} \beta_k(a, c, d, e) = (1+a-d+2 k)(a+d-e+k) \\ +\frac{(1-c+k)(1+c-d+k)(e-d+k)}{1+2 k} .\end{gathered}$$ Let us consider the Taylor expansion at $(a,c,d,e)=(1/2,1/2,1,3/2)$. By searching the linear combination of all Taylor coefficients of order $4$, we find that on the left hand side, $$8 [de^3] + 4 [d^2 e^2]+[ade^2] = \frac{1}{8} \sum_{k=1}^\infty \frac{{2k \choose k}^5} {(-4096)^k} (20k^2+8k+1) (64 H_{2k}^{(4)} - 3H_k^{(4)})$$ While on the right hand side, it can be compute by `Maple` that $$8 [de^3] + 4 [d^2 e^2]+[ade^2] = - \frac{7}{45} \pi^2.$$ Therefore, [\[eq-bi5-12\]](#eq-bi5-12){reference-type="eqref" reference="eq-bi5-12"} follows immediately. \(ii\) We quote Identity (3.6) from [@WX2308]: $$\begin{gathered} \sum_{k=0}^{\infty} (-1)^k \frac{(a)_k(b)_k(c-b)_k(d-c)_k(1-b)_k(1+b-c)_k(1+c-d)_k}{(1)_{2 k}(1+a-b)_{2 k}(1+a+b-c)_{2 k}(1+a+c-d)_{2 k}} \\ \times \frac{(1+a-c)_k(1+a-b+c-d)_k(1+a+b-d)_k}{(2+a-d)_{2 k}} \theta_k(a, b, c, d) \\ = \frac{\Gamma(1+a-b) \Gamma(1+a+b-c) \Gamma(1+a+c-d) \Gamma(2+a-d)}{\Gamma(1+a) \Gamma(1+a-c) \Gamma(1+a-b+c-d) \Gamma(1+a+b-d)},\end{gathered}$$ where $$\begin{gathered} \theta_k(a, b, c, d) = \frac{2 k(1+2 a-d+3 k)}{a} +\frac{(a+k)(1+a-c+k)}{a(1+a-b+2 k)} \\ \quad \times \frac{(1+a-b+c-d+k)(1+a+b-d+k)(2+a-b+c-d+3 k)}{(1+a+c-d+2 k)(2+a-d+2 k)} \\ +\frac{(a+k)(c-b+k)(1-b+k)(1+c-d+k)}{a(1+2 k)(1+a-b+2 k)(1+a+b-c+2 k)} \\ \quad \times \frac{(1+a-c+k)(1+a-b+c-d+k)(1+a+b-d+k)}{(1+a+c-d+2 k)(2+a-d+2 k)} .\end{gathered}$$ Let us consider the Taylor expansion at $(a,b,c,d) = (1/2,1/2,1,3/2)$. By investigating all Taylor coefficients of order $4$, we find the following linear combination $$16 [d^4]+12[cd^3]+4[c^2 d^2]+8[ad^3]+4[acd^2]+4[a^2d^2]+2[a^2cd]+2[a^3d]+[a^4].$$ On the left hand side, the coefficient is $$\frac{1}{64} \sum_{k=0}^\infty \frac{{2k \choose k}^5} {(-2^{20})^k} \left( (820k^2+180k+13) (49 H_{2k}^{(4) } - 3H_k^{(4)} + 16) - \frac{195}{(2k+1)^2} \right).$$ While on the right hand side, the coefficient is $$-\frac{14}{45} \pi^2 + \frac{32}{\pi^2} .$$ Notice that Guillera have shown in [@Gui02] that $$\sum_{k=0}^\infty \frac{{2k \choose k}^5} {(-2^{20})^k} (820k^2+180k+13) = \frac{128}{\pi^2}.$$ Therefore, [\[eq-bi5-12\]](#eq-bi5-12){reference-type="eqref" reference="eq-bi5-12"} follows immediately. In view of the above, the proof of Theorem [Theorem 3](#bi5-12){reference-type="ref" reference="bi5-12"} is now complete. ------------------------------------------------------------------------ Setting $e \to -\infty$ in [@ChZh Theorem 9], we get $$\begin{aligned} & \sum_{k=0}^\infty (-1)^k (a+2k) \frac{(b)_k (c)_k (d)_k}{(1+a-b)_k (1+a-c)_k (1+a-d)_k} \nonumber \\ = &\sum_{k=0}^{\infty} \frac{ \alpha_k(a; b, c, d)}{(1+a-b)_{2k}} \cdot \frac{(c)_k (d)_k (1+a-b-c)_k (1+a-b-d)_k}{(1+a-c)_k(1+a-d)_k} ,\label{eq-9}\end{aligned}$$ where $$\alpha_k(a; b, c, d)=\frac {{a}^{2}-ab+4\,ak-2\,bk-cd-ck-kd+3\,{k}^{2}+a+2\,k}{1+a-b+2\,k}$$ Let us consider the linear combination $2[a]+2[b]+[d]$ of Taylor coefficients at $(a,b,c,d)=(1,1,1,3/2)$. The coefficient of the right hand side of [\[eq-9\]](#eq-9){reference-type="eqref" reference="eq-9"} is $$-\frac{1}{8} \sum_{k=1}^{\infty} \frac{16^k(6 k-1)}{(2 k-1) k^2{2k \choose k}{4k \choose 2k}} (4H_{2k-1} - H_{k-1} - 6).$$ Recall the known identity [\[Ex84\]](#Ex84){reference-type="eqref" reference="Ex84"}. While the coefficient of the right hand side is $$2\, \sum_{k=0}^\infty \left( -1 \right) ^{k} {\frac { \left( 6\,k+1 \right)}{ \left( 1+2 \,k \right) ^{3}}}$$ \ By `MultipleZetaValues`, this sum reduces to $6G-\frac{\pi^3}{8}$. Therefore, [\[eq-16-b2b4\]](#eq-16-b2b4){reference-type="eqref" reference="eq-16-b2b4"} follows immediately. ------------------------------------------------------------------------ Recall that [@ChZh Theorem 14] $$\begin{aligned} & \sum_{k=0}^\infty (a+2k) \frac{(b)_k (c)_k (d)_k (e)_k}{(1+a-b)_k (1+a-c)_k (1+a-d)_k (1+a-e)_k} \nonumber \\ = &\sum_{k=0}^{\infty} \beta_k(a ; b, c, d, e) \frac{(1+a-b-d)_{2 k}}{(1+a-c_k (1+a-e)_k} \nonumber \\ & \times \frac{(c)_k (e)_k (1+a-b-c)_k (1+a-b-e)_k (1+a-c-d)_k (1+a-d-e)_k}{(1+a-b)_{2k} (1+a-d)_{2k} (1+2 a-b-c-d-e)_{2 k}} ,\label{eq-14}\end{aligned}$$ where $$\begin{aligned} & \beta_k(a ; b, c, d, e)=\frac{(1+2 a-b-c-d+3 k)(a-e+k)}{1+2 a-b-c-d-e+2 k} + (e+k)(1+a-b-c+k) \\ & \qquad \times \frac{(1+a-b-d+2 k)(1+a-c-d+k)(2+2 a-b-d-e+3 k)}{(1+a-b+2 k)(1+a-d+2 k)(1+2 a-b-c-d-e+2 k)_2}. \end{aligned}$$ Let us consider the Taylor expansion at $(a,b,c,d,e)=(2,1,1,1,1)$. By investigating all Taylor coefficients of order $3$, we find the following linear combination $$\begin{gathered} +[a^2d]+2[a^2e]+[abd]+[ace]-[ad^2]+2[ade]+2[ae^2]+[bd^2]+2[bde]\\ +[cde]+2[ce^2]-2[d^3]-2[d^2e]+2[de^2]+2[e^3].\end{gathered}$$ The coefficient of the right hand side of [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} is $$\frac{1}{16}\sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{8(30k-11)}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{k^4} \right).$$ While the coefficient of the right hand side is $$\begin{gathered} -\frac{1}{3} \sum_{k=0}^\infty \left( \frac { \left( -8\,{k}^{3}-24\,{k}^{2}-24\,k-8 \right) H_k^3}{ \left( 1+k \right) ^{6}} + \frac { \left( -12\,{k}^{3}-30\,{k}^{2}-24\,k-6 \right) H_k^2}{ \left( 1+k \right) ^{6}} \right. \\ + \frac{-12\,{k}^{3}-30\,{k}^{2}-18\,k}{ \left( 1+k \right) ^{6}} H_k +\frac{ \left( 12\,{k}^{3}+36\,{k}^{2}+ 36\,k+12 \right) H_k^{(2)} H_k} { \left( 1+k \right) ^{ 6}} \\ \left. + \frac { \left( 6\,{ k}^{3}+15\,{k}^{2}+12\,k+3 \right) H^{(2)}_k }{ \left( 1+k \right) ^{6}} + \frac { \left( -4\,{k}^{3}-12\,{k}^{2 }-12\,k-4 \right)H^{(3)}_k }{ \left( 1+k \right) ^ {6}} +\frac {3\,{k}^{2}+9\,k}{ \left( 1+k \right) ^{6}} \right).\end{gathered}$$\ By `MultipleZetaValues`, this sum reduces to $2 \zeta(3)^2$. Therefore, [\[eq-bi2\]](#eq-bi2){reference-type="eqref" reference="eq-bi2"} follows immediately. ------------------------------------------------------------------------ Recall that [@ChZh Theorem 18] $$\begin{aligned} & \sum_{k=0}^\infty (a+2k) \frac{(b)_k (c)_k (d)_k (e)_k}{(1+a-b)_k (1+a-c)_k (1+a-d)_k (1+a-e)_k} \nonumber \\ = &\sum_{k=0}^{\infty} \gamma_k(a ; b, c, d, e) \frac{(-1)^k}{(1+a-b)_{3 k}} \frac{(c)_k (d)_k (1+a-b-e)_k (1+a-c-d)_k}{(1+a-e)_k} \nonumber \\ & \times \frac{ (e)_{2k} (1+a-b-c)_{2k} (1+a-b-d)_{2k}}{ (1+a-c, 1+a-d)_{2k} (1+2 a-b-c-d-e)_{2 k}} ,\label{eq-18}\end{aligned}$$ where $$\begin{aligned} & \gamma_k(a ; b, c, d, e)= \frac{(1+2 a-b-c-d+4 k)(a-e+k)}{1+2 a-b-c-d-e+2 k} + \frac{ (1+a-b-c+2 k) }{(1+a-b+3 k) } \nonumber \\ & \times \frac{(e+2 k)(1+a-b-d+2 k) (1+a-c-d+k) (2+2 a-b-d-e+3 k)}{(1+a-d+2 k) (1+2 a-b-c-d-e+2 k)_2} \nonumber \\ & + \frac{(c+k) (e+2 k) (1+a-b-c+2 k)(1+a-b-e+k) (1+a-c-d+k) }{(1+a-c+2 k) (1+a-d+2 k) } \nonumber \\ & \times \frac{ (1+a-b-d+2 k)_2}{(1+a-b+3 k)_2 (1+2 a-b-c-d-e+2 k)_2} \end{aligned}.$$ To prove the first identity, we consider Taylor expansion at $(a,b,c,d,e)=(2,1,1,1,1)$ and the linear combination $$2 [a] + 2 [b] + [e].$$ The value for the right hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^3 {2k \choose k} {3k \choose k}} \left( - 2 \frac{56k^2-32k+5}{(2k-1)^2} H_{2k-1} +{\frac {280\,{k}^{ 3}-168\,{k}^{2}+33\,k-2}{2 k \left( 2\,k-1 \right) ^{2}}} \right).$$ Notice that $${\frac {280\,{k}^{3}-168\,{k}^{2}+33\,k-2}{2 k \left( 2\,k-1 \right) ^{2}}}= \frac{5}{12} \left( \frac {6(56\,{k}^{2}-32\,k+5)}{ \left( 2\,k-1 \right) ^{2}} \right) - \frac{1}{k}.$$ The value for the left hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\sum_{k=0}^\infty \left( -4\,{\frac {H_k }{ \left( 1+k \right) ^{3}}}+ 2\,{\frac {5\,k+1}{ \left( 1+k \right) ^{4}}} \right),$$ which can be evaluated by `MultipleZetaValues`. Its value is $-\pi^4/10 + 10 \zeta(3)$. Recall the known identity [\[Ex21\]](#Ex21){reference-type="eqref" reference="Ex21"}, we derive [\[eq-Chu21\]](#eq-Chu21){reference-type="eqref" reference="eq-Chu21"}. We also utilize Equation [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} and consider the Taylor expansion at $(a,b,c,d,e)=(2,1,1,1,1)$. This time, we consider the linear combination $$21[a]+19[b]+\frac{115}{3}[d]+\frac{80}{3}[e].$$ The coefficient of the right hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} simplifies to the left hand of [\[eq-1.12\]](#eq-1.12){reference-type="eqref" reference="eq-1.12"}. While the coefficient of the left hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} becomes $$\sum_{k=0}^\infty \left( \frac{168 H_k}{(k+1)^3}+ \frac{21}{(k+1)^4} \right),$$ which equals $7 \pi^4/10$ by `MultipleZetaValues`. This completes the proof of [\[eq-1.12\]](#eq-1.12){reference-type="eqref" reference="eq-1.12"}. ------------------------------------------------------------------------ To prove the first identity, we investigate the Taylor expansion at $(a,b,c,d,e)=(2,1,1,1,3/2)$ and the linear combination $[a]+[b]+[d]$. The value for the right hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\sum_{k=1}^\infty \frac{(-1)^k}{k^2 {2k \choose k} {3k \choose k}} \left( 8 \frac{7k-2}{2k-1} (2 H_{2k-1} - H_{k-1}) - \frac{1}{2} \frac{336k^3-160k^2+16k-1}{k (2k-1)^2} \right) .$$ Notice that $$\begin{gathered} - \frac{1}{2} \frac{336k^3-160k^2+16k-1}{k (2k-1)^2} \\ = \frac{1}{4} \cdot \frac{-6(56\,{k}^{2}-32\,k+5)}{k \left( 2\,k-1 \right) ^{2}} + 2 \cdot \frac{-6(7\,k-2)}{2\,k-1} + \frac{8}{k}\end{gathered}$$ In addition to [\[Ex21\]](#Ex21){reference-type="eqref" reference="Ex21"}, we also have [\[Ex24\]](#Ex24){reference-type="eqref" reference="Ex24"}. While the value for the left hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\sum_{k=0}^\infty \left( {4\frac {H_k }{ \left( 1+k \right) ^{2}}}-{4\frac {H_{2k} }{ \left( 1+k \right) ^{2}}} +{\frac {12\,{k}^{2}+12\,k+1}{ \left( 1+2\,k \right) \left( 1+k \right) ^{3}}} \right).$$ By `MultipleZetaValues`, this sum reduces to $\pi^2-6 \zeta(3)^2$. Therefore, [\[eq-Chu24a\]](#eq-Chu24a){reference-type="eqref" reference="eq-Chu24a"} follows immediately. Finally, we consider the Taylor expansion at $(a,b,c,d,e)=(2,1,1,1,3/2)$ and the linear combination $$[e^2] - 2 [de] - 8 [d^2] + 6 [cd].$$ The value for the right hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\sum_{k=1}^\infty \frac{(-1)^k}{k^2 {2k \choose k} {3k \choose k}} \left( -8\,{\frac { \left( 7\,k-2 \right)H_{k-1}^{(2)} }{2\,k-1}}-{\frac {336\,{k}^{4}-320\,{k}^{3}+56\,{k}^{2}+19\,k-6}{{k}^{2} \left( 2\,k-1 \right) ^{2}}}. \right)$$ Notice that $$\begin{gathered} -{\frac {336\,{k}^{4}-320\,{k}^{3}+56\,{k}^{2}+19\,k-6}{{k}^{2} \left( 2\,k-1 \right) ^{2}}} \\ = -\frac{1}{6} \cdot \frac{-6(56\,{k}^{2}-32\,k+5)}{k \left( 2\,k-1 \right) ^{2}} + 4 \cdot \frac {-6(7\,k-2)}{2\,k-1} + \frac{6}{k^2}.\end{gathered}$$ While the value for the left hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\begin{gathered} \sum_{k=0}^\infty \left( 36\,{\frac {H_k^{2}}{ \left( 1+k \right) ^{2}}}-24\,{\frac { \left( (4\, {k}^{2}+6\,k+2) H_{2k} -2\,{k}^{2}-3\,k \right) H_k }{ \left( 1+2\,k \right) \left( 1+k \right) ^{3}}}+16\,{ \frac { H_{2k} ^{2}}{ \left( 1+k \right) ^{2}}} \right. \\ \left. -16\,{\frac { \left( 3+2\,k \right) kH_{2k} }{ \left( 1+2\,k \right) \left( 1+k \right) ^{ 3}}}+4\,{\frac {k \left( 12\,{k}^{3}+36\,{k}^{2}+27\,k+4 \right) }{ \left( 1+2\,k \right) ^{2} \left( 1+k \right) ^{4}}}\right).\end{gathered}$$ By `MultipleZetaValues`, this sum reduces to $2 \pi^2 - \frac{\pi^4}{90} - 4 \zeta(3)$. Therefore, [\[eq-Chu24b\]](#eq-Chu24b){reference-type="eqref" reference="eq-Chu24b"} follows immediately. ------------------------------------------------------------------------ *Proof of Theorem [Theorem 8](#th-1.9){reference-type="ref" reference="th-1.9"}.* Let us consider the Taylor expansion at $(a,b,c,d,e)=(1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$ and the linear combination $$-32[a]-48[b]-16[e].$$ The coefficient of the right hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\begin{aligned} & \sum_{k=1}^\infty \frac{(-16)^k \big( (20k^2-8k+1) (2H_{4k-1}-9H_{2k-1}) + 160k^2-72k+8 \big)}{(2k-1)^2 k^3 {2k \choose k} {4k \choose 2k}} \\ = & \sum_{k=1}^\infty \frac{(-16)^k \big( (20k^2-8k+1) (2H_{4k-1}-9H_{2k-1}) -8k \big)}{(2k-1)^2 k^3 {2k \choose k} {4k \choose 2k}} \\ & + 8 \sum_{k=1}^\infty \frac{(-16)^k (20k^2-8k+1) }{(2k-1)^2 k^3 {2k \choose k} {4k \choose 2k}}.\end{aligned}$$ Recall the known identity [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"}. While the coefficient of the left hand of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} simplifies to $$- 32 \sum_{k=0}^\infty \frac{8k+1}{(2k+1)^4},$$ which equals $\pi^4 - 112 \zeta(3)$ by `MultipleZetaValues`. This completes the proof of [\[eq-1.13\]](#eq-1.13){reference-type="eqref" reference="eq-1.13"}. ------------------------------------------------------------------------ Let us recall the transformation [@ChZh Theorem 24] $$\begin{aligned} & \sum_{k=0}^\infty (a+2k) \frac{(b)_k (c)_k (d)_k (e)_k}{(1+a-b)_k (1+a-c)_k (1+a-d)_k (1+a-e)_k} \nonumber \\ = &\sum_{k=0}^{\infty} \mu_k(a ; b, c, d, e) \frac{(-1)^k}{(1+a-b)_{3 k}} \frac{(c)_k (1+a-b-d)_k (1+a-b-e)_k}{(d+e-a)_k (1+a-d)_k (1+a-e)_k} \nonumber \\ & \times \frac{(d)_{2k} (e)_{2k} (1+a-b-c)_{2k}}{(1+2a-b-c-d-e)_k (1+a-c)_{2k}},\label{eq-24}\end{aligned}$$ where $$\begin{gathered} \mu_k(a ; b, c, d, e)= \frac{(1 + 2a - b - c - d +3k)(a - e + k)}{1+2a - b - c - d - e + k} \\ + \frac{(e +2k)(1+ a - b - c +2k)(1+ a - b - d + k)}{(1 + a - b +3k)(1+2a - b - c - d - e + k) } \\ + \frac{(c+k)(d+2k)(e+2k)(1+a-b-c+2k)(1+a-b-d+k)(1+a-b-e+k)}{(d+e-a+k)(1+a-c+2k)(1+2a-b-c-d-e+k)(1+a-b+3k)_2} . \end{gathered}$$ Firstly, we consider the Taylor expansion at $(a,b,c,d,e)=(3/2,1,-\infty,1,1)$ and the linear combination $$128[a] +768[b]-768[d].$$ The coefficient of the left hand of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is $$\begin{gathered} \sum_{k=1}^\infty \frac{256^k \big( (22k-1) \mathcal H(k) +132k-56 \big)}{(2k-1) k^2 {3k \choose k}{6k \choose 3k}} \\ = \sum_{k=1}^\infty \frac{256^k \big( (22k-1) \mathcal H(k) -50 \big)}{(2k-1) k^2 {3k \choose k}{6k \choose 3k}} + 6 \sum_{k=1}^\infty \frac{256^k (22k-1)}{(2k-1) k^2 {3k \choose k}{6k \choose 3k}}.\end{gathered}$$ Recall the known identity [\[Ex50\]](#Ex50){reference-type="eqref" reference="Ex50"}. Denote the series on the left hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} by $\Omega(a;b,c,d,e)$. We have the transformation formula [@ChZh Theorem 5]: $$\begin{gathered} \label{eq-tr} \Omega(a;b,c,d,e) = \\ \frac{a-c}{1+2a-b-c-d-e} \Omega(1+2a-b-d-e;1+a-b-e, 1+a-b-d, 1+a-d-e, c).\end{gathered}$$ The coefficient of the right hand of [\[eq-tr\]](#eq-tr){reference-type="eqref" reference="eq-tr"} is $$256 \sum_{k=0}^\infty (-1)^k \frac{6k+1}{(2k+1)^3},$$ which equals $768 G -16 \pi^3$ by `MultipleZetaValues`. This completes the proof of [\[eq-1.14\]](#eq-1.14){reference-type="eqref" reference="eq-1.14"}. We use [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} and fix the Taylor expansion at $(a,b,c,d,e)=(\frac{3}{2}, 1, \frac{1}{2}, 1, 1)$. The first combination we consider is $$-128([a]+[b]+[c]+2[e]).$$ The coefficient of the right hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is the same as the left hand side of [\[eq-4.33\]](#eq-4.33){reference-type="eqref" reference="eq-4.33"}. By aid of the transform [\[eq-tr\]](#eq-tr){reference-type="eqref" reference="eq-tr"} once again, we find that the coefficient of the left hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is $$-768\sum_{k=0}^\infty \frac{1}{(2k+1)^4}.$$ By `MultipleZetaValues`, it equals $-8\pi^4$. This completes the proof of [\[eq-4.33\]](#eq-4.33){reference-type="eqref" reference="eq-4.33"}. The second combination we consider is $$-128(4[a]+5[b]+4[c]+7[e]).$$ By a similar argument as before, we find that the series on the left hand side of [\[eq-4.34\]](#eq-4.34){reference-type="eqref" reference="eq-4.34"} is equal to $$-3072\sum_{k=0}^\infty \frac{1}{(2k+1)^4},$$ which completes the proof of [\[eq-4.34\]](#eq-4.34){reference-type="eqref" reference="eq-4.34"}. The third combination we consider is $$128([a^2]+[ab]+[ac]+[ae]+[b^2]+[bc]+[be]+[c^2]+[ce]+[e^2]).$$ By a similar argument as before, we find that the series on the left hand side of [\[eq-4.35\]](#eq-4.35){reference-type="eqref" reference="eq-4.35"} is equal to $$1024\sum_{k=0}^\infty \frac{1}{(2k+1)^5},$$ which equals $992\zeta(5)$, completing the proof of [\[eq-4.35\]](#eq-4.35){reference-type="eqref" reference="eq-4.35"}. Finally, we consider the combination $$\begin{gathered} -128 \big( [a^3]+[a^2b]+[a^2c]+[a^2e]+[ab^2]+[abc]+[abe]+[ac^2]+[ace]+[ade]-[ae^2]+[b^3]\\ +[b^2c]+[b^2e]+[bc^2]+[bce]+[bde]-[be^2]+[c^3]+[c^2e]+[cde]-[ce^2]+[de^2]-2[e^3] \big).\end{gathered}$$ By a similar argument as before, we find that the series on the left hand side of [\[eq-4.36\]](#eq-4.36){reference-type="eqref" reference="eq-4.36"} is equal to $$-512 \sum_{k=0}^\infty \frac{8H^{(3)}_{2k}-H^{(3)}_{k}}{(2k+1)^3} + 2048 \sum_{k=0}^\infty \frac{1}{(2k+1)^6},$$ which equals $-1568 \zeta(3)^2$, completing the proof of [\[eq-4.36\]](#eq-4.36){reference-type="eqref" reference="eq-4.36"}. ------------------------------------------------------------------------ We utilize the transformation [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} and consider the Taylor expansion at $(a,b,c,d,e)=(\frac{3}{2}, 1, \frac{1}{2}, \frac{1}{2}, 1)$. The first combination is $$8[a]+8[b]+6[d]+11[e].$$ We find that the series on the left hand side of [\[eq-1.11a\]](#eq-1.11a){reference-type="eqref" reference="eq-1.11a"} equals $$\sum_{k=0}^\infty \left( \frac{9}{2} \frac{(4k+3)H_k}{(2k+1)^2(k+1)^2} + \frac{9}{2} \frac{(4k+3)(2H_{2k}-H_k)}{(2k+1)^2(k+1)^2} + \frac{16k^3+24k^2+21k+8}{(2k+1)^3(k+1)^3} \right).$$ By `MultipleZetaValue`, it equals $11\zeta(3)$, completing the proof of [\[eq-1.11a\]](#eq-1.11a){reference-type="eqref" reference="eq-1.11a"}. The second combination is $$36[a]+40[b]+28[d]+46[e].$$ We find that the series on the left hand side of [\[eq-1.11b\]](#eq-1.11b){reference-type="eqref" reference="eq-1.11b"} equals $$\sum_{k=0}^\infty \left( \frac{21(4k+3)H_k}{(2k+1)^2(k+1)^2} + \frac{21(4k+3)(2H_{2k}-H_k)}{(2k+1)^2(k+1)^2} + \frac{2(32k^3+48k^2+45k+18)}{(2k+1)^3(k+1)^3} \right).$$ By `MultipleZetaValue`, it equals $50\zeta(3)$, completing the proof of [\[eq-1.11b\]](#eq-1.11b){reference-type="eqref" reference="eq-1.11b"}. ------------------------------------------------------------------------ We utilize the transformation [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} and consider the Taylor expansion at $(a,b,c,d,e)=(1, 1/2, 1, 1/2, 1/2)$. The first combination is $$-[c]+[d] -[e].$$ The coefficient of the right hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is exactly the same as the left hand side of [\[eq-1.12a\]](#eq-1.12a){reference-type="eqref" reference="eq-1.12a"}. While coefficient of the left hand sider of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$-2 \sum_{k=0}^\infty \frac{H_k}{(2k+1)^2},$$ which equals $\frac{\pi^2}{2} \log 2- \frac{7}{2} \zeta(3)$ by `MultipleZetaValue`. This proves [\[eq-1.12a\]](#eq-1.12a){reference-type="eqref" reference="eq-1.12a"}. The second combination is $$-2[b]+2[c]$$ The coefficient of the right hand side of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\begin{gathered} \sum_{k=1}^\infty \frac{(-16)^{k} \big( P(k) (2H_{6k-1}-H_{3k-1}-3H_{k-1}) -224k^3+168k^2-24k-5/2 \big)}{k^2(2k-1)(4k-1)(4k-3) \binom{3k}k\binom{6k}{3k}} \\ = \sum_{k=1}^\infty \frac{(-16)^{k} \big( P(k) (2H_{6k-1}-H_{3k-1}-3H_{k-1}) -64k^2+46k -17/2 \big)}{k^2(2k-1)(4k-1)(4k-3) \binom{3k}k\binom{6k}{3k}} \\ -2 \sum_{k=1}^\infty \frac{(-16)^{k} P(k) }{k^2(2k-1)(4k-1)(4k-3) \binom{3k}k\binom{6k}{3k}} \end{gathered}$$ While coefficient of the left hand sider of [\[eq-18\]](#eq-18){reference-type="eqref" reference="eq-18"} is $$\sum_{k=0}^\infty \left( \frac{4H_k}{(1+2k)^2}- \frac{4(2 H_{2k}-H_k)}{(1+2k)^2}+\frac{8k}{(1+2k)^3} \right),$$ which equals $\pi^2/2 - 2 \pi^2 \log(2) + 7 \zeta(3)$ by `MultipleZetaValue`. Noting the formula [\[Ex25\]](#Ex25){reference-type="eqref" reference="Ex25"}, we derive [\[eq-1.12b\]](#eq-1.12b){reference-type="eqref" reference="eq-1.12b"}. ------------------------------------------------------------------------ We utilize [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} and consider the Taylor expansion at $(a,b,c,d,e)=(3/2, 1, 1, 1, 1/2)$. The first combination is $$[a]+[c]+2[d]+[e].$$ The coefficient of the right hand side of [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} is $$\sum_{k=1}^\infty (a_{2k-1}+a_{2k}) = \sum_{k=1}^\infty a_k,$$ where $a_k$ is the summand of the left hand side of [\[eq-1.13a\]](#eq-1.13a){reference-type="eqref" reference="eq-1.13a"}. By the transformation [\[eq-tr\]](#eq-tr){reference-type="eqref" reference="eq-tr"}, the coefficient of the left hand side of [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} is $$\sum_{k=0}^\infty \frac{6}{(2k+1)^4} = \frac{\pi^4}{16},$$ completing the proof of [\[eq-1.13a\]](#eq-1.13a){reference-type="eqref" reference="eq-1.13a"}. The second combination is $$[a^2] +[ac] + [ad]+ [ae]+[c^2]+[cd]+[ce]+[d^2]+[de]+[e^2].$$ The coefficient of the right hand side of [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} is $$\sum_{k=1}^\infty (a_{2k-1}+a_{2k}) = \sum_{k=1}^\infty a_k,$$ where $a_k$ is the summand of the left hand side of [\[eq-1.13b\]](#eq-1.13b){reference-type="eqref" reference="eq-1.13b"}. By the transformation [\[eq-tr\]](#eq-tr){reference-type="eqref" reference="eq-tr"}, the coefficient of the left hand side of [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} is $$\sum_{k=0}^\infty \frac{8}{(2k+1)^5} = \frac{31}{4} \zeta(5),$$ completing the proof of [\[eq-1.13b\]](#eq-1.13b){reference-type="eqref" reference="eq-1.13b"}. The third combination is $$\begin{gathered} +[a^2c]+[a^2d]+[a^2e]+[abd]+[ac^2]+[acd]+[ace]-[ad^2]+[ade]+[ae^2]+[bcd]\\ +[bd^2]+[bde]+[c^3]+[c^2d]+[c^2e]-[cd^2]+[cde]+[ce^2]-2[d^3]-[d^2e]+[de^2]+[e^3].\end{gathered}$$ The coefficient of the right hand side of [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} is $$\sum_{k=1}^\infty (a_{2k-1}+a_{2k}) = \sum_{k=1}^\infty a_k,$$ where $a_k$ is the summand of the left hand side of [\[eq-1.13c\]](#eq-1.13c){reference-type="eqref" reference="eq-1.13c"}. By the transformation [\[eq-tr\]](#eq-tr){reference-type="eqref" reference="eq-tr"}, the coefficient of the left hand side of [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} is $$\sum_{k=0}^\infty \left( \frac{4(8H^{(3)}_{2k}-H^{(3)}_k)}{(2k+1)^3} + \frac{16}{(2k+1)^6} \right)= \frac{49}{4} \zeta(3)^2,$$ completing the proof of [\[eq-1.13c\]](#eq-1.13c){reference-type="eqref" reference="eq-1.13c"}. We utilize [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} and consider the Taylor expansion at $(a,b,c,d,e)=(1,1/2,-\infty,1/2,1)$. The first combination is $$-8([a]+[b]).$$ The coefficient of the right hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is $$\begin{aligned} & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} - (88 k^3 -136 k^2+68 k-9)/(2k-1) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ = & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} + (6k-1)(4k-3)/(2k-1) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ & - 2 \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} Q(k) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ = & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} + (6k-1)(4k-3)/(2k-1) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} - 4 \pi.\end{aligned}$$ The coefficient of the left hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is $$\sum_{k=0}^\infty \left( \frac{8 (-1)^k H_k}{2k+1} - \frac{8 (-1)^k (4 k+1)}{(2k+1)^2} \right) = 16 G - 4 \pi - 4 \pi \log 2,$$ completing the proof of [\[eq-1.14a\]](#eq-1.14a){reference-type="eqref" reference="eq-1.14a"}. The second combination is $$-8 ( [a]+[b]- [d]/3).$$ The coefficient of the right hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is $$\begin{aligned} & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} - (264 k^3-342 k^2+157 k-21)/(6k-3) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ = & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} + (6k-1)(k-3)/(6k-3) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ & - 2 \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} Q(k) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ = & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} + (6k-1)(4k-3)/(2k-1) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} - 4 \pi.\end{aligned}$$ The coefficient of the left hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is $$\sum_{k=0}^\infty \left( \frac{8 (-1)^k H_k}{3(2 k+1)} - \frac{8 (-1)^k (4 k+1)}{(2k+1)^2} \right) = \frac{32}{4} G - 4 \pi - \frac{4}{3} \pi \log 2.$$ completing the proof of [\[eq-1.14b\]](#eq-1.14b){reference-type="eqref" reference="eq-1.14b"}. The third combination is $$-8 (2[a]+[b]+2[d]/3).$$ $$\begin{aligned} & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} - 2(396 k^3-540 k^2+247 k-33)/(6k-3) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ = & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} + 4(18 k^2- 20 k+3)/(6k-3) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ & - 6 \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} Q(k) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} \\ = & \sum_{k=1}^\infty \frac{16^k {4k \choose 2k} \big( Q(k) H_{k-1} + (6k-1)(4k-3)/(2k-1) \big) }{k (4k-1)(4k-3){3k \choose k} {6k \choose 3k}} - 12 \pi.\end{aligned}$$ The coefficient of the left hand side of [\[eq-24\]](#eq-24){reference-type="eqref" reference="eq-24"} is $$\sum_{k=0}^\infty \left( \frac{(-1)^k (32 H_{2k} - 32 H_k/3)}{2k+1} - \frac{16 (-1)^k (1+6 k)}{(2 k+1)^2} \right) = \frac{64}{4} G - 12 \pi + \frac{4}{3} \pi \log 2.$$ completing the proof of [\[eq-1.14c\]](#eq-1.14c){reference-type="eqref" reference="eq-1.14c"}. ------------------------------------------------------------------------ We utilize [\[eq-14\]](#eq-14){reference-type="eqref" reference="eq-14"} and consider the Taylor expansion at $(a,b,c,d,e)=(3/2,1/2,1,1/2,1)$. Since the proof is similar to that of Theorem [Theorem 14](#th-1.14){reference-type="ref" reference="th-1.14"}, we only list the combinations. For [\[eq-1.15a\]](#eq-1.15a){reference-type="eqref" reference="eq-1.15a"}, the combination is $$4[a]+4[d]+10[e].$$ For [\[eq-1.15b\]](#eq-1.15b){reference-type="eqref" reference="eq-1.15b"}, the combination is $$3[a] +2 [d] + 7[e].$$ For [\[eq-1.15c\]](#eq-1.15c){reference-type="eqref" reference="eq-1.15c"}, the combination is $$-8[a^2] -8[ad]-16[ae]-6[bd]-17[ce]/2+4[d^2]-16[de]-15[e^2].$$ For [\[eq-1.15c\]](#eq-1.15c){reference-type="eqref" reference="eq-1.15c"}, the combination is $$2[a^2]+2[ad]+4[ae]+4[bd]+[ce]-6[d^2]+4[de]+6[e^2]. \tag*{\hfill \rule{4pt}{7pt}}$$ 99 T. Amdeberhan and D. Zeilberger, Hypergeometric series acceleration via the WZ method, Electron. J. Comb. **4** (1997) R3. J. Au, Colored multiple zeta values, WZ-pairs and some infinite sums, arXiv: 2212.02986. W. Chu, Hypergeometric series and the Riemann zeta function, Acta Arith. **82** (1997) 103--118. W. Chu and W. Zhang, Accelerating Dougall's 5F4-sum and infinite series involving $\pi$, Math. Comput. **83** (2014) 475--512. J. Guillera, Some binomial series obtained by the WZ-method, Adv. Appl. Math. **29** (2002), 599--603. J. Guillera, Hypergeometric identities for $10$ extended Ramanujan-type series, Ramanujan J. **15** (2008), 219--234. C. Li and W. Chu, Infinite series about harmonic numbers inspired by Ramanujan-like formulae, Electr. Res. Arch. **31(8)** (2023) 4611--4636. S.B. Ekhad and D. Zeilberger, A WZ proof of Ramanujan's formula for $\pi$, In: Geometry, Analysis, and Mechanics, J.M. Rassias Ed., World Scientific, Singapore, 1994, pp. 107--108. M. Petkovsěk, H.S. Wilf, and D. Zeilberger, A=B, A K Peters, 1996. K.H. Pilehrood and T.H. Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, Electr. J. Combin. **18** (2012), no. 2, P35. Z.-W. Sun, Super congruences and Euler numbers, Sci. China Math. **54** (2011), 2509--2535. Z.-W. Sun, Series with summands involving harmonic numbers, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory, Springer, to appear. See also arXiv:2210.07238. Z.-W. Sun, New series involving binomial coefficients, arXiv: 2307.03086. C. Wei, Some fast convergent series for the mathematical constants $\zeta(4)$ and $\zeta(5)$, arXiv: 2303.07887. C. Wei, On some conjectural series containing binomial coefficients and harmonic numbers, arXiv: 2306.02641. C. Wei and C. Xu, On some conjectural series containing harmonic numbers of $3$-order, arXiv: 2308.06440.

arxiv_math

--- author: - "[Abhinandan]{.smallcaps}" bibliography: - references.bib title: "[Crystalline representations and Wach modules in the relative case II]{.smallcaps}" --- =0.3em [Abstract.]{.smallcaps} We study relative Wach modules generalising our previous works on this subject. Our main result shows a categorical equivalence between relative integral Wach modules and lattices inside relative crystalline representations. Using this result, we deduce a purity statement for relative crystalline representations and provide a criteria for checking crystallinity of relative $p\textrm{-adic}$ representations. Furthermore, we interpret relative Wach modules as modules with $q\textrm{-connections}$. Moreover, for a crystalline representation, we show that its associated Wach module (together with the Nygaard filtration) is the canonical $q\textrm{-deformation}$ of the filtered $(\varphi,\partial)\textrm{-module}$ associated to the representation. # Introduction The study of arithmetic Wach modules and their relationship to crystalline representations is of classical nature, having been taken up in the works of Fontaine [@fontaine-phigamma], Wach [@wach-free; @wach-torsion], Colmez [@colmez-hauteur] and Berger [@berger-limites]. More explicitly, in op. cit. the authors studied the situation of an absolutely unramified extension of $\mathbb{Q}_p$ with perfect residue field. In [@abhinandan-relative-wach-i] we defined a similar concept in the relative case, i.e. for certain étale algebras over a formal torus (see §[1.5](#subsec:setup_nota){reference-type="ref" reference="subsec:setup_nota"} for precise setup) and showed that such objects give rise to crystalline representations of the fundamental group of the generic fiber. On the other hand, in [@abhinandan-imperfect-wach], we generalised the theory of Wach modules and their relationship to crystalline representations, to the imperfect residue field case. In this article, we combine these two generalisations of the classical theory, to discuss the equivalence in its most natural generality. In addition, we show that Wach modules are $q\textrm{-deformations}$ of lattices inside the filtered $(\varphi, \partial)\textrm{-module}$ attached to crystalline representations. Our motivation for studying relative Wach modules is twofold, largely stemming from geometry. In [@abhinandan-syntomic], for smooth ($p\textrm{-adic}$ formal) schemes, we defined the notion of crystalline syntomic complex with coefficients in global relative Fontaine-Laffaille modules. Moreover, [@abhinandan-syntomic Theorem 1.15] showed that such a complex is naturally comparable to the complex of $p\textrm{-adic}$ nearby cycles of the associated crystalline $\mathbb{Z}_p\textrm{-local system}$ on the (rigid analytic) generic fiber of the (formal) scheme. The work in loc. cit. was motivated by the results of [@fontaine-messing], [@tsuji-syntomic], [@tsuji-cst] and [@colmez-niziol], and the proof of [@abhinandan-syntomic Theorem 1.15] follows via careful computations in the local setting in which relative Wach modules play a pivotal role (see [@abhinandan-syntomic Corollary 1.12]). To generalise these results beyond the Fontaine-Laffaille case, it is therefore necessary to understand the relationship between crystalline representations of the fundamental group and general relative Wach modules (see Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}). On the other hand, in [@bhatt-morrow-scholze-2], for smooth $p\textrm{-adic}$ formal schemes, the authors defined a prismatic syntomic complex and compared it to the complex of $p\textrm{-adic}$ nearby cycles integrally. In the same vein, comparison results beyond the smooth case, have also been obtained in [@antieau-mathew-morrow-nikolaus] and [@bhatt-mathew], where the latter uses the theory of prismatic cohomology from [@bhatt-scholze-prisms]. The aforementioned results were in the case of constant coefficients, and it is natural to ask if [@bhatt-morrow-scholze-2 Theorem 10.1] could be generalised to non-constant coefficients, i.e. prismatic $F\textrm{-crystals}$. It is reasonable to expect that Wach modules will play an important role in resolving these questions. Another motivation for considering Wach modules is to construct a deformation of crystalline cohomology, i.e. the functor $\mathbf{D}_{\textup{cris}}$ from classical $p\textrm{-adic}$ Hodge theory, to better capture mixed characteristic information. In [@fontaine-phigamma §B.2.3] Fontaine expressed similar expectations which were verified by Berger in [@berger-limites Théorème III.4.4] and generalised to finer integral conjectures in [@scholze-q-deformations §6]. Some conjectures of [@scholze-q-deformations] were resolved by the introduction of prismatic cohomology [@bhatt-scholze-prisms]. It is also worth mentioning that the proof of the result on prismatic syntomic complex mentioned above, i.e. [@bhatt-morrow-scholze-2 Theorem 10.1], relies on local computation of prismatic cohomology using the $q\textrm{-de Rham}$ complex, i.e. $q\textrm{-deformation}$ of the de Rham complex. In addition, the importance of $q\textrm{-de Rham}$ cohomology in computation of prismatic cohomology has also been emphasised in [@bhatt-lurie Example 1.3.3]. In this paper, we interpret Wach modules as $q\textrm{-de Rham}$ complexes (see Theorem [Theorem 11](#intro_thm:qdeformation_dcrys){reference-type="ref" reference="intro_thm:qdeformation_dcrys"}). Moreover, we show that such an object is the $q\textrm{-deformation}$ of a lattice inside the filtered $(\varphi, \partial)\textrm{-module}$ attached to a crystalline representation. In a subsequent work (cf. [@abhinandan-prismatic-wach]), we will show that in our setting, a relative Wach module can be regarded as the evaluation of a prismatic $F\textrm{-crystal}$ over a covering (by a suitable $q\textrm{-de Rham}$ prism) of the final object of the prismatic topos. Hence, from these apparent tight connections between Wach modules and prismatic $F\textrm{-crystals}$ and $p\textrm{-adic}$ crystalline representations, we expect these objects to play an important role in the study of $p\textrm{-adic}$ nearby cycles of crystalline $\mathbb{Z}_p\textrm{-local systems}$ (for smooth formal schemes) and its comparison to prismatic syntomic complex with coefficients. In summary, within the overarching program sketched above, this paper realises two of our goals (see Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} and Theorem [Theorem 11](#intro_thm:qdeformation_dcrys){reference-type="ref" reference="intro_thm:qdeformation_dcrys"}). Additionally, we provide interesting applications of our results to purity statements in $p\textrm{-adic}$ Hodge theory (see Theorem [Theorem 9](#intro_thm:purity_crystalline){reference-type="ref" reference="intro_thm:purity_crystalline"} and Corollary [Corollary 10](#intro_cor:purity_crystalline){reference-type="ref" reference="intro_cor:purity_crystalline"}). ## The arithmetic case {#subsec:arithmetic_case} Let $p$ be a fixed prime number and $\kappa$ a perfect field of characteristic $p$; set $O_F := W(\kappa)$ to be the ring of $p\textrm{-typical}$ Witt vectors with coefficients in $\kappa$ and $F := O_F[1/p]$. We fix an algebraic closure $\overline{F}$ of $F$, and let $G_F := \textup{Gal}(\overline{F}/F)$ denote the absolute Galois group of $F$. Moreover, set $F_{\infty}:= \cup_n F(\mu_{p^n})$, let $F_{\infty}^{\flat}$ denote the tilt of $F_{\infty}$ (see §[1.5](#subsec:setup_nota){reference-type="ref" reference="subsec:setup_nota"}) and denote by $\Gamma_F := \textup{Gal}(F_{\infty}/F)$. Furthermore, fix $\varepsilon := (1, \zeta_p, \zeta_{p^2}, \ldots) \in O_{F_{\infty}}^{\flat}$, $\mu := [\varepsilon]-1, [p]_q := \tilde{\xi}:= \varphi(\mu)/\mu \in \mathbf{A}_{\inf}(O_{F_{\infty}}) := W(O_{F_{\infty}}^{\flat})$, the ring of $p\textrm{-typical}$ Witt vectors with coefficients in $O_{F_{\infty}}^{\flat}$. Let $\mathbf{A}_F^+ := O_F\llbracket \mu \rrbracket \subset \mathbf{A}_{\inf}(O_{F_{\infty}})$, which is stable under the $(\varphi, \Gamma_F)\textrm{-action}$ on $\mathbf{A}_{\inf}(O_{F_{\infty}})$. We equip $\mathbf{A}_F^+$ with induced structures and note the following: **Definition 1**. Let $a, b \in \mathbb{Z}$ with $b \geq a$. A *Wach module* over $\mathbf{A}_F^+$ with weights in the interval $[a, b]$ is a finite free $\mathbf{A}_F^+\textrm{-module}$ $N$ satisfying the following: 1. $N$ is equipped with a continuous and semilinear action of $\Gamma_F$ such that the induced action is trivial on $N/\mu N$. 2. There is a Frobenius-semilinear operator $\varphi: N[1/\mu] \rightarrow N[1/\varphi(\mu)]$ commuting with the action of $\Gamma_F$ such that $\varphi(\mu^b N) \subset \mu^b N$, and the map $(1 \otimes \varphi) : \varphi^{\ast}(\mu^b N) \rightarrow \mu^b N$ is injective and its cokernel is killed by $[p]_q^{b-a}$. Denote by $(\varphi, \Gamma_F)\textup{-Mod}_{\mathbf{A}_F^+}^{[p]_q}$ the category of Wach modules over $\mathbf{A}_F^+$, with morphisms between objects being $\mathbf{A}_F^+\textrm{-linear}$, $\varphi\textrm{-equivariant}$ (after inverting $\mu$) and $\Gamma_F\textrm{-equivariant}$ morphisms. Let $\textup{Rep}_{\mathbb{Z}_p}^{\textup{cris}}(G_F)$ denote the category of $\mathbb{Z}_p\textrm{-lattices}$ inside $p\textrm{-adic}$ crystalline representations of $G_F$. To any $T$ in $\textup{Rep}_{\mathbb{Z}_p}^{\textup{cris}}(G_K)$ one can functorially attach the Wach module $\mathbf{N}_F(T)$ over $\mathbf{A}_F^+$ as in [@berger-limites]. From [@wach-free; @colmez-hauteur; @berger-limites] we have the following: **Theorem 2**. *The Wach module functor induces an equivalence of categories $$\begin{aligned} \textup{Rep}_{\mathbb{Z}_p}^{\textup{cris}}(G_F) &\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}(\varphi, \Gamma_F)\textup{-Mod}_{\mathbf{A}_F^+}^{[p]_q}\\ T &\longmapsto \mathbf{N}_F(T), \end{aligned}$$ with a quasi-inverse given as $N \mapsto \big(W(\mathbb{C}_p^{\flat}) \otimes_{\mathbf{A}_F^+} N\big)^{\varphi=1}$, where $\mathbb{C}_p := \widehat{\overline{F}}$.* *Remark 3*. From [@abhinandan-imperfect-wach Theorem 1.5], a statement analogous to Theorem [Theorem 2](#intro_thm:crystalline_wach_classical){reference-type="ref" reference="intro_thm:crystalline_wach_classical"} is true in the greater generality of absolutely unramified extension of $\mathbb{Q}_p$ with imperfect residue field, for example, the field $L$ described in §[1.2](#subsec:base_relative){reference-type="ref" reference="subsec:base_relative"} below. ## The relative case {#subsec:base_relative} Let $d \in \mathbb{N}$ and $X_1, X_2, \ldots, X_d$ be some indeterminates. Set $O_F\langle X_1^{\pm 1}, \ldots, X_d^{\pm 1}\rangle$ to be the $p\textrm{-adic}$ completion of Laurent polynomial ring $O_F[X_1^{\pm 1}, \ldots, X_d^{\pm 1}]$, and let $R$ denote the $p\textrm{-adic}$ completion of an étale algebra over $O_F\langle X_1^{\pm 1}, \ldots, X_d^{\pm 1}\rangle$ with non-empty and geometrically integral special fiber. Let $G_R$ denote the étale fundamental group of $R[1/p]$ and $\Gamma_{R}$ the Galois group of $R_{\infty}[1/p]$ over $R[1/p]$, where $R_{\infty}$ is obtained by adjoining to $R$ all $p\textrm{-power}$ roots of unity and all $p\textrm{-power}$ roots of $X_i$ for all $1 \leq i \leq d$. We have $\Gamma_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbb{Z}_p(1)^d \rtimes \mathbb{Z}_p^{\times}$ (see §[2](#sec:period_rings_padic_reps){reference-type="ref" reference="sec:period_rings_padic_reps"} for precise definitions). Set $O_L := (R_{(p)})^{\wedge}$ to be a complete discrete valuation ring with uniformiser $p$, residue field a finite étale extension of $\kappa(X_1, \ldots, X_d)$ and set $L = O_L[1/p]$. Let $G_L$ denote the absolute Galois group of $L$ such that we have a continuous homomorphism $G_L \rightarrow G_R$; let $\Gamma_L$ denote the Galois group of the $L_{\infty}$ over $L$, where $L_{\infty}$ is obtained by adjoining to $L$ all $p\textrm{-power}$ roots of unity and all $p\textrm{-power}$ roots of $X_i$ for all $1 \leq i \leq d$. The homomorphism $G_L \rightarrow G_R$ induces a continuous isomorphism $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$. In this setting, we have the theory of crystalline representations of $G_R$ [@brinon-relatif] and the theory of étale $(\varphi, \Gamma)\textrm{-modules}$ [@andreatta-phigamma; @andreatta-brinon]. ### Relative Wach modules For $1 \leq i \leq d$, fix $X_i^{\flat} := (X_i, X_i^{1/p}, \ldots) \in R_{\infty}^{\flat}$ and $[X_i^{\flat}] \in \mathbf{A}_{\inf}(R_{\infty})$. Let $\mathbf{A}_R^+$ denote the $(p, \mu)\textrm{-adic}$ completion of the unique extension of the $(p, \mu)\textrm{-adic}$ completion of $O_F[\mu, [X_1^{\flat}]^{\pm 1}, \ldots, [X_d^{\flat}]^{\pm 1}]$ along the $p\textrm{-adically}$ completed étale map $O_F\langle X_1^{\pm 1}, \ldots, X_d^{\pm 1}\rangle \rightarrow R$ (see §[1.5](#subsec:setup_nota){reference-type="ref" reference="subsec:setup_nota"} and §[2.2](#subsec:ainf_relative){reference-type="ref" reference="subsec:ainf_relative"}). The ring $\mathbf{A}_R^+$ is equipped with a Frobenius endomorphism $\varphi$ and a continuous action of $\Gamma_R$, set $\mathbf{A}_L^+$ to be the $(p, \mu)\textrm{-adic}$ completion of $(\mathbf{A}_R^+)_{(p, \mu)}$ equipped with induced Frobenius endomorphism $\varphi$ and continuous action of $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$. We define the following: **Definition 4**. Let $a, b \in \mathbb{Z}$ with $b \geq a$. A *Wach module* over $\mathbf{A}_R^+$ with weights in the interval $[a, b]$ is a finitely generated $\mathbf{A}_R^+\textrm{-module}$ $N$ satisfying the following assumptions: 1. The sequences $\{p, \mu\}$ and $\{\mu, p\}$ are regular on $N$. 2. $N$ is equipped with a continuous and semilinear action of $\Gamma_{R}$ such that the induced action is trivial on $N/\mu N$. 3. There is a Frobenius-semilinear operator $\varphi: N[1/\mu] \rightarrow N[1/\varphi(\mu)]$ commuting with the action of $\Gamma_R$ such that $\varphi(\mu^b N) \subset \mu^b N$, and the map $(1 \otimes \varphi) : \varphi^{\ast}(\mu^b N) \rightarrow \mu^b N$ is injective and its cokernel is killed by $[p]_q^{b-a}$. Denote by $(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q}$ the category of Wach modules over $\mathbf{A}_R^+$, with morphisms between objects being $\mathbf{A}_R^+\textrm{-linear}$, $\varphi\textrm{-equivariant}$ (after inverting $\mu$) and $\Gamma_R\textrm{-equivariant}$ morphisms. *Remark 5*. The condition (1) in Definition [Definition 4](#intro_defi:wach_mods_relative){reference-type="ref" reference="intro_defi:wach_mods_relative"} is completely new and relaxes finite projectivity assumption of relative Wach modules in [@abhinandan-relative-wach-i Definition 4.8]. Moreover, condition (1) above is equivalent to the vanishing of local cohomology of $N$ with respect to the ideal $(p, \mu) \subset \mathbf{A}_R^+$ in degree 1 (see Lemma [Lemma 31](#lem:local_coh){reference-type="ref" reference="lem:local_coh"} and Remark [Remark 32](#lem:compsupp_coh){reference-type="ref" reference="lem:compsupp_coh"}). Furthermore, one can also show that $N[1/p]$ is finite projective over $\mathbf{A}_R^+[1/p]$ (see Proposition [Proposition 73](#prop:finiteproj_torus){reference-type="ref" reference="prop:finiteproj_torus"}), $N[1/\mu]$ is finite projective over $\mathbf{A}_R^+[1/\mu]$ (see Proposition [Proposition 35](#prop:wachmod_proj_pmu){reference-type="ref" reference="prop:wachmod_proj_pmu"}) and $N = N[1/p] \cap N[1/\mu] \subset N[1/p, 1/\mu]$ (see Lemma [Lemma 33](#lem:wach_saturated){reference-type="ref" reference="lem:wach_saturated"}). *Remark 6*. Definition [Definition 4](#intro_defi:wach_mods_relative){reference-type="ref" reference="intro_defi:wach_mods_relative"} may be adapted to the case of a field, i.e. over the ring $\mathbf{A}_F^+$ (resp. $\mathbf{A}_L^+$). In such cases, from the assumptions of Definition [Definition 4](#intro_defi:wach_mods_relative){reference-type="ref" reference="intro_defi:wach_mods_relative"} it follows that a Wach module over $\mathbf{A}_F^+$ (resp. $\mathbf{A}_L^+$) is necessarily finite free. Indeed, if $N$ is a Wach module over $\mathbf{A}_F^+$ (resp. $\mathbf{A}_L^+$), in the sense of Definition [Definition 4](#intro_defi:wach_mods_relative){reference-type="ref" reference="intro_defi:wach_mods_relative"}, then one first observes that $N$ is in fact torsion-free since $N \subset N[1/p]$ and the latter is finite free over $\mathbf{A}_F^+[1/p]$ (resp. $\mathbf{A}_L^+[1/p]$) by [@abhinandan-imperfect-wach Lemma 2.14]. Then using [@fontaine-phigamma §B.1.2.4 Proposition] it follows that $N$ is finite free (resp. Lemma [Lemma 33](#lem:wach_saturated){reference-type="ref" reference="lem:wach_saturated"} and [@abhinandan-imperfect-wach Remark 2.15]). In particular, over $\mathbf{A}_F^+$ (resp. $\mathbf{A}_L^+$), Definition [Definition 4](#intro_defi:wach_mods_relative){reference-type="ref" reference="intro_defi:wach_mods_relative"} is equivalent to Definition [Definition 1](#defi:wach_mod_point){reference-type="ref" reference="defi:wach_mod_point"} (resp. [@abhinandan-imperfect-wach Definition 1.3]). Set $\mathbf{A}_R:= \mathbf{A}_R^+[1/\mu]^{\wedge}$ as the $p\textrm{-adic}$ completion, equipped with induced Frobenius endomorphism $\varphi$ and continuous action of $\Gamma_R$, similarly set $\mathbf{A}_L:= \mathbf{A}_L^+[1/\mu]^{\wedge}$ equipped with induced Frobenius endomorphism $\varphi$ and continuous action of $\Gamma_L$. Let $T$ be a finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$, then one can functorially attach to $T$ a finite projective étale $(\varphi, \Gamma_R)\textrm{-module}$ $\mathbf{D}_R(T)$ over $\mathbf{A}_R$ of rank $=\textup{rk}_{\mathbb{Z}_p} T$, equipped with a semilinear and continuous action of $\Gamma_R$ and a Frobenius-semilinear operator $\varphi$ commuting with the action of $\Gamma_R$. In fact, the preceding functor induces a categorical equivalence between the category of finite free $\mathbb{Z}_p\textrm{-representations}$ of $G_R$ and the category of finite projective étale $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{A}_R$ (see [@andreatta-phigamma Theorem 7.11]). Additionally, the category of Wach modules over $\mathbf{A}_R^+$ can be realised as a full subcategory of étale $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{A}_R$ (see Proposition [Proposition 38](#prop:wach_etale_ff_relative){reference-type="ref" reference="prop:wach_etale_ff_relative"}). ### Main results Let $\textup{Rep}_{\mathbb{Z}_p}^{\textup{cris}}(G_R)$ denote the category of $\mathbb{Z}_p\textrm{-lattices}$ inside $p\textrm{-adic}$ crystalline representations of $G_R$. For $T$ a $\mathbb{Z}_p\textrm{-lattice}$ inside a $p\textrm{-adic}$ crystalline representation of $G_R$, we construct a Wach module $\mathbf{N}_R(T)$ over $\mathbf{A}_R^+$, functorial in $T$, and contained in $\mathbf{D}_R(T)$ (see Theorem [Theorem 53](#thm:crys_fh_relative){reference-type="ref" reference="thm:crys_fh_relative"}). The main result of this article is as follows: **Theorem 7** (Corollary [Corollary 55](#cor:crystalline_wach_equivalence_relative){reference-type="ref" reference="cor:crystalline_wach_equivalence_relative"}). *The Wach module functor induces an equivalence of categories $$\begin{aligned} \textup{Rep}_{\mathbb{Z}_p}^{\textup{cris}}(G_R) &\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q}\\ T &\longmapsto \mathbf{N}_R(T), \end{aligned}$$ with a quasi-inverse given as $N \mapsto \mathbf{T}_R(N) := \big(W\big(\overline{R}^{\flat}[1/p^{\flat}]\big) \otimes_{\mathbf{A}_R^+} N\big)^{\varphi=1}$.* *Remark 8*. In Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} we do not expect the functor $\mathbf{N}_R$ to be an exact equivalence. However, note that, after inverting $p$, the Wach module functor induces an exact equivalence between $\otimes\textrm{-categories}$ $\textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{B}_R^+}^{[p]_q}$, via $V \mapsto \mathbf{N}_R(V)$ where $\mathbf{B}_R^+ = \mathbf{A}_R^+[1/p]$, and with an exact $\otimes\textrm{-compatible}$ quasi-inverse given as $M \mapsto \mathbf{V}_R(M) := \big(W(\overline{R}^{\flat}[1/p^{\flat}]) \otimes_{\mathbf{A}_R^+} M\big)^{\varphi=1}$ (see Corollary [Corollary 56](#cor:crystalline_wach_rat_equivalence_relative){reference-type="ref" reference="cor:crystalline_wach_rat_equivalence_relative"}). As an application of Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}, we obtain the following purity statement: **Theorem 9** (Theorem [Theorem 57](#thm:purity_crystalline){reference-type="ref" reference="thm:purity_crystalline"}). *Let $V$ be a $p\textrm{-adic}$ representation of $G_R$. Then $V$ is crystalline as a representation of $G_R$ if and only if it is crystalline as a representation of $G_L$.* For a $p\textrm{-adic}$ representation $V$ of $G_R$, let $\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ denote the associated filtered $(\varphi, \partial)\textrm{-module}$ over $R[1/p]$ (see [@brinon-relatif §8.2]). We deduce the following criterion for checking crystallinity: **Corollary 10** (Theorem [Theorem 57](#thm:purity_crystalline){reference-type="ref" reference="thm:purity_crystalline"} & Corollary [Corollary 58](#cor:odcrys_functoriality){reference-type="ref" reference="cor:odcrys_functoriality"}). *Let $V$ be a $p\textrm{-adic}$ representation of $G_R$. Then $V$ is crystalline if and only if $\textup{rk}_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) = \dim_{\mathbb{Q}_p} V$. Moreover, under these equivalent conditions, there exists a natural isomorphism $L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, L}(V)$ of filtered $(\varphi, \partial)\textrm{-modules}$ over $L$.* Important inputs for the proof of Corollary [Corollary 10](#intro_cor:purity_crystalline){reference-type="ref" reference="intro_cor:purity_crystalline"} are Theorem [Theorem 9](#intro_thm:purity_crystalline){reference-type="ref" reference="intro_thm:purity_crystalline"} and a careful study of period rings for the localisation of $\overline{R}$ at its minimal primes above $(p) \subset R$ (see §[2.1](#subsec:localisation){reference-type="ref" reference="subsec:localisation"}). ### Strategy for the proof of Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} {#strategy-for-the-proof-of-theorem-intro_thmcrystalline_wach_equivalence} The proof of Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} crucially uses analogous results obtained in the imperfect residue field case (see [@abhinandan-imperfect-wach Theorem 1.5]). Starting with a Wach module $N$ over $\mathbf{A}_R^+$, we use ideas from [@abhinandan-relative-wach-i Theorem 4.25 & Proposition 4.28], the observation that $\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} N$ is a Wach module over $\mathbf{A}_L^+$ as well as [@abhinandan-imperfect-wach Lemma 3.6 & Theorem 3.12], to establish that $\mathbf{T}_R(N)$ is a $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ such that $\mathbf{T}_R(N)[1/p]$ is crystalline (see Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"}). Conversely, starting with a $\mathbb{Z}_p\textrm{-lattice}$ $T$ inside a $p\textrm{-adic}$ crystalline representation of $G_R$, we observe that $T[1/p]$ is a $p\textrm{-adic}$ crystalline representation of $G_L$, and we use [@abhinandan-imperfect-wach Theorem 4.1] to obtain a unique Wach module $\mathbf{N}_L(T)$ over $\mathbf{A}_L^+$. Moreover, from the theory of $(\varphi, \Gamma)\textrm{-modules}$ we have an étale $(\varphi, \Gamma_R)\textrm{-module}$ $\mathbf{D}_R(T)$ over $\mathbf{A}_R$ (see [@andreatta-phigamma]). We set $\mathbf{N}_R(T) := \mathbf{N}_L(T) \cap \mathbf{D}_R(T) \subset \mathbf{D}_L(T)$ as an $\mathbf{A}_R^+\textrm{-module}$, where $\mathbf{D}_L(T)$ is the $(\varphi, \Gamma_L)\textrm{-module}$ over $\mathbf{A}_L$, associated to $T$. Then using the compatible Frobenius-semilinear endomorphism $\varphi$ and continuous action of $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$ on $\mathbf{N}_L(T)$ and $\mathbf{D}_R(T)$, we equip the $\mathbf{A}_R^+\textrm{-module}$ $\mathbf{N}_R(T)$ with a natural $(\varphi, \Gamma_R)\textrm{-action}$. Finally, we use properties of $\mathbf{N}_L(T)$ and $\mathbf{D}_R(T)$ to show that $\mathbf{N}_R(T)$ is the unique Wach module associated to $T$ (similar to the relative Breuil-Kisin case in [@du-liu-moon-shimizu]). This completes a sketch of our proof of Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}. ## Wach modules as $q\textrm{-deformations}$ {#subsec:q_deformation} In §[5](#sec:wachmod_qconnection){reference-type="ref" reference="sec:wachmod_qconnection"} we recall the definition of $q\textrm{-connection}$ axiomatically, following [@morrow-tsuji]. Moreover, we show that a Wach module $N$ over $\mathbf{A}_R^+$ can also be seen as a module with $q\textrm{-connection}$. More precisely, let $A := \mathbb{Z}_p\llbracket \mu \rrbracket$, and let $\{\gamma_1, \ldots, \gamma_d\}$ be topological generators of the geometric part of $\Gamma_R$, i.e. $\Gamma_R'$ (see §[2](#sec:period_rings_padic_reps){reference-type="ref" reference="sec:period_rings_padic_reps"}). Then in Proposition [Proposition 67](#defi:wachmod_qconnection){reference-type="ref" reference="defi:wachmod_qconnection"} we show that the $q\textrm{-connection}$ defined as $$\nabla_q : N \longrightarrow N \otimes_{\mathbf{A}_R^+} \Omega^1_{\mathbf{A}_R^+/A}, \hspace{5mm} x \longmapsto \textstyle\sum_{i=1}^d \tfrac{\gamma_i(x)-x}{\mu} \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}([X_i^{\flat}]),$$ describes $(N, \nabla_q)$ as a $\varphi\textrm{-module}$ with $(p, [p]_q)\textrm{-adically}$ quasi-nilpotent flat $q\textrm{-connection}$ over $\mathbf{A}_R^+$. We equip $N$ with the Nygaard filtration as in Definition [Definition 47](#defi:nygaard_fil){reference-type="ref" reference="defi:nygaard_fil"}. Then using [@morrow-tsuji Remarks 2.4 & 2.10], it follows that $N/\mu N$ is a $\varphi\textrm{-module}$ over $R$ equipped with a $p\textrm{-adically}$ quasi-nilpotent connection and we further equip it with a filtration $\textup{Fil}^k(N/\mu N)$ given as the image of $\textup{Fil}^k N$ under the surjection $N \twoheadrightarrow N/\mu N$. We equip $N[1/p]/\mu N[1/p] = (N/\mu N)[1/p]$ with induced structures, in particular, it is a filtered $(\varphi, \partial)\textrm{-module}$ over $R[1/p]$. **Theorem 11** (Theorem [Theorem 70](#thm:qdeformation_dcrys){reference-type="ref" reference="thm:qdeformation_dcrys"}). *Let $N$ be a Wach module over $\mathbf{A}_R^+$ and $V := \mathbf{T}_R(N)[1/p]$, the associated crystalline representation from Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}. Then we have $(N/\mu N)[1/p] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ as filtered $(\varphi, \partial)\textrm{-modules}$.* Note that $\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ denotes the filtered $(\varphi, \partial)\textrm{-module}$ over $R[1/p]$ associated to $V$ (see [@brinon-relatif §8.2]). Proof of the theorem follows from computations done for the proof of Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"} (based on ideas developed in the proof of [@abhinandan-relative-wach-i Theorem 4.25 & Proposition 4.28] and [@abhinandan-imperfect-wach Theorem 1.7]). Finally, let us summarise the relationship between various categories considered in Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} and Theorem [Theorem 11](#intro_thm:qdeformation_dcrys){reference-type="ref" reference="intro_thm:qdeformation_dcrys"}. Recall that $\textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R)$ is the category of $p\textrm{-adic}$ crystalline representations of $G_R$, and let $\textup{MF}_R(\varphi, \partial)$ denote the category of filtered $(\varphi, \partial)\textrm{-modules}$ over $R[1/p]$. From [@brinon-relatif §8.2] we have a $\otimes\textrm{-compatible}$ functor $\pazocal{O}\mathbf{D}_{\textup{cris}, R}: \textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R) \rightarrow \textup{MF}_R(\varphi, \partial)$, and let $\textup{MF}_R^{\textup{ad}}(\varphi, \partial)$ denote its essential image. Then from [@brinon-relatif Théorème 8.5.1] we have an exact equivalence of $\otimes\textrm{-categories}$ $\pazocal{O}\mathbf{D}_{\textup{cris}, R}: \textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\textup{MF}_R^{\textup{ad}}(\varphi, \partial)$, with an exact $\otimes\textrm{-compatible}$ quasi-inverse $\pazocal{O}\mathbf{V}_{\textup{cris}, R}$ (see §[2.6](#subsec:relative_padicreps){reference-type="ref" reference="subsec:relative_padicreps"}). Remark [Remark 8](#intro_rem:crystalline_wach_equivalence_rational){reference-type="ref" reference="intro_rem:crystalline_wach_equivalence_rational"} and Theorem [Theorem 11](#intro_thm:qdeformation_dcrys){reference-type="ref" reference="intro_thm:qdeformation_dcrys"} can be summarised as follows: **Corollary 12** (Corollary [Corollary 72](#cor:cat_equiv_diagram){reference-type="ref" reference="cor:cat_equiv_diagram"}). *Functors in the following diagram induce exact equivalence of $\otimes\textrm{-categories}$* ## Relation to previous works Our first main result, Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}, is a generalisation of Theorem [Theorem 2](#intro_thm:crystalline_wach_classical){reference-type="ref" reference="intro_thm:crystalline_wach_classical"} from [@wach-free; @colmez-hauteur; @berger-limites] and [@abhinandan-imperfect-wach Theorem 1.5]. That said, the methods of op. cit. do not directly apply to our current situtation. In fact, the proof of Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} uses crucial inputs of results and ideas from [@abhinandan-relative-wach-i] and [@abhinandan-imperfect-wach]. Recent developments in the theory of prismatic $F\textrm{-crystals}$ in [@bhatt-scholze-crystals; @du-liu-moon-shimizu; @guo-reinecke] would suggest that there is a categorical equivalence between the category of Wach modules over $\mathbf{A}_R^+$ and (completed/analytic) prismatic $F\textrm{-crystals}$ on the absolute prismatic site $(\textup{Spf }R)_{{\mathlarger{\mathbbl{\Delta}}}}$. From that perspective, Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} could be seen as an analogue of [@du-liu-moon-shimizu Theorem 1.2 & Proposition 1.4], i.e. the categorical equivalence between Kisin descent data (up to twisting) and lattices inside crystalline representations of $G_R$. However, a key difference with loc. cit. is that Wach modules admit a natural action of $\Gamma_R$, instead of a descent datum. Nonetheless, as our base ring $R$ is absolutely unramified (at $p$), the action of $\Gamma_R$ is rich enough to recover a $\mathbb{Z}_p\textrm{-lattice}$ inside a $p\textrm{-adic}$ crystalline representation of $G_R$, thus establising Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}. In the current paper, we provide two applications of Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}. The first application, i.e. Theorem [Theorem 9](#intro_thm:purity_crystalline){reference-type="ref" reference="intro_thm:purity_crystalline"} establishes a certain purity statement for crystalline representations. Our result is similar to the purity statement for Hodge-Tate representations in [@tsuji-purity-hodge-tate Theorem 9.1] and rigidity of de Rham local systems in [@liu-zhu Theorem 1.3]. It should be noted that the purity result in Theorem [Theorem 9](#intro_thm:purity_crystalline){reference-type="ref" reference="intro_thm:purity_crystalline"} can also be obtained by combining [@liu-zhu Theorem 1.3] and unpublished results of [@tsuji-crystalline Theorem 5.4.8]. Moreover, the result of loc. cit. works for general ramified (at $p$) small base. A similar statement has been obtained in [@moon Theorem 1.4] using the results of [@du-liu-moon-shimizu]. The second application of Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"} is given in Corollary [Corollary 10](#intro_cor:purity_crystalline){reference-type="ref" reference="intro_cor:purity_crystalline"}. Our result provides a new criterion for checking crystallinity of a $p\textrm{-adic}$ representation of $G_R$. Note that the analogous statement for de Rham representations is true from the results of [@liu-zhu]. However, our result in crystalline case is entirely new and uses Theorem [Theorem 9](#intro_thm:purity_crystalline){reference-type="ref" reference="intro_thm:purity_crystalline"} as an important input. At this point, for expert readers, it is worth mentioning that for general ramified (at $p$) small base, a statement analogous to Corollary [Corollary 10](#intro_cor:purity_crystalline){reference-type="ref" reference="intro_cor:purity_crystalline"} appears to be true. In particular, we expect that one can deduce the statement using [@liu-zhu Theorem 1.3], [@tsuji-crystalline Theorem 5.4.8] and employing arguments similar to our proof of Theorem [Theorem 57](#thm:purity_crystalline){reference-type="ref" reference="thm:purity_crystalline"}. For our second main result, Theorem [Theorem 11](#intro_thm:qdeformation_dcrys){reference-type="ref" reference="intro_thm:qdeformation_dcrys"}, the motivation for interpreting a Wach module as a $q\textrm{-de Rham}$ complex and as $q\textrm{-deformation}$ of crystalline cohomology, i.e. $\pazocal{O}\mathbf{D}_{\textup{cris}}$, comes from [@fontaine-phigamma §B.2.3], [@berger-limites Théorème III.4.4] and [@scholze-q-deformations §6]. In particular, we provide a direct generalisation of [@berger-limites Théorème III.4.4], as well as verify expectations put forth in [@abhinandan-relative-wach-i Remark 4.48] and [@abhinandan-imperfect-wach Remark 1.8] (see Remark [Remark 71](#rem:qdeformation_dcrys_imperfect){reference-type="ref" reference="rem:qdeformation_dcrys_imperfect"} for the latter). ## Setup and notations {#subsec:setup_nota} In this section we will describe our setup and fix some notations, which are essentially the same as in [@abhinandan-relative-wach-i §1.4]. We will work under the convention that $0 \in \mathbb{N}$, the set of natural numbers. Let $p$ be a fixed prime number, $\kappa$ a perfect field of characteristic $p$, $O_F := W(\kappa)$ the ring of $p\textrm{-typical}$ Witt vectors with coefficients in $\kappa$. Then $O_F$ is a complete discrete valuation ring with uniformiser $p$ and set $F := O_F[1/p]$ to be the fraction field of $O_F$. Let $\overline{F}$ denote a fixed algebraic closure of $F$ so that its residue field, denoted as $\overline{\kappa}$, is an algebraic closure of $\kappa$. Furthermore, denote the absolute Galois group of $F$ to be $G_F := \textup{Gal}(\overline{F}/F)$. *Notation 1*. Let $\Lambda$ be an $I\textrm{-adically}$ complete algebra for a finitely generated ideal $I \subset \Lambda$. Let $Z := (Z_1, \ldots, Z_s)$ denote a set of indeterminates and $\mathbf{k}:= (k_1, \ldots, k_s) \in \mathbb{N}^s$ be a multi-index, then we write $Z^{\mathbf{k}} := Z_1^{k_1} \cdots Z_s^{k_s}$. For $\mathbf{k}\rightarrow +\infty$ we will mean that $\sum k_i \rightarrow +\infty$. Define $$\Lambda\langle Z\rangle := \Big\{\sum_{\mathbf{k}\in \mathbb{N}^s} a_{\mathbf{k}} Z^{\mathbf{k}}, \hspace{1mm} \textrm{where} \hspace{1mm} a_{\mathbf{k}} \in \Lambda \hspace{1mm} \textrm{and} \hspace{1mm} a_{\mathbf{k}} \rightarrow 0 \hspace{1mm} I\textrm{-adically} \hspace{1mm} \textrm{as} \hspace{1mm} \mathbf{k}\rightarrow +\infty\Big\}.$$ We fix $d \in \mathbb{N}$ and let $X := (X_1, X_2, \ldots, X_d)$ be some indeterminates. Let $R$ be the $p\textrm{-adic}$ completion of an étale algebra over $R^{\square}:= O_F\langle X, X^{-1}\rangle$, with non-empty geometrically integral special fiber. We fix an algebraic closure $\overline{\textup{Frac}(R)}$ of $\textup{Frac}(R)$ containing $\overline{F}$. Let $\overline{R}$ denote the union of finite $R\textrm{-subalgebras}$ $S \subset \overline{\textup{Frac}(R)}$, such that $S[1/p]$ is étale over $R[1/p]$. Let $\overline{\eta}$ denote the fixed geometric point of the generic fiber $\textup{Spec}\hspace{0.5mm}R[1/p]$ (defined by $\overline{\textup{Frac}(R)}$), and let $G_R := \pi_1^{\textup{\'et}}\big(\textup{Spec}\hspace{0.5mm}R[1/p], \overline{\eta}\big)$ denote the étale fundamental group. We can write this étale fundamental group as the Galois group (of the fraction field of $\overline{R}[1/p]$ over the fraction field of $R[1/p]$), i.e. $G_R = \pi_1^{\textup{\'et}}(\textup{Spec}\hspace{0.5mm}(R[1/p]), \overline{\eta}) = \textup{Gal}(\overline{R}[1/p] / R[1/p])$. For $k \in \mathbb{N}$, let $\Omega^k_R$ denote the $p\textrm{-adic}$ completion of module of $k\textrm{-differentials}$ of $R$ relative to $\mathbb{Z}$. Then, we have $\Omega^1_R = \oplus_{i=1}^d R \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}X_i$, and $\Omega^k_R = \wedge_R^k \Omega^1_R$. Let $\varphi$ denote an endomorphism of $R^{\square}$ which extends the natural Frobenius on $O_F$ by setting $\varphi(X_i) = X_i^p$ for all $1 \leq i \leq d$. The morphism $\varphi : R^{\square}\rightarrow R^{\square}$ is flat by [@brinon-relatif Lemma 7.1.5], and it is faithfully flat since $\varphi(\mathfrak{m}) \subset \mathfrak{m}$ for any maximal ideal $\mathfrak{m}\subset R^{\square}$. Moreover, using Nakayama Lemma and the fact that the absolute Frobenius on $R^{\square}/p$ is evidently of degree $p^d$, it easily follows that $\varphi$ on $R^{\square}$ is finite of degree $p^d$. Recall that the $O_F\textrm{-algebra}$ $R$ is given as $p\textrm{-adic}$ completion of an étale algebra $R^{\square}$, therefore, the Frobenius endomorphism $\varphi$ on $R^{\square}$ admits a unique extension $\varphi : R \rightarrow R$ such that the induced map $\varphi : R/p \rightarrow R/p$ is the absolute Frobenius $x \mapsto x^p$ (see [@colmez-niziol Proposition 2.1]). Similar to above, again note that the endomorphism $\varphi : R \rightarrow R$ is faithfully flat and finite of degree $p^d$. Let $O_L := (R_{(p)})^{\wedge}$, where ${}^{\wedge}$ denotes the $p\textrm{-adic}$ completion. Let $\overline{L}$ denote a fixed algebraic closure of $L$ with ring of integers $O_{\overline{L}}$ such that we have an embedding $\overline{R}\rightarrow O_{\overline{L}}$. Then we get a continuous homomorphism $G_L := \textup{Gal}(\overline{L}/L) \rightarrow G_R$, inducing an isomorphism $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$. The Frobenius on $R$ extends to a unique Frobenius endomorphism $\varphi : O_L \rightarrow O_L$, lifting the absolute Frobenius on $O_L/pO_L$ (see [@colmez-niziol Proposition 2.1]). Similar to above, $\varphi$ on $O_L$ is faithfuly flat and finite of degree $p^d$. Let $S$ be a commutative ring with $\pi := p^{1/p} \in S$ such that $S$ is $\pi\textrm{-adically}$ complete and $\pi\textrm{-torsion free}$, for example, $S = O_{F_{\infty}}, O_{L_{\infty}}, O_{\overline{F}}, O_{\overline{L}}, R_{\infty}, \overline{R}$. Then the tilt of $S$ is defined as $S^{\flat} := \lim_{\varphi} S/p$ and the tilt of $S[1/p]$ is defined as $S[1/p]^{\flat} := S^{\flat}[1/p^{\flat}]$, where $p^{\flat} := (1, p^{1/p}, \ldots) \in S^{\flat}$ (see [@fontaine-pdivisibles Chapitre V, §1.4] and [@bhatt-morrow-scholze-1 §3]). Finally, consider a $\mathbb{Z}_p\textrm{-algebra}$ $A$ equipped with a lift of the absolute Frobenius on $A/p$, i.e. an endomorphism $\varphi : A \rightarrow A$ such that $\varphi$ modulo $p$ is the absolute Frobenius. Then for any $A\textrm{-module}$ $M$ we write $\varphi^*(M) := A \otimes_{\varphi, A} M$. **Outline of the paper.** This article consists of four main sections. In §[2](#sec:period_rings_padic_reps){reference-type="ref" reference="sec:period_rings_padic_reps"} we collect relevant results in relative $p\textrm{-adic}$ Hodge theory. In §[2.1](#subsec:localisation){reference-type="ref" reference="subsec:localisation"} we consider localisations of $\overline{R}$ at minimal primes above $(p) \subset R$ and study their properties. Then in §[2.2](#subsec:ainf_relative){reference-type="ref" reference="subsec:ainf_relative"}, §[2.3](#subsec:derham_relative){reference-type="ref" reference="subsec:derham_relative"} & §[2.4](#subsec:crystalline_relative){reference-type="ref" reference="subsec:crystalline_relative"} we define relative period rings and study their localisations at primes of $\overline{R}$ above $(p) \subset R$. In §[2.5](#subsec:phigamma_mod_rings){reference-type="ref" reference="subsec:phigamma_mod_rings"} we quickly recall important rings from the theory of relative $(\varphi, \Gamma)\textrm{-modules}$ and in §[2.6](#subsec:relative_padicreps){reference-type="ref" reference="subsec:relative_padicreps"} we recall the relation between $(\varphi, \Gamma)\textrm{-module}$ theory and $p\textrm{-adic}$ representations, as well as, definition and properties of crystalline representations. The aim of §[3](#sec:wach_modules){reference-type="ref" reference="sec:wach_modules"} is to define and study properties of a Wach module in the relative case and the associated representation of $G_R$. In §[3.1](#subsec:wach_mod_props){reference-type="ref" reference="subsec:wach_mod_props"} we define relative Wach modules, study its properties and relate it to étale $(\varphi, \Gamma)\textrm{-modules}$ (see Proposition [Proposition 38](#prop:wach_etale_ff_relative){reference-type="ref" reference="prop:wach_etale_ff_relative"}). Then in §[3.2](#subsec:wach_mod_rep){reference-type="ref" reference="subsec:wach_mod_rep"}, we functorially attach a $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ to a relative Wach module and in §[3.3](#subsec:finite_pqheight_reps){reference-type="ref" reference="subsec:finite_pqheight_reps"} we show that such representations are closely related to finite $[p]_q\textrm{-height}$ representations studied in [@abhinandan-relative-wach-i]. In §[3.4](#subsec:nygaard_fil_wach_mod){reference-type="ref" reference="subsec:nygaard_fil_wach_mod"} we study the Nygaard filtration on relative Wach modules. Finally, in §[3.5](#subsec:wachmod_crystalline){reference-type="ref" reference="subsec:wachmod_crystalline"} we show that the $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ associated to a relative Wach module, as in §[3.2](#subsec:wach_mod_rep){reference-type="ref" reference="subsec:wach_mod_rep"}, is a lattice inside a $p\textrm{-adic}$ crystalline representation of $G_R$ (see Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"}). In §[4](#sec:crystalline_finite_height){reference-type="ref" reference="sec:crystalline_finite_height"} we prove our main result, i.e. Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}. Before proving the theorem, we draw some important conclusions from the statement, in particular, in §[4.1](#subsec:crys_wach_equiv_consequence){reference-type="ref" reference="subsec:crys_wach_equiv_consequence"} we prove Theorem [Theorem 9](#intro_thm:purity_crystalline){reference-type="ref" reference="intro_thm:purity_crystalline"} and Corollary [Corollary 10](#intro_cor:purity_crystalline){reference-type="ref" reference="intro_cor:purity_crystalline"}. Finally, in §[4.2](#subsec:crys_wach_equiv_proof){reference-type="ref" reference="subsec:crys_wach_equiv_proof"} we construct the promised relative Wach module and prove Theorem [Theorem 7](#intro_thm:crystalline_wach_equivalence){reference-type="ref" reference="intro_thm:crystalline_wach_equivalence"}. In §[5](#sec:wachmod_qconnection){reference-type="ref" reference="sec:wachmod_qconnection"}, we state and prove our second main result, i.e. Theorem [Theorem 11](#intro_thm:qdeformation_dcrys){reference-type="ref" reference="intro_thm:qdeformation_dcrys"}. In §[5.1](#subsec:qconnection_formalism){reference-type="ref" reference="subsec:qconnection_formalism"}, we recall the formalism of $q\textrm{-connection}$. Then in §[5.2](#subsec:wachmod_qdeformation){reference-type="ref" reference="subsec:wachmod_qdeformation"}, we show that a Wach module can be interpreted as a module with $q\textrm{-connection}$ (see Proposition [Proposition 67](#defi:wachmod_qconnection){reference-type="ref" reference="defi:wachmod_qconnection"}). Finally, using the computations done in the proof of Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"} we prove Theorem [Theorem 11](#intro_thm:qdeformation_dcrys){reference-type="ref" reference="intro_thm:qdeformation_dcrys"}. **Acknowledgements.** I would like to sincerely thank Takeshi Tsuji for discussing many ideas during the course of this project, reading an earlier version of the article and suggesting improvements. I would also like to thank Yong Suk Moon, Koji Shimizu and Alex Youcis for helpful remarks. This research is supported by JSPS KAKENHI grant numbers 22F22711 and 22KF0094. # Period rings and $p\textrm{-adic}$ representations {#sec:period_rings_padic_reps} We will use the setup and notations from §[1.5](#subsec:setup_nota){reference-type="ref" reference="subsec:setup_nota"}. Recall that $R$ is the $p\textrm{-adic}$ completion of an étale algebra over $O_F\langle X_1^{\pm 1}, \ldots, X_d^{\pm 1} \rangle$ and $O_L := (R_{(p)})^{\wedge}$. Set $R_{\infty}:= \cup_{i=1}^d R[\mu_{p^{\infty}}, X_i^{1/p^{\infty}}]$ and $\overline{R}$ is the union of finite $R\textrm{-subalgebras}$ $S$ in a fixed algebraic closure $\overline{\textup{Frac}(R)} \supset \overline{F}$, such that $S[1/p]$ is étale over $R[1/p]$. We have (see [@abhinandan-relative-wach-i §2 & §3]) $$\begin{aligned} G_R &:= \textup{Gal}(\overline{R}[1/p]/R[1/p]), \hspace{1mm} H_R = \textup{Gal}(\overline{R}[1/p]/R_{\infty}[1/p]),\\ \Gamma_R &:= G_R/H_R = \textup{Gal}(R_{\infty}[1/p]/R[1/p]) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbb{Z}_p(1)^d \rtimes \mathbb{Z}_p^{\times}, \\ \Gamma'_R &:= \textup{Gal}(R_{\infty}[1/p]/R(\mu_{p^{\infty}})[1/p]) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbb{Z}_p(1)^d, \hspace{1mm} \textup{Gal}\big(R(\mu_{p^{\infty}})[1/p]/R[1/p]) = \Gamma_R/\Gamma'_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbb{Z}_p^{\times}.\end{aligned}$$ We fixed $\overline{L}$ as an algebraic closure of $L := O_L[1/p]$ with ring of integers $O_{\overline{L}}$ such that we have an embedding $\overline{R}\rightarrow O_{\overline{L}}$. Then we have a continuous homomorphism of groups $G_L := \textup{Gal}(\overline{L}/L) \rightarrow G_R$ inducing an isomorphism $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$. For $1 \leq i \leq d$, we fix $X_i^{\flat} := (X_i, X_i^{1/p}, X_i^{1/p^2}, \ldots) \in R_{\infty}^{\flat}$ and take $\{\gamma_0, \gamma_1, \ldots, \gamma_d\}$ to be topological generators of $\Gamma_R$ such that $\{\gamma_1, \ldots, \gamma_d\}$ are topological generators of $\Gamma'_R$ and $\gamma_0$ is a topological generator of $\Gamma_R/\Gamma'_R$ and $\gamma_j(X_i^{\flat}) = \varepsilon X_i^{\flat}$ if $i=j$ and $X_i^{\flat}$ otherwise. ## Localisation {#subsec:localisation} Let $\pazocal{S}$ denote the set of minimal primes of $\overline{R}$ above $pR \subset R$. The set $\pazocal{S}$ is equipped with a transitive action of $G_R$ (see [@matsumura Theorem 9.3]). For each prime $\mathfrak{p}\in \pazocal{S}$, set $G_R(\mathfrak{p}):= \{g \in G_R \textrm{ such that } g(\mathfrak{p}) = \mathfrak{p}\}$ as the decomposition group of $G$ at $\mathfrak{p}$. Recall that $O_L = (R_{(p)})^{\wedge}$ and $L = O_L[1/p]$. For each $\mathfrak{p}\in \pazocal{S}$, let $\overline{L}(\mathfrak{p})$ denote an algebraic closure of $L$ with ring of integers $O_{\overline{L}(\mathfrak{p})}$ containing $(\overline{R})_{\mathfrak{p}}$. Set $\widehat{G}_R(\mathfrak{p}):= \textup{Gal}(\overline{L}(\mathfrak{p})/L)$ and we have a natural homomorphism $\widehat{G}_R(\mathfrak{p})\rightarrow G_R$ which factors as $\widehat{G}_R(\mathfrak{p})\twoheadrightarrow G_R(\mathfrak{p})\subset G_R$ (see [@brinon-relatif Lemme 3.3.1]). Note that for each $\mathfrak{p}\in \pazocal{S}$ we have a natural embedding $\overline{R}\subset O_{\overline{L}(\mathfrak{p})}$ and hence we have a (non-canonical) isomorphism of Galois groups $\widehat{G}_R(\mathfrak{p})\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}G_L$. Now, for each $\mathfrak{p}\in \pazocal{S}$, let $\mathbb{C}^+_{\mathfrak{p}}$ denote the $p\textrm{-adic}$ completion of $O_{\overline{L}(\mathfrak{p})}$ and let $\mathbb{C}_{\mathfrak{p}}:= \textup{Frac}(\mathbb{C}^+_{\mathfrak{p}})$. Then $\mathbb{C}_{\mathfrak{p}}$ is an algebraically closed valuation field equipped with a continuous action of $\widehat{G}_R(\mathfrak{p})$ and $(\mathbb{C}^+_{\mathfrak{p}})^{\widehat{G}_R(\mathfrak{p})} = O_L$ (see [@hyodo Theorem 1]). Furthermore, let $\mathbb{C}^+(\mathfrak{p})$ denote the $p\textrm{-adic}$ completion of $(\overline{R})_{\mathfrak{p}}$ and let $\mathbb{C}(\mathfrak{p}) := \mathbb{C}^+(\mathfrak{p})[1/p]$ equipped with a continuous action of $G_R(\mathfrak{p})$. **Lemma 13**. *For each $\mathfrak{p}\in \pazocal{S}$, we have $(\overline{R})_{\mathfrak{p}}\subset \mathbb{C}^+(\mathfrak{p})$ and $(\overline{R})_{\mathfrak{p}}\cap p\mathbb{C}^+(\mathfrak{p})= p(\overline{R})_{\mathfrak{p}}$. Moreover, $(\overline{R})_{\mathfrak{p}}\cap p O_{\overline{L}(\mathfrak{p})}= p(\overline{R})_{\mathfrak{p}}$.* *Proof.* The proof is similar to [@brinon-relatif Proposition 2.0.3]. Let $\mathfrak{p}\in \pazocal{S}$ and $x \in (\overline{R})_{\mathfrak{p}}$. Then there exists a finite normal $R\textrm{-subalgebra}$ $S \subset \overline{R}$ such that $S[1/p]$ is étale over $R[1/p]$ and $\mathfrak{q}:= \mathfrak{p}\cap S$ is a height 1 prime ideal of $S$ with $p \in \mathfrak{q}$ (since $\overline{R}$ is integral over $S$), such that $x \in S_{\mathfrak{q}}$. Moreover, $S_{\mathfrak{q}}$ is a 1-dimensional normal noetherian domain, in particular, a discrete valuation ring. Now if the image of $x$ is zero in $\mathbb{C}^+(\mathfrak{p})$, then we have that $x \in p^n(\overline{R})_{\mathfrak{p}}\cap S_{\mathfrak{q}} = p^nS_{\mathfrak{q}}$ for each $n \in \mathbb{N}$ since $S_{\mathfrak{q}}$ is normal. So $x$ must be zero since $S_{\mathfrak{q}}$ is $p\textrm{-adically}$ separated. This shows the first claim. For the second claim, let $x = py$ for some $y \in \mathbb{C}^+(\mathfrak{p})$. We have that $y \in S_{\mathfrak{q}}[1/p]$ and we need to show that $y \in S_{\mathfrak{q}}$. Let $\widehat{S}_{\mathfrak{q}}$ denote the completion of $S_{\mathfrak{q}}$ for the valuation (say $\upsilon_{\mathfrak{q}}$) described above. Then $\widehat{S}_{\mathfrak{q}}[1/p]$ is a finite separable extension of $L$ and $\widehat{S}_{\mathfrak{q}}$ embeds into $\mathbb{C}^+_{\mathfrak{p}}$. Moreover, the image of $\mathbb{C}^+(\mathfrak{p})$ in $\mathbb{C}_{\mathfrak{p}}$ is contained in $\mathbb{C}^+_{\mathfrak{p}}$, therefore $\upsilon_{\mathfrak{q}}(y) \geq 0$, i.e. $y \in S_{\mathfrak{q}}[1/p] \cap \widehat{S}_{\mathfrak{q}} = S_{\mathfrak{q}}$ as desired. Finally, let $x = pz$ for some $z \in O_{\overline{L}(\mathfrak{p})}$. Then similar to above, we have $z \in S_{\mathfrak{q}}[1/p]$ and $\upsilon_{\mathfrak{q}}(z) \geq 0$, so $z \in S_{\mathfrak{q}}$. This shows the third claim. ◻ All rings discussed above are $p\textrm{-torsion free}$, so from Lemma [Lemma 13](#lem:prbarp){reference-type="ref" reference="lem:prbarp"}, it easily follows that $\mathbb{C}^+(\mathfrak{p})\subset \mathbb{C}^+_{\mathfrak{p}}$ compatible with $\widehat{G}_R(\mathfrak{p})\textrm{-action}$, where the action of $\widehat{G}_R(\mathfrak{p})$ on left hand term factors through $\widehat{G}_R(\mathfrak{p})\twoheadrightarrow G_R(\mathfrak{p})$. In particular, we get that $\mathbb{C}^+(\mathfrak{p})^{G_R(\mathfrak{p})} = O_L$ (see [@brinon-relatif p. 24]). Now note that we have natural injective maps $\overline{R}\rightarrow (\overline{R})_{\mathfrak{p}}\rightarrow O_{\overline{L}(\mathfrak{p})}$. Upon passing to $p\textrm{-adic}$ completions and setting $\mathbb{C}^+(\overline{R}) := \widehat{\overline{R}}$, we obtain natural maps $\mathbb{C}^+(\overline{R}) \rightarrow \mathbb{C}^+(\mathfrak{p})\rightarrow \mathbb{C}^+_{\mathfrak{p}}$, where the first map need not be injective. However, recall that $\overline{R}$ is a direct limit of finite and normal $R\textrm{-algebras}$, therefore the natural map $\overline{R}/p^n \rightarrow \oplus_{\mathfrak{p}\in \pazocal{S}} (\overline{R})_{\mathfrak{p}}/p^n$ is injective. Passing to the limit over $n$, we obtain injective maps $$\label{eq:cplus_gequiv} \mathbb{C}^+(\overline{R}) \longrightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})\longrightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+_{\mathfrak{p}}.$$ Note that in [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"} the left term admits a natural action of $G_R$, the middle term admits a natural action of $\prod_{\mathfrak{p}\in \pazocal{S}}G_R(\mathfrak{p})$ and the right term admits a natural action of $\prod_{\mathfrak{p}\in \pazocal{S}} \widehat{G}_R(\mathfrak{p})$. The two homomorphisms in [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"} are compatible with these actions. Moreover, from [@brinon-relatif Remarque 3.3.2] the middle term of [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"} can be equipped with an action of $G_R$ and the left homomorphism is equivariant with respect to this action of $G_R$. *Remark 14*. Note that $\mathbb{C}^+(\mathfrak{p})$ is an $O_L\textrm{-algebra}$ for each $\mathfrak{p}\in \pazocal{S}$, so the maps in [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"} extend to injective maps $O_L \otimes_R \mathbb{C}^+(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})\rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+_{\mathfrak{p}}$ (see [@brinon-relatif Proposition 3.3.3]). **Lemma 15**. *The $O_L\textrm{-algebra}$ $\mathbb{C}^+(\mathfrak{p})$ is perfectoid in the sense of [@bhatt-morrow-scholze-1 Definition 3.5].* *Proof.* Note that we have $\pi := p^{1/p} \in \overline{R}\subset (\overline{R})_{\mathfrak{p}}\subset \mathbb{C}^+(\mathfrak{p})$ and $\pi^p = p$ divides $p$. Moreover, it is clear that $\mathbb{C}^+(\mathfrak{p})$ is $\pi\textrm{-adically}$ complete. Now consider the following commutative diagram where the left and right vertical arrows are injective by Lemma [Lemma 13](#lem:prbarp){reference-type="ref" reference="lem:prbarp"} and the middle vertical arrow is also injective by a similar argument. So it follows that the top right horizontal arrow is injective as well. From [@bhatt-morrow-scholze-1 Lemma 3.9 and Lemma 3.10] we are left to show that $\varphi : \mathbb{C}^+(\mathfrak{p})/p = (\overline{R})_{\mathfrak{p}}/p \rightarrow (\overline{R})_{\mathfrak{p}}/p = \mathbb{C}^+(\mathfrak{p})/p$ is surjective. So let $x \in (\overline{R})_{\mathfrak{p}}/p$ and take a lift $y \in (\overline{R})_{\mathfrak{p}}$. Then there exists an $a \in \overline{R}\setminus \mathfrak{p}$ such that $ay \in \overline{R}$. Now from [@brinon-relatif Proposition 2.0.1] there exists $z, w \in \overline{R}$ such that $ay = z^p + pw$. Moreover, there exists $b \in \overline{R}\setminus \mathfrak{p}$ and $c \in \overline{R}$ such that $a = b^p + pc$. Then we can write $b^py + pcy = z^p + pw$, or equivalently, $y = (z/b)^p + p(cy+w)/b^p$ with $(z/b)^p \in (\overline{R})_{\mathfrak{p}}$ and $p(cy+w)/b^p \in p(\overline{R})_{\mathfrak{p}}$. Hence, $x = (z/b)^p \mod p(\overline{R})_{\mathfrak{p}}$ proving that $\varphi : (\overline{R})_{\mathfrak{p}}/p \rightarrow (\overline{R})_{\mathfrak{p}}/p$ is surjective. ◻ ## The period ring $\mathbf{A}_{\inf}$ {#subsec:ainf_relative} In this subsection we briefly recall the relative version of $\mathbf{A}_{\inf}$ to be used in the sequel (see [@abhinandan-relative-wach-i §2 and §3] for details). Let $\mathbf{A}_{\inf}(R_{\infty}) := W(R_{\infty}^{\flat})$ and $\mathbf{A}_{\inf}(\overline{R}) := W(\overline{R}^{\flat})$ admitting the Frobenius on Witt vectors and continuous $G_R\textrm{-action}$ (for the weak topology). Moreover, we have $\mathbf{A}_{\inf}(R_{\infty}) = \mathbf{A}_{\inf}(\overline{R})^{H_R}$ (see [@andreatta-phigamma Proposition 7.2]). Let $\varepsilon := (1, \zeta_p, \zeta_{p^2}, \ldots)$, $\overline{\mu}:= \varepsilon-1 \in O_{F_{\infty}}^{\flat}$ and set $\mu := [\varepsilon] - 1, \xi := \mu/\varphi^{-1}(\mu) \in \mathbf{A}_{\inf}(O_{F_{\infty}})$. Let $\chi$ denote the $p\textrm{-adic}$ cyclotomic character, then for $g \in G_R$ we have $g(1+\mu) = (1+\mu)^{\chi(g)}$. Additionally, we have a $G_R\textrm{-equivariant}$ surjection $\theta : \mathbf{A}_{\inf}(\overline{R}) \rightarrow \mathbb{C}^+(\overline{R})$ and $\textup{Ker }\theta = \xi \mathbf{A}_{\inf}(\overline{R})$. The map $\theta$ further induces a $\Gamma_R\textrm{-equivariant}$ surjection $\theta : \mathbf{A}_{\inf}(R_{\infty}) \rightarrow \widehat{R}_{\infty}$. Let $\pazocal{S}$ denote the set of minimal primes of $\overline{R}$ above $pR \subset R$ and for each prime $\mathfrak{p}\in \pazocal{S}$ let $\mathbb{C}_{\mathfrak{p}}$ denote the valuation field described in §[2.1](#subsec:localisation){reference-type="ref" reference="subsec:localisation"} and $\mathbb{C}^+_{\mathfrak{p}}$ its ring of integers. Moreover, from Lemma [Lemma 15](#lem:rbarphat_perfectoid){reference-type="ref" reference="lem:rbarphat_perfectoid"} we have that $\mathbb{C}^+(\mathfrak{p})$ is a perfectoid algebra. So we set $\mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) := W(\mathbb{C}^{+, \flat}_{\mathfrak{p}})$ (resp. $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) := W(\mathbb{C}^+(\mathfrak{p})^{\flat})$) admitting the Frobenius on Witt vectors and continuous $\widehat{G}_R(\mathfrak{p})\textrm{-action}$ (resp. $G_R(\mathfrak{p})\textrm{-action}$). Similar to above, we have a $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ surjection $\theta : \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) \rightarrow \mathbb{C}^+_{\mathfrak{p}}$ with $\textup{Ker }\theta = \xi \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})$ (resp. a $G_R(\mathfrak{p})\textrm{-equivariant}$ surjection $\theta : \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbb{C}^+(\mathfrak{p})$ with $\textup{Ker }\theta = \xi \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))$). **Lemma 16**. *For each $\mathfrak{p}\in \pazocal{S}$ we have $(\varphi, \widehat{G}_R(\mathfrak{p})\textrm{-equivariant})$ embeddings $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})$ and $W(\mathbb{C}(\mathfrak{p})^{\flat}) \rightarrow W(\mathbb{C}_{\mathfrak{p}}^{\flat})$, where the action of $\widehat{G}_R(\mathfrak{p})$ on left hand terms factors through $\widehat{G}_R(\mathfrak{p})\twoheadrightarrow G_R(\mathfrak{p})$. Moreover, we have a $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ identification $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) = \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) \cap W(\mathbb{C}(\mathfrak{p})^{\flat}) \subset W(\mathbb{C}_{\mathfrak{p}}^{\flat})$.* *Proof.* From the discussion before [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"} we have a $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ injective map $\mathbb{C}^+(\mathfrak{p})\rightarrow \mathbb{C}^+_{\mathfrak{p}}$. Applying the tilting functor we further obtain a $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ commutative diagram of rings $$\label{eq:cpplus_cplusp_flat} \begin{tikzcd} \mathbb{C}^+(\mathfrak{p})^{\flat} \arrow[r] \arrow[d] & \mathbb{C}^{+, \flat}_{\mathfrak{p}}\arrow[d]\\ \mathbb{C}(\mathfrak{p})^{\flat} \arrow[r] & \mathbb{C}_{\mathfrak{p}}^{\flat}, \end{tikzcd}$$ where the vertical arrows are injective. Note that the natural map $\mathbb{C}^+(\mathfrak{p})/p = (\overline{R})_{\mathfrak{p}}/p \rightarrow O_{\overline{L}(\mathfrak{p})}/p = \mathbb{C}^+_{\mathfrak{p}}/p$ is injective, so by left exactness of $\lim_{\varphi}$, we obtain that in [\[eq:cpplus_cplusp_flat\]](#eq:cpplus_cplusp_flat){reference-type="eqref" reference="eq:cpplus_cplusp_flat"} the top horizontal arrow and hence the bottom left horizontal arrow are injective. Since $\mathbb{C}_{\mathfrak{p}}^{\flat}$ is a valuation field, let $\upsilon_{\mathfrak{p}}^{\flat}$ denote the normalised valuation on it such that $\upsilon_{\mathfrak{p}}^{\flat}(p^{\flat}) = 1$. Then we have that $x \in \mathbb{C}^{+, \flat}_{\mathfrak{p}}$ if and only if $\upsilon_{\mathfrak{p}}^{\flat}(x) \geq 0$. Moreover, we have $\mathbb{C}(\mathfrak{p})^{\flat} = \mathbb{C}^+(\mathfrak{p})^{\flat}[1/p^{\flat}]$ and $\mathbb{C}_{\mathfrak{p}}^{\flat} = \mathbb{C}^{+, \flat}_{\mathfrak{p}}[1/p^{\flat}]$. So from [\[eq:cpplus_cplusp_flat\]](#eq:cpplus_cplusp_flat){reference-type="eqref" reference="eq:cpplus_cplusp_flat"} and injectivity of its arrows it follows that for $x \in \mathbb{C}(\mathfrak{p})^{\flat}$ we have $x \in \mathbb{C}^+(\mathfrak{p})^{\flat}$ is and only if $\upsilon_{\mathfrak{p}}^{\flat}(x) \geq 0$. In particular, $$\label{eq:cplusp_intersect} \mathbb{C}^+(\mathfrak{p})^{\flat} = \mathbb{C}(\mathfrak{p})^{\flat} \cap \mathbb{C}^{+, \flat}_{\mathfrak{p}}\subset \mathbb{C}_{\mathfrak{p}}^{\flat}.$$ Furthermore, recall that the $p\textrm{-typical}$ Witt vector functor is left exact since it is right adjoint to the forgetful functor from the category of rings to the category of $\delta\textrm{-rings}$. Therefore, all maps in the following natural $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ commutative diagram are injective Hence, from [\[eq:cplusp_intersect\]](#eq:cplusp_intersect){reference-type="eqref" reference="eq:cplusp_intersect"} it follows that $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) = W(\mathbb{C}(\mathfrak{p})^{\flat}) \cap \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) \subset W(\mathbb{C}_{\mathfrak{p}}^{\flat})$. ◻ *Remark 17*. From Lemma [Lemma 16](#lem:cplusp_in_cpplus_ainf){reference-type="ref" reference="lem:cplusp_in_cpplus_ainf"}, the discussion preceding it and the fact that $\mathbb{C}^+(\mathfrak{p})\subset \mathbb{C}^+_{\mathfrak{p}}$, it easily follows that $\xi \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) = \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \cap \xi \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) \subset \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})$. *Remark 18*. By functoriality of tilting construction and Witt vector construction, we note that the action of $G_R$ on $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})$ described after [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"} (see [@brinon-relatif Remarque 3.3.2]), extends to a natural action of $G_R$ on $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))$ and $\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})$. **Lemma 19**. *In the notations described above, we have $(\varphi, G_R)\textrm{-equivariant}$ embeddings $\mathbf{A}_{\inf}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))$ and $W(\mathbb{C}(\overline{R})^{\flat}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})$, where right hand terms are equipped with a $G_R\textrm{-action}$ as described in Remark [Remark 18](#rem:gr_act_prodainf){reference-type="ref" reference="rem:gr_act_prodainf"}. Moreover, we have a $(\varphi, G_R)\textrm{-equivariant}$ identification $\mathbf{A}_{\inf}(\overline{R}) = W(\mathbb{C}(\overline{R})^{\flat}) \cap \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \subset \prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})$.* *Proof.* From [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"} recall that we have injective maps $\mathbb{C}^+(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} (\overline{R})_{\mathfrak{p}}^{\wedge}$. Applying the tilting functor we further obtain a $(\varphi, G_R)\textrm{-equivariant}$ commutative diagram $$\label{eq:cc_in_prod_cp} \begin{tikzcd} \mathbb{C}^+(\overline{R})^{\flat} \arrow[r] \arrow[d] & \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})^{\flat} \arrow[d]\\ \mathbb{C}(\overline{R})^{\flat} \arrow[r] & \Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})^{\flat}\Big)\big[\tfrac{1}{p^{\flat}}\big] \arrow[r] & \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}(\mathfrak{p})^{\flat}, \end{tikzcd}$$ where the bottom right horizontal arrow and vertical arrows are injective. From the injectivity of $\overline{R}/p \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})/p$ and left exactness of $\lim_{\varphi}$, we obtain that in [\[eq:cc_in_prod_cp\]](#eq:cc_in_prod_cp){reference-type="eqref" reference="eq:cc_in_prod_cp"} the top horizontal arrow is injective and since $\mathbb{C}(\overline{R})^{\flat} = \mathbb{C}^+(\overline{R})^{\flat}[1/p^{\flat}]$ it also follows that the bottom left horizontal arrow is injective. Now let $\upsilon_{\mathfrak{p}}^{\flat}$ denote the valuation on $\mathbb{C}_{\mathfrak{p}}^{\flat}$ intriduced in the proof of Lemma [Lemma 16](#lem:cplusp_in_cpplus_ainf){reference-type="ref" reference="lem:cplusp_in_cpplus_ainf"}. Then under the composition of left vertical and bottom horizontal arrows of [\[eq:cc_in_prod_cp\]](#eq:cc_in_prod_cp){reference-type="eqref" reference="eq:cc_in_prod_cp"}, it follows that for $x \in \mathbb{C}(\overline{R})^{\flat}$ we have $x \in \mathbb{C}^+(\overline{R})$ if and only if $\upsilon_{\mathfrak{p}}^{\flat}(x) \geq 0$ for each $\mathfrak{p}\in \pazocal{S}$. In particular, $$\label{eq:tilt_intersect_injective} \mathbb{C}^+(\overline{R})^{\flat} = \mathbb{C}(\overline{R})^{\flat} \cap \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})^{\flat} \subset \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}(\mathfrak{p})^{\flat}.$$ Furthermore, recall that the $p\textrm{-typical}$ Witt vector functor is left exact since it is right adjoint to the forgetful functor from the category of rings to the category of $\delta\textrm{-rings}$. Therefore, all maps in the following natural $(\varphi, G_R)\textrm{-equivariant}$ commutative diagram are injective Then from [\[eq:tilt_intersect_injective\]](#eq:tilt_intersect_injective){reference-type="eqref" reference="eq:tilt_intersect_injective"} we obtain $\mathbf{A}_{\inf}(\overline{R}) = W(\mathbb{C}(\overline{R})^{\flat}) \cap \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \subset \prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})$. ◻ ## de Rham period rings {#subsec:derham_relative} In this subsection we recall de Rham period rings (see [@abhinandan-relative-wach-i §2.1]). Note that the $\Gamma_R\textrm{-equivariant}$ map $\theta : \mathbf{A}_{\inf}(R_{\infty}) \twoheadrightarrow \widehat{R}_{\infty}$ described in §[2.2](#subsec:ainf_relative){reference-type="ref" reference="subsec:ainf_relative"} extends to a surjective map $\theta : \mathbf{A}_{\inf}(R_{\infty})[1/p] \twoheadrightarrow \widehat{R}_{\infty}[1/p]$. We set $\mathbf{B}_{\textup{dR}}^+(R_{\infty}) := \lim_n (\mathbf{A}_{\inf}(R_{\infty})[1/p])/\xi^n$. Let $t := \log(1+\mu) \in \mathbf{B}_{\textup{dR}}^+(R_{\infty})$, then $\mathbf{B}_{\textup{dR}}^+(R_{\infty})$ is $t\textrm{-torsion free}$ and we set $\mathbf{B}_{\textup{dR}}(R_{\infty}) := \mathbf{B}_{\textup{dR}}^+(R_{\infty})[1/t]$. Furthermore, one can define period rings $\pazocal{O}\mathbf{B}_{\textup{dR}}^+(R_{\infty})$ and $\pazocal{O}\mathbf{B}_{\textup{dR}}(R_{\infty})$. These rings are equipped with a $\Gamma_R\textrm{-action}$, an appropriate extension of the map $\theta$ and a decreasing filtration. Rings with a prefix "$\pazocal{O}$" are further equipped with an integrable connection satisfying Griffiths transversality with respect to the filtration. One can define variations of these rings over $\overline{R}$ as well. Now let $\pazocal{S}$ denote the set of minimal primes of $\overline{R}$ above $pR \subset R$ as in §[2.1](#subsec:localisation){reference-type="ref" reference="subsec:localisation"}. Similar to above for each $\mathfrak{p}\in \pazocal{S}$ we set $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}}) := \lim_n (\mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})[1/p])/(\textup{Ker }\theta)^n$ and $\mathbf{B}_{\textup{dR}}(\mathbb{C}^+_{\mathfrak{p}}) := \mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}})[1/t]$ equipped with a $\widehat{G}_R(\mathfrak{p})\textrm{-action}$ (resp. $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p})) := \lim_n (\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))[1/p])/(\textup{Ker }\theta)^n$ as well as $\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) := \mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p}))[1/t]$ equipped with a $G_R(\mathfrak{p})\textrm{-action}$), an appropriate extension of the map $\theta$ and a decreasing filtration. **Lemma 20**. *The $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ embedding $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})$ of Lemma [Lemma 16](#lem:cplusp_in_cpplus_ainf){reference-type="ref" reference="lem:cplusp_in_cpplus_ainf"} extends to a $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ embedding $\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{B}_{\textup{dR}}(\mathbb{C}^+_{\mathfrak{p}})$.* *Proof.* Note that by definition, the $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ embedding $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})$ induces a $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ map $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}})$. Then from Remark [Remark 17](#rem:xiainf_intersect){reference-type="ref" reference="rem:xiainf_intersect"} and the fact that $\lim$ is a left exact functor on the category of abelian groups, we get that the map $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}})$ is injective. The claim now follows since $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}})$ and $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p}))$ are $t\textrm{-torsion}$ free (see [@brinon-relatif Proposition 5.1.4]). ◻ Moreover, for each $\mathfrak{p}\in \pazocal{S}$ we have big period rings $\pazocal{O}\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}})$ and $\pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+_{\mathfrak{p}})$ equipped with a $\widehat{G}_R(\mathfrak{p})\textrm{-action}$ (resp. $\pazocal{O}\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p}))$ and $\pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ equipped with a $G_R(\mathfrak{p})\textrm{-action}$), an appropriate extension of the map $\theta$, a decreasing filtration and a connection. From [@brinon-relatif §5.2 & §5.3], in particular, from the alternative description of $\pazocal{O}\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}})$ (resp. $\pazocal{O}\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p}))$) as a power series ring over $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+_{\mathfrak{p}})$ (resp. $\mathbf{B}_{\textup{dR}}^+(\mathbb{C}^+(\mathfrak{p}))$) and using Lemma [Lemma 20](#lem:cplusp_in_cpplus_bdr){reference-type="ref" reference="lem:cplusp_in_cpplus_bdr"}, the following is obvious: **Lemma 21**. *The $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ embedding $\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{B}_{\textup{dR}}(\mathbb{C}^+_{\mathfrak{p}})$ of Lemma [Lemma 20](#lem:cplusp_in_cpplus_bdr){reference-type="ref" reference="lem:cplusp_in_cpplus_bdr"} extends to an $L\textrm{-linear}$ $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ embedding $\pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+_{\mathfrak{p}})$ compatible with respective filtrations and connections.* *Remark 22*. Recall that product is an exact functor on the category of abelian groups. So the natural embeddings $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ for each $\mathfrak{p}\in \pazocal{S}$, extend to embeddings $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$. By an argument similar to [@brinon-relatif Remarque 3.3.2] the products $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ and $\prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ can be equipped with an action of $G_R$ extending the $G_R\textrm{-action}$ on $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))$ (see Remark [Remark 18](#rem:gr_act_prodainf){reference-type="ref" reference="rem:gr_act_prodainf"}), in particular, the embedding $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ are $G_R\textrm{-equivariant}$. **Lemma 23**. *In the notations described above, we have an $R[1/p]\textrm{-linear}$ $G_R\textrm{-equivariant}$ embedding $\pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$, where the right hand term is equipped with a $G_R\textrm{-action}$ as described in Remark [Remark 22](#rem:gr_act_prodbdr){reference-type="ref" reference="rem:gr_act_prodbdr"}. Moreover, for each $\mathfrak{p}\in \pazocal{S}$, the induced natural map $\pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \rightarrow \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ is compatible with respective filtrations and connections.* *Proof.* From Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"} and Remark [Remark 22](#rem:gr_act_prodbdr){reference-type="ref" reference="rem:gr_act_prodbdr"}, we have $G_R\textrm{-equivariant}$ injective maps $\mathbf{A}_{\inf}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$. Then from the definition of $\mathbf{B}_{\textup{dR}}(\overline{R})$ and $\pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R})$, the preceding maps naturally induce an $R[1/p]\textrm{-linear}$ and $G_R\textrm{-equivariant}$ commutative diagram $$\label{eq:bdr_embedding} \begin{tikzcd} \mathbf{B}_{\textup{dR}}(\overline{R}) \arrow[r] \arrow[d] & \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \arrow[d] \arrow[r] & \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \arrow[d]\\ \pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \arrow[r] & \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \arrow[r] & \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})), \end{tikzcd}$$ where the vertical maps are injective, with the leftmost and rightmost arrows being compatible with respective filtrations and connections for each $\mathfrak{p}\in \pazocal{S}$. We need to show that the top left and bottom left horizontal arrows are injective. But first, let us note that from the explicit description of filtration on $\mathbf{B}_{\textup{dR}}$ and $\pazocal{O}\mathbf{B}_{\textup{dR}}$ in [@brinon-relatif §5.2], it easily follows that compositions of horizontal arrows in [\[eq:bdr_embedding\]](#eq:bdr_embedding){reference-type="eqref" reference="eq:bdr_embedding"} are compatible with filtrations and connections, i.e. for each $k \in \mathbb{Z}$, the respective images of $\textup{Fil}^k \mathbf{B}_{\textup{dR}}(\overline{R})$ and $\textup{Fil}^k \pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R})$ are contained in $\textup{Fil}^k \mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ and $\textup{Fil}^k \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ under the composition of top and bottom horizontal arrows, respectively. Similarly, from the explicit description of connection on $\pazocal{O}\mathbf{B}_{\textup{dR}}$ in [@brinon-relatif §5.3], it easily follows that the composition of bottom horizontal arrows in [\[eq:bdr_embedding\]](#eq:bdr_embedding){reference-type="eqref" reference="eq:bdr_embedding"} is further compatible with connections, for each $\mathfrak{p}\in \pazocal{S}$. It remains to show that the lower horizontal arrow in [\[eq:bdr_embedding\]](#eq:bdr_embedding){reference-type="eqref" reference="eq:bdr_embedding"} is injective, for which our argument is similar to [@brinon-relatif Proposition 6.2.6]. Note that the filtration on $\pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R})$ and $\pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ for each $\mathfrak{p}\in \pazocal{S}$ is separated. Therefore, it is enough to show that the induced map on grading of the filtration is injective. From [@brinon-relatif Proposition 5.2.7] recall that $\textup{gr}^{\bullet} \pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbb{C}^+(\overline{R})[z_1, \ldots, z_d, t^{\pm 1}]$, where $z_i$ denotes the image of $(X_i-[X_i]^{\flat})/t$ in $\textup{gr}^0 \pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbb{C}^+(\overline{R})[z_1, \ldots, z_d]$. Similarly, we have $\textup{gr}^{\bullet} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p})) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbb{C}^+(\mathfrak{p})[z_1, \ldots, z_d, t^{\pm 1}]$ for each $\mathfrak{p}\in \pazocal{S}$. The claim now follows from injectivity of the natural map $\mathbb{C}^+(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbb{C}^+(\mathfrak{p})$ (see [\[eq:cplus_gequiv\]](#eq:cplus_gequiv){reference-type="eqref" reference="eq:cplus_gequiv"}). ◻ *Remark 24*. The $G_R\textrm{-equivariant}$ embedding $\pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ of Lemma [Lemma 23](#lem:bdr_embedding){reference-type="ref" reference="lem:bdr_embedding"} admits a natural $L\textrm{-linear}$ and $G_R\textrm{-equivariant}$ extension to an embedding $L \otimes_{R[1/p]} \pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$. Indeed, this follows from an argument similar to Lemma [Lemma 23](#lem:bdr_embedding){reference-type="ref" reference="lem:bdr_embedding"} or directly from [@brinon-relatif Proposition 6.2.6]. Furthermore, from Lemma [Lemma 23](#lem:bdr_embedding){reference-type="ref" reference="lem:bdr_embedding"} it also follows that for each $\mathfrak{p}\in \pazocal{S}$, the induced natural map $L \otimes_{R[1/p]} \pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R}) \rightarrow \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ is compatible with filtrations and connections, where the left hand term is equipped with filtration on $\pazocal{O}\mathbf{B}_{\textup{dR}}(\overline{R})$ and tensor product connection. ## Crystalline period rings {#subsec:crystalline_relative} In this subsection we recall crystalline period rings (see [@abhinandan-relative-wach-i §2.2]). We set $\mathbf{A}_{\textup{cris}}(R_{\infty}) := \mathbf{A}_{\inf}(R_{\infty})\langle \xi^k/k!, k \in \mathbb{N}\rangle$ and we have $t = \log(1+\mu) \in \mathbf{A}_{\textup{cris}}(O_{F_{\infty}})$, so we set $\mathbf{B}_{\textup{cris}}^+(R_{\infty}) := \mathbf{A}_{\textup{cris}}(R_{\infty})[1/p]$ and $\mathbf{B}_{\textup{cris}}(R_{\infty}) := \mathbf{B}_{\textup{cris}}^+(R_{\infty})[1/t]$. Furthermore, one can define period rings $\pazocal{O}\mathbf{A}_{\textup{cris}}(R_{\infty})$, $\pazocal{O}\mathbf{B}_{\textup{cris}}^+(R_{\infty})$ and $\pazocal{O}\mathbf{B}_{\textup{cris}}(R_{\infty})$. These rings are equipped with a continuous $\Gamma_R\textrm{-action}$, a Frobenius endomorphism and a natural extension of the map $\theta$. Rings with a subscript "cris" are equipped with a natural decreasing filtration and rings with a prefix "$\pazocal{O}$" are additionally equipped with an integrable connection satisfying Griffiths transversality with respect to the filtration. Moreover, we have $G_R\textrm{-equivariant}$ and filtration compatible natural inclusions $\mathbf{B}_{\textup{cris}}(R_{\infty}) \subset \mathbf{B}_{\textup{dR}}(R_{\infty})$ and $\pazocal{O}\mathbf{B}_{\textup{cris}}(R_{\infty}) \subset \pazocal{O}\mathbf{B}_{\textup{dR}}(R_{\infty})$. One can define variations of these rings over $\overline{R}$ as well. From [@morrow-tsuji Corollary 4.34] we have a $(\varphi, \Gamma_R)\textrm{-equivariant}$ isomorphism $\pazocal{O}\mathbf{A}_{\textup{cris}}(R_{\infty}) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{A}_{\textup{cris}}(\overline{R})^{H_R}$. As in §[2.1](#subsec:localisation){reference-type="ref" reference="subsec:localisation"}, $\pazocal{S}$ denotes the set of minimal primes of $\overline{R}$ above $pR \subset R$. Similar to above, for each $\mathfrak{p}\in \pazocal{S}$, we have rings $\mathbf{A}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$, $\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$, $\pazocal{O}\mathbf{A}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$ and $\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$ equipped with a $\widehat{G}_R(\mathfrak{p})\textrm{-action}$ (resp. $\mathbf{A}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$, $\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$, $\pazocal{O}\mathbf{A}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ and $\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ equipped with a $G_R(\mathfrak{p})\textrm{-action}$), an appropriate extension of the map $\theta$,a decreasing filtration and a connection. Then we have the following: **Lemma 25**. *The $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ embedding $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})$ of Lemma [Lemma 16](#lem:cplusp_in_cpplus_ainf){reference-type="ref" reference="lem:cplusp_in_cpplus_ainf"} extends to $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ embeddings $\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$ and $\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$, where the latter is $L\textrm{-linear}$ and compatible with respective filtrations and connections.* *Proof.* By definition, the $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ embedding $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}})$ extends to $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ maps $\mathbf{A}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \mathbf{A}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$ and $\pazocal{O}\mathbf{A}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \pazocal{O}\mathbf{A}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$, where the latter is $O_L\textrm{-linear}$ and compatible with connections. Now consider the following $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ commutative diagram where all horizontal arrows are injective and the right vertical arrow is injective and compatible with connections. Therefore, it follows that the left and middle vertical arrows are injective and compatible with connections. Finally, the claims for $\mathbf{B}_{\textup{cris}}$ and $\pazocal{O}\mathbf{B}_{\textup{cris}}$ follow after inverting $t$. ◻ *Remark 26*. From Remark [Remark 22](#rem:gr_act_prodbdr){reference-type="ref" reference="rem:gr_act_prodbdr"} it is easy to see that we have injective maps $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$, where the first two maps are compatible with Frobenius. By an argument similar to [@brinon-relatif Remarque 3.3.2] the products $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ and $\prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ are stable under the $G_R\textrm{-action}$ on $\prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ (see Remark [Remark 22](#rem:gr_act_prodbdr){reference-type="ref" reference="rem:gr_act_prodbdr"}) and we equip them with induced action. Then it follows that the injective maps $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{dR}}(\mathbb{C}^+(\mathfrak{p}))$ are $G_R\textrm{-equivariant}$ as well. **Lemma 27**. *In the notations described above, we have an $R[1/p]\textrm{-linear}$ $(\varphi, G_R)\textrm{-equivariant}$ embedding $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$, where the right hand term is equipped with a $G_R\textrm{-action}$ as described in Remark [Remark 26](#rem:gr_act_prodbcrys){reference-type="ref" reference="rem:gr_act_prodbcrys"}. Moreover, for each $\mathfrak{p}\in \pazocal{S}$, the induced natural map $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ is compatible with respective Frobenii, filtrations and connections.* *Proof.* From Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"} and Remark [Remark 26](#rem:gr_act_prodbcrys){reference-type="ref" reference="rem:gr_act_prodbcrys"}, we have $(\varphi, G_R)\textrm{-equivariant}$ injective maps $\mathbf{A}_{\inf}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$. Then from the definition of $\pazocal{O}\mathbf{B}_{\textup{cris}}$, preceding maps naturally induce an $R[1/p]\textrm{-linear}$ and $(\varphi, G_R)\textrm{-equivariant}$ map $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$. The claim on injectivity of the preceding map follows in a manner similar to [@brinon-relatif Proposition 6.2.6]. Indeed, consider the following natural diagram where the left and right vertical arrows are natural inclusions and the bottom arrow is injective from Lemma [Lemma 23](#lem:bdr_embedding){reference-type="ref" reference="lem:bdr_embedding"}. The diagram commutes since the top and bottom horizontal arrows are defined using the embedding $\mathbf{A}_{\inf}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))$ of Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"}. So the injectivity of the top arrow follows, proving the first claim. Finally, for each $\mathfrak{p}\in \pazocal{S}$, the induced natural map $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ is tautologically compatible with respective Frobenii and the claims on filtrations and connections follow from the corresponding claims on $\pazocal{O}\mathbf{B}_{\textup{dR}}$ in Lemma [Lemma 23](#lem:bdr_embedding){reference-type="ref" reference="lem:bdr_embedding"}. ◻ *Remark 28*. The $(\varphi, G_R)\textrm{-equivariant}$ embedding $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ of Lemma [Lemma 27](#lem:bcrys_embedding){reference-type="ref" reference="lem:bcrys_embedding"} admits a natural $L\textrm{-linear}$ and $(\varphi, G_R)\textrm{-equivariant}$ extension to an embedding $L \otimes_{R[1/p]} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$. Indeed, this follows from an argument similar to Lemma [Lemma 27](#lem:bcrys_embedding){reference-type="ref" reference="lem:bcrys_embedding"} or directly from [@brinon-relatif Proposition 6.2.6]. Furthermore, from Lemma [Lemma 27](#lem:bcrys_embedding){reference-type="ref" reference="lem:bcrys_embedding"} it also follows that the induced natural map $L \otimes_{R[1/p]} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$ is compatible with respective Frobenii, filtrations and connections, where the left hand term is equipped with filtration on $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$ and tensor product Frobenius and connection. ## Rings of $(\varphi, \Gamma)\textrm{-modules}$ {#subsec:phigamma_mod_rings} Let us fix Teichmüller lifts $[X_i^{\flat}] \in \mathbf{A}_{\inf}(R_{\infty})$ for $1 \leq i \leq d$. Let $A_{\square}^+$ denote the $(p, \mu)\textrm{-adic}$ completion of $O_F[\mu, [X_1^{\flat}]^{\pm 1}, \ldots, [X_d^{\flat}]^{\pm 1}]$. There exists a natural embedding $A_{\square}^+ \subset \mathbf{A}_{\inf}(R_{\infty})$ and its image is stable under the Witt vector Frobenius endomorphism $\varphi$ and the $\Gamma_R\textrm{-action}$ on $\mathbf{A}_{\inf}(R_{\infty})$ (see [@abhinandan-relative-wach-i §3]); we equip $A_{\square}^+$ with induced structures. Furthermore, note that we have an embedding $\iota : R^{\square}\rightarrow A_{\square}^+$ defined by the map $X_i \mapsto [X_i^{\flat}]$ and it is easy to see that it extends to give an isomorphism of rings $R^{\square}\llbracket \mu \rrbracket \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}A_{\square}^+$. Extend the Frobenius endomorphism on $R^{\square}$ to a Frobenius endomorphism $\varphi$ on $R^{\square}\llbracket \mu \rrbracket$ by setting $\varphi(\mu) = (1+\mu)^p-1$. Then the Frobenius on $R^{\square}\llbracket \mu \rrbracket$ is finite and faithfully flat of degree $p^{d+1}$ . Moreover, it follows that the embedding $\iota$ and the isomorphism $R^{\square}\llbracket \mu \rrbracket \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}A_{\square}^+$ are Frobenius-equivariant. Let $\mathbf{A}_R^+$ denote the $(p, \mu)\textrm{-adic}$ completion of the unique extension of the embedding $A_{\square}^+ \rightarrow \mathbf{A}_{\inf}(R_{\infty})$ along the $p\textrm{-adically}$ completed étale map $R^{\square}\rightarrow R$ (see [@abhinandan-relative-wach-i §3.3.2] and [@colmez-niziol Proposition 2.1]). Then there exists a natural embedding $\mathbf{A}_R^+ \subset \mathbf{A}_{\inf}(R_{\infty})$ and its image is stable under the Witt vector Frobenius and $\Gamma_R\textrm{-action}$ on $\mathbf{A}_{\inf}(R_{\infty})$; we equip $\mathbf{A}_R^+$ with induced structures. Furthermore, the embedding $\iota : R^{\square}\rightarrow A_{\square}^+ \subset \mathbf{A}_R^+$ and the isomorphism $R^{\square}\llbracket \mu \rrbracket \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}A_{\square}^+ \subset \mathbf{A}_R^+$ naturally extend to a unique embedding $\iota: R \rightarrow \mathbf{A}_R^+$ and an isomorphism of rings $R\llbracket \mu \rrbracket \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+$. Extend the Frobenius endomorphism on $R$ to a Frobenius endomorphism $\varphi$ on $R\llbracket \mu \rrbracket$ by setting $\varphi(\mu) = (1+\mu)^p-1$. Then the Froebnius on $R\llbracket \mu \rrbracket$ is finite and faithfully flat of degree $p^{d+1}$ . Moreover, the embedding $\iota$ and the isomorphism $R\llbracket \mu \rrbracket \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+$ are Frobenius-equivariant. In particular, the induced Frobenius endomorphism $\varphi$ on $\mathbf{A}_R^+$ is finite and faithfully flat of degree $p^{d+1}$ and we have $\varphi^*(\mathbf{A}_R^+) := \mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{A}_R^+ \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\oplus_{\alpha} \varphi(\mathbf{A}_R^+) u_{\alpha}$, where $u_{\alpha} := (1+\mu)^{\alpha_0} [X_1^{\flat}]^{\alpha_1} \cdots [X_d^{\flat}]^{\alpha_d}$ for $\alpha = (\alpha_0, \alpha_1, \ldots, \alpha_d) \in \{0, 1, \ldots, p-1\}^{[0, d]}$. Set $\mathbf{A}_R:= \mathbf{A}_R^+[1/\mu]^{\wedge}$ and the Frobenius endomorphism $\varphi$ and continuous action of $\Gamma_R$ on $\mathbf{A}_R^+$ naturally extend to $\mathbf{A}_R$. Similar to above, the induced Frobenius endomorphism $\varphi$ on $\mathbf{A}_R$ is finite and faithfully flat of degree $p^{d+1}$ and $\varphi^*(\mathbf{A}_R) := \mathbf{A}_R\otimes_{\varphi, \mathbf{A}_R} \mathbf{A}_R\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\oplus_{\alpha} \varphi(\mathbf{A}_R) u_{\alpha} = (\oplus_{\alpha} \varphi(\mathbf{A}_R^+) u_{\alpha}) \otimes_{\varphi(\mathbf{A}_R^+)} \varphi(\mathbf{A}_R) \xleftarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{A}_R$. Let $\mathbb{C}(\overline{R}):= \widehat{\overline{R}}[1/p]$, $\tilde{\mathbf{A}}:= W\big(\mathbb{C}(\overline{R})^{\flat}\big)$ and $\tilde{\mathbf{B}}:= \tilde{\mathbf{A}}[1/p]$ admitting the Frobenius on Witt vectors and continuous $G_R\textrm{-action}$ (for the weak topology). We have natural Frobenius and $\Gamma_R\textrm{-equivariant}$ embeddings $\mathbf{A}_R^+ \subset \mathbf{A}_{\inf}(R_{\infty})$ which extends to an embedding $\mathbf{A}_R\subset \tilde{\mathbf{A}}^{H_R}$ and we set $\mathbf{B}_R:= \mathbf{A}_R[1/p]$. Take $\mathbf{A}$ to be the $p\textrm{-adic}$ completion of the maximal unramified extension of $\mathbf{A}_R$ inside $\tilde{\mathbf{A}}$ and set $\mathbf{B}:= \mathbf{A}[1/p] \subset \tilde{\mathbf{B}}$. The rings $\mathbf{A}$ and $\mathbf{B}$ are stable under the $G_R\textrm{-action}$ and Frobenius endomorphism on $\tilde{\mathbf{B}}$ and we equip $\mathbf{A}$ and $\mathbf{B}$ with induced structures. We have $\mathbf{A}_R= \mathbf{A}^{H_R}$ and $\mathbf{B}_R= \mathbf{B}^{H_R}$. Moreover, set $\mathbf{A}^+ := \mathbf{A}_{\inf}(\overline{R}) \cap \mathbf{A}\subset \tilde{\mathbf{A}}$ and $\mathbf{B}^+ := \mathbf{A}^+[1/p]$ stable under the Frobenius and $G_R\textrm{-action}$ on $\mathbf{B}$, then we have $\mathbf{A}_R^+ = (\mathbf{A}^+)^{H_R}$ and $\mathbf{B}_R^+ = (\mathbf{B}^+)^{H_R}$. Via the identification $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$, we have a $(\varphi, \Gamma_L)\textrm{-equivariant}$ isomorphism $\mathbf{A}_L^+ \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}((\mathbf{A}_R^+)_{(p, \mu)})^{\wedge}$, where ${ }^{\wedge}$ denotes the $(p, \mu)\textrm{-adic}$ completion. The preceding isomorphism extends to an isomorphism $\mathbf{A}_L\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}((\mathbf{A}_R)_{(p)})^{\wedge}$, where ${}^{\wedge}$ denotes the $p\textrm{-adic}$ completion. Then we have $\mathbf{A}_R^+ = \mathbf{A}_L^+ \cap \mathbf{A}_R\subset \mathbf{A}_L$ and $\mathbf{B}_R^+ := \mathbf{A}_R^+[1/p] = \mathbf{B}_L^+ \cap \mathbf{B}_R\subset \mathbf{B}_L$. ## $p\textrm{-adic}$ representations {#subsec:relative_padicreps} Let $T$ be a finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$. By the theory of étale $(\varphi, \Gamma)\textrm{-modules}$ (see [@fontaine-phigamma] and [@andreatta-phigamma]) one can functorially associate to $T$ a finite projective étale $(\varphi, \Gamma_R)\textrm{-module}$ $\mathbf{D}_R(T) := (\mathbf{A}\otimes_{\mathbb{Z}_p} T)^{H_R}$ over $\mathbf{A}_R$ of rank $= \textup{rk}_{\mathbb{Z}_p} T$. Moreover, $\tilde{\mathbf{D}}_R(T) := (\tilde{\mathbf{A}}\otimes_{\mathbb{Z}_p} T)^{H_R} \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\tilde{\mathbf{A}}^{H_R} \otimes_{\mathbf{A}_R} \mathbf{D}_R(T)$ and we have a natural $(\varphi, \Gamma_R)\textrm{-equivariant}$ isomorphism $$\label{eq:phigamma_comp_iso_relative} \mathbf{A}\otimes_{\mathbf{A}_R} \mathbf{D}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}\otimes_{\mathbb{Z}_p} T.$$ These constructions are functorial in $\mathbb{Z}_p\textrm{-representations}$ and induce an exact equivalence of $\otimes\textrm{-categories}$ (see [@andreatta-phigamma Theorem 7.11]) $$\label{eq:rep_phigamma_relative} \textup{Rep}_{\mathbb{Z}_p}(G_R) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R}^{\textup{\'et}},$$ with an exact $\otimes\textrm{-compatible}$ quasi-inverse given as $\mathbf{T}_R(D) := (\mathbf{A}\otimes_{\mathbf{A}_R} D)^{\varphi=1} = (\tilde{\mathbf{A}}\otimes_{\mathbf{A}_R} D)^{\varphi=1}$. Similar statements are also true for $p\textrm{-adic}$ representations of $G_R$. Furthermore, let $\mathbf{D}_R^+(T) := (\mathbf{A}^+ \otimes_{\mathbb{Z}_p} T)^{H_R}$ be the $(\varphi, \Gamma_R)\textrm{-module}$ over $\mathbf{A}_R^+$ associated to $T$ and for $V := T[1/p]$ let $\mathbf{D}_R^+(V) := \mathbf{D}_R^+(T)[1/p]$ be the $(\varphi, \Gamma_R)\textrm{-module}$ over $\mathbf{B}_R^+$ associated to $V$. Let $V$ be a $p\textrm{-adic}$ representation of $G_R$. From $p\textrm{-adic}$ Hodge theory of $G_R$ (see [@brinon-relatif]), one can attach to $V$ a filtered $(\varphi, \partial)\textrm{-module}$ over $R[1/p]$ of rank $\leq \dim_{\mathbb{Q}_p} V$ given by the functor $$\begin{aligned} \pazocal{O}\mathbf{D}_{\textup{cris}, R}: \textup{Rep}_{\mathbb{Q}_p}(G_R) &\longrightarrow \textup{MF}_R(\varphi, \partial)\\ V &\longmapsto (\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbb{Q}_p} V)^{G_R}.\end{aligned}$$ The representation $V$ is said to be crystalline if the natural map $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbb{Q}_p} V$ is an isomorphism, in particular, if $V$ is crystalline then $\textup{rk}_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) = \dim_{\mathbb{Q}_p} V$. Restricting $\pazocal{O}\mathbf{D}_{\textup{cris}, R}$ to the category of crystalline representations of $G_R$ and writing $\textup{MF}_R^{\textup{ad}}(\varphi, \partial)$ for the essential image of restricted functor, we have an exact equivalence of $\otimes\textrm{-categories}$ (see [@brinon-relatif Théorème 8.5.1]) $$\label{eq:odcrysr_func} \pazocal{O}\mathbf{D}_{\textup{cris}, R}: \textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\textup{MF}_R^{\textup{ad}}(\varphi, \partial),$$ with an exact $\otimes\textrm{-compatible}$ quasi-inverse given as $\pazocal{O}\mathbf{V}_{\textup{cris}, R}(D) := (\textup{Fil}^0(\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{R[1/p]} D))^{\partial=0, \varphi=1}$. Furthermore, we have a continuous homomorphism $G_L \rightarrow G_R$, i.e. $V$ is also a $p\textrm{-adic}$ representation of $G_L$. Base changing the isomorphism $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbb{Q}_p} V$ along $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(O_{\overline{L}})$, we obtain a $G_L\textrm{-equivariant}$ isomorphism $\pazocal{O}\mathbf{B}_{\textup{cris}}(O_{\overline{L}}) \otimes_L \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{B}_{\textup{cris}}(O_{\overline{L}}) \otimes_{\mathbb{Q}_p} V$, i.e. $V$ is a crystalline representation of $G_L$. Taking $G_L\textrm{-invariants}$ in the preceding isomorphism we further obtain a natural isomorphism $L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, L}(V)$ compatible with Frobenius, filtration and connection. # Relative Wach modules {#sec:wach_modules} In this section we will describe relative Wach modules and finite $[p]_q\textrm{-height}$ representations of $G_R$ and relate them to crystalline representations. We start with the notion of Wach modules. ## Wach modules over $\mathbf{A}_R^+$ {#subsec:wach_mod_props} In $\mathbf{A}_{\inf}(O_{F_{\infty}})$ we fix $q := [\varepsilon]$, $\mu := [\varepsilon]-1 = q-1$ and $[p]_q := \tilde{\xi} := \varphi(\mu)/\mu$. **Definition 29**. Let $N$ be a finitely generated $\mathbf{A}_R^+\textrm{-module}$. The sequence $\{p, \mu\}$ in $\mathbf{A}_R^+$ is said to be *$N\textrm{-regular}$* if $N$ is $p\textrm{-torsion free}$ and $N/pN$ is $\mu\textrm{-torsion free}$. Similarly, $\{\mu, p\}$ is $N\textrm{-regular}$ if $N$ is $\mu\textrm{-torsion free}$ and $N/\mu N$ is $p\textrm{-torsion free}$. The sequence $\{p, \mu\}$ in $\mathbf{A}_R^+$ is said to be *strictly $N\textrm{-regular}$* if both $\{p, \mu\}$ and $\{\mu, p\}$ are $N\textrm{-regular}$. *Remark 30*. In Definition [Definition 29](#defi:strict_regular_seq){reference-type="ref" reference="defi:strict_regular_seq"} note that $\{p, \mu\}$ is strictly $N\textrm{-regular}$ if and only if $N$ is $a\textrm{-torsion}$ free for each nonzero $a$ in the ideal $(p, \mu) \subset \mathbf{A}_R^+$ and $N/\mu N$ is $p\textrm{-torsion}$ free. Indeed, one direction is obvious and for the converse one needs to check that $N/pN$ is $\mu\textrm{-torsion}$ free. So let $x \in N$ such that $\mu x = p y$ for some $y \in N$; we claim that $x \in pN$. Reducing the preceding equality modulo $p$ and using $(N/\mu N)[p] = 0$, we get that $y = \mu z$ for some $z \in \mathbb{N}$. From $\mu\textrm{-torsion freeness}$ of $N$, it follows that $x = pz$, as claimed. **Lemma 31**. *Let $N$ be a finitely generated $\mathbf{A}_R^+\textrm{-module}$ and consider the complex $$\mathcal{C}^{\bullet} : N \xrightarrow{(p, \mu)} N \oplus N \xrightarrow{(\mu, -p)} N,$$ where the first map is given by $x \mapsto (px, \mu x)$ and the second map is given by $(x, y) \mapsto \mu x - py$. Then $\{p, \mu\}$ is strictly $N\textrm{-regular}$ if and only if $H^1(\mathcal{C}^{\bullet}) = 0$. Moreover, under these equivalent conditions $H^0(\mathcal{C}^{\bullet}) = 0$.* *Proof.* If $\{p, \mu\}$ is strictly $N\textrm{-regular}$ then $(N/p)[\mu] = (N/\mu)[p] = 0$. This implies $H^0(\mathcal{C}^{\bullet}) = H^1(\mathcal{C}^{\bullet}) = 0$. For the converse, consider the following diagram $$\label{eq:nmodpmu} \begin{tikzcd} & N[p, \mu] \arrow[r] \arrow[d] & N[p] \arrow[r, "\mu"] \arrow[d] & N[p] \arrow[r] \arrow[d] & (N/\mu)[p] \arrow[d]\\ 0 \arrow[r] & N[\mu] \arrow[r] \arrow[d, "p"] & N \arrow[r, "\mu"] \arrow[d, "p"] & N \arrow[r] \arrow[d, "p"] & N/\mu \arrow[r] \arrow[d, "p"] & 0\\ 0 \arrow[r] & N[\mu] \arrow[r] \arrow[d] & N \arrow[r, "\mu"] \arrow[d] & N \arrow[r] \arrow[d] & N/\mu \arrow[r] \arrow[d] & 0\\ & (N/p)[\mu] \arrow[r] & N/p \arrow[r, "\mu"] & N/p \arrow[r] & N/(p, \mu). \end{tikzcd}$$ Since $H^1(\mathcal{C}^{\bullet}) = 0$, we get that the top right and bottom left corners of [\[eq:nmodpmu\]](#eq:nmodpmu){reference-type="eqref" reference="eq:nmodpmu"} are zero, i.e. $(N/\mu)[p] = (N/p)[\mu] = 0$. Now let $x \in N[\mu]$, then from the surjectivity of the leftmost vertical arrow from second to third row it follows that there exists $x_1 \in N[\mu]$ such that $x = px_1$. Proceeding by induction it is easy to see that $x \in p^n N[\mu] \subset p^n N$ for all $n \in \mathbb{N}$. But since $N$ is finitely generated over $\mathbf{A}_R^+$, which is $(p, \mu)\textrm{-adically}$ complete, it follows that $N$ is $p\textrm{-adically}$ separated, i.e. $x = 0$, in particular, $N[\mu] = 0$. A similar argument shows that $N[p] = 0$, in particular, $N[p, \mu] = 0$. This proves both claims in the lemma. ◻ *Remark 32*. The complex $\mathcal{C}^{\bullet}$ in Lemma [Lemma 31](#lem:local_coh){reference-type="ref" reference="lem:local_coh"} computes local cohomology of $N$ with respect to the ideal $(p, \mu) \subset \mathbf{A}_R^+$ (see [@weibel Theorem 4.6.8]). So if we set $Z := V(p, \mu) \subset \textup{Spec}\hspace{0.5mm}(\mathbf{A}_R^+) =: X$ as a closed subset, then one also says that $\mathcal{C}^{\bullet}$ computes cohomology with compact support along $Z$, written as $H^i_Z(X, N)$ (see [@weibel Generalization 4.6.2]). **Lemma 33**. *Let $N$ be a finitely generated $\mathbf{A}_R^+\textrm{-module}$ such that $\{p, \mu\}$ is strictly $N\textrm{-regular}$. Then we have $N = N[1/p] \cap N[1/\mu] \subset N[1/p, 1/\mu]$ as $\mathbf{A}_R^+\textrm{-modules}$. Moreover, $N = N[1/p] \cap N[1/\mu]^{\wedge} \subset N[1/\mu]^{\wedge}[1/p]$, where ${}^{\wedge}$ denotes the $p\textrm{-adic}$ completion.* *Proof.* Note that from definitions we have $(N/p)[\mu] = (N/\mu)[p] = 0$ and $(N[1/\mu])/p = (N/p)[1/\mu]$. So it follows that $N/p^nN \subset (N/p^n)[1/\mu]$ for all $n \in \mathbb{N}$ and therefore $N \cap p^n N[1/\mu] = p^n N$. Hence, $N[1/p] \cap N[1/\mu] = N$. Furthermore, since $(N[1/\mu]^{\wedge})/p^n = (N[1/\mu])/p^n = (N/p^n)[1/\mu]$, similar to above we also get that $N \cap p^n N[1/\mu]^{\wedge} = p^n N$ for all $n \in N$. Hence, $N[1/p] \cap N[1/\mu]^{\wedge} = N$. ◻ **Definition 34**. Let $a, b \in \mathbb{Z}$ with $b \geq a$. A *Wach module* over $\mathbf{A}_R^+$ with weights in the interval $[a, b]$ is a finitely generated $\mathbf{A}_R^+\textrm{-module}$ $N$ equipped with a continuous and semilinear action of $\Gamma_{R}$ satisfying the following: 1. The sequence $\{p, \mu\}$ is strictly $N\textrm{-regular}$. 2. The action of $\Gamma_R$ is trivial on $N/\mu N$. 3. There is a Frobenius-semilinear operator $\varphi: N[1/\mu] \rightarrow N[1/\varphi(\mu)]$ commuting with the action of $\Gamma_{R}$ such that $\varphi(\mu^b N) \subset \mu^b N$, the induced map $(1 \otimes \varphi) : \varphi^{\ast}(\mu^b N) \rightarrow \mu^b N$ is injective and its cokernel is killed by $[p]_q^{b-a}$. We define the $[p]_q\textit{-height}$ of $N$ to be the largest value of $-a$ for $a \in \mathbb{Z}$ as above. The module $N$ is said to be *effective* if we can take $b = 0$ and $a \leq 0$. A Wach module over $\mathbf{B}_R^+$ is a finitely generated module $M$ equipped with an action of $\Gamma_R$ and a Frobenius-semilinear operator $\varphi: M[1/\mu] \rightarrow M[1/\varphi(\mu)]$ commuting with $\Gamma_R\textrm{-action}$ such that there exists a $\Gamma_R\textrm{-stable}$ and $\varphi\textrm{-stable}$ (after inverting $\mu$) $\mathbf{A}_R^+\textrm{-submodule}$ $N \subset M$ and equipped with induced $(\varphi, \Gamma_R)\textrm{-action}$ $N$ is a Wach module over $\mathbf{A}_R^+$ and $N[1/p] = M$. Denote by $(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q}$, the category of Wach modules over $\mathbf{A}_R^+$ with morphisms between objects being $\mathbf{A}_R^+\textrm{-linear}$ $\varphi\textrm{-equivariant}$ (after inverting $\mu$) and $\Gamma_R\textrm{-equivariant}$ morphisms. Next, we note some structural properties of Wach modules. **Proposition 35**. *Let $N$ be a Wach module over $\mathbf{A}_R^+$. Then $N[1/p]$ is finite projective over $\mathbf{A}_R^+[1/p]$ and $N[1/\mu]$ is finite projective over $\mathbf{A}_R^+[1/\mu]$.* *Proof.* For $r \in \mathbb{N}$ large enough, note that the Wach module $\mu^r N(-r)$ is always effective. So without loss of generality, we may assume that $N$ is effective. Then the first claim follows from Proposition [Proposition 73](#prop:finiteproj_torus){reference-type="ref" reference="prop:finiteproj_torus"}. For the second claim, note that $N$ is $p\textrm{-torsion}$ free, so $\mathbf{A}_R\otimes_{\mathbf{A}_R^+} N$ is a $p\textrm{-torsion}$ free étale $(\varphi, \Gamma_R)\textrm{-module}$ over $\mathbf{A}_R$ and therefore finite projective by [@andreatta-phigamma Lemma 7.10]. Since $\mathbf{A}_R^+[1/\mu]$ is noetherian, we have $N[1/\mu]^{\wedge} \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R\otimes_{\mathbf{A}_R^+[1/\mu]} N[1/\mu] = \mathbf{A}_R\otimes_{\mathbf{A}_R^+} N$, where ${}^{\wedge}$ denotes the $p\textrm{-adic}$ completion. Moreover, the natural map $\textup{Spec}\hspace{0.5mm}(\mathbf{A}_R^+[1/\mu]^{\wedge}) \cup \textup{Spec}\hspace{0.5mm}(\mathbf{A}_R^+[1/\mu, 1/p]) \rightarrow \textup{Spec}\hspace{0.5mm}(\mathbf{A}_R^+[1/\mu])$ is a flat cover. Therefore, by faithfully flat descent it follows that $N[1/\mu]$ is finite projective over $\mathbf{A}_R^+[1/\mu]$. ◻ *Remark 36*. Note that for a Wach module $N$ over $\mathbf{A}_R^+$, we have that $N$ is $p\textrm{-torsion}$ free, in particular, $N \subset N[1/p]$. Since $N[1/p]$ is projective over $\mathbf{A}_R^+[1/p]$ by Proposition [Proposition 35](#prop:wachmod_proj_pmu){reference-type="ref" reference="prop:wachmod_proj_pmu"}, we obtain that $N$ is a torsion free $\mathbf{A}_R^+\textrm{-module}$. **Lemma 37**. *Let $N$ be a Wach module over $\mathbf{A}_R^+$, then we have $N = (\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} N) \cap (\mathbf{A}_R\otimes_{\mathbf{A}_R^+} N) \subset \mathbf{A}_L\otimes_{\mathbf{A}_R^+} N$ as $\mathbf{A}_R^+\textrm{-modules}$.* *Proof.* Let $N_R := N$, $N_L := \mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} N$ and $D_R := \mathbf{A}_R\otimes_{\mathbf{A}_R^+} N$. Note that $N_R[1/p]$ is finite projective over $\mathbf{B}_R^+$ with $N_L[1/p] = \mathbf{B}_L^+ \otimes_{\mathbf{B}_R^+} N_R[1/p]$ and $D_R[1/p] = \mathbf{B}_R\otimes_{\mathbf{B}_R^+} N_R[1/p]$, therefore $N_L[1/p] \cap D_R[1/p] = (\mathbf{B}_L^+ \cap \mathbf{B}_R) \otimes_{\mathbf{B}_R^+} N_R[1/p] = N_R[1/p]$. Moreover, we have $N_L \cap D_R \subset N_L[1/p] \cap D_R[1/p] = N_R[1/p]$ and using Lemma [Lemma 33](#lem:wach_saturated){reference-type="ref" reference="lem:wach_saturated"} we see that $N_L \cap D_R = N_L \cap D_R \cap N_R[1/p] = N_R$. ◻ From the proof of Proposition [Proposition 35](#prop:wachmod_proj_pmu){reference-type="ref" reference="prop:wachmod_proj_pmu"}, it is clear that extending scalars along $\mathbf{A}_R^+ \rightarrow \mathbf{A}_R$ induces a functor $(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q} \rightarrow (\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R}^{\textup{\'et}}$ and we make the following claim: **Proposition 38**. *The following natural functor is fully faithful $$\begin{aligned} (\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q} &\longrightarrow (\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R}^{\textup{\'et}}\\ N &\longmapsto \mathbf{A}_R\otimes_{\mathbf{A}_R^+} N. \end{aligned}$$* *Proof.* Let $N, N'$ be two Wach modules over $\mathbf{A}_R^+$. Write $N_R := N$, $N_L := \mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} N$, $D_L := \mathbf{A}_R\otimes_{\mathbf{A}_R^+} N$ and similarly for $N'$. We need to show that for Wach modules $N_R$ and $N_R'$, we have $$\label{eqref:homset_bijection_relative} \textup{Hom}_{(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q}}(N_R, N_R') \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\textup{Hom}_{(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R}^{\textup{\'et}}}(D_R, D_R')$$ Note that $\mathbf{A}_R^+ \rightarrow \mathbf{A}_R= \mathbf{A}_R^+[1/\mu]^{\wedge}$ is injective, in particular, the map in [\[eqref:homset_bijection_relative\]](#eqref:homset_bijection_relative){reference-type="eqref" reference="eqref:homset_bijection_relative"} is injective. To check that [\[eqref:homset_bijection_relative\]](#eqref:homset_bijection_relative){reference-type="eqref" reference="eqref:homset_bijection_relative"} is surjective take an $\mathbf{A}_R\textrm{-linear}$ and $(\varphi, \Gamma_R)\textrm{-equivariant}$ map $f : D_R \rightarrow D_R'$. We need to show that $f(N_R) \subset N_R'$. Base changing $f$ along $\mathbf{A}_R\rightarrow \mathbf{A}_L$ and using the isomorphism $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$ induces an $\mathbf{A}_L\textrm{-linear}$ and $(\varphi, \Gamma_L)\textrm{-equivariant}$ map $f : D_L \rightarrow D_L'$. Then from [@abhinandan-imperfect-wach Proposition 3.3] we have $f(N_L) \subset N_L'$. Using Lemma [Lemma 37](#lem:wach_intersection_lemma){reference-type="ref" reference="lem:wach_intersection_lemma"} we get that inside $D_L'$ we have $f(N_R) = f(N_L \cap D_R) = f(N_L) \cap f(D_R) \subset N_L' \cap D_R' = N_R'$. ◻ Analogous to above, one can define categories $(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{B}_R^+}^{[p]_q}$ and $(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{B}_R}^{\textup{\'et}}$ and a functor from the former to latter by extending scalars along $\mathbf{B}_R^+ \rightarrow \mathbf{B}_R$. Then passing to associated isogeny catgeories in Proposition [Proposition 38](#prop:wach_etale_ff_relative){reference-type="ref" reference="prop:wach_etale_ff_relative"} we get the following: **Corollary 39**. *The natural functor $(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{B}_R^+}^{[p]_q} \rightarrow (\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{B}_R}^{\textup{\'et}}$ is fully faithful.* ## $G_R\textrm{-representations}$ attached to Wach modules {#subsec:wach_mod_rep} Composing the functor in Proposition [Proposition 38](#prop:wach_etale_ff_relative){reference-type="ref" reference="prop:wach_etale_ff_relative"} with the equivalence in [\[eq:rep_phigamma_relative\]](#eq:rep_phigamma_relative){reference-type="eqref" reference="eq:rep_phigamma_relative"}, we obtain a fully faithful functor $$\begin{aligned} \label{eq:wach_reps_relative} \begin{split} \mathbf{T}_R: (\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q} &\longrightarrow \textup{Rep}_{\mathbb{Z}_p}(G_R)\\ N &\longmapsto \big(\mathbf{A}\otimes_{\mathbf{A}_R^+} N\big)^{\varphi = 1} \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\big(W(\mathbb{C}(\overline{R})^{\flat}) \otimes_{\mathbf{A}_R^+} N\big)^{\varphi = 1}. \end{split}\end{aligned}$$ **Proposition 40**. *Let $N$ be a Wach module over $\mathbf{A}_R^+$ and $T := \mathbf{T}_R(N)$ the associated finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$. Then we have a natural $G_R\textrm{-equivariant}$ comparison isomorphism $$\label{eq:wachmod_comp_relative_ainf} \mathbf{A}_{\inf}(\overline{R})[1/\mu] \otimes_{\mathbf{A}_R^+} N \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_{\inf}(\overline{R})[1/\mu] \otimes_{\mathbb{Z}_p} T.$$* *Proof.* Note that for $T = \mathbf{T}_R(N)$ from the equivalence in [\[eq:rep_phigamma_relative\]](#eq:rep_phigamma_relative){reference-type="eqref" reference="eq:rep_phigamma_relative"} we have $\mathbf{D}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R\otimes_{\mathbf{A}_R^+} N$ as étale $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{A}_R$. Then extending scalars of the isomorphism in [\[eq:phigamma_comp_iso_relative\]](#eq:phigamma_comp_iso_relative){reference-type="eqref" reference="eq:phigamma_comp_iso_relative"} along $\mathbf{A}\rightarrow W(\mathbb{C}(\overline{R})^{\flat})$ gives $(\varphi, G_R)\textrm{-equivariant}$ isomorphism $$\label{eq:phigamma_comp_tilde_relative} W(\mathbb{C}(\overline{R})^{\flat}) \otimes_{\mathbf{A}_R^+} N \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}W(\mathbb{C}(\overline{R})^{\flat}) \otimes_{\mathbb{Z}_p} T.$$ Now for $r \in \mathbb{N}$ large enough, the Wach module $\mu^r N (-r)$ is always effective and we have $\mathbf{T}_R(\mu^rN(-r)) = T(-r)$ (the twist $(-r)$ denotes the Tate twist on which $\Gamma_R$ acts via the cyclotomic character). Therefore, it is enough to show the claim for effective Wach modules, in particular, in the rest of the proof we will assume $N$ is effective. Let $\pazocal{S}$ denote the set of minimal primes of $\overline{R}$ above $pR \subset R$ as in §[2.1](#subsec:localisation){reference-type="ref" reference="subsec:localisation"}. Recall that for each $\mathfrak{p}\in \pazocal{S}$ we have $\overline{L}(\mathfrak{p}) \subset \mathbb{C}_{\mathfrak{p}}$ denotes an algebraic closure of $L$ containing $(\overline{R})_{\mathfrak{p}}$ and we have $\widehat{G}_R(\mathfrak{p})= \textup{Gal}(\overline{L}(\mathfrak{p})/L)$. Moreover, we have an identification $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$ and for each prime $\mathfrak{p}\in \pazocal{S}$ let $\mathbf{A}_L^+(\mathfrak{p})$ denote the base ring for Wach modules in the imperfect residue field case (see [@abhinandan-imperfect-wach §2.1.2]). To avoid confusion let us write $N_R := N$ and $N_L(\mathfrak{p}):= \mathbf{A}_L^+(\mathfrak{p})\otimes_{\mathbf{A}_R^+} N$, in particular, $N_L(\mathfrak{p})$ is a Wach module over $\mathbf{A}_L^+(\mathfrak{p})$ finite free of rank $= \textup{rk}_{\mathbb{Z}_p} T$. From [@abhinandan-imperfect-wach Lemma 3.6] recall that we have $\widehat{G}_R(\mathfrak{p})\textrm{-equivariant}$ inclusions for each $\mathfrak{p}\in \pazocal{S}$ $$\label{eq:wachmod_almost_comp_imperfect} \mu^s \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) \otimes_{\mathbb{Z}_p} T \subset \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) \otimes_{\mathbf{A}_L^+(\mathfrak{p})} N_L(\mathfrak{p})\subset \mathbf{A}_{\inf}(\mathbb{C}^+_{\mathfrak{p}}) \otimes_{\mathbb{Z}_p} T.$$ Now note that the $(\varphi, G_R(\mathfrak{p}))\textrm{-equivariant}$ composition $\mathbf{A}_R^+ \rightarrow W(\mathbb{C}(\overline{R})^{\flat}) \rightarrow W(\mathbb{C}(\mathfrak{p})^{\flat})$ factors as $(\varphi, G_R(\mathfrak{p}))\textrm{-equivariant}$ maps $\mathbf{A}_R^+ \rightarrow \mathbf{A}_L^+(\mathfrak{p})\rightarrow W(\mathbb{C}(\mathfrak{p})^{\flat})$. So base changing the $(\varphi, G_R)\textrm{-equivariant}$ isomorphism in [\[eq:phigamma_comp_tilde_relative\]](#eq:phigamma_comp_tilde_relative){reference-type="eqref" reference="eq:phigamma_comp_tilde_relative"} along the $(\varphi, G_R(\mathfrak{p}))\textrm{-equivariant}$ map $W(\mathbb{C}(\overline{R})^{\flat}) \rightarrow W(\mathbb{C}(\mathfrak{p})^{\flat})$ gives a $(\varphi, G_R(\mathfrak{p}))\textrm{-equivariant}$ isomorphism $$\label{eq:phigamma_comp_tildep_relative} W(\mathbb{C}(\mathfrak{p})^{\flat}) \otimes_{\mathbf{A}_L^+(\mathfrak{p})} N_L(\mathfrak{p})\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}W(\mathbb{C}(\mathfrak{p})^{\flat}) \otimes_{\mathbb{Z}_p} T.$$ All terms in [\[eq:wachmod_almost_comp_imperfect\]](#eq:wachmod_almost_comp_imperfect){reference-type="eqref" reference="eq:wachmod_almost_comp_imperfect"} and [\[eq:phigamma_comp_tildep_relative\]](#eq:phigamma_comp_tildep_relative){reference-type="eqref" reference="eq:phigamma_comp_tildep_relative"} admit compatible $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ embedding into $W(\mathbb{C}_{\mathfrak{p}}^{\flat}) \otimes_{\mathbf{A}_L^+(\mathfrak{p})} N_L(\mathfrak{p})\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}W(\mathbb{C}_{\mathfrak{p}}^{\flat}) \otimes_{\mathbb{Z}_p} T$, where the action of $\widehat{G}_R(\mathfrak{p})$ on [\[eq:phigamma_comp_tildep_relative\]](#eq:phigamma_comp_tildep_relative){reference-type="eqref" reference="eq:phigamma_comp_tildep_relative"} factors through $\widehat{G}_R(\mathfrak{p})\twoheadrightarrow G_R(\mathfrak{p})$. So taking the intersection of [\[eq:wachmod_almost_comp_imperfect\]](#eq:wachmod_almost_comp_imperfect){reference-type="eqref" reference="eq:wachmod_almost_comp_imperfect"} with [\[eq:phigamma_comp_tildep_relative\]](#eq:phigamma_comp_tildep_relative){reference-type="eqref" reference="eq:phigamma_comp_tildep_relative"} inside $W(\mathbb{C}_{\mathfrak{p}}^{\flat}) \otimes_{\mathbf{A}_L^+(\mathfrak{p})} N_L(\mathfrak{p})\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}W(\mathbb{C}_{\mathfrak{p}}^{\flat}) \otimes_{\mathbb{Z}_p} T$ and using Lemma [Lemma 16](#lem:cplusp_in_cpplus_ainf){reference-type="ref" reference="lem:cplusp_in_cpplus_ainf"}, for each $\mathfrak{p}\in \pazocal{S}$, we obtain $(\varphi, G_R(\mathfrak{p}))\textrm{-equivariant}$ inclusions $$\label{eq:wachmod_almost_comp_p} \mu^s \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Z}_p} T \subset \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_L^+(\mathfrak{p})} N_L(\mathfrak{p})\subset \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Z}_p} T,$$ where the middle term can be written as $\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_L^+(\mathfrak{p})} N_L(\mathfrak{p})= \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_R^+} N_R$. Now from Remark [Remark 18](#rem:gr_act_prodainf){reference-type="ref" reference="rem:gr_act_prodainf"} recall that $\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))$ is equipped with an action of $G_R$ and from Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"} we have a $(\varphi, G_R)\textrm{-equivariant}$ embedding $\mathbf{A}_{\inf}(\overline{R}) \subset \prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))$. Then we can equip $\prod_{\mathfrak{p}\in \pazocal{S}} (\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Z}_p} T) = (\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))) \otimes_{\mathbb{Z}_p} T$ with the diagonal action of $(\varphi, G_R)$ and similarly for $\prod_{\mathfrak{p}\in \pazocal{S}} (\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_L^+(\mathfrak{p})} N_L(\mathfrak{p})) = \prod_{\mathfrak{p}\in \pazocal{S}} (\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_R^+} N_R) = (\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))) \otimes_{\mathbf{A}_R^+} N_R$, where the second equality follows from the fact that product is an exact functor on the category of $\mathbf{A}_R^+\textrm{-modules}$ and $N_R$ is finitely presented over the noetherian ring $\mathbf{A}_R^+$ (see [@stacks-project [Tag 059K](https://stacks.math.columbia.edu/tag/059K)]). So taking the product of [\[eq:wachmod_almost_comp_p\]](#eq:wachmod_almost_comp_p){reference-type="eqref" reference="eq:wachmod_almost_comp_p"} over all $\mathfrak{p}\in \pazocal{S}$, we obtain $(\varphi, G_R)\textrm{-equivariant}$ inclusions $$\label{eq:wachmod_prod_almost_comp} \mu^s \prod_{\mathfrak{p}\in \pazocal{S}} \big(\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Z}_p} T\big) \subset \prod_{\mathfrak{p}\in \pazocal{S}} \big(\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_R^+} N_R\big) \subset \prod_{\mathfrak{p}\in \pazocal{S}} \big(\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Z}_p} T\big).$$ Inverting $\mu$ in [\[eq:wachmod_prod_almost_comp\]](#eq:wachmod_prod_almost_comp){reference-type="eqref" reference="eq:wachmod_prod_almost_comp"} and from the discussion above we get a $G_R\textrm{-equivariant}$ isomorphism $$\label{eq:wachmod_prod_muinverse_comp} \Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{\mu}\big]\otimes_{\mathbf{A}_R^+[1/\mu]} N_R\big[\tfrac{1}{\mu}\big]\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{\mu}\big]\otimes_{\mathbb{Z}_p} T.$$ Furthermore, the $(\varphi, G_R)\textrm{-equivariant}$ isomorphism in [\[eq:phigamma_comp_tilde_relative\]](#eq:phigamma_comp_tilde_relative){reference-type="eqref" reference="eq:phigamma_comp_tilde_relative"} can be written as $$\label{eq:phigamma_comp_muinverse} W(\mathbb{C}(\overline{R})^{\flat}) \otimes_{\mathbf{A}_R^+[1/\mu]} N_R\big[\tfrac{1}{\mu}\big]\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}W(\mathbb{C}(\overline{R})^{\flat}) \otimes_{\mathbb{Z}_p} T.$$ Using Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"}, all terms in [\[eq:wachmod_prod_muinverse_comp\]](#eq:wachmod_prod_muinverse_comp){reference-type="eqref" reference="eq:wachmod_prod_muinverse_comp"} and [\[eq:phigamma_comp_muinverse\]](#eq:phigamma_comp_muinverse){reference-type="eqref" reference="eq:phigamma_comp_muinverse"} admit an embedding into $\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big) \otimes_{\mathbf{A}_R^+} N_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big) \otimes_{\mathbb{Z}_p} T$ compatible with $G_R\textrm{-action}$. Note that $N_R[1/\mu]$ is finite projective over $\mathbf{A}_R^+[1/\mu]$ (see Proposition [Proposition 35](#prop:wachmod_proj_pmu){reference-type="ref" reference="prop:wachmod_proj_pmu"}), so the intersection of left hand terms in [\[eq:wachmod_prod_muinverse_comp\]](#eq:wachmod_prod_muinverse_comp){reference-type="eqref" reference="eq:wachmod_prod_muinverse_comp"} and [\[eq:phigamma_comp_muinverse\]](#eq:phigamma_comp_muinverse){reference-type="eqref" reference="eq:phigamma_comp_muinverse"}, inside $\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big) \otimes_{\mathbf{A}_R^+} N_R$, gives $$\begin{aligned} \Big(W(\mathbb{C}(\overline{R})^{\flat}) \otimes_{\mathbf{A}_R^+[1/\mu]} N_R\big[\tfrac{1}{\mu}\big]\Big) &\cap \Big(\Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{\mu}\big]\otimes_{\mathbf{A}_R^+[1/\mu]} N_R\big[\tfrac{1}{\mu}\big]\Big)\\ &= \mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{\mu}\big]\otimes_{\mathbf{A}_R^+[1/\mu]} N_R\big[\tfrac{1}{\mu}\big]= \mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{\mu}\big]\otimes_{\mathbf{A}_R^+} N_R, \end{aligned}$$ where the first equality follows from Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"}. Similarly, the intersection of right hand terms in [\[eq:wachmod_prod_muinverse_comp\]](#eq:wachmod_prod_muinverse_comp){reference-type="eqref" reference="eq:wachmod_prod_muinverse_comp"} and [\[eq:phigamma_comp_muinverse\]](#eq:phigamma_comp_muinverse){reference-type="eqref" reference="eq:phigamma_comp_muinverse"}, inside $\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big) \otimes_{\mathbb{Z}_p} T$, gives $$\begin{aligned} \Big(W(\mathbb{C}(\overline{R})^{\flat}) \otimes_{\mathbb{Z}_p} T\Big) \cap \Big(\Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{\mu}\big]\otimes_{\mathbb{Z}_p} T\Big) = \mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{\mu}\big]\otimes_{\mathbb{Z}_p} T, \end{aligned}$$ where the equality again follows from Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"}. Since [\[eq:wachmod_prod_muinverse_comp\]](#eq:wachmod_prod_muinverse_comp){reference-type="eqref" reference="eq:wachmod_prod_muinverse_comp"} and [\[eq:phigamma_comp_muinverse\]](#eq:phigamma_comp_muinverse){reference-type="eqref" reference="eq:phigamma_comp_muinverse"} are isomorphisms, we get a natural $G_R\textrm{-equivariant}$ isomorphism $\mathbf{A}_{\inf}(\overline{R})[1/\mu] \otimes_{\mathbf{A}_R^+} N_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_{\inf}(\overline{R})[1/\mu] \otimes_{\mathbb{Z}_p} T$. ◻ **Corollary 41**. *Let $N$ be a Wach module over $\mathbf{A}_R^+$ and $T := \mathbf{T}_R(N)$ the associated finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$. Then we have a natural $G_R\textrm{-equivariant}$ comparison isomorphism $$\mathbf{A}^+[1/\mu] \otimes_{\mathbf{A}_R^+} N \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}^+[1/\mu] \otimes_{\mathbb{Z}_p} T.$$* *Proof.* Since $N[1/\mu]$ is finite projective over $\mathbf{A}_R^+[1/\mu]$, taking the intersection of the isomorphism in Proposition [Proposition 40](#prop:wachmod_comp_relative){reference-type="ref" reference="prop:wachmod_comp_relative"} with the isomorphism in [\[eq:phigamma_comp_iso_relative\]](#eq:phigamma_comp_iso_relative){reference-type="eqref" reference="eq:phigamma_comp_iso_relative"}, inside $\tilde{\mathbf{A}}\otimes_{\mathbb{Z}_p} T$, we obtain $\mathbf{A}^+[1/\mu] \otimes_{\mathbf{A}_R^+[1/\mu]} N[1/\mu] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}^+[1/\mu] \otimes_{\mathbb{Z}_p} T$. ◻ **Proposition 42**. *Let $N$ be an effective Wach module over $\mathbf{A}_R^+$ and $T := \mathbf{T}_R(N)$ the associated finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$. Then we have $(\varphi, \Gamma_R)\textrm{-equivariant}$ inclusions $\mu^s \mathbf{D}_R^+(T) \subset N \subset \mathbf{D}_R^+(T)$ (see §[2.6](#subsec:relative_padicreps){reference-type="ref" reference="subsec:relative_padicreps"} for notations).* *Proof.* The proof follows in a manner similar to the proof of Proposition [Proposition 40](#prop:wachmod_comp_relative){reference-type="ref" reference="prop:wachmod_comp_relative"}, so we will freely use the notation of that proof. Inverting $p$ in [\[eq:wachmod_prod_almost_comp\]](#eq:wachmod_prod_almost_comp){reference-type="eqref" reference="eq:wachmod_prod_almost_comp"} we get $(\varphi, G_R)\textrm{-equivariant}$ inclusions $$\begin{aligned} \label{eq:wachmod_prod_pinverse_almost_comp} \begin{split} \mu^s \Big(\prod_{\mathfrak{p}\in \pazocal{S}} \big(\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Z}_p} T\big)\Big)\big[\tfrac{1}{p}\big]&\subset \Big(\prod_{\mathfrak{p}\in \pazocal{S}} \big(\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_R^+} N_R\big)\Big)\big[\tfrac{1}{p}\big]\\ &\subset \Big(\prod_{\mathfrak{p}\in \pazocal{S}} \big(\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Z}_p} T\big)\Big)\big[\tfrac{1}{p}\big]. \end{split} \end{aligned}$$ The last term of [\[eq:wachmod_prod_pinverse_almost_comp\]](#eq:wachmod_prod_pinverse_almost_comp){reference-type="eqref" reference="eq:wachmod_prod_pinverse_almost_comp"} can be written as $\Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{p}\big]\otimes_{\mathbb{Q}_p} V$ and similarly for the first term. Moreover, we have $\prod_{\mathfrak{p}\in \pazocal{S}} (\mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbf{A}_R^+} N_R) = (\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))) \otimes_{\mathbf{A}_R^+} N_R$, so the middle term of [\[eq:wachmod_prod_pinverse_almost_comp\]](#eq:wachmod_prod_pinverse_almost_comp){reference-type="eqref" reference="eq:wachmod_prod_pinverse_almost_comp"} can be written as $\Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{p}\big]\otimes_{\mathbf{B}_R^+} N_R\big[\tfrac{1}{p}\big]$. Furthermore, by inverting $p$ in [\[eq:phigamma_comp_tilde_relative\]](#eq:phigamma_comp_tilde_relative){reference-type="eqref" reference="eq:phigamma_comp_tilde_relative"}, we have a $(\varphi, G_R)\textrm{-equivariant}$ comparison isomorphism $$\label{eq:phigamma_comp_pinverse} W(\mathbb{C}(\overline{R})^{\flat})\big[\tfrac{1}{p}\big]\otimes_{\mathbf{B}_R^+} N_R\big[\tfrac{1}{p}\big]\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}W(\mathbb{C}(\overline{R})^{\flat})\big[\tfrac{1}{p}\big]\otimes_{\mathbb{Q}_p} V.$$ Using Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"}, we embed all terms in [\[eq:wachmod_prod_pinverse_almost_comp\]](#eq:wachmod_prod_pinverse_almost_comp){reference-type="eqref" reference="eq:wachmod_prod_pinverse_almost_comp"} and [\[eq:phigamma_comp_pinverse\]](#eq:phigamma_comp_pinverse){reference-type="eqref" reference="eq:phigamma_comp_pinverse"} inside $\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big)[1/p] \otimes_{\mathbf{B}_R^+} N_R[1/p] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big)[1/p] \otimes_{\mathbb{Q}_p} V$. Since $N_R[1/p]$ is finite projective over $\mathbf{B}_R^+$, the intersection of middle term in [\[eq:wachmod_prod_pinverse_almost_comp\]](#eq:wachmod_prod_pinverse_almost_comp){reference-type="eqref" reference="eq:wachmod_prod_pinverse_almost_comp"} and left hand term in [\[eq:phigamma_comp_pinverse\]](#eq:phigamma_comp_pinverse){reference-type="eqref" reference="eq:phigamma_comp_pinverse"}, inside $\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big)[1/p] \otimes_{\mathbf{B}_R^+} N_R[1/p]$, gives $$\begin{aligned} \Big(W(\mathbb{C}(\overline{R})^{\flat})\big[\tfrac{1}{p}\big]\otimes_{\mathbf{B}_R^+} N_R\big[\tfrac{1}{p}\big]\Big) &\cap \Big(\Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{p}\big]\otimes_{\mathbf{B}_R^+} N_R\big[\tfrac{1}{p}\big]\Big) = \mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{p}\big]\otimes_{\mathbf{B}_R^+} N_R\big[\tfrac{1}{p}\big]. \end{aligned}$$ where the equality follows from Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"}. Similarly, the intersection of right hand terms in [\[eq:wachmod_prod_muinverse_comp\]](#eq:wachmod_prod_muinverse_comp){reference-type="eqref" reference="eq:wachmod_prod_muinverse_comp"} and [\[eq:phigamma_comp_pinverse\]](#eq:phigamma_comp_pinverse){reference-type="eqref" reference="eq:phigamma_comp_pinverse"}, inside $\big(\prod_{\mathfrak{p}\in \pazocal{S}} W(\mathbb{C}(\mathfrak{p})^{\flat})\big)[1/p] \otimes_{\mathbb{Q}_p} V$, gives $$\begin{aligned} \Big(W(\mathbb{C}(\overline{R})^{\flat})\big[\tfrac{1}{p}\big]\otimes_{\mathbb{Q}_p} V\Big) \cap \Big(\Big(\prod_{\mathfrak{p}\in \pazocal{S}} \mathbf{A}_{\inf}(\mathbb{C}^+(\mathfrak{p}))\Big)\big[\tfrac{1}{p}\big]\otimes_{\mathbb{Q}_p} V\Big) = \mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{p}\big]\otimes_{\mathbb{Q}_p} V, \end{aligned}$$ where the equality again follows from Lemma [Lemma 19](#lem:ainf_intersection){reference-type="ref" reference="lem:ainf_intersection"}. Therefore, from [\[eq:wachmod_prod_pinverse_almost_comp\]](#eq:wachmod_prod_pinverse_almost_comp){reference-type="eqref" reference="eq:wachmod_prod_pinverse_almost_comp"} and $(\varphi, G_R)\textrm{-equivariance}$ of [\[eq:phigamma_comp_pinverse\]](#eq:phigamma_comp_pinverse){reference-type="eqref" reference="eq:phigamma_comp_pinverse"}, we obtain $(\varphi, G_R)\textrm{-equivariant}$ inclusions $$\label{eq:wachmodrat_almost_comp} \mu^s \big(\mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{p}\big]\otimes_{\mathbb{Q}_p} V\big) \subset \mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{p}\big]\otimes_{\mathbf{B}_R^+} N_R\big[\tfrac{1}{p}\big]\subset \mathbf{A}_{\inf}(\overline{R})\big[\tfrac{1}{p}\big]\otimes_{\mathbb{Q}_p} V.$$ Inverting $p$ in the isomorphism of Corollary [Corollary 41](#cor:wachmod_comp_relative){reference-type="ref" reference="cor:wachmod_comp_relative"} and taking its intersection with [\[eq:wachmodrat_almost_comp\]](#eq:wachmodrat_almost_comp){reference-type="eqref" reference="eq:wachmodrat_almost_comp"}, inside $W(\mathbb{C}(\overline{R})^{\flat})[1/p] \otimes_{\mathbf{B}_R^+} N_R[1/p] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}W(\mathbb{C}(\overline{R})^{\flat})[1/p] \otimes_{\mathbb{Q}_p} V$, we obtain $(\varphi, G_R)\textrm{-equivariant}$ inclusions $$\mu^s \big(\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V\big) \subset \mathbf{B}^+ \otimes_{\mathbf{B}_R^+} N_R\big[\tfrac{1}{p}\big]\subset \mathbf{B}^+ \otimes_{\mathbb{Q}_p} V.$$ Taking $H_R\textrm{-invariants}$ and taking the intersection with $\mathbf{D}_R(T) = N_R[1/\mu]^{\wedge}$ in the preceding equation, inside $\mathbf{D}_R(V)$, we obtain $\mu^s \mathbf{D}_R^+(T) \subset N_R \subset \mathbf{D}_R^+(T)$ since $N_R = N_R[1/p] \cap N_R[1/\mu]^{\wedge}$ from Lemma [Lemma 33](#lem:wach_saturated){reference-type="ref" reference="lem:wach_saturated"} and $\mathbf{D}_R^+(T) = \mathbf{D}_R(T) \cap \mathbf{D}_R^+(V) \subset \mathbf{D}_R(V)$ by definition. ◻ ## Finite $[p]_q\textrm{-height}$ representations {#subsec:finite_pqheight_reps} In this section we generalize the definition of finite $[p]_q\textrm{-height}$ representations from [@abhinandan-relative-wach-i Definition 4.9] in the relative case. **Definition 43**. A finite $[p]_q\textrm{-height}$ $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ is a finite free $\mathbb{Z}_p\textrm{-module}$ $T$ admitting a linear and continuous action of $G_R$ such that there exists a finitely generated torsion-free $\mathbf{A}_R^+\textrm{-submodule}$ $\mathbf{N}_R(T) \subset \mathbf{D}_R(T)$ satisfying the following: 1. $\mathbf{N}_R(T)$ is a Wach module in the sense of Defintion [Definition 34](#defi:wach_mods_relative){reference-type="ref" reference="defi:wach_mods_relative"}. 2. $\mathbf{A}_R\otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T)$. The height of $T$ is defined to be the height of $\mathbf{N}_R(T)$. Say that $T$ is *positive* if $\mathbf{N}_R(T)$ is effective. A finite $[p]_q\textrm{-height}$ $p\textrm{-adic}$ representation of $G_R$ is a finite dimensional $\mathbb{Q}_p\textrm{-vector space}$ admitting a linear and continuous action of $G_R$ such that there exists a $G_R\textrm{-stable}$ $\mathbb{Z}_p\textrm{-lattice}$ $T \subset V$ with $T$ of finite $[p]_q\textrm{-height}$. We set $\mathbf{N}_R(V) := \mathbf{N}_R(T)[1/p]$ satisfying properties analogous to (1) and (2) above. The height of $V$ is defined to be the height of $T$. Say that $V$ is positive if $\mathbf{N}_R(V)$ is effective. **Lemma 44**. *Let $T$ be a finite $[p]_q\textrm{-height}$ $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ then the $\mathbf{A}_R^+\textrm{-module}$ $\mathbf{N}_R(T)$ is unique.* *Proof.* By definition $\mathbf{A}_R\otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T)$ and this scalar extension induces a fullly faithful functor in Proposition [Proposition 38](#prop:wach_etale_ff_relative){reference-type="ref" reference="prop:wach_etale_ff_relative"}. So from [\[eq:rep_phigamma_relative\]](#eq:rep_phigamma_relative){reference-type="eqref" reference="eq:rep_phigamma_relative"} we obtain the uniqueness of $\mathbf{N}(T)$. Alternatively, the uniquess can also be deduced using Proposition [Proposition 42](#prop:wachmod_almost_comp){reference-type="ref" reference="prop:wachmod_almost_comp"} and [@abhinandan-relative-wach-i Proposition 4.13]. ◻ *Remark 45*. Let $V$ be a finite $[p]_q\textrm{-height}$ $p\textrm{-adic}$ representation of $G_R$ and $T \subset V$ a finite $[p]_q\textrm{-height}$ $G_R\textrm{-stable}$ $\mathbb{Z}_p\textrm{-lattice}$. Then we have $\mathbf{N}_R(V) = \mathbf{N}_R(T)[1/p]$ and from Proposition [Proposition 42](#prop:wachmod_almost_comp){reference-type="ref" reference="prop:wachmod_almost_comp"} we get that if $V$ is positive then $\mu^s \mathbf{D}_R^+(V) \subset \mathbf{N}_R(V) \subset \mathbf{D}_R^+(V)$. Moreover, similar to [@abhinandan-relative-wach-i Remark 3.10] we can show that $\mathbf{N}_R(V)$ is unique, in particular, it is independent of choice of the lattice $T$ by Corollary [Corollary 39](#cor:wach_etale_ff_relative){reference-type="ref" reference="cor:wach_etale_ff_relative"}. *Remark 46*. By the definition of finite $[p]_q\textrm{-height}$ representations, Lemma [Lemma 44](#lem:wach_unique_relative){reference-type="ref" reference="lem:wach_unique_relative"} and the fully faithful functor in [\[eq:wach_reps_relative\]](#eq:wach_reps_relative){reference-type="eqref" reference="eq:wach_reps_relative"} it follows that the data of a finite height representation is equivalent to the data of a Wach module. ## Nygaard filtration on Wach modules {#subsec:nygaard_fil_wach_mod} In this section we consider the Nygaard filtration on Wach modules as follows: **Definition 47**. Let $N$ be a Wach module over $\mathbf{A}_R^+$. Define a decreasing filtration on $N$ called the *Nygaard filtration*, for $k \in \mathbb{Z}$, as $$\textup{Fil}^k N := \{ x \in N \textrm{ such that } \varphi(x) \in [p]_q^k N\}.$$ From the definition it is clear that $N$ is effective if and only if $\textup{Fil}^0 N = N$. Similarly, we define Nygaard filtration on $M := N[1/p]$ and it easily follows that $\textup{Fil}^k M = (\textup{Fil}^k N)[1/p]$. The reason to consider the Nygaard filtration as above filtration is the following: note that $\mathbf{A}_{\textup{cris}}(\overline{R})$ is equipped with a filtration by divided power ideals and the embedding $\mathbf{A}_{\inf}(\overline{R}) \subset \mathbf{A}_{\textup{cris}}(\overline{R})$ induces a filtration on $\mathbf{A}_{\inf}(\overline{R})$ given as $\textup{Fil}^k \mathbf{A}_{\inf}(\overline{R}) := \xi^k \mathbf{A}_{\inf}(\overline{R})$ for $k \in \mathbb{N}$. We equip $\mathbf{A}^+$ with the induced filtration $\textup{Fil}^k \mathbf{A}^+ := \mathbf{A}^+ \cap \textup{Fil}^k \mathbf{A}_{\inf}(\overline{R}) \subset \mathbf{A}_{\inf}(\overline{R})$. **Lemma 48**. *Let $N$ be an effective Wach module over $\mathbf{A}_R^+$ and $T := \mathbf{T}_R(N)$ the associated $\mathbb{Z}_p\textrm{-representation}$ of $G_R$. Then for $k \in \mathbb{N}$ we have $(\textup{Fil}^k\mathbf{A}^+ \otimes_{\mathbb{Z}_p} T) \cap N = \textup{Fil}^k N$.* *Proof.* Let $V := T[1/p]$, $M := N[1/p]$ and $\textup{Fil}^k \mathbf{B}^+ := (\textup{Fil}^k \mathbf{A}^+)[1/p]$. Then it is enough to show that $(\textup{Fil}^k\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V) \cap M = \textup{Fil}^k M$. Indeed, from Definition [Definition 47](#defi:nygaard_fil){reference-type="ref" reference="defi:nygaard_fil"} we have $\textup{Fil}^k N := \textup{Fil}^k M \cap N = (\textup{Fil}^k\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V) \cap M \cap N = (\textup{Fil}^k\mathbf{A}^+ \otimes_{\mathbb{Z}_p} T) \cap N$ since $\textup{Fil}^k \mathbf{B}^+ \cap \mathbf{A}^+ = \textup{Fil}^k \mathbf{A}^+$. Now the inclusion $\textup{Fil}^k M \subset (\textup{Fil}^k \mathbf{B}^+ \otimes_{\mathbb{Q}_p} V)$ is obvious and for the converse it is enough to show $([p]_q^k\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V) \cap M = [p]_q^k M$. Indeed, if we have $x \in (\textup{Fil}^k\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V) \cap M$ then $\varphi(x) \in ([p]_q^k\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V) \cap M = [p]_q^k M$, i.e. $x \in \textup{Fil}^k M$. For the reduced claim, the inclusion $[p]_q^k M \subset ([p]_q^k\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V) \cap M$ is obvious. To show the converse, let $x \in \mathbf{B}^+ \otimes_{\mathbb{Q}_p} V$ such that $[p]_q^k x \in M$, in particular, $x \in M[1/[p]_q]$. Then it follows that $h(x) = x$ for all $h \in H_R$, i.e. $x \in (\mathbf{B}^+ \otimes_{\mathbb{Q}_p} V)^{H_R} =: \mathbf{D}_R^+(V) = \mathbf{D}_R^+(T)[1/p]$. From Proposition [Proposition 42](#prop:wachmod_almost_comp){reference-type="ref" reference="prop:wachmod_almost_comp"} recall that $\mu^s \mathbf{D}_R^+(V) \subset M$, where $s$ is the $[p]_q\textrm{-height}$ of $N$. So we get that $\mu^s x \in M$, in particular, $x \in M[1/\mu]$. Combining this with the previous observation we get that $x \in M[1/\mu] \cap M[1/[p]_q] \subset \mathbf{B}_R\otimes_{\mathbf{B}_R^+} M$. But from Proposition [Proposition 35](#prop:wachmod_proj_pmu){reference-type="ref" reference="prop:wachmod_proj_pmu"} we know that $M$ is finite projective over $\mathbf{B}_R^+$ and note that $\mathbf{B}_R^+ = \mathbf{B}_R^+[1/\mu] \cap \mathbf{B}_R^+[1/[p]_q] \subset \mathbf{B}_R$ since $[p]_q = p \mod \mu \mathbf{B}_R^+$. Hence, it follows that we have $x \in M[1/\mu] \cap M[1/[p]_q] = M$ as desired. ◻ ## Wach modules are crystalline {#subsec:wachmod_crystalline} The goal of this section is to prove Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"}. In order to prove our results we need auxiliary period rings $\mathbf{A}_{R,\varpi}^{\textup{PD}}$ and $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ from [@abhinandan-relative-wach-i §4.3.1]. We briefly recall their definitions. Let $\varpi := \zeta_p-1$ and set $\mathbf{A}_{R,\varpi}^+ := \mathbf{A}_R^+[\varphi^{-1}(\mu)] \subset \mathbf{A}_{\inf}(R_{\infty})$ stable under the $(\varphi, \Gamma_R)\textrm{-action}$ on the latter. By restricting to $\mathbf{A}_{R,\varpi}^+$, the map $\theta$ on $\mathbf{A}_{\inf}(R_{\infty})$ (see §[2.2](#subsec:ainf_relative){reference-type="ref" reference="subsec:ainf_relative"}), we obtain a surjective ring homomorphism $\theta: \mathbf{A}_{R,\varpi}^+ \twoheadrightarrow R[\varpi]$. We define $\mathbf{A}_{R,\varpi}^{\textup{PD}}$ to be the $p\textrm{-adic}$ completion of the divided power envelope of the map $\theta$ with respect to $\textup{Ker }\theta$. Furthemore, the map $\theta$ extends $R\textrm{-linearly}$ to a surjective ring homomorphism $\theta_R : R \otimes_{\mathbb{Z}} \mathbf{A}_{R,\varpi}^+ \twoheadrightarrow R[\varpi]$ given as $x \otimes y \mapsto x\theta(y)$. Similar to above, we define $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ to be the $p\textrm{-adic}$ completion of the divided power envelope of the map $\theta_R$ with respect to $\textup{Ker }\theta_R$. The morphism $\theta$ and $\theta_R$ naturally extend to respective surjections $\theta : \mathbf{A}_{R,\varpi}^{\textup{PD}} \twoheadrightarrow R[\varpi]$ and $\theta_R : \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \twoheadrightarrow R[\varpi]$. Now from loc. cit. we have natural inclusions $\mathbf{A}_{R,\varpi}^{\textup{PD}} \subset \mathbf{A}_{\textup{cris}}(R_{\infty})$ and $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \subset \pazocal{O}\mathbf{A}_{\textup{cris}}(R_{\infty})$, and it is easy to verify that the former rings are stable under $(\varphi, \Gamma_R)\textrm{-action}$ on the latter rings. Therefore, we equip $\mathbf{A}_{R,\varpi}^{\textup{PD}}$ and $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ with induced structures, in particular, a filtration and an $\mathbf{A}_{R,\varpi}^{\textup{PD}}\textrm{-linear}$ connection $\partial_A$ on $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ satisfying Griffiths transversality with respect to the filtration, and it is easy to show that $(\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}})^{\partial_A=0} = \mathbf{A}_{R,\varpi}^{\textup{PD}}$. Note that the aforementioned filtration on $\mathbf{A}_{R,\varpi}^{\textup{PD}}$ and $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ coincide with the divided power filtration by $\textup{Ker }\theta$ and $\textup{Ker }\theta_R$ respectively (see [@abhinandan-relative-wach-i Remark 4.23]). **Theorem 49**. *Let $N$ be a Wach module over $\mathbf{A}_R^+$ and let $T := \mathbf{T}_R(N)$ be the associated finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$. Then $V := T[1/p]$ is a $p\textrm{-adic}$ crystalline representation of $G_R$.* *Proof.* For $r \in \mathbb{N}$ large enough, the Wach module $\mu^r N (-r)$ is always effective and $\mathbf{T}_R(\mu^rN(-r)) = \mathbf{T}_R(N)(-r)$ (the twist $(-r)$ denotes a Tate twist on which $\Gamma_R$ acts via $\chi^{-r}$, where $\chi$ is the $p\textrm{-adic}$ cyclotomic character). Therefore, it follows that it is enough to show the claim for effective Wach modules. So in the rest of the proof we will assume that $N$ is effective. Now take $D_R := (\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} N[1/p])^{\Gamma_R} \subset \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ and from Proposition [Proposition 50](#prop:oarpd_comparison){reference-type="ref" reference="prop:oarpd_comparison"} note that $D_R$ is a finite projective $R[1/p]\textrm{-module}$ of rank $= \textup{rk}_{\mathbf{B}_R^+} N[1/p]$. Moreover, $D_R$ is equipped with tensor product Frobenius and a filtration defined as $\textup{Fil}^k D_R := \sum_{i+j=k} (\textup{Fil}^i \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} \textup{Fil}^j N[1/p])^{\Gamma_R}$ where $N[1/p]$ is equipped with the Nygaard filtration of Definition [Definition 47](#defi:nygaard_fil){reference-type="ref" reference="defi:nygaard_fil"}. Furthermore, $D_R$ is equipped with a connection induced from the connection on $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ and satisfying Griffiths transversality with respect to the filtration described above. Using Proposition [Proposition 50](#prop:oarpd_comparison){reference-type="ref" reference="prop:oarpd_comparison"}, note that we have a natural isomorphism $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_R D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} N[1/p]$. Now consider the following diagram where the left vertical arrow is extension of the $R[1/p]\textrm{-linear}$ injective map $D_R \rightarrow \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ from [\[eq:dr_in_dcrys\]](#eq:dr_in_dcrys){reference-type="eqref" reference="eq:dr_in_dcrys"} along the faithfully flat ring homomorphism $R[1/p] \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$ (see [@brinon-relatif Thèoréme 6.3.8]), the top horizontal arrow is extension along $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}[1/p] \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$ of the isomorphism [\[eq:oarpd_comparison\]](#eq:oarpd_comparison){reference-type="eqref" reference="eq:oarpd_comparison"} in Proposition [Proposition 50](#prop:oarpd_comparison){reference-type="ref" reference="prop:oarpd_comparison"}, the right vertical arrow is extension along $\mathbf{A}^+[1/\mu] \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$ of the isomorphism in Proposition [Proposition 40](#prop:wachmod_comp_relative){reference-type="ref" reference="prop:wachmod_comp_relative"} and the bottom horizontal arrow is the natural injective map (see [@brinon-relatif Proposition 8.2.6]). Commutativity of the diagram above and compatibility of its arrows with $(\varphi, G_R)\textrm{-action}$ and connection follows from [\[eq:dr_in_dcrys\]](#eq:dr_in_dcrys){reference-type="eqref" reference="eq:dr_in_dcrys"}. Since the top horizontal and right vertical arrows are bijective, we conclude that the left vertical arrow and bottom horizontal arrow are also bijective. Therefore, $V$ is a $p\textrm{-adic}$ crystalline representation of $G_R$ and by taking $G_R$ fixed part of the left vertical arrow we obtain an isomorphism of $R[1/p]\textrm{-modules}$ $$\label{eq:dr_dcrys_comp} D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$$ compatible with Frobenius and connection. Moreover, from [@brinon-relatif Proposition 8.4.3] recall that the bottom horizontal arrow of the diagram above is compatible with filtrations, so using [@abhinandan-relative-wach-i Proposition 4.49] it is easy to show that [\[eq:dr_dcrys_comp\]](#eq:dr_dcrys_comp){reference-type="eqref" reference="eq:dr_dcrys_comp"} is compatible with filtrations, where we consider Hodge filtration on $\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$. Indeed, the claim has been proven in loc. cit. under the assumption that $\mathbf{N}_R(T)$ becomes free after extension of scalars along $\mathbf{A}_R\rightarrow \mathbf{A}_{R'}^+$ (in the notation of loc. cit.). This assumption does not always hold in our current case, however, in [@abhinandan-relative-wach-i §4.5.1] we use the preceding assumption only in the proof of [@abhinandan-relative-wach-i Lemma 4.53]. So in our current setting we can replace [@abhinandan-relative-wach-i Lemma 4.53] by Lemma [Lemma 48](#lem:nygaardfil_induced){reference-type="ref" reference="lem:nygaardfil_induced"}. Then the results of [@abhinandan-relative-wach-i §4.5.1] remain valid for our setting as well, in particular, we get that [\[eq:dr_dcrys_comp\]](#eq:dr_dcrys_comp){reference-type="eqref" reference="eq:dr_dcrys_comp"} is compatible with filtrations. Finally, from [@abhinandan-relative-wach-i Corollary 4.54] it follows that the isomorphism in Proposition [Proposition 50](#prop:oarpd_comparison){reference-type="ref" reference="prop:oarpd_comparison"} is compatible with filtrations. ◻ **Proposition 50**. *Let $N$ be an effective Wach module over $\mathbf{A}_R^+$, then $D_R := \big(\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} N[1/p]\big)^{\Gamma_R}$ is a finite projective $R[1/p]\textrm{-module}$ of rank $= \textup{rk}_{\mathbf{B}_R^+} N[1/p]$ equipped with Frobenius, filtration and a connection satisfying Griffiths transversality with respect to the filtration. Moreover, we have a natural comparison isomorphism $$\label{eq:oarpd_comparison} f : \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_R D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} N[1/p],$$ compatible with Frobenius, filtration, connection and $\Gamma_R\textrm{-action}$.* *Proof.* We will adapt the proof of [@abhinandan-relative-wach-i Proposition 4.28]. Recall the following rings from [@abhinandan-relative-wach-i §4.4.1]: for $n \in \mathbb{N}$, define a $p\textrm{-adically}$ complete ring $S_n^{\textup{PD}} := \mathbf{A}_R^+\big\langle\tfrac{\mu}{p^n}, \tfrac{\mu^2}{2!p^{2n}}, \ldots, \tfrac{\mu^k}{k!p^{kn}}, \ldots\big\rangle$. We have a Frobenius homomorphism $\varphi : S_n^{\textup{PD}}\rightarrow S_{n-1}^{\textup{PD}}$, in particular, $\varphi^n(S_n^{\textup{PD}}) \subset \mathbf{A}_{R,\varpi}^{\textup{PD}}$ and the ring $S_n^{\textup{PD}}$ is further equipped with a continuous (for the $p\textrm{-adic}$ topology) action of $\Gamma_R$. The reader should note that in [@abhinandan-relative-wach-i §4.4.1] we consider a further completion of $S_n^{\textup{PD}}$ with respect to certain filtration by PD-ideals which we denote as $\widehat{S}_n^{\textup{PD}}$ in loc. cit. However, such a completion is not strictly necessary and all proofs of loc. cit. can be carried out without it. In particular, many good properties of $\widehat{S}_n^{\textup{PD}}$ restrict to good properties on $S_n^{\textup{PD}}$ as well (for example, $(\varphi, \Gamma_R)\textrm{-action}$ above). Let us now consider the $O_F\textrm{-linear}$ homomorphism of rings $\iota : R \rightarrow S_n^{\textup{PD}}$ defined by sending $X_j \mapsto [X_j^{\flat}]$ for $1 \leq j \leq d$. Using $\iota$ we can define an $O_F\textrm{-linear}$ homomorphism of rings $f : R \otimes_{O_F} S_n^{\textup{PD}}\rightarrow S_n^{\textup{PD}}$ sending $a \otimes b \mapsto \iota(a) b$. Let $\pazocal{O}S_n^{\textup{PD}}$ denote the $p\textrm{-adic}$ completion of the divided power envelope of $R \otimes_{O_F} S_n^{\textup{PD}}$ with respect to $\textup{Ker }f$. The tensor product Frobenius induces $\varphi : \pazocal{O}S_n^{\textup{PD}}\rightarrow \pazocal{O}S_{n-1}^{\textup{PD}}$ such that $\varphi^n(\pazocal{O}S_n^{\textup{PD}}) \subset \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ and the action of $\Gamma_R$ extends to a continuous (for $p\textrm{-adic}$ topology) action on $\pazocal{O}S_n^{\textup{PD}}$. Moreover, we have a $(\varphi, \Gamma_R)\textrm{-equivariant}$ embedding $S_n^{\textup{PD}}\subset \pazocal{O}S_n^{\textup{PD}}$ and the latter is equipped with a $\Gamma_R\textrm{-equivariant}$ $\pazocal{O}S_n^{\textup{PD}}\textrm{-linear}$ integrable connection given as the universal continuous $S_n^{\textup{PD}}\textrm{-linear}$ de Rham differential $d : \pazocal{O}S_n^{\textup{PD}}\rightarrow \Omega^1_{\pazocal{O}S_n^{\textup{PD}}/S_n^{\textup{PD}}}$. Furthermore, we have $R = (\pazocal{O}S_n^{\textup{PD}})^{\Gamma_R}$ and with $V_j = \frac{X_j \otimes 1}{1 \otimes [X_j^{\flat}]}$ for $1 \leq j \leq d$, we have $p\textrm{-adically}$ closed divided power ideals of $\pazocal{O}S_n^{\textup{PD}}$ as follows: $$J^{[i]}\pazocal{O}S_n^{\textup{PD}}:= \Big\langle \tfrac{\mu^{[k_0]}}{p^{nk_0}} \prod_{j=1}^d (1-V_j)^{[k_j]}, \hspace{1mm} \mathbf{k}= (k_0, k_1, \ldots, k_d) \in \mathbb{N}^{d+1} \hspace{1mm} \textrm{such that} \hspace{1mm} \sum_{j=0}^d k_j \geq i\Big\rangle.$$ We equip $\pazocal{O}S_n^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N$ with tensor product Frobenius, tensor product filtration and the connection on $\pazocal{O}S_n^{\textup{PD}}$ induces a $S_n^{\textup{PD}}\textrm{-linear}$ integrable connection on $\pazocal{O}S_n^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N$. Then $D_n := \big(\pazocal{O}S_n^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N[1/p]\big)^{\Gamma_R}$ is an $R[1/p]\textrm{-module}$ equipped with a Frobenius $\varphi : D_n \rightarrow D_{n-1}$ and an integrable connection. In particular, it follows that $\varphi^n(D_n) \subset D_R = \big(\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} N[1/p]\big)^{\Gamma_R} \subset \big(\pazocal{O}\mathbf{A}_{\textup{cris}}(\overline{R}) \otimes_{\mathbf{A}_R^+} N[1/p]\big)^{H_R}$, where we have $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \subset \pazocal{O}\mathbf{A}_{\textup{cris}}(R_{\infty}) = \pazocal{O}\mathbf{A}_{\textup{cris}}(\overline{R})^{H_R}$ (see [@morrow-tsuji Corollary 4.34] for the equality). Let $T := \mathbf{T}_R(N)$ denote the finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ associated to $N$ and set $V := T[1/p]$, then we have $$\begin{aligned} \label{eq:dr_in_dcrys} \begin{split} D_R \subset \big(\pazocal{O}\mathbf{B}_{\textup{cris}}^+(\overline{R}) \otimes_{\mathbf{B}_R^+} N[1/p]\big)^{G_R} \subset &\big(\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbf{B}_R^+} N[1/p]\big)^{G_R}\\ &\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\big(\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbb{Q}_p} V\big)^{G_R} = \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V), \end{split} \end{aligned}$$ where the isomorphism follows by taking $G_R\textrm{-fixed}$ elements after extension of scalars of the isomorphism in Proposition [Proposition 40](#prop:wachmod_comp_relative){reference-type="ref" reference="prop:wachmod_comp_relative"} along $\mathbf{A}_{\inf}(\overline{R})[1/\mu] \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$. Since $\varphi^n(D_n) \subset D_R$, or equivalently, the $R[1/p]\textrm{-linear}$ map $1 \otimes \varphi^n : R[1/p] \otimes_{\varphi^n, R[1/p]} D_n \rightarrow D_R$ is injective, we get that $R[1/p] \otimes_{\varphi^n, R[1/p]} D_n$ is a finitely generated $R[1/p]\textrm{-module}$. Moreover, recall that $\varphi^n : R[1/p] \rightarrow R[1/p]$ is finite flat (see §[1.5](#subsec:setup_nota){reference-type="ref" reference="subsec:setup_nota"}), so it follows that $D_n$ is finitely generated over the source of $\varphi^n$, i.e. $D_n$ is a finitely generated $R[1/p]\textrm{-module}$ equipped with an integrable connection, in particular, it is finite projective over $R[1/p]$ by [@brinon-relatif Proposition 7.1.2]. Furthermore, recall that $N[1/p]$ is a finite projective $\mathbf{B}_R^+\textrm{-module}$ (see Proposition [Proposition 35](#prop:wachmod_proj_pmu){reference-type="ref" reference="prop:wachmod_proj_pmu"}), therefore $\pazocal{O}S_n^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N[1/p]$ is a finite projective $\pazocal{O}S_n^{\textup{PD}}[1/p]\textrm{-module}$ and from [@abbes-gros-tsuji Lemma IV.3.2.2] it follows that $\pazocal{O}S_n^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N$ is $p\textrm{-adically}$ complete. Then for $n \geq 1$, similar to the proof of [@abhinandan-relative-wach-i Lemmas 4.32 & 4.36], it is easy to show that $\log \gamma_i := \sum_{k \in \mathbb{N}} (-1)^k \frac{(\gamma_i-1)^{k+1}}{k+1}$ converge as a series of operators on $\pazocal{O}S_n^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N$, where $\{\gamma_0, \gamma_1, \ldots, \gamma_d\}$ are topological generators of $\Gamma_R$ (see §[2](#sec:period_rings_padic_reps){reference-type="ref" reference="sec:period_rings_padic_reps"}). **Lemma 51**. *Let $m \geq 1$ (let $m \geq 2$ if $p=2$), then we have a $\Gamma_R\textrm{-equivariant}$ isomorphism via the natural map $a \otimes b \otimes x \mapsto ab \otimes x$: $$\label{eq:osmpd_dm_n} \pazocal{O}S_m^{\textup{PD}}\otimes_R D_m \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}S_m^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N[1/p].$$* *Proof.* The map in [\[eq:osmpd_dm_n\]](#eq:osmpd_dm_n){reference-type="eqref" reference="eq:osmpd_dm_n"} is obviously compatible with $\Gamma_R\textrm{-action}$, so it remains to check that it is bijective. Let us first check the injectivity of [\[eq:osmpd_dm_n\]](#eq:osmpd_dm_n){reference-type="eqref" reference="eq:osmpd_dm_n"}. We have a composition of injective homomorphisms $\pazocal{O}S_m^{\textup{PD}}[1/p] \xrightarrow{\hspace{1mm} \varphi^m \hspace{1mm}} \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}[1/p] \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$. As $D_m$ is finite projective over $R[1/p]$, the map $$\label{eq:osmpd_obcrys_dm} \pazocal{O}S_m^{\textup{PD}}\otimes_R D_m = \pazocal{O}S_m^{\textup{PD}}[1/p] \otimes_{R[1/p]} D_m \longrightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\varphi^m, R[1/p]} D_m,$$ is injective Next, we have $V = T[1/p]$ and we consider the composition $$\label{eq:dm_obcrys_dcrys} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\varphi^m, R[1/p]} D_m \xrightarrow{\hspace{1mm} 1 \otimes \varphi^m \hspace{1mm}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{R[1/p]} D_R \longrightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V).$$ As $R[1/p] \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$ is faithfully flat (see [@brinon-relatif Théorème 6.3.8]), so in [\[eq:dm_obcrys_dcrys\]](#eq:dm_obcrys_dcrys){reference-type="eqref" reference="eq:dm_obcrys_dcrys"}, the second map is injective since [\[eq:dr_in_dcrys\]](#eq:dr_in_dcrys){reference-type="eqref" reference="eq:dr_in_dcrys"} is injective and the first map is injective because $1 \otimes \varphi^m : R[1/p] \otimes_{\varphi^m, R[1/p]} D_m \rightarrow D_R$ is injective, in particular, [\[eq:dm_obcrys_dcrys\]](#eq:dm_obcrys_dcrys){reference-type="eqref" reference="eq:dm_obcrys_dcrys"} is injective. Moreover, since $N[1/p]$ is a finite projective $\mathbf{B}_R^+\textrm{-module}$, similiar to [\[eq:osmpd_obcrys_dm\]](#eq:osmpd_obcrys_dm){reference-type="eqref" reference="eq:osmpd_obcrys_dm"} it can be shown that the map $\pazocal{O}S_m^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N[1/p] = \pazocal{O}S_m^{\textup{PD}}[1/p] \otimes_{\mathbf{B}_R^+} N[1/p] \longrightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\varphi^m, \mathbf{B}_R^+} N[1/p]$ is injective. Furthermore, from the definition of Wach modules (see Definition [Definition 34](#defi:wach_mods_relative){reference-type="ref" reference="defi:wach_mods_relative"}), we have an isomorphism $1 \otimes \varphi : \mathbf{B}_R^+ \otimes_{\varphi, \mathbf{B}_R^+} N[1/p, 1/[p]_q] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}N[1/p, 1/[p]_q]$. Therefore, we obtain an isomorphism $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\varphi^m, \mathbf{B}_R^+} N[1/p] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbf{B}_R^+} N[1/p]$ since $[p]_q$ is unit in $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})$. From the preceding observations, it follows that the composition $$\label{eq:osmpd_obcrys_n} \pazocal{O}S_m^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N[1/p] \longrightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\varphi^m, \mathbf{B}_R^+} N[1/p] \xrightarrow[\sim]{\hspace{1mm} 1 \otimes \varphi^m \hspace{1mm}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbf{B}_R^+} N[1/p],$$ is injective Let us now consider the diagram where the right vertical arrow is the natural injective map (see [@brinon-relatif Proposition 8.2.6]). From definitions, it easily follows that the diagram commutes, therefore the left vertical arrow, i.e. [\[eq:osmpd_dm_n\]](#eq:osmpd_dm_n){reference-type="eqref" reference="eq:osmpd_dm_n"} is injective. Now let us check the surjectivity of the map [\[eq:osmpd_dm_n\]](#eq:osmpd_dm_n){reference-type="eqref" reference="eq:osmpd_dm_n"}. We define the following operators on $\pazocal{O}N_m^{\textup{PD}}:= \pazocal{O}S_m^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N[1/p]$, $$\partial_i := \left\{ \begin{array}{ll} -(\log \gamma_0)/t & \textrm{for} \hspace{1mm} i = 0,\\ (\log \gamma_i)/(tV_i) & \textrm{for} \hspace{1mm} 1 \leq i \leq d, \end{array} \right.$$ where $V_i = \frac{X_i \otimes 1}{1 \otimes [X_i^{\flat}]}$ for $1 \leq i \leq d$ (see [@abhinandan-relative-wach-i §4.4.2]). Note that for $g \in \Gamma_R$ and $x \in \pazocal{O}S_m^{\textup{PD}}\otimes_{\mathbf{A}_L^+} N$ we have $(g-1)(ax) = (g-1)a \cdot x + g(a)(g-1)x$. Then from the identity $\log(\gamma_i) = \lim_{n \rightarrow +\infty} (\gamma_i^{p^n}-1)/p^n$, it easily follows that $\partial_i$ satisfy the Leibniz rule for all $0 \leq i \leq d$. In particular, the operator $\partial : \pazocal{O}N_m^{\textup{PD}}\rightarrow \pazocal{O}N_m^{\textup{PD}}\otimes_{\pazocal{O}S_m^{\textup{PD}}} \Omega^1_{\pazocal{O}S_m^{\textup{PD}}/R}$ given by $x \mapsto \partial_0(x) dt + \sum_{i=1}^d \partial_i(x) d [X_i^{\flat}]$ defines a connection on $\pazocal{O}N_m^{\textup{PD}}$. Furthermore, from [@abhinandan-relative-wach-i Lemma 4.38] the operators $\partial_i$ commute with each other, so the connection $\partial$ is integrable and using the finite $[p]_q\textrm{-height}$ property of $N$, similar to [@abhinandan-relative-wach-i Lemma 4.39], it is easy to show that $\partial$ is $p\textrm{-adically}$ quasi-nilpotent. Then similar to the proof of [@abhinandan-relative-wach-i Lemma 4.39 & Lemma 4.41], it follows that for $x \in N[1/p]$ the following sum converges in $D_m = (\pazocal{O}N_m^{\textup{PD}})^{\Gamma_R} = (\pazocal{O}N_m^{\textup{PD}})^{\partial=0}$ $$\label{eq:horizontal_elems} y = \sum_{\mathbf{k}\in \mathbb{N}^{d+1}} \partial_0^{k_0} \circ \partial_1^{k_1} \circ \cdots \circ \partial_d^{k_d} (x) \tfrac{t^{[k_0]}}{p^{mk_0}} (1-V_1)^{[k_1]} \cdots (1-V_d)^{[k_d]}.$$ Using the construction above we define a linear transformation $\alpha$ on the finite projective $\pazocal{O}S_m^{\textup{PD}}[1/p]\textrm{-module}$ $\pazocal{O}N_m^{\textup{PD}}$ and claim that $\alpha$ is an automorphism of $\pazocal{O}N_m^{\textup{PD}}$. Indeed, first choose a presentation $\pazocal{O}N_m^{\textup{PD}}\oplus N' = (\pazocal{O}S_m^{\textup{PD}})^r$ for some $r \in \mathbb{N}$. Then on a chosen basis of $(\pazocal{O}S_m^{\textup{PD}})^r$ we can define a linear transformation $\beta$ using [\[eq:horizontal_elems\]](#eq:horizontal_elems){reference-type="eqref" reference="eq:horizontal_elems"} over $\pazocal{O}N_m^{\textup{PD}}$ and identity on $N'$. Note that the transformation $\beta$ preserves $\pazocal{O}N_m^{\textup{PD}}$ and we set $\det \alpha = \det \beta$ which is independent of the chosen presentation (see [@goldman Proposition 1.2]). Now by an argument similar to the proof of [@abhinandan-relative-wach-i Lemma 4.43] it easily follows that for some $N \in \mathbb{N}$ large enough one can write $p^N \det \alpha = p^N \det \beta \in 1 + J^{[1]}\pazocal{O}S_m^{\textup{PD}}$, in particular, $\det \alpha$ is a unit in $\pazocal{O}S_m^{\textup{PD}}[1/p]$, therefore $\alpha$ defines an automorphism of $\pazocal{O}N_m^{\textup{PD}}$ (see [@goldman Proposition 1.3]). Since the formula considered in [\[eq:horizontal_elems\]](#eq:horizontal_elems){reference-type="eqref" reference="eq:horizontal_elems"} converges in $D_m$, we conclude that $\pazocal{O}S_m^{\textup{PD}}\otimes_R D_m \rightarrow \pazocal{O}S_m^{\textup{PD}}\otimes_{\mathbf{A}_R^+} N[1/p]$ is a surjective map. Hence, [\[eq:osmpd_dm_n\]](#eq:osmpd_dm_n){reference-type="eqref" reference="eq:osmpd_dm_n"} is bijective, proving the lemma. ◻ Recall that $D_R$ is an $R[1/p]\textrm{-module}$ equipped with an integrable connection and finite over $R[1/p]$ since we have an inclusion $D_R \subset \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ of $R[1/p]\textrm{-modules}$ from [\[eq:dr_in_dcrys\]](#eq:dr_in_dcrys){reference-type="eqref" reference="eq:dr_in_dcrys"}. In particular, $D_R$ is a finite projective module over $R[1/p]$ by [@brinon-relatif Proposition 7.1.2]. Moreover, $D_R$ is equipped with a Frobenius-semilinear operator $\varphi$ and a filtration given as $\textup{Fil}^k D_R = \sum_{i+j=k} (\textup{Fil}^i \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} \textup{Fil}^j N[1/p])^{\Gamma_R}$, where $N[1/p]$ is equipped with the Nygaard filtration of Definition [Definition 47](#defi:nygaard_fil){reference-type="ref" reference="defi:nygaard_fil"}. Recall that the connection on $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ satisfies Griffiths transversality with respect to the filtration, so the connection on $D_R$ induced from the connection on $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ also satisfies Griffiths transversality. Now consider the following diagram: $$\label{eq:oarpipd_commdiag} \begin{tikzcd} \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\varphi^m, R} D_m \arrow[r, "1 \otimes \varphi^m"] \arrow[d, "\eqref{eq:osmpd_dm_n}"', "\wr"] & \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_R D_R \arrow[d, "\eqref{eq:oarpd_comparison}"']\\ \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\varphi^m, \mathbf{A}_R^+} N[1/p] \arrow[r, "\sim"] & \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} N[1/p], \end{tikzcd}$$ where the left vertical arrow is extension along $\varphi^m : \pazocal{O}S_m^{\textup{PD}}\rightarrow \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$ of the isomorphism [\[eq:osmpd_dm_n\]](#eq:osmpd_dm_n){reference-type="eqref" reference="eq:osmpd_dm_n"} in Lemma [Lemma 51](#lem:osmpd_comp){reference-type="ref" reference="lem:osmpd_comp"} and the bottom horizontal isomorphism follows from an argument similar to [@abhinandan-relative-wach-i Lemma 4.46]. By description of the arrows it follows that the diagram is $(\varphi, \Gamma_R)\textrm{-equivariant}$ and commutative. Taking $\Gamma_R\textrm{-invariants}$ of the diagram [\[eq:oarpipd_commdiag\]](#eq:oarpipd_commdiag){reference-type="eqref" reference="eq:oarpipd_commdiag"}, we obtain an isomorphism of $1 \otimes \varphi^m : R[1/p]\textrm{-modules}$ $R \otimes_{\varphi^m, R[1/p]} D_m \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}D_R$. In particular, it follows that the top horizontal arrow of [\[eq:oarpipd_commdiag\]](#eq:oarpipd_commdiag){reference-type="eqref" reference="eq:oarpipd_commdiag"} is an isomorphism. Hence, we conclude that the right vertical arrow of [\[eq:oarpipd_commdiag\]](#eq:oarpipd_commdiag){reference-type="eqref" reference="eq:oarpipd_commdiag"} is bijective as well, in particular, the comparison in [\[eq:oarpd_comparison\]](#eq:oarpd_comparison){reference-type="eqref" reference="eq:oarpd_comparison"} is an isomorphism compatible with Frobenius, connection and $\Gamma_R\textrm{-action}$. Compatibilty of [\[eq:oarpd_comparison\]](#eq:oarpd_comparison){reference-type="eqref" reference="eq:oarpd_comparison"} with filtrations follows from [@abhinandan-relative-wach-i Corollary 4.54] (also see the discussion after [\[eq:dr_dcrys_comp\]](#eq:dr_dcrys_comp){reference-type="eqref" reference="eq:dr_dcrys_comp"}, the argument is not circular since the proof of Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"} only uses bijectivity of $f$). This finishes our proof. ◻ *Remark 52*. In the basis $\{\hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(X_1), \ldots, \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(X_d)\}$ of $\Omega^1_R$, let $\partial_{A, i}$ denote the $i^{\textrm{th}}$ component of the connection on $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$, for $1 \leq i \leq d$, and let $\partial_{D, i}$ denote the induced operator on $D_R$. Moreover, employing arguments similar to [@abhinandan-syntomic Lemmas 4.12, 5.17 & 5.18] we can show that for $1 \leq i \leq d$, the operator $\nabla_i = (\log \gamma_i)/t = \frac{1}{t}\sum_{k \in \mathbb{N}} (-1)^k \frac{(\gamma_i-1)^{k+1}}{k+1}$ converges as a series of operators on $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_L^+} N$. Now using [\[eq:horizontal_elems\]](#eq:horizontal_elems){reference-type="eqref" reference="eq:horizontal_elems"} and the top horizontal arrow in diagram [\[eq:oarpipd_commdiag\]](#eq:oarpipd_commdiag){reference-type="eqref" reference="eq:oarpipd_commdiag"} we note that for any $x \in N[1/p]$ there exists $w \in D_R$ and $z \in (\textup{Fil}^1\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} N[1/p]$ such that $x = f(w) + z$, where $f$ is the isomorphism in [\[eq:oarpd_comparison\]](#eq:oarpd_comparison){reference-type="eqref" reference="eq:oarpd_comparison"}. Then an easy computation shows that $\nabla_i(x) - f(\partial_{D, i}(w)) = \nabla_i(z) + \partial_{A, i}(z) \in (\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} N[1/p]$. # Crystalline implies finite height {#sec:crystalline_finite_height} The goal of this section is to prove the following claim: **Theorem 53**. *Let $T$ be a finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ such that $V := T[1/p]$ is a $p\textrm{-adic}$ crystalline representation of $G_R$. Then there exists a unique Wach module $\mathbf{N}_R(T)$ over $\mathbf{A}_R^+$. In other words, $T$ is of finite $[p]_q\textrm{-height}$.* *Proof.* For a $p\textrm{-adic}$ representation the property of being crystalline and of finite $[p]_q\textrm{-height}$ is invariant under twisting the representation by $\chi^r$, where $\chi$ is the $p\textrm{-adic}$ cyclotomic character and $r \in \mathbb{N}$. Therefore, we can assume that $V$ is positive crystalline. Note that $V$ is also a positive crystalline representation of $G_L$ and hence positive finite $[p]_q\textrm{-height}$ as a $p\textrm{-adic}$ representation of $G_L$ (see [@abhinandan-imperfect-wach Definition 3.7]). This implies that $T$ is positive and of finite $[p]_q\textrm{-height}$ as a $\mathbb{Z}_p\textrm{-representation}$ of $G_L$. In particular, we have the Wach module $\mathbf{N}_L(T)$ over $\mathbf{A}_L^+$ and we set $\mathbf{N}_R(T) := \mathbf{N}_L(T) \cap \mathbf{D}_R(T) \subset \mathbf{D}_L(T)$ as an $\mathbf{A}_R^+\textrm{-module}$. From Proposition [Proposition 59](#prop:wach_module_constr){reference-type="ref" reference="prop:wach_module_constr"}, the module $\mathbf{N}_R(T)$ satisfies axioms of Definitions [Definition 34](#defi:wach_mods_relative){reference-type="ref" reference="defi:wach_mods_relative"} & [Definition 43](#defi:finite_pqheight){reference-type="ref" reference="defi:finite_pqheight"}. Hence, it follows that $\mathbf{N}_R(T)$ is the unique Wach module attached to $T$, or equivalently, $T$ is of finite $[p]_q\textrm{-height}$. ◻ *Remark 54*. In Theorem [Theorem 53](#thm:crys_fh_relative){reference-type="ref" reference="thm:crys_fh_relative"}, we also have that $T$ is a $\mathbb{Z}_p\textrm{-representation}$ of $G_L$ such that $V := T[1/p]$ is crystalline for $G_L$. Then from [@abhinandan-imperfect-wach Theorem 4.1] it follows $T$ is of finite $[p]_q\textrm{-height}$ as a representation of $G_L$, i.e. there exists a unique Wach module $\mathbf{N}_L(T)$ over $\mathbf{A}_L^+$. However, note that $\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T)$ is also a Wach module over $\mathbf{A}_L^+$, where we use $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$. Now using Proposition [Proposition 64](#prop:wach_phigamm_comp){reference-type="ref" reference="prop:wach_phigamm_comp"} we conclude that $\mathbf{A}_L\otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_L\otimes_{\mathbf{A}_R} \mathbf{D}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_L(T)$ as étale $(\varphi, \Gamma_L)\textrm{-modules}$ over $\mathbf{A}_L$. Hence, by the uniqueness of Wach module attached to $T$ over $\mathbf{A}_L^+$ in [@abhinandan-imperfect-wach Lemma 3.9] it follows that $\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_L(T)$ as $(\varphi, \Gamma_L)\textrm{-modules}$ over $\mathbf{A}_L^+$. ## Consequences of Theorem [Theorem 53](#thm:crys_fh_relative){reference-type="ref" reference="thm:crys_fh_relative"} {#subsec:crys_wach_equiv_consequence} Let $\textup{Rep}_{\mathbb{Z}_p}^{\textup{cris}}(G_R)$ denote the category of $\mathbb{Z}_p\textrm{-lattices}$ inside $p\textrm{-adic}$ crystalline representations of $G_R$. Then combining Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"} and Theorem [Theorem 53](#thm:crys_fh_relative){reference-type="ref" reference="thm:crys_fh_relative"}, we obtain the following: **Corollary 55**. *The Wach module functor induces an equivalence of categories $$\begin{aligned} \textup{Rep}_{\mathbb{Z}_p}^{\textup{cris}}(G_R) &\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{A}_R^+}^{[p]_q}\\ T &\longmapsto \mathbf{N}_R(T), \end{aligned}$$ with a quasi-inverse given as $N \mapsto \mathbf{T}_R(N) := \big(W\big(\overline{R}^{\flat}[1/p^{\flat}]\big) \otimes_{\mathbf{A}_R^+} N\big)^{\varphi=1}$.* Passing to associated isogeny categories, we obtain the following: **Corollary 56**. *The Wach module functor induces an exact equivalence between $\otimes\textrm{-categories}$ $\textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}(\varphi, \Gamma_R)\textup{-Mod}_{\mathbf{B}_R^+}^{[p]_q}$ via $V \mapsto \mathbf{N}_R(V)$ and with an exact $\otimes\textrm{-compatible}$ quasi-inverse given as $M \mapsto \mathbf{V}_R(M) := \big(W\big(\overline{R}^{\flat}[1/p^{\flat}]\big) \otimes_{\mathbf{A}_R^+} M\big)^{\varphi=1}$.* *Proof.* The equivalence of categories follows from Theorem [Theorem 53](#thm:crys_fh_relative){reference-type="ref" reference="thm:crys_fh_relative"}. For the rest of the proof, let us remark that for a $p\textrm{-adic}$ crystalline representation $V$ of $G_R$, from Proposition [Proposition 59](#prop:wach_module_constr){reference-type="ref" reference="prop:wach_module_constr"} we have $\mathbf{N}_R(V) = \mathbf{N}_L(V) \cap \mathbf{D}_R(V) \subset \mathbf{D}_L(V)$ as finite projective $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{B}_R^+$. Moreover, from Proposition [Proposition 64](#prop:wach_phigamm_comp){reference-type="ref" reference="prop:wach_phigamm_comp"} and Remark [Remark 54](#rem:wachmod_ar_al){reference-type="ref" reference="rem:wachmod_ar_al"}, note that $\mathbf{B}_L^+ \otimes_{\mathbf{B}_R^+} \mathbf{N}_R(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_L(V)$ and $\mathbf{B}_R\otimes_{\mathbf{B}_R^+} \mathbf{N}_R(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(V)$ compatible with natural $(\varphi, \Gamma_R)\textrm{-actions}$. Now let $V_1$ and $V_2$ be two crystalline representations of $G_R$, then $V_1 \otimes_{\mathbb{Q}_p} V_2$ is again crystalline (see [@brinon-relatif Théorèm 8.4.2]). We have $$\begin{aligned} \mathbf{N}_R(V_1) \otimes_{\mathbf{B}_R^+} \mathbf{N}_R(V_2) &= \mathbf{N}_R(V_1) \otimes_{\mathbf{B}_R^+} (\mathbf{N}_L(V_2) \cap \mathbf{D}_R(V_2))\\ &= (\mathbf{N}_R(V_1) \otimes_{\mathbf{B}_R^+} \mathbf{N}_L(V_2)) \cap (\mathbf{N}_R(V_1) \otimes_{\mathbf{B}_R^+} \mathbf{D}_R(V_2))\\ &= (\mathbf{N}_L(V_1) \otimes_{\mathbf{B}_L^+} \mathbf{N}_L(V_2)) \cap (\mathbf{D}_R(V_1) \otimes_{\mathbf{B}_L^+} \mathbf{D}_R(V_2))\\ &= \mathbf{N}_L(V_1 \otimes_{\mathbb{Q}_p} V_2) \cap \mathbf{D}_R(V_1 \otimes_{\mathbb{Q}_p} V_2) = \mathbf{N}_R(V_1 \otimes V_2), \end{aligned}$$ where the first equality follows from the discussion above, the second equality follows since $\mathbf{N}_R(V_1)$ is projective, the third equality again follows from the discussion above and the last equality follows from [@abhinandan-imperfect-wach Corollary 4.3] and [\[eq:rep_phigamma_relative\]](#eq:rep_phigamma_relative){reference-type="eqref" reference="eq:rep_phigamma_relative"}. This shows the compatibility of $\mathbf{N}_R$ with tensor products. Conversely, let $N_1$ and $N_2$ be two Wach modules over $\mathbf{A}_R^+$ and set $N_3 := (N_1 \otimes_{\mathbf{A}_R^+} N_2)/(p\textrm{-torsion})$ as a finitely generated $\mathbf{A}_R^+\textrm{-module}$. Then note that we have $N_3 \subset N_3[1/p] = N_1[1/p] \otimes_{\mathbf{B}_R^+} N_2[1/p]$, where the right hand term is a projective $\mathbf{B}_R^+\textrm{-module}$. Therefore, $N_3$ is torsion free and by definition $N_3/\mu$ is also $p\textrm{-torsion free}$, in particular, the sequence $\{p, \mu\}$ is strictly $N_3\textrm{-regular}$ by Remark [Remark 30](#rem:strict_regular_ideal){reference-type="ref" reference="rem:strict_regular_ideal"}. Furthermore, assumptions for the $(\varphi, \Gamma_R)\textrm{-action}$ on $N_3$, as in Definition [Definition 34](#defi:wach_mods_relative){reference-type="ref" reference="defi:wach_mods_relative"}, can be verified similar to [@abhinandan-relative-wach-i Proposition 4.14]. So it follows that $N_3$ is a Wach module over $\mathbf{A}_R^+$. Since, $N_3[1/p] = N_1[1/p] \otimes_{\mathbf{B}_R^+} N_2[1/p]$, compatibility of the functor $\mathbf{V}_R$ with tensor products now follows from [\[eq:rep_phigamma_relative\]](#eq:rep_phigamma_relative){reference-type="eqref" reference="eq:rep_phigamma_relative"}. It remains to show the exactness of $\mathbf{N}_R$ since exactness of the quasi-inverse functor $\mathbf{V}_R$ follows from Proposition [Proposition 38](#prop:wach_etale_ff_relative){reference-type="ref" reference="prop:wach_etale_ff_relative"} and the exact equivalence in [\[eq:rep_phigamma_relative\]](#eq:rep_phigamma_relative){reference-type="eqref" reference="eq:rep_phigamma_relative"}. Now consider an exact sequence of $p\textrm{-adic}$ crystalline representations of $G_R$ as $0 \rightarrow V_1 \rightarrow V_2 \rightarrow V_3 \rightarrow 0$ and we want to show that the sequence $$\label{eq:nrv_exact} 0 \longrightarrow \mathbf{N}_R(V_1) \longrightarrow \mathbf{N}_R(V_2) \longrightarrow \mathbf{N}_R(V_3) \longrightarrow 0,$$ is exact. Let $T_2 \subset V_2$ a $G_R\textrm{-stable}$ $\mathbb{Z}_p\textrm{-lattice}$, then $T_1 := V_1 \cap T_2 \subset V_2$ is a $G_R\textrm{-stable}$ $\mathbb{Z}_p\textrm{-lattice}$ inside $V_1$ and set $T_3 := T_2/T_1 \subset V_3$ as a $G_R\textrm{-stable}$ $\mathbb{Z}_p\textrm{-lattice}$. By definition, we have Wach modules $\mathbf{N}_R(T_1)$, $\mathbf{N}_R(T_2)$ and $\mathbf{N}_R(T_3)$ and we set $N := \mathbf{N}_R(T_2)/\mathbf{N}_R(T_1)$ as a finitely generated $\mathbf{A}_R^+\textrm{-module}$ equipped with a Frobenius $\varphi : N[1/\mu] \rightarrow N[1/\varphi(\mu)]$ and a continuous action of $\Gamma_R$ induced from the corresponding structures on $\mathbf{N}_R(T_2)$. We claim that $N[1/p] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_R(V_3)$ as $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{B}_R^+$. Recall that $\mathbf{D}_R$ is an exact functor from the category of $\mathbb{Z}_p\textrm{-representations}$ of $G_R$ to the category of étale $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{A}_R$ (see §[2.6](#subsec:relative_padicreps){reference-type="ref" reference="subsec:relative_padicreps"}). So we get that the natural map $N = \mathbf{N}_R(T_2)/\mathbf{N}_R(T_1) \rightarrow \mathbf{N}_R(T_3)$ is injective and since $\mathbf{A}_R^+ \rightarrow \mathbf{A}_R$ is flat we have $\mathbf{A}_R\otimes_{\mathbf{A}_R^+} N \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T_2)/\mathbf{D}_R(T_1) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T_3) \xleftarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R\otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T_3)$ as étale $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{A}_R$. Moreover, since $\varphi : \mathbf{A}_R^+ \rightarrow \mathbf{A}_R^+$ is faithfully flat, from the finite $[p]_q\textrm{-height}$ property of $\mathbf{N}_R(T_2)$ we get that $N$ is of finite $[p]_q\textrm{-height}$, i.e. $1 \otimes \varphi : (\varphi^*N)[1/[p]_q] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}N[1/[p]_q]$. In particular, $N[1/p]$ is finite projective over $\mathbf{B}_R^+$ by Proposition [Proposition 73](#prop:finiteproj_torus){reference-type="ref" reference="prop:finiteproj_torus"}. For $i = 1, 2, 3$ considering $V_i$ as a $p\textrm{-adic}$ crystalline representation of $G_L$, from [@abhinandan-imperfect-wach Corollary 4.3] we have an exact sequence $0 \rightarrow \mathbf{N}_L(V_1) \rightarrow \mathbf{N}_L(V_2) \rightarrow \mathbf{N}_L(V_3) \rightarrow 0$ of Wach modules over $\mathbf{B}_L^+ = \mathbf{A}_L^+[1/p]$. Note that the natural map $\mathbf{B}_R^+ \rightarrow \mathbf{B}_L^+$ is flat so we get that $\mathbf{B}_L^+ \otimes_{\mathbf{B}_R^+} N[1/p] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_L(V_3)$ as $(\varphi, \Gamma_L)\textrm{-modules}$ over $\mathbf{B}_L^+$. Moreover, from Remark [Remark 54](#rem:wachmod_ar_al){reference-type="ref" reference="rem:wachmod_ar_al"} we have $\mathbf{B}_L^+ \otimes_{\mathbf{B}_R^+} \mathbf{N}_R(V_i) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_L(V_i)$ for $i = 1, 2, 3$. Now since $N[1/p]$ is projective over $\mathbf{B}_R^+$, inside $\mathbf{D}_L(V_3)$ we obtain an isomorphism of $(\varphi, \Gamma_R)\textrm{-modules}$ over $\mathbf{B}_R^+$ as $$N[1/p] = (\mathbf{B}_L^+ \otimes_{\mathbf{B}_R^+} N[1/p]) \cap (\mathbf{B}_R\otimes_{\mathbf{B}_R^+} N[1/p]) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_L(V_3) \cap \mathbf{D}_R(V_3) \xleftarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_R(V_3).$$ Hence, [\[eq:nrv_exact\]](#eq:nrv_exact){reference-type="eqref" reference="eq:nrv_exact"} is exact, concluding our proof. ◻ We obtain applications of Theorem [Theorem 53](#thm:crys_fh_relative){reference-type="ref" reference="thm:crys_fh_relative"} as follows: **Theorem 57**. *Let $V$ be a $p\textrm{-adic}$ representation of $G_R$. Then the following are equivalent:* 1. *$V$ is crystalline as a representation of $G_R$;* 2. *$V$ is crystalline as a representation of $G_L$;* 3. *$\textup{rk}_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) = \dim_{\mathbb{Q}_p} V$.* *Proof.* Let $V \in \textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R)$, then obviously we have $V \in \textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_L)$. Conversely, let $V \in \textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_L)$ and choose a $G_R\textrm{-stable}$ $\mathbb{Z}_p\textrm{-lattice}$ $T \subset V$ such that $T$ is of finite $[p]_q\textrm{-height}$ as a representation of $G_L$. Then from Proposition [Proposition 59](#prop:wach_module_constr){reference-type="ref" reference="prop:wach_module_constr"} note that $T$ is of finite $[p]_q\textrm{-height}$ as a representation of $G_R$. Therefore, $V = T[1/p]$ is a crystalline representation of $G_R$ by Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"}. This shows the equivalence of (1) and (2). Next, if $V \in \textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R)$ then $\textup{rk}_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) = \dim_{\mathbb{Q}_p} V$ (see §[2.6](#subsec:relative_padicreps){reference-type="ref" reference="subsec:relative_padicreps"}), proving that (1) implies (3). Finally we will show that (3) implies (2). Let $V$ be a $p\textrm{-adic}$ representation of $G_R$ such that $\textup{rk}_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) = \dim_{\mathbb{Q}_p} V$. From [@brinon-imparfait §3.3] recall that $V$ is crystalline for $G_L$ if and only if $\dim_L \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V) = \dim_{\mathbb{Q}_p} V$. So we will show that $\dim_L \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V) = \dim_{\mathbb{Q}_p} V$ by constructing a natural isomorphism of $L\textrm{-vector spaces}$ $L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, L}(V)$. Since $\dim_L \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V) \leq \dim_{\mathbb{Q}_p} V$, it is enough to construct a natural $L\textrm{-linear}$ injective map $L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \rightarrow \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V)$ and the claim would follow by considering $L\textrm{-dimensions}$. From Remark [Remark 28](#lem:bcrys_embedding_l){reference-type="ref" reference="lem:bcrys_embedding_l"} we have a natural $(\varphi, G_R)\textrm{-equivariant}$ $L\textrm{-linear}$ injective map $L \otimes_{R[1/p]} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))$. Tensoring this map with $V$ and considering diagonal action of $G_R$, we obtain a $(\varphi, G_R)\textrm{-equivariant}$ injective map $$\label{eq:lbcrysv_embed} L \otimes_{R[1/p]} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbb{Q}_p} V \longrightarrow \Big(\prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p}))\Big) \otimes_{\mathbb{Q}_p} V = \prod_{\mathfrak{p}\in \pazocal{S}} (\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V).$$ The map in [\[eq:lbcrysv_embed\]](#eq:lbcrysv_embed){reference-type="eqref" reference="eq:lbcrysv_embed"} further induces a natural map $L \otimes_{R[1/p]} \pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbb{Q}_p} V \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V$, compatible with respective Frobenii, filtrations and connections (see Remark [Remark 28](#lem:bcrys_embedding_l){reference-type="ref" reference="lem:bcrys_embedding_l"}). Now we take the $G_R\textrm{-invariant}$ part of [\[eq:lbcrysv_embed\]](#eq:lbcrysv_embed){reference-type="eqref" reference="eq:lbcrysv_embed"} and note that product commutes with left exact functors, in particular, with taking $G_R\textrm{-invariants}$. So we obtain $\varphi\textrm{-equivariant}$ $L\textrm{-linear}$ injective maps $$\begin{aligned} \label{eq:ldcrysrv_embed} \begin{split} L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) &\longrightarrow \Big(\prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V\Big)^{G_R}\\ &= \hspace{3mm} \prod_{\mathfrak{p}\in \pazocal{S}} (\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V)^{G_R}\\ &\longrightarrow \prod_{\mathfrak{p}\in \pazocal{S}} (\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V)^{G_R(\mathfrak{p})}, \end{split} \end{aligned}$$ where note that the last arrow is injective since $G_R(\mathfrak{p})\subset G_R$ is a subgroup. Moreover, since $G_R$ acts transitively on $\pazocal{S}$, it transitively permutes the components of $\prod_{\mathfrak{p}\in \pazocal{S}} (\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V)^{G_R(\mathfrak{p})}$, i.e. if $0 \neq x \in L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$, then its image $(x_{\mathfrak{p}})_{\mathfrak{p}\in \pazocal{S}}$ under the composition [\[eq:ldcrysrv_embed\]](#eq:ldcrysrv_embed){reference-type="eqref" reference="eq:ldcrysrv_embed"} satisfies $x_{\mathfrak{p}} \neq 0$ for all $\mathfrak{p}\in \pazocal{S}$. Therefore, for each $\mathfrak{p}\in \pazocal{S}$ composing [\[eq:ldcrysrv_embed\]](#eq:ldcrysrv_embed){reference-type="eqref" reference="eq:ldcrysrv_embed"} with the natural $\varphi\textrm{-equivariant}$ $L\textrm{-linear}$ projection $\prod_{\mathfrak{p}\in \pazocal{S}} (\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V)^{G_R(\mathfrak{p})} \rightarrow (\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V)^{G_R(\mathfrak{p})}$ gives a natural $\varphi\textrm{-equivariant}$ $L\textrm{-linear}$ injective map $$\label{eq:ldrcysrv_embed_p} L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \longrightarrow (\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V)^{G_R(\mathfrak{p})},$$ compatible with filtrations and connections (see above and Remark [Remark 28](#lem:bcrys_embedding_l){reference-type="ref" reference="lem:bcrys_embedding_l"}), where the left hand term is equipped with filtration on $\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ and tensor product Frobenius and connection. Finally, from Lemma [Lemma 25](#lem:cplusp_in_cpplus_bcrys){reference-type="ref" reference="lem:cplusp_in_cpplus_bcrys"}, recall that we have a natural $L\textrm{-linear}$ $(\varphi, \widehat{G}_R(\mathfrak{p}))\textrm{-equivariant}$ injective map $\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$ compatible with respective filtrations and connections and where the $\widehat{G}_R(\mathfrak{p})\textrm{-action}$ on the left term factors through $\widehat{G}_R(\mathfrak{p})\twoheadrightarrow G_R(\mathfrak{p})$. Tensoring the preceding injective map with $V$, equipping each term with diagonal action of $\widehat{G}_R(\mathfrak{p})$ and taking $\widehat{G}_R(\mathfrak{p})\textrm{-invariants}$ produces a natural $L\textrm{-linear}$ injective map $(\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \otimes_{\mathbb{Q}_p} V)^{G_R(\mathfrak{p})} \rightarrow \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V)$ compatible with respective Frobenii, filtrations and connections. Composing [\[eq:ldrcysrv_embed_p\]](#eq:ldrcysrv_embed_p){reference-type="eqref" reference="eq:ldrcysrv_embed_p"} with the preceding $L\textrm{-linear}$ map gives a natural $L\textrm{-linear}$ injective map $$\label{eq:odcrys_functoriality} L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \rightarrow \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V),$$ compatible with respective Frobenii, filtrations and connections. By considering $L\textrm{-dimension}$, it follows that [\[eq:odcrys_functoriality\]](#eq:odcrys_functoriality){reference-type="eqref" reference="eq:odcrys_functoriality"} is bijective (see Corollary [Corollary 58](#cor:odcrys_functoriality){reference-type="ref" reference="cor:odcrys_functoriality"} for a stronger statement). Hence, $\dim_L \pazocal{O}\mathbf{D}_{\textup{cris}, L}(V) = \textup{rk}_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) = \dim_{\mathbb{Q}_p} V$, showing that (3) implies (2). This concludes our proof. ◻ **Corollary 58**. *Let $V$ be a $p\textrm{-adic}$ representation of $G_R$. Under the equivalent conditions of Theorem [Theorem 57](#thm:purity_crystalline){reference-type="ref" reference="thm:purity_crystalline"}, we have a natural isomorphism $L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, L}(V)$ of filtered $(\varphi, \partial)\textrm{-modules}$ over $L$.* *Proof.* Assume that $V$ is a crystalline representation of $G_R$, in particular, we have a natural $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R})\textrm{-linear}$ isomorphism $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \otimes_{\mathbb{Q}_p} V$ compatible with respective Frobenii, filtrations, connections and $G_R\textrm{-actions}$ (see [@brinon-relatif Proposition 8.4.3]). For any $\mathfrak{p}\in \pazocal{S}$, base changing the preceding isomorphism along the composition $\pazocal{O}\mathbf{B}_{\textup{cris}}(\overline{R}) \rightarrow \prod_{\mathfrak{p}\in \pazocal{S}} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+(\mathfrak{p})) \rightarrow \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})$, we get a $\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}})\textrm{-linear}$ isomorphism $$\label{eq:obcrysp_odcrys_iso} \pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}}) \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{B}_{\textup{cris}}(\mathbb{C}^+_{\mathfrak{p}}) \otimes_{\mathbb{Q}_p} V$$ compatible with respective Frobenii, filtrations, connections and $\widehat{G}_R(\mathfrak{p})\textrm{-actions}$. In [\[eq:obcrysp_odcrys_iso\]](#eq:obcrysp_odcrys_iso){reference-type="eqref" reference="eq:obcrysp_odcrys_iso"}, by taking $\widehat{G}_R(\mathfrak{p})\textrm{-invariants}$ we get [\[eq:odcrys_functoriality\]](#eq:odcrys_functoriality){reference-type="eqref" reference="eq:odcrys_functoriality"}, i.e. $L \otimes_{R[1/p]} \pazocal{O}\mathbf{D}_{\textup{cris}, R}(V) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, L}(V)$, and by construction the preceding isomorphism is compatible with respective Frobenii, filtrations and connections. Hence, the claim follows. ◻ ## Main ingredients for the proof of Theorem [Theorem 53](#thm:crys_fh_relative){reference-type="ref" reference="thm:crys_fh_relative"} {#subsec:crys_wach_equiv_proof} In this subsection, let $T$ be a finite free $\mathbb{Z}_p\textrm{-representation}$ of $G_R$ such that $T$ is a finite $[p]_q\textrm{-height}$ representation of $G_L$ (see Definition [Definition 43](#defi:finite_pqheight){reference-type="ref" reference="defi:finite_pqheight"} and [@abhinandan-imperfect-wach Definition 3.7]). In particular, we can functorially attach to $T$ a $(\varphi, \Gamma_R)\textrm{-module}$ $\mathbf{D}_R(T)$ over $\mathbf{A}_R$ as well as a Wach module $\mathbf{N}_L(T)$ over $\mathbf{A}_L^+$. In this subsection we will prove the following claim: **Proposition 59**. *The $\mathbf{A}_R^+\textrm{-module}$ $\mathbf{N}_R(T) := \mathbf{N}_L(T) \cap \mathbf{D}_R(T) \subset \mathbf{D}_L(T)$ satisfies all the axioms of Definition [Definition 43](#defi:finite_pqheight){reference-type="ref" reference="defi:finite_pqheight"}. In particular, $T$ is a finite $[p]_q\textrm{-height}$ representation of $G_R$.* *Proof.* It is immediate that $\mathbf{N}_R(T)$ is $p\textrm{-torsion}$ free and $\mu\textrm{-torsion}$ free. From Lemma [Lemma 60](#lem:modp_fingen){reference-type="ref" reference="lem:modp_fingen"} note that $\mathbf{N}_R(T)$ is finitely generated over $\mathbf{A}_R^+$ and from its proof we have $\mathbf{N}_R(T)/p \subset (\mathbf{N}_L(T)/p) \cap (\mathbf{D}_R(T)/p) \subset \mathbf{D}_L(T)/p$, in particular, $\mathbf{N}_R(T)/p$ is $\mu\textrm{-torsion}$ free. Next, from Lemma [Lemma 61](#lem:finite_height){reference-type="ref" reference="lem:finite_height"} we know that $\mathbf{N}_R(T)$ is of finite $[p]_q\textrm{-height}$, i.e. cokernel of the injective map $1 \otimes \varphi : \varphi^*(\mathbf{N}_R(T)) \rightarrow \mathbf{N}_R(T)$ is killed by $[p]_q^s$ where $s$ is the height of $\mathbf{N}_L(T)$. Furthermore, from Proposition [Proposition 64](#prop:wach_phigamm_comp){reference-type="ref" reference="prop:wach_phigamm_comp"} we have $\mathbf{A}_R\otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T)$. Finally, recall that the action of $\Gamma_L$ is trivial on $\mathbf{N}_L(T)/\mu \mathbf{N}_L(T)$ and $\Gamma_L \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\Gamma_R$, so for any $g \in \Gamma_R$ we have $(g-1)\mathbf{N}_L(T) \subset \mu \mathbf{N}_L(T)$. Therefore, we get that $(g-1)\mathbf{N}_R(T) \subset (\mu \mathbf{N}_L(T)) \cap \mathbf{D}_R(T) = \mu \mathbf{N}_R(T)$, so it follows that $\Gamma_R$ acts trivially on $\mathbf{N}_R(T)/\mu \mathbf{N}_R(T)$. This concludes our proof. ◻ **Lemma 60**. *The $\mathbf{A}_R^+\textrm{-module}$ $\mathbf{N}_R(T) := \mathbf{N}_L(T) \cap \mathbf{D}_R(T)$ is finitely generated.* *Proof.* We first claim that for each $n \in \mathbb{N}_{\geq 1}$, the natural $\mathbf{A}_R^+/p^n\textrm{-linear}$ map $\mathbf{N}_R(T)/p^n \rightarrow (\mathbf{N}_L(T)/p^n) \cap (\mathbf{D}_R(T)/p^n) \subset \mathbf{D}_L(T)/p^n$ is injective and $(\mathbf{N}_L(T)/p^n) \cap (\mathbf{D}_R(T)/p^n)$ is a finitely generated $\mathbf{A}_R^+/p^n\textrm{-module}$. Since $\mathbf{N}_R(T)$, $\mathbf{N}_L(T)$ and $\mathbf{D}_R(T)$ are $p\textrm{-torsion free}$, it is enough to show the claim for $n=1$ and the claim for $n \geq 1$ can be deduced by an easy induction. So we are reduced to showing that $\mathbf{N}_R(T)/p \rightarrow (\mathbf{N}_L(T)/p) \cap (\mathbf{D}_R(T)/p) \subset \mathbf{D}_L(T)/p$ is injective and $(\mathbf{N}_L(T)/p) \cap (\mathbf{D}_R(T)/p)$ is finitely generated $\mathbf{A}_R^+/p =: \mathbf{E}_R^+\textrm{-module}$. Note that we have $p \mathbf{N}_L(T) \cap \mathbf{N}_R(T) \subset p \mathbf{D}_L(T) \cap \mathbf{D}_R(T) = p \mathbf{D}_R(T)$, so $$p \mathbf{N}_R(T) \subset p\mathbf{N}_L(T) \cap \mathbf{N}_R(T) \subset p \mathbf{N}_L(T) \cap p \mathbf{D}_R(T) = p\mathbf{N}_R(T),$$ therefore $\mathbf{N}_R(T)/p \subset \mathbf{N}_L(T)/p$. Moreover, $p \mathbf{D}_R(T) \cap \mathbf{N}_R(T) \subset p \mathbf{D}_L(T) \cap \mathbf{N}_L(T) = p \mathbf{N}_L(T)$, so $$p \mathbf{N}_R(T) \subset p\mathbf{D}_R(T) \cap \mathbf{N}_R(T) \subset p \mathbf{D}_R(T) \cap p \mathbf{N}_L(T) = p\mathbf{N}_R(T),$$ therefore $\mathbf{N}_R(T)/p \subset \mathbf{D}_R(T)/p$. Next, we will show that $(\mathbf{N}_L(T)/p) \cap (\mathbf{D}_R(T)/p)$ is a finitely generated $\mathbf{E}_R^+\textrm{-module}$. Assume that $\overline{D}_R:= \mathbf{D}_R(T)/p$ is finite free (a priori it is finite projective) of rank $h$ over $\mathbf{E}_R:= \mathbf{A}_R/p$. Let $\mathbf{e}= \{e_1, \ldots, e_h\}$ be a basis of $\overline{N}_L:= \mathbf{N}_L(T)/p$ over $\mathbf{E}_L^+ := \mathbf{A}_L^+/p$ and $\mathbf{f}= \{f_1, \ldots, f_h\}$ a basis of $\overline{D}_R$ over $\mathbf{E}_R$. Then, for $\mathbf{E}_L:= \mathbf{A}_L/p$, we have $\mathbf{f}= A \mathbf{e}$ for $A := (a_{i j}) \in \textup{GL}(h, \mathbf{E}_L)$ and write $A^{-1} = (b_{ij}) \in \textup{GL}(h, \mathbf{E}_L)$. Set $M := \oplus_{i=1}^h \mathbf{E}_R^+ f_i$, so that $M[1/\mu] = \overline{D}_R$. Let $x \in M[1/\mu] \cap \overline{N}_L$ and write $x = \sum_{i=1}^h c_i e_i = \sum_{i=1}^h d_i f_i$ with $c_i \in \mathbf{E}_L^+$ and $d_i \in \mathbf{E}_R$ for all $1 \leq i \leq h$. So we obtain that $d_i = \sum_{j=1}^h b_{j i} c_j$ for all $1 \leq i \leq h$. Therefore, we have $d_i \in \mu^{-k} \mathbf{E}_L^+$ for all $1 \leq i \leq h$ for $k$ large enough. Note that we have $\mu^{-k} \mathbf{E}_L^+ \cap \mathbf{E}_R= \mu^{-k} \mathbf{E}_R^+$, so we obtain that $d_i \in \mu^{-k} \mathbf{E}_R^+$. Hence, $M[1/\mu] \cap \overline{N}_L\subset \mu^{-k} M$, in particular, $M[1/\mu] \cap \overline{N}_L= \overline{D}_R\cap \overline{N}_L$ is finitely generated over $\mathbf{E}_R^+$. In general, when $\overline{D}_R$ is finite projective, we choose an $\mathbf{E}_R^+\textrm{-module}$ $D'$ such that $\overline{D}_R\oplus D' = \mathbf{E}_R^{\oplus k}$ for some $k \in \mathbb{N}$. Let $D_L' := \mathbf{E}_L\otimes_{\mathbf{E}_R} \overline{D}_R$, so we have that $\overline{D}_L\oplus D_L' = \mathbf{E}_L^{\oplus k}$. Note that since $\mathbf{E}_L$ is a field with ring of integers $\mathbf{E}_L^+$, we can choose a lattice of $D_L'$ over $\mathbf{E}_L^+$, i.e. there exists a free $\mathbf{E}_L^+\textrm{-submodule}$ $N_L' \subset D_L'$ such that $N_L'[1/\mu] = D_L'$. Then we get that $\overline{N}_L\oplus N_L'$ is a free $\mathbf{E}_L^+\textrm{-module}$ such that $\mathbf{E}_L\otimes_{\mathbf{E}_L^+} (\overline{N}_L\oplus N_L') = \overline{D}_L\oplus D_L' = \mathbf{E}_L^{\oplus k}$. Inside $\mathbf{E}_L^{\oplus k}$, consider the inclusion of $\mathbf{E}_R^+\textrm{-modules}$ $$\label{eq:modp_fingen} (\overline{D}_R\cap \overline{N}_L) \oplus (D' \cap N_L') = (\overline{D}_R\oplus D') \cap (\overline{N}_L\oplus N_L') \subset \mathbf{E}_R^{\oplus k} \cap (\overline{N}_L\oplus N_L').$$ Using the conclusion in free case from the previous paragraph, we get that the last term in [\[eq:modp_fingen\]](#eq:modp_fingen){reference-type="eqref" reference="eq:modp_fingen"} is a finite $\mathbf{E}_R^+\textrm{-module}$. Hence, $\overline{D}_R\cap \overline{N}_L$ is also a finite $\mathbf{E}_R^+\textrm{-module}$, proving the claim. To prove the lemma, it remains to show that $\mathbf{N}_R(T)$ is $p\textrm{-adically}$ complete. Indeed, from the claim above note that for all $n \in \mathbb{N}_{\geq 1}$, $\mathbf{N}_R(T)/p^n$ is a finitely genearted $\mathbf{A}_R^+/p^n\textrm{-module}$. Since $\mathbf{A}_R^+$ is noetherian, for each $n \in \mathbb{N}$ and $k \in \mathbb{N}$ as in the previous paragraph, we have a presentation $0 \rightarrow M_n \rightarrow (\mathbf{A}_R^+/p^n)^{\oplus k} \rightarrow \mathbf{N}_R(T)/p^n \rightarrow 0$, where $M_n$ is a finitely generated $\mathbf{A}_R^+/p^n\textrm{-module}$. By taking a finite presentation of $M_n$ as an $\mathbf{A}_R^+/p^n\textrm{-module}$ it is easy to see that the system $\{M_n\}_{n \in \mathbb{N}_{\geq 1}}$ is Mittag-Leffler. In particular, it follows that $\lim_n \mathbf{N}_R(T)/p^n$ is a finitely generated $\lim_n \mathbf{A}_R^+/p^n = \mathbf{A}_R^+\textrm{-module}$. Now consider the following natural $\mathbf{A}_R^+\textrm{-linear}$ maps: $$\begin{aligned} f : \mathbf{N}_R(T) \longrightarrow \lim_n \mathbf{N}_R(T)/p^n &\longrightarrow \lim_n ((\mathbf{N}_L(T)/p^n) \cap (\mathbf{D}_R(T)/p^n))\\ &\longrightarrow (\lim_n\mathbf{N}_L(T)/p^n) \cap (\lim_n \mathbf{D}_R(T)/p^n)\\ &\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_L(T) \cap \mathbf{D}_R(T) = \mathbf{N}_R(T), \end{aligned}$$ where the first arrow is the natural projection map, the second arrow is injective by the claim proved above, the third arrow is injective by definition and the fourth arrow is bijective since $\mathbf{N}_L(T)$ and $\mathbf{D}_R(T)$ are $p\textrm{-adically}$ complete. Chasing an element of $x \in \mathbf{N}_R(T)$ through the composition, we see that $f(x) = x$. Hence, we get that $\mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\lim_n \mathbf{N}_R(T)/p^n$, in particular, finitely generated $\mathbf{A}_R^+\textrm{-module}$. ◻ **Lemma 61**. *The $\mathbf{A}_R^+\textrm{-module}$ $\mathbf{N}_R(T)$ is of finite $[p]_q\textrm{-height}$, i.e. cokernel of the injective map $1 \otimes \varphi : \varphi^*(\mathbf{N}_R(T)) \rightarrow \mathbf{N}_R(T)$ is killed by $[p]_q^s$ for some $s \in \mathbb{N}$.* *Proof.* Note that $\varphi : \mathbf{A}_R^+ \rightarrow \mathbf{A}_R^+$ is finite and faithfully flat of degree $p^{d+1}$ (see §[2.2](#subsec:ainf_relative){reference-type="ref" reference="subsec:ainf_relative"}). Moreover, from §[2.2](#subsec:ainf_relative){reference-type="ref" reference="subsec:ainf_relative"} we have $\varphi^*(\mathbf{A}_R) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{A}_R$ and $\varphi^*(\mathbf{A}_L^+) := \mathbf{A}_L^+ \otimes_{\varphi, \mathbf{A}_L^+} \mathbf{A}_L^+ \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\oplus_{\alpha} \varphi(\mathbf{A}_L^+) u_{\alpha} = (\oplus_{\alpha} \varphi(\mathbf{A}_R^+) u_{\alpha}) \otimes_{\varphi(\mathbf{A}_R^+)} \varphi(\mathbf{A}_L^+) \xleftarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{A}_L^+$. Therefore, we also obtain that $\varphi^*(\mathbf{N}_L(T)) := \mathbf{A}_L^+ \otimes_{\varphi, \mathbf{A}_L^+} \mathbf{N}_L(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{N}_L(T)$ and $\varphi^*(\mathbf{D}_R(T)) := \mathbf{A}_R\otimes_{\varphi, \mathbf{A}_R} \mathbf{D}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{D}_R(T)$. Hence, inside $\varphi^*(\mathbf{D}_L(T))$ we conclude that $$\begin{aligned} \varphi^*(\mathbf{N}_R(T)) &:= \mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{N}_R(T) = \mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} (\mathbf{N}_L(T) \cap \mathbf{D}_R(T))\\ &= (\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{N}_L(T)) \cap (\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} \mathbf{D}_R(T)) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\varphi^*(\mathbf{N}_L(T)) \cap \varphi^*(\mathbf{D}_R(T)). \end{aligned}$$ Since cokernel of the injective map $(1 \otimes \varphi) : \varphi^*(\mathbf{N}_L(T)) \rightarrow \mathbf{N}_L(T)$ is killed by $[p]_q^s$ for some $s \in \mathbb{N}$ and $(1 \otimes \varphi) : \varphi^*(\mathbf{D}_R(T)) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T)$, it easily follows that the cokernel of $(1 \otimes \varphi) : \varphi^*(\mathbf{N}_R(T)) \rightarrow \mathbf{N}_R(T)$ is killed by $[p]_q^s$ as well. ◻ Finally, we will show that $\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{N}_L(T)$ and $\mathbf{A}_R\otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T)$. For $n \in \mathbb{N}_{\geq 1}$ let $N_{R,n}:= \mathbf{N}_R(T)/p^n$, $D_{R,n}:= \mathbf{D}_R(T)/p^n$, $N_{L,n}:= \mathbf{N}_L(T)/p^n$, $D_{L,n}:= \mathbf{D}_L(T)/p^n$ and $M_n := N_{L,n}\cap D_{R,n}\subset D_{L,n}$. We have a commutative diagram where the vertical arrows are natural inclusions, the bottom horizontal arrow $f_n$ is the natural projection map and the top arrow is the induced map. We have a similar diagram with the bottom row replaced by $N_{L,n}\twoheadrightarrow N_{L, 1}$. **Lemma 62**. *We have* 1. *$M_n$ is a finitely generated $\mathbf{A}_R^+/p^n\textrm{-module}$ and $\mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\lim_n M_n$.* 2. *$M_n$ is of finite $[p]_q\textrm{-height}$ $s$ for $s \in \mathbb{N}$ as in Lemma [Lemma 61](#lem:finite_height){reference-type="ref" reference="lem:finite_height"}.* 3. *$M_n[1/\mu] = \mathbf{A}_R\otimes_{\mathbf{A}_R^+} M_n \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}D_{R,n}$ and $\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} M_n \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}N_{L,n}$.* *Proof.* The claim in (1) follows from the proof of Lemma [Lemma 60](#lem:modp_fingen){reference-type="ref" reference="lem:modp_fingen"} and the claim in (2) follows similar to Lemma [Lemma 61](#lem:finite_height){reference-type="ref" reference="lem:finite_height"}. As the maps $\mathbf{A}_R^+ \rightarrow \mathbf{A}_R$ and $\mathbf{A}_R^+ \rightarrow \mathbf{A}_L^+$ are flat, the last claim follows from the equalities below: $$\begin{aligned} \mathbf{A}_R\otimes_{\mathbf{A}_R^+} M_n &= (\mathbf{A}_R\otimes_{\mathbf{A}_R^+} D_{R,n}) \cap (\mathbf{A}_R\otimes_{\mathbf{A}_R^+} N_{L,n}) = (\mathbf{A}_R\otimes_{\mathbf{A}_R^+} D_{R,n}) \cap (\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} D_{R,n}) = D_{R,n},\\ \mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} M_n &= (\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} D_{R,n}) \cap (\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} N_{L,n}) = (\mathbf{A}_R\otimes_{\mathbf{A}_R^+} N_{L,n}) \cap (\mathbf{A}_L^+ \otimes_{\mathbf{A}_R^+} N_{L,n}) = N_{L,n}. \end{aligned}$$ Hence, the lemma is proved. ◻ Let $\pazocal{S}$ denote the set of $\mathbf{A}_R^+\textrm{-submodules}$ $M' \subset M_1$ stable under the action of $\varphi$, is of finite height $s$ and satisfies $M'[1/\mu] = M_1[1/\mu] = D_{R,1} = \mathbf{D}_R(T)/p$. Set $M^{\circ}:= \cap_{M' \in \pazocal{S}} M' \subset M_1$. **Lemma 63**. *We have $M^{\circ}\in \pazocal{S}$ and $f_n(M_n) \in \pazocal{S}$ for all $n \in \mathbb{N}_{\geq 1}$.* *Proof.* The idea of the proof is motivated from [@du-liu-moon-shimizu Lemma 4.25]. Let $M' \in \pazocal{S}$. For the first claim we need to show that there exists $r \in \mathbb{N}$ such that $\mu^r M_1 \subset M' \subset M_1$. Let $M'' := M_1/M'$ such that $M'' \neq 0$ and let $k = p(p-1)s \in \mathbb{N}$. Also, let $\varphi^*(M'') := \varphi^*(M_1)/\varphi^*(M')$ and let $1 \otimes \varphi_{M''} : \varphi^*(M'') \rightarrow M''$ denote the map induced from $1 \otimes \varphi_M$. Since $M_1$ (resp. $M'$) is of finite $[p]_q\textrm{-height}$ $k$ (since $s < k$), we define $\psi_M : M_1 \xrightarrow{\mu^k} \mu^k M_1 \rightarrow \varphi^*(M_1)$ (resp. $\psi_{M'} : M' \xrightarrow{\mu^k} \mu^k M' \rightarrow \varphi^*(M')$) to be the unique $\mathbf{A}_R^+/p\textrm{-linear}$ map such that $\psi_M \circ (1 \otimes \varphi_M) = \mu^k \textup{Id}_{\varphi^*_M}$ (resp. $\psi_{M'} \circ (1 \otimes \varphi_{M'}) = \mu^k \textup{Id}_{\varphi^*_{M'}}$). Let $\psi_{M''} : M'' \rightarrow \varphi^*(M'')$ denote the map induced from $\psi_M$. Now consider the following commutative diagram: Note that $[p]_q = \mu^{p-1} \mod p$, $\varphi(\mu) = \mu^p \mod p$ and $\varphi([p]_q) = \mu^{p(p-1)} \mod p$. Since $M_1[1/\mu] = M'[1/\mu]$, let $i \in \mathbb{N}_{\geq 1}$ such that $\mu^{pi} M'' = 0$ and $\mu^{p(i-1)} M'' \neq 0$. Let $x \in M''$ such that $\mu^{pi} x \neq 0$ and set $y = 1 \otimes x \in \varphi^*(M'')$. Then $\varphi(\mu^{pi}) y = 1 \otimes \mu^{pi} x = 0$ but $\mu^{p^2(i-1)}y = \varphi(\mu^{p(i-1)}) y = 1 \otimes \mu^{p(i-1)} x \neq 0$. Let $z = (1 \otimes \varphi_{M''}) y \in M''$, then $\mu^{pi} z = 0$. So we have $0 = \psi_{M''}(\mu^{pi} z) = \mu^{pi} (\psi_{M''} \circ (1 \otimes \varphi_{M''}) y) = \mu^{pi+k} y$. Therefore, we get that $pi+k = pi+p(p-1)s > p^2 (i-1)$, i.e. $i < s + \frac{p}{p-1}$. Hence, $\mu^{s+1} M'' = 0$. Since the constant obtained is independent of $M'$ we also get that $\mu^{s+1} M_1 \subset M^{\circ}\subset M_1$ and $M^{\circ}[1/\mu] = M_1[1/\mu]$. Next, we will show that $M^{\circ}$ is of finite height $s$. Let $x \in M^{\circ}$ then $x \in M'$ for each $M' \in \pazocal{S}$ and there exists some $y \in \varphi^*(M') \subset \varphi^*(M_1)$ such that $(1 \otimes \varphi)y = [p]_q^s x$. Note that $y$ is unique in $\varphi^*(M_1)$ and since $\varphi: \mathbf{A}_R^+ \rightarrow \mathbf{A}_R^+$ is flat, we get that $y \in \cap_{M' \in \pazocal{S}} (\mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} M') = \mathbf{A}_R^+ \otimes_{\varphi, \mathbf{A}_R^+} (\cap_{M' \in \pazocal{S}} M') = \varphi^*(M^{\circ})$. Therefore, we get that $M^{\circ}\in \pazocal{S}$. For the second part of the claim note that $M_n[1/\mu] = D_{R,n}$ and $f_n(D_{R,n}) = D_{R,n}/p = \mathbf{D}_R(T)/p$ (see Lemma [Lemma 62](#lem:Mn_large){reference-type="ref" reference="lem:Mn_large"}). So we get that $f_n\big(M_n[1/\mu]\big) = \mathbf{D}_R(T)/p$ and we are left to show that $f_n(M_n)$ is of finite height $s$. Note that we have a commutative diagram with exact rows: The rightmost vertical arrow is injective since $f_n(M_n) \subset D_{R,n}$ and the cokernel of the middle vertical arrow is killed by $[p]_q^s$ by Lemma [Lemma 62](#lem:Mn_large){reference-type="ref" reference="lem:Mn_large"}. Hence, the cokernel of the rightmost vertical arrow is killed by $[p]_q^s$. ◻ **Proposition 64**. *We have $\mathbf{A}_R\otimes_{\mathbf{A}_R^+} \mathbf{N}_R(T) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{D}_R(T)$.* *Proof.* Since everything is $p\textrm{-adically}$ complete and $\mathbf{D}_R(T)$ and $\mathbf{N}_R(T)$ are $p\textrm{-torsion}$ free, it is enough to show the claim modulo $p$. Recall that we have $\mathbf{N}_R(T)/p \subset M_1 = \mathbf{D}_R(T)/p \cap \mathbf{N}_L(T)/p \subset \mathbf{D}_L(T)/p$ and from Lemma [Lemma 63](#lem:mcirc_props){reference-type="ref" reference="lem:mcirc_props"} we have $M^{\circ}\subset \mathbf{N}_R(T)/p$. Therefore, we get that $\mathbf{D}_R(T)/p = M^{\circ}[1/\mu] \subset \mathbf{A}_R/p \otimes_{\mathbf{A}_R^+/p} \mathbf{N}_R(T)/p \subset M_1[1/\mu] = \mathbf{D}_R(T)/p$. ◻ # Wach modules and $q\textrm{-connections}$ {#sec:wachmod_qconnection} In this section we will interpret Wach modules over $\mathbf{A}_R^+$ (resp. $\mathbf{B}_R^+$) as modules with $q\textrm{-connection}$ and show that Wach modules over $\mathbf{B}_R^+$ can be seen as $q\textrm{-deformation}$ of filtered $(\varphi, \partial)\textrm{-modules}$ over $R[1/p]$ coming from $p\textrm{-adic}$ crystalline representations of $G_R$ (see Theorem [Theorem 70](#thm:qdeformation_dcrys){reference-type="ref" reference="thm:qdeformation_dcrys"}). We will follow [@morrow-tsuji §2] for our definitions with slight modifications. ## Formalism on $q\textrm{-connection}$ {#subsec:qconnection_formalism} Let $D$ be a commutative ring and consider a $D\textrm{-algebra}$ $A$ equipped with $d$ commuting $D\textrm{-algebra}$ automorphisms $\gamma_1 \ldots, \gamma_d$, i.e. an action of $\mathbb{Z}^d$. Moreover, fix an element $q \in D$ such that $q-1$ is a nonzerodivisor of $D$ and $\gamma_i = 1 \textup{ mod }(q-1)A$ for all $1 \leq i \leq d$. Assume that we have units $U_1, \ldots, U_d \in A^{\times}$ such that $\gamma_i(U_j) = q U_j$ if $i=j$ or $U_j$ if $i \neq j$. We fix these choices for the rest of the section. **Definition 65** ([@morrow-tsuji Definition 2.1]). Let $q\Omega^{\bullet}_{A/D} := \oplus_{k=0}^d q\Omega^k_{A/D}$ be a differential graded $D\textrm{-algebra}$ defined as: - $q\Omega^0_{A/D} := A$ and $q\Omega^1_{A/D}$ is a free left $A\textrm{-module}$ on formal basis elements $\hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i)$. - The right $A\textrm{-module}$ structure on $q\Omega^1_{A/D}$ is twisted by the rule $\hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i) \cdot f = \gamma_i(f) \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i)$. - $\hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i) \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_j) = - \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_j) \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i)$ if $i \neq j$ and $0$ if $i=j$. - The following map, where $I_k = \{\mathbf{i}= (i_1, \ldots, i_k) \in \mathbb{N}^k \textrm{ such that } 1 \leq i_1 < \cdots < i_k \leq d\}$, is an isomorphism of $A\textrm{-modules}$ $$\begin{aligned} \oplus_{\mathbf{i}\in I_k} A &\xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}q\Omega^k_{A/D}\\ (f_{\mathbf{i}}) &\longmapsto \textstyle\sum_{\mathbf{i}\in I_k} f_{\mathbf{i}} \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_{i_1}) \cdots \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_{i_k}). \end{aligned}$$ - The $0^{\textrm{th}}$ differential $d_q : A \rightarrow \Omega^1_{A/D}$ is given as $f \mapsto \sum_{i=1}^d \frac{\gamma_i(f)-f}{q-1} \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i)$. - The elements $\hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i) \in q\Omega^1_{D/A}$ are cocycles for all $1 \leq i \leq d$. The data $d_q : A \rightarrow q\Omega^1_{A/D}$ forms a differential ring over $D$, i.e. $q\Omega^1_{A/D}$ is an $D\textrm{-bimodule}$ and $d_q$ is $D\textrm{-linear}$ satisfying the Leibniz rule $d_q(fg) = d_q(f)g + fd_q(g)$ (see [@andre §II.1.2.1]). **Definition 66** ([@morrow-tsuji Definition 2.2]). A module with $\textit{q-connection}$ over $A$ is a right $A\textrm{-module}$ $N$ equipped with a $D\textrm{-linear}$ map $\nabla_q : N \rightarrow N \otimes_A q\Omega^1_{A/D}$ satisfying the Leibniz rule $\nabla_q(xf) = \nabla_q(x)f + x \otimes d_q(f)$ for all $f \in A$ and $x \in N$. The $q\textrm{-connection}$ $\nabla_q$ extends uniquely to a map of graded $D\textrm{-modules}$ $\nabla_q : N \otimes_A q\Omega^{\bullet}_{A/D} \rightarrow N \otimes_A q\Omega^{\bullet+1}_{A/D}$ satisfying $\nabla_q((n \otimes \omega) \cdot \omega') = \nabla_q(n \otimes \omega) \cdot \omega' + (-1)^{\textrm{deg } \omega} (n \otimes \omega) \cdot d_q(\omega')$. The $q\textrm{-connection}$ $\nabla_q$ is said to be *flat* or *integrable* if $\nabla_q \circ \nabla_q = 0$. Now assume that $D$ is equipped with an endomorphism $\varphi : D \rightarrow D$ such that it is a lift of absolute Frobenius on $D/p$ and $\varphi(q) = q^p$. Further, assume that $A$ is equipped with a compatible (with $\varphi$ on $D$) endomorphism $\varphi : A \rightarrow A$ such that it is a lift of absolute Frobenius on $A/p$ and commutes with the action of $\gamma_1, \ldots, \gamma_d$ on $A$. The endomorphism $\varphi$ induces an endomorphism $\varphi_{\Omega}$ on $q\Omega^1_{A/D}$ given as $\varphi_{\Omega}(\sum_{i=1}^d f_i \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i)) = [p]_q \sum_{i=1}^d \varphi(f_i) \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}(U_i)$. In particular, from [@morrow-tsuji Lemma 2.12] the following diagram commutes It follows that given a $q\textrm{-connection}$ $(N, \nabla_q)$ we can define the base change via Frobenius of the $q\textrm{-connection}$ denoted $\varphi^*\nabla_q$ on $\varphi^*N := N \otimes_{A, \varphi} A$ as $$\begin{aligned} \varphi^* \nabla_q : \varphi^* N &\longrightarrow N \otimes_{A, \varphi} q\Omega^1_{A/D} = \varphi^* N \otimes q\Omega^1_{A/D}\\ x \otimes f &\longmapsto (1 \otimes \varphi_{\Omega})(\nabla_q(x)) \cdot f + n \otimes d_q(f).\end{aligned}$$ A $\varphi\textrm{-module}$ with $q\textrm{-connection}$ is a pair $(N, \nabla_q)$ as above equipped with an $A\textrm{-linear}$ isomorphism $\varphi_N : (\varphi^*N)[1/[p]_q] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}N[1/[p]_q]$ such that the following diagram commutes $$\label{eq:qconnection_horizontal_frob} \begin{tikzcd} (\varphi^*N)[1/[p]_q] \arrow[r, "\varphi^* \nabla_q"] \arrow[d, "\varphi_N"'] & (\varphi^*N)[1/[p]_q] \otimes q\Omega^1_{A/D} \arrow[d, "\varphi_N \otimes 1"]\\ N[1/[p]_q] \arrow[r, "\nabla_q"] & N[1/[p]_q] \otimes q\Omega^1_{A/D}. \end{tikzcd}$$ ## Wach module as $q\textrm{-deformation}$ {#subsec:wachmod_qdeformation} In this subsection, we take $D := \mathbb{Z}_p\llbracket \mu \rrbracket$, $A := \mathbf{A}_R^+$ equipped with the action of $\Gamma_R$ and $\{\gamma_1, \ldots, \gamma_d\}$ as topological generators of $\Gamma_R'$, the geometric part of $\Gamma_R$ (see §[2](#sec:period_rings_padic_reps){reference-type="ref" reference="sec:period_rings_padic_reps"}). Then setting $q := 1+\mu$ and $U_i := [X_i^{\flat}]$ for $1 \leq i \leq d$, we have $\gamma_i = 1 \textup{ mod }\mu \mathbf{A}_R^+$ for all $1 \leq i \leq d$. In particular, $\mathbf{A}_R^+$ satisfies the hypotheses of Definition [Definition 65](#defi:qdeRham_complex){reference-type="ref" reference="defi:qdeRham_complex"}. Moreover, the Frobenius endomorphism on $\mathbf{A}_R^+$ extends the Frobenius on $D$ given by identity on $\mathbb{Z}_p$ and $\varphi(\mu) = (1+\mu)^p-1$. Furthermore, in this case, $q\Omega^1_{\mathbf{A}_R^+/D}$ identifies with $\Omega^1_{\mathbf{A}_R^+/D}$ given as $(p, \mu)\textrm{-adic}$ completion of module of Kähler differentials of $\mathbf{A}_R^+$ with respect to $D$. Note that we have Frobenius-equivariant isomorphism of rings $\mathbf{A}_R^+/\mu \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}R$, so from [@morrow-tsuji Remarks 2.4 & 2.10] reduction modulo $q-1$ of the differential ring $d_q : \mathbf{A}_R^+ \rightarrow \Omega^1_{\mathbf{A}_R^+/D}$ is the usual de Rham differential $d : R \rightarrow \Omega^1_R$. Similarly, the reduction modulo $q-1$ of a module with $q\textrm{-connection}$ over $\mathbf{A}_R^+$ (Definition [Definition 66](#defi:qconnection){reference-type="ref" reference="defi:qconnection"}) is an $R\textrm{-module}$ with connection. We say that $q\textrm{-connection}$ is $(p, [p]_q)\textit{-adically}$ *quasi-nilpotent* (equivalently, $(p, q-1)\textit{-adically}$ quasi-nilpotent) if $\nabla_q \textup{ mod }q-1$ is $p\textrm{-adically}$ quasi-nilpotent. **Proposition 67**. *Let $N$ be a Wach module over $\mathbf{A}_R^+$. Then the geometric $q\textrm{-connection}$ $$\begin{aligned} \nabla_q : N &\longrightarrow N \otimes_{\mathbf{A}_R^+} \Omega^1_{\mathbf{A}_R^+/D}\\ x &\longmapsto \sum_{i=1}^d \tfrac{\gamma_i(x)-x}{\mu} \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}([X_i^{\flat}]), \end{aligned}$$ describes $(N, \nabla_q)$ as a $\varphi\textrm{-module}$ equipped with $(p, [p]_q)\textrm{-adically}$ quasi-nilpotent flat $q\textrm{-connection}$ over $\mathbf{A}_R^+$.* *Proof.* Flatness of the $q\textrm{-connection}$ $\nabla_q$ follows from the first part of the proof of [@morrow-tsuji Proposition 2.6]. From Definition [Definition 34](#defi:wach_mods_relative){reference-type="ref" reference="defi:wach_mods_relative"} note that we have $\varphi \otimes 1: (N \otimes_{\mathbf{A}_R^+, \varphi} \mathbf{A}_R^+)[1/[p]_q] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}N[1/[p]_q]$. Now note that $(N, \nabla_q)$ is a $\varphi\textrm{-module}$ with $q\textrm{-connection}$ over $\mathbf{A}_R^+$ satisfying the assumptions of [@morrow-tsuji Lemma 2.24], therefore from loc. cit. it follows that $\nabla_q$ is $(p, [p]_q)\textrm{-adically}$ quasi-nilpotent. Finally, since the action of $\varphi$ and $\Gamma_R'$ commute on $N$, it follows that the corresponding diagram [\[eq:qconnection_horizontal_frob\]](#eq:qconnection_horizontal_frob){reference-type="eqref" reference="eq:qconnection_horizontal_frob"} is commutative. This concludes our proof. ◻ *Remark 68*. In Proposition [Proposition 67](#defi:wachmod_qconnection){reference-type="ref" reference="defi:wachmod_qconnection"} we call the $q\textrm{-connection}$ "geometric" because in the definition we only use the geometric part of $\Gamma_R$, i.e. $\Gamma_R'$. *Remark 69*. From §[3.5](#subsec:wachmod_crystalline){reference-type="ref" reference="subsec:wachmod_crystalline"} recall that we have the ring $\mathbf{A}_{R,\varpi}^{\textup{PD}} \subset \mathbf{A}_{\textup{cris}}(R_{\infty})$ stable under Frobenius and $\Gamma_R\textrm{-action}$. For $R = O_F$ we denote the aforementioned ring by $D^{\textup{PD}}$ and for general $R$, we denote it by $A^{\textup{PD}} := \mathbf{A}_{R,\varpi}^{\textup{PD}}$ (we do not use $D$ and $A$ for these rings to avoid confusion with preceding assumptions). Then it is easy to see that the hypotheses of Definition [Definition 65](#defi:qdeRham_complex){reference-type="ref" reference="defi:qdeRham_complex"} are satisfied for $D^{\textup{PD}}$, $A^{\textup{PD}}$ with $\Gamma_R\textrm{-action}$ and $U_i := [X_i^{\flat}]$. Now given a Wach module $N$ over $\mathbf{A}_R^+$, similar to Propostion [Proposition 67](#defi:wachmod_qconnection){reference-type="ref" reference="defi:wachmod_qconnection"}, one can show that for $N^{\textup{PD}} := A^{\textup{PD}} \otimes_{\mathbf{A}_R^+} N$, the $q\textrm{-connection}$ $$\nabla_q : N^{\textup{PD}} \longrightarrow N^{\textup{PD}} \otimes_{A^{\textup{PD}}} \Omega^1_{A^{\textup{PD}}/D^{\textup{PD}}}, \hspace{5mm} x \longmapsto \textstyle\sum_{i=1}^d \tfrac{\gamma_i(x)-x}{\mu} \hspace{0.2mm}d\textup{\hspace{0.3mm}log\hspace{0.3mm}}([X_i^{\flat}]),$$ describes $(N^{\textup{PD}}, \nabla_q)$ as a $\varphi\textrm{-module}$ with $p\textrm{-adically}$ quasi-nilpotent flat $q\textrm{-connection}$ over $A^{\textup{PD}}$. Set $\nabla_{q, i} := (\gamma_i-1)/\mu$ for $1 \leq i \leq d$. Furthermore, employing arguments similar to [@abhinandan-syntomic Lemmas 4.12, 5.17 & 5.18] we can show that for $1 \leq i \leq d$, the operator $\nabla_i := (\log \gamma_i)/t = \frac{1}{t}\sum_{k \in \mathbb{N}} (-1)^k \frac{(\gamma_i-1)^{k+1}}{k+1}$ converges as a series of operators on $N^{\textup{PD}}$. So using the explicit formulas described above, it is easy to see that for $x \in N$, we have $\nabla_{q, i}(x) - \nabla_i(x) = (\frac{\gamma_i-1}{\mu} - \frac{\log\gamma_i}{t})(x) \in (\textup{Fil}^1 A^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} N$, since $t/\mu$ is a unit in $A^{\textup{PD}}$ by [@abhinandan-relative-wach-i Lemma 3.14]. We are now ready to state the main result of this section. Let $N$ be a Wach module over $\mathbf{A}_R^+$ equipped with $q\textrm{-connection}$ as in Proposition [Proposition 67](#defi:wachmod_qconnection){reference-type="ref" reference="defi:wachmod_qconnection"} and a Nygaard filtration as in Definition [Definition 47](#defi:nygaard_fil){reference-type="ref" reference="defi:nygaard_fil"}. Then from the discussion preceding Proposition [Proposition 67](#defi:wachmod_qconnection){reference-type="ref" reference="defi:wachmod_qconnection"}, we note that $N/\mu N$ is a $\varphi\textrm{-module}$ over $R$ equipped with a $p\textrm{-adically}$ quasi-nilpotent connection and a filtration $\textup{Fil}^k(N/\mu N)$ given as the image of $\textup{Fil}^k N$ under the surjection $N \twoheadrightarrow N/\mu N$. We equip $N[1/p]/\mu N[1/p] = (N/\mu N)[1/p]$ with induced structures, in particular, it is a filtered $(\varphi, \partial)\textrm{-module}$ over $R[1/p]$. **Theorem 70**. *Let $N$ be a Wach module over $\mathbf{A}_R^+$ and $V := \mathbf{T}_R(N)[1/p]$ the associated crystalline representation from Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"}. Then we have $(N/\mu N)[1/p] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ as filtered $(\varphi, \partial)\textrm{-modules}$ over $R[1/p]$.* *Proof.* For $r \in \mathbb{N}$ large enough, the Wach module $\mu^r N (-r)$ is always effective and we have $\mathbf{T}_R(\mu^rN(-r)) = \mathbf{T}_R(N)(-r)$ (the twist $(-r)$ denotes a Tate twist on which $\Gamma_R$ acts via $\chi^{-r}$ where $\chi$ is the $p\textrm{-adic}$ cyclotomic character). Therefore, it is enough to show both the claims for effective Wach modules. So we assume that $N$ is effective and set $M := N[1/p]$ equipped with induced $\Gamma_R\textrm{-action}$, Frobenius-semilinear operator $\varphi$ and the Nygaard filtration. It follows that the finite projective $R[1/p]\textrm{-module}$ $M/\mu$ is equipped with a Frobenius-semilinear operator $\varphi$ induced from $M$. Note that $[p]_q = p \mod \mu \mathbf{A}_R^+$, therefore we have $1 \otimes \varphi : \varphi^*(M/\mu) \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}M/\mu$ . Furthemrmore, the filtration $\textup{Fil}^k (M/\mu)$ is defined to be the image of $\textup{Fil}^k M$ under the surjective map $M \twoheadrightarrow M/\mu$. Now recall that we have a period ring $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \subset \pazocal{O}\mathbf{A}_{\textup{cris}}(R_{\infty})$ equipped with compatible Frobenius endomorphism $\varphi$, $\Gamma_R\textrm{-action}$, filtration and a connection satisfying Griffiths transversality with respect to the filtration (see §[3.5](#subsec:wachmod_crystalline){reference-type="ref" reference="subsec:wachmod_crystalline"}). Moreover, from Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"} we have $D_R := (\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} M)^{\Gamma_R}$ equipped with a Frobenius-semilinear operator $\varphi$, filtration and a connection satisfying Griffiths transversality with respect to the filtration; we have $D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$ compatible with supplementary structures (see [\[eq:dr_dcrys_comp\]](#eq:dr_dcrys_comp){reference-type="eqref" reference="eq:dr_dcrys_comp"}). Let us consider the following diagram with exact rows Note that $(\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} M) \cap M = (\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \cap \mathbf{A}_R^+) \otimes_{\mathbf{A}_R^+} M = \mu M$, then from the exactness of the second row it follows that the vertical maps from first to second row are natural inclusions. The middle vertical arrow from third to second row is the isomorphism [\[eq:oarpd_comparison\]](#eq:oarpd_comparison){reference-type="eqref" reference="eq:oarpd_comparison"} in Proposition [Proposition 50](#prop:oarpd_comparison){reference-type="ref" reference="prop:oarpd_comparison"} from which it can easily be shown that the left vertical arrow is an isomorphism as well, so it follows that the right vertical arrow is also an isomorphism. Taking the $\textup{Gal}(R[1/p][\varpi]/R[1/p]) = \textup{Gal}(F(\zeta_p)/F)\textrm{-invariants}$ of the right vertical arrows gives $M/\mu \xleftarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, R}(V)$, where the last isomorphism is compatible with filtration, Frobenius and connection as explained in the proof of Theorem [Theorem 49](#thm:fh_crys_relative){reference-type="ref" reference="thm:fh_crys_relative"} (see [\[eq:dr_dcrys_comp\]](#eq:dr_dcrys_comp){reference-type="eqref" reference="eq:dr_dcrys_comp"}). Note that the isomorphism $D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}M/\mu$ is compatible with Frobenius and we need to check the compatibility between respective filtrations and connections. Let us compare the filtrations first. In the commutative diagram above, filtration on the middle term of the second row is given by the tensor product filtration, so it is easy to see that under the surjective map from second to third term, the image of $\textup{Fil}^k(\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} M)$ is given as $R[\varpi] \otimes_R \textup{Fil}^k(M/\mu)$. Similarly, the middle term of the third row is equipped with tensor product filtration so the image of $\textup{Fil}^k(\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_R D_R)$ under the surjective map from second to third term is given as $R[\varpi] \otimes_R \textup{Fil}^k D_R$. Recall that the isomorphism $f$ in [\[eq:oarpd_comparison\]](#eq:oarpd_comparison){reference-type="eqref" reference="eq:oarpd_comparison"} of Proposition [Proposition 50](#prop:oarpd_comparison){reference-type="ref" reference="prop:oarpd_comparison"} is compatible with filtrations, so we conclude that $R[\varpi] \otimes_R \textup{Fil}^k D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}R[\varpi] \otimes_R \textup{Fil}^k(M/\mu)$. Now taking invariants under the natural $\textup{Gal}(F(\zeta_p)/F)\textrm{-action}$ on both terms of the preceding isomorphism gives $\textup{Fil}^k D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\textup{Fil}^k(M/\mu)$. Next, note that the connection on $M/\mu$ is obtained by first taking reduction modulo $\mu = q-1$ of the $q\textrm{-connection}$ $\nabla_q$ on $N$ and then inverting $p$. On the other hand, the connection $\partial_D$ on $D_R = (\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_{\mathbf{A}_R^+} M)^{\Gamma_R}$ is induced from the natural connection on $\pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}$. Let $\nabla_{q, i}$ and $\partial_{D, i}$ respectively denote the $i^{\textrm{th}}$ component of the $q\textrm{-connection}$ on $N$ and the connection on $D_R$. Now let $x \in M$, then from Remark [Remark 52](#rem:relate_con_qcon){reference-type="ref" reference="rem:relate_con_qcon"} there exists $w \in \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}} \otimes_R D_R$ such that $x = f(w) \textup{ mod }(\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} M$, where $f$ is the isomorphism in [\[eq:oarpd_comparison\]](#eq:oarpd_comparison){reference-type="eqref" reference="eq:oarpd_comparison"}. Then it follows that to check the compatibility of the isomorphism $D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}M/\mu$ with connections, it is enough to show that for $x \in M$ we have $\nabla_{q, i}(x) - f(\partial_{D, i}(w)) \in (\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} M$. From Remark [Remark 52](#rem:relate_con_qcon){reference-type="ref" reference="rem:relate_con_qcon"} for $\nabla_i = (\log \gamma_i)/t$, we know that $\nabla_i(x) - f(\partial_{D, i}(w)) \in (\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} M$. Furthermore, from Remark [Remark 69](#rem:qconnection_arpipd){reference-type="ref" reference="rem:qconnection_arpipd"} we have $\nabla_{q, i}(x) - \nabla_i(x) \in (\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} M$. Combining the two we get $\nabla_{q, i}(x) - f(\partial_{D_i}(w)) \in (\textup{Fil}^1 \pazocal{O}\mathbf{A}_{R,\varpi}^{\textup{PD}}) \otimes_{\mathbf{A}_R^+} M$, i.e. the isomorphism $D_R \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}M/\mu$ is also compatible with connections. This concludes our proof. ◻ *Remark 71*. Obvious variation of Theorem [Theorem 70](#thm:qdeformation_dcrys){reference-type="ref" reference="thm:qdeformation_dcrys"} is also true in the imperfect residue field case. Indeed, for $O_L$, recall that all compatibilities except for the connection part was already proven in [@abhinandan-imperfect-wach Corollary 3.15]. To verify the compatibility of connections, similar to Proposition [Proposition 67](#defi:wachmod_qconnection){reference-type="ref" reference="defi:wachmod_qconnection"}, we can define a $q\textrm{-connection}$ over a Wach module over $\mathbf{A}_L^+$. Then using the results of [@abhinandan-imperfect-wach §3.3] one obtains obvious variation of Remark [Remark 69](#rem:qconnection_arpipd){reference-type="ref" reference="rem:qconnection_arpipd"} over $\mathbf{A}_{L,\varpi}^{\textup{PD}}$. Finally, proceeding exactly as in the proof of Theorem [Theorem 70](#thm:qdeformation_dcrys){reference-type="ref" reference="thm:qdeformation_dcrys"} (after replacing each object by analogous object for $L$), we obtain the desired isomorphism of filtered $(\varphi, \partial)\textrm{-modules}$ over $L$. Let us summarise the relationship between various categories considered in [\[eq:odcrysr_func\]](#eq:odcrysr_func){reference-type="eqref" reference="eq:odcrysr_func"}, Corollary [Corollary 56](#cor:crystalline_wach_rat_equivalence_relative){reference-type="ref" reference="cor:crystalline_wach_rat_equivalence_relative"} and Theorem [Theorem 70](#thm:qdeformation_dcrys){reference-type="ref" reference="thm:qdeformation_dcrys"}. Recall that $\textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R)$ is the category of $p\textrm{-adic}$ crystalline representations of $G_R$ and $\textup{MF}_R^{\textup{ad}}(\varphi, \partial)$ denotes the essential image of the functor $\pazocal{O}\mathbf{D}_{\textup{cris}, R}$ restricted to $\textup{Rep}_{\mathbb{Q}_p}^{\textup{cris}}(G_R)$. **Corollary 72**. *Functors in the following diagram induce exact equivalence of $\otimes\textrm{-categories}$* *Proof.* The exact equivalence induced by functors $\mathbf{N}_R$ and $\mathbf{V}_R$ is from Corollary [Corollary 56](#cor:crystalline_wach_rat_equivalence_relative){reference-type="ref" reference="cor:crystalline_wach_rat_equivalence_relative"} and the exact equivalence induced by $\pazocal{O}\mathbf{D}_{\textup{cris}, R}$ and $\pazocal{O}\mathbf{V}_{\textup{cris}, R}$ is from [@brinon-relatif Théorème 8.5.1]. Moreover, from Theorem [Theorem 70](#thm:qdeformation_dcrys){reference-type="ref" reference="thm:qdeformation_dcrys"} note that for a Wach module $M$ over $\mathbf{B}_R^+$ we have $M/(q-1) = M/\mu \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\pazocal{O}\mathbf{D}_{\textup{cris}, R}(\mathbf{V}_R(M))$. Hence, from the preceding exact equivalence of $\otimes\textrm{-categories}$, it follows that the slanted arrow labeled "$q \mapsto 1$" is also an exact equivalence of $\otimes\textrm{-categories}$. ◻ # Structure of $\varphi$-modules We will use setup and notations from §[1.5](#subsec:setup_nota){reference-type="ref" reference="subsec:setup_nota"} and the rings defined in §[2.2](#subsec:ainf_relative){reference-type="ref" reference="subsec:ainf_relative"}. Let $q$ be an indeterminate, then we have a Frobenius-equivariant isomorphism $R\llbracket q-1 \rrbracket \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}\mathbf{A}_R^+$ via the map $X_i \mapsto [X_i^{\flat}]$ and $q \mapsto 1+\mu$. We will show the following structural result: **Proposition 73**. *Let $N$ be a finitely generated $\mathbf{A}_R^+\textrm{-module}$ and suppose that $N$ is equipped with a Frobenius-semilinear endomorphism $\varphi : N \rightarrow N$ such that $1 \otimes \varphi : \varphi^*(N)[1/[p]_q] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}N[1/[p_q]]$. Then $N[1/p]$ is finite projective over $\mathbf{B}_R^+$.* *Proof.* The proof is essentially the same as [@du-liu-moon-shimizu Proposition 4.13]. Compared to loc. cit. the Frobenius endomorphism on $\mathbf{A}_R^+$ and finite height assumption on $N$ are different and we do not assume $N$ to be torsion free. However, one observes that torsion freeness of $N$ is not used in the proof and one can use [@abhinandan-imperfect-wach Lemma 2.14] and [Lemma 74](#lem:finiteproj_regularlocal){reference-type="ref" reference="lem:finiteproj_regularlocal"} instead of [@bhatt-morrow-scholze-1 Proposition 4.3] and [@du-liu-moon-shimizu Lemma 4.12]. ◻ **Lemma 74**. *Let $k$ be a perfect field of characteristic $p$ and $S := W(k)\llbracket u_1, \ldots, u_m \rrbracket$ equipped with a Frobenius endomorphism $\varphi$ extending the Witt vector Frobenius on $W(k)$ such that $\varphi(u_i) \in S$ has zero constant term for each $1 \leq i \leq m$. Let $A := S \llbracket q-1 \rrbracket$ equipped with a Frobenius endomorphism extending the one on $S$ by $\varphi(q) = q^p$ and let $N$ be a finitely generated $A\textrm{-module}$ equipped with a Frobenius-semilinear endomorphism such that $1 \otimes \varphi : \varphi^*(N)[1/[p]_q] \xrightarrow{\hspace{0.5mm} \sim \hspace{0.5mm}}N[1/[p]_q]$. Then $N[1/p]$ is finite projective over $A[1/p]$.* *Proof.* The proof is essentially the same as [@du-liu-moon-shimizu Lemma 4.12], except for a few changes. One proceeds by induction on $m$. The case $m=0$ follows from [@abhinandan-imperfect-wach Lemma 2.14], so let $m \geq 1$. Take $J$ to be the smallest non-zero Fitting ideal of $N$ over $A$. It suffices to show that $JA[1/p] = A[1/p]$. Compatibility of Fitting ideals under base change implies that $JA[1/[p]_q] = \varphi(J)A[1/[p]_q]$ as ideals of $A[1/[p]_q]$, therefore $(A/J)[1/[p]_q] = (A/\varphi(J))[1/[p]_q]$. Let us assume $JA[1/p] \neq A[1/p]$ and we will reach a contradiction. In our setting, the Frobenius endomorphism on $A$ and the finite height condition are different from [@du-liu-moon-shimizu Lemma 4.12]. Therefore, we need some modifications in the arguments of loc. cit.; let us point out the differences in terms of their notations. Let $K = W(k)[1/p]$, fix $\overline{K}$ as an algebraic closure of $K$. Consider the $\overline{K}\textrm{-valued}$ points of $\textup{Spec}\hspace{0.5mm}(A[1/p]/J)$ and let $Z = \{(|u_1|, \ldots, |u_m|, |q-1|) \in \mathbb{R}^{m+1}\}$ be the corresponding set of $(m+1)\textrm{-tuple}$ norms. Define the set $Z' = \{(|u_1|, \ldots, |u_m|, |q-1|) \in \mathbb{R}^{m+1} \textrm{ such that } (|\varphi(u_1), \ldots |\varphi(u_m)|, |q^p-1|) \in Z\}$ and take $\zeta_p-1$ as the chosen uniformizer. Then one proceeds as in loc. cit. to show that $JA[1/p] \subset (u_1, \ldots, u_m, q-1)A[1/p]$ and $JA[1/p] \not\subset IA[1/p]$ where $I = (u_1, \ldots, u_m) \subset A[1/p]$. Finally, consider the Frobenius-equivariant projection $A \rightarrow \overline{A}= A/I = W(k)\llbracket q-1 \rrbracket$ and let $\overline{J}\subset \overline{A}$ denote the image of $J$. Since $JA[1/p] \not\subset IA[1/p]$, we get that $\overline{J}\neq 0$. Moreover, $\overline{J}\overline{A}[1/p] \neq \overline{A}[1/p]$ since $JA[1/p] \subset (u_1, \ldots, u_m, q-1)A[1/p]$. However, the equality $(A/J)[1/[p]_q] = (A/\varphi(J))[1/[p]_q]$ implies that $(\overline{A}/\overline{J})[1/[p]_q] = (\overline{A}/\varphi(\overline{J}))[1/[p]_q]$, i.e. $\overline{J}\overline{A}[1/p] = \overline{A}[1/p]$ by inductive hypothesis (see [@abhinandan-imperfect-wach Lemma 2.14]). This gives a contradiction. Hence, we must have $JA[1/p] = A[1/p]$, thus proving the lemma. ◻ ------------------------------------------------------------------------ [Abhinandan]{.smallcaps} [University of Tokyo, 3 Chome-8-1, Komaba, Meguro City, Tokyo, Japan]{.smallcaps} *E-mail*: [abhi\@ms.u-tokyo.ac.jp](abhi@ms.u-tokyo.ac.jp), *Web*: <https://abhinandan.perso.math.cnrs.fr/>

arxiv_math

--- abstract: | We investigate a polynomial factorization problem which naturally arises in Hartley's factorization condition on the Alexander polynomial of freely periodic knots. We give a number-theoretic interpretation of this factorization condition, which allows for efficient computation. As an application, we prove that any polynomial which is not a product of cyclotomic polynomials can be the Alexander polynomial of a freely $p$-periodic knot for only finitely many $p$. As a demonstration of the computational efficiency of these methods, we also show that the Alexander polynomial of any freely-periodic L-space knot with genus at most 16 must be a product of cyclotomic polynomials. We conjecture that any periodic or freely periodic L-space knot must be an iterated torus knot. address: - Department of Mathematics, University of British Columbia, Canada - Department of Mathematics, University of British Columbia, Canada author: - Keegan Boyle - Nicholas Rouse bibliography: - bibliography.bib title: Obstructions to free periodicity and symmetric L-space knots --- # Introduction The Alexander polynomial of a knot is quite good at obstructing some types of symmetry. Perhaps most famously, Murasugi proved that the Alexander polynomial of a periodic knot satisfies a factorization condition [@MR292060]. Murasugi's theorem has received significant attention in the literature; see [@MR639581] for the version we state here. **Theorem 1**. *Let $K$ be an $n$-periodic knot with quotient knot $\overline{K}$, and let $A$ be the axis of symmetry. Then $$\Delta_K(t) = \Delta_{\overline{K}}(t) \prod_{i = 1}^{n-1}\Delta_{A \cup \overline{K}}(\zeta_n^i,t),$$ where $\zeta_n$ is a primitive $n^{th}$ root of unity.* A standard use of this theorem is to show that a particular knot is not $n$-periodic for any $n$. Periodic knots are one of two types of knots with cyclic symmetries which preserve the orientations of the knot and $S^3$; see for example [@boyle2023classification]. The other type (FPer in [@boyle2023classification]) contains the class of freely periodic knots, for which there is also a factorization condition on the Alexander polynomial due to Hartley [@MR608857]. **Theorem 2**. *Let $K$ be a freely $n$-periodic knot with Alexander polynomial $\Delta_K(t)$. Then $$\Delta_K(t^n) = \prod_{i=0}^{n-1}f(\zeta_n^i t),$$ where $\zeta_n$ is a primitive $n^{th}$ root of unity, and $f$ is an Alexander polynomial of a knot.* Unlike with Murasugi's condition, where a finite computation can check the factorization for all $n$, with Hartley's condition it seems necessary to check the condition one integer $n$ at a time, since the polynomial being factored depends on $n$. Although for specific knots there are bounds on the order of the free periodicity coming from hyperbolic geometry, when working with only the Alexander polynomials there is no such luxury. The main motivation and application of this paper is this latter situation. We study polynomials satisfying a slightly generalized version of Hartley's condition [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"} for various $n$ wherein we allow a factor of $-1$ to appear in the factorization. We refer to such polynomials as ***n*-Hartley**. Note that when $\Delta_K(t^n)$ is irreducible, it clearly is not $n$-Hartley. Previous work, for example [@MR2852954] and [@SchinzelPolynomials §4.1], has studied whether $\Delta_K(t^n)$ is irreducible; however, these results do not apply in the generality we need. For example, we do not know of any existing results which apply to reducible, monic polynomials---a class of particular interest for our applications to symmetric L-space knots as in Conjecture [Conjecture 7](#conj:iteratedtorus){reference-type="ref" reference="conj:iteratedtorus"} below. ## Results Our first application will be to compute exactly the infinite set of integers for which the cyclotomic polynomials satisfy Hartley's factorization condition [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"}. **Proposition 3**. *Let $\Phi_m(t)$ be the $m$th cyclotomic polynomial. Then $\Phi_m(t)$ satisfies Hartley's factorization condition [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"} for $n$ if and only if $n$ is coprime to $m$.* In contrast we prove that there are only finitely many such integers for polynomials which are not products of cyclotomic polynomials. **Theorem 4**. *Let $\Delta(t)$ be a polynomial that is not a product of cyclotomic polynomials. Then the set of $n$ for which $\Delta(t)$ satisfies Hartley's condition [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"} is finite and effectively computable.* This theorem will follow immediately from the results in Section [2](#sec:nt){reference-type="ref" reference="sec:nt"}. **Corollary 5**. *Let $\Delta(t)$ be a polynomial that is not a product of cyclotomic polynomials. Then there are only finitely many $n$ for which there is a freely $n$-periodic knot with Alexander polynomial equal to $\Delta(t)$.* Our core results are entirely number-theoretic, and so may be of independent interest. The main technique is to define an integer associated to a polynomial $\Delta(t)$ called $E(\Delta(t))$ (Definition [Definition 18](#defn:schinzelDegree){reference-type="ref" reference="defn:schinzelDegree"}) which is defined in terms of the irreducible factors of $\Delta(t)$ and their multiplicities. This $E(\Delta(t))$ is effectively computable, and we prove that $\Delta(t)$ is $n$-Hartley if and only if $n$ divides $E(\Delta(t))$ (see Theorem [Theorem 21](#thm:pDividesSchinzelDegree){reference-type="ref" reference="thm:pDividesSchinzelDegree"}) as long as $\Delta(t)$ is not a product of cyclotomic polynomials or divisible by a prime integer. The main difficulty treated in Section [2](#sec:nt){reference-type="ref" reference="sec:nt"} is to understand how Hartley's condition behaves under factorization. As any polynomial to the $n$th power is $n$-Hartley, care must be paid to the multiplicities of the irreducible factors. Since a polynomial is $n$-Hartley if and only if it is $p^r$-Hartley for each prime power dividing $n$, we state our result in terms of prime powers. **Theorem 6**. *Let $p$ be a prime, $\Delta(t)$ be $p^r$-Hartley and primitive, and $f_1(t)$ an irreducible factor of $\Delta(t)$. Then $f_1(t)$ is $p^s$-Hartley form some $s \geq 0$ and appears with multiplicity $mp^k$ for some $m \geq 1$ coprime to $p$ with $s+k \geq r$.* The two extremal cases covered by Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"} are when $s=0$ in which case $f_1(t)$ appears with multiplicity divisible by $p^r$ and when $k=0$ in which case $f_1(t)$ is $p^r$-Hartley. We remark that our results not only imply an effective algorithm but also often a practical one. We resolve Conjecture [Conjecture 8](#conj:polynomial){reference-type="ref" reference="conj:polynomial"} below for all knots of genus less than $13$ unconditionally and less than $17$ assuming the L-space conjecture. This computation involved calculating $E(\Delta(t))$ for over $65,000$ polynomials. ## Conjectures on symmetric L-space knots As an application, we provide some computational evidence for some conjectures about symmetric L-space knots. L-space knots are those knots which have have a surgery that is an L-space, a 3-manifold with a particularly simple form of Heegaard-Floer homology [@MR2168576 Definition 1.1]. Famously, the L-space conjecture states that a 3-manifold has a left-orderable fundamental group if and only if it is not an L-space [@MR3072799]. One motivation for the conjectures we present here is that it seems difficult to construct polynomials which simultaneously satisfy the restrictions on the Alexander polynomial of an L-space knot (see for example [@MR2168576], and [@MR3782416]), and satisfy either the factorization condition from Theorem [Theorem 1](#thm:Murasugi){reference-type="ref" reference="thm:Murasugi"} or from Theorem [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"}. **Conjecture 7**. *Every periodic or freely periodic L-space knot is an iterated torus knot.* **Conjecture 8**. *Let $K$ be a periodic or freely periodic L-space knot. Then $\Delta_K(t)$ is a product of cyclotomic polynomials.* **Conjecture 9**. *Let $K$ be a periodic L-space knot with quotient knot $\overline{K}$. Then $\overline{K}$ is an L-space knot.* **Remark 10**. Conjecture [Conjecture 7](#conj:iteratedtorus){reference-type="ref" reference="conj:iteratedtorus"} implies Conjectures [Conjecture 8](#conj:polynomial){reference-type="ref" reference="conj:polynomial"} and [Conjecture 9](#conj:quotient){reference-type="ref" reference="conj:quotient"}, since the Alexander polynomial of an iterated torus knot is a product of cyclotomic polynomials (this follows from the cabling formula for the Alexander polynomial), and the quotient of a periodic iterated torus knot is again an interated torus knot and hence is an L-space knot by [@MR3546454]. **Remark 11**. There is also a conjecture of Li and Ni [@MR3485330 Conjecture 1.3] that L-space knots with Alexander polynomials which are products of cyclotomic polynomials are iterated torus knots. This would imply that Conjecture [Conjecture 8](#conj:polynomial){reference-type="ref" reference="conj:polynomial"} is equivalent to Conjecture [Conjecture 7](#conj:iteratedtorus){reference-type="ref" reference="conj:iteratedtorus"}. **Remark 12**. It was once conjectured that L-space knots must be strongly invertible; see [@MR3649241 Section 6]. However, Baker and Luecke have produced examples of L-space knots with no symmetries [@MR4194294 Theorem 1.2]. Conjecture [Conjecture 7](#conj:iteratedtorus){reference-type="ref" reference="conj:iteratedtorus"} is then a partial converse, claiming that L-space knots cannot have more than one strong inversion. Finally, we prove that the L-space conjecture implies Conjecture [Conjecture 9](#conj:quotient){reference-type="ref" reference="conj:quotient"}, and we observe that Conjecture [Conjecture 7](#conj:iteratedtorus){reference-type="ref" reference="conj:iteratedtorus"} is true for the $P(-2,3,2q+1)$ pretzel knots by [@MR891592]. ## Acknowledgments We would like to thank Liam Watson for helpful conversations. The second author is supported by the Pacific Institute for Mathematical Sciences (PIMS). The research and findings may not reflect those of the institute. # Number-theoretic interpretation of Hartley's condition {#sec:nt} Throughout this section, $\zeta_n$ will be a primitive $n$th root of unity. We define a slightly more general condition that agrees with Hartley's factorization condition when the polynomial in question is an Alexander polynomial. **Definition 13**. *Let $\Delta(t)$ be a polynomial with rational integer coefficients and $n \in \mathbf{Z}_{\geq 2}$. We say that $\Delta(t)$ is **n-Hartley** if there exists $g(t) \in \mathbf{Z}[t]$ such that $$\Delta(t^n) = \pm \prod_{i=0}^{n-1}g(\zeta_n^i t).$$* We prove now that every Alexander polynomial that is $n$-Hartley and is normalized to have positive constant term satisfies Hartley's factorization condition [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"}. **Proposition 14**. *Let $\Delta(t)$ be $n$-Hartley with positive constant term. Then $\Delta(t)$ satisfies Hartley's factorization condition [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"}.* *Proof.* Suppose that $\Delta(t^n)$ factors as $$\Delta(t^n) = (-1)^j \prod_{i=0}^{n-1}g(\zeta_n^i t)$$ for $j=0$ or $1$. First observe that if $j=1$ and $n$ is odd, then we can replace $g(t)$ with $-g(t)$ to obtain $j=0$, so suppose $n$ is even. Then $\Delta(0) = (-1)^j g(0)^n$, so $j=0$. ◻ One justification for this generalized definition is that there are polynomials which factor with the minus sign but enjoy almost all the same algebraic properties as those that factor with the positive sign. **Counterexample 15**. Let $K$ be the knot `K14n26330`. Its Alexander polynomial is $$\Delta(t) = 4t^6 - 17t^5 + 38t^4 - 51t^3 + 38t^2 - 17t + 4,$$ so that $$\Delta(t^2) = (2t^6 - 3t^5 - 2t^4 + 7t^3 - 2t^2 - 3t + 2)(2t^6 + 3t^5 - 2t^4 - 7t^3 - 2t^2 + 3t + 2),$$ so $\Delta(t)$ is $2$-Hartley. However $f(t)$ factors into irreducibles as $$\Delta(t) = (t^3 - 3t^2 + 5t - 4)(4t^3 - 5t^2 + 3t - 1),$$ and neither irreducible factor satisfies Hartley's condition with the positive sign and $p=2$. If we write $f_1(t)$ and $f_2(t)$ for the irreducible factors of $\Delta(t)$, then $f_1(t^2)$ factors as $-h_1(t)h_1(-t)$ for $h_1(t) = (t^3 - t^2 - t + 2)$, and $f_2(t^2)$ factors as $-h_2(t)h_2(-t)$ for $h_2(t) = 2t^3 - t^2 - t + 1$. Then $\Delta(t^2) = f_1(t^2)f_2(t^2) = h_1(t)h_2(t)h_1(-t)h_2(-t)$, and $\Delta(t)$ does satisfy Hartley's condition with $p=2$. Most of our results hinge on the following characterization of $n$-Hartley polynomials. By an irreducible polynomial in this section we shall mean a polynomial that is irreducible as an element of $\mathbf{Z}[t]$, so, in particular, irreducible polynomials are primitive. **Proposition 16**. *Let $\alpha$ be a root of an irreducible $\Delta(t) \in \mathbf{Z}[t]$. Then $\Delta(t)$ is $n$-Hartley if and only if $\alpha$ is an $n$th power in $\mathbf{Q}(\alpha)$.* Before proving Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"} we give a lemma that we will use implicitly throughout this section. **Lemma 17**. *Suppose that $\Delta(t)$ is irreducible, and that $\Delta(t^n)$ admits a factorization of the form $$\Delta(t^p) = \pm \prod_{i=0}^{n-1}g(\zeta_n^it).$$ Then $g(x)$ is irreducible.* *Proof.* Let $m$ be the degree of $\Delta(t)$ so that the degree of $\Delta(x^n)$ is $nm$. It follows that the degree of $g(t)$ is also $m$. The roots of $g(t)$ are $n$th roots of roots of $\Delta(t)$. Their degree over $\mathbf{Q}$ is therefore bounded below by $m$. Hence $g(t)$ is the minimal polynomial for some $n$th root of a root of $f(t)$. ◻ *Proof of Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"}.* Suppose that we have a factorization as in Definition [Definition 13](#defn:Hartley){reference-type="ref" reference="defn:Hartley"}. We claim that $g(x)$ has an $n$th root of a root of $\Delta(t)$ as a root. Let $\beta$ be a root of $g(t)$ so that $\Delta(\beta^n) = 0$. That is, $\beta^n = \alpha$ for some root $\alpha$ of $\Delta(t)$. The degree of $g(t)$ is that of $\Delta(t)$, and the field generated by $\beta$ contains $\mathbf{Q}(\alpha)$, so we conclude $\mathbf{Q}(\alpha) = \mathbf{Q}(\beta)$. As the various roots of $\Delta(t)$ are all either $n$th powers or all not $n$th powers, the result follows. Now suppose that $\alpha$ is a $n$th power in $\mathbf{Q}(\alpha)$. Then the minimal polynomial $g(t)$ of an $n$th root of $\alpha$, $\sqrt[n]{\alpha}$, has the same degree as $\Delta(t)$. The other $n$th roots of $\alpha$ are roots of $g(\zeta_n^it)$. Since the roots of $\Delta(t^n)$ contain all the $n$th roots of $\alpha$, we have $$\Delta(t^n) = C\prod_{i=0}^{n-1}g(\zeta_n^i t)$$ for some constant $C$. Since $\Delta(t^n)$ is primitive, we have $C = \pm 1$. ◻ We can now prove Proposition [Proposition 3](#prop:cycloHartley){reference-type="ref" reference="prop:cycloHartley"} as an easy corollary of Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"}. *Proof of Proposition [Proposition 3](#prop:cycloHartley){reference-type="ref" reference="prop:cycloHartley"}.* Primitive $m$th roots of unity are $n$th powers if and only if $n$ coprime to $m$, so $\Phi_m(x)$ is $n$-Hartley if and only if $\gcd(n,m) =1$. To handle the sign, we note that cyclotomic polynomials have positive constant term for $m > 1$ and apply Proposition [Proposition 14](#prop:nHartleyImpliesHartley){reference-type="ref" reference="prop:nHartleyImpliesHartley"}. ◻ From Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"} and the fact that the only algebraic numbers which are infinitely divisible in the multiplicative group are roots of unity, it is already clear that a given irreducible, noncyclotomic polynomial can only be $n$-Hartley for finitely many $n$. However, it is not obvious how to compute the set of such $n$, nor is it clear how to treat reducible polynomials. We now develop explicit and practical methods for computing this set. Then we will proceed to reducible polynomials. In view of Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"}, we focus on the case of $n$ a prime power. Our methods are inspired by those in [@SchinzelPolynomials §4.1], but we do not directly use any result from there. **Definition 18**. *Let $\alpha$ be an algebraic number. If $\alpha$ is a root of unity, define $E(\alpha) = 0$. Otherwise, define $$E(\alpha) = \mathrm{sup}\{E \> | \> \alpha = \theta^E \text{ for some } \theta \in \mathbf{Q}(\alpha)\}.$$ For a polynomial $\Delta(t) \in \mathbf{Q}[t]$ whose distinct roots $\alpha_i$ appear with multiplicity $m_i$, define $$E(\Delta(t)) = \mathrm{gcd}(m_iE(\alpha_i)).$$* From this definition it is clear that if $\alpha$ is an $n$th power then $n \mid E(\alpha)$ and that if $\sigma$ is a field automorphism of $\mathbf{Q}(\alpha)$ fixing $\mathbf{Q}$ then $E(\alpha) = E(\sigma(\alpha))$. Also note that $E(\Delta(t)) = 0$ if and only if $\Delta(t)$ is a product of cyclotomic polynomials. Our next proposition shows that $E(\Delta(t))$ is finite and that there is an effectively computable upper bound. The explicit formula will not be used in this section, but we do leverage it for the calculations in the next section. **Proposition 19**. *Let $\alpha$ be an algebraic number. If $\alpha$ is not an algebraic unit, then $E(\alpha)$ divides the gcd of the maximal exponents $i$ for which $\alpha \in \mathfrak{l}^i$ for prime ideals $\mathfrak{l}$ of $\mathbf{Q}(\alpha)$. That is, $$E(\alpha) \mid \gcd\limits_{\mathfrak{l}}(\left| v_{\mathfrak{l}}(\alpha) \right|\}).$$ If $\alpha$ is an algebraic unit that is not a root of unity and is written as $\alpha = \zeta_q^{a_0}\epsilon_1^{a_i}\cdots\epsilon_m^{a_m}$ for $\zeta_q$ a generator for the torsion part of $\mathcal{O}_K^{\times}$, $\epsilon_i$ fundamental units, and $a= \mathrm{gcd}(a_1, \dots, a_m)$ then $$E(\alpha) = \prod_{\substack{p \textup{ prime } \\ p \mid a}} \begin{cases} p^{v_p(a)} & \mathrm{gcd}(p,q) = 1 \\ \mathrm{gcd}(q, a_0, p^{v_p(a)}) & \mathrm{gcd}(p,q) \neq 1, \end{cases}$$ where $v_p(a)$ is the $p$-adic valuation of $a$.* *Proof.* If $\alpha$ is not a unit in the ring of integers, and is a $p^r$th power then valuation of $\alpha$ with respect to any finite prime $\mathfrak{l}$ is divisible by $p^r$, and the result follows. Now let $\alpha$ be a unit. First note that if $\alpha$ is a $p^r$th power, then the exponent appearing for each fundamental unit $\epsilon_i$ is divisible by $p^r$. That is $a$ is divisible by $p^t$. The converse holds if $\mathrm{gcd}(p,q) = 1$ because $\zeta_q^{a_0}$ is $p^r$th power for every $p$ coprime to $q$ and every $t$. Finally suppose that $p$ and $q$ are not coprime. By the Chinese remainder theorem, we may assume without loss of generality $q$ is a power of $p$. Then $\zeta_{q}^{a_0}$ will be a $p^r$th power if and only if $a_0 \in p^r \mathbf{Z}/ q\mathbf{Z}$. This value $r$ is the smaller of $p$-adic valuations of $a_0$ and $q$. Since $\gcd(q,a_0, p^{v_p(a)}) = p^{\min(v_p(q), v_p(a_0), v_p(a))}$, the result follows. ◻ **Remark 20**. When $\alpha$ is a unit but not a root of unity, we always have the condition $$E(\alpha) \mid \prod_{p \mid a}p^{v_p(a)},$$ and this also gives a practical upper bound. However, computing the system of fundamental units is much more expensive than computing the torsion part, so in implementing algorithms it makes sense to use the sharper result. We also remark that computing a system of fundamental units is a computable problem that has been implemented in modern software packages. See Section [3](#sec:computations){reference-type="ref" reference="sec:computations"}. For an irreducible Alexander polynomial $\Delta_K(t)$, combining Proposition [Proposition 19](#prop:SchinzelDegreeCalculation){reference-type="ref" reference="prop:SchinzelDegreeCalculation"} with Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"} allows us to restrict the order of possible free periodicities with a finite computation and without any extra information about the knot by computing $E(\Delta_K(t))$. The remainder of the section is devoted to removing the irreducibility hypothesis and justifying factoring the Alexander polynomial and applying Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"} to each irreducible power. Our goal is the following theorem. **Theorem 21**. *Let $n \in \mathbf{Z}_{\geq 1}$ and $\Delta(t) \in \mathbf{Z}[t]$ primitive and not a product of cyclotomic polynomials. Then $\Delta(t)$ is $n$-Hartley if and only if $n \mid E(\Delta(t))$.* We remark that the primitivity hypothesis is necessary but not especially interesting. The content of $\Delta(t)$ must be an $n$th power for $\Delta(t)$ to be $n$-Hartley; moreover, Alexander polynomials of knots are primitive. The cyclotomic factors can also be handled by Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"} and Proposition [Proposition 3](#prop:cycloHartley){reference-type="ref" reference="prop:cycloHartley"}. Our next step to proving Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"} is to handle powers of irreducible polynomials. **Lemma 22**. *Suppose that $\Delta(t) \in \mathbf{Z}[t]$ is irreducible and that $\Delta(t)^m$ is $n$-Hartley for $m$ coprime to $n$. Then $\Delta(t)$ is $n$-Hartley.* *Proof.* It suffices to show the result for each prime power dividing $n$, so let $p$ be a prime dividing $n$ $r$ times, and let $q=p^r$. Write $$\label{eq:m_power_hartley} \Delta(t^q)^m = \pm \prod_{i=0}^{q-1} g(\zeta_q^i t).$$ Write $\alpha_1, \dots, \alpha_s$ for the $s$ distinct roots of $\Delta(t)$. Fixing a $p$th root $\beta_j$ of $\alpha_j$, the $p$th roots of $\Delta(t^p)$ are $\{\zeta_q^i\beta_j\}$. Note that these are all distinct as $\zeta_q^i\beta_j = \zeta_q^k \beta_l$ implies $\beta_j^p = \beta_l^p$ which implies $\alpha_j = \alpha_l$. Without loss of generality, let $\beta_1$ be a root of $g(t)$. We claim that $\zeta_q^i \beta_1$ is not a root of $g(t)$ for any $i \neq 0$. Let $h(t)$ be the irreducible factor over $\mathbf{Z}[t]$ of $g(t)$ having $\beta_1$ as a root. We first claim that no $\zeta_q^i\beta_1$ is a root of $h(t)$. If $K$ is the splitting field of $h(t)$, it follows that there is an automorphism $\varphi \in \mathop{\mathrm{Gal}}(K/\mathbf{Q})$ taking $\beta_1$ to $\zeta^i_q\beta_1$. Arranging $i$ to be minimal we obtain that $i$ is $p^u$ for some $0 \leq u < r$. By repeatedly applying this automorphism, we find that $\zeta_q^{vp^u} \beta_1$ is a root of $h(t)$ for all $0 \leq v < p^{r-u}$. If $\beta_1, \dots, \beta_m$ are the roots of $h(t)$, we then have that $$\label{eq:h_expr} h(t) = \prod_{\substack{0 \leq v < p^{r-u} \\ 1 \leq j \leq m}}(x-\zeta_p^{vp^u}\beta_j).$$ From Equation [\[eq:h_expr\]](#eq:h_expr){reference-type="ref" reference="eq:h_expr"}, it is clear that $h(t) = h(\zeta_q^{vp^u} t)$ for all $0 \leq v < p^{r-u}$, so $$\label{eq:h_factored} \Delta(t^p)^m = \pm h(t)^{kp^{r-u}}\prod_{v=0}^{p^{r-u}-1} g_1(\zeta_q^{vp^u}t)\prod_{\substack{i \nmid p^u \\ 0 < i \leq q - 1}} g(\zeta_q^it).$$ where $g_1(x)$ is the quotient of $g(t)$ by $h(t)^k$, and $k$ is maximal. The maximality of $k$ implies that $h(t)$ and $g_1(t)$ are coprime. Note that $\beta_1$ cannot be a root of any $g_1(\zeta_q^it)$, for if it were we would have that $\zeta_q^i \beta_1$ is a root of $g_1(t)$, but $\zeta_q^i \beta_1$ is a root of $h(t)$, and $g_1(t)$ and $h(t)$ are coprime. Furthermore, the minimality hypothesis on $u$ implies that $\beta_1$ is not a root of any polynomial in the right product of Equation [\[eq:h_factored\]](#eq:h_factored){reference-type="ref" reference="eq:h_factored"}. Then Equation [\[eq:h_factored\]](#eq:h_factored){reference-type="ref" reference="eq:h_factored"} implies that the multiplicity of a root of $h(t)$ is divisible by $p$ when the multiplicity is exactly $m$. Hence $\zeta_q^i\beta_1$ is not a root of $h$. It remains to show that $\zeta_q^i\beta_1$ cannot be a root of some other irreducible factor of $g(t)$. Supposing that $h_1(t)$ is an irreducible factor of $g(t)$ having $\zeta_q^i\beta_1$ as a root, we may write $g(t) = h(t)h_1(t)g_2(t)$ or $g_1(t) = h_1(t)g_2(t)$ where $g_1(t) \in \mathbf{Z}[t]$ is as in Equation [\[eq:h_factored\]](#eq:h_factored){reference-type="ref" reference="eq:h_factored"}. Note that $\zeta_q^i \beta_j$ is a root of $h_1(t)$ for all $j$, so $\beta_j$ is a root of $h_1(\zeta_q^i t)$ is a root of $h_1(t)$ for all $j$. If some other $\zeta_q^l \beta_j$ is a root of $h_1(\zeta_q^i t)$ then $\zeta_q^{i+l} \beta_j$ is a root of $h(t)$, which we previously ruled out. We've shown that the roots of $h_1(\zeta_q^i t)$ are exactly $\beta_1, \dots, \beta_j$ which in turn are the roots of $h(t)$, so $h_1(\zeta_q^i t)$ and $h(t)$ differ by a constant. As $h(0), h_1(0) \in \mathbf{Z}$, the constant must be an integer. This in turn implies that $h_1(\zeta_q^i t) \in \mathbf{Z}[t]$ which is only possible if $h_1(\zeta_q^i t) = \pm h_1(t)$ (in fact $h_1(\zeta_q^i(t)) = h_1(t)$ if $q$ is odd). This is a contradiction since $h(t)$ and $h_1(t)$ were coprime. We may now assume that the roots of $g(\zeta_q^it)$ are exactly $\{\zeta_q^{-i}\beta_1,\dots,\zeta_q^{-i}\beta_n\}$ and that these sets partition the roots of $\Delta(t^q)$. Restricting attention to $g(t)$, we see that its roots are $\{\beta_1, \dots, \beta_n\}$. Since all of these are roots of $\Delta(t^q)$ with multiplicity exactly $m$, we find that $g(t) = w(t)^m$ for some polynomial $w(t) \in \mathbf{Z}[t]$. It follows that $$\Delta(t^q)^m = \pm \prod_{i=0}^{q-1}w(\zeta_q^it)^m,$$ and hence $$\Delta(t^q) = \xi_{2m} \prod_{i=0}^{q-1}w(\zeta_q^it)$$ for some not necessarily primitive $2m$th root of unity $\xi_{2m}$. However, $\xi_{2m}$ must be in $\mathbf{Z}$, so $\xi_{2m} = \pm 1$. ◻ We can now prove Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"}. *Proof of Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"}.* Write $\Delta(x) = f_1(x)^{mp^k} f_2(x)$ for $f_1$ irreducible, $f_1(t)$ and $f_2(t)$ coprime and $m$ and $p$ also coprime. By Lemma [Lemma 22](#lem:hartley_power){reference-type="ref" reference="lem:hartley_power"}, it suffices to show that $f_1(t)^m$ is $p^s$-Hartley. Let $q=p^s$, and let $g(t)$ be as in Equation [\[eq:m_power_hartley\]](#eq:m_power_hartley){reference-type="ref" reference="eq:m_power_hartley"}. As before write $\{\beta_1, \dots, \beta_n\}$ for the $q$th roots of roots of $f_1(t)$, so that $\{\zeta_q^i\beta_j\}_{1 \leq j \leq n}^{0 \leq i < q}$ form a complete set of roots of $f_1(t^q)$. The same argument as in Lemma [Lemma 22](#lem:hartley_power){reference-type="ref" reference="lem:hartley_power"} shows that $\{\beta_1, \dots, \beta_n\}$ are roots of $g(t)$ (after possibly relabeling). Note that $g(t)$ may have other roots besides these. It follows that the common roots of $f_1(t^q)$ and $g(\zeta_q^it)$ are pairwise disjoint for varying $i$ and that the Galois orbit (with group the Galois group of the splitting field of $g$) of $\beta_1$ is $\{\beta_1, \dots, \beta_n\}$. Since this is true for the Galois orbit of each common root of $g(t)$ and $f_1(t^q)$, we see that $\{\beta_1, \dots, \beta_n\}$ is Galois stable. Hence there is an integer polynomial $w(t)$ dividing $g(t)$ having $\{\beta_1, \dots, \beta_n\}$ as roots. Similarly $w(\zeta_p^it)$ divides $g(\zeta_p^it)$ and has $\{\zeta_p^{-i}\beta_1, \dots \zeta_p^{-i}\beta_n\}$ as roots. It follows that $$f_1(t^q) = C\prod_{i=0}^{q-1} w(\zeta_p^i t)$$ for some $C \in \mathbf{Z}$. Since $\Delta(t)$ is primitive, we must have $C = \pm 1$. ◻ **Lemma 23**. *Let $p$ be a prime and $\Delta(t)$ an irreducible $p^r$-Hartley polynomial. Then $\Delta(t)^{p^s}$ is $p^{r+s}$-Hartley.* *Proof.* We have $$\Delta(t^{p^r}) = \pm \prod_{i=0}^{p^r-1}g(\zeta_{p^r}^i t).$$ The role of $g(t)$ in the definition of $p^{r+s}$-Hartley for $\Delta(t)^{p^s}$ will be played by $h(t) = g(t^{p^s})$. We compute $$\begin{aligned} \prod_{i=0}^{p^{r+s} - 1} h\left(\zeta_{p^{r+s}}^i t\right) &= \prod_{i=0}^{p^{r+s} - 1} g\left( \left( \zeta_{p^{r+s}}^i t \right)^{p^s} \right) \\ &= \prod_{i=0}^{p^{r+s} - 1} g\left( \zeta_{p^r}^i t^{p^s} \right) \\ &= \left(\prod_{i=0}^{p^r - 1} g\left( \zeta_{p^r}^i t^{p^s} \right) \right)^{p^s} \\ &= \left(\pm \Delta\left(t^{{(p^r)}^{(p^s)}}\right)\right)^{p^s} = \pm \Delta(t^{p^{r+s}})^{p^s}. \end{aligned}$$ ◻ *Proof of Theorem [Theorem 21](#thm:pDividesSchinzelDegree){reference-type="ref" reference="thm:pDividesSchinzelDegree"}.* Let $\Delta(t) = f_1(t)^{a_1} \cdots f_n(t)^{a_n}$ be the factorization of $\Delta(t)$ into powers of irreducible polynomials with $f_i$ pairwise coprime. Suppose that $\Delta(t)$ is $p^r$-Harltey. Let $\alpha_i$ be a root of $f_i$. It suffices to show that $p^r$ divides $a_i E(\alpha_i)$ for all $i$. By Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"}, we have that each $f_i$ is $p^s$-Hartley with $a_i = mp^k$ with $m$ coprime to $p$ and $s+k = r$, so by Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"}, $\alpha_i$ is a $p^s$th power. Hence $p^s \mid E(\alpha_i)$, and $p^r \mid a_i E(\alpha_i)$. Conversely if $p^r \mid a_i E(\alpha_i)$ for all $i$, then each $f_i(t)$ has a root that is a $p^{s}$th power and appears with multiplicity $p^k$ so that $s+k = r$. By Proposition [Proposition 16](#prop:pthPowerIffpHartley){reference-type="ref" reference="prop:pthPowerIffpHartley"}, each $f_i(t)$ is $p^s$-Hartley, and by Lemma [Lemma 23](#lem:powerUpHartley){reference-type="ref" reference="lem:powerUpHartley"} each $f_i(t)^{a_i}$ is $p^r$-Hartley. By combining the factorizations of each $f_i(t)^{a_i}$, it follows that $\Delta(t)$ is $p^r$-Hartley. ◻ We conclude with a proof of Theorem [Theorem 4](#thm:main){reference-type="ref" reference="thm:main"}. *Proof of Theorem [Theorem 4](#thm:main){reference-type="ref" reference="thm:main"}.* Since $\Delta(t)$ is not a product of cyclotomic polynomials, it has an irreducible factor $f(t)$ which is not cyclotomic. Then by Proposition [Proposition 19](#prop:SchinzelDegreeCalculation){reference-type="ref" reference="prop:SchinzelDegreeCalculation"} and Theorem [Theorem 21](#thm:pDividesSchinzelDegree){reference-type="ref" reference="thm:pDividesSchinzelDegree"}, $f(t)$ is $n$-Hartley for only finitely many $n$. Then by Theorem [Theorem 6](#thm:HartleyFactorization){reference-type="ref" reference="thm:HartleyFactorization"}, $\Delta(t)$ is $n$-Hartley for finitely many $n$. ◻ # Applications and examples {#sec:computations} We include a code snippet for computing the quantity $E(\Delta(t))$ from the previous section. ``` {.python} ``` We used SageMath [@sagemath] (the `pari` functions are wrappers around PARI [@PARI2]), and the above code to show that with the exception of two polynomials which are handled in Example [Example 27](#exmp:deg30){reference-type="ref" reference="exmp:deg30"}, every polynomial satisfying the conditions from [@MR2168576 Corollary 1.3] and [@MR3782416 Corollary 9] with degree at most 32 does not factor as in Theorem [Theorem 2](#thm:Hartley){reference-type="ref" reference="thm:Hartley"} or Theorem [Theorem 1](#thm:Murasugi){reference-type="ref" reference="thm:Murasugi"}. This proves the following theorem. **Theorem 24**. *Conjecture [Conjecture 8](#conj:polynomial){reference-type="ref" reference="conj:polynomial"} is true for all knots $K$ with genus $g(K) < 13$. If Conjecture [Conjecture 9](#conj:quotient){reference-type="ref" reference="conj:quotient"} is true, then Conjecture [Conjecture 8](#conj:polynomial){reference-type="ref" reference="conj:polynomial"} is true for all knots $K$ with $g(K) < 16$.* In order to handle the two exceptional polynomials, we use the following results. **Proposition 25**. *Let $K$ be an $n$-periodic knot with quotient knot $\overline{K}$ and suppose that $\pi_1(S^3_{p/q}(\overline{K}))$ is left-orderable. Then $\pi_1(S^3_{np/q}(K))$ is also left-orderable.* *Proof.* Let $K$ be an $n$-periodic with quotient knot $\overline{K}$, and suppose that $\pi_1(S^3_{p/q}(\overline{K}))$ is left-orderable. The periodic symmetry on $K$ extends to $S^3_{np/q}(K)$, and the quotient manifold is $S^3_{p/q}(\overline{K})$ (see for example [@MR1755823 Section 2.1]), so that there is a non-trivial homomorphism $\pi_1(S^3_{np/q}(K)) \to \pi_1(S^3_{p/q}(\overline{K}))$. Then by [@MR2141698 Theorem 3.2], $\pi_1(S^3_{np/q}(K))$ is left-orderable since it has a non-trivial homomorphism into a left-orderable group. ◻ **Corollary 26**. *The L-space conjecture implies Conjecture [Conjecture 9](#conj:quotient){reference-type="ref" reference="conj:quotient"}.* *Proof.* Let $K$ be an $n$-periodic L-space knot, and suppose that the quotient knot $\overline{K}$ is not an L-space knot. Then there is a surgery $S^3_{p/q}(K)$ which is an L-space, and the corresponding surgery $S^3_{p/(nq)}(\overline{K})$ is not an L-space. By the L-space conjecture [@MR3072799 Conjecture 3], $\pi_1(S^3_{p/q}(K))$ is not left-orderable, while $\pi_1(S^3_{p/(nq)}(\overline{K})$ is left orderable, which contradicts Proposition [Proposition 25](#prop:periodic-orderable){reference-type="ref" reference="prop:periodic-orderable"}. ◻ **Example 27**. Consider the polynomial $$\Delta(t) = t^{30} - t^{29} + t^{28} - t^{26} + t^{25} - t^{24} + t^{18} - t^{17} + t^{15} - t^{13} + t^{12} - t^6 + t^5 - t^4 + t^2 - t + 1,$$ which is not a product of cyclotomic polynomials, but appears to be a possible Alexander polynomial of an L-space knot. We do not know if $\Delta(t)$ is the Alexander polynomial of an L-space knot, but it does satisfy Murasugi's condition with $n = 2$ and $$\Delta_{\overline{K}}(t) = t^{14} + t^{12} - t^8 - t^7 - t^6 + t^2 + 1.$$ Note that this $\Delta_{\overline{K}}(t)$ cannot be the Alexander polynomial of an L-space knot since its coefficients are not alternating [@MR2168576 Corollary 1.3]. In particular, if $\Delta(t)$ is the Alexander polynomial of a periodic L-space knot which has a quotient with Alexander polynomial $\Delta_{\overline{K}}(t)$, then Conjectures [Conjecture 7](#conj:iteratedtorus){reference-type="ref" reference="conj:iteratedtorus"}, [Conjecture 8](#conj:polynomial){reference-type="ref" reference="conj:polynomial"}, and [Conjecture 9](#conj:quotient){reference-type="ref" reference="conj:quotient"} (and hence the L-space conjecture) are all false. The polynomial $$\Delta(t) = t^{26} - t^{25} + t^{24} - t^{23} + t^{20} - t^{19} + t^{14} - t^{13} + t^{12} - t^{7} + t^{6} - t^{3} + t^{2} - t + 1$$ is similar. These are the only two such examples through degree $32$. **Example 28**. A standard example of a family of L-space knots is the pretzel knots $$P(-2,3,2q+1).$$ In [@MR891592 Section 2], it is shown that for $q > 2$, these knots have only a strong inversion and no other symmetries. In particular, this provides some evidence for Conjecture [Conjecture 7](#conj:iteratedtorus){reference-type="ref" reference="conj:iteratedtorus"}, since none of these knots are periodic or freely periodic. Note that $P(-2,3,3) = T(3,4)$ and $P(-2,3,5) = T(3,5)$ are torus knots; hence the restriction $q > 2$.

arxiv_math

--- abstract: | We prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains noncollapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such flow converges to free boundary embedded geodesic in infinite time, or shrinks to a round half-point on the boundary. As a consequence, we prove the existence of two free boundary embedded geodesics on a Riemannian $2$-disk with a strictly convex boundary. Moreover, we prove that there exists a simple closed geodesic with Morse Index $1$ and $2$. This settles the free boundary analog of Grayson's theorem. address: Department of Mathematics, Rutgers University - New Brunswick, Piscataway, NJ 08854 author: - Dongyeong Ko title: Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary --- # Introduction The search of closed geodesics on surfaces by min-max methods was initiated by Birkhoff [@B]. The following classical theorem on the existence of three embedded geodesics on spheres was initially proven by Lusternik-Schnirelmann [@LS], and the embeddedness part is proven by Grayson [@Gr] by curve shortening flow. **Theorem 1** (Lusternik-Schnirelmann [@LS], Grayson [@Gr]). *A closed Riemannian $2$-sphere $(S^{2},g)$ contains at least three simple closed geodesics.* In the free boundary setting, Lusternik-Schnirelmann [@LS] proved that a bounded domain in $\mathbb{R}^{n}$ with a convex and smooth boundary admits at least $n$ distinct geodesic chords which meet the boundary orthogonally. Bos [@Bo] extended the existence of orthogonal geodesic chords to Riemannian manifolds via Birkhoff's discrete curve shortening process. He also proved the necessity of the convexity assumption by giving an example of a non-convex domain in $\mathbb{R}^{2}$ (See Figure 1 in [@Bo]). The construction of free boundary (immersed) geodesics in various settings have been developed by Gluck-Ziller [@GZ] and Zhou [@Z] by the discrete curve shortening process. Li [@L] introduced the chord shortening flow and provided a simpler approach to prove Lusternik-Schnirelmann's existence result. Moreover, the recent work of Donato-Montezuma [@DM] constructed a free boundary embedded geodesic or a geodesic loop in the nonnegative sectional curvature scenario via Almgren-Pitts min-max theory (See also [@AMS]). However, to the author's knowledge, the existence of free boundary embedded geodesics on Riemannian $2$-disks with convex boundary is not known. The following is our first main result on this: **Theorem 2**. *There are at least $2$ free boundary embedded geodesics on a Riemannian $2$-disk $(D^{2},\partial D^{2},g)$ with a strictly convex boundary.* For $a>b>0$, since the ellipse $E := \{ (x,y) | x^{2}/a^{2} + y^{2}/b^{2}=1 \}$ achieves only two free boundary geodesics, the existence theorem of two free boundary embedded geodesics (Theorem 1.1) is optimal. A Riemannian metric $g$ is called *bumpy* if every closed geodesic is nondegenerate i.e. there is no free boundary geodesic that admits a non-trivial Jacobi field. The arguments in Abraham [@Ab] and Ambrozio-Carlotto-Sharp [@ACS] shows that the set of bumpy metrics is generic in $C^{r}$-sense for $r \ge 5$. We have **Corollary 3**. *There are at least two free boundary embedded geodesics with distinct length on the Riemannian $2$-disk endowed with a bumpy metric and strictly convex boundary.* We note here that we do not need to impose the condition ensures the nonexistence of interior simple closed geodesics because the mass cancellation such as neck-pinching over the limiting process does not happen in the curve setting. This is different from the min-max construction of free boundary minimal disks in Haslhofer-Ketover [@HK]. We compare our result with the problem on its higher dimensional analogue. There is a conjecture on the existence of three free boundary minimal disks on Riemannian $3$-balls with strictly convex boundary and nonnegative Ricci curvature. Struwe [@St] constructed one free boundary immersed minimal disk via mapping approach. Grüter and Jost [@GJ] constructed a free boundary minimal surface by developing Simon-Smith min-max theory in the free boundary setting (See also Jost [@J]). We refer to [@DR], [@F], [@LP], [@Li2],[@LiZ], [@LSZ] for seminal works on the construction of free boundary minimal disks. Recently, Haslhofer-Ketover [@HK] proved this conjecture for generic metric and obtained at least $2$ free boundary minimal disks by combining the ideas from min-max theory, free boundary mean curvature flow and degree theory. To obtain the existence of a free boundary embedded geodesic, we apply the free boundary curve shortening flow for the tightening procedure. The study of free boundary curve shortening flow (FBCSF), the Neumann boundary problem of curve shortening flow, dates back to the work of Huisken [@H1] on free boundary mean curvature flow on the graph setting in general dimensions. After this work, free boundary mean curvature flow is extensively studied in [@AW], [@St1] and [@St2]. One of the most recent development of free boundary curve shortening flow is the full determination of the long time behavior of the free boundary curve shortening flow on convex domain in $\mathbb{R}^{2}$. We generalize the result of Langford-Zhu to the surfaces with strictly convex boundary. Huisken [@H2] found a simpler proof of the long-time behavior of curve shortening flows for closed and embedded curves by developing the idea of distance comparison principle. The essence of their idea is to prove that the ratio between extrinsic and intrinsic distance has monotonicity so that does not collapse along the flow (See also [@AB]). Edelen [@E] extended Huisken's work to the surface setting. He proved that the ratio along the flow decreases at worst exponentially while the ratio may not have the monotonicity. Langford-Zhu [@LZ] developed the distance comparison principle in the free boundary setting by introducing the notion of reflected chord-arc profile. Our generalization of the idea of reflected chord-arc profile to the surface setting proves that the chord-arc profile achieves the lower bound weighted by the exponential terms by time over the lower bound in the Euclidean setting, where our approach relies on Edelen's work. The noncollapsing property gives the following: **Theorem 4**. *Let $(N,\partial N, g)$ be a closed Riemannian surface with strictly convex boundary and $\{ \Gamma_{t} \}_{t \in [0,T)}$ be a maximal free boundary curve shortening flow starting from a properly embedded closed interval $\Gamma_{0}$ in $N$. Then either:* 1. *$T = \infty$, in which case $\Gamma_{t}$ converges smoothly as $t \rightarrow \infty$ to an embedded geodesic in $N$ which meets $\partial N$ orthogonally; or* 2. *$T< \infty$, in which case $\Gamma_{t}$ converges uniformly to some single round half-point $z \in \partial N$ smoothly in the sense of the blow up limit of the curve converges to the unit semi-circle.* Marques-Neves ([@MN1], [@MN2]) proved the upper and lower bound of Morse Index of min-max minimal hypersurfaces on closed manifolds with dimension $3 \le n+1 \le 7$ by proving deformation theorems. In the smooth setting of curves, we are forced to construct the interpolation deformation between two families of simple closed curves with controlled length, where the deformation arises from the curve shortening flow. The author [@K1] proved the Morse Index bound of the simple closed geodesics on Riemannian $2$-spheres (see also [@DMMS]). Moreover, the author [@K2] proved the Morse Index bound of min-max capillary embedded geodesics on certain Riemannian $2$-disks by constructing the interpolation based on the curve shortening flow with a strictly fixed boundary. **Theorem 5**. *Suppose $(D^{2},\partial D^{2}, g)$ is a Riemannian $2$-disk with convex boundary endowed with a bumpy metric. Then for each $k=1,2$, there exists a free boundary embedded geodesic $\gamma_{k}$ with $$index(\gamma_{k}) = k$$ and these two geodesics satisfy $|\gamma_{1}| <|\gamma_{2}|$.* **Corollary 6**. *For a $2$-Riemannian disk $(D^{2},\partial D^{2},g)$ with a convex boundary, for $k=1,2$, there exists a free boundary embedded geodesic $\gamma_{k}$ with $$index(\gamma_{k}) \le k \le index(\gamma_{k})+ nullity (\gamma_{k}).$$* By the work of Smale [@Sm] on the diffeomorphisms of $2$-sphere, the space of embedded intervals on $D^{2}$ after identifying the point curves retract onto $\mathbb{R}P^{2}$. Lusternik-Schnirelmann arguments, for instance in [@CLOT], suggests that the number of critical points of a smooth real-valued function has the lower bound of one plus the maximal cup-length. In our setting, the cohomology ring of $\mathbb{R}P^{2}$ has a maximal length $2$, there are at least $3$ critical points. The point curves can be regarded as a stable critical point and we expect there are at least $2$ free boundary embedded geodesics. We prove the existence of two free boundary embedded geodesics by considering one and two parameter min-max construction. The free boundary curve shortening flow provides the way to tight the minimizing sequences and the classical Lusternik-Schnirelmann arguments gives the existence. Our interpolation arguments rely on the quantitative estimate of $F$-distance bound in [@K1] along the squeezing homotopy arising from the free boundary curve shortening flow. The main modification we needed to make was the construction of free boundary mean convex foliation when the geodesic is strictly stable. We obtain a free boundary mean convex foliation by the first eigenfunction of the stability operator with Robin boundary condition (See also Proposition 2.4 in [@HK]). The organization of the paper is as follows. In Section 2, we collect the basic notions on free boundary curve shortening flow, the reflected chord-arc profile and the second variation of a free boundary embedded geodesic. In Section 3, we discuss the variation of the chord-arc profile on surfaces. In Section 4, we prove the noncollapsedness of the free boundary curve shortening flow at finite times. In Section 5, we consider the min-max construction of free boundary embedded geodesics. In Section 6, we prove the Morse Index bound of free boundary embedded geodesics. # Acknowledgments {#acknowledgments .unnumbered} The author wishes to thank his advisor Daniel Ketover for his constant support and valuable discussions. The author also thanks to Otis Chodosh for helpful explanations relating to this work. The author also thanks to Jonathan Zhu for valuable discussion relating to this work. The author was partially supported by NSF grant DMS-1906385. # Preliminaries ## Free boundary curve shortening flow We consider an oriented Riemannian surface with boundary $(N^{2},\partial N^{2},g)$ with a convex $C^{2}$ boundary and a properly immersed family of curves with boundary $\gamma: M^{1} \times I \rightarrow N^{2}$. We denote $J$ by the counterclockwise rotation by $\pi/2$ and take the convention that the orientation of the unit normal vector field on $\nu$ is given to satisfy $\gamma'/|\gamma'|=J\nu$. We call a properly immersed family of curves with boundary $\{ \Gamma_{t} \}$ satisfies a *free boundary curve shortening flow* if $\gamma: M^{1} \times I \rightarrow N^{2}$ with $\Gamma_{t} = \gamma(M,t)$ satisfies $$\begin{aligned} \begin{cases} \partial_{t} \gamma = \kappa \nu \text{ in } \mathring{M} \times I \\ \langle \nu, \mathcal{N}^{\partial N} \rangle = 0 \text{ on } \partial M \times I, \end{cases}\end{aligned}$$ where $\kappa(\cdot,t)$ is a geodesic curvature of $\Gamma_{t}$ with respect to the unit normal vector field $\nu$, and $\mathcal{N}^{\partial N}$ is the inward unit normal vector field on $\partial N$. We define $\kappa^{\partial N}$ to be a geodesic curvature of boundary $\partial N$. We consider the setting that $\gamma(\cdot,t)$ are embeddings and have two boundary points on $\partial N$. Since $\partial N$ is convex, there is no loss of generality to assume that $\Gamma_{t}$ does not have touching points in the interior of the curve on the boundary by the maximum principle. We recall some notions related to *completed chord-arc profile* in [@LZ]. For a given (unit length) parametrization of curve $\gamma: M \rightarrow N$ and $x,y \in \gamma$, denote $d(x,y)$ and $l(x,y)$ to be a distance in $N$ and the arclength between $x$ and $y$, respectively. We define the *reflected distance* $\tilde{d}(x,y)$ and *reflected arclength* $\tilde{l}(x,y)$ between two points in $x, y \in M$ by $$\tilde{d}(x,y) = \min_{z \in \partial N} (d(x,z)+d(y,z))$$ and $$\tilde{l}(x,y) = \min_{s \in \partial \gamma} (l(x,s)+l(y,s)).$$ The *reflected chord-arc profile* $\tilde{\psi}_{\gamma}$ of $\gamma$ is denoted by $$\tilde{\psi}_{\gamma}(\delta) = \inf \{ \tilde{d}(x,y) : x,y \in \gamma, \tilde{l}(x,y) = \delta \},$$ and the extended chord-arc profile $\bm{\psi}_{\gamma}$ is defined by $$\bm{\psi}_{\gamma}(\delta) = \min \{ \psi_{\gamma}(\delta),\tilde{\psi}_{\gamma}(\delta) \}.$$ We consider a connected, properly immersed curve-with-boundary $\gamma$ in $N$ whose endpoints are on $\partial N$. Also in case of we need the doubling of $N$, then we denote the doubling of $N$ by $\tilde{N}$. We denote the formal double $\bm{M} = (M \sqcup M)/ \partial M$ and write $\bm{x} = (x,sign(\bm{x})) \in \bm{M}$ where $sign(\bm{x})$ distinguishes to which copy of $M$ it belongs. We also denote continuous curve $\bm{\gamma} : \bm{M} \rightarrow N$ by $\bm{\gamma}(\bm{x}) = \gamma(x)$. Then we define the *completed arclength* by $$\bm{l}(\bm{x},\bm{y}) = \begin{cases} l(\gamma(x),\gamma(y)), \text{ if } sign(\bm{x}) = sign(\bm{y}) \\ \tilde{l}(\gamma(x),\gamma(y)), \text{ if } sign(\bm{x}) \neq sign(\bm{y}). \end{cases}$$ Also we define the *completed distance* function $\bm{d}(\bm{x},\bm{y})$ on $\bm{M} \times \bm{M}$ by $$\bm{d}(\bm{x},\bm{y}) = \begin{cases} d(\gamma(x),\gamma(y)), \text{ if } sign(\bm{x}) = sign(\bm{y}) \\ \tilde{d}(\gamma(x),\gamma(y)), \text{ if } sign(\bm{x}) \neq sign(\bm{y}). \end{cases}$$ Then we now define the *completed chord-arc profile* $\bm{\psi}$ of $\Gamma$ by $$\bm{\psi}(\delta) = \inf \{ \bm{d}(\bm{x},\bm{y}) : \bm{x},\bm{y} \in \bm{M}, \bm{l}(\bm{x},\bm{y}) = \delta \}.$$ Also we denote $\bm{\psi}(\delta,t)$ by the completed chord-arc profile of $\Gamma_{t} = \gamma(M,t)$. We call the completed chord-arc profile $\bm{\phi}$ as *a classical profile* if $sign(\bm{x}) = sign(\bm{y})$ and *a reflected profile* otherwise. We define $L = |\gamma|$ and $\bm{L} = 2|\gamma|$, and we denote the length functions depending on time $t$ by $L(t) = |\Gamma_{t}|$ and $\bm{L}(t) = 2|\Gamma_{t}|$. We also follow the setting of an auxiliary function introduced in [@LZ]. First, we will obtain the control of the chord-arc profile by a $C^{2}$-function $\varphi \in C^{2}([0,1])$ satisfying the following: 1. $\varphi(1-\eta) = \varphi(\eta)$ for all $\eta \in [0,1]$. 2. $0<\varphi'<1$. 3. $\varphi$ is strictly concave. Note that $\varphi(\bm{l}/\bm{L})$ is smooth away from the diagonal $\bm{D}$ in $\bm{M} \times \bm{M}$ and well-defined since $0<\bm{l}(\bm{x},\bm{y})/\bm{L} \le 1/2$ for $\bm{x}, \bm{y} \in \bm{M}$. In case of considering a time dependent auxiliary function, we consider the following function; We define a $C^{2}$-function $\varphi: [0,1] \times [0,T) \rightarrow \mathbb{R}$ satisfying the following conditions for every time $t \in [0, T)$. 1. $\varphi(1-\eta,t) = \varphi(\eta,t)$ for all $\eta \in [0,1]$. 2. $|\partial_{\eta}\varphi(\eta,t)|<1$ for $\eta \in [0,1]$ and $t \in [0,T)$. 3. $\varphi(\cdot,t)$ is strictly concave. We consider the auxiliary functions $Z$ and $\tilde{Z}$ on $M \times M$ given by $$\begin{aligned} Z(x,y) = d(\gamma(x),\gamma(y)) - \bm{L} \varphi \Big( \frac{l(\gamma(x),\gamma(y))}{\bm{L}} \Big), \\ \tilde{Z}(x,y) = \tilde{d}(\gamma(x),\gamma(y)) - \bm{L} \varphi \Big( \frac{\tilde{l}(\gamma(x),\gamma(y))}{\bm{L}} \Big).\end{aligned}$$ If we consider the auxiliary function on $M \times M \times \partial N$ given by $$\overline{Z}(x,y,z) = d(\gamma(x),z) + d(\gamma(y),z) - \bm{L} \varphi \Big( \frac{\tilde{l}(\gamma(x),\gamma(y))}{\bm{L}} \Big)$$ then $\tilde{Z}(x,y) = \min_{z \in \partial N} \overline{Z}(x,y,z)$. We denote our completed auxiliary function $\bm{Z}$ on $\bm{M} \times \bm{M}$ by $$\bm{Z}(\bm{x},\bm{y}) = \bm{d}(\bm{x},\bm{y}) - \bm{L} \varphi \Big( \frac{\bm{l}(\bm{x},\bm{y})}{\bm{L}} \Big) = \begin{cases} Z(\bm{x},\bm{y}) \text{ if } sign(\bm{x}) = sign(\bm{y}) \\ \tilde{Z}(\bm{x},\bm{y}) \text{ if } sign(\bm{x}) \neq sign(\bm{y}). \end{cases}$$ We also denote $d(\cdot,\cdot,t)$, $\tilde{d}(\cdot,\cdot,t)$, $\bm{d}(\cdot,\cdot,t)$ and $l(\cdot,\cdot,t)$, $\tilde{l}(\cdot,\cdot,t)$, $\bm{l}(\cdot,\cdot,t)$ by the distances, lengths of $\Gamma_{t}$. We consider the auxiliary functions at time $t$ as $$\begin{aligned} Z(x,y,t) = d(\gamma(x),\gamma(y),t) - \bm{L}(t) \varphi \Big( \frac{l(\gamma(x),\gamma(y),t)}{\bm{L}} \Big), \\ \tilde{Z}(x,y,t) = \tilde{d}(\gamma(x),\gamma(y),t) - \bm{L}(t) \varphi \Big( \frac{\tilde{l}(\gamma(x),\gamma(y),t)}{\bm{L}} \Big).\end{aligned}$$ Then the auxiliary function $\overline{Z}$ at time $t$ is defined by $$\overline{Z}(x,y,z,t) = d(\gamma(x),z,t) + d(\gamma(y),z,t) - \bm{L} \varphi \Big( \frac{\tilde{l}(\gamma(x),\gamma(y),t)}{\bm{L}} \Big).$$ Then the completed auxiliary function $\bm{Z}$ at time $t$ is $$\bm{Z}(\bm{x},\bm{y},t) = \bm{d}(\bm{x},\bm{y},t) - \bm{L} \varphi \Big( \frac{\bm{l}(\bm{x},\bm{y},t)}{\bm{L}} \Big) = \begin{cases} Z(x,y,t) \text{ if } sign(\bm{x}) = sign(\bm{y}) \\ \tilde{Z}(x,y,t) \text{ if } sign(\bm{x}) \neq sign(\bm{y}). \end{cases}$$ ## Second Variation, Morse Index and the first eigenfunction We consider a free boundary embedded geodesic $\gamma$ on a Riemannian $2$-disk $(D^{2},\partial D^{2},g)$. Let us denote $f$ by a smooth section of the normal bundle of $\gamma$. Then the second variation of the length of $\gamma$ is defined as $$\begin{aligned} \nonumber Q(f,f) &:= \int_{\gamma} (|\nabla_{\gamma}f|^{2} - K f^{2}) ds - (\kappa(p_{1})f^{2}(p_{1}) + \kappa(p_{2})f^{2}(p_{2})) \\ &= - \int_{\gamma} (f Lf) ds + f(p_{1})(-\nabla_{\gamma}f(p_{1})-\kappa(p_{1})f(p_{1})) + f(p_{2})(\nabla_{\gamma}f(p_{2})-\kappa(p_{2})f(p_{2})).\end{aligned}$$ where $K$ is a Gaussian curvature on $D^{2}$, $\kappa$ is a geodesic curvature of $\partial D^{2}$ and $L = \Delta_{\gamma} + K$ is the Jacobi operator of $\gamma$. The boundary condition of the Jacobi operator is $$\nabla_{\gamma}f(p_{i})-\kappa(p_{i})f(p_{i})=0$$ for $i=1,2$. Let us consider the increasing sequence of eigenvalues $\{ \lambda_{i} \}$ and associated eigenfunctions $\{ \phi_{i} \}$ of the following equation with Robin boundary condition which correspond to the eigenvalues of the stability operator $Q$: $$\begin{cases} L \phi_{i} + \lambda_{i} \phi_{i} = 0 \text{ on }\gamma\\ \frac{\partial \phi_{i}}{\partial \eta} -\kappa \phi_{i} = 0 \text{ on } \partial \gamma, \end{cases}$$ where $\eta$ is an outward unit vector on $\gamma$. We define Morse index of $\gamma$ to be a maximal dimension of a subspace of a space of $C^{\infty}(\gamma)$ where $Q$ is negative definite. i.e. the number of negative eigenvalues of the stability operator $Q$. Note that the first eigenfunction $\phi_{1}$ of (4) can be chosen to be strictly positive by the standard elliptic theory. # The spatial variation of the chord-arc profile on surfaces In this section, we calculate the variation of the chord-arc profile on surfaces, in particular, we find inequalities which holds if the auxiliary function $\bm{Z}$ achieves a zero minimizer at some pair of points $(\bm{x}_{0},\bm{y}_{0}) \in (\bm{M} \times \bm{M}) \setminus \bm{D}$. This is a generalization of Section 4 of [@LZ] to surfaces. We denote $\alpha_{(\bm{x},\bm{y})}$ by the (possibly broken) geodesic realizing $\bm{d}(\bm{x},\bm{y})$ parametrized (with a constant speed) by $\alpha_{(\bm{x},\bm{y})}(0) = \bm{\gamma}(\bm{x})$ and $\alpha_{(\bm{x},\bm{y})}(1) = \bm{\gamma}(\bm{y})$ between $\bm{\gamma}(\bm{x})$ and $\bm{\gamma}(\bm{y})$. We omit the subscription $(\bm{x},\bm{y})$ for simplicity and denote this curve by $\alpha$. Let us consider the doubled curve $\bm{\alpha}$ of $\alpha$ in $\tilde{N}$ if $sign(\bm{x}) \neq sign(\bm{y})$, namely the curve connecting $\bm{\gamma}(\bm{x})$ and $z$, and $z$ and $\bm{\gamma}(\bm{y})$ in $\tilde{N}$, and $\bm{\alpha} = \alpha$ otherwise. Also denote $\partial_{s}$ by the arclength parameter of $\bm{\gamma}$. Also denote $[\bm{\gamma}(\bm{x}):\bm{\gamma}(\bm{y})]$ as the shorter portion of $\bm{\gamma} \setminus \{ \bm{\gamma}(\bm{x}), \bm{\gamma}(\bm{y}) \}$. Before deducing the inequalities, we observe the nonexistence of the intersection point between $\bm{\alpha}$ and $[\bm{\gamma}(\bm{x}):\bm{\gamma}(\bm{y})]$ other than boundary points, which follows from the arguments in the proof of Theorem 4.1 in [@E]. Note that two curves $\bm{\alpha}$ and $[\bm{\gamma}(\bm{x}):\bm{\gamma}(\bm{y})]$ represent $\bm{d}(\bm{x},\bm{y})$ and $\bm{l}(\bm{x},\bm{y})$, respectively. We can directly apply the argument in the reflected profile cases. **Proposition 7**. *There is no interior intersection point between $[\bm{\gamma}(\bm{x}):\bm{\gamma}(\bm{y})]$ and $\bm{\alpha}$.* We denote the region bounded by $\bm{\alpha}$ and $[\bm{\gamma}(\bm{x}):\bm{\gamma}(\bm{y})]$ by $A_{(\bm{x},\bm{y})}$. By Proposition 3.1, $A_{(\bm{x},\bm{y})}$ is a topological disk in $\tilde{N}$. We first consider a classical profile case. We follow and modify the arguments in the proof of Theorem 4.1 of [@E] and Proposition 4.2 of [@LZ]. **Proposition 8**. *Suppose $0 = \min _{(x,y) \in M \times M \setminus D} Z(x,y) = Z(x_{0},y_{0})$. At $(x_{0},y_{0})$, we have $$0 \le -4 \frac{\varphi''}{\bm{L}}- \frac{\kappa(\gamma(x_{0}))}{d} \langle \alpha'(0),\nu(\gamma(x_{0})) \rangle +\frac{\kappa(\gamma(y_{0}))}{d} \langle \alpha'(1), \nu(\gamma(y_{0}))\rangle - ( 1- \varphi'^{2} ) \int_{\alpha} K.$$* **Proof.* Consider a variation $\alpha_{x,y}$ of $\alpha$ which are family of curves moved by $\partial_{x}$ and $\partial_{y}$ at endpoints $\gamma(x_{0})$ and $\gamma(y_{0})$ with corresponding vector fields $V$ and $W$, respectively, satisfies the following: $$\begin{aligned} V(0) = -\partial_{s}(\gamma(x_{0})) \text{, } V(1) = 0 \\ W(0) = 0 \text{, } W(1) = \partial_{s}(\gamma(y_{0})). \end{aligned}$$ Denote $\rho(\gamma(x),\gamma(y)) = |\alpha_{x,y}|$. Then by definition, $\rho(\gamma(x),\gamma(y)) \ge d(\gamma(x),\gamma(y))$. We define $\hat{Z}(x,y)$ by $$\hat{Z}(x,y) = \rho(\gamma(x),\gamma(y)) - \bm{L} \varphi \Big( \frac{l(\gamma(x),\gamma(y))}{\bm{L}} \Big).$$ Note that $\hat{Z} \ge Z$ for every $(x,y) \in (M \times M) \setminus D$ and $\hat{Z}(x_{0},y_{0}) = Z(x_{0},y_{0})$. Moreover, $\hat{Z}$ achieves a local minimizer and $$\partial_{x} \hat{Z}= \partial_{y} \hat{Z} = 0$$ at $(x_{0},y_{0})$. (6) yields $$- \partial_{x} \rho = \partial_{y} \rho = \varphi' \Big( \frac{l(\gamma(x_{0}),\gamma(y_{0}))}{\bm{L}} \Big)$$ and this implies $$\Big\langle -V(0), -\frac{\alpha'(0)}{d}\Big\rangle = \Big\langle W(1), \frac{\alpha'(1)}{d}\Big\rangle = \varphi' \Big( \frac{l(x_{0},y_{0})}{\bm{L}} \Big).$$ Then let us consider the second variation of $\hat{Z}$ by the second variation formula of the length functional: $$\begin{aligned} \partial_{x}^{2} \hat{Z} &= \partial_{x}^{2} \rho - \frac{1}{\bm{L}} \varphi'' \\ &= \frac{1}{d} \Big \{ \int_{0}^{1} ( \langle (V^{\perp})', (V^{\perp})' \rangle - Rm(V^{\perp},\alpha',V^{\perp},\alpha') ) + \langle \nabla_{V}V, \alpha'\rangle|_{0}^{1} \Big\} - \frac{1}{\bm{L}} \varphi''\\ &= \frac{1}{d} (I(V^{\perp},V^{\perp}) -\kappa(\gamma(x_{0})) \langle \alpha'(0),\nu(\gamma(x_{0})) \rangle) - \frac{1}{\bm{L}} \varphi''. \end{aligned}$$ By the similar way, we have $$\begin{aligned} \partial_{x} \partial_{y} \hat{Z} &= \frac{1}{d} I(V^{\perp},W^{\perp}) + \frac{1}{\bm{L}} \varphi'' \\ \partial_{y}^{2} \hat{Z} &= \frac{1}{d} (I(W^{\perp},W^{\perp}) +\kappa(\gamma(y_{0})) \langle \alpha'(0),\nu(\gamma(y_{0})) \rangle) - \frac{1}{\bm{L}} \varphi''. \end{aligned}$$ By (7), we first have $$|(V^{\perp} \pm W^{\perp})(0)| = |(V^{\perp} \pm W^{\perp})(1)| = \sqrt{1 - \varphi'^{2}}.$$ By Proposition 3.1, there is no interior intersection point between $\alpha$ and $[\gamma(x):\gamma(y)]$, and note that $A_{(x,y)}$ is a topological disk. Since $\gamma$ is embedded and separates $N$ into two topological disks, we see that $$\langle \nu (\gamma(x_{0})), \alpha'(0) \rangle = \langle \nu (\gamma(y_{0})), - \alpha'(1) \rangle$$ and so two terms in (9) have the same sign. From (8) and (9), we can set $(V^{\perp}-W^{\perp})(t)$ to be a parallel transport of $(V^{\perp}-W^{\perp})(0)$ along $\alpha$ for $t \in [0,1]$. Now Since $\hat{Z}$ achieves local minimum at $(x_{0},y_{0})$, we have $$\begin{aligned} \nonumber 0 &\le (\partial_{x} - \partial_{y})^{2} \hat{Z} \\ &= \frac{1}{d} (I(V^{\perp}-W^{\perp},V^{\perp}-W^{\perp}) -\kappa(\gamma(x_{0})) \langle \alpha'(0),\nu(\gamma(x_{0})) \rangle +\kappa(\gamma(y_{0})) \langle \alpha'(1),\nu(\gamma(y_{0})) \rangle) -4 \frac{\varphi''}{\bm{L}}. \end{aligned}$$ Then we have $$\frac{1}{d(x_{0},y_{0})} I(V^{\perp}-W^{\perp},V^{\perp}-W^{\perp}) = - (1- \varphi'^{2}) \int_{\alpha} K$$ since $(V^{\perp}-W^{\perp})(t)$ is a parallel transport over $\alpha$. By applying (11) into (10), we obtain (5). ◻* Now we see the case of the reflected profile. **Proposition 9**. *Suppose $\bm{Z} \ge 0$ with $0 = \tilde{Z}(x_{0},y_{0})= \overline{Z}(x_{0},y_{0},z_{0})$ for some $((x_{0},y_{0}),z_{0}) \in (( \mathring{M} \times \mathring{M}) \setminus D) \times N$. Let $t_{0} =\alpha^{-1}(z_{0})$. Then at $((x_{0},y_{0}),z_{0})$, $$\begin{aligned} 0 \le &- (1- \varphi'^{2}) \int_{\alpha} K - \frac{\kappa(\gamma(x_{0}))}{\tilde{d}} \langle \alpha'(0),\nu(\gamma(x_{0})) \rangle + \frac{\kappa(\gamma(y_{0}))}{\tilde{d}} \langle \alpha'(1),\nu(\gamma(y_{0})) \rangle \\ \nonumber &- \frac{2\kappa^{\partial N}(z_{0})} {\Big\langle \frac{\alpha'_{+}(t_{0})-\alpha'_{-}(t_{0})}{\tilde{d}},\mathcal{N}^{\partial N}(z_{0}) \Big\rangle} -\frac{4}{\bm{L}} \varphi''.\end{aligned}$$* **Proof.* Note that, since $\partial N$ is convex, both pieces $\alpha([0,t_{0}])$ and $\alpha([t_{0},1])$ are transversal to $\partial N$. We take the orientation of $\partial_{z}$ near $z_{0} \in \partial N$ which satisfies $$\Big\langle \partial_{z}(z_{0}), \frac{\alpha'_{-}(t_{0})}{\tilde{d}}\Big\rangle >0$$ on $\partial N$. Now we consider a variation $\alpha_{x,y,z}$ of $\alpha$ which are family of curves moved by $\partial_{x}$, $\partial_{y}$ and $\partial_{z}$ near $\gamma(x_{0})$, $\gamma(y_{0})$ and $z_{0}$ with the corresponding vector fields $V$, $W$ and $X$ respectively, which satisfy the following: $$\begin{aligned} &V(0) = \partial_{s}(\gamma(x_{0})) \text{, } V(t) = 0 \text{ for } t \in [t_{0},1], \\ &W(t) = 0 \text{ for } t \in [0,t_{0}]\text{, } W(1) = \partial_{s}(\gamma(y_{0})),\\ &X(0) = X(1) =0 \text{, } X(t_{0}) = \partial_{z}(z_{0}). \end{aligned}$$ We define $\rho(x,y,z) = |\alpha_{x,y,z}|$ and define $\hat{Z}(x,y,z)$ for $(x,y,z) \in (( \mathring{M} \times \mathring{M}) \setminus D) \times N$ by $$\hat{Z}(x,y,z) = \rho(x,y,z) - \bm{L} \varphi \Big( \frac{\tilde{l}(\gamma(x),\gamma(y))}{\bm{L}} \Big).$$ Then $\hat{Z} \ge Z$ for every $(x,y,z) \in (( \mathring{M} \times \mathring{M}) \setminus D) \times N$ and $\hat{Z}(x_{0},y_{0},z_{0}) = Z(x_{0},y_{0},z_{0})$. Moreover, $\hat{Z}(x_{0},y_{0},z_{0})$ is a local minimizer of $\hat{Z}$ and $\partial_{x}\hat{Z} = \partial_{y}\hat{Z} = \partial_{z}\hat{Z} =0$. Let us first consider the first variations. This gives $$\begin{aligned} \partial_{x} \rho = \partial_{y} \rho = \varphi' \\ \partial_{z} \rho = 0\end{aligned}$$ and we have $$\begin{aligned} \Big\langle V(0), -\frac{\alpha'(0)}{\tilde{d}}\Big\rangle = \Big\langle W(1), \frac{\alpha'(1)}{\tilde{d}}\Big\rangle &= \varphi' \Big( \frac{l(x_{0},y_{0})}{\bm{L}} \Big) \\ \Big\langle X(t_{0}), \frac{\alpha'_{-}(t_{0})-\alpha'_{+}(t_{0})}{\tilde{d}}\Big\rangle &= 0. \end{aligned}$$ Then by (13) and (17), there exists $\theta_{0} \in (0, \pi/2)$ such that $$\Big\langle X(t_{0}), \frac{\alpha'_{-}(t_{0})}{\tilde{d}}\Big\rangle = \Big\langle X(t_{0}), \frac{\alpha'_{+}(t_{0})}{\tilde{d}}\Big\rangle = \cos \theta_{0}.$$ We calculate the second variation by $x$: $$\begin{aligned} \nonumber \partial_{x}^{2} \hat{Z} &= \partial_{x}^{2} \rho - \frac{1}{\bm{L}} \varphi'' \\ \nonumber &= \frac{1}{\tilde{d}} \Big \{ \int_{0}^{t_{0}} ( \langle (V^{\perp})', (V^{\perp})' \rangle - Rm(V^{\perp},\alpha',V^{\perp},\alpha') ) + \langle \nabla_{V}V, \alpha'\rangle|_{0}^{t_{0}} \Big\} - \frac{1}{\bm{L}} \varphi''\\ &= \frac{1}{\tilde{d}} (I(V^{\perp},V^{\perp}) -\kappa(\gamma(x_{0})) \langle \alpha'(0),\nu(\gamma(x_{0})) \rangle) - \frac{1}{\bm{L}} \varphi''. \end{aligned}$$ Similarly we have $$\begin{aligned} \partial_{y}^{2} \hat{Z} &= \frac{1}{\tilde{d}} (I(W^{\perp},W^{\perp}) +\kappa(\gamma(y_{0})) \langle \alpha'(1),\nu(\gamma(y_{0})) \rangle) - \frac{1}{\bm{L}} \varphi'' \\ \partial_{x} \partial_{y} \hat{Z} &= \frac{1}{\tilde{d}} I(V^{\perp},W^{\perp}) - \frac{1}{\bm{L}} \varphi'' \\ \partial_{z}^{2} \hat{Z} &= \frac{1}{\tilde{d}} (I(X^{\perp},X^{\perp})+ \kappa^{\partial N}(z) \langle \alpha'_{-}(t_{0})-\alpha'_{+}(t_{0}),\mathcal{N}^{\partial N}(z_{0}) \rangle). \end{aligned}$$ Also we obtain $$\begin{aligned} \nonumber \partial_{x} \partial_{z} \hat{Z} &= \partial_{x} \partial_{z} \rho \\ \nonumber &= \frac{1}{\tilde{d}} \Big \{ \int_{0}^{t_{0}} ( \langle (V^{\perp})', (X^{\perp})' \rangle - Rm(V^{\perp},\alpha',X^{\perp},\alpha') ) + \langle \nabla_{V}X, \alpha'\rangle|_{0}^{t_{0}} \Big\}\\ &= \frac{1}{\tilde{d}} I(V^{\perp},X^{\perp}), \\ \partial_{y} \partial_{z} \hat{Z} &= \frac{1}{\tilde{d}} I(W^{\perp},X^{\perp}). \end{aligned}$$ For the brevity of notation, we put $c = \sqrt{1- \varphi'^{2}}/ \sin \theta_{0}$. From (16) and (18), we obtain $$|(V^{\perp} \pm W^{\perp}+cX^{\perp})(0)| = |(V^{\perp} \pm W^{\perp}+cX^{\perp})(t_{0})| = |(V^{\perp} \pm W^{\perp}+cX^{\perp})(1)| = \sqrt{1-\varphi'^{2}}.$$ By applying Proposition 3.1, we can argue the same reasoning as in the proof of Proposition 3.2. Namely, we obtain $$\langle \nu (\gamma(x_{0})), \alpha'(0) \rangle = \langle \nu (\gamma(y_{0})), - \alpha'(1) \rangle$$ and we know two terms in (26) have the same sign again. Hence by (25) and (26), we can take $V, W, X$ on $[0,1]$ such that $(V^{\perp}+W^{\perp}+cX^{\perp})(t)$ to be a parallel transport of $(V^{\perp}+W^{\perp}+cX^{\perp})(0)$ and $(V^{\perp}+W^{\perp}+cX^{\perp})(t_{0})$ on $[0,t_{0})$ and $[t_{0}, 1]$, respectively. Then we have $$\frac{1}{\tilde{d}} I(V^{\perp}+W^{\perp}+cX^{\perp},V^{\perp}+W^{\perp}+cX^{\perp}) = - (1- \varphi'^{2}) \int_{\alpha} K$$* *Since $(x_{0},y_{0},z_{0})$ is a minimizer of $\hat{Z}$, we have $$\begin{aligned} \nonumber 0 &\le (\partial_{x} + \partial_{y} + c \partial_{z})^{2} \hat{Z}|_{(x_{0},y_{0},z_{0})} \\ &= \frac{1}{\tilde{d}} ( I(V^{\perp}+W^{\perp}+cX^{\perp},V^{\perp}+W^{\perp}+cX^{\perp})- \kappa(\gamma(x_{0})) \langle \alpha'(0),\nu(\gamma(x_{0})) \rangle \\ \nonumber &+ \kappa(\gamma(y_{0})) \langle \alpha'(1),\nu(\gamma(y_{0})) \rangle + c^{2} \kappa^{\partial N}(z) \langle \alpha'_{-}(t_{0})-\alpha'_{+}(t_{0}),\mathcal{N}^{\partial N}(z_{0}) \rangle)-\frac{4}{\bm{L}} \varphi'' \\ &= - (1- \varphi'^{2}) \int_{\alpha} K - \frac{\kappa(\gamma(x_{0}))}{\tilde{d}} \langle \alpha'(0),\nu(\gamma(x_{0})) \rangle + \frac{\kappa(\gamma(y_{0}))}{\tilde{d}} \langle \alpha'(1),\nu(\gamma(y_{0})) \rangle \\ \nonumber &+ \frac{c^{2} \kappa^{\partial N}(z)}{\tilde{d}} \langle \alpha'_{-}(t_{0})-\alpha'_{+}(t_{0}),\mathcal{N}^{\partial N}(z_{0}) \rangle-\frac{4}{\bm{L}} \varphi'', \end{aligned}$$ where $c = \sqrt{1- \varphi^{2}}/ \sin \theta_{0}$. (28) follows by summing up (19)-(24) and (29) comes from (17). Notice that $$\frac{1}{2}\Big\langle \frac{\alpha'_{+}(t_{0})-\alpha'_{-}(t_{0})}{\tilde{d}},\mathcal{N}^{\partial N}(z_{0}) \Big\rangle = \sin \theta_{0}.$$ By applying (30) to (29), we obtain (12). ◻* Now we consider the completed profile. We can still apply the arguments in [@LZ] to prove that the first derivatives of $\bm{Z}$ vanishes even it achieves the minimum at boundary points. We directly apply Lemma 4.4 and Lemma 4.5 in [@LZ] to our estimates in Proposition 3.1 and Proposition 3.2. **Proposition 10**. *Suppose $0 = \min_{\bm{M} \times \bm{M}} \bm{Z} = \bm{Z}(\bm{x}_{0}, \bm{y}_{0})$ for some $(\bm{x}_{0}, \bm{y}_{0}) \in (\bm{M} \times \bm{M}) \setminus \bm{D}$, then there exists $(\bm{x},\bm{y}) \in (\bm{M} \times \bm{M}) \setminus \bm{D}$ such that $\bm{Z}(\bm{x},\bm{y})=0$ and either of the following holds:* 1. *$\text{sign}(\bm{x}) = \text{sign}(\bm{y})$ and $$0 \le -4 \frac{\varphi''}{\bm{L}}- \frac{\kappa(\gamma(x))}{d} \langle \alpha'(0),\nu(\gamma(x)) \rangle +\frac{\kappa(\gamma(y))}{d} \langle \alpha'(1), \nu(\gamma(y))\rangle - ( 1- \varphi'^{2} ) \int_{\alpha} K$$ or* 2. *$\text{sign}(\bm{x}) \neq \text{sign}(\bm{y})$, $\bm{x}, \bm{y} \in \mathring{\bm{M}}$, $\bm{Z}(\bm{x},\bm{y}) = \tilde{Z}(x,y)= \overline{Z}(x,y,z)$, $t_{0} = \alpha^{-1}(z)$. $$\begin{aligned} 0 \le &- (1- \varphi'^{2}) \int_{\alpha} K - \frac{\kappa(\gamma(x))}{\tilde{d}} \langle \alpha'(0),\nu(\gamma(x)) \rangle + \frac{\kappa(\gamma(y))}{\tilde{d}} \langle \alpha'(1),\nu(\gamma(y)) \rangle \\ &- \frac{2\kappa^{\partial N}(z)} {\Big\langle \frac{\alpha'_{+}(t_{0})-\alpha'_{-}(t_{0})}{\tilde{d}},\mathcal{N}^{\partial N}(z) \Big\rangle} -\frac{4}{\bm{L}} \varphi''. \end{aligned}$$* # Noncollapsing and the long time behavior of the flow Based on the estimates we obtained in Proposition 3.4, we obtain the non-collapsing properties of complete chord-arc profile under the free boundary curve shortening flow on surfaces, which are generalizations of Theorem 5.3 and Theorem 5.4 in [@LZ]. We define $[\Gamma_{t}(x):\Gamma_{t}(y)]$ as the portion of $\Gamma_{t}$ connecting $\Gamma_{t}(x)$ and $\Gamma_{t}(y)$ for $x,y \in M$. Also denote $[\bm{\Gamma}_{t}(\bm{x}):\bm{\Gamma}_{t}(\bm{y})]$ as the shorter portion of $\bm{\Gamma}_{t} \setminus \{ \bm{\Gamma}_{t}(\bm{x}), \bm{\Gamma}_{t}(\bm{y}) \}$ as in Section 3. The same arguments with the proof of Proposition 5.1 in [@LZ] follows the evolution of the chord-arc profile on surfaces. **Proposition 11**. *Assume that $\bm{Z}(\cdot,\cdot,0) \ge 0$ and $\bm{Z}(\cdot,\cdot,0) >0$ on off-diagonal points. We denote $t_{0} = \sup\{ t \in [0,T):Z(\cdot,\cdot,t) \ge 0 \}< T$. Then there exist $\bm{x}, \bm{y} \in (\bm{M} \times \bm{M}) \setminus \bm{D}$ such that $\bm{Z}(\bm{x},\bm{y},t_{0}) = 0$ and either of the following holds:* 1. *$\text{sign}(\bm{x}) = \text{sign}(\bm{y})$ and $$0 \ge 4 \frac{\varphi''}{\bm{L}} + ( 1- \varphi'^{2} ) \int_{\alpha} K + 2 \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) \int_{\Gamma_{t}} \kappa^{2} ds + \varphi' \int_{[\Gamma_{t}(x):\Gamma_{t}(y)]} \kappa^{2} ds - \bm{L} \partial_{t} \varphi$$ or* 2. *$\text{sign}(\bm{x}) \neq \text{sign}(\bm{y})$, $\bm{x}, \bm{y} \in \mathring{\bm{M}}$, $\bm{Z}(\bm{x},\bm{y}) = \tilde{Z}(x,y)= \overline{Z}(x,y,z)$. $$\begin{aligned} 0 \ge &4 \frac{\varphi''}{\bm{L}} + ( 1- \varphi'^{2} ) \int_{\alpha} K + 2 \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) \int_{\Gamma_{t}} \kappa^{2} ds + \varphi' \int_{[\bm{\Gamma}_{t}(x):\bm{\Gamma}_{t}(y)]} \kappa^{2} ds - \bm{L} \partial_{t} \varphi \\ \nonumber &+ \frac{2\kappa^{\partial N}(z)} {\Big\langle \frac{\alpha'_{+}(t_{0})-\alpha'_{-}(t_{0})}{\tilde{d}},\mathcal{N}^{\partial N}(z) \Big\rangle}. \end{aligned}$$* We now discuss the lower bounds of the chord-arc profile on surfaces. The following lemma gives rise to the estimate of total curvature between two points along the boundary when the distance between two points is small. For $z_{1}, z_{2} \in \partial N$, denote $[z_{1}:z_{2}]$ as the smaller portion of $\partial N \setminus \{ z_{1}, z_{2} \}$. **Lemma 12**. *There exists $\epsilon_{0} = \epsilon_{0}(N)>0$ such that the following holds: Suppose $\epsilon \in (0, \epsilon_{0})$. Let $\Gamma$ be a curve in $(N,\partial N,g)$ which meets $\partial N$ orthogonally at $\partial \Gamma = \{ z_{0}, z_{1} \}$ with length $L$. Denote $C = \sup_{\partial N \cap B_{3L}(\partial \Gamma)} \kappa^{\partial N}$. If $L(1+C) \le \min (\frac{\epsilon}{100}, \frac{|\partial N|}{8})$ and $z \in \partial N$ is a point achieving $\tilde{d}(x,y) = d(x,z)+d(z,y)$ for some $x,y \in \Gamma$, then $$\int_{[z_{0}:z_{1}]} \kappa \le \frac{2\epsilon}{5} \text{ and } \int_{[z_{0}:z]} \kappa \le \frac{2\epsilon}{5}.$$* **Proof.* We regard $N$ as a convex domain of a closed manifold $\hat{N}$. We parametrize $\Gamma$ with arclength by $\Gamma : [0,|\Gamma|] \rightarrow N$. Fix a point $z_{0} \in \partial N \cap \Gamma$ and consider an exponential map $\exp_{z_{0}}: B_{a}(0) \rightarrow \hat{N}$ for some $a>0$. We also consider a geodesic normal coordinate $x = (x_{1},x_{2})$ such that $x := X \circ \exp_{z_{0}}^{-1}$ where $X = (X_{1},X_{2})$ is an Euclidean coordinate. In the geodesic normal coordinate, note that $g_{ij} = \delta_{ij} + O(r^{2})$ and Christoffel symbols satisfy $\Gamma_{ij}^{k} = O(r)$ where $r = \sqrt{X_{1}^{2}+X_{2}^{2}}$. Denote $\theta(s) = \tan^{-1}(\dot{x}_{2}(\Gamma(s))/\dot{x}_{1}(\Gamma(s)))$ as an angle in the geodesic normal coordinate. We consider the local geodesic equation $|\nabla_{\dot{\Gamma}}\dot{\Gamma}|^{\perp} = \kappa_{\Gamma} N_{\Gamma}$ in terms of local coordinates: $$\sum_{k=1,2} \Big( \ddot{x}_{k}+\sum_{i,j=1,2}\dot{x}_{i}\dot{x}_{j}\Gamma_{ij}^{k} \Big)\frac{\partial}{\partial x_{k}} = (\kappa+O(r)) \Big(-\dot{x}_{2} \frac{\partial}{\partial x_{1}}+\dot{x}_{1} \frac{\partial}{\partial x_{2}}\Big).$$ From (34), we obtain $$\frac{\partial \theta(s)}{\partial s} = \frac{\ddot{x}_{2} \dot{x}_{1} - \ddot{x}_{1} \dot{x}_{2}}{\dot{x}^{2}_{1}+\dot{x}^{2}_{2}} = \kappa(s) + O(r).$$ By (35) and the definition of the geodesic normal coordinate, there exists $\epsilon_{0} = \epsilon_{0}(N)>0$ such that the arguments in Lemma 3.5 in [@ACGL] to bound the length of $[z_{0}:z_{1}]$ in terms of $d(z_{0},z_{1})$ works directly. We follow the arguments in the proof of Lemma 5.2 in [@LZ] and have $$|[z_{0}:z_{1}]| \le \frac{2\epsilon}{5C} \text{ and } |[z_{0}:z]| \le \frac{2\epsilon}{5C}.$$ Note that the constant in the proof of Lemma 5.2 in [@LZ] is not sharp and we took a more strict upper bound. Then we finally obtain $$\int_{[z_{0}:z_{1}]} \kappa^{\partial N} ds \le C |[z_{0}:z_{1}]| \le C \cdot \frac{2\epsilon}{5C} = \frac{2\epsilon}{5}$$ and another inequality of (33) also follows. ◻* Note that there exists $L_{0} = L_{0}(N,g)$ such that for a topological disk $A \subseteq N$ with $|\partial A| \le L_{0}$, the isoperimetric inequality holds $C' |A| \le |\partial A|^{2}$ for some explicit constant $C'$ by Proposition 2.1 in [@BR]. Denote $K_{0} = \sup_{x \in N} |K|$. **Theorem 13**. *Suppose $L(t) \rightarrow 0$ as $t \rightarrow T$. Take $\epsilon_{1}= \epsilon_{1}(N)>0$. Given any $\epsilon \in (0,\epsilon_{1})$, there exists $c_{\epsilon}$ such that the following holds: let $\{ \Gamma_{t} \}_{t \in [0,T)}$ be a free boundary curve shortening flow on $N$. Suppose $L(0) (1+C) \le \min(\frac{\epsilon}{100},\frac{L_{0}}{3})$, where $C = \sup_{\partial N} \kappa^{\partial N}$. Given any $c_{0} \in (0,c_{\epsilon})$, if the inequality $$\bm{\psi}(\delta,t) \ge \begin{cases} c_{0} \bm{L}(t) \Big\{ \sin \Big((\pi-\epsilon) \frac{\delta}{\bm{L}(t)} + \frac{\epsilon}{2} \Big)+ 64 (\frac{\delta}{\bm{L}(t)} -\frac{1}{4})^{3} \sin \frac{\epsilon}{2} \Big\} e^{-K_{0}t} &\text{ if }0 \le \delta \le \frac{\bm{L}(t)}{4} \\ c_{0} \bm{L}(t) \sin \Big((\pi-\epsilon) \frac{\delta}{\bm{L}(t)} + \frac{\epsilon}{2} \Big) e^{-K_{0}t} &\text{ otherwise}. \end{cases}$$ holds at $t=0$, then it holds for all $t \in [0,T)$.* **Proof.* We take $\epsilon_{1} = \min(\epsilon_{0},L_{0},\frac{2}{C'K_{0}},\frac{\pi}{20})$. Take $\varphi \in C^{2}([0,\frac{1}{2}] \times [0,T))$ as following: $$\varphi(\zeta,t) = \begin{cases} c_{0} \{ \sin ((\pi-\epsilon) \zeta + \frac{\epsilon}{2} )+ 64 (\zeta -\frac{1}{4})^{3} \sin \frac{\epsilon}{2} \} e^{-K_{0}t} &\text{ if }\zeta \in [0,\frac{1}{4}]\\ c_{0} \{ \sin ((\pi-\epsilon) \zeta + \frac{\epsilon}{2} )\} e^{-K_{0}t} &\text{ otherwise} \end{cases}$$ and extend (38) to $\varphi \in C^{2}([0,1] \times [0,T))$ to satisfy $\varphi(1-\zeta,t) = \varphi(\zeta,t)$ for $\zeta \in [0,1/2]$ and $t \in [0,T)$. We will choose $c_{0}>0$ later in the proof. Note that $\varphi(0,t) = 0$ and $\partial_{\zeta} \varphi (1/2,t) =0$ for every $t \in [0,T)$. We define the auxiliary function $\bm{Z}$ with $\varphi$ defined in (38). Notice that the initial condition holds by (37) and we argue by contradiction. Denote $t_{0}:= \sup \{ t \in [0,T): \bm{Z}(\cdot,\cdot,t) \ge 0 \}$ and assume $t_{0} <T$. Suppose $\bm{Z}(\bm{x},\bm{y},t_{0})=0$ and note that this satisfies the conditions of Proposition 4.1. Denote $\Gamma_{t_{0}} \cap \partial N = \{ z_{0}, z_{1} \}$. First we estimate $$\Theta :=\int_{\Gamma_{t_{0}}} \kappa ds \text{ and } \omega:=\int_{[\bm{\Gamma}_{t_{0}}(x):\bm{\Gamma}_{t_{0}}(y)]} \kappa ds.$$ Denote the region surrounded by $[z_{0}:z_{1}]$ and $\Gamma_{t_{0}}$ by $A_{t_{0}}$.* *Since $\Gamma_{t_{0}}$ meets $\partial N$ orthogonally, by Gauss-Bonnet theorem we have $$\begin{aligned} \nonumber \Theta &= 2 \pi - \int_{[z_{0}:z_{1}]} \kappa^{\partial N} - \int_{A_{t_{0}}} K - 2 \cdot \frac{\pi}{2} \\ &\ge \pi - \frac{2\epsilon}{5} - \int_{A_{t_{0}}} K , \end{aligned}$$ where we obtain (39) by applying Lemma 4.2.* *Now we deduce the lower bound of $\omega$ and we consider the case of $\text{sign}(\bm{x}) = \text{sign}(\bm{y})$ first. By Proposition 3.1, $A_{(\bm{x},\bm{y})}$ is a topological disk. We denote $\beta = \cos ^{-1} \varphi'$ and $\beta$ is an interior angle between two curves at $x$ and $y$ of $A_{(\bm{x},\bm{y})}$ by (7). We obtain the following by Gauss-Bonnet theorem: $$\begin{aligned} \nonumber \omega=\int_{[\bm{\Gamma}_{t_{0}}(\bm{x}):\bm{\Gamma}_{t_{0}}(\bm{y})]} \kappa ds &= 2 \pi - 2(\pi-\beta) - \int_{A_{(\bm{x},\bm{y})}} K \\ &= 2 \cos ^{-1} \varphi' - \int_{A_{(\bm{x},\bm{y})}} K. \end{aligned}$$* *Now we consider the case $\text{sign}(\bm{x}) \neq \text{sign}(\bm{y})$. We apply Proposition 3.1 again and obtain that $A_{(\bm{x},\bm{y})}$ is a topological disk in $\tilde{N}$. Moreover, denote $\beta = \cos ^{-1} \varphi'$ to be an interior angle between two curves $[\bm{\Gamma}_{t_{0}}(\bm{x}):\bm{\Gamma}_{t_{0}}(\bm{y})]$ and $\bm{\alpha}$ at $\bm{x}$ and $\bm{y}$ by (16). We separate $A_{\bm{\Gamma}_{t_{0}}}$ by $$A_{1} = A_{(\bm{x},\bm{y})} \cap N \text{ and } A_{2} = A_{(\bm{x},\bm{y})} \cap (\tilde{N} \setminus N).$$Note that $A_{1}$ and $A_{2}$ are both topological disks. Without loss of generality, $z_{0} \in [\Gamma_{t_{0}}(x):\Gamma_{t_{0}}(y)]$. Since $\Gamma_{t_{0}}$ orthogonally meets $\partial N$, by applying (18) and Gauss-Bonnet theorem we have $$\begin{aligned} \nonumber \omega=\int_{[\bm{\Gamma}_{t_{0}}(x):\bm{\Gamma}_{t_{0}}(y)]} \kappa ds &= \int_{[\bm{\Gamma}_{t_{0}}(x):z_{0}]} \kappa ds + \int_{[z_{0}:\bm{\Gamma}_{t_{0}}(y)]} \kappa ds \\ \nonumber &= \Big( 2 \pi - \theta_{0} - \pi/2 - (\pi-\beta) - \int_{A_{1}} K - \int_{[z_{0}:z]} \kappa^{\partial N} \Big) \\ \nonumber &+ \Big( 2 \pi - (\pi-\theta_{0}) - \pi/2 - (\pi-\beta) - \int_{A_{2}} K - \int_{[z_{0}:z]} \kappa^{\partial N} \Big) \\ \nonumber &= 2 \cos ^{-1} \varphi' - \int_{A_{(\bm{x},\bm{y})}} K - 2 \int_{[z_{0}:z]} \kappa^{\partial N} \\ &\ge 2 \cos ^{-1} \varphi' - \int_{A_{(\bm{x},\bm{y})}} K - \frac{4\epsilon}{5}, \end{aligned}$$ where we obtain (42) by applying Lemma 4.2.* *By the isoperimetric inequality and our choice of $L_{0}$, we obtain the following estimate of the area of $A_{t_{0}}$ and $A_{(\bm{x},\bm{y})}$: $$\begin{aligned} \nonumber |A_{t_{0}}| &\le C' ( L(t_{0}) +|[z_{0}:z_{1}]| )^{2} \\ &\le C' (\frac{\epsilon}{100} + \frac{\epsilon}{2})^{2} \le C' \epsilon^{2}, \\ |A_{(\bm{x},\bm{y})}| &\le C' ( 2L(t_{0}))^{2} \le \frac{C' \epsilon^{2}}{100}. \end{aligned}$$ We obtain (42) by applying (36) and (43) by applying $|\alpha| \le [\bm{\Gamma}_{t_{0}}(x):\bm{\Gamma}_{t_{0}}(y)] \le L(t_{0})$. Moreover, by (42) and (43), we have the control of total Gaussian curvature of $A_{t_{0}}$ and $A_{(\bm{x},\bm{y})}$: $$\begin{aligned} \int_{A_{t_{0}}} K &\le K_{0} |A_{t_{0}}| \le C'K_{0}\epsilon^{2}, \\ \int_{A_{(\bm{x},\bm{y})}} K &\le K_{0} |A_{(\bm{x},\bm{y})}| \le \frac{1}{100}C'K_{0}\epsilon^{2}. \end{aligned}$$ By our choice of $\epsilon_{1}$, (39)-(41), (44) and (45), we have $$\Theta \ge \pi -\epsilon \text{ and } \omega \ge 2 \Big(\cos ^{-1} \varphi'- \frac{\epsilon}{2}\Big).$$ By our choice of $\varphi$, $|\varphi'| \le c_{0} \{ (\pi- \epsilon) + 12 \sin \frac{\epsilon}{2} \}$ holds. By taking sufficiently small $c_{0}$, we have $\cos ^{-1} \varphi' \ge \frac{\epsilon}{2}$ and our estimate of $\omega$ in (46) is proper. We apply Hölder inequality and obtain $$\begin{aligned} \int_{\Gamma_{t}} \kappa^{2} ds &\ge |\Gamma_{t}|^{-1} \Big( \int_{\Gamma_{t}} |\kappa| ds \Big)^{2} \ge \frac{2}{\bm{L}} \Theta^{2} \ge \frac{2}{\bm{L}} (\pi-\epsilon)^{2} \end{aligned}$$ and $$\begin{aligned} \nonumber \int_{[\bm{\Gamma}_{t}(x):\bm{\Gamma}_{t}(y)]} \kappa^{2} ds &\ge |[\bm{\Gamma}_{t}(x):\bm{\Gamma}_{t}(y)]|^{-1} \Big( \int_{[\bm{\Gamma}_{t}(x):\bm{\Gamma}_{t}(y)]} |\kappa| ds \Big)^{2} \\ &\ge \frac{1}{l} \omega^{2} \ge \frac{4}{l} \Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2}. \end{aligned}$$ Note that We apply (47) and (48) to (31) and (32), in either case we have $$\begin{aligned} \nonumber 0 &\ge 4 \frac{\varphi''}{\bm{L}} + ( 1- \varphi'^{2} ) \int_{\alpha} K + 2 \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) \int_{\Gamma_{t}} \kappa^{2} ds + \varphi' \int_{[\Gamma_{t}(x):\Gamma_{t}(y)]} \kappa^{2} ds - \bm{L} \partial_{t} \varphi \\ \nonumber &\ge 4 \frac{\varphi''}{\bm{L}} + ( 1- \varphi'^{2} ) \int_{\alpha} K + \frac{4}{\bm{L}} \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) ( \pi-\epsilon)^{2} + \frac{4}{l} \Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2} \varphi' - \bm{L} \partial_{t} \varphi \\ \nonumber &\ge 4 \frac{\varphi''}{\bm{L}} - K_{0} d + \frac{4}{\bm{L}} \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) (\pi-\epsilon)^{2} + \frac{4}{l} \Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2} \varphi' - \bm{L} \partial_{t} \varphi \\&= 4 \frac{\varphi''}{\bm{L}} - K_{0} \bm{L} \varphi + \frac{4}{\bm{L}} \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) (\pi-\epsilon)^{2} + \frac{4}{l} \Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2} \varphi' - \bm{L} \partial_{t} \varphi. \end{aligned}$$ If $\epsilon \in (0,\pi/20)$, then the following holds by direct calculation: $$(\pi-\epsilon) \cos \Big( (\pi-\epsilon)\frac{1}{4} + \frac{\epsilon}{2} \Big) > 12 \sin \frac{\epsilon}{2}.$$ Moreover, since $\cos ^{-1} \varphi' \rightarrow \pi/2$ uniformly as $c_{0} \rightarrow 0$ in $[0,1/4]$, we can take $c_{0}$ sufficiently small to satisfy $$\Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2}-(\pi-\epsilon)^{2} \zeta^{2} \ge 1$$ for $\zeta \in [0,1/4]$. Let us consider the following term for $\zeta \in (0,1/4]$: $$\begin{aligned} \nonumber &\varphi''+ (\pi-\epsilon)^{2} \varphi + \zeta^{-1} \varphi'\Big( \Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2}-(\pi-\epsilon)^{2} \zeta^{2} \Big) \\ \nonumber &= c_{0} e^{-K_{0}t}\Big( 384 \Big(\zeta-\frac{1}{4}\Big)\sin \frac{\epsilon}{2} \Big) + c_{0} \zeta^{-1} e^{-K_{0}t} \Big( (\pi-\epsilon) \cos \Big( (\pi-\epsilon)\zeta + \frac{\epsilon}{2} \Big) + 192 \Big(\zeta-\frac{1}{4}\Big)^{2}\sin \frac{\epsilon}{2}\Big)\\ \nonumber&\Big( \Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2}-(\pi-\epsilon)^{2} \zeta^{2} \Big) \\ &> c_{0} e^{-K_{0}t} \Big( 384 \Big(\zeta-\frac{1}{4}\Big)\sin \frac{\epsilon}{2} \Big) + c_{0} \zeta^{-1} e^{-K_{0}t} \Big( 12 \sin \frac{\epsilon}{2} + 192 \Big(\zeta-\frac{1}{4}\Big)^{2}\sin \frac{\epsilon}{2}\Big) \\ \nonumber &\ge c_{0} e^{-K_{0}t} \Big( 384 \Big(\zeta-\frac{1}{4}\Big)\sin \frac{\epsilon}{2} \Big) + 4c_{0} e^{-K_{0}t} \Big( 12 \sin \frac{\epsilon}{2} + 192 \Big(\zeta-\frac{1}{4}\Big)^{2}\sin \frac{\epsilon}{2}\Big) \\ &= 48 c_{0} e^{-K_{0}t} \sin \frac{\epsilon}{2} \Big( 4 \Big(\zeta-\frac{1}{4}\Big) + 1\Big)^{2} \ge 0.\end{aligned}$$ (52) comes from (50), (51) and the monotone decreasing property of the Cosine function. For $\zeta \in [1/4,1/2]$, both of the following holds for small $c_{0}>0$: $$\begin{aligned} \varphi''+ (\pi-\epsilon)^{2} \varphi &=0 \\ \Big( \cos ^{-1} \varphi'- \frac{\epsilon}{2} \Big)^{2}-(\pi-\epsilon)^{2} \zeta^{2} &>0.\end{aligned}$$* *We also have the following identity in time derivative terms: $$K_{0} \partial_{t} \varphi + \varphi =0.$$ By applying (53)-(56) into (49), we obtain the contradiction. ◻* **Theorem 14**. *Suppose $L(t) \nrightarrow 0$ as $t \rightarrow T$ where $T < \infty$. Let $\{ \Gamma_{t} \}_{t \in [0,T)}$ be a free boundary curve shortening flow on $N$. If $\bm{L}_{T} := \lim_{t \rightarrow T} \bm{L}(t)>0$, then if the inequality $$\bm{\psi}(\delta,t) \ge c_{0} \bm{L}(t) e^{\big(-\frac{4 \pi^{2}}{L_{T}^{2}} - K_{0}\big)t} \sin \Big(\frac{\pi \delta}{\bm{L}(t)} \Big)$$ holds at $t=0$, then it holds for all $t \in [0,T)$.* **Proof.* We modify the proof of Theorem 5.4 in [@LZ]. We adopt the modified time coordinate $\tau := \int^{t}_{0} \frac{1}{\bm{L}(s)^{2}} ds$. We take $\varphi: [0,1] \times [0,T)$ by $$\varphi(\zeta,t) = c_{0} e^{-4\pi^{2}\tau(t)-K_{0}t}\sin (\pi \zeta).$$ As before, denote $t_{0}:= \sup \{ t \in [0,T): \bm{Z}(\cdot,\cdot,t) \ge 0 \}$ and assume $t_{0}<T$. Then there exists $(\bm{x} , \bm{y}) \in (\bm{M} \times \bm{M} ) \setminus \bm{D}$ such that $\bm{Z}(\bm{x}_{0},\bm{y}_{0},t_{0}) = \min_{(\bm{x},\bm{y}) \in (\bm{M} \times \bm{M} ) \setminus \bm{D}} \bm{Z}(\bm{x},\bm{y},t_{0})$. By Proposition 4.1, in either case we have $$\begin{aligned} \nonumber 0 &\ge 4 \frac{\varphi''}{\bm{L}} + ( 1- \varphi'^{2} ) \int_{\alpha} K + 2 \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) \int_{\Gamma_{t}} \kappa^{2} ds + \varphi' \int_{[\Gamma_{t}(x):\Gamma_{t}(y)]} \kappa^{2} ds - \bm{L} \partial_{t} \varphi \\ \nonumber &\ge 4 \frac{\varphi''}{\bm{L}} - K_{0}\bm{L}\varphi + 2 \Big( \varphi - \varphi' \frac{\bm{l}}{\bm{L}} \Big) \int_{\Gamma_{t}} \kappa^{2} ds + \varphi' \int_{[\Gamma_{t}(x):\Gamma_{t}(y)]} \kappa^{2} ds - \bm{L} \partial_{t} \varphi \\ &> 4 \frac{\varphi''}{\bm{L}} - K_{0}\bm{L}\varphi - \bm{L} \partial_{t} \varphi,\end{aligned}$$ where (57) follows from the strict concavity of $\varphi(\cdot,t)$. But our choice of $\varphi$ gives $$\begin{aligned} \nonumber 4 \frac{\varphi''}{\bm{L}} - K_{0}\bm{L}\varphi - \bm{L} \partial_{t} \varphi &= -4 \pi^{2} \frac{\varphi}{\bm{L}} - K_{0}\bm{L}\varphi + \bm{L}(4 \pi^{2} \tau'(t) +K_{0}) \varphi \\ &= -4 \pi^{2} \frac{\varphi}{\bm{L}} - K_{0}\bm{L}\varphi + \bm{L}\Big(4 \pi^{2} \frac{1}{\bm{L}^{2}} +K_{0}\Big) \varphi =0\end{aligned}$$ (57) and (58) give the contradiction. And since $\bm{L}$ is a decreasing function in time $t$, the claim of the Theorem follows. ◻* Let us denote $\lambda : M \rightarrow \mathbb{R}$ by the arclength to the nearer endpoint in the sense of internal distance in $\Gamma_{t}$. Together with the proof of Proposition 5.5 in [@LZ], Theorem 4.3 and Theorem 4.4 gives the following boundary avoidance estimate. **Proposition 15**. *Let $\{ \Gamma_{t} \}_{t \in [0,T)}$ be a free boundary curve shortening flow on $N$. Given any $\delta>0$, there exists $\epsilon = \epsilon(\Gamma_{0},N,\delta)>0$ such that $$\lambda(x,t) > \delta \Rightarrow d(\gamma(x,t), \partial N)> \epsilon.$$* As in [@LZ], Proposition 4.5 and arguments of the proof of Theorem 6.1 in [@LZ] gives the following long-time behavior. **Theorem 16**. *Let $(N,\partial N, g)$ be a closed Riemannian surface with convex boundary and $\{ \Gamma_{t} \}_{t \in [0,T)}$ be a maximal free boundary curve shortening flow starting from a properly embedded closed interval $\Gamma_{0}$ in $N$. Then either:* 1. *$T = \infty$, in which case $\Gamma_{t}$ converges smoothly as $t \rightarrow \infty$ to an embedded geodesic in $N$ which meets $\partial N$ orthogonally; or* 2. *$T< \infty$, in which case $\Gamma_{t}$ converges uniformly to some single half-round point $z \in \partial N$ smoothly in the sense of the blow up limit of the curve converges to the unit semi-circle.* # The family of curves and tightening procedure In this section, we formulate the min-max construction of free boundary embedded geodesics on Riemannian $2$-disks with convex boundary. We discuss the smooth min-max setting and obtain the existence result via proper pull-tight procedure. Theorem B in [@Sm] (see also Appendix of Hatcher's work [@Ha]) proves that the space of (unparametrized) embedded intervals on $D^{2}$ whose endpoints are on $\partial D^{2}$ relative to the space of point curves retracts onto $\mathbb{R}P^{2}$. Let us denote the space of embedded curves by $\Sigma$ and denote by $\Sigma_{0}$ the space of point curves. We denote $\mathcal{S} = \Sigma/\Sigma_{0}$. We consider two distinct nontrivial relative homology classes $h_{1}$ and $h_{2}$ on the space of embedded intervals: $$h_{i} := H_{i}(\mathcal{S}, \mathbb{Z}_{2}) = \mathbb{Z}_{2}.$$ Let us consider the $\mathbb{Z}_{2}$-Cohomology ring $H^{*}(\mathcal{S},\mathbb{Z}_{2})$ of $\mathcal{S}$. We denote $\alpha$ to be a generator of the first cohomology ring. Then the cohomology ring is: $$H^{*}(\mathcal{S},\mathbb{Z}_{2}) = \mathbb{Z}_{2}[\alpha]/\alpha^{2}.$$ Let $IV_{1}(D^{2})$ be a space of integral varifolds on $(D^{2},g)$ and endow an $F$-metric on the space of varifolds as in 2.1(19)(20) in [@P]. For each $i=1,2$, we now define the $i$-sweepout. We denote an $i$-dimensional simplicial complex by $X$. If $$\Phi^{*} (\omega) \neq 0,$$ then we say that $\Phi:X \rightarrow \mathcal{S}$ detects $\omega \in H^{i}(\mathcal{S}, \mathbb{Z}_{2})$. We define $\Phi$ to be an *$i$-sweepout* endowed with a smooth topology if $\Phi$ detects $i$-th cup product $\alpha^{i} \in H^{i}(\mathcal{S},\mathbb{Z}_{2})$. We define the *width* of $i$-parameter sweepouts as $$\omega_{i}(D^{2}) : = \inf _{\Phi \in S_{i}} \sup_{x \in X} |\Phi(x)| = L_{i},$$ for $i \in \{ 1, 2 \}$. By the definition of $i$-sweepouts in (59), $\omega_{1}(D^{2}) \le \omega_{2}(D^{2})$ holds. We define a *minimizing sequence* to be the sequence of $i$-sweepouts such that $\lim_{j \rightarrow \infty} \sup_{x \in X} |\Phi_{j}(x)| = L_{i}$. We denote sequence of curves $\Phi_{j}(x_{j})$ to be a min-max sequence if $|\Phi_{j}(x_{j})|$ converges to $L_{i}$, where $x_{j} \in X$ and $\{ \Phi_{j}(x) \}$ is a minimizing sequence. We define the *critical set* $\Lambda(\{ \Phi_{j} \})$ to be a set of stationary varifolds achieved by the limit of min-max sequence induced by $\{ \Phi_{j}(x) \}$. We denote the set of critial geodesic by $W_{L_{i}}$ which is a set of stationary varifolds whose support is a free boundary embedded geodesic and length is $L_{i}$. We follow Abraham's proof of the bumpy metric theorem for curves [@Ab] and Theorem 9 in Ambrozio-Carlotto-Sharp [@ACS] of the free boundary minimal surface version which proves the genericity of bumpy metric in $C^{r}$-Baire sense. The compactness of free boundary embedded geodesics with bounded length follows from arguments in Appendix A in [@K2]. By tightening argument via free boundary curve shortening flow of Theorem 4.6 on long-time behavior of free boundary curve shortening flow, and applying the arguments in the proof of Theorem 2.1 in [@K1], we have the following existence of free boundary embedded geodesics achieving the width: **Theorem 17**. *Suppose $(D^{2},\partial D^{2},g)$ to be a smooth Riemannian $2$-disk with a strictly convex boundary. For $i=1,2$ and any minimizing sequence $\{ \Phi_{j} \}$ of $i$-sweepouts, there is a deformed minimizing sequence $\{\hat{ \Phi}_{j} \}$ of $\{ \Phi_{j} \}$ satisfying the following property. For any $s>0$, there is some $0<a<L_{i}$ satisfying $$\{ \hat{\Phi}_{j}(x) \in IV_{1}(D^{2}) : |\hat{\Phi}_{j}(x)| \ge L_{i}-a \} \subset \bigcup_{\gamma \in \Lambda(\{ \Phi_{j} \})\cap W_{L_{i}}} B^{F}_{s}(\gamma)$$ for all sufficiently large $j$, where $B^{F}_{s}(\gamma)$ is a $F$-metric ball with center $\gamma$. Moreover, the multiplicity of geodesics in the critical set is $1$.* **Remark 18**. *We still can run the tightening procedure even initial curves do not meet orthogonally with the boundary $\partial D^{2}$. Geometrically, we can slightly deform the curves near the boundary to make the intersection angles the right angle. Moreover, we can apply the flow in the 'weak' sense. We may consider the free boundary curve shortening flow as a Neumann boundary problems for nonlinear parabolic PDEs (See Chapter 10 of [@Fr]). Then the weak solution of the flow has a orthogonal boundary condition and smoothness at any positive time before the maximum existence time $t \in (0,T)$. The flow is still a $C^{0}$-solution at $t \in [0,T)$.* By applying the classical Lusternik-Schnirelmann argument, we prove that if two widths are the same, then there exists an $S^{1}$-cycle of free boundary embedded geodesics. This gives the existence of two free boundary geodesics on any Riemannian disk with a strictly convex boundary. Moreover, if the metric on $(D^{2},\partial D^{2},g)$ is bumpy, then we deduce the existence of two free boundary embedded geodesics with distinct lengths. The argument in the proof of Corollary 2.2 in [@K1] applies directly. **Theorem 19**. *Suppose $(D^{2},\partial D^{2},g)$ to be a Riemannian $2$-disk with strictly convex boundary. If $w_{1}(D^{2}) = w_{2}(D^{2})$, then there exist infinitely many distinct free boundary embedded geodesics in $(D^{2},\partial D^{2}, g)$.* **Corollary 20**. *On the Riemannian disk endowed with a bumpy metric and strictly convex boundary, there are at least two free boundary embedded geodesics with length $L_{1}$ and $L_{2}$.* # Morse Index Bound In this section, we obtain the generic Morse Index bound of free boundary embedded geodesics on Riemannian $2$-disk. We mainly follow the idea in Section 6 and Section 7 of [@K1] which proves the Morse Index of simple closed geodesics on bumpy spheres based on the interpolation technique based on quantitative $F$-distance estimate. We focus on the necessary modification to prove the Morse Index bound in the free boundary setting. Throughout this section, we suppose that $(D^{2},\partial D^{2},g)$ is endowed with a bumpy metric. Suppose $D^{2}$ is embedded in some closed surface $\tilde{D}^{2}$. Let us fix a free boundary embedded geodesic $\gamma$. We adopt the Fermi coordinate $c: [0,L] \times (-h,h) \rightarrow D^{2}$ on $N_{h}(\gamma)$ on the tubular neighborhood $N_{h}(\gamma)$ of the fixed geodesic $\gamma$. Moreover, we adopt the metric perturbation in Section 5.2 in [@K1] and follow the notation therein. By choosing a sufficiently small $\beta>0$ and sufficiently large $M > \sup_{D^{2}} |K|$ in Proposition 5.2 of [@K1] and applying the corresponding deformation (17) in [@K1], we obtain the strict stability of the free boundary embedded geodesic $\gamma_{g_{\beta}}$ in the perturbed metric. **Proposition 21**. *There exists small $\beta>0$ and $M>\sup_{D^{2}}|K|$ satisfying the following: $\gamma_{g_{\beta}}$ is a strictly stable geodesic and the ambient Gaussian curvature is strictly negative.* **Proof.* Denote $\partial \gamma_{g_{\beta}} = \{p_{1},p_{2} \}$. Then by the change of the second fundamental form by conformal deformation following [@Be], for $i=1,2$, we have a geodesic curvature of boundary on $\kappa^{\partial D^{2}}_{g_{\beta}}$ as $$\kappa^{\partial D^{2}}_{g_{\beta}}(p_{i}) = e^{-\phi_{\beta}} \Big( \kappa^{\partial D^{2}}_{g}(p_{i}) - \frac{\partial \phi_{\beta}}{\partial \nu }\Big) = e^{-\phi_{\beta}} \kappa^{\partial D^{2}}_{g}(p_{i}).$$ Also note that $K_{g_{\beta}}(x) = K(x)-M<0$ on $x \in \gamma_{g_{\beta}}$ by (19) of [@K1] and this proves the latter conclusion of the claim. Now we prove that we can choose $\beta$ and $M$ such that the second variation (3) of $\gamma_{g_{\beta}}$ is positive definite. It suffices to show that by taking the suitable $\beta$ and $M$, the eigenvalue $\{ \lambda_{k} \}$ with associated eigenfunctions $\{ \phi_{k}\}$ of the following equation with Robin boundary condition from (4) are all positive: $$\begin{cases} (\Delta_{\gamma_{g_{\beta}}} + K-M) \phi_{k} + \lambda_{k} \phi_{k} = 0 \text{ on }\gamma_{g_{\beta}}\\ \frac{\partial \phi_{k}}{\partial \eta}(p_{i}) -e^{-\phi_{\beta}}\kappa (p_{i}) \phi_{k} (p_{i}) = 0 \text{ for } i=1,2. \end{cases}$$ By the condition (iii) in Proposition 5.2 in [@K1], we can take $e^{-\phi_{\beta}}$ to be arbitrarily close to $1$ by taking sufficiently small $\beta$. By taking sufficiently large $M$ and sufficiently small $\beta$, we can obtain all the eigenvalues $\{ \lambda_{k} \}$ to be strictly positive by standard elliptic theory. ◻* We now discuss the free boundary mean convex neighborhood of the geodesic $\gamma$ which is strictly stable and with negative ambient Gaussian curvature to adopt the squeezing lemma in [@K1]. We adopt the idea in Proposition 2.4 of [@HK] which constructs the free boundary mean convex neighborhood of the free boundary embedded minimal surface in $3$-ball with a strict convex boundary. Consider the first eigenvalue $\lambda_{1} >0$ and the associated eigenfunction $\phi_{1} \in C^{\infty}(\gamma)$ with $\int_{\gamma} \phi_{1}^{2} ds=1$ which is a solution of the equation (4). Without loss of generality, we can set $\phi_{1}$ to be strictly positive in $\gamma$. We define $$C_{\gamma} := \frac{\max_{\gamma} \phi_{1}}{\min_{\gamma} \phi_{1}} \ge 1$$ and call $C_{\gamma}$ by *Harnack constant* of $\gamma$. **Proposition 22**. *Suppose $\gamma$ is a strictly stable geodesic on a Riemannian $2$-disk $(D^{2},\partial D^{2},g)$ with strictly convex boundary and with negative ambient Gaussian curvature. Then there is a neighborhood $N_{h}(\gamma)$ foliated by a free boundary mean convex foliation $\{\gamma_{t} \}_{t \in [-\epsilon,\epsilon]}$ of $\gamma$ satisfying the following:* 1. *$\gamma_{0} = \gamma$,* 2. *$\gamma_{t}$ has a mean curvature vector towards $\gamma_{0}$ for $t \in [-\epsilon,\epsilon]$,* 3. *$C_{t}:= \max{d_{x \in \gamma_{t}}(x,\gamma)}/\min{d_{x \in \gamma_{t}}(x,\gamma)} \le 2C_{\gamma}$ for $t \in [-\epsilon,0) \cup (0,\epsilon]$.* **Proof.* Let us denote the unit normal vector on $\gamma$ by $\nu$. Then we consider normal vector field $\phi_{1} \nu$ generated by the first eigenfuntion $\phi_{1}$ on $\gamma$ and extend to $X \in \mathcal{X}_{\tan} (D^{2})$. Then denote $\psi_{t}$ to be a flow generated by $X$ and define $\psi_{t}(\gamma) = \gamma_{t}$. For small $|t|$, we can expand the geodesic curvature $$\kappa_{\gamma_{t}} = \lambda_{1} \phi_{1} |t| + O(t^{2})$$ toward $\gamma$. Moreover, by the definition of $C_{\gamma}$ in (61) and our setting of $X$, $$\lim _{t \rightarrow 0} C_{t} = C_{\gamma}.$$ By (62)-(63), there exists $\epsilon>0$ such that $\{ \gamma_{t} \}_{t \in [-\epsilon,\epsilon]}$ is a foliation satisfying the conditions (2)-(3) in the statement. The Robin boundary condition in the equation (4) gives the orthogonality at boundary of mean convex foliation. ◻* Now we can apply the ideas to construct the squeezing homotopy via flow by Theorem 4.6 and its pullback homotopy in the general geodesic case. We obtain the following lemma: **Lemma 23**. *Let $(D^{2},\partial D^{2},g)$ be a Riemannian $2$-disk with a strictly convex boundary endowed with a bumpy metric. Let $\gamma$ be a free boundary embedded geodesic, and $X$ be a simplicial complex with finite dimension $k$. There exists $\delta_{0} = \delta_{0}((D^{2},\partial D^{2},g), \gamma)>0$ with the following property:* *For $0<\delta<\delta_{0}$, if $\Phi : X \rightarrow IV_{1}(D^{2})$ is a continuous map in the smooth topology such that $$\sup \{ F(\Phi(x), \gamma) : x \in X \} < \delta,$$ then there is a homotopy $H:[0,1] \times X \rightarrow IV_{1}(D^{2})$ such that $H(0,x) = \Phi(x)$ and $H(1,x) = \gamma$ so that $\Phi$ is nullhomotopic.* Now we discuss the modification in the quantitative $F$-distance estimate along the squeezing map. We only need to modify the arguments of Lemma 6.2 in [@K1] which proves the upper bound of $F$-distance along the squeezing homotopy in Lemma 5.4 of [@K1] when $\gamma$ is a strictly stable geodesic with ambient negative curvature. We rewrite the free boundary version of Lemma 6.1 in [@K1] first. **Lemma 24** (Lemma 6.1, [@K1]). *Suppose $\gamma$ is a strictly stable geodesic on Riemannian $2$-disk $(D^{2},\partial D^{2},g)$ with a strictly convex boundary which achieves negative ambient Gaussian curvature in $N_{h}(\gamma)$. Then there exists $C=C(|\gamma|)>0$ satisfying the following property: For $0<\epsilon<h^{2}$, if an embedded curve $\alpha$ whose each boundary point is on the each component of $\partial D^{2} \cap N_{h}(\gamma)$ satisfies $|\alpha|<|\gamma|+\epsilon$, then $$F(\alpha,\gamma)< C(|\gamma|)(h+ \sqrt{\epsilon}).$$* For a given free boundary mean convex foliation $\{ \gamma_{t} \}_{t \in [-z,z]}$, we denote $\gamma_{\le z'}$ by $$\gamma_{\le z'} := \{ x \in D^{2} | x \in \gamma_{t} \text{ for some } |t| \le z' \}.$$ **Lemma 25**. *Let $\gamma$ be a strictly stable geodesic on $(D^{2},\partial D^{2},g)$ with a strictly convex boundary, Gaussian curvature $K(z)<0$ for $z \in N_{h}(\gamma)$, and $X$ be a $k$-dimensional simplicial complex. Then there exists $\epsilon_{0}= \epsilon_{0}(\gamma)>0$ and $C = C(\gamma)>0$ satisfying the following property: For $0<\epsilon<\epsilon_{0}$, if $\Phi:X \rightarrow IV_{1}(D^{2})$ is a continuous map in the smooth topology such that $$\sup \{ F(\Phi(x), \gamma) : x \in X \} < \epsilon,$$ then there is a homotopy $H:[0,1] \times X \rightarrow IV_{1}(D^{2})$ such that $H(0,x) = \Phi(x)$, $H(1,x) = \gamma$ and the following $F$-distance estimate holds along $H$: $$\sup \{ F(H(t,x),\gamma) : x \in X \text{ and } t \in [0,1] \} < C(\gamma) \sqrt{\epsilon}.$$* **Proof.* We consider the $N_{h_{0}}(\gamma)$ can be foliated by a free boundary mean convex foliation $\{ \gamma_{t} \}_{t \in [-z_{0},z_{0}]}$ of $\gamma$ thanks to Proposition 6.1 and suppose the ambient Gaussian curvature in $N_{h_{0}}(\gamma)$ is negative. Take $\epsilon_{0} = h_{0}^{2}/10$. Then by Lemma 3.1 of [@K1], Hausdorff distance between $\Phi(x)$ and $\gamma$ is smaller than $h_{0}$ and $\Phi(x)$ is supported in $N_{h_{0}}(\gamma)$.* *Now we assume the condition (64). By Lemma 3.1 in [@K1], $\Phi(x)$ is supported in $N_{\sqrt{10 \epsilon}}(\gamma)$. We take $z \in (0,z_{0})$ by $z:= \inf \{z' \in (0,z) \,|\, N_{\sqrt{10 \epsilon}}(\gamma) \subseteq \gamma_{\le z'}\}$. Note that $\gamma_{\le z} \subseteq N_{2C_{\gamma} \sqrt{10 \epsilon}}(\gamma)$ by Proposition 6.2. Moreover, $|\Phi(x)|< |\gamma|+ \epsilon$ by the definition of the $F$-distance.* *We consider the squeezing homotopy $H$ which is the composition of free boundary curve shortening flow and squeezing map of graphical curves in the smaller scale. The length is monotonically decreasing along the free boundary curve shortening flow part in Lemma 6.3 and we still can apply the length bound of (40) and (41) in [@K1] for the length bound over the squeezing map. Saying again, $$|H(t,x)| <|\gamma| + \epsilon$$ for $x \in X$ and $t \in [0,1]$.* *Recall $supp(\Phi(x)) \subset N_{\sqrt{10 \epsilon}} \subseteq \gamma_{\le z}$. We apply the avoidance principle of the free boundary curve shortening flow between $\gamma_{z}$ and $\Phi(x)$. Then we have $$supp(H(t,x))\subset \gamma_{\le z} \subseteq N_{2C_{\gamma} \sqrt{10 \epsilon}}(\gamma).$$ By applying Lemma 6.4 together with (66) and (67), we obtain the $F$-distance upper bound (65) along the homotopy. ◻* We denote $W_{L_{i},j}$ to be the set of the elements in $W_{L_{i}}$ whose support has Morse index less than or equal to $j$. We also have the pulling-tight procedure toward free boundary embedded geodesics with Morse Index upper bound, which is the free boundary version of Theorem 2.3 of [@K1]: **Theorem 26**. *Let $(D^{2},\partial D^{2},g)$ be a Riemannian $2$-disk with a strictly convex boundary endowed with a bumpy metric. For any minimizing sequence $\{ \Phi_{j} \}$ which is an $i$-sweepout, there is a deformed minimizing sequence $\{\hat{ \Phi}_{j} \}$ of $\{ \Phi_{j} \}$ satisfying the following property. For any small $s>0$, there is some $0<a<L_{i}$ satisfying $$\{ \hat{\Phi}_{j}(x) \in IV_{1}(D^{2}) : |\hat{\Phi}_{j}(x)| \ge L_{i}-a \} \in \bigcup_{\gamma \in \Lambda(\{ \Phi_{j} \}) \cap W_{L_{i},i}} B^{F}_{s}(\gamma)$$ for all sufficiently large $j$. Moreover, the multiplicity of geodesics in the critical set is $1$.* By applying the arguments in the remaining parts of Section 6 and Section 7 in [@K1], we obtain the following Morse Index bound of free boundary embedded geodesic obtained by smooth min-max construction and free boundary curve shortening flow on surfaces. **Theorem 27**. *Suppose $(D^{2},\partial D^{2}, g)$ is a Riemannian $2$-disc with convex boundary endowed with a bumpy metric. Then for each $k=1,2$, there exists a free boundary embedded geodesic $\gamma_{k}$ with $$index(\gamma_{k}) = k$$ and these two geodesics satisfy $|\gamma_{1}| <|\gamma_{2}|$.* **Corollary 28**. *For a $2$-Riemannian disk $(D^{2},\partial D^{2},g)$ with a convex boundary, for $k=1,2$, there exists a free boundary embedded geodesic $\gamma_{k}$ with $$index(\gamma_{k}) \le k \le index(\gamma_{k})+ nullity (\gamma_{k}).$$* 20 R. Abraham, Bumpy metrics, In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pages 1--3. Amer. Math. Soc., Providence, R.I., 1970. S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differ. Equ. 2, 1 (1994), 101--111. L. Ambrozio, A. Carlotto, and B. Sharp, Compactness analysis for free boundary minimal hypersurfaces, Calc. Var. Partial Differential Equations 57 (2018), no. 1, 57:22. MR3740402 L. Ambrozio, R. Montezuma, and R. Santos, The Width of curves in Riemannian manifolds, arXiv: 2307.12939 (2023). B. Andrews and P. Bryan, Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem. J. Reine Angew. Math. 653 (2011), 179--187. B. Andrews, B. Chow, C. Guenther, and M. Langford, Extrinsic Geometric Flows, first ed., vol. 206 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2020. V. Bayle and C. Rosales. Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds. Indiana Univ. Math. J., 54(5):1371--1394, 2005. A. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag (1987). G. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199-300. W. Bos, Kritische Sehnen auf Riemannschen Elementarraumstücken, Math. Ann. 151 (1963), 431--451. O. Cornea, G. Lupton, J. Oprea, and D. Tanre. Lusternick-Schnirelman category. Mathematical Surveys and Monographs, AMS, 103, 2003. C. De Lellis and J. Ramic. Min-max theory for minimal hypersurfaces with boundary. Ann. Inst. Fourier (Grenoble), 68(5):1909--1986, 2018. G. De Philippis, M. Marini, M. Mazzucchelli, and S. Suhr, Closed geodesics on reversible Finsler 2-spheres, J. Fixed Point Theory Appl. 24 (2022), no. 2, Paper No. 19. S. Donato and R. Montezuma, The first width of non-negatively curved surfaces with convex boundary, preprint arXiv:2211.14815 (2022). N. Edelen, Noncollapsing of curve-shortening flow in surfaces, Int. Math. Res. Not., 2015, No. 20, pp. 10143-10153. A. Fraser, On the free boundary variational problem for minimal disks. Comm. Pure Appl. Math., 53(8):931--971, 2000. A. Friedman, Partial differential equations of parabolic type, Courier Dover Publications, 2008. H. Gluck and W. Ziller, Existence of periodic motions of conservative systems, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 65--98. M. Grayson, Shortening embedded curves, Ann. of Math. (2) 129, no. 1 (1989) 71-111. M. Grüter and J. Jost. On embedded minimal disks in convex bodies. Ann. Inst. H. Poincaré Anal. Non Linéaire, 3(5):345--390, 1986. A. Hatcher, A proof of the Smale conjecture, $\text{Diff}(S^{3}) \equiv O(4)$. Ann. of Math. (2), 117(3):553--607, 1983. R. Haslhofer and D. Ketover, Free boundary minimal disks in convex balls, preprint, arXiv:2307.01828. G. Huisken, Nonparametric mean curvature evolution with boundary conditions. J. Differential Equations 77, 2 (1989), 369--378. G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2, 127--133 (1998) J. Jost. Existence results for embedded minimal surfaces of controlled topolog-ical type. II. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13(3):401--426, 1986. D. Ko, Morse Index bound and Morse inequalities of simple closed geodesics on $2$-spheres, arXiv:2303.00644 (2023). D. Ko, Min-max construction of two capillary embedded geodesics on Riemannian $2$-disks, arXiv:2308.14180 (2023). M. Langford and J. Zhu, A distance comparison principle for curve shortening flow with free boundary, arXiv:2302.14258 (2023). P. Laurain and R. Petrides. Existence of min-max free boundary disks realizing the width of a manifold. Adv. Math., 352:326--371, 2019. M. Li. A general existence theorem for embedded minimal surfaces with free boundary. Comm. Pure Appl. Math., 68(2):286--331, 2015. M. Li, Chord shortening flow and a theorem of Lusternik and Schnirelmann. Pacific J. Math. 299 (2019), no. 2, 469-488. M. Li and X. Zhou. Min-max theory for free boundary minimal hypersurfaces I---Regularity theory. J. Differential Geom., 118(3):487--553, 2021. L. Lin, A. Sun, and X. Zhou. Min-max minimal disks with free boundary in Riemannian manifolds. Geom. Topol., 24(1):471--532, 2020. L. Lusternik and L. Schnirelmann. Topological methods in variational problems and their application to the differential geometry of surfaces. Uspehi Matem. Nauk (N.S.), 2(1(17)):166--217, 1947. F. Marques and A. Neves, Morse index and multiplicity of min-max minimal hypersurfaces, Camb. J. Math. 4 (2016), no. 4, 463--511. MR3572636 F. Marques and A. Neves, Morse index of multiplicity one min-max minimal hypersurfaces, Advances in Mathematics 378 (2021), 107527. J. T. Pitts, Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Princeton University Press, 1981. S. Smale, Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc., 10:621--626, 1959. A. Stahl, Convergence of solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differential Equations 4, 5 (1996), 421--441. A. Stahl, Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var. Partial Differential Equations 4, 4 (1996), 385--407. M. Struwe. On a free boundary problem for minimal surfaces. Invent. Math., 75(3):547--560, 1984. X. Zhou, On the free boundary min-max geodesics. Int. Math. Res. Not. IMRN, (5):1447--1466, 2016.

arxiv_math

--- author: - | Rui Song[^1]  \ School of Mathematics and Information Engineering, Longdong University,\ Qingyang, Gansu 745000, P.R.China\ title: On the third ABC index of trees and unicyclic graphs --- Let $G=(V,E)$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. The third atom-bond connectivity index, $ABC_3$ index, of $G$ is defined as $ABC_3(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}$, where eccentricity $e(u)$ is the largest distance between $u$ and any other vertex of $G$, namely $e(u)=\max\{d(u,v)|v\in V(G)\}$. This work determines the maximal $ABC_3$ index of unicyclic graphs with any given girth and trees with any given diameter, and characterizes the corresponding graphs. **Keywords:** eccentricity; third $ABC$ index; unicyclic graph; tree **AMS Classification:** 05C50 # Introduction Molecular descriptors are playing significant role in the chemistry, pharmacology and biology. Among them, topological indices have a prominent place [@1]. One of the best known and widely used indices is the connectivity index $\chi$ introduced in 1975 by Milan Randić [@2]. For any graph $G=(V,E)$ with vertex set $V(G)$ and edge set $E(G)$, let $|V(G)|=n$ and $|E(G)|=m$ and $d(u)$ denote the degree of the vertex $u\in V(G)$. The distance $d_{G}(u,v)$ between vertices $u$ and $v$ is defined as the length of any shortest path in $G$ connecting $u$ and $v$. For a vertex $u\in V(G)$, its eccentricity $e_{G}(u)$, or $e(u)$ for short, is the largest distance between $u$ and any other vertex of $G$. The diameter $d(G)$ of $G$ is defined as the maximum eccentricity among the vertices of $G$, similarly, the radius $r(G)$ is defined as the minimum eccentricity among the vertices of $G$. There are some topological indices which are related to the eccentricity, for example, the eccentric connectivity index, Zagreb eccentricity indices, eccentric distance sum. Ernesto Estrada et al. [@3] proposed a topological index named atom-bond connectivity index, or $ABC$ index for short, using a modification of the Randić connectivity index, as follows. $$ABC(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}.$$ $ABC$ index has been proven to be a valuable predictive index in the study of the heat formation of alkanes [@3; @4], some of its mathematical properties are obtained in [@5; @6]. A. Graovac and M. Ghorbani [@7] proposed a new index, nowadays known as the second $ABC$ index, as $$ABC_{2}(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{n(u)+n(v)-2}{n(u)n(v)}},$$ where $n(u)$ is the number of vertices of $G$ whose distance to the vertex $u$ is smaller than the distance to the vertex $v$. In the chemistry applications, it is used to model both the boiling point and melting point of the molecules. Some papers have given a lot of properties and upper bounds for the second $ABC$ index [@8; @9; @10; @11; @12]. In 2016, Dae-won Lee [@13] defined the third $ABC$ index, $ABC_3$ index for short, of graph $G$ as $$ABC_{3}(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}},$$ some lower and upper bounds on third $ABC$ index of graphs were obtained therein. Let $U_{n}(k)$ be the set of $n$-vertex unicyclic graphs with girth $k$ and $U_{n}$ be the set of all unicyclic grahs of order $n$. In section 2, we investigate the $ABC_{3}$ index of unicyclic graphs in $U_{n}(k)$ and characterize the extremal graphs with the maximal $ABC_{3}$ index in $U_{n}$; in section 3, we determine the second maximal $ABC_{3}$ index among all unicyclic graphs in $U_{n}$. In section 4, we characterize the extremal trees with the maximal $ABC_{3}$ index with any given diameter. # The maximal third $ABC$ index of unicyclic graphs If $k=n,n-1$, then $U_{n}(k)$ is unique. So in what follows we assume that $3\leq k\leq n-2$. The following lemma is obviously true, the readers can find its proof in [@14]. **Lemma 1** ([@14]). *If $u,v$ are two adjacent vertices of graph $G$, then $|e_{G}(u)-e_{G}(v)|\leq 1$.* For any real numbers $x,y\geq 1$, let $f(x,y)=\sqrt{\frac{x+y-2}{xy}}$. Then $f(x+1,x)=\sqrt{\frac{2x-1}{x^{2}+x}}$, $f(x,x)=\sqrt{\frac{2x-2}{x^{2}}}$. **Lemma 2** ([@15]). *$f(x,1)$ is strictly increasing for $x$, $f(x,2)=\sqrt{\frac{1}{2}}$, and $f(x,y)$ is strictly decreasing for $x$ if $y\geq 3$.* **Lemma 3**. *If $x\geq 2$, then $f(x+1,x)$ and $f(x,x)$ are strictly decreasing.* *Proof.* Let $g(x)=\frac{2x-1}{x^2+x}$. Then $\frac{dg(x)}{dx}=\frac{-2x^2+2x+1}{(x^2+x)^2}<0$ since $x\geq 2$, and so $f(x+1,x)$ is strictly decreasing. Similarly, $f(x,x)$ is also strictly decreasing when $x\geq 2$. ◻ Let $G_{0}$ be a connected graph with at least two vertices and $u_{0}$ be one of its vertex. Let $G_{1}$ be the graph obtained by identifying vertex $u_{0}$ of $G_{0}$ and a pendent vertex of a star $S_{t+2}$, and $G_{2}$ be the graph obtained by identifying vertex $u_{0}$ of $G_{0}$ and the center of a star $S_{t+2}$. **Lemma 4**. *Let $t\geq2$ be an integer then $ABC_{3}(G_{1})\leq ABC_{3}(G_{2})$, with equality holding if and only if $G_0=S_n$ and $u_{0}$ is its center.* *Proof.* Let $d_0=e_{G_0}(u_0)=\max\{d_{G_0}(u,u_0)|u\in V(G_0)\}$. Then $$\begin{aligned} ABC_{3}(G_{1})-ABC_{3}(G_{2}) &=&\sum\limits_{uv\in E(G_{0})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f( e_{G_{2}}(u),e_{G_{2}}(v))\right) \nonumber \\ & &+\sum\limits_{uv\in E(S_{t+2})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f(e_{G_{2}}(u),e_{G_{2}}(v))\right). \nonumber\end{aligned}$$ Since $G_0$ is a connected graph with $|G_0|\geq 2$, it follows that $d(G_0)\geq d_0\geq 1$. **Case 1.** $d(G_0)=d_0$. If $d(G_0)=d_0=1$, then $G_0$ is a complete graph $K_n$. In this subcase, for each edge $uv$ of $G_1$ with $e_{G_1}(u)=3=e_{G_1}(v)$, we deduce that $uv\in E(G_0)$ and $e_{G_2}(u)=2=e_{G_2}(v)$. Hence, $$\begin{aligned} &&ABC_{3}(G_{1})-ABC_{3}(G_{2})\nonumber \\ &=&\sum\limits_{uv\in E(G_{0})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f(e_{G_{2}}(u),e_{G_{2}}(v))\right) \nonumber \\ & &+\sum\limits_{uv\in E(S_{t+2})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f(e_{G_{2}}(u),e_{G_{2}}(v))\right) \nonumber \\ &=&(\frac{n(n-1)}{2}-(n-1))(f(3,3)-f(2,2))+(n-1)(f(3,2)-f(2,1))\nonumber \\ &&+(f(2,2)-f(2,1))+t(f(3,2)-f(2,1))\nonumber \\ &=&\frac{(n-1)(n-2)}{2}(f(3,3)-f(2,2))\nonumber \\ &\leq&0,\nonumber\end{aligned}$$ where the last equality follows from Lemma [Lemma 2](#lem-2-2){reference-type="ref" reference="lem-2-2"}. The equality of the last inequality holds if and only if $n=2$, namely $G_0=K_2$. Now we consider the second subcase when $d(G_0)=d_0\geq 2$. Since $\frac{d(G)}{2}\leq r(G)\leq d(G)$ it follows that $\lceil\frac{d_0}{2}\rceil\leq e_{G_0}(u)\leq d_0$ for any $u\in V(G_0)$. Notice that if $e_{G_0}(u)\geq d(u,u_0)+2$ then $e_{G_1}(u)=e_{G_2}(u)=e_{G_0}(u)$; if $e_{G_0}(u)\leq d(u,u_0)+1$ then $e_{G_1}(u)=e_{G_2}(u)+1=d(u,u_0)+2$. Furthermore, there is a vertex $u\in V(G_0)$ such that $d(u,u_0)=d_0=d(G_0)$, and so $e_{G_1}(u)=e_{G_2}(u)+1$. Hence, $$\begin{aligned} &&ABC_{3}(G_{1})-ABC_{3}(G_{2})\nonumber \\ &=&\sum\limits_{uv\in E(G_{0})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f(e_{G_{2}}(u),e_{G_{2}}(v))\right) \nonumber \\ & &+\sum\limits_{uv\in E(S_{t+2})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f(e_{G_{2}}(u),e_{G_{2}}(v))\right) \\ &<&\sum\limits_{uv\in E(S_{t+2})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f(e_{G_{2}}(u),e_{G_{2}}(v))\right) \\ &=&(f(d_0,d_0+1)-f(d_0,d_0+1))+t(f(d_0+1,d_0+2)-f(d_0,d_0+1)) \\ &<&0.\end{aligned}$$ **Case 2.** $d_0=d(G_0)-1$. If $d_0=d(G_0)-1=1$, then every vertex of $V(G_0)$ is adjacent to $u_0$. And so, $e_{G_{1}}(u)=e_{G_{2}}(u)+1$ for any vertex $u\in V(G_0)$. By formula (1), we have $$\begin{aligned} &&ABC_{3}(G_{1})-ABC_{3}(G_{2})\nonumber \\ &\leq&\sum\limits_{uv\in E(S_{t+2})}\left(f(e_{G_{1}}(u),e_{G_{1}}(v))-f(e_{G_{2}}(u),e_{G_{2}}(v))\right) \nonumber \\ &=&(f(2,2)-f(1,2))+t(f(2,3)-f(1,2)) \nonumber \\ &=&0.\nonumber\end{aligned}$$ The equality in last inequality holds if and only if $G_0=S_n$ with center $u_0$, since if $G_0$ is not such graph then it has an edge $xy$ not incident with $u_0$, and so $e_{G_1}(x)=e_{G_1}(y)=3$ but $e_{G_2}(x)=e_{G_2}(y)=2$, which implies formula (2). In the second subcase when $d_0=d(G_0)-1\geq 2$, we have that $e_{G_1}(u_0)=e_{G_2}(u_0)=e_{G_0}(u_0)=d_0\geq 2$, and that $e_{G_1}(x)\geq e_{G_2}(x)\geq 2$ for every vertex $x\in V(G_0)-\{u_0\}$. Furthermore, for the vertex $u\in V(G_0)$ with $d(u_0,u)=d_0\geq 2$ we deduce that $e_{G_1}(u)=d_0+2\geq 4$ and $e_{G_1}(x)\geq 3$ for any neighbor of $u$. Hence, formula (2) follows by the last two results of Lemma [Lemma 2](#lem-2-2){reference-type="ref" reference="lem-2-2"}. Therefore, formulas (3) and (4) are also true in this subcase. **Case 3.** $d_0\leq d(G_0)-2$. Since $d_0\geq 1$, it follows that $d_0\geq \lceil\frac{d(G_0)}{2}\rceil\geq 2$ in this case. And so, $e_{G_1}(x)\geq e_{G_2}(x)\geq 2$ for any vertex $x\in V(G_0)$. Furthermore, for the vertex $u\in V(G_0)$ with $d(u,u_0)=d_0$ it has a neighbor $v$ such that $e_{G_1}(u)\geq 4$, $e_{G_2}(u)\geq 3$, $e_{G_1}(v)\geq 3$ and $e_{G_2}(v)\geq 2$. Therefore, formula (2) follows by the last two results of Lemma [Lemma 2](#lem-2-2){reference-type="ref" reference="lem-2-2"}, which implies formulas (3) and (4). ◻ Let $H(n,k;n_{1},n_{2},\ldots, n_{k})$ be a unicyclic graph of order $n$ obtained from a cycle $C_{k}=v_{1}v_{2}\cdots v_{k}$ by attaching $n_{i}$ new vertices to vertex $v_{i}$ for every $i=1,2,\ldots,k$, where $\sum\limits^{k}_{i=1}n_{i}=n-k$. **Lemma 5**. *Let $G\in U_n(k)$ be a unicyclic graph of order $n\geq 6$. If $k=3$ then $ABC_3(G)\leq n\sqrt{\frac{1}{2}}$, with equality holding if and only if $G=H(n,3;n_1,n_2,n_3)$; if $k\geq 4$ then $$\begin{aligned} &&ABC_3(G) \nonumber \\ &\leq& \left\{ \begin{array}{ll} (n-k+2)f(\frac{k}{2}+1,\frac{k}{2})+(k-2)f(\frac{k}{2},\frac{k}{2}) \quad \mbox{if $k$ is even }; \\ (n-k+2)f(\frac{k+1}{2},\frac{k-1}{2})+(k-3)f(\frac{k-1}{2},\frac{k-1}{2})+f(\frac{k+1}{2},\frac{k+1}{2}) \quad \mbox{if $k$ is odd with $k\geq 5$}, \end{array} \right.\end{aligned}$$ with equality holding if and only if $G=H(n,k; n-k,0,\ldots,0)$.* *Proof.* Let $G\in U_n(k)$ be a unicyclic graph with the maximum $ABC_3$ index. By Lemma [Lemma 4](#lem-2-4){reference-type="ref" reference="lem-2-4"}, $G$ must be of the form $H(n,k;n_1,n_2,\ldots,n_k)$. And so, $$ABC_3(G)=\sum\limits_{v_iv_j\in E(C_k)}f(e_G(v_i),e_G(v_j))+\sum\limits_{\substack{v_ix\in E(G)\\ d(x)=1}}f(e_G(v_i),e_G(x)).$$ Firstly, we consider the case when $k=3$. If $n_1,n_2,n_3\geq 1$ then $e_{G}(v_i)=2$ for every vertex $v_i$ in the cycle and $e_G(x)=3$ for every pendent vertex of $G$. From the last two results of Lemma [Lemma 2](#lem-2-2){reference-type="ref" reference="lem-2-2"} it follows that $$ABC_3(G)=3\sqrt{\frac{1}{2}}+\sum\limits^3_{i=1}k_if(2,3)=n\sqrt{\frac{1}{2}}.$$ Since $n\geq 6$ and $k=3$, it follows that at least one of $n_1,n_2$ and $n_3$ is not zero. And so, for every edges of $G$, at least one of its end has eccentricity two. Hence, the above formula is true in any case. Secondly, we consider the case when $k\geq 4$. In this case, one can deduce easily that $e_G(x)\geq 2$ for every vertex $x$ of graph $G$. By Lemma [Lemma 1](#lem-2-1){reference-type="ref" reference="lem-2-1"} and [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"}, to maximize $ABC_3(G)$ it suffices to maximize $f(e_G(u),e_G(u)+1)$ for every edge $uv$ with $e_G(v)=e_G(u)+1$, and to maximize $f(e_G(u),e_G(u))$ for every edge $uv$ with $e_G(v)=e_G(u)$. Let $C=v_1v_2\cdots v_kv_1$ be the cycle of graph $G$. Notice that $e_G(v_i)\leq e_C(v_i)+1$ for every vertex $v_i\in V(C)$, and that $e_G(x)=e_C(v_i)+1$ or $e_C(v_i)+2$ for every pendent vertex $x$ adjacent to vertex $v_i\in V(C)$, with $e_G(x)=e_C(v_i)+1$ if and only if $n_{i+\frac{k}{2}(\text{mod}\ k)}=0$ when $k$ is even, and $n_{i+\frac{k-1}{2}(\text{mod}\ k)}=n_{i+\frac{k+1}{2}(\text{mod}\ k)}=0$ when $k$ is odd. In conclusion, to maximize $ABC_3(G)$ if and only if to attach every pendent vertex of graph $G$ to a unique vertex of cycle $C$, and so the extremal graph is as prescribed in this lemma. In the extremal graph $G$, only one vertex of $C$ has eccentricity $\frac{k}{2}+1$ when $k$ is even, and exactly two vertices of $C$ has eccentricity $\frac{k+1}{2}$ when $k$ is odd. Therefore, when $k$ is even $$\begin{aligned} ABC_3(G)&=&(n-k)f(\frac{k}{2}+1,\frac{k}{2})+2f(\frac{k}{2}+1,\frac{k}{2})+(k-2)f(\frac{k}{2},\frac{k}{2}) \nonumber \\ &=&(n-k+2)f(\frac{k}{2}+1,\frac{k}{2})+(k-2)f(\frac{k}{2},\frac{k}{2});\nonumber\end{aligned}$$ when $k$ is odd, $$\begin{aligned} ABC_3(G)&=&(n-k)f(\frac{k+1}{2},\frac{k-1}{2})+f(\frac{k+1}{2},\frac{k+1}{2})\nonumber \\ &&+2f(\frac{k+1}{2},\frac{k-1}{2})+(k-3)f(\frac{k-1}{2},\frac{k-1}{2}) \nonumber \\ &=&(n-k+2)f(\frac{k+1}{2},\frac{k-1}{2})+(k-3)f(\frac{k-1}{2},\frac{k-1}{2}) \nonumber \\ &&+f(\frac{k+1}{2},\frac{k+1}{2}).\nonumber\end{aligned}$$ And so, the lemma follows. ◻ **Theorem 1**. *If $G$ is a unicyclic graph of order $n\geq 6$ then $ABC_3(G)\leq n\sqrt{\frac{1}{2}}$, with equality holding if and only if $G\in \{H(n,3;n_1,n_2,n_3),H(n,4;n-4,0,\ldots,0)\}$.* *Proof.* If $G$ has even girth $k$ with $6\leq k\leq n-1$, let $$f(k)=(n-k+2)f(\frac{k}{2}+1,\frac{k}{2})+(k-2)f(\frac{k}{2},\frac{k}{2}),$$ then $$f(k-2)=(n-k+4)f(\frac{k}{2},\frac{k}{2}-1)+(k-4)f(\frac{k}{2}-1,\frac{k}{2}-1).$$ From Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} it follows that $f(k)<f(k-2)$. Hence, by the second results of Lemma [Lemma 5](#lem-2-5){reference-type="ref" reference="lem-2-5"} we have $$\begin{aligned} ABC_3(G)&\leq&ABC_3(H(n,k;n-k,0,\ldots,0))\nonumber\\ &<&ABC_3(H(n,k-2;n-k+2,0,\ldots,0))<\cdots\nonumber\\ &<&ABC_3(H(n,4;n-4,0,\ldots,0))=n\sqrt{\frac{1}{2}}.\nonumber\end{aligned}$$ If $k$ is odd with $5\leq k\leq n-1$, let $$g(k)=(n-k+2)f(\frac{k+1}{2},\frac{k-1}{2})+(k-3)f(\frac{k-1}{2},\frac{k-1}{2})+f(\frac{k+1}{2},\frac{k+1}{2}),$$ then $g(k)$ has similar property as $f(k)$. And so, $g(k)<ABC_3(H(n,3;n_1,n_2,n_3))=n\sqrt{\frac{1}{2}}$. The theorem follows. ◻ # The second maximal $ABC_3$ index of unicyclic graphs For any integer $t\geq 1$ and $n_1\geq t+1$, let $H^t(n,3;n_1-t,n_2,n_3)$ be the unicyclic graph obtained from $H(n,3;n_1-t,n_2,n_3)$ by attaching $t$ new vertices to any given pendent vertex adjacent to $v_1$ of $H(n,3;n_1-t,n_2,n_3)$. Similarly, $H^t(n,k;n-k-t,0,\ldots,0)$ is the unicyclic graph obtained from $H(n-t,k;n-k-t,0,\ldots,0)$ by identifying one of its pendent vertex with the center of star $S_{t+1}$. This section will characterize the unicyclic graph with second maximal $ABC_3$ index. **Lemma 6**. *If integer $t\geq 1$ then $ABC_3(H^t(n,3;n-3-t,0,0))=(n-1)\sqrt{\frac{1}{2}}+f(3,3)$; if integers $t\geq 2$ and $k\geq 4$ then $ABC_3(H^t(n,k;n-k-t,0,\ldots,0))<ABC_3(H^1(n,k;n-k-1,0,\ldots,0))$.* *Proof.* Let $w$ be any pendent vertex of $H^t(n,k;n-k-t,0,\ldots,0)$ that has maximum eccentricity, and $u$ be its neighbor. Noticing that the cycle $C$ of $H^t(n,k;n-k-t,0,\ldots,0)$ has unique vertex $v$ with $e_{H^t(n,k;n-k-t,0,\ldots,0)}(v)=e_{H^t(n,k;n-k-t,0,\ldots,0)}(w)=\frac{k}{2}+2$ if $k$ is even, and that $C$ has exactly two vertices $v_i$ and $v_j$ with maximum eccentricity $\frac{k+3}{2}$ if $k$ is odd, when $k$ is even we have $$\begin{aligned} &&ABC_3(H^t(n,k;n-k-t,0,\ldots,0))\nonumber\\ &=&(t+2)f(\frac{k}{2}+2,\frac{k}{2}+1)+(n-k-t+2)f(\frac{k}{2}+1,\frac{k}{2})+(k-4)f(\frac{k}{2},\frac{k}{2}).\nonumber\end{aligned}$$ Since $k\geq 4$, from Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} it follows that $f(\frac{k}{2}+1,\frac{k}{2})>f(\frac{k}{2}+2,\frac{k}{2}+1)$. And so, $$\begin{aligned} &&ABC_3(H^t(n,k;n-k-t,0,\ldots,0))-ABC_3(H^1(n,k;n-k-1,0,\ldots,0))\nonumber\\ &=&(t-1)(f(\frac{k}{2}+2,\frac{k}{2}+1)-f(\frac{k}{2}+1,\frac{k}{2}))\nonumber\\ &<&0.\nonumber\end{aligned}$$ Similarly, when $k\geq 5$ is odd we have $$\begin{aligned} &&ABC_3(H^t(n,k;n-k-t,0,\ldots,0))\nonumber\\ &=&(t+2)f(\frac{k+3}{2},\frac{k+1}{2})+(n-k-t+2)f(\frac{k+1}{2},\frac{k-1}{2})\nonumber\\ &&+f(\frac{k+3}{2},\frac{k+3}{2})+(k-5)f(\frac{k-1}{2},\frac{k-1}{2}).\nonumber\end{aligned}$$ And so, from Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} it follows that $$\begin{aligned} &&ABC_3(H^t(n,k;n-k-t,0,\ldots,0))-ABC_3(H^1(n,k;n-k-1,0,\ldots,0))\nonumber\\ &=&(t-1)(f(\frac{k+3}{2},\frac{k+1}{2})-f(\frac{k+1}{2},\frac{k-1}{2}))\nonumber\\ &<&0.\nonumber\end{aligned}$$ Finally, when $k=3$, since $t\geq 1$ by direct calculation we have that $ABC_3(H^t(n,3;n-3-t,0,0))=(n-1)\sqrt{\frac{1}{2}}+f(3,3)$. And so, the lemma follows. ◻ **Lemma 7**. *Let $G$ be an $n$-vertex unicyclic graph with $n\geq 6$ and girth $k\geq 3$. If $k=3$ and $G\neq H(n,3;n_1,n_2,n_3)$ then $ABC_3(G)\leq (n-1)\sqrt{\frac{1}{2}}+f(3,3)$, with equality holding if and only if $G=H^t(n,3;n-3-t,0,0)$; if $k=4$ and $G\neq H(n,4;n-4,0,0,0)$ then $ABC_3(G)\leq (n-1)\sqrt{\frac{1}{2}}+f(3,3)$, with equality holding if and only if $G=H(n,4;n_1,n-4-n_1,0,0)$ with $n_1\geq 1$ and $n-n_1-4\geq 1$.* *Proof.* Let $G$ be an extremal unicyclic graph with girth $k$ and second maximal $ABC_3$. When $k=3$, since $G\neq H(n,3;n_1,n_2,n_3)$ it follows from Lemma [Lemma 4](#lem-2-4){reference-type="ref" reference="lem-2-4"} that $G= H^t(n,3;n_1-t,n_2,n_3)$ with $t,n_1-t\geq 1$. If $n_2\geq 1$ or $n_3\geq 1$, noticing that $n_1+n_2+n_3=n$, by Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} we have $$\begin{aligned} &&ABC_3(H^t(n,3;n_1-t,n_2,n_3))\nonumber\\ &=&(n_2+n_3+t)f(3,4)+(n_1-t+2)f(2,3)+f(3,3)\nonumber\\ &<&(n-1)\sqrt{\frac{1}{2}}+f(3,3)\nonumber\\ &=&ABC_3(H^t(n,3;n-t-3,0,0)).\nonumber\end{aligned}$$ This contradiction and Lemma [Lemma 6](#lem-3-1){reference-type="ref" reference="lem-3-1"} imply that $n_2=n_3=0$, namely $G=H^t(n,3;n-t-3,0,0)$ for any $1\leq t\leq n_1-1$. And so, the first statement of Lemma [Lemma 7](#lem-3-2){reference-type="ref" reference="lem-3-2"} is true. When $k=4$, since $G\neq H(n,4;n-4,0,0,0)$, from Lemma [Lemma 4](#lem-2-4){reference-type="ref" reference="lem-2-4"} it follows that $G\in \{H(n,4;n_1,\ldots,n_4),H^t(n,4;n-4-t,0,0,0)\}$, where $|\{i:n_i\geq 1,i=1,2,3,4\}|\geq 2$ and $t\geq 1$. By Lemma [Lemma 6](#lem-3-1){reference-type="ref" reference="lem-3-1"}, we deduce that $t=1$. Now, by direct calculation one can obtain easily that $$\begin{aligned} ABC_3(H^1(n,4;n-5,0,0,0)) &=&f(3,4)+(n-1)\sqrt{\frac{1}{2}}\nonumber\\ &<&f(3,3)+(n-1)\sqrt{\frac{1}{2}}\nonumber\\ &=&ABC_3(H(n,4;n_1,n-n_1-4,0,0)).\nonumber\end{aligned}$$ Similarly, one can deduce by direct calculation that $ABC_3(H(n,4;n_1,n_2,n_3,n_4))\leq ABC_3(H(n,4;n_1,n-n_1-4,0,0))$ if $|\{i:n_i\geq 1,i=1,2,3,4\}|\geq 2$, with equality holding if and only if $H(n,4;n_1,n_2,n_3,n_4)$ is isomorphic to $H(n,4;n_1,n-n_1-4,0,0)$. Hence, the second statement of this lemma is also true. ◻ **Theorem 2**. *If $G\notin \{H(n,4;n-4,0,0,0),H(n,3;n_1,n_2,n_3\}$ is a unicyclic graph of order $n\geq 6$ then $ABC_3(G)\leq(n-1)\sqrt{\frac{1}{2}}+f(3,3)$, with equality holding if and only if $G\in \{H(n,5;n-5,0,0,0,0),H(n,4;n_1,n-n_1-4,0,0),H^t(n,3;n-t-3,0,0)\}$, where $n_1,n-n_1-4\geq 1$ for $H(n,4;n_1,n-n_1-4,0,0)$, and $t,n-t-3\geq 1$ for $H^t(n,3;n-t-3,0,0)$.* *Proof.* Let $G\notin \{H(n,4;n-4,0,0,0),H(n,3;n_1,n_2,n_3)\}$ be an extremal unicyclic graph with second maximal $ABC_3$ index. By Lemma [Lemma 7](#lem-3-2){reference-type="ref" reference="lem-3-2"}, the theorem is true when graph $G$ has girth $k=3$ or $4$. When $k\geq 5$, by Lemma [Lemma 5](#lem-2-5){reference-type="ref" reference="lem-2-5"} we have that $G=H(n,k;n-k,0,\ldots,0)$. If $k\geq 5$ is even, from Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} and [Lemma 5](#lem-2-5){reference-type="ref" reference="lem-2-5"} it follows that $$\begin{aligned} ABC_3(G)&=&(n-k+2)f(\frac{k}{2}+1,\frac{k}{2})+(k-2)f(\frac{k}{2},\frac{k}{2})\nonumber\\ &\leq&(n-k+2)f(4,3)+(k-2)f(3,3)\nonumber\\ &<&(n-1)\sqrt{\frac{1}{2}}+f(3,3)\nonumber\\ &=&ABC_3(H(n,4;n_1,n-n_1-4,0,0)) \ \text{where}\ n_1,n-n_1-4\geq 1.\nonumber\end{aligned}$$ This contradiction shows that $G$ cannot has even girth $k\geq 5$. If $G$ has odd girth $k\geq 5$, by Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} and [Lemma 5](#lem-2-5){reference-type="ref" reference="lem-2-5"} we deduce that $$\begin{aligned} &&ABC_3(G)\nonumber\\ &=&(n-k+2)f(\frac{k+1}{2},\frac{k-1}{2})+(k-3)f(\frac{k-1}{2},\frac{k-1}{2})+f(\frac{k+1}{2},\frac{k+1}{2})\nonumber\\ &\leq&(n-k+2)f(3,2)+(k-3)f(2,2)+f(3,3)\nonumber\\ &=&(n-1)\sqrt{\frac{1}{2}}+f(3,3),\nonumber\end{aligned}$$ where the equality hold in the inequality if and only if $k=5$. Therefore, the unique extremal graph in this case is $H(n,5;n-5,0,0,0,0)$. And so, the theorem follows. ◻ # Maximal $ABC_3$ index of trees Let $C(a_1,a_2,\ldots,a_{d-1})$ be a caterpillar, which is obtained from path $P_d=v_0v_1\cdots v_d$ by attaching $a_i$ isolated vertices to vertex $v_i$ for every $i=1,2,\ldots,d-1$. Clearly, $C(a_1,a_2,\ldots,a_{d-1})$ has diameter $d$ and order $n=\sum\limits_{i=1}^{d-1}a_i+d+1$. For simplicity, let $C_{n,d}$ denote the set of graph $C(0,\ldots,0,a_{\frac{d}{2}},0,\ldots,0)$ when $d$ is even, and $C_{n,d}$ denote the set of all such graphs $C(0,\ldots,0,a_{\frac{d-1}{2}},a_{\frac{d+1}{2}},0,\ldots,0)$ when $d$ is odd. **Lemma 8**. *If $T$ is an $n$-vertex tree with diameter $d\geq 2$, then $$\begin{aligned} \nonumber ABC_3(G) \leq \left\{ \begin{array}{ll} (n-1)\sqrt{\frac{1}{2}}\quad \mbox{$d=2,3$}; \\ (n-d-1)f(\frac{d}{2}+1,\frac{d}{2})+ABC_3(P_d) \quad \mbox{if $d$ is even with $d\geq 4$}; \\ (n-d-1)f(\frac{d+3}{2},\frac{d+1}{2})+ABC_3(P_d) \quad \mbox{if $d$ is odd with $d\geq 5$}, \end{array} \right.\end{aligned}$$ with the equality holding if and only if $T\in C_{n,d}$.* *Proof.* The lemma is obviously true when $2\leq d\leq 3$, since $T$ is the prescribed caterpillar and its $ABC_3$ index can be obtained by direct calculation. So, we assume in what follows that $d\geq 4$, and $T$ has maximal $ABC_3$ index. Let $P_d=v_0v_1\cdots v_d$ be a longest path of $T$. By Lemma [Lemma 4](#lem-2-4){reference-type="ref" reference="lem-2-4"}, we deduce that $T=C(a_1,a_2\ldots, a_{d-1})$. When $d$ is even, $e_{P_d}(v_r)>e_{P_d}(v_{\frac{d}{2}})=\frac{d}{2}\geq 2$ for $v_r\in V(P_d)\backslash v_{\frac{d}{2}}$. And so, by Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"}, if we delete the $a_r$ pendent vertices adjacent to $v_r$ and add $a_r$ pendent vertices adjacent to the center vertex $v_{\frac{d}{2}}$, the $ABC_3$ index will increase. This observation shows that $T\in C_{n,d}$ in this case. Similarly, one can deduce with case that $T\in C_{n,d}$ when $d\geq 5$ is odd. Their $ABC_3$ index can also be easily obtained as are listed in the lemma. ◻ **Theorem 3**. *If $T$ is an $n$-vertex tree with diameter $d\geq 2$ then $ABC_3(T)\leq (n-1)\sqrt{\frac{1}{2}}$, with equality holding if and only if $T\in C_{n,3}\cup \{S_n\}$. Furthermore, $C_{n,4}$ has second maximal $ABC_3$ index.* *Proof.* Firstly, by direct calculation, we have $$\begin{aligned} \nonumber ABC_3(P_d)= \left\{ \begin{array}{ll} 2(f(\frac{d}{2}+1,\frac{d}{2})+f(\frac{d}{2}+2,\frac{d}{2}+1)+\cdots+f(d,d-1)) \quad \mbox{if $d$ is even with $d\geq 4$}; \\ f(\frac{d+1}{2},\frac{d+1}{2})+2(f(\frac{d+3}{2},\frac{d+1}{2})+f(\frac{d+5}{2},\frac{d+3}{2})+\cdots+f(d,d-1)) \quad \mbox{if $d$ is odd with $d\geq 5$}. \end{array} \right.\end{aligned}$$ For simplicity, the following symbol $C_{n,d}$ denotes any graph in it. When $d\geq 4$ is even, we can deduce by Lemma [Lemma 8](#lem-4-1){reference-type="ref" reference="lem-4-1"} and Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} that $$\begin{aligned} &&ABC_3(C_{n,d+1})-ABC_3(C_{n,d}) \nonumber\\ &=&(n-d-2)f(\frac{d}{2}+2,\frac{d}{2}+1)+f(\frac{d}{2}+1,\frac{d}{2}+1)+2(f(\frac{d}{2}+2,\frac{d}{2}+1)\nonumber\\ &&+f(\frac{d}{2}+3,\frac{d}{2}+2)+\cdots+f(d+1,d))-(n-d-1)f(\frac{d}{2}+1,\frac{d}{2})\nonumber\\ &&-2(f(\frac{d}{2}+1,\frac{d}{2})+f(\frac{d}{2}+2,\frac{d}{2}+1)+\cdots+f(d,d-1))\nonumber\\ &<&0.\nonumber\end{aligned}$$ Similarly, when $d\geq 5$ is odd, by Lemma [Lemma 8](#lem-4-1){reference-type="ref" reference="lem-4-1"} and Lemma [Lemma 3](#lem-2-3){reference-type="ref" reference="lem-2-3"} we have $$\begin{aligned} &&ABC_3(C_{n,d+1})-ABC_3(C_{n,d}) \nonumber\\ &=&(n-d-2)f(\frac{d+1}{2}+1,\frac{d+1}{2})+2(f(\frac{d+1}{2}+1,\frac{d+1}{2}) +f(\frac{d+1}{2}+2,\frac{d+1}{2}+1)\nonumber\\ &&+\cdots+f(d+1,d))-(n-d-1)f(\frac{d+3}{2},\frac{d+1}{2})-f(\frac{d+1}{2},\frac{d+1}{2})\nonumber\\ &&-2(f(\frac{d+3}{2},\frac{d+1}{2})+f(\frac{d+5}{2},\frac{d+3}{2})+\cdots+f(d,d-1))\nonumber\\ &=&(n-d-2)f(\frac{d+1}{2}+1,\frac{d+1}{2})+2f(d+1,d)\nonumber\\ &&-(n-d-1)f(\frac{d+3}{2},\frac{d+1}{2})-f(\frac{d+1}{2},\frac{d+1}{2})\nonumber\\ &<&0.\nonumber\end{aligned}$$ These two observation show that if $d\geq 5$ then $ABC_3(C_{n,d})<ABC_3(C_{n,4})$. Since $$\begin{aligned} ABC_3(C_{n,4})&=&(n-4-1)f(\frac{4}{2}+1,\frac{4}{2})+ABC_3(P_4)\nonumber\\ &=&(n-5)f(2,2)+2f(4,3)+2f(3,2)\nonumber\\ &=&(n-3)\sqrt{\frac{1}{2}}+2f(4,3)\nonumber\\ &<&(n-1)\sqrt{\frac{1}{2}},\nonumber\end{aligned}$$ it follows that the theorem is true. ◻ # Data availability {#data-availability .unnumbered} No data was used for the research described in the article. # Acknowledgments {#acknowledgments .unnumbered} This work is supported by National Natural Science Foundation of China (Grant No. 12061074), the Doctoral Research Foundation of Longdong University (No. XYBYZK2306). 00 R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH. Weinheim. 2000. M. Randić, On characterization of molecular branching, J. Am, Chem. Soc. 97(1975)6609-6615. E. Estrada, L. Torres, L. Rodriguez, I. Gutman, An Atom-bond connectivity index: Modeling the enthalpy of formation of alkanes, Indian J. Chem. Sect. A 37(1998)849-855. E. Estrada, Atom-bond connectivity index and the energetic of branched alkanes. Phys. Lett. 463(2008)422-425. K. C. Das, Atom-bond connectivity index of graphs, Discrete Appl. Math. 158(2010)1181-1188. K. C. Das, I. Gutman, B. Furtula, On atom-bond connectivity index, Chem. Phys. Lett. 463(2011)452-454. A. Graovac, M. Ghorbani, A new version of atom-bond connectivity index, Acta Chim. Slov. 57(2010)609-612. K. C. Das, K. Xu, A. Graovac, Maximal unicyclic graphs with respect to new atom-bond connectivity index, Acta Chim. Slov. 60(2013)34-42. W. Gao, W. Wang, Second atom-bond connectivity index of special chemical molecular structures, Journal of Chemistry, 2014, http://dx.doi.org/10/1155/2014/906254. M. Rostami, M. Sohrabi-Haghighat, Further results on new version of atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 71(2014)21-32. Z. Tang, Y. Hou, Note on the second atom-bond connectivity index, Journal of Natural Science of Hunan Normal University. 38(2015)80-84. B. Furtula, Atom-bond connectivity index versus Graovac-Ghorbani analog, MATCH Commun. Math. Comput. Chem. 75(2016)233-242. Dae-Won Lee, Some lower and supper bounds on the third index, AKCE International Journal of Graphs and Combinatorics. 13(2016)11-15. D. Vukicevic, A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chim. Slov. 57(2010)524-528. R. Xing, B. Zhou, F. Dong, On atom-bond connectivity index of connected graphs, Discrete Appl. Math. 159(2011)1617-1630. [^1]: Corresponding author. 15819931748\@163.com(R. Song).

arxiv_math

--- abstract: | We study sparse domination for operators defined with respect to an atomic filtration on a space equipped with a general measure $\mu$. In the case of Haar shifts, $L^p$-boundedness is known to require a weak regularity condition, which we prove to be sufficient to have a sparse domination-like theorem. Our result allows us to characterize the class of weights where Haar shifts are bounded. A surprising novelty is that said class depends on the complexity of the Haar shift operator under consideration. Our results are qualitatively sharp. address: - José M. Conde Alonso Departamento de Matemáticas Universidad Autónoma de Madrid C/ Francisco Tomás y Valiente sn Madrid, Spain - Jill Pipher Department of Mathematics Brown University Thayer Street Providence, RI, 02912 USA - Nathan A. Wagner Department of Mathematics Brown University Thayer Street Providence, RI, 02912 USA author: - José M. Conde Alonso - Jill Pipher - Nathan A. Wagner date: today title: Balanced measures, sparse domination and complexity-dependent weight classes --- # Introduction {#introduction .unnumbered} Probabilistic techniques and methods are ubiquitous in harmonic analysis and especially so in Calderón-Zygmund theory. Often these methods make use of the filtration of $\sigma$-algebras generated by the dyadic system of cubes $$\mathscr{D}:= \bigcup_{k\in\mathbb{Z}} \mathscr{D}_k = \left\{2^{-k}[j_1,j_1+1) \times [j_2,j_2+1) \times \ldots [j_n,j_n+1) : j \in \mathbb{Z}^n \right\}.$$ Probabilistic stopping-time arguments have a long history as a critical tool in analysis; for example, in the Calderón-Zygmund decomposition, which is used to prove a weak-type estimate for singular integrals. Some fundamental insights into the close connection between certain function spaces and operators in analysis and their probabilistic counterparts were established in: [@GJ1982], which explored the relationship between BMO and dyadic BMO, and later in the dyadic representation theorems of [@P2000; @Hyt2012]. In particular, the main result in [@P2000] recovers the Hilbert transform as an average of dyadic operators: $$\label{eq:PetermichlRep} \tag{HRep} \mathcal{H}f(x) = c_0 \int_\Omega \Sh_{\mathscr{D}^\omega}f(x) dP(\omega),$$ where $\mathcal{H}$ is the Hilbert transform, $(\Omega,P)$ is a probability space, $\mathscr{D}^\omega$ is a different dyadic system for each value of $\omega$ and the dyadic Hilbert transform is the operator $$\label{eq:DyadicHilbert} \tag{HDyad} \Sh_{\mathscr{D}^\omega}f(x) = \sum_{I \in \mathscr{D}^\omega} \langle f, h_I\rangle(h_{I_-}(x)-h_{I_+}(x)),$$ where $h_I=2^{-1}(|I_-|^{-1/2}1_{I_-}-|I_+|^{-1/2}1_{I_+})$ denotes the Haar function associated with the interval $I$, and $I_-$ (resp. $I_+$) is the left (resp. right) dyadic child of $I$. The main result in [@Hyt2012] expands on [\[eq:PetermichlRep\]](#eq:PetermichlRep){reference-type="eqref" reference="eq:PetermichlRep"} by showing that any Calderón-Zygmund operator on $\mathbb{R}^n$ can be written as an average of higher-dimensional generalizations of $\Sh$. Once it has been established a continuous operator can be recovered from probabilistic counterparts, one can often more readily derive quantitative consequences, like weighted estimates. One efficient way of doing so is the sparse domination technique, initiated in [@Ler2013] and applied to get pointwise estimates for Calderón-Zygmund operators via estimates for the dyadic objects that represent that operator [@CR2016; @LN2019]. Sparse domination is not the only possible path: direct estimates are possible, like those in [@Lacey2017] or using the one-third trick as in [@Mei2003]. The setting of the results discussed above is $\mathbb{R}^n$, equipped with the Lebesgue measure, which is a natural one for the problems under study. However, other measures $\mu$ on $\mathbb{R}^n$ also appear naturally, arising from questions in geometry or partial differential equations. For example, the class of representing measures associated to elliptic or parabolic divergence form operators are all "doubling\". A measure $\mu$ is said to be doubling if there exists a constant $c_\mu$ such that for all balls $B$ centered at its support, there holds $$\label{eq:doubling} \tag{Db} \mu(2B) \leq c_\mu \mu(B).$$ Another important class of measures is the class of measures of *polynomial growth*. These may fail [\[eq:doubling\]](#eq:doubling){reference-type="eqref" reference="eq:doubling"}. Despite the lack of the doubling property, using mainly geometric techniques, a very complete Calderón-Zygmund theory has been developed for measures satisfying the polynomial growth condition in [@NTV1998; @To2001; @To2001b; @NTV2003]. These developments led to profound applications, among which are: the solution of the Painlevé problem [@To2003], the $n-1$ case of the David-Semmes problem [@NTV2014], and the characterization of the rectifiability of harmonic measure [@AHMMMTV2016]. In martingale language, [\[eq:doubling\]](#eq:doubling){reference-type="eqref" reference="eq:doubling"} is closely related to the regularity of the filtration. Regularity of the filtration is equivalent, in the case of the dyadic filtration on $(\mathbb{R}^n,\mu)$, to the condition $\mu(Q) \sim \mu(\widehat{Q})$ for all cubes in $\mathscr{D}$, where $\hat{Q}$ is the "parent\" dyadic cube of $Q$. Probabilistic tools are usually versatile enough to be applicable to $\sigma$-finite filtered spaces with highly non-regular structure, where the most important martingale inequalities hold true [@BG1970; @Dav1970]. Therefore, one might be tempted to think that the techniques and results described above are still powerful even when the geometric and measure theoretic properties of the ambient space are not tightly coupled, even if [\[eq:doubling\]](#eq:doubling){reference-type="eqref" reference="eq:doubling"} fails. However, when the classical doubling condition of the underlying measure fails, the situation becomes significantly more complicated. The one-third trick is no longer useful in the usual way, and there are limited alternatives available [@C2020]. Moreover, direct arguments of sparse domination for Calderón-Zygmund operators [@CP2019; @VZ2018] yield quantitative estimates that seem far from tight. To start, just as in the doubling setting, they do imply weighted inequalities for Calderón-Zygmund operators. Recall that a weight is an a.e. positive function $w\in L_{\mathrm{loc}}^1(d\mu)$ that we identify with the measure $d\nu=w d\mu$. The results in [@CP2019; @VZ2018] imply that a Calderón-Zygmund operator is bounded on $L^2(w d\mu)$ whenever $w\in A_2(\mu)$, that is, the following holds: $$\label{eq:A2cond} \tag{A2} [w]_{A_2(\mu)} :=\sup_{B \; \mathrm{ball}} \frac{w(B)w^{-1}(B)}{\mu(B)^2} < \infty.$$ However, in [@To2007], the necessary and sufficient condition that weights must satisfy so that Calderón-Zygmund operators are bounded on $L^2(w d\mu)$, termed $\mathcal{Z}_2(\mu)$, was shown to be strictly weaker than the classical $A_2(\mu)$ condition. Moreover, the issue of a representation theorem along the lines of [\[eq:PetermichlRep\]](#eq:PetermichlRep){reference-type="eqref" reference="eq:PetermichlRep"} or [@Hyt2012] remained wide open. In that direction, the study of the operators that arise in the representation formulas is an interesting problem that can inform what, if any, representation theorems one can expect when the measure under consideration is not Lebesgue. Exploring the feasibility of extending this program to the non-doubling setting was one of the motivations of this work. We begin this study on $\mathbb{R}$, and point out that there will be additional considerations needed to extend the forthcoming results to higher dimensions. ## Sparse domination for Haar shifts with respect to $\mu$ {#sparse-domination-for-haar-shifts-with-respect-to-mu .unnumbered} Let $\mu$ be a Radon, atomless measure on $\mathbb{R}$. We use the standard notation for dyadic ancestors: $I^{(j)}$ is the interval from $\mathscr{D}$ that contains $I$ and has sidelength equal to $2^j \ell(I)$. We denote $$\mathscr{D}_j(I)=\{J \in \mathscr{D}: J^{(j)} = I\}, \;\mbox{ and also } \mathscr{D}_{\leq j}(I)= \bigcup_{k=0}^j \mathscr{D}_k(I).$$ Given an $I \in \mathscr{D}$, the Haar function associated to $I$ is constant on the dyadic children of $I$, normalized in $L^2(\mu)$, and has mean value zero: $$h_I(x) = \sqrt{\frac{\mu(I_-)\mu(I_+)}{\mu(I)}}\left(\frac{1_{I-}(x)}{\mu(I_-)}-\frac{1_{I+}(x)}{\mu(I_+)}\right).$$ Given nonnegative integers $s$ and $t$, a Haar shift of complexity $(s,t)$ is an operator of the form $$T^{s,t}f(x)=\sum_{I \in \mathscr{D}} \sum_{J \in \mathscr{D}_r(I)} \sum_{K \in \mathscr{D}_s(I)} \alpha_{J,K}^{I} \langle f, h_J \rangle h_K,$$ where the pairing $\langle \cdot, \cdot\rangle$ (and integral averages $\langle \cdot \rangle_I$) are taken with respect to $\mu$: $$\langle f,g\rangle = \int_{\mathbb{R}^n} f(x) g(x) d\mu(x), \quad \langle f \rangle_Q = \frac{1}{\mu(Q)} \int_Q f(x) d\mu(x).$$ In this language, $\Sh$ is an operator of complexity $(0,1)$. If $\alpha \in \ell^\infty$, then $T^{s,t}$ is bounded on $L^2(\mu)$ regardless of the properties of $\mu$. We will always assume that $\|\alpha\|_{\ell^\infty} \leq 1$. In [@LMP2014], a condition is identified that characterizes $L^p(\mu)$ and weak type $(1,1)$ boundedness of $T^{s,t}$ when $r$ and $s$ are nonzero: if $$m(I) = \frac{\mu(I_-)\mu(I_+)}{\mu(I)},$$ then $T^{s,t}$ is of weak type $(1,1)$ if and only if, for all $I\in\mathscr{D}$, $$\label{eq:mBalanced} \tag{Balance} m(I) \sim m(\widehat{I}),$$ where $\widehat{I}$ denotes the dyadic parent of $I$. When [\[eq:mBalanced\]](#eq:mBalanced){reference-type="eqref" reference="eq:mBalanced"} holds and $\mu$ is free of atoms, we will say that $\mu$ is *balanced*[^1]. The first question that we address in this paper is whether the operators $T^{s,t}$ satisfy sparse bounds for balanced measures $\mu$. In the case of the Lebesgue measure, this is contained in [@Hyt2012; @CR2016]. We first consider estimates in the dual sense: we look for an inequality like $$\label{eq:sparseNo} \tag{ClasSp} \langle T^{s,t}f_1,f_2 \rangle \lesssim \sum_{I \in \mathcal{S}} \langle f_1 \rangle_Q \langle f_2 \rangle_Q \mu(Q) =: \mathcal{A}_\mathcal{S}(f_1,f_2),$$ for nice enough $f_1$ and $f_2$ and a family $\mathcal{S}=\mathcal{S}(f_1,f_2) \subset \mathscr{D}$ which is sparse in the usual sense: for each $I\in \mathcal{S}$, there exists $E_I \subset I$ such that $\mu(E_I) \sim \mu(I)$ and the family $\{E_I\}_{I \in \mathcal{S}}$ is pairwise disjoint. The question is natural, since sparse domination methods are usually well adapted to dyadic operators. Notably, the simple argument in [@CDO2018] might be expected to go through in the balanced case once the classical Calderón-Zygmund decomposition employed there is replaced by the one in [@LMP2014], (or the streamlined version from [@CCP2022]). In this direction, our first result is a negative one: we shall show that the classical sparse domination inequality, [\[eq:sparseNo\]](#eq:sparseNo){reference-type="eqref" reference="eq:sparseNo"}, must fail, by constructing a balanced measure $\mu$ on the interval $[0,1]$ so that $\Sh$ fails sparse domination. Of course, similar examples can be constructed to disprove sparse domination in general for higher complexity dyadic shifts. ## Modified sparse forms {#modified-sparse-forms .unnumbered} The above discussion shows that sparse domination, in the classical formulation, must fail for some balanced measures $\mu$. To remedy that, we propose a variant of the sparse form in the following way: $$\mathcal{C}_{\mathcal{S}}(f_1,f_2) := \sum_{I \in \mathcal{S}} \sum_{J \in \mathcal{S}\cap \mathscr{D}_{\leq 2}(\widehat{I})} \left[\langle f_1 \rangle_I \langle f_2 \rangle_{J} + \langle f_2 \rangle_I \langle f_1 \rangle_{J}\right] m(I).$$ The role of the second sum in the definition of $\mathcal{C}_\mathcal{S}$ takes into account the interaction of intervals $I$ and its dyadic neighbors, which is necessary for the sparse domination argument to work when the complexity is nonzero. On the other hand, replacing $\mu(I)$ by $m(I)$ on the right hand side helps one to obtain quantitative bounds for $\mathcal{C}_\mathcal{S}$, since we have $m(I) \ll \mu(I)$ in general. With this definition, the result that we obtain reads as follows: **Theorem 1**. *Let $\mu$ be balanced, and let $T$ be a Haar shift of complexity $(s,t)$ with $s + t \leq 1$. Then for each pair of compactly supported, bounded nonnegative functions $f_1,f_2$ there exists a sparse collection $\mathcal{S}\subset \mathscr{D}$ such that $$| \langle Tf_1,f_2\rangle | \lesssim \mathcal{A}_\mathcal{S}(f_1,f_2) + \mathcal{C}_\mathcal{S}(f_1,f_2).$$* We postpone the full statement of Theorem [Theorem 1](#th:thmA){reference-type="ref" reference="th:thmA"} for Haar shifts of higher complexity to Section [2](#sec:section2){reference-type="ref" reference="sec:section2"}. As is standard, the proof of Theorem [Theorem 1](#th:thmA){reference-type="ref" reference="th:thmA"} uses the weak $(1,1)$ bound for the maximal operator $\mathcal{M}$ associated to the dyadic grid $\mathscr{D}$. On the other hand, both forms $\mathcal{A}_\mathcal{S}$ and $\mathcal{C}_\mathcal{S}$ satisfy the expected weak-type and $L^p(\mu)$ estimates, which is evidence that this definition of the sparse forms is the right one in the context of balanced measures. The broad scheme of the proof of theorem [Theorem 1](#th:thmA){reference-type="ref" reference="th:thmA"} is similar to arguments that were used in the context of doubling measures like [@BFP2016; @CCDO2017], and is most similar to [@CDO2018]. However, we need to use a Calderón-Zygmund decomposition adapted to general measures, originally inspired by Gundy's decomposition for martingales [@LMP2014; @CCP2022]. An additional feature that we need to use is a $\mathrm{BMO}$ estimate for the good part of the decomposition, which is a technical novelty key to our approach. ## A weighted theory for balanced measures {#a-weighted-theory-for-balanced-measures .unnumbered} The main applications of sparse bounds are weighted estimates. The classical sparse domination yields quantitatively sharp estimates for doubling measures. In the case of the Lebesgue measure, both Calderón-Zygmund operators and Haar shifts are bounded on $L^2(w)$ if and only if $w \in A_2$. The corresponding condition with respect to a general measure $\mu$ is just [\[eq:A2cond\]](#eq:A2cond){reference-type="eqref" reference="eq:A2cond"}, where we should take the sup with respect to all $Q \in \mathscr{D}$ in the Haar shift case (instead of over balls). As we mentioned before, in Calderón-Zygmund case the $A_2(\mu)$ condition is not necessary in general, but it is certainly sufficient [@To2007]. In the dyadic setting, the situation is different: we can show that the $A_2(\mu)$ condition is not sufficient for the boundedness of $\Sh$, by constructing a pair $(\mu,w)$ where $\mu$ is balanced, $w\in A_2(\mu)$ and $\Sh$ is not bounded on $L^2(w d\mu)$. Our main weighted result is the identification of the right class of weights that governs the boundedness of Haar shifts. A notable feature of this identification is the fact that it takes into account the higher complexity of the operators that we consider. Moreover, unlike the sparse form, the weight class is the same for all operators of complexity greater than or equal to 1. **Theorem 2**. *Let $\mu$ be balanced. There exists a collection of weights $A_2^{b}$ (balanced $A_2$) with the following properties:* - *$A_2^{b}\subseteq A_2(\mu)$.* - *If $(s,t) \neq (0,0)$, all operators $T^{s,t}$ are bounded on $L^2(w d\mu)$ if and only if $w \in A_2^{b}$.* *Moreover, there exists a balanced, nondoubling measure $\mu$ so that $A_2^{b}\subsetneq A_2(\mu).$* We postpone, until section [3](#sec:section3){reference-type="ref" reference="sec:section3"}, the precise definition of the class $A_2^{b}(\mu)$, which can be described as a variant of [\[eq:A2cond\]](#eq:A2cond){reference-type="eqref" reference="eq:A2cond"} involving averages over dyadic intervals $I, J$ where $J \in \mathscr{D}_{\leq 2}(\widehat{I})$. As one would expect, in case $\mu$ is doubling, we have that $A_2^{b}(\mu)=A_2(\mu)$. Our weighted results can be readily generalized to $L^p(w d\mu)$, with a straightforward modification of our $A_2$-type class. Finally, we have a maximal function characterization. We formulate it by identifying a family of variants of the usual dyadic maximal function, that we denote by $\mathcal{M}^j$. $\mathcal{M}^j$ is $L^p(\mu)$-bounded and weak-type $(1,1)$, and satisfies the same sparse domination as the Haar shifts $T^{s,t}$ whenever $r+s=j$. We then use each $\mathcal{M}^j$ to define a family of $A_p$-type classes of weights: we say that $w\in A_1^j(\mu)$ if $\mathcal{M}^jw(x) \lesssim w(x)$, while we say that $w\in A_p^j(\mu)$ if $\mathcal{M}^j:L^p(w d\mu) \to L^p(w d\mu)$. Even though the operators $\mathcal{M}^j$ are not equivalent, the weight classes that they determine are the same; that is, $A_p^j(\mu) = A_p^{b}(\mu)$ for all $j \geq 1$. The rest of the paper is organized as follows: Section [1](#sec:section1){reference-type="ref" reference="sec:section1"} contains an example that shows that sparse domination in the usual sense must fail for Haar shifts and some balanced measures, and another one of $w \in A_2(\mu)$ \|with $\mu$ balanced\| so that $\Sh$ is not bounded on $L^2(w d\mu)$. Section [2](#sec:section2){reference-type="ref" reference="sec:section2"} contains the proof of Theorem [Theorem 1](#th:thmA){reference-type="ref" reference="th:thmA"} and our sparse domination result for general complexity shifts on $\mathbb{R}$. Section [3](#sec:section3){reference-type="ref" reference="sec:section3"} contains the weighted theory, including the proof of Theorem [Theorem 2](#th:thmB){reference-type="ref" reference="th:thmB"}. ## Acknowledgements {#acknowledgements .unnumbered} The authors would like to thank Sergei Treil for several valuable discussions. The first named author has been supported by grants CNS2022-135431 and RYC2019-027910-I (Ministerio de Ciencia, Spain). The third named author is supported by grant DMS-2203272 (National Science Foundation). # Failure of usual sparse domination {#sec:section1} We shall work with Borel measures $\mu$ on $\mathbb{R}$ that we will always assume to be balanced. This implies for us that $\mu$ has no atoms -which we shall need in order to use the Carleson packing characterization of the sparse condition below-. Fix $0<\eta<1$. A collection $\mathcal{S}\subset \mathscr{D}$ is called $\eta$-sparse (or just sparse) if for each $I \in \mathcal{S}$ there exists $E_I \subset I$ so that $E_I \cap E_{I'} = \emptyset$ if $I\neq I'$ and $\mu(E_I) \geq \eta \mu(I)$. According to [@Han2018 Theorem 1.3], the family $\mathcal{S}$ is sparse if and only if it satisfies a the Carleson packing condition $$\label{eq:CarlesonPacking} \sum_{\substack{J \subset I:\\ J \in \mathcal{S}}} \mu(J) \leq C_\mu \mu(I)$$ for all $I \in \mathscr{D}$. One important consequence of [\[eq:CarlesonPacking\]](#eq:CarlesonPacking){reference-type="eqref" reference="eq:CarlesonPacking"} is the fact that the union of two sparse families is again sparse (with a possibly different constant), and this observation we shall use throughout. The sparse operator associated with $\mathcal{S}$ is $$\mathcal{A}_\mathcal{S}f(x) = \sum_{I \in \mathcal{S}} \langle f \rangle_I 1_I(x),$$ while the corresponding bilinear form already appeared in the Introduction: $$\mathcal{A}_\mathcal{S}(f,g) = \langle \mathcal{A}_\mathcal{S}f,g\rangle = \sum_{I \in \mathcal{S}} \langle f \rangle_I \langle g \rangle_I \mu(I).$$ We start constructing such a measure $\mu$ so that $\Sh$ (we omit the dyadic grid dependence from the notation) does not admit sparse domination in the usual dual sense. **Proposition 1**. Fix $0 < \eta < 1$. There exists a balanced measure $\mu$ and two sequences of compactly supported functions $\{f_j\},\{g_j\} \subset L^2(\mu)$ such that for all $\eta$-sparse families $\mathcal{S}\subset \mathscr{D}$ $$\label{eq:sparsefailure} |\langle \Sh f_j, g_j \rangle | \gtrsim j |\mathcal{A}_\mathcal{S}(f_j,g_j)|.$$ *Proof.* We take the measure $\mu$ to be the one constructed in [@LMP2014]\*Section 4, (a). In particular, $\mu$ is supported on the unit interval[^2] and is defined inductively as follows: let $a_k=1-1/k$ and $b_k=1/k$ for $k \geq 2$. Set $I_k=[0,2^{-k}]$ for $k \geq 0$, and denote by $I_k^b=[2^{-k},2^{-k+1}]$ its dyadic sibling. Set $\mu(I_0)=1$, $\mu(I_1)=\mu(I_1^b)=\frac{1}{2}$ and for $k \geq 2,$ $\mu(I_k)= a_k \mu(I_{k-1})= a_k \mu(\widehat{I}_k)$, so $\mu(I_k^b)=b_k \mu(I_{k-1})$. The construction is finished by declaring that the density of $\mu$ is taken to be uniform on $I_k^b$. $\mu$ is non-dyadically doubling because $$\frac{\mu(\widehat{I}_k)}{\mu(I_k^b)}=\frac{1}{b_k}=k \rightarrow \infty \mbox{ as } k \rightarrow \infty.$$ Clearly, we have $\mu(I_k) \sim \mu(\widehat{I}_k)$ since the sequence $a_k$ is bounded from above and below. Straightforward computations lead to $$m(I_k^b)= \frac{1}{4} b_k \mu(\widehat{I}_k) \sim \frac{\mu(\widehat{I}_k)}{k},\quad m(I_k)= a_{k+1} b_{k+1} a_k \mu(\widehat{I}_k) \sim \frac{\mu(\widehat{I}_k)}{k},$$ which show that $\mu$ is balanced. We also have the estimates $$\mu(I_k) \sim \frac{1}{k}, \quad \mu(I_k^b) \sim \frac{1}{k^2}.$$ One can also show that $\mu$ does not have any point masses. Indeed, the only possible point mass is $\delta_0$, but this does not occur since $\mu(I_k) \rightarrow 0$ as $k \rightarrow \infty$ and so $\mu(\{0\})=0$. To prove [\[eq:sparsefailure\]](#eq:sparsefailure){reference-type="eqref" reference="eq:sparsefailure"}, we assume $\eta \in (0,1)$ has been fixed and suppress dependence on $\eta$ in the computations that follow. For $j \geq 2$, we take $f_j= 1_{I_{j-1}^b}$ and $g_j= 1_{I_{j}^b}.$ As a preliminary, we notice $f_j$ and $g_j$ have disjoint supports and if $I \in \mathscr{D}$ intersects both $\text{supp} (f_1)$ and $\text{supp} (f_2)$ non-trivially, we must have $I=I_k$ with $0 \leq k \leq j-2.$ We also observe $$\mathcal{M}f_j(x) = \frac{\mu(I_{j-1}^b)}{\mu(I_{k-1})}, \, \mathcal{M}g_j(x) =\frac{\mu(I_{j}^b)}{\mu(I_{k-1})}, \, x \in I_{k}^b \text{ for } 0 < k \leq j-2; \label{MaximalPointwiseBound1}$$ $$\mathcal{M}f_j(x) = 1, \, \mathcal{M}g_j(x) = \frac{\mu(I_{j}^b)}{\mu(I_{j-2})}, \, x \in I_{j-1}^b ; \label{MaximalPointwiseBound2}$$ $$\mathcal{M}f_j(x) = \frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})}, \, \mathcal{M}g_j(x) = 1, \, x \in I_{j}^b ; \label{MaximalPointwiseBound3}$$ $$\mathcal{M}f_j(x) = \frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})}, \, \mathcal{M}g_j(x) =\frac{\mu(I_{j}^b)}{\mu(I_{j-1})}, \, x \in I_{k}^b \text{ for } k > j \label{MaximalPointwiseBound4}$$ Using the sparsity of $\mathcal{S}$ and equations [\[MaximalPointwiseBound1\]](#MaximalPointwiseBound1){reference-type="eqref" reference="MaximalPointwiseBound1"} through [\[MaximalPointwiseBound4\]](#MaximalPointwiseBound4){reference-type="eqref" reference="MaximalPointwiseBound4"}, we can estimate the sum directly: $$\begin{aligned} \sum_{I \in \mathcal{S}} \langle f_j \rangle_{I} \langle g_j \rangle_{I} \mu(I) & \lesssim \sum_{I \in \mathcal{S}} \langle f_j \rangle_{I} \langle g_j \rangle_{I} \mu(E_I) \\ & \leq \int_{[0,1]} \mathcal{M}f_j \cdot \mathcal{M}g_j\, d\mu \\ & \leq \sum_{k=1}^{\infty} \int_{I_k^b} \mathcal{M}f_j \cdot \mathcal{M}g_j \, d\mu \\ & = \sum_{k=1}^{j-2} \left( \frac{\mu(I_{j-1}^b)}{\mu(I_{k-1})} \right) \cdot \left(\frac{\mu(I_{j}^b)}{\mu(I_{k-1})}\right) \cdot \mu(I_k^b) + \left( \frac{\mu(I_j^b)}{\mu(I_{j-2})}\right) \cdot \mu(I_{j-1}^b)\\ & + \left( \frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})}\right) \cdot \mu(I_{j}^b) + \sum_{k=j+1}^{\infty} \left( \frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})} \right) \cdot \left(\frac{\mu(I_{j}^b)}{\mu(I_{j-1})}\right) \cdot \mu(I_k^b) \\ & \lesssim \left(\frac{1}{j^4} \sum_{k=0}^{j-2} 1\right) + \frac{1}{j^3} + \frac{1}{j^3}+ \left(\frac{1}{j^2}\sum_{k=j+1}^{\infty} \frac{1}{k^2} \right) \lesssim \frac{1}{j^3}. \end{aligned}$$ Now we turn to a lower bound for $|\langle \Sh f_j, g_j \rangle|.$ Given $I \in \mathscr{D}$, denote by $S_I$ the simple shift $S_I f= \langle f, h_I \rangle (h_{I_{-}}-h_{I_{+}}).$ Then, by definition and using the support properties of $f_j$ and $g_j,$ we have $\langle S_I f_j, g_j \rangle= \langle f_j, h_I \rangle \cdot \langle h_{I_{-}}-h_{I_{+}}, g_j \rangle \neq 0$ if and only if $I= I_k$ with $0 \leq k \leq j-2.$ Therefore, we may write $$\langle \Sh f_j, g_j \rangle = \sum_{k=0}^{j-2} \langle S_{I_k} f_j, g_j \rangle = \sum_{k=0}^{j-2} \langle f_j, h_{I_k} \rangle \cdot \langle h_{I_{k+1}}- h_{I_{k+1}^b}, g_j \rangle.$$ For the last term in the sum, we have $$\langle f_j, h_{I_{j-2}} \rangle = -\sqrt{m(I_{j-2})} \frac{\mu(I_{j-1}^b)}{\mu(I_{j-1}^b)}=-\sqrt{m(I_{j-2})},$$ while $$\langle h_{I_{j-1}}- h_{I_{j-1}^b}, g_j \rangle = \langle h_{I_{j-1}}, g_j \rangle= - \sqrt{m(I_{j-1})}.$$ Therefore, $$\langle f_j, h_{I_{j-2}} \rangle \cdot \langle h_{I_{j-1}}- h_{I_{j-1}^b}, g_j \rangle= \sqrt{m(I_{j-1})m(I_{j-2}}) \sim \frac{\mu(I_j)}{j} \sim \frac{1}{j^2}.$$ Notice that all the other terms in the summation will similarly be positive, because if $k<j-2,$ $f_j$ is supported entirely on the left half of $I_k$ and $g_j$ is supported on the left half of $I_{k+1}.$ Therefore, we need not consider the other terms and [\[eq:sparsefailure\]](#eq:sparsefailure){reference-type="eqref" reference="eq:sparsefailure"} follows immediately. ◻ **Remark 2**. If one were instead to consider the measure $\mu'$ defined by $b_k'=\frac{1}{k^2}$ with parallel construction to Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"}, then $\mu'$ would have a point mass at $0$. In this case one can actually prove the much stronger result that the bilinear form $|\langle \Sh f_1, f_2 \rangle|$ cannot be controlled by $\sum_{I \in \mathcal{S}} \langle f_1 \rangle_I \langle f_2 \rangle_I \mu(I)$ for *any* dyadic collection $\mathcal{S}$, let alone a sparse one. Our next construction uses the same measure $\mu$ that we used in Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"} to show that the class $A_2(\mu)$ is not the right one to study $L^2$-boundedness for higher complexity shifts. Again, we exemplify using $\Sh$. **Proposition 3**. There exists a balanced measure $\mu$ and a weight $w \in A_2(\mu)$ so that $\Sh$ is unbounded on $L^2(w d \mu)$. *Proof.* Define $\mu$ as in the proof of Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"}. Define the weight $w$ as follows: $$w(x)= \begin{cases} 2^{-\frac{k}{2}}, & x \in I_{2^k}^b \text{ for $k \geq 1$,} \\ 1, & \text{otherwise.} \end{cases}$$ We claim $w \in A_2(\mu)$. To see this, first note that if $I$ is contained in $I_{k}^b$ for some $k$, then $\langle w \rangle_{I} \langle w^{-1} \rangle_{I}=1.$ The only other case to consider is $I=I_\ell$ for some $\ell$. We will use the simple facts that $\sum_{j > \ell}^{\infty} \mu(I_j^b)= \mu(I_{\ell})$ and $\mu(I_j) \leq \mu(I_k)$ if $j \geq k.$ We estimate $$\begin{aligned} \langle w^{-1} \rangle_{I_\ell} & = \frac{ \sum_{ \substack{j> \ell: \\ j \neq 2^k \text{ for any $k$} }} \mu(I_{j}^b)+ \sum_{ \substack{k:\\ 2^k > \ell }} 2^{\frac{k}{2}} \mu(I_{2^k}^b)}{\mu(I_\ell)}\\ & \leq 1+ \frac{ \sum_{ \substack{k:\\ 2^k > \ell }} 2^{-\frac{k}{2}} \mu(I_{2^k-1})}{\mu(I_{\ell})} \lesssim 1,\end{aligned}$$ using $\ell\mu(I_\ell)=\mu(I_{\ell+1})$. The implicit constant is independent of $\ell.$ The estimate $\langle w \rangle_{I_{\ell}} \lesssim 1$ is even easier, and so $\langle w \rangle_{I_\ell} \langle w^{-1} \rangle_{I_\ell} \lesssim 1$ in this case as well, yielding the claim. However, notice that $$\langle w^{-1} \rangle_{I_{2^k}^b} \langle w \rangle_{I_{2^k+1}^b}=2^{\frac{k}{2}} \rightarrow \infty \mbox{ as } k \rightarrow \infty.$$ This behavior of the weight will lead to a contradiction: consider test functions $f_k \equiv w^{-1} 1_{I_{2^k}^b}$ and $g_k \equiv 1_{I_{2^k+1}^b}.$ We compute $$\|f_k\|_{L^2(w d\mu)}= \left(\int_{I_{2^k}^b} w^{-1} \, d\mu\right)^{\frac{1}{2}}= 2^{\frac{k}{4}} \mu(I_{2^k}^b)^{\frac{1}{2}},$$ and similarly we have $$\|g_k\|_{L^2(w d\mu)}= \mu(I_{2^k+1}^b)^{\frac{1}{2}}.$$ We now obtain a lower bound for $|\langle \Sh f_k, w g_k \rangle |$ using similar considerations as in the proof of Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"}: $$\begin{aligned} |\langle \Sh f_k, w \, g_k \rangle | & = \left| \sum_{j=0}^{2^k-1} \langle S_{I_j} f_k, w \, g_k \rangle \right| \\ & \geq |\langle f_k, h_{I_{2^k-1}}\rangle| \cdot | \langle h_{I_{2^k}}- h_{I_{2^k}^b} , w \, g_k \rangle|\\ & = 2^{\frac{k}{2}} \sqrt{m(I_{2^k-1})} \cdot \sqrt{m(I_{2^k})}\\ & \sim 2^{\frac{k}{2}} \mu(I_{2^k}^b)^{\frac{1}{2}} \mu(I_{2^k+1}^b)^{\frac{1}{2}}\\ & = 2^{\frac{k}{4}} \|f_k\|_{L^2(w d\mu)} \|g_k\|_{L^2(w d\mu)}. \end{aligned}$$ This shows that $\Sh$ is unbounded on $L^2(w d\mu).$ ◻ # Modified Sparse Domination and Consequences {#sec:section2} ## Modified Sparse forms and Calderón-Zygmund estimates Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"} shows that usual sparse forms are not enough to achieve sparse domination for Haar shifts in general. To remedy this, we introduce other forms that reflect the complexity of the operator that we want to dominate. The dyadic distance between two intervals $I, J \in \mathscr{D}$ that share a common ancestor is defined as follows: $$%$\label{eq:DyadicDist} \operatorname{dist}_{\mathscr{D}}(I,J):= \min_{(s,t): I^{(s)}= J^{(t)}} (s+t).$$ If $I,J$ do not share a common ancestor, then by convention we let $\operatorname{dist}_{\mathscr{D}}(I,J)=\infty.$ Notice that $\operatorname{dist}_{\mathscr{D}}(I,J)=0$ if and only if $I=J$. For any $I \in \mathscr{D}$, we let $C_N$ the number of $J\in \mathscr{D}$ such that $\operatorname{dist}_{\mathscr{D}}(I,J) \leq N+2$. $C_N$ does not depend on $I$. For convenience, we enumerate the dyadic intervals $J \in \mathscr{D}$ that satisfy $2<\operatorname{dist}_{\mathscr{D}}(I,J) \leq N+2$ and $J \cap I = \emptyset$ as $c_1(I),c_2(I),\cdots, c_{N'}(I)$, where $N'$ is a constant depending only on $N$. We will suppose that we have fixed an arbitrary enumeration function for each $I \in \mathscr{D}$ and abuse notation by writing $c_j$ to represent all enumerations for all intervals. Note that given a fixed $J \in \mathscr{D}$, there are at most $N'$ intervals $I$ satisfying $c_j(I)=J$ for some $j$. This fact will be used in various estimates without further comment. Given $\mathcal{S}\subset \mathscr{D}$, $N \in \mathbb{N}$, and $f_1,f_2 \in L^2(\mu)$, we define $$%\label{ComplexSparseForm} \mathcal{C}_{\mathcal{S}}^N(f_1,f_2):= \sum_{\substack{J, K \in \mathcal{S}:\\ 2<\operatorname{dist}_{\mathscr{D}}(J,K) \leq N+2,\\ J \cap K= \emptyset}} \langle f_1 \rangle_{J} \langle f_2 \rangle_{K}\, \, \sqrt{m(J)} \sqrt{m(K)}.$$ We will later show that the forms $\mathcal{C}_{\mathcal{S}}^N$ are $L^p$-bounded and of weak-type $(1,1)$. What we show next is a technical lemma that justifies considering only intervals at distance greater than $2$ and disjoint in the definition. **Lemma 4**. Let $N\in\mathbb{N}$ and $\mathcal{S}\subset \mathscr{D}$ be $\eta$-sparse. There exist $0<\eta'\leq \eta$ and $\mathcal{S}' \supseteq \mathcal{S}$ $\eta'$-sparse such that $$\sum_{\substack{J, K \in \mathcal{S}:\\ 1 \leq \operatorname{dist}_{\mathscr{D}}(J,K) \leq N+2}} \langle f_1 \rangle_{J} \langle f_2 \rangle_{K}\, \, \sqrt{m(J)} \sqrt{m(K)} \lesssim \mathcal{A}_{\mathcal{S'}}(f_1, f_2)+ \mathcal{C}_{\mathcal{S'}}^N(f_1,f_2)$$ *Proof.* Since we are going to construct $\mathcal{S}'$ so that $\mathcal{S}\subseteq \mathcal{S}'$, we only need to bound terms arising from pairs $(J,K)$ such that either $\operatorname{dist}_{\mathscr{D}}(J,K)=2$ and $J \cap K=\emptyset$, or $J \cap K \in \{J,K\}$ and $\operatorname{dist}_{\mathscr{D}}(J,K)\leq N+2$. In the latter case, if $J=K^{(k)}$ for some $k \leq N+2$ -say-, we estimate $$\langle f_1 \rangle_{J} \langle f_2 \rangle_{K}\, \, \sqrt{m(J)} \sqrt{m(K)} \lesssim_k \langle f_1 \rangle_{J} \, \langle f_2 \rangle_{J} \, \, \mu(J),$$ because $m(J) \sim m(K)$ and $m(J) \leq \mu(J)$. In the former case, we must have $K=J^b$. If we suppose $J$ and $J^b$ both belong to $\mathcal{S}$, then $$\langle f_1 \rangle_{J} \langle f_2 \rangle_{J^b}\, \, \sqrt{m(J)} \sqrt{m(J^b)} \lesssim \langle f_1 \rangle_{\widehat{J}} \, \langle f_2 \rangle_{\widehat{J}} \, \, \mu(\widehat{J}),$$ where the implicit constant is independent of $J$ because each dyadic interval has exactly two children. We define now $\mathcal{S}'$ as the union of the family $\mathcal{S}$ and the intervals $\widehat{J}$ for each $J \in \mathcal{S}$ such that $J^b \in \mathcal{S}$. It can be seen, using [\[eq:CarlesonPacking\]](#eq:CarlesonPacking){reference-type="eqref" reference="eq:CarlesonPacking"} that $\mathcal{S}'$ is $\eta'$-sparse for some $\eta'=\eta'(\eta,N)$, and the assertion follows. ◻ To prove Theorem [Theorem 1](#th:thmA){reference-type="ref" reference="th:thmA"} we will perform Calderón-Zygmund decompositions at different heights for two functions $f_1$ and $f_2$ simultaneously. In the Lebesgue measure case, the Calderón-Zygmund decomposition yields $L^\infty$ estimates for the good part of the function, but that is unavailable in our case, so we must content ourselves with a $\mathrm{BMO}$ estimate. We use the standard martingale $\mathrm{BMO}$ norm (associated with the filtration generated by $\mathscr{D}$), which is $$\|f\|_{\mathrm{BMO}}:=\sup_{I \in \mathscr{D}} \frac{1}{\mu(I)} \int_{I} |f -\langle f \rangle_{\widehat{I}}| \, d\mu.$$ **Lemma 5**. Let $0\leq f_1, f_2 \in L^1(\mu)$ be supported on $I \in \mathscr{D}$ and $\lambda_1, \lambda_2>0$. Then there exist functions $g_j,b_j$ such $f_j=g_j+b_j$ for $j=1,2$ and satisfying the following properties: 1. [\[item:1\]]{#item:1 label="item:1"} There exist pairwise disjoint, dyadic intervals $\{I_k\} \subset \mathscr{D}(I)$ so that $b_j= \sum_{k=1}^{\infty}b_{j,k}$ where each $b_{j,k}$ is supported on $\widehat{I}_k$, $\int_{\widehat{I}_k}b_{j,k}=0$, and $\|b_{j,k}\|_{L^1(\mu)} \lesssim \int_{I_k} |f_j| \,d \mu$ for $j=1,2.$ In particular, for $j=1,2$ we have $$b_{j,k}= f_j 1_{I_k}- \langle f_j 1_{I_k} \rangle_{\widehat{I}_k} 1_{\widehat{I}_k}.$$ 2. [\[item:2\]]{#item:2 label="item:2"} For $j=1,2$, the function $g_j \in L^p(\mu)$ for all $1 \leq p< \infty$ and satisfies $\|g_j\|_{L^p(\mu)}^p \leq C_p \lambda_j^{p-1} \|f_j\|_{L^1(\mu)}.$ 3. [\[item:3\]]{#item:3 label="item:3"} For $j=1,2$ the function $g_j \in \mathrm{BMO}$ and $\|g_j\|_{\mathrm{BMO}}\leq \lambda_j$. *Proof.* The statement of the decomposition is contained in [@LMP2014; @CCP2022], except for [\[item:3\]](#item:3){reference-type="eqref" reference="item:3"}, which is new and for which we give full details below. The intervals $I_k$ are selected using the usual stopping-time argument: let $\{I_k\}$ be the set of maximal dyadic intervals contained in $I$ so that either $\langle f_{1} \rangle_{I_k}> \lambda_1$ or $\langle f_{2} \rangle_{I_k} >\lambda_2$. Then [\[item:1\]](#item:1){reference-type="eqref" reference="item:1"} follows. We now turn to [\[item:3\]](#item:3){reference-type="eqref" reference="item:3"}, which by interpolation proves [\[item:2\]](#item:2){reference-type="eqref" reference="item:2"}. For $j=1,2$, we have $$g_j= f_j 1_{\left(\cup_{k} I_k\right)^c}+\sum_{k} \langle f_j 1_{I_k} \rangle_{{\widehat{I}_k}} 1_{\widehat{I}_k} := g_{j,1}+g_{j,2}.$$ By the Lebesgue Differentiation theorem, for almost every point $x \in \left(\cup_{j} I_j\right)^c$ there is a sequence of unselected cubes shrinking down to $x$ and we have $|f_j(x)| \leq \lambda_j,$ so $\|g_{j,1}\|_{L^\infty}\leq \lambda_j.$ It easily follows that $\|g_{j,1}\|_{\mathrm{BMO}}\leq 2 \lambda_j$ for $j=1,2.$ It therefore suffices to show $\|g_{j,2}\|_{\mathrm{BMO}} \lesssim \lambda_j$ for $j=1,2.$ If $I=I_k$ is a selected interval, we have $$\begin{aligned} \langle g_{j,2} \rangle_{\widehat{I}_k} 1_{I_k}& = \left(\sum_{\ell: \widehat{I}_\ell \subsetneq \widehat{I}_k} \frac{\mu(\widehat{I}_\ell)}{\mu(\widehat{I}_k)} \langle f_j 1_{I_\ell} \rangle_{\widehat{I}_\ell} + \sum_{\ell: \widehat{I}_k \subset \widehat{I}_\ell} \langle f_j 1_{I_\ell} \rangle_{\widehat{I}_\ell} \right) 1_{I_k}\\ & = \frac{ \int_{E_k} f_j \, d \mu}{\mu(\widehat{I}_k)} 1_{I_k} + \left(\sum_{\ell: \widehat{I}_k \subset\widehat{I}_\ell} \langle f_j 1_{I_\ell} \rangle_{\widehat{I}_\ell}\right) 1_{I_k},\end{aligned}$$ where $$E_k= \bigcup_{\ell: \widehat{I}_\ell \subsetneq \widehat{I}_k} I_{\ell}.$$ On the other hand, notice that by definition and the disjointness of the selected intervals, $g_{j,2} 1_{I_k}= \left(\sum_{\ell} \langle f_j 1_{I_{\ell}} \rangle_{\widehat{I}_\ell}1_{\widehat{I}_\ell}\right) 1_{I_k}= \left(\sum_{\ell: \widehat{I}_k \subset\widehat{I}_\ell} \langle f_j 1_{I_\ell} \rangle_{\widehat{I}_\ell}\right) 1_{I_k}$ and therefore $$\frac{1}{\mu(I_k)} \int_{I_k}|g_{j,2}- \langle g_{j,2} \rangle_{\widehat{I}_k}| \, d \mu = \frac{ \int_{E_k} f_j \, d \mu}{\mu(\widehat{I}_k)} \leq \langle f_j \rangle_{\widehat{I}_k} \leq \lambda_j.$$ If $I$ is not a selected interval, we can similarly argue $$\langle g_{j,2} \rangle_{\widehat{I}} 1_{I}= \frac{ \int_{E_I} f_j \, d \mu}{\mu(\widehat{I})} 1_{I} + \left(\sum_{\ell: \widehat{I}_ \subset\widehat{I}_{\ell}} \langle f_j 1_{I_{\ell}} \rangle_{\widehat{I}_\ell}\right) 1_{I},$$ where $E_I= \bigcup_{\ell: \widehat{I}_\ell \subsetneq \widehat{I}} I_{\ell}.$ However, in this case it is possible that $I= \widehat{I}_\ell$ for some $\ell,$ so we have $g_2 1_{I}= \left(\sum_{\ell: \widehat{I}_ \subset\widehat{I}_\ell} \langle f_j 1_{I_\ell} \rangle_{\widehat{I}_\ell}\right) 1_{I}+ \left(\sum_{\ell:I= \widehat{I}_\ell} \langle f_j 1_{I_\ell} \rangle_{\widehat{I}_\ell}\right) 1_{I} .$ However, there are at most two terms in the second summation, and each average in the sum is controlled by $\lambda_j$, so the $\mathrm{BMO}$ norm can be estimated as before. ◻ We next introduce notation about collections of intervals. Suppose functions $f_1,f_2$ and an initial interval $I_0 \in \mathscr{D}$ have been fixed as in Lemma [Lemma 5](#CZD){reference-type="ref" reference="CZD"}. For $I \subseteq I_0$, let $\mathcal{B}(I)$ denote the collection of selected intervals in Lemma [Lemma 5](#CZD){reference-type="ref" reference="CZD"} applied to $f_j 1_{I}$ at heights $\lambda_j= 16 \langle f_j \rangle_{I}$, $j=1,2$. We then inductively define, for $k \in \mathbb{N}$: $$\label{eq:notationGB} \mathcal{B}_0(I)= \mathcal{B}(I), \quad \mathcal{B}_1(I)= \bigcup_{J \in \mathcal{B}(I)} \mathcal{B}(J), \quad \mathcal{B}_k(I)=\bigcup_{J \in \mathcal{B}_{k-1}(I)} \mathcal{B}(J).$$ Moreover, given $N \in \mathbb{N}$, we set $$\mathcal{B}^N(I):= \bigcup_{k=0}^{N} \mathcal{B}_k(I).$$ Finally, we set $$\mathcal{G}(I):= \{J \in \mathscr{D}(I): J \not \subset K \text{ for any } K \in \mathcal{B}(I)\}, \quad \mathcal{O}(I)=\{J \in \mathscr{D}: J \supsetneq I\}.$$ ## Haar shifts Let $\alpha=\{\alpha_{J,K}^I\}_{I, J, K \in \mathscr{D}}$ be a (triply indexed) sequence with $\|\alpha\|_\infty \leq 1$. For any two integers $s,t \in \mathbb{N}$, we define a *Haar shift of complexity $(s,t)$* by $$T_\alpha^{s,t}f:= \sum_{I \in \mathscr{D}} \sum_{J \in \mathscr{D}^s(I)} \sum_{K \in \mathscr{D}^t(I)} \alpha_{J,K}^{I} \langle f, h_J \rangle h_K, \quad f \in L^2(\mu)$$ The above sum converges in $L^2(\mu)$ and $\mu$-almost everywhere. The family of Haar shifts of complexity $(s,t)$ will be denoted by $$\mathrm{HS}(s,t) = \left\{T_\alpha^{s,t}: \|\alpha\|_{\infty} \leq 1 \right\}.$$ It is immediate from the fact that $\{h_I\}_{I \in \mathscr{D}}$ forms an orthonormal basis for $L^2(\mu)$ that for any complexity $(s,t)$ and any measure $\mu$, any $T\in \mathrm{HS}(s,t)$ is bounded on $L^2(\mu)$ with operator norm only depending on the complexity. If $\mu$ is balanced, then $T$ is bounded on $L^p(\mu)$ and of weak-type $(1,1)$, as shown in [@LMP2014]. We are now ready to state our main theorem. **Theorem 6**. *Let $\mu$ be balanced and $N\in \mathbb{N}$. There exists $\eta \in (0,1)$ such that if $(s,t)$ satisfies $s+t \leq N$, the following holds: for all $T \in \mathrm{HS}(s,t)$ and each pair of compactly supported, bounded nonnegative functions $f_1,f_2$ there exists an $\eta$-sparse collection $\mathcal{S}\subset \mathscr{D}$ such that $$\begin{aligned} \mathlarger{|} \langle T f_1, f_2 \rangle \mathlarger{|} & \lesssim \left( \mathcal{A}_{\mathcal{S}}(f_1, f_2)+ \mathcal{C}_{\mathcal{S}}^N(f_1,f_2) \right).\end{aligned}$$* For the remainder of this subsection we assume that the pair $(s,t)$ has been fixed, and the two functions $f_1$ and $f_2$ are supported on a fixed dyadic interval $I_0$. We also suppose that the sequence $\alpha$ that defines the Haar shift $T$ is nonzero except for finitely many indices, since the estimates we obtain will be uniform over all finite sums, and that $\alpha$ has been fixed as well. It will be useful to introduce the following notation: let $I=[2^{-k} p, (p+1) 2^{-k}) \in \mathscr{D}$, where $k, p \in \mathbb{Z}.$ Given fixed $s, t \in \mathbb{N}$, let $0 \leq m \leq 2^{s}-1$ and $0 \leq n \leq 2^{t}-1$ be integers. We index the intervals $J \in \mathscr{D}^{s}(I)$ via $$I_{s}^m=[2^{-k}p+m 2^{-k-s}, 2^{-k} p+(m+1)2^{-k-s}),$$ and we define $I_{t}^n$ in an analogous manner. With this notation, the two dyadic children of a given interval $I_s^m$ are $I_{s+1}^{2m}$ and $I_{s+1}^{2m+1}.$ The following lemma contains the key iteration step in the sparse domination argument. **Lemma 7**. Let $0\leq m < 2^s$, $0\leq n < 2^t$ be integers and set $N=s+t$. If $\beta = \{\beta_I\}_{I\in\mathscr{D}}$ is a sequence with $\|\beta\|_\infty \leq 1$ which is nonzero for finitely many indices, then $$\left| \sum_{J \in \mathcal{G}(I)} \beta_J\langle f_1, h_{J_{s}^m} \rangle \langle h_{J_{t}^n}, f_2 \rangle \right| \lesssim \bigg( \langle f_1 \rangle_{I} \langle f_2 \rangle_{I} \mu(I) + \sum_{\substack{S\in \mathcal{B}^s(I), \, T \in \mathcal{B}^t(I): \\ \operatorname{dist}_{\mathscr{D}}(S,T) \leq N+2}} \langle f_1 \rangle_{S} \langle f_2 \rangle_{T} \sqrt{m(S)} \sqrt{m(T)} \bigg).$$ *Proof.* Fix $I \subseteq I_0.$ We begin with the usual splitting with $f_j 1_{I}=g_j+b_j$ with respect to the Calderón-Zygmund decomposition applied on $I$: $$\begin{aligned} \left| \sum_{J \in \mathcal{G}(I)} \beta_J\langle f_1, h_{J_{s}^m} \rangle \langle h_{J_{t}^n}, f_2 \rangle \right| & \leq \left| \sum_{J \in \mathcal{G}(I)} \beta_J\langle g_1, h_{J_{s}^m} \rangle \langle h_{J_{t}^n}, g_2 \rangle \right|+ \left| \sum_{J \in \mathcal{G}(I)} \beta_J\langle g_1, h_{J_{s}^m} \rangle \langle h_{J_{t}^n}, b_2 \rangle \right|\\ & + \left| \sum_{J \in \mathcal{G}(I)} \beta_J\langle b_1, h_{J_{s}^m} \rangle \langle h_{J_{t}^n}, g_2 \rangle \right|+ \left| \sum_{J \in \mathcal{G}(I)} \beta_J\langle b_1, h_{J_{s}^m} \rangle \langle h_{J_{t}^n}, b_2 \rangle \right| \\ & := \mathrm{(I)} + \mathrm{(II)} + \mathrm{(III)} +\mathrm{(IV)}.\end{aligned}$$ We handle the four terms in order. We estimate, using the fact that the $h_J$ are orthonormal, the $L^2$ control on the good functions, and the fact that $\|\beta\|_{\ell^\infty} \leq 1$: $$\begin{aligned} (I) & \leq \sum_{J \in \mathcal{G}(I)}| \langle g_1, h_{J_m^s}\rangle| \cdot | \langle h_{J_n^t}, g_2 \rangle|\\ & \leq \left( \sum_{K \in \mathscr{D}}| \langle g_1, h_{K}\rangle|^2\right)^{1/2} \left( \sum_{L \in \mathscr{D}}| \langle h_L, g_2 \rangle|^2\right)^{1/2} \\ & \leq \|g_1\|_{L^2(\mu)} \|g_2\|_{L^2(\mu)}\\ & \lesssim \left( \langle f_1 \rangle_{I}\right)^{1/2} \left(\int_{I} f_1 \, d\mu\right)^{1/2} \left( \langle f_2 \rangle_{I}\right)^{1/2} \left(\int_{I} f_2 \, d\mu\right)^{1/2} = \langle f_1 \rangle_{I} \langle f_2 \rangle_{I} \mu(I).\end{aligned}$$ This proves the desired estimate for (I). For (II), we have $$\mathrm{(II)} \leq \sum_{J \in \mathcal{G}(I)} \sum_{k} |\langle g_1, h_{J_s^m} \rangle| | \langle h_{J_t^n},b_{2,k} \rangle|.$$ However, many of the terms in this summation are zero due to the cancellation/support of the $b_{2,k}$ and the constraint $J \in \mathcal{G}(I).$ Indeed, the cancellation on the $b_{2,k}$ implies $J_t^n \subsetneq I_k^{(2)}$, while the condition $J \in \mathcal{G}(I)$ together with the support of $b_{2,k}$ implies that either $I_k \subsetneq J$ or $J\subset I_k^b.$ But in the latter case, $b_{2,k}$ is a constant function on the support of $h_{J_t^n}$ (which is contained in $I_k^b$), so these terms may be eliminated as well by the cancellation of $h_{J_t^n}.$ Note that given any fixed $I_k$, there are finitely many intervals $J$ that can satisfy these constraints, and the number of intervals only depends on $t$, not $k.$ Indeed, we may write $I_k \subsetneq J \subsetneq I_k^{(2+t)}.$ Therefore, after eliminating these terms, the double summation is equal to: $$\sum_{k} \sum_{\substack{J \in \mathcal{G}(I):\\ I_k \subsetneq J \subsetneq I_k^{(2+t)}}} |\langle g_1, h_{J_s^m} \rangle| | \langle h_{J_t^n},b_{2,k} \rangle|.$$ Fix $J \in \mathcal{G}(I)$ satisfying the additional two conditions in the summation. We will obtain a uniform estimate on the inner summation. On the one hand, we estimate $$\begin{aligned} | \langle g_1, h_{J_s^m} \rangle| & = \sqrt{m(J_s^m)} \left| \langle g_1 \rangle_{J_{s+1}^{2m}}- \langle g_1 \rangle_{J_{s+1}^{2m+1}} \right|\\ & \leq \sqrt{m(J_s^m)} \left( \left| \langle g_{1} \rangle_{J_{s+1}^{2m}}- \langle g_{1} \rangle_{J_s^m}\right|+ \left|\langle g_{1} \rangle_{J_s^m} - \langle g_1 \rangle_{J_{s+1}^{2m+1}} \right| \right) \\ & \leq 2 \sqrt{m(J_s^m)} \|g_1\|_{\mathrm{BMO}} \lesssim \sqrt{m(J_s^m)} \langle f_1 \rangle_{I},\end{aligned}$$ by Lemma [Lemma 5](#CZD){reference-type="ref" reference="CZD"}, [\[item:3\]](#item:3){reference-type="eqref" reference="item:3"}. On the other hand, using the $L^\infty$ estimate of the Haar functions, the $L^1$ control on the bad function, and the fact that $J_s^m$ and $J_t^n$ share the common ancestor $J$, we estimate: $$\begin{aligned} | \langle h_{J_t^n}, b_{2,k} \rangle| & \leq \frac{1}{\sqrt{m(J_t^n)}} \|b_{2,k}\|_{L^1(\mu)}\\ & \lesssim \frac{1}{\sqrt{m(J_t^n)}} \int_{I_k}|f_2| \, d \mu. \\ & \lesssim \frac{1}{\sqrt{m(J_s^m)}} \int_{I_k}|f_2| \, d \mu,\end{aligned}$$ where in the last line we used our assumption that $\mu$ is balanced applied at most $N$ times. Thus, in particular, for fixed $k$, we have a bound $$|\langle g_1, h_{J_s^m} \rangle| | \langle h_{J_t^n},b_{2,k} \rangle| \lesssim \langle f_1 \rangle_{I} \int_{I_k}|f_2| \, d \mu,$$ that is independent of $J.$ Putting these two estimates together and summing, we have $$\begin{aligned} \mathrm{(II)} & \leq \sum_{k} \sum_{\substack{J \in \mathcal{G}(I):\\ I_k \subsetneq J \subsetneq I_k^{(2+t)}}} |\langle g_1, h_{J_s^m} \rangle| | \langle h_{J_t^n},b_{2,k} \rangle|\\ & \lesssim \sum_{k} \sum_{\substack{J \in \mathcal{G}(I):\\ I_k \subsetneq J \subsetneq I_k^{(2+t)}}} \langle f_1 \rangle_{I} \int_{I_k}|f_2| \, d \mu\\ & \lesssim \sum_k \langle f_1 \rangle_{I} \int_{I_k}|f_2| \, d \mu \lesssim \langle f_1 \rangle_{I} \langle f_2 \rangle_{I} \mu(I),\end{aligned}$$ where in the penultimate estimate the implicit constant depends only on $t$ and in the last step we used the fact that the $I_k \in \mathcal{B}(I)$ are disjoint and contained in $I$ for all $k$. Term (III) is handled similarly and we omit the details, so we are left with (IV). We estimate $$\mathrm{(IV)} \leq \sum_{J \in \mathcal{G}(I)} |\langle b_1, h_{J_{s}^m} \rangle| \cdot |\langle h_{J_{t}^n}, b_2 \rangle| .$$ So, we need to handle terms of the form $|\langle b_{1,j}, h_{J_s^m} \rangle| \cdot |\langle h_{J_t^n}, b_{2,k}\rangle|$ where $b_{1,j}$ is supported on $I_j$, $b_{2,k}$ is supported on $I_k$, and both $I_j$ and $I_k$ are selected intervals. Again, there are some constraints on the interval $J$. In particular, we must have $I_j \subsetneq J \subsetneq I_j^{(2+s)}$ and $I_k \subsetneq J \subsetneq I_k^{(2+t)}$, as we determined previously. So for fixed $j,k$, there are finitely many intervals $J$ that contribute to the sum, where the number of terms only depends on $s,t$. Therefore, similar to before, it suffices to consider the sum $$\sum_{j} \sum_{k} \sum_{\substack{J \in \mathcal{G}(I):\\ I_j \subsetneq J \subsetneq I_j^{(2+s)} \text{ and } I_k \subsetneq J \subsetneq I_k^{(2+t)}} } |\langle b_{1,j}, h_{J_{s}^m} \rangle| \cdot |\langle h_{J_{t}^n}, b_{2,k} \rangle|$$ and show it is controlled by the right hand side in the statement. Let us fix $J, j,k$ as in the summation for the time being, and consider a term $|\langle b_{1,j}, h_{J_{s}^m} \rangle| \cdot |\langle h_{J_{t}^n}, b_{2,k} \rangle|.$ Note that either $J_s^m= \widehat{I}_j$ or $J_s^m \subset I_j.$ In the former case, we can control the inner product as follows: $$\begin{aligned} |\langle b_{1,j}, h_{J_{s}^m} \rangle| & = \sqrt{m(\widehat{I}_j)} \left| \langle b_{1,j} \rangle_{I_j}- \langle b_{1,j} \rangle_{I_j^b}\right|\\ & = \sqrt{m(\widehat{I}_j)}\left| \langle f_1 \rangle_{I_j}- \langle f_{1} 1_{I_j} \rangle_{\widehat{I}_j} + \langle f_{1} 1_{I_j} \rangle_{\widehat{I}_j}- \langle f_1 1_{I_j} \rangle_{I_j^b}\right|\\ & \sim \sqrt{m(I_j)} \langle f_1 \rangle_{I_j}.\end{aligned}$$ If $J_t^n=\widehat{I}_k$ as well, then we obviously have the analogous estimate and therefore: $$|\langle b_{1,j}, h_{J_{s}^m} \rangle| \cdot |\langle h_{J_{t}^n}, b_{2,k} \rangle| \lesssim \langle f_1 \rangle_{I_j} \langle f_2 \rangle_{I_k} \sqrt{m(I_j)} \sqrt{m(I_k)} \label{BB1}$$ Moreover, notice that $\operatorname{dist}_{\mathscr{D}}(I_j,I_k) \leq N+2$ and by definition, $I_j, I_k \in \mathcal{B}(I)$ and are disjoint unless $I_j=I_k$. In the latter case when $J_s^m \subset I_j$, we see that $$\begin{aligned} |\langle b_{1,j}, h_{J_{s}^m} \rangle| & = \sqrt{m(J_s^m)} \left| \langle b_{1,j} \rangle_{J_{s-}^{m}}- \langle b_{1,j} \rangle_{J_{s+}^{m}}\right| \nonumber\\ & =\sqrt{m(J_s^m)} \left| \langle f_1 \rangle_{J_{s-}^{m}}- \langle f_{1} 1_{I_j} \rangle_{\widehat{I}_j} + \langle f_{1} 1_{I_j} \rangle_{\widehat{I}_j}- \langle f_1 \rangle_{J_{s+}^{m}}\right| \nonumber\\ & \leq \sqrt{m(J_s^m)} \langle f_1 \rangle_{J_{s-}^{m}}+ \sqrt{m(J_{s}^m)} \langle f_1 \rangle_{J_{s+}^{m}} \label{BB2} .\end{aligned}$$ It suffices to control $\sqrt{m(J_{s}^m)} \langle f_1 \rangle_{J_{s-}^{m}}$; the other term is obviously handled in the same way. If $J_{s-}^{m} \in \mathcal{B}(I_j)$, we leave the estimate as is. So suppose that $J_{s-}^{m} \notin \mathcal{B}(I_j).$ Then either $\langle f_1 \rangle_{J_{s-}^{m}} \leq 16 \langle f_1 \rangle_{I_j}$ or there exists $K \subsetneq I_j$ with $K \in \mathcal{B}(I_j)$ and $J_s^m \subsetneq K.$ In the former case, we have the estimate: $$\sqrt{m(J_s^m)} \langle f_1 \rangle_{J_{s-}^{m}} \lesssim \sqrt{m(I_j)} \langle f_1 \rangle_{I_j} , \label{BB3}$$ using the balanced property of $\mu.$ In the latter case, we apply the same procedure, letting $K$ assume the role of $I_j.$ That is, either $J_{s-}^{m} \in \mathcal{B}(K),$ in which case we stop and leave the original estimate as is, or we have the control $\langle f_1 \rangle_{J_{s-}^{m}} \leq 16 \langle f_1 \rangle_{K}$, or there exists an interval $K' \in \mathcal{B}(K)$ so that $J_s^{m-} \subsetneq K',$ and we apply the procedure again to $K'.$ Notice that $\operatorname{dist}_{\mathscr{D}}(J_{s-}^{m},I_j)\leq s.$ Iterating this procedure at most $s$ times and using the balanced condition, we can find an interval $S \in \mathcal{B}^s(I)$ (more precisely, $\mathcal{B}^{s-1}(I_j)$) containing $J_{s-}^{m}$ and satisfying $$\sqrt{m(J_s^m)} \langle f_1 \rangle_{J_{s-}^{m}} \lesssim \sqrt{m(S)} \langle f_1 \rangle_{S} , \label{BB4}$$ where the implicit constant depends only on $\mu$ and s. Moreover, the same reasoning applies to $J_t^n$, giving us an estimate $$\sqrt{m(J_{t}^n)} \langle f_1 \rangle_{J_{t-}^{n}} \lesssim \sqrt{m(T)} \langle f_1 \rangle_{T} \label{BB5} .$$ for some $T \in \mathcal{B}^{t-1}(I_k)$ (or $T=I_k$) with $J_{t-}^{n} \subset T.$ Let $S,T$ be as above. We observe that $\operatorname{dist}_{\mathscr{D}}(S,T) \leq N+2$, since both have common ancestor $J.$ Examining estimates [\[BB1\]](#BB1){reference-type="eqref" reference="BB1"} through [\[BB5\]](#BB5){reference-type="eqref" reference="BB5"} and using the fact that the number of intervals $J$ satisfying these constraints only depends on $s,t$, we deduce that for fixed $j,k$, we have the bound: $$\begin{aligned} & \sum_{\substack{J \in \mathcal{G}(I):\\ I_j \subsetneq J \subsetneq I_j^{(2+s)} \text{ and } I_k \subsetneq J \subsetneq I_k^{(2+t)}} } |\langle b_{1,j}, h_{J_{s}^m} \rangle| \cdot |\langle h_{J_{t}^n}, b_{2,k} \rangle| \\ & \lesssim \bigg(\langle f_1 \rangle_{I_j} \langle f_2 \rangle_{I_k} \sqrt{m(I_j)} \sqrt{m(I_k)} +\sum_{\substack{S \in \mathcal{B}^s(I_j), \, T \in \mathcal{B}^t(I_k):\\ \operatorname{dist}_{\mathscr{D}}(S,T) \leq N+2}} \langle f_1 \rangle_{S} \langle f_2 \rangle_{T} \sqrt{m(S)} \sqrt{m(T)} \bigg),\end{aligned}$$ where the implicit constant depends on $s,t$. Summing over $j,k$ (which correspond to disjoint intervals $I_j, I_k$) then completes the proof. ◻ We can now turn to the proof of Theorem [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"}. *Proof of Theorem [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"}.* We first describe the construction of the sparse collection $\mathcal{S}$ and then show how Lemma [Lemma 7](#lem:IterationLemma){reference-type="ref" reference="lem:IterationLemma"} leads to the desired domination. With the notation introduced in [\[eq:notationGB\]](#eq:notationGB){reference-type="eqref" reference="eq:notationGB"}, we assume that $f_1,f_2$ are supported on an interval $\tilde{I}_0\in \mathscr{D}$, we set $I_0=\tilde{I}_0^{\max\{s,t\}+1}$ and $$\mathcal{S}:= I_0 \cup \bigcup_{k=0}^\infty \mathcal{B}_k(I_0).$$ $\mathcal{S}$ is independent of $s$ and $t$ and only depends on $f_1$ and $f_2.$ We need to show that it is sparse, and we do so showing that it satisfies [\[eq:CarlesonPacking\]](#eq:CarlesonPacking){reference-type="eqref" reference="eq:CarlesonPacking"}. By an inductive argument, it suffices to show that for a given $I \in \mathcal{B}_j(I_0),$ there holds $$\sum_{\substack{J \in \mathcal{B}_{j+1}(I_0):\\J \subset I}} \mu(J) \leq \frac{\mu(I)}{2}.$$ For each such interval $J$, select $s(J) \in \{1,2\}$ such that $\langle f_{s(J)} \rangle_{J} \geq 16 \langle f_{s(J)} \rangle_{I}.$ We estimate, using the fact that the intervals in the sum are pairwise disjoint: $$\begin{aligned} \sum_{\substack{J \in \mathcal{B}_{j+1}(I_0):\\J \subset I}} \mu(J) & \leq \sum_{\substack{J \in \mathcal{B}_{j+1}(I_0):\\J \subset I}} \frac{\int_{J} f_{s(J)} \, d\mu}{16 \langle f_{s(J)} \rangle_{I}} \\ & \leq \frac{\mu(I)}{16} \sum_{\substack{J \in \mathcal{B}_{j+1}(I_0):\\J \subset I}} \left( \frac{\int_{J} f_{1} \, d\mu}{\int_{I} f_{1} \, d\mu}+ \frac{\int_{J} f_{2} \, d\mu}{\int_{I} f_{2} \, d \mu} \right)\\ & \leq \frac{\mu(I)}{8}. \end{aligned}$$ The claim then follows. We now turn to actually proving the domination. Since $f_1$ and $f_2$ are supported on $I_0$, we may assume that $\alpha_{J,K}^{I}=0$ if $I\cap I_0=\emptyset$. Therefore, we may decompose $$\begin{aligned} \langle Tf_1,f_2\rangle = \sum_{I \in \mathcal{O}(I_0)} \sum_{J \in \mathscr{D}_s(I)}\sum_{K \in \mathscr{D}_t(I)}\alpha_{J,K}^{I} \langle f_1, h_J \rangle \langle f_2, h_K \rangle \\ + \sum_{m=0}^{2^s-1} \sum_{n=0}^{2^t-1} \sum_{I \in \mathscr{D}(I_0)} \alpha_{I_m^s,I_n^t}^{I} \langle f_1, h_{I_m^s} \rangle \langle f_2, h_{I_n^t} \rangle \\ =: \langle T_0 f_1, f_2 \rangle + \sum_{m=0}^{2^s-1} \sum_{n=0}^{2^t-1} \langle T_{m,n} f_1, f_2 \rangle.\end{aligned}$$ Since we have the corona-type decomposition $$\mathscr{D}(I_0)= \bigcup_{J \in \mathcal{S}} \mathcal{G}(J),$$ we may apply Lemma [Lemma 7](#lem:IterationLemma){reference-type="ref" reference="lem:IterationLemma"} to each $I\in \mathcal{S}$ and sum to get $$|\langle T_{m,n} f_1, f_2 \rangle| \lesssim \mathcal{A}_{\mathcal{S}}(f_1,f_2) + \sum_{\substack{J,K \in \mathcal{S}: \\ 1 \leq \operatorname{dist}_{\mathscr{D}}(J,K) \leq N+2}} \langle f_1 \rangle_{J} \langle f_2 \rangle_{K} \sqrt{m(J)} \sqrt{m(K)}.$$ after summing over $I\in\mathcal{S}$. Lemma [Lemma 4](#lem:SparseReduction){reference-type="ref" reference="lem:SparseReduction"} then completes the estimate for the operators $T_{m,n}$ after passing to a second sparse collection $\mathcal{S}'$. Finally, because of our choice of $I_0$, if $I_0 \subsetneq I$ then $h_J$ is constant on any interval contained in $\tilde{I}_0$ if $J\in \mathscr{D}_s(I) \cup \mathscr{D}_t(I)$. Therefore, $$\begin{aligned} | \langle T_0 f_1, f_2 \rangle | & \leq \sum_{I \in \mathcal{O}(I_0)} \sum_{J \in \mathscr{D}_s(I)}\sum_{K \in \mathscr{D}_t(I)} |\langle f_1, h_J \rangle| |\langle f_2, h_K \rangle| \\ & = \sum_{I \in \mathcal{O}(I_0)} \sum_{J \in \mathscr{D}_s(I)}\sum_{K \in \mathscr{D}_t(I)} |\langle \langle f_1\rangle_{\tilde{I}_0} 1_{\tilde{I}_0}, h_J \rangle| |\langle \langle f_2\rangle_{\tilde{I}_0} 1_{\tilde{I}_0}, h_K \rangle| \\ & \lesssim_{s,t} \left( \sum_{J\in \mathscr{D}} |\langle \langle f_1\rangle_{\tilde{I}_0} 1_{\tilde{I}_0}, h_J \rangle|^2\right)^{\frac12} \left( \sum_{K\in \mathscr{D}} |\langle \langle f_2\rangle_{\tilde{I}_0} 1_{\tilde{I}_0}, h_K \rangle|^2\right)^{\frac12} \\ & = \left\| \langle f_1\rangle_{\tilde{I}_0} 1_{\tilde{I}_0}\right\|_2 \left\| \langle f_2\rangle_{\tilde{I}_0} 1_{\tilde{I}_0}\right\|_2 = \langle f_1\rangle_{\tilde{I}_0} \langle f_2\rangle_{\tilde{I}_0} \mu(\tilde{I}_0), \\\end{aligned}$$ and the result follows. ◻ **Remark 8**. Notice that $\mathcal{C}_{\mathcal{S}}^0(f_1,f_2)\equiv 0$ for any pair $f_1,f_2$. Therefore, Theorem [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"} with $(s,t)=(0,0)$ recovers the known result for Haar multipliers, which admit the usual sparse domination by $\mathcal{A}_{\mathcal{S}}(f_1,f_2)$ (see [@Lacey2017]). **Remark 9**. Theorem [Theorem 1](#th:thmA){reference-type="ref" reference="th:thmA"} is an immediate consequence of Theorem [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"} when we take $N=1$. Note that this provides a sparse domination for the dyadic Hilbert transform $\Sha$ and -the same one- for its adjoint $\Sha^*$. In addition, by following the proof of Theorem [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"}, one can obtain a refined sparse domination for $\Sha$ alone of the form $$| \langle \Sha f_1, f_2 \rangle| \lesssim C \left(\sum_{I \in \mathcal{S}} \langle f_1 \rangle_I \langle f_2 \rangle_{I} \mu(I) + \sum_{\substack{I \in \mathcal{S}:\\ I^b_{-} \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{I^b_{-}} m(I) + \sum_{\substack{I \in \mathcal{S}:\\ I^b_{+} \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{I^b_{+}} m(I) \right)$$ We leave the details to the interested reader. ## A maximal function In what follows, given a pair of dyadic intervals $I,J$ and $p \in [1,\infty)$, let $$c_p^b(I,J)= \begin{cases} 1 & I=J \\ \left( \frac{m(I)^{p/2} m(J)^{p/2}}{\mu(J) \mu(I)^{p-1}}\right) & \text{otherwise}. \end{cases} \label{ModApCorFactor}$$ The following modified maximal function that depends on $N\in\mathbb{N}$ is intimately related to sparse domination for $T\in \mathrm{HS}(s,t)$ for $s+t=N$. **Definition 10**. Given $N \in \mathbb{N}$, define the following maximal dyadic operator for $f \in L^1(\mu)$: $$\mathcal{M}^Nf(x):= \sup_{\substack{I, J \in \mathscr{D}:\\ \operatorname{dist}_{\mathscr{D}}(I,J)\leq N+2}} c_1^b(I,J) \langle |f| \rangle_{I} 1_{J}(x)$$ Under the assumption that $\mu$ is balanced, the operator $\mathcal{M}^N$ enjoys similar boundedness properties as the classical Hardy-Littlewood maximal function. **Lemma 11**. $\mathcal{M}^N$ is bounded on $L^p(\mu)$ for $1<p\leq \infty$ and is of weak-type $(1,1).$ *Proof.* $\mathcal{M}^N$ is bounded on $L^\infty(\mu)$ with constant depending on the parameter $N$ and the balancing constants of $\mu$, so it suffices to show $\mathcal{M}^N$ is weak-type $(1,1)$ and then interpolate. Take $0\leq f \in L^1(\mu)$, and for $\lambda>0$ let $$E_\lambda=\{x: \mathcal{M}^N_{\mathscr{D}}f(x)>\lambda\}.$$The following countable collection of intervals then clearly covers $E_\lambda$: $$\mathscr{C}_{\lambda}=\left\{J \in \mathscr{D}: \exists \, I \in \mathscr{D}\text{ so } \operatorname{dist}_{\mathscr{D}}(I,J)\leq N+2 \text{ and } c_1^b(I,J) \langle f\rangle_{I} > \lambda \right\}.$$ By selecting the maximal dyadic intervals in $\mathscr{C}_\lambda$, we obtain a countable, pairwise disjoint collection of intervals $\{J_j\}_{j=1}^{\infty}$ that covers $E_\lambda$, together with an associated collection of dyadic intervals $\{I_j\}_{j=1}^{\infty}$ satisfying $$c_1^b(I_j,J_j) \langle f \rangle_{I_j}> \lambda \label{LargeAverage}$$ for all $j.$ The intervals $\{I_j\}_{j=1}^{\infty}$ need not be pairwise disjoint. However, we can select a subsequence of maximal ones that are pairwise disjoint, which we denote by $\{I_{j_k}\}_{k=1}^{\infty}$. For fixed $k \in \mathbb{N}$, define the following index collections: $$A_k=\{j \in \mathbb{N}: I_j \subset I_{j_k}\}, \quad B_k=\{j \in A_k: J_j \cap I_{j_k} \neq \emptyset\}.$$ $\{A_k\}_{k=1}^{\infty}$ partitions $\mathbb{N}.$ First, we claim that $$| A_k \setminus B_k|\leq C_N. \label{CardBound}$$ To see this, notice that if $j \in A_k \setminus B_k$, then $I_j \subset I_{j_k}$ and $J_j \cap I_{j_k}= \emptyset.$ Notice that if $K$ is a common dyadic ancestor to $I_j$ and $J_j$, it must contain $I_{j_k}$, for otherwise it is contained in $I_{j_k}$, and thus $I_{j_k}$ contains $J_j$, a contradiction. Thus $K$ is also a common dyadic ancestor of $I_{j_k}$ and $J_j$. But then, by the definition of dyadic distance, we have $\operatorname{dist}_{\mathscr{D}}(J_j,I_{j_k}) \leq\operatorname{dist}_{\mathscr{D}}(J_j,I_{j}) \leq C_N$, which implies [\[CardBound\]](#CardBound){reference-type="eqref" reference="CardBound"}. Also, notice that for any $j \in A_k \setminus B_k$, by [\[LargeAverage\]](#LargeAverage){reference-type="eqref" reference="LargeAverage"} and the balanced hypothesis we have the uniform control $$\begin{aligned} \mu(J_j) & \leq \frac{1}{\lambda} \left(\frac{\mu(J_j)}{\mu(I_j)}\right) \left(\frac{\sqrt{m(I_{j}) m(J_{j})}}{\mu(J_{j})}\right) \int_{I_j} f \, d\mu \nonumber \\ & \lesssim \frac{1}{\lambda} \int_{I_{j_k}} f \, d\mu \label{UniformWeakType} . \end{aligned}$$ On the other hand, if there exists $j \in B_k$ that satisfies $J_j \supset I_{j_k}$, then $J_j$ is the only element of $B_k$, since the $J_j$ are pairwise disjoint, and moreover $\mu(J_j)$ is controlled by [\[UniformWeakType\]](#UniformWeakType){reference-type="eqref" reference="UniformWeakType"} by the same argument as above. In this case, $|A_k|\leq C_N+1$ and we can redefine $B_k= \emptyset$ in the argument that follows. In the second case, we have $J_j \subsetneq I_{j_k}$ for all $j \in B_k$, but this time $B_k$ can be infinite. We have $$\bigcup_{j \in B_k} J_j \subset I_{j_k}, \label{WeakTypeContainment}$$ and moreover, by the definition of $I_{j_k}$ and using the fact that $m(I_{j_k}) \sim m(J_{j_k})$ by the balanced condition with implicit constant depending only on $N$: $$\begin{aligned} \mu(I_{j_k}) & \leq \frac{1}{\lambda} \frac{\sqrt{m(I_{j_k}) m(J_{j_k})}}{\mu(J_{j_k})} \int_{I_{j_k}} f \, d\mu \nonumber\\ & \lesssim \frac{1}{\lambda} \int_{I_{j_k}} f \, d\mu \label{UniformWeak2}. \end{aligned}$$ Collecting these facts together, we then have $$\begin{aligned} \mu(E_\lambda) & \leq \sum_{j=1}^{\infty} \mu(J_j)\\ & = \sum_{k=1}^{\infty} \sum_{j \in A_k \setminus B_k} \mu(J_j) +\sum_{k=1}^{\infty} \sum_{j \in B_k} \mu(J_{j}) \\ & := \mathrm{(I)} + \mathrm{(II)}. \end{aligned}$$ Note that for $\mathrm{(I)}$, using [\[CardBound\]](#CardBound){reference-type="eqref" reference="CardBound"} and [\[UniformWeakType\]](#UniformWeakType){reference-type="eqref" reference="UniformWeakType"} and recalling that the $I_{j_k}$ are pairwise disjoint, we have the control $$\begin{aligned} \sum_{k=1}^{\infty} \sum_{j \in A_k \setminus B_k} \mu(J_j) & \lesssim \frac{1}{\lambda} \sum_{k=1}^{\infty} \sum_{j \in A_k \setminus B_k} \int_{I_{j_k}} f \, d\mu\\ & \leq \frac{(C_N+1)}{\lambda} \sum_{k=1}^{\infty} \int_{I_{j_k}} f \, d\mu \\ & \lesssim \frac{\|f\|_{L^1(\mu)}}{\lambda},\end{aligned}$$ which is the bound we want. We bound $\mathrm{(II)}$ using the pairwise disjointess of the $J_j$, [\[WeakTypeContainment\]](#WeakTypeContainment){reference-type="eqref" reference="WeakTypeContainment"}, and [\[UniformWeak2\]](#UniformWeak2){reference-type="eqref" reference="UniformWeak2"}: $$\begin{aligned} \sum_{k=1}^{\infty} \sum_{j \in B_k} \mu(J_{j}) & \leq \sum_{k=1}^{\infty} \mu(I_{j_k})\\ & \lesssim \frac{1}{\lambda} \sum_{k=1}^{\infty} \int_{I_{j_k}} f \, d\mu \\ & \lesssim \frac{\|f\|_{L^1(\mu)}}{\lambda},\end{aligned}$$ which completes the proof. ◻ $\mathcal{M}^N$ admits the same sparse domination as any $T\in\mathrm{HS}(s,t)$ for $N=s+t$. As before, we prove the sparse domination for the bi-sublinear form defined by it. **Theorem 12**. *Let $N \in \mathbb{N}.$ There exists $\eta \in (0,1)$ so that for each pair of compactly supported, bounded, nonnegative functions $f_1, f_2 \in L^1(\mu)$, there exists an $\eta$-sparse collection $\mathcal{S}\subset \mathscr{D}$ so that* *$$\mathlarger{|} \left \langle \mathcal{M}^N f_1, f_2 \right \rangle \mathlarger{|} \lesssim (\mathcal{A}_{\mathcal{S}}(f_1,f_2)+ \mathcal{C}_{\mathcal{S}}^N(f_1,f_2)). \label{MaximalSparseBound}$$* *Proof.* Fix $f_1,f_2$ as in the statement. By Lemma [Lemma 4](#lem:SparseReduction){reference-type="ref" reference="lem:SparseReduction"}, it suffices to show there exists $\eta \in (0,1)$ and an $\eta$-sparse collection $\mathcal{S}$ so that $$\mathlarger{|} \left \langle \mathcal{M}^N f_1, f_2 \right \rangle \mathlarger{|} \lesssim \mathcal{A}_{\mathcal{S}}(f_1,f_2)+ \sum_{\substack{J, K \in \mathcal{S}:\\ 1 \leq \operatorname{dist}_{\mathscr{D}}(J,K) \leq N+2}}\langle f_1 \rangle_{J} \langle f_2 \rangle_{K}\, \, \sqrt{m(J)} \sqrt{m(K)}. \label{MaximalSparseRedu}$$ We will use a level set argument to prove [\[MaximalSparseRedu\]](#MaximalSparseRedu){reference-type="eqref" reference="MaximalSparseRedu"}. For $n \in \mathbb{Z}$ set $$E_n=\{x \in \mathbb{R}: \mathcal{M}^Nf_1(x)>2^n\}.$$ Cover $E_n \setminus E_{n+1}$ by maximal dyadic intervals $\{J_n^j\}_{j=1}^{\infty}$ so that for each $J_n^j$, there exists a corresponding $I_n^j \in \mathscr{D}$ satisfying $$2^n < c_1^b(I_n^j,J_n^j) \langle f_1 \rangle_{I_n^j} \leq 2^{n+1}, \quad \operatorname{dist}_{\mathscr{D}}(I_n^j,J_n^j) \leq N+2.$$ For fixed $n$, $\{J_n^j\}$ is a pairwise disjoint collection of intervals, although this is no longer necessarily the case when one varies the parameter $n$ and moves up and down level sets. However, even for fixed $n$, the collection $\{I_n^j\}$ need not be pairwise disjoint. Even so, by using an argument similar to that in Lemma [Lemma 11](#ModMaxBounds){reference-type="ref" reference="ModMaxBounds"}, in the bound we obtain we can replace $\{I_n^j\}$ with a subcollection of intervals which satisfy a finite overlap condition. We next describe the details of passing to this subcollection and obtaining the sparse bound. First, for fixed $n$, select a maximal subsequence $\{I_n^{j_k}\}_{k=1}^{\infty}$ among the $I_n^j.$ This collection will be pairwise disjoint. Then, for each $k$, define index collections $A_n^k$ and $B_n^k$ as in the proof of Lemma [Lemma 11](#ModMaxBounds){reference-type="ref" reference="ModMaxBounds"}. Recall we argued that for each $k$, we have $|A_n^{k} \setminus B_n^k|\leq C_N$, and this bound is independent of $n$ as well. If $|B_n^k|=1$, let $\ell(k) \in B_n^k$ denote the unique index so that $I_{n}^{\ell(k)} \subset I_{n}^{j_k}$ and $J_{n}^{\ell(k)} \cap I_{n}^{j_k} \neq \emptyset.$ Let $C_n$ denote the collection of indices $k$ so that $|B_n^k|=1$ and $J_{n}^{\ell(k)} \supset I_{n}^{j_k}.$ Set $$\mathcal{S}_n^1:= \{J_n^j\}_{j=1}^{\infty}, \quad \mathcal{S}_n^2:= \{I_n^{j_k}\}_{k=1}^{\infty}, \quad \mathcal{S}_n^3:=\bigcup_{k=1}^{\infty} \{I_n^j\}_{j \in A_n^k \setminus B_{n}^k}, \quad \mathcal{S}_n^4:= \{I_n^{\ell(k)}\}_{k \in C_n} .$$ The collections $\mathcal{S}_n^1$, $\mathcal{S}_n^1$, and $\mathcal{S}_n^4$ are pairwise disjoint by construction, while the collection $\mathcal{S}_{n}^3$ has finite overlap. Next, we estimate: $$\begin{aligned} \mathlarger{|} \left \langle \mathcal{M}^N f_1, f_2 \right \rangle \mathlarger{|} & = \int_{\mathbb{R}} \mathcal{M}^N f_1 \cdot f_2 \, d \mu\\ & \sim \sum_{n =-\infty}^{\infty} 2^n \int_{E_n \setminus E_{n+1}} f_2 \, d\mu\\ & \leq \sum_{n =-\infty}^{\infty} 2^n \sum_{j=1}^{\infty}\int_{J_n^j} f_2 \, d\mu\\ & = \sum_{n =-\infty}^{\infty} 2^n \sum_{k=1}^{\infty} \sum_{j \in A_n^k \setminus B_n^k}\int_{J_n^j} f_2 \, d\mu + \, \sum_{n =-\infty}^{\infty} 2^n \sum_{k=1}^{\infty} \sum_{j \in B_n^k}\int_{J_n^j} f_2 \, d\mu. \end{aligned}$$ For the first sum, we have: $$\begin{aligned} \sum_{n =-\infty}^{\infty} 2^n \sum_{k=1}^{\infty} \sum_{j \in A_n^k \setminus B_n^k}\int_{J_n^j} f_2 \, d\mu & \lesssim \sum_{n =-\infty}^{\infty} \sum_{k=1}^{\infty} \sum_{j \in A_n^k \setminus B_n^k} c_1^b(I_n^j,J_n^j) \langle f_1 \rangle_{I_n^j} \langle f_2 \rangle_{J_n^j} \mu(J_n^j)\\ & \lesssim \sum_{n =-\infty}^{\infty} \sum_{\substack{I \in \mathcal{S}_n^3,\, J \in \mathcal{S}_n^1:\\ \operatorname{dist}_{\mathscr{D}(I,J) \leq N+2}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{J} \sqrt{m(I)m(J)}.\end{aligned}$$ For the second sum, we first consider the case when $k \in C_n$. Then it is easy to see we arrive at the estimate: $$\sum_{n =-\infty}^{\infty} 2^n \sum_{k \in C_n} \int_{J_n^{\ell(k)}} f_2 \, d\mu \lesssim \sum_{n =-\infty}^{\infty} \sum_{\substack{I \in \mathcal{S}_n^4, \, J \in \mathcal{S}_n^1:\\ \operatorname{dist}_{\mathscr{D}(I,J) \leq N+2}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{J} \sqrt{m(I)m(J)}.$$ If $k \not\in C_n$, then any $j' \in B_n^k$ must satisfy $J_n^{j'} \subset I_{n}^{j_k}.$ Then we have, using the pairwise disjointness of the $J_n^{j'}$: $$\begin{aligned} \sum_{n =-\infty}^{\infty} 2^n \sum_{k \not \in C_n} \sum_{j' \in B_n^k} \int_{J_n^{j'}} f_2 \, d\mu & \lesssim \sum_{n =-\infty}^{\infty} \sum_{k \not\in C_n} c_1^b(I_n^{j_k}, J_n^{j_k}) \langle f_1 \rangle_{I_n^{j_k}} \int_{I_n^{j_k} } f_2 \, d\mu\\ & \lesssim \sum_{n =-\infty}^{\infty} \sum_{I \in \mathcal{S}_n^2} \langle f_1 \rangle_{I} \langle f_2 \rangle_{I} \mu(I).\end{aligned}$$ Put $$\mathcal{S}^j= \bigcup_{n=-\infty}^{\infty} \mathcal{S}_n^j, \quad \mbox{ and } \quad \mathcal{S}= \bigcup_{j=1}^{4} \mathcal{S}^j.$$ Altogether, we have proven the desired estimate [\[MaximalSparseRedu\]](#MaximalSparseRedu){reference-type="eqref" reference="MaximalSparseRedu"} for the collection $\mathcal{S}.$ We claim that the collection $\mathcal{S}$ is sparse. For readability, we separate out this technical section of the proof into a dedicated lemma. The proof of Theorem [Theorem 12](#MaximalSparse){reference-type="ref" reference="MaximalSparse"} is complete, up to the verification of the sparsity of $\mathcal{S}$. ◻ **Lemma 13**. There exists $\eta \in (0,1)$ so that the collections $\mathcal{S}_j$, constructed in the proof of Theorem [Theorem 12](#MaximalSparse){reference-type="ref" reference="MaximalSparse"}, are individually $\eta$-sparse, for $j \in \{1,2,3, 4\}.$ Therefore, $\mathcal{S}$ is $\eta'$-sparse for some $0<\eta'<\eta$. *Proof.* We first argue for the sparsity of $\mathcal{S}^1.$ It is enough to show that $\mathcal{S}^1$ is a union of finitely many sparse collections. To begin with, note that if $J_{n'}^{j'} \subset J_n^j$ for some indices $n,j,n',j'$, we necessarily have $n' \geq n$, since $J_n^j \subset E_n$ and $J_{n'}^{j'} \cap E_{n'+1}^c \neq \emptyset$ by construction. By a standard argument, it suffices to show that there exists a sufficiently large positive integer $m$ (depending only on $N$) so that for $n'=n+m$, we have, for each interval $J_n^j \in \mathcal{S}_n^1$, $$\sum_{j':\, J_{n'}^{j'} \subset J_n^j}\mu(J_{n'}^{j'}) \leq \frac{1}{2} \mu(J_n^j) \label{CPackingMaximal}.$$ First, we split the left hand side of [\[CPackingMaximal\]](#CPackingMaximal){reference-type="eqref" reference="CPackingMaximal"} as follows: $$\begin{aligned} & \sum_{\substack{j':\, J_{n'}^{j'} \subset J_n^j\\ I_{n'}^{j'} \not\subset J_n^j}}\mu(J_{n'}^{j'}) + \sum_{\substack{j':\, J_{n'}^{j'} \subset J_n^j\\ I_{n'}^{j'} \subset J_n^j}}\mu(J_{n'}^{j'})\\ & := \mathrm{(I)}+\mathrm{(II)}.\end{aligned}$$ To estimate $\mathrm{(I)}$, notice that if $I_{n'}^{j'} \not\subset J_n^j$, we have $\operatorname{dist}_{\mathscr{D}}(J_{n'}^{j'}, J_n^j) \leq \operatorname{dist}_{\mathscr{D}}(J_{n'}^{j'}, I_{n'}^{j'})\leq N+2$, so the summation actually has at most $C_N$ terms. Moreover, we also have $\operatorname{dist}_{\mathscr{D}}(I_{n'}^{j'}, J_n^j) \leq N+2$. Then we estimate, using these facts, the balanced hypothesis, and the fact that $J_n^j$ intersects $E_{n+1}^c$: $$\begin{aligned} \mathrm{(I)} & \leq \frac{1}{2^{n'}} \sum_{\substack{j':\, J_{n'}^{j'} \subset J_n^j\\ I_{n'}^{j'} \not\subset J_n^j}} \sqrt{m(I_{n'}^{j'}) m(J_{n'}^{j'})} \langle f_1 \rangle_{I_{n'}^{j'}} \\ & \lesssim \frac{1}{2^{n'}} \sum_{\substack{j':\, J_{n'}^{j'} \subset J_n^j\\ I_{n'}^{j'} \not\subset J_n^j}} \sqrt{m(I_{n'}^{j'}) m(J_{n}^{j})} \langle f_1 \rangle_{I_{n'}^{j'}}\\ & \leq \frac{1}{2^{m-1}} \sum_{\substack{j':\, J_{n'}^{j'} \subset J_n^j\\ I_{n'}^{j'} \not\subset J_n^j}} \mu(J_n^j) = \frac{C_N}{2^{m-1}} \mu(J_n^j),\end{aligned}$$ so we clearly have the desired control on $\mathrm{(I)}$ assuming a sufficiently large choice of $m.$ To control $\mathrm{(II)}$, we begin by observing that by definition, $$\langle f_1 \rangle_{J_n^j}= c_1^b(J_n^j,J_n^j) \langle f_1 \rangle_{J_n^j} \leq 2^{n+1} , \quad \text{and} \quad \bigcup_{\substack{j':\, J_{n'}^{j'} \subset J_n^j\\ I_{n'}^{j'} \subset J_n^j}} J_{n'}^{j'}\subset \left\{x \in J_n^j: \mathcal{M}^N(1_{J_n^j}f_1)(x) > 2^{n'} \right \}.$$ Then we may estimate, using the pairwise disjointness of $J_{n'}^{j'}$ and the weak-type estimate in the proof of Lemma [Lemma 11](#ModMaxBounds){reference-type="ref" reference="ModMaxBounds"}: $$\begin{aligned} \mathrm{(II)} & \leq \mu\left( \left\{x \in J_n^j: \mathcal{M}^N(1_{J_n^j}f_1)(x) > 2^{n'} \right \}\right)\\ & \leq \frac{\|\mathcal{M}^N\|_{L^1(\mu) \rightarrow L^{1,\infty}(\mu)}}{2^{n'}} \int_{J_n^j}f_1 \, d \mu \\ & \leq \frac{\|\mathcal{M}^N\|_{L^1(\mu) \rightarrow L^{1,\infty}(\mu)}}{2^{m-1}} \mu(J_n^j),\end{aligned}$$ which also establishes the necessary control for $\mathrm{(II)}$ as long as $m$ is chosen large enough. Thus, we conclude the collection $\mathcal{S}^1$ is sparse. Next, is not difficult to show that if we prove the collection $\mathcal{S}^2$ is sparse, then this will also imply that collections $\mathcal{S}^3$ and $\mathcal{S}^4$ are sparse, so we focus on checking the Carleson packing condition for each $I=I_{n}^{j_k} \in \mathcal{S}^2.$ To begin with, choose $a$ to be the unique positive integer satisfying $$2^a<\langle f_1 \rangle_{I_n^{j_k}}= c_1^b(I_n^{j_k},I_{n}^{j_k})\langle f_1 \rangle_{I_n^{j_k}} \leq 2^{a+1}.$$ This implies that $I_{n}^{j_k} \subset E_a.$ Now, we split $$\sum_{\substack{K \in \mathcal{S}^2:\\ K \subsetneq I_n^{j_k}}} \mu(K)= \sum_{n'< a} \sum_{\substack{K \in \mathcal{S}_{n'}^2:\\ K \subsetneq I_n^{j_k}}} \mu(K)+ \sum_{n' \geq a} \sum_{\substack{K \in \mathcal{S}_{n'}^2:\\ K \subsetneq I_n^{j_k}}} \mu(K). \label{InteriorSplit}$$ Take $K=I_{n'}^{j_\ell'}$ as in the first sum. Note we must have $J_{n'}^{j_{\ell}'} \cap I_{n}^{j_k}= \emptyset.$ Indeed, $I_{n}^{j_k}$ is contained in $E_{a}$, while $J_{n'}^{j_{\ell}'}$ nontrivially intersects $E_{n'+1}^c$, which is disjoint from $E_a$ when $n'< a.$ Therefore, $\operatorname{dist}_{\mathscr{D}}(I_{n'}^{j_{\ell}'}, I_n^{j_k}) \leq \operatorname{dist}_{\mathscr{D}}(J_{n'}^{j_{\ell}'}, I_{n'}^{j_\ell'}) \leq N+2$, so the first sum in [\[InteriorSplit\]](#InteriorSplit){reference-type="eqref" reference="InteriorSplit"} is finite and in fact is bounded by $C_N \mu(I_{n}^{j_k}).$ The second summation may be estimated as follows, using the pairwise disjointness of each collection $\mathcal{S}_{n'}^2$ and our choice of $a$: $$\begin{aligned} \sum_{n' \geq a} \sum_{\substack{I_{n'}^{j_\ell'} \in \mathcal{S}_{n'}^2:\\ I_{n'}^{j_\ell'} \subsetneq I_n^{j_k}}} \mu(I_{n'}^{j_\ell'}) & \leq \sum_{n' \geq a} \frac{1}{2^{n'}} \sum_{\substack{I_{n'}^{j_\ell'} \in \mathcal{S}_{n'}^2:\\ I_{n'}^{j_\ell'} \subsetneq I_n^{j_k}}} C_{1}(I_{n'}^{j_\ell'}, J_{n'}^{j_\ell'}) \int_{I_{n'}^{j_\ell'}} f_1 \, d \mu \\ & \lesssim \sum_{n' \geq a} \frac{1}{2^{n'}} \int_{I_n^{j_k}} f_1 \, d \mu \\ & \leq \sum_{n' \geq a} \frac{1}{2^{n'}} 2^{a+1} \mu(I_{n}^{j_k}) \\ & \lesssim \mu(I_{n}^{j_k}).\end{aligned}$$ This bound proves the Carleson packing condition for $I$ and completes the proof. ◻ **Remark 14**. The sparse domination given in Theorem [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"} depends on the complexity in an essential way. To prove this, we next show that a complexity $2$ Haar shift cannot be dominated by a complexity $1$ sparse form by a straightforward adaptation of Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"}. Define $$Tf:= \sum_{I \in \mathscr{D}} \langle f, h_I \rangle h_{I_{--}}, \quad f \in L^2(\mu).$$ Fix $0 < \eta < 1$ and let $\mu$ be as in Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"}. We will show there exist two sequences of compactly supported functions $\{f_j\},\{g_j\} \subset L^2(\mu)$ such that for all $\eta$-sparse families $\mathcal{S}\subset \mathscr{D}$ $$\label{sparsefailure2} |\langle T f_j, g_j \rangle | \gtrsim j (|\mathcal{A}_\mathcal{S}(f_j,g_j)| +|\mathcal{C}_{\mathcal{S}}^1(f_j,g_j)| ).$$ For $j \geq 2$, we take $f_j= 1_{I_{j-1}^b}$ and $g_j= 1_{I_{j+1}^b}.$ As a preliminary, we notice $f_j$ and $g_j$ have disjoint supports and if $I \in \mathscr{D}$ intersects both $\text{supp} (f_j)$ and $\text{supp} (g_j)$ non-trivially, we must have $I=I_k$ with $0 \leq k \leq j-2.$ We also observe $$\mathcal{M}^1(f_j)(x) \lesssim \frac{\mu(I_{j-1}^b)}{\mu(I_{k-1})}, \, \mathcal{M}(g_j)(x) =\frac{\mu(I_{j+1}^b)}{\mu(I_{k-1})}, \, x \in I_{k}^b \text{ for } 0 < k \leq j-3 ; \label{Maximal1PointwiseBound1}$$ $$\mathcal{M}^1(f_j)(x) \lesssim 1, \, \mathcal{M}(g_j)(x) = \frac{\mu(I_{j+1}^b)}{\mu(I_{k-1})}, \, x \in I_{k}^b \text{ for } j-2 \leq k \leq j ; \label{Maximal1PointwiseBound2}$$ $$\mathcal{M}^1(f_j)(x)\lesssim \frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})}, \, \mathcal{M}(g_j)(x) = 1, \, x \in I_{j+1}^b ; \label{Maximal1PointwiseBound3}$$ $$\mathcal{M}^1(f_j)(x) \lesssim \frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})}, \, \mathcal{M}(g_j)(x) =\frac{\mu(I_{j+1}^b)}{\mu(I_{j})}, \, x \in I_{k}^b \text{ for } k > j+1 \label{Maximal1PointwiseBound4}$$ Using the sparsity of $\mathcal{S}$ and equations [\[Maximal1PointwiseBound1\]](#Maximal1PointwiseBound1){reference-type="eqref" reference="Maximal1PointwiseBound1"} through [\[Maximal1PointwiseBound4\]](#Maximal1PointwiseBound4){reference-type="eqref" reference="Maximal1PointwiseBound4"}, we can estimate the sums on the right hand side of [\[sparsefailure2\]](#sparsefailure2){reference-type="eqref" reference="sparsefailure2"} directly: $$\begin{aligned} \sum_{I \in \mathcal{S}} \langle f_j \rangle_{I} \langle g_j \rangle_{I} \mu(I) & + \sum_{\substack{J, K \in \mathcal{S}:\\ \operatorname{dist}_{\mathscr{D}}(J,K) = 3,\\ J \cap K= \emptyset}} \langle f_j \rangle_{J} \langle g_j \rangle_{K}\, \, \sqrt{m(J)} \sqrt{m(K)} \\ & \lesssim \sum_{I \in \mathcal{S}} \langle f_j \rangle_{I} \langle g_j \rangle_{I} \mu(E_I) + \sum_{\substack{J, K \in \mathcal{S}:\\ \operatorname{dist}_{\mathscr{D}}(J,K) = 3,\\ J \cap K= \emptyset}} \langle f_j \rangle_{J} \langle g_j \rangle_{K}\, \, \mu(E_K) \frac{\sqrt{m(J)} \sqrt{m(K)}}{\mu(K)} \\ & \lesssim \int_{[0,1]} \mathcal{M}^{1} f_j \cdot \mathcal{M}g_j\, d\mu \\ & \leq \sum_{k=1}^{\infty} \int_{I_k^b} \mathcal{M}^1 f_j \cdot \mathcal{M}g_j \, d\mu \\ & \lesssim \sum_{k=1}^{j-3} \left( \frac{\mu(I_{j-1}^b)}{\mu(I_{k-1})} \right) \cdot \left(\frac{\mu(I_{j+1}^b)}{\mu(I_{k-1})}\right) \cdot \mu(I_k^b) + \sum_{k=j-2}^{j} \left(\frac{\mu(I_{j+1}^b)}{\mu(I_{k-1})}\right) \cdot \mu(I_k^b) \\ & + \left(\frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})}\right) \cdot \mu(I_{j+1}^b) + \sum_{k=j+2}^{\infty}\left(\frac{\mu(I_{j-1}^b)}{\mu(I_{j-2})}\right) \cdot \left(\frac{\mu(I_{j+1}^b)}{\mu(I_{j})}\right) \cdot\mu(I_k^b) \\ & \lesssim \left(\frac{1}{j^4} \sum_{k=1}^{j-3} 1\right) + \frac{1}{j^3} + \frac{1}{j^3}+ \left(\frac{1}{j^2}\sum_{k=j+2}^{\infty} \frac{1}{k^2} \right) \lesssim \frac{1}{j^3}. \end{aligned}$$ Now we turn to a lower bound for $|\langle T f_j, g_j \rangle|.$ Given $I \in \mathscr{D}$, denote by $S_I$ the simple shift $S_I f= \langle f, h_I \rangle h_{I_{--}}.$ Then, by definition and using the support properties of $f_j$ and $g_j,$ we have $\langle S_I f_j, g_j \rangle= \langle f_j, h_I \rangle \cdot \langle h_{I_{--}}, g_j \rangle \neq 0$ if and only if $I= I_k$ with $0 \leq k \leq j-2.$ Therefore, we may write $$\langle T f_j, g_j \rangle = \sum_{k=0}^{j-2} \langle S_{I_k} f_j, g_j \rangle = \sum_{k=0}^{j-2} \langle f_j, h_{I_k} \rangle \cdot \langle h_{I_{k+2}}, g_j \rangle.$$ For the last term in the sum, computations similar to Proposition [Proposition 1](#prop:SparseFailure){reference-type="ref" reference="prop:SparseFailure"} give $$\langle f_j, h_{I_{j-2}} \rangle \cdot \langle h_{I_{j}}, g_j \rangle= \sqrt{m(I_{j})m(I_{j-2}}) \sim \frac{1}{j^2}.$$ As before, all the other terms in the summation will similarly be positive, so [\[sparsefailure2\]](#sparsefailure2){reference-type="eqref" reference="sparsefailure2"} follows and the proof is complete. Of course, one can adapt this argument to higher complexities in an obvious way. The key point is that the higher complexity maximal function $\mathcal{M}^N$ only needs to be applied to one of the functions $f_j, g_j$, which leads to a tighter upper bound than if it were applied to both. **Remark 15**. We can obtain a similar result as Remark [Remark 14](#SparseDiffComplex){reference-type="ref" reference="SparseDiffComplex"} for the sparse domination of the maximal function $M^N$ in Theorem [Theorem 12](#MaximalSparse){reference-type="ref" reference="MaximalSparse"}. In particular, we can show that it is impossible to dominate $\langle M^{N} f, g \rangle$ by a complexity $N '$ sparse form where $N'<N$. The proof is very similar to the Haar shift case. ## $L^p$ and weak-type estimates We can now turn to estimates for modified sparse forms. We start with an $L^p$ one. **Lemma 16**. Let $\mathcal{S}$ be sparse and $N\in\mathbb{N}$. For all nonnegative $f_1 \in L^p(\mu)$ and $f_2 \in L^{p'}(\mu)$, there holds $$\mathcal{C}_{\mathcal{S}}^N(f_1,f_2) \lesssim_p \|f_1\|_{L^p(\mu)} \|f_2\|_{L^{p'}(\mu)}.$$ *Proof.* Fix an integer $j$ satisfying $1 \leq j \leq N'.$ It clearly suffices to prove $$\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{c_j(I)} \sqrt{m(I) m(c_j(I))} \lesssim \|f_1\|_{L^p(\mu)} \|f_2\|_{L^{p'}(\mu)},$$ which we do as follows: $$\begin{aligned} \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{c_j(I)} \sqrt{m(I) m(c_j(I))} & \lesssim \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{c_j(I)} m(I)^{1/p} m(c_j(I))^{1/p'} \\ & \lesssim \left( \sum_{I \in \mathcal{S}} (\langle f_1 \rangle_{I})^p \mu(I) \right)^{1/p} \left( \sum_{J \in \mathcal{S}} (\langle f_2 \rangle_{J})^{p'} \mu(J) \right)^{1/p'} \\ & \lesssim \left( \sum_{I \in \mathcal{S}} (\langle f_1 \rangle_{I})^p \mu(E_I) \right)^{1/p} \left( \sum_{J \in \mathcal{S}} (\langle f_2 \rangle_{J})^{p'} \mu(E_J) \right)^{1/p'}\\ & \leq \|\mathcal{M}f_1\|_{L^p(\mu)} \|\mathcal{M}f_2\|_{L^{p'}(\mu)}\\ & \leq \| f_1\|_{L^p(\mu)} \| f_2\|_{L^{p'}(\mu)}.\end{aligned}$$ where we have used the fact that $\mu$ is balanced in the first step. This finishes the proof. ◻ **Proposition 17**. Let $\mathcal{S}$ be sparse and $N\in\mathbb{N}$. Then $$\sup_{\substack{f_1: \\ \|f_1\|_{L^1(\mu)} \leq 1}} \sup_{\substack{G \subset \mathbb{R}}} \inf_{\substack{G': \\ \mu(G) \leq 2 \mu(G')}} \sup_{f_2: |f_2| \leq 1_{G'}} \mathcal{C}_{\mathcal{S}}^N(|f_1|, |f_2|) < \infty.$$ *Proof.* Let $G$ be an arbitrary Borel set, and $f_1$ with $\|f_1\|_{L^1(\mu)}\leq 1$. We define the following set: $$H= \left\{x \in \mathbb{R}: \mathcal{M}^N_{\mathscr{D}}f_1(x)> \frac{C_0}{\mu(G)}\right\}$$ where we will choose the value of $C_0$ momentarily. By Lemma [Lemma 11](#ModMaxBounds){reference-type="ref" reference="ModMaxBounds"}, we get $$\mu(H) \leq \|\mathcal{M}^{N}_{\mathscr{D}}\|_{L^1(\mu) \rightarrow L^{1,\infty}(\mu)} \frac{\mu(G)}{C_0} \leq \frac{\mu(G)}{2},$$ after choosing $C_0 = 2 \|\mathcal{M}^{N}_{\mathscr{D}}\|_{L^1(\mu) \rightarrow L^{1,\infty}(\mu)}$. Take $G'= G \setminus H.$ The above estimate shows $\mu(G) \leq 2 \mu(G')$, so $G'$ is a contender in the infimum. For any $f_2$ with $|f_2| \leq 1_{G'}.$, it is enough to prove $$\sum_{\substack{I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{c_j(I)} \sqrt{m(I)} \sqrt{m(c_j(I))} \lesssim 1$$ for each $j \in \{1, \cdots, N'\}$. Let $I \in \mathcal{S}$ be such that $c_j(I) \in \mathcal{S}.$ If we have $$\frac{\sqrt{m(I) m(c_j(I))}}{\mu(c_j(I))} \langle f_1 \rangle_{I}> \frac{C_0}{\mu(G)},$$ then by definition $c_j(I) \subset H.$ But since $f_2$ is supported in $H^c,$ in this case we have $\langle f_2 \rangle_{c_j(I)}=0$. Therefore, we may assume without loss of generality that if $I \in \mathcal{S}$ is such that $c_j(I) \in \mathcal{S}$, then we have the estimate $$\langle f_1 \rangle_{I} \leq \frac{\mu(c_j(I))}{\sqrt{m(I) m(c_j(I))}} \frac{C_0}{\mu(G)}. \label{AvgControl}$$ We then estimate, using Cauchy-Schwarz and the $L^2$ bound for the ordinary maximal function: $$\begin{aligned} \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle f_2 \rangle_{c_j(I)} \sqrt{m(I) m(c_j(I))} & \lesssim \left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 m(I) \left(\frac{m(c_j(I))}{\mu(c_j(I))} \right) \right)^{1/2} \left(\sum_{J \in \mathcal{S}} (\langle f_2 \rangle_{J})^2 \mu(J) \right)^{1/2}\\ & \lesssim \left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) \mu(c_j(I)) \right)^{1/2} \left(\sum_{J \in \mathcal{S}} (\langle f_2 \rangle_{J})^2 \mu(E_J) \right)^{1/2} \\ & \lesssim \left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) \mu(E_{c_j(I)}) \right)^{1/2} \mu(G)^{1/2}.\end{aligned}$$ We will now show $$\left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) \mu(E_{c_j(I)}) \right)^{1/2} \lesssim \mu(G)^{-1/2},$$ which is enough to conclude. Let $$\widetilde{f}= \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} c_1^b(I,c_j(I)) \, \langle f_1 \rangle_{I} 1_{E_{c_j(I)}},$$ and notice that we have $$\|\widetilde{f}\|_{L^2(\mu)} \sim \left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) \mu(E_{c_j(I)}) \right)^{1/2}.$$ Moreover, we have the $L^\infty$ estimate, using [\[AvgControl\]](#AvgControl){reference-type="eqref" reference="AvgControl"}: $$\| \widetilde{f}\|_{L^\infty(\mu)} \lesssim \frac{C_0}{\mu(G)}.$$ Also notice that for almost every $x \in \mathbb{R}$, we have $$|\widetilde{f}(x)| \lesssim \mathcal{M}^N f_1(x).$$ Therefore, we can write using the distribution function, the weak-type estimate for $\mathcal{M}^N$ again, and the $L^1$ normalization of $f_1$: $$\begin{aligned} \| \widetilde{f} \|_{L^2(\mu )}^2 & = 2\int_{0}^\infty \lambda \mu(\{|\widetilde{f}|> \lambda\}) \, d \lambda \\ & \lesssim \int_{0}^{\frac{C}{\mu(G)}} \lambda \mu(\{ x: \mathcal{M}^N_{\mathscr{D}} f_1(x)> \lambda \}) \, d \lambda.\\ & \lesssim \|f_1\|_{L^1(\mu)} \int_{0}^{\frac{C'}{\mu(G)}} 1 \, d \lambda\\ & \lesssim \frac{\|f_1\|_{L^1(\mu)}}{\mu(G)}\leq \frac{1}{\mu(G)},\end{aligned}$$ proving the required estimate. ◻ **Remark 18**. As a consequence of Lemma [Lemma 16](#lem:LpBounds){reference-type="ref" reference="lem:LpBounds"}, Proposition [Proposition 17](#prop:WeakType){reference-type="ref" reference="prop:WeakType"} and the corresponding ones for $\mathcal{A}_\mathcal{S}$, we recover the known results for the $L^p$ bounds and weak-type estimates for $\Sha$ and other Haar shifts (see [@LMP2014]). **Remark 19**. We leave as an interesting open question whether pointwise sparse estimates can be obtained, instead of the bilinear ones proved in this section. # Weighted Estimates {#sec:section3} We now consider weights $w$ on $\mathbb{R}$ with respect to a fixed balanced measure $\mu$. In what follows, we write $$\langle f \rangle_{I}^w=\frac{1}{w(I)}\int_{I} f(x) w(x) \, d\mu(x),$$ and $\mathcal{M}^{w}$ to indicate the dyadic Hardy-Littlewood maximal operator with respect to the measure $w d\mu$. We now turn to natural question of weighted $L^p$ estimates for operators $T \in \mathrm{HS}(s,t)$. We first define the appropriate weight classes and quantities which reflect the complexity of the dyadic shift. ## A new class of weights For $p \in (1,\infty)$, we say $w \in A_p^{b}(\mu)$ if $$[w]_{A_p^{b}(\mu)}:= \sup_{\substack{I, J \in \mathscr{D}:\\ J \in \{I, I_{-}^b, I_{+}^b\} \text{ or }\\ I \in \{J_{-}^b,J_{+}^b\} }} c_p^b(I,J) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{J}^{p-1} < \infty.$$ If $p=1$, we say $w \in A_1^{b}(\mu)$ if $$[w]_{A_1^{b}(\mu)}:= \sup_{\substack{I, J \in \mathscr{D}:\\ J \in \{I, I_{-}^b, I_{+}^b\} \text{ or }\\ I \in \{J_{-}^b,J_{+}^b\}}} c_1^b(I,J) \langle w \rangle_{I} \|w^{-1}\|_{L^\infty(J)}\leq C.$$ *Remark 20*. If $w \in A_p^{b}(\mu)$, $[w]_{A_p(\mu)} \leq [w]_{A_p^{b}(\mu)}.$ If $\mu$ is dyadically doubling, $A_p(\mu) = A_p^{b}(\mu)$. Proposition [Proposition 3](#BadWeight){reference-type="ref" reference="BadWeight"} shows that, for general balanced measures, the inclusion $A_p^{b}(\mu) \subset A_p(\mu)$ is proper. As one would expect, we have the duality relationship $w \in A_p^{b}(\mu)$ if and only if $w^{1-p'} \in A_{p'}^{b}(\mu).$ We are going to use a different quantitative characterization of the classes $A_p^{b}(\mu)$ that will be useful to study the boundedness of $\mathcal{M}^N$. If $p \in (1,\infty)$ and $N \in \mathbb{N}$, we set $$[w]_{A_p^N(\mu)}:= \sup_{\substack{I, J \in \mathscr{D}:\\ 0 \leq \operatorname{dist}_{\mathscr{D}}(J,K) \leq N+2}} c_p^b(I,J) \left(\langle w \rangle_{I} \langle w^{1-p'} \rangle_{J}^{p-1}\right).$$ For $p=1$, we set $$[w]_{A_1^N(\mu)}:= \sup_{\substack{I, J \in \mathscr{D}:\\ \operatorname{dist}_{\mathscr{D}}(I,J)\leq N+2}} c_1^b(I,J) \langle w \rangle_{I} \|w^{-1}\|_{L^\infty(J)}.$$ **Remark 21**. $[w]_{A_1^N(\mu)}<\infty$ if and only if there exists $C$ such that $$\mathcal{M}^N w(x) \leq C w(x).$$ Moreover, $[w]_{A_1^N(\mu)}$ is equal to the infimum of all such $C$. **Lemma 22**. Let $\mu$ be balanced. For $N \in\mathbb{N}$ and $1 \leq p< \infty$, $$[w]_{A_p^{b}(\mu)} \lesssim [w]_{A_p^N(\mu)} \lesssim_N \left([w]_{A_p^{b}(\mu)}\right)^{2^{N-1}}.$$ In particular, $[w]_{A_p^N(\mu)} <\infty$ exactly when $[w]_{A_p^{b}(\mu)}<\infty$. *Proof.* We use induction on $N$. For the base case $N=1$, it is obvious that $[w]_{A_p^{b}} \lesssim [w]_{A_p^1(\mu)}.$ Conversely, if $w \in A_p^{b}(\mu)$ and $I \subset J$, for $p>1$ we have $$\begin{aligned} \left( \frac{m(I)^{p/2} m(J)^{p/2}}{\mu(J) \mu(I)^{p-1}}\right) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{J}^{p-1} & \leq \left( \frac{m(I)^{p/2} m(J)^{p/2}}{\mu(I)^{p}}\right) \langle w \rangle_{J} \langle w^{1-p'} \rangle_{J}^{p-1}\\ & \lesssim [w]_{A_p^{b}},\end{aligned}$$ An entirely analogous argument works when $J \subset I$. The only other case to consider is when $J=I^b$, in which case we have the control: $$\left( \frac{m(I)^{p/2} m(I^b)^{p/2}}{\mu(I^b) \mu(I)^{p-1}}\right) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{I^b}^{p-1} \lesssim \langle w \rangle_{\widehat{I}} \langle w^{1-p'} \rangle_{\widehat{I}}^{p-1} .$$ The case $p=1$ can be dealt with with straightforward modifications. This completes the proof of the base case. Now, assume that the statement holds for some positive integer $j$. The inequality $$[w]_{A_p^j(\mu)} \leq [w]_{A_p^{j+1}(\mu)}$$ is immediate. Additionally, it is enough to consider disjoint intervals $I,J$ when bounding $[w]_{A_p^{j+1}(\mu)}$, so suppose $I,J$ are disjoint with $\operatorname{dist}_{\mathscr{D}}(I,J)=j+3.$ We may assume $\mu(I)< \frac{1}{2} \mu(\widehat{I})$ and $\mu(J)< \frac{1}{2} \mu(\widehat{J})$. For example, if $\mu(I)\geq \frac{1}{2} \mu(\widehat{I})$, then in fact $\mu(I) \sim \mu(\widehat{I})$ and we would have the control $$\left( \frac{m(I)^{p/2} m(J)^{p/2}}{\mu(J) \mu(I)^{p-1}}\right) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{J}^{p-1} \lesssim \left( \frac{m(\widehat{I})^{p/2} m(J)^{p/2}}{\mu(J) \mu(\widehat{I})^{p-1}}\right) \langle w \rangle_{\widehat{I}} \langle w^{1-p'} \rangle_{J}^{p-1} \leq [w]_{A_p^j(\mu)}, \label{DoublingReduct}$$ since $\operatorname{dist}_{\mathscr{D}}(\widehat{I},J)=j+2.$ Therefore, under this assumption we have $m(I) \sim \mu(I)$ and $m(J) \sim \mu(J)$, since $m(\widehat{I}) \sim \min \{\mu(I), \mu(I^b)\}$ for any dyadic interval $I$ and $\mu$ is balanced. Let $K$ denote the minimal common ancestor of $I$ and $J$. It is not difficult to see that either $I^{(2)} \subset K$ or $J^{(2)} \subset K$. Assume without loss of generality that $I^{(2)} \subset K.$ We consider two cases. If $\mu(\widehat{I}^b) \geq \frac{1}{2} \mu(I^{(2)})$, then we have $$\mu(\widehat{I}) \sim m(I^{(2)}) \sim m(I) \sim \mu(I),$$ and we can then use the doubling argument as in [\[DoublingReduct\]](#DoublingReduct){reference-type="eqref" reference="DoublingReduct"}. Otherwise, $\mu(\widehat{I}^b) \sim m(\widehat{I}^b)$ and we may estimate by Hölder's inequality: $$\begin{aligned} \left( \frac{m(I)^{p/2} m(J)^{p/2}}{\mu(J) \mu(I)^{p-1}}\right) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{J}^{p-1} & \leq \left( \frac{m(I)^{p/2} m(J)^{p/2}}{\mu(J) \mu(I)^{p-1}}\right) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{\widehat{I}^b}^{p-1} \langle w \rangle_{\widehat{I}^b}\langle w^{1-p'} \rangle_{J}^{p-1} \\ & \sim \left[ \left( \frac{m(I)^{p/2} m(\widehat{I}^b)^{p/2}}{\mu(\widehat{I}^b) \mu(I)^{p-1}}\right) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{\widehat{I}^b}^{p-1} \right]\\ & \times \left[ \left( \frac{m(\widehat{I}^b)^{p/2} m(J)^{p/2}}{\mu(J) \mu(\widehat{I}^b)^{p-1}}\right) \langle w \rangle_{\widehat{I}^b} \langle w^{1-p'} \rangle_{J}^{p-1} \right]\\ & \leq [w]_{A_p^j(\mu)}^2,\end{aligned}$$ since $\operatorname{dist}_{\mathscr{D}}(I, \widehat{I}^b)=3 \leq j+2$ and $\operatorname{dist}_{\mathscr{D}}(\widehat{I}^b,J)=j+2$. This completes the induction. ◻ **Remark 23**. Regardless of the sharpness of the inequalities in the statement of Lemma [Lemma 22](#lem:ApbvsApN){reference-type="ref" reference="lem:ApbvsApN"}, it is probably not the case that $[w]_{A_p^j(\mu)} \sim [w]_{A_p^k(\mu)}$ with implicit constants independent of $w,j,k$. More compelling evidence to support the fact that the characteristics are not quantitatively equivalent comes from the sparse domination in Theorems [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"} and [Theorem 12](#MaximalSparse){reference-type="ref" reference="MaximalSparse"}. The sparse domination depends in an essential way on the complexity. Remark [Remark 14](#SparseDiffComplex){reference-type="ref" reference="SparseDiffComplex"} shows that, in general, a complexity one sparse form $\mathcal{C}_{\mathcal{S}}^1$ cannot possibly dominate a dyadic shift with complexity $2$, and so on for higher complexities. Therefore, obtaining sharp quantitative weighted estimates for dyadic shift operators defined with respect to balanced measures remains an open problem, but one would expect the quantitative bounds to depend on the complexity of the operator. The proof of our next result justifies the introduction of the quantities $[w]_{A_p^N(\mu)}$. **Proposition 24**. Let $p \in (1, \infty)$ and $N \in \mathbb{N}.$ The operator $\mathcal{M}^N$ is bounded on $L^p(w d\mu)$ if and only if $w \in A_p^{b}(\mu)$. $\mathcal{M}^N$ is bounded from $L^1(wd \mu)$ to $L^{1,\infty}(w d\mu)$ if and only if $w \in A_1^{b}(\mu).$ *Proof.* We apply Lemma [Lemma 22](#lem:ApbvsApN){reference-type="ref" reference="lem:ApbvsApN"} to use the quantity $[w]_{A_p^N(\mu)}$ instead of $[w]_{A_p^{b}(\mu)}$. To begin with, fix $1<p<\infty$ and suppose that $\|\mathcal{M}^N\|_{L^p(w d\mu) \rightarrow L^p(wd\mu)}=C.$ Let $I \in \mathscr{D}$ be arbitrary. Suppose that $w^{1-p'}$ is locally integrable (otherwise, consider $(w+\varepsilon)^{1-p'}$ and use a limiting argument). Let $J$ be any interval satisfying $0 \leq \operatorname{dist}(I,J)\leq N+2$ and take $f= w^{1-p'} 1_J.$ A computation shows that $$\|f\|_{L^p(wd \mu)}= \left( \int_{J} w^{1-p'} \, d \mu\right)^{1/p}.$$ Using the above and our hypothesis we estimate: $$\begin{aligned} C^p \left( \int_{J} w^{1-p'} \, d \mu \right)& \geq \|\mathcal{M}^Nf\|_{L^p(w d\mu)}^p\\ & \geq [c_1^b(J,I)]^p \int_{I} \left( \langle w^{1-p'} \rangle_J\right)^p w d\mu \\ & = c_p^b(I,J) \frac{\mu(J)}{\mu(I)} \int_{I} \left( \langle w^{1-p'} \rangle_J\right)^p w d\mu.\end{aligned}$$ Simple rearrangement then yields $$c_p^b(I,J) \langle w \rangle_{I} \langle w^{1-p'} \rangle_{J}^{p-1} \leq C^p,$$ which means $[w]_{A_p^N(\mu)} < \infty$, since $I,J$ were arbitrary. We prove the necessity when $p=1$ with an argument similar to [@M1972]. Put now $$\|\mathcal{M}^N\|_{L^1(w d\mu) \rightarrow L^{1,\infty}(w d\mu)} =C,$$ and let $I, J \in \mathscr{D}$ satisfy $\operatorname{dist}_{\mathscr{D}}(I,J) \leq N+2.$ Let $\varepsilon>0$ be arbitrary and let $$A=\{x \in J: w(x)< \operatorname{ess \, inf}_{J}w+\varepsilon\}.$$ Note that $\mu(A)>0.$ Take $f=1_{A}.$ Then, by definition we have $$I \subset \left\{x: \mathcal{M}^Nf(x) \geq \frac{\mu(A)}{\mu(J)} c_1^b(J,I) \right \}.$$ It then follows from the supposed weak-type estimate that $$\begin{aligned} w(I) & \leq C \frac{\mu(J)}{c_1^b(J,I)\mu(A)} \int_{A} w(x) d \mu(x)\\ & \leq C \frac{\mu(J)}{c_1^b(J,I)} (\operatorname{ess \, inf}_{J}w+\varepsilon).\end{aligned}$$ Since $\varepsilon$ was arbitrary and $C$ is independent of $\varepsilon$, simple rearrangment yields $$c_1^b(I,J) \langle w \rangle_{I} \leq C (\operatorname{ess \, inf}_{J}w),$$ which is precisely what we wanted to show. We next turn to the sufficiency of the condition $[w]_{A_p^N(\mu)} < \infty$. For $p>1$, it is immediately given by a weighted bound for the sparse form, which we postpone to the proof of Corollary [Corollary 28](#WeightedBounds){reference-type="ref" reference="WeightedBounds"}, which is obtained independently from the maximal function, together with Theorem [Theorem 12](#MaximalSparse){reference-type="ref" reference="MaximalSparse"}. When $p=1$, we argue like in Lemma [Lemma 11](#ModMaxBounds){reference-type="ref" reference="ModMaxBounds"}. Suppose $w \in A_1^{b}(\mu)$ and take $f \in L^1(w d \mu).$ Put $E_\lambda=\{x: \mathcal{M}^Nf(x)>\lambda\}$, cover $E_\lambda$ by disjoint maximal intervals $\{J_j\}$ with corresponding averaging intervals $\{I_j\}$. Select the maximal dyadic intervals from $\{I_j\}$, label them $\{I_{j_k}\}$, and define the index sets $A_k$, $B_k$ as in the proof of Lemma [Lemma 11](#ModMaxBounds){reference-type="ref" reference="ModMaxBounds"}. We may write $$\begin{aligned} w(E_\lambda) & \leq \sum_{j=1}^{\infty} \langle w \rangle_{J_j} \mu(J_j)\\ & = \sum_{k=1}^{\infty} \sum_{j \in A_k \setminus B_k} \langle w \rangle_{J_j} \mu(J_j) +\sum_{k=1}^{\infty} \sum_{j \in B_k} \langle w \rangle_{J_j} \mu(J_{j}) \\ & := \mathrm{(I)} + \mathrm{(II)}. \end{aligned}$$ We control $\mathrm{(I)}$ as follows, noticing that $c_1^b(J_j,I_j) \langle w \rangle_{J_j} \leq \mathcal{M}^Nw(x) 1_{I_j}(x)$ if $x \in I_j$: $$\begin{aligned} \sum_{k=1}^{\infty} \sum_{j \in A_k \setminus B_k} \langle w \rangle_{J_j} \mu(J_j) & \lesssim \frac{1}{\lambda} \sum_{k=1}^{\infty} \sum_{j \in A_k \setminus B_k} \langle w \rangle_{J_j} c_1^b(J_j,I_j)\int_{I_j} f \, d\mu\\ & \leq\frac{1}{\lambda }\sum_{k=1}^{\infty} \sum_{j \in A_k \setminus B_k} \int_{I_j} f \cdot \mathcal{M}^N w \, d\mu\\ & \leq \frac{(C_N+1)[w]_{A_1^N(\mu)}}{\lambda} \sum_{k=1}^{\infty} \int_{I_{j_k}} f \, wd\mu \\ & \leq \frac{(C_N+1)[w]_{A_1^N(\mu)}}{\lambda} \|f\|_{L^1(w d\mu)}.\end{aligned}$$ For $\mathrm{(II)}$, we can assume without loss of generality $J_j \subsetneq I_{j_k}$ for all $j \in B_k$. We then have $$\begin{aligned} \sum_{k=1}^{\infty} \sum_{j \in B_k} w(J_{j}) & \leq \sum_{k=1}^{\infty} \langle w \rangle_{I_{j_k}} \mu (I_{j_k})\\ & \lesssim \frac{1}{\lambda} \sum_{k=1}^{\infty} \langle w \rangle_{I_{j_k}}\int_{I_{j_k}} f \, d\mu \\ & \leq \frac{1}{\lambda} \sum_{k=1}^{\infty} \int_{I_{j_k}} f (\mathcal{M}w)\, d\mu \\ & \leq \frac{[w]_{A_1^N(\mu)}}{\lambda} \sum_{k=1}^{\infty} \int_{I_{j_k}} f w d\mu \\ & \lesssim \frac{[w]_{A_1^N(\mu)}}{\lambda} \|f\|_{L^1(w d\mu)}.\end{aligned}$$ ◻ ## Haar Shifts We now turn to weighted estimates for Haar shift operators. For convenience in what follows, we prove all weighted estimates for $p=2.$ However, all of the proofs carry over to the other values of $p$ with minor modifications. We first record the following standard lemma, which is an analog of an $A_\infty$ property we will need in the sequel. **Lemma 25**. Let $w \in A_p(\mu).$ Then for any $I \in \mathscr{D}$ and $E_I \subset I$ with $\mu(E_I) \geq \eta (\mu(I))$ for some $\eta \in (0,1)$, $$w(E_I) \geq \frac{\eta^p}{[w]_{A_p(\mu)}} w(I).$$ **Proposition 26**. Let $\mathcal{S}$ be sparse and $N\in \mathbb{N}$. Then $$\|\mathcal{C}_\mathcal{S}^N\|_{L^p(w d\mu) \rightarrow L^p(w d\mu)} \lesssim_{N,p} [w]_{A_p(\mu)}^{\frac{(p-1)^2+1}{p(p-1)}}[w]_{A_p^{N}(\mu)}^{1/p}.$$ *Proof.* We only prove the result for $p=2$, and we proceed by duality: we need to show that for all bounded, compactly supported nonnegative $f_1 , f_2 \in L^2(w d\mu)$, $$|\mathcal{C}_\mathcal{S}^N f_1, w f_2 \rangle| \lesssim [w]_{A_2(\mu)}[w]_{A_2^{N}(\mu)}^{1/2} \|f_1\|_{L^2(w d\mu)}\|f_2\|_{L^2(w d\mu)}.$$ As in the proof of Theorem [Lemma 16](#lem:LpBounds){reference-type="ref" reference="lem:LpBounds"}, we only need to show the following estimate: $$\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle w f_2 \rangle_{c_j(I)} \sqrt{m(I) m(c_j(I))} \leq C [w]_{A_2(\mu)} [w]_{A_2^{N}(\mu)}^{1/2} \|f_1\|_{L^2(w)} \|f_2\|_{L^{2}(w)}.$$ We estimate as follows: $$\begin{aligned} \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} &\langle w f_2 \rangle_{c_j(I)} \sqrt{m(I) m(c_j(I))} = \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle w f_1 \rangle_{I}^{w^{-1}} \langle f_2 \rangle_{c_j(I)}^{w} \langle w^{-1} \rangle_{I} \langle w \rangle_{c_j(I)} \sqrt{m(I) m(c_j(I))} \\ & \sim \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle w f_1 \rangle_{I}^{w^{-1}} \langle f_2 \rangle_{c_j(I)}^{w} \sqrt{\frac{m(I) m(c_j(I))}{\mu(I)\mu(c_j(I))}} \left(\langle w^{-1} \rangle_{I} \langle w \rangle_{c_j(I)}\right)^{1/2} (w^{-1}(I))^{1/2} w(c_j(I))^{1/2} \\ & \lesssim [w]_{A_2^{N}(\mu)}^{1/2} \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle w f_1 \rangle_{I}^{w^{-1}} \langle f_2 \rangle_{c_j(I)}^{w} (w^{-1}(I))^{1/2} w(c_j(I))^{1/2} \\ & \lesssim [w]_{A_2^{N}(\mu)}^{1/2} \left( \sum_{ I \in \mathcal{S}} (\langle w f_1 \rangle_{I}^{w^{-1}})^2 w^{-1}(I) \right)^{1/2} \left(\sum_{ J \in \mathcal{S}} (\langle f_2 \rangle_{J}^{w})^2 w(J) \right)^{1/2}\\ & \lesssim [w]_{A_2(\mu)}[w]_{A_2^{N}(\mu)}^{1/2} \left( \sum_{ I \in \mathcal{S}} (\langle w f_1 \rangle_{I}^{w^{-1}})^2 w^{-1}(E_I) \right)^{1/2} \left(\sum_{ J \in \mathcal{S}} (\langle f_2 \rangle_{J}^{w})^2 w(E_J) \right)^{1/2}\\ & \leq [w]_{A_2(\mu)} [w]_{A_2^{N}(\mu)}^{1/2} \|\mathcal{M}^{w^{-1}}(f_1 w)\|_{L^2(w^{-1} d \mu)} \|\mathcal{M}^w(f_2 )\|_{L^2(w d \mu)} \\ & \lesssim [w]_{A_2(\mu)} [w]_{A_2^{N}(\mu)}^{1/2} \|f_1 w\|_{L^2(w^{-1} d \mu)} \|f_2 \|_{L^2(w d \mu)} \\ & = [w]_{A_2(\mu)} [w]_{A_2^{N}(\mu)}^{1/2} \|f_1\|_{L^2(w d \mu)} \|f_2 \|_{L^2(w d \mu)},\\\end{aligned}$$ where we used Lemma [Lemma 25](#FairDivision){reference-type="ref" reference="FairDivision"} in the fifth line. This completes the proof. ◻ We can also obtain a weighted weak-type estimate for the sparse form when $p=1.$ **Proposition 27**. Let $\mathcal{S}$ be sparse and $N \in \mathbb{N}$. Then $$\|\mathcal{C}_\mathcal{S}^N \|_{L^1(w d\mu) \rightarrow L^{1,\infty}(w d \mu) } \lesssim_N [w]_{A_1^N(\mu)}^{2}.$$ *Proof.* It is enough to show $$\sup_{\substack{f_1: \\ \|f_1\|_{L^1(w d \mu)} \leq 1}} \sup_{\substack{G \subset \mathbb{R}}} \inf_{\substack{G': \\ w(G) \leq 2 w(G')}} \sup_{f_2: |f_2| \leq 1_{G'}} \mathcal{C}_{\mathcal{S}}^N(|f_1|, w|f_2|) \lesssim [w]_{A_1^N(\mu)}^{2}.$$ We follow the general scheme of the proof of Theorem [Proposition 17](#prop:WeakType){reference-type="ref" reference="prop:WeakType"}. Let $G$ be an arbitrary Borel set, and $f_1$ with $\|f_1\|_{L^1(w d \mu)}\leq 1$. Define $$H= \left\{x \in \mathbb{R}: \mathcal{M}^N f_1(x)> \frac{C_0}{w(G)}\right\},$$ where $C_0$ is some sufficiently large positive constant. By Proposition [Proposition 24](#MaximalChar){reference-type="ref" reference="MaximalChar"}, $w(H) \leq w(G)/2$ as long as $C_0 \geq C_0' [w]_{A_1^N(\mu)}$, where $C_0'$ depends only on $\mu$ and $N$. As before, we take $G'= G \setminus H.$ Let $f_2$ be arbitrary with $|f_2| \leq 1_{G'}.$ It is enough to show, for $1 \leq j \leq C_N$, that we have $$\sum_{\substack{I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle w f_2 \rangle_{c_j(I)} \sqrt{m(I)} \sqrt{m(c_j(I))} \lesssim [w]_{A_1^N(\mu)}^{2}$$ By the same argument as the proof of Proposition [Proposition 24](#MaximalChar){reference-type="ref" reference="MaximalChar"}, we may assume without loss of generality that if $I \in \mathcal{S}$ is such that $c_j(I) \in \mathcal{S}$, then we have the estimate $$\langle f_1 \rangle_{I} \leq \frac{\mu(c_j(I))}{\sqrt{m(I) m(c_j(I))}} \frac{C_0' [w]_{A_1^N(\mu)}}{w(G)}.$$ We then estimate, using Cauchy-Schwartz inequality, Lemma [Lemma 25](#FairDivision){reference-type="ref" reference="FairDivision"}, the fact that $[w]_{A_2(\mu)} \leq [w]_{A_1^N(\mu)}$, and the weighted $L^2$ bound for $\mathcal{M}^w$: $$\begin{aligned} \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} \langle f_1 \rangle_{I} \langle w f_2 \rangle_{c_j(I)} &\sqrt{m(I) m(c_j(I))} \\ &\lesssim \left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) w(c_j(I)) \right)^{1/2} \left(\sum_{J \in \mathcal{S}} (\langle wf_2 \rangle_{J})^2 \frac{\mu(J)^2}{w(J)} \right)^{1/2}\\ & \lesssim [w]_{A_1^N(\mu)} \left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) w(E_{c_j(I)}) \right)^{1/2} \left(\sum_{J \in \mathcal{S}} (\langle f_2 \rangle_{J}^w)^2 w(E_J) \right)^{1/2} \\ & \lesssim [w]_{A_1^N(\mu)} \left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) w(E_{c_j(I)}) \right)^{1/2} w(G)^{1/2}.\end{aligned}$$ We will now show $$(\ast):=\left(\sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} (\langle f_1 \rangle_{I})^2 \left(\frac{m(I) m(c_j(I))}{\mu(c_j(I))^2}\right) w(E_{c_j(I)}) \right)^{1/2} \lesssim [w]_{A_1^N(\mu)} w(G)^{-1/2}.$$ To see this, let $$\widetilde{f}= \sum_{ \substack{ I \in \mathcal{S}: \\ c_j(I) \in \mathcal{S}}} c_1^b(I,c_j(I)) \, \langle f_1 \rangle_{I} 1_{E_{c_j(I)}},$$ which satisfies $$\|\widetilde{f}\|_{L^\infty(\mu)} \lesssim \frac{C_0}{w(G)}, \quad |\widetilde{f}| \lesssim \mathcal{M}^N f_1, \quad \mbox{and} \quad \|\widetilde{f}\|_{L^2(w d\mu)} \sim (\ast).$$ This, combined with the weighted weak-type estimate for $\mathcal{M}^N$, and the $L^1(w d \mu)$ normalization of $f_1$, yields $$\begin{aligned} \| \widetilde{f} \|_{L^2(w d \mu )}^2 & = 2\int_{0}^\infty \lambda w(\{|\widetilde{f}|> \lambda\}) \, d \lambda \\ & \lesssim \int_{0}^{\frac{C[w]_{A_1^N(\mu)}}{w(G)}} \lambda w (\{ x: \mathcal{M}^N f_1(x)> \lambda\}) \, d \lambda.\\ & \lesssim [w]_{A_1^N(\mu)} \|f_1\|_{L^1(w d \mu)} \int_{0}^{\frac{C[w]_{A_1^N(\mu)}}{w(G)}} 1 \, d \lambda\\ & \lesssim \frac{[w]_{A_1^N(\mu)}^2}{w(G)},\end{aligned}$$ proving the required estimate. ◻ Using Theorem [Theorem 6](#th:ModifiedSparse){reference-type="ref" reference="th:ModifiedSparse"} and the estimates for $\mathcal{A}_\mathcal{S}$ (which are analogous to those in the case of the Lebesgue measure), we get the following consequence: **Corollary 28**. Let $N\in \mathbb{N}$. If $T\in\mathrm{HS}(s,t)$ and $s+t \leq N$ or $T=\mathcal{M}^N$, then - If $p>1$, $$\|T\|_{L^p(w d\mu) \rightarrow L^p(w d\mu)} \lesssim [w]_{A_p(\mu)}^{\max\{1, \frac{1}{p-1}\}}+[w]_{A_p(\mu)}^{\frac{(p-1)^2+1}{p(p-1)}}[w]_{A_p^{N}(\mu)}^{1/p}.$$ - The following weak-type estimate holds: $$\|T\|_{L^1(w d\mu) \rightarrow L^{1,\infty}(w d \mu) } \lesssim [w]_{A_1^N(\mu)}^{2}.$$ *Remark 29*. One can follow similar arguments to obtain "one sided" weight conditions that, for example, are sufficient for the boundedness on $L^2(w d\mu)$ (or $L^p(w d\mu)$) of $\Sha$ or other particular operators $T\in\mathrm{HS}(s,t)$. Indeed, one can check that the condition $$_{A_2^{0,1}(\mu)}:= \sup_{\substack{J \in \mathscr{D}:\\ K \in \{ J, J^{b}_{-}, J^{b}_{+}\}}}c_2^b(K,J) \langle w \rangle_{K} \langle w^{-1} \rangle_{J} < \infty \label{HilbertWeight}$$ is sufficient for the boundedness of $\Sha$. However, such conditions are less natural in the sense that they are asymmetric with respect to $w$ and $w^{-1}$ and are specialized to the form of the particular operator under consideration. We finally consider the necessity of the $A_p^{b}(\mu)$ condition. **Lemma 30**. Let $N \in \mathbb{N}$ and $p \in (1,\infty)$. Suppose there exists a constant $C$ so that $$\max_{\substack{s,t \in \mathbb{N}: \\ s+t \leq N }} \sup_{T\in\mathrm{HS}(s,t)} \|T \|_{L^p(w d\mu) \rightarrow L^p(w d\mu) } \leq C.$$ Let $J,K \in \mathscr{D}$ be such that $2<\operatorname{dist}_{\mathscr{D}}(J,K)\leq N+2$ and $J \cap K= \emptyset.$ Then $$c_p^b(J,K)) \left(\langle w \rangle_{J} \langle w^{1-p'} \rangle_{K}^{p-1}\right) \lesssim C^p,$$ where the implicit constant is independent of the particular intervals $J,K$. *Proof.* By definition, $J, K$ have a unique (minimal) common dyadic ancestor interval $L$, and there exist nonnegative integers $s,t$ satisfying $s+t \leq N$ so that $\operatorname{dist}_{\mathscr{D}}(K,L)=s+1$ and $\operatorname{dist}_{\mathscr{D}}(J,L)=t+1$. Without loss of generality, we may suppose $K=L_{s -}^{m}$ and $J=L_{t -}^{n}.$ Again, for simplicity we only prove the result for $p=2.$ We consider test functions $f_1= w^{-1} 1_{L_{s-}^m}$ and $f_2= 1_{L_{t-}^n}.$ Straightforward computations yield $$\|f_1\|_{L^2(w d\mu)}^2 = w^{-1}(L_s^{m-}), \quad \|f_2\|_{L^2(w d \mu)}^2 =w(L_t^{n-}).$$ Choose the sequence $\alpha$ to be the unique sequence of signs satisfying $\alpha^I_{I_s^m, I_t^n} \langle f_1, h_{I_s^m} \rangle \langle h_{I_t^n}, w f_2 \rangle \geq 0$ for all $I \in \mathscr{D}$ (and $\alpha_{J,K}^{I}$ otherwise), and consider the operator $T\in\mathrm{HS}(s,t)$ defined by $\alpha$. Then we estimate, using the hypothesis : $$\begin{aligned} C \|f_1\|_{L^2(w d\mu)} \|f_2\|_{L^2(w d\mu)} & \geq \langle |T f_1,w f_2 \rangle| \\ & = \left| \sum_{I \in \mathscr{D}} \alpha^I_{ I_s^m, I_t^n} \langle f_1, h_{I_s^m} \rangle \langle h_{I_t^n}, w f_2 \rangle\right|\\ & = \sum_{I \in \mathscr{D}} |\langle f_1, h_{I_s^m} \rangle| |\langle h_{I_t^n}, w f_2 \rangle|\\ & \geq | \langle f_1, h_{L_s^m} \rangle| |\langle h_{L_t^n}, w f_2 \rangle|\\ & = \sqrt{m(L_s^m) m(L_t^n)} \left(\frac{w^{-1}(L_{s-}^{m})}{\mu(L_{s-}^{m})}\right) \left(\frac{w(L_{t-}^{n})}{\mu(L_{t-}^{n})}\right)\\ & \sim \sqrt{\frac{m(J) m(K)}{\mu(J) \mu(K)}} \left(\langle w \rangle_{J} \langle w^{-1} \rangle_{K}\right)^{1/2} |f_1\|_{L^2(w)} \|f_2 \|_{L^2(w)}.\end{aligned}$$ Squaring both sides and cancelling the $L^2(w d\mu)$ norms then produces the desired estimate. ◻ The above result is the remaining bit that allows us to characterize the class of weights for which Haar shifts are $L^p$-bounded. **Corollary 31**. Let $p \in (1,\infty)$ and $N\in\mathbb{N}$. There exists a constant $C$ such that all operators $$T \in \bigcup_{\substack{s,t \in \mathbb{N}: \\ s+t \leq N}} \mathrm{HS}(s,t)$$ satisfy $$\|T\|_{L^p(w d\mu) \rightarrow L^p(w d\mu) } \leq C$$ if and only if $w \in A_p^{b}(\mu)$. *Proof.* One direction immediately follows from Corollary [Corollary 28](#WeightedBounds){reference-type="ref" reference="WeightedBounds"}. For the other, we need to recall a relevant fact concerning dyadic operators of total complexity $0.$ In particular, given a sequence $\alpha=\{ \alpha_J\}_{J \in \mathscr{D}} \in \ell^\infty$, define the *Haar multiplier* $\Pi_{\mathscr{D}}^\alpha$ as follows: $$\Pi_{\mathscr{D}}^\alpha(f):= \sum_{J \in \mathscr{D}} \alpha_J \langle f, h_J \rangle h_J$$ It is immediate that $\Pi_{\mathscr{D}}^\alpha$ is bounded on $L^2(\mu).$ It is also well-known that for any Borel measure $\mu$, such an operator is bounded on $L^p(\mu)$ for $1<p<\infty$. In fact, these operators admit sparse domination in the usual sense, see [@Lacey2017]. Moreover, the collection of such operators with $\|\alpha\|_{\ell^\infty} \leq 1$ is uniformly bounded on $L^p(w d\mu)$ if and only if $w \in A_p(\mu)$, see [@TTV] for $p=2$. For $p \neq 2$, the sufficiency of the $A_p(\mu)$ condition may be obtained via the sparse domination, while the necessity may be obtained in the same way as $p=2$, see the proof of [@TTV]\*Proposition 2.3. This fact together with Lemma [Lemma 30](#PairBoundBelow){reference-type="ref" reference="PairBoundBelow"} completes the proof. ◻ We end this subsection observing that Theorem [Theorem 2](#th:thmB){reference-type="ref" reference="th:thmB"} follows from Proposition [Proposition 3](#BadWeight){reference-type="ref" reference="BadWeight"} and Corollary [Corollary 31](#Weighted Characterization){reference-type="ref" reference="Weighted Characterization"}. ## Higher-Dimensional Haar Shifts It is a natural question whether all results in this paper can be generalized to Haar shift operators defined on $\mathbb{R}^n$, $n>1$. A central point of departure from the one-dimensional case occurs in constructing the Haar system (say according to the abstract definition in [@LMP2014]), as there are more degrees of freedom. We make a couple of remarks concerning the higher dimensional case. **Remark 32**. The higher dimensional dyadic case for non-doubling measures was partially addressed in [@LMP2014], where sufficient conditions were given for the weak $(1,1)$ bound of dyadic shifts defined with respect to two generalized Haar systems. In such a general context, we do not know if there are appropriate generalizations of the modified sparse forms and weight classes so that our sparse domination or weighted estimates remain true. It would be very interesting to investigate this question, as the higher dimensional theory could be significantly more complicated. The general martingale setting, with possibly nonatomic filtrations replacing the one generated by $\mathscr{D}$, is also open. **Remark 33**. In the special case that the Haar system $\{h_Q\}_{Q \in \mathscr{D}}$ consists of functions taking only two values, we expect there to be a parallel theory based on the condition $$\sup_{Q} \|h_Q\|_{L^1(\mu)} \|h_{Q'}\|_{L^{\infty}(\mu)} \lesssim 1,$$ for $Q, Q'$ with prescribed dyadic distance, which is the right higher dimensional analog of balanced. This case had already been identified in [@LMP2014]. Although we have not checked the details, the two-value Haar functions can be essentially regarded as one-dimensional objects, which is why we are confident this generalization will work without much modification to the arguments. [^1]: In [@LMP2014], the authors term these measures $m$-equilibrated, but we prefer the shorter term *balanced*. [^2]: If we want to consider measures supported on all of $\mathbb{R}$, we can just take $\mu$ to be uniform with density $1$ outside $[0,1]$ and the argument still works.

arxiv_math

--- abstract: | In this paper, we present a new generalized Hughes model designed to intelligently depict pedestrian congestion dynamics, allowing pedestrian groups to either navigate through or circumvent high-density regions. First, we describe the microscopic settings of the model. The corresponding optimization problems are deterministic and can be formulated by a closed-loop model predictive control strategy. This microscopic setup leads in the mean-field limit to the generalized Hughes model which is a class of non-separable mean field games system, i.e., Fokker-Planck equation and viscous Hamilton-Jacobi Bellman equation are coupled in a forward-backward structure. We give an overview on the mean field games in connection to our intelligent fluid model. Therefore, we show the existence of weak solutions to the generalized Hughes model and analyze the vanishing viscosity limit of weak solutions. Finally, we illustrate the generalized Hughes model with various numerical experiments. Crowd dynamics, generalized Hughes model, non-separable mean field games, existence of weak solutions, vanishing viscosity limit. author: - "Mohamed Ghattassi [^1] Nader Masmoudi [^2]" bibliography: - HughesmodelingRefs.bib title: "Non-separable Mean Field Games for Pedestrian Flow: Generalized Hughes Model" --- # Introduction In recent years, the study of pedestrian dynamics has become a focal point for numerous researchers in various scientific disciplines. Originating from the realm of traffic engineering, the movement and behavior of pedestrian crowds have garnered increasing attention from the mathematics community. For an in-depth exploration of modeling and analytical challenges in this domain, please refer to [@bellomo2022towards; @bellomo2023human]. Modeling pedestrian dynamics can be classified crowd by three possible modeling scales (i.e. microscopic, mesoscopic, macroscopic) as follows: - **Microscopic (individual-based) Models:** force-based models such as the social force model, see [@helbing1995social; @helbing2000simulating; @aurell2018mean; @aurell2020behavior; @li2020approach], cellular automata approaches, [@burstedde2001simulation; @burger2011continuous]. - **Mesoscopic (Kinetic) Models:** The interactions between individuals are modeled by the collisions where the ideas are initially used on the gas kinetics; see [@bellomo2013microscale; @liao2023kinetic; @bellomo2023human] and references therein. - **Macroscopic (hydrodynamic) Models:** The macroscopic model uses an Eulerian-type hydrodynamic description of a system of crowd dynamics, see [@lachapelle2011mean; @bellomo2015multiscale; @borzi2020fokker; @liang2021continuum; @bellomo2023human], and some variants of the popular Hughes model[@huang2009revisiting; @burger2014mean; @carrillo2016improved; @fischer2020micro; @herzog2020optimal; @pietschmann2023numerical]. In 2002, Roger Hughes proposed a macroscopic model for crowd dynamics in which individuals seek to minimize their travel time by avoiding regions of high density. The celebrated Hughes model describes fast exit and evacuation scenarios, where a group of people wants to leave a domain $\Omega \subset\mathbb{R}^{2}$ with one or several exits/doors and/or obstacles as fast as possible. The macroscopic model for pedestrian dynamics in [@Hughes2002continuum] is based on a continuity equation (describing the evolution of crowd density) and an Eikonal equation (giving the shortest weighted distance to an exit). The model is given by $$\begin{aligned} \partial_{t}\rho -div(\rho f^{2}(\rho)\nabla \phi )&=0 \label{FluidEq}\\ |\nabla\phi| &=\frac{1}{f(\rho)}\label{EikEq}\\ \rho(t=0,x)&= \rho_{in}(x),\label{FluidIn}\end{aligned}$$ where $x\in \Omega$ denotes the position in space, $t\in (0,T]$, $T>0$, the time variable and $\nabla$ the gradient with respect to the space variable $x$. The function $\rho$ corresponds to the pedestrian density, $\rho_{in}$ is the initial pedestrian density and $\phi$ the weighted shortest distance to the exit. $f$ is a function introducing saturation effects such as $f(\rho)=\rho_{m}-\rho$, where $\rho_{m}$ corresponds to the maximum scaled pedestrian density. Hughes model [\[FluidEq\]](#FluidEq){reference-type="eqref" reference="FluidEq"}-[\[FluidIn\]](#FluidIn){reference-type="eqref" reference="FluidIn"} is supplemented with different boundary conditions for the walls and the exits. We assume that the boundary $\partial \Omega$ of our domain is subdivided into three parts: inflow $\Gamma$, outflow $\Sigma$ and insulation $\Gamma_{a}=\partial \Omega / (\Gamma \cup\Sigma)$ with $\Gamma \cap \Sigma =\emptyset$. We assume that particles enter the domain $\Omega$ on $\Gamma$ with boundary pedestrian density $\rho_{b}$. On the remaining part of the boundary $\Gamma_{a}$, we impose no flux conditions $$\begin{aligned} \rho &= \rho_{b} \qquad (t,x)\in [0,T]\times\Gamma\label{BGam}\\ \nabla\rho\cdot {\bf n(x)}&=0 \qquad (t,x)\in [0,T]\times\Gamma_{a}\label{BGamx}\end{aligned}$$ where **n(x)** denotes the outer normal vector to the boundary. For the Eikonal equation [\[EikEq\]](#EikEq){reference-type="eqref" reference="EikEq"}, we consider the non-homogenous Dirichlet boundary condition $$\phi = \phi_{b} \qquad (t,x)\in [0,T]\times\partial\Omega \label{EikonalBC}$$ where the function $\phi_{b}$ takes zeros on the exit $\Sigma$. The system [\[FluidEq\]](#FluidEq){reference-type="eqref" reference="FluidEq"}-[\[FluidIn\]](#FluidIn){reference-type="eqref" reference="FluidIn"} is a highly nonlinear coupled system of partial differential equations (PDE). Few analytic results are available, all of them restricted to spatial dimension one. The main difficulty comes from the low regularity of the potential $\phi$, which is only Lipschitz continuous. For existence and uniqueness results of a regularized problem in 1D, we refer the reader to [@burger2014mean; @amadori2023mathematical] and references therein. In this paper, we present a generalization of the Hughes model, which tries to avoid the high-density with local vision via partial knowledge of the pedestrian density. First, we discuss the proper modeling setup, the microscopic description, and the derivation of the macroscopic generalized Hughes (GH) model. The generalization of the model consists of taking on consideration human behavioral by avoiding or traveling to the high-density regions. Quantification of properties related to pedestrian crowds plays a pivotal role in comprehending pedestrian flows and implementing real-time control mechanisms. Often, crowd models rooted in physics and fluid dynamics represent individuals as particles, yielding satisfactory results under specific conditions. However, these models exhibit limitations in their general applicability, underscoring the importance of gaining a deeper understanding of the cognitive and psychological facets of human motion within crowds. In our present study, we propose the introduction of a new parameter, denoted as $\beta$, into the Hughes model, serving as a safety crowd parameter. The parameter $\beta$ is versatile and can be tailored to various characteristics of pedestrians, such as psychological profiles, walking speeds, perception abilities, traversal abilities, physical sizes, etc. We establish the relationship between the GH model and the mean-field games (MFGs for short) system, providing an overview of recent developments in the field of mean field games. Additionally, we give proof of the existence of weak solutions for the GH model and, under appropriate monotonicity conditions, demonstrate uniqueness results. Subsequently, we study the vanishing viscosity limit of weak solutions. Finally, we present a set of numerical results aimed at dissecting the impact of the parameter $\beta$ within our intelligent fluid model. For these numerical experiments, we employ the finite element method (FEM), implemented through the FreeFem++ library. It is worth noting that this choice significantly simplifies the computational complexity for the authors; however, it necessitates the inclusion of some viscosity within the model to ensure numerical convergence. Additionally, alternative numerical approaches such as the Finite Volume Scheme (FVM), which has been successfully applied to the viscous-hughes model in prior work [@twarogowska2014macroscopic], can offer valuable insights and comparisons. In fact, in a related study [@twarogowska2014macroscopic], the authors conducted a comparative analysis of solution behaviors between the Hughes model and a second-order model based on a mixed finite volume method. Furthermore, we explore the application of the WENO scheme for the density equation and the sweeping WENO method for the HJB-type equation, as demonstrated in previous research by Huang et al. [@huang2009revisiting]. However, we have the finite difference method for MFGs introduced in [@achdou2010mean] for stationary and time-dependent MFGs. The Lax-Friedrichs or Godunov-type schemes are introduced to approximate Hamiltonians. The Kolmogorov-type equation is discretized in such a way that it preserves the adjoint structure in the MFG. We also refer the readers to [@achdou2016convergence; @bakaryan2023discrete] and the references therein for numerical methods and convergence results of different discrete schemes for second-order mean field games. This paper is structured as follows. We start in Section [2](#sec0){reference-type="ref" reference="sec0"} with a microscopic description of the GH model. Then, this microscopic setup leads in the mean-field limit to the GH model which is a class of MFGs systems. An overview of the results for the existence and uniqueness of the MFG system is presented in Section [2.3](#sec3){reference-type="ref" reference="sec3"}. In Section [3](#sec4){reference-type="ref" reference="sec4"}, we give the existence and uniqueness of weak solution for our fluid model. We then analyze the vanishing viscosity limit of weak solutions. Finally, in Section [4](#sec6){reference-type="ref" reference="sec6"} we show some numerical experiments based on the FEM discussing the effect of the parameter $\beta$ and viscosity in the GH model. # Generalized Hughes Model {#sec0} ## Microscopic description: Mean-field limit {#sec1} The objective of this section is to give a derivation of our GH model from the microscopic level and its mean-field limit. We consider a system of controlled individuals (also referred to as particles or players) in mutual interaction, each pedestrian (or player) controls his own private state $x(t)$ at time $t \in [0,T]$, when $T>0$, driven by the vector-valued drift function $b$ value in some subset $\mathcal{A}$ of an euclidian space. We assume that $x(t)$ is a Markov diffusion process governed by the following stochastic differential equation (SDE): $$\label{EQStoc} dx(t)=b(t,x(t),u(t,x(t))) dt + \sqrt{2\sigma} dW(t)$$ where $dW(t) \in \mathbb{R}^{2}$ is a $2d$ brownian motion process, with stochastically independent components. $x(t)$ denotes the position of a pedestrian in a domain $\Omega \subset \mathbb{R}^{2}$, and $b = (b_{1}, b_{2})$ denotes the velocity field of motion of the pedestrian. For simplicity, we assume that the dispersion coefficient $\sigma$ is a constant but it can depend on the control strategy. In the control theory framework, the function $u$ uses the current value $x(t)$ to affect the dynamics of the stochastic process by adjusting the drift function. Similarly to [\[EQStoc\]](#EQStoc){reference-type="eqref" reference="EQStoc"}, we consider the following *action functional* $$\label{EQFUNctional} J\left(t,x,u \right)=\mathbb{E} \Big [ \int_{t}^{T} \mathbf{L}\left(x(s),u(s,x(s))\right) ds+g(x(T))\,\,\, | \,\,\, x(t)=x \Big ],$$ which is an expectation to the process $x(t)$ taking the value $x$ at time $t$. We refer to the functions $\mathbf{L}$ and $g$ as the running cost and the terminal cost functions, respectively. Now, the optimal control $\bar{u}$ that minimizes $J\left(t,x,u \right)$ for the process [\[EQStoc\]](#EQStoc){reference-type="eqref" reference="EQStoc"} is given by $$\label{OptimalcontrolU} \bar{u}=\text{argmin}_{u\in \mathcal{A}} J\left(t,x,u \right).$$ Furthermore, we define the value function $\phi$ of the stochastic dynamics control problem as $$\label{Valuefun} \phi(t,x)=\min_{u\in \mathcal{A}} J\left(t,x,u \right)=J\left(t,x,\bar{u} \right).$$ The value function is a critical element in this framework. For each player, it is the cost functional of the agent determined in the Nash equilibrium. Isaacs in [@isaacsdifferential] was the first to link the value function in these differential games to the Hamilton-Jacobi equations. Crandall, Ishii, and Lions then further developed this concept in their work on viscosity solutions; see [@ishii1990viscosity; @crandall1992user] for a comprehensive overview. We now state the theorem showing that the value function $\phi$ is the solution to the Hamilton-Jacobi-Bellman (HJB) equation. **Theorem 1**. *Assume that $x(t)$, for $t\in[0,T]$ solving [\[EQStoc\]](#EQStoc){reference-type="eqref" reference="EQStoc"} with a control function $u$ and that the function $\phi$ defined by [\[Valuefun\]](#Valuefun){reference-type="eqref" reference="Valuefun"} is bounded and smooth. Then, the value function $\phi$ verifies the following HJB equation. $$\label{Hamiltoneq} \begin{cases} -\partial_{t}\phi-H(t,x,\nabla_{x}\phi,\nabla^{2}_{x}\phi)=0,\qquad (t,x)\in [0,T)\times \Omega\\ \phi(t,x)=g(x)\qquad\qquad\qquad \qquad\qquad x\in \partial\Omega \end{cases}$$ where the Hamiltonian function is defined by $$\label{EQHAM} H(t,x, \nabla_{x}\phi,\nabla^{2}_{x}\phi) := \min_{v \in \mathcal{A}} \left( \sum_{i=1}^{2}b_{i}(t,x,u(t,x))\partial_{x_{i}}\phi+\sigma\Delta \phi(t,x) +\mathbf{L}(x,u(t,x)) \right) .$$* For more details we refer the reader to [@zbMATH05032360 Theorem 3.1 page 157] By assuming differentiability with respect to the control function $u$ in [\[EQHAM\]](#EQHAM){reference-type="eqref" reference="EQHAM"}, then the optimal control $\bar{u}$ should satisfy at each time $t$ and for each point $x$ the optimality condition as follows: $$\label{OptimalityHAM} \sum_{i=1}^{2}\partial_{u} b_{i}(t,x,u(t,x))\partial_{x_{i}}\phi +\partial_{u}\mathbf{L}(x,u(t,x)) =0.$$ Then by using the assumption that this process has an absolutely continuous probability measure, we can formulate the expectation in [\[EQFUNctional\]](#EQFUNctional){reference-type="eqref" reference="EQFUNctional"} in terms of probability density functions (PDF) governed by the Kolmogorov or Fokker-Planck (FP) equation problem with initial density distribution $\rho_{0}=\delta_{x(t)-x}$ at time $t=0$. Thus, the functional [\[EQFUNctional\]](#EQFUNctional){reference-type="eqref" reference="EQFUNctional"} becomes $$\label{EQFUNctionalcont} J\left(u,\rho \right)= \int_{t}^{T}\int_{\Omega}\mathbf{L}\left(x,u(t,x)\right) dx ds + \int_{\Omega}g(x)\rho(T,x) dx$$ Therefore, we can state the optimization problem [\[OptimalcontrolU\]](#OptimalcontrolU){reference-type="eqref" reference="OptimalcontrolU"} as an FP optimal control problem where an optimal control $u$ in the admissible set $\mathcal{A}$ sought that minimizes the functional [\[EQFUNctionalcont\]](#EQFUNctionalcont){reference-type="eqref" reference="EQFUNctionalcont"}. We are identifying the chosen admissible set of Markov control policies to the admissible set of controls in the FP optimal control formulation. To characterize the optimal FP solution to this problem, we introduce the following Lagrange function. $$\begin{aligned} \mathrm{L}(\rho, \phi,u) :=&\int_{0}^{T}\int_{\Omega}\mathbf{L}\left(x,u(t,x)\right) dx ds + \int_{\Omega}g(x)\rho(T,x) dx+\int_{0}^{T}\int_{\Omega}\phi(s,x)\Big[-\partial_{s}\rho(s,x) \\& - \sum_{i=1}^{2}\partial_{x_{i}}\left(b_{i}(t,x,u(t,x))\rho(t,x)\right)+\sigma\Delta\rho(t,x) \Big] dx ds \end{aligned}$$ Consequently, the optimal control solution is characterized as the solution to the following optimality system $$\label{XZX0} \partial_{\phi}\mathrm{L}(\rho, \phi,u)=0 \iff \quad \begin{cases} -\partial_{t}\rho-\sum_{i=1}^{2}\partial_{x_{i}}\left(b_{i}(x,u(t,x))\right)\rho(t,x)+\sigma\Delta\rho(t,x)=0,\\ \rho(t=0,x)=\rho_{0}(x), \end{cases}$$ $$\label{XZX1} \partial_{\rho}\mathrm{L}(\rho, \phi,u)=0 \iff \quad \begin{cases} \partial_{t}\phi +\sum_{i=1}^{2}b_{i}(x,u(t,x))\partial_{x_{i}}\phi+\sigma\Delta\phi(t,x) +\mathbf{L}(x,u(t,x))=0,\\ \phi(T,x)=g(x), \end{cases}$$ and the optimality condition $$\label{OptimaltyCond} \partial_{u}\mathrm{L}(\rho, \phi,u)=0 \iff \quad \rho(t,x)\left( \sum_{i=1}^{2} \partial_{u}b_{i}(x,u(t,x)) \partial_{x_{i}}\phi +\partial_{u}\mathbf{L}(x,u(t,x)) \right) =0.$$ A sufficient condition to ensure that [\[OptimaltyCond\]](#OptimaltyCond){reference-type="eqref" reference="OptimaltyCond"} holds is to satisfy the optimality condition [\[OptimalityHAM\]](#OptimalityHAM){reference-type="eqref" reference="OptimalityHAM"} within the context of the Hamilton-Jacobi-Bellman (HJB) formulation. ## Macroscopic description of the GH model {#sec21} This section is dedicated to the formal derivation of both the generalized and classical Hughes models within a macroscopic context, deriving them from the system defined by equations [\[XZX0\]](#XZX0){reference-type="eqref" reference="XZX0"}-[\[OptimaltyCond\]](#OptimaltyCond){reference-type="eqref" reference="OptimaltyCond"}. To begin, we will present the derivation of the Hughes model. We first take the following. $$b=u,\quad \quad \mathbf{L}(x,u)=\frac{u^{2}}{2f^{2}(\rho)} +\frac{1}{2},$$ where $f(\rho)=(\rho_{max}-\rho)_{+}$. Consequently, from the optimality condition [\[OptimaltyCond\]](#OptimaltyCond){reference-type="eqref" reference="OptimaltyCond"}, we get a feedback in this form $u=-f^{2}(\rho)\nabla \phi$. Then, system [\[XZX0\]](#XZX0){reference-type="eqref" reference="XZX0"} -[\[OptimaltyCond\]](#OptimaltyCond){reference-type="eqref" reference="OptimaltyCond"} becomes $$\label{XZXx1} \begin{cases} \partial_{t}\rho -div(\rho f^{2}(\rho)\nabla \phi )-\sigma\Delta \rho=0\\ -\partial_{t}\phi +f^{2}(\rho)|\nabla\phi|^{2}-\sigma\Delta\phi-\frac{1}{2}f^{2}(\rho)|\nabla\phi|^{2} -\frac{1}{2}=0. \end{cases}$$ Our core assumption is that the pedestrian lacks the ability to foresee the future evolution of the population. Instead, they base their strategic decisions solely on real-time information, without the ability to predict forthcoming events. This strategic approach serves as a foundational principle supporting a particular class of quasi-stationary Mean Field Games (MFG) systems, as discussed in [@mouzouni2020quasi] and [@camilli2023quasi]. In this context, individuals operate within a static environment. At each moment, they determine their course of action solely based on current information, devoid of any foresight into the future. Consequently, this leads to the stationary HJB equation coupled with an evolutive FP equation. Moreover, if we assume a complete absence of viscosity ($\sigma = 0$) within the model, we can deduce the classical Hughes model [\[FluidEq\]](#FluidEq){reference-type="eqref" reference="FluidEq"}-[\[EikonalBC\]](#EikonalBC){reference-type="eqref" reference="EikonalBC"}. In this macroscopic framework, individuals strive to minimize their travel time while conscientiously avoiding high-density regions. However, we are now introducing a pioneering model that explicitly incorporates the element of congestion. To achieve this, we introduce a novel parameter, denoted as $\beta$, into the running cost term denoted by $\mathbf{L}$ (here, we adopt the notation $\mathcal{L}$ to reflect its dependence on the spatial variable $x$, density $\rho$, and the potential field $v$). This leads to the redefinition of the functional $\mathbf{L}$ as follows: $$\label{runningFunc} \mathbf{L}(x,u)=\mathcal{L}(x,\rho,v)=\frac{1}{2} f^{-\beta}(\rho)\vert v\vert^{2} +\frac{1}{2} f^{2-\beta}(\rho).$$ Therefore from the optimality condition [\[OptimaltyCond\]](#OptimaltyCond){reference-type="eqref" reference="OptimaltyCond"}, the optimal control situation is given by $u=-f^{\beta}(\rho)\nabla \phi$. Then [\[XZX0\]](#XZX0){reference-type="eqref" reference="XZX0"} -[\[XZX1\]](#XZX1){reference-type="eqref" reference="XZX1"} becomes $$\label{sSa} \begin{cases} \partial_{t}\rho- div(\rho f^{\beta}(\rho)\nabla \phi )-\sigma\Delta \rho=0\\ -\partial_{t}\phi +\frac{1}{2} f^{\beta}(\rho)|\nabla\phi|^{2}-\sigma\Delta\phi -\frac{1}{2} f^{2-\beta}(\rho)=0. \end{cases}$$ Using the same argument that the generic pedestrian cannot predict the evolution of the population, but chooses its strategy only on the basis of the information available at the given instant of time, without anticipating, we can consider the quasi-stationary HJB equation as follows $$\label{QuasiStat} \begin{cases} \partial_{t}\rho- div(\rho f^{\beta}(\rho)\nabla \phi )-\sigma\Delta \rho=0\\ \frac{1}{2} f^{\beta}(\rho)|\nabla\phi|^{2}-\sigma\Delta\phi -\frac{1}{2} f^{2-\beta}(\rho)=0. \end{cases}$$ Furthermore, assuming the absence of viscosity ($\sigma = 0$) in the model, we derive the following system $$\label{XZX1G} \begin{cases} \partial_{t}\rho -div(\rho f^{\beta}(\rho)\nabla \phi )=0\\ f^{2\beta -2}(\rho)|\nabla\phi|^{2} =1. \end{cases}$$ When $\beta = 2$, the Hughes model is obtained, in which people try to stay away from areas of high density. However, when $\beta =0$, the HJB equation suggests a connection between velocity and density, with $|\nabla\phi| =f(\rho)$. This leads to a situation in which individuals are drawn to areas of high density, a phenomenon known as concentration. This is in stark contrast to the Hughes model, where people usually try to avoid high-density areas. An example of this behavior can be seen during the Tawaf ritual at the Kaaba, where Muslim pilgrims circle the Kaaba and many attempt to pause and kiss the Al Hajaru al Aswad. When $\beta = 1$, the speed of pedestrians remains constant regardless of the density of the area they are in. This parameter is heavily influenced by the psychological state of the pedestrians, which is determined by the event or location. The value of $\beta$ reflects the adaptive walking strategies of pedestrians, depending on their psychological state in response to different environmental conditions or events. As an example, consider the Tawaf ritual at the Kaaba, which is performed during Hajj or Umrah. When a worshiper enters the mosque precinct. This leads to a defined area known as the *Sahan*. Worshipers in this area can be divided into two groups: those who want to get close to the Kaaba and kiss the Black Stone (*Al Hajar Aswad*), and those who want to avoid densely populated areas, often due to age or health issues. For the first group, their main goal is to reach the *Al Hajar Aswad*, so they are not concerned with the crowd density. In this case, when $\beta = 1$, it implies that $\nabla\phi = 1$. On the other hand, the second group of worshipers is more focused on navigating through the crowd density, as they want to avoid the hustle and bustle typically associated with densely populated areas around the Kaaba. This group includes people of advanced age, those with health concerns, expectant mothers, and others. In this case, $\beta$ remains consistently greater than 1, specifically within the range $\beta \in (1, 2]$. For $\beta \in [0, 1)$, there are at least three different scenarios to consider. To begin with, there is the case of *curiosity-driven crowds*, such as those attending superstar concerts or political gatherings, who are drawn to high-density locations out of interest. Second, there is the scenario of *panic*, where people flock to high-density regions because they believe that safety or something valuable can be found there. Lastly, there is the case of *fear*, where individuals, when feeling scared, tend to seek proximity to high-density areas as a response. These three scenarios represent different motivations for individuals in situations with $\beta \in [0, 1)$. We now assume that the curve is described parametrically in two-dimensional (2D) by the function ${\bf x}(s) = \left(x_{1}(s),x_{2}(s)\right)$. We define ${\bf z}(s)=\phi({\bf x}(s) )$, and $${\bf p}(s)=\nabla_{{\bf x}}\phi({\bf x}(s) )= \left(p_{1}(s),p_{2}(s)\right), \quad \text{such that} \quad p_{i}(s)=\phi_{x_{i}}({\bf x}(s) ), \qquad i\in\{1,2\},$$ where ${\bf z}(.)$ gives the values of $\phi$ along the curve ${\bf x}(.)$, ${\bf p}(.)$ records the values of the gradient $\nabla_{{\bf x}}\phi$ and we define ${\bf q}(s)=\rho({\bf x}(s) )$ . We set $$\dot{ {\bf p}}(s)=D_{v}\mathcal{L}({\bf x}(s),{\bf q}(s),{\bf p}(s)),\qquad (0\leqslant s\leqslant t)$$ where ${\bf p}(.)$ is called the generalized momentum corresponding to the position ${\bf x}(.)$ and the velocity $\dot{ {\bf x}}$. Now we make the same assumption as [@evans2010partial (9), p.118]: $$\label{HamLag} \begin{cases} \text{Suppose for all }\,\,\,\,\, x,p \in\mathbb{R}^{2}\,\,\, \text{ that the equation} \\ p=D_{v}\mathcal{L}(x,q,p),\\ \text{can be uniquely solved for}\,\, v \,\, \text{as a smooth function of}\,\, p \,\, \text{and}\,\, x, v= {\bf v}(x,p). \end{cases}$$ The Hamiltonian $\mathcal{H}$ associated with the Lagrangian operator $v\longmapsto \mathcal{L}(x,\rho,v)$, is given by $$\label{EqHamil0} \mathcal{H}(x,\rho,p)= p\cdot {\bf v} - \mathcal{L}(x,\rho,{\bf v}),$$ where the function ${\bf v}(.)$ is defined implicitly by [\[HamLag\]](#HamLag){reference-type="eqref" reference="HamLag"}. Formally, we assume that $\mathcal{H}$ only depends on $p$, not on $v$. $$\label{EqHamil1} D_{v}\mathcal{H}(x,\rho,p) = 0 = p- D_{v}\mathcal{L}(x,\rho,{\bf v})$$ Then, we have ${ p =f^{-\beta}(\rho){\bf v}}$. Replacing the expression of ${\bf v}$ in [\[EqHamil0\]](#EqHamil0){reference-type="eqref" reference="EqHamil0"}, we get $$\label{EqHamil} \begin{aligned} \mathcal{H}(x,\rho,p)=& p\cdot {\bf v} -\left( \frac{1}{2} f^{-\beta}(\rho)\vert {\bf v}\vert^{2} +\frac{1}{2} f^{2-\beta}(\rho)\right)\\& =f^{\beta}(\rho)|p|^{2} -\left( \frac{1}{2} f^{-\beta}(\rho) f^{2\beta}(\rho)\vert p\vert^{2} +\frac{1}{2} f^{2-\beta}(\rho)\right)\\& =\frac{1}{2} f^{\beta}(\rho)|p|^{2} -\frac{1}{2} f^{2-\beta}(\rho), \end{aligned}$$ we denote by $\mathcal{H}$ (or $\mathcal{L}^{\ast}$) the Legendre transform defined by $$\label{EqHamilX} \mathcal{H}(x,\rho,p)=\mathcal{L}^{\ast}(x,\rho,p)=\sup_{q\in \mathbb{R}^{2}}[q\cdot p -\mathcal{L}(x,\rho,q)],$$ and $$\label{EqHamilY} \mathcal{L}(x,\rho,{\bf v})=\mathcal{H}^{\ast}(x,\rho,{\bf v})=\mathcal{H}_{p}(x,\rho,p)\cdot p-\mathcal{H}(x,\rho,p),\quad \text{where}\,\,\,\,{\bf v}=\mathcal{H}_{p}(x,\rho,p),$$ for more details we refer to [@evans2010partial Chapter 3]. Now, we can write system [\[sSa\]](#sSa){reference-type="eqref" reference="sSa"} in the structure of second order MFGs system as follows $$\begin{aligned} \partial_t\rho - \nabla \cdot \left( \rho \mathcal{H}_{p}(x,\rho,\nabla \phi)\right) -\sigma \Delta \rho= 0, &\qquad t>0, \quad x\in\Omega \label{EqFluidHd0}\\ -\partial_t\phi + \mathcal{H}(x,\rho,\nabla \phi) - \sigma \Delta \phi=0, &\qquad t>0, \quad x\in\Omega \label{EqEikonalHd0}\\ \rho(t=0,x)= \rho_{0}(x),&\quad x\in\Omega \label{FluidIn2}\\ \phi(t=T,x)= \phi_{T}(x),&\quad x\in\Omega, \label{EikonalIn2}\end{aligned}$$ where $\mathcal{H}$ is a convex function of $p$ and $\mathcal{H}_{p}$ represents $\frac{\partial\mathcal{H}}{\partial p}( x, \rho, p)$. The following boundary conditions for the density equation are given by $$\begin{aligned} \rho = 0,&\qquad t>0, \quad x\in\Sigma \label{Boundrho1}\\ \left(\sigma\nabla\rho +\rho\mathcal{H}_{p}(x,\rho,\nabla \phi)\right)\cdot n(x)= 0,&\qquad t>0, \quad x\in\Gamma_{a} \label{Boundrho2}\\ \rho=\rho_{b}, &\qquad t>0, \quad x\in\Gamma, \label{Boundar3}\end{aligned}$$ and for the HJB equation we consider the following boundary conditions $$\begin{aligned} \phi = 0,&\qquad t>0, \quad x\in\Sigma \label{Boundphi1}\\ \nabla \phi\cdot n(x)= 0,&\qquad t>0, \quad x\in\Gamma_{a} \label{Boundphi2}\\ \phi=\phi_{b}, &\qquad t>0, \quad x\in\Gamma. \label{Boundar3}\end{aligned}$$ ## Review of the literature in connection to the Mean-Field Games {#sec3} Previously, we have already shown that the GH model has the structure of a non-separable MFGs. In this section, we present an overview of the recent advanced results on the MFGs system and its derivation. MFGs have been introduced in the mathematics literature by Lasry and Lions as limits of problems from game theory, as the number of agents tends to infinity [@lasry2006jeux; @lasry2006jeux1; @lasry2007mean]. From a control theory perspective, mean field games were also introduced around the same time by Huang, Caines, and Malhame [@huang2007large; @caines2006large]. This theory is motivated by problems in economics and engineering and with the goal of approximating Nash equilibria of games with a large number of symmetric agents. Since their introduction, MFGs have been extensively studied in the literature and several research topics have been addressed, from both theoretical and applied perspectives. The main goal is typically to study the equilibria of such games, which are usually characterized as solutions of a system of PDEs. There are two classes of MFGs that are characterized by the separable and non-separable Hamiltonian $\mathcal{H}$. This means that it is assumed that there exist $H$ and $F$ such that $\mathcal{H}(t, x,\rho,p) = H(t,x,p)+ F(t,x,\rho)$. In such a case, the function $H$ is still known as the Hamiltonian, but $F$ is then referred to the coupling term. However, in applications, non-separable Hamiltonians are frequently of interest for example in economics, which does not tend to have this separable structure [@moll2016mean; @achdou2014partial; @achdou2022income] which is also the case of our intelligent fluid model [\[EqFluidHd0\]](#EqFluidHd0){reference-type="eqref" reference="EqFluidHd0"}-[\[EikonalIn2\]](#EikonalIn2){reference-type="eqref" reference="EikonalIn2"} and [\[Boundrho1\]](#Boundrho1){reference-type="eqref" reference="Boundrho1"}-[\[Boundar3\]](#Boundar3){reference-type="eqref" reference="Boundar3"}. In the case of the separable Hamiltonian, there are a number of works that prove the existence of solutions. The separability assumption, as well as other structural assumptions, such as the convexity of $H$ and the monotonicity of $F$, are helpful in the analysis of such models. For example the case of the second-order quadratic MFGs with aggregation force $$\begin{aligned} \partial_tm- \nabla \cdot \left( m\nabla u\right) - \Delta m= 0, &\qquad t>0, \quad x\in\mathbb{T}^{d} \label{EqFluidHd0C}\\ -\partial_t u + \frac{1}{2}|\nabla u|^{2}- \Delta u =-f(m) +V(x), &\qquad t>0, \quad x\in\mathbb{T}^{d} \label{EqEikonalHd0C}\\ m(t=0,x)= m_{0}(x),&\quad x\in\mathbb{T}^{d} \label{FluidInC}\\ u(t=T,x)= u_{T}(x),&\quad x\in\mathbb{T}^{d} \label{EikonalInC}\end{aligned}$$ with $\mathbb{T}^{d}$ is standard flat torus (the dimension space $d$ ), $m$ is the population density, $u$ the potential, $m_{0}$ is a smooth probability density, $u_{T}$ a smooth final cost, $V$ is a bounded potential, and $f$ is chosen $$f(m)=\pm\gamma m^{\alpha},$$ where $\gamma$ and $\alpha$ are related to the aggregation force, for more details see [@cirant2022existence] and references therein. Moreover, in [@porretta2014planning; @porretta2015weak] Porretta proved the existence of weak solutions of the second-order MFGs system. Then, recently in [@griffin2022variational] Griffin-Pickering and Mészáros studied weak solutions to a first-order mean-field games system involving kinetic transport operators. Gomes and Pimentel, in [@gomes2015time], showed the existence of a strong solution with logarithmic coupling. Gomes, Pimentel, and Sanchez-Morgado proved in [@gomes2015time0; @gomes2016time] the existence of a strong solution for the case of super-quadratic and subquadratic Hamiltonians. However, for more references on the existence of solutions for MFGs, we refer to [@ferreira2018existence; @ferreira2019existence; @cirant2021maximal] and the references therein. To our knowledge, the first work on the existence of strong solution to the non-separable Hamiltonian is proved by Ambrose in [@ambrose2018strong] . The author presented an existence for strong solutions of the non-separable MFG system. First, a smallness condition on the data is used, and a small parameter in front of the Hamiltonian is used to compensate for the nonlinear term. We also found the work of Cirant, Gianni, and Mannucci [@cirant2020short] on MFGs with non-separable Hamiltonians. The authors proved an existence theorem for non-separable mean field games in Sobolev spaces, under a smallness condition on the time horizon. In [@Lions2023], P.-L.Lions gives the general structural conditions yielding the uniqueness for the nonseparable MFG systems with local coupling $$\begin{aligned} \partial_t m - \sigma\Delta m -div(m\mathcal{H}_{p}(x,m, \nabla u ))= 0, &\qquad t>0, \quad x\in\Omega \label{EqFluidHd0Cun}\\ -\partial_t u - \sigma\Delta u +\mathcal{H}(x,m, \nabla u ) = F(t,x,m), &\qquad t>0, \quad x\in\Omega \label{EqEikonalHd0Cun} %m(t=0,x)= m_{0}(x),&\quad x\in\Omega \label{FluidInCun}\\ %u(t=T,x)= u_{T}(x),&\quad x\in\Omega \label{EikonalInCun}\end{aligned}$$ namely, that $F$ and $G$ will increase w.r.t. $m$ and that the Lasry--Lions monotonicity condition $$\label{uniq} \mathbf{H}=\begin{pmatrix} -\frac{2}{m}\mathcal{H}_{m}(t, x,m, p)& \mathcal{H}_{m,p}(t, x,m, p )\\ \mathcal{H}_{m,p}(t, x,m, p)& 2\mathcal{H}_{p,p}(t, x,m, p )\\ \end{pmatrix} \geqslant 0,$$ is satisfied, where $\mathcal{H}_{m}$ stands for $\frac{\partial\mathcal{H}}{\partial m}(x, m, p)$ , where $\mathcal{H}_{m,p}$ stands for $\frac{\partial\mathcal{H}}{\partial m \partial p}( x, \rho, p)$, for all $x\in \Omega$, $m>0$ and $p \in \mathbb{R}^{2}$. Recently, the system extended MFGs (EMFGs) was introduced by Lions and Souganidis in [@lions2020extended] who coined the term extended MFGs, to simultaneously study several MFGs type problems for which, in contrast to the case of standard MFG, the vector field $\mathcal{B}$ does not necessarily equal $\rho\mathcal{H}_{p}(x,m ,\nabla \phi )$ $$\begin{aligned} \partial_t m - \nabla \cdot \left(\mathcal{B}(x,m ,\nabla u )\right) = 0, &\qquad t>0, \quad x\in\Omega \label{EqFluidHd0xx}\\ -\partial_tu + \mathcal{H}(x,m ,\nabla u )=0, &\qquad t>0, \quad x\in\Omega \label{EqEikonalHd0xx} %\rho (t=0,x)= \rho_{0}(x),&\quad x\in\Omega \label{FluidInxx}\\ %u (t=T,x)= u_{T}(x),&\quad x\in\Omega \label{EikonalInxx}\end{aligned}$$ where it was shown that (EMFGs) has at most one classical solution under some sufficient condition. It was demonstrated in [@munoz2022classical; @munoz2023classical] that classical solutions to EMFGs in any dimension exist when the initial density has a lower bound and the following blow-up assumption is satisfied $$\label{Eqcor} \lim_{m \to 0^{+}}\mathcal{H}(x,m,\nabla u) = +\infty.$$ Recently, [@porretta2023regularizing] proved a similar regularity result for the separable Hamiltonian $\mathcal{H}(x,m,p) = H(x,p)-f(m)$ with a lower bound on the terminal density $m_{T}$. We should also mention that Lasry and Lions [@Lions2023] introduced the *master equation* in order to reformulate the original MFG system in order to take into consideration a much more complex stochastic situation; the authors introduce the *master equation* in order to encompass complex situations. However, the master equation can be understood as a nonlinear transport equation in the space of probability measures. The global well-posedness of master equations requires the uniqueness of mean field equilibrium, typically under certain monotonicity conditions, we refer to [@cardaliaguet2019master] for more details. # A weak solution for generalized Hughes Model {#sec4} This section is devoted to the proof of the existence and uniqueness of weak solutions for system [\[EqFluidHd0\]](#EqFluidHd0){reference-type="eqref" reference="EqFluidHd0"}-[\[EikonalIn2\]](#EikonalIn2){reference-type="eqref" reference="EikonalIn2"}. For simplicity we assume that the equation takes place in a standard flat torus $\Omega= \mathbb{T}^{2}$. The function $\mathcal{H}(x, \rho, p)$ is assumed to be measurable with respect to $(t, x)$, continuous with respect to $\rho$ and $C^{1}$with respect to p. We recall our fluid model [\[EqFluidHd0\]](#EqFluidHd0){reference-type="eqref" reference="EqFluidHd0"}-[\[EikonalIn2\]](#EikonalIn2){reference-type="eqref" reference="EikonalIn2"} in the flat torus $\Omega$ $$\begin{aligned} \partial_t\rho^{\sigma} - \nabla \cdot \left( \rho^{\sigma} \mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\right) -\sigma \Delta \rho^{\sigma}= 0, &\qquad (t,x)\in\mathrm{Q}_{T}= (0,T)\times\Omega \label{EqFluidHd0x}\\ -\partial_t\phi^{\sigma} + \mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma}) - \sigma \Delta \phi^{\sigma}=0, &\qquad (t,x)\in\mathrm{Q}_{T} \label{EqEikonalHd0x}\\ \rho^{\sigma}(t=0,x)= \rho_{0}(x),&\quad x\in\Omega \label{FluidInx}\\ \phi^{\sigma}(t=T,x)= \phi_{T}(x),&\quad x\in\Omega.\label{EikonalInx}\end{aligned}$$ Here, we denote the initial data as $\rho_{0}$ and $\phi_{T}$, while the Hamiltonian $\mathcal{H}$ is represented by [\[EqHamil\]](#EqHamil){reference-type="eqref" reference="EqHamil"}. To illustrate the dependence on the parameter $\sigma$, we utilize the index ${\sf ^{\sigma}}$ in the context of $\left(\rho^{\sigma},\phi^{\sigma}\right)$. Now we start by giving some assumptions on the initial data. $$\text{{\bf (AS)}} \begin{cases} &\text{Let}\,\,\,\, \rho_{0}\in C(\Omega), \,\,\rho_{0} \in (0,\rho_{m}).\\ % \rho_{0}\in L^{2}_{\infty}=L^{2}(\Omega)\cap L^{\infty}(\Omega), \,\,\rho_{0}>0.\\ &\text{Let}\,\,\,\,\phi_{T} \in C(\Omega),\,\,\text{and there exists}\,\,\, c_{0} \in \mathbb{R}\,\,\,\text{such that} \,\, \phi_{T}\geqslant c_{0}. \end{cases}$$ We will start by defining a weak solution for the system of equations [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}. **Definition 2**. *A pair $(\rho^{\sigma},\phi^{\sigma}) \in L^{1}(\mathrm{Q}_{T}) \times L^{1}(\mathrm{Q}_{T})$ is a weak solution to [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"} if* (i) : *$\rho^{\sigma} \in C(0,T;L^{1}(\Omega))$, $\rho^{\sigma}\geqslant 0$, $\phi^{\sigma} \in L^{\infty}(0,T;L^{1}(\Omega)) \in L^{\infty}(0,T;L^{1}(\Omega)) \cap L^{q}(0,T;W^{1,q}(\Omega))$ for every $q<\frac{4}{3}$.* (ii) : *$\rho^{\sigma} f^{\beta}(\rho^{\sigma}) |\nabla \phi^{\sigma}|^{2} \in L^{1}(\mathrm{Q}_{T})$ and $f^{\beta}(\rho^{\sigma}) |\nabla \phi|^{2} \in L^{1}(\mathrm{Q}_{T})$* (iii) : *$\phi^{\sigma} \in L^{\infty}(0,T;L^{1}(\Omega))$ is a solution of the HJB type equation in the sense of distributions $$\label{weakphi} \int_{0}^{T}\int_{\Omega} \phi^{\sigma} \partial_{t}\varphi dx dt - \sigma \int_{0}^{T}\int_{\Omega} \phi^{\sigma} \Delta \varphi dx dt + \int_{0}^{T}\int_{\Omega} \mathcal{H}(x, \rho^{\sigma},\nabla \phi^{\sigma}) \varphi dx dt = \int_{\Omega} u_{T}\varphi(T) dx,$$ for every $\varphi \in C_{c}^{\infty}((0,T]\times\Omega)$.* (iv) : *$\rho^{\sigma} \in L^{\infty}(0,T;L^{1}(\Omega)) \cap L^{2}(0,T;H^{1}(\Omega))$ is a solution of the Kolmogorov equation in the sense of distributions $$\label{weakrho} \int_{0}^{T}\int_{\Omega} \rho^{\sigma} \left( -\partial_{t}\psi - \sigma \Delta \psi + \mathcal{H}_{p}(x, \rho^{\sigma},\nabla \phi^{\sigma}) \cdot \nabla \psi \right) dx dt = \int_{\Omega} \rho_{0}\psi(0) dx,$$ for every $\psi \in C_{c}^{\infty}([0,T)\times\Omega)$.* A solution $\phi^{\sigma}$ of equation [\[weakphi\]](#weakphi){reference-type="eqref" reference="weakphi"} belongs to $C^{0}(0,T;L^{1}(\Omega))$. However, we also need to work with the subsolutions of the same equation, for which this kind of continuity may not hold. We recall a result in [@cardaliaguet2015second; @achdou2018mean] showing in the weak sense the continuity for sub-solutions. **Lemma 3**. *Let $\phi \in L^{\infty}(0,T;L^{1}(\Omega))$ satisfy for some function $h\in L^{1}(\mathrm{Q}_{T})$ and $g \in L^{1}(\Omega)$ $$\int_{0}^{T}\int_{\Omega}\phi \partial_{t}\varphi dx dt - \sigma \int_{0}^{T}\int_{\Omega}\phi \Delta \varphi dx dt \leqslant\int_{0}^{T}\int_{\Omega}h\varphi dx dt + \int_{\Omega}g(x)\varphi(T,x) dx$$ for every nonnegative function $\varphi \in C_{c}^{\infty}((0,T]\times\Omega)$. Then, for any Lipschitz continuous map $\xi : \Omega \to \mathbb{R}$, the map $t \longmapsto \int_{\Omega} \xi(x)\phi(t,x)dx$ has BV representative on \[0,T\]. Moreover, if we note its right limit at $t\in [0, T)$, then the map $\xi \longmapsto \int_{\Omega} \xi(x)\phi(t^{+}, x) dx$ can be extended to a linear bounded form on $C(\Omega)$.* As a consequence of Lemma [Lemma 3](#lemma33){reference-type="ref" reference="lemma33"}, for any sub-solution $\phi^{\sigma}$ of [\[weakphi\]](#weakphi){reference-type="eqref" reference="weakphi"} we can define $\phi^{\sigma}(0^{+})$ as a bounded Radon measure on $\Omega$. For simplicity, we note $\phi^{\sigma}(0) =\phi^{\sigma}(0^{+})$. ## Approximation of [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"} $$\begin{aligned} \partial_t\rho^{\sigma}_{\epsilon} - \nabla \cdot \left( \rho^{\sigma}_{\epsilon} \mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})\right) -\sigma \Delta \rho^{\sigma}_{\epsilon} = 0, &\qquad (t,x)\in\mathrm{Q}_{T} \label{EqFluidHd0xa}\\ -\partial_t\phi^{\sigma}_{\epsilon} + \mathcal{H}(x, \mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon} ) - \sigma \Delta \phi^{\sigma}_{\epsilon} =0, &\qquad (t,x)\in\mathrm{Q}_{T} \label{EqEikonalHd0xa}\\ \rho^{\sigma}_{\epsilon}(t=0,x)= \rho_{\epsilon 0}(x),&\quad x\in\Omega \label{FluidInxa}\\ \phi^{\sigma}_{\epsilon}(t=T,x)= \phi_{\epsilon T}(x),&\quad x\in\Omega \label{EikonalInxa}\end{aligned}$$ where $\mathcal{T}_{\epsilon}\rho=\mathcal{T}_{\epsilon}(\rho)= \min(\rho,\frac{\rho_{m}}{1+\epsilon})$, $\rho_{\epsilon 0}=m^{\epsilon}\ast \rho_{0}$. Here, $\ast$ denotes the convolution in the spatial variable and $m^{\epsilon}$ is a standard symmetric mollifier, i.e., $m^{\epsilon}(x)= \frac{1}{\epsilon^{2}} m(\frac{x}{\epsilon})$ for a non-negative function $m\in C_{c}^{\infty}(\mathbb{R}^{2})$ such that $\int_{\mathbb{R}^{2}} m(x) dx=1$. **Lemma 4**. *Let $(\rho^{\sigma}_{\epsilon},\phi^{\sigma}_{\epsilon})$ be a solution to the system [\[EqFluidHd0xa\]](#EqFluidHd0xa){reference-type="eqref" reference="EqFluidHd0xa"}-[\[EikonalInxa\]](#EikonalInxa){reference-type="eqref" reference="EikonalInxa"}. Then, we have $$\label{EstimateA0} \phi^{\sigma}_{\epsilon} \geqslant c_{0},$$ $$\label{EstimateA00} \phi^{\sigma}_{\epsilon} \in L^{\infty}(0,T;L^{1}(\Omega)),\quad \rho^{\sigma}_{\epsilon} \in L^{2}(0,T;H^{1}(\Omega)),$$* *$$\label{EstimateA001} \mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \in L^{1}(\mathrm{Q}_{T}),$$* *and $$\label{EstimateA} \begin{aligned} \int_{\Omega}\phi_{\epsilon T}\rho^{\sigma}_{\epsilon}(T)dx &+\int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon}\left(\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})\cdot \nabla \phi^{\sigma}_{\epsilon} -\mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \right) dx dt \\&+ \frac{1}{2} \int_{0}^{T}\int_{\Omega}f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt \leqslant C, \end{aligned}$$ for a positive constant $C$ depending on $\phi_{T}$, $\rho_{m}$, $\beta$, $\rho_{0}$ and $\sigma$.* *Proof.* We multiply the equation of $\phi^{\sigma}_{\epsilon}$ by $\rho^{\sigma}_{\epsilon}$ and equation of $\rho^{\sigma}_{\epsilon}$, by $\phi^{\sigma}_{\epsilon}$, integrate by parts some terms and subtract; we get $$\int_{\Omega}\phi^{\sigma}_{\epsilon}(0) \rho_{\epsilon 0} dx - \int_{\Omega}\phi_{\epsilon T}\rho^{\sigma}_{\epsilon}(T)dx - \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon}\left(\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})\cdot \nabla \phi^{\sigma}_{\epsilon} -\mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \right) dx dt =0,$$ then $$\int_{\Omega}\phi^{\sigma}_{\epsilon}(0) \rho_{\epsilon 0} dx = \int_{\Omega}\phi_{\epsilon T}\rho^{\sigma}_{\epsilon}(T)dx +\frac{1}{2} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt + \frac{1}{2} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{2-\beta}(\rho^{\sigma}_{\epsilon}) dx dt$$ which implies that $$\label{Eqestimate0} \begin{aligned} \int_{\Omega}\big[\phi^{\sigma}_{\epsilon}(0)\big]^{-} \rho_{\epsilon 0} dx +\int_{\Omega}\phi_{\epsilon T}\rho^{\sigma}_{\epsilon}(T)dx +\frac{1}{2} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt &+ \frac{1}{2} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{2-\beta}(\rho^{\sigma}_{\epsilon}) dx dt \\& \leqslant C \|\rho_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}\big[\phi^{\sigma}_{\epsilon}(0)\big]^{+} dx . \end{aligned}$$ From the definition of the Hamiltonian $\mathcal{H}$ and and $\phi_{T}\geqslant c_{0}$, we can deduce by comparison that $\phi^{\sigma}_{\epsilon}(t) \geqslant c_{0} + c_{0}(T-t)$ and then there exists an absolute constant where $$\label{labboubdn} \big[\phi^{\sigma}_{\epsilon}(0)\big]^{-}\leqslant C_{1}.$$ We integrate the equation of $\phi^{\sigma}_{\epsilon}$, and using [\[labboubdn\]](#labboubdn){reference-type="eqref" reference="labboubdn"}, we get $$\label{eqX0} \int_{\Omega}\int_{\Omega}\big[\phi^{\sigma}_{\epsilon}(0)\big]^{+} dx +\frac{1}{2} \int_{0}^{T}\int_{\Omega}f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt \leqslant\int_{\Omega} \phi_{\epsilon T} dx + \frac{1}{2} \int_{0}^{T}\int_{\Omega} f^{2-\beta}(\rho^{\sigma}_{\epsilon}) dx dt +C_{1}.$$ Therefore, we have $$\label{eqX0ZA} \int_{\Omega}\int_{\Omega}\big[\phi^{\sigma}_{\epsilon}(0)\big]^{+} dx +\frac{1}{2} \int_{0}^{T}\int_{\Omega}f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt \leqslant C(\rho_{m},\phi_{T},\Omega,T)+C_{1}.$$ Combining [\[eqX0ZA\]](#eqX0ZA){reference-type="eqref" reference="eqX0ZA"} with [\[Eqestimate0\]](#Eqestimate0){reference-type="eqref" reference="Eqestimate0"} we get $$\label{Eqestimate} \begin{aligned} \int_{\Omega}\phi_{\epsilon T}\rho^{\sigma}_{\epsilon}(T)dx &+\frac{1}{2} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt + \frac{1}{2} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{2-\beta}(\rho^{\sigma}_{\epsilon}) dx dt \\&+ \frac{1}{2} \int_{0}^{T}\int_{\Omega}f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt \leqslant C(\rho_{m}, \|\rho_{0}\|_{L^{\infty}}, \|\phi_{T}\|_{L^{\infty}},\Omega,T,C_{1}), \end{aligned}$$ which can be written as follows $$\label{EqestimateX} \begin{aligned} \int_{\Omega}\phi_{\epsilon T}\rho^{\sigma}_{\epsilon}(T)dx &+\int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon}\left(\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})\cdot \nabla \phi^{\sigma}_{\epsilon} -\mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \right) dx dt \\&+ \frac{1}{2} \int_{0}^{T}\int_{\Omega}f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon}|^{2} dx dt \leqslant C(\rho_{m}, \|\rho_{0}\|_{L^{\infty}}, \|\phi_{T}\|_{L^{\infty}},\Omega,T,C_{1}). \end{aligned}$$ We integrate $\phi^{\sigma}_{\epsilon}$ equation in $(t,T)\times \Omega$, we get that $$\int_{\Omega} \phi^{\sigma}_{\epsilon}(t,x) dx +\frac{1}{2}\int_{t}^{T}\int_{\Omega} \mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) dx dt = \int_{\Omega} \phi_{\epsilon T} dx.$$ Consequently, from [\[labboubdn\]](#labboubdn){reference-type="eqref" reference="labboubdn"}-[\[eqX0ZA\]](#eqX0ZA){reference-type="eqref" reference="eqX0ZA"} and the definition of $f$ we deduce that $$\mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \in L^{1}(\mathrm{Q}_{T}),$$ and then we have $$\int_{\Omega} \phi^{\sigma}_{\epsilon}(t,x) dx \leqslant C.$$ Finally, we have already proved that $\phi^{\sigma}_{\epsilon}$ has a lower-bound, we deduce that $\phi^{\sigma}_{\epsilon} \in L^{\infty}(0,T;L^{1}(\Omega))$. We multiply the equation [\[EqFluidHd0xa\]](#EqFluidHd0xa){reference-type="eqref" reference="EqFluidHd0xa"} with $(\rho_{m}-\rho^{\sigma}_{\epsilon})$ and we integrate over space and time, we get $$\int_{0}^{T}\int_{\Omega}\partial_t\rho^{\sigma}_{\epsilon}(\rho_{m}-\rho^{\sigma}_{\epsilon}) - \int_{0}^{T}\int_{\Omega}\nabla \cdot \left( \rho^{\sigma}_{\epsilon} \mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})\right)(\rho_{m}-\rho^{\sigma}_{\epsilon}) dx dt -\sigma \int_{0}^{T}\int_{\Omega} \Delta \rho^{\sigma}_{\epsilon} (\rho_{m}-\rho^{\sigma}_{\epsilon}) dx dt = 0,$$ it yields that $$\begin{aligned} -\frac{1}{2}\int_{0}^{T}\int_{\Omega}\partial_t(\rho_{m}-\rho^{\sigma}_{\epsilon})^{2} dx dt &-\int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} \mathcal{H}_{p}\left(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}\right)\cdot \nabla \rho^{\sigma}_{\epsilon}dx dt \\&-\sigma \int_{0}^{T}\int_{\Omega} |\nabla \rho^{\sigma}_{\epsilon}|^{2} dx dt = 0, \end{aligned}$$ then $$\begin{aligned} \frac{1}{2}\|(\rho_{m}-\rho^{\sigma}_{\epsilon})(t)\|^{2}_{L^{2}} &+ \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} (\rho_{m}-\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon})^{\beta}\nabla \phi^{\sigma}_{\epsilon}\cdot \nabla \rho^{\sigma}_{\epsilon}dx dt \\&+\sigma \int_{0}^{T}\int_{\Omega} |\nabla \rho^{\sigma}_{\epsilon}|^{2} dx dt = \frac{1}{2}\|(\rho_{m}-\rho^{\sigma}_{\epsilon})(0)\|^{2}_{L^{2}}. \end{aligned}$$ By Young inequality and [\[eqX0ZA\]](#eqX0ZA){reference-type="eqref" reference="eqX0ZA"} it yields that $$\begin{aligned} \frac{1}{2}\|(\rho_{m}-\rho^{\sigma}_{\epsilon})(t)\|^{2}_{L^{2}} &+\sigma \int_{0}^{T}\int_{\Omega} |\nabla \rho^{\sigma}_{\epsilon}|^{2} dx dt \leqslant C(\rho_{m}, \|\rho_{0}\|_{L^{\infty}}, \|\phi_{T}\|_{L^{\infty}},\Omega,T,C_{1}) +\frac{1}{2}\|(\rho_{m}-\rho^{\sigma}_{\epsilon})(0)\|^{2}_{L^{2}} \end{aligned}$$ which gives that $\rho^{\sigma}_{\epsilon} \in L^{2}(0,T;H^{1}(\Omega))$ and $\rho_{m}-\rho^{\sigma}_{\epsilon} \in L^{\infty}(0,T;L^{2}(\Omega))$ which completes the proof. ◻ Now, inspired by [@achdou2018mean; @porretta2015weak], we show some compactness properties for the family $\left( \rho^{\sigma}_{\epsilon}, \phi^{\sigma}_{\epsilon}\right)$ solution to the system [\[EqFluidHd0xa\]](#EqFluidHd0xa){reference-type="eqref" reference="EqFluidHd0xa"}-[\[EikonalInxa\]](#EikonalInxa){reference-type="eqref" reference="EikonalInxa"}. **Theorem 5**. *Under the assumptions , it follows that* \(1\) : *There exist $\rho^{\sigma} \in C(0,T;L^{1}(\Omega))$ and $\phi^{\sigma} \in L^{\infty}(0,T;L^{1}(\Omega)) \cap L^{q}(0,T;W^{1,q}(\Omega))$ for every $q<\frac{4}{3}$ such that we can extract $\rho^{\sigma}_{\epsilon} \to \rho$ in $L^{1}(\mathrm{Q}_{T}))$. Moreover, we have $$\phi^{\sigma}_{\epsilon} \to \phi^{\sigma}\,\,\, \text{and} \,\,\, \nabla \phi^{\sigma}_{\epsilon} \to \nabla \phi^{\sigma}, \quad \text{ almost everywhere.}$$ Furthermore, $\phi^{\sigma}$ and $\rho^{\sigma}$ satisfy* *$$\label{EstimateAlim} \begin{aligned} \int_{0}^{T}\int_{\Omega} \rho^{\sigma} f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}|^{2} dx dt + & \int_{0}^{T}\int_{\Omega} \rho^{\sigma} f^{2-\beta}(\rho^{\sigma}) dx dt\\&+ \frac{1}{2} \int_{0}^{T}\int_{\Omega}f^{\beta}(\rho^{\sigma})|\nabla \phi^{\sigma}|^{2} dx dt \leqslant C, \end{aligned}$$ for a positive constant C depending on $\phi_{T}$,$\rho_{m}$, $\beta$ and $\rho_{0}$.* \(2\) : *We have $\rho^{\sigma}_{\epsilon} \to \rho^{\sigma}$ in $C(0,T;L^{1}(\Omega))$ for every $t\in [0,T]$ and [\[weakrho\]](#weakrho){reference-type="eqref" reference="weakrho"} holds for every $\psi \in C_{c}^{\infty}([0,T)\times\Omega)$.* *Proof.* By Lemma [Lemma 4](#Lemmaex){reference-type="ref" reference="Lemmaex"}, we have $\mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})$ is bounded in $L^{1}(\mathrm{Q}_{T})$ uniformly in $\epsilon$. Therefore, $\left(-\partial_{t}\phi^{\sigma}_{\epsilon}-\sigma\Delta\phi^{\sigma}_{\epsilon}\right)_{\epsilon}$ is bounded in $L^{1}(\mathrm{Q}_{T})$. By the classical results on the parabolic equation with $L^{1}$ data it follows the compactness of $\phi^{\sigma}_{\epsilon}$ and ${\nabla \phi^{\sigma}_{\epsilon}}$ in $L^{1}(\mathrm{Q}_{T})$, see [@boccardo1997nonlinear Thoerem 2.1 and Theorem 3.3]. Then, there exists $\phi^{\sigma} \in L^{\infty}(0,T;L^{1}(\Omega)) \cap L^{q}(0,T;W^{1,q}(\Omega))$ for every $q<\frac{4}{3}$ such we can extract a subsequence such that $\phi^{\sigma}_{\epsilon} \to \phi^{\sigma}$ and $\nabla\phi^{\sigma}_{\epsilon} \to \nabla \phi^{\sigma}$ in $L^{1}(\mathrm{Q}_{T})$ almost everywhere. Now for the density $\rho^{\sigma}_{\epsilon}$, from Theorem [Lemma 4](#Lemmaex){reference-type="ref" reference="Lemmaex"} [\[EstimateA00\]](#EstimateA00){reference-type="eqref" reference="EstimateA00"}, we have $\rho^{\sigma}_{\epsilon} \in L^{2}(0,T;H^{1}(\Omega))$. Then ,we can apply the compactness results in [@porretta2015weak Theorem 6.1]. We can extract a subsequence converges in $L^{1}(\mathrm{Q}_{T})$ such that $\rho^{\sigma}_{\epsilon} \to \rho^{\sigma}$. Moreover, from the estimate [\[EstimateA\]](#EstimateA){reference-type="eqref" reference="EstimateA"}, and Fatou's Lemma, we get the estimate [\[EstimateAlim\]](#EstimateAlim){reference-type="eqref" reference="EstimateAlim"}. Now, we prove that $\rho^{\sigma}_{\epsilon}\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})$ strongly converges in $L^{1}(\mathrm{Q}_{T})$. Indeed, we have $$\begin{aligned} \int_{0}^{T}\int_{\Omega} | \rho^{\sigma}_{\epsilon}\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) | dx dt \leqslant& \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{\beta}(\rho^{\sigma}_{\epsilon})|\nabla \phi^{\sigma}_{\epsilon} | dx dt \\& \leqslant\left(\int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{\beta}(\rho^{\sigma}_{\epsilon}) dx dt \right)^{\frac{1}{2}} \left(\int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} f^{\beta}(\rho^{\sigma}_{\epsilon}) |\nabla \phi^{\sigma}_{\epsilon} |^{2} \right)^{\frac{1}{2}} check \end{aligned}$$ Therefore, from [\[EstimateA\]](#EstimateA){reference-type="eqref" reference="EstimateA"}, we can conclude that $\rho^{\sigma}_{\epsilon}\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \to \rho^{\sigma}\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})$ in $L^{1}(\mathrm{Q}_{T})$. As consequence, [\[weakrho\]](#weakrho){reference-type="eqref" reference="weakrho"} holds for any $\varphi \in C^{\infty}_{c}((0,T)\times\Omega)$. Moreover, we can apply [@porretta2015weak Theorem 6.1], to deduce the strong convergence of $\rho^{\sigma}_{\epsilon}$ in $C([0,T];L^{1}(\Omega))$. ◻ ## Existence and uniqueness for the system of [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"} The objective of this section is to establish the existence and uniqueness of a weak solution for the system given in [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}. **Lemma 6**. *Let $(\rho^{\sigma},\phi^{\sigma})$ be given by Theorem [Theorem 5](#Thmlim){reference-type="ref" reference="Thmlim"}. Then $$\label{WeakEq} \int_{0}^{T}\int_{\Omega}\phi^{\sigma}\partial_{t}\varphi dx dt - \sigma \int_{0}^{T}\int_{\Omega} \phi^{\sigma}\Delta\varphi dx dt + \int_{0}^{T}\int_{\Omega} \mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma}) \varphi dx dt \leqslant\int_{0}^{T}\int_{\Omega}\phi_{T}\varphi(T) dx$$ for every nonnegative function $\varphi \in C_{c}^{\infty}((0,T]\times\Omega)$. Moreover, we have $\phi_{T}\rho^{\sigma}(T) \in L^{1}(\Omega)$ and $$\int_{\Omega}\phi_{T}\rho^{\sigma}(T) dx \leqslant C$$ for some constant $C$ depending on $\phi_{T}$, $\rho_{m}$, $\beta$, and $\rho_{0}$.* *Proof.* From [\[EstimateA001\]](#EstimateA001){reference-type="eqref" reference="EstimateA001"} and the convergence of $\rho^{\sigma}$, we can apply Fatou's lemma and we get for any $\varphi \in C_{c}^{\infty}((0,T]\times\Omega)$, $\varphi\geqslant 0$, $$\int_{0}^{T}\int_{\Omega} \mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma}) \varphi dx dt \leqslant\liminf_{\epsilon \to 0 }\int_{0}^{T}\int_{\Omega} \mathcal{H}(x,\mathcal{T}_{\epsilon}\rho_{\epsilon}^{\sigma},\nabla \phi_{\epsilon}^{\sigma}) \varphi dx dt.$$ Now, we will deal with the boundary condition at $t=T$, by [\[EstimateA\]](#EstimateA){reference-type="ref" reference="EstimateA"} we obtain $$\lim_{\epsilon \to 0 }\int_{\Omega}\phi_{\epsilon}^{\sigma}(T,x)\varphi(T,x)dx =\lim_{\epsilon \to 0 }\int_{\Omega}\phi_{\epsilon}^{\sigma}(T,x)\varphi_{\epsilon}(T,x)dx =\int_{\Omega}\phi_{T}(x)\varphi(T,x)dx$$ Combining the previous estimates, we deduce [\[WeakEq\]](#WeakEq){reference-type="eqref" reference="WeakEq"}. ◻ **Lemma 7**. *Let $(\rho_{\epsilon}^{\sigma},\phi_{\epsilon}^{\sigma})$ be given by Theorem [Theorem 5](#Thmlim){reference-type="ref" reference="Thmlim"}. Then, we can extract a subsequence, ${\phi_{\epsilon}^{\sigma}}_{|_{t=0}}$ converges weakly $^{\ast}$ to a bounded measure $\bar{\bar{\phi}}_{0}$ on $\Omega$ and $\phi^{\sigma}(0) \geqslant\bar{\bar{\phi}}_{0}$.* For the proof of Lemma [Lemma 7](#lemmaA){reference-type="ref" reference="lemmaA"}, we send the reader to [@achdou2018mean Lemma 4.7]. Now we give a useful lemma for the uniqueness results. **Lemma 8**. *Under the assumptions , let $\phi^{\sigma} \in L^{\infty}(0,T;L^{1}(\Omega))$ satisfies [\[WeakEq\]](#WeakEq){reference-type="eqref" reference="WeakEq"} and let $\rho^{\sigma}$ be a solution of [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"} and we assume that $(\rho^{\sigma},\phi^{\sigma})$ satisfy Definition [Definition 2](#defweaksig){reference-type="ref" reference="defweaksig"}. Then, we have $$\label{EqX0} \langle\phi^{\sigma}(0),\rho_{0}\rangle \leqslant\int_{\Omega} \phi_{T} \tilde{\rho}^{\sigma}(T)dx + \int_{0}^{T}\int_{\Omega} \tilde{\rho}\mathcal{H}_{p}(x,\tilde{\rho}^{\sigma},\nabla \tilde{\phi}^{\sigma})\cdot \nabla \phi^{\sigma} -\tilde{\rho}^{\sigma}\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})$$ for any $(\tilde{\rho}^{\sigma},\tilde{\phi}^{\sigma})$ verifying the same condition as $(\rho^{\sigma},\phi^{\sigma})$.* *Proof.* Let $m_{\delta}(.)$ be s sequence of standard symmetric mollifiers in $\mathbb{R}^{2}$ and set $$\tilde{\rho}^{\sigma}_{\delta}(t,x)=\tilde{\rho}^{\sigma}_{\delta}\star m_{\delta} = \int_{\mathbb{R}^{2}} \tilde{\rho}^{\sigma}_{\delta}(t,y) m_{\delta}(x-y) dy.$$ Since $\tilde{\rho}^{\sigma} \in L^{\infty}(0,T;L^{1}(\Omega))$ ,we deduce that $\tilde{\rho}^{\sigma}_{\delta}$, $\nabla \tilde{\rho}^{\sigma}_{\delta} \in L^{\infty}(\mathrm{Q}_{T})$. We consider a sequence of $1-d$ mollifiers $\xi_{\epsilon}(t)$ such that $\text{supp} (\xi_{\epsilon}) \subset (-\epsilon,0)$, and we set $$\tilde{\rho}^{\sigma}_{\delta,\epsilon} := \int_{0}^{T}\xi_{\epsilon}(s-t)\tilde{\rho}^{\sigma}_{\delta}(s) ds$$ Notice that this function vanishes at $t=0$, so we can consider it as a test function in the inequality satisfied by $\phi^{\sigma}$. We get $$\label{EsA} \int_{0}^{T}\int_{\Omega}\phi^{\sigma} \left(\partial_{t}\tilde{\rho}^{\sigma}_{\delta,\epsilon} - \sigma\Delta \tilde{\rho}^{\sigma}_{\delta,\epsilon} \right) dx dt + \int_{0}^{T}\int_{\Omega}\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma}_{\delta,\epsilon} dx dt \leqslant\int_{\Omega} \phi_{T} \tilde{\rho}^{\sigma}_{\delta,\epsilon} dx$$ Let ${\phi^{\sigma}_{\delta,\epsilon}(s,y)= \int_{0}^{T}\int_{\Omega} \phi^{\sigma}_{\delta,\epsilon}\xi_{\epsilon}(s-t) m_{\delta}(x-y) dt dx }$, then $$\int_{0}^{T}\int_{\Omega}\phi^{\sigma} \left(\partial_{t}\tilde{\rho}^{\sigma}_{\delta,\epsilon} - \sigma\Delta \tilde{\rho}^{\sigma}_{\delta,\epsilon} \right) dx dt = \int_{0}^{T}\int_{\Omega} \left(-\partial_{s}\phi^{\sigma}_{\delta,\epsilon} - \sigma\Delta \phi^{\sigma}_{\delta,\epsilon} \right) \tilde{\rho}^{\sigma}_{\delta,\epsilon} (s,y)dy ds.$$ Moreover, from the equation of $\tilde{\rho}^{\sigma}$ we obtain $$\begin{aligned} \int_{0}^{T}\int_{\Omega} &\left(-\partial_{s}\phi^{\sigma}_{\delta,\epsilon} - \sigma\Delta \phi^{\sigma}_{\delta,\epsilon} \right) \tilde{\rho}^{\sigma}_{\delta,\epsilon} (s,y)dy ds \\& =-\int_{0}^{T}\int_{\Omega} \tilde{\rho}^{\sigma}_{\delta,\epsilon} (s,y) \tilde{b}_{\delta,\epsilon} (s,y)\cdot \nabla \phi^{\sigma}_{\delta,\epsilon} dy ds + \int_{\Omega}\tilde{\rho}^{\sigma}_{0}(y)\phi^{\sigma}_{\delta,\epsilon}(0) dy \end{aligned}$$ where $\tilde{b}_{\delta,\epsilon} = \mathcal{H}_{p}(x,\tilde{\rho}^{\sigma},\nabla \tilde{\phi}^{\sigma})$. We have $$\begin{aligned} -\int_{0}^{T}\int_{\Omega}\nabla \phi^{\sigma} \cdot \tilde{w}_{\delta,\epsilon} &+ \int_{0}^{T}\int_{\Omega}\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma}_{\delta,\epsilon} dx dt + \int_{\Omega} \tilde{\rho}^{\sigma}_{0}\star m_{\delta}\int_{0}^{T}\phi^{\sigma}(t)\xi_{\epsilon}(-t) dt dx \\& \leqslant\int_{\Omega}\phi_{T} \rho^{\sigma}_{\delta,\epsilon} dx \end{aligned}$$ where here we denote $\tilde{w}_{\delta} = \left(\left(\tilde{b}\tilde{\rho}^{\sigma}\right)\star m_{\delta} \right)$ and $\tilde{w}_{\delta,\epsilon}= \int_{0}^{T} \tilde{w}_{\delta}(s)\xi_{\epsilon}(s-t) ds$. Now from the definition [Definition 2](#defweaksig){reference-type="ref" reference="defweaksig"}, we can use Fatou's lemma, and we deduce $$\begin{aligned} \liminf_{\epsilon \to 0}& \int_{0}^{T}\int_{\Omega} \left(\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma}_{\delta,\epsilon} - \nabla \phi^{\sigma} \cdot \tilde{w}_{\delta,\epsilon} \right)dx dt\\ & =\liminf_{\epsilon \to 0} \int_{0}^{T}\int_{\Omega} \int_{0}^{T} \left(\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma}_{\delta} - \nabla \phi^{\sigma} \cdot \tilde{w}_{\delta} \right) \xi_{\epsilon}(s-t) ds dx dt\\ \qquad & \geqslant\int_{0}^{T}\int_{\Omega} \left(\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma}_{\delta} - \nabla \phi^{\sigma} \cdot \tilde{w}_{\delta} \right) dx dt \end{aligned}$$ and in the same way we get $$\label{EqRes0} \begin{aligned} \liminf_{\delta \to 0}& \int_{0}^{T} \int_{\Omega} \left(\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma}_{\delta} - \nabla \phi^{\sigma} \cdot \tilde{w}_{\delta} \right) dx dt\\& \geqslant\int_{0}^{T}\int_{\Omega} \left(\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma} - \nabla \phi^{\sigma} \cdot\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma} \right)dx dt \end{aligned}$$ For $t=0$, we have $\int_{\Omega} \phi^{\sigma}(t)(\tilde{\rho_{0}}\star m_{\delta}) dx$ has a trace at $t=0$ from Lemma [Lemma 3](#lemma33){reference-type="ref" reference="lemma33"} and this trace is continuous as $\delta \to 0$ since $\tilde{\rho}_{0}$ is continuous. Then as $\epsilon \to 0$, by [\[EsA\]](#EsA){reference-type="eqref" reference="EsA"} we get $$\label{EqRes} \begin{aligned} \int_{0}^{T} \int_{\Omega}&\left(\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\tilde{\rho}^{\sigma}_{\delta} - \nabla \phi^{\sigma} \cdot \tilde{w}_{\delta} \right) dx dt + \int_{\Omega} \phi^{\sigma}(t)(\tilde{\rho_{0}}\star m_{\delta}) dx \leqslant& \int_{\Omega}\phi_{T}\star m_{\delta}\tilde{\rho}_{\delta}(T)dx \end{aligned}$$ Passing now to the limit in [\[EqRes\]](#EqRes){reference-type="eqref" reference="EqRes"} and using [\[EqRes0\]](#EqRes0){reference-type="eqref" reference="EqRes0"} we deduce [\[EqX0\]](#EqX0){reference-type="eqref" reference="EqX0"}. ◻ **Lemma 9**. *Consider a subsequence $(\rho^{\sigma}_{\epsilon},\phi^{\sigma}_{\epsilon})$ converging to $(\rho^{\sigma},\phi^{\sigma})$ as in Theorem [Theorem 5](#Thmlim){reference-type="ref" reference="Thmlim"}. Then, $(\rho^{\sigma},\phi^{\sigma})$ satisfies the following energy identity $$\label{Eqenerg} \langle\rho^{\sigma}(T),\phi_{T}\rangle + \int_{0}^{T}\int_{\Omega} \rho\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})= \langle\phi^{\sigma}(0),\rho_{0}\rangle$$* *Proof.* We star by the energy identity for the system [\[EqFluidHd0xa\]](#EqFluidHd0xa){reference-type="eqref" reference="EqFluidHd0xa"}-[\[EikonalInxa\]](#EikonalInxa){reference-type="eqref" reference="EikonalInxa"} $$\label{EqX} \int_{\Omega}\phi^{\sigma}_{\epsilon}(0) \rho_{\epsilon 0} dx = \int_{\Omega}\phi_{\epsilon T}\rho^{\sigma}_{\epsilon}(T)dx + \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon}\left(\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})\cdot \nabla \phi^{\sigma}_{\epsilon} -\mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \right) dx dt.$$ Using ${\left(\mathcal{H}_{p}(x,\rho, p)\cdot p -\mathcal{H}(x,\rho, p) \right) \geqslant 0}$, estimate [\[EstimateA\]](#EstimateA){reference-type="eqref" reference="EstimateA"} and Fatou's lemma ,we get $$\label{EqX1} \begin{aligned} \lim_{\epsilon \to 0}\inf \int_{0}^{T}\int_{\Omega} \rho^{\sigma}_{\epsilon} &\left(\mathcal{H}_{p}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon})\cdot \nabla \phi^{\sigma}_{\epsilon} -\mathcal{H}(x,\mathcal{T}_{\epsilon}\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) \right) dx dt \\& \geqslant\int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left(\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma}) \right) dx dt. \end{aligned}$$ From Lemma [Lemma 7](#lemmaA){reference-type="ref" reference="lemmaA"}, we can assume that ${\phi^{\sigma}_{\epsilon}}_{t=0}$ converges weakly $\star$ to a bounded measure $\bar{\bar{\phi}}_{0}$ on $\Omega$, and $\bar{\bar{\phi}}_{0} \leqslant\phi^{\sigma}(0)$. Since $\rho_{0}\in C(\Omega)$, we deduce $$\lim _{\epsilon \to 0}\int_{\Omega}\phi^{\sigma}_{\epsilon}(0)\rho_{\epsilon 0} dx \leqslant\langle \phi^{\sigma}_{\epsilon}(0),\rho_{0}\rangle.$$ Combining the previous results we get from [\[EqX\]](#EqX){reference-type="eqref" reference="EqX"} $$\label{Eqss} \begin{aligned} \limsup_{\epsilon \to 0}&\int_{\Omega}\phi_{T}\rho_{\epsilon}^{\sigma}(T) dx \leqslant\langle\phi^{\sigma}(0),\rho_{0}\rangle - \int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left(\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma}) \right) dx dt \end{aligned}$$ We use [\[EqX0\]](#EqX0){reference-type="eqref" reference="EqX0"} in Lemma [Lemma 8](#lemmaIneq){reference-type="ref" reference="lemmaIneq"} for $\rho^{\sigma}=\tilde{\rho}^{\sigma}$ and $\phi^{\sigma}=\tilde{\phi}^{\sigma}$ $$\label{Eqss} \begin{aligned} \limsup_{\epsilon \to 0}\int_{\Omega}\phi_{\epsilon T}\rho_{\epsilon}^{\sigma}(T) dx \leqslant& \langle\phi^{\sigma}(0),\rho_{0}\rangle - \int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left(\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma}) \right) dx dt\\ & \leqslant\int_{\Omega}\phi_{T}\rho^{\sigma}(T) dx \end{aligned}$$ Then, we can conclude [\[Eqenerg\]](#Eqenerg){reference-type="eqref" reference="Eqenerg"}. ◻ Now, we will give the existence and uniqueness results **Proposition 10**. *Consider a subsequence $(\rho^{\sigma}_{\epsilon},\phi^{\sigma}_{\epsilon})$ converging to the $(\rho^{\sigma}, \phi^{\sigma} )$ as in Theorem [Theorem 5](#Thmlim){reference-type="ref" reference="Thmlim"}. Then, $(\rho^{\sigma}, \phi^{\sigma} )$ is a weak solution of [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}. Moreover, for any weak solution $(\rho^{\sigma}, \phi^{\sigma} )$ to [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}, the energy identity [\[Eqenerg\]](#Eqenerg){reference-type="eqref" reference="Eqenerg"} remains true. Furthermore, under the following monotonicity condition $$\label{Ourassumption} \mathbf{H}_{c}=\begin{pmatrix} -\frac{2}{\rho}\mathcal{H}_{\rho}(x,\rho^{\sigma}, p)& \mathcal{H}_{\rho,p}^{T}( x,\rho^{\sigma}, p )\\ \mathcal{H}_{\rho,p}( x,\rho^{\sigma}, p )& 2\mathcal{H}_{p,p}( x,\rho^{\sigma}, p )\\ \end{pmatrix} \geqslant 0, \quad \rho>0,$$ there exists a unique weak solution of the system [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"} in sense of Definition [Definition 2](#defweaksig){reference-type="ref" reference="defweaksig"}. In particular,* 1. *If $\beta=2$ and ${\rho^{\sigma} \leqslant\frac{\rho_{m}}{2}}$, there exists a unique weak solution of the system [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}.* 2. *If $\beta=0$ the monotonicity condition [\[Ourassumption\]](#Ourassumption){reference-type="eqref" reference="Ourassumption"} is not fulfilled.* *Proof.* By Theorem [Theorem 5](#Thmlim){reference-type="ref" reference="Thmlim"}, we deduce that $(\rho^{\sigma}, \phi^{\sigma} )$ is a weak solution of [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}. Moreover, from Lemma [Lemma 8](#lemmaIneq){reference-type="ref" reference="lemmaIneq"} $$\label{EqX0a} \langle\phi^{\sigma}(0),\rho_{0}\rangle \leqslant\int_{\Omega} \phi_{T} \rho^{\sigma}(T)dx + \int_{0}^{T}\int_{\Omega} \rho\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -\rho^{\sigma}\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})$$ Now , let us consider $\phi_{k}:= \min(\phi^{\sigma},k)$. We have $-\partial_{t} \phi^{\sigma} - \sigma \Delta \phi^{\sigma} \in L^{1}(\mathrm{Q}_{T})$ and by Kato's inequality we obtain $$-\partial_{t} \phi_{k} - \sigma \Delta \phi_{k} + \mathcal{H}(x,\rho^{\sigma}_{\epsilon},\nabla \phi^{\sigma}_{\epsilon}) 1_{\{\phi^{\sigma}<k \}} \geqslant 0.$$ Since, $\partial_{t}\rho^{\sigma} - \sigma \Delta \rho^{\sigma} - \operatorname{div}(\rho^{\sigma}b ) =0$ for some $b$ such that $\rho^{\sigma}|b|^2 \in L^{1}(\mathrm{Q}_{T})$, by [@porretta2015weak Theorem 3.6], we have $\rho^{\sigma} \in C^{0}(0,T;L^{1}(\Omega))$ is also a normalized solution and we have $$\partial_{t}S_{n}(\rho^{\sigma}) - \sigma \Delta S_{n}(\rho^{\sigma}) - div(S^{'}_{n}(\rho^{\sigma})\rho^{\sigma}\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma} )= R_{n}$$ where $S_{n}(\rho^{\sigma})$ is a suitable $C^{1}$ truncation and $R_{n} \to 0$ in $L^{1}(\mathrm{Q}_{T})$, for more details see [@porretta2015weak]. Now, we multiply the equation of $S_{n}(\rho^{\sigma})$ by $\phi_{k}$ we get $$\begin{aligned} \int_{\Omega} S_{n}(\rho_{0})\phi_{k}(0) dx - & \int_{\Omega} S_{n}(\rho^{\sigma}(T))\phi_{k}(T) dx \geqslant- \int_{0}^{T}\int_{\Omega} R_{n} \phi_{k} dx dt \\ & \int_{0}^{T}\int_{\Omega} \left(S^{'}_{n}(\rho^{\sigma})\rho^{\sigma}\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -S_{n}(\rho^{\sigma})\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\right)\mathrm{1}_{\{\phi^{\sigma}<k \}} dx dt \end{aligned}$$ We have $\phi_{k}$ is bounded then the first term in the right-hand side vanishes as $n \to +\infty$. Then we pass to the limit as $n \to +\infty$, we obtain $$\begin{aligned} \int_{\Omega} \rho_{0}\phi_{k}(0) dx - & \int_{\Omega} \rho^{\sigma}(T)\phi_{k}(T) dx \\ & \geqslant\int_{0}^{T}\int_{\Omega} \left(\rho^{\sigma}\mathcal{H}_{p}(x,\rho^{\sigma},\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -\rho^{\sigma}\mathcal{H}(x,\rho^{\sigma},\nabla \phi^{\sigma})\right) \mathrm{1}_{\{\phi^{\sigma}<k \}} dx dt \end{aligned}$$ Finally, by letting $k \to +\infty$ and with [\[EqX0a\]](#EqX0a){reference-type="eqref" reference="EqX0a"} we deduce that the energy identity [\[Eqenerg\]](#Eqenerg){reference-type="eqref" reference="Eqenerg"} remains true. We now process for proving the uniqueness results. Let $(\rho^{}, \phi)$ and $(\tilde{\rho}, \tilde{\phi})$ be two solution of [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"} both solutions satisfy the energy identity [\[Eqenerg\]](#Eqenerg){reference-type="eqref" reference="Eqenerg"}. First, using the inequality [\[EqX0\]](#EqX0){reference-type="eqref" reference="EqX0"} from Lemma [Lemma 8](#lemmaIneq){reference-type="ref" reference="lemmaIneq"} we get $$\langle\phi(0),\rho_{0}\rangle \leqslant\langle\tilde{\rho}(T),\phi_{T}\rangle + \int_{0}^{T}\int_{\Omega} \tilde{\rho}\mathcal{H}_{p}(x,\tilde{\rho},\nabla \tilde{\phi})\cdot \nabla \phi -\tilde{\rho}\mathcal{H}(x,\rho,\nabla \phi)$$ $$\langle\tilde{\phi}(0),\rho_{0}\rangle \leqslant\langle\rho(T),\phi_{T}\rangle + \int_{0}^{T}\int_{\Omega} \rho\mathcal{H}_{p}(x,\rho,\nabla \phi)\cdot \nabla \tilde{\phi} -\rho\mathcal{H}(x,\tilde{\rho},\nabla \tilde{\phi})$$ Moreover, by the energy identity [\[Eqenerg\]](#Eqenerg){reference-type="eqref" reference="Eqenerg"} we obtain $$\langle\phi(0),\rho_{0}\rangle =\langle\rho(T),\phi_{T}\rangle + \int_{0}^{T}\int_{\Omega} \rho\mathcal{H}_{p}(x,\rho,\nabla \phi)\cdot \nabla \phi -\rho\mathcal{H}(x,\rho,\nabla \phi)$$ $$\langle\tilde{\phi}(0),\rho_{0}\rangle =\langle\tilde{\rho}(T),\phi_{T}\rangle + \int_{0}^{T}\int_{\Omega} \tilde{\rho}\mathcal{H}_{p}(x,\tilde{\rho},\nabla \tilde{\phi})\cdot \nabla \tilde{\phi} -\rho\mathcal{H}(x,\tilde{\rho},\nabla \tilde{\phi}),$$ and therefore we get $$\begin{aligned} \int_{0}^{T}\int_{\Omega} &\tilde{\rho}\left(\mathcal{H}(x,\rho,\nabla \phi)-\mathcal{H}(x,\tilde{\rho},\nabla \tilde{\phi})-\mathcal{H}_{p}(x,\tilde{\rho},\nabla \tilde{\phi})\cdot \left(\nabla \phi-\nabla \tilde{\phi}\right) \right) dx dt\\ +&\int_{0}^{T}\int_{\Omega} \rho\left(\mathcal{H}(x,\tilde{\rho},\nabla \tilde{\phi})-\mathcal{H}(x,\rho,\nabla \phi )-\mathcal{H}_{p}(x,\rho,\nabla \phi)\cdot \left(\nabla \tilde{\phi}-\nabla \phi\right) \right) dx dt =0. \end{aligned}$$ Let us define $$\begin{aligned} E(x,\rho_{1},p_{1},\rho_{2},p_{2}) =&-\left( \mathcal{H}(x,\rho_{1}, p_{1}) - \mathcal{H}(x,\rho_{2}, p_{2})\right)\left( \rho_{1} - \rho_{2}\right) \\ & +\left( \rho_{1} \mathcal{H}_{p}(x,\rho_{1}, p_{1}) -\rho_{2} \mathcal{H}_{p}(x,\rho_{2}, p_{2}) \right)\left(p_{1} - p_{2}\right)\\ &=\begin{pmatrix} \rho_{2}- \rho_{1}\\ p_{2}- p_{1} \end{pmatrix}\cdot \begin{pmatrix} \mathcal{H}(x,\rho_{1}, p_{1}) - \mathcal{H}(x,\rho_{1}, p_{1})\\ \rho_{2} \mathcal{H}_{p}(x,\rho_{2}, p_{2}) -\rho_{1} \mathcal{H}_{p}(x,\rho_{1}, p_{1}) \end{pmatrix}\\&=\begin{pmatrix} \rho_{2}- \rho_{1}\\ p_{2}- p_{1} \end{pmatrix}\cdot \left( \Theta\begin{pmatrix} \rho_{2}\\ p_{2} \end{pmatrix} -\Theta \begin{pmatrix} \rho_{1}\\ p_{1} \end{pmatrix}\right) \end{aligned}$$ with $\Theta\begin{pmatrix} \rho\\ p \end{pmatrix} =\begin{pmatrix} - \mathcal{H}(x,\rho, p)\\ \rho \mathcal{H}_{p}(x,\rho, p) \end{pmatrix}$. Therefore, the positivity of $E$ is guaranteed if and only if $\Theta$ monotone which is equivalent to [\[Ourassumption\]](#Ourassumption){reference-type="eqref" reference="Ourassumption"}. The preceding monotonicity condition aligns with the condition [\[uniq\]](#uniq){reference-type="eqref" reference="uniq"}, which is also the Lasry--Lions monotonicity condition used by P.L. Lions in his work on solution uniqueness, as demonstrated in [@Lions2023]. In addition, assuming that all the following differentiations are allowed, we see that the Hessian $$\mathbf{H}_{c}=\begin{pmatrix} -\frac{2}{\rho}\mathcal{H}_{\rho}(x,\rho^{\sigma}, p)& \mathcal{H}_{\rho,p}^{T}( x,\rho^{\sigma}, p )\\ \mathcal{H}_{\rho,p}( x,\rho^{\sigma}, p )& 2\mathcal{H}_{p,p}( x,\rho^{\sigma}, p )\\ \end{pmatrix}$$ and $\mathcal{H}_{\rho}(x,\rho, p ) =- \frac{\beta}{2} \left(\rho_{m}-\rho\right)^{\beta-1}p^2 + \frac{2-\beta}{2}\left(\rho_{m}-\rho \right)^{1-\beta}$, $\mathcal{H}_{p}(x,\rho, p ) = \left(\rho_{m}-\rho\right)^{\beta}p$, $\mathcal{H}_{\rho,p}(x,\rho, p ) =-\beta \left(\rho_{m}-\rho\right)^{\beta-1}p$ and $\mathcal{H}_{p,p}(x,\rho,p ) = \left(\rho_{m}-\rho\right)^{\beta}$. For $\rho>0$, the matrix $\mathbf{H}_{c}$ is given by $$\mathbf{H}_{c}=\begin{pmatrix} -\frac{2}{\rho}\left( - \frac{\beta}{2} \left(\rho_{m}-\rho\right)^{\beta-1}p^2 + \frac{2-\beta}{2}\left(\rho_{m}-\rho \right)^{1-\beta} \right)& -\beta \left(\rho_{m}-\rho\right)^{\beta-1}p\\ -\beta \left(\rho_{m}-\rho\right)^{\beta-1}p)& 2 \left(\rho_{m}-\rho\right)^{\beta}\ \end{pmatrix}.$$ Taking $z=(z_{1},z_{2}) \in \mathbb{R}^{2}$. Therefore, the matrix $\mathbf{H}_{c}$ is positive semi-definite if and only if $$z^{T}\mathbf{H}_{c}z = -\frac{2}{\rho}\left( - \frac{\beta}{2} \left(\rho_{m}-\rho\right)^{\beta-1}p^2 + \frac{2-\beta}{2}\left(\rho_{m}-\rho \right)^{1-\beta} \right) z_{1}^{2} -2\beta \left(\rho_{m}-\rho\right)^{\beta-1}p z_{1}z_{2} +2 \left(\rho_{m}-\rho\right)^{\beta}z_{2}^{2} \geqslant 0.$$ In particular, when we consider the case of $\beta =2$, we have $$\begin{aligned} z^{T}\mathbf{H}_{c} z = &\frac{2}{\rho}\left( \left(\rho_{m}-\rho\right)p^2 \right) z_{1}^{2} -4 \left(\rho_{m}-\rho\right)p z_{1}z_{2} +2 \left(\rho_{m}-\rho\right)^{2}z_{2}^{2} \\ = &\frac{2}{\rho} \left(\rho_{m}-\rho\right)p^2 z_{1}^{2} -2p^{2} z_{1}^{2} +2\left(p z_{1} -\left(\rho_{m}-\rho\right)z_{2}\right)^{2}\\ = &2p^{2} z_{1}^{2}\left(\frac{\left(\rho_{m}-\rho\right)}{\rho} -1\right) +2\left(p z_{1} -\left(\rho_{m}-\rho\right)z_{2}\right)^{2} \end{aligned}$$ Consequently, the assumption $0 <\rho \leqslant\frac{\rho_{m}}{2}$ is a sufficient condition to guarantee that the matrix $\mathbf{H}_{c}$ is positive semi-definite. Now, when we consider the case of $\beta =0$, we have $$z^{T}\mathbf{H}_{c} z = -\frac{2}{\rho}\left(\rho_{m}-\rho \right) z_{1}^{2} +2 z_{2}^{2},$$ which is not positive for all $z \in \mathbb{R}^{2}$. Finally, we conclude with the desired uniqueness results. ◻ ****Remark** 11**. *The Larsy-Lions monotonicity condition suggests that pedestrians tend to avoid densely populated areas and prefer to be more spread out. This is evident in our model when $\beta = 2$, as pedestrians actively avoid high-density regions. The condition $0 < \rho \leqslant\frac{\rho_{m}}{2}$ is essential for the uniqueness of the solution, which is logical as people tend to distribute themselves more evenly when an area is close to half its capacity. This condition not only ensures that the solution is unique, but also that it is stable against perturbations, as discussed in [@GMP2023preprint].* ## Vanishing viscosity limit In this section, we will analyze the vanishing viscosity limit of weak solutions. Here, we will utilize the connection between the weak sub-solution for the HJB equation and the weak solution for the Kolmogorov-type equation. The analysis in this section was inspired by [@achdou2020introduction Section 1.3.7, p.83] **Definition 12**. *Under the assumptions , Given $h\in L^{1}(\mathrm{Q}_{T})$, and $\rho\in L^{1}(\mathrm{Q}_{T})$. A function $\phi \in L^{\infty}(0,T;L^{1}(\Omega)) \cap L^{q}(0,T;W^{1,q}(\Omega))$ for every $q<\frac{4}{3}$, is a weak sub-solution of $$\begin{aligned} -\partial_t\phi - \mathcal{H}(x,\rho,\nabla \phi) =h, & \label{HJBEq0}\\ \phi(t=T,x)= \phi_{T}(x),& \label{HJBEqb}\end{aligned}$$ if it satisfies $$\begin{aligned} \int_{0}^{T}\int_{\Omega} \phi \partial_{t}\varphi dx dt + \int_{0}^{T}\int_{\Omega} \mathcal{H}(x,\rho,\nabla \phi) \varphi dx dt \leqslant& \int_{0}^{T}\int_{\Omega} h \varphi dx dt +\int_{\Omega} \phi_{T}(x)\varphi(T) dx \\& \forall \varphi \in C^{1}_{c}((0,T]\times \Omega),\,\,\, \varphi \geqslant 0 \end{aligned}$$ We will shortly denotes $-\partial_t\phi + \mathcal{H}(x,\rho,\nabla \phi) \leqslant h$ and $\phi(T) \leqslant\phi_{T}$, the definition of a weak sub-solution.* Now we state the existence of a weak solution for the Kolmogorov-type equation. We denote by $\mathcal{P}(\Omega)$ the set of Borel probability measures on $\Omega$. **Definition 13**. *Let $\rho_{0} \in \mathcal{P}(\Omega)$, Given a measurable vector field $b : \mathrm{Q}_{T} \longrightarrow \mathbb{R}$ , a function $\rho \in L^{1}(\mathrm{Q}_{T})$, is a weak solution of $$\label{contEq0} \begin{cases} \partial_t \rho - div(\rho b) =0, & \\\ \rho(t=0,x)= \rho_{0}(x),& \end{cases}$$ if $\rho \in C^{0}([0,T]; \mathcal{P}(\Omega))$, $\int_{0}^{T}\int_{\Omega}\rho|b|^{2} dx dt <\infty$ and we have $$\label{WeakfomulationA} -\int_{0}^{T}\int_{\Omega}\rho \partial_{t}\varphi dx dt + \int_{0}^{T}\int_{\Omega} \rho b\cdot \nabla\varphi dx dt = \int_{\Omega}\rho_{0}\varphi(0) \quad \forall \varphi \in C^{1}_{c}([0,T)\times \Omega).$$* Now, we give a definition of the weak solution for the first-order MFGs system. **Definition 14**. *Under the assumptions , a pair $(\rho, \phi) \in L^{1}(\mathrm{Q}_{T})\times L^{\infty}(0,T;L^{1}(\Omega))$ is a weak solution to the following first-order MFG system $$\begin{aligned} \partial_t\rho - \nabla \cdot \left( \rho \mathcal{H}_{p}(x, \rho,\nabla \phi)\right) = 0, &\qquad (t,x)\in\mathrm{Q}_{T} \label{EqFluidHd0sb}\\ \rho(t=0,x)= \rho_{0}(x),&\quad x\in\Omega \label{FluidInsb}\end{aligned}$$ $$\begin{aligned} -\partial_t\phi + \mathcal{H}(x, \rho,\nabla \phi ) =0, &\qquad (t,x)\in\mathrm{Q}_{T} \label{EqEikonalHd0sb}\\ \phi(t=T,x)= \phi_{T}(x),&\quad x\in\Omega \label{EikonalInsb}\end{aligned}$$ if* (i) : *$\phi_{T}\rho(T) \in L^{1}(\mathrm{Q}_{T})$, $\rho f^{\beta}(\rho) |\nabla \phi|^{2} \in L^{1}(\mathrm{Q}_{T})$ and $f^{\beta}(\rho) |\nabla \phi|^{2} \in L^{1}(\mathrm{Q}_{T})$* (ii) : *$\phi$ is a weak sub-solution of [\[EqEikonalHd0sb\]](#EqEikonalHd0sb){reference-type="eqref" reference="EqEikonalHd0sb"}-[\[EikonalInsb\]](#EikonalInsb){reference-type="eqref" reference="EikonalInsb"}, $\rho \in C^{0}(o,T;\mathcal{P}(\Omega))$ is a weak solution of [\[EqFluidHd0sb\]](#EqFluidHd0sb){reference-type="eqref" reference="EqFluidHd0sb"}-[\[FluidInsb\]](#FluidInsb){reference-type="eqref" reference="FluidInsb"}.* (iii) : *$\phi$ and $\rho$ satisfy the following identity $$\label{WeakfomulationF} \int_{\Omega} \rho_{0}\phi(0) dx = \int_{\Omega} \rho(T)\phi_{T} dx + \int_{0}^{T}\int_{\Omega} \rho \left( \mathcal{H}_{p}(x, \rho ,\nabla \phi)\cdot \nabla \phi -\mathcal{H}(x, \rho,\nabla \phi )\right) dx dt ,$$* **Lemma 15**. *Under assumptions , let $\phi$ be a weak subsolution of [\[HJBEq0\]](#HJBEq0){reference-type="eqref" reference="HJBEq0"}-[\[HJBEqb\]](#HJBEqb){reference-type="eqref" reference="HJBEqb"}and $\rho$ be a weak solution of [\[contEq0\]](#contEq0){reference-type="eqref" reference="contEq0"}. Then, we have $\rho |\nabla \phi |^{2} \in L^{1}(\mathrm{Q}_{T})$ and $\rho(0)\phi(0) \in L^{1}(\Omega)$ and $$\label{EsR} \int_{\Omega} \rho_{0}\phi(0) dx \leqslant\int_{\Omega} \phi_{T}\rho(T) dx + \int_{\Omega}\rho \left(-\nabla \phi \cdot b + \mathcal{H}(x, \rho ,\nabla \phi)\right) dx dt.$$* *Proof.* Let $m_{\delta}(.)$ be s sequence of standard symmetric mollifiers in $\mathbb{R}^{2}$ and we set $\rho_{\delta}(t,x)= \rho(t,.)\star m_{\delta}$. We consider a sequence of $1-d$ mollifiers $\xi_{\epsilon}(t)$ such that $\text{supp} (\xi_{\epsilon}) \subset (-\epsilon,0)$, and we set $$\rho_{\delta,\epsilon} := \int_{0}^{T}\xi_{\epsilon}(s-t)\rho_{\delta}(s) ds.$$ We have $$\label{EQS} \int_{0}^{T}\int_{\Omega}\phi \partial_{t}\rho_{\delta,\epsilon} dx dt + \int_{0}^{T}\int_{\Omega}\mathcal{H}(x, \rho,\nabla \phi ) \rho_{\delta,\epsilon} dx dt \leqslant\int_{\Omega}\phi_{T}\rho_{\delta,\epsilon} dx.$$ Let ${\phi_{\delta,\epsilon}(s,y)= \int_{0}^{T}\int_{\Omega} \phi_{\delta,\epsilon}\xi_{\epsilon}(s-t) m_{\delta}(x-y) dx dt }$, then $$\int_{0}^{T}\int_{\Omega}\phi^{\sigma}\partial_{t}\tilde{\rho}_{\delta,\epsilon} dx dt = \int_{0}^{T}\int_{\Omega} \left(-\partial_{s}\phi_{\delta,\epsilon} \right) \tilde{\rho}^{\sigma}_{\delta,\epsilon} (s,y)dy ds.$$ We observe that $\phi_{\delta,\epsilon}$ vanishes near $s=T$, so it can be used as a test function in [\[WeakfomulationA\]](#WeakfomulationA){reference-type="eqref" reference="WeakfomulationA"}. It follows that $$\begin{aligned} \int_{0}^{T}\int_{\Omega}&\phi^{\sigma}\partial_{t}\tilde{\rho}_{\delta,\epsilon} dx dt \\ &= -\int_{0}^{T}\int_{\Omega} \partial_{s}\phi_{\delta,\epsilon} \tilde{\rho}^{\sigma}_{\delta,\epsilon} (s,y)dy ds\\& =- \int_{0}^{T}\int_{\Omega} \rho(s,y) b(s,y)\cdot \nabla\phi_{\delta,\epsilon} dx dt + \int_{\Omega}\rho_{0}(y)\phi_{\delta,\epsilon}(0) \end{aligned}$$ Now we shift the convolution kernels from $\phi$ to $\rho$ and we use it in [\[EQS\]](#EQS){reference-type="eqref" reference="EQS"}, we get $$\label{QA} \begin{aligned} -\int_{0}^{T}\int_{\Omega}\nabla \phi \cdot w_{\delta,\epsilon} dx dt &+ \int_{0}^{T}\int_{\Omega} \mathcal{H}(x, \rho ,\nabla \phi)\rho_{\delta,\epsilon} dx dt + \int_{\Omega} \rho_{0}\star m_{\delta}\left(\int_{0}^{T}\phi(t)\xi_{\epsilon}(-t) dt\right) dx \\& \leqslant\int_{\Omega}\phi_{T}\rho_{\delta,\epsilon}(T) dx \end{aligned}$$ where we denote $w_{\delta}= [(b\rho)\star m_{\delta}]$ and ${w_{\delta,\epsilon}(s,y)= \int_{0}^{T}w_{\delta}(x)\xi_{\epsilon}(s-t) dt }$. Since $\phi$ is bounded below, we have $$\liminf_{\delta \to 0}\liminf_{\epsilon \to 0} \int_{\Omega}\rho_{0} \star m_{\delta} \left(\phi(t) \xi_{\epsilon}(-t) dx \right) dx \geqslant\int_{\Omega} \rho_{0}\phi(0) dx$$ Moreover we have at $t=T$ $$\label{EQestimate0} \lim_{\epsilon \to 0}\int_{\Omega} \phi_{T}\rho_{\delta,\epsilon}(T) dx =\int_{\Omega} \phi_{T}\rho_{\delta}(T)\star m_{\delta} dx$$ Then, we see that $\phi_{T}$ is bounded below, and then we deduce $|\phi_{T}\rho_{\delta}(T)\star m_{\delta}|$ is dominated in $L^{1}(\Omega)$. Therefore, we have $$\label{EQestimate} \lim_{\delta \to 0}\lim_{\epsilon \to 0} \int_{\Omega}\phi_{T}\rho_{\delta,\epsilon}(T) dx =\int_{\Omega} \phi_{T}\rho(T) dx$$ We know that $$\rho_{\delta,\epsilon}\mathcal{H}(x, \rho ,\nabla \phi) = \rho_{\delta,\epsilon}\left(\frac{1}{2}f^{\beta}(\rho)|\nabla \phi|^{2} -\frac{1}{2}f^{2-\beta}(\rho) \right)$$ and by Young inequality we get $$\nabla \phi \cdot w_{\delta,\epsilon} \leqslant\frac{1}{2}\rho_{\delta,\epsilon} |\nabla \phi |^{2} + \frac{1}{2}\frac{|w_{\delta,\epsilon}|^{2}}{\rho_{\delta,\epsilon}},$$ then $$-\nabla \phi \cdot w_{\delta,\epsilon} + \mathcal{H}(x, \rho ,\nabla \phi) \rho_{\delta,\epsilon} \geqslant\frac{1}{2}\rho_{\delta,\epsilon}f^{\beta}(\rho) |\nabla \phi |^{2} - \frac{1}{2}\rho_{\delta,\epsilon}f^{2-\beta}(\rho) - \frac{1}{2}\rho_{\delta,\epsilon} |\nabla \phi |^{2} - \frac{1}{2}\frac{|w_{\delta,\epsilon}|^{2}}{\rho_{\delta,\epsilon}}$$ Now we introduce the lower semi-continuous function $\Phi$ on $\mathbb{R}^{2}\times\mathbb{R}$ by $$\label{functionPsi} \Phi(w,\rho) = \begin{cases} \frac{|w|^{2}}{\rho} & \text{if}\,\, \rho>0 \\ 0 & \text{if}\,\, \rho =0 \,\, \text{and }\, w=0 \\ +\infty \,\,\,\,\,\,\quad \text{otherwise.} \end{cases}$$ We observe that $\Phi$ is convex in the couple $(w,\rho)$. Then ,by Jensen inequality we have ${\frac{|w_{\delta,\epsilon}|^{2}}{\rho_{\delta,\epsilon}} \leqslant\frac{|w|^{2}}{\rho}} \star m_{\delta,\epsilon}$ ( recalling that $w=b\rho$ hence $\frac{|w|^{2}}{\rho}= \rho|b|^{2}$), the we deduce that $$\label{QAs} -\nabla \phi \cdot w_{\delta,\epsilon} + \mathcal{H}(x, \rho ,\nabla \phi) \rho_{\delta,\epsilon} \geqslant\frac{1}{2}\rho_{\delta,\epsilon}f^{\beta}(\rho) |\nabla \phi |^{2} - \frac{1}{2}\rho_{\delta,\epsilon}f^{2-\beta}(\rho) - \frac{1}{2}\rho_{\delta,\epsilon} |\nabla \phi |^{2} - \frac{1}{2}\left(\rho|b|^{2}\star m_{\delta,\epsilon}\right)$$ From [\[QA\]](#QA){reference-type="eqref" reference="QA"}-[\[EQestimate\]](#EQestimate){reference-type="eqref" reference="EQestimate"} we will use Fatou's Lemma we get $$\label{QA} \begin{aligned} \liminf_{\delta \to 0}\liminf_{\epsilon \to 0} & \int_{\Omega}\left(-\nabla \phi \cdot w_{\delta,\epsilon} + \mathcal{H}(x, \rho ,\nabla \phi)\rho_{\delta,\epsilon}\right) dx dt \\& \geqslant\int_{\Omega}\rho \left(-\nabla \phi \cdot b + \mathcal{H}(x, \rho ,\nabla \phi)\right) dx dt \end{aligned}$$ We deduce from [\[QAs\]](#QAs){reference-type="eqref" reference="QAs"} that $\rho |\nabla \phi |^{2} \in L^{1}(\mathrm{Q}_{T})$. Then, we obtain [\[EsR\]](#EsR){reference-type="eqref" reference="EsR"} by combining [\[EQestimate0\]](#EQestimate0){reference-type="eqref" reference="EQestimate0"} , [\[EQestimate\]](#EQestimate){reference-type="eqref" reference="EQestimate"} and [\[QA\]](#QA){reference-type="eqref" reference="QA"}. ◻ **Theorem 16**. *Under the assumptions , let $\left(\rho^{\sigma}, \phi^{\sigma}\right)$ be a weak solution of [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}. Then, there exists a subsequence and a couple $\left(\rho, \phi\right) \in L^{1}(\mathrm{Q}_{T})\times L^{\infty}(0,T;L^{1}(\Omega))$ such that $\left(\rho^{\sigma}, \phi^{\sigma}\right) \to \left(\rho, \phi\right)$ in $L^{1}(\mathrm{Q}_{T})$ and $\left(\rho, \phi\right)$ is a weak solution to [\[EqFluidHd0sb\]](#EqFluidHd0sb){reference-type="eqref" reference="EqFluidHd0sb"}-[\[EikonalInsb\]](#EikonalInsb){reference-type="eqref" reference="EikonalInsb"} in the sense of Definiton [Definition 14](#WeakSubSolSystem){reference-type="ref" reference="WeakSubSolSystem"}. Moreover, under the condition [\[Ourassumption\]](#Ourassumption){reference-type="eqref" reference="Ourassumption"}, system [\[EqFluidHd0sb\]](#EqFluidHd0sb){reference-type="eqref" reference="EqFluidHd0sb"}-[\[EikonalInsb\]](#EikonalInsb){reference-type="eqref" reference="EikonalInsb"} admits a unique weak solution. In particular,* 1. *If $\beta=2$ and ${\rho \leqslant\frac{\rho_{m}}{2}}$, there exists a unique weak solution of the system [\[EqFluidHd0sb\]](#EqFluidHd0sb){reference-type="eqref" reference="EqFluidHd0sb"}-[\[EikonalInsb\]](#EikonalInsb){reference-type="eqref" reference="EikonalInsb"}.* 2. *If $\beta=0$, the monotonicity condition [\[Ourassumption\]](#Ourassumption){reference-type="eqref" reference="Ourassumption"} is not fulfilled.* *Proof.* Since, we have $p \longmapsto \mathcal{H}(x, \rho,p)$ is convex, then by weak lower semi-continuity we deduce that $\phi$ verifies $$\begin{aligned} -\partial_t\phi + \mathcal{H}(x, \rho,\nabla \phi ) \leqslant 0, & \label{EqEikonalHd0sbx}\\ \phi(T)\leqslant\phi_{T}(x).& \label{EikonalInsbx}\end{aligned}$$ By Theorem [Theorem 5](#Thmlim){reference-type="ref" reference="Thmlim"}, we have $\rho^{\sigma}\mathcal{H}_{p}(x, \rho^{\sigma},\nabla \phi^{\sigma})$ is equi-integrable and therefore weakly converges to $w$ in $L^{1}(\mathrm{Q}_{T})$. From the definition of function $\Phi$ [\[functionPsi\]](#functionPsi){reference-type="eqref" reference="functionPsi"} Then, we have $$\int_{0}^{T}\int_{\Omega}\Phi(w,\rho) dx dt \leqslant\liminf_{\sigma \to 0}\int_{0}^{T}\int_{\Omega}\Phi(w,\rho) dx dt = \liminf_{\sigma \to 0}\int_{0}^{T}\int_{\Omega} \rho^{\sigma}|\mathcal{H}_{p}(x, \rho^{\sigma},\nabla \phi^{\sigma})|^{2}dx dt \leqslant C,$$ hence, we have $\Phi(w,\rho) \in L^{1}(\mathrm{Q}_{T})$. In particular we can set $b:= \frac{w}{\rho}1_{\rho>0}$, then $\rho$ is a weak solution of $$\begin{aligned} \partial_t\rho + div(b\rho)=0, &\qquad (t,x)\in\mathrm{Q}_{T} \label{KOlm1}\\ \rho(t=0,x) =\rho_{0}(x),&\quad x\in\Omega, \label{kolm2}\end{aligned}$$ with $\rho|b|^{2}\in L^{1}(\mathrm{Q}_{T})$. Collecting all the above properties, we get the following inequality. $$\label{Myeq} \int_{\Omega}\rho_{0}\phi(0) dx \leqslant\int_{\Omega}\phi_{T}\rho(T) dx + \int_{0}^{T}\int_{\Omega} \rho \left( b\cdot \nabla \phi - \mathcal{H}(x, \rho,\nabla \phi)\right)$$ From system [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}, we get $$\int_{\Omega} \phi_{T}(x)\rho^{\sigma}(T) dx + \int_{0}^{T}\rho^{\sigma} \left( \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma} )\cdot \nabla \phi^{\sigma} -\mathcal{H}(x, \rho^{\sigma},\nabla \phi^{\sigma} )\right) = \int_{\Omega} \phi^{\sigma}(0)\rho_{0} dx$$ We observe that $\phi^{\sigma}(0)$ is equi-integrable. In addition, let $k>0$, $\left(\rho^{\sigma} -k \right)_{+}$ be a sub-solution of the HJB-type equation, then $$\begin{aligned} \int_{\Omega} \left( \phi^{\sigma}(0) - k\right)_{+} dx &+ \int_{0}^{T}\int_{\Omega}\mathcal{H}(x, \rho^{\sigma} ,\nabla \phi^{\sigma} ) 1_{\phi^{\sigma}>k}\\& \leqslant\int_{\Omega} \left( \phi_{T} - k \right)_{+} dx \end{aligned}$$ which implies from the integrability of $\phi_{T}$ that $\int_{\Omega} \left( \phi^{\sigma}(0) - k\right)_{+} dx \to 0$ as $k \to +\infty$ uniformly with respect to $\sigma$. Then, $\phi^{\sigma}(0)$ is weakly converges in $L^{1}(\mathrm{Q}_{T})$ to some function $\tilde{\phi}_{0}$. Therefore we pass to the limit in HJB-type equation [\[EqEikonalHd0x\]](#EqEikonalHd0x){reference-type="eqref" reference="EqEikonalHd0x"} we obtain $$\int_{\Omega}\varphi(0)\tilde{\phi}_{0} dx + \int_{0}^{T}\int_{\Omega} \phi \partial_{t}\varphi dx + \int_{0}^{T}\int_{\Omega} \mathcal{H}(x, \rho ,\nabla \phi)\varphi \leqslant\int_{\Omega}\phi_{T}\varphi(T), \quad \forall \varphi \in C^{1}(\overline{\mathrm{Q}_{T}}), \,\, \varphi \geqslant 0.$$ Then, taking a sequence $\varphi_{i}$ such that $\varphi_{i}(0)=1$ and $\varphi_{i}$ approximate the Dirac mass at $t=0$ we deduce that $\tilde{\phi}_{0} \leqslant\phi(0)$ where $\phi(0)$ is the trace of $\phi$ at time $t=0$. $$\begin{aligned} \int_{\Omega}\phi_{T}\rho^{\sigma}(T) dx &+ \int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left( \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})\cdot \nabla \phi^{\sigma} -\mathcal{H}(x, \rho^{\sigma} ,\nabla \phi^{\sigma}) \right)dx dt \\ & \int_{\Omega}\rho_{0}\tilde{\phi}_{0} dx \leqslant\int_{\Omega}\rho_{0}\phi(0)dx \end{aligned}$$ Now, using [\[Myeq\]](#Myeq){reference-type="eqref" reference="Myeq"}, we get $$\label{bound0} \begin{aligned} \limsup_{\sigma \to 0} &\Big (\int_{\Omega}\phi_{T}\rho^{\sigma}(T) dx + \int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left( \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})\cdot\nabla \phi^{\sigma} - \mathcal{H}(x, \rho^{\sigma} ,\nabla \phi^{\sigma}) \right)dx dt \Big)\\ & \leqslant\int_{\Omega}\phi_{T}\rho(T) dx + \int_{0}^{T}\int_{\Omega} \rho \left( b\cdot \nabla\phi - \mathcal{H}(x, \rho ,\nabla \phi) \right) dx dt \end{aligned}$$ We denote $w^{\sigma}:= \rho^{\sigma} \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})$ where $w$ its weak limit in $L^{1}(\mathrm{Q}_{T})$. Moreover, we have the following $$\begin{aligned} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}&\left( \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})\cdot\nabla \phi^{\sigma} - \mathcal{H}(x, \rho^{\sigma} ,\nabla \phi^{\sigma}) \right)dx dt \\&= \int_{0}^{T}\int_{\Omega}\left(\frac{1}{2}\rho^{\sigma} f^{\beta}(\rho^{\sigma})|\nabla \phi^{\sigma}|^{2} + \frac{1}{2} \rho^{\sigma} f^{2-\beta}(\rho^{\sigma})\right) dx dt \\&= \int_{0}^{T}\int_{\Omega} \rho^{\sigma} \mathcal{H}^{\ast}(x, \rho^{\sigma} ,\mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})) dx dt \\&= \int_{0}^{T}\int_{\Omega} \rho^{\sigma} \mathcal{H}^{\ast}(x, \rho^{\sigma} ,\frac{w^{\sigma}}{\rho^{\sigma}}) dx dt, \end{aligned}$$ where $\mathcal{H}^{\ast}$ (or $\mathcal{L}$) is the Legendre transform of $\mathcal{H}$ as defined in [\[EqHamil\]](#EqHamil){reference-type="eqref" reference="EqHamil"}-[\[EqHamilY\]](#EqHamilY){reference-type="eqref" reference="EqHamilY"}. From Fatou's lemma, we deduce $$\label{bound} \begin{aligned} \liminf_{\sigma \to 0} &\int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left( \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})\cdot\nabla \phi^{\sigma} - \mathcal{H}(x, \rho^{\sigma} ,\nabla \phi^{\sigma}) \right)dx dt \\ &=\liminf_{\sigma \to 0} \int_{0}^{T}\int_{\Omega} \left(\frac{1}{2}\rho^{\sigma} f^{\beta}(\rho^{\sigma})|\nabla \phi^{\sigma}|^{2} + \frac{1}{2} \rho^{\sigma} f^{2-\beta}(\rho^{\sigma})\right) dx dt \\& \geqslant\int_{0}^{T}\int_{\Omega}\left(\frac{1}{2}\rho f^{\beta}(\rho)|\nabla \phi|^{2} + \frac{1}{2} \rho f^{2-\beta}(\rho)\right) dx dt =\int_{0}^{T}\int_{\Omega} \rho \mathcal{H}^{\ast}(x, \rho ,\frac{w}{\rho}) dx dt . \end{aligned}$$ We have $$\int_{0}^{T} \int_{\Omega} \rho\left(b\cdot\nabla \phi - \mathcal{H}(x, \rho ,\nabla \phi) \right) dx dt \leqslant\liminf_{\sigma \to 0} \int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left( \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})\cdot\nabla \phi^{\sigma} - \mathcal{H}(x, \rho^{\sigma} ,\nabla \phi^{\sigma}) \right)dx dt,$$ then we can deduce that $$\int_{0}^{T} \int_{\Omega} \rho\left(b\cdot\nabla \phi - \mathcal{H}(x, \rho ,\nabla \phi) \right) dx dt - \int_{0}^{T}\int_{\Omega} \rho \mathcal{H}^{\ast}(x, \rho ,\frac{w}{\rho}) dx dt \leqslant 0.$$ Therefore, from [\[bound0\]](#bound0){reference-type="ref" reference="bound0"} and [\[bound\]](#bound){reference-type="ref" reference="bound"} we obtain $$\label{Eqss} \begin{aligned} \limsup_{\sigma \to 0}&\int_{\Omega}\phi_{T}\rho^{\sigma}(T) dx \leqslant\int_{\Omega}\phi_{T}\rho(T) dx + \\& \int_{0}^{T} \int_{\Omega} \rho\left(b\cdot\nabla \phi - \mathcal{H}(x, \rho ,\nabla \phi) \right) dx dt - \int_{0}^{T}\int_{\Omega} \rho \mathcal{H}^{\ast}(x, \rho ,\frac{w}{\rho}) dx dt \\&\leqslant\int_{\Omega}\phi_{T}\rho(T) dx, \end{aligned}$$ it follows that the limit of the right-hand side and the left-hand side coincides, then we conclude $$\int_{0}^{T} \int_{\Omega} \rho\left(\frac{w}{\rho}\cdot\nabla \phi - \mathcal{H}(x, \rho ,\nabla \phi) - \mathcal{H}^{\ast}(x, \rho ,\frac{w}{\rho}) \right) dx dt =0,$$ We have $$\begin{aligned} \rho&\left(\frac{w}{\rho}\cdot\nabla \phi - \mathcal{H}(x, \rho ,\nabla \phi) - \mathcal{H}^{\ast}(x, \rho ,\frac{w}{\rho}) \right) \\&=\left(w\cdot\nabla \phi - \frac{1}{2}\rho f^{\beta}(\rho)|\nabla \phi|^{2} + \frac{1}{2} \rho f^{2-\beta}(\rho) - \frac{1}{2}\rho f^{\beta}(\rho)|\nabla \phi|^{2} - \frac{1}{2} \rho f^{2-\beta}(\rho)\right)\\&=\left(w\cdot\nabla \phi -\rho f^{\beta}(\rho)|\nabla \phi|^{2} \right), \end{aligned}$$ then $$\int_{0}^{T} \int_{\Omega} \left( w\cdot\nabla \phi -\rho f^{\beta}(\rho)|\nabla \phi|^{2} \right) dx dt =0,$$ which implies that $w=\rho f^{\beta}(\rho)\nabla \phi =\rho \mathcal{H}_{p}(x, \rho ,\nabla \phi)$. Therefore, we have [\[bound\]](#bound){reference-type="eqref" reference="bound"} holds when $w=\rho \mathcal{H}_{p}(x, \rho ,\nabla \phi)$. then ,we have $$\label{boundw} \begin{aligned} \int_{\Omega}\phi_{T}\rho(T) dx &+ \int_{0}^{T}\int_{\Omega}\rho\left( \mathcal{H}_{p}(x, \rho ,\nabla \phi)\cdot\nabla \phi - \mathcal{H}(x, \rho ,\nabla \phi) \right)dx dt \\ & \leqslant\liminf_{\sigma \to 0} \Big (\int_{\Omega}\phi_{T}\rho^{\sigma}(T) dx + \int_{0}^{T}\int_{\Omega} \rho^{\sigma}\left( \mathcal{H}_{p}(x, \rho^{\sigma} ,\nabla \phi^{\sigma})\cdot\nabla \phi^{\sigma} - \mathcal{H}(x, \rho^{\sigma} ,\nabla \phi^{\sigma}) \right)dx dt \Big)\\ & \leqslant\int_{\Omega}\phi(T)\rho(T) dx. \end{aligned}$$ Combining the previous inequality with [\[Myeq\]](#Myeq){reference-type="eqref" reference="Myeq"} its yields the energy equality [\[WeakfomulationF\]](#WeakfomulationF){reference-type="eqref" reference="WeakfomulationF"} and we deduce that $(\rho,\phi)$ is a weak solution of the GH system [\[EqFluidHd0sb\]](#EqFluidHd0sb){reference-type="eqref" reference="EqFluidHd0sb"}[\[EikonalInsb\]](#EikonalInsb){reference-type="eqref" reference="EikonalInsb"} in the sense of Definition [Definition 14](#WeakSubSolSystem){reference-type="ref" reference="WeakSubSolSystem"}. Finally, the proof of the uniqueness results follows from Proposition [Proposition 10](#propoexun){reference-type="ref" reference="propoexun"}. ◻ ****Remark** 17**. - *We have performed the analysis above with a specified terminal condition, namely the boundary conditions [\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}, which represent a special case of the more general payoff problem. This more general problem employs the following expression: $$\phi^{\sigma}(t=T,x)= G(x,\rho^{\sigma}(T)),\quad x\in\Omega,$$ where $G$ is known as the payoff function. In this context, the Young measures serve as the main tools for proving the existence of the solution, as discussed in[@achdou2018mean; @achdou2020introduction].* - *We can show the existence and uniqueness of classical solutions to the system [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"} under the monotonicity condition by using the results of S. Munoz [@munoz2022classical; @munoz2023classical]. In fact, we assume that the initial data $\rho_{0}$ is in $C^{4}(\Omega)$ and $\rho_{0} \in(0,\rho_{m})$. Then, there exists a unique classical solution $(\rho,\phi)\in C^{3,\alpha}(\bar{\mathrm{Q}_{T}}) \times C^{2,\alpha}(\bar{\mathrm{Q}_{T}})$ to [\[EqFluidHd0x\]](#EqFluidHd0x){reference-type="eqref" reference="EqFluidHd0x"}-[\[EikonalInx\]](#EikonalInx){reference-type="eqref" reference="EikonalInx"}, for $0<\alpha<1$.* # Numerical experiments {#sec6} This section is devoted to the numerical experiments for solving our quasi-stationary fluid [\[QuasiStat\]](#QuasiStat){reference-type="eqref" reference="QuasiStat"} by FEM. The computations have been performed with the FreeFem++ library[^3]. It is important to highlight that the presence of viscosity within the model plays a pivotal role in enhancing the numerical stability of our computational scheme. Without viscosity, we would be dealing with a nonlinear hyperbolic system, which falls outside the scope of FreeFem++'s capabilities for solving such complex systems effectively. For the numerical example, we consider the following domain $\Omega= [0,L]\times[0,R]$ where $L=4$ and $R=2$. A periodic boundary condition for the density $\rho$ in the $x-$direction, that is, $\rho(t,0,y)=\rho(t,L,y)$ for all $y\in [0,R]$, and the y direction we consider the Neumann boundary condition. The viscosity $\sigma \geqslant 0$ is sufficiently small, $\sigma =0.01$. The time of horizon $T$ is sufficiently large fixed to $T=50$, with a time step of $dt = 5\times10^{-2}$. We have built triangular meshes $\mathcal{T}_{h}$ of characteristic size $h=1/40$. For initial and boundary data, we have considered an initial pedestrian density $\rho_{0}=\rho_{0i}+\rho_{0p}$, where $\rho_{0i}=0.5$ and $\rho_{0p}$ is a Gaussian distribution localized at the point $(x,y)=(1,1)$ given by $$\rho_{0p}=0.2\times\exp{(-10\times((x-1)^2 +(y-1)^2)}.$$ $\rho_{0p}$ is the perturbation of the initial data $\rho_{0i}$. We also consider a periodic speed $\nabla \phi$. We assume that $\bar{\rho}=0.5$. Our particular solution of the GH model is given by $$\left(\bar{\rho},\bar{\phi}\right)=\left(\bar{\rho},f^{1-\beta}(\bar{\rho})(L-x)\right), \quad \quad \forall \beta \in [0,2].$$ Therefore, we assume that $\bar{\rho}=0.5$, and $$\phi= 0 \qquad (x,y)\in \Sigma\qquad \phi= f^{1-\beta}(\bar{\rho})L. \qquad (x,y)\in \Gamma.$$ Analyzing the Hughes model with $\beta = 2$, as depicted in the velocity fields, see Figures [2](#fsofigbeta20){reference-type="ref" reference="fsofigbeta20"}-[4](#fsofigbeta22){reference-type="ref" reference="fsofigbeta22"}, we observe a clear trend where pedestrians actively avoid high-density regions. The numerical results, illustrated in Figure [4](#fsofigbeta22){reference-type="ref" reference="fsofigbeta22"}, offer insights into the long-time behavior of the model, proving the presence of dispersion and diffusion phenomena over time. In the scenario where $\beta = 1$, pedestrians are indifferent to the density of the regions because their objective is to reach a specific target. This situation is analogous to the case of pedestrain want to be close to *Hajar Aswad* as explained in section [2.2](#sec21){reference-type="ref" reference="sec21"}, see Figures [6](#fsofigbeta10){reference-type="ref" reference="fsofigbeta10"}-[8](#fsofigbeta12){reference-type="ref" reference="fsofigbeta12"}. However, when we set $\beta = 0$, we observe a different behavior: pedestrians tend to move toward densely populated areas, a behavior that contradicts the classical Hughes model where individuals typically steer clear of high-density regions. The formation of concentration phenomena becomes evident, as seen in Figures [10](#fsofigbeta00){reference-type="ref" reference="fsofigbeta00"}-[12](#fsofigbeta02){reference-type="ref" reference="fsofigbeta02"}. Figure [12](#fsofigbeta02){reference-type="ref" reference="fsofigbeta02"} displays results up to a horizon time of $T=13.5$. Beyond this point, we begin to notice some instability in the model, possibly arising from the concentration phenomena within the system or due to the sensitivty of numerical scheme to small viscosity values in the FreeFem++ code. In our concluding remarks, it becomes evident that our numerical simulations not only confirm the validity of the Hughes model, but also shed light on the pivotal role that the parameter $\beta$ can play as a safety measure within the model, contributing to the effective regulation of pedestrian evacuation dynamics. This dual outcome emphasizes the capacity of the model to not only replicate real-world pedestrian behavior but also to serve as a versatile tool for enhancing safety protocols in crowded environments. Finally, our work opens the door to exploration of stability/instability and long-time behavior within the MFG system. In our forthcoming paper [@GMP2023preprint], we present a comprehensive analysis of the stability of the GH model. ------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------- {#fsofigbeta20} {#fsofigbeta20} \(a\) \(b\) ------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------- {#fsofigbeta22} {#fsofigbeta22} \(a\) \(b\) ----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------- {#fsofigbeta10} {#fsofigbeta10} \(a\) \(b\) ------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------- {#fsofigbeta12} {#fsofigbeta12} \(a\) \(b\) ----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------ {#fsofigbeta00} {#fsofigbeta00} \(a\) \(b\) -------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------- {#fsofigbeta02} {#fsofigbeta02} \(a\) \(b\) ---------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------- # Acknowledgments The work of the authors is supported by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. M.G. and N.M. thank Prof. Diogo Gomes (KAUST) for discussing the problem during his visit to NYU Abu Dhabi. Special thanks are extended to Prof. Diogo Gomes (KAUST) and the entire KAUST team for their warm hospitality, greatly appreciated by M.G. Furthermore, M.G. expresses gratitude to Prof. Alessio Porretta (Università di Roma Tor Vergata) for bringing the reference [@achdou2020introduction] to our attention. [^1]: NYUAD Research Institute, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates, mg6888\@nyu.edu [^2]: Department of Mathematics, New York University in Abu Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates-- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA, nm30\@nyu.edu [^3]: https://freefem.org

arxiv_math

--- address: Department of Mathematics, I.I.T. Jodhpur, NH 65, Nagaur Road, Karwar, Jodhpur-342030, INDIA author: - Sushmita Yadav and Puneet Sharma bibliography: - xbib.bib title: On Equicontinuity and Related Notions in Nonautonomous Dynamical Systems --- # Abstract {#abstract .unnumbered} In this work, we investigate the dynamics of a general non-autonomous system generated by a commutative family of homeomorphisms. In particular, we investigate properties such as periodicity, equicontinuity, minimality and transitivity for a general non-autonomous dynamical system. In [@sk2], the authors derive necessary and sufficient conditions for a system to be minimal. We claim the result to be false and provide an example in support of our claim. Further, we correct the result to derive necessary and sufficient conditions for a non-autonomous system to be minimal. We prove that for an equicontinuous flow generated by a commutative family, while the system need not exhibit almost periodic points, if $x$ is almost periodic then every point in $\overline{\mathcal{O}_H(x)}$ is almost periodic. We further prove that in such a case, the set $\overline{\mathcal{O}_H(x)}$ is uniformly almost periodic and hence provide an analogous extension to a result known for the autonomous systems. We prove that a system generated by a commutative family is transitive if and only if it exhibits a point with dense orbit. We also prove that any minimal system generated by commutative family is either equicontinuous or has a dense set of sensitive points. # Introduction {#introduction .unnumbered} Dynamical systems have been long used to investigate various natural and physical processes around us. While mathematical investigations have enriched the literature with qualitative results determining long term behavior of such systems, the field has also found a variety of applications in areas such as complex systems, control theory, biomechanics and cognitive sciences [@beer; @Hamill; @kohmoto]. Although the theory has been used extensively in various fields, most of problems have been approximated using autonomous systems (systems with time invariant governing rule). However, as governing rule varies with time in many natual processes around us, better approximations can be obtained by allowing the governing rule to be time variant. While some investigations for such a setting in the discrete case have been made, many questions for such a setting are still unanswered. In [@sk1], the authors investigate the topological entropy when the family $\mathbb{F}$ is equicontinuous or uniformly convergent. In [@sk2], ther authors discusses minimality conditions for a non-autonomous system on a compact Hausdorff space while focussing on the case when the non-autonomous system is defined on a compact interval of the real line. In [@pm], the authors investigate a non-autonomous system generated by a finite family of continuous self maps. In the process, they study properties such as transitivity, weak mixing, topological mixing, existence of periodic points, various forms of sensitivities and Li-Yorke chaos. In [@jd], the authors establish that if $f_n\rightarrow f$, there is no relation between chaotic behavior of the non-autonomous system generated by $f_n$ and the chaotic behavior of $f$. Before we move further, we give some of the basic concepts and definitions required.\ Let $(X,d)$ be a compact metric space and let $\mathbb{F}=\{f_n: n \in \mathbb{N}\}$ be a family of homeomorphisms on $X$. For any given initial state of the system $x_0$, any such family generates a *non-autonomous* dynamical system via the relation $x_{n} = \left\{ \begin{array}{lr} {f_n(x_{n-1})} & : n\geq 1, \\ {f_n^{-1}(x_{n+1})} & : n<0. \\ \end{array} \right.$ In other words, the non-autonomous system generated by the family $\mathbb{F}$ can be visualized as orbit of $x_0$ under the ordered set $\{\ldots,f_2^{-1},f_1^{-1},I,f_1,f_2,\ldots,\}$. For a given initial state $x_0$ of the system, let $\omega_n(x_0)$ denote the state of the system at time $n$. The set $\mathcal{O}(x)=\{\omega_n(x): n\in \mathbb{Z}\}$ is called the *orbit* of any point $x$ in $X$. Further, we refer to the set $\mathcal{O}_H(x)=\{(\omega_{k_n}\circ\omega_{k_{n-1}}\circ\ldots\circ\omega_{k_1})(x): k_i\in \mathbb{Z}, n\in \mathbb{N}\}$ as the orbital hull of the point $x$. Let $\mathcal{O}_H^k(x)=\{\omega_{r_n}\omega_{r_{n-1}}...\omega_{r_2}\omega_{r_1}(x) : n\in \mathbb{N}, r_i\in\{-k,-k+1,\ldots,1,2,...,k\}\}$ denote the truncation (of order $k$) of the orbitall hull of the point $x$. It may be noted that orbital hull of a point $x$ is the smallest invariant set containing $x$.\ A point $x\in X$ is said to be *periodic* of period $n\in \mathbb{N}$ if $\omega_{nk}(x)=x$ $\forall k \in \mathbb{Z}$. A point $x\in X$ is called *almost periodic* if for any $\epsilon>0$, the set $\{n\in \mathbb{Z}: d(\omega_n(x),x)<\epsilon\}$ is syndetic. If every point $x\in X$ is *almost periodic* then $(X,\mathbb{F})$ is said to be *pointwise almost periodic*. $(X,\mathbb{F})$ is said to be *uniformly almost periodic* if for every $\epsilon>0$ there exists $M>0$ such that the set $\{n\in Z : d(\omega_n(x),x)<\epsilon \}$ is $M$-syndetic for all $x\in X$. A set $Y\subseteq X$ is said to be *invariant* if $\omega_k(Y)\subseteq Y$, for all $k\in \mathbb{Z}$. We say $(Y,\mathbb{F})$ to be a *minimal subsystem* of $(X,\mathbb{F})$ if it is a non-empty, closed, invariant subsystem of $(X,\mathbb{F})$ with no proper non-empty subset having these properties. A system $(X,\mathbb{F})$ is said to be *equicontinuous* if for each $\epsilon>0$, there exists $\delta>0$ such that $d(x, y) < \delta$ implies $d(\omega_n(x), \omega_n(y)) < \epsilon$ for all $n\in \mathbb{Z}$, $x, y \in X$. A pair $(x, y)$ is *proximal* for $(X,\mathbb{F})$ if $\liminf\limits_n ~~d(\omega_n(x), \omega_n(y)) = 0$. Let $P(X)$ denote the set of proximal pairs of system $(X,\mathbb{F})$, then $(X,\mathbb{F})$ is said to be *distal* if $P(X)=\Delta$, where $\Delta$ denotes the diagonal in the space $X\times X$. A system $(X,\mathbb{F})$ is said to be *point transitive* if there exists a point $x\in X$ such that $\overline{\mathcal{O}(x)}=X$. In this case, the point $x$ is referred as a *transitive point*. The system is said to be *topologically transitive* if for every pair of non-empty open sets $U, V$ in $X$, there exists $k\in \mathbb{Z}$ such that $\omega_k(U)\cap V \neq \emptyset$. The system $(X,\mathbb{F})$ is called $r$-transitive if the system generated by the family $\mathbb{F}_r=\{f_{(k+1)r}\circ f_{(k+1)r-1}\circ\ldots\circ f_{kr+1}:k\in\mathbb{N}\}$ is transitive. A system $(X,\mathbb{F})$ is totally transitive if it is $r$-transitive for all $r\in\mathbb{N}$. A system $(X,\mathbb{F})$ is said to be *sensitive* at a point $x$ if there exists $\delta_x>0$ such that for each neighborhood $U_x$ of $x$ there exists $k\in\mathbb{Z}$ such that $diam(\omega_k(U_x))>\delta_x$. A system $(X,\mathbb{F})$ is said to be *sensitive* if there exists $\delta>0$ such that for each $x\in X$ and each neighborhood $U$ of $x$ there exists $k\in\mathbb{Z}$ such that $diam(\omega_k(U))>\delta$. It may be noted that in case the $f_n$'s coincide, the above definitions coincide with the known notions of an autonomous dynamical system [@bc; @bs; @de]. Some basic concepts and recent works in this area can be found in literature [@kol; @of; @jd; @sk1; @sk2; @pm].\ In this paper, we investigate properties such as periodicity, equicontinuity, minimality and transitivity for a non-autonomous system generated by a commutative family of homeomorphisms. We prove that every point in the orbital hull of periodic point is periodic. We give example to show that a pair of periodic points may form a Li-Yorke pair and hence need not exhibit simple dynamical behavior. In [@sk2], the authors claim that system is minimal if and only if orbit of each point is dense in $X$ (page $84$, line $-7$). Also, the authors claim that a non-autonomous system is minimal if and only if for non-empty every open set $U$ in $X$, there exists $k\in\mathbb{N}$ such that trajectory of every point meets $U$ in at most $k$ iterations (Lemma $2.2$). We establish both the claims to be false. While we provide an example of a minimal system void of any points with dense orbit, we correct the result to derive necessary and sufficient conditions for a non-autonomous system to be minimal. We prove that a non-autonomous system is minimal if and only if for non-empty every open set $U$ in $X$, there exists $k\in\mathbb{N}$ such that $\mathcal{O}_H^k(x)$ meets $U$ for every point $x\in X$. We prove that for equicontinuous systems, if $x$ is almost periodic then every point in $\overline{\mathcal{O}_H(x)}$ is almost periodic. In such a setting, we establish $\overline{\mathcal{O}_H(x)}$ to be uniformly almost periodic. We prove that a system is transitive if and only if it exhibits points with dense orbit. We prove that while minimal systems need not be transitive, an equicontinuous transitive system is necessarily minimal. We also prove that a minimal system is either equicontinuous or has a dense set of points of sensitivity. # Main Results {#main-results .unnumbered} **Proposition 1**. *For any system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, $x$ is periodic for $(X,\mathbb{F}) \implies$ each point of $\overline{\mathcal{O}_H(x)}$ is periodic (with same period).* *Proof.* Let $(X,\mathbb{F})$ be generated by a commutative family of homeomorphisms and let $x$ be periodic for $(X,\mathbb{F})$ (with period r). Then, $\omega_{nr}(x)=x~~$ for all $n\in\mathbb{Z}$. For any point $\omega_{r_n}\circ\omega_{r_{n-1}}\circ\ldots\omega_{r_1}(x)$ in orbital hull of $x$, as $\omega_{nr}(\omega_{r_n}\circ\omega_{r_{n-1}}\circ\ldots\omega_{r_1}(x))=\omega_{r_n}\circ\omega_{r_{n-1}}\circ\ldots\omega_{r_1}(\omega_{nr}(x))=\omega_{r_n}\circ\omega_{r_{n-1}}\circ\ldots\omega_{r_1}(x)$ (as $\mathbb{F}$ is commutative), every point in $\mathcal{O}_H(x)$ is periodic (with same period). Finally, as the limit of periodic points of period $k$ is a periodic point of period $k$, each point of $\overline{\mathcal{O}_H(x)}$ is periodic point of period $k$ and the proof is complete. ◻ **Remark 1**. *The above result establishes the periodicity of the elements of the closure of the orbital hull of a point $x$, when the point $x$ is itself periodic. It may be noted that as the governing rule is time variant, the periodicity of $x$ need not guarantee the periodicity of the members of the orbital hull (or even orbit itself). Also, if the governing rule is time variant, a periodic point may have infinite orbit and hence need not attribute to simpler dynamical behavior. In fact, if the governing rule is time variant, a pair of periodic points may behave in an unexpected manner and form a Li-Yorke pair. We now give examples in support of our claim.* **Example 1**. *Let $X=[0,1]$ be the unit interval and let $f_n:X\rightarrow X$ be defined as $f_{2n-1}(x) = \left\{ \begin{array}{lr} \frac{x}{2} & : 0\leq x \leq \frac{1}{2}, \\ {\frac{3}{2}x-\frac{1}{2}} & : \frac{1}{2}\leq x\leq 1 \\ \end{array} \right\}. ~~~ where~ k\in \mathbb{N}$ and $f_{2n}(x)=1-\sqrt{x}$ for all $n\in\mathbb{N}$.* *Then, $\frac{1}{2}\in X$ is a periodic point of period 2, but $f_1(\frac{1}{2})=\frac{1}{4}$ is not periodic for $(X,\mathbb{F})$. Thus, periodicity need not be preserved in the elements of the orbital hull when the generating maps do not commute.* **Example 2**. *Let $X=[0,1]$ and define $f_n:X\rightarrow X$ such that $f_{2n-1}(x)=x^{2n}$ and $f_{2n}(x)=x^{\frac{1}{2n}}$ for $n\in \mathbb{N}$. Then, every point is periodic (with period $2$). Also as $x^n$ converges to $0$ for all $x$ in $(0,1)$, the pair $(x,y)$ forms a Li-Yorke pair for all $x,y\in (0,1)$. Consequently, periodic points may form a Li-Yorke pair and hence need not attribute to simpler dynamical behavior.* **Proposition 2**. *For any system $(X,\mathbb{F})$, $(X,\mathbb{F})$ is minimal if and only if $\overline{\mathcal{O}_H(x)}=X$ for all $x\in X$.* *Proof.* As orbital hull of $x$ is smallest invariant subset of $X$ containing $x$, $(X,\mathbb{F})$ is minimal if and only if $\overline{\mathcal{O}_H(x)}=X$ for all $x\in X$. ◻ **Remark 2**. *The above result provides necessary and sufficient criteria for a non-autonomous system to be minimal. It may be noted that $X$ is itself invariant for the system generated by the family $\mathbb{F}$, a simple application of Zorn's lemma establishes the existence of minimal sets for non-autonomous systems. In [@sk2], the authors claim that system is minimal if and only if orbit of each point is dense in $X$ (page $84$, line $-7$). Further, the authors claim that a non-autonomous system is minimal if and only if for non-empty every open set $U$ in $X$, there exists $k\in\mathbb{N}$ such that trajectory of every point meets $U$ in at most $k$ iterations (Lemma $2.2$). However, we claim that both of the observations fail to hold. It may be noted that as minimality in non-autonomous systems is equivalent to orbital hull of each point being dense in $X$, minimality of a system does not guarantee orbit of each point to be dense in $X$ (Example [Example 3](#Example 2){reference-type="ref" reference="Example 2"}). Also, as orbit of a point is a non-invariant set, the second assertion also fails to hold good (Example [Example 3](#Example 2){reference-type="ref" reference="Example 2"}). In fact, as orbital hull of $x$ is the smallest invariant set containing $x$, a non-autonomous system is minimal if and only if for non-empty every open set $U$ in $X$, there exists $k\in\mathbb{N}$ such that $\mathcal{O}_H^k(x)$ meets $U$ for every point $x\in X$ (Proposition [Proposition 3](#mo){reference-type="ref" reference="mo"}). Also, while minimality of a system ensures each of its points to be almost periodic in the autonomous case, non-invariance of the governing rule forces such an implication not to hold true for non-autonomous systems. The proof follows from the fact that if the governing rule varies with time, any initial point $x_0$ may fail to return to its neighborhood even in the absence of proper invariant sets. In fact contrary to the autonomous case, a minimal set in a non-autonomous system may contain periodic points in the non-trivial sense (periodic points whose orbit is proper in the minimal set). We now give examples in support of our claim.* **Example 3**. *Let $X=\mathbb{S}^1$ be the unit circle and let $f_1(\theta)=\theta+\frac{1}{2}$, $f_2(\theta)=\theta-\frac{1}{2^2}$. For $n\geq 3$, define $f_n(\theta) = \left\{ \begin{array}{lr} {\theta+\frac{1}{2^{k}}} & :n=2k+1, \\ {\theta-\frac{1}{2^k}-\frac{1}{2^{k+1}}} & : n=2k. \\ \end{array} \right. ~~~ where~ k\in \mathbb{N}$* *As closure of the orbital hull of each $x$ is $X$, the non-autonomous system generated by the family $(f_n)$ is minimal. However as each point settles at the diametrically opposite end (of the initial point $x_0$), none of the points in the system are almost periodic. Consequently, almost periodic points are not guaranteed to exist in the non-autonomous setting.* **Example 4**. *Define $f_n: \mathbb{S}^1\rightarrow \mathbb{S}^1$ as follows: $$f_n(\theta) = \left\{ \begin{array}{lr} {\theta+\frac{1}{2^{k}}} & :n=4k ~or~ 4k-3, \\ {\theta-\frac{1}{2^k}} & : n=4k-1 ~or~ 4k-2. \\ \end{array} \right. ~~~ where~ k\in \mathbb{N}$$ It is clear that every element in $\mathbb{S}^1$ is periodic with period 2. However as orbital hull of every point in dense in $S^1$, the system $(X,\mathbb{F})$ is minimal and contains periodic points in the non-trivial sense.* **Proposition 3**. *Any system $(X,\mathbb{F})$ is minimal if and only if for every non-empty open set $U$ in $X$ there exists $k \in \mathbb{N}$ such that set $\mathcal{O}_H^k(x)\cap U\neq \emptyset$ for all $x\in X$.* *Proof.* Let $(X,\mathbb{F})$ be minimal and let $U$ be a non-empty open set in $X$. Firstly, note that if the claim does not hold then for each $k\in\mathbb{N}$ there exists $x_k$ such that $\mathcal{O}_H^k(x_k)\cap U=\emptyset$. Let $x$ be a limit point of $(x_k)$ and let $(r_1,r_2,\ldots,r_m)$ be any tuple (of any fixed length $m$). Let $p=\max \{r_i:i=1,2,\ldots,m\}$. As $\mathcal{O}_H^k(x_k)\cap U=\emptyset$, we have $(\omega_{r_m}\circ\omega_{r_{m-1}}\circ\ldots\circ\omega_{r_1})(x_n)\notin U$ for all $n\geq p$ and hence $\omega_{r_m}\circ\omega_{r_{m-1}}\circ\ldots\circ\omega_{r_1}(x)\notin U$. As the argument holds for any tuple $(r_1,r_2,\ldots,r_m)$, we have $\mathcal{O}_H(x)\cap U=\emptyset$ (which contradicts minimality of $X$). Consequently, there exists $k \in \mathbb{N}$ such that set $\mathcal{O}_H^k(x)\cap U\neq \emptyset$ for all $x\in X$ and the proof of forward part is complete.\ Conversely, if there exists $k \in \mathbb{N}$ such that set $\mathcal{O}_H^k(x)\cap U\neq \emptyset$ for all $x\in X$, then orbital hull of any point $x$ intersects every non-empty open set $U$ and hence $X$ is minimal. ◻ **Proposition 4**. *For any system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, $(X,\mathbb{F})$ is equicontinuous $\implies$ $(X,\mathbb{F})$ is distal.* *Proof.* Let $(X,\mathbb{F})$ be equicontinuous and $x$ and $y$ be proximal. Then thee exists sequence $(n_k)$ of integers such that $\lim \limits_{k\rightarrow \infty} d(\omega_{n_k}(x), \omega_{n_k}(y))=0$. For any $\epsilon>0$, there exists $\delta>0$ such that $d(a,b)<\delta \implies d(\omega_n(a),\omega_n(b))<\epsilon ~~\forall n \in \mathbb{Z}$. As $d(\omega_{n_r}(x), \omega_{n_r}(y))<\delta$ (for some $n_r$), we have $d(x,y)<\epsilon$. As the argument holds for any $\epsilon>0$, we have $x=y$ and hence the system $(X,\mathbb{F})$ is distal. ◻ **Proposition 5**. *For any non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, if $x$ is almost periodic then every point in $\mathcal{O}(x)$ is almost periodic.* *Proof.* Let $(X,\mathbb{F})$ be generated by a commutative family of homeomorphisms and let $x$ be an almost periodic point. Let $k\in \mathbb{Z}$ and let $U$ be a neighborhood of $\omega_k(x)$. Then as $x$ is almost periodic, the set $\{r\in\mathbb{Z}:\omega_r(x)\in \omega_k^{-1}(U)\}$ is syndetic and hence the set $\{r\in\mathbb{Z}:(\omega_k\circ\omega_r)(x)\in U\}$ is syndetic. Consequently, the set $\{r\in\mathbb{Z}:\omega_r(\omega_k)(x)\in U\}$ is syndetic and thus $\omega_k(x)$ is almost periodic. As the argument holds for any $k$, every point in the orbit of $x$ is almost periodic. ◻ **Remark 3**. *The above remark establishes the almost periodicity of elements in the orbit when the initial point $x$ is almost periodic. The proof uses the fact that if the initial point $x$ is almost periodic then the syndetic bound of some neighborhood of $x$ carries forward to the neighborhood of the given point in the orbit. As the arguments establish the almost periodicity of elements in the orbit of $\omega_k(x)$, a similar argument ensures almost periodicity of elements in the orbit of $\omega_k(x)$. Thus, almost periodicity of $x$ ensures almost periodicity of elements of $\mathcal{O}_H(x)$. Further for any equicontinuous system, a similar argument establishes almost periodicity of any limit of almost periodic points and hence we get the following result.* **Proposition 6**. *For any non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, if $x$ is almost periodic then every point in $\mathcal{O}_H(x)$ is almost periodic. Further, if $(X,\mathbb{F})$ is equicontinuous then every point in $\overline{\mathcal{O}_H(x)}$ is almost periodic.* *Proof.* Let $(X,\mathbb{F})$ be a non-autonomous system and let $x\in X$. As almost periodicity of $x$ guarantees $\omega_k(x)$ to be almost periodic for all $k\in\mathbb{Z}$ (Proposition [Remark 3](#ap){reference-type="ref" reference="ap"}), every point in $\mathcal{O}_H(x)$ is almost periodic. Further, let $(X,\mathbb{F})$ be equicontinuous, $y\in\overline{\mathcal{O}_H(x)}\setminus\mathcal{O}_H(x)$ be fixed and $\epsilon>0$ be given. As $(X,\mathbb{F})$ is equicontinuous, there exists $\delta<\frac{\epsilon}{3}$ such that $d(x,y)<\delta$ $\implies$ $d(\omega_k(x),\omega_k(y))<\frac{\epsilon}{3}$ for all $k\in\mathbb{Z}$. Also as $y\in\overline{\mathcal{O}_H(x)}\setminus\mathcal{O}_H(x)$, there exists $p(x)=\omega_{k_t}\omega_{k_{t-1}}...\omega_{k_2}\omega_{k_1}(x)\in \mathcal{O}_H(x)$ such that $d(p(x),y)<\delta$. As $p(x)$ is almost periodic, the set $\{k_r : d(\omega_{k_r}(p(x)), p(x))<\frac{\epsilon}{3}\}$ is syndetic. Further, as $d(\omega_{k_r}(y),y)\leq d(\omega_{k_r}(y),\omega_{k_r}(p(x)))+d(\omega_{k_r}(p(x)),p(x))+d(p(x),y)<\epsilon$ the set $\{m : d(\omega_{m}(y), y)<\epsilon\}$ is syndetic. Consequently every point in $\overline{\mathcal{O}_H(x)}$ is almost periodic and the proof is complete. ◻ **Proposition 7**. *For any equicontinuous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, if $x\in X$ is an almost periodic point then $\overline{\mathcal{O}_H(x)}$ is uniformly almost periodic.* *Proof.* Let $\epsilon>0$ and $u\in \overline{\mathcal{O}_H(x)}$ be any arbitrary element. As $(X,\mathbb{F})$ is equicontinuous, there exists a $\delta>0$ such that $d(x,y)<\delta\implies d(\omega_k(x),\omega_k(y))<\frac{\epsilon}{4},~ \forall x,y\in X, k\in \mathbb{Z}$. Let $\eta=min\{\frac{\epsilon}{4},\delta\}$ and let $F=\{x_1,x_2,...x_n\}$ be a $\eta$-dense set in $\overline{\mathcal{O}_H(x)}$. As $F$ is $\eta$-dense, there exists $x_r\in F$ such that $d(u,x_r)<\eta$ and hence $d(\omega_k(x_r),\omega_k(u))<\frac{\epsilon}{4}$ for all $k\in \mathbb{Z}$. Consequently, if orbit of $x_r$ returns to its $\frac{\epsilon}{4}$-neighborhood syndetically (at times $(n_r)$), $d(u,\omega_{n_r}(u))\leq d(u,x_i)+d(x_i,\omega_{n_r}(x_i))+d(\omega_{n_r}(x_i),\omega_{n_r}(u))<\epsilon$ and hence orbit of $u$ returns to its $\epsilon$-neighborhood syndetically (at same set of times $(n_r)$). As $(X,\mathbb{F})$ is equicontinuous, there exists a common syndetic set for $\{x_1,x_2,\ldots,x_r\}$ and hence every point returns to its $\epsilon$-neighborhood sydetically with the same syndetic set. Thus, $\overline{\mathcal{O}_H(x)}$ is uniformly almost periodic and the proof is complete. ◻ **Proposition 8**. *For any non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, if $(X,\mathbb{F})$ is equicontinuous then $\overline{\mathcal{O}_H(x)}=\overline{\mathcal{O}_H(y)}$ for all $y\in\overline{\mathcal{O}(x)}$.* *Proof.* Let $(X,\mathbb{F})$ be an equicontinuous system generated by a commutative family of homeomorphisms and let $x\in X$. Let $y\in \overline{\mathcal{O}(x)}$ and $\epsilon>0$ be given. As $(X,\mathbb{F})$ is equicontinuous, there exists $\delta>0$ such that $d(a,b)<\delta$ ensures $d(\omega_{n}(x),\omega_{n}(y))<\epsilon~~ \forall n\in \mathbb{Z}$. Also $y\in \overline{\mathcal{O}(x)}$ forces some $n_k\in\mathbb{Z}$ such that $d(\omega_{n_k}(x),y)<\delta$ and hence $d(x,\omega_{-n_k}(y))<\epsilon$. As the argument holds for any $\epsilon>0$, we have $\overline{\mathcal{O}(x)}= \overline{\mathcal{O}(y)}$. Further, as $x\in \overline{\mathcal{O}(y)}\subset \overline{\mathcal{O}_H(y)}$ implies $\overline{\mathcal{O}_H(x)}\subset \overline{\mathcal{O}_H(y)}$ (as orbital hull of $x$ is the smallest invariant set containing $x$), $x$ and $y$ have identical orbital hulls and hence elements in the orbit closures generate orbital hulls identical to the original point and the proof is complete. ◻ **Remark 1**. The above result establishes that if the system $(X,\mathbb{F})$ is equicontinuous, $\overline{\mathcal{O}_H(y)}=\overline{\mathcal{O}_H(x)}$ for any point $y$ in $\overline{\mathcal{O}(x)}$. As the arguments can repeated for elements of the orbit, $\overline{\mathcal{O}_H(y)}=\overline{\mathcal{O}_H(x)}$ for any point $y$ in $\mathcal{O}_H(x)$. It may be noted that if $\{(\omega_{k_n}\circ\omega_{k_{n-1}}\circ\ldots\circ\omega_{k_1}): k_i\in \mathbb{Z}, n\in \mathbb{N}\}$ is an equicontinuous family, then similar arguments establish $\overline{\mathcal{O}_H(y)}=\overline{\mathcal{O}_H(x)}$ for any point $y$ in $\overline{\mathcal{O}_H(x)}$. Consequently, all elements in $\overline{\mathcal{O}_H(x)}$ generate the same set (equal to $\overline{\mathcal{O}_H(x)}$) and hence the system is minimal. However, if the family under discussion fails to be equicontinuous, the non-autonomous system may fail to be minimal (even when the generating family is commutative). Thus we get the following results.\ **Corollary 1**. *For any non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, if $\{(\omega_{k_n}\circ\omega_{k_{n-1}}\circ\ldots\circ\omega_{k_1}): k_i\in \mathbb{Z}, n\in \mathbb{N}\}$ is equicontinuous then every point in $X$ generates a minimal subsystem of $(X,\mathbb{F})$.* *Proof.* The proof follows from discussions in Remark [Remark 1](#oh){reference-type="ref" reference="oh"}. ◻ **Example 5**. *Let $X=[0,1]$ and define $f_n:X\rightarrow X$ such that $f_{2n-1}(x)=x^{2}$ and $f_{2n}(x)=\sqrt{x}$ for $n\in \mathbb{N}$. Then as every point is periodic (with finite orbit), the system $(X,\mathbb{F})$ is equicontinuous. However, as $0$ is a fixed point such that $0\in \overline{\mathcal{O}_H(\frac{1}{2})}$, we have $\overline{\mathcal{O}_H(0)}\neq \overline{\mathcal{O}_H(\frac{1}{2})}$ and the above result cannot be generalized to elements in $\overline{\mathcal{O}_H(x)}$.* **Proposition 9**. *For any non-autonomous dynamical system $(X,\mathbb{F})$ generated by a commutative family $\mathbb{F}$, $(X,\mathbb{F})$ is transitive if and only if it has a point with dense orbit.* *Proof.* Let $(X,\mathbb{F})$ be a transitive system such that no point in $X$ has dense orbit in $(X,\mathbb{F})$. As $X$ is compact, for each $k\in\mathbb{N}$ there exists a finite subset $F_k$ such that $F_k$ is $\frac{1}{k}$-dense in $X$. As no point in $X$ has dense orbit, for each $x\in X$ there exists $r\in\mathbb{N}$ and $x_r\in F_r$ such that $\mathcal{O}(x)\cap S(x_r,\frac{1}{r})=\emptyset$. However, as $(X,\mathbb{F})$ is transitive, $O_{x,r}=\bigcup\limits_{n=0}^{\infty} \omega^{-1}_n (S(x_r,\frac{1}{r}))$ is an open dense subset of $X$. Thus $C_{x,r}=O_{x,r}^c$ is a non-empty closed nowhere dense subset of $X$. As choices of $x_r$ are countable, $X$ is countable union nowhere dense subsets of $X$ which contradicts compactness of $X$ and hence $X$ must have a point with dense orbit.\ Conversely, let $x\in X$ such that $\mathcal{O}(x)$ is dense in $X$ and let $U$ and $V$ be non-empty open subsets of $X$. As orbit of $x$ is dense, there exists $k\in\mathbb{N}$ such that $\omega_k(x)\in U$. Further, note that $\mathcal{O}(\omega_k(x))=\{\omega_n(\omega_k(x)):n\in\mathbb{N}\} =\{\omega_k(\omega_n(x)):n\in\mathbb{N}\} =\omega_k(\mathcal{O}(x))$ is dense in $X$ (as $\mathbb{F}$ is commutative and surjective). Thus there exists $r\in\mathbb{N}$ such that $\omega_r(\omega_k(x))\in V$ and hence $\omega_r(U)\cap V\neq\emptyset$. Consequently, $(X,\mathbb{F})$ is transitive and the proof is complete. ◻ **Remark 2**. The above proof establishes a necessary and sufficient criteria to establish transitivity of a non-autonomous dynamical system $(X,\mathbb{F})$. In particular, the result establishes existence of a dense orbit to be an equivalent criteria for a non-autonomous system to be transitivity. In addition, if the system is equicontinuous then the points in the space move in a synchronized manner (which may be visualized better via uniform almost periodicity) and hence denseness of an orbit (in $X$) forces denseness of all the orbits (in $X$). Hence we get the following result. **Proposition 10**. *For any equicontinuous non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, $(X,\mathbb{F})$ is transitive if and only if $\mathcal{O}(x)$ is dense in $X$ for all $x\in X$ (and hence $(X,\mathbb{F})$ is minimal).* *Proof.* Let $(X,\mathbb{F})$ be an equicontinuous system generated by a commutative family of homeomorphisms and let $(X,\mathbb{F})$ be transitive. By Proposition [Proposition 9](#st1){reference-type="ref" reference="st1"}, there exists $x\in X$ such that $\mathcal{O}(x)$ is dense in $X$. Let $y\in X$ be arbitrary and let $U=S(z,\epsilon)$ be any non-empty open subset of $X$. As $(X,\mathbb{F})$ is equicontinuous, there exists $\delta>0$ such that $d(a,b)<\delta$ ensures $d(\omega_k(a),\omega_k(b))<\frac{\epsilon}{4}$ for all $a,b\in X~~, k\in \mathbb{Z}$. As $\mathcal{O}(x)$ is dense in $X$, there exists $u\in \mathcal{O}(x)$ such that $d(u,y)<\delta$ and hence $d(\omega_k(u),\omega_k(y))<\frac{\epsilon}{4}$ for all $k\in\mathbb{Z}$. As denseness of $\mathcal{O}(x)$ (in $X$) forces $\overline{\mathcal{O}(u)}=X$, we have $d(z,\omega_r(u))<\frac{\epsilon}{4}$ for some $r\in \mathbb{Z}$. Thus, $d(z,\omega_r(y))<d(z,\omega_r(u))+d(\omega_r(u),\omega_r(y))<\epsilon$ and hence orbit of $y$ intersects $S(z,\epsilon)$. As the argument holds for any $y\in X$, orbit of any point is dense in $X$. As the proof of converse is trivial, $\mathcal{O}(x)$ is dense in $X$ for some $x\in X$ if and only if $\mathcal{O}(x)$ is dense in $X$ for all $x\in X$. Finally, as denseness of orbit of a point forces denseness of orbital hull (of the same point), $(X,\mathbb{F})$ is minimal and the proof is complete. ◻ **Remark 3**. The above proof establishes a necessary and sufficient criteria for an equicontinuous system to be transitive. It may be noted that as minimality of a system does not guarantee transitivity in the non-autonomous case, a minimal equicontinuous system mail fail to be transitive (as shown in Example [Example 4](#ex4){reference-type="ref" reference="ex4"}). Further, as the above arguments hold good when the element of the orbit is replaced by element of the orbital hull, a similar set of arguments establish equivalence of denseness of orbital hulls (among elements of the space $X$). Also, for autonomous systems, it is known that a transitive system with dense set of periodic points is necessarily sensitive. However, as the governing rule may vary significantly in a non-autonomous setting, an analogous version of the result fails to hold good in a non-autonomous setting. However, as totally transitive systems guarantee visiting times to intersect multiples of each integer, totally transitive systems with dense set of transitive points are necessarily sensitive. We now establish our claims below. **Proposition 11**. *For any equicontinuous non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, if $\mathcal{O}_H(x)$ is dense (in $X$) for some $x\in X$ then $\mathcal{O}_H(x)$ is dense (in $X$) for all $x\in X$* *Proof.* The proof follows from discussions in Remark [Proposition 12](#tth){reference-type="ref" reference="tth"}. ◻ **Example 6**. *Let $S^1$ be the unit circle and let the sequence $(f_n)$ be defined as $f_n(\theta) = \left\{ \begin{array}{lr} {\theta+ 2\pi \sum\limits_{i=1}^k \frac{1}{i}} & :n=2k-1, \\ {\theta- 2\pi \sum\limits_{i=1}^k \frac{1}{i}} & : n=2k. \\ \end{array} \right.$* *Firstly, note that as $\sum\limits_{i=1}^{\infty} \frac{1}{i}=\infty$, any point traverses the circle infinitely often (and hence passes across the origin infinitely often). Also, at the end of $n=2k+1$ iterations, any point $\theta$ rotates effectively by an angle ($2\pi \sum\limits_{i=1}^{k+1} \frac{1}{i}) (mod 2\pi)$. As rotations after $n$ iterations are of magnitude less than $\frac{1}{n}$, orbit of any point is $\frac{1}{n}$-dense in $X$ (for any $n\in\mathbb{N}$) and hence dense in $X$ and thus the system is transitive. Also, as $\omega_{2n}(x)=x$ for all $n\in\mathbb{Z}$, every point in $S^1$ is periodic (of period $2$). However, as the maps involved are isometries, the system is equicontinuous. Thus, transitive system with dense set of periodic points need not exhibit sensitive dependence on initial conditions.* **Proposition 12**. *For any non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, if $(X,\mathbb{F})$ is totally transitive with dense set of periodic points then $(X,\mathbb{F})$ is sensitive.* *Proof.* Let $(X,\mathbb{F})$ be totally transitive with dense set of periodic points, $x\in X$ and $U$ be any $\frac{1}{n}$-neighborhood of $X$. As set of periodic points is dense, $U$ contains a periodic point $p$ (say of order $r$). Also, if $diam(X)>k$ there exists $y\in X$ such that $d(x,y)>\frac{k}{2}$. As $(X,\mathbb{F})$ is $r$-transitive, there exists $u\in U,~~n_y\in\mathbb{Z}$ such that $d(\omega_{rn_y}(u),y)<\frac{1}{n}$. Also as $\omega_{rn_y}(p)=p$, for a sufficiently large $n$, we have $d(\omega_{rn_y}(u),\omega_{rn_y}(p))>\frac{k}{4}$. Consequently, neighborhood of any point in $X$ expands to diameter greater than $\frac{k}{4}$ and hence $(X,\mathbb{F})$ is sensitive. ◻ **Proposition 13**. *For any minimal non-autonomous system $(X,\mathbb{F})$ generated by a commutative family of homeomorphisms, $(X,\mathbb{F})$ is either equicontinuous or exhibits a dense set of sensitive points.* *Proof.* Let $(X,\mathbb{F})$ be minimal system generated by a commutative family of homeomorphisms. If $(X,\mathbb{F})$ is equicontinuous then the result holds trivially. If not, let $x$ be a point of sensitivity (with sensitivity constant $\eta$) and let $k\in \mathbb{Z}$ be fixed. Let $\epsilon>0$ be given and let $U=S(\omega_k(x),\epsilon)$. As $\omega_k$ is continuous, there exists $\delta>0$ such that $\omega_k(S(x,\delta))\subset S(\omega_k(x),\epsilon)$. As each $\omega_k$ is a homeomorphism, there exists $\eta'>0$ such that $d(a,b)\geq\eta$ implies $d(\omega_k(a),\omega_k(b))\geq\eta'$. Further, as $x$ is a point of sensitivity, there exists $y\in S(x,\delta)$ and $r\in\mathbb{Z}$ such that $d(\omega_r(x),\omega_r(y))\geq\eta$. Thus, we have $d(\omega_k(\omega_r(x)),\omega_k(\omega_r(y)))\geq\eta'$ and hence $d(\omega_r(\omega_k(x)),\omega_r(\omega_k(y)))>\eta'$ (as $\mathbb{F}$ is commutative). As $\omega_k(y)\in U$, $\omega_k(x)$ is a point of sensitivity and hence sensitivity of a point ensures sensitivity at each element of the orbit. As the arguments can repeated for elements of the orbit, sensitivity at a point $x$ ensures sensitivity at elements of the orbital hull. Finally, as $X$ is minimal, orbital hull of $x$ is dense in $X$ and the proof is complete. ◻ # Acknowledgement {#acknowledgement .unnumbered} The first author thanks MHRD for financial support. The second author thanks National Board for Higher Mathematics (NBHM) for financial support. 99 **Akin E., Kolyada S.**, Li-Yorke Sensitivity, *Nonlinearity,* 16 (2003), 1421-1433. **Balibrea F., Oprocha P.**, Weak mixing and chaos in nonautonomous discrete systems,*Applied Mathematics Letters,* 25(2012), 1135-1141. **Beer, R.D.**, Dynamical approaches to cognitive science *Trends in cognitive sciences*, Vol. 4, No. 3, March 2000. **Block L., Coppel W.**, Dynamics in one dimension, *Springer-Verlag, Berlin Hiedelberg* (1992). **Brin Michael, Stuck Garrett**, Introduction to dynamical systems, *Cambridge University Press* (2002). **Devaney Robert L.**, Introduction to chaotic dynamical systems, *Addison Wesley* (1986). **Dvorakova J.**, Chaos in nonautonomous discrete dynamical systems, *Communications in Nonlinear Science and Numerical Simulation* 17 (2012), 4649-4652. **Hamill J., Emmerik R., Heiderscheit B., Li L.**, A dynamical systems approach to lower extremity running injuries *Clinical biomechanics*, 14(1999), 297-308. **Kolyada S., Snoha L.**, Topological entropy of Nonautonomous Dynamical Systems, *Random and Computational Dynamics,* 4(2&3) (1996), 205-233. **Kolyada S., Snoha L., Trofimchuk S.**, On minimality of Non-autonomous Dynamical Systems, *Nonlinear Oscillations*, vol. 7, Issue 1 (2004),83-89. **Oono Y., Kohmoto M.**, Discrete model of chemical turbulence, *Physical review letters* vol. 55 no.27 2927-2931. **Sharma P., Raghav M.**, Dynamics Of Non-Autonomous Discrete Dynamical Systems, *Topology Proceedings* vol. 52 (2018), 45-59.

arxiv_math

--- abstract: | In this paper, a class of anisotropic weights having the form of $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ in dimensions $n\geq2$ is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find the optimal range of $(\theta_{1},\theta_{2},\theta_{3})$ such that this type of weights belongs to the Muckenhoupt class $A_{q}$. Then we further study its doubling property, which shows that it provides an example of a doubling measure but is not in $A_{q}$. As a consequence, we obtain anisotropic weighted Poincaré and Sobolev inequalities, which are used to study the local behavior for solutions to non-homogeneous weighted $p$-Laplace equations. address: - 1. Beijing Computational Science Research Center, Beijing 100193, China. - 2. Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088, China. - Beijing Computational Science Research Center, Beijing 100193, China. author: - Changxing Miao - Zhiwen Zhao title: On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations --- # Introduction Nowdays, there has been a surge of interest in studying a class of anisotropic weights comprising of different power-type weights due to their extensive and significant applications in composite materials, fluid mechanics and diffusion problems of the flows etc. To be specific, for $n\geq2$ and $\theta_{i}\in\mathbb{R}$, $i=1,2,3,$ the weight considered in this paper has the form of $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$. Here and throughout this paper, we denote the $(n-1)$-dimensional variables and domains by adding superscript prime, such as $x'=(x_{1},...,x_{n-1})$ and $B_{1}'(0)$. Note that these three component parts in the weight are of different dimensions if $n\geq3$. Furthermore, different from the fact that $|x|^{\theta_{2}}$ exhibits singular or degenerate behavior only at the origin, $|x'|^{\theta_{1}}$ and $|x_{n}|^{\theta_{3}}$ represent singular or degenerate line and plane, respectively. When $\theta_{2}\neq0$ and $\theta_{1}=\theta_{3}=0$, $|x|^{\theta_{2}}$ is a well-known isotropic weight due to the famous work [@CKN1984], where Caffarelli, Kohn and Nirenberg [@CKN1984] established the following interpolation inequalities $$\begin{aligned} \label{CKN90} \||x|^{\gamma_{1}}u\|_{L^{s}(\mathbb{R}^{n})}\leq C\||x|^{\gamma_{2}}u\|^{a}_{L^{p}(\mathbb{R}^{n})}\||x|^{\gamma_{3}}u\|^{1-a}_{L^{q}(\mathbb{R}^{n})},\quad\text{for any }u\in C^{\infty}_{0}(\mathbb{R}^{n}),\end{aligned}$$ and found the necessary and sufficient condition such that it holds under natural assumptions on the parameters. The Caffarelli-Kohn-Nirenberg weight has been widely applied to the study on weighted Sobolev spaces and relevant PDEs. In particular, Dong, Li and Yang [@DLY2022; @DLY2021] recently revealed the optimal gradient blow-up rate for solution to the linear insulated conductivity problem arising from composite materials by precisely capturing the asymptotic behavior of solution to the weighted elliptic equation as follows: $$\begin{aligned} \mathrm{div}(\kappa(x)|x|^{2}\nabla u)=0,\quad\text{in }B_{1},\end{aligned}$$ where $\tau^{-1}\leq\kappa\leq\tau$ in $B_{1}$ for some $\tau>0$, and $\int_{\mathbb{S}^{n-1}}\kappa x_{i}=0$, $i=1,2,...,n$. Under these assumed conditions, they made use of spherical harmonic expansion to obtain $$\begin{aligned} u(x)=u(0)+O(1)|x|^{\alpha},\quad\alpha=2^{-1}\big(-n+\sqrt{n^{2}+4\lambda_{1}}\big),\quad\mathrm{in}\;B_{1/2}.\end{aligned}$$ Here $\lambda_{1}\leq n-1$ represents the first nonzero eigenvalue for the following eigenvalue problem: $-\mathrm{div}_{\mathbb{S}^{n-1}}(\kappa(\xi)\nabla_{\mathbb{S}^{n-1}}u(\xi))=\lambda \kappa(\xi) u(\xi)$, $\xi\in\mathbb{S}^{n-1}.$ Especially when $\kappa=1$, we have $\lambda_{1}=n-1$. See Lemma 2.2 in [@DLY2021] and Lemmas 2.2 and 5.1 in [@DLY2022] for more details. Before that, the local behavior for the solution to weighted elliptic problem has been investigated in the earlier work [@FKS1982]. However, the value of Hölder exponent $\alpha$ captured in [@FKS1982] is not explicit. In fluid mechanics, this weight was also utilized to describe the singular behavior of Landau solutions [@L1944] with the singularity of order $O(|x|^{-1})$. When $\theta_{1}\neq0$, $\theta_{2}\in\mathbb{R}$ and $\theta_{3}=0$, the weight $|x'|^{\theta_{1}}|x|^{\theta_{2}}$ is anisotropic. This type of weights has been used to classify the singularities of $(-1)$-homogeneous axisymmetric no-swirl solutions to the three-dimensional stationary Navier-Stokes equations with a singular ray. Specifically, besides Landau solutions with single point singularity, Li and Yan [@LY2021] classified these other homogeneous solutions found in [@LLY201801; @LLY201802] into two types of singular solutions possessing the singularities of order $O(|x|^{-1}\ln|x'|^{-1}|x|)$ and $O(|x'|^{-1})$, respectively. With regard to the singular solutions of order $O(|x|^{-1}\ln|x'|^{-1}|x|)$, Li and Yan [@LY2021] further studied their asymptotic stability to non-stationary Navier-Stokes equations. The key to their proof lies in establishing an extended Hardy-type inequality with anisotropic weights, see Theorem 1.3 in [@LY2021]. Stimulated by this problem, Li and Yan [@LY2023] recently extended the Caffarelli-Kohn-Nirenberg inequalities in [\[CKN90\]](#CKN90){reference-type="eqref" reference="CKN90"} to the anisotropic case. When $\theta_{1}=\theta_{2}=0$ and $\theta_{3}\neq0$, the weight $|x_{n}|^{\theta_{3}}$ is closely linked to the establishment for the global regularity for diffusion equations. To be specific, Jin and Xiong [@JX2019] established the boundary estimates for the linearized weighted fast diffusion equation such that the implicit function theorem can be utilized to produce a short time regular solution $u$ of fast diffusion equation. Based on the obtained short time regular solution, they further bootstrap its regularity to be smooth provided that the Hölder continuity of $\partial_{t}u/u$ is proved. This difficulty was overcame in [@JX2019] by proving arbitrarily high integrability of $\partial_{t}u/u$. Different from this idea, Jin and Xiong [@JX2022] reduced this difficulty to establishing the Hölder estimates of $u/x_{n}$ in $B_{1/2}^{+}\times(-1/4,0]$, which is achieved by studying the Hölder regularity for a weighted nonlinear parabolic equation. Their regularity results are optimal, which answers a open problem proposed by Berryman and Holland [@BH1980]. Subsequently, Jin, Ros-Oton and Xiong [@JRX2023] further extended the results to the case of porous medium equations. Recently, Jin and Xiong [@JX2023] considered the general non-homogeneous linearized weighted fast diffusion equations and porous medium equations and established the Hölder estimates for their weak solutions, which also motivates our study on the local behavior for non-homogeneous $p$-Laplace equations with more general anisotropic weights in Section [3](#SEC03){reference-type="ref" reference="SEC03"} below. With regard to more investigations on the regularity for PDEs with the weight $x_{n}$, we refer to [@DPT2023; @FP2013; @DP2023; @DPT202302] and the references therein. In order to better apply the weight $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ to geometric and functional inequalities and relevant PDEs, we need first make clear its own properties, especially finding the largest range of $(\theta_{1},\theta_{2},\theta_{3})$ such that it becomes an $A_{q}$-weight. In fact, $A_{q}$-weights play a significant role in fourier analysis and nonlinear potential theory, see e.g. [@HKM2006; @G2014] for systematic investigations on the Muckenhoupt class $A_{q}$. The rest of the paper is organized as follows. Section [2](#SEC02){reference-type="ref" reference="SEC02"} is dedicated to the finding of optimal range of $(\theta_{1},\theta_{2},\theta_{3})$ which makes $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ be in the Muckenhoupt class $A_{q}$. Furthermore, its doubling property is proved to be satisfied in a larger range. This implies that the weight $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ provides an example of a doubling measure but is not of the Muckenhoupt class $A_{q}$. As a direct application of the theories of $A_{q}$-weights, we obtain the anisotropic weighted Sobolev and Poincaré inequalities, which are subsequently used to investigate the local behavior for solutions to non-homogeneous $p$-Laplace equations with anisotropic weights in Section [3](#SEC03){reference-type="ref" reference="SEC03"}. Finally, we further discuss their applications in doubly non-linear parabolic equations and some intriguing problems are naturally raised in Section [4](#SEC05){reference-type="ref" reference="SEC05"}. # Anisotropic Muckenhoupt weights consisting of three different power-type weights {#SEC02} For $q>1$, define $$\begin{aligned} \begin{cases} \mathcal{A}=\{(\theta_{1},\theta_{2},\theta_{3}): \theta_{1}>-(n-1),\,\theta_{2}\geq0,\,\theta_{3}>-1\},\\ \mathcal{B}=\{(\theta_{1},\theta_{2},\theta_{3}):\theta_{1}>-(n-1),\,\theta_{2}<0,\,\theta_{3}>-1,\,\theta_{1}+\theta_{2}+\theta_{3}>-n\},\\ \mathcal{C}_{q}=\{(\theta_{1},\theta_{2},\theta_{3}):\theta_{1}<(n-1)(q-1),\,\theta_{2}\leq0,\,\theta_{3}<q-1\},\\ \mathcal{D}_{q}=\{(\theta_{1},\theta_{2},\theta_{3}):\theta_{1}<(n-1)(q-1),\,\theta_{2}>0,\,\theta_{3}<q-1,\,\theta_{1}+\theta_{2}+\theta_{3}<n(q-1)\}. \end{cases}\end{aligned}$$ Introduce the definition of $A_{q}$-weight as follows. **Definition 1**. Set $1<q<\infty$. We say that $w$ is an $A_{q}$-weight (or $w$ is of the Muckenhoupt class $A_{q}$), provided that there exists some constant $C=C(n,q,w)>0$ such that $$\begin{aligned} \mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B}wdx\left(\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B}w^{-\frac{1}{q-1}}dx\right)^{q-1}\leq C(n,q,w),\quad\mathrm{with}\;\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B}=\frac{1}{|B|}\int_{B},\end{aligned}$$ for any ball $B$ in $\mathbb{R}^{n}$. In the following, we use $\omega_{n}$ to represent the volume of unit ball in $\mathbb{R}^{n}$. The notation $a\sim b$ implies that $\frac{1}{C}b\leq a\leq Cb$ for some constant $C=C(n,\theta_{1},\theta_{2},\theta_{3})>0$. The first step is to find the largest range of $(\theta_{1},\theta_{2},\theta_{3})$ ensuring that the weight $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ is a Radon measure, which is a necessary condition such that it further becomes an $A_{q}$-weight. **Lemma 2**. *$d\mu:=wdx=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx$ becomes a Radon measure iff $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{A}\cup\mathcal{B}$. Furthermore, $\mu(B_{r})\sim r^{n+\theta_{1}+\theta_{2}+\theta_{3}}$ for any $r>0$.* *Proof.* Let $\theta_{2}\geq0$. Then we obtain $$\begin{aligned} \mu(B_{r})=&2\int_{B_{r}\cap\{x_{n}>0\}}|x'|^{\theta_{1}}|x|^{\theta_{2}}x_{n}^{\theta_{3}}dx\notag\\ \sim&\int_{B_{r}'}|x'|^{\theta_{1}}dx'\int_{0}^{\sqrt{r^{2}-|x'|^{2}}}(|x'|^{\theta_{2}}+x_{n}^{\theta_{2}})x_{n}^{\theta_{3}}dx_{n}\notag\\ \sim&\int^{r}_{0}\left(s^{n+\theta_{1}+\theta_{2}-2}(r^{2}-s^{2})^{\frac{\theta_{3}+1}{2}}+s^{n+\theta_{1}-2}(r^{2}-s^{2})^{\frac{\theta_{2}+\theta_{3}+1}{2}}\right)ds\notag\\ \sim&\,r^{n+\theta_{1}+\theta_{2}+\theta_{3}}\int^{1}_{0}\left(s^{n+\theta_{1}+\theta_{2}-2}(1-s)^{\frac{\theta_{3}+1}{2}}+s^{n+\theta_{1}-2}(1-s)^{\frac{\theta_{2}+\theta_{3}+1}{2}}\right)ds,\end{aligned}$$ where in the second line we require $\theta_{3}>-1$ and utilized the following elemental inequalities: for $a,b\geq0$, $$\begin{aligned} \label{INE001} \begin{cases} \frac{a^{q}+b^{q}}{2}\leq(a+b)^{q}\leq a^{q}+b^{q},&0\leq q\leq1,\\ a^{q}+b^{q}\leq(a+b)^{q}\leq2^{q-1}(a^{q}+b^{q}),&q>1. \end{cases}\end{aligned}$$ Combining the properties of Beta function, we see that this integration makes sense iff $n+\theta_{1}+\theta_{2}>1$, $n+\theta_{1}>1$ and $\theta_{3}>-1$. Hence the conclusion holds under the condition of $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{A}$. Let $\theta_{2}<0$. From [\[INE001\]](#INE001){reference-type="eqref" reference="INE001"}, we have $$\begin{aligned} \label{WDZM001} \mu(B_{r})=&2\int_{B_{r}\cap\{x_{n}>0\}}|x'|^{\theta_{1}}|x|^{\theta_{2}}x_{n}^{\theta_{3}}dx\notag\\ \sim&\int_{B_{r}'}|x'|^{\theta_{1}}dx'\int^{\sqrt{r^{2}-|x'|^{2}}}_{0}\frac{x_{n}^{\theta_{3}}}{|x'|^{-\theta_{2}}+x_{n}^{-\theta_{2}}}dx_{n}.\end{aligned}$$ The last integration term in [\[WDZM001\]](#WDZM001){reference-type="eqref" reference="WDZM001"} can be decomposed as follows: $$\begin{aligned} I_{1}=&\int_{B'_{\frac{1}{\sqrt{2}}r}}|x'|^{\theta_{1}}dx'\int^{|x'|}_{0}\frac{x_{n}^{\theta_{3}}}{|x'|^{-\theta_{2}}+x_{n}^{-\theta_{2}}}dx_{n},\notag\\ I_{2}=&\int_{B'_{\frac{1}{\sqrt{2}}r}}|x'|^{\theta_{1}}dx'\int^{\sqrt{r^{2}-|x'|^{2}}}_{|x'|}\frac{x_{n}^{\theta_{3}}}{|x'|^{-\theta_{2}}+x_{n}^{-\theta_{2}}}dx_{n},\notag\\ I_{3}=&\int_{B_{r}'\setminus B'_{\frac{1}{\sqrt{2}}r}}|x'|^{\theta_{1}}dx'\int^{\sqrt{r^{2}-|x'|^{2}}}_{0}\frac{x_{n}^{\theta_{3}}}{|x'|^{-\theta_{2}}+x_{n}^{-\theta_{2}}}dx_{n}.\end{aligned}$$ As for the first term $I_{1}$, since $|x'|^{-\theta_{2}}\leq|x'|^{-\theta_{2}}+x_{n}^{-\theta_{2}}\leq2|x'|^{-\theta_{2}}$ if $0\leq x_{n}\leq |x'|$, we have $$\begin{aligned} I_{1}\sim&\int_{B'_{\frac{1}{\sqrt{2}}r}}|x'|^{\theta_{1}+\theta_{2}}dx'\int^{|x'|}_{0}x_{n}^{\theta_{3}}dx_{n}\sim\int^{\frac{r}{\sqrt{2}}}_{0}s^{n+\theta_{1}+\theta_{2}+\theta_{3}-1}ds\notag\\ \sim&r^{n+\theta_{1}+\theta_{2}+\theta_{3}}\int^{1}_{0}s^{n+\theta_{1}+\theta_{2}+\theta_{3}-1}ds.\end{aligned}$$ This integration is valid iff $n+\theta_{1}+\theta_{2}+\theta_{3}>0$ and $\theta_{3}>-1$. For the second term $I_{2}$, in the following we divide into two subcases to discuss. **Case 1.** When $\theta_{2}+\theta_{3}=-1$, we have $$\begin{aligned} I_{2}\sim&\int_{B'_{\frac{1}{\sqrt{2}}r}}|x'|^{\theta_{1}}dx'\int^{\sqrt{r^{2}-|x'|^{2}}}_{|x'|}x_{n}^{-1}dx_{n}\sim\int^{\frac{r}{\sqrt{2}}}_{0}s^{n+\theta_{1}-2}\ln\frac{\sqrt{r^{2}-s^{2}}}{s}ds\notag\\ \sim&\,r^{n+\theta_{1}-1}\int^{\frac{1}{\sqrt{2}}}_{0}s^{n+\theta_{1}-2}\ln\frac{\sqrt{1-s^{2}}}{s}ds\sim-r^{n+\theta_{1}-1}\int^{\frac{1}{\sqrt{2}}}_{0}s^{n+\theta_{1}-2}\ln s\,ds\notag\\ \sim&-r^{n+\theta_{1}+\theta_{2}+\theta_{3}}\bigg(s^{n+\theta_{1}+\theta_{2}+\theta_{3}}\ln s\Big|^{\frac{1}{\sqrt{2}}}_{0}-\int^{\frac{1}{\sqrt{2}}}_{0}s^{n+\theta_{1}+\theta_{2}+\theta_{3}-1}ds\bigg).\end{aligned}$$ This is valid iff $n+\theta_{1}+\theta_{2}+\theta_{3}>0$. **Case 2.** When $\theta_{2}+\theta_{3}\neq-1$, we have $$\begin{aligned} I_{2}\sim&\int_{B'_{\frac{1}{\sqrt{2}}r}}|x'|^{\theta_{1}}dx'\int^{\sqrt{r^{2}-|x'|^{2}}}_{|x'|}x_{n}^{\theta_{2}+\theta_{3}}dx_{n}\notag\\ \sim&\int^{\frac{1}{\sqrt{2}}r}_{0}s^{n+\theta_{1}-2}\left((r^{2}-s^{2})^{\frac{\theta_{2}+\theta_{3}+1}{2}}-s^{\theta_{2}+\theta_{3}+1}\right)ds\notag\\ \sim&\,r^{n+\theta_{1}+\theta_{2}+\theta_{3}}\int^{\frac{1}{\sqrt{2}}}_{0}s^{n+\theta_{1}-2}\left((1-s^{2})^{\frac{\theta_{2}+\theta_{3}+1}{2}}-s^{\theta_{2}+\theta_{3}+1}\right)ds.\end{aligned}$$ Observe that $\min\{1,2^{-\frac{\theta_{2}+\theta_{3}+1}{2}}\}\leq(1-s^{2})^{\frac{\theta_{2}+\theta_{3}+1}{2}}\leq\max\{1,2^{-\frac{\theta_{2}+\theta_{3}+1}{2}}\}$ in $[0,\frac{1}{\sqrt{2}}].$ Therefore, it is valid iff $\theta_{1}>-(n-1)$ and $n+\theta_{1}+\theta_{2}+\theta_{3}>0$. We proceed to calculate the last term $I_{3}$. Due to the fact that $|x'|\geq\sqrt{r^{2}-|x'|^{2}}\geq x_{n}$ if $\frac{1}{\sqrt{2}}r\leq|x'|\leq r$ and $0\leq x_{n}\leq\sqrt{r^{2}-|x'|^{2}}$, we obtain $$\begin{aligned} I_{3}\sim&\int_{B_{r}'\setminus B'_{\frac{1}{\sqrt{2}}r}}|x'|^{\theta_{1}+\theta_{2}}dx'\int^{\sqrt{r^{2}-|x'|^{2}}}_{0}x_{n}^{\theta_{3}}dx_{n}\notag\\ \sim&\int_{\frac{1}{\sqrt{2}}r}^{r}s^{n+\theta_{1}+\theta_{2}-2}(r^{2}-s^{2})^{\frac{\theta_{3}+1}{2}}ds\sim r^{n+\theta_{1}+\theta_{2}+\theta_{3}}\int_{\frac{1}{\sqrt{2}}}^{1}s^{n+\theta_{1}+\theta_{2}-2}(1-s^{2})^{\frac{\theta_{3}+1}{2}}ds\notag\\ \sim&\,r^{n+\theta_{1}+\theta_{2}+\theta_{3}}.\end{aligned}$$ This integration is valid iff $\theta_{3}>-1$. A combination of these above facts shows that Lemma [Lemma 2](#MWQAZ090){reference-type="ref" reference="MWQAZ090"} holds. ◻ Based on the measure condition revealed in Lemma [Lemma 2](#MWQAZ090){reference-type="ref" reference="MWQAZ090"}, we now make clear its range which lets $w=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ become an $A_{q}$-weight as follows. **Theorem 3**. *Set $1<q<\infty$. Then $w=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ is an $A_{q}$-weight if $(\theta_{1},\theta_{2},\theta_{3})\in(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})$.* **Remark 4**. In fact, the range of $(\theta_{1},\theta_{2},\theta_{3})\in(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})$ captured in Theorem [Theorem 3](#THM000Z){reference-type="ref" reference="THM000Z"} is optimal. This is due to the fact that the uniform lower bound of $\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}wdx\big(\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}w^{-\frac{1}{q-1}}dx\big)^{q-1}\geq C(n,q,\theta_{1},\theta_{2},\theta_{3})$ for any $B_{R}(\bar{x})\subset\mathbb{R}^{n}$ can be established by following the proof of Theorem [Theorem 7](#Lem009){reference-type="ref" reference="Lem009"} below with a slight modification. **Remark 5**. In the case when $\theta_{1}=\theta_{3}=0$, see pages 505-506 in [@G2014] for the corresponding results. With regard to the case of $\theta_{3}=0$, see Theorems 2.3 and 2.6 in [@MZ2023]. *Proof.* For any $\bar{x}=(\bar{x}',\bar{x}_{n})\in\mathbb{R}^{n}$ and $R>0$, we prepare to complete the proof by dividing into the following three cases. Let $|\bar{x}_{n}|\geq3R$. Observe first that $$\begin{aligned} \label{ZQ00} &\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\notag\\ &\leq (|\bar{x}|+R\mathrm{sgn}(\theta_{2}))^{\theta_{2}}(|\bar{x}_{n}|+R\mathrm{sgn}(\theta_{3}))^{\theta_{3}}\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}dx\notag\\ &\leq C(n,\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}|\bar{x}_{n}|^{\theta_{3}}R\int_{B_{R}'(\bar{x}')}|x'|^{\theta_{1}}dx',\end{aligned}$$ where $\mathrm{sgn}$ is the sign function as follows: $$\begin{aligned} \label{SGN0} \mathrm{sgn}(\theta)=& \begin{cases} 1,&\text{if }\theta>0,\\ 0,&\text{if }\theta=0,\\ -1,&\text{if }\theta<0. \end{cases}\end{aligned}$$ Note that if $|\bar{x}'|\geq3R$, then $$\begin{aligned} \label{ZQ091} \int_{B'_{R}(\bar{x}')}|x'|^{\theta_{1}}dx'\leq&\omega_{n-1}R^{n-1} \begin{cases} \big(|\bar{x}'|+R\big)^{\theta_{1}},&\text{if }\theta_{1}\geq0,\\ \big(|\bar{x}'|-R\big)^{\theta_{1}},&\text{if }\theta_{1}<0 \end{cases}\notag\\ \leq& C(n,\theta_{1}) R^{n-1}|\bar{x}'|^{\theta_{1}},\end{aligned}$$ while, if $|\bar{x}'|<3R$, $$\begin{aligned} \label{ZQ092} \int_{B'_{R}(\bar{x}')}|x'|^{\theta_{1}}dx'\leq&\int_{B'_{4R}(0')}|x'|^{\theta_{1}}dx'=\frac{(n-1)\omega_{n-1}}{n-1+\theta_{1}}(4R)^{n-1+\theta_{1}}.\end{aligned}$$ Inserting [\[ZQ091\]](#ZQ091){reference-type="eqref" reference="ZQ091"}--[\[ZQ092\]](#ZQ092){reference-type="eqref" reference="ZQ092"} into [\[ZQ00\]](#ZQ00){reference-type="eqref" reference="ZQ00"}, we obtain that if $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{A}\cup\mathcal{B}$, $$\begin{aligned} \mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\leq C(n,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}|\bar{x}_{n}|^{\theta_{3}} \begin{cases} |\bar{x}'|^{\theta_{1}},&\text{for }|\bar{x}'|\geq3R,\\ R^{\theta_{1}},&\text{for }|\bar{x}'|<3R. \end{cases}\end{aligned}$$ In exactly the same way, we deduce that if $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{C}_{q}\cup\mathcal{D}_{q}$, $$\begin{aligned} &\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}|x'|^{-\frac{\theta_{1}}{q-1}}|x|^{-\frac{\theta_{2}}{q-1}}|x_{n}|^{-\frac{\theta_{3}}{q-1}}dx\notag\\ &\leq C(n,q,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{-\frac{\theta_{2}}{q-1}}|\bar{x}_{n}|^{-\frac{\theta_{3}}{q-1}} \begin{cases} |\bar{x}'|^{-\frac{\theta_{1}}{q-1}},&\text{for }|\bar{x}'|\geq3R,\\ R^{-\frac{\theta_{1}}{q-1}},&\text{for }|\bar{x}'|<3R. \end{cases}\end{aligned}$$ Let $|\bar{x}_{n}|<3R$ and $|\bar{x}|\geq3R$. Then we have $$\begin{aligned} \label{ZW865} &\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\notag\\ &\leq (|\bar{x}|+R\mathrm{sgn}(\theta_{2}))^{\theta_{2}}\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x_{n}|^{\theta_{3}}dx\notag\\ &\leq C(n,\theta_{2})|\bar{x}|^{\theta_{2}}\int_{B_{R}'(\bar{x}')}|x'|^{\theta_{1}}dx'\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}.\end{aligned}$$ On one hand, if $|\bar{x}_{n}|\leq\sqrt{R^{2}-|x'-\bar{x}'|^{2}}$, then $$\begin{aligned} \label{ZQ095} &\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}\notag\\ &=\frac{\big(\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}\big)^{\theta_{3}+1}+\big(\sqrt{R^{2}-|x'-\bar{x}'|^{2}}-\bar{x}_{n}\big)^{\theta_{3}+1}}{\theta_{3}+1}\notag\\ &\leq\frac{2^{\theta_{3}+2}\big(R^{2}-|x'-\bar{x}'|^{2}\big)^{\frac{\theta_{3}+1}{2}}}{\theta_{3}+1}\leq\frac{2^{\theta_{3}+2}R^{\theta_{3}+1}}{\theta_{3}+1}.\end{aligned}$$ On the other hand, if $|\bar{x}_{n}|>\sqrt{R^{2}-|x'-\bar{x}'|^{2}}$, in light of $|\bar{x}_{n}|<3R$, we have $$\begin{aligned} \label{ZQ096} &\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}\notag\\ &=\frac{\big(|\bar{x}_{n}|+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}\big)^{\theta_{3}+1}-\big(|\bar{x}_{n}|-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}\big)^{\theta_{3}+1}}{\theta_{3}+1}\notag\\ &\leq\frac{(4R)^{\theta_{3}+1}}{\theta_{3}+1}.\end{aligned}$$ Substituting [\[ZQ091\]](#ZQ091){reference-type="eqref" reference="ZQ091"}--[\[ZQ092\]](#ZQ092){reference-type="eqref" reference="ZQ092"} and [\[ZQ095\]](#ZQ095){reference-type="eqref" reference="ZQ095"}--[\[ZQ096\]](#ZQ096){reference-type="eqref" reference="ZQ096"} into [\[ZW865\]](#ZW865){reference-type="eqref" reference="ZW865"}, it follows that if $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{A}\cup\mathcal{B}$, $$\begin{aligned} \label{DA9032} \mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\leq C(n,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}R^{\theta_{3}} \begin{cases} |\bar{x}'|^{\theta_{1}},&\text{for }|\bar{x}'|\geq3R,\\ R^{\theta_{1}},&\text{for }|\bar{x}'|<3R. \end{cases}\end{aligned}$$ By the same arguments, if $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{C}_{q}\cup\mathcal{D}_{q}$, then we obtain $$\begin{aligned} &\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}|x'|^{-\frac{\theta_{1}}{q-1}}|x|^{-\frac{\theta_{2}}{q-1}}|x_{n}|^{-\frac{\theta_{3}}{q-1}}dx\notag\\ &\leq C(n,q,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{-\frac{\theta_{2}}{q-1}}R^{-\frac{\theta_{3}}{q-1}} \begin{cases} |\bar{x}'|^{-\frac{\theta_{1}}{q-1}},&\text{for }|\bar{x}'|\geq3R,\\ R^{-\frac{\theta_{1}}{q-1}},&\text{for }|\bar{x}'|<3R. \end{cases}\end{aligned}$$ Let $|\bar{x}_{n}|<3R$ and $|\bar{x}|<3R$. It then follows from Lemma [Lemma 2](#MWQAZ090){reference-type="ref" reference="MWQAZ090"} that if $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{A}\cup\mathcal{B}$, $$\begin{aligned} \mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\leq 4^{n}\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{4R}(0)}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\leq C(n,\theta_{1},\theta_{2},\theta_{3})R^{\theta_{1}+\theta_{2}+\theta_{3}},\end{aligned}$$ and, if $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{C}_{q}\cup\mathcal{D}_{q}$, $$\begin{aligned} &\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{R}(\bar{x})}|x'|^{-\frac{\theta_{1}}{q-1}}|x|^{-\frac{\theta_{2}}{q-1}}|x_{n}|^{-\frac{\theta_{3}}{q-1}}dx\notag\\ &\leq 4^{n}\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{4R}(0)}|x'|^{-\frac{\theta_{1}}{q-1}}|x|^{-\frac{\theta_{2}}{q-1}}|x_{n}|^{-\frac{\theta_{3}}{q-1}}dx\leq C(n,q,\theta_{1},\theta_{2},\theta_{3})R^{-\frac{\theta_{1}+\theta_{2}+\theta_{3}}{q-1}}.\end{aligned}$$ The proof is complete. ◻ In order to understand the Radon measure $d\mu=wdx$ more clearly, we proceed to investigate its doubling property. From Chapters 1 and 15 of [@HKM2006], we see that the Radon measure $d\mu=wdx$ formed by the $A_{q}$-weight $|x'|^{\theta}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ is doubling if $(\theta_{1},\theta_{2},\theta_{3})\in(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})$. In the following we prove that this range is just a sufficient condition but not the necessary condition. The definition of doubling is now stated as follows. **Definition 6**. A Radon measure $d\mu$ is called doubling provided that there is some constant $0<C<\infty$ such that $\mu(B_{2R}(\bar{x}))\leq C\mu(B_{R}(\bar{x}))$ for any $\bar{x}\in\mathbb{R}^{n}$ and $R>0$. **Theorem 7**. *The Radon measure $d\mu=wdx$ is doubling when $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{A}\cup\mathcal{B}$.* **Remark 8**. The range of $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{A}\cup\mathcal{B}$ is the necessary and sufficient condition to make $d\mu=wdx$ satisfy the doubling property, since this range is the largest to ensure that $d\mu$ is a Radon measure. *Proof.* For any given $\bar{x}=(\bar{x}',\bar{x}_{n})\in\mathbb{R}^{n}$ and $R>0$, we divide into three subparts to complete the proof as follows. Let $|\bar{x}_{n}|\geq 3R$. On one hand, similar to the proof of Case 1 in Theorem [Theorem 3](#THM000Z){reference-type="ref" reference="THM000Z"}, we have $$\begin{aligned} \label{M098} &\int_{B_{2R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\notag\\ &\leq C(n,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}|\bar{x}_{n}|^{\theta_{3}}R^{n} \begin{cases} |\bar{x}'|^{\theta_{1}},&\text{for }|\bar{x}'|\geq3R,\\ R^{\theta_{1}},&\text{for }|\bar{x}'|<3R. \end{cases}\end{aligned}$$ On the other hand, $$\begin{aligned} \label{DA90} &\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\notag\\ &\geq (|\bar{x}|-R\mathrm{sgn}(\theta_{2}))^{\theta_{2}}(|\bar{x}_{n}|-R\mathrm{sgn}(\theta_{3}))^{\theta_{3}}\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}dx\notag\\ &\geq C(n,\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}|\bar{x}_{n}|^{\theta_{3}}R\int_{B_{R/2}'(\bar{x}')}|x'|^{\theta_{1}}dx',\end{aligned}$$ where the notation $\mathrm{sgn}$ is defined by [\[SGN0\]](#SGN0){reference-type="eqref" reference="SGN0"} and in the last inequality we utilized the fact that $$\begin{aligned} \int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}dx=&2\int_{B_{R}'(\bar{x}')}|x'|^{\theta_{1}}\sqrt{R^{2}-|x'-\bar{x}'|^{2}}dx'\geq\sqrt{3}R\int_{B'_{R/2}(\bar{x}')}|x'|^{\theta_{1}}dx'.\end{aligned}$$ Since $|x'|^{\theta_{1}}$ increases radially for $\theta_{1}\geq0$ and it decreases radially if $\theta_{1}<0$, it then follows that if $|\bar{x}'|\geq\frac{3}{2}R$, $$\begin{aligned} \label{DQ806} \int_{B'_{R/2}(\bar{x}')}|x'|^{\theta_{1}}dx'\geq&\omega_{n-1}\Big(\frac{R}{2}\Big)^{n-1} \big(|\bar{x}'|-\mathrm{sgn}(\theta_{1})R/2\big)^{\theta_{1}}\notag\\ \geq& C(n,\theta_{1})R^{n-1}|\bar{x}'|^{\theta_{1}},\end{aligned}$$ while, if $|\bar{x}'|<\frac{3}{2}R,$ $$\begin{aligned} \label{DQ809} \int_{B'_{R/2}(\bar{x}')}|x'|^{\theta_{1}}dx'\geq& \begin{cases} \int_{B'_{R/2}(0')}|x'|^{\theta_{1}}dx',&\text{if }\theta_{1}\geq0,\vspace{0.3ex}\\ \int_{B'_{R/2}(\frac{3}{2}R\frac{\bar{x}'}{|\bar{x}'|})}|x'|^{\theta_{1}}dx',&\text{if }\theta_{1}<0 \end{cases}\notag\\ \geq&\omega_{n-1}R^{n-1+\theta_{1}} \begin{cases} \frac{n-1}{2^{n-1+\theta_{1}}(n-1+\theta_{1})},&\text{if }\theta_{1}\geq0,\vspace{0.3ex}\\ \frac{1}{2^{n-1}},&\text{if }\theta_{1}<0. \end{cases}\end{aligned}$$ Substituting these two relations into [\[DA90\]](#DA90){reference-type="eqref" reference="DA90"}, we derive $$\begin{aligned} &\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq C(n,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}|\bar{x}_{n}|^{\theta_{3}}R^{n} \begin{cases} |\bar{x}'|^{\theta_{1}},&\text{if }|\bar{x}'|\geq\frac{3}{2}R,\\ R^{\theta_{1}},&\text{if }|\bar{x}'|<\frac{3}{2}R, \end{cases}\end{aligned}$$ which, together with [\[M098\]](#M098){reference-type="eqref" reference="M098"}, reads that Theorem [Theorem 7](#Lem009){reference-type="ref" reference="Lem009"} holds in the case of $|\bar{x}_{n}|\geq 3R$. Let $|\bar{x}_{n}|<3R$ and $|\bar{x}|\geq3R$. To begin with, we have $$\begin{aligned} \label{WM810} &\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\notag\\ &\geq (|\bar{x}|-R\mathrm{sgn}(\theta_{2}))^{\theta_{2}}\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x_{n}|^{\theta_{3}}dx\notag\\ &\geq C(n,\theta_{2})|\bar{x}|^{\theta_{2}}\int_{B_{R/2}'(\bar{x}')}|x'|^{\theta_{1}}dx'\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}.\end{aligned}$$ Utilizing [\[INE001\]](#INE001){reference-type="eqref" reference="INE001"}, we deduce that for $x'\in B'_{R/2}(\bar{x}'),$ $(i)$ if $|\bar{x}_{n}|\leq\sqrt{R^{2}-|x'-\bar{x}'|^{2}}$, $$\begin{aligned} \label{DQ812} &\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}\notag\\ &=\frac{\big(\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}\big)^{\theta_{3}+1}+\big(\sqrt{R^{2}-|x'-\bar{x}'|^{2}}-\bar{x}_{n}\big)^{\theta_{3}+1}}{\theta_{3}+1}\notag\\ &\geq\frac{2^{\theta_{3}+1}\min\{1,2^{-\theta_{3}}\}}{\theta_{3}+1}(R^{2}-|x'-\bar{x}'|^{2})^{\frac{\theta_{3}+1}{2}}\geq\frac{3^{\frac{\theta_{3}+1}{2}}\min\{1,2^{-\theta_{3}}\}R^{\theta_{3}+1}}{\theta_{3}+1};\end{aligned}$$ $(ii)$ if $|\bar{x}_{n}|>\sqrt{R^{2}-|x'-\bar{x}'|^{2}}$, $$\begin{aligned} \label{DQ815} &\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}\notag\\ &=\frac{\big(|\bar{x}_{n}|+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}\big)^{\theta_{3}+1}-\big(|\bar{x}_{n}|-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}\big)^{\theta_{3}+1}}{\theta_{3}+1}\notag\\ &\geq(R^{2}-|x'-\bar{x}'|^{2})^{\frac{\theta_{3}+1}{2}} \begin{cases} \frac{2^{\theta_{3}+1}}{\theta_{3}+1},&\text{for }\theta_{3}\geq0,\\ 2((1+\sqrt{3})^{\theta_{3}+1}-3^{\frac{\theta_{3}+1}{2}}),&\text{for }-1<\theta_{3}<0 \end{cases}\notag\\ &\geq R^{\theta_{3}+1} \begin{cases} \frac{3^{\frac{\theta_{3}+1}{2}}}{\theta_{3}+1},&\text{for }\theta_{3}\geq0,\\ 2((3+\sqrt{3})^{\theta_{3}+1}-3^{\theta_{3}+1}),&\text{for }-1<\theta_{3}<0, \end{cases}\end{aligned}$$ where we also utilized the following inequality with $t=\frac{|\bar{x}_{n}|-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}{2\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}$: for $-1<\theta_{3}<0$, $$\begin{aligned} (1+t)^{\theta_{3}+1}-t^{\theta_{3}+1}\geq(1+\sqrt{3})^{\theta_{3}+1}-3^{\frac{\theta_{3}+1}{2}},\quad\text{for }0<t\leq\sqrt{3}.\end{aligned}$$ Therefore, inserting [\[DQ806\]](#DQ806){reference-type="eqref" reference="DQ806"}--[\[DQ809\]](#DQ809){reference-type="eqref" reference="DQ809"} and [\[DQ812\]](#DQ812){reference-type="eqref" reference="DQ812"}--[\[DQ815\]](#DQ815){reference-type="eqref" reference="DQ815"} into [\[WM810\]](#WM810){reference-type="eqref" reference="WM810"}, we deduce $$\begin{aligned} &\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq C(n,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}R^{n+\theta_{3}} \begin{cases} |\bar{x}'|^{\theta_{1}},&\text{if }|\bar{x}'|\geq\frac{3}{2}R,\\ R^{\theta_{1}},&\text{if }|\bar{x}'|<\frac{3}{2}R. \end{cases}\end{aligned}$$ Applying [\[DA9032\]](#DA9032){reference-type="eqref" reference="DA9032"} with $R$ replaced by $2R$, we have $$\begin{aligned} \int_{B_{2R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\leq C(n,\theta_{1},\theta_{2},\theta_{3})|\bar{x}|^{\theta_{2}}R^{n+\theta_{3}} \begin{cases} |\bar{x}'|^{\theta_{1}},&\text{for }|\bar{x}'|\geq3R,\\ R^{\theta_{1}},&\text{for }|\bar{x}'|<3R. \end{cases}\end{aligned}$$ Combining these two above relations, we obtain that Theorem [Theorem 7](#Lem009){reference-type="ref" reference="Lem009"} holds if $|\bar{x}_{n}|<3R$ and $|\bar{x}|\geq3R$. Let $|\bar{x}_{n}|<3R$ and $|\bar{x}|<3R$. First, observe from Lemma [Lemma 2](#MWQAZ090){reference-type="ref" reference="MWQAZ090"} that $$\begin{aligned} \int_{B_{2R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\leq C(n,\theta_{1},\theta_{2},\theta_{3})R^{n+\theta_{1}+\theta_{2}+\theta_{3}}.\end{aligned}$$ We now proceed to calculate the same order lower bound of $\int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx$ with respect to $R$. In the following we divide into five subcases to complete the proof. **Case 1.** If $\theta_{1}<0$ and $\theta_{3}\leq0$, it follows that $$\begin{aligned} \int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq&\int_{B_{R}(\bar{x})}|x|^{\theta_{1}+\theta_{2}+\theta_{3}}dx\notag\\ \geq& \begin{cases} \int_{B_{R}(0)}|x|^{\theta_{1}+\theta_{2}+\theta_{3}}dx,&\text{if }\theta_{1}+\theta_{2}+\theta_{3}\geq0,\\ \int_{B_{R}(3R\frac{\bar{x}}{|\bar{x}|})}|x|^{\theta_{1}+\theta_{2}+\theta_{3}}dx,&\text{if }\theta_{1}+\theta_{2}+\theta_{3}<0 \end{cases}\notag\\ \geq&R^{n+\theta_{1}+\theta_{2}+\theta_{3}} \begin{cases} \frac{n\omega_{n}}{n+\theta_{1}+\theta_{2}+\theta_{3}},&\text{if }\theta_{1}+\theta_{2}+\theta_{3}\geq0,\\ 2^{\theta_{1}+\theta_{2}+\theta_{3}}\omega_{n},&\text{if } \theta_{1}+\theta_{2}+\theta_{3}<0. \end{cases}\end{aligned}$$ **Case 2.** If $\theta_{1}<0$, $\theta_{3}>0$, and $\theta_{1}+\theta_{2}\geq0$, then $$\begin{aligned} \int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq&\int_{B_{R}(\bar{x})}|x|^{\theta_{1}+\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq\int_{B_{R}(\bar{x})}|x_{n}|^{\theta_{1}+\theta_{2}+\theta_{3}}dx\notag\\ \geq&\int_{B'_{R/2}(\bar{x}')}dx'\int_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{1}+\theta_{2}+\theta_{3}}dx_{n}.\end{aligned}$$ For $x'\in B_{R/2}'(\bar{x}')$, we deduce from [\[INE001\]](#INE001){reference-type="eqref" reference="INE001"} that when $|\bar{x}_{n}|\leq \sqrt{R^{2}-|x'-\bar{x}'|^{2}}$, $$\begin{aligned} &\int_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{1}+\theta_{2}+\theta_{3}}dx_{n}\notag\\ &=\frac{(\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}})^{\theta_{1}+\theta_{2}+\theta_{3}+1}+(\sqrt{R^{2}-|x'-\bar{x}'|^{2}}-\bar{x}_{n})^{\theta_{1}+\theta_{2}+\theta_{3}+1}}{\theta_{1}+\theta_{2}+\theta_{3}+1}\notag\\ &\geq\frac{2(R^{2}-|x'-\bar{x}'|^{2})^{\frac{\theta_{1}+\theta_{2}+\theta_{3}+1}{2}}}{\theta_{1}+\theta_{2}+\theta_{3}+1}\geq\frac{3^{\frac{\theta_{1}+\theta_{2}+\theta_{3}+1}{2}}R^{\theta_{1}+\theta_{2}+\theta_{3}+1}}{2^{\theta_{1}+\theta_{2}+\theta_{3}}(\theta_{1}+\theta_{2}+\theta_{3}+1)},\end{aligned}$$ and, when $|\bar{x}_{n}|>\sqrt{R^{2}-|x'-\bar{x}'|^{2}}$, $$\begin{aligned} &\int_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{1}+\theta_{2}+\theta_{3}}dx_{n}\notag\\ &=\frac{(|\bar{x}_{n}|+\sqrt{R^{2}-|x'-\bar{x}'|^{2}})^{\theta_{1}+\theta_{2}+\theta_{3}+1}-(|\bar{x}_{n}|-\sqrt{R^{2}-|x'-\bar{x}'|^{2}})^{\theta_{1}+\theta_{2}+\theta_{3}+1}}{\theta_{1}+\theta_{2}+\theta_{3}+1}\notag\\ &\geq\frac{2^{\theta_{1}+\theta_{2}+\theta_{3}+1}(R^{2}-|x'-\bar{x}'|^{2})^{\frac{\theta_{1}+\theta_{2}+\theta_{3}+1}{2}}}{\theta_{1}+\theta_{2}+\theta_{3}+1}\geq\frac{3^{\frac{\theta_{1}+\theta_{2}+\theta_{3}+1}{2}}R^{\theta_{1}+\theta_{2}+\theta_{3}+1}}{\theta_{1}+\theta_{2}+\theta_{3}+1}.\end{aligned}$$ A combination of these above facts shows that $$\begin{aligned} \int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq C(n,\theta_{1},\theta_{2},\theta_{3})R^{n+\theta_{1}+\theta_{2}+\theta_{3}}.\end{aligned}$$ **Case 3.** If $\theta_{1}<0$, $\theta_{3}>0$, and $\theta_{1}+\theta_{2}<0$, it follows from [\[DQ812\]](#DQ812){reference-type="eqref" reference="DQ812"}--[\[DQ815\]](#DQ815){reference-type="eqref" reference="DQ815"} that $$\begin{aligned} \int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq&\int_{B_{R}(\bar{x})}|x|^{\theta_{1}+\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq(4R)^{\theta_{1}+\theta_{2}}\int_{B_{R}(\bar{x})}|x_{n}|^{\theta_{3}}dx\notag\\ \geq&(4R)^{\theta_{1}+\theta_{2}}\int_{B_{R/2}'(\bar{x}')}dx'\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}\notag\\ \geq& C(n,\theta_{1},\theta_{2},\theta_{3})R^{n+\theta_{1}+\theta_{2}+\theta_{3}},\end{aligned}$$ where we also made use of the fact that $|x|\leq|x-\bar{x}|+|\bar{x}|<4R$. **Case 4.** If $\theta_{1}\geq0$ and $\theta_{2}\geq0$, applying the proofs of [\[DQ806\]](#DQ806){reference-type="eqref" reference="DQ806"}--[\[DQ809\]](#DQ809){reference-type="eqref" reference="DQ809"} with minor modification and making use of [\[DQ812\]](#DQ812){reference-type="eqref" reference="DQ812"}--[\[DQ815\]](#DQ815){reference-type="eqref" reference="DQ815"} again, we obtain $$\begin{aligned} \int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq&\int_{B'_{R/2}(\bar{x}')}|x'|^{\theta_{1}+\theta_{2}}dx'\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}\notag\\ \geq& C(n,\theta_{1},\theta_{2},\theta_{3})R^{n+\theta_{1}+\theta_{2}+\theta_{3}}.\end{aligned}$$ **Case 5.** If $\theta_{1}\geq0$ and $\theta_{2}<0$, it follows from [\[DQ806\]](#DQ806){reference-type="eqref" reference="DQ806"}--[\[DQ809\]](#DQ809){reference-type="eqref" reference="DQ809"} and [\[DQ812\]](#DQ812){reference-type="eqref" reference="DQ812"}--[\[DQ815\]](#DQ815){reference-type="eqref" reference="DQ815"} that $$\begin{aligned} \int_{B_{R}(\bar{x})}|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx\geq&(4R)^{\theta_{2}}\int_{B'_{R/2}(\bar{x}')}|x'|^{\theta_{1}}dx'\int^{\bar{x}_{n}+\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}_{\bar{x}_{n}-\sqrt{R^{2}-|x'-\bar{x}'|^{2}}}|x_{n}|^{\theta_{3}}dx_{n}\notag\\ \geq& C(n,\theta_{1},\theta_{2},\theta_{3})R^{n+\theta_{1}+\theta_{2}+\theta_{3}},\end{aligned}$$ where we also utilized the fact that $|x|\leq|x-\bar{x}|+|\bar{x}|<4R$. A consequence of these above facts shows that Theorem [Theorem 7](#Lem009){reference-type="ref" reference="Lem009"} holds. ◻ According to the theories of $A_{q}$-weights, we directly obtain the following anisotropic weighted Sobolev and Poincaré inequalities, which are two fundamental tools to study relevant Sobolev spaces and partial differential equations. Combining Theorems 1.8 and 15.23 in [@HKM2006] and Theorem [Theorem 3](#THM000Z){reference-type="ref" reference="THM000Z"} above, we have the following anisotropic weighted Sobolev embedding theorem. **Corollary 9**. *Let $n\geq2$, $\sum^{3}_{i=1}\theta_{i}\geq n(q-1)$, $1<q<p<nq$, and $(\theta_{1},\theta_{2},\theta_{3})\in(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})$. Then we obtain that for any $u\in W_{0}^{1,p}(B_{R}(x_{0}),w)$, $$\begin{aligned} \label{QE90} \|u\|_{L^{p\chi}(B_{R}(x_{0}),w)}\leq C(n,p,q,\theta_{1},\theta_{2},\theta_{3})\|\nabla u\|_{L^{p}(B_{R}(x_{0}),w)},\end{aligned}$$ where $w=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ and $\chi$ is given by $$\begin{aligned} \label{CHI} \chi:=\frac{n+\sum^{3}_{i=1}\theta_{i}}{n+\sum^{3}_{i=1}\theta_{i}-p}. %\begin{cases} %\frac{n}{n-p/q},&\text{if }p=n+\sum^{3}_{i=1}\theta_{i},\vspace{0.3ex}\\ %\min\big\{\frac{n}{n-p/q},\frac{n+\sum^{3}_{i=1}\theta_{i}}{n+\sum^{3}_{i=1}\theta_{i}-p}\big\},&\text{if }p<n+\sum^{3}_{i=1}\theta_{i}. %\end{cases}\end{aligned}$$* **Remark 10**. The condition of $\sum^{3}_{i=1}\theta_{i}\geq n(q-1)$, $1<q<p<nq$ is assumed to make $\chi\leq\frac{n}{n-p/q}$ for the purpose of meeting the requirement of Theorem 15.23 in [@HKM2006]. **Remark 11**. When $\theta_{3}=0$, making use of Theorem 1 in [@LY2021], we can enlarge the range of $(\theta_{1},\theta_{2},\theta_{3})$ which lets [\[QE90\]](#QE90){reference-type="eqref" reference="QE90"} hold, especially removing the restrictive condition $\mathcal{C}_{q}\cap\mathcal{D}_{q}$. Therefore, it is significant to extend the anisotropic Caffarelli-Kohn-Nirenberg type inequalities in [@LY2021] to the case of $\theta_{3}>-1$. More importantly, anisotropic Caffarelli-Kohn-Nirenberg type inequalities can provide a fine parabolic Sobolev inequality with anisotropic weights, which allows us to extend the following regularity results to weighted fast diffusion equations as shown in [@MZ2023]. For more related investigations on Sobolev and Poincaré inequalities with $A_{q}$-weights covering power-type weights, see [@BCG2006; @BT2002; @CR2013; @CRS2016; @CW1985; @L1986; @NS2018; @NS2019; @DD2023; @CW2006] and the references therein. For $1<q_{0}<\frac{n}{n-1}$, define $$\begin{aligned} \begin{cases} \mathcal{F}_{q_{0}}=\{(\theta_{1},\theta_{2},\theta_{3}):\,\theta_{1}>-\frac{(n-1)(q_{0}-1)}{q_{0}},\,\theta_{2}\geq0,\,\theta_{3}>-\frac{q_{0}-1}{q_{0}}\},\vspace{0.3ex}\\ \mathcal{G}_{q_{0}}=\{(\theta_{1},\theta_{2},\theta_{3}):\,\theta_{1}>-\frac{(n-1)(q_{0}-1)}{q_{0}},\,\theta_{2}<0,\,\theta_{1}+\theta_{2}+\theta_{3}>-\frac{n(q_{0}-1)}{q_{0}}\}. \end{cases}\end{aligned}$$ We now state the anisotropic weighted Poincaré inequalities as follows. **Lemma 12**. *Let $n\geq2$, $1<q<\infty$, and $1<q_{0}<\frac{n}{n-1}$. Then we have* *$(i)$ if $(\theta_{1},\theta_{2},\theta_{3})\in[(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})]\cup\{\theta_{1}=\theta_{3}=0,\,\theta_{2}>-n\}$, then for any $B:=B_{R}(\bar{x})\subset\mathbb{R}^{n}$, $R>0$, and $u\in W^{1,q}(B,w)$, $$\begin{aligned} \label{A065} \int_{B}|u-u_{B}|^{q}d\mu\leq C(n,q,\theta_{1},\theta_{2},\theta_{3})R^{q}\int_{B}|\nabla u|^{q}d\mu;\end{aligned}$$* *$(ii)$ if $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{F}_{q_{0}}\cup\mathcal{G}_{q_{0}}$, then for any $B:=B_{R}\subset\mathbb{R}^{n}$ and $u\in W^{1,1}(B)$, $$\begin{aligned} \label{A095} \int_{B}|u-u_{B}|d\mu\leq C(n,q_{0},\theta_{1},\theta_{2},\theta_{3})R^{1+\theta_{1}+\theta_{2}+\theta_{3}}\int_{B}|\nabla u|dx,\end{aligned}$$ where $d\mu:=wdx=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx$ and $u_{B}=\frac{1}{\mu(B)}\int_{B}u d\mu.$* **Remark 13**. It is worth pointing out that we use the theories of quasiconformal mappings rather than that of $A_{q}$-weights to keep [\[A065\]](#A065){reference-type="eqref" reference="A065"} holding for the weight $|x|^{\theta_{2}}$ if $\theta_{2}\geq n(q-1)$, see Corollary 15.35 in [@HKM2006]. This fact also indicates that $A_{q}$-weight is not a necessary condition to ensure that [\[A065\]](#A065){reference-type="eqref" reference="A065"} holds. Moreover, in contrast to [\[A065\]](#A065){reference-type="eqref" reference="A065"}, the result in [\[A095\]](#A095){reference-type="eqref" reference="A095"} holds without restriction in terms of the upper bound on the ranges of $\theta_{i}$, $i=1,2,3$. Then it is natural to ask if [\[A065\]](#A065){reference-type="eqref" reference="A065"} still holds when we get rid of restrictive condition $\mathcal{C}_{q}\cup \mathcal{D}_{q}$. *Proof.* On one hand, a combination of Theorem 15.21 and Corollary 15.35 in [@HKM2006] and Theorem [Theorem 3](#THM000Z){reference-type="ref" reference="THM000Z"} above shows that [\[A065\]](#A065){reference-type="eqref" reference="A065"} holds. On the other hand, by scaling, it suffices to demonstrate that [\[A095\]](#A095){reference-type="eqref" reference="A095"} holds when $R=1$. Making use of Lemma 1.4 of Fabes-Kenig-Serapioni [@FKS1982], we have $$\begin{aligned} \left|u(x)-\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}udy\right|\leq\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}|u(x)-u(y)|dy\leq C\int_{B_{1}}\frac{|\nabla u(y)|}{|x-y|^{n-1}}dy,\quad\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}:=\frac{1}{|B_{1}|}\int_{B_{1}},\end{aligned}$$ which reads that $$\begin{aligned} \label{A091} \int_{B_{1}}\left|u(x)-\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}udy\right|wdx\leq C\int_{B_{1}}\int_{B_{1}}\frac{|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}}{|x-y|^{n-1}}|\nabla u(y)| dxdy.\end{aligned}$$ In light of $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{F}_{q_{0}}\cup\mathcal{G}_{q_{0}}$, we obtain from Hölder's inequality and Lemma [Lemma 2](#MWQAZ090){reference-type="ref" reference="MWQAZ090"} that for $1<q_{0}<\frac{n}{n-1}$, $$\begin{aligned} &\int_{B_{1}}\frac{|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}}{|x-y|^{n-1}}dx\notag\\ &\leq\bigg(\int_{B_{2}(y)}\frac{1}{|x-y|^{q_{0}(n-1)}}dx\bigg)^{\frac{1}{q_{0}}}\left(\int_{B_{1}}|x'|^{\frac{\theta_{1}q_{0}}{q_{0}-1}}|x|^{\frac{\theta_{2}q_{0}}{q_{0}-1}}|x_{n}|^{\frac{\theta_{3}q_{0}}{q_{0}-1}}dx\right)^{\frac{q_{0}-1}{q_{0}}}\leq C.\end{aligned}$$ Substituting this into [\[A091\]](#A091){reference-type="eqref" reference="A091"}, we have $$\begin{aligned} \int_{B_{1}}\left|u(x)-\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}udy\right|wdx\leq C\int_{B_{1}}|\nabla u(x)|dx.\end{aligned}$$ It then follows that $$\begin{aligned} \int_{B_{1}}|u(x)-u_{B_{1}}|wdx\leq&\int_{B_{1}}\left|u(x)-\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}u\right|wdx+\int_{B_{1}}\left|\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}u-u_{B_{1}}\right|wdx\notag\\ \leq&2\int_{B_{1}}\left|u(x)-\mathchoice {{\setbox 0=\hbox{$\displaystyle{\textstyle-}{\int}$} \vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\textstyle{\scriptstyle-}{\int}$} \vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% {{\setbox 0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$} \vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}% \!\int_{B_{1}}u\right|wdx\leq C\int_{B_{1}}|\nabla u|dx,\end{aligned}$$ where $u_{B_{1}}=\frac{1}{\mu(B_{1})}\int_{B_{1}}ud\mu$. Then [\[A095\]](#A095){reference-type="eqref" reference="A095"} holds. ◻ # Applications to anisotropic weighted $p$-Laplace equations {#SEC03} $p$-Laplace equation comes up in a large variety of physical contexts and involves wide applications from game theory to nonlinear flow in porous media, mechanics and image processing etc, see e.g. [@DLK2013; @K1996; @L2019]. These important applications stimulate us to investigate the local behavior of solutions to anisotropic weighted $p$-Laplace equations with non-homogeneous terms in the following. We first introduce some notations for later use. For $\bar{x}\in\mathbb{R}^{n}$ and $R>0$, define $B^{+}_{R}(\bar{x})=B_{R}(\bar{x})\cap\{(x',x_{n}):x_{n}>0\}$. For brevity, we simplify the notation $B^{+}_{R}(\bar{x})$ as $B^{+}_{R}$ if $\bar{x}=0$. Denote $\partial'B^{+}_{R}(\bar{x})=B_{R}(\bar{x})\cap\{x_{n}=0\}$ and $\partial''B_{R}^{+}(\bar{x})=\partial B_{R}^{+}(\bar{x})\setminus\partial'B^{+}_{R}(\bar{x})$. Given a weight $w$, we denote by $L^{p}(B^{+}_{R}(\bar{x}),w)$ and $W^{1,p}(B^{+}_{R}(\bar{x}),w)$ the weighted $L^{p}$ spaces and weighted Sobolev spaces with their norms, respectively, represented as $$\begin{aligned} \begin{cases} \|u\|_{L^{p}(B^{+}_{R}(\bar{x}),w)}=\Big(\int_{B^{+}_{R}(\bar{x})}|u|^{p}wdx\Big)^{\frac{1}{p}},\\ \|u\|_{W^{1,p}(B^{+}_{R}(\bar{x}),w)}=\Big(\int_{B^{+}_{R}(\bar{x})}|u|^{p}wdx\Big)^{\frac{1}{p}}+\left(\int_{B^{+}_{R}(\bar{x})}|\nabla u|^{p}wdx\right)^{\frac{1}{p}}. \end{cases}\end{aligned}$$ For $\phi_{0}\in L^{\infty}(\partial'' B^{+}_{1})$ and $p>1$, we consider the general non-homogeneous weighted $p$-Laplace equations satisfying the Dirichlet condition as follows: $$\begin{aligned} \label{PROBLEM001} \begin{cases} -\mathrm{div}(w|\nabla u|^{p-2}\nabla u)=f_{0}+wf_{1},& \mathrm{in}\;B_{1}^{+},\\ u=0,&\text{on }\partial'B^{+}_{1},\\ u=\phi_{0},&\text{on }\partial'' B^{+}_{1}. \end{cases}\end{aligned}$$ Here $w=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$, the ranges of $\theta_{i}$, $i=1,2,3$ are prescribed in the following theorems, $f_{0}\in L^{\frac{mp\chi}{m(\chi-1)+\chi(p-1)}}(B_{1}^{+},w^{-\frac{\chi(m-1)(p-1)+m}{m(\chi-1)+\chi(p-1)}})$, and $f_{1}\in L^{\frac{mp\chi}{m(\chi-1)+\chi(p-1)}}(B^{+}_{1},w)$, where $\chi$ is given by [\[CHI\]](#CHI){reference-type="eqref" reference="CHI"} and $m>\frac{n+\sum^{3}_{i=1}\theta_{i}}{p}$. We say that $u\in W^{1,p}(B_{1}^{+},w)$ is termed a weak solution of [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"} provided that for any $\varphi\in W_{0}^{1,p}(B_{1}^{+},w)$, $$\begin{aligned} \int_{B_{1}^{+}}w|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi dx=\int_{B_{1}^{+}}(f_{0}+wf_{1})\varphi.\end{aligned}$$ With regard to the local behavior of solution to problem [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"} near the origin, we have **Theorem 14**. *Assume that $n\geq2$, $\sum^{3}_{i=1}\theta_{i}\geq n(q-1)$, $1<q<p<nq$, $(\theta_{1},\theta_{2},\theta_{3})\in(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})$, and $m>\frac{n+\sum^{3}_{i=1}\theta_{i}}{p}$. Let $u$ be the weak solution of problem [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"}. Then there exists a small constant $0<\alpha=\alpha(n,m,p,q,\theta_{1},\theta_{2},\theta_{3})<1$ such that for any $\gamma>0,$ $$\begin{aligned} \label{QNAW001ZW} u(x)=u(0)+O(1)(\|u\|_{L^{\gamma}(B_{1}^{+},w)}+\mathcal{H})|x|^{\alpha},\quad\text{for any }x\in B^{+}_{1/2},\end{aligned}$$ where $O(1)$ satisfies that $|O(1)|\leq C=C(n,m,p,q,\gamma,\theta_{1},\theta_{2},\theta_{3}),$ and $$\begin{aligned} \label{F09} \mathcal{H}:=&\|f_{0}\|^{\frac{1}{p-1}}_{L^{\frac{mp\chi}{m(\chi-1)+\chi(p-1)}}(B_{1}^{+},w^{-\frac{\chi(m-1)(p-1)+m}{m(\chi-1)+\chi(p-1)}})}+\|f_{1}\|^{\frac{1}{p-1}}_{L^{\frac{mp\chi}{m(\chi-1)+\chi(p-1)}}(B_{1}^{+},w)}.\end{aligned}$$* **Remark 15**. First, our results in Theorems [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"} and [Theorem 16](#THM002){reference-type="ref" reference="THM002"} can be easily extended to general degenerate elliptic equations with anisotropic weights. Second, when we let $B^{+}_{R_{0}}$ substitute for $B^{+}_{1}$, it follows from the proofs with a slight modification that [\[QNAW001ZW\]](#QNAW001ZW){reference-type="eqref" reference="QNAW001ZW"} and [\[Z010\]](#Z010){reference-type="eqref" reference="Z010"} hold with $B^{+}_{1/2},B^{+}_{1/4}$ replaced by $B^{+}_{R_{0}/2}$ and $B^{+}_{R_{0}/4}$, respectively. Furthermore, the constant $C$ will be dependent on $R_{0}$, but the exponents $\alpha$ and $\tilde{\alpha}$ not. The result in [\[QNAW001ZW\]](#QNAW001ZW){reference-type="eqref" reference="QNAW001ZW"} naturally leads to an interesting problem in terms of pursuing a sharp decay rate exponent $\alpha$, which may have a potential application in predicting the optimal gradient blow-up rate for solution to the nonlinear insulated conductivity problem governed by $p$-Laplacian. When the weight $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ becomes a single power-type weight, that is, only one of $\theta_{i}$, $i=1,2,3$ is nonzero, we establish the following Hölder estimates. **Theorem 16**. *Let $n\geq2$, $\sum^{3}_{i=1}\theta_{i}\geq n(q-1)$, $1<q<p<nq$, $(\theta_{1},\theta_{2},\theta_{3})\in(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})$, and $m>\frac{n+\sum^{3}_{i=1}\theta_{i}}{p}$. Assume that $u$ is the weak solution of problem [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"}. Then there exists some constant $0<\tilde{\alpha}<\frac{\alpha}{1+\alpha}$ with $\alpha$ determined by Theorem [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"}, such that when only one of $\theta_{i}$, $i=1,2,3$ is nonzero, $$\begin{aligned} \label{Z010} |u(x)-u(y)|\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B^{+}_{1})}+\mathcal{H})|x-y|^{\tilde{\alpha}},\quad\text{for all}\;x,y\in B^{+}_{1/8},\end{aligned}$$ where $\mathcal{H}$ is defined by [\[F09\]](#F09){reference-type="eqref" reference="F09"}, the positive constants $\tilde{\alpha}$ and $C$ depend only on $n,m,p,q$ and $\max\limits_{1\leq i\leq3}\theta_{i}$.* In the following, we will adopt the classical De Giorgi trunction method created in [@D1957] to study the boundedness and local regularity for solution to problem [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"}. ## Boundedness of weak solutions For simplicity, in the following we still denote $$d\mu=wdx=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx.$$ For $E\subset B^{+}_{1}$, let $|E|_{\mu}=\int_{E}wdx$. We start by establishing the Caccioppoli inequality for the truncated solution as follows. **Lemma 17**. *Assume as above. Let $B_{R}^{+}(x_{0})\subset B^{+}_{1}$. For any $\xi\in C^{\infty}(B_{1})$ satisfying that $0\leq\xi\leq1$ in $B_{1}$ and $\xi=0$ on $\overline{B_{1}^{+}}\setminus (B_{R}^{+}(x_{0})\cup\partial'B_{1}^{+})$, we derive $$\begin{aligned} &\int_{B^{+}_{R}(x_{0})}|\nabla(v\xi)|^{p}wdx\leq C\bigg(\int_{B^{+}_{R}(x_{0})}|v\nabla\xi|^{p}w+\mathcal{H}^{p}|\{u>k\}\cap B^{+}_{R}(x_{0})|_{\mu}^{1-\frac{1}{m}}\bigg),\end{aligned}$$ where $v=(u-k)^{\pm}$, $k\geq0$, $u$ is the solution of [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"}, and $\mathcal{H}$ is given by [\[F09\]](#F09){reference-type="eqref" reference="F09"}.* **Remark 18**. When $B_{R}^{+}(x_{0})$ is chosen to be $B_{1}^{+}$ and $v=(\pm u-k)^{+}$ with $k\geq\|\phi_{0}\|_{L^{\infty}(\partial'' B^{+}_{1})}$, Lemma [Lemma 17](#lem003){reference-type="ref" reference="lem003"} holds for any $\xi\in C^{\infty}(B_{1})$ which may not vanish on $\partial'' B^{+}_{1}$, since $v=0$ on $\partial'' B^{+}_{1}$ in this case. *Proof.* By picking test function $\varphi=v\xi^{p}$, we have $$\begin{aligned} \int_{B^{+}_{R}(x_{0})}w|\nabla u|^{p-2}\nabla u\cdot\nabla(v\xi^{p}) dx=\int_{B^{+}_{R}(x_{0})}(f_{0}+wf_{1})v\xi^{p}dx.\end{aligned}$$ Observe first from Hölder's inequality and Young's inequality that $$\begin{aligned} \label{KM01} &\int_{B^{+}_{R}(x_{0})}|\nabla u|^{p-2}\nabla u\nabla(v\xi^{p})w=\int_{B^{+}_{R}(x_{0})}|\nabla v|^{p-2}\nabla v\nabla(v\xi^{p})w\notag\\ &\geq\int_{B^{+}_{R}(x_{0})}|\nabla v|^{p}\xi^{p}w-\int_{B^{+}_{R}(x_{0})}pv\xi^{p-1}|\nabla\xi||\nabla v|^{p-1}w\notag\\ &\geq\int_{B^{+}_{R}(x_{0})}|\nabla v|^{p}\xi^{p}w-p\bigg(\int_{B^{+}_{R}(x_{0})}|\xi\nabla v|^{p}w\bigg)^{\frac{p-1}{p}}\bigg(\int_{B^{+}_{R}(x_{0})}|v\nabla\xi|^{p}w\bigg)^{\frac{1}{p}}\notag\\ &\geq\frac{1}{2}\int_{B^{+}_{R}(x_{0})}|\nabla v|^{p}\xi^{p}w-C(p)\int_{B^{+}_{R}(x_{0})}|v\nabla\xi|^{p}w.\end{aligned}$$ Perform zero extension for $v$ in $B_{1}\setminus B_{1}^{+}$ and still write the extended function as $v$. A consequence of Hölder's inequality, Young's inequality and Corollary [Corollary 9](#CO901){reference-type="ref" reference="CO901"} leads to that $$\begin{aligned} \label{KM02} &\int_{B^{+}_{R}(x_{0})}(f_{0}+wf_{1})v\xi^{p}dx\notag\\ &\leq\|v\xi\|_{L^{p\chi}(B_{R}(x_{0}),w)}\Big(\|f_{0}\|_{L^{\frac{p\chi}{p\chi-1}}(B^{+}_{R}(x_{0}),w^{-\frac{1}{p\chi-1}})}+\|f_{1}\|_{L^{\frac{p\chi}{p\chi-1}}(B^{+}_{R}(x_{0}),w)}\Big)\notag\\ &\leq C\|\nabla(v\xi)\|_{L^{p}(B_{R}(x_{0}),w)}\Big(\|f_{0}\|_{L^{\frac{p\chi}{p\chi-1}}(B^{+}_{R}(x_{0}),w^{-\frac{1}{p\chi-1}})}+\|f_{1}\|_{L^{\frac{p\chi}{p\chi-1}}(B^{+}_{R}(x_{0}),w)}\Big)\notag\\ &\leq\frac{1}{3}\|\nabla(v\xi)\|_{L^{p}(B^{+}_{R}(x_{0}),w)}^{p}+C\Big(\|f_{0}\|^{\frac{p}{p-1}}_{L^{\frac{p\chi}{p\chi-1}}(B^{+}_{R}(x_{0}),w^{-\frac{1}{p\chi-1}})}+\|f_{1}\|^{\frac{p}{p-1}}_{L^{\frac{p\chi}{p\chi-1}}(B^{+}_{R}(x_{0}),w)}\Big)\notag\\ &\leq \frac{1}{3}\|\nabla(v\xi)\|_{L^{p}(B^{+}_{R}(x_{0}),w)}^{p}+C\mathcal{H}^{p}|\{u>k\}\cap B^{+}_{R}(x_{0})|_{\mu}^{1-\frac{1}{m}},\end{aligned}$$ where $\chi$ and $\mathcal{H}$ are defined by [\[CHI\]](#CHI){reference-type="eqref" reference="CHI"} and [\[F09\]](#F09){reference-type="eqref" reference="F09"}, respectively. Combining [\[KM01\]](#KM01){reference-type="eqref" reference="KM01"}--[\[KM02\]](#KM02){reference-type="eqref" reference="KM02"} and the fact that $|\nabla(v\xi)|^{p}\leq2^{p-1}(|\xi\nabla v|^{p}+|v\nabla\xi|^{p})$, we obtain that Lemma [Lemma 17](#lem003){reference-type="ref" reference="lem003"} holds. ◻ Utilizing Lemma [Lemma 17](#lem003){reference-type="ref" reference="lem003"} and Remark [Remark 18](#RE06){reference-type="ref" reference="RE06"}, we can obtain the following global $L^{\infty}$ estimate. **Theorem 19**. *Assume as above. Let $u$ be the solution of [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"}. Then $$\begin{aligned} \|u\|_{L^{\infty}(B_{1}^{+})}\leq\|\phi_{0}\|_{L^{\infty}(\partial'' B^{+}_{1})}+C\mathcal{H},\end{aligned}$$ where $C=C(n,p,m,\theta_{1},\theta_{2},\theta_{3})$ and $\mathcal{H}$ is defined by [\[F09\]](#F09){reference-type="eqref" reference="F09"}.* *Proof.* Take $k_{i}=\|\phi_{0}\|_{L^{\infty}(\partial'' B^{+}_{1})}+\mathcal{M}-\frac{\mathcal{M}}{2^{i}}$, $i\geq0$ and $\xi\equiv1$ in Lemma [Lemma 17](#lem003){reference-type="ref" reference="lem003"}, where $\mathcal{M}$ is determined in the following. Extend $u$ to be identically zero in $B_{1}\setminus B_{1}^{+}$ and still denote the extended function as $u$. A direct application of Corollary [Corollary 9](#CO901){reference-type="ref" reference="CO901"} and Remark [Remark 18](#RE06){reference-type="ref" reference="RE06"} reads that $$\begin{aligned} &(k_{i+1}-k_{i})^{p}|\{u>k_{i+1}\}\cap B^{+}_{1}|_{\mu}^{\frac{1}{\chi}}\leq\bigg(\int_{B_{1}}|(u-k_{i})^{+}|^{p\chi}wdx\bigg)^{\frac{1}{\chi}}\notag\\ &\leq C\int_{B_{1}}|\nabla(u-k_{i})^{+}|^{p}wdx\leq C\mathcal{H}^{p}|\{u>k_{i}\}\cap B^{+}_{1}|_{\mu}^{1-\frac{1}{m}},\end{aligned}$$ where $\chi$ and $\mathcal{H}$ are, respectively, given by [\[CHI\]](#CHI){reference-type="eqref" reference="CHI"} and [\[F09\]](#F09){reference-type="eqref" reference="F09"}. Denote $F_{i}:=|\{u>k_{i}\}\cap B^{+}_{1}|_{\mu}.$ Then we have $$\begin{aligned} F_{i+1}\leq&\left(\frac{C\mathcal{H}^{p}2^{(i+1)p}}{\mathcal{M}^{p}}\right)^{\chi}F_{i}^{\frac{\chi(m-1)}{m}}\notag\\ =&\prod_{s=0}^{i}\left[\left(\frac{C\mathcal{H}^{p}2^{(i+1-s)p}}{\mathcal{M}^{p}}\right)^{\frac{m}{m-1}}\right]^{\big(\frac{\chi(m-1)}{m}\big)^{s+1}}F_{0}^{\big(\frac{\chi(m-1)}{m}\big)^{i+1}}\notag\\ \leq&\left[\left(\frac{C_{0}^{p}\mathcal{H}^{p}}{\mathcal{M}^{p}}\right)^{\frac{m\chi}{\chi(m-1)-m}}F_{0}\right]^{\big(\frac{\chi(m-1)}{m}\big)^{i+1}},\end{aligned}$$ where $\frac{\chi(m-1)}{m}>1$. Picking $\mathcal{M}=C_{0}\mathcal{H}(2|B^{+}_{1}|_{\mu})^{\frac{\chi(m-1)-m}{mp\chi}}$, we deduce $$\begin{aligned} F_{i+1}\leq2^{-\big(\frac{\chi(m-1)}{m}\big)^{i+1}}\rightarrow0,\quad\text{as }i\rightarrow\infty,\end{aligned}$$ which implies that $$\begin{aligned} \sup_{B^{+}_{1}}u\leq\|\phi_{0}\|_{L^{\infty}(\partial'' B^{+}_{1})}+C\mathcal{H}.\end{aligned}$$ Applying the above arguments to the equation of $-u$, we complete the proof. ◻ We now proceed to give the local $L^{\infty}$ estimate of the solution as follows. **Theorem 20**. *Assume as above. Suppose that $u$ is the solution of [\[PROBLEM001\]](#PROBLEM001){reference-type="eqref" reference="PROBLEM001"}. Then we obtain that for any $\gamma>0$, $$\begin{aligned} \|u\|_{L^{\infty}(B^{+}_{3/4})}\leq C(\|u\|_{L^{\gamma}(B^{+}_{1},w)}+\mathcal{H}),\end{aligned}$$ where $\mathcal{H}$ is given by [\[F09\]](#F09){reference-type="eqref" reference="F09"} and $C=C(n,p,m,\gamma,\theta_{1},\theta_{2},\theta_{3})$.* *Proof.* For any given $\tau\in(0,1)$, set $$\begin{aligned} r_{i}=\tau+2^{-i}(1-\tau),\quad k_{i}=k(2-2^{-i}),\quad i\geq0,\end{aligned}$$ where $k>0$ is determined in the following. Pick a nonnegative cut-off function $\xi_{i}\in C^{\infty}(\mathbb{R}^{n})$ satisfying that $\xi_{i}=1$ in $B^{+}_{r_{i+1}}$, $\xi=0$ in $\mathbb{R}^{n}\setminus B_{r_{i}}$, and $|\nabla\xi_{i}|\leq C(n)(r_{i}-r_{i+1})^{-1}$. Denote $A(k,r):=\{u>k\}\cap B^{+}_{r}$ and $|A(k,r)|_{\mu}:=\int_{A(k,r)}wdx$ for $k,r>0$. Extend $u$ to be identically zero in $B_{1}\setminus B_{1}^{+}$ and still write the extended function as $u$. It follows from Hölder's inequality, Corollary [Corollary 9](#CO901){reference-type="ref" reference="CO901"} and Lemma [Lemma 17](#lem003){reference-type="ref" reference="lem003"} that $$\begin{aligned} &\|(u-k_{i+1})^{+}\|_{L^{p}(B^{+}_{r_{i+1}},w)}^{p}\notag\\ &\leq\|\xi_{i}(u-k_{i+1})^{+}\|_{L^{p\chi}(B_{r_{i}},w)}^{p}|A(k_{i+1},r_{i})|_{\mu}^{1-\frac{1}{\chi}}\notag\\ &\leq C\|\nabla(\xi_{i}(u-k_{i+1})^{+})\|_{L^{p}(B_{r_{i}},w)}^{p}|A(k_{i+1},r_{i})|_{\mu}^{1-\frac{1}{\chi}}\notag\\ &\leq C\left(\frac{2^{p(i+1)}}{(1-\tau)^{p}}\|(u-k_{i+1})^{+}\|_{L^{p}(B^{+}_{r_{i}},w)}^{p}+\mathcal{H}^{p}|A(k_{i+1},r_{i})|_{\mu}^{1-\frac{1}{m}}\right)|A(k_{i+1},r_{i})|_{\mu}^{1-\frac{1}{\chi}},\end{aligned}$$ where $\chi$ is given by [\[CHI\]](#CHI){reference-type="eqref" reference="CHI"}. Note that $$\begin{aligned} \|(u-k_{i})^{+}\|^{p}_{L^{p}(B^{+}_{r_{i}},w)}\geq&(k_{i+1}-k_{i})^{p}|A(k_{i+1},r_{i})|_{\mu}.\end{aligned}$$ For $i\geq0$, denote $F_{i}:=k^{-p}\|(u-k_{i})^{+}\|_{L^{p}(B^{+}_{r_{i}},w)}^{p}$ and choose $k\geq\|u\|_{L^{p}(B^{+}_{1},w)}+\mathcal{H}$. It then follows that $$\begin{aligned} F_{i+1}\leq&\frac{C2^{p(i+1)(2-\frac{1}{\chi})}}{(1-\tau)^{p}}F_{i}^{2-\frac{1}{\chi}}+C\mathcal{H}^{p}2^{(i+1)p(2-\frac{1}{\chi}-\frac{1}{m})}k^{-p}F_{i}^{2-\frac{1}{\chi}-\frac{1}{m}}\notag\\ \leq&\prod\limits_{s=0}^{i}\left(\frac{C2^{p(i+1-s)(2-\frac{1}{\chi})}}{(1-\tau)^{p}}\right)^{(2-\frac{1}{\chi}-\frac{1}{m})^{s}}F_{0}^{(2-\frac{1}{\chi}-\frac{1}{m})^{i+1}}\notag\\ \leq&\left(\frac{\overline{C}_{0}^{p}F_{0}}{(1-\tau)^{^{p(1-\frac{1}{\chi}-\frac{1}{m})}}}\right)^{(2-\frac{1}{\chi}-\frac{1}{m})^{i+1}},\quad\overline{C}_{0}=\overline{C}_{0}(n,p,m,\theta_{1},\theta_{2},\theta_{3}).\end{aligned}$$ By choosing $k=\frac{2\overline{C}_{0}}{(1-\tau)^{1-\frac{1}{\chi}-\frac{1}{m}}}\|u\|_{L^{p}(B^{+}_{1},w)}+\mathcal{H}$, we have $$\begin{aligned} F_{i+1}\leq2^{-p(2-\frac{1}{\chi}-\frac{1}{m})^{i+1}}\rightarrow0,\quad\text{as } i\rightarrow\infty,\end{aligned}$$ which implies that $$\begin{aligned} \sup_{B^{+}_{\tau}}u\leq2k\leq\frac{4\overline{C}_{0}}{(1-\tau)^{1-\frac{1}{\chi}-\frac{1}{m}}}\|u\|_{L^{p}(B^{+}_{1},w)}+2\mathcal{H}.\end{aligned}$$ Repeating the above arguments with $u$ replaced by $-u$, we deduce that for any $\tau\in(0,1),$ $$\begin{aligned} \label{WE806} \|u\|_{L^{\infty}(B^{+}_{\tau})}\leq\frac{4\overline{C}_{0}}{(1-\tau)^{1-\frac{1}{\chi}-\frac{1}{m}}}\|u\|_{L^{p}(B^{+}_{1},w)}+2\mathcal{H}.\end{aligned}$$ For $R\in(0,1]$, let $\tilde{u}(x)=u(Rx)$, $\tilde{f}_{0}(x)=f_{0}(Rx)$ and $\tilde{f}_{1}(x)=f_{1}(Rx)$. Then we have $$\begin{aligned} -\mathrm{div}(w|\nabla\tilde{u}|^{p-2}\nabla\tilde{u})=R^{p-\sum^{3}_{i=1}\theta_{i}}\tilde{f}_{0}+R^{p}w\tilde{f}_{1},\quad\mathrm{in}\;B^{+}_{1}.\end{aligned}$$ Then applying [\[WE806\]](#WE806){reference-type="eqref" reference="WE806"} to $\tilde{u}$ and rescaling back to $u$, we deduce $$\begin{aligned} \|u\|_{L^{\infty}(B^{+}_{\tau R})}\leq&\frac{4\overline{C}_{0}}{(1-\tau)^{1-\frac{1}{\chi}-\frac{1}{m}}R^{\frac{n+\sum^{3}_{i=1}\theta_{i}}{p}}}\|u\|_{L^{p}(B^{+}_{R},w)}+2R^{\frac{m_{1}p-n-\sum^{3}_{i=1}\theta_{i}}{m_{1}(p-1)}}\mathcal{H}\notag\\ \leq&\frac{4\overline{C}_{0}}{(1-\tau)^{1-\frac{1}{\chi}-\frac{1}{m}}R^{\frac{n+\sum^{3}_{i=1}\theta_{i}}{p}}}\|u\|_{L^{p}(B^{+}_{R},w)}+2\mathcal{H},\end{aligned}$$ where we also utilized the fact that $m>\frac{n+\sum^{3}_{i=1}\theta_{i}}{p}$. For any $\gamma>0$, it follows from Hölder's inequality that if $\gamma\geq p$, $$\begin{aligned} \|u\|_{L^{p}(B^{+}_{R},w)}\leq \begin{cases} \|u\|_{L^{\gamma}(B^{+}_{R},w)},\quad\text{for }\gamma=p,\\ C\|u\|_{L^{\gamma}(B^{+}_{R},w_{2})},\quad\text{for }\gamma>p, \end{cases}\end{aligned}$$ while, if $0<\gamma<p$, by choosing $\bar{\gamma}=\frac{\gamma}{p}$, we have from Young's inequality that $$\begin{aligned} \|u\|_{L^{p}(B^{+}_{R},w)}\leq&\|u\|_{L^{\infty}(B^{+}_{R})}^{\frac{p-\bar{\gamma}}{p}}\left(\int_{B^{+}_{R}}|u|^{\bar{\gamma}}w\right)^{\frac{1}{p}}\leq C\|u\|_{L^{\infty}(B^{+}_{R})}^{\frac{p-\bar{\gamma}}{p}}\|u\|_{L^{\gamma}(B^{+}_{R},w)}^{\frac{\bar{\gamma}}{p}}\notag\\ \leq&\frac{(1-\tau)^{1-\frac{1}{\chi}-\frac{1}{m}}R^{\frac{n+\sum^{3}_{i=1}\theta_{i}}{p}}}{8\overline{C}_{0}}\|u\|_{L^{\infty}(B^{+}_{R})}\notag\\ &+\frac{C}{(1-\tau)^{\frac{(1-\frac{1}{\chi}-\frac{1}{m})(p-\bar{\gamma})}{\bar{\gamma}}}R^{\frac{(n+\sum^{3}_{i=1}\theta_{i})(p-\bar{\gamma})}{p\bar{\gamma}}}}\|u\|_{L^{\gamma}(B^{+}_{R},w)}.\end{aligned}$$ Combining these above facts, we obtain that for any $\gamma>0,$ $$\begin{aligned} \|u\|_{L^{\infty}(B^{+}_{\tau R})}\leq\frac{1}{2}\|u\|_{L^{\infty}(B^{+}_{R})}+\frac{C}{(R-\tau R)^{\beta}}\|u\|_{L^{\gamma}(B^{+}_{R},w)}+2\mathcal{H},\end{aligned}$$ where $$\begin{aligned} \beta:=\frac{1}{\bar{\gamma}}\max\bigg\{p\Big(1-\frac{1}{\chi}-\frac{1}{m}\Big),n+\sum^{3}_{i=1}\theta_{i}\bigg\}.\end{aligned}$$ It then follows from Lemma 1.1 in [@GG1982] that for any $0<\tau<1$ and $\gamma>0$, $$\begin{aligned} \|u\|_{L^{\infty}(B^{+}_{\tau})}\leq\frac{C}{(1-\tau)^{\beta}}\|u\|_{L^{\gamma}(B^{+}_{1},w)}+C\mathcal{H}.\end{aligned}$$ The proof is complete. ◻ ## Local regularity for weak solutions To complete the proofs of Theorems [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"} and [Theorem 16](#THM002){reference-type="ref" reference="THM002"}, the key is to establish the following two De Giorgi lemmms including the improvement on local oscillation of the solution in Lemma [Lemma 21](#LEM0035ZZW){reference-type="ref" reference="LEM0035ZZW"} and the explicit decay estimates for the distribution function of the solution in Lemma [Lemma 24](#lem005ZZW){reference-type="ref" reference="lem005ZZW"}. **Lemma 21**. *Assume as in Theorems [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"} and [Theorem 16](#THM002){reference-type="ref" reference="THM002"}. For $R\in(0,1)$, let $0\leq\sup\limits_{B_{R}^{+}}u\leq M\leq\|u\|_{L^{\infty}(B_{1}^{+})}$. Then there exists a small constant $0<\tau_{0}<1$, depending only on $n,m,p,\theta_{1},\theta_{2},\theta_{3}$, such that for $0\leq k<M$ and $0<\tau<\tau_{0}$, if $$\begin{aligned} \label{E90} \delta:=M-k\geq\mathcal{H}R^{1-\frac{n+\sum^{3}_{i=1}\theta_{i}}{mp}},\end{aligned}$$ and $$\begin{aligned} \label{ZWZ007ZZW} |\{x\in B_{R}^{+}: u>M-\delta\}|_{\mu}\leq\tau|B_{R}^{+}|_{\mu},\end{aligned}$$ then $$\begin{aligned} \label{DZ001ZZW} u\leq M-\frac{\delta}{2},\quad\mathrm{in}\; B^{+}_{R/2}.\end{aligned}$$* *Proof.* For $\delta>0$ and $i=0,1,2,...,$ denote $$\begin{aligned} r_{i}=\frac{R}{2}+\frac{R}{2^{i+1}},\quad k_{i}=M-\delta+\frac{\delta}{2}(1-2^{-i}).\end{aligned}$$ Pick a cut-off function $\xi_{i}\in C_{0}^{\infty}(B_{r_{i}})$ satisfying that $\xi_{i}=1\;\mathrm{in}\;B^{+}_{r_{i+1}}$, $0\leq\xi_{i}\leq1,$ and $|\nabla\xi_{i}|\leq C(r_{i}-r_{i+1})^{-1}$. For $k\in[0,M]$ and $r\in(0,R]$, let $v_{i}=(u-k_{i})^{+}$ and $A(k,r)=\{x\in B^{+}_{r}: u>k\}$. Carry out zero extension for $u$ in $B_{1}\setminus B_{1}^{+}$ and still use $u$ to represent the extended function. From Corollary [Corollary 9](#CO901){reference-type="ref" reference="CO901"} and Lemma [Lemma 17](#lem003){reference-type="ref" reference="lem003"}, we obtain $$\begin{aligned} &(k_{i+1}-k_{i})^{p}|A(k_{i+1},r_{i+1})|_{\mu}^{\frac{1}{\chi}}\leq\bigg(\int_{B_{R}}|\xi_{i}v_{i}|^{p\chi}w\bigg)^{\frac{1}{\chi}}\leq C\int_{B_{R}}|\nabla(\xi_{i}v_{i})|^{p}w\notag\\ &\leq C\bigg(\int_{B_{R}^{+}}|\nabla\xi_{i}|v_{i}^{p}w+\mathcal{H}^{p}|A(k_{i},r_{i})|_{\mu}^{1-\frac{1}{m}}\bigg)\notag\\ &\leq C\left(\frac{(M-k_{i})^{p}}{(r_{i}-r_{i+1})^{p}}|A(k_{i},r_{i})|_{\mu}+\mathcal{H}^{p}|A(k_{i},r_{i})|_{\mu}^{1-\frac{1}{m}}\right),\end{aligned}$$ where $\chi$ and $\mathcal{H}$ are, respectively, defined by [\[CHI\]](#CHI){reference-type="eqref" reference="CHI"} and [\[F09\]](#F09){reference-type="eqref" reference="F09"}. In light of [\[E90\]](#E90){reference-type="eqref" reference="E90"}, we have $$\begin{aligned} |A(k_{i+1},r_{i+1})|_{\mu}\leq& C\left(\frac{4^{(i+2)p}}{R^{p}}|A(k_{i},r_{i})|_{\mu}+\frac{2^{(i+2)p}\mathcal{H}^{p}}{\delta^{p}}|A(k_{i},r_{i})|^{1-\frac{1}{m}}_{\mu}\right)^{\chi}\notag\\ \leq&\left(\frac{C2^{(i+2)p}}{R^{p-\frac{n+\sum^{3}_{i=1}\theta_{i}}{m}}}|A(k_{i},r_{i})|^{1-\frac{1}{m}}_{\mu}\right)^{\chi}.\end{aligned}$$ Define $$\begin{aligned} F_{i}:=\frac{|A(k_{i},r_{i})|_{\mu}}{|B^{+}_{R}|_{\mu}}.\end{aligned}$$ Then we have $$\begin{aligned} F_{i+1}\leq&(C4^{(i+2)p})^{\chi} F_{i}^{\frac{\chi(m-1)}{m}}\leq\prod\limits^{i}_{s=0}\left[(C4^{p(i+2-s)})^{\chi}\right]^{\left(\frac{\chi(m-1)}{m}\right)^{s}}F_{0}^{\left(\frac{\chi(m-1)}{m}\right)^{i+1}}\notag\\ \leq&(C^{\ast}F_{0})^{\left(\frac{\chi(m-1)}{m}\right)^{i+1}}.\end{aligned}$$ Take $\tau_{0}=(C^{\ast})^{-1}$. Hence, it follows that if [\[ZWZ007ZZW\]](#ZWZ007ZZW){reference-type="eqref" reference="ZWZ007ZZW"} holds for any $0<\tau<\tau_{0}$, then $C^{\ast}F_{0}<1$ and $F_{i+1}\rightarrow0$, as $i\rightarrow\infty$. Therefore, [\[DZ001ZZW\]](#DZ001ZZW){reference-type="eqref" reference="DZ001ZZW"} holds. ◻ Before establishing the decaying estimates for the distribution function of the solution $u$, we first give the required anisotropic weighted isoperimetric inequalities. Similar to the proof of Proposition 2.10 in [@MZ2023], we have from Lemma [Lemma 12](#QWZM090){reference-type="ref" reference="QWZM090"} that **Lemma 22**. *Let $n\geq2$ and $1<q<\infty$. If $(\theta_{1},\theta_{2},\theta_{3})\in[(\mathcal{A}\cup\mathcal{B})\cap(\mathcal{C}_{q}\cup\mathcal{D}_{q})]\cup\{\theta_{1}=\theta_{3}=0,\,\theta_{2}>-n\}$, then for any $R>0$, $l>k$ and $u\in W^{1,q}(B_{R},w)$, $$\begin{aligned} &(l-k)^{q}\bigg(\int_{\{u\geq l\}\cap B_{R}}wdx\bigg)^{q}\int_{\{u\leq k\}\cap B_{R}}wdx\notag\\ &\leq C(n,q,\theta_{1},\theta_{2},\theta_{3})R^{q(n+\theta_{1}+\theta_{2}+\theta_{3}+1)}\int_{\{k<u<l\}\cap B_{R}}|\nabla u|^{q}wdx,\end{aligned}$$ and $$\begin{aligned} &(l-k)^{q}\bigg(\int_{\{u\leq k\}\cap B_{R}}wdx\bigg)^{q}\int_{\{u\geq l\}\cap B_{R}}wdx\notag\\ &\leq C(n,q,\theta_{1},\theta_{2},\theta_{3})R^{q(n+\theta_{1}+\theta_{2}+\theta_{3}+1)}\int_{\{k<u<l\}\cap B_{R}}|\nabla u|^{q}wdx,\end{aligned}$$ where $w=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}.$* **Remark 23**. If $(\theta_{1},\theta_{2},\theta_{3})\in\mathcal{F}_{q_{0}}\cup\mathcal{G}_{q_{0}}$, applying the proof of Proposition 2.10 in [@MZ2023] again, it follows from [\[A095\]](#A095){reference-type="eqref" reference="A095"} that for any $R>0$, $l>k$ and $u\in W^{1,1}(B_{R})$, $$\begin{aligned} &(l-k)\int_{\{u\geq l\}\cap B_{R}}wdx\int_{\{u\leq k\}\cap B_{R}}wdx\notag\\ &\leq C(n,q_{0},\theta_{1},\theta_{2},\theta_{3})R^{n+2\theta_{1}+2\theta_{2}+2\theta_{3}+1}\int_{\{k<u<l\}\cap B_{R}}|\nabla u|dx.\end{aligned}$$ This type of anisotropic weighted isoperimetric inequality can simplify the computations in solving the similar decaying estimates in the next lemma. We leave it to interested readers. Using Lemma [Lemma 22](#prop002){reference-type="ref" reference="prop002"}, we now present the desired decaying estimates as follows. **Lemma 24**. *Assume as in Theorems [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"} and [Theorem 16](#THM002){reference-type="ref" reference="THM002"}. For $R\in(0,1/2)$, let $0<\gamma<1$ and $0\leq\sup\limits_{B^{+}_{2R}}u\leq M\leq\|u\|_{L^{\infty}(B^{+}_{1})}$. Then for every $j\geq0$, there holds either $$\begin{aligned} \label{VD008} M\leq2^{j}\mathcal{H}R^{1-\frac{n+\sum^{3}_{i=1}\theta_{i}}{mp}},\end{aligned}$$ or $$\begin{aligned} \label{DEC001ZZW} \frac{|\{x\in B^{+}_{R}:u>M-\frac{M}{2^{j}}\}|_{\mu}}{|B^{+}_{R}|_{\mu}}\leq\frac{C_{0}}{j^{\frac{p-q}{pq}}},\end{aligned}$$ where $C_{0}=C_{0}(n,m,p,q,\theta_{1},\theta_{2},\theta_{3})$.* *Proof.* Perform zero extension for $u$ in $B_{1}\setminus B_{1}^{+}$ and still write the extended function as $u$. For $i\geq0$, set $k_{i}=M-\frac{M}{2^{i}}$ and $A(k_{i},R)=B_{R}\cap\{u>k_{i}\}.$ Using Lemma [Lemma 22](#prop002){reference-type="ref" reference="prop002"}, we see that for $q>1,$ $$\begin{aligned} \label{WMZ001ZZW} &(k_{i+1}-k_{i})^{q}|A(k_{i+1},R)|_{\mu}^{q}|B_{R}\setminus A(k_{i},R)|_{\mu}\notag\\ &\leq CR^{q(n+\sum^{3}_{i=1}\theta_{i}+1)}\int_{A(k_{i},R)\setminus A(k_{i+1},R)}|\nabla u|^{q}wdx.\end{aligned}$$ Note that $$\begin{aligned} |B_{R}\setminus A(k_{i},R)|_{\mu}\geq|B_{R}\setminus B^{+}_{R}|_{\mu}\geq C R^{n+\sum^{3}_{i=1}\theta_{i}}.\end{aligned}$$ Substituting this into [\[WMZ001ZZW\]](#WMZ001ZZW){reference-type="eqref" reference="WMZ001ZZW"}, we derive $$\begin{aligned} &|A(k_{i+1},R)|_{\mu}\leq\frac{C2^{i+1}}{M}R^{\frac{(n+\theta_{1}+\theta_{2}+\theta_{3})(q-1)}{q}+1}\bigg(\int_{A(k_{i},R)\setminus A(k_{i+1},R)}|\nabla u|^{q}wdx\bigg)^{\frac{1}{q}}.\end{aligned}$$ In view of $1<q<p<nq$, it follows from Hölder's inequality that $$\begin{aligned} &\int_{A(k_{i},R)\setminus A(k_{i+1},R)}|\nabla u|^{q}wdx\notag\\ &\leq\bigg(\int_{A(k_{i},R)\setminus A(k_{i+1},R)}|\nabla u|^{p}wdx\bigg)^{\frac{q}{p}}\bigg(\int_{A(k_{i},R)\setminus A(k_{i+1},R)}wdx\bigg)^{\frac{p-q}{p}}\notag\\ &\leq\bigg(\int_{B^{+}_{R}}|\nabla(u-k_{i})^{+}|^{p}wdx\bigg)^{\frac{q}{p}}|A(k_{i},R)\setminus A(k_{i+1},R)|_{\mu}^{\frac{p-q}{p}}.\end{aligned}$$ Choose a cut-off function $\xi\in C_{0}^{\infty}(B_{2R})$ satisfying that $$\begin{aligned} \xi=1\;\mathrm{in}\;B_{R},\quad0\leq\xi\leq1,\;|\nabla\xi|\leq\frac{C(n)}{R}\;\,\mathrm{in}\;B_{2R}.\end{aligned}$$ A consequence of Lemma [Lemma 17](#lem003){reference-type="ref" reference="lem003"} shows that $$\begin{aligned} &\bigg(\int_{B^{+}_{R}}|\nabla (u-k_{i})^{+}|^{p}wdx\bigg)^{\frac{1}{p}}\notag\\ &\leq C\bigg(\int_{B^{+}_{2R}}|(u-k_{i})^{+}|^{p}|\nabla\xi|^{p}wdx+\mathcal{H}^{p}|A(k_{i},2R)|_{\mu}^{1-\frac{1}{m}}\bigg)^{\frac{1}{p}}\notag\\ &\leq C\bigg(\frac{M^{p}}{2^{ip}}R^{n+\sum^{3}_{i=1}\theta_{i}-p}+\mathcal{H}^{p}R^{(n+\sum^{3}_{i=1}\theta_{i})(1-\frac{1}{m})}\bigg).\end{aligned}$$ If [\[VD008\]](#VD008){reference-type="eqref" reference="VD008"} fails for some $j$, it then follows that for $i\leq j$, $$\begin{aligned} \bigg(\int_{B^{+}_{R}}|\nabla (u-k_{i})^{+}|^{p}wdx\bigg)^{\frac{1}{p}}\leq\frac{CM}{2^{i}}R^{\frac{n+\sum^{3}_{i=1}\theta_{i}-p}{p}}.\end{aligned}$$ We have from these above facts that $$\begin{aligned} |A(k_{i+1},R)|_{\mu}\leq&CR^{(n+\sum^{3}_{i=1}\theta_{i})(1-\frac{p-q}{pq})}|A(k_{i},R)\setminus A(k_{i+1},R)|_{\mu}^{\frac{p-q}{pq}}.\end{aligned}$$ This leads to that for $j\geq1$, $$\begin{aligned} j|A(k_{j},R)|_{\mu}^{\frac{pq}{p-q}}\leq&\sum^{j-1}_{i=0}|A(k_{i+1},R)|_{\mu}^{\frac{pq}{p-q}}\notag\\ \leq& CR^{(n+\sum^{3}_{i=1}\theta_{i})(\frac{pq}{p-q}-1)}|B_{R}|_{\mu}\leq C|B_{R}|_{\mu}^{\frac{pq}{p-q}}.\end{aligned}$$ Then [\[DEC001ZZW\]](#DEC001ZZW){reference-type="eqref" reference="DEC001ZZW"} is proved. ◻ We are now ready to prove Theorems [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"} and [Theorem 16](#THM002){reference-type="ref" reference="THM002"} in turn. *Proof of Theorem [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"}.* For $0<R\leq\frac{1}{2}$, denote $$\begin{aligned} \overline{\mu}(R)=\sup_{B^{+}_{R}}u,\quad\underline{\mu}(R)=\inf_{B^{+}_{R}}u,\quad \omega(R)=\overline{\mu}(R)-\underline{\mu}(R).\end{aligned}$$ Take a sufficiently large integer $j_{0}$ such that $C_{0}j_{0}^{-\frac{p-q}{pq}}\leq\tau_{0},$ where $\tau_{0}$ and $C_{0}$ are, respectively, given by Lemma [Lemma 21](#LEM0035ZZW){reference-type="ref" reference="LEM0035ZZW"} and [\[DEC001ZZW\]](#DEC001ZZW){reference-type="eqref" reference="DEC001ZZW"}. It then follows from Lemmas [Lemma 21](#LEM0035ZZW){reference-type="ref" reference="LEM0035ZZW"} and [Lemma 24](#lem005ZZW){reference-type="ref" reference="lem005ZZW"} that there holds $$\begin{aligned} \mathrm{either}\;\;\overline{\mu}(R)\leq2^{j_{0}}\mathcal{H}R^{1-\frac{n+\sum^{3}_{i=1}\theta_{i}}{mp}},\quad\mathrm{or}\;\;\overline{\mu}(R/4)\leq\overline{\mu}(R)-\frac{\overline{\mu}(R)}{2^{j_{0}+1}}.\end{aligned}$$ Using the above arguments for $-u$, we obtain $$\begin{aligned} \mathrm{either}\;\;\underline{\mu}(R)\geq-2^{j_{0}}\mathcal{H}R^{1-\frac{n+\sum^{3}_{i=1}\theta_{i}}{mp}},\quad\mathrm{or}\;\;\underline{\mu}(R/4)\geq\underline{\mu}(R)-\frac{\underline{\mu}(R)}{2^{j_{0}+1}}.\end{aligned}$$ In either case, we all have $$\begin{aligned} \omega(R/4)\leq(1-2^{-j_{0}-1})\omega(R)+2^{j_{0}+1}\mathcal{H}R^{1-\frac{n+\sum^{3}_{i=1}\theta_{i}}{mp}}=\frac{1}{4^{\beta_{1}}}\omega(R)+2^{j_{0}+1}\mathcal{H}R^{\beta_{2}},\end{aligned}$$ where $\beta_{1}=-\frac{\ln(1-2^{-j_{0}-1})}{\ln4}$ and $\beta_{2}=1-\frac{n+\sum^{3}_{i=1}\theta_{i}}{mp}$. For every $0<R\leq\frac{1}{2}$, there exists an integer $k\geq1$ such that $4^{-(k+1)}\cdot2^{-1}<R\leq4^{-k}\cdot2^{-1}$. Due to the fact that $\omega(R)$ is nondecreasing in $R$, it follows from Theorem [Theorem 20](#THE621){reference-type="ref" reference="THE621"} that for any $\gamma>0$, $$\begin{aligned} \omega(R)\leq&4^{-k\beta_{1}}\omega(2^{-1})+2^{j_{0}+1-\beta_{2}}4^{(1-k)\beta_{2}}\mathcal{H}\sum^{k-1}_{i=0}4^{i(\beta_{2}-\beta_{1})}\notag\\ \leq& C(\|u\|_{L^{\gamma}(B^{+}_{1},w)}+\mathcal{H})R^{\min\{\beta_{1},\beta_{2}\}},\end{aligned}$$ where $C=C(n,m,p,q,\gamma,\theta_{1},\theta_{2},\theta_{3}).$ ◻ *Proof of Theorem [Theorem 16](#THM002){reference-type="ref" reference="THM002"}.* Take the proof corresponding to the case of $\theta_{1}\neq0$ and $\theta_{2}=\theta_{3}=0$ for example. The other two cases are the same and thus omitted. Applying the proof of Theorem [Theorem 14](#ZWTHM90){reference-type="ref" reference="ZWTHM90"} with a slight modification, we obtain that there exists two constants $0<\alpha<1$ and $C>1$, depending only on $n,m,p,q,\theta_{1},$ such that for any fixed $\bar{x}\in\{(0',\bar{x}_{n}):0\leq\bar{x}_{n}\leq1/8\}$, $$\begin{aligned} \label{QM916} |u(x)-u(\bar{x})|\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|x|^{\alpha},\quad\text{for all }x\in B^{+}_{1/4}(\bar{x}).\end{aligned}$$ For $R\in(0,1/8)$ and $y\in B^{+}_{1/R}$, let $u_{R}(y)=u(Ry)$, $f_{0,R}(y)=f_{0}(Ry)$ and $f_{1,R}(y)=f_{1}(Ry)$. Then $u_{R}$ solves $$\begin{aligned} -\mathrm{div}(|y'|^{\theta_{1}}|\nabla u_{R}|^{p-2}\nabla u_{R})=R^{p-\theta_{1}}f_{0,R}+R^{p}|y'|^{\theta_{1}}f_{1,R},\quad\mathrm{in}\;B^{+}_{1/R}.\end{aligned}$$ For any given $x,\tilde{x}\in B^{+}_{1/8},$ assume without loss of generality that $|\tilde{x}'|\leq|x'|$. Let $R=|x'|$. Then using Theorem [Theorem 19](#CORO06){reference-type="ref" reference="CORO06"} and the interior Hölder estimates for degenerate elliptic equations, we deduce that there exist two constants $0<\beta<1$ and $C>1$, both depending only on $n,m,p,q,\theta_{1}$, such that for any fixed $\bar{y}\in B^{+}_{1/(4R)}\cap\{|y'|=1\}$, $$\begin{aligned} \label{WAQA001} |u_{R}(y)-u_{R}(\bar{y})|\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|y-\bar{y}|^{\beta},\quad\text{for every }y\in B_{1/2}^{+}(\bar{y}).\end{aligned}$$ Note that $$\begin{aligned} |u(x)-u(\tilde{x})|\leq|u(x',x_{n})-u(x',\tilde{x}_{n})|+|u(x',\tilde{x}_{n})-u(\tilde{x}',\tilde{x}_{n})|.\end{aligned}$$ Set $\lambda>1$. If $|x_{n}-\tilde{x}_{n}|\leq R^{\lambda}$, it follows from [\[WAQA001\]](#WAQA001){reference-type="eqref" reference="WAQA001"} that $$\begin{aligned} &|u(x',x_{n})-u(x',\tilde{x}_{n})|\leq\left|u_{R}(x'/R,x_{n}/R)-u_{R}(x'/R,\tilde{x}_{n}/R)\right|\notag\\ &\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|(x_{n}-\tilde{x}_{n})/R|^{\beta}\notag\\ &\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|x_{n}-\tilde{x}_{n}|^{\frac{(\lambda-1)\beta}{\lambda}}.\end{aligned}$$ By contrast, if $|x_{n}-\tilde{x}_{n}|>R^{\lambda}$, using [\[QM916\]](#QM916){reference-type="eqref" reference="QM916"}, we obtain $$\begin{aligned} &|u(x',x_{n})-u(x',\tilde{x}_{n})|\notag\\ &\leq|u(x',x_{n})-u(0',x_{n})|+|u(0',x_{n})-u(0',\tilde{x}_{n})|+|u(0',\tilde{x}_{n})-u(x',\tilde{x}_{n})|\notag\\ &\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})(|x'|^{\alpha}+|x_{n}-\tilde{x}_{n}|^{\alpha})\notag\\ &\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|x_{n}-\tilde{x}_{n}|^{\frac{\alpha}{\lambda}}.\end{aligned}$$ Similarly, if $|x'-\tilde{x}'|\leq R^{\lambda}$, we have from [\[WAQA001\]](#WAQA001){reference-type="eqref" reference="WAQA001"} that $$\begin{aligned} &|u(x',\tilde{x}_{n})-u(\tilde{x}',\tilde{x}_{n})|=\left|u_{R}(x'/R,\tilde{x}_{n}/R)-u_{R}(\tilde{x}'/R,\tilde{x}_{n}/R)\right|\notag\\ &\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|(x'-\tilde{x}')/R|^{\beta}\notag\\ &\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|x'-\tilde{x}'|^{\frac{(\lambda-1)\beta}{\lambda}},\end{aligned}$$ while, if $|x-\tilde{x}|>R^{\lambda}$, it follows from [\[QM916\]](#QM916){reference-type="eqref" reference="QM916"} that $$\begin{aligned} &|u(x',\tilde{x}_{n})-u(\tilde{x}',\tilde{x}_{n})|\leq|u(x',\tilde{x}_{n})-u(0',\tilde{x}_{n})|+|u(0',\tilde{x}_{n})-u(\tilde{x}',\tilde{x}_{n})|\notag\\ &\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})\big(R^{\alpha}+|\tilde{x}'|^{\alpha}\big)\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|x'-\tilde{x}'|^{\frac{\alpha}{\lambda}}.\end{aligned}$$ Combining these above facts, we choose $\lambda=1+\frac{\alpha}{\beta}$ such that $\frac{(\lambda-1)\beta}{\lambda}=\frac{\alpha}{\lambda}$. In fact, this choice is best, since it achieves the maximum Hölder regularity exponent in virtue of the fact that $\frac{\lambda-1}{\lambda}$ is increasing in $\lambda$ and $\frac{1}{\lambda}$ is decreasing in $\lambda$. Therefore, we obtain that for any $x,\tilde{x}\in B_{1/8}^{+},$ $$\begin{aligned} |u(x)-u(\tilde{x})|\leq C(\|\phi_{0}\|_{L^{\infty}(\partial''B_{1}^{+})}+\mathcal{H})|x-\tilde{x}|^{\frac{\alpha\beta}{\alpha+\beta}},\end{aligned}$$ which implies that Theorem [Theorem 16](#THM002){reference-type="ref" reference="THM002"} holds. ◻ # Further discussions and some open problems {#SEC05} Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$. Denote $\Omega_{T}:=\Omega\times(-T,0]$, $T>0$. In this section, we mainly focus on the nonnegative solutions for doubly non-linear parabolic equations with anisotropic weights as follows: $$\begin{aligned} \label{PROBLEM06} \begin{cases} w_{1}\partial_{t}u^{q}-\mathrm{div}(w_{2}|\nabla u|^{p-2}\nabla u)=0,& \mathrm{in}\;\Omega_{T},\\ u=0,&\mathrm{on}\;\partial\Omega\times(-T,0],\\ u(x,0)=\phi_{0}\geq0,& \end{cases}\end{aligned}$$ where $\phi_{0}\in L^{\infty}(\Omega)$, $w_{1}=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ and $w_{2}=|x'|^{\theta_{4}}|x|^{\theta_{5}}|x_{n}|^{\theta_{6}}$. A nonnegative function $u$ is said to be a weak solution of [\[PROBLEM06\]](#PROBLEM06){reference-type="eqref" reference="PROBLEM06"} if $$u\in C([0,T];L^{q+1}(\Omega,w_{1}))\cap L^{p}(0,T;W_{0}^{1,p}(\Omega,w_{2}))$$ satisfies $$\begin{aligned} \int_{\Omega}w_{1}u^{q}\varphi dx\Big|_{t_{1}}^{t_{2}}+\int_{t_{1}}^{t_{2}}\int_{\Omega}(-w_{1}u^{q}\partial_{t}\varphi+w_{2}|\nabla u|^{p-2}\nabla u\nabla\varphi)dxdt=0,\end{aligned}$$ for any $-T\leq t_{1}<t_{2}\leq0$ and $\varphi\in W^{1,q+1}(0,T;L^{q+1}(\Omega,w_{1}))\cap L^{p}(0,T;W_{0}^{1,p}(\Omega,w_{2}))$. The equation in [\[PROBLEM06\]](#PROBLEM06){reference-type="eqref" reference="PROBLEM06"} is frequently utilized to describe various diffusion phenomena occurring in heat flow, chemical concentration, gas-kinetics, plasmas and thin liquid film dynamics. The problem of precisely analyzing the diffusion processes of the flows is critical to their applications in industry. For more related physical interpretations and derivations of physical models, we refer to [@DK2007; @V2007; @BDGLS2023] and the references therein. According to the diffusion mechanism of the flows, the equation in [\[PROBLEM06\]](#PROBLEM06){reference-type="eqref" reference="PROBLEM06"} can be classified into the following three types. If $q>p-1$, it is called fast diffusion equation and its solution will extinct in finite time. By contrast, the case of $q<p-1$ corresponds to slow diffusion equation and its solution always decays with power-function to the stable state. The equation in borderline case $q=p-1$ is called Trudinger's equation, which was originally observed by Trudinger [@T1968]. Its solution decays faster than slow diffusion equation and decreases exponentially in time variable. Moreover, there is a special feature for Trudinger's equation that it can be linked to extremals of Poincaré inequalities, see [@LL2022] for more details. As for the existence of weak solutions to problem [\[PROBLEM06\]](#PROBLEM06){reference-type="eqref" reference="PROBLEM06"}, see Theorem 1.7 in [@AL1983], Theorem 1.3 in [@BDMS2018] and Theorem 5.1 in [@BDMS201802]. The uniqueness result can be seen in Corollary 4.17 of [@BDGLS2023]. With regard to the study on Hölder regularity of weak solutions to problem [\[PROBLEM06\]](#PROBLEM06){reference-type="eqref" reference="PROBLEM06"} without weights, the well-studied ranges contain the following two cases: one is that $q>p-1$ and $1<p\leq2$; another is that $q\leq p-1$ and $p\geq2$. Especially when the considered diffusion operator is linear, there has a long list of literature devoted to studying the regularity (see e.g. [@JX2019; @JX2022; @S1983; @K1988; @BV2010; @DK2007; @DK1992; @DGV2012; @DKV1991; @JRX2023; @BFR2017; @BFV2018]) and the asymptotic behavior (see e.g. [@BH1980; @BGV2012; @FS2000; @BF2021; @A2021; @JRX2023; @AP1981; @BSV2015]). When the diffusion operator is nonlinear, see [@BDGLS2023] for systematic introduction on doubly non-linear parabolic equations without weights. However, to the best of our knowledge, less work is dedicated to investigating the weighted cases in [\[PROBLEM06\]](#PROBLEM06){reference-type="eqref" reference="PROBLEM06"}. When the diffusion operate is linear, that is, when $p=2$, we refer to [@MZ2023; @JX2022] for weighted fast diffusion equations and [@JRX2023] for weighted porous medium equations. From Theorem 3.11 in [@JX2022], Theorem 1.6 in [@MZ2023] and Theorem 2.10 in [@JRX2023], we see that the decay rate exponent $\alpha$ of the solution near the degenerate or singular points of the weights is just known to be in $(0,1)$ but its value is not explicit. Therefore, a natural problem is to solve the optimal decay rate of the solution near these degenerate or singular points of the weights. In fact, we can use the standard separation of variables method to obtain exact solution for heat conduction equation with monomial weight $|x|^{\theta}$ in cylindrical domain. The value of $\theta$ can affect the diffusion rate in time variable with either enhancement or reduction. Based on these above facts, the problem of accurately quantifying the enhancement and reduction effect induced by weighted diffusion operators may have a potential application in manufacturing porous medium materials with special permeation rates and controlling diffusion processes according to the needs of the industry. Denote $$\begin{aligned} &d\mu_{1}=w_{1}dx=|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}dx,\quad d\mu_{2}=w_{2}dx=|x'|^{\theta_{4}}|x|^{\theta_{5}}|x_{n}|^{\theta_{6}}dx, \notag\\ &d\mu_{3}=w_{3}dx=|x'|^{\theta_{7}}|x|^{\theta_{8}}|x_{n}|^{\theta_{9}}dx.\end{aligned}$$ Based on the investigations in this paper, we now sum up the following three intriguing problems for readers' convenience. **P1.** Find the optimal ranges of parameters $s,p,q,\beta$ and $\theta_{i}$, $i=1,2,...,9$ such that the following interpolation inequality holds: for any $u\in C^{\infty}_{0}(\mathbb{R}^{n})$, $$\begin{aligned} \|u\|_{L^{s}(\mathbb{R}^{n},w_{1})}\leq C\|u\|^{\beta}_{L^{p}(\mathbb{R}^{n},w_{2})}\|u\|^{1-\beta}_{L^{q}(\mathbb{R}^{n},w_{3})},\end{aligned}$$ where the constant $C$ depends only on $n,s,p,q,\beta$ and $\theta_{i}$, $i=1,2,...,9$. **P2.** Find the optimal ranges of parameters $p,\gamma$ and $\theta_{i}$, $i=1,2,...,6$ such that the following anisotropic weighted Poincaré inequality holds: for any $u\in W^{1,p}(B_{R},w_{2})$, $R>0$, $$\begin{aligned} \int_{B_{R}}\bigg|u-\frac{1}{|B_{R}|_{\mu_{1}}}\int_{B_{R}}ud\mu_{1}\bigg|^{p}d\mu_{1}\leq CR^{\gamma}\int_{B_{R}}|\nabla u|^{p}d\mu_{2},\end{aligned}$$ for some constant $C$ depending only on $n,p$ and $\theta_{i},$ $i=1,2,...,6$. **P3.** Solve the optimal decay rate of the solution to problem [\[PROBLEM06\]](#PROBLEM06){reference-type="eqref" reference="PROBLEM06"} near the singular or degenerate points of the weights and meanwhile classify the enhancement and reduction effect on the diffusion arising from anisotropic weights. This work was supported in part by the National Key research and development program of China (No. 2022YFA1005700 and 2020YFA0712903). C. Miao was partially supported by the National Natural Science Foundation of China (No. 12371095 and 12071043). Z. Zhao was partially supported by China Postdoctoral Science Foundation (No. 2021M700358). 99 G. Akagi, Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods. Arch. Ration. Mech. Anal. 247 (2023), no. 2, Paper No. 23, 38 pp. H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), no. 3, 311-341. D.G. Aronson and L.A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains. J. Differential Equations 39 (1981), no. 3, 378-412. M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163 (2002), no. 4, 259-293. H. Bahouri, J.-Y. Chemin and I. Gallagher, Refined Hardy inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 3, 375-391. J.B. Berryman and C.J. Holland, Stability of the separable solution for fast diffusion. Arch. Rational Mech. Anal. 74 (1980), 379-388. V. Bögelein, F. Duzaar, P. Marcellini and C. Scheven, Doubly nonlinear equations of porous medium type. Arch. Ration. Mech. Anal. 229 (2018), no. 2, 503-545. V. Bögelein, F. Duzaar, P. Marcellini and C. Scheven, A variational approach to doubly nonlinear equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 4, 739-772. V. Bögelein, F. Duzaar, U. Gianazza, N. Liao and C. Scheven, Hölder Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations. arXiv:2305.08539. M. Bonforte, A. Figalli and X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains. Comm. Pure Appl. Math. 70 (2017), no. 8, 1472-1508. M. Bonforte, A. Figalli and J.L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Anal. PDE 11 (2018), no. 4, 945-982. M. Bonforte and A. Figalli, Sharp extinction rates for fast diffusion equations on generic bounded domains. Comm. Pure Appl. Math. 74 (2021), no. 4, 744-789. M. Bonforte, G. Grillo and J.L. Vázquez, Behaviour near extinction for the Fast Diffusion Equation on bounded domains. J. Math. Pures Appl. (9) 97 (2012), no. 1, 1-38. M. Bonforte, J.L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations. Adv. Math. 223 (2010), no. 2, 529-578. M. Bonforte, Y. Sire and J.L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5725-5767. L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Composition Math. 53 (1984), 259-275. X. Cabré and X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights. J. Differential Equations 255 (2013), no. 11, 4312-4336. X. Cabré, X. Ros-Oton and J. Serra, Sharp isoperimetric inequalities via the ABP method. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 12, 2971-2998. S. Chanillo and R.L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. Amer. J. Math. 107 (1985), no. 5, 1191-1226. S.-K. Chua and R.L. Wheeden, Estimates of best constants for weighted Poincaré inequalities on convex domains. Proc. London Math. Soc. (3) 93 (2006), no. 1, 197-226. P. Daskalopoulos and C. Kenig, Degenerate diffusions. Initial value problems and local regularity theory. EMS Tracts in Mathematics, 1. European Mathematical Society (EMS), Zürich, 2007. P.M.N. Feehan and C.A. Pop, A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients. J. Differential Equations 254 (2013), no. 12, 4401-4445. M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31-46. R. Duarte and J. Drumond Silva, Weighted Gagliardo-Nirenberg interpolation inequalities. J. Funct. Anal. 285 (2023), no. 5, Paper No. 110009, 49 pp. E. De Giorgi, Sulla differenziabilit$\grave{\mathrm{a}}$ e l'analiticit$\grave{\mathrm{a}}$ delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25-43. H.J. Dong, Y.Y. Li and Z.L. Yang, Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two. arXiv:2110.11313, J. Eur. Math. Soc., to appear. H.J. Dong, Y.Y. Li and Z.L. Yang, Gradient estimates for the insulated conductivity problem: the non-umbilical case. arXiv:2203.10081. H.J. Dong and T. Phan, Weighted mixed-norm $L_{p}$ estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem. J. Funct. Anal. 285 (2023), no. 2, Paper No. 109964, 43 pp. H.J. Dong, T. Phan and H.V. Tran, Degenerate linear parabolic equations in divergence form on the upper half space. Trans. Amer. Math. Soc. 376 (2023), no. 6, 4421-4451. H.J. Dong, T. Phan and H.V. Tran, Nondivergence form degenerate linear parabolic equations on the upper half space. arXiv:2306.11567. E. DiBenedetto and Y.C. Kwong, Harnack estimates and extinction profile for weak solution of certain singular parabolic equations. Trans. Amer. Math. Soc. 330 (2) (1992), 783-811. E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's inequality for degenerate and singular parabolic equations. Springer Monographs in Mathematics. Springer, New York, 2012. E. DiBenedetto, Y.C. Kwong and V. Vespri, Local space-analyticity of solutions of certain singular parabolic equations. Indiana Univ. Math. J. 40 (2) (1991), 741-765. L. Diening, P. Lindqvist and B. Kawohl, Mini-Workshop: The $p$-Laplacian operator and applications. Oberwolfach Rep. 10 (2013), no. 1, 433-482. E.B. Fabes, C.E. Kenig and R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7 (1982), no. 1, 77-116. E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimension. J. Dynam. Differential Equations 12 (2000), 647-673. L. Grafakos, Classical Fourier analysis. Third edition. Graduate Texts in Mathematics, 249. Springer, New York, 2014. J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations. Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola, NY, 2006. T.L. Jin and J.G. Xiong, Optimal boundary regularity for fast diffusion equations in bounded domains. Amer. J. Math. 145 (2023), no. 1, 151-219. T.L. Jin and J.G. Xiong, Regularity of solutions to the Dirichlet problem for fast diffusion equations. arXiv:2201.10091. T.L. Jin, X. Ros-Oton and J.G. Xiong, Optimal regularity and fine asymptotics for the porous medium equation in bounded domains. arXiv:2211.06124. T.L. Jin and J.G. Xiong, Hölder regularity for the linearized porous medium equation in bounded domains. arXiv:2303.00321. Y.C. Kwong, Interior and boundary regularity of solutions to a plasma type equation. Proc. Amer. Math. Soc. 104 (1988), no. 2, 472-478. P.K. Kythe, Fundamental solutions for differential operators and applications. Birkhäuser Boston, Inc., Boston, MA, 1996. xxiv+414 pp. L. Landau, A new exact solution of Navier-Stokes equations. C. R. (Doklady) Acad. Sci. URSS (N.S.) 43, (1944), 286-288. C.S. Lin, Interpolation inequalities with weights. Comm. Partial Differential Equations 11 (1986), no. 14, 1515-1538. P. Lindqvist, Notes on the stationary $p$-Laplace equation. SpringerBriefs in Mathematics. Springer, Cham, 2019. E. Lindgren and P. Lindqvist, On a comparison principle for Trudinger's equation. Adv. Calc. Var. 15 (2022), no. 3, 401-415. L. Li, Y.Y. Li, and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. I. One singularity. Arch. Ration. Mech. Anal. 227 (2018), no. 3, 1091-1163. L. Li, Y.Y. Li, and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. II. Classification of axisymmetric no-swirl solutions. J. Differential Equations 264 (2018), no. 10, 6082-6108. Y.Y. Li and X.K. Yan, Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations. J. Differential Equations 297 (2021), 226-245. Y.Y. Li and X.K. Yan, Anisotropic Caffarelli-Kohn-Nirenberg type inequalities. Adv. Math. 419 (2023), Paper No. 108958, 44 pp. C.X. Miao and Z.W. Zhao, Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights. Proc. Edinb. Math. Soc. (2) 66 (2023), no. 2, 391-436. H. Nguyen and M. Squassina, Fractional Caffarelli-Kohn-Nirenberg inequalities. J. Funct. Anal. 274 (2018), no. 9, 2661-2672. H. Nguyen and M. Squassina, On Hardy and Caffarelli-Kohn-Nirenberg inequalities. J. Anal. Math. 139 (2019), no. 2, 773-797. P.E. Sacks, Continuity of solutions of a singular parabolic equation. Nonlinear Anal. 7 (1983), no. 4, 387-409. N.S. Trudinger, Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math. 21 (1968), 205-226. J.L. Vázquez, The porous medium equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

arxiv_math

--- abstract: | We consider the problem of reconstructing inhomogeneities in an isotropic elastic body using time harmonic waves. Here we extend the so called monotonicity method for inclusion detection and show how to determine certain types of inhomogeneities in the Lamé parameters and the density. We also included some numerical tests of the method. address: - Institute of Mathematics, Goethe University Frankfurt, Germany - Institute of Mathematics, Goethe University Frankfurt, Germany author: - Sarah Eberle-Blick - Valter Pohjola title: The monotonicity method for inclusion detection and the time harmonic elastic wave equation --- # Introduction We study the problem of reconstructing the shape of inhomogeneities in elastic bodies from fixed frequency oscillations. This problem is of importance in nondestructive testing, and has potential applications in engineering, geoscience and medical imaging. Our main aim is to formulate a monotonicity based reconstruction method. The elastic properties of an isotropic elastic body $\Omega \subset \mathbb{R}^3$ can in the linear regime be described by means of the Lamé parameters $\lambda$ and $\mu$. The effect of a time harmonic oscillation in such a body $\Omega$ is described by the Navier equation, which gives the behaviour of the displacement field $u : \Omega \to \mathbb{R}^3$, $u \in H^1(\Omega)^3$ of the solid body $\Omega$. Here we consider $\Omega \subset \mathbb{R}^3$ with Lipschitz boundary, and the Navier equation in terms of the following boundary value problem $$\begin{aligned} \label{eq_bvp1} \begin{cases} \nabla \cdot (\mathbb{C}\, \hat \nabla u ) + \omega^2\rho u &= 0, \quad\text{in}\,\,\Omega,\\ \hspace{1.8cm}(\mathbb{C}\, \hat \nabla u ) \nu &= g, \quad\text{on}\,\,\Gamma_N, \\ \hphantom{\hspace{1.8cm}(\mathbb{C}\, \hat \nabla u ) } u &= 0, \quad\text{on}\,\,\Gamma_D, %\\ \end{cases}\end{aligned}$$ where $\Gamma_N$ and $\Gamma_D$ are such that $$\Gamma_N, \Gamma_D \subset \partial\Omega \text{ are open }, \qquad \Gamma_N \neq \emptyset, \qquad \partial\Omega = \overline{\Gamma}_N \cup \overline{\Gamma}_D,$$ and where $\hat \nabla u = \frac{1}{2}(\nabla u + (\nabla u)^T)$ is the symmetrization of the Jacobian (or the strain tensor), and $\mathbb{C}$ is the 4th order tensor defined by $$(\mathbb{C}A)_{ij} = 2\mu A_{ij} + \lambda \operatorname{tr}(A) \delta_{ij}, \quad \text{ where } A \in \mathbb{R}^{3 \times 3},$$ and $\delta_{ij}$ is the Kronecker delta. Here $\lambda,\mu \in L^\infty_+(\Omega)$ are scalar functions called the Lamé parameters, that describe the elastic properties of the material, $\rho \in L^\infty_+(\Omega)$ is the density of the material, and $\omega \neq 0$ the angular frequency of the oscillation, and $\nu$ is the outward pointing unit normal vector to the boundary $\partial\Omega$. See sections [2](#sec_prelim){reference-type="ref" reference="sec_prelim"} and [3.1](#sec_wellposed){reference-type="ref" reference="sec_wellposed"} for more details and definitions. The vector field $g \in L^2(\Gamma_N)^3$ acts as the source of the oscillation, and since $\mathbb{C}\,\hat \nabla u$ equals by Hooke's law to the Cauchy stress tensor, we see that the boundary condition $g$ specifies the traction on the surface $\partial\Omega$. We also make the standing assumption that $\omega \in \mathbb{R}$ is not a resonance frequency. By this we mean that zero is not an eigenvalue for the mixed eigenvalue value problem [\[eq_Neumann\]](#eq_Neumann){reference-type="eqref" reference="eq_Neumann"}. When this assumption holds, the problem [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} admits a unique solution for given boundary condition $g\in L^2(\Gamma_N)^3$. See Corollary [Corollary 6](#cor_ZeroEigenvalue){reference-type="ref" reference="cor_ZeroEigenvalue"}. We can thus define the Neumann-to-Dirichlet map $\Lambda:L^2(\Gamma_N)^3 \to L^2(\Gamma_N)^3$, as $$\Lambda: g \mapsto u|_{\Gamma_N}.$$ Thus $\Lambda$ maps the traction to the displacement $u|_{\Gamma_N}$ on the boundary. In this paper we deal with the shape reconstruction problem, which is also known as the inclusion detection problem. We are more specifically interested in determining the region where the material properties $\lambda$, $\mu$ or $\rho$ differ from some known constant background values $\lambda_0$, $\mu_0$ or $\rho_0$, when given a set of boundary measurements in the form of a Neumann-to-Dirichlet map $\Lambda$. This problem corresponds physically to determining the inhomogeneous regions in the body $\Omega$ from boundary measurements that give the displacement due to some tractions applied at the boundary. Here we more specifically assume that $\lambda_0,\mu_0,\rho_0 > 0$ are some known constants, and that there is a jump in the material parameters $$\begin{aligned} \lambda &= \lambda_0 + \chi_{D_1} \psi_1 ,\quad D_1 \Subset \Omega, \\ \mu &= \mu_0 + \chi_{D_2} \psi_{2}, \quad D_2 \Subset \Omega, \\ \rho &= \rho_0 - \chi_{D_3} \psi_{3}, \quad D_3 \Subset \Omega, \end{aligned}$$ where $\chi_{D_j}$ are the characteristic functions of the sets $D_j$ and $\psi_j |_{D_j} \in L_+^\infty(D_j)$, so that there is a jump in the material parameters at the boundaries $\partial D_j$ of the regions where the material parameters differ from the background values. We study this problem using the so called monotonicity method. The monotonicity method is a shape reconstruction method first formulated in [@TR02; @Ta06] and also [@Ge08], for the conductivity equation. In [@HPS19b] the method was extended to the analysis of time harmonic waves in the context of a Helmholtz type equation. In [@EH21] the monotonicity method was formulated for the recovery of inhomogeneities in an isotropic elastic body in the stationary case. In addition, the monotonicity methods can be applied for realistic data as shown, e.g., for the stationary elastic problem in [@EM21]. Our aim here is to study this problem in the non-stationary, or time harmonic case, of the Navier equation. The work here thus combines the ideas of [@HPS19b] and [@EH21]. Our main result is the justification of the shape reconstruction procedure outlined in Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"}. This is done in Theorems [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} and [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"}. To understand the algorithm consider the set $D \subset \Omega$ given by $$D = \operatorname{supp}(\lambda - \lambda_0) \cup \operatorname{supp}(\mu-\mu_0) \cup \operatorname{supp}(\rho-\rho_0).$$ The set $D$ is the region where the material parameters differ from background. Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} approximates the set $\operatorname{osupp}(D)$, which stands for the outer support of the set $D$[^1]. See definition [\[eq_def_osupp\]](#eq_def_osupp){reference-type="eqref" reference="eq_def_osupp"}. The set $\operatorname{osupp}(D)$ coincides with $D$ if $\Omega \setminus D$ is connected, and if not, then $\operatorname{osupp}(D)$ corresponds intuitively to $D$ and the union of all its internal cavities. Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} generates a collection of subsets $\mathcal A$, such that $\cup \mathcal{A}$ approximates the set $\operatorname{osupp}(D)$. This gives a good approximation of the shape of the inhomogeneous region $D$, disregarding any internal cavities. Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} works roughly by choosing a collection of subsets $\mathcal{B} = \{ B \subset \Omega\}$, and building an approximating collection $\mathcal{A}$ by choosing $B$, such that $B \subset \operatorname{osupp}(D)$. An approximation of $\operatorname{osupp}(D)$ is then obtained as $\cup \mathcal{A}$, from which we have removed any internal cavities. Note that in Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} $\Lambda$ denotes the measured Neumann-to-Dirichlet map, and $\Lambda^\flat$ a test Neumann-to-Dirichlet map built using the set $B$ and the prescripts in Theorems [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"} or [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"}. [\[alg_shapeInclusion\]]{#alg_shapeInclusion label="alg_shapeInclusion"} Choose a collection of sets $\mathcal{B} = \{ B \subset \Omega\}$. Set the approximating collection $\mathcal{A} = \{\}$. Compute $M_0$ using Theorem [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"}. Compute $N_B := \sum_{\sigma_k < 0} 1$, where $\sigma_k$ are the eigenvalues of $\Lambda - \Lambda^\flat$ (where the eigenvalues are counted as many times as its multiplicity indicates). Add $B$ to the approximating collection $\mathcal{A}$, since by Theorem [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"} we have that $B \subset \operatorname{osupp}(D)$. Discard $B$, since by Theorem [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"} $B \not \subset D_j$, $j=1,2,3$. Compute the union of all elements in $\mathcal{A}$ and all components of $\Omega \setminus \cup \mathcal{A}$ not connected to $\partial\Omega$. The resulting set is an approximation of $\operatorname{osupp}(D)$. The main novelty in Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} is that it extends the reconstruction method for Helmholtz type equations formulated in [@HPS19b] to the Navier equation, thus generalizing the work in [@HPS19b] and [@EH21]. Monotonicity methods for shape reconstruction have in general various advantages. Some of the benefits of the monotonicity methods are that it is simple to formulate and implement, it can be implemented efficiently using a linearization of Neumann-to-Dirichlet maps, and that indefinite inhomogeneities can be recovered. The shape reconstruction and related reconstruction methods for the Navier equation have received a fair amount of attention. This is partly due to the importance of this problem in geophysical and engineering applications. Several methods have been used to analyze the shape reconstruction problem for the Navier equation and related equations. For the Navier equation we have e.g. [@HKS12; @HLZ13; @EH19; @GK08], which are based on the factorization method. For work based on iterative and other methods see e.g. [@SFHC14; @BYZ19; @BHQ13; @BC05] and the references therein. There are also a number of works related to the shape reconstruction problem in elasticity in the stationary case $\omega =0$. See e.g. [@EH21; @II08; @Ik99]. The shape reconstruction problem has also been extensively studied for scalar equations, in particular for electrical impedance tomography, and acoustic and electromagnetic inverse problems. Several methods have been formulated. For this see e.g. [@Ta06; @Ge08; @GK08; @Ik90; @HPS19b; @CDGH20]. Shape reconstruction using the monotonicity method for time harmonic waves have in particular been explored by [@HPS19b; @HPS19a; @GH18; @Fu20a; @Fu20b; @AG23]. Calderón type inverse problems for isotropic linear elasticity have more generally been the subject of a number of studies. Early work on this problem include [@Ik90] that deals with the linearized problem, and [@ANS91] which deals with determining the coefficients at the boundary. One of the first more comprehensive uniqueness results in the three dimensional case was obtained in [@NU94] and also in [@ER02], where it is shown that the Lamé parameters $\mu$ and $\lambda$ are uniquely determined, in the stationary case by the Dirichlet-to-Neumann map assuming that $\mu$ is close to a constant. It should be noted that the general uniqueness problem is still open. There are a number of earlier and subsequent works related to the Calderón type inverse problem. We also present a preliminary numerical implementation of Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} in section [7](#sec_numerics){reference-type="ref" reference="sec_numerics"}. We want to remark that Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} is in many ways an idealized formulation, and that it holds for a sufficiently fine discretization. Since our computations do not have enough precision, we introduce a new discrete bound $\tilde{M}$, which is motivated due to our numerical experiments (see section [7](#sec_numerics){reference-type="ref" reference="sec_numerics"}). All in all, we end up with first numerical results for the elasto-oscillatory case which shows us that in principle the monotonicity tests work. We will now comment on the proofs and the main ideas that they involve in more detail. The analysis of the monotonicity method for time harmonic wave phenomena in [@HPS19b] is largely based on spectral theoretic considerations. Here we thus need at various points to fairly carefully analyze the spectral properties of both the direct problem [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} in the form of [\[eq_Neumann\]](#eq_Neumann){reference-type="eqref" reference="eq_Neumann"}, and the Neumann-to-Dirichlet map as compact operator on $L^2(\Gamma_N)^3$. One of the main ingredients of the monotonicity method is the use of localized solutions. This was originally achieved in [@Ge08] by the use of a range inclusion argument, which was also used to analyze the stationary case in [@EH21]. Here we explore an alternative approach of localizing solutions using Runge approximation as in [@HPS19b]. Runge approximation was originally formulated in the context of elliptic PDEs in [@La56] (for more on Runge approximation in the current context, see e.g. [@La56; @HPS19b; @HLL18; @Po22]). The Runge approach to localizing solutions is in principle straight forward, see e.g. [@HLL18]. In the monotonicity method for time harmonic waves we however need to work modulo a finite dimensional subspace as in Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"} and in [@HPS19b]. To create localized solutions we need to approximate certain candidate solutions on subsets of $D_1,D_2\subset\Omega$, with desired properties. For this we have to find enough candidate solutions with non-zero divergence. One central problem is for us thus to find a suitable criterion on a boundary condition of the Navier equation that guarantees non-zero divergence. This is one of the main challenges in making our Runge argument work, and this analysis is contained in subsection [5.1](#sec_div){reference-type="ref" reference="sec_div"}. The main idea here is to relate the Navier equation to the time harmonic Maxwell equation, and utilize the theory related to Maxwell's equations. This connection can also be used to create solutions when $\mu$ and $\rho$ are constant, with no divergence. The localized solutions also need to have a non-zero strain tensor $\hat \nabla u$. Achieving this is simpler, and for this we use a variant of the Korn inequality, see equation [\[eq_korn2_2\]](#eq_korn2_2){reference-type="eqref" reference="eq_korn2_2"} and Lemma [Lemma 4](#lem_symGradient){reference-type="ref" reference="lem_symGradient"}. A major obstacle in recovering multiple coefficients from a single frequency $\omega$ is that uniqueness is expected to fail when a density term is added to the stationary equation. One expects there to be several distinct tuples of coefficients that match the given boundary measurements, in analogy with the scalar case where this happens. A straight forward mechanism for obtaining counter examples to uniqueness in the scalar case of optical tomography was given in [@AL98], where it was shown that one cannot recover both a diffusion and an absorption coefficient from boundary measurements. This phenomenon also holds more widely, and thus a similar form of non-uniqueness should effect the corresponding inverse problem for the Navier equation. We note in this context that our shape reconstruction method works in the general case only for inhomogeneities where the density either remains constant or becomes smaller than the background density. One should also note that that two different frequencies are enough to recover multiple coefficients in the closely related acoustic problem, see [@Na88]. One can also impose special conditions on the coefficients, such as them being piecewise constant or analytic in order to recover several coefficients from a single frequency, see [@Ha09]. Piecewise constant coefficients are in the case of the Navier equation explored in [@BHFVZ17], where stability and uniqueness and local reconstruction are considered. Several aspects of inverse problems are left unstudied in the current paper. We only consider the specific form of perturbations, where the Lamé parameters increases and the density decreases in the inhomogeneous regions, or where the Lamé parameters remain constant and the density increases. Natural questions which we do not address here is if the case where all parameters increase in the homogeneity can somehow be handled, and if we can get some improvements when setting some of the coefficients as constant. Our numerical implementation is also in some aspects preliminary. We do not consider a regularized method that could handle noisy data nor a linearized version that would be more efficient. The paper is structured as follows. In section [2](#sec_prelim){reference-type="ref" reference="sec_prelim"} we review some definitions and properties that will be used in the rest of the paper. In section [3](#sec_direct_spec){reference-type="ref" reference="sec_direct_spec"} we study the direct problem and the mixed eigenvalue problem related to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}. Section [4](#sec_mono_ineq){reference-type="ref" reference="sec_mono_ineq"} contains the monotonicity inequality that underlies our analysis. In section [5](#sec_localization){reference-type="ref" reference="sec_localization"} we create the localized solutions that become small or large in certain prescribed subsets. In section [6](#sec_shape){reference-type="ref" reference="sec_shape"} we justify Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"}. In the last section, we conduct some numerical tests of the algorithm. # Preliminaries {#sec_prelim} In this section we will review some notations, definitions, and preliminary results that are used throughout the paper. We begin by reviewing the definitions related to function spaces. We define $$L^\infty_+(\Omega) := \big\{ f \in L^\infty(\Omega) \;:\; \operatorname{essinf}_\Omega f > 0 \big\}.$$ The space $H^1(\Omega)$ denotes the $L^2(\Omega)$ based Sobolev space with one weak derivative. If $X$ is a function space, then $Z^n := Z \times \dots \times Z$, where the right hand side contains $n$ copies of $Z$. The $L^2$-inner product is denoted by $(\cdot,\cdot)_{L^2}$, so that $$(u,v)_{L^2(\Omega)^n} := \int_\Omega u \cdot v\,dx, \quad u,v \in L^2(\Omega)^n.$$ We use the notation $\perp$ for orthogonality with respect to the $L^2$-inner product, unless otherwise stated, so that $$u \perp v \qquad \Leftrightarrow \qquad (u,v)_{L^2(\Omega)^n} = 0, \quad \text{ when } u,v \in L^2(\Omega)^n.$$ Next we will review some definitions related to matrices and vectors. The Frobenius inner product $A:B$ is defined as $$A:B = \sum_{ij} A_{ij}B_{ij}, \qquad A,B \in \mathbb{R}^{m\times n}.$$ We denote the Euclidean norm on $\mathbb{R}^{m \times n}$, $m,n \in \mathbb{N}$, by $$|A| = (A:A)^{1/2},\qquad \text{ when } A \in \mathbb{R}^{m \times n}.$$ Note that this definition applies also to vectors and scalars. The matrix divergence is give by $$(\nabla \cdot A)_{i} = (\nabla \cdot (\operatorname{row}_i A))_i, \qquad A \in H^1(\Omega)^{3 \times 3}.$$ We will use a tensor form of the divergence theorem, that states $$\int_\Omega \nabla \cdot A u \,dx = - \int_\Omega A:\hat \nabla u \,dx + \int_{\partial\Omega} A \nu \cdot u\,dS,$$ for $A \in H^1(\Omega)^{n\times n}$. See [@Ci88] p. xxix. Note that this gives the tensor form of the divergence theorem, since $A$ can here be interpreted as a 2nd order tensor, provided that the coefficients are regular enough. The bilinear form related to equation [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} is given by $$\begin{aligned} \label{eq_weak} B(u,v) := -\int_\Omega 2 \mu \hat \nabla u :\hat \nabla v + \lambda \nabla \cdot u \nabla \cdot v - \omega^2\rho u\cdot v\,dx, \end{aligned}$$ for all $u,v \in H^1(\Omega)^3$. For the precise definition of a weak solution that we employ here see section [3.1](#sec_wellposed){reference-type="ref" reference="sec_wellposed"}. We will also use the notations $$\begin{aligned} \label{eq_L} L_{\lambda,\mu,\rho} u := \nabla \cdot (\mathbb{C}\hat \nabla u ) + \omega^2\rho u = \nabla \cdot ( 2 \mu \hat \nabla u + \lambda (\nabla \cdot u) I ) + \omega^2\rho u,\end{aligned}$$ where $I$ is the identity matrix. When the Lamé parameters and density are regular, and $u$ solves [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} with $g$, and $v$ solves [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} with $h$, we see by integrating by parts that $$\begin{aligned} \label{eq_NDmap} B(u,v) = -\int_{ \Gamma_N} g \cdot v \,dS = -( g, \Lambda h)_{L^2(\Gamma_N)^3}. \end{aligned}$$ As it will later turn out, this holds also for a weak solution $u \in H^1(\Omega)^3$ by definition [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"}. Let $\Gamma \subset \partial\Omega$ be open. We abbreviate the boundary condition in [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} by $$\begin{aligned} \gamma_{\mathbb{C},\Gamma} u = (\mathbb{C}\, \hat \nabla u ) \nu |_{\Gamma},\end{aligned}$$ or with $\gamma_\mathbb{\mathbb{C}} u$ if the boundary is clear from the context. Note that these notations are formal when $u \in H^1(\Omega)^3$, since we cannot in general take the trace of a $L^2(\Omega)^3$ function. In the low regularity case we understand the boundary condition in a weak sense as suggested by [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"}. We can also define $\gamma_{\mathbb{C}} u \in L^2(\Gamma_N)^3$, provided that $u \in H^1(\Omega)^3$ solves [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}. In this case we use duality, so that $$\begin{aligned} -(\gamma_{\mathbb{C}} u, \, \varphi|_{\Gamma_N} )_{L^2(\Gamma_N)^3} = B(u,\varphi), \qquad \forall \varphi \in H^1(\Omega)^3. \end{aligned}$$ When the boundary value problem [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} admits a unique solution for all $g \in L^2(\Gamma_N)^3$, then the Neumann-to-Dirichlet map $\Lambda$ is well-defined.\ \ The localized solutions we consider in section [5](#sec_localization){reference-type="ref" reference="sec_localization"}, are built using the unique continuation principle, which is hence also an important tool in the sequel. We need a version of the unique continuation principle that applies to the Navier equation. For this we use Theorem 1.2 in [@LNUW11]. Notice that these results require the Lamé parameter $\mu$ to be Lipschitz continuous. **Proposition 1**. *Assume that $\Omega$ is open and connected. Let $\lambda,\rho \in L^\infty(\Omega)$ and $\mu \in W^{1,\infty}(\Omega)$, be such that $$\begin{aligned} &\mu(x),\; 2\lambda(x) + \mu(x) > \delta_0,\quad \forall \text{ a.e. } x \in \Omega, \\ &\| \mu \|_{ W^{1,\infty}(\Omega) } + \| \lambda \|_{ L^\infty(\Omega) } \leq M_0,\quad \|\rho \|_{ L^\infty(\Omega) } \leq M_0,\end{aligned}$$ where $M_0,\delta_0 > 0$. Assume that $u$, solves the equation $L_{\lambda,\mu,\rho} u = 0$ in $\Omega$ and $u = 0$ in some open subset $U \subset \Omega$, then $u=0$ in $\Omega$.* *Proof.* This is a direct consequence of the strong unique continuation principle of Theorem 1.2 in [@LNUW11]. ◻ **Proposition 2**. *Assume that $\partial\Omega$ is $C^{1,1}$ smooth and that $\lambda,\rho \in L^\infty(\Omega)$ and $\mu \in W^{1,\infty}(\Omega)$ are as in Proposition [Proposition 1](#prop_UCP){reference-type="ref" reference="prop_UCP"}. Suppose $u$ is such that $$L_{\lambda,\mu,\rho} u = 0, \quad \text{ in } \Omega, \qquad u|_\Gamma = 0, \qquad \gamma_\mathbb{C}u |_\Gamma = 0,$$ where $\Gamma \subset \partial\Omega$ is open and non-empty, then $u=0$ in $\Omega$.* *Proof.* We can show that the claim follows from Proposition [Proposition 1](#prop_UCP){reference-type="ref" reference="prop_UCP"}. If $x_0 \in \Gamma$, then we can extend the solution $u$ by zero, to $B(x_0,r) \setminus \Omega$, where $B(x_0,r)$ is a ball, with sufficiently small $r>0$. The claim follows now from Proposition [Proposition 1](#prop_UCP){reference-type="ref" reference="prop_UCP"}. ◻ # The direct problem and spectral properties {#sec_direct_spec} In the next two subsections we will investigate the existence and uniqueness of the solutions to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} and a related mixed eigenvalue problem in some detail. We furthermore derive a norm estimate where the norm of the strain tensor controls the norm of the Jacobian, and use spectral representation to derive a convenient formula for the expression $B(w,w)$. These results will be of importance in the subsequent sections. ## A variational argument {#sec_wellposed} In this subsection we investigate the well-posedness of the inhomogeneous problem [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"}, and further derive a useful norm estimate. In the next subsection we use these results in conjunction with Fredholm theory to show that the boundary value problem [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} is well-posed, and investigate the related eigenvalue problem. Through out this subsection we assume that $\Omega \subset \mathbb{R}^3$ is a bounded Lipschitz domain and that $\lambda,\mu,\rho \in L_+^\infty(\Omega)$. Recall moreover that we assume that $$\Gamma_N, \Gamma_D \subset \partial\Omega \text{ are open }, \qquad \Gamma_N \neq \emptyset, \qquad \partial\Omega = \overline{\Gamma}_N \cup \overline{\Gamma}_D.$$ We will first show that there exists a unique weak solution $u \in H^1(\Omega)^3$ to the mixed boundary value problem $$\begin{aligned} \label{eq_bvp2} \begin{cases} \nabla \cdot (\mathbb{C}\, \hat \nabla u ) + \omega^2\rho u + \tau u&= F, \\ \;\quad\quad\quad\quad\quad\quad(\gamma_\mathbb{C}u ) |_{ \Gamma_N} &= g, \\ \;\quad\quad\quad\quad\quad\quad\quad\quad u |_{\Gamma_D} &= 0, %\\ \end{cases}\end{aligned}$$ for suitable $\tau \in \mathbb{R}$, when $g \in L^2(\Gamma_N)^3$ and $F \in L^2(\Omega)^{3}$. In order to specify what we mean by a weak solution to [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"}, we define the bilinear form $$B_\tau(u,v) := B(u,v) + \tau (u,v)_{L^2(\Omega)^3}, \quad u,v \in \mathcal{V},$$ where $\mathcal{V} \subset H^1(\Omega)^3$ is the closed subspace $$\mathcal{V} := \{ u \in H^1(\Omega)^3 \,:\, u|_{\Gamma_D} = 0 \}.$$ The weak solution $u \in \mathcal{V}$ to [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"} is a vector field that satisfies $$\begin{aligned} \label{eq_weak2} B_\tau(u,v) = - \int_{\Gamma_N} g\cdot v \,dS + \int_\Omega F \cdot v \,dx\end{aligned}$$ for every $v \in \mathcal{V}$.[^2] One of the main tools for obtaining existence and uniqueness results in linear elasticity is the second Korn inequality. We will need the following version of this inequality $$\begin{aligned} \label{eq_korn2_1} \| u \|_{H^1(\Omega)^3} \leq C \big(\| \hat \nabla u \|_{L^2(\Omega)^{3 \times 3}} + \| u \|_{ L^2(\Omega)^3}\big), \quad u \in H^1(\Omega)^3, \end{aligned}$$ where $C >0$ is a constant. A proof of this can be found in [@OSY92], see Theorem 2.4 p.17. Next we prove existence and uniqueness for solutions to the boundary value problem [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"}. **Proposition 3**. *There exists a $\tau_0 \leq 0$, for which the boundary value problem in [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"} admits a unique weak solution $u \in H^1(\Omega)^3$, that satisfies $$\begin{aligned} \label{eq_aprioriEst} \| u \|_{ H^1(\Omega)^3} \leq C \big(\| g \|_{ L^2(\Gamma_N)^3}+ \| F \|_{ L^2(\Omega)^3 } \big).\end{aligned}$$* *Proof.* We use the Lax-Milgram Lemma to prove the uniqueness and existence of a weak solution (see e.g. Theorem 1.3 in [@OSY92]). We need to show that the bilinear form $B_\tau: \mathcal{V}\times \mathcal{V}\to \mathbb{R}$ is coercive and continuous for some $\tau \leq 0$. For coercivity we need to show that $$\begin{aligned} \label{eq_Btau_coer} |B_{\tau_0}(u,u)| \geq c \| u \|^2_{ H^1(\Omega)^3}, \quad c>0,\end{aligned}$$ for some $\tau_0 \leq 0$. Let $\tau \leq 0$. The Korn inequality [\[eq_korn2_1\]](#eq_korn2_1){reference-type="eqref" reference="eq_korn2_1"} gives that $$\begin{aligned} |B_\tau(u,u)| &\geq \int_\Omega 2 \mu \hat \nabla u :\hat \nabla u + \lambda \nabla \cdot u \nabla \cdot u -\tau \int_\Omega u^2\,dx - \omega^2 \| \rho \|_{L^\infty}\int_\Omega u^2\,dx \\ %\| u \|^2_{L^2(\Omega)^3} \\ &\geq C \big(\| \hat \nabla u \|^2_{L^2(\Omega)^{3\times 3}} +\| \nabla \cdot u \|^2_{L^2(\Omega)} \big) -(\tau + \omega^2\| \rho \|_{L^\infty})\| u \|^2_{L^2(\Omega)^3} \\ &\geq C \| u \|^2_{H^1(\Omega)^3} -(C+\tau + \omega^2\| \rho \|_{L^\infty})\| u \|^2_{L^2(\Omega)^3}. \end{aligned}$$ Thus if we choose a $|\tau|$ large enough with $\tau\leq 0$, the 2nd term is positive and can be dropped. We thus see that we can choose a $\tau_0 \leq 0$ so that [\[eq_Btau_coer\]](#eq_Btau_coer){reference-type="eqref" reference="eq_Btau_coer"} holds. The continuity of $B_\tau$ follows from the estimate $$\begin{aligned} |B_\tau(u,v)| &= \Bigg|\int_\Omega 2 \mu \hat \nabla u :\hat \nabla v + \lambda \nabla \cdot u \nabla \cdot v - (\tau + \omega^2\rho) u\cdot v\,dx \Bigg| \\ &\leq C (\| \hat \nabla u \|_{L^2(\Omega)^{3\times 3}} \| \hat \nabla v \|_{L^2(\Omega)^{3\times 3}} +\| \nabla \cdot u \|_{L^2(\Omega)} \| \nabla \cdot v \|_{L^2(\Omega)} +\| u \|_{L^2(\Omega)^3} \| v \|_{L^2(\Omega)^3}) \\ &\leq C \| u \|_{H^1(\Omega)^3} \| v \|_{ H^1(\Omega)^3}.\end{aligned}$$ The last step before applying the Lax-Milgram Lemma is to check that the left side of [\[eq_weak\]](#eq_weak){reference-type="eqref" reference="eq_weak"} gives a continuous functional on $\mathcal{V}$. To this end note that by the Cauchy-Schwarz and trace inequalities $$\begin{aligned} \label{eq_rhs_cont} - \int_{\Gamma_N } g\cdot v \,dS \leq \big\| g \,\big\|_{ L^2(\Gamma_N)^3 } \big\| v|_{\partial\Omega} \big\|_{ L^2(\partial\Omega)^3} \leq C \| v \|_{ H^1(\Omega)^3},\end{aligned}$$ and similarly by the Cauchy-Schwarz inequality $$\begin{aligned} \label{eq_rhs_cont_2} \int_\Omega F \cdot v \,dx \leq C \| v \|_{ H^1(\Omega)^3}.\end{aligned}$$ The Lax-Milgram Lemma gives us thus the existence of a unique $u \in \mathcal{V} \subset H^1(\Omega)^3$ for which [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"} holds for all $v \in \mathcal{V}$, and when $\tau \leq 0$, $|\tau|$ is large enough. We have thus found a weak solution in accordance with [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"}. The final step is to verify the estimate of the claim. By [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"}, [\[eq_Btau_coer\]](#eq_Btau_coer){reference-type="eqref" reference="eq_Btau_coer"}, [\[eq_rhs_cont\]](#eq_rhs_cont){reference-type="eqref" reference="eq_rhs_cont"} and [\[eq_rhs_cont_2\]](#eq_rhs_cont_2){reference-type="eqref" reference="eq_rhs_cont_2"} we have that $$\| u \|_{H^1(\Omega)^3 } \leq C (\| g \,\|_{ L^2(\Gamma_N)^3 } + \| F \|_{ L^2(\Omega)^{3}}).$$ ◻ Later we will be interested in solutions, where the strain tensor $\hat \nabla u$ controls the entire Jacobian $\nabla u$. In order to investigate this we need to introduce some additional concepts. The space of rigid motions $\mathcal{R}$ is given by $$\mathcal{R} :=\big \{ \eta : \mathbb{R}^3 \to \mathbb{R}^3 \,:\, \eta(x) = Ax + c,\; A \in \mathbb{R}^{3\times 3} \text{ is antisymmetric }, c \in \mathbb{R}^3\big \}.$$ (See e.g. p.19 in [@OSY92].) With this subspace we can formulate another version of the second Korn inequality that applies to $u \in W$, where $W \subset H^1(\Omega)^3$ is a closed subspace and $W\cap\mathcal{R} = \{0\}$. For such $u$ we have that $$\begin{aligned} \label{eq_korn2_2} \| u \|_{H^1(\Omega)^3} \leq C \| \hat \nabla u \|_{L^2(\Omega)^{3\times 3}}, \end{aligned}$$ where $C>0$ is a constant. For a proof of this see [@OSY92] theorem 2.5 p.19. We can use [\[eq_korn2_2\]](#eq_korn2_2){reference-type="eqref" reference="eq_korn2_2"} to prove the following Lemma, which gives a criterion on the boundary condition $g$ that allows for controlling the $H^1$-norm of the solution by the strain tensor, and which we use in section [5](#sec_localization){reference-type="ref" reference="sec_localization"}. **Lemma 4**. *Suppose $u \in H^1(\Omega)^3$ solves [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"} with $F=0$, $\Gamma_D = \emptyset$, $\Gamma_N = \partial\Omega$ and let $g \in L^2(\Omega)^3$ be such that $$g \perp \mathcal{R},\qquad \text{ in the $L^2(\partial\Omega)^3$-inner product}.$$ then we have the estimate $$\begin{aligned} \label{eq_JacobianEst} \| u \|_{H^1(\Omega)^3} \leq C \| \hat \nabla u \|_{L^2(\Omega)^{3\times 3}}, \end{aligned}$$ where $C>0$ is a constant.* *Proof.* According to [\[eq_korn2_2\]](#eq_korn2_2){reference-type="eqref" reference="eq_korn2_2"} it is enough to show that $$\mathcal{W} \cap \mathcal{R} = \{0\},$$ where $\mathcal{W} \subset H^1(\Omega)^3$ is the closed subspace given by $$\mathcal{W} := \big\{ u \in H^1(\Omega)^3 \;:\; u \text{ solves \eqref{eq_bvp2} with } F=0, g \perp \mathcal{R} \big \}.$$ It is straight forward to check that this is indeed a closed subspace. To check that the condition holds, assume that $u \in \mathcal{W} \cap \mathcal{R}$. Then $u = \tilde r$, $\tilde r \in \mathcal{R}$. Since $u$ is a weak solution we have by [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"} that $$B(\tilde r,r) = (g,r)_{L^2(\partial\Omega)^3}=0, \qquad \forall r\in \mathcal{R}.$$ Choosing $r = \tilde r$ and since $\tilde r = Ax +c$ where $A$ is antisymmetric, we have that $\hat \nabla \tilde r=0$ and $\nabla \cdot \tilde r = 0$, and thus $$B(\tilde r ,\tilde r ) = k^2\int_\Omega \rho \tilde r^2 \,dx.$$ Combining the two previous equations and using the fact that $\rho$ is strictly positive, gives that $u = \tilde r = 0$, which is what we wanted to show. ◻ ## The spectrum, well-posedness and a representation formula Next we will use Fredholm theory to describe the eigenvalues and eigenfunctions related to the boundary value problem [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"}. This is a mixed eigenvalue problem sometimes called the Zaremba eigenvalue problem, see [@Za10]. Note that if $\Gamma_D = \emptyset$, then we are dealing with the Neumann eigenvalue problem. In Corollary [Corollary 6](#cor_ZeroEigenvalue){reference-type="ref" reference="cor_ZeroEigenvalue"} we show that the boundary value problem [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} is well-posed. At the end of this subsection we will also derive a spectral representation of the expression $B(w,w)$. The next Proposition encapsulates the main spectral theoretic facts related to the direct problem that are needed in the sequel. **Proposition 5**. *There exists a sequence of $\sigma_k \in \mathbb{R}$, such that $$\sigma_1 \geq \sigma_2 \geq \sigma_3 \geq \dots \to - \infty$$ and such that the mixed eigenvalue problem $$\begin{aligned} \label{eq_Neumann} \begin{cases} \nabla \cdot (\mathbb{C}\, \hat \nabla \varphi_k ) + \omega^2\rho\varphi_k &= \sigma_k \varphi_k,\\ \quad\quad\quad\quad\quad\quad \gamma_\mathbb{C} \varphi_k |_{\Gamma_N} &= 0, \\ \quad\quad\quad\quad\quad\quad\quad \varphi_k |_{\Gamma_D} &= 0, \end{cases}\end{aligned}$$ has a non-zero solution $\varphi_k \in H^1(\Omega)^3$. We have moreover that:* 1. *The eigenvalues $\{\sigma_k\}$ are of finite multiplicity and the eigenfunctions $\{\varphi_k\}$ form an orthonormal basis in $L^2(\Omega)^3$.\ * 2. *There are only finitely many positive eigenvalues $\{\sigma_1,..,\sigma_K \}$.\ * 3. *The boundary value problem $$\begin{aligned} \label{eq_bvpLsource} \begin{cases} \nabla \cdot (\mathbb{C}\, \hat \nabla u ) + \omega^2\rho u + \tau u&= F, \\ %L_{\lambda,\mu,\rho}w, \\ \;\quad\quad\quad\quad\quad\quad\quad \gamma_\mathbb{C}u |_{\Gamma_N} &= 0, \\ \quad\quad\quad\quad\quad\quad\quad\quad u |_{\Gamma_D} &= 0, \end{cases}\end{aligned}$$ admits a unique solution $u \in H^1(\Omega)^3$ when $-\tau \notin \{ \sigma_1,\sigma_2,\dots \}$, and $F \in L^2(\Omega)^3$.\ * 4. *Suppose $0 \in \{ \sigma_1,\sigma_2,\dots \}$. Then there exists a unique solution $u \in H^1(\Omega)^3 / \mathcal{N}_0$[^3] to the boundary value problem [\[eq_bvpLsource\]](#eq_bvpLsource){reference-type="eqref" reference="eq_bvpLsource"} when $\tau=0$, provided that $$(F,\psi)_{L^2(\Omega)^3} = 0,\qquad \forall \psi \in \mathcal{N}_0,$$ where $\mathcal{N}_0$ is the eigenspace corresponding to zero.* *Proof.* Pick $g=0$ in [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"}, and let $\tau_0$ be as in Proposition [Proposition 3](#prop_Wellposedness){reference-type="ref" reference="prop_Wellposedness"}. Then Proposition [Proposition 3](#prop_Wellposedness){reference-type="ref" reference="prop_Wellposedness"} gives the right inverse operator $$K_{\tau_0} := (L_{\lambda,\mu,\rho} + \tau_0)^{-1} : L^2(\Omega)^3 \to H^1(\Omega)^3,$$ that maps a source term $F$ to the corresponding weak solution. Note that this is a compact operator on $L^2(\Omega)^3$, since the inclusion $H^1(\Omega)^3 \subset L^2(\Omega)^3$ is compact. $K_{\tau_0}$ is also self-adjoint, since $K_{\tau_0}$ is bounded and since $$\begin{aligned} (K_{\tau_0}F,G)_{L^2(\Omega)^3} = B_{\tau_0}(K_{\tau_0}F,K_{\tau_0}G) = B_{\tau_0}(K_{\tau_0}G,K_{\tau_0}F) = (F,K_{\tau_0}G)_{L^2(\Omega)^3}. \end{aligned}$$ The spectral Theorem for compact self-adjoint operators, see [@RSI] Theorem VI.16, shows that the spectrum of $K_{\tau_0}$ consists of eigenvalues $\tilde \sigma_k \in \mathbb{R}$, such that $$\begin{aligned} \label{eq_neg} K_{\tau_0} \tilde \varphi_k = \tilde \sigma_k \tilde \varphi_k, \qquad \tilde \sigma_k \to 0, \text{ as } k \to \infty,\end{aligned}$$ and that the corresponding eigenfunctions $\tilde \varphi_k$ form an orthonormal basis of $L^2(\Omega)^3$. Notice also that all the eigenvalues are negative. Using the fact that $K_{\tau_0}$ is the right inverse and orthonormality, we have that $$\begin{aligned} \label{eq_sign} C\| \tilde \varphi_k \|^2_{H^1(\Omega)^3} \leq -B_{\tau_0}(\tilde \varphi_k, \tilde \varphi_k) =\leq -\tfrac{1}{\tilde \sigma_k} B_{\tau_0}(K_{\tau_0}\tilde \varphi_k, \tilde \varphi_k) = -\tfrac{1}{\tilde \sigma_k} (\tilde \varphi_k,\,\tilde \varphi_k)_{L^2(\Omega)^3} = -\tfrac{1}{\tilde \sigma_k}.\end{aligned}$$ To obtain a solution to the eigenvalue problem [\[eq_Neumann\]](#eq_Neumann){reference-type="eqref" reference="eq_Neumann"}, notice that $$\tilde \sigma_k B_{\tau_0}(\tilde \varphi_k, w) = B_{\tau_0}(K_{\tau_0} \tilde \varphi_k, w) = (\tilde \varphi_k,\,w)_{L^2(\Omega)^3}.$$ So if we define $\sigma_k := \tilde \sigma_k^{-1} - \tau_0$ and $\varphi_k := \tilde \varphi_k$, then we have that $$B(\varphi_k,w) = B_{\tau_0}(\tilde \varphi_k, w) - \tau_0 (\tilde \varphi_k,w)_{L^2(\Omega)^3} = \sigma_k (\varphi_k,w)_{L^2(\Omega)^3}.$$ Thus $\varphi_k$ and $\sigma_k$ solve the weak form of the eigenvalue problem and by [\[eq_sign\]](#eq_sign){reference-type="eqref" reference="eq_sign"} we see that the eigenvalues converge to $-\infty$. This proves the first part of the claim. Part $(1)$ follows from the fact that $\tilde \varphi_k$ is an orthonormal basis in $L^2(\Omega)^3$. The second part $(2)$ follows now from [\[eq_neg\]](#eq_neg){reference-type="eqref" reference="eq_neg"} and [\[eq_sign\]](#eq_sign){reference-type="eqref" reference="eq_sign"}. We will use the Fredholm alternative to prove claim $(3)$. Let $F \in L^2(\Omega)^3$, $F\neq0$. The existence of a weak solution $u$ to [\[eq_Neumann\]](#eq_Neumann){reference-type="eqref" reference="eq_Neumann"} means that $$\begin{aligned} \label{eq_weakB} B_\tau(u,v) = (F,v)_{L^2(\Omega)^3}, \text{ for all } v \in \mathcal{V}. %H^1(\Omega)^3,\end{aligned}$$ Since $B_{\tau_0}(u,v) = (F+(\tau_0-\tau) u ,v)_{L^2(\Omega)^3}$ if and only if $u = K_{\tau_0}(F + (\tau_0-\tau) u)$, we have that $$\begin{aligned} \label{eq_intEq} u - (\tau_0-\tau) K_{\tau_0} u = K_{\tau_0} F.\end{aligned}$$ By the Fredholm alternative for a compact operator $T$ on $L^2(\Omega^3)$ we know that either $$\begin{aligned} \label{eq_nonHomogeneous} \exists ! u \in L^2(\Omega)^3 , \quad u - T u = h, \quad \forall h \in L^2(\Omega)^3, \end{aligned}$$ or the homogeneous equation has a non trivial solution, i.e. $$\begin{aligned} \label{eq_Homogeneous} \exists u \in L^2(\Omega)^3, u \neq 0,\quad u - T u = 0.\end{aligned}$$ See p. 203 in [@RSI]. Thus by the dichotomy between [\[eq_nonHomogeneous\]](#eq_nonHomogeneous){reference-type="eqref" reference="eq_nonHomogeneous"} and [\[eq_Homogeneous\]](#eq_Homogeneous){reference-type="eqref" reference="eq_Homogeneous"} applied to $T = (\tau_0-\tau)K_{\tau_0}$, we see that [\[eq_intEq\]](#eq_intEq){reference-type="eqref" reference="eq_intEq"} has a unique solution if and only if $$1 \notin \operatorname{Spec}((\tau_0 - \tau)K_{\tau_0}) \quad\Leftrightarrow\quad \frac{1}{\tau_0 -\tau} \neq \tilde \sigma_k \quad\Leftrightarrow\quad -\tau \neq \frac{1}{\tilde \sigma_k} - \tau_0 = \sigma_k.$$ It thus follows that [\[eq_weakB\]](#eq_weakB){reference-type="eqref" reference="eq_weakB"} has a unique weak solution $u$ if and only if $-\tau \notin \{\sigma_1,\sigma_2,\sigma_3,..\}$. The last step is to prove (4). We will again use Fredholm theory. Note firstly that now $\tau_0 \neq 0$, since $0$ is an eigenvalue of the boundary value problem. Since we assume that $\sigma_k = 0$, for some $k$, we have that $\tilde \sigma_k = \tau_0^{-1}$ is an eigennvalue of $K_{\tau_0}$. And thus there is a non trivial $w\neq 0$, solving $$\begin{aligned} \label{eq_intEq_2} w - \tau_0 K_{\tau_0} w = 0. %K_{\tau_0} F.\end{aligned}$$ The Fredholm alternative of Theorem 2.27 in [@Mc00] p.37--38, implies that the eigenspace of $K_{\tau_0}$ corresponding to $\tilde \sigma_k = \tau_0^{-1}$, which we denote by $\mathcal{M}_k$ and to which $w$ belongs, is such that $$\dim\big(\mathcal{M}_k\big) < \infty.$$ Since $K_{\tau_0}$ is self-adjoint, Theorem 2.27 in [@Mc00] gives furthermore that $$\begin{aligned} \label{eq_Kh} u - \tau_0 K_{\tau_0} u = h %K_{\tau_0} F,\end{aligned}$$ is solvable if and only if $$(h,\psi)_{L^2(\Omega)^3} = 0, \qquad \forall \psi \in \mathcal{M}_k.$$ Now since $K_{\tau_0}$ is self-adjoint, we have for $\psi \in \mathcal{M}_k$ that $$\begin{aligned} (F,\psi)_{L^2(\Omega)^3} = 0 \quad\Leftrightarrow\quad (K_{\tau_0}F,\psi)_{L^2(\Omega)^3} = 0,\end{aligned}$$ so that [\[eq_Kh\]](#eq_Kh){reference-type="eqref" reference="eq_Kh"} is solvable for $h = K_{\tau_0}F$, if $F \perp \mathcal{M}_k$. It follows that there exists a weak solution $u$ to $$\begin{aligned} \label{eq_weakB_2} B_0(u,v) = (F,v)_{L^2(\Omega)^3}, \text{ for all } v \in \mathcal{V}, %H^1(\Omega)^3,\end{aligned}$$ when $F \perp \mathcal{M}_k$. It is easy to check that this solution is unique in $H^1(\Omega)^3 / \mathcal{M}_k$. By definition we also have $\mathcal{M}_k = \mathcal{N}_0$ where $\mathcal{N}_0$ denotes the eigenspace of zero for the problem [\[eq_Neumann\]](#eq_Neumann){reference-type="eqref" reference="eq_Neumann"}. ◻ As a corollary to the previous Lemma we get the well-posedness of the boundary value problem of [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}, assuming that $\omega$ is not a resonance frequency. **Corollary 6**. *Let $\{\sigma_1,\sigma_2,..\}$ be the eigenvalues given by Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"}. For the boundary value problem $$\begin{aligned} \label{eq_bvp_inh} \begin{cases} \nabla \cdot (\mathbb{C}\, \hat \nabla u ) + \omega^2\rho u &= 0, \\ \;\quad\quad\quad\quad\quad \gamma_\mathbb{C}u |_{\Gamma_N} &= g, \\ \;\quad\quad\quad\quad\quad \quad u |_{\Gamma_D} &= 0, %\\ \end{cases}\end{aligned}$$ with $g \in L^2(\Gamma_N)^3$, the following holds:* 1. *The problem [\[eq_bvp_inh\]](#eq_bvp_inh){reference-type="eqref" reference="eq_bvp_inh"} admits a unique solution if $0 \notin \{ \sigma_1,\sigma_2,..\}$.* 2. *If zero is an eigenvalue, i.e. $0 \in \{ \sigma_1,\sigma_2,..\}$, then [\[eq_bvp_inh\]](#eq_bvp_inh){reference-type="eqref" reference="eq_bvp_inh"} admits a unique solution $u \in H^1(\Omega)^3 \,/\mathcal{N}_0$ for all $g\in L^2(\Gamma_N)^3$ that satisfy $$(g,\psi)_{L^2(\Gamma_N)^ 3} = 0,\quad \forall \psi \in \mathcal{N}_0,$$ where $\mathcal{N}_0$ is the eigenspace of zero.* *Proof.* Using Proposition [Proposition 3](#prop_Wellposedness){reference-type="ref" reference="prop_Wellposedness"} we can define an inverse trace operator $E_{\tau_0}:L^2(\Gamma_N)^3 \to H^1(\Omega)^3$ as $$E_{\tau_0}g := v,$$ where $v$ is the unique solution to [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"}, with $F=0$, and where $\tau_0$ is as in Proposition [Proposition 3](#prop_Wellposedness){reference-type="ref" reference="prop_Wellposedness"}. Let us prove part (2) of the claim. The proof of (1) follows similar lines. We can prove (2) by reducing it to finding a solution $w$ to the inhomogeneous case $$\begin{aligned} \label{eq_bvp_inh_2} \begin{cases} L_{\lambda,\mu,\rho} w &= \widetilde F, \\ \quad \gamma_\mathbb{C}w |_{\Gamma_N} &= 0, \\ \quad \quad w |_{\Gamma_D} &= 0, %\\ \end{cases}\end{aligned}$$ and applying Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"}. To solve [\[eq_bvp_inh\]](#eq_bvp_inh){reference-type="eqref" reference="eq_bvp_inh"} it is sufficient to solve [\[eq_bvp_inh_2\]](#eq_bvp_inh_2){reference-type="eqref" reference="eq_bvp_inh_2"} with $$\widetilde F:=-L_{\lambda,\mu,\rho}E_{\tau_0}g = \tau_0 v,$$ where the last equality holds, since $E_{\tau_0}g=v$ solves by definition $(L_{_{\lambda,\mu,\rho}} + \tau_0) E_{\tau_0}g = 0$. Note firstly that since $v \in H^1(\Omega)^3$, also $\widetilde F \in H^1(\Omega)^3$. By Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"} part (4) we know that [\[eq_bvp_inh_2\]](#eq_bvp_inh_2){reference-type="eqref" reference="eq_bvp_inh_2"} admits a unique solution in $H^1(\Omega)^3/\mathcal{N}_0$. To complete the proof, we need to check that we can rewrite the condition appearing in part (4) of Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"}, in the form appearing in (2) of the claim. To this end note that since a $\psi \in \mathcal{N}_0$ is a weak solution to the mixed eigenvalue problem for zero, and since $v$ is a solution to [\[eq_bvp2\]](#eq_bvp2){reference-type="eqref" reference="eq_bvp2"} with $g$ as Neumann data and $\tau = \tau_0$, we have that $$\begin{aligned} 0 = (\widetilde F , \psi)_{L^2(\Omega)^3} = \int_\Omega \tau_0 v \cdot \psi \,dx = \int_{\Omega} -\mathbb{C}\hat \nabla \psi : \hat \nabla v + \omega^2 \rho\psi \cdot v \,dx + \int_\Omega \tau_0 v\cdot \psi \,dx = -\int_{\Gamma_N} g \cdot \psi \,dS.\end{aligned}$$ We thus have the condition $$\begin{aligned} \label{eq_f_ortho} 0 = (g,\psi)_{L^2(\Gamma_N)^ 3}.\end{aligned}$$ And hence we see that [\[eq_bvp_inh_2\]](#eq_bvp_inh_2){reference-type="eqref" reference="eq_bvp_inh_2"} is solvable when this condition is satisfied. The solution to [\[eq_bvp_inh\]](#eq_bvp_inh){reference-type="eqref" reference="eq_bvp_inh"} is then given by $$u = w - E_{\tau_0} g.$$ We thus see that $u$ satisfies [\[eq_bvp_inh\]](#eq_bvp_inh){reference-type="eqref" reference="eq_bvp_inh"} when $g$ satisfies [\[eq_f\_ortho\]](#eq_f_ortho){reference-type="eqref" reference="eq_f_ortho"}, and consequently that (2) holds. ◻ We now turn to the final matter in this subsection. The eigenvalues $\{ \sigma_k\}$ and eigenfunctions $\{ \varphi_k\}$ allow us to 'diagonalize' the differential operator $L_{\lambda,\mu,\rho}$ in [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}. This gives us the identity $$(L_{\lambda,\mu,\rho}w,w)_{L^2(\Omega)^3} = \sum_k \sigma_k c_k^2, \quad \text{ where } \quad w = \sum_k c_k \varphi_k,$$ for any $w \in \mathcal{V}$. In the rest of this subsection we will derive some results that gives us a weak version of this that holds when $\lambda,\mu \in L_+^\infty(\Omega)$. By the coercivity and continuity of the bilinear form $B_{\tau_0}$, for suitable ${\tau_0} \leq 0$, which are guaranteed to exist by Proposition [Proposition 3](#prop_Wellposedness){reference-type="ref" reference="prop_Wellposedness"}, we have that $$(u,v)_{B_{\tau_0}} := -B_{\tau_0}(u,v), \qquad u,v \in \mathcal{V}, %:= \{u \,:\, u \in H^1(\Omega)^3 \},$$ gives an inner product on $\mathcal{V}$, and a corresponding norm $\| u \|^2_{ B_{\tau_0} } := (u,u)_{B_{\tau_0}}$. This allows us to extend the result of Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"}. **Corollary 7**. *Suppose $\varphi_k$ and $\sigma_k$ are the eigenfunctions given by Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"} and let ${\tau_0} \leq 0$ be as in Proposition [Proposition 3](#prop_Wellposedness){reference-type="ref" reference="prop_Wellposedness"}. Then we have that $$\psi_k := \frac{\varphi_k}{[-(\sigma_k + {\tau_0})]^{1/2}} , \qquad k=1,2,..$$ form an orthonormal basis of $\mathcal{V}$ in the inner product $(\cdot, \cdot)_{B_{\tau_0}}$ and corresponding norm.* *Proof.* To prove that the set $\{\psi_k \}$ is orthonormal, notice that $$\begin{aligned} (\psi_k,\psi_j)_{B_{\tau_0}} = -B(\psi_k,\psi_j) - {\tau_0}(\psi_k,\psi_j)_{L^2(\Omega)^3} = \frac{-B(\varphi_k,\varphi_j) - {\tau_0}(\varphi_k,\varphi_j)_{L^2(\Omega)^3}}{[-(\sigma_k + {\tau_0})]^{1/2}[-(\sigma_j + {\tau_0})]^{1/2}} = \delta_{jk},\end{aligned}$$ where we used the weak definition of the eigenvalue problem and the orthogonality of the functions $\varphi_k$. It remains to check that the set is complete. Suppose that $u \in \mathcal{V}$. Completeness follows if we can show that the condition $(u,\psi_k)_{B_{\tau_0}} = 0$, for every $k$, implies that $u = 0$. To this end we note that if $$0 = (u,\psi_k)_{B_{\tau_0}} = \frac{(u,\varphi_k)_{B_{\tau_0}}}{[-(\sigma_k + {\tau_0})]^{1/2}}$$ for every $k$, then $$0 = (u,\varphi_k)_{B_{\tau_0}} = -B(u,\varphi_k) - {\tau_0}(u,\varphi_k)_{L^2(\Omega)^3} = -(\sigma_k + {\tau_0})(u,\varphi_k)_{L^2(\Omega)^3}.$$ We know that $(\sigma_k + {\tau_0}) < 0$, since ${\tau_0}$ makes $B_{\tau_0}$ coercive, so that $(u,\varphi_k)_{L^2(\Omega)^3}=0$, for every $k$. The functions $\varphi_k$ are an orthonormal basis, and thus $u=0$. ◻ **Lemma 8**. *Suppose that $w \in \mathcal{V}$, then $$B(w,w) = \sum_k \sigma_k c_k^2,$$ where $w=\sum_k c_k \varphi_k$ is the representation via the orthonormal basis $\lbrace\varphi_k\rbrace$ by Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"}.* *Proof.* According to Corollary [Corollary 7](#cor_onorm){reference-type="ref" reference="cor_onorm"} we have the representation $$w=\sum_k c'_k \psi_k, \quad \psi_k := \frac{\varphi_k}{[-({\tau_0}+\sigma_k)]^{1/2}} , \quad c'_k := (w,\psi_k)_{B_{\tau_0}},$$ where $(u,v)_{B_{\tau_0}} = -B_{\tau_0}(u,v)$ and where the series converges in the corresponding $\|\cdot\|_{B_{\tau_0}}$-norm. We now have that $$\begin{aligned} -B_{\tau_0}(w,w) = -B_{{\tau_0}}\Big(\sum_k c'_k \psi_k ,\, \sum_j c'_j \psi_j\Big) = \sum_k |c'_k|^2.\end{aligned}$$ On the other hand we have that $$-B_{\tau_0}(w,w) = -B(w,w) - {\tau_0} (w,w)_{L^2(\Omega)^3},$$ so that $$B(w,w) = -\sum_k |c'_k|^2 - {\tau_0} (w,w)_{L^2(\Omega)^3}.$$ Let us compute the coefficients $c'_k$ in terms of $c_k$. We have that $$\begin{aligned} c'_k = (w,\psi_k)_{B_{\tau_0}} &= -B(w,\psi_k) - {\tau_0} (w,\psi_k)_{L^2(\Omega)^3} =\frac{-\sigma_k c_k - {\tau_0} c_k}{[-({\tau_0}+\sigma_k)]^{1/2}} = [-(\sigma_k + {\tau_0})]^{1/2} c_k.\end{aligned}$$ From the earlier equality we get that $$B(w,w) = -\sum_k [-(\sigma_k + {\tau_0})] c^2_k - {\tau_0} \sum_k c^2_k = \sum_k \sigma_k c^2_k,$$ which is what we wanted to prove. ◻ # Monotonicity inequalities {#sec_mono_ineq} In this section we derive some monotonicity relations that are of fundamental importance in justifying monotonicity based shape reconstruction, and will be needed in the later sections. The use of this type of inequalities goes back to [@KSS97] and [@Ik98]. It will be convenient to use the quantity $d(\lambda,\mu,\rho)$, which we define as $$\begin{aligned} \label{eq_def_d} d(\lambda,\mu,\rho) := \text{ the number of $\sigma_k > 0$ in problem \eqref{eq_Neumann} counted with multiplicity.}\end{aligned}$$ The following Lemma is the main monotonicity inequality we will be using. **Lemma 9**. *Let $\mu_j,\lambda_j,\rho_j \in L^\infty_+(\Omega)$, for $j=1,2$ and $\omega \neq 0$. Let $u_j$ denote the solution to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} where $\mu=\mu_j, \lambda= \lambda_j$ and $\rho=\rho_j$, with the boundary value $g$. There exists a finite dimensional subspace $V\subset L^2( \Gamma_N)^3$, such that $$\begin{aligned} %\label{eq_mono1} \big( (\Lambda_2 - \Lambda_1)g, \,g \big )_{L^2(\Gamma_N)^3} \geq \int_\Omega 2(\mu_1-\mu_2 ) |\hat \nabla u_1 |^2 + (\lambda_1 - \lambda_2 ) |\nabla \cdot u_1|^2 + \omega^2(\rho_2-\rho_1) |u_1|^2 \,dx, \end{aligned}$$ when $g \in V^\perp$. We have moreover that $\dim(V) \leq d(\lambda_2,\mu_2,\rho_2)$.* *Proof.* Let $B_j$ and $\Lambda_j$ be the bilinear form and Neumann-to-Dirichlet map given in [\[eq_weak\]](#eq_weak){reference-type="eqref" reference="eq_weak"} and [\[eq_NDmap\]](#eq_NDmap){reference-type="eqref" reference="eq_NDmap"}, with $\mu=\mu_j, \lambda= \lambda_j$ and $\rho=\rho_j$. Notice firstly that from [\[eq_weak\]](#eq_weak){reference-type="eqref" reference="eq_weak"} we can deduce that $$\begin{aligned} % \label{eq_Bchange} B_1(u_1,u_1) = -\int_{\Gamma_N} g\cdot u_1 \,dS = B_2(u_2,u_1). \end{aligned}$$ In particular we have that $$B_1(u_1,u_1) = - B_1(u_1,u_1) +2 B_2(u_2,u_1).$$ This and the symmetry of $B$, gives that $$\begin{aligned} \big( (\Lambda_2 - \Lambda_1)g, \,g \big )_{L^2(\Gamma_N)^3} &= B_1(u_1,u_1) - B_2(u_2,u_2) \\ &= -B_1(u_1,u_1) + 2 B_2(u_2,u_1) - B_2(u_2,u_2) \pm B_2(u_1,u_1) \\ &= -B_1(u_1,u_1) + B_2(u_1,u_1) - \big( B_2(u_2,u_2) - 2 B_2(u_2,u_1) + B_2(u_1,u_1) \big) \\ &= B_2(u_1,u_1) - B_1(u_1,u_1) - B_2(w,w),\end{aligned}$$ where $w = u_2-u_1$. The claim follows if we can show that there exists a finite dimensional subspace $V$, such that $$B_2(w,w) \leq 0 \quad \text{ for } g \in V^\perp,$$ since then we have that $$\begin{aligned} \big( (\Lambda_2 - \Lambda_1)g, \,g \big )_{L^2(\Gamma_N)^3} \geq B_2(u_1,u_1) - B_1(u_1,u_1) \quad \text{ for } g \in V^\perp,\end{aligned}$$ and since this is equivalent to the claim. Let $\{\sigma_k\}$ and $\{\varphi_k\}$ denote the eigenvalues and eigenfunctions of the mixed eigenvalue problem for the operator $L_{\mu_2, \lambda_2,\rho_2}$, that are given by Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"}. The positive eigenvalues are finite, because of Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"} part (2). Suppose they are enumerated as $$\sigma_1 \geq \dots \geq \sigma_{K} \geq 0.$$ Assume now that $w \perp \{\varphi_1,..,\varphi_K\}$ in the $L^2(\Gamma_N)^3$-inner product. Clearly we have that $w \in \mathcal{V}$. By the diagonalization result of Lemma [Lemma 8](#lem_diagonaliztion){reference-type="ref" reference="lem_diagonaliztion"}, we have, when writing $w = \sum_k c_k \varphi_k$, that $$\begin{aligned} B_2(w,w) = \sum_{k > K} \sigma_k c_k^2 \leq 0,\end{aligned}$$ since $\sigma_k < 0$, when $k\geq K$. The last step of the proof is to write the condition $w \in \operatorname{span}\{ \varphi_1,..,\varphi_{K} \}^\perp =: W^\perp$ in terms of the boundary condition $g$. Let $S_j : L^2(\Gamma_N)^3 \to L^2(\Omega)^3$ be the solution operators given by Corollary [Corollary 6](#cor_ZeroEigenvalue){reference-type="ref" reference="cor_ZeroEigenvalue"}, for which $$S_j:g\mapsto u_j, \quad j=1,2.$$ Now we have that $$\big(w,\varphi_k\big)_{L^2(\Omega)^3} =\big((S_2-S_1)g,\varphi_k\big)_{L^2(\Omega)^3} =\big(g,(S_2-S_1)^*\varphi_k\big)_{L^2(\Gamma_N)^3}.$$ And thus that $$w = (S_2-S_1)g \in W^\perp \quad\Leftrightarrow\quad g \in ((S_2-S_1)^*W)^\perp.$$ Note that $\dim ((S_2-S_1)^*W) < \dim(W)$, and that $\dim(\mathcal{W}) = d(\lambda_2, \mu_2, \rho_2)$. The claim of the lemma holds when we choose $V=(S_2-S_1)^*W$. ◻ As a direct consequence of Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"} we have the following. **Lemma 10**. *Assume that $\mu_1 \geq \mu_2$, $\lambda_1 \geq \lambda_2$ and $\rho_2 \geq \rho_1$. Then there exists a finite dimensional subspace $V \subset L^2(\Gamma_N)^3$ such that $$(\Lambda_2 g,g)_{L^2(\Gamma_N)^3} \geq (\Lambda_1 g,g)_{L^2(\Gamma_N)^3}, \qquad \forall g \in V^\perp.$$* *Proof.* Notice that Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"} gives directly that $$\begin{aligned} %\label{eq_mono1} \big( (\Lambda_2 - \Lambda_1)g, \,g \big )_{L^2(\Gamma_N)^3} \geq \int_\Omega 2(\mu_1-\mu_2 ) |\hat \nabla u_1|^2 + (\lambda_1 - \lambda_2 ) |\nabla \cdot u_1|^2 + \omega^2(\rho_2-\rho_1) |u_1|^2 \,dx \geq 0,\end{aligned}$$ for $g \in V^\perp$, which implies the claim. ◻ # Localization by Runge approximation {#sec_localization} We will prove localization results for solutions to the boundary value problem in [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}. The use of localized solutions go back to [@Ge08], and plays a fundamental part in justifying monotonicity based shape reconstruction. Here we use an approach based on Runge approximation as in [@HPS19b]. The localized solutions will later be used in conjunction with the monotonicity inequality of Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}. The main challenge in this is to employ Runge approximation with solutions which on the boundary lie in $V^\perp$, where $V$ is the subspace in Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}. We deal with this as in [@HPS19b]. Here we however have the extra complication of localizing $\nabla \cdot u$ and $\hat \nabla u$ suitably. We begin by showing how to localize the solutions assuming we have the approximation results of Lemma [Lemma 17](#lem_blowupCand){reference-type="ref" reference="lem_blowupCand"} and Lemma [Lemma 16](#lem_approxCrit){reference-type="ref" reference="lem_approxCrit"} at hand. We will postpone the proofs of Lemmas [Lemma 17](#lem_blowupCand){reference-type="ref" reference="lem_blowupCand"} and [Lemma 16](#lem_approxCrit){reference-type="ref" reference="lem_approxCrit"} to subsections [5.1](#sec_div){reference-type="ref" reference="sec_div"} and [5.2](#sec_runge){reference-type="ref" reference="sec_runge"}, as these need more a detailed analysis. We will localize a solution so that it is small in some set $D_1 \subset \Omega$ and large in another set $D_2 \subset \Omega$. We will assume that $\partial D_1$ is Lipschitz and that $\partial D_2$ is smooth, and moreover that $$\begin{aligned} \label{eq_Dassump} D_1 \cap D_2 = \emptyset, \qquad \Omega \setminus (D_1 \cup D_2) \text{ is connected}, \qquad \overline{\Omega \setminus (D_1 \cup D_2)}\cap\Gamma_N \neq \emptyset.\end{aligned}$$ **Proposition 11**. *Assume that $D_1,D_2 \subset \Omega$ are as in [\[eq_Dassump\]](#eq_Dassump){reference-type="eqref" reference="eq_Dassump"}. Let $V\subset L^2(\Gamma_N)^3$ be a subspace with $\dim (V)<\infty$, then there exists a sequence $g_j \in L^2(\Gamma_N)^3$ s.t. $g_j \perp V$ in the $L^2(\Gamma_N)^3$-norm, such that $$\| u_j \|_{L^2(D_1)^3 } \to 0, \qquad \| u_j \|_{L^2(D_2)^3 } \to \infty,$$ as $j \to \infty$, and where $u_j$ solves [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}, with the boundary conditions $g_j$.* *Proof.* By Lemma [Lemma 17](#lem_blowupCand){reference-type="ref" reference="lem_blowupCand"} and Lemma [Lemma 16](#lem_approxCrit){reference-type="ref" reference="lem_approxCrit"} there is a function $$w = \begin{cases} w \quad\text{ in }D_2 \quad\text{ with }\| w \|_{ L^2(D_2)^3 } \neq 0, \\ 0 \quad\,\,\text{ in }D_1. \end{cases}$$ which can be approximated by solutions $w_j$ to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} with the boundary condition $\tilde g_j \in V^\perp$, such that $$\| w_j - w \|_{L^2(\Omega)^3 } \leq \tfrac{1}{j^2}, \quad j \in \mathbb{N}.$$ We now set $u_j := jw_j$ and $g_j := j\tilde g_j$, then $$\| u_j - jw \|_{L^2(\Omega)^3 } \leq \tfrac{1}{j}.$$ It is easy to see that $u_j$ satisfy the desired norm estimates. The claim follows from this. ◻ The next Proposition extends Proposition [Proposition 11](#prop_localization1){reference-type="ref" reference="prop_localization1"} by showing that we can also localize the derivatives in a suitable way. **Proposition 12**. *Assume that $D_1,D_2 \subset \Omega$ are as in [\[eq_Dassump\]](#eq_Dassump){reference-type="eqref" reference="eq_Dassump"}, and that $D'_i \Subset D_i$, $i=1,2$ are open and non-empty. Let $V\subset L^2(\Gamma_N)^3$ be a subspace with $\dim (V)<\infty$, then there exists a sequence $g_j \in L^2(\Gamma_N)^3$, such that $g_j \perp V$ in the $L^2(\Gamma_N)^3$-norm, and for which $$\| u_j \|_{ L^2(D_1)^3 }, \;\| \hat \nabla u_j \|_{ L^2(D'_1)^{3\times 3} }, \;\| \nabla \cdot u_j \|_{ L^2(D'_1)} \to 0,$$ and for which $$\| u_j \|_{ L^2(D_2)^3 }, \;\| \hat \nabla u_j \|_{ L^2(D'_2)^{3\times 3} }, \;\| \nabla \cdot u_j \|_{ L^2(D'_2)} \to \infty,$$ as $j \to \infty$, and where $u_j$ solves [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}, with the boundary conditions $g_j$.* *Proof.* Let $u_j$ and $w$ be as in the proof of Proposition [Proposition 11](#prop_localization1){reference-type="ref" reference="prop_localization1"}. By Lemma [Lemma 17](#lem_blowupCand){reference-type="ref" reference="lem_blowupCand"} we know that we can pick these, so that $$\begin{aligned} \;\| \hat \nabla (jw) \|_{ L^2(D_2)^{3\times 3} }, \;\| \nabla \cdot (jw) \|_{ L^2(D_2)} \to \infty, \end{aligned}$$ as $j \to \infty$, and so that $$\begin{aligned} \;\| \hat \nabla (jw) \|_{ L^2(D_1)^{3\times 3} }, \;\| \nabla \cdot (jw) \|_{ L^2(D_1)} = 0. \end{aligned}$$ By elliptic regularity we have that $$\begin{aligned} \| \hat \nabla u_j - \hat \nabla (jw) \|_{L^2(D'_i)^{3\times 3}} \leq \| u_j - jw \|_{H^1(D'_i)^3} \leq C \| u_j - jw \|_{L^2(D_i)^3} \to 0,\end{aligned}$$ as $j \to \infty$. It follows that $$\;\| \hat \nabla u_j \|_{ L^2(D'_2)^{3\times 3}} \to \infty \qquad \;\| \hat \nabla u_j \|_{ L^2(D'_1)^{3\times 3}} \to 0.$$ The claim for $\nabla \cdot u_j$ can be proved in a similar manner. ◻ To complete the proof of Propositions [Proposition 11](#prop_localization1){reference-type="ref" reference="prop_localization1"} and [Proposition 12](#prop_loc2){reference-type="ref" reference="prop_loc2"} we need to prove the lemmas that they rely on. We will do this in the next two subsections. ## Boundary conditions for divergence {#sec_div} In this subsection we start proving Lemmas [Lemma 16](#lem_approxCrit){reference-type="ref" reference="lem_approxCrit"} and [Lemma 17](#lem_blowupCand){reference-type="ref" reference="lem_blowupCand"} that were needed by Propositions [Proposition 11](#prop_localization1){reference-type="ref" reference="prop_localization1"} and [Proposition 12](#prop_loc2){reference-type="ref" reference="prop_loc2"} in the previous part. One of the main challenges in making the Runge approximation argument work in the next subsection is to find a general enough criterion that guarantees that a solution to the Navier equation has divergence. We will look for a criterion on the boundary condition of the solution. The complicating factor is that the criterion has to be such that it works even when we consider boundary conditions in the orthogonal complement of some finite dimensional subspace. The following Lemma gives a characterization of zero divergence solutions to the Navier equation. The Lemma says essentially that a zero divergence solution to the Navier equation is already uniquely determined by the tangential part of a Dirichlet condition (modulo some finite dimensional subspace, which appears if $\omega$ is a type of a resonance frequency). The second lemma uses this result to formulate a condition of the Dirichlet value of a solution that guarantees zero divergence. The main idea behind the lemma is to relate a zero divergence solution of the Navier equation to a second order Maxwell type system of the form $$\nabla \times (\nabla \times u) + \dots = 0,$$ where the dots indicate lower order terms. The theory related to Maxwell's equations tells us that the boundary condition $\nu \times u |_{\partial D} =f$ is enough to determine the solution $u$ uniquely. Note that this is less information than in a Dirichlet condition $u|_{\partial\Omega} = f$ that is needed for the Navier equation, since the first condition does not determine the normal component of $u$. **Lemma 13**. *Assume that $\lambda, \mu, \rho \in C^\infty(D) \cap L^\infty_+(\Omega)$, where $D \subset \mathbb{R}^3$ is a bounded domain with smooth boundary. Suppose $u \in H^1(D)^3$ solves $$\begin{aligned} \label{eq_bvp_tang} \nabla \cdot (\mathbb{C}\hat \nabla u ) + \omega^2 \rho u &= 0, \quad \text{ in } D.\end{aligned}$$ And suppose that $u$ is such that $$(u \times \nu)|_{\partial D} = 0 \quad \text{and } \quad \nabla \cdot u = 0,\quad \text{ in } D.$$ Then there exists a subspace $\mathcal{N} \subset H^1(D)^3$, with $\dim(\mathcal{N}) < \infty$, and for which $u=0$ as an element in $H^1(D)^3/\mathcal{N}$.* *Proof.* Assume that $u\in H^1(D)^3$ is a weak solution of [\[eq_bvp_tang\]](#eq_bvp_tang){reference-type="eqref" reference="eq_bvp_tang"} and $\nabla \cdot u = 0$. Then $$\begin{aligned} \label{eq_B0} B(u,v) = 0 \qquad \forall v \in C^\infty_0(D)^3.\end{aligned}$$ We can choose a sequence $u_k \in C^\infty(D)^3$, with $\nabla \cdot u_k = 0$, such that $u_k \to u$, in the $H^1(D)^3$-norm as $k \to \infty$. Integrating by parts gives that $$\begin{aligned} -B(u_k,v) = \int_D 2 \mu \hat \nabla u_k :\hat \nabla v - \omega^2\rho u_k \cdot v\,dx = \int_D -\nabla \cdot ( 2 \mu \hat \nabla u_k ) \cdot v - \omega^2\rho u_k \cdot v\,dx. \end{aligned}$$ We rewrite the 2nd order term. Notice first that by a direct computation we have that $$2 \nabla \cdot \hat \nabla u_k - \nabla \cdot ((\nabla \cdot u_k) I) = \Delta u_k.$$ Using this and that $\nabla \cdot u_k = 0$, we can add zero suitable so that $$\begin{aligned} \nabla \cdot ( 2 \mu \hat \nabla u_k ) = 2 \mu \nabla \cdot \hat \nabla u_k + 2 \nabla \mu \hat \nabla u_k = \mu \Delta u_k + 2 \nabla \mu \hat \nabla u_k.\end{aligned}$$ From this and the identity $\Delta u = \nabla(\nabla \cdot u) - \nabla \times (\nabla \times u)$, we get that $$\begin{aligned} \nabla \cdot ( 2 \mu \hat \nabla u_k ) = -\mu \nabla \times (\nabla \times u_k) + 2 \nabla \mu \hat \nabla u_k.\end{aligned}$$ Using this and the integration by parts formula for the curl, we get that $$\begin{aligned} % \label{eq_curlCurl} -B(u_k,v) &= \int_D (\mu \nabla \times (\nabla \times u_k) - 2 \nabla \mu \hat \nabla u_k) \cdot v - \omega^2\rho u_k \cdot v\,dx \\ &= \int_D \nabla \times u_k \cdot \nabla \times (\mu v) - 2 \nabla \tilde \mu \hat \nabla u_k \cdot (\mu v) - \omega^2\tilde \rho u_k \cdot (\mu v)\,dx, %=: B'(u_k, \mu v).\end{aligned}$$ where we define for convenience $\tilde \mu := \log(\mu)$ and $\tilde \rho := \frac{\rho}{2\mu}$. It is straight forward to see that, when defining a bilinear form $B'$, and taking the limit $k \to \infty$ of the previous equation and using [\[eq_B0\]](#eq_B0){reference-type="eqref" reference="eq_B0"}, that $$\begin{aligned} % \label{eq_curlCurl} B'(u,\mu v) := \int_D \nabla \times u \cdot \nabla \times (\mu v) - 2 \nabla \tilde \mu \hat \nabla u \cdot (\mu v) - \omega^2\tilde \rho u \cdot (\mu v)\,dx = 0, \end{aligned}$$ for all $v \in C^\infty_0(D)^3$. By writing $w = \mu v$, for some $v \in C^\infty_0(D)^3$, we see that in fact $$\begin{aligned} \label{eq_Bprime} B'(u,w) = 0, \qquad \forall w \in C^\infty_0(D)^3. \end{aligned}$$ Now consider the space $$H_0(\operatorname{curl}; \,D) := \big\{ u \in L^2(D)^3 \,:\, \nabla \times u \in L^2(D)^3,\, (\nu \times u)|_{\partial D} = 0 \big\},$$ equipped with the norm $\| v \|^2_{ H(\operatorname{curl};\,D) }:= \| \nabla \times v \|^2_{ L^2(D)^3 }+\| v \|^2_{ L^2(D)^3 }$. The set $C_0^\infty(\Omega)^3$ is dense in $H_0(\operatorname{curl}; \,D)$. Thus it follows from [\[eq_Bprime\]](#eq_Bprime){reference-type="eqref" reference="eq_Bprime"} with straight forward estimates that $$\begin{aligned} % \label{eq_Bprime} B'(u,w) = 0, \qquad \forall w \in H_0(\operatorname{curl}; \,D)\end{aligned}$$ also holds. Consider the closed subspace $X \subset H^1(D)^3$, defined by $$X := \big\{ u \in H^1(D)^3 \,:\, \nabla \cdot u = 0,\, (\nu \times u)|_{\partial D} = 0 \big\}.$$ Firstly since $X \subset H_0(\operatorname{curl}; \,D)$ we also have that $$\begin{aligned} \label{eq_Bhomogeneous} B'(u,w) = 0, \qquad \forall w \in X.\end{aligned}$$ To show that $u$ is zero modulo some finite dimensional subspace we consider the inhomogeneous problem, related to [\[eq_Bhomogeneous\]](#eq_Bhomogeneous){reference-type="eqref" reference="eq_Bhomogeneous"}, of finding a $u_F \in X$ for which $$\begin{aligned} \label{eq_weakApprox} B'(u_F,v) = \int_{D} F\cdot v \,dx, \quad \forall v \in X,\end{aligned}$$ for a given $F \in L^2(D)^3$.[^4] We will apply the Lax-Milgram Lemma. First we check that $B'(u,u)+\tau(u,u)_{L^2}$, is coercive on $X$, for suitable $\tau \geq 0$. To this end note that by the triangle inequality, we get that $$\begin{aligned} |B' (v,v) | &= \Big|\int_{D} \nabla \times v \cdot \nabla \times v \,dx - \int_D \nabla \tilde \mu \hat \nabla v \cdot v - \omega^2 \tilde \rho v \cdot v \,dx\Big| \\ &\geq \| \nabla \times v \|^2_{L^2(D)^3} - \Big| \int_D \nabla \tilde \mu \hat \nabla u \cdot v \, dx \Big| - C \| v \|^2_{ L^2(D)^3 }.\end{aligned}$$ To obtain a lower bound in the $H^1$-norm we will use Friedrich's or Gaffney inequality. From this it follows that $$\label{eq_friedrichs} \begin{aligned} \| v \|_{ H^1(D)^3} &\sim \| \nabla \times v \|_{ L^2(D)^3 }+\| \nabla \cdot v \|_{ L^2(D) }+\| v \|_{ L^2(D)^3 } + \| \nu \times v \|_{H^{1/2}(\partial D)^3} \\ &= \| \nabla \times v \|_{ L^2(D)^3 }+\| v \|_{ L^2(D)^3 }, \end{aligned}$$ since $v \in X$. See e.g. [@Ce96] Corollary 5 on p.51. From [\[eq_friedrichs\]](#eq_friedrichs){reference-type="eqref" reference="eq_friedrichs"} we get now that $$\begin{aligned} | B' (v,v) | &\geq \| v \|^2_{H^1(D)^3} - \Big| \int_D \nabla \tilde \mu \hat \nabla v \cdot v \, dx \Big| - (C+1) \| v \|^2_{ L^2(D)^3 }.\end{aligned}$$ To estimate the middle term notice that $$\begin{aligned} \Big| \int_D \nabla \tilde \mu \hat \nabla v \cdot v \, dx \Big| \leq C \| v \|_{ H^1(D)^3 }\| v \|_{ L^2(D)^3} \leq \epsilon\| v \|^2_{ H^1(D)^3 } + \frac{C}{\epsilon} \| v \|^2_{ L^2(D)^3}, \end{aligned}$$ with $\epsilon>0$. The two previous inequalities give the estimate $$\begin{aligned} | B' (v,v) | &\geq (1 - \epsilon) \| v \|^2_{H^1(D)^3} - \frac{C'}{\epsilon} \| v \|^2_{ L^2(D)^3 },\end{aligned}$$ for small enough $\epsilon> 0$. By choosing a large enough $\tau \geq 0$ we see that $B'(u,u) + \tau(u,u)_{L^2}$ is coercive on $X$. It is straight forward to see that $$\begin{aligned} |B' (v,u) | \leq \|u \|_{H^1(D)^3 } \|v \|_{H^1(D)^3}\end{aligned}$$ holds, and that $B'(u,u) + \tau(u,u)_{L^2}$ is also continuous. Let $\tau_0 \geq 0$ be such that $B'_{\tau_0} := B' + \tau_0(\cdot,\cdot)_{L^2}$ is coercive and continuous. Then the Lax-Milgram lemma gives us a unique $u_F \in X$, such that $$B'_{\tau_0}(u_F,v) = (F,v)_{L^2(D)^3}, \qquad \forall v \in X.$$ Thus there is a $\tau_0 \geq 0$ such that we may define the inverse operator $K'_{\tau_0}$, as $$K'_{\tau_0}: L^2(D)^3 \to L^2(D)^3,\qquad K'_ {\tau_0}F := u_F.$$ It is illustrative to first consider the case when $\tau_0 = 0$. Since $u$ solves [\[eq_Bhomogeneous\]](#eq_Bhomogeneous){reference-type="eqref" reference="eq_Bhomogeneous"}, we have thus by uniqueness $u=0$. In this case the claim of the lemma holds, with the choice $\mathcal{N} =\{0\}$. Consider now the case where we cannot choose $\tau_0 = 0$. In this case we will generally have a non trivial eigenspace $\mathcal{N}$ corresponding to the eigenvalue zero of $K'_{\tau_0}$. The operator $K'_{\tau_0}$ is compact on $X$ by Sobolev embedding, since $u_F \in H^1(D)^3$. Since $u_F$ solves [\[eq_weakApprox\]](#eq_weakApprox){reference-type="eqref" reference="eq_weakApprox"} with $F=0$, we see that $$B'(u_F,v) = 0,\qquad \forall v \in X.$$ We will show that this implies that $u_F=0$ in $X / \mathcal{N}$, for some $\mathcal{N} \subset X$, with $\dim(\mathcal{N}) < \infty$. From the above equation we get that $$B'_{\tau_0}(u_F,v) = \tau_0(u_F,v),\qquad \forall v \in X,$$ and therefore $$\begin{aligned} \label{eq_KKer} u_F = \tau_0 K'_{\tau_0} u_F. %\quad\Leftrightarrow\quad \Big(K_{\tau_0} - \tau_0^{-1} \Big)u = 0.\end{aligned}$$ We now use Fredholm theory. By Theorem 2.22 in [@Mc00] p.35 we see that $I - \tau_0 K'_{\tau_0}$ is a Fredholm operator of index $0$. By Theorem 2.27 in p.37-38 in [@Mc00], we have that for any operator $A:X \to Y$ of index $0$, we have that $$\dim (\operatorname{Ker}A) = p < \infty.$$ Applying this to $A = I - \tau_0 K'_{\tau_0}$, gives that $$\dim(\operatorname{Ker}(I - \tau_0 K'_{\tau_0})) = p < \infty.$$ By [\[eq_KKer\]](#eq_KKer){reference-type="eqref" reference="eq_KKer"} we have that $u_F \in \operatorname{Ker}\big(I - \tau_0 K'_{\tau_0}\big)$ so $$u_F = 0 \qquad\text{ in }\quad X / \mathcal{N},$$ when we choose $\mathcal{N} := \operatorname{Ker}(I - \tau_0 K'_{\tau_0})$. ◻ **Remark 14**. *If $\mu$ and $\rho$ are constant then it seems that the argument could with some modifications be used to construct solutions $u$ with $\nabla \cdot u = 0$.* We use Lemma [Lemma 13](#lem_tang_unique){reference-type="ref" reference="lem_tang_unique"} to find a criteria for when a solution has divergence. The essential content of the next Lemma is that a non-zero Dirichlet condition, with a tangential component that is zero, will always create a solution with non-zero divergence. **Corollary 15**. *Assume that $\lambda, \mu, \rho \in C^\infty(D)\cap L^\infty_+(\Omega)$, where $D \subset \mathbb{R}^3$ is a bounded domain with smooth boundary. Suppose that $u \in H^1(D)^3$ solves $$\begin{aligned} \label{eq_bvp5} \begin{cases} \nabla \cdot (\mathbb{C}\hat \nabla u ) + \omega^2 \rho u &= 0, \quad \text{ in } D\\ \;\quad\quad\quad \quad\quad u |_{\partial D} &= f, %\\ \end{cases}\end{aligned}$$ where $f \in H^{1/2}(D)^3$ and $f \neq 0$. Then there exists a finite dimensional subspace $\mathcal{N} \subset L^2(\partial D)^3$ such that if $$\nu \times f |_{\partial D} = 0 \quad \text{ and }\quad f \perp \mathcal{N},$$ then $\nabla \cdot u \neq 0$ in $D$.* *Proof.* Let $\mathcal{N}'$ denote the subspace $\mathcal{N}$ in Lemma [Lemma 13](#lem_tang_unique){reference-type="ref" reference="lem_tang_unique"}. Here we define $\mathcal{N} \subset L^2(\partial D)^3$, as the subspace $$\mathcal{N} := \{ (\nu\cdot \psi|_{\partial D})\nu \,:\, \psi \in \mathcal{N}' \}.$$ Now suppose that the claim does not hold, i.e. $f\neq0$ and $\nu \times f |_{\partial D} = 0,$ and $f \perp \mathcal{N}$ but $\nabla \cdot u \equiv 0$ in $D$. By Lemma [Lemma 13](#lem_tang_unique){reference-type="ref" reference="lem_tang_unique"} we have that $u = 0$ in $H^1(D)^3 / \mathcal{N'}$, so that we can express $u$ as $$u = \sum_{k=1}^K c_k \psi_k,\qquad \psi_k \in \mathcal{N}'.$$ We can decompose $u$ on $\partial D$ as $u = (u\cdot \nu) \nu - \nu \times \nu \times u$. This and the orthogonality condition on $f$ implies that $$(f , f)_{L^2(\partial D)^3} = (f , u|_{\partial D})_{L^2(\partial D)^3} = \big(f, (\nu \cdot u)\nu|_{\partial D} \big)_{L^2(\partial D)^3}=\sum_{k=1}^K c_k (f, \big(\nu \cdot \psi_k)\nu|_{\partial D}\big)_{L^2(\partial D)^3} = 0.$$ This is a contradiction, since we assumed that $f \neq 0$. ◻ ## A Runge type argument {#sec_runge} Propositions [Proposition 11](#prop_localization1){reference-type="ref" reference="prop_localization1"} and [Proposition 12](#prop_loc2){reference-type="ref" reference="prop_loc2"} are proved using the fact that a certain function $w \in H^1(D_1 \cup D_2)^3$ is such that it can be approximated by solutions to the Navier equation on $\Omega$. In this subsection we will construct the desired $w$, by using a Runge type argument. Let $D \Subset \Omega$ with Lipschitz boundary. We start with proving a criterion that tells us when a subspace $\mathcal{S}_D$ of solutions of the Navier equation on $D$, can be approximated in $L^2(D)^3$, by solutions in $$\mathcal{S}_\Omega := \{ u \in H^1(\Omega)^3 \,:\, u \text{ in solves \eqref{eq_bvp1} with } g \in V^\perp\},$$ where $V$ is a given finite dimensional subspace of $L^2(\Gamma_N)^3$. **Lemma 16**. *Every $\varphi \in \mathcal{S}_D$, where $$\mathcal{S}_D := \{ \varphi \in H^1(D)^3 \,:\, L_{\lambda,\mu,\rho} \varphi = 0 \text{ in } D ,\, \varphi \perp \mathcal{F}_V \},$$ can be approximated by elements in $\mathcal{S}_\Omega$ in the $L^2(D)^3$-norm, where $\mathcal{F}_V$ is given by $$\mathcal{F}_V = \operatorname{span}\big\{ v \in L^2(D)^3 \,:\, ( S(\chi_D v), \gamma_\mathbb{C} \varphi )_{L^2(\Gamma_N )^3} = 0, \; \forall \varphi \in \mathcal{S}_\Omega \big\},$$ where $S$ is the solution operator mapping the source term $F$ to the corresponding solution of the boundary value problem [\[eq_bvpLsource\]](#eq_bvpLsource){reference-type="eqref" reference="eq_bvpLsource"} with $\tau=0$.* *Proof.* Denote the restrictions of elements of $\mathcal{S}_\Omega$ by $$\mathcal{R}_D := \{ \varphi|_D \,:\, \varphi \in \mathcal{S}_\Omega \}.$$ Now consider $v \in \mathcal{R}_D^\perp \subset L^2(D)^3$, and let $\varphi|_D \in \mathcal{R}_D$. Define $w$ as the weak solution to the source problem $$\begin{aligned} \label{eq_bvp_w} \begin{cases} L_{\lambda,\mu,\rho} w &= \chi_D v , \quad \text{ in } \Omega,\\ \quad \gamma_\mathbb{C}w |_{\Gamma_N} &= 0, \\ \quad \quad w |_{\Gamma_D} &= 0. %\\ \end{cases}\end{aligned}$$ Now since $v \in \mathcal{R}_D^\perp$, and since both $w$ and $\varphi$ are weak solutions on $\Omega$ (to slightly different equations), we have that $$\begin{aligned} 0 = (v,\varphi|_D)_{L^2(D)^3} = - \int_{\Omega} 2 \mu \hat \nabla w :\hat \nabla \varphi + \lambda \nabla \cdot w \nabla \cdot \varphi - \omega^2\rho w \cdot \varphi\,dx = - (w,\gamma_\mathbb{C}\varphi)_{L^2(\Gamma_N)^3}.\end{aligned}$$ We can write $w = S(\chi_D v)$, where $S$ is the solution operator implicitly given by Proposition [Proposition 5](#prop_Neumann_spectrum){reference-type="ref" reference="prop_Neumann_spectrum"} and the fact that we assume that $\omega$ is a non resonance frequency. Thus we have that $$v \in \mathcal{R}_D^\perp \quad \Leftrightarrow \quad v \in \operatorname{span}\big\{ v \in L^2(D)^3 \,:\, ( S(\chi_D v), \gamma_\mathbb{C} \varphi )_{L^2(\Gamma_N )^3} = 0, \; \forall \varphi \in \mathcal{S}_\Omega \big\}.$$ We thus see that $$\begin{aligned} u \in \overline{\mathcal{R}_D} = \mathcal{R}_D^{\perp \perp} &\;\Leftrightarrow\; u \perp \mathcal{R}_D^\perp \\ &\;\Leftrightarrow\; u \perp \operatorname{span}\big\{ v \in L^2(D)^3 \,:\, ( S(\chi_D v), \gamma_\mathbb{C} \varphi )_{L^2(\Gamma_N )^3} = 0, \; \forall \varphi \in \mathcal{S}_\Omega \big\},\end{aligned}$$ which proves the claim. ◻ We can now use Lemma [Lemma 16](#lem_approxCrit){reference-type="ref" reference="lem_approxCrit"} and Corollary [Corollary 15](#cor_divNonZero){reference-type="ref" reference="cor_divNonZero"} to construct a solution $w$ to the Navier equation, that we can Runge approximate with the restricted boundary data, and which has non-zero divergence in a specified subset. **Lemma 17**. *Let $D_1$ and $D_2$ be as in [\[eq_Dassump\]](#eq_Dassump){reference-type="eqref" reference="eq_Dassump"} and $\mathcal{S}_D$ as in Lemma [Lemma 16](#lem_approxCrit){reference-type="ref" reference="lem_approxCrit"}. There exists a solution $w \in \mathcal{S}_D$, such that $w=0$ in $D_1$ and $$\| w \|_{ L^2(D_2)^3 }, \;\| \hat \nabla w \|_{ L^2(D_2)^{3\times 3} }, \;\| \nabla \cdot w \|_{ L^2(D_2)} \neq 0.$$* *Proof.* Let $D = D_1 \cup D_2$. We will show that we can pick the required $w \neq 0$ as $$L_{\lambda,\mu,\rho}w_{\tilde g} = 0,\quad \text{ in } D,$$ where we set $w_{\tilde g} \equiv 0$, in $D_1$, so that $$\begin{aligned} \label{eq_wBVP0} L_{\lambda,\mu,\rho} w_{\tilde g} = 0 \quad \text{ in } D_1,\qquad \gamma_\mathbb{C} w_{\tilde g} = 0\quad \text { on } \partial D_1,\end{aligned}$$ and in $D_2$ we let $w_{\tilde g}$ solve the boundary value problem $$\begin{aligned} \label{eq_wBVP} L_{\lambda,\mu,\rho} w_{\tilde g} = 0 \quad \text{ in } D_2,\qquad \gamma_\mathbb{C} w_{\tilde g} = {\tilde g}\quad \text { on } \partial D_2.\end{aligned}$$ We derive four conditions on the boundary value ${\tilde g} \in L^2(\partial D_2)^3$, which will guarantee that we have the desired properties, and then show that these conditions can be satisfied. *Condition 1.* It might happen that zero is a Neumann eigenvalue on $D_2$ we thus require that $$\begin{aligned} \label{eq_condN} {\tilde g} \perp \mathcal{N}_0, %\quad \text{ in the $L^2(\p D_2)^3$-norm}.\end{aligned}$$ where $\mathcal{N}_0$ is the finite dimensional subspace of Corollary [Corollary 6](#cor_ZeroEigenvalue){reference-type="ref" reference="cor_ZeroEigenvalue"} and otherwise we set $\mathcal{N}_0 = \{0\}$. This guarantees that we have a unique solution, and that we can define the Neumann-to-Dirichlet map on $\partial D_2$. *Condition 2.* The second condition on ${\tilde g}$ will guarantee that $w_{\tilde g} \in \mathcal{S}_D$. We will derive this next. In order to obtain a $w_{\tilde g} \in \mathcal{S}_D$, we know by the definition of $\mathcal{S}_D$ in Lemma [Lemma 16](#lem_approxCrit){reference-type="ref" reference="lem_approxCrit"} that it is enough to pick a boundary condition ${\tilde g}\neq 0$ resulting in a $w_{\tilde g}$ that satisfies the condition $$w_{\tilde g} \perp \mathcal{F}_V.$$ The first step is to rewrite this as a condition on the boundary value ${\tilde g}$. Suppose that the condition holds for $w_{\tilde g}$ solving [\[eq_wBVP0\]](#eq_wBVP0){reference-type="eqref" reference="eq_wBVP0"}--[\[eq_wBVP\]](#eq_wBVP){reference-type="eqref" reference="eq_wBVP"}. For all $v \in \mathcal{F}_V$ and corresponding $w$ that solve [\[eq_bvp_w\]](#eq_bvp_w){reference-type="eqref" reference="eq_bvp_w"}, we have that $$\begin{aligned} 0 = (w_{\tilde g}, v)_{L^2(D)^3} &= -\int_{D_2} 2 \mu \hat \nabla w_{\tilde g} : \hat \nabla w + \lambda (\nabla \cdot w_{\tilde g}) (\nabla \cdot w) - \omega^2 \rho w_{\tilde g} \cdot w \, dx \\ &\quad + \int_{\partial D_2} \gamma_\mathbb{C}w \cdot w_{\tilde g} \,dS\end{aligned}$$ where we used the fact that $w$ restricted to $D_2$ is a solution in $D_2$ to the boundary value problem implied by [\[eq_wBVP0\]](#eq_wBVP0){reference-type="eqref" reference="eq_wBVP0"}. Furthermore if we let $\Lambda_{D_2}$ be the Neumann-to-Dirichlet map on $D_2$, then $$\int_{\partial D_2} \gamma_\mathbb{C}w \cdot w_{\tilde g} \,dS = \big( \gamma_\mathbb{C}w ,\,\Lambda_{D_2} \tilde g \big)_{L^2(\partial D_2)^3} = \big( \Lambda_{D_2} \gamma_\mathbb{C}w ,\, \tilde g \big)_{L^2(\partial D_2)^3}.$$ Using this and the fact that $w_{\tilde g}$ is a solution in $D_2$ with the boundary condition $\tilde g$, we get from the above that $$0 = \big( \Lambda_{D_2} \gamma_\mathbb{C}w ,\, \tilde g \big)_{L^2(\partial D_2)^3} -\big( w ,\, \tilde g \big)_{L^2(\partial D_2)^3} = \big( \Lambda_{D_2} \gamma_\mathbb{C}S(v) - S(v) ,\, \tilde g \big)_{L^2(\partial D_2)^3},$$ where we write $w = S(v)$, using $S$ which is the solution operator of [\[eq_bvp_w\]](#eq_bvp_w){reference-type="eqref" reference="eq_bvp_w"}. It follows that ${\tilde g}$ should satisfy the constraint $$\begin{aligned} \label{eq_condW} \tilde g \perp \mathcal{W} \,:=\, \operatorname{span}\{\,(\Lambda_{D_2}\gamma_{\mathbb{C}, \partial D_2} - \operatorname{Tr}_{\partial D_2}) \circ \,S(v) \,:\, v \in \mathcal{F}_V \},\end{aligned}$$ where $\operatorname{Tr}$ denotes the trace operator. Next we show that this is a finite dimensional constraint on ${\tilde g}$, in the sense that $\dim{\mathcal{W}}< \infty$. It is enough to show that $$\begin{aligned} \label{eq_Sv_dim} N:= \dim \big(\operatorname{span}\{ S(v)|_{\Omega\setminus D_2} \,:\, v \in \mathcal{F}_V \} \big) < \infty,\end{aligned}$$ since $\mathcal{W}$ can be obtained from this by a linear map. Note firstly that by definition $$\begin{aligned} v \in \mathcal{F}_V \quad \Rightarrow \quad S(v)|_{\Gamma_N} \perp V^\perp \quad\Rightarrow \quad S(v)|_{\Gamma_N} \in V^{\perp\perp} = V,\end{aligned}$$ so that $S(\cdot)|_{\Gamma_N}$ here maps into $V$. Suppose that $N > \dim(V)$. Then we have an $S(v)|_{\Omega\setminus D_2} \neq 0$, such that $$S(v)|_{\Gamma_N} = 0$$ since the trace map is linear. Moreover since $S(v)$ has a zero Neumann condition, we have that $\gamma_{\mathbb{C}} S(v)|_{\Gamma_N} = 0$. Thus $$S(v)|_{\Gamma_N} = 0, \qquad \gamma_{\mathbb{C}} S(v)|_{\Gamma_N} = 0.$$ The unique continuation principle of Proposition [Proposition 2](#prop_bndryUCP){reference-type="ref" reference="prop_bndryUCP"} give now that $S(v_0) \equiv 0$, which is a contradiction. *Condition 3:* The third condition on ${\tilde g}$ will guarantee that $\nabla \cdot w_{\tilde g} \neq 0$. For this we will use Corollary [Corollary 15](#cor_divNonZero){reference-type="ref" reference="cor_divNonZero"}. We require that $$\begin{aligned} {\tilde g} \in \mathcal{T} :&= \{ g \in \mathcal{N}_0^\perp \;:\; \nu \times \Lambda_{D_2} g = 0 ,\; \Lambda_{D_2}g \perp \mathcal{N} \},\end{aligned}$$ where $\Lambda_{D_2}$ is the Neumann-to-Dirichlet map on $\partial D_2$ and $\mathcal{N}$ is as in Corollary [Corollary 15](#cor_divNonZero){reference-type="ref" reference="cor_divNonZero"}. Notice that since $w_{\tilde g}|_{\partial D_2} = \Lambda_{D_2} \tilde g$, this implies by Corollary [Corollary 15](#cor_divNonZero){reference-type="ref" reference="cor_divNonZero"} that $\nabla \cdot w_{\tilde g} \neq 0$, when $\tilde g \neq 0$. We claim that $\mathcal{T}$ is infinite dimensional. To see this consider the space of Dirichlet conditions $$\mathcal{D} := \{ f \in H^{1/2}(\partial D_2)^3\;:\; \nu \times f = 0 ,\; f \perp \mathcal{N} \}.$$ Clearly $\dim( \mathcal{D}) = \infty$. Any (modulo a finite dimensional subspace, in case zero is a Dirichlet eigenvalue) $f_k \in \mathcal{D}$ corresponds to a $g_k \in \mathcal{T}$, such that $$\Lambda_{D_2} g_k = f_k.$$ Since $\Lambda_{D_2}$ restricted to $\mathcal{T}$ has thus an image that is infinite dimensional, we have that $\dim( \mathcal{T}) = \infty$. *Condition 4.* The final requirement on ${\tilde g}$ will be that $$\begin{aligned} \label{eq_condR} {\tilde g} \perp \mathcal{R}, %\quad \text{ in the $L^2(\p D_2)^3$-norm}.\end{aligned}$$ where $\mathcal{R}$ is the space of rigid motions. The requirements on $\tilde g$ in [\[eq_condR\]](#eq_condR){reference-type="eqref" reference="eq_condR"}, [\[eq_condN\]](#eq_condN){reference-type="eqref" reference="eq_condN"} and [\[eq_condW\]](#eq_condW){reference-type="eqref" reference="eq_condW"} can be written as $${\tilde g} \in \mathcal{T}\qquad \text{ and }\qquad {\tilde g} \;\perp\; \mathcal{N}_0 \oplus \mathcal{W} \oplus \mathcal{R}.$$ These conditions can be met since $\dim (\mathcal{N}_0 \oplus \mathcal{W} \oplus \mathcal{R}) < \infty$ is and $\dim(\mathcal{T})= \infty$. The final step is to check that $w_{\tilde g}$ satisfies the desired properties. Since ${\tilde g} \perp \mathcal{W}$, we have that $w_{\tilde g} \in \mathcal{S}_D$. Because $\tilde g \in \mathcal{T}$ we have that $\nabla \cdot w_{\tilde g} \neq 0$ and thus $\| \nabla \cdot w_{\tilde g} \|_{ L^2(D)^3 } \neq 0$ and $\| w_{\tilde g} \|_{ L^2(D)^3 } \neq 0$. Moreover we have that ${\tilde g} \perp \mathcal{R}$ in $L^2(D)^3$, hence we get by Lemma [Lemma 4](#lem_symGradient){reference-type="ref" reference="lem_symGradient"} that $$\| \nabla w_{\tilde g} \|_{ L^2(D)^3 } \leq c \| \hat \nabla w_{\tilde g} \|_{ L^2(\Omega)^3 }.$$ This implies that $\| \hat \nabla w_{\tilde g} \|_{ L^2(\Omega)^{3 \times 3} } \neq 0$. ◻ # Shape reconstruction {#sec_shape} In this section we will justify the individual steps in Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"}. We will first quickly in subsection [6.1](#sec_density_perturbation){reference-type="ref" reference="sec_density_perturbation"} consider the simpler case of a perturbation in the density and Lamé parameters that are known. The main arguments are in subsection [6.2](#sec_mult){reference-type="ref" reference="sec_mult"}, where we consider the case in which multiple parameters may have been perturbed. We begin however with establishing some preliminaries. We extend the use of the term *outer support* that was formulated in [@HU13] for measurable functions, to measurable sets. The outer support (with respect to $\partial\Omega$) of a measurable function $f: \Omega \to \mathbb{R}$ is defined as $$\operatorname{osupp} (f) := \Omega \setminus \bigcup \, \big \{ U \subset \Omega \,:\, U \text{ is relatively open and connected to $\partial\Omega$}, f|_U \equiv 0 \big\}.$$ It will be convenient to extend this definition to sets. We define the outer support of a measurable set $D \subset \Omega$ (with respect to $\partial\Omega$) as $$\begin{aligned} \label{eq_def_osupp} \operatorname{osupp}(D) := \operatorname{osupp}(\chi_D)\end{aligned}$$ where $\chi_D$ is the characteristic function of the set $D$. We will also use the following elementary lemma for compact self-adjoint operators, which we will later apply to the differences between Neumann-to-Dirichlet maps. **Lemma 18**. *Suppose $\mathcal{H}$ is a separable Hilbert space and $T:\mathcal{H} \to \mathcal{H}$ is a compact self-adjoint operator.* 1. *If $T$ has finitely many negative eigenvalues, then there exists a subspace $W \subset \mathcal{H}$, such that $\dim (W)<\infty$, and $$(Tg,g)_{\mathcal{H}} \geq 0, \qquad g \in W^\perp.$$* 2. *Assume $W \subset \mathcal{H}$ is a subspace, with $\dim(W) < \infty$ and that $$(Tg,g)_{\mathcal{H}} \geq 0, \qquad g \in W^\perp.$$ Then $T$ has finitely many negative eigenvalues $\tau_k$. Furthermore $$\dim(W) \geq \dim( \operatorname{span}\{ \varphi_k \,:\, \tau_k < 0 \})$$ where $\varphi_k$ are the corresponding eigenfunctions.* *Proof.* Note first that according to the spectral Theorem for compact self-adjoint operators, see [@RSI] Theorem VI.16, the eigenfunctions $\varphi_k$ are an orthonormal basis, and $$T g = \sum_k \tau_k c_k \varphi_k,\quad \text{ where } \quad c_k = (\varphi_k,g)_{\mathcal{H}}, \quad g \in \mathcal{H},$$ and where $\tau_k$ are the eigenvalues of $T$. We now prove $(a)$. Set $$W = \operatorname{span}\{ \varphi_k \;:\; \tau_k < 0 \} = \{\varphi_1,\dots,\varphi_N\}.$$ Suppose that $g \in W^\perp$. Then $(\varphi_k,g)_{\mathcal{H}} = 0$ for $k=1,..,N$, so that $$g = \sum_{k=N+1}^\infty c_k \varphi_k.$$ To see that the claim holds for $W$, notice that $$\begin{aligned} (Tg,g)_{\mathcal{H}} = \Big(\sum_{k\geq N+1} \tau_k c_k \varphi_k,\, \sum_{j\geq N+1} c_j \varphi_j \Big)_{\mathcal{H}} = \sum_{k=N+1} \tau_k c_k^2 \geq 0,\end{aligned}$$ since $\tau_k \geq 0$ for $k \geq N+1$. Next we prove $(b)$. Let $\mathcal{N} = \operatorname{span}\{\varphi_k \,:\, \tau_k < 0 \}$ and assume that $\dim(\mathcal{N}) = \infty$. Consider the projection $P_\mathcal{N} W \subset \mathcal {N}$, i.e. the projection of the subspace $W$ to $\mathcal{N}$. This is a subspace and notice that $\dim (P_\mathcal{N} W) < \dim(W) = M < \infty$. Since $\dim (\mathcal{N}) = \infty$, we can find a non-zero vector $u \in \mathcal{N} \cap (P_\mathcal{N}W)^\perp$[^5]. A projection operator is a symmetric operator, and since $u = P_\mathcal{N}u$ we have for all $w \in W$, that $$(u,w)_\mathcal{H} =(P_\mathcal{N}u,w)_\mathcal{H} =(u,P_\mathcal{N}w)_\mathcal{H} = 0.$$ Thus $u \in W^\perp$. We can furthermore write $u$ as $$u = \sum_{\varphi_k \in \mathcal{N}} c_k \varphi_k.$$ We now have that $$\begin{aligned} (Tu,u)_{\mathcal{H}} = \Big(\sum_{\varphi_k \in \mathcal{N}} \tau_k c_k \varphi_k,\, \sum_{\varphi_j \in \mathcal{N}} c_j \varphi_j \Big)_{\mathcal{H}} = \sum_{\varphi_k \in \mathcal{N}} \tau_k c_k^2 < 0.\end{aligned}$$ This is a contradiction since we assumed that if $g \in W^\perp$, then $(Tg,g)_{\mathcal{H}} \geq 0$. As the last step we will derive the lower bound for $\dim(W)$. Assume the bound does not hold. Then there is a $w \in \mathcal N := \operatorname{span}\{ \varphi_k \,:\, \tau_k < 0 \}$, such that $w \in W^\perp$. Using the fact that $\{\varphi_k\}$ is an orthonormal basis, we have that $$\begin{aligned} (Tw,w)_{\mathcal{H}} = \Big(\sum_{\varphi_k \in \mathcal{N}} \tau_k c_k \varphi_k,\, \sum_{\varphi_j \in \mathcal{N}} c_j \varphi_j \Big)_{\mathcal{H}} = \sum_{\varphi_k \in \mathcal{N}} \tau_k c_k^2 < 0.\end{aligned}$$ where $w = \sum_{\varphi_k \in \mathcal{N}} c_k \varphi_k$. This is a contradiction, since $w \in W^\perp$ and by the assumption $$(Tw,w)_{\mathcal{H}} \geq 0.$$ ◻ In the sequel it will be important that $\Lambda$ is compact and self-adjoint, as the following Lemma shows. This will in particular guarantee that $\Lambda$ has the correct spectral properties. **Lemma 19**. *The mapping $\Lambda:L^2(\Gamma_N)^3 \to L^2(\Gamma_N)^3$ is compact and self-adjoint.* *Proof.* We begin by proving that $\Lambda$ is compact. Let $g \in L^2(\Gamma_N)^3$. The solution $u$ to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} depends continuously on the boundary data, and by the apriori estimate [\[eq_aprioriEst\]](#eq_aprioriEst){reference-type="eqref" reference="eq_aprioriEst"}, we have that $$\| u \|_{H^1(\Omega)^3} \leq C \| g \|_{ L^2(\Gamma_N)^3}.$$ This together with the continuity of the trace, gives that $$\| \Lambda g \|_{ H^{1/2}(\Gamma_N)^3 } = \| u |_{\partial\Omega} \|_{ H^{1/2}(\partial\Omega)^3 } \leq C \| u \|_{H^1(\Omega)^3} \leq C \| g \|_{L^2(\Gamma_N)^3}.$$ Thus $\Lambda : L^2(\Gamma_N)^3 \to H^{1/2}(\Gamma_N)^3$ is continuous. It follows that $\Lambda : L^2(\Gamma_N)^3 \to L^{2}(\Gamma_N)^3$ is compact, since the inclusion mapping $i:H^{1/2}(\Gamma_N)^3 \to L^{2}(\Gamma_N)^3$ is compact. Next we will show that $\Lambda$ is a self-adjoint operator. It is enough to check that $\Lambda$ is symmetric, since $\Lambda$ is bounded on $L^2(\Gamma_N)^3$. Let $f,g \in L^2(\Gamma_N)^3$. Let $u_g \in\mathcal{V}$ be a weak solution to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} with boundary data $g$, and $u_f\in\mathcal{V}$ be a weak solution to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} with boundary data $f$. By the definition of weak solutions [\[eq_weak2\]](#eq_weak2){reference-type="eqref" reference="eq_weak2"}, we have that $$B(u_f,u_g) = -\int_{\Gamma_N} f \cdot u_g \,dS = -\big(f\,,\Lambda g\big)_{L^2(\Gamma_N)^3}.$$ Likewise we have that $$B(u_g,u_f) = -\int_{\Gamma_N} g \cdot u_f \,dS = -\big(g\,,\Lambda f\big)_{L^2(\Gamma_N)^3}.$$ By the symmetry of $B$ we have thus that $$\big(\Lambda f\,,g\big)_{L^2(\Gamma_N)^3} = \big(f,\,\Lambda g\big)_{L^2(\Gamma_N)^3},$$ and we thus see that $\Lambda$ is symmetric. ◻ ## Recovering inclusions in the density {#sec_density_perturbation} In this short subsection we study the shape reconstruction problem in the case when only the density is perturbed. The purpose of this subsection is twofold. Firstly we show that we can derive an inclusion detection test for an inhomogeneity with increasing[^6] density, when the Lamé parameters are known but not necessarily constant. The argument here also partly serves to illustrate the procedure in the next subsection in a simpler context. Here we only derive the analogue of Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} and omit proving the analogue of Theorem [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"}. Let $D \Subset \Omega$ with Lipschitz boundary. We will now assume that[^7] $\lambda,\mu \in W^{1,\infty}(\Omega) \cap L^\infty_+(\Omega)$ and $\rho_0 > 0$. Now consider $\rho \in L_+^\infty(\Omega)$ which is such that $$\rho(x) = \rho_0 + \chi_D(x) \psi(x), \quad \psi \in L^\infty(\Omega),\quad \psi(x) > m_0 > 0.$$ Here $\psi\chi_D$ models an inhomogeneity in an otherwise homogeneous background density. The following Theorem gives a criterion for when a set $B\subset \Omega$ is a subset of $\operatorname{osupp}(D)$. Note that $B$ might be misscharacterized, if $B$ is a subset of a component of $\Omega \setminus D$ that is not connected to the boundary, so that internal cavities in the inhomogeneous region are not necessarily classified correctly. For a more elaborate explanation on how to use the following theorem for inclusion detection see the next section. **Theorem 20**. *Let $B \subset \Omega$ be an open set and $\alpha > 0$. Let the Neumann-to-Dirichlet maps $$\Lambda := \Lambda_{\lambda,\mu,\rho} \qquad \text{ and } \qquad \Lambda^\flat := \Lambda_{\lambda,\mu, \rho_0 + \alpha \chi_B}$$ where the r.h.s. Neumann-to-Dirichlet maps are those of boundary value problem [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}, with coefficients indicated by the indices. The following holds:\ * 1. *If $B\subset D$, then for all $\alpha\leq m_0$ the map $\Lambda - \Lambda^\flat$ has finitely many negative eigenvalues.\ * 2. *If $B\not \subset \operatorname{osupp}(D)$, then $\Lambda - \Lambda^\flat$ has infinitely many negative eigenvalues for all $\alpha >0$.* *Proof.* Notice firstly that $\Lambda - \Lambda^\flat$ is by Lemma [Lemma 19](#lem_ND_cmpct_SA){reference-type="ref" reference="lem_ND_cmpct_SA"} a compact self-adjoint operator. We begin by proving $(2)$. We can assume that $B$ is a ball and that $B \cap \operatorname{osupp}(D) = \emptyset$, by choosing a smaller ball inside $B$. Assume that the claim is false and that there are only finitely many negative eigenvalues. By Lemma [Lemma 18](#lem_finV){reference-type="ref" reference="lem_finV"} there is a finite dimensional subspace $V_1 \subset L^2(\Gamma_N)^3$, for which $$\begin{aligned} \label{eq_pos1} \big( (\Lambda - \Lambda^\flat)g,\,g \big)_{L^2(\Gamma_N)^3} \geq 0, \qquad \text{ for } g \in V_1^\perp.\end{aligned}$$ Choosing $\rho_1 = \rho$, $\rho_2 = \rho_0 + \alpha \chi_B$ and $\lambda_1=\lambda_2=\lambda$ and $\mu_1=\mu_2=\mu$, in Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}, gives another finite dimensional subspace $V_2$ such that when we rearrange the inequality of the lemma, we have that $$\begin{aligned} \big( (\Lambda - \Lambda^\flat)g, \,g \big)_{L^2(\Gamma_N)^3} &\leq \int_\Omega \omega^2( \rho_1 - \rho_2) |u_1|^2 \,dx \\ &\leq \int_\Omega \omega^2(\psi \chi_D - \alpha \chi_B) |u_1|^2 \,dx \\ &\leq \omega^2\int_D \psi |u_1|^2 \,dx - \omega^2\int_B \alpha |u_1|^2 \,dx, \end{aligned}$$ where $u_1$ solves [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} with coefficients $\lambda$, $\mu$ and $\rho$, with a boundary conditions that satisfies $g \in V_2^\perp$. Now by Lemma [Proposition 11](#prop_localization1){reference-type="ref" reference="prop_localization1"} we can choose a sequence of solutions $u_{1,j}$ such that $$\| u_{1,j} \|_{L^2(\operatorname{osupp}(D))^3} \to 0,\qquad \| u_{1,j} \|_{L^2(B)^3} \to \infty,$$ as $j \to \infty$ and where $g_{1,j} := \gamma_\mathbb{C}u_{1,j}|_{\Gamma_N} \in (V_1 \oplus V_2)^\perp$, since $\dim(V_1\oplus V_2) < \infty$. The previous estimate gives us then that $$\begin{aligned} \big( (\Lambda - \Lambda^\flat)g_j, \,g_j \big)_{L^2(\Gamma_N)^3} \leq - \tfrac{1}{2} \omega^2 \int_B \alpha |u_{1,j}|^2 \,dx < 0,\end{aligned}$$ for big enough $j$. This is in contradiction with [\[eq_pos1\]](#eq_pos1){reference-type="eqref" reference="eq_pos1"}, since $$g_j \in (V_1 \oplus V_2)^\perp \subset V_1^\perp,$$ part $(2)$ of the claim is thus proven. Next we prove part $(1)$. Assume that $\alpha \leq m_0$. We choose $\rho_1 = \rho_0 + \alpha \chi_B$ and $\rho_2 = \rho$ in Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}. According to Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"} there exists a finite dimensional subspace $V \subset L^2(\Gamma_N)^3$, such that for $g \in V^\perp$ we have that $$\begin{aligned} %\label{eq_mono1} \big( (\Lambda - \Lambda^\flat)g, \,g \big )_{L^2(\Gamma_N)^3} \geq \int_\Omega \omega^2(\rho - \rho_0 - \alpha \chi_B) |u_1|^2 \,dx \geq \int_D \omega^2(\psi - \alpha \chi_B) |u_1|^2 \,dx \geq 0,\end{aligned}$$ since $\psi-\alpha\chi_B \geq m_1 -\alpha \geq 0$. We thus have that $$\big( (\Lambda - \Lambda^\flat)g, \,g \big )_{L^2(\Gamma_N)^3} \geq 0, \qquad g \in V^\perp,$$ and by Lemma [Lemma 18](#lem_finV){reference-type="ref" reference="lem_finV"} that $\Lambda - \Lambda^\flat$ has finitely many negative eigenvalues, which proves part $(1)$ of the claim. ◻ ## Recovery of inclusions in multiple coefficients {#sec_mult} Here we prove Theorems [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} and [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"}, and discuss how they can be used to reconstruct the outer support of an inhomogeneities in the material parameters. This will justify Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"}. We will consider inhomogeneities in the material parameters of the following type. Let $D_1, D_2, D_3 \Subset \Omega$. We will now assume that $\lambda,\mu, \rho \in L_+^\infty(\Omega)$ are such that $$\label{eq_lambdaMuRho} \begin{aligned} \lambda(x) &= \lambda_0 + \chi_{D_1}(x) \psi_\lambda (x), \qquad \psi_\lambda \in L^\infty(\Omega), \quad \psi_\lambda(x) > m_1, \\ \mu(x) &= \mu_0 + \chi_{D_2}(x) \psi_\mu(x), \qquad \psi_\mu \in L^\infty(\Omega),\quad \psi_\mu(x) > m_2, \\ \rho(x) &= \rho_0 - \chi_{D_3}(x) \psi_\rho(x), \qquad \psi_\rho \in L^\infty(\Omega),\quad m_3 < \psi_\rho(x) < M_3, \end{aligned}$$ where the constants $\lambda_0,\mu_0,\rho_0 > 0$ and the bounds $m_1,m_2, m_3 > 0$ and $\rho_0 > M_3$. The coefficients $\lambda,\mu$ and $\rho$ model inhomogeneities in an otherwise homogeneous background medium given by the coefficients $\lambda_0,\mu_0$ and $\rho_0$. Next we define the test coefficients $\lambda^\flat,\mu^\flat$ and $\rho^\flat$. Let $B \Subset \Omega$ be a ball. We let $$\label{eq_testCoeff} \begin{aligned} \lambda^\flat(x) &= \lambda_0 + \alpha_1 \chi_{B}(x), \\ \mu^\flat(x) &= \mu_0 + \alpha_2\chi_{B}(x), \\ \rho^\flat(x) &= \rho_0 - \alpha_3\chi_{B}(x), \\ \end{aligned}$$ where $\alpha_j \geq 0$ are constants and $\chi_{D_j}$ are characteristic functions, for $j=1,2,3$ . The following proposition gives a method for recovering $\operatorname{osupp}(D_1\cup D_2\cup D_3)$ from the Neumann-to-Dirichlet map, and thus the shape of the region where the coefficients differ from the background coefficients $\lambda_0,\mu_0$ and $\rho_0$. **Theorem 21**. *Let $D := D_1 \cup D_2 \cup D_3$ where the sets are as in [\[eq_lambdaMuRho\]](#eq_lambdaMuRho){reference-type="eqref" reference="eq_lambdaMuRho"} and $B \subset \Omega$ and $\alpha_j > 0$ be as in [\[eq_testCoeff\]](#eq_testCoeff){reference-type="eqref" reference="eq_testCoeff"}, and set $\alpha:=(\alpha_1,\alpha_2,\alpha_3)$. The following holds:\ * 1. *Assume that $B \subset D_j$, for $j \in I$, for some $I \subset\{1,2,3\}$. Then for all $\alpha_j$ with $\alpha_j \leq m_j$, $j \in I$, and $\alpha_j = 0$, $j \notin I$, the map $\Lambda^\flat - \Lambda$ has finitely many negative eigenvalues.\ * 2. *If $B \not \subset \operatorname{osupp}(D)$, then for all $\alpha$, $|\alpha| \neq 0$, the map $\Lambda^\flat - \Lambda$ has infinitely many negative eigenvalues.* *where $\Lambda$ is the Neumann-to-Dirichlet map the coefficients in [\[eq_lambdaMuRho\]](#eq_lambdaMuRho){reference-type="eqref" reference="eq_lambdaMuRho"} and $\Lambda^\flat$ is the Neumann-to-Dirichlet map for the coefficients in [\[eq_testCoeff\]](#eq_testCoeff){reference-type="eqref" reference="eq_testCoeff"}.* *Proof.* Notice firstly that $\Lambda - \Lambda^\flat$ and $\Lambda - \Lambda^\sharp$ are by Lemma [Lemma 19](#lem_ND_cmpct_SA){reference-type="ref" reference="lem_ND_cmpct_SA"} compact self-adjoint operators. We begin by proving $(2)$. We can assume that $B \cap \operatorname{osupp}{D} = \emptyset$, by making the ball $B$ smaller if needed. Assume that the claim is false and thus that there are only finitely many negative eigenvalues. We will use the localized solutions of Proposition [Proposition 12](#prop_loc2){reference-type="ref" reference="prop_loc2"}. The test coefficients $\lambda^\flat,\mu^\flat$ and $\rho^\flat$ are however not smooth enough. To deal with this we choose a ball $B_0 \Subset B$ and a $\phi \in C^\infty(\Omega)$. $$\phi(x) = 1, \; x \in B_0, \quad \phi(x) = 0, \; x \in \Omega \setminus B, \quad 0 \leq \phi(x) \leq 1,\; x \in \Omega.$$ Now define $$\label{eq_testCoeff_smooth} \begin{aligned} \lambda^\sharp(x) &= \lambda_0 + \alpha_1 \phi(x), \\ \mu^\sharp(x) &= \mu_0 + \alpha_2\phi(x), \\ \rho^\sharp(x) &= \rho_0 - \alpha_3\phi(x), \\ \end{aligned}$$ where $\alpha_j$ are as in [\[eq_testCoeff\]](#eq_testCoeff){reference-type="eqref" reference="eq_testCoeff"}. Clearly we have that $$\lambda^\flat \geq \lambda^\sharp,\quad \mu^\flat \geq \mu^\sharp, \quad \rho^\sharp \geq \rho^\flat.$$ Thus by Lemma [Lemma 10](#lem_LambdaMono){reference-type="ref" reference="lem_LambdaMono"} there is a subspace $V_1$ for which $$\begin{aligned} \label{eq_LambdaMono2} ((\Lambda^\sharp-\Lambda) g,g)_{L^2(\Gamma_N)^3} \geq ((\Lambda^\flat -\Lambda) g,g)_{L^2(\Gamma_N)^3}, \qquad \forall g \in V_1^\perp,\end{aligned}$$ where $\Lambda^\flat$ is the Neumann-to-Dirichlet map corresponding to the coefficients in [\[eq_testCoeff\]](#eq_testCoeff){reference-type="eqref" reference="eq_testCoeff"} and where $\Lambda^\sharp$ corresponds to the coefficients in [\[eq_testCoeff_smooth\]](#eq_testCoeff_smooth){reference-type="eqref" reference="eq_testCoeff_smooth"}. We assumed that $\Lambda^\flat - \Lambda$ has finitely many negative eigenvalues. According to Lemma [Lemma 18](#lem_finV){reference-type="ref" reference="lem_finV"}, we then have that there is a finite dimensional subspace $V_2 \subset L^2(\Gamma_N)^3$, such that $$\begin{aligned} \label{eq_pos} \big( (\Lambda^\flat - \Lambda)g,\,g \big)_{L^2(\Gamma_N)^3} \geq 0 \qquad \text{ for } g \in V_2^\perp.\end{aligned}$$ To obtain a contradiction we consider Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}, where $\Lambda_1 = \Lambda^\sharp$ and $\Lambda_2 = \Lambda$ and which is rearranged to give $$\begin{aligned} \big( (\Lambda^\sharp - \Lambda)g, \,g \big)_{L^2(\Gamma_N)^3} &\leq \int_\Omega 2(\mu-\mu^\sharp ) |\hat \nabla u_1 |^2 + (\lambda - \lambda^\sharp ) |\nabla \cdot u_1|^2 + \omega^2(\rho^\sharp - \rho ) |u_1|^2 \,dx \\ &\leq \int_\Omega 2(\psi_\mu\chi_{D_2} - \alpha_2 \phi )|\hat \nabla u_1|^2 +(\psi_\lambda\chi_{D_1} - \alpha_1 \phi )|\nabla \cdot u_1|^2 \\ &\qquad+ \omega^2(-\alpha_3 \phi + \psi_\rho\chi_{D_3} ) |u_1|^2 \,dx,\end{aligned}$$ where $u_1$ solves [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"} with coefficients given by [\[eq_testCoeff_smooth\]](#eq_testCoeff_smooth){reference-type="eqref" reference="eq_testCoeff_smooth"} boundary condition $g \in V_3^\perp$, where $V_3$ is the subspace given by Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}. By Proposition [Proposition 12](#prop_loc2){reference-type="ref" reference="prop_loc2"} we can choose a sequence $g_j = (\gamma_\mathbb{C}u_{1,j})|_{\partial\Omega} \in (V_1 \oplus V_2 \oplus V_3)^\perp$ of boundary data that give the solutions $u_{1,j}$ to [\[eq_bvp1\]](#eq_bvp1){reference-type="eqref" reference="eq_bvp1"}, with the coefficients [\[eq_testCoeff_smooth\]](#eq_testCoeff_smooth){reference-type="eqref" reference="eq_testCoeff_smooth"}, such that $$\| u_{1,j} \|_{ L^2(D^+)^3 }, \;\| \hat \nabla u_{1,j} \|_{ L^2(\operatorname{osupp}(D))^{3\times 3} }, \;\| \nabla \cdot u_{1,j} \|_{ L^2(\operatorname{osupp}(D))} \to 0,$$ where $\operatorname{osupp}(D) \Subset D^+$, and $D^+ \cap B = \emptyset$, and such that $$\| u_{1,j} \|_{ L^2(B)^3 }, \;\| \hat \nabla u_{1,j} \|_{ L^2(B_0)^{3\times 3} }, \;\| \nabla \cdot u_{1,j} \|_{ L^2(B_0)} \to \infty,$$ as $j \to \infty$. Inserting these solutions to the previous inequality and using that $B \cap \operatorname{osupp}(D) = \emptyset$, we get that $$\begin{aligned} \big( (\Lambda^\sharp - \Lambda)g_j, \,g_j \big)_{L^2(\Gamma_N)^3} &\leq C \int_{D} |\hat \nabla u_{1,j}|^2 +|\nabla \cdot u_{1,j}|^2 +|u_{1,j}|^2 \,dx \\ &\quad- \int_{B_0} \alpha_1 |\nabla \cdot u_{1,j}|^2 + \alpha_2 |\hat \nabla u_{1,j}|^2 + \alpha_3 |u_{1,j}|^2 \,dx.\end{aligned}$$ Since $|\alpha| \neq 0$ and $\alpha_j \geq 0$, we see that the last integral becomes large and increasingly negative while the first integral vanishes as $j$ grows, and thus $$\begin{aligned} \big( (\Lambda^\sharp - \Lambda)g_j, \,g_j \big)_{L^2(\Gamma_N)^3} < 0,\end{aligned}$$ for large enough $j$. This is in contradiction with [\[eq_pos\]](#eq_pos){reference-type="eqref" reference="eq_pos"} and [\[eq_LambdaMono2\]](#eq_LambdaMono2){reference-type="eqref" reference="eq_LambdaMono2"}, since $g_j \in (V_1\oplus V_2 \oplus V_3)^\perp \subset (V_1 \oplus V_2 )^\perp$. Part $(2)$ of the claim thus holds. Next we prove part $(1)$. Assume that $B \subset D_j$ for $j \in I$. Choose $0 \leq \alpha_j \leq m_j$ for $j \in I$ and $\alpha_j = 0$ for $j \notin I$. Moreover choose $\Lambda_1 = \Lambda$ and $\Lambda_2 = \Lambda^\flat$ in Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}. According to Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"} there exists a finite dimensional subspace $V \subset L^2(\Gamma_N)^3$, such that if $g \in V^\perp$, then $$\begin{aligned} %\label{eq_mono1} \big( (\Lambda^\flat - \Lambda)g, \,g \big )_{L^2(\Gamma_N)^3} &\geq \int_\Omega 2(\mu -\mu^\flat ) |\hat \nabla u_1|^2 + (\lambda - \lambda^\flat ) |\nabla \cdot u_1|^2 + \omega^2(\rho^\flat -\rho) |u_1|^2 \,dx \\ &\geq \int_{D_2} 2( m_2 - \alpha_2 \chi_{B} ) |\hat \nabla u_1|^2 \,dx + \int_{D_1} ( m_1 - \alpha_1 \chi_{B} )|\nabla \cdot u_1|^2 \,dx\\ &\quad + \int_{D_3} \omega^2(-\alpha_3\chi_{B} + m_3) |u_1|^2 \,dx \\ &\geq 0,\end{aligned}$$ where we use the properties in [\[eq_lambdaMuRho\]](#eq_lambdaMuRho){reference-type="eqref" reference="eq_lambdaMuRho"}. We have thus shown that for $\alpha_j$, chosen as $0 \leq \alpha_j \leq m_j$ for $j \in I$ and $\alpha_j = 0$ for $j \notin I$, we have that $$\big( (\Lambda^\flat - \Lambda)g, \,g \big )_{L^2(\Gamma_N)^3} \geq 0, \qquad \forall g \in V^\perp.$$ By Lemma [Lemma 18](#lem_finV){reference-type="ref" reference="lem_finV"} we have that $\Lambda^\flat - \Lambda$ has finitely many negative eigenvalues, which proves part $(1)$. ◻ We can use the previous theorem to give a reconstruction procedure for the outer support of the inhomogeneous region $D$ as follows. Choose an open cover $\mathcal{B} = \{ B \}$ of $\Omega$. By computing $N_B$ which stands for the lowest number of negative eigenvalues of the $\Lambda^\flat - \Lambda$ corresponding to a $B$, when varying the parameters as suggested by Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"}, we can classify the sets in $\mathcal{B}$. We let $$\mathcal{A} := \{ B \in \mathcal{B} \,:\, N_B < \infty \}.$$ By Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} part (2) we have that $$B \in \mathcal{A} \quad\Rightarrow\quad B \subset \operatorname{osupp}(D),$$ and by Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} part (1) we have that $$B \not\in \mathcal{A} \quad\Rightarrow\quad B \not \subset D_j,\; j=1,2,3.$$ So all $B \in \mathcal{A}$ are such that $B \subset \operatorname{osupp}(D)$, and for all $B \in \mathcal{B}$ with $$B \subset D_j, \quad \text{ for some }j,$$ we have that $B \in \mathcal{A}$. Supposing we have chosen $\mathcal{B}$ with enough precision, so that it contains an exact sub cover of $D$, then $$D = \bigcup_{j=1}^3 D_j \subset \cup \mathcal{A}.$$ On the other hand $$\cup\mathcal{A} \subset \operatorname{osupp}(D).$$ The set $\cup\mathcal{A}$ gives thus an approximation that contains the inhomogeneous region $D$ and is contained in $\operatorname{osupp}(D)$. This approximation may differ from $\operatorname{osupp}(D)$, in that sets $B$ that are contained in components of $\Omega \setminus D$ that are not connected to $\partial\Omega$ (i.e. $B$ that are in internal cavities in the inhomogeneity) can be classified incorrectly. Note however that, since these possibly misclassified regions are contained in $\operatorname{osupp}(D)$ and are disconnected from $\partial\Omega$, we can as a last step topologically inspect $\mathcal{A}$ and fill in any internal holes, to obtain an approximation of $\operatorname{osupp}(D)$, that becomes arbitrarily good as we use more precise covers $\mathcal{B}$ of $\Omega$. The criterion offered by Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} and Theorem [Theorem 20](#thm_inclusionDetection){reference-type="ref" reference="thm_inclusionDetection"} suffer from the problem that part (2) of the claims involve an infinite bound. We can however replace the bound in Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} (and Theorem [Theorem 20](#thm_inclusionDetection){reference-type="ref" reference="thm_inclusionDetection"}), with a finite bound, which we can compute from background coefficients. More specifically we can derive the following version of Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"}. Note that the bound in the next Theorem is in all likelihood not optimal. For more on this see Theorem 1 in [@HPS19a]. **Theorem 22**. *Suppose the assumptions of Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} hold. Let $M_0\in \mathbb{R}$ be defined as $$M_0 := d(\lambda_0,\mu_0,\rho_{0}),$$ where $d(\lambda_0,\mu_0,\rho_{0})$ is the number of positive eigenvalues of $L_{\lambda_0,\mu_0,\rho_{0}}$ as defined in [\[eq_def_d\]](#eq_def_d){reference-type="eqref" reference="eq_def_d"}. Then* 1. *Assume that $B \subset D_j$, for $j \in I$, for some $I \subset\{1,2,3\}$. Then for all $\alpha_j$ with $\alpha_j \leq m_j$, $j \in I$, and $\alpha_j = 0$, $j \notin I$, the map $\Lambda^\flat - \Lambda$ has at most $M_0$ negative eigenvalues.\ * 2. *If $B \not \subset \operatorname{osupp}(D)$, then for all $\alpha$, $|\alpha| \neq 0$, the map $\Lambda^\flat - \Lambda$ has more than $M_0$ negative eigenvalues,* *where $\Lambda$ is the Neumann-to-Dirichlet map the coefficients in [\[eq_lambdaMuRho\]](#eq_lambdaMuRho){reference-type="eqref" reference="eq_lambdaMuRho"} and $\Lambda^\flat$ is the Neumann-to-Dirichlet map for [\[eq_testCoeff\]](#eq_testCoeff){reference-type="eqref" reference="eq_testCoeff"}, and where the eigenvalues of $\Lambda^\flat - \Lambda$ are counted with multiplicity.* *Proof.* Part (2) of the claim can be proven as part (2) of Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"}. It remains to prove (1). As in the proof of (1) in Theorem [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} we have that $$\big( (\Lambda^\flat - \Lambda)g, \,g \big )_{L^2(\Gamma_N)^3} \geq 0, \qquad g \in V^\perp.$$ By Lemma [Lemma 18](#lem_finV){reference-type="ref" reference="lem_finV"} we have that $\Lambda^\flat - \Lambda$ has less or equally as many negative eigenvalues as $\dim(V)$, when counted with multiplicity. Combining this with the bound on $\dim(V)$ in Lemma [Lemma 9](#lem_monotonicity_ineq1){reference-type="ref" reference="lem_monotonicity_ineq1"}, we get that $$\text{number of negative eigenvalues of $\Lambda^\flat - \Lambda$ } \leq \dim(V ) \leq d(\lambda^\flat,\mu^\flat,\rho^\flat).$$ We can use the min-max principle to give an upper bound on $\dim(V)$ using $d(\lambda_0,\mu_0,\rho_{0})$ which we can easily compute, and which will prove the claim. We do this by showing that $$\begin{aligned} \label{eq_dd} d(\lambda^\flat,\mu^\flat,\rho^\flat) \leq d(\lambda_0,\mu_0,\rho_{0}).\end{aligned}$$ Let $-B^\flat_{\tau_0}$ and $- B^0_{\tau_0}$ be the bilinear forms related to the coefficients $\lambda^\flat,\mu^\flat,\rho^\flat$ and $\lambda_0,\mu_0,\rho_{0}$ respectively. Let $\tau_0 \leq 0$ be such that both $-B^\flat_{\tau_0}$ and $- B^0_{\tau_0}$ are positive definite bilinear forms. Note that Proposition [Proposition 3](#prop_Wellposedness){reference-type="ref" reference="prop_Wellposedness"} guarantees that such a $\tau_0 \leq 0$ exists. Then by the min-max principle of Theorem 3.2 on p. 97 in [@Ba80], we have that $$\begin{aligned} -\sigma^\flat_k - \tau_0 = \min_{S_k \subset H^1(\Omega)^3} \max_{v \in S_k} \frac{-B^\flat_{\tau_0}(v,v)}{(v,v)_{L^2}}, \qquad -\sigma^0_k - \tau_0 = \min_{S_k \subset H^1(\Omega)^3} \max_{v \in S_k} \frac{- B^0_{\tau_0}(v,v)}{(v,v)_{L^2}}, \end{aligned}$$ where $S_k \subset H^1(\Omega)^3$ is an arbitrary $k$ dimensional subspace, and where $\sigma^\flat_k$ are the eigenvalues related to $L_{\lambda^\flat,\mu^\flat,\rho^\flat}$ and $\sigma^0_k$ are the eigenvalues related to $L_{\lambda_0,\mu_0,\rho_{0}}$. From definition [\[eq_testCoeff\]](#eq_testCoeff){reference-type="eqref" reference="eq_testCoeff"} we easily see that $$- B^0_{\tau_0}(v,v) \leq -B^\flat_{\tau_0}(v,v).$$ Thus $-\sigma^0_k - \tau_0 \leq -\sigma^\flat_k - \tau_0$, and hence $\sigma^0_k \geq \sigma^\flat_k$. It follows that $L_{\lambda_0,\mu_0,\rho_{0}}$ has more or equally as many positive eigenvalues as $L_{\lambda^\flat,\mu^\flat,\rho^\flat}$, which shows that [\[eq_dd\]](#eq_dd){reference-type="eqref" reference="eq_dd"} holds. ◻ **Remark 23**. *We need more information than in Theorem [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"} to derive a similar bound $M_0$ for Theorem [Theorem 20](#thm_inclusionDetection){reference-type="ref" reference="thm_inclusionDetection"}. This is because in Theorem [Theorem 20](#thm_inclusionDetection){reference-type="ref" reference="thm_inclusionDetection"} the perturbation in the density has a larger value than the homogeneous background density. In this case we need to have an apriori bound $\rho \leq \rho_M$, and use $\rho_M$ instead of $\rho_0$ when computing $M_0$.* **Remark 24**. *A natural extension of Theorems [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} and [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"} is the case where there is an increase in the density $\rho$ and a decrease in the Lamé parameters $\lambda$ and $\mu$ (as in [@HPS19b] Theorems 6.1 and 6.2). We do not however pursue this here as the arguments are similar.* **Remark 25**. *If not all of the parameters $\lambda,\mu$ and $\rho$ are perturbed, and we assume that we know the unperturbed coefficients, then we could derive versions of Theorems [Theorem 21](#thm_inclusionDetection_rho_mu){reference-type="ref" reference="thm_inclusionDetection_rho_mu"} and [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"}, where the unperturbed coefficients are non-constant as in Theorem [Theorem 20](#thm_inclusionDetection){reference-type="ref" reference="thm_inclusionDetection"}.* # A numerical realization {#sec_numerics} In this section, we give the background for the implementation of the monotonicity tests and present initial numerical results. In more detail, we introduce a test model with two inclusions and consider different frequencies for the reconstruction of these inclusions.\ \ We start with the background of the discretization. Let $G=\lbrace g_1,\ldots, g_m\rbrace$ be an orthogonal system of boundary loads. We discretize the Neumann-to-Dirichlet operator by $$\begin{aligned} \overline{\Lambda}=\left(\left(\Lambda g_i, g_j\right)\right)_{i,j=1,\ldots,m}.\end{aligned}$$\ We base the discretization on Theorem [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"}. However, we want to remark, that in our discritization we pick an ad hoc chosen $\tilde M$ that is smaller than the $M_0$ given by Theorem [Theorem 22](#thm_inclusionDetection_EVbound){reference-type="ref" reference="thm_inclusionDetection_EVbound"} and is chosen by inspecting the number of negative eigenvalues. The $M_0$ is too large for the accuracy of our current tests. We hope to remedy the situation in future work by linearizing and regularizing the method.\ \ **Discritization:**\ Let $D:=D_1 \cup D_2 \cup D_3$, where the sets are as in ([\[eq_lambdaMuRho\]](#eq_lambdaMuRho){reference-type="ref" reference="eq_lambdaMuRho"}) and $B\subset \Omega$. We chose a nonnegative constant $\tilde{M} \leq M_0$ and mark - $B\subset osupp(D)$ if $\overline{\Lambda^\flat}-\overline{\Lambda}$ has at most $\tilde{M}$ many negative eigenvalues for some $\alpha$, $|\alpha|\neq 0$, - $B\not\subset D$ if $\overline{\Lambda^\flat}-\overline{\Lambda}$ has more than $\tilde{M}$ negative eigenvalues for all $\alpha=(\alpha_1,\alpha_2,\alpha_3)$, $0\leq \alpha_j\leq m_j$, $j=1,2,3$ with $|\alpha|\neq 0$, where $\overline{\Lambda}$ is the discrete Neumann-to-Dirichlet operator of the coefficients in ([\[eq_lambdaMuRho\]](#eq_lambdaMuRho){reference-type="ref" reference="eq_lambdaMuRho"}) and $\overline{\Lambda^\flat}$ is the Neumann-to-Dirichlet map for ([\[eq_testCoeff\]](#eq_testCoeff){reference-type="ref" reference="eq_testCoeff"}).\ \ Based on this, we present some numerical tests and examine an artificial test object with two inclusions (blue) shown in Figure [1](#testobject){reference-type="ref" reference="testobject"}. The size of our test object is $1\,m^3$. ![Cube with two inclusions (blue) (similar as the model in [@EH21]).](geometry_new.png){#testobject width="32%"} The parameters of the corresponding materials are given in Table [1](#lame_parameter_mono){reference-type="ref" reference="lame_parameter_mono"} material $\lambda_i$ $\mu_i$ $\rho_i$ ------------------- --------------- --------------- --------------- $i=0$: background $6\cdot 10^5$ $6\cdot 10^3$ $3\cdot 10^3$ $i=1$: inclusion $2\cdot 10^6$ $2\cdot 10^4$ $10^3$ : Lamé parameter $\lambda$ and $\mu$ in \[$Pa$\] and density $\rho$ in \[$kg/m^3$\]. Given an angular frequency $\omega$, the $s$-wavelength and $p$-wavelength (see, e.g., [@H01]) for the homogeneous background material are defined via $$\begin{aligned} l_p= 2\pi \dfrac{v_p}{k} \quad\text{and}\quad l_s= 2\pi \dfrac{v_s}{k}\end{aligned}$$ with the velocities $$\begin{aligned} v_p=\sqrt{\dfrac{\lambda_0+2\mu_0}{\rho_0}} \quad\text{and}\quad v_s=\sqrt{\dfrac{\mu_0}{\rho_0}}.\end{aligned}$$\ In order to perform the reconstruction algorithm (see Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"}), we have to define the required input. We consider $125$ ($5\times5\times 5$) test inclusions, i.e. cubes. We want to remark that these test inclusions are chosen in such a way that they perfectly fit into the unknown inclusions.\ \ Further on, we choose for the upper bounds $$\begin{aligned} &\alpha_1=\lambda_1 -\lambda_0=1.4\cdot 10^6 \, Pa, \\ &\alpha_2=\mu_1 -\mu_0=1.4\cdot 10^4 \, Pa, \\ &\alpha_3=\rho_0 -\rho_1=2\cdot 10^3 \, kg/m^3.\end{aligned}$$ For the boundary loads $g_j$, we divide each surface of the cube except its bottom, which denotes the Dirichlet boundary into $25$ $(5 \times 5)$ squares of equal size. On each of those $125$ squares, we consecutively apply a boundary load of $100\, N/m^2$ in the normal direction resulting in $125$ boundary loads $g_j$. We want to remark that the simulations are performed with COMSOLMultiphysics with LiveLink for Matlab.\ \ In the following we take a look at different values for $\omega$ and start with the cases $\omega=10\,rad/s,50\,rad/s$ and the aforementioned $125$ boundary loads. Figure [2](#plot_eig_k_10){reference-type="ref" reference="plot_eig_k_10"} - Figure [8](#plot_eig_k_50){reference-type="ref" reference="plot_eig_k_50"} show the number of the negative eigenvalues of the corresponding test inclusions. ## Case: $\omega=10$ \ Figure [2](#plot_eig_k_10){reference-type="ref" reference="plot_eig_k_10"} depicts the number of eigenvalues for each test inclusion, where the values of the $x$-axis indicate the index of the $125$ test inclusions and the $y$-axis gives to the number of negative eigenvalues. {#plot_eig_k_10 width="37%"} {#eig_k_10 width="45%"} {#eig_k_10 width="45%"}\ {#eig_k_10 width="45%"} {#eig_k_10 width="45%"}\ {#eig_k_10 width="45%"} We observe that there are distinct spikes in the eigenvalue plots (see Figure [2](#plot_eig_k_10){reference-type="ref" reference="plot_eig_k_10"}) which corresponds to test blocks inside the unknown inclusions. Hence, the eigenvalue plots suggest a choice of $\tilde{M}$ in the range between $4$ to $12$ which is less than $M_0=24$ calculated for $\omega=10\,rad/s$. For more details, the test blocks and their eigenvalues are depicted in Figure [7](#eig_k_10){reference-type="ref" reference="eig_k_10"}, where the blocks which lie inside the unknown inclusions are clearly visible. ## Case: $\omega=50$ \ Similar as before, Figure [8](#plot_eig_k_50){reference-type="ref" reference="plot_eig_k_50"} gives us the required information concerning the choice of $\tilde{M}$. Thus, we set $\tilde{M}=50$, where the theoretical $M_0$ was given by $M_0=90$. {#plot_eig_k_50 width="37%"} \ The algorithm then compares the calculated positive eigenvalues with the $\tilde{M}$ as stated in the **Discritization**.\ \ The calculated values $M_0$ as well as the numerically chosen $\tilde{M}$ for the frequencies $\omega$ under investigation are given in Table [2](#k_M_0){reference-type="ref" reference="k_M_0"}. We also added the special case $\omega=0$ (see Figure [9](#plot_eig_k_0){reference-type="ref" reference="plot_eig_k_0"}), which corresponds to the stationary case as considered in [@EH21]. {#plot_eig_k_0 width="37%"} $k$ $M_0$ $\tilde{M}$ ---------------------- ---------- ---------------- $\omega =0\,$ $M_0=0$ $\tilde{M}=0$ $\omega =10\, rad/s$ $M_0=24$ $\tilde{M}=7$ $\omega =50\, rad/s$ $M_0=90$ $\tilde{M}=50$ : Calculated values $M_0$ and chosen $\tilde{M}$. All in all, applying Algorithm [\[alg_shapeInclusion\]](#alg_shapeInclusion){reference-type="ref" reference="alg_shapeInclusion"} with the corresponding $\tilde{M}$ given in Table [2](#k_M_0){reference-type="ref" reference="k_M_0"} results in the following reconstructions for $\omega=0,10, 50 \,rad/s$ (see Figure [10](#reconst_0_10_50){reference-type="ref" reference="reconst_0_10_50"}). {reference-type="ref" reference="k_M_0"}.](freq=50_M=50.png){#reconst_0_10_50 width="40%"} \ It should be noted that for high frequencies $\omega$, e.g., $\omega=50\,rad/s$, finding a suitable $\tilde{M}$, which detects the inclusions correctly, gets more and more difficult. For $k=10$ we could chose $\tilde{M}$ in the range of $4$ to $12$, but for $k=50$, we can only choose one possible $\tilde{M}$, namely $\tilde{M}=50$.\ \ To rectify this, the number of Neumann boundary loads $g_j$ to discretize the operator has to be increased to differentiate the blocks better. We want to discuss two possible ways to do that. The first is to increase the number of normal boundary loads from $125$ to $500$ and end up with the eigenvalues shown in Figure [11](#plot_eig_k_50_10_x_10){reference-type="ref" reference="plot_eig_k_50_10_x_10"}. {#plot_eig_k_50_10_x_10 width="37%"} \ This allows us to choose $\tilde{M}$ in the range of $63$ to $112$ instead of a single value as we have for the case of $125$ boundary loads.\ \ Note that we only considered traction in the normal direction in the previous numerics, which neglects all tangential tractions. One motivation for this is that that in an practical lab experiment, it is easier to test an object in this manner. We however also considered the case of $125$ normal tractions plus $250$ tangential tractions resulting in $375$ boundary loads $g_j$, since this provides a more complete description of the degrees of freedom. The results can be seen in Figure [12](#plot_eig_k_50_normal_tangential){reference-type="ref" reference="plot_eig_k_50_normal_tangential"}. Comparing Figure [11](#plot_eig_k_50_10_x_10){reference-type="ref" reference="plot_eig_k_50_10_x_10"} and Figure [12](#plot_eig_k_50_normal_tangential){reference-type="ref" reference="plot_eig_k_50_normal_tangential"}, we see that the distriubution of negative eigenvalues is nearly the same with Figure [11](#plot_eig_k_50_10_x_10){reference-type="ref" reference="plot_eig_k_50_10_x_10"} covering a slightly wider range. In the case for $375$ boundary loads, we can choose $\tilde{M}$ in the range of $63$ to $100$. {#plot_eig_k_50_normal_tangential width="37%"} \ All in all, taking more boundary loads will stabilize the reconstruction but increases the computation time, respectively.\ \ Finally, we want to remark that our numerical simulations are very preliminary but show that in principle, the monotonicity tests work. Thus, this will build the basis for further examinations and the development of the \"linearized monotonicity methods\" (see [@EH21]) for the stationary case) which will allow us a faster implementation and as such, a testing with more test inclusions. A further extension could be the consideration of a monotonicity-based regularization as well as the examination of the resolution guarantee (similar as considered in for the stationary elastic inverse problem in [@EH22] and [@EBH23], respectively).\ \ **Acknowledgement**\ The first author thanks the German Research Foundation (DFG) for funding the project \"Inclusion Reconstruction with Monotonicity-based Methods for the Elasto-oscillatory Wave Equation\" (reference number 499303971). + A. Albicker, R. Griesmaier, Monotonicity in inverse scattering for Maxwell's equations, Inverse Probl. and Imaging, 17, 68-105, 2023. M. Akamatsu, G. Nakamura, S. Steinberg, Identification of Lamé coefficients from boundary observations *Inverse Problems*, 7, 335--354, 1991. S. Arridge, W. Lionheart, Nonuniqueness in diffusion-based optical tomography, Opt. Lett. 23, 1998. C. Bandle, Isoperimetric Inequalities and Applications, Pitman Publishing, Boston, 1980. G. Bao, T. Yin, F. Zeng, Multifrequency Iterative Methods for the Inverse Scattering Medium Problems in Elasticity, SIAM Journal on Scientific Computing, 41(4): B721--B745, 2019. E. Beretta, M. V de Hoop, E. Francini, S. Vessella, J. Zhai, Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, Inverse Problems, 33, 035013, 2017. E. Beretta,M. de Hoop M, L. Qiu, Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, part I: recovery of the density, Proceedings of the Project Review, Geo-Mathematical Imaging Group (Purdue University, West Lafayette IN), 1, 263-272, 2013. Inverse problems in elasticity, M. Bonnet, A. Constantinescu,Inverse Problems 21, 2005. V. Candiani, J. Darde, H. Garde, N. Hyvönen, Monotonicity-Based Reconstruction of Extreme Inclusions in Electrical Impedance Tomography, SIAM J. Math. Anal., 52, 6, 2020. M. Cessenat, Mathematical Methods in Electromagnetism, World Scientific, Singapore, 1996. P. Ciarlet, Mathematical Elasticity, Volume: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988. S. Eberle, B. Harrach, Shape reconstruction in linear elasticity: Standard and linearized monotonicity method, Inverse Problems, 37(4):045006, 2021. S. Eberle, B. Harrach, Monotonicity-Based Regularization for Shape Reconstruction in Linear Elasticity, Comput. Mech., 69, 1069-1086, 2022. S. Eberle-Blick, B. Harrach, Resolution guarantees for the reconstruction of inclusions in linear elasticity based on monotonicity methods, Inverse Problems, 39, 075006, 2023. J. Elschner, G. Hu, Uniqueness and factorization method for inverse elastic scattering with a single incoming wave, Inverse Problems, 35(9): 094002, 2019. S. Eberle, J. Moll, Experimental detection and shape reconstruction of inclusions in elastic bodies via a monotonicity method, Int. J. Solids Struct., 233, 111169, 2021. G. Eskin, J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18, 907--922, 2002. T. Furuya, The factorization and monotonicity method for the defect in an open periodic waveguide, Journal of Inverse and Ill-posed Problems, vol. 28, no. 6, pp. 783-796, 2020. T. Furuya, The monotonicity method for the inverse crack scattering problem, Inverse Problems in Science and Engineering 28(11):1-12, 2020. B. Gebauer, Localized potentials in electrical impedance tomography, *Inverse Probl. Imaging* 2, 251--269, 2008. N. Grinberg, A. Kirsch, The factorization method for inverse problems, Oxford University Press, USA, 2008. R. Griesmaier, B. Harrach, Monotonicity in inverse medium scattering on unbounded domains, SIAM J. Appl. Math. 78, 2533--2557, 2018. B. Harrach, On uniqueness in diffuse optical tomography, Inverse problems, 25, 2009. B. Harrach, Y. Lin, H. Liu, On Localizing and Concentrating Electromagnetic Fields, SIAM J. Appl. Math., 78, 5, 10.1137/18M1173605, 2018. B. Harrach, V. Pohjola, M. Salo, Dimension bounds in monotonicity methods for the Helmholtz equation, *SIAM J. Math. Anal.* 51-4, pp. 2995-3019, 2019. B. Harrach, V. Pohjola, M. Salo, Monotonicity and local uniqueness for the Helmholtz equation in a bounded domain, *Anal. PDE*, Vol. 12, No. 7, 1741--1771, 2019. B. Harrach, M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, *SIAM J. Math. Anal.* Vol. 45, No. 6, pp. 3382--3403, 2013. J.Harris, Linear Elastic Waves. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2001. G. Hu, A. Kirsch, M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles. Inverse Problems, 29(1): 015009, 2012. G. Hu, Y. Lu, B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29(11): 115005, 2013. M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl. 6:2, 127--140, 1998. M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, *SIAM J. Appl. Math.* 50, 1635--1644, 1990. M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, Journal of Inverse and Ill-Posed Problems, 7(3):255--271, 1999. M. Ikehata, H. Itou, Enclosure method and reconstruction of a linear crack in an elastic body, Journal of Physics: Conference Series 135, 012052, 2008. H. Kang, J. K. Seo, D. Sheen, The inverse conductivity problem with one measurement: stability and estimation of size, SIAM J. Math. Anal. 28:6, 1389--1405, 1997. P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pure Appl. Math. 9, no. 4, 747--66, 1956. C. Lin, G. Nakamura, G. Uhlmann, J. Wang, Quantitative strong unique continuation for the Lamé system with less regular coefficients, Methods and Applications of Analysis, 18(1):85--92, 2011. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. A. Nachman, Reconstructions from boundary measurements Ann. Math. 128, 531--76, 1988. G. Nakamura, G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity, *Invent. Math.* 152, 205--207, 2003 (Erratum to Invent. Math. 118 (3) (1994) 457--474). V. Pohjola, On quantitative Runge approximation for the time harmonic Maxwell equations, *Trans. Amer. Math. Soc.* 375 , 5727-5751, 2022. O. Oleinik, S.Shamaev, G. Yosifian, Mathematical problems inelasticity and homogenization, North-Holland, Amsterdam, 1992. M. Reed, B. Simon, Methods of Modern Mathematical Physics, Functional Analysis, Vol. 1, Academic Press, London, 1980. J. Shi, F. Faucher , M. de Hoop, H. Calandra, Multi-level elastic full waveform inversion in isotropic media via quantitative Lipschitz stability estimates. Proceedings of the Project Review, Geo-Mathematical Imaging Group, Chicago, United States. 1:1-34, 2014. A. Tamburrino and G. Rubinacci, A new non-iterative inversion method for electrical resistance tomography, *Inverse Problems*, 6, 18, 1809--1829, 2002. A. Tamburrino, Monotonicity based imaging methods for elliptic and parabolic inverse problems. *Journal of Inverse and Ill-posed Problems*, 14(6):633--642, 2006. S. Zaremba, Sur un problème mixte relatif à l'équation de Laplace, Bull. Acad. Sci. Cracovie, 314--344, 1910. [^1]: We extend the terminology of [@HU13] to sets. [^2]: Note that it easy to check that the existence of a weak solution in this sense, implies the existence of a strong solution with the desired boundary conditions, when given enough regularity. [^3]: *This means the quotient space.* [^4]: Note that this corresponds to showing that a weak approximate solution $w$ in the subspace $X$ to $$\begin{aligned} %\label{eq_bvp_tang_2} \begin{cases} \nabla \times (\nabla \times w) - \nabla \tilde \mu \hat \nabla w - \omega^2 \tilde \rho w &= 0, \\ \;\quad\quad\quad\quad\quad\quad\quad\quad\quad (\nu \times w ) |_{\partial\Omega} &= 0, %\\ \end{cases}\end{aligned}$$ needs to be zero modulo some finite dimensional subspace. [^5]: where the orthogonal complements and inner products are induced by the $\mathcal{H}$ inner product. [^6]: A similar result could be established for decreasing density as in the next subsection but we do not pursue this here. [^7]: This assumption can be relaxed to $\lambda,\mu \in L^\infty_+(\Omega)$ by the smoothing argument in the next subsection.

arxiv_math

--- abstract: | We describe a construction for the full $\mathrm{C}^*$-algebra of a possibly unsaturated Fell bundle over a possibly non-Hausdorff locally compact étale groupoid without appealing to Renault's disintegration theorem. This construction generalises the standard construction given by Muhly and Williams. address: Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India. author: - Rohit Dilip Holkar - Md Amir Hossain title: $\mathrm{C}^*$-algebras of Fell bundles over étale groupoids --- # Introduction {#sec:intro .unnumbered} The $\mathrm{C}^*$-algebra of a locally compact groupoid was defined by Renault [@Renault1980Gpd-Cst-Alg] as the completion of $\mathrm{C_c}(G)$ with respect to the full norm. The full norm of $f\in \mathrm{C_c}(G)$ is given by supremum of $\lvert\!\lvert \pi(f)\rvert\!\rvert$ over all $I$-norm bounded representation $\pi$ of $\mathrm{C_c}(G)$. Exel in [@Exel2008Inverse-semigroup-combinotorial-Cst-alg] introduced a new approach to defining the $\mathrm{C}^*$-algebras of ${\'e}tale$ groupoid and proved that the algebra he defined is same as that of Renault's. Exel worked with *non-Hausdorff* étale groupoids. In [@Sims-Szabo-Williams2020Operator-alg-book] and [@Sims2017Etale-gpd], Sims prove that for an étale groupoid any representation of $\mathrm{C_c}(G)$ is $I$-norm bounded and continuous in inductive limit topology. Clark and Zimmerman [@Clark-Zimmerman2022A-steinberg-Appro-to-etale-gpd-alg] proved similar results as Sims for second countable groupoids which can be, possibly, *non-Hausdorff*. Fell bundles over locally compact groupoids seems to be first introduced by Yamagami [@Yamagami1990On-primitive-ideal-spac-gpd] and Kumjian [@Kumjian1998Fell-bundles-over-gpd]. The Fell bundles over groupoids is a natural generalisation of $\mathrm{C}^*$-dynamical system. One may construct Fell bundles from group actions on $\mathrm{C}^*$-algebras [@Exel1997Twisted-Partial-Acts-Fell-Bundle]. Conversely, Exel [@Exel1997Twisted-Partial-Acts-Fell-Bundle] shows that a certain class of Fell bundles essentially corresponds to twisted partial actions of locally compact groups on $\mathrm{C}^*$-algebras. As a consequence, the $\mathrm{C}^*$-algebras of *nice* Fell bundles constitute a large class of $\mathrm{C}^*$-algebras associated with $\mathrm{C}^*$-dynamical systems. Turning our attention to $\mathrm{C}^*$-algebras of Fell bundles, the $\mathrm{C}^*$-algebra of a Fell bundle is typically defined as the completion of $\mathrm{C_c}(G;\mathcal{A})$ with respect to the full norm; the existence of this full norm needs a justification. Muhly and Williams [@Muhly-Williams2008FellBundle-ME] offer an exhaustive construction of the (full) $\mathrm{C}^*$-algebra of a Fell bundle over a *locally compact groupoid with a Haar system*. Kumjian [@Kumjian1998Fell-bundles-over-gpd] describes the reduced $\mathrm{C}^*$-algebra of a Fell bundle. We indent to encounter a technical issue in the construction of the full $\mathrm{C}^*$-algebra of a Fell bundle. The proof of Renault's disintegration in [@Muhly-Williams2008FellBundle-ME] uses the fact that the Fell bundle is *saturated*; the Hausdorffness of the underlying groupoid is also used in the proof. On the other hand, the partial actions of groupoids provide a large class of *unsaturated* Fell bundles [@Exel2017PDS-Fell-bundles; @Exel1997Twisted-Partial-Acts-Fell-Bundle; @Anantharaman-D:2020Partial-Action-of-Gpd]. We propose a method to construct the $\mathrm{C}^*$-algebras of these *unsaturated* Fell bundles over *étale* groupoids. This construction basically uses the convolution $^*$-algebra of the groupoid which can be defined without bothering about the saturation of the Fell bundle. Even the proofs involved are algebraic in nature. Therefore, the saturation of the Fell bundle does not come into the picture while construction the $\mathrm{C}^*$-algebra of the Fell bundle. In fact, since the convolution algebra can defined for a nonHausdorff groupoid, the proposed construction works for unsaturated Fell bundles over non-Hausdorff étale groupoids. For a Fell bundle $p\colon \mathcal{A}\to G$ over an étale groupoid, one can consider four types of representations of $\mathrm C(G;\mathcal{A})$ on a Hilbert space: $^*$-representations (no continuity condition imposed), $^*$-representations continuous in the inductive limit topology or the $I$-norm, and the pre-representations; see page  for details. The ideas for proving that the classes of first three representations are are same, are coming from the construction of the $\mathrm{C}^*$-algebra of the full $\mathrm{C}^*$-algebra of an étale groupoid introduced by Exel[@Exel2008Inverse-semigroup-combinotorial-Cst-alg], Sims [@Sims-Szabo-Williams2020Operator-alg-book] and Clark--Zimmerman [@Clark-Zimmerman2022A-steinberg-Appro-to-etale-gpd-alg]; the motivation of this article comes from these works . Showing that the fourth representations are equivalent to any of the earlier ones requires more work. We do not use the Disintegration theorem. We show that our construction of $\mathrm{C}^*(G;\mathcal{A})$ is equivalent to that of Muhly--Williams' [@Muhly-Williams2008FellBundle-ME] when the Fell bundle is saturated and underlying locally compact groupoid is Hausdorff and second countable. #### *Structure of the article:* Section [1](#sec:prelim){reference-type="ref" reference="sec:prelim"} establishes basic ideas and definitions about groupoids, bundle of $\mathrm{C}^*$-algebras and Fell bundles over groupoids. Section [2](#sec:rep-of-Cc-G-A){reference-type="ref" reference="sec:rep-of-Cc-G-A"}, discusses the four notions of representations of $\mathrm{C_c}(G;\mathcal{A})$. The equivalence of the first three norms is discussed in Section [2.2](#sec:pre-rep){reference-type="ref" reference="sec:pre-rep"}. Section [2.1](#sec:equiv-three-repr){reference-type="ref" reference="sec:equiv-three-repr"} contains the proof that pre-represetations are also equivalent to the other three; Theorem [Theorem 1](#thm:norm-equi){reference-type="ref" reference="thm:norm-equi"} summarises the main results. Many remarks after this theorem are given which point out certain properties of $\mathrm{C}^*(G; \mathcal{A})$. Examples are discussed at the end. #### *Acknowledgement:* This work was supported by SERB's grant MTR/2020/000198 of the first author and the CSIR grant 09/1020(0159)/2019-EMR-I of the second one. We are thankful to the funding institutions. # Preliminaries {#sec:prelim} We assume that the reader is familiar with locally compact étale groupoids and construction of their $\mathrm{C}^*$-algebras. Our references for locally compact groupoids and their $\mathrm{C}^*$-algebras are [@Exel2008Inverse-semigroup-combinotorial-Cst-alg; @Renault1980Gpd-Cst-Alg; @Sims-Szabo-Williams2020Operator-alg-book]. The *spaces* considered in this article are locally compact and Hausdorff, and groupoids are mostly locally compact (but not necessarily Hausdorff). For the sake of clarification, we call a space *locally compact* if every point of the space has a Hausdorff neighbourhood with the property that every open cover of the neighbourhood admits a finite subcover. For a groupoid $G$, we denote its space of units by $G^{(0)}$; the range and the source maps by $r$ and $s$, respectively; and the set of composable pairs, $\{(\gamma,\eta)\colon G\times G : s(\gamma)=r(\eta)\}$, by $G^{(2)}$. For $\gamma\in G$, $\gamma^{-1}$ denotes its inverse. The groupoid $G$ is called topological if it has a topology that makes the range, source, inversion and multiplication maps continuous when ${G}^{(0)}\subseteq G$ and $G^{(2)}\subseteq G\times G$ are given subspace topologies. ## Étale groupoids {#sec:etale-groupoids} A topological groupoids $G$ is called *étale* if its space of units ${G}^{(0)}$ is locally compact Hausdorff and the range map (equivalently, the source map) $G\to {G}^{(0)}$ is a local homeomorphism in the subspace topology ${G}^{(0)}\subseteq G$. Thus ${G}^{(0)}\subseteq G$ is open with the subspace topology. Assume that $G$ is an étale groupoid. Then the space of units ${G}^{(0)}\subseteq G$ is an open subspace [@Exel2008Inverse-semigroup-combinotorial-Cst-alg Proposition 3.2]. The space of units is closed in $G$ *iff* $G$ is Hausdorff [@Sims-Szabo-Williams2020Operator-alg-book Lemma 8.3.2]. Since the groupoid is locally homeomorphic to its space of units, which is locally compact Hausdorff, the groupoid itself is locally compact hence locally Hausdorff. In fact, every open Hausdorff subspace of $G$ is locally compact [@Exel2008Inverse-semigroup-combinotorial-Cst-alg Proposition 3.7]. Since $G$ is not Hausdorff in general, $\mathrm{C_c}(G)$ is defined in the sense of Connes [@Connes1982A-Survey-of-folliantion-and-OP-Alg]. Thus $\mathrm{C_c}(G)$ is span of those functions $f$ on $G$ which are supported on a compact set in an open Hausdorff subset $V$ of $G$ and $f|_V$ is continuous. Thus, the functions in $\mathrm{C_c}(G)$ need not be continuous. Nonetheless, $\mathrm{C_c}(G)$ is an involutive algebra. A particular type of basic open sets, called bisections, in an étale groupoid prove very useful in computation. An open subset $U\subseteq G$ is called a *bisection* if the restrictions of the range and source maps to $U$ are homeomrphisms on open subsets of ${G}^{(0)}$. Clearly, every bisection is locally compact Hausdorff subspace of $G$. And the topology of étale groupoid $G$ has a basis consisting of bisections; as a consequence $\mathrm{C_c}(G)$ can be realised as a spanned of $\mathrm{C_c}(U)$ for bisections $U$ [@Exel2008Inverse-semigroup-combinotorial-Cst-alg Proposition 3.10]. We denote the basis consisting of bisections by $G^{\mathrm{op}}$. Continuing the last discussion, since $r$ is a local homeomorphism, it follows that $G^x$ is discrete for all $x\in G^{(0)}$. Let $\lambda_x$ be a counting measure on $G^x$. The family of measure $\{\lambda_x\}_{x\in G^{0}}$ constitutes a Haar system for the étale groupoid $G$, see [@Renault1980Gpd-Cst-Alg Defintion I.2.2]. ## $\mathrm{C}^*$-algebras and Hilbert modules {#sec:cst-algebras-hilbert} Our references for $\mathrm{C}^*$-algebras are Davidson's [@Davidson1996Cst-by-Examples] book and that of Murhpy's [@Murphy-Book]; for Hilbert modules we refer to Lance's monograph [@Lance1995Hilbert-modules]. Let $A$ be a $\mathrm{C}^*$-algebra. By a (right) Hilbert module over $A$ we mean a Banach space $\mathcal H$ that is a right $A$-module equipped with an $A$-valued *inner product* $\langle, \rangle_A$. This inner product is an $A$-valued *positive definite sesquilinear form* on $\mathcal H$ that is linear in the second variable and antilinear in the first one. The positive definiteness has the standard $\mathrm{C}^*$-algebraic meaning here, namely, $\langle a, a\rangle\in A$ is positive and equals $0$ *iff* $a=0$. It is assumed that the Banach norm on $\mathcal H$ is given by $a\mapsto \lvert\!\lvert \langle a, a\rangle\rvert\!\rvert^{1/2}$ where $a\in \mathcal H$. A left Hilbert $A$-module is defined similarly. We reserve the name *Hilbert* module for a *right* Hilbert module; a left one shall be explicitly pointed out. *Example 1*. Let $A$ be a $\mathrm{C}^*$-algebra and $I\subseteq A$ a closed ideal. Then $\langle a, b\rangle_A \mathrel{\vcentcolon=}a^*b$ and $_A\langle a, b\rangle \mathrel{\vcentcolon=}ab^*$, respectively, define left and right $A$-valued inner products on $I$. In fact they make $I$ $A$-module. With these inner products $I$ also becomes Hilbert $A$-$A$-module. This Hilbert module is not full unless $I=A$. **Lemma 1** ([@Lance1995Hilbert-modules Page 5]). *Assume that $X$ is a Hilbert $A$-module and $I\subseteq A$ the closed ideal generated by $\langle X, X\rangle$. Let $(u_i)_{i}$ be an approximate identity of $I$. Then $\lvert\!\lvert x u_i- x \rvert\!\rvert\to 0$ for $x\in X$.* **Lemma 1**. *For an element $a$ of a unital $\mathrm{C}^*$-algebra $A$, $\lvert\!\lvert a\rvert\!\rvert^2 - a^*a$ is a positive element.* *Proof.* The $\mathrm{C}^*$-subalgebra of $A$ generated by 1 and $a^*a$ is $\mathrm C(\sigma(a^*a))$. As the spectral radius of $a^*a$ is $\lvert\!\lvert a\rvert\!\rvert^2$, $\lvert\!\lvert a\rvert\!\rvert^2 - a^*a$ is a positive element in this $\mathrm{C}^*$-subalgebra. ◻ ## Bundles of $\mathrm{C}^*$-algebras {#sec:upper-semic-bundl} We refer the reader to Appendix C of [@Williams2007Crossed-product-Cst-Alg] for the basics of the bundles of Banach spaces and upper semicontinuous bundles of $\mathrm{C}^*$-algebras over locally compact Hausdorff spaces. However, we shall keep recalling some results whenever required. Following [@Fell1988Representation-of-Star-Alg-Banach-bundles Defintion I.13.4], an *upper semicontinuous bundle of Banach spaces* is an open mapping of spaces $p\colon \mathcal{B} \to X$ and following hold: each *fibre* $\mathcal{B}_x \mathrel{\vcentcolon=}p^{-1}(x)$ for $x\in X$ has a complex Banach space structure and the [norm]{.roman} from $\mathcal{B}\to \mathbb{R}$ by $b\mapsto \lvert\!\lvert b\rvert\!\rvert$ is an upper semicontinuous function; the scalar multiplication[^1] $\mathbb{C} \times \mathcal{B} \to \mathcal{B}$ is also a continuous function. Moreover, the addition is continuous from $\mathcal{B}^{(2)} \to \mathcal{B}$ where $\mathcal{B}^{(2)}$ is the fibre product $\{(b,b')\in \mathcal{B}\times \mathcal{B} : p(b) = p(b')\}$. Finally, one more condition regarding the topology on $X$ and the norms on fibres is satisfied. We call $\mathcal{B}$ is the *total space*, $X$ is the *base space* and $p$ is the *bundle map*. An upper semicontinuous Banach bundle $p\colon \mathcal{A}\to X$ is called an upper semicontinuous $\mathrm{C}^*$-bundle if each of the fibre is a $\mathrm{C}^*$-algebra, and the following conditions hold: the fibrewise multiplication is a continuous mapping $\mathcal{A}^{(2)} \to \mathcal{A}$ and the fibrewise involution, $a \mapsto a^*$ from $\mathcal{A}\to \mathcal{A}$ is continuous. The base spaces in all our bundles shall be assumed to be locally compact and Hausdorff. It is customary to write either $p$ or $\mathcal{A}$ instead of $p\colon \mathcal{A}\to X$ when the base space and the bundle map in a bundle are clear. An upper semicontinuous bundle of $\mathrm{C}^*$-algebras shall be simply called a bundle of $\mathrm{C}^*$-algebras or a $\mathrm{C}^*$-bundle. Given a subspace $Y\subseteq X$, $p\colon \mathcal{A}|_Y \to Y$ denotes the restricted bundle; note that the bundle map is still denoted by $p$. Assume that a locally compact Hausdorff and second countable space $X$ is given. A bundle of $\mathrm{C}^*$-algebras $p\colon \mathcal{A}\to X$ is called *separable* if each fibre in $p$ is separable. In this case, the section algebra $\mathrm{C_0}(X;\mathcal{A})$ is separable. Given a bundle $p\colon \mathcal{A}\to X$, a continuous section has standard meaning and we shall simply call a continuous section a *section*. If $K\subseteq X$ is compact, $\mathrm C(K; \mathcal{A}|_K)$ is the $\mathrm{C}^*$-algebra of continuous sections of $\mathcal{A}|_K \to K$. A bundle $p\colon \mathcal{A}\to X$ is said to have *enough sections* if for $x \in X$ and $a\in \mathcal{A}_x$, there is a section $f$ with $f(x)=a$. If the base space $X$ is locally compact and Hausdorff, then the bundle $p\colon \mathcal{A}\to X$ has enough section [@Hofmann1977Bundles-and-sheaves-equi-in-Cat-Ban]. In general, Fell and Doran [@Fell1988Representation-of-Star-Alg-Banach-bundles Vol. 1, C. 16] prove the existence of enough sections for an upper semicontinuous bundle of Banach spaces over a paracompact base space. For a bundle of $\mathrm{C}^*$-algebras $p\colon \mathcal{A}\to X$, $\mathrm{C_0}(X;\mathcal{A})$ denotes the $\mathrm{C}^*$-algebra of sections vanishing at infinity. $\mathrm{C_c}(X; \mathcal{A})$ denotes the dense two-sided ideal of $\mathrm{C_0}(X;\mathcal{A})$ consisting of continuous sections with compact support. **Lemma 1** (Proposition 1.7.1 in [@Dixmier1977Cst-Alg-Enlglish]). *Let $A$ be a $\mathrm{C}^*$-algebra, and $m$ a two-sided ideal of $A$ which is dense in $A$. Then there is an increasing approximate identity of $A$ consisting of elements of $m$. If $A$ is separable, this approximate identity can be taken to be indexed by $\{1,2, \dots\}$.* **Corollary 1** (Corollary of Lemma [Lemma 1](#lem:dixmier-approx-id){reference-type="ref" reference="lem:dixmier-approx-id"}). *Assume an upper semicontinuous $\mathrm{C}^*$-bundle $p\colon \mathcal{A}\to X$ over a locally compact Hausdorff space $X$ is given. Then $\mathrm{C_0}(X;\mathcal{A})$ has a approximate identity consisting of elements in $\mathrm{C_c}(X;\mathcal{A})$. Moreover, if the bundle is separable, then a countable approximate can be chosen.* **Lemma 1**. *Let $p\colon \mathcal{A}\to X$ be an upper semicontinuous bundle of $\mathrm{C}^*$-algebras over a locally compact Hausdorff space. Assume that a nondegenerate $^*$-homomorphism $L\colon \mathrm{C_c}(X;\mathcal{A}) \to \mathbb B(\mathcal H)$ is given. Then $L$ is continuous in the inductive limit topology on $\mathrm{C_c}(X;\mathcal{A})$ and norm topology on $\mathbb B(\mathcal H)$.* *Proof.* Let $\mathcal{C}$ denote the directed set consisting of compact subsets of $X$ as in Corollary [Corollary 1](#cor:cc-apprx-unit-general-case){reference-type="ref" reference="cor:cc-apprx-unit-general-case"}. For $K\in \mathcal{C}$, let $\mathrm C(K; \mathcal{A}|_K)$ be the Banach space consists of continuous section of $p\colon \mathcal{A}\to X$ vanishing outside $K$. The inductive limit topology on $\mathrm{C_c}(X;\mathcal{A})$ is induced by $\{\mathrm C(K; \mathcal{A}|_K)\}_{K\in \mathcal{C}}$. Since each $\mathrm C(K;\mathcal{A}|_K)$ is in fact a $\mathrm{C}^*$-algebra, the restriction of $L|_{\mathrm C(K;\mathcal{A}|_K)}$ is a homomorphism of $\mathrm{C}^*$-algebras, hence a contractive map. As $L|_{\mathrm C(K;\mathcal{A}|_K)}$ is continuous for every relatively compact open set $K \subseteq X$, so is $L$. ◻ Notice the proof of last lemma implies that for $f\in \mathrm C(K;\mathcal{A}|_K)$, $\lvert\!\lvert L(f)\rvert\!\rvert\leq \lvert\!\lvert f\rvert\!\rvert_\infty$ for $L|_{\mathrm C(K;\mathcal{A}|_K)}$ is a contraction. ## Fell bundles By now, Fell bundles are established objects in Operator Algebras which are being widely studied. Therefore, we shall *briefly* describe them. Our main references for Fell bundles are [@Muhly-Williams2008FellBundle-ME; @Kumjian1998Fell-bundles-over-gpd]. Many of our notation and elementary discussion of Fell bundles is adopted from [@Muhly-Williams2008FellBundle-ME]. We consider the definition of Fell bundles from [@Muhly-Williams2008FellBundle-ME] for étale groupoids as we defined; thus our groupoids are locally compact and not necessarily Hausdorff. Our use of this definition is justified if one sees the standard construction of Fell bundles from partial dynamical systems. Moreover, the Fell bundles we consider are not necessarily *saturated*. **Definition 1** (Fell bundle [@Muhly-Williams2008FellBundle-ME]). A Fell bundle over an étale groupoid $G$ is an upper semicontinuous bundle of Banach spaces $p\colon \mathcal{A}\to G$ equipped with a continuous 'multiplication' map $m\colon \mathcal{A}^{(2)}\to \mathcal{A}$ and an involution map $\mathcal{A}\to \mathcal{A}, a\mapsto a^*$ for $a\in \mathcal{A}$ which satisfy the following axioms: 1. [\[item:Fell-1\]]{#item:Fell-1 label="item:Fell-1"} $p(m(a,b)) =p(a)p(b)$ for all $(a,b) \in \mathcal{A}^{(2)}$; 2. [\[item:Fell-2\]]{#item:Fell-2 label="item:Fell-2"} the induced map from $\mathcal{A}_{\gamma_1}\times \mathcal{A}_{\gamma_2} \to \mathcal{A}_{\gamma_1 \gamma_2}$, $(a,b) \mapsto m(a,b)$ is bilinear for all $(\gamma_1, \gamma_2)\in G^{(2)}, a\in \mathcal{A}_{\gamma_1}$ and $b\in \mathcal{A}_{\gamma_2}$; 3. [\[item:Fell-3\]]{#item:Fell-3 label="item:Fell-3"} $m(m(a,b),c) =m(a, m(b,c))$ for $(a,b),(b,c)\in \mathcal{A}^{(2)}$; 4. [\[item:Fell-4\]]{#item:Fell-4 label="item:Fell-4"} $\lvert\!\lvert m(a,b)\rvert\!\rvert \leq \lvert\!\lvert a\rvert\!\rvert\lvert\!\lvert b\rvert\!\rvert$ for all $(a,b) \in \mathcal{A}^{(2)}$; 5. [\[item:Fell-5\]]{#item:Fell-5 label="item:Fell-5"} $p(a^*)=p(a)^{-1}$ for all $a\in \mathcal{A}$; 6. [\[item:Fell-6\]]{#item:Fell-6 label="item:Fell-6"} the induced map $\mathcal{A}_\gamma \to \mathcal{A}_{\gamma^{-1}}$, $a\mapsto a^*$ is conjugate linear for all $\gamma \in G$; 7. [\[item:Fell-7\]]{#item:Fell-7 label="item:Fell-7"} $(a^*)^*=a$ for all $a\in \mathcal{A}$; 8. [\[item:Fell-8\]]{#item:Fell-8 label="item:Fell-8"} $m(a,b)^*=m(b^*,a^*)$ for all $(a,b)\in \mathcal{A}^{(2)}$; 9. [\[item:Fell-9\]]{#item:Fell-9 label="item:Fell-9"} $\lvert\!\lvert m(a^*,a)\rvert\!\rvert=\lvert\!\lvert m(a,a^*)\rvert\!\rvert=\lvert\!\lvert a\rvert\!\rvert^2$ for all $a\in \mathcal{A}$; 10. [\[item:Fell-10\]]{#item:Fell-10 label="item:Fell-10"} $m(a^*,a) \geq 0$ for all $a\in \mathcal{A}$. Note that the last definition makes sense for non-étale groupoids as well. In the above definiton $\mathcal{A}^{2}$ is given by the set $\{(a,b) : (s(p(a)), r(p(b))) \in G^{(2)}\}$. Given $(a,b) \in \mathcal{A}^{(2)}$, we use the shorthand $ab$ instead of $m(a, b)$. For two composition arrows $\gamma_1,\gamma_2\in G$, define the subset $$\mathcal{A}_{\gamma_1} \cdot \mathcal{A}_{\gamma_2} = \textup{span} \{ab: a\in \mathcal{A}_{\gamma_1}, b\in \mathcal{A}_{\gamma_2}\} \subseteq \mathcal{A}_{\gamma_1\gamma_2}.$$ The Fell bundle is called *saturated* if $\mathcal{A}_{\gamma_1} \cdot \mathcal{A}_{\gamma_2}$ is dense in $\mathcal{A}_{\gamma_1 \gamma_2}$ for all $(\gamma_1,\gamma_2) \in G^{(2)}$. For $x\in G^{(0)}$, Conditions [\[item:Fell-2\]](#item:Fell-2){reference-type="eqref" reference="item:Fell-2"}--[\[item:Fell-9\]](#item:Fell-9){reference-type="eqref" reference="item:Fell-9"} in Definition [Definition 1](#def:fell-bundle){reference-type="ref" reference="def:fell-bundle"} make the fibre $\mathcal{A}_x$ a $\mathrm{C}^*$-algebra with the obvious norm, multiplication and involution. Furthermore, if $\gamma \in G$ and $a,b\in \mathcal{A}_\gamma$, then $\mathcal{A}_\gamma$ is a right Hilbert $\mathcal{A}_{s(\gamma)}$-module when equipped with the inner product $\langle a, b\rangle_{\mathcal{A}_{s(\gamma)}}\mathrel{\vcentcolon=}a^*b$; and $\mathcal{A}_\gamma$ is a left Hilbert $\mathcal{A}_{r(\gamma)}$-module if the inner product is defined by ${}_{\mathcal{A}_{r(\gamma)}}\langle a, b\rangle \mathrel{\vcentcolon=}ab^*$. Thus, the Fell bundle $p\colon \mathcal{A}\to G$ is saturated if and only if $\mathcal{A}_\gamma$ is an $\mathcal{A}_{r(\gamma)}$-$\mathcal{A}_{s(\gamma)}$-imprimitivity bimodule for each $\gamma \in G$. For a subset  $X\subseteq G$, by $\mathcal{A}|_X$ we denote the restriction of $\mathcal{A}$ to $X$; but we still denote the bundle map $\mathcal{A}|_X\to X$ by $p$ rather than $p|_{\mathcal{A}|_X}$. We define $\mathrm{C_c}(G; \mathcal{A})$ analogous to $\mathrm{C_c}(G)$ as in [@Connes1982A-Survey-of-folliantion-and-OP-Alg]. Thus, the elements of $\mathrm{C_c}(G;\mathcal{A})$ are not necessarily continuous. With this definition of $\mathrm{C_c}(G;\mathcal{A})$, the bundle $p$ has enough sections: for this let $\gamma\in G$ and $\xi\in \mathcal{A}_\gamma$. For a locally compact Hausdorff neighbourhood $X$ of $\gamma$ consider the restricted bundle $\mathcal{A}|_X \to X$. Then this bundle has a section $f$ which is continuous on $X$ has compact support in $X$ and $f(\gamma) = \xi$. Then $f\in \mathrm{C_c}(G; \mathcal{A})$ (in fact, $f$ belongs to the set which is typically denoted by $\mathrm C^0_{\mathrm{c}}(G;\mathcal{A})$, see [@Exel2008Inverse-semigroup-combinotorial-Cst-alg Definition 3.9] whose span is $\mathrm{C_c}(G;\mathcal{A})$). In any case, we assume that our bundle has enough sections. **Lemma 1**. *Let $G$ be a locally compact Hausdorff groupoid. Let $p\colon \mathcal{A}\to G$ be a *saturated* Fell bundle. Then each fibre of $\mathcal{A}$ is separable *iff* for each $x\in {G}^{(0)}$ the fibre $\mathcal{A}_x$ is a separable $\mathrm{C}^*$-algebra.* *Proof.* One implication is clear. For the converse, let $\gamma\in G$. Then $A_\gamma$ is an $A_{r(\gamma)}$-$A_{s(\gamma)}$-equivalence. Recall from [@Jensen-Thomsen1991KK-theory-book Corollary 1.1.25] that a Hilbert module is countably generated *iff* the $\mathrm{C}^*$-algebra of the compact operators on it is $\sigma$-unital. Since both $A_{r(\gamma)}$ and $A_{s(\gamma)}$ are separable, $A_\gamma$ is countably generated, hence separable. ◻ Last lemma says that, a *saturated* Fell bundle over a locally compact second countable Hausdorff groupoid is separable *iff* the $\mathrm{C}^*$-bundle over the unit space is separable. # Representations of $\mathrm{C_c}(G;\mathcal{A})$ {#sec:rep-of-Cc-G-A} Consider a Fell bundle $p\colon \mathcal{A}\to G$ over an étale groupoid $G$. As it is done for groups or groupoids, one can construst the *full* $\mathrm{C}^*$-algebra $\mathrm{C}^*(G;\mathcal{A})$ of this Fell bundle. One way to define this $\mathrm{C}^*$-algebras is to announce it the enveloping $\mathrm{C}^*$-algebra of the $^*$-algebra $\mathrm{C_c}(G;\mathcal{A})$. The existence of the enveloping algebra needs to be justified. Another way, similar to locally compact groups, is to define it to be the universal $\mathrm{C}^*$-algebra using the unitary representation *of the Fell bundle*. In [@Muhly-Williams2008FellBundle-ME], Muhly and Williams define $\mathrm{C}^*(G;\mathcal{A})$ in both the ways for a groupoid with a *Haar system*; they assume that the groupoid is locally compact second countable and Hausdorff and the Fell bundle is saturated. In the same article, they also prove Renault's integration and disintegration theorems which establishes the equivalence of the two methods of defining $\mathrm{C}^*(G;\mathcal{A})$. Third interesting, and practically useful way to construct representations of Fell bundles is to use their *pre-representations* [@Muhly-Williams2008FellBundle-ME Definitino 4.1]. All in all, in case of *étale* groupoids, we may consider following four types of *representations* of the $^*$-algebra $\mathrm{C_c}(G;\mathcal{A})$ on a Hilbert space $\mathcal H$: 1. [\[it:alg-rep\]]{#it:alg-rep label="it:alg-rep"} a nondegenerate $^*$-homomorphism $\pi \colon \mathrm{C_c}(G;\mathcal{A}) \to \mathbb B(\mathcal H)$ (without any continuity condition); 2. [\[it:i-norm-rep\]]{#it:i-norm-rep label="it:i-norm-rep"} an $I$-norm bounded nondegenerate $^*$-homomorphism $\pi \colon \mathrm{C_c}(G;\mathcal{A}) \to \mathbb B(\mathcal H)$; 3. [\[it:ilt-rep\]]{#it:ilt-rep label="it:ilt-rep"} a nondegenerate $^*$-homomorphism $\pi \colon \mathrm{C_c}(G;\mathcal{A}) \to \mathbb B(\mathcal H)$ continuous in the inductive limit topology; 4. [\[it:pre-rep\]]{#it:pre-rep label="it:pre-rep"} a *pre-representation* $\pi \colon \mathrm{C_c}(G;\mathcal{A}) \to \textup{Lin}(\mathcal H_0)$ on a dense subspace $\mathcal H_0\subseteq \mathcal H$. Our goal is proving Theorem [Theorem 1](#thm:norm-equi){reference-type="ref" reference="thm:norm-equi"} which shows that all four notions above are equivalent for Fell bundles over *étale* groupoids *without* appealing to the technical machinery of the disintegration theorem in [@Muhly-Williams2008FellBundle-ME]. Nonetheless, extending a pre-representation [\[it:pre-rep\]](#it:pre-rep){reference-type="ref" reference="it:pre-rep"} to a one that is continuous in the inductive limit topology [\[it:ilt-rep\]](#it:ilt-rep){reference-type="ref" reference="it:ilt-rep"} involves technicalities. The equivalence of all four representations above allows us to use any one of these representations can be used to define the full $\mathrm{C}^*$-algebra $\mathrm{C}^*(G;\mathcal{A})$ for a Fell bundle $\mathcal{A}$ over an étale groupoid $G$. The upcoming content is planned as follows: we start the discussion by defining some standard ideas which we have already mentioned above. Then we quickly recall the $^*$-algebra $\mathrm{C_c}(G;\mathcal{A})$ from [@Muhly-Williams2008FellBundle-ME]. Then we recall the notions of representations described above. In Subsection [2.1](#sec:equiv-three-repr){reference-type="ref" reference="sec:equiv-three-repr"}, we first prove the equivalence of the three representations [\[it:alg-rep\]](#it:alg-rep){reference-type="ref" reference="it:alg-rep"}--[\[it:ilt-rep\]](#it:ilt-rep){reference-type="ref" reference="it:ilt-rep"} on page . In in Section [2.2](#sec:pre-rep){reference-type="ref" reference="sec:pre-rep"}, we prove equivalence of pre-representations with the rest ones. Now on, we fix $G$ to be a locally compact étale groupoid (as in § [1.1](#sec:etale-groupoids){reference-type="ref" reference="sec:etale-groupoids"}) and a Fell bundle $p\colon \mathcal{A}\to G$ (the Fell bundle need not be *saturated or separable*). To start with, make $\mathrm{C_c}(G;\mathcal{A})$ into a $^*$-algebra using the following convolution and involution: $$f*g(\gamma)= \sum_{\alpha\beta=\gamma}f(\alpha)g(\beta) \quad \textup{and} \quad f^*(\gamma)=f(\gamma^{-1})^*$$ where $f,g\in \mathrm{C_c}(G;\mathcal{A})$. It is a standard fact that last operations make $\mathrm{C_c}(G;\mathcal{A})$ a $^*$-algebra. The convolution is continuous in the inductive limit topology on $\mathrm{C_c}(G;\mathcal{A})$. Since $G$ is étale, ${G}^{(0)}$ is an open Hausdorff subset of $G$. Then by definition of $\mathrm{C_c}(G;\mathcal{A})$, we can say that $\mathrm{C_c}({G}^{(0)},\mathcal{A}|_{{G}^{(0)}})$ is a $^*$-subalgebra of $\mathrm{C_c}(G;\mathcal{A})$. Note that if $f$ is a continuous section vanishing outside ${G}^{(0)}$ and $g\in \mathrm{C_c}(G;\mathcal{A})$, then $$\label{eq:conv} f*g(\gamma) = f(r(\gamma)) g(\gamma), \quad g*f(\gamma) = g(\gamma) f(s(\gamma))$$ for $\gamma\in G$. As $f\circ r$ is continuous section on $G$, $f\circ r\cdot g\in \mathrm{C_c}(G; \mathcal{A})$. Similarly, $g\cdot f\circ s\in \mathrm{C_c}(G; \mathcal{A})$. Define the *$I$-norm* on $\mathrm{C_c}(G;\mathcal{A})$ by $$\lvert\!\lvert f\rvert\!\rvert_I= \max \biggl\{\sup_{x\in G^{(0)}}\sum_{\gamma\in r^{-1}(x)} \lvert\!\lvert f(\gamma)\rvert\!\rvert,\sup_{x\in G^{(0)}}\sum_{\gamma\in s^{-1}(x)} \lvert\!\lvert f(\gamma)\rvert\!\rvert\biggr\}.$$ *Example 1*. Let $f\in \mathrm{C_c}(G;\mathcal{A})$ be supported on a bisection $U$. Then $\lvert\!\lvert f\rvert\!\rvert_I = \lvert\!\lvert f\rvert\!\rvert_\infty$. Because $$\sup_{x\in G^{(0)}}\sum_{\gamma\in r^{-1}(x)} \lvert\!\lvert f(\gamma)\rvert\!\rvert = \lvert\!\lvert f\circ s|_U^{-1}\rvert\!\rvert_{\infty} = \lvert\!\lvert f\rvert\!\rvert_\infty = \lvert\!\lvert f\circ r|_U^{-1}\rvert\!\rvert_{\infty}= \sup_{x\in G^{(0)}}\sum_{\gamma\in s^{-1}(x)} \lvert\!\lvert f(\gamma)\rvert\!\rvert.$$ A *representation* of the $^*$-algebra $\mathrm{C_c}(G; \mathcal{A})$ on a Hilbert space $\mathcal H$ is a non-degenerate $^*$-homomorphism $L\colon \mathrm{C_c}(G;\mathcal{A})\to \mathbb B(\mathcal H)$ (without any continuity condition on $L$). The representation $L$ is called *$I$-norm bounded* if $\lvert\!\lvert L(f)\rvert\!\rvert\leq \lvert\!\lvert f\rvert\!\rvert_I$ for all $f\in \mathrm{C_c}(G;\mathcal{A})$. We shall also discuss representations $\mathrm{C_c}(G;\mathcal{A}) \to \mathbb B(\mathcal H)$ which are *continuous in the inductive limit topology* on $\mathrm{C_c}(G; \mathcal{A})$. Next is the definition of a *pre-representation*. Let $\mathcal H_0$ be a dense subspace of a Hilbert space $\mathcal H$, and denote by $\textup{Lin}(\mathcal H_0)$ the collection of all linear mappings of the vector space $\mathcal H_0$ to itself. **Definition 1** ([@Muhly-Williams2008FellBundle-ME Definition 4.1]). A pre-representation of $\mathrm{C_c}(G;\mathcal{A})$ on a Hilbert space $\mathcal H$ is a pair $(\mathcal H_0, L)$ consisting of a linear subspace $\mathcal H_0 \subseteq \mathcal H$ and a homomorphism of complex algebras $L\colon \mathrm{C_c}(G;\mathcal{A}) \to \textup{Lin}(\mathcal H_0)$ such that the following conditions holds for given $\xi, \eta \in \mathcal H_0$: 1. (Continuity) $f\mapsto \langle\eta, L(f)\xi\rangle$ is continuous in the inductive limit topology on $\mathrm{C_c}(G;\mathcal{A})$; 2. (Adjoinatability) $\langle\eta, L(f)\xi\rangle = \langle L(f^*)\eta, \xi\rangle$; 3. [\[it:H00\]]{#it:H00 label="it:H00"} (Nondegeneracy) $\mathcal H_{00}\mathrel{\vcentcolon=}\textup{span}\{L(f)\xi : f\in \mathrm{C_c}(G,\mathcal{A}), \xi \in \mathcal H_0 \}$ is dense in $\mathcal H$. The nomenclature of properties above is not standard; we coin this nomenclature as it facilitate latter writing. We frequently refer $L$ as the pre-representation on $\mathcal H$ assuming that the representation subspace is $\mathcal H_0$. Moreover, $\mathcal H_{00}$ shall have the same meaning as above. Note that $\mathcal H_{00}\subseteq \mathcal H_0$, and the nondegeneracy condition implies that $\mathcal H_{0}$ is dense in $\mathcal H$. ## Equivalence of the first three representations {#sec:equiv-three-repr} In current section, we prove equivalence of the representations [\[it:alg-rep\]](#it:alg-rep){reference-type="ref" reference="it:alg-rep"}--[\[it:ilt-rep\]](#it:ilt-rep){reference-type="ref" reference="it:ilt-rep"} on page . Since the inductive limit topology is weaker than the $I$-norm topology, a representation that is continuous in the inductive limit topology is also continuous in the $I$-norm topology. And a representation in $I$-topology is obviously a nondegenerate $^*$-homomorphism. Therefore, among the definitions on page , [\[it:ilt-rep\]](#it:ilt-rep){reference-type="ref" reference="it:ilt-rep"}$\implies$[\[it:i-norm-rep\]](#it:i-norm-rep){reference-type="ref" reference="it:i-norm-rep"}$\implies$ [\[it:alg-rep\]](#it:alg-rep){reference-type="ref" reference="it:alg-rep"}. The upcoming Proposition [Proposition 1](#prop:three-reps-are-equi){reference-type="ref" reference="prop:three-reps-are-equi"} proves that a nondegenerate $^*$-homomorphism is, in fact, continuous in the inductive limit topology; this justifies the equivalence of the first three notions of representations. The idea of this proof is coming from Section 3 in Exel's article [@Exel2008Inverse-semigroup-combinotorial-Cst-alg]; it can also be found in Sims' book [@Sims-Szabo-Williams2020Operator-alg-book § 9.2]. Consider the Fell bundle $p\colon \mathcal{A}\to G$ over an étale groupoid $G$ as fixed earlier. Recall from preliminaries that $G^{\mathrm{op}}$ denotes the set of bisections in $G$. For two bisections $U$ and $V$ in $G^{\mathrm{op}}$, define the sets $$U^{-1} = \{u^{-1} : u\in U\} \quad \text{ and } \quad UV = \{ uv : u\in U, v\in V , (u,v) \in G^{(2)} \}.$$ Then $U^{-1}, UV\in G^{\mathrm{op}}$; here $VU$ could be empty. In particular, it can be readily seens that the bisections $UU^{-1} = r(U)$ and $U^{-1}U = s(U)$ are subsets of ${G}^{(0)}$. In fact, $G^{\mathrm{op}}$ is an inverse semigroup with the inverses and products of subsets, see Proposition 2.2.4 in [@PatersonA1999Gpd-InverseSemigps-Operator-Alg]. For a bisection $U\subseteq G$, we consider $\mathrm{C_c}(U;\mathcal{A}|_U)$ as a subset of $\mathrm{C_c}(G;\mathcal{A})$ in the obvious way. Following lemma is an analogue of Proposition 3.11 in [@Exel2008Inverse-semigroup-combinotorial-Cst-alg] for Fell bundles. **Lemma 1**. *Let $U$ and $V$ be bisections of $G$.* 1. *[\[lem:convulu-supp\]]{#lem:convulu-supp label="lem:convulu-supp"} If $f \in \mathrm{C_c}(U;\mathcal{A}|_{U})$ and $g\in \mathrm{C_c}(V;\mathcal{A}|_{V})$, then $f*g \in \mathrm{C_c}(UV;\mathcal{A}|_{UV})$.* 2. *[\[lem:involu-supp\]]{#lem:involu-supp label="lem:involu-supp"} If $f \in \mathrm{C_c}(U;\mathcal{A}|_{U})$, then $f^* \in \mathrm{C_c}(U^{-1}; \mathcal{A}|_{U^{-1}})$.* 3. *[\[lem:inv-with-adj\]]{#lem:inv-with-adj label="lem:inv-with-adj"} If $f\in \mathrm{C_c}(U;\mathcal{A}|_{U})$, then $f*f^*$ and $f^**f \in \mathrm{C_c}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$. In fact, $f*f^* \in \mathrm{C_c}(r(U); \mathcal{A}|_{r(U)})$ and $f^**f \in \mathrm{C_c}(s(U) ; \mathcal{A}|_{s(U)})$.* 4. *[\[lem:uni-norm-bisec\]]{#lem:uni-norm-bisec label="lem:uni-norm-bisec"} If $f\in \mathrm{C_c}(U;\mathcal{A}|_{U})$, then $\lvert\!\lvert f*f^*\rvert\!\rvert_\infty = \lvert\!\lvert f^**f\rvert\!\rvert_\infty = \lvert\!\lvert f\rvert\!\rvert_\infty^2$.* The proof of ([\[lem:convulu-supp\]](#lem:convulu-supp){reference-type="ref" reference="lem:convulu-supp"})--([\[lem:inv-with-adj\]](#lem:inv-with-adj){reference-type="ref" reference="lem:inv-with-adj"}) above lemma closely follow that of [@Exel2008Inverse-semigroup-combinotorial-Cst-alg Proposition 3.11], hence we skip them. The last claim follows from the following observation: let $U$ be a bisection and $f\in \mathrm{C_c}(U;\mathcal{A}|_{U})$. If $x\in r(U)$, then $$\lvert\!\lvert f*f^*(x)\rvert\!\rvert = \lvert\!\lvert f*f^*(\gamma \gamma^{-1})\rvert\!\rvert = \lvert\!\lvert f(\gamma)\rvert\!\rvert^2$$ where $\gamma\in r^{-1}(x)$. And $f*f^*(x) = 0$ if $x\notin r(U)$. The claim follows by taking supremum in the last equation. **Proposition 1**. *Let $p\colon \mathcal{A}\to G$ be a Fell bundle over an étale groupoid $G$. Let $L\colon \mathrm{C_c}(G;\mathcal{A}) \to \mathbb B(\mathcal H)$ be a nondegenerate $^*$-homomorphism. Then $L$ is continuous in the inductive limit topology on $\mathrm{C_c}(G;\mathcal{A})$, and hence for the $I$-norm on $\mathrm{C_c}(G;\mathcal{A})$.* *Proof.* First of all, note that since $\mathrm{C_c}({G}^{(0)}; \mathcal{A}|_{{A}^{(0)}})\subseteq \mathrm{C_c}(G;\mathcal{A})$ is a $^*$-subalgebra, the restriction $L|_{\mathrm{C_c}({G}^{(0)}; \mathcal{A}|_{{A}^{(0)}})}$ is continuous in the inductive limit topology on $\mathrm{C_c}({G}^{(0)}; \mathcal{A}|_{{A}^{(0)}})$ due to Lemma [Lemma 1](#lem:local-cont-of-alg-rep){reference-type="ref" reference="lem:local-cont-of-alg-rep"}. Now, let $U\subseteq G$ be an open bisection. Let $f\in \mathrm{C_c}(U;\mathcal{A}|_{U})$. So that $f^**f\in \mathrm{C_c}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$ due to Lemma [Lemma 1](#lem:convo-func:supp:bisec){reference-type="ref" reference="lem:convo-func:supp:bisec"}(3). With this in mind, we compute $$\label{equ:rep:contractive:bise} \lvert\!\lvert L(f)\rvert\!\rvert^2 = \lvert\!\lvert L(f)^* L(f)\rvert\!\rvert = \lvert\!\lvert L(f^**f)\rvert\!\rvert\leq \lvert\!\lvert f^**f\rvert\!\rvert_{\infty} = \lvert\!\lvert f\rvert\!\rvert^2_{\infty}$$ where the inequality follows from the remark after the proof of Lemma [Lemma 1](#lem:local-cont-of-alg-rep){reference-type="ref" reference="lem:local-cont-of-alg-rep"}, and the last equality is due to Lemma [Lemma 1](#lem:convo-func:supp:bisec){reference-type="ref" reference="lem:convo-func:supp:bisec"}([\[lem:uni-norm-bisec\]](#lem:uni-norm-bisec){reference-type="ref" reference="lem:uni-norm-bisec"}). In general, let $f\in \mathrm{C_c}(G;\mathcal{A})$ with $f = \sum_{i=1}^{n}f_i$ where each ${U_i}$ is a bisection and $f_i\in \mathrm{C_c}(U_i;\mathcal{A}|_{U_i})$. Using Equation [\[equ:rep:contractive:bise\]](#equ:rep:contractive:bise){reference-type="eqref" reference="equ:rep:contractive:bise"}, we see that $$\lvert\!\lvert L(f)\rvert\!\rvert \leq \sum_{i=1}^n\lvert\!\lvert L(f_i)\rvert\!\rvert \leq \sum_{i=1}^n \lvert\!\lvert f_i\rvert\!\rvert_\infty \leq n \lvert\!\lvert f\rvert\!\rvert_\infty.$$ ◻ Next lemma, meant for a later use, concludes this section. **Lemma 1**. *Let $(u_{\lambda})_{\lambda \in \Lambda}$ be an approximate unit of $\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$ where $u_{\lambda}\in \mathrm{C_c}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$, see Corollary [Corollary 1](#cor:cc-apprx-unit-general-case){reference-type="ref" reference="cor:cc-apprx-unit-general-case"}.* 1. *For $f\in \mathrm{C_c}(G;\mathcal{A})$, $u_{\lambda}*f\to f$ in the inductive limit topology.* 2. *Let $P$ be the vector subspace of $\mathrm{C_c}(G;\mathcal{A})$ spanned by the set $\mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})*\mathrm{C_c}(G;\mathcal{A}) \mathrel{\vcentcolon=}\{\phi* f : \phi \in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}}) \text{ and } f\in \mathrm{C_c}(G;\mathcal{A})\}$. Then $P$ is uniformly dense in $\mathrm{C_c}(G;\mathcal{A})$.* *Proof.* (1): As $\mathrm{C_c}(G;\mathcal{A})$ is an inductive limit of $\mathrm{C_c}(U;\mathcal{A}|_U)$ for bisections $U$, proving the claim for $\mathrm{C_c}(U; \mathcal{A}|_U)$ is sufficient where $U$ is a bisection. Fix such $U$ and let $f\in \mathrm{C_c}(U; \mathcal{A}|_{U})$. Note that ${\textup{supp}}(u_{\lambda}*f)\subseteq {\textup{supp}}(u_{\lambda}){\textup{supp}}(f) \subseteq {\textup{supp}}(f)$. We apply Lemma [Lemma 1](#lem:convo-func:supp:bisec){reference-type="ref" reference="lem:convo-func:supp:bisec"}(4) for the bisection supported section $u_{\lambda}*f -f$ and get $$\begin{gathered} \lvert\!\lvert u_{\lambda}*f-f\rvert\!\rvert^2_\infty = \lvert\!\lvert (u_{\lambda}*f-f)*(u_{\lambda}*f-f)^*\rvert\!\rvert_\infty = \lvert\!\lvert (u_{\lambda}*f-f)*(f^**u_{\lambda}-f^*)\rvert\!\rvert_\infty \\ = \lvert\!\lvert u_{\lambda}*(f*f^*)*u_{\lambda} - u_{\lambda}*(f*f^*) - (f*f^*)*u_{\lambda} - f* f^*\rvert\!\rvert_\infty. \end{gathered}$$ As $f*f^*\in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$ and $(u_{\lambda})_{\lambda \in \Lambda}$ is an approximate unit of $\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$, the limit of last term is $0$. Therefore, $u_{\lambda}*f\to f$ in the inductive limit topology. \(2\) Follows immediately from first claim of current lemma as $u_{\lambda}*f \in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})*\mathrm{C_c}(G;\mathcal{A})$ for all $f\in \mathrm{C_c}(G;\mathcal{A})$ and $\lambda \in \Lambda$. ◻ The second claim of last lemma can also be proved using [@Williams2007Crossed-product-Cst-Alg Proposition C24] without appealing to the approximate identity argument. ## Pre-representations of $\mathrm{C_c}(G;\mathcal{A})$ {#sec:pre-rep} A $^*$-representation of $\mathrm{C_c}(G;\mathcal{A})$ that is continuous in the inductive limit topology is a obviously a pre-representation. Conversely, assume that a pre-representation $L\colon \mathrm{C_c}(G;\mathcal{A}) \to \mathrm{Lin}(\mathcal H_{0})$ of $\mathrm{C_c}(G;\mathcal{A})$ on a Hilbert space $\mathcal H$ is given. We shall extend $L$ to a $^*$-representation of $\mathrm{C_c}(G;\mathcal{A})$ on $\mathcal H$ that is continuous in the inductive limit topology in Proposition [Proposition 1](#prop:pre-rep-to-rep-of-C0-X){reference-type="ref" reference="prop:pre-rep-to-rep-of-C0-X"} of current section. This shall show the all four notions of representation of $\mathrm{C_c}(G;\mathcal{A})$, described in page , are equivalent. Here is a notation: let $1_{\mathcal M(\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}}))}$ denote the identity in the multiplier algebra of $\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$. We fix an approximate unit $(u_{\lambda})_{\lambda \in \Lambda}$ of $\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$ consisting of compactly supported sections. *Observation 1*. Let $\xi,\eta\in \mathcal H_{0}$, and $f,g,k, h\in \mathrm{C_c}(G;\mathcal{A})$, then $$\begin{aligned} \lim_{\lambda} \bigl\langle L(u_{\lambda}*f)\xi, L(u_{\lambda}*k)\eta\bigr\rangle &= \bigl\langle L(f)\xi, L(k)\eta\bigr\rangle \label{obs:pre-rep-computation-obs-1}\\ \lim_{\lambda} \bigl\langle L(f*u_{\lambda})\xi, L(k*u_{\lambda})\eta\bigr\rangle &= \bigl\langle L(f)\xi, L(k)\eta\bigr\rangle \label{obs:pre-rep-computation-obs-2}\\ \lim_{\lambda} \bigl\langle L(f*u_{\lambda}*g)\xi, L(k*u_{\lambda}*h)\eta\bigr\rangle &= \bigl\langle L(f*g)\xi, L(k*h)\eta\bigr\rangle.\label{obs:pre-rep-computation-obs-3} \end{aligned}$$ For the first equation, we note that due to the adjointability of a pre-representation, the left hand side equals $\lim_{\lambda} \langle L((u_{\lambda}*g)^**(u_{\lambda}*f))\xi, \eta\rangle$. Lemma [Lemma 1](#lem:mult-by-bdd-sections){reference-type="ref" reference="lem:mult-by-bdd-sections"}(1) implies that $u_{\lambda}*f\to f$ and $g^**u_{\lambda}\to g$ in the inductive limit topology. Plus, the convolution is continuous in the inductive limit topology. Therefore, the continuity of a pre-representation implies that $\lim_{\lambda} \bigl\langle L((u_{\lambda}*g)^**(u_{\lambda}*f))\xi, \eta\bigr\rangle = \bigl\langle L( g^* *f)\xi, \eta\bigr\rangle = \bigl\langle L(f)\xi, L(g)\eta\bigr\rangle$. Equation [\[obs:pre-rep-computation-obs-2\]](#obs:pre-rep-computation-obs-2){reference-type="eqref" reference="obs:pre-rep-computation-obs-2"} is justified similarly. The last one, Equation [\[obs:pre-rep-computation-obs-3\]](#obs:pre-rep-computation-obs-3){reference-type="eqref" reference="obs:pre-rep-computation-obs-3"}, follows from Equation [\[obs:pre-rep-computation-obs-2\]](#obs:pre-rep-computation-obs-2){reference-type="eqref" reference="obs:pre-rep-computation-obs-2"} by noticing that $$\bigl\langle L(f*u_{\lambda}*g)\xi, L(k*u_{\lambda}*h)\eta\bigr\rangle = \bigl\langle L(f*u_{\lambda}) \bigl(L(g)\xi\bigr), L(k*u_{\lambda}) \bigl(L(h)\eta\bigr)\bigr\rangle$$ where $L(g)\xi, L(h)\eta\in \mathcal H_{00}\subseteq \mathcal H_0$. **Lemma 1**. *For a section $h\in \mathrm{C_c}(U;\mathcal{A}|_{U})$ where $U$ is a bisection, define $$q \mathrel{\vcentcolon=}1_{ \mathcal M(\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}}))} \cdot \lvert\!\lvert h \rvert\!\rvert^2_\infty - (h^** h) \quad \text {and } \quad q_{\lambda} = u_{\lambda} q u_{\lambda}$$ where $q\in \mathcal M(\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}}))$ and $q_{\lambda}\in \mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$. The following statements hold:* 1. *$q$ is a positive element in $\mathcal M(\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}}))$.* 2. *$q_{\lambda}$ is a positive element in $\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$ for $\lambda \in \Lambda$.* 3. *For a fixed $v\in \mathcal H_{00}$, $$\lim_{\lambda} (\lvert\!\lvert L(\lvert\!\lvert h\rvert\!\rvert_\infty u_{\lambda})v\rvert\!\rvert^2 - \lvert\!\lvert L(h*u_{\lambda})v\rvert\!\rvert^2) = (\lvert\!\lvert h\rvert\!\rvert_\infty \lvert\!\lvert v\rvert\!\rvert)^2 - \lvert\!\lvert L(h)v\rvert\!\rvert^2.$$* *Proof.* (1): Lemma [Lemma 1](#lem:convo-func:supp:bisec){reference-type="ref" reference="lem:convo-func:supp:bisec"}([\[lem:inv-with-adj\]](#lem:inv-with-adj){reference-type="ref" reference="lem:inv-with-adj"})--([\[lem:uni-norm-bisec\]](#lem:uni-norm-bisec){reference-type="ref" reference="lem:uni-norm-bisec"}) implies that $h^**h\in \mathrm{C_c}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$ with $\lvert\!\lvert h^* * h\rvert\!\rvert_\infty = \lvert\!\lvert h\rvert\!\rvert_\infty^2$. Now Lemma [Lemma 1](#lem:positive-element){reference-type="ref" reference="lem:positive-element"} implies that $q = 1_{ \mathcal M(\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}}))} \cdot \lvert\!\lvert h^**h\rvert\!\rvert_\infty -(h^**h)$ is a positive element in $\mathcal M(\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}}))$. (2): Let $k$ be the positive square root of $q$ in the multiplier algebra. Then $u_{\lambda}qu_{\lambda} = (k u_{\lambda})^*(ku_{\lambda})$. (3): For $v\in \mathcal H_{00}$, compute the difference $$\begin{gathered} \lvert\!\lvert L(\lvert\!\lvert h\rvert\!\rvert_\infty u_{\lambda})v\rvert\!\rvert^2 - \lvert\!\lvert L(h*u_{\lambda})v\rvert\!\rvert^2 \\= \big\langle L(\lvert\!\lvert h\rvert\!\rvert_\infty u_{\lambda})v \mathbin{,} L(\lvert\!\lvert h\rvert\!\rvert_\infty u_{\lambda})v \big\rangle - \bigl\langle L(h*u_{\lambda})v, L(h*u_{\lambda})v\bigr\rangle. \end{gathered}$$ As $v\in \mathcal H_{00}$, we can write $v=\sum_{i=1}^k L(f_i)\xi_i$ for some $f_i\in \mathrm{C_c}(G;\mathcal{A})$, $\xi_i\in \mathcal H_0$ and $k\in \mathbb{N}$. Substitute this value of $v$ in last subtraction, and compute it further: $$\begin{gathered} \bigl\langle L(\lvert\!\lvert h\rvert\!\rvert_\infty u_{\lambda}) \sum_{i=1}^k L(f_i)\xi_i, L(\lvert\!\lvert h\rvert\!\rvert_\infty u_{\lambda}) \sum_{i=1}^k L(f_i)\xi_i\bigr\rangle \\ - \bigl\langle L(h*u_{\lambda}) \sum_{i=1}^k L(f_i)\xi_i , L(h*u_{\lambda})\sum_{i=1}^k L(f_i)\xi_i\bigr\rangle\\ = \sum_{i,j} \Bigl(\lvert\!\lvert h\rvert\!\rvert_\infty^2\bigl\langle L(u_{\lambda} * f_i)\xi_i, L(u_{\lambda}*f_j)\xi_j\bigr\rangle - \bigl\langle L(h*u_{\lambda}* f_i)\xi_i, L(h*u_{\lambda} * f_j)\xi_j\bigr\rangle\Bigr). \end{gathered}$$ Using Equations [\[obs:pre-rep-computation-obs-1\]](#obs:pre-rep-computation-obs-1){reference-type="eqref" reference="obs:pre-rep-computation-obs-1"} and [\[obs:pre-rep-computation-obs-3\]](#obs:pre-rep-computation-obs-3){reference-type="eqref" reference="obs:pre-rep-computation-obs-3"}, the limit of preceding subtraction over $\Lambda$ can be computed to be $$\begin{gathered} \sum_{i,j} \Big(\lvert\!\lvert h\rvert\!\rvert_\infty^2 \bigl\langle L(f_i)\xi_i, L(f_j)\xi_j\bigr\rangle - \bigl\langle L(h*f_i)\xi_i, L(h*f_j)\xi_j\bigr\rangle\Bigr)\\ = \sum_{i,j} \Big(\lvert\!\lvert h\rvert\!\rvert_\infty^2 \bigl\langle L(f_i)\xi_i, L(f_j)\xi_j\bigr\rangle - \bigl\langle L(h) L(f_i)\xi_i, L(h)L(f_j)\xi_j\bigr\rangle\Bigr)\\ = \lvert\!\lvert h\rvert\!\rvert_\infty^2 \Bigl\langle\sum_i^kL(f_i)\xi_i, \sum_i^kL(f_i)\xi_i\Bigr\rangle - \Bigl\langle L(h) \sum_i^kL(f_i)\xi_i, L(h) \sum_i^kL(f_i)\xi_i\Bigr\rangle \end{gathered}$$ which proves the claim. ◻ **Proposition 1**. *Suppose that a Fell bundle $p\colon \mathcal{A}\to G$ over an étale groupoid $G$ is given. Suppose that a pre-representation $L\colon \mathrm{C_c}(G;\mathcal{A}) \to \mathrm{Lin}(\mathcal H_0)$ is given where $\mathcal H_0 \subseteq \mathcal H$ is a dense subspace. Then $L$ can be extended to a representation $M\colon \mathrm{C_c}(G, \mathcal{A}) \to \mathbb B(\mathcal H)$ that is continuous in the inductive limit topology. To be precise, the following holds: let $h\in \mathrm{C_c}(G; \mathcal{A})$ and $\mathcal H_{00}$ have the same meaning as in Definition [Definition 1](#def:pre-rep){reference-type="ref" reference="def:pre-rep"}. For this $h$, define the linear operator $M'(h)$ on the inner product space $\mathcal H_{00}$ which is given on generating vectors $L(f)\xi\in \mathcal H_{00}$ by $$M'(h)(L(f)\xi) = L( h * f)\xi\label{eq:rep-from-pre-rep};$$ here $f\in \mathrm{C_c}(G; \mathcal{A})$ and $\xi\in \mathcal H_0$. Then:* 1. *the linear operator $M'(h)$ extends to a bounded linear operator $M(h)\colon \mathcal H\to \mathcal H$.* 2. *$M\colon \mathrm{C_c}(G;\mathcal{A}) \to \mathbb B(\mathcal H)$ given by $M\colon h \mapsto M(h)$ is a nondegenerate representation that is continuous in the inductive limit topology on $\mathrm{C_c}(G;\mathcal{A})$.* 3. *$M$ is an extension of $L$, that is, $M(h)|_{\mathcal H_{0}} = L(h)$.* *Proof.* (1): We prove it in two steps. Step I: Assume that ${\textup{supp}}(h)$ lies inside an open bisection. We intend to prove that $$\lvert\!\lvert M'(h)v\rvert\!\rvert\leq \lvert\!\lvert h\rvert\!\rvert_\infty \lvert\!\lvert v\rvert\!\rvert \quad \text{ for } v \in \mathcal H_{00}.\label{eq:bdd-op-on-M00}$$ If last inequality is proved, then due to the density of $\mathcal H_{00}\subseteq \mathcal H$, $M'(h)$ can be extended to a bounded operator $M(h)\colon \mathcal H\to \mathcal H$. For proving Equation [\[eq:bdd-op-on-M00\]](#eq:bdd-op-on-M00){reference-type="eqref" reference="eq:bdd-op-on-M00"}, consider the given approximate unit $(u_{\lambda})_{\lambda \in \Lambda}$ of $\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$ where $u_{\lambda}\in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$ (cf. Corollary [Corollary 1](#cor:cc-apprx-unit-general-case){reference-type="ref" reference="cor:cc-apprx-unit-general-case"}). Let $v\in \mathcal H_{00}$, so that $v = \sum_{i=1}^k L(f_i)\xi_i$ for some $k\in \mathbb{N}$, $f_i\in \mathrm{C_c}(G;\mathcal{A})$ and $\xi_i\in \mathcal H_0$. Note that Lemma [Lemma 1](#lem:sqrt-of-q){reference-type="ref" reference="lem:sqrt-of-q"}(3) implies that $$(\lvert\!\lvert h\rvert\!\rvert_\infty \lvert\!\lvert v\rvert\!\rvert)^2 - \lvert\!\lvert M'(h)v\rvert\!\rvert^2 = \lim_{\lambda} (\lvert\!\lvert h\rvert\!\rvert_\infty \lvert\!\lvert L(u_{\lambda})v\rvert\!\rvert)^2 - \lvert\!\lvert L(h*u_{\lambda})v\rvert\!\rvert^2.$$ Therefore, for proving Equation [\[eq:bdd-op-on-M00\]](#eq:bdd-op-on-M00){reference-type="eqref" reference="eq:bdd-op-on-M00"}, it suffices to show that $(\lvert\!\lvert h\rvert\!\rvert_\infty \lvert\!\lvert L(u_{\lambda})v\rvert\!\rvert)^2 - \lvert\!\lvert L(h*u_{\lambda})v\rvert\!\rvert^2 \geq 0$ for all $\lambda \in \Lambda$. For $\lambda \in \Lambda$, we compute this differences as follows: $$\begin{gathered} (\lvert\!\lvert h\rvert\!\rvert_\infty \lvert\!\lvert L(u_{\lambda})v\rvert\!\rvert)^2 - \lvert\!\lvert L(h*u_{\lambda})v\rvert\!\rvert^2 = \lvert\!\lvert h\rvert\!\rvert^2_\infty \lvert\!\lvert \sum_{i=1}^k L(u_{\lambda}*f_i)\xi_i\rvert\!\rvert^2 - \lvert\!\lvert \sum_{i=1}^k L(h*u_{\lambda} * f_i)\xi_i\rvert\!\rvert^2\\ = \Bigl\langle\lvert\!\lvert h\rvert\!\rvert^2_\infty \left(\sum_{i=1}^k L(u_{\lambda}*f_i)\xi_i\right), \sum_{i=1}^k L(u_{\lambda} *f_i)\xi_i\Bigr\rangle - \Bigl\langle \sum_{i=1}^k L(h * u_{\lambda}* f_i)\xi_i, \sum_{i=1}^k L(h * u_{\lambda}* f_i)\xi_i\Bigr\rangle. \end{gathered}$$ Now use the adjointability of the pre-representation, so that the last term becomes   $$\begin{gathered} \label{eq:pre-rep-to-rep-of-C0-X-1} \sum_{i, j=1}^k \Bigl\langle ( L(\lvert\!\lvert h\rvert\!\rvert^2_\infty u_{\lambda}*u_{\lambda}* f_i))\xi_i, L(f_j)\xi_j\Bigr\rangle\\ - \sum_{i, j=1}^k \Bigl\langle L(u_{\lambda}*h^* * h * u_{\lambda} * f_i)\xi_i, L(f_j)\xi_j \Bigr\rangle. \end{gathered}$$ Notice that $h^**h \in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$. As a consequence $u_{\lambda}*(h^**h) *u_{\lambda} = u_{\lambda}\cdot (h^**h) \cdot u_{\lambda}\in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$. Moreover, Equation [\[eq:conv\]](#eq:conv){reference-type="eqref" reference="eq:conv"} implies that the convolutions $$\begin{aligned} ( u_{\lambda}*(h^**h)*u_{\lambda}) * f_i &= (u_{\lambda}\cdot (h^**h) \cdot u_{\lambda})\circ r \cdot f_i = (u_{\lambda}\circ r \cdot (h^**h)\circ r \cdot u_{\lambda}\circ r) \cdot f_i\; ;\\ \text{and }\quad u_{\lambda}^2 * f_i &= (u_{\lambda}^2 \circ r) \cdot f_i = (u_{\lambda} \circ r)^2 \cdot f_i. \end{aligned}$$ Therefore, the difference [\[eq:pre-rep-to-rep-of-C0-X-1\]](#eq:pre-rep-to-rep-of-C0-X-1){reference-type="eqref" reference="eq:pre-rep-to-rep-of-C0-X-1"} can be written as $$\begin{gathered} \sum_{i, j=1}^k \Bigl\langle L( \lvert\!\lvert h\rvert\!\rvert^2_\infty \cdot (u_{\lambda}\circ r)^2\cdot f_i)\xi_i, L(f_j)\xi_j\Bigr\rangle \\- \sum_{i, j=1}^k \Bigl\langle L( (u_{\lambda}\circ r)\cdot (h^**h)\circ r \cdot(u_{\lambda}\circ r) \cdot f_i)\xi_i, L(f_j)\xi_j\Bigr\rangle \end{gathered}$$ Using the bilinearity of the inner product, the last term can be written as $$\begin{gathered} \sum_{i, j=1}^k \Bigl\langle L\left(\left(\lvert\!\lvert h\rvert\!\rvert^2_\infty \cdot (u_{\lambda}\circ r)^2 - (u_{\lambda}\circ r)\cdot (h^**h)\circ r \cdot(u_{\lambda}\circ r) \right)\cdot f_i\right)\xi_i, L(f_j)\xi_j\Bigr\rangle \\ = \sum_{i, j=1}^k \Bigl\langle L\left(\left(u_{\lambda}\circ r \left(\lvert\!\lvert h\rvert\!\rvert^2_\infty\cdot 1_{ \mathcal M(\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}}))} - (h^**h) \circ r \right) u_{\lambda}\circ r\right) \cdot f_i\right)\xi_i, L(f_j)\xi_j\Bigr\rangle \end{gathered}$$ where $1_{\mathcal M(\mathrm{C_0}({G}^{(0)}); \mathcal{A}|_{{G}^{(0)}})}\in \mathcal M(\mathrm{C_0}({G}^{(0)}); \mathcal{A}|_{{G}^{(0)}})$ is the unit in the multiplier algebra. Following Lemma [Lemma 1](#lem:sqrt-of-q){reference-type="ref" reference="lem:sqrt-of-q"}, put $q =\lvert\!\lvert h\rvert\!\rvert^2_\infty\cdot 1_{ \mathcal M(\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}}))} - (h^**h) \in \mathcal M(\mathrm{C_0}({G}^{(0)}); \mathcal{A}|_{{G}^{(0)}})$ and $q_{\lambda} = u_{\lambda} q u_{\lambda}\in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$. Then the last term in above computation becomes $$\sum_{i, j=1}^k \bigl\langle L\left((q_{\lambda} \circ r) \cdot f_i\right)\xi_i, L(f_j)\xi_j\bigr\rangle.$$ As Lemma [Lemma 1](#lem:sqrt-of-q){reference-type="ref" reference="lem:sqrt-of-q"} says, $q_{\lambda}$ is a positive element in $\mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}}) \subseteq \mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$. Let $s_{\lambda}\in \mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$ be its unique positive square root; then $s_{\lambda}\in \mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$. By substituting $q_{\lambda}=s_{\lambda}^2$ last term of main computation becomes $$\begin{gathered} \sum_{i, j=1}^k \bigl\langle L((s_{\lambda}\circ r) \cdot f_i)\xi_i, L((s_{\lambda}\circ r) \cdot f_j)\xi_j\bigr\rangle = \sum_{i, j=1}^k \bigl\langle L(s_{\lambda}* f_i)\xi_i, L(s_{\lambda} * f_j)\xi_j\bigr\rangle\\ = \bigl\langle\sum_{i=1}^nL(s_{\lambda}* f_i)\xi_i, \sum_{i=1}^nL(s_{\lambda} * f_i)\xi_i\bigr\rangle \geq 0. \end{gathered}$$ Thus Equation [\[eq:bdd-op-on-M00\]](#eq:bdd-op-on-M00){reference-type="eqref" reference="eq:bdd-op-on-M00"} is proved. Step II: Now, suppose $h\in \mathrm{C_c}(G, \mathcal{A})$ is given. Write $h= \sum_{i=1}^p h_i$ where each $h_i$ is supported in a bisection. Now, Equation [\[eq:bdd-op-on-M00\]](#eq:bdd-op-on-M00){reference-type="eqref" reference="eq:bdd-op-on-M00"} implies that $\lvert\!\lvert M'(h)v\rvert\!\rvert \leq (\sum_{i=1}^p \lvert\!\lvert h_i\rvert\!\rvert_\infty ) \lvert\!\lvert v\rvert\!\rvert \leq p \lvert\!\lvert h\rvert\!\rvert_\infty \lvert\!\lvert v\rvert\!\rvert$ for $v\in \mathcal H_{00}$; here the number $p$ depends on a representation of $h$ as a sum of $h_i$'s. But for the fixed representation, $p$ is fixed. Now we can extend $M'(h)$ to a bounded operator $M(h)\colon \mathcal H\to \mathcal H$ using the density of $\mathcal H_0\subseteq \mathcal H$. Finally, note that $M(h)\leq p\lvert\!\lvert h\rvert\!\rvert_\infty$. Therefore, $M\colon \mathrm{C_c}(G;\mathcal{A}) \to \mathbb B(\mathcal H)$ is continuous in the inductive limit topology. (3): By construction of $M(h)$ in (1) of the current lemma, it is clear that $M(h)|_{\mathcal H_{00}} = L(h)|_{\mathcal H_{00}}$. Since $\mathcal H_{00}\subseteq \mathcal H_0$ is dense, $M(h)|_{\mathcal H_0} = L(h)$. (2): Let $h\in \mathrm{C_c}(G;\mathcal{A})$. Then due to (3) and the adjointability of a pre-representation, $M(h^*) = M(h)^*$. Finally, the nondegeneracy of $M$ follows from the nondegeneracy of a pre-representation and the fact that the linear subspace of $\mathrm{C_c}(G;\mathcal{A})$ spanned by $\mathrm{C_c}(G;\mathcal{A})*\mathrm{C_c}(G;\mathcal{A})$ is uniformly dense, cf. Lemma [Lemma 1](#lem:mult-by-bdd-sections){reference-type="ref" reference="lem:mult-by-bdd-sections"}(2). ◻ For the Fell bundle $p\colon \mathcal{A}\to G$, define the following collections: 1. $\mathcal{R}_R$, the collection of representations; 2. $\mathcal{R}_{H}$, the collection of $I$-norm bounded representations; 3. $\mathcal{R}_{\mathrm{In}}$ the collection of representations continuous in the inductive limit topology; 4. $\mathcal{R}_P$, the collection of all representations induced from pre-representations as in Proposition [Proposition 1](#prop:pre-rep-to-rep-of-C0-X){reference-type="ref" reference="prop:pre-rep-to-rep-of-C0-X"}. With this notation, we can state the following theorem: **Theorem 1**. *Let $p\colon \mathcal{A}\to G$ be a (not necessarily saturated) Fell bundle over an étale groupoid $G$ (not necessarily Hausdorff). For $f\in \mathrm{C_c}(G;\mathcal{A})$, define[^2] $$\begin{aligned} \lvert\!\lvert f\rvert\!\rvert_R &= \sup_{\rho\in \mathcal{R}_R}\{\lvert\!\lvert \rho(f)\rvert\!\rvert \}, & \lvert\!\lvert f\rvert\!\rvert_{H} &= \sup_{\rho\in \mathcal{R}_{H}}\{\lvert\!\lvert \rho(f)\rvert\!\rvert \},\\ \lvert\!\lvert f\rvert\!\rvert_{\mathrm{In}} &= \sup_{\rho\in \mathcal{R}_{\mathrm{In}}}\{\lvert\!\lvert \rho(f)\rvert\!\rvert \}, & \lvert\!\lvert f\rvert\!\rvert_P &= \sup_{\rho\in \mathcal{R}_P}\{\lvert\!\lvert \rho(f)\rvert\!\rvert \}. \end{aligned}$$* *Then $\lvert\!\lvert f\rvert\!\rvert_R = \lvert\!\lvert f\rvert\!\rvert_{H} = \lvert\!\lvert f\rvert\!\rvert_{\mathrm{In}}= \lvert\!\lvert f\rvert\!\rvert_{P}$. And $\lvert\!\lvert f\rvert\!\rvert \mathrel{\vcentcolon=}\lvert\!\lvert f\rvert\!\rvert_R$ is a $\mathrm{C}^*$-norm on $\mathrm{C_c}(G;\mathcal{A})$. Moreover, the $\mathrm{C}^*$-algebra $\overline{\mathrm{C_c}(G;\mathcal{A})}^{\lvert\!\lvert \cdot\rvert\!\rvert_R}$ is same as the $\mathrm{C}^*$-algebra that Muhly--Williams define in [@Muhly-Williams2008FellBundle-ME] for saturated Fell bundle over locally compact Hausdorff second countable groupoid.* *Proof.* The equality $\lvert\!\lvert f\rvert\!\rvert_R =\lvert\!\lvert f\rvert\!\rvert_{H} = \lvert\!\lvert f\rvert\!\rvert_{\mathrm{In}} = \lvert\!\lvert f\rvert\!\rvert_P$ follows from equivalence of the four notions of representations, Propositions [Proposition 1](#prop:three-reps-are-equi){reference-type="ref" reference="prop:three-reps-are-equi"} and [Proposition 1](#prop:pre-rep-to-rep-of-C0-X){reference-type="ref" reference="prop:pre-rep-to-rep-of-C0-X"}. The claim that $\lvert\!\lvert \cdot\rvert\!\rvert_R$ is a $\mathrm{C}^*$-norm, follows from a standard argument, for example, see [@Sims-Szabo-Williams2020Operator-alg-book Theorem 9.2.3]. Muhly and Williams use $\lvert\!\lvert \cdot\rvert\!\rvert_{\mathrm{In}}$ and $\lvert\!\lvert \cdot\rvert\!\rvert_P$ to define Fell bundle $\mathrm{C}^*$-algebra; we show that these norms agree with $\lvert\!\lvert \cdot\rvert\!\rvert_R$; this justifies the last claim. ◻ The $\mathrm{C}^*$-algebra $\overline{\mathrm{C_c}(G;\mathcal{A})}^{\lvert\!\lvert \cdot\rvert\!\rvert}$ is the *full* $\mathrm{C}^*$-algebra of $p\colon \mathcal{A}\to G$. This $\mathrm{C}^*$-algebra is denoted by $\mathrm{C}^*(G;\mathcal{A})$. Assume that in Theorem [Theorem 1](#thm:norm-equi){reference-type="ref" reference="thm:norm-equi"}, $G$ is locally compact Hausdorff and second countable étale; and $\mathcal{A}$ saturated and separable. Then Muhly and Williams define the full $\mathrm{C}^*$-algebra of $\mathcal{A}$ using $\mathcal{R}_H, \mathcal{R}_{\mathrm{In}}$ and $\mathcal{R}_P$ (and unitary representations of $p$). They show that these three norms are same. We have added one more norm $\mathrm{R}_R$. The $\mathrm{C}^*$-algebra given by last theorem is same as that of Muhly and Williams. For all following remarks a Fell bundle $p\colon \mathcal{A}\to G$ over an étale groupoid is fixed. *Remark 1*. Consider Theorem [Theorem 1](#thm:norm-equi){reference-type="ref" reference="thm:norm-equi"} for the trivial Fell bundle over an étale groupoid. Then $\mathcal{R}_R$ is the norm on $\mathrm{C_c}(G)$ that Exel defines in [@Exel2008Inverse-semigroup-combinotorial-Cst-alg]; $\mathcal{R}_I$ is discussed in [@Renault1980Gpd-Cst-Alg] by Renault or [@Sims-Szabo-Williams2020Operator-alg-book] by Sims; and $\mathcal{R}_{\mathrm{In}}$ and $\mathcal{R}_{P}$ are the norms Renault defines in  [@Renault1985Representations-of-crossed-product-of-gpd-Cst-Alg] (and proves are equal). Theorem [Theorem 1](#thm:norm-equi){reference-type="ref" reference="thm:norm-equi"} shows that all these norms, and hence corresponding $\mathrm{C}^*$-algebras are same. *Remark 1*. Now $\mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$ is a $^*$-algebra of $\mathrm{C_c}(G;\mathcal{A})$. Due to the uniqueness of $\mathrm{C}^*$-norm, $\overline{\mathrm{C_c}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})}^{\lvert\!\lvert \cdot\rvert\!\rvert} \subseteq \mathrm{C}^*(G,\mathcal{A})$ must be $\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$. Thus, the $\mathrm{C_0}({G}^{(0)})$-algebra $\mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$ is a $\mathrm{C}^*$-subalgebra of $\mathrm{C}^*(G;\mathcal{A})$. In particular, for $f\in \mathrm{C_0}({G}^{(0)};\mathcal{A}|_{{G}^{(0)}})$, the $\mathrm{C}^*$-norm $\lvert\!\lvert f\rvert\!\rvert = \lvert\!\lvert f\rvert\!\rvert_\infty$. *Remark 1*. For $f\in \mathrm{C_c}(G;\mathcal{A})$ be supported in a bisection, the $\mathrm{C}^*$-norm $$\lvert\!\lvert f\rvert\!\rvert^2 = \lvert\!\lvert f*f^*\rvert\!\rvert = \lvert\!\lvert f*f^*\rvert\!\rvert_\infty = \lvert\!\lvert f\rvert\!\rvert_\infty^2$$ where the first equality is the $\mathrm{C}^*$-identity, the second equality is because $f*f^* \in \mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$ and the last one is Lemma [Lemma 1](#lem:convo-func:supp:bisec){reference-type="ref" reference="lem:convo-func:supp:bisec"}(4). Moreover, Example [Example 1](#exa:I-norm-on-bisection){reference-type="ref" reference="exa:I-norm-on-bisection"} says that $\lvert\!\lvert f\rvert\!\rvert_I = \lvert\!\lvert f\rvert\!\rvert_\infty$. Therefore, for a section supported on a bisection, the full $\mathrm{C}^*$-norm equals the $I$-norm or uniform norm. *Remark 1*. Let $(u_\lambda)$ be an approximate unit of $\mathrm{C_0}({G}^{(0)}; \mathcal{A}|_{{G}^{(0)}})$ consisting of compactly supported sections. Remark [Remark 1](#rem:section-alg-of-base){reference-type="ref" reference="rem:section-alg-of-base"} and Lemma [Lemma 1](#lem:mult-by-bdd-sections){reference-type="ref" reference="lem:mult-by-bdd-sections"}(1) imply that $(u_\lambda)$ is also an approximate unit of $\mathrm{C}^*(G;\mathcal{A})$. *Example 1*. Let $G$ be locally compact Hausdorff étale groupoid and $\mathcal{A}\to {G}^{(0)}$ an upper semicontinuous bundle of $\mathrm{C}^*$-algebras. Consider the bundle $\mathcal{B}\to G$ which is $\mathcal{A}$ over ${G}^{(0)}$ and has the zero fibres otherwise. Then $\mathcal{B}$ is upper semicontinuous, and, in fact, a Fell bundle over $G$. The Fell bundle $\mathcal{B}$ is not a saturated bundle unless $\mathcal{A}$ consists of the zero $\mathrm{C}^*$-algebras. In this case, $\mathrm{C_c}(G; \mathcal{B}) = \mathrm{C_c}({G}^{(0)}; \mathcal{A})$ and $\mathrm{C}^*(G; \mathcal{B}) \simeq\mathrm{C_0}({G}^{(0)}; \mathcal{A})$. *Example 1*. Let $G$ be the groupoid of trivial equivalence relation on the set $\{0, 1\}$, that is, $G = \{(0, 0), (0,1), (1,0),(1,1)\}$. Then $G$ is an étale groupoid. Let $A$ and $B$ be two $\mathrm{C}^*$-algebras and $X$ a (not necessarily full) Hilbert $A$-$B$-bimodule. Let $X^*$ be the dual module of $X$. Consider the Fell bundle $p\colon \mathcal{A}\to G$ as follows: the fibres are $\mathcal{A}_{(0,0)} = A, \mathcal{A}_{(0,1)} = X, \mathcal{A}_{(1,0)} = X^*$ and $\mathcal{A}_{(1,1)} = B$. The multiplications of Fell bundle $\mathcal{A}_{(0,1)} \times \mathcal{A}_{(1,0)} \to \mathcal{A}_{(0,0)}$ is defined by $\xi \cdot \eta \mathrel{\vcentcolon=}{}_A \langle\xi, \eta^*\rangle$ and $\mathcal{A}_{(1,0)} \times \mathcal{A}_{(0,1)} \to \mathcal{A}_{(1,1)}$ is defined by $\eta \cdot \xi \mathrel{\vcentcolon=}\langle\eta^*, \xi\rangle_B$, where $\xi \in X$ and $\eta \in X^*$. This Fell bundle is unsaturated unless $X$ is an *imprimitivity bimodule*. Let $L$ be the linking algebra associated with the Hilbert bimodule $X$ defined by Brown, Mingo and Shen [@Brown-Mingo-Shen1994Quasi-mult-enbed-Hilbert-mod]. Define $\phi \colon \mathrm{C}(G;\mathcal{A}) \to L$ by $$\phi(f) = \begin{pmatrix} f(0,0) & f(0,1) \\ f(1,0) & f(1,1) \end{pmatrix}$$ for $f\in \mathrm{C}(G;\mathcal{A})$. Then $\phi$ is a bijective $^*$-homomorphism. Since $L$ is a $\mathrm{C}^*$-algebra, by the uniqueness of the $\mathrm{C}^*$-norm $\mathrm{C}^*(G;\mathcal{A}) \simeq L$. This example is an *unsaturated* version of Kumjian's example [@Kumjian1998Fell-bundles-over-gpd Example 3.5(ii)]. *Example 1* (Fell bundle associated with partial actions). One may associate Fell bundles to partial actions of groups or groupoids; and these Fell bundles are the prototypical examples of unsaturated Fell bundles. Assume that a discrete group  $\Gamma$ acts partially on a unital $\mathrm{C}^*$-algebra $A$ (see [@Exel2017PDS-Fell-bundles Chapter 11]). Then Exel proved [@Exel2017PDS-Fell-bundles Chapter 16] that the Fell bundle $\mathcal{B}$ associated with the partial action is an *unsaturated* one. Moreover, the $\mathrm{C}^*$-algebra of the Fell bundle $\mathcal{B}$ is naturally isomorphic to the partial crossed product $A\rtimes \Gamma$ (Proposition 16.28 of [@Exel2017PDS-Fell-bundles]). Anantharaman-Delaroche generalises Exel's idea of constructing a Fell bundle out of group partial action [@Exel2017PDS-Fell-bundles Chapter 16] to groupoids in [@Anantharaman-D:2020Partial-Action-of-Gpd Section 4]. In particular, let $G$ be a locally compact Hausdorff étale groupoid with unit space ${G}^{(0)}$. Let $A$ be a $\mathrm{C_0}({G}^{(0)})$-algebra on which $G$ is acting partially. In [@Anantharaman-D:2020Partial-Action-of-Gpd Section 4], Anantharaman-Delaroche defines the Fell bundle $p\colon \mathcal{B}\to G$ associated with this action. She constructs $\mathrm{C}^*(G;\mathcal{B})$ using the $I$-norm bounded representation. Since we have assumed $G$ to be étale, her $\mathrm{C}^*(G;\mathcal{B})$ is same as the Fell bundle $\mathrm{C}^*$-algebra Theorem [Theorem 1](#thm:norm-equi){reference-type="ref" reference="thm:norm-equi"} leads to. As Anantharaman-Delaroche notices [@Anantharaman-D:2020Partial-Action-of-Gpd Remark 4.5], the Fell bundle $\mathrm{C}^*$-algebra is isomorphic to the $\mathrm{C}^*$-algebra of the partial action $G\curvearrowright A$. One can note that her construction and last remark hold even if the étale groupoid $G$ is not Hausdorff as in our sense. [^1]: Fell writes an equivalent condition that *looks* weaker, cf. [@Fell1988Representation-of-Star-Alg-Banach-bundles II.13.10] [^2]: *For "RRR" fans, $\rho\in \mathcal{R}_R$ is not a coincidence!*

arxiv_math

--- abstract: | We define a $\mathbb{Q}$-valued deformation invariant of certain complete Riemann-Finsler manifolds. It is proved that every rational number is the value of this invariant for some compact Riemannian manifold. We use this to prove (possibly non-compact but complete) fibration generalizations of Preissman's theorem on non-existence of negative sectional curvature metrics on compact products. Along the way, we also prove that sky catastrophes of smooth dynamical systems are not geodesible by a certain class of forward complete Riemann-Finsler metrics, in particular by Riemannian metrics with non-positive sectional curvature. This partially answers a question of Fuller and gives important examples for our theory here. In a sister paper  [@cite_SavelyevEllipticCurvesLcs], we study a direct generalization of this metric invariant, by lifting the count of geodesics to a Gromov-Witten count of elliptic curves in an associated locally conformally symplectic manifold. address: Faculty of Science, University of Colima, Mexico author: - Yasha Savelyev bibliography: - "C:/Users/yasha/texmf/bibtex/bib/link.bib" title: Gromov-Witten invariants of Riemann-Finsler manifolds --- # Introduction We will define certain rational number valued deformation invariants for certain complete Riemann-Finsler manifolds, in particular for complete Riemannian metrics with non-positive sectional curvature. These invariants can be directly interpreted as a part of certain elliptic Gromov-Witten invariants in an associated lcs manifold,  [@cite_SavelyevEllipticCurvesLcs]. However, in the more basic setting here, we can reduce the invariants to counts of closed geodesic strings (equivalence classes of closed unit speed geodesics up to reparametrization $S ^{1}$ action), via Fuller index  [@cite_FullerIndex]. And so this self contained more elementary story is developed separately here. **Terminology 1**. All our metrics are Riemann-Finsler metrics unless specified to be Riemannian, and usually denoted by just $g$. Completeness, always means forward completeness. Curvature always means sectional curvature in the Riemannian case and flag curvature in the Finsler case. Thus we will usually just say complete metric $g$, for a forward complete Riemann-Finsler metric. A reader may certainly choose to interpret all metrics as Riemannian metrics, completeness as standard completeness, and curvature as sectional curvature. In what follows $\pi _{1} (X)$ denotes the set of free homotopy classes of continuous maps $o: S ^{1} \to X$. **Definition 1**. *Let $X$ be a smooth manifold. Fix an exhaustion by nested compact sets $\bigcup _{i \in \mathbb{N}} K _{i} = X$, $K _{i} \supset K _{i-1}$ for all $i \geq 1$. We say that a class $\beta \in \pi _{1} (X)$ is boundary compressible if $\beta$ is in the image of $$inc _{*}: \pi _{1}(X - K _{i}) \to \pi _{1} (X)$$ for all $i$, where $inc: X - K _{i} \to X$ is the inclusion map. We say that $\beta$ is ***boundary incompressible*** if it is not boundary compressible.* Let $\pi _{1} ^{inc} (X)$ denote the set of such boundary incompressible classes. When $X$ is compact, we set $\pi _{1} ^{inc} (X) := \pi _{1} (X) - const$, where $const$ denotes the set of homotopy classes of constant loops. It is easily seen that the above is well defined and moreover any homeomorphism $X _{1} \to X _{2}$ of a pair of manifolds induces a set isomorphism $\pi _{1} ^{inc} (X _{1}) \to \pi _{1} ^{inc} (X _{2})$. Denote by $L _{\beta } X$ the class $\beta \in \pi _{1} % TODO: {Lbeta} ^{inc}(X)$ component of the free loop space of $X$, with its compact open topology. Let $g$ be a complete metric on $X$, and let $S (g, \beta) \subset L _{\beta } X$ denote the subspace of all unit speed parametrized, smooth, closed $g$-geodesics in class $\beta$. **Definition 1**. *We say that a metric $g$ on $X$ is ***$\beta$-taut*** if $g$ is complete and $S (g, \beta )$ is compact. We will say that $g$ is ***taut*** if it is $\beta$-taut for each $\beta \in \pi _{1} ^{inc} (X)$.* A complete metric all of whose boundary incompressible closed geodesics are minimizing in their homotopy class is taut, see Lemma [Lemma 1](#lemma_compact){reference-type="ref" reference="lemma_compact"}. Thus, by the Cartan-Hadamard theorem  [@cite_ChernBook], a basic example of a taut metric is a complete metric with non-positive curvature. It should be emphasized that taut metrics form a much larger class of metrics then just non-positive curvature metrics. One class of examples comes by way of Lemma [Lemma 1](#lemma_betataut){reference-type="ref" reference="lemma_betataut"} ahead, but this only scratches the surface. **Definition 1**. *Let $\beta \in \pi_{1} ^{inc}(X)$, and let $g _{0}, g _{1}$ be a pair of $\beta$-taut metrics on $X$. A ***$\beta$-taut deformation*** between $g _{0}, g _{1}$, is a continuous (in the topology of $C ^{0}$ convergence on compact sets) family $\{g _{t}\}$, $t \in [0,1]$ of complete metrics on $X$, s.t. $$S (\{g _{t}\}, \beta ) := \{(o,t) \in L _{\beta }X \times [0,1] \,|\, o \in S (g _{t}, \beta )\}$$ is compact. We say that $\{g _{t}\}$ is a ***taut deformation*** if it is $\beta$-taut for each $\beta \in \pi _{1} ^{inc}(X)$. The above definitions of tautness are extended naturally to the case of a smooth fibration $X \hookrightarrow P \to [0,1]$, with a smooth fiber-wise family of metrics.* A useful criterion for $\beta$-tautness is the following. **Theorem 1**. *Let $\{g _{t}\} _{t \in [0,1]}$ be a continuous family of complete metrics on $X$. Suppose that: $$\sup _{t} |\max _{o \in S (g _{t}, \beta )} l _{g _{t}} (o) - \min _{o \in S (g _{t}, \beta )} l _{g _{t}} (o)| < \infty,$$ where $l _{g _{t}}$ is the length functional with respect to $g _{t}$, then $\{g _{t}\}$ is $\beta$-taut. It follow that sky catastrophes of vector fields on closed manifolds are not geodesible by metrics all of whose geodesics are minimal, Appendix [7.1](#appendix_bluesky){reference-type="ref" reference="appendix_bluesky"}.* For example, the hypothesis is trivially satisfied if $g _{t}$ have the property that all their class $\beta$ geodesics are minimal. In particular, if $g _{t}$ have non-positive curvature then $\{g _{t}\}$ is taut, again by the Cartan-Hadamard theorem. Fuller at the end of  [@cite_FullerIndex] has asked for any metric conditions on vector fields to rule out sky catastrophes, see Appendix [7.1](#appendix_bluesky){reference-type="ref" reference="appendix_bluesky"}. By the above, non-positivity of curvature is one such condition. *Remark 1*. Note that if sky catastrophes were never geodesible then the geodesible Seifert conjecture would follow, by the main result of  [@cite_SavelyevFuller]. Hence, this is a subtle situation and it now seems to be plausible that sky catastrophes for geodesible and Reeb families of vector fields do exist. The qualitative structure of such potential geodesible or Reeb sky catastrophes is somewhat understood by results in  [@cite_SavelyevFuller], and  [@cite_SavelyevFuller Theorem 1.10] in particular. But this does not greatly aid constructing potential examples, which must be topologically very complex, (there are necessarily infinitely many suitably synchronized bifurcation events). No results prior to the theorem above are known to me aside from those mentioned by Fuller himself in  [@cite_FullerIndex]. Let $\mathcal{G} (X)$ be the set of equivalence classes of taut metrics $g$, where $g _{0}$ is equivalent to $g _{1}$ whenever there is a taut deformation between them. We may denote an equivalence class by its representative $g$ by a slight abuse of notation. **Theorem 1**. *For each manifold $X$, there is a natural, functional $$\operatorname {F}: \mathcal{G} (X) \times \pi _{1} ^{inc} (X) \to \mathbb{Q}.$$* The value $\operatorname {F} (g, \beta )$ can be interpreted as a count of the set of closed $g$-geodesic strings in class $\beta$. But one must take care of exactly how to count, as in general this set should be understood as an orbifold or rather a Kuranishi space of a certain kind, hence this is why $\operatorname {F}$ is $\mathbb{Q}$ valued. *Question 1*. Do there exist a pair of taut metrics $g _{1}, g _{2}$ on a manifold $X$ which are not taut homotopic? Probably both possibilities are interesting. If the answer is 'no' then we may obtain much sharper applications, particularly in the setup of  [@cite_SavelyevEllipticCurvesLcs]. On the other hand, if the answer is 'yes', then the previous Theorem [Theorem 1](#thm:invariantF){reference-type="ref" reference="thm:invariantF"} becomes far more intriguing. **Definition 1**. *Let $\beta \in \pi _{1} (X)$. For any based point $x _{0} \in \mathop{\mathrm{\mathrm{image}}}\beta \subset X$ (for $\mathop{\mathrm{\mathrm{image}}}\beta$ the image of some representative of $\beta$) there is a naturally determined element $\beta _{x _{0}} \in \pi _{1} (X, x _{0})$ well defined up to an inner automorphism, (concatenate a representative of $\beta$ with a path from $x _{0}$ to a point in $\mathop{\mathrm{\mathrm{image}}}\beta$). We say that a class $\beta \in \pi _{1} (X, x _{0})$ is at ***most a $k$-power***, if whenever $\beta = \alpha ^{n}$, with $n>0$ then $n \leq k$. Similarly, $\beta \in \pi _{1} (X)$ is at most a $k$-power if for any $x _{0}$ as above, $\beta _{x _{0}}$ is at most a $k$-power. We say that $\beta \in \pi _{1} (X)$ is ***not a power*** if it is at most a $1$-power.* *Example 1*. Let $g$ be a Riemannian metric with negative sectional curvature on a closed manifold $X$ and $\beta \in \pi _{1} (X)$ a class represented by a multiplicty $n$ closed geodesic, then $$\label{equation_frac1n} F (g, \beta ) = \frac{1}{n}.$$ In particular, if $\beta$ is not a power then $F (g, \beta ) = {1}$. More generally, [\[equation_frac1n\]](#equation_frac1n){reference-type="eqref" reference="equation_frac1n"} holds whenever $g$ has a unique and non-degenerate closed geodesic string in class $\beta$. Here and throughout the paper, a closed geodesic string is ***non-degenerate*** if the corresponding $S ^{1}$ family of closed geodesics is Morse-Bott non-degenerate. **Theorem 1**. *Every rational number has the form $F (g, \beta )$ for some $\beta$-taut Riemannian $g$ on some compact manifold $X$ and for some $\beta \in \pi _{1} ^{inc} (X)$.* If $\beta \in \pi _{1} ^{inc} (X)$ is not a power, then it is easy to see that that the reparametrization $S ^{1}$ action on $L _{\beta }X$ is free (see Appendix [7](#appendix:Fuller){reference-type="ref" reference="appendix:Fuller"}), so that $H ^{S ^{1}} _{*} (L _{\beta}X, \mathbb{Z}) \simeq H _{*} (L _{\beta}X/ {S ^{1}}, \mathbb{Z})$, where $H ^{S ^{1}} _{*} (L _{\beta}X, \mathbb{Z})$ denotes the $S ^{1}$-equivariant homology. Moreover, we have: **Theorem 1**. *Suppose that $\beta \in \pi _{1} ^{inc} (X)$ is not a power, and $X$ admits a $\beta$-taut metric, then $H ^{S ^{1}} _{*} (L _{\beta}X, \mathbb{Z})$ is finite dimensional. Denote by $\chi ^{S ^{1}} (L _{\beta} X)$ the Euler characteristic of this homology. Then for any $\beta$-taut metric $g$ on $X$: $$F (g, \beta ) = \chi ^{S ^{1}}(L _{\beta }X).$$* Explicit examples for the theorem above can be found by the proof of Theorem [Theorem 1](#theorem_valuesOfInvariant){reference-type="ref" reference="theorem_valuesOfInvariant"}. For these types of examples any negative integer may appear as the value of $F (g, \beta )$. We leave out the details. *Remark 1*. If $\beta$ is a power, the idea behind Theorem [Theorem 1](#theorem_Eulercharacteristic){reference-type="ref" reference="theorem_Eulercharacteristic"} breaks down, as the $S ^{1}$-equivariant homology of $L _{\beta }X$ may then be infinite dimensional even if $X$ admits a $\beta$-taut $g$. As a trivial example this homology is already infinite dimensional when $g$ is negatively curved, and the class $\beta$ geodesic is $k$-covered, as then this homology is the group homology of $\mathbb{Z} _{k}$. In particular the connection with the Euler characteristic a priori breaks down. It is thus an interesting open problem if the functional $F$ remains topological, this is related to question [Question 1](#question_tauthomotopy){reference-type="ref" reference="question_tauthomotopy"}. A celebrated theorem of Preissman  [@cite_PreissmanNegativeCurvatureProducts] says that there are no negative sectional curvature metrics on compact products. Fibration counterexamples to Preissman's product theorem certainly exist. In fact, every closed 3-manifold $X ^{3}$, for which there is no injection $\mathbb{Z} ^{2} \to \pi _{1} (X, x _{0})$, and which fibers over a circle has a hyperbolic structure $g _{h}$, Thurston [@cite_ThurstonClassificationSurfaceDiffeo]. We are going to give a certain generalization of Preissman's theorem to fibrations, with possibly non-compact fibers, also replacing the negative sectional curvature condition by much weaker tautness conditions. **Definition 1**. *Let $Z \hookrightarrow X \xrightarrow{p} Y$ be a smooth fiber bundle with $X$ having a $\beta$-taut Riemannian metric $g$, for $\beta \in \pi_{1} ^{inc}(X)$, and let $g _{Y}$ be a metric on $Y$. Suppose that* 1. *The fibers $Z _{y} = p ^{-1} (y)$ are totally $g$-geodesic, for closed geodesics in class $\beta$. We denote by $g _{y}$ the metric $g$ restricted to $Z _{y}$.* 2. *The fibers are parallel (the distribution $T ^{vert}X = \ker p_*$ is parallel along any smooth curve in $X$ with respect to the Levi-Civita connection of $g$).* 3. *For any pair of fibers $(Z _{y_0}, g _{y _{0}})$, $(Z _{y_1}, g _{y _{1}})$, and a path $\gamma: [0,1]$ from $y _{0}$ to $y _{1}$ the fiber family $\{(Z _{\gamma (t)}, g _{\gamma t})\}$ furnishes a taut deformation. [\[item_tauthomotopyfibers\]]{#item_tauthomotopyfibers label="item_tauthomotopyfibers"}* 4. *$p$ projects $g$-geodesics to geodesics of $Y,g _{Y}$.* *We then call $p: X \to Y$ a ***$\beta$-taut submersion***, with the metrics $g,g _{Y}$ and $g _{Z}$ all possibly implicit.* **Definition 1**. *For $Z \hookrightarrow X \to Y$ as above, we say that $\beta \in \pi _{1} (X)$ is a ***fiber class*** if it is in the image of the inclusion $i _{Z}: \pi _{1} (Z) \to \pi _{1} (X)$.* In the above definition of a taut submersion and the following theorem we need the auxiliary metric $g$ on $X$ to be Riemannian, and there is no obvious extension of the theorem to the Riemann-Finsler case. However, the conclusions of the theorem are for Riemann-Finsler metrics. **Theorem 1**. *Let $p: (X,g) \to (Y, g _{Y})$ be a $\beta$-taut submersion, where $\beta \in \pi_{1} ^{inc}(X)$ is a fiber class. Suppose further that $Y$ is connected and closed, and is such that all smooth closed contractible $g _{Y}$ geodesics in $Y$ are constant, then the following holds.* - *If $\chi (Y) \neq \pm 1$ and $\beta$ is at most a $k$-power then $X$ does not admit a complete Riemann-Finsler metric with a unique and non-degenerate class $\beta$ geodesic string. [\[part_chipm1\]]{#part_chipm1 label="part_chipm1"}* - *If $\chi (Y) \neq \pm 1$ then $X$ does not admit a complete Riemann-Finsler metric with negative curvature.* Note that $\chi(Y) \neq 1$ is of course essential, as the trivial fibration $X \to \{pt\}$, with $X$ admitting a complete negatively curved metric, will satisfy the hypothesis. The condition that there is a fiber class $\beta \in \pi _{1} ^{inc} (X)$ is also essential, for any vector bundle over a manifold admitting a Riemannian metric of negative curvature admits a metric of negative curvature, Anderson  [@cite_AndersonNegativeCurvaturefibration]. As one corollary we may partially generalize Preissman's theorem (by removing the compactness assumption, but adding other assumptions), a more interesting fibration variation of this corollary is further ahead. **Corollary 1**. *Let $X = Z \times Y$ where $Z,Y$ admit complete Riemannian metrics with non-positive sectional curvature, $Y$ is closed and $\beta \in \pi _{1} ^{inc} (X)$ is in the image of the inclusion $\pi _{1} (Z) \to \pi _{1} (X)$.* 1. *If $\chi(Y) \neq \pm 1$ and $\beta$ is at most a $k$-power then $X$ does not admit a complete (Riemann-Finsler) metric with a unique and non-degenerate class $\beta$ geodesic string.* 2. *If $\chi(Y) \neq \pm 1$ then $X$ does not admit a complete metric of negative curvature.* A basic set of examples for the theorem is obtained by starting with any homomorphism $$\label{equation_phi} \phi: \pi _{1} (Y, y _{0}) \to \operatorname {Isom} (Z, g _{Z}), \quad \text{(the group of all isometries).}$$ where $g _{Z}$ is a taut Riemannian metric, and there is a class $\beta _{Z} \in \pi_{1} ^{inc}({Z})$. Suppose further: 1. The orbit $$O := \bigcup _{\gamma \in \pi _{1} (Y, y _{0}) } {\phi} _{*} (\gamma ) (\beta _{Z})$$ is finite. [\[item_finiteorbit\]]{#item_finiteorbit label="item_finiteorbit"} 2. $Y$ is closed and connected. 3. All contractible $g _{Y}$ geodesics in $Y$ are constant. We have the obvious induced diagonal action $$\pi _{1} (Y, y _{0}) \to \operatorname {Diff} (Z \times \widetilde{Y}), \text{ (the group of all diffeomorphisms)},$$ $$\gamma \mapsto ((z, y) \mapsto (\phi (\gamma) (z), \gamma \cdot y)),$$ for $\widetilde{Y}$ the universal cover of $Y$. Taking the quotient of $Z \times \widetilde{Y}$ by this action, we get an associated "flat" bundle $Z \hookrightarrow X _{\phi} \xrightarrow{p} Y$, with a metric $g _{\phi }$ induced from the product metric $\widetilde{g} = g _{Z} \oplus g _{Y}$, on the covering space $q: Z \times \widetilde{Y} \to Z \times Y$. **Lemma 1**. *Let $p: (X _{\phi}, g _{\phi }) \to (Y, g _{Y})$ be as above, then this is a $\beta$-taut submersion, where $\beta = i _{*} (\beta _{Z})$, for $i _{*}: \pi_{1} ^{inc}(Z) \to \pi_{1} ^{inc}(X _{\phi})$ induced by inclusion.* By the lemma above, $p: (X _{\phi}, g _{\phi }) \to (Y, g _{Y})$ satisfies the hypothesis of the theorem above. Yet more concretely: *Example 2*. Suppose we have $\beta _{Z} \in \pi_{1} ^{inc}({Z})$, and let $\phi: Z \to Z$ be an isometry of a taut metric $g _{Z}$. Then by the construction above, the mapping torus $(Z, g _{Z}) \hookrightarrow (X _{\phi}, g _{\phi}) \xrightarrow{\pi _{} } S ^{1}$ has the structure of a $\beta$-taut submersion, satisfying the hypothesis of the theorem, for $\beta = i _{*} (\beta _{Z})$ as above. The next corollary of Theorem [Theorem 1](#thm:Fibration){reference-type="ref" reference="thm:Fibration"} is immediate. **Corollary 1**. *Let $(X _{\phi}, g _{\phi}) \to (Y, g _{Y})$ be as in the construction above for $Z, g _{Z}$ having non-positive curvature, and let $\beta _{Z} \in \pi_{1} ^{inc}({Z})$. Then* 1. *If $\chi(Y) \neq \pm 1$, $X _{\phi }$ does not admit a complete Riemann-Finsler metric with negative curvature.* 2. *If $\chi(Y) \neq \pm 1$, and $\beta _{Z}$ is at most a $k$-power, then $X _{\phi }$ does not admit a Riemann-Finsler metric with a unique and non-degenerate class $\beta$ geodesic string, for $\beta = i _{*} (\beta _{Z})$ as above.* *As a special case, this applies to the mapping tori $X _{\phi }$, for $\phi: Z \to Z$ an isometry of a complete Riemannian non-positively curved metric on $Z$, satisfying the finiteness condition [\[item_finiteorbit\]](#item_finiteorbit){reference-type="ref" reference="item_finiteorbit"}. (The non-positive curvature hypothesis is for concreteness we may of course replace this condition by tautness.)* In the special case $Z$ is a compact, and if restrict to Riemannian metrics rather than Finsler, the above corollary readily follows by Preissman's theorem (specifically, because of the condition [\[item_finiteorbit\]](#item_finiteorbit){reference-type="ref" reference="item_finiteorbit"}). # Proof of Theorem [Theorem 1](#thm:notgeodesible){reference-type="ref" reference="thm:notgeodesible"} {#proof-of-theorem-thmnotgeodesible} The first part of the theorem clearly follows by the second part. So let $\{g _{t}\}$, $t \in [0,1]$ be as in the hypothesis, with $$\sup _{t} |\max _{o \in S (g _{t}, \beta )} l _{g _{t}} (o) - \min _{o \in S (g _{t}, \beta )} l _{g _{t}} (o)| < c,$$ and suppose that $$\sup _{(o,t) \in \mathcal{O} (\{g _{t}\},\beta )} l _{g _{t}} (o) = \infty.$$ Then we have a sequence $\{o _{k}\}$, $k \in \mathbb{N}$, of closed class $\beta$ $g _{t _{k}}$-geodesics in $X$, satisfying: 1. $\lim _{k \to \infty } t _{k} = t _{\infty } \in [0,1].$ [\[cond:1\]]{#cond:1 label="cond:1"} 2. $\lim _{k \to \infty }l _{g _{t _{k}}} (o _{k}) = \infty$, where $l _{g _{t _{k}}} (o _{t _{k}})$ is the length with respect to $g _{t _{k}}$. [\[cond:2\]]{#cond:2 label="cond:2"} Let $o _{\infty}$ be a minimal, class $\beta$, $g _{\infty } = g _{t _{\infty }}$ geodesic in $X$, with $g _{\infty }$ length $L$. Let $g _{aux}$ be a fixed auxiliary metric on $X$, and let $L _{aux}$ be the $g _{aux }$ length of $o _{\infty }$. Define a pseudo-metric on the space of metrics on $X$ as follows. Let $K \subset X$ be a fixed compact set containing $\mathop{\mathrm{\mathrm{image}}}o _{\infty }$, and set $$V \subset TX = \{v \in TX \,|\, \pi ({v}) \in K \text{ for $\pi _{}: TX \to X$ the canonical projection, and $|v| _{aux} = 1$} \},$$ where $|v| _{aux}$ is the norm taken with respect to $g _{aux}$. Then define: $$d _{C ^{0}} (g _{1}, g _{2}) = \sup _{v \in V} | |v| _{g _{1}} - |v| _{g _{2}}|.$$ By properties [\[cond:1\]](#cond:1){reference-type="ref" reference="cond:1"} and [\[cond:2\]](#cond:2){reference-type="ref" reference="cond:2"} we may find a $k > 0$ such that $$\label{eq:lessepsilon} d _{C ^{0}} (g _{t _{k}}, g _{t _{\infty }}) < \epsilon$$ and $$l _{g _{t _{k}}} (o _{k}) > c + L + L _{aux} \cdot \epsilon.$$ As $L + L _{aux} \cdot \epsilon \geq l _{g _{t _{k}}} (o _{\infty})$ by [\[eq:lessepsilon\]](#eq:lessepsilon){reference-type="eqref" reference="eq:lessepsilon"}, we have: $$l _{g _{t _{k}}} (o _{k}) > l _{g _{t _{k}}} (o _{\infty }) +c.$$ Since we may find a closed $g _{t _{k}}$-geodesic $o'$ satisfying $l _{g _{t _{k}}} (o') \leq l _{g _{t _{k}}} (o _{\infty })$, we get that $$|\max _{o \in S (g _{t _{k}}, \beta)} l _{g _{t _{k}}} (o) - \min _{o \in S (g _{t _{k}}, \beta)} l _{g _{t _{k}}} (o)|> c,$$ and so we are in contradiction. Thus, $$\sup _{(o,t) \in \mathcal{O} (\{g _{t}\},\beta )} l _{g _{t}} (o) < \infty.$$ It follows, by an analogue of Lemma [Lemma 1](#lemma_containedinCompact){reference-type="ref" reference="lemma_containedinCompact"}, that the images of all elements $o \in S (\{g _{t} \}, {\beta} )$ are contained in a fixed compact ${K} \subset X$. Compactness of $S (\{g _{t} \}, {\beta} )$ then follows by the Arzella-Ascolli theorem. 0◻ # Proof of Lemma [Lemma 1](#lemma_betataut){reference-type="ref" reference="lemma_betataut"} {#sec_Proof of Lemma lemma_betatau} Let ${\phi}_*: \pi _{1} (Y, y _{0}) \to \operatorname {Aut} (\pi_{1} ^{inc}(Z))$ be the natural induced action, where $\operatorname {Aut} (\pi_{1} ^{inc}(Z))$ denotes the group of set isomorphisms of $\pi_{1} ^{inc}(Z))$. And such that the orbit $$O := \bigcup _{\gamma \in \pi _{1} (Y, y _{0}) } {\phi} _{*} (\gamma ) (\beta _{Z})$$ is finite. As $g _{Z}$ is taut, $S (g _{Z}, {\phi} _{*} (\gamma ) (\beta _{Z}))$ is compact for each $\gamma$, where $S (g _{Z}, {\phi} _{*} (\gamma ) (\beta _{Z}))$ is the space of geodesics as in Definition [Definition 1](#definition_betataut){reference-type="ref" reference="definition_betataut"}. By the condition on contractible geodesics of $g _{Y}$, we get: $$\begin{aligned} S (g _{\phi }, \beta ) & = q_* (S (g _{Z} \oplus g _{Y}, \beta)) ) \\ & = \bigcup _{\beta \in O } q_*(S (g _{Z}, \beta ) \times \widetilde{Y}), \end{aligned}$$ for $q _{*}: L(Z \times \widetilde{Y}) \to L(Z \times Y)$ induced by the quotient map $q: Z \times \widetilde{Y} \to Z \times Y$, (as in the preamble to the statement of the lemma) and where $S (g _{Z}, \gamma) \times \widetilde{Y}$ is identified as a subset $S (g _{Z}, \gamma) \times \widetilde{Y} \subset L (Z) \times \widetilde{Y} \subset L (Z \times \widetilde{Y})$. Given that $O$ is finite, this then readily implies our claim. 0◻ # Preliminaries on Reeb flow {#sec_Some preliminaries on Reeb dynamics} Let $(C ^{2n+1}, \lambda )$ be a contact manifold with $\lambda$ a contact form, that is a one form s.t. $\lambda \wedge (d \lambda) ^{n} \neq 0$. Denote by $R ^{\lambda}$ the Reeb vector field satisfying: $$d\lambda (R ^{\lambda}, \cdot ) = 0, \quad \lambda (R ^{\lambda}) = 1.$$ Recall that a ***closed $\lambda$-Reeb orbit*** (or just Reeb orbit when $\lambda$ is implicit) is a smooth map $$o: (S ^{1} = \mathbb{R} / \mathbb{Z}) \to C$$ such that $$\dot o (t) = c R ^ {\lambda} (o (t)),$$ with $\dot o (t)$ denoting the time derivative, for some $c>0$ called period. Let $S (R ^{\lambda }, \beta )$ denote the space of all closed ${\lambda }$-Reeb orbits in free homotopy class $\beta$, with its compact open topology. And set $$\mathcal{O} (R ^{\lambda}, \beta ) = S (R ^{\lambda }, \beta )/S ^{1},$$ where $S ^{1} = \mathbb{R} ^{} /\mathbb{Z}$ acts by reparametrization $t \cdot o(\tau) = o (t + \tau)$. # Definition of the functional $\operatorname {F}$ and proofs of auxiliary results {#sec:Definition of F} Let $X$ be a manifold with a taut metric $g$. Let $C$ be the unit cotangent bundle of $X$, with its Louiville contact 1-form $\lambda _{g}$. If $o: S ^{1} = \mathbb{R} ^{}/\mathbb{Z} \to X$ is a unit speed closed geodesic, it has a canonical lift $\widetilde{o}: S ^{1} \to C$. If $\beta \in \pi _{1} ^{inc} (X)$, let $\widetilde{\beta} \in \pi _{1} (C)$ denote class $[\widetilde{o} ] \in \pi _{1} (C)$, where $o$ is a unit speed closed geodesic representing $\beta$. Let $S (R ^{\lambda _{g}}, \widetilde{\beta})$ be the orbit space as in Section [\[sec_Some preliminaries on Reeb dynamics\]](#sec_Some preliminaries on Reeb dynamics){reference-type="ref" reference="sec_Some preliminaries on Reeb dynamics"}, for the Reeb flow of the contact form $\lambda _{g}$. And set $$\mathcal{O} _{g, \beta} = \mathcal{O} (R ^{\lambda _{g}}, \widetilde{\beta } ) := S (R ^{\lambda _{g}}, \widetilde{\beta})/S ^{1},$$ i.e. this can be identified with the space of class ${\beta }$ $g$-geodesic strings. By the tautness assumptions $\mathcal{O} _ {g, \beta }$ is compact. We then define $$F (g, \beta ) = i (\mathcal{O} _{g, \beta }, R ^{\lambda _{g}}, \widetilde{\beta}) \in \mathbb{Q}$$ where the right hand side is the Fuller index of $R ^{\lambda _{g}}$ in class $\widetilde{\beta}$. As a basic example we have: **Lemma 1**. *Suppose that $g$ is a complete metric on $X$, all of whose boundary incompressible geodesics are minimal, then $g$ is taut.* *Proof.* First we state a more basic lemma. **Lemma 1**. *Suppose that $g$ is a complete metric on $X$, $\beta \in \pi _{1} ^{inc} (X)$ and let $S \subset L _{\beta} X$ be a subset on which the $g$-length functional is bounded from above. Then the images in $X$ of elements of $S$ are contained in a fixed compact subset of $X$.* *Proof.* Suppose otherwise. Fix an exhaustion by nested compact sets $$\bigcup _{i \in \mathbb{N} } K _{i} = X, \quad K _{i} \supset K _{i-1}.$$ Then either there is sequence $\{o _{i}\} _{i \in \mathbb{N} }$, $o _{i} \in S$ s.t. $o _{i} \in K _{i} ^{c}$, for $K _{i} ^{c}$ the complement of $K _{i}$, which contradicts the fact that $\beta$ is incompressible. Or there is a sequence $\{o _{k} \} _{k \in \mathbb{N} }$, $o _{k} \in S$ s.t.: 1. Each $o _{k}$ intersects $K _{i _{0}}$ for some $i _{0}$ fixed. [\[condition_intersects\]]{#condition_intersects label="condition_intersects"} 2. For each $i \in \mathbb{N}$ there is a $k _{i} >i$ s.t. $o _{k _{i}}$ is not contained in $K _{i}$. [\[condition_notcontained\]]{#condition_notcontained label="condition_notcontained"} Now if $\operatorname {diam} (o _{k})$ is bounded in $k$, then condition [\[condition_intersects\]](#condition_intersects){reference-type="ref" reference="condition_intersects"} implies that $o _{k}$ are contained in a set of bounded diameter. (Here $\operatorname {diam} (o _{k})$ denotes the diameter of $\mathop{\mathrm{\mathrm{image}}}o _{k}$.) Consequently, by Hopf-Rinow theorem  [@cite_ChernBook], $o _{k}$ are contained in a compact set. But this contradicts condition [\[condition_notcontained\]](#condition_notcontained){reference-type="ref" reference="condition_notcontained"}, and the fact that $K _{i}$ form an exhaustion of $X$. Thus, we conclude that $\operatorname {diam} (o _{k})$ is unbounded, but this contradicts the hypothesis. ◻ Returning to the proof of the main lemma. By assumption, closed, class $\beta \in \pi _{1} ^{inc} (X)$ geodesics are $g$-minimizing in their homotopy class and in particular have fixed length. By the lemma above there is a fixed $K \subset X$ s.t. every class $\beta$ closed geodesic has image contained in $K$. Then compactness of $S (g, \beta )$ follows by Arzella-Ascolli theorem. ◻ *Proof Theorem [Theorem 1](#thm:invariantF){reference-type="ref" reference="thm:invariantF"}.* Let $\beta \in \pi _{1} ^{inc} (X)$, be given and let $g$ be $\beta$-taut. We just need to prove that $F (g, \beta)$ is invariant under a $\beta$-taut deformation of $g$. So let $\{g _{t}\}$, $t \in [0,1]$ be a $\beta$-taut deformation of metrics on a compact manifold $X$. Let $R ^{\lambda _{g _{t}}}$ be the geodesic flow on the $g _{t}$ unit cotangent bundle $C _{t}$. Trivializing the family $\{C _{t}\}$ we get a family $\{R _{t}\}$ of flows on $C \simeq C _{t}$, with $R _{t}$ conjugate to $R ^{\lambda _{g _{t}}}$. Let $\mathcal{O} (\{R _{t} \}, \widetilde{\beta})$ be the cobordism as in [\[equation_Ohomotopy\]](#equation_Ohomotopy){reference-type="eqref" reference="equation_Ohomotopy"}, where $\widetilde{\beta} \in \pi _{1} (C)$ is as above. Then $\mathcal{O} (\{R _{t} \}, \widetilde{\beta})$ is compact as $S (\{g _{t}\}, \beta )$ is compact by assumption. Basic invariance of the Fuller index, that is [\[eq_basicinvariance\]](#eq_basicinvariance){reference-type="eqref" reference="eq_basicinvariance"}, immediately yields: $F (g _{0}, \beta) = F (g _{1}, \beta )$. ◻ *Proof of Theorem [Theorem 1](#theorem_Eulercharacteristic){reference-type="ref" reference="theorem_Eulercharacteristic"}.* This is an application of Morse theory. As $g$ is $\beta$-taut, and so $S (g, \beta )$ is compact, we may find a $C ^{\infty }$-nearby metric $g'$, s.t. $g'$ has finitely many class $\beta$ closed geodesic strings, all of which are non-degenerate. The notation $L _{\beta } X$ now denotes the Hilbert manifold of $H ^{1}$ loops, as used for example in the classical work of Gromoll-Meyer [@cite_GromollMeyerGeodesics]. This Hilbert manifold is well known to be homotopy equivalent to the standard free loop space with its compact open topology. The energy function $e _{g'}: L _{\beta } X \to S ^{1}$, $$\label{equation_energyfunction} e _{g'} (o) = \int_{S ^{1}}^{} \langle \dot o (t), \dot o (t) \rangle _{g'} dt$$ is smooth, $S ^{1}$ invariant and satisfies the Palais-Smale condition. The flow for its negative gradient vector field $V$ is complete, and we can do Morse theory mostly as usual. This is understood starting with the work of Klingenberg [@cite_KlingenbergClosedGeodesics], with the framework of Palais and Smale  [@cite_PalaisSmale]. In our case, $e _{g'}$ is moreover a Morse-Bott function with critical manifolds $C _{o}$ corresponding to $S ^{1}$ families of geodesics, for each closed geodesic string $o$. There is an induced Morse-Bott cell decomposition on $L _{\beta }X$, meaning a stratification formed by $V$ unstable manifolds of the above mentioned critical manifolds $C _{o}$. This is Bott's extension of the fundamental Morse decomposition theorem. Now the $S ^{1}$ action on $L _{\beta }X$ is free by the condition that $\beta$ is not a power. This action is not smooth, but is continuous and so taking the topological $S ^{1}$ quotient we get a standard CW cell decomposition of $L _{\beta }X/S ^{1}$ with one $k$-cell for each closed $g'$-geodesic string $o$ of index $\operatorname {morse} (o) = k$. (The "index" means the Morse-Bott index of the critical manifold $C _{o}$.) All of the above is understood, see for instance  [@cite_GromollMeyerGeodesics]. From the above cell decomposition, we readily get that the homology $H_*(L _{\beta }X/S ^{1}, \mathbb{Z}) = H _{*} (L _{\beta } ^{S ^{1}} X, \mathbb{Z})$ is finite dimensional. We also get that: $$\begin{aligned} \chi(L _{\beta }X/S ^{1}) & = \displaystyle \sum_{o \in \mathcal{O} (R ^{\lambda_{g'}}, \beta)} {-1} ^{\operatorname {morse} (o)} \quad \text{(immediate from the cell decomposition)} \\ & = F (g', \beta) \quad \text{(basic properties of the fixed point index, see for instance ~\cite[Section 2]{cite_SavelyevFuller})} \\ & = F (g, \beta ) \quad \text{(by the local invariance \eqref{eq_basicinvariance2} of the Fuller index)}.\end{aligned}$$ ◻ # Proof of Theorem [Theorem 1](#thm:Fibration){reference-type="ref" reference="thm:Fibration"} and its corollaries {#sec:Proof of Theorem fibration} We first prove: **Theorem 1**. *Let $p: X \to Y$ be a $\beta$-taut submersion as in the statement of Theorem [Theorem 1](#thm:Fibration){reference-type="ref" reference="thm:Fibration"} and $\beta \in \pi _{1} ^{inc} (X)$ a fiber class. Then $$\label{equation_mainformula} F (g, {\beta}) = card \cdot \chi (Y) \cdot F (g _{Z}, \beta _{Z}),$$ where $card \in \mathbb{N} - \{0\}$ is the cardinality of a certain orbit of the holonomy group (as explained in the proof), and where $\beta _{Z}$ is as in Lemma [Lemma 1](#lemma_betataut){reference-type="ref" reference="lemma_betataut"}.* *Proof.* We have a natural subset of $\mathcal{O}' \subset \mathcal{O} {g, \beta }$, consisting of all vertical geodesics, that is $g$-geodesics contained in fibers $p ^{-1} (y) = Z _{y}$. In fact, $\mathcal{O}' = \mathcal{O} {g, \beta}$, for if $o$ is any class $\beta$ geodesic, the projection $p (o)$ is a contractible $g _{Y}$ geodesic, and by assumptions is constant. In particular there a natural continuous projection $\widetilde{p}: \mathcal{O} {g, \beta} \to Y$, $\widetilde{p}(o) = y$ where $y$ is determined by the condition that $Z _{y} \supset \mathop{\mathrm{\mathrm{image}}}o$. We will use this to construct a suitable (in a sense abstract i.e. not Reeb) perturbation of the vector field $R ^{\lambda _{g}}$, using which we can calculate the invariant $F (g, \beta )$. Fix a Morse function on $f$ on $Y$, let $C= S ^{*} X$ denote the $g$-unit cotangent bundle of $X$. For $v \in T _{x} X$ let $\langle v|$ denote the functional $T _{x} X \to \mathbb{R} ^{}$, $w \mapsto \langle v, w \rangle _{g}$. Define $\widetilde{f}: C \to \mathbb{R} ^{}$ by $\widetilde{f} (\langle v|) := f (p (v))$, also define $P: C \to \mathbb{R} ^{}$ by $P (\langle v|) := |P^{vert} (v)| ^{2} _{g}$, where $P ^{vert} (v)$ denotes the $g$-orthogonal projection of $v$ onto the $T ^{vert} _{x} X \subset T _{x}X$, for $T ^{vert} X$ the vertical tangent bundle of $X$, i.e. the kernel of the map $p_*: TX \to TY$. Next define $F: C \to \mathbb{R} ^{}$ by: $$F (\langle v|) := P (\langle v|) + \widetilde{f} (\langle v|).$$ Set $$V _{t} = R ^{\lambda _{g}} - t\operatorname {grad} _{g _{S}} F,$$ where the gradient is taken with respect to the Sasaki metric $g _{S}$ on $C$  [@cite_SasakiMetric] induced by $g$. The latter Sasaki metric is the natural metric for which we have an orthogonal splitting $TC = T ^{vert}C \oplus T ^{hor} C$, where $T ^{vert} C$ is the kernel of $pr_*: TC \to TX$, induced by the natural projection $pr: C \to X$, and where $T ^{hor} C$ is the $g$ Levi-Civita horizontal sub-bundle. Set $\mathcal{O} _{t} = \mathcal{O} (V _{t}, \widetilde{\beta} )$, where $\widetilde{\beta }$ is as in Section [5](#sec:Definition of F){reference-type="ref" reference="sec:Definition of F"}. **Lemma 1**. 1. *For all $t \in [0, 1]$, $N_t := \mathcal{O} _{t} \cap \mathcal{O} _{g, \beta }$ is open and closed in $\mathcal{O} _{t}$.* 2. *For all $t \in (0, 1]$, $N _{t} = \cup _{y \in \operatorname {crit} (f)} \widetilde{p} ^{-1} (y),$ where $\operatorname {crit} (f)$ is the set of critical points of $f$.* *Proof.* It is easy to see that $V _{t}$ is complete and without zeros. Suppose that $t>0$. Let $\langle v _{\tau} |$, $\tau \in \mathbb{R} ^{}$ be the flow line of $V _{t}$, through $\langle v _{0} |$, i.e. $\langle v _{\tau} | = \phi _{\tau} (\langle v _{0} | )$, for $\phi _{\tau}$ the time $\tau$ flow map of $V _{t}$. By the fact that the fibers of $p$ are assumed to be parallel, we have that $${R ^{\lambda _{g}}} (P) =0, \quad \text{using the derivation notation}.$$ Also, $$\operatorname {grad} _{g _{S}} \widetilde{f} (P) =0,$$ which readily follows by the conjunction of $g _{S}$ being Sasaki and the fibers of $p$ being parallel. Consequently, the function $$\tau \mapsto P (\langle v _{\tau}|) = |P^{vert} (v _{\tau})| ^{2} _{g}$$ is monotonically decreasing unless either: 1. $v _{0}$ is tangent to $T ^{vert} X$, in which case for all $\tau$, $v _{\tau}$ are tangent to $T ^{vert} X$ and $|P^{vert} (v _{\tau})| ^{2} _{g} =1$. 2. For all $\tau$, $|P^{vert} (v _{\tau})| ^{2} _{g} =0$. In particular, the closed orbits of $V _{t}$ split into two types. 1. Closed orbits $o (\tau) = \langle v _{\tau} |$ with $v _{\tau}$ always tangent to $T ^{vert}X$. In this case we may immediately, conclude that $o$ is a lift to $C$ of a closed $g$-geodesic contained in the fiber over a critical point of $f$. 2. Closed orbits $o (\tau) = \langle v _{\tau} |$ for which $v _{\tau}$ is always $g$-orthogonal to $T ^{vert}X$. Clearly, the conclusion follows. ◻ *Remark 1*. It would be very fruitful to remove the condition on the fibers of $p$ being parallel. But our argument would need to substantially change. We return to the proof of the theorem. Set $$\widetilde{N} = \{(o,t) \in L _{\widetilde{\beta} } C \times [0, \epsilon] \,|\, o \in {N} _{t} \},$$ where $L _{\widetilde{\beta} } C$ denotes the $\widetilde{\beta }$ component of the free loop space as previously. By part I of Lemma [Lemma 1](#lemma_epsilon){reference-type="ref" reference="lemma_epsilon"}, this is an open compact subset of $\mathcal{O} (\{V _{t}\}, \widetilde{\beta })$ s.t. $$\widetilde{N} \cap (L _{\widetilde{\beta} } C \times \{0\}) = \mathcal{O} (R ^{\lambda _{g}}, \widetilde{\beta}),$$ (equalities throughout are up to natural set theoretic identifications.) By definitions: $$N _{t} = \widetilde{N} \cap (L _{\widetilde{\beta} } C \times \{t\}).$$ Now the invariance of the Fuller index gives: $$i(N _{0}, R ^{\lambda _{g}}, \widetilde{\beta }) = i (N _{1}, V _{1}, \widetilde{\beta } ).$$ We proceed to compute the right hand side. Fix any smooth Ehresmann connection $\mathcal{A}$ on the fiber bundle $p: X \to Y$. This induces a holonomy homomorphism: $$hol _{y}: \pi _{1} (Y, y) \to \operatorname {Aut} \pi _{1} (Z _{y}) \text{ (the right-hand side is the group of set automorphisms)},$$ with image denoted $\mathcal{H} _{y} \subset \operatorname {Aut} \pi _{1} (Z _{y})$. Let $\beta _{Z}$ denote a class in $\pi _{1} (Z _{y})$ s.t. $(i _{Z _{y}})_* (\beta _{Z}) = \beta$, for $i _{Z _{y}}: Z _{y} \to X$ the inclusion map. Set $$S _{y} := \bigcup _{g \in \mathcal{H} _{y}} g ( \beta _{Z}) \subset \pi _{1} (Z _{y}).$$ Then for another $y' \in Y$, $$\label{equation_Sy} h_*: S _{y'} \to S _{y},$$ is an isomorphism, where $h: Z _{y} \to Z _{y'}$ is a smooth map given by the $\mathcal{A}$-holonomy map determined by some path from $y$ to $y'$, and where $h _{*}$ is the naturally induced map. Denoting by $g _{y}$ the restriction of $g$ to the fiber $Z _{y}$, let $R ^{{y}}$ denote the $\lambda _{g _{{y}}}$ Reeb vector field on the $g _{Z _{y}}$-unit cotangent bundle $C _{y}$ of $Z _{y}$. The cardinality $card$ of $S _{y}$ is finite, as otherwise we get a contradiction to the compactness of $S (g, {\beta} )$. Now $$\widetilde{p} ^{-1} (y) = \bigcup _{\alpha \in S _{y}} \mathcal{O} (R ^{\lambda _{y}}, \alpha).$$ From part 2 of Lemma [Lemma 1](#lemma_epsilon){reference-type="ref" reference="lemma_epsilon"} and by straightforward index computations we get: $$i (N _{1}, V _{1}, \widetilde{\beta } ) = \sum_{y \in \operatorname {crit} (f)} (-1)^{\operatorname {morse} (y)} \cdot i (\widetilde{p} ^{-1} (y), R ^{\lambda _{y}}, \widetilde{\beta } ),$$ where $\operatorname {morse} (y)$ denote the Morse index of $y$. Now $$\begin{aligned} i (\widetilde{p} ^{-1} (y), R ^{\lambda _{y}}, \widetilde{\beta }) & = \sum _{\alpha \in S _{y}} i (\mathcal{O} (R ^{\lambda _{y}}), R ^{\lambda _{y}}, {\widetilde{\alpha}}) \\ & = \sum_{\alpha \in S _{y}} F (g _{y}, \alpha). \\ & = card \cdot F (g _{Z}, \beta _{Z}),\end{aligned}$$ where the last equality follows by [\[equation_Sy\]](#equation_Sy){reference-type="eqref" reference="equation_Sy"}, and by the condition [\[item_tauthomotopyfibers\]](#item_tauthomotopyfibers){reference-type="ref" reference="item_tauthomotopyfibers"} in the Definition [Definition 1](#definition_betataut){reference-type="ref" reference="definition_betataut"}. And so the result follows. ◻ Now returning to the proof of the main theorem. As $\beta$ is at most a $k$-power, $\beta = \alpha ^{n}$, for some $n$ where $\alpha$ is not a power. By the assumption that all contractible $g _{Y}$ geodesics are constant, classical Morse theory Milnor [@cite_MilnorMorsetheory] tells us that $Y$ has vanishing higher homotopy group $\pi _{k} (Y, y _{0})$, $k \geq 2$, and in particular $i _{Z, *}: \pi _{1} (Z, p _{0}) \to \pi _{1} (X, p _{0})$ is a group injection, by the long exact sequence of a fibration. It follows that $\alpha \in \pi _{1} ^{inc} (X)$ is also a fiber class. Now, any $\alpha$-class closed $g$-geodesic must be tangent a fiber of $p$, as otherwise clearly $p$ would not be $\beta$-taut. It follows that $p: X \to Y$ is also $\alpha$-taut. Now, to prove the first part of the conclusion, if $\chi(Y) \neq \pm 1$ then by [\[equation_mainformula\]](#equation_mainformula){reference-type="eqref" reference="equation_mainformula"} $F (g, \alpha ) \neq \pm 1$, since $F (g _{Z}, \alpha _{Z})$ is an integer by Theorem [Theorem 1](#theorem_Eulercharacteristic){reference-type="ref" reference="theorem_Eulercharacteristic"}. And also by Theorem [Theorem 1](#theorem_Eulercharacteristic){reference-type="ref" reference="theorem_Eulercharacteristic"} $$F (g, \alpha ) = \chi ^{S ^{1}}(L _{\alpha}X),$$ so that $X$ cannot admit a complete metric with a unique and non-degenerate class $\alpha$-geodesic string as otherwise $\chi ^{S ^{1}}(L _{\alpha}X) = 1$ (see Proof of Theorem [Theorem 1](#theorem_Eulercharacteristic){reference-type="ref" reference="theorem_Eulercharacteristic"}). This also readily implies that $X$ does not admit a metric with a unique and non-degenerate class $\beta$-geodesic string. So that the first conclusion follows. To prove the second conclusion, it is enough to show that if $X$ admits a complete metric $g'$ of negative curvature then a class $\beta \in \pi ^{inc} _{1} (X)$ must be at most a $k$-power for some $k$. As then the needed result follows by the first conclusion. Let then $o$ be the unique, closed, class $\beta$ $g'$-geodesic string $X$. Then $o$ covers a simple, closed, fiber class geodesic string $\widetilde{o}$ in class $\widetilde{\beta } \in \pi _{1} ^{inc} (X)$. We claim that $\widetilde{\beta }$ is not a power. Suppose otherwise, so that $\widetilde{\beta} _{x _{0}} = \alpha ^{k}$ for $k>1$ and $\alpha \in \pi _{1} (X, x _{0})$, ($\beta _{x _{0}}$ is as in Definition [Definition 1](#definition_indivisible){reference-type="ref" reference="definition_indivisible"}). Let $u$ be the unique, class $\alpha$, closed $g'$-geodesic string (where $\alpha$ also denotes the free homotopy class corresponding to the based class $\alpha$.) It is immediate that the $k$ cover of $u$, $u ^{k}$ represents $\widetilde{\beta}$ and is a closed $g$-geodesic string. As $g'$ is negatively curved, we must have $u ^{k} = o$, which gives a contradiction. 0◻ *Proof of Corollary [Corollary 1](#corollary_product){reference-type="ref" reference="corollary_product"}.* Let $g _{Z}, g _{Y}$ be complete Riemannian metrics on $Z$, $Y$ with non-positive curvature. Take the product metric $g = g _{Z} \times g _{Y}$ on $X = Z \times Y$. By Lemma [Lemma 1](#lemma_betataut){reference-type="ref" reference="lemma_betataut"} and the Cartan-Hadamard theorem, $g$ is taut. Moreover, for a class $\beta \in \pi _{1} (Z)$ in the image of the inclusion $\pi _{1} (Z) \to \pi _{1} (X)$ the natural projection $X \to S ^{1}$ is automatically a $\beta$-taut submersion. Then the conclusion readily follows from Theorem [Theorem 1](#thm:Fibration){reference-type="ref" reference="thm:Fibration"}. ◻ *Proof of Theorem [Theorem 1](#theorem_valuesOfInvariant){reference-type="ref" reference="theorem_valuesOfInvariant"}.* By Theorem [Theorem 1](#thm:EulerProduct){reference-type="ref" reference="thm:EulerProduct"} 0 is certainly a value of the invariant $F$. We first prove that every negative rational number is the value of the invariant. Let $p,q$ be positive integers. Let $Y$ be a closed surface of genus $(p +1) >1$ with a hyperbolic metric $g _{Y}$, let $Z$ be the genus 2 closed surface with a hyperbolic metric $g _{Z}$ and let $\beta _{Z} \in \pi _{1} ^{inc} (Z)$ be the class represented by a $2 \cdot q$-fold covering of a simple loop representing a generator of the fundamental group of $Z$. Let $X = Y \times Z$ with the product metric $g = g _{Y} \times g _{Z}$ and $p: X \to Y$ the canonical projection. By Theorem [Theorem 1](#thm:EulerProduct){reference-type="ref" reference="thm:EulerProduct"} $$F (g, \beta) = \chi(Y) \cdot F (g _{Z}, \beta _{Z}) = ( - 2p) \cdot \frac{1}{2q} = - \frac{p}{q},$$ where $\beta$ is as in Lemma [Lemma 1](#lemma_betataut){reference-type="ref" reference="lemma_betataut"}. So we proved our first claim. Let again $p,q$ be positive integers. Let $Y$ be closed surface of genus $2$, with a hyperbolic metric $g _{Y}$. And let $Z$ be a manifold satisfying $F (g _{Z}, \beta _{Z}) = -\frac{p}{2q}$ for some $\beta _{Z}$-taut metric $g _{Z}$ on $Z$ and for some class $\beta _{Z} \in \pi _{1} ^{inc} (Z)$. This exist by the discussion above. Let $g = g _{Y} \times g _{Z}$ be the product metric on $Y \times Z$, and $\beta$ as above. Analogously to the discussion above we get: $$F (g, \beta) = \chi(Y) \cdot F (g _{Z}, \beta _{Z}) = ( -2) \cdot \frac{-p}{2q} = \frac{p}{q}.$$ ◻ # Fuller index and sky catastrophes {#appendix:Fuller} Let $X$ be a complete vector field without zeros on a manifold $M$. Set $$S (X, \beta) = \{o \in L _{\beta} M \,\left . \right | \exists p \in (0, \infty), \, \text{ $o: \mathbb{R}/\mathbb{Z} \to M $ is a periodic orbit of $p X $} \}.$$ The above $p$ is uniquely determined and we denote it by $p (o)$ called the period of $o$. There is a natural $S ^{1}$ reparametrization action on $S (X, \beta)$: $t \cdot o$ is the loop $t \cdot o(\tau) = o (t + \tau)$. The elements of $\mathcal{O} (X, \beta) := S (X, \beta)/S ^{1}$ will be called ***orbit strings*** or just closed orbits. Slightly abusing notation we just write $o$ for the equivalence class of $o$. The multiplicity $m (o)$ of an orbit string is the ratio $p (o) /l$ for $l>0$ the period of a simple orbit string covered by $o$. We want a kind of fixed point index which counts orbit strings $o$ with certain weights. Assume for simplicity that $N \subset \mathcal{O} (X, \beta)$ is finite. (Otherwise, for a general open compact $N \subset \mathcal{O} (X, \beta)$, we need to perturb.) Then to such an $(N,X, \beta)$ Fuller associates an index: $$i (N,X, \beta) = \sum _{o \in N}\frac{1}{m (o)} i (o),$$ where $i (o)$ is the fixed point index of the time $p (o)$ return map of the flow of $X$ with respect to a local surface of section in $M$ transverse to the image of $o$. Fuller then shows that $i (N, X, \beta )$ has the following invariance property. For a continuous homotopy $\{X _{t} \}$, $t \in [0,1]$ set $$S (\{X _{t}\}, \beta) = \{(o, t) \in L _{\beta} M \times [0,1] \,|\, \text{$o \in S (X _{t})$}\}.$$ And given a continuous homotopy $\{X _{t} \}$, $X _{0} =X$, $t \in [0,1]$, suppose that $\widetilde{N}$ is an open compact subset of $$\label{equation_Ohomotopy} \mathcal{O} (\{X _{t}\}, \beta ) := S (\{X _{t} \}, \beta ) / S ^{1},$$ such that $$\widetilde{N} \cap \left (L _{\beta }M \times \{0\} \right) / S ^{1} =N.$$ Then if $$N_1 = \widetilde{N} \cap \left (L _{\beta }M \times \{1\} \right) / S ^{1}$$ we have $$\label{eq_basicinvariance} i (N, X, \beta ) = i (N_1, X_1, \beta).$$ We call this ***basic invariance***. In the case $\mathcal{O} (X _{0}, \beta )$ is compact, $\mathcal{O} (X _{1}, \beta )$ is compact for any sufficiently $C ^{0}$ nearby $X _{1}$, and in this case basic invariance implies (see for instance  [@cite_SavelyevFuller Proof of Lemma 1.6]): $$\label{eq_basicinvariance2} i (\mathcal{O} (X _{0}, \beta ), X, \beta ) = i (\mathcal{O} (X _{1}, \beta ), X_1, \beta).$$ ## Blue sky catastrophes {#appendix_bluesky} **Definition 1** (Preliminary). *A ***sky catastrophe*** for a smooth family $\{X _{t} \}$, $t \in [0,1]$, of non-vanishing vector fields on a closed manifold $M$ is a continuous family of closed orbit strings $\tau \mapsto o _{t _{\tau} }$, $o _{t _{\tau} }$ is an orbit string of $X _{t _{\tau} }$, $\tau \in [0, \infty)$, such that the period of $o _{t _{\tau} }$ is unbounded from above.* A sky catastrophe as above was initially constructed by Fuller  [@cite_FullerBlueSky]. Or rather his construction essentially contained this phenomenon. A more general definition appears in  [@cite_SavelyevFuller], we slightly extend it here to the case of non-compact manifolds. All these definitions become equivalent given certain regularity conditions on the family $\{X _{t}\}$ and assuming $M$ is compact. **Definition 1**. *Let $\{X _{t} \}$, $t \in [0,1]$ be a continuous family of non-zero, complete smooth vector fields on a manifold $M$ and $\beta \in \pi _{1} ^{inc} (X)$.* *We say that $\{X _{t}\}$ has a ***catastrophe in class $\beta$***, if there is an element $$y \in \mathcal{O} (X _{0}, \beta) \sqcup \mathcal{O} (X _{1}, \beta) \subset \mathcal{O} (\{X _{t}\}, \beta)$$ such that there is no open compact subset of $\mathcal{O} (\{X _{t}\}, \beta)$ containing $y$.* A vector field $X$ on $M$ is ***geodesible*** if there exists a metric $g$ on $M$ s.t. every flow line of $X$ is a unit speed $g$-geodesic. A family $\{X _{t}\}$ is ***geodesible*** if there is a continuous family $\{g _{t}\}$ of metrics, with $X _{t}$ geodesible with respect to $g _{t}$ for each $t$. A family $\{X _{t}\}$ is ***geodesible*** if there is a continuous family $\{g _{t}\}$ of metrics with $X _{t}$ geodesible with respect to $g _{t}$ for each $t$. A ***geodesible sky catastrophe*** is a geodesible family $\{X _{t}\}$ with a sky catastrophe. A ***Reeb sky catastrophe*** is a family of Reeb vector fields $\{X _{t}\}$ with a sky catastrophe. # Acknowledgements Larry Guth for related discussions. Dennis Sullivan and Misha Gromov for much welcome supportive comments.

arxiv_math

--- abstract: | In root finding and optimization, there are many cases where there is a closed set $A$ one likes that the sequence constructed by one's favourite method will not converge to A (here, we do not assume extra properties on $A$ such as being convex or connected). For example, if one wants to find roots, and one chooses initial points in the basin of attraction for 1 root $x^*$ (a fact which one may not know before hand), then one will always end up in that root. In this case, one would like to have a mechanism to avoid this point $z^*$ in the next runs of one's algorithm. In this paper, we propose two new methods aiming to achieve this. In the first method, we divide the cost function by an appropriate power of the distance function to $A$. This idea is inspired by how one would try to find all roots of a function in 1 variable. In the second method, which is more suitable for constrained optimization, we redefine the value of the function to be a big constant on $A$. We also propose, based on this, an algorithm to escape the basin of attraction of a component of positive dimension to reach another component. As an application, we prove a rigorous guarantee for finding roots of a meromorphic function of 1 complex variable in a given domain. Along the way, we compare with main existing relevant methods in the current literature. We provide several examples in various different settings to illustrate the usefulness of the new approach. address: University of Oslo, 0316 Oslo, Norway author: - Tuyen Trung Truong title: Creating walls to avoid unwanted points in root finding and optimization --- # The problem, motivation and main result Here we present the problem and motivation, a brief literature survey, a main theoretical result on finding roots of meromorphic functions in a given domain, and the plan for the remaining of the paper. ## The problem and motivation Solving equations is an important task one usually encounters in research and applications. Some examples are: finding periodic points of a map, finding the trajectory of an object (e.g. robot) obeying a certain system of equations coming from physical laws. Ever since the time of Abel, Ruffini and Galois, it is clear that one cannot find precise roots of a simple polynomial in 1 complex variable, and hence must utilise approximative methods. In this paper, we will concentrate on iterative algorithms, which are very easy to implement and use in practice. One can treat solving a system of equations $F(x)=0$, where $F:\mathbf{R}^m\rightarrow \mathbf{R}^k$, as a global optimization problem by the following common trick. Define $f(x)=||F(x)||^2/2:~\mathbf{R}^m\rightarrow \mathbf{R}$. Then finding roots to $F=0$ is equivalent to the following two problems (which an iterative method may be able to solve simultaneously): 1) show that $\min _{x\in \mathbf{R}^m}f(x)=0$, and 2) find global minimizers for $f$. Even if $F$ has no root, finding global minimizers of $f$ still makes sense and has important applications (e.g. in the Least Square Fit problem in statistics). Since finding global minimizer is NP-hard, research on effective numerical methods for global optimization can be helpful with . This paper considers the following general problem: **Problem:** Let $X$ be a (complete) metric space, and let $f:X\rightarrow \mathbf{R}$ a (smooth) cost function. Assume that an iterative method IM, aiming to find global minimizers of $f$, is used, which has the property that whenever an initial point $x_0\in X$, then a sequence $x_{n+1}=IM(x_n,f)$ is generated, with $x_{n}\in X$ for all $n$. Can we have a way to construct sequences which avoid the set $A$ while still having a big chance to converge to global minima? In other words, can we use IM to solve the global optimization problem $f$ on the non-complete set $X\backslash{A}$? For this problem to make sense, the method IM must itself have big chance of convergence to global minimizer on the complete set $X$. Below we list some examples for applying this problem. **Application 1: Avoiding known roots.** Assume that one wants to find roots to a system $F(x)=0$, with the associated cost function $f(x)=||F(x)||^2/2$. Assume that in previous runs of IM one already found several roots $z_1^*,z_2^*,\ldots ,z_j^*$. One chooses the initial point for to run IM in a fixed domain $B$, and one does not know whether with probability 1, sequences constructed by IM with initial point in B will always converge to one of the known roots $z_1^*,z_2^*,\ldots ,z_j^*$. Then one can define the close set $A=\{z_1^*,\ldots ,z_j^*\}$ and apply Problem. In the case where the variable x is in 1 dimension, an alternative way is to divide $F(x)$ by $(x-z_1^*)\times \ldots \times (x-z_j^*)$. However, one cannot do this in higher dimension. Even in dimension 1, there are situations when one would need to use Problem. For example, one would like to find roots inside a certain domain $C$ (e.g. $C=\{z\in \mathbf{C}: ~|z|<1\}$). Then one can choose $A=\partial C$. **Application 2: Avoiding known local minima.** Assume that one wants to solve a global optimization problem for a cost function $f(x)$. Assume that in previous runs of IM one already found local minima $z_1^*,z_2^*,\ldots ,z_j^*$. One does not know whether these are global minimizers (in particular, one does not want to stuck at bad local minima). Then one can define the close set $A=\{z_1^*,\ldots ,z_j^*\}$ and apply Problem. **Application 3: Constrained optimization.** Assume that one has a function $f:X\rightarrow \mathbf{R}$, but only wants to solve the optimization problem in a smaller closed subset $Y$. Then one can choose $A$ to be the boundary of $A$ and apply Problem. **Application 4: Constrained optimization, version 2.** Assume that one has a function $f:X\rightarrow \mathbf{R}$, but only wants to solve the optimization problem in a smaller domain $Y$. Different from the choice in Application 3, here one chooses $A=X\backslash Y_0$, where $Y_0$ is the interior of $Y$. **Application 5: Finding different components of a variety.** Assume the set of solutions to $F=0$ has different connected components $C_1,\ldots, C_k$, each of them may have positive dimension. Assume that beforehand one does not know any of them, and one will choose initial points for IM in a fixed domain $B$. After many runs of IM, one finds many roots $z_1^*,\ldots ,z_j^*$ but they seem to be close together and seem to belong to the same component $C_i$. (For example, this is the case if $B$ happens to belong to the basin of attraction of $C_i$). One hopes that if $j$ is big enough and one can avoid all of the points $z_1^*,\ldots ,z_j^*$ , then one can also avoid $C_i$ and finds another component of $\{F=0\}$. ## Survey of main existing relevant methods in the current literature The relevant methods which we are aware of are mostly in the setting of constrained optimization. **Approach 1: New metric.** An approach is to redefine the metric on $X\backslash A$, such that the points in $A$ become infinity in the new metric. (This fits particularly well if $X$ is a Riemannian manifold.) However, if indeed the minimum of $f$ occurs in $A$, creating this new metric does not guarantee that the constructed sequence will not converge to $A$. **Approach 2: Linear Programming.** If the cost function and all constraints are linear functions, then this is treated effectively in the well known Linear programming, see e.g. [@dantzig-thapa]. An experiment presented later will test whether our method can find good approximates of the optimizers for Linear Programming problems. **Approach 3: Algebraic methods.** If the cost function and all constraints are polynomials, then there are algebraic methods aiming to solve it (e.g. [@lasserre], [@pham]). We have a couple of remarks here. First, these methods are usually not iterative in nature (hence, implementation them in computers can be difficult). Second, they can - usually - only find the value $\min f(x)$ but not the points $z^*$ which minimize $f$. A combination between these and iterative methods may be more useful. **Approach 4: Projected methods.** If $X=\mathbf{R}^m$ and the constrained set $Y$ is a closed convex subset, then for each $x\in X$ there is a unique point $y\in Y$, denoted $y=pr_Y(x)$, so that $||x-y||=$ the distance from $x$ to $Y$. One can redefine the iterative scheme as follows: $x_{n+1}=pr_Y(IM(x_n,f))$. However, if $A=\partial Y$ does not contain any global minimum of $f$, and if an open neighbourhood $U$ of $A$ belongs to the basin of attraction for a point in $X\backslash Y$, then choosing a random initial point $x_0\in U$ and apply the projected method one ends up at a sequence in $A$ which can never converge to a global minimum of $f$. (An explicit example relating to this point was given in [@nouiehed-etal] . We will revisit this example in the experiments presented later.) **Approach 5: Lagrange multiplier/Karush-Kuhn-Tucker conditions.** If the constraints are given by equations $h_1,\ldots ,h_j$, then Lagrange's multiplier method looks for critical points of a new function $F(x,\lambda _1,\ldots ,\lambda _j)=$ $f(x)-\lambda _1h_1(x)-\ldots \lambda _jh_j(x)$. This method works well if most of the critical points of $F$ corresponds to global minima of $f$, otherwise a lot of the work is wasted. Karush-Kuhn-Tuckter conditions [@karush][@kuhn-tucker]: this extends Lagrange's method to the case where constraints also include inequalities. Then one looks for saddle points of a similar function. Again, the same comment as above for Lagrange's multiplier method can be applied here, and in this case the issue may be more serious since saddle points are more dominant in higher dimensions [@bray-dean]. **Approach 6: Interior-point/Penalty methods.** The main idea of these methods, see e.g. [@wright], is to consider a new unconstrained problem $G(x)=f(x)-\epsilon \rho _A(x)$, where $\rho _A(x)$ is a function which is infinity on the boundary $A$, and $\epsilon >0$ is a parameter. It is unclear which $\epsilon$ is good to use. A usual choice of $\rho _A(x)$ is $\rho _A(x)=\log d(x,A)$, where $d(.,.)$ is the distance function. The idea is that when $\epsilon$ becomes smaller and smaller, the method IM will find (for cost function $G(x)$) points which are closer and closer to global minimizers of $f(x)$. However, when $\epsilon$ is too small, what can happen is that it may cancel the effect of $\rho _A(x)$ to a certain level that the constructed sequence by applying IM to $F(x)$ will behave similarly to the constructed sequence by applying IM to the original function $f(x)$. We will demonstrate this in an experiment later. **Approach 7: Tunnelling/Deflation method.** This method deals with the case where $A=\{z_1^*,\ldots ,z_j^*\}$ is a finite set. In the case of minimizing a function $f(x)$, it considers the new function $f(x)/[d(x,z_1)^{N_1}\ldots d(x,z_j)^{N_j}]$ for some appropriate choices of $N_1,\ldots ,N_j$. It uses Backtracking line search to have descent property. In the case of solving a system of equations $F=0$, it applies Newton's method directly to the system, and again uses Backtracking line search to have descent property. Hence, it can show that the sequence constructed will avoid the finite set $A$. However, descent property alone does not guarantee strong convergence. For a constrained optimization problem, it does not work directly like our new methods below, but rather similar to the Lagrange multiplier/Karush-Kuhn-Tucker conditions. See [@levy-gomez][@farrell-etal][@farrell-etal2] for more details, where discussions on related methods are also given. In an experiment later, we see that using this function $f(x)/[d(x,z_1)^{N_1}\ldots d(x,z_j)^{N_j}]$ may make it difficult to escape a component of the solution set to another component. ## Main theoretical result: Finding roots of a meromorphic function of 1 complex variable in a given domain Here we present a result which guarantees finding roots of a meromorphic function of 1 complex variable in a given domain. Finding roots of a meromorphic function of 1 complex variable is a topic intensively studied. Many special and interesting functions are meromorphic functions, like Gamma function, Bessel function, and Riemann zeta function. The special question of finding roots of polynomial functions is a main subject in the field of Complex Dynamics. Often, one has the need of finding roots in a given domain $U\subset \mathbf{C}$, for example inside the unit disk. Then, starting from a point in that domain $U$ and using an iterative method, it is difficult to know before hand if the sequence one constructed will converge to a root inside $U$. The reason is that, first of all the sequence constructed may not converge to any root at all, or it may converge to a root outside of $U$. Basins of attraction of the roots can be too complicated to allow a good guess of where the sequence will go. Here, we illustrate the use of the approach proposed in this paper towards this question. For the description and properties of Backtracking New Q-Newton's method, see [@truong]. **Theorem 1**. *Let $g(z)$ be a non-constant meromorphic function in 1 complex variable $z$. Assume that $g$ is generic, in the sense that $\{z\in \mathbf{C}:~g(z)g"(z)=g'(z)=0\}=\emptyset$.* *Let $f(x,y)=|g(x+iy)|^2/2$. Let $U\subset \mathbf{C}$ be an open subset. There is a set $\mathcal{E}$ with Lebesgue measure 0 such that the following hold. Let $M$ be a positive number and define a new function $F(x,y)$ by the following formula: $F(x,y)=f(x,y)$ if $(x,y)\in U$, and $F(x,y)=M$ if $(x,y)\in \mathbf{C}$.* *1) If $z_0=(x_0,y_0)\in U\backslash \mathcal{E}$, and $f(x_0,y_0)<M$, then Backtracking New Q-Newton's method applied to $F(x,y)$ with initial point $(x_0,y_0)$ will produce a sequence $\{z_n=(x_n,y_n)\}$ which must satisfy one of the three options: a) $\{z_n\}$ converges to a root of $g(z)$ inside $U$ and the rate of convergence is quadratic, b) $\lim _{n\rightarrow \infty}|z_n|=\infty$, or c) all cluster points of $\{z_n\}$ are on $\partial U$.* *2) Assume moreover that $U$ is bounded and $\inf _{z\in \partial U}f(z) >f(z_0)$. Then the constructed sequence (Backtracking New Q-Newton's method applied to F(x,y)) converges to a root of $g(z)$ inside $U$, with a quadratic rate of convergence.* *[\[Theorem\]]{#Theorem label="Theorem"}* *Proof.* 1) This follows easily from Theorem [\[Theorem1\]](#Theorem1){reference-type="ref" reference="Theorem1"} below, in combination with Theorem 3.3 in [@truong-etal]. 2\) Since $U$ is bounded, option b) in part 1) cannot happen, given that the sequence $\{f(z_n)\}$ is non-increasing. The same reason gives that $\{z_n\}$ cannot have any cluster point on $\partial U$. Hence, only option a) can happen. ◻ ## Acknowledgements and the plan of the paper Several main ideas presented here were initiated in the inspiring environment of the ICIAM 2023 conference in Tokyo. The author would like to thank specially the organisers and participants of the minisymposium at ICIAM "Theory and applications of random/non-autonomous dynamical systems\" and its satellite conference, for support and interesting discussions, in particular to Hiroki Sumi, Tayuki Watanabe and Mark Comerford. The author also would like to thank Xiao Wang on his informative presentation on convergence of iterative methods in a conference at University of Dalat in July 2023, and for pointing out the reference [@nouiehed-etal]. We thank Patrick Farrell for information about the tunnelling/deflation method. The author is partially supported by Young Research Talents grant 300814 from Research Council of Norway. The remaining of the paper is as follows. In Section 2 we present our two new methods. In Section 3 we present some illustrating experiments, including constrained optimization problems. In the last section we present some conclusions. # Two new methods: Creating walls to avoid unwanted points The main idea is to create walls at $A$, that is to make values of the functions at $A$ so large that it will unlikely for a sequence constructed from a point outside of $A$ can converge to or cross $A$. We propose here two new methods to create such walls. ## Multiplying poles at $A$ The first method is to multiplying the cost function with a function having poles at $A$. We recall the setting: We are finding global minima of a function $f:X\rightarrow \mathbf{R}$, while wanting to avoid a closed subset $A\subset X$. We will present the method first in a special case, and then in the general case. [**Special case: $\min _{x\in X}f(x)=0$**]{.ul}. In this case, we choose an appropriate positive integer $N$ and consider a new function $G(x)=f(x)/d(x,A)^N$, where $d(.,.)$ is the distance function. The number $N$ depends on the "multiplicity" of $f(x)$ along $A$. The following lemma shows that at least we are on the right track with this method. **Lemma 2**. *Assume that $f:X\rightarrow \mathbf{R}$ is such that $\min _{x\in X}f(x)=0$. Let $x^*\in X\backslash A$ be a root of $f(x)$ on $X$. Then $x^*$ is also a global minimizer of the new function $G(x)$. [\[Lemma1\]]{#Lemma1 label="Lemma1"}* *Proof.* Indeed, since $f(x)$ is non-negative, the same is true for $G(x)$. Since $x^*\in X\backslash A$ we have $d_A(x^*)\not= 0$. Hence, since $f(x^*)=0$, we have $G(x^*)=0$ as well. Therefore, $x^*$ is a global minimizer of $G$. ◻ In the case of finding roots of a function $F(x)=0$ for $F:\mathbf{C}\rightarrow \mathbf{C}$, then this method is almost the same as what one usually does. Assume that one already found the roots $z_1^*,\ldots ,z_j^*$ of $F(x)$. Then the usual way to find new roots is to consider a new function $F(x)/[(x-z_1^*)\ldots (x-z_j^*)]$. In our method, we will try to find global minima of $f(x)=||F(x)||^2/2$, and will consider a new cost function $G(x)=f(x)/d(x,A)^2$. Here $d(x,A)$ has the following explicit formula: $d(x,A)=\min \{|x-z_1^*|,\ldots ,|x-z_j^*|\}$. Numerically, the complexity of computing $\nabla G$ and $\nabla ^2G$, as well as higher derivatives, is not increased when we increase the number of points $z_1,\ldots ,z_j$ to avoid. In the usual approach, if the function $f$ has multiplicity $N$ at a root $z_i^*$, one should divide by $(x-z_i^*)^N$. Similarly one should use $f(x)/d(x,A)^{2N}$ in our approach. \[Note that since $f(x)=||F(x)||^2$, a root of multiplicity $N$ of $F$ will become a root of "multiplicity\" $2N$ for $f(x)$.\] The tunnelling/deflation method is probably the one closest to our method of multiplying with poles at $A$. However, there are some differences. First, tunnelling/deflation method only applies for the case where $A=\{z_1,\ldots , z_j\}$, while our method can apply to any closed set $A$. This is because we use a more flexible function $d(x,A)$, instead of the function $d(x,z_1)\ldots d(x,z_j)$. Second, while Backtracking line search is also used in tunnelling/deflation, it seems that they only use to guarantee descent property, and not Armijo's condition. Hence, while avoidance of the finite set $A$ is guaranteed, the convergence issue is not (except the case where some restrictive conditions are assumed). (Note that strong convergence guarantees for Armijo's Backtracking line search for general cost functions only appear quite recently in the literature.) Third, for to solve a system of equations $F=0$, the tunnelling/deflation method directly applies Newton's method for the system, while we work with the associated cost function $f=||F||^2$. Here we also note the relation between our method and the Penalty method recalled in the previous section. Since here the function $f(x)$ is non-negative, one can consider a new cost function $h(x)=\log f(x)$. Then, the Penalty method will require to consider the function $H(x)=\log f(x) - \epsilon \log d(x,A)$, where $\epsilon >0$ is **small enough**. Taking logarithm of the function $G(x)=f(x)/d(x,A)^N$, we get another function $\log G(x)=\log f(x)-N \log d(x,A)$. Hence, the forms of $H(x)$ and $\log G(x)$ are the same, except a difference in detail: In $H(x)$ the factor $\epsilon$ before $\log d(x,A)$ is to be small, while in $\log G(x)$ the factor $N$ before $\log d(x,A)$ is to be big (at least $\geq 1$). Note also that generally our method using $G(x)$ is more numerically stable than using the Penalty method using $H(x)$, if IM at least uses gradients of the cost functions. Indeed, let $x^*\in X\backslash A$ be a point where $f(x^*)=0$ Then, near $x^*$ we have that $d(x,A)$ is strictly positive. Hence $\nabla G(x)$ is of the same size as $\nabla f(x)$, while $\nabla H(x)$ is of the same size as $(\nabla f(x))/f(x)$. Hence, near $x^*$, the gradient $\nabla H(x)$ is very big. Similar observation applies to points near $A$, where $d(x,A)$ is $0$. An experiment showing the difference between these two approaches will be presented later. **A heuristic argument.** While finding global minima is NP-hard, the following heuristic argument supports that our method will work well generally. Assume that $IM$ has the descent property for function values, that is whenever it is applied for a function $g$, then $g(x_{n+1})\leq g(x_n)$ for all $n$. There are several methods to ensure this property: Line search, Trust region, and Backtracking line search (Armijo or Frank-Wolfe). Among these, Armijo's Backtracking line search is the most flexible while have strong theoretical guarantees when applied to many large and useful classes of cost functions. (For example, functions with at most countably many critical points, or functions satisfying Lojasiewicz gradient inequality. The first class includes Morse functions, which is a dense class in the closed-open topology. The second class includes semi-analytic functions.) For some recent results on using Armijo's method on general functions, for both first and second order methods, see [@truong-nguyen1][@truong-nguyen2][@truong-etal][@truong]. There, one also finds a brief comparison of several different methods. We also assume that with an appropriate choice of $N$, we obtain that $G(x)=\infty$ if and only if $x\in A$. Then for any choice of the initial point $x_0\in X\backslash A$, because IM has the descent property, it follows easily that the sequence $\{x_n\}$ constructed by IM from $x_0$ will belong to $X\backslash A$. Moreover, any cluster point of $\{x_n\}$ must be in $X\backslash A$. Now we explain why it is more likely that if $x_0$ is in a connected component of $X\backslash A$, then the whole sequence $\{x_n\}$ should be also in $X\backslash A$. We first present a rigorous argument, which is easy to prove. **Lemma 3**. *Assume further that IM is such that there is a constant $r>0$ such that $d(x_n,x_{n+1})<r$ for all constructed sequence $\{x_n\}$. Let $A_r=\{x\in X:~d(x,A)\leq r\}$. Assume that $x_0\in X$ is such that $G(x_0)< \inf _{x\in A_r}G(x)$. Then the sequence $\{x_n\}$ constructed by IM will belong to the same connected component of $X\backslash A$ as $x_0$. [\[Lemma2\]]{#Lemma2 label="Lemma2"}* *Proof.* Indeed, assume by contradiction that some points in the sequence $\{x_n\}$ belongs to another component of $X\backslash A$. Let $N$ be the one index for which $x_N$ is still in the same connected component of $X\backslash A$ as $x_0$, but $x_{N+1}$ belongs to another component. Then since $d(x_N,x_{N+1})\leq r$, we must have $x_N\in A_r$. However, this contradicts the facts that IM is descent (hence, $G(x_N)\leq G(x_0)$) and $G(x_0)< \inf _{x\in A_r}G(x)$. ◻ Even if the assumptions in Lemma [\[Lemma2\]](#Lemma2){reference-type="ref" reference="Lemma2"} are not satisfied, we have the following heuristic argument. When we use Backtracking line search, then heuristically since on $A$ the function $G$ is $+\infty$, $A$ will play the roll of a wall which $\{x_n\}$ cannot cross. When $x_n$ is closer to $A$, the distance $d(x_n,x_{n+1})$ should become smaller in such a way that $x_{n+1}$ is always kept in the same connected component as $x_n$. This is how far heuristic arguments go, given that finding global minima are NP-hard. There are experiments which show that indeed there are cases where the sequence $\{x_n\}$ can cross to another connected component. Even so, we observe that by using the function $G(x)$, the chance for $\{x_n\}$ to stay in the same connected component indeed increases. [**The general case**]{.ul}. We now consider a general function $f:X\rightarrow \mathbf{R}$, without assuming that it is non-negative (in particular, without assuming that $\min _{x}f(x)=0$). In theory, this general case and the special case are equivalent, since if $f(x)$ achieves minimum on $X$, then we can replace $f$ by the function $f(x)-\min _{x}f(x)$ and one is reduced to the special case. However, finding $\min _{x}f(x)$ may not be easy, and hence in such cases one would like to be able to proceed without the hurdle to know that $\min _{x}f(x)=0$ (or even knowing if the function is non-negative). Some remarks are in order, before we continue. First, note that in this case the Penalty method which uses $\log f(x)$ is formally inapplicable, since the logarithm of a negative number is not defined (as a real number). Second, note that if the setting is semi-algebraic, one may use algebraic methods to find the minimum value of $f(x)$, and hence can reduce to the special case discussed before. To deal with this case where we do not know the minimum value of $f$, we consider the following function: $G_{\gamma}(x)=(f(x)-\gamma )/d(x,A)^N$, where $\gamma$ is an approximate lower bound for $min _{x}f(x)$ obtained "on the fly\". More precisely, we will proceed as follows. We will do many runs of the method IM for $G_{\gamma}(x)$, and in each run will try to obtain a better value for $\gamma$. In each run, we will do the following steps: Step 1: Choose a random initial point $x_0$. Step 2: Apply IM to $G_{\gamma}(x)$ and construct a sequence $\{x_n\}$, where we will stop when some conditions are satisfied (for example, when $||\nabla G_{\gamma}(x_n)||$ is smaller than a prescribed threshold or when the number of iterates exceeds a given bound). Step 3: If $\gamma > \min _n f(x_n)$, then we replace $\gamma$ by $\min _n f(x_n)$. If one wants to do optimization on a connected component $B$ of $X\backslash A$ only, then in Step 1 one chooses a random point $x_0\in B$, and one can modify Step 3 above in the following manner: Step 3': If $\gamma > \min _{x_n\in B}f(x_n)$, then we replace $\gamma$ by $\min _n f(x_n)$. An example presented later will illustrate how this algorithm is used in practice. [**To escape the basin of attraction of a positive dimension component.**]{.ul} We now consider the following question: Assume we apply the iterative method IM to a non-negative cost function $f(x)$, whose zero set has different components $C_1,C_2,\ldots ,C_j$. If we found only points in $C_1$, can we escape it to reach another component? Here we present an idea to use our method of multiplying poles, in combination of a careful choice of the next initial point, to resolve this question. Assume that we already found a sequence of points (close to points) on $C_1$, called $p_1,\ldots ,p_k$. As before, we will consider a new cost function $f(x)/d(x,A)^N$. Now, for the next initial point, we will not choose it randomly in a prescribed domain, but to close to the point $p_k$. This way, it can help us to quickly move away from $C_1$ and hopefully reach another component. Here is a heuristic for why this idea can help. Because the component we attempt to escape has positive dimension, it can happen that even if we already created many poles, still in the next run we may end up at another point in that same component. When we creating more poles, we are also somehow creating a wall preventing initial points to escape to the other side. Hence, if the new poles are in between the domain where we choose our initial points and the component we want to reach, then may be we would not be able to reach that component as we wanted. To choose the initial point for the next pole close to the point (which now becomes a new pole) we found in the previous run, it is more likely that we avoid better the affect of the wall created by the poles. An experiment later will illustrate how this is used. [**Constrained optimization.**]{.ul} This is a more precise form of the idea in Applications 3 and 4. In the above, we assumed that $G=+\infty$ on $A$ so that we can avoid $A$ (this is the case if $f>0$ on $A$). However, if $f<0$ at some points of $A$, the sequence we construct may converge to that point and we discover a possible minimum point. This way, we still can apply to constrained optimization. If the constraints contain an equation $p(x)=0$, we replace it by two inequalties $p(x)\geq -\epsilon$ and $p(x)\leq \epsilon$, where $\epsilon >0$. The idea is that if $\epsilon$ goes to $0$, then the minimizers we found will converge to a minimizer of the original question. Numerically, with $\epsilon$ small enough, the minimizer we found for the new question should be very close to a minimizer of the original question. Recall about the theorems for Backtracking New Q-Newton's method. Note that if a function has compact sublevels, then when adding poles this way the new function will also has compact sublevels. ## Making the function to be a big constant on $A$ Another way to make walls at $A$ is to define the function $f$ to be a very large constant on $A$. More precise, we choose a very big number $R>0$, and then define a new function: $g(x)=f(x)$ if $x\in X\backslash A$, and $g(x)=R$ if $x\in A$. While this method is very simple, it is guaranteed to avoid the interior of $A$. It also has the good property that it preserves critical points of $f$ inside $X\backslash A$ (this property, while simple, does not hold for all other approaches). **Theorem 4**. *Assume that the iterative method IM has the descent property. Assume that IM is applied to $g$ at an initial point $x_0$ so that $f(x_0)<R$. Then no cluster point of the constructed sequence belongs to the interior of $A$.* *The critical points (minima, maxima, saddle points) inside $X\backslash A$ of $f$ and $g$ are the same.* *[\[Theorem1\]]{#Theorem1 label="Theorem1"}* *Proof.* Indeed, let $x_n$ be the constructed sequence. Then $f(x_n)\leq f(x_0)<R$ for all $n$. Hence $\{x_n\}\subset X\backslash A$, therefore no cluster point of $\{x_n\}$ belongs to the interior of $A$. ◻ Some experiments presented later illustrate that this method works well for constrained optimization. Note that while the function $g$ is not continuous on the boundary $\partial A$ of $A$, usually $\partial A$ has zero Lebesgue measure and does not affect in numerical calculations. ## Convergence issue Since we will use Backtracking New Q-Newton's method, which has strong convergence guarantees see [@truong-etal][@truong], the issue of global convergence is less serious than if we use other methods, like Newton's method in the tunnelling/deflation method where a lot of care is needed. # Experimental results We now present several experimental results illustrating the usefulness of the approach proposed in this paper. The setting of the experimental results is as follows. In a run for a cost function $g$, we will stop until either $||\nabla g||$ is smaller than a small threshold (e.g. $1e-6$) or if the number of iterates exceeds $10 000$. In experiments where basins of attractions are drawn, we choose initial points on a square grid where the centre of the grid is chosen randomly in some domains. An initial point $x_0$ is considered to be in the basin of attraction of a point $z^*$ if $d(x_0,z^*)$ is smaller than a small threshold (e.g. $1e-5$). Usually we choose the initial point to be random, but there are cases where we fix the initial point. Except Example 1- (which tests with methods Gradient Descent, Backtracking Gradient Descent (i.e. Armijo's Backtracking line search for Gradient Descent) and Backtracking New Q-Newton's method [@truong] (developed from an earlier algorithm in [@truong-etal]) - all other examples test only Backtracking New Q-Newton's method. Backtracking New Q-Newton's method is found to be very much faster than the other two methods in the examples considered here. The exception is that Example 1 is taken from [@nouiehed-etal], where Projected Gradient Descent was tested and found to fail to avoid the saddle point. Therefore, it is reasonable to test Gradient Descent and Backtracking Gradient Descent for the function $G(x)$ in Example 1 also. ## Example 1 In this example, taken from [@nouiehed-etal], we consider the following constrained optimization problem: $arg\min _S f (x,y)$, where $S=\{(x,y)\in \mathbf{R}^2:~x+y\leq 0\}$ and $f(x,y)=-xye^{-x^2-y^2}+y^2/2$. One can check that the critical points of $f$ are (approximate to) the following: $p_1=(0,0)$, $p_2=(0.7071067,0.3128011)$ and $p_3=(-0.7071067,-0.3128011 )$. The point $(0,0)$ is a saddle point with function value $0$, while the other two points are global minimizer with function value (close to) $-0.0727279$. Therefore, $\gamma _0=-0.0727280$ is a good lower bound for $\min _Sf(x,y)$. In [@nouiehed-etal], it was shown that Projected Gradient Descent applied to this constrained optimization problem fails. More precisely, there is a small open set $U$ in $S$ which touches the point $(0.5,-0.5)$ on the boundary for which if we apply Projected Gradient Descent with a learning rate $0<\alpha <2/3$ with an initial point in $U$, then the constructed sequence will converge to the saddle point $(0,0)$. Here we draw basins of attraction for different methods for the function $G(x,y)=(f(x,y)-\gamma _0)/d((x,y),A)$ where $A=\partial S$. More explicitly, $d((x,y),A)=|x+y|$. We will also consider 2 functions $H_1(x,y)= (f(x,y)-\gamma _0) -\epsilon \log d((x,y),A)$, and $H_2(x,y)=\log (f(x,y)-\gamma _0) -\epsilon \log d((x,y),A)$, where $\epsilon >0$ is a constant. $H_1$ is the Penalty method applied to the function $f(x,y)-\gamma _0$, and $H_2(x,y)$ is the Penalty method applied to the function $\log (f(x,y)-\gamma _0)$. {#fig:pic1} {#fig:pic2} {#fig:pic3} {#fig:pic4} {#fig:pic4.1} {#fig:pic4.2} {#fig:pic4.3} We have some remarks about experiments in Example 1. Figures [1](#fig:pic1){reference-type="ref" reference="fig:pic1"}, [2](#fig:pic2){reference-type="ref" reference="fig:pic2"} and [3](#fig:pic3){reference-type="ref" reference="fig:pic3"} show that indeed using the function $G(x,y)$ helps improve the convergence to global minima. In particular, the phenomenon mentioned in [@nouiehed-etal] for Projected Gradient Descent does not seem to occur here. The pictures for methods using Armijo's Backtracking line search look better than that for Gradient Descent. Figures [4](#fig:pic4){reference-type="ref" reference="fig:pic4"}, [5](#fig:pic4.1){reference-type="ref" reference="fig:pic4.1"}, [6](#fig:pic4.2){reference-type="ref" reference="fig:pic4.2"} and [7](#fig:pic4.3){reference-type="ref" reference="fig:pic4.3"} show that using Penalty method with $\log f(x,y)$, with small $\epsilon$ does not improve the performance. For $H_1(x,y)$, as we expected in the above, the performance will be like that for the function $f(x,y)$ when $\epsilon$ is small. Note that Figure [5](#fig:pic4.1){reference-type="ref" reference="fig:pic4.1"} is reminiscent of the phenomenon in Schroder's theorem for Newton's method for a polynomial of degree 2, which is also numerically observed for Backtracking New Q-Newton's method see [@truong]. The yellow part in Figure [5](#fig:pic4.1){reference-type="ref" reference="fig:pic4.1"} is close to the bisector of 2 points $p_2$ and $p_3$. Figures [6](#fig:pic4.2){reference-type="ref" reference="fig:pic4.2"} and [7](#fig:pic4.3){reference-type="ref" reference="fig:pic4.3"} show that, while formally optimising a non-negative function $f$ or its logarithm should be similar, there are enough differences (both theoretical and numerical), which lead to very different behaviours in experiments. We also remark that for $H_1$ and $H_2$, if we choose $\epsilon$ bigger, for example $\epsilon =0.01$, then we encounter errors such as NAN. ## Example 2 In this example, we find roots of a polynomial of degree 5 in 1 complex variable. More precisely, the polynomial is $F(z)=z^5-3iz^3-(5+2i)z^2+3z+1$. This polynomial has the following (approximate roots): $p_1\sim -1.28992-1.87357i$, $p_2\sim -0.824853+1.17353i$, $p_3\sim -0.23744+0.0134729i$, $p_4\sim 0.573868-0.276869i$, and $p_5\sim 1.77834+0.963437i$. The associated cost function is $f(x,y)=|F(x+iy)|^2/2$, where $x,y\in \mathbf{R}$. The following is a picture of basins of attractions when applying Backtracking New Q-Newton's method (it is drawn in a bigger domain than that in [@truong]). {#fig:pic5} We see from Figure [8](#fig:pic5){reference-type="ref" reference="fig:pic5"} that the basin of attraction for $p_3$ seems to be smaller and surrounded by the basins of attraction of other points. Hence, it may be difficult to find the root $p_3$. We will consider the following 4 closed sets: $A_1=\{p_1,p_2,p_4,p_5\}$, $A_2=\{p_1,p_2,p_4\}$, $A_3=\{p_1,p_2\}$ and $A_4=\{p_1\}$. Correspondingly, we consider 4 functions: $G_1(.)=f(.)/d(.,A_1)^2$, $G_2(.)=f(.)/d(.,A_2)^2$, $G_3(.)=f(.)/d(.,A_3)^2$, and $G_4(.)=f(.)/d(.,A_4)^2$. {#fig:pic6} {#fig:pic7} {#fig:pic8} {#fig:pic9} ## Example 3 We consider now finding roots of a polynomial in 2 real variables $F(x,y)=(y-x^2-2)\times (y+x^4+2)\times (x^2+(y-1)^2)\times ((x-1)^2+(y+1)^2)\times ((x+1)^2+(y-4)^2)$. The solution set has 5 connected components, where 2 of them are curves: $C_1=\{(x,y)\in \mathbf{R}^2:~y-x^2-2=0\}$, $C_2=\{(x,y)\in \mathbf{R}^2:~y+x^4+2=0\}$, $C_3=(0,1)$, $C_4=(1,-1)$, $C_5=(-1,4)$. The first two components are curves, and the remaining components are points. Component $5$ is separated components $C_2,C_3$ and $C_4$ by component $C_1$. In this experiment, we test if the initial point is in the set $y>x^2+2$, whether it is possible for an iterative use of the method in paper will allow to reach the components $C_2, C_3$ or $C_4$. Since we want to solve equation, we choose our cost function as follows: $f(x,y)=(y-x^2-2)^2\times (y+x^4+2)^2\times (x^2+(y-1)^2)\times ((x-1)^2+(y+1)^2)\times ((x+1)^2+(y-4)^2)$ a\) We test with our method using $f(x,y)/d((x,y),A)^2$. We always choose the initial point to be $(0.1,3.1)$. Run Backtracking New Q-Newton's method for $f(x,y)$, we end up at the point\ $p_1=$ $(0.02299946, 2.00052898)$, which is close to a point on $C_1$. Set $A_1=\{p_1\}$. Run Backtracking New Q-Newton's method for $f(x,y)/d((x,y),A_1)^2$, we end up at the point $p_2=(-0.2120813,2.04497848)$, which is again close to a point on $C_2$. Set $A_2=\{p_1,p_2\}$. Run Backtracking New Q-Newton's method for $f(x,y)/d((x,y),A_2)^2$, we end up at the point $p_3=(-0.2559445,2.06550759)$, which is again close to a point on $C_2$. Set $A_3=\{p_1,p_2,p_3\}$. Run Backtracking New Q-Newton's method for $f(x,y)/d((x,y),A_3)^2$, we end up at the point $C_3$. Thus, this example shows that it is possible to use our method iteratively to escape the basin of attraction for a big component and move to another component. b\) We next test if using $\prod _{p_j\in A}d((x,y),p_j)^2$, as in the tunnelling/deflation method, can help escape to another component. Again, we always choose the initial point to be $(0.1,3.1)$, which lies inside the domain $y>x^2+2$. Run Backtracking New Q-Newton's method for $f(x,y)$, we end up at the point\ $p_1=$ $(0.02299946, 2.00052898)$, which is close to a point on $C_1$. Set $A_1=\{p_1\}$. Run Backtracking New Q-Newton's method for $f(x,y)/d((x,y),A_1)^2$, we end up at the point $p_2=(-0.2120813,2.04497848)$, which is again close to a point on $C_2$. Set $A_2=\{p_1,p_2\}$. Run Backtracking New Q-Newton's method for $f(x,y)/\prod _{p_j\in A}d((x,y),p_j)^2$ $=$\ $f(x,y)/[d((x,y),p_1)^2d((x,y),p_2)^2]$, we end up at the point $(-1,4)$, which is the component $C_5$. So we cannot escape the domain $y>x^2+2$. ## Example 4 The test in this experiment is like that in Example 3, but now we look for roots of the real elliptic curve $E=\{y^2=x^3-x\}\subset \mathbf{R}^2$. This curve has two components $C_1=E\cap \{x\leq 0\}$ and $C_2=E\cap \{x\geq 1\}$. Since we want to find roots, we choose the cost function to be $f(x,y)=(y^2-x^3+x)^2$. We would like to start from a point close to $C_1$, and hope to reach $C_2$. In this example, there are some differences to Example 3. First, if we only divide by $d(.,A)^2$, it is not enough to escape $C_1$. Hence, we change to $d(.,A)^4$. Second, we find that if we choose the initial point in a prescribed small open set near $C_1$, we may not be able to reach $C_2$ but may wander around in $\mathbf{R}^2$. Therefore, we will use the idea how to escape a component of positive dimension in Section 0.3. For example, assume that our random initial point happens to be $(-0.9, 0.1)$, which is inside the open set bound by $C_1$! After running Backtracking New Q-Newton's method for $f$, we end at the point $p=(-0.14223901, 0.3733089)$, which is (close to) a point on $C_1$. We now consider the new cost function $G(.)=f(.)/d(.,p)^4$. We now run Backtracking New Q-Newton's method for $G$, with the initial point $(-0.142, 0.373)$ which is close to the point $p$. We end up at the point $(1.01894051,-0.19739787)$, which is (close to) a point on the component $C_2$, as wanted! ## Example 5 In this experiment, we illustrate how the method in Section 0.3 applies for the case we do not know about $\min f(x)$ (and hence, not know if that value is actually $0$, which is the best case to use the method). We revisit the question in Example 1: Consider the following constrained optimization problem: $arg\min _S f (x,y)$, where $S=\{(x,y)\in \mathbf{R}^2:~x+y\leq 0\}$ and $f(x,y)=-xye^{-x^2-y^2}+y^2/2$. Now, we do not before hand try to determine the critical points (and in particular, the minimum value) of the function $f$. By the ideas in Section 0.3, we will start from $M=0$ and update $M$ to have better estimates for a lower bound for $\min f$ on the fly. Accordingly, the new cost function which we will apply Backtracking New Q-Newton's method to is $G_M(x,y)=(f(x,y)-M)/d((x,y),A)$, here $A=\{x+y=0\}\cup \{x=-10\} \cup \{x=10\} \cup \{y=1-0\} \cup \{y=10\}$. The reason to choose $A$ much bigger than the line $\{x+y=0\}$ is because it can easily check that the function $f$, while having only 3 critical points, has its gradient converges to $0$ for certain sequences of points going to $\infty$. Hence (and this is observed in practice), the sequence constructed by an iterative algorithm can diverge to infinity. To prevent this, we create new walls surrounding a bounded domain. Here, it turns out that always choosing a fixed point as the initial point for every run does not yield desired result. Hence, we switch to choosing our initial point randomly inside the domain $x+y<0$. Specifically, we will choose our random initial point in this way. We choose $x_0$ as a random point between $(-1,1)$ and also choose a random number $err\in (0,1)$. Then our initial point is $(x_0,y_0)=(x_0,-x_0-err)$. Beware that if $M\not= \min f(x)$, then global minimzer of $f$ may not be critical points of $G_M$! Therefore, applying an iterative method to $G_M$ may not be able to find a global minimizer of $f$. Here we report such an experiment. Start with $M_0=0$: The random initial point is $(0.23201923, -0.29366141)$. Running Backtracking New Q-Newton's method for $G_{M_0}(x,y)$, we get the sequence of points $(0.20221512,0.3141085)$, $(-0.04991277, 0.02411369)$, $(-0.05235699, 0.01419474)$, $(0.05467527, -0.01150241)$,\ $(-0.70066302, -0.14982963)$, $(-0.52160544, -0.19009722)$, $(-0.51322439, -0.2418863)$,\ $(-0.52258626, -0.24766466)$, $(-0.52258709, -0.24768905)$, $(-0.52258709, -0.24768905)$,\ $(-0.52258709, -0.24768905)$, $(-0.5225871, -0.24768905)$, $(-0.5225871, -0.24768905)$,\ $(-0.5225871, -0.24768905)$, $(-0.5225871, -0.24768905)$. The end point here is in the domain $x+y<0$, but is not close to any critical point of $f$. We find that the value of $f(-0.5225871, -0.24768905)-M_0$ $=$ $-0.06196899168272818$. Therefore, we can choose $M_1=M_0-0.061968992$ $=$ $-0.061968992$ as the next lower bound estimate for $\min f$ in the concerned domain. Continue with $M_1=$ $-0.06196899168272818$: The random initial point is $(0.67180663, -1.47719597)$. Running Backtracking New Q-Newton's method for $G_{M_1}$, we get the sequence of points:\ $(-0.21324179, -0.40600826)$, $(-0.44163373, -0.22503746)$, $(-0.60779332, -0.29120854)$,\ $(-0.67567597, -0.30618429)$, $(-0.68489123, -0.30625665)$, $(-0.68507292, -0.30623803)$,\ $(-0.68507296, -0.30623802)$, $(-0.685073, -0.30623802)$, $(-0.685073 ,-0.30623802)$. Again, the end point here is in the domain $x+y<0$, but is not close to any critical point of $f$. We find that the value of $f(-0.685073 ,-0.30623802)-M_1$ $=$ $-0.01060545418333153$. Therefore, we can choose $M_2=M_1-0.01060545418333153$ $=$ $-0.07257444619$ as the next lower bound estimate for $\min f$ in the concerned domain. Continue with $M_2=-0.07257444619$: The random initial point is $(0.68565678, -0.90941219)$. Running Backtracking New Q-Newton's method for $G_{M_2}$, we get the sequence of points:\ $(-0.1515453, -0.07688046)$, $(-0.23533302, -0.11778859)$, $(-0.36993082, -0.18298619)$,\ $(-0.5502725, -0.26540423)$, $(-0.67244411, -0.30841356)$, $(-0.70462466, -0.31280385)$,\ $(-0.70678697, -0.31271293)$, $(-0.70679752, -0.31271166)$. Again, the end point here is in the domain $x+y<0$. Now, it gets fairly close to the global minimizer of $f$ in this domain. We find that the value of $f(-0.70679752, -0.31271166)-M_2$ $=$ $-0.00015342244591928789$. Therefore, we can choose $M_3=M_2-0.00015342244592$ $=$ $-0.07272786863$ as the next lower bound estimate for $\min f$ in the concerned domain. Continue with $M_3=-0.07272786863$: The random initial point is $(-0.65260407 -0.16301343)$. Running Backtracking New Q-Newton's method for $G_{M_2}$, we get the sequence of points:\ $(-0.66547817, -0.28644322)$, $(-0.70420061, -0.31143069)$, $(-0.70709159, -0.31279723)$,\ $(-0.70710672, -0.31280114)$. Again, the end point is in the domain $x+y<0$, and indeed it is very close to the global minimizer in the domain. We observe that the value of $f(-0.70710672, -0.31280114)-M_3$ is, while still $<0$, very small. Hence, we can conclude (correctly) that the end point is a global minimizer. We check this by running one more time with the new value $M_4=M_3-3.1e-08$. Running Backtracking New Q-Newton's method for the function $G_{M_4}$, from the randomly chosen initial point $(0.37597523, -0.91234048)$, we get the sequence of points:\ $(-0.44234965, -0.06819544)$, $(-0.5447063, -0.19358355)$, $(-0.66585017, -0.28453739)$,\ $(-0.70420049, -0.31125032)$, $(-0.7070916, -0.31279619)$, $(-0.70710678, -0.31280116)$. ## Example 6 In this example, we test our methods with a linear programming problem. The problem is as follows: Find $arg\min_{(x,y)\in S}f(x,y)$ where $f(x,y)=-40x-30y$, and $$S=\{(x,y)\in \mathbf{R}^2:~x+y\leq 12,~2x+y\leq 6, ~x\geq 0, ~y\geq 0 \}.$$ The global minimum is $-400$, obtained for $(x,y)=(4,8)$. Linear programming are usually difficult for iterative methods (in particular, Newton's type method), since the gradient of the cost function is a constant and the Hessian of the cost function is $0$. For this problem, we find that the method of dividing with $d((x,y),\partial S)^2$ can lead to sequences which cross $\partial S$. We find that with the method of changing the function $f$ to be a big constant on $\mathbf{R}^2\backslash S$ indeed can produce sequences which approximate the global minimizers. Below we report experiments for the new function $g(x,y)=f(x,y)$ if $(x,y)\in S$, and $g(x,y)=1000$ if $(x,y)\in \mathbf{R}^2\backslash S$. Running Backtracking New Q Newton's method for the function $g(x,y)$, with initial point $(0.1,0.1)$ we obtain the sequence of points (all in S): $(4.56428571, 3.44821429)$, $(5.68035714, 4.28526786)$, $(5.75011161, 4.33758371)$, $(5.75011161, 4.33758371)$, $(5.80072885, 4.39576044)$, $(5.80159205, 4.39583457)$, $(5.80202368, 4.3958531)$, $(5.80205066, 4.39585426)$, $(5.80206414, 4.39585484)$, $(5.80207089, 4.39585513)$, $(5.80207173, 4.39585516)$, $(5.80207215, 4.39585518)$, $(5.80207236, 4.39585519)$, $(5.80207239, 4.39585519)$. The function value at the last point is $f(5.80207239, 4.39585519)=-363.95855134613464$. Running Backtracking New Q Newton's method for the function $g(x,y)$, with initial point $(1,2)$ we obtain the sequence of points (all in S): $(3.23214286, 3.67410714)$, $(4.34821429, 4.51116071)$, $(4.34832006, 6.18830041)$, $(4.37053239, 6.88502747)$, $(4.4401263, 6.95855197)$, $(4.48607869, 6.99847415)$, $(4.48773112, 7.02416628)$, $(4.48774402 7,.02436737)$, $(4.48775047, 7.02446792)$, $(4.48775209, 7.02449305)$, $(4.48775219, 7.02449462)$, $(4.48775224, 7.02449541)$, $(4.48775224, 7.02449551)$. The function value at the last point is $f(4.48775224, 7.02449551)=-390.2449550156782$. Running Backtracking New Q Newton's method for the function $g(x,y)$, with initial point $(2,1)$ we obtain the sequence of points (all in S): $(4.23214286, 2.67410714)$, $(3.82457748, 7.44440135)$, $(4.01653468, 7.71777055)$, $(4.07413609, 7.77070227)$, $(4.09326204, 7.78987838)$, $(4.09733609, 7.79377355)$, $(4.09771509, 7.80059851)$, $(4.09873397, 7.80157261)$, $(4.09878135, 7.80242612)$, $(4.09878172, 7.80243279)$, $(4.09878191, 7.80243612)$, $(4.09878191, 7.80243618)$. The function value at the last point is $f(4.09878191, 7.80243618)=-398.0243616122982$, extremely close to the global minimum value! ## Example 7 In this example we consider a constrained problem with a quadratic cost function [@gould-toint]. The question is: Find $arg\min _{(x,y)\in S}f(x,y)$ where $f(x,y)=-2(x-0.25)^2+2(y-0.5)^2$, and $$S=\{(x,y)\in \mathbf{R}^2:~x+y\leq 1, ~6x+2y\leq 3,~x,y\geq 0\}.$$ This has a global minimum at the point $(0,0.5)$, with function value $f(0,0.5)=-0.125$. In [@farrell-etal], the tunnelling/deflation method was used to solve the associated equations coming from the Karush-Kuhn-Tucker optimality conditions. We can also use our method to solve these equations, similar to above. Here, we illustrate how to use our method directly for the cost function $f$, without going through the Karush-Kuhn-Tucker optimality conditions. We consider a new cost function $F(x,y)=f(x,y)$ if $(x,y)\in S$, and $F(x,y)=1000$ when $(x,y)\notin S$. Here are some runs. Running Backtracking New Q-Newton's method for the function $F(x,y)$, with initial point $(0.1,0.1)$, we get a sequence: $(0.025, 0.35)$, $(0.00625, 0.35)$, $(0.00117188, 0.35)$, $(0.00104637, 0.5)$, $(2.68394854e-04, 5.00000000e-01)$. The last point is very close to the global minimum in $S$. Running Backtracking New Q-Newton's method for the function $F(x,y)$, with initial point $(0.2,0.3)$, we get a sequence: $(0.15, 0.5)$, $(0.05, 0.5)$, $(0.025, 0.5)$, $(0.0109375, 0.5)$, $(0.0034668, 0.5)$, $(0.00154076, 0.5)$, $(0.00076432, 0.5)$, $(0.00076432, 0.5)$. Again, the last point is very close to the global minimum. Running Backtracking New Q-Newton's method for the function $F(x,y)$, with initial point $(0.24,0.48)$, we get a sequence: $(0.23, 0.5)$, $(0.21, 0.5)$, $(0.17, 0.5)$, $(0.09, 0.5)$, $(0.01, 0.5)$, $(0.00142857, 0.5 )$, $(4.57589286e-04, 5.00000000e-01)$. Again, the last point is very close to the global minimum. ## Example 8 In this example we find roots of a special function in a bounded domain. The question is: Find roots of the Bessel function $jv(1,z)$ inside the domain $S=\{z=x+iy:~-5\leq x,y\leq 5\}$. We use the library for special functions in Python to do computations with the Bessel function. We consider the cost function $f(x,y)=|jv(1,x+iy)|^2/2$. We define a new cost function $F(x,y)=f(x,y)$ if $(x,y)\in S$, $F(x,y)=1000$ if $(x,y)\notin F$. Here are some experiments (initial points are randomly generated inside $S$). Running Backtracking New Q-Newton's method for the function $F(x,y)$, with initial point $(3.61713097, 1.21693436)$, we get a sequence: $(3.81002318, 0.70240853)$, $(3.80130537, 0.23407724)$, $(3.82500859, 0.01478695)$, $(3.83166043e+00, 2.93087041e-05)$, $(3.83170597e+00, 3.48394778e-10)$, $(3.83170597e+00, 1.74197389e-10)$, $(3.83170597e+00, -6.20221574e-20)$. The last point is very close to a root of $jv(1,z)$ in $S$. Running Backtracking New Q-Newton's method for the function $F(x,y)$, with initial point $(0.77926808, 3.75383432)$, we get a sequence: $(0.68962951, 3.20441492)$, $(0.22533733, 2.64615371)$, $(-0.28003512, 2.08563803)$, $(0.10219435, 1.52027522)$, $(-0.0071105, 0.95589959)$, $(7.82278286e-05, 4.21454021e-01)$, $(-4.48653604e-08, 6.18471202e-02)$, $(1.34987291e-14, 2.35499690e-04)$, $(-3.02313546e-13, 1.30608358e-11)$, $(-2.10926406e-12, 4.01821256e-13)$. The last point is very close to $(0,0)$, which is a root of $jv(1,z)$ inside $S$. Running Backtracking New Q-Newton's method for the function $F(x,y)$, with initial point $(-2.1267499, -0.96193073)$, we get a sequence: $(-2.44714686, -0.43462968)$, $(-3.44602951, -0.09106988)$, $(-3.64119905, -0.05036513)$, $(-3.82515578e+00, -2.52956231e-03)$, $(-3.83168871e+00, -4.33099427e-06)$, $(-3.83170597e+00, -1.95131964e-11)$, $(-3.83170597e+00, -9.75659827e-12)$. This is close to a root of $jv(1,z)$. We have done many runs, and haven't seen a case where the sequence has a cluster point on $\partial S$. This is supported by Theorem [\[Theorem\]](#Theorem){reference-type="ref" reference="Theorem"}. In contrast, if we use the method of dividing $f$ by $d(.,\partial S)^2$, we often see the case where the constructed sequence converges to a root outside of $S$. ## Example 9 We revisit the function in Example 1 and Example 5. Consider the following constrained optimization problem: $arg\min _S f (x,y)$, where $S=\{(x,y)\in \mathbf{R}^2:~x+y\leq 0\}$ and $f(x,y)=-xye^{-x^2-y^2}+y^2/2$. We consider a new cost function $F(x,y)=f(x,y)$ if $(x,y)\in S$, $F(x,y)=1000$ if $(x,y)\notin S$. Running Backtracking New Q-Newton's method for the function $F(x,y)$, with initial point $(0.5, -0.5003)$, we get a sequence: $(-0.57530003, -0.28065)$, $(-0.64608507, -0.30173188)$,\ $(-0.67661335, -0.30806708)$,$(-0.69177324, -0.31061912)$, $(-0.70704079, -0.31289215)$,\ $(-0.70710676, -0.31280116)$. The last point is very close to the global minimum inside $S$. An interesting point here is that the point $(0.5,-0.5003)$ is in the basin of attraction of the global minimum **outside of S** of the dynamics associated to Backtracking New Q-Newton's method applied to $f(x,y)$. By a simple change of the cost function, it belongs to the basin of attraction of the basin of attraction of the global minimum **inside S** of the dynamics associated to Backtracking New Q-Newton's method applied to $F(x,y)$! The way we run experiments here is much simpler that that in Example 5, where we need to take care that the minimum of the function $f(x,y)-M$ is as close to $0$ as possible, and have to update the estimate $M$ for the global minimum of $f(x,y)$. # Conclusions In this paper we introduced two new methods to avoid a given closed set $A$. The first method is to divide the cost function $f$ by $d(x,A)^N$ for a suitable exponent $A$. This method is suitable in case one wants to avoid known points, to hop to a new component of the solution set, or to solve constrained optimization problems with no (local) minima on $A$. The second method is to change the value of $f$ on $A$ to a big constant. This method is more suitable for constrained optimization with (local) minima on the boundary. Experiments illustrate that the new methods are promising. Combination of these methods with each other, or with other methods can yield better performances. For example, our method can be used to quickly find a good initial point for Linear Programming problems, at which Linear Programming methods can be used to find exact global minima. XXXXX A. J. Bray and and D. S. Dean, *Statistics of critical points of gaussian fields on large-dimensional spaces*, Physics Review Letter, 98, 150201. G. B. Dantzig and M. N. Thapa, Linear programming 2: Theory and extensions, Springer-Verlag, 2003. P. E. Farrell, A. Birkisson and S. W. Funke, Deflation techniques for finding distinct solutions of nonlinear Partial Differential Equations, SIAM Journal on Scientific Computing, vol 37, no 4, pp A2026--A2045, 2015. P. E. Farrell, M. Croci and T. M. Surowiec, Deflation for semismooth equations, Optimization Methods and Software, 35:6, 1248--1271, 2015. N. I. M. Gould and P. L. Toint, The state-of-the-art in numerical methods for quadratic programming, 19th Biennial conference on Numerical Analysis, Dundee, Scotland, 2002. W. Karush, Minima of functions of several variables with inequalities as side constraints, M.Sc. thesis, Department of Mathematics, University of Chiago, USA, 1939. H. W. Kuhn and A. W. Tucker, Nonlinear programming, Proceedings of 2nd Berkeley Symposium, University of Berkeley, USA, 1951. J.-B. Lasserre, An introduction to polynomial and semi-algebraic optimization, Cambridge University Press, 2015. A. V. Levy and S. Gomez, The tunnelling method applied to global optimization, in Numerical Optimization 1984, P. T. Boggs, ed., SIAM, Philadelphia, 1985. M. Nouiehed, J. Lee and M. Razaviyayn, Convergence to second-order stationarity for constrained non-convex optimization, arXiv:1810.02024. T.-S. Pham, Tangencies and polynomial optimization, Mathematical Programming 199, 1239--1272, 2023. T. T. Truong, Backtracking New Q-Newton's method: a good algorithm for optimization and solving systems of equations, arXiv:2209.05378. T. T. Truong, Tat Dat To, Hang-Tuan Nguyen, Thu Hang Nguyen, Hoang Phuong Nguyen and Maged Helmy, A fast and simple modification of Newton?s method avoiding saddle points, Journal of Optimization Theory and Applications, Published online 18 July 2023. https://doi.org/10.1007/s10957-023-02270-9 T. T. Truong and H.-T. Nguyen, Backtracking gradient descent method and some applications in large scale optimisation, Part 1: Theory, Minimax Theory and its Applications, 7 (2022), no 1, 79-108. T. T. Truong and H.-T. Nguyen, Backtracking gradient descent method and some applications in large scale optimisation, Part 2: Algorithms and experiments, Applied Mathematics and Optimization, 84 (2021), no 3, 2557-2586. M. H. Wright, The interior-point revolution in optimization: History, recent developments, and lasting consequences, Bulletin of the American Mathematical Society, 42, 39--57, 2004.

arxiv_math

--- abstract: | Let $G$ be a classical group defined over a local field $F$ of characteristic zero. For any irreducible admissible representation $\pi$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean, we extend the study of spectral decomposition of local descents in [@JZ18] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field $F$. In particular, if $\pi$ has a generic local $L$-parameter, we introduce the *spectral first occurrence index* ${\mathfrak {f}}_{\mathfrak {s}}(\pi)$ and the *arithmetic first occurrence index* ${\mathfrak {f}}_{{\mathfrak {a}}}(\pi)$ of $\pi$ and prove in Theorem [\[thm:SA-foi\]](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"} that ${\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi)$. Based on the theory of consecutive descents of enhanced $L$-parameters developed in [@JLZ22], we are able to show in Theorem [\[thm:ADS\]](#thm:ADS){reference-type="ref" reference="thm:ADS"} that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result ([@JZ18 Theorem 1.7]) to the great generality. address: - School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA - School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA - School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, Zhejiang, China - Department of Mathematics, National University of Singapore, Singapore 119076 author: - Cheng Chen - Dihua Jiang - Dongwen Liu - Lei Zhang title: "Arithmetic Branching Law and Generic $L$-Packets" --- [^1] # Introduction Let $F$ be a local field of characteristic zero. Let $G_n^*$ be $F$-quasisplit classical groups and $G_n$ be an $F$-pure inner form of $G_n^*$ (Section [2.1](#ssec-CGIF){reference-type="ref" reference="ssec-CGIF"}). Let $\Pi_F(G_n)$ be the set of equivalence classes of irreducible admissible representations $\pi$ of $G_n(F)$, which are of Casselman-Wallach type if $F$ is archimedean. The local Gan-Gross-Prasad conjecture for classical groups detects certain information on the branching law for either Bessel setting or Fourier-Jacobi setting by means of certain relations of the relevant local root numbers on the arithmetic side of the local Langlands correspondence for the classical groups over $F$ ([@GGP12] and [@GGP20]). More precisely, for an integer $p_1>0$, which is even if $G_n(F)$ is symplectic or metaplectic; and is odd if $G_n$ is orthogonal, there is a twisted Jacquet module ${\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)$ of either Bessel type or Fourier-Jacobi type for $\pi\in\Pi_F(G_n)$ as defined in Section [2.3](#ssec-SFO){reference-type="ref" reference="ssec-SFO"}, which is a representation of $H_{{\mathcal {O}}_{p_1}}(F)$. The $F$-quasisplit form of $H_{{\mathcal {O}}_{p_1}}(F)$ that will be explicitly explained in [\[relevantpair\]](#relevantpair){reference-type="eqref" reference="relevantpair"} is of the form: $$\label{rp} (G_n^*, H_{\lfloor(\frak{n}-p_1)/2\rfloor}^*)= \begin{cases} ({\mathrm{SO}}_\frak{n}, {\mathrm{SO}}_{\frak{n}-p_1}), & \textrm{if }E=F\textrm{ and }\epsilon=1,\\ ({\mathrm{Sp}}_{2n}, {\mathrm{Mp}}_{2n-p_1}) \textrm{ or }({\mathrm{Mp}}_{2n}, {\mathrm{Sp}}_{2n-p_1}), & \textrm{if }E=F\textrm{ and }\epsilon=-1,\\ ({\mathrm {U}}_\frak{n}, {\mathrm {U}}_{\frak{n}-p_1}), & \textrm{if }E=F(\delta). \end{cases}$$ If $p_1={\mathfrak {n}}$, we regard $H_{{\mathcal {O}}_{p_1}}^*$ as the trivial group. If $(V,q)$ is a non-degenerate $\epsilon$-Hermitian vector space such that $G_n:={\mathrm{Isom}}(V,q)^\circ$, then we must have $p_1\leq {\mathfrak {n}}=\dim V$. Here ${\mathcal {O}}_{p_1}$ is an $F$-rational nilpotent orbit in ${\mathfrak {g}}_n(F)$, with ${\mathfrak {g}}_n$ the Lie algebra of $G_n$, which is contained in the $F$-stable nilpotent orbit ${\mathcal {O}}_{p_1}^{\mathrm{st}}$ associated with partition $[p_1,1^{{\mathfrak {n}}-p_1}]$. See Section [2.3](#ssec-SFO){reference-type="ref" reference="ssec-SFO"} for more details. The branching problem related to the local Gan-Gross-Prasad conjecture is to detect such representations $\sigma\in\Pi_F(H_{{\mathcal {O}}_{p_1}})$ that $$\label{GGP-BL} {\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat\otimes\,\sigma, \mathbbm{1})\neq 0$$ by means of the enhanced $L$-parameters of $\pi$ and $\sigma$. Here and thereafter, $\mathbbm{1}$ denotes the trivial representation. In this paper, we continue our previous work ([@JZ18] and [@JLZ22]) to obtain much refined structures for the branching problem in this setting, by using the method of descents of enhanced $L$-parameters as developed in [@JZ18] and [@JLZ22]. We denote by ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$ the set of the contragredient representations $\sigma^\vee\in \Pi_F(H_{{\mathcal {O}}_{p_1}})$ with [\[GGP-BL\]](#GGP-BL){reference-type="eqref" reference="GGP-BL"} true for $\sigma$, and call ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$ the spectrum of $\pi$ along ${\mathcal {O}}_{p_1}$ or the spectrum of the module ${\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)$. We define the spectrum of $\pi$ at $p_1$ to be $$\label{specp1} {\mathrm{Spec}}_{p_1}(\pi):=\bigcup_{{\mathcal {O}}_{p_1}\subset{\mathcal {O}}_{p_1}^{\mathrm{st}}}{\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi).$$ **Question 1** (Existence of Spectrum). *For a given $\pi\in\Pi_F(G_n)$, is the spectrum ${\mathrm{Spec}}_{p_1}(\pi)$ nonempty for some choice of integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$?* By using the Vogan version of the local Langlands correspondence (Section [3.1](#ssec-LVP){reference-type="ref" reference="ssec-LVP"}), the local Gan-Gross-Prasad conjecture (Section [3.2](#ssec-LGGP){reference-type="ref" reference="ssec-LGGP"}) and the descent of enhanced $L$-parameters (Section [3.4](#ssec-DELP){reference-type="ref" reference="ssec-DELP"}), we are able to prove that if a $\pi\in\Pi_F(G_n)$ has a generic local $L$-parameter, then there exists at least one integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$, such that the spectrum ${\mathrm{Spec}}_{p_1}(\pi)$ is not empty (Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}). As a basic question in the general branching problem (see [@Ko19]), it is interesting to ask the following question. **Question 2** (Discrete Series Spectrum). *For any $\pi\in\Pi_F(G_n)$, does there exist an integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$ such that if the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$ is not empty for some $F$-rational nilpotent orbit ${\mathcal {O}}_{p_1}$, then ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$ consists only of irreducible discrete series representations $\sigma^\vee$ of $H_{{\mathcal {O}}_{p_1}}(F)$?* In order to understand Questions [Question 1](#ques:NE){reference-type="ref" reference="ques:NE"} and [Question 2](#ques:DSS){reference-type="ref" reference="ques:DSS"}, we introduce a notion of *spectral first occurrence index*, which is denoted by ${\mathfrak {f}}_{\mathfrak {s}}(\pi)$ for any $\pi\in\Pi_F(G_n)$ and is defined by $${\mathfrak {f}}_{\mathfrak {s}}(\pi):=\max\{p_1\mid 0<p_1\leq{\mathfrak {n}}, {\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)\neq\emptyset\ {\rm for\ some}\ {\mathcal {O}}_{p_1}\subset{\mathcal {O}}_{p_1}^{\mathrm{st}}\}.$$ In this case, we call the spectrum ${\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {s}}}(\pi)$ the *first descent spectrum* of $\pi$ if ${\mathfrak {f}}_{\mathfrak {s}}={\mathfrak {f}}_{\mathfrak {s}}(\pi)$. One of the main results of this paper is the following theorem, which provides a positive answer to a stronger version of Question [Question 2](#ques:DSS){reference-type="ref" reference="ques:DSS"}. **Theorem 3**. *If $\pi\in\Pi_F(G_n)$ has a generic $L$-parameter, then the first descent spectrum ${\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {s}}}(\pi)$ consists only of discrete series representations, which are representations of $H_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}}}(F)$ for some $F$-raitonal nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}}\subset{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}}^{\mathrm{st}}$, where ${\mathfrak {f}}_{\mathfrak {s}}={\mathfrak {f}}_{\mathfrak {s}}(\pi)$ is the spectral first occurrence index of $\pi$.* In order to prove Theorem [Theorem 3](#thm:DSS){reference-type="ref" reference="thm:DSS"}, we use the Vogan version of the local Langlands correspondence to obtain a unique enhanced $L$-parameter $(\varphi,\mu)$ for each $\pi\in\Pi_F(G_n)$. Here $\varphi$ is a generic $L$-parameter of an $F$-quasisplit classical group $G_n^*$, of which $G_n$ is an $F$-pure inner form, and is $G_n$-relevant, so that $\pi\in\Pi_\varphi(G_n)$, the local $L$-packet associated with $\varphi$. And $\mu$ is a character of the component group ${\mathcal {S}}_\varphi$ of $\varphi$ determined by $\pi$ and $G_n$. We refer to Section [3.1](#ssec-LVP){reference-type="ref" reference="ssec-LVP"} for the definition of ${\mathcal {S}}_\varphi$. Finally, the uniqueness of the correspondence between $\pi$ and $(\varphi,\mu)$ depends on a choice of the Whittaker datum that is determined by a number $a\in F^\times$ (or $a\in{\mathcal {Z}}$ as in [\[CZ\]](#CZ){reference-type="eqref" reference="CZ"}, more precisely). In this way, we write $$\pi=\pi_a(\varphi,\mu)\ {\rm and}\ \iota_a(\pi)=(\varphi,\mu)$$ which realizes a unique one-to-one correspondence between the local Vogan packet associated with $\varphi$, which is denoted by $\Pi_\varphi[G_n^*]$, and the character group (the Pontryagin dual) $\widehat{{\mathcal {S}}_\varphi}$ of the component group ${\mathcal {S}}_\varphi$ associated with $\varphi$. Here the local Vogan packet $\Pi_\varphi[G_n^*]$ is defined to be $$\Pi_\varphi[G_n^*]:=\bigcup_{G_n}\Pi_\varphi(G_n)$$ where $G_n$ runs over all $F$-pure inner forms of the $F$-quasisplit group $G_n^*$. The local $L$-packet $\Pi_\varphi(G_n)$ is defined to be empty if $\varphi$ is not $G_n$-relevant. The method of descents of enhanced $L$-parameters $(\varphi,\mu)$ was first studied in [@JZ18] and is further developed in [@JLZ22]. This method defines the descent of an enhanced $L$-parameter $(\varphi,\mu)$ and the notion of *first occurrence index* of $(\varphi,\mu)$, which is denoted by ${\mathfrak {l}}_0(\varphi,\mu)$. The descent of $(\varphi,\mu)$ at the first occurrence index ${\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi,\mu)$ is denoted by ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi,\mu)$ as in Definition [Definition 9](#defn:pd){reference-type="ref" reference="defn:pd"}. We call ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi,\mu)$ the *first descent* of $(\varphi,\mu)$, which by definition consists of some enhanced $L$-parameters of the $F$-quasisplit $H_{\lfloor(\frak{n}-{\mathfrak {l}}_0)/2\rfloor}^*(F)$ (as in [\[rp\]](#rp){reference-type="eqref" reference="rp"} with $p_1={\mathfrak {l}}_0$). Theorem [Theorem 12](#thm:DFD){reference-type="ref" reference="thm:DFD"}, which is [@JLZ22 Theorem 5.3], asserts that the first descent ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi,\mu)$ of $(\varphi,\mu)$ consists only such enhanced $L$-parameters $(\phi,\nu)$ of $H_{\lfloor(\frak{n}-{\mathfrak {l}}_0)/2\rfloor}^*(F)$ that $\phi$ are discrete $L$-parameters of $H_{\lfloor(\frak{n}-{\mathfrak {l}}_0)/2\rfloor}^*(F)$. This motivates the following theorem, which is one of the basic properties that relate the descent on the representation side (or the spectral side) and that on the enhanced $L$-parameter side (or arithmetic side). **Theorem 4** (Spectral-Arithmetic). *Given any generic $L$-parameter $\varphi$ of $G_n^*(F)$, for any $\pi\in\Pi_F[G_n^*]$, if $$\pi=\pi_a(\varphi,\mu)$$ holds for some $\mu\in\widehat{{\mathcal {S}}_\varphi}$, then the spectral first occurrence index ${\mathfrak {f}}_{\mathfrak {s}}(\pi)$ is equal to the arithmetic first occurrence index ${\mathfrak {f}}_{\mathfrak {a}}(\pi)$, i.e. $${\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{\mathfrak {a}}(\pi),$$ where ${\mathfrak {f}}_{\mathfrak {a}}(\pi):={\mathfrak {l}}_0(\varphi,\mu)$, the first occurrence index of $(\varphi,\mu)$.* We are going to prove Theorem [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"} in Section [4.3](#sec-PC-SAFOI){reference-type="ref" reference="sec-PC-SAFOI"}. Combining Theorem [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"} with Theorem [Theorem 15](#thm:AFD){reference-type="ref" reference="thm:AFD"}, we obtain the following discreteness of the first descent spectrum, which is clearly a refinement of Theorem [Theorem 3](#thm:DSS){reference-type="ref" reference="thm:DSS"}. **Theorem 5** (Discreteness of the First Descent Spectrum). *Given any generic $L$-parameter $\varphi$ of $G_n^*(F)$, for any $\pi\in\Pi_F[G_n^*]$, if $$\pi=\pi_a(\varphi,\mu)$$ holds for some $\mu\in\widehat{{\mathcal {S}}_\varphi}$, then the first descent spectrum ${\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {s}}}(\pi)$ consists exactly of the discrete series representations whose enhanced $L$-parameters belong to the first descent ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi,\mu)$ of $(\varphi,\mu)$.* As a consequence, we are able to establish the following submodule theorem, whose proof is given in Section [4.4](#ssec-ST){reference-type="ref" reference="ssec-ST"}. **Theorem 6** (Submodule). *For any given $\pi\in\Pi_F(G_n)$ with generic $L$-parameter, there exists an integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$, such that the representation $\pi$ can be embedded as an irreducible submodule into the following induced representation $${\mathrm{Ind}}^{G_n(F)}_{R_{{\mathcal {O}}_{p_1}}(F)}(\sigma^\vee\otimes\theta_{{\mathcal {O}}_{p_1}}),$$ where $\sigma$ is an irreducible discrete series representation of $H_{{\mathcal {O}}_{p_1}}(F)$, and $(R_{{\mathcal {O}}_{p_1}}, \theta_{{\mathcal {O}}_{p_1}})$ is given by [\[eq:submodule\]](#eq:submodule){reference-type="eqref" reference="eq:submodule"}.* Theorem [Theorem 5](#thm:ADS){reference-type="ref" reference="thm:ADS"} is a main result of this paper and is the extension of the main result ([@JZ18 Theorem 1.7]) to all classical groups (considered in the paper) over all local fields of characteristic zero. We would like to point out that the method used in this paper to prove Theorem [Theorem 5](#thm:ADS){reference-type="ref" reference="thm:ADS"} is different from that used in the proof of [@JZ18 Theorem 1.7]. Just as in [@JZ18] and in the local Gan-Gross-Prasad conjecture, our results determine the structure of the representations occurring in the first descent spectrum ${\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {s}}}(\pi)$, without full information about the twisted Jacquet modules ${\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)$, which is expected to be a difficult problem in the understanding of the branching problem for infinite-dimensional representations. The paper is organized as follows. Section [2](#sec-CGWF){reference-type="ref" reference="sec-CGWF"} studies the local descent theory for representation data. Section [2.1](#ssec-CGIF){reference-type="ref" reference="ssec-CGIF"} recalls the classical groups and their pure inner forms for the setting of the local Gan-Gross-Prasad conjecture from [@GGP12; @JZ18; @JZ20; @JLZ22]. Section [2.2](#ssec-TJM-p1){reference-type="ref" reference="ssec-TJM-p1"} recalls from [@JLZ22] the definition of twisted Jacquet modules related to the local descent theory. Based on the non-vanishing in Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}, we are able to define the notion of the *spectral first occurrence index* of $\pi$. In Section [2.4](#ssec-SPI){reference-type="ref" reference="ssec-SPI"}, we discuss the relation between local descent and parabolic induction and formulate the most technical result of the paper in Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"}. Section [3](#sec-AD){reference-type="ref" reference="sec-AD"} considers the local descent theory for arithmetic data. Section [3.1](#ssec-LVP){reference-type="ref" reference="ssec-LVP"} recalls from [@JLZ22] the basic structure of enhanced local $L$-parameters involved in this paper and basic facts related to the local Vogan packets. Section [3.2](#ssec-LGGP){reference-type="ref" reference="ssec-LGGP"} recalls from [@GGP12] the local Gan-Gross-Prasad conjecture for all classical groups and from [@JLZ22] the relevant explicit arithmetic data associated with the local Gan-Gross-Prasad conjecture. The relevant twisted distinguished characters are the most important ingredient in the local Gan-Gross-Prasad conjecture and are recalled in Section [3.3](#ssec-DC){reference-type="ref" reference="ssec-DC"}. The descent of the enhanced local $L$-parameters that was first considered in [@JZ18] for special orthogonal groups over $p$-adic local fields and was developed in [@JLZ22] for all classical groups over all local fields of characteristic zero is recalled in Section [3.4](#ssec-DELP){reference-type="ref" reference="ssec-DELP"}. Based on the tower property (Proposition [Proposition 11](#prop:TP){reference-type="ref" reference="prop:TP"}), we are able to define the notion of the *arithmetic first index* of $\pi$. The main result in the local descent theory of arithmetic data is Theorem [Theorem 12](#thm:DFD){reference-type="ref" reference="thm:DFD"}. which was proved in [@JLZ22]. Section [3.5](#ssec-PSNE){reference-type="ref" reference="ssec-PSNE"} proves Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}. Section [4](#sec-SDAFD){reference-type="ref" reference="sec-SDAFD"} proves the main results of the paper. Section [4.1](#ssec-afd){reference-type="ref" reference="ssec-afd"} establishes the discreteness of the arithmetic first descent (Theorem [Theorem 15](#thm:AFD){reference-type="ref" reference="thm:AFD"}). Combined with Theorem [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"}, which is proved in Section [4.3](#sec-PC-SAFOI){reference-type="ref" reference="sec-PC-SAFOI"}, we establish Theorem [Theorem 5](#thm:ADS){reference-type="ref" reference="thm:ADS"}. Finally, Section [4.4](#ssec-ST){reference-type="ref" reference="ssec-ST"} proves Theorem [Theorem 6](#thm:SM){reference-type="ref" reference="thm:SM"}. The last section (Section [5](#sec-PPI){reference-type="ref" reference="sec-PPI"}) is devoted to the proof of Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"}. In fact, we prove a stronger result, which is Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"}. According to [@MW12; @GI16; @Ch23], it remains to prove Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} in the Fourier-Jacobi cases when $F$ is non-archimedean, which is done in Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}, following the method of [@MW12]. # Classical Groups and Algebraic Wavefront Sets {#sec-CGWF} ## Classical groups and their pure inner forms {#ssec-CGIF} The classical groups considered in this paper include the unitary groups ${\mathrm {U}}_{{\mathfrak {n}}}$, special orthogonal groups ${\mathrm{SO}}_{{\mathfrak {n}}}$, symplectic groups ${\mathrm{Sp}}_{2n}$ and metaplectic groups ${\mathrm{Mp}}_{2n}$, following the notations in [@JLZ22] and also in [@JZ18] and [@JZ20], which are compatible with those in [@GGP12]. Let $F$ be a local field of characteristic zero. As in [@JLZ22], we assume that $F\neq {\mathbb {C}}$. Thus if $F$ is archimedean, we mean that $F$ is real; otherwise, we mean that $F$ is $p$-adic, which is a finite extension of the field ${\mathbb {Q}}_p$ of $p$-adic numbers for some prime $p$. Let $F(\delta)$ be a quadratic field extension of $F$, with $\delta=\sqrt{d}$ for a non-square $d\in F^\times$. Let $E$ be either $F$ or $F(\delta)$. Denote by ${\mathfrak {c}}\colon x\mapsto \bar{x}$ [\[pg:iota\]]{#pg:iota label="pg:iota"} the unique nontrivial element in the Galois group ${\mathrm{Gal}}(E/F)$ if $E\ne F$; and ${\mathfrak {c}}={\rm id}_E$, $\bar{x}=x$ if $E=F$. Let $(V,q_V)$ be an ${\mathfrak {n}}$-dimensional vector space $V$ over $E$, equipped with a non-degenerate $\epsilon$-Hermitian form $q=q_V$, where $\epsilon=\pm 1$. More precisely, if $E=F$, $q$ is symmetric or symplectic; and if $E=F(\delta)\neq F$, $q$ is Hermitian or skew-Hermitian. Write ${\epsilon}_q=-1$ if $q$ is symplectic or skew-Hermitian; otherwise ${\epsilon}_q=1$. Denote by $G_n={\mathrm{Isom}}(V,q)^\circ$ (or ${\mathrm{Mp}}_{2n}(F)$ in the metaplectic case) the identity connected component of the isometry group of the space $(V,q)$, with $n=\lfloor\frac{{\mathfrak {n}}}{2}\rfloor$. Let ${\mathfrak {r}}$ be the Witt index of $(V,q)$ and $(V_{{\mathrm{an}}},q)$ be the $F$-anisotropic kernel of $(V,q)$ with dimension ${\mathfrak {d}}_0$. Then the $F$-rank of $G_n$ is ${\mathfrak {r}}$ and ${\mathfrak {d}}_0={\mathfrak {n}}-2{\mathfrak {r}}$. As in [@JLZ22 Section 2], the following $$V=V^{+}\oplus V_{{\mathrm{an}}}\oplus V^{-}$$ is a polar decomposition for $V$, where $V^{\pm}$ are maximal totally isotropic subspaces and dual to each other. Take a basis $\{e_{\pm 1},\dots,e_{\pm {\mathfrak {r}}}\}$ for $V^{\pm}$ such that $$q(e_{i},e_{-j})=\delta_{i,j}$$ for all $1\leq i,j\leq{\mathfrak {r}}$. Choose an orthogonal basis $\{e'_{1},\dots,e'_{{\mathfrak {d}}_0}\}$ of $V_{{\mathrm{an}}}$ and put $$d_i=q(e'_{i},e'_{i})\in F^\times \text{ or }\delta\cdot F^\times, \text{ for }1\leq i\leq {\mathfrak {d}}_0.$$ Note that if $(V,q)$ is a Hermitian space, then $(V,\delta \cdot q)$ is skew-Hermitian and ${\mathrm{Isom}}(V,q)={\mathrm{Isom}}(v,\delta\cdot q)$ gives us the identical unitary groups. However, for the induction purpose, we will consider both cases as in [@JLZ22]. If $(V,q)$ is symplectic, then ${\mathfrak {n}}=2n=2{\mathfrak {r}}$ and $V_{{\mathrm{an}}}=\{0\}$. We put the above bases together in the following order to form a basis of $(V,q)$ $$\label{eq:basis} {\mathfrak {B}}\colon\quad e_{1},\dots,e_{{\mathfrak {r}}},e'_{1},\dots,e'_{{\mathfrak {d}}_0},e_{-{\mathfrak {r}}},\dots,e_{-1},$$ and fix the following full isotropic flag in $(V,q)$: $${\mathrm{Span}}\{e_{1}\}\subset{\mathrm{Span}}\{e_{1},e_{2}\}\subset \cdots\subset {\mathrm{Span}}\{e_{1},\dots,e_{{\mathfrak {r}}}\},$$ which defines a minimal parabolic $F$-subgroup $P_0$ of $G_n$. With respect to the order of the basis in [\[eq:basis\]](#eq:basis){reference-type="eqref" reference="eq:basis"}, the group $G_n$ is also defined with respect to the following matrix: $$\label{eq:J} J_{{\mathfrak {r}}}^{\mathfrak {n}}=\begin{pmatrix} &&w_{\mathfrak {r}}\\&J_{0}^{{\mathfrak {d}}_0}&\\ \epsilon_q w_{{\mathfrak {r}}}&& \end{pmatrix}_{{\mathfrak {n}}\times{\mathfrak {n}}} \text{ where } w_{\mathfrak {r}}=\begin{pmatrix} &w_{{\mathfrak {r}}-1}\\1& \end{pmatrix}_{{\mathfrak {r}}\times {\mathfrak {r}}},$$ which is defined inductively, and $J_{0}^{{\mathfrak {d}}_0}={\rm diag}\{d_{1},\dots,d_{{\mathfrak {d}}_0}\}$. Moreover, the Lie algebra ${\mathfrak {g}}_n$ of $G_n$ is defined by $${\mathfrak {g}}_n=\{A\in {\rm {End}}_E(V) \mid \bar{A}^{t}J_{{\mathfrak {r}}}^{\mathfrak {n}}+ J_{{\mathfrak {r}}}^{\mathfrak {n}}A=0\},$$ with the Lie bracket $[A,B]=AB-BA$. For simplicity, set $A^*=(J_{{\mathfrak {r}}}^{\mathfrak {n}})^{-1} \bar{A}^{t} J_{{\mathfrak {r}}}^{\mathfrak {n}}$. As in [@JLZ22 Section 2.1], we take the following ${\mathrm{Ad}}(G_n)$-invariant non-degenerate $F$-bilinear form $\kappa$ on ${\mathfrak {g}}_n\times {\mathfrak {g}}_n$: $$\kappa(A,B)={\mathrm{tr}}(AB^*)/2={\mathrm{tr}}(A^*B)/2.$$ Alternatively, $\kappa$ can be computed by $$\label{eq:kappa} 2\kappa(A,B)= \sum_{i=1}^{{\mathfrak {r}}}\left\langle {A(e_{i}),B(e_{-i})} \right\rangle+\epsilon_q \left\langle {A(e_{-i}),B(e_{i})} \right\rangle+\sum_{j=1}^{{\mathfrak {d}}_0}\frac{\left\langle {A(e'_{j}),B(e'_{j})} \right\rangle}{\left\langle {e'_{j},e'_{j}} \right\rangle}.$$ Note that this ${\mathrm{Ad}}(G_n)$-invariant symmetric bilinear form is the same as the one in [@GZ14] and is proportional to the Killing form. For an ${\epsilon}$-Hermitian space $V$ of dimension ${\mathfrak {n}}$, we define its discriminant by $${\mathrm{disc}}(V)=(-1)^{\frac{{\mathfrak {n}}({\mathfrak {n}}-1)}{2}}\det(V) \in \left\{ \begin{array}{ll} \delta^{\mathfrak {n}}\cdot F^\times /{\mathbb {N}}E^\times, & \textrm{if }E=F(\delta)\textrm{ and }\epsilon = -1, \\ F^\times / {\mathbb {N}}E^\times, & \textrm{otherwise,} \end{array}\right.$$ where ${\mathbb {N}}E^\times:=\{x\bar{x} \mid x\in E^\times\}$, unless $E=F$ and $V$ is a symplectic space in which case this definition is not applicable. ## Twisted Jacquet modules associated with $[p_1,1^{{\mathfrak {n}}-p_1}]$ {#ssec-TJM-p1} We review here the discussion in [@JLZ22 Section 4.2], which specializes to the partition of type $[p_1,1^{{\mathfrak {n}}-p_1}]$. Denote by ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ the unique $F$-stable nilpotent orbit associated with the partition $[p_1,1^{{\mathfrak {n}}-p_1}]$. To parameterize the $F$-rational nilpotent orbits in ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$, we need to assign an ${\epsilon}_1$-Hermitian form $(V^h_{(p_1)},q^h_{(p_1)})$ and an ${\epsilon}$-Hermitian form $(V_{(1)},q_{(1)})$ where ${\epsilon}_1=(-1)^{p_1-1}{\epsilon}$ and $\dim V^h_{(p_1)}=1$. By definition, $V_{(p_1)}$ has the polar decomposition $V_{(p_1)}^+ \oplus {\mathfrak {e}}\oplus V_{(p_1)}^-$, where ${\mathfrak {e}}$ is an anisotropic line if $p_1$ is odd, and is zero otherwise, and $V_{(p_1)}^+$ and $V_{(p_1)}^-$ are dual to each other. In particular when $p_1$ is odd, $q_{(p_1)}$ can be determined by $q_{(p_1)}\vert_{{\mathfrak {e}}}=q^h_{(p_1)}$. The constructed Young tableaux for $(V,q)$ with the partition $[p_1,1^{{\mathfrak {n}}-p_1}]$ is admissible if and only if $$\label{eq:admissible-Young-p1} (V,q)\cong (V_{(p_1)}^+\oplus {\mathfrak {e}}\oplus V_{(p_1)}^-,q_{(p_1)})\oplus (V_{(1)},q_{(1)}).$$ We refer to [@GZ14 Section 3.1] and [@JLZ22 Section 2.2] for the detailed discussion in general situation. From [@JLZ22 Section 4.2], in order for ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ to be defined over $F$ (i.e., ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}\cap {\mathcal {N}}_F({\mathfrak {g}}_n)\ne \varnothing$), the following conditions must be satisfied: - If $V$ is orthogonal and ${\mathfrak {n}}\ne 2{\mathfrak {r}}$, then $p_1$ is odd, and ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ is defined over $F$ if and only if $p_1\leq 2{\mathfrak {r}}+1$. - If $V$ is orthogonal and ${\mathfrak {n}}=2{\mathfrak {r}}$, then $p_1$ is odd, and ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ is defined over $F$ if and only if $p_1\leq 2{\mathfrak {r}}-1$. - If $V$ is symplectic, then $p_1$ is even, and ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ is defined over $F$ for all $p_1\leq {\mathfrak {n}}$. - If $G_n$ is unitary, then there is no constrain on $p_1$, and ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ is defined over $F$ if and only if $p_1\leq 2{\mathfrak {r}}+1$. From [@JLZ22 Section 4.2], there is an explicit choice of ${\frak {sl}}_2$-triple $\{X,\hbar, Y\}$ for each $F$-rational orbit in ${\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ with $X\in {\mathcal {O}}^{{\mathrm{st}}}_{p_1}$. With the basis given in [\[eq:basis\]](#eq:basis){reference-type="eqref" reference="eq:basis"}, the following is a choice of $\hbar\in {\mathfrak {g}}_n$, for $x\in {\mathfrak {B}}$, $$\hbar(x)= \begin{cases} \mp(p_1-2i+1)e_{\pm i} &\text{ if $x=e_{\pm i}$ and } 1\leq i\leq \lfloor\frac{p_1}{2}\rfloor,\\ 0 &\text{ otherwise,} \end{cases}$$ which is uniform for all $F$-rational orbits. With such a chosen $\hbar$, we obtain the following eigenspaces $$\begin{aligned} \label{eigensp} V_j=\{v\in V\ \mid \hbar(v)=jv\}\end{aligned}$$ for $j\in{\mathbb {Z}}$. For $v,w\in V$, define the element $A_{v,w}$ in ${\mathfrak {g}}_n\subset{\mathrm{End}}(V)$ by $$A_{v,w}(x)= {\epsilon}q(x,v)w- q(x,w)v.$$ Throughout this paper, we choose and fix a vector $e$ in $V$ based on the parity of $p_1$: - if $p_1$ is odd, we take $e$ to be an anisotropic vector in $V_0$; - if $p_1$ is even, we take $e=\frac{{\epsilon}}{2}e_{-m}$, where $m=\lfloor\frac{p_1}{2}\rfloor$. Note that if $p_1$ is odd, then $V$ is not symplectic and there always exists anisotropic vectors if $V_{0}$ is not zero. With the fixed vector $e$, [@JLZ22 (4.4)] defines $$\label{xe,var} X_{e,\varsigma}=A_{ e, \bar{\varsigma} e_{-m}}+\sum_{i=1}^{m-1}A_{ e_{i+1}, \bar{\varsigma} e_{-i}}.$$ for $\varsigma\in E^\times$, which implies that $X_{e,\varsigma}(e_m)= \bar{\varsigma} e_{-m}$. After completing $X_{e,\varsigma}$ and $\hbar$ to an ${\frak {sl}}_2$-triple in ${\mathfrak {g}}_n$, we have that $$V_{(p_1)}= {\rm {Span}}\{e_1,e_2,\dots,e_{m}\}\oplus Ee \oplus {\rm {Span}}\{e_{-m},e_{-(m-1)},\dots,e_{-1}\}$$ and $V_{(1)}=V_{(1)}^h=V_{0}\cap e^\perp$. Note that $V_{(p_1)}^h=Ee_{-1}$ and $$\label{eq:X-highest-weight} X^{p_1-1}(e_{1})=\begin{cases} (-\varsigma\bar{\varsigma})^{m-1}\bar{\varsigma} e_{-1} &\text{ if $p_1$ is even,}\\ {\epsilon}\cdot(-\varsigma\bar{\varsigma})^m\left\langle {e,e} \right\rangle e_{-1} &\text{ if $p_1$ is odd.}\\ \end{cases}$$ From [\[eq:X-highest-weight\]](#eq:X-highest-weight){reference-type="eqref" reference="eq:X-highest-weight"}, we deduce that $q^{h}_{(p_1)}(e_{-1},e_{-1})$ is equivalent to the one-dimensional $(-1)^{p_1-1}{\epsilon}$-Hermitian form defined by $$\label{q'-form} q'_{e,\varsigma}(e_{-1},e_{-1}):= \begin{cases} (-1)^{m-1}\varsigma &\text{ if $p_1$ is even,}\\ (-1)^m \left\langle {e,e} \right\rangle &\text{ if $p_1$ is odd.}\\ \end{cases}$$ Denote the resulting $\epsilon$-Hermitian form on $V_{(p_1)}\simeq Ee_{-1}\otimes F^{p_1}$ by $$\label{x-form} q_{(p_1), e, \varsigma}:=q'_{e,\varsigma}\otimes q_{p_1}.$$ Therefore, we obtain the admissible Young tableau associated to $X_{e,\varsigma}$: $$\label{eq:X-p-1} ([p_1,1^{{\mathfrak {n}}-p_1}],\{(Ee_{-1},q'_{e,\varsigma}),(V_{0}\cap e^{\perp},q_V\vert_{V_{0}\cap e^{\perp}})\}).$$ We refer to [@JLZ22 Section 2.2] for the details in the general situation. The twisted Jacquet module associated to the partition of type $[p_1,1^{{\mathfrak {n}}-p_1}]$ splits into two types: the Bessel type and the Fourier-Jacobi type, according to $p_1$ is odd and even, respectively. Note that when $p_1=1$ the corresponding stable nilpotent orbit is the zero nilpotent orbit. However, the twisted Jacquet modules of the Bessel type and Fourier-Jacobi type for the classical groups can still be defined for $p_1=1$ ([@GGP12]). For $X_{e,\varsigma}$ as in [\[xe,var\]](#xe,var){reference-type="eqref" reference="xe,var"}, following [@JLZ22 (4.8)], we define $$\label{psix} \psi_{X_{e,\varsigma},2}(\exp (u))=\psi_0(\kappa(u, X_{e,\varsigma})),\quad u\in {\mathfrak {u}}_{X_{e,\varsigma}, 2}:=\bigoplus_{i\leq -2}{\mathfrak {g}}_{n, i}^{\hbar},$$ where ${\mathfrak {g}}_{n, i}^{\hbar}$ denotes the $i$-eigenspace of ${\rm ad}(\hbar)$ acting on ${\mathfrak {g}}_n$. Following [@JLZ22 Lemma 4.2] for the explicit computation of $\kappa(u, X_{e,\varsigma})$, we obtain that $$\label{Mx} M_{X_{e,\varsigma}}={\mathrm{Isom}}(V_{(1)}, q),$$ where $V_{(1)}=V_0\cap e^\perp$. Define $${\mathfrak {u}}_{X_{e,\varsigma}}:=\bigoplus_{i\leq -1}{\mathfrak {g}}_{n,i}^{\hbar},\quad U_{X_{e,\varsigma}}:=\exp({\mathfrak {u}}_{X_{e,\varsigma}}),$$ and we specify a representation $\omega_{\psi_{X_{e,\varsigma}}}$ of $U_{X_{e,\varsigma}}$ below. The *twisted Jacquet modules of Bessel type* are given as follows. If $\epsilon=1$ and $p_1$ is odd, then ${\mathfrak {u}}_{X_{e,\varsigma}}={\mathfrak {u}}_{X_{e,\varsigma},2}$ and we let $\omega_{\psi_{X_{e,\varsigma}}}=\psi_{X_{e,\varsigma},2}$. For a smooth representation $(\pi, V_\pi)$ of $G_n(F)$, we define its twisted Jacquet module with respect to $(U_{X_{e,\varsigma}}, \psi_{X_{e, \varsigma}})$ $$\label{BF-coe} {\mathcal {J}}_{X_{e, \varsigma}}(\pi)=V_\pi /\overline{{\mathrm{Span}}\{\pi(u)v-\psi_{X_{e, \varsigma},2}(u)v \mid u\in U_{X_{e, \varsigma}}, v\in V_\pi\}},$$ as a smooth module of $M^\circ_{X_{e,\varsigma}}$. The *twisted Jacquet modules of Fourier-Jacobi type* are given as follows. If $\epsilon=-1$ and $p_1$ is even, then we have the Heisenberg-oscillator representation, denoted by $(\omega_{\psi_{X_{e,\varsigma}}}, V_{\psi_{X_{e, \varsigma}}})$, of $\widetilde{M}_{X_{e,\varsigma}}\ltimes U_{X_{e,\varsigma}}$. For a smooth representation $(\pi, V_\pi)$ of $G_n(F)$ (or $\widetilde{G}_n(F)$), we define the $U_{X_{e, \varsigma}}$-coinvariants $$\label{FJ-coe} {\mathcal {J}}_{X_{e, \varsigma}}(\pi)=V_\pi\,\widehat{\otimes}\, V^\vee_{\psi_{X_{e,\varsigma}}}/\overline{{\mathrm{Span}}\{\pi\otimes \omega^\vee_{\psi_{X_{e,\varsigma}}}(u)v-v \mid u\in U_{X_{e, \varsigma}}, v\in V_\pi \,\widehat{\otimes}\, V^\vee_{\psi_{X_{e,\varsigma}}}\}},$$ as a smooth module of $\widetilde{M}_{X_{e,\varsigma}}$ (or $M_{X_{e,\varsigma}}$). In the skew-Hermitian case, choose a splitting character $\xi: E^\times/{\mathbb {N}}E^\times \to {\mathbb {C}}^\times$ whose restriction to $F^\times$ is the quadratic character $\omega_{E/F}$. Then we may and do remove the double cover and obtain the Heisenberg-oscillator representation $(\omega_{\psi_{X_{e,\varsigma}},\xi},V_{\psi_{X_{e,\varsigma}},\xi})$ of the unitary group $M_{X_{e,\varsigma}}$ in the above. ## Spectral first occurrence {#ssec-SFO} From the construction of the twisted Jacquet modules associated with the partition $[p_1,1^{{\mathfrak {n}}-p_1}]$, the equivalence class of the module ${\mathcal {J}}_{X_{e, \varsigma}}(\pi)$ depends only on the $F$-rational nilpotent orbit ${\mathcal {O}}_{e, \varsigma}\subset{\mathcal {O}}_{p_1}^{\mathrm{st}}$. We may write ${\mathcal {O}}_{p_1}={\mathcal {O}}_{e, \varsigma}$ for some pair $(e, \varsigma)$, and write $$\label{LD-module} {\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)={\mathcal {J}}_{X_{e, \varsigma}}(\pi).$$ To uniform the notations, we write $H_{{\mathcal {O}}_{p_1}}$ for either $M_{X_{e,\varsigma}}$ or $\widetilde{M}_{X_{e,\varsigma}}$, whose $F$-quasisplit forms are given as follows: $$\label{relevantpair} (G_n^*, H_{\lfloor(\frak{n}-p_1)/2\rfloor}^*)= \begin{cases} ({\mathrm{SO}}_\frak{n}, {\mathrm{SO}}_{\frak{n}-p_1}), & \textrm{if }E=F\textrm{ and }\epsilon=1,\\ ({\mathrm{Sp}}_{2n}, {\mathrm{Mp}}_{2n-p_1}) \textrm{ or }({\mathrm{Mp}}_{2n}, {\mathrm{Sp}}_{2n-p_1}), & \textrm{if }E=F\textrm{ and }\epsilon=-1,\\ ({\mathrm {U}}_\frak{n}, {\mathrm {U}}_{\frak{n}-p_1}), & \textrm{if }E=F(\delta). \end{cases}$$ If $p_1={\mathfrak {n}}$, we regard $H_{{\mathcal {O}}_{p_1}}^*$ as the trivial group. We also write $$\label{eq:submodule} (R_{{\mathcal {O}}_{p_1}}, \theta_{{\mathcal {O}}_{p_1}}):= (H_{{\mathcal {O}}_{p_1}}\ltimes U_{X_{e,\varsigma}}, \omega_{\psi_{X_{e,\varsigma}}}).$$ For any $\pi\in\Pi_F(G_n)$, we consider the spectral structure of the module ${\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)$ as representation of $H_{{\mathcal {O}}_{p_1}}(F)$. For any $\sigma\in\Pi_F(H_{{\mathcal {O}}_{p_1}})$, we say that $\sigma^\vee$ *occurs* in ${\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)$ if $$\label{spec} {\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat\otimes\,\sigma, \mathbbm{1})\neq 0.$$ Denote by ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$ the *spectrum* of the module ${\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)$, which is a set consisting of all $\sigma^\vee\in\Pi_F(H_{{\mathcal {O}}_{p_1}})$ that occur in ${\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)$. **Proposition 7**. *If a representation $\pi\in\Pi_F(G_n)$ has a generic $L$-parameter, then there exists at least one integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$, such that the set ${\mathrm{Spec}}_{p_1}(\pi)$ that is the spectrum of $\pi$ at $p_1$ as defined in [\[specp1\]](#specp1){reference-type="eqref" reference="specp1"} is not empty.* The proof needs preparation of arithmetic data associated with $\pi\in\Pi_F(G_n)$ and will be given in Section [3.5](#ssec-PSNE){reference-type="ref" reference="ssec-PSNE"}. Assume that for a $\pi\in\Pi_F(G_n)$, there exists an integer $p_1$ with $0<p_1\leq {\mathfrak {n}}$ such that the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$ is non-empty. We define the *spectral first occurrence index* ${\mathfrak {f}}_{\mathfrak {s}}={\mathfrak {f}}_{\mathfrak {s}}(\pi)$ for such a $\pi$ to be $${\mathfrak {f}}_{\mathfrak {s}}={\mathfrak {f}}_{\mathfrak {s}}(\pi)=\max\{p_1\mid 0<p_1\leq{\mathfrak {n}},\ {\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)\neq \emptyset\}.$$ If the spectral first occurrence index ${\mathfrak {f}}_{\mathfrak {s}}={\mathfrak {f}}_{\mathfrak {s}}(\pi)$ exists, we call the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}}}(\pi)$ the *spectral first descent* of $\pi$ (along the orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}}$). We are going to understand the structure of the spectral first descent ${\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}}}(\pi)$ of $\pi$ by the descent of enhanced $L$-parameters as developed in [@JZ18] and [@JLZ22]. ## Spectrum and parabolic induction {#ssec-SPI} Let $p_1$ be an integer with $0<p_1\leq{\mathfrak {n}}$. We take $\sigma\in\Pi_F(H_{\lfloor(\frak{n}-p_1)/2\rfloor})$, where $H_{\lfloor(\frak{n}-p_1)/2\rfloor}$ is an $F$-pure inner form of $H_{\lfloor(\frak{n}-p_1)/2\rfloor}^*$. When $H_{\lfloor(\frak{n}-p_1)/2\rfloor}$ is not metaplectic, write $\sigma$ as the Langlands quotient ${\mathcal {L}}(\underline{s},\tau_1,\dots,\tau_t,\sigma_0)$ of the following standard module of $H_{\lfloor(\frak{n}-p_1)/2\rfloor}(F)$ $$\label{SM1} {\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0):= {\mathrm{Ind}}^{H_{\lfloor(\frak{n}-p_1)/2\rfloor}(F)}_{P(F)} (|\det|^{s_1}\tau_1\otimes\cdots\otimes|\det|^{s_t}\tau_t\otimes\sigma_0)$$ where the Levi part of the standard parabolic subgroup $P$ is isomorphic to $${\mathrm{GL}}_{n_1}\times\cdots\times{\mathrm{GL}}_{n_t}\times H_{\lfloor(\frak{n}-p_1)/2\rfloor -p_0}$$ with $p_0=n_1+\cdots+n_t$, and $\tau_j$ is an irreducible tempered representation of ${\mathrm{GL}}_{n_j}(F)$ for each $j=1,\dots,t$, $\sigma_0$ is an irreducible tempered representation of $H_{\lfloor(\frak{n}-p_1)/2\rfloor-p_0}(F)$, and $s_j\in{\mathbb {R}}$ with $s_1 \geq \dots \geq s_t \geq 0$. When $H_{\lfloor(\frak{n}-p_1)/2\rfloor}$ is metaplectic, from [@AB98][@GS12], we can write $\sigma$ as the unique quotient ${\mathcal {L}}(\underline{s},\tau_1,\dots,\tau_t,\sigma_0)$ of the following standard module: $$\label{SM-Mp} {\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0):= {\mathrm{Ind}}^{H_{\lfloor(\frak{n}-p_1)/2\rfloor}(F)}_{P(F)} (|\det|^{s_1}\widetilde{\tau}_1\otimes\cdots\otimes|\det|^{s_t}\widetilde{\tau}_t\otimes\sigma_0)$$ where the Levi part of the standard parabolic subgroup $P$ is isomorphic to $$\widetilde{{\mathrm{GL}}}_{n_1}\times_{\pm 1}\cdots\times_{\pm1}\widetilde{{\mathrm{GL}}}_{n_t}\times_{\pm1} H_{\lfloor(\frak{n}-p_1)/2\rfloor -p_0}$$ with $p_0=n_1+\cdots+n_t$, and $\widetilde{\tau}_j$ is the genuine representation representation of $\widetilde{{\mathrm{GL}}}_{n_j}(F)$ associated to an irreducible tempered representation $\tau_j$ of ${\mathrm{GL}}_{n_j}(F)$ with respect to the fixed character $\psi_F$ of $F$ for each $j=1,\dots,t$, $\sigma_0$ is an irreducible genuine tempered representation of $H_{\lfloor(\frak{n}-p_1)/2\rfloor-p_0}(F)$, and $s_j\in{\mathbb {R}}$ with $s_1 \geq \dots \geq s_t \geq 0$. **Proposition 8**. *Assume that a representation $\pi\in\Pi_F(G_n)$ has a generic $L$-parameter. Let $p_1$ be an integer with $0<p_1\leq{\mathfrak {n}}$. If the standard module $${\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0):= {\mathrm{Ind}}^{H_{\lfloor(\frak{n}-p_1)/2\rfloor}(F)}_{P(F)} (|\det|^{s_1}\tau_1\otimes\cdots\otimes|\det|^{s_t}\tau_t\otimes\sigma_0)$$ as given in [\[SM1\]](#SM1){reference-type="eqref" reference="SM1"} has the property that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat\otimes\,{\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0) , \mathbbm{1})\neq 0$$ for the $F$-rational nilpotent orbit ${\mathcal {O}}_{p_1}$, then $\sigma_0^\vee \in{\mathrm{Spec}}_{{\mathcal {O}}_{p_1+2p_0}}(\pi)$ for some $F$-rational nilpotent orbit ${\mathcal {O}}_{p_1+2p_0}$.* If the standard module ${\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0)$ is irreducible, the result of this proposition is a consequence of the local Gan-Gross-Prasad conjecture. The point is to prove this proposition when the standard module ${\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0)$ is reducible. In this case, we have to show that if $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat\otimes\,{\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0) , \mathbbm{1})\neq 0$$ for the $F$-rational nilpotent orbit ${\mathcal {O}}_{p_1}$, then we must have that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1+2p_0}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat\otimes\,\sigma_0,\mathbbm{1})\neq 0$$ for some $F$-rational nilpotent orbit ${\mathcal {O}}_{p_1+2p_0}$. A complete proof of Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"} will be given in Section [5](#sec-PPI){reference-type="ref" reference="sec-PPI"}. # Arithmetic Data {#sec-AD} ## Enhanced $L$-parameters and local Vogan packets {#ssec-LVP} We recall from [@JLZ22 Section 3.1] the definition of enhanced $L$-parameters and their associated local Vogan packets. Let ${\mathcal {W}}_E$ be the local Weil group of $E$. Then ${\mathcal {W}}_{\mathbb {C}}={\mathbb {C}}^\times$, ${\mathcal {W}}_{\mathbb {R}}={\mathbb {C}}^\times\cup j{\mathbb {C}}^\times$, where $j^2=-1$ and $j z j^{-1}=\bar{z}$ for $z\in {\mathbb {C}}^\times$; and when $E$ is $p$-adic, ${\mathcal {W}}_E={\mathcal {I}}_E\rtimes \left\langle {\rm {Frob}} \right\rangle$ is the semi-direct product of the inertia group ${\mathcal {I}}_E$ of $E$ and a geometric Frobenius element ${\rm {Frob}}$. The local Langlands group (also called the Weil-Deligne group) of $E$ is defined by $$\begin{aligned} \label{LLG} {\mathcal {L}}_E=\begin{cases} {\mathcal {W}}_E &\text{if $E$ is archimedean},\\ {\mathcal {W}}_E\times{\mathrm{SL}}_2({\mathbb {C}}) &\text{if $E$ is $p$-adic}. \end{cases} \end{aligned}$$ We take the Galois version of the $L$-group ${^LG}:=\widehat G \rtimes {\mathrm{Gal}}(E/F)$ of $G$, where $\widehat G$ is the complex dual group of $G$. Note that if $G={\mathrm{Mp}}_{2n}$, then $\widehat G={\mathrm{Sp}}_{2n}({\mathbb {C}})$, and if $G$ is $F$-split, the action of ${\mathrm{Gal}}(E/F)$ on $\widehat G$ is trivial. When $G_n=G_n^*$ is $F$-quasisplit, an $L$-parameter $\varphi\colon{\mathcal {L}}_F\to{^LG_n}$ of $G^*_n$ is attached to a datum $(L^*,\varphi^{L^*},\underline{\beta})$ with the following properties: 1. $L^*$ is a Levi subgroup of $G_n$ of the form $$L^*={\mathrm{GL}}_{n_1}(E)\times\cdots\times{\mathrm{GL}}_{n_t}(E)\times G_{n_0};$$ 2. $\varphi^{L^*}$ is a local $L$-parameter of $L^*$ given by $$\varphi^{L^*}:=\varphi_1\oplus\cdots\oplus\varphi_t\oplus\varphi_0\ :\ {\mathcal {L}}_{F}\rightarrow {^L\!L^*},$$ where $\varphi_j\colon{\mathcal {L}}_E\to{\mathrm{GL}}_{n_j}({\mathbb {C}})$ is a local tempered $L$-parameter of ${\mathrm{GL}}_{n_j}(E)$ for $j=1,2,\cdots,t$, and $\varphi_0$ is a local tempered $L$-parameter of $G_{n_0}^*$; 3. $\underline{\beta}:=(\beta_1,\cdots,\beta_t)\in {\mathbb {R}}^t$, such that $\beta_1>\beta_2>\cdots>\beta_t>0$; and 4. the $L$-parameter $\varphi$ can be expressed as $$\varphi=(\varphi_1\otimes|\cdot|^{\beta_1}\oplus{}^c \varphi_1^\vee \otimes|\cdot|^{-\beta_1})\oplus\cdots\oplus (\varphi_t\otimes|\cdot|^{\beta_t}\oplus {}^c \varphi_t^\vee \otimes|\cdot|^{-\beta_t})\oplus\varphi_0,$$ where $|\cdot|$ is the absolute value of $E$ and ${}^c\varphi_j^\vee$ denotes the conjugate dual of $\varphi_j$ for $j=1,2,\cdots,t$. Following [@Ar13], [@GS12], [@MW12], [@KMSW14], [@Mok15], and [@AG17a], with a choice of a Whittaker datum, one can form a local $L$-packet ${\Pi}_\varphi(G_n)$ for each $L$-parameter $\varphi$, which is a finite subset of $\Pi_F(G_n)$ satisfies certain constraints in the local Langlands conjecture for $G_n$ over $F$. A local $L$-parameter $\varphi$ is called *generic* if the local $L$-packet $\Pi_\varphi(G_n)$ contains a generic member, i.e. a member with a non-zero Whittaker model with respect to the given Whittaker datum. One may define ${\Pi}_\varphi[G_n^*]$ to be the local Vogan packet associated to each $L$-parameter $\varphi$, as in [@GGP12 Sections 9--11], for the classical groups under consideration: $$\label{lvp} {\Pi}_\varphi[G_n^*]=\bigcup_{G_n}{\Pi}_\varphi(G_n)$$ where $G_n$ runs over all pure inner $F$-forms of the given $F$-quasisplit $G_n^*$ and the local $L$-packet ${\Pi}_\varphi(G_n)$ is the subset of representations of $G_n$. The $L$-packet ${\Pi}_\varphi(G_n)$ is defined to be *empty* if the parameter $\varphi$ is not $G_n$-relevant. In the case that $G^*_n={\mathrm{SO}}_{2n}$, we use the weak local Langlands correspondence by [@AG17a] in the above and replace $\Pi_\varphi(G_n)$ by the equivalence classes $\Pi_\varphi(G_n)/\sim_{\rm c}$ under the outer action of ${\mathrm {O}}_{2n}$, i.e. conjugation by an element ${\rm c}\in {\mathrm {O}}_{2n}\setminus {\mathrm{SO}}_{2n}$ with $\det(\rm{c})=-1$. The set of all generic local $L$-parameters of $G_n^*$, up to equivalence, is denoted by ${\Phi}_{\mathrm{gen}}(G_n^*)$. For an $L$-parameter $\varphi$, denote $S_\varphi$ to be the centralizer of the image of $\varphi$ in $\widehat G$, and $S^\circ_\varphi$ to be its identity connected component group. Then the component group ${\mathcal {S}}_\varphi:=S_\varphi/S_\varphi^\circ$ is an abelian $2$-group for the classical groups under consideration. For $\varphi\in{\Phi}_{\mathrm{gen}}(G_n^*)$, the group ${\mathcal {S}}_\varphi$ has been computed explicitly in [@GGP12 Section 8] and also in [@JLZ22 Section 3.2], which is recalled briefly as follows. Write $$\label{decomp} \varphi=\oplus_{i\in {\mathrm {I}}}m_i\varphi_i,$$ which is the decomposition of $\varphi$ into simple and generic ones. The simple, generic local $L$-parameter $\varphi_i$ can be written as $\rho_i\otimes\mu_{b_i}$, where $\rho_i$ is an $a_i$-dimensional irreducible representation of ${\mathcal {W}}_F$ and $\mu_{b_i}$ is the irreducible representation of ${\mathrm{SL}}_2({\mathbb {C}})$ of dimension $b_i$. In the decomposition [\[decomp\]](#decomp){reference-type="eqref" reference="decomp"}, $\varphi_i$ is called *of good parity* if $\varphi_i$ is of the same type of $\varphi$. We denote by ${\mathrm {I}}_{{\mathrm{gp}}}$ the subset of ${\mathrm {I}}$ consisting of indices $i$ such that $\varphi_i$ is of good parity; and by ${\mathrm {I}}_{{\mathrm{bp}}}$ the subset of ${\mathrm {I}}$ consisting of indices $i$ such that $\varphi_i$ is (conjugate) self-dual, but not of good parity. We set ${\mathrm {I}}_{{\mathrm{nsd}}}:={\mathrm {I}}-({\mathrm {I}}_{{\mathrm{gp}}}\cup{\mathrm {I}}_{{\mathrm{bp}}})$ for the indexes of $\varphi_i$'s which are not (conjugate) self-dual. Hence we may write $\varphi\in{\Phi}_{\mathrm{gen}}(G_n^*)$ in the following more explicit way: $$\label{edec} \varphi=(\oplus_{i\in {\mathrm {I}}_{\mathrm{gp}}}m_i\varphi_i)\oplus(\oplus_{j\in {\mathrm {I}}_{\mathrm{bp}}}2m'_j\varphi_j)\oplus(\oplus_{k\in {\mathrm {I}}_{\mathrm{nsd}}}m_k(\varphi_k\oplus{}^c \varphi_k^\vee)),$$ where $2m'_j=m_j$ in [\[decomp\]](#decomp){reference-type="eqref" reference="decomp"} for $j\in {\mathrm {I}}_{{\mathrm{bp}}}$. According to this explicit decomposition, it is easy to know that $$\label{2grp} {\mathcal {S}}_\varphi\cong{\mathbb {Z}}_2^{\#{\mathrm {I}}_{\mathrm{gp}}} \text{ or } {\mathbb {Z}}_2^{\#{\mathrm {I}}_{\mathrm{gp}}-1}.$$ The latter case occurs if $G_n$ is even orthogonal and some orthogonal summand $\varphi_i$ for $i\in {\mathrm {I}}_{\mathrm{gp}}$ has odd dimension, or if $G_n$ is symplectic. As in [@JLZ22 (3.5)], for any $\varphi\in{\Phi}_{\mathrm{gen}}(G_n^*)$ we write elements of ${\mathcal {S}}_\phi$ in the following form $$\label{eq:e-i} (e_i)_{i\in {\mathrm {I}}_{{\mathrm{gp}}}}\in {\mathbb {Z}}_{2}^{\#{\mathrm {I}}_{{\mathrm{gp}}}}, \text{ (or simply denoted by $(e_{i})$)}, \ {\rm with}\ e_i\in\{0,1\},$$ where each $e_i$ corresponds to $\varphi_i$-component in the decomposition [\[edec\]](#edec){reference-type="eqref" reference="edec"} for $i\in {\mathrm {I}}_{\mathrm{gp}}$. In the case that $\widehat{G}={\mathrm{SO}}_N({\mathbb {C}})$, denote by $A_\varphi$ the component group ${\rm Cent}_{{\mathrm {O}}_N({\mathbb {C}})}(\varphi)/{\rm Cent}_{{\mathrm {O}}_N({\mathbb {C}})}(\varphi)^\circ$. Then ${\mathcal {S}}_\varphi$ consists of elements in $A_\varphi$ with determinant $1$ and is a subgroup of index $1$ or $2$. Also write elements of $A_\varphi$ in the form $(e_i)$ with $e_i\in\{0,1\}$ corresponding to the $\varphi_i$-component in the decomposition [\[edec\]](#edec){reference-type="eqref" reference="edec"} for $i\in {\mathrm {I}}_{\mathrm{gp}}$. When $G_n$ is even orthogonal and some $\varphi_i$ for $i\in {\mathrm {I}}_{\mathrm{gp}}$ has odd dimension or $G_n$ is symplectic, $(e_i)_{i\in {\mathrm {I}}_{{\mathrm{gp}}}}$ is in ${\mathcal {S}}_\varphi$ if and only if $\sum_{i\in{\mathrm {I}}_{\mathrm{gp}}}e_i\dim\varphi_i$ is even. The Vogan version of the local Langlands correspondence for the classical groups under consideration asserts that for any $\varphi\in{\Phi}_{\mathrm{gen}}(G_n^*)$ the local Vogan packet ${\Pi}_\varphi[G_n^*]$ is in one-to-one correspondence with the character group $\widehat{{\mathcal {S}}_\varphi}$ (the Pontryagin dual) of the component group ${\mathcal {S}}_\varphi$. The parameterization depends on a choice of the Whittaker datum on $G_n^*(F)$, which as explained in [@JLZ22 Section 3.2], is completely determined by a number $a$ belonging to a group ${\mathcal {Z}}$. As in [@JLZ22 Section 3.2.1], this group ${\mathcal {Z}}$ is defined to be $$\label{CZ} {\mathcal {Z}}=F^\times/ {\mathbb {N}}E^\times \cong \begin{cases} F^\times/ F^{\times2}, & \textrm{if }E=F,\\ \textrm{Gal}(E/F), & \textrm{if }E=F(\delta). \end{cases}$$ We refer to [@JLZ22 Section 3.2.1] for a detailed discussion of the $F$-rationality of the local Langlands correspondence and the relation between the Whittaker data on $G_n^*(F)$ and the ${\mathcal {Z}}$-action on the character group $\widehat{{\mathcal {S}}_\varphi}$ of the component group ${\mathcal {S}}_\varphi$. With a fixed $a\in{\mathcal {Z}}$, which determines the local Whittaker datum and the $F$-rationality of the local Langlands correspondence, for any $\pi\in {\Pi}_\varphi[G_n^*]$, there exists a unique character $\mu\in\widehat{{\mathcal {S}}_\varphi}$, such that the enhanced $L$-parameter $(\varphi,\mu)$ corresponds to $\pi$. In this case, we write $$\label{pia} \pi=\pi_a(\varphi,\mu),\quad \iota_a(\pi)=(\varphi,\mu).$$ ## Local Gan-Gross-Prasad conjecture {#ssec-LGGP} The local Gan-Gross-Prasad conjecture for classical groups and generic $L$-parameters ([@GGP12]) is one of the main inputs in the theory of *arithmetic wavefront sets* as developed in [@JLZ22] and in what we are discussing in this paper. It is important to point out that the local Gan-Gross-Prasad conjecture ([@GGP12]) is now a well-established theorem for classical groups and generic local $L$-parameters. More precisely, it is proved for all $p$-adic cases by J.-L. Waldspurger ([@W10; @W12a; @W12b]), C. Mœglin and J.-L. Waldspurger ([@MW12]), R. Beuzart-Plessis ([@BP16]), W. T. Gan and A. Ichino ([@GI16]), and H. Atobe ([@At18]). It is proved for real unitary groups by R. Beuzart-Plessis [@BP20], (H. He ([@He17]) for some special cases) and H. Xue ([@X1; @X2]). Finally, it is proved for real special orthogonal groups by Z. Luo ([@Lu20]), C. Chen ([@Ch21]), and Chen-Luo ([@CL22]). Among many other results, the rank one real orthogonal groups are also treated in the work of T. Kobayashi and B. Speh [@KS15; @KS18]. For convenience, we recall from [@JLZ22 Section 4.4] the explicit arithmetic data that we need for this paper from the local Gan-Gross-Prasad conjecture ([@GGP12]). They are given case by case, in order to take into account the compatibility of the local conjectures ([@GGP12 §18]) in our formulation through the rationality of nilpotent orbits. ### Special orthogonal groups {#sec:GGP-SO} Let $(\varphi, \phi) \in \Phi_{\mathrm{gen}}({\mathrm{SO}}(V^*)\times {\mathrm{SO}}(W^*))$, where $V^*\supset W^*$ is a relevant pair of orthogonal spaces of dimensions ${\mathfrak {n}}>{\mathfrak {n}}'$ with opposite parities. Then the local Gan-Gross-Prasad conjecture asserts that up to isometry there is a unique relevant pure inner form ${\mathrm{SO}}(V)\times {\mathrm{SO}}(W)$ and a unique pair of representations $(\pi, \pi')\in \Pi_\varphi({\mathrm{SO}}(V))\times \Pi_\phi({\mathrm{SO}}(W))$ up to $\sim_{\rm c}$ equivalence, such that $${\mathrm{Hom}}_{P_{X_{e,\varsigma}}}(\pi\widehat{\otimes} \pi', \psi_{X_{e,\varsigma}})\neq 0,% \blue{(\varsigma \textrm{ irrelevant here?})}$$ with $p_1={\mathfrak {n}}-{\mathfrak {n}}'$, $m=\lfloor p_1/2\rfloor$, $W^\perp={\mathbb {H}}^m+F e$, $X_{e, \varsigma}\in {\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ and $P_{X_{e,\varsigma}} = M_{X_{e,\varsigma}}^\circ\ltimes U_{X_{e, \varsigma}}$. Here and below, ${\mathbb {H}}$ denotes a hyperbolic plane (over $E$). Moreover, the pair $(\pi, \pi')$ is determined by $$\pi=\pi_a(\varphi, \chi_{\varphi,\phi}),\quad \pi'=\pi_a(\phi,\chi_{\phi,\varphi}),$$ where $(\chi_{\varphi, \phi}, \chi_{\phi, \varphi})$ is given by [\[char-O\]](#char-O){reference-type="eqref" reference="char-O"}, and the weak local Langlands correspondence for the even orthogonal group is via $\iota_a$ as in [\[pia\]](#pia){reference-type="eqref" reference="pia"} with $$a=-{\mathrm{disc}}(V){\mathrm{disc}}(W)=(-1)^{\mathfrak {n}}\langle e, e\rangle.$$ ### Symplectic-metaplectic groups Let $(\varphi, \phi) \in \Phi_{\mathrm{gen}}({\mathrm{Sp}}(V)\times {\mathrm{Mp}}(W))$ or $\Phi_{\mathrm{gen}}({\mathrm{Mp}}(V)\times {\mathrm{Sp}}(W))$, where $V\supset W$ is a pair of symplectic spaces of dimensions $2n\geq 2n'$. Then the local Gan-Gross-Prasad conjecture asserts that for $\varsigma\in F^\times$, there is a unique pair of representations $(\pi, \pi')\in \Pi_\varphi({\mathrm{Sp}}(V))\times \Pi_\phi({\mathrm{Mp}}(W))$ or $\Pi_\varphi({\mathrm{Mp}}(V))\times \Pi_\phi({\mathrm{Sp}}(W))$ such that $${\mathrm{Hom}}_{\widetilde{P}_{X_{e,\varsigma}}}(\pi\widehat{\otimes} \pi', \omega_{\psi_{X_{e, \varsigma}}})\neq 0,$$ with $p_1=2n-2n'$, $m=p_1/2$, $W^\perp ={\mathbb {H}}^m$, $X_{e, \varsigma}\in {\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ and $\widetilde{P}_{X_{e,\varsigma}} = \widetilde{M}_{X_{e,\varsigma}}\ltimes U_{X_{e, \varsigma}}$. Moreover, the pair $(\pi, \pi')$ is determined by $$\pi=\pi_1(\varphi, \chi^a_{\varphi,\phi}),\quad \pi'=\pi_1(\phi,\chi^a_{\phi,\varphi}),$$ where $(\chi^a_{\varphi, \phi}, \chi^a_{\phi, \varphi})$ is given by [\[char-spmp\]](#char-spmp){reference-type="eqref" reference="char-spmp"}, $a=(-1)^{m-1}\varsigma$, and the local Langlands correspondence is via $\iota_1$. The above is a reformulation of [@GGP12 Prop. 18.1]. ### Unitary groups: Bessel case Let $(\varphi, \phi) \in \Phi_{\mathrm{gen}}({\mathrm {U}}(V^*)\times {\mathrm {U}}(W^*))$, where $V^*\supset W^*$ is a relevant pair of Hermitian spaces of dimensions ${\mathfrak {n}}>{\mathfrak {n}}'$ with opposite parities. Then the local Gan-Gross-Prasad conjecture asserts that for a trace zero element $\varsigma\in E^\times$, up to isometry there is a unique relevant pure inner form ${\mathrm {U}}(V)\times {\mathrm {U}}(W)$ and a unique pair of representations $(\pi, \pi')\in \Pi_\varphi({\mathrm {U}}(V))\times \Pi_\phi({\mathrm {U}}(W))$, such that $${\mathrm{Hom}}_{P_{X_{e,\varsigma}}}(\pi\widehat{\otimes} \pi', \psi_{X_{e,\varsigma}})\neq 0,$$ with $p_1={\mathfrak {n}}-{\mathfrak {n}}'$, $m=\lfloor p_1/2\rfloor$, $W^\perp={\mathbb {H}}^m+E e$, $X_{e, \varsigma}\in {\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ and $P_{X_{e,\varsigma}} = M_{X_{e,\varsigma}}\ltimes U_{X_{e, \varsigma}}$. Moreover, the pair $(\pi, \pi')$ is determined by $$\pi=\pi_a(\varphi, \chi_{\varphi,\phi}),\quad \pi'=\pi_{-a}(\phi,\chi_{\phi,\varphi}),$$ where $(\chi_{\varphi, \phi}, \chi_{\phi, \varphi})$ is given by [\[char-U\]](#char-U){reference-type="eqref" reference="char-U"}, and $$a=(-1)^{{\mathfrak {n}}-1}{\mathrm{disc}}(V){\mathrm{disc}}(W) = \langle e,e \rangle.$$ ### Unitary groups: Fourier-Jacobi case Fix a splitting character $\xi$ that is used to define the Heisenberg-oscillator representation, namely $\xi$ is a character of $E^\times$ such that $\xi|_{F^\times}=\omega_{E/F}$ is the quadratic character of ${\mathcal {Z}}=F^\times/{\mathbb {N}}E^\times$ given by the local class field theory. Let $(\varphi, \phi) \in \Phi_{\mathrm{gen}}({\mathrm {U}}(V^*)\times {\mathrm {U}}(W^*))$, where $V^*\supset W^*$ is a relevant pair of skew-Hermitian spaces of dimensions ${\mathfrak {n}}\geq {\mathfrak {n}}'$ with the same parity. Then the local Gan-Gross-Prasad conjecture asserts that for $\varsigma\in F^\times$, up to isometry there is a unique relevant pure inner form ${\mathrm {U}}(V)\times {\mathrm {U}}(W)$ and a unique pair of representations $(\pi, \pi')\in \Pi_\varphi({\mathrm {U}}(V))\times \Pi_\phi({\mathrm {U}}(W))$, such that $${\mathrm{Hom}}_{P_{X_{e,\varsigma}}}(\pi\widehat{\otimes} \pi', \omega_{\psi_{X_{e,\varsigma}},\xi})\neq 0,$$ with $p_1={\mathfrak {n}}-{\mathfrak {n}}'$, $m=p_1/2$, $W^\perp ={\mathbb {H}}^m$, $X_{e, \varsigma}\in {\mathcal {O}}^{{\mathrm{st}}}_{p_1}$ and $P_{X_{e,\varsigma}} = M_{X_{e,\varsigma}}\ltimes U_{X_{e, \varsigma}}$. Moreover, the pair $(\pi, \pi')$ is determined by $$\pi=\pi_1(\varphi, \chi_{\varphi(\xi^{-1}),\phi}\cdot \eta_a),\quad \pi'=\pi_1(\phi,\chi_{\phi,\varphi(\xi^{-1})}\cdot\eta_a),$$ where $(\chi_{\varphi(\xi^{-1}), \phi}, \chi_{\phi, \varphi(\xi^{-1})})$ is given by [\[char-U\]](#char-U){reference-type="eqref" reference="char-U"}, $a= (-1)^{m-1}\varsigma$, and the local Langlands correspondence is via $\iota_1$. ## Twisted distinguished characters {#ssec-DC} Assume that $\varphi$ and $\phi$ are two (conjugate) self-dual representations of the local Langlands group ${\mathcal {L}}_E$ (see [\[LLG\]](#LLG){reference-type="eqref" reference="LLG"} for definition) of opposite types, written in the form [\[edec\]](#edec){reference-type="eqref" reference="edec"} with index sets ${\mathrm {I}}_{\mathrm{gp}}$ and ${\mathrm {I}}_{\mathrm{gp}}'$, respectively. The pair of twisted characters $(\chi_{\varphi,\phi},\chi_{\phi,\varphi})$ is the key arithmetic ingredient in the local Gan-Gross-Prasad conjecture as discussed in Section [3.2](#ssec-LGGP){reference-type="ref" reference="ssec-LGGP"}. We recall from [@JLZ22 Section 3.4] the explicit expressions of $(\chi_{\varphi,\phi},\chi_{\phi,\varphi})$ in each case. We write the elements of the corresponding component groups ${\mathcal {S}}_\varphi$ and ${\mathcal {S}}_\phi$ as $(e_i)_{i\in {\mathrm {I}}_{\mathrm{gp}}}$ and $(e_{i'})_{i'\in {\mathrm {I}}_{\mathrm{gp}}'}$ respectively. Let $\psi_0$ be a fixed nontrivial additive character of $F$ and $\psi$ be the character of $E$ given by $$\label{psiE} \psi(x):=\psi_0(\frac{1}{2}{\mathrm{tr}}_{E/F}(\delta\cdot x))$$ with $E=F(\delta)$ as in Section [2.1](#ssec-CGIF){reference-type="ref" reference="ssec-CGIF"}. For $a\in \mathcal{Z}$, define a character $\eta_a$ of ${\mathcal {S}}_\varphi$ by[^2] $$\label{eta} \eta_a((e_i)_{i\in {\mathrm {I}}_{{\mathrm{gp}}}})= \begin{cases} \prod_{i\in {\mathrm {I}}_{{\mathrm{gp}}}}(\det \varphi_i, a)_F^{e_i}, & \textrm{if }E=F\textrm{ and }G_n^*\neq {\mathrm{Mp}}_{2n},\\ \prod_{i\in {\mathrm {I}}_{\mathrm{gp}}} \left(\varepsilon(\varphi_i)\varepsilon(\varphi_i(a))(a,-1)_F^{\dim \phi_i/2}\right)^{e_i}, & \textrm{if }E=F\textrm{ and }G_n^*={\mathrm{Mp}}_{2n},\\ \prod_{i\in {\mathrm {I}}_{{\mathrm{gp}}}}\omega_{E/F}(a)^{e_i\dim \varphi_i}, & \textrm{if }E=F(\delta), \end{cases}$$ where $(\cdot, \cdot)_F$ is the Hilbert symbol defined over $F$, and $\omega_{E/F}$ is the quadratic character of ${\mathcal {Z}}$ given by the local classfield theory. Note that $\eta_a$ is trivial if $G_n$ is special odd orthogonal. Then we have a ${\mathcal {Z}}$-action on $\widehat{{\mathcal {S}}}_\varphi$ by $$\begin{aligned} \label{Z-action} a\ \colon\ \widehat{{\mathcal {S}}_\varphi} \to \widehat{{\mathcal {S}}_\varphi},\quad \mu\mapsto \mu\cdot\eta_a\end{aligned}$$ for every $a\in{\mathcal {Z}}$. It clear that one may define a character $\eta_a$ for $\phi$ and define an action of ${\mathcal {Z}}$ on $\widehat{{\mathcal {S}}_\phi}$. ### If $E=F$, $\varphi$ and $\phi$ are self-dual of even dimensions, define a pair of characters $(\chi_{\varphi,\phi}, \chi_{\phi,\varphi})\in \widehat{{\mathcal {S}}}_\varphi\times\widehat{{\mathcal {S}}}_\phi$ by $$\label{char-O} \chi_{\varphi,\phi}((e_i)_{i\in {\mathrm {I}}_{\mathrm{gp}}}):= \prod_{i\in {\mathrm {I}}_{\mathrm{gp}}}\left(\varepsilon(\varphi_i\otimes\phi) (\det \varphi_i)(-1)^{\dim \phi/2}(\det \phi)(-1)^{\dim\varphi_i/2}\right)^{e_i},$$ and a similar formula defines $\chi_{\phi,\varphi}$. Note that the local root number $$\varepsilon(\varphi_i\otimes\phi):=\varepsilon(\varphi_i\otimes\phi, \psi_0)$$ does not depend on $\psi_0$. In this case, for $z\in {\mathcal {Z}}$ define $$(\chi_{\varphi, \phi}^z, \chi_{\phi,\varphi}^z) = (\chi_{\varphi, \phi}\cdot \eta_z, \chi_{\phi,\varphi}\cdot \eta_z),$$ where $\eta_z$ is given by [\[eta\]](#eta){reference-type="eqref" reference="eta"}. ### If $E=F$, $\varphi$ is odd orthogonal and $\phi$ is symplectic, define $(\chi_{\varphi,\phi}, \chi_{\phi,\varphi})$ to be the restriction of $$(\chi_{\varphi_1,\phi},\chi_{\phi,\varphi_1})\in \widehat{{\mathcal {S}}}_{\varphi_1}\times\widehat{{\mathcal {S}}}_\phi$$ to ${\mathcal {S}}_{\varphi}\times {\mathcal {S}}_\phi$, where $$\varphi_1:=\varphi\oplus{\mathbb {C}}.$$ We also consider the twist $\varphi(z)$ with $z\in {\mathcal {Z}}$, and make the canonical identification ${\mathcal {S}}_{\varphi(z)}\cong {\mathcal {S}}_\varphi$. Then we define $$\label{char-spmp} (\chi^z_{\varphi, \phi}, \chi^z_{\phi,\varphi})=(\chi_{\varphi(z), \phi} \cdot \eta_z, \chi_{\phi, \varphi(z)})\in \widehat{{\mathcal {S}}}_\varphi\times\widehat{{\mathcal {S}}}_\phi.$$ ### If $E=F(\delta)$, $\varphi$ and $\phi$ are conjugate self-dual, define $(\chi_{\varphi,\phi}, \chi_{\phi,\varphi})\in \widehat{{\mathcal {S}}}_\varphi\times\widehat{{\mathcal {S}}}_\phi$ by $$\label{char-U} \chi_{\varphi,\phi}((e_i)_{i\in {\mathrm {I}}_{\mathrm{gp}}})= \prod_{i\in {\mathrm {I}}_{\mathrm{gp}}}\varepsilon(\varphi_i\otimes\phi, \psi) ^{e_i},$$ and a similar formula defines $\chi_{\phi,\varphi}$, where $\psi$ is given by [\[psiE\]](#psiE){reference-type="eqref" reference="psiE"}. Put ${\mathfrak {l}}= \dim \varphi - \dim\phi$. In this case, for $z\in {\mathcal {Z}}$ define $$\label{chi-a} (\chi^z_{\varphi,\phi}, \chi^z_{\phi,\varphi})= (\chi_{\varphi, \phi}\cdot \eta_z, \chi_{\phi,\varphi}\cdot \eta_{(-1)^{{\mathfrak {l}}}z}).$$ By convention, in the above cases we allow $\varphi$ or $\phi$ to be zero. In such a case we formally take $\chi_{\varphi, \phi}$ and $\chi_{\phi,\varphi}$ to be the trivial characters of ${\mathcal {S}}_\varphi$ and ${\mathcal {S}}_\phi$ respectively. ## Descent of enhanced $L$-parameters {#ssec-DELP} In order to unify the notations, we denote $$\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)=\Phi_{\mathrm{gen}}(G_n^*)$$ if $E=F$, and $$\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)= \Phi_{\mathrm{gen}}(G_n^*)\cup\Phi_{\mathrm{gen}}(G_n^*)\otimes\xi$$ the set of all $\frak{n}$-dimensional conjugate self-dual $L$-parameters if $E=F(\delta)$, where $\xi$ is any character of $E^\times$ such that $\xi|_{F^\times}=\omega_{E/F}$. Note that if $E=F(\delta)$ then $\Phi_{\mathrm{gen}}(G_n^*)$ consists of conjugate self-dual $L$-parameters of sign $(-1)^{{\mathfrak {n}}-1}$. This modification is made to avoid the choice of a splitting character $\xi$ in the Fourier-Jacobi model of unitary groups. Let $\varphi\in\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$ be an $L$-parameter given by the form [\[edec\]](#edec){reference-type="eqref" reference="edec"}. For $\mu\in \widehat{{\mathcal {S}}}_\varphi$, define its ${\mathcal {Z}}$-orbit $${\mathscr {O}}_{\mathcal {Z}}(\mu)=\{\mu\cdot\eta_z \in \widehat{{\mathcal {S}}}_\varphi \mid z\in{\mathcal {Z}}\}.$$ We recall from [@JLZ22 Section 5.1] the definition and properties of the descents of enhanced $L$-parameters $(\varphi,\mu)$ with $\varphi\in \widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$ and $\chi\in \widehat{{\mathcal {S}}}_\varphi$, along with an ${\mathfrak {n}}$-dimensional $\epsilon$-Hermitian vector space $(V, q)$ over $F$. As in Section [2.1](#ssec-CGIF){reference-type="ref" reference="ssec-CGIF"}, let $(V, q)$ be an ${\mathfrak {n}}$-dimensional $\epsilon$-Hermitian vector space over $F$. Let $0<{\mathfrak {l}}\leq {\mathfrak {n}}$ be such that $(-1)^{{\mathfrak {l}}-1}=\epsilon$ if $E=F$. There is no restriction on the parity of ${\mathfrak {l}}$ if $E=F(\delta)$. We obtain quasi-split relevant pairs of classical groups $(G_n^*, H_{\lfloor(\frak{n}-{\mathfrak {l}})/2\rfloor}^*)$ as given in [\[relevantpair\]](#relevantpair){reference-type="eqref" reference="relevantpair"}. **Definition 9** (Definition 5.1 of [@JLZ22]). *For any given enhanced $L$-parameter $(\varphi,\mu)$ with $\varphi\in \widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$ and $\mu\in\widehat{{\mathcal {S}}_\varphi}$, the ${\mathfrak {l}}$-th descent ${\mathfrak {D}}_{{\mathfrak {l}}}^z(\varphi,\mu)$ of $(\varphi,\mu)$ along $z\in {\mathcal {Z}}$ is the set of contragredients $(\widehat{\phi}, \widehat{\nu})$ of all enhanced $L$-parameters $(\phi,\nu)$ with $\phi\in \widetilde{\Phi}_{\mathrm{gen}}(H_{\lfloor(\frak{n}-{\mathfrak {l}})/2\rfloor}^*)$ and $\nu\in\widehat{{\mathcal {S}}_\phi}$ satisfying the following conditions:* 1. *the type of $\phi$ is different from that of $\varphi$, and* 2. *the following $$(\chi^z_{\varphi,\phi}, \chi^z_{\phi, \chi} ) = (\mu, \nu),$$ holds, where $(\chi^z_{\varphi,\phi},\chi^z_{\phi,\varphi})$ is defined, case by case, as in Section [3.3](#ssec-DC){reference-type="ref" reference="ssec-DC"}.* *The ${\mathfrak {l}}$-th descent of the enhanced $L$-parameter $(\varphi,\mu)$ is defined to be $$\label{Fl-pd} {\mathfrak {D}}_{\mathfrak {l}}(\varphi, \mu) := \bigcupdot_{z\in {\mathcal {Z}}} {\mathfrak {D}}_{\mathfrak {l}}^z(\varphi, \mu).$$* The contragredient $(\widehat\phi,\widehat{\nu})$ of $(\phi,\nu)$ is defined in [@JLZ22 Section 3.3] and is explicitly given in [@JLZ22 Proposition 3.2] for the classical groups under consideration. **Proposition 10** (Proposition 3.2 of [@JLZ22]). *For an enhanced $L$-parameter $(\varphi,\mu)$ with $\varphi\in \widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$ and $\mu\in\widehat{{\mathcal {S}}_\varphi}$, the contragredient $(\widehat{\varphi}, \widehat{\mu})$ can be written, case by case, as follows:* - *$G^*={\mathrm{SO}}_{\mathfrak {n}}^*$, $(\widehat\varphi, \widehat\mu) = (\varphi, \mu)$.* - *$G^*={\mathrm{Sp}}_{2n}$, $(\widehat\varphi, \widehat\mu) = (\varphi, \mu \cdot \eta_{-1})$.* - *$G^*={\mathrm{Mp}}_{2n}$, $(\widehat\varphi, \widehat\mu) = (\varphi(-1), \mu\cdot \eta_{-1})$.* - *$G^* ={\mathrm {U}}_{\mathfrak {n}}^*$, $(\widehat\varphi, \widehat\mu) = (\varphi^\vee, \mu\cdot\eta)$ where $\eta$ is trivial if ${\mathfrak {n}}$ is odd and $\eta = \eta_{-1}$ if ${\mathfrak {n}}$ is even.* *In all cases as listed above, the following equality $$(\widehat\varphi, \widehat{\mu\cdot \eta_a}) = (\widehat\varphi, \widehat\mu\cdot \eta_a)$$ holds for any $a\in {\mathcal {Z}}$.* We prove the tower property of the descents of enhanced $L$-parameters, which can be stated as follows. **Proposition 11**. *For any enhanced $L$-parameter $(\varphi,\mu)$ with $\varphi\in\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$ and $\mu\in\widehat{{\mathcal {S}}_\varphi}$, if there exists an integer ${\mathfrak {l}}_1$ with $0<{\mathfrak {l}}_1\leq{\mathfrak {n}}$ such that the ${\mathfrak {l}}_1$-th descent ${\mathfrak {D}}_{{\mathfrak {l}}_1}(\varphi, \mu)$ is non-empty, then for any integer ${\mathfrak {l}}$ with $0<{\mathfrak {l}}\leq{\mathfrak {l}}_1$ and ${\mathfrak {l}}\equiv {\mathfrak {l}}_1 \text{ mod }2$, the ${\mathfrak {l}}$-th descent ${\mathfrak {D}}_{\mathfrak {l}}(\varphi, \mu)$ is non-empty.* *Proof.* Let $\phi_0$ be a tempered $L$-parameter of dimension $({\mathfrak {l}}_1-{\mathfrak {l}})/2$ which contains no irreducible (conjugate) self-dual summands of the same type as $\varphi$. For instance, one can choose $\phi_0$ to be a direct sum of unitary characters of the Weil-Deligne group which are not quadratic and not conjugate self-dual. Assume that $(\widehat\phi,\widehat\nu)\in {\mathfrak {D}}_{{\mathfrak {l}}_1}^z(\varphi,\mu)$. Let $\phi'=\phi\oplus\phi_0\oplus\phi_0^\vee$. Then ${\mathcal {S}}_{\phi'}={\mathcal {S}}_\phi$ and it is clear that $(\widehat{\phi'},\widehat\nu)\in {\mathfrak {D}}_{{\mathfrak {l}}}^z(\varphi,\mu)$. Thus ${\mathfrak {D}}_{\mathfrak {l}}(\varphi, \mu)\neq\varnothing$. ◻ By [@JLZ22 Proposition 5.2], for any enhanced $L$-parameter $(\varphi,\mu)$ with $\varphi\in\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$ and $\mu\in\widehat{{\mathcal {S}}_\varphi}$, there exists an integer ${\mathfrak {l}}$ with $0<{\mathfrak {l}}\leq{\mathfrak {n}}$ such that the ${\mathfrak {l}}$-th descent ${\mathfrak {D}}_{\mathfrak {l}}(\varphi, \mu)$ is non-empty. This leads to a definition of the *first occurrence index* of the enhanced $L$-parameter $(\varphi, \mu)$, as in [@JLZ22 (5.3)], $$\label{FO} {\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi, \mu):=\max\{0<{\mathfrak {l}}\leq {\mathfrak {n}}\mid {\mathfrak {D}}_{\mathfrak {l}}(\varphi, \mu)\neq \emptyset\};$$ and the ${\mathfrak {l}}_0$-th descent ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi, \mu)$ is called the *first descent* of the enhanced $L$-parameter $(\varphi,\mu)$. The following is the first important result on descents of enhanced $L$-parameters. **Theorem 12** (Theorem 5.3 of [@JLZ22]). *For any enhanced $L$-parameter $(\varphi,\mu)$ with $\varphi\in\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$ and $\mu\in\widehat{{\mathcal {S}}_\varphi}$, the first descent ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi, \mu)$ at the first occurrence index ${\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi, \mu)$ contains only of discrete enhanced $L$-parameters.* ## Proof of Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"} {#ssec-PSNE} We are ready to prove Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}, that is, for any $\pi\in\Pi_\varphi[G_n^*]$ with $\varphi$ generic, there exists an integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$, such that the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$ is non-empty for some $F$-rational nilpotent orbit ${\mathcal {O}}_{p_1}$ in the $F$-stable nilpotent orbit ${\mathcal {O}}_{p_1}^{\mathrm{st}}$ associated with the partition $[p_1,1^{{\mathfrak {n}}-p_1}]$. By the Vogan version of the local Langlands correspondence $\iota_a$ for some fixed $a\in{\mathcal {Z}}$, we write $$\pi=\pi_a(\varphi,\mu)$$ for some character $\mu\in\widehat{{\mathcal {S}}_\varphi}$, which is unique. By [@JLZ22 Proposition 5.2], there exists an integer ${\mathfrak {l}}$ with $0<{\mathfrak {l}}\leq{\mathfrak {n}}$, such that the descent ${\mathfrak {D}}_{\mathfrak {l}}(\varphi,\mu)$ is non empty. This means that there exists an enhanced $L$-parameter $(\phi,\nu)$ of $H_{\lfloor(\frak{n}-{\mathfrak {l}})/2\rfloor}^*(F)$ with $\phi$ generic, such that the contragredient $(\widehat\phi,\widehat\nu)$ belongs to the ${\mathfrak {l}}$-th descent ${\mathfrak {D}}_{\mathfrak {l}}(\varphi,\mu)$. From Definition [Definition 9](#defn:pd){reference-type="ref" reference="defn:pd"}, there exists a number $z\in{\mathcal {Z}}$, such that $$(\chi^z_{\varphi,\phi}, \chi^z_{\phi,\varphi})=(\mu,\nu),$$ where $(\chi^z_{\varphi,\phi},\chi^z_{\phi,\varphi})$ is defined, case by case, as in Section [3.3](#ssec-DC){reference-type="ref" reference="ssec-DC"}. By the Vogan version of the local Langlands correspondence, there exists an $F$-pure inner form $H_{\lfloor(\frak{n}-{\mathfrak {l}})/2\rfloor}$ of $H_{\lfloor(\frak{n}-{\mathfrak {l}})/2\rfloor}^*$ and a $\sigma\in\Pi_\phi(H_{\lfloor(\frak{n}-{\mathfrak {l}})/2\rfloor})$, such that the pair $(\pi,\sigma)$ is the distinguished pair in the local Vogan packet $\Pi_{\varphi\otimes\phi}[G_n^*\times H_{\lfloor(\frak{n}-{\mathfrak {l}})/2\rfloor}^*]$. Hence there exists an $F$-rational nilpotent orbit ${\mathcal {O}}_{\mathfrak {l}}$, associated with the partition $[{\mathfrak {l}},1^{{\mathfrak {n}}-{\mathfrak {l}}}]$, such that $\sigma^\vee$ belongs to the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{\mathfrak {l}}}(\pi)$. Finally, by taking $p_1={\mathfrak {l}}$, we finish the proof of Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}. From the proof of Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}, we obtain a stronger version of Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}. **Corollary 13**. *If a representation $\pi\in\Pi_F(G_n)$ has a generic local $L$-parameter, then there exists at least one integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$, such that the spectrum ${\mathrm{Spec}}_{p_1}(\pi)$ contains at least one $\sigma^\vee$, which has a generic $L$-parameter.* By the relation between parabolic induction and the multiplicities, we are able to prove the following even stronger result. **Proposition 14**. *If a representation $\pi\in\Pi_F(G_n)$ has a generic local $L$-parameter, then there exists at least one integer $p_1$ with $0<p_1\leq{\mathfrak {n}}$, such that the spectrum ${\mathrm{Spec}}_{p_1}(\pi)$ contains at least one $\sigma^\vee$, which is tempered.* *Proof.* This follows directly from Propositions [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"} and [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"}. ◻ # Spectrum and Arithmetic First Descent {#sec-SDAFD} ## Arithmetic first descent {#ssec-afd} We take the local Langlands correspondence $\iota_a$ with the $F$-rationality determined by $a\in{\mathcal {Z}}$. For any generic $L$-parameter $\varphi\in\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$, any member $\pi\in\Pi_F[G_n^*]$ corresponds with a unique $\mu\in\widehat{{\mathcal {S}}_\varphi}$ such that $\pi=\pi_a(\varphi,\mu)$. By [@JLZ22 Proposition 5.2], the first occurrence index ${\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi,\mu)$ of the enhanced $L$-parameter $(\varphi,\mu)$ exists. In this case, we define the *arithmetic first occurrence index* ${\mathfrak {f}}_{\mathfrak {a}}(\pi)$ of $\pi$ to be the first occurrence index ${\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi,\mu)$ of $(\varphi,\mu)$: $$\label{afoi} {\mathfrak {f}}_{\mathfrak {a}}(\pi):={\mathfrak {l}}_0(\varphi,\mu).$$ The spectral descent ${\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}(\pi)$ of $\pi$ at the arithmetic first occurrence index ${\mathfrak {f}}_{\mathfrak {a}}(\pi)$ is called the *arithmetic first descent* of $\pi$. ## Discreteness of arithmetic first descent {#ssec-AFD} By Theorem [Theorem 12](#thm:DFD){reference-type="ref" reference="thm:DFD"} and the local Gan-Gross-Prasad conjecture, we obtain the following discreteness result. **Theorem 15** (Discreteness). *Given any generic $L$-parameter $\varphi\in\widetilde{\Phi}_{\mathrm{gen}}(G_n^*)$, for any $\pi\in\Pi_\varphi[G_n^*]$ with $\pi=\pi_a(\varphi,\mu)$ for some $\mu\in\widehat{{\mathcal {S}}_\varphi}$, the arithmetic first descent ${\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}(\pi)$ consists exactly of the discrete series representations $\sigma^\vee$ whose enhanced $L$-parameters are given in the first descent ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi, \mu)$ at the first occurrence index ${\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi, \mu)$.* *Proof.* For any $\sigma^\vee\in {\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}(\pi)$ with ${\mathfrak {f}}_{\mathfrak {a}}(\pi)={\mathfrak {l}}_0(\varphi, \mu)$, there exists an $F$-rational nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}$, such that $\sigma^\vee\in{\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}}(\pi)$. By definition, we must have that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}}(\pi)\,\widehat\otimes\,\sigma, \mathbbm{1})\neq 0.$$ We realize $\sigma$ as the Langlands quotient ${\mathcal {L}}(\underline{s},\tau_1,\dots,\tau_t,\sigma_0)$ of the standard module ${\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0)$ as displayed in [\[SM1\]](#SM1){reference-type="eqref" reference="SM1"}, and obtain that $$\label{pia3} {\mathrm{Hom}}_{H_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}}(\pi)\,\widehat\otimes\,{\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0) , \mathbbm{1})\neq 0,$$ for the $F$-rational nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}$. By Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"}, we obtain that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)+2p_0}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)+2p_0}}(\pi)\,\widehat\otimes\,\sigma_0, \mathbbm{1})\neq 0$$ for some $F$-rational nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)+2p_0}$, which implies that the irreducible tempered representation $\sigma_0^\vee$ belongs to the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)+2p_0}}(\pi)$. By the Vogan version of the local Langlands correspondence, we write $$\sigma_0=\sigma_a(\phi,\nu)$$ for some generic (tempered) $L$-parameter $\phi$ of $H_{\lfloor(\frak{n}-{\mathfrak {f}}_{\mathfrak {a}}(\pi)-2p_0)/2\rfloor}^*$ and $\nu\in\widehat{{\mathcal {S}}_\phi}$. Since $(\pi,\sigma_0)$ is the unique distinguished pair in the local Vogan packet $\Pi_{\varphi\otimes\phi}[G_n^*\times H_{\lfloor(\frak{n}-{\mathfrak {f}}_{\mathfrak {a}}(\pi)-2p_0)/2\rfloor}^*]$, we must have, from the descent of the enhanced $L$-parameter $(\varphi,\mu)$ (Definition [Definition 9](#defn:pd){reference-type="ref" reference="defn:pd"}), that $(\widehat\phi,\widehat\nu)$ belongs to the $({\mathfrak {f}}_{\mathfrak {a}}(\pi)+2p_0)$-th descent ${\mathfrak {D}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)+2p_0}(\varphi,\mu)$ of $(\varphi,\mu)$. Since ${\mathfrak {f}}_{\mathfrak {a}}(\pi)={\mathfrak {l}}_0(\varphi, \mu)$ is the first occurrence index of the enhanced $L$-parameter $(\varphi,\mu)$, we must have that $${\mathfrak {f}}_{\mathfrak {a}}(\pi)+2p_0\leq{\mathfrak {f}}_{\mathfrak {a}}(\pi)={\mathfrak {l}}_0(\varphi, \mu).$$ Hence we obtain that $p_0=0$ and $\sigma=\sigma_0$ is tempered. Moreover, we have that $(\widehat\phi,\widehat\nu)$ belongs to the first descent ${\mathfrak {D}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}(\varphi,\mu)$ of $(\varphi,\mu)$. By Theorem [Theorem 12](#thm:DFD){reference-type="ref" reference="thm:DFD"}, the enhanced $L$-parameter $(\phi,\nu)$ must be discrete, and hence $\sigma$ must be a discrete series representation. Conversely, if $\sigma$ is a discrete series representation with enhanced $L$-parameter $(\phi,\nu)$, whose contragredient $(\widehat\phi,\widehat\nu)$ belongs to the first descent ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi, \mu)$ at the first occurrence index ${\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi, \mu)$, we are going to show that $\sigma^\vee\in{\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}(\pi)$. In fact, since the discrete enhanced $L$-parameter $(\phi,\nu)$ of $\sigma$ has the property that its dual $(\widehat\phi,\widehat\nu)$ belongs to the first descent ${\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi, \mu)$, by the definition of descent of enhanced $L$-parameters (Definition [Definition 9](#defn:pd){reference-type="ref" reference="defn:pd"}) and the local Gan-Gross-Prasad conjecture, $(\pi,\sigma)$ must be the distinguished pair in the local Vogan packet $\Pi_{\varphi\otimes\phi}[G_n^*\times H_{\lfloor(\frak{n}-{\mathfrak {f}}_{\mathfrak {a}}(\pi))/2\rfloor}^*]$. Thus, $\sigma^\vee\in{\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}}(\pi)$ for some $F$-rational nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}$. This proves that $\sigma^\vee\in{\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {a}}(\pi)}(\pi)$. ◻ ## Proof of Theorem [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"} {#sec-PC-SAFOI} Theorem [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"} asserts that for any given generic $L$-parameter $\varphi$ of $G_n^*(F)$ and for any $\pi\in\Pi_\varphi[G_n^*]$, if $\pi$ has its enhanced $L$-parameter $(\varphi,\mu)$, i.e. $$\label{pia1} \pi=\pi_a(\varphi,\mu)$$ holds for some $\mu\in\widehat{{\mathcal {S}}_\varphi}$ (which is unique), then the spectral first occurrence index ${\mathfrak {f}}_{\mathfrak {s}}(\pi)$ is equal to the arithmetic first occurrence index ${\mathfrak {f}}_{\mathfrak {a}}(\pi)$, i.e. $$\label{sa-foi} {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{\mathfrak {a}}(\pi),$$ where ${\mathfrak {f}}_{\mathfrak {a}}(\pi):={\mathfrak {l}}_0(\varphi,\mu)$, the first occurrence index of $(\varphi,\mu)$. We are going to prove the identity in [\[sa-foi\]](#sa-foi){reference-type="eqref" reference="sa-foi"}. First, we prove that ${\mathfrak {f}}_{\mathfrak {a}}(\pi)\leq{\mathfrak {f}}_{\mathfrak {s}}(\pi)$. For the given $(\varphi,\mu)$ as in [\[pia1\]](#pia1){reference-type="eqref" reference="pia1"}, the first occurrence index ${\mathfrak {l}}_0={\mathfrak {l}}_0(\varphi,\mu)$, which is equal to ${\mathfrak {f}}_{\mathfrak {a}}(\pi)$, is an integer with $0<{\mathfrak {l}}_0\leq{\mathfrak {n}}$. Take an enhanced $L$-parameter $(\phi,\nu)$ with $\phi$ generic, such that $$(\widehat\phi,\widehat\nu)\in{\mathfrak {D}}_{{\mathfrak {l}}_0}(\varphi,\mu).$$ As in the proof of Proposition [Proposition 7](#prop:sne){reference-type="ref" reference="prop:sne"}, the contragredient $\sigma^\vee$ of the representation $$\sigma=\sigma_a(\phi,\nu)$$ belongs to the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {l}}_0}}(\pi)$. In particular, the spectrum ${\mathrm{Spec}}_{{\mathfrak {l}}_0}(\pi)$ is not empty. By definition of the spectral first occurrence index, we must have $$\label{pia2} {\mathfrak {f}}_{\mathfrak {a}}(\pi)={\mathfrak {l}}_0(\varphi,\mu)\leq{\mathfrak {f}}_{\mathfrak {s}}(\pi).$$ Now we prove the converse: ${\mathfrak {f}}_{\mathfrak {a}}(\pi)\geq{\mathfrak {f}}_{\mathfrak {s}}(\pi)$. If $\sigma^\vee\in{\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}(\pi)$, then there exists an $F$-rational nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}$, such that $\sigma^\vee\in{\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}}(\pi)$. By definition, we must have that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}}(\pi)\,\widehat\otimes\,\sigma, \mathbbm{1})\neq 0.$$ We realize $\sigma$ as the Langlands quotient ${\mathcal {L}}(\underline{s},\tau_1,\dots,\tau_t,\sigma_0)$ of the standard module ${\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0)$ as in [\[SM1\]](#SM1){reference-type="eqref" reference="SM1"}, we obtain that $$\label{pia3} {\mathrm{Hom}}_{H_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}}(\pi)\,\widehat\otimes\,{\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0) , \mathbbm{1})\neq 0,$$ for the $F$-rational nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}$. By Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"}, we obtain that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)+2p_0}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)+2p_0}}(\pi)\,\widehat\otimes\,\sigma_0,\mathbbm{1})\neq 0$$ for some $F$-rational nilpotent orbit ${\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)+2p_0}$, which implies that the irreducible tempered representation $\sigma_0^\vee$ belongs to the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)+2p_0}}(\pi)$, and in particular the spectrum ${\mathrm{Spec}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)+2p_0}(\pi)$ is not empty. Since ${\mathfrak {f}}_{\mathfrak {s}}(\pi)$ is the spectral first occurrence index of $\pi$, we must have $${\mathfrak {f}}_{\mathfrak {s}}(\pi)\geq {\mathfrak {f}}_{\mathfrak {s}}(\pi)+2p_0,$$ which implies that $p_0=0$, and $\sigma=\sigma_0$ is tempered. By the Vogan version of the local Langlands correspondence, we obtain that $$\sigma=\sigma_a(\phi,\nu)$$ for some generic (tempered) $L$-parameter $\phi$ of $H_{\lfloor(\frak{n}-{\mathfrak {f}}_{\mathfrak {s}}(\pi))/2\rfloor}^*$ and $\nu\in\widehat{{\mathcal {S}}_\phi}$. Since $(\pi,\sigma)$ is the unique distinguished pair in the local Vogan packet $\Pi_{\varphi\otimes\phi}[G_n^*\times H_{\lfloor(\frak{n}-{\mathfrak {f}}_{\mathfrak {s}}(\pi))/2\rfloor}^*]$, we must have, from the descent of the enhanced $L$-parameter $(\varphi,\mu)$ (Definition [Definition 9](#defn:pd){reference-type="ref" reference="defn:pd"}), that $(\widehat\phi,\widehat\nu)$ belongs to the ${\mathfrak {f}}_{\mathfrak {s}}(\pi)$-th descent ${\mathfrak {D}}_{{\mathfrak {f}}_{\mathfrak {s}}(\pi)}(\varphi,\mu)$ of $(\varphi,\mu)$. By the definition of the first occurrence index ${\mathfrak {l}}_0(\varphi,\mu)$, we obtain that $${\mathfrak {f}}_{\mathfrak {s}}(\pi)\leq {\mathfrak {l}}_0(\varphi,\mu)={\mathfrak {f}}_{\mathfrak {a}}(\pi).$$ Combining with [\[pia2\]](#pia2){reference-type="eqref" reference="pia2"}, we prove that ${\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{\mathfrak {a}}(\pi)$. This proves Theorem [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"}. It is important to mention that Theorems [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"} and [Theorem 15](#thm:AFD){reference-type="ref" reference="thm:AFD"} imply the main result of this paper (Theorem [Theorem 5](#thm:ADS){reference-type="ref" reference="thm:ADS"}), which extends the main result ([@JZ18 Theorem 1.7]) to the great generality. ## Submodule Theorem {#ssec-ST} We are going to prove Theorem [Theorem 6](#thm:SM){reference-type="ref" reference="thm:SM"} in this section. For any given $\pi\in\Pi_F(G_n)$ with a generic $L$-parameter, By Theorems [Theorem 4](#thm:SA-foi){reference-type="ref" reference="thm:SA-foi"} and [Theorem 15](#thm:AFD){reference-type="ref" reference="thm:AFD"}, we have that by taking $$p_1:={\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{\mathfrak {a}}(\pi)$$ there exists an irreducible discrete series representation $\sigma$ of $H_{{\mathcal {O}}_{p_1}}$ for some $F$-rational nilpotent orbit ${\mathcal {O}}_{p_1}$, such that $\sigma^\vee$ belongs to the spectrum ${\mathrm{Spec}}_{{\mathcal {O}}_{p_1}}(\pi)$. This means that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat\otimes\,\sigma, \mathbbm{1})\neq 0.$$ **Lemma 16**. *For any $\pi\in\Pi_F(G_n)$ with a generic $L$-parameter, if $\sigma$ is an irreducible smooth representation of $H_{{\mathcal {O}}_{p_1}}(F)$, then $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat\otimes\,\sigma,\mathbbm{1}) \cong {\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi),\sigma^\vee)$$ where $\sigma^\vee$ is the contragredient of $\sigma$.* *Proof.* The lemma is clear when $F$ is non-archimedean. For $F$ archimedean, it follows from [@Ch23 Lemma 2.2.22]. ◻ By Lemma [Lemma 16](#frob){reference-type="ref" reference="frob"}, we have that $${\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi),\sigma^\vee)\neq 0.$$ Hence we have $${\mathrm{Hom}}_{G_n(F)}(\pi,{\mathrm{Ind}}^{G_n(F)}_{R_{{\mathcal {O}}_{p_1}}(F)}(\sigma^\vee\otimes\psi_{{\mathcal {O}}_{p_1}}))\neq 0.$$ We are done. # Proof of Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"} {#sec-PPI} We are going to complete the proof of Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"} in this section by establishing a multiplicity formula. For any $\pi\in \Pi_F(G_n)$ and $\sigma\in \Pi_F(H_{\lfloor ({\mathfrak {n}}-p_1)/2 \rfloor})$, we define the multiplicity $m(\pi,\sigma)$ or $m(\sigma,\pi)$ the dimension $$\label{equ: def of multiplicity} \dim{\mathrm{Hom}}_{H_{{\mathcal {O}}_{p_1} }(F)}({\mathcal {J}}_{{\mathcal {O}}_{p_1}}(\pi)\,\widehat{\otimes}\,\sigma, \mathbbm{1}).$$ In order to prove Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"}, it suffices to show that when $\pi$ is in a generic $L$-packet and $\sigma={\mathrm {I}}(\underline{s},\tau_1,\dots\tau_t,\sigma_0)$, where $\tau_i,\sigma_0$ are tempered representations, we have $$\label{equ: multiplicity ineq} m(\pi,\sigma)\leq m(\pi,\sigma_0).$$ When $F$ is non-archimedean, it was proved for special orthogonal groups in [@MW12], for unitary groups in [@GI16], and for symplectic groups and metaplectic groups in [@At18] that for every representation $\pi$ in a generic $L$-packet, we have $$\pi={\mathrm {I}}(\underline{s}',\rho_1,\dots\rho_{t'},\pi_0)$$ with certain $\underline{s}'\in {\mathbb {R}}^{t'}$ and tempered $\rho_i$, $\pi_0$. When $F$ is archimedean, this result was shown in [@X2 §1.1], [@Ch21 §4] and [@Ch23 Theorem 5.2.1]. With this classification, the following proposition implies the inequality of multiplicities in [\[equ: multiplicity ineq\]](#equ: multiplicity ineq){reference-type="eqref" reference="equ: multiplicity ineq"}. **Proposition 17**. *Let $\underline{s}=(s_1,\cdots,s_t)\in {\mathbb {R}}^t$ and $\underline{s}'=(s_1',\cdots,s_{t'}')\in {\mathbb {R}}^{t'}$, satisfying $$s_1\geq \cdots \geq s_t\geq 0\quad\text{and} \quad s_1'\geq \cdots \geq s_{t'}'\geq 0.$$ Suppose that $\rho_i,\tau_i$ are tempered representations of ${\mathrm{GL}}_{m_i}(E)$, ${\mathrm{GL}}_{m_i'}(E)$ and $\pi_0,\sigma_0$ are tempered representations of $G_{n},H_{\lfloor (n-p_1)/2 \rfloor}$ respectively. Then for $$\pi={\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0)\quad \text{and} \quad \sigma={\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0),$$ we have that $$\label{equ: multiplicity formula} m(\pi,\sigma)=m(\pi_0,\sigma_0).$$* In the setting of Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"}, we denote by $$n^+=n+\sum_{i=1}^t{m_i},\quad n^{\prime,+}=\lfloor ({\mathfrak {n}}-p_1)/2\rfloor+\sum_{i=1}^{t'}m_i'.$$ When applied to the setting of Proposition [Proposition 8](#prop:PI){reference-type="ref" reference="prop:PI"}, we always have that $$%n^+\geq n^{\prime,+}\text{ in the Bessel cases},\quad n^+\neq n^{\prime,+} \text{ in the Fourier-Jacobi cases}.$$ It is worth mentioning that Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} indeed holds in a more general context, and we implement the definition for $m(\pi,\sigma)$ in the Fourier-Jacobi case with $n^+=n^{\prime,+}$ as following $$m(\pi,\sigma)=\dim {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \,\widehat{\otimes}\, \sigma \,\widehat{\otimes}\, \omega_{n^+,\psi_F},\mathbbm{1}).$$ Here the representation $\omega_{n^+,\psi_F}$ is the Heisenberg-oscillator representation of $$\label{equ: define wtG} \widetilde{G}^J_{n^+}:=\widetilde{G}_{n^+}\ltimes {\mathcal {H}}_{n^+},\quad \text{ where } \widetilde{G}_{n^+}:=\begin{cases} {\mathrm {U}}_{{\mathfrak {n}}^+}& \text{ when }G_{n^+}={\mathrm {U}}_{{\mathfrak {n}}^+} \text{ and }n^+=\lfloor{\mathfrak {n}}^+/2\rfloor,\\ {\mathrm{Mp}}_{2n^+}& \text{ when }G_{n^+}={\mathrm{Sp}}_{2n^+},{\mathrm{Mp}}_{2n^+} \end{cases}$$ associated to $\psi_F$, where ${\mathcal {H}}_{n^+}$ is the Heisenberg group ${\mathrm{Res}}_{E/F}V\oplus {\mathbb {G}}_{a,F}$ for the $\epsilon$-Hermitian space $V$ associated to $G_{n^+}$, and $\pi, \sigma$ in the completed tensor product are inflations of the corresponding representations of $G_{n^+},H_{n^+}$ to $\widetilde{G}_{n^+}^J$ respectively. We first point out that Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} is known in the following situations. When $F$ is non-archimedean, Propostion [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} follows from [@MW12 Proposition 1.3] when $G_n, H_{\lfloor ({\mathfrak {n}}-p_1)/2 \rfloor}$ are special orthogonal groups, and it follows from [@GI16 Proposition 9.4] when $G_n, H_{\lfloor ({\mathfrak {n}}-p_1)/2 \rfloor}$ are unitary groups. When $F$ is archimedean, Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} follows from [@Ch23 Theorem A]. Hence, it remains to prove Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} in the Fourier-Jacobi cases when $F$ is non-archimedean. We present a proof of this situation by following [@MW12] and using the tempered local Gan-Gross-Prasad conjecture as proved in [@GI16; @At18]. In the setting of Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"}, we call the inequality $$\label{equ: first inequality} m(\pi,\sigma)\leq m(\pi_0,\sigma_0)$$ "the first inequality\" and call the inequality $$\label{equ: second inequality} m(\pi,\sigma)\geq m(\pi_0,\sigma_0)$$ "the second inequality\". Following the framework of Mœglin and Waldspurger in [@MW12], there are three key steps in the proof of the inequalities: reduction to basic cases, basic forms of the first inequality, and basic forms of the second inequality. In the setting of non-archimedean Fourier-Jacobi cases, we state them in the following proposition. **Proposition 18**. *In the setting of Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"}, we denote by $$\sigma_{t'-1}={\mathrm {I}}((s_2',\cdots,s_{t'}'),\tau_2,\dots,\tau_{t'},\sigma_0).$$ Then we have the following results.* 1. ***(Reduction to basic cases)** When $n^+=n^{\prime,+}$ and $\tau_1$ is a supercuspidal representation, then $$m(\pi,\sigma)=m(\pi,\sigma_{t'-1})$$ if $\pi^{\vee}$ does not belong to the Bernstein component associated to $|\det|^{s_1'}\tau_1 \otimes \sigma'$, where $\sigma'$ is any supercuspidal representation of a Levi subgroup of $H_{n^{\prime,+}-m_1'}$.* 2. ***(Basic forms of the first inequality)** When $n^+=n^{\prime,+}$ and $s_1'\geq s_1$, $$m(\pi,\sigma)\leq m(\pi,\sigma_{t'-1}).$$* 3. ***(Basic forms of the second inequality)** When $n^+=n^{\prime,+}$, $$m(\pi,\sigma)\geq m(\pi_0,\sigma_0).$$* Since Part (1) was proved in [@GGP12 Theorem 16.1], we prove parts (2) and (3) in the section. We will apply Parts (1) and (2) to prove the first inequality of Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} and apply Parts (1) and (3) to prove the second inequality of Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"}. ## The first inequality In this section, we aim to prove the first inequality [\[equ: first inequality\]](#equ: first inequality){reference-type="eqref" reference="equ: first inequality"} for Fourier-Jacobi models over non-archimedean local fields. ### Basic forms of the first inequality First, we prove Part (2) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"} following [@MW12 §1.4]. See [@Ch23 §4.1] for the archimedean counterpart. For this purpose, we study $m(\pi,\sigma)$ with $n^+=n^{\prime,+}$. Recall that in this situation, $$m(\pi,\sigma)=\dim {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \sigma \otimes \omega_{n^+,\psi_F}, \mathbbm{1}).$$ Since $n^+=n^{\prime,+}$, $G_{n^+}$ and $H_{n^{\prime,+}}$ are defined as the isometric groups on the same $\epsilon$-hermitian space $V$. We fix an $m_1'$-dimensional totally isotropic space $X_{m_1'}$ over $E$. We denote by $P_{X_{m_1'}}$, $P'_{X_{m_1'}}$ and $\widetilde{P}_{X_{m_1'}}$ the respective parabolic subgroups of $G_{n^+}$, $H_{n^{\prime,+}}$ and $\widetilde{G}_{n^+}$ stabilizing $X_{m_1'}'$. The Levi decomposition gives $P_{X_{m_1'}}=L_{X_{m_1'}}\ltimes N_{X_{m_1'}}$ with $L_{X_{m_1'}}={\mathrm{Res}}_{E/F}{\mathrm{GL}}_{m_1'}\times G_{n^+-m_1'}$. For $i=1,\cdots,m_1'$, we let ${\mathrm{GL}}^{(i)}_{m_1'}$ be the semidirect product of ${\mathrm{Res}}_{E/F}{\mathrm{GL}}(X_{m_1'-i})$ and the unipotent part $N^{(i)}_{m_1'}$ of the parabolic subgroup of ${\mathrm{Res}}_{E/F}{\mathrm{GL}}(X_{m_1'})$ stabilizing $X_{m_1'-i}\subset X_{m_1'}$ and a full flag of $X_{m_1'}/X_{m_1'-i}$. We let $$P_{X_{m_1'}}^{(i)}=({\mathrm{GL}}_{m_1'}^{(i)}\times G_{n^+-m_1'})\ltimes N_{X_{m_1'}}^{(i)}\subset P_{X_{m_1'}}.$$ By definition, we have $$\sigma ={\mathrm {I}}(\underline{s}',\tau_1,\dots,\tau_{t'},\sigma_0)={\mathrm{ind}}_{P_{X_{m_1'}}'(F)}^{H_{n^{\prime,+}}(F)}(\delta_{P_{X_{m_1'}}'}^{1/2}|\det|^{s_1'}\tau_1 \otimes \sigma_{t-1}).$$ Here and thereafter, we denote by ${\mathrm{ind}}_{H}^{G}$ the (unnormalized) compact induction from $H$ to $G$. From the mixed model of the Weil representation, we have $$\omega_{n^+,\psi_F}={\mathrm{ind}}^{\widetilde{G}_{n^+}\ltimes {\mathcal {H}}_{n^+}(F)}_{\widetilde{P}_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)}(|\det|^{1/2} \otimes \omega_{n^+-m_1',\psi_F}).$$ Following a similar computation as in [@LS13 p3336], we have that $$\sigma \otimes \omega_{n^{\prime,+},\psi_F}={\mathrm{ind}}^{G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F)}_{P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)}(\delta_{P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})}^{1/2}|\det|^{s_1'}\tau_1 \otimes (\sigma_{t-1} \otimes\omega_{n^+-m_1',\psi_F})).$$ Here $X_{m_1'}^{\perp}$ is the complement of $X_{m_1'}$ in $V$ with respect to the $\epsilon$-hermitian form, ${\mathcal {H}}(X_{m_1'}^{\perp})$ is the subgroup ${\mathrm{Res}}_{E/F}X_{m_1'}^{\perp}\oplus {\mathbb {G}}_{a,F}$ of the Heisenberg group ${\mathcal {H}}_{n^{+}}$. We deduce that $$\begin{aligned} m(\pi,\sigma)&=\dim {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes {\mathrm{ind}}^{G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F)}_{P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)}(\delta_{P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})}^{1/2}|\det|^{s_1'}\tau_1 \otimes (\sigma_{t-1} \otimes \omega_{n^+-m_1',\psi_F})), \mathbbm{1}). \end{aligned}$$ Using Mackey's theory, we will study the multiplicity based on the structure of the double cosets $$P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\backslash G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F)/ G_{n^+}(F).$$ From the computation in [@GRS11 §6] , we have the following lemma on the structure of the above double cosets. **Lemma 19**. *With notations as given above, the following hold.* 1. *The set $$P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\backslash G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F)/ G_{n^+}(F)$$ contains an open double coset $P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\gamma_{\mathrm{open}} G_{n^+}(F)$ and a closed double coset $P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\gamma_{\mathrm{closed}} G_{n^+}(F)$.* 2. *One has that $$\gamma_{\mathrm{open}}^{-1}P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})\gamma_{\mathrm{open}}\cap G_{n^+}=P_{X_{m_1'}}^{(1)},$$* 3. *One has that $$\gamma_{\mathrm{closed}}^{-1}P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})\gamma_{\mathrm{closed}}\cap G_{n^+}=P_{X_{m_1'}}.$$* We define that $\sigma^+:=\delta_{P_{X_{m_1'}}\ltimes X_{m_1'}^{\perp}}^{1/2}|\det|^{s_1'}\tau_1 \otimes (\sigma_{t-1} \otimes \omega_{n^+-m_1',\psi_F})$. Then the left-hand side of the inequality can be expressed as $$\label{equ: multiplicity big} m(\pi,\sigma)=\dim {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes {\mathrm{ind}}_{P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)}^{G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F)}(\sigma^+),\mathbbm{1}).$$ By definition, we have that $${\mathrm{ind}}_{P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)}^{G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F)}(\sigma^+)=\Gamma^{{\mathcal {C}}}(P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\backslash G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F),{\mathcal {E}}_{\sigma^+}),$$ that is, the space of compact-supported sections on the bundle ${\mathcal {E}}_{\sigma^+}$, where $${\mathcal {E}}_{\sigma^+} := P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\backslash( (G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F))\times \sigma^+).$$ Here the left $P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)$-action on $(G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F))\times \sigma^+$ is given by $p.(g,v)=(pg,\sigma^+(p)v)$. We set $$\begin{aligned} & {\mathcal {X}}=P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\backslash G_{n^+}\ltimes {\mathcal {H}}_{n^+}(F),\\ & {\mathcal {U}}=P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\backslash P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\gamma_{\mathrm{open}}G_{n^+}(F), \\ & {\mathcal {Z}}={\mathcal {X}}-{\mathcal {U}}=P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\backslash P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})(F)\gamma_{\mathrm{closed}}G_{n^+}(F) \end{aligned}$$ and obtain an exact sequence $$0\to \Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+})\to \Gamma^{{\mathcal {C}}}({\mathcal {X}},{\mathcal {E}}_{\sigma^+})\to \Gamma^{{\mathcal {C}}}({\mathcal {Z}},{\mathcal {E}}_{\sigma^+})\to 0,$$ which implies an exact sequence $$\label{equ: exact 1} \begin{aligned} 0 & \to {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma^{{\mathcal {C}}}({\mathcal {Z}},{\mathcal {E}}_{\sigma^+}), \mathbbm{1}) \to {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma^{{\mathcal {C}}}({\mathcal {X}},{\mathcal {E}}_{\sigma^+}), \mathbbm{1}) \\ & \to{\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+}), \mathbbm{1}). \end{aligned}$$ Moreover, from Lemma [Lemma 19](#lem: double cosets){reference-type="ref" reference="lem: double cosets"}, we have $$\label{equ: open double coset} \Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+})={\mathrm{ind}}_{P_{X_{m_1'}}^{(1)}}^{G_{n^+}}(\sigma^+|_{P_{X_{m_1'}}^{(1)}})={\mathrm{ind}}_{P_{X_{m_1'}}^{(1)}}^{G_{n^+}}(|\det|^{s_1'+s_0}\tau_1|_{{\mathrm{GL}}_{m_1'}^{(1)}} \otimes \sigma_{t'-1})$$ Here $s_0$ is the constant satisfying $\delta_{P_{X_{m_1'}}\ltimes {\mathcal {H}}(X_{m_1'}^{\perp})}^{1/2}=|\det|^{s_0}$. For $1\leq i\leq m_1'$, we set $\mu_{i}$ be the character on the unipotent radical of ${\mathrm{GL}}_{X_{m_1'}}^{(i)}(F)$ and $P_{X_{m_1'}}^{(i)}(F)$ obtain from the restriction of $\psi_{X_{m_1'}}$defined in ([\[psix\]](#psix){reference-type="ref" reference="psix"}). The right side of the inequality in Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}(2) is $$\begin{aligned} m(\pi,\sigma_{t'-1})=\dim {\mathrm{Hom}}_{H_{{\mathcal {O}}_{m_1'}}(F)}({\mathcal {J}}_{{\mathcal {O}}_{m_1'}}(\pi) \otimes\sigma_{t'-1}, \mathbbm{1}) \end{aligned}$$ where the ${\mathrm{Hom}}$-space on the right-hand side is equal to $${\mathrm{Hom}}_{G_{n^+-m_1'}(F)}(\delta_{P_{X_{m_1'}}^{(m_1')}}\otimes V_{\pi}/{\mathrm{Span}}\{\pi(u)v-\mu_{m_1'}(u)v\mid u\in N_{X_{m_1'}}^{(m_1')}(F),v\in V_{\pi}\}\otimes \sigma_{t'-1} \otimes \omega_{n^+-m_1',\psi_F}, \mathbbm{1}),$$ which can be written as $${\mathrm{Hom}}_{P_{X_{m_1'}}^{(m_1')}(F)}(\delta_{P_{X_{m_1'}}^{(m_1')}}\otimes\pi \otimes \sigma_{t'-1} \otimes \omega_{n^+-m_1',\psi_F},\mu_{m_1'}).$$ Hence we obtain that $$\begin{aligned} \label{equ: rhs expansion} m(\pi,\sigma_{t'-1})=\dim {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+})^{(m_1')},\mathbbm{1}),\end{aligned}$$ where $$\Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+})^{(m_1')}={\mathrm{ind}}_{P_{X_{m_1'}}^{(m_1')}(F)}^{G_{n^+}(F)}(\mu_{m_1'}^{-1}\otimes \sigma_{t'-1} \otimes \omega_{n^+,\psi_F}|_{G_{n-m_1'}}).$$ We set $$\Gamma_{\mathrm{open}}=\Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+})/ \Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+})^{(m_1')}.$$ Then by [\[equ: open double coset\]](#equ: open double coset){reference-type="eqref" reference="equ: open double coset"}, we have that $$\Gamma_{\mathrm{open}}={\mathrm{ind}}_{P_{X_{m_1'}}^{(1)}(F)}^{G_{n^+}(F)}((|\det|^{s_1'+s_0}\tau_1|_{{\mathrm{GL}}_{m_1'}^{(1)}}/{\mathrm{ind}}_{{\mathrm{GL}}_{m_1'}^{(m_1')}}^{{\mathrm{GL}}_{m_1'}^{(1)}}(\mu_{m_1'}^{-1}))\otimes\sigma_{t'-1}).$$ Therefore, we obtain an exact sequence $$\label{equ: open contribution} \begin{aligned} 0 \to {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma_{\mathrm{open}},\mathbbm{1})&\to {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes\Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+}),\mathbbm{1}) \\ & \to{\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma^{{\mathcal {C}}}({\mathcal {U}},{\mathcal {E}}_{\sigma^+})^{(m_1')},\mathbbm{1}). \end{aligned}$$ The structure of $|\det|^{s_1'}\tau_1|_{{\mathrm{GL}}_{m_1'}^{(1)}}/{\mathrm{ind}}_{{\mathrm{GL}}_{m_1'}^{(m_1')}(F)}^{{\mathrm{GL}}_{m_1'}^{(1)}(F)}(\mu_{m_1'}^{-1})$ can be computed with the derivative theory in [@BZ77]. I n particular, we recall the following result from [@BZ77 §4.3]. **Lemma 20**. *For an irreducible admissible representation $\tau$ of ${\mathrm{GL}}_n(F)$, there is a filtration $$1=\tau^{0}\subset \tau^{(1)}\subset \cdots \subset \tau^{(n-1)}=\tau|_{{\mathrm{GL}}_{n}^{(1)}(F)}$$ such that $$\tau^{(i+1)}/\tau^{(i)}\cong {\mathrm{ind}}_{{\mathrm{GL}}_n^{(k)}(F)}^{{\mathrm{GL}}_n^{(1)}(F)}(\Delta^k\tau\otimes \mu_k^{-1})$$ where $\Delta^k$ denotes the $k$-th Bernstein-Zelevinsky derivative.* Therefore, we obtain a filtration in $\sigma_{\mathrm{open}}$ with graded pieces $$\label{equ: graded pieces} {\mathrm{ind}}_{P_{X_{m_1'}}^{(i)}(F)}^{G_{n^+}(F)}(|\det|^{s_1'+s_0}\Delta^k\tau_1 \otimes \sigma_{t-1}\otimes\mu_k^{-1}),\quad k=1,\cdots,m_1'.$$ **Lemma 21** (Vanishing results). *Let $\sigma\in\Pi_F({\mathrm{Res}}_{E/F}{\mathrm{GL}}_{r})$ be a tempered representation and $\pi_1$ be a smooth representation of $G_{n^+-r}(F)$. Let $\pi_2={\mathrm {I}}(\underline{s},\tau_1,\cdots,\tau_t,\pi_0)$ be a representation of $G_{n^+}(F)$ where $\underline{s}=(s_1,\cdots,s_t)$ for $s_1\geq s_2\geq \cdots \geq s_t\geq 0$. Then $${\mathrm{Hom}}_{G_{n^+}(F)}({\mathrm {I}}(s',\sigma, \pi_1) \otimes \pi_2, \mathbbm{1})=0$$ when $s'>s_1$.* *Proof.* Our proof follows from that in [@MW12 p177]. From the second adjointness theorem, we have $${\mathrm{Hom}}_{G_{n^+}(F)}({\mathrm {I}}(s',\sigma, \pi_1),\pi_2^{\vee})={\mathrm{Hom}}_{M_{X_{m_1}}(F)}(|\det|^{s'}\sigma \otimes \pi_1,{\mathrm{Jac}}_{P^-}(\pi_2^{\vee})),$$ where $M_{X_{m_1}} = {\mathrm{Res}}_{E/F}{\mathrm{GL}}_{m_1} \times G_{n^+-m_1}$ and $P^-$ is the opposite parabolic subgroup with Levi subgroup $M_{X_{m_1}}$. Suppose that ${\mathrm{Hom}}_{G_{n^+}(F)}({\mathrm {I}}(s',\sigma, \pi_1),\pi_2^{\vee})\neq 0$. Since ${\mathrm{Jac}}_{P^-}(\pi_2^{\vee})$ is admissible of finite length, there is an irreducible admissible representation $\pi_1'$ such that $${\mathrm{Hom}}_{M_{X_{m_1}}(F)}(|\det|^{s'}\sigma \otimes \pi_1',{\mathrm{Jac}}_{P^-}(\pi_2^{\vee}))\neq 0,$$ which leads to a contradiction when comparing the exponent ([@Ch23 Lemma A.0.4]). ◻ *Proof for Part (2) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}.* First, we take both sides of the inequality as the dimensions studied in ([\[equ: multiplicity big\]](#equ: multiplicity big){reference-type="ref" reference="equ: multiplicity big"}) and ([\[equ: rhs expansion\]](#equ: rhs expansion){reference-type="ref" reference="equ: rhs expansion"}). Based on the structures computed in ([\[equ: exact 1\]](#equ: exact 1){reference-type="ref" reference="equ: exact 1"}), ([\[equ: open contribution\]](#equ: open contribution){reference-type="ref" reference="equ: open contribution"}) and ([\[equ: graded pieces\]](#equ: graded pieces){reference-type="ref" reference="equ: graded pieces"}), it suffices to show that 1. ${\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma^{{\mathcal {C}}}({\mathcal {Z}},{\mathcal {E}}_{\sigma^+}),\mathbbm{1})=0$, when $s_1'\geq s_1$, and 2. ${\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes {\mathrm{ind}}_{P_{X_{m_1'}}^{(i)}(F)}^{G_{n^+}(F)}(|\det|^{s_1'+s_0}\Delta^i\tau_1 \otimes\sigma_{t-1}\otimes \mu_i^{-1}),\mathbbm{1})=0$ for all $1\leq i\leq m_1'$, when $s_1'\geq s_1$. On the one hand, for $g\in P_{X_{m_1'}}(F)$, $x_{m_1'}\in X_{m_1'}$,$$\delta_{P_{X_{m_1'}}\ltimes X_{m_1'}}(g\ltimes x_{m_1'})=|\det(g_{{\mathrm{GL}}(X_{m_1'})})|\delta_{P_{X_{m_1'}}}(g)$$ Here $g_{{\mathrm{GL}}(X_{m_1'})}$ is the ${\mathrm{GL}}(X_{m_1'})$-factor of $g$ in the Levi component. Then we have $$\begin{aligned} \Gamma^{{\mathcal {C}}}({\mathcal {Z}},{\mathcal {E}}_{\sigma^+})={\mathrm{ind}}_{P_{X_{m_1'}}(F)}^{G_{n^+}(F)}(\sigma^+|_{ P_{X_{m_1'}}})={\mathrm {I}}(s_1'+\frac{1}{2},\tau_1,\sigma_{t-1} \otimes \omega_{n^+-m_1',\psi_F}|_{G_{n-m_1'}}). \end{aligned}$$ Then from Lemma [Lemma 21](#lem: vanishing){reference-type="ref" reference="lem: vanishing"}, we have $$\label{equ: vanishing closed} {\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes \Gamma^{{\mathcal {C}}}({\mathcal {Z}},{\mathcal {E}}_{\sigma^+}),\mathbbm{1})=0,\quad \text{when } s_1'\geq s_1.$$ On the other hand, for $1\leq i\leq m_1'$, $$\begin{aligned} &{\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes {\mathrm{ind}}_{P_{X_{m_1'}}^{(i)}(F)}^{G_{n^+}(F)}(|\det|^{s_1'+s_0}\Delta^i\tau_1 \otimes\sigma_{t-1}\otimes\mu_i^{-1}),\mathbbm{1})\\ =\,&{\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes {\mathrm {I}}(s_1'+\frac{i}{2},\Delta^i\tau_1, {\mathrm{ind}}_{G_{n^+-m_1'+i}\cap P_{X_{m_1'}}^{(i)}(F)}^{G_{n^+-m_1+i}(F)}(\sigma_{t'-1} \otimes \omega_{n^+-m_1',\psi_F}\otimes \mu_i^{-1})),\mathbbm{1}). \end{aligned}$$ Then from Lemma [Lemma 21](#lem: vanishing){reference-type="ref" reference="lem: vanishing"}, we have $${\mathrm{Hom}}_{G_{n^+}(F)}(\pi \otimes {\mathrm{ind}}_{P_{X_{m_1'}}^{(i)}(F)}^{G_{n^+}(F)}(|\det|^{s_1'+s_0}\Delta^i\tau_1 \otimes \sigma_{t-1}\otimes \mu_i^{-1}),\mathbbm{1})=0,\quad \text{when } s_1'\geq s_1.$$ This completes the proof for Part (2) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}. ◻ ### Mathematical induction We are ready to prove the first inequality $$\label{equ: first inequality} m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0))\leq m(\pi_0,\sigma_0)$$ in Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"} using Parts (1) and (2) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}. First, we reduce it to the equal-rank cases, that is, the situations with $n^+=n^{\prime,+}$. When $n^+\neq n^{\prime,+}$, say $n^+>n^{\prime,+}$, we may choose a supercuspidal representation $\rho_0$ of ${\mathrm{GL}}_{n^+-n^{\prime,+}}(E)$ such that by Part (1) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}, we have $$m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0))=m({\mathrm {I}}(\underline{s}^+,\rho_0,\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0))$$ with $\underline{s}^+=(0,s_1,\cdots,s_{t})$. The comparison of $m({\mathrm {I}}(\underline{s}^+,\rho_0,\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0))$ with $m(\pi_0,\sigma_0)$ is an equal-rank case, which can be proved by induction on $N= N({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0))$, the number of nonzero elements among $s_1,\cdots,s_t,s_1',\cdots,s_{t'}'$. When $N({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0))=0$, the $s_i$ and $s_i'$ are all equal to zero. By the full decomposition of $\pi={\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0)$ and $\sigma={\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0)$ computed in [@Ar13] and [@Mok15] (see [@At16 Desideratum 2.1(4)]) and then apply the local Gan-Gross-Prasad conjecture for tempered $L$-parameters for each component, we have $m(\pi_0,\sigma_0)=1$ if and only if there exactly one pair of $(\pi',\sigma')$ such that $m(\pi',\sigma')=1$, where $\pi',\sigma'$ are irreducible components of $\pi,\sigma$. When $N({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0))=k\geq 1$, suppose that the inequality holds for all $N=k-1$ situations. We may assume that $$s_1\geq \cdots \geq s_t\geq 0,\quad \text{and}\quad s_1'\geq \cdots \geq s_{t'}'\geq 0.$$ 1. If $s_1'\geq s_1$, then from Part (2) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}, we have $$\begin{aligned} & m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0)) \\ \leq \, & m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}((s_2',\cdots,s_{t'}'),\tau_2,\dots\tau_{t'},\sigma_0)). \end{aligned}$$ By Part (1) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}, we can choose a supercuspidal representation $\tau_0$ of ${\mathrm{GL}}_{m_1'}(E)$ such that $$\begin{aligned} & m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}((0,s_2',\cdots,s_{t'}'),\tau_0,\tau_2,\dots\tau_{t'},\sigma_0)) \\ =\, & m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}((s_2',\cdots,s_{t'}'),\tau_2,\dots\tau_{t'},\sigma_0)), \end{aligned}$$ noting that $$N({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}((0,s_2',\cdots,s_{t'}'),\tau_0,\tau_2,\dots\tau_{t'},\sigma_0))=k-1.$$ 2. If $s_1'<s_1$, then from Part (2) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}, we have $$\begin{aligned} & m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0)) \\ =\, &m({\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_{t'},\sigma_0),{\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0))\\ \leq\, &m({\mathrm {I}}(\underline{s}',\tau_1,\dots,\tau_{t'},\sigma_0),{\mathrm {I}}((s_2,\cdots,s_{t}),\rho_2,\dots\rho_t,\pi_0)). \end{aligned}$$ By Part (1) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}, we can choose a supercuspidal representation $\tau_0$ of ${\mathrm{GL}}_{n_1}(E)$ such that $$\begin{aligned} & m({\mathrm {I}}(\underline{s}',\tau_1,\dots,\tau_{t'},\sigma_0),{\mathrm {I}}((s_2,\cdots,s_{t}),\rho_1,\dots\rho_t,\pi_0)) \\ =\, &m({\mathrm {I}}(\underline{s}',\tau_1,\tau_2\dots\tau_{t'},\pi_0),{\mathrm {I}}((0,s_2,\cdots,s_{t}),\rho_0,\rho_2,\dots\rho_{t},\sigma_0))\\ =\, & m({\mathrm {I}}((0,s_2,\cdots,s_{t}),\rho_0,\rho_2,\dots\rho_{t},\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_t,\sigma_0)), \end{aligned}$$ noting that $$N({\mathrm {I}}((0,s_2,\cdots,s_{t}),\rho_0,\rho_2,\dots\rho_{t},\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_t,\sigma_0))=k-1.$$ Therefore, in both situations, we can reduce ([\[equ: first inequality\]](#equ: first inequality){reference-type="ref" reference="equ: first inequality"}) to the inequality for $$m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}((0,s_2',\cdots,s_{t'}'),\tau_0,\tau_2,\dots\tau_{t'},\sigma_0))$$ or $$m({\mathrm {I}}((0,s_2,\cdots,s_{t}),\rho_0,\rho_2,\dots\rho_{t},\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots\tau_t,\sigma_0)),$$ which can be obtained by the induction hypothesis. This completes the mathematical induction. ## The second inequality In this section, we prove the basic forms of second inequality (Part (3) of Proposition [Proposition 18](#prop: three ingredients){reference-type="ref" reference="prop: three ingredients"}) following the integral method in [@MW12]. ### Construction of the integral We work in the situation when $n^+=n^{\prime,+}$, under the assumption that $m(\pi_0,\sigma_0)\neq 1$. We will prove by construction that $$m({\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),{\mathrm {I}}(\underline{s}',\tau_1,\dots,\tau_{t'},\sigma_0))\neq 0,$$ We first define $\pi_{\underline{s}} (\underline{s}\in {\mathbb {C}}^{t})$ using induction from $P$ in [\[SM1\]](#SM1){reference-type="eqref" reference="SM1"} when ${\mathrm {G}}_{n^+}$ is not metaplectic and in [\[SM-Mp\]](#SM-Mp){reference-type="eqref" reference="SM-Mp"} when ${\mathrm {G}}_{n^+}$ is metaplectic. We define $\sigma_{\underline{s}'} (\underline{s}'\in {\mathbb {C}}^{t'})$ similarly using induction from $P'$. Then $$\pi_{\underline{s}}={\mathrm {I}}(\underline{s},\rho_1,\dots\rho_t,\pi_0),\quad \sigma_{\underline{s}'}={\mathrm {I}}(\underline{s}',\tau_1,\dots,\tau_{t'},\sigma_0),\quad \underline{s}\in {\mathbb {R}}^{t},\underline{s}'\in {\mathbb {R}}^{t'}$$ Let $K$ and $K'$ be maximal compact subgroups of $G_{n^+}(F)$ and $H_{n^+}(F)$, which are in good position with respect to $P$ and $P'$ respectively. Then the underlying spaces of $\pi_{\underline{s}}$ and $\sigma_{\underline{s'}}$ can be realized as $$\pi_K={\mathrm{Ind}}_{P\cap K}^{K}(\rho_1 \otimes \cdots \otimes \rho_{t} \otimes \pi_0)\quad \text{and}\quad \sigma_{K'}={\mathrm{Ind}}_{P'\cap K'}^{K'}(\tau_1 \otimes \cdots \otimes \tau_{t'} \otimes \sigma_0)$$ respectively, and the actions of $\pi_{\underline{s}}$ and $\sigma_{\underline{s}'}$ are given by $$\pi_{\underline{s}}(pk)v=\left(\delta_{P}^{1/2}|\det|^{s_1}\rho_1 \otimes \cdots \otimes |\det|^{s_t}\rho_t \otimes \pi_0\right)(p)\pi_K(k)v, \text{ for } p\in P, k\in K,$$ $$\sigma_{\underline{s}'}(p'k')v'=\left(\delta_{P'}^{1/2}|\det|^{s_1'}\tau_1 \otimes \cdots \otimes |\det|^{s_{t'}'}\tau_{t'} \otimes \sigma_0\right)(p')\sigma_{K'}(k')v', \text{ for } p'\in P', k'\in K'.$$ Following [@MW12], we define the integral $$\begin{aligned} {\mathrm {I}}_{v,v^*,v'\otimes w,v^{\prime,*}\otimes w^*}(\underline{s},\underline{s}')&=\int_{G_{n^+}(F)}\langle\pi_{\underline{s}}(g)v, v^*\rangle \langle(\sigma_{\underline{s'}}\otimes \omega_{n^+, \psi_F})(g,0)v'\otimes w,v^{\prime,*}\otimes w^*\rangle dg\\ &=\int_{G_{n^+}(F)}\langle\pi_{\underline{s}}(g)v, v^*\rangle \langle\sigma_{\underline{s'}}(g)v',v^{\prime,*}\rangle\langle \omega_{n^+,\psi_F}(g,0)w, w^*\rangle dg, \end{aligned}$$ where $v\otimes v' \otimes w\in \pi_K\otimes \sigma_{K'}\otimes \omega_{n^+, \psi_F}$ and $v^*\otimes v^{\prime, *} \otimes w^* \in \pi_K^\vee\otimes \sigma_{K'}^\vee \otimes \omega_{n^+,\psi_F}^\vee$. **Proposition 22**. *When $m(\pi_0,\sigma_0)\neq 0$, the following hold.* 1. *There is a constant $c>0$ such that the integral ${\mathrm {I}}_{v,v^*,v'\otimes w,v^{\prime,*}\otimes w^*}(\underline{s},\underline{s}')$ is absolutely convergent when ${\mathrm{Re}}(s_i)<c$;* 2. *There exists $(\underline{s},\underline{s}')\in (i{\mathbb {R}})^{t+t'}$ such that $${\mathrm {I}}_{v,v^*,v'\otimes w,v^{\prime,*}\otimes w^*}(\underline{s},\underline{s}')\neq 0;$$* 3. *The integral ${\mathrm {I}}_{v,v^*,v'\otimes w,v^{\prime,*}\otimes w^*}(\underline{s},\underline{s}')$ has a meromorphic continuation to ${\mathbb {C}}^{r}\times {\mathbb {C}}^{r'}$.* We can prove the second inequality $$m(\pi_{\underline{s}_0},\sigma_{\underline{s}'_0})\geq m(\pi_0,\sigma_0)$$ using this proposition. Since $\pi_0$ and $\sigma_0$ are irreducible, we have $m(\pi_0,\sigma_0)\leq 1$ by the multiplicity-one theorem ([@AGRS10; @Su12]). When $m(\pi_0,\sigma_0)=0$, the inequality always holds. When $m(\pi_0,\sigma_0)=1$, from Proposition [Proposition 22](#pro: integral){reference-type="ref" reference="pro: integral"}, we obtain a nonzero meromorphic family in $${\mathrm{Hom}}_{G_{n^+}(F)}(\pi_{\underline{s}} \otimes \sigma_{\underline{s}'} \otimes \omega_{n^+,\psi_F}, \mathbbm{1}).$$ By taking the principal term at $\underline{s}=\underline{s_0}$, $\underline{s}'=\underline{s}_0'$, we obtain a nonzero element in $${\mathrm{Hom}}_{G_{n^+}(F)}(\pi_{\underline{s}_0} \otimes \sigma_{\underline{s}'_0} \otimes \omega_{n^+,\psi_F},\mathbbm{1}).$$ Therefore, $$m(\pi_{\underline{s}_0},\sigma_{\underline{s}'_0})\geq 1=m(\pi_0,\sigma_0).$$ This completes the proof of Proposition [Proposition 17](#pro: multiplicity){reference-type="ref" reference="pro: multiplicity"}. ### Estimates and Tempered intertwinings We use some estimates to prove Proposition [Proposition 22](#pro: integral){reference-type="ref" reference="pro: integral"}. For two functions $f_1$ and $f_2$ on $G$, we denote by $f_1\ll f_2$ if there is a constant $C>0$ such that $$| f_1(g) | <C | f_2(g) |,\quad \text{ for all }g\in G.$$ For a reductive algebraic group $G$ over $F$, we denote by $\Xi^G(g)$ the Harish-Chandra $\Xi$-function of $G(F)$, and denote by $\iota_G(g)$ the log-norm on $G(F)$. When $G$ is unitary or symplectic, we fix a minimal parabolic subgroup $P_0=M_0\ltimes N$ of $G_{n^+}$ and let $A_0$ be the maximal split subtorus of $M_0$. Let $r$ be the split rank of $A_0$. The Cartan decomposition of $G_{n^+}$ gives $$G_{n^+}(F)=K_{G_{n^+}}A^+_{G_{n^+}}K_{G_{n^+}}, \quad A^+_{G_{n^+}}=\set{a\in A_0(F) | |\alpha(a)|\geq 1,\alpha\in R(A_0,P_0)},$$ where $R(A_0,P_0)$ is the set of roots of $A_0$ in the unipotent radical of $P_0$. When taking $P_0$ to be upper-triangular, we have $$\label{equ: Cartan parameterize 1} A^+_{G_{n^+}}=\set{ {\rm diag}(\varpi^{n_1}_E,\cdots,\varpi^{n_r}_E) | n_1\geq n_2\geq \cdots n_r\geq 0},$$ where $\varpi_E$ is the uniformizer of ${\mathfrak {o}}_E$. When $G$ is metaplectic, if we take $K_{G_{n^+}}$, $A^+_{G_{n^+}}$ to be the inverse images of $K_{{\mathrm{Sp}}(2n^+)}$, $A^+_{{\mathrm{Sp}}(2n^+)}$, then $$G_{n^+}(F)=K_{G_{n^+}}A^+_{G_{n^+}}K_{G_{n^+}}.$$ From now on, we lift representations of $G_{n^+}(F)$ and $H_{n^+}(F)$ to representations of $\widetilde{G}_{n^+}$ ([\[equ: define wtG\]](#equ: define wtG){reference-type="eqref" reference="equ: define wtG"}). For preimages $\widetilde{a}_1$ and $\widetilde{a}_2$ of $a\in A_{{\mathrm{Sp}}(2n^+)}^+$, $$K_{G_{n^+}}\widetilde{a}_1K_{G_{n^+}}=K_{G_{n^+}}\widetilde{a}_2K_{G_{n^+}}.$$ We write this space of $K_{G_{n^+}}a K_{G_{n^+}}$ for simplicity. **Lemma 23**. *Given $v\otimes v' \otimes w\in \pi_K\otimes \sigma_{K'}\otimes \omega_{n^+,\psi_F}$ and $v^*\otimes v^{\prime, *} \otimes w^* \in \pi_K^\vee\otimes \sigma_{K'}^\vee \otimes \omega_{n^+,\psi_F}^\vee$, there exists $\epsilon>0$ such that $$\langle\pi_{\underline{0}}(g)v, v^*\rangle \langle(\sigma_{\underline{0}'}\otimes \omega_{n^+, \psi_F})(g,0)v'\otimes w,v^{\prime,*}\otimes w^*\rangle \ll \Xi^{G_{n^+}}(g)^2 %\iota_G(g)^{-d} \exp(-\epsilon\iota_{G_{n^+}}(g)).$$* *Proof.* We take $\widetilde{g}$ to be a preimage of $g$ in $\widetilde{G}_{n^+}(F)$. Since $\pi_{\underline{0}}$ and $\sigma_{\underline{0}'}$ are tempered and the matrix coefficients of tempered representations are weakly Harish-Chandra Schwartz ([@W03 III.2], [@Li12 Proposition 2.3.1]), there exists $d_1,d_2>0$ such that $$\langle\pi_{\underline{0}}(\widetilde{g})v, v^*\rangle \ll \Xi^{\widetilde{G}_{n^+}}(\widetilde{g})\iota_{\widetilde{G}_{n^+}}(\widetilde{g})^{d_1}$$ $$\langle\sigma_{\underline{0}'}(\widetilde{g})v', v^{\prime,*}\rangle \ll \Xi^{\widetilde{G}_{n^+}}(\widetilde{g})\iota_{\widetilde{G}_{n^+}}(\widetilde{g})^{d_2}.$$ It is known and easy to show that (cf. proof of [@GI11 Lemma 9.1]), $$\langle\omega_{n^+,\psi_F}(\widetilde{a}) w,w^*\rangle\ll |a_1\cdots a_r|^{1/2}, %\iota_{G_{n^+}}(\wt{a})^{-d'}, \quad a ={\rm diag}(a_1,\cdots ,a_r)\in A^+_{G_{n^+}},$$ where we fix a splitting of the metaplectic cover over $A_0(F)$. To prove this estimation, we take a polarization of the underlying symplectic or skew-Hermitian space $V = X \oplus V_0 \oplus X^\vee$ which is preserved by $A_0(F)$, with $V_0$ the anisotropic kernel and $X\cong E^r$ maximal totally isotropic. The Weil representation $\omega_{n^+,\psi_F}$ can be realized as the mixed model $\mathcal{S}(X)\otimes \omega_{V_0, \psi_F}$, where $\mathcal{S}(X)$ is the space of Bruhat-Schwartz functions on $X$ and $\omega_{V_0,\psi_F}$ is the Weil representation associated to $V_0$. Then $a =(a_1,\ldots,a_r)\in A_0(F)$ acts on $w= \phi\otimes\phi_0\in \mathcal{S}(X)\otimes \omega_{V_0, \psi_F}$ by $$\omega_{n^+,\psi_F}(\widetilde{a})w = \pm|a_1\cdots a_r|^{1/2}\phi(\cdot\, a) \otimes \phi_0.$$ For Bruhat-Schwartz functions $\phi, \phi'\in \mathcal{S}(X)$, one has that $$\int_{X}\phi(xa) \overline{\phi'(x)}dx \leq C_{\phi, \phi'},\quad \widetilde{a}\in A^+_{\widetilde{G}_{n^+}}$$ for some constant $C_{\phi,\phi'}$. This easily implies the required estimation $$\langle\omega_{n^+,\psi_F}(\widetilde{a}) w,w^*\rangle\ll |a_1\cdots a_r|^{1/2},\quad a \in A^+_{\widetilde{G}_{n^+}}.$$ From the Cartan decomposition of $\widetilde{G}_{n^+}$, there exists $\epsilon>0$ such that $$\langle\omega_{n^+,\psi_F}(\widetilde{g}) w,w^*\rangle\ll %\iota_{G_{n^+}}(\wt{g})^{-d'} \exp(-\epsilon \iota_{G_{n^+}}(\widetilde{g})).$$ The lemma follows easily. ◻ From [@W03 Lemme II.1.5] and [@BP20 Proposition 1.5.1 (v)], there exists $d>0$ such that the integral $$\int_{G_{n^+}(F)}\Xi^{G_{n^+}}(g)^2\iota_{G_{n^+}}(g)^{-d}dg$$ is absolutely convergent. Using this and Lemma [Lemma 23](#lem: estimate of matrix coefficient){reference-type="ref" reference="lem: estimate of matrix coefficient"}, we obtain Proposition [Proposition 22](#pro: integral){reference-type="ref" reference="pro: integral"} (1). For $\underline{s}\in (i{\mathbb {R}})^t$ and $\underline{s}'\in (i{\mathbb {R}})^{t'}$, the tempered intertwining ${\mathcal {L}}_{\pi}$ is defined as $${\mathcal {L}}_{\pi_{\underline{s}},\sigma_{\underline{s}'}}(v\otimes v'\otimes w,v^*\otimes v^{\prime,*}, w^*)={\mathrm {I}}_{v,v^*,v'\otimes w,v^{\prime,*}\otimes w^*}(\underline{s},\underline{s}').$$ The following theorem is crucial for many approaches to the local Gan-Gross-Prasad conjecture for tempered $L$-parameters. This result has been established in all the other situations. We refer to [@W12a Proposition 5.7] for non-archimedean Bessel cases, to [@BP20 Proposition 7.2.1] for archimedean Bessel cases and to [@X3 Theorem 3.2] for archimedean Fourier-Jacobi cases. **Theorem 24**. *For tempered representations $\pi_V,\pi_W$ of $G_{n^+},H_{n^+}$, if $m(\pi_V,\pi_W)\neq 0$, then ${\mathcal {L}}_{\pi_V\widehat{\otimes} \pi_W}\neq 0$.* *Proof.* The proof follows from the arguments in [@X3 §3] verbatim based on the estimate in Lemma [Lemma 23](#lem: estimate of matrix coefficient){reference-type="ref" reference="lem: estimate of matrix coefficient"}. ◻ *Proof for (2) of Proposition [Proposition 22](#pro: integral){reference-type="ref" reference="pro: integral"}.* We choose $\underline{s}\in (i{\mathbb {R}})^t$, $\underline{s}'\in (i{\mathbb {R}})^{t'}$ such that $s_i+s_j\neq 0$ for all $1\leq i\leq j\leq t$ and $s_i'+s_j'\neq 0$ for all $1\leq i\leq j \leq t'$, then $\pi_{\underline{s}},\sigma_{\underline{s'}}$ are irreducible and have the same $\chi$-parameter as $\pi_0,\sigma_0$, respectively. From the local Gan-Gross-Prasad conjecture for tempered parameters, $m(\pi_0,\sigma_0)\neq 0$ implies $m(\pi_{\underline{s}},\sigma_{\underline{s'}})\neq 0$. From Theorem [Theorem 24](#thm: tempered intertwining nonzero){reference-type="ref" reference="thm: tempered intertwining nonzero"}, we have ${\mathcal {L}}_{\pi_{\underline{s}},\sigma_{\underline{s}'}}\neq 0$, so there exist $v\otimes v' \otimes w\in \pi_K\otimes \sigma_{K'}\otimes \omega_{n^+,\psi_F}$ and $v^*\otimes v^{\prime, *} \otimes w^* \in \pi_K^\vee\otimes \sigma_{K'}^\vee \otimes \omega_{n^+,\psi_F}^\vee$ such that $${\mathrm {I}}_{v,v^*,v'\otimes w,v^{\prime,*}\otimes w^*}(\underline{s},\underline{s}')\neq 0.$$ ◻ ### Casselman's canonical pairing One key ingredient of Mœglin and Waldspurger's proof is Casselman's canonical pair for irreducible admissible representations. Since we are treating the Fourier-Jacobi models, we need the corresponding results for Heisenberg-oscillator representations. Let $\widetilde{P}=\widetilde{M}\ltimes N$ be the parabolic subgroup of $\widetilde{G}_{n^+}$ stabilizing a totally isotropic space flag $$X_{m_1}\subset \cdots \subset X_{m_t},$$ and we denote by $\widetilde{P}^-=\widetilde{M}\ltimes N^-$ the opposite parabolic subgroup of $\widetilde{P}$, which stabilizes a totally isotropic flag $$Y_{m_1}\subset \cdots \subset Y_{m_t}.$$ For $\pi_{n^+}\in \Pi_F(\widetilde{G}_{n^+})$, we denote by ${\mathrm{Jac}}_{\widetilde{P}}(\pi_{n^+})$ the Jacquet module of $\pi_{n^+}$ with respect to $\widetilde{P}$ and by $p_{N}$ the projection from $\pi_{n^+}$ to ${\mathrm{Jac}}_{\widetilde{P}}(\pi_{n^+})$. **Lemma 25**. *With the notations above, the following hold.* 1. *The coinvariant $$(\omega_{n^+,\psi_F})_{{N}\ltimes X_{m_t}}=V_{\omega_{n^+,\psi_F}}/{\mathrm{Span}}\{\omega_{n^+,\psi_F}(g)v-v\mid g\in N(F)\ltimes X_{m_t}(F),v\in V_{\omega_{n^+,\psi_F}}\}$$ is equal to $\omega_{n^+-m_t,\psi_F}$ is $\widetilde{M}(F)$-representation such that ${\mathrm{GL}}_{m_{i+1}-m_i}(E)$ acts trivially on it and $G_{n^+-m_t}(F)$ acts as Heisenberg-oscillator representation.* 2. *There is a nondegenerate $\widetilde{M}(F)$-equivariant pairing $$(\omega_{n^+,\psi_F})_{{N}\ltimes X_{m_t}}\times (\omega_{n^+,\psi_F^{-1}})_{N^-\ltimes Y_{m_t}}\to {\mathbb {C}}.$$* *Proof.* From [@GKT Lemma 3.55], we have $$(\omega_{n^+,\psi_F})_{X_{m_t}}=\omega_{n^+-m_t,\psi_F}.$$ With the mixed model of the Weil representation $\omega_{n^+,\psi_F}$ that is realized on $\mathcal{S}(X_{m_t}) \otimes \omega_{n^+-m_t,\psi_F}$, it is clear that ${N}(F)$ acts trivially on $\omega_{n^+-m_t,\psi_F}$. This proves Part (1). Part (2) follows from the nondegenerate $\widetilde{G}_{n^+-m_t}(F)$-equivariant bilinear pairing $$\omega_{n^+-m_t,\psi_F}\times \omega_{n^+-m_t,\psi_F^{-1}}\to {\mathbb {C}}.$$ ◻ Now, we are ready to do the explicit computation by using the parameterization $$\label{equ: Cartan} A^+_{G_{n^+}}=\set{{\varpi}^{\underline{n}}={\rm diag}(\varpi^{n_1}_E,\cdots,\varpi_E^{n_r}) | \underline{n}\in {{\mathcal {N}}}^+},\text{ where } {{\mathcal {N}}}^+=\set{(n_1,\cdots,n_r) | n_1\geq\cdots\geq n_{r}\geq 0 }.$$ For fixed $v,v^*,v',w,v^{\prime,*},w^*$, the smoothness of the representations implies the existence of an open compact subgroup $K_0$ of $K_{\widetilde{G}_{n^+}}$ such that $$\pi_{K}(K_0)v=v,\quad \sigma_{K}(K_0)v'=v',\quad \omega_{n^+,\psi_F}(K_0,0)w=w,$$ $$\pi_K^*(K_0)v^*=v^*,\quad \sigma_{K}(K_0)v^{\prime,*}=v^{\prime,*},\quad \omega_{n^+,\psi_F^{-1}}(K_0,0)w^*=w^*.$$ From the Cartan decomposition, $$\widetilde{G}_{n^+}(F)=\bigcup_{\underline{n}\in {\mathcal {N}}^+}K_{\widetilde{G}_{n^+}}\varpi^{\underline{n}} K_{\widetilde{G}_{n^+}}.$$ Then we express the integral as the infinite sum $$\label{equ: infinite sum}{\mathrm {I}}_{v,v^*,v'\otimes w,v^{\prime,*}\otimes w^*}(\underline{s},\underline{s}')=\frac{1}{[K_{\widetilde{G}_n}:K_0]^2}\sum_{\underline{n}\in {\mathcal {N}}^+}\sum_{x,x^*\in K_{\widetilde{G}_n}/K_0 } F_{\underline{n},x,x'}(\underline{s},\underline{s'}),$$ where $$\begin{aligned} F_{\underline{n},x,x'}(\underline{s},\underline{s'}) &= \mathrm{meas}(K_{\widetilde{G}_{n^+}}{\varpi}^{\underline{n}}K_{\widetilde{G}_{n^+}})\langle\pi_{\underline{s}}(\widetilde{{\varpi}^{\underline{n}}})\pi_K(x)v, \pi_K^*(x^*)v^*\rangle \langle\sigma_{\underline{s'}}(\widetilde{{\varpi}^{\underline{n}}})\sigma_K(x)v',\sigma_K^*(x^*)v^{\prime,*}\rangle\\ &\qquad\cdot\langle \omega_{n^+,\psi_F}(\widetilde{{\varpi}^{\underline{n}}},0)\omega_{n^+,\psi_F}(x)w, \omega_{n^+,\psi_F^{-1}}(x^*,0)w^*\rangle. \end{aligned}$$ From Casselman's canonical pairing for $\pi_{\underline{s}}$, $\sigma_{\underline{s}'}$ and Lemma [Lemma 25](#lem: Casselman's){reference-type="ref" reference="lem: Casselman's"}, we have that $$\begin{aligned} F_{\underline{n},x,x'}(\underline{s},\underline{s'}) &=\mathrm{meas}(K_{\widetilde{G}_{n^+}}{\varpi}^{\underline{n}}K_{\widetilde{G}_{n^+}}) \langle\pi_{\underline{s}}(\widetilde{{\varpi}^{\underline{n}}})p_{\widetilde{N}}(\pi_K(x)v),p^*_{\widetilde{N}^-}( \pi_K^*(x^*)v^*)\rangle \\ &\qquad\cdot\langle\sigma_{\underline{s'}}(\widetilde{{\varpi}^{\underline{n}}})p_{\widetilde{N}}(\sigma_K(x)v'),p^*_{\widetilde{N}^-}(\sigma_K^*(x^*)v^{\prime,*})\rangle\\ &\qquad\qquad \cdot\langle \omega_{n^+,\psi_F}(\widetilde{{\varpi}^{\underline{n}}},0)p_{X_{m_t}}(\omega_{n^+,\psi_F}(x,0)w), \omega_{n^+,\psi_F^{-1}}(x^*,0)p^*_{Y_{m_t}}(w^*)\rangle. \end{aligned}$$ For fixed $x,x'\in K_{\widetilde{G}_{n^+}}/K_0$, we have $$F_{\underline{n},x,x'}(\underline{s},\underline{s'})=C_{x,x'}\mathrm{meas}(K_{\widetilde{G}_{n^+}}{\varpi}^{\underline{n}}K_{\widetilde{G}_{n^+}})\chi_{\underline{s}}(\varpi^{\underline{n}})\chi_{\underline{s'}}(\varpi^{\underline{n}}),$$ where $\chi_{\underline{s}}$ and $\chi_{\underline{s}'}$ are the central characters of the representations $$\delta_{P}^{1/2}|\det|^{s_1}\rho_1 \otimes \cdots \otimes |\det|^{s_t}\rho_t \otimes \pi_0 \quad\text{and}\quad \delta_{P'}^{1/2}|\det|^{s_1'}\tau_1 \otimes \cdots \otimes |\det|^{s_{t'}'}\tau_t \otimes \sigma_0$$respectively. From [@M71 Proposition (3.2.15)], we have $$\mathrm{meas}(K_{\widetilde{G}_{n^+}}\varpi^{\underline{n}}K_{\widetilde{G}_{n^+}})=\delta_{P_0}(\varpi^{\underline{n}})Q^{\underline{n}}(q^{-1}),$$ where $Q^{\underline{n}}$ only depends on the set $S_{\underline{n}}=\set{1\leq i\leq r | n_i=n_{i+1}}$, here we take $n_{r+1}=0$. When $$|{\mathrm{Re}}(s_i)|,\ |{\mathrm{Re}}(s_i')|<\frac{1}{4r}$$ for every $1\leq i\leq r$, the infinite sum $$\sum_{\substack{\underline{n}\in{\mathcal {N}}^+\\S_{\underline{n}}=S}} F_{\underline{n},x,x'}(\underline{s},\underline{s'})=\sum_{\substack{\underline{n}\in{\mathcal {N}}^+\\S_{\underline{n}}=S}}C_{x,x'} \delta_{P_0}({\varpi}^{\underline{n}})\chi_{\underline{s}}(\varpi^{\underline{n}})\chi_{\underline{s'}}(\varpi^{\underline{n}})$$ converges to a rational function. From ([\[equ: infinite sum\]](#equ: infinite sum){reference-type="ref" reference="equ: infinite sum"}), we conclude Part (3) of Proposition [Proposition 22](#pro: integral){reference-type="ref" reference="pro: integral"}. 0000000000 J. Adams, D. Barbasch, *Genuine representations of the metaplectic group*. Compositio Mathematica, 1998, 113(1): 23-66. A. Aizenbud, D. Gourevitch, S. Rallis, and G. Schiffmann. Multiplicity one theorems. Annals of Mathematics, pages 1407--1434, 2010. J. Arthur, *The endoscopic classification of representations. Orthogonal and symplectic groups,* American Mathematical Society Colloquium Publications, 61. American Mathematical Society, Providence, RI, 2013. H. Atobe, On the Uniqueness of Generic Representations in an L-Packet, International Mathematics Research Notices, Volume 2017, Issue 23, December 2017, Pages 7051--7068. H. Atobe, *The local theta correspondence and the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case,* Math. Ann. **371** (2018), no. 1-2, 225--295. H. Atobe and W. T. Gan, *On the local Langlands correspondence for quasi-split even orthogonal groups,* Represent. Theory **21** (2017), 354--415. J. Bernstein and B. Krötz, *Smooth Fréchet globalizations of Harish-Chandra modules*, Israel J. Math. **199** (2014), no. 1, 45--111. I. Bernstein, A. Zelevinsky, *Induced representations of reductive ${\mathfrak {p}}$-adic groups. I,* Ann. Soc. ENS 10 (1977) 441--472 R. Beuzart-Plessis, *La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires,* Mém. Soc. Math. Fr. (N.S.) 2016, no. 149, vii+191 pp. R. Beuzart-Plessis, *A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean case,* Astérisque No. 418 (2020), viii + 299 pp. C. Chen, *The Local Gan-Gross-Prasad Conjecture for Special Orthogonal Groups over Archimedean Local Fields,* [arXiv:2102.11404](https://arxiv.org/abs/2102.11404). C. Chen, *Multiplicity formula for induced representations: Bessel and Fourier-Jacobi models over Archimedean local fields*, [arXiv:2308.02912](https://arxiv.org/abs/2308.02912). C. Chen and Z. Luo, *The local Gross-Prasad conjecture over ${\mathbb {R}}$: Epsilon dichotomy*, [arXiv:2204.01212](https://arxiv.org/abs/2204.01212). W. T. Gan, B. Gross, and D. Prasad, *Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups,* Sur les conjectures de Gross et Prasad. I. Astérisque No. 346 (2012), 1--109. W. T. Gan, B. Gross, and D. Prasad, *Branching laws for classical groups: the non-tempered case,* Compos. Math. **156** (2020), no. 11, 2298--2367. W. T. Gan and A. Ichino, *On endoscopy and the refined Gross-Prasad conjecture for $({\mathrm{SO}}_5,{\mathrm{SO}}_4)$*, J. Inst. Math. Jussieu 10 (2011), no. 2, 235--324. W. T. Gan and A. Ichino, *The Gross-Prasad conjecture and local theta correspondence,* Invent. Math. **206** (2016), no. 3, 705--799. W. T. Gan, S. Kudla, and S. Takeda. The Local Theta Correspondence ([prelimiary version)](https://sites.google.com/view/grothendieck-jr/theta-book). W. T. Gan and G. Savin, *Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence,* Compos. Math. **148** (2012), no. 6, 1655--1694. D. Ginzburg, S. Rallis, and D. Soudry, *The descent map from automorphic representations of $\mathrm{GL}(n)$ to classical groups*. World Scientific, 2011 R. Gomez and C. B. Zhu, *Local theta lifting of generalized Whittaker models associated to nilpotent orbits,* Geom. Funct. Anal. **24** (2014), no. 3, 796--853. H. He, *On the Gan-Gross-Prasad conjecture for $U(p,q)$,* Invent. Math. **209** (2017), no. 3, 837--884. D. Jiang, D. Liu, and L. Zhang, *Arithmetic wavefront sets and generic $L$-packets*, [arXiv:2207.04700](https://arxiv.org/abs/2207.04700). D. Jiang and L. Zhang, *Local root numbers and spectrum of the local descents for orthogonal groups: p-adic case,* Algebra Number Theory **12** (2018), no. 6, 1489--1535. D. Jiang and L. Zhang, *Arthur parameters and cuspidal automorphic modules of classical groups,* Ann. of Math. (2) **191** (2020), no. 3, 739--827. T. Kaletha, A. Minguez, S. W. Shin, and P.-J. White, *Endoscopic Classification of Representations: Inner Forms of Unitary Groups,* [arXiv:1409.3731](https://arxiv.org/abs/1409.3731). T. Kobayashi, *Recent developments in branching problems of representation theory*, Sūgaku 71 (2019), no. 4, 388--416; translated in [arXiv:2112.00642](https://arxiv.org/abs/2112.00642). T. Kobayashi and B. Speh, *Symmetry breaking for representations of rank one orthogonal groups,* Mem. Amer. Math. Soc. **238** (2015), no. 1126, v+110 pp. T. Kobayashi and B. Speh, *Symmetry breaking for representations of rank one orthogonal groups II,* Lecture Notes in Mathematics, 2234. Springer, Singapore, 2018. xv+342 pp. R. Langlands, *On the classification of irreducible representations of real algebraic groups*, Representation theory and harmonic analysis on semisimple Lie groups, 101--170, Math. Surveys Monogr., 31, Amer. Math. Soc., Providence, RI, 1989. W.-W. Li, *La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale*, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 5, 787--859. Y. Liu and B. Sun. *Uniqueness of Fourier--Jacobi models: the archimedean case,* Journal of Functional Analysis, **265** (2013), no.12, 3325--3344. Z. Luo, *A Local Trace Formula for the Local Gan-Gross-Prasad Conjecture for Special Orthogonal Groups,* [arXiv:2009.13947](https://arxiv.org/abs/2009.13947). I. G. Macdonald, *Spherical functions on a group of p-adic type,* Publ. Ramanujan Inst. 2 (1971). C. Mœglin, J.-L. Waldspurger, *La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général,* Sur les conjectures de Gross et Prasad. II. Astérisque No. 347 (2012), 167--216. C. Mok, *Endoscopic classification of representations of quasi-split unitary groups,* Mem. Amer. Math. Soc. **235** (2015), no. 1108. B.-Y. Sun. Multiplicity one theorems for Fourier-Jacobi models. American Journal of Mathematics, 134(6):1655--1678, 2012. D. Vogan, *The local Langlands conjecture*, Representation theory of groups and algebras, 305--379, Contemp. Math., 145, Amer. Math. Soc., Providence, RI, 1993. J-L. Waldspurger, *La formule de Plancherel pour les groupes p-adiques. D'apres Harish-Chandra.* Journal of the Institute of Mathematics of Jussieu 2.2 (2003): 235--333. J.-L. Waldspurger, *Une formule intégrale reliée à la conjecture locale de Gross-Prasad,* Compos. Math. **146** (2010), no. 5, 1180--1290. J.-L. Waldspurger, *Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2e partie: extension aux représentations tempérées,* Sur les conjectures de Gross et Prasad. I. Astérisque No. 346 (2012), 171--312. J.-L. Waldspurger, *La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux,* Sur les conjectures de Gross et Prasad. II. Astérisque No. 347 (2012), 103--165. N. Wallach, *Real reductive groups. II*, Pure and Applied Mathematics, 132-II. Academic Press, Inc., Boston, MA, 1992. H. Xue, *Bessel models for real unitary groups: the tempered case,* Duke Math. J. 172 (2023), no. 5, 995--1031. H. Xue, *Bessel models for unitary groups and Schwartz homology,* [preprint](https://www.math.arizona.edu/~xuehang/lggp_generic_v1.pdf). H. Xue, *Fourier-Jacobi models for real unitary groups,* [preprint](https://www.math.arizona.edu/~xuehang/local_FJ_u.pdf). [^1]: The research of C. Chen and D. Jiang is supported in part by the NSF Grant DMS--2200890; that of D. Liu is supported in part by National Key R&D Program of China No. 2022YFA1005300 and National Natural Science Foundation of China No. 12171421; that of L. Zhang is supported by AcRF Tier 1 grants A-0004274-00-00 and A-0004279-00-00 of the National University of Singapore. [^2]: For ${\mathrm{Mp}}_{2n}$ the character $\eta_a$ is denoted by $\eta[a]$ in [@GGP12]; we change the notation for uniformity.

arxiv_math

--- abstract: | This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank and size, together with information on the modular pairs of cyclic flats. We provide a formula which allows us to calculate $f$-vectors without the need of taking convex hulls or computing face lattices. We discuss the particular cases of sparse paving matroids and rank two matroids, which are of independent interest due to their appearances in other combinatorial and geometric settings. address: Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden author: - Luis Ferroni - Benjamin Schröter bibliography: - bibliography.bib title: Enumerating the faces of split matroid polytopes --- # Introduction To every matroid $\mathsf{M}$ one may associate its base polytope $\mathscr{P}(\mathsf{M})$, carrying all the information of the matroid. Not only this polytope plays prominent roles in combinatorial otpimization [@schrijver], but also is of fundamental importance in tropical geometry [@MaclaganSturmfels; @Joswig_book], the theory of valuations [@derksen-fink; @ardila-sanchez], combinatorial Hodge theory, and the study of matroid invariants [@berget-eur-spink-tseng; @eur-huh-larson; @ferroni-schroter]. A question that arises naturally in the study of a convex polytope $\mathscr{P}\subseteq \mathbb{R}^n$ is how many faces of each dimension $\mathscr{P}$ has. The ***f*-vector** of $\mathscr{P}$ is defined by $$f(\mathscr{P}) := (f_0, f_1,\ldots, f_{d-1},f_d),$$ where $f_i := \#\{\text{$i$-dimensional faces of $\mathscr{P}$}\}$ for each $i\in \{0,\ldots, d\}$ and $d:=\dim \mathscr{P}$. In particular, the number of vertices of $\mathscr{P}$ is just $f_0$, the number of facets of $\mathscr{P}$ is $f_{d-1}$, and $f_d=1$. The difficulty of calculating the $f$-vector may vary drastically depending on the polytope $\mathscr{P}$, on the properties it possesses, or on how it is described; for some concrete examples of the computation of $f$-vectors and certain related problems, see [@ziegler]. The family of possible vectors arising as the $f$-vector of a polytope is notoriously hard, and their classification is open in dimensions as low as four, see [@ziegler-2007]. Even in the case of $0/1$-polytopes of fixed dimension, although the set of possible $f$-vectors is finite, much remains to be discovered, see [@ziegler01]. In this article we will initiate the study of the explicit face enumeration of matroid polytopes, by focusing on the well-structured subclass of **split matroids**. The face structure of some special classes as positroids and lattice path matroids appeared in previous work, however without an explicit enumeration. The class of split matroids was introduced by Joswig and Schröter in [@joswig-schroter] to study tropical linear spaces. They have received considerable attention in the past few years, including a forbidden minor characterization [@cameron-mayhew], hypergraphs descriptions [@berczi], Tutte polynomial inequalities [@ferroni-schroter0], subdivisions and computation of valuations [@ferroni-schroter], and conjectures about exchange properties on the bases [@berczi-schwarcz] which are related to White's conjecture. Even though the $f$-vector of the matroid base polytope constitutes an invariant of the matroid $\mathsf{M}$ under isomorphisms, it is not valuative; see Example [Example 2](#example:f-non-valuative){reference-type="ref" reference="example:f-non-valuative"} below. This makes its computation considerably subtler and difficult. In particular, for the case of split matroids we require a non-trivial modification of the machinery presented in [@ferroni-schroter]. One important reason why split matroids deserve to be studied is that they encompass the classes of paving and copaving matroids. A long-standing conjecture often attributed to Crapo and Rota, appearing in print in [@mayhew], predicts that asymptotically almost all matroids are sparse paving. There is some evidence supporting this assertion [@pendavingh-vanderpol], but another intriguing conjecture affirms that even restricting to the enumeration of *non sparse paving* matroids, the class of split matroids will continue to be predominant [@ferroni-schroter Conjecture 4.15]. As of today, the problem of face enumeration of matroid polytopes has not been approached systematically in the literature, and to the best of our knowledge there are no prior articles addressing their computation. Nonetheless, there are some results that could be of interest in the study of $f$-vectors of certain classes of matroids. The computation of the cd-index of matroid polytopes of rank two appears in work by Kim [@kim]. In [@an-jung-kim] An, Jung and Kim investigated the lattice of faces of the base polytopes of lattice path matroids. In [@ardila-rincon-williams] Ardila, Rincón and Williams approached the lattice of faces of positroids, whereas in [@oh-xiang] Oh and Xiang studied the facets of positroid polytopes. In [@grande-sanyal] Grande and Sanyal used the faces of matroid polytopes to characterize their $k$-levelness. In all of the aforementioned cases, although combinatorial descriptions and properties of the faces of the polytope are provided, an explicit enumeration of them does not seem direct or easy. In [@postnikov-reiner-williams] Postnikov, Reiner and Williams described the $h$-vector of simple generalized permutohedra; however, although the class of generalized permutohedra encompasses the family of matroid base polytopes, these fail to be simple when the rank or the corank are greater than one. In particular, perhaps as a reminiscence of the situation for polytopes in general (and even for $0/1$-polytopes), questions about properties of $f$-vectors of matroid polytopes are widely open. ## Main results {#main-results .unnumbered} As mentioned before, the fact that the face numbers are not valuations makes the computation of the $f$-vector of matroid polytopes a delicate task. In the case of split matroids, we need more data than just the number of cyclic flats of each rank and size. Some information on their pairwise intersection is necessary. In order to express the $f$-vector of a polytope $\mathscr{P}$ in a more compact fashion, we will often refer to the ***f*-polynomial**, which is defined via: $$f_{\mathscr{P}}(t) := \sum_{i=0}^d f_i\cdot t^i.$$ Following the notation and terminology of [@ferroni-schroter], whenever we have a matroid $\mathsf{M}$ of rank $k$ and cardinality $n$, we will denote by $\uplambda_{r,h}$ the number of stressed subsets with non-empty cusp that $\mathsf{M}$ has --- for a connected split matroid, the number $\uplambda_{r,h}$ counts the number of proper non-empty cyclic flats of rank $r$ and size $h$ that $\mathsf{M}$ has. Although one of the main results of that article establishes that the numbers $\uplambda_{r,h}$ are enough to compute any valuative invariant on $\mathsf{M}$, we need further data to compute the $f$-vector. For a matroid $\mathsf{M}$ as before, we will denote by $\upmu_{\alpha,\beta,a,b}$ the number of **modular pairs** of cyclic flats $\{F_1,F_2\}$ such that $a = |F_1\smallsetminus F_2|$, $b = |F_2\smallsetminus F_1|$, $\alpha = \mathop{\mathrm{rk}}(F_1)-\mathop{\mathrm{rk}}(F_1\cap F_2)$, and $\beta = \mathop{\mathrm{rk}}(F_2)-\mathop{\mathrm{rk}}(F_1\cap F_2)$; see also equation [\[eq:star-property-pairs-of-flats\]](#eq:star-property-pairs-of-flats){reference-type="eqref" reference="eq:star-property-pairs-of-flats"} below. The following constitutes the main result of this article and is stated as Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"} further below. It tells us that the numbers $\upmu_{\alpha,\beta,a,b}$ are the precise additional datum needed to perform the computation of the $f$-vector of a split matroid polytope. Moreover, the statement tells us concretely how to calculate the number of faces of given dimension. **Theorem 1**. *Let $\mathsf{M}$ be a connected split matroid of rank $k$ on $n$ elements. The number of faces of its base polytope $\mathscr{P}(\mathsf{M})$ is given by the polynomial $$f_{\mathscr{P}(\mathsf{M})}(t) = f_{\Delta_{k,n}}(t)-\sum_{r,h} \uplambda_{r,h}\cdot u_{r,k,h,n}(t)-\sum_{\alpha,\beta,a,b} \upmu_{\alpha,\beta,a,b}\cdot w_{\alpha,\beta,a,b}(t)$$ where the first sum ranges over all values with $0<r<h<n$ and the second sum ranges over the values $0<\alpha<a$, $0<\beta<b$ for which either $a<b$ or $a=b$ and $\alpha\leq\beta$.* In the above theorem, the expressions $u_{r,k,h,n}(t)$ and $w_{\alpha,\beta,a,b}(t)$ are polynomials which depend only on their subindices. We present in Propositions [Proposition 7](#prop:nice-formula-w){reference-type="ref" reference="prop:nice-formula-w"} and [Proposition 8](#prop:nice-formula-u){reference-type="ref" reference="prop:nice-formula-u"} explicit (but complicated) formulas for them which allow us or a computer to calculate the face numbers effortlessly. A formula for the $f$-vector of the hypersimplex $\Delta_{k,n}$ is also given explicitly in Example [Example 1](#ex:hypersimplex){reference-type="ref" reference="ex:hypersimplex"}. In particular, the entire calculation can be done without the necessity of building costly face lattices or computing convex hulls. As two direct but interesting application of our result, we provide closed expressions for the $f$-vector of sparse paving matroids, a class that made a prominent appearance in the theory of the extension complexity of independence polytopes [@rothvoss]. We also prove a fairly explicit formula for the $f$-vector of arbitrary rank two matroids, which is of independent interest due to the connection of these polytopes with edge polytopes of complete multipartite graphs [@ohsugi-hibi]. # The number of faces of split matroids {#sec:two} ## The set up Throughout this paper we will assume that the reader is familiar with the usual terminology and notation in matroid theory. For the notions and machinery introduced very recently, in particular about **stressed subsets**, **relaxations**, and **cuspidal matroids** we refer the reader to our previous article [@ferroni-schroter Sections 3--4]. Regarding **split matroids** and **elementary split matroids** the reader can consult the same article as well as [@joswig-schroter; @berczi]. However, basic knowledge on polytopes should be enough to follow the arguments and methods in this manuscript. For a $d$-dimensional polytope $\mathscr{P}$ we denote by $f(\mathscr{P}):=(f_0,\ldots,f_d)$ its *$f$-vector*, and by $$f_{\mathscr{P}}(t) := \sum_{i=0}^{d} f_i\ t^i$$ its *$f$-polynomial*. In both cases, $f_i$ denotes the number of $i$-dimensional faces of $\mathscr{P}$. Notice that we omit the inclusion of $f_{-1}:=1$ for the empty set in both the $f$-vector and the $f$-polynomial, but we do include $f_d=1$ for the polytope itself. A basic property of $f$-polynomials that we will use without explicitly mention is the fact that it behaves multiplicatively under cartesian products of polytopes, i.e., $f_{\mathscr{P}_1\times \mathscr{P}_2}(t) = f_{\mathscr{P}_1}(t) \cdot f_{\mathscr{P}_2}(t)$. Another basic property that we use implicitly is that a matroid on a ground set of size $n$ with exactly $c$ connected components has a base polytope of dimension $\dim \mathscr{P}(\mathsf{M}) = n - c$. **Essential notation 1**. Following [@ferroni-schroter], whenever we have a matroid $\mathsf{M}$, unless specified otherwise, the rank of $\mathsf{M}$ is denoted by $k$ and the size of its ground set is denoted by $n$. We reserve the letters $r$ and $h$ for the rank and the size of stressed subsets that $\mathsf{M}$ may possess. Our aim is to find formulas for the number of faces a matroid base polytope $\mathscr{P}(\mathsf{M})$ whenever the matroid $\mathsf{M}$ is connected, i.e., $\mathscr{P}(\mathsf{M})\subseteq \mathbb{R}^n$ is of dimension $n-1$, and split. Note that under the assumption of connectedness the classes of split matroids and elementary split matroids coincide [@berczi Theorem 11]. Since the base polytope of a direct sum of matroids $\mathsf{M}_1\oplus \mathsf{M}_2$ is the cartesian product of $\mathscr{P}(\mathsf{M}_1)$ and $\mathscr{P}(\mathsf{M}_2)$, the $f$-vector of any disconnected split matroid can be recovered from the $f$-vector of the connected components, all of which are split as well. The most basic example of a matroid polytope is the hypersimplex $\Delta_{k,n}$, the matroid base polytope of the uniform matroid $\mathsf{U}_{k,n}$ of rank $k$ on $n$ elements. **Example 1**. The face enumeration of hypersimplices is encoded in the following $f$-polynomial: $$\begin{aligned} f_{\mathscr{P}(\mathsf{U}_{k,n})}(t) = f_{\Delta_{k,n}}(t) %&= \binom{n}{k} + \sum^{n-1}_{d=1} \sum_{j = k-d}^{k-1} \binom{n}{j} \binom{n-j}{n-d-j-1} t^d\\ &= \binom{n}{k} + \sum^{n-1}_{i=1} \binom{n}{i+1}\sum_{j = 1}^{i} \binom{n-i-1}{k-j}\cdot t^i \enspace . \end{aligned}$$ This formula can be obtained by contracting and deleting the elements of $\mathsf{U}_{k,n}$. That is, by intersecting with hyperplanes of the form $x_i=0$ or $x_i=1$ (and forgetting the coordinate $i$) which leads to lower dimensional hypersimplices. For a detailed proof see for example [@hibi-li-ohsugi Corollary 1.4]. The next example is similar to the one in [@ferroni2 Remark 5.9] and gives a glimpse of the subtlety of the $f$-vector as a matroid invariant. In general, we see that the assignment $\mathsf{M}\mapsto f_{\mathscr{P}(\mathsf{M})}(t)$ is an invariant of the matroid $\mathsf{M}$ that fails to be valuative. Hence its computation is a more delicate task, even for the case of paving or split matroids. In these cases, we cannot rely on the strength of [@ferroni-schroter Theorem 5.40] --- that result asserts that the evaluation of a valuative invariant on a split matroid $\mathsf{M}$ can be achieved by knowing relatively little about the matroid $\mathsf{M}$, consisting in its rank $k$, its size $n$, and parameters $\uplambda_{r,h}$ which denote the number of stressed subsets with non-empty cusp of rank $r$ and size $h$ that $\mathsf{M}$ has. If one is interested in knowing the $f$-vector of $\mathscr{P}(\mathsf{M})$, the matroidal information we just mentioned is far from enough. One of the main difficulties in order to carry out the enumeration of the faces of $\mathscr{P}(\mathsf{M})$ consists of first identifying what matroid data we need in addition to the parameters mentioned before. **Example 2**. Consider the four matroids $\mathsf{U}_{3,6}$, $\mathsf{M}$, $\mathsf{N}_1$ and $\mathsf{N}_2$ with ground set $\{1,\ldots,6\}$ and rank three, whose families of bases are given as follows: $$\begin{aligned} \mathscr{B}(\mathsf{U}_{3,6}) &:= \binom{[6]}{3}, &\mathscr{B}(\mathsf{N}_1) &:= \binom{[6]}{3} \smallsetminus \{\{1,2,3\}, \{4,5,6\}\}\\ \mathscr{B}(\mathsf{M}) &:= \binom{[6]}{3} \smallsetminus \{\{1,2,3\}\}, &\mathscr{B}(\mathsf{N}_2) &:= \binom{[6]}{3} \smallsetminus \{\{1,2,3\}, \{3,4,5\}\}. \end{aligned}$$ The $f$-vectors of their base polytopes are respectively: $$\begin{aligned} f(\mathscr{P}(\mathsf{U}_{3,6})) &= (20, 90, 120, 60, 12, 1), &f(\mathscr{P}(\mathsf{N}_1)) &= (18, 72, 102, 60, 14, 1),\\ f(\mathscr{P}(\mathsf{M})) &= (19, 81, 111, 60, 13, 1), &f(\mathscr{P}(\mathsf{N}_2)) &= (18, 72, 101, 59, 14, 1). \end{aligned}$$ All of these matroids are sparse paving. In particular, the two matroids $\mathsf{N}_1$ and $\mathsf{N}_2$ have, e.g., the same Tutte polynomial and the same Ehrhart polynomial --- in fact, via [@ferroni-schroter Corollary 5.41] any valuative invariant on these two matroids yields the same result. Yet, observe that their $f$-vectors differ in the third and the fourth entries. ## Cuspidal matroids and a technical lemma By using [@ferroni-schroter Corollary 5.4], we see that the intersection of the hypersimplex $\Delta_{k,n}$ with the half-space of a single split hyperplane leads to the polytope: $$\label{eq:cuspidal} \mathscr{P}(\mathsf{\Lambda}_{k-r,k,n-h,n}) = \left\{x\in\Delta_{k,n}\vphantom{\sum_{i=1}^h x_i\leq r}\ :\ \sum_{i=1}^h x_i\leq r\vphantom{x\in\Delta_{k,n}}\right\} \enspace .$$ for appropriate values $r$ and $h$. This is the base polytope of the cuspidal matroid $\mathsf{\Lambda}_{k-r,k,n-h,n}$, a matroid having exactly three cyclic flats: the empty set, the entire ground set, and one proper cyclic flat having size $h$ and rank $r$. For the purposes of this paper, the reader may regard equation [\[eq:cuspidal\]](#eq:cuspidal){reference-type="eqref" reference="eq:cuspidal"} as the definition of cuspidal matroids. Let us introduce some notation that will help us formulate later our main results in a more compact way: $$\label{eq:definition-of-u} u_{r,k,h,n}(t) := f_{\Delta_{k,n}}(t)-f_{\mathscr{P}(\mathsf{\Lambda}_{k-r,k,n-h,n})}(t).$$ The $i$-th coefficient of this polynomial is the difference between the number of $i$-dimensional faces of the hypersimplex $\Delta_{k,n}$ and the number of $i$-dimensional faces of the cuspidal matroid $\mathsf{\Lambda}_{k-r,k,n-h,n}$. A non-obvious property is that some of these coefficients may be negative while other are positive --- moreover, the actual sign of each individual coefficient a priori depends on the four parameters $r,k,h,n$. Before we go on with let us introduce a second polynomial, which will play an important role in the sequel. For fixed numbers $0< \alpha < a$ and $0<\beta < b$ let us define, $$\begin{aligned} w_{\alpha,\beta,a,b}(t) &:= f_{\Delta_{\alpha+\beta,a+b}}(t)-f_{\Delta_{\alpha,a}}(t)\cdot f_{\Delta_{\beta,b}}(t)-u_{\alpha,\alpha+\beta,a,a+b}(t)-u_{\beta,\alpha+\beta,b,a+b}(t)\\ &\; = f_{\mathscr{P}(\mathsf{\Lambda}_{\beta,\alpha+\beta,b,a+b})}(t) + f_{\mathscr{P}(\mathsf{\Lambda}_{\alpha,\alpha+\beta,a,a+b})}(t) - f_{\Delta_{\alpha+\beta,a+b}}(t) - f_{\Delta_{\alpha,a}}(t)\cdot f_{\Delta_{\beta,b}}(t).\end{aligned}$$ Later, in Proposition [Proposition 7](#prop:nice-formula-w){reference-type="ref" reference="prop:nice-formula-w"}, we provide a compact formula for the polynomials $w_{\alpha,\beta,a,b}(t)$ and a formula for the polynomials $u_{r,k,h,n}(t)$ in Proposition [Proposition 8](#prop:nice-formula-u){reference-type="ref" reference="prop:nice-formula-u"} both of which can be used to compute these polynomials, bypassing the computation of $f$-vectors of cuspidal matroids using the polytopes themselves. **Remark 3**. The intuition of why it is reasonable to consider and define the complicated expression above stems from [@ferroni-schroter Example 5.39]. As follows from the explanation there, if the assignment $\mathsf{M}\mapsto f_{\mathscr{P}(\mathsf{M})}(t)$ were valuative, then the defining formula for $w_{\alpha,\beta,a,b}(t)$ would actually be identically zero. The polynomial $w_{\alpha,\beta,a,b}(t)$ quantifies (in a certain way) how far the map $\mathsf{M}\mapsto f_{\mathscr{P}(\mathsf{M})}(t)$ is from being valuative. The following lemma is the key in the proof of our main result. Its proof constitutes arguably the most technical part of the paper. In the Lemma and the text below we denote the cuspidal matroid of rank $k$ on $n$ elements with cyclic flat $F$ by $\mathsf{\Lambda}_{k,n}^{F}$. This matroid is isomorphic to the matroid $\mathsf{\Lambda}_{k-\operatorname{rk}F,k,n-|F|,n}$. Similarly, for a set $N$ of coordinates we use the notation $\Delta_{k,N}\cong\Delta_{k,|N|}$ to indicate the coordinates of the hypersimplex. **Lemma 4**. *Let $\mathsf{N}$ be a rank $k$ split matroid on $[n]$ whose cyclic flats are the four sets $\varnothing$, $F$, $G$, and $F\cup G=[n]$ of rank $0$, $r_F$, $r_G$, and $k$, respectively. Then $$\begin{aligned} \label{eq:twosplits} f_{\mathscr{P}(\mathsf{N})}(t) &= f_{\mathscr{P}(\mathsf{\Lambda}_{k,n}^{F})}(t) + f_{\mathscr{P}(\mathsf{\Lambda}_{k,n}^{G})}(t) - f_{\Delta_{k,n}}(t) \end{aligned}$$ if $|F\cap G| + k < r_F+r_G$, and $$\begin{aligned} \label{eq:twosplits_w} f_{\mathscr{P}(\mathsf{N})}(t) &= f_{\mathscr{P}(\mathsf{\Lambda}_{k,n}^{F})}(t) + f_{\mathscr{P}(\mathsf{\Lambda}_{k,n}^{G})}(t) - f_{\Delta_{k,n}}(t) - w_{r_F-c,r_G-c,|F|-c,|G|-c}(t) \end{aligned}$$ where $c = |F\cap G|$ if $|F\cap G| + k = r_F+r_G$.* *Proof.* Note either the matroid $\mathsf{N}$ is connected or it is the direct sum $\mathsf{U}_{r_F,|F|}\oplus\mathsf{U}_{r_G,|G|}$. The matroid polytope of the latter is $\mathscr{P}(\mathsf{U}_{r_F,|F|}\oplus\mathsf{U}_{r_G,|G|}) = \Delta_{r_F,|F|}\times\Delta_{r_G,|G|}$ with $f$-polynomial $f_{\Delta_{r_F,|F|}}(t)\cdot f_{\Delta_{r_G,|G|}}(t)$. Moreover, in this case $|F\cap G| = 0$ and $k=r_F+r_G$. Thus formula [\[eq:twosplits_w\]](#eq:twosplits_w){reference-type="eqref" reference="eq:twosplits_w"} applies by definition of $w_{r_F,r_G,|F|,|G|}$. From now on, we assume that $\mathsf{N}$ is connected. We will compare the faces of the polytope $\mathscr{P}(\mathsf{N})$ with those of $\Delta_{k,n}$. By [@joswig-schroter Proposition 7] every $d$-face of $\mathscr{P}(\mathsf{N})=\mathscr{P}(\mathsf{\Lambda}_{k,n}^{F})\cap\mathscr{P}(\mathsf{\Lambda}_{k,n}^{G})$, and also $\mathscr{P}_F:=\mathscr{P}(\mathsf{\Lambda}_{k,n}^{F})$ or $\mathscr{P}_G:=\mathscr{P}(\mathsf{\Lambda}_{k,n}^{G})$ lies in a $d$-face of the hypersimplex $\Delta_{k,n}$ or in at least one of the hyperplanes $H_F=\{x\in\mathbb{R}^d\,|\,\sum_{i\in F} x_i = r_F\}$ and $H_G=\{x\in\mathbb{R}^d\,|\,\sum_{i\in G} x_i = r_G\}$. Notice further that by [@joswig-schroter Proposition 14] we have that $|F\cap G| + k \leq r_F+r_G$. Thus the statement of the lemma covers all possible cases, and there is no point in $\Delta_{k,n}$ violating both of the inequalities $\sum_{i\in F} x_i \leq r_F$ and $\sum_{i\in G} x_i \leq r_G$ simultaneously. Therefore, the two hyperplanes $H_F$ and $H_G$ split a face of $\Delta_{k,n}$ in at most three maximal dimensional polytopes. Now, let us analyze the various cases. If $Q'$ is a $d$-dimensional face of $\mathscr{P}(\mathsf{N})$ and not contained in a $d$-face of the hypersimplex $\Delta_{k,n}$. Then either either $Q'$ lies in exactly one of the hyperplanes $H_F$ and $H_G$, or in both. If it is in exactly one of them, say $H_F$, then $Q'$ is a $d$-dimensional face of $\mathscr{P}_F$ and not a $d$-dimensional face of $\mathscr{P}_G$. If the $d$-dimensional face $Q'$ lies in both hyperplanes then $Q'$ lies in a $(d+1)$-dimensional face $Q$ of $\Delta_{k,n}$ which is subdivided into two parts. We discuss this situation in detail further below in this proof. Let us now assume $Q$ is a $d$-dimensional face of $\Delta_{k,n}$. We will go through the three possibilities of how many cells $Q$ is subdivided into by $H_F$ and $H_G$. If $Q$ remains undivided then either it is a face of $\mathscr{P}(\mathsf{N})$ and also of both $\mathscr{P}_F$ and $\mathscr{P}_G$, or it is not a face of $\mathscr{P}(\mathsf{N})$ and thus a face of exactly one of the two polyhedra $\mathscr{P}_F$ and $\mathscr{P}_G$. In the case that the face $Q$ is subdivided into three parts and of dimension $d$, it contributes a $d$-face to each of the four polytopes as well. It remains to analyze the situation in which $Q$ is subdivided into two polytopes $Q'$ and $Q''$. It cannot happen that both of these polytopes are faces of $\mathscr{P}(\mathsf{N})$. If one of them, say $Q'$, is a face of $\mathscr{P}(\mathsf{N})$ then $Q'$ is a face of one of the polytopes $\mathscr{P}_F$ and $\mathscr{P}_G$, while $Q$ is a face of the other. In total $Q$ and $Q'$ contribute a $d$-dimensional face to each of the four polytopes. If neither $Q'$ nor $Q''$ is a face of $\mathscr{P}(\mathsf{N})$ then $Q'$ and $Q''$ meet in a $(d-1)$-face $\widetilde{Q}\subsetneq Q$ which is a face of the three polytopes $\mathscr{P}(\mathsf{N})$, $\mathscr{P}_F$ and $\mathscr{P}_G$. Moreover, the $d$-dimensional face $Q'$ is a face of one of the two polytopes $\mathscr{P}_F$ and $\mathscr{P}_G$ while $Q''$ is a $d$-dimensional face of the other one. The $(d-1)$-dimensional face $\widetilde{Q}$ is not a face of the hypersimplex $\Delta_{k,n}$ and sits in both hyperplanes $H_F$ and $H_G$. Every $(d-1)$-dimensional face that is not a face of $\Delta_{k,n}$ but lies in both $H_F$ and $H_G$ must be contained in a $d$-face of $\Delta_{k,n}$ as we mentioned at the beginning of this proof. Let us investigate the situation further. There must exist four pairwise disjoint sets $A,B,C,D$ such that $$Q = \Delta_{|C|,C}\times \Delta_{k-|C|,A\cup B}\times\Delta_{0,|D|}$$ where $C$ consists of all coordinates which are one and $D$ of those which are $0$, and $$\widetilde{Q} = \Delta_{|C|,C}\times \Delta_{\alpha,A}\times\Delta_{\beta,B}\times\Delta_{0,|D|}$$ where $\alpha = r_F-|C\cap F|$, $\beta = r_G-|C\cap G|$, $A\subseteq F\smallsetminus G$ and $B\subseteq G\smallsetminus F$. The inclusions follow (up to interchanging the roles of $A$ and $B$) from the fact that both hyperplanes $H_F$ and $H_G$ induce the same split of $\Delta_{\alpha+\beta,A\cup B}$. From this we get $k-|C| = \alpha + \beta =r_F-|F\cap C|+r_G-|G\cap C|$ and hence $$\begin{aligned} |F\cap G| \leq r_F+r_G-k = |F\cap C| + |G\cap C| - |C| = |F\cap G\cap C| - |C\smallsetminus(F\cup G)|\enspace . \end{aligned}$$ We obtain $F\cap G \subseteq C \subseteq F\cup G$, and also the equality $|F\cap G|+k = r_F+r_G$. Furthermore, every choice of such $C$, $A\subseteq F\smallsetminus G$ and $B\subseteq G\smallsetminus F$ leads to a face that sits in both hyperplanes and in a higher dimensional face of $\Delta_{k,n}$. Hence, for the inclusion-wise maximal such face we have to pick $C=F\cap G$ and $D=[n]\smallsetminus(F\cup G)$. We see that the expression $$w_{r_F-|C|,r_G-|C|,|F|-|C|,|G|-|C|}(t)$$ counts exactly the two types of faces. In summary, our comparison verifies the formulas of equations [\[eq:twosplits\]](#eq:twosplits){reference-type="eqref" reference="eq:twosplits"} and [\[eq:twosplits_w\]](#eq:twosplits_w){reference-type="eqref" reference="eq:twosplits_w"} and thus the proof is complete. ◻ ## Face counting of split matroids For a connected split matroid $\mathsf{M}$, let us define the following numbers that we have already mentioned in the introduction. The number of stressed subsets with non-empty cusp having rank $r$ and size $h$, denoted $\uplambda_{r,h}$ --- recall that by [@ferroni-schroter Proposition 3.10], in a connected split matroid this is the same as the number of proper non-empty cyclic flats of rank $r$ and size $h$. We also need the numbers $\upmu_{\alpha,\beta,a,b}$ of (unordered) modular pairs $\{F_1,F_2\}$ of proper non-empty cyclic flats, i.e., $F_1$ and $F_2$ fulfilling the *modularity property*, $$\label{eq:star-property-pairs-of-flats} \operatorname{rk}(F_1) + \operatorname{rk}(F_2)= \operatorname{rk}(F_1\cap F_2) + \operatorname{rk}(F_1\cup F_2) , \tag{$\star$}$$ where the indices denote the following quantities: $$\begin{aligned} a &=|F_1\smallsetminus F_2|,& \alpha &= \operatorname{rk}F_1 - \operatorname{rk}(F_1\cap F_2)\\ b &=|F_2\smallsetminus F_1|,& \beta&=\operatorname{rk}F_2 - \operatorname{rk}(F_1\cap F_2) \enspace .\end{aligned}$$ Note that the set $F_1\cap F_2 \subsetneq F_1 \subsetneq [n]$ can not contain a circuit if $\mathsf{M}$ is a connected split matroid, thus it is an independent set, i.e., $\operatorname{rk}(F_1\cap F_2)=|F_1\cap F_2|$. In terms of these numbers and variables, our main result is the following theorem. **Theorem 5**. *Let $\mathsf{M}$ be a connected split matroid of rank $k$ on $n$ elements. The number of faces of its base polytope $\mathscr{P}(\mathsf{M})$ is given by the polynomials $$\label{eq:main-result} f_{\mathscr{P}(\mathsf{M})}(t) = f_{\Delta_{k,n}}(t)-\sum_{r,h} \uplambda_{r,h}\cdot u_{r,k,h,n}(t)-\sum_{\alpha,\beta,a,b} \upmu_{\alpha,\beta,a,b}\cdot w_{\alpha,\beta,a,b}(t)$$ where the first sum ranges over all values with $0<r<h<n$ and the second sum ranges over the values $0<\alpha<a$, $0<\beta<b$ for which either $a<b$ or $a=b$ and $\alpha\leq\beta$.* Before moving towards the proof of this result, let us digress about the meaning of its statement. On one hand, note that the polynomials $f_{\Delta_{k,n}}(t)$, $u_{r,k,h,n}(t)$ and $w_{\alpha,\beta,a,b}(t)$ can be precomputed for all the occurring instances of the variables which appear as subindices. The first non-trivial fact that is deduced by our statement is that in addition to the parameters $\uplambda_{r,h}$, which always appear in the computation of a valuative invariant, the precise additional matroidal datum needed to compute the $f$-vector consists of the numbers $\upmu_{\alpha,\beta,a,b}$. Surprisingly, the last sum in equation [\[eq:main-result\]](#eq:main-result){reference-type="eqref" reference="eq:main-result"} does not take into consideration the rank nor the size of the matroid $\mathsf{M}$ itself, only the intersection data for the modular pairs of flats. The second non-trivial fact is that it explains how to put together this information in order to effectively computing the $f$-vector of $\mathscr{P}(\mathsf{M})$ for a split matroid, circumventing the necessity of constructing the polytope. *Proof of Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"}..* We follow the guidance of the proof of Lemma [Lemma 4](#lem:key){reference-type="ref" reference="lem:key"} and compare the faces of $\mathscr{P}(\mathsf{M})$ with those of $\Delta_{k,n}$ taking into account the polytopes $\mathscr{P}(\mathsf{\Lambda}_{k,n}^{F_j})$ for the various cyclic flats $F_j$. We recall that there is no point in $\Delta_{k,n}$ that violates more than one of the inequalities $\sum_{i\in F_j} x_i \leq \operatorname{rk}F_j$. Now, if $Q$ is a $d$-face of $\Delta_{k,n}$ that contains a $d$-face of $\mathscr{P}(\mathsf{M})$ then every polytope $\mathscr{P}(\mathsf{\Lambda}_{k,n}^{F_i})$ has a $d$-face that is contained in $Q$. Hence, these faces do not contribute to any $u_{r,k,h,n}$ or $w_{\alpha,\beta,a,b}$. If $Q$ is a $d$-face of $\Delta_{k,n}$ whose interior has no point in common with $\mathscr{P}(\mathsf{M})$, then all these points lie beyond exactly one of the split hyperplanes $\sum_{i\in F_j} x_i = \operatorname{rk}F_j$. Therefore, $Q$ contributes $1$ to $f_{\Delta_{k,n}}(t)$ and to $u_{\operatorname{rk}_{F_j},k,|F_j|,n}(t)$, but not to the other polynomials. It remains again to look at the $(d-1)$-faces $\widetilde{Q}$ of $\mathscr{P}(\mathsf{M})$ that are contained in the interior of $d$-face $Q$ of $\Delta_{k,n}$. Note that $\widetilde{Q}$ is a hyperplane in $Q$ that separates two of the vertices of $Q$. Each of the two vertices can only violate one of the inequalities, thus $\widetilde{Q}$ is a face of exactly two of the polytopes $\mathscr{P}(\mathsf{\Lambda}_{k,n}^{F_j})$. Let us assume that $\overline{Q}$ is inclusion-wise maximal among all these $(d-1)$-faces of $\mathscr{P}(\mathsf{M})$ in the relative interior of $d$-faces of $\Delta_{k,n}$. Then there must be pairwise disjoint sets $C$ and $D$ in the complement of $F\cup G$ such that $\overline{Q}$ is a face of $$\Delta_{|C|,C}\times\Delta_{k-|C|,F\cup G}\times\Delta_{0,D}$$ where $F$ and $G$ are the two cyclic flats for which $\overline{Q}$ is a face of $\mathsf{\Lambda}_{k,n}^{F}$ and $\mathsf{\Lambda}_{k,n}^{G}$. It follows that $k-|C|\leq \operatorname{rk}_{\mathsf{M}}(F\cup G)$ as $\overline{Q}$ is a face of $\mathscr{P}(\mathsf{M})$. Furthermore, applying Lemma [Lemma 4](#lem:key){reference-type="ref" reference="lem:key"} to the matroid $(\mathsf{M}\smallsetminus D) / C$, i.e., the contraction of $C$ and the deletion of $D$ in $\mathsf{M}$, which is a split matroid of rank $k-|C|$ on $F\cup G$, yields that the hyperplanes of $F$ and $G$ coincide only if $k-|C|=\operatorname{rk}F + \operatorname{rk}G - |F\cap G|$. Moreover, because $\mathsf{M}$ is a connected split matroid we get $$|F\cap G| + \operatorname{rk}(F\cup G) \leq |F\cap G| + k \leq \operatorname{rk}F + \operatorname{rk}G \enspace .$$ Thus $k-|C| = \operatorname{rk}F\cup G$ and $|F\cap G| = \operatorname{rk}F + \operatorname{rk}G - \operatorname{rk}(F\cup G)$. Furthermore, Lemma [Lemma 4](#lem:key){reference-type="ref" reference="lem:key"} shows also that the faces $\widetilde{Q}$ and $Q$ contribute a summand $t^{d-1}$ and $t^{d}$ to the corresponding $w_{\alpha,\beta,a,b}(t)$. This completes our proof. ◻ **Example 6**. Let us take a look again at Example [Example 2](#example:f-non-valuative){reference-type="ref" reference="example:f-non-valuative"}. The matroids $\mathsf{N}_1$ and $\mathsf{N}_2$ are sparse paving, have rank $k=3$ and size $n=6$. In each case the proper non-empty cyclic flats are exactly the non-bases, yielding for both matroids $\uplambda_{2,3,3,6}=2$. One can compute the corresponding polynomial, $u_{2,3,3,6}(t) = 1+9t+9t^2-t^4$. In $\mathsf{N}_1$, the intersection of the only pair of proper non-empty cyclic flats, $F_1=\{1,2,3\}$ and $F_2=\{4,5,6\}$, does not satisfy the property [\[eq:star-property-pairs-of-flats\]](#eq:star-property-pairs-of-flats){reference-type="eqref" reference="eq:star-property-pairs-of-flats"}, because $\mathop{\mathrm{rk}}(F_1\cap F_2)+\mathop{\mathrm{rk}}(F_1\cup F_2)=0+3$, whereas $\mathop{\mathrm{rk}}(F_1)+\mathop{\mathrm{rk}}(F_2) = 2 + 2 = 4$. For $\mathsf{N}_2$, the situation is different, as $F_1 = \{1,2,3\}$ and $F_2=\{3,4,5\}$ indeed satisfy [\[eq:star-property-pairs-of-flats\]](#eq:star-property-pairs-of-flats){reference-type="eqref" reference="eq:star-property-pairs-of-flats"}, and we have $a = |F_1\smallsetminus F_2|=2$, $b = |F_2\smallsetminus F_1|=2$, $\alpha = \mathop{\mathrm{rk}}(F_1)-|F_1\cap F_2| = 2 - 1 = 1$, and $\beta=\mathop{\mathrm{rk}}(F_2)-|F_1\cap F_2| = 2 - 1=1$, so that $\upmu_{1,1,2,2}=1$ and we need to subtract $w_{1,1,2,2}(t) = t^2+t^3$ to obtain the correct $f$-polynomial, as we expected. ## Explicit formulas The polynomials $u_{r,k,h,n}(t)$ and $w_{\alpha,\beta,a,b}(t)$ in Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"} are defined in terms of $f$-vectors of specific matroid polytopes. In this subsection we will present explicit descriptions for these polynomials, enabling us to do the face enumeration of a split matroid polytope, without any convex hull or face lattice computation. To express the formulas in a compact form, we will make use of multinomial coefficients. Let $i,j,\ell$ be non negative integers, then $$\binom{i+j+\ell}{i,j} := \binom{i+j+\ell}{i,j,\ell} = \frac{(i+j+\ell)!}{i!j!\ell!}\enspace .$$ We begin with an explicit formula for the polynomials $w_{\alpha,\beta,a,b}(t)$. **Proposition 7**. *For any $0 < \alpha < a$ and $0 < \beta < b$, the following formula holds: $$w_{\alpha,\beta,a,b}(t) = \sum_{i=0}^{a-\alpha-1} \sum_{j=0}^{\alpha-1} \sum_{i'=0}^{b-\beta-1} \sum_{j'=0}^{\beta-1} \binom{a}{i,j}\binom{b}{i',j'} \cdot (1+t) \cdot t^{a+b-i-j-i'-j'-2} \enspace .$$* *Proof.* We are revisiting the proof of Lemma [Lemma 4](#lem:key){reference-type="ref" reference="lem:key"}. We observe the following: the faces counted by $w_{\alpha,\beta,a,b}(t)$ come in pairs, namely the faces of $\Delta_{\alpha,a}\times\Delta_{\beta,b}$ that do not lie in the boundary of $\Delta_{\alpha+\beta,a+b}$ and the faces of $\Delta_{k,n}$ one dimension higher. Furthermore, these faces are obtained by deleting and contracting elements such that the hyperplane $\sum_{\ell=1}^a x_\ell = \alpha$ still induces a split. Deleting $i$ and contracting $j$ of the first $a$ elements as well as deleting $i'$ and contracting $j'$ of the other $b$ elements then leads to such a pair of faces of dimension $a+b-i-j-i'-j'-2$ and $a+b-i-j-i'-j'-1$. Expressing the ways of choosing the elements with multinomial coefficients leads to the desired formula. ◻ For the polynomials $u_{r,k,h,n}(t)$ we provide the following formula. **Proposition 8**. *For any $0<r<k<n$ and $r<h<n$ the following formula holds $$u_{r,k,h,n}(t) = p_{r,k,h,n}(t) - p'_{r,h}(t)\cdot p'_{k-r,n-h}(t)\cdot(1+t) + \sum_{i = r+1}^k \binom{h}{i}\binom{n-h}{k-i}$$ where $p_{r,h}'(t) =f_{\Delta_{r,h}}(t)-\binom{h}{r}$ and $$p_{r,k,h,n}(t) = \sum_{j=0}^{h-r-1} \sum_{i=0}^{\min\{h-j,k-1\}} \sum_{\ell = 0}^{\min\{k-i-1,k-r-1\}} \sum_{m=0}^{\min\{n-h-\ell,n-k-j-1\}} \binom{h}{i,j}\binom{n-h}{\ell,m} \cdot t^{n-1-i-j-\ell-m}.$$* *Proof.* We begin by counting the faces of $\Delta_{k,n}$ that are not faces of $\mathscr{P}(\mathsf{\Lambda}_{k-r,k,n-h,n})$. There are two types of such faces. Those that are entirely beyond the splitting hyperplane, and the ones that are separated by the hyperplane. Clearly there are $\sum_{i=r+1}^k\binom{h}{i}\binom{n-h}{k-i}$ vertices of $\Delta_{k,n}$ that are cut off by the hyperplane $\sum_{i=1}^h x_i = r$. The other faces are precisely those containing such a vertex. Hence we count faces for which more than $r$ of the first $h$ coordinates can be chosen to be one, where in total we have $k$ coordinates equal to one. We do so by picking coordinates whose value we fix, the remaining coordinates can either be zero or one, and the total number of ones is precisely $k$. Moreover, we do not want to fix all coordinates, i.e., we can fix at most $k-1$ coordinates to be equal to one and at most $n-k-1$ of them to be zero. Say we fix $i$ coordinates in $\{1,\ldots, h\}$ to be one and $j$ to be zero. Then $0 \leq j< h-r$ and $0\leq i < k$ as well as $i+j\leq h$, since there are only $h$ coordinates to select from. Similarly, we fix $\ell$ ones in the last $n-h$ coordinates and $m\geq 0$ zeros. Then $0\leq\ell < k-i$ and $\ell<k-r$ as it must be allowed to move another $1$ to the first coordinates to get there more than $r$ out of the $k$ ones. Furthermore, $\ell+m\leq n-h$ as we select $\ell+m$ out of $\{h+1,\ldots,n\}$ and $m+j < n-k$ because otherwise we would have fixed all zeros and hence also the ones. Clearly, there are $\binom{h}{i,j}$ ways to select the $i$ and $j$ first coordinates and $\binom{h}{\ell,m}$ ways to fix the $\ell$ ones and $m$ zeros in the last coordinates. Every fixed coordinate reduces the dimension and hence counts $p_{r,k,h,n}(t)$ these type of faces that are not vertices. Now we count the $(d-1)$-faces of $\mathsf{\Lambda}_{r,k,h,n}$ that split a $d$-face of $\Delta_{k,n}$. These are exactly the faces of $\Delta_{r,h}\times\Delta_{k-r,n-h}$ that contain a product of edges, i.e., are not of the from $\Delta_{r,h}\times\{v\}$ or $\{w\}\times\Delta_{k-r,n-h}$ for vertices $v$ of $\Delta_{k-r,n-h}$ and $w$ of $\Delta_{k,h}$. Thus these faces are enumerated by the polynomial $p'_{r,h}(t)\times p'_{k-r,n-h}(t)$. Each of these faces contributes $-t^{d-1}$ to $u_{r,k,h,n}(t)$. Furthermore, the $d$-faces of $\Delta_{k,n}$ that are separated by such a split do not contribute to $u_{r,k,h,n}(t)$. Therefore we subtract $p'_{r,h}(t)\cdot p'_{k-r,n-h}(t)\cdot(1+t)$ from $p_{r,k,h,n}(t)$ to obtain the polynomial $u_{r,k,h,n}(t)$. ◻ In the following we will specialize Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"} to some common and interesting classes of matroids. We begin with an example. **Example 9**. Let $\mathsf{M}$ be the projective geometry $\mathrm{PG}(2,3)$. This is a matroid on $n=13$ elements of rank $k=3$. It is split as it is in fact paving. This matroid has $13$ stressed hyperplanes, i.e., rank $k-1=2$ flats, all of which have cardinality $h=4$. In other words, we have $\uplambda_{2,4} = 13$. In particular, to use the formula of Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"}, the polynomial $$\begin{aligned} u_{2,3,4,13}(t) &= -\, t^{11} - 11\, t^{10} - 54\,t^{9} - 156\,t^{8} - 294\,t^{7} - 378\,t^{6}\\ &\qquad - 336\,t^{5} - 195\,t^{4} + \, t^{3} + 166\,t^{2} + 114\,t + \,4 \end{aligned}$$ is required. Since projective geometries are modular matroids, any pair of distinct proper non-empty cyclic flats fulfills the property [\[eq:star-property-pairs-of-flats\]](#eq:star-property-pairs-of-flats){reference-type="eqref" reference="eq:star-property-pairs-of-flats"}. Also, every pair of them intersect in a single element. Moreover, for every pair of these cyclic flats we have $a = |F_1\smallsetminus F_2| = 3$, and by symmetry $b = |F_1\smallsetminus F_2| = 3$. Additionally, $\alpha=\mathop{\mathrm{rk}}(F_1) - |F_1\cap F_2| = 2 - 1 = 1$ and again by symmetry $\beta=\mathop{\mathrm{rk}}(F_2) - |F_1\cap F_2| = 1$. Therefore there is a single non-vanishing coefficient $\upmu_{\alpha,\beta,a,b}$ which is $$\upmu_{1,1,3,3} = \binom{13}{2} = 78 \enspace.$$ It remains to compute: $$w_{1,1,3,3}(t) = t^5 + 7 t^4 + 15 t^3 + 9 t^2\enspace .$$ Now applying Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"}, we obtain: $$\begin{aligned} f_{\mathscr{P}(\mathrm{PG}(2,3))}(t) &= f_{\Delta_{3,13}}(t) - 13 \, u_{2,3,4,13}(t) - 78 \, w_{1,1,3,3}(t)\\ &= t^{12} + 39 \, t^{11} + 455\, t^{10} + 2704\, t^{9} + 9893\, t^{8} + 24414\, t^{7} + 42666\, t^{6} + \\ &\qquad 54054\, t^{5} + 49608\, t^{4} + 31707\, t^{3} + 12870\, t^{2} + 2808\, t + 234 \enspace . \end{aligned}$$ ## Face numbers of sparse paving matroids As mentioned in the introduction, it is conjectured that almost all matroids are sparse paving; see [@mayhew] for the details. Furthermore, many famous examples of matroids fall into this class; notable examples are the Fano matroid, the Vámos matroid, the complete graph on four vertices, and the duals of each of them. Sparse paving and paving matroids are split, so we can make use of our main result. For sparse paving matroids all the proper cyclic flats are circuit hyperplanes, i.e., of rank $r=k-1$ and size $h=k$. Using these parameters, Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"} simplifies to the following statement. **Corollary 10**. *Let $\mathsf{M}$ be a connected sparse paving matroid of rank $k$ on $n$ elements having exactly $\uplambda$ circuit-hyperplanes, and let $\upmu$ count the pair of circuit-hyperplanes which have $k-2$ elements in common. Then $$f_{\mathscr{P}(\mathsf{M})}(t) = f_{\Delta_{k,n}}(t)-\uplambda\cdot u(t)- \upmu\cdot (t^2+t^3)$$ where $u(t)$ is given by $$\begin{aligned} %&1 - \left((t+1)^k-1\right)\cdot\left((t+1)^{n-k}-1\right)\cdot t^{-2}+\left((n-k)\cdot(t+1)^k+k\cdot(t+1)^{n-k}-k(n-k)t\right)\cdot(t^{-1}+1)\\ %=& 1-k\cdot(n-k)\cdot(t+1)+\left((n-k)\cdot (t+1)^{k+1} + k\cdot(t+1)^{n-k+1}-n\cdot(t+1)\right)\cdot t^{-1}\\ +\left((t+1)^k+(t+1)^{n-k}-(t+1)^n-1\right)\cdot t^{-2}\enspace . \end{aligned}$$* *Proof.* By substituting $r$ by $k-1$ and $h$ with $k$ in Proposition [Proposition 8](#prop:nice-formula-u){reference-type="ref" reference="prop:nice-formula-u"} we obtain $$u(t) = u_{k-1,k,k,n}(t) = p_{k-1,k,k,n}(t)-p'_{k-1,k}(t)\cdot p'_{1,n-k}(t)\cdot(1+t)+1$$ where $p_{k-1,k,k,n}(t)$ is equal to $$\begin{aligned} \sum_{i=0}^{k-1}\binom{k}{i}\sum_{m=0}^{n-k-1} \binom{n-k}{m}\cdot t^{n-k-m}\cdot t^{k-i}\cdot t^{-1} &= \left((t+1)^k-1\right)\cdot\left((t+1)^{n-k}-1\right)\cdot t^{-1} \enspace . \end{aligned}$$ Furthermore, the polytope $\Delta_{k-1,k}$ is a affine transformation of $\Delta_{1,k}$ hence $$p'_{k-1,k}(t) = p'_{1,k}(t) = %\sum_{d=1}^{k-1} \binom{k}{d+1} \sum_{j=1}^d\binom{k-d-1}{1-j} t^d = \sum_{i=2}^k \binom{k}{i}\cdot t^{i-1} = \left((t+1)^k-1-k\cdot t\right)\cdot t^{-1}\enspace .$$ Using the same formula for $p'_{1,n-k}(t)$ leads to the desired formula. ◻ As was mentioned earlier, there are many famous matroids that are sparse paving. We saw four sparse paving matroids in the running Example [Example 2](#example:f-non-valuative){reference-type="ref" reference="example:f-non-valuative"}. In the next example we take a look at family of sparse paving matroids with maximum number of circuit-hyperplanes. This example demonstrates that one can derive a bit more from our calculations. **Example 11**. Let $m\geq 3$. Consider the rank $k=4$ matroid $\mathsf{M}$ whose bases are the quadruples of affinely independent binary vectors of length $m$. This is a sparse paving matroid on $n=2^m$ elements with the maximum number $\uplambda= \frac{1}{4}\binom{n}{3}$ of circuit-hyperplanes, as every three points define a circuit. A simple double counting argument reveals that the number of pairs of these circuit-hyperplanes that share two elements is $\upmu=\frac{3n-12}{8}\binom{n}{3}$. The polytope $\mathscr{P}(\mathsf{M})$ has $\frac{6n^2-57n+132}{8}\binom{n}{3}$ many square faces all of which are induced by split hyperplanes, and hence can be read off from the quadratic term of the product $p'_{3,4}(t)\cdot p'_{1,n-4}(t)$, and $\left(4\,\binom{n-3}{2}-\binom{n-4}{4}+\binom{n}{4}-3\,n^2+17\,n-20\right)\frac{1}{4}\,\binom{n}{3}$ many triangular faces that are also faces of the hypersimplex $\Delta_{4,n}$, e.g., for $m=3$ the matroid is the binary affine cube and its polytope has $420$ square and $448$ triangular faces. ## Face numbers of rank two matroids A loopless matroid of rank two is trivially paving, and hence a split matroid. This allows us to use the strength of Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"} to compute their $f$-vectors. The key is the following elementary observation. The hyperplanes, i.e., the flats of rank one, of a loopless matroid of rank two form a partition of the ground, and conversely, any partition of the ground set defines precisely a single rank two matroid having each part as a flat. The bases of the matroid are obtained by taking two elements of the ground set, not belonging to the same part. Base polytopes of matroids of rank two have made prominent appearances throughout algebraic combinatorics, under various guises. Notably, as is pointed out in [@ferroni-higashitani Section 6.1], they coincide with edge polytopes of complete multipartite graphs --- we refer to that paper for the precise definition of edge polytopes and a short overview of them. In this vein, the work of Ohsugi and Hibi [@ohsugi-hibi] addresses the edge polytopes of complete multipartite graphs, motivated both from toric geometry and graph theory. In particular, the content of [@ohsugi-hibi Theorem 2.5] provides a formula for the $f$-vector of the edge polytope of an arbitrary complete multipartite graph, and thus for general rank two matroid polytopes. Let us point out that there appears to be an error in the formula as they stated it --- in particular within the quantity they denote by $\alpha_i$. As an application of Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"} we can give another formula for the $f$-vector of these polytopes. **Corollary 12**. *Let $\mathsf{M}$ be a loopless matroid of rank two having $s$ hyperplanes with cardinalities $h_1,\ldots, h_s$. Then, the number of $i$-dimensional faces of $\mathscr{P}(\mathsf{M})$ or, equivalently, the edge polytope of a complete multipartite graph with parts of sizes $h_1,\ldots,h_s$ is given by: $$f_i(\mathscr{P}(\mathsf{M})) = \binom{n+1}{i+2}+(s-1)\binom{n}{i+2} - \sum_{j<\ell} \binom{h_j+h_\ell+1}{i+2} + (s-2)\sum_{j=1}^s \binom{h_j+1}{i+2} - \sum_{j=1}^s \binom{n-h_j}{i+2}.$$* *Proof.* It is straightforward to check the formula for the case $s=2$, that is $\mathscr{P}(\mathsf{M}) = \Delta_{1,h_1}\times \Delta_{1,h_2}$. If $s>2$ then $\mathsf{M}$ is connected and we may apply Theorem [Theorem 5](#thm:main){reference-type="ref" reference="thm:main"}. Observe that all pairs of flats are trivially modular, and they are pairwise disjoint. Thus, the non-vanishing coefficients are $\upmu_{1,1,h_j,h_\ell}$. In the following we omit parts of the long and tedious calculations but indicate the main steps. For $k=2$ and $r=1$ using some change of summation and the Vandermonde identity we obtain $$\begin{aligned} \sum_{j<l} w_{1,1,h_j,h_\ell}(t) &= \sum_{i=2}^{n-1} \left( \sum_{j<\ell} \binom{h_j+h_\ell}{i+2} -\sum_{j=1}^s\left((s-1)\binom{h_j}{i+2}+(n-h_j)\binom{h_j}{i+1}\right)\right)\cdot t^i\, (1+t) \enspace . \end{aligned}$$ Similarly, we get this formula for $p'_{1,h_j}(t)\cdot p'_{1,n-h_j}(t)\cdot (1+t)$ $$\begin{aligned} \sum_{i=2}^{n-1} \left(\binom{n}{i+2}-\binom{n-h_j}{i+2}-h\binom{n-h_j}{i+1}-(n-h_j)\binom{h_j}{i+1}-\binom{h_j}{i+2}\right)\cdot t^i\,(1+t)\end{aligned}$$ and for $p_{1,2,h_j,n}(t)$ the expression $$\begin{aligned} \sum_{i=1}^{n-1}\left(\binom{n}{i+2}-\binom{n-h_j}{i+2}-h\binom{n-h_j}{i+1}\right)t^{i+1} +\sum_{i=1}^{n-1}\left(h_j\binom{n-1}{i+1}-h_j\binom{n-h_j}{i+1}\right)\cdot t^i\end{aligned}$$ where the two sums correspond to the indices $i=0$ and $i=1$ in the definition of $p_{r,k,h,n}(t)$. This gives us $u_{1,2,h_j,n}(t) = p_{1,2,h_j,n}(t)-p'_{1,h_j}(t)p'_{1,n-h_j}(t)(t+1)+\binom{h_j}{2}$ and hence $\sum_{j=1}^s u_{1,2,h_j,n}$. We also need $$\begin{aligned} f_{\Delta_{2,n}}(t) &= \binom{n}{2} + \sum_{i=1}^{n-1} \binom{n}{i+1}\sum_{j=1}^i \binom{n-i-1}{2-j}\cdot t^i = \binom{n}{2} + 3\binom{n}{3} \, t +\sum_{i=2}^{n-1}\binom{n}{i+1}(n-i) \cdot t^i \enspace . \end{aligned}$$ Now putting the pieces together, applying Pascal's identity, and cancelling many terms, one may obtain the desired formula of the statement (where additional effort is required for the constant, linear and quadratic term). ◻ **Remark 13**. Kim described in [@kim] how the cd-indices of polytopes change when under hyperplane splits. From this, he derived a formula for the cd-index of rank two matroid base polytopes. This formula depends on the cd-index of polytopes of rank two matroids with $1$, $2$ and $3$ hyperplanes. While there are explicit expressions for the cd-index when the matroid has $1$ or $2$ hyperplanes, the case with $3$ hyperplanes remains unsolved. We point out that the cd-index contains more information than the $f$-vector --- however, recovering the $f$-vector from the cd-index is often a very laborious task. # Final remarks and open problems Steinitz characterized $f$-vectors of $3$-polytopes. Since then the $f$-vectors of $4$-polytopes have been intensively studied. In spite of the great interest from mathematicians, only little is known about the face numbers of higher dimensional polytopes [@ziegler-2007] or $0/1$-polytopes [@ziegler01]. The contribution of this paper is an explicit formula for $f$-vectors of split matroids. However, many questions and problems regarding $f$-vectors of $0/1$-polytopes, or even the class of matroid polytopes itself, remain broadly open. This last section aims to propose a few problems and questions in this underexplored area of discrete geometry. ## Explicit formulas for other polytopes Besides the base polytope studied in this paper, another famous polytope that one may associate to a matroid is the so-called *independence polytope*. This polytope is defined as the convex hull of the indicator vectors of all the independent sets, i.e., subsets of bases, of the matroid, and it contains the base polytope as a facet. A natural challenge that we raise is to provide a good way of computing the $f$-vectors of these polytopes. **Problem 14**. Find a formula for the $f$-vector of the independent set polytope of a (connected) split matroid. We speculate that one can approach this problem by using modifications of the ideas and the techniques that we presented in this article. On the other hand, it is also natural to ask about the problem of finding a formula for $f$-vectors of matroid base polytopes or independence polytopes for *arbitrary matroids*. These two problems appear to be considerably more difficult. Nonetheless, we pose the following broad question. **Question 15**. Can the approach of splitting polytopes, or treating the $f$-vector as a valuation with an error term, be used to obtain an explicit formula for the $f$-vector of an arbitrary matroid polytope? Independently of the details of the computation, what is the precise matroidal data that one needs in order to recover the $f$-vector? Note that all matroid polytopes are cut out of the hypersimplex by intersecting it with split hyperplanes --- or, more precisely, with half-spaces whose boundary is a split hyperplane --- which are weakly compatible, i.e., that may intersect in the interior of the hypersimplex. Also, every matroid base polytopes can be obtained by slicing pieces off of a unit cube. Unit cubes themselves possess many split hyperplanes; see [@JoswigHerrmann] for more details. Therefore, it seems reasonable to attempt to generalize the techniques and ideas of the present paper beyond the class of matroid polytopes, in order to include other classes of $0/1$-polytopes. **Problem 16**. Describe the $f$-vectors of $0/1$-polytopes in terms of their supporting split hyperplanes. ## The shape of *f*-vectors of matroid polytopes A recent trend in matroid theory is that of proving unimodal and log-concave inequalities for various vectors of numbers associated to matroids. A finite sequence of numbers $(a_0,\ldots,a_n)$ is said to be *unimodal* if there exists some index $0\leq j\leq n$ with the property that $$a_0\leq \cdots \leq a_{j-1}\leq a_j\geq a_{j+1}\geq \cdots \geq a_n.$$ If all the $a_i$'s are positive, a stronger condition is that of *log-concavity*, which asserts that for each index $1\leq j\leq n-1$ the inequalities $a_j^2\geq a_{j-1}a_{j+1}$ hold. There are two objects that, in other contexts, can be referred to as "the $f$-vector of a matroid," though they hold no direct relation with the $f$-vector we studied in this paper. The first of them comes from considering a matroid as a pure simplicial complex with the property that each vertex-induced subcomplex is shellable [@bjorner]. In this case, one can define the $f$-vector of a matroid to be the $f$-vector of this simplicial complex. This has been object of intensive research and several open problems and conjectures exist in the literature regarding this $f$-vector, for instance Stanley's conjecture asserting that the corresponding $h$-vector is a pure $O$-sequence (see [@stanley-green]). In [@adiprasito-huh-katz] these $f$-vectors are proved to be log-concave. The second object that sometimes is referred to as "the $f$-vector of a matroid" is the vector having as coordinates the number of flats of the matroid of each rank, i.e., the Whitney numbers of the second kind of $\mathsf{M}$. This object has also received some attention in the recent years; one of the main open problems regarding this $f$-vector was the so-called "Top-Heavy Conjecture," which was settled in [@braden-huh-matherne-proudfoot-wang]. It remains as an open problem to prove or disprove the unimodality of the Whitney numbers of the second kind. In a similar vein, it is quite inviting to ask the following question. **Question 17**. Are the $f$-vectors of matroid base polytopes unimodal, or even log-concave? It is known that there are simplicial polytopes having a non-unimodal $f$-vector; see [@ziegler Chapter 8.6]. However, let us point out that, within the existing literature, we were not able to find any examples of non-unimodal $f$-vectors for the general class of *all* $0/1$-polytopes. We have been able to verify the log-concavity of the $f$-vectors of the following classes of matroids, in some cases relying critically on the results of this paper: - All matroids on a ground set of size at most $9$. - Split matroids on a ground set of size at most $15$. - Sparse paving matroids on a ground set of size at most $40$. - Split matroids with four cyclic flats as in Lemma [Lemma 4](#lem:key){reference-type="ref" reference="lem:key"} of size at most $50$. - Cuspidal matroids on a ground set of size at most $100$. - Lattice path matroids on a ground set of size at most $13$. - Rank two matroids on a ground set of size at most $60$. Matroid base polytopes have many remarkable properties. One that is particularly relevant is that their vertex-edge graph, i.e., the $1$-skeleton of the polytope, determines the entire polytope up to a rigid transformation; see [@holzmann-norton-tobey; @SchroeterPineda] for more precise statements. Similarly, the data of the $2$-skeleton of a matroidal subdivision describes the subdivision completely; this technique has been used in several articles, see for example [@HerrmannJoswigSpeyer]. It is tempting to ask whether the enumerative information encoded in the first few entries of the $f$-vector of a matroid base polytope is already sufficient to derive the remaining entries. More precisely, we ask for the following constant. **Problem 18**. Is it true that there exists a number $c$ such that the first $c$ entries of the $f$-vector of some matroid $\mathsf{M}$, i.e., $f_0, \ldots, f_{c-1}$, are enough to determine the complete $f$-vector of $\mathsf{M}$? If it is true, what is the smallest such $c$? Our experiments indicate that $c=7$ suffices for all matroids on up to nine elements. ## A digression on the extension complexity of split matroids A motivation to study the face enumeration of matroid polytopes stems from the work of Rothvoss [@rothvoss], Conforti, Kaibel, Walter and Weltge [@ConfortiKaibelWalterWeltge], and Kaibel, Lee, Walter and Weltge [@KaibelLeeWalterWeltge] on the extension complexity of matroid polytopes. The special case of hypersimplices is discussed in work by Grande, Padrol and Sanyal [@GrandePadrolSanyal]. Given a lattice polytope $\mathscr{P}\subseteq\mathbb{R}^n$, an *extended formulation* of $\mathscr{P}$ is another lattice polytope $\mathscr{Q}\subseteq\mathbb{R}^m$ together with a projection map $\pi :\mathbb{R}^m \to \mathbb{R}^n$ which projects $\mathscr{Q}$ onto $\mathscr{P}$. The *complexity* of an extended formulation is the number of facets of the polytope $\mathscr{Q}$. The *extension complexity* of $\mathscr{P}$, denoted $\operatorname{xc}(\mathscr{P})$, is the minimum complexity of an extended formulation of $\mathscr{P}$. Rothvoss [@rothvoss Corollary 6] proved[^1] that for all $n$ there exists a matroid $\mathsf{M}$ on $n$ elements whose base polytope has extension complexity $\operatorname{xc}(\mathscr{P}(\mathsf{M})) \in \Omega\left(\frac{2^{n/2}}{n^{5/4} \sqrt{\log(2n)}}\right)$. Moreover, Rothvoss' proof is non-constructive and relies only on an enumerative result of matroids, that therefore guarantees that whatever these examples are, they must belong to the class of sparse paving matroids, and are therefore split matroids. One of the main tools finding a lower bound on the extension complexity are rectangular covering numbers, but these numbers grow at most quadratically in the size of the ground set [@KaibelLeeWalterWeltge Proposition 2]. Furthermore, also the well studied extension complexity of regular matroids is polynomial [@aprile-fiorini]. Thus it remains a notorious open problem to find an explicit family of matroids having exponential extension complexity. In fact, having one would yield an explicit infinite family of Boolean functions requiring superlogarithmic depth circuits, according to an observation attributed to Göös in [@aprile-fiorini Section 8] and [@Matroid-union-post]; see also the relevant [@goos-jain-watson]. Although we cannot compute explicitly the extension complexity of split matroid polytopes, we conjecture that the following natural family of sparse paving matroids might have the desired exponential extension complexity. **Conjecture 19**. *For each positive integer $n$, let us denote $\mathsf{S}_n$ the sparse paving matroid on $[n]$ of rank $\lfloor\frac{n}{2}\rfloor$, having the maximal possible number of circuit-hyperplanes, and whose set of bases is lexicographically minimal[^2]. Then $$\operatorname{xc}(\mathscr{P}(\mathsf{S}_n)) \in \Omega\left(\frac{2^{n/2}}{n^{5/4} \sqrt{\log(2n)}}\right).$$* Observe that the explicit determination of the matroid $\mathsf{S}_n$ is yet another open problem, related to the construction of binary codes with constant weight and Hamming distance $4$, as well as stable subsets of Johnson graphs. In particular, we point out that it remains an open problem to determine the matroid $\mathsf{S}_{20}$ --- in fact, its number of circuit-hyperplanes seems to be unknown, though it is between $13452$ and $16652$ (see [@agrell Table I]). Although we cannot prove this conjecture, at least we can prove that the number of facets of $\mathscr{P}(\mathsf{S}_n)$ is indeed exponential. **Proposition 20**. *Let $\mathsf{S}_n$ the matroid described in the prior statement. The number of facets of $\mathscr{P}(\mathsf{S}_n)$ satisfies: $$f_{n-2}(\mathsf{S}_{n}) \geq 2n + \frac{1}{n}\binom{n}{\lfloor\frac{n}{2}\rfloor} \sim c \,\frac{2^n}{n^{3/2}}.$$* *Proof.* By Corollary [Corollary 10](#coro:sparse-paving){reference-type="ref" reference="coro:sparse-paving"} the number of facets of $\mathsf{S}_n$ is $2n+\uplambda$ (whenever $n>4$) where $\uplambda$ is the number of circuit-hyperplanes of $\mathsf{S}_n$. As follows from a result of Graham and Sloane in [@graham-sloane], the maximal possible value of $\uplambda$ is at least $\frac{1}{n}\binom{2n}{n}$; see for example [@ferroni-schroter Lemma 4.6]. ◻ **Remark 21**. Even though the number of facets of a matroid polytope can be exponential, for arbitrary $0/1$-polytopes in $\mathbb{R}^n$ it is known that the number of facets can be larger than $\left(\frac{cn}{\log n}\right)^{n/4}$, via a random construction [@barany-por]. # Acknowledgments {#acknowledgments .unnumbered} The authors want to thank Günter Ziegler for useful email exchanges regarding $f$-vectors of $0/1$-polytopes. [^1]: To be precise, Rothvoss proved that the extension complexity of the *independence polytope* of some matroid is exponential, but an elementary reasoning shows that this is equivalent to an analogous statement for the base polytope. See for example the short explanation in [@aprile-fiorini p. 1]. [^2]: *In the sense that if we write each individual basis $B\in\mathscr{B}$ with their elements in increasing order, and sort the set $\mathscr{B}$ lexicographically, then $\mathscr{B}$ is minimal.*

arxiv_math

--- abstract: | The BG-rank BG($\pi$) of an integer partition $\pi$ is defined as $$\text{BG}(\pi) := i-j$$ where $i$ is the number of odd-indexed odd parts and $j$ is the number of even-indexed odd parts of $\pi$. In a recent work, Fu and Tang ask for a direct combinatorial proof of the following identity of Berkovich and Uncu $$B_{2N+\nu}(k,q)=q^{2k^2-k}\left[\begin{matrix}2N+\nu\\N+k\end{matrix}\right]_{q^2}$$ for any integer $k$ and non-negative integer $N$ where $\nu\in \{0,1\}$, $B_N(k,q)$ is the generating function for partitions into distinct parts less than or equal to $N$ with BG-rank equal to $k$ and $\left[\begin{matrix}a+b\\b\end{matrix}\right]_q$ is a Gaussian binomial coefficient. In this paper, we provide a combinatorial proof of Berkovich and Uncu's identity along the lines of Fu and Tang's idea. address: - Department of Mathematics, University of Florida, Gainesville FL 32611, USA - Department of Mathematics, University of Florida, Gainesville FL 32611, USA author: - Aritram Dhar - Avi Mukhopadhyay date: September 14, 2023 title: Combinatorial Proof of an Identity of Berkovich and Uncu --- # Introduction {#s1} An integer partition is a non-increasing finite sequence $\pi = (\lambda_1,\lambda_2,\ldots)$ of non-negative integers where $\lambda_i$'s are called the parts of $\pi$. We denote the number of parts of $\pi$ by $\#(\pi)$ and the largest part of $\pi$ by $l(\pi)$. The size of $\pi$ is the sum of the parts of $\pi$ and is denoted by $|\pi|$. We say that $\pi$ is a partition of $n$ if $|\pi| = n$. $\lambda_{2i-1}$ (resp. $\lambda_{2i}$) are called odd-indexed (resp. even-indexed) parts of $\pi$.\ In [@Ber-Gar06] and [@Ber-Gar08], Berkovich and Garvan defined the BG-rank of a partition $\pi$, denoted by $BG(\pi)$, as $$\begin{aligned} BG(\pi) := \sum_{i=1}^{k}(-1)^{i+1}\text{par}(\lambda_i)\end{aligned}$$ where $\pi=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ and par($\lambda$) denotes the parity of an integer $\lambda$ which is defined as $\text{par}(\lambda) = 1$ if $\lambda$ is odd and $0$, otherwise. It is then easy to see that $$\begin{aligned} BG(\pi) := i-j\end{aligned}$$ where $i$ is the number of odd-indexed odd parts and $j$ is the number of even-indexed odd parts. The BG-rank of a partition $\pi$ can also be represented by $2$-residue Ferrers diagram of $\pi$. The $2$-residue Ferrers diagram of a partition $\pi$ is represented by writing the ordinary Ferrers diagram with boxes instead of dots and filling the boxes using alternate $0$'s and $1$'s starting from $0$ on odd-indexed parts and $1$ on even-indexed parts. In [@Ber-Gar08], Berkovich and Garvan show that $$\begin{aligned} BG(\pi) = r_0-r_1\end{aligned}$$ where $r_0$ (resp. $r_1$) is the number of $0$'s (resp. $1$'s) in the $2$-residue Ferrers diagram of $\pi$. For example, Figure [\[fig1\]](#fig1){reference-type="ref" reference="fig1"} depicts the $2$-residue Ferrers diagram for the partition $\pi = (10,7,4,2)$ and so, $BG(\pi) = r_0-r_1 = 11-12 = -1$. 0101010101,1010101,0101,10 \* 10,7,4,2 \ Let $L,m,n$ be non-negative integers. We now recall some notations from the theory of $q$-series that can be found in [@And98]. $$\begin{aligned} (a)_L = (a;q)_L &:= \prod_{k=0}^{L-1}(1-aq^k);\\ (a)_{\infty} = (a;q)_{\infty} &:= \lim_{L\rightarrow \infty}(a)_L\,\,\text{where}\,\,\lvert q\rvert<1.\end{aligned}$$ We define the $q$-binomial (Gaussian) coefficient as $$\begin{aligned} \left[\begin{matrix}m\\n\end{matrix}\right]_q := \Bigg\{\begin{array}{lr} \dfrac{(q)_m}{(q)_n(q)_{m-n}}\quad\text{for } m\ge n\ge 0,\\ 0\qquad\qquad\quad\text{otherwise}.\end{array}\end{aligned}$$ *Remark 1*. For $m, n\ge 0$, $\left[\begin{matrix}m+n\\n\end{matrix}\right]_q$ is the generating function for partitions into at most $n$ parts each of size at most $m$ (see [@And98 Chapter $3$]). Note that $$\lim_{m\longrightarrow\infty}\left[\begin{matrix}m+n\\n\end{matrix}\right]_q = \dfrac{1}{(q)_n}.$$ We call a partition into distinct parts a *strict* partition. If $B_N(k,q)$ denotes the generating function for the number of strict partitions into parts less than or equal to $N$ with BG-rank equal to $k$, then Berkovich and Uncu [@Ber-Unc16] showed that for any non-negative integer $N$ and any integer $k$, $$\begin{aligned} \label{eq1} B_{2N+\nu}(k,q) = q^{2k^2-k}\left[\begin{matrix}2N+\nu\\N+k\end{matrix}\right]_{q^2}\end{aligned}$$ where $\nu\in\{0,1\}$. Letting $N\rightarrow \infty$ in ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}), we have $$\begin{aligned} \label{eq2} \sum_{n=0}^{\infty}p_k^d(n)q^n = \dfrac{q^{2k^2-k}}{(q^2;q^2)_{\infty}}\end{aligned}$$ where $p_k^d(n)$ denotes the number of strict partitions of $n$ with BG-rank equal to $k$. Note that ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) is exactly Conjecture 1 in [@Van10] where Vandervelde defined a partition statistic called characteristic, denoted by $\chi(\pi)$, which is related to BG-rank as $$\begin{aligned} \chi(\pi) = -BG(\pi).\end{aligned}$$ In [@Fu-Tang20 Remark $3.9$], Fu and Tang mention that ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) can also be derived from the work of Boulet [@Bou06]. Setting $a=d=qz$, $b=c=q/z$, and $z=1$ in Corollary 2 in [@Bou06], we get ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}). In [@Van10], Vandervelde provided a bijective proof of ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) with $k = 0$. More precisely, [@Van10 Theorem $1$] states $$\begin{aligned} \label{eq3} \sum_{n=0}^{\infty}p_0^d(n)q^n = \dfrac{1}{(q^2;q^2)_{\infty}}.\end{aligned}$$ Building upon Vandervelde's bijection, Fu and Tang [@Fu-Tang20] provided a bijective proof of ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) for all integers $k$ using certain unimodal sequences whose alternating sum equals zero. In their paper [@Fu-Tang20 Remark $3.9$], Fu and Tang ask for a direct combinatorial proof of ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}). The main aim of this paper is to provide such a combinatorial proof.\ The rest of the paper is organized as follows. In Section [2](#s2){reference-type="ref" reference="s2"}, we present Fu and Tang's bijection. In Section [3](#s3){reference-type="ref" reference="s3"}, we present the proof of Berkovich and Uncu's identity ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}). In Section [4](#s4){reference-type="ref" reference="s4"}, we provide some examples to illustrate the combinatorial proof of ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}). We conclude with a few remarks in Section [5](#s5){reference-type="ref" reference="s5"} to motivate further investigation.\ # Fu and Tang's bijection {#s2} ## $(a,b)$-sequences {#s21} \ First, we will define a certain type of unimodal sequence called an $(a,b)$-sequence introduced by Fu and Tang [@Fu-Tang20 Definition $2.1$]. **Definition 1**. For some non negative integer $a$ and an integer $1\leq b\leq l$, we call a sequence of $l$ positive integers $\{d_1,\ldots,d_l\}$ an $(a,b)$-*sequence of length* $l$ if the following conditions hold: 1. $d_{i} = a+i$ for $1\leq i\leq b$, 2. $d_{i}$ forms a non-increasing sequence of positive integers for $i\geq b$, and 3. $\sum\limits_{i=1}^{l} (-1)^{i}d_{i}=0$.\ We denote the collection of all such sequences by $\mathcal{S}_{a,b}$ and define $\mathcal{S} := (\bigcup_{a\geq 0, b\geq 1} \mathcal{S}_{a,b})\cup\{\varepsilon\}$ where $\varepsilon$ is the empty sequence. For $\Delta = \{d_1,\ldots,d_l\}$, we denote $l(\Delta) = l$, $|\Delta| = \sum\limits_{i=1}^ld_i$, and $|\Delta|_{\text{alt}} = \sum\limits_{i=1}^l(-1)^id_i$. If $\Delta\in\mathcal{S}_{a,b}$, we denote $a(\Delta) = a$ and $b(\Delta) = b$.\ **Example 2**. $\{5,6,7,8,3,3,2,2,2,1,1\}$ is a $(4,4)$-sequence of length $11$.\ ## The Bijection {#s22} \ According to Chu [@Chu93], a $k$-Durfee rectangle for the Young diagram of a partition is an $i\times (i+k)$ rectangle (having $i$ rows and $i+k$ columns) which is obtained by choosing the largest possible $i$ such that the $i\times (i+k)$ rectangle is contained in the Young diagram for a fixed integer $k$. It is to be noted that Fu and Tang [@Fu-Tang20] mention that this notion of Durfee rectangle is different from the generalization by Andrews in [@And71].\ For integers $a \geq 0$ and $b \geq 1$, we consider a map $\phi_a : \mathcal{S}_{a,b}\to\mathcal{P}_{a,b}$ where $\mathcal{P}_{a,b}$ is the set of all integer partitions $\lambda = (\lambda_{1},\lambda_{2},\ldots)$ whose $a$-Durfee rectangle has size $\lceil\frac{b}{2}\rceil\times(\lceil\frac{b}{2}\rceil + a)$ and $\lambda_{\frac{b}{2}} > a + b/2$ if $b$ is even or $\lambda_{\frac{b+1}{2}} = a + (b+1)/2$ if $b$ is odd.\ Now, we will define the map $\phi_a$. Consider a sequence $\Delta = \{d_1,d_2,\ldots,d_l\}\in \mathcal{S}_{a,b}$. The aim is to use the sequence $\Delta\in\mathcal{S}_{a,b}$ to *double cover* the block diagram configuration shown in Figure [1](#fig2){reference-type="ref" reference="fig2"}. The notion of double covering of the cells in the block diagram configuration is equivalent to labeling the cells by a '$1$' first and then re-labeling them again by a '$2$' so that in the end, all the cells are labeled by '$2$' which we portray in green color throughout from now onwards in the rest of the paper. This is exactly the reason why the base in the $q$-binomial coefficient in ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}) is $q^2$ instead of just $q$ as we are counting the cells twice.\ We then call a block diagram *doubly covered* when all the cells are colored green. The doubly covered block diagram will then be the Young diagram of a partition in $\mathcal{P}_{a,b}$. {#fig2 width=".4\\linewidth"} Following Fu and Tang [@Fu-Tang20 Fig. $2$], in the block diagram configuration (see Figure [1](#fig2){reference-type="ref" reference="fig2"} above), we call the $i$th labeled block $B_i$. From now onwards, we label all the cells contained in $B_i$ by $\mathcal{B}_i$. $B_i$ has size $1\times \left(a + \dfrac{i+1}{2}\right)$ (resp. $\dfrac{i}{2}\times 1$) if $i$ is odd (resp. even). We denote the area of $B_i$ , i.e., the number of cells labeled $\mathcal{B}_i$ by $b_i$. So, $b_1 = a+1,\,b_2 = 1,\,b_3 = a+2,\,b_4 = 2$, and so on. We obtain $\phi_{a}(\Delta)$ by performing the following operations: 1. Fill up $B_1$ in the block diagram Figure [1](#fig2){reference-type="ref" reference="fig2"} with $d_1 = a+1$ cells which is equivalent to labeling the $a+1$ cells in $B_1$ with '$1$'. 2. Use $d_i$ cells first to *double cover* the already existing cells in $B_{i-1}$ for $2\leq i\leq l$ and then use the remaining cells to fill $B_i$. This is equivalent to using $d_i$ cells to re-label the already existing $b_{i-1}$ cells in $B_{i-1}$ by '$2$' first for $2\leq i\leq l$ and then labeling the remaining $d_i-b_{i-1}$ cells by '$1$' to fill $B_i$. 3. Filling of $B_i$'s (labeling by '$1$' and re-labeling by '$2$') are done from left to right if $i$ is odd and from top to bottom if $i$ is even. 4. After having used up all the $d_i$'s where $1\leq i\leq l$, the *doubly covered* cells (cells which are labeled by '$2$') form the Young diagram of a partition (say) $\lambda = \phi_a(\Delta)$.\ **Theorem 3**. *([@Fu-Tang20 Theorem $2.5$])[\[ftbij\]]{#ftbij label="ftbij"} For a fixed $a\geq 0$ and any $b\geq 1$, the map $\phi_a$ defined above is a bijection from $\mathcal{S}_{a,b}$ to $\mathcal{P}_{a,b}$, such that $|\Delta| = 2|\phi_a(\Delta)|$, for any $\Delta\in\mathcal{S}_{a,b}$.\ * **Example 4**. Suppose $a = 3$, $b = 2$, and $\Delta = \{4,5,2,1\}\in \mathcal{S}_{3,2}$. Then following steps ($1$) to ($4$) above, we have $\lambda = \phi_3(\Delta) = (5,1)\in\mathcal{P}_{3,2}$ and $|\lambda| = |\Delta|/2 = 6$. For an illustration, see Figure [\[fig3\]](#fig3){reference-type="ref" reference="fig3"} below where the intermediate steps are denoted by arrows from left to right. All the cells labeled $\mathcal{B}_i$ form the $i$th block $B_i$ and $b_i$ is the number of cells labeled $\mathcal{B}_i$ for $i\in\{1,2,3\}$. Here, $b_1 = 4$, $b_2 = 1$, and $b_3 = 1$. All *singly covered* (equivalent to being labeled by '$1$' or counted once) cells are colored yellow and all *doubly covered* (equivalent to being labeled by '$2$' or counted twice) cells are colored green. \_1\_1\_1\_1 \* 4 \* \[\*(yellow)\]4 $\longrightarrow$ \_1\_1\_1\_1\_2 \* 5 \* \[\*(green)\]4\* \[\*(yellow)\]1+4 $\longrightarrow$ \_1\_1\_1\_1\_2,\_3 \* 5,1 \* \[\*(green)\]5\* \[\*(yellow)\]0,1 $\longrightarrow$\ \ \ \_1\_1\_1\_1\_2,\_3 \* 5,1 \* \[\*(green)\]5,1 ## Application of Fu and Tang's bijection to strict partitions {#s23} \ First, we consider the map $\iota:\mathcal{D}\to\mathcal{T}\times\mathcal{S}$ where $\mathcal{D}$ is the set of all strict partitions and $\mathcal{T}$ is the set of all triangular numbers, i.e., $\mathcal{T} := \left\{\dfrac{n(n+1)}{2} : n\in\mathbb{Z}\right\}$. Fu and Tang [@Fu-Tang20 Lemma $3.1$] proved that $\iota$ is infact an injection.\ For any strict partition $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_r)\in\mathcal{D}$, we consider the shifted Young diagram of $\lambda$ (for instance, see the Young diagram in Figure [\[fig4\]](#fig4){reference-type="ref" reference="fig4"} whose cells are colored orange) and consider the sequence of column lengths (read from left to right) of its shifted Young diagram. These column lengths form a unimodal sequence $c(\lambda) = \{c_1,c_2,\ldots,c_{\lambda_1}\}$. For example, for the shifted Young diagram shown in Figure [\[fig4\]](#fig4){reference-type="ref" reference="fig4"}, $c(\lambda) = \{1,2,3,4,2,2,1,1\}$. ,,, \* 8,5,2,1 \* \[\*(orange)\]8,1+5,2+2,3+1 Fu and Tang [@Fu-Tang20 Lemma $3.1$] proved that there exists a unique integer $0\le m\le r\le \lambda_1$ such that $\sum\limits_{i=1}^m(-1)^ic_i = \sum\limits_{i=1}^{\lambda_1}(-1)^ic_i$, i.e., $|\Delta|_\text{alt} = 0$ where $\Delta := \{c_{m+1},c_{m+2},\ldots,c_{\lambda_1}\}$ and so, $\Delta\in\mathcal{S}$.\ Now, define $\iota(\lambda) = (t,\Delta)$ where $t = 1+2+\ldots+m = \binom{m+1}{2}$. Clearly, $\iota(\lambda)\in\mathcal{T}\times \mathcal{S}$ and so, $|\lambda| = \lambda_1+\lambda_2+\ldots+\lambda_r = c_1+c_2+\ldots+c_{\lambda_1} = t+|\Delta|$. Fu and Tang [@Fu-Tang20 Lemma $3.1$] proved that $(t,\Delta)\in\iota(\mathcal{D})$ if and only if any one of the following conditions hold 1. $a(\Delta) = m$, or 2. $a(\Delta)\le m-1$ and $b(\Delta) = 1$, or 3. $\Delta = \varepsilon$. $\iota$ is one-one simply because the pre-image of any $(t,\Delta)\in\mathcal{T}\times\mathcal{S}$ satisfying either (1) or (2) or (3) mentioned above can be constructed uniquely by appending columns of length $1,2,\ldots,m$ to the left of the columns of length given by the elements of $\Delta$ and obtaining a shifted Young diagram.\ # Combinatorial proof of Berkovich and Uncu's identity ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}) {#s3} We now present the statement of the main result which we prove in this section.\ **Theorem 5**. *Let $\nu\in\{0,1\}$, $N$ be a non-negative integer, and $k$ be any integer. Then, for any positive integer $n$, the number of strict partitions $\pi_d$ of $n$ with BG-rank equal to $k$ and $l(\pi_d)\le 2N+\nu$ is equal to the number of partitions $\pi$ of $\frac{n-2k^2+k}{2}$ where $l(\pi)\le N+\nu-k$, $\#(\pi)\le N+k$ if $k\le 0$ and $l(\pi)\le N+k$, $\#(\pi)\le N+\nu-k$ if $k>0$.\ * Note that Theorem [Theorem 5](#combbu){reference-type="ref" reference="combbu"} together with Remark [Remark 1](#rmk1){reference-type="ref" reference="rmk1"} implies Berkovich and Uncu's identity ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}). We will now provide a combinatorial proof of Theorem [Theorem 5](#combbu){reference-type="ref" reference="combbu"}.\ *Proof.* Let the set of all strict partitions $\pi_d$ of $n$ having BG-rank equal to $k$ and $l(\pi_d)\le 2N+\nu$ be denoted by $\mathcal{SP}_{n,k}^{N,\nu}$, the set of all partitions $\pi$ of $n$ with $l(\pi)\le L$ and $\#(\pi)\le m$ be denoted by $\mathcal{P}_{n,L,m}$, and $T_i = i(i+1)/2$ be the $i$th triangular number for any integer $i$. Clearly, $\mathcal{SP}_{n,k}^{N,\nu}\subset \mathcal{D}$ and $T_i\in \mathcal{T}$.\ Consider any $\pi_d\in\mathcal{SP}_{n,k}^{N,\nu}$. First, construct the shifted Young diagram of $\pi_d = (\lambda_1,\ldots,\lambda_r)$ $(\lambda_1\le 2N+\nu)$ and then form the unimodal sequence $c(\pi_d) = \{c_1,\ldots,c_{\lambda_1}\}$ where $c_i$ is the length of the $i$th column of the shifted Young diagram of $\pi_d$. There exists $0\le a\le r$ such that $\sum\limits_{i=1}^a(-1)^ic_i = \sum\limits_{i=1}^{\lambda_1}(-1)^ic_i$ and so, $\Delta := \{c_{a+1},c_{a+2},\ldots,c_{\lambda_1}\}\in\mathcal{S}_{a,b}\subset\mathcal{S}$ for some integer $b\ge 1$.\ **Lemma 6**. *For the $\Delta\in\mathcal{S}_{a,b}$ obtained from the shifted Young diagram of $\pi_d\in\mathcal{SP}_{n,k}^{N,\nu}$, $$a = a(\Delta) = \Bigg\{\begin{array}{lr} -2k\quad\quad\text{if}\,\,k\le 0,\\ 2k-1\quad\text{if}\,\,k>0.\end{array}$$* *Proof.* The triangular part obtained from the shifted Young diagram of $\pi_d$ is $\frac{a(a+1)}{2}$ which is exactly equal to $2k^2-k$. Therefore, on solving the following equation for $a$ $$\begin{aligned} \dfrac{a(a+1)}{2} = 2k^2-k, \end{aligned}$$ the two solutions for $a$ are $-2k$ and $2k-1$. If $k\le 0$, $a = -2k$ since $a$ is a non-negative integer and if $k > 0$, $a = 2k-1$. ◻ We will now show that for $k\le 0$, $\mathcal{SP}_{n,k}^{N,\nu}$ is in bijection with $\mathcal{P}_{\frac{n-2k^2+k}{2},N+\nu-k,N+k}$ and for $k > 0$, $\mathcal{SP}_{n,k}^{N,\nu}$ is in bijection with $\mathcal{P}_{\frac{n-2k^2+k}{2},N+k,N+\nu-k}$.\ - \ One can now easily verify that for Fu and Tang's map $\iota$, ${% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter } : \mathcal{SP}_{n,k}^{N,\nu}\longrightarrow\{T_{-2k}\}\times\mathcal{S}_{-2k,b}$ is a bijection where ${% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }(\pi_d) = (T_{-2k},\Delta)$ with $T_{-2k} = 2k^2-k\in\mathcal{T}$ and $\Delta\in\mathcal{S}_{-2k,b}$.\ Now, recall Fu and Tang's bijection $\phi_a$. Consider the map $\chi_{-} : \{T_{-2k}\}\times\mathcal{S}_{-2k,b}\longrightarrow\{T_{-2k}\}\times\mathcal{P}_{\frac{n-2k^2+k}{2},N+\nu-k,N+k}$ defined as $$\chi_{-}(T_{-2k},\Delta) := (T_{-2k},{% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \phi_{-2k} % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{S}_{-2k,b}} % this is the delimiter }(\Delta)).$$ Therefore, we have $\chi_{-}(T_{-2k},\Delta) = (T_{-2k},\pi)$ where $\pi\in\mathcal{P}_{\frac{n-2k^2+k}{2},N+\nu-k,N+k}$. Thus, $\chi_{-}$ is a bijection.\ Next, consider the map $\psi_{-} : \mathcal{SP}_{n,k}^{N,\nu}\longrightarrow\{T_{-2k}\}\times\mathcal{P}_{\frac{n-2k^2+k}{2},N+\nu-k,N+k}$ defined as $$\psi_{-} := \chi_{-}\circ{% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }.$$ So, for any $\pi_d\in\mathcal{SP}_{n,k}^{N,\nu}$, we have $$\psi_{-}(\pi_d) := \chi_{-}\left({% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }(\pi_d)\right) = \chi_{-}(T_{-2k},\Delta) = (T_{-2k},\pi)$$ where $\pi\in\mathcal{P}_{\frac{n-2k^2+k}{2},N+\nu-k,N+k}.$\ Clearly, $\psi_{-}$ is an invertible map since it is the composition of two invertible maps ${% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }$ and $\chi_{-}$ where $\psi_{-}^{-1}$ is given by $$\psi_{-}^{-1} = \left({% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }\right)^{-1}\circ\chi_{-}^{-1}.$$ - \ Again, it can be verified that for Fu and Tang's map $\iota : \mathcal{D}\longrightarrow\mathcal{T}\times\mathcal{S}$, ${% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter } : \mathcal{SP}_{n,k}^{N,\nu}\longrightarrow\{T_{2k-1}\}\times\mathcal{S}_{2k-1,b}$ is a bijection where ${% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }(\pi_d) = (T_{2k-1},\Delta)$ with $T_{2k-1} = 2k^2-k\in\mathcal{T}$ and $\Delta\in\mathcal{S}_{2k-1,b}$.\ Now, recall Fu and Tang's bijection $\phi_a$. Analogous to $\chi_{-}$, consider the map $\chi_{+} : \{T_{2k-1}\}\times\mathcal{S}_{2k-1,b}\longrightarrow\{T_{2k-1}\}\times\mathcal{P}_{\frac{n-2k^2+k}{2},N+k,N+\nu-k}$ defined as $$\chi_{+}(T_{2k-1},\Delta) := (T_{2k-1},{% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \phi_{2k-1} % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{S}_{2k-1,b}^{N,\nu}} % this is the delimiter }(\Delta)).$$ Therefore, we have $\chi_{+}(T_{2k-1},\Delta) = (T_{2k-1},\pi)$ where $\pi\in\mathcal{P}_{\frac{n-2k^2+k}{2},N+k,N+\nu-k}$. Thus, $\chi_{+}$ is a bijection.\ Next, consider the map $\psi_{+} : \mathcal{SP}_{n,k}^{N,\nu}\longrightarrow\{T_{2k-1}\}\times\mathcal{P}_{\frac{n-2k^2+k}{2},N+k,N+\nu-k}$ defined as $$\psi_{+} := \chi_{+}\circ{% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }.$$ So, for any $\pi_d\in\mathcal{SP}_{n,k}^{N,\nu}$, we have $$\psi_{+}(\pi_d) := \chi_{+}\left({% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }(\pi_d)\right) = \chi_{+}(T_{2k-1},\Delta) = (T_{2k-1},\pi)$$ where $\pi\in\mathcal{P}_{\frac{n-2k^2+k}{2},N+k,N+\nu-k}.$\ Clearly, $\psi_{+}$ is an invertible map since it is the composition of two invertible maps ${% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }$ and $\chi_{+}$ where $\psi_{+}^{-1}$ is given by $$\psi_{+}^{-1} = \left({% we make the whole thing an ordinary symbol \left.\kern-\nulldelimiterspace % automatically resize the bar with \right \iota % the function \mathchoice{\vphantom{\big|}}{}{}{}% pretend it's a little taller at normal size \right|_{\mathcal{SP}_{n,k}^{N,\nu}} % this is the delimiter }\right)^{-1}\circ\chi_{+}^{-1}.$$ For a detailed illustration of how the forward (resp. inverse) map $\psi$ (resp. $\psi^{-1}$) works, see the examples listed in Section [4](#s4){reference-type="ref" reference="s4"}.\ Thus, it is clear that for any strict partition $\pi_d\in\mathcal{SP}_{n,k}^{N,\nu}$ of size $|\pi_d| = n$, the image partition $\pi$ has size $|\pi| = \dfrac{n-\delta}{2}$ where $\delta = 2k^2-k$ and vice-versa.\ Finally, we focus our attention on obtaining the bounds on the largest part and the number of parts of $\pi$ under the action of $\psi$. We first present a lemma which lies at the heart of obtaining the required bounds.\ **Lemma 7**. *The index of the last block present in the block diagram representation of the Young diagram of $\pi$ is at most $l(\pi_d)-a-1$.\ * *Proof.* In the shifted Young diagram of $\pi_d$, the length of the unimodal sequence whose alternating sum is zero is equal to $l(\pi_d)-a$. So, the number of blocks that can be *doubly covered* by the elements of this sequence is at most $l(\pi_d)-a-1$. ◻ Now, we consider two cases according to the sign of the BG-rank $k$ of $\pi_d\in\mathcal{SP}_{n,k}^{N,\nu}$. In both the cases, we also show that the we can retrieve back the bound on the largest part of $\pi_d$ upon the action of $\psi^{-1}$ on $\pi$.\ - \ If $k\le 0$, then from Lemma ([Lemma 6](#relka){reference-type="ref" reference="relka"}), we have $a = -2k$, i.e, $a$ is even.\ Let $I$ be the index of the last present block in the block diagram representation of the Young diagram of $\pi$. From Lemma ([Lemma 7](#maxblock){reference-type="ref" reference="maxblock"}), we know that $$\begin{aligned} I & \le l(\pi_d)-a-1\\ & \le 2N+\nu-a-1\\ & = 2N+\nu+2k-1\\ & = 2(N+k)+\nu-1\\ & = \Bigg\{\begin{array}{lr} 2(N+k)-1\,\,\text{if}\,\,\nu = 0,\\ 2(N+k)\quad\quad\text{if}\,\,\nu = 1.\end{array} \end{aligned}$$ Therefore, $\#(\pi)\le N+k$.\ Now, let $E$ be the number of even-indexed blocks present in the block diagram representation of the Young diagram of $\pi$. Then, it is clear that $$E\le \sum\limits_{\substack{i=2 \\ 2\mid i}}^{l(\pi_d)-a-1}1.$$ Again from the block diagram representation of the Young diagram of $\pi$, we have $$\begin{aligned} l(\pi) & = a+1+E\\ & \le a+1+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{l(\pi_d)-a-1}1\\ & \le a+1+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N+\nu-a-1}1\\ & = -2k+1+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N+\nu+2k-1}1\\ & = \Bigg\{\begin{array}{lr} -2k+1+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N+2k-1}1\quad\,\text{if}\,\,\nu = 0,\\ -2k+1+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N+2k}1\quad\quad\text{if}\,\,\nu = 1\end{array}\\ & = \Bigg\{\begin{array}{lr} -2k+1+N+k-1\,\,\,\,\text{if}\,\,\nu = 0,\\ -2k+1+N+k\quad\quad\,\,\text{if}\,\,\nu = 1\end{array}\\ & = \Bigg\{\begin{array}{lr} N-k\quad\quad\,\,\text{if}\,\,\nu = 0,\\ N-k+1\,\,\,\,\text{if}\,\,\nu = 1.\end{array} \end{aligned}$$ Hence, $l(\pi)\le N+\nu-k$.\ For the reverse direction, since $k\le 0$, we know that $a = -2k$, $l(\pi)\le N+\nu-k$, and $\#(\pi)\le N+k$. Clearly, $l(\pi_d) = -2k+l(\Delta) = -2k+I+1$ where $I$ is the index of the last present block in the block diagram representation of the Young diagram of $\pi$. Now, we consider two sub-cases regarding the parity of $I$:\ - \ Since $\#(\pi)\le N+k$, $$\begin{aligned} I&\le 2(N+k)-1\\ &= 2N+2k-1\\ &\le 2N+\nu+2k-1\addtocounter{equation}{1}\tag{\theequation}\label{eq31} \end{aligned}$$ where ([\[eq31\]](#eq31){reference-type="ref" reference="eq31"}) follows from the fact that $\nu\in\{0,1\}$.\ Therefore, from ([\[eq31\]](#eq31){reference-type="ref" reference="eq31"}), it follows that $l(\pi_d) = -2k+I+1\le 2N+\nu$.\ - \ Since $l(\pi)\le N+\nu-k$, $$\begin{aligned} I&\le 2((N+\nu-k)-(a+1))\\ &= 2N+2\nu-2k-2a-2\\ &= 2N+2\nu+2k-2\\ &= 2N+\nu+2k-1+\nu-1\\ &\le 2N+\nu+2k-1+\nu-1+1-\nu\addtocounter{equation}{1}\tag{\theequation}\label{eq32}\\ &= 2N+\nu+2k-1\addtocounter{equation}{1}\tag{\theequation}\label{eq33} \end{aligned}$$ where ([\[eq32\]](#eq32){reference-type="ref" reference="eq32"}) follows from the fact that $1-\nu\in\{0,1\}$.\ Therefore, from ([\[eq33\]](#eq33){reference-type="ref" reference="eq33"}), it follows that $l(\pi_d) = -2k+I+1\le 2N+\nu$.\ - \ If $k>0$, then from Lemma ([Lemma 6](#relka){reference-type="ref" reference="relka"}), we have $a = 2k-1$, i.e, $a$ is odd.\ Let $I$ be the index of the last present block in the block diagram representation of the Young diagram of $\pi$. From Lemma ([Lemma 7](#maxblock){reference-type="ref" reference="maxblock"}), we know that $$\begin{aligned} I & \le l(\pi_d)-a-1\\ & \le 2N+\nu-a-1\\ & = 2N+\nu-2k\\ & = 2(N+\nu-k)-\nu\\ & = \Bigg\{\begin{array}{lr} 2(N+\nu-k)\quad\quad\text{if}\,\,\nu = 0,\\ 2(N+\nu-k)-1\,\,\text{if}\,\,\nu = 1.\end{array} \end{aligned}$$ Therefore, $\#(\pi)\le N+\nu-k$.\ If $E$ is the number of even-indexed blocks present in the block diagram representation of the Young diagram of $\pi$, $$E\le \sum\limits_{\substack{i=2 \\ 2\mid i}}^{l(\pi_d)-a-1}1.$$ Again, from the block diagram representation of the Young diagram of $\pi$, we have $$\begin{aligned} l(\pi) & = a+1+E\\ & \le a+1+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{l(\pi_d)-a-1}1\\ & \le a+1+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N+\nu-a-1}1\\ & = 2k+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N+\nu-2k}1\\ & = \Bigg\{\begin{array}{lr} 2k+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N-2k}1\quad\quad\text{if}\,\,\nu = 0,\\ 2k+\sum\limits_{\substack{i=2 \\ 2\mid i}}^{2N-2k+1}1\quad\,\text{if}\,\,\nu = 1\end{array}\\ & = \Bigg\{\begin{array}{lr} 2k+N-k\,\,\,\,\text{if}\,\,\nu = 0,\\ 2k+N-k\,\,\,\,\text{if}\,\,\nu = 1.\end{array} \end{aligned}$$ Hence, $l(\pi)\le N+k$.\ For the reverse direction, since $k > 0$, we know that $a = 2k-1$, $l(\pi)\le N+k$, and $\#(\pi)\le N+\nu-k$. Clearly, $l(\pi_d) = 2k-1+l(\Delta) = 2k-1+I+1 = 2k+I$ where $I$ is the index of the last present block in the block diagram representation of the Young diagram of $\pi$. Now, we consider two sub-cases regarding the parity of $I$:\ - \ Since $\#(\pi)\le N+\nu-k$, $$\begin{aligned} I&\le 2(N+\nu-k)-1\\ &= 2N+2\nu-2k-1\\ &= 2N+\nu-2k+\nu-1\\ &\le 2N+\nu-2k+\nu-1+1-\nu\addtocounter{equation}{1}\tag{\theequation}\label{eq34}\\ &= 2N+\nu-2k\addtocounter{equation}{1}\tag{\theequation}\label{eq35} \end{aligned}$$ where ([\[eq34\]](#eq34){reference-type="ref" reference="eq34"}) follows from the fact that $1-\nu\in\{0,1\}$.\ Therefore, from ([\[eq35\]](#eq35){reference-type="ref" reference="eq35"}), it follows that $l(\pi_d) = 2k+I\le 2N+\nu$.\ - \ Since $l(\pi)\le N+k$, $$\begin{aligned} I&\le 2((N+k)-(a+1))\\ &= 2N+2k-2a-2\\ &= 2N-2k\\ &= 2N+\nu-2k-\nu\\ &\le 2N+\nu-2k-\nu+\nu\addtocounter{equation}{1}\tag{\theequation}\label{eq36}\\ &= 2N+\nu-2k\addtocounter{equation}{1}\tag{\theequation}\label{eq37} \end{aligned}$$ where ([\[eq36\]](#eq36){reference-type="ref" reference="eq36"}) follows from the fact that $\nu\in\{0,1\}$.\ Therefore, from ([\[eq37\]](#eq37){reference-type="ref" reference="eq37"}), it follows that $l(\pi_d) = 2k+I\le 2N+\nu$.\ Thus, we conclude that in the forward direction, $\#(\pi)\le N+k$, $l(\pi)\le N+\nu-k$ if $k\le 0$ and $\#(\pi)\le N+\nu-k$, $l(\pi)\le N+k$ if $k > 0$ and in the reverse direction, $l(\pi_d)\le 2N+\nu$ irrespective of the sign of $k$. This completes the proof of Theorem [Theorem 5](#combbu){reference-type="ref" reference="combbu"}. ◻ # Examples illustrating Theorem [Theorem 5](#combbu){reference-type="ref" reference="combbu"} {#s4} In this section, we present four different examples where we show the correspondence $\pi_d\underset{\psi^{-1}}{\stackrel{\psi}{\rightleftarrows}} (T_a,\pi)$. Here, $\pi_d\in\mathcal{SP}_{n,k}^{N,\nu}$ is a strict partition with fixed $BG$-rank $k$ and $l(\pi_d)\le 2N+\nu$, $T_a = \frac{a(a+1)}{2}$ is the triangular part where $a = a(\Delta)$ with $\Delta = \{d_1,d_2,\ldots,d_{l(\Delta)}\}\in\mathcal{S}_{a,b}$ obtained from the shifted Young diagram of $\pi_d$, $\pi\in\mathcal{P}_{\frac{n-2k^2+k}{2},N+\nu-k,N+k}$ is a partition where $l(\pi)\le N+\nu-k$, $\#(\pi)\le N+k$ if $k\le 0$, and $\pi\in\mathcal{P}_{\frac{n-2k^2+k}{2},N+k,N+\nu-k}$ is a partition where $l(\pi)\le N+k$, $\#(\pi)\le N+\nu-k$ if $k > 0$.\ In examples [Example 8](#eg1){reference-type="ref" reference="eg1"}, [Example 9](#eg2){reference-type="ref" reference="eg2"}, [Example 10](#eg3){reference-type="ref" reference="eg3"}, and [Example 11](#eg4){reference-type="ref" reference="eg4"}, all *singly covered* (equivalent to being labeled by '$1$' or counted once) cells are colored yellow and all *doubly covered* (equivalent to being labeled by '$2$' or counted twice) cells are colored green. The cells labeled $\mathcal{B}_i$ form the $i$th block $B_i$ and $b_i$ is the number of cells labeled $\mathcal{B}_i$ for $i\in\{1,2,3,\ldots\}$. In example [Example 8](#eg1){reference-type="ref" reference="eg1"}, we show all the intermediate steps (denoted by arrows from left to right) for the forward map in detail. However, in examples [Example 9](#eg2){reference-type="ref" reference="eg2"}, [Example 10](#eg3){reference-type="ref" reference="eg3"}, and [Example 11](#eg4){reference-type="ref" reference="eg4"}, we just portray the strict partition $\pi_d$ and the image $(T_a,\pi)$ without displaying the intermediate steps.\ **Example 8**. \ \ Let $\pi_d = (9,7,5,4,1)\in\mathcal{SP}_{26,2}^{4,1}$ so that $l(\pi_d) = 9\le 2N+\nu = 9$. Since $k = 2 > 0$, by Lemma [Lemma 6](#relka){reference-type="ref" reference="relka"}, $a = 2k-1 = 3$ which implies $T_3 = 6$ is the triangular part. From the shifted Young diagram of $\pi_d$, $c(\pi_d) = \{1,2,3,4,5,4,4,2,1\}$, and $\Delta = \{4,5,4,4,2,1\}$. $\psi_{+}(\pi_d) = (T_3,\pi)$. So, $b_1 = 4$, $b_2 = 1$, $b_3 = 3$, $b_4 = 1$, and $b_5 = 1$ which implies $\pi = (6,3,1)$. Clearly, $l(\pi) = 6 = N+k$ and $\#(\pi) = 3 = N+\nu-k$. Hence, $\pi\in\mathcal{P}_{10,6,3}$.\ Now, for the reverse direction, we are given $T_3 = 6$ and $\pi = (6,3,1)\in\mathcal{P}_{10,6,3}$. So, the solutions to $2k^2-k = 6$ are $k = 2$ and $k = -\frac{3}{2}$. Since $k\in\mathbb{Z}$, $k = 2 > 0$. On solving $N+k = 6$ and $N+\nu-k = 3$, we have $(N,\nu) = (4,1)$. On solving $\frac{n-6}{2} = 10$, we have $|\pi_d| = n = 26$. Now, $a = 2\cdot 2-1 = 3$ since $k = 2 > 0$ which implies $b_1 = a+1 = 4$, $b_2 = 1$, $b_3 = 3$, $b_4 = 1$, and $b_5 = 1$ following the block diagram configuration in Figure [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}. Now, we obtain $d_1 = b_1 = 4$, $d_2 = b_1+b_2 = 5$, $d_3 = b_2+b_3 = 4$, $d_4 = b_3+b_4 = 4$, $d_5 = b_4+b_5 = 2$, and $d_6 = b_5+b_6 = 1$ since $b_6 = 0$. Thus, we obtain the sequence $\{4,5,4,4,2,1\}$ which we write column-wise and if we append columns of length $\{1,2,3\}$ to the left of the column of length $4$, we retrieve back the shifted Young diagram of the partition $\pi_d = (9,7,5,4,1)\in\mathcal{SP}_{26,2}^{4,1}$.\ \ \ ,,, \* 9,7,5,4,1 $\longrightarrow$ $\longrightarrow$\ \ \ , $\longrightarrow$ \* \[\*(yellow)\]3,1+2,2+1 , \_1\_1\_1\_1 \* 4 \* \[\*(yellow)\]4 $\longrightarrow$\ \ \ \* \[\*(yellow)\]3,1+2,2+1 , \_1\_1\_1\_1\_2 \* 5 \* \[\*(green)\]4\* \[\*(yellow)\]1+4 $\longrightarrow$ \* \[\*(yellow)\]3,1+2,2+1 , \_1\_1\_1\_1\_2,\_3\_3\_3 \* 5,3 \* \[\*(green)\]5\* \[\*(yellow)\]0,3 $\longrightarrow$\ \ \ \* \[\*(yellow)\]3,1+2,2+1 , \_1\_1\_1\_1\_2\_4,\_3\_3\_3 \* 6,3 \* \[\*(green)\]5\* \[\*(yellow)\]1+5\* \[\*(green)\]0,3 $\longrightarrow$ \* \[\*(yellow)\]3,1+2,2+1 , \_1\_1\_1\_1\_2\_4,\_3\_3\_3,\_5 \* 6,3,1 \* \[\*(green)\]6\* \[\*(green)\]0,3\* \[\*(yellow)\]0,0,1\ \ \ $\longrightarrow$ \* \[\*(yellow)\]3,1+2,2+1 , \_1\_1\_1\_1\_2\_4,\_3\_3\_3,\_5 \* 6,3,1 \* \[\*(green)\]6,3,1\ \ \ **Example 9**. \ \ Let $\pi_d = (12,11,6,4,2)\in\mathcal{SP}_{35,-1}^{6,0}$ so that $l(\pi_d) = 12\le 2N+\nu = 12$. Since $k = -1 <= 0$, by Lemma [Lemma 6](#relka){reference-type="ref" reference="relka"}, $a = -2k = 2$ which implies $T_2 = 3$ is the triangular part. From the shifted Young diagram of $\pi_d$, $c(\pi_d) = \{1,2,3,4,5,5,4,3,2,2,2,2\}$, and $\Delta = \{3,4,5,5,4,3,2,2,2,2\}$. $\psi_{-}(\pi_d) = (T_2,\pi)$. So, $b_1 = 3$, $b_2 = 1$, $b_3 = 4$, $b_4 = 1$, $b_5 = 3$, $b_6 = 0$, $b_7 = 2$, $b_8 = 0$, and $b_9 = 2$ which implies $\pi = (5,4,3,2,2)$. Clearly, $l(\pi) = 5 < N+\nu-k = 7$ and $\#(\pi) = 5 = N+k$. Hence, $\pi\in\mathcal{P}_{16,7,5}$.\ Now, for the reverse direction, we are given $T_2 = 3$ and $\pi = (5,4,3,2,2)\in\mathcal{P}_{16,7,5}$. So, the solutions to $2k^2-k = 3$ are $k = -1$ and $k = \frac{3}{2}$. Since $k\in\mathbb{Z}$, $k = -1\le 0$. On solving $N+\nu-k = 7$ and $N+k = 5$, we have $(N,\nu) = (6,0)$. On solving $\frac{n-3}{2} = 16$, we have $|\pi_d| = n = 35$. Now, $a = -2\cdot (-1) = 2$ since $k = -1\le 0$ which implies $b_1 = a+1 = 3$, $b_2 = 1$, $b_3 = 4$, $b_4 = 1$, $b_5 = 3$, $b_6 = 0$, $b_7 = 2$, $b_8 = 0$, and $b_9 = 2$ following the block diagram configuration in Figure [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}. Now, we obtain $d_1 = b_1 = 3$, $d_2 = b_1+b_2 = 4$, $d_3 = b_2+b_3 = 5$, $d_4 = b_3+b_4 = 5$, $d_5 = b_4+b_5 = 4$, $d_6 = b_5+b_6 = 3$, $d_7 = b_6+b_7 = 2$, $d_8 = b_7+b_8 = 2$, $d_9 = b_8+b_9 = 2$, and $d_{10} = b_9+b_{10} = 2$ since $b_{10} = 0$. Thus, we obtain the sequence $\{3,4,5,5,4,3,2,2,2,2\}$ which we write column-wise and if we append columns of length $\{1,2\}$ to the left of the column of length $3$, we retrieve back the shifted Young diagram of the partition $\pi_d = (12,11,6,4,2)\in\mathcal{SP}_{35,-1}^{6,0}$.\ \ \ ,,, \* 12,11,6,4,2 ${}\underset{\psi^{-1}}{\stackrel{\psi}{\rightleftarrows}}{}$ \* \[\*(yellow)\]2,1+1 , \_1\_1\_1\_2\_4,\_3\_3\_3\_3,\_5\_5\_5,\_7\_7,\_9\_9 \* 5,4,3,2,2 \* \[\*(green)\]5,4,3,2,2\ \   **Example 10**. \ \ Let $\pi_d = (11,8,6,5,4,3,2,1)\in\mathcal{SP}_{40,-2}^{5,1}$ so that $l(\pi_d) = 11\le 2N+\nu = 11$. Since $k = -2 <= 0$, by Lemma [Lemma 6](#relka){reference-type="ref" reference="relka"}, $a = -2k = 4$ which implies $T_4 = 10$ is the triangular part. From the shifted Young diagram of $\pi_d$, $c(\pi_d) = \{1,2,3,4,5,6,7,8,2,1,1\}$, and $\Delta = \{5,6,7,8,2,1,1\}$. $\psi_{-}(\pi_d) = (T_4,\pi)$. So, $b_1 = 5$, $b_2 = 1$, $b_3 = 6$, $b_4 = 2$, $b_5 = 0$, and $b_6 = 1$ which implies $\pi = (8,7)$. Clearly, $l(\pi) = 8 = N+\nu-k$ and $\#(\pi) = 2 < N+k = 3$. Hence, $\pi\in\mathcal{P}_{15,8,3}$.\ Now, for the reverse direction, we are given $T_4 = 10$ and $\pi = (8,7)\in\mathcal{P}_{15,8,3}$. So, the solutions to $2k^2-k = 10$ are $k = -2$ and $k = \frac{5}{2}$. Since $k\in\mathbb{Z}$, $k = -2\le 0$. On solving $N+\nu-k = 8$ and $N+k = 3$, we have $(N,\nu) = (5,1)$. On solving $\frac{n-10}{2} = 15$, we have $|\pi_d| = n = 40$. Now, $a = -2\cdot (-2) = 4$ since $k = -2\le 0$ which implies $b_1 = a+1 = 5$, $b_2 = 1$, $b_3 = 6$, $b_4 = 2$, $b_5 = 0$, and $b_6 = 1$ following the block diagram configuration in Figure [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}. Now, we obtain $d_1 = b_1 = 5$, $d_2 = b_1+b_2 = 6$, $d_3 = b_2+b_3 = 7$, $d_4 = b_3+b_4 = 8$, $d_5 = b_4+b_5 = 2$, $d_6 = b_5+b_6 = 1$, and $d_7 = b_6+b_7 = 1$ since $b_7 = 0$. Thus, we obtain the sequence $\{5,6,7,8,2,1,1\}$ which we write column-wise and if we append columns of length $\{1,2,3,4\}$ to the left of the column of length $5$, we retrieve back the shifted Young diagram of the partition $\pi_d = (11,8,6,5,4,3,2,1)\in\mathcal{SP}_{40,-2}^{5,1}$.\ \ \ ,,, \* 11,8,6,5,4,3,2,1 ${}\underset{\psi^{-1}}{\stackrel{\psi}{\rightleftarrows}}{}$ \* \[\*(yellow)\]4,1+3,2+2,3+1 , \_1\_1\_1\_1\_1\_2\_4\_6,\_3\_3\_3\_3\_3\_3\_4 \* 8,7 \* \[\*(green)\]8,7\ \ \ **Example 11**. \ \ Let $\pi_d = (11,8,7,4,3,1)\in\mathcal{SP}_{34,2}^{6,1}$ so that $l(\pi_d) = 11\le 2N+\nu = 13$. Since $k = 2 > 0$, by Lemma [Lemma 6](#relka){reference-type="ref" reference="relka"}, $a = 2k-1 = 3$ which implies $T_3 = 6$ is the triangular part. From the shifted Young diagram of $\pi_d$, $c(\pi_d) = \{1,2,3,4,5,6,5,3,3,1,1\}$, and $\Delta = \{4,5,6,5,3,3,1,1\}$. $\psi_{+}(\pi_d) = (T_3,\pi)$. So, $b_1 = 4$, $b_2 = 1$, $b_3 = 5$, $b_4 = 0$, $b_5 = 3$, $b_6 = 0$, and $b_7 = 1$ which implies $\pi = (5,5,3,1)$. Clearly, $l(\pi) = 5 < N+k = 8$ and $\#(\pi) = 4 < N+\nu-k = 5$. Hence, $\pi\in\mathcal{P}_{14,8,5}$.\ Now, for the reverse direction, we are given $T_3 = 6$ and $\pi = (5,5,3,1)\in\mathcal{P}_{14,8,5}$. So, the solutions to $2k^2-k = 6$ are $k = 2$ and $k = -\frac{3}{2}$. Since $k\in\mathbb{Z}$, $k = 2 > 0$. On solving $N+k = 8$ and $N+\nu-k = 5$, we have $(N,\nu) = (6,1)$. On solving $\frac{n-6}{2} = 14$, we have $|\pi_d| = n = 34$. Now, $a = 2\cdot 2-1 = 3$ since $k = 2 > 0$ which implies $b_1 = a+1 = 4$, $b_2 = 1$, $b_3 = 3$, $b_4 = 0$, $b_5 = 3$, $b_6 = 0$, and $b_7 = 1$ following the block diagram configuration in Figure [\[fig1\]](#fig1){reference-type="ref" reference="fig1"}. Now, we obtain $d_1 = b_1 = 4$, $d_2 = b_1+b_2 = 5$, $d_3 = b_2+b_3 = 6$, $d_4 = b_3+b_4 = 5$, $d_5 = b_4+b_5 = 3$, $d_6 = b_5+b_6 = 3$, $d_7 = b_6+b_7 = 1$, and $d_8 = b_7+b_8 = 1$ since $b_8 = 0$. Thus, we obtain the sequence $\{4,5,6,5,3,3,1,1\}$ which we write column-wise and if we append columns of length $\{1,2,3\}$ to the left of the column of length $4$, we retrieve back the shifted Young diagram of the partition $\pi_d = (11,8,7,4,3,1)\in\mathcal{SP}_{34,2}^{6,1}$.\ \ \ ,,, \* 11,8,7,4,3,1 ${}\underset{\psi^{-1}}{\stackrel{\psi}{\rightleftarrows}}{}$ \* \[\*(yellow)\]3,1+2,2+1 , \_1\_1\_1\_1\_2,\_3\_3\_3\_3\_3,\_5\_5\_5,\_7 \* 5,5,3,1 \* \[\*(green)\]5,5,3,1\ \ \ # Concluding remarks {#s5} We get the bounds on the largest part and the number of parts of the image partition arising from the $q$-binomial coefficient on the right-hand side of ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}). However, the bounds on the largest part and the number of parts of the image partition become exact equalities under certain conditions on the pre-image partition which we state as a proposition below whose proof we leave as an exercise to the reader. **Proposition 12**. *In Theorem [Theorem 5](#combbu){reference-type="ref" reference="combbu"}, irrespective of the sign of $k$, all the inequalities on the right-hand side become equalities simultaneously if and only if the following conditions hold* - *the largest part of $\pi_d$, i.e., $l(\pi_d)$ is equal to $2N+\nu$, and* - *the second largest part of $\pi_d$ is equal to $2N+\nu-2$.* It will be interesting to investigate the conditions for which just any one of the inequalities on the right-hand side become an equality. It will also be worth finding an exact formula (or at least the generating function) of the number of strict partitions of an integer $N$ with fixed BG-rank $k$, fixed largest part $L$, and fixed number of parts $M$.\ # Acknowledgments The authors would like to thank Alexander Berkovich for encouraging them to prove ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}) using combinatorial methods and for his very helpful comments and suggestions. The authors would also like to thank Ali Uncu for previewing a preliminary draft of this paper and for his helpful suggestions. 10 G. E. Andrews, *Generalizations of the Durfee square*, J. Lond. Math. Soc. **3 (2)** (1971) 563--570. G. E. Andrews, *The Theory of Partitions*, Cambridge University Press, 1998. A. Berkovich and F. Garvan, *On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo $5$ and generalizations*, Trans. Amer. Math. Soc. **358 (2)** (2006) 703--726. A. Berkovich and F. Garvan, *The BG-rank of a partition and its appications*, Adv. in Appl. Math. **40 (3)** (2008) 377--400. A. Berkovich and A. K. Uncu, *On partitions with fixed number of even-indexed and odd-indexed odd parts*, J. Number Theory **167** (2016) 7--30. C. E. Boulet, *A four-parameter partition identity*, Ramanujan J. **12 (3)** (2006) 315--320. W. Chu, *Durfee rectangles and the Jacobi triple product identity*, Acta Math. Sin. New Ser. **9 (1)** (1993) 24--26. S. Fu and D. Tang, *On certain unimodal sequences and strict partitions*, Discrete Math. **343 (2)** (2020) 111650. S. Vandervelde, *Balanced partitions*, Ramanujan J. **23 (1)** (2010) 297--306.

arxiv_math

**The Lotka-Volterra Dynamical System and its Discretization** **Márcia Lemos-Silva**\ `marcialemos@ua.pt`\ Center for Research and Development in Mathematics\ and Applications (CIDMA), Department of Mathematics,\ University of Aveiro, 3810-193 Aveiro, Portugal\ <https://orcid.org/0000-0001-5466-0504> **Delfim F. M. Torres**\ `delfim@ua.pt`\ Center for Research and Development in Mathematics\ and Applications (CIDMA), Department of Mathematics,\ University of Aveiro, 3810-193 Aveiro, Portugal\ <https://orcid.org/0000-0001-8641-2505> **Abstract**. Dynamical systems are a valuable asset for the study of population dynamics. On this topic, much has been done since Lotka and Volterra presented the very first continuous system to understand how the interaction between two species -- the prey and the predator -- influences the growth of both populations. The definition of time is crucial and, among options, one can have continuous time and discrete time. The choice of a method to proceed with the discretization of a continuous dynamical system is, however, essential, because the qualitative behavior of the system is expected to be identical in both cases, despite being two different temporal spaces. In this work, our main goal is to apply two different discretization methods to the classical Lotka--Volterra dynamical system: the standard progressive Euler's method and the nonstandard Mickens' method. Fixed points and their stability are analyzed in both cases, proving that the first method leads to dynamic inconsistency and numerical instability, while the second is capable of keeping all the properties of the original continuous model.\ **Keywords:** dynamical systems; Lotka--Volterra prey-predator model; stability; Euler's discretization; Mickens' discretization. # Introduction to the Lotka--Volterra model {#LST:section01} Prey-predator equations intend to describe the dynamics of an ecological system where two species interact with each other. Alfred J. Lotka (1880--1949) introduced such equations in 1925 [@LST:lotka]; and Vito Volterra (1860-1940) studied them, independently [@LST:volterra]. For this reason, these equations are known as the Lotka--Volterra equations. On this topic, much has been done since Lotka and Volterra presented the very first continuous system to understand how the interaction between two species -- the prey and the predator -- influences the growth of both populations [@LST:MR4509368; @LST:MR4512320; @LST:MR4509395]. For a review of some recent advances, we refer the reader to [@LST:review]. Here we consider the classical model of Lotka--Volterra, which is composed by two autonomous and nonlinear differential equations given by $$\label{LST:lotka_volterra} \begin{cases} \dot{x} = \alpha x - \beta xy,\\ \dot{y} = -\delta y + \gamma xy, \end{cases}$$ where $x(t)$ and $y(t)$ represent the size at time $t$ of prey and predator populations, respectively. Moreover, all the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$ are assumed to be positive. If the density of both species reach the zero value at any moment $t$, then they will remain there indefinitely, which represents the natural extinction of both species. The absence of prey leads to the extinction of predators since in that case $y(t)$ converges to 0 when $t \rightarrow +\infty$. On the other hand, the absence of predators leads to exponential growth of prey, since $x(t) \rightarrow +\infty$ when $t \rightarrow +\infty$. From an ecological point of view, population densities must always be nonnegative, restricting the system trajectories to $\mathbb{R}^2_+$. In fact, from the equations of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}), we have $$\begin{cases} \left.\dot{x}\right|_{x = 0} = 0,\\ \left.\dot{y}\right|_{y = 0} = 0, \end{cases}$$ from which, according to Lemma 2 of [@LST:invariant], we can conclude that the solution of the system is nonnegative, meaning that $\mathbb{R}^2_+$ is the invariant domain of the system. This system has equilibria at two different points: $p_1 = (0,0)$ and $p_2 = \left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right)$. To observe the approximate behavior of the solutions over time near these equilibrium points, we start by computing the Jacobian matrix of the system, which is given by $$\label{LST:jacobian_cont} J(x,y) = \begin{pmatrix} \alpha - \beta y & -\beta x \\ \gamma y & \gamma x - \delta \end{pmatrix}.$$ The Jacobian matrix ([\[LST:jacobian_cont\]](#LST:jacobian_cont){reference-type="ref" reference="LST:jacobian_cont"}), evaluated at the equilibrium $(0,0)$, is given by $$J(0,0) = \begin{pmatrix} \alpha & 0 \\ 0 & - \delta \end{pmatrix}.$$ The corresponding eigenvalues are $\lambda_1 = \alpha$ and $\lambda_2 = -\delta$ and, as $\alpha, \delta > 0$, it turns out that $(0,0)$ is a saddle point. In contrast, the Jacobian matrix ([\[LST:jacobian_cont\]](#LST:jacobian_cont){reference-type="ref" reference="LST:jacobian_cont"}) evaluated at the coexistence equilibrium point $p_2$ is $$J \left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right) = \begin{pmatrix} 0 & -\frac{\beta\delta}{\gamma} \\ \frac{\alpha\gamma}{\beta} & 0 \end{pmatrix},$$ for which eigenvalues are pure imaginary: $\lambda = \pm i\sqrt{\alpha\delta}$. This means that $p_2$ is a stable center in the linearized system. However, with this analysis, nothing can be concluded regarding the stability for the nonlinear system at this equilibrium. To investigate the phase portrait of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}), we start by drawing the two lines $$\begin{gathered} x = \frac{\delta}{\gamma} \quad,\quad y = \frac{\alpha}{\beta}.\end{gathered}$$ By doing so, the first quadrant of the $xy$-plane is divided into four different regions, as shown in Figure [1](#LST:regions){reference-type="ref" reference="LST:regions"}. {#LST:regions} In each region, the signs of $\dot{x}$ and $\dot{y}$ determine the behavior of the solution of the system. By analyzing the equations of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}), the following result holds. **Proposition 1**. *The trajectory $x$ of the system will* - *decrease in region I and II, since $\dot{x} < 0$;* - *increase in region III and IV, since $\dot{x} > 0$.* *Regarding the trajectory $y$ of the system, it will* - *decrease in region II and III, since $\dot{y} < 0$;* - *increase in region I and IV, since $\dot{y} > 0$.* Proposition [Proposition 1](#LST:trajectory_cont){reference-type="ref" reference="LST:trajectory_cont"} suggests that the curve of the system in the phase plane will be counterclockwise around the equilibrium point $p_2$, but that it is not enough to conclude whether the trajectory spiral towards $p_2$; spiral out, towards infinity; or it is a closed curve. Despite this, it has already been explained that this equilibrium is, in fact, a center in the nonlinear system, meaning the trajectories will be closed curves. This allows to write the following result. **Proposition 2**. *Except for those beginning at the equilibrium $p_2$ or at a coordinate axes, every trajectory of the system is a closed orbit that turns counterclockwise around the equilibrium point $p_2$.* The behavior described by Proposition [Proposition 2](#LST:closed){reference-type="ref" reference="LST:closed"} can be seen in Figure [2](#LST:phase){reference-type="ref" reference="LST:phase"}, for several different initial conditions. ![Phase portrait of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}) with $\alpha = 1$, $\beta = 0.1$, $\gamma = 0.075$, and $\delta = 0.75$.](diagrama_fases.pdf){#LST:phase} From Proposition [Proposition 2](#LST:closed){reference-type="ref" reference="LST:closed"}, it comes directly that the densities of predators and prey will oscillate periodically, as can be seen in Figure [3](#LST:oscillations){reference-type="ref" reference="LST:oscillations"}, with the amplitude and frequency of oscillations depending only on the considered initial conditions. ![Oscillations of prey and predator densities for system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}) with $\alpha = 1$, $\beta = 0.1$, $\gamma = 0.075$, and $\delta = 0.75$.](oscilacoes.pdf){#LST:oscillations} All the results described so far are well-known. In particular, both Propositions [Proposition 1](#LST:trajectory_cont){reference-type="ref" reference="LST:trajectory_cont"} and [Proposition 2](#LST:closed){reference-type="ref" reference="LST:closed"} can be found, e.g., in [@LST:lotka:theorems]. In Sections [2](#LST:section02){reference-type="ref" reference="LST:section02"} and [3](#LST:section03){reference-type="ref" reference="LST:section03"} we provide new insights. # Discretization by Euler's Method {#LST:section02} There are several methods for converting continuous systems into discrete counterparts. The most conventional way to do so is to implement a standard difference scheme, the most classical one being the progressive Euler's method. However, it is known that this method can raise several problems such as lack of dynamical consistency, even when applied to the simplest systems [@LST:consistent]. A discrete-time model is said to be dynamically consistent with its continuous analog if they both exhibit the same dynamical behavior, namely the stability behavior of fixed points, bifurcation, and chaos. In [@LST:mickens:book], Mickens points out that the fundamental reason for the existence of numerical instabilities is that discrete models have a larger parameter space than the corresponding differential equations: one has the step size $h$ as an additional parameter. Nevertheless, this step $h$ is, obviously, inherent to any discretization and any discrete dynamical system. Therefore, it is crucial to consider a numerical method that is able to overcome this setback during discretization. Here we prove that Euler's method applied to the Lotka--Volterra model brings a discrete system that is not dynamically consistent with its continuous counterpart. Applying the progressive Euler's method to both equations of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}), we obtain that $$\label{LST:euler} \begin{cases} x_{i+1} = x_i + h(\alpha x_i - \beta x_iy_i),\\ y_{i+1} = y_i + h(\gamma x_i y_i - \delta y_i), \end{cases}$$ where $h$ denotes the step size and $x_i$ and $y_i$ define the density of the prey and predators' populations, respectively, at time $i$. The fixed points of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) are $p_1 = (0,0)$ and $p_2 = \left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right)$. To determine the nature of the fixed points, one must compute the Jacobian matrix $J(x,y)$ of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}). This matrix is given by $$\label{LST:jacobian_euler} J (x,y) = \begin{pmatrix} -\beta hy + \alpha h + 1 & -\beta hx \\ \gamma hy & \gamma hx - \delta h + 1 \end{pmatrix}.$$ Follows our first result. **Theorem 1**. *The fixed point $(0,0)$ of system [\[LST:euler\]](#LST:euler){reference-type="eqref" reference="LST:euler"} is* - *a saddle point if $h \in \left]0, \frac{2}{\delta}\right[$;* - *a source if $h \in \left]\frac{2}{\delta}, +\infty\right[$.* *Proof.* The Jacobian matrix ([\[LST:jacobian_euler\]](#LST:jacobian_euler){reference-type="ref" reference="LST:jacobian_euler"}) evaluated at the fixed point $(0,0)$ is $$J(0,0) = \begin{pmatrix} \alpha h + 1 & 0 \\ 0 & -\delta h + 1 \end{pmatrix},$$ whose eigenvalues are $\lambda_1 = -\delta h + 1$ and $\lambda_2 = \alpha h + 1$. As all the parameters are positive, one can easily conclude that $\lvert \lambda_2 \rvert > 1$. On the other hand, $\lvert\lambda_1\rvert$ can either be greater or less that one. In particular, $$\begin{aligned} \lvert \lambda_1 \rvert < 1 &\Leftrightarrow -\delta h + 1 < 1 \wedge -\delta h + 1 > -1\\ &\Leftrightarrow h > 0 \wedge h < \frac{2}{\delta}, \end{aligned}$$ and $$\begin{aligned} \lvert \lambda_1 \rvert > 1 &\Leftrightarrow -\delta h + 1 > 1 \vee -\delta h + 1 < -1\\ &\Leftrightarrow h < 0 \vee h > \frac{2}{\delta}. \end{aligned}$$ Therefore, the fixed point $(0,0)$ is a saddle point if $h \in \left]0, \frac{2}{\delta}\right[$ or a source if $h \in \left]\frac{2}{\delta}, +\infty\right[$. As $h$ is strictly positive, the condition $h < 0$ is not considered. ◻ For both possibilities of Theorem [Theorem 1](#LST:thm:01){reference-type="ref" reference="LST:thm:01"}, the point $p_1 = (0,0)$ is unstable, which brings no major changes to what is obtained in the continuous case, as described in Section [1](#LST:section01){reference-type="ref" reference="LST:section01"}. We now study what happens with the second fixed point $p_2$. **Theorem 2**. *The fixed point $\left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right)$ is an unstable focus.* *Proof.* The Jacobian matrix ([\[LST:jacobian_euler\]](#LST:jacobian_euler){reference-type="ref" reference="LST:jacobian_euler"}) evaluated at the fixed point $\left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right)$ is given by $$J\left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right) = \begin{pmatrix} 1 & -\frac{\beta\delta h}{\gamma} \\ \frac{\alpha\gamma h}{\beta} & 1 \end{pmatrix},$$ whose eigenvalues are the complex conjugates $\lambda = 1 \pm \sqrt{\alpha\delta}h$. As $\alpha$, $\delta$, and $h$ are strictly positive, it is clear that $\lvert \lambda \rvert > 1$, meaning that the fixed point $\left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right)$ is an unstable focus. ◻ Theorem [Theorem 2](#LST:thm:uf){reference-type="ref" reference="LST:thm:uf"} asserts that the orbits of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) near the fixed point $p_2$ will not be closed, but spirals that spiral out toward infinity. Through a simple analysis of the system equations, taking into consideration the four regions defined in Figure [1](#LST:regions){reference-type="ref" reference="LST:regions"}, it is possible to understand the direction of the solution in those regions. **Theorem 3**. *The trajectory $x$ of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) will* - *decrease in region I and II, i.e., $x_{i+1} < x_i$;* - *increase in region III and IV, i.e., $x_{i+1} > x_i$.* *Regarding the trajectory $y$ of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}), it will* - *decrease in region II and III, i.e., $y_{i+1} < y_i$;* - *increase in region I and IV, i.e., $y_{i+1} > y_i$.* *Proof.* We start by analyzing the trajectory of $x$ by looking to the first equation of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}). In regions I and II, $y_i > \frac{\alpha}{\beta}$. This means that $$h(\alpha x_i - \beta x_iy_i) < 0,$$ and $$x_i + h(\alpha x_i - \beta x_iy_i) < x_i \Rightarrow x_{i+1} < x_i.$$ On the other hand, in regions III and IV, we have $y_i < \frac{\alpha}{\beta}$. In this case, $$h(\alpha x_i - \beta x_iy_i) > 0,$$ from which we can conclude that $$x_i + h(\alpha x_i - \beta x_iy_i) > x_i \Rightarrow x_{i+1} > x_i.$$ Through the second equation of the system, by an analogous reasoning, the intended conclusions are obtained for $y$. ◻ Theorem [Theorem 3](#LST:trajectory_euler){reference-type="ref" reference="LST:trajectory_euler"} implies a counterclockwise displacement of the system. Precisely, it follows directly from Theorem [Theorem 3](#LST:trajectory_euler){reference-type="ref" reference="LST:trajectory_euler"} that a trajectory of ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) near the fixed point $p_2$ will spiral in a counterclockwise direction, as can be seen in Figure [4](#LST:diagrama_euler){reference-type="ref" reference="LST:diagrama_euler"}. {#LST:diagrama_euler} Since the trajectories of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) are not closed curves, they cease to be periodic orbits. Here, the trajectories are expansive, not converging to a particular fixed point, and the amplitude of the curves does not remain constant. In this case, the amplitude increases over time: see Figure [5](#LST:oscilacoes_euler){reference-type="ref" reference="LST:oscilacoes_euler"}. ![Oscillations of prey and predator densities for system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) with $\alpha = 1$, $\beta = 0.1$, $\gamma = 0.075$, $\delta = 0.75$, $h = 0.02$, and $x_0 = y_0 = 5$.](euler.pdf){#LST:oscilacoes_euler} In addition to the aforementioned dynamic inconsistency, caused by the progressive Euler method, the considered discrete system also makes it possible to predict negative population densities, even when all parameters and initial conditions are taken positive. Furthermore, it is also possible to prove that, under some circumstances, negative solutions can return to positive values. Although mathematically possible, these two possibilities do not make any sense, neither in the context of the problem (the problem is defined only in $\mathbb{R}^2_+$) nor from an ecological point of view. However, as we shall prove analytically, and geometrically, under Euler's method both mentioned situations are indeed possible. This shows the inconsistency of the discrete-time system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}). Next we study such situations in detail. According to the orientation of the solution pointed out in Theorem [Theorem 3](#LST:trajectory_euler){reference-type="ref" reference="LST:trajectory_euler"}, system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) can only predict negative solutions in two different cases and, in each of them, for only one of the variables. 1. Let $(x_i,y_i)$ be a point in region II of Figure [1](#LST:regions){reference-type="ref" reference="LST:regions"}. Here we have $y_i > \frac{\alpha}{\beta}$, $h(\alpha x_i - \beta x_iy_i) < 0$, and $x_{i+1} < x_i$. From the first equation of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}), $x_{i+1}$ can assume a negative value if $$x_i < -h(\alpha x_i - \beta x_iy_i).$$ Now, two situations can also occur. When the trajectory crosses the positive semi-axis $yy$, predicting a $x_{i+1} < 0$, this intersection can happen in such a way that $y_i$ remains greater than $\frac{\alpha}{\beta}$ or $y_i$ becomes less than that same value. We now note that, with $x_i < 0$, the first equation of the system can be rewritten as $$x_{i+1} = -x_i + h(-\alpha x_i + \beta x_iy_i) \quad \text{with} \quad x_i,y_i > 0.$$ - If $y_i > \frac{\alpha}{\beta}$, then we have $h(-\alpha x_i + \beta x_iy_i) > 0$. Thus, $x_{i+1}$ can assume a positive value, as long as $h(-\alpha x_i + \beta x_iy_i) > x_i$. If this happens, the system enters region III of Figure [1](#LST:regions){reference-type="ref" reference="LST:regions"}. Otherwise, the system goes outside the four mentioned regions, resulting in negative values for prey density. - If $y_i < \frac{\alpha}{\beta}$, then $h(\alpha x_i - \beta x_iy_i) < 0$, which leads to $$-x_i + h(-\alpha x_i + \beta x_i y_i) < 0 \Rightarrow x_{i+1} < 0,$$ meaning that the system will go outside the four admissible regions. 2. Let $(x_i,y_i)$ be a point in region III of the Figure [1](#LST:regions){reference-type="ref" reference="LST:regions"}. Here $x_i < \frac{\delta}{\gamma}$, $h(\gamma x_iy_i - \delta y_i) < 0$, and $y_{i+1} < y_i$. According to the second equation of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}), $y_{i+1}$ can assume a negative value if $$y_i < -h(\gamma x_iy_i - \delta y_i).$$ When the trajectory of the system crosses the positive semi-axis $xx$, obtaining $y_{i+1} < 0$, one can continue to have $x_i < \frac{\delta}{\gamma}$ or there can be a change in its value such that $x_i > \frac{\delta}{\gamma}$. Rewriting the second equation of system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}), knowing that now $y_i < 0$, we obtain $$y_{i+1} = -y_i + h(-\gamma x_iy_i + \delta y_i) \quad \text{with} \quad x_i,y_i > 0.$$ - If $x_i < \frac{\delta}{\gamma}$, then $h(\gamma x_iy_i - \delta y_i) > 0$. Consequently, $y_{i+1}$ may be positive as long as $h(\gamma x_iy_i - \delta y_i) > y_i$. In this case, the trajectory of the system will enter in region IV. Otherwise, it will remain outside the four regions under study, with negative values for the density of predators. - If, on the other hand, $x_i > \frac{\delta}{\gamma}$, then $h(\gamma x_iy_i - \delta y_i) < 0$. Thus, $$-y_i + h(-\gamma x_iy_i + \delta y_i) < 0 \Rightarrow y_{i+1} < 0,$$ which means that the system obtained by Euler's method gives negative values for $y_{i+1}$, with values outside the four admissible regions. By way of example, changing the value of $h$ from 0.02 to 0.03, it is possible to observe negative values for the variable $x_i$, as seen in Figure [6](#LST:oscilacoes_euler_negativo){reference-type="ref" reference="LST:oscilacoes_euler_negativo"}. In addition, it is verified that, after some time, the solutions that were previously negative return to positive values. ![Oscillations of prey and predator densities for system ([\[LST:euler\]](#LST:euler){reference-type="ref" reference="LST:euler"}) with $\alpha = 1$, $\beta = 0.1$, $\gamma = 0.075$, $\delta = 0.75$, $h = 0.03$, and $x_0 = y_0 = 5$.](euler_negativo.pdf){#LST:oscilacoes_euler_negativo} # Discretization by Mickens' Method {#LST:section03} As seen in Section [2](#LST:section02){reference-type="ref" reference="LST:section02"}, the progressive Euler method, when applied to the classical Lotka--Volterra system [\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="eqref" reference="LST:lotka_volterra"}, has the particularity of losing the periodic solutions, which correspond to closed curves in the phase space. In [@LST:mickens:lotka], Mickens points out that the most likely reason for the loss of periodic solutions is the fact that the Lotka--Volterra system is not structurally stable, i.e., a small perturbation in the equations of the system may change its topological properties. In particular, it can change the closed curves into ones that can spiral into or out of the fixed point. It is known that the application of most classical numerical methods to a system with periodic solutions, transforms the original system into a very close one whose trajectories are not closed [@LST:sanz]. Here we intend to demonstrate that a nonstandard finite difference scheme, as generated according to the rules suggested by Mickens [@LST:mickens:paper], can be applied consistently to a structurally unstable dynamical system such as the one of Lotka--Volterra. In addition to proving that this scheme preserves the periodic solutions, it is also ensured that the positivity of the system is kept unchanged. In [@LST:mickens:lotka], Mickens suggests a discretization of the Lotka--Volterra model where, for simplicity, it is considered that all parameters -- $\alpha$, $\beta$, $\delta$, and $\gamma$ -- are equal to one. Here, the same strategy suggested by Mickens is followed, with the difference that the parameters are general, assuming any value in $\mathbb{R}_+$. Following the rules stated by Mickens, the first-order derivatives are approximated by $$\dot{x} \rightarrow \frac{x_{i+1} - x_i}{\phi}$$ and $$\dot{y} \rightarrow \frac{y_{i+1} - y_i}{\phi},$$ where in both cases $\phi$ is such that $\phi(h) = h + \mathcal{O}(h^2)$. Starting with the first equation of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}), the linear and nonlinear terms are all substituted by nonlocal forms given by $$\begin{gathered} \alpha x = 2\alpha x - x \rightarrow 2\alpha x_i - \alpha x_{i+1},\\ -\beta x y \rightarrow -\beta x_{i+1}y_i.\end{gathered}$$ Thus, through the above substitutions, the first equation of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}) can be rewritten as $$\frac{x_{i+1} - x_i}{\phi} = 2\alpha x_i - \alpha x_{i+1} - \beta x_{i+1}y_i,$$ which is equivalent to $$\label{LST:xmickens} x_{i+1} = \frac{x_i(2\alpha\phi + 1)}{1 + \alpha\phi + \beta\phi y_i}.$$ Regarding the second equation of system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}), the following substitutions are proposed: $$\begin{gathered} \gamma x y = 2\gamma x y - \gamma x y \rightarrow 2\gamma x_{i+1}y_i - \gamma x_{i+1}y_{i+1}, \\ - \delta y \rightarrow -\delta y_{i+1}.\end{gathered}$$ Applying the two substitutions above, the second equation of the system is defined as $$\frac{y_{i+1} - y_i}{\phi} = 2\gamma x_{i+1}y_i - \gamma x_{i+1}y_{i+1} - \delta y_{i+1},$$ which is equivalent to $$\label{LST:ymickens} y_{i+1} = \frac{y_i(2\gamma\phi x_{i+1} + 1)}{1 + \gamma \phi x_{i+1} + \delta\phi}.$$ Substituting ([\[LST:xmickens\]](#LST:xmickens){reference-type="ref" reference="LST:xmickens"}) into ([\[LST:ymickens\]](#LST:ymickens){reference-type="ref" reference="LST:ymickens"}), and joining both equations, we obtain the Lotka--Volterra model discretized by the Mickens method as $$\label{LST:mickens} \begin{cases} x_{i+1} = \displaystyle \frac{x_i(2\alpha\phi + 1)}{1 + \alpha\phi + \beta\phi y_i},\\[0.3cm] y_{i+1} = \displaystyle \frac{2\gamma\phi x_i y_i(2\alpha\phi + 1) + y_i(1 + \alpha\phi + \beta\phi y_i)}{(1 + \delta\phi)(1 + \alpha\phi + \beta\phi y_i) + \gamma\phi x_i(2\alpha\phi + 1)}, \end{cases}$$ which, as we shall show next, recovers the periodic solutions and ensure that the positivity property of the Lotka--Volterra system is maintained. In concrete, through a simple analysis of the equations of system ([\[LST:mickens\]](#LST:mickens){reference-type="ref" reference="LST:mickens"}), it is clear that the Mickens method guarantees that the positivity property is maintained. Indeed, by choosing $(x_0,y_0) \in \mathbb{R}^2_+$, and as a consequence of all the parameters being positive, it is impossible to have negative values for any of the variables, since both equations will be quotients of strictly positive quantities. The fixed points of system ([\[LST:mickens\]](#LST:mickens){reference-type="ref" reference="LST:mickens"}) coincide with the ones of Sections [1](#LST:section01){reference-type="ref" reference="LST:section01"} and [2](#LST:section02){reference-type="ref" reference="LST:section02"}: $p_1 = (0,0)$ and $p_2 = \left(\frac{\delta}{\gamma}, \frac{\alpha}{\beta}\right)$. Given the complexity of the system [\[LST:mickens\]](#LST:mickens){reference-type="eqref" reference="LST:mickens"} under study, we make use of the free open-source mathematics software system [SageMath]{.sans-serif} [@LST:sage] to analyze the nature of each one of the fixed points. For this purpose, we start by computing the Jacobian matrix of the system ([\[LST:mickens\]](#LST:mickens){reference-type="ref" reference="LST:mickens"}) in an arbitrary point $(x,y)$. This matrix is given by $$\label{LST:jacobian_mickens} Jf_{(x,y)} = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$$ where $$\begin{aligned} a =& \frac{2 \alpha \phi + 1}{\beta \phi y + \alpha \phi + 1},\\ b =& -\frac{{\left(2 \alpha \phi + 1\right)} \beta \phi x}{{\left(\beta \phi y + \alpha \phi + 1\right)}^{2}},\\ c =& \frac{2 {\left(2 \alpha \phi + 1\right)} \gamma \phi y}{{\left(2 \alpha \phi + 1\right)} \gamma \phi x + {\left(\beta \phi y + \alpha \phi + 1\right)} {\left(\delta \phi + 1\right)}} \\ & - \frac{{\left(2 {\left(2 \alpha \phi + 1\right)} \gamma \phi x y + {\left(\beta \phi y + \alpha \phi + 1\right)} y\right)} {\left(2 \alpha \phi + 1\right)} \gamma \phi}{{\left({\left(2 \alpha \phi + 1\right)} \gamma \phi x + {\left(\beta \phi y + \alpha \phi + 1\right)} {\left(\delta \phi + 1\right)}\right)}^{2}},\\ d =& -\frac{{\left(2 {\left(2 \alpha \phi + 1\right)} \gamma \phi x y + {\left(\beta \phi y + \alpha \phi + 1\right)} y\right)} {\left(\delta \phi + 1\right)} \beta \phi}{{\left({\left(2 \alpha \phi + 1\right)} \gamma \phi x + {\left(\beta \phi y + \alpha \phi + 1\right)} {\left(\delta \phi + 1\right)}\right)}^{2}} \\ &+ \frac{2 {\left(2 \alpha \phi + 1\right)} \gamma \phi x + 2 \beta \phi y + \alpha \phi + 1}{{\left(2 \alpha \phi + 1\right)} \gamma \phi x + {\left(\beta \phi y + \alpha \phi + 1\right)} {\left(\delta \phi + 1\right)}}. \end{aligned}$$ **Theorem 4**. *The fixed point $(0,0)$ of system [\[LST:mickens\]](#LST:mickens){reference-type="eqref" reference="LST:mickens"} is a saddle point.* *Proof.* The Jacobian matrix ([\[LST:jacobian_mickens\]](#LST:jacobian_mickens){reference-type="ref" reference="LST:jacobian_mickens"}) evaluated at the fixed point $(0,0)$ is $$Jf_{(0,0)} = \begin{pmatrix} \dfrac{2 \alpha \phi + 1}{\alpha \phi + 1} & 0 \\ 0 & \dfrac{1}{\delta \phi + 1} \end{pmatrix},$$ whose eigenvalues are $\lambda_1 = \dfrac{1}{\delta\phi + 1}$ and $\lambda_2 = \dfrac{2 \alpha \phi + 1}{\alpha \phi + 1}$. From these results, it is possible to draw the following conclusions: - Since $\delta,\phi > 0$, it follows that $\delta\phi + 1 > 1$. Thus, $\lambda_1$ is always less than one, regardless of the values of $\delta$ and $\phi$. Moreover, by the positivity of the parameters, it is clear that $\lambda_1$ is always greater than zero. Thereby, $\lvert \lambda_1\rvert < 1$. - On the other hand, since $\alpha,\phi > 0$, then $\alpha\phi + 1$ is always less than $2\alpha\phi + 1$. For this reason, $\lambda_2 > 1$, which leads to $\lvert \lambda_2 \rvert > 1$. Thus, $p_1$ is a saddle point and, therefore, unstable. ◻ In contrast, the Jacobian matrix ([\[LST:jacobian_cont\]](#LST:jacobian_cont){reference-type="ref" reference="LST:jacobian_cont"}) evaluated at the coexistence equilibrium point $p_2$ is $$Jf_{\left(\frac{\delta}{\gamma},\frac{\alpha}{\beta}\right)} =\begin{pmatrix} 1 & -\dfrac{\beta \delta \phi}{{\left(2 \alpha \phi + 1\right)} \gamma} \\ \dfrac{\alpha \gamma \phi}{2 \beta \delta \phi + \beta} & \dfrac{3 \alpha \delta \phi^{2} + 2 {\left(\alpha + \delta\right)} \phi + 1}{4 \alpha \delta \phi^{2} + 2 {\left(\alpha + \delta\right)} \phi + 1} \end{pmatrix},$$ whose eigenvalues are complex conjugates $$\lambda = \frac{7 \alpha \delta \phi^{2} + 4 {\left(\alpha + \delta\right)}\phi + 2 \pm i\phi \sqrt{\displaystyle 15 \alpha^{2} \delta^{2} \phi^{2} + 4 \alpha \delta + 8 {\left(\alpha^{2} \delta + \alpha \delta^{2}\right)} \phi}}{2 {\left(4 \alpha \delta \phi^{2} + 2 {\left(\alpha + \delta\right)} \phi + 1\right)}}.$$ With the help of [SageMath]{.sans-serif}, it is easily verified that $|\lambda| = 1$, which means that the point $p_2$ is a center point in the linearized system, while nothing can be concluded regarding the stability for the nonlinear system at this equilibrium. However, it is possible to verify numerically that the orbits are periodic, corresponding to closed curves in the phase space, meaning that, at least for the indicated parameter values, $p_2$ is, in fact, a center. This effect can be seen in Figures [7](#LST:vs_presas){reference-type="ref" reference="LST:vs_presas"} and [8](#LST:vs_predadores){reference-type="ref" reference="LST:vs_predadores"}, which simultaneously show the results obtained here and those obtained in the continuous case. ![Oscillations of preys for the system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}) versus system ([\[LST:mickens\]](#LST:mickens){reference-type="ref" reference="LST:mickens"}) with $\alpha = 1$, $\beta = 0.1$, $\gamma = 0.075$, and $\delta = 0.75$.](vs_presas.pdf "fig:"){#LST:vs_presas}\ ![Oscillations of predators for the system ([\[LST:lotka_volterra\]](#LST:lotka_volterra){reference-type="ref" reference="LST:lotka_volterra"}) versus system ([\[LST:mickens\]](#LST:mickens){reference-type="ref" reference="LST:mickens"}) with $\alpha = 1$, $\beta = 0.1$, $\gamma = 0.075$, and $\delta = 0.75$.](vs_predadores.pdf "fig:"){#LST:vs_predadores}\ Despite the fact that one of Mickens' rules mention that a more complex expression should be used for the step function $\phi(h)$, it appears that all the results achieved are valid regardless of the expression used for $\phi(h)$. Accordingly, in our simulations we chose the simplest function given by $\phi(h) = h$. In particular, a step size given by $h = 0.01$ was considered. In both Figures [7](#LST:vs_presas){reference-type="ref" reference="LST:vs_presas"} and [8](#LST:vs_predadores){reference-type="ref" reference="LST:vs_predadores"}, it is observed that the periodic oscillations of the discrete system practically overlap those of the original continuous system. Logically, the smaller the value of the chosen step $h$, the more superimposed the curves for each of the systems will be. To complete the analysis of system ([\[LST:mickens\]](#LST:mickens){reference-type="ref" reference="LST:mickens"}), we end by proving that Theorem [Theorem 3](#LST:trajectory_euler){reference-type="ref" reference="LST:trajectory_euler"} remains valid in this case, from which one can conclude that the direction of the trajectories of the Mickens' discrete system will continue to be counterclockwise. Consider the first equation of system ([\[LST:mickens\]](#LST:mickens){reference-type="ref" reference="LST:mickens"}). Equivalently, one can write that $$\frac{x_{i+1}}{x_i} = \frac{2\alpha\phi + 1}{1 + \alpha\phi + \beta\phi y_i}.$$ - Let $y_i$ be a value that either belongs to regions I or II considered in Figure [1](#LST:regions){reference-type="ref" reference="LST:regions"}. Then, we have $y_i > \frac{\alpha}{\beta}$. For this reason, $$\alpha\phi < \beta\phi y_i \Rightarrow 2\alpha\phi < \alpha\phi + \beta\phi y_i.$$ Thus, $$2\alpha\phi + 1 < 1 + \alpha\phi + \beta\phi y_i \Rightarrow \frac{x_{i+1}}{x_i} < 1 \Leftrightarrow x_{i+1} < x_i.$$ - Now, let $y_i$ be a value that either belongs to regions III or IV. In both cases, we have $y_i < \frac{\alpha}{\beta}$, which leads to $$\alpha\phi > \beta\phi y_i \Rightarrow 2\alpha\phi > \alpha\phi + \beta\phi y_i.$$ Finally, $$2\alpha\phi + 1 > 1 + \alpha\phi + \beta\phi y_i \Rightarrow \frac{x_{i+1}}{x_i} > 1 \Leftrightarrow x_{i+1} > x_i.$$ We now consider equation ([\[LST:ymickens\]](#LST:ymickens){reference-type="ref" reference="LST:ymickens"}) that is equivalent to $$\frac{y_{i+1}}{y_i} = \frac{2\gamma\phi x_{i+1} + 1}{1 + \gamma\phi x_{i+1} + \delta\phi}.$$ - Let $x_{i+1}$ be a value that either belongs to regions II or III of Figure [1](#LST:regions){reference-type="ref" reference="LST:regions"}. In there we have $x_{i+1} < \frac{\delta}{\gamma}$, which is equivalent to $\delta > \gamma x_{i+1}$. In this way, $$\delta\phi > \gamma\phi x_{i+1} \Rightarrow \gamma\phi x_{i+1} + \delta\phi > 2\gamma\phi x_{i+1}.$$ Therefore, $$2\gamma\phi x_{i+1} + 1 < 1 + \gamma\phi x_{i+1} + \delta\phi \Rightarrow \frac{y_{i+1}}{y_i} < 1 \Leftrightarrow y_{i+1} < y_i.$$ - On the other hand, let $x_{i+1}$ be a value that either belongs to regions I or IV. In this case, $x_{i+1} > \frac{\delta}{\gamma}$, which means that $\delta < \gamma x_{i+1}$, and we obtain $$\delta\phi < \gamma\phi x_{i+1} \Rightarrow \gamma\phi x_{i+1} + \delta\phi < 2\gamma\phi x_{i+1}.$$ Finally, $$2\gamma\phi x_{i+1} + 1 > 1 + \gamma\phi x_{i+1} + \delta\phi \Rightarrow \frac{y_{i+1}}{y_i} > 1 \Leftrightarrow y_{i+1} > y_i.$$ We conclude that Theorem [Theorem 3](#LST:trajectory_euler){reference-type="ref" reference="LST:trajectory_euler"} also holds for system [\[LST:mickens\]](#LST:mickens){reference-type="eqref" reference="LST:mickens"}. # Conclusion In this work our goal was to show that the choice of the numerical method for the discretization of a continuous dynamical system is crucial in order to obtain consistent results. It was proved that the progressive Euler method, although appealing for its simplicity, is not able to deal with structurally unstable systems, making the solutions of the classical Lotka--Volterra model, that should be closed curves in phase space, become spirals. Furthermore, Euler's discretization does not take into account special fundamental properties of the systems, such as positivity. On the other hand, Mickens' method, despite generating an apparently more complex system, manages to guarantee that the qualitative behavior of the system, in a neighborhood of the fixed points, is identical to the one found in its continuous counterpart. Additionally, this method takes into account basic rules so that positivity is never compromised. Our conclusions open the possibility of applying Mickens' method to other structurally unstable dynamical systems of particular interest, recovering properties that may have been lost through different standard discretizations. We also concluded that the Computer Algebra System [SageMath]{.sans-serif} is a strong tool that allows to do computations in a reliable way, serving as a good support when the systems under study are complex. In addition, it produces numerical simulations of good quality and in a very simple way. All the figures were generated with [SageMath]{.sans-serif}. # Acknowledgments {#acknowledgments .unnumbered} The authors were partially supported by the Portuguese Foundation for Science and Technology (FCT) through the Center for Research and Development in Mathematics and Applications (CIDMA), projects UIDB/04106/2020 and UIDP/04106/2020. 10 É. Diz-Pita and M. V. Otero-Espinar. Predator--prey models: A review of some recent advances. , 9(15):Art. 1783, 2021. M. W. Hirsch and S. Smale. . Pure and Applied Mathematics, Vol. 60. Academic Press \[Harcourt Brace Jovanovich, Publishers\], New York-London, 1974. S. Li, S. Yuan, and H. Wang. Disease transmission dynamics of an epidemiological predator-prey system in open advective environments. , 28(2):1480--1502, 2023. A. J. Lotka. . Williams and Wilkins, Baltimore, 1925. R. E. Mickens. . World Scientific Publishing Co., Inc., River Edge, NJ, 1994. R. E. Mickens. Nonstandard finite difference schemes for differential equations. , 8(9):823--847, 2002. R. E. Mickens. A nonstandard finite-difference scheme for the Lotka-Volterra system. , 45(2-3):309--314, 2003. B. T. Mulugeta, L. Yu, Q. Yuan, and J. Ren. Bifurcation analysis of a predator-prey model with strong Allee effect and Beddington-DeAngelis functional response. , 28(3):1938--1963, 2023. J. M. Sanz-Serna. An unconventional symplectic integrator of W. Kahan. , 16(1-2):245--250, 1994. M. S. Shabbir, Q. Din, M. Safeer, M. A. Khan, and K. Ahmad. A dynamically consistent nonstandard finite difference scheme for a predator-prey model. , pages Paper No. 381, 17, 2019. V. Volterra. Variations and fluctuations of the number of individuals in animal species living together. , 3(1):3--51, 1928. D. Yan, Y. Yuan, and X. Fu. Asymptotic analysis of an age-structured predator-prey model with ratio-dependent Holling III functional response and delays. , 12(1):391--414, 2023. X. Yang, L. Chen, and J. Chen. Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models. , 32(4):109--116, 1996. P. Zimmermann, A. Casamayou, N. Cohen, G. Connan, T. Dumont, L. Fousse, F. Maltey, M. Meulien, M. Mezzarobba, C. Pernet, et al. . SIAM, Philadelphia, PA, 2018.

arxiv_math

--- abstract: | Cluster exchange groupoids are introduced by King-Qiu as an enhancement of cluster exchange graphs to study stability conditions and quadratic differentials. In this paper, we introduce the exchange groupoid for any finite Coxeter-Dynkin diagram $\Delta$ and show that the fundamental group of which is isomorphic to the corresponding braid group associated with $\Delta$. address: - "Hz: School of Mathematics and Statistics Henan University 475004 Kaifeng China" - "Hp: Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, 101408 Beijing, China" - "Qy: Yau Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China. & Beijing Institute of Mathematical Sciences and Applications, Yanqi Lake, Beijing, China" author: - Zhe Han, Ping He and Yu Qiu title: Cluster braid groups of Coxeter-Dynkin diagrams --- # Introduction Coxeter groups considered as reflection groups have a vibrant structure of geometry and algebra. Coxeter groups of finite types are classified Coxeter-Dynkin diagrams $\Delta$ [@Bo68]. A presentation of a Coxeter group is encoded in the Coxeter diagram. For each Coxeter group, the (Artin's) braid group is presented by the same generators $s_i$ and relations except $s^2_i=1$. Thus Coxeter groups are quotient groups of the corresponding braid groups. It is well-known that the braid groups are the fundamental groups of the space of regular orbits for which Coxeter groups are the corresponding complex reflection groups [@Br72; @De; @V83]. The braid groups also appear in the theory of cluster algebras and stability conditions on triangulated categories [@ST; @Q15; @QW; @Q16; @KQ2]. In [@KQ2], the braid group corresponding to a simply laced Dynkin diagram is realized as the cluster braid group, which is constructed using the combinatorial of the cluster category. In this paper, we focus on the braid group associated to any finite type *weighted graph/Coxeter graph* $\Delta$. We give an alternative realization of this group by the corresponding cluster braid group. Given an ordinary quiver with potential $(Q, W)$, let $\Lambda$ be the mutation equivalent class of $(Q,W)$. There is a Ginzburg dg algebra $\Gamma:=\Gamma(Q,W)$. The *cluster category* $\mathcal{C}( \Lambda)\colon=\operatorname{per}(\Gamma)/\operatorname{pvd}(\Gamma)$ has an exchange graph $\underline{\operatorname{CEG}}(\Lambda)$ with vertices corresponding to cluster tilting objects and unoriented edges corresponding to mutations between cluster tilting objects, cf. [@IY; @K12]. In many classes of examples, $\underline{\operatorname{CEG}}(\Lambda)$ can be decomposed into squares and pentagons (i.e., [@FST] for the surface case and [@Q15] for the simply laced Dynkin case). This is related to the pentagon identity of (quantum) dilogarithms, cf. [@K11]. From the point view of tilting theory (i.e., simple HRS-tilting of hearts in $\operatorname{pvd}(\Gamma)$), it is natural to consider the oriented version of $\underline{\operatorname{CEG}}(\Lambda)$ [@KQ] by replacing each edge with an oriented 2-cycle. The cluster exchange groupoid $\operatorname{\mathcal{CEG}}(\Lambda)$ is the quotient of the path groupoid of $\operatorname{CEG}(\Lambda)$ by the square and pentagon relations (and an extra hexagon relation in general). The cluster braid group $\operatorname{CT}_{\Lambda}(\mathbf{Y})$ is defined to be the fundamental group of $\operatorname{\mathcal{CEG}}(\Lambda)$ based at a vertex $\mathbf{Y}$ in $\operatorname{CEG}(\Lambda)$ (). The generators of which are 2-cycles (called local twists) mentioned above. Cluster categories of simply laced (ADE) Dynkin type correspond to finite root systems and thus to finite crystallographic Coxeter groups [@FZ2]. For a Coxeter diagram of type BCFG, one could define the cluster category using folding techniques, cf. [@CQ]. However, for finite non-crystallographic Coxeter graphs (i.e., of types H and I), there is no automorphism of the diagram inducing the embedding of the corresponding root systems, cf. [@Dy; @Lu]. Thus there is no cluster category and cluster exchange graph of these types yet, from a categorical point of view. In papers [@DT; @DT2], the authors extend the cluster theory to the quivers of types H and I by mutations of matrices with real entries. Their construction is based on the weighted folding technique, cf. [@Cr] for more details. In this paper, we introduce and study cluster exchange groupoid $\operatorname{\mathcal{CEG}}$ for all finite-type Coxeter-Dynkin diagrams. Given a weighted graph $\overrightarrow \Delta$, one could construct a simplicial map $f\colon \overrightarrow \Lambda\to \overrightarrow \Delta$ from a simply laced quiver $\overrightarrow \Lambda$ which is called a weighted folding (see Definition [\[def:fold\]](#def:fold){reference-type="ref" reference="def:fold"}). For each finite weighted folding $f: \overrightarrow \Lambda\to \overrightarrow \Delta$, we define the vertex set of $\operatorname{CEG}(\Delta)$ as a subset of $\operatorname{CEG}(\Lambda)$. More precisely, each vertex is a cluster tilting object (CTO) $\mathbf{Y}$ in $\mathcal{C}(\Lambda)$, which is compatible with weighted folding, and hence is called a weighted CTO. The edges of $\operatorname{CEG}(\Delta)$ correspond to paths in $\operatorname{\mathcal{CEG}}(\Lambda)$. In the finite type case, a key point is that the definition of $\operatorname{CEG}(\Delta)$ is independent of the choice of $f$. To obtain $\operatorname{\mathcal{CEG}}(\Delta)$, we need to add an additional $(m+2)$-gon relation for each edge with weight $m$, which generalizes the square and pentagon relations [@KQ2] in the simply laced case. Along the way, we also show that $\underline{\operatorname{CEG}}(\Delta)$ decomposes into various $(m+2)$-gons. The *cluster braid group* $\operatorname{CT}_\Delta(\mathbf{Y})$ is the subgroup of the fundamental group $\pi_1(\operatorname{\mathcal{CEG}}(\Delta),\mathbf{Y})$, which is generated by local twists $t_\mathbf{i}$ (indexed by $\mathbf{i}\in \Lambda_0$). On the other hand, for a weighted cluster tilting object (weighted CTO for short) $\mathbf{Y}$, there is an associated weighted quiver with potential $(\mathbf{Q}_\mathbf{Y},\mathbf{W}_\mathbf{Y})$. We define the associated braid group $\operatorname{Br}(\mathbf{Q}_\mathbf{Y},\mathbf{W}_\mathbf{Y})$ given by an explicit presentation with generators $b_\mathbf{i}$ corresponding to vertices $\mathbf{i}\in \mathbf{Q}_0$ of $\mathbf{Q}$ and relations corresponding to edges and terms in the potential $\mathbf{W}$, similarly to [@QZy]. Our main theorem is the following, which generalizes [@KQ2 Thm. 2.16] for the ADE case. **Theorem 1** (). *Let $\overrightarrow{\Delta}$ be a finite weighted quiver. For any weighted CTO $\mathbf{Y}$ in $\operatorname{CEG}(\Delta)$, there is a natural isomorphism $$\begin{gathered} \begin{array}{rcl} \Psi: \operatorname{CT}_\Delta(\mathbf{Y}) & \longrightarrow & \operatorname{Br}(\mathbf{Q}_\mathbf{Y},\mathbf{W}_\mathbf{Y}) \\ t_\mathbf{i}& \mapsto & b_\mathbf{i} \end{array}\end{gathered}$$ where $t_\mathbf{i}$ (resp. $b_\mathbf{i}$) is the generator of $\operatorname{CT}_\Delta(\mathbf{Y})$ (resp. $\operatorname{Br}(\mathbf{Q}_\mathbf{Y},\mathbf{W}_\mathbf{Y})$) corresponding to $\mathbf{i}\in\Delta_0$.* The paper is organized as follows. In Section 2, we recall some basic notations and results on weighted graphs and Artin braid groups. In Section 3, we introduce braid groups with finite presentations for weighted quivers with potential of types H and I and show that the presentation is compatible with the mutation. In Section 4, we define the cluster exchange groupoid for a finite weighted quiver. We prove that its fundamental group is isomorphic to the corresponding Artin braid group. ## Acknowledgment {#acknowledgment .unnumbered} Zhe Han would like to thank Department of Mathematical Sciences of Tsinghua University for its support and hospitality during the visit in 2023. In the twin paper \[QZx\], the fusion-stable interpretation of cluster categories/exchange graphs is given. This work is inspired by the works of Duffield and Tumarkin. Zhe Han is supported by the National Natural Science Foundation of China (No. 12001164). Yu Qiu is supported by the National Key R$\&$D Program of China (No.2020 YFA0713000) and National Natural Science Foundation of China (No.12031007 and No.12271279). # Artin's braid groups associated to weighted graphs A *weighted graph* (or *Coxeter graph*) $\Delta$ is a graph $(\Delta_0,\Delta_1)$ with at most one edge $\epsilon_{ij}$ between any two vertices $i,j\in\Delta_0$ and a weight function $\mathbf{w}:\Delta_1\to\mathbb{Z}_{\geq 2}$. For convenience, we use $\epsilon_{ij}=\emptyset$ to denote the empty edge with weight $\mathbf{w}(\epsilon_{ij})=2$. We will regard a usual graph as a weighted graph with trivial weights $\mathbf{w}(\epsilon_{ij})\equiv3$ for $\epsilon_{ij}\ne\emptyset$. The map could be encoded in Coxeter matrix $M=(m_{ij}=\mathbf{w}(\epsilon_{ij}))$ satisfies $m_{ii}=2$ and $m_{ij}=m_{ji}$. For each positive integer $k\geq 2$, there is a relation $$\operatorname{Br}^k(a,b)\colon\;\underbrace{aba\cdots}_{k}=\underbrace{bab\cdots}_{k}.$$ Here, the composition is from left to right. For simplicity, we write $\operatorname{Co}(a,b)=\operatorname{Br}^2(a,b)$ and $\operatorname{Br}(a,b)=\operatorname{Br}^3(a,b)$. Throughout, we will use the conjugation notation $a^b=b^{-1}ab$, and thus $\operatorname{Br}(a,b)$ is equivalent to $a^b=b^a$. **Definition 2**. Let $\Delta$ be a weighted graph with $|\Delta_0|=n$. The *(Artin) braid group* $\operatorname{Br}\Delta$ associated with $\Delta$ is a group with generators $b_1,\cdots,b_n$ and defined as the following $$\operatorname{Br}\Delta= \langle b_1, \ldots, b_n \mid \operatorname{Br}^{\mathbf{w}(\epsilon_{ij})}(b_i,b_j), i \ne j \rangle$$ The *Coxeter group* $W(\Delta)$ of $\Delta$ is defined to be the quotient group of $\operatorname{Br}\Delta$ by the normal subgroup generated by $b^2_i, 1\le i\leq n$. A weighted graph $\Delta$ is called *finite* if its Coxeter group $W(\Delta)$ is a finite group. The list in provides a complete and non-redundant enumeration of finite Coxeter graphs. Note that types GHI can be simplified to $H_{3,4}$ and $I_2(m\ge5)$. $$\begin{gathered} \renewcommand{\arraystretch}{2} \begin{array}{llr} A_{n}\ (n\geq 1): \quad & \xymatrix{1 \ar@{-}[r]& 2 \ar@{-}[r]& \cdots \ar@{-}[r]&n-1\ar@{-}[r]& n } \\ B_{n}=C_n\ (n\geq 2): \quad & \xymatrix{1 \ar@{-}[r]& 2 \ar@{-}[r]& \cdots\ar@{-}[r]&n-1 \ar@{-}[r]^{\quad 4}& n } \\ D_{n}\ (n\geq 4): \quad & \xymatrix@R=0.4pc{ 1 \\ & 3 \ar@{-}[ul]\ar@{-}[dl] \ar@{-}[r]& 4 \ar@{-}[r]& \cdots \ar@{-}[r]&n-1\ar@{-}[r]& n\\ 2\\} \\ E_{6,7,8}: \quad & \xymatrix@R=1.5pc{ && 4 \\ 1 \ar@{-}[r]& 2 \ar@{-}[r]& 3 \ar@{-}[r]\ar@{-}[u]& 5 \ar@{-}[r]& 6 \ar@{.}[r]& 7 \ar@{.}[r]& 8} \vspace{1mm}\\ F_4: \quad & \xymatrix@R=1.5pc{ 1 \ar@{-}[r]& 2 \ar@{-}[r]^4& 3 \ar@{-}[r]&4}\\ G_2: \quad & \xymatrix{1\ar@{-}[r]^-{6}&2}\\ H_{2,3,4}: \quad &\xymatrix@R=1.5pc{ 1 \ar@{-}[r]^5& 2 \ar@{.}[r]& 3\ar@{.}[r]&4}\\ I_2(m)\ (m\geq 7): \quad & \xymatrix@R=1.5pc{ 1 \ar@{-}[r]^m& 2} \end{array} \end{gathered}$$ The non-simply laced weighted graphs can be obtained from the simply laced ones via the so-called *weighted folding* operation defined as follows. **Definition 3**. [@Cr][\[def:fold\]]{#def:fold label="def:fold"} Let $\Lambda$ and $\Delta$ be two weighted graphs with $\Lambda$ a simply laced graph. A simplicial map $f:\Lambda\to\Delta$ is called a *weighted folding* if for each edge $\epsilon=\mathbf{i}\overset{m}{-}\mathbf{j}$ in $\Delta$ the restriction $f_\epsilon$ of $f$ to $f^{-1}(\epsilon)$ is of one of the following types. - $f_\epsilon$ is a *k-fold trivial folding*, i.e., $f^{-1}(\epsilon)=\epsilon^{\bigsqcup k}, k\in\mathbb{Z}_{\ge 0}$ and $f_\epsilon$ maps each copy identically onto $\epsilon$; - $f_\epsilon$ is a *dihedral folding*, i.e., $m>3$ and $f^{-1}(\epsilon)$ is a bipartite finite Coxeter graph of Coxeter number $m$ such that $f^{-1}(\mathbf{i})$ and $f^{-1}(\mathbf{j})$ are the two parts of $f^{-1}(\epsilon)$; - $f_\epsilon$ is a *composite folding*, i.e., $f_\epsilon=f'_\epsilon g$ where $f'_\epsilon$ is a k-trivial folding onto $\epsilon$ and $g$ is a folding onto $\epsilon^{\coprod k}$ which restricts to trivial and dihedral foldings. A weighted folding is called *finite* if $\Delta$ is a finite weighted graph. **Remark 4**. Given a weighted graph $\Delta$, there is a general construction of its folding $f\colon \Lambda\to \Delta$ in [@CP § 6]. We remark that an unfolded graph $\Lambda$ of a weighted graph is possibly non-connected. However, each connected component of the unfolded graph is also an unfolding of the weighted graph. Indeed, let $f:\Lambda\to\Delta$ be a weighted folding and $\Lambda$ be a disjoint union $\Lambda_1\cup\cdots\cup\Lambda_s$ of connected components $\Lambda_i,1\leq i\leq s$. Then for each $\epsilon\in\Delta$, $f^{-1}(\epsilon)=\coprod^s_{i=1}f^{-1}_i(\epsilon)$ is a disjoint union of finite Coxeter graphs, where $f_i$ is the restriction of $f$ on $\Lambda_i$. Since $f$ is a folding, we have that $f_i:f^{-1}_i(\epsilon)\to\epsilon$ is of one of the three types in Definition [\[def:fold\]](#def:fold){reference-type="ref" reference="def:fold"}. So the claim follows. In the rest of this paper, we always assume unfolded graphs $\Lambda$ are connected without mentioning them. For a weighted folding $f\colon \Lambda\to \Delta$, it induces a group homomorphism between corresponding braid groups $$\label{eq:cri} \begin{array}{rcl} \iota_f:\operatorname{Br}\Delta & \longrightarrow & \operatorname{Br}\Lambda\\ b_\mathbf{i}& \mapsto & \prod_{j\in f^{-1}(\mathbf{i})}b_j, \end{array}$$ where $b_\mathbf{i}$ (resp. $b_j,j\in f^{-1}(\mathbf{i})$) is the generator of $\operatorname{Br}\Delta$ (resp. $\operatorname{Br}\Lambda$) given by vertex $\mathbf{i}\in\Delta_0$ (resp. $j\in\Lambda_0$). It is conjectured that the map $\iota_f$ is injective [@CP] and one has the following. **Theorem 5** ([@Cr Prop. 4.3 and Thm. 1.4]). *Let $f:\Lambda\to\Delta$ be a finite weighted folding. Then $\iota_f$ in  [\[eq:cri\]](#eq:cri){reference-type="eqref" reference="eq:cri"} is injective.* **Example 6** (Finite weighted foldings). Let $\Delta$ be a finite weighted graph. For each edge $\epsilon \in\Delta_1$ of weight $m$, there are only finitely many simply laced graphs of Coxeter number $m$. So there are only finitely many unfoldings of $\Delta$. Finite weighted foldings of Dynkin diagrams are well-known. $$\begin{array}{ccccc} A_{2n-1}: & \begin{tikzcd}[row sep=small,column sep=small] 1^+\ar[r,no head]&\cdots\ar[r,no head]&(n-1)^+\ar[dr,no head] &\\ &&&n\\ 1^-\ar[r,no head]&\cdots\ar[r,no head]&(n-1)^-\ar[ur,no head] & \end{tikzcd} & \overset{f}{\longrightarrow} & C_n: & \begin{tikzcd}[row sep=.2em,column sep=small] \mathbf{1}\ar[r,no head]&\cdots\ar[r,no head]&\mathbf{n-1}\ar[r, no head,"4"]&\mathbf{n} \end{tikzcd}\\ D_{n+1}:&\begin{tikzcd}[row sep=small,column sep=small] &&&n^+\\ 1\ar[r,no head] &\cdots\ar[r,no head]&n-1\ar[ur,no head]\ar[dr,no head]&\\ & & & n^- \end{tikzcd}& \overset{f}{\longrightarrow} &B_n: &\begin{tikzcd}[row sep=small,column sep=small] \mathbf{1}\ar[no head,r]&\cdots\ar[r,no head]&\mathbf{n-1}\ar[r,no head,"4"]&\mathbf{n} \end{tikzcd}\\ E_6:&\begin{tikzcd}[row sep=small,column sep=small] 4^+\ar[no head, r]& 3^+\ar[no head,dr] & & \\ &&2\ar[no head,r]&1\\ 4^-\ar[no head,r]&3^-\ar[no head,ur]&& \end{tikzcd}& \overset{f}{\longrightarrow} &F_4:&\begin{tikzcd}[row sep=small,column sep=small] \mathbf{4}\ar[no head,r]&\mathbf{3}\ar[no head,r,"4"]&\mathbf{2} \ar[no head,r]&\mathbf{1} \end{tikzcd}\\ D_4:&\begin{tikzcd}[row sep=small,column sep=small] &&1^+\\ 2\ar[no head, urr]\ar[no head,drr]\ar[no head,rr]&&1^\circ\\ &&1^- \end{tikzcd}& \overset{f}{\longrightarrow} &G_2:&\begin{tikzcd}[row sep=small,column sep=small] \mathbf{2}\ar[r,no head,"6"]&\mathbf{1} \end{tikzcd} \end{array}$$ Finite weighted foldings $f:\Lambda\to\Delta$ of non-crystallographic Coxeter diagrams $\Delta$ are listed in the following. - The graph $\Lambda$ is of type $E_8$ and $\Delta$ is of type $H_4$ while $$f(i^\pm)=\mathbf{i}, 1\leq i\leq 4,\text{ and } f(\epsilon_{st})=\epsilon_{f(s)f(t)}, s,t\in(E_8)_0.$$ Note that the restriction of $f$ onto the subgraph $\{i^\pm|1\leq i\leq 3\}$ gives a folding from $D_6$ to $H_3$. - The graph $\Lambda$ is simply laced Dynkin of Coxeter number $m$ and $\Delta$ is of type $I_2(m)$ while $f$ maps bullets in the first (resp. second) row of $\Lambda$ to $\mathbf{1}$ (resp. $\mathbf{2}$) and all edges to the unique edge in $I_2(m)$   # Presentation of a weighted quiver with potential An ordinary *quiver* $Q$ is a quadruple $(Q_0,Q_1,s,t)$, where $Q_0$ denotes the set of vertices, $Q_1$ denotes the set of arrows, and $s$ and $t$ are two maps that associate an arrow with its start and end points, respectively. A *potential* $W$ of $Q$ is a linear combination (possibly infinite) of cyclic paths of $Q$, with the property that no two different cycles in $W$ with non-zero coefficient can be obtained from each other by rotations. The pair $(Q,W)$ is called a *quiver with potential* (QP for short). Mutation of QP is developed by Derksen-Weyman-Zelevinsky in [@DWZ], which is an involution on a QP $(Q,W)$ and results in a new quiver with QP $(Q',W')$. We follow the formulations in [@DWZ]. A *weighted quiver* is an oriented weighted graph. A weighted quiver with potential (weighted QP for short) is a pair $(\mathbf{Q},\mathbf{W})$ where $\mathbf{Q}$ is a weighted quiver and $\mathbf{W}$ is a linear combination of cycles in $\mathbf{Q}$, with the property that no two different cycles in $W$ with non-zero coefficient can be obtained from each other by rotations. A map between $(Q,W)$ and $(\mathbf{Q},\mathbf{W})$ is called a *folding* if it is a folding between the underlying graphs that is compatible with the orientation, and sends $W$ to $\mathbf{W}$. In the rest of this paper, when we have a finite weighted folding $f:\Lambda\to\Delta$ in , we fix an orientation $\overrightarrow{\Lambda}$ (resp. $\overrightarrow{\Delta}$) of the finite graph $\Lambda$ (resp. $\Delta$), and $$\label{eq:f} f\colon \overrightarrow{\Lambda}\longrightarrow \overrightarrow{\Delta}$$ represents the folding $(\overrightarrow{\Lambda},0)\to (\overrightarrow{\Delta},0)$. For any $\mathbf{i}\in\mathbf{Q}_0$, we call $\mu_\mathbf{i}=\prod_{i\in f^{-1}(\mathbf{i})}\mu_i$ a *weighted mutation* with respect to $f$. We denote $$\mu_{\mathbf{i}_1,\cdots,\mathbf{i}_n}:=\mu_{\mathbf{i}_1}\ldots\mu_{\mathbf{i}_n}$$ for the sequence of repeated weighted mutations. **Definition 7**. For any mutation sequence $\mu_{\mathbf{i}_1,\cdots,\mathbf{i}_n}$, define $$\mu_{\mathbf{i}_1,\cdots,\mathbf{i}_n}(\overrightarrow{\Delta})=f\circ\mu_{\mathbf{i}_1,\cdots,\mathbf{i}_n}(\overrightarrow{\Lambda})$$ called the *mutation* of the weighted quiver $\overrightarrow{\Delta}$. **Lemma 8**. *The mutation of a weighted quiver is well-defined.* *Proof.* For $\Delta$ of types $B_n, C_n, F_4$ and $G_2$, it is obtained from $\Lambda$ by a group action, which is compatible with mutations, see [@CQ] for the categorical constructions. For $\Delta$ of types $H$ and $I$, the assertion follows by a direct check, or [@DT; @DT2], which also alternatively follows from a categorical interpretation via fusion actions, cf. [@QZx]. ◻ Two weighted QPs are said to be *mutation-equivalent* if a sequence of mutations relates them. A weighted QP is called *finite* if it is mutation-equivalent to $\overrightarrow{\Delta}$ with $\Delta$ a finite weighted graph. **Definition 9**. Let $(\mathbf{Q},\mathbf{W})$ be a finite weighted QP. We define the associated braid group $\operatorname{Br}(\mathbf{Q},\mathbf{W})$ by the following presentation. - Generators: vertices of $\mathbf{Q}$. - Relations $\operatorname{Br}^m(b_\mathbf{i},b_\mathbf{j})$ if there is exactly one arrow between $\mathbf{i}$ and $\mathbf{j}$ of weight $m$; - Relations from potential terms: 1. $\operatorname{Co}(b_\mathbf{2},b_\mathbf{1}^{b_{\mathbf{l}} b_{\mathbf{l}-1} \cdots b_\mathbf{3}})$ if there is a $l$-cycle, $l\geq3$, between $\mathbf{1},\mathbf{2},\cdots,\mathbf{l}$ which contributes a term in $\mathbf{W}$ and has all arrows of weight 3, see Figure [\[fig:cycle-present\]](#fig:cycle-present){reference-type="ref" reference="fig:cycle-present"} I; 2. $\operatorname{Co}(b_\mathbf{1},b^{b_\mathbf{2}}_\mathbf{3})$ if there is a 3-cycle between $\mathbf{1},\mathbf{2}$ and $\mathbf{3}$ which contributes a term in $\mathbf{W}$ such that the two arrows incident to $\mathbf{1}$ are of weights $m$ for $m\in\{4,5\}$ and the arrow opposite to $\mathbf{1}$ is of weight 3, see Figure [\[fig:cycle-present\]](#fig:cycle-present){reference-type="ref" reference="fig:cycle-present"} II; 3. $\operatorname{Co}(b_\mathbf{3},b^{b_\mathbf{1}b_\mathbf{4}}_\mathbf{2})$ if there is a 4-cycle between $\mathbf{1},\mathbf{2},\mathbf{3}$ and $\mathbf{4}$ which contributes a term in $\mathbf{W}$ such that the two arrows end at $1$ and $3$ are of weight $m$ for $m\in\{4,5\}$ while the two arrows start at $1$ and $3$ are of weight $3$, see Figure [\[fig:cycle-present\]](#fig:cycle-present){reference-type="ref" reference="fig:cycle-present"} III; 4. $\operatorname{Co}(b_\mathbf{1},b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{2})$ and $\operatorname{Br}(b_\mathbf{1},b^{b_\mathbf{2}}_\mathbf{3})$ if there is a 3-cycle between $\mathbf{1},\mathbf{2}$ and $\mathbf{3}$ which contributes a term in $\mathbf{W}$ and has all arrows of weight 5, see Figure [\[fig:cycle-present\]](#fig:cycle-present){reference-type="ref" reference="fig:cycle-present"} IV. **Remark 10**. Note that $\operatorname{Br}(\mathbf{Q},0)$ is exactly $\operatorname{Br}\mathbf{Q}$ for any finite weighted quiver $\mathbf{Q}$. Moreover, when $(\mathbf{Q},\mathbf{W})$ mutation-equivalent to a Dynkin quiver, our definition coincides with [@Q16 Def. 10.1], [@GM Def. 2.2] and [@HHLP Def. 4.5]. We remark that relations from potential terms have equivalent descriptions as the following (cf. Figure [\[fig:cycle-present\]](#fig:cycle-present){reference-type="ref" reference="fig:cycle-present"}). - For type I, we have the following equivalences of relations $$\begin{tikzcd}[row sep=small,column sep=huge] \operatorname{Co}(b_\mathbf{1},b^{b_{\mathbf{l}-\mathbf{1}}\cdots b_\mathbf{2}}_\mathbf{l})\ar[Leftrightarrow]{r}{(-)^{\underline{b_\mathbf{2}}}}& \operatorname{Co}(b^{\underline{b_\mathbf{2}}}_\mathbf{1},b^{b_{\mathbf{l}-\mathbf{1}}\cdots b_\mathbf{3}}_\mathbf{l})\ar[Leftrightarrow]{r}{\operatorname{Br}(b_\mathbf{1},b_\mathbf{2})}& \operatorname{Co}(b^{b_\mathbf{1}}_\mathbf{2},b^{b_{\mathbf{l}-\mathbf{1}}\cdots b_\mathbf{3}}_\mathbf{l})\\ \ar[Leftrightarrow,shorten=5.3mm,xshift=3.5ex]{r}{\underline{b_\mathbf{1}}} & \operatorname{Co}(b_\mathbf{2},b^{b_{\mathbf{l}-\mathbf{1}}\cdots b_\mathbf{3}\underline{b_\mathbf{1}}}_\mathbf{l}) \ar[Leftrightarrow]{r}{\operatorname{Co}(b_\mathbf{3},b_\mathbf{1})} & \operatorname{Co}(b_\mathbf{2},b^{b_{\mathbf{l}-\mathbf{1}}\cdots \underline{b_\mathbf{1}}b_\mathbf{3}}_\mathbf{l})\\ \ar[Leftrightarrow,shorten=11mm,xshift=0ex]{r}{\operatorname{Co}(b_{\mathbf{4}},b_\mathbf{1})} & \ldots \ar[Leftrightarrow,shorten=5.8mm,xshift=3.5ex]{r}{\operatorname{Co}(b_{\mathbf{l}-\mathbf{1}},b_\mathbf{1})} & \operatorname{Co}(b_\mathbf{2},b^{\underline{b_\mathbf{1}}b_{\mathbf{l}-\mathbf{1}}\cdots b_\mathbf{3}}_\mathbf{l})\\ \ar[Leftrightarrow,shorten=5.8mm,xshift=3.5ex]{r}{\operatorname{Br}(b_\mathbf{1},b_\mathbf{l})} & \operatorname{Co}(b_\mathbf{2},b^{b_\mathbf{l}b_{\mathbf{l}-\mathbf{1}}\cdots b_\mathbf{3}}_\mathbf{1}) \end{tikzcd}$$ which are also equivalent to any $\operatorname{Co}(b_{\mathbf{i}+1},b_{\mathbf{i}}^{b_{\mathbf{i}-1}\cdots b_{\mathbf{i}+1}})$ (with the convention $\mathbf{i}=\mathbf{l}+\mathbf{i}$ here), see also [@Q16 Lem. 10.2] and [@QZy Lem. 5.4]; - For type II, we have that $$\begin{tikzcd}[row sep=small, column sep=huge] \operatorname{Co}(b_\mathbf{1},b^{b_\mathbf{2}}_\mathbf{3})\ar[Leftrightarrow]{r}{\operatorname{Br}(b_\mathbf{2},b_\mathbf{3})} & \operatorname{Co}(b_\mathbf{1},b^{\underline{b_\mathbf{3}}}_\mathbf{2})\ar[Leftrightarrow]{r}{(-)^{b_\mathbf{3}}}& \operatorname{Co}(b_\mathbf{2},b^{b_\mathbf{3}}_\mathbf{1}). \end{tikzcd}$$ - For type III, we have that $$\begin{tikzcd}[row sep=small,column sep=huge] \operatorname{Co}(b_\mathbf{1},b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{4})\ar[Leftrightarrow]{r}{\operatorname{Br}(b_\mathbf{4},b_\mathbf{3})} & \operatorname{Co}(b_\mathbf{1},b^{\underline{b_\mathbf{4}}b_\mathbf{2}}_\mathbf{3})\ar[Leftrightarrow]{r}{(-)^{\underline{b_\mathbf{2}}b_\mathbf{4}}} & \operatorname{Co}(b^{\underline{b_\mathbf{2}}b_\mathbf{4}}_\mathbf{1},b_\mathbf{3})\\ \ar[Leftrightarrow,shorten=5mm,xshift=3.2ex]{r}{\operatorname{Br}(b_\mathbf{1},b_\mathbf{2})} & \operatorname{Co}(b^{b_\mathbf{1}b_\mathbf{4}}_\mathbf{2},b_\mathbf{3}). \end{tikzcd}$$ - For type IV, we have $$\begin{tikzcd}[row sep=small,column sep=huge] \operatorname{Co}(b_\mathbf{1},b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{2})\ar[Leftrightarrow]{r}{\operatorname{Br}^5(b_\mathbf{2},b_\mathbf{3})} & \operatorname{Co}(b_\mathbf{1},b^{\underline{b_\mathbf{2}}\underline{b_\mathbf{3}}}_\mathbf{3})\ar[Leftrightarrow]{r}{(-)^{b_\mathbf{3}b_\mathbf{2}}} & \operatorname{Co}(b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{1},b_\mathbf{3})\\ \ar[Leftrightarrow,shorten=4.5mm,xshift=3.5ex]{r}{\operatorname{Br}(b_\mathbf{2},b^{b_\mathbf{3}}_\mathbf{1})} & \operatorname{Co}(b^{\underline{b_\mathbf{3}}\underline{b_\mathbf{1}}b_\mathbf{3}}_\mathbf{2},b_\mathbf{3})\ar[Leftrightarrow]{r}{(-)^{\underline{b_\mathbf{3}}b_\mathbf{1}b_\mathbf{3}}} & \operatorname{Co}(b_\mathbf{2}, b^{b_\mathbf{1}b_\mathbf{3}}_\mathbf{3}). \end{tikzcd}$$ On the other hand, we have $$\begin{tikzcd}[row sep=small,column sep=huge] \operatorname{Co}(b_\mathbf{1},b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{2})\ar[Leftrightarrow]{r}{\operatorname{Br}^5(b_\mathbf{2},b_\mathbf{3})} & \operatorname{Co}(b_\mathbf{1},b^{\underline{b_\mathbf{2}}\underline{b_\mathbf{3}}}_\mathbf{3})\ar[Leftrightarrow]{r}{(-)^{b_\mathbf{3}b_\mathbf{2}}} & \operatorname{Co}(b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{1},b_\mathbf{3}). \end{tikzcd}$$ So $$\begin{tikzcd}[row sep=small,column sep=huge] \operatorname{Br}(b_\mathbf{1},b^{b_\mathbf{2}}_\mathbf{3})\ar[Leftrightarrow]{r}{\operatorname{Br}^5(b_\mathbf{2},b_\mathbf{3})} & \operatorname{Br}(b_\mathbf{1},b^{\underline{b_\mathbf{3}}}_\mathbf{2})\ar[Leftrightarrow]{r}{(-)^{b_\mathbf{3}}} & \operatorname{Br}(b^{b_\mathbf{3}}_\mathbf{1},b^{\underline{b_\mathbf{3}}\underline{b_\mathbf{2}}}_\mathbf{2})\\ \ar[Leftrightarrow,shorten=4mm,xshift=3.2ex]{r}{(-)^{b_\mathbf{2}b_\mathbf{3}}} & \operatorname{Br}(b^{b_\mathbf{3}b_\mathbf{2}b_\mathbf{3}}_\mathbf{1},b_\mathbf{2})\ar[Leftrightarrow]{r}{\operatorname{Co}(b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{1},b_\mathbf{3})} & \operatorname{Br}(b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{1},b_\mathbf{2}) \\ \ar[Leftrightarrow,shorten=4.8mm,xshift=3ex]{r}{(-)^{\underline{b_\mathbf{2}}}} & \operatorname{Br}(b^{b_\mathbf{3}}_\mathbf{1},b_\mathbf{2}). \end{tikzcd}$$ A similar discussion shows that $\operatorname{Co}(b_\mathbf{2},b^{b_\mathbf{1}b_\mathbf{3}}_\mathbf{3})$ and $\operatorname{Br}(b_\mathbf{2},b^{b_\mathbf{3}}_\mathbf{1})$ are equivalent to $\operatorname{Co}(b_\mathbf{3},b^{b_\mathbf{2}b_\mathbf{1}}_\mathbf{1})$ and $\operatorname{Br}(b_\mathbf{3},b^{b_\mathbf{1}}_\mathbf{2})$. Let $(\mathbf{Q},\mathbf{W})$ be a finite weighted QP. For each $\mathbf{i}\in\mathbf{Q}_0$, denote by $\mathbf{Q}'=\mu_\mathbf{i}(\mathbf{Q})$ and by $b'_\mathbf{i},\mathbf{i}\in\mathbf{Q}_0$ the generators of $B_{\mathbf{Q}'}$. Let $$\begin{aligned} \theta^\flat_\mathbf{i}:\operatorname{Br}(\mathbf{Q},\mathbf{W})\longrightarrow\operatorname{Br}(\mathbf{Q}',\mathbf{W}')\\ \theta^\sharp_\mathbf{i}:\operatorname{Br}(\mathbf{Q}',\mathbf{W}')\longrightarrow\operatorname{Br}(\mathbf{Q},\mathbf{W})\end{aligned}$$ satisfying $$\begin{aligned} \label{eq:flat} \theta^\flat_\mathbf{i}(b_\mathbf{j})=\begin{cases} (b'_\mathbf{j})^{b'_\mathbf{i}} &\text{if there is an arrow from $\mathbf{j}$ to $\mathbf{i}$ in $\mathbf{Q}$,}\\ b'_\mathbf{j}& \text{otherwise;} \end{cases}\end{aligned}$$ $$\begin{aligned} \label{eq:sharp} \theta^\sharp_\mathbf{i}(b'_\mathbf{j})=\begin{cases} (b_\mathbf{j})^{\underline{b_\mathbf{i}}} &\text{if there is an arrow from $\mathbf{i}$ to $\mathbf{j}$ in $\mathbf{Q}'$,}\\ b_\mathbf{j}& \text{otherwise.} \end{cases}\end{aligned}$$ **Theorem 11**. *For any finite weighted QP $(\mathbf{Q},\mathbf{W})$, $\theta^\sharp_\mathbf{i}$ and $\theta^\flat_\mathbf{i},\mathbf{i}\in\mathbf{Q}_0$ are mutually inverse group isomorphisms.* *Proof.* The assertion follows by [@GM] (see also [@QZy]) for simply laced Dynkin quivers and by [@HHLP Thm. 6.1] for weighted quivers of types $B_n=C_n,F_4,G_2$. We only prove weighted QPs of types $H$ and $I_2(m)$. Formulas  [\[eq:flat\]](#eq:flat){reference-type="eqref" reference="eq:flat"} and  [\[eq:sharp\]](#eq:sharp){reference-type="eqref" reference="eq:sharp"} define two homomorphisms $\theta^\sharp_\mathbf{i}$ and $\theta^\flat_\mathbf{i}$ between the groups that are freely generated by the vertices of $\mathbf{Q}$ and $\mathbf{Q}'$, respectively. Moreover, $\theta^\sharp_\mathbf{i}$ is the composition of $\theta^\flat_\mathbf{i}$ with the conjugation by $b_\mathbf{i}$, and $\theta^\sharp_\mathbf{i}$ and $\theta^\flat_\mathbf{i}$ are mutually inverse. So we only need to show $\theta^\sharp_\mathbf{i}$ preserves the relations in the presentation of $\operatorname{Br}(\mathbf{Q}',\mathbf{W}')$. For any vertex $j$, denote by $t_\mathbf{j}=\theta^\sharp_\mathbf{i}(b'_\mathbf{j})$. Then we have $$\operatorname{Br}^m(b'_\mathbf{i},b'_\mathbf{j})\xLongleftrightarrow{\theta^\sharp_\mathbf{i}} \operatorname{Br}^m(t_\mathbf{i},t_\mathbf{j})\xLongleftrightarrow{~\eqref{eq:sharp}} \begin{cases} \operatorname{Br}^m(b_\mathbf{i},(b_\mathbf{j})^{\underline{b_\mathbf{i}}})\xLongleftrightarrow{(-)^{\underline{b_\mathbf{i}}}} \operatorname{Br}^m(b_\mathbf{i},b_\mathbf{j}) & \text{if }\mathbf{i}\to\mathbf{j}\textbf{ in }\mathbf{Q}',\\ \operatorname{Br}^m(b_\mathbf{i},b_\mathbf{j}) & \text{otherwise.} \end{cases}$$ where $m$ is the weight of the edge in $(\mathbf{Q}',\mathbf{W}')$ between $\mathbf{i}$ and $\mathbf{j}$. A similar discussion shows that $\operatorname{Br}^k(t_\mathbf{j},t_\mathbf{l})$ holds for any $\mathbf{j},\mathbf{l}\neq\mathbf{i}$ such that there is no cycles between $\mathbf{i},\mathbf{j},\mathbf{l}$ in $(\mathbf{Q},\mathbf{W})$ and $(\mathbf{Q}',\mathbf{W}')$, where $k$ is the weight of the edge in $(\mathbf{Q}',\mathbf{W}')$ between $\mathbf{j}$ and $\mathbf{l}$. It remains to show that $\theta^\sharp_\mathbf{i}$ preserves the relations of cycles of $\operatorname{Br}(\mathbf{Q}',\mathbf{W}')$. A direct calculation shows that, up to the opposite of weighted QPs, all possible cycles and related mutations are listed below. The images of the relations in $\operatorname{Br}(\mathbf{Q}',\mathbf{W}')$ under $\theta^\sharp_\mathbf{i}$ are equivalent to the relations in $\operatorname{Br}(\mathbf{Q},\mathbf{W})$, where $t_\mathbf{j}=\theta^\sharp_\mathbf{i}(b'_\mathbf{j})$. In each case of the list, the left weight QP is (the full subquiver of) $(\mathbf{Q}',\mathbf{W}')$ and the right weighted QP is (the corresponding full subquiver of) $(\mathbf{Q},\mathbf{W})$, and an equivalence may use equivalences above it. $$\begin{gathered} \begin{tikzpicture}[scale=0.5, arrow/.style={->,>=stealth}, equalto/.style={double,double distance=2pt}, mapto/.style={|->}] \node (x2) at (0,2){}; \node (x1) at (2,-1){}; \node (x3) at (-2,-1){}; \node at (x1){$\mathbf{1}$}; \node at (x2){$\mathbf{2}$}; \node at (x3){$\mathbf{3}$}; \foreach \mathbf{n}/\m in {3/2,2/1}{\draw[arrow] (x\mathbf{n}) to (x\m);} \draw(-1.3,.1)node[above]{\tiny{$3$}}(1.3,.1)node[above]{\tiny{$5$}}; \draw(4,1)node[above]{$\mu_\mathbf{2}$}node[below]{\tiny{$t_\mathbf{1}=b_\mathbf{1}^{\underline{b_\mathbf{2}}}$}}; \draw(4,-.4)node{\tiny{$t_\mathbf{2}=b_\mathbf{2}$}}(4,-1)node{\tiny{$t_\mathbf{3}=b_\mathbf{3}$}}; \draw[arrow,decorate, decoration={snake}](3,1)to(5,1); \node (x2) at (0+8,2){}; \node (x1) at (2+8,-1){}; \node (x3) at (-2+8,-1){}; \node at (x1){$\mathbf{1}$}; \node at (x2){$\mathbf{2}$}; \node at (x3){$\mathbf{3}$}; \foreach \mathbf{n}/\m in {1/2,2/3,3/1} {\draw[arrow] (x\mathbf{n}) to (x\m);} \draw(6.7,.1)node[above]{\tiny{$3$}}(9.4,.1)node[above]{\tiny{$5$}}(8,-1.3)node{\tiny{5}}; \draw(17,.5)node{ $\begin{array}{ccl} \operatorname{Co}(t_\mathbf{1},t_\mathbf{3})&\Longleftrightarrow&\operatorname{Co}(b_\mathbf{1},b_\mathbf{3}^{b_\mathbf{2}})\\ \operatorname{Br}^5(t_\mathbf{1},t_\mathbf{2})&\Longleftrightarrow&\operatorname{Br}^5(b_\mathbf{1},b_\mathbf{2})\\ \operatorname{Br}(t_\mathbf{2},t_\mathbf{3})&\Longleftrightarrow&\operatorname{Br}(b_\mathbf{2},b_\mathbf{3}) \end{array}$ }; \end{tikzpicture}\end{gathered}$$ $$\begin{gathered} \begin{tikzpicture}[scale=0.5, arrow/.style={->,>=stealth}, equalto/.style={double,double distance=2pt}, mapto/.style={|->}] \node (x) at (-3,2){}; \node (x2) at (0,2){}; \node (x1) at (2,-1){}; \node (x3) at (-2,-1){}; \node at (x1){$\mathbf{1}$}; \node at (x2){$\mathbf{2}$}; \node at (x3){$\mathbf{3}$}; \foreach \mathbf{n}/\m in {3/2,2/1,1/3}{\draw[arrow] (x\mathbf{n}) to (x\m);} \draw(-1.3,.1)node[above]{\tiny{$5$}}(1.3,.1)node[above]{\tiny{$5$}}(0,-1.3)node{\tiny{$3$}}; \draw(4,1)node[above]{$\mu_\mathbf{2}$}node[below]{\tiny{$t_\mathbf{1}=b_\mathbf{1}^{\underline{b_\mathbf{2}}}$}}; \draw(4,-.4)node{\tiny{$t_\mathbf{2}=b_\mathbf{2}$}}(4,-1)node{\tiny{$t_\mathbf{3}=b_\mathbf{3}$}}; \draw[arrow,decorate, decoration={snake}](3,1)to(5,1); \node (x2) at (0+8,2){}; \node (x1) at (2+8,-1){}; \node (x3) at (-2+8,-1){}; \node at (x1){$\mathbf{1}$}; \node at (x2){$\mathbf{2}$}; \node at (x3){$\mathbf{3}$}; \foreach \mathbf{n}/\m in {1/2,2/3,3/1} {\draw[arrow] (x\mathbf{n}) to (x\m);} \draw(6.7,.1)node[above]{\tiny{$5$}}(9.4,.1)node[above]{\tiny{$5$}}(8,-1.3)node{\tiny{5}}; \draw(17.5,.5)node{ $\begin{array}{ccl} \operatorname{Co}(t_\mathbf{1},t^{t_\mathbf{3}}_\mathbf{2})&\Longleftrightarrow&\operatorname{Co}(b_\mathbf{1},b_\mathbf{2}^{b_\mathbf{3}b_\mathbf{2}})\\ \operatorname{Br}(t_\mathbf{1},t_\mathbf{3})&\Longleftrightarrow&\operatorname{Br}(b_\mathbf{1},b^{b_\mathbf{2}}_\mathbf{3})\\ \operatorname{Br}^5(t_\mathbf{1},t_\mathbf{2})&\Longleftrightarrow&\operatorname{Br}^5(b_\mathbf{1},b_\mathbf{2})\\ \end{array}$ }; \end{tikzpicture}\end{gathered}$$ $$\begin{gathered} \begin{tikzpicture}[yscale=0.6,xscale=0.5, arrow/.style={->,>=stealth}, equalto/.style={double,double distance=2pt}, mapto/.style={|->}] \node (x2) at (-4,2){}; \node (x3) at (0,2){}; \node (x4) at (0,-1){}; \node (x1) at (-4,-1){}; \node at (x1){$\mathbf{1}$}; \node at (x2){$\mathbf{2}$}; \node at (x3){$\mathbf{3}$}; \node at (x4){$\mathbf{4}$}; \foreach \mathbf{n}/\m in {2/1,3/2,1/3,4/1,3/4} {\draw[arrow] (x\mathbf{n}) to (x\m);} \draw(-4,.5)node[left]{\tiny{$3$}}(0,.5)node[right]{\tiny{$3$}}(-2,-1)node[below]{\tiny{$5$}}(-2,2)node[above]{\tiny{$5$}}(-2.2,.7)node{\tiny{$5$}}; \draw(2.5,1.5)node[above]{$\mu_\mathbf{2}$}node[below]{\tiny{$t_\mathbf{1}=b_\mathbf{1}^{\underline{b_\mathbf{2}}}$}}; \draw(2.5,.1)node{\tiny{$t_\mathbf{2}=b_\mathbf{2}$}}(2.5,-.5)node{\tiny{$t_\mathbf{3}=b_\mathbf{3}$}}(2.5,-1.1)node{\tiny{$t_\mathbf{4}=b_\mathbf{4}$}}; \draw[arrow,decorate, decoration={snake}](1.5,1.5)to(3.5,1.5); \node (x2) at (5,2){}; \node (x3) at (9,2){}; \node (x4) at (9,-1){}; \node (x1) at (5,-1){}; \node at (x1){$\mathbf{1}$}; \node at (x2){$\mathbf{2}$}; \node at (x3){$\mathbf{3}$}; \node at (x4){$\mathbf{4}$}; \foreach \mathbf{n}/\m in {1/2,2/3,3/4,4/1}{\draw[arrow] (x\mathbf{n}) to (x\m);} \draw(5,.5)node[left]{\tiny{$3$}}(9,.5)node[right]{\tiny{$3$}}(7,-1)node[below]{\tiny{$5$}}(7,2)node[above]{\tiny{$5$}}; \draw(16,.5)node{ $\begin{array}{ccl} %&\Co(b'_\2,b'_\4)&\\ \operatorname{Br}(t_\mathbf{1},t_\mathbf{2})&\Longleftrightarrow&\operatorname{Br}(b_\mathbf{1},b_\mathbf{2})\\ \operatorname{Co}(t_\mathbf{3},t^{t_\mathbf{2}}_\mathbf{1})&\Longleftrightarrow&\operatorname{Co}(b_\mathbf{1},b_\mathbf{3})\\ \operatorname{Br}(t_\mathbf{3},t_\mathbf{4})&\Longleftrightarrow&\operatorname{Br}(b_\mathbf{3},b_\mathbf{4})\\ \operatorname{Co}(t_\mathbf{3},t^{t_\mathbf{4}}_\mathbf{1})&\Longleftrightarrow&\operatorname{Co}(b_\mathbf{3},b_\mathbf{2}^{b_\mathbf{1}b_\mathbf{4}})\\ \operatorname{Br}^5(t_\mathbf{1},t_\mathbf{3})&\Longleftrightarrow&\operatorname{Br}^5(b_\mathbf{2},b_\mathbf{3}) \end{array}$ }; \end{tikzpicture}\end{gathered}$$ ◻ # Cluster braid groups: weighted case ## Cluster exchange graphs   Let $(Q, W)$ be a quiver with potential that mutation equivalent to $\Lambda$. Denote by $\Gamma(Q,W)$ be the *Ginzburg dg algebra* associated to $(Q,W)$ (c.f. [@K12 §7.2] for the definition). Let $\operatorname{pvd}(\Lambda)$ be the *finite-dimensional (or perfectly valued) derived category* of $\Gamma(Q,W)$, and $\operatorname{per}(\Lambda)$ be the *perfect derived category* of $\Gamma(Q,W)$. The Verdier quotient $\mathcal{C}(\Lambda):=\operatorname{per}(\Lambda)/\operatorname{pvd}(\Lambda)$ is known as the *cluster category* of $\Lambda$. An object $\mathbf{Y}$ in $\mathcal{C}(\Lambda)$ is said to be *rigid* if it has no self-extensions, and is a *cluster tilting object* (CTO for short) if, furthermore, $\mathbf{Y}$ is basic and maximal with respect to this property. **Definition 12** ([@IY]). Let $\mathbf{Y}=\oplus^n_{i=1}\mathbf{Y}_i$ be a basic CTO in $\mathcal{C}(\Lambda)$. The *forward mutation* $\mu^\sharp_i(\mathbf{Y})$ of $\mathbf{Y}$ at $\mathbf{Y}_i$ is defined as the CTO obtained by replacing $\mathbf{Y}_i$ with $\mathbf{Y}^\sharp_i$, where $$\mathbf{Y}^\sharp_i=\operatorname{Cone}(\mathbf{Y}_i\to\oplus_{j\neq i}\operatorname{Irr}(\mathbf{Y}_i,\mathbf{Y}_j)^*\otimes \mathbf{Y}_j)$$ with $\operatorname{Irr}(\mathbf{Y}_i,\mathbf{Y}_j)$ the space of irreducible morphisms between $\mathbf{Y}_i$ and $\mathbf{Y}_j$. The *oriented cluster exchange graph* $\operatorname{CEG}(\Lambda)$ of the cluster category $\mathcal{C}(\Lambda)$ is the graph whose vertices are CTOs and whose edges correspond to forward mutations. The *unoriented cluster exchange graph* $\underline{\operatorname{CEG}}(\Lambda)$ of the cluster category $\mathcal{C}(\Lambda)$ is the oriented graph obtained from $\operatorname{CEG}(\Lambda)$ by replacing each 2-cycle given by $\mu^\sharp_i$ with a single edge $\mu_i$. **Lemma 13** ([@Q15 Prop. 4.5]). *$\pi_1(\underline{\operatorname{CEG}}(\Lambda))$ is generated by squares and pentagons.* For each $\mathbf{i}\in\Delta_0$, the *weighted forward mutation* is defined as $\mu^\sharp_\mathbf{i}=\prod_{i\in f^{-1}(\mathbf{i})}\mu^\sharp_i$, which, as the categorical interpretation of $\mu_\mathbf{i}(\Delta)$, is well-defined. The *weighted cluster tilting object* (weighted CTO for short) is the object $\mathbf{Y}$ in $\mathcal{C}(\Delta)$ obtained from the initial CTO $\mathbf{Y}_\Delta:=\mathbf{k}\overrightarrow{\Delta}$ via weighted-mutations. The oriented cluster exchange graph $\operatorname{CEG}(\Delta,f)$ of $\Delta$ has weighted CTOs as vertices and weighted mutations as edges. The unoriented cluster exchange graph $\underline{\operatorname{CEG}}(\Delta,f)$ is obtained from $\operatorname{CEG}(\Delta,f)$ by replacing each 2-cycle given by $\mu^\sharp_\mathbf{i}$ with a single edge $\mu_\mathbf{i}$. **Remark 14**. The construction of generalized associahedron in [@FZ1] is valid for the Coxeter-Dynkin diagram of type $H$ and $I$. The 1-skeleton of the associahedron is exactly the cluster exchange graph. In [@FR § 5.3], the 1-skeleton of the associahedron of type $H_3$ was constructed. In [@DT; @DT2], there is a categorical approach to construct the cluster exchange graph. **Lemma 15** ([@QZx Thm. 5.2]). *The unoriented cluster exchange graph $\underline{\operatorname{CEG}}(\Delta,f)$ is a generalized associahedron of $\overrightarrow{\Delta}$ in [@FZ1], i.e. independent of $f$.* In the rest of this paper, we may use $\underline{\operatorname{CEG}}(\Delta)$ (resp. $\operatorname{CEG}(\Delta)$) to denote the unoriented (resp. oriented) cluster exchange graph of $\Delta$. **Example 16**. A direct calculation shows that $\underline{\operatorname{CEG}}(I_2(m))$ is an $(m+2)$-gon as shown by the first picture in Figure [\[fig:ceg i2\]](#fig:ceg i2){reference-type="ref" reference="fig:ceg i2"}, whose oriented version is as shown by the right picture. For each weighted CTO $\mathbf{Y}$ in $\operatorname{CEG}(\Delta)$, denote by $(\mathbf{Q}_\mathbf{Y},\mathbf{W}_\mathbf{Y})$ the weighted QP corresponds to $\mathbf{Y}$, which is folded from $(Q_\mathbf{Y},W_\mathbf{Y})$ in $\operatorname{CEG}(\Lambda)$. When $\Delta=I_2(m)$, we have $\mathbf{Q}_\mathbf{Y}=I_2(m)$ for all $\mathbf{Y}$ in $\operatorname{CEG}(\Delta)$. Denote by $\mathbf{x}$ (resp. $\mathbf{y}$) the weighted forward mutation at the object that corresponds to the sink (resp. source) of $I_2(m)$ (cf. the right picture in Figure [\[fig:ceg i2\]](#fig:ceg i2){reference-type="ref" reference="fig:ceg i2"}). The *polygonal/(m+2)-gon relation* is defined to be $$\label{eq:xy} \mathbf{x}^2=\mathbf{y}^m.$$ **Example 17**. The unoriented cluster exchange graph of $A_3, B_3/C_3, H_3$ is as shown by Figure [\[fig:ABH\]](#fig:ABH){reference-type="ref" reference="fig:ABH"}, where - $\underline{\operatorname{CEG}}(A_3)$, as shown by the first picture, has three squares and six pentagons; - $\underline{\operatorname{CEG}}(B_3)$, as shown by the second picture, has four hexagons (two in the front and two in the back) and eight squares/pentagons alternatively around them; - $\underline{\operatorname{CEG}}(H_3)$ has six heptagons (three in the front and three in the back) and twelve squares/pentagons alternatively around them. **Remark 18**. For each weighted CTO $\mathbf{Y}$ in $\underline{\operatorname{CEG}}(\Delta)$ and any arrow $\epsilon:\mathbf{i}\xrightarrow{m}\mathbf{j}$ in $\mathbf{Q}_\mathbf{Y}$, there is a mutation sequence of alternatively mutating $\mu^\sharp_\mathbf{i}$ and $\mu^\sharp_\mathbf{j}$. Using Calabi-Yau reduction (cf. [@IY Thm. 4.2] and [@KV Thm. 7.4]), one gets a full subgraph of $\underline{\operatorname{CEG}}(\Delta)$ that is isomorphic to $\underline{\operatorname{CEG}}(I_2(m))$, i.e., an $(m+2)$-gon in Example [Example 16](#ex:i2-gon){reference-type="ref" reference="ex:i2-gon"}. We prove an analogue of [@Q15 Prop. 4.5] in the case of Coxeter-Dynkin diagrams about the fundamental group of $\underline{\operatorname{CEG}}$. **Proposition 19**. *For any finite weighted quiver $\overrightarrow{\Delta}$, $\pi_1(\underline{\operatorname{CEG}}(\Delta))$ is generated by $(m+2)$-gons of the form in , for $m\in\{\mathbf{w}(\alpha)|\alpha\in\overrightarrow{\Delta}_1\}$.* *Proof.* The assertion is followed by a similar discussion as in [@Q15 Prop. 4.5]. More precisely, fixing a finite weighted folding $f:\overrightarrow{\Lambda}\to\overrightarrow{\Delta}$, we have the following. - Recall that $\underline{\operatorname{CEG}}(\Delta)$ can be embedded in $\underline{\operatorname{CEG}}(\Lambda)$ in the sense that the vertex set of $\underline{\operatorname{CEG}}(\Delta)$ is a subset of $\underline{\operatorname{CEG}}(\Lambda)$ and edges of $\underline{\operatorname{CEG}}(\Delta)$ correspond to paths in $\underline{\operatorname{CEG}}(\Lambda)$. Alternatively, if the finite weighted folding $f:\overrightarrow{\Lambda}\to\overrightarrow{\Delta}$ comes from a fusion action (cf. [@QZx § 3]), $\underline{\operatorname{CEG}}(\Delta)$ is exactly the fusion-stable part of $\underline{\operatorname{CEG}}(\Lambda)$. - The same assertion holds for the silting exchange graph. Namely, if we consider simultaneously multi-mutation in [@DT] (or fusion-stable mutation in [@QZx]), one can introduce and embed the silting exchange graph $\operatorname{SEG}(\Delta)$ into $\operatorname{SEG}(\Lambda)$. - The cluster exchange graph $\underline{\operatorname{CEG}}(\Lambda)$ can be identified with the (underlying graph of) the interval $[\mathbf{k}\overrightarrow{\Lambda},\mathbf{k}\overrightarrow{\Lambda}[1]]$ of the silting exchange graph $\operatorname{SEG}(\Lambda)$, cf. [@KQ Thm. 5.14]. The same statement holds for $\underline{\operatorname{CEG}}(\Delta)$. Thus, it is enough to show that the fundamental group of $\operatorname{SEG}(\Delta)$ is generated by oriented $(m+2)$-gons. - Similar to , there are indeed oriented $(m+2)$-gons in $\operatorname{SEG}(\Delta)$ via silting reduction (cf. [@Q15 Lem. 4.3] or [@QW Lem. 6.1] for the dual statement of exchange graphs of hearts). Note that any such oriented $(m+2)$-gon is of the form [\[eq:xy\]](#eq:xy){reference-type="eqref" reference="eq:xy"}, in the sense that there exists a source and a sink of the $(m+2)$-gon and the two paths from the source to the sink have length 2 and $m$ respectively. Finally, one can follow the proof strategy of [@Q15 Prop. 4.5] to show that $\pi_1(\operatorname{SEG}(\Delta))$ is generated by oriented $(m+2)$-gons.  ◻ ## Cluster braid groups for weighted QPs   Given a weighted graph $\Delta$, denote by $\mathcal{W}(\operatorname{CEG}(\Delta))$ the *path groupoid* of $\operatorname{CEG}(\Delta)$, i.e., the category whose objects are vertices in $\operatorname{CEG}(\Delta)$ and whose generating morphisms are edges of $\operatorname{CEG}(\Delta)$ and their formal inverse. Recall that $\mathbf{x}$ (resp. $\mathbf{y}$) is the weighted forward mutation at the object that corresponds to the sink (resp. source) of a $I_2(m)$ subgraph of $\Delta$. There is a polygonal/(m+2)-gon relation of the form $\mathbf{x}^2=\mathbf{y}^m$ [\[eq:xy\]](#eq:xy){reference-type="eqref" reference="eq:xy"}. By , fundamental group $\pi_1(\underline{\operatorname{CEG}}(\Delta))$ is generated by (m+2)-gon relations. We could construct a groupoid from $\operatorname{CEG}(\Delta)$ by adding these (m+2)-gon relations. **Definition 20**. The *oriented cluster exchange groupoid* $\operatorname{\mathcal{CEG}}(\Delta)$ is defined to be the quotient of $\mathcal{W}(\operatorname{CEG}(\Delta))$ by all polygonal relations. For each weighted CTO $\mathbf{Y}$ and $\mathbf{i}\in\mathbf{Q}_{\mathbf{Y}}$, the *local twist* $t^\mathbf{Y}_\mathbf{i}$ is the 2-cycle in $\mathcal{W}(\operatorname{CEG}(\Delta))$ corresponding to the weighted forward mutation $\mu^\sharp_{\mathbf{i}}$ at $\mathbf{Y}$. The *cluster braid group* $\operatorname{CT}_\Delta(\mathbf{Y})$ is defined to be the subgroup of $\pi_1(\operatorname{\mathcal{CEG}}(\Delta),\mathbf{Y})$ generated by all local twists $t^\mathbf{Y}_\mathbf{i},\mathbf{i}\in\Delta_0$. Let $\mathbf{Y}$ be a weighted CTO in $\operatorname{CEG}(\Delta)$ and $f^{-1}(\mathbf{i})=\{i_1,\cdots,i_p\}$. Denote by $\mathbf{x}$ (resp. $\mathbf{y}$) the weighted forward mutation of $\mathbf{Y}$ (resp. $\mathbf{Y}'=\mu_\mathbf{i}(\mathbf{Y})$) at $\mathbf{i}$. Denote by $x_{i_j}$ (resp. $y_{i_j}$), $1\leq j\leq p$ the forward mutation of the corresponding object at $i_j$, see Figure [\[fig:local twist\]](#fig:local twist){reference-type="ref" reference="fig:local twist"}. By [@KQ2 Prop. 2.8] and commutativity between mutations, we have $$\label{eq:local twist} t^\mathbf{Y}_\mathbf{i}=\mathbf{x}\mathbf{y}=(x_{i_1}\cdots x_{i_p})(y_{i_p}\cdots y_{i_1})=t^\mathbf{Y}_{i_p}\cdots t^\mathbf{Y}_{i_1}=\prod_{k\in f^{-1}(\mathbf{i})}t^\mathbf{Y}_k.$$ Similarly to [@KQ2 § 2.4], we define the conjugation map $$\begin{gathered} \begin{array}{rcl} \operatorname{ad}_\mathbf{i}:\pi_1(\operatorname{\mathcal{CEG}}(\Lambda),\mathbf{Y}) & \longrightarrow & (\operatorname{\mathcal{CEG}}(\Delta),\mathbf{Y}')\\ t & \mapsto & \mathbf{x}^{-1} t\mathbf{x}. \end{array}\end{gathered}$$ Note that $$\label{eq:ad} \operatorname{ad}_\mathbf{i}(t)=\mathbf{x}^{-1}t\mathbf{x}=(x^{-1}_{i_p}\cdots x^{-1}_{i_2}x^{-1}_{i_1})t(x_{i_1}x_{i_2}\cdots x_{i_p})=\operatorname{ad}_{i_p}\circ\cdots\circ\operatorname{ad}_{i_2}\circ\operatorname{ad}_{i_1}(t).$$ Under the above setting, we have the analogue of [@KQ2 Prop. 2.9]. **Proposition 21**. *Let* *be a subgraph in $\operatorname{CEG}(\Delta)$ with $\mathbf{x}$ (resp. $\mathbf{y}$) the weighted forward mutation of $\mathbf{Y}$ (resp, $\mathbf{Y}'$) at $\mathbf{i}\in\Delta_0$. Then $$\operatorname{ad}_\mathbf{i}(t_\mathbf{j})=\begin{cases} (t'_\mathbf{i})^{-1}t'_\mathbf{j}t'_\mathbf{i}&\text{ if there are arrows from $\mathbf{j}$ to $\mathbf{i}$ in $\overrightarrow{\Delta}_1$,}\\ t'_\mathbf{j}&\text{otherwise,} \end{cases}$$ where $t_\mathbf{j}$ (resp. $t'_\mathbf{j}$), $\mathbf{j}\in\Delta_0$, is the local twist based at $\mathbf{Y}$ (resp. $\mathbf{Y}'$). Hence $\operatorname{ad}_\mathbf{i}$ induces an isomorphism between $\operatorname{CT}_\Delta(\mathbf{Y})$ and $\operatorname{CT}_\Delta(\mathbf{Y}')$. In particular, we have $\pi_1(\operatorname{\mathcal{CEG}}(\Delta),\mathbf{Y})\cong\operatorname{CT}_\Delta(\mathbf{Y})$.* *Proof.* If there is no arrow in $\overrightarrow{\Delta}$ from $\mathbf{j}$ to $\mathbf{i}$, then there is no arrows in $\overrightarrow{\Lambda}$ from $f^{-1}(\mathbf{j})$ to $f^{-1}(\mathbf{i})$. Let $f^{-1}(\mathbf{i})=\{i_1,i_2,\cdots,i_p\}$ and $f^{-1}(\mathbf{j})=\{j_1,j_2,\cdots,j_q\}$. By equation [\[eq:local twist\]](#eq:local twist){reference-type="eqref" reference="eq:local twist"},  [\[eq:ad\]](#eq:ad){reference-type="eqref" reference="eq:ad"} and [@KQ2 Prop. 2.8], we have $$\operatorname{ad}_\mathbf{i}(t_\mathbf{j})=\operatorname{ad}_{i_p}\circ\cdots\circ\operatorname{ad}_{i_2}\circ\operatorname{ad}_{i_1}(t_{j_q}\cdots t_{j_2}t_{j_1})=(t'_{j_q}\cdots t'_{j_2}t'_{j_1})=t'_\mathbf{j}.$$ Similarly, if there is an arrow from $\mathbf{j}$ to $\mathbf{i}$ then we have $$\begin{aligned} \operatorname{ad}_\mathbf{i}(t_\mathbf{j}){}&=\operatorname{ad}_\mathbf{i}(t_{j_q})\cdots\operatorname{ad}_\mathbf{i}(t_{j_2})\operatorname{ad}_\mathbf{i}(t_{j_1}){}\\ &=(\operatorname{ad}_{i_1}\circ\cdots\circ\operatorname{ad}_{i_p})(t_{j_q})\cdots(\operatorname{ad}_{i_1}\circ\cdots\circ\operatorname{ad}_{i_p})(t_{j_1}){}\\ &=\big((t_{i_p}\cdots t_{i_1})^{-1}t_{j_q}(t_{i_p}\cdots t_{i_1})\big)\cdots\big((t_{i_p}\cdots t_{i_1})^{-1}t_{j_1}(t_{i_p}\cdots t_{i_1})\big){}\\ &=(t_{i_p}\cdots t_{i_1})^{-1}(t_{j_q}\cdots t_{j_1})(t_{i_p}\cdots t_{i_1})=(t'_\mathbf{i})^{-1}t'_\mathbf{j}t'_\mathbf{i} \end{aligned}$$ where the third equality follows from [@KQ2 Lem. 2.7]. By Proposition [Proposition 19](#prop:fund gp){reference-type="ref" reference="prop:fund gp"}, $\pi_1(\underline{\operatorname{\mathcal{CEG}}}(\Delta),\mathbf{Y})$ is trivial. A similar argument to [@KQ2 Prop. 2.9] shows that $\pi_1(\operatorname{\mathcal{CEG}}(\Delta),\mathbf{Y})\cong\operatorname{CT}_\Delta(\mathbf{Y})$. So the last assertion follows. ◻ ## Cluster braid groups as braid groups   We simply denote $\operatorname{CT}(\Delta)$ the cluster braid group $\operatorname{CT}_\Delta(\mathbf{Y}_\Delta)$. **Lemma 22**. *The following map is a group isomorphism $$\begin{gathered} \begin{array}{rcl} \operatorname{Br}I_2(m)&\longrightarrow&\operatorname{CT}(I_2(m)) \\ b_\mathbf{i}&\mapsto&t_\mathbf{i}, \end{array} \end{gathered}$$ where $b_\mathbf{i}$ (resp. $t_\mathbf{i}$) is the generator of $\operatorname{Br}I_2(m)$ (resp. $\operatorname{CT}(I_2(m))$) corresponds to the vertex $\mathbf{i}$ of $I_2(m)$.* *Proof.* Recall that $\operatorname{Br}I_2(m)$ has a canonical presentation $$\label{eq:br} \langle b_\mathbf{1},b_\mathbf{2} \mid \operatorname{Br}^m(b_\mathbf{1},b_\mathbf{2}) \rangle.$$ There is an alternative presentation of $\operatorname{Br}I_2(m)$: - If $2\nmid m$, then the presentation [\[eq:br\]](#eq:br){reference-type="ref" reference="eq:br"} is equivalent to $$\langle u,v\mid u^2=v^m\rangle$$ for $$\begin{cases} u=\underbrace{b_\mathbf{1}b_\mathbf{2}b_\mathbf{1}\ldots b_\mathbf{1}}_m\\ v=b_\mathbf{1}b_\mathbf{2} \end{cases}.$$ - If $2\mid m$, then the presentation [\[eq:br\]](#eq:br){reference-type="ref" reference="eq:br"} is equivalent to $$\langle \sigma,\tau\mid (\sigma\tau)^{m/2}=(\tau\sigma)^{m/2}\rangle$$ for $$\begin{cases} \sigma=b_\mathbf{1} b_\mathbf{2} b_\mathbf{1}\\ \tau=b_\mathbf{1}^{-1} \end{cases}.$$ Recall that the groupoid $\operatorname{\mathcal{CEG}}(I_2(m))$ is constructed from $(m+2)$-gon by replacing each edge by a 2-cycle and adding the polygon relation $x^2=y^m$. By , the group $\operatorname{CT}(I_2(m))$ is isomorphic to the fundamental group of $\operatorname{\mathcal{CEG}}(I_2(m))$. Moreover, local twists $t_{\mathbf{1}}$ and $t_{\mathbf{2}}$ are generators of $\pi_1(\operatorname{\mathcal{CEG}}(I_2(m)))$. **Case 1.** If $2\nmid m$, we take two new generators $$u=x_1\ldots x_{m+1}x_{m+2}=\underbrace{t_{\mathbf{1}}t_{\mathbf{2}}t_{\mathbf{1}}\ldots t_\mathbf{1}}_m\text{ and } v=y_{m+2}\ldots y_2y_1=t_{\mathbf{1}}t_{\mathbf{2}}.$$ By a direct calculation, $u^2=v^m$. Moreover, this is the only relation for $\pi_1(\operatorname{\mathcal{CEG}}(I_2(m)))$, since $u^2$ and $v^m$ form the boundary of the only 2-cell in $\operatorname{\mathcal{CEG}}(I_2(m))$, see Figure [\[fig:modd\]](#fig:modd){reference-type="ref" reference="fig:modd"}. Therefore $\operatorname{Br}I_2(m)=\operatorname{CT}(I_2(m))$. **Case 2.** If $2\mid m$, we take two generators $\sigma=t_{\mathbf{1}}v$ and $\tau=t_{\mathbf{1}}^{-1}$, where $v=y_{m+2}\ldots y_2y_1=t_{\mathbf{2}}t_{\mathbf{1}}$ and $t_{\mathbf{1}}=x_1y_1$. Since $t_{\mathbf{2}}=\tau\sigma\tau$ and $t_{\mathbf{1}}=\tau^{-1}$. A direct calculation shows that $(\sigma\tau)^{m/2}=(\tau\sigma)^{m/2}$. Moreover, this is the unique relation for $\pi_1(\operatorname{\mathcal{CEG}}(I_2(m)))$. For $m=4$, see Figure [\[fig:meven\]](#fig:meven){reference-type="ref" reference="fig:meven"}. Thus $\operatorname{Br}I_2(m)=\operatorname{CT}(I_2(m))$ in this case. It follows from the definition that $t_{\mathbf{1}}=xy, t_\mathbf{2}=yx$ and $x^2=y^m$, see the right picture in Figure [\[fig:ceg i2\]](#fig:ceg i2){reference-type="ref" reference="fig:ceg i2"}. The following two equations show that $t_{\mathbf{1}}$ and $t_{\mathbf{2}}$ satisfy the braid relation. $$\begin{gathered} \underbrace{t_{\mathbf{1}}t_\mathbf{2} t_{\mathbf{1}}\cdots}_m=\begin{cases} y^{m(l+1)}, & m=2l,\\ xy^{m(l+1)}, & m=2l+1; \end{cases} \end{gathered}$$ and $$\begin{gathered} \underbrace{t_\mathbf{2} t_{\mathbf{1}}t_\mathbf{2}\cdots}_m=\begin{cases} y^{m(l+1)}, & m=2l,\\ y^{m(l+1)}x, & m=2l+1. \end{cases} \end{gathered}$$ Thus $t_{\mathbf{1}}=xy, t_\mathbf{2}=yx$ are the generators of $\operatorname{CT}(I_2(m))$ which satisfy the braid relation. $b_\mathbf{i}\mapsto t_\mathbf{i}, \mathbf{i}\in\{\mathbf{1},\mathbf{2}\}$ gives an isomorphism between $\operatorname{Br}I_2(m)$ and $\operatorname{CT}(I_2(m))$ ◻ The proof of Proposition [Proposition 19](#prop:fund gp){reference-type="ref" reference="prop:fund gp"} shows that each edge $\epsilon:\mathbf{i}\overset{m}{-}\mathbf{j}$ in $\Delta$ gives a doubled $(m+2)$-gon at $\mathbf{Y}_\Delta$ with an $(m+2)$-gon relation. By Lemma [Lemma 22](#lem:surj){reference-type="ref" reference="lem:surj"}, the relation $\operatorname{Br}^m(t_\mathbf{i},t_\mathbf{j})$ holds in $\operatorname{CT}(\Delta)$, and hence we have a group surjection $$\begin{gathered} \begin{array}{rcl} \Psi:\operatorname{Br}\Delta&\longrightarrow&\operatorname{CT}(\Delta) \\ b_\mathbf{i}&\mapsto&t_\mathbf{i}, \end{array}\end{gathered}$$ where $b_\mathbf{i}$ (resp. $t_\mathbf{i}$) is the generator of $\operatorname{Br}\Delta$ (resp. $\operatorname{CT}(\Delta)$) corresponding to the vertex $\mathbf{i}\in\Delta_0$. Now we state our main theorem which shows that the morphism above is indeed an isomorphism. **Theorem 23**. *Let $\overrightarrow{\Delta}$ be a finite weighted quiver. For any weighted CTO $\mathbf{Y}$ in $\operatorname{CEG}(\Delta)$, the morphism $$\begin{gathered} \label{eq:psi} \begin{array}{rcl} \Psi:\operatorname{Br}(\mathbf{Q}_\mathbf{Y},\mathbf{W}_\mathbf{Y})&\longrightarrow&\operatorname{CT}_\Delta(\mathbf{Y})\\ b_\mathbf{i}&\mapsto& t_\mathbf{i} \end{array}\end{gathered}$$ is a group isomorphism, sending the standard generators to the standard ones (for $\mathbf{i}\in \mathbf{Q}_{\mathbf{Y}}$).* *Proof.* By Proposition [Proposition 21](#lem:conj){reference-type="ref" reference="lem:conj"} and Theorem [Theorem 11](#thm:mu-present){reference-type="ref" reference="thm:mu-present"}, it is enough to show the isomorphism for the initial weighted CTO $\mathbf{Y}_\Delta$. By the previous discussion, we only need to show that $\Psi$ is injective. Recall that there is a natural embedding from $\operatorname{CEG}(\Delta)$ to $\operatorname{CEG}(\Lambda)$. Now, we would like to make it more precise, that there is an injective map/functor $$\begin{gathered} \label{eq:f*} f^*\colon\operatorname{\mathcal{CEG}}(\Delta)\to\operatorname{\mathcal{CEG}}(\Lambda)\end{gathered}$$ sending $\mu^\sharp_\mathbf{i}$ to the path $\prod_{i\in f^{-1}(\mathbf{i})}\mu^\sharp_i$ (to up homotopy). Note that on the level of vertices/objects and arrows/morphisms, $f^*$ is an injective map. The only thing we need to check is that the $(m+2)$-gon relations in $\operatorname{\mathcal{CEG}}(\Delta)$ hold in $\operatorname{\mathcal{CEG}}(\Lambda)$. We use a similar method as in the proof of . Choose any vertex $\mathbf{Y}$ in $\operatorname{\mathcal{CEG}}(\Delta)$ and edge $\mathbf{j}\xrightarrow{m}\mathbf{i}\in\mathbf{Q}_{\mathbf{Y}}$, we have the corresponding $(m+2)$-gon relation. We need to show that it is generated by oriented square and pentagon relations in $\operatorname{CEG}(\Lambda)$. By [@KQ2 Thm. 2.10], $\operatorname{CEG}(\Lambda)=\operatorname{EG}\operatorname{\mathcal{D}}_3(\Lambda)/\operatorname{Br}\Lambda$, where $\operatorname{\mathcal{D}}_3(\Lambda)$ is the 3-Calabi-Yau category associated to $\overrightarrow{\Lambda}$. A fundamental domain of $\operatorname{EG}\operatorname{\mathcal{D}}_3(\Lambda)/\operatorname{Br}\Lambda$ is the interval $[\mathcal{H}_{\Lambda},\mathcal{H}_{\Lambda}[1]]$ for the canonical heart $\mathcal{H}_{\Lambda}$ of $\operatorname{\mathcal{D}}_3(\Lambda)$. Thus, $\mathbf{Y}$ corresponds to a unique heart $\mathcal{H}_{\mathbf{Y}}$ in $[\mathcal{H}_{\Lambda},\mathcal{H}_{\Lambda}[1]]$. Moreover, by [@KQ § 10] (cf. [@QZx Thm. 4.7]), the interval $[\mathcal{H}_{\mathbf{Y}},\mathcal{H}_{\mathbf{Y}}[1]]$ can be regarded as a fundamental domain of $\operatorname{EG}\operatorname{\mathcal{D}}_3(\Lambda)/\operatorname{Br}\Lambda$. Thus the oriented $(m+2)$-gon in $\operatorname{CEG}(\Delta)$ lifts to a cycle in $[\mathcal{H}_{\mathbf{Y}},\mathcal{H}_{\mathbf{Y}}[1]]$. By [@QW Lem. 6.1], such cycle decomposes into squares and pentagons as required. In particular, [\[eq:f\*\]](#eq:f*){reference-type="eqref" reference="eq:f*"} induces a map, still denoted by $f^*$, between the corresponding fundamental groups, which fits into the following diagram: $$\label{eq:last} \begin{tikzcd} \operatorname{Br}\Delta\ar[r,"\Psi_\Delta"]\ar[d,"\iota_f"',hookrightarrow] & \operatorname{CT}(\Delta)\ar[d,"f^\ast"]\\ \operatorname{Br}\Lambda\ar[r,"\Psi_\Lambda"']&\operatorname{CT}(\Lambda), \end{tikzcd}$$ where $\Psi_\Lambda$ is the isomorphism in [@KQ2 Thm 2.16] and $\iota_f$ is the group injection in Theorem [Theorem 5](#thm:cri){reference-type="ref" reference="thm:cri"}. Examining the standard generators, one gets that such a diagram is commutative. Thus, $\Psi_\Delta$ is injective because $\Psi_\Lambda\circ\iota_f$ is, which completes the proof. ◻ **Example 24**. Consider the weighted graph $\Delta=H_3$, we have the cluster exchange graph $\underline{\operatorname{CEG}}(H_3)$ in Figure [\[fig:ABH\]](#fig:ABH){reference-type="ref" reference="fig:ABH"}. Denote $\mathbf{Y}$ the common vertex of three heptagons in the front of $\underline{\operatorname{CEG}}(H_3)$. The corresponding quiver $\mathbf{Q}_\mathbf{Y}$ is the weighted quiver IV in Figure [\[fig:cycle-present\]](#fig:cycle-present){reference-type="ref" reference="fig:cycle-present"}. The potential $\mathbf{W}_\mathbf{Y}$ is the 3-cycle. Then we have $$\operatorname{CT}_{H_3}(\mathbf Y)\cong\operatorname{Br}(\mathbf{Q}_\mathbf{Y},\mathbf{W}_\mathbf{Y})$$ which is generated by $b_\mathbf{1},b_\mathbf{2},b_\mathbf{3}$ and the following relations $$\operatorname{Br}^5(b_\mathbf{1},b_\mathbf{2}),\quad \operatorname{Br}^5(b_\mathbf{2},b_\mathbf{3}), \quad \operatorname{Br}^5(b_\mathbf{1},b_\mathbf{3}),\quad \operatorname{Co}(b_\mathbf{1},b^{b_\mathbf{3}b_\mathbf{2}}_\mathbf{2}),\quad \operatorname{Br}(b_\mathbf{1},b^{b_\mathbf{2}}_\mathbf{3}).$$ **Remark 25**. A final remark is that we expect [\[eq:psi\]](#eq:psi){reference-type="eqref" reference="eq:psi"} holds for any finite mutation type quivers with potential.

arxiv_math

--- abstract: | Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\mathfrak a$, is non-zero. It is known that $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\mathfrak a$, is zero for all $i>p+c$. In this paper, we obtain the coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we give a necessary and sufficient condition for the vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules. address: Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran. author: - Ali Fathi title: Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules --- (Originally published in Persian: A. Fathi, *Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules*, J. Adv. Math. Model. **13**(2) (2023) 250--258. DOI:10.22055/JAMM.2023.43821.2166) # **Introduction** Throughout this paper, let $R$ be a commutative Noetherian ring with non-zero identity. Let $\mathfrak a$ be an ideal of $R$ and $N$ be an $R$-module. The $i$-th local cohomology module of $N$ with respect to $\mathfrak a$ was defined by Grothendieck as follows: $$\operatorname{H}^{i}_{\mathfrak a}(N):={\underset{n\in\mathbb N}{\varinjlim}\operatorname{Ext}^i_R(R/\mathfrak a^n, N)};$$ see [@bs] for more details. For a pair of $R$-modules $M$ and $N$, the $i$-generalized local cohomology module of $M, N$ with respect to $\mathfrak a$ was introduced by Herzog [@h] as follows: $$\operatorname{H}^{i}_{\mathfrak a}(M,N):=\underset{n\in\mathbb N}{\varinjlim}\operatorname{Ext}^i_R(M/\mathfrak a^n M, N);$$ see [@h; @b] for more details. It is clear that $\operatorname{H}^{i}_{\mathfrak a}(R,N)=\operatorname{H}^{i}_{\mathfrak a}(N)$. The cohomological dimension of $N$ with respect to $\mathfrak a$ and the cohomological dimension of $M, N$ with respect to $\mathfrak a$ are defined, respectively, as follow: $$\operatorname{cd}_{\mathfrak a}(N):=\sup\{i\in\mathbb N_0: \operatorname{H}^{i}_{\mathfrak a}(N)\neq 0\}$$ and $$\operatorname{cd}_{\mathfrak a}(M,N):=\sup\{i\in\mathbb N_0: \operatorname{H}^{i}_{\mathfrak a}(M,N)\neq 0\}.$$ Assume that $N$ is finitely generated with finite dimension $d$. For each $i>d$, $\operatorname{H}^{i}_{\mathfrak a}(N)=0$ [@bs Theorem 6.1.2] (in other words, $\operatorname{cd}_{\mathfrak a}(N)\leq d$) and $\operatorname{H}^{d}_{\mathfrak a}(N)$ is Artinian [@bs Exercise 7.1.7]. When $R$ is local, Dibaei and Yassemi as a main result proved in [@dy Theorem A] that $$\operatorname{Att}_R(\operatorname{H}^{d}_{\mathfrak a}(N))=\{\mathfrak p\in\operatorname{Ass}_R(N): \operatorname{cd}_{\mathfrak a}(R/\mathfrak p)=d\}.$$ This equality also holds without the hypothesis that $R$ is local (see [@d Theorem 2.5]). Also, if $M$ is finitely generated with finite projective dimension $p$, then $\operatorname{H}^{i}_{\mathfrak a}(M,N)=0$ for all $i>p+d$ [@b Lemma 5.1] (i. e., $\operatorname{cd}_{\mathfrak a}(M,N)\leq p+d$) and $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)$ is Artinian (see for example [@maf Theorem 2.9] or [@hv Proposition 3.1]). When $R$ is local, as a generalization of the theorem of Dibaei and Yassemi, Gu and Chu show in [@gc Theorem 2.3] that $$\operatorname{Att}_R(\operatorname{H}^{p+d}_{\mathfrak a}(M,N))=\{\mathfrak p\in\operatorname{Ass}_R(N): \operatorname{cd}_{\mathfrak a}(M,R/\mathfrak p)=p+d\}.$$ In [@f2015 Theorem 5.3], Fathi, Tehranian and Zakeri proved this equality in the case when $R$ is not necessarily local. They also show in [@f2015 Theorem 5.6] that $$\tag{\dag} \operatorname{Att}_R(\operatorname{H}^{p+d}_{\mathfrak a}(M,N))=\operatorname{Supp}_R(\operatorname{Ext}^{p}_{R}(M,R))\cap\operatorname{Att}_R(\operatorname{H}^{d}_{\mathfrak a}(N))$$ whenever $R/ \operatorname{Ann}_R(\operatorname{H}^{d}_{\mathfrak a}(N))$ is a complete semilocal ring. This equality allows us to compute the set of attached prime ideals of the top generalized local cohomology module $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)$ from the set of attached prime ideals of the top local cohomology module $\operatorname{H}^{d}_{\mathfrak a}(N)$. Now we set $c:=\operatorname{cd}_{\mathfrak a}(N)$. For all $i>p+c$, we have $\operatorname{H}^{i}_{\mathfrak a}(M,N)=0$; see [@hv Proposition 2.8]. Since $c\leq d$, $p+c$ yields a sharper upper bound for $\operatorname{cd}_{\mathfrak a}(M,N)$. Note that $\operatorname{H}^{c}_{\mathfrak a}(N)$ and $\operatorname{H}^{p+c}_{\mathfrak a}(M,N)$ are not necessarily Artinian. In Theorem [Theorem 3](#coass){reference-type="ref" reference="coass"}, using the set of coassociated prime ideals of $\operatorname{H}^{c}_{\mathfrak a}(N)$, we compute the set of coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M,N)$. More precisely, we show that $$\begin{aligned} &\operatorname{Coass}_R\left(\operatorname{H}^{p+c}_{\mathfrak a}(M,N)\right)\\ &=\{\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Coass}_R\left(\operatorname{H}^{c}_{\mathfrak a}(N)\right): \operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)=p\}. \end{aligned}$$ As a consequence of this equality, we prove in Corollary [Corollary 4](#att){reference-type="ref" reference="att"} that the equality () holds even if $R/ \operatorname{Ann}_R(\operatorname{H}^{d}_{\mathfrak a}(N))$ is not a complete semilocal ring, and we show that $$\begin{aligned} \tag{\ddag}&\operatorname{Att}_R\left(\operatorname{H}^{p+d}_{\mathfrak a}(M,N)\right)\\ \nonumber& =\{\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Ass}_R(N): \operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)=p, \operatorname{cd}_{\mathfrak a}(R/\mathfrak p)=d\}. \end{aligned}$$ In particular, if $R$ is local and $M$ is Cohen-Macaulay, then it is shown in Corollary [Corollary 5](#cm){reference-type="ref" reference="cm"} that $$\operatorname{Att}_R(\operatorname{H}^{p+d}_{\mathfrak a}(M,N))=\{\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Ass}_R(N): \operatorname{cd}_{\mathfrak a}(R/\mathfrak p)=d\}.$$ Finally, using the equality (), we extend the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules. More precisely, when $R$ is a local ring, we prove in Theorem [Theorem 7](#lh){reference-type="ref" reference="lh"} that $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)=0$ if and only if for all $\mathfrak P\in\operatorname{Supp}_{\widehat R}(\widehat M)\cap\operatorname{Ass}_{\widehat R}(\widehat N)$ with $\dim_{\widehat R}(\widehat R/\mathfrak P)=d$ and $\operatorname{proj\,dim}_{\widehat R_\mathfrak P}(\widehat M_{\mathfrak P})=p$, $\dim_{\widehat R}(\widehat R/(\mathfrak a\widehat R+\mathfrak P))>0$. # **Preliminaries** Let $M$ be an $R$-module. We denote the localization of $M$ at $\mathfrak p$ by $M_\mathfrak p$, and the set of all prime ideals $\mathfrak p$ of $R$ such that $M_\mathfrak p$ is nonzero is called the support of $M$ and denoted by $\operatorname{Supp}_R(M)$. Also, the annihilator of $M$ in $R$, denoted by $\operatorname{Ann}_R(M)$, is defined to be the set $\{r\in R: rx=0 \textrm{ for all } x\in M\}$. If $\mathfrak p:=\operatorname{Ann}_R(Rx)$ is a prime ideal of $R$ for some $x\in M$, then $\mathfrak p$ is called an associated prime ideal of $M$, and we denote the set of all associated prime ideals of $M$ by $\operatorname{Ass}_R(M)$. We will denote the set of all positive integers (respectively, non-negative integers) by $\mathbb N$ (respectively, $\mathbb {N}_0$). The concepts of attached prime ideal and secondary representation as the dual of the concepts of associated prime ideal and primary decomposition were introduced by Macdonald in [@m]. An $R$-module $M$ is said to be secondary if $M\neq 0$ and, for each $r\in R$, the endomorphism $\mu_r:M\rightarrow M$ defined by $\mu_r(x)=rx$ (for $x\in M$) is either surjective or nilpotent. If $M$ is secondary, then $\mathfrak p:=\sqrt{\operatorname{Ann}_R(M)}$ is a prime ideal and $M$ is said to be $\mathfrak p$-secondary. A prime ideal $\mathfrak p$ is called an attached prime ideal of $M$ if $M$ has a $\mathfrak p$-secondary quotient. We denote the set of all attached prime ideals of $M$ by $\operatorname{Att}_R(M)$. If $M$ can be written as a finite sum of its secondary submodules, then we say that $M$ has a secondary representation. Such a secondary representation $$M=M_1+\dots+M_t\quad {\rm with }\ \sqrt{\operatorname{Ann}_R(M_i)}=\mathfrak p_i\ {\rm for }\ i=1,\dots, t$$ of $M$ is said to be minimal when none of the modules $M_i$ ($1\leq i\leq t$) is redundant and the prime ideals $\mathfrak p_1,\dots,\mathfrak p_t$ are distinct. Since the sum of two $\mathfrak p$-secondary submodules of $M$ is again $\mathfrak p$-secondary, so if $M$ has a secondary representation, then it has a minimal one. When the above secondary representation is minimal, then $\operatorname{Att}_R(M)=\{\mathfrak p_1,\dots, \mathfrak p_t\}$, and hence $t$ and the set $\{\mathfrak p_1,\dots, \mathfrak p_t\}$ are independent of the choice of minimal secondary representation of $M$. Artinian modules have secondary representation. Yassemi [@y] has introduced the coassociated prime ideal as a dual of associated prime ideal. In Yassemi's definition, we do not need to assume that the module has a secondary representation, and if a module has a secondary representation, then its sets of coassociated prime ideals and attached prime ideals are same (see [@y Theorem 1.14]). **Definition 1**. We say that an $R$-module $M$ is cocyclic when $M$ is a submodule of $E(R/\mathfrak m)$ for some maximal ideal $\mathfrak m$ of $R$, where $E(R/\mathfrak m)$ denotes the injective envelope of $R/\mathfrak m$. **Definition 2**. We say that a prime ideal $\mathfrak p$ of $R$ is a coassociated prime ideal of an $R$-module $M$ when there exists a cocyclic homomorphic image $L$ of $M$ such that $\mathfrak p=\operatorname{Ann}_R(L)$. We denote by $\operatorname{Coass}_R(M)$ the set of all coassociated prime ideals of $M$. # **Main results** In the following theorem, using the set of coassociated prime ideals of the top local cohomology module, we compute the set of coassociated prime ideals of the top generalized local cohomology module. **Theorem 3**. *Let $\mathfrak a$ be an ideal of $R$ and $M$ be a non-zero finitely generated $R$-module with finite projective dimension $p$. Let $N$ be an $R$-module such that $N\neq \mathfrak aN$ and $c:=\operatorname{cd}_{\mathfrak a}(N)$. Then for each $n>p+c$, $\operatorname{H}^{n}_{\mathfrak a}(M,N)=0$ and $$\operatorname{H}^{p+c}_{\mathfrak a}(M,N)\cong\operatorname{Ext}^{p}_{R}(M,R)\otimes_R\operatorname{H}^{c}_{\mathfrak a}(N).$$ In particular, $$\begin{aligned} &\operatorname{Coass}_R\left(\operatorname{H}^{p+c}_{\mathfrak a}(M,N)\right)\\ &=\{\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Coass}_R\left(\operatorname{H}^{c}_{\mathfrak a}(N)\right): \operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)=p\}. \end{aligned}$$* *Proof.* Hassanzadeh and Vahidi, in [@hv Proposition 2.8], show that $\operatorname{H}^{n}_{\mathfrak a}(M,N)=0$ for all $n>p+c$ and $$\operatorname{H}^{p+c}_{\mathfrak a}(M,N)\cong \operatorname{Ext}^{p}_{R}(M,\operatorname{H}^{c}_{\mathfrak a}(N)).$$ Now the functor $\operatorname{Ext}^{p}_{R}(M,\cdot)$ is additive and right exact. Also since $M$ is a finitely generated module over a Noetherian ring, it follows from [@e Lemma 3.1.16] that $\operatorname{Ext}^{p}_{R}(M,\cdot)$ preserves direct sums and so [@ro Theorem 5.45] implies that $\operatorname{Ext}^{p}_{R}(M,\cdot)\cong \operatorname{Ext}^{p}_{R}(M,R)\otimes_R(\cdot).$ Therefore $$\operatorname{H}^{p+c}_{\mathfrak a}(M,N)\cong\operatorname{Ext}^{p}_{R}(M,R)\otimes_R\operatorname{H}^{c}_{\mathfrak a}(N).$$ Hence, by [@y Theorem 1.21], we have $$\operatorname{Coass}_R\left(\operatorname{H}^{p+c}_{\mathfrak a}(M,N)\right) =\operatorname{Supp}_R\left(\operatorname{Ext}^{p}_{R}(M,R)\right)\cap\operatorname{Coass}_R\left(\operatorname{H}^{c}_{\mathfrak a}(N)\right).$$ Thus to complete the proof, it is sufficient for us to show that $$\operatorname{Supp}_R(\operatorname{Ext}^{p}_{R}(M,R))=\{\mathfrak p\in\operatorname{Supp}_R(M): \operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)=p\}.$$ Suppose that $\mathfrak p\in\operatorname{Supp}_R(\operatorname{Ext}^{p}_{R}(M,R))$. Since $M$ is a finitely generated module over the Noetherian ring $R$, it follows from [@ro Proposition 7.39] that $$\operatorname{Ext}^{p}_{R_\mathfrak p}(M_\mathfrak p,R_\mathfrak p)\cong(\operatorname{Ext}^{p}_{R}(M,R))_\mathfrak p\neq 0.$$ Therefore $\mathfrak p\in\operatorname{Supp}_R(M)$ and $\operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)\geq p$. Thus $\operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)=p$. Conversely, assume that $\mathfrak p\in\operatorname{Supp}_R(M)$ and $\operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)=p$. Therefore $\operatorname{Ext}^{p}_{R_\mathfrak p}(M_\mathfrak p,R_\mathfrak p)\neq 0$ by [@mat Section 19, Lemma 1(iii)]. Hence $(\operatorname{Ext}^{p}_{R}(M,R))_\mathfrak p\neq 0$, and this completes the proof. ◻ Let the notations and assumptions be as in Corollary [Corollary 4](#att){reference-type="ref" reference="att"}. Fathi, Tehranian and Zakeri, in [@f2015 Theorem 5.6], proved that $$\begin{aligned} \tag{\dag} \operatorname{Att}_R(\operatorname{H}^{p+d}_{\mathfrak a}(M,N))=\operatorname{Supp}_R(\operatorname{Ext}^{p}_{R}(M,R))\cap\operatorname{Att}_R(\operatorname{H}^{d}_{\mathfrak a}(N))\end{aligned}$$ whenever $B:=R/ \operatorname{Ann}_R(\operatorname{H}^{d}_{\mathfrak a}(N))$ is a complete semilocal ring. In the following corollary it is shown that the equality () holds without the hypothesis that $B$ is a complete semilocal ring. **Corollary 4**. *Let $\mathfrak a$ be an ideal of $R$ and let $M, N$ be non-zero finitely generated $R$-modules such that $p:=\operatorname{proj\,dim}_R(M)<\infty$ and $d:=\dim_R(N)<\infty$. Then $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)$ is Artinian and $$\begin{aligned} \tag{\ddag} &\operatorname{Att}_R(\operatorname{H}^{p+d}_{\mathfrak a}(M,N))\\ &=\{\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Ass}_R(N) : \operatorname{proj\,dim}_{R_\mathfrak p}( M_\mathfrak p)=p, \operatorname{cd}_{\mathfrak a}(R/\mathfrak p)=d \}. \end{aligned}$$* *Proof.* We set $c:=\operatorname{cd}_{\mathfrak a}(N)$. By Grothendieck's vanishing theorem [@bs Theorem 6.1.2], $c\leq d$. Now if $c<d$, then by the previous theorem $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)=0$, and so its set of attached prime ideals is empty. On the other hand, by [@dnt Theorem 1.2], for each $\mathfrak p\in\operatorname{Ass}_R(N)$, we have $\operatorname{cd}_{\mathfrak a}(R/\mathfrak p)\leq c<d$. Hence the set in the right hand side of () is also empty and so the equality () holds in this case. We may therefore assume that $c=d$. By [@bs Exersise 7.1.7], $\operatorname{H}^{d}_{\mathfrak a}(N)$ is Artinian and since $\operatorname{Ext}^{p}_{R}(M,R)$ is finitely generated, $\operatorname{Ext}^{p}_{R}(M,R)\otimes_R\operatorname{H}^{d}_{\mathfrak a}(N)$ is Artinian. By Theorem [Theorem 3](#coass){reference-type="ref" reference="coass"}, this module is isomorphic to $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)$, and so $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)$ is Artinian (Artinianess of $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)$ is not a new result, see for example [@maf Theorem 2.9] or [@hv Proposition 3.1]). Hence the sets of attached prime ideals of $\operatorname{H}^{d}_{\mathfrak a}(N)$ and $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)$ coincide with their sets of coassociated prime ideals [@y Theorem 1.14]. Therefore the equality () follows from [@d Theorem 2.5] and the last part of Theorem [Theorem 3](#coass){reference-type="ref" reference="coass"}. ◻ **Corollary 5**. *Let $R$ be a local ring and $\mathfrak a$ be an ideal of $R$. Let $M, N$ be non-zero finitely generated $R$-modules such that $M$ is Cohen-Macaulay, $p:=\operatorname{proj\,dim}_R(M)<\infty$ and $d:=\dim_R(N)$. Then we have $$\begin{aligned} \operatorname{Att}_R\operatorname{H}^{p+d}_{\mathfrak a}(M,N) =\{\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Ass}_R(N) : \operatorname{cd}_{\mathfrak a}(R/\mathfrak p)=d \}. \end{aligned}$$* *Proof.* By [@bh Corollary 9.46, Remark 9.4.8(a)] and the Auslander---Buchsbaum formula [@bh Theorem 1.3.3], we have $$\dim_R(R)\leq\operatorname{proj\,dim}_R(M)+\dim_R(M)=\operatorname{depth}_R(R).$$ Thus $R$ is Cohen-Macaulay. If $\mathfrak p\in\operatorname{Supp}_R(M)$, then $\dim_R(R/\mathfrak p)=\dim_R(M/\mathfrak pM)$, and so the Auslander-Buchsbaum formula and [@bh Theorem 2.1.3(b)] imply that $$\begin{aligned} \operatorname{proj\,dim}_{R_\mathfrak p}(M_\mathfrak p)&=\dim_{R_\mathfrak p} ({R_\mathfrak p})-\dim_{R_\mathfrak p}( {M_\mathfrak p})\\ &=(\dim_R(R)-\dim_R(R/\mathfrak p))-(\dim_R(M)-\dim_R(M/\mathfrak pM))\\ &=\dim_R(R)-\dim_R(M)\\ &=p.\end{aligned}$$ Now the assertion follows from the previous corollary. ◻ In Theorem [Theorem 7](#lh){reference-type="ref" reference="lh"}, we are going to prove the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules. Before that the following lemma which extends the flat base change theorem [@bs Theorem 4.3.2] for generalized local cohomology modules is needed. This lemma is stated in [@dst Lemma 2.1(ii)] without proof. Here we give a proof for the readers' convenience. Also, we note that in our proof we need to assume that the first module in the generalized local cohomology module is finitely generated but in [@dst Lemma 2.1(ii)] there is no such a restriction on the module. **Lemma 6** ([@dst Lemma 2.1(ii)]). *Let $\mathfrak a$ be an ideal of $R$, $M$ be a finitely generated $R$-module and $N$ be an arbitrary $R$-module. Let $B$ be a flat $R$-algebra. Then for each $i\in\mathbb N_0$, we have $$B\otimes_R\operatorname{H}^{i}_{\mathfrak a}(M,N)\cong \operatorname{H}^{i}_{\mathfrak aB}(B\otimes_RM,B\otimes_RN).$$* *Proof.* For each $n\in\mathbb N_0$, we have $$\begin{aligned} B\otimes_R(M/\mathfrak a^n M)&\cong B\otimes_RM\otimes_R(R/\mathfrak a^n)\\ &\cong(B\otimes_RM)\otimes_BB\otimes_R(R/\mathfrak a^n)\\ &\cong(B\otimes_RM)\otimes_B(B/\mathfrak a^n B)\\ &\cong(B\otimes_RM)\otimes_B(B/(\mathfrak aB)^n)\\ &\cong(B\otimes_RM)/(\mathfrak aB)^n(B\otimes_R M).\end{aligned}$$ Now it follows from [@ro Theorem 5.27], [@e Theorem 3.2.5] and the above isomorphism that $$\begin{aligned} B\otimes_R\operatorname{H}^{i}_{\mathfrak a}(M,N)&\cong B\otimes_R\underset{n}{\varinjlim}\operatorname{Ext}^{i}_{R}(M/\mathfrak a^nM,N)\\ &\cong\underset{n}{\varinjlim}(B\otimes_R\operatorname{Ext}^{i}_{R}(M/\mathfrak a^nM,N))\\ &\cong\underset{n}{\varinjlim}(\operatorname{Ext}^{i}_{B}((B\otimes_RM)/(\mathfrak aB)^n(B\otimes M),B\otimes_RN)\\ &\cong \operatorname{H}^{i}_{\mathfrak aB}(B\otimes_RM,B\otimes_RN).\end{aligned}$$ ◻ **Theorem 7** (The Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules). *Let $(R, \mathfrak m)$ be a local ring and $\mathfrak a$ be a proper ideal of $R$. Let $M, N$ be non-zero finitely generated $R$-modules such that $p:=\operatorname{proj\,dim}_R(M)<\infty$ and $d:=\dim_R (N)$. Then the following statements are equivalent:* 1. *$\operatorname{H}^{p+d}_{\mathfrak a}(M,N)=0$;* 2. *for each $\mathfrak P\in\operatorname{Supp}_{\widehat R}(\widehat M)\cap\operatorname{Ass}_{\widehat R}(\widehat N)$ satisfying $\operatorname{proj\,dim}_{\widehat R_\mathfrak P}(\widehat M_{\mathfrak P})=p$ and $\dim_{\widehat R}(\widehat R/\mathfrak P)=d$, we have $\dim_{\widehat R}(\widehat R/(\mathfrak a\widehat R+\mathfrak P))>0$.* *Proof.* $\widehat R$ is a Noetherian local ring with maximal ideal $\mathfrak m\widehat R$ (see [@mat Theorem 8.12]) and for a finitely generated $R$-module $L$ we have $L\otimes_R\widehat{R}=\widehat{L}$ (see [@mat Theorem 8.7]). Since $\widehat R$ is a flat $R$-algebra (see [@mat Theorem 8.8]), by Lemma [Lemma 6](#flat){reference-type="ref" reference="flat"}, we have $${\widehat {R}}\otimes_R\operatorname{H}^{i}_{\mathfrak a}(M,N)\cong\operatorname{H}^{i}_{\mathfrak a\widehat R}(\widehat M,\widehat N).$$ The above isomorphism implies that $\operatorname{H}^{i}_{\mathfrak a}(M,N)=0$ if and only if $\operatorname{H}^{i}_{\mathfrak a\widehat R}(\widehat M,\widehat N)=0$ because $\widehat R$ is a faithfully flat $R$-module (see [@mat Theorem 8.14]). Also, for a finitely generated $R$-module $L$, by [@bh Corollary 2.1.8(a)], we have $$\operatorname{depth}_{\widehat R}(\widehat L)=\operatorname{depth}_R(L),\ \dim_{\widehat R}(\widehat L)=\dim_R(L).$$ By hypothesis, $M$ has a finite free resolution. Tensoring a finite free resolution of $M$ by $\widehat R$ yields a finite free resolution for $\widehat M$ over $\widehat R$, and so $\operatorname{proj\,dim}_{\widehat R}(\widehat M)<\infty$. Now, by the above equality and the Auslander-Buchsbaum formula, we obtain $$\operatorname{proj\,dim}_{\widehat R}(\widehat M)=\operatorname{proj\,dim}_R(M).$$ Therefore we can (and do) replace $M$, $N$ and $R$ by $\widehat M$, $\widehat N$ and $\widehat R$ and assume henceforth in this proof that $R$ is complete. (i)$\Rightarrow$(ii). Assume that $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)=0$ and $\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Ass}_R(N)$ is such that $\dim_R(R/\mathfrak p)=d$ and $\operatorname{proj\,dim}_{R_\mathfrak p}( M_{\mathfrak p})=p$. Since $\operatorname{Att}_R(\operatorname{H}^{p+d}_{\mathfrak a}(M,N))=\emptyset$, by Corollary [Corollary 4](#att){reference-type="ref" reference="att"}, we have $\operatorname{cd}_{\mathfrak a}(R/\mathfrak p)<d$. It follows from the independence theorem [@bs Theorem 4.2.1] that $\operatorname{H}^{d}_{\mathfrak a+\mathfrak p}(R/\mathfrak p)\cong\operatorname{H}^{d}_{\mathfrak a}(R/\mathfrak p)=0$. Since $\dim_R(R/\mathfrak p)=d$, Grothendieck's non-vanishing theorem [@bs Theorem 6.1.4] implies that the proper ideal $\mathfrak a+\mathfrak p$ is not $\mathfrak m$-primary and so $\dim_R(R/(\mathfrak a+\mathfrak p))>0$. (ii)$\Rightarrow$(i). Assume that the condition (ii) holds. If $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)\neq 0$, then $\operatorname{Att}_R(\operatorname{H}^{p+d}_{\mathfrak a}(M,N))$ is not empty and hence, by Corollary [Corollary 4](#att){reference-type="ref" reference="att"}, there exists $\mathfrak p\in\operatorname{Supp}_R(M)\cap\operatorname{Ass}_R(N)$ such that $\operatorname{cd}_{\mathfrak a}(R/\mathfrak p)=d$ and $\operatorname{proj\,dim}_{R_\mathfrak p}(M_{\mathfrak p})=p$. By Grothendieck's vanishing theorem [@bs Theorem 6.1.2], $\operatorname{cd}_{\mathfrak a}(R/\mathfrak p)\leq \dim_R(R/\mathfrak p)$. Thus $\dim_R(R/\mathfrak p)=d$ and the statement (ii) yields $\dim_R(R/(\mathfrak a+\mathfrak p))>0$. We set $\bar{R}:=R/\mathfrak p$. Now $\bar R$ is a complete local domain of dimension $d$ and $\dim_{\bar R}(\bar R/\mathfrak a\bar R)=\dim_R(R/(\mathfrak a+\mathfrak p))>0$. Thus the Lichtenbaum-Hartshorne vanishing theorem (see [@bs Theorem 8.2.1 ]) for the ring $\bar R$ implies that $\operatorname{H}^{d}_{\mathfrak a\bar R}(\bar R)=0$. Hence, by the independence theorem, $\operatorname{H}^{d}_{\mathfrak a}(R/\mathfrak p)\cong\operatorname{H}^{d}_{\mathfrak a\bar R}(\bar R)=0$. Since $\operatorname{cd}_{\mathfrak a}(R/\mathfrak p)\leq\operatorname{cd}_{\mathfrak a}(N)\leq d$, we obtain $\operatorname{cd}_{\mathfrak a}(R/\mathfrak p)<d$, which is a contradiction. Therefore $\operatorname{H}^{p+d}_{\mathfrak a}(M,N)=0$. ◻ **Remark 8**. *Let $(R, \mathfrak m)$ be a local ring. Let $M, N$ be non-zero finitely generated $R$-modules such that $p:=\operatorname{proj\,dim}_R(M)<\infty$ and $d:=\dim_R(N)$. By Grothendieck's vanishing and non-vanishing theorems [@bs Theorems 6.1.2 and 6.1.4], we have $\operatorname{cd}_{\mathfrak m}(N)=\dim_R(N)$. The exact value of $\operatorname{cd}_{\mathfrak m}(M,N)$ is unknown under the above assumptions. However, if in addition $R$ is Cohen-Macaulay, then Divaani-Aazar and Hajikarimi in [@dh Theorem 3.5] proved that $$\operatorname{cd}_{\mathfrak m}(M,N)=\dim_R(R)-\operatorname{grade}_{R}(\operatorname{Ann}_R(N),M).$$ We know that $p+d$ is an upper bound for $\operatorname{cd}_{\mathfrak m}(M,N)$. If we set $\mathfrak a:=\mathfrak m$ in Theorem [Theorem 7](#lh){reference-type="ref" reference="lh"}, then it is not true to say that since $\dim_{\widehat R}(\widehat R/\mathfrak m\widehat R+\mathfrak P)=0$ for each prime ideal $\mathfrak P$ of $\widehat R$, $\operatorname{H}^{p+d}_{\mathfrak m}(M,N)$ is non-zero and so $\operatorname{cd}_{\mathfrak m}(M,N)=p+d$. The following example shows that $p+d$ can be a strict upper bound for $\operatorname{cd}_{\mathfrak m}(M,N)$. In fact, when there is not a prime ideal $\mathfrak P$ in $\operatorname{Supp}_{\widehat R}(\widehat M)\cap\operatorname{Ass}_{\widehat R}(\widehat N)$ satisfying $\dim_{\widehat R}(\widehat R/\mathfrak P)=d$ and $\operatorname{proj\,dim}_{\widehat R_\mathfrak P}(\widehat M_{\mathfrak P})=p$, then the statement (ii) in Theorem [Theorem 7](#lh){reference-type="ref" reference="lh"} is true and hence $\operatorname{H}^{p+d}_{\mathfrak m}(M,N)=0$.* **Example 9**. *Let $K$ be a field and $R:=K[[x, y]]$ be the ring of formal power series over $K$ in indeterminates $x,y$. $R$ is a complete regular local ring of dimension 2 with maximal ideal $\mathfrak m:=(x, y)$. We set $M:=R/(x^2, xy)$. It follows from $\operatorname{Ass}_R(M)=\{(x), (x, y)\}$ that $\operatorname{depth}_R(M)=0$ and $\dim_R(M)=\dim_R(R/(x))=1$. Since $R$ is regular, all modules have finite projective dimension and so the Auslander-Buchsbaum formula gives $\operatorname{proj\,dim}_R(M)=2$. Therefore $\operatorname{proj\,dim}_R(M)+\dim_R(R)=4$. Now since $\operatorname{Ass}_R(R)=\{0\}$, $\operatorname{Supp}_R(M)\cap\operatorname{Ass}_R(R)=\emptyset$ and so, by Theorem [Theorem 7](#lh){reference-type="ref" reference="lh"} or Corollary [Corollary 4](#att){reference-type="ref" reference="att"}, we obtain $\operatorname{H}^{4}_{\mathfrak m}(M,R)=0$. Hence $$\operatorname{cd}_{\mathfrak m}(M,R)<\operatorname{proj\,dim}_R(M)+\dim_R(R).$$ Furthermore, since $R$ is Cohen-Macaulay and $M$ has a finite projective dimension, the Divaani Azar--Hajikarimi formula implies that $$\operatorname{cd}_{\mathfrak m}(M,R)=\dim_R(R)-\operatorname{grade}_{R}(\operatorname{Ann}_R(R),M)=2-0=2.$$* # Acknowledgements {#acknowledgements .unnumbered} The author would like to thank the referees for careful reading of the Persian version of this paper and for helpful suggestions. 100 M. H. Bijan-Zadeh, *A common generalization of local cohomology theories*, Glasgow Math. J. **21**(2) (1980) 173--181. M. P. Brodmann and R. Y. Sharp, *Local Cohomology: An Algebraic Introduction with Geometric Applications*, Cambridge Studies in Advanced Mathematics **60** (Cambridge University Press, Cambridge, 1998). W. Bruns and J. Herzog, *Cohen-Macaulay Rings*, Cambridge Studies in Advanced Mathematics **39** (Cambridge University Press, Cambridge, 1993). M. T. Dibaei and S. Yassemi, *Attached primes of the top local cohomology modules with respect to an ideal*, Arch. Math. (Basel) **84**(4) (2005) 292--297. K. Divaani-Aazar, *Vanishing of the top local cohomology modules over Noetherian rings*, Proc. Indian Acad. Sci. Math. Sci. **119**(1) (2009) 23--35. K. Divaani-Aazar and A. Hajikarimi, *Generalized local cohomology modules and homological Gorenstein dimensions*, Comm. Algebra **39**(6) (2011) 2051--2067. K. Divaani-Aazar, R. Naghipour and M. Tousi, *Cohomological dimension of certain algebraic varieties*, Proc. Amer. Math. Soc. **130**(12) (2002) 3537--3544. K. Divaani-Aazar, R. Sazeedeh and M. Tousi, *On vanishing of generalized local cohomology modules*, Algebra Colloq. **12**(2) (2005) 213--218. E. E. Enochs and O. M. G. Jenda, *Relative Homological Algebra*, Volume 1. Second revised and extended edition. De Gruyter Expositions in Mathematics **30** (Walter de Gruyter GmbH & Co. KG, Berlin, 2011). A. Fathi, A. Tehranian and H. Zakeri, *Filter regular sequences and generalized local cohomology modules*, Bull. Malays. Math. Sci. Soc. **38**(2) (2015) 467--482. Y. Gu and L. Chu, *Attached primes of the top generalized local cohomology modules*, Bull. Aust. Math. Soc. **79**(1) (2009) 59--67. S. H. Hassanzadeh and A. Vahidi, *On vanishing and cofiniteness of generalized local cohomology modules*, Comm. Algebra **37**(7) (2009) 2290--2299. J. Herzog, *Komplexe, Auflösungen und Dualität in der Localen Algebra* (Habilitationsschrift, Universität Regensburg, 1970). I. G. Macdonald, *Secondary representation of modules over a commutative ring*, Symp. Math. **11** (1973) 23--43. A. Mafi, *On the associated primes of generalized local cohomology modules*, Comm. Algebra **34**(7) (2006) 2489--2494. H. Matsumura, *Commutative Ring Theory*, Cambridge Studies in Advanced Mathematics **8** (Cambridge University Press, Cambridge, 1986). J. J. Rotman, *An Introduction to Homological Algebra*, Second edition, Universitext (Springer, New York, 2009). S. Yassemi, *Coassociated primes*, Comm. Algebra **23**(4) (1995) 1473--1498.

arxiv_math

--- abstract: | The first named author introduced the notion of upper stability for metric spaces in [@baudier:22] as a relaxation of stability. The motivation was a search for a new invariant to distinguish the class of reflexive Banach spaces from stable metric spaces in the coarse and uniform category. In this paper we show that property $Q$ does in fact imply upper stability. We also provide a direct proof of the fact that reflexive spaces are upper stable by relating the latter notion to the asymptotic structure of Banach spaces. address: - Department of Mathematics, Texas A&M University, College Station, TX 77843, USA - Department of Mathematics, Texas A&M University, College Station, TX 77843, USA and Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27, Prague, Czech Republic - Peterhouse, Cambridge CB2 1RD and Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom author: - F. Baudier - Th. Schlumprecht - A. Zsák title: On stability of metric spaces and Kalton's property $Q$ --- # Introduction In [@kalton:07], Kalton proved two landmark results about the coarse and uniform geometry of infinite-dimensional Banach spaces. The first result was a nonlinear embedding result that states that every stable metric space admits a coarse and uniform embedding into a reflexive Banach space. While uniform embeddings have long been considered in nonlinear Banach space geometry (see [@ben-lin:00] for an extensive account), the notion of coarse embedding was introduced more recently by Gromov (under a different terminology) in [@gromov:93] as a notion of metric faithfulness that turned out to be of paramount importance in geometric group theory and noncommutative geometry. We refer to the monograph [@now-yu:12] for a further discussion of coarse embeddings. Kalton's embedding result was quantitatively refined in [@bau-lan:15], where it was observed that the embeddings possess even better metric faithfulness properties: they can be arbitrarily close to isometric embeddings. Krivine and Maurey introduced the notion of stability for separable Banach spaces [@kri-mau:81] in order to extend a result of Aldous [@aldous:81] by showing that every stable Banach space admits an isomorphic copy of $\ell_p$ for some $p\in[1,\infty)$. Garling [@garling:82] then studied this isometric Banach space notion in the more general context of arbitrary metric spaces. **Definition 1**. A metric space $(M,d_M)$ is *stable* if for any pair $\ensuremath{\mathcal U},\ensuremath{\mathcal V}$ of free ultrafilters on $\ensuremath{\mathbb N}$ and for any two bounded sequences $(x_n),(y_n)$ in $M$ we have $$\lim_{m,\ensuremath{\mathcal U}}\lim_{n,\ensuremath{\mathcal V}}d_M(x_m,y_n)=\lim_{n,\ensuremath{\mathcal V}}\lim_{m,\ensuremath{\mathcal U}}d_M(x_m,y_n)\ .$$ A close connection between stability and reflexivity in the category of metrizable topological spaces and continuous maps was known since the original work of Krivine and Maurey, and the new insight of Kalton was to establish another connection in the category of metric spaces and Lipschitz maps. In the latter connection, certain spaces of Lipschitz functions play a decisive role. Both connections rely on an important factorization result for weakly compact operators via reflexive spaces [@dav-fig-joh-pel:74]. Kalton raised a few natural problems. **Problem 1** ([@kalton:07]\*Problem 6.1). *Does every (separable) reflexive Banach space coarsely (or uniformly) embed into a stable metric space? Does every reflexive Banach space admit an embedding, which is simultaneously coarse and uniform, into a stable metric space?* Kalton explicitly suspected that Problem [Problem 1](#problem:kalton1){reference-type="ref" reference="problem:kalton1"} might have a negative solution, but that Problem [Problem 2](#problem:kalton2){reference-type="ref" reference="problem:kalton2"} below could have a positive solution. **Problem 2** ([@kalton:07]\*Problem 6.2). *If $X$ is a separable reflexive Banach space, does $B_X$ embed uniformly into a stable metric space?* We will now mainly focus on the coarse case of Problem [Problem 1](#problem:kalton1){reference-type="ref" reference="problem:kalton1"} (as the discussion of the uniform case is similar). Recall that a metric space $(M,d_M)$ *coarsely embeds* into a metric space $(N,d_N)$ if there exist a map $f\colon M\to N$ and non-decreasing functions $\omega,\rho\colon[0,\infty)\to[0,\infty)$ with $\lim_{t\to\infty}\rho(t)=\infty$ such that for all $x,y\in M$ we have $$\label{eq:coarse-embedding} \rho\big(d_M(x,y)\big)\leqslant d_N\big(f(x),f(y)\big)\leqslant\omega\big(d_M(x,y)\big)\ .$$ A time-tested method to provide a negative answer to Problem [Problem 1](#problem:kalton1){reference-type="ref" reference="problem:kalton1"} is to find a coarse invariant, *i.e.,* a property that is preserved under coarse embeddings, that is possessed by every stable metric space but fails for at least one separable reflexive Banach space. In [@kalton:07], Kalton introduced several variants of a concentration property for Lipschitz maps defined on certain infinite graphs, namely the *interlacing graphs*. Given an infinite subset $L$ of $\ensuremath{\mathbb N}$ and an integer $k\in \ensuremath{\mathbb N}$, we consider the set ${[L]}^{k}$ of all subsets of $L$ of size $k$ as the vertex set of a graph equipped with the interlacing adjacency relation, where two vertices $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}=\{m_1,\dots,m_k\}$ and $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}=\{n_1,\dots,n_k\}$ in ${[L]}^{k}$ with $m_1<\dots<m_k$ and $n_1<\dots<n_k$ are adjacent if and only if they *interlace*, *i.e.,*  $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}\neq\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ and either $m_1\leqslant n_1 \leqslant m_2\leqslant n_2\leqslant\dots\leqslant m_k\leqslant n_k$ or $n_1\leqslant m_1\leqslant n_2\leqslant m_2\leqslant\dots\leqslant n_k\leqslant m_k$. The graph metric induced by the interlacing adjacency relation will be denoted by $d_I$. **Definition 2**. A metric space $(M,d_M)$ is said to have Kalton's *property $Q$* if there is a constant $C\geqslant 0$ such that for all $k\in\ensuremath{\mathbb N}$ and for all bounded functions $f\colon({[\ensuremath{\mathbb N}]}^{k},d_I)\to(M,d_M)$, there exists $L\in{[\ensuremath{\mathbb N}]}^{\omega}$ such that $$\sup_{\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[L]}^{k}} d_M\big(f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\big)\leqslant C\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)\ .$$ The infimum of all constants $C$ satisfying the above will be denoted $Q_M$ and called the *property $Q$ constant of $M$*. Here ${[\ensuremath{\mathbb N}]}^{\omega}$ denotes the set of all infinite subsets of $\ensuremath{\mathbb N}$ and $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)$ is the Lipschitz constant of $f$. Kalton's original definition of property $Q$ for Banach spaces was given in terms of a modulus [@kalton:07]\*pages 403--404, but eventually took the form above in [@kalton:13]\*page 1057 (see also [@kalton:12]\*page 1283). The study of concentration inequalities for Lipschitz maps defined on infinite graphs with countable degree, which began with property $Q$ in [@kalton:07] for interlacing graphs and continued shortly thereafter in [@kal-ran:08] for Hamming graphs, was a genuine breakthrough towards unlocking many mysteries regarding the nonlinear geometry of Banach spaces in terms of their infinite-dimensional and asymptotic properties (see ). Kalton observed [@kalton:07]\*Proposition 3.1 that every stable metric space has property $Q$, and that if a Banach space $X$ admits a coarse embedding into a metric space with property $Q$, then $X$ has property $Q$. The interlacing graph metric can be seen as being induced by the summing basis of $\mathrm{c}_0$, and thus every Banach space containing an isomorphic copy of $\mathrm{c}_0$ contains bi-Lipschitz copies of the sequence $\big({[\ensuremath{\mathbb N}]}^{k},d_I\big)_{k=1}^\infty$ of interlacing graphs with uniformly bounded distortion. It follows that $\mathrm{c}_0$ does not embed coarsely into any metric space with property $Q$. The second remarkable result of Kalton from [@kalton:07] is that reflexive Banach spaces also have property $Q$. An important consequence is that $\mathrm{c}_0$ does not coarsely embed into any reflexive Banach space, thereby solving a longstanding open question in the coarse geometry of Banach spaces. Kalton also showed that property $Q$ is a uniform as well as coarse invariant for Banach spaces: if a Banach space $X$, or its closed unit ball $B_X$, admits a uniform embedding into a metric space with property $Q$, then $X$ has property $Q$. It follows that $\mathrm{c}_0$, or indeed the closed unit ball of $\mathrm{c}_0$, does not uniformly embed into any reflexive Banach space, solving yet another old open problem in the nonlinear theory of Banach spaces. The fact that reflexive spaces have property $Q$ rules out property $Q$ as a potential coarse invariant to resolve negatively Problem [Problem 1](#problem:kalton1){reference-type="ref" reference="problem:kalton1"}. In an attempt to attack Problem [Problem 1](#problem:kalton1){reference-type="ref" reference="problem:kalton1"}, the first author introduced in [@baudier:22] another invariant called upper stability. This bi-Lipschitz invariant is a natural relaxation of the notion of stability in light of a characterization of stability due to Raynaud [@raynaud:83] (see Theorem [Theorem 1](#thm:raynaud){reference-type="ref" reference="thm:raynaud"} below). Upper stability was shown to imply property $Q$ and to be preserved under coarse embeddings (for Banach space domains and metric space targets akin to property $Q$). **Definition 3**. Let $(M,d_M)$ be a metric space and $C$ be a non-negative real number. We say that $M$ is *$C$-upper stable* if for all integers $1\leqslant l<k$, bounded functions $f\colon{[\ensuremath{\mathbb N}]}^{l}\to M$ and $g\colon{[\ensuremath{\mathbb N}]}^{k-l}\to M$, infinite sets $L\subset\ensuremath{\mathbb N}$ and permutations $\pi\in S_k$ preserving the order on $\{1,\dots,l\}$ and on $\{l+1,\dots,k\}$, $$\begin{gathered} \inf_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[L]}^{k}} d_M\big(f(n_1,\dots,n_l),g(n_{l+1},\dots,n_k)\big) \leqslant\\ C\cdot\sup_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[L]}^{k}} d_M\big(f(n_{\pi(1)},\dots,n_{\pi(l)}),g(n_{\pi(l+1)},\dots,n_{\pi(k)})\big)\ . \end{gathered}$$ We say that $M$ is *upper stable* if it is $C$-upper stable for some $C$. There are good reasons to think that an approach to solving Problem [Problem 1](#problem:kalton1){reference-type="ref" reference="problem:kalton1"} via upper stability would work. Indeed, if $X$ is a Banach space containing no copies of $\ell_p$ for $1\leqslant p<\infty$, then $X$ cannot isomorphically embed into a stable Banach space because of the result of Krivine and Maurey mentioned above. If in addition $X$ is separable, then $X$ is not even bi-Lipschitzly embeddable into a reflexive stable Banach space due to a result of Heinrich and Mankiewicz [@hei-man:82]\*Theorem 3.5. Thus, any such space $X$ (Tsirelson's space, Schlumprecht's space, Gowers-Maurey space, *etc.*) could be a candidate for a reflexive space failing upper stability. In this paper, we dash this hope in two different ways, and at the same time, we further our understanding of the connection between stability, reflexivity and property $Q$. In Section [3](#sec:upper-stability-and-asymptotic-structure){reference-type="ref" reference="sec:upper-stability-and-asymptotic-structure"}, we prove that every reflexive Banach space is upper stable. Since upper stability implies property $Q$ (see [@baudier:22]), we thus recover the fact, originally proved by Kalton in [@kalton:07], that reflexive spaces have property $Q$. Kalton's original proof relies on some clever "calculus" for certain limiting operators in reflexive spaces. In order to prove the a priori stronger property of upper stability, we take a more conceptual approach as we relate upper stability with the asymptotic structure of Banach spaces in the sense of Maurey, Milman and Tomczak-Jaegermann [@mau-mil-tom:95]. First, however, we prove in Section [2](#sec:upper-stability-and-property-Q){reference-type="ref" reference="sec:upper-stability-and-property-Q"} the more general result that property $Q$ and upper stability are equivalent notions (up to the value of the constants involved) in the category of metric spaces. # Upper stability and property $Q$ {#sec:upper-stability-and-property-Q} We begin by stating the following characterization of stability, which combines work of Krivine and Maurey [@kri-mau:81] and Raynaud [@raynaud:83] (see also [@baudier:22]). **Theorem 1**. *The following are equivalent for a metric space $(M,d_M)$.* * * *$M$ is stable.* *For all $1\leqslant l<k$, bounded functions $f\colon{[\ensuremath{\mathbb N}]}^{l}\to M$ and $g\colon{[\ensuremath{\mathbb N}]}^{k-l}\to M$, free ultrafilters $\ensuremath{\mathcal U}_1,\dots,\ensuremath{\mathcal U}_k$ and permutations $\pi\in S_k$ preserving the order on $\{1,\dots,l\}$ and on $\{l+1,\dots,k\}$, $$\begin{gathered} \lim_{n_1,\ensuremath{\mathcal U}_1}\dots\lim_{n_k,\ensuremath{\mathcal U}_k} d_M\big(f(n_1,\dots,n_l),g(n_{l+1},\dots,n_k)\big) =\\ \lim_{n_{\pi^{-1}(1)},\ensuremath{\mathcal U}_{\pi^{-1}(1)}}\dots\lim_{n_{\pi^{-1}(k)},\ensuremath{\mathcal U}_{\pi^{-1}(k)}} d_M\big(f(n_1,\dots,n_l),g(n_{l+1},\dots,n_k)\big)\ . \end{gathered}$$* *For all $1\leqslant l<k$, bounded functions $f\colon{[\ensuremath{\mathbb N}]}^{l}\to M$ and $g\colon{[\ensuremath{\mathbb N}]}^{k-l}\to M$, free ultrafilters $\ensuremath{\mathcal U}$ and permutations $\pi\in S_k$ preserving the order on $\{1,\dots,l\}$ and on $\{l+1,\dots,k\}$, $$\begin{gathered} \lim_{n_1,\ensuremath{\mathcal U}}\dots\lim_{n_k,\ensuremath{\mathcal U}} d_M\big(f(n_1,\dots,n_l),g(n_{l+1},\dots,n_k)\big) =\\ \lim_{n_1,\ensuremath{\mathcal U}}\dots\lim_{n_k,\ensuremath{\mathcal U}} d_M\big(f(n_{\pi(1)},\dots,n_{\pi(l)}),g(n_{\pi(l+1)},\dots,n_{\pi(k)})\big)\ . \end{gathered}$$* *For all $1\leqslant l<k$, bounded functions $f\colon{[\ensuremath{\mathbb N}]}^{l}\to M$ and $g\colon{[\ensuremath{\mathbb N}]}^{k-l}\to M$, infinite sets $L\subset\ensuremath{\mathbb N}$ and permutations $\pi\in S_k$ preserving the order on $\{1,\dots,l\}$ and on $\{l+1,\dots,k\}$, $$\begin{gathered} \inf_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[L]}^{k}} d_M\big(f(n_1,\dots,n_l),g(n_{l+1},\dots,n_k)\big) \leqslant\\ \sup_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[L]}^{k}} d_M\big(f(n_{\pi(1)},\dots,n_{\pi(l)}),g(n_{\pi(l+1)},\dots,n_{\pi(k)})\big)\ . \end{gathered}$$* In light of the above characterization, the definition of upper stability given in the Introduction is a very natural relaxation of stability. There is a certain asymmetry in part (iv) above, and hence in the definition of upper stability, in the sense that the permutation is applied only to the right-hand side of the defining inequality. We will now introduce an a priori stronger version of upper stability by "symmetrizing" this inequality. We will also introduce a different formulation in terms of cuts instead of permutations. Fix $k,l,m\in\ensuremath{\mathbb N}$ with $k=l+m$. By a *cut of $\{1,\dots,k\}$ of size $l$* we mean simply a subset $P$ of $\{1,\dots,k\}$ of size $l$. If $P=\{p_1,\dots,p_l\}$ with $p_1<\dots<p_l$, then for $n_1<\dots<n_k$ in $\ensuremath{\mathbb N}$, we write $(n_i:\,i\in P)$ for $(n_{p_1},\dots,n_{p_l})$. The proof of Raynaud's characterization of stability shows the following. **Proposition 2**. *For a metric space $(M,d_M)$ and a constant $C\geqslant 1$, the following are equivalent.* * * *For all $k,l,m\in\ensuremath{\mathbb N}$ with $k=l+m$, free ultrafilters $\ensuremath{\mathcal U}$, bounded functions $f\colon{[\ensuremath{\mathbb N}]}^{l}\to M$ and $g\colon{[\ensuremath{\mathbb N}]}^{m}\to M$ and cuts $P,Q$ of $\{1,\dots,k\}$ of size $l$, we have $$\begin{gathered} \lim_{n_1,\ensuremath{\mathcal U}}\dots\lim_{n_k,\ensuremath{\mathcal U}} d_M\big(f(n_i:\,i\in P),g(n_i:\,i\in P^c)\big)\leqslant\\ C\cdot \lim_{n_1,\ensuremath{\mathcal U}}\dots\lim_{n_k,\ensuremath{\mathcal U}} d_M\big(f(n_i:\,i\in Q),g(n_i:\,i\in Q^c)\big)\ . \end{gathered}$$* *For all $k,l,m\in\ensuremath{\mathbb N}$ with $k=l+m$, bounded functions $f\colon{[\ensuremath{\mathbb N}]}^{l}\to M$ and $g\colon{[\ensuremath{\mathbb N}]}^{m}\to M$, infinite sets $L\subset\ensuremath{\mathbb N}$ and cuts $P,Q$ of $\{1,\dots,k\}$ of size $l$, we have $$\begin{gathered} \inf_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[L]}^{k}}d_M\big(f(n_i:\,i\in P),g(n_i:\,i\in P^c)\big)\leqslant\\ C\cdot\sup_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[L]}^{k}}d_M\big(f(n_i:\,i\in Q),g(n_i:\,i\in Q^c)\big)\ . \end{gathered}$$* **Definition 4**. We say a metric space $(M,d_M)$ is *$C$-cut stable* if it satisfies one of the two equivalent conditions in Proposition [Proposition 2](#prop:cut-stable-characterization){reference-type="ref" reference="prop:cut-stable-characterization"}. We say $M$ is *cut stable* if it is $C$-cut stable for some $C\geqslant 1$. *Remarks 1*. We recover the notion of upper stability by taking $P$ to be the trivial cut $\{1,2,\dots,l\}$ above. We will later see that every upper stable metric space is cut stable, although this is not entirely trivial. It is clear that upper stability is a Lipschitz invariant, and hence in particular an isomorphic invariant for Banach spaces. In contrast, stability is an isometric invariant. It is easy to find equivalent norms on the stable Banach space $\ell_1$, for instance, which is not stable. As mentioned in the Introduction, Kalton's original definition of property $Q$ was given in terms of a modulus, which we now recall. **Definition 5** (Kalton [@kalton:07]). Let $(M,d_M)$ be a metric space. Given $\varepsilon>0$ and $\delta\geqslant 0$, we say that $M$ has the *$Q(\varepsilon,\delta)$-property* if for all $k\in\ensuremath{\mathbb N}$ and for all functions $f\colon\big({[\ensuremath{\mathbb N}]}^{k},d_I\big)\to M$ with $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)\leqslant\delta$, there exists $L\in{[\ensuremath{\mathbb N}]}^{\omega}$ such that $$d_M(f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}))<\varepsilon\qquad\text{for all }\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\text{ in }{[L]}^{k}\ .$$ For $\varepsilon>0$ we define $$\Delta_M(\varepsilon)=\sup\{\delta\geqslant 0:\,M \text{ has the $Q(\varepsilon,\delta)$-property}\}\ ,$$ and set $q_M$ to be the supremum of all constants $c\geqslant 0$ for which $\Delta_M(\varepsilon)\geqslant c\varepsilon$ for all $\varepsilon>0$. *Remark 1*. It is immediate from above that $M$ has property $Q$ if and only if $q_M>0$ in which case $q_M^{-1}\leqslant Q_M\leqslant 2q_M^{-1}$. Kalton proved in [@kalton:07] that for a stable metric space $M$, $q_M\geqslant 1$, and for a reflexive Banach space $X$, $q_X\geqslant 1/2$ (and $Q_X\leqslant 2$). Note that for any infinite-dimensional Banach space $X$, $q_X\leqslant 1$. Indeed, for any $\delta>1$ there exists $f\colon\ensuremath{\mathbb N}\to X$ with $1<\lVert f(m)-f(n)\rVert<\delta$ for all $m<n$, and thus $X$ does not have the $Q(1,\delta)$-property. *Remark 2*. Let $(e_i)$ and $(s_i)$ denote the unit vector basis and, respectively, the summing basis of $\mathrm{c}_0$. Thus, $s_n=\sum_{i=1}^ne_i$, $n\in\ensuremath{\mathbb N}$. For $k\in\ensuremath{\mathbb N}$ define $f\colon({[\ensuremath{\mathbb N}]}^{k},d_I)\to\mathrm{c}_0$ by $f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})=\sum_{i=1}^ks_{n_i}$. Note that $\lVert f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert=d_I(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})=1$ for adjacent vertices, and hence $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)=1$. Moreover, $\lVert f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert=k$ for all $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ in ${[\ensuremath{\mathbb N}]}^{k}$. Thus $\mathrm{c}_0$ has the worst possible behaviour for property $Q$. Indeed, this behaviour of the summing basis motivated the proof by Kalton that $\mathrm{c}_0$ does not uniformly or coarsely embed into a reflexive space. It was also the motivation behind the interlacing graph. Indeed, $$\lVert f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert\leqslant d_I(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\leqslant 2\lVert f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert \qquad\text{for all }\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[\ensuremath{\mathbb N}]}^{k}$$ and for every $k\in\ensuremath{\mathbb N}$. *Remark 3*. If follows easily by scaling that for a Banach space $X$, the function $\Delta_X$ is linear, and thus $\Delta_X(\varepsilon)=q_X\varepsilon$ for all $\varepsilon>0$. Another observation of Kalton seems slightly less obvious, so we provide a proof next. **Proposition 3**. *For a Banach space $X$, $\Delta_X(\varepsilon)=\Delta_{B_X}(\varepsilon)$ for all $0<\varepsilon<1$.* *Proof.* Clearly, $\Delta_X(\varepsilon)\leqslant\Delta_{B_X}(\varepsilon)$ for any $\varepsilon>0$. For the converse, we will first show that $\Delta_{B_X}$ is "linear" on $(0,1)$. Fix $b$ with $0<b<1$. Given $$0<\varepsilon_1<\varepsilon_2\leqslant b\ ,$$ write $\varepsilon_2=a\varepsilon_1$ for suitable $a>1$. We show that $\Delta_{B_X}(\varepsilon_2)=a\Delta_{B_X}(\varepsilon_1)$. This will show linearity on $(0,b]$, and hence on $(0,1)$. To show that $\Delta_{B_X}(\varepsilon_2)=a\Delta_{B_X}(\varepsilon_1)$, we may clearly assume that $a\leqslant b^{-1}$ as the general case will then follow by repeated applications of the special case. The inequality $\Delta_{B_X}(\varepsilon_2)\geqslant a\Delta_{B_X}(\varepsilon_1)$ follows by simple scaling. Conversely, assume that $B_X$ has the $Q(\varepsilon_2,\delta)$-property. Let $k\in\ensuremath{\mathbb N}$ and $f\colon\big({[\ensuremath{\mathbb N}]}^{k},d_I\big)\to B_X$ be a function with $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)\leqslant\delta/a$. Then in particular, $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)\leqslant\delta$, and hence there exists $L\in{[\ensuremath{\mathbb N}]}^{\omega}$ such that $\lVert f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert<\varepsilon_2$ for all $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ in ${[L]}^{k}$. Enumerate $L$ as $l_1<l_2<\dots$ and set $\ensuremath{\accentset{\rule{.4em}{.6pt}}{l}}=\{l_1,\dots,l_k\}$. Define $g\colon\big({[\ensuremath{\mathbb N}]}^{k},d_I\big)\to X$ by $$g(n_1,\dots,n_k)=\big(f(l_{k+n_1},\dots,l_{k+n_k})-f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{l}})\big)\cdot a\ .$$ Since $a\varepsilon_2\leqslant ab\leqslant 1$, it follows that $g$ takes values in $B_X$. Moreover, $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(g)\leqslant a\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)\leqslant\delta$, and hence there exists $N\in{[\ensuremath{\mathbb N}]}^{\omega}$ such that $\lVert g(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-g(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert<\varepsilon_2$ for all $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ in ${[N]}^{k}$. It follows that $\lVert f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert<\varepsilon_1$ for all $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ in ${[P]}^{k}$, where $P=\{l_{k+n}:\,n\in N\}$. We have proved that $B_X$ has the $Q(\varepsilon_1,\delta/a)$-property, and hence $a\cdot\Delta_{B_X}(\varepsilon_1)\geqslant\Delta_{B_X}(\varepsilon_2)$, as required. Having shown linearity of $\Delta_{B_X}$ on $(0,1)$, we now finish the proof by showing that $\Delta_{B_X}(\varepsilon)\leqslant\Delta_X(\varepsilon)$ for any $0<\varepsilon<1$. Assume that $0\leqslant\delta<\Delta_{B_X}(\varepsilon)$ for some $\delta$ and $0<\varepsilon<1$. Let $k\in\ensuremath{\mathbb N}$ and $f\colon\big({[\ensuremath{\mathbb N}]}^{k},d_I\big)\to X$ be a function with $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)\leqslant\delta$. Then for some $n\in\ensuremath{\mathbb N}$, the function $n^{-1}f$ takes values in $B_X$, and moreover $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(n^{-1}f)\leqslant{}n^{-1}\delta<n^{-1}\Delta_{B_X}(\varepsilon)=\Delta_{B_X}(n^{-1}\varepsilon)$. Hence there exists $L\in{[\ensuremath{\mathbb N}]}^{\omega}$ such that $\lVert n^{-1}f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}})-n^{-1}f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})\rVert<n^{-1}\varepsilon$ for all $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ in ${[L]}^{k}$. It follows that $X$ has the $Q(\varepsilon,\delta)$-property, and thus $\Delta_X(\varepsilon)\geqslant\Delta_{B_X}(\varepsilon)$ as required. ◻ The following result shows that property $Q$ is preserved under coarse and uniform embeddings with Banach space domain and metric space target. This is a trivial extension of [@kalton:07]\*Theorem 4.2. We give the short proof for convenience. Let us first recall that for a function $f\colon(M,d_M)\to(N,d_N)$ between metric spaces, its *modulus of continuity $\omega_f$* and *compression modulus $\rho_f$* are defined, respectively, by $$\omega_f(t)=\sup \{ d_N(f(x),f(y)):\,d_M(x,y)\leqslant t\}$$ and $$\rho_f(t)=\inf \{ d_N(f(x),f(y)):\,d_M(x,y)\geqslant t\}$$ for $t\in[0,\infty)$. Note that $f$ is a uniform embedding if and only if $\lim_{t\to0^+}\omega_f(t)=0$ and $\rho_f(t)>0$ for all $t>0$, and that $f$ is a coarse embedding if and only if $\lim_{t\to\infty}\rho_f(t)=\infty$ and $\omega_f(t)<\infty$ for all $t>0$. Note also that if $M$ is a graph and $d_M$ is the graph distance, then $\mathop{\mathrm{\ensuremath{\mathrm{Lip}}}}(f)=\omega_f(1)$. **Proposition 4**. *Let $h\colon (M,d_M)\to(N,d_N)$ be a function between metric spaces, and assume that $N$ has property $Q$. Then for all $\varepsilon>0$, $$\Delta_M(\varepsilon)\geqslant\sup\{\delta\geqslant 0:\, q_N\cdot\rho_h(\varepsilon)>\omega_h(\delta)\}\ .$$ This has the following consequences.* * * *If $h$ is a uniform embedding, then $\Delta_M(\varepsilon)>0$ for all $\varepsilon>0$.* *If $h$ is a coarse embedding, then $\Delta_M(\varepsilon)\to\infty$ as $\varepsilon\to\infty$.* *In particular, if a Banach space $X$ embeds coarsely into a metric space with property $Q$ or if the unit ball $B_X$ embeds uniformly into a metric space with property $Q$, then $X$ has property $Q$.* *Proof.* Let $\delta\geqslant 0$ and $\varepsilon>0$ be such that $M$ fails the $Q(\varepsilon,\delta)$-property. Then there exist $k\in\ensuremath{\mathbb N}$ and a function $f\colon{[\ensuremath{\mathbb N}]}^{k}\to M$ such that $\omega_f(1)\leqslant\delta$ and for all $L\in{[\ensuremath{\mathbb N}]}^{\omega}$ there exist $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ in ${[L]}^{k}$ with $d_M(f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}))\geqslant\varepsilon$. Then $\omega_{hf}(1)\leqslant\omega_h(\delta)$ and for all $L\in{[\ensuremath{\mathbb N}]}^{\omega}$ there exist $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ in ${[L]}^{k}$ with $d_N(hf(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),hf(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}))\geqslant\rho_h(\varepsilon)$. It follows that $N$ does not have the $Q(\rho_h(\varepsilon),\omega_h(\delta))$-property, and hence $$\omega_h(\delta)\geqslant\Delta_N(\rho_h(\varepsilon))\geqslant q_N\cdot\rho_h(\varepsilon)\ .$$ The result follows. ◻ Recall that for $k\in\ensuremath{\mathbb N}$, the interlacing graph on the vertex set ${[\ensuremath{\mathbb N}]}^{k}$ joins two vertices $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}=\{m_1,\dots,m_k\}$ and $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}=\{n_1,\dots,n_k\}$ if and only if $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}$ and $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ are interlacing, *i.e.,* either $m_1\leqslant n_1\leqslant m_2\leqslant n_2\leqslant\dots\leqslant m_k\leqslant n_k$ or $n_1\leqslant m_1\leqslant n_2\leqslant m_2\leqslant\dots\leqslant n_k\leqslant m_k$. *Remark 4*. Using the above notation, let $m(x)=\lvert \{i:\,m_i\leqslant x\}\rvert$, $x\in\ensuremath{\mathbb N}$. The function $x\mapsto m(x)-n(x)$ measures the discrepancy between $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}$ and $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$, which in turn measures the graph distance between $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}$ and $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$: $$d_I(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})=\max_{x\in\ensuremath{\mathbb N}}\big(m(x)-n(x)\big)- \min_{x\in\ensuremath{\mathbb N}}\big(m(x)-n(x)\big)\ .$$ It follows that the diameter of the interlacing graph on ${[\ensuremath{\mathbb N}]}^{k}$ is $k$, and moreover, $d_I(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})=k$ if and only if either $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}$ lies strictly between two consecutive elements of $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ including the degenerate cases $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}<\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}$ (which means that $m_k<n_1$) and $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}<\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}$, or vice versa. As noted earlier, the summing basis of $\mathrm{c}_0$ provides another description of the graph distance $d_I$. We will need the following stronger version of the interlacing adjacency relation. **Definition 6**. Given $k\in\ensuremath{\mathbb N}$ and $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[\ensuremath{\mathbb N}]}^{k}$, we say that the pair $(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})$ is *strongly interlacing* if either $m_1\leqslant n_1<m_2\leqslant n_2<\dots<m_k\leqslant n_k$ or $n_1\leqslant m_1<n_2\leqslant m_2<\dots< n_k\leqslant m_k$. **Lemma 5**. *Given $k\in\ensuremath{\mathbb N}$ and an interlacing pair $(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})$ in ${[2\ensuremath{\mathbb N}]}^{k}$, there exists $\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}}\in{[\ensuremath{\mathbb N}]}^{k}$ such that the pairs $(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}})$ and $(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}})$ are strongly interlacing.* *Proof.* Without loss of generality, we may assume that $$m_1\leqslant n_1\leqslant m_2\leqslant n_2\leqslant\dots\leqslant m_k\leqslant n_k\ .$$ Define $\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}}=\{p_1,\dots,p_k\}$ in ${[\ensuremath{\mathbb N}]}^{k}$ by $$p_i=\begin{cases}% n_i-1 & \text{if }n_i=m_{i+1}\ ,\\ n_i & \text{if }n_i<m_{i+1}\ . \end{cases}$$ It is straightforward to verify that this works. ◻ We are now ready to state and prove the main result of this section. **Theorem 6**. *A metric space with property $Q$ is cut stable.* *Proof.* Let $(M,d_M)$ be a metric space with property $Q$. We will show that $M$ is $C$-cut stable with $C=8Q_M+1$. Fix integers $1\leqslant l<k$, bounded functions $f\colon{[\ensuremath{\mathbb N}]}^{l}\to M$, $g\colon{[\ensuremath{\mathbb N}]}^{k-l}\to M$ and cuts $P,Q$ of $\{1,\dots,k\}$ of size $l$. Set $$\begin{aligned} A &= \inf_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[\ensuremath{\mathbb N}]}^{k}} d_M\big(f(n_i:\,i\in P),g(n_i:\,i\in P^c)\big)\\ B &=\sup_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[\ensuremath{\mathbb N}]}^{k}} d_M\big(f(n_i:\,i\in Q),g(n_i:\,i\in Q^c)\big)\ . \end{aligned}$$ Put $L=\{2nk:\,n\in\ensuremath{\mathbb N}\}$ and let $L_f$ and $L_g$ be the Lipschitz constants of $f\ensuremath{\!\!\restriction}_{{[L]}^{l}}$ and of $g\ensuremath{\!\!\restriction}_{{[L]}^{k-l}}$, respectively. We first show that $L_f\leqslant 4B$ and $L_g\leqslant 4B$. To see this, set $L'=\{nk:\,n\in\ensuremath{\mathbb N}\}$ and let $(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})$ be an interlacing pair in ${[L]}^{l}$. By Lemma [Lemma 5](#lem:strongly-interlacing){reference-type="ref" reference="lem:strongly-interlacing"} there exists $\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}}\in{[L']}^{l}$ such that the pairs $(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}})$ and $(\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})$ are strongly interlacing. We can then choose $m_1<\dots<m_k$ and $p_1<\dots<p_k$ in $L'$ such that $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}=\{m_i:\,i\in Q\}$, $\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}}=\{p_i:\,i\in Q\}$ and $m_i=p_i$ for all $i\in Q^c$. It follows that $$\begin{gathered} d_M(f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}}))\leqslant d_M\big(f(m_i:\,i\in Q),g(m_i:\,i\in Q^c)\big)\\ + d_M\big(f(p_i:\,i\in Q),g(p_i:\,i\in Q^c)\big) \leqslant 2B\ . \end{gathered}$$ Similarly, we get $d_M(f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{p}}),f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}))\leqslant 2B$, and hence $d_M(f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}))\leqslant 4B$, as required. A similar argument shows that $L_g\leqslant 4B$. Now fix $C>Q_M$. Since $M$ has property $Q$, there exists $N\in{[L]}^{\omega}$ such that $d_M(f(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),f(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}))\leqslant CL_f$ for all $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[N]}^{l}$ and $d_M(g(\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}}),g(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}))\leqslant CL_g$ for all $\ensuremath{\accentset{\rule{.8em}{.6pt}}{m}},\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[N]}^{k-l}$. Now fix any $n_1<\dots<n_k$ in $N$. We get $$\begin{aligned} A &\leqslant& d_M\big(f(n_i:\,i\in P),g(n_i:\,i\in P^c)\big)\\ &\leqslant& d_M\big(f(n_i:i\in P),f(n_i:i\in Q)\big)\\ && + d_M\big(f(n_i:i\in Q),g(n_i:i\in Q^c)\big)\\ && + d_M\big(g(n_i:i\in Q^c),g(n_i:i\in P^c)\big) \leqslant(8C+1)B\ . \end{aligned}$$ The result follows. ◻ The implication that an upper stable metric space has property $Q$ was shown in [@baudier:22]. Combining this with Theorem [Theorem 6](#thm:property-Q-implies-cut-stable){reference-type="ref" reference="thm:property-Q-implies-cut-stable"}, we obtain **Corollary 7**. *The following are equivalent for a metric space $(M,d_M)$.* * * *$M$ has property $Q$.* *$M$ is cut stable.* *$M$ is upper stable.* Combining this with Proposition [Proposition 4](#prop:inheriting-Q){reference-type="ref" reference="prop:inheriting-Q"}, we obtain the following. **Corollary 8**. *If a Banach space $X$ embeds coarsely into an upper stable metric space, or if the unit ball $B_X$ embeds uniformly into an upper stable metric space, then $X$ is upper stable.* # Reflexive spaces, upper stability and asymptotic structure {#sec:upper-stability-and-asymptotic-structure} Since reflexive spaces have property $Q$ (Kalton [@kalton:07]), the following is an immediate consequence of Theorem [Theorem 6](#thm:property-Q-implies-cut-stable){reference-type="ref" reference="thm:property-Q-implies-cut-stable"}. **Corollary 9**. *Every reflexive Banach space is upper stable.* In this section, we give a direct proof of Corollary [Corollary 9](#cor:reflexive-upper-stable){reference-type="ref" reference="cor:reflexive-upper-stable"} without using property $Q$. Instead, we shall relate stability and asymptotic structure. We begin by recalling the notion of asymptotic structure introduced by Maurey, Milman and Tomczak-Jaegermann [@mau-mil-tom:95]. For each $k\in\ensuremath{\mathbb N}$, let $\ensuremath{\mathcal E}_k$ denote the set of norms on $\ensuremath{\mathbb R}^k$ with respect to which the standard basis $(e_i)_{i=1}^k$ is a normalized, monotone basis. Note that $\ensuremath{\mathcal E}_k$ is a compact subset of $C\big([-1,1]^k\big)$ by the Arzelà--Ascoli theorem. Fix an infinite-dimensional Banach space $X$ with norm $\lVert \cdot\rVert$, and consider the following two-player game between $\ensuremath{\boldsymbol{\mathrm{S}}}$ (subspace chooser) and $\ensuremath{\boldsymbol{\mathrm{P}}}$ (point chooser). At each move of the game $\ensuremath{\boldsymbol{\mathrm{S}}}$ first picks a finite-codimensional subspace $Y$ of $X$ and $\ensuremath{\boldsymbol{\mathrm{P}}}$ responds with a vector $x\in Y$ with $\lVert x\rVert=1$. The game finishes after exactly $k$ moves producing a sequence $Y_1,x_1,Y_2,x_2,\dots,Y_k,x_k$. Given a norm $N$ on $\ensuremath{\mathbb R}^k$ and $\delta>0$, we say that *$\ensuremath{\boldsymbol{\mathrm{P}}}$ wins the $(N,\delta)$-game* if $$\label{eq:within-delta-of-norm} \Big\lvert N\big(\textstyle\sum_{i=1}^ka_ie_i\big)-\big\lVert \textstyle\sum_{i=1}^ka_ix_i\big\rVert\Big\rvert <\delta\qquad \text{for all }(a_i)_{i=1}^k\in[-1,1]^k\ .$$ The *$k^{\text{th}}$ asymptotic structure of $X$* is the set $\{X\}_k$ of all norms $N$ on $\ensuremath{\mathbb R}^k$ for which $\ensuremath{\boldsymbol{\mathrm{P}}}$ has a winning strategy for the $(N,\delta)$-game for all $\delta>0$. One of the results in [@mau-mil-tom:95] states that $\{X\}_k$ is a non-empty closed subset of $\ensuremath{\mathcal E}_k$, and moreover it is the smallest closed subset $\ensuremath{\mathcal F}$ of $\ensuremath{\mathcal E}_k$ such that for every $\delta>0$, $\ensuremath{\boldsymbol{\mathrm{S}}}$ has a winning strategy for forcing $\ensuremath{\mathcal P}$ to choose vectors $x_1,\dots,x_k$ such that [\[eq:within-delta-of-norm\]](#eq:within-delta-of-norm){reference-type="eqref" reference="eq:within-delta-of-norm"} holds for some $N\in\ensuremath{\mathcal F}$. Winning strategies in the game described above naturally correspond to trees. For $k\in\ensuremath{\mathbb N}$ and $L\subset\ensuremath{\mathbb N}$ we denote by ${[L]}^{\leqslant k}$ the set of all subsets of $L$ of size at most $k$. By a *(countably branching) tree in $X$ of height $k$* we mean a function $t\colon{[\ensuremath{\mathbb N}]}^{\leqslant k}\to X$ or $t\colon{[\ensuremath{\mathbb N}]}^{\leqslant k}\setminus\{\emptyset\}\to X$; in the former case $t$ is a *rooted* tree with root $t(\emptyset)$, and in the latter case $t$ is an *unrooted* tree. We say $t$ is *normalized* if it takes values in the unit sphere $S_X$ of $X$. Given a sequence $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}=(\ensuremath{\mathcal U}_1,\dots,\ensuremath{\mathcal U}_k)$ of free ultrafilters, we say that $t$ is *$\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}$-weakly null* if for each $1\leqslant i\leqslant k$ we have $$\forall \,_{\ensuremath{\mathcal U}_1}n_1\ \forall \,_{\ensuremath{\mathcal U}_2}n_2\ \dots\ \forall \,_{\ensuremath{\mathcal U}_{i-1}}n_{i-1}\quad \mathop{\mathrm{w\mathrm{-}\lim}}_{n,\ensuremath{\mathcal U}_i}t(n_1,\dots,n_{i-1},n)=0\ .$$ In the unrooted case, given $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}=\{n_1,\dots,n_k\}\in{[\ensuremath{\mathbb N}]}^{k}$ with $n_1<\dots<n_k$, the sequence $\big(t(n_1,\dots,n_i)\big)_{i=1}^k$ is called a *branch* of $t$. Since every weakly null sequence is eventually almost contained in any finite-codimensional subspace, it follows that for every $k\in\ensuremath{\mathbb N}$ and $\delta>0$, every normalized, weakly null (with respect to some sequence of free ultrafilters) unrooted tree in $X$ of height $k$ has a branch $(x_1,\dots,x_k)$ satisfying [\[eq:within-delta-of-norm\]](#eq:within-delta-of-norm){reference-type="eqref" reference="eq:within-delta-of-norm"} for some $N\in\{X\}_k$. We shall need a somewhat stronger and more precise formulation of this statement. Let us fix $k\in\ensuremath{\mathbb N}$, a sequence $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}=(\ensuremath{\mathcal U}_1,\dots,\ensuremath{\mathcal U}_k)$ of free ultrafilters and a normalized, $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}$-weakly null tree $t$ in $X$. Define $$\begin{aligned} N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}} \colon X\oplus\ensuremath{\mathbb R}^k &\to&\ensuremath{\mathbb R}\\ N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big(x\oplus\textstyle\sum_{i=1}^ka_ie_i\Big) &=& \lim_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}} \Big\lVert x+\textstyle\sum_{i=1}^ka_it(n_1,\dots,n_i)\Big\rVert\end{aligned}$$ where $\lim_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}$ is an abbreviation for the iterated limit $\lim_{n_1,\ensuremath{\mathcal U}_1}\dots\lim_{n_k,\ensuremath{\mathcal U}_k}$. It is clear that $N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}$ is a seminorm on $X\oplus\ensuremath{\mathbb R}^k$. **Lemma 10**. *In the situation described above, we have $$N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big(x\oplus\textstyle\sum_{i=1}^ja_ie_i\Big) \leqslant N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big(x\oplus\textstyle\sum_{i=1}^ka_ie_i\Big)$$ for all $0\leqslant j\leqslant k$, $x\in X$ and $(a_i)_{i=1}^k\in\ensuremath{\mathbb R}^k$. It follows that $$N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big(0\oplus\textstyle\sum_{i=1}^ka_ie_i\Big) \leqslant 2 N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big(x\oplus\textstyle\sum_{i=1}^ka_ie_i\Big)$$ for all $x\in X$ and $(a_i)_{i=1}^k\in\ensuremath{\mathbb R}^k$.* *Proof.* The second inequality follows from the first one by triangle inequality. To see the first inequality, fix $x\in X$, $(a_i)_{i=1}^k\in\ensuremath{\mathbb R}^k$ and $0\leqslant j\leqslant k$. For every $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[\ensuremath{\mathbb N}]}^{j}$, fix a norming functional $z^*(\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}})$ for $x+\sum_{i=1}^ja_it(n_1,\dots,n_i)$. It follows that $$\begin{aligned} N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big(x\oplus\textstyle\sum_{i=1}^ja_ie_i\Big) &= \lim_{n_1,\ensuremath{\mathcal U}_1}\dots\lim_{n_j,\ensuremath{\mathcal U}_j} \Big\lVert x+\textstyle\sum_{i=1}^ja_it(n_1,\dots,n_i)\Big\rVert \\ &= \lim_{n_1,\ensuremath{\mathcal U}_1}\dots\lim_{n_j,\ensuremath{\mathcal U}_j} z^*(n_1,\dots,n_j) \Big(x+\textstyle\sum_{i=1}^ja_it(n_1,\dots,n_i)\Big) \\ &= \lim_{n_1,\ensuremath{\mathcal U}_1}\dots\lim_{n_k,\ensuremath{\mathcal U}_k} z^*(n_1,\dots,n_j) \Big(x+\textstyle\sum_{i=1}^ka_it(n_1,\dots,n_i)\Big) \\ &\leqslant\lim_{n_1,\ensuremath{\mathcal U}_1}\dots\lim_{n_k,\ensuremath{\mathcal U}_k} \Big\lVert x+\textstyle\sum_{i=1}^ka_it(n_1,\dots,n_i)\Big\rVert \\ &= N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big(x\oplus\textstyle\sum_{i=1}^ka_ie_i\Big)\ . \end{aligned}$$ ◻ *Remark 5*. Lemma [Lemma 10](#lem:asymptotic-norm){reference-type="ref" reference="lem:asymptotic-norm"} shows that $N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}$ is in fact a norm on $X\oplus\ensuremath{\mathbb R}^k$ extending the norm of $X$, and the unit vector basis of $\ensuremath{\mathbb R}^k$ is a normalized, monotone basis with respect to $N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}$. By an earlier remark, the restriction of $N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}$ to $\ensuremath{\mathbb R}^k$ is in the $k^{\text{th}}$ asymptotic structure of $X$. It is not very hard to show that if $X$ is separable and reflexive, then the converse also holds: every member of $\{X\}_k$ is of the form $N_t^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}$ for some normalized, weakly null tree $t$ in $X$ with $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}$ being arbitrary. Our next result approximates bounded functions, like those appearing in Raynaud's characterization of stability, by weakly null trees. **Lemma 11**. *Assume $X$ is reflexive. Let $k\in\ensuremath{\mathbb N}$, $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}=(\ensuremath{\mathcal U}_1,\dots,\ensuremath{\mathcal U}_k)$ be a sequence of free ultrafilters and $f\colon{[\ensuremath{\mathbb N}]}^{k}\to X$ a bounded function. Then there exist a normalized, $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}$-weakly null rooted tree $t$ in $X$ and non-negative scalars $a_0,a_1,\dots,a_k$ such that $$\lim_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big\lVert f(n_1,\dots,n_k)-\textstyle\sum_{i=0}^ka_it(n_1,\dots,n_i)\Big\rVert=0\ .$$* *Proof.* Define for each $1\leqslant j\leqslant k$ the function $$\begin{aligned} \partial^j f\colon{[\ensuremath{\mathbb N}]}^{k-j} &\to& X\\ \partial^j f(n_1,\dots,n_{k-j}) &=& \mathop{\mathrm{w\mathrm{-}\lim}}_{n_{k-j+1},\ensuremath{\mathcal U}_{k-j+1}}\dots\mathop{\mathrm{w\mathrm{-}\lim}}_{n_k,\ensuremath{\mathcal U}_k} f(n_1,\dots,n_k)\ . \end{aligned}$$ We will identify the function $\partial^kf$ on ${[\ensuremath{\mathbb N}]}^{0}=\{\emptyset\}$ with its image in $X$. Note that for each $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[\ensuremath{\mathbb N}]}^{k}$ we have $$\label{eq:bdd-fn-to-tree} f(n_1,\dots,n_k)=\partial^kf+\sum_{i=1}^k \big(\partial^{k-i}f(n_1,\dots,n_i)-\partial^{k-i+1}f(n_1,\dots,n_{i-1})\big)\ .$$ Define an unrooted tree $a$ in $\ensuremath{\mathbb R}$ of height $k$ by setting $$a(n_1,\dots,n_i)= \big\lVert \partial^{k-i}f(n_1,\dots,n_i)-\partial^{k-i+1}f(n_1,\dots,n_{i-1})\big\rVert$$ for $1\leqslant i\leqslant k$ and $n_1<\dots<n_i$ in $\ensuremath{\mathbb N}$. We next define $t(\emptyset)\in S_X$ and non-negative scalars $a_0,a_1,\dots,a_k$ by $$\begin{aligned} \partial^kf &= a_0t(\emptyset)\\ a_i &= \lim_{n_1,\ensuremath{\mathcal U}_1}\dots\lim_{n_i,\ensuremath{\mathcal U}_i} a(n_1,\dots,n_i) \qquad\text{for }1\leqslant i\leqslant k\ . \end{aligned}$$ We now define our rooted tree $t$ in $S_X$ as follows. We have already defined $t(\emptyset)$. Fix $1\leqslant i\leqslant k$. If $a_i=0$, then for any $n_1<\dots<n_{i-1}$ in $\ensuremath{\mathbb N}$, we choose $\big(t(n_1,\dots,n_{i-1},n)\big)_{n=n_{i-1}+1}^\infty$ to be an arbitrary normalized weakly null sequence. If $a_i\neq0$, then for $n_1<\dots<n_i$ we set $$t(n_1,\dots,n_i)=% \begin{cases} \frac{\partial^{k-i}f(n_1,\dots,n_i)-\partial^{k-i+1}f(n_1,\dots,n_{i-1})}{a(n_1,\dots,n_i)} & \text{if }a(n_1,\dots,n_i)\neq0\ ,\\ \text{an arbitrary norm-$1$ vector} & \text{otherwise.} \end{cases}$$ If $a_i\neq0$, then $\forall \,_{\ensuremath{\mathcal U}_1}n_1\dots\forall \,_{\ensuremath{\mathcal U}_i}n_i$ $a(n_1,\dots,n_i)>a_i/2$, from which it follows that $t$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}$-weakly null. Now observe that for $n_1<\dots<n_k$ we have $$\begin{gathered} f(n_1,\dots,n_k)-\sum_{i=0}^ka_it(n_1,\dots,n_i)\\ =\sum_{i=1}^k \Big[ \big(\partial^{k-i}f(n_1,\dots,n_i)-\partial^{k-i+1}f(n_1,\dots,n_{i-1})\big) - a(n_1,\dots,n_i)t(n_1,\dots,n_i) \Big]\\ +\sum_{i=1}^k\big(a(n_1,\dots,n_i)-a_i\big)t(n_1,\dots,n_i)\ , \end{gathered}$$ where we used [\[eq:bdd-fn-to-tree\]](#eq:bdd-fn-to-tree){reference-type="eqref" reference="eq:bdd-fn-to-tree"}. It follows immediately that $$\lim_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\Big\lVert f(n_1,\dots,n_k)-\textstyle\sum_{i=0}^ka_it(n_1,\dots,n_i)\Big\rVert=0\ ,$$ as required. ◻ An important tool in showing that reflexive spaces are cut stable will be the following operation on trees. **Definition 7**. Let $k,l,m\in\ensuremath{\mathbb N}$ with $k=l+m$. Let $s$ and $t$ be trees in $X$ of height $l$ and $m$, respectively. For a cut $P$ of $\{1,\dots,k\}$ of size $l$, the *$P$-intertwining of $s$ and $t$* is the unrooted tree $s\text{\textcircled z}_P t$ in $X$ of height $k$ defined by $$(s\text{\textcircled z}_P t)(n_1,\dots,n_j)=% \begin{cases} s(n_i:\,i\in P,\ i\leqslant j) & \text{if }j\in P\ ,\\ t(n_i:\,i\in P^c,\ i\leqslant j) & \text{if }j\in P^c \end{cases}$$ for $1\leqslant j\leqslant k$ and $n_1<\dots<n_j$ in $\ensuremath{\mathbb N}$. Note that if $P=\{p_1,\dots,p_l\}$ with $p_1<\dots<p_l$ and $P^c=\{q_1,\dots,q_m\}$ with $q_1<\dots<q_m$, then $$\begin{aligned} (s\text{\textcircled z}_P t)(n_1,n_2,\dots,n_{p_i}) &= s(n_{p_1},\dots,n_{p_i}) && \text{for }1\leqslant i\leqslant l\ ,\\ (s\text{\textcircled z}_P t)(n_1,n_2,\dots,n_{q_j}) &= t(n_{q_1},\dots,n_{q_j}) && \text{for }1\leqslant j\leqslant m\ .\end{aligned}$$ Given a sequence $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}=(\ensuremath{\mathcal U}_1,\dots,\ensuremath{\mathcal U}_k)$ of free ultrafilters, let $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P=(\ensuremath{\mathcal U}_{p_1},\dots,\ensuremath{\mathcal U}_{p_l})$ and $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{P^c}=(\ensuremath{\mathcal U}_{q_1},\dots,\ensuremath{\mathcal U}_{q_m})$. Note that if $s$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P$-weakly null and $t$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{P^c}$-weakly null, then $s\text{\textcircled z}_Pt$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}$-weakly null. Although the standard basis of $\ensuremath{\mathbb R}^k$ is not in general unconditional with respect to a norm in the $k^{\text{th}}$ asymptotic structure of a Banach space, we have the following result. **Proposition 12**. *Assume $X$ is reflexive. Let $k,l,m\in\ensuremath{\mathbb N}$ with $k=l+m$ and $P$ be a cut of $\{1,\dots,k\}$ of size $l$. Let $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}=(\ensuremath{\mathcal U}_1,\dots,\ensuremath{\mathcal U}_k)$ be a sequence of free ultrafilters. Let $s$ and $t$ be normalized trees in $X$ of height $l$ and $m$, respectively, such that $s$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P$-weakly null and $t$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{P^c}$-weakly null. Then $$\label{eq:project-P} N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_{s\text{\textcircled z}_P t}\Big(\textstyle\sum_{i\in P}a_ie_i\Big) \leqslant N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_{s\text{\textcircled z}_P t}\Big(\textstyle\sum_{i=1}^ka_ie_i\Big)$$ and $$\label{eq:project-PC} N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_{s\text{\textcircled z}_P t}\Big(\textstyle\sum_{i\in P^c}a_ie_i\Big) \leqslant N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_{s\text{\textcircled z}_P t}\Big(\textstyle\sum_{i=1}^ka_ie_i\Big)$$ for all $(a_i)_{i=1}^k\in\ensuremath{\mathbb R}^k$.* *Proof.* Write $P$ as $P=\{p_1,\dots,p_l\}$ with $p_1<\dots<p_l$ and $P^c$ as $P^c=\{q_1,\dots,q_m\}$ with $q_1<\dots<q_m$. Put $u=s\text{\textcircled z}_Pt$. We claim that $$\label{eq:one-step-projection} N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u \Big(\textstyle\sum_{i=1}^la_{p_i}e_{p_i} +\textstyle\sum_{j=1}^{m-1}a_{q_j}e_{q_j}\Big) \leqslant N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u \Big(\textstyle\sum_{i=1}^ka_ie_i\Big)\ .$$ for all $(a_i)_{i=1}^k\in\ensuremath{\mathbb R}^k$. Using the proof of this claim, we can repeatedly eliminate coefficients to deduce [\[eq:project-P\]](#eq:project-P){reference-type="eqref" reference="eq:project-P"}. The proof of [\[eq:project-PC\]](#eq:project-PC){reference-type="eqref" reference="eq:project-PC"} is similar. To see the claim, fix $\varepsilon>0$, $n_1<\dots<n_{q_m-1}$ in $\ensuremath{\mathbb N}$ and $L\in\ensuremath{\mathcal U}_{q_m}$ such that $$0\in\overline{\{u(n_1,n_2,\dots,n_{q_m}):\,n_{q_m}\in L\}}^w$$ and for all $n_{q_m}\in L$ and for all $(a_i)_{i=1}^k\in[-1,1]^k$, we have $$\begin{gathered} \bigg\lvert N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u\Big(\textstyle\sum_{i=1}^ka_ie_i\Big) -\\ \lim_{n_{q_m+1},\ensuremath{\mathcal U}_{q_m+1}}\dots\lim_{n_k,\ensuremath{\mathcal U}_k} \Big\lVert \sum_{i=1}^ka_i u(n_1,\dots,n_i)\Big\rVert\bigg\rvert <\varepsilon\ . \end{gathered}$$ Note that $\forall \,_{\ensuremath{\mathcal U}_1}n_1\dots\forall \,_{\ensuremath{\mathcal U}_{q_m-1}}n_{q_m-1}$ such $L\in\ensuremath{\mathcal U}_{q_m}$ exists. By Mazur's theorem there is a finite set $F\subset L$ and a convex combination $$\sum_{n\in F}\tau_n u(n_1,n_2,\dots,n_{q_m-1},n)$$ with norm $$\Big\lVert \textstyle\sum_{n\in F}\tau_n u(n_1,n_2,\dots,n_{q_m-1},n)\Big\rVert<\varepsilon\ .$$ Since $F$ is finite, there exist $\max F<n_{q_m+1}<\dots<n_k$ such that $$\bigg\lvert N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u\Big(\textstyle\sum_{i=1}^ka_ie_i\Big) -\\ \Big\lVert \sum_{i=1}^ka_i u(n_1,\dots,n_i)\Big\rVert\bigg\rvert <2\varepsilon$$ for all $n_{q_m}\in F$ and for all $(a_i)_{i=1}^k\in[-1,1]^k$. Now fix $(a_i)_{i=1}^k\in[-1,1]^k$ and observe that $$\begin{aligned} N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u \Big(\textstyle\sum_{i=1}^la_{p_i}e_{p_i} &+\textstyle\sum_{j=1}^{m-1}a_{q_j}e_{q_j}\Big)\\ &\leqslant \Big\lVert \textstyle\sum_{i=1}^{q_m-1}a_iu(n_1,\dots,n_i)+\textstyle\sum_{i=q_m+1}^ka_iu(n_1,\dots,n_i)\Big\rVert +2\varepsilon\\ &\leqslant\Big\lVert \textstyle\sum_{n_{q_m}\in F}\tau_{n_{q_m}}\textstyle\sum_{i=1}^ka_iu(n_1,\dots,n_i)\Big\rVert+3\varepsilon\\ &\leqslant N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u \Big(\textstyle\sum_{i=1}^ka_ie_i\Big)+5\varepsilon\ . \end{aligned}$$ Here it is crucial that for $i>q_m$, $u(n_1,\dots,n_i)$ does not depend on $n_{q_m}$. This completes the proof of the claim. ◻ We are now ready to prove the main result of this section. **Theorem 13**. *Every reflexive Banach space is $5$-cut stable, and thus, in particular, upper stable.* *Proof.* Fix $k,l,m\in\ensuremath{\mathbb N}$ with $k=l+m$, a sequence $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}=(\ensuremath{\mathcal U}_1,\dots,\ensuremath{\mathcal U}_k)$ of free ultrafilters and cuts $P,Q$ of $\{1,\dots,k\}$ each of size $l$. Let $X$ be an infinite-dimensional, reflexive Banach space and $f\colon{[\ensuremath{\mathbb N}]}^{l}\to X$ and $g\colon{[\ensuremath{\mathbb N}]}^{m}\to X$ be bounded functions. We will show that $$\begin{gathered} \label{eq:cut-stable-claim} \lim_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\lVert f(n_i:\,i\in P)+g(n_i:\,i\in P^c)\rVert\\ \leqslant 5 \lim_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\lVert f(n_i:\,i\in Q)+g(n_i:\,i\in Q^c)\rVert \end{gathered}$$ in the special case when the ultrafilters $\ensuremath{\mathcal U}_i$ are all the same. This will then complete the proof. Write $P=\{p_1\,\dots,p_l\}$ with $p_1<\dots<p_l$ and $P^c=\{q_1,\dots,q_m\}$ with $q_1<\dots<q_m$. By Lemma [Lemma 11](#lem:bdd-fn-as-tree){reference-type="ref" reference="lem:bdd-fn-as-tree"} we may assume that there exist normalized, rooted trees $s$ and $t$ in $X$ of height $l$ and $m$, respectively, such that $s$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P$-weakly null, $t$ is $\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{P^c}$-weakly null, and there exist non-negative scalars $a_0,a_1,\dots,a_l$ and $b_0,b_1,\dots,b_m$ such that $$f(n_1,\dots,n_l)=a_0s(\emptyset)+\sum_{i=1}^la_is(n_1,\dots,n_i) \qquad\text{for }n_1<\dots<n_l$$ and $$g(n_1,\dots,n_m)=b_0t(\emptyset)+\sum_{j=1}^mb_jt(n_1,\dots,n_j) \qquad\text{for }n_1<\dots<n_m\ .$$ Let $u=s\text{\textcircled z}_Pt$. Set $c_{p_i}=a_i$ for $1\leqslant i\leqslant l$ and $c_{q_j}=b_j$ for $1\leqslant j\leqslant m$. We shall frequently use the fact that $$N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u\Big(\textstyle\sum_{i\in P}c_ie_i\Big)=N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P}_s\Big(\textstyle\sum_{i=1}^la_ie_i\Big)\ .$$ Now observe that for $\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}}\in{[\ensuremath{\mathbb N}]}^{k}$ we have $$f(n_i:\,i\in P)+g(n_i:\,i\in P^c)=a_0s(\emptyset)+b_0t(\emptyset)+\sum_{i=1}^k c_i u(n_1,\dots,n_i)\ .$$ It follows that $$\label{eq:sum-as-intertwined-limit} \lim_{\ensuremath{\accentset{\rule{.4em}{.6pt}}{n}},\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}\lVert f(n_i:\,i\in P)+g(n_i:\,i\in P^c)\rVert=N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u \Big( \big(a_0s(\emptyset)+b_0t(\emptyset)\big)\oplus \textstyle\sum_{i=1}^k c_ie_i\Big)\ .$$ By the triangle inequality, we have $$\begin{gathered} \label{eq:upper-bound} N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u \Big( \big(a_0s(\emptyset)+b_0t(\emptyset)\big)\oplus \textstyle\sum_{i=1}^k c_ie_i\Big)\\ \leqslant\lVert a_0s(\emptyset)+b_0t(\emptyset)\rVert + N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P}_s\Big(\textstyle\sum_{i=1}^la_ie_i\Big) + N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{P^c}}_t\Big(\textstyle\sum_{j=1}^mb_je_j\Big) \end{gathered}$$ On the other hand, it follows from Lemma [Lemma 10](#lem:asymptotic-norm){reference-type="ref" reference="lem:asymptotic-norm"} and Proposition [Proposition 12](#prop:intertwining-and-unconditionality){reference-type="ref" reference="prop:intertwining-and-unconditionality"} that $$\begin{gathered} \label{eq:lower-bound} \lVert a_0s(\emptyset)+b_0t(\emptyset)\rVert + N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P}_s\Big(\textstyle\sum_{i=1}^la_ie_i\Big) + N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{P^c}}_t\Big(\textstyle\sum_{j=1}^mb_je_j\Big) \\ \leqslant 5\cdot N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}}_u \Big(\big(a_0s(\emptyset)+b_0t(\emptyset)\big)\oplus \textstyle\sum_{i=1}^k c_ie_i\Big)\ . \end{gathered}$$ When the ultrafilters $\ensuremath{\mathcal U}_i$ are all identical, then $$N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_P}_s\Big(\textstyle\sum_{i=1}^la_ie_i\Big) = N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_Q}_s\Big(\textstyle\sum_{i=1}^la_ie_i\Big)$$ and $$N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{P^c}}_t\Big(\textstyle\sum_{j=1}^mb_je_j\Big) = N^{\ensuremath{\accentset{\rule{.6em}{.6pt}}{\mathcal U}}_{Q^c}}_t\Big(\textstyle\sum_{j=1}^mb_je_j\Big)\ .$$ Hence [\[eq:cut-stable-claim\]](#eq:cut-stable-claim){reference-type="eqref" reference="eq:cut-stable-claim"} follows immediately from [\[eq:sum-as-intertwined-limit\]](#eq:sum-as-intertwined-limit){reference-type="eqref" reference="eq:sum-as-intertwined-limit"}, [\[eq:upper-bound\]](#eq:upper-bound){reference-type="eqref" reference="eq:upper-bound"} and [\[eq:lower-bound\]](#eq:lower-bound){reference-type="eqref" reference="eq:lower-bound"} applied to $u=s\text{\textcircled z}_Pt$ or $u=s\text{\textcircled z}_Qt$ as appropriate. ◻

arxiv_math

--- abstract: | Markov chain Monte Carlo (MCMC) algorithms are based on the construction of a Markov Chain with transition probabilities $P_\mu(x,\cdot)$, where $\mu$ indicates an invariant distribution of interest. In this work, we look at these transition probabilities as functions of their invariant distributions, and we develop a notion of *derivative in the invariant distribution* of a MCMC kernel. We build around this concept a set of tools that we refer to as *Markov Chain Monte Carlo Calculus*. This allows us to compare Markov chains with different invariant distributions within a suitable class via what we refer to as mean value inequalities. We explain how MCMC Calculus provides a natural framework to study algorithms using an approximation of an invariant distribution, also illustrating how it suggests practical guidelines for MCMC algorithms efficiency. We conclude this work by showing how the tools developed can be applied to prove convergence of interacting and sequential MCMC algorithms, which arise in the context of particle filtering. author: - Rocco Caprio, Adam M Johansen bibliography: - bib.bib date: | Department of Statistics, University of Warwick\ title: | A calculus for Markov chain Monte Carlo:\ studying approximations in algorithms --- # Introduction ## Notation It it convenient to begin by introducing some notation that will be used throughout. $(\Omega,\mathcal{F},\mathbb{P})$ is some underlying probability space. $(\mathsf{X}\xspace,\mathcal{B}(\mathsf{X}\xspace))$ is a Polish space equipped with metric $\varrho$ for which we assume that $x\mapsto \varrho(x,y)$ is continuously differentiable for all $y\neq x$, and endowed with the associated Borel $\sigma$-field. $B(x,r)$ denotes a ball in $\mathsf{X}\xspace$ centered at $x$ and having radius $r>0$. We will slightly abuse notation and use $1$ to denote the unit function on any space. $\mathcal{M}(\mathsf{X}\xspace)$ denotes the set of finite signed measures on $\mathsf{X}\xspace$. $\mathcal{M}_{\lambda,0}(\mathsf{X}\xspace)\subset\mathcal{M}(\mathsf{X}\xspace)$ denotes the set of $0$--mass signed measures with continuous bounded Radon-Nikodým derivatives w.r.t. a reference $\sigma$-finite measure $\lambda$. $\mathcal{P}(\mathsf{X}\xspace)\subset\mathcal{M}(\mathsf{X}\xspace)$ denotes the set of probability measures on $\mathsf{X}\xspace$, and similarly $\mathcal{P}_\lambda(\mathsf{X}\xspace)\subset \mathcal{P}(\mathsf{X}\xspace)$ denotes the set of probability measures with continuous bounded densities w.r.t. $\lambda$. $\Rightarrow$ denotes weak convergence. For $\chi\in\mathcal{M}(\mathsf{X}\xspace)$ and a measurable function $f$, we often write $\chi(f):=\int f(x)\chi(dx)$. When $\chi\in\mathcal{M}_\lambda(\mathsf{X}\xspace)$ for some $\lambda$ which is implied by the context, we often abuse notation and just write $\chi(x)$ for the value of $d\chi/d\lambda$ at $x\in\mathsf{X}\xspace$. For $V:\mathsf{X}\xspace\rightarrow [1,\infty)$, we define the $V$-norm of a measurable function $f:\mathsf{X}\xspace\mapsto \mathbb{R}$ as $$\left\vert f\right\vert_{V}:=\sup_{x\in \mathsf{X}\xspace} \frac{\vert f(x) \vert}{V(x)}.$$ $\mathbf{B}_V(\mathsf{X}\xspace)$ denotes the space of measurable functions with $V$-norm lower or equal than 1; $\mathbf{B}_1(\mathsf{X}\xspace)$ that of measurable functions with supremum norm no greater than 1. The $V$-norm of a signed measure $\chi\in\mathcal{M}(\mathsf{X}\xspace)$ is instead defined by $$\norm{\chi}_{V}:=\sup_{f \in \mathbf{B}_V(\mathsf{X}\xspace)}\left\vert \int fd\chi\right\vert.$$ When $V=1$, the $V$-norm of a function and of a signed measure are just the supremum norm $\left\vert\cdot\right\vert_{\infty}$ and the total variation norm $\norm{\cdot}_{\rm tv}$, respectively. $\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ is the space of continuously differentiable functions with $V$-norm lower or equal than 1. We endow $\mathcal{P}(\mathsf{X}\xspace)$ with the $\sigma$-algebra generated by $\mathbf{B}_1(\mathsf{X}\xspace)$. A Markov kernel on $\mathsf{X}\xspace\times \mathcal{B}(\mathsf{X}\xspace)$ is a function $P:\mathsf{X}\xspace\times\mathcal{B}(\mathsf{X}\xspace)\mapsto [0,1]$ such that (i) for each $A\in\mathcal{B}(\mathsf{X}\xspace)$, $P(\cdot,A)$ is a non-negative measurable function on $\mathsf{X}\xspace$ and (ii) for each $x\in\mathsf{X}\xspace$, $P(x,\cdot)\in\mathcal{P}(\mathsf{X}\xspace)$. $P$ acts on bounded measurable functions via $P(x,f):=Pf(x):=\int f(y)P(x,dy)$, and on probability measures via $\rho P(dx):=P(\rho,dx):=\int \rho(dy)P(y,dx)\in\mathcal{P}(\mathsf{X}\xspace)$. Whenever a probability measure $\mu$ on $\mathsf{X}\xspace$ satisfies $\mu(dx)=P(\mu,dx)$, we say that $\mu$ is an invariant distribution for $P$. A Markov kernel having an invariant distribution $\mu$ will be denoted $P_\mu$ whenever the emphasis is important. ## Markov chain Monte Carlo with approximations and calculus Markov chain Monte Carlo (MCMC) algorithms are based on the construction of a Markov Chain with transition probabilities $P_\mu(x,\cdot)$, where $\mu$ indicates an invariant density of interest one can evaluate pointwise (up to a possibly unknown normalizing constant). It often happens, however, that $\mu$ is intractable, or just very hard to target directly. In these situations, possible and popular remedies include simply using an approximation $\mu_n$ of $\mu$ and hence simulating a homogeneous Markov Chain with transitions $P_{\mu_n}$ , or considering an evolving sequence $\{\mu_n\}_{n\in\mathbb{N}}$ converging to $\mu$ in some sense, and simulating an inhomogeneous Markov Chain $\{P_{\mu_n}\}_{n\in\mathbb{N}}$ perhaps in a tempering-like context. In these situations, one usually has control over how good the approximation is, and it might be of interest to ensure that the "approximate" chain $P_{\mu_n}$ moves similarly to the "limiting" chain $P_\mu$, and perhaps to quantify how the "degree of similarity" depends on the quality of our approximation. Roughly speaking, the ultimate goal of this paper is to look at Markov kernels as functions of their invariant distribution, and to be able to say that $P_{\mu_n}$ and $P_\mu$ are similar whenever we can say both that $\mu$ and $\mu_n$ are similar and that the derivative in the invariant distribution of $P_\cdot$ at $\mu$ towards $\mu_n$ is bounded. In this paper, we shall make sense of this sentence, first by introducing a notion of a family of Markov kernels, where one has a mapping from the invariant distribution to the Markov kernel ([2.1](#sec:markovfamilies){reference-type="ref" reference="sec:markovfamilies"}), and then of derivative in the invariant distribution and of boundedness thereof ([\[sec:derivative,sec:ftc\]](#sec:derivative,sec:ftc){reference-type="ref" reference="sec:derivative,sec:ftc"}). We shall then provide a "Fundamental theorem of MCMC Calculus'' to compare Markov chains with different invariant distribution within the same family. When $\mu,\nu$ have positive densities, this result will allow us to easily derive *mean-value* type inequalities $$\norm{P_\mu(\rho,\cdot)-P_\nu(\rho,\cdot)}_{\rm tv} \leq \norm{\mu-\nu}_{\rm tv}\int_0^1 D_{\rho,t}dt$$ when $\rho$ also has a positive density, and when $\rho=\delta_x$, $$\norm{P_\mu(x,\cdot)-P_\nu(x,\cdot)}_{\rm tv} \leq \norm{\mu-\nu}_{\rm tv} \int_0^1 D_{1,t,x}dt + |\mu(x)-\nu(x)|\int_0^1 D_{2,t,x}dt,$$ for Hastings type and Gibbs kernels, with explicit values for the "Lipschitz constants" $\int_0^1 D_{\rho,t}dt$, $\int_0^1 D_{1,t,x}dt$ and $\int_0^1 D_{2,t,x}dt$ ([\[prop:MHboundedder,prop:Gibbsboundedder\]](#prop:MHboundedder,prop:Gibbsboundedder){reference-type="ref" reference="prop:MHboundedder,prop:Gibbsboundedder"}). When $\{\mu_n\}_{n\in\mathbb{N}}$ is a sequence of random measures and which has limiting Gaussian fluctuations around $\mu$, in a sense we shall make more precise later, we will see that provided the Markov kernel $P_\cdot$ is differentiable in the invariant distribution, then $P_{\mu_n}$ will have also have limiting Gaussian fluctuations around $P_\mu$, with the magnitude of the fluctuations depending explicitly on its derivative ([Proposition 10](#prop:weaklaw){reference-type="ref" reference="prop:weaklaw"}). These results are immediately useful when one is using an approximate Chain $P_{\mu_n}$ and wants it to move like $P_{\mu}$: it suffices to pick a Markov kernel with finite Lipschitz constants (which will mean that the kernel has a bounded derivative in the invariant distribution) or to minimize fluctuations around $P_\mu$. In [3](#sec:applications){reference-type="ref" reference="sec:applications"} we use our calculus approach to study the efficiency of approximation based MCMC algorithms like Sequential MCMC or Interacting MCMC in terms of their asymptotic variance. Our approach allow for a very natural study of these, allowing us to state conditions for convergence simply in terms of good enough approximation quality, boundedness of the derivative, and enough uniformity of ergodic constants along the approximation sequence, and we improve some existent convergence results in some cases. # Markov Chain Monte Carlo Calculus {#sec:mcmccalc} ## Families of Markov kernels {#sec:markovfamilies} To compare Markov kernels with different invariant distributions, our main tool will be the derivative in the invariant distribution of a Markov kernel, which we introduce in the next section. This quantity will describe the variation of the transition probabilities of a Markov Chain when $\mu$ is perturbed. Certainly, there are many possible transition probabilities associated with each invariant distribution, so to accomplish this we first have to restrict the class of Markov kernels considered. We introduce the notion of *family of Markov kernels*---which is a collection of Markov kernels indexed by a convex subset of invariant distributions: **Definition 1** (Family of Markov kernels). *Let $\mathcal{I}$ be a convex subset of $\mathcal{P}(\mathsf{X}\xspace)$. A family of Markov kernels $\{P_\star\}$ is a set $\{P_\mu:\mathsf{X}\xspace\times \mathcal{B}(\mathsf{X}\xspace)\mapsto[0,1] \vert \mu\in\mathcal{I} \}$, where for all $\mu \in \mathcal{I}$ one has $\mu P_\mu = \mu$.* According to our definition, if for some $\mu,\nu\in\mathcal{P}(\mathsf{X}\xspace)$, $P_\mu, P_\nu \in\{P_\star\}$, then defining the curve $\mu_t:=(1-t)\mu + t\nu$, as the map $t\in[0,1]\mapsto \mu_t \in \mathcal{P}(\mathsf{X}\xspace)$, interpolating $\mu$ and $\nu$, we have that $P_{\mu_t}\in \{ P_\star\}$ for any $t\in[0,1]$. $\mu_t$ and $P_{\mu_t}$ will play an important role. The following two examples describe some Markov kernel families. **Example 1** (The Hastings family). Let $\mu\in\mathcal{P}(\mathsf{X}\xspace)$ be a probability measure dominated by some measure $\lambda$ in that $\mu \ll \lambda$, and write $\mu(y)$ for $d\mu/d\lambda(y)$. Typically, $\lambda$ will be the Lebesgue measure when $\mathsf{X}\xspace=\mathbb{R}^d$ and the counting measure when $\mathsf{X}\xspace=\mathbb{Z}$. Let $Q$ be a *proposal* Markov kernel such that $Q(x,dy)=q(x,y)\lambda(dy)$ for all $x\in\mathsf{X}\xspace$ and let $g$ be a *balancing function* satisfying $g(x)=xg(1/x)$, $g(x)\leq 1$. Consider the Markov kernels of the form $$\label{eq:MHtypekernels} P_\mu(x,f) := \int f(y)Q(x,dy)g(r_\mu(x,y)) + f(x)\left(1-\int Q(x,dy)g(r_\mu(x,y))\right)$$ where $$r_\mu(x,y) := \begin{cases} \frac{\mu(y)q(y,x)}{\mu(x)q(x,y)} \quad &\text{if} \quad \mu(x)q(x,y)>0, \\ 1 \quad &\text{if} \quad \mu(x)q(x,y)=0. \end{cases}$$ These kernels have $\mu$ as invariant distribution and are indexed by the convex set of all probability distributions $\mu$ with a density w.r.t. $\lambda$ on $\mathsf{X}\xspace$, or any convex subset thereof. That is, they form proper Markov families as per [Definition 1](#def:markovfamily){reference-type="ref" reference="def:markovfamily"}. For generic $Q$ and $g$, we refer to the *Hastings family*, as [@hastings1970mcmc] introduced kernels of this general form. When, for instance, $Q$ is a Gaussian Random Walk (RW) and $g(x)=\min(1,x)$ one obtain the *RW Metropolis family* [@metropolis1953mcmc] over the same set of invariant distributions. If $Q$ is a particular Gaussian RW and $g(x)=x/(1+x)$, they form the *RW Barker family* [@barker1965mcmc] etc. In general, our framework imposes no requirement that $Q$ be fixed across the family as described in this simple example and in general it can depend upon $\mu$. **Example 2** (The Gibbs family). Suppose $\mu$ is the probability distribution of a random vector $(U_1,U_2)$. Let $P_{1|2}$ and $P_{2|1}$ two Markov kernels on $\mathsf{X}\xspace\times\mathcal{B}(\mathsf{X}\xspace)$ representing the conditional distributions of $U_1$ given $U_2$ and $U_2$ given $U_1$, respectively. The Markov kernels on $\mathsf{X}\xspace^2\times\mathcal{B}(\mathsf{X}\xspace^2)$ given by $$\label{eq:DSGibbskernel} P_\mu((x,x'),f):=\int P_{1|2}(x',dy)P_{2|1}(y,dy')f(y,y')$$ are $\mu$-invariant. Kernels of the type [\[eq:DSGibbskernel\]](#eq:DSGibbskernel){reference-type="eqref" reference="eq:DSGibbskernel"} form a proper Markov family---the (two-stage) deterministic scan Gibbs family. We considered here the two-stage Gibbs family purely for ease for exposition and simplicity of notation; corresponding results can be obtained for much more general Gibbs samplers. Considering other Markov kernel families is possible. For instance, if in [Example 1](#ex:MHtypekernels){reference-type="ref" reference="ex:MHtypekernels"} we also allow $Q$ to depend on the gradient of $\mu$ as in the Metropolis-Adjusted Langevin Algorithm (MALA; [@roberts1996MALA]), one could obtain the MALA family (over the probability distributions with differentiable densities). If $P_{1|2}$, $P_{2|1}$ in [Example 2](#ex:Gibbsfamily){reference-type="ref" reference="ex:Gibbsfamily"} instead of being conditional distributions are Markov kernels invariant for the same, one could obtain a Metropolis-within-Gibbs family etc. It is also possible to consider a Markov family that is using a given type of Markov kernel if the invariant distribution belong to a certain set (e.g. MALA if the invariant density is differentiable), and another Markov kernel otherwise (e.g. RW Metropolis if it is not). For Hastings-type and Gibbs-type algorithms of [\[ex:MHtypekernels,ex:Gibbsfamily\]](#ex:MHtypekernels,ex:Gibbsfamily){reference-type="ref" reference="ex:MHtypekernels,ex:Gibbsfamily"} above, checking that they form a Markov family in the sense of [Definition 1](#def:markovfamily){reference-type="ref" reference="def:markovfamily"} is easily done because the dependence on the invariant distribution is somehow explicit. In principal one could define Markov families in which the dependence of kernels on the invariant distribution is not made so explicit, but checking [Definition 1](#def:markovfamily){reference-type="ref" reference="def:markovfamily"} would be harder. ## The derivative in the invariant distribution {#sec:derivative} Let $\{P_\star\}$ be a Markov family indexed by the convex set of probability measures admitting densities w.r.t. some $\sigma$-finite measure $\lambda$, or any convex subset thereof. We will define the derivative with respect the invariant distribution of $P_\cdot$ at a point $\mu$ indexed by the family by first accurately defining the derivative of the functional $P_\cdot(\rho,f):\mu\in\mathcal{P}(\mathsf{X}\xspace)\mapsto P_\mu(\rho,f)\in\mathbb{R}$, for all $(\rho,f)$ in some suitable class. The derivative of such functional will first be defined for all $\rho$ in a subset of $\mathcal{P}_\lambda(\mathsf{X}\xspace)$, and then defined for initial distributions in its completion by extension. Throughout this section, we fix a continuously differentiable function $V:\mathsf{X}\xspace\mapsto [1,\infty)$, and consider $f \in \mathbf{C}^1_{V}(\mathsf{X}\xspace)$. This function directly influences how stringent the differentiability statement is, but also what we are able to say with the objects we introduce next, and its role will become clear when we introduce mean value inequalities later. **Definition 2**. *For $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$, the derivative of $P_\cdot(\rho,f)$ in the invariant distribution at $\mu\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ is the linear functional $\partial_\pi P_{\mu}(\rho,f)[\cdot]:\mathcal{M}_{\lambda,0}(\mathsf{X}\xspace)\mapsto \mathbb{R}$ such that for all $\nu\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$, $$\label{eq:derivative} \frac{d}{dt} P_{\mu+t(\nu-\mu)}(\rho,f)\biggr|_{t=0} = \partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$$* We denote with $\mathcal{D}_{\mu,\lambda}(P,f)$ the set of $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ for which $P_\cdot(\rho,f)$ is differentiable at $\mu$. In all our examples, for $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$, the action of the functional $\partial_\pi P_{\mu}(\rho,f)$ will be expressible in integral form. In particular, for a measurable function abusively denoted as $\partial_\pi P_{\mu}(\rho,f)(\cdot):\mathsf{X}\xspace\mapsto \mathbb{R}$ we will have $$\begin{aligned} \partial_\pi P_{\mu}(\rho,f)[\nu-\mu] = \int (\nu(y)-\mu(y))\partial_\pi P_{\mu}(\rho,f)(y)\lambda(dy) \quad \text{and} \label{eq:derivativefirstvariation} \\ \mu(\partial_\pi P_{\mu}(\rho,f)(\cdot))=0. \label{eq:derivativecentering}\end{aligned}$$ We sometimes distinguish the functional $\partial_\pi P_{\mu}(\rho,f)[\cdot]$ from the function of its integral representation only via the brackets that follow. The context will always make obvious to which object we are referring to. The function $\partial_\pi P_{\mu}(\rho,f)(\cdot)$ will be referred to as density of the derivative in the invariant distribution. The density of $\partial_\pi P_{\mu}(\rho,f)$ corresponds to the *first variation* of $P_\cdot(\rho,f)$, and its analogues appears for instance in [@cardaliaguet2019mastereqn] in the context of Mean Field Games and in Optimal Transport in [@santambrogio2015ot]. Under some conditions on $\mathsf{X}\xspace$ and $\mathcal{P}(\mathsf{X}\xspace)$, these can be related to notions of derivatives in the Wasserstein geometry [@ambrosio2005gradientflows], see also [Theorem 2](#prop:FTC){reference-type="ref" reference="prop:FTC"}. Here, [\[eq:derivativecentering\]](#eq:derivativecentering){reference-type="eqref" reference="eq:derivativecentering"} is a centering condition, and it guarantees that the integral representation is unique. Without, it would be unique only up to constants.\ We defined the derivative in general as a linear functional $\partial_\pi P_{\mu}(\rho,f)[\cdot]$ despite in practice it admits an integral representation for $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ because this allows us an immediate extension to general $\rho\in\mathcal{P}(\mathsf{X}\xspace)$ for which such representation will not in general be available. Let $d_p$ be a distance function metrizing the weak topology on $\mathcal{P}(\mathsf{X}\xspace)$. $d_p$ could be the Lévy-Prokhorov or the Bounded Lipschitz metric [@dudley2002analysis Chapter 11]. If the functional $\rho\mapsto \partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$ is uniformly continuous w.r.t. $d_p$ on some subset $\mathcal{A}$ of $\mathcal{D}_{\mu,\lambda}(P,f)$, we can uniquely extend $\partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$ on the weak completion $\Tilde{\mathcal{A}}$ of $\mathcal{A}$ in a very natural way via continuity. **Proposition 1**. *If $\rho\mapsto \partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$ is uniformly continuous w.r.t. $d_p$ on some subset $\mathcal{A}$ of $\mathcal{P}_\lambda(\mathsf{X}\xspace)$, we can uniquely extend it on its weak completion $\Tilde{\mathcal{A}}$ by defining $$\partial_\pi P_{\mu}(\rho,f)[\nu-\mu]:=\lim_{j\rightarrow\infty}\partial_\pi P_{\mu}(\rho_j,f)[\nu-\mu], \quad \text{ where } \{\rho_j\}_{j\in\mathbb{N}}\in\mathcal{A}\Rightarrow \rho\in\Tilde{\mathcal{A}}\setminus \mathcal{A}.$$ Furthermore, the extension is uniformly continuous w.r.t. $d_p$ on $\Tilde{\mathcal{A}}$.* *Proof.* See e.g. [@kaplansky1972settheorymetric Theorem 52]. ◻ We use this proposition to define the derivative of $P_\cdot(\rho,f)$ for initial distributions that are not in $\mathcal{P}_\lambda(\mathsf{X}\xspace)$, but on which we have sufficient regularity to allow an extension. Let $\Tilde{\mathcal{A}}$ be the weak completion of a subset $\mathcal{A}$ of $\mathcal{D}_{\mu,\lambda}(P,f)$ for which $\rho\mapsto \partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$ is uniformly continuous w.r.t. $d_p$ for all $\nu\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$. **Definition 3**. *For $\rho\in\Tilde{\mathcal{A}}\setminus \mathcal{A}$, the derivative of $P_\cdot(\rho,f)$ in the invariant distribution at $\mu$ is the linear functional $\partial_\pi P_{\mu}(\rho,f)[\cdot]:\mathcal{M}_{\lambda,0}(\mathsf{X}\xspace)\mapsto \mathbb{R}$ defined by $$\label{eq:derivativex} \partial_\pi P_{\mu}(\rho,f)[\nu-\mu]:=\lim_{j\rightarrow\infty}\partial_\pi P_{\mu}(\rho_{j},f)[\nu-\mu], \quad \text{where } \{\rho_j\}_{j\in\mathbb{N}}\in\mathcal{A}\Rightarrow \rho\in\Tilde{\mathcal{A}}\setminus \mathcal{A}.$$ for all $\nu\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$, where $\partial_\pi P_{\mu}(\rho_{j},f)$ is the derivative in the invariant distribution of $P_\cdot(\rho_j,f)$ as per [Definition 2](#def:derivative){reference-type="ref" reference="def:derivative"}.* The case we are mostly interested is defining $\partial_\pi P_{\mu}(\delta_x,f)=:\partial_\pi P_{\mu}(x,f)$, the derivative of $P_\cdot(x,f)$ in the invariant distribution at $\mu$. The results above supply its definition and its uniqueness whenever $\delta_x$ is in the weak completion of a subset of $\mathcal{P}_\lambda(\mathsf{X}\xspace)$ where $\rho\mapsto \partial_\pi P_{\mu}(\rho,f)$ is uniformly continuous. We denote with $\mathcal{D}_{\mu}(P,f)$ the set of initial distributions $\rho$ for which $P_\cdot(\rho,f)$ is differentiable at $\mu$. This comprise all initial distributions for which we may also obtain a definition by extension, and it has $\mathcal{D}_{\mu,\lambda}(P,f)$ as subset. It might happen that $\rho$ is in the weak completion of two different sets $\mathcal{A}$ and $\mathcal{A}'$ on which the derivative is uniformly continuous. In this case the definition of $\partial_\pi P_{\mu}(\rho,f)$ can be defined by either the limit of $\lim_{j\rightarrow\infty}\Pder{\mu}{\rho_{j}}$ with $\rho_j$ in $\mathcal{A}$ or the limit $\lim_{j\rightarrow\infty}\Pder{\mu}{\rho_{j}'}$ with $\rho'_j$ in $\mathcal{A}'$. We have not sought to establish that these limits coincide, but that is no cause of concern for our purposes. Any such derivative will satisfy the properties and equations we investigate next. We can now leverage these quantities to define the derivative of transition probabilities. **Definition 4**. *The derivative of $P_\cdot(\rho,\cdot)$ in the invariant distribution at $\mu$ is the operator $\partial_\pi P_{\mu}(\rho,\cdot)[\cdot]:f\in \mathbf{C}^1_{V}(\mathsf{X}\xspace)\mapsto \partial_\pi P_{\mu}(\rho,f)[\cdot]$, where $\partial_\pi P_{\mu}(\rho,f)[\cdot]$ is defined in [\[def:derivative,def:derivativex\]](#def:derivative,def:derivativex){reference-type="ref" reference="def:derivative,def:derivativex"}.* The differentiability of $P_\cdot(\rho,\cdot)$ amounts to ask that $P_\cdot(\rho,f)$ is differentiabile in the sense of [\[def:derivative,def:derivativex\]](#def:derivative,def:derivativex){reference-type="ref" reference="def:derivative,def:derivativex"} for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$. Hence, $\mathcal{D}_{\mu}(P)=\bigcap_{f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)}\mathcal{D}_{\mu}(P,f)$. We are now in a position to naturally define the derivative of a Markov kernel itself. **Definition 5**. *The derivative of $P_\cdot$ in the invariant distribution at $\mu$ is the operator $\partial_\pi P_{\mu}(\cdot,\cdot)[\cdot]:(\rho,f)\in \mathcal{D}_{\mu}(P)\times\mathbf{C}^1_{V}(\mathsf{X}\xspace)\mapsto \partial_\pi P_{\mu}(\rho,f)[\cdot]$, where $\partial_\pi P_{\mu}(\rho,f)[\cdot]$ is defined in [\[def:derivative,def:derivativex\]](#def:derivative,def:derivativex){reference-type="ref" reference="def:derivative,def:derivativex"}.* We will sometimes refer to the differential and the derivative in the invariant measure of a Markov Kernel simply as "kernel differentials" and "kernel derivative", respectively. Since that is defined indirectly for each point $(\rho,f)\in\mathcal{D}_{\mu}(P)\times\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, we sometimes also refer to $\partial_\pi P_{\mu}(\rho,f)$ as a kernel derivative. Let us come to the interpretation of these derivatives and differentials at $\mu$: they provide a map from the initial distribution to signed integral operators which capture the variation of the transitions associated with Markov kernels within a particularly family with small perturbations of the associated invariant measure, $\mu$. In other words, the derivative in the invariant measure quantifies how much the conditional expectation $P_{\mu}(\rho,\cdot)$ changes when the invariant distribution $\mu$ is perturbed by the $0$-mass measure $\chi=\nu-\mu$ or---alternatively---since $\mu+t\chi=(1-t)\mu + t\nu$, when $\mu$ is contaminated infinitesimally by any $\nu$. In a spatial perspective, it quantifies the variation in mean displacement of a particle in Markovian motion $P_\mu$ initially distributed as $\rho$ when $\mu$ is perturbed by $\nu$. The density of the kernel derivative $\partial_\pi P_{\mu}(\rho,f)(y)$---when it exists---can be heuristically interpreted as the change in the transition probabilities when $\mu$ is contaminated infinitesimally by a point mass $\delta_y$ for all $y\in\mathsf{X}\xspace$. In fact, using [\[eq:derivative\]](#eq:derivative){reference-type="eqref" reference="eq:derivative"}-[\[eq:derivativecentering\]](#eq:derivativecentering){reference-type="eqref" reference="eq:derivativecentering"} we informally see that $$\partial_\pi P_{\mu}(\rho,f)(y)=\partial_\pi P_{\mu}(\rho,f)[\delta_y-\mu] = \int \partial_\pi P_{\mu}(\rho,f)(y')d(\delta_y-\mu)(y') = \frac{d}{dt} P_{\mu+t(\delta_y-\mu)}(\rho,f)\bigg|_{t=0}.$$ This is similar to the role played by Influence functions in robust statistics, which describe variations in estimators when the data distribution is contaminated - see [@huber2009robust]. In the remainder of this section we develop some basic properties to build up our intuition and understanding of kernel derivatives. In the end of the section we investigate the differentiability of Hastings and Gibbs families. The next proposition supplies a recursive formula for the derivative of iterated kernels. These recursions are really expression of the Chapman--Kolmogorov equations. Consider the conventions $P^0_\mu(\cdot,f):=f$ and $P^0_\mu(\rho,\cdot):=\rho$. **Proposition 2**. *Let $k\in\mathbb{N}$ and assume that $P^{j}_\mu(\rho,\cdot)\in\mathcal{D}_{\mu,\lambda}(P,f)$ and that $P^j_\mu(\cdot,f)\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ for all $j\leq k$. For all $k\in\mathbb{N}$ the derivative in the invariant distribution of an iterated Markov kernel satisfy the formulæ $$\partial_\pi P_{\mu}^{k}(\rho,f) = \sum_{j=0}^{k-1} \partial_\pi P_{\mu}(P^{k-j-1}_\mu(\rho,\cdot),P^j_\mu(\cdot,f)).$$ provided one can differentiate inside the integral $P^k_{\mu+t(\nu-\mu)}(\rho,f)$ at $t=0$.* *Proof.* Let $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ and $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, we have $$\begin{aligned} \frac{d}{dt}P^k_{\mu+t(\nu-\mu)}(\rho,f)\bigg|_{t=0} &= \frac{d}{dt}\int P_{\mu+t(\nu-\mu)}(\rho,dz)P^{k-1}_{\mu_t}(z,f)\bigg|_{t=0} \\ &= \frac{d}{dt} P_{\mu+t(\nu-\mu)}(\rho,P^{k-1}_{\mu}(\cdot,f))\bigg|_{t=0} + \frac{d}{dt} P_{\mu+t(\nu-\mu)}^{k-1}(P_\mu(\rho,\cdot),f)\bigg|_{t=0} \\ &= \partial_\pi P_{\mu}(\rho,P^{k-1}_{\mu}(\cdot,f))[\nu-\mu] + \partial_\pi P_{\mu}^{k-1}(P_\mu(\rho,\cdot),f)[\nu-\mu] \end{aligned}$$ therefore, exploiting this recursion yields, $$\begin{aligned} \partial_\pi P_{\mu}^{k}(\rho,f) &= \partial_\pi P_{\mu}(\rho,P^{k-1}_{\mu}(\cdot,f)) + \partial_\pi P_{\mu}^{k-1}(P_\mu(\rho,\cdot),f) = \sum_{j=0}^{k-1} \partial_\pi P_{\mu}(P^{k-j-1}_\mu(\rho,\cdot),P^j_\mu(\cdot,f)). \end{aligned}$$ ◻ Ergodicity can give us some information on how the derivative of iterated kernels behave when $k\rightarrow \infty$. A kernel derivative quantifies variations in transition probabilities $P^k_\mu(\rho,f)$ when the invariant changes is perturbed infinitesimally by $\chi:=\nu-\mu$. Under ergodicity for large $k$ we will have $P^k_\mu(\rho,f)\approx \mu(f)$, hence, intuitively, in the limit the kernel derivative ought to be $\chi(f)$. Define $$\mathcal{N}(\mu,\nu):=\{\mu_t\in\mathcal{P}(\mathsf{X}\xspace): \mu_t=(1-t)\mu+t\nu, t\in[0,1]\}.$$ This set is known as *contamination neighbourhood* in robust statistics [@huber2009robust], although it is not a neighbourhood in the sense of the total variation or the weak topology. For some starting distribution $\rho$, say that a Markov kernel is ergodic uniformly in $\mathcal{N}(\mu,\nu)$ if $\sup_{t\in[0,1]}|P^k_{\mu_t}(\rho,f)-\mu_t(f)|\rightarrow 0$ as $k\to\infty$. We note that this is a definition of ergodicity uniform over a collection of invariant distributions and is quite distinct from the notion of uniform ergodicity of an individual Markov kernel. **Proposition 3**. *For all starting distributions $\rho\in\mathcal{D}_{\mu,\lambda}(P,f)$ for which the Markov kernel $P$ is ergodic uniformly in $\mathcal{N}(\mu,\nu)$, $$\lim_{k\rightarrow\infty} \partial_\pi P_{\mu}^{k}(\rho,f)[\nu-\mu] = [\nu-\mu](f).$$* *Proof.* We compute directly $$\begin{aligned} \lim_{k\rightarrow\infty} \partial_\pi P_{\mu}^{k}(\rho,f)[\nu-\mu] &= \lim_{k\rightarrow\infty} \lim_{t\rightarrow0} \frac{P^k_{\mu+t(\nu-\mu)}(\rho,f)-P_\mu^k(\rho,f)}{t} \\ &= \lim_{t\rightarrow0} \lim_{k\rightarrow\infty} \frac{P^k_{\mu+t(\nu-\mu)}(\rho,f)-P_\mu^k(\rho,f)}{t} \\ &= \lim_{t\rightarrow0} \frac{(\mu+t(\nu-\mu))(f)-\mu(f)}{t} = [\nu-\mu](f), \end{aligned}$$ where the limit exchange if justified by the uniformity across the neighbourhood. ◻ Hence, informally, its density in the limit is given by $\lim_k \partial_\pi P_{\mu}^{k}(\rho,f)(y)=f(y)-\mu(f)$. It is interesting to point out another connection between derivatives in the invariant distributions and ergodicity. $\partial_\pi P_{\mu}(\mu,f)$ is understood as the infinitesimal variation of the transition probability $P_\mu(X,f)$ when the invariant measure is perturbed with a 0--mass measure, the starting point $X$ follows the (non-perturbed) original distribution $\mu$. The density of the derivative takes a very natural expression in this case that does not require further computations. Denote $\mu_t := (1-t)\mu + t\nu$. Write $$P_{\mu_t}(\mu_t,f) = \mu_t(f).$$ Differentiating this expression w.r.t. $t$ in $t=0$ one obtains, provided $\frac{d}{dt} P_{\mu_t}(\mu,f)$ exists at $t=0$, $$\frac{d}{dt} P_{\mu_t}(\mu,f)\Bigg|_{t=0} + P_{\mu_t}(\chi,f)\Bigg|_{t=0} + t\frac{d}{dt} P_{\mu_t}(\chi,f)\Bigg|_{t=0} = \chi(f)$$ Therefore, $\frac{d}{dt} P_{\mu_t}(\mu,f)\big|_{t=0}=\chi(f-P_\mu f)$, which via [\[eq:derivative\]](#eq:derivative){reference-type="eqref" reference="eq:derivative"} and [\[eq:derivativecentering\]](#eq:derivativecentering){reference-type="eqref" reference="eq:derivativecentering"} imply $$\label{eq:Gener=Deriv} \partial_\pi P_{\mu}(\mu,f)(y) = f(y)-P_\mu(y,f).$$ This relationship can also be heuristically proved using our interpretation of derivative in the invariant measure at a point $y$ as the variation in conditional expectation caused by a point mass $\delta_y$. We have then, up to constants, $$\partial_\pi P_{\mu}(\mu,f)(y) = \lim_{t\rightarrow 0} \frac{P_{\mu+t(\delta_y-\mu)}(\mu,f)-\mu(f)}{t},$$ but $P_{\mu_t}(\mu,f)=\mu_t-tP_{\mu_t}(y,f)+tP_{\mu_t}(\mu,f)$ so that $P_{\mu_t}(\mu,f)=(\mu_t-tP_{\mu_t}(y,f))/(1-t)$. Hence, the right hand side of the expression above becomes $$\begin{aligned} \lim_{t\rightarrow 0}\frac{\mu_t(f)-tP_{\mu_t}(y,f)-(1-t)\mu(f)}{(1-t)t} = \lim_{t\rightarrow 0}\frac{f(y) - P_\mu f(y)}{1-t} = f(y)-P_\mu(y,f).\end{aligned}$$ This points out to an interesting connection with discrete-time generators and Foster-Lyapunov type arguments. One way to exploit this identity is simply by restating classical Markov Chains regularity conditions, very often expressed and verified in terms of drifts. The result below is a (partial) restatement of [@meyntweedie2012 Theorem 13.0.1]. **Theorem 1** (Ergodic Theorem with kernel derivatives). *Suppose that $\{X_n\}_{n\in\mathbb{N}}$ is an aperiodic, $\mu$-irreducible Markov Chain with transition probabilities $P_\mu$ and invariant distribution $\mu$. If there exists some petite set $C$, some $b<\infty$ and a non-negative finite function $f$ bounded on $C$ such that $$-\partial_\pi P_{\mu}(\mu,f)(x)\leq -1 +b 1_C(x), \quad x\in\mathsf{X}\xspace$$ whenever such kernel derivative exists, then for all $x\in\mathsf{X}\xspace$, as $k\rightarrow\infty$, $$\norm{P_\mu^k(x,\cdot)-\mu}_{\rm tv}\rightarrow 0.$$* Many other regularity conditions can be expressed in terms of drift conditions, and therefore in terms of $\partial_\pi P_{\mu}(\mu,f)$. The main applications of kernel derivatives in this work will be however in the context of comparing Markov chains with different invariant distributions within the same family, and in particular in what we will refer to as mean value inequalities. First, for illustration, we compute the derivative in the invariant distribution for Markov kernels in the Hastings and Gibbs families of [\[ex:MHtypekernels,ex:Gibbsfamily\]](#ex:MHtypekernels,ex:Gibbsfamily){reference-type="ref" reference="ex:MHtypekernels,ex:Gibbsfamily"} for $V=1$. The next example also illustrates why we defined $\partial_\pi P_{\mu}(x,\cdot)$ only as a weak limit. **Example 3** (Derivatives of Hastings kernels). Consider the Hastings family of [Example 1](#ex:MHtypekernels){reference-type="ref" reference="ex:MHtypekernels"} with $x\mapsto g(x)$ is twice differentiable, with $g'$ and $g''$ bounded. This class of kernel includes, for instance, the Barker form of the acceptance probability ($g(x)=x/(1+x)$), but not the Metropolis-Hastings form ($g(x)=\min(1,x)$). Further consider the following assumption which we discuss after. 1. The indexed invariant distributions have upper bounded continuously differentiable positive densities. The proposal has an upper bounded continuously differentiable positive densities in both arguments. For all $x\in\mathsf{X}\xspace$ there exist an $r>0$ and $L^1(\lambda)$ functions $\psi_1,\psi_2$ such that $$\sup_{z\in B(x,r)}|\partial_z(q(z,y)g'(r_{\mu}(z,y)))|\leq \psi_1(y) \textrm{ and } \sup_{z\in B(x,r)}|\partial_z(q(y,z)g'(r_{\mu}(z,y)))|\leq \psi_2(y). \label{hp:HastingsHP}$$ Let $\mathcal{W}_{H}:=\{\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace):d\rho/d\mu^2 \text{ is bounded for all indexed $\mu$}\}$. **Proposition 4**. *The Hastings kernel $P_\cdot(\rho,\cdot)$ is differentiable in the invariant distribution at $\mu$ for all $\rho\in\mathcal{W}_{H}$ provided only the proposal is upper bounded. In this case, the derivative $\partial_\pi P_{\mu}(\rho,\cdot)$ admits an integral representation, with its density given for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ by $$\begin{aligned} \label{eq:MHderivativedensity} \partial_\pi P_{\mu}(\rho,f)(y) = \int(f(y)-f(z))q(y,z)&\frac{\rho(z)}{\mu(z)}g'(r_\mu(z,y))\lambda(dz) \\ &- \frac{\rho(y)}{\mu(y)^2}\int (f(z)-f(y))q(z,y)g'(r_\mu(y,z))\mu(z)\lambda(dz) \addtocounter{equation}{1}\tag{\theequation}. \end{aligned}$$ Furthermore, if [\[hp:HastingsHP\]](#hp:HastingsHP){reference-type="ref" reference="hp:HastingsHP"} holds, $P_\cdot(\rho,\cdot)$ is differentiable at $\mu$ for all $\rho$ in the weak completion of the set $\mathcal{A}_x:=\{\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace):\mathrm{supp}(\rho)\subseteq B(x,r)\}$. In particular, $P_\cdot(x,\cdot)$ is also differentiable for all $x\in\mathsf{X}\xspace$. In this case, $\partial_\pi P_{\mu}(x,\cdot)$ is given for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ and $\nu\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ by $$\begin{aligned} \label{eq:MHderivativex} \partial_\pi P_{\mu}(x,f)[\nu-\mu] = \int (f(y)-f(x))&\frac{g'(r_{\mu}(x,y))q(y,x)}{\mu(x)}(\nu-\mu)(y)\lambda(dy) \nonumber\\ &- \frac{(\nu-\mu)(x)}{\mu(x)^2}\int (f(y)-f(x))q(y,x)g'(r_{\mu}(x,y))\mu(y)\lambda(dy). \end{aligned}$$* *Proof.* If we can interchange differentiation and integration, one has, with $\chi:=\nu-\mu$, $$\begin{aligned} \frac{d}{dt}P_{\mu+t(\nu-\mu)}(\rho ,f)\bigg|_{t=0} &= \int \int \rho(du) f(w)Q(u,dw)g'(r_\mu(u,w)) \frac{d}{dt} r_{\mu_t}(u,w)\bigg|_{t=0} \\ &+ \int \rho(du)f(u)\left(1-\int Q(u,dw)g'(r_\mu(u,w) \frac{d}{dt} r_{\mu_t}(u,w)\bigg|_{t=0}\right) \\ &= \int \int (f(w)-f(u))\frac{q(w,u)g'(r_\mu(u,w))}{\mu(u)}\rho(u)\chi(w)\lambda(du)\lambda(dw) \\ &- \int \int(f(w)-f(u)) \frac{q(w,u)g'(r_\mu(u,w))}{\mu(u)^2}\mu(w)\rho(u)\chi(u)\lambda(du)\lambda(dw)\\ &= \int \left[\int (f(w)-f(u)) \frac{q(w,u)g'(r_\mu(u,w))}{\mu(u)}\rho(u) \lambda(du)\right] \chi(dw) \\ &- \int \left[\int (f(w)-f(u)) \frac{q(w,u)g'(r_\mu(u,w))\mu(w)}{\mu(u)^2} \lambda(dw)\right]\rho(u)\chi(du). \end{aligned}$$ Hence, using the expressions above and [\[eq:derivative\]](#eq:derivative){reference-type="eqref" reference="eq:derivative"} one identifies [\[eq:MHderivativedensity\]](#eq:MHderivativedensity){reference-type="eqref" reference="eq:MHderivativedensity"}. Integrating the latter w.r.t. $\mu$ shows that [\[eq:derivativecentering\]](#eq:derivativecentering){reference-type="eqref" reference="eq:derivativecentering"} is also satisfied. [5.2](#app:technicalcondboundder){reference-type="ref" reference="app:technicalcondboundder"} shows that the the limit exchange performed is justified under our assumptions. Now consider $\mathcal{A}_x$. In [5.2](#app:technicalcondboundder){reference-type="ref" reference="app:technicalcondboundder"} we show that $\rho\mapsto \partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$ is uniformly continuous on $\mathcal{A}_x$ for all $\nu\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$. We can then apply [Proposition 1](#prop:derextension){reference-type="ref" reference="prop:derextension"} to prove the differentiability of $P_\cdot(\rho,\cdot)$ for all $\rho\in\Tilde{\mathcal{A}}_x$. Since clearly $\delta_x\in\Tilde{\mathcal{A}}_x$, this also proves the differentiability of $P_\cdot(x,\cdot)$ and the expression [\[eq:MHderivativex\]](#eq:MHderivativex){reference-type="eqref" reference="eq:MHderivativex"} follows from the direct computation $\partial_\pi P_{\mu}(x,f)[\nu-\mu] = \lim_{j} \partial_\pi P_{\mu}(\rho_{j,x},f)[\nu-\mu]$ for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, where $\rho_{j,x}\Rightarrow\delta_x$. ◻ Let us notice that the operator $\partial_\pi P_{\mu}(x,f)$ has no integral representation. The boundedness of $\rho/\mu^2$ is a warm-start type condition. If $\rho$ has lighter tails than $\mu^2$, it will hold. In particular, if $\rho$ is compactly supported, $P_\cdot(\rho,\cdot)$ is differentiable at all $\mu$. On the other hand, if $\rho$ is heavy tailed, $P_\cdot(\rho,\cdot)$ will seldom be differentiable. The assumptions in [\[hp:HastingsHP\]](#hp:HastingsHP){reference-type="ref" reference="hp:HastingsHP"} are rather weak and often verified in practice. The requirement that $|\partial_z(q(y,z)g'(r_{\mu}(z,y)))|$ is uniformly bounded by a $L^1(\lambda)$ function on $z\in B(x,r)$ is a requirement on the regularity of the proposal and the balancing function around the point $x\in\mathsf{X}\xspace$ where we want to show $P_\cdot(x,\cdot)$ is differentiable at $\mu$, and it is very often satisfied. For instance, for the Gaussian Random Walk we have $$\begin{aligned} |\partial_z(q(y,z)g'(r_{\mu}(z,y)))| &\propto |y-z|q(y,z)g'(r_{\mu}(z,y)) + q(y,z)g''(r_{\mu}(z,y))\mu(y)\partial_z \mu(z)/\mu(z)^2 \\ &\leq q(y,z)(|y|g'(r_{\mu}(z,y)) + |z|g'(r_{\mu}(z,y)) + g''(r_{\mu}(z,y))\mu(y)\partial_z \mu(z)/\mu(z)^2) \\ &\leq (|y| + |r| + \mu(y)c)\sup_{z\in B(x,r)}q(z,y) \end{aligned}$$ where $c:=\sup_{z\in B(x,r)}\partial_z \mu(z)/\mu(z)^2<\infty$ due our positivity and continuous differentiability of the target assumption, and where we used $g',g''\leq 1$. The other domination condition on $|\partial_z(q(z,y)g'(r_{\mu}(z,y)))|$ --which is not used here but later to prove a Feller-type property-- is equivalent to the former in the Random Walk case. It is possible to find kernel derivatives with other assumptions on the starting distributions, on the invariant distributions, the proposal, the balancing function, and for general $V$ by adapting the arguments used above and in the relevant appendixes. For instance, for an unbounded $V$ one would additionally require that $\mu(V)<\infty$ for all $\mu$ indexed by the family to perform the limit exchange justified in [5.2](#app:technicalcondboundder){reference-type="ref" reference="app:technicalcondboundder"}. This naturally (and notationally) suggests the use of a $V$ that is a suitable Lypunov function for the different distributions indexed by the family---see [3](#sec:applications){reference-type="ref" reference="sec:applications"}. The requirement that $g$ is differentiable however cannot be immediately relaxed. Although the result above then does not guarantee differentiability of Metropolis-Hastings kernels, we will be able to obtain results for the Metropolis choice indirectly later. In the example below, a probability distribution with a numerical subscript denotes the marginal in the indicated coordinate of the underlying probability distribution, we write $y=(y_1,y_2)$ for a point $y\in\mathsf{X}\xspace^2$, $\lambda^2$ for $\lambda\otimes\lambda$ and a single integral sign for multiple integrals when no confusion is possible. **Example 4** (Derivatives of Gibbs kernels). Consider the MCMC kernels of [Example 2](#ex:Gibbsfamily){reference-type="ref" reference="ex:Gibbsfamily"} with $P_{1|2}$ and $P_{2|1}$ admitting a common dominating measure with $\mu_1$ and $\mu_2$. Further consider the following assumption. 1. The indexed invariant distributions have upper bounded continuously differentiable and positive densities. The full conditionals are bounded in the starting point and for all $x\in\mathsf{X}\xspace$ they satisfy $\sup_{z\in B(x,r)}|\partial_z P_{1|2}(z,y)|\leq \psi(y)$ for some function $\psi\in L^1(\lambda)$. [\[hp:GibbsHP\]]{#hp:GibbsHP label="hp:GibbsHP"} Let $\mathcal{W}_G:=\{\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace):d\rho_2/d\mu_2 \text{ is bounded for all indexed $\mu$}\}$. **Proposition 5**. *The Gibbs kernel $P_\cdot(\rho,\cdot)$ is differentiable in the invariant distribution at $\mu$ for all $\rho\in\mathcal{W}_{G}$. In this case, the derivative $\partial_\pi P_{\mu}(\rho,\cdot)$ admits an integral representation, with its density given for all $f\in\mathbf{C}_{V}^1(\mathsf{X}\xspace^2)$ by $$\begin{aligned} \label{eq:DSGibbsderivative} \partial_\pi P_{\mu}(\rho,f)(y) &= \int f(y_1,w_2)\frac{\rho(y_2)}{\mu_2(y_2)}P_{2|1}(y_1,dw_2) - \int f(w_1,w_2) \frac{\rho(y_2)}{\mu_2(y_2)}P_{1|2}(y_2,dw_1) P_{2|1}(w_1,dw_2) \\ &+ \int f(y_1,y_2) \frac{\rho(dw_2)}{\mu_1(y_1)} P_{1|2}(w_2,y_1) - \int f(y_1,w_2)\frac{\rho(du_2)}{\mu_1(y_1)} P_{1|2}(u_2,y_1) P_{2|1}(y_1,dw_2) \addtocounter{equation}{1}\tag{\theequation}. \end{aligned}$$ Furthermore, if [\[hp:GibbsHP\]](#hp:GibbsHP){reference-type="ref" reference="hp:GibbsHP"} holds, $P_\cdot(\rho,\cdot)$ is differentiable at $\mu$ for all $\rho$ in the weak completion of the set $\mathcal{A}_x:=\{\rho\in\mathcal{P}_{\lambda^2}(\mathsf{X}\xspace^2):\mathrm{supp}(\rho)\subseteq B(x,r)\}$. In particular, $P_\cdot(x,\cdot)$ is also differentiable for all $x\in\mathsf{X}\xspace^2$. In this case, $\partial_\pi P_{\mu}(x,\cdot)$ is given for all $f\in\mathbf{C}_{V}^1(\mathsf{X}\xspace^2)$ and $\nu\in\mathcal{P}_{\lambda^2}(\mathsf{X}\xspace^2)$ by $$\begin{aligned} \label{eq:Gibbsderivativex} &\partial_\pi P_{\mu}(x,f)[\nu-\mu]= \nonumber \\ &=\int\bigg(\int f(y_1,w_2)\frac{P_{2|1}(y_1,dw_2)}{\mu_2(x_2)} - \int f(w_1,w_2)\frac{P_{1|2}(x_2,dw_1)P_{2|1}(w_1,dw_2)}{\mu_2(x_2)}\bigg)(\nu-\mu)(y_1,x_2)\lambda(dy_1) \nonumber\\ &+\int\bigg(f(y_1,y_2)\frac{P_{1|2}(x_2,y_1)}{\mu_1(y_1)} - \int f(y_1,w_2)\frac{P_{1|2}(x_2,y_1)P_{2|1}(y_1,dw_2)}{\mu_1(y_1)}\bigg)(\nu-\mu)(y)\lambda^2(dy). \end{aligned}$$* *Proof.* This is similar the proof of [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"}. First we notice that for the deterministic scan we can just write the full conditionals in terms of $\mu$ as $P_{1|2}(x',y)=\mu(y,x')/\int \mu(z,x')\lambda(dz)$ and similarly for $P_{2|1}$, hence $$P_\mu((x_1,x_2),f)= \int \int f(w_1,w_2) \frac{\mu(w_1,x_2)}{\int \mu(z,x_2)\lambda(dz)}\frac{\mu(w_1,w_2)}{\int \mu(w_1,z)\lambda(dz)}\lambda^2(dw).$$ Then, by exploiting the fact that the transition probabilities do not depend on the first coordinate of the starting point we compute $$\begin{aligned} \frac{d}{dt}P_{\mu+t(\nu-\mu)}(\rho ,f)\bigg|_{t=0} &= \int \int \chi(dw_1,du_2) \int f(w_1,w_2)\frac{\rho_2(u_2)}{\mu_2(u_2)} P_{2|1}(w_1,dw_2) \\ &- \int\int \chi(dz,du_2) \int \int f(w_1,w_2)\frac{\rho_2(u_2)}{\mu_2(u_2)} P_{1|2}(u_2,dw_1) P_{2|1}(w_1,dw_2) \\ &+ \int \int \chi(dw_1,dw_2) \int f(w_1,w_2)\frac{\rho_2(du_2)}{\mu_1(w_1)} P_{1|2}(u_2,dw_1) \\ &- \int \int \chi(dw_1,dz) \int \int f(w_1,w_2)\frac{\rho_2(du_2)}{\mu_1(w_1)}P_{1|2}(u_2,dw_1)P_{2|1}(w_1,dw_2). \end{aligned}$$ Using the expression above and [\[eq:derivative\]](#eq:derivative){reference-type="eqref" reference="eq:derivative"} one identifies [\[eq:DSGibbsderivative\]](#eq:DSGibbsderivative){reference-type="eqref" reference="eq:DSGibbsderivative"}. Integrating the latter w.r.t. $\mu$ shows that [\[eq:derivativecentering\]](#eq:derivativecentering){reference-type="eqref" reference="eq:derivativecentering"} is also satisfied. The details on the limit exchange are in [5.2](#app:technicalcondboundder){reference-type="ref" reference="app:technicalcondboundder"}. To show differentiability for all starting distribution in the weak completion of $\mathcal{A}_x$ we show that $\rho\mapsto \partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$ is uniformly continuous on $\mathcal{A}_x$ for all $\nu\in\mathcal{P}_{\lambda^2}(\mathsf{X}\xspace^2)$. This is similar to the proof of [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"} - the details are in [5.2](#app:technicalcondboundder){reference-type="ref" reference="app:technicalcondboundder"}. ◻ Again, the boundedness of $d\rho_2/d\mu_2$ is a warm-start type condition, which we note is in general weaker than the warm-start required to differentiate Hastings kernels. We again note that the assumption [\[hp:GibbsHP\]](#hp:GibbsHP){reference-type="ref" reference="hp:GibbsHP"} is rather weak, and it is somehow the analogue of the domination condition required in [\[hp:HastingsHP\]](#hp:HastingsHP){reference-type="ref" reference="hp:HastingsHP"} discussed above. Let us notice that also here the operator $\partial_\pi P_{\mu}(x,f)$ has no integral representation. Using the fact that the balancing function $g$ has to satisfy $g(x)=xg'(x)+g'(1/x)$, it is possible to check that when $\rho=\mu$, the derivative in the invariant measure of the Hastings kernels at $y\in\mathsf{X}\xspace$ becomes (minus) the generator of the Markov chain, and the same is true for Gibbs kernels derivatives, as expected. ## FTC Formula and bounded derivatives {#sec:ftc} We now state a formula that, under a weak-Feller type condition on which we comment below, allow us to compare Markov Chains with different invariant distributions belonging to the same Markov family in terms of the derivative in the invariant distribution. This is a Fundamental Theorem of Calculus-like formula, whose analogue appears in Robust Statistics [@huber2009robust Chapter 2] and Mean Field Games [@cardaliaguet2019mastereqn]. 1. The Markov kernels family $\{P_\star\}$ satisfies a Feller-type condition: $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ implies $P_\mu f\in\mathbf{C}^1_1(\mathsf{X}\xspace)$ for all $P_\mu\in\{P_\star\}$. [\[hp:Fellertypechain\]]{#hp:Fellertypechain label="hp:Fellertypechain"} **Theorem 2** (Fundamental Theorem of MCMC Calculus Formula). *For all $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ for which $P_\cdot(\rho,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$, i.e. $\rho\in\bigcap_{t=0}^1 \mathcal{D}_{\mu_t,\lambda}(P)$, $$\label{eq:FTC} P_\mu(\rho,f)-P_\nu(\rho,f)= \int_0^1 \partial_{\pi}P_{\mu_t}(\rho,f)[\nu-\mu] dt.$$ for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$.* *If [\[hp:Fellertypechain\]](#hp:Fellertypechain){reference-type="ref" reference="hp:Fellertypechain"} holds and $\partial_\pi P_{\mu_t}(\rho,f)[\nu-\mu]$ is bounded in $t$, [\[eq:FTC\]](#eq:FTC){reference-type="eqref" reference="eq:FTC"} is true also for all $\rho\in\bigcap_{t=0}^1 \mathcal{D}_{\mu_t}(P)$.* *Furthermore, if $\partial_{\pi}P_{\mu_t}(\rho,f)$ admits a continuously differentiable density and $\nu$ can be expressed via some pushforward $T$ from $\mu$ in that $\nu=T_* \mu$ we also have $$\begin{aligned} \label{eq:FTCintrinsic} P_\mu(\rho ,f)-P_\nu(\rho,f) &= \int_0^1 \int \int^y_{T(y)} D_\pi P_{\mu_t}(\rho,f) (s) dsd\mu(y) dt \\ &= \int_0^1 \int (T(y)-y) \int_0^1 D_\pi P_{\mu_t}(\rho,f) (s(T(y)-y)+y)dsd\mu(y) dt, \addtocounter{equation}{1}\tag{\theequation} \end{aligned}$$ where $D_\pi P_{\mu}(\rho,f) (y):=(d/dy)\partial_\pi P_{\mu}(\rho,f)(y)$.* *Proof.* Let $(\rho,f)\in\mathcal{P}_\lambda(\mathsf{X}\xspace)\times\mathbf{C}^1_{V}(\mathsf{X}\xspace)$. Write $\mu_t = (1-t)\mu+t\nu$, so that $\mu_1=\nu$ and $\mu_0=\mu$. Then, $$P_\mu(\rho ,f)-P_\nu(\rho ,f) = \int_0^1 \frac{d}{dt} P_{\mu_t}(\rho ,f) dt = \int_0^1 \lim_{s\rightarrow0}\frac{P_{\mu_{t+s}}(\rho ,f)-P_{\mu_{t}}(\rho ,f)}{s}dt$$ Now, since for all fixed $t$, $\mu_{t+s}=\mu_t+s(\mu_1-\mu_t)/(1-t)=\mu_t+s((\mu_1-t\mu_t)/(1-t)-\mu_t)$, and then using [\[eq:derivative\]](#eq:derivative){reference-type="eqref" reference="eq:derivative"} with $(\mu_1-\mu_t)/(1-t)\in \mathcal{M}_{\lambda,0}(\mathsf{X}\xspace)$ being a 0--mass perturbation one can write under our hypotheses that for $t\in[0,1)$ $$\begin{aligned} P_\mu(\rho ,f)-P_\nu(\mu ,f) = \int_0^1 \partial_\pi P_{\mu_t}(\rho,f)\left[\frac{\mu_1-\mu_t}{1-t}\right]dt = \int_0^1 \partial_\pi P_{\mu_t}(\rho,f)[\nu-\mu]dt, \end{aligned}$$ with last equality following because $\mu_1-\mu_t=\nu-\mu_t=(1-t)(\nu-\mu)$. This proves the claim for all $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ for which the differentiability requirement is met. Now assume [\[hp:Fellertypechain\]](#hp:Fellertypechain){reference-type="ref" reference="hp:Fellertypechain"} and consider a $\rho\in\bigcap_{t=0}^1 \mathcal{D}_{\mu_t}(P)$ not necessarily in $\mathcal{P}_\lambda(\mathsf{X}\xspace)$. By definition, $\rho$ is in the weak completion $\Tilde{\mathcal{A}}$ of some set $\mathcal{A}\subset\mathcal{P}_\lambda(\mathsf{X}\xspace)$ on which $\rho\mapsto \partial_\pi P_{\mu_t}(\rho,f)[\nu-\mu]$ is uniformly continuous for all $t\in[0,1]$. Considering a sequence $\rho_j\in\mathcal{A}\Rightarrow \rho\in\Tilde{\mathcal{A}}$ under [\[hp:Fellertypechain\]](#hp:Fellertypechain){reference-type="ref" reference="hp:Fellertypechain"} we compute $$\begin{aligned} \label{eq:FTClimitpush} P_\mu(\rho,f)-P_\nu(\rho,f) &= \lim_{j\rightarrow\infty} P_\mu(\rho_{j},f)-P_\nu(\rho_{j},f) = \lim_{j\rightarrow\infty} \int_0^1 \partial_\pi P_{\mu_t}(\rho_{j},f)[\nu-\mu]dt \nonumber \\ &=: \int_0^1 \partial_\pi P_{\mu_t}(\rho,f)[\nu-\mu]dt, \end{aligned}$$ where the limit exchange follows from the assumed boundedness of $t\mapsto \partial_\pi P_{\mu_t}(\rho,f)[\nu-\mu]$ via dominated convergence. To prove [\[eq:FTCintrinsic\]](#eq:FTCintrinsic){reference-type="eqref" reference="eq:FTCintrinsic"}, use [\[eq:FTC\]](#eq:FTC){reference-type="eqref" reference="eq:FTC"} with $\mu_t:=(1-t)\mu+tT_*\mu$ to obtain $$\begin{aligned} P_{\mu}(\rho,f)-P_{T_*\mu}(\rho,f) &= \int_0^1 \int \partial_\pi P_{\mu_t}(\rho,f)(y)d(T_*\mu-\mu)(y)dt \\ &= \int_0^1 \int \left[\partial_\pi P_{\mu_t}(\rho,f)(T(y))- \partial_\pi P_{\mu_t}(\rho,f)(y)\right]d\mu(y) dt. \end{aligned}$$ Since $y\mapsto \partial_\pi P_{\mu_t}(\rho,f)(y)$ is continuously differentiable by assumption, an application of the Fundamental Theorem of Calculus completes the proof. ◻ **Remark 1**. We have seen in [\[ex:MHtypekernelsdiff,ex:Gibbsderivatives\]](#ex:MHtypekernelsdiff,ex:Gibbsderivatives){reference-type="ref" reference="ex:MHtypekernelsdiff,ex:Gibbsderivatives"} that for Hastings and Gibbs type kernels in fact, under regularity conditions, such $\bigcap_{t=0}^1 \mathcal{D}_{\mu_t}(P)$ contains at least the set $\{\delta_x\}_{x\in\mathsf{X}\xspace}$. **Remark 2**. When $V$ is bounded, [\[hp:Fellertypechain\]](#hp:Fellertypechain){reference-type="ref" reference="hp:Fellertypechain"} is mild and satisfied by many MCMC kernel used in practice, including the Hastings and Gibbs families under assumption [\[hp:HastingsHP,hp:GibbsHP\]](#hp:HastingsHP,hp:GibbsHP){reference-type="ref" reference="hp:HastingsHP,hp:GibbsHP"}. For instance, for the Hastings chain, if $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ then $Pf$ is clearly bounded, and to show it is also continuously differentiable in all $x\in\mathsf{X}\xspace$, we can show that each $x$ is in a neighbourhood $B(x,r)$ in which $Pf$ is continuously differentiable. Under our assumptions $q(x,y)g(r_{\mu}(x,y))$ is continuously differentiable and by the (first) domination condition we can conclude that $\int f(y)q(x,y)g(r_{\mu}(x,y))\lambda(dy)$ and $\int q(x,y)g(r_{\mu}(x,y))\lambda(dy)$ (and so $Pf$) are too at $x$ by e.g. [@klenke2013prob Theorem 6.28]. When $V$ is not bounded, $Pf$ might not be bounded. Thus to obtain [\[eq:FTC\]](#eq:FTC){reference-type="eqref" reference="eq:FTC"} for $\rho\in\bigcap_{t=0}^1 \mathcal{D}_{\mu_t}(P)$ not necessarily in $\mathcal{P}_\lambda(\mathsf{X}\xspace)$, we can instead require that for $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, $Pf$ is continuously differentiable and it satisfies a suitable uniform integrability condition. Call $\mathcal{A}$ the subset of $\mathcal{P}_\lambda(\mathsf{X}\xspace)$ that has $\rho$ in its weak completion, and consider a sequence $\rho_j\in\mathcal{A}\Rightarrow\rho$. If $\sup_{j\geq 0}\int |Pf(y)|^{1+\epsilon}\rho_{j}(dy)<\infty$ and $\sup_{j\geq 0}\int |P_\nu f(y)|^{1+\epsilon}\rho_{j}(dy)<\infty$ for some $\epsilon>0$ for all such sequences $\{\rho_j\}_{j\in\mathbb{N}}$, we would still obtain [\[eq:FTClimitpush\]](#eq:FTClimitpush){reference-type="eqref" reference="eq:FTClimitpush"} (e.g. [@billingsley1999cpm Section 3.5]). For instance, using the preceding remark, we have seen that $\delta_x\in\bigcap_{t=0}^1 \mathcal{D}_{\mu_t}(P)$, and that $\delta_x$ is the weak completion of $\mathcal{A}_x:=\{\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace):\mathrm{supp}(\rho)\subseteq B(x,r)\}$. Clearly for any sequence $\rho_j$ in $\mathcal{A}_x$, for $\sup_{j\geq 0}\int |P_\mu f(y)|^{1+\epsilon}\rho_{j}(dy)<\infty$ to hold it is sufficient that $P_\mu f$ is bounded in the compact $B(x,r)$, which is very often the case for Hastings and Gibbs kernels. In particular, the Hastings and Gibbs families under assumption [\[hp:HastingsHP,hp:GibbsHP\]](#hp:HastingsHP,hp:GibbsHP){reference-type="ref" reference="hp:HastingsHP,hp:GibbsHP"} satisfy this property. **Remark 3**. The assumption of boundedness of $t\mapsto \partial_\pi P_{\mu_t}(\rho,f)[\nu-\mu]$ is simply verified simply whenever $\partial_\pi P_{\mu}(\rho,f)[\nu-\mu]$ and $\partial_\pi P_{\nu}(\rho,f)[\nu-\mu]$ exist for our Hastings and Gibbs families. We put together the last three Remark for reference. **Lemma 1**. *For the Hastings and Gibbs families [\[eq:FTC\]](#eq:FTC){reference-type="eqref" reference="eq:FTC"} is true for all $\rho\in\mathcal{W}_H$ and $\rho\in\mathcal{W}_G$, respectively, and if [\[hp:HastingsHP,hp:GibbsHP\]](#hp:HastingsHP,hp:GibbsHP){reference-type="ref" reference="hp:HastingsHP,hp:GibbsHP"} hold, also for all $\rho\in\{\delta_x\}_{x\in\mathsf{X}\xspace}$.* **Remark 4**. The quantity $D_\pi P_{\mu}(\rho,f)$ can be related to rigorous notions of derivatives in the Wasserstein geometry studied in [@ambrosio2005gradientflows]. It describes variations of the transition probabilities when $\mu$ is perturbed in the direction $\nu$ along the curves that are not simple interpolations like $\mu_t$ but rather *transport maps*. In fact, one can substitute $T$ with $\ensuremath{\textrm{Id}}\xspace-hT$ in [\[eq:FTCintrinsic\]](#eq:FTCintrinsic){reference-type="eqref" reference="eq:FTCintrinsic"} to obtain $$\begin{aligned} P_{\mu}(\rho,f)-P_{T_*\mu}(\rho,f) &= h \int_0^1 \int T(y) \int_0^1 D_\pi P_{\mu_t}(\rho,f) (sh+y)dsd\mu(y) dt.\end{aligned}$$ Dividing by $h$ and letting $h\rightarrow 0$ we get $$\lim_{h\rightarrow 0}\frac{P_{(\ensuremath{\textrm{Id}}\xspace-hT)_*\mu}(\rho,f)-P_\mu(\rho,f)}{h} = \int D_\pi P_{\mu}(\rho,f) (y) T(y)d\mu(y).$$ We will employ [Theorem 2](#prop:FTC){reference-type="ref" reference="prop:FTC"} mainly as a way to obtain mean value type inequalities in the next section. However, formulæ such as [\[eq:FTC\]](#eq:FTC){reference-type="eqref" reference="eq:FTC"} or [\[eq:FTCintrinsic\]](#eq:FTCintrinsic){reference-type="eqref" reference="eq:FTCintrinsic"} could be used in general to obtain a possibly hard to guess expression for the difference in expectations given by two Markov kernels with different invariant distributions within the same family. We will often think of one of them as being an approximation of the other. In this context, we think of $T$ in [\[eq:FTCintrinsic\]](#eq:FTCintrinsic){reference-type="eqref" reference="eq:FTCintrinsic"} as a map that makes the distribution $\mu$ easier to work with. For instance, for diffeomorphisms $T_1,...,T_j$, one can let $T=T_1\circ....\circ T_j$ obtaining a so-called normalizing flow, that can be used to smooth out complicated distributions when sampling via MCMC---see [@gabrie2021normalizingflows; @papamakarios2021normalizingflows] Notice that with $\rho=\nu$ one also obtains $$P_\mu(\nu ,f)-\nu(f) = \int_0^1 \int \partial_\pi P_{\mu_t}(\nu,f)(y)d(\nu-\mu)(y)dt;$$ an expression for the difference in expectation with respect to a Markov kernel applied to an initial distribution to that with respect to the distribution itself. ## Bounded kernel derivatives and Mean-Value inequalities {#sec:mvi} We want to argue that if the derivative in the invariant distribution at $\mu$ towards $\nu$ is bounded in some sense, then the Markov Chains associated with kernels $P_\mu$ and $P_\nu$ move similarly whenever $\mu$ and $\nu$ are close. We define the concept of bounded derivative in the invariant distribution accordingly. We aim to provide a statement similar to the fact that on $\mathbb{R}$ a differentiable function has a bounded derivative if and only if it is Lipschitz. **Definition 6**. *We say that a differentiable Markov kernel $P_\cdot(\rho,\cdot)$ has a bounded derivative at $\mu$ towards $\nu$ if there exist constants $M_{1,\rho},M_{2,\rho}<\infty$ such that $$\label{eq:boundedderimplicationall} \norm{P_{\mu}(\rho,\cdot)-P_{\nu}(\rho,\cdot)}_{V} \leq M_{1,\rho} \norm{\mu-\nu}_{V}+M_{2,\rho}\rho(|\mu-\nu|).$$* Noting that $d(\mu,\nu)=||\mu-\nu||_V + \rho(|\mu-\nu|)$ is itself a metric, this could be written as a standard statement of Lipschitz continuity with respect to that distance, with constant given by the larger of $M_{1,\rho}$ and $M_{2,\rho}$. However, the form we have here allow us to separate the role of two separate contributions and we believe that this refinement can be informative. With this in mind, we will sometimes refer to $M_{1,\rho},M_{2,\rho}$ as Lipschitz constants, and to inequalities of the form [\[eq:boundedderimplicationall\]](#eq:boundedderimplicationall){reference-type="eqref" reference="eq:boundedderimplicationall"} a mean value inequalities. The next Propositions show that [Definition 6](#def:boundedderivative){reference-type="ref" reference="def:boundedderivative"} work as intended: we will see that if $P_\cdot(\rho,\cdot)$ is differentiable in $\mu_t=(1-t)\mu+t\nu$ for all $t\in[0,1]$ and a boundedness condition on the derivative holds, then $P_\cdot(\rho,\cdot)$ will have a bounded derivative in the invariant distribution at $\mu$ towards $\nu$. Of course the boundedness depend on $V$, but we often silence this dependence because it is implicit in the context. When $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$, this [Definition 6](#def:boundedderivative){reference-type="ref" reference="def:boundedderivative"} is equivalent to the existence of a constant $M_\rho<\infty$ such that $$\label{eq:boundedderimplication} \norm{P_{\mu}(\rho,\cdot)-P_{\nu}(\rho,\cdot)}_{V} \leq M_{\rho} \norm{\mu-\nu}_{V}.$$ i.e. to the operator $P_\cdot(\rho,\cdot)$ from $\mathcal{P}(\mathsf{X}\xspace)$ (via the invariant distribution) to $\mathcal{P}(\mathsf{X}\xspace)$ (the corresponding transition probabilities) being Lipschitz continuous w.r.t. $\norm{\cdot}_{V}$. In fact, obviously if [\[eq:boundedderimplication\]](#eq:boundedderimplication){reference-type="eqref" reference="eq:boundedderimplication"} is true then [\[eq:boundedderimplicationall\]](#eq:boundedderimplicationall){reference-type="eqref" reference="eq:boundedderimplicationall"} holds with $M_{2,\rho}:=0$, and on the other hand when $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ we can bound $M_{2,\rho}\rho(|\mu-\nu|)\leq M_{2,\rho}\sup(\rho/V)\norm{\mu-\nu}_{V}$, and [\[eq:boundedderimplication\]](#eq:boundedderimplication){reference-type="eqref" reference="eq:boundedderimplication"} holds true with $M_\rho:=M_{1,\rho}+M_{2,\rho}\sup(\rho/V)$. If $\rho=\delta_x$, [\[eq:boundedderimplicationall\]](#eq:boundedderimplicationall){reference-type="eqref" reference="eq:boundedderimplicationall"} becomes $$\label{eq:boundedderimplicationx} \norm{P_{\mu}(x,\cdot)-P_{\nu}(x,\cdot)}_{V} \leq M_{1,x} \norm{\mu-\nu}_{V}+M_{2,x}|\mu(x)-\nu(x)|.$$ This express the fact that as for many kernels employed in MCMC settings in particular, $P_\cdot(x,\cdot)$ typically depends on the values of the density of the invariant distribution at $x\in\mathsf{X}\xspace$, one cannot expect $P_\mu(x,\cdot)$ to be close to $P_\nu(x,\cdot)$ unless $\mu(x)$ is close to $\nu(x)$, since the closedness of $\mu$ to $\nu$ in $V$-distance does not imply that their densities functions at single points are close, at least without further assumptions. Sometimes, like in the Gibbs case, $P_\cdot(x,\cdot)$ depends on the values of the invariant at some coordinate of the vector $x$ only. In this situation, if $\mathcal{J}$ is an index set, $x_\mathcal{J}$ denote the element of the vector $x$ indicated by $\mathcal{J}$ and a sub-scripted probability measures indicate the marginal in the indicated coordinate, a more sensible definition of bounded derivative of $P_\cdot(x,\cdot)$ would be the existence of $M_{1,x},M_{2,x}<\infty$ such that $$\norm{P_{\mu}(x,\cdot)-P_{\nu}(x,\cdot)}_{V} \leq M_{1,x} \norm{\mu-\nu}_{V} + M_{2,x}|\mu_\mathcal{J}(x_\mathcal{J})-\nu_\mathcal{J}(x_\mathcal{J})|.$$ where $dx_{\mathcal{J}}$ indicates that we integrate out all the coordinates in some index set $\mathcal{J}$. The case $\mathcal{J}=\emptyset$ corresponds to [\[eq:boundedderimplicationx\]](#eq:boundedderimplicationx){reference-type="eqref" reference="eq:boundedderimplicationx"}. This relaxed definition, although useful in certain contexts, is slightly cumbersome so we opted for [\[eq:boundedderimplicationx\]](#eq:boundedderimplicationx){reference-type="eqref" reference="eq:boundedderimplicationx"} to illustrate concepts parsimoniously. Notice that if a Markov kernel has a bounded derivative in the invariant distribution at $\mu$ towards $\nu$ and $\mu$ is close to $\nu$, the Markov Chains following $P_\mu(x,\cdot)$ and $P_\nu(x,\cdot)$ never move too differently starting from $x$, in that their $V$-distance is small. We can similarly consider other integral probability metrics and obtain related inequalities. Here we focused on the $V$-norm due its popularity within the MCMC literature arising from its close connection with Lyapunov functions for ergodicity [@R&R2004mcmc; @meyntweedie2012]. In [3](#sec:applications){reference-type="ref" reference="sec:applications"} we exploit these connections. We now introduce a stronger form of boundedness. **Definition 7**. *We say that a Markov kernel $P_\cdot(\rho,\cdot)$ has a $\mathcal{Q}$-uniformly bounded derivative at $\mu$ towards $\nu$ if there exist constants $M_1,M_2<\infty$ such that for all $\rho\in\mathcal{Q}$: $$\norm{P_{\mu}(\rho,\cdot)-P_{\nu}(\rho,\cdot)}_{V} \leq M_{1} \norm{\mu-\nu}_{V}+M_{2}\rho(|\mu-\nu|).$$* If $\mathcal{Q}=\{\delta_x\}_{x\in\mathsf{X}\xspace}$ the chains move alike uniformly in the space if their invariant distributions are close at all points. Notice that a Markov kernel has an $\mathcal{Q}$-uniformly bounded derivative at $\mu$ towards $\nu$ if it has bounded derivative and the associated Lipschitz constants can be bounded uniformly over $\rho\in\mathcal{Q}$. See at the end of this section for such an example where $\mathcal{Q}$ is the set of warm-start distributions. Finding conditions for $P_\cdot(\rho,\cdot)$ to have a bounded derivative in the invariant distribution when $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ is rather immediate from [Theorem 2](#prop:FTC){reference-type="ref" reference="prop:FTC"}. **Proposition 6**. *For all $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ for which the Markov kernel $P_\cdot(\rho,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$ with its derivative admitting a density, and $$M_\rho:=\sup_{f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)} \left\vert\int_0^1\partial_\pi P_{\mu_t}(\rho,f)(\cdot)dt\right\vert_{V}<\infty,$$ $P_\cdot(\rho,\cdot)$ has a bounded derivative at $\mu$ towards $\nu$ with Lipschitz constant $M_\rho$, i.e. $$\label{eq:PLipschitz} \norm{P_\mu(\rho,\cdot)-P_\nu(\rho,\cdot)}_{V} \leq \norm{\mu-\nu}_{V} M_\rho$$* *Proof.* For all $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ obeying the differentiability requirement and $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, from [\[eq:FTC\]](#eq:FTC){reference-type="eqref" reference="eq:FTC"} we obtain $$\begin{aligned} |P_\mu(\rho,f)-P_\nu(\rho,f)| &= \left| \int \int_0^1 \partial_\pi P_{\mu_t}(\rho,f)(y)dt d(\nu-\mu)(y) \right|. \end{aligned}$$ However, for any $f \in \mathbf{C}^1_{V}(\mathsf{X}\xspace)$, we have $$\left|\int\int_0^1\partial_\pi P_{\mu_t}(\rho,f)(y)dt d(\nu-\mu)(y) \right| \leq \norm{\mu-\nu}_{V} \left\vert\int_0^1\partial_\pi P_{\mu_t}(\rho,f)(\cdot)dt\right\vert_{V},$$ and the claim follows by [Lemma 3](#app:lemma:VnormsupCb){reference-type="ref" reference="app:lemma:VnormsupCb"} upon taking the supremum over functions in $\mathbf{C}^1_{V}(\mathsf{X}\xspace)$. ◻ In other words, the mapping $P_\cdot(\rho,\cdot):\mu\in(\mathcal{P}(\mathsf{X}\xspace),\norm{\cdot}_{V})\mapsto P_\mu(\rho,\cdot)\in (\mathcal{P}(\mathsf{X}\xspace),\norm{\cdot}_{V})$ is a Lipschitz continuous function of its invariant distribution with Lipschitz constant $M_{\rho}$. [Proposition 6](#prop:boundedderivativerho){reference-type="ref" reference="prop:boundedderivativerho"} provides a condition for a Markov kernel to have bounded derivative when $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$. Proving boundedness for more general $\rho$ and obtaining mean value inequalities is harder, and there is not in general a direct analogue to [Proposition 6](#prop:boundedderivativerho){reference-type="ref" reference="prop:boundedderivativerho"} since in this case $\partial_\pi P_{\mu}(\rho,\cdot)$ is only defined weakly, and one has to retrieve Mean-Value inequalities on a case to case basis. In the next example we illustrate this point for $\rho=\delta_x$ and we study when the kernel derivatives of the Hastings and Gibbs type kernels are bounded for $V=1$. **Example 5** (Bounded derivatives and Mean Value inequalities for Hastings kernels). **Proposition 7**. *Let $\{P_\star\}$ be a Hastings family.* 1. *Let $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$. If the Markov kernel $P_\cdot(\rho,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$, and $M_\rho:=\int_0^1 D_{t,\rho}dt<\infty$ with $$\label{eq:MHMVIlipconst} D_{t,\rho}:= 2 \sup_y\left|\int \frac{\rho(z)}{\mu_t(z)}q(y,z)g'(r_{\mu_t}(z,y))\lambda(dz) + \frac{\rho(y)}{\mu_t(y)^2}\int {\mu_t}(z)q(z,y)g'(r_{\mu_t}(y,z))\lambda(dz)\right|,$$ then $P_\cdot(\rho,\cdot)$ has a bounded derivative at $\mu$ towards $\nu$ with Lipschitz constant $M_\rho$ and we have the mean value inequality $$\label{eq:MHMVI} \norm{P_\mu(\rho,\cdot)-P_\nu(\rho,\cdot)}_{\rm tv} \leq \norm{\mu-\nu}_{\rm tv} \int_0^1 D_{t,\rho} dt.$$* 2. *If [\[hp:Fellertypechain\]](#hp:Fellertypechain){reference-type="ref" reference="hp:Fellertypechain"} holds, the Markov kernel $P_\cdot(x,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$, and $M_{1,x}:=\int_0^1 D_{1,t,x}dt<\infty, M_{2,x}:=\int_0^1 D_{2,t,x}dt<\infty$ with $$\begin{aligned} \label{eq:MHMVIxlipconst} D_{1,t,x}:=2\sup_y\left|\frac{g'(r_{\mu_t}(x,y))q(y,x)}{\mu_t(x)}\right|, \quad \text{and} \quad D_{2,t,x}:=2\int\frac{\mu_t(z)}{\mu_t(x)^2}q(z,x)g'(r_{\mu_t}(x,z))\lambda(dz), \end{aligned}$$ then $P_\cdot(x,\cdot)$ has a bounded derivative at $\mu$ towards $\nu$ with Lipschitz constants $M_{1,x}.M_{2,x}$ and we have the mean value inequality $$\label{eq:MHMVIx} \norm{P_\mu(x,\cdot)-P_\nu(x,\cdot)}_{\rm tv} \leq \norm{\mu-\nu}_{\rm tv} \int_0^1 D_{1,t,x}dt + |\mu(x)-\nu(x)|\int_0^1 D_{2,t,x}dt.$$* *Furthermore, if $\mathcal{Q}$ is a subset of $\mathcal{P}(\mathsf{X}\xspace)$ such that $M_\rho$ can be bounded uniformly in $\rho\in\mathcal{Q}$, or $\mathcal{Q} = \{ \delta_x : x \in \mathfrak{X}\}$ and $M_{1,x},M_{2,x}$ can be bounded uniformly in $x\in \mathfrak{X}$, then $P_\cdot(\rho,\cdot)$ has a $\mathcal{Q}$-uniformly bounded derivative at $\mu$ towards $\nu$.* *Proof.* The expression for $D_{t,\rho}$ and [\[eq:MHMVI\]](#eq:MHMVI){reference-type="eqref" reference="eq:MHMVI"} are immediate from [\[prop:boundedderivativerho,prop:MHderivatives\]](#prop:boundedderivativerho,prop:MHderivatives){reference-type="ref" reference="prop:boundedderivativerho,prop:MHderivatives"} and the fact that if $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, $|f(y)-f(z)|\leq 2$ when $V=1$. To prove the inequality [\[eq:MHMVIx\]](#eq:MHMVIx){reference-type="eqref" reference="eq:MHMVIx"}, notice that under our assumptions we can use [Theorem 2](#prop:FTC){reference-type="ref" reference="prop:FTC"} and then [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"} to write $$\begin{aligned} \left|P_\mu(x,f)-P_\nu(x,f)\right| &= \left|\int_0^1 \partial_\pi P_{\mu_t}(x,f)[\nu-\mu]dt\right| \leq \norm{\mu-\nu} \int_0^1 D_{1,f,t,x}dt + |\mu(x)-\nu(x)|\int_0^1 D_{2,f,t,x}dt, \end{aligned}$$ with $$\begin{aligned} D_{1,f,t,x} := \sup_y\left|{(f(y)-f(x))\frac{g'(r_{\mu_t}(x,y))q(y,x)}{\mu_t(x)}}\right|\leq 2\sup_y\left|{\frac{g'(r_{\mu_t}(x,y))q(y,x)}{\mu_t(x)}}\right|=:D_{1,t,x}, \end{aligned}$$ $$\begin{aligned} D_{2,f,t,x} := \int (f(z)-f(x)) \frac{\mu_t(z)}{\mu_t(x)^2}q(z,x)g'(r_{\mu_t}(x,z))\lambda(dz) \leq 2\int\frac{\mu_t(z)}{\mu_t(x)^2}q(z,x)g'(r_{\mu_t}(x,z))\lambda(dz)=:D_{2,t,x}. \end{aligned}$$ Taking the supremum over functions in $\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ and using [Lemma 3](#app:lemma:VnormsupCb){reference-type="ref" reference="app:lemma:VnormsupCb"} one obtains [\[eq:MHMVIx\]](#eq:MHMVIx){reference-type="eqref" reference="eq:MHMVIx"}. ◻ Putting together [\[prop:MHderivatives,prop:MHboundedder,lemma:HastingsGibbsFTCisOK\]](#prop:MHderivatives,prop:MHboundedder,lemma:HastingsGibbsFTCisOK){reference-type="ref" reference="prop:MHderivatives,prop:MHboundedder,lemma:HastingsGibbsFTCisOK"} yields the following. **Corollary 1**. *Let $\rho\in\mathcal{W}_{H}$. For all indexed $\mu,\nu$, the Hastings kernel $P_\cdot(\rho,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$ and $M_{\rho}<\infty$, where $M_{\rho}$ is given in [\[eq:MHMVIlipconst\]](#eq:MHMVIlipconst){reference-type="eqref" reference="eq:MHMVIlipconst"}. Therefore, $P_\cdot(\rho,\cdot)$ has a bounded derivative in the invariant distribution at $\mu$ towards $\nu$, and the mean value inequality [\[eq:MHMVI\]](#eq:MHMVI){reference-type="eqref" reference="eq:MHMVI"} holds.* **Corollary 2**. *Assume [\[hp:HastingsHP\]](#hp:HastingsHP){reference-type="ref" reference="hp:HastingsHP"}. For all indexed $\mu,\nu$, the Hastings kernel $P_\cdot(x,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$ and $M_{1,x},M_{2,x}<\infty$, where $M_{1,x},M_{2,x}$ are given in [\[eq:MHMVIxlipconst\]](#eq:MHMVIxlipconst){reference-type="eqref" reference="eq:MHMVIxlipconst"}. Therefore, $P_\cdot(\rho,\cdot)$ has a bounded derivative in the invariant distribution at $\mu$ towards $\nu$, and the mean value inequality [\[eq:MHMVIx\]](#eq:MHMVIx){reference-type="eqref" reference="eq:MHMVIx"} holds.* Although [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"} does not guarantee the differentiability of Metropolis-Hastings kernels in the invariant distribution, we can still obtain Lipschitz inequalities for such kernels indirectly by looking at Metropolis-Hastings kernels as limit of a sequence of differentiable ones. The function $g_j(x):=(x+\dots+x^j)/(1+x+\dots+x^j)$ is a proper balancing function, it yields a differentiable Markov family, and it satisfies $g_j(x)\rightarrow \min(1,x)=:g(x)$ for all $x>0$ [@agrawal2023optimal]. Let $\{P_\star\}$ be now a Metropolis-Hastings family, with the same restrictions on the indexed invariants and $q$ as above. **Proposition 8**. *For all $\rho\in\mathcal{W}_{H}$ the Lipschitz inequality [\[eq:MHMVI\]](#eq:MHMVI){reference-type="eqref" reference="eq:MHMVI"} holds with $$\label{eq:MHrealMVI} D_{t,\rho}:= 2 \sup_y\left|\int \frac{\rho(z)}{\mu_t(z)}q(y,z)\lambda(dz) + \frac{\rho(y)}{\mu_t(y)^2}\int {\mu_t}(z)q(z,y)\lambda(dz)\right|.$$ For all $x\in\mathsf{X}\xspace$ the Lipschitz inequality [\[eq:MHMVIx\]](#eq:MHMVIx){reference-type="eqref" reference="eq:MHMVIx"} holds with $$\label{eq:MHrealMVIx} D_{1,t,x}:= 2\sup_y \left|\frac{q(y,x)}{\mu_t(x)}\right| \quad \text{and} \quad D_{2,t,x}:= 2\int \frac{\mu_t(z)}{\mu_t(x)^2}q(z,x)1_{\{z:r_{\mu_t}(x,z)\leq 1\}}(z)\lambda(dz).$$* *Proof.* Let $P_{\cdot,j}$ be the differentiable Hastings kernel employing the balancing function $g_j$. For $\rho\in\mathcal{W}_{2,H}$ proceeding as in [\[prop:boundedderivativerho,prop:MHboundedder\]](#prop:boundedderivativerho,prop:MHboundedder){reference-type="ref" reference="prop:boundedderivativerho,prop:MHboundedder"} we obtain $|P_{\mu,j}(\rho,f)-P_{\nu,j}(\rho,f)| \leq \norm{\mu-\nu}_{\rm tv}\int_0^1 D_{t,\rho,j}dt$, where $D_{t,\rho,j}$ is as in [\[eq:MHMVIlipconst\]](#eq:MHMVIlipconst){reference-type="eqref" reference="eq:MHMVIlipconst"} with balancing function $g_j$. Noting then that $\sup_j g_j \leq 1$ we can take the limit $j\rightarrow \infty$ both sides. At the left hand side we obtain $|P_{\mu}(\rho,f)-P_{\nu}(\rho,f)|$ via bounded convergence. At the right hand side we can use $\sup_j |g'_j|\leq 1$ to bound $D_{t,\rho,j}\leq D_{t,\rho}$ and prove the first claim upon taking the supremum across $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$. The second claim is proved very similarly by following the second part of [Proposition 7](#prop:MHboundedder){reference-type="ref" reference="prop:MHboundedder"} and using $g_j$ as above, however, upon taking the limit $j\rightarrow \infty$ at the left hand side we also use the fact that $g'_j(x)\rightarrow 1_{\{x\leq 1\}}(x)$ and the bounded convergence theorem to obtain the displayed Lipschitz constants. ◻ Roughly speaking, we have shown that we can ensure that if $\mu$ and $\nu$ are close, two Hastings chain $P_\mu$ and $P_\nu$ move similarly with some proper choice of the initial distribution, the proposal distribution and the balancing function. [Proposition 7](#prop:MHboundedder){reference-type="ref" reference="prop:MHboundedder"} also allows to easily find conditions for $\mathcal{Q}$-uniformly bounded derivatives. If we let $\mathcal{Q}_{H}:=\{\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace):d\rho/d\mu^2, d\rho/d\nu^2\leq C\}$ for some finite $C$, then of course $d\rho/d\mu_t^2\leq C$ for all $t\in[0,1]$ and it is easy to see that we can bound $D_{t,\rho}$ uniformly in $\rho\in\mathcal{Q}_{H}$. This means that we can get two Hastings chain $P_\mu$ and $P_\nu$ to move similarly, uniformly in the starting distribution, provided that a warm-start condition holds, i.e. that they are not initialized too far into the tails of their invariant distributions. On the other hand, if we start our Markov chains from a point $x\in\mathsf{X}\xspace$, to get $P_\mu$ and $P_\nu$ to move alike uniformly in the starting point $x\in\mathsf{X}\xspace$ seems to be much harder on non-compact spaces, and conditions such as $\sup_x q(y,x)/\mu(x)^2, \sup_x q(y,x)/\mu(x)^2<\infty$ seem to be required. **Example 6** (Bounded derivatives and Mean Value inequalities for Gibbs kernels). **Proposition 9**. *Let $\{P_\star\}$ be a deterministic scan Gibbs family.* 1. *Let $\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$. If the Markov kernel $P_\cdot(\rho,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$, and it holds that $M_{\rho}:=\int_0^1 D_{t,\rho}dt<\infty$ with $$\begin{aligned} \label{eq:GibbsMVIlipconst} D_{\rho,t}:=\sup_y&\bigg(\int\frac{\rho_2(y_2)}{\mu_{2,t}(y_2)}P_{2|1,t}(y_1,dw_2)+ \int\frac{\rho_2(y_2)}{\mu_{2,t}(y_2)}P_{1|2,t}(y_2,dw_1)P_{2|1,t}(w_1,dw_2). \nonumber \\ & +\int \frac{\rho_2(du_2)}{\mu_{1,t}(y_1)}P_{1|2,t}(u_2,y_1) + \int \frac{\rho_2(du_2)}{\mu_{1,t}(y_1)} P_{1|2,t}(u_2,y_1)P_{2|1,t}(y_1,dw_2) \bigg) \end{aligned}$$ then $P_\cdot(\rho,\cdot)$ has a bounded derivative at $\mu$ towards $\nu$ with Lipschitz constant $M_\rho$ and we have the mean value inequality $$\label{eq:GibbsMVI} \norm{P_\mu(\rho,\cdot)-P_\nu(\rho,\cdot)}_{\rm tv} \leq \norm{\mu-\nu}_{\rm tv}\int_0^1 D_{\rho,t}dt.$$* 2. *If [\[hp:Fellertypechain\]](#hp:Fellertypechain){reference-type="ref" reference="hp:Fellertypechain"} holds, the Markov kernel $P_\cdot(x,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$, and it holds that $M_{1,x}:=\int_0^1 D_{1,t,x}dt<\infty$ and $M_{2,x}:=\int_0^1 D_{2,t,x,z}dt<\infty$ with $$\label{eq:GibbsMVIlipconstx1} D_{1,t,x}:=4\sup_y\bigg(\frac{P_{1|2,t}(x_2,y_1)}{\mu_{1,t}(y_1)}\bigg),$$ $$\label{eq:GibbsMVIlipconstx2} D_{2,t,x}:=2\bigg(\frac{1}{\mu_{2,t}(x_2)} + \int \frac{P_{1|2,t}(x_2,dw_1)P_{2|1,t}(w_1,dw_2)}{\mu_{2,t}(x_2)}\bigg),$$ then, $P_\cdot(x,\cdot)$ has a bounded derivative in the invariant distribution at $\mu$ towards $\nu$ with Lipschitz constants $M_{1,x},M_{2,x,z}$ and we have the mean value inequality $$\label{eq:GibbsMVIx} \norm{P_\mu(x,\cdot)-P_\nu(x,\cdot)}_{\rm tv} \leq \norm{\mu-\nu}_{\rm tv}\int_0^1 D_{1,t,x}dt + |\mu_2(x_2)-\nu_2(x_2)|\int_0^1 D_{2,t,x}dt$$* *Furthermore, if $\mathcal{Q}$ is a subset of $\mathcal{P}(\mathsf{X}\xspace)$ such that $M_\rho$ can be bounded uniformly in $\rho\in\mathcal{Q}$, or $\mathcal{Q} = \{ \delta_x : x \in \mathfrak{X}\}$ and $M_{1,x},M_{2,x}$ can be bounded uniformly in $x\in \mathfrak{X}$, then $P_\cdot(\rho,\cdot)$ has a $\mathcal{Q}$-uniformly bounded derivative at $\mu$ towards $\nu$.* *Proof.* This follows by basically the same arguments used in [Proposition 7](#prop:MHboundedder){reference-type="ref" reference="prop:MHboundedder"} by using [Proposition 5](#prop:Gibbsderivatives){reference-type="ref" reference="prop:Gibbsderivatives"} in place of [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"}. ◻ Combining [\[prop:Gibbsderivatives,prop:Gibbsboundedder,lemma:HastingsGibbsFTCisOK\]](#prop:Gibbsderivatives,prop:Gibbsboundedder,lemma:HastingsGibbsFTCisOK){reference-type="ref" reference="prop:Gibbsderivatives,prop:Gibbsboundedder,lemma:HastingsGibbsFTCisOK"} yields the following. The claim on the finiteness of the Lipschitz constants follows from Bayes' formula. **Corollary 3**. *Let $\rho\in\mathcal{W}_G$. For all indexed $\mu,\nu$, the Gibbs kernel $P_\cdot(\rho,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$ and $\int_0^1 D_{t,\rho}dt<\infty$, where $D_{t,\rho}$ is given in [\[eq:GibbsMVIlipconst\]](#eq:GibbsMVIlipconst){reference-type="eqref" reference="eq:GibbsMVIlipconst"}. Therefore, $P_\cdot(\rho,\cdot)$ has a bounded derivative in the invariant distribution at $\mu$ towards $\nu$, and the mean value inequality [\[eq:GibbsMVI\]](#eq:GibbsMVI){reference-type="eqref" reference="eq:GibbsMVI"} holds.* Similarly to the Hastings case, we can also envisage conditions to bound the Lipschitz constants uniformly in the starting distributions. In fact, recently, [@ascolani2023gibbs Proposition 2.4] derived the inequality $$\label{eq:ascolanizanellamvi} \norm{P_\mu(\rho,\cdot)-P_\nu(\rho,\cdot)}_{\rm tv} \leq 2 K M \norm{\mu-\nu}_{\rm tv}$$ for the deterministic scan Gibbs kernel, where $K$ represents the number of stages, for $\rho$ having a density and satisfying the warm-start conditions $d\rho_2/d\mu_2\leq M$, $d\rho_2/d\nu_2\leq M$ for some $M>0$. In fact, we can verify using Bayes formula that for all initial distributions with warm-start $\rho\in\mathcal{Q}_G:=\{\rho\in\mathcal{P}_\lambda(\mathsf{X}\xspace):d\rho_2/d\mu_2,d\rho_2/d\nu_2\leq M\}$ we have $M_{\rho}<\infty$, with $M_{\rho}$ given in [Proposition 9](#prop:Gibbsboundedder){reference-type="ref" reference="prop:Gibbsboundedder"} above, and that in particular $\sup_{\rho\in \mathcal{W}}M_{\rho}\leq 4M$. That is, by [Proposition 9](#prop:Gibbsboundedder){reference-type="ref" reference="prop:Gibbsboundedder"}, the deterministic Gibbs kernel has a $\mathcal{Q}_G$-uniformly bounded derivative at $\mu$ towards $\nu$ and the mean value inequality [\[eq:ascolanizanellamvi\]](#eq:ascolanizanellamvi){reference-type="eqref" reference="eq:ascolanizanellamvi"} holds, providing an alternative proof of the result in [@ascolani2023gibbs] in the case $K=2$ studied here (and the argument could be extended to the general case at the expense of introducing some slightly cumbersome notation). As noted in [@ascolani2023gibbs], under warm-start we do not have to require any other regularity from the full conditionals $P_{1|2},P_{2|1}$ to obtain this mean value inequality. If we also require some extra regularity we can also obtain the following. **Corollary 4**. *Assume [\[hp:GibbsHP\]](#hp:GibbsHP){reference-type="ref" reference="hp:GibbsHP"}. For all indexed $\mu,\nu$, the Gibbs kernel $P_\cdot(x,\cdot)$ is differentiable at $\mu_t:=(1-t)\mu+t\nu$ for all $t\in[0,1]$ and $\int_0^1 D_{1,t,x}dt,\int_0^1 D_{2,t,x}dt<\infty$, where $D_{1,t,\rho},D_{2,t,\rho}$ are given in [\[eq:GibbsMVIlipconstx1\]](#eq:GibbsMVIlipconstx1){reference-type="eqref" reference="eq:GibbsMVIlipconstx1"} and [\[eq:GibbsMVIlipconstx2\]](#eq:GibbsMVIlipconstx2){reference-type="eqref" reference="eq:GibbsMVIlipconstx2"}. Therefore, $P_\cdot(\rho,\cdot)$ has a bounded derivative in the invariant distribution at $\mu$ towards $\nu$, and the mean value inequality [\[eq:MHMVIx\]](#eq:MHMVIx){reference-type="eqref" reference="eq:MHMVIx"} holds.* As in the Hastings case, requiring uniformity in the starting point will typically restrict the results applicability to compact spaces. ### Fluctuations of Markov kernels One can use the kernel derivatives defined to infer weak laws for Markov kernels using random invariant distributions without much trouble. This result is potentially useful to design efficient MCMC algorithms when using approximations instead of an invariant distribution of interest. In the preceding section, we argued that one can ensure that the approximated chain move similarly to the limiting one by choosing a Markov kernel with a bounded derivative. An alternative is to minimize fluctuations of the approximated kernel around the limiting one. These fluctuations depend directly on the derivative in the invariant distribution. **Proposition 10**. *Let $\mathcal{G}(\mathsf{X}\xspace)$ be some function class. Let $P_\cdot(\rho,f)$ be a Markov kernel differentiable in $\mu$, with $\partial_\pi P_{\mu}(\rho,f)$ admitting a density. Suppose that $\{\mu_n\}_{n\in\mathbb{N}} \in \mathcal{P}(\mathsf{X}\xspace)$ are random measures w.r.t. $\mathbb{P}$ and suppose that the random field $\{\sqrt{n}(\mu_n-\mu)(f);f\in\mathcal{G}(\mathsf{X}\xspace)\}$ converges in law to a Gaussian random field $\{ N(f); f\in\mathcal{G}(\mathsf{X}\xspace)\}$ specified by $\mathbb{E}(N(f)) = m(f)$ and $\mathbb{E}(N(f)N(g))=v(f,g)$. Then, if $\partial_\pi P_{\mu}(\rho,f)\in\mathcal{G}(\mathsf{X}\xspace)$, $$\label{eq:Pweaklawviadiff} \sqrt{n}(P_{\mu_{n}}(\rho,f) - P_\mu(\rho,f)) \Rightarrow N\left(\partial_\pi P_{\mu}(\rho,f)\right).$$ I.e. the random field $\{\sqrt{n}(P_{\mu_{n}}(\rho,f) - P_\mu(\rho,f)),f\in\mathcal{G}(\mathsf{X}\xspace)\}$ converges in law to a Gaussian random field $\{ N'(f); f\in\mathcal{G}(\mathsf{X}\xspace)\}$ specified by $\mathbb{E}(N'(f)) = m\left(\partial_\pi P_{\mu}(\rho,f)\right)$ and $\mathbb{E}(N'(f)N'(g))=v\left(\partial_\pi P_{\mu}(\rho,f),\partial_\pi P_{\mu}(\rho,g)\right)$.* *Proof.* Denoting $\mu_{t,n}:=(1-t)\mu+t\mu_n$, we have by definition $$|P_{\mu_{t,n}}(\rho,f) - P_\mu(\rho,f) - \partial_\pi P_{\mu}(\rho,f)[\mu-\mu_{t,n}]| = o(t)$$ as $t\rightarrow0$. Now, whenever $\partial_\pi P_{\mu}(\rho,f)$ admits a density such that $\partial_\pi P_{\mu}(\rho,f)(\cdot)\in\mathcal{G}(\mathsf{X}\xspace)$ one has $$\sqrt{n}\partial_\pi P_{\mu}(\rho,f)[\mu_n-\mu] = \sqrt{n}\left(\mu_n-\mu\right)\left(\partial_\pi P_{\mu}(\rho,f)\right) \Rightarrow N\left(\partial_\pi P_{\mu}(\rho,f)\right),$$ and the claim follows. ◻ # Applications to Monte Carlo {#sec:applications} We now demonstrate the applications of the tools developed in the previous section to Monte Carlo Methods. The tools we developed allow an easy comparison between Markov Chains of the same family with different invariant distributions. Usually we think of one of these distributions as an approximation of the other. Therefore, other than being of theoretical interest, this framework is naturally appealing to study the use of approximations in Markov Chain Monte Carlo. In this context, we compare a Markov Chain targeting a limiting distribution of interest, $\mu$, with a Markov Chain targeting an approximation thereof. Using an MCMC kernel with an approximation is common and can occur in a number of contexts. For instance, when $\mu$ is intractable, or when $\mu$ is tractable, but rather inefficient to target directly, so one targets an approximation instead. For instance, (i) Griddy Gibbs Sampling [@griddygibbs1992] proposes to use "griddy" approximations of the full conditionals when sampling from the latter is not possible; (ii) [@papamakarios2021normalizingflows] proposes to use Normalizing Flows to smooth away distributions that are complicated to sample from via MCMC---see also [@gabrie2021normalizingflows] for an application; (iii) [@marzouk2007stochasticspectr; @marzouk2009stochasticcollocation] employ a stochastic collocation approach to approximate complicate target distribution to target via Metropolis-Hastings. These approximations could be evolved or not as the Markov Chain runs, or the output of the resulting MCMC can be used to approximate the invariant distribution of another MCMC, giving rise to the framework for Sequential MCMC and Interacting MCMC we study in [\[sec:smcmc,sec:imcmc\]](#sec:smcmc,sec:imcmc){reference-type="ref" reference="sec:smcmc,sec:imcmc"}. In these two settings belong (iv) Sequential Particle Filters [@berzuini1997smcmc; @golightly2006smcmc; @septier2009mcmc; @li2023smcmc] and [Example 7](#avar:ex:smcmc){reference-type="ref" reference="avar:ex:smcmc"}, based on running a sequence of MCMC algorithms, each with an approximate invariant distribution obtained with another MCMC output; and (v) Interacting Particle Filters [@andrieu2007nonlinearmcmc; @atchade2010cautionarytale; @brockwell2010simcmc; @delmoraldoucet2010iMCMC; @fort2011convergence] and [Example 8](#avar:ex:imcmc){reference-type="ref" reference="avar:ex:imcmc"} where we run an inhomogeneous Markov Chain of which invariant distribution evolves, perhaps depending on another Markov Chain running on another level. Another natural context would be the Uncertainty Quantification literature, where the ideal distribution $\mu$ to target as a distribution over a function space, and one instead targets $\mu_n$ being measures over discretisations. Let us denote with $\nu$ an approximation of $\mu$. It might be of interest to ensure that the Markov chains $P_\mu$ and $P_{\nu}$ move similarly. We have seen that this can be ensured when the derivative in the invariant distribution at $\mu$ towards $\nu$ is bounded. Also, it might be of interest to study the rate of convergence of $P_\mu$ to $P_{\nu}$, and this can be done via the Mean-Value inequalities of [2.4](#sec:mvi){reference-type="ref" reference="sec:mvi"}. If $\nu$ is random, perhaps is also of interest to minimize the fluctuations of the kernel $P_{\nu}$ around $P_\mu$---see [Proposition 10](#prop:weaklaw){reference-type="ref" reference="prop:weaklaw"}. For Hastings type kernels, these strategies can be for instance implemented by choosing the initial distribution, the proposal $q$ and the balancing function $g$ accordingly, see [Example 5](#ex:MHboundedder){reference-type="ref" reference="ex:MHboundedder"}. As an illustration of why it might be of interest to ensure that $P_\mu$ and $P_{\nu}$ move similarly, we show in the next section that (under suitable regularity conditions) this ensures that estimators associated with the two associated Markov chains achieve similar asymptotic variances, up to an additional variability due to the fluctuations of $\nu$ around $\mu$. ## Efficiency of MCMC with approximations {#sec:AVconv} In this section we combine MCMC Calculus tools with results from [@fort2011convergence] and [@fort2014clt] to study the efficiency of Markov chains employing approximations in place of a limiting distribution of interest. Let $\{P_\star\}$ be a Markov Family. Consider a "limiting" Markov chain $\{X_k\}_{k\in\mathbb{N}}$ with transitions $X_{k+1}|X_{k}=x_{k} \sim P_\mu(x_{k},\cdot)$, targeting a distribution $\mu$ of interest, that might be intractable or inefficient to target directly. We consider two approaches to approximate $\{X_k\}_{k\in\mathbb{N}}$. In the first we consider a situation in which one has access to a sequence $\{\mu_n\}$ of increasingly good approximations of $\mu$ and study the convergence of a sequence of approximating chains which make use of this sequence as its invariant measures; that is, it comprises a triangular array of Markov chains $\{Y_{n,k}\}_{k\leq n}$, where for each $k\leq n$, $Y_{n,k+1}|Y_{n,k}=y_{n,k} \sim P_{\mu_n}(y_{n,k},\cdot)$. Since in our examples below $\mu_n$ comes from a previously run chain, we refer to this scheme as Sequential MCMC (sMCMC). In this framework belong [@berzuini1997smcmc; @golightly2006smcmc; @septier2009mcmc; @li2023smcmc] among others. The second approach to approximate $\{X_k\}_{k\in\mathbb{N}}$ is to consider an (inhomogeneous) Markov chain $\{Z_k\}_{k\in\mathbb{N}}$ targeting at each $k$ a distribution $\mu_k$ that is refined at each step and that is getting closer to $\mu$ i.e. with transitions $Z_{k+1}|Z_{k}=z_{k} \sim P_{\mu_{k}}(z_{k},\cdot)$. We refer to this scheme as Interacting MCMC (iMCMC). In this framework belong [@andrieu2007nonlinearmcmc; @atchade2010cautionarytale; @brockwell2010simcmc; @delmoraldoucet2010iMCMC; @fort2011convergence] among others. We are interested in studying the fluctuations of $n^{-1/2}\sum_{i=1}^n f(Y_{n,i})$ and $n^{-1/2}\sum_{i=1}^n f(Z_i)$ around $\mu(f)$ for some test functions $f$ to be specified. Studying these fluctuations desirable as it provides information on the properties of sampled values viewed as estimators of expectations w.r.t. $\mu$ and characterizes the additional variability of estimators arising from using Markov kernels targeting (random) approximations of distributions rather than the distributions themselves. Fluctuations for these algorithms have already been studied in some of the papers cited above, but we express the results somehow naturally in terms of boundedness of the derivatives in the invariant distributions and the quality of the approximation scheme $\mu_n\rightarrow \mu$, and in some cases we improve on existing results. For both sMCMC and iMCMC, we assume that $\mu_n\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$ for each $n\in\mathbb{N}$, and we employ the following standard (if strong) assumptions. 1. 1. For all $n\geq 1$, $P_{\mu_n}$ are $\mu_n$-irreducible, aperiodic Markov kernels. Furthermore, there exists a function $V:\mathsf{X}\xspace\mapsto [1,\infty)$ and constants $b<\infty$, $\lambda\in (0,1)$ such that for all $n \geq 1$: $$P_{\mu_n}V \leq \lambda V(x) + b1_C(x),$$ where $C:=\{x:V(x)\leq d\}$ for some $d\geq b/(2(1-\lambda))-1$. [\[avar:hp:drift\]]{#avar:hp:drift label="avar:hp:drift"} 2. There exist an integer $j\geq 1$, a positive constant $\kappa_n$ with $\inf_n \kappa_n>0$ and a probability measure $\upsilon_n$ such that for all $x\in C$, $P^j_{\mu_n}(x,A)\geq \kappa_n \upsilon_n(A)$ for all $A\in\mathcal{B}(\mathsf{X}\xspace)$. [\[avar:hp:minorization\]]{#avar:hp:minorization label="avar:hp:minorization"} [\[avar:hp:unifergodicity\]]{#avar:hp:unifergodicity label="avar:hp:unifergodicity"} 2. There exist an integer $j>1$ such that $\sup_n \mu_n(V^j)<\infty$, $\sup_n \mathbb{E}(V^j(Z_n))<\infty$ (for $\{Z_k\}_{k\in\mathbb{N}}$) and $\sup_{n,k} \mathbb{E}(V^j(Y_{n,k}))<\infty$ (for $\{Y_{n,k}\}_{k\leq n}$). [\[avar:hp:Vgrowth\]]{#avar:hp:Vgrowth label="avar:hp:Vgrowth"} [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} is an uniform (in the sequence of invariant distributions $\{\mu_n\}_{n\in\mathbb{N}}$) geometric drift and minorization assumption, similar to the conditions considered in [@andrieu2001sa] and in the adaptive MCMC literature (e.g. [@fort2011convergence; @fort2014clt]). For the algorithms under examination, we can in fact just look at the invariant distributions as an adaptation parameter living in the infinite dimensional space $\mathcal{P}(\mathsf{X}\xspace)$. Using the techniques in [@fort2011convergence; @fort2014clt] it is also possible to let $(\lambda,b)$ depend on $n$ in some judicious way, but we require their uniformity w.r.t. $n$ to allow the arguments which follow to focus upon the use of MCMC Calculus tools rather than technical details. Verifying [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} requires that $\mu_n$ does not behave too differently with $n$. By way of an illustration, to verify the minorization condition [\[avar:hp:minorization\]](#avar:hp:minorization){reference-type="ref" reference="avar:hp:minorization"}, following [@mengersen1996rates Lemma 1.2] one has $P_{\mu_n}(x,A)\geq (\varepsilon/b_n) \mu_n(A)$, with $\varepsilon:=\inf_{x,y\in B}q(x,y)$, $b_n:=\sup_{x\in B}\mu_n(x)$, $B$ being any compact set with positive Lebesgue measure. Therefore, it suffices to require that the approximation sequence satisfies $\inf_n \sup_{x\in B}\mu_n(x)>0$, which is arguably not hard to verify in practice. If this holds, then any compact set is small, and adapting [@jarner2000geometric Lemma 3.5] one can verify [\[avar:hp:drift\]](#avar:hp:drift){reference-type="ref" reference="avar:hp:drift"} whenever $$\sup_n \limsup_{|x|\rightarrow \infty} \frac{P_{\mu_n}V(x)}{V(x)}<1 \quad \text{and} \quad \sup_n \sup_{x\in\mathsf{X}\xspace} \frac{P_{\mu_n}V(x)}{V(x)}< \infty$$ Which, adapting the proof of [@mengersen1996rates Theorem 3.2], can be verified with the Lyapunov function $V(x)=\exp(\gamma|x|)$ for Random Walk Metropolis (and other Random Walk Hastings kernels) if all $\mu_n$ are uniformly log-concave in the tails i.e. when for all $n\geq 1$ there exist a $\gamma>0$ and $z>0$ such that $$\label{eq:uniflogconcave1} \log(\mu_n(x))-\log(\mu_n(y))\geq \gamma(y-x) \quad \text{for } y\geq x\geq z$$ $$\label{eq:uniflogconcave2} \log(\mu_n(x))-\log(\mu_n(y))\geq \gamma(x-y) \quad \text{for } y\leq x\leq -z.$$ See [Example 7](#avar:ex:smcmc){reference-type="ref" reference="avar:ex:smcmc"} later for a concrete setting where we verify this. In the case where $P_{\mu_n}$ is a Hastings Independence sampler $Q(x,dy)=Q(dy)$ and the proposal has heavier tail than any distribution in the approximation sequence in that $\inf_n dQ/d\mu_n(y) \geq \beta$ for some $\beta>0$, then the whole state space $\mathsf{X}\xspace$ is small for any $P_{\mu_n}$ simultaneously, and the drift condition will hold with a bounded $V$. [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"} is a growth condition which can actually always be verified with some drift function if [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} holds. In fact, via the drift condition [\[avar:hp:drift\]](#avar:hp:drift){reference-type="ref" reference="avar:hp:drift"} we have $$\mathbb{E}(V(Z_n)) \leq \lambda\mathbb{E}(V(Z_{n-1})) + b \leq \lambda^n \mathbb{E}(V(Z_{0})) + b \sum^{n-1}_{i=0}\lambda^i$$ where we just iterated the first inequality in $n$, and upon taking the supremum in $n$ we obtain that $\sup_n \mathbb{E}(V(Z_n))<\infty$ and also $\sup_n \mu_n(V)<\infty$ via [Lemma 4](#avar:lemma:geomerg){reference-type="ref" reference="avar:lemma:geomerg"} (i.e. [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"} holds with $j=1$). On the other hand, for $j>1$, we can adapt the proof of [@rosenthal2023geometric Proposition 6.5] and define $\Tilde{V}:=V^{1/j}$, so that $\sup_n \mu_n(\Tilde{V}^j)=\sup_n \mu_n(V)<\infty$. By Jensen's inequality, $$\begin{aligned} P_{\mu_n}\Tilde{V}\leq (P_{\mu_n}V)^{1/j} \leq (\lambda V+b1_C)^{1/j} \leq \Tilde{\lambda}V + \Tilde{b}1_C\end{aligned}$$ with $\Tilde{\lambda}:=\lambda^{1/j}$ and $\Tilde{b}:=b^{1/j}$. Hence, $\Tilde{V}$ also satisfies [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} and arguing as above we may verify $\sup_n \mathbb{E}(V^j(Z_n))<\infty$ and $\sup_n \mu_n(V^j)<\infty$. For $\{Y_{n,k}\}_{k\leq n}$ it is similar. ### Sequential MCMC {#sec:smcmc} For the Sequential MCMC sampling scheme $\{Y_{n,k}\}_{k\leq n}$ we employ the following assumptions. Let $V$ satisfy [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"}. 1. $\mu_n(x)\rightarrow\mu(x)$ $\mathbb{P}$ for all $x\in\mathsf{X}\xspace$. [\[avar:hp:ptwiseapprox\]]{#avar:hp:ptwiseapprox label="avar:hp:ptwiseapprox"} 2. The Markov kernel $P_\cdot(x,\cdot)$ is differentiable in the invariant distribution at $\mu$, and has a $V$-bounded derivative at $\mu$ towards every $\mu_n$. [\[avar:hp:boundedder\]]{#avar:hp:boundedder label="avar:hp:boundedder"} 3. As $n\rightarrow 0$, for all $f$ in some functions class $\mathcal{G}(\mathsf{X}\xspace)$ and for some variance functional $v$, $$n^{-1/2}[\mu_n-\mu](f) \Rightarrow N(0,v(f)).$$ [\[avar:hp:Yapproxnormality\]]{#avar:hp:Yapproxnormality label="avar:hp:Yapproxnormality"} [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} is a requirement on the nature of the approximation scheme, which has to occur pointwise. [\[avar:hp:boundedder\]](#avar:hp:boundedder){reference-type="ref" reference="avar:hp:boundedder"} will ensure that the Markov Chains using $\mu$ and $\mu_n$ will in fact not move too differently when [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} holds. In fact, it is sufficient that a mean value inequality [\[eq:boundedderimplicationx\]](#eq:boundedderimplicationx){reference-type="eqref" reference="eq:boundedderimplicationx"} holds, and by [Proposition 8](#prop:MHrealMVI){reference-type="ref" reference="prop:MHrealMVI"} also the Metropolis-Hastings kernels satisfy that under the boundedness conditions stated therein. [\[avar:hp:Yapproxnormality\]](#avar:hp:Yapproxnormality){reference-type="ref" reference="avar:hp:Yapproxnormality"} is a mild assumption that says that if the employed approximation is random w.r.t. some $\mathbb{P}$, it also has a limiting Gaussian fluctuations around $\mu$, and it will be fulfilled by most approximation schemes used in practice. Let $\sigma^2$ denote the asymptotic variance achieved by $\{X_k\}_{k\in\mathbb{N}}$. **Theorem 3**. *Assume [\[hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:ptwiseapprox,avar:hp:boundedder\]](#hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:ptwiseapprox,avar:hp:boundedder){reference-type="ref" reference="hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:ptwiseapprox,avar:hp:boundedder"}. For all $f\in \mathbf{C}^1_{V}(\mathsf{X}\xspace)$, $$\label{avar:eq:YrandomcenterCLT} n^{-1/2}\sum_{i=0}^n f(Y_{n,i})-\mu_n(f) \Rightarrow N(0,\sigma^2(f))$$ and if [\[avar:hp:Yapproxnormality\]](#avar:hp:Yapproxnormality){reference-type="ref" reference="avar:hp:Yapproxnormality"} holds too, for $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)\cap\mathcal{G}(\mathsf{X}\xspace)$, $$\label{avar:eq:YdetcenterCLT} n^{-1/2}\sum_{i=0}^n f(Y_{n,i})-\mu(f) \Rightarrow N(0,\sigma^2(f)+v^2(f)).$$* [Theorem 3](#avar:thm:YCLT){reference-type="ref" reference="avar:thm:YCLT"} tells us in fact $\{Y_{n,k}\}_{k\leq n}$ possess a limiting Normal law around $\mu$ as $n\rightarrow\infty$, with the asymptotic variance equal to the asymptotic variance of an ideal MCMC scheme targeting $\mu$ *plus* the variability due of the fluctuations of $\mu_n$. **Example 7** (Sequential MCMC for Feynman-Kac flow). [@berzuini1997smcmc; @golightly2006smcmc; @septier2009mcmc; @li2023smcmc] among others consider the following algorithm, recently theoretically analyzed by [@finke2020smcmc]. Let $\{G^{(p)}\}_{p\in\mathbb{N}}$ be a sequence of non-negative functions, and $\{M^{(p)}\}_{p\in\mathbb{N}}$ be a sequence of Markov kernel, and $\eta^{(1)}$ an initial distribution. [\[alg:mcmcpf\]]{#alg:mcmcpf label="alg:mcmcpf"} For $p=1$, 1. Simulate $Y^{(1)}_{i+1}\sim P_{\eta^{(1)}}(Y^{(1)}_{i},\cdot)$ for $i=0,\dots,n$. Set $\eta^{(1)}_n:=n^{-1}\sum_{i=0}^n \delta_{Y^{(1)}_i}$.\ For $p\geq 2$, 1. Simulate $Y^{(p)}_{n,i+1}\sim P_{\Phi(\eta^{(p-1)}_n)}(Y^{(p)}_{n,i},\cdot)$ for $i=0,\dots,n$. Set $\eta^{(p)}_n:=n^{-1}\sum_{i=0}^n \delta_{Y^{(p)}_{n,i}}$.\ In the algorithm above, at each level $p$, the random variable $Y^{(p)}_0$ is sampled according to some initial distribution possibly depending on $p$. Here, $\Phi(\eta^{(p)})(dx):= \int\eta(dy)G^{(p)}(y)M^{(p)}(y,dx)/\eta(G^{(p)})$ is the so-called Boltzmann-Gibbs transformation associated to the Feynman-Kac model $\{G^{(p)},M^{(p)}\}_{p\in\mathbb{N}}$. This is very flexible class of models, used perhaps most popularly in the filtering context within state space models, where one has an underlying unobserved Markovian state space process with some transition kernels $m^{(p)}$ at time $p$, and an observable process having likelihood $g^{(p)}$ at $p$. $\{G^{(p)},M^{(p)}\}_{p\in\mathbb{N}}$ can be identified with different relevant quantities related to $m^{(p)}$ and $g^{(p)}$, giving rise to different filters. The classic but usually inefficient choice $G^{(p)}=g^{(p)}$ and $M^{(p)}=m^{(p)}$ for all $p$ yields the Bootstrap particle filter, see [@delmoral2004fk; @chopin2020smc]. Sampling from $\Phi(\eta^{(p)})$ with a particle filter amounts to being able to evaluate pointwise $G$ and sample i.i.d. from $M$. When one tries to implement more sophisticated filters, this might not be possible. In such case, a natural choice is to try to target $\Phi(\eta^{(p)})$ with a MCMC algorithm, giving rise to the algorithm above. By targeting otherwise non implementable filters, this class of algorithms can outperform standard particle filters in some settings---see [@finke2020smcmc], which gives the only CLT for $\sqrt{n}[\eta^{(p)}_n-\eta^{(p)}](f)$ of which we are aware. This CLT holds under conditions that are both restrictive and difficult to verify, and that have only been shown to hold for independence samplers in finite spaces. We can extend their results using our theory. We prove below a CLT for bounded test functions for ease of exposition, but one can do much better using the theorems above. We consider the following very weak assumption regarding the Feynman-Kac flow. 1. $G^{(p)}$ is positive and $M^{(p)}(x,\cdot)$ has a bounded density for all $x\in\mathsf{X}\xspace$ and $p\in\mathbb{N}$. [\[avar:hp:FKflow\]]{#avar:hp:FKflow label="avar:hp:FKflow"} Provided an LLN holds for the first Markov Chain $\{Y^{(1)}_k\}_{k\in\mathbb{N}}$, for each subsequent Chain $\{Y^{(p)}_{n,k}\}_{k\leq n},p>1$ we are in the setting studied for the process $\{Y_{n,k}\}_{k\leq n}$ earlier. In particular, suppose that [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} holds. If follows that the Markov Chain $\{Y^{(1)}_{k}\}_{k\in\mathbb{N}},t>1$ satisfies a LLN and a CLT in that $\eta^{(1)}_n(f)\rightarrow\eta^{(1)}(f)$ $\mathbb{P}$ and $\sqrt{n}[\eta^{(1)}_n-\eta^{(1)}](f)\Rightarrow N(0,\sigma_{\eta^{(1)}}(f))$ for all $f\in\mathbf{C}^1_1$. By the decomposition $$\label{avar:eq:FKdec} [\Phi(\eta^{(1)}_n)-\Phi(\eta^{(1)})](f) = [\eta^{(1)}_n-\eta^{(1)}](\bar{Q}^{(1)}(f-\Phi(\eta^{(1)}_n)(f)))$$ with the integral operator $\bar{Q}^{(p)}(x,f):=G^{(p)}(x) M^{(p)}(x,f)/\eta^{(p)}(G^{(p)})$, under [\[avar:hp:FKflow\]](#avar:hp:FKflow){reference-type="ref" reference="avar:hp:FKflow"} one then also has $\Phi(\eta^{(1)}_n)(x)\rightarrow \Phi(\eta^{(1)})(x)$ $\mathbb{P}$ for all $x\in\mathsf{X}\xspace$, $\Phi(\eta^{(1)}_n)(x)\rightarrow \Phi(\eta^{(1)})(x)$ for all $x\in\mathsf{X}\xspace$ $\mathbb{P}$ by separability and $\sqrt{n}[\Phi(\eta^{(1)}_n)-\Phi(\eta^{(1)})](f)\Rightarrow N(0,\sigma^2_{\eta^{(1)}}(\bar{Q}^{(1)}(f-\Phi(\eta^{(1)})(f)))$. If $P$ has a bounded derivative at $\Phi(\eta^{(1)})$ towards every $\Phi(\eta^{(1)}_n)$ then [Theorem 3](#avar:thm:YCLT){reference-type="ref" reference="avar:thm:YCLT"} shows $$(f)\Rightarrow N\left(0,\sigma^2_{\eta^{(2)}}\left(f-\eta^{(2)}(f)\right)+\sigma^2_{\eta^{(1)}}\left(\bar{Q}^{(1)}(f-\eta^{(2)}(f)\right)\right).$$ Repeating this argument allows us to establish the following result by induction. **Theorem 4**. *Let $p\in\mathbb{N}$ and $f\in\mathbf{C}^1_1$. Let $\{P_\star\}$ be a Markov family such that $\Phi(\eta^{(j)}),\Phi(\eta^{(j)}_n)$ for all $j\leq p$ belong to its index set, and for which [\[hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:Vgrowth\]](#hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:Vgrowth){reference-type="ref" reference="hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:Vgrowth"} hold. If $P_\cdot$ has a bounded derivative at $\Phi(\eta^{(j)})$ towards every $\Phi(\eta^{(j)}_n)$ for all $j\leq p$ and if the Feynman-Kac model $\{G^{(p)},M^{(p)}\}_{p\in\mathbb{N}}$ satisfies [\[avar:hp:FKflow\]](#avar:hp:FKflow){reference-type="ref" reference="avar:hp:FKflow"} it holds $$(f)\Rightarrow N\left(0,\sum_{j=1}^p \sigma^2_{\eta^{(j)}}\left(\bar{Q}^{(j:p)}(f-\eta^{(j)}(f)\right)\right).$$ where we defined $\bar{Q}^{(j:p)}(f):=[\bar{Q}^{(j+1)} \circ \cdots \circ \bar{Q}^{(p)}](f)$ for $j<p$ and $Q^{(j:p)}(f)=\ensuremath{\textrm{Id}}\xspace$ for $j=p$.* For a concrete setting where this theorem is applicable, consider the following simple example, which can be easily extended. Let $p\in\mathbb{N}$ and let $\{P_\star\}$ be the Metropolis-Hastings kernel. Consider the State-Space model given by the latent process $W_{j+1}|W_{j}\sim N(w_{j+1};\varphi(w_j),1/2)$ and the observable $S_{j+1}|W_{j+1}\sim N(s_{j+1};w_{j+1},1/2)$, where $\varphi$ is a function bounded away from $-\infty$ and $+\infty$, say $-\bar{\varphi}\leq \varphi(x)\leq \bar{\varphi}$ for some finite and positive $\bar{\varphi}$. State-space models provide a very flexible way to model quantities that evolve in time, and that are widely used in engineering, physics, quantitative finance and other fields, see e.g. [@chopin2020smc] for more details. Let us consider a Bootstrap interpretation of the Feynman-Kac flow, by identifying the Markov kernels $M^{(p)}$ with $W$'s transitions, and the potential functions $G^{(p)}$ with the likelihoods of the observable. With this choice, it is easy to see that [\[avar:hp:FKflow\]](#avar:hp:FKflow){reference-type="ref" reference="avar:hp:FKflow"} always hold and that $\Phi(\eta^{(j)}),\Phi(\eta^{(j)}_n)$ are for all $n\geq1$, $j\leq p$, continuously differentiable bounded above densities, and bounded away from 0 on any compact set. Hence, by [Proposition 8](#prop:MHrealMVI){reference-type="ref" reference="prop:MHrealMVI"}, $P_\cdot(x,\cdot)$ satisfies a mean value inequality at $\Phi(\eta^{(j)})$ towards every $\Phi(\eta^{(j)}_n)$ for all $j\leq p$. We now verify [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} with $V(x)=\exp(\gamma|x|)$ for any $\gamma>0$ by verifying [\[eq:uniflogconcave1\]](#eq:uniflogconcave1){reference-type="eqref" reference="eq:uniflogconcave1"} and [\[eq:uniflogconcave2\]](#eq:uniflogconcave2){reference-type="eqref" reference="eq:uniflogconcave2"} hence showing that $\Phi(\eta^{(j)}_n)$ are for all $n,j\geq1$ uniformly log-concave in the tails. Let $z:=\bar{\varphi}+\gamma/2$. If $y\geq x\geq z$, for $x,y\in\mathsf{X}\xspace$, we compute $$\frac{\Phi(\eta^{(j)}_n)(x)}{\Phi(\eta^{(j)}_n)(y)} = \frac{\sum_{i=1}^n e^{-(s_j-Y^{(j)}_i)^2}e^{-(x-\varphi(Y^{(j)}_i))^2}}{\sum_{i=1}^n e^{-(s_j-Y^{(j)}_i)^2}e^{-(y-\varphi(Y^{(j)}_i))^2}} = e^{-x^2+y^2} \frac{\sum_{i=1}^n e^{-(s_j-Y^{(j)}_i)^2+2x\varphi(Y^{(j)}_i) - \varphi(Y^{(j)}_i)^2}}{\sum_{i=1}^n e^{-(s_j-Y^{(j)}_i)^2+2y\varphi(Y^{(j)}_i) - \varphi(Y^{(j)}_i)^2}}.$$ Now, since $e^{-(s_j-Y^{(j)}_i)^2+2x\varphi(Y^{(j)}_i) - \varphi(Y^{(j)}_i)^2} / e^{-(s_j-Y^{(j)}_i)^2+2y\varphi(Y^{(j)}_i) - \varphi(Y^{(j)}_i)^2} = e^{2\varphi(Y^{(j)}_i)(x-y)}\geq e^{(2z-\gamma)(x-y)}$, where the last inequality follows from $\varphi(Y^{(j)}_i)\leq z-\gamma/2$, we have $$\frac{\Phi(\eta^{(j)}_n)(x)}{\Phi(\eta^{(j)}_n)(y)} \geq e^{-x^2+y^2} e^{(2z-\gamma)(x-y)} = e^{(y-x)(y+x-2z+\gamma)} \leq e^{\gamma (y-x)},$$ thus verifying [\[eq:uniflogconcave1\]](#eq:uniflogconcave1){reference-type="eqref" reference="eq:uniflogconcave1"}. [\[eq:uniflogconcave2\]](#eq:uniflogconcave2){reference-type="eqref" reference="eq:uniflogconcave2"} follows from analogous computations when considering the case $y\leq x\leq -z$. ### Interacting MCMC {#sec:imcmc} For $\{Z_k\}_{k\in\mathbb{N}}$ we strengthen the assumptions to 1. $n^{-1/2}\sum_{k=1}^{n} \sup_x|\mu_{k}(x)-\mu_{k-1}(x)| \rightarrow 0$ and $n^{-1/2}\sum_{k=1}^{n} \norm{\mu_k-\mu_{k-1}}_{V} \rightarrow 0$ $\mathbb{P}$ [\[avar:hp:unifapprox\]]{#avar:hp:unifapprox label="avar:hp:unifapprox"} 2. The Markov kernel $P_\cdot(x,\cdot)$ is differentiable in the invariant distribution at $\mu$, and has a $(V,\{\delta_x\}_{x\in\mathsf{X}\xspace})$-uniformly bounded derivative at $\mu_n$ towards $\mu_{n-1}$ for all $n\geq 1$. [\[avar:hp:unifboundedder\]]{#avar:hp:unifboundedder label="avar:hp:unifboundedder"} 3. As $n\rightarrow 0$, for all $f$ in some functions class $\mathcal{G}(\mathsf{X}\xspace)$ and for some variance functional $w$, $$n^{-1/2}\sum_{k=1}^{n}[\mu_k-\mu](f) \Rightarrow N(0,w(f)).$$ [\[avar:hp:Zapproxnormality\]]{#avar:hp:Zapproxnormality label="avar:hp:Zapproxnormality"} The first condition [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"} strengthen [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} requiring that the approximation schemes is uniformly convergent on the space, and that the convergence occurs fast-enough. We believe that the second condition of [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"} will often follow from the first, just as [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} implies $\norm{\mu_n-\mu}_{V}\rightarrow0$ $\mathbb{P}$(see [Lemma 6](#app:lemma:ptwisetoV){reference-type="ref" reference="app:lemma:ptwisetoV"}) and this is quite immediate if $\mathsf{X}\xspace$ is compact. [\[avar:hp:unifboundedder\]](#avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifboundedder"} strengthens [\[avar:hp:boundedder\]](#avar:hp:boundedder){reference-type="ref" reference="avar:hp:boundedder"} to require that the "Lipschitz constants" of the differentiable Markov kernel $P$ are bounded in the starting point as per [Definition 7](#def:unifboundedderivative){reference-type="ref" reference="def:unifboundedderivative"}. [\[avar:hp:Zapproxnormality\]](#avar:hp:Zapproxnormality){reference-type="ref" reference="avar:hp:Zapproxnormality"} is an asymptotic normality assumption concerning the approximation scheme that will often hold in the same situations as those in which [\[avar:hp:Yapproxnormality\]](#avar:hp:Yapproxnormality){reference-type="ref" reference="avar:hp:Yapproxnormality"} does---see [Example 8](#avar:ex:imcmc){reference-type="ref" reference="avar:ex:imcmc"}. **Theorem 5**. *Let $\alpha\in(0,1/2)$. If [\[hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:Vgrowth\]](#hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:Vgrowth){reference-type="ref" reference="hp:Fellertypechain,avar:hp:unifergodicity,avar:hp:Vgrowth"}, as well as [\[avar:hp:unifapprox,avar:hp:unifboundedder\]](#avar:hp:unifapprox,avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifapprox,avar:hp:unifboundedder"}, hold, then for all $f\in \mathbf{B}_{V^\alpha}(\mathsf{X}\xspace)$, $$\label{avar:eq:ZrandomcenterCLT} n^{-1/2}\sum_{i=1}^n f(Z_i)-\mu_{i-1}(f) \Rightarrow N\left(0,\sigma^2(f)\right);$$ and if [\[avar:hp:Zapproxnormality\]](#avar:hp:Zapproxnormality){reference-type="ref" reference="avar:hp:Zapproxnormality"} holds too, for $f\in\mathbf{B}_{V^\alpha}(\mathsf{X}\xspace)\cap\mathcal{G}(\mathsf{X}\xspace)$, $$\label{avar:eq:ZdetcenterCLT} n^{-1/2}\sum_{i=1}^n f(Z_i)-\mu(f) \Rightarrow N\left(0,\sigma^2(f)+w^2(f)\right).$$* [Theorem 5](#avar:thm:ZCLT){reference-type="ref" reference="avar:thm:ZCLT"} tells us that $\{Z_k\}_{k\in\mathbb{N}}$ also possess a limiting Normal law, with asymptotic variance equal to that of an ideal MCMC scheme targeting $\mu$ plus someadditional variability due the approximation. This latter is often greater than the one we obtain with a Sequential MCMC scheme, as in the Example below. **Example 8** (Interacting MCMC for Feynman-Kac flow). [@andrieu2007nonlinearmcmc; @atchade2010cautionarytale; @brockwell2010simcmc; @delmoraldoucet2010iMCMC; @fort2011convergence] study the following type of algorithm. [\[alg:imcmc\]]{#alg:imcmc label="alg:imcmc"} For $n=1,...$ 1. Simulate $Z^{(1)}_n\sim P_{\eta^{(1)}}(Z^{(1)}_{n-1},\cdot)$, set $\eta^{(1)}_n:= n^{-1}\sum_{i=1}^n \delta_{Z^{(1)}_i}$ 2. Simulate $Z^{(2)}_n\sim P_{\Phi(\eta^{(1)}_n)}(Z_{n-1}^{(2)},\cdot)$, set $\eta^{(2)}_n:= n^{-1}\sum_{i=1}^n \delta_{Z^{(2)}_i}$ 3. ⋮ 4. Simulate $Z^{(p)}_n\sim P_{\Phi(\eta^{(p-1)}_n)}(Z_{n-1}^{(p)},\cdot)$, set $\eta^{(p)}_n:= n^{-1}\sum_{i=1}^n \delta_{Z^{(p)}_i}$ At each level $p$, the random variable $Z^{(p)}_0$ is sampled according to some initial distribution possibly depending on $p$. Each $\{Z^{(p)}_k\}_{k\in\mathbb{N}},p>1$ is in the setting studied for the process $\{Z_k\}_{k\in\mathbb{N}}$. On the top of [\[avar:hp:FKflow\]](#avar:hp:FKflow){reference-type="ref" reference="avar:hp:FKflow"}, consider 1. $y\mapsto M(x,y)$ is continuous on $\mathsf{X}\xspace$ $\mathbb{P}$ for all $x\in\mathsf{X}\xspace$. [\[avar:hp:FKflow2\]]{#avar:hp:FKflow2 label="avar:hp:FKflow2"} Under [\[avar:hp:FKflow\]](#avar:hp:FKflow){reference-type="ref" reference="avar:hp:FKflow"}, as argued in [Example 7](#avar:ex:smcmc){reference-type="ref" reference="avar:ex:smcmc"}, one has $\Phi(\eta^1_n)(x)\rightarrow \Phi(\eta^1)(x)$ for all $x\in\mathsf{X}\xspace$ $\mathbb{P}$. If $\mathsf{X}\xspace$ is compact and [\[avar:hp:FKflow2\]](#avar:hp:FKflow2){reference-type="ref" reference="avar:hp:FKflow2"} holds too, the convergence can be made uniform, verifying [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"}. To prove that [\[avar:hp:Zapproxnormality\]](#avar:hp:Zapproxnormality){reference-type="ref" reference="avar:hp:Zapproxnormality"} holds too, use [\[avar:eq:FKdec\]](#avar:eq:FKdec){reference-type="eqref" reference="avar:eq:FKdec"} to write $$\begin{aligned} n^{-1/2}\sum_{k=1}^{n}&[\Phi(\eta^{(1)}_k)-\Phi(\eta^{(1)})](f) = \sum_{k=1}^{n} \frac{n^{-1/2}}{k} \sum_{j=1}^k [\bar{Q}^{(1)}(f-\Phi(\eta_k^{(1)})(f))(Z^{(1)}_j)-\eta^{(1)}(\bar{Q}^{(1)}(f-\Phi(\eta_k^{(1)})(f))] \\ &= \int_0^1 \frac{n^{-1/2}}{s} \sum_{j=1}^{sk}[\bar{Q}^{(1)}(f-\Phi(\eta_k^{(1)})(f))(Z^{(1)}_j)-\eta^{(1)}(\bar{Q}^{(1)}(f-\Phi(\eta_k^{(1)})(f))]ds + o_{\mathbb{P}}(1) \\ &\Rightarrow \sigma_{\eta^{(1)}}(\bar{Q}^{(1)}(f-\Phi(\eta^{(1)})(f)) \int_0^1 \frac{B_s}{s} ds,\end{aligned}$$ where in the last line we used the fact that a (functional) CLT holds for $\{Z^{(1)}_n;n\geq 0\}$, and where $B$ is a standard Brownian Motion. The right hand side is a Gaussian random variable with variance $w(f):=2\sigma^2_{\eta^{(1)}}(\bar{Q}^{(1)}(f-\Phi(\eta^{(1)})(f))$. [Theorem 5](#avar:thm:ZCLT){reference-type="ref" reference="avar:thm:ZCLT"} then applies to prove **Theorem 6**. *Let $f\in\mathbf{B}_1$. Assume that $\mathsf{X}\xspace$ is compact, and that the Markov family $\{P_\star\}$ [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} holds and that $P$ has an uniformly bounded derivative at $\Phi(\eta^{(p)})$ towards every $\Phi(\eta^{(p)}_n)$ for all $p\leq 2$. Then, if the Feynman-Kac flow $(G,M)$ satisfies [\[avar:hp:FKflow\]](#avar:hp:FKflow){reference-type="ref" reference="avar:hp:FKflow"} and [\[avar:hp:FKflow2\]](#avar:hp:FKflow2){reference-type="ref" reference="avar:hp:FKflow2"} it holds $$(f)\Rightarrow N(0,\sigma^2_{\eta^{(2)}}\left(f-\eta^{(2)}(f))+2\sigma^2_{\eta^{(1)}}(\bar{Q}^{(1)}(f-\eta^{(2)}(f))\right),$$ where we defined the operator $\bar{Q}$ as in [Theorem 4](#avar:thm:sMCMCFK){reference-type="ref" reference="avar:thm:sMCMCFK"}.* The case $p>2$ can be dealt with using [Theorem 5](#avar:thm:ZCLT){reference-type="ref" reference="avar:thm:ZCLT"}, but it also requires complex multilevel expansion formulæ---we refer to [@bercu2012iMCMCfluctuations]. # Closing remarks and research directions {#sec:discussion} We have developed herein a basic theory for methods allowing an easy and natural comparison between Markov Chains of the same family with different invariant distributions, by deriving analogues of the Fundamental Theorem of Calculus and Mean-Value Inequality for MCMC kernels viewed as functions of their invariant distributions. These tools allow us to study when Markov chains with different invariant distributions will move alike. For instance, under regularity conditions, we have seen that this can be ensured for Hastings and Gibbs samplers if the Markov chain is started under warm-start [\[cor:MHMVI,cor:MHMVIx,cor:GibbsMVI,cor:GibbsMVIx\]](#cor:MHMVI,cor:MHMVIx,cor:GibbsMVI,cor:GibbsMVIx){reference-type="ref" reference="cor:MHMVI,cor:MHMVIx,cor:GibbsMVI,cor:GibbsMVIx"}, with a slightly "warmer" condition required for Hastings chains. Other than being of theoretical interest to derive interesting relationships and inequality per se (see also the discussion in [Example 6](#ex:Gibbsderivatives){reference-type="ref" reference="ex:Gibbsderivatives"}), this framework is naturally appealing to study the use of approximations of invariant distributions in Markov Chain Monte Carlo, as shown in [3](#sec:applications){reference-type="ref" reference="sec:applications"}, where, roughly speaking, we have seen that under boundedness of the kernel derivatives estimators associated with a Markov chain employing an approximation $\mu_n$ of the invariant distribution $\mu$ achieve a similar asymptotic variances to the limiting chain employing $\mu$, up to an additional variability due to the fluctuations of the approximation. In particular, we used our framework and the derived inequalities to prove Central Limit Theorems for Sequential and Interacting MCMC algorithms. We also notice that the techniques developed can also be adapted can to study Markov chain Monte Carlo kernels as functions of their proposal, for instance. We believe that interesting research directions include extending the methods developed in this article to compare Markov Chains in different but somehow related Markov families, explore the other notions of gradient and derivatives in $\mathcal{P}(\mathsf{X}\xspace)$, making further connections with the theory in [@ambrosio2005gradientflows], exploring the use of higher-order derivatives, and developing methods to study Markov families where the dependence on the invariant distribution is not explicit. Furthermore, it might be of interest to explore other applications of these tools, perhaps in the contexts discussed in [3](#sec:applications){reference-type="ref" reference="sec:applications"}. # Acknowledgmenets {#acknowledgmenets .unnumbered} RC was funded by the UK Engineering and Physical Sciences Research Council (EPSRC) via studentship 2585619 as part of grant number EP/W523793/1; AMJ was partially funded by the EPSRC (grant numbers EP/R034710/1 and EP/T004134/1). For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. Data sharing is not applicable to this article as no new data were created or analyzed in this study. # Appendix to [2](#sec:mcmccalc){reference-type="ref" reference="sec:mcmccalc"} {#appendix-to-secmcmccalc} ## Auxiliary Lemmas We collect here some technical results which are used throughout our proofs. **Lemma 2**. *Let $\mu,\nu\in\mathcal{P}(\mathsf{X}\xspace)$. We have $$\begin{aligned} \norm{\mu-\nu}_{V}&:=\sup_{f:|f|\leq V}|\mu(f)-\nu(f)| =(\mu-\nu)_+V-(\mu-\nu)_-V = \int V(x)|\mu(x)-\nu(x)|\lambda(dx), \end{aligned}$$ with the last equality holding only if $\mu,\nu\in\mathcal{P}_\lambda(\mathsf{X}\xspace)$, and where $(\mu-\nu)_+,(\mu-\nu)_-$ is the Hahn-Jordan decomposition of $\mu-\nu$.* *Proof.* This is immediate from the definition, and well known---see, e.g., equation $9$ in [@heidergott2010vnorm]. ◻ **Lemma 3**. *Let $\chi$ be a finite regular signed measure on $(\mathsf{X}\xspace,\mathcal{B}(\mathsf{X}\xspace))$. If $V$ is a continuously differentiable $\chi$-integrable function we have $$\norm{\chi}_{V}:=\sup_{f\in\mathbf{B}_V(\mathsf{X}\xspace)}|\chi(f)|=\sup_{f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)}|\chi(f)|.$$* *Proof.* The inequality $\sup_{f\in\mathbf{B}_V(\mathsf{X}\xspace)}|\chi(f)|\geq \sup_{f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)}|\chi(f)|$ is immediate, so we focus on the converse. By [Lemma 2](#app:lemma:Vnormrepr){reference-type="ref" reference="app:lemma:Vnormrepr"}, $\norm{\chi}_{V}=\chi(V1_{\mathsf{X}\xspace^+})-\chi(V1_{\mathsf{X}\xspace^-})$, where $\mathsf{X}\xspace=\mathsf{X}\xspace^+\cup \mathsf{X}\xspace^-$ is an Hahn decomposition w.r.t. $\chi$. Thus it suffices to prove that the latter is equal to $\lim_{r\rightarrow \infty}|\chi(\varphi_{r})|$ for some sequence $\{\varphi_{r}\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)\}_{r\in\mathbb{N}}$ as $\lim_{r\rightarrow \infty}|\chi(\varphi_{r})|\leq \sup_{f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)}|\chi(f)|$. Since $\chi$ is regular, one can find closed sets $(\Tilde{\mathsf{X}\xspace}^+,\Tilde{\mathsf{X}\xspace}^-)$ such that $\chi(1_{\mathsf{X}\xspace^+\setminus\Tilde{\mathsf{X}\xspace}^+})$ and $\chi(1_{\mathsf{X}\xspace^-\setminus\Tilde{\mathsf{X}\xspace}^-})$ are both as small as we wish. By the integrability of $V$ one then has that for arbitrarily small $\epsilon>0$, $\chi(V1_{\mathsf{X}\xspace^+/\Tilde{\mathsf{X}\xspace}^+}) < \epsilon/2$ and $\chi(V1_{\mathsf{X}\xspace^-/\Tilde{\mathsf{X}\xspace}^-})<\epsilon/2$. Recall that $\varrho$ denotes the (assumed continuously differentiable) metric on $\mathsf{X}\xspace$. Employing the notation $(a)_+ = \min(1,\max(0,3a^2-2a^3))$ to denote a Hermite interpolant acting as a smooth, monotonic approximation of the positive part function on $(-\infty,1]$, we define $$\varphi_{r}^+:=V1_{\Tilde{\mathsf{X}\xspace}^+} + V(1-\varrho(\cdot,\Tilde{\mathsf{X}\xspace}^+)/r)_+ 1_{\mathsf{X}\xspace/\Tilde{\mathsf{X}\xspace}^+}, \quad \varphi_{r}^-:=V1_{\Tilde{\mathsf{X}\xspace}^-} + V(1-\varrho(\cdot,\Tilde{\mathsf{X}\xspace}^-)/r)_+ 1_{\mathsf{X}\xspace/\Tilde{\mathsf{X}\xspace}^-}$$ and $\varphi_{r}:=\varphi_{r}^+-\varphi_{r}^-$. In fact, $\varphi_{r}\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ and $\varphi_{r}\rightarrow V1_{\Tilde{\mathsf{X}\xspace}^+}-V1_{\Tilde{\mathsf{X}\xspace}^-}$ pointwise almost everywhere as $r\rightarrow \infty$. Therefore $\chi(\varphi_{r})\rightarrow \chi(V1_{\Tilde{\mathsf{X}\xspace}^+})-\chi(V1_{\Tilde{\mathsf{X}\xspace}^-})$ by dominated convergence and then by choice of $(\Tilde{\mathsf{X}\xspace}^+,\Tilde{\mathsf{X}\xspace}^-)$, $$\left|\lim_{r\rightarrow 0}\chi(\varphi_{r})-\norm{\chi}_{V}\right|\leq \epsilon.$$ We conclude by letting $\epsilon\rightarrow 0$. ◻ ## Technical conditions in [\[prop:MHderivatives,prop:Gibbsderivatives\]](#prop:MHderivatives,prop:Gibbsderivatives){reference-type="ref" reference="prop:MHderivatives,prop:Gibbsderivatives"} {#app:technicalcondboundder} ### Technical conditions in [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"}. {#technical-conditions-in-propmhderivatives.} #### Limit exchange. To justify the interchange of differentiation and integration performed in [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"} one can simply show that for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, $\varphi_t(u,w):=|(d/dt)f(w)g(r_{\mu_t}(u,w))q(u,w)\rho(u)|$ is uniformly bounded in $t\in[0,1]$ by an $L^1(\lambda(du)\otimes\lambda(dw))$ function (e.g. [@klenke2013prob Theorem 6.28]). Since $|f|\leq V=1$ and $g'$ is bounded, say by $\bar{g}'$, $$\begin{aligned} \varphi_t(u,w)&= f(w)g'(r_{\mu_t}(u,w))\frac{|\chi(w)q(w,u)\mu_t(u)q(u,w)-\chi(u)q(u,w)\mu_t(w)q(w,u)|}{(\mu_t(u)q(u,w))^2}q(u,w)\rho(u) \\ &\leq \bar{g}'\frac{|\chi(w)\mu_t(u)-\chi(u)\mu_t(w)|}{\mu_t(u)^2}q(w,u)\rho(u) = \bar{g}'\frac{|\mu(w)\nu(u)-\mu(u)\nu(w)|}{\mu_t(u)^2}q(w,u)\rho(u) \\ &\leq \bar{g}'|\mu(w)\nu(u)+\mu(u)\nu(w)|a b\end{aligned}$$ where $a:=\sup_{x,y}q(x,y), b:=\sup_x \rho(x)/\min(\mu(x),\nu(x))^2<\infty$ by our assumptions. #### Uniform continuity of $\partial_\pi P_{\mu}(\rho,f)$. We need to show that for all $\epsilon>0$ there exists a $\delta>0$ such that for every $\rho,\rho'\in\mathcal{A}_x$ with $d_p(\rho,\rho')<\delta$ we have $|\partial_\pi P_{\mu}(\rho,f)[\nu-\mu]-\partial_\pi P_{\mu}(\rho',f)[\nu-\mu]|<\epsilon$. Let $\rho,\rho'\in\mathcal{A}_x$, and let $d_p$ be the Bounded-Lipschitz metric [@dudley2002analysis Chapter 11] $d_p(\rho,\rho'):=\sup |\int fd(\rho-\rho')|$, the supremum being taken across functions $f$ on $B(x,r)$ with bounded Lipschitz norm $|f|_{bl}:=|f|_{l}+\left\vert f\right\vert_{\infty}$ no greater than 1, $|f|_{l}$ denoting the Lipschitz seminorm. Notice that for all $\rho\in\mathcal{A}_x$ we have $\partial_\pi P_{\mu}(\rho,f)[\nu-\mu]=\rho(h_1-h_2)$, where $$\begin{aligned} h_1(z):=&\mu(z)^{-1}\int(f(y)-f(z))q(y,z)g'(r_{\mu}(z,y))(\nu-\mu)(dy)\\ \textrm{and } h_2(z):=&(\nu-\mu)(z)\mu(z)^{-2}\int(f(z)-f(y))\mu(y)q(y,z)g'(r_{\mu}(z,y))\lambda(dy).\end{aligned}$$ Since $|h_1-h_2|_{bl}\leq |h_1|_{bl}+|h_2|_{bl}$, the claim follows with the choice $\delta=\epsilon/(|h_1|_{bl}+|h_2|_{bl})$ provided $h_1,h_2$ have finite bounded Lipschitz norm on $B(x,r)$. Since $g'$ is bounded, and $q,\mu$ are continuous and bounded away from $0$ on the (compact closure of) $B(x,r)$, it follows immediately that both $h_1$ and $h_2$ are bounded on $B(x,r)$. To show that they are Lipschitz, we show they are continuously differentiable on $B(x,r)$. Since $\mu,\nu$ are assumed to be continuously differentiable and $f$ is too, we only need to show that $z\mapsto \int f(y)q(y,z)g'(r_{\mu}(z,y))\mu(dy)$ and $z\mapsto \int f(y)q(y,z)g'(r_{\mu}(z,y))(\nu-\mu)(dy)$ are continuously differentiable on $B(x,r)$. Whenever the assumed domination condition hold, we have $$\begin{aligned} \partial_z \int f(y)q(y,z)g'(r_{\mu}(z,y))&\mu(dy) = \int f(y) \partial_z(q(y,z)g'(r_{\mu}(z,y)) \mu(dy) \end{aligned}$$ (e.g. [@klenke2013prob Theorem 6.28]) which is a continuous function of $z$ on $B(x,r)$ too, because the integrand is continuous in $z$ and we have shown above it can be bounded uniformly in $z$ by a $\mu$-integrable function ([@klenke2013prob Theorem 6.27]). This proves that $z\mapsto \int f(y)q(y,z)g'(r_{\mu}(z,y))\mu(dy)$ is continuously differentiable on $B(x,r)$. For $z\mapsto \int f(y)q(y,z)g'(r_{\mu}(z,y))(\nu-\mu)(dy)$ the proof is analogous. ### Technical conditions in [Proposition 5](#prop:Gibbsderivatives){reference-type="ref" reference="prop:Gibbsderivatives"}. {#technical-conditions-in-propgibbsderivatives.} #### Limit exchange. Much as in the proof of uniform continuity above, it suffices to show that for all $f\in\mathbf{C}_{V}^1(\mathsf{X}\xspace^2)$, $\varphi_t(u,w):=|(d/dt)f(w_1,w_2)P_{1|2,t}(u_2,w_1)P_{2|1,t}(w_1,w_2)\rho_2(u_2)|$ is uniformly bounded in $t\in[0,1]$ by an $L^1(\lambda(du)\otimes\lambda^2(dw))$ function. Write $\varphi_t(u,w)=\varphi_{1,t}(u,w)+\varphi_{2,t}(u,w)$, and compute, recalling that we take $V=1$: $$\begin{aligned} \varphi_{1,t}(u,w)&=f(w_1,w_2) \rho_2(u_2)\frac{|\chi(w_1,u_2)\mu_{2,t}(u_2)-\chi_2(u_2)\mu_t(w_1,u_2)|}{\mu_{2,t}(u_2)^2}P_{2|1,t}(w_1,w_2) \\ &\leq \rho_2(u_2)\frac{|\chi(w_1,u_2)-\chi_2(u_2)P_{1|2,t}(u_2,w_1))|}{\min(\mu_2(u_2),\nu_2(u_2))}P_{2|1,t}(w_1,w_2) \\ &\leq b(|\chi(w_1,u_2)|+|\chi_2(u_2)|P_{1|2,t}(u_2,w_1)))P_{2|1,t}(w_1,w_2) \\ &\leq b(|\chi(w_1,u_2)|+|\chi_2(u_2)|(P_{1|2,0}(u_2,w_1))+P_{1|2,1}(u_2,w_1)))(P_{2|1,0}(w_1,w_2)+P_{2|1,1}(w_1,w_2)) \intertext{where $b:=\sup_x \rho_2(x)/\min(\mu_2(x),\nu_2(x))<\infty$ by assumption. To bound $\psi_{2,t}(u,w)$ we use Bayes' formula to write} \psi_{2,t}(u,w)&=f(w_1,w_2) \rho_2(u_2)\frac{\chi(w_1,w_2)\mu_{1,t}(w_1)-\chi_1(w_1)\mu_t(w_1,w_2)}{\mu_{1,t}(w_1)^2}P_{1|2,t}(u_2,w_1) \\ &=f(w_1,w_2) \rho_2(u_2)\frac{\chi(w_1,w_2)\mu_{1,t}(w_1)-\chi_1(w_1)\mu_t(w_1,w_2)}{\mu_{1,t}(w_1)\mu_{2,t}(u_2)}P_{2|1,t}(w_1,u_2)\end{aligned}$$ and then we proceed as for $\psi_{1,t}(u,w)$. #### Uniform continuity of $\partial_\pi P_{\mu}(\rho,f)$. This is almost analogous to the proof of [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"}. We notice that for all $\rho\in\mathcal{A}_x$ we have $\partial_\pi P_{\mu}(\rho,f)[\nu-\mu]=\rho(h_1-h_2+h_3-h_4)$, where $$\begin{aligned} h_1(z)&:=\int f(y_1,w_2)\frac{P_{2|1}(y_1,dw_2)}{\mu_2(z)}(\nu-\mu)(y_1,z)\lambda(dy_1), \\ h_2(z)&:=\int f(w_1,w_2)\frac{P_{1|2}(z,dw_1)P_{2|1}(w_1,dw_2)}{\mu_2(z)}(\nu-\mu)(y_1,z)\lambda(dy_1), \\ h_3(z)&:= \int f(y_1,y_2)\frac{P_{1|2}(z,dy_1)}{\mu_1(y_1)}(\nu-\mu)(y_1,y_2)\lambda(dy_2), \quad \text{and} \\ h_4(z)&:= \int f(y_1,w_2)\frac{P_{1|2}(z,dy_1)P_{2|1}(y_1,dw_2)}{\mu_1(y_1)}(\nu-\mu)(y_1,y_2)\lambda(dy_2).\end{aligned}$$ Hence, following the same reasoning, we just need to show that these functions are bounded and Lipschitz on $B(x,r)$. Since $\mu$ being continuous, positive and bounded implies that the marginal densities $\mu_1,\mu_2$ are bounded away from 0 on $B(x,r)$ and since we assumed that the full conditionals are upper bounded, we check that $h_1(z),h_2(z)\leq (\nu_2-\mu_2)(z)/\mu_2(z)$ are in fact bounded, we then use Bayes' formula to compute $$\begin{aligned} h_3(z)&=\int f(y_1,y_2)\frac{P_{2|1}(y_1,z)}{\mu_2(z)}(\nu-\mu)(y_1,y_2)\lambda^2(dy), \\ h_4(z)&=\int f(y_1,w_2)\frac{P_{2|1}(y_1,z)P_{2|1}(y_1,dw_2)}{\mu_2(z)}(\nu-\mu)(y_1,y_2)\lambda^2(dy)\end{aligned}$$ after which we can use our upper bounds on the full conditionals and lower bounds on the marginals on $B(x,r)$. To prove they are Lipschitz on $B(x,r)$, we show again they are continuously differentiable. This follows immediately using our domination assumption similarly to the proof of [Proposition 4](#prop:MHderivatives){reference-type="ref" reference="prop:MHderivatives"}. # Appendix to [3](#sec:applications){reference-type="ref" reference="sec:applications"} {#appendix-to-secapplications} Before the proofs of [\[avar:thm:YCLT,avar:thm:ZCLT\]](#avar:thm:YCLT,avar:thm:ZCLT){reference-type="ref" reference="avar:thm:YCLT,avar:thm:ZCLT"} can be given, a number of auxiliary Lemmas and results are required. We use an approach based on the Poisson equations and resolvents. For a Markov kernel $P_\upsilon$ having $\upsilon$ as invariant measure, the Poisson equation is $$\left\{ \begin{array}{rl} (P_\upsilon-\ensuremath{\textrm{Id}}\xspace)R_\upsilon =& \upsilon - \ensuremath{\textrm{Id}}\xspace, \\ \upsilon R_\upsilon =& 0, \end{array} \right.$$ with $R_\upsilon$ being the resolvent operator defined by $R_\upsilon f(x) := \sum_{k=0}^\infty [P_\upsilon^k(x,f)-\upsilon(f)].$ The following Lemma is an immediate consequence of [@fort2011convergence Lemma 2.3] **Lemma 4**. *If [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} holds, for all $n\in\mathbb{N}$ the Markov chains $\{P_{\mu_n}\}_{n\in\mathbb{N}}$ are $V$-geometrically ergodic in that for all $x\in\mathsf{X}\xspace$, for some finite $C_n$ and $\rho_n\in (0,1)$, $$\norm{P^k_{\mu_n}(x,\cdot)-\mu_n}_{V} \leq C_n \rho_n^k V(x).$$ Furthermore, $L:=\sup_n L_n<\infty$, where $L_n:=C_n \vee (1-\rho_n)^{-1}$.* Using the Lemma above and the definition of the Poisson resolvent we also immediately obtain that for $f\in\mathbf{B}_V(\mathsf{X}\xspace)$, $|R_{\mu_n}f(x)|\leq \sum_{k=1}^\infty |P^k_{\mu_n}(x,f)-\mu_n(f)|\leq \left\vert f\right\vert_{V}V(x)C_n\sum_{k=0}^\infty\rho^k_n$ and hence the following. **Lemma 5**. *If [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"} holds then, for all $n\geq 1$ and $\alpha\in(0,1]$, $|R_{\mu_n} f(x)|\leq V^\alpha(x)L^2\left\vert f\right\vert_{V^\alpha}$. In particular, $\sup_n \norm{R_{\mu_n}f}_{V^\alpha}<\infty$.* The role of $\alpha$ will become clear later. **Lemma 6**. *If Assumptions [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"}, [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"} and [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} hold, then $\norm{\mu_n-\mu}_{V}\rightarrow 0$.* *Proof.* Since $\norm{\mu_n-\mu}_{V}=\int V(x)|\mu_n(x)-\mu(x)|\lambda(dx)$ by [Lemma 2](#app:lemma:Vnormrepr){reference-type="ref" reference="app:lemma:Vnormrepr"}, by Scheffé's lemma, and the pointwise convergence given by [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"}, it suffices to prove that $\mu_n(V)\rightarrow \mu(V)$. The convergence of $V$-moments is then verified by uniform integrability (e.g. [@billingsley1999cpm Theorem 3.5]) via [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"}. ◻ The following Theorem gives the Law of Large Numbers we will need to prove the Central Limit Theorems for sMCMC. The proof is similar to [@fort2011convergence Theorem 2.7] but in a sequential rather an interacting setting. This result extends the LLN of [@finke2020smcmc] under much less stringent conditions, although it does not supply non-asymptotic estimates. **Theorem 7**. *Let $F:\mathsf{X}\xspace\times\mathcal{P}(\mathsf{X}\xspace)\mapsto \mathbb{R}^d$ be a measurable function satisfying $\sup_n \left\vert F(\cdot,\mu_n)\right\vert_{V}<\infty$ and assume $\lim_n \int F(x,\mu_n)\mu_n(dx)$ exists and that [\[avar:hp:unifergodicity,avar:hp:Vgrowth\]](#avar:hp:unifergodicity,avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:unifergodicity,avar:hp:Vgrowth"} hold. Then, in probability, $$\label{app:eq:Ylln} \lim_n n^{-1} \sum_{i=1}^n F(Y_{n,i},\mu_n) = \lim_n \int F(x,\mu_n)\mu_n(dx).$$ If [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} also holds and $F(\cdot,\mu):=\lim_n F(\cdot,\mu_n)$ exists, the latter limit is equal to $\int F(x,\mu)\mu(dx).$* *Proof.* Write $F_n:=F(\cdot,\mu_n), R_n:=R_{\mu_n}, P_n:=P_{\mu_n}$. We first use the Poisson equation to obtain the following decomposition $$n^{-1} \sum_{i=1}^n F_n(Y_{n,i}) - \mu_n(F_n) = M_n + r_n$$ where $M_n:= n^{-1} \sum_{i=1}^n R_n F_n (Y_{n,i}) - P_n R_n F_n (Y_{n,i-1}); r_n:=n^{-1}[P_nR_nF_n(Y_{n,0})-P_nR_nF_n(Y_{n,n})]$. The result follows if we show that both $M_n,r_n$ converge to 0 in probability. Let $U_{n,i}:=n^{-1}[R_n F_n (Y_{n,i}) - P_n R_n F_n (Y_{n,i-1})]$. $\{U_{n,i};i\leq n\}$ is a triangular array of martingale difference sequences with respect to the triangular array of $\sigma$-fields $\{\mathcal{F}_{n,i}\}$, where $\mathcal{F}_{n,i}:=\bigvee_{m \leq n} \sigma(Y_{m,j}; j \leq m) \vee \sigma(Y_{n,j}; j \leq i)$. $M_n\rightarrow_{\mathbb{P}} 0$ then follows by [@douc2007limit Lemma A.1] if we estabish the asymptotic negligibility of individual increments, i.e that for all $\epsilon>0$, $$\sum_{i=1}^n \mathbb{E}[|U_{n,i}|1_{|U_{n,i}|\geq \epsilon}|\mathcal{F}_{n,i-1}] \rightarrow_{\mathbb{P}} 0.$$ For any $\gamma>1$ and $\epsilon>0$ we compute $$\begin{aligned} \mathbb{E}[|U_{n,i}|1_{|U_{n,i}|\geq \epsilon}|\mathcal{F}_{n,i-1}] &\leq \epsilon^{-\gamma+1}\mathbb{E}[|U_{n,i}|^\gamma|\mathcal{F}_{n,i-1}]\\ &\leq \epsilon(n\epsilon)^{-\gamma}\mathbb{E}[|R_n F_n (Y_{n,i}) - P_n R_n F_n (Y_{n,i-1})|^\gamma|\mathcal{F}_{n,i-1}] \\ &\leq \epsilon(n\epsilon)^{-\gamma} 2^{\gamma-1}\mathbb{E}[|R_n F_n (Y_{n,i})|^\gamma + |P_n R_n F_n (Y_{n,i-1})|^\gamma|\mathcal{F}_{n,i-1}] \\ &\leq \epsilon(n\epsilon)^{-\gamma} 2^{\gamma} L^{2\gamma} \left\vert F_n(\cdot)\right\vert_{V}^\gamma P_n V^\gamma(Y_{n,i})\\ &\leq \epsilon(n\epsilon)^{-\gamma} 2^{\gamma}L^{2\gamma} \sup_n \left\vert F_n(\cdot)\right\vert_{V}^\gamma P_n V^\gamma(Y_{n,i}) \end{aligned}$$ where in the third line we used the $C_p$ inequality, and in the fourth [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"}. [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"} guarantees that, for some $\gamma > 1$, $P_n V^\gamma(Y_{n,i})$ is uniformly bounded in both $i,n$ and $M_n\rightarrow_{\mathbb{P}} 0$ follows. To prove $r_n\rightarrow_{\mathbb{P}} 0$ we use again [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"} to estimate $$\begin{aligned} |r_n|\leq n^{-1}[|P_nR_nF_n(Y_{n,0})|+|P_nR_nF_n(Y_{n,n})|] \leq n^{-1}\sup_n \left\vert F_n(\cdot)\right\vert_{V} L^2(P_nV(Y_{n,0})+P_nV(Y_{n,n})). \end{aligned}$$ By [Lemma 4](#avar:lemma:geomerg){reference-type="ref" reference="avar:lemma:geomerg"}, $P_n V(Y_{n,n}), P_nV(Y_{n,0})$ are both bounded in $n$, and [\[app:eq:Ylln\]](#app:eq:Ylln){reference-type="eqref" reference="app:eq:Ylln"} is proved. Now let $F:=F(\cdot,\mu)$, by the triangle inequality, $$|\mu_n(F_n)-\mu(F)|\leq |\mu_n(F_n)-\mu(F_n)|+|\mu(F_n)-\mu(F)|\leq \norm{\mu_n-\mu}_{V}\left\vert F_n\right\vert_{V} +|\mu(F_n-F)|.$$ Since $\sup_n \left\vert F_n\right\vert_{V}<\infty$, under [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} the first term vanishes. Furthermore, we also have $|F_n-F|\leq 2V$, and the fact $\mu(V)<\infty$ guarantees that the second term also vanishes by the Dominated Convergence Theorem. It follows that $\lim_n \int F(x,\mu_n)\mu_n(dx) = \int F(x,\mu)\mu(dx)$. ◻ The next result is instead a Law of Large Numbers for iMCMC. This follows straight from [@fort2011convergence Theorem 2.7] together with a mean value inequality. **Theorem 8**. *Assume that Assumptions [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"}, [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"}, [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"} and [\[avar:hp:unifboundedder\]](#avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifboundedder"} hold. Let $F:\mathsf{X}\xspace\times\mathcal{P}(\mathsf{X}\xspace)\mapsto \mathbb{R}^d$ be a measurable function satisfying $\sup_n \norm{F(\cdot,\mu_n)}_{V}<\infty$ and $\sum_{i=1}^\infty i^{-1}\sup_x \left\vert F(\cdot,\mu_i)-F(\cdot,\mu_{i-1})\right\vert_{V}<\infty$. Then, if $\lim_n \int F(x,\mu_n)\mu_n(dx)$ exists, in probability, $$\label{app:eq:Zlln} \lim_n n^{-1} \sum_{i=1}^n F(Z_i,\mu_i) = \lim_n \int F(x,\mu_n)\mu_n(dx).$$ If $F(\cdot,\mu):=\lim_n F(\cdot,\mu_n)$ exists, the latter limit is equal to $\int F(x,\mu)\mu(dx).$* *Proof.* [\[app:eq:Zlln\]](#app:eq:Zlln){reference-type="eqref" reference="app:eq:Zlln"} follows from [@fort2011convergence Theorem 2.7]. By [Lemma 4](#avar:lemma:geomerg){reference-type="ref" reference="avar:lemma:geomerg"}, [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"} and the implied fact that $\sup_n V(Z_n)$ is $\mathbb{P}$-finite, their assumptions their assumptions A3, A5 are all verified. To verify their A4 we apply a mean value inequality and the assumed uniformly bounded derivative in the invariant distribution [\[avar:hp:unifboundedder\]](#avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifboundedder"} to write, for some $M_1,M_2<\infty$, $$\begin{aligned} \sum_{i=1}^\infty i^{-1}(L_{\mu_i}\wedge L_{\mu_{i-1}})^6&\sup_x \norm{P_{\mu_i}(x,\cdot)-P_{i-1}(x,\cdot)}_{V} V(Z_i) \\ &\leq \sum_{i=1}^\infty i^{-1}(L_{\mu_i}\wedge L_{\mu_{i-1}})^6[M_1\norm{\mu_i-\mu_{i-1}}_{V}+M_2\sup_x|\mu_i(x)-\mu_{i-1}(x)|]V(Z_i) \end{aligned}$$ and the last expression is finite by $\sup_n L_{\mu_n}<\infty$, [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"} and the fact that $\sup_n V(Z_n)$ is $\mathbb{P}$-finite by [Lemma 4](#avar:lemma:geomerg){reference-type="ref" reference="avar:lemma:geomerg"}, verifying the conditions in [@fort2011convergence Theorem 2.7] and proving the claim. The fact that under [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} or [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"} $\lim_n \int F(x,\mu_n)\mu_n(dx) = \int F(x,\mu)\mu(dx)$ follows from the same arguments used in [Theorem 7](#app:thm:Yllns){reference-type="ref" reference="app:thm:Yllns"}. ◻ In the proof below, for $\nu,\mu\in\mathcal{P}(\mathsf{X}\xspace)$, we employ the operator identity $$\label{avar:eq:PoiResolventkernelID} R_\nu-R_\mu = R_\mu(P_\nu-P_\mu)R_\nu + (\mu - \nu)R_\nu.$$ The equation above is proved in [@bercu2012iMCMCfluctuations proof of Proposition 3.1]---we repeat the argument here for completeness. In operator sense, $P_\mu R_\mu = R_\mu P_\mu$ Therefore it holds that $R_\mu(P_\mu-\ensuremath{\textrm{Id}}\xspace)=(P_\mu-\ensuremath{\textrm{Id}}\xspace)R_\mu=\mu-\ensuremath{\textrm{Id}}\xspace$ by the Poisson equation, but then $$R_\mu(P_\mu-\ensuremath{\textrm{Id}}\xspace)R_\nu = (\mu-\ensuremath{\textrm{Id}}\xspace)R_\nu$$ and from the other side, again from an application of the Poisson identities, and since $R_\mu \nu(f)=0$, $$R_\mu(P_\nu-\ensuremath{\textrm{Id}}\xspace)R_\nu = R_\mu(\nu-\ensuremath{\textrm{Id}}\xspace) = -R_\mu$$ It follows that $R_\mu(P_\mu-(+\ensuremath{\textrm{Id}}\xspace-\ensuremath{\textrm{Id}}\xspace)-P_\nu)R_\nu=R_\mu(P_\mu-\ensuremath{\textrm{Id}}\xspace)R_\nu+R_\mu(\ensuremath{\textrm{Id}}\xspace-P_\nu)R_\nu=(\mu-\ensuremath{\textrm{Id}}\xspace)R_\nu+R_\mu = (\mu-\nu)R_\nu - (R_\nu-R_\mu)$, where in the last line we used $\nu R_\nu = 0$. Hence, [\[avar:eq:PoiResolventkernelID\]](#avar:eq:PoiResolventkernelID){reference-type="eqref" reference="avar:eq:PoiResolventkernelID"} follows. For brevity, we will denote $P_n=P_{\mu_n}, R_{\mu_n}=R, P=P_\mu, R = R_\mu$ in what follows. **Lemma 7**. *Assume Assumptions [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"}, [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"}, [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"} and [\[avar:hp:boundedder\]](#avar:hp:boundedder){reference-type="ref" reference="avar:hp:boundedder"} hold. Then, for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$, $R_nf(x)\rightarrow R_{\mu}f(x)$ for all $x\in\mathsf{X}\xspace$ $\mathbb{P}$.* *Proof.* From [\[avar:eq:PoiResolventkernelID\]](#avar:eq:PoiResolventkernelID){reference-type="eqref" reference="avar:eq:PoiResolventkernelID"} we have that $R_nf-Rf = R(P_n-P)R_nf + (\mu-\mu_n)R_nf$. Since $R_nf$ has uniformly bounded $V$-norm by [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"}, the second term vanishes with $n\rightarrow\infty$ under [\[avar:hp:ptwiseapprox\]](#avar:hp:ptwiseapprox){reference-type="ref" reference="avar:hp:ptwiseapprox"}, and we just need to show that $R(P_n-P)R_nf(x)\rightarrow 0$ for all $x\in\mathsf{X}\xspace$ $\mathbb{P}$. Since $(P_n-P)R_nf\in\mathbf{B}_V(\mathsf{X}\xspace)$ by [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"}, the series $$R(P_n-P)R_nf(x) = \sum_{k=1}^\infty[P^k(x,(P_n-P)R_nf)-\mu((P_n-P)R_nf)]$$ must be convergent, again by [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"}. In particular for any given $\varepsilon>0$ there exists a finite $\bar{k}$ independent of $n$ such that $$R(P_n-P)R_nf(x) = \sum_{k=\bar{k}}^\infty[P^k(x,(P_n-P)R_nf)-\mu((P_n-P)R_nf)] < \varepsilon.$$ Therefore, we just need to show that $$\label{avar:eq:lemmaproofconvk} P^k(x,(P_n-P)R_nf)-\mu((P_n-P)R_nf) \rightarrow 0$$ for $k=1,...\bar{k}$. Since $\bar{k}$ is finite and also independent of $n$, it is enough to show that [\[avar:eq:lemmaproofconvk\]](#avar:eq:lemmaproofconvk){reference-type="eqref" reference="avar:eq:lemmaproofconvk"} holds for a fixed finite $k$. In fact, by the derivative in the invariant measure being bounded by [\[avar:hp:boundedder\]](#avar:hp:boundedder){reference-type="ref" reference="avar:hp:boundedder"}, by mean value, for all $x\in\mathsf{X}\xspace$ we have some finite positive $M_{1,x},M_{2,x}$ such that $$|(P-P_n)(x,R_nf)|\leq \left\vert R_nf\right\vert_{V}[M_{1,x}\norm{\mu_n-\mu}_{V}+M_{2,x}|\mu_n(x)-\mu(x)|]$$ and the right hand side goes to 0 $\mathbb{P}$ by [\[app:lemma:resolventbound,avar:hp:ptwiseapprox,avar:hp:boundedder\]](#app:lemma:resolventbound,avar:hp:ptwiseapprox,avar:hp:boundedder){reference-type="ref" reference="app:lemma:resolventbound,avar:hp:ptwiseapprox,avar:hp:boundedder"} for all $x\in\mathsf{X}\xspace$. Also, $|(P_-P_n)(x,R_nf)|$ is uniformly bounded in $n$ by a function in $\mathbf{B}_V(\mathsf{X}\xspace)$ by [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"}. Because $P^k(x,V)<\infty$ ([Lemma 4](#avar:lemma:geomerg){reference-type="ref" reference="avar:lemma:geomerg"}), an application of the Dominated Convergence Theorem shows that [\[avar:eq:lemmaproofconvk\]](#avar:eq:lemmaproofconvk){reference-type="eqref" reference="avar:eq:lemmaproofconvk"} indeed holds $\mathbb{P}$ for all $x\in\mathsf{X}\xspace$. This proves $R_nf(x)\rightarrow Rf(x)$ $\mathbb{P}$ for all $x\in\mathsf{X}\xspace$. We can now use a standard separability argument to prove that $R_nf(x)\rightarrow Rf(x)$ for all $x\in\mathsf{X}\xspace$ $\mathbb{P}$. Let $\{x_j\}$ be a countable dense subset of $\mathsf{X}\xspace$, its existence guaranteed by $\mathsf{X}\xspace$ being separable. Notice that then it is enough for the claim to prove that $R_nf(x)\rightarrow Rf(x)$ for all $x\in\mathsf{X}\xspace$ if and only if $R_nf(x_j)\rightarrow Rf(x_j)$ for all $x_j\in\{x_j\}$. This because $\{x_j\}$ is countable, and because the intersection of countably many probability 1 events has probability 1. The only if direction is trivial. To prove the if direction, we can write, for some $j$, $$|R_nf(x)-Rf(x)|\leq |R_nf(x)-R_nf(x_j)|+|Rf(x)-Rf(x_j)| + |R_nf(x_j) + Rf(x_j)|$$ The last term is small with $n$ by hypothesis. Since $\{x_j\}$ is dense in $\mathsf{X}\xspace$, we can pick a $x_j$ sufficiently close to $x$ so that the continuity of $x\mapsto R_nf(x)$ and $x\mapsto Rf(x)$ (which follows from [\[hp:Fellertypechain\]](#hp:Fellertypechain){reference-type="ref" reference="hp:Fellertypechain"} and the definition of the Poisson resolvent) ensure the first two terms are small, too. The claim follows. ◻ For a kernel $K$ on $(\mathsf{X}\xspace,\mathcal{B}(\mathsf{X}\xspace))$ define $\norm{K}_{V}:=\sup_x \norm{K(x,\cdot)}_{V}$. The next lemma is a more quantitative version of the one just proved. **Lemma 8**. *Assume that Assumptions [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"}, [\[avar:hp:Vgrowth\]](#avar:hp:Vgrowth){reference-type="ref" reference="avar:hp:Vgrowth"}, [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"} and [\[avar:hp:unifboundedder\]](#avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifboundedder"} hold. Then, for $\alpha\in(0,1)$, we have $n^{-1/2}\sum_{i=1}^n \norm{R_i-R_{i-1}}_{V^\alpha}\rightarrow 0$ $\mathbb{P}$.* *Proof.* Using [@fort2014clt Lemma A.1] and the hypothesis of uniformly bounded derivative in the invariant distribution [\[avar:hp:unifboundedder\]](#avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifboundedder"} we have $$\begin{aligned} \norm{R_{i}-R_{i-1}}_{V^\alpha} &\leq 3(L_{\mu_i}\wedge L_{\mu_{i-1}})^6 \mu_{i}(V^\alpha)\sup_x \norm{P_{i}(x,\cdot)-P_{i-1}(x,\cdot)}_{V^\alpha} \\ &\leq \sup_i \mu_{i}(V^\alpha) \sup_i L_{\mu_i}^6[M_1\norm{\mu_{i}-\mu_{i-1}}_{V^\alpha}+M_2\sup_x|\mu_i(x)-\mu_{i-1}(x)|] \end{aligned}$$ and the claim follows by [\[avar:lemma:geomerg,avar:hp:unifapprox\]](#avar:lemma:geomerg,avar:hp:unifapprox){reference-type="ref" reference="avar:lemma:geomerg,avar:hp:unifapprox"}. ◻ #### Central Limit Theorem for sMCMC. We come to the proof of [Theorem 3](#avar:thm:YCLT){reference-type="ref" reference="avar:thm:YCLT"}. Let $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$ and denote $$S_n(f) := n^{-1/2}\sum_{i=0}^n f(Y_{n,i})-\mu(f) = S_{1,n}(f) + S_{2,n}(f)$$ where $S_{1,n}(f)= n^{-1/2}\left( \sum_{i=0}^n f(Y_{n,i})-\mu_n(f)\right)$ and $S_{2,n}(f)=n^{-1/2} (\mu_n-\mu)(f)$. *Proof of [Theorem 3](#avar:thm:YCLT){reference-type="ref" reference="avar:thm:YCLT"}.* We first show that $\mathbb{E}[\exp{(iuS_{1,n})}|\mathcal{F}_{n,0}]\rightarrow \exp(-(u^2/2)\sigma^2)$, $u\in\mathbb{R}^d$. Similarly to [Theorem 7](#app:thm:Yllns){reference-type="ref" reference="app:thm:Yllns"}, we write $n^{-1/2} \sum_{i=1}^n f(Y_{n,i}) - \mu_n(f) = \sum_{i=1}^n U_{n,i} + r_n$, where $U_{n,i}:=n^{-1/2}[R_n f(Y_{n,i}) - P_n R_n f(Y_{n,i-1})]$, $r_{n}:=n^{-1/2}[P_nR_nf(Y_{n,0})-P_nR_nf(Y_{n,n})]$. $r_n\rightarrow_{\mathbb{P}}0$ follows from the same arguments used in the proof of [Theorem 7](#app:thm:Yllns){reference-type="ref" reference="app:thm:Yllns"}. To verify the claim it then suffices to check $\mathbb{E}[\exp{(iu\sum_{i=1}^n U_{n,i})}|\mathcal{F}_{n,0}]\rightarrow \exp(-(u^2/2)\sigma^2)$, and we aim to apply [@douc2007limit Theorem A.3]. We need to check that $$\sum_{i=1}^N \mathbb{E}[U_{n,i}^2|\mathcal{F}_{n,i-1}] \rightarrow_{\mathbb{P}} \sigma^2 \quad \text{and} \quad \sum_{i=1}^n \mathbb{E}[|U_{n,i}|^21_{|U_{n,i}|\geq \epsilon}|\mathcal{F}_{n,i-1}]\rightarrow_{\mathbb{P}} 0.$$ The second condition follows using basically the same techniques we used in the proof of [Theorem 7](#app:thm:Yllns){reference-type="ref" reference="app:thm:Yllns"} to check $\mathbb{E}[|U_{n,i}|1_{|U_{n,i}|\geq \epsilon}|\mathcal{F}_{n,i-1}]\rightarrow_{\mathbb{P}} 0$. To prove the first, notice that $\sum_{i=1}^N \mathbb{E}[U_{n,i}^2|\mathcal{F}_{n,i-1}]=n^{-1}\sum_{i=1}^N F_n(Y_{n,i})$, where $F_{n}(y):=P_{n}(R_{n}f(y))^2-(P_{n}R_{n}f(y))^2$ and that $\sigma^2=\mu(F)$, with $F(y):=P(Rf(y)^2)-(PRf(y))^2$. We prove this via the LLN [Theorem 7](#app:thm:Yllns){reference-type="ref" reference="app:thm:Yllns"}. Since by [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"} $\sup_n \left\vert F_n\right\vert_{V^\alpha}<\infty$, we only need to check that $F_n(y)\rightarrow F(y)$ for all $f\in\mathbf{C}^1_{V}(\mathsf{X}\xspace)$. By the triangle inequality, $$\begin{aligned} \label{app:eq:Fnbound1} |F_n(y)-F(y)|&\leq |P_{n}(R_{n}f(y)^2)-P(R_{n}f(y)^2)|+|(P_{n}R_{n}f(y))^2-(PR_{n}f(y))^2| \nonumber \\ &+ |(PR_{n}f(y))^2-(PRf(y))^2|+ |P(R_{n}f(y)^2)-P(Rf(y)^2)|, \end{aligned}$$ and we can check that, indeed, since the derivative in the invariant distribution of $P$ is bounded at $\mu$ towards every $\mu_n$, $$\begin{aligned} \label{app:eq:Fnbound2} &|P_{n}(R_{n}f(y)^2)-P(R_{n}f(y)^2)| \leq \left\vert R_{n}f^2\right\vert_{V^\alpha}[M_{1,y}\norm{\mu_n-\mu}_{V^\alpha}+M_{2,y}|\mu_n(y)-\mu(y)|]. \end{aligned}$$ The right hand side goes to zero under our assumptions by [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"}. Similarly, the second term of the triangle inequality vanishes because $$\begin{aligned} \label{app:eq:Fnbound3} &|(P_{n}R_{n}f(y))^2-(PR_{n}f(y))^2| =|P_{n}R_{n}f(y)+PR_{n}f(y)||P_{n}R_{n}f(y)-PR_{n}f(y)| \nonumber \\ &\leq |P_{n}R_{n}f(y)+PR_{n}f(y)|\left\vert R_{n}f\right\vert_{V^\alpha}[M_{1,y}\norm{\mu_n-\mu}_{V^\alpha}+M_{2,y}|\mu_n(y)-\mu(y)|] \nonumber \\ &\leq 2\left\vert R_{n}f\right\vert_{V^\alpha}^2[M_{1,y}\norm{\mu_n-\mu}_{V^\alpha}+M_{2,y}|\mu_n(y)-\mu(y)|] \end{aligned}$$ Also, since $|P(R_{n}f(y)^2)-P(Rf(y)^2)| = |P(y,R_{n}f^2-R f^2)|$, the third and the fourth vanish by Bounded Convergence Theorem and [\[app:lemma:resolventconv,app:lemma:resolventbound\]](#app:lemma:resolventconv,app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventconv,app:lemma:resolventbound"}. Thus, we have proved $$\mathbb{E}(\exp (iuS_{1,n}(f))|\mathcal{F}_{n,0})\rightarrow_{\mathbb{P}} \exp \left(-u^2\sigma^2(f)/2\right), \quad u\in\mathbb{R}^d.$$ Furthermore, by [\[avar:hp:Yapproxnormality\]](#avar:hp:Yapproxnormality){reference-type="ref" reference="avar:hp:Yapproxnormality"}, for $f\in\mathcal{G}(\mathsf{X}\xspace)$, $$\mathbb{E}(\exp (iuS_{2,n}(f)))\rightarrow \exp \left(-u^2v^2(f)/2\right), \quad u\in\mathbb{R}^d$$ for $f\in\mathcal{G}(\mathsf{X}\xspace)$. It follows that for $f\in \mathbf{C}^1_{V}(\mathsf{X}\xspace)\cap \mathcal{G}(\mathsf{X}\xspace)$, $$\begin{aligned} \mathbb{E} (\exp(iuS_n(f))) &= \mathbb{E}((\mathbb{E}(\exp (iuS_{1,n}(f))|\mathcal{F}_{n,0})-\exp(-u^2\sigma^2(f)/2))\exp(iuS_{2,n}(f)))) \\ &+\exp(-u^2\sigma^2(f)/2) \mathbb{E}(\exp (iuS_{2,n}(f))) \rightarrow_{\mathbb{P}} \exp (-u^2(v^2(f)+\sigma^2(f))/2). \end{aligned}$$ ◻ #### Central Limit Theorem for iMCMC. We come to the proof of [Theorem 5](#avar:thm:ZCLT){reference-type="ref" reference="avar:thm:ZCLT"}. Let $f\in\mathbf{B}_{V^\alpha}(\mathsf{X}\xspace)$ and denote $$S_n(f) := n^{-1/2}\sum_{i=1}^n f(Z_i)-\mu(f) = S_{1,n}(f) + S_{2,n}(f)$$ where $S_{1,n}(f)= n^{-1/2}\sum_{i=1}^n f(Y_i)-\mu_{i-1}(f)$ and $S_{2,n}(f)=n^{-1/2}\sum_{i=1}^n(\mu_{i-1}-\mu)(f)$. *Proof of [Theorem 5](#avar:thm:ZCLT){reference-type="ref" reference="avar:thm:ZCLT"}.* We first show that $\mathbb{E}[\exp{(iuS_{1,n})}|\mathcal{F}_{n,0}]\rightarrow \exp(-(u^2/2)\sigma^2)$, $u\in\mathbb{R}^d$. To do that, we check the conditions of [@fort2014clt Theorem 2.2] to prove that $S_{1,n}\Rightarrow N(0,\sigma^2(f))$. Their A1 and A2 are satisfied under our assumption by [\[avar:hp:unifergodicity\]](#avar:hp:unifergodicity){reference-type="ref" reference="avar:hp:unifergodicity"}. To check their A3 we need to check that $$\label{app:eq:ZverifyA3} \frac{1}{\sqrt{n}} \sum_{i=1}^n \left\vert P_{i}R_{i}f-P_{i-1}R_{i-1}f\right\vert_{V^\alpha}V^\alpha(Z_i) \rightarrow_{\mathbb{P}} 0$$ and $$\frac{1}{n^{1/2\alpha}} \sum_{i=1}^n L_{i}^{2/\alpha}P_{i}V(Z_i) \rightarrow_{\mathbb{P}} 0.$$ The last condition is satisfied by [Lemma 4](#avar:lemma:geomerg){reference-type="ref" reference="avar:lemma:geomerg"} since $\sup_n V(Z_n)$ is $\mathbb{P}$-finite and $\sup_n L_n<\infty$. To verify the first, we use [@fort2014clt Lemma A.1] and then the uniformly bounded derivative [\[avar:hp:unifboundedder\]](#avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifboundedder"} with a mean value inequality to write $$\begin{aligned} \left\vert P_{i}R_{i}f-P_{i-1}R_{i-1}f\right\vert_{V^\alpha} &\leq \left\vert f\right\vert_{V^\alpha}5L^6\mu_{i}(V^\alpha)\sup_x \norm{P_{i}(x,\cdot)-P_{{i-1}}(x,\cdot)}_{V^\alpha} \\ &\leq \left\vert f\right\vert_{V^\alpha}5L^6\mu_{i}(V^\alpha)[M_1\norm{\mu_i-\mu_{i-1}}_{V^\alpha}+M_2\sup_x|\mu_i(x)-\mu_{i-1}(x)|] \end{aligned}$$ for some $M_1,M_2<\infty$. Since [Lemma 4](#avar:lemma:geomerg){reference-type="ref" reference="avar:lemma:geomerg"} shows that $\limsup_n \mu_n(V)<\infty$, under [\[avar:hp:unifapprox\]](#avar:hp:unifapprox){reference-type="ref" reference="avar:hp:unifapprox"}, [\[app:eq:ZverifyA3\]](#app:eq:ZverifyA3){reference-type="eqref" reference="app:eq:ZverifyA3"} is verified. We now need to check A4 in [@fort2014clt Theorem 2.2], namely, that $n^{-1}\sum_{i=1}^n F_{i}(Z_i)\rightarrow_{\mathbb{P}}\sigma^2(f)=\mu(F)$, where $F_{n}(y):=P_{n}(R_{n}f(y))^2-(P_{n}R_{n}f(y))^2$ and $F(y):=P(R f(y)^2)-(PRf(y))^2$. For this we use [Theorem 8](#app:thm:Zllns){reference-type="ref" reference="app:thm:Zllns"}. By [Lemma 5](#app:lemma:resolventbound){reference-type="ref" reference="app:lemma:resolventbound"}, $\sup_n \left\vert F_n\right\vert_{V^\alpha}<\infty$, and by using the same arguments of the proof of [Theorem 3](#avar:thm:YCLT){reference-type="ref" reference="avar:thm:YCLT"}, we see that $F_n(y)\rightarrow F(y)$ for all $f\in\mathbf{B}_{V^\alpha}(\mathsf{X}\xspace)$. Hence we only need to verify that $\sum_{i=1}^\infty i^{-1}\sup_x \left\vert F(\cdot,\mu_i)-F(\cdot,\mu_{i-1})\right\vert_{V}<\infty$. Since $$\begin{aligned} |(P_{{i-1}}R_if(x))^2&-(P_{i-1}R_{i-1}f(x))^2+ |P_{i-1}(R_if(y)^2)-P_{i-1}(R_{i-1}f(y)^2)| \\ &\leq \sup_x|R_if(x)+R_{i-1}f(x)||R_if(x)-R_{i-1}f(x)|+ \sup_x |R_if^2(x)-R_{i-1}f^2(x)| \\ &\leq \norm{R_i-R_{i-1}}_{V^\alpha}c_i \end{aligned}$$ with $c_i:=[|f^2|_{V^\alpha}+|f|_{V^\alpha}\sup_x|R_if(x)+R_{i-1}f(x)|]$, using again the mean value estimates [\[app:eq:Fnbound1\]](#app:eq:Fnbound1){reference-type="eqref" reference="app:eq:Fnbound1"}-[\[app:eq:Fnbound3\]](#app:eq:Fnbound3){reference-type="eqref" reference="app:eq:Fnbound3"} with $\mu_{i-1}$ instead of $\mu$ and $M_1, M_2$ instead of $M_{1,y}, M_{2,y}$ (in light of [\[avar:hp:unifboundedder\]](#avar:hp:unifboundedder){reference-type="ref" reference="avar:hp:unifboundedder"}) we obtain $$\begin{aligned} \sum_{i=1}^\infty i^{-1}\sup_x \left\vert F(\cdot,\mu_i)-F(\cdot,\mu_{i-1})\right\vert_{V} \leq \sum_{i=1}^\infty &i^{-1}[M_{1}\norm{\mu_i-\mu_{i-1}}_{V^\alpha}+M_{2}\sup_x|\mu_i(x)-\mu_{i-1}(x)|]d_i \\ &+\norm{R_i-R_{i-1}}_{V^\alpha}c_i \end{aligned}$$ where $d_i:=2|R_if^2|_{V^\alpha}+ 2|R_if|^2_{V^\alpha}$. Since $\sup_n c_n,d_n<\infty$ by [\[avar:hp:unifapprox,app:lemma:resolventconvfast\]](#avar:hp:unifapprox,app:lemma:resolventconvfast){reference-type="ref" reference="avar:hp:unifapprox,app:lemma:resolventconvfast"} the condition is verified. This proves that $S_{1,n}\Rightarrow N(0,\sigma^2(f))$ and then [\[avar:eq:ZrandomcenterCLT\]](#avar:eq:ZrandomcenterCLT){reference-type="eqref" reference="avar:eq:ZrandomcenterCLT"}. The proof for [\[avar:eq:ZdetcenterCLT\]](#avar:eq:ZdetcenterCLT){reference-type="eqref" reference="avar:eq:ZdetcenterCLT"} from [\[avar:eq:ZrandomcenterCLT\]](#avar:eq:ZrandomcenterCLT){reference-type="eqref" reference="avar:eq:ZrandomcenterCLT"} follows the same steps as the one for [\[avar:eq:YdetcenterCLT\]](#avar:eq:YdetcenterCLT){reference-type="eqref" reference="avar:eq:YdetcenterCLT"} from [\[avar:eq:YrandomcenterCLT\]](#avar:eq:YrandomcenterCLT){reference-type="eqref" reference="avar:eq:YrandomcenterCLT"}. ◻

arxiv_math

--- abstract: | This paper explores an iterative coupling approach to solve thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative coupling technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot's poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess. author: - Francesco Ballarin - Sanghyun Lee - Son-Young Yi bibliography: - references.bib title: Projection-based reduced order modeling of an iterative coupling scheme for thermo-poroelasticity --- thermo-poroelasticity ,iterative coupling ,fixed-stress ,reduced order modelling ,proper orthogonal decomposition # Introduction Thermo-hydro-mechanical (THM) processes refer to the coupled interactions between temperature, fluid flow, and mechanical deformation that occur in a wide range of natural and engineered systems. These processes are relevant in various fields, including environmental science, civil engineering, and material science, to name a few. Therefore, the ability to understand and predict tightly coupled THM processes in natural and engineering systems has significant impacts, for example, on the environment, public safety, and the economy. One widely employed mathematical model for describing THM processes is based on Biot's non-isothermal consolidation theory [@biot1956thermo], known as the thermo-poroelasticity model. This model extends Biot's poroelasticity model [@biot1941general], which characterizes the interaction between a deformable porous medium and fluid flow within it under isothermal conditions. The governing system of partial differential equations (PDEs) of the thermo-poroelasticity model consists of a heat transfer equation, mass conservation equation, and momentum conservation equation. These equations are fully coupled through linear and nonlinear coupling terms. For simplicity, this paper focuses on a fully-coupled, linearized thermo-poroelasticity model. Due to the complex nature of the mathematical model for coupled multi-physics thermo-poroelasticity, it is highly challenging to develop an accurate and computationally efficient numerical method. There are various approaches to handling the coupled nature of the problems. First, one can solve the coupled system using a monolithic numerical method, where all subproblems (PDE equations) are solved simultaneously in each time step. In general, monolithic schemes are unconditionally stable but computationally expensive and may require some special linear or nonlinear solvers, often making them less practical. Another approach is to use a sequential method, where the subproblems are separated and solved sequentially in each time step. This approach is much more economical compared to the monolithic method. However, a sequential method may not converge to the right solution. To overcome this limitation, one can resort to an iterative coupling approach, which is a staggered yet tightly coupled scheme. In iterative coupling schemes, the subproblems are solved in a staggered way while ensuring full convergence of the solution in each time step. Iterative coupling schemes for poroelasticity have been extensively studied in recent years [@Park83; @FarhatEtAl91; @HuangZienk98; @SettariWalters99; @wheeler2020ipacs; @Bause17; @BauseElAl17; @BauseElAl21]. In particular, one of the most successful approaches is based on the fixed-stress spitting method [@Wheeler07; @kim2011stability; @mikelic2013convergence; @YiBean17; @lee2017iterative]. In this fixed-stress scheme, one solves the flow problem while fixing the volumetric stress (mean stress), subsequently solving the mechanics problem using the updated pressure solution. The convergence and stability of this fixed-stress splitting method have been studied in the context of various numerical methods, including CG, DG, and mixed finite element methods. In contrast to poroelasticity, the subject of iterative methods for thermo-poroelasticity has not been heavily studied. Some recent papers on this topic include [@brun2020monolithic; @kim2018unconditionally]. This paper first describes an iterative coupling scheme for solving the thermal-poroelasticity model based on the classical continuous Galerkin finite element method, which is considered to be the high-fidelity solver or offline solver in the context of the reduced order modeling approach. We especially utilize the fixed stress iterative method and derive the stability and convergence properties that are close to those of the monolithic approach. Next, we utilize this fixed stress iterative scheme to train a projection-based reduced order model (ROM) as outlined in [@hesthaven2016certified; @quarteroni2015reduced]. ROMs generally consist of two distinct phases: the offline (or training) stage, and the online (or evaluation) phase. In the offline stage, a high-fidelity scheme is employed to generate a dataset containing solutions for the specific problem. In our context, we utilize the fixed stress iterative method, and this dataset is compressed using proper orthogonal decomposition [@berkooz1993proper], yielding a reduced basis space of significantly lower dimension. During the online phase, the ROM incorporates a Galerkin method that leverages the reduced basis space, in contrast to the original high-fidelity space. If the reduced basis space possesses dimensions substantially smaller than its high-fidelity counterpart, and if the problem operators can be suitably pre-assembled, then the ROM evaluation is anticipated to be significantly more computationally efficient than assessing the high-fidelity scheme. The application of projection-based ROMs to THM systems is still limited, see in particular [@larion2020building; @florez2022model]. Furthermore, the use of iterative schemes as high-fidelity solvers in ROMs [@boon2023reduced; @NoninoBallarinRozzaMaday2022; @NoninoBallarinRozza2021; @BallarinRozzaMaday2017; @ShahGirfoglioQuintelaRozzaLengominBallarinBarral2021; @zancanaro2022segregated; @ngan2023reducedorder; @kadeethum2022enhancing] is not as common as monolithic ones, and none of the aforementioned references uses a scheme with the specific properties of the fixed-stress iterative scheme. Therefore, by applying a fixed-stress iterative ROM to a THM problem, this work advances the state-of-the-art and could pave the way for applications of the proposed ROM in geothermal flows. The rest of the paper is organized as follows. We describe the governing equations for the thermo-poroelasticity model in Section [2](#sec: govern){reference-type="ref" reference="sec: govern"}. We then introduce the high-fidelity discretization based on the fixed-stress iterative scheme in Section [3](#sec: hf){reference-type="ref" reference="sec: hf"}. In there, we also prove that the solution of the iterative scheme converges to the monolithic scheme. In Section [4](#sec: rom){reference-type="ref" reference="sec: rom"}, we introduce the projection-based ROM, which builds upon the fixed-stress iterative scheme, and show that its solution converges to the one of the ROM trained on the monolithic scheme. Finally, we present several numerical experiments in Section [5](#sec: num_exp){reference-type="ref" reference="sec: num_exp"} to validate and present the capabilities of the proposed algorithm. # Governing Equations {#sec: govern}  Let $\Omega$ be a bounded, connected, and Lipschitz domain in $\mathbb{R}^d$, $d=2, 3$, with the boundary $\partial \Omega$ and let $\mathbb{I} = (0, T]$ with $T >0$. Then, let ${\bf u}: \Omega \rightarrow \mathbb{R}^d$ be the vector-valued displacement of the solid, $p: \Omega \rightarrow \mathbb{R}$ the scalar-valued fluid pressure, and $\theta: \Omega \rightarrow \mathbb{R}$ the scalar-valued temperature. Then, governing equations for thermo-poroelasticity are derived by coupling momentum balance for mechanics based on linear elasticity, mass balance for the pressure, and energy balance for the temperature as follows: [\[sys: thm\]]{#sys: thm label="sys: thm"} $$\begin{aligned} {2} & - \nabla \cdot( {\boldsymbol \sigma}({\bf u}) - \alpha p {\bf I}- 3 \alpha_T K_{dr} \theta {\bf I})= \mathbf{f}&&\quad \text{ in } \Omega \times \mathbb{I}, \label{thm-m}\\ & \dfrac{\partial {}}{\partial {t}}(c_0p + \alpha \nabla \cdot{\bf u}- 3\alpha_m \theta ) - \nabla \cdot({\bf K}\nabla p) = g && \quad \text{ in } \Omega \times \mathbb{I}, \label{thm-h} \\ & \dfrac{\partial}{\partial t} \left(C_d\theta + 3 \alpha_T K_{dr}\theta_0\nabla \cdot {\bf u}- 3 \alpha_m \theta_0p \right) - \nabla \cdot ({\bf D}\nabla \theta) = \eta && \quad \text{ in } \Omega \times \mathbb{I}, \label{thm-t}\end{aligned}$$ In the momentum balance equation [\[thm-m\]](#thm-m){reference-type="eqref" reference="thm-m"}, ${\boldsymbol \sigma}({\bf u})$ is the standard stress tensor from linear elasticity. It satisfies the constitutive equation ${\boldsymbol \sigma}({\bf u}) := 2\mu {\boldsymbol \epsilon}({\bf u}) + \lambda ( \nabla \cdot{\bf u}) {\bf I}$, where ${\boldsymbol \epsilon}({\bf u}) := \frac{1}{2} [ \nabla {\bf u}+ (\nabla {\bf u})^T ]$ is the strain tensor, ${\bf I}$ is the $d\times d$ identity tensor, and $\mu, \lambda$ are the Lamé constants. The Lamé constants are assumed to be in the range $\mu \in [\mu_0, \mu_1]$ and $\lambda \in [0, \infty)$ for some $0 < \mu_0 < \mu_1 < \infty$. Also, $\mathbf{f}$ is the body force, $\alpha$ is the *Biot-Willis* constant, $K_{dr}:={(d\lambda+2\mu)}/d$ is the drained isothermal bulk modulus, and $\alpha_T$ is the volumetric skeleton thermal dilation coefficient. The total stress tensor is given by $\tilde{{\boldsymbol \sigma}}({\bf u},p,\theta)= {\boldsymbol \sigma}({\bf u}) - \alpha p {\bf I}- 3 \alpha_T K_{dr} \theta {\bf I}$. The second equation [\[thm-h\]](#thm-h){reference-type="eqref" reference="thm-h"} is the mass balance equation for the fluid, assuming the Darcy law for the volumetric fluid flux: ${\bf q}= -{\bf K}\nabla p$. We ignore the gravity effect and set the fluid viscosity to be one here for a simple presentation of the numerical method. However, including the gravity term and the fluid viscosity in the numerical formulation is straightforward. Here, ${\bf K}\in \mathbb{R}^{d \times d}$ is the permeability tensor, which is symmetric and uniformly positive-definite and satisfies the following assumption: there exist positive constants $k_{\text{min}}, k_{\text{max}}$ such that for any ${\bf x}\in \Omega$, $$\label{Kelliptic_1} k_{\text{min}} {\boldsymbol \xi}^T {\boldsymbol \xi}\leq {\boldsymbol \xi}^T {\bf K}({\bf x}) {\boldsymbol \xi}\leq k_{\text{max}} {\boldsymbol \xi}^T {\boldsymbol \xi}, \quad \forall {\boldsymbol \xi}\in \mathbb{R}^d.$$ In addition, $c_0 = 1/M$, where $M$ is Biot's modulus, $3\alpha_m$ is the thermal dilation coefficient, and $g$ is the volumetric fluid source/sink term. Finally, the energy balance equation, or the heat transfer equation [\[thm-t\]](#thm-t){reference-type="eqref" reference="thm-t"}, is obtained by assuming local thermal equilibrium between solid and fluid in pores. Therefore, this energy balance equation is expressed in terms of a single temperature variable $\theta$ with the effective total heat conductivity $C_d$. Also, $\eta$ is the volumetric heat source/sink term and $\theta_0$ is a reference temperature and is assumed to be nonzero. The use of this reference temperature in the heat equation is justified due to small magnitudes of $\alpha_T$ and $\alpha_m$ [@zimmerman2000coupling; @cacace2017flexible]. In addition, the bulk thermal conductivity tensor ${\bf D}$ is symmetric and uniformly positive-definite and assumed to satisfy the following: there exist positive constants $d_{\text{min}}, d_{\text{max}}$ such that for any ${\bf x}\in \Omega$, $$\label{Kelliptic_2} d_{\text{min}} {\boldsymbol \xi}^T {\boldsymbol \xi}\leq {\boldsymbol \xi}^T {\bf D}({\bf x}) {\boldsymbol \xi}\leq d_{\text{max}} {\boldsymbol \xi}^T {\boldsymbol \xi}, \quad \forall {\boldsymbol \xi}\in \mathbb{R}^d.$$ To complete the system of governing equations [\[sys: thm\]](#sys: thm){reference-type="eqref" reference="sys: thm"}, we have to provide initial conditions and boundary conditions. As our main focus here is on the iterative coupling scheme, we will consider the pure homogeneous Dirichlet boundary conditions for all three variables, whereas the initial conditions are given as $$\label{eqn: ic} {\bf u}( \cdot, 0) = {\bf u}^0, \quad p(\cdot, 0) = p^0, \quad \theta(\cdot, 0) = \theta^0 \quad \forall x \in \Omega.$$ # High-fidelity Discretization {#sec: hf} In this section, we present the discretization schemes employed during the reduced order model training in section [4](#sec: rom){reference-type="ref" reference="sec: rom"}, which will be referred to as *high-fidelity schemes* therein. We discuss first the monolithic scheme (section [3.1](#sec: coupled){reference-type="ref" reference="sec: coupled"}), then introduce the proposed fixed-stress iterative scheme (section [3.2](#sec: fs){reference-type="ref" reference="sec: fs"}). In section [3.3](#sec: convergence){reference-type="ref" reference="sec: convergence"}, we prove a convergence result for the proposed fixed-stress iterative scheme by utilizing linear continuous Galerkin finite element methods. ## Monolithic high-fidelity (M-HF) method {#sec: coupled}  In this section, we present a weak formulation and a fully discrete continuous Galerkin method for the model problem [\[sys: thm\]](#sys: thm){reference-type="eqref" reference="sys: thm"}. The standard notation for the $L^2$- and Sobolev spaces and their associated inner products and norms will be used here. In particular, $\left(\cdot, \cdot\right)$ and $\| \cdot\|$ denote the $L^2$ inner product and $L^2$-norm, respectively, and $\left(\cdot, \cdot\right)_1$ and $\| \cdot\|_1$ denote the (full) $H^1$-inner product and (full) $H^1$-norm, respectively. To derive a weak formulation, we multiply [\[thm-m\]](#thm-m){reference-type="eqref" reference="thm-m"}, [\[thm-h\]](#thm-h){reference-type="eqref" reference="thm-h"}, and [\[thm-t\]](#thm-t){reference-type="eqref" reference="thm-t"} by ${\bf v}\in [H^1_{0}(\Omega)]^d$, $w \in H^1_{0}(\Omega)$, and $s \in H^1_{0}(\Omega)$, respectively, and integrate by parts, resulting in the following weak problem: At every $t \in (0, T]$, find $({\bf u}, p, \theta) \in [H^1_{0}(\Omega)]^d\times H^1_{0}(\Omega)\times H^1_{0}(\Omega)$ such that [\[variational\]]{#variational label="variational"} $$\begin{aligned} {2} 2\mu(\varepsilon({\bf u}), \varepsilon({\bf v})) + \lambda ( \nabla \cdot{\bf u}, \nabla \cdot{\bf v}) - \alpha(p, \nabla \cdot{\bf v}) - 3\alpha_T K_{dr} (\theta , \nabla \cdot {\bf v}) &= (\mathbf{f}, {\bf v}), \label{variational-elasticity} \\ c_0( p_t, w ) + \alpha(\nabla \cdot{\bf u}_t, w ) - 3\alpha_m (\theta_t, w) + ( {\bf K}\nabla p, \nabla w) & = (g, w), \label{variational-flow} \\ C_d( \theta_t, s ) + 3 \alpha_T K_{dr} \theta_0( \nabla \cdot {\bf u}_t, s ) - 3\alpha_m \theta_0(p_t, s) + ({\bf D}\nabla \theta, \nabla s) &= (\eta, s), \label{variational-thermal}\end{aligned}$$ for any $({\bf v}, w, s) \in [H^1_{0}(\Omega)]^d\times H^1_{0}(\Omega)\times H^1_{0}(\Omega)$. To present a fully-discrete method, first let $\Delta t> 0$ be a time step size and $t^n = n \Delta t$, where $n = 0, \ldots, N$ with $t^N = T$. For temporal discretization, we consider the backward Euler time-stepping scheme for simplicity. For spatial discretization, let us consider a shape-regular triangulation $\mathcal{T}_h$ of $\Omega$. Then, we let $\boldsymbol{V}_h, W_h, {\Theta}_h$ be finite-dimensional subspaces of $[H^1_{0}(\Omega)]^d, H^1_{0}(\Omega), H^1_{0}(\Omega)$, respectively, on the mesh $\mathcal{T}_h$. Then, the fully-discrete monolithic CG method for [\[variational\]](#variational){reference-type="eqref" reference="variational"} reads as follows: Given $({\bf u}_h^n, p_h^n, \theta_h^n) \in \boldsymbol{V}_h\times W_h\times {\Theta}_h$, find $({\bf u}_h^{n+1}, p_h^{n+1}, \theta_h^{n+1})\in \boldsymbol{V}_h\times W_h\times {\Theta}_h$ such that [\[sys: cg\]]{#sys: cg label="sys: cg"} $$\begin{aligned} &2\mu(\varepsilon({\bf u}_h^{n+1}), \varepsilon({\bf v})) + \lambda ( \nabla \cdot{\bf u}_h^{n+1}, \nabla \cdot{\bf v}) - \alpha(p_h^{n+1}, \nabla \cdot{\bf v}) - 3\alpha_T K_{dr} (\theta_h^{n+1}, \nabla \cdot {\bf v}) \notag \\ & \qquad \qquad = (\mathbf{f}^{n+1}, {\bf v}), \label{eqn: cg-elasticity} \\ &c_0\left( \frac{p_h^{n+1} - p_h^{n}}{\Delta t}, w \right) + \alpha\left(\frac{\nabla \cdot{\bf u}_h^{n+1} - \nabla \cdot{\bf u}_h^{n} }{\Delta t}, w \right) - 3\alpha_m \left( \frac{\theta_h^{n+1}-\theta_h^{n}}{\Delta t}, w \right) \notag \\ & \qquad \qquad + ( {\bf K}\nabla p_h^{n+1}, \nabla w) = (g^{n+1}, w), \label{eqn: cg-flow} \\ &C_d\left( \frac{\theta_h^{n+1}-\theta^{n}}{\Delta t}, s \right) + 3 \alpha_T K_{dr} \theta_0\left( \frac{\nabla \cdot{\bf u}_h^{n+1} - \nabla \cdot{\bf u}_h^{n} }{\Delta t}, s \right) - 3\alpha_m \theta_0\left(\frac{p_h^{n+1} - p_h^{n}}{\Delta t}, s \right) \notag \\ & \qquad \qquad + ({\bf D}\nabla \theta_h^{n+1}, \nabla s) = (\eta^{n+1}, s), \label{eqn: cg-thermal}\end{aligned}$$ for any $({\bf v}, w, s) \in \boldsymbol{V}_h\times W_h\times {\Theta}_h$. The well-posedness and convergence of the CG method for the same model as [\[sys: thm\]](#sys: thm){reference-type="eqref" reference="sys: thm"} but with an extra nonlinear coupling term $-{\bf K}\nabla p^n \cdot \nabla \theta^{n+1}$ in the temperature equation were studied in [@ZhangRui2022]. ## Fixed-stress high-fidelity (FS-HF) iterative scheme {#sec: fs} The M-HF method [\[sys: cg\]](#sys: cg){reference-type="eqref" reference="sys: cg"} is unconditionally stable under certain conditions as studied in [@ZhangRui2022]. However, solving the monolithic system is very expensive computationally. Our goal is to develop an iteratively coupled method whose solutions converge to the solution of the M-HF method. To this end, we will employ a version of the fixed-stress splitting scheme. The main idea behind the fixed-stress split is to fix (or lag) the total mean stress while solving the flow and heat problems. Here, the total mean stress, $\overline{{\boldsymbol \sigma}}$, is defined as $\overline{{\boldsymbol \sigma}}= \mathrm{tr}(\tilde{{\boldsymbol \sigma}})/d$, where $\mathrm{tr}(\cdot)$ is the trace operator on tensors. By noting the relation between volumetric stress and strain, we can see that $$\overline{{\boldsymbol \sigma}}= K_{dr} \nabla \cdot {\bf u}- \alpha p - 3 \alpha_T K_{dr} \theta.$$ In light of this relationship, let $\overline{{\boldsymbol \sigma}}_{h}^{n+1}$ be the approximate total mean stress at time $t^{n+1}$, defined by $$\overline{{\boldsymbol \sigma}}_{h}^{n+1} = K_{dr} \nabla \cdot {\bf u}_h^{n+1} - \alpha p_h^{n+1} - 3 \alpha_T K_{dr} \theta_h^{n+1}.$$ Now, to present our fixed-stress iterative scheme, we will denote by $\phi^{n+1, i}$ the $i$-th iterate of $\phi^{n+1}$ for any variable $\phi = {\bf u}_h, p_h, \theta_h$. To derive a fixed-stress iterative scheme, we want the condition $$\overline{{\boldsymbol \sigma}}_{h}^{n+1, i+1} = \overline{{\boldsymbol \sigma}}_h^{n+1, i}, \quad i = 0, 1, 2, \ldots$$ or, equivalently, $$\label{eqn: divu} \nabla \cdot {\bf u}_h^{n+1, i+1} = \frac{\alpha}{K_{dr}} \left( p_h^{n+1, i+1} -p_h^{n+1, i} \right) + 3\alpha_T \left(\theta^{n+1, i+1} - \theta^{n+1, i} \right) + \nabla \cdot {\bf u}_h^{n+1, i},$$ while solving the flow and heat equations. Note that the flow and heat equations can be solved simultaneously or sequentially, leading to different versions of the fixed-stress scheme. In this section, we consider a method where we solve these two problems sequentially, solving the flow equation first, then the heat problem, before solving the mechanics problem. Since we will solve the flow and heat equations sequentially, we will use a modified version of [\[eqn: divu\]](#eqn: divu){reference-type="eqref" reference="eqn: divu"} when solving them. Specifically, we will assume that the temperature and pressure variables, besides the total mean stress, are fixed while solving the flow and heat problems, respectively. This means that we use $$\label{eqn: divu2} \nabla \cdot {\bf u}_h^{n+1, i+1} = \frac{\alpha}{K_{dr}} \left( p_h^{n+1, i+1} -p_h^{n+1, i} \right) + \nabla \cdot {\bf u}_h^{n+1, i},$$ when solving the flow problem and $$\label{eqn: divu3} \nabla \cdot {\bf u}_h^{n+1, i+1} = 3\alpha_T \left(\theta^{n+1, i} - \theta^{n+1, i} \right) + \nabla \cdot {\bf u}_h^{n+1, i},$$ when solving the heat problem. Given an initialization ${\bf u}_h^{n+1, 0}$, $p_h^{n+1,0}$ and $\theta_h^{n+1,0}$, the following is our FS-HF iterative scheme at each time step $n+1$ to generate infinite sequences $\{ {\bf u}_h^{n+1, i+1} \}_{i=0}^\infty, \{ p_h^{n+1, i+1} \}_{i=0}^\infty$, and $\{ \theta_h^{n+1, i+1} \}_{i=0}^\infty$. Note that $L > 0$ to appear in the algorithm is a stabilization coefficient. If we drive a scheme using [\[eqn: divu2\]](#eqn: divu2){reference-type="eqref" reference="eqn: divu2"} and [\[eqn: divu\]](#eqn: divu){reference-type="eqref" reference="eqn: divu"}, then $L = 1$. However, one can consider tuning this coefficient to achieve faster convergence.\ **Step 1.** Given $({\bf u}_h^{n+1, i}, p_h^{n+1, i},\theta_h^{n+1, i}) \in \boldsymbol{V}_h\times W_h\times {\Theta}_h$, solve the flow problem for $p_h^{n+1, i+1} \in W_h$: [\[sys: fs\]]{#sys: fs label="sys: fs"} $$\begin{aligned} \label{eqn: fs-flow} & c_0\left( \frac{p_h^{n+1, i+1} - p_h^{n}}{\Delta t}, w \right) + ( {\bf K}\nabla p_h^{n+1, i+1}, \nabla w) + L \frac{\alpha^2}{K_{dr}} \left(\frac{p_h^{n+1, i+1} -p_h^{n+1, i}}{\Delta t}, w \right) \notag \\ & \quad = (g^{n+1}, w) - \alpha \left(\nabla \cdot\left( \frac{{\bf u}_h^{n+1, i} - {\bf u}_h^{n} }{\Delta t} \right), w \right) + 3\alpha_m \left( \frac{\theta_h^{n+1, i}-\theta_h^{n}}{\Delta t}, w \right) \quad \forall w \in W_h.\end{aligned}$$ **Step 2.** Given $({\bf u}_h^{n+1, i}, p_h^{n+1, i},\theta_h^{n+1, i}) \in \boldsymbol{V}_h\times W_h\times {\Theta}_h$, solve the heat problem for $\theta_h^{n+1, i+1} \in {\Theta}_h$: $$\begin{aligned} \label{eqn: fs-thermal} &C_d\left( \frac{\theta_h^{n+1, i+1}-\theta_h^{n}}{\Delta t}, s \right) + ( {\bf D}\nabla \theta_h^{n+1, i+1}, \nabla s) + 9 L \alpha_T^2 K_{dr}\theta_0\left( \frac{\theta_h^{n+1, i+1}- \theta_h^{n+1, i}}{\Delta t}, s \right) \notag \\ & \quad = (\eta^{n+1}, s) + 3 \alpha_m\theta_0\left(\frac{p_h^{n+1, i} - p_h^{n}}{\Delta t}, s \right) - 3 \alpha_TK_{dr}\theta_0\left(\nabla \cdot\left( \frac{{\bf u}_h^{n+1, i} - {\bf u}_h^{n} }{\Delta t} \right), s \right) \quad \forall s \in {\Theta}_h.\end{aligned}$$ **Step 3.** Given $({\bf u}_h^{n+1, i}, p_h^{n+1, i+1},\theta_h^{n+1, i+1}) \in \boldsymbol{V}_h\times W_h\times {\Theta}_h$, solve the mechanics problem for ${\bf u}_h^{n+1, i+1} \in \boldsymbol{V}_h$: $$\begin{aligned} \label{eqn: fs-elasticity} &2\mu(\varepsilon({\bf u}_h^{n+1, i+1}), \varepsilon({\bf v})) + \lambda ( \nabla \cdot{\bf u}_h^{n+1, i+1}, \nabla \cdot{\bf v}) \notag \\ & = (\mathbf{f}^{n+1}, {\bf v}) + \alpha(p_h^{n+1, i+1}, \nabla \cdot{\bf v}) + 3\alpha_T K_{dr} (\theta_h^{n+1, i+1}, \nabla \cdot {\bf v}) \quad \forall {\bf v}\in \boldsymbol{V}_h.\end{aligned}$$ Steps 1, 2, and 3 are repeated by increasing $i$ to $i+1$ until an appropriate stopping criterion is satisfied. The specific choice of the stopping criterion depending on a given problem, as well as the procedure to initialize ${\bf u}_h^{n+1,0}$, $p_h^{n+1,0}$, and $\theta_h^{n+1,0}$, will be discussed in Section [5](#sec: num_exp){reference-type="ref" reference="sec: num_exp"}. ## Convergence analysis of the FS-HF scheme {#sec: convergence} In this section, we will prove that the solution of the FS-HF scheme, [\[sys: fs\]](#sys: fs){reference-type="eqref" reference="sys: fs"}, converges to that of the M-HF scheme, [\[sys: cg\]](#sys: cg){reference-type="eqref" reference="sys: cg"}. To facilitate the analysis, we introduce some notations for the errors in the iterates. Let $e_{{\bf u}}^{n+1, i} = {\bf u}_h^{n+1, i} - {\bf u}_h^{n+1}$ be the error in the $i$-th iterate of ${\bf u}_h^{n+1}$ for $1 \leq n \leq N$ and $i \ge 1$. Likewise, $e_p^{n+1, i} = p_h^{n+1, i} - p_h^{n+1}$ and $e_\theta^{n+1, i} = \theta_h^{n+1, i} - \theta_h^{n+1}$ are the errors in the $i$-th iterates of $p_h^{n+1}$ an $\theta_h^{n+1}$, respectively. To derive a system of error equations, subtract [\[eqn: cg-elasticity\]](#eqn: cg-elasticity){reference-type="eqref" reference="eqn: cg-elasticity"}, [\[eqn: cg-flow\]](#eqn: cg-flow){reference-type="eqref" reference="eqn: cg-flow"}, and [\[eqn: cg-thermal\]](#eqn: cg-thermal){reference-type="eqref" reference="eqn: cg-thermal"} from [\[eqn: fs-elasticity\]](#eqn: fs-elasticity){reference-type="eqref" reference="eqn: fs-elasticity"}, [\[eqn: fs-flow\]](#eqn: fs-flow){reference-type="eqref" reference="eqn: fs-flow"}, and [\[eqn: fs-thermal\]](#eqn: fs-thermal){reference-type="eqref" reference="eqn: fs-thermal"}, respectively, and do some algebraic manipulations. We note that the resulting system involves quantities only at the current time step $n+1$. Therefore, in the following, we will use superscripts indicating only the iteration numbers. Then, the system of error equations reads as follows: [\[sys: err\]]{#sys: err label="sys: err"} $$\begin{aligned} &2\mu(\varepsilon(e_{{\bf u}}^{i+1}), \varepsilon({\bf v})) + \lambda ( \nabla \cdot e_{{\bf u}}^{i+1}, \nabla \cdot{\bf v}) = \alpha(e_p^{i+1}, \nabla \cdot{\bf v}) + 3\alpha_T K_{dr} (e_\theta^{i+1}, \nabla \cdot {\bf v}), \label{eqn: err-elasticity} \\ & c_0\left( \frac{e_p^{i+1}}{\Delta t}, w \right) + ( {\bf K}\nabla e_p^{i+1}, \nabla w) + L \frac{\alpha^2}{K_{dr}} \left(\frac{e_p^{i+1} -e_p^{i}}{\Delta t}, w \right) \notag \\ & \qquad = - \alpha \left(\frac{\nabla \cdot e_{{\bf u}}^{i}}{\Delta t} , w \right) + 3\alpha_m \left( \frac{e_\theta^{i}}{\Delta t}, w \right), \label{eqn: err-flow} \\ &C_d\left( \frac{e_\theta^{i+1}}{\Delta t}, s \right) + ( {\bf D}\nabla e_\theta^{i+1}, \nabla s) + 9 L \alpha_T^2 K_{dr}\theta_0\left( \frac{e_\theta^{i+1}- e_\theta^{i}}{\Delta t}, s \right) \notag \\ & \qquad = 3 \alpha_m\theta_0\left(\frac{e_p^{i}}{\Delta t}, s \right) - 3 \alpha_TK_{dr}\theta_0\left( \frac{\nabla \cdot e_{{\bf u}}^{i} }{\Delta t}, s \right), \label{eqn: err-thermal}\end{aligned}$$ for any $({\bf v}, w, s) \in \boldsymbol{V}_h\times W_h\times {\Theta}_h$. For the subsequent analysis, we provide two preliminary results here. **Lemma 1**. *The following coercivity condition is satisfied for any ${\bf v}\in \boldsymbol{V}_h$. $$\label{eqn: coer} K_{dr}\| \nabla \cdot{\bf v}\|^2 \leq 2 \mu \| \varepsilon({\bf v})\|^2 + \lambda \| \nabla \cdot{\bf v}\|^2.$$* ***Proof.*** It is easy to check that $\| \nabla \cdot{\bf v}\|^2 \leq d \| \varepsilon({\bf v})\|^2$ using Young's inequality. Then, [\[eqn: coer\]](#eqn: coer){reference-type="eqref" reference="eqn: coer"} follows from the definition of $K_{dr}$. 0◻ **Lemma 2**. *The following inequality holds: $$\begin{aligned} \label{eqn: eu} & 2\mu \| \varepsilon(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})\|^2 + \lambda \| \nabla \cdot(e_{{\bf u}}^{i+1}- e_{{\bf u}}^{i}) \|^2 \notag \\ & \quad \leq \frac{3}{2} \left( \frac{\alpha^2}{K_{dr}} \| e_p^{i+1} - e_p^i\|^2 + 9 \alpha_T^2 K_{dr}\| e_\theta^{i+1} - e_\theta^i\|^2 \right).\end{aligned}$$* ***Proof.*** First, subtract the mechanics equation [\[eqn: err-elasticity\]](#eqn: err-elasticity){reference-type="eqref" reference="eqn: err-elasticity"} at the iteration $i$ from the iteration $i+1$: $$\begin{aligned} \label{eqn: err-elas1} &2\mu(\varepsilon(e_{{\bf u}}^{i+1}- e_{{\bf u}}^{i}), \varepsilon({\bf v})) + \lambda ( \nabla \cdot(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i}), \nabla \cdot{\bf v})\notag \\ &\quad = \alpha(e_p^{i+1}- e_p^{i}, \nabla \cdot{\bf v}) + 3\alpha_T K_{dr} (e_\theta^{i+1} - e_\theta^{i}, \nabla \cdot {\bf v}). \end{aligned}$$ Then, take ${\bf v}= e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i}$ in [\[eqn: err-elas1\]](#eqn: err-elas1){reference-type="eqref" reference="eqn: err-elas1"} and use the coercivity condition [\[eqn: coer\]](#eqn: coer){reference-type="eqref" reference="eqn: coer"} on the left-hand side and the Cauchy-Schwarz inequality on the right-hand side to obtain $$\begin{aligned} \label{eqn: err-elas2} & K_{dr}\| \nabla \cdot(e_{{\bf u}}^{i+1}- e_{{\bf u}}^{i}) \|^2 \notag \\ & \leq 2\mu \| \varepsilon(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})\|^2 + \lambda \| \nabla \cdot(e_{{\bf u}}^{i+1}- e_{{\bf u}}^{i}) \|^2 \notag \\ & \leq \left( \alpha \| e_p^{i+1}- e_p^{i}\|+ 3\alpha_T K_{dr} \| e_\theta^{i+1} - e_\theta^{i}\|\right) \| \nabla \cdot(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})\|,\end{aligned}$$ which leads to $$\label{eqn: err-elas3} \| \nabla \cdot(e_{{\bf u}}^{i+1}- e_{{\bf u}}^{i}) \| \leq \frac{\alpha}{K_{dr}} \| e_p^{i+1} - e_p^{i}\| + 3\alpha_T \| e_\theta^{i+1}- e_\theta^{i}\|.$$ Using [\[eqn: err-elas3\]](#eqn: err-elas3){reference-type="eqref" reference="eqn: err-elas3"} back in [\[eqn: err-elas2\]](#eqn: err-elas2){reference-type="eqref" reference="eqn: err-elas2"}, we now get $$\begin{aligned} & 2\mu \| \varepsilon(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})\|^2 + \lambda \| \nabla \cdot(e_{{\bf u}}^{i+1}- e_{{\bf u}}^{i}) \|^2 \notag \\ & \leq \left( \alpha \| e_p^{i+1}- e_p^{i}\|+ 3\alpha_T K_{dr} \| e_\theta^{i+1} - e_\theta^{i}\|\right) \left(\frac{\alpha}{K_{dr}} \| e_p^{i+1} - e_p^{i}\| + 3\alpha_T \| e_\theta^{i+1}- e_\theta^{i}\| \right),\end{aligned}$$ from which we can get the desired result [\[eqn: eu\]](#eqn: eu){reference-type="eqref" reference="eqn: eu"} after multiplying out the terms and using Young's inequality. 0◻ **Theorem 1**. *Assume that $$\label{eqn: assump1} c_0> 3 \alpha_m, \quad C_d> 3 \alpha_m\theta_0,$$ and $$\label{eqn: assump2} L \ge \frac{3}{2} \delta \quad \text{for some } \delta \ge \frac{1}{2}.$$ For a fixed $n + 1$, assume that the FS-HF solution at time step $n + 1$ is initialized to the M-HF solution at time step $n$, i.e. $${\bf u}_h^{n+1, 0} = {\bf u}_h^{n}, \quad p_h^{n+1, 0} = p_h^{n}, \quad \theta_h^{n+1, 0} = \theta_h^{n}.$$ Then, the FS-HF solution defined in [\[sys: fs\]](#sys: fs){reference-type="eqref" reference="sys: fs"} converges to the solution of the M-HF method [\[sys: cg\]](#sys: cg){reference-type="eqref" reference="sys: cg"}: $$\| {\bf u}_h^{n+1, i} - {\bf u}_h^{n+1}\|_1 \to 0, \quad \| p_h^{n+1, i} - p_h^{n+1}\| \to 0,\quad \| \theta_h^{n+1, i} - \theta_h^{n+1}\| \to 0$$ as $i \to \infty$.* ***Proof.*** We first recall that $$\label{eqn: i+1-i} (e^{i+1}-e^i, e^{i+1}) = \frac{1}{2} \left( \| e^{i+1} - e^i\|^2 + \| e^{i+1}\|^{2} - \| e^i\|^{2} \right)$$ holds for any $e^i, e^{i+1}$ in $L^2(\Omega)$. Now, take ${\bf v}= e_{{\bf u}}^{i+1}, w = \Delta t\ e_p^{i+1}, s = \Delta t\ e_\theta^{i+1}/\theta_0$ in [\[sys: err\]](#sys: err){reference-type="eqref" reference="sys: err"}, add the three equations, and apply [\[eqn: i+1-i\]](#eqn: i+1-i){reference-type="eqref" reference="eqn: i+1-i"} to obtain $$\begin{aligned} & 2\mu \| \varepsilon(e_{{\bf u}}^{i+1})\|^2 + \lambda \| \nabla \cdot e_{{\bf u}}^{i+1}\|^2 + c_0\| e_p^{i+1}\|^2 + \frac{C_d}{\theta_0}\| e_\theta^{i+1}\|^2 \notag \\ & \quad + \Delta t\left(\| {\bf K}^\frac{1}{2}\nabla e_p^{i+1}\|^2 + \frac{1}{\theta_0} \| {\bf D}^\frac{1}{2}\nabla e_\theta^{i+1}\|^2 \right) + \frac{L}{2} \frac{\alpha^2}{K_{dr}} \left( \| e_p^{i+1} - e_p^i\|^2 + \| e_p^{i+1}\|^{2} - \| e_p^i\|^{2} \right) \notag \\ & \qquad + \frac{9}{2 } L \alpha_T^2 K_{dr}\left( \| e_\theta^{i+1} - e_\theta^i\|^2 + \| e_\theta^{i+1}\|^2 - \| e_\theta^i\|^2 \right) - \alpha (e_p^{i+1}, \nabla \cdot(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})) \notag \\ & \qquad \quad - 3 \alpha_TK_{dr}(e_\theta^{i+1}, \nabla \cdot(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})) = 3 \alpha_m( e_\theta^{i}, e_p^{i+1}) + 3 \alpha_m(e_p^{i}, e_\theta^{i+1}).\end{aligned}$$ The last two terms on the left-hand side of the above equation can be replaced by $$- 2\mu(\varepsilon(e_{{\bf u}}^{i+1}), \varepsilon(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})) - \lambda ( \nabla \cdot e_{{\bf u}}^{i+1}, \nabla \cdot(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})),$$ which is obtained by taking ${\bf v}= -(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})$ in [\[eqn: err-elasticity\]](#eqn: err-elasticity){reference-type="eqref" reference="eqn: err-elasticity"}. Then, applying Young's inequality to the last two terms on the left-hand side and to the two terms on the right-hand side, we obtain $$\begin{aligned} & 2\mu \| \varepsilon(e_{{\bf u}}^{i+1})\|^2 + \lambda \| \nabla \cdot e_{{\bf u}}^{i+1}\|^2 + c_0\| e_p^{i+1}\|^2 + \frac{C_d}{\theta_0}\| e_\theta^{i+1}\|^2 + \Delta t\left( \| {\bf K}^\frac{1}{2}\nabla e_p^{i+1}\|^2 + \frac{1}{\theta_0}\| {\bf D}^\frac{1}{2}\nabla e_\theta^{i+1}\|^2 \right) \notag \\ & \ + \frac{L}{2} \frac{\alpha^2}{K_{dr}} \left( \| e_p^{i+1} - e_p^i\|^2 + \| e_p^{i+1}\|^{2} - \| e_p^i\|^{2} \right) + \frac{9}{2} L \alpha_T^2 K_{dr}\left( \| e_\theta^{i+1} - e_\theta^i\|^2 + \| e_\theta^{i+1}\|^{2} - \| e_\theta^i\|^{2} \right) \notag \\ & \ \ - \frac{\delta}{2}\left( 2\mu \| \varepsilon(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})\|^2 + \lambda \| \nabla \cdot(e_{{\bf u}}^{i+1} - e_{{\bf u}}^{i})\|^2 \right) - \frac{1}{2 \delta}\left( 2\mu \| \varepsilon(e_{{\bf u}}^{i+1})\|^2 + \lambda \| \nabla \cdot e_{{\bf u}}^{i+1}\|^2 \right) \notag \\ & \leq \frac{3 \alpha_m}{2}\left(\| e_p^{i+1} \|^2 + \| e_\theta^{i+1} \|^2 + \| e_p^{i} \|^2 + \| e_\theta^{i} \|^2\right),\end{aligned}$$ where $\delta>0$ is a positive constant. Using the assumptions in [\[eqn: assump1\]](#eqn: assump1){reference-type="eqref" reference="eqn: assump1"}, the uniform ellipticity [\[Kelliptic_1\]](#Kelliptic_1){reference-type="eqref" reference="Kelliptic_1"} and [\[Kelliptic_2\]](#Kelliptic_2){reference-type="eqref" reference="Kelliptic_2"}, Poincaré inequality, and [\[eqn: eu\]](#eqn: eu){reference-type="eqref" reference="eqn: eu"} to the flow and heat flux terms in the left-hand side of the above inequality and moving some terms around, we can get the following inequality: $$\begin{aligned} & \left( 1 - \frac{1}{2\delta}\right) \left( 2\mu \| \varepsilon(e_{{\bf u}}^{i+1})\|^2 + \lambda \| \nabla \cdot e_{{\bf u}}^{i+1}\|^2 \right) + \left( \frac{3\alpha_m}{2} + \frac{L}{2}\frac{\alpha^2}{K_{dr}} + \frac{k_{min}}{C_\Omega} \Delta t\right) \| e_p^{i+1}\|^2 \notag \\ & \quad +\left(\frac{3\alpha_m}{2} + \frac{L}{2} 9 \alpha_T^2 K_{dr}+ \frac{d_{min}}{C_\Omega} \frac{\Delta t}{\theta_0} \right) \| e_\theta^{i+1}\|^2 + \left( \frac{L}{2} -\frac{3 \delta}{4 }\right) \frac{\alpha^2}{K_{dr}} \left( \| e_p^{i+1} - e_p^i\|^2 \right) \notag \\ & \qquad + \left( \frac{L}{2} -\frac{3 \delta}{4 } \right)9 \alpha_T^2 K_{dr}\left( \| e_\theta^{i+1} - e_\theta^i\|^2 \right) \notag \\ & \leq \left(\frac{3 \alpha_m}{2} + \frac{L}{2}\frac{\alpha^2}{K_{dr}} \right) \| e_p^i\|^2 + \left(\frac{3 \alpha_m}{2} + \frac{9}{2} L \alpha_T^2 K_{dr}\right) \| e_\theta^i\|^2,\end{aligned}$$ where $C_\Omega$ is a constant from the Poincaré inequality. Now, due to the assumptions in [\[eqn: assump2\]](#eqn: assump2){reference-type="eqref" reference="eqn: assump2"}, both $\left( 1 - \frac{1}{2\delta}\right)$ and $\left( \frac{L}{2} -\frac{3 \delta}{4 }\right)$ are nonnegative, hence we have $$\begin{aligned} & \left( 3\alpha_m+ L\frac{\alpha^2}{K_{dr}} + \frac{2 k_{min}}{C_\Omega}\Delta t\right) \| e_p^{i+1}\|^2 +\left(3\alpha_m+ 9 L \alpha_T^2 K_{dr}+ \frac{2 d_{min}}{C_\Omega} \frac{\Delta t}{\theta_0} \right) \| e_\theta^{i+1}\|^2 \notag \\ & \leq \left( 3\alpha_m+ L \frac{\alpha^2}{K_{dr}} \right) \| e_p^i\|^2 + \left( 3\alpha_m+ 9 L \alpha_T^2 K_{dr}\right) \| e_\theta^i\|^2.\end{aligned}$$ Now, let $$\label{eqn: tildeC} \tilde{C} = \text{min}\left \{\frac{2 k_{min} \Delta t}{C_\Omega(3\alpha_m+ L\frac{\alpha^2}{K_{dr}})}, \, \frac{2 d_{min} \Delta t}{C_\Omega \theta_0(3\alpha_m+ 9 L \alpha_T^2 K_{dr})} \right \}.$$ Then, we have $$\begin{aligned} \label{eqn: err-flowheat} & ( 1 + \tilde{C}) \left[ \left( 3\alpha_m+ L\frac{\alpha^2}{K_{dr}} \right) \| e_p^{i+1}\|^2+ (3\alpha_m+ 9 L \alpha_T^2 K_{dr})\| e_\theta^{i+1}\|^2 \right ] \notag \\ & \leq \left( 3\alpha_m+ L \frac{\alpha^2}{K_{dr}} \right) \| e_p^i\|^2 + \left( 3\alpha_m+ 9 L \alpha_T^2 K_{dr}\right) \| e_\theta^i\|^2.\end{aligned}$$ If we define a norm $% \left\vert\kern-0.9pt\left\vert\kern-0.9pt\left\vert \cdot \right\vert\kern-0.9pt\right\vert\kern-0.9pt\right\vert$ in $W_h\times {\Theta}_h$ by $$% \left\vert\kern-0.9pt\left\vert\kern-0.9pt\left\vert (w, s) \right\vert\kern-0.9pt\right\vert\kern-0.9pt\right\vert^2 = \left( 3\alpha_m+ L \frac{\alpha^2}{K_{dr}} \right) \| w\|^2 + \left( 3\alpha_m+ 9 L \alpha_T^2 K_{dr}\right) \| s\|^2 \quad \forall (w, s) \in W_h\times {\Theta}_h,$$ then, [\[eqn: err-flowheat\]](#eqn: err-flowheat){reference-type="eqref" reference="eqn: err-flowheat"} can be rewritten as $$\label{eqn: contract} % \left\vert\kern-0.9pt\left\vert\kern-0.9pt\left\vert (e_p^{i+1},e_\theta^{i+1}) \right\vert\kern-0.9pt\right\vert\kern-0.9pt\right\vert^2 \leq \frac{1}{1 + \tilde{C}}% \left\vert\kern-0.9pt\left\vert\kern-0.9pt\left\vert (e_p^{i},e_\theta^{i}) \right\vert\kern-0.9pt\right\vert\kern-0.9pt\right\vert^2.$$ Therefore, $% \left\vert\kern-0.9pt\left\vert\kern-0.9pt\left\vert (e_p^{i},e_\theta^{i}) \right\vert\kern-0.9pt\right\vert\kern-0.9pt\right\vert \to 0$ as $i \to \infty$, which also implies that $\| e_p^{i}\| \to 0$ and $\| e_\theta^{i}\| \to 0$. In order to prove the convergence of $\| e_{{\bf u}}^{i}\|_1 \to 0$, first obtain $$\label{eqn: eu2} 2\mu \| \varepsilon(e_{{\bf u}}^{i+1})\|^2 + \lambda \| \nabla \cdot(e_{{\bf u}}^{i+1}) \|^2 \leq \frac{3}{2} \left( \frac{\alpha^2}{K_{dr}} \| e_p^{i+1}\|^2 + 9 \alpha_T^2 K_{dr}\| e_\theta^{i+1}\|^2 \right)$$ in the same manner as in the proof of Lemma [Lemma 2](#lem: eu){reference-type="ref" reference="lem: eu"}, and recall Korn's inequality, stating that there is a constant $C_{korn} >0$ such that $$C_{korn} \| {\bf v}\|_1^2 \leq \| \varepsilon({\bf v})\|^2 \quad \forall {\bf v}\in [H^1_{0}(\Omega)]^d.$$ Using these two results, we have $$\| e_{{\bf u}}^{i+1}\|_1^2 \leq \frac{3}{4 \mu C_{korn}} \left( \frac{\alpha^2}{K_{dr}} \| e_p^{i+1}\|^2 + 9 \alpha_T^2 K_{dr}\| e_\theta^{i+1}\|^2 \right).$$ As the quantity on the right-hand side goes to $0$ as $i \to \infty$, $\| e_{{\bf u}}^{i}\|_1$ also goes to $0$ as $i \to \infty$. 0◻ # Projection-Based Reduced Order Models {#sec: rom} In this section, we discuss projection-based reduced order models (ROMs) based on compression of the time history of the solutions provided by one of the high-fidelity methods (either M-HF or FS-HF method) from Section [3](#sec: hf){reference-type="ref" reference="sec: hf"}. Such compression comprises the offline stage (or training) of the ROM and will result in a reduced basis of small dimension $r$. The reduced basis will be obtained by means of the proper orthogonal decomposition (POD) algorithm, see Section [4.1](#sec: POD){reference-type="ref" reference="sec: POD"}. The online stage (or evaluation) of the ROM will then consist of a Galerkin method applied on either the fully-coupled scheme (Section [4.2](#sec: coupled ROM){reference-type="ref" reference="sec: coupled ROM"}) or the fixed-stress iterative scheme (Section [4.3](#sec: fs ROM){reference-type="ref" reference="sec: fs ROM"}). Convergence of the fixed-stress iterative ROM scheme to the solution of the fully-coupled ROM scheme will be proven in section [4.4](#sec: convergence ROM){reference-type="ref" reference="sec: convergence ROM"}. ## Proper orthogonal decomposition {#sec: POD} Let $\phi = {\bf u}_h, p_h, \theta_h$ denote one of the components of the solution field, the displacement, pressure, and temperature, respectively. Then, $\left\{\phi^{n}\right\}_{n = 0}^{N}$ denote the sequence obtained by collecting such a component at different times, $t^0, \hdots, t^N$. The proper orthogonal decomposition [@berkooz1993proper; @quarteroni2015reduced; @hesthaven2016certified] is a data analysis method that aims to obtain an equivalent representation of the sequence $\left\{\phi^{n}\right\}_{n = 0}^{N}$ as a linear combination of at most $N + 1$ orthogonal modes $\varphi_{0}, \hdots, \varphi_{N}$ and define information content indices $\nu_{0}, \hdots, \nu_{N}$ associated to each corresponding mode. Assuming the latter to be sorted as $\nu_{0} \geq \nu_{1} \geq \hdots \geq \nu_{N}$, POD is typically used as a data *compression* tool to obtain an approximate representation in the linear space $\Phi_r = \mathrm{span}(\varphi_{0}, \hdots, \varphi_{r - 1})$, truncating the combination to the first $r$ modes for some $r \leq N$ and neglecting the modes $\varphi_{r}, \hdots, \varphi_{N}$ associated to indices $\nu_{r}, \hdots, \nu_{N}$ with small information content. In the context of ROMs, the primary interest of the application of POD is in the generation of the space $\Phi_r$, to be called the *reduced basis space*. To this end, we will focus on the so-called *method of snapshots*, even though alternative presentations are possible, for instance by means of the singular value decomposition or the principal component analysis [@berkooz1993proper; @quarteroni2015reduced; @hesthaven2016certified]. First of all, the method of snapshots requires solving the following eigenvalue problem $$\label{eq:POD_eigenvalues} \mathbf{C}^{\phi} \underline{\boldsymbol{\mathbf{v}}} = \nu \ \underline{\boldsymbol{\mathbf{v}}}, \text{ where } \left[\mathbf{C}^{\phi}\right]_{nm} = (\phi^n, \phi^m)_1 \text{ for every } n, m = 0, \hdots, N.$$ Since the matrix $\mathbf{C}^{\phi} \in \mathbb{R}^{(N + 1) \times (N + 1)}$ is symmetric positive definite by construction, its eigenvalues $\nu_{0}, \hdots, \nu_{N}$ are real and positive and, without loss of generality, can be assumed to be sorted in decreasing order. Denoting by $\underline{\boldsymbol{\mathbf{v}}}_n \in \mathbb{R}^{N + 1}$ the eigenvector associated with the eigenvalue $\nu_n$, the $n$-th POD mode is then computed as $$\label{eq: POD mode} \varphi_n = \frac{1}{\sqrt{\nu_n}} \sum_{\beta = 0}^{N} \left[\underline{\boldsymbol{\mathbf{v}}}_n\right]_{\beta} \phi^{\beta}.$$ Indeed, the obtained POD modes are $(\cdot, \cdot)_1$-orthonormal. To see this, first note that $\underline{\boldsymbol{\mathbf{v}}}_n^T \underline{\boldsymbol{\mathbf{v}}}_m = \delta_{nm}$, where $\delta_{nm}$ is the Kronecker delta function. Then, we obtain $$\begin{aligned} \left(\varphi_n, \varphi_m\right)_1 &= \frac{1}{\sqrt{\nu_n \nu_m}} \left(\sum_{\beta = 0}^{N} \left[\underline{\boldsymbol{\mathbf{v}}}_n\right]_{\beta} \phi^{\beta}, \sum_{\gamma = 0}^{N} \left[\underline{\boldsymbol{\mathbf{v}}}_m\right]_{\gamma} \phi^{\gamma}\right)_1 = \frac{1}{\sqrt{\nu_n \nu_m}} \sum_{\beta = 0}^{N} \sum_{\gamma = 0}^{N} \left[\underline{\boldsymbol{\mathbf{v}}}_n\right]_{\beta} \left[\underline{\boldsymbol{\mathbf{v}}}_m\right]_{\gamma} %\left(\phi^{\beta}, \phi^{\gamma}\right)_1\\ \left[\mathbf{C}^{\phi}\right]_{\beta, \gamma}\notag\\ &= \frac{1}{\sqrt{\nu_n \nu_m}} \underline{\boldsymbol{\mathbf{v}}}_n^T \mathbf{C}^{\phi} \underline{\boldsymbol{\mathbf{v}}}_m = \frac{\nu_m}{\sqrt{\nu_n \nu_m}} \underline{\boldsymbol{\mathbf{v}}}_n^T \underline{\boldsymbol{\mathbf{v}}}_m = \delta_{nm}.\label{eq: POD orthonormal}\end{aligned}$$ Finally, we notice that POD modes $\varphi_{0}, \hdots, \varphi_{N}$ satisfy homogeneous Dirichlet boundary conditions if the input sequence $\phi^{0}, \hdots, \phi^{N}$ is zero on $\partial\Omega$. This easily follows by taking the trace of [\[eq: POD mode\]](#eq: POD mode){reference-type="eqref" reference="eq: POD mode"} on $\partial\Omega$. Therefore, any element in $\Phi_r$ is by construction equal to zero on $\partial\Omega$. ## Monolithic reduced order model (M-ROM) {#sec: coupled ROM} In this section, we describe a projection-based ROM, which builds upon the fully-coupled monolithic (M-HF) scheme introduced in Section [3.1](#sec: coupled){reference-type="ref" reference="sec: coupled"} as a high-fidelity discretization. During the offline stage, the high-fidelity discretization is queried to obtain three sequences, $\left\{{\bf u}_h^{n}\right\}_{n = 0}^{N}$, $\left\{p_h^{n}\right\}_{n = 0}^{N}$, and $\left\{\theta_h^{n}\right\}_{n = 0}^{N}$, representing the time history of the displacement, pressure, and temperature fields, respectively. Upon selecting a reduced basis size $r \leq N$ and applying POD to each sequence $\left\{{\bf u}_h^{n}\right\}_{n = 0}^{N}$, $\left\{p_h^{n}\right\}_{n = 0}^{N}$, and $\left\{\theta_h^{n}\right\}_{n = 0}^{N}$ as discussed in Section [4.1](#sec: POD){reference-type="ref" reference="sec: POD"}, we obtain the following reduced basis spaces $$\boldsymbol{V}_r= \mathrm{span}\left\{\varphi_1^{{\bf u}}, \hdots, \varphi_r^{{\bf u}}\right\}, \,\, W_r= \mathrm{span}\left\{\varphi_1^{p}, \hdots, \varphi_r^{p}\right\}, \,\, {\Theta}_r= \mathrm{span}\left\{\varphi_1^{\theta}, \hdots, \varphi_r^{\theta}\right\},$$ respectively. During the online stage, the ROM is a Galerkin method for problem [\[variational\]](#variational){reference-type="eqref" reference="variational"} on the reduced basis space $\boldsymbol{V}_r\times W_r\times {\Theta}_r$. Therefore, the fully-coupled monolithic ROM (M-ROM) scheme for [\[variational\]](#variational){reference-type="eqref" reference="variational"} reads as follows: Given $({\bf u}_r^n, p_r^n, \theta_r^n) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$, find $({\bf u}_r^{n+1}, p_r^{n+1}, \theta_r^{n+1}) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$ such that [\[sys: cg ROM\]]{#sys: cg ROM label="sys: cg ROM"} $$\begin{aligned} &2\mu(\varepsilon({\bf u}_r^{n+1}), \varepsilon({\bf v})) + \lambda ( \nabla \cdot{\bf u}_r^{n+1}, \nabla \cdot{\bf v}) - \alpha(p_r^{n+1}, \nabla \cdot{\bf v}) - 3\alpha_T K_{dr} (\theta_r^{n+1}, \nabla \cdot {\bf v}) \notag \\ & \qquad \qquad = (\mathbf{f}^{n+1}, {\bf v}), \label{eqn: cg-elasticity-ROM} \\ &c_0\left( \frac{p_r^{n+1} - p_r^{n}}{\Delta t}, w \right) + \alpha\left(\frac{\nabla \cdot{\bf u}_r^{n+1} - \nabla \cdot{\bf u}_r^{n} }{\Delta t}, w \right) - 3\alpha_m \left( \frac{\theta_r^{n+1}-\theta_r^{n}}{\Delta t}, w \right) \notag \\ & \qquad \qquad + ( {\bf K}\nabla p_r^{n+1}, \nabla w) = (g^{n+1}, w), \label{eqn: cg-flow-ROM} \\ &C_d\left( \frac{\theta_r^{n+1}-\theta^{n}}{\Delta t}, s \right) + 3 \alpha_T K_{dr} \theta_0\left( \frac{\nabla \cdot{\bf u}_r^{n+1} - \nabla \cdot{\bf u}_r^{n} }{\Delta t}, s \right) - 3\alpha_m \theta_0\left(\frac{p_r^{n+1} - p_r^{n}}{\Delta t}, s \right) \notag \\ & \qquad \qquad + ({\bf D}\nabla \theta_r^{n+1}, \nabla s) = (\eta^{n+1}, s), \label{eqn: cg-thermal-ROM}\end{aligned}$$ for any $({\bf v}, w, s) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$. Even though [\[sys: cg\]](#sys: cg){reference-type="eqref" reference="sys: cg"} and [\[sys: cg ROM\]](#sys: cg ROM){reference-type="eqref" reference="sys: cg ROM"} are both obtained by means of a Galerkin method, the fundamental difference is that the reduced basis space $\boldsymbol{V}_r\times W_r\times {\Theta}_r$ employed in [\[sys: cg ROM\]](#sys: cg ROM){reference-type="eqref" reference="sys: cg ROM"} is of small dimension $3r$, owing to the compression carried out by POD. In order to highlight further differences between [\[sys: cg\]](#sys: cg){reference-type="eqref" reference="sys: cg"} and [\[sys: cg ROM\]](#sys: cg ROM){reference-type="eqref" reference="sys: cg ROM"}, let $\underline{\boldsymbol{\mathbf{u}}}_r^{n+1} \in \mathbb{R}^r$ be the vector whose components are the M-ROM degrees of freedom for ${\bf u}_r^{n+1} \in \boldsymbol{V}_r$, i.e. $${\bf u}_r^{n+1} = \sum_{\beta = 0}^r [\underline{\boldsymbol{\mathbf{u}}}_r^{n+1}]_\beta \varphi_\beta^{{\bf u}}.$$ Similarly, let $\underline{\boldsymbol{\mathbf{p}}}_r^{n+1} \in \mathbb{R}^r$ and $\underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1} \in \mathbb{R}^r$ be the vector whose components are the M-ROM degrees of freedom for $p_r^{n+1} \in W_r$ and $\theta_r^{n+1} \in {\Theta}_r$, respectively. The system [\[sys: cg ROM\]](#sys: cg ROM){reference-type="eqref" reference="sys: cg ROM"} can thus be written in the following block-matrix form $$\label{eq: block cg ROM} \begin{bmatrix} \mathbf{A}_r^{{\bf u}{\bf u}} & \mathbf{A}_r^{{\bf u}p} & \mathbf{A}_r^{{\bf u}\theta}\\ \mathbf{M}_r^{p {\bf u}} & \mathbf{M}_r^{p p} + \mathbf{A}_r^{p p} & \mathbf{M}_r^{p \theta}\\ \mathbf{M}_r^{\theta {\bf u}} & \mathbf{M}_r^{\theta p} & \mathbf{M}_r^{\theta \theta} + \mathbf{A}_r^{\theta \theta},\\ \end{bmatrix} \begin{bmatrix} \underline{\boldsymbol{\mathbf{u}}}_r^{n+1}\\ \underline{\boldsymbol{\mathbf{p}}}_r^{n+1}\\ \underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1} \end{bmatrix} = \begin{bmatrix} \textbf{0} & \textbf{0} & \textbf{0}\\ \mathbf{M}_r^{p {\bf u}} & \mathbf{M}_r^{p p} & \mathbf{M}_r^{p \theta}\\ \mathbf{M}_r^{\theta {\bf u}} & \mathbf{M}_r^{\theta p} & \mathbf{M}_r^{\theta \theta}\\ \end{bmatrix} \begin{bmatrix} \underline{\boldsymbol{\mathbf{u}}}_r^{n}\\ \underline{\boldsymbol{\mathbf{p}}}_r^{n}\\ \underline{\boldsymbol{\mathbf{\theta}}}_r^{n} \end{bmatrix} + \begin{bmatrix} \underline{\boldsymbol{\mathbf{f}}}_r^{n+1}\\ \underline{\boldsymbol{\mathbf{g}}}_r^{n+1}\\ \underline{\boldsymbol{\mathbf{\eta}}}_r^{n+1} \end{bmatrix}$$ where the expressions of matrices and vectors appearing in the block formulation are summarized in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"}. Indeed, matrices in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"} can be assembled once and for all at the end of the offline stage since such computations involve integration over the finite element mesh $\mathcal{T}_h$. Similarly, vectors in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"} can be pre-assembled at the end of the offline stage for every $n = 1, \hdots, N$ and stored. During the online stage, iterating in time through [\[eq: block cg ROM\]](#eq: block cg ROM){reference-type="eqref" reference="eq: block cg ROM"} can finally be carried out at a vastly decreased computational cost since i) the assembly of the block system only requires loading $r \times r$ matrices and vectors of dimension $r$ in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"} from storage without necessitating any operation involving the finite element $\mathcal{T}_h$, and ii) the resulting linear system is of small size $3r \times 3r$. **ROM matrices** **ROM vectors** ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------ $[\mathbf{A}_r^{{\bf u}{\bf u}}]_{\beta\gamma} = 2\mu(\varepsilon(\varphi_\gamma^{{\bf u}}), \varepsilon(\varphi_\beta^{{\bf u}})) + \lambda ( \nabla \cdot\varphi_\gamma^{{\bf u}}, \nabla \cdot\varphi_\beta^{{\bf u}})$, $[\underline{\boldsymbol{\mathbf{f}}}_r^{n+1}]_\beta = (\mathbf{f}^{n+1}, \varphi_\beta^{{\bf u}})$, $[\mathbf{A}_r^{\theta \theta}]_{\beta\gamma} = ({\bf D}\nabla \varphi_\gamma^{\theta} , \nabla \varphi_\beta^{\theta})$, $[\mathbf{A}_r^{p p}]_{\beta\gamma} = ( {\bf K}\nabla \varphi_\gamma^{p}, \nabla \varphi_\beta^{p})$, $[\underline{\boldsymbol{\mathbf{g}}}_r^{n+1}]_\beta = (g^{n+1}, \varphi_\beta^{p})$, $[\mathbf{A}_r^{{\bf u}\theta}]_{\beta\gamma} = - 3\alpha_T K_{dr} (\varphi_\gamma^{\theta}, \nabla \cdot \varphi_\beta^{{\bf u}})$, $[\mathbf{A}_r^{{\bf u}p}]_{\beta\gamma} = - \alpha(\varphi_\gamma^{p}, \nabla \cdot\varphi_\beta^{{\bf u}})$, $[\underline{\boldsymbol{\mathbf{\eta}}}_r^{n+1}]_\beta = (\eta^{n+1}, \varphi_\beta^{\theta}).$ $[\mathbf{M}_r^{\theta \theta}]_{\beta\gamma} = $[\mathbf{M}_r^{p p}]_{\beta\gamma} = c_0\left( \frac{\varphi_\gamma^{p}}{\Delta t}, \varphi_\beta^{p} \right)$, C_d\left( \frac{\varphi_\gamma^{\theta}}{\Delta t}, \varphi_\beta^{\theta} \right)$, $[\mathbf{M}_r^{\theta {\bf u}}]_{\beta\gamma} = $[\mathbf{M}_r^{\theta p}]_{\beta\gamma} = 3 \alpha_T K_{dr} \theta_0\left( \frac{\nabla \cdot\varphi_\gamma^{{\bf u}} }{\Delta t}, \varphi_\beta^{\theta} \right)$, - 3\alpha_m \theta_0\left(\frac{\varphi_\gamma^{p}}{\Delta t}, \varphi_\beta^{\theta} \right)$, $[\mathbf{M}_r^{p {\bf u}}]_{\beta\gamma} = \alpha\left(\frac{\nabla \cdot\varphi_\gamma^{{\bf u}}}{\Delta t}, \varphi_\beta^{p} \right)$, $[\mathbf{M}_r^{p \theta}]_{\beta\gamma} = - 3\alpha_m \left( \frac{\varphi_\gamma^{\theta}}{\Delta t}, \varphi_\beta^{p} \right)$, $[\mathbf{S}_r^{\theta \theta}]_{\beta\gamma} = $[\mathbf{S}_r^{p p}]_{\beta\gamma} = L \frac{\alpha^2}{K_{dr}} \left(\frac{\varphi_\gamma^{p}}{\Delta t}, \varphi_\beta^{p} \right)$. 9 L \alpha_T^2 K_{dr}\theta_0\left( \frac{\varphi_\gamma^{\theta}}{\Delta t}, \varphi_\beta^{\theta} \right)$, : Definitions of ROM matrices and vectors in [\[eq: block cg ROM\]](#eq: block cg ROM){reference-type="eqref" reference="eq: block cg ROM"}. Here $\beta, \gamma = 1, \hdots, r$. ## Fixed-stress reduced order model (FS-ROM) {#sec: fs ROM} In this section, we further describe a projection-based ROM built upon the fixed-stress high-fidelity (FS-HF) iterative scheme described in Section [3.2](#sec: fs){reference-type="ref" reference="sec: fs"}. During the offline stage, the FS-HF scheme is queried to obtain the time evolution $\left\{{\bf u}_h^{n,\infty}\right\}_{n = 0}^{N}$, $\left\{p_h^{n, \infty}\right\}_{n = 0}^{N}$ and $\left\{\theta_h^{n, \infty}\right\}_{n = 0}^{N}$ of displacement, pressure and temperature fields, respectively, where the superscript ${n,\infty}$ denotes the converged solutions at time step $n$. We notice that in practice the number of iterations will not be $\infty$ upon defining suitable stopping criteria in Section [5.1](#sec: num){reference-type="ref" reference="sec: num"}. Those sequences are then compressed by means of POD to obtain the reduced basis spaces $\boldsymbol{V}_r^{FS}, W_r^{FS}$ and ${\Theta}_r^{FS}$ upon proceeding as in Section [4.2](#sec: coupled ROM){reference-type="ref" reference="sec: coupled ROM"}. We notice that the obtained reduced basis spaces $\boldsymbol{V}_r^{FS}, W_r^{FS}$ and ${\Theta}_r^{FS}$ are in principle different from $\boldsymbol{V}_r, W_r$ and ${\Theta}_r$ obtained in Section [4.2](#sec: coupled ROM){reference-type="ref" reference="sec: coupled ROM"} since the former is obtained by applying POD to solutions of the FS-HF scheme, while the latter is the result of a compression of solutions computed by the M-HF scheme. For the sake of a simpler notation, however, we will drop the suffix $FS$ from the reduced basis spaces $\boldsymbol{V}_r^{FS}, W_r^{FS}$, and ${\Theta}_r^{FS}$, still understanding that reduced basis spaces employed in this section are generated from the FS-HF scheme, rather than the M-HF one. During the online stage, given an initialization $({\bf u}_r^{n+1, 0}, p_r^{n+1,0}, \theta_r^{n+1,0}) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$, the ROM uses a Galerkin projection of a fixed-stress iterative scheme to generate infinite sequences $\{ {\bf u}_r^{n+1, i+1} \}_{i=0}^\infty \subset \boldsymbol{V}_r, \{ p_r^{n+1, i+1} \}_{i=0}^\infty \subset W_r$, and $\{ \theta_r^{n+1, i+1} \}_{i=0}^\infty \subset {\Theta}_r$.\ **Step 1-ROM.** Given $({\bf u}_r^{n+1, i}, p_r^{n+1, i},\theta_r^{n+1, i}) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$, solve the flow problem for $p_r^{n+1, i+1} \in W_r$: [\[sys: fs ROM\]]{#sys: fs ROM label="sys: fs ROM"} $$\begin{aligned} \label{eqn: fs-flow ROM} & c_0\left( \frac{p_r^{n+1, i+1} - p_r^{n}}{\Delta t}, w \right) + ( {\bf K}\nabla p_r^{n+1, i+1}, \nabla w) + L \frac{\alpha^2}{K_{dr}} \left(\frac{p_r^{n+1, i+1} -p_r^{n+1, i}}{\Delta t}, w \right) \notag \\ & \quad = (g^{n+1}, w) - \alpha \left(\nabla \cdot\left( \frac{{\bf u}_r^{n+1, i} - {\bf u}_r^{n} }{\Delta t} \right), w \right) + 3\alpha_m \left( \frac{\theta_r^{n+1, i}-\theta_r^{n}}{\Delta t}, w \right) \quad \forall w \in W_r.\end{aligned}$$ **Step 2-ROM.** Given $({\bf u}_r^{n+1, i}, p_r^{n+1, i},\theta_r^{n+1, i}) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$, solve the heat problem for $\theta_r^{n+1, i+1} \in {\Theta}_r$: $$\begin{aligned} \label{eqn: fs-thermal ROM} &C_d\left( \frac{\theta_r^{n+1, i+1}-\theta_r^{n}}{\Delta t}, s \right) + ( {\bf D}\nabla \theta_r^{n+1, i+1}, \nabla s) + 9 L \alpha_T^2 K_{dr}\theta_0\left( \frac{\theta_r^{n+1, i+1}- \theta_r^{n+1, i}}{\Delta t}, s \right) \notag \\ & = (\eta^{n+1}, s) + 3 \alpha_m\theta_0\left(\frac{p_r^{n+1, i} - p_r^{n}}{\Delta t}, s \right) - 3 \alpha_TK_{dr}\theta_0\left(\nabla \cdot\left( \frac{{\bf u}_r^{n+1, i} - {\bf u}_r^{n} }{\Delta t} \right), s \right) \quad \forall s \in {\Theta}_r.\end{aligned}$$ **Step 3-ROM.** Given $({\bf u}_r^{n+1, i}, p_r^{n+1, i+1},\theta_r^{n+1, i+1}) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$, solve the mechanics problem for ${\bf u}_r^{n+1, i+1} \in \boldsymbol{V}_r$: $$\begin{aligned} \label{eqn: fs-elasticity ROM} &2\mu(\varepsilon({\bf u}_r^{n+1, i+1}), \varepsilon({\bf v})) + \lambda ( \nabla \cdot{\bf u}_r^{n+1, i+1}, \nabla \cdot{\bf v}) \notag \\ & \quad = (\mathbf{f}^{n+1}, {\bf v}) + \alpha(p_r^{n+1, i+1}, \nabla \cdot{\bf v}) + 3\alpha_T K_{dr} (\theta_r^{n+1, i+1}, \nabla \cdot {\bf v}) \quad \forall {\bf v}\in \boldsymbol{V}_r.\end{aligned}$$\ Steps 1-ROM, 2-ROM, 3-ROM are repeated by increasing $i$ to $i +1$ until appropriate stopping criteria are satisfied. The specific choice of the stopping criteria, as well as the procedure to initialize ${\bf u}_r^{n+1, 0}$, $p_r^{n+1,0}$ and $\theta_r^{n+1,0}$, will be discussed in Section [5.1](#sec: num){reference-type="ref" reference="sec: num"}. The FS-ROM scheme can be equivalently reformulated in matrix form as follows. It generates the sequences $\{ \underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i+1} \}_{i=0}^\infty \subset \mathbb{R}^r, \{ \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i+1} \}_{i=0}^\infty \subset \mathbb{R}^r$, and $\{ \underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i+1} \}_{i=0}^\infty \subset \mathbb{R}^r$ containing their ROM degrees of freedom.\ **Step i-ROM.** Given $(\underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i}, \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i},\underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i}) \in \mathbb{R}^r \times \mathbb{R}^r \times \mathbb{R}^r$, solve the flow problem for $\underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i+1} \in \mathbb{R}^r$: [\[sys: fs ROMeq\]]{#sys: fs ROMeq label="sys: fs ROMeq"} $$\begin{aligned} \label{eqn: fs-flow ROMeq} &\left(\mathbf{M}_r^{p p} + \mathbf{A}_r^{p p} + \mathbf{S}_r^{p p}\right) \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i + 1} = \underline{\boldsymbol{\mathbf{g}}}_r^{n+1} - \mathbf{M}_r^{p p} \underline{\boldsymbol{\mathbf{p}}}_r^{n} - \mathbf{S}_r^{p p} \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i} \notag\\ &\quad - \mathbf{M}_r^{p {\bf u}} \left(\underline{\boldsymbol{\mathbf{u}}}_r^{n+1,i} - \underline{\boldsymbol{\mathbf{u}}}_r^{n}\right) - \mathbf{M}_r^{p \theta} \left(\underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i} - \underline{\boldsymbol{\mathbf{\theta}}}_r^{n}\right).\end{aligned}$$ **Step ii-ROM.** Given $(\underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i}, \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i},\underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i}) \in \mathbb{R}^r \times \mathbb{R}^r \times \mathbb{R}^r$, solve the heat problem for $\underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i+1} \in \mathbb{R}^r$: $$\begin{aligned} \label{eqn: fs-thermal ROMeq} &\left(\mathbf{M}_r^{\theta \theta} + \mathbf{A}_r^{\theta \theta} + \mathbf{S}_r^{\theta \theta}\right) \underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1,i+1} = \underline{\boldsymbol{\mathbf{\eta}}}_r^{n+1} - \mathbf{M}_r^{\theta \theta} \underline{\boldsymbol{\mathbf{\theta}}}_r^{n} - \mathbf{S}_r^{\theta \theta} \underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1,i}\notag\\ &\quad- \mathbf{M}_r^{\theta {\bf u}}\left(\underline{\boldsymbol{\mathbf{u}}}_r^{n+1, i} - \underline{\boldsymbol{\mathbf{u}}}_r^{n}\right) - \mathbf{M}_r^{\theta p} \left(\underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i} - \underline{\boldsymbol{\mathbf{p}}}_r^{n}\right).\end{aligned}$$ **Step iii-ROM.** Given $(\underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i}, \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i+1},\underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i+1}) \in \mathbb{R}^r \times \mathbb{R}^r \times \mathbb{R}^r$, solve the mechanics problem for $\underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i+1} \in \mathbb{R}^r$: $$\begin{aligned} \label{eqn: fs-elasticity ROMeq} &\mathbf{A}_r^{{\bf u}{\bf u}} \underline{\boldsymbol{\mathbf{u}}}_r^{n+1,i + 1} = \underline{\boldsymbol{\mathbf{f}}}_r^{n+1} - \mathbf{A}_r^{{\bf u}p}\underline{\boldsymbol{\mathbf{p}}}_r^{n+1,i+1} - \mathbf{A}_r^{{\bf u}\theta}\underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1,i+1}.\end{aligned}$$ To conclude, we note that [\[sys: fs ROMeq\]](#sys: fs ROMeq){reference-type="eqref" reference="sys: fs ROMeq"} requires solving three linear systems of dimension $r \times r$. Matrices and vectors appearing in [\[sys: fs ROMeq\]](#sys: fs ROMeq){reference-type="eqref" reference="sys: fs ROMeq"} can be precomputed as in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"}, where the final row of the table contains two further matrices associated with the stabilization terms that were not required by M-ROM. ## Convergence analysis of the FS-ROM {#sec: convergence ROM} In this section, we retrace the convergence analysis carried out in Section [3.3](#sec: convergence){reference-type="ref" reference="sec: convergence"} in order to adapt it to the FS-ROM introduced in Section [4.3](#sec: fs ROM){reference-type="ref" reference="sec: fs ROM"}. **Theorem 2**. *Assume that [\[eqn: assump1\]](#eqn: assump1){reference-type="eqref" reference="eqn: assump1"} and [\[eqn: assump2\]](#eqn: assump2){reference-type="eqref" reference="eqn: assump2"} hold. For a fixed $n + 1$, assume that the FS-ROM solution at time step $n + 1$ is initialized to the M-ROM solution at time step $n$, i.e. $${\bf u}_r^{n+1, 0} = {\bf u}_r^{n}, \quad p_r^{n+1, 0} = p_r^{n}, \quad \theta_r^{n+1, 0} = \theta_r^{n}.$$* *Then, the FS-ROM solution defined in [\[sys: fs ROM\]](#sys: fs ROM){reference-type="eqref" reference="sys: fs ROM"} converges to the solution of the M-ROM method [\[sys: cg ROM\]](#sys: cg ROM){reference-type="eqref" reference="sys: cg ROM"}: $$\| {\bf u}_r^{n+1, i} - {\bf u}_r^{n+1}\|_1 \to 0, \quad \| p_r^{n+1, i} - p_r^{n+1}\| \to 0,\quad \| \theta_r^{n+1, i} - \theta_r^{n+1}\| \to 0$$ as $i \to \infty$.* ***Proof.*** Let $e_{{\bf u}}^{i} = {\bf u}_r^{n+1, i} - {\bf u}_r^{n+1} \in \boldsymbol{V}_r$, $e_p^{i} = p_r^{n+1, i} - p_r^{n+1} \in W_r$ and $e_\theta^{i} = \theta_r^{n+1, i} - \theta_r^{n+1} \in \boldsymbol{V}_r$. Upon subtracting [\[eqn: cg-elasticity-ROM\]](#eqn: cg-elasticity-ROM){reference-type="eqref" reference="eqn: cg-elasticity-ROM"}, [\[eqn: cg-flow-ROM\]](#eqn: cg-flow-ROM){reference-type="eqref" reference="eqn: cg-flow-ROM"}, and [\[eqn: cg-thermal-ROM\]](#eqn: cg-thermal-ROM){reference-type="eqref" reference="eqn: cg-thermal-ROM"} from [\[eqn: fs-elasticity ROM\]](#eqn: fs-elasticity ROM){reference-type="eqref" reference="eqn: fs-elasticity ROM"}, [\[eqn: fs-flow ROM\]](#eqn: fs-flow ROM){reference-type="eqref" reference="eqn: fs-flow ROM"}, and [\[eqn: fs-thermal ROM\]](#eqn: fs-thermal ROM){reference-type="eqref" reference="eqn: fs-thermal ROM"} we obtain again [\[sys: err\]](#sys: err){reference-type="eqref" reference="sys: err"}, with the only exception that it must be now intended for any $({\bf v}, w, s) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$. Also, Lemma [Lemma 1](#lem: coer){reference-type="ref" reference="lem: coer"} implies that the coercivity condition [\[eqn: coer\]](#eqn: coer){reference-type="eqref" reference="eqn: coer"} also holds for any ${\bf v}\in \boldsymbol{V}_r$, since $\boldsymbol{V}_r\subset \boldsymbol{V}_h$. Even with the modified definitions of $e_{{\bf u}}^{i}, e_p^{i}$ and $e_\theta^{i}$ defined above, the proof of Lemma [Lemma 2](#lem: eu){reference-type="ref" reference="lem: eu"} and the proof of the convergence in Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} follow in the same manner as the original proofs. 0◻ **Remark 1**. *We notice that we cannot prove that $$\label{eq:fsROM_to_cg} ({\bf u}_r^{n+1, i}, p_r^{n+1, i}, \theta_r^{n+1, i}) \longrightarrow ({\bf u}_h^{n+1}, p_h^{n+1}, \theta_h^{n+1}) \text{ as } i \to \infty$$ with the same technique used in the proofs of Lemma [Lemma 2](#lem: eu){reference-type="ref" reference="lem: eu"} and Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}. Indeed, if we define our errors as $e_{{\bf u}}^{i} = {\bf u}_r^{n+1, i} - {\bf u}_h^{n+1} \in \boldsymbol{V}_h$, $e_p^{i} = p_r^{n+1, i} - p_h^{n+1} \in W_h$ and $e_\theta^{i} = \theta_r^{n+1, i} - \theta_h^{n+1} \in \boldsymbol{V}_h$, then we can obtain the same error equation, [\[sys: err\]](#sys: err){reference-type="eqref" reference="sys: err"}, as before but for any $({\bf v}, w, s) \in \boldsymbol{V}_r\times W_r\times {\Theta}_r$. However, since $(e_{{\bf u}}^{i}, e_p^{i}, e_\theta^{i}) \in \boldsymbol{V}_h\times W_h\times {\Theta}_r$, we cannot use these errors as test functions. As a result, the proofs of Lemma [Lemma 2](#lem: eu){reference-type="ref" reference="lem: eu"} and Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} would not follow through. Instead, we will empirically show that [\[eq:fsROM_to_cg\]](#eq:fsROM_to_cg){reference-type="eqref" reference="eq:fsROM_to_cg"} holds by means of the numerical experiments in the next section.* # Numerical Experiments {#sec: num_exp} In this section, we provide numerical experiments to validate and demonstrate the computational efficiency of the proposed algorithm. All the following computations are conducted by utilizing `FEniCSx`[^1] for HF computations, and `RBniCSx` [^2] for ROM computations. `FEniCSx` and `RBniCSx` rely on `PETSc`[^3] for sparse and dense linear algebra, respectively. `RBniCSx` further queries `LAPACK`[^4] for the solution of the dense eigenproblem [\[eq:POD_eigenvalues\]](#eq:POD_eigenvalues){reference-type="eqref" reference="eq:POD_eigenvalues"}. ## Example 1. Validation of ROM on test cases with smooth analytical solution {#sec: num} In this example, we validate our numerical methods against four different test cases with the given analytical solution, namely [\[eq:analytical solution\]]{#eq:analytical solution label="eq:analytical solution"} $$\begin{aligned} &{\bf u}(x, y; t) = \left[ \sin(\pi x t)\cos(\pi y t), \cos(\pi x t) \sin(\pi y t) \right]^T x y (1 - x) (1 - y),\\ &p(x, y; t) = \cos(t + x -y) x y (1 - x) (1 - y),\\ &\theta(x, y; t) = \sin(t + x - y) x y (1 - x) (1 - y),\end{aligned}$$ where $(x,y) \in \Omega = (0, 1)^2$ and $t \in \mathbb{I} = (0, T]$ for some $T > 0$ to be specified in each test case. We note that the factor $x y (1 - x) (1 - y)$ in each solution variable guarantees that the solution satisfies homogeneous Dirichlet boundary conditions, in agreement with the assumptions in the previous sections. The right-hand side functions $\mathbf{f},g,$ and $\eta$ in [\[sys: thm\]](#sys: thm){reference-type="eqref" reference="sys: thm"} are obtained from the expressions in [\[eq:analytical solution\]](#eq:analytical solution){reference-type="eqref" reference="eq:analytical solution"}. Furthermore, the physical parameters are chosen to be $$\begin{aligned} &\lambda = 10^2, \ \mu = 10^2, \ c_0 = 1, \ \alpha = 1, \ K_{dr} = \mu +\lambda = 2 \cdot 10^2, \ {\bf K}= 10^{-5},\\ &C_d = 1, \ \alpha_T=10^{-3}, \ \theta_0= 1, \ \alpha_m= 10^{-5}, \text{ and } {\bf D}= 10^{-5}.\end{aligned}$$ To compute the initial conditions at $n=0$, the M-HF and FS-HF schemes employ the linear interpolation of [\[eq:analytical solution\]](#eq:analytical solution){reference-type="eqref" reference="eq:analytical solution"}, while M-ROM and FS-ROM employ the $L^2$-projection of [\[eq:analytical solution\]](#eq:analytical solution){reference-type="eqref" reference="eq:analytical solution"} on the reduced basis spaces. The stabilization parameter $L=1$ is employed in the FS-HF and FS-ROM schemes. Initialization of the iterations at time $n + 1$ is based on the assignment of the converged solution at time $n$. Iterations for FS-HF will continue until the following conditions are satisfied $$\begin{aligned} &\| {\bf u}_h^{n+1, i+1} - {\bf u}_h^{n+1, i}\|_1 \leq \epsilon \| {\bf u}_h^{n+1, i+1}\|_1,\\ &\| p_h^{n+1, i+1} - p_h^{n+1, i}\|_1 \leq \epsilon \| p_h^{n+1, i+1}\|_1,\\ &\| \theta_h^{n+1, i+1} - \theta_h^{n+1, i}\|_1 \leq \epsilon \| \theta_h^{n+1, i+1}\|_1,\end{aligned}$$ where $\epsilon = 10^{-10}$. Similarly, the FS-ROM iterations will continue until $$\begin{aligned} &\| \underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i+1} - \underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i}\|_{\mathbb{R}^r} \leq \epsilon \| \underline{\boldsymbol{\mathbf{{\bf u}}}}_r^{n+1, i+1}\|_{\mathbb{R}^r},\\ &\| \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i+1} - \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i}\|_{\mathbb{R}^r} \leq \epsilon \| \underline{\boldsymbol{\mathbf{p}}}_r^{n+1, i+1}\|_{\mathbb{R}^r},\\ &\| \underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i+1} - \underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i}\|_{\mathbb{R}^r} \leq \epsilon \| \underline{\boldsymbol{\mathbf{\theta}}}_r^{n+1, i+1}\|_{\mathbb{R}^r};\end{aligned}$$ we note that such conditions still represent a stopping criteria on the relative $\| \cdot\|_1$-norm of the increment because, due to [\[eq: POD orthonormal\]](#eq: POD orthonormal){reference-type="eqref" reference="eq: POD orthonormal"}, the matrix representing the $(\cdot, \cdot)_1$ inner product on $\boldsymbol{V}_r, W_r$ and ${\Theta}_r$ is the identity. Furthermore, we set the maximum iteration number as $20$ throughout this example. ### Example 1A. Validation of HF and ROM solvers by a mesh refinement analysis {#fig:test1_convergence_analysis_wrt_exact_u_H1_cg width="\\textwidth"} {#fig:test1_convergence_analysis_wrt_exact_u_L2_cg width="\\textwidth"} \ \ {#fig:test1_convergence_analysis_wrt_exact_u_H1_fs width="\\textwidth"} {#fig:test1_convergence_analysis_wrt_exact_u_L2_fs width="\\textwidth"} \ \ As the first test case in Example 1, we validate the four distinct solvers discussed in the previous sections: M-HF (Section [3.1](#sec: coupled){reference-type="ref" reference="sec: coupled"}), FS-HF (Section [3.2](#sec: fs){reference-type="ref" reference="sec: fs"}), M-ROM (Section [4.2](#sec: coupled ROM){reference-type="ref" reference="sec: coupled ROM"}), and FS-ROM (Section [4.3](#sec: fs ROM){reference-type="ref" reference="sec: fs ROM"}). In the domain $\Omega$ of the unit square, we employ an initial spatial discretization of $h=0.25$ and a temporal discretization of $\Delta t= 0.0025$. Subsequently, with each cycle, we reduce $h$ by half and divide $\Delta t$ by four. The convergence rate is then computed by employing corresponding norms to calculate the errors (maximum over $(0, T]$) against the provided analytical solutions in [\[eq:analytical solution\]](#eq:analytical solution){reference-type="eqref" reference="eq:analytical solution"}. To validate the ROM algorithm, for each cycle, we initially run the HF solver to generate sequences $\left\{{\bf u}_h^{n}\right\}_{n = 0}^{N}$, $\left\{p_h^{n}\right\}_{n = 0}^{N}$, $\left\{\theta_h^{n}\right\}_{n = 0}^{N}$ in time. Then, we employ the POD algorithm as discussed in Section [4.1](#sec: POD){reference-type="ref" reference="sec: POD"} to compress these sequences, focusing on retaining the first five modes i.e., $r \leq 5$. The ROM's performance is evaluated across five distinct scenarios, specifically for values of $r$ equal to 1, 2, 3, 4, and 5. Results of the convergence analyses are plotted in Figure [\[fig:test1_convergence_analysis_wrt_exact_cg\]](#fig:test1_convergence_analysis_wrt_exact_cg){reference-type="ref" reference="fig:test1_convergence_analysis_wrt_exact_cg"} for the monolithic schemes (M-HF and M-ROM) and in Figure [\[fig:test1_convergence_analysis_wrt_exact_fs\]](#fig:test1_convergence_analysis_wrt_exact_fs){reference-type="ref" reference="fig:test1_convergence_analysis_wrt_exact_fs"} for the fixed-stress iterative schemes (FS-HF and FS-ROM). Specifically, the left column illustrates results in the $\| \cdot\|_1$-norm, whereas the panels on the right show results in the $\| \cdot\|$-norm. The convergence analysis for the displacement ${\bf u}$, pressure $p$, and temperature $\theta$ are arranged in three rows within the figure. The first row pertains to the displacement ${\bf u}$, the second row to the pressure $p$, and the third row to the temperature $\theta$. For the M-HF scheme we observe a convergence rate of 2 for errors in the $\| \cdot\|$-norm and of 1 for errors in the $\| \cdot\|_1$-norm. Furthermore, since the solution of FS-HF converges to that of M-HF, as presented in Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}, FS-HF is expected to converge at the same rate as M-HF. Indeed, numerical results show that FS-HF achieves the same convergence rates. Furthermore, with the exception of $r = 1$, the ROM scheme also achieves identical convergence rates to the HF scheme used in its training. {#fig:test1_CPU_time_cg width="\\textwidth"} {#fig:test1_CPU_time_fs width="\\textwidth"} {#fig:test1_iterations width="48%"} Figure [\[fig:test1_CPU_time\]](#fig:test1_CPU_time){reference-type="ref" reference="fig:test1_CPU_time"} highlights one of the main advantages of the ROM schemes, which is the reduction of computational cost. This advantage becomes evident through the significant reduction in the number of degrees of freedom achieved by the ROM schemes in comparison to the HF schemes. In addition, this reduction is accompanied by the precomputation of matrices and vectors in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"}. Consequently, ROMs exhibit a remarkable speedup of two orders of magnitude when compared to the underlying HF discretization for both monolithic and fixed-stress cases. We further remark that, while decreasing the mesh size $h$ demands a higher CPU time for solving with the HF method, the results in Figure  [\[fig:test1_CPU_time\]](#fig:test1_CPU_time){reference-type="ref" reference="fig:test1_CPU_time"} show that the ROM CPU time is not affected by mesh refinements. Indeed, the evaluation of the ROM only requires solutions of $r \times r$ linear systems for the FS-ROM or $3 r \times 3 r$ linear systems for the M-ROM regardless of the value of $h$. Moreover, increasing $r$ from 1 to 5 does not seem to increase the required CPU time either, since the solution of linear systems of dimension at most $15 \times 15$ is computationally inexpensive. Finally, Figure [7](#fig:test1_iterations){reference-type="ref" reference="fig:test1_iterations"} demonstrates that the average number of iterations necessary by FS-ROM for any given value of $r$ falls within the range of 5 to 6. Moreover, except for the case when $r = 1$, the FS-ROM exhibits an equivalent number of iterations as the FS-HF solver. ### Example 1B. Comparison of HF and ROM solvers in time In the second test case in Example 1, we focus on the fixed stress iteration scheme and compare the errors between HF and ROM solvers. Here, we use a mesh size of $h=1/16$ and a time step size of $\Delta t= 0.001$. We set the final time to be $T = 1$, which is larger than the one used in Example 1A. We then train both M-ROM and FS-ROM schemes, compute at most $10$ POD modes, and evaluate the ROMs on the same time interval $(0, T]$ with the same time step size $\Delta t$. The larger number of POD modes selected in this test case compared to Example 1A is due to the fact that the final time is now ten times larger than the one employed in Example 1A. First, Figure [\[fig:test2_relative_error_wrt_exact_fs\]](#fig:test2_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_exact_fs"} illustrates the relative errors with respect to the analytical solutions in [\[eq:analytical solution\]](#eq:analytical solution){reference-type="eqref" reference="eq:analytical solution"} for the fixed-stress schemes. In particular, the left column presents the results in the $\| \cdot\|_1$-norm, whereas the right column shows the results in the $\| \cdot\|$-norm. The first row pertains to the displacement ${\bf u}$, the second row to the pressure $p$, and the third row to the temperature $\theta$. We observe that the ROM for $r = 1$ is inaccurate, with relative errors up to $100\%$ over the time interval $(0, T]$, regardless of the solution component ${\bf u}, p$ or $\theta$. For $r \geq 3$, the ROM solutions are as accurate as the HF results, with only minor discrepancies appearing at the beginning of the time interval $(0, T]$ in Figures [\[fig:test2_relative_error_wrt_exact_p\_L2_fs\]](#fig:test2_relative_error_wrt_exact_p_L2_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_exact_p_L2_fs"} and [\[fig:test2_relative_error_wrt_exact_theta_L2_fs\]](#fig:test2_relative_error_wrt_exact_theta_L2_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_exact_theta_L2_fs"}. Furthermore, the error rises as time progresses. {#fig:test2_relative_error_wrt_exact_u_H1_fs width="\\textwidth"} {#fig:test2_relative_error_wrt_exact_u_L2_fs width="\\textwidth"} \ \ {#fig:test2_relative_error_wrt_FE_u_H1_fs width="\\textwidth"} {#fig:test2_relative_error_wrt_FE_u_L2_fs width="\\textwidth"} \ \ \ {#fig:test2_CPU_time_cg width="\\textwidth"} {#fig:test2_CPU_time_fs width="\\textwidth"} {#fig:test2_iterations width="\\textwidth"} {#fig:test2_condition_numbers width="\\textwidth"} \ {#fig:test2_POD_eigenvalues width="\\textwidth"} Next, Figure [\[fig:test2_relative_error_wrt_FE_fs\]](#fig:test2_relative_error_wrt_FE_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_fs"} demonstrates the performance of the FS-ROM scheme by comparing its solutions to those of the FS-HF scheme. Figures [10](#fig:test2_relative_error_wrt_FE_u_H1_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_u_H1_fs"}-[11](#fig:test2_relative_error_wrt_FE_u_L2_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_u_L2_fs"} show that ${\bf u}_r$ converges very fast to ${\bf u}_h$ in the $\| \cdot\|_1$ and $\| \cdot\|$-norm as $r$ increases. In particular, we observe an error reduction of five orders of magnitude when increasing the value of $r$ from $r = 1$ to $r = 5$. Similarly, Figures [\[fig:test2_relative_error_wrt_FE_p\_H1_fs\]](#fig:test2_relative_error_wrt_FE_p_H1_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_p_H1_fs"}-[\[fig:test2_relative_error_wrt_FE_p\_L2_fs\]](#fig:test2_relative_error_wrt_FE_p_L2_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_p_L2_fs"} show a decrease of more than one order of magnitude for $p$ within the first five POD modes, and Figures [\[fig:test2_relative_error_wrt_FE_theta_H1_fs\]](#fig:test2_relative_error_wrt_FE_theta_H1_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_theta_H1_fs"}-[\[fig:test2_relative_error_wrt_FE_theta_L2_fs\]](#fig:test2_relative_error_wrt_FE_theta_L2_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_theta_L2_fs"} a decrease of almost two orders of magnitude for $\theta$. Furthermore, Figure [\[fig:test2_CPU_time\]](#fig:test2_CPU_time){reference-type="ref" reference="fig:test2_CPU_time"} reports the comparisons of the total CPU time for both HF and ROM solvers. Analogous to Example 1A, the ROM approach ensures a speedup of over two orders of magnitude, regardless of the specific value of $r$. Figure [\[fig:test2_CPU_time\]](#fig:test2_CPU_time){reference-type="ref" reference="fig:test2_CPU_time"} also depicts the CPU time necessary for precomputation of the ROM operators summarized in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"}. This observation leads us to conclude that the precomputation of the ROM operators at the end of the offline stage is essential to achieve the desired speedup. This becomes particularly crucial for larger values of $r$ as attempting to carry out ROM assembly during an online ROM solve would result in an online evaluation that is nearly as computationally intensive as the HF solve itself. Lastly, we present the fixed-stress iteration counts for the FS-ROM in Figure [14](#fig:test2_iterations){reference-type="ref" reference="fig:test2_iterations"}. Similar to Example 1A, the FS-ROM exhibits an identical number of iterations when compared to the FS-HF solver. In addition, Figure [15](#fig:test2_condition_numbers){reference-type="ref" reference="fig:test2_condition_numbers"} presents the condition numbers for each matrix in Step i-ROM, Step ii-ROM, and Step iii-ROM. We observe that the condition numbers increase mildly while $r$ increases. Furthermore, Figure [16](#fig:test2_POD_eigenvalues){reference-type="ref" reference="fig:test2_POD_eigenvalues"} presents the normalized eigenvalues of the POD compression, where $\nu_0$ is the largest eigenvalue; the fast decay of the eigenvalues justifies limiting the ROM dimension to $r = 10$ at most. ### Example 1C. ROM on a larger time interval than the HF solver {#fig:test3_relative_error_wrt_exact_u_H1_fs width="\\textwidth"} {#fig:test3_relative_error_wrt_exact_u_L2_fs width="\\textwidth"} \ \ \ {#fig:test3_iterations width="\\textwidth"} {#fig:test3_condition_numbers width="\\textwidth"} \ {#fig:test3_POD_eigenvalues width="\\textwidth"} {#fig:test3_CPU_time_fs width="\\textwidth"} Employing ROM in Examples 1A and 1B serves well for validation purposes; however, its practical utility remains limited. This is because the ROM, when trained on the time interval $(0, T]$, merely replicates pre-existing solutions within the same time span. In the context of the third test case, we delve into the ROM's extrapolation potential. Here, we train the model on the time span $(0, 0.1]$ and assess its performance on a more extensive interval, $(0, 1]$. First, Figure [\[fig:test3_relative_error_wrt_exact_fs\]](#fig:test3_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test3_relative_error_wrt_exact_fs"} illustrates the relative errors with respect to the analytical solutions in [\[eq:analytical solution\]](#eq:analytical solution){reference-type="eqref" reference="eq:analytical solution"} for the FS-HF and FS-ROM schemes in the time interval $(0, 1]$. As in Figure [\[fig:test2_relative_error_wrt_FE_fs\]](#fig:test2_relative_error_wrt_FE_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_FE_fs"}, the left column presents the results in the $\| \cdot\|_1$-norm, whereas the right column shows the results in the $\| \cdot\|$-norm. The first row pertains to the displacement ${\bf u}$, the second row to the pressure $p$, and the third row to the temperature $\theta$. We note that the errors for the FS-HF solver are now limited to the interval $(0, 0.1]$ but the errors displayed in the rest of the time interval $[0.1, 1]$ is only reported for comparison purposes from the previous Example 1B. Comparing the results in Figure [\[fig:test3_relative_error_wrt_exact_fs\]](#fig:test3_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test3_relative_error_wrt_exact_fs"} to those obtained in Figure [\[fig:test2_relative_error_wrt_exact_fs\]](#fig:test2_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_exact_fs"} for Example 1B, one can notice that a ROM with $r = 5$ is not as accurate in Example 1B due to the errors from extrapolation. Still, increasing the reduced basis size to $r = 7$ allows us to obtain a solution that is as accurate as the solution of the FS-HF scheme. Although this test case demonstrates a successful extrapolation in accuracy, such an achievement introduces additional challenges for the ROM. Figure [19](#fig:test3_iterations){reference-type="ref" reference="fig:test3_iterations"} indicates that the average number of fixed-stress iterations from the FS-ROM is greater than that of the FS-HF solver for $r \geq 7$. In addition, for $r \geq 9$, the FS-ROM scheme surpasses the maximum allowable limit of 20 iterations at each time step. It is important to emphasize that this behavior does not contravene Theorem [Theorem 2](#thm:main ROM){reference-type="ref" reference="thm:main ROM"}; rather, it stems from limitations in arithmetic precision. To support this statement, we plot the condition numbers of the matrices appearing on the left-hand side of Step i-ROM, Step ii-ROM, and Step iii-ROM (from the algorithm [\[sys: fs ROMeq\]](#sys: fs ROMeq){reference-type="eqref" reference="sys: fs ROMeq"}) in Figure [20](#fig:test3_condition_numbers){reference-type="ref" reference="fig:test3_condition_numbers"}. We notice that the condition numbers of the matrices from Step iii-ROM is above $10^9$ for $r \geq 7$ and the condition numbers of all matrices from Steps i-ROM, ii-ROM, and iii-ROM are above $10^9$ for $r \geq 9$. From Figure [21](#fig:test3_POD_eigenvalues){reference-type="ref" reference="fig:test3_POD_eigenvalues"} we understand that this behavior is due to having added to the reduced basis some trailing POD modes associated with eigenvalues which, in machine precision, are zero. Since such eigenvalues appear on the denominator of [\[eq: POD mode\]](#eq: POD mode){reference-type="eqref" reference="eq: POD mode"}, the amplification of numerical errors causes the trailing POD modes not to satisfy [\[eq: POD orthonormal\]](#eq: POD orthonormal){reference-type="eqref" reference="eq: POD orthonormal"} anymore and may result in linearly dependent POD modes. This behavior can be attributed to the use of a very small time interval during the training process. To observe this, compare Figure [20](#fig:test3_condition_numbers){reference-type="ref" reference="fig:test3_condition_numbers"} with Figure [15](#fig:test2_condition_numbers){reference-type="ref" reference="fig:test2_condition_numbers"}, as well as Figure [21](#fig:test3_POD_eigenvalues){reference-type="ref" reference="fig:test3_POD_eigenvalues"} with Figure [16](#fig:test2_POD_eigenvalues){reference-type="ref" reference="fig:test2_POD_eigenvalues"}. The increase in the number of fixed stress iterations also results in a mild increase of the CPU time required by the ROM, as shown in Figure [22](#fig:test3_CPU_time_fs){reference-type="ref" reference="fig:test3_CPU_time_fs"} for $r \geq 7$. ### Example 1D. ROM using a larger time step than the HF solver {#fig:test4_relative_error_wrt_exact_u_H1_fs width="\\textwidth"} {#fig:test4_relative_error_wrt_exact_u_L2_fs width="\\textwidth"} \ \ {#fig:test4_CPU_time_cg width="\\textwidth"} {#fig:test4_CPU_time_fs width="\\textwidth"} {#fig:test4_iterations width="48%"} For the last test case in Example 1, we explore the possibility of employing different time step sizes $\Delta t$ between the HF and the ROM solvers. Indeed, since the evaluation of the ROM still relies on time stepping through the interval $(0, T]$, adopting a larger time step presents a way to further enhance the speed of online evaluation as it entails a reduction in the number of time instances. Here, we set $\Delta t= 0.001$ for FS-HF as in Example 1B, but FS-ROM with $\Delta t= 0.01$, i.e. ten times larger than the one in Example 1B. We consider $T=1$. Figure [\[fig:test4_relative_error_wrt_exact_fs\]](#fig:test4_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test4_relative_error_wrt_exact_fs"} displays the relative errors of the displacement, pressure, and temperature (in $\|\cdot\|_1$ and $\| \cdot \|$ norms) with respect to the analytical solutions in [\[eq:analytical solution\]](#eq:analytical solution){reference-type="eqref" reference="eq:analytical solution"} for the FS-HF and FS-ROM schemes. Comparing Figures [\[fig:test2_relative_error_wrt_exact_fs\]](#fig:test2_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test2_relative_error_wrt_exact_fs"} and [\[fig:test4_relative_error_wrt_exact_fs\]](#fig:test4_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test4_relative_error_wrt_exact_fs"}, we conclude that employing the FS-ROM scheme with a larger time step introduces a larger time discretization error in the approximation of the pressure $p$ and the temperature $\theta$. However, in Figure [\[fig:test4_relative_error_wrt_exact_fs\]](#fig:test4_relative_error_wrt_exact_fs){reference-type="ref" reference="fig:test4_relative_error_wrt_exact_fs"}, the errors in $p_r$ and $\theta_r$ still remain in the same order of magnitude as those for $p_h$ and $\theta_h$, thus offering an approximation that could be accurate enough in several practical scenarios. Furthermore, the displacement ${\bf u}$, which exhibits minimal variation over time in this setup, can be precisely approximated even by a ROM that employs a larger time step for time integration. In fact, there are negligible distinctions observed between ${\bf u}_r$ and ${\bf u}_h$. As expected, choosing a larger time step decreases the online computational cost. Comparing Figure [\[fig:test4_CPU_time\]](#fig:test4_CPU_time){reference-type="ref" reference="fig:test4_CPU_time"} to Figure [\[fig:test2_CPU_time\]](#fig:test2_CPU_time){reference-type="ref" reference="fig:test2_CPU_time"}, we realize that the CPU time is decreased by a factor of ten since the number of time instances is divided by ten. Finally, choosing a larger time step does not have a detrimental effect on the average number of FS-ROM iterations, which are still equal to the ones of the FS-HF scheme, see Figure [27](#fig:test4_iterations){reference-type="ref" reference="fig:test4_iterations"}. ## Example 2. A domain with parametric heterogeneities {#sec:appl} The purpose of this example is to assess the capabilities of the proposed FS-ROM in a parametric setting with a more realistic application problem. The domain $\Omega=(0,1)^2$ is divided into two subdomains, $\Omega_1$ and $\Omega_2$, with $\Omega_1$ being characterized by higher permeability and conductivity than $\Omega_2$. The heterogeneity of the domain is due to the differences in the values of ${\bf D}$ and ${\bf K}$ in $\Omega_1$ and $\Omega_2$, and it will be represented by introducing a parameter $\omega_1$. Specifically, ${\bf D}= 1$ and ${\bf K}= 10^{-1}$ in $\Omega_1$, while ${\bf D}= 10^{\omega_1}$ and ${\bf K}= 10^{\omega_1-1}$ in $\Omega_2$. See Figure [\[fig:ex3_setup_a\]](#fig:ex3_setup_a){reference-type="ref" reference="fig:ex3_setup_a"} for illustration. We introduce another parameter $\omega_2$ for the order of magnitude of the elastic properties of the material in $\Omega$, where $\lambda = \mu = 10^{\omega_2}$. The remaining physical properties are independent of $\omega_1$ or $\omega_2$ and fixed as $\alpha = 1$, $\alpha_T = \num{e-4}$, $\alpha_M = \num{e-6}$, $c_0 = \num{e-2}$, $C_d = \num{1}$, and $\theta_0 = 1$. The injection and production of the pressure and temperature are realized by the functions defined by $$g(x,y) = \eta(x,y) =10^{-2} \left( e^{(-1000 (x-x_1)^2 - 1000(y-y_1)^2)} - e^{(-1000 (x-x_2)^2 - 1000(y-y_2)^2)}\right),$$ where $(x_1,y_1) = (0.25, 0.5)$ and $(x_2,y_2) = (0.75, 0.5)$ [@YiLee23]. The body force $\mathbf{f}$ and the initial conditions for the displacement, pressure, and temperature are all set to zero. As for the boundary conditions, we impose homogeneous Dirichlet boundary conditions for the displacement and homogeneous Neumann boundary conditions for the pressure and temperature on the entire boundary, $\partial \Omega$. Additionally, we choose a mesh size $h=0.01$ and a time step size $\Delta t = 0.1$ with the final time $T = 2$, and the stabilization parameter is again set to $L=1$. {#fig:application_POD_eigs_cg width="\\textwidth"} {#fig:application_POD_eigs_fs width="\\textwidth"} Let ${\boldsymbol \omega}:= (\omega_1, \omega_2)$. During the training of the ROM, the parameter pair ${\boldsymbol \omega}= (\omega_1, \omega_2)$ is varied in a given training set $\mathbb{P}_{\text{train}} = [-3, 0] \times [-1, 1]$. Then, to assess the extrapolation capabilities of M-ROM and FS-ROM, the parameter ${\boldsymbol \omega}$ is allowed to have a larger variation during the evaluation of the ROM. More specifically, ${\boldsymbol \omega}$ is chosen from the test parameter space $\mathbb{P}_{\text{test}} = [-4, 1] \times [-2, 2]$, which is larger than $\mathbb{P}_{\text{train}}$. See Figure [\[fig:ex3_setup_b\]](#fig:ex3_setup_b){reference-type="ref" reference="fig:ex3_setup_b"} for illustration. On the other hand, we let iterations continue until the relative increment in the $\| \cdot\|_1$-norm becomes less than $\epsilon = 10^{-3}$ or it reaches a maximum of $20$ iterations, whichever is met first. We note that the value of $\epsilon$ is considerably larger than the one employed in the Examples 1A-1D. Even with this larger $\epsilon$, however, the FS-HF scheme converges for the most challenging parametric configuration (i.e., ${\boldsymbol \omega}= (-3, -1))$ within the maximum allowed number of iterations. Since FS-HF will need to be queried for several values of ${\boldsymbol \omega}$, by requiring a tolerance $\epsilon$ as tight as in the previous validation (and, thus, increasing the maximum allowed number of iterations) we would make the training process extremely expensive. We then define a discrete training set by sampling 25 equidistant points on a $5 \times 5$ uniform grid on $\mathbb{P}_{\text{train}}$ and proceed to train both M-ROM and FS-ROM. The inclusion of the parameter ${\boldsymbol \omega}$ in this example necessitates a slight modification to the POD method presented in Section [4.1](#sec: POD){reference-type="ref" reference="sec: POD"}. This alteration is introduced to facilitate the applicability of the ROM for parameter values that may deviate from those employed during the training phase. Consequently, the construction of the reduced basis spaces $\boldsymbol{V}_r$, $W_r$, and ${\Theta}_r$ involves compressing the parametric and temporal evolution of the M-HF and FS-HF solutions at the same time. Notably, since each parametric evaluation entails 21 time steps, this compression encompasses a sequence of $25 \times 21 = 525$ M-HF and FS-HF solutions. Figure [\[fig:application_POD_eigs\]](#fig:application_POD_eigs){reference-type="ref" reference="fig:application_POD_eigs"} illustrates the POD normalized eigenvalues resulting from the training of the M-ROM and FS-ROM schemes. Two main differences compared to Examples 1A-1D are observed here. First, the decay of the POD eigenvalues is considerably slower than those observed in Examples 1A-1D due to the higher complexity of this application. Still, the $60$-th, $80$-th and $100$-th POD normalized eigenvalue are less than $10^{-12}$ for ${\bf u}$, $p$ and $\theta$, respectively. We thus set the maximum value of $r = 100$. Though $r$ is larger than the one used previously, $100$ ROM degrees of freedom are still a very large reduction compared to the $20402$, $10201$, and $10201$ HF degrees of freedom required by $\boldsymbol{V}_h$, $W_h$ and ${\Theta}_h$, respectively. Secondly, comparing Figure [28](#fig:application_POD_eigs_cg){reference-type="ref" reference="fig:application_POD_eigs_cg"} and Figure [29](#fig:application_POD_eigs_fs){reference-type="ref" reference="fig:application_POD_eigs_fs"} we note that the eigenvalue decay obtained by applying POD from M-HF data and FS-HF data, respectively, is slightly different, with the pressure presenting the most noticeable difference. This is due to the relatively large value of $\epsilon$ employed in the FS-HF, due to which FS-HF solutions are not as accurate as M-HF ones. Next, Figure [\[fig:application\]](#fig:application){reference-type="ref" reference="fig:application"} demonstrates the assessment of the ROM through $\| \cdot\|_1$-norm relative errors. This evaluation is conducted on three distinct test cases, each representing progressively more demanding conditions for the ROM. - Case i) The first testing set is composed of the same 25 parameter values used while training the ROM. These cases are represented by star-shaped markers ($\mbox{\fontsize{10pt}{\baselineskip}\selectfont\FiveStar}$) in Figure [\[fig:application\]](#fig:application){reference-type="ref" reference="fig:application"}. - Case ii) The second testing set is obtained by taking a finer $7 \times 7$ uniform grid (i.e., 49 parameters) on $\mathbb{P}_{\text{train}}$ and discarding any parameter values already present in the training set (e.g., a parameter ${\boldsymbol \omega}= (-3, -1)$), resulting in 40 parameter values. This set is used to evaluate the accuracy of the ROM on parameters that were not seen during training but still belong to the training range. The results are depicted in Figure [\[fig:application\]](#fig:application){reference-type="ref" reference="fig:application"} by diamond-shaped markers ($\blacklozenge$). - Case iii) Finally, the third testing set amounts to sampling a $7 \times 7$ uniform grid on $\mathbb{P}_{\text{test}}$ and discarding parameter values lying in $\mathbb{P}_{\text{train}}$, for a total of 40 parameter configurations in $\mathbb{P}_{\text{test}} \setminus \mathbb{P}_{\text{train}}$. These values are employed to assess the accuracy of the ROM while extrapolating outside of the training range and are reported with plus-shaped markers () in Figure [\[fig:application\]](#fig:application){reference-type="ref" reference="fig:application"}. The evaluation of M-ROM and FS-ROM follows through as described in Section [4](#sec: rom){reference-type="ref" reference="sec: rom"}, with a minor modification regarding the assembly of the ROM operators in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"}. Here, we consider only $[\mathbf{A}_r^{\theta \theta}]_{\beta\gamma} = ({\bf D}\nabla \varphi_\gamma^{\theta} , \nabla \varphi_\beta^{\theta})$ for the sake of exposition. Since now ${\bf D}= {\bf D}({\boldsymbol \omega})$, the resulting $\mathbf{A}_r^{\theta \theta}({\boldsymbol \omega})$ cannot be preassembled as reported in Table [1](#table:rom_operators){reference-type="ref" reference="table:rom_operators"} without knowing the value of ${\boldsymbol \omega}$ a priori. To retain the computational efficiency, the ROM makes use of the so-called *affine parametric dependence* or *parameter separability* [@hesthaven2016certified; @quarteroni2015reduced] and rewrites $[\mathbf{A}_r^{\theta \theta}]_{\beta\gamma}$ as $$\begin{aligned} _{\beta\gamma}({\boldsymbol \omega}) &= ({\bf D}({\boldsymbol \omega}) \nabla \varphi_\gamma^{\theta} , \varphi_\beta^{\theta}) = ({\bf D}({\boldsymbol \omega}) \nabla \varphi_\gamma^{\theta} , \nabla \varphi_\beta^{\theta})_{\Omega_1} + ({\bf D}({\boldsymbol \omega}) \nabla \varphi_\gamma^{\theta} , \nabla \varphi_\beta^{\theta})_{\Omega_2}\\ & = 1 \ (\nabla \varphi_\gamma^{\theta} , \nabla \varphi_\beta^{\theta})_{\Omega_1} + 10^{\omega_1} \ (\nabla \varphi_\gamma^{\theta} , \nabla \varphi_\beta^{\theta})_{\Omega_2} := [\mathbf{A}_r^{\theta \theta, (1)}]_{\beta\gamma} + 10^{\omega_1} [\mathbf{A}_r^{\theta \theta, (2)}]_{\beta\gamma}.\end{aligned}$$ The matrices $\mathbf{A}_r^{\theta \theta, (1)}$ and $\mathbf{A}_r^{\theta \theta, (2)}$ are now independent of ${\boldsymbol \omega}$ and can be preassembled at the end of the offline stage. {#fig:application_u_H1_relative_errors width="23.5%"} {#fig:application_u_H1_relative_errors width="21.2%"} {#fig:application_u_H1_relative_errors width="21.2%"} {#fig:application_u_H1_relative_errors width="28.7%"}\ {#fig:application_u_H1_relative_errors width="23.5%"} {#fig:application_u_H1_relative_errors width="21.2%"} {#fig:application_u_H1_relative_errors width="21.2%"} {#fig:application_u_H1_relative_errors width="28.7%"} \ {#fig:application_p_H1_relative_errors width="23.5%"} {#fig:application_p_H1_relative_errors width="21.2%"} {#fig:application_p_H1_relative_errors width="21.2%"} {#fig:application_p_H1_relative_errors width="28.7%"}\ {#fig:application_p_H1_relative_errors width="23.5%"} {#fig:application_p_H1_relative_errors width="21.2%"} {#fig:application_p_H1_relative_errors width="21.2%"} {#fig:application_p_H1_relative_errors width="28.7%"} \ {#fig:application_theta_H1_relative_errors width="23.5%"} {#fig:application_theta_H1_relative_errors width="21.2%"} {#fig:application_theta_H1_relative_errors width="21.2%"} {#fig:application_theta_H1_relative_errors width="28.7%"}\ {#fig:application_theta_H1_relative_errors width="23.5%"} {#fig:application_theta_H1_relative_errors width="21.2%"} {#fig:application_theta_H1_relative_errors width="21.2%"} {#fig:application_theta_H1_relative_errors width="28.7%"} {#fig:application_iterations width="23%"} {#fig:application_iterations width="20.8%"} {#fig:application_iterations width="20.8%"} {#fig:application_iterations width="29.5%"} \ {#fig:application_speedup width="23.7%"} {#fig:application_speedup width="21.4%"} {#fig:application_speedup width="21.4%"} {#fig:application_speedup width="28%"}\ {#fig:application_speedup width="23.7%"} {#fig:application_speedup width="21.4%"} {#fig:application_speedup width="21.4%"} {#fig:application_speedup width="28%"} Figure [\[fig:application\]](#fig:application){reference-type="ref" reference="fig:application"} is divided into five panels, representing five different quantities of interest. The first three panels, Figures [37](#fig:application_u_H1_relative_errors){reference-type="ref" reference="fig:application_u_H1_relative_errors"}--[53](#fig:application_theta_H1_relative_errors){reference-type="ref" reference="fig:application_theta_H1_relative_errors"}, depict the $\| \cdot\|_1$-norm relative errors for ${\bf u}_r$, $p_r$, and $\theta_r$ with respect to ${\bf u}_h$, $p_h$ and $\theta_h$, respectively. In each panel, the top row displays the results obtained from the M-ROM, while the bottom row illustrates those from the FS-ROM. Each row contains four images, corresponding to increasing values of $r = 10$, $r = 30$, $r = 60$ and $r = 90$. Markers are positioned at the provided ${\boldsymbol \omega}= (\omega_1, \omega_2)$ coordinates, signifying the errors for the three distinct cases: case i) denoted by $\mbox{\fontsize{10pt}{\baselineskip}\selectfont\FiveStar}$, case ii) by $\textstyle \blacklozenge$, and case iii) by . In the first column of Figures [37](#fig:application_u_H1_relative_errors){reference-type="ref" reference="fig:application_u_H1_relative_errors"}, [45](#fig:application_p_H1_relative_errors){reference-type="ref" reference="fig:application_p_H1_relative_errors"} and [53](#fig:application_theta_H1_relative_errors){reference-type="ref" reference="fig:application_theta_H1_relative_errors"}, it is evident that both the M-ROM and HF-ROM with $r = 10$ exhibit inaccuracies. The relative errors are approximately $10^{-2}$ within the training range (inside the inner rectangular region) and can escalate to as much as $1$ outside of this range. The accuracy progressively increases when increasing $r$ to $30, 60$ and $90$. In particular, when $r = 90$ we observe that both the M-ROM and HF-ROM become more accurate within the training range, including both seen (i.e., in the training set) and unseen data (i.e., not in the training set). We note that, at $r = 90$, the temperature ($\theta$) exhibits the lowest error within the training range, reaching a maximum of $10^{-6}$ for both the M-ROM and FS-ROM. This outcome aligns with the observation that the POD eigenvalues associated with temperature were the ones that exhibited the most rapid decrease. On the other hand, the displacement ${\bf u}$ and pressure $p$ show differences of up to an order of magnitude between the M-ROM and FS-ROM for parameter values within the training range. Nevertheless, the resulting relative errors remain at most $10^{-4}$, a value that falls below the tolerance $\epsilon$ established by the FS-HF solver. We also used both M-ROM and FS-ROM for extrapolation outside of the training range, which resulted in larger relative errors, especially in the bottom left of $\mathbb{P}_{\text{test}}$. The highest relative error is observed at ${\boldsymbol \omega}= (-4, -2)$ and is of the order of $10^{-2}$ for FS-ROM and $10^{-1}$ for M-ROM for the pressure approximation, i.e. the variable that was characterized by the slowest POD decay. Finally, Figures [57](#fig:application_iterations){reference-type="ref" reference="fig:application_iterations"} and [65](#fig:application_speedup){reference-type="ref" reference="fig:application_speedup"} discuss the number of iterations and the computational efficiency of the ROM. Specifically, Figure [57](#fig:application_iterations){reference-type="ref" reference="fig:application_iterations"} illustrates the ratio of the total number of FS-ROM iterations to FS-HF iterations. Apart from the smallest values, $r = 10$ and $r = 30$, in the extrapolation region, FS-ROM converges in a comparable number of iterations to FS-HF across the entire parameter range. Also, Figure [65](#fig:application_speedup){reference-type="ref" reference="fig:application_speedup"} depicts the speedup offered by the ROM, representing the ratio of CPU time required for a HF solve compared to a ROM solve. The magnitude of the speedup for the M-ROM diminishes from $10^4$ (for $r = 10$) to $10^3$ (for $r = 90$) as the reduced basis size $r$ increases. On the other hand, FS-ROM exhibits a smaller speedup, approximately on the order of $10^2$, which remains relatively consistent across varying $r$ values. In summary, any ROM yields query times that are at least 100 times faster than the corresponding HF scheme. # Conclusions This paper introduces a novel approach involving a fixed-stress iterative coupling for the solution of thermo-poroelasticity problems in conjunction with reduced order modeling techniques. The approach is validated by means of several numerical examples, which also illustrate its computational capabilities. The benefits of this methodology are two-fold. Firstly, the utilization of fixed-stress iterations aids in the management of complex multi-physics coupling scenarios. Secondly, the incorporation of reduced order modeling significantly boosts computational efficiency. In future works we plan to demonstrate the versatility of the proposed techniques by employing them in conjunction with various numerical discretization techniques, including mixed finite elements, as well as discontinuous or enriched Galerkin finite element methods. # Acknowledgments {#acknowledgments .unnumbered} Francesco Ballarin acknowledges the PRIN 2022 PNRR project "ROMEU: Reduced Order Models for Environmental and Urban flows", and the INdAM-GNCS projects "Metodi numerici per lo studio di strutture geometriche parametriche complesse" (CUP E53C22001930001) and "Metodi di riduzione computazionale per le scienze applicate: focus su sistemi complessi" (CUP E55F22000270001). The work of Sanghyun Lee was supported by the US National Science Foundation under Grant DMS-2208402. The work of Son-Young Yi was supported by the US National Science Foundation under Grant DMS-2208426. [^1]: `https://github.com/FEniCS/dolfinx` [^2]: `https://github.com/RBniCS/RBniCSx` [^3]: `https://petsc.org` [^4]: `https://www.netlib.org/lapack/`

arxiv_math

--- abstract: | We describe how previously known methods for determining the number of decimation classes of density $\delta$ binary vectors can be extended to nonnegative integer vectors, where the vectors are indexed by a finite abelian group $G$ of size $\ell$ and exponent ${\ell^*}$ such that $\delta$ is relatively prime to ${\ell^*}$. We extend the previously discovered theory of multipliers for arbitrary *subsets* of finite abelian groups, to arbitrary *multisets* of finite abelian groups. Moreover, this developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing. address: - Department of Mathematics, University of Findlay, Ohio 45840, USA - | Department of Mathematics and Statistics, Air Force Institute of Technology,\ Wright-Patterson Air Force Base, Ohio 45433, USA author: - Daniel M. Baczkowski - Dursun A. Bulutoglu title: Decimation classes of nonnegative integer vectors using multisets --- Bracelet; Lattice of subgroups; Legendre pair; Multiplier group; Necklace; Recursion # Introduction Theorems and algorithms were developed in [@decimation] for counting the decimation classes of binary vectors. We closely follow this work done for binary vectors and show similar methods apply when replacing binary vectors with nonnegative integer vectors. Many of the proofs from this work hold for the setting of nonnegative integer vectors. Similar algorithms apply and some results are simplified within this more general context of nonnegative integer vectors. A *multiset* is a collection of elements where an element is allowed to occur more than once. Such an occurrence is not allowed for sets. For a finite set $S=\{ s_1,\ldots,s_n \}$, let $I=\{ s_1^{\mathop{\mathrm{mult}}(s_1)},\ldots,s_n^{\mathop{\mathrm{mult}}(s_n)} \}$ denote the multiset with $s_i$ having multiplicity $\mathop{\mathrm{mult}}(s_i)=\mathop{\mathrm{mult}}_I(s_i)$ for each $i=1,\ldots,n$. To prevent confusion we often write $\mathop{\mathrm{mult}}_I(\cdot)$ to clarify the underlying multiset $I$. The cardinality of a multiset is the sum of its multiplicities. We will often let $\sigma$ denote the sum of all elements in a finite multiset; that is, $\sigma=\sum_{i\in I} i = \sum_{s\in S} \mathop{\mathrm{mult}}_I(s)\,s$. Let $x_i=\mathop{\mathrm{mult}}_I(s_i)$ for each $i=1,\ldots,n$. The number of all such possible multisets $I$ with cardinality $k$ is equal to the number of solutions to $$\label{2.1.1} \sum_{i=1}^n x_i =k,\quad x_i\in\mathbb{Z}_{\ge 0},$$ i.e., the number of nonnegative integer vectors whose components sum to $k$. It is well-known that the number of solutions to ([\[2.1.1\]](#2.1.1){reference-type="ref" reference="2.1.1"}) is $$\binom{n+k-1}{k}.$$ Throughout this work we will use the notation $$\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{n}{k}\right)\kern-.3em\right)} = \binom{n+k-1}{k}$$ for the number of multisets of cardinality $k$ with elements from a set $S$ with $|S|=n$. Throughout, we consider vectors indexed by a finite abelian group $G$ of order $\ell$. The density of a vector $\boldsymbol{v}\in \{\mathbb{Z}_{\ge 0}\}^G$ is defined to be $\delta=\sum_{g \in G}v_{g}$. We use $|\cdot|$ to denote the order of a group or the cardinality of a multiset. For an isomorphism $\Phi: G_1\rightarrow G_2$ and a multiset $I$ with elements from $G_1$, $\Phi(I)=\{\Phi(i)^{\mathop{\mathrm{mult}}(i)}\mid i \in I\}$. We assume that $G$ is a finite abelian group that has the operation $+$ unless specified otherwise. It is well known that such a group is isomorphic to $\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$ for some $\ell_1,\ell_2,\ldots,\ell_r \in \mathbb{Z}_{\geq 1}$. Again, we let $\mathbb{Z}_{\ell}^\times=\{j \in \mathbb{Z}_{\ell}\, |\, \gcd(j,\ell)=1\}$. Let ${\ell^*}$ denotes the *exponent* of the group $G\cong\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, i.e., the smallest positive integer $n$ such that $ng=0$ for all $g \in G$. Any subgroup $H$ or $K$ of $\mathbb{Z}_{\ell^*}^\times$ is always written multiplicatively with its identity element equal to $1$. A *decimation* of a vector $\boldsymbol{v}\in \{\mathbb{Z}_{\ge 0}\}^G$ by $j\in \mathbb{Z}_{{\ell^*}}^\times$, denoted by $d_j(\boldsymbol{v})$, is defined to be $\left(d_j(\boldsymbol{v})\right)_{g}=v_{j*g}$ for each $g\in G$. It is easy to see for $G\cong\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$ that ${\ell^*}=\mathop{\mathrm{lcm}}(\ell_1,\ldots,\ell_r)$ and $|G|=\ell=\prod_{i=1}^r\ell_i$. The *necklace* of a vector $\boldsymbol{v}\in\{\mathbb{Z}_{\ge 0}\}^G$ of length $\ell$ is the orbit of $\boldsymbol{v}$ under circulant shifts, whereas the *bracelet* of $\boldsymbol{v}$ is the orbit of $\boldsymbol{v}$ under circulant shifts and decimation by $-1$ (*reversals*) [@Dokovic2015; @Sawada2013]. Throughout the paper, we denote the necklace containing $\boldsymbol{v}$ (the necklace of $\boldsymbol{v}$) by $U_{\boldsymbol{v}}$. The *decimation class* of a vector is the orbit of the vector under circulant shifts and decimations [@Fletcher2001]. For a finite abelian group $G$ with exponent ${\ell^*}$, let $\mathcal{I}_{\delta,G}$ be the collection of all multisets with elements from $G$ with cardinality $\delta$. Then $G$ acts on $\mathcal{I}_{\delta,G}$ by $I\rightarrow I+g$ for each $I \in \mathcal{I}_{\delta,G}$ and $g \in G$, where ${I+g =\{(i+g)^{\mathop{\mathrm{mult}}_I(i)} \mid i \in I\}}$. The orbit of $I$ under this action is called the *necklace* of $I$. Similarly, $G\rtimes\mathbb{Z}_{{\ell^*}}^{\times}$ acts on $\mathcal{I}_{\delta,G}$ by $I\rightarrow hI+g$ for each $I \in \mathcal{I}_{\delta,G}$ and $(g,h) \in G\rtimes\mathbb{Z}_{{\ell^*}}^{\times}$, where ${hI=\{(hi)^{\mathop{\mathrm{mult}}_I(i)} \mid i \in I\}}$ and $\rtimes$ is the semidirect product [@RotmanBook p. 167]. The orbit of $I$ under the action of $G\rtimes\mathbb{Z}_{{\ell^*}}^{\times}$ is called the *decimation class* of $I$. The *bracelet* of $I$ is defined to be the orbit of $I$ under the action of $G \rtimes\{-1,1\}$. There is a one-to-one correspondence between decimation classes of nonnegative integer vectors indexed by $G$ with density $\delta$ and the decimation classes of elements in $\mathcal{I}_{\delta,G}$ given by $${\color{black}\boldsymbol{v}\rightarrow I=\{i^{v_i} \mid i \in G\} }$$ for each $\boldsymbol{v}\in \{\mathbb{Z}_{\ge 0}\}^G$ with $\sum_{i \in G}v_i=\delta$. Hence, it suffices to count the distinct decimation classes in $\mathcal{I}_{\delta,G}$ to count decimation classes of nonnegative integer vectors indexed by $G$ with density $\delta$. We assume this one-to-one correspondence throughout this work and do not differentiate between decimation classes of density $\delta$ nonnegative integer vectors and those of the elements in $\mathcal{I}_{\delta,G}$. The multiset $I$ with elements from $G$ is said to be *non-periodic* if there exists no $g \in G$, $g\neq 0$, such that $I+g=I$. To avoid trivialities, we always assume that $I$ is nonempty, and we often assume $|I|$ is finite. The following provides a sufficient condition for $I$ to be non-periodic. The proof is identical to that found in [@decimation], which used that $I$ had a finite number of elements. **Lemma 1**. *Let $G$ be a finite abelian group with exponent ${\ell^*}$. Let $I$ be a multiset with elements from $G$ such that $|I|=\delta$ and $\gcd(\delta,{\ell^*})=1$. Then $I$ is non-periodic and there are $|G|$ distinct multisets with cardinality $\delta$ in each necklace in $\mathcal{I}_{\delta,G}$.* The following several results are analogous to those found in [@decimation]. They are restatements in the context of multisets. Their proofs, now for multisets (rather than just sets), are identical to those of Lemmas 2-5 and Theorem 1 found in [@decimation], since they depend on $I$ being finite or are general group results. Lemma [Lemma 2](#lem:unique){reference-type="ref" reference="lem:unique"} is a consequence of $I$ being non-periodic. The proof of Lemma [Lemma 4](#lem:|D|){reference-type="ref" reference="lem:|D|"} depends on $I$ being finite. The proofs of Lemmas [Lemma 3](#lma:stab2){reference-type="ref" reference="lma:stab2"} and [Lemma 5](#lem:frob){reference-type="ref" reference="lem:frob"} and Theorem [Theorem 1](#thm:gcd){reference-type="ref" reference="thm:gcd"} use group properties. **Lemma 2**. *Let $I$ be a non-periodic multiset with elements from $G$, where $G$ is a finite abelian group with exponent ${\ell^*}$. Let $t \in \mathbb{Z}_{{\ell^*}}^\times$ be a multiplier of $I$ with $tI=I+g$ for some $g \in G$. Then $g$ is unique.* **Lemma 3**. *Let $\mathcal{I} = \mathcal{I}_G = \cup_{\delta\ge 0} \mathcal{I}_{\delta,G}$ denote the set of all multisets with elements from $G$. The group $\mathbb{Z}^\times_{{\ell^*}}$ acts on elements in $\{U_I \mid I\in\mathcal{I} \}$ by multiplication, where $jU_I=U_{jI}$ for each $j\in\mathbb{Z}_{{\ell^*}}^\times$. For a fixed $I \in\mathcal{I}$, let $H$ be the multiplier group of $I$ and $\mathop{\mathrm{Orb}}(U_I)$ be the orbit of $U_I$ under the action of $\mathbb{Z}_{{\ell^*}}^\times$. Then $H=\mathop{\mathrm{Stab}}(U_I)$ and $|\mathop{\mathrm{Orb}}(U_I)|=\phi({\ell^*})/|\mathop{\mathrm{Stab}}(U_I)|$.* **Lemma 4**. *Let $I$ be a finite multiset with elements from $G$, where $G$ is a finite abelian group with exponent ${\ell^*}$. Let $t \in \mathbb{Z}_{{\ell^*}}^{\times}$ be such that $tI=I+g$ for some $g \in G$ and $\sigma=\sum_{i \in I}i$. Then $|I| g=(t-1)\sigma$. If $\gcd(|I|,{\ell^*})=1$, then $g=|I|^{\phi({\ell^*})-1}(t-1)\sigma$.* **Lemma 5**. *Let $G$ be a finite abelian group with $\Phi(G)=\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, where $\Phi$ is an isomorphism. Let ${\ell^*}$ be the exponent of $G$, $a\in G$, and $m\in\mathbb{Z}$. Then the number of solutions to $mx=a$ for $x\in G$ is either $0$ or $\prod_{1\le i\le r} \gcd(m,\ell_i)$. Moreover, a solution exists if and only if $\Phi(a)_k \in \gcd(m,\ell_k)\mathbb{Z}_{\ell_k}$ for each $1\le k\le r$.* **Theorem 1**. *Let $G$ be a finite abelian group with $\Phi(G)=\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, where $\Phi$ is an isomorphism. Let $I$ be a multiset with elements from $G$ that is non-periodic and $t$ be a multiplier of $I$ such that $tI=I+g$. Then, there exists $z\in G$ such that $t(I+z) = I+z$ if and only if $\Phi(g)_k \in \gcd(t-1,\ell_k) \mathbb{Z}_{\ell_k}$ for each $1 \le k \le r$. Moreover, if such a $z$ exists, there are $\prod_{1\le i\le r} \gcd(t-1,\ell_i)$ such $z$'s.* The next theorem also has an identical proof with the exception that there is a one-to-one correspondence between the translates of $\Phi(I)$ fixed by $t$ and the translates of $I$ fixed by $t$. (It is not needed and it is not used that $\Phi(I)\subseteq \mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$.) **Theorem 2**. *Let $G$ be a finite abelian group with exponent ${\ell^*}$ and $\Phi(G)=\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, where $\Phi$ is an isomorphism. Let $I$ be a multiset with elements from $G$ that is non-periodic with multiplier group $H$. Let $K=\langle t_1,\ldots, t_m\rangle\leq H$, $t_iI=I+g_i$ for $1\le i\le m$, and $C=\gcd(t_1-1,\ldots,t_m-1,{\ell^*})$. Then the following hold.* 1. *If $C=1$, then there exists a translate $I+z$ fixed by all multipliers $t\in K$.[\[cond:unique\]]{#cond:unique label="cond:unique"}* 2. *For each $1 \le j \le r$, let $1 \le i_j \le \gcd(C,\ell_j)$ and $\Phi( h_j' )$ be defined such that $$\langle \Phi(h'_j)\rangle= \langle0\rangle\times\cdots\times\mathbb{Z}_{\ell_j}\times\cdots\times\langle0\rangle.$$ If a $z'_0\in G$ exists such that $I+z_0'$ is fixed by $K$, then $I+z'$ is fixed by $K$ for $$z'=z'_0+h'_1i_1\frac{\ell_1}{\gcd(C,\ell_1)}+\cdots+h'_ri_r\frac{\ell_r}{\gcd(C,\ell_r)}.$$ Moreover, these are the precisely $\prod_{1\le i\le r}\gcd(C,\ell_i)$ distinct such $z'\in G$. [\[cond:count\]]{#cond:count label="cond:count"}* The following results are the multiset versions of those found in [@decimation]. The proofs are identical for finite multisets with the same properties. **Lemma 6**. *Let $G$ be a finite abelian group with exponent ${\ell^*}$ and $\Phi(G) = \mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, where $\Phi$ is an isomorphism. Let $I$ be a finite, non-periodic multiset with elements from $G$ and $\sigma=\sum_{i\in I} i$. Let $t\in \mathbb{Z}_{\ell^*}^\times$ be such that $tI = I + g_0$ for some $g_0\in G$. For each $1 \le k \le r$, let $d_k = \gcd(|I|,\ell_k)$ and $e_k$ be the $k$'th column of the $r\times r$ identity matrix. Then, $(t-1)\Phi(\sigma)_k / d_k \in \mathbb{Z}$ for each $1\le k\le r$. Moreover, $g_0$ is unique and one of the $\prod_{1\le i\le r} \gcd(|I|,\ell_i)$ solutions to $|I| g = (t-1)\sigma$ for $g\in G$, namely $$g = \sum_{1\le k\le r} \left(\left( \frac{| I | }{d_k} \right)^{\phi\left(\frac{\ell_k}{d_k}\right)-1} \frac{(t-1) \Phi(\sigma)_k}{d_k} + \frac{\ell_k}{d_k} \, j_k \right) \Phi^{-1}(e_k)$$where each choice of $j_k$ satisfies $0\le j_k < d_k$ for each $1\le k\le r$.* Based on Lemmas [Lemma 2](#lem:unique){reference-type="ref" reference="lem:unique"} and [Lemma 6](#lem:|D|dan){reference-type="ref" reference="lem:|D|dan"}, we have the following definition. **Definition 1**. Let $G$ be a finite abelian group with exponent ${\ell^*}$ and $\Phi(G) = \mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, where $\Phi$ is an isomorphism. Let $I$ be a finite, non-periodic multiset with elements from $G$ and $\sigma=\sum_{i\in I} i$. Let $t\in \mathbb{Z}_{\ell^*}^\times$ be such that $tI = I + g_0$ for some $g_0\in G$. For each $1 \le k \le r$, let $d_k = \gcd(|I|,\ell_k)$ and $e_k$ be the $k$'th column of the $r\times r$ identity matrix. Then, define $j_1(t),\ldots,j_r(t)$ to be the necessarily unique integers (by Lemmas [Lemma 2](#lem:unique){reference-type="ref" reference="lem:unique"} and [Lemma 6](#lem:|D|dan){reference-type="ref" reference="lem:|D|dan"}) such that $0\leq j_k<d_k$ for each $1 \leq k\leq r$ and $$g_0=\sum_{1\le k\le r}\left(\left(\frac{|I|}{d_k}\right)^{\phi\left(\frac{\ell_k}{d_k}\right)-1}\frac{(t-1)\Phi(\sigma)_k}{d_k} + \frac{\ell_k}{d_k}j_k(t)\right)\Phi^{-1}(e_k).$$ **Theorem 3**. *Let $G$ be a finite abelian group with exponent ${\ell^*}$ and $\Phi(G) = \mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, where $\Phi$ is an isomorphism. Let $I$ be a finite, non-periodic multiset with elements from $G$. Let $t\in \mathbb{Z}_{\ell^*}^\times$ be a multiplier of $I$ and $\sigma=\sum_{i\in I} i$. For each $1 \le k \le r$, let $d_k = \gcd(|I|,\ell_k)$ and $e_k$ be the $k$'th column of the $r\times r$ identity matrix. Let $j_1(t),\ldots,j_r(t)$ be as in Definition [Definition 1](#dfn:Jt){reference-type="ref" reference="dfn:Jt"}. Then, $z\in G$ is a solution to $t(I+z)=I+z$ if and only if $z\in G$ is a solution to $$\label{tgt} (t-1)z = -\sum_{1\le k\le r}\left(\left(\frac{|I|}{d_k}\right)^{\phi\left(\frac{\ell_k}{d_k}\right)-1}\frac{(t-1)\Phi(\sigma)_k}{d_k} + \frac{\ell_k}{d_k}{j_k}(t)\right)\Phi^{-1}(e_k).$$* **Corollary 1**. *Let $G$ be a finite abelian group with exponent ${\ell^*}$ and $\Phi(G) = \mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, where $\Phi$ is an isomorphism. Let $I$ be a finite, non-periodic multiset with elements from $G$. Let $t\in \mathbb{Z}_{\ell^*}^\times$ be a multiplier of $I$ and $\sigma=\sum_{i\in I} i$. For each $1 \le k \le r$, let $d_k = \gcd(|I|,\ell_k)$ and $e_k$ be the $k$'th column of the $r\times r$ identity matrix. Let $j_1(t),\ldots,j_r(t)$ be as in Definition [Definition 1](#dfn:Jt){reference-type="ref" reference="dfn:Jt"}. Then, $t$ fixes at least one translate $I + z$ for some $z\in G$ if and only if $$\label{div-cond} d_k\cdot \gcd(t-1,\ell_k) {\rm~divides~} \left( \frac{| I |}{d_k} \right)^{\phi\left(\frac{\ell_k}{d_k}\right)-1} (t-1) \Phi(\sigma)_k + \ell_k \, j_k(t)$$ for each $1\le k\le r$.* **Theorem 4**. *Let $G\cong\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$ have exponent ${\ell^*}$. Let $I$ be a finite, non-periodic multiset with elements from $G$ such that $\gcd(|I|,{\ell^*})=1$, and let $t\in \mathbb{Z}_{\ell^*}^\times$ be a multiplier of $I$. Then, there are exactly $\prod_{1\le i\le r} \gcd(t-1,\ell_i)$ distinct $z\in G$ such that $t(I+z)=I+z$. Moreover, $z = -|I|^{\phi({\ell^*})-1} (\sum_{i \in I}i)$ is a solution that does not depend on $t$. Hence, there is a $z\in G$ such that $I+z$ is fixed by the multiplier group $H$ of $I$.* # A sufficient condition for fixed translates from the adjacency matrix {#sec:adjacencymatrix} A multiplier $t$ of $I$ is called *translate fixing* if there exists at least one $z \in G$ such that $t(I+z)=I+z$. First, we introduce the concept of the adjacency matrix of a multiset $I$, where $I$ has elements from a finite abelian group $G$. This is then used to derive a sufficient condition for each multiplier of $I$ to be translate fixing. For a finite multiset $I$ with elements from $G$, the *adjacency matrix* of $I$, denoted by $\mathbf{T}_I$, is defined by $$\label{eqn:IncidenceMatrix} \mathbf{T}_I(i,j)=\mathop{\mathrm{mult}}_{I+g_i}(g_j)$$ for $0\le i,j\le |G|-1$; that is, $\mathbf{T}_I(i,j)$ is defined to be the multiplicity of $g_j$ in $I+g_i$. Let $\mathbf{P}_g$ and $\mathbf{Q}_t$ be permutation matrices such that $$\label{PQeqns} \mathbf{P}_g\mathbf{T}_I = \mathbf{T}_{I+g}~\quad\text{and}\quad~~\mathbf{Q}_t^\top\mathbf{T}_I\mathbf{Q}_t = \mathbf{T}_{tI}.$$ These permutation matrices have the following properties, which will be used repeatedly. - $\mathbf{P}_g\mathbf{P}_h=\mathbf{P}_{h+g}=\mathbf{P}_h\mathbf{P}_g$ - $\mathbf{Q}_s^\top\mathbf{Q}_t^\top\mathbf{T}_I\mathbf{Q}_t\mathbf{Q}_s = \mathbf{Q}_{st}^\top\mathbf{T}_I\mathbf{Q}_{st} = \mathbf{Q}_t^\top\mathbf{Q}_s^\top\mathbf{T}_I\mathbf{Q}_s\mathbf{Q}_t$ - $\mathbf{Q}_t^\top\mathbf{P}_g\mathbf{Q}_t=\mathbf{P}_{tg}$ - $\mathbf{P}_{g}^\top=\mathbf{P}_{g}^{-1}=\mathbf{P}_{-g}; \quad \mathbf{Q}_{t}^\top=\mathbf{Q}_{t}^{-1}=\mathbf{Q}_{t^{-1}}$ The following lemmas demonstrate how the permutation matrices $\mathbf{P}_g$ and $\mathbf{Q}_t$ act on $\mathbf{T}_I$. The proof of the next result is the same as that found in [@decimation]. **Lemma 7**. *Let $G$ be a finite abelian group with exponent ${\ell^*}$. Let $I$ be a finite multiset with elements from $G$. If $t\in \mathbb{Z}^\times_{{\ell^*}}$ is a multiplier of $I$, then $t$ is a multiplier of every translate of $I$.* The proof of the next result is the same as that found in [@decimation] with the exception that the $r$th row of $\mathbf{T}_I$ is identified by a translate of $I$, and each multiset identifying a row of $\mathbf{T}_I\boldsymbol{q}_t$ is obtained by multiplying a multiset identifying a row of $\mathbf{T}_I$ by $t$ in $\mathbb{Z}^{\times}_{{\ell^*}}$. The remaining results in this seciton have identical proofs. **Theorem 5**. *Let $G$ be a finite abelian group with exponent ${\ell^*}$, $I$ be a finite multiset with elements from $G$, and $\mathbf{T}_I$ be the adjacency matrix from ([\[eqn:IncidenceMatrix\]](#eqn:IncidenceMatrix){reference-type="ref" reference="eqn:IncidenceMatrix"}). If $\mathbf{T}_I$ is invertible, then each multiplier of $I$ is translate fixing.* **Theorem 6**. *Let $G$ be a cyclic group with exponent ${\ell^*}$, $I$ be a finite multiset with elements from $G$, and $\mathbf{T}_I$ be the adjacency matrix from ([\[eqn:IncidenceMatrix\]](#eqn:IncidenceMatrix){reference-type="ref" reference="eqn:IncidenceMatrix"}). Suppose $[c_{g_0},c_{g_1},\ldots,c_{g_{\ell-1}}]$ is the first column of the adjacency matrix $\mathbf{T}_I$. Then, $\mathbf{T}_I$ is invertible if and only if $\gcd(\sum_{0\le j\le\ell-1}c_{g_j}x^j,x^\ell-1)$ in $\mathbb{Q}[X]$ is constant.* **Corollary 2**. *Let $G$ be a cyclic group with exponent ${\ell^*}$, $I$ be a finite multiset with elements from $G$, and $[c_{g_0},c_{g_1},\ldots,c_{g_{\ell-1}}]$ be the first column of the adjacency matrix $\mathbf{T}_I$. If $\gcd(\sum_{0\le j\le\ell-1}c_jx^j,x^\ell-1)$ in $\mathbb{Q}[X]$ is constant, then each multiplier of $I$ is translate fixing.* # Counting necklaces and bracelets {#sec:CountingNecks} Let $N(\ell,{\ell^*},\delta)$ be the number of necklaces in $\mathcal{I}_{\delta,G}$ for a finite abelian group $G$ of order $\ell$ and exponent ${\ell^*}$. Then, by Lemma [Lemma 1](#lem:necklace){reference-type="ref" reference="lem:necklace"}, $N(\ell,{\ell^*}, \delta)=\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{\ell}{\delta}\right)\kern-.3em\right)}/\ell$ whenever $\gcd(\delta,{\ell^*})=1$. Thus, if $\gcd({\ell^*},\delta)=1$, then each necklace is guaranteed to contain exactly $\ell$ vectors. Since each necklace contains $|G|$ vectors and each bracelet contains at most two necklaces, each bracelet contains at most $2|G|$ vectors. A vector $\boldsymbol{v}$ is called *symmetric* if there exists some $j_0\in G$ such that $${v_{j_0+k\,}=v_{j_0-k\, } \ \forall \ k\in G}.$$ Such a $j_0$ is called the *index of symmetry* of $\boldsymbol{v}$. If an index is not an index of symmetry, then it is called an *index of non-symmetry*. It is possible for a vector to have more than one index of symmetry. Observe that indices of symmetry of a vector $\boldsymbol{v}$ are the only indices $i$ such that $v_i=v_{i'}$ does not necessarily hold for some $i' \in G$ with $i' \neq i$. If a vector within a necklace is symmetric, then all other vectors in the necklace are also symmetric. Such a necklace is defined to be *symmetric*. A bracelet contains a single necklace if and only if that necklace is symmetric. Hence we get the following fact. **Fact 1**. The number of density $\delta$ symmetric necklaces of length $\ell$ is the same as the number of density $\delta$ symmetric bracelets of length $\ell$. The following lemma determines the form and the number of indices of symmetry for a vector $\boldsymbol{v}$ indexed by a finite abelian group $G$. The proof from [@decimation] holds for such vectors. **Lemma 8**. *Let $G\cong \mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$ be a finite abelian group of order $\ell=\ell_1 \cdots\ell_r$ with exponent ${\ell^*}=\mathop{\mathrm{lcm}}(\ell_1,\ldots,\ell_r)$. Let $\boldsymbol{v}$ be a vector indexed by $G$ and symmetric. Then the number of indices of symmetry for $\boldsymbol{v}$ is $\prod_{i=1}^r\gcd(2,\ell_i)$. Moreover, if $j_1$ is an index of symmetry for $\boldsymbol{v}$, then every other index of symmetry $j_2$ has the form $j_2=j_1+\Delta$, where $\Delta$ is a solution to the equation $$\label{eqn:Delta} 2\Delta=0, \quad \Delta \in G.$$* The following theorem determines the number of binary symmetric necklaces of length $\ell$ and density $\delta$ when $\gcd({\ell^*},\delta)=1$. It generalizes the previous result found in [@decimation]. **Theorem 7**. *Let $G\cong \mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$, $|G|=\ell=\prod_{i=1}^r\ell_i$, ${\ell^*}=\mathop{\mathrm{lcm}}(\ell_1,\ldots,\ell_r)$ be the exponent of $G$, and $\theta=\prod_{i=1}^r\gcd(2,\ell_i)$. Then the number of density $\delta$ symmetric necklaces in $\{\mathbb{Z}_{\ge 0}\}^G$ is $$\label{eqn:NumSymNecks} \sum_{\delta_1+2\delta_2=\delta} \ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{\frac{\ell-\theta}{2}}{\delta_2}\right)\kern-.3em\right)} \ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{\theta}{\delta_1}\right)\kern-.3em\right)}$$ where the sum is over all $\delta_1,\delta_2\in\mathbb{Z}_{\ge 0}$ such that $\delta_1+2\delta_2=\delta$.* *Proof.* First, by Fact [Fact 1](#fact:symnecklaces){reference-type="ref" reference="fact:symnecklaces"}, counting symmetric necklaces of length $\ell$ is the same as counting symmetric bracelets of length $\ell$. Let $\boldsymbol{v}\in \{\mathbb{Z}_{\ge 0}\}^G$ and have density $\delta$. By Lemma [Lemma 8](#lem:indsym){reference-type="ref" reference="lem:indsym"}, the number of indices of symmetry for $\boldsymbol{v}$ is $\theta=\prod_{i=1}^r\gcd(2,\ell_i)$. Let $j_1,\ldots,j_{\theta}$ be all the indices of symmetry of $\boldsymbol{v}$. Let $I_{non}=G \setminus \{j_1,\ldots j_{\theta}\}$. Let $I^{1/2}_{non}\subseteq I_{non}$ be such that $I^{1/2}_{non} \cap -I^{1/2}_{non}=\emptyset$ and $|I^{1/2}_{non}|=(\ell-\theta)/2$. Then the entries of $\boldsymbol{v}$ on $I^{1/2}_{non}$ determine the entries of $\boldsymbol{v}$ on $-I^{1/2}_{non}$ and $I^{1/2}_{non} \cup -I^{1/2}_{non}=I_{non}$. Thus, $\boldsymbol{v}$ is completely determined by the $\theta$ indices of symmetry and the indices in $I^{1/2}_{non}$. For a vector $\boldsymbol{v}$, let $\delta_1$ and $\delta_2$ be the sum of multiplicities of indices of symmetry and non-symmetry in $I^{1/2}_{non}$, respectively. More precisely, as $\delta = \sum_{i\in G}v_i$, $$\label{deltas} \delta_1 = \sum_{i=1}^{\theta}v_{j_i} \quad\text{and}\quad \delta_2 = \sum_{i\in I^{1/2}_{non}} v_i.$$ Then the density of $\boldsymbol{v}$ is $\delta=\delta_1+2\delta_2$. By the comment including ([\[2.1.1\]](#2.1.1){reference-type="ref" reference="2.1.1"}) and the fact that $|I^{1/2}_{non}|=(\ell-\theta)/2$, there are $\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{\theta}{\delta_1}\right)\kern-.3em\right)} \ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{\frac{\ell-\theta}{2}}{\delta_2}\right)\kern-.3em\right)}$ choices of vectors $\boldsymbol{v}$ with density $\delta$ such that the equations in ([\[deltas\]](#deltas){reference-type="ref" reference="deltas"}) hold. Hence, as $\delta_1$ and $\delta_2$ vary, the number of density $\delta$ symmetric necklaces in $\{\mathbb{Z}_{\ge 0}\}^G$ given in equation ([\[eqn:NumSymNecks\]](#eqn:NumSymNecks){reference-type="ref" reference="eqn:NumSymNecks"}) holds. ◻ The next corollary follows immediately from Theorem [Theorem 7](#thm:numSymNecks){reference-type="ref" reference="thm:numSymNecks"}. **Corollary 3**. *Let $G\cong \mathbb{Z}_{\ell}$. Then the number of density $\delta$ symmetric necklaces in $\{\mathbb{Z}_{\ge 0}\}^G$ is $$\label{eqn:NumSymNeckscrly} \eta= {\left\lfloor\frac{\ell-1}{2}\right\rfloor \choose \left\lfloor\frac{\delta}{2}\right\rfloor}.$$* # $H$-orbits {#sec:Ringcosets} For a subgroup $K=\{t_1,\ldots, t_{|K|}\}\leq\mathbb{Z}^\times_{{\ell^*}}$ and $s\in G$, let $sK=\{st_1,\ldots, st_{|K|}\}$ be the *$K$-orbit of $s$*. The next theorem follows from Theorem [Theorem 4](#thm:Dplusalpha){reference-type="ref" reference="thm:Dplusalpha"}. **Theorem 8**. *Let $G\cong\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$ have exponent ${\ell^*}$, $K\leq\mathbb{Z}^\times_{{\ell^*}}$ be a subgroup of the group of all multipliers of a finite multiset $I$ with elements from $G$, and $\sigma=\sum_{i \in I}i$. If $\gcd(|I|,{\ell^*})=1$, then $$\label{eqn:union} I-|I|^{\phi({\ell^*})-1}\sigma=\alpha_1 s_1K\cup \alpha_2 s_2K\cup \cdots \cup \alpha_r s_rK$$ for some $s_1, s_2, \ldots, s_r \in G$ and $\alpha_1, \alpha_2, \ldots, \alpha_r \in \mathbb{Z}_{\ge0}$, where the union in equation ([\[eqn:union\]](#eqn:union){reference-type="ref" reference="eqn:union"}) is disjoint.* *Proof.* By Theorem [Theorem 4](#thm:Dplusalpha){reference-type="ref" reference="thm:Dplusalpha"}, $$\label{eqn:unionAll} t(I-|I|^{\phi({\ell^*})-1}\sigma)= I-|I|^{\phi({\ell^*})-1}\sigma \quad \text{for all } t \in K,$$ and $K$ acts on the elements in $I-|I|^{\phi({\ell^*})-1}\sigma$. Then equation ([\[eqn:union\]](#eqn:union){reference-type="ref" reference="eqn:union"}) is the decomposition of $I-|I|^{\phi({\ell^*})-1}\sigma$ into disjoint union of orbits under the action of $K$, where each $\alpha_i$ is the multiplicity over $I-|I|^{\phi({\ell^*})-1}\sigma$ of the elements is $s_i K$. ◻ The following lemma is an application of the orbit-stabilizer theorem. Its proof is identical to that found in [@decimation], which relied only on the group action. **Lemma 9**. *If $K\leq\mathbb{Z}^\times_{{\ell^*}}$ and $s \in G$, then $|s K|$ divides $|K|$.* The following definition relates vectors to multisets and allows the extension of the algorithm from [@decimation] to work for nonnegative integer vectors. **Definition 2**. Let $G$ be a finite abelian group. An integer $t\in \mathbb{Z}_{{\ell^*}}^\times$ is called a *multiplier of a vector* $\boldsymbol{v}\in\{\mathbb{Z}_{\ge 0}\}^G$ if $t$ is a multiplier of the multiset $I_{\boldsymbol{v}}=\{i^{v_i}\mid i \in G\}$ which contains $i\in G$ $v_i$ times. It follows from Definition [Definition 2](#dfn:mul_vect){reference-type="ref" reference="dfn:mul_vect"} that $t$ is a multiplier of $\boldsymbol{v}$ if and only if $d_{t^{-1}}(\boldsymbol{v})\in U_{\boldsymbol{v}}$. Since the set of multipliers of vectors forms a subgroup of $\mathbb{Z}_{{\ell^*}}^\times$, $t$ is a multiplier of $\boldsymbol{v}$ if and only if $t^{-1}=t^{\phi({\ell^*})-1}$ is a multiplier of $\boldsymbol{v}$. Then $t$ is a multiplier of $\boldsymbol{v}$ if and only if $d_{t}(\boldsymbol{v})\in U_{\boldsymbol{v}}$. Next, given a potential multiplier group $H \le \mathbb{Z}_{{\ell^*}}^\times$, we determine the number of necklaces $U_{\boldsymbol{v}}$ with multiplier group $H$. By Lemma [Lemma 3](#lma:stab2){reference-type="ref" reference="lma:stab2"} and Theorem [Theorem 8](#crly:union){reference-type="ref" reference="crly:union"}, finding each necklace $U_{\boldsymbol{v}}$ with multiplier group $\mathop{\mathrm{Stab}}(U_{\boldsymbol{v}})= H$ is equivalent to finding each collection of $H$-orbits whose combined size is $|I_{\boldsymbol{v}}|$. Let $G$ be a finite abelian group, and consider any $\boldsymbol{v}\in\{\mathbb{Z}_{\ge 0}\}^G$ with multiplier group $H\leq\mathbb{Z}^\times_{{\ell^*}}$ such that $\gcd(\delta,{\ell^*})=1$ where $\delta := |I_{\boldsymbol{v}}|$. Since $I_{\boldsymbol{v}}$ is a multiset with elements from $G$ and $\gcd(|I_{\boldsymbol{v}}|,{\ell^*})=1$, by Theorem [Theorem 4](#thm:Dplusalpha){reference-type="ref" reference="thm:Dplusalpha"}, there exists a translate of $I_{\boldsymbol{v}}$ fixed by $H$. Thus, when searching for necklace representatives, it suffices to search for vectors $\boldsymbol{v}$ such that $I_{\boldsymbol{v}}$ is fixed by $H$ by replacing $\boldsymbol{v}$ with $\boldsymbol{v}'$ where $v_g'=\mathop{\mathrm{mult}}_{I'}(g)$ and $I'=I_{\boldsymbol{v}} - | I_{\boldsymbol{v}} |^{\phi({\ell^*})-1}(\sum_{i\in I_{\boldsymbol{v}}} i)$. The finite abelian group $G\cong\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$ has a decomposition into $H$-orbits; that is, $G=\bigcup_{i=1}^{e}s_iH$ is a disjoint union of $H$-orbits. Since the $H$-orbits are disjoint, there exists $\{f_1,f_2,\ldots, f_b\}\subseteq \{1,2,\ldots, e\}$ such that $$I_{\boldsymbol{v}}=r_1s_{f_1}H\cup r_2s_{f_2}H\cup \cdots \cup r_b s_{f_b}H$$ for some positive integers $b$ and $r_1,r_2,\ldots,r_b\in\mathbb{Z}_{\ge 1}$ by Theorem [Theorem 8](#crly:union){reference-type="ref" reference="crly:union"}. In general, let $$x_{s_i}=\begin{cases} r_i \quad \text{if $v_j=r_i$ for all $j\in s_iH$}\\ 0 \quad \text{otherwise} \end{cases}$$ and $a_i=|s_iH|$ for each $1 \le i \le e$. Then by Theorem [Theorem 8](#crly:union){reference-type="ref" reference="crly:union"}, the number of solutions to the binary integer linear program for $H$ (ILP$_H$) $$\label{ILP} \begin{array}{rl} \min & 0 \\ \mbox{subject to:} & \sum_{i=1}^ea_ix_{s_i}=\delta,~x_{s_i}\in\mathbb{Z}_{\ge 0} \end{array}$$ is equal to the number of possible $I_{\boldsymbol{v}}$ whose multiplier group contains $H$ and satisfies $\delta=|I_{\boldsymbol{v}}|$. Here the zero objective function is chosen to cast the problem of finding the number of possible $I_{\boldsymbol{v}}$ whose multiplier group contains $H$ and satisfies $\delta=|I_{\boldsymbol{v}}|$ as the problem of counting the number of solutions to a ILP. ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) is a formulation of a sum problem (SP). Determining whether ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) is feasible is known to be NP-complete [@Alfonsin1998]. Finding all solutions of ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) is NP-hard. The set of solutions to ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) potentially has multiple $j$ such that $I_{c_j(\boldsymbol{v})}$ satisfies equation ([\[eqn:union\]](#eqn:union){reference-type="ref" reference="eqn:union"}) for some $\boldsymbol{v}$. The repetitions of such solutions are due to the translates of $I$ fixed by $H$. By Theorem [Theorem 2](#thm:OneFixAll-dan){reference-type="ref" reference="thm:OneFixAll-dan"}, the number of such translates is $\prod_{1\le i\le r} \gcd(C,\ell_i)$, where $C=\gcd(t_1-1,t_2-1,\ldots,t_m-1,{\ell^*})$, $H=\langle t_1,t_2,\dots,t_r\rangle$, and ${\ell^*}$ is the exponent of $H$. Only the number of solutions to ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) is needed for the purpose of counting decimation classes. Since $|\mathbb{Z}_{{\ell^*}}^\times/H|=|\mathbb{Z}_{{\ell^*}}^\times|/|H|$ and $|s_iH|=|H|$ if $s_i\in \mathbb{Z}_{{\ell^*}}^\times$, for each multiplier group $H$, there exists at least $|\mathbb{Z}_{{\ell^*}}^\times|/|H|$ $H$-orbits of size $|H|$. Furthermore, each $H$-orbit has size dividing $|H|$. Hence, there exists significant duplicity among SP set values, i.e., among the elements of the sequence $\{a_i\}_{i=1}^e$. The number of solutions to ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) is computed more efficiently by determining only the number of solutions which are unique up to permutations of the variables of ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) with the same constraint coefficients. We call this reformulated problem the *unique sum problem* (USP). In Algorithm [\[alg:recursion\]](#alg:recursion){reference-type="ref" reference="alg:recursion"} (Recursion), $\text{zeros}(n,1)$ is a vector of all zeros of length $n$. For a vector $\boldsymbol{x}$, let $\{\boldsymbol{x}\}$ be the set of entries of $\boldsymbol{x}$ without repetitions. The vector $\boldsymbol{f}_{H}$ stores the sizes of all possible $H$-orbits of $s$ for the multiplier group $H$ and $s \in G$ sorted in ascending order. In Method [\[meth:USP\]](#meth:USP){reference-type="ref" reference="meth:USP"}, called *unique sum* (US), $\boldsymbol{q}_H=\text{uniquesort}(\boldsymbol{f}_{H})$ is a vector whose entries are all the elements of the set $\{\boldsymbol{f}_{H}\}$ sorted in ascending order, i.e., $\boldsymbol{q}_H$ is obtained by removing the repeat values in the vector $\boldsymbol{f}_{H}$. The length of a vector $\boldsymbol{x}$ is $\text{length}(\boldsymbol{x})$ and $\sum_{i}[f_H(i)=q_H(j)]$ is the number of entries of $\boldsymbol{f}_H$ that are equal to $q_H(j)$. For any ordered solution $\boldsymbol{p}$ to the US we must have $\{\boldsymbol{p}\}\subseteq \{\boldsymbol{f}_H\}$ and satisfy the additional constraint, $p_{s_i}\leq p_{s_{i+1}}$ for each $p_{s_i}\in \{\boldsymbol{p}\}$. Each $p_{s_i}\in \{\boldsymbol{p}\}$ corresponds to some $a_jx_{s_j}$ in ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) such that $x_{s_j}=r_j$. Such $H$-orbit combinations are obtained in Algorithm [\[alg:recursion\]](#alg:recursion){reference-type="ref" reference="alg:recursion"} via recursion using the vector $\boldsymbol{q}_H$ and their corresponding duplicity in $\{\boldsymbol{f}_H\}$, denoted by $\boldsymbol{r}_H$. At the $k$'th stage of the recursion, the desired sum is $\mu$ and the index of $\boldsymbol{q}_H$ being considered for addition is $k$. $\boldsymbol{sols}:= \text{zeros}(r_H(k),1$); $\nu := \mu-j_k*q_H(k)$; $sols(j_k):=\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{r_H(k)}{j_k}\right)\kern-.3em\right)}$; break; break; $sols(j_k) := \text{Recursion}(\boldsymbol{q}_H,\boldsymbol{r}_H,\nu,k+1)$; $sols(j_k) := sols(j_k)*\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{r_H(k)}{j_k}\right)\kern-.3em\right)}$; $\sum_{j_k}sols(j_k)$; $\boldsymbol{q}_H := \text{uniquesort}(\boldsymbol{f}_H)$; $m:=\text{length}(\boldsymbol{q}_H)$; $\boldsymbol{r}_H:=\text{zeros}(m,1)$; $r_H(j):=\sum_{i}[f_H(i)=q_H(j)]$; $z:=\,$Recursion($\boldsymbol{q}_H,\boldsymbol{r}_H,\delta,0$); At recursion $k$ in Algorithm [\[alg:recursion\]](#alg:recursion){reference-type="ref" reference="alg:recursion"}, the number of solutions is multiplied by $\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{r_H(k)}{j_k}\right)\kern-.3em\right)}$. Implementing this operation at each $k$ inflates the number of unique solutions to the number of ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}) solutions. This is the number of ways of choosing the number of duplicates of each element appearing within the US solution from the number of duplicates available. Hence, Method [\[meth:USP\]](#meth:USP){reference-type="ref" reference="meth:USP"} is designed to compute $$\sum_{\substack{0\le i \le m-1 \\[1pt] 0\le j_i\le \lfloor \frac{\delta}{q_H(i)} \rfloor \\ j_0q_{H}(0)+\cdots+j_{m-1}q_{H}(m-1)=\delta}}\prod_{i=0}^{m-1} \ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{r_H(i)}{j_i}\right)\kern-.3em\right)}.$$ # A method for determining the number of decimation classes {#sec:CountExam} The following lemma is used to develop a method for finding the number of decimation classes. Its proof is identical to that found in [@decimation]. **Lemma 10**. *Let $G\cong\mathbb{Z}_{\ell_1}\times\cdots \times\mathbb{Z}_{\ell_r}$ have exponent ${\ell^*}$ and $\boldsymbol{u},\boldsymbol{v}\in \mathbb{Z}_{\ge 0}^{G}$ be two vectors in the same decimation class. Let $H_{\boldsymbol{u}}, H_{\boldsymbol{v}} \leq \mathbb{Z}^\times_{{\ell^*}}$ be the multiplier groups of $\boldsymbol{u}$ and $\boldsymbol{v}$, respectively. Then $H_{\boldsymbol{u}}=H_{\boldsymbol{v}}$.* The method for finding the number of decimation classes from [@decimation] now carries over for nonnegative integer vectors. This method requires generating the lattice of all subgroups of a finite abelian group. The following lemma was proven in [@decimation] and aids in generating the lattice of all subgroups of an abelian group. **Lemma 11**. *Let $K=\langle t_1,\dots,t_k\rangle$ be a finite abelian group, where $k$ is the smallest integer such that $K=\langle x_1,\dots,x_k\rangle$ for some $x_1,\ldots, x_k \in G$. Then each subgroup $J$ of $K$ can be generated by no more than $k$ elements.* For any two subgroups, $K<L$, $K$ is called a *maximal subgroup* of $L$ if $K <L$ and there is no subgroup $J$ such that $K<J<L$. *Cyclic extension* is a method for finding all subgroups of a finite abelian group $L=\langle u_1,\dots,u_k\rangle$, and constructing the lattice of all subgroups $\mathbb{L}$. The lattice $\mathbb{L}$ is presented as a graph, where each distinct subgroup labels a distinct vertex, and an edge from $J$ to $K$ exists if and only if $J$ is a maximal subgroup of $K$. Cyclic extension first generates all of $L$'s cyclic subgroups $\langle v_i\rangle$ for each $1\le i\le m.$ It then recursively combines them to generate all subgroups of $L$. Let $\alpha_1$ be the number of subgroups of $\langle v_1\rangle$, and for each $2\le j\le m$ let $\alpha_j$ be the number of subgroups of $\langle v_j\rangle$ each of which is not a subgroup of any of the subgroups $\langle v_1\rangle, \langle v_2\rangle,\ldots, \langle v_{j-1}\rangle$. By Lemma [Lemma 11](#lma:SNF){reference-type="ref" reference="lma:SNF"}, it suffices to combine cyclic subgroups of at most $k$ cyclic subgroups of $L$. Cyclic extension computes $$\sum_{1\le i_1 \le m} \alpha_{i_1} + \sum_{1\le i_1<i_2 \le m} \alpha_{i_1}\alpha_{i_2} + \cdots + \sum_{1\le i_1<\cdots<i_k \le m} \alpha_{i_1}\cdots\alpha_{i_k}$$ number of all such subgroups to construct $\mathbb{L}$. Cyclic extension uses a breadth-first search to minimize the number of groups generated. *Lattice insertion* is a method used within cyclic extension for iteratively constructing $\mathbb{L}$. Let $\mathbb{L}_i$ be the interim lattice containing only $i$ subgroups of $L$. The next group generated, $L_{i+1}$, is compared against each subgroup $K$ in $\mathbb{L}_i$ with $|K|$ dividing $|L_{i+1}|$ and $J$ in $\mathbb{L}_i$ such that $|J|$ is divisible by $|L_{i+1}|$. If $K$ is a $\mathbb{L}_i$-maximal subgroup of $L_{i+1}$, i.e., there is no subgroup $K'$ in $\mathbb{L}_i$ such that $K<K'<L_{i+1}$, then an edge is added from $L_{i+1}$ to $K$. Moreover, each edge from $J$ to $K$ in $\mathbb{L}_i$ such that $K<L_{i+1}<J$ is replaced with an edge from $J$ to $L_{i+1}$. Each interim lattice $\mathbb{L}_i$ is guaranteed to be connected in this method as $\mathbb{L}_0$ contains only the trivial subgroup, $\langle 1 \rangle$, and at each step of the method, each newly introduced subgroup $L_{i+1}$ is either equal to a subgroup in $\mathbb{L}_i$ and discarded or one of the subgroups in $\mathbb{L}_i$ is an $\mathbb{L}_i$-maximal subgroup of $L_{i+1}$. Method [\[meth:basic\]](#meth:basic){reference-type="ref" reference="meth:basic"} (Count) determines the number of decimation classes of density $\delta$ nonnegative integer vectors indexed by a finite abelian group, of odd order $\ell$ and exponent ${\ell^*}$, such that $\gcd({\ell^*},\delta)=1$. the set of all subgroups $\mathcal{H}$ of $\mathbb{Z}^\times_{{\ell^*}}$ along with $g_1,\ldots, g_{\gamma(H)}$ such that $\langle g_1,\ldots, g_{\gamma(H)}\rangle=H$, where $\gamma(H)$ is preferably small for each $H \in \mathcal{H}$;[\[step:start\]]{#step:start label="step:start"} $\mathbb{L}$, the lattice of subgroups for $\mathbb{Z}_{{\ell^*}}^\times$;[\[step:construct\]]{#step:construct label="step:construct"} $N'_{\langle1\rangle}=\frac{{\ell \choose \delta}}{\ell}$[\[step:initial\]]{#step:initial label="step:initial"}; all $H$-orbits; Method [\[meth:USP\]](#meth:USP){reference-type="ref" reference="meth:USP"} to find the number of solutions $nsol$ to ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"});[\[step:ILP\]]{#step:ILP label="step:ILP"} $N'_H:=\frac{nsol}{\prod_{1\le i\le r} \gcd(C,\ell_i)}$, where $C=\gcd(g_1-1,g_2-1,\ldots,g_{\gamma(H)}-1,{\ell^*})$ to be the number of necklaces with multiplier groups containing $H$; [\[step:multiplier\]]{#step:multiplier label="step:multiplier"} $N'_H$, by the number of necklaces whose multiplier groups strictly contain $H$ from the top down in $\mathbb{L}$; [\[step:discount2\]]{#step:discount2 label="step:discount2"} the resulting value from Step [\[step:discount2\]](#step:discount2){reference-type="ref" reference="step:discount2"} to be $N_H$, i.e., the number of necklaces whose multiplier groups are equal to $H$; $numD_H:=\frac{(N_H)*|H|}{\phi({\ell^*})}$;[\[step:laststatement\]]{#step:laststatement label="step:laststatement"} $N_{\langle1\rangle}=N'_{\langle1\rangle}-\sum_{H\in \mathcal{H}\setminus \{\langle 1\rangle \}} N_H$;[\[step:N1\]]{#step:N1 label="step:N1"} $numD_{\langle 1\rangle} = \frac{N_{\langle1\rangle}}{\phi({\ell^*})}$;[\[step:numD1\]]{#step:numD1 label="step:numD1"}\ $\sum_{H\in \mathcal{H}} numD_H$;[\[step:sum\]]{#step:sum label="step:sum"} In Step 2, we used cyclic extension for generating the lattice of all subgroups $\mathbb{L}$ of $\mathbb{Z}^\times_{{\ell^*}}$ for $\ell={\ell^*}$ and $3 \le {\ell^*}\le 121$. While generating $\mathbb{L}$, all subgroup members are stored to improve efficiency. For example, if the generators of subgroup $K$ exist within subgroup $H$ and $|K|<|H|$, then $K<H$. This also improved efficiency of subsequently generating the $H$-orbits by multiplying elements of $G=\mathbb{Z}_{\ell}$. The sizes and duplicities of these orbits are recorded in $\boldsymbol{r}_H$ and $\boldsymbol{q}_H$, respectively, to solve the corresponding US. Step [\[step:initial\]](#step:initial){reference-type="ref" reference="step:initial"} computes the number of necklaces containing $\langle 1\rangle$ based on Lemma [Lemma 1](#lem:necklace){reference-type="ref" reference="lem:necklace"}. Step [\[step:multiplier\]](#step:multiplier){reference-type="ref" reference="step:multiplier"} is necessary by Theorem [Theorem 2](#thm:OneFixAll-dan){reference-type="ref" reference="thm:OneFixAll-dan"} for obtaining the number of necklaces containing $H$ from the number of solutions to ILP$_H$ ([\[ILP\]](#ILP){reference-type="ref" reference="ILP"}). Let $\mathcal{H}$ be the set of all subgroups of $H$. In Step [\[step:discount2\]](#step:discount2){reference-type="ref" reference="step:discount2"}, our method of counting necklaces with a given multiplier group, $H \in \mathcal{H} \setminus \{ \langle 1 \rangle\}$, discounts the number of necklaces whose multiplier groups strictly contain $H$ from that of necklaces whose multiplier groups contain $H$. The resulting count is the number of necklaces with multiplier group $H$. Steps [\[step:laststatement\]](#step:laststatement){reference-type="ref" reference="step:laststatement"} and [\[step:numD1\]](#step:numD1){reference-type="ref" reference="step:numD1"} reduce necklace counts from Steps [\[step:discount2\]](#step:discount2){reference-type="ref" reference="step:discount2"} and [\[step:N1\]](#step:N1){reference-type="ref" reference="step:N1"} to decimation counts by applying the last statement of Lemmas [Lemma 3](#lma:stab2){reference-type="ref" reference="lma:stab2"} and [Lemma 10](#lma:HuHv){reference-type="ref" reference="lma:HuHv"}. Step [\[step:sum\]](#step:sum){reference-type="ref" reference="step:sum"} sums counts across all multiplier groups to get the total number of decimation classes. 99 J. L. R. Alfonsın. On variations of the subset sum problem. Discrete Appl. Math., 81(1-3):1--7, 1998. D. Djokovic, I. Kotsireas, D. Recoskie, and J. Sawada. Charm bracelets and their application to the construction of periodic Golay pairs. Discrete Applied Mathematics, 188(1):32--40, 2015. R. J. Fletcher, M. Gysin, and J. Seberry. Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices. Australas. J. Combin., 23:75--86, 2001. J. J. Rotman. An Introduction to the Theory of Groups. Springer-Verlag, New York, NY, USA, 4th edition, 1994. J. Sawada and A. Williams. A Gray code for fixed-density necklaces and Lyndon words in constant amortized time. Theoret. Comput. Sci., 502:46--54, 2013. J. S. Turner, D. A. Bulutoglu, I. Kotsireas, D. Baczkowski, and Geyer A. J. Counting the decimation classes of binary vectors with relatively prime length and density. J. Algebr. Comb., 55:61--87, 2022. 12

arxiv_math

--- abstract: | Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a high implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver leads to a higher number of iterations to get the same level of convergence as the monolithic one. The objective of this paper is to accelerate the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. We also consider deflation of approximate eigenvectors in conjunction with advanced inner-outer Krylov solvers for the fluid block equations. We demonstrate the benefit of these techniques by computing the coupled derivatives of an aeroelastic configuration of the ONERA-M6 fixed wing in transonic flow. For this exercise the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. All computations are performed using RANS flow modeling and a fully linearized one-equation Spalart-Allmaras turbulence model. Numerical simulations show up to 39% reduction in matrix-vector products for GCRO-DR and up to 19% for the nested FGCRO-DR solver. author: - Christophe Blondeau - Mehdi Jadoui bibliography: - Paper_for_arXiv_Recycling_Subspaces_for_coupledAdjoint_v2.bib title: Recycling Krylov Subspaces for Efficient Partitioned Solution of Aerostructural Adjoint Systems --- partitioned solver, subspace recycling, coupled adjoint, GCRO-DR # Introduction We are interested in robust and efficient solvers for the solution of the coupled-adjoint linear system which is crucial to successful aerostructural optimization. Fine control of an aerodynamic shape or of a structural layout leads to a high-dimensional parameter space. For high-fidelity simulations, gradient-based optimizers in conjunction with the adjoint approach are the methods of choice. However, the coupled adjoint linear system is inherently ill-conditioned as it embeds matrix blocks of different scales and structures. In addition, the fluid block coming from the exact linearization of the RANS equations associated to a turbulence model is often very stiff. Besides, a strong level of fluid-structure interaction is known to be detrimental to the robustness and efficiency of existing solution techniques. For such linear systems, monolithic and partitioned (or segregated) strategies can be applied. The former approach is expected to offer better robustness and efficiency for strong fluid-structure interactions [@heil2004; @heil2008]. However, it requires a high implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver leads to a higher number of iterations to get the same level of convergence as the monolithic one. A review of partitioned simulations of fluid-structure interactions involving black-box solvers is proposed in [@degroote2013partitioned]. The partitioned approach simply consists in solving, in an alternating way, the aerodynamic and the structural sub-problems by applying the Linear Block Gauss-Seidel algorithm (LBGS). It accounts for the interdisciplinary coupling by adding a source term to the right-hand side of each set of disciplinary adjoint equations. The modularity of this approach makes it rather interesting since it takes advantage of the specific routines designed for each sub-problem and does not require a high implementation cost. Nevertheless, this approach becomes rapidly inefficient and could even diverge for strong fluid-structure coupling even though the addition of some level of relaxation helps to mitigate this issue. On the other hand, the monolithic approach consists in solving the fluid and structural equations simultaneously making it more robust in the sense that it is less sensitive to the strength of the fluid-structure interaction. The coupled adjoint system is generally solved by using Krylov subspace methods. The challenging aspect of such an approach is then to develop advanced preconditioning strategies combined with numerical ingredients so that Krylov methods reach the best performances in terms of robustness and efficiency. The objective of this work is to improve the efficiency of the existing partitioned solver [@Achard2018; @Jadoui_AIAA2022] by considering techniques borrowed from Krylov subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides [@parks2006recycling]. We will demonstrate the benefit of these advanced techniques by computing the coupled derivatives for the ONERA-M6 fixed wing in transonic flow [@AileM6]. For this exercise the fluid grid is coupled to a structural model specifically designed to exhibit a high flexibility. All computations are performed using RANS flow modeling and a fully linearized one-equation Spalart-Allmaras turbulence model. As an example, Figure [2](#fig:FGMRESDR_70_10_35_SAlin_LUSGS_6_CFL_effect_iterations){reference-type="ref" reference="fig:FGMRESDR_70_10_35_SAlin_LUSGS_6_CFL_effect_iterations"} illustrates the performance of the existing embedded coupled-adjoint segregated solver for structured grids in a former version of the elsA CFD code [^1], applied to the M6 wing test case. {#fig:FGMRESDR_70_10_35_SAlin_LUSGS_6_CFL_effect_iterations width="95%"} {#fig:FGMRESDR_70_10_35_SAlin_LUSGS_6_CFL_effect_iterations width="95%"} These convergence curves will serve as a reference for comparison with the improved solver efficiency allowed by the developments performed in this work. A flexible GMRES Krylov solver with deflated restarting is used with an outer Krylov basis of 70 vectors, an inner Krylov basis of size 10 and a deflation subspace of 35 vectors (between fluid cycles). At the beginning of this work, the only available preconditioner for structured grids in elsA was a combination of a Restrictive Additive Schwarz (RAS) domain decomposition method coupled with a Lower-Upper Symmetric Gauss Seidel (LU-SGS) relaxation. This preconditioner is controlled by two parameters: the number of relaxations and the CFL coefficient. The number of relaxations is fixed at 6 and the CFL is varied. In [@jadoui2022comparative] we point out that a proper tuning of the CFL coefficient is crucial for this type of preconditioners and how new block ILU type preconditioners accelerate the rate of convergence. Also, after each fluid-structure coupling, a cold restart is performed and the spectral information from the previous fluid-structure cycle is discarded. As a consequence, a plateau is observed after each restart which dramatically hampers convergence in the first cycles. We will demonstrate that well-chosen subspace recycling strategies will eliminate these convergence stagnations. In a high-fidelity aerostructural optimization context, Zhang and Zingg [@zhang2018efficient] implemented a robust monolithic solution method for both aerostructural analysis and coupled adjoint problem. A three-field formulation was adopted involving the mesh, the flow and the structural states. The performance of the monolithic method as well as the partitioned one was investigated through a comparative study by varying the level of fluid-structure coupling. For the coupled adjoint problem solution, a GCROT [@de1999truncation] Krylov solver has been used in conjunction with a block Gauss-Seidel preconditioner. The monolithic adjoint solution has been 60$\%$ more efficient than the partitioned one for strong coupling. For weak fluid-structure coupling, the monolithic solution still outperformed the partitioned one with a better efficiency of 40$\%$. In terms of computational time, the monolithic method showed 50 $\%$ to over 60 $\%$ faster than the partitioned method. A similar comparative study of both monolithic and partitioned approaches was performed by Kenway et al. [@kenway2014scalable] except that a Block Jacobi preconditioner was applied to the coupled adjoint system. The aerodynamic and structural block preconditioners were solved by a preconditioned Krylov method (restarted Generalized Minimal RESidual - GMRES [@saad1986]) and a direct factorization method respectively. A Flexible Krylov method (e.g FGMRES [@saad1993flexible]) has been used for the coupled adjoint system solution. The Common Research Model (CRM) wing-body-tail configuration was sized by considering two critical load cases: 1$g$ cruise condition with moderate elastic deformation and a 2.5$g$ pull-up with significantly more deflection. For the same memory footprint, the best monolithic solution seems to outperform the best partitioned one by reducing the time by 19$\%$ for the 1$g$ load and by 29$\%$ for the 2.5$g$ case. These numerical experiments demonstrate the great benefit of using monolithic approach in the strong coupling case but at the price of a robust preconditioner for the Krylov solver. We note however that both studies only considered inviscid flow modeling. These conclusions about the monolithic solver efficiency might be mitigated by the added stiffness of adjoint system matrices produced by a RANS fluid model associated to a linearized turbulence model. Although the satisfactory performance of monolithic solvers, advanced strategies that could accelerate the partitioned algorithm using black-box solvers have received less attention. We recall that a partitioned algorithm consists in approximately solving the aerodynamic adjoint block at each fluid-structure iteration resulting in a sequence of adjoint linear systems with varying right-hand sides. As already mentioned, the structural source term that affects the right-hand side of the fluid block has the nice property of rapidly converging to a constant value. The corollary of this property is that after several fluid-structure couplings, the subsequent fluid systems should greatly benefit from recycling spectral information from the previous fluid-structure cycles. At the start of this work, the current partitioned adjoint solver did not take advantage of recycling and at each update of the structural source term the Krylov solver did a cold start from the previous solution. The principle of deflation is to remove the influence of a system's subspace on the iterative process. This is usually beneficial when directions of certain subspaces hamper convergence. Deflation of an eigenspace can be performed in two ways: the linear system (matrix and right-hand side) is left-multiplied by a projector $\mathbf{P}$, i.e. deflation by projection [@vuik1999efficient; @tang2007deflation; @Gaul2014], or some eigenvectors are added to the Krylov subspace, i.e. deflation by augmentation [@morgan2002gmres]. A survey of deflation and augmentations techniques can be found in [@coulaud:hal-00803225]. A specific type of deflation preconditioning aims at solving a rank-deficient projected system, using a Krylov solver, in a certain subspace outside of the problematic subspace. The solution is then complemented with the solution in the latter subspace. In [@Gaul2014] the author makes the post-correction superfluous by using a projection as right-preconditioner instead. For deflation by augmentation, adding eigenvectors to the Krylov subspace can effectively deflate corresponding eigenvalues from the spectrum because when these directions are included in the solution approximation, the convergence of the Krylov solver continues according to the modified spectrum. The deflation by augmentation led to the well-known FGMRES-DR solver [@giraud2010] and its extension to inner-outer Krylov solver [@simoncini2002flexible; @jadoui2022comparative]. Unfortunately, the deflated restarted GMRES framework based on subspace augmentation is not adapted for solving sequences of linear systems [@parks2006recycling]. Fortunately, some authors have proposed new strategies in order to reuse information accumulated in previous fluid-structure cycles and use it to accelerate the solution of the next linear system. Krylov subspaces recycling methods seem to be the suitable choice. Historically, De Sturler suggested the Generalized Conjugate Residuals with inner Orthogonalization (GCRO) method [@de1996nested], an improved version of the recursive GMRES (GMRESR) solver [@van1994gmresr] by maintaining an orthogonality condition between the outer and the inner spaces generated by GMRESR. This way, it provides the optimal correction to the solution in a global search space. Later, Parks et al. formulated the GCRO-DR algorithm [@parks2006recycling] that combines GCRO and deflation techniques by augmentation introduced by Morgan [@morgan2002gmres]. They demonstrated better performances of GCRO-DR compared to the GMRES-DR in a long sequence of linear systems from a fracture mechanics problem. Carvhalo et al. extended GCRO-DR to the flexible case (FGCRO-DR) [@carvalho2011flexible] and they conducted in-depth analysis of both flexible methods. In particular, they showed that both methods can be algebraically equivalent if a certain colinearity condition is satisfied at each cycle. In 2013, Niu et al. introduced Loose GCRO-DR (LGCRO-DR) [@niu2013accelerated] for improving the convergence of GCRO-DR by recycling both spectral information and approximate error information. The error is defined as the distance between the current iterate and the exact solution of the system. It is not known by definition but a fair approximation to it can be computed. This idea was initially proposed by Baker et al. [@baker2005technique] and mimics the idea behind GMRESR of including approximations to the error in the current approximation space. This error information is interesting since it represents in some sense the previous Krylov space generated in the previous cycle and subsequently discarded. In addition to that, LGCRO-DR is straightforward and economic to implement. In this work, we investigate advanced Krylov subspace methods using subspace recycling strategies for accelerating the partitioned solver applied to the linear coupled-adjoint system. More specifically, we compare GMRES-DR and FGMRES-DR to GCRO-DR and FGCRO-DR with and without subspace recycling. This work benefits from the recent achievements to improve efficiency of the fluid adjoint solution by applying nested Krylov subspace methods [@jadoui2022comparative]. The numerical experiments are performed on an aeroelastic configuration of the ONERA M6 fixed wing in transonic viscous flow. This paper is organized as follows. In section [2](#Aerostructuraladjointsystem){reference-type="ref" reference="Aerostructuraladjointsystem"} we briefly recall the theoretical background of aeroelastic and coupled-adjoint equations. The partitioned algorithm is also outlined. The aeroelastic numerical test case is presented in section [3](#ONERA_M6){reference-type="ref" reference="ONERA_M6"}. To support later comparison with GCRO-DR anf FGCRO-DR, we then review the fundamentals of FGMRES-DR in section [4](#FGMRESDR_Review){reference-type="ref" reference="FGMRESDR_Review"} with a focus on numerical implementation and application to the solving of the fluid and coupled adjoint systems. Section [5](#GCRODR_theory){reference-type="ref" reference="GCRODR_theory"} is devoted to the description of the GCRO algorithm with adaptations related to variable preconditioning and subspace recycling. We also take the opportunity to give some insights related to an efficient implementation of the deflation and recycling strategy. The numerical experiments are then repeated with the GCRO-DR solver and show very promising reduction in terms of matrix-vector products compared to the standard implementation. Finally, the flexible case is presented in section [6](#FGCRODR_theory){reference-type="ref" reference="FGCRODR_theory"}. # Aerostructural adjoint system {#Aerostructuraladjointsystem} ## Aeroelastic equilibrium Let us denote the state variables of the coupled system $\mathbf{W}$ and $\mathbf{U}$, representing the fluid conservative variables and the structural displacements respectively. At the aeroelastic equilibrium, the state variables and the meshes satisfy the discretized equations of fluid and structural mechanics simultaneously: $$\left\{ \begin{aligned} \mathbf{R}_a(\mathbf{X}_a,\mathbf{W},\mathbf{U}) &= \mathbf{0} \\ \mathbf{R}_s(\mathbf{X}_s,\mathbf{W},\mathbf{U}) &= \mathbf{0} \end{aligned} \right. \label{eq1}$$ where $\mathbf{R}_a$ is the discrete aerodynamic residual and $\mathbf{R}_{s}$ the discrete structural residual. These two blocks of equations are coupled through aerodynamic forces $\mathbf{Q}_{a}$ loading the skin of the structure and the structural displacements $\mathbf{U}$ deforming the fluid mesh. The structural mesh is noted $\mathbf{X}_s$. In the following we introduce two aerodynamic grids $\mathbf{X}_a$ and $\mathbf{X}_{a0}$. $\mathbf{X}_a$ denotes the deformed aerodynamic grid at the aeroelastic equilibrium at the outcome of the aeroelastic analysis. $\mathbf{X}_{a0}$ is called the reference mesh which supports the aerodynamic shape parametrization. Typically for an aircraft design study the reference mesh is chosen as the jig shape or the flight shape in reference nominal flight conditions. The load, displacement and mesh deformation operators then merely depend on $\mathbf{X}_{a0}$ for an aeroelastic or coupled-adjoint analysis. The structural loads $\mathbf{Q}_s$ are obtained with a suitable load transfer technique applied to $\mathbf{Q}_a$ such that $$\mathbf{Q}_s(\mathbf{Q}_a(\mathbf{W},\mathbf{X}_a),\mathbf{X}_{a0},\mathbf{X}_s) = \mathbf{T}^{Q}_{surf}(\mathbf{X}_{a0},\mathbf{X}_s) \mathbf{Q}_a(\mathbf{W},\mathbf{X}_a)$$ where $\mathbf{T}^{Q}_{surf}$ represents a linear load transfer operator. The subscript $surf$ stipulates that the associated linear operators or data relate to the fluid-structure interface. The structural displacements alter the fluid grid locations through the relation: $$\mathbf{X}_{a} = \mathbf{X}_{a0} + \delta\mathbf{X}_{a}(\delta\mathbf{X}_{a,surf},\mathbf{X}_{a0}) = \mathbf{X}_{a0} + \mathbf{T}_{vol}(\mathbf{X}_{a0})\delta\mathbf{X}_{a,surf}$$ with $\mathbf{T}_{vol}(\mathbf{X}_{a0})$ the volume operator performing the deformation of the fluid domain. The vector $\delta\mathbf{X}_{a,surf}$ corresponds to the displacements of the fluid nodes at the fluid-structure interface. $$\delta\mathbf{X}_{a,surf} = \delta\mathbf{X}_{a,surf}(\mathbf{X}_{a0},\mathbf{X}_{s},\mathbf{U}) = \mathbf{T}^{U}_{surf}(\mathbf{X}_{a0},\mathbf{X}_{s})\mathbf{U}$$ where $\mathbf{T}^{U}_{surf}(\mathbf{X}_{a0},\mathbf{X}_{s})$ represents a linear displacement transfer operator. ## Partitioned strategy for the coupled adjoint system Let us consider a scalar aeroelastic objective function $J(\mathbf{W},\mathbf{U},\mathbf{X}_{a},\mathbf{X}_{s})$ and a design parameter $p$. One way to obtain the coupled adjoint equations is to formulate an augmented objective function by adding the total variation of the residuals $\mathbf{R}_{s}$ and $\mathbf{R}_{a}$ to the total derivative $dJ/dp$. More specifically, we define $d\tilde{J}/dp$ as $$\begin{aligned} %\tilde{J} &= J + \lambda^{T}_{a} \mathbf{R}_{a} + \lambda^{T}_{s} \mathbf{R}_{s}, \\ \frac{d\tilde{J}}{dp} &= \frac{dJ}{dp} + \lambda^{T}_{a} \frac{d \mathbf{R}_{a}}{dp} + \lambda^{T}_{s} \frac{d\mathbf{R}_{s}}{dp}, \label{eq2}\end{aligned}$$ where $$\frac{dJ}{dp} = \frac{\partial J}{\partial \mathbf{W}} \frac{d \mathbf{W}}{dp} + \frac{\partial J}{\partial \mathbf{X}_{a}} \frac{d \mathbf{X}_{a}}{dp} + \frac{\partial J}{\partial \mathbf{X}_{s}} \frac{d \mathbf{X}_{s}}{dp} + \frac{\partial J}{\partial \mathbf{U}} \frac{\partial \mathbf{U}}{\partial p}. \label{eq3}$$ In Eq. ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) the total variations of residuals are exactly zero since they represent constraints related to the satisfaction of the equilibrium equations at the outcome of the aeroelastic analysis. For simplicity, we restrict here to the specific case of a shape design parameter not affecting the structural geometry nor the structural stiffness. In addition, the explicit dependency of the objective function with respect to the structural states is dropped, i.e., we consider only derivatives of aerodynamic coefficients. The full derivation for the general case can be found in [@Achard2017]. It is worth to mention that these assumptions do not lead to any loss of generality of the work presented in this paper since we focus on solution techniques for the adjoint system. As $\mathbf{X}_{s}$ does not depend on the design parameter $p$, we have $d \mathbf{X}_{s}/dp = \mathbf{0}$. Under the same assumption we also have $d\mathbf{K}/dp=\mathbf{0}$. After some algebra manipulation we end up with the following expression for $d\tilde{J}/dp$ in which the total derivatives $d\mathbf{W}/dp$ and $d\mathbf{U}/dp$ have been factored out: $$\begin{split} \frac{d \tilde{J}}{dp} & = \left( \frac{\partial J}{\partial \mathbf{W}} + \lambda^{T}_{a} \frac{\partial \mathbf{R}_{a}}{\partial \mathbf{W}} - \lambda^{T}_s\mathbf{C} \right) \frac{d \mathbf{W}}{dp} + \left( \frac{\partial J}{\partial \mathbf{X}_{a}} \mathbf{A} + \lambda^{T}_{a} \frac{\partial \mathbf{R}_{a}}{\partial \mathbf{X}_{a}}\mathbf{A} + \lambda^{T}_{s}(\mathbf{K} - \mathbf{D})\right) \frac{d \mathbf{U}}{dp} \\ & + \left( \frac{\partial J}{\partial \mathbf{X}_{a}} + \lambda^{T}_{a} \frac{\partial \mathbf{R}_{a}}{\partial \mathbf{X}_{a}} \right) \mathbf{B} \frac{d \mathbf{X}_{a0}}{dp}-\lambda^{T}_{s}\mathbf{E}\frac{d \mathbf{X}_{a0}}{dp} \end{split} \label{eq4}$$ Constant matrices $\mathbf{A}$ to $\mathbf{E}$ are defined analytically with the following formulas (see [@Achard2017; @Achard2018]): $$\begin{aligned} \mathbf{A} &= \mathbf{T}_{vol} \mathbf{T}^{U}_{surf} \\ \mathbf{B} &= \frac{\partial \mathbf{X}_a}{\partial \mathbf{X}_{a0}} = \mathbf{I} + \frac{\partial \mathbf{A}}{\partial \mathbf{X}_{a0}} \mathbf{U}\\ \mathbf{C} &= \mathbf{T}^{Q}_{surf} \frac{\partial \mathbf{Q}_a}{\partial \mathbf{W}} \\ \mathbf{D} &= \mathbf{T}^{Q}_{surf} \frac{\partial \mathbf{Q}_a}{\partial \mathbf{X}_a} \mathbf{T}^{U}_{surf} \\ \mathbf{E} &= \mathbf{T}^{Q}_{surf} \frac{\partial \mathbf{Q}_a}{\partial \mathbf{X}_a}\mathbf{B} + \frac{\partial \mathbf{Q}_s}{\partial \mathbf{X}_{a0}} \label{eq5}\end{aligned}$$ The coupled adjoint linear system is obtained by canceling factors related to $d\mathbf{W}/dp$ and $d\mathbf{U}/dp$ in Eq. ([\[eq4\]](#eq4){reference-type="ref" reference="eq4"}) to give $$\begin{bmatrix} \left[ {\frac{\partial \mathbf{R}_{a}}{\partial \mathbf{W}}}\right]^{T} & -\mathbf{C}^{T} \\ & \\ \mathbf{A}^{T}\left[\frac{\partial \mathbf{R}_{a}}{\partial \mathbf{X}_{a}}\right]^{T} & \mathbf{K}^{T} - \mathbf{D}^{T} \end{bmatrix} \begin{bmatrix} \lambda_{a} \\ \\ \lambda_{s} \end{bmatrix} = \begin{bmatrix} - \left[ \frac{\partial J}{\partial \mathbf{W}} \right]^{T} \\ \\ -\mathbf{A}^{T} \left[ \frac{\partial J}{\partial \mathbf{X}_{a}} \right]^{T} \end{bmatrix} \label{Coupled_Adjoint_system}$$ The process for solving the adjoint system follows an iterative block scheme. Algorithm [\[alg:LBGS-algorithm\]](#alg:LBGS-algorithm){reference-type="ref" reference="alg:LBGS-algorithm"} details the Linear Block Gauss-Seidel (LBGS) scheme applied for the solution of system ([\[Coupled_Adjoint_system\]](#Coupled_Adjoint_system){reference-type="ref" reference="Coupled_Adjoint_system"}). In this derivation, we use the structural flexibility $\mathbf{S}$ which is a small reduced matrix relating the set of structural forces to the set of structural displacments pertaining to the fluid-structure coupling. The relaxation factor $\theta_{s}$ has been introduced on the adjoint vector $\lambda_{s}$. Assuming that the coupled system is solved to machine accuracy, the total derivative reconstruction is given by $$\frac{d J}{dp} = \left( \frac{\partial J}{\partial \mathbf{X}_{a}} + \lambda^{T}_{a} \frac{\partial \mathbf{R}_{a}}{\partial \mathbf{X}_{a}} \right) \mathbf{B} \frac{d \mathbf{X}_{a0}}{dp}-\lambda^{T}_{s}\mathbf{E}\frac{d \mathbf{X}_{a0}}{dp} \label{dJdp}$$ In the expression above the computation of the product of the geometrical sensitivities with the matrix $\mathbf{B}$ is not trivial. If one already has at hand a linearized version of the operator $\mathbf{A}$, i.e. of $\mathbf{T}_{surf}^U$ and $\mathbf{T}_{vol}$, it can be applied to $d \mathbf{X}_{a0}/{dp}$ as many times as the number of design variables. This is the most straightforward manner but the benefit of the adjoint formulation is then mitigated by the cost of the gradient assembly. The other way is to transpose the first term in the right-hand side of Eq. ([\[dJdp\]](#dJdp){reference-type="ref" reference="dJdp"}) and compute products like $[ \partial\mathbf{T}_{surf}^U / \partial\mathbf{X}_{a0} ]^T \mathbf{v}$ and $[ \partial\mathbf{T}_{vol} / \partial\mathbf{X}_{a0} ]^T \mathbf{v}$, where $\mathbf{v}$ has fluid grid size. We call this mode the geometrical adjoint of the mesh deformation and displacement transfer operators. These two modes of gradient assembly have been implemented in the coupled-adjoint module of the elsA software. The iterations stop after a maximum number $n_{cpl}$ of fluid-structure couplings or when the relative fluid and structural residuals get lower than a prescribed tolerance: $r_A \le \epsilon_A$ and $r_S \le \epsilon_S$. Typically we choose $\epsilon_A = \epsilon_S = 10^{-6}$. $\mathbf{RHS}_{stru} \gets \mathbf{0}$ $\mathbf{RHS}_{stru} \gets \left( \mathbf{T}^{Q}_{surf} \frac{\partial \mathbf{Q}_{a}}{\partial \mathbf{W}} \right)^T \mathbf{S}^{T}\lambda^{0}_{s}$ $\left[ \frac{\partial \mathbf{R}_a}{\partial \mathbf{W}} \right]^T \lambda^{0}_{a} = - \left[ \frac{\partial J}{\partial \mathbf{W}} \right]^{T} + \mathbf{RHS}_{stru}$ $A_{Xs,surf} \leftarrow \left( \frac{\partial \mathbf{Q}_a}{\partial \mathbf{X}_{a,surf}} \right)^{T}\left( \mathbf{T}^{Q}_{surf} \right)^T \mathbf{S}^{T}\lambda^{k-1}_{s}$ $A_{Xa} \leftarrow - \left( [\lambda^{k-1}_{a}]^{T} \frac{\partial \mathbf{R}_{a}}{\partial \mathbf{X}_{a}} + \frac{\partial J}{\partial \mathbf{X}_{a}} \right)^{T}$ $A_{Xa,surf} \leftarrow (\mathbf{T}_{vol})^{T} \mathbf{A}_{Xa}$ $\lambda^{k}_{s} \leftarrow (\mathbf{T}^{U}_{surf})^{T}(\mathbf{A}_{Xs,surf} + \mathbf{A}_{Xa,surf})$ $\lambda^{k}_{s} \leftarrow (1-\theta_{s}) \lambda^{k-1}_{s} + \theta_{s} \lambda^{k}_{s}$ [\[alg:Relaxation_step\]]{#alg:Relaxation_step label="alg:Relaxation_step"} $\mathbf{RHS}_{stru} \leftarrow \left( \mathbf{T}^{Q}_{surf} \frac{\partial \mathbf{Q}_{a}}{\partial \mathbf{W}} \right)^{T}\mathbf{S}^{T} \lambda^{k}_{s}$ $\left[ \frac{\partial \mathbf{R}_a}{\partial \mathbf{W}} \right]^T \lambda^{k}_{a} = - \left[ \frac{\partial J}{\partial \mathbf{W}} \right]^{T} + \mathbf{RHS}_{stru}$ goto 14 $\frac{dJ}{dp} \leftarrow \left( \frac{\partial J}{\partial \mathbf{X}_{a}} + \lambda^{T}_{a} \frac{\partial \mathbf{R}_{a}}{\partial \mathbf{X}_{a}} \right) \mathbf{B} \frac{d \mathbf{X}_{a0}}{dp}-\lambda^{T}_{s}\mathbf{E}\frac{d \mathbf{X}_{a0}}{dp}$ [\[assembly\]]{#assembly label="assembly"} # ONERA-M6 wing aeroelastic analysis {#ONERA_M6} In this study the numerical experiments have been performed with the well known ONERA-M6 fixed wing configuration which has been extensively used for CFD solvers validation in transonic flow conditions. In this work we use the RANS solver provided by elsA for the steady rigid and aeroelastic analyses [@cambier:hal-01293795; @cambier_elsA_2014]. The elsA adjoint and coupled-adjoint solvers have also been used for the computation of rigid and flexible derivatives. The latest improvements to the Krylov solvers for the solution of the adjoint linear system are described in [@jadoui2022comparative]. A multi-block structured mesh featuring a C-H topology is used (Fig. [4](#fig:Meshes){reference-type="ref" reference="fig:Meshes"}). It consists of 3.8 million grids divided into 42 blocks. The flight conditions are a free-stream Mach number of 0.84 at an angle of attack of 3.06 degrees. The convective fluxes are discretized by an upwind Roe scheme associated to a Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) reconstruction and a Van Albada flux limiter. The one-equation Spalart-Allmaras turbulence model is selected. The surface contours in the bottom plot of Fig. [6](#fig:Cp_contours){reference-type="ref" reference="fig:Cp_contours"} below show typical results for the ONERA-M6 wing. The pressure coefficient contours identify a lambda-shock along the mid-chord of the wing. For the aeroelastic analysis a simple but realistic finite element model has been designed (Fig. [4](#fig:Meshes){reference-type="ref" reference="fig:Meshes"}). The stiffness of this model can be easily tuned to get stronger or weaker fluid-structure interaction. The pressure coefficient contours at the aeroelastic equilibrium are plotted in the upper part of Fig. [6](#fig:Cp_contours){reference-type="ref" reference="fig:Cp_contours"} and can be compared to the rigid contours. The maximum vertical displacement is 0.14 meters corresponding to 11.7 % of the wing span. To get a better insight of the effect of flexibility on the pressure distribution, we report in Fig. [8](#fig:Cp_sections){reference-type="ref" reference="fig:Cp_sections"} the Cp distributions for two sections at $y=0.60$ m and $y=1.12$ m. The vertical displacement distributions associated to the front and rear spars as well as the twist increment distribution are plotted in Fig. [9](#fig:DispTwistAEL){reference-type="ref" reference="fig:DispTwistAEL"}. The rigid analysis results in a lift coefficient $C_L = 0.27$ whereas the aeroelastic analysis, at the same angle of attack, results in a lower lift coefficient $C_L = 0.23$. {#fig:Meshes width="49%"} {#fig:Meshes width="49%"} {#fig:Cp_contours width="70%"} {#fig:Cp_contours width="70%"} {#fig:Cp_sections width="49%"} {#fig:Cp_sections width="49%"} {#fig:DispTwistAEL width="60%"} # Minimal residual Krylov subspace methods combined with spectral deflation {#FGMRESDR_Review} In this section we focus on a particular minimal residual norm Krylov subspace method for the solution of linear systems with a non-symmetric real coefficient matrix of type $$\label{eq:Linear_System} \mathbf{Ax} = \mathbf{b}, \qquad \mathbf{A} \in \mathbb{R}^{N \times N}; \quad \mathbf{b}, \hspace{0.1cm} \mathbf{x} \hspace{0.1cm} \in \mathbb{R}^N,$$ with the initial guess $\mathbf{x}_0$ and the associated residual $\mathbf{r}_0 = \mathbf{b} - \mathbf{Ax}_0$. The GMRES method [@saad1986] computes the correction $\mathbf{z}_i$ in the $i$th Krylov subspace $\mathcal{K}_i(\mathbf{A},\mathbf{r}_0) \equiv \operatorname{span}\{\mathbf{r}_0,\mathbf{Ar}_0,\mathbf{A}^2\mathbf{r}_0,\cdots,\mathbf{A}^{i-1}\mathbf{r}_0\}$ that minimizes the norm of the residual $\mathbf{r}_i = \mathbf{b} - \mathbf{A}(\mathbf{x}_0+\mathbf{z}_i) = \mathbf{r}_0 - \mathbf{Az}_i$. The relation between the minimal residual correction $\mathbf{z}_i$ and the orthogonality of the new residual $\mathbf{r}_i$ to the *shifted Krylov space* $\mathcal{AK}_i(\mathbf{A},\mathbf{r}_0) \equiv \operatorname{span}\{\mathbf{Ar}_0,\mathbf{A}^2\mathbf{r}_0,\cdots,\mathbf{A}^{i}\mathbf{r}_0\}$ is given by the following theorem [@saad2003]: **Theorem 1**. *The vector $\mathbf{z}_i \in \mathcal{K}_i(\mathbf{A},\mathbf{r}_0)$ satisfies $\mathbf{z}_i = \underset{\mathbf{z} \in \mathcal{K}_i(\mathbf{A},\mathbf{r}_0)}{\operatorname{argmin}} \| \mathbf{r}_0 - \mathbf{Az} \|_2 \Leftrightarrow \mathbf{r}_i \; \bot \; \mathcal{AK}_i(\mathbf{A},\mathbf{r}_0)$* This is known as the optimality property of the residual. In this work, a right-preconditioned system is considered so that ([\[eq:Linear_System\]](#eq:Linear_System){reference-type="ref" reference="eq:Linear_System"}) becomes $$\begin{aligned} \label{eq:Variable_Precond_System} \mathbf{A}\mathcal{M}(\mathbf{t}) = \mathbf{b}, \\ \mathbf{x} = \mathcal{M}(\mathbf{t})\end{aligned}$$ with $\mathbf{t} \in \mathbb{R}^{N}$ and $\mathcal{M}:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ the preconditioning operator which may be a nonlinear function. ## Flexible GMRES algorithm with right preconditioning Saad proposed a minimal residual norm subspace method based on the standard GMRES approach [@saad1986] that allows a variable nonlinear preconditioning function $\mathcal{M}_{j}:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ at each iteration $j$ [@saad1993flexible]. Starting from an initial guess $\mathbf{x}_{0}$, the flexible Arnoldi relation is written as: $$\label{eq:Flexible_Arnoldi} \mathbf{AZ}_{m} = \mathbf{V}_{m+1}\Bar{\mathbf{H}}_{m},$$ where the matrices $\mathbf{V}_{m+1} \in \mathbb{R}^{N \times (m+1)}, \mathbf{Z}_{m} \in \mathbb{R}^{N \times m}$ and $\bar{\mathbf{H}}_{m} \in \mathbb{R}^{(m+1) \times m}$ stand for the orthonormal basis of the Krylov space, the solution space and the upper Hessenberg matrix respectively. The approximate solution is written as $\mathbf{x}_{m} = \mathbf{x}_{0} + \mathbf{Z}_{m}\mathbf{y}_{m}$ where $\mathbf{y}_{m}$ minimizes $|| \mathbf{r}_{0} -\mathbf{AZ}_{m}\mathbf{y} ||_{2}$ over $\mathbf{x}_{0} + \operatorname{span}\lbrace \mathbf{Z}_{m} \rbrace$, with $\mathbf{Z}_{m}=\mathbfcal{M}(\mathbf{V}_{m})=\left[\mathcal{M}_{1}(\mathbf{v}_{1}),\cdots,\mathcal{M}_{m}(\mathbf{v}_{m}) \right]$ and both $\mathbf{Z}_{m}$ and $\mathbf{V}_{m}$ need to be stored. We point out that the operator $\mathbfcal{M}$ represents the action of the nonlinear operators $\mathcal{M}_{j}$ on the set of basis vectors $\mathbf{v}_{j}$. The restarted FGMRES($m$, $m_{i}$) pseudocode is presented in Algorithm [\[alg:FGMRES\]](#alg:FGMRES){reference-type="ref" reference="alg:FGMRES"}. We denote by $m_{i}$ the size of the Krylov subspace associated to the GMRES solver devoted to the inner linear system. We point out that the stopping criterion is essentially based on the least-squares relative residual $||\mathbf{c} - \bar{\mathbf{H}}_{m}\mathbf{y}_{m}||/||\mathbf{b}||$, with $\mathbf{c}=\|\mathbf{r}_0\|\mathbf{e}_1$ (see [@fraysse2008algorithm] for several definitions of stopping criteria and some practical considerations for the implementation of the key points of the algorithm). The latter is a cheap approximation of the true residual $\| \mathbf{Ax}-\mathbf{b} \|/\|\mathbf{b}\|$, assuming $\mathbf{V}_{m+1}$ is orthonormal. In an inner-outer FGMRES the preconditioning operation $\mathbf{z}_{j} = \mathcal{M}_{j}(\mathbf{v}_{j})$ of step [\[alg:innerGMRES\]](#alg:innerGMRES){reference-type="ref" reference="alg:innerGMRES"} can be thought of as a means of approximately solving $\mathbf{Az}_j=\mathbf{v}_j$ where $\mathbf{M}^{-1}_j \approx \mathbf{A}^{-1}$ is the inner preconditioner. To prevent unnecessary propagation of rounding errors, the relative true residual is only computed at the end of each cycle and is used to construct the first vector of the next Krylov subspace basis. In the case of large and ill-conditioned linear systems, least-squares and true residuals may differ due to loss of orthogonality during the construction of the Krylov basis. A standard way to tackle such a phenomenon is to ask for a second iteration of the Modified Gram-Schmidt algorithm (loop from line 6 to 9) in order to strengthen the orthogonality of the Krylov basis. In this work, we consider the standard GMRES with stationary right-preconditioning on one hand, and the specific class of flexible nested GMRES strategy on the other hand where an inner GMRES acts as an iterative preconditioner. Two stationary preconditioning strategies are considered. The first one consists in a block version of a standard LU-SGS iterative solver. LU-SGS is applied to a first order diagonally dominant upwind approximation of the flux Jacobian matrix inspired by [@yoon1988]. This operator is based on a first order spatial discretization of the convective and of the viscous fluxes using a simplifying thin layer assumption [@peter2007large]. This strategy leads to a very compact stencil for the preconditioning matrix which will be denoted by $\textbf{J}^{APP}_{O1}$ in the sequel of this document. [\[alg:FGMRES\]]{#alg:FGMRES label="alg:FGMRES"} Choose an initial guess $\mathbf{x}_{0}$, a convergence threshold $\epsilon$ and krylov size $m$ Compute $\mathbf{r}_{0} = \mathbf{b}-\mathbf{Ax}_{0}$, $\beta =||\mathbf{r}_{0}||$, $\mathbf{c}=\beta\mathbf{e}_1$ and $\mathbf{v}_{1} = \mathbf{r}_{0}/\beta$ $\mathbf{z}_{j} = \mathcal{M}_{j}(\mathbf{v}_{j})$ [\[alg:innerGMRES\]]{#alg:innerGMRES label="alg:innerGMRES"} $\mathbf{w} = \mathbf{Az}_{j}$ [\[alg:matvecprod\]]{#alg:matvecprod label="alg:matvecprod"} $h_{i,j} = \mathbf{v}_{i}^{T}\mathbf{w}$ [\[alg:scalprod\]]{#alg:scalprod label="alg:scalprod"} $\mathbf{w} = \mathbf{w} -h_{i,j}\mathbf{v}_{i}$ $h_{j+1,j} =||\mathbf{w}||_2$ and $\mathbf{v}_{j+1} = \mathbf{w}/h_{j+1,j}$ Solve the least-squares problem $min_{\mathbf{y}} ||\mathbf{c} - \bar{\mathbf{H}}_{j}\mathbf{y}||_2$ for $\mathbf{y}_{j}$ Exit if $||\mathbf{c} - \bar{\mathbf{H}}_{j}\mathbf{y}_{j}||/||\mathbf{b}|| \le \epsilon$ Compute $\mathbf{x}_{m} = \mathbf{x}_{0} + \mathbf{Z}_{m}\mathbf{y}_{m}$ where $\mathbf{Z}_{m} = [\mathbf{z}_{1},...,\mathbf{z}_{m}]$ Set $\mathbf{x}_{0} = \mathbf{x}_{m}$ and go to 2 The second one is a Block Incomplete LU (BILU($k$)) factorization applied to either an approximate or exact first-order flux Jacobian matrix. For the so-called first-order exact Jacobian matrix $\textbf{J}^{EXA}_{O1}$ a first-order spatial Roe scheme is used for the discretization of the mean-flow convective fluxes and a 5-point corrected centered discretization scheme is used for the diffusive fluxes. More specifically, the spatial gradients at the cell interfaces are modified to avoid high frequency oscillations (see [@resmini2015] or [@puigt2014]). The BILU($k$) preconditioner will be applied either to the first-order approximate Jacobian matrix $\textbf{J}^{APP}_{O1}$, or to the first-order exact Jacobian matrix $\textbf{J}^{EXA}_{O1}$. $\textbf{J}^{EXA}_{O1}$ is different from $\textbf{J}^{APP}_{O1}$ when it comes to memory footprint. Indeed, $\textbf{J}^{EXA}_{O1}$ has a 9-point stencil in 2D whereas a 5-point stencil is associated with $\textbf{J}^{APP}_{O1}$. In 3D, we have a 7-point stencil for $\textbf{J}^{APP}_{O1}$ and a stencil of 19 points for $\textbf{J}^{EXA}_{O1}$. Consequently, a better robustness is achieved but at the price of about twice the storage for $\textbf{J}^{EXA}_{O1}$ compared to $\textbf{J}^{APP}_{O1}$. For a better understanding, we reproduce the 3D stencils in Fig. [12](#fig:stencils){reference-type="ref" reference="fig:stencils"} below: {#fig:stencils width="0.9\\linewidth"} a) Exact $2^{nd}$ order (size 25) [\[fig:stencil_JO2\]]{#fig:stencil_JO2 label="fig:stencil_JO2"} {#fig:stencils width="0.85\\linewidth"} b) Exact $1^{st}$ order (size 19) [\[fig:Stencil_JO1_exa\]]{#fig:Stencil_JO1_exa label="fig:Stencil_JO1_exa"} {#fig:stencils width="0.70\\linewidth"} c) Approx. $1^{st}$ order (size 7) [\[fig:Stencil_JO1_app\]]{#fig:Stencil_JO1_app label="fig:Stencil_JO1_app"} We point out that the relevant numerical ingredients that characterize the GMRES algorithm are the matrix product (step [\[alg:matvecprod\]](#alg:matvecprod){reference-type="ref" reference="alg:matvecprod"} in algorithm [\[alg:FGMRES\]](#alg:FGMRES){reference-type="ref" reference="alg:FGMRES"}), the preconditioning strategy (step [\[alg:innerGMRES\]](#alg:innerGMRES){reference-type="ref" reference="alg:innerGMRES"}) and the scalar product (step [\[alg:scalprod\]](#alg:scalprod){reference-type="ref" reference="alg:scalprod"}). These algebraic operations are global in conjunction with a domain decomposition method. More specifically, the globalization of the preconditioner (step [\[alg:innerGMRES\]](#alg:innerGMRES){reference-type="ref" reference="alg:innerGMRES"}) is achieved with a Restricted Additive Schwarz method [@cai1999]. In addition, the product by the operator $\mathbf{A}$ (step [\[alg:matvecprod\]](#alg:matvecprod){reference-type="ref" reference="alg:matvecprod"}) is exact. We thus get a global and parallel FGMRES($m$,$m_{i}$). ## Deflated restarting {#DeflatedRestarting_FGMRESDR} The main drawback of the restarted GMRES($m$) and FGMRES($m$, $m_i$) is the loss of spectral information contained in the current Krylov subspace during the restarting procedure. Let us recall the definition of a Ritz pair [@morgan1998] as it plays an important role in the strategy of deflated restarting. [\[Ritzpair\]]{#Ritzpair label="Ritzpair"} Consider a subspace $\mathcal{U}$ of $\mathbb{R}^N$. Given an operator $\mathbf{B} \in \mathbb{R}^{N \times N}$, a scalar $\lambda \in \mathbb{R}$ and $\mathbf{y} \in \mathbb{R}^N$, ($\lambda$, $\mathbf{y}$) is a Ritz pair of $\mathbf{B}$ with respect to $\mathcal{U}$ if and only if the residual of the eigenvalue problem $\mathbf{By} = \lambda \mathbf{y}$ satisfies the following orthogonality condition: $$\label{eq:Eigenvalue_equation} (\mathbf{By} - \lambda \mathbf{y}) \perp \mathcal{U} \quad \forall \mathbf{y} \in \mathcal{U}.$$ Choosing $\mathcal{U} \equiv \mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0})$ and $\mathbf{B} \equiv \mathbf{A}$, we have $$(\mathbf{Ay} -\lambda \mathbf{y}) \perp \mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0}) \quad \forall \mathbf{y} \in \mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0}). \label{eq:Ritz_problem}$$ Recalling $\mathbf{V}_{m}$ is an orthonormal basis of $\mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0})$, we can write $\mathbf{y}=\mathbf{V}_{m}\mathbf{g}$, $\mathbf{g} \in \mathbb{R}^m$, and using the standard Arnoldi relation $\mathbf{AV}_{m} = \mathbf{V}_{m+1}\bar{\mathbf{H}}_{m}$ in ([\[eq:Ritz_problem\]](#eq:Ritz_problem){reference-type="ref" reference="eq:Ritz_problem"}), we obtain the standard eigenvalue problem $$\mathbf{H}_{m}\mathbf{g} = \lambda \mathbf{g}, \label{eq:Ritz_spectral}$$ where $\mathbf{H}_{m} = \left[ \mathbf{I}_m \ \mathbf{0}_{m \times 1} \right] \bar{\mathbf{H}}_{m}$. Thus, the spectral residual norm of the Ritz pair $\left\{\lambda,\mathbf{y}=\mathbf{V}_{m}\mathbf{g}\right\}$ satisfies: $$\label{eq:Residual_eigenvalue_norm} \| \mathbf{A}(\mathbf{V}_{m}\mathbf{g}) - \lambda (\mathbf{V}_{m}\mathbf{g}) \| = \| \mathbf{A}\mathbf{V}_{m}\mathbf{g}-\mathbf{V}_{m}\mathbf{H}_{m}\mathbf{g}\| = \| \mathbf{V}_{m+1}\Bar{\mathbf{H}}_{m}\mathbf{g} - \mathbf{V}_{m}\mathbf{H}_{m}\mathbf{g} \| = | h_{m+1,m} | |\mathbf{e}_{m}^{T}\mathbf{g}|.$$ From a small value of $|h_{m+1,m}| |\mathbf{e}_{m}^{T}\mathbf{g}|$, the Arnoldi method takes the Ritz pair as a good approximation of the eigenvalue-eigenvector pair of the operator $\mathbf{A}$ [@arnoldi1951; @saad1980]. Indeed, neglecting the last row of the rectangular upper Hessenberg matrix $\Bar{\mathbf{H}}_{m}$ leads to: $$\label{eq:Hessenberg_Restriction_operator} \mathbf{H}_{m} = \mathbf{V}_{m}^{T}\mathbf{AV}_{m}.$$ Therefore, the spectrum of $\mathbf{H}_{m}$ naturally approximates a part of the spectrum of $\mathbf{A}$. The idea of deflation techniques is to keep relevant spectral information from $\mathbf{H}_{m}$ in the search space of the next cycle to expect a better convergence of the Krylov iterative methods. In [@giraud2010], Giraud et al. take into account both smallest and largest eigenvalues to maximize the deflation effect. In contrast, Morgan [@morgan2002gmres] only deflates the smallest ones. This last strategy will be adopted for our numerical experiments. Actually, Ritz values of the operator $\mathbf{A}$ give a good approximation of its exterior eigenvalues, i.e., of largest magnitude. Unfortunately, interior eigenvalues are of greater interest because they are generally responsible for the convergence stagnation. An alternative which does better at finding eigenvalues nearest zero was proposed by Morgan [@morgan1991computing; @morgan1995restarted] who introduced the harmonic Ritz values of $\mathbf{A}$ with respect to $\mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0})$, which are equivalent to the Ritz values of $\mathbf{A}^{-1}$ with respect to $\mathcal{AK}_{m}(\mathbf{A},\mathbf{r}_{0})$. In practice we will use the following definition [\[HarmonicRitzpair\]]{#HarmonicRitzpair label="HarmonicRitzpair"} Consider a subspace $\mathcal{U}$ of $\mathbb{R}^N$. Given an operator $\mathbf{B} \in \mathbb{R}^{N \times N}$, a scalar $\lambda \in \mathbb{R}$ and $\mathbf{y} \in \mathbb{R}^N$, ($\lambda$, $\mathbf{y}$) is a harmonic Ritz pair of $\mathbf{B}$ with respect to $\mathcal{U}$ if and only if the residual of the eigenvalue problem $\mathbf{By} = \lambda \mathbf{y}$ satisfies the following orthogonality condition: $$\label{eq:Eigenvalue_equation} (\mathbf{By} - \lambda \mathbf{y}) \perp \mathbf{B}\mathcal{U} \quad \forall \mathbf{y} \in \mathcal{U}.$$ Choosing $\mathcal{U} \equiv \mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0})$ and $\mathbf{B} \equiv \mathbf{A}$, we have, $$(\mathbf{Ay} -\lambda \mathbf{y}) \perp \mathcal{AK}_{m}(\mathbf{A},\mathbf{r}_{0}) \quad \forall \mathbf{y} \in \mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0}). \label{eq:Harmonic_Ritz_problem}$$ Recalling $\mathbf{V}_{m}$ is an orthonormal basis of $\mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0})$, we can write $\mathbf{y}=\mathbf{V}_{m}\mathbf{g}$, $\mathbf{g} \in \mathbb{R}^m$, and using the flexible Arnoldi relation [\[eq:Flexible_Arnoldi\]](#eq:Flexible_Arnoldi){reference-type="eqref" reference="eq:Flexible_Arnoldi"} in ([\[eq:Harmonic_Ritz_problem\]](#eq:Harmonic_Ritz_problem){reference-type="ref" reference="eq:Harmonic_Ritz_problem"}), we obtain the following generalized eigenvalue problem $$\begin{aligned} & \label{eq:Generalized_eigenvalue_problem_a} (\mathbf{AZ}_{m})^T (\mathbf{AZ}_{m} \mathbf{g} - \lambda \mathbf{V}_{m} \mathbf{g}) = \mathbf{0} \\ \Leftrightarrow \quad & \Bar{\mathbf{H}}_{m}^{T} \Bar{\mathbf{H}}_{m}\mathbf{g} = \lambda \bar{\mathbf{H}}^T_{m} \mathbf{V}^T_{m+1} \mathbf{V}_m \mathbf{g} \nonumber \\ % \Leftrightarrow \quad & \Bar{\mathbf{H}}_{m}^{T} \Bar{\mathbf{H}}_{m}\mathbf{g} = \lambda \bar{\mathbf{H}}^T_{m} \left[ \mathbf{I}_m \ \mathbf{0}_{m \times 1} \right]^T \mathbf{g} \nonumber \\ \label{eq:Generalized_eigenvalue_problem_b} \Leftrightarrow \quad & \boxed{\Bar{\mathbf{H}}_{m}^{T} \Bar{\mathbf{H}}_{m}\mathbf{g} = \lambda \mathbf{H}_{m}^{T} \mathbf{g}}\end{aligned}$$ After some algebraic manipulations, ([\[eq:Generalized_eigenvalue_problem_b\]](#eq:Generalized_eigenvalue_problem_b){reference-type="ref" reference="eq:Generalized_eigenvalue_problem_b"}) can be reformulated as a standard eigenvalue problem (see [@morgan1998] for this formula): $$\label{eq:Standard_eigenvalue_problem} (\mathbf{H}_{m} + h_{m+1,m}^{2}\mathbf{H}_{m}^{-T}\mathbf{e}_{m}\mathbf{e}_{m}^{T})\mathbf{g} = \lambda \mathbf{g},$$ where $\lambda$ is a harmonic Ritz value and the corresponding harmonic Ritz vector is $\mathbf{y}=\mathbf{V}_{m}\mathbf{g}$. To follow definition [\[HarmonicRitzpair\]](#HarmonicRitzpair){reference-type="ref" reference="HarmonicRitzpair"}, we use the identity $\mathbf{V}_m^T \mathbf{V}_m = \mathbf{I}$ into equation [\[eq:Generalized_eigenvalue_problem_a\]](#eq:Generalized_eigenvalue_problem_a){reference-type="eqref" reference="eq:Generalized_eigenvalue_problem_a"} which gives $$((\mathbf{AZ}_{m}\mathbf{V}_m^T) \mathbf{V}_m)^T ((\mathbf{AZ}_{m}\mathbf{V}_m^T) \mathbf{V}_m \mathbf{g} - \lambda \mathbf{V}_{m} \mathbf{g}) = \mathbf{0}.$$ Thus, $\mathbf{Y}_m = \{ \mathbf{y}_1, \cdots, \mathbf{y}_m \}$ corresponds to harmonic Ritz vectors of $\mathbf{AZ}_{m}\mathbf{V}_m^T$ with respect to $\operatorname{range}(\mathbf{V}_m)$. To underline the link with FGCRO-DR in section [6](#FGCRODR_theory){reference-type="ref" reference="FGCRODR_theory"}, we denote by *strategy B* this deflation strategy. Obviously, in exact arithmetic solutions to ([\[eq:Generalized_eigenvalue_problem_b\]](#eq:Generalized_eigenvalue_problem_b){reference-type="ref" reference="eq:Generalized_eigenvalue_problem_b"}) and ([\[eq:Standard_eigenvalue_problem\]](#eq:Standard_eigenvalue_problem){reference-type="ref" reference="eq:Standard_eigenvalue_problem"}) are identical. But it is not the case in finite precision since the operator $\Bar{\mathbf{H}}_{m}^{T} \Bar{\mathbf{H}}_{m}$ is usually ill-conditioned in the fully linearized turbulence case. Therefore, the accurate estimation of the eigenvectors could be strongly altered and lead to stagnation of the relative true residual of the GMRES process. As mentioned by Giraud et al. [@giraud2010], the flexible Arnoldi relation obtained at each restart within the FGMRES with deflated restarting (FGMRES-DR) framework given by $$\label{eq:Flexible_Arnoldi_Realtion_Deflation} \mathbf{AZ}_{k} = \mathbf{V}_{k+1}\bar{\mathbf{H}}_{k},$$ holds with $\mathbf{Z}_{k} = \mathbf{Z}_{m}\mathbf{P}_{k}$, $\mathbf{V}_{k+1} = \mathbf{V}_{m+1}\mathbf{P}_{k+1}$ and $\bar{\mathbf{H}}_{k} = \mathbf{P}_{k+1}\bar{\mathbf{H}}_{m}\mathbf{P}_{k}$ where $\mathbf{P}_{k} \in \mathbb{R}^{m \times k}$ corresponds to the orthonormal matrix whose columns are spanned by the set of the $k$ retained eigenvectors of ([\[eq:Standard_eigenvalue_problem\]](#eq:Standard_eigenvalue_problem){reference-type="ref" reference="eq:Standard_eigenvalue_problem"}). Also, the right-hand side of the least-squares problem is computed at each restart as $\mathbf{c} = \mathbf{V}_{m+1}^{T}\mathbf{r}_{m}$ which requires $(2N-1)(m+1)$ operations. Rollins and Fichtner [@rollin2008] have proposed an efficient way to compute $\mathbf{c}$ so that we can save some inner products. Indeed, they notice that the residual $\mathbf{r}_{m}$ is a linear combination of the columns of the deflation subspace $\mathbf{V}_{k+1}$. Consequently, we compute $\mathbf{c} = [ (\mathbf{r}_{m}^T \mathbf{V}_{k+1})^T \quad \mathbf{0}_{1 \times (m-k)} ]^T$ with $(2N-1)(k+1)$ operations only. Also, they improved the construction of $\mathbf{P}_{k+1} \in \mathbb{R}^{(m+1)\times(k+1)}$ and in particular the last column of this matrix which is usually chosen as the vector $\mathbf{c} - \bar{\mathbf{H}}_{m}\mathbf{y}_{m}$. More specifically, they demonstrated that the vector $\mathbf{c} - \bar{\mathbf{H}}_{m}\mathbf{y}_{m}$ is colinear to the vector $[-\delta \mathbf{f}^{T} \; 1]^{T}$ with $\mathbf{f}=\mathbf{H}^{-T}_{m}\mathbf{e}_{m}$ and $\delta=h_{m+1,m}$. The explicit relation is given below: $$\label{eq:Rollin_vector} \mathbf{c} - \bar{\mathbf{H}}_{m}\mathbf{y}_{m} = \begin{pmatrix} -\delta \mathbf{f} \\ 1 \end{pmatrix} \left( \frac{\omega - \delta \mathbf{f}^{T}\mathbf{v}}{1+\delta^{2}\mathbf{f}^{T}\mathbf{f}} \right),$$ where $\mathbf{v}$ and $\omega$ are the first $m$ rows of $\mathbf{c}$ and the last element of $\mathbf{c}$ respectively. This formulation reduces the propagation of rounding errors in the Arnoldi relation at restart. Choose $\mathbf{x}_{0}$, $m$, $m_i$, $k$, $tol$. Let $\mathbf{r}_{0} = \mathbf{b}-\mathbf{Ax}_{0}$, $\beta=\|\mathbf{r}_0\|$, $\mathbf{v}_1=\mathbf{r}_0/\beta$ and $\mathbf{c}=\beta\mathbf{e}_1 \in \mathbb{R}^{m+1}$ Apply a full cycle of FGMRES($m$,$m_{i}$) (algorithm [\[alg:FGMRES\]](#alg:FGMRES){reference-type="ref" reference="alg:FGMRES"}) to build $\mathbf{V}_{m+1}$, $\mathbf{Z}_{m}$, $\bar{\mathbf{H}}_{m}$. Compute $\mathbf{x}_m = \mathbf{x}_0 + \mathbf{V}_m \mathbf{y}_m$, $\rho_m = \| \mathbf{c}-\bar{\mathbf{H}}_m \mathbf{y}_m \|/\|\mathbf{b}\|$, where $\mathbf{y}_m = {\operatorname{argmin}}_{\mathbf{y}} \| \mathbf{c}-\bar{\mathbf{H}}_m \mathbf{y} \|$ Set $\mathbf{x}_0 = \mathbf{x}_m$ Solve $\mathbf{f}=\mathbf{H}^{-T}_{m}\mathbf{e}_{m}$. Compute the first $k$ eigenvectors $\mathbf{p}_{i}$ of $\mathbf{H}_{m} + h^{2}_{m+1,m}\mathbf{f} \, \mathbf{e}_{m}^{T}$ associated to smallest eigenvalues $\theta_i$. Form $\mathbf{P}_{k} = \left[\mathbf{p}_1,\cdots,\mathbf{p}_k \right]$ and $\mathbf{P}_{k+1} = \left[ \begin{bmatrix} \mathbf{P}_k \\\mathbf{0}_{1 \times k} \end{bmatrix}, \begin{bmatrix} -\beta \mathbf{f} \\ 1 \end{bmatrix} \right ]$ Obtain the reduced QR decomposition of $\mathbf{P}_{k+1} = \bar{\mathbf{P}}_{k+1} \Gamma_{k+1}$, $\bar{\mathbf{P}}_{k+1} \in \mathbb{R}^{(m+1) \times (k+1)}$ Form $\mathbf{V}_{k+1}^{new} = \mathbf{V}_{m+1}\Bar{\mathbf{P}}_{k+1}$, $\mathbf{Z}^{new}_{k} = \mathbf{Z}_{m}\Bar{\mathbf{P}}_{k}$ and $\Bar{\mathbf{H}}_{k}^{new} = \Bar{\mathbf{P}}_{k+1}^{T}\Bar{\mathbf{H}}_{m}\Bar{\mathbf{P}}_{k}$. [\[alg:step1\]]{#alg:step1 label="alg:step1"} Let $\Bar{\mathbf{H}}_{k} = \Bar{\mathbf{H}}_{k}^{new}$, $\mathbf{V}_{k+1} = \mathbf{V}_{k+1}^{new}$ and $\mathbf{Z}_{k} =\mathbf{Z}^{new}_{k}$. [\[alg:step2\]]{#alg:step2 label="alg:step2"} Apply ($m$-$k$) steps of the Arnoldi process from $\mathbf{V}_{k+1}$, $\mathbf{Z}_{k}$, and $\Bar{\mathbf{H}}_{k}$ to build $\mathbf{V}_{m+1}$, $\mathbf{Z}_{m}$ and $\Bar{\mathbf{H}}_{m}$. Set $\mathbf{c} = \mathbf{V}^{T}_{k+1}\mathbf{r}_{0}$, with $\mathbf{r}_{0} = \mathbf{b} - \mathbf{Ax}_0$. Solve the least-squares problem $\mathbf{y}_m = {\operatorname{argmin}}_{\mathbf{y}} \| \mathbf{c}-\bar{\mathbf{H}}_m \mathbf{y} \|$ Compute $\mathbf{x}_{m} = \mathbf{x}_{0} + \mathbf{Z}_{m}\mathbf{y}_{m}$ and $\rho_m = \| \mathbf{c}-\bar{\mathbf{H}}_m \mathbf{y}_m \|/\|\mathbf{b}\|$ Set $\mathbf{x}_0 = \mathbf{x}_m$ [\[alg:FGMRES_DR\]]{#alg:FGMRES_DR label="alg:FGMRES_DR"} ## Alternative deflation strategy for FGMRES-DR {#StrategyA_FGMRESDR} From the expression of the solution $\mathbf{x}_{m} = \mathbf{x}_{0} + \mathbf{Z}_{m}\mathbf{y}_{m}$, where $\mathbf{Z}_{m} = [\mathbf{z}_{1},...,\mathbf{z}_{m}]$, it seems legitimate to write the Harmonic Ritz vectors as $\mathbf{y}=\mathbf{Z}_{m}\mathbf{g}$, $\mathbf{g} \in \mathbb{R}^m$. From [\[eq:Eigenvalue_equation\]](#eq:Eigenvalue_equation){reference-type="eqref" reference="eq:Eigenvalue_equation"} and [\[eq:Harmonic_Ritz_problem\]](#eq:Harmonic_Ritz_problem){reference-type="eqref" reference="eq:Harmonic_Ritz_problem"} we have $$\begin{aligned} &(\mathbf{Ay} -\lambda \mathbf{y}) \perp \operatorname{span}(\mathbf{AZ}_{m}) \quad \forall \mathbf{y} \in \operatorname{span}(\mathbf{Z}_{m}) \nonumber \\ \Leftrightarrow \quad & (\mathbf{AZ}_{m})^T (\mathbf{AZ}_{m} \mathbf{g} - \lambda \mathbf{Z}_{m} \mathbf{g}) = \mathbf{0} \nonumber \\ \Leftrightarrow \quad & \boxed{\Bar{\mathbf{H}}_{m}^{T} \Bar{\mathbf{H}}_{m}\mathbf{g} = \lambda \bar{\mathbf{H}}^T_{m} \mathbf{V}^T_{m+1} \mathbf{Z}_m \mathbf{g}} \label{eq:Alternative_deflation_strategy_b}\end{aligned}$$ Thus, $\mathbf{Y}_m = \{ \mathbf{y}_1, \cdots, \mathbf{y}_m \}$ corresponds to harmonic Ritz vectors of $\mathbf{A}$ or equivalently of $\mathbf{AZ}_{m} \mathbf{Z}_m^{\dagger}$ with respect to $\operatorname{range}(\mathbf{Z}_m)$. Noticing that the last row of $\bar{\mathbf{H}}_m$ is null except the rightmost element $h_{m+1,m}$, and defining $\mathbf{f} = \mathbf{H}_m^{-T} \mathbf{e}_m$, we can reformulate [\[eq:Alternative_deflation_strategy_b\]](#eq:Alternative_deflation_strategy_b){reference-type="eqref" reference="eq:Alternative_deflation_strategy_b"} as $$\mathbf{g} = \lambda [ \mathbf{I}_m \quad h_{m+1,m}\mathbf{f} ] \mathbf{V}^T_{m+1} \mathbf{Z}_m \mathbf{g}.$$ The product $\mathbf{V}^T_{m+1} \mathbf{Z}_m$ brings an additional cost to the deflation process. A block form of this term can be found as $$\label{eq:FGMRESDR_block_deflation} \mathbf{V}^T_{m+1} \mathbf{Z}_m = \begin{bmatrix} \mathbf{V}^T_{k+1} \mathbf{Z}_k & \mathbf{V}^T_{k+1} \mathbf{Z}_{m-k} \\ \mathbf{V}^T_{m-k} \mathbf{Z}_k & \mathbf{V}^T_{m-k} \mathbf{Z}_{m-k} \end{bmatrix}\!.$$ Thanks to steps [\[alg:step1\]](#alg:step1){reference-type="ref" reference="alg:step1"} and [\[alg:step2\]](#alg:step2){reference-type="ref" reference="alg:step2"} in algorithm [\[alg:FGMRES_DR\]](#alg:FGMRES_DR){reference-type="ref" reference="alg:FGMRES_DR"} the leading block $\mathbf{V}^T_{k+1} \mathbf{Z}_k$ can be written as $$(\mathbf{V}^T_{k+1} \mathbf{Z}_k)^{(i)} = \Bar{\mathbf{P}}_{k+1}^{T} (\mathbf{V}^T_{m+1} \mathbf{Z}_m)^{(i-1)} \Bar{\mathbf{P}}_{k},$$ where the superscript is related to the cycle index. Thus storing $\mathbf{V}^T_{m+1} \mathbf{Z}_m$ at the end of each cycle allows us to compute at a cheap cost the $(k+1)\times k$ block $\mathbf{V}^T_{k+1} \mathbf{Z}_k$ for the next cycle. To underline the link with FGCRO-DR in section [6](#FGCRODR_theory){reference-type="ref" reference="FGCRODR_theory"}, we denote by *strategy A* this formulation. ## Application of GMRES-DR and FGMRES-DR to the solution of the fluid adjoint system {#GMRESDR_Fluid_Adjoint_System} In this section we apply the GMRES-DR Krylov solver and its flexible variant (variable preconditioning), to the solution of the fluid adjoint system of the M6 wing. We recall that the discretization of the fluid RANS equations relies on an upwind Roe scheme associated to a Van Albada limiter and that the Spalart Allmaras turbulence model is fully linearized. Various stationary preconditioners are considered. First, we consider the legacy RAS+LU-SGS(nrelax, CFL), with nrelax = 6 and CFL = 100 selected as the best options, applied to the first-order approximate Jacobian operator. Then, the BILU(k) decomposition type preconditioners are used. More specifically, BILU(0) is applied to the first-order approximate flux Jacobian operators (approximate and exact) and BILU(1) is applied to the first-order approximate Jacobian matrix only. We start with a GMRES($m$, $k$) solver with a krylov basis of size $m$ = 120 and a deflation subspace of size $k$ = 40. Figure [13](#fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_cvg_erratic){reference-type="ref" reference="fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_cvg_erratic"} plots the convergence of the relative least-squares residual for the above preconditioning strategies. The least-squares residuals all converge up to a prescribed threshold of $10^{-8}$. However, below $10^{-6}$ some erratic behavior happens for the BILU preconditioner applied to the first-order approximate Jacobian operator. The square symbols correspond to the true residuals computed at the end of each GMRES cycle. Except for the preconditioning strategy based on the BILU(0) factorization of the first-order Jacobian matrix, some residual stagnation is observed. This is explained by the propagation of rounding errors in the Arnoldi relation during the restart procedure. A simple remedy for this is to perform a cold restart when the true and least-squares residuals differ too much. In this work, we set this criterion at $\epsilon$ = 5% of relative discrepancy: $\epsilon= | \mathbf{r}_{true} - \mathbf{r}_{leastsq} | / \mathbf{r}_{true}$ = 0.05. Convergence improvements are clearly illustrated in Figure [14](#fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_resratio){reference-type="ref" reference="fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_resratio"} where the true and least-squares residuals now match accurately. Also, it is not surprising to see a plateau appear when restarting. The second set of numerical experiments consisted in the solution of the adjoint linear system but with a nested GMRES strategy where the inner GMRES is right preconditioned by the same variants of stationary preconditioners. Figure [16](#fig:cvg_AOC_FGMRESDR_70_10_35_SAlin_impact_preconditioning){reference-type="ref" reference="fig:cvg_AOC_FGMRESDR_70_10_35_SAlin_impact_preconditioning"} shows the corresponding convergence histories of the relative and least-squares residuals in terms of number of iterations and matrix-vector products. The FGMRES-DR($m$, $m_i$, $k$) solver is applied with a size of Krylov basis $m$ = 70, a size $m_i$ = 10 of the inner Krylov basis and a size $k$ = 40 for the deflation subspace. This time, the true and least-squares residuals always match which confirms the superior numerical stability of the flexible Krylov solver. For both standard and flexible GMRES-DR, the best convergence rate is obtained using the BILU(0) factorization of the exact first-order Jacobian matrix. FGMRES-DR reaches the tolerance threshold in 175 iterations compared to 1500 for GMRES-DR. However, the number of matrix-vector products is lower for the latter (1500 compared to 1920), illustrating a higher computational cost associated to FGMRES-DR for this specific numerical test. {#fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_cvg_erratic width="99%"} {#fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_resratio width="99%"} {#fig:cvg_AOC_FGMRESDR_70_10_35_SAlin_impact_preconditioning width="99%"} {#fig:cvg_AOC_FGMRESDR_70_10_35_SAlin_impact_preconditioning width="99%"} ## Application of GMRES-DR and FGMRES-DR to the solution of the fluid-structure coupled-adjoint system We are now interested in the application of GMRES-DR and FGMRES-DR to the solution of the coupled-adjoint system [\[Coupled_Adjoint_system\]](#Coupled_Adjoint_system){reference-type="ref" reference="Coupled_Adjoint_system"}. The solution process is presented in algorithm [\[alg:LBGS-algorithm\]](#alg:LBGS-algorithm){reference-type="ref" reference="alg:LBGS-algorithm"}. The fluid-structure coupling is triggered when the ratio $\rho = \mathbf{r}_i / \mathbf{r}_{i-1}$ between the residual $\mathbf{r}_i$ at the end of the current cycle and the residual $\mathbf{r}_{i-1}$ at the end of the previous cycle is less than a prescribed threshold. Relaxation of the structural adjoint vector (parameter $\theta_s$ at step [\[alg:Relaxation_step\]](#alg:Relaxation_step){reference-type="ref" reference="alg:Relaxation_step"} in algorithm [\[alg:LBGS-algorithm\]](#alg:LBGS-algorithm){reference-type="ref" reference="alg:LBGS-algorithm"}) can be activated to stabilize the convergence of the partitioned solver. A popular choice is to adapt $\theta_s$ dynamically by using an autoregressive predictor such as the well-known Aitken strategy [@irons1969version; @kuttler2008fixed]. In order to select appropriate values for $\rho$ ans $\theta_s$, we carried out a parameter study with $\rho \in [0.4,0.5,0.6,0.7]$ in combination with a constant relaxation parameter $\theta_s=1.0$ on one hand, and a dynamic relaxation parameter $\theta_s^{*}$, with $\theta_s^0=1.0$, on the other hand. The associated convergence curves are presented in Figure [17](#fig:cvg_AOC_GMRESDR_120_40_relaxation_parameter_study){reference-type="ref" reference="fig:cvg_AOC_GMRESDR_120_40_relaxation_parameter_study"}. For a constant relaxation parameter, the best convergence rate is achieved for a residual ratio $\rho=0.6$ (plain black line). The impact of dynamic relaxation (dash-dotted lines) is systematically negative except for $\rho=0.7$ where the efficiency of the combination of $\rho=0.6$ with $\theta_s=1.0$ is retrieved. In light of these results, it seems more appropriate to adopt a constant parameter relaxation strategy for our specific test case. However, higher-order Aitken techniques could be applied to better capture the nonlinear profile of the convergence, but this is out of the scope of this paper. Consequently, in the remaining of this document we will select $\rho=0.6$ and $\theta_s=1.0$ for the partitioned solver. {#fig:cvg_AOC_GMRESDR_120_40_relaxation_parameter_study width="60%"} In Figure [18](#fig:cvg_AOC_GMRESDR_120_40_SAlin_impact_preconditioning_matvecprod){reference-type="ref" reference="fig:cvg_AOC_GMRESDR_120_40_SAlin_impact_preconditioning_matvecprod"} we plot the convergence of the fluid relative least-squares residual for GMRES-DR(120,40) associated to several preconditioners. The preconditioners RAS+LUSGS(6,100) and BILU($1$) both applied to the approximate first-order Jacobian operator perform similarly. As before, BILU(0) applied to the exact first-order Jacobian shows the best efficiency. Contrary to the purely fluid case, the cycles of the coupled solver are often shorter which disadvantages the less efficient preconditioners. The same computation is repeated using a FGMRES-DR(70,10,35) Krylov solver. This time, BILU(1) applied to the approximate first-order Jacobian operator is not competitive and BILU(0) applied to the exact first-order Jacobian far outperforms other strategies. Again, the nested solver requires roughly 4 times less iterations than its non-flexible counterpart whereas the number of matrix-vector products is 10800 compared to 4400. {#fig:cvg_AOC_GMRESDR_120_40_SAlin_impact_preconditioning_matvecprod width="60%"} {#fig:cvg_AOC_FGMRESDR_70_10_35_SAlin_impact_preconditioning_iterations width="99%"} {#fig:cvg_AOC_FGMRESDR_70_10_35_SAlin_impact_preconditioning_iterations width="99%"} # Generalized Conjugate Residual with inner Orthogonalization and Deflated Restarting: GCRO-DR {#GCRODR_theory} As already explained, the current implementation of our partitioned solver does not take advantage of spectral information produced during the Krylov solver instances applied to the sequence of preceding right-hand sides. An advanced inner-outer GMRES-DR solver is used for the approximate solution of the fluid block between consecutive fluid-structure couplings but a simple restart is performed when the structural source term is updated. Unfortunately, deflated restarting based on subspace augmentation by appending approximate Ritz vectors to the Krylov subspace is not suitable for sequences of linear systems. Indeed, if the Ritz vectors are obtained from a previous linear system with another matrix or even another right-hand side, the concatenation of the recycled subspace to the current Krylov subspace does not form a Krylov subspace for the current problem. It is necessary to introduce a new Krylov solver that uses recycling of any given subspace without restriction. A famous one is the generalized conjugate residual with inner orthogonalization (GCRO). It belongs to the family of inner-outer methods [@axelsson1996iterative] where the outer method is based on GCR, a minimum residual norm method proposed by Eisenstat, Elman and Schultz [@eisenstat1983variational]. The inner solver is GMRES applied to a projected system matrix. ## The GCRO Krylov solver As previously mentioned, we focus on the solution of a sequence of linear systems with a varying right-hand side only. This section briefly introduces the Generalized Conjugate Residual with inner Orthogonalization (GCRO) algorithm with deflated restarting. In this framework, deflation can re-use spectral information from a previous cycle or from a previous linear system. We start by recalling the original formulation of the generalized conjugate residual method (GCR)[@eisenstat1983variational]. The idea is to introduce the concept of optimality for the solution residual. We want to solve $$\mathbf{A}\mathbf{x}^{(s)} = \mathbf{b}^{(s)}, \quad s=1,2,... \label{eq7}$$ where $\mathbf{A} \in \mathbb{R}^{n \times n}$ and $\mathbf{b}^{(s)} \in \mathbb{R}^{n}$ changes from one system to the next. The GCR method relies on a given full-rank matrix $\mathbf{U}_{k} \in \mathbb{R}^{n \times k}$ and an orthonormal matrix $\mathbf{C}_{k} \in \mathbb{R}^{n \times k}$ as the image of $\mathbf{U}_{k}$ by $\mathbf{A}$ satisfying the relations $$\begin{aligned} \mathbf{A}\mathbf{U}_{k} &= \mathbf{C}_{k}, \label{eq8} \\ \mathbf{C}^{T}_{k}\mathbf{C}_{k} &= \mathbf{I}_{k}. \label{eq9}\end{aligned}$$ For the sake of understanding, suppose that we want to solve the first system of the sequence of linear systems, i.e., $s=1$. Given an initial guess $\mathbf{x}_{0}^{(1)}$, the principle is to compute an approximation to the solution $\mathbf{x}^{(1)} \in \mathbf{x}_{0}^{(1)} + \operatorname{range}(\mathbf{U}_{k})$ that minimizes the corresponding residual norm over the approximation space $\operatorname{range}(\mathbf{U}_{k})$. More precisely, GCR solves the following minimization problem $$\mathbf{x}^{(1)} = \underset{\mathbf{x} \in \mathbf{x}^{(1)}_{0} + \operatorname{range}(\mathbf{U}_{k})}{\textnormal{argmin}} \| \mathbf{b}^{(1)} - \mathbf{Ax} \|_2, \label{eq10}$$ The optimal solution of ([\[eq10\]](#eq10){reference-type="ref" reference="eq10"}) over the subspace $\mathbf{x}^{(1)}_{0} + \operatorname{range}(\mathbf{U}_{k})$ is defined by $$\mathbf{x}^{(1)} = \mathbf{x}^{(1)}_{0} + \mathbf{U}_{k}\mathbf{C}^{T}_{k}\mathbf{r}_{0}^{(1)} \label{eq11}$$ where $\mathbf{r}_{0}^{(1)} = \mathbf{b}^{(1)} - \mathbf{A}\mathbf{x}^{(1)}_{0}$. Consequently, the corresponding residual vector satisfies $$\mathbf{r}^{(1)}_k = \mathbf{b}^{(1)}-\mathbf{A}\mathbf{x}^{(1)} = (\mathbf{I} - \mathbf{A}\mathbf{U}_k\mathbf{C}_k^T)\mathbf{r}^{(1)}_{0} = (\mathbf{I} - \mathbf{C}_{k}\mathbf{C}^{T}_{k})\mathbf{r}^{(1)}_{0}, \quad \mathbf{r}^{(1)}_k \perp \operatorname{range}(\mathbf{C}_{k}). \label{eq12}$$ The orthogonality of the residual $\mathbf{r}^{(1)}_k$ to the subspace $\mathcal{AK}_{k}(\mathbf{A},\mathbf{r}^{(1)}_{0})$ spanned by the columns of $\mathbf{C}_{k}$ is known as the optimality property of the residual. In practice, GCR is not considered as a means to solve the linear system. Instead, for the first system or when no spectral information is available from a previous solve, it is replaced by GMRES which computes an implicit representation of the matrices $\mathbf{U}_{k}$ and $\mathbf{C}_{k}$ [@de1999truncation]. In order to simplify the notations, we temporarily omit the index of the current system in the equations. Given an orthonormal basis $\boldsymbol{\operatorname{C}}_{k}$ of an outer subspace and the corresponding residual $\mathbf{r}_k=(\boldsymbol{\operatorname{I}}- \boldsymbol{\operatorname{C}}_{k}\boldsymbol{\operatorname{C}}^{T}_{k})\mathbf{r}_0$ after $k$ steps of GCR, GCRO($m$) obtains the next iterates $\mathbf{r}_{k+1}$ and $\mathbf{x}_{k+1}$ by performing $m$ steps of GMRES applied to the projected operator $\boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k}=(\boldsymbol{\operatorname{I}}- \boldsymbol{\operatorname{C}}_{k}\boldsymbol{\operatorname{C}}^{T}_{k})\boldsymbol{\operatorname{A}}$, thereby maintaining optimality of the inner residual with respect to the outer space. Let $\boldsymbol{\operatorname{V}}_{m+1} \in \mathbb{R}^{n \times (m+1)}$ be an orthonormal basis for $\mathcal{K}_{m+1}(\boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k},\mathbf{r}_{k})$ with $\mathbf{v}_1=\mathbf{r}_{k}/\| \mathbf{r}_{k} \|$, if we apply an inner GMRES we have the following Arnoldi relation: $$\boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k}\boldsymbol{\operatorname{V}}_m = (\boldsymbol{\operatorname{I}}- \boldsymbol{\operatorname{C}}_{k}\boldsymbol{\operatorname{C}}^{T}_{k})\boldsymbol{\operatorname{A}}\boldsymbol{\operatorname{V}}_m = \boldsymbol{\operatorname{V}}_{m+1}\bar{\boldsymbol{\operatorname{H}}}_{m} \quad \text{with} \quad \bar{\boldsymbol{\operatorname{H}}}_{m} = \boldsymbol{\operatorname{V}}_{m+1}^{T} \boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k}\boldsymbol{\operatorname{V}}_m, \quad \bar{\boldsymbol{\operatorname{H}}}_{m} \in \mathbb{R}^{(m+1) \times m} \label{eq:ArnoldiGCRO}$$ This equation can be expanded as $$\mathbf{A}\mathbf{V}_m = \mathbf{C}_{k}\mathbf{B}_m + \mathbf{V}_{m+1}\bar{\mathbf{H}}_{m} \quad \text{with} \quad \mathbf{B}_m = \mathbf{C}_k^{T} \mathbf{A} \mathbf{V}_m, \quad \boldsymbol{\operatorname{B}}_{m} \in \mathbb{R}^{k \times m} \label{eq:ExpandedArnoldiGCRO}$$ At the end of the inner GMRES cycle, using [\[eq:ArnoldiGCRO\]](#eq:ArnoldiGCRO){reference-type="eqref" reference="eq:ArnoldiGCRO"}, the residual can be written as $$\mathbf{r}_{k+1} = \mathbf{r}_k - \mathbf{A}_{\mathbf{C}_k}\mathbf{V}_{m}\mathbf{y}_m = \mathbf{r}_{k} - \mathbf{V}_{m+1}\bar{\mathbf{H}}_{m}\mathbf{y}_m, \label{eq:GCRO_residual}$$ and the corresponding solution reads $$\mathbf{x}_{k+1} = \mathbf{x}_{k} + \mathbf{A}^{-1}\mathbf{A}_{\mathbf{C}_{k}}\mathbf{V}_{m}\mathbf{y}_{m}, \label{eq:GCRO_solution}$$ where, using [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"}, we have $\mathbf{A}^{-1}\mathbf{A}_{\mathbf{C}_{k}}=\boldsymbol{\operatorname{I}}-\mathbf{U}_k \mathbf{C}^{T}_{k} \mathbf{A}$. Using [\[eq11\]](#eq11){reference-type="eqref" reference="eq11"} and [\[eq:GCRO_solution\]](#eq:GCRO_solution){reference-type="eqref" reference="eq:GCRO_solution"} we obtain the expression for the solution at the end of the first cycle: $$\begin{aligned} \mathbf{x}_{k+1} &= \mathbf{x}^{(1)}_{0} + \mathbf{U}_{k}\mathbf{C}^{T}_{k}\mathbf{r}_{0}^{(1)} + (\boldsymbol{\operatorname{I}}-\mathbf{U}_k \mathbf{C}^{T}_{k} \mathbf{A}) \mathbf{V}_m \mathbf{y}_m \nonumber \\ &= \mathbf{x}^{(1)}_{0} + \mathbf{U}_{k}\mathbf{C}^{T}_{k} \left( \mathbf{r}_{0}^{(1)} - \mathbf{A} \mathbf{V}_m \mathbf{y}_m \right) + \mathbf{V}_m \mathbf{y}_m \nonumber \\ &= \mathbf{x}^{(1)}_{0} + \mathbf{U}_{k} \mathbf{z}_k + \mathbf{V}_m \mathbf{y}_m, \label{solution_GCR}\end{aligned}$$ with $$\mathbf{z}_{k} = \mathbf{C}^{T}_{k}\mathbf{r}_{0}^{(1)} - \mathbf{B}_m \mathbf{y}_m, \quad \mathbf{B}_m = \mathbf{C}^{T}_{k}\mathbf{AV}_m.$$ We easily show that the residual computed in the inner GMRES equals the true outer residual: $$\mathbf{r}_{k+1} = \mathbf{b} - \mathbf{Ax}_{k+1} = \mathbf{b} - \mathbf{Ax}_{k} - \mathbf{A}_{\mathbf{C}_k}\mathbf{V}_{m}\mathbf{y}_m = \mathbf{r}_{k} - \mathbf{V}_{m+1}\bar{\mathbf{H}}_{m}\mathbf{y}_m.$$ Because $\operatorname{range}(\mathbf{C}_{k}) \bot \operatorname{range}(\mathbf{V}_{m+1})$ by construction, it follows that $\mathbf{x}_{k+1} \in \mathbf{x}^{(1)}_{0} + \operatorname{range}(\mathbf{U}_{k}) \oplus \operatorname{range}(\mathbf{V}_m)$ is the unique solution of the global residual minimization problem over the subspace spanned by the complementary subspaces $\operatorname{range}(\mathbf{U}_{k})$ and $\operatorname{range}(\mathbf{V}_m)$ (see Theorem 2.2 and its proof in [@de1996nested]). Now, introducing the orthogonal decomposition $\bar{\mathbf{H}}_m = \bar{\mathbf{Q}}_m \mathbf{R}_m$, the vector of reduced coordinates $\mathbf{y}_m$ is such that $$\mathbf{y}_m = \underset{\mathbf{y} \in \mathbb{R}^m}{\operatorname{argmin}} \| \mathbf{r}_{k+1} \|_2 = \underset{\mathbf{y} \in \mathbb{R}^m}{\operatorname{argmin}} \| \mathbf{r}_k - \mathbf{A}_{\mathbf{C}_k}\mathbf{V}_{m}\mathbf{y} \|_2 = \mathbf{R}_m^{-1}\bar{\mathbf{Q}}_m^{T} \| \mathbf{r}_k \|_2 \mathbf{e}_1$$ In the above Arnoldi process [\[eq:ExpandedArnoldiGCRO\]](#eq:ExpandedArnoldiGCRO){reference-type="eqref" reference="eq:ExpandedArnoldiGCRO"} of the inner GMRES($m$), the vectors $\mathbf{Av}_i$ are first orthogonalized against $\mathbf{C}_k$, thus constructing $\mathbf{V}_{m+1}$ such that $\mathbf{C}_k^{T}\mathbf{V}_{m+1}=\mathbf{0}$. ## The GCRO-DR Krylov solver So far, we have not yet explained how the matrix $\mathbf{C}_{k}$ is built. A fundamental difference between GCRO and GCRO-DR is that GCRO is a nested Krylov solver where the outer spaces $\mathbf{U}_k$ and $\mathbf{C}_k$ keep growing and GMRES($m$) is applied at each iteration $k$, whereas GCRO-DR supplements given deflation subspaces derived from a previous cycle with ($m-k$) iterations of GMRES. To comply to the GCRO-DR formalism, we first slightly change the notation in [\[eq:ArnoldiGCRO\]](#eq:ArnoldiGCRO){reference-type="eqref" reference="eq:ArnoldiGCRO"} and [\[eq:ExpandedArnoldiGCRO\]](#eq:ExpandedArnoldiGCRO){reference-type="eqref" reference="eq:ExpandedArnoldiGCRO"} to point out that we perform $m-k$ steps (instead of $m$) of inner GMRES. This produces the Arnoldi relation $$\mathbf{A}_{\mathbf{C}_k}\mathbf{V}_{m-k} = \mathbf{V}_{m-k+1}\bar{\mathbf{H}}_{m-k} \quad \Leftrightarrow \quad \bar{\boldsymbol{\operatorname{H}}}_{m-k} = \boldsymbol{\operatorname{V}}_{m-k+1}^{T} \boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k}\boldsymbol{\operatorname{V}}_{m-k}, \quad \bar{\boldsymbol{\operatorname{H}}}_{m-k} \in \mathbb{R}^{(m-k+1) \times (m-k)} \label{eq:FlexibleGCRO1}$$ Combining [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"} and [\[eq:FlexibleGCRO1\]](#eq:FlexibleGCRO1){reference-type="eqref" reference="eq:FlexibleGCRO1"} we have $$\mathbf{A} \left[ \mathbf{U}_k \quad \mathbf{V}_{m-k} \right] = [ \mathbf{C}_k \quad \mathbf{V}_{m-k+1} ] \begin{bmatrix} \mathbf{I}_{k} & \mathbf{B}_{m-k} \\ \mathbf{0} & \bar{\mathbf{H}}_{m-k} \end{bmatrix}, \label{eq:ExpandedArnoldiGCRO1}$$ with $\mathbf{B}_{m-k} = \mathbf{C}_k^{T} \mathbf{AV}_{m-k}, \mathbf{B}_{m-k} \in \mathbb{R}^{k \times (m-k)}$. Numerical tests suggest that the rightmost matrix in [\[eq:ExpandedArnoldiGCRO1\]](#eq:ExpandedArnoldiGCRO1){reference-type="eqref" reference="eq:ExpandedArnoldiGCRO1"} is ill-conditioned. To reduce unnecessary ill-conditioning we can compute the diagonal matrix $\mathbf{D}_k = \operatorname{diag}(\|\textbf{u}_1\|^{-1}, \|\textbf{u}_2\|^{-1},\ldots, \|\textbf{u}_k\|^{-1})$ such that $\tilde{\mathbf{U}}_k = \mathbf{U}_k \mathbf{D}_k$, has unit columns. Defining $$\hat{\mathbf{V}}_m = [\tilde{\mathbf{U}}_k \quad \mathbf{V}_{m-k}], \quad \hat{\mathbf{W}}_{m+1} = [\mathbf{C}_k \quad \mathbf{V}_{m-k+1}], \quad \bar{\mathbf{H}}_m = \begin{bmatrix} \mathbf{D}_k & \mathbf{B}_{m-k} \\ \mathbf{0} & \bar{\mathbf{H}}_{m-k} \end{bmatrix},$$ the Arnoldi relation [\[eq:ExpandedArnoldiGCRO1\]](#eq:ExpandedArnoldiGCRO1){reference-type="eqref" reference="eq:ExpandedArnoldiGCRO1"} can be cast into $$\mathbf{A}\hat{\mathbf{V}}_m = \hat{\mathbf{W}}_{m+1} \bar{\mathbf{H}}_m. \label{eq:GeneralizedArnoldi}$$ From [\[solution_GCR\]](#solution_GCR){reference-type="eqref" reference="solution_GCR"} we have $\mathbf{x}_1 = \mathbf{x}_0 + \tilde{\mathbf{U}}_k \mathbf{z}_k + \mathbf{V}_{m-k} \mathbf{y}_{m-k} = \mathbf{x}_0 + \hat{\mathbf{V}}_m \mathbf{y}_m$ with $\mathbf{y}_m = {\operatorname{argmin}}_{\mathbf{y}} \|\bar{\mathbf{H}}_{m}\mathbf{y}-\hat{\mathbf{W}}_{m+1}^T \mathbf{r}_0\|_2$. The expanded form of this least-squares problem reads $$\mathbf{y}_m = [ \mathbf{z}_k, \mathbf{y}_{m-k} ] = {\operatorname{argmin}}_{\mathbf{z,y}} \left\| \begin{bmatrix} \mathbf{D}_k & \mathbf{B}_{m-k} \\ \mathbf{0} & \bar{\mathbf{H}}_{m-k} \end{bmatrix} \begin{bmatrix} \mathbf{z} \\ \mathbf{y} \end{bmatrix} - \begin{bmatrix} \mathbf{C}_k^T\mathbf{r}_0 \\ \mathbf{V}_{m-k+1}^T \mathbf{r}_0 \end{bmatrix} \right\|_2 \label{GCRO_LeastSquares}$$ In [@de1996nested] and [@soodhalter2020] the authors suggest to solve [\[GCRO_LeastSquares\]](#GCRO_LeastSquares){reference-type="eqref" reference="GCRO_LeastSquares"} blockwise, first for $\mathbf{y}_{m-k}$ and then set $\mathbf{z}_k = \mathbf{D}_k^{-1} [\mathbf{C}_k^T \mathbf{r}_0 - \mathbf{B}_{m-k}\mathbf{y}_{m-k}]$. Recall that the first Krylov vector in $\mathbf{V}_{m-k+1}$ is $\mathbf{v}_{k+1}=(\mathbf{I}-\mathbf{C}_k\mathbf{C}_k^T)\mathbf{r}_0$, which simplifies the expression for the right-hand side of the least-squares sub-problem to $\mathbf{V}_{m-k+1}^T \mathbf{r}_0 = \mathbf{V}_{m-k+1}^T (\mathbf{I}-\mathbf{C}_k\mathbf{C}_k^T) \mathbf{r}_0 = \beta \mathbf{e}_1^{m-k+1}$, where $\beta = \| (\mathbf{I}-\mathbf{C}_k\mathbf{C}_k^T)\mathbf{r}_0 \|_2$ and $\mathbf{e}_1^{m-k+1}$ is the first unit vector of $\mathbb{R}^{m-k+1}$. We have now completed the process for the first linear system which formulates the solution from the complementary subspaces $\tilde{\mathbf{U}}_k$ and $\mathbf{V}_{m-k}$. At this stage, we introduce deflated restarting within the GCRO framework to build the outer space $\mathbf{C}_k$ for the next cycle. Like GMRES-DR, the deflation subspace is spanned by the $k$ harmonic Ritz vectors corresponding to harmonic Ritz values of smallest magnitude. Using definition [\[HarmonicRitzpair\]](#HarmonicRitzpair){reference-type="ref" reference="HarmonicRitzpair"} with $\mathcal{U} \equiv \operatorname{span}\lbrace \hat{\mathbf{V}}_m \rbrace$ and $\mathbf{B} \equiv \mathbf{A}$ we can write $\mathbf{y} = \hat{\mathbf{V}}_m \mathbf{g}$ which gives the orthogonality relation $$(\mathbf{A}\hat{\mathbf{V}}_m)^T(\mathbf{A}\hat{\mathbf{V}}_m \mathbf{g} -\theta \hat{\mathbf{V}}_m \mathbf{g}) = \mathbf{0}. \label{eq:Ritz_problem_GCRO}$$ Now using [\[eq:GeneralizedArnoldi\]](#eq:GeneralizedArnoldi){reference-type="eqref" reference="eq:GeneralizedArnoldi"} in [\[eq:Ritz_problem_GCRO\]](#eq:Ritz_problem_GCRO){reference-type="eqref" reference="eq:Ritz_problem_GCRO"} and recalling that $\hat{\mathbf{W}}_{m+1}$ is orthonormal, we obtain the generalized eigenvalue problem $$\label{eq:GCRO_eigenvalue_problem} \boxed{\bar{\mathbf{H}}^T_m \bar{\mathbf{H}}_m \mathbf{g} = \theta \bar{\mathbf{H}}^T_m \hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m \mathbf{g}}$$ In practice, for better numerical accuracy, instead of computing the smallest eigenvalues of [\[eq:GCRO_eigenvalue_problem\]](#eq:GCRO_eigenvalue_problem){reference-type="eqref" reference="eq:GCRO_eigenvalue_problem"} we prefer to compute the largest eigenvalues $\theta^{-1}$ of $\bar{\mathbf{H}}^T_m \hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m \mathbf{g} = \theta^{-1} \bar{\mathbf{H}}^T_m \bar{\mathbf{H}}_m \mathbf{g}$. Since $\hat{\mathbf{W}}_{m+1} = [\mathbf{C}_k \quad \mathbf{V}_{m-k+1}]$, the matrix product $\hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m$ has the following block structure: $$\label{eq:GCRO_block_structure1} \hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m = \begin{bmatrix} \mathbf{C}_k^T \tilde{\mathbf{U}}_k & \mathbf{0}_{k \times (m-k)} \\ \mathbf{V}_{m-k+1}^T \tilde{\mathbf{U}}_k & \begin{bmatrix} \mathbf{I}_{m-k} \\ \mathbf{0}_{1 \times (m-k)} \end{bmatrix} \end{bmatrix} \!\!.$$ We can further simplify this expression. Let $\mathbf{v}_{k+1}=\mathbf{r}_k/\| \mathbf{r}_k \|$, we have $\hat{\mathbf{V}}_m = [\tilde{\mathbf{U}}_k, \mathbf{v}_{k+1}, \mathbf{V}_{m-k-1}]$ and $\hat{\mathbf{W}}_{m+1} = [\mathbf{C}_k, \mathbf{v}_{k+1},\mathbf{V}_{m-k}]$. It can also be shown that $\operatorname{range}(\tilde{\mathbf{U}}_k, \mathbf{v}_{k+1}) = \operatorname{range}(\mathbf{C}_k, \mathbf{v}_{k+1}) = \operatorname{range}(\mathbf{Y}_k, \mathbf{v}_{k+1})$, $\mathbf{Y}_k = \hat{\mathbf{V}}_m \mathbf{P}_k$ being the deflation subspace spanned by the Ritz vectors (see [@carvalho2011flexible]). It follows that the bottom left $(m-k) \times (k+1)$ block in [\[eq:GCRO_block_structure1\]](#eq:GCRO_block_structure1){reference-type="eqref" reference="eq:GCRO_block_structure1"} is null, which gives $$\label{eq:GCRO_block_structure2} \hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m = \left[ \begin{array}{cc|c} \mathbf{C}_k^T \tilde{\mathbf{U}}_k & \mathbf{0}_{k \times 1} & \mathbf{0}_{k \times (m-k-1)} \\ \mathbf{v}_{1}^T \tilde{\mathbf{U}}_k & 1 & \mathbf{0}_{1 \times (m-k-1)} \\ % \hline \cmidrule[.5pt]{1-3} % \midrule % \specialrule{\cmidrulewidth}{0pt}{0pt} \multicolumn{2}{c|}{\mathbf{V}_{m-k}^T [\tilde{\mathbf{U}}_k, \mathbf{r}_k/\| \mathbf{r}_k \|]} & \left[ \! \begin{array}{c} \mathbf{I}_{m-k-1} \\ \mathbf{0}_{1 \times (m-k-1)} \end{array} \! \right] \end{array} \right] = \hfill \left[ \begin{array}{cc|c} \mathbf{C}_k^T \tilde{\mathbf{U}}_k & \mathbf{0}_{k \times 1} & \mathbf{0}_{k \times (m-k-1)} \\ \mathbf{v}_{1}^T \tilde{\mathbf{U}}_k & 1 & \mathbf{0}_{1 \times (m-k-1)} \\ \cmidrule[.5pt]{1-3} \multicolumn{2}{c|}{\mathbf{0}_{(m-k) \times (k+1)}} & \left[ \! \begin{array}{c} \mathbf{I}_{m-k-1} \\ \mathbf{0}_{1 \times (m-k-1)} \end{array} \! \right] \end{array} \right] \!\!.$$ In [@carvalho2011flexible] the authors suggest a number of improvements, mainly based on recursion formulas, to save computational cost in forming the block $\mathbf{C}_k^T \tilde{\mathbf{U}}_k$ in [\[eq:GCRO_block_structure2\]](#eq:GCRO_block_structure2){reference-type="eqref" reference="eq:GCRO_block_structure2"} at a cost independent of the problem size. The deflation subspace spanned by the Ritz vectors $\mathbf{Y}_k = \hat{\mathbf{V}}_m \mathbf{P}_k$ is then shifted by $\mathbf{A}$ to give $$\label{eq:shifted_ritz_subspace} \mathbf{A} \mathbf{Y}_k = \hat{\mathbf{W}}_{m+1} \bar{\mathbf{H}}_m \mathbf{P}_k.$$ Introducing in [\[eq:shifted_ritz_subspace\]](#eq:shifted_ritz_subspace){reference-type="eqref" reference="eq:shifted_ritz_subspace"} the reduced QR factorization $\bar{\mathbf{H}}_m \mathbf{P}_k = \bar{\mathbf{Q}}_m \mathbf{R}$, $\bar{\mathbf{Q}}_m \in \mathbb{R}^{(m+1) \times k}$, we obtain $\mathbf{A} \mathbf{Y}_k = ( \hat{\mathbf{W}}_{m+1} \bar{\mathbf{Q}}_m) \mathbf{R} = \mathbf{C}_k \mathbf{R}$. The outer residual subspace for the next cycle is then given by $$\label{eq:new_Ck} \mathbf{C}_k = \hat{\mathbf{W}}_{m+1} \bar{\mathbf{Q}}_m,$$ and, from $\mathbf{C}_k = \mathbf{A} \mathbf{Y}_k \mathbf{R}^{-1} = \mathbf{A} \mathbf{U}_k$, it follows directly the definition of the corresponding solution subspace $$\label{eq:new_Uk} \mathbf{U}_k = \mathbf{Y}_k \mathbf{R}^{-1}.$$ We note that [\[eq:new_Ck\]](#eq:new_Ck){reference-type="eqref" reference="eq:new_Ck"} implies that $\mathbf{C}_k$ is an orthonormal matrix since $\hat{\mathbf{W}}_{m+1}$ and $\bar{\mathbf{Q}}_m$ are also orthonormal. We are now in a position to perform the next cycle $(i+1)$ and obtain the next iterates $\mathbf{r}_{i+1}^{(s)}$ and $\mathbf{x}_{i+1}^{(s)}$ for the current system $(s)$ by performing $m-k$ steps of GMRES applied to the projected operator $\boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k}=(\boldsymbol{\operatorname{I}}- \boldsymbol{\operatorname{C}}_{k}\boldsymbol{\operatorname{C}}^{T}_{k})\boldsymbol{\operatorname{A}}$. We still have to detail the construction of the deflation subspaces $\mathbf{U}_k$ and $\mathbf{C}_k$ at the end of the initial GMRES cycle on one hand, and at the beginning of a subsequent linear system on the other hand. ## [First cycle of initial linear system]{.ul} {#first-cycle-of-initial-linear-system .unnumbered} After $m$ iterations of GMRES we generate $\mathbf{V}_{m+1}$ and $\bar{\mathbf{H}}_{m}$ which are related by the Arnoldi relation $\mathbf{AV}_{m} = \mathbf{V}_{m+1}\bar{\mathbf{H}}_{m}$. We then solve the standard eigenvalue problem [\[eq:Standard_eigenvalue_problem\]](#eq:Standard_eigenvalue_problem){reference-type="eqref" reference="eq:Standard_eigenvalue_problem"} to obtain the matrix $\mathbf{P}_{k} \in \mathbb{R}^{m \times k}$ which columns correspond to the retained eigenvectors used for the construction of the harmonic Ritz vectors $\mathbf{Y}_k = \mathbf{V}_m \mathbf{P}_k$. Defining $[\mathbf{Q}, \mathbf{R}]$ as the reduced QR-factorization of $\bar{\mathbf{H}}_{m} \mathbf{P}_k$, we have $$\mathbf{AY}_k = \mathbf{AV}_m \mathbf{P}_k = \mathbf{V}_{m+1} \mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_m \mathbf{P}_k = (\mathbf{V}_{m+1}\mathbf{Q})\mathbf{R}. \label{eq:AVmPk_CkR}$$ Since $\mathbf{V}_{m+1}$ and $\mathbf{Q}$ are orthonormal matrices and we are looking for a relation of the type [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"}, we directly deduce from [\[eq:AVmPk_CkR\]](#eq:AVmPk_CkR){reference-type="eqref" reference="eq:AVmPk_CkR"} the following identities to be used for the next cycle: $$\begin{aligned} \mathbf{C}_k &= \mathbf{V}_{m+1} \mathbf{Q} \\ \mathbf{U}_k &= \mathbf{Y}_k \mathbf{R}^{-1}\end{aligned}$$ At the end of the GMRES($m$) cycle, the optimality property $\mathbf{r}_1^{(1)} \; \bot \; \mathcal{AK}_m(\mathbf{A},\mathbf{r}_0^{(1)})$ of the residual $\mathbf{r}_1^{(1)}=\mathbf{r}_0^{(1)}-\mathbf{AV}_m \mathbf{y}_m$ can be easily verified. Recall that the reduced coordinates $\mathbf{y}_m$ solve the least-squares problem $\mathbf{y}_m = \underset{\mathbf{y} \in \mathbb{R}^m}{\operatorname{argmin}} \|\mathbf{V}_{m+1}^T \mathbf{r}_0^{(1)} - \bar{\mathbf{H}}_{m}\mathbf{y}\|_2$, which gives $\mathbf{y}_m = ( \bar{\mathbf{H}}_m^T \bar{\mathbf{H}}_m )^{-1} \bar{\mathbf{H}}_m^T \mathbf{V}_{m+1}^T \mathbf{r}_0^{(1)} = \bar{\mathbf{H}}_m^{\dagger} \mathbf{V}_{m+1}^T \mathbf{r}_0^{(1)} = \beta \bar{\mathbf{H}}_m^{\dagger} \mathbf{e}_1$, with $\beta=\|\mathbf{r}_0\|$. Then, it follows that $$\begin{aligned} (\mathbf{AV}_m)^{T} \mathbf{r}^{(1)}_1 &= (\mathbf{AV}_m)^{T} \mathbf{V}_{m+1} ( \beta \mathbf{e}_1 - \bar{\mathbf{H}}_m \mathbf{y}_m ) \nonumber \\ &= \bar{\mathbf{H}}_m^T \mathbf{V}_{m+1}^T \mathbf{V}_{m+1} ( \beta \mathbf{e}_1 - \bar{\mathbf{H}}_m \mathbf{y}_m ) \nonumber \\ &= \bar{\mathbf{H}}_m^T ( \beta \mathbf{e}_1 - \beta \bar{\mathbf{H}}_m \bar{\mathbf{H}}_m^{\dagger} \mathbf{e}_1 ) \nonumber \\ \Rightarrow (\mathbf{AV}_m)^{T} \mathbf{r}^{(1)}_1 &= \mathbf{0}. \label{eq:optimality_GMRES1}\end{aligned}$$ Equation [\[eq:optimality_GMRES1\]](#eq:optimality_GMRES1){reference-type="eqref" reference="eq:optimality_GMRES1"} can be written for any system $(s)$, i.e., $(\mathbf{AV}_m)^{T} \mathbf{r}^{(s)}_1 = \mathbf{0}, s=1,2,\cdots$. Now, starting from [\[eq:optimality_GMRES1\]](#eq:optimality_GMRES1){reference-type="eqref" reference="eq:optimality_GMRES1"} we obtain $$\begin{aligned} \mathbf{P}_k^T (\mathbf{AV}_m)^{T} \mathbf{r}^{(1)}_1 &= \mathbf{0} \nonumber \\ ( \mathbf{V}_{m+1} \bar{\mathbf{H}}_m \mathbf{P}_k )^T \mathbf{r}^{(1)}_1 &= \mathbf{0} \nonumber \\ \Rightarrow \mathbf{R}^T \mathbf{C}_k^T \mathbf{r}^{(1)}_1 &= \mathbf{0}, \label{eq:optimality_GMRES2}\end{aligned}$$ which shows that $\mathbf{C}_k^T \mathbf{r}^{(1)}_1 = \mathbf{0}$ since $\mathbf{R}$ is supposed to be nonsingular. Similarly, this is valid for any system in the sequence: $\mathbf{C}_k^T \mathbf{r}^{(s)}_1 = \mathbf{0}, s=1,2,\cdots$. Incidentally, comparing [\[eq:optimality_GMRES1\]](#eq:optimality_GMRES1){reference-type="eqref" reference="eq:optimality_GMRES1"} and [\[eq:optimality_GMRES2\]](#eq:optimality_GMRES2){reference-type="eqref" reference="eq:optimality_GMRES2"} shows that $\mathcal{AK}_m(\mathbf{A},\mathbf{r}_0^{(1)})$ is spanned by the columns of $\mathbf{C}_k$. ## [First cycle of subsequent linear systems]{.ul} {#first-cycle-of-subsequent-linear-systems .unnumbered} We now assume that two subspaces $\mathbf{U}_k^{(s-1)}$ and $\mathbf{C}_k^{(s-1)}$ are available from a previous linear system $(s-1)$ and we want to solve a new system $\mathbf{A} \mathbf{x}^{(s)} = \mathbf{b}^{(s)}$. In this work, we deal with the special case where the system matrix remains unchanged from one linear system to the next. This implies $\mathbf{U}_k^{(s)} = \mathbf{U}_k^{(s-1)}$ and $\mathbf{C}_k^{(s)} = \mathbf{C}_k^{(s-1)}$. In order to apply $m-k$ steps of GMRES to the projected operator $\boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k^{(s)}}=(\boldsymbol{\operatorname{I}}- \mathbf{C}_k^{(s)} \mathbf{C}_k^{(s)T})\boldsymbol{\operatorname{A}}$, we need to compute first the projected residual $\mathbf{r}_k^{(s)} = (\mathbf{I} - \mathbf{C}_k^{(s)} \mathbf{C}_k^{(s)T}) \mathbf{r}_0^{(s)}$ and the associated solution vector $\mathbf{x}_k^{(s)} = \mathbf{x}_0^{(s)} + \mathbf{U}_k^{(s)} \mathbf{C}_k^{(s)T}\mathbf{r}_0^{(s)}$, where $\mathbf{x}_0^{(s)}$ and $\mathbf{r}_0^{(s)}$ denote the solution and residual vector at convergence of the previous linear system. Once the Arnoldi process [\[eq:GeneralizedArnoldi\]](#eq:GeneralizedArnoldi){reference-type="eqref" reference="eq:GeneralizedArnoldi"} is completed, the solution at the end of the first cycle of system $(s)$ reads (see [\[eq:GCRO_residual\]](#eq:GCRO_residual){reference-type="eqref" reference="eq:GCRO_residual"} and [\[eq:GCRO_solution\]](#eq:GCRO_solution){reference-type="eqref" reference="eq:GCRO_solution"}): $$\begin{aligned} \mathbf{r}_{1}^{(s)} &= \mathbf{r}_{k}^{(s)} - \mathbf{V}_{m-k+1}\bar{\mathbf{H}}_{m-k}\mathbf{y}_{m-k} \\ \mathbf{x}_{1}^{(s)} &= \mathbf{x}_{k}^{(s)} + \mathbf{A}^{-1}\mathbf{A}_{\mathbf{C}_{k}^{(s)}}\mathbf{V}_{m-k}\mathbf{y}_{m-k} \nonumber \\ & = \mathbf{x}_{k}^{(s)} + (\boldsymbol{\operatorname{I}}-\mathbf{U}_k^{(s)}\mathbf{C}^{(s)T}_{k} \mathbf{A}) \mathbf{V}_{m-k} \mathbf{y}_{m-k} \nonumber \\ & = \mathbf{x}_{k}^{(s)} + (\mathbf{V}_{m-k}-\mathbf{U}_k^{(s)}\mathbf{B}_{m-k}) \mathbf{y}_{m-k}.\end{aligned}$$ The above expressions make explicitly appear the correction to the outer iterates $\mathbf{x}_k^{(s)}$ and $\mathbf{r}_k^{(s)}$. Equivalent formulas can be obtained by considering the generalized Arnoldi relation [\[eq:GeneralizedArnoldi\]](#eq:GeneralizedArnoldi){reference-type="eqref" reference="eq:GeneralizedArnoldi"}: $$\begin{aligned} \mathbf{r}_{1}^{(s)} &= \mathbf{r}_{0}^{(s)} - \hat{\mathbf{W}}_{m+1} \bar{\mathbf{H}}_m\mathbf{y}_{m} \\ \mathbf{x}_{1}^{(s)} &= \mathbf{x}_{0}^{(s)} + \hat{\mathbf{V}}_{m} \mathbf{y}_{m}\end{aligned}$$ The vector of reduced coordinates $\mathbf{y}_m$ is the solution of [\[GCRO_LeastSquares\]](#GCRO_LeastSquares){reference-type="eqref" reference="GCRO_LeastSquares"} and its partition $\mathbf{y}_{m-k}$ is the solution of ${\operatorname{argmin}}_{\mathbf{y}} \left\| \bar{\mathbf{H}}_{m-k} \mathbf{y} -\beta \mathbf{e}_1^{m-k+1} \right\|_2$, $\beta = \| (\mathbf{I}-\mathbf{C}_k^{(s)}\mathbf{C}_k^{(s)T})\mathbf{r}_0^{(s)} \|_2$. ## [Reformulation of the generalized eigenvalue problem]{.ul} {#reformulation-of-the-generalized-eigenvalue-problem .unnumbered} At the end of each cycle of GCRO-DR we compute the Ritz eigenpairs from the following generalized eigenvalue problem: $$\bar{\mathbf{H}}^T_m \bar{\mathbf{H}}_m \mathbf{g} = \theta \bar{\mathbf{H}}^T_m \hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m \mathbf{g} \label{GCRODR_Ritz_eigenproblem}$$ However, direct construction of both sides of this equation as product of matrices may lead to highly ill-conditioned operators. In the following we propose to reformulate [\[GCRODR_Ritz_eigenproblem\]](#GCRODR_Ritz_eigenproblem){reference-type="eqref" reference="GCRODR_Ritz_eigenproblem"} by exploiting the specific structure of $\bar{\mathbf{H}}_m$ and $\hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m$. First, noticing that the last row of $\bar{\mathbf{H}}_m$ is null except the last term $h_{m+1,m}$, we can write $\bar{\mathbf{H}}_m^T = [ \mathbf{H}_m^T \quad h_{m+1,m}\mathbf{e}_m ]$. Thus, we have $\bar{\mathbf{H}}^T_m \bar{\mathbf{H}}_m = \mathbf{H}^T_m \mathbf{H}_m + h_{m+1,m}^2\mathbf{e}_m \mathbf{e}_m^T$. Inserting these relations in [\[GCRODR_Ritz_eigenproblem\]](#GCRODR_Ritz_eigenproblem){reference-type="eqref" reference="GCRODR_Ritz_eigenproblem"} gives $$\mathbf{g} = \theta [ \mathbf{H}_m^T \quad h_{m+1,m}\mathbf{e}_m ] \hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m \mathbf{g}.$$ Left multiplying by $\mathbf{H}_m^{-T}$ and defining $\mathbf{f} = \mathbf{H}_m^{-T} \mathbf{e}_m$, we obtain $$\mathbf{g} = \theta [ \mathbf{I}_m \quad h_{m+1,m}\mathbf{f} ] \hat{\mathbf{W}}^T_{m+1} \hat{\mathbf{V}}_m \mathbf{g}.$$ From [\[eq:GCRO_block_structure2\]](#eq:GCRO_block_structure2){reference-type="eqref" reference="eq:GCRO_block_structure2"} we know that the last row of $\hat{\mathbf{W}}_{m+1}^T \hat{\mathbf{V}}_{m}$ is null which finally leads to the following eigenvalue problem $$\label{eq:Reformulated_GCRO_eigenproblem} \boxed{[ \mathbf{H}_m + h_{m+1,m}^2 \mathbf{f} \, \mathbf{e}_m^T ] \mathbf{g} = \theta \left[ \begin{array}{cc} \hat{\mathbf{W}}_{k+1}^T \hat{\mathbf{V}}_{k+1} & \mathbf{0}_{(k+1) \times (m-k-1)} \\ \mathbf{0}_{(m-k-1) \times (k+1)} & \mathbf{I}_{m-k-1} \end{array} \right] \mathbf{g}},$$ where the structure of the block $\hat{\mathbf{W}}_{k+1}^T \hat{\mathbf{V}}_{k+1}$ has been highlighted in [\[eq:GCRO_block_structure2\]](#eq:GCRO_block_structure2){reference-type="eqref" reference="eq:GCRO_block_structure2"}. [\[GCRODR_algorithm\]]{#GCRODR_algorithm label="GCRODR_algorithm"} Choose $m$, the size of the Krylov subspace and $k$, the size of the deflated/recycled subspace. Let *tol* be the relative convergence threshold. Choose an initial guess $\mathbf{x}_{0}$. Compute $\mathbf{r}_{0}=\mathbf{b}-\mathbf{Ax}_{0}$, and set $i=1$. $\mathbf{x}_{1} = \mathbf{x}_{0} + \mathbf{U}_{k}\mathbf{C}^{T}_{k}\mathbf{r}_{0}$ $\mathbf{r}_{1} = \mathbf{r}_{0} - \mathbf{C}_{k}\mathbf{C}^{T}_{k}\mathbf{r}_{0}$ $\mathbf{v}_{1} = \mathbf{r}_{0}/\|\mathbf{r}_{0}\|$ $\mathbf{c} = \|\mathbf{r}_{0}\| \mathbf{e}_{1}$ Apply $m$ steps of GMRES to generate $\mathbf{V}_{m+1}$ and $\bar{\mathbf{H}}_{m}$. solve $\operatorname{min}_{\mathbf{y}} \|\mathbf{c} -\mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_{m}\mathbf{y}\|_{2}$ for $\mathbf{y}_m$ $\mathbf{x}_{1} = \mathbf{x}_{0} + \mathbf{V}_{m}\mathbf{y}_{m}$ $\mathbf{r}_{1} = \mathbf{V}_{m+1}(\mathbf{c}-\bar{\mathbf{H}}_{m}\mathbf{y}_{m})$ Select the $k$ deflation eigenvectors of $\mathbf{H}_{m}\mathbf{g} = \lambda \mathbf{g}$ associated to the smallest eigenvalues and concatenate into $\mathbf{P}_k$ Let \[Q,R\] be the reduced QR factorization of $\bar{\mathbf{H}}_{m}\mathbf{P}_{k}$. $\mathbf{C}_{k} = \mathbf{V}_{m+1}\mathbf{Q}$ $\mathbf{U}_{k} = \hat{\mathbf{V}}_{m}\mathbf{P}_{k}\mathbf{R}^{-1}$ $i = i+1$ Apply ($m-k$) steps of Arnoldi with the projected operator $(\mathbf{I}- \mathbf{C}_{k}\mathbf{C}^{T}_{k})\mathbf{A}$ starting with $\mathbf{v}_{1}=\mathbf{r}_{i-1}/\|\mathbf{r}_{i-1}\|_{2}$ to generate $\mathbf{V}_{m-k+1}$, $\bar{\mathbf{H}}_{m-k}$, and $\mathbf{B}_{m-k}$. $\hat{\mathbf{W}}_{m+1} = [\mathbf{C}_{k} \quad \mathbf{V}_{m-k+1}]$ $\hat{\mathbf{V}}_m = [\tilde{\mathbf{U}}_k \quad \mathbf{V}_{m-k}]$ $\bar{\mathbf{H}}_{m} = \begin{bmatrix} \mathbf{D}_{k} & \mathbf{B}_{m-k} \\ \mathbf{0} & \bar{\mathbf{H}}_{m-k} \end{bmatrix}$ $\mathbf{y}_m = {\operatorname{argmin}}_{\mathbf{y}} \|\bar{\mathbf{H}}_{m}\mathbf{y}-\hat{\mathbf{W}}_{m+1}^T \mathbf{r}_{i-1}\|_2$ $\mathbf{x}_{i} = \mathbf{x}_{i-1} + \hat{\mathbf{V}}_m \mathbf{y}_m$ $\mathbf{r}_{i} = \mathbf{r}_{i-1} - \hat{\mathbf{W}}_{m+1}\bar{\mathbf{H}}_m\mathbf{y}_{m}$ Solve $\mathbf{f}=\mathbf{H}^{-T}_{m}\mathbf{e}_{m}$ Select the $k$ deflated eigenvectors of $[ \mathbf{H}_m + h_{m+1,m}^2 \mathbf{f} \, \mathbf{e}_m^T ] \mathbf{g} = \theta \begin{bmatrix} \hat{\mathbf{W}}_{k+1}^T \hat{\mathbf{V}}_{k+1} & \mathbf{0}_{(k+1) \times (m-k-1)} \\ \mathbf{0}_{(m-k-1) \times (k+1)} & \mathbf{I}_{m-k-1} \end{bmatrix} \mathbf{g}$ associated to the smallest eigenvalues and concatenate into $\mathbf{P}_k$ $\mathbf{Y}_k = \hat{\mathbf{V}}_{m} \mathbf{P}_{k}$ Let \[Q,R\] be the reduced QR factorization of $\bar{\mathbf{H}}_{m}\mathbf{P}_{k}$. $\mathbf{C}_k = \hat{\mathbf{V}}_{m+1} \mathbf{Q}$ $\mathbf{U}_{k} = \mathbf{Y}_k\mathbf{R}^{-1}$ Recycle $\mathbf{U}_{k}$ and $\mathbf{C}_{k}$ for the next linear system ## Application of GCRO-DR to the solution of the fluid adjoint system As for GMRES-DR in section [4.4](#GMRESDR_Fluid_Adjoint_System){reference-type="ref" reference="GMRESDR_Fluid_Adjoint_System"}, we first apply GCRO-DR(120,40) to the purely fluid adjoint system. The convergence curves are plotted in Figure [\[fig:cvg_GCRODR_120_40_impact_preconditioning_strategy_matvecprod\]](#fig:cvg_GCRODR_120_40_impact_preconditioning_strategy_matvecprod){reference-type="ref" reference="fig:cvg_GCRODR_120_40_impact_preconditioning_strategy_matvecprod"}. This Figure should be compared to the corresponding Figure [13](#fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_cvg_erratic){reference-type="ref" reference="fig:cvg_GMRESDR_120_40_impact_preconditioning_strategy_matvecprod_cvg_erratic"} for GMRES-DR. First, we note that GCRO-DR performs much better than GMRES-DR in terms of numerical robustness since the true and least-squares relative residuals match perfectly except for the preconditioner BILU(0) applied to the approximate first-order Jacobian matrix. However, the discrepancy only appears below a threshold of $5\times10^{-7}$. Again, this loss of accuracy can be suppressed by resorting to a cold restart (see Figure [22](#fig:cvg_GCRODR_120_40_impact_preconditioning_strategy_matvecprod_cvg_resratio){reference-type="ref" reference="fig:cvg_GCRODR_120_40_impact_preconditioning_strategy_matvecprod_cvg_resratio"}). In theory, GMRES-DR and GCRO-DR are algebraically equivalent [@carvalho2011flexible] but in practice our numerical experiments show that this equivalence is only observed for relative residuals above $1\times10^{-5}$, except for the most accurate preconditioner BILU(0) applied to the exact first-order Jacobian matrix. This perfect numerical equivalence is illustrated in Figure [23](#fig:cvg_OPT_comparison_GMRESDR_GCRODR){reference-type="ref" reference="fig:cvg_OPT_comparison_GMRESDR_GCRODR"}. {#fig:cvg_GCRODR_120_40_impact_preconditioning_strategy_matvecprod_cvg_erratic width="99%"} {#fig:cvg_GCRODR_120_40_impact_preconditioning_strategy_matvecprod_cvg_resratio width="99%"} {#fig:cvg_OPT_comparison_GMRESDR_GCRODR width="60%"} ## Application of GCRO-DR to the solution of the fluid-structure coupled-adjoint system As exposed in section [5](#GCRODR_theory){reference-type="ref" reference="GCRODR_theory"}, unlike GMRES-DR which is based on an augmentation deflation strategy, GCRO-DR is able to accelerate the solution of a sequence of linear systems with a slowly varying system matrix and/or right-hand side. The strategy relies on recycling an approximate invariant subspace from one system to the next. In the context of coupled-adjoint linear systems, only the right-hand side of the fluid block of equations changes during the partitioned solution strategy. In Figure [24](#fig:parameter_study_GCRODR_subspace_recycling_matvecprod){reference-type="ref" reference="fig:parameter_study_GCRODR_subspace_recycling_matvecprod"} we illustrate the acceleration of convergence for different recycling strategies. The black plain line corresponds to the convergence of the standard GMRES-DR or GCRO-DR solver preconditioned by BILU(0) applied to the first-order exact Jacobian matrix. For each new fluid-structure cycle a cold restart is performed, discarding any previous spectral information. The corresponding plateaus are clearly observable at the beginning of each cycle. The remaining curves present the convergence for a subspace recycling activated from an increasing fluid-structure cycle index. As can be seen, even recycling starting from the second cycle improves convergence compared to the standard Krylov solver. The best option seems here to recycle from the second fluid-structure coupling. This is very promising because one would have expected that the quality of the spectral information to be recycled, i.e., the distance between the true invariant subspace and its approximation, would not be small enough to get such an acceleration. A maximum saving of 1780 matrix-vector products out of 4580 ($\sim$`<!-- -->`{=html}39%) is achieved. {#fig:parameter_study_GCRODR_subspace_recycling_matvecprod width="63%"} In an attempt to understand the convergence of the approximate invariant subspace, we propose to compute the distance between $\mathbf{C}^{(i-1)}$ and $\mathbf{C}^{(i)}$ which is the best estimation of the true distance to the invariant subspace of the system matrix that we can compute. Since the size of the recycling subspace varies (in this case 30% of the size of the Krylov space for the last fluid cycle) we use the Grassmann distance formula between vector spaces of different dimensions [@ye2016schubert]: $$d_{p}(\mathbf{C}^{(i-1)}, \mathbf{C}^{(i)}) = \left( \sum^{p}_{i=1} \theta^{2}_i \right)^{1/2}, \quad p = \textrm{min}(k_{i-1}, k_{i}), \label{eq:Grassmann_distance}$$ where $\theta_i$, the principal angles between columns of $\mathbf{C}^{(i-1)}$ and $\mathbf{C}^{(i)}$, are computed via the singular value decomposition of ${\mathbf{C}^{(i-1)}}^T \mathbf{C}^{(i)}$. In order to interpret this distance criterion during convergence, we plot a normalized version of [\[eq:Grassmann_distance\]](#eq:Grassmann_distance){reference-type="eqref" reference="eq:Grassmann_distance"}: $\tilde{d}_{p}=d_{p}/\sqrt{p}$. In Figures [25](#fig:GCRO-DR_120_40_distances){reference-type="ref" reference="fig:GCRO-DR_120_40_distances"} and [26](#fig:GCRO-DR_120_40_Ck_vectors){reference-type="ref" reference="fig:GCRO-DR_120_40_Ck_vectors"} we report the histories of the subspace distance $d_p$ and of the smaller subspace dimension $p$. In case of no recycling, the distances show higher values meaning that the Krylov solver continues to search after an optimal projection space up to convergence. When recycling is activated, the last distances are about three times lower which confirms that spectral information was actually propagated between cycles. {#fig:GCRO-DR_120_40_distances width="99%"} {#fig:GCRO-DR_120_40_Ck_vectors width="99%"} To conclude our numerical experiments, Figure [27](#fig:parameter_study_GCRODR_subspace_recycling_matvecprod_BILU0_vs_BILU1){reference-type="ref" reference="fig:parameter_study_GCRODR_subspace_recycling_matvecprod_BILU0_vs_BILU1"} compares two instances of GCRO-DR with subspace recycling combined to the BILU(1) preconditioner applied to the first-order approximate Jacobian operator on one hand, and to BILU(0) applied to the first-order exact flux Jacobian operator on the other hand. Even for the simpler preconditioner a remarkable acceleration is observed. {#fig:parameter_study_GCRODR_subspace_recycling_matvecprod_BILU0_vs_BILU1 width="65%"} # Flexible Generalized Conjugate Residual with inner Orthogonalization and Deflated Restarting: FGCRO-DR {#FGCRODR_theory} ## The FGCRO-DR Krylov solver Similar to GCRO-DR, FGCRO-DR relies on a given full-rank matrix $\mathbf{Z}_{k} \in \mathbb{R}^{n \times k}$ and an orthonormal matrix $\mathbf{C}_{k} \in \mathbb{R}^{n \times k}$ as the image of $\mathbf{Z}_{k}$ by $\mathbf{A}$ satisfying the relations $$\begin{aligned} \mathbf{A}\mathbf{Z}_{k} &= \mathbf{C}_{k}, \\ \mathbf{C}^{T}_{k}\mathbf{C}_{k} &= \mathbf{I}_{k}.\end{aligned}$$ The first cycle of FGCRO-DR for solving the initial linear system in the sequence consists in applying $m$ steps of the flexible Arnoldi process to build $\mathbf{V}_{m+1}$, $\mathbf{Z}_{m}$ and $\Bar{\mathbf{H}}_{m}$ (see algorithm [\[alg:FGMRES\]](#alg:FGMRES){reference-type="ref" reference="alg:FGMRES"}). At this point, we assume that the following flexible Arnoldi relation exists: $\mathbf{AZ}_{m} = \mathbf{V}_{m+1}\bar{\mathbf{H}}_{m}$. Then, we solve the least-squares problem $\mathbf{y}_m = {\operatorname{argmin}}_{\mathbf{y}} \| \mathbf{c}-\bar{\mathbf{H}}_m \mathbf{y} \|$ with $\mathbf{c} = \| \mathbf{r}_0 \| \mathbf{e}_1$ and compute the solution $\mathbf{x}_{m} = \mathbf{x}_{0} + \mathbf{Z}_{m}\mathbf{y}_{m}$. To initiate the next cycle, we solve the standard eigenvalue problem [\[eq:Standard_eigenvalue_problem\]](#eq:Standard_eigenvalue_problem){reference-type="eqref" reference="eq:Standard_eigenvalue_problem"} and retain the eigenpairs associated to the $k$ eigenvalues with smallest magnitudes. The corresponding eigenvectors form the columns of the matrix $\mathbf{P}_k$. Following definition [\[HarmonicRitzpair\]](#HarmonicRitzpair){reference-type="ref" reference="HarmonicRitzpair"}, we then define $\mathbf{Y}_k$ as the set of $k$ harmonic Ritz vectors of $\mathbf{AZ}_{m}\mathbf{V}_m^T$ with respect to $\operatorname{range}(\mathbf{V}_m)$. Defining $[\mathbf{Q}, \mathbf{R}]$ as the reduced QR-factorization of $\bar{\mathbf{H}}_{m} \mathbf{P}_k$, we have $$\mathbf{AZ}_m \mathbf{P}_k = \mathbf{V}_{m+1} \mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_m \mathbf{P}_k = (\mathbf{V}_{m+1}\mathbf{Q})\mathbf{R}. \label{eq:AZmPk_CkR}$$ We define the subspaces for the next cycle as $\mathbf{C}_{k} = \mathbf{V}_{m+1}\mathbf{Q}$ and $\mathbf{Z}_{k} = \mathbf{Z}_m \mathbf{P}_k \mathbf{R}^{-1}$, assuming that $\mathbf{R}$ is nonsingular. Since $\mathbf{V}_{m+1}$ and $\mathbf{Q}$ are orthonormal matrices, so is $\mathbf{C}_k$. At the end of the FGMRES($m$) cycle, the optimality property $\mathbf{r}_1^{(1)} \; \bot \; \operatorname{span}\lbrace \mathbf{AZ}_m \rbrace$ of the residual $\mathbf{r}_1^{(1)}=\mathbf{r}_0^{(1)}-\mathbf{AZ}_m \mathbf{y}_m$ can be easily verified. The following theorem holds: **Theorem 2**. *The vector $\mathbf{z}_i \in \operatorname{span}\lbrace \mathbf{Z}_m \rbrace$ satisfies $\mathbf{z}_i = \underset{\mathbf{z}}{\operatorname{argmin}} \| \mathbf{r}_0 - \mathbf{Az} \|_2 \Leftrightarrow \mathbf{r}_i \; \bot \; \operatorname{span}\lbrace \mathbf{AZ}_m \rbrace$* Combining this result with relation [\[eq:AZmPk_CkR\]](#eq:AZmPk_CkR){reference-type="eqref" reference="eq:AZmPk_CkR"}, we have $$\begin{aligned} \mathbf{P}_k^T (\mathbf{AZ}_m)^{T} \mathbf{r}^{(1)}_1 &= \mathbf{0} \nonumber \\ ( \mathbf{V}_{m+1} \bar{\mathbf{H}}_m \mathbf{P}_k )^T \mathbf{r}^{(1)}_1 &= \mathbf{0} \nonumber \\ \Rightarrow \mathbf{R}^T \mathbf{C}_k^T \mathbf{r}^{(1)}_1 &= \mathbf{0},\end{aligned}$$ which shows that $\mathbf{C}_k^T \mathbf{r}^{(1)}_1 = \mathbf{0}$, with $\mathbf{C}_{k} = \mathbf{V}_{m+1}\mathbf{Q}$. To complete the next cycle $(i+1)$ and obtain the next iterates $\mathbf{r}_{i+1}^{(s)}$ and $\mathbf{x}_{i+1}^{(s)}$ for the current system $(s)$ we perform $m-k$ steps of FGMRES applied to the projected operator $\boldsymbol{\operatorname{A}}_{\boldsymbol{\operatorname{C}}_k}=(\boldsymbol{\operatorname{I}}- \boldsymbol{\operatorname{C}}_{k}\boldsymbol{\operatorname{C}}^{T}_{k})\boldsymbol{\operatorname{A}}$, i.e., we solve the projected system $\textbf{PA}\textbf{x} = \textbf{Pb}$ where $\textbf{P}=\textbf{I} -\textbf{C}_k\textbf{C}^{T}_k$ is the orthogonal projector on $\textrm{range}(\textbf{C}_k)^{\perp}$. This results in the generalized Arnoldi relation $$\textbf{AZ}_m = \textbf{V}_{m+1} \mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\textbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_m, \label{eq:flexible_generalized_Arnoldi_relation}$$ where $$\mathbf{Z}_m = \left[ \mathbf{Z}_k \quad \mathbf{Z}_{m-k} \right], \quad \mathbf{V}_{m+1} = \left[ \mathbf{C}_k \quad \mathbf{V}_{m-k+1} \right] \quad\textrm{and} \quad \mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_{m} = \begin{bmatrix} \mathbf{I}_k & \mathbf{B}_{m-k} \\ \mathbf{0} & \mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_{m-k} \end{bmatrix}\!\!, \label{eq:flexible_generalized_Arnoldi_matrices}$$ with $\mathbf{B}_{m-k} = \mathbf{C}_k^{T} \mathbf{AZ}_{m-k}, \mathbf{B}_{m-k} \in \mathbb{R}^{k \times (m-k)}$. To build $\mathbf{V}_{m+1}$ the Arnoldi recurrence restarts from $\mathbf{v}_{k+1} = \mathbf{r}_1 / \| \mathbf{r}_1 \|$ with $\mathbf{r}_1 = (\mathbf{I} - \mathbf{C}_{k}\mathbf{C}^{T}_{k})\mathbf{r}_{0}$. To initiate the subsequent cycle or to solve the next linear system, we compute $k$ eigenvectors of a generalized eigenvalue problem that we will specify in section [6.2](#Flexible_deflation_FGCRODR){reference-type="ref" reference="Flexible_deflation_FGCRODR"} hereafter. These eigenvectors are concatenated in matrix $\mathbf{P}_k$. Again, introducing the reduced QR-factorization of $\bar{\mathbf{H}}_{m} \mathbf{P}_k$, we define the new residual and solution subspaces $$\begin{aligned} \mathbf{C}_k &= \mathbf{V}_{m+1} \mathbf{Q} \\ \mathbf{Z}_k &= \mathbf{Z}_m \mathbf{P}_k \mathbf{R}^{-1}. \label{eq:Zk}\end{aligned}$$ To initiate the solving of the next linear system we compute the projected initial solution and residual with $$\begin{aligned} \mathbf{x}_{1} &= \mathbf{x}_{0} + \mathbf{Z}_{k}\mathbf{C}^{T}_{k}\mathbf{r}_{0} \\ \mathbf{r}_{1} &= (\mathbf{I} - \mathbf{C}_{k}\mathbf{C}^{T}_{k})\mathbf{r}_{0}\end{aligned}$$ ## Flexible deflation strategies {#Flexible_deflation_FGCRODR} From the definition of harmonic Ritz vectors, there exits different ways of deriving the deflated subspace according to the choice of operator $\textbf{B}$ and vector space $\mathcal{U}$. In the technical report [@carvalho2010report], Carvalho et al. proposed three formulations (labeled A, B and C) for the projected generalized eigenvalue problem. We review these variants below and give additional insight for strategy B.\ [**Strategy A**]{.ul}: This strategy has already been considered for FGMRES-DR in section [4.3](#StrategyA_FGMRESDR){reference-type="ref" reference="StrategyA_FGMRESDR"}, see relation [\[eq:Alternative_deflation_strategy_b\]](#eq:Alternative_deflation_strategy_b){reference-type="eqref" reference="eq:Alternative_deflation_strategy_b"}. Defining $\mathbf{B} \equiv \mathbf{AZ}_{m} \mathbf{Z}_m^{\dagger}$ and $\mathcal{U} \equiv \operatorname{range}(\mathbf{Z}_m)$, the harmonic Ritz pair $(\lambda,\mathbf{y}=\mathbf{Z}_m \mathbf{g})$ satisfies $$\label{eq:FGCRODR_StrategyA} ((\mathbf{AZ}_{m} \mathbf{Z}_m^{\dagger}) \mathbf{Z}_m)^T ((\mathbf{AZ}_{m} \mathbf{Z}_m^{\dagger}) \mathbf{Z}_m \mathbf{g} - \lambda \mathbf{Z}_{m} \mathbf{g}) = \mathbf{0}.$$ Thus, $\mathbf{Y}_m = \{ \mathbf{y}_1, \cdots, \mathbf{y}_m \}$ corresponds to the harmonic Ritz vectors of $\mathbf{AZ}_{m} \mathbf{Z}_m^{\dagger}$ with respect to $\operatorname{range}(\mathbf{Z}_m)$. Also, inserting [\[eq:flexible_generalized_Arnoldi_relation\]](#eq:flexible_generalized_Arnoldi_relation){reference-type="eqref" reference="eq:flexible_generalized_Arnoldi_relation"} in [\[eq:FGCRODR_StrategyA\]](#eq:FGCRODR_StrategyA){reference-type="eqref" reference="eq:FGCRODR_StrategyA"}, the eigenpair $(\lambda, \mathbf{g})$ satisfies the generalized eigenvalue problem $$\boxed{\Bar{\mathbf{H}}_{m}^{T} \Bar{\mathbf{H}}_{m}\mathbf{g} = \lambda \bar{\mathbf{H}}^T_{m} \mathbf{V}^T_{m+1} \mathbf{Z}_m\mathbf{g}}.$$ [**Strategy B**]{.ul}: This strategy has also already been considered for FGMRES-DR in section [4.2](#DeflatedRestarting_FGMRESDR){reference-type="ref" reference="DeflatedRestarting_FGMRESDR"}, see relations [\[eq:Generalized_eigenvalue_problem_a\]](#eq:Generalized_eigenvalue_problem_a){reference-type="eqref" reference="eq:Generalized_eigenvalue_problem_a"} and [\[eq:Generalized_eigenvalue_problem_b\]](#eq:Generalized_eigenvalue_problem_b){reference-type="eqref" reference="eq:Generalized_eigenvalue_problem_b"}. Letting $\mathbf{V}_{m} = \left[ \textbf{C}_k \quad \textbf{V}_{m-k} \right]$ and choosing $\mathbf{B} \equiv \mathbf{AZ}_{m}\mathbf{V}_m^{T}$ and $\mathcal{U} \equiv \operatorname{range}(\mathbf{V}_m)$, the harmonic Ritz pair $(\lambda,\mathbf{y}=\mathbf{V}_m \mathbf{g})$ satisfies $$((\mathbf{AZ}_{m}\mathbf{V}_m^{T}) \mathbf{V}_m)^T ((\mathbf{AZ}_{m}\mathbf{V}_m^{T}) \mathbf{V}_m \mathbf{g} - \lambda \mathbf{V}_m \mathbf{g}) = \mathbf{0}.$$ Thus, $\mathbf{Y}_m = \{ \mathbf{y}_1, \cdots, \mathbf{y}_m \}$ corresponds to the harmonic Ritz vectors of $\mathbf{AZ}_{m}\mathbf{V}_m^{T}$ with respect to $\operatorname{range}(\mathbf{V}_m)$. Also, the eigenpair $(\lambda, \mathbf{g})$ satisfies the generalized eigenvalue problem $$\boxed{\Bar{\mathbf{H}}_{m}^{T} \Bar{\mathbf{H}}_{m}\mathbf{g} = \lambda \mathbf{H}_{m}^{T} \mathbf{g}}. \label{eq:FGCRODR_generalized_eigenvalue_problem}$$ This generalized eigenvalue problem can be reformulated as a standard one, see equation [\[eq:Standard_eigenvalue_problem\]](#eq:Standard_eigenvalue_problem){reference-type="eqref" reference="eq:Standard_eigenvalue_problem"}. We have $\hat{\mathbf{H}}_m\mathbf{g}=\lambda\mathbf{g}$, with $\hat{\mathbf{H}}_m = [ \mathbf{H}_m + h^2 \mathbf{f} \, \mathbf{e}_m^T ]$ and $\mathbf{H}_m = [\mathbf{I}_m \quad \mathbf{0}_{m \times 1}]\mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_m$. The block upper triangular structure of $\mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_m$ is highlighted in [\[eq:flexible_generalized_Arnoldi_matrices\]](#eq:flexible_generalized_Arnoldi_matrices){reference-type="eqref" reference="eq:flexible_generalized_Arnoldi_matrices"} with a leading $k \times k$ identity block. More specifically we have $$\hat{\mathbf{H}}_m = \begin{bmatrix} \mathbf{I}_k & \tilde{\mathbf{B}}_{m-k} \\ \mathbf{0} & \tilde{\mathbf{H}}_{m-k} \end{bmatrix} \label{eq:FGCRODR_generalized_matrix}$$ with $\tilde{\mathbf{B}}_{m-k} = \mathbf{B}_{m-k} + h^2 \mathbf{f}_{[1:k]} \mathbf{e}_{m-k}^T$ and $\tilde{\mathbf{H}}_{m-k} = \mathbf{H}_{m-k} + h^2 \mathbf{f}_{[k+1:m]} \mathbf{e}_{m-k}^T$, $\mathbf{e}_{m-k} = [0 \cdots 0, 1]^T \in \mathbb{R}^{m-k}$. We assume here that the Hessenberg sub-matrix $\mskip.5\thinmuskip\overline{\mskip-.5\thinmuskip {\mathbf{H}} \mskip-.5\thinmuskip}\mskip.5\thinmuskip_{m-k}$ is unreduced, meaning that there are no zero elements on the subdiagonal (the problem of multiple eigenvalues is discussed in [@morgan_Implicit_2000]). The eigenvalues of [\[eq:FGCRODR_generalized_matrix\]](#eq:FGCRODR_generalized_matrix){reference-type="eqref" reference="eq:FGCRODR_generalized_matrix"} are the combined eigenvalues of the diagonal blocks of $\hat{\mathbf{H}}_m$ and therefore satisfy $\operatorname{det}(\hat{\mathbf{H}}_m - \lambda \mathbf{I}_m) = \operatorname{det}((1-\lambda)\mathbf{I}_k) \operatorname{det}(\tilde{\mathbf{H}}_{m-k} -\lambda\mathbf{I}_{m-k}) = 0$. Thus, we have a unit eigenvalue of algebraic multiplicity $k$. Now, if $\lambda=1$ is an eigenvalue of the upper diagonal block $\mathbf{I}_k$, with associated eigenvectors $\mathbf{g}_k=\mathbf{I}_k$, then it is also an eigenvalue of the full matrix $\hat{\mathbf{H}}_m$, with the same eigenvectors augmented with zeros, which gives $\mathbf{g}_m^{(\lambda=1)} = [ \mathbf{g}_k \quad \mathbf{0}_{(m-k) \times k}]^T = [ \mathbf{I}_k \quad \mathbf{0}_{(m-k) \times k}]^T$. This shows that the $k$ eigenvectors associated to the unit eigenvalues are linearly independent. We can also obtain an explicit expression for the complementary eigenvectors associated to the eigenvalues of the lower diagonal block of $\hat{\mathbf{H}}_m$. Let $(\lambda_{m-k},\mathbf{g}_{m-k})$ be an eigenpair of $\tilde{\mathbf{H}}_{m-k}$. The full eigenvalue problem reads $$\begin{bmatrix} \mathbf{I}_k & \tilde{\mathbf{B}}_{m-k} \\ \mathbf{0} & \tilde{\mathbf{H}}_{m-k} \end{bmatrix} \begin{pmatrix} \mathbf{x}_k \\ \mathbf{g}_{m-k} \end{pmatrix} = \begin{pmatrix} \mathbf{x}_k +\tilde{\mathbf{B}}_{m-k} \mathbf{g}_{m-k} \\ \lambda_{m-k} \mathbf{g}_{m-k} \end{pmatrix}\!\!.$$ We can make $\mathbf{x}_k +\tilde{\mathbf{B}}_{m-k} \mathbf{g}_{m-k} = \lambda_{m-k} \mathbf{x}_k$ by choosing $\mathbf{x}_k=-(1-\lambda_{m-k})^{-1} \tilde{\mathbf{B}}_{m-k} \mathbf{g}_{m-k}$. By assumption $\lambda_{m-k} \neq 1$ since it is not an eigenvalue of the leading block of $\hat{\mathbf{H}}_m$. Thanks to these relations, the deflated subspace associated to strategy B is obtained efficiently. In light of this attractive property, Jolivet and Tournier proposed a block version of GCRO-DR combined with this deflation strategy [@jolivet2016block].\ [**Strategy C**]{.ul}: This third strategy was adopted in [@carvalho2011flexible] where the FGCRO-DR algorithm maintains an additional set of vectors $\mathbf{W}_m$ such that the deflated harmonic Ritz vectors read $\mathbf{Y}_k = \mathbf{W}_m \mathbf{P}_k$. Similar to $\mathbf{Z}_k$ in [\[eq:Zk\]](#eq:Zk){reference-type="eqref" reference="eq:Zk"}, a set $\mathbf{W}_k = \mathbf{W}_m \mathbf{P}_k \mathbf{R}^{-1}$ is propagated between cycles. The complete subspace is built by appending the Krylov vectors generated during the Arnoldi process: $\mathbf{W}_{m} = \left[ \textbf{W}_k \quad \textbf{V}_{m-k} \right]$. This particular subspace was used to prove the algebraic equivalence of FGMRES-DR and FGCRO-DR under a specific colinearity constraint. As a consequence, strict algebraic equivalence of GMRES-DR and GCRO-DR was also demonstrated by the authors. Note that this property was stated in [@parks2006recycling] without demonstration. Using definition [\[HarmonicRitzpair\]](#HarmonicRitzpair){reference-type="ref" reference="HarmonicRitzpair"} with $\mathcal{U} \equiv \operatorname{range}(\mathbf{W}_m)$ and $\mathbf{B} \equiv \mathbf{AZ}_{m}\mathbf{W}_m^{\dagger}$ we can write $\mathbf{y} = \mathbf{W}_m \mathbf{g}$ which gives the orthogonality constraint $$((\mathbf{AZ}_{m}\mathbf{W}_m^{\dagger}) \mathbf{W}_m)^T ((\mathbf{AZ}_{m}\mathbf{W}_m^{\dagger}) \mathbf{W}_m \mathbf{g} - \lambda \mathbf{W}_m \mathbf{g}) = \mathbf{0}.$$ Now, using [\[eq:flexible_generalized_Arnoldi_relation\]](#eq:flexible_generalized_Arnoldi_relation){reference-type="eqref" reference="eq:flexible_generalized_Arnoldi_relation"} and recalling that $\mathbf{V}_{m+1}$ is orthonormal, we obtain the generalized eigenvalue problem $$% \label{eq:GCRO_eigenvalue_problem} \boxed{\bar{\mathbf{H}}^T_m \bar{\mathbf{H}}_m \mathbf{g} = \lambda \bar{\mathbf{H}}^T_m \mathbf{V}^T_{m+1} \mathbf{W}_m \mathbf{g}}$$ To reduce ill-conditioning, an alternative formulation for this generalized eigenvalue problem has been proposed in equation [\[eq:Reformulated_GCRO_eigenproblem\]](#eq:Reformulated_GCRO_eigenproblem){reference-type="eqref" reference="eq:Reformulated_GCRO_eigenproblem"}.\ ## Application of FGCRO-DR to the fluid adjoint system Our first numerical experiments compare the three deflation strategies detailed in section [6.2](#Flexible_deflation_FGCRODR){reference-type="ref" reference="Flexible_deflation_FGCRODR"} for the solving of the fluid adjoint linear system. We consider increasing sizes of Krylov bases of 30, 50 and 70 vectors. The deflated subspace size is half of the Krylov basis size except for the smaller basis where we keep only one third of the approximation subspace. The variable preconditioner consists in a non-restarted inner GMRES Krylov solver with a basis of size 10. The innermost stationary preconditioner is BILU(0) applied to the first-order exact Jacobian matrix. The corresponding convergence histories are presented in the left-hand side of Figure [\[fig:cvg_OPT_FGCRODR_impact_deflation_strategy_various_sizes_Krylov_space\]](#fig:cvg_OPT_FGCRODR_impact_deflation_strategy_various_sizes_Krylov_space){reference-type="ref" reference="fig:cvg_OPT_FGCRODR_impact_deflation_strategy_various_sizes_Krylov_space"}. Clearly, strategy C seems the most effective regardless the size of the Krylov space. However, when the number of vector increases, the three deflation strategies perform similarly. Indeed, the right-hand side of Figure [\[fig:cvg_OPT_FGCRODR_impact_deflation_strategy_various_sizes_Krylov_space\]](#fig:cvg_OPT_FGCRODR_impact_deflation_strategy_various_sizes_Krylov_space){reference-type="ref" reference="fig:cvg_OPT_FGCRODR_impact_deflation_strategy_various_sizes_Krylov_space"} shows very close convergence profiles of FGCRO-DR(70,35) for the three deflation strategies. There is no theoretical evidence that a deflation strategy should perform better than another one. As a consequence, these conclusions are likely to be case dependent. {#fig:cvg_OPT_FGCRODR_impact_deflation_strategy_various_sizes_Krylov_space_matvecprod width="99%"} {#fig:cvg_OPT_FGCRODR_70_10_35_impact_deflation_strategy_matvecprod width="99%"} ## Application of FGCRO-DR to the solution of the fluid-structure coupled-adjoint system In this section we study the impact of recycling spectral information between fluid-structure cycles in the context of GCRO-DR with variable preconditioning. From our previous numerical experiments with FGMRES-DR and FGCRO-DR applied to the fluid adjoint system, we choose to recycle a subspace of size half the size of the Krylov subspace, i.e., the same size as the one of the deflated subspace. In [@carvalho2010report] the authors suggest that larger recycled subspaces give better performance in the context of a flexible Krylov solver used to solve linear systems arising from a specific class of problems in quantum chromodynamics. For instance, for an outer space of size $m=20$, they carried out a parameter study by varying the number of deflated vectors from 1 to $m-1$. Regardless of the selected recycling strategy, the lowest computational cost was achieved for a size of the recycled subspace higher than 50% of the size of the Krylov subspace. In another technical report [@carvalho2011report] the same authors solve a sequence of twelve elliptic partial differential equation problems for an increasing grid size. Their numerical experiments show that a reduction in the range of 40% to 45% can be achieved by recycling an approximate invariant subspace of half the size of the Krylov vector space compared to a standard FGMRES-DR applied to each system separately. These results seem to confirm the similar gains observed for the non-flexible GCRO-DR Krylov solver. We recall that in our context, the sequence of linear adjoint systems corresponds to a varying right-hand side arising from the update of the structural source term during the partitioned fluid-structure solution strategy. After a number of fluid-structure cycles, the right-hand side is expected to converge and we end up with a constant linear system to solve. We then apply FGCRO-DR($m=70$,$m_i=10$,$k=35$), $k$ being the number of deflated and recycled vectors, combined with the three deflation strategies A, B and C. Table [1](#tab:strategies_deflation_performances){reference-type="ref" reference="tab:strategies_deflation_performances"} collects the corresponding numbers of matrix-vector products. Using recycling helps to improve the convergence rate of flexible GCRO-DR in this application since a reduction of approximately 16% to 19% in terms of matrix-vector products is obtained. -------------------------- ---------- ---------- ---------- Starting recycling cycle \#Mvps \#Mvps \#Mvps N/A **8940** **8754** **9108** **8790** *12018* **9250** -------------------------- ---------- ---------- ---------- : Total number of matrix-vector products \#Mvps for strategies A, B and C, indexed by the starting recycling cycle. The lowest \#Mvps in each row is highlighted in bold and should be compared to the reference \#Mvps=10815. Figures [\[fig:res_hist_M6W_AOC_fgcrodr_strategy_A\_MGS2\]](#fig:res_hist_M6W_AOC_fgcrodr_strategy_A_MGS2){reference-type="ref" reference="fig:res_hist_M6W_AOC_fgcrodr_strategy_A_MGS2"}, [\[fig:FGMRES-DR_70_10_35_subspace_recycling_monitoring\]](#fig:FGMRES-DR_70_10_35_subspace_recycling_monitoring){reference-type="ref" reference="fig:FGMRES-DR_70_10_35_subspace_recycling_monitoring"} and [\[fig:res_hist_M6W_AOC_fgcrodr_strategy_B\_MGS2\]](#fig:res_hist_M6W_AOC_fgcrodr_strategy_B_MGS2){reference-type="ref" reference="fig:res_hist_M6W_AOC_fgcrodr_strategy_B_MGS2"} show the convergence histories of FGCRO-DR(70,10,35) for strategy A, B and C respectively. We have reported convergences for a varying starting recycling fluid-structure cycle. Regardless the deflation strategy, the plateau following a fluid-structure coupling restart is always noticeably reduced. A comparison of distances between FGCRO-DR without recycling and FGCRO-DR with a subspace recycling starting from cycle 5 is reported in Figure [32](#fig:FGMRES-DR_70_10_35_Ck_distances){reference-type="ref" reference="fig:FGMRES-DR_70_10_35_Ck_distances"}. Clearly the subspace distance is much smaller when recycling is activated, whereas it keeps constant for the solver with cold restart after a fluid-structure coupling. The dimension $p$ in [\[eq:Grassmann_distance\]](#eq:Grassmann_distance){reference-type="eqref" reference="eq:Grassmann_distance"} is plotted if Figure [33](#fig:FGMRES-DR_70_10_35_Ck_vectors){reference-type="ref" reference="fig:FGMRES-DR_70_10_35_Ck_vectors"}. With recycling, the dimension of the approximation Krylov space is larger which favors a better convergence. {#fig:cvg_AOC_FGCRODR_70_35_RATIO_060_MGS2_impact_deflation_strategy_A_iterations width="99%"} {#fig:cvg_AOC_FGCRODR_70_35_RATIO_060_MGS2_impact_deflation_strategy_A_matvecprod width="99%"} {#fig:FGMRES-DR_70_10_35_Ck_distances width="99%"} {#fig:FGMRES-DR_70_10_35_Ck_vectors width="99%"} {#fig:cvg_AOC_FGCRODR_70_35_RATIO_060_MGS2_impact_deflation_strategy_B_iterations width="99%"} {#fig:cvg_AOC_FGCRODR_70_35_RATIO_060_MGS2_impact_deflation_strategy_B_matvecprod width="99%"} {#fig:cvg_AOC_FGCRODR_70_35_RATIO_060_MGS2_impact_deflation_strategy_C_iterations width="99%"} {#fig:cvg_AOC_FGCRODR_70_35_RATIO_060_MGS2_impact_deflation_strategy_C_matvecprod width="99%"} # Conclusion In this paper we have accelerated the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging. This recycling and deflation strategies considered in this work were inspired by the theoretical developments related to GCRO-DR and its flexible variant detailed in [@carvalho2010report; @carvalho2011flexible]. Our objective was to make this paper as self-contained as possible and in that respect, we choose to recall theoretical details of both GCRO-DR and FGCRO-DR. It also gave us the opportunity to point out practical implementation details mainly to make the computation of harmonic Ritz vector more stable. We demonstrate the benefit of these techniques by computing the coupled derivatives for an aeroelastic configuration of the ONERA-M6 fixed wing in transonic flow representative of the numerical complexity of realistic industrial applications. For this exercise the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. All computations were performed using RANS flow modeling and a fully linearized one-equation Spalart-Allmaras turbulence model, which typically results in stiff linear systems. For the non flexible GCRO-DR solver, subspace recycling between fluid-structure cycles achieved a reduction of 39% in terms of matrix-vector products with respect to the legacy solver. Even recycling early from the second cycle led to a reduction in computational cost of 37% showing the robustness of the proposed strategy. The conclusions with respect to the Flexible GCRO-DR are more mitigated, even if recycling almost systematically ended up with a reduction of the total number of matrix-vector products. Indeed, gains were achieved when recycling was activated from the fourth fluid-structure cycle only, regardless of the deflation strategy A, B or C. Deflation strategy A seems to be more consistent when comparing convergence profiles for various starting recycling cycles. For our application, a reduction of approximately 16% to 19% in terms of matrix-vector products was still obtained. However, the convergence seems less smooth exhibiting some local stagnations. This suggests that the recycled subspace may not always be suitable and a dynamic recycling approach would certainly be beneficial in this case. This will be part of a future work. [^1]: elsA and is the joint property of ONERA and Safran.

arxiv_math

--- abstract: | We study Constraint Satisfaction Problems (CSPs) in an infinite context. We show that the dichotomy between easy and hard problems --established already in the finite case-- presents itself as the strength of the corresponding De Bruijin-Erdős-type compactness theorem over ZF. More precisely, if $\mathcal{D}$ is a structure, let $K_\mathcal{D}$ stand for the following statement: for every structure $\mathcal{X}$ if every finite substructure of $\mathcal{X}$ admits a solution to $\mathcal{D}$, then so does $\mathcal{X}$. We prove that if $\mathcal{D}$ admits no cyclic polymorphism, and thus it is NP-complete by the CSP Dichotomy Theorem, then $K_\mathcal{D}$ is equivalent to the Boolean Prime Ideal Theorem (BPI) over ZF. Conversely, we also show that if $\mathcal{D}$ admits a cyclic polymorphism, and thus it is in P, then $K_\mathcal{D}$ is strictly weaker than BPI. address: - Eötvös Loránd University, Institute of Mathematics, Pázmány Péter stny. 1/C, 1117 Budapest, Hungary - Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15., 1053 Budapest, Hungary - Eötvös Loránd University, Institute of Mathematics, Pázmány Péter stny. 1/C, 1117 Budapest, Hungary author: - Tamás Kátay - László Márton Tóth - Zoltán Vidnyánszky bibliography: - bibliography.bib title: The CSP Dichotomy, the Axiom of Choice, and Cyclic Polymorphisms --- [^1] # Introduction By *structure* we mean a relational structure with finitely many relations. For structures $\mathcal{X}$ and $\mathcal{D}$ we say that $\mathcal{X}$ is an *instance of the $\mathcal{D}$-homomorphism problem*, or, in short, an *instance of $\mathcal{D}$* if their signature coincides. If $\mathcal{D}$ is a finite structure, the *(finite) $\mathcal{D}$-homomorphism problem* is the problem to decide whether a given (finite) instance of $\mathcal{D}$ admits a homomorphism to $\mathcal{D}$. These homomorphism problems are also called *Constraint Satisfaction Problems* (CSPs). Examples of CSPs include the problem of finding a proper $n$-coloring of a graph, graph homomorphism problems in general, but also solving various types of SAT problems or systems of linear equations over finite fields (see, e.g., [@feder1998computational]). It is easy to check that CSPs are always in $NP$, and after decades of concerted efforts toward understanding the computational complexity of CSPs, Bulatov [@bulatov2017dichotomy] and Zhuk [@zhuk2020proof] independently proved the remarkable CSP Dichotomy Theorem (conjectured by Feder and Vardi [@feder1998computational]): **Theorem 1** (Bulatov, Zhuk). *The $\mathcal{D}$-homomorphism problem is either in $P$ or $NP$-complete. More specifically:* 1. *if $\mathcal{D}$ admits a nontrivial polymorphism, then the $\mathcal{D}$-homomorphism problem is in $P$;* 2. *if $\mathcal{D}$ does not admit a nontrivial polymorphism, then the $\mathcal{D}$-homomorphism problem is $NP$-complete.* A *polymorphism* is a homomorphism $\phi: \mathcal{D}^n \to \mathcal{D}$, where $\mathcal{D}^n$ is the categorical power of the structure $\mathcal{D}$. We say it is *nontrivial* if it is not essentially a projection to one of the coordinates (see Definition [Definition 6](#d:tractability){reference-type="ref" reference="d:tractability"}). In this paper, we prove an infinitary version of the CSP Dichotomy Theorem, inspired by a remark of Halbeisen. Let $K_{\mathcal{D}}$ stand for the following statement: for every (possibly infinite) instance $\mathcal{X}$ of the $\mathcal{D}$-homomorphism problem, if every finite substructure of $\mathcal{X}$ admits a homomorphism to $\mathcal{D}$, then so does $\mathcal{X}$. For example, if $\mathcal{D}=K_n$, the complete graph on $n$ vertices, then the statement $K_{K_n}$ is the De Bruijn-Erdős theorem, namely, if every finite subgraph of a graph is $n$-colorable, then so is the whole graph. **Theorem 2**. *(ZF) [\[t:main\]]{#t:main label="t:main"}* 1. *[\[c:maineasy\]]{#c:maineasy label="c:maineasy"} If $\mathcal{D}$ admits a nontrivial polymorphism, then $K_{\mathcal{D}}$ is strictly weaker than BPI. In fact, there is a model of ZF in which $K_{\mathcal{D}}$ is true for exactly those $\mathcal{D}$ that admit nontrivial polymorphisms.* 2. *[\[c:mainhard\]]{#c:mainhard label="c:mainhard"} If $\mathcal{D}$ does not admit a nontrivial polymorphism, then $K_{\mathcal{D}} \iff BPI$.* This theorem generalizes results of Lévy, Mycielski, and Läuchli [@levy1963remarks; @mycielski1961some; @lauchli1971coloring; @cowen1990two], and verifies a conjecture posed in [@cowen1990two (7) p. 238]. The key idea of our proof is to use the theory of cyclic polymorphisms (see Section [2](#s:detailed){reference-type="ref" reference="s:detailed"}). Our arguments are significantly softer than those of Bulatov and Zhuk in their proofs of the CSP Dichotomy Theorem, hence the following natural question arises. **Question 3**. *Is it possible to adapt some ideas from the proof of Theorem [\[t:main\]](#t:main){reference-type="ref" reference="t:main"} to give a novel proof of the CSP Dichotomy Theorem?* In Section [2](#s:detailed){reference-type="ref" reference="s:detailed"} we provide a brief history of the CSP Dichotomy Theorem and some of the possible infinitary directions. We also give an overview of the proof ideas of our result. We formulate a black box theorem, which can be used without familiarity with forcing or abstract set theory. In Sections [3](#s:hard){reference-type="ref" reference="s:hard"} and [4](#s:easy){reference-type="ref" reference="s:easy"} we prove the two parts of the main theorem. The formal proof of the former statement is rather heavy in notation, so we postpone it to the Appendix. Finally, we conclude the paper by listing a couple of open problems. # Context and proof overview {#s:detailed} ## The CSP Dichotomy Theorem It was observed early on that in every known case the $\mathcal{D}$-homomorphism problem is either in $P$ or $NP$-complete. Schaefer [@schaefer1978complexity] proved this to be the case when $D$ has size at most two[^2]. Later, Hell and Nesetřil [@hell1990complexity] gave a complete characterization when $\mathcal{D}$ is a graph: they showed that if $\mathcal{D}$ is bipartite, then the problem is in $P$, and otherwise it is $NP$-complete. Based on the mounting evidence, Feder and Vardi [@feder1998computational] formulated the statement which became known as the CSP Dichotomy Conjecture: $$\text{The $\mathcal{D}$-homomorphism problem is either in $P$ or $NP$-complete}.$$ Note that by a classical theorem of Ladner [@ladner1975structure], if $P \neq NP$ then there are problems in $NP$ which are neither in $P$ nor $NP$-complete. Since the formulation of the conjecture, an immense amount of effort has been put into trying to verify it (some of the outstanding results are [@maroti2008existence; @kun2012linear; @barto2009constraint; @barto2012absorbing; @barto2009csp], see [@brady2022notes] for a comprehensive introduction). Bulatov, Jeavons and Krokhin [@bulatov2005classifying] isolated an algebraic condition: the existence of nontrivial polymorphisms of $\mathcal{D}$ that seemed to characterize the easy problems. They proved that if $\mathcal{D}$ has no nontrivial polymorphisms, then the $\mathcal{D}$-homomorphism problem is $NP$-complete and conjectured that conversely, if $\mathcal{D}$ admits a nontrivial polymorphism, then the problem is in $P$. Finally, as a culmination of all these works, Bulatov [@bulatov2017dichotomy] and Zhuk [@zhuk2020proof] independently proved the conjecture, see Theorem [Theorem 1](#t:dichotomy){reference-type="ref" reference="t:dichotomy"} above. ## Cyclic polymorphisms The key algebraic tool for the investigation of CSPs turned out to be polymorphisms. Recall that a(n) *($n$-ary) polymorphism* is a homomorphism $\phi: \mathcal{D}^n \to \mathcal{D}$. Here $\mathcal{D}^n$ is the categorical power of the structure $\mathcal{D}$, i.e. if $R$ is a $k$-ary relation of $\mathcal{D}$, then it is interpreted on $\mathcal{D}^n$ as follows: let $\overline{x}_i=\big(\overline{x}_i(1), \ldots, \overline{x}_i(n)\big) \in D^n$ for $i=1, \ldots,k$, then $(\overline{x}_1, \ldots, \overline{x}_k) \in R^{\mathcal{D}^n} \iff \big(\overline{x}_1(j), \ldots \overline{x}_k(j)\big) \in R^{\mathcal{D}}$ for all $1\leq j\leq n$. A useful way to look at, say, $n$-ary polymorphisms is that they combine $n$-many homomorphisms to $\mathcal{D}$ into a new one. Clearly, projection maps are always polymorphisms. There are several results which show that the existence of essentially non-projective polymorphisms imply the existence of ones of special forms. In our considerations the following type will play a crucial role. **Definition 4**. A polymorphism $\phi: \mathcal{D}^n \to \mathcal{D}$, is called *cyclic*, if it satisfies $$\phi(x_0,x_1,\dots,x_{n-1})=\phi(x_1,x_2,\dots,x_{n-1},x_0),$$ for all $(x_0,\dots,x_{n-1}) \in D^n$. One of the fundamental results of the area is the following. **Theorem 5** ([@barto2012absorbing; @siggers2010strong]). *The following are equivalent for a structure $\mathcal{D}$:* 1. *$\mathcal{D}$ admits an identity of polymorphisms (i.e., a multivariable system of equations) not satisfied by projections.* 2. *[\[c:important\]]{#c:important label="c:important"} $\mathcal{D}$ admits a cyclic polymorphism of every large enough prime arity.* 3. *$\mathcal{D}$ admits a polymorphism $f:\mathcal{D}^4 \to \mathcal{D}$ with $$f(r,a,r,e)=f(a,r,e,a),$$ for all $a,e,r\in\mathcal{D}$.* **Definition 6**. $(*)_\mathcal{D}$ will denote the statement that some (all) of the above conditions hold. ## Infinite versions of the CSP Dichotomy Let us briefly mention that there are several infinite versions of the CSP dichotomy problem. A flourishing direction is to allow $\mathcal{D}$ to be infinite, while still requiring $\mathcal{X}$ to be finite and ask about the computational complexity of the problem. In this case a rich structure theory emerges, with a myriad of questions yet to be answered (see, e.g., [@bodirsky2012complexity; @bodirsky2006constraint; @bodirsky2015schaefer; @bodirsky2015topological]). Another direction, which has been recently initiated by Thornton [@riley] is to keep $\mathcal{D}$ finite, and require the homomorphisms and the instance to be Borel. An advantage of this approach is that in the Borel context, it can be proved that say, $2$-coloring is easier than $3$-coloring, see [@toden]. It seems however, that solving systems of linear equations over finite fields is already hard in the Borel context [@grebikv], thus, the split between hard and easy problems occurs at a different place. The direction explored in this paper is to keep $\mathcal{D}$ finite and investigate the strength of the statement $K_{\mathcal{D}}$ for infinite $\mathcal{X}$. We build on the work of Lévy, Mycielski, and Läuchli [@levy1963remarks; @mycielski1961some; @lauchli1971coloring], who showed that over ZF, for $n \geq 3$ the statement $K_{K_n}$ is equivalent to BPI and $K_{K_2}$ is significantly weaker (see also [@howard1998consequences; @halbeisen2012combinatorial]). Our Theorem [\[t:main\]](#t:main){reference-type="ref" reference="t:main"} shows that the same split between easy and hard problems occurs in the infinite context as in the CSP Dichotomy. That is, $K_\mathcal{D}$ is equivalent to BPI if $(*)_\mathcal{D}$ does not hold, while otherwise it is a weaker statement. Note also that in our case this is a dichotomy in the usual sense, unlike the finite case, where if $P=NP$ then the two cases collapse into one. In the rest of the paper, we work over ZF, unless specified otherwise. ## Hard problems. First, let us discuss our proof of the second statement of Theorem [\[t:main\]](#t:main){reference-type="ref" reference="t:main"}. We will build on the work Bulatov-Jeavons-Krokhin [@bulatov2005classifying] and Thornton [@riley]. They provide a constructive way of reduction between homomorphism problems of structures for which $(*)_{\mathcal{D}}$ fails. This reduction turns out to be essentially sufficient for our purposes as well. Motivated by the statements $K_{\mathcal{D}}$, we define a new notion of reduction between problems. **Definition 7**. Let $\mathcal{D}$ and $\mathcal{E}$ be finite structures. We say that the *$\mathcal{E}$-homomorphism problem finitely reduces to the $\mathcal{D}$-homomorphism problem*, or in short, *$\mathcal{E}$ finitely reduces to $\mathcal{D}$*, if there exist operations $\Gamma$, $\Phi$ and $\Psi$ such that: \(1\) if $\mathcal{X}$ is an instance of $\mathcal{E}$, then $\Gamma(\mathcal{X})$ is an instance of $\mathcal{D}$; \(2\) for every instance $\mathcal{X}$ of $\mathcal{E}$ the operation $\Phi$ maps $\mathcal{X}\to\mathcal{E}$ homomorphisms to $\Gamma(\mathcal{X})\to\mathcal{D}$ homomorphisms and $\Psi$ maps $\Gamma(\mathcal{X})\to\mathcal{D}$ homomorphisms to $\mathcal{X}\to\mathcal{E}$ homomorphisms; \(3\) if there exists a finite substructure $\mathcal{H}$ of $\Gamma(\mathcal{X})$ that does not admit a homomorphism to $\mathcal{D}$, then there exists a finite substructure $\mathcal{F}$ of $\mathcal{X}$ that does not admit a homomorphism to $\mathcal{E}$. **Remark 8**. Observe that finite reducibility is a transitive relation. **Remark 9**. Assuming that (1) and (2) of Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"} are satisfied, to check (3) it suffices to verify the following. For every finite substructure $\mathcal{H}$ of $\Gamma(\mathcal{X})$ there exists a finite substructure $\mathcal{F}$ of $\mathcal{X}$ such that there exists an $\mathcal{H}\to \Gamma(\mathcal{F})$ homomorphism. *Proof.* Assume that a finite substructure $\mathcal{H}$ of $\Gamma(\mathcal{X})$ does not admit a homomorphism to $\mathcal{D}$. Since there is a finite substructure $\mathcal{F}$ of $\mathcal{X}$ and an $\mathcal{H}\to\Gamma(\mathcal{F})$ homomorphism, $\Gamma(\mathcal{F})$ cannot admit a homomorphism to $\mathcal{D}$. Then, by (2) of Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"}, $\mathcal{F}$ cannot admit a homomorphism to $\mathcal{E}$. ◻ The definition of finite reducibility is tailored to suit the next statement. **Proposition 10**. *For finite structures $\mathcal{D}$ and $\mathcal{E}$, if the $\mathcal{E}$-homomorphism problem finitely reduces to the $\mathcal{D}$-homomorphism problem, then $K_\mathcal{D}\implies K_\mathcal{E}$.* *Proof.* Assume that $\mathcal{E}$ finitely reduces to $\mathcal{D}$ and $K_\mathcal{D}$ holds. Let $\mathcal{X}$ be an instance of $\mathcal{E}$, and suppose that every finite substructure $\mathcal{F}$ of $\mathcal{X}$ admits a homomorphism to $\mathcal{E}$. Consider $\Gamma(\mathcal{X})$ (given by Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"}), which is an instance of $\mathcal{D}$. By (3) of Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"}, every finite substructure of $\Gamma(\mathcal{X})$ admits a homomorphism to $\mathcal{D}$. Thus, by $K_\mathcal{D}$, there exists a $\Gamma(\mathcal{X})\to\mathcal{D}$ homomorphism. Now (2) of Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"} provides an $\mathcal{X}\to\mathcal{E}$ homomorphism. ◻ We need two more definitions to describe how homomorphism problems can be reduced to each other. **Notation.** Let $\Sigma_\mathcal{D}$ denote the signature of the structure $\mathcal{D}$, i.e., the set of relations of $\mathcal{D}$. **Definition 11**. For finite structures $\mathcal{D}$ and $\mathcal{E}$ we say that *$\mathcal{E}$ is a $pp$-power of $\mathcal{D}$* if for some $n\in\mathbb{N}$ we have $E=D^n$ and for every $k\in\mathbb{N}$ and relation symbol $R$ of arity $k$ in $\Sigma_\mathcal{E}$ there exist $m_R\in\mathbb{N}$ and relation symbols $\alpha_{R,1},\ldots,\alpha_{R,m_R}$[^3] in $\Sigma_\mathcal{D}\cup\{=\}$ such that $$\label{e:simp_def_witness_2} (\overline z_1,\ldots,\overline z_k)\in R^\mathcal{E}\iff \exists \overline w\ \bigwedge_{i=1}^{m_R} {\alpha_{R,i}}^\mathcal{D}(\overline z_1,\ldots,\overline z_k,\overline w).$$ **Definition 12**. Two structures $\mathcal{D}$ and $\mathcal{E}$ are *homomorphically equivalent* if there exist $\mathcal{D}\to\mathcal{E}$ and $\mathcal{E}\to\mathcal{D}$ homomorphisms. Now we can state our result. **Theorem 13**. *For finite structures $\mathcal{D}$ and $\mathcal{E}$ we have the following:* *(A) If $\mathcal{D}$ and $\mathcal{E}$ are homomorphically equivalent, then they finitely reduce to each other.* *(B) If $\mathcal{E}$ is a $pp$-power of $\mathcal{D}$, then $\mathcal{E}$ finitely reduces to $\mathcal{D}$.* *Proof.* See Section [3](#s:hard){reference-type="ref" reference="s:hard"}. ◻ To exploit this result, we use the following theorem which follows from the work of Barto, Kozik, and Pinsker [@barto2018wonderland] together with the classical results of Taylor [@taylor1977varieties] (see also [@riley Theorem 2.6]). **Theorem 14**. *Assume that $\mathcal{D},\mathcal{E}$ are finite structures so that $\lnot (*)_\mathcal{D}$. Then $\mathcal{E}$ is homomorphically equivalent to a pp-power of $\mathcal{D}$.* This yields a complete characterization of the strength of the statement $K_\mathcal{D}$ in case $(*)_\mathcal{D}$ fails to hold. **Corollary 15**. *If $\lnot (*)_\mathcal{D}$, then $K_\mathcal{D} \iff K_{K_3} \iff BPI$.* *Proof.* To see the first equivalence, by Proposition [Proposition 10](#pr:finitary_red){reference-type="ref" reference="pr:finitary_red"}, it suffices to prove that $K_\mathcal{D}$ and $K_{K_3}$ finitely reduce to each other. By Theorems [Theorem 13](#t:fin_const_fin_red){reference-type="ref" reference="t:fin_const_fin_red"} and [Theorem 14](#t:intract_constructs_all){reference-type="ref" reference="t:intract_constructs_all"}, this follows from the fact that both $(*)_\mathcal{D}$ and $(*)_{K_3}$ fail. The second equivalence follows from the work of Lévy, Mycielski, and Läuchli [@levy1963remarks; @mycielski1961some; @lauchli1971coloring]. ◻ ## Easy problems. Second, let us give a high level overview of the main ideas of the proof of [\[c:maineasy\]](#c:maineasy){reference-type="eqref" reference="c:maineasy"} of Theorem [\[t:main\]](#t:main){reference-type="ref" reference="t:main"}. Call an instance $\mathcal{X}$ of $\mathcal{D}$ *finitely solvable*, if every finite substructure of $\mathcal{X}$ admits a homomorphism to $\mathcal{D}$. In order to construct models in which, say $K_{K_3}$ fails, we use classical ideas of Mostowski and Fraenkel. They constructed so-called *permutation models*, that is, models in which all of the axioms of ZF hold except for the Axiom of Equality: in addition to the $\emptyset$, there is a collection of *atoms* which do not have elements. The corresponding axiom system is denoted by ZFA. While these models are not models of ZF, forcing arguments using similar ideas often -- this happens in our case as well -- yield ZF models with analogous properties (see, e.g., [@jech2003set Chapter 15]). We fix some subgroup $\Gamma$ of the permutation group of atoms and elements of this model will be required to be invariant under some further "large\" subgroups $\Gamma'<\Gamma$. (The action on atoms naturally extends to an action on the sets containing atoms, sets of atoms, etc.) In such a way we will ensure that the atoms are somewhat indistinguishable from each other within the model. Now, for a fixed structure $\mathcal{D}$ with $(*)_\mathcal{D}$ we take a prime $p$ and a cyclic polymorphism $\phi$ of arity $p$ (see [Definition 6](#d:tractability){reference-type="ref" reference="d:tractability"}). We define a graph on the atoms, consisting of disjoint cycles of size $p$, and choose the group to be the one generated by rotations of single cycles. It will easily follow that such a graph cannot have a $3$-coloring in the model (i.e., an invariant one), but every finite subgraph has one, that is, $\lnot K_{K_3}$. In order to show that $K_{\mathcal{D}}$ holds in this model, we have to prove that given a finitely solvable instance $\mathcal{X}$ of the $\mathcal{D}$-homomorphism problem, it has a solution (i.e. an $\mathcal{X}\to \mathcal{D}$ homomorphism) in the model. By the definition of the model, $\mathcal{X}$ is invariant under some subgroup $\Gamma'$, and it suffices to construct a $\Gamma'$-invariant solution. The main observation is that this can be done starting from an arbitrary solution, outside the model, which exists by finite solvability and compactness (using AC). We can then use compactness (AC) again to find an invariant solution inside the model. Since we have compactness, it suffices to construct solutions that are invariant under any finite subset of $\Gamma'$. This can be done using cyclic polymorphisms: if $h_0$ is a homomorphism, $\alpha$ is a group element with $\alpha^p=1$, and $\phi$ is a cyclic polymorphism of $\mathcal{D}$ of arity $p$, then $h=\phi(h_0,\alpha \cdot h_0, \dots, \alpha^{p-1} \cdot h_0)$ is a solution invariant under $\alpha$. To make this intuition precise, we proceed to describe a general theorem, which can be applied without familiarity with forcing or abstract set theory. Let $\Gamma$ be a group. A collection $\mathcal{F}$ of subgroups of $\Gamma$ is called a *filter* if - $\{1\} \not \in \mathcal{F}$, $\Gamma \in \mathcal{F}$ - $\Delta, \Delta' \in \mathcal{F}$ implies $\Delta \cap \Delta' \in \mathcal{F}$, - $\Delta \in \mathcal{F}$, $\Delta <\Delta'$ implies $\Delta' \in \mathcal{F}$, - $\Delta \in \mathcal{F}$, $\gamma \in \Gamma$ implies $\gamma^{-1} \Delta \gamma \in \mathcal{F}$. The main example of a filter to be kept in mind for a group acting on an infinite set $T$ is the collection of subgroups that contain the *pointwise stabilizer* $\mathrm{Stab}_{pw}(F)$ of some finite set $F \subset T$. If $\cdot: \Gamma \times X \to X$ is an action of $\Gamma$ on a set $X$ and $Y_i \subset X$ for $i \in I$, then the setwise *stabilizer of $(Y_i)_{i \in I}$* is the subgroup $$\mathrm{Stab}((Y_i)_{i \in I})=\{\gamma \in \Gamma: \forall i \in I \ \gamma \cdot Y_i=Y_i\}.$$ The action $\cdot$ extends to $X^n$, $\bigcup_n X^n$ coordinate-wise. Similarly, if $D$ is any set on which $\Gamma$ does not act (or equivalently, assumed to act trivially), we can extend the action to $X \times D$. We also use $\cdot$ to denote these actions. Note that this allows us to talk about the stabilizers of structures on $X$ (as they are collections of subsets of $\bigcup_n X^n$) and functions $X \to D$, as they are subsets of $X \times D$. For $S \subset \Gamma$, an action of $\Gamma$ on a set $Z$, a set $Y \subset Z$ is called *$S$-invariant* if $S \subset \mathrm{Stab}(Y)$. **Definition 16**. Assume that $\Gamma$ acts on $X$, the universe of some structure $\mathcal{X}$. We say that $\mathcal{X}$ is *$\mathcal{F}$-symmetric (w.r.t. the given action of $\Gamma$)* if $\mathrm{Stab}(\mathcal{X}) \in \mathcal{F}$. In Section [4](#s:easy){reference-type="ref" reference="s:easy"}, we prove the following theorem, which one can use as a black box, without familiarity with abstract set theory. **Theorem 17**. *Let $\Gamma$ be a group and $\mathcal{F}$ be a filter of subgroups of $\Gamma$. Assume that for some structures $\mathcal{D},\mathcal{E}$ the below statements can be proved in ZFC:* 1. *For every finitely solvable $\mathcal{F}$-symmetric $\mathcal{D}$-instance $\mathcal{X}$ there exists a $\mathcal{F}$-symmetric homomorphism from $\mathcal{X}$ to $\mathcal{D}$.* *[\[c:forevery\]]{#c:forevery label="c:forevery"}* 2. *[\[c:example\]]{#c:example label="c:example"} There exists a finitely solvable $\mathcal{F}$-symmetric $\mathcal{E}$-instance $\mathcal{Y}$ with $\mathrm{Stab}(F) \in \mathcal{F}$ for all finite $F \subset Y$, which does not admit an $\mathcal{F}$-symmetric homomorphism to $\mathcal{E}$.* *Then there exists a model of ZFA, in which $K_{\mathcal{D}}$ holds but $K_{\mathcal{E}}$ fails.* *Proof.* See Section [4](#s:easy){reference-type="ref" reference="s:easy"}. ◻ We immediately obtain a more combinatorial form of the theorem using a compactness argument. **Corollary 18**. *Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"} holds even if we replace [\[c:forevery\]](#c:forevery){reference-type="eqref" reference="c:forevery"} by* 1. *For every $\mathcal{F}$-symmetric finitely solvable $\mathcal{D}$-instance $\mathcal{X}$, there exists some $\Gamma' \in \mathcal{F}$ such that for every finite $S \subset \Gamma'$ and $\{x_1,\dots,x_n\} \subset X$ there exists an $S$-invariant partial homomorphism of $\mathcal{X} \upharpoonright \{x_1,\dots,x_n\}$ to $\mathcal{D}$.* *In particular, to check that [\[c:forevery\]](#c:forevery){reference-type="eqref" reference="c:forevery"} holds it suffices to find a subgroup $\Gamma' \in \mathcal{F}$ such that for all finite $S \subset \Gamma'$ there is an $S$-invariant homomorphism from $\mathcal{X}$ to $\mathcal{D}$.* *Proof of Corollary [Corollary 18](#c:finitecheck){reference-type="ref" reference="c:finitecheck"}.* We show that [\[c:forevery\]](#c:forevery){reference-type="eqref" reference="c:forevery"} of Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"} holds using a compactness argument. Take the compact topological space $D^X$, where $D$ is endowed with the discrete topology. The following subsets are closed in this space for all $\gamma \in \Gamma, x_i \in X,n \in \mathbb{N}$: $C_\gamma=\{h:h \text{ is $\gamma$-invariant}\}$, $C_{x_1,\dots,x_n}=\{h: h \upharpoonright \{x_1,\dots,x_n\} \text{ is a partial homomorphism}\}$. By our assumption, the intersection of any finite collection from $(C_\gamma)_{\gamma\in \Gamma'}, (C_{x_1,\dots,x_n})_{x_i \in X}$ is nonempty, hence there is some $h$ in the intersection of all of these sets (note that we work in ZFC, so compactness can be used), and such an $h$ is a $\Gamma'$-invariant, and thereby $\mathcal{F}$-symmetric homomorphism $\mathcal{X} \to \mathcal{D}$. ◻ Finally, using forcing arguments, we obtain a ZF result as well. The more precise version of our main theorem reads as follows. **Theorem 19**. *There exists a model of ZF in which $K_{\mathcal{\mathcal{D}}}$ holds precisely if $(*)_{\mathcal{D}}$ does.* # Hard problems are hard {#s:hard} In this section, we show Theorem [Theorem 13](#t:fin_const_fin_red){reference-type="ref" reference="t:fin_const_fin_red"}: Theorem [Theorem 13](#t:fin_const_fin_red){reference-type="ref" reference="t:fin_const_fin_red"} For finite structures $\mathcal{D}$ and $\mathcal{E}$ we have the following: \(A\) If $\mathcal{D}$ and $\mathcal{E}$ are homomorphically equivalent, then they finitely reduce to each other. \(B\) If $\mathcal{E}$ is a $pp$-power of $\mathcal{D}$, then $\mathcal{E}$ finitely reduces to $\mathcal{D}$. *Proof.* (A) Now the structures $\mathcal{D}$ and $\mathcal{E}$ are of the same signature and there are homomorphisms $\theta_1:\mathcal{D}\to\mathcal{E}$ and $\theta_2:\mathcal{E}\to\mathcal{D}$. Set $\Gamma(\mathcal{X})=\mathcal{X}$, $\Phi(\varphi)=\theta_2\circ\varphi$ and $\Psi(\psi)=\theta_1\circ\psi$. This clearly works. \(B\) **Sketch.** The precise argument is quite heavy on notation, so we give an informal outline. Details can be found in the appendix. By definition, we have $E=D^n$, and by ([\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="ref" reference="e:simp_def_witness_2"}), $(\overline z_1,\ldots,\overline z_k)\in R^\mathcal{E}$ comes with some number of witnesses in $D$. Given an instance $\mathcal{X}$ of $\mathcal{E}$, we build the $\mathcal{D}$-instance $\Gamma(\mathcal{X})$ by adding the following elements: - $n$ "formal coordinates" for every $x \in X$; - the appropriate number of "formal witnesses of relation" for every relation symbol $R$ in $\Sigma_{\mathcal{E}}$ and tuple $\overline x \in R^\mathcal{X}$, arising from [\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="eqref" reference="e:simp_def_witness_2"}. We must carefully manage using the equality sign in ([\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="ref" reference="e:simp_def_witness_2"}). First, we can assume without loss of generality that no witness appears in any of the equalities. Second, the appropriate formal coordinates of those $x \in X$ that appear in relations $R^{\mathcal{X}}$ whose $pp$-definitions include equalities need to be identified. (Formally, we quotient out by the generated equivalence relation.) We need to transfer homomorphisms (to $\mathcal{E}$ and $\mathcal{D}$) between $\mathcal{X}$ and $\Gamma(\mathcal{X})$. On the one hand, a homomorphism $\varphi: \mathcal{X}\to \mathcal{E}$ gives rise to a map $\Phi(\varphi):\Gamma(\mathcal{X}) \to \mathcal{D}$ in a straightforward way, sending formal coordinates of $x \in X$ to actual coordinates of $\varphi(x)$ (and by necessity this factors through the quotient map). The relations on $\Gamma(\mathcal{X})$ are tailored to ensure that $\Phi(\varphi)$ is a homomorphism. On the other hand, if $\psi:\Gamma(\mathcal{X}) \to \mathcal{D}$ is a homomorphism, evaluating $\psi$ on all formal coordinates of each $x \in X$ gives rise to a map $\Psi(\psi): \mathcal{X}\to D^n=E$. The images of formal witnesses serve as actual witnesses in $D$, and the identification of the necessary formal coordinates ensures that all equalities in ([\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="ref" reference="e:simp_def_witness_2"}) are satisfied. The two together imply that $\Psi(\psi)$ is a homomorphism to $\mathcal{E}$. We also need to check (3) of Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"}, regarding the finite substructures. We do this using Remark [Remark 9](#r:suff_for_fin_red){reference-type="ref" reference="r:suff_for_fin_red"}, by collecting necessary elements of $X$ into a finite set $F$ to provide enough relations in $\Gamma(\mathcal{F})$ for the images of all related tuples of $\mathcal{H}$. Again, we have to take into account the identification of formal coordinates, which forces us to add extra elements to $F$ to ensure that all the necessary identifications already happen in $\Gamma(\mathcal{F})$. We make choices at multiple points along the construction but these are always possible in ZF because we always choose either finitely many times or from some fixed finite set. ◻ Let us remark that the first version of this manuscript used the technique developed in [@riley] to deal with the case of equality. However, Thornton pointed out to us that this is not necessary. # Easy problems are easy {#s:easy} In this section, we show Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"} and the remaining part of our main result, Theorem [\[t:main\]](#t:main){reference-type="ref" reference="t:main"}. ## ZFA results. First, as a warm-up, we prove the following general theorem about ZFA models. Let $\Gamma$ be a group and $\mathcal{F}$ be a filter of subgroups of $\Gamma$. Assume that for some structures $\mathcal{D},\mathcal{E}$ the below statements can be proved in ZFC: 1. For every finitely solvable $\mathcal{F}$-symmetric $\mathcal{D}$-instance $\mathcal{X}$ there exists a $\mathcal{F}$-symmetric homomorphism from $\mathcal{X}$ to $\mathcal{D}$. [\[c:forevery1\]]{#c:forevery1 label="c:forevery1"} 2. [\[c:example1\]]{#c:example1 label="c:example1"} There exists a finitely solvable $\mathcal{F}$-symmetric $\mathcal{E}$-instance $\mathcal{Y}$ with $\mathrm{Stab}(F) \in \mathcal{F}$ for all finite $F \subset Y$, which does not admit an $\mathcal{F}$-symmetric homomorphism to $\mathcal{E}$. Then there exists a model of ZFA in which $K_{\mathcal{D}}$ holds, but $K_{\mathcal{E}}$ fails. In fact, the model depends only on $\Gamma$, $\mathcal{F}$ and $\mathcal{Y}$. *Proof.* Let $\mathcal{Y}$ be the structure from [\[c:example1\]](#c:example1){reference-type="eqref" reference="c:example1"}. We build a permutation submodel of the universe $V$ as in [@jech2003set 15.48]. Let the set of atoms $Y'$ be chosen so that there is a bijection $b:Y' \to Y$ and define $\mathcal{Y}'$ to be a structure on $Y'$ by pulling back the relations form $\mathcal{Y}$. Similarly, define the $\Gamma$ action on $Y'$ by $\gamma \cdot y'=b^{-1}(\gamma \cdot b(y')).$ Let $U$ be the permutation model corresponding to the set of atoms $Y'$ and the action $\cdot$ defined above. By the $\mathcal{F}$-symmetricity of $\mathcal{Y}$, $\mathcal{Y}' \in U$, and by the nonexistence of an $\mathcal{F}$-symmetric homomorphism to $\mathcal{E}$, the $\mathcal{E}$-instance $\mathcal{Y}'$ is not solvable in $U$. As the stabilizer of every finite set $F \subset Y'$ is in the filter, every partial map $Y' \to E$ with a finite domain is in $U$, in particular, $\mathcal{Y}'$ is finitely solvable. Thus, $U \models \lnot K_\mathcal{E}$. Now, let $\mathcal{X} \in U$ be a finitely solvable $\mathcal{D}$-instance. We check that it is such an $\mathcal{F}$-symmetric instance in $V$ as well. Indeed, $\mathcal{X}$ is hereditarily $\mathcal{F}$-symmetric in $V$. To see that it is finitely solvable, just note that as $\mathcal{F}$ is a filter, any finite set containing only hereditarily $\mathcal{F}$-symmetric elements is in $U$, so every finite substructure of $\mathcal{X}$ in $V$ is also in $U$. By our assumptions, it admits an $\mathcal{F}$-symmetric homomorphism $h$ to $\mathcal{D}$ in $V$. But, as all elements of $X$ are hereditarily symmetric, so are all elements of $h$. Thus, $h$ is hereditarily symmetric as well, in particular, $h \in U$. ◻ Now we apply the above theorem in our particular case. **Theorem 20**. *There exists a model of ZFA in which $K_{\mathcal{D}}$ if $(*)_\mathcal{D}$ does, and $K_{\mathcal{K}_3}$ fails.* Let us remark that a careful examination of the constructions presented in Section [3](#s:hard){reference-type="ref" reference="s:hard"} show that in the model below, $K_\mathcal{D}$ fails for all $\mathcal{D}$ with $\lnot(*)_\mathcal{D}$. The model is a straightforward modification of the model $\mathcal{N}2^*(3)$ from [@howard1998consequences], which one could call $\mathcal{N}2^*(\text{Prime})$. **Definition 21**. Let $\Gamma=\oplus_{p \text{ prime}} \mathbb{Z}_p$, with the standard generating set $(\gamma_{p_i})_{p_i}$. Then $\Gamma$ acts on the set $$Y=\bigcup_{\text{$p_i$ is the $i$th prime}}\{i\} \times \{0,\dots,p_{i}-1\},$$ by $\gamma_{p_i} \cdot (i,j)=(i,j+1 \mod p_{i})$ and fixing every other element of $Y$. Let $\mathcal{Y}$ be the Schreier graph of $\Gamma$'s action on $Y$ (w.r.t. the generating set $(\gamma_p)_p$). Let $\mathcal{F}$ be the filter generated by the subgroups $\langle \gamma_{p_i}:i \geq n \rangle$ for $n \in \mathbb{N}$. *Proof.* We apply Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"}, using the group $\Gamma$ defined above. *First*, we show that the "in particular\" part of Corollary [Corollary 18](#c:finitecheck){reference-type="ref" reference="c:finitecheck"} holds, thereby guaranteeing [\[c:forevery\]](#c:forevery){reference-type="eqref" reference="c:forevery"} of Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"}. Let $\mathcal{D}$ be a structure with $(*)_\mathcal{D}$ and assume that $\mathcal{X}$ is an $\mathcal{F}$-symmetric finitely solvable instance of $\mathcal{D}$. Let $\Gamma'$ witness that $\mathcal{X}$ is $\mathcal{F}$-symmetric, fix an $n \in \mathbb{N}$, a sequence $(\phi_p)_{p \geq p_n}$ of cyclic polymorphisms of arity $p$ as in Definition [Definition 6](#d:tractability){reference-type="ref" reference="d:tractability"}. We start with an easy observation. **Claim 22**. *Let $\gamma \in \Gamma'$ with $\gamma^p=1$ and $h_0:\mathcal{X} \to \mathcal{D}$ be a homomorphism. Then the map $$h(x)=\phi_p(h_0(x),h_0(\gamma^{-1} \cdot x), h_0(\gamma^{-2}\cdot x), \dots,h_0 (\gamma^{-p+1}\cdot x))$$ is a $\{\gamma\}$-invariant homomorphism. Moreover, if $h_0$ is $\{\gamma'\}$-invariant for some $\gamma' \in \Gamma'$, then so is $h$.* *Proof.* First note that by the $\Gamma'$-invariance of $\mathcal{X}$, for any $\delta \in \Gamma'$ the map $x \mapsto h_0(\delta^{-1} \cdot x)$ is a homomorphism. Thus, $h$ is a homomorphism, and by the cyclicity of $\phi_p$ we get $$(\gamma \cdot h)(x)=h(\gamma^{-1} \cdot x)=$$$$\phi_p(h_0(\gamma^{-1} \cdot x),h_0(\gamma^{-2} \cdot x),\dots, h_0(\gamma^{-p} \cdot x))=$$ $$\phi_p(h_0(\gamma^{-p} \cdot x),h_0(\gamma^{-1} \cdot x),\dots, h_0(\gamma^{-p+1} \cdot x))=h(x).$$ In order to check the $\{\gamma'\}$-invariance, just observe that $\Gamma'$ is abelian, hence $$(\gamma' \cdot h)(x)= h(\gamma'^{-1} \cdot x)=$$ $$\phi_p(h_0(\gamma'^{-1} \cdot x),h_0(\gamma^{-1}\cdot \gamma'^{-1} \cdot x), \dots,h_0 (\gamma^{-p+1}\cdot \gamma'^{-1} \cdot x))=$$ $$\phi_p(h_0(\gamma'^{-1} \cdot x),h_0(\gamma'^{-1}\cdot \gamma^{-1} \cdot x), \dots,h_0 (\gamma'^{-1}\cdot \gamma^{-p+1}\cdot x))=$$ $$\phi_p(h_0(x),h_0(\gamma^{-1}(x)), h_0(\gamma^{-2}\cdot x), \dots,h_0 (\gamma^{-p+1}\cdot x))=h(x).$$ ◻ Let $$\Gamma''=\Gamma' \cap \mathrm{Stab}_{pw}(\{(i,j): i \leq n\}),$$ which is in $\mathcal{F}$, by the definition of $\mathcal{F}$. Let $S \subset \Gamma''$ be finite. We have to show that there is an $S$-invariant homomorphism $h:\mathcal{X} \to \mathcal{D}$. Pick a subsequence sequence $(\gamma_{p_i})_{n \leq i \leq k}$ such that $S \subset \langle \gamma_{p_i}:n \leq i \leq k\rangle$. Since $\mathcal{X}$ is finitely solvable, there is a homomorphism $h_0:\mathcal{X} \to \mathcal{D}$. Applying Claim [Claim 22](#cl:obvious){reference-type="ref" reference="cl:obvious"} inductively to $(\gamma_{p_i}), (\phi_{p_i})_{n \leq i \leq k}$ starting from $h_0$, we get a homomorphism invariant under each $(\gamma_{p_i})_{n \leq i \leq k}$, in turn under all elements of $S$. This shows that Corollary [Corollary 18](#c:finitecheck){reference-type="ref" reference="c:finitecheck"} holds which implies the first part of Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"}. *Second*, we check [\[c:example\]](#c:example){reference-type="eqref" reference="c:example"} of Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"} for $\mathcal{E}=K_3$, i.e., $3$-coloring. Let $\mathcal{Y}$ be graph from Definition [Definition 21](#d:model){reference-type="ref" reference="d:model"}, it is clear that $\mathcal{Y}$ is $\Gamma$-invariant. Moreover, $\Gamma$ is the vertex disjoint union of cycles, hence it admits a $3$-coloring. Now, if $c$ was an $\mathcal{F}$-symmetric $3$-coloring, then we could find a finite set $F \subset Y$ such that for every $\gamma \in \mathrm{Stab}_{pw}(F)$ we had $\gamma \cdot c=c$. In particular, for every large enough $p_i$ we had $\gamma_{p_i} \cdot c=c$, which means that $c$ is constant on the cycle corresponding to $p_i$, a contradiction. To finish the proof of Theorem [Theorem 20](#c:ZFA){reference-type="ref" reference="c:ZFA"} observe that we used the same $\Gamma$, $\mathcal{F}$ and $\mathcal{Y}$ for all $\mathcal{D}$, hence in the model guaranteed by Theorem [Theorem 17](#t:forcingblackbox){reference-type="ref" reference="t:forcingblackbox"} the statement $K_\mathcal{D}$ holds if $(*)_\mathcal{D}$ is true. ◻ ## ZF results. Now we are ready to prove the main result of the paper: There is a model of ZF, in which $K_{\mathcal{D}}$ holds precisely when $(*)_\mathcal{D}$ does. The argument below is the forcing version of the one presented in Corollary [Theorem 20](#c:ZFA){reference-type="ref" reference="c:ZFA"}. *Proof of Theorem [Theorem 19](#t:forcing){reference-type="ref" reference="t:forcing"}.* We build a symmetric submodel of a generic extension as in [@jech2003set Chapter 15]. In order to make the forcing argument work, we need to consider a slightly modified version of the group from Definition [Definition 21](#d:model){reference-type="ref" reference="d:model"}. Let $\mathcal{Y}$ be the graph from Definition [Definition 21](#d:model){reference-type="ref" reference="d:model"}, define $\Gamma_{i}=\oplus_{n \in \mathbb{N}} \mathbb{Z}_{p_i}$ the infinite fold direct sum of the group $\mathbb{Z}_{p_i}$ with itself, and $\Delta=\oplus_i \Gamma_i$. In other words, $\Delta$ is the sum of $\mathbb{Z}_p$s, where each cyclic group appears infinitely often. Fix an enumeration of the generators $(\gamma_{p_i,n})_{n \in \mathbb{N}, p_i \text{ prime}}$, and let $\mathcal{H}$ be the filter generated by the subgroups $\langle \gamma_{p_i,n}:n \in \mathbb{N}, i \geq l\rangle$, for $l \in \mathbb{N}$. To each element of $Y$, we will add a countable set of Cohen reals on which sets $\Delta$ is going to act. To achieve this, we define an action of $\Delta$ on $Y \times \mathbb{N}\times \mathbb{N}$ as follows. For each $i$, fix a bijection $b_i:\mathbb{N}\to \Gamma_i$. A generator $\gamma_{p_i,n}$ acts on the $Y$ coordinate as in Definition [Definition 21](#d:model){reference-type="ref" reference="d:model"}, while for each $Y \ni v=(i,j)$ the corresponding first $\mathbb{N}$ is identified with $\Gamma_i$ via $b_i$, and there $\gamma_{p_i,n}$ acts by translation. Formally, for every $\gamma_{p_i,n}$, let $$\gamma_{p_i,n}\cdot ((i,j),m,l):=((i,j+1 \mod p_i),b_i^{-1}(\gamma_{p_i,n}b_i(m)),l),$$ and let $\gamma_{p_i,n}$ fix every other element. Let us collect the most important observations about this action. **Claim 23**. *Let $F \subset Y \times \mathbb{N}\times \mathbb{N}$ be finite. Then* 1. *$\mathrm{Stab}_{pw}(F) \in \mathcal{H}$.* 2. *$\forall i \ \exists n \ \forall ((i,j),m,l) \in F \ \gamma_{p_i,n} \cdot ((i,j),m,l) \not \in F$.* 3. *$\forall i, n,m \ $ $$\gamma_{p_i,n} \cdot \{((i,j),m,l):l \in \mathbb{N}\}=\{((i,j+1),b_i^{-1}(\gamma_{p_{i},n}b_i(m)),l):l \in \mathbb{N}\}.$$* *Proof.* The first and last statements are clear. For the second one, define a partial map $b'_i((i,j),m,l)=b_i(m)$, and let $n$ be such that $\gamma_{p_{i},n}$ does not appear in $b'_i(F)$, that is, the support of every element of $b'_i(F)$ is disjoint from the copy of $\mathbb{Z}_{p_i}$ corresponding to $\gamma_{p_{i},n}$. Then $\gamma_{p_i,n}$ appears in every element of $b'_i (\gamma_{p_i,n} \cdot F)$, in particular, there is no $((i,j),m,l) \in F$ with $\gamma_{p_i,n}\cdot ((i,j),m,l) \in F$. ◻ Define $P$ to be the forcing notion consisting of partial functions of $Y \times \mathbb{N}\times \mathbb{N}\to 2$ with finite domain ordered by reverse inclusion. The $\Delta$ action on $Y \times \mathbb{N}$ gives rise to a $\Delta$ action on $P$. **Claim 24**. *For every $q \in P$* 1. *$\mathrm{Stab}_{pw}(q) \in \mathcal{H}$* 2. *$\forall i \ \exists n$ such $\gamma_{p_i,n} \cdot q$ and $q$ are compatible.* *Proof.* Applying Claim [Claim 23](#cl:action){reference-type="ref" reference="cl:action"} to $\mathop{\mathrm{dom}}(q)$ yields the desired conclusions. ◻ Let $B(P)$ be the complete Boolean algebra to which $P$ densely embeds (see [@jech2003set Corollary 14.12]). We will identify $P$ with its copy in $B(P)$. Let $M$ be the ground model. The action of $\Delta$ on $P$ gives rise to an action on $B(P)$ and on the collection of names $M^{B(P)}$ (see [@jech2003set 14.38]). Let $HS$ be the collection of hereditarily $\mathcal{H}$-symmetric names, $G$ be a generic filter on $P$, and $M[G]$ be the forcing extension. Define $$N=\{\dot{x}^G:\dot{x} \in HS\}.$$ Then $N \subset M[G]$ is a model of ZF by [@jech2003set Lemma 15.51]. The most important technical tool to be used is the Symmetry Lemma [@jech2003set 14.37]: for every $\gamma \in \Delta$, $q \in B(P)$, formula $\phi$ and names $\dot{x}_0,\dots,\dot{x}_{k-1}$ we have $$q \Vdash \phi(\dot{x}_0,\dots,\dot{x}_{k-1}) \iff \gamma \cdot q \Vdash \phi(\gamma \cdot \dot{x}_0,\dots,\gamma \cdot \dot{x}_{k-1}).$$ Let us first show the easier part of the theorem. **Proposition 25**. *$K_{\mathcal{K}_3}$ fails in $N$. In particular, by Corollary [Corollary 15](#c:hardness){reference-type="ref" reference="c:hardness"}, if $\lnot (*)_\mathcal{E}$ then $N \models \lnot K_{\mathcal{E}}$.* *Proof.* The poset $P$ adds a collection of Cohen reals indexed by $Y \times \mathbb{N}$, let $(\dot{r}_{(v,m)})_{v,m}$ be names for these. Let $\dot{A}_v$ contain all these names, i.e., $\dot{A}_v(\dot{r}_{v,m})=1,$ and no other name appears in the domain of $\dot{A}_v$. Observe that by Claim [Claim 23](#cl:action){reference-type="ref" reference="cl:action"} for all $\gamma_{p_i,n}$ and $v=(i',j)$ we have $\gamma_{p_i,n} \cdot \dot{A}_{i',j}= \dot{A}_{i',j+1 \mod p_i}$ if $i=i'$ and $\gamma_{p_i,n} \cdot \dot{A}_{i',j}= \dot{A}_{i',j}$ otherwise. In particular, all names $\dot{A}_v$ are in $HS$. These will play the role of atoms in $N$. Let $\dot{\mathcal{Y}}'$ be the name for a graph on the vertices $\dot{A}_v$, where $\dot{A}_v$ and $\dot{A}_{v'}$ are adjacent precisely if $v$ and $v'$ are in $\mathcal{Y}$. Note that $\dot{\mathcal{Y}}'$ is in $HS$, thus, $\dot{\mathcal{Y}}'^G$ is in $N$. Clearly, $\dot{\mathcal{Y}}'^G$ is isomorphic to $\mathcal{Y}$, therefore, every finite subgraph admits a $3$-coloring in $N$. Now, assume that $\mathcal{Y}'$ admits a $3$-coloring in $N$. Then, for some $q \in G$ and $\dot{c} \in HS$ we have $$q \Vdash \dot{c} \text{ is a homomorphism } \dot{\mathcal{Y}}' \to K_3.$$ Therefore, there is an $i$ such that for each $\gamma_{p_i,n}$ we have $\gamma_{p_i,n} \cdot \dot{c}=\dot{c}$ and $\gamma_{p_i,n} \cdot q=q$. Also, there exists some $q' \leq q$ with $q' \in P$ and $d \in D$ such that $$q' \Vdash \dot{c}(\dot{A}_{(i,0)})=d.$$ By Claim [Claim 24](#cl:action2){reference-type="ref" reference="cl:action2"} there exists some $n$ with $\gamma_{p_i,n} \cdot q'$ compatible with $q'$. Then by the Symmetry Lemma $$\gamma_{p_i,n} \cdot q' \Vdash (\gamma_{p_i,n} \cdot\dot{c})(\gamma_{p_i,n} \cdot \dot{A}_{(i,0)})=d,$$ hence, $$q' \vee \gamma_{p,i,n} \cdot q' \Vdash \dot{c}(\dot{A}_{(i,0)})=d=\dot{c}(\dot{A}_{(i,1)}).$$ As $\gamma_{p_i,n} \cdot q=q$, we have $q' \vee \gamma_{p_i,n} \cdot q' \leq q$, contradicting that $q$ forces that $\dot{c}$ is a $3$-coloring. ◻ Now we turn to the proof of the more involved part: **Proposition 26**. *If $(*)_\mathcal{D}$ holds then $N \models K_\mathcal{D}$.* *Proof.* Fix a $\mathcal{D}$ with $(*)_\mathcal{D}$ and let $(\phi_p)_{p \geq p^*}$ be the collection of the corresponding cyclic polymorphisms. Assume that we are given a finitely solvable instance $\mathcal{X}$ in $N$. Then $\mathcal{X}$ is finitely solvable in $M[G]$, so as it is a model of ZFC, it admits a homomorphism to $\mathcal{D}$ in $M[G]$. In particular, for some $q \in G$, name $\dot{\mathcal{X}} \in HS$ with $\dot{\mathcal{X}}^G=\mathcal{X}$ there is a name $\dot{h}$ such that $$q \Vdash \dot{h} \text{ is a homomorphism from $\dot{\mathcal{X}}$ to $\mathcal{D}$}.$$ The overall strategy is simple: using the cyclic polymorphisms and compactness, from $\dot{h}$ we construct a name $\in HS$ for a homomorphism. As $q \in P$ and $\dot{\mathcal{X}} \in HS$ we can find an $i_0 \geq p^*$ large enough $$\langle \gamma_{p_i,n}:n \in \mathbb{N},p_i \geq i_0 \rangle \subseteq \mathrm{Stab}_{pw}(p) \cap \mathrm{Stab}(\dot{\mathcal{X}}),$$ and let $\Delta'=\langle \gamma_{p_i,n}:n \in \mathbb{N},p_i \geq i_0 \rangle$. First we show some technical lemmas. For $q \in B(P)$, $\gamma \in \Delta$ define $Homo_q$ to be the collection of names $\dot{h}$ for which $\mathop{\mathrm{dom}}(\dot{h}) \subseteq \dot{X},$ $$q \Vdash \dot{h} \text{ is a homomorphism from $\dot{\mathcal{X}}$ to $\mathcal{D}$},$$ and for $\gamma \in \Delta$ let $Inv_{q,\gamma}$ to be the collection of the ones $\dot{h}$ with $$q \Vdash \gamma \cdot \dot{h}=\dot{h}.$$ Let us show now the forcing analogue of Claim [Claim 22](#cl:obvious){reference-type="ref" reference="cl:obvious"}. **Lemma 27**. *Assume that $\dot{h}_0 \in Homo_q$ and $\gamma \in \Delta'$ with $\gamma^p=1$ with $p \geq p_{i_0}$ prime. Then there exists a name $\dot{h} \in Homo_q \cap Inv_{q,\gamma}$. Moreover, if $\dot{h}_0 \in Inv_{q,\gamma'}$ then $\dot{h} \in Inv_{q,\gamma'}$ for any $\gamma' \in \Delta'$.* *Proof.* By the Symmetry Lemma and the fact that $\gamma \in \mathrm{Stab}(\dot{X})$, we have that $\dot{h}_0 \in Homo_{p}$ is equivalent to $\gamma \cdot \dot{h}_0 \in Homo_{\gamma \cdot q}.$ By $\gamma \in \mathrm{Stab}_{pw}(q)$ these are also equivalent to $\gamma \cdot \dot{h}_0 \in Homo_{q}.$ Therefore, all names $\dot{h}_0,\gamma \cdot \dot{h}_0,\dots, \gamma^{p-1} \cdot \dot{h}_0$ are forced to be homomorphisms by $q$. Hence, $$q \Vdash \phi_p(\dot{h}_0,\gamma \cdot \dot{h}_0,\dots, \gamma^{p-1} \cdot \dot{h}_0) \text{ is a homomorphism from $\dot{\mathcal{X}}$ to $\mathcal{D}$}.$$ Let $\dot{h}$ be a name for the composition $\phi_p(\dot{h}_0,\gamma \cdot \dot{h}_0,\dots, \gamma^{p-1} \cdot \dot{h}_0)$. Then $\dot{h} \in Homo_q$ and the calculations similar to the ones in Claim [Claim 22](#cl:obvious){reference-type="ref" reference="cl:obvious"} show both remaining statements of the lemma: $$q \Vdash \dot{h}=\phi_p(\dot{h}_0,\gamma \cdot \dot{h}_0,\dots, \gamma^{p-1} \cdot \dot{h}_0)$$ is equivalent to $$\gamma \cdot q \Vdash \gamma \cdot\dot{h}=\gamma \cdot\phi_p(\gamma \cdot\dot{h}_0,\gamma^2 \cdot \dot{h}_0,\dots, \gamma^{p} \cdot \dot{h}_0),$$ which implies $$q \Vdash \gamma \cdot\dot{h}=\phi_p(\dot{h}_0,\gamma \cdot \dot{h}_0,\dots, \gamma^{p-1} \cdot \dot{h}_0)=\dot{h},$$ by the cyclicity of $\phi_p$ and the $\gamma$-invariance of $q$. The moreover part follows from the fact that $\Delta$ is abelian. ◻ Enumerate the generators in $\Delta'$ as $(\gamma_n)_{n \in \mathbb{N}}$. Observe that by the choice of $i_0$, every such $\gamma_n$ has some prime order $p \geq p^*$. Applying Lemma [Lemma 27](#l:invariance){reference-type="ref" reference="l:invariance"} repeatedly, starting with $\dot{h}$, we obtain a sequence of names $(\dot{h}_n)_{n \in \mathbb{N}}$, with $\dot{h}_n \in Inv_{q,\gamma_i} \cap Homo_q$ for all $i<n$. Using the compactness of $\mathcal{D}^\mathcal{X}$, in $M[G]$, we have that $$M[G] \models \exists h \in \mathcal{D}^{\mathcal{X}} \ h \in \bigcap_n \overline{\{\dot{h}^G_i:i \geq n}\}.$$ Hence there is some name $\dot{h}$ and $q' \leq q$ form $G$ with $$q' \Vdash \dot{h} \in \bigcap_n \overline{\{\dot{h}_i:i \geq n\}}.$$ **Lemma 28**. *$\dot{h} \in Homo_{q'} \cap Inv_{q',\gamma}$, for all $\gamma \in\mathrm{Stab}_{pw}(q') \cap \Delta'$.* *Proof.* We have to show that $$q' \Vdash \dot{h} \text{ is a homomorphism from $\dot{\mathcal{X}}$ to $\mathcal{D}$}$$ and that for all $\gamma \in \mathrm{Stab}_{pw}(q') \cap \Delta'$, we have $$q' \Vdash \gamma \cdot\dot{h}=\dot{h}.$$ The first statement is clear, since $q'$ forces that $\dot{h}$ is in the closure of a set of homomorphisms. In order to see the second statement, we need the next observation. **Claim 29**. *For any names $\dot{x},\dot{y}$ we have $$q' \Vdash \forall n_0 \ \exists n \geq n_0 \ ((\dot{x} \in \dot{h} \iff \dot{x} \in \dot{h}_n) \land (\dot{y} \in \dot{h} \iff \dot{y} \in \dot{h}_n)).$$* *Proof.* Fix $\dot{x},\dot{y}$. For any $G' \ni q'$ generic by the definition of the product topology, we have that $$M[G'] \vDash \forall n_0 \ \exists n \geq n_0 \ ( (\dot{x}^{G'} \in \dot{h}^{G'} \iff \dot{x}^{G'} \in \dot{h}^{G'}_n) \land (\dot{y}^{G'} \in \dot{h}^{G'} \iff \dot{y}^{G'} \in \dot{h}^{G'}_n)),$$ so the claim follows. ◻ Now we show that for every $\dot{x}$ we have $q' \Vdash (\dot{x} \in \gamma \cdot \dot{h} \iff \dot{x} \in \dot{h})$. By the definition of the action on names, and the facts that $\dot{h}_n \in Inv_{\gamma,q'}$ and $\gamma \cdot q'=q'$ we get $$q' \Vdash\dot{x} \in \dot{h}_n \iff \dot{x} \in \gamma \cdot \dot{h}_n \iff \gamma^{-1} \cdot \dot{x} \in \dot{h}_n.$$ Since the construction of the sequence $(\dot{h}_n)$ has been done in $M$, we get that $$q' \Vdash \forall n \ \dot{x} \in \dot{h}_n \iff \gamma^{-1} \cdot \dot{x} \in \dot{h}_n.$$ Applying the claim to $\dot{x}$ and $\gamma^{-1} \cdot \dot{x}$ we get $$q' \Vdash \forall n_0 \ \exists n \geq n_0 \ ((\dot{x} \in \dot{h} \iff \dot{x} \in \dot{h}_n) \land (\gamma^{-1}\cdot \dot{x} \in \dot{h} \iff \gamma^{-1}\cdot \dot{x}\in \dot{h}_n)).$$ Combining the last two relations, we get $$q' \Vdash \dot{x} \in \dot{h} \iff \gamma^{-1} \cdot \dot{x}\in \dot{h},$$ so, using the definition of the action and $\gamma \cdot q'=q'$ again we obtain $$q' \Vdash \dot{x} \in \dot{h} \iff \dot{x} \in \gamma \cdot \dot{h}.$$ Thus, $q' \Vdash \dot{h}=\gamma \cdot \dot{h}.$ ◻ Now we show that there exists a name $\dot{h'} \in HS\cap Homo_{q'}$. For each $\dot{x} \in \mathop{\mathrm{dom}}(\dot{h})$ define $\dot{h}'(\dot{x})$ to be $\dot{h}(\dot{x}) \wedge q'$. We claim that $\dot{h}'$ is a suitable choice. First, by Lemma [Lemma 28](#l:invarianceh){reference-type="ref" reference="l:invarianceh"} it is clear that $$\dot{h}' \in \bigcap_{\gamma \in \Delta'\cap \mathrm{Stab}_{pw}(q')} Inv_{q',\gamma} \cap Homo_{q'},$$ since $q' \Vdash \dot{h}'=\dot{h}.$ Assume that for some name $\dot{x}$ we had $\dot{h}'(\dot{x}) \neq (\gamma \cdot \dot{h}')(\dot{x})$. Note that $(\gamma \cdot \dot{h}')(\dot{x})=(\gamma \cdot \dot{h}')(\gamma^{-1} \cdot\dot{x})\leq \gamma \cdot q'=q'$. Thus, there was a $q'' \leq q'$ with $q''\leq \dot{h}'(\dot{x})$ and $q'' \wedge (\gamma \cdot \dot{h}')(\dot{x})=0$, or $q''\leq \gamma \cdot \dot{h}'(\dot{x})$ and $q'' \wedge \dot{h}'(\dot{x})=0$. But then $q'' \Vdash \dot{x} \in \gamma \cdot \dot{h}'\setminus \dot{h}' \cup \dot{h}' \setminus \gamma \cdot \dot{h}'$, contradicting that $\dot{h}' \in Inv_{q',\gamma}$. Hence, for each $\gamma \in \Delta' \cap \mathrm{Stab}_{pw}(q') \in \mathcal{H}$ we have $\gamma \cdot \dot{h}'=\dot{h}'$, showing that $\dot{h}' \in HS$. By $\dot{h}' \in HS$, we have $(\dot{h}')^G \in N$ and it follows from $q' \in G$ and $\dot{h}' \in Homo_{q'}$ that $(\dot{h}')^G$ is a homomorphism from $\dot{\mathcal{X}}^G$ to $\mathcal{D}$, concluding the proof of Proposition [Proposition 26](#pr:mainhard){reference-type="ref" reference="pr:mainhard"}. ◻ Propositions [Proposition 26](#pr:mainhard){reference-type="ref" reference="pr:mainhard"} and [Proposition 25](#p:fails){reference-type="ref" reference="p:fails"} yield Theorem [Theorem 19](#t:forcing){reference-type="ref" reference="t:forcing"}. ◻ # Open problems As mentioned above, finite reducibility yields a complexity hierarchy on homomorphism problems. **Problem 30**. *Describe the hierarchy of finite reducibility.* It would be also interesting to see the relationship of reduction to other well-known forms of reducibility. While our work detects a distinction between easy and hard problems, the difference between the strength of different easy problems is yet to be investigated. For example, let $K_{3LIN_2}$ stand for the compactness principle for systems of linear equations over $\mathbb{F}_2$. **Problem 31**. *Is there a model of ZF in which $K_{K_2}$ holds but $K_{3LIN_2}$ does not?* Symmetric models are the least sophisticated way to get independence results over ZF. Correspondingly, even very weak forms of choice fail in them. It would be extremely interesting to construct models where some choice principles remain true, but the same split between easy and hard problems can be detected. A deep theory has been developed in the past couple of years to construct models (see, e.g., [@larson2020geometric]). **Problem 32**. *Is there a model of ZF+DC in which $K_\mathcal{D}$ holds exactly when $(*)_\mathcal{D}$ is true?* **Acknowledgements.** We would like to thank Lorenz Halbeisen for pointing out the similarity between Lévy's and Mycielski's theorems with results obtained in [@riley; @toden]. We are very grateful to Riley Thornton for suggesting the currently included proof of Theorem [Theorem 13](#t:fin_const_fin_red){reference-type="ref" reference="t:fin_const_fin_red"}. We are also thankful to Amitayu Banerjee, Jan Grebík, Paul Howard, Asaf Karagila, Gábor Kun, Michael Pinsker, Eleftherios Tachtsis, and Jindřich Zapletal for their useful comments and enlightening discussions. # Appendix {#appendix .unnumbered} *Proof of Theorem [Theorem 13](#t:fin_const_fin_red){reference-type="ref" reference="t:fin_const_fin_red"} (B).* By definition, we have $E=D^n$, and we recall from ([\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="ref" reference="e:simp_def_witness_2"}) that for every $k\in\mathbb{N}$ and relation symbol $R$ in $\Sigma_\mathcal{E}$ there are relations symbols $\alpha_{R,i}$ in $\Sigma_\mathcal{D}\cup\{=\}$ such that $$%\label{e:simp_def_witness_2} (\overline z_1,\ldots,\overline z_k)\in R^\mathcal{E}\iff \exists \overline w \bigwedge_{i=1}^{m_R}{\alpha_{R,i}}^\mathcal{D}(\overline z_1,\ldots,\overline z_k,\overline w).$$ Here the $\overline z_j$ are $n$-tuples, and $\overline w$ is an $r$-tuple of variables (where $r$ depends on $R$). Recall that the $\alpha_{R,i}$ are not necessarily $(kn+r)$-ary relations. (e.g., some of them may be the equality). We may assume that for every $s\leq k$, $j\leq n$, and $l,l'\leq r$ none of the $\alpha_{R,i}$ are of the form $\overline z_s(j)=\overline w(l)$ or $w(l)=w(l')$ because otherwise we could eliminate the variable $\overline w(l)$. Let $\mathcal{X}$ be an instance of $\mathcal{E}$. We need to construct $\Gamma(\mathcal{X})$. First, let us define an instance $\widetilde\mathcal{X}$ of $\mathcal{D}$. Let $\widetilde X$ contain the following elements: - an element $\overline x(j)$ for every $x\in X$ and $j\leq n$; - an element $\overline w_{R,\overline x}(j)$ for every relation symbol $R$ in $\Sigma_\mathcal{E}$, tuple $\overline x\in R^\mathcal{X}$, and $j\leq r$, where $r$ is the arity of $\overline w$ in ([\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="ref" reference="e:simp_def_witness_2"}). Now we define the relations. For every $k\in\mathbb{N}$, relation symbol $R\in\Sigma_\mathcal{E}$ of arity $k$, tuple $(x_1,\ldots,x_k)\in R^\mathcal{X}$, and index $i\leq m_R$, if $\alpha_{R,i}$ is not the equality and it depends (formally) exactly on the variables $\overline z_{s_1}(j_1),\ldots,\overline z_{s_l}(j_l),\overline w(t_1),\ldots,\overline w(t_v)$, then let $$\left(\overline x_{s_1}(j_1),\ldots,\overline x_{s_l}(j_l),\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right)$$ be an element of ${\alpha_{R,i}}^{\widetilde\mathcal{X}}$. Relation symbols not occurring this way are interpreted in $\widetilde\mathcal{X}$ as empty relations. Let us define a graph $G_\mathcal{X}$ on $\widetilde X$. For every relation $R^\mathcal{X}$ on $\mathcal{X}$ and tuple $(x_1,\ldots,x_k)\in R^\mathcal{X}$ we join $\overline x_s(j)$ and $\overline x_t(l)$ with an edge if one of the relations $\alpha_{R,i}$ is the equality $\overline z_s(j)=\overline z_t(l)$. For $x,y\in \widetilde X$ let $x\sim_\mathcal{X}y$ if and only if $x$ and $y$ are in the same connected component of $G_\mathcal{X}$. This is an equivalence relation. Let $[x]$ denote the $\sim_\mathcal{X}$-class of $x$. Note that elements of the form $w_{R,\overline x}(j)$ are alone in their $\sim_\mathcal{X}$-class. Now we define $\Gamma(\mathcal{X})$. Let $\mathop{\mathrm{dom}}(\Gamma(\mathcal{X}))$ contain the following elements: - the equivalence class $[\overline x(j)]$ for each $x\in X$ and $j\leq n$, - the elements of the form $\overline w_{R,\overline x}(j)$ from $\widetilde\mathcal{X}$. For every relation symbol $R$ in $\Sigma_\mathcal{E}$ and $i\leq m_R$, if $\alpha_{R,i}$ is not the equality, then let ${\alpha_{R,i}}^{\Gamma(\mathcal{X})}$ be the following $$\left\{ \begin{array}{ll} \left([\overline x_{s_1}(j_1)],\ldots,[\overline x_{s_l}(j_l)],\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right):\ \\ \left(\overline x_{s_1}(j_1),\ldots,\overline x_{s_l}(j_l),\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right)\in{\alpha_{R,i}}^{\widetilde\mathcal{X}} \end{array} \right\}.$$ We interpret the remaining relation symbols of $\Sigma_\mathcal{D}$ as empty relations. Clearly, (1) of Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"} is satisfied. To check (2) we need to transfer homomorphisms. Let $\varphi:\mathcal{X}\to \mathcal{E}$ be a homomorphism. First, let us define a map $\widetilde\varphi:\widetilde X\to\mathcal{D}$. Let $u\in\widetilde X$. **Case 1.** We have $u=\overline x(j)$ for some $x\in \mathcal{X}$ and $j\leq n$. Then set $\widetilde\varphi(\overline x(j))=\varphi(x)(j)$. **Case 2.** We have $u=\overline w_{R,(x_1,\ldots,x_k)}(j)$ for some $R\in\Sigma_\mathcal{E}$, $(x_1,\ldots,x_k)\in R^{\mathcal{X}}$, and $j\leq r$. Since $\varphi$ is a homomorphism, we have $(\varphi(x_1),\ldots,\varphi(x_k))\in R^\mathcal{E}$, hence we can choose a tuple $\overline w_{R,(\varphi(x_1),\ldots,\varphi(x_k))}$ that witnesses this in ([\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="ref" reference="e:simp_def_witness_2"}). Note that this choice can be made in ZF because $\mathcal{E}$ is finite and $\overline w_{R,(\varphi(x_1),\ldots,\varphi(x_k))}$ depends only on $(\varphi(x_1),\ldots,\varphi(x_k))$, not $(x_1,\ldots,x_k)$. Set $\widetilde\varphi(u)=\overline w_{R,(\varphi(x_1),\ldots,\varphi(x_k))}(j)$. **Claim 1.** The map $\widetilde\varphi$ is $\sim_\mathcal{X}$-invariant. Fix $\overline x(j),\overline y(l)\in \widetilde\mathcal{X}$ with $\overline x(j)\sim_\mathcal{X}\overline y(l)$. Then there is a path $\overline x(j)=\overline x_0(j_0), \overline x_1(j_1),\ldots,\overline x_n(j_n)=\overline y(l)$ in $G_\mathcal{X}$. By the definition of $G_\mathcal{X}$, for each $0\leq q<n$ there is an arity $k$, a relation $R$ of arity $k$, a tuple $(x^*_1,\ldots,x^*_k)\in R^{\mathcal{X}}$, an index $0\leq i<m_R$, and coordinates $s,t\leq k$ such that $x_q=x^*_s$, $x_{q+1}=x^*_t$ and $\alpha_{R,i}$ is $\overline z_s(j_q)=\overline z_t(j_{q+1})$. Since $\varphi$ is a homomorphism, we have $(\varphi(x^*_1),\ldots,\varphi(x^*_k))\in R^\mathcal{E}$. Then $$\widetilde\varphi(\overline x_q(j_q))=\widetilde\varphi(\overline x^*_s(j_q))=\varphi(x^*_s)(j_q)=$$ $$=\varphi(x^*_t)(j_{q+1})=\widetilde\varphi(\overline x^*_t(j_{q+1}))=\widetilde\varphi(\overline x_{q+1}(j_{q+1})),$$ where the third equality follows from ([\[e:simp_def_witness_2\]](#e:simp_def_witness_2){reference-type="ref" reference="e:simp_def_witness_2"}) and the fact that $\alpha_{R,i}$ is $\overline z_s(j_q)=\overline z_t(j_{q+1})$. Applying this to every $0\leq q<n$ we conclude that $\widetilde\varphi(\overline x(j))=\widetilde\varphi(\overline y(l))$, which proves Claim 1. Now we define the homomorphism $\Phi(\varphi):\Gamma(\mathcal{X})\to\mathcal{D}$. Let $u\in\Gamma(\mathcal{X})$. **Case 1.** We have $u=[\overline x(j)]$ for some $x\in\mathcal{X}$ and $j\leq n$. Set $\Phi(\varphi)(u)=\widetilde\varphi(\overline x(j))=\varphi(x)(j)$. This is well-defined by Claim 1. Alternatively, to see that *we do not need to choose* a representative element from $[\overline x(j)]$, we can set $\Phi(\varphi)(u)=\bigcup\widetilde\varphi([\overline x(j)])$, that is, the element of the singleton $\widetilde\varphi([\overline x(j)])$. **Case 2.** We have $u=\overline w_{R,(x_1,\ldots,x_k)}$ for some $R\in\Sigma_\mathcal{E}$, $(x_1,\ldots,x_k)\in R^\mathcal{X}$, and $j\leq n$. Set $\Phi(\varphi)(u)=\widetilde\varphi(u)=\overline w_{R,(\varphi(x_1),\ldots,\varphi(x_k))}(j)$ We need to verify that $\Phi(\varphi)$ is a homomorphism. It suffices to check those relation symbols in $\Sigma_\mathcal{D}$ that are of the form $\alpha_{R,i}$ since other relation symbols are interpreted in $\Gamma(\mathcal{X})$ as empty relations. The image of a tuple $$\left([\overline x_{s_1}(j_1)],\ldots,[\overline x_{s_l}(j_l)],\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right)\in{\alpha_{R,i}}^{\Gamma(\mathcal{X})}$$ is, by definition, $$(\varphi(x_{s_1})(j_1),\ldots,\varphi(x_{s_l})(j_l),\overline w_{R,(\varphi(x_1),\ldots,\varphi(x_k))}(t_1),\ldots,\overline w_{R,(\varphi(x_1),\ldots,\varphi(x_k))}(t_v)),$$ which is in ${\alpha_{R,i}}^\mathcal{D}$ by the choice of the tuple $\overline w_{R,(\varphi(x_1),\ldots,\varphi(x_k))}$. Now let $\psi:\Gamma(\mathcal{X})\to\mathcal{D}$ be a homomorphism. Set $\Psi(\psi)(x)=(\psi([\overline x(1)]),\ldots,\psi([\overline x(n)]))$ for every $x\in \mathcal{X}$. Let $R$ be a relation symbol in $\Sigma_\mathcal{E}$, and $(x_1,\ldots,x_k)\in R^\mathcal{X}$. Then, for every relation symbol of the form $\alpha_{R,i}$ that is not the equality the tuple $$\label{e:rel_in_gamma_X} \left([\overline x_{s_1}(j_1)],\ldots,[\overline x_{s_l}(j_l)],\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right)\tag{$\star$}$$ is in ${\alpha_{R,i}}^{\Gamma(\mathcal{X})}$ by the definition of ${\alpha_{R,i}}^{\Gamma(\mathcal{X})}$. If, on the other hand, $\alpha_{R,i}$ is of the form $\overline z_s(j)=\overline z_t(l)$ for some $s,t\leq k$ and $j,l\leq n$, then $\overline x_s(j)$ and $\overline x_t(l)$ are joined by an edge in $G_\mathcal{X}$. In particular, $[\overline x_s(j)]=[\overline x_t(l)]$, hence ([\[e:rel_in_gamma_X\]](#e:rel_in_gamma_X){reference-type="ref" reference="e:rel_in_gamma_X"}) is in ${\alpha_{R,i}}^{\Gamma(\mathcal{X})}$ again. Since $\psi$ is a homomorphism, we obtain that $$(\psi([\overline x_{s_1}(j_1)]),\ldots,\psi([\overline x_{s_l}(j_l)]),\psi(\overline w_{R,(x_1,\ldots,x_k)}(t_1)),\ldots,\psi(\overline w_{R,(x_1,\ldots,x_k)}(t_v)))$$ is in ${\alpha_{R,i}}^\mathcal{D}$ for every $i\leq m_R$. (Here the indices $s_1,\ldots,s_l,j_1,\ldots,j_l, t_1,\ldots,t_v$ depend on $i$.) That is, $\psi(\overline w_{R,(x_1,\ldots,x_k)}(1)),\ldots,\psi(\overline w_{R,(x_1,\ldots,x_k)}(r))$ witness that $(\Psi(\psi)(x_1),\ldots,\Psi(\psi)(x_k))\in R^\mathcal{E}$. It remains to prove that the above constructed $\Gamma$, $\Phi$ and $\Psi$ satisfy (3) of Definition [Definition 7](#d:finitely_reduces){reference-type="ref" reference="d:finitely_reduces"}. By Remark [Remark 9](#r:suff_for_fin_red){reference-type="ref" reference="r:suff_for_fin_red"}, it suffices to show that for every finite substructure $\mathcal{H}$ of $\Gamma(\mathcal{X})$ there is a finite substructure $\mathcal{F}$ of $\mathcal{X}$ such that there exists an $\mathcal{H}\to\Gamma(\mathcal{F})$ homomorphism. Fix a finite substructure $\mathcal{H}$ of $\Gamma(\mathcal{X})$. First, we define a finite set $F'\subseteq\widetilde X$. All relations of $\mathcal{H}$ are of the form ${\alpha_{R,i}}^{\Gamma(\mathcal{X})}|_\mathcal{H}$. For each $\alpha_{R,i}$ and $\left([\overline x_{s_1}(j_1)],\ldots,[\overline x_{s_l}(j_l)],\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right)\in{\alpha_{R,i}}^\mathcal{H}$ we add elements to $F'$ as follows. If $v\geq 1$, then we add $\overline x_1(1),\ldots,\overline x_1(n),\ldots,\overline x_k(1),\ldots,\overline x_k(n)$ to $F'$ (we read them off from the index of $\overline w_{R,(x_1,\ldots,x_k)}(t_1)$). If $v=0$, then we *choose* elements $\overline x_1(1),\ldots,\overline x_1(n),\ldots,\overline x_k(1),\ldots,\overline x_k(n)$ such that $(x_1,\ldots,x_k)\in R^\mathcal{X}$ and $\overline x_{s_1}(j_1),\ldots,\overline x_{s_l}(j_l)$ represent the classes $[\overline x_{s_1}(j_1)],\ldots,[\overline x_{s_l}(j_l)]$ respectively. **Remark.** We need to treat these cases separately because whenever witnesses are present they correspond to specific representatives, and we need to pick the right representatives for them to be in relation with the witnesses. Reading the representatives off from the index of $\overline w$ is not done to avoid Choice, it is done to ensure the tuple is related in $\widetilde\mathcal{X}$. Now $F'$ has finitely many elements. Add finitely many more elements to $F'$ to ensure that for every $\overline x(j),\overline y(l)\in F'$ if $\overline x(j)\sim_\mathcal{X}\overline y(l)$, then there is a path from $\overline x(j)$ to $\overline y(l)$ in $F'$. Thus, the definition of $F'$ involves choosing finitely many elements from $\mathcal{X}$ and $\widetilde\mathcal{X}$, which can be done in ZF. Finally, we define $F$. First, let $F$ contain every element $x\in\mathcal{X}$ for which there is $j\leq n$ such that $x(j)\in F'$. Second, if two elements $\overline x(j),\overline y(l)\in F'$ are connected by an edge in $G_\mathcal{X}$, we add finitely many more elements to $F$ to ensure that they are connected in $G_\mathcal{F}$ as well (see the definition of $G_\mathcal{X}$). This involves choosing finitely many elements from $\mathcal{X}$. Now we define a homomorphism $\vartheta:\mathcal{H}\to\Gamma(\mathcal{F})$. Let $u\in\mathcal{H}$. **Case 1.** The element $u$ does not occur as a coordinate in any tuple in any of the relations ${\alpha_{R,i}}^\mathcal{H}$. Then pick $\vartheta(u)\in\Gamma(\mathcal{F})$ arbitrarily. This involves choosing finitely many elements from $\Gamma(\mathcal{F})$. **Case 2.** For some relation symbol $\alpha_{R,i}$ the element $u$ occurs as a coordinate of a tuple $$\left([\overline x_{s_1}(j_1)],\ldots,[\overline x_{s_l}(j_l)],\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right)\in{\alpha_{R,i}}^\mathcal{H}.$$ **Subcase 2.1.** We have $v\geq 1$. If $u=\overline w_{R,(x_1,\ldots,x_k)}(t_m)$ for some $m\leq v$, then set $\vartheta(u)=u$, which is an element of $\Gamma(\mathcal{F})$ since $(x_1,\ldots,x_k)\in R^\mathcal{X}$. If $u=[\overline x_{s_p}(j_p)]$ for some $p\leq l$, then set $\vartheta(u)=[\overline x_{s_p}(j_p)]_\mathcal{F}$, which is the $\sim_\mathcal{F}$-class of $\overline x_{s_p}(s_p)$. As before, this does not involve choice since we simply read off the representatives from the index of $\overline w_{R,(x_1,\ldots,x_k)}$. The map $\vartheta$ is well-defined by the construction of $F$: $\sim_\mathcal{X}$-equivalent elements in $F'$ are $\sim_\mathcal{F}$-equivalent as well. **Subcase 2.2.** We have $v=0$. Then $u=[\overline x_{s_p}(j_p)]$ for some $p\leq l$. Set $\vartheta(u)=[\overline x_{s_p}(j_p)]_\mathcal{F}$, where $\overline x_{s_p}(j_p)$ is the representative already chosen in the definition of $F'$ in the case $v=0$. Again, this is well-defined because $\sim_\mathcal{X}$-equivalent elements in $F'$ are $\sim_\mathcal{F}$-equivalent as well. Now the image of a tuple $$\left([\overline x_{s_1}(j_1)],\ldots,[\overline x_{s_l}(j_l)],\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right)\in{\alpha_{R,i}}^\mathcal{H}$$ under $\vartheta$ is $$\left([\overline x_{s_1}(j_1)]_\mathcal{F},\ldots,[\overline x_{s_l}(j_l)]_\mathcal{F},\overline w_{R,(x_1,\ldots,x_k)}(t_1),\ldots,\overline w_{R,(x_1,\ldots,x_k)}(t_v)\right),$$ which is in ${\alpha_{R,i}}^{\Gamma(\mathcal{F})}$ because of the definition of $\Gamma$ and the fact that $(x_1,\dots,x_k)\in R^\mathcal{F}$. (If $v\geq 1$, then $(x_1,\ldots,x_k)$ comes from the index of $\overline w_{R,(x_1,\ldots,x_k)}(t_1)$, otherwise it was chosen in the definition of $F'$.) Thus $\vartheta$ is a homomorphism. The proof of Theorem [Theorem 13](#t:fin_const_fin_red){reference-type="ref" reference="t:fin_const_fin_red"} is now complete. ◻ [^1]: [^2]: We follow the convention of using calligraphic letters to denote structures and plain upper case letters to denote their universe. [^3]: The $\alpha_{R,i}$ may have different arity. In the displayed formula we treat them as $(kn+r)$-ary relations only for notational simplicity.

arxiv_math

--- abstract: | We consider a toy model of rate independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. Taking advantage of two notions of weak solutions, energy-based and comparison-principle-based, we study the dynamic contact angle of moving contact lines and the geometry of de-pinning. We show that these two notions of solutions coincide in a star-shaped setting, where we show (almost) optimal regularity of the contact line and the convergence of a minimizing movements scheme. In a general setting, the notions differ essentially in how they handle jumps, but both are shown to satisfy a weak motion law. address: - Department of Mathematics, University of Utah, Salt Lake City, Utah, USA - Department of Mathematics, University of California, Los Angeles, California, USA - Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan author: - William M Feldman - Inwon C Kim - Norbert Požár bibliography: - monotone-articles.bib title: On the geometry of rate independent droplet evolution --- # Introduction In this work we consider a toy model, based on the one-phase Bernoulli free boundary problem, for the rate independent motion of droplets on a solid surface with the effects of contact angle hysteresis. We study two related notions of weak solution which handle time jump discontinuities differently: an energetic notion and a notion based on comparison principle and a free boundary obstacle problem. We study the geometry of the contact line and the dynamic contact angle of moving contact lines for these solutions. Our first main result is in a star-shaped setting. We show equivalence between the typical minimizing movements energy solutions and the obstacle solutions based on a novel comparison principle. We also show the optimal $C^{1,\frac{1}{2}-}$ regularity of the contact line. Our second main result is in a general setting. We show that both energy solutions and obstacle solutions satisfy a weak motion law essentially reducing the difference of these two evolutions to their jump laws. To our knowledge these are the first rigorous results establishing these properties for a general class of weak solutions to a rate independent model of droplet evolution. It is well understood in the physics and engineering literature that capillary droplets on many solid surfaces experience a phenomenon known as *contact angle hysteresis* [@EralMannetjeOh]. Rather than a single contact angle specified by the material properties, as would be predicted by the classical Young's law in capillarity theory [@KarimReview], there is a range of stable contact angles. Consequently droplets can "stick\" to the solid surface; the contact line (the curve separating the wet and dry regions) does not move under small applied forces although the free surface (the liquid-air interface) does move. The origin and appropriate modeling of contact angle hysteresis is the subject of much debate in the engineering literature [@EralMannetjeOh; @KarimReview]. Similar phenomena occur in the context of two phase flow in a porous medium, called capillary pressure hysteresis in this context [@ZMSNHJ; @DPW; @SS]. Typically the flow velocity in each phase is determined by Darcy's law and the phase interface moves with velocity proportional to a pressure differential. However, when capillarity forces at the interface, where individual grains of the medium are only partially wetted, are of the same order as the pressure differential, pinning can occur. The origin of this pinning is exactly the microscopic surface roughness of the matrix medium and the associated contact angle hysteresis. The phase interface only advances or recedes when the pressure differential exceeds a certain threshold. The present paper is on the qualitative features of a macroscopic model for these types of interface pinning phenomena. Our model is inspired by the one proposed by DeSimone, Grunewald, and Otto [@DeSimoneGrunewaldOtto], with theory developed by Alberti and DeSimone [@alberti2011]. Instead of the capillarity energy with slowly varying volume constraint, as considered in [@DeSimoneGrunewaldOtto; @alberti2011], we study a quasi-static and rate independent evolution associated with the Alt-Caffarelli one-phase energy functional under a slowly varying Dirichlet forcing. The mathematical reason for considering the one-phase problem instead of the capillarity problem is to simplify various technical aspects of the problem, in particular giving us access to the more developed regularity theory of the one-phase free boundary problem. Specifically, the work of Chang-Lara and Savin [@ChS] on Bernoulli obstacle problems and its extension by Ferreri and Velichkov [@FerreriVelichkov] will play a major role. We emphasize that, as it concerns capillarity, the models we consider in this work are *toy models*. Nonetheless, as is typically the purpose of toy models, we expect the mathematical philosophies developed here to be relevant to the true capillarity model. Our contribution in this work is in connecting several energetic and geometric notions of weak solution, to study the pointwise qualitative description of those solutions: (1) *global energetic solutions* (which we just refer to as "energy solutions\"), (2) *obstacle evolution solutions*, and (3) *motion law solutions*, see Section [1.1](#s.intro-problem-details){reference-type="ref" reference="s.intro-problem-details"} below for definitions. Each of these notions of weak solution has different advantages, and only by connecting the concepts can we obtain a more complete understanding of the model. Energy solutions can be defined in a quite general setting, but are difficult to work with in terms of geometric properties and their jump law can be un-physical. Among the general energy solutions, minimizing movements solutions generated by a time incremental scheme behave especially well. Obstacle solutions on the other hand are easier to work with in terms of geometry, and have a desirable jump law, jumping "as late and as little as possible\". However the obstacle notion only makes sense in the important special case of piecewise time monotone forcing and it is difficult to derive energetic properties. These two evolutions have different jump laws, but conceptually are linked by the idea that their motion at non-jump times is closely related. Both evolutions, at least formally, satisfy the *dynamic slope condition* which governs the continuous motion but only partially specifies the jumps. In what follows we we introduce the notion of solutions that we will consider in the paper, and then state our main results. ## One-phase free boundary problem {#s.intro-problem-details} Consider a connected domain $U$ in $\mathbb{R}^d$ with compact complement. For $u: U \to [0,\infty)$ consider the Alt--Caffarelli one-phase free boundary energy functional $$\label{e.energy-intro} \mathcal{J}(u) = \int_U |\nabla u|^2 + {\bf 1}_{\{u>0\}} \ dx,$$ where $\mathbf{1}_{\{u > 0\}}$ is the indicator function of the set $\Omega(u) := \{u>0\}$. In the context of capillarity modelling the droplet's free surface (liquid--gas) is given by the graph of $u$ on $U$, and the positivity set $\Omega(u)$ is the wetted area: the contact between the liquid and solid phase; see Figure [\[f.one-phase\]](#f.one-phase){reference-type="ref" reference="f.one-phase"}. The triple junction or contact line (liquid-gas-solid) is located at the free boundary $\partial\Omega(u) \cap U$. The Dirichlet energy $\int_U |\nabla u|^2 \,dx$ comes from a linearization of the surface area. We recall that the Euler-Lagrange equation associated with the energy functional $\mathcal{J}$ is the one-phase Bernoulli free boundary problem $$\label{e.euler-lagrange-bernoulli} \begin{cases} \Delta u = 0 & \hbox{in } \Omega(u) \cap U\\ |\nabla u|^2 = 1 &\hbox{on } \partial \Omega(u) \cap U. \end{cases}$$ We are interested in rate independent evolutions associated with the energy functional $\mathcal{J}$, driven by a Dirichlet boundary data $F(t) : [0,T] \to (0,\infty)$ acting as the external forcing. We augment the energy functional ([\[e.energy-intro\]](#e.energy-intro){reference-type="ref" reference="e.energy-intro"}) by adding an energy dissipation in a form of a *dissipation distance*: for any pair of sets $\Omega_0$ and $\Omega_1$ in $\mathbb{R}^d$ we define $$\operatorname{Diss}(\Omega_0,\Omega_1) = \mu_+|\Omega_1 \setminus \Omega_0| + \mu_-|\Omega_0 \setminus \Omega_1|.$$ This non-symmetric distance measures the energy dissipated by the motion of the free interface under (monotone) motion of the positive phase from state $\Omega_0$ to state $\Omega_1$. The coefficients $\mu_+>0$ and $\mu_- \in (0,1)$ can be viewed as the friction forces per unit length of the free interface, respectively for advancing and receding regimes. In what follows we will abuse notation and also write $\operatorname{Diss}(u,v) = \operatorname{Diss}(\Omega(u),\Omega(v))$ for the dissipation distance between the positive phase of two profiles $u$ and $v$. In the simple model we assume that free interface can move only if the "force" $|\nabla u|^2 - 1$ per unit length on the interface coming from the first variation of potential energy ([\[e.energy-intro\]](#e.energy-intro){reference-type="ref" reference="e.energy-intro"}) can overcome the static friction force $\mu_+$ or $\mu_-$, depending whether the contact line advances or recedes. Furthermore, the scale at which the contact line moves is much faster than the scale on which we observe the state of the system, and therefore at each time $t$ the interface is assumed to be in an equilibrium configuration and cannot move. This state $u(t)$ can be characterized as a local minimizer of the augmented energy functional $$\begin{aligned} \label{e.E-def-intro} \mathcal{E}(u, u') := \mathcal{J}(u') + \operatorname{Diss}(u, u'),\end{aligned}$$ that expresses the total of the potential energy $\mathcal{J}(u')$ in an alternative state $u'$ and the energy dissipated by the friction forces on the contact line required to reach the alternative state. The formal first variation of the augmented energy function $\mathcal E(u,\cdot)$ results in the pinned one-phase free boundary problem $$\label{e.stability-condition-intro} \begin{cases} \Delta u= 0 & \hbox{ in } \ \{u>0\} \cap U,\\ 1-\mu_- \leq |\nabla u|^2 \leq 1+\mu_+ & \hbox{on } \partial \{u>0\} \cap U. \end{cases}$$ Now the gradient is allowed to take a range of values on the interface, this is the manifestation of contact angle hysteresis in this model as "gradient hysteresis\". The reason why the state might change with $t$ is that it is subject to the forcing by Dirichlet boundary condition $$\begin{aligned} u(t) = F(t) \qquad \text{on } \partial U\end{aligned}$$ at each $t \geq 0$. Since we can observe the state only in an equilibrium, the evolution is rate-independent in the sense that the path does not depend on how fast $F$ changes, that is, the time variable can be monotonically reparametrized yielding an equivalent evolution. Varying $F(t)$ pulls up (or down) the profile $u(t)$ but the free boundary remains pinned as long as the gradient at the free boundary is within the pinning interval in ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}). Once the gradient saturates one of the endpoints in the pinning condition ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}) somewhere, the interface needs to move. And, indeed, the free boundary *only* advances or recedes when the gradient saturates the corresponding endpoint of the pinning interval. This heuristic suggests the *dynamic slope condition* $$\label{e.dynamic-slope-condition-intro} |\nabla u(t)|^2 = 1\pm \mu_\pm \quad \hbox{ if } \quad \pm V_n(\Omega(u(t)),x) >0 \quad \hbox{on } \ \partial \Omega(u(t)) \cap U$$ where $V_n(\Omega(u(t)),x)$ is the outward normal velocity of $\Omega(u(t))$ at $x \in \partial \Omega(u(t))$. This condition is analogous to the dynamic contact angle condition which is studied in the physics of capillarity. ## Rate independent evolution rules Now we proceed to detailed descriptions of the (1) energy solutions, (2) obstacle solutions, and (3) motion law solutions. ## Energetic weak solutions {#energetic-weak-solutions .unnumbered} The view of the evolution as a continuum of stable states of the energy functional $\mathcal{E}$ motivates the first notion of solutions of the rate-independent evolution of the Bernoulli functional. However, the local minimality is strengthened to a global one, as is common in the theory of rate-independent systems [@mielke2015book; @alberti2011]. Additionally, an evolution between any two states at two times along the solution must be energetically allowed: the dissipated energy required to reach the second state from the first cannot be larger than the difference of potential energies of the two states together with the work done by the external forcing. **Definition 1**. A measurable $u : [0,T] \to H^1(U)$ is a *energy solution* [([E](#d.energy_solution))]{.upright} of the quasistatic evolution problem driven by Dirichlet forcing $F$ if the following hold: 1. (*Forcing*) For all $t \in [0,T]$ $$u(t) = F(t) \ \hbox{ on } \partial U$$ 2. (*Global stability*) The solution $u(t)\in H^1(U)$ and satisfies for all $t \in [0,T]$: $$\label{e.stability} \mathcal{J}(u(t)) \leq \mathcal{J}(u') + \operatorname{Diss}(u(t),u') \qquad \text{for all }u' \in u(t)+ H^1_0(U).$$ 3. (*Energy dissipation inequality*) For every $0 \leq t_0 \leq t_1 \leq T$ it holds $$\label{e.diss-inequality} \mathcal{J}(u({t_0}))-\mathcal{J}(u({t_1})) + \int_{t_0}^{t_1}2 \dot{F}(t)P(t) \ dt \geq \operatorname{Diss}(u(t_0),u(t_1)).$$ Here $P(t) = P(u(t))= \int_{\partial U} \frac{\partial u(t)}{\partial n} \ dS$ is an associated pressure. In some works, for example [@alberti2011], energy solutions are required to satisfy an *energy dissipation balance* condition instead of the inequality we use. That notion is equivalent to ours, see Remark [Remark 50](#r.energy-dissipation-equality){reference-type="ref" reference="r.energy-dissipation-equality"} later for more details. The Euler-Lagrange equations for the global stability condition ([\[e.stability\]](#e.stability){reference-type="ref" reference="e.stability"}) is the pinned one-phase problem ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}). Taking the time derivative in the energy dissipation balance, see ([\[e.energy-diss-eq\]](#e.energy-diss-eq){reference-type="ref" reference="e.energy-diss-eq"}) and ([\[e.formal-energy-argument-conclusion\]](#e.formal-energy-argument-conclusion){reference-type="ref" reference="e.formal-energy-argument-conclusion"}) below for details of the computation, shows that smooth energy solutions satisfy the dynamic slope condition ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}). Justifying this formal computation for general energy solutions is quite difficult and is the content of Theorem [Theorem 6](#t.main-2){reference-type="ref" reference="t.main-2"} below. ### Minimizing movements scheme {#minimizing-movements-scheme .unnumbered} The most typical way to construct an energy solution [([E](#d.energy_solution))]{.upright} is based on a time-incremental or minimizing movements scheme. Consider the time-discrete approximation scheme $$\label{e.minimizing-movement-scheme} u_{\delta}^k \in \mathop{\textup{argmin}} \left\{ \mathcal{J}(w) + \operatorname{Diss}(u^{k-1}_\delta, w): w\in F(k\delta)+ H^1_0(U)\right\}.$$ Using piecewise constant interpolation define, for all $t \in [0,T]$, $$u_{\delta}(t):= u_{\delta}^k \ \hbox{ and } \ F_\delta(t) = F(k\delta) \ \hbox{ if } \ t\in [k\delta, (k+1)\delta).$$ The time incremental scheme does turn out to produce energy solutions via a compactness idea introduced by Mainik and Mielke [@MainikMielke]. **Definition 2**. Say that $u : [0,T] \to H^1(U)$ measurable is a *minimizing movements energy solution* if there is $u_\delta(t)$ solving the scheme ([\[e.minimizing-movement-scheme\]](#e.minimizing-movement-scheme){reference-type="ref" reference="e.minimizing-movement-scheme"}) and a subsequence $\delta_k \to 0$ so that $u_{\delta_k}(t) \to u(t)$ in $L^2(U)$ for *every* $t \in [0,T]$. Note that, despite the terminology, it definitely requires proof to show that minimizing movements energy solutions actually solve [([E](#d.energy_solution))]{.upright}. Our results for minimizing movements solutions are stronger than for general energy solutions of [([E](#d.energy_solution))]{.upright}, see Theorem [Theorem 5](#t.main-1){reference-type="ref" reference="t.main-1"}. ## Geometric weak solutions: obstacle solution {#geometric-weak-solutions-obstacle-solution .unnumbered} Next we introduce an, apparently, quite different notion of rate independent evolution associated with the one-phase problem which we call the *obstacle solution*. We will see that this notion is physically quite natural, but it is difficult to classify the evolution in terms of energetics, see Section [1.4](#s.discussion){reference-type="ref" reference="s.discussion"} for discussion of possibly related energetic notions. This notion of solution does rely on a certain monotonicity hypothesis on the forcing. More precisely, we assume that the forcing $F : [0,T] \to (0,\infty)$ is Lipschitz regular and only changes monotonicity on a finite set $Z \subset [0,T]$, that is, for any $(s, t) \subset [0, T] \setminus Z$ the forcing $F$ is either strictly increasing or strictly decreasing on $[s, t]$. **Definition 3**. We say that $u: [0, T] \times \overline U \to [0, \infty)$ is a *obstacle solution* [([O](#viscosity_solution))]{.upright} in $U$ driven by $F$ on $\partial U$ if 1. (*Initial data*) $u(0)$ is a viscosity solution of the local stability conditions ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}). 2. (*Dirichlet forcing*) For all $t \in [0,T]$ $$\label{e.dirichletforcing-01} u(t) = F(t) \ \hbox{ on } \ \partial U.$$ 3. (*Obstacle condition*) For every $(s,t) \cap Z = \emptyset$, so that $F$ is monotone on $[s,t]$, $u(t)$ is the minimal supersolution of ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}) and ([\[e.dirichletforcing-01\]](#e.dirichletforcing-01){reference-type="ref" reference="e.dirichletforcing-01"}) above $u(s)$ when $F$ is increasing on $[s,t]$ (resp. maximal subsolution below $u(s)$ when $F$ is decreasing). The notions of minimal supersolution and maximal subsolution above are in the viscosity solutions / Perron's method sense. The stability condition on the initial data is for convenience, otherwise the solution would jump at the initial time resulting in a "replacement\" initial data arising from a Bernoulli obstacle problem. ## Geometric weak solutions: dynamic slope condition {#geometric-weak-solutions-dynamic-slope-condition .unnumbered} Although the obstacle solution [([O](#viscosity_solution))]{.upright} and the energetic solution [([E](#d.energy_solution))]{.upright} have different jump laws, our philosophy is that they should "only\" differ due to the jump laws. The commonality of these two will be expressed in the notion of motion law solution. The motion law solution property imposes a local stability condition and a dynamic condition on the slope. In this model the local evolution law can be naturally interpreted in the framework of viscosity solutions (i.e. solutions based on the comparison principle). This motivates the notion of *motion-law viscosity solution*. **Definition 4**. $u : [0,T] \to C(\overline U)$ is a *motion-law viscosity solution* [([M](#d.MVS))]{.upright} if 1. (*Forcing*) For all $t \in [0,T]$ $$u(t) = F(t) \ \hbox{ on } \partial U.$$ 2. (*Local stability condition*) For all $t \in [0,T]$ the function $u(t)\in C(\overline U)$ is a continuous viscosity solution of [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"} in the semicontinuous envelope sense. 3. (*Dynamic slope condition*) $u$ is a viscosity solution of ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}) on $[0,T] \times U$ in the semicontinuous envelope sense. We recall viscosity solutions in Section [4](#s.comparison){reference-type="ref" reference="s.comparison"}, and in particular the precise meaning of the local stability condition in Definition [Definition 26](#d.local-stability-visc){reference-type="ref" reference="d.local-stability-visc"} and the dynamic slope condition is given in Definition [Definition 30](#d.dynamic-slope-condition-visc){reference-type="ref" reference="d.dynamic-slope-condition-visc"}. ## Statements of main results Our first collection of results is on the uniqueness and regularity properties of the three evolutions in the case of strongly star-shaped data. Star-shapedness is a natural topological hypothesis in the context of free boundary problems which often allows for uniqueness and regularity and prevents topological changes. In our context star-shapedness also rules out jumps and so we can show a significant (but not total) overlap of the three notions of solution considered in this work. **Theorem 5** (see Theorem [Theorem 32](#t.equivalence){reference-type="ref" reference="t.equivalence"} and Theorem [Theorem 67](#t.oqe-MM-relation){reference-type="ref" reference="t.oqe-MM-relation"}). *Suppose that $\mathbb{R}^d \setminus U$ and $\Omega_0$ are strongly star-shaped and bounded, $\Omega_0$ is $C^{1,\alpha}$, and $F: [0,T] \to (0, \infty)$ is Lipschitz and only changes monotonicity finitely many times. Let $u$ be the unique obstacle solution [([O](#viscosity_solution))]{.upright} on $[0,T]$ with a strongly-star-shaped initial data $u(0)$, and $u(0)$ satisfies [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"}. Then* 1. *[\[part.main1-p1\]]{#part.main1-p1 label="part.main1-p1"} The profile $u(t)$ is strongly star-shaped for each time, and it is the unique motion law viscosity solution [([M](#d.MVS))]{.upright} with initial data $u(0)$.* 2. *[\[part.main1-p2\]]{#part.main1-p2 label="part.main1-p2"} The profile $u(t)$ and positivity set $\Omega(u(t))$ have the regularity $C^{0,1}_tC_x \cap L^\infty_tC^{1,\min\{\frac{1}{2},\alpha\}-}_x$.* 3. *[\[part.main1-p3\]]{#part.main1-p3 label="part.main1-p3"} The unique [([M](#d.MVS))]{.upright} solution $u(t)$ is also the unique minimizing movement energy solution with initial data $u(0)$. Moreover the solutions of the discrete-time minimizing scheme converge uniformly to $u(t)$ with a uniform rate that only depends on $F$, $\mu_{\pm}$ and $d$.* Besides the regularity theory developed in Section [3](#s.bernoulli-reg){reference-type="ref" reference="s.bernoulli-reg"}, the novel comparison principle for solutions of [([M](#d.MVS))]{.upright}, Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"}, also plays a central role in this theorem. The convergence result for minimizing movements solutions part [\[part.main1-p3\]](#part.main1-p3){reference-type="ref" reference="part.main1-p3"} follows a similar idea to Chambolle's proof of convergence of the Almgren-Taylor-Wang / Luckhaus-Sturzenhecker schemes for mean curvature flow [@chambolle04; @ATW; @LS]. We establish that the discrete-time minimizing movements satisfy [([M](#d.MVS))]{.upright} and we use the aforementioned comparison principle. The star-shapedness is a key hypothesis in Theorem [Theorem 5](#t.main-1){reference-type="ref" reference="t.main-1"}. Perhaps most importantly, the star-shaped geometry is used to ensure that there is no jump in the obstacle solution [([O](#viscosity_solution))]{.upright}. This is crucial to obtain part [\[part.main1-p3\]](#part.main1-p3){reference-type="ref" reference="part.main1-p3"}, since the energy solutions and obstacle solutions have different jump laws, see Section [2.2](#s.example-jump-conditions){reference-type="ref" reference="s.example-jump-conditions"}. The occurrence of jumps is also exactly related to non-uniqueness of the exterior obstacle Bernoulli problem and the [([M](#d.MVS))]{.upright} problem. The local cone monotonicity implied by star-shapedness is also used to obtain the regularity result in [\[part.main1-p2\]](#part.main1-p2){reference-type="ref" reference="part.main1-p2"}. Cone monotonicity allows us to show, in Section [3](#s.bernoulli-reg){reference-type="ref" reference="s.bernoulli-reg"}, that all blow-ups at the free boundary are half-plane solutions. Then we can invoke the "flat means smooth\" results of [@ChS; @FerreriVelichkov] for Bernoulli obstacle problems. All these properties are consistent with the idea that the energy landscape on the star-shaped sets has a convexity property, it would be interesting to make this precise. Next we address the general case without the topological restrictions of the star-shaped setting. In Section [4](#s.comparison){reference-type="ref" reference="s.comparison"} we show that all non-degenerate [([O](#viscosity_solution))]{.upright} solutions, regardless of geometry, satisfy [([M](#d.MVS))]{.upright}, see Lemma [Lemma 35](#l.OvsMvs){reference-type="ref" reference="l.OvsMvs"}. But whether general energy solutions are [([M](#d.MVS))]{.upright} is unclear: since jumps are now possible, [([O](#viscosity_solution))]{.upright} and [([E](#d.energy_solution))]{.upright} are typically different evolutions. Nonetheless we still show that energy solutions satisfy [([M](#d.MVS))]{.upright} in some sense. More precisely, our second main result says that general energy solutions still satisfy the dynamic slope condition in a geometric measure theoretic sense **Theorem 6** (see Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} and Theorem [Theorem 68](#t.ae-viscosity-prop){reference-type="ref" reference="t.ae-viscosity-prop"}). *Suppose $u$ is an energy solution on $[0,T]$. Then* 1. *[\[part.main2-p0\]]{#part.main2-p0 label="part.main2-p0"} *(Basic regularity properties)* The states $u(t)$ are uniformly Lipschitz and non-degenerate and $\mathcal{H}^{d-1}(\partial\Omega(t))$ is uniformly bounded in time. Also $t \mapsto \Omega(u(t))$ is in $BV([0,T];L^1(\mathbb{R}))$ and $u(t)$ has left and right limits in uniform metric at every time, denoted $u_\ell(t)$ and $u_r(t)$.* 2. *[\[part.main2-p1\]]{#part.main2-p1 label="part.main2-p1"} *(Upper and lower envelopes)* The upper and lower semicontinuous envelopes of $u$, called $u^*$ and $u_*$, are themselves energy solutions and actually $u^*(t) = \max\{u_\ell(t), u_r(t)\}$ and $u_*(t) = \min\{u_\ell(t) , u_r(t)\}$.* 3. *[\[part.main2-p2\]]{#part.main2-p2 label="part.main2-p2"} *(Dynamic slope condition a.e.)* For all $t \in [0,T]$ the function $u(t)$ satisfies the stability condition ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}) and satisfies (in terms of $u^*$ and $u_*$) the dynamic slope condition ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}) at $\mathcal{H}^{d-1}$ almost every point of its free boundary $\partial \Omega(u(t)) \cap U$.* The regularity properties in [\[part.main2-p0\]](#part.main2-p0){reference-type="ref" reference="part.main2-p0"} follow from relatively standard energy arguments, we are recalling them here for context. The precise description of the semicontinuous envelopes [\[part.main2-p1\]](#part.main2-p1){reference-type="ref" reference="part.main2-p1"} is a somewhat unusual result in the context of literature on rate-independent motion. Our analysis of the energetic properties of the upper and lower envelopes in [\[part.main2-p1\]](#part.main2-p1){reference-type="ref" reference="part.main2-p1"}, and see Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} below, is more typical of the theory of discontinuous viscosity solutions [@usersguide] and plays an important role in the proof of the dynamic slope condition. ## Discussion {#s.discussion} In order to contextualize our results we now explain further about the advantages and disadvantages of the different notions of solution and their interrelations. ### Discussion of the energy solution notion {#discussion-of-the-energy-solution-notion .unnumbered} Energy solutions have the advantage of "baking in\" the energy dissipation balance and this notion of solution generalizes better to the more physically relevant case of volume constrained evolution. There may also be advantages of the variational framework in studying questions related to homogenization. On the other hand it is difficult to derive geometric properties of the evolution purely from the energetic structure. We are not aware of any purely energetic method to obtain higher regularity of the free boundary. Even seemingly simple properties are difficult to prove just from energetics, for example it is not clear whether arbitrary [([E](#d.energy_solution))]{.upright} solutions must respect time monotonicity of the Dirichlet forcing. The central difficulty, in general, with energy solutions is to find a suitable globally defined dissipation distance. In the isotropic case that we consider in this paper there is a simple and natural dissipation distance. Such a quantity is not always available. For example, it is not clear how to define any global dissipation distance in the case of anisotropic media arising from periodic homogenization [@Feldman21; @FeldmanSmart; @Kim08; @CaffarelliMellet2]. Furthermore, even with a mathematically elegant choice of global dissipation distance, as we have, the energy evolution handles jump discontinuities in a "non-physical\" way. This is a well-known and common shortcoming of the, so-called, global energetic solutions in the dissipative evolution framework [@MielkeRossiSavare1]. One of main proposals, within the energetic framework, for improving the "locality\" of the jumps is the notion of *balanced viscosity solution* [@DalMasoToader; @MielkeRossiSavare1; @MielkeRossiSavare2; @MinottiSavare] (different from the comparison viscosity solutions we consider below) and see the book [@mielke2015book Chapter 3.8.2]. This notion is based on a modification of the time incremental minimization scheme ([\[e.minimizing-movement-scheme\]](#e.minimizing-movement-scheme){reference-type="ref" reference="e.minimizing-movement-scheme"}), adding an $L^2$-type distance term in the incremental minimization problem which strongly penalizes jumps. The balanced viscosity approach leads to a notion of energetic solution which jumps "as late and as little as possible\". In the context of the literature on rate independent motions our work is novel in that we can take advantage of certain monotonicity and comparison structures in the problem in order to (1) give a simple non-energetic characterization of the solutions which jump "as late and as little as possible\", (2) display the optimal regularity of energy solutions induced by the dynamic slope condition. ### Discussion of the obstacle solution notion {#discussion-of-the-obstacle-solution-notion .unnumbered} The obstacle solution [([O](#viscosity_solution))]{.upright} notion is well suited to study the regularity theory and qualitative geometric features of the free boundary. In particular using this notion, in Section [3](#s.bernoulli-reg){reference-type="ref" reference="s.bernoulli-reg"} and Section [4](#s.comparison){reference-type="ref" reference="s.comparison"}, we can study the shape of the free boundary as it de-pins. We will use regularity results for Bernoulli obstacle problems due to Chang-Lara and Savin [@ChS] and very recently extended to the obstacle from below case by Ferreri and Velichkov [@FerreriVelichkov]. The regularity at de-pinning turns out to be governed, at first order in the flat asymptotic expansion, by a thin obstacle problem. The thin obstacle problem is well known to have $C^{1,\frac{1}{2}}$ optimal regularity and we inherit $C^{1,\frac{1}{2}-}$ regularity for obstacle solutions in the star-shaped case, see Theorem [Theorem 5](#t.main-1){reference-type="ref" reference="t.main-1"} below. The other advantage of the obstacle solution, even outside of the star-shaped case, is that it handles time jump discontinuities well. Specifically, the obstacle solution jumps "as late and as little as possible\", which is regarded as a more physical jump condition the same as the notion of balanced viscosity solution, see for example [@MielkeRossiSavare1] or [@mielke2015book Chapter 3.8.2]. The obstacle solution dissipates the "right\" amount of energy on its jumps, but does not obviously yield an energetic notion of jump dissipation. It would be interesting to study the possible connection of the balanced viscosity notion with our obstacle solution [([O](#viscosity_solution))]{.upright}, but we do not address that in this work. Finally we mention that the problem [([O](#viscosity_solution))]{.upright} has an easy uniqueness property from its definition, although establishing local conditions equivalent to the minimality / maximality properties in [([O](#viscosity_solution))]{.upright} is not so easy in general, see Section [4](#s.comparison){reference-type="ref" reference="s.comparison"}. ### Discussion of the motion law solution notion {#discussion-of-the-motion-law-solution-notion .unnumbered} The concept of the motion law solution property, which we can partially but not fully justify in this work, is that the motion law viscosity solution property contains all the information in [([E](#d.energy_solution))]{.upright} or [([O](#viscosity_solution))]{.upright} *except for the jump law*. We do not have a way to make this concept fully precise, but our results are in support of this notion. The motion-law viscosity solution property, philosophically speaking, contains significantly less information than either the notion of energy solution or the notion of obstacle solution. The [([M](#d.MVS))]{.upright} property only imposes a very mild condition on the jumps via ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}). Specifically that the dynamic slope is saturated at any point that has jumped strictly inwards or outwards. This property is indeed satisfied by both energy solutions and obstacle solutions. Despite this philosophy it is still highly nontrivial to actually prove the [([M](#d.MVS))]{.upright} property, especially for general energy solutions. In Lemma [Lemma 35](#l.OvsMvs){reference-type="ref" reference="l.OvsMvs"} we show that [([O](#viscosity_solution))]{.upright} solutions do satisfy [([M](#d.MVS))]{.upright}. And our second main result, Theorem [Theorem 6](#t.main-2){reference-type="ref" reference="t.main-2"}, shows that the dynamic slope condition of [([M](#d.MVS))]{.upright} at least holds in the surface measure almost everywhere sense. ## Acknowledgments W. Feldman was partially supported by the NSF grant DMS-2009286. I. Kim was partially supported by the NSF grant DMS-2153254. N. Pozar was partially supported by JSPS KAKENHI Kiban C Grant No. 23K03212. ## Notations and conventions We list several notations and conventions which will be in force through the paper. 1. We call a constant *universal* if it only depends on $d$ and $\mu_+>0$, $\mu_- \in (0,1)$, a constant is called *dimensional* if it only depends on the dimension. 2. We will refer to universal constants by $C \geq1$ and $0<c \leq 1$ and allow such constants to change from line to line of the computation. 3. Denote $a \vee b = \max\{a, b)$ and $a \wedge b = \min(a, b)$, respectively, for the maximum and minimum of two real numbers $a$ and $b$. 4. We often abuse notation and write $\Omega(t)$ instead of $\Omega(u(t))$ etc. 5. $u^*$ and $u_*$ denote the upper-semicontinuous envelope and the lower-semicontinuous envelope of $u$, respectively, see [\[envelopes\]](#envelopes){reference-type="eqref" reference="envelopes"}. We write USC and LSC respectively as shorthand for upper semicontinuous and lower semicontinuous. # Motivating examples {#s.examples} In order to introduce the problem and motivate the phenomena we will study in the paper we present several examples with analytical computations and numerical simulations. ## Numerical simulations The obstacle solutions [([O](#viscosity_solution))]{.upright} can be relatively easily approximated by a large time limit ($\tau \to \infty$) of a dynamic contact angle problem: Suppose that $F$ is increasing on $[s, t]$. Given $u(s)$, $u(t)$ can be found as the limit $\tau \to \infty$ of the unique, monotone solution of the free boundary problem $$\begin{aligned} \left\{\begin{aligned} - \Delta w(\tau) &= 0, &&\text{in } \{w(\tau) > 0\} \cap U,\\ V_n &= \max(|\nabla w| - (1 + \mu_+)^{1/2}, 0) && \text{on } \partial\{w(\tau) > 0\} \cap U,\\ w(\tau) &= F(t), && \text{on } \partial U,\\ \{w(0) > 0\} &= \{u(s) > 0\}. \end{aligned}\right.\end{aligned}$$ Analogously, for $F$ decreasing on $[s, t]$ we replace the velocity law by $V_n = \min(|\nabla w| - (1 - \mu_-)^{1/2}, 0)$. A numerical solution can be found by adapting the level set method introduced in [@Gibou02], stopping at $\tau$ when $|V_n| < \varepsilon$ for some small parameter $\varepsilon> 0$. Plots in Figure [4](#f.convexity-loss){reference-type="ref" reference="f.convexity-loss"} were produced this way. ## Jump conditions for (global) energetic vs obstacle solutions {#s.example-jump-conditions} Next we consider an example where the jump time for the energy solutions is different from the jump time for the obstacle evolution. Heuristically speaking the obstacle evolution solutions jump as late and as little as possible, while the (global) energy solutions jump whenever it becomes energetically favorable. Consider a domain $U$ which is the complement of two disjoint closed disks and initial data given by two disjoint annuli. Then under increasing $F(t)$ the [([O](#viscosity_solution))]{.upright} solution will consist of two disjoint annuli until the value of the forcing when the boundaries of the two annuli meet at a single point. As $F$ continues increasing past that critical value the solution will need to jump outwards to a new state. This situation is depicted in the simulations in Figure [\[f.two-discs\]](#f.two-discs){reference-type="ref" reference="f.two-discs"}. The energy solution with the same data and forcing must jump before the value of the forcing when the two annuli touch. One way to see this is that the blow-up at the touching point of the annuli is a two plane solution of the form $$v(x) = (1+\mu_+)^{1/2} |x \cdot e|.$$ However this blow-up does not satisfy the global stability condition in [([E](#d.energy_solution))]{.upright}, since the harmonic replacement in any open region has the same positivity set but lower Dirichlet energy. Since any blow-up of a globally stable state is also globally stable we conclude that no energy solution can coincide with the obstacle solution all the way to the jump time. ## Convexity is not preserved under the obstacle evolution {#s.ex-non-convexity} Consider the Bernoulli free boundary problem in the complement of a convex obstacle $K$ $$- \Delta u = 0 \ \hbox{ in } \ \{u>0\} \setminus K, \ \hbox{ with } \ |\nabla u| =1 \ \hbox{ on } \ \partial \{u>0\} \setminus K.$$ It is known, see Henrot and Shahgholian [@HenrotShahgholian], that the (unique) compactly supported solution of this problem has convex super-level sets. In particular if one considers the obstacle solution [([O](#viscosity_solution))]{.upright} without pinning $\mu_\pm = 0$, then the solution $u(t)$ of the obstacle evolution (which depends only on the current value $F(t)$ due to the lack of hysteresis) is convex at all times. It is natural to ask whether convexity is still preserved under [([O](#viscosity_solution))]{.upright} in the case of nontrivial pinning interval $\mu_\pm >0$. In fact it is not. We give a simulation of a counterexample in Figure [4](#f.convexity-loss){reference-type="ref" reference="f.convexity-loss"} and present a sketched proof here. Consider the case of $K = B_1$, $F(0) = 1$, and an initial region $\Omega_0$ which is a "stadium\" type initial data which is a large portion of the strip $-a < x_2 < a$ capped off by two circles of radius $a$ centered at $(\pm b,0)$. Here $0 < a-1 \ll 1$ and $b \gg 1$. The initial region $\Omega_0$ and $K$ uniquely determine the initial profile $u_0$, which does have convex super-level sets due to [@CaffarelliSpruck]. Fix the pinning interval by the relations $$\mu_+ := \max_{\partial \Omega_0} (|\nabla u_0|^2-1) \ \hbox{ and } \ \mu_-:=\max_{\partial \Omega_0} (1 - |\nabla u_0|^2).$$ By symmetry considerations the first maximum is achieved at $(0,\pm a)$ and the second at $(\pm (b+a),0)$. We can guarantee that both maxima are positive by choosing the $a$ close to $1$ and $b$ large. Now increasing $F(t)$ slightly above $1$ the free boundary needs to immediately move outwards near $(0,\pm a)$ since the slope there is already saturated at $|\nabla u_0(0,a)|^2 = 1+\mu_+$, but since this is exactly the maximum value of the slope the outwards movement will only be in a small neighborhood of those two points. This motion must produce nonconvexity because the domain $\Omega(t)$ will have outward normal $e_2$ both at some point $(0,a+\delta(t))$ and at $(b,a)$. This argument could be justified rigorously using Theorem [Theorem 5](#t.main-1){reference-type="ref" reference="t.main-1"}. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- )]{.upright} simulations. Solid curves represent $\partial \Omega(t) \cap U$ plotted for evenly spaced values of $F(t)$, with the initial shape dashed. $\partial U$ is given by a dotted curve. Top left: Disconnected annuli initial data, jump discontinuity on touching. Top right: Receding situation (decreasing $F(t)$) with initial data given by the last step of the top left image. Note that the jump occurs at a different configuration, as late as possible. Bottom left: Different radius annuli, free boundary peels from the larger annulus after the jump. Bottom right: Stadium type initial data, convexity is not preserved.](figures/twodisks-levelset-n256s128-cab44d4.pdf){#f.convexity-loss width="50%"} )]{.upright} simulations. Solid curves represent $\partial \Omega(t) \cap U$ plotted for evenly spaced values of $F(t)$, with the initial shape dashed. $\partial U$ is given by a dotted curve. Top left: Disconnected annuli initial data, jump discontinuity on touching. Top right: Receding situation (decreasing $F(t)$) with initial data given by the last step of the top left image. Note that the jump occurs at a different configuration, as late as possible. Bottom left: Different radius annuli, free boundary peels from the larger annulus after the jump. Bottom right: Stadium type initial data, convexity is not preserved.](figures/twodisks_reverse-levelset-n256s128-cab44d4.pdf){#f.convexity-loss width="50%"} )]{.upright} simulations. Solid curves represent $\partial \Omega(t) \cap U$ plotted for evenly spaced values of $F(t)$, with the initial shape dashed. $\partial U$ is given by a dotted curve. Top left: Disconnected annuli initial data, jump discontinuity on touching. Top right: Receding situation (decreasing $F(t)$) with initial data given by the last step of the top left image. Note that the jump occurs at a different configuration, as late as possible. Bottom left: Different radius annuli, free boundary peels from the larger annulus after the jump. Bottom right: Stadium type initial data, convexity is not preserved.](figures/twodiffdisks-levelset-n256s64-61bda9a.pdf){#f.convexity-loss width=".5\\textwidth"} )]{.upright} simulations. Solid curves represent $\partial \Omega(t) \cap U$ plotted for evenly spaced values of $F(t)$, with the initial shape dashed. $\partial U$ is given by a dotted curve. Top left: Disconnected annuli initial data, jump discontinuity on touching. Top right: Receding situation (decreasing $F(t)$) with initial data given by the last step of the top left image. Note that the jump occurs at a different configuration, as late as possible. Bottom left: Different radius annuli, free boundary peels from the larger annulus after the jump. Bottom right: Stadium type initial data, convexity is not preserved.](figures/stadium-levelset-n256-8a03334.pdf){#f.convexity-loss width=".5\\textwidth"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [\[f.two-discs\]]{#f.two-discs label="f.two-discs"} [\[f.convexity-loss\]]{#f.convexity-loss label="f.convexity-loss"} ## Hysteresis {#s.ex-hysteresis} In order to understand the macroscopic effect of the hysteresis inherent in the slope pinning condition we consider a simple explicitly solvable radially symmetric example in dimension $d=2$. See Figure [\[f.hysteresis-loop\]](#f.hysteresis-loop){reference-type="ref" reference="f.hysteresis-loop"} which will be further explained below. -- -- -- -- Denote the family of radially symmetric solutions $V_{\lambda,F}$ of $$\left\{\begin{aligned} \Delta V_{\lambda,F} &= 0 \ \hbox{ in } \ \Omega(V_{\lambda,F}) \setminus \overline{B_1}, \\ V_{\lambda,F} &= F \ \hbox{ on } \ \partial B_1,\\ |\nabla V_{\lambda,F}| &= \lambda \ \hbox{ on } \ \partial \Omega(V_{\lambda,F}) \setminus \overline{B_1}. \end{aligned}\right.$$ These can be explicitly computed as $$V_{\lambda,F}(x) = F\left(1-\frac{\log |x|}{\log \zeta(\lambda^{-1}F)}\right)_+$$ where $\zeta : (0,\infty) \to (1,\infty)$ strictly monotone increasing is the inverse of $R \mapsto R \log R$ i.e. $$\zeta(s)\log \zeta(s) = s \ \hbox{ for } \ s > 0.$$ Note that the radius of the support of $V_{\lambda,F}$ is $R = \zeta(\lambda^{-1}F)$. Now let's consider the radially symmetric solution of [([M](#d.MVS))]{.upright} in $U = \mathbb{R}^2 \setminus B_1$ with some forcing $F(t)$ and $$u_0 = V_{(1-\mu_-)^{1/2},F_0}.$$ The solution must be of the form $$u(t) = V_{\lambda(t),F(t)} \ \hbox{ with } \ \lambda(t)^2 \in [1-\mu_-,1+\mu_+]$$ and the dynamic slope condition becomes $$\pm\frac{d}{dt}R(t) = \pm\frac{d}{dt}\zeta(\lambda(t)^{-1}F(t)) >0 \ \hbox{ implies } \ \lambda(t) = 1\pm\mu_\pm.$$ In other words $$\pm\frac{d}{dt}R(t) > 0 \ \hbox{ implies } \ R(t) = \zeta((1\pm\mu_\pm)^{1/2}F(t)).$$ or viewing matters in the $(R,F)$ plane the state of the system is always in the region $$\mathcal{S} = \bigg\{(R,F) \in [1,\infty)\times (0,\infty) : \ \zeta((1+\mu_+)^{-1/2}F) \leq R \leq \zeta((1-\mu_-)^{-1/2}F)\bigg\}$$ and $R$ can only increase/decrease while on the respective boundary curves $$\gamma_\pm = \{(R,F): \ R = \zeta((1\pm\mu_\pm)^{-1}F)\}.$$ # Regularity theory of Bernoulli obstacle problems {#s.bernoulli-reg} In this section we consider a pair of obstacle problems for the Bernoulli free boundary problem, one with an obstacle from above and the other with an obstacle from below. The two problems are similar but not exactly symmetric, as we will see below in the analysis. The regularity theory of the problem with obstacle from above has been developed by Chang-Lara and Savin [@ChS], and as we were finishing preparing this paper their theory has been extended to the obstacle from below case by Ferreri and Velichkov [@FerreriVelichkov]. We present the problems, recall the flat implies smooth regularity results from the above-mentioned works, and then show how to achieve the initial flatness and full regularity under a cone monotonicity hypothesis which is appropriate for our work. ## Bernoulli obstacle problems Let $U$ be an open region, the domain. We say that $u$ is a solution / supersolution / subsolution of the (unconstrained) Bernoulli free boundary problem in $U$ if $$\label{e.bernoulli-unconstrained} \begin{cases} \Delta u = 0 &\hbox{in } \{u>0\} \cap U\\ |\nabla u| = 1 &\hbox{on } \ \partial \{u>0\} \cap U. \end{cases}$$ Let the *obstacle* $O$ be another open region with $C^{1,\alpha}$ boundary. **Definition 7**. A function $u \in C(\overline{U})$ is a solution of the Bernoulli problem in $U$ with obstacle $O$ from below if $$\label{e.obstacle-below} \begin{cases} \Delta u = 0 &\hbox{in } \{u>0\} \cap U\\ u>0 &\hbox{in } O \cap U \\ |\nabla u| = 1 &\hbox{on } \ (\partial \{u>0\} \setminus \overline{O}) \\ |\nabla u| \leq 1 &\hbox{on } \Lambda:=\ (\partial\{u>0\} \cap \partial O) \end{cases}$$ **Definition 8**. A function $u \in C(\overline{U})$ is a solution of the Bernoulli problem in $U$ with obstacle $O$ from above if $$\label{e.obstacle-above} \begin{cases} \Delta u = 0 &\hbox{in } \{u>0\} \cap U\\ u=0 &\hbox{in } \ \overline{U} \setminus O\\ |\nabla u| = 1&\hbox{on } \ (\partial \{u>0\} \cap O) \\ |\nabla u| \geq 1 &\hbox{on } \Lambda:= \ (\partial\{u>0\} \cap \partial O). \end{cases}$$ See Figure [\[f.obstacle-pics\]](#f.obstacle-pics){reference-type="ref" reference="f.obstacle-pics"} for depictions of obstacle solutions. ## Additional hypotheses The obstacle below (resp. above) problems do not provide any a-priori bound on the slope at the free boundary from below (resp. above). Given a regular domain $O$ and a specific boundary data on $\partial U$ one could establish such bounds on the interior. However it is more convenient for us to just list these additional bounds as hypotheses which will be in force for some (but not all) of the statements below. Let $0 < \kappa < 1$ and consider the hypothesis: in the obstacle from below case $$\label{e.obstacle-below-hyp} |\nabla u| \geq \kappa \ \hbox{ in the viscosity sense on } \ \partial \{u>0\} \cap U,$$ and in the obstacle from above case $$\label{e.obstacle-above-hyp} |\nabla u| \leq \kappa^{-1} \ \hbox{ in the viscosity sense on } \ \partial \{u>0\} \cap U.$$ ## Existence and typical examples An indicative example of $u$ solving a Bernoulli problem with obstacle from below [\[e.obstacle-below\]](#e.obstacle-below){reference-type="eqref" reference="e.obstacle-below"} is the Perron's method obstacle minimal supersolution $$\label{e.minimal-example} u(x) := \inf\{v(x): \hbox{$v$ is a supersolution of (\ref{e.bernoulli-unconstrained}), $v = g$ on $\partial U$, and } v>0 \hbox{ in } O \cap \overline{U}\}.$$ Here $g \in C(\partial U)$ is some boundary condition. In order for the minimal supersolution to also be positive on $O$ we need to put some additional condition on $g$. Notice that every supersolution in ([\[e.minimal-example\]](#e.minimal-example){reference-type="ref" reference="e.minimal-example"}) is above $w$, the solution of the Dirichlet problem $$\Delta w = 0 \ \hbox{ in } \ O \cap U, \ w = 0 \ \hbox{ on } \ \partial O \cap U, \ \hbox{ and } \ w = g \ \hbox{ on }\ \partial U \cap \overline{O}.$$ So if $w >0$ in $O$ then $u$ will be positive in $O$ as well. In particular it would suffice to assume that $$g > 0 \ \hbox{ on } \ O \cap \partial U.$$ The obstacle solution property follows from the typical Perron's method arguments. Local upward perturbations are possible everywhere in $U$, so $u$ is a supersolution everywhere, while local downward perturbations are possible only away from $\partial \{u>0\} \cap \overline{O}$ limiting the subsolution property. An indicative example of $u$ solving a Bernoulli problem with obstacle from above ([\[e.obstacle-above\]](#e.obstacle-above){reference-type="ref" reference="e.obstacle-above"}) is the obstacle maximal subsolution $$\label{e.maximal-example} u(x) := \sup\{v(x): \hbox{$v$ a subsolution of (\ref{e.bernoulli-unconstrained}) in $U$, $v = g$ on $\partial U$, and } \{v>0\} \subset O\}.$$ For consistency the positivity set of $g$ on $\partial U$ should be contained in $O \cap \partial U$. As before Perron's method applies, arbitrary downward perturbations are possible so $u$ is a subsolution everywhere, but local upward perturbations are only allowed away from $\partial \{u>0\} \cap \overline{O}$. **Remark 9**. As in [@ChS] it is also possible to construct solutions to ([\[e.obstacle-above\]](#e.obstacle-above){reference-type="ref" reference="e.obstacle-above"}) by minimal supersolution Perron's method, or by energy minimization. However, it is the maximal subsolutions that we will encounter in this work. This is something we need to be careful with, since non-degeneracy at the free boundary is a more delicate issue for maximal subsolutions. ## Flat implies $C^{1,\frac{1}{2}-}$ and regularity of cone monotone solutions First let us recall the flat implies $C^{1,1-}$ regularity away from the obstacle. This result is originally due to Caffarelli [@Caffarelli1; @Caffarelli2]. A more recent alternative proof by De Silva [@DeSilva] has motivated the techniques used to study the obstacle problems we are considering.. **Theorem 10**. *(flat implies $C^{1,1-}$ for unconstrained Bernoulli [@Caffarelli1; @Caffarelli2]) Let $u$ solves ([\[e.bernoulli-unconstrained\]](#e.bernoulli-unconstrained){reference-type="ref" reference="e.bernoulli-unconstrained"}) in $B_1$ and $0 \in \partial \{u>0\}$. For any $\beta \in (0,1)$ there is $\varepsilon_0>0$ and $C \geq 1$ universal so that if $$(x \cdot e - \varepsilon)_+ \leq u(x) \leq (x\cdot e + \varepsilon)_+ \ \hbox{ in } \ B_1$$ for some $\varepsilon\leq \varepsilon_0$, then for all $0 < r < 1$ and some $|\nabla u(0)| = 1$ $$(x \cdot \nabla u(0) - C\varepsilon r^{1+\beta})_+ \leq u(x) \leq (x\cdot \nabla u(0) + C\varepsilon r^{1+\beta})_+ \ \hbox{ in } \ B_r.$$* Next we recall a similar flat implies smooth result on the contact set between the solution and the obstacle. Consider the contact set of the free boundary with the obstacle $\Lambda := \partial \{u>0\} \cap \partial O$. Let us denote $\partial'\Lambda$ to be the boundary of $\Lambda$ relative to $\partial O$. **Theorem 11**. *(flat implies $C^{1,\frac{1}{2}-}$ at the contact set [@ChS; @FerreriVelichkov]) Let $u$ solves either ([\[e.obstacle-above\]](#e.obstacle-above){reference-type="ref" reference="e.obstacle-above"}) or ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}) in $B_1$ and $0 \in \partial' \Lambda$, $O$ a $C^{1,\alpha}$ obstacle, and $e$ the inward normal to $O$ at $0$. For any $0 < \beta < \min\{\alpha,\frac{1}{2}\}$ there is $\varepsilon_0>0$ and $C \geq 1$ universal so that if $$(x \cdot e - \varepsilon)_+ \leq u(x) \leq (x\cdot e + \varepsilon)_+ \ \hbox{ in } \ B_1$$ for some $\varepsilon\leq \varepsilon_0$, then for all $0 < r < 1$ $$(x \cdot e - C\varepsilon r^{1+\beta})_+ \leq u(x) \leq (x\cdot e + C\varepsilon r^{1+\beta})_+ \ \hbox{ in } \ B_r.$$* In both the obstacle above and obstacle below cases the proof is based on establishing an asymptotic expansion for flat solutions of the form $$u(x) \sim (x \cdot e + \varepsilon w(x) + o(\varepsilon))_+$$ where $w$ is a solution of a certain Signorini or thin obstacle problem and $\varepsilon$ is the flatness in $B_1$ as in the hypothesis of Theorem [Theorem 11](#t.flat-implies-regular-obstacle){reference-type="ref" reference="t.flat-implies-regular-obstacle"}. The $C^{1,\frac{1}{2}}$ optimal regularity for the Signorini problem is the reason for the $C^{1,\frac{1}{2}-}$ regularity which appears here, and is likely (almost) optimal. The idea of the argument, which is based on a compactness principle using a special Harnack inequality for flat solutions, goes back to the work of De Silva [@DeSilva]. It is important for us to obtain a full regularity result without the flatness assumption. Of course some assumption is still necessary, singular solutions exist in higher dimensions, and fitting with the strongly star-shaped geometry we study later in the paper we will consider obstacle solutions satisfying a cone monotonicity condition. For the statement define $$\textup{Cone}(e, \theta):= \{e': e'\cdot e \geq 1-\theta\} \ \hbox{ for } \ e \in S^{d-1} \ \hbox{ and } \ \theta \in (0,1).$$ **Theorem 12**. *Suppose $O$ is a bounded set with $C^{1,\alpha}$ boundary, and that $u$ solves either ([\[e.obstacle-above\]](#e.obstacle-above){reference-type="ref" reference="e.obstacle-above"}) and ([\[e.obstacle-above-hyp\]](#e.obstacle-above-hyp){reference-type="ref" reference="e.obstacle-above-hyp"}) or ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}) and ([\[e.obstacle-below-hyp\]](#e.obstacle-below-hyp){reference-type="ref" reference="e.obstacle-below-hyp"}) in $B_1$. Assume also that there are $e \in S^{d-1}$ and $1\geq c_0>0$ so that $u$ is monotone increasing in the directions of the cone $\textup{Cone}(e, c_0)$.* *Then for any $0 < \beta < \min\{\alpha,\frac{1}{2}\}$ the positivity set $\{u>0\}$ is a $C^{1,\beta}$ domain in $B_{1/2}$ and $u\in C^{1,\beta}(\overline{\{u>0\}} \cap B_{1/2})$. The $C^{1,\beta}$ norms depend only on $\alpha$, $\kappa$, $c_0$, $d$, and the $C^{1,\alpha}$ norm of $\partial O$.* **Remark 13**. Chang-Lara and Savin [@ChS] actually prove *optimal* $C^{1,\frac{1}{2}}$ regularity. This matches the regularity of the thin obstacle / Signorini problem, which appears as the first order term in the asymptotic expansion for $\varepsilon$-flat solutions in the flatness parameter $\varepsilon$. Since this optimal regularity requires significantly more work, and it is not necessary for our purposes here, we will not address that question for the obstacle from below problem. We need to establish the initial flatness in order to apply Theorem [Theorem 10](#t.flat-implies-regular-unconstrained){reference-type="ref" reference="t.flat-implies-regular-unconstrained"} and/or Theorem [Theorem 11](#t.flat-implies-regular-obstacle){reference-type="ref" reference="t.flat-implies-regular-obstacle"} [@ChS]. This is a bit different in our case from the way that initial flatness is established in [@ChS] (minimal supersolutions) or [@FerreriVelichkov] (one-sided energy minimizers). The first difference is in the obstacle from below case. This issue appears in Lemma [Lemma 14](#l.contact-blowup){reference-type="ref" reference="l.contact-blowup"}, where we establish the initial flatness of $u$ at points of $\partial'\Lambda$ based on the existence of a non-tangential slope. In the case of the obstacle from above ([\[e.obstacle-above\]](#e.obstacle-above){reference-type="ref" reference="e.obstacle-above"}) the positive set of $u$ lies inside of $O$ which has smooth boundary. This means that a non-tangential gradient of $u$ at the free boundary point in contact with $\partial O$ can describe the leading-order behavior of $u$ near the point in the entire positive set. This is no longer true with ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}), where the positive set contains $O$. This motivates the cone monotonicity hypothesis on $u$ in the obstacle from below case. The other difference is in the obstacle from above case. Unlike [@ChS], who consider minimal supersolutions and energy minimizers, we are interested to study maximal subsolutions. Non-degeneracy at the free boundary is not known in general for maximal subsolutions. However, in the case when the free boundary is a Lipschitz graph, in particular in the cone monotone case, non-degeneracy does hold for all viscosity solutions including the maximal subsolution, see Lemma [Lemma 75](#l.lip-bdry-nondegen){reference-type="ref" reference="l.lip-bdry-nondegen"} below. This motivates the cone monotonicity hypothesis on $u$ in the obstacle from above case. In both cases the cone monotonicity hypothesis is probably overly strong, however it suffices for the applications in this paper. The possibility of a more general regularity result is an open question. We also remark that Lipschitz regularity is easier than non-degeneracy and just follows from the viscosity supersolution property ([\[e.obstacle-above-hyp\]](#e.obstacle-above-hyp){reference-type="ref" reference="e.obstacle-above-hyp"}), see Lemma [Lemma 47](#l.initial-fb-regularity){reference-type="ref" reference="l.initial-fb-regularity"} below. ## Initial free boundary flatness In order to obtain the initial flatness we will show the blow-up limit at a free boundary point on the contact set $\Lambda$ exists and is a half-planar supersolution of the unconstrained problem ([\[e.bernoulli-unconstrained\]](#e.bernoulli-unconstrained){reference-type="ref" reference="e.bernoulli-unconstrained"}). **Lemma 14**. *Assume that $u$ is a solution of ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}) and ([\[e.obstacle-below-hyp\]](#e.obstacle-below-hyp){reference-type="ref" reference="e.obstacle-below-hyp"}) (or ([\[e.obstacle-above\]](#e.obstacle-above){reference-type="ref" reference="e.obstacle-above"}) and ([\[e.obstacle-above-hyp\]](#e.obstacle-above-hyp){reference-type="ref" reference="e.obstacle-above-hyp"})) in $B_1$ which is monotone with respect to the directions of a cone $\textup{Cone}(e,c_0)$, then $$\label{e.loc-unif-convergence-blowup} \lim_{r \to 0} \frac{u(rx)}{r} = (\nabla u(0) \cdot x)_+ \ \hbox{ locally uniformly in $\mathbb{R}^d$}$$ with some gradient $c(d,\kappa)< |\nabla u(0)| \leq 1$ (resp. $1 \leq |\nabla u(0)|\leq C(d,\kappa)$). In particular, for any $\varepsilon>0$ there exists $r_0>0$ sufficiently small (depending on $u$) so that for any $r \leq r_0$ $$\label{e.initial-flatness1} |\nabla u|(0)(x_d - \varepsilon r)_+ \leq u(x) \leq |\nabla u|(0) (x_d + \varepsilon r)_+ \ \hbox{ in } \ B_{r}.$$* *Proof.* We will just consider the obstacle from below ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}), the obstacle above case is in [@ChS Lemma 2.6]. The idea is that the interior ball condition furnished by the regular obstacle gives a non-tangential blow-up limit and then the cone monotonicity upgrades this to a local uniform limit on the whole space. Note that $u$ is harmonic in its positivity set, $\{u>0\} \supset O$ in $B_1$ and $O$ is a half-space in $B_1$. Therefore $0$ is an inner regular point so there is a slope $\nabla u(0)$, parallel to the inward normal $e_d$ to $O$ at $0$, such that $$u(x) = (\nabla u(0) \cdot x)_+ + o(|x|) \hbox{ as } x\to 0 \hbox{ non-tangentially in $O$. }$$ See [@CaffarelliSalsa Lemma 11.17] for the proof, which does extend to the case of Hölder continuous coefficients. By non-degeneracy, Lemma [Lemma 75](#l.lip-bdry-nondegen){reference-type="ref" reference="l.lip-bdry-nondegen"}, and the Lipschitz estimate, standard recalled in Lemma [Lemma 47](#l.initial-fb-regularity){reference-type="ref" reference="l.initial-fb-regularity"} below, $|\nabla u(0)|$ is bounded from below away from zero, and from above. Here is where we need the cone monotonicity property. The non-tangential information, by itself, does not give us any control in $B_1 \setminus O$. From the non-tangential limit, for any $\varepsilon>0$, there is $r_0>0$ such that $$u(x) \leq 2|\nabla u(0)|\varepsilon r \ \hbox{ on } \ \{x_d = \varepsilon r\} \cap B_r \ \hbox{ for all } \ r \leq r_0.$$ Since $u$ is monotone increasing in the $e_d$ direction also $$u(x) \leq 2|\nabla u(0)|\varepsilon r \ \hbox{ on } \ \{x_d\leq \varepsilon r\} \cap B_r \ \hbox{ for all } \ r \leq r_0.$$ See Figure [\[f.cone-mon-app\]](#f.cone-mon-app){reference-type="ref" reference="f.cone-mon-app"}. Thus $\lim_{r \to 0} \frac{u(rx)}{r} = 0$ for $x_d \leq 0$ and we have upgraded the non-tangential limit to local uniform convergence of the blow-up sequence ([\[e.loc-unif-convergence-blowup\]](#e.loc-unif-convergence-blowup){reference-type="ref" reference="e.loc-unif-convergence-blowup"}). Then the uniform stability of viscosity solutions implies that $|\nabla u(0)| \leq 1$. Using non-degeneracy again we can prove the convergence of the free boundaries as well ([\[e.initial-flatness1\]](#e.initial-flatness1){reference-type="ref" reference="e.initial-flatness1"}). ◻ If we use Lemma [Lemma 14](#l.contact-blowup){reference-type="ref" reference="l.contact-blowup"} as stated then the initial flatness radius $r_0$ would depend on $u$ is a non-universal way. In turn the $C^{1,\beta}$ norm of the solution depends on the initial flatness, and would also depend on $u$ in a non-universal way. Next we show, using the flat implies smooth results, that $r_0$ is actually universal. **Lemma 15**. *Assume that $u$ is a solution of ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}) and ([\[e.obstacle-below-hyp\]](#e.obstacle-below-hyp){reference-type="ref" reference="e.obstacle-below-hyp"}) (or ([\[e.obstacle-above\]](#e.obstacle-above){reference-type="ref" reference="e.obstacle-above"}) and ([\[e.obstacle-above-hyp\]](#e.obstacle-above-hyp){reference-type="ref" reference="e.obstacle-above-hyp"})) in $B_1$ which is monotone with respect to the directions of a cone $\textup{Cone}(e,c_0)$. For any $\varepsilon>0$ there exists $r_0>0$ sufficiently small, depending only on $\varepsilon$, $c_0$, $\kappa$, $d$ and the $C^{1,\alpha}$ property of $O$, so that for any $r \leq r_0$ there is $\kappa \leq {q} \leq 1$ (resp. $1 \leq {q} \leq \kappa^{-1}$) so that $$\label{e.initial-flatness2} {q} (x_d - \varepsilon r)_+ \leq u(x) \leq {q} (x_d + \varepsilon r)_+ \ \hbox{ in } \ B_{r}.$$* *Proof.* We just consider the obstacle from below case, the obstacle from above case is similar. The proof is by compactness. Suppose otherwise, then there is $\varepsilon_0>0$, a sequence of $u_k$ solving ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}) in $B_1$, a sequence of obstacles $O_k$ all uniformly $C^{1,\alpha}$ regular, and a sequence of radii $r_k \to 0$ so that ([\[e.initial-flatness2\]](#e.initial-flatness2){reference-type="ref" reference="e.initial-flatness2"}) fails with $r = r_k$ and $\varepsilon= \varepsilon_0$. By uniform Lipschitz regularity and non-degeneracy we can assume that, on compact subsets of $B_1$, the $u_k$ converge uniformly to some $u$, the $\partial \{u_k>0\}$ converge in Hausdorff distance to $\partial \{u>0\}$, and the $O_k$ and $\partial O_k$ also converge in Hausdorff distance to another $C^{1,\alpha}$ domain $O$. Standard viscosity solution arguments show that $u$ solves ([\[e.obstacle-below\]](#e.obstacle-below){reference-type="ref" reference="e.obstacle-below"}) in $B_1$. This implies that $u$ is in $C^{1,\beta}(\overline{\{u>0\}} \cap B_{1/2})$ so for any $\varepsilon>0$ there is $r_1>0$ so that ([\[e.initial-flatness2\]](#e.initial-flatness2){reference-type="ref" reference="e.initial-flatness2"}) holds for $r \leq r_1$. Let $\varepsilon_1>0$ sufficiently small so that Theorem [Theorem 10](#t.flat-implies-regular-unconstrained){reference-type="ref" reference="t.flat-implies-regular-unconstrained"} and Theorem [Theorem 11](#t.flat-implies-regular-obstacle){reference-type="ref" reference="t.flat-implies-regular-obstacle"} hold, and then $r_1>0$ small so that $$|\nabla u|(0)(x_d - \tfrac{1}{2}\varepsilon_1 r_1)_+ \leq u(x) \leq |\nabla u|(0) (x_d + \tfrac{1}{2}\varepsilon_1 r_1)_+ \ \hbox{ in } \ B_{r_1}$$ which implies, for $k$ large enough, $$|\nabla u|(0)(x_d - \varepsilon_1 r_1)_+ \leq u_k(x) \leq |\nabla u|(0) (x_d + \varepsilon_1 r_1)_+ \ \hbox{ in } \ B_{r_1}.$$ Then, applying either Theorem [Theorem 10](#t.flat-implies-regular-unconstrained){reference-type="ref" reference="t.flat-implies-regular-unconstrained"} or Theorem [Theorem 11](#t.flat-implies-regular-obstacle){reference-type="ref" reference="t.flat-implies-regular-obstacle"} as the case may be, implies that that for all $0 < r \leq r_1$ $$|\nabla u_k|(0)(x_d - C\varepsilon_1 r^{1+\beta})_+ \leq u(x) \leq |\nabla u_k|(0) (x_d + C\varepsilon_1 r^{1+\beta})_+ \ \hbox{ in } \ B_{r}$$ which contradicts the hypothesis for large $k$. ◻ # Local laws: weak solutions, regularity, and comparison principle {#s.comparison} In this section we study the viscosity solution notions [([O](#viscosity_solution))]{.upright} and [([M](#d.MVS))]{.upright}, especially in the star-shaped setting. We establish regularity of [([O](#viscosity_solution))]{.upright} solutions using the results of Section [3](#s.bernoulli-reg){reference-type="ref" reference="s.bernoulli-reg"}. The central element of a viscosity solutions theory is always the comparison principle. One of the main results of this section is a comparison principle in the star-shaped setting, Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"}, between the [([O](#viscosity_solution))]{.upright} solution and an arbitrary [([M](#d.MVS))]{.upright} solution. ## Definitions and main results Let $F = F(t)$ be a strictly positive $C^1$ function on the time interval $[0,T]$ with a discrete set of critical points, or more precisely, $$\begin{aligned} \label{F-hyp} F \in C^1([0, T]) \quad \text{and} \quad Z := \{t \in (0, T): F'(t) = 0\} \text{ is discrete}.\end{aligned}$$ In other words, $F$ only changes monotonicity at most finitely many times on any bounded interval. We start by introducing a notion of solution that states explicitly what equations the obstacle solution [([O](#viscosity_solution))]{.upright} satisfies, see Lemma [Lemma 33](#l.OqeOvs){reference-type="ref" reference="l.OqeOvs"} below. **Definition 16**. A function $u: [0, T) \times \overline U \to [0, \infty)$ is an *obstacle viscosity solution* [([OVS](#d.OVS))]{.upright} if $u(t)$ is continuous for each $t \in [0,T]$, satisfies $u(t) = F(t)$ on $\partial U$, and the following holds for any $(s,t) \subset (0,T) \setminus Z$: - If $F$ is monotone increasing on $[s,t]$ then $u(t)$ solves in the viscosity sense $$\begin{cases}\Delta u(t) = 0 &\hbox{in } \Omega(u(t)) \cap U\\ u(t)>0 &\hbox{in } \ \Omega(u(s)) \cap U\\ |\nabla u(t)|^2 \leq 1+\mu_+ &\hbox{on } \ (\partial \Omega(u(t)) \cap \partial \Omega(u(s))) \cap U\\ |\nabla u(t)|^2 = 1+\mu_+ &\hbox{on } \ (\partial \Omega(u(t)) \setminus \overline{\Omega(u(s))}) \cap U \end{cases}$$ - If $F$ is monotone decreasing on $[s,t]$ then $u(t)$ solves in the viscosity sense $$\begin{cases}\Delta u(t) = 0 &\hbox{in } \Omega(u(t)) \cap U\\ u(t)=0 &\hbox{in } \ \overline{U} \setminus \Omega(u(s))\\ |\nabla u(t)|^2 = 1-\mu_- &\hbox{on } \ (\partial \Omega(u(t)) \cap \Omega(u(s))) \cap U\\ |\nabla u(t)|^2 \geq 1-\mu_- &\hbox{on } \ (\partial\Omega(u(t)) \cap \partial \Omega(u(s))) \cap U.\end{cases}$$ **Remark 17**. Note in particular in case of (a) $u$ is monotone increasing on $[s,t]$, and in case of (b) $u$ is monotone decreasing in $[s,t]$. Given the condition on $F$, it follows that $u$ also only changes monotonicity finitely many times on any bounded interval. ### Slope condition in the comparison sense The slope conditions are interpreted in the comparison / viscosity sense which we make precise in two ways. First we define a notion of sub and superdifferential which is suited to the problem. This is also called the first order semi-jet $\mathcal J^{+,1}_{\overline{\{u > 0\}}}$ in [@usersguide]. **Definition 18**. Given a non-negative continuous function $u$ on $U$ define, for each $x_0 \in \partial \Omega(u)\cap U$ $$D_+ u(x_0) = \left\{ p \in \mathbb{R}^d : u(x) \leq p \cdot (x-x_0) + o(|x-x_0|) \ \hbox{ in } \ \overline{\Omega(u)}\right\}$$ **Definition 19**. Given a non-negative continuous function $u$ on $U$ define, for each $x_0 \in \partial \Omega(u) \cap U$ $$D_- u(x_0) = \left\{ p \in \mathbb{R}^d : u(x) \geq p \cdot (x-x_0) - o(|x-x_0|)\right\}$$ This leads to a comparison notion of slope conditions. **Definition 20**. Suppose $u: U \to [0,\infty)$ is continuous and is subharmonic in $\Omega(u) \cap U$. Then $u$ is a subsolution of $$|\nabla u|^2 \geq {Q} \ \hbox{ on } \ \partial \Omega(u) \cap U$$ if, for every $x \in \partial \Omega(u)$ and every $p \in D_+u(x)$, $$|p|^2 \geq {Q}.$$ Similarly, if $u: U \to [0,\infty)$ is continuous and is superharmonic in $\Omega(u) \cap U$, then $u$ is a supersolution of $$|\nabla u|^2 \leq {Q} \ \hbox{ on } \ \partial \Omega(u) \cap U$$ if, for every $x \in \partial \Omega(u)$ and every $p \in D_-u(x)$, $$|p|^2 \leq {Q}.$$ In this section we will define the spatial slope conditions in terms of the sub and superdifferential notions above. However, in later sections we prefer to use the usual touching test functions notion which is more convenient to argue with in many cases. **Definition 21**. Suppose $u: U \to [0,\infty)$ is continuous and is subharmonic in $\Omega(u) \cap U$. Then $u$ is a subsolution of $$|\nabla u|^2 \geq {Q} \ \hbox{ on } \ \partial \Omega(u) \cap U$$ if, whenever $\varphi$ is a smooth test function such that $\varphi_+$ touches $u$ from above at $x \in \partial \Omega(u) \cap U$ with $\Delta \varphi(x) < 0$, $$|\nabla\varphi(x)|^2 \geq {Q}.$$ Similarly, suppose $u: U \to [0,\infty)$ is continuous and is superharmonic in $\Omega(u) \cap U$. Then $u$ is a supersolution of $$|\nabla u|^2 \leq {Q} \ \hbox{ on } \ \partial \Omega(u) \cap U$$ if, whenever $\varphi$ is a smooth test function touching $u$ from above at $x \in \partial \Omega(u) \cap U$ with $\Delta \varphi(x) > 0$, $$|\nabla\varphi(x)|^2 \leq {Q}.$$ These two notions of viscosity solution are equivalent. **Lemma 22**. *Suppose that $u: U \to [0,\infty)$ is continuous and subharmonic (resp. superharmonic) in $\Omega(u) \cap U$. Then $u$ is a viscosity subsolution (resp. supersolution) in the sense of Definition [Definition 20](#d.subdifferentials-vs-def){reference-type="ref" reference="d.subdifferentials-vs-def"} if and only if it is a viscosity subsolution (resp. supersolution) in the sense of Definition [Definition 21](#d.test-fcns-vs-def){reference-type="ref" reference="d.test-fcns-vs-def"}.* We provide a proof sketch in the appendix for convenience, see Section [8.4](#s.app-vs-equivalence){reference-type="ref" reference="s.app-vs-equivalence"}. **Remark 23**. Note that this equivalence requires both solution properties to hold *everywhere*. If $\varphi$ touches $u$ from above at $x_0 \in \partial \Omega(u) \cap U$ then $\nabla\varphi(x_0) \in D_+u(x_0)$. However the reverse is not true, if $p \in D_+u(x_0)$ it does not necessarily guarantee that $u$ can be touched from above by a smooth test function at the *same point* with $\nabla\varphi(x_0) \approx p$, rather one can find a nearby point where the touching occurs. In Section [7](#s.energy-soln-motion-law){reference-type="ref" reference="s.energy-soln-motion-law"} this distinction is important since we only prove a sub/superdifferential notion $\mathcal{H}^{d-1}$-almost everywhere. ### Dynamic slope condition in the comparison sense In order to formulate the dynamic slope condition ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}) we need a weak pointwise sense of positive and negative normal velocity. We will use a notion based on space-time "light cones\" in place of standard barrier functions, see Figure [\[f.cone-pics\]](#f.cone-pics){reference-type="ref" reference="f.cone-pics"} for a sketch of the geometry. This notion of viscosity solutions is stronger than the usual comparison definition of level set velocity, since we test more free boundary points that have weaker space-time regularity. Nonetheless, this stronger notion of normal velocity fits well with other notion of solutions that we use. **Definition 24**. Given a time varying family of domains $\Omega(t)$ such that $\chi_{\Omega(t)}$ is upper semi-continuous, and for given $c,r_0>0$, we say that the *outward normal velocity* at $x_0 \in \partial \Omega(t_0)$ is at least $c$ at scale $r_0$ and write $$V^{r_0}(t_0,x_0) \geq c$$ if the *positive cone condition* holds, namely $$\label{cone_positive} \{x: \ |x-x_0| \leq c(t_0-t)\} \subset \Omega(t)^{\complement} \ \hbox{ for } \ \ t_0> t \geq t_0 - r_0.$$ Define $$V^+(t_0,x_0) := 0 \vee \sup \{c: \exists r_0>0 \ \hbox{ so that } \ V^{r_0}(t_0,x_0) \geq c\}.$$ We write $V(t_0,x_0) > 0$ if $V^+(t_0,x_0) > 0$, i.e., there are $c>0$ and $r_0>0$ so that $V^{r_0}(t_0,x_0) \geq c$. If we need to clearly specify the domain we write $V(\cdot;\Omega)$, $V^{r_0}(\cdot;\Omega)$, etc. **Definition 25**. Given a time varying family of domains $\Omega(t)$ such that $\chi_{\Omega(t)}$ is lower semi-continuous, and for given $c,r_0>0$, we say that the *inward normal velocity* at $x_0 \in \partial \Omega(t_0)$ is at least $c$ at scale $r_0$ and write $$V^{r_0}(t_0,x_0) \leq -c$$ if the *negative cone condition* holds, namely $$\label{e.cone_negative} \{x: \ |x-x_0| \leq c(t_0-t)\} \subset \Omega(t) \ \hbox{ for } \ \ t_0> t \geq t_0 - r_0.$$ Define $$V^-(t_0,x_0) := 0 \wedge \inf\{-c <0 : \exists r_0>0 \ \hbox{ so that } \ V^{r_0}(t_0,x_0) \leq -c\}.$$ We write $V(t_0,x_0) < 0$ if $V^-(t_0, x_0) < 0$, i.e., there are $c>0$ and $r_0>0$ so that $V^{r_0}(t_0,x_0) \leq -c$. If we need to clearly specify the domain we write $V(\cdot;\Omega)$, $V^{r_0}(\cdot;\Omega)$, etc. Finally, let us recall the semicontinuous envelopes $u^*$ and $u_*$ of a function $u: [0, T] \times \overline{U} \to \mathbb{R}$: $$\begin{aligned} \label{envelopes} u^*(t,x) := \limsup_{(s,y) \to (t,x)} u(s,y),\qquad u_*(t,x) := \liminf_{(s,y) \to (t,x)} u(s,y),\end{aligned}$$ where $(s,y)$ is always assumed to be in $[0, T] \times \overline U$. Recall that $u^*$ is USC and $u_*$ is LSC. Equipped with the notions of nonzero normal velocity and subdifferentials we can now precisely define the meaning of the local stability condition and the dynamic slope condition of [([M](#d.MVS))]{.upright} in Definition [Definition 4](#d.MVS){reference-type="ref" reference="d.MVS"}: **Definition 26**. We say that a bounded map $u: [0, T] \to C(\overline U)$ is a *viscosity solution of [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"} on $[0,T] \times U$ in the semicontinuous envelope sense* if 1. [\[part.stab-part-std\]]{#part.stab-part-std label="part.stab-part-std"} $u(t)$ satisfies ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}) for every $t \in [0, T]$. 2. [\[part.local-stab-sub-part\]]{#part.local-stab-sub-part label="part.local-stab-sub-part"} If $p \in D_+ u^*(t_0, x_0)$ for some $x_0 \in \partial\Omega(u^*(t_0))$ then $|p|^2 \geq 1 - \mu_-$. **Remark 27**. Note that ([\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="ref" reference="e.stability-condition-intro"}) and uniform boundedness imply that $u(t)$ are uniformly Lipschitz continuous in compact subsets of $U$ by Lemma [Lemma 47](#l.initial-fb-regularity){reference-type="ref" reference="l.initial-fb-regularity"}. Therefore the upper and lower envelopes $u^*(t)$ and $u_*(t)$ are continuous in $x$ for each $t$. It is also standard that $u^*(t)$ and $u_*(t)$ are respectively subharmonic and superharmonic in their positivity sets. **Remark 28**. Note that if $u: [0, T] \to C(\overline{U})$ such that $u(t)$ is non-degenerate uniformly in $t$ then [\[part.local-stab-sub-part\]](#part.local-stab-sub-part){reference-type="ref" reference="part.local-stab-sub-part"} in Definition [Definition 26](#d.local-stability-visc){reference-type="ref" reference="d.local-stability-visc"} follow from [\[part.stab-part-std\]](#part.stab-part-std){reference-type="ref" reference="part.stab-part-std"}. Indeed, say that $u^*(t) - \phi$ has a strict local maximum at $x_0$ in $\overline{\Omega(u^*(t))}$. By the uniform nondegeneracy we can deduce that there exists a sequence $t_n \to t$ and $x_n \to x_0$ such that $u(t_n) - \phi$ has a local maximum at $x_n$ in $\overline{\Omega(u(t_n))}$, from which it follows that $u^*(t)$ satisfies [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"}. **Remark 29**. A similar argument to Remark [Remark 28](#r.envelopes-stability){reference-type="ref" reference="r.envelopes-stability"} can be done for $u_*$ but non-degeneracy is not necessary because the test functions touch from below in $\mathbb{R}^d$ instead of in $\Omega(u(t))$. So [\[part.stab-part-std\]](#part.stab-part-std){reference-type="ref" reference="part.stab-part-std"} actually directly implies the supersolution analogue to [\[part.local-stab-sub-part\]](#part.local-stab-sub-part){reference-type="ref" reference="part.local-stab-sub-part"}: if $p \in D_- u_*(t_0, x_0)$ for some $x_0 \in \partial\Omega(u_*(t_0))$, then $|p|^2 \leq 1 + \mu_+$. **Definition 30**. We say that $u: [0, T] \to C(\overline U)$ is a *viscosity solution of ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}) on $[0,T] \times U$ in the semicontinuous envelope sense* if 1. [\[part.advancing-cond\]]{#part.advancing-cond label="part.advancing-cond"} If $V(t_0,x_0;\Omega(u^*)) >0$ for some $x_0 \in \partial\Omega(u^*(t_0))$, then any $p\in D_+u^*(t_0,x_0)$ satisfies $|p|^2 \geq 1+\mu_+.$ 2. [\[part.receding-cond\]]{#part.receding-cond label="part.receding-cond"} If $V(t_0,x_0;\Omega(u_*)) <0$ for some $x_0 \in \partial\Omega(u_*(t_0))$, then any $p\in D_-u_*(t_0, x_0)$ satisfies $|p|^2 \leq 1-\mu_-$. in $U\times (0,T]$. ### Star-shaped comparison principle / equivalence of solution notions We will show the equivalence of the notions in a strongly star-shaped setting. **Definition 31**. In the following we say that a set is *strongly star-shaped* if it is star-shaped with respect to all $x$ in a neighborhood of the origin. A function $u: U \supset\mathbb{R}^d \to \mathbb{R}$ is strongly star-shaped if each of its superlevel sets $\{u > \eta\} \cup U^\complement$ are strongly star-shaped. The important property is that a strongly star-shaped set has a Lipschitz boundary. This can be seen by drawing the interior and exterior cones created by drawing the rays from every star-shaped center though a point on the boundary, see for example [@FeldmanKim14 Figure 2]. The main result of this section is the following theorem that asserts that in the strongly star-shaped setting all three notions of solutions that we introduced turn out to be equivalent. **Theorem 32**. *If $\mathbb{R}^d \setminus U$ and $\Omega_0$ are both bounded strongly star-shaped sets, $\Omega_0$ is a $C^{1, \alpha}$ domain for some $\alpha > 0$, $F$ satisfies [\[F-hyp\]](#F-hyp){reference-type="eqref" reference="F-hyp"} and $u(0) \in C(\overline U)$ satisfies [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"}, then $$u \hbox{ is \textup{(\hyperref[viscosity_solution]{O})}\ solution } \Leftrightarrow u \hbox{ is \textup{(\hyperref[d.OVS]{OVS})}\ solution} \Leftrightarrow u \hbox{ is \textup{(\hyperref[d.MVS]{M})}{} solution}.$$ Furthermore $\Omega(u(t))$ are $C^{1,\beta}$ domains uniformly in $t \in [0,T]$ for some $\beta > 0$.* *Proof.* To prove this theorem, we first show the easy equivalence of [([O](#viscosity_solution))]{.upright} and [([OVS](#d.OVS))]{.upright} that follows from a standard viscosity solution argument and the uniqueness of an obstacle Bernoulli problem in the star-shaped setting (Lemma [Lemma 33](#l.OqeOvs){reference-type="ref" reference="l.OqeOvs"}). The fact that an obstacle solution [([O](#viscosity_solution))]{.upright} is also motion-law viscosity solution [([M](#d.MVS))]{.upright} is shown in Lemma [Lemma 35](#l.OvsMvs){reference-type="ref" reference="l.OvsMvs"}. Then we deduce the regularity in both space and time of [([O](#viscosity_solution))]{.upright}/[([OVS](#d.OVS))]{.upright} solutions (Lemma [Lemma 37](#l.space-regularity){reference-type="ref" reference="l.space-regularity"}, Corollary [Corollary 40](#c.derivative-time-cont){reference-type="ref" reference="c.derivative-time-cont"}). Finally showing that [([M](#d.MVS))]{.upright} implies the other two notions (Corollary [Corollary 42](#c.MvsOvs){reference-type="ref" reference="c.MvsOvs"}) follows a comparison-type viscosity argument using sup- and inf-convolutions between a regular [([M](#d.MVS))]{.upright} solution and a general [([M](#d.MVS))]{.upright} solution (Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"}). ◻ **Lemma 33**. *The obstacle solution [([O](#viscosity_solution))]{.upright} is also an obstacle viscosity solution [([OVS](#d.OVS))]{.upright}. Under the strong star-shapedness assumption of Theorem [Theorem 32](#t.equivalence){reference-type="ref" reference="t.equivalence"} this is the unique [([OVS](#d.OVS))]{.upright} solution.* *Proof.* [([O](#viscosity_solution))]{.upright} $\Rightarrow$ [([OVS](#d.OVS))]{.upright} can be verified by a standard viscosity solution argument and does not require star-shapedness. [([OVS](#d.OVS))]{.upright} $\Rightarrow$ [([O](#viscosity_solution))]{.upright} on the other hand follows from the uniqueness of the obstacle Bernoulli problem in the strongly star-shaped case: [([O](#viscosity_solution))]{.upright} always exists and by the previous step, it is also [([OVS](#d.OVS))]{.upright}, and hence by Theorem [Theorem 78](#t.bernoulli-obstacle-uniqueness){reference-type="ref" reference="t.bernoulli-obstacle-uniqueness"} it must be the unique [([OVS](#d.OVS))]{.upright}. ◻ **Remark 34**. Let us point out that, while most of the analysis in this section regards star-shaped initial data, for general initial data we still have a unique [([O](#viscosity_solution))]{.upright} solution by its definition, and this solution is also an [([OVS](#d.OVS))]{.upright} by the lemma above. We do not know whether [([OVS](#d.OVS))]{.upright} would imply [([M](#d.MVS))]{.upright}, or vice versa, for general initial data. However formally both notions [([M](#d.MVS))]{.upright} and [([OVS](#d.OVS))]{.upright} come with the same dynamic slope condition at space-time regular points of the free boundary. ## Obstacle solutions are motion law solutions Next we show that, in a general setting without star-shaped assumption, obstacle solutions of [([O](#viscosity_solution))]{.upright} solve the motion law [([M](#d.MVS))]{.upright}. **Lemma 35**. *If $u: [0,T] \to C(\overline{U})$ is an obstacle solution [([O](#viscosity_solution))]{.upright} on $[0,T]$ that is uniformly non-degenerate at its free boundary, then $u$ solves [([M](#d.MVS))]{.upright}.* *Proof.* The state $u(t)$ is harmonic in $\Omega(u(t))$ by definition. By Remark [Remark 27](#r.immediate-envelope-properties){reference-type="ref" reference="r.immediate-envelope-properties"} $u^*$ and $u_*$ are respectively sub and superharmonic in their positivity sets. Now we aim to check the stability conditions in Definition [Definition 26](#d.local-stability-visc){reference-type="ref" reference="d.local-stability-visc"}[\[part.stab-part-std\]](#part.stab-part-std){reference-type="ref" reference="part.stab-part-std"}. We first check the lower bound condition $|\nabla u(t)|^2 \geq 1 - \mu_-$ in the viscosity sense on $\partial \Omega(t) \cap U$ by induction on the monotonicity intervals of $F$ provided by assumption [\[F-hyp\]](#F-hyp){reference-type="eqref" reference="F-hyp"}. We know that the condition is satisfied at $t = 0$ by the compatibility of the initial data. Suppose now that we know that $u$ satisfies this condition up to $s \in Z \cup \{0\}$ and consider $t > s$ such that $(s, t) \cap Z = \emptyset$. - If $F$ is decreasing on $[s, t]$ then the lower bound on $|\nabla u(t)|^2$ follows from [([OVS](#d.OVS))]{.upright}-(b) immediately. - Otherwise $F$ is increasing on $[s, t]$. In this case $u(t)$ are minimal supersolutions above $u(s)$ for $t_0 \in (s,t]$ and so, by Lemma [Lemma 75](#l.lip-bdry-nondegen){reference-type="ref" reference="l.lip-bdry-nondegen"}, they are uniformly non-degenerate. On $(\partial \Omega(t) \setminus \overline{\Omega(s)}) \cap U$ we get the same bound from [([OVS](#d.OVS))]{.upright}-(a) since $|\nabla u(t)|^2 = 1 + \mu_+ \geq 1- \mu_-$. Finally, on $(\partial \Omega(t) \cap \partial \Omega(s)) \cap U$, as $u(t) \geq u(s)$, any test function touching $u(t)$ from above in $\overline{\{u(t) > 0\}}$ at $x \in (\partial \Omega(t) \cap \partial \Omega(s))\cap U$ also touches $u(s)$ and hence $D_+ u(t, x) \subset D_+ u(s, x)$. Hence $u(t)$ is a solution $|\nabla u(t)|^2 \geq 1 - \mu_-$ at $x$ in the viscosity sense since $u(s)$ is by the induction hypothesis. By induction we can therefore extend the property to all $t \in [0, T]$ since $Z$ is discrete. An analogous argument gives $|\nabla u(t)|^2 \leq 1 + \mu_+$. By the assumption of uniform non-degeneracy, Remark [Remark 28](#r.envelopes-stability){reference-type="ref" reference="r.envelopes-stability"} yields Definition [Definition 26](#d.local-stability-visc){reference-type="ref" reference="d.local-stability-visc"}[\[part.local-stab-sub-part\]](#part.local-stab-sub-part){reference-type="ref" reference="part.local-stab-sub-part"}. We note that the assumption is actually needed only when $F$ is decreasing. When $F$ is increasing, $u(t)$ is a minimal supersolution above $u(s)$ and hence non-degenerate by Lemma [Lemma 75](#l.lip-bdry-nondegen){reference-type="ref" reference="l.lip-bdry-nondegen"}. We now check the remaining dynamic slope conditions. First consider monotone increasing interval $[s,t]$ of $F$ and we aim to check the advancing condition Definition [Definition 30](#d.dynamic-slope-condition-visc){reference-type="ref" reference="d.dynamic-slope-condition-visc"} part [\[part.advancing-cond\]](#part.advancing-cond){reference-type="ref" reference="part.advancing-cond"}. Suppose the positive cone condition [\[cone_positive\]](#cone_positive){reference-type="eqref" reference="cone_positive"} holds for $x_0 \in \partial \Omega(u^*(t_0)) \cap U$ for some $s < t_0 \leq t$. By the positive cone condition and monotonicity there is a neighborhood $B_{r}(x_0) \subset \overline{\Omega(u(s))}^\complement$, and thus we have $|\nabla u|^2 \geq 1+\mu_+$ on $\partial \Omega(u)$ in $(s,t] \times B_r(x_0)$ by [([OVS](#d.OVS))]{.upright}-(a). If $t_0 = t$ then $u^*(t) = u(t)$ because $u(t)$ is a local maximum by increasing monotonicity to the left and decreasing monotonicity to the right, so we can reduce to $t_0 \in (s,t)$. Then, by uniform non-degeneracy of minimal supersolutions Lemma [Lemma 75](#l.lip-bdry-nondegen){reference-type="ref" reference="l.lip-bdry-nondegen"} and Remark [Remark 28](#r.envelopes-stability){reference-type="ref" reference="r.envelopes-stability"}, we can conclude that $|\nabla u^*|^2 \geq 1+\mu_+$ on $\partial \Omega(u^*)$ in $(s,t) \times B_r(x_0)$, in particular it also holds at $(t_0,x_0)$. Next we check that the receding condition Definition [Definition 30](#d.dynamic-slope-condition-visc){reference-type="ref" reference="d.dynamic-slope-condition-visc"} part [\[part.receding-cond\]](#part.receding-cond){reference-type="ref" reference="part.receding-cond"} holds for monotone decreasing interval $[s,t]$ of $F$. The argument is mostly the same with a subtle difference which we need to point out. Suppose the negative cone condition ([\[e.cone_negative\]](#e.cone_negative){reference-type="ref" reference="e.cone_negative"}) holds for $x_0 \in \partial \Omega(u_*(t_0)) \cap U$ for some $s < t_0 \leq t$. As before there is a neighborhood $B_{r}(x_0) \subset \Omega(u(s))$, and thus we have $|\nabla u|^2 \leq 1-\mu_-$ on $\partial \Omega(u)$ in $(s,t] \times B_r(x_0)$ by [([OVS](#d.OVS))]{.upright}-(b). If $t_0 = t$ then $u_*(t) = u(t)$ so we reduce to the case $t_0 \in (s,t)$. Then by Remark [Remark 29](#r.envelopes-stability-supersoln){reference-type="ref" reference="r.envelopes-stability-supersoln"}, we can conclude that $|\nabla u_*| \leq 1 - \mu_-$ on $\partial \Omega(u^*)$ in $(s,t] \times B_r(x_0)$, in particular it also holds at $(t_0,x_0)$. Notice that we do *not* need uniform non-degeneracy, which is not known in general for maximal subsolutions, because we are testing a supersolution condition. ◻ ## Spatial regularity We start with star-shapedness of [([O](#viscosity_solution))]{.upright}. **Lemma 36**. *Suppose that $u$ solves [([O](#viscosity_solution))]{.upright}, $K = \mathbb{R}^d \setminus U$ and $\Omega_0 = \{u_0>0\}$ are strongly star-shaped, $u_0$ satisfies [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"} and $F$ satisfies [\[F-hyp\]](#F-hyp){reference-type="eqref" reference="F-hyp"}. Then $u(t)$ is strongly star-shaped for every $t \geq 0$ with respect to the same set of centers as $K$ and $\Omega_0$.* *Proof.* We show the star-shapedness only with respect to the origin assuming that $K$ and $\Omega_0$ are star-shaped with respect to the origin. The proof with respect to other centers can be done analogously by translation. We will use induction on the monotonicity intervals of $F$. Suppose that $t > 0$ with $(0, t) \cap Z \neq \emptyset$. Consider the case $F(t) > F(0)$. First note that for $\lambda < 1$ $$v(x) = \begin{cases} \min\big(u(t, \lambda x), F(0)\big) & x \in \lambda^{-1} U,\\ F(0) & x \in U \setminus \lambda^{-1} U, \end{cases}$$ is a supersolution of [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"}. Since $\{u(0)>0\}$ is star-shaped and $\{u(t)>0\} \supset \{u(0)>0\}$ also $$\lambda^{-1}\{u(t) >0\} \supset \{u(0) >0\}.$$ Thus $v$ is a supersolution of [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"} above $u(0)$ and so $$u(t, \lambda \cdot) \geq u(t) \qquad \text{in }\lambda^{-1} U$$ by the minimality of $u(t)$ in the definition of [([O](#viscosity_solution))]{.upright}. Since $\lambda>0$ was arbitrary, $u(t)$ is star-shaped (as are all of its super-level sets) with respect to $0$. In the case $F(t) < F(0)$ we consider the dilation with $\lambda > 1$ and easily check that $u(t, \lambda \cdot) \leq u(0)$ in $U$ and $u(t, \lambda \cdot)$ is a subsolution of [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"} and hence $u(t, \lambda \cdot) \leq u(t)$ in $U$ by the maximality of $u(t)$ in the definition of [([O](#viscosity_solution))]{.upright}. In particular, $u(t)$ is star-shaped with respect to $0$. By translation we have star-shapedness at any point at which $K$ and $\{u_0 > 0\}$ are star-shaped. Finally by induction on the monotonicity intervals of $F$, we can continue the star-shapedness property up to any $t > 0$. ◻ A solution of [([O](#viscosity_solution))]{.upright} is a solution of the Bernoulli obstacle problem on each monotonicity interval of $F$, where the obstacle is the solution at the beginning of the interval. This allows us to use the regularity of the obstacle problem, Section [3](#s.bernoulli-reg){reference-type="ref" reference="s.bernoulli-reg"}, only a finite number of times to establish the regularity of [([O](#viscosity_solution))]{.upright} at an arbitrary time. **Lemma 37**. *Suppose that $u$ solves [([O](#viscosity_solution))]{.upright} with $F$ satisfying [\[F-hyp\]](#F-hyp){reference-type="eqref" reference="F-hyp"}, $K = \mathbb{R}^d \setminus U$ has $C^2$ boundary, $\Omega_0$ has a $C^{1,\alpha}$ boundary, and both $K$ and $\Omega_0$ are strongly star-shaped. Then for any $0 < \beta < \min\{\alpha,\frac{1}{2}\}$ the domains $\{u(t) > 0\}$ are $C^{1,\beta}$ and $u(t) \in C^{1, \beta}(\overline{\{u(t) > 0\} \cap U})$ uniformly in $t \in [0, T]$.* *Proof.* Due to the regularity of $K = \mathbb{R}^d \setminus U$ the function $u(t)$ is in $C^2(\{u(t) > 0\} \cap \partial U)$. We need to show the regularity at the free boundary of $u(t)$. By Lemma [Lemma 36](#l.star-shapedness){reference-type="ref" reference="l.star-shapedness"} the sets $\{u(t) > 0\}$ are uniformly strongly star-shaped, and therefore they are uniformly Lipschitz in $[0, T]$. Let $t > 0$ be such that $(0, t) \cap Z = \emptyset$. Therefore by the regularity of the Bernoulli obstacle problem Theorem [Theorem 12](#t.bernoulli-reg){reference-type="ref" reference="t.bernoulli-reg"} with $C^{1, \alpha}$ obstacle $\{u_0 > 0\}$ and any $0 < \beta_0 < \min\{\alpha,\frac{1}{2}\}$ the domain $\{u(t) > 0\}$ is $C^{1,\beta_0}$ and $u(t) \in C^{1, \beta_0}(\overline{\{u(t) > 0\} \cap U})$. Consider now $t > 0$ such that $(0, t) \cap Z = \{s_1\}$. By the previous step $\{u({s_1}) > 0\} \in C^{1, \beta_0}$. Applying the regularity of the Bernoulli obstacle problem with $\alpha = \beta_0$ for any $0 < \beta_1 < \beta_0$ we find that $u(t)$ has regularity $C^{1, \beta_1}$. We can continue inductively on the monotonicity intervals of $F$ up to $t = T$, just taking a strictly decreasing sequence of $\min\{\alpha,\frac{1}{2}\} > \beta_0>\beta_1> \ldots > \beta_{\# (0, T) \cap Z} = \beta$. ◻ ## Time regularity The regularity in space Lemma [Lemma 37](#l.space-regularity){reference-type="ref" reference="l.space-regularity"} of a star-shaped [([O](#viscosity_solution))]{.upright} solution yields in particular that $u(t)$ is Lipschitz locally uniformly in time. This in particular implies (with the nondegeneracy of the solutions of the Bernoulli problem) that its support moves in a Lipschitz way as well. **Lemma 38**. *Suppose that $u$ solves [([O](#viscosity_solution))]{.upright}, $K = \mathbb{R}^d \setminus U$ has $C^2$ boundary, and $K$ and $\Omega_0 = \{u_0>0\}$ are strongly star-shaped and bounded, and $u_0$ satisfies [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"}. Then $$\begin{aligned} \label{lip-in-time} |u(t, x) - u(s, x)| \leq C \|F'\|_\infty |t - s| \ \hbox{ in } \ U\end{aligned}$$ where $C = C(L, R, \min F)$ with $L$ the Lipschitz constant of $u$ in space and $R := \max_{x \in K \cup \Omega_0} |x|$.* *Proof.* *Step 1.* By the minimality of $u(t)$, including $u(0)$ by [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"}, $$\begin{aligned} u(t, x) \leq \Big(F(t) + (1 + \mu_+) (R - |x|)\Big)_+, \qquad t \in [0, T], x \in \overline U,\end{aligned}$$ since the right-hand side is a supersolution. Therefore $$\begin{aligned} \label{Omegat-radius} x \in \Omega(t) \quad \Rightarrow \quad |x| \leq \frac{F(t)}{1 + \mu_+} + R.\end{aligned}$$ *Step 2.* Fix $(s, t) \subset \mathbb{R}\setminus Z$ with $F(s) < F(t)$. Let $\lambda := F(s)/F(t) < 1$. Define $$\begin{aligned} v(x) := \begin{cases} \lambda^{-1} u(s, \lambda x), &x \in \lambda^{-1} \overline U\\ F(t), &\text{otherwise}. \end{cases}\end{aligned}$$ By the choice of $\lambda$, $v$ is continuous and hence a supersolution on $U$ with boundary data $F(t)$. By minimality, $u(t) \leq v$. Fix $\lambda x \in \Omega(s) \cap U$. [\[Omegat-radius\]](#Omegat-radius){reference-type="eqref" reference="Omegat-radius"} implies $|x| \leq \lambda^{-1} (\frac{F(s)}{1 + \mu_+} + R)$. Recall also that $u(s, \lambda x) \leq F(s)$. We have $$\begin{aligned} v(x) &= \lambda^{-1} u(s, \lambda x) = u(s, \lambda x) + (\lambda^{-1} - 1) u(s, \lambda x)\\ &\leq u(s, x) + \|\nabla u(s)\|_\infty (1 -\lambda) |x| + (\lambda^{-1} - 1)F(s)\\ &\leq u(s, x) + (\lambda^{-1} - 1) \left(L (\frac{F(s)}{1 + \mu_+} + R) + F(s)\right).\end{aligned}$$ Since $u_{t} \leq v$ we conclude that $$\begin{aligned} 0 \leq u(t, x) - u(s, x) \leq \left(\frac{L}{1+ \mu_+} + 1 + \frac{R}{F(s)}\right) (F(t) - F(s)).\end{aligned}$$ For $x \in U \setminus \lambda^{-1} U$ we note that $\operatorname{dist}(x, \partial U) \leq (\lambda^{-1} - 1) R = \frac{R}{F(s)} (F(t) - F(s))$ and therefore $$\begin{aligned} 0 \leq u(t, x) - u(s, x) &\leq F(t) - F(s) + \|\nabla u(s)\|_\infty \operatorname{dist}(x, \partial U)\\ &\leq \left(1 + \frac{LR}{F(s)}\right) (F(t) - F(s)).\end{aligned}$$ *Step 3.* If $(s, t) \subset \mathbb{R}\setminus Z$ with $F(s) > F(t)$, we have $\lambda = F(s) / F(t) > 1$ and $v \leq F(t)$. Hence $v$ is a subsolution and therefore by maximality $u(t) \geq v$. We only need to consider $x \in \Omega(s) \cap U$. This time $$\begin{aligned} v(x) &= \lambda^{-1} u(s, \lambda x) = u(s, \lambda x) + (\lambda^{-1} - 1) u(s, \lambda x)\\ &\geq u(s,x) - \|\nabla u(s)\|_\infty (\lambda - 1) |x| - (1 - \lambda^{-1})F(s)\\ &\leq u(s,x) - \left(L (\frac{F(s)}{1 + \mu_+} + R\right)\frac{F(s) - F(t)}{F(t)} - F(t) + F(s).\end{aligned}$$ Hence by $u(t) \geq v$ $$\begin{aligned} 0 \geq u(t,x) - u(s,x) \geq L \left(\frac{F(s)}{(1 + \mu_+)F(t)} + 1 + \frac{R}{F(t)}\right)(F(s) - F(t)).\end{aligned}$$ For general $(s, t)$, we recover the estimate [\[lip-in-time\]](#lip-in-time){reference-type="eqref" reference="lip-in-time"} by triangle inequality. ◻ Using the nondegeneracy of cone monotone solutions of the Bernoulli problem, Lemma [Lemma 75](#l.lip-bdry-nondegen){reference-type="ref" reference="l.lip-bdry-nondegen"}, we can deduce Lipschitz regularity of $\Omega(u(t))$ in time. **Corollary 39**. *Under the assumptions of Lemma [Lemma 38](#l.time-regularity){reference-type="ref" reference="l.time-regularity"}, the sets $\Omega(u(t))$ are locally Lipschitz in time with respect to the Hausdorff distance.* *Proof.* Let us fix $s \neq t$. Let $z \in \partial \Omega(u(t))$ be the point that maximizes the distance from $\partial \Omega(u(s))$. Let us denote $r := \operatorname{dist}(z, \partial \Omega(u(s)))$. Suppose that $z \notin \Omega(u(s))$. We have $B_r(z) \cap \Omega(u(s)) = \emptyset$ and hence $u_s = 0$ on $B_r(z)$. On the other hand, the free boundary is locally a Lipschitz graph and so the non-degeneracy (note that $u(t)$ satisfies [\[e.stability-condition-intro\]](#e.stability-condition-intro){reference-type="eqref" reference="e.stability-condition-intro"} by Lemma [Lemma 35](#l.OvsMvs){reference-type="ref" reference="l.OvsMvs"}) for the Bernoulli problem in this setting Lemma [Lemma 75](#l.lip-bdry-nondegen){reference-type="ref" reference="l.lip-bdry-nondegen"} yields $\sup_{B_r(z)} u(t) \geq c_0 r$. By the Lipschitz continuity of $u$ in time, we deduce $$\begin{aligned} c_0 r \leq\sup_{B_r(z)} u(t) \leq \tilde L |t - s|,\end{aligned}$$ where $\tilde L$ is the Lipschitz constant in [\[lip-in-time\]](#lip-in-time){reference-type="eqref" reference="lip-in-time"}. In particular, $r \leq \frac{\tilde L}{c_0} |t - s|$. If $z \in \Omega(u(s))$ then $B_r(z) \subset \Omega(u(s))$. Using the non-degeneracy, Harnack inequality and the Lipschitz regularity, we have $$\begin{aligned} C_H c_0 \frac r2 \leq C_H \sup_{B_{r/2}(z)} u(s) \leq u(s,z) \leq \tilde L |t - s|,\end{aligned}$$ which yields $r \leq \frac{2 \tilde L}{C_H c_0} |t - s|$. ◻ Finally we mention that the time regularity and space regularity can be interpolated to get time regularity of $\nabla u$. **Corollary 40**. *Under the assumptions of Lemma [Lemma 37](#l.space-regularity){reference-type="ref" reference="l.space-regularity"} for any $0 < \beta < \min\{\alpha,\frac{1}{2}\}$ $$u\in C^{0,1}_tC_x \cap L^\infty_t C^{1,\beta}_x \ \hbox{ so by interpolation } \ u \in C^{\frac{\beta}{1+\beta}}_tC_x^1.$$ In particular, if $\alpha \geq \frac{1}{2}$ then $\nabla u \in C^{\frac{2}{3}-}_tC^0_x$.* *Proof.* Since $\nabla u(t)$ are in $C^\beta(\overline{\Omega(u(t))})$ uniformly in $t$ we can use the uniform convergence of the difference quotients and the Lipschitz continuity in time of $u(t)$ from Lemma [Lemma 38](#l.time-regularity){reference-type="ref" reference="l.time-regularity"} $$\begin{aligned} (\nabla u(t,x) - \nabla u(s,x)) \cdot e &= \frac{1}{h}(u(t,x+he) - u(t,x)) + \frac{1}{h}(u(s,x+he) - u(s,x)) +O(h^\beta) \\ &= O(h^{-1}|t-s|)+O(h^\beta).\end{aligned}$$ choose $h = |t-s|^{\frac{1}{1+\beta}}$ to match the orders of the two terms. ◻ ## Comparison principle in star-shaped setting Now we state a comparison principle that closes our discussion on equivalences of notions. We will utilize this comparison principle later on in Section [6](#s.energy-soln-existence){reference-type="ref" reference="s.energy-soln-existence"}, when we discuss a specific energy solution generated by minimizing movements. **Proposition 41**. *Suppose that $w$ and $u$ are respectively a subsolution and a LSC supersolution (or a supersolution and an USC subsolution) of [([M](#d.MVS))]{.upright} in $[0,T]$ with strongly star-shaped initial data, with the ordering $w(0) \leq u(0)$ (or $w(0) \geq u(0)$). Further suppose that $w$ satisfies the following:* - *$\{w(t)>0\}$ is bounded and strongly star-shaped at each time, and its boundary is uniformly $C^{1,\alpha}$ in space and Lipschitz in time.* - *There exists a finite partition $\{t_i\}_{i=0}^{i=n}$ of $[0,T]$ such that $w$ is monotone in $[t_i, t_{i+1}]$.* *Then $w\leq u$ (or $w \geq u$).* As is typical with viscosity solutions, one of the main difficulties in the comparison principle of weak solutions is the lack of regularity. We follow a typical approach based on sup / inf convolutions in order to regularize at the point where the first touching occurs. The novel difficulty has to do with the weak notion of positive velocity and the difficulties stemming from crossings which occur with zero velocity. We can use sup / inf convolutions with decreasing radius to accelerate the free boundary velocity, but a delicate argument relying on the regularity of the [([O](#viscosity_solution))]{.upright} solution is still needed to actually locate a positive velocity first touching point. Let us mention that an [([OVS](#d.OVS))]{.upright} solution with star-shaped initial data satisfies (a) in Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"} due to Lemma [Lemma 37](#l.space-regularity){reference-type="ref" reference="l.space-regularity"} and Lemma [Lemma 38](#l.time-regularity){reference-type="ref" reference="l.time-regularity"} and (b) due to Remark [Remark 17](#monotonicity){reference-type="ref" reference="monotonicity"}. It is also an $\textup{(\hyperref[d.MVS]{M})}{}$ by Lemma [Lemma 35](#l.OvsMvs){reference-type="ref" reference="l.OvsMvs"}. Thus, from the comparison principle Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"}, we can derive the uniqueness: **Corollary 42**. *Any motion law solution [([M](#d.MVS))]{.upright} with a strongly star-shaped initial data is the unique star-shaped obstacle solution [([O](#viscosity_solution))]{.upright} with the same initial data.* *Proof of Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"}.* Let us assume that $w$ is a [([M](#d.MVS))]{.upright} subsolution with regularity properties, and $u$ is a LSC [([M](#d.MVS))]{.upright} supersolution. We will show that $$w(t,\lambda \cdot) \leq u(t,\cdot) \hbox{ for } t>0 \hbox{ and } \lambda>1,$$ which yields $w \leq u$ in the limit $\lambda \searrow 1$. The proof for the other half of the statement is parallel, arguing the opposite inequality with $\lambda < 1$. Fix $\lambda > 1$. Define $$W(t,x) := \sup_{y \in \overline B_{\rho(t)}(x)} w(t,\lambda y).$$ Here $\rho(t) = \rho_0 e^{-t}$, and $\rho_0$ is chosen proportional to $(\lambda-1)$ so that $W(0) < u(0)$ in $\overline{\{W(0) > 0\}}$, which is possible by strong star-shapedness. The main motivation for the sup-convolution is the effective reduction of the normal velocity of $W$ by $r'(t) < 0$ compared to $w$. Hence if $W$ and $u$ touch each other, either the normal velocity of $u$ is negative or $w$ has positive normal velocity near the contact, and therefore the dynamic slope condition can be used in either situation to arrive at a contradiction. Recall that $W$ is subharmonic in its positive set. We would like to show that $W(t) < u(t)$ in $\overline{\{W(t) > 0\}}$ for all $t \geq 0$. Define the first contact time $$\begin{aligned} t_0 = \inf\big\{t > 0: W(t) \not< u(t) \text{ in $\overline{\{W(t) > 0\}}$}\big\}.\end{aligned}$$ Clearly $t_0 > 0$ by lower semi-continuity of $u$ in time. If $t_0$ is finite, we want to get a contradiction. The only non-standard part here lies in the fact that the support of $u$ may jump in time, so let us discuss this point carefully. Since we assume that $W$ is star-shaped, $W$ and its support continuously decrease as $\lambda$ increases. The support of $\{W(t_0)>0\}$ indeed can be made arbitrarily close to $\overline B_{\rho(t_0)}(0)$, in particular a subset of the fixed boundary $U^\complement$ for $u$, if $\lambda$ is sufficiently large. Thus if $\{W(\cdot,t_0)>0\} \not\subset\{u(\cdot, t_0)>0\}$ due to discontinuity in time in $u$ at $t_0$, we increase $\lambda$ in the definition of $W$ until $\{W(t_0)>0\}$ precisely touches $\{u(t_0)>0\}$ from inside, so that there exists a point $x_0 \in \partial\{W(t_0)>0\} \cap \partial\{u(t_0)>0\}$. We point out that $\rho_0$ does not need to be changed even if we need to increase $\lambda$. Note that $W\leq u$ for $t\leq t_0$ for this larger choice of $\lambda$ due to the definition of $t_0$ and $W(t_0) - u(t_0)$ being subharmonic in $\{W(t_0) > 0\}$. In particular $u(t_0) - W(t_0)$ has a minimum at $x_0$. By the definition of $W$ there is a point $y_0 \in \partial \{w(t_0, \lambda \cdot) > 0\}$ with $y_0 \in \overline{B_{\rho(t_0)}}(x_0)$, and $$\begin{aligned} W(t_0, x) \geq w(t_0, \lambda (x - x_0 + y_0)) =: \phi(x).\end{aligned}$$ As $u(t_0) - \phi$ has a minimum at $x_0$, we deduce that $$\begin{aligned} \label{gradw-inD-} \lambda \nabla w(t_0, \lambda y_0) = \nabla \phi(x_0) \in D_- u(x_0).\end{aligned}$$ By our assumption (b), we know that $w$ is either monotone increasing or decreasing in a closed time interval $[t_{-1}, t_0]$ with $t_{-1}<t_0$. If $w$ is monotone decreasing in $[t_{-1},t_0]$, then $W$ has strictly decreasing support in $[t_{-1},t_0]$, namely for $c = -r'(t_0)>0$ we have $$\label{cone} B_{c(t_0-t)}(x_0) \subset\{W(t)>0\} \subset\{u(t)>0\} \hbox{ for } t_{-1}\leq t<t_0.$$ In particular the negative cone condition ([\[e.cone_negative\]](#e.cone_negative){reference-type="ref" reference="e.cone_negative"}) is satisfied for $u$ at $(t_0, x_0)$, and so as $u$ is a [([M](#d.MVS))]{.upright} supersolution we conclude by [\[gradw-inD-\]](#gradw-inD-){reference-type="eqref" reference="gradw-inD-"} that $\lambda^2 |\nabla w|^2 (t_0, \lambda y_0) \leq 1 - \mu_-$. However as $w$ is itself [([M](#d.MVS))]{.upright} subsolution, the stability condition yields $\lambda^2 |\nabla w|^2(t_0, \lambda y_0) \geq \lambda^2(1 - \mu_-)$, a contradiction. Now suppose $w$ is monotone increasing in $[t_{-1},t_0]$. It is possible that in this case $u$ still satisfies the negative cone condition ([\[e.cone_negative\]](#e.cone_negative){reference-type="ref" reference="e.cone_negative"}) for some $c<0$ and $r_0>0$, in which case one can still proceed as in the above case. So now suppose that ([\[e.cone_negative\]](#e.cone_negative){reference-type="ref" reference="e.cone_negative"}) is not true for $c := -\frac{1}{3}\rho'(t_0) > 0$ and any $r_0>0$. This means that along an increasing time sequence $t_n$ converging to $t_0$, there is $x_n$ such that $$x_n \in \{u(t_n)=0\}\cap B_{c(t_0-t_n)}(x_0).$$ -- -- -- -- Now we consider what this means in terms of $w$. Recall $y_0$ above and consider $y$ such that $|y-y_0| \leq 2c(t_0-t_n)$. Then $$\begin{aligned} |y- x_n| &\leq |y-y_0| + |y_0 - x_0| + |x_0 - x_n| \\ &\leq 2c(t_0-t_n) + \rho(t_0) + c(t_0-t_n) \\ &\leq \rho(t_n),\end{aligned}$$ where the last inequality is by convexity of $\rho(t)$. Thus, by definition of $W$, $w(t_n, \lambda y) \leq W(t_n, x_n) \leq u(t_n, x_n) = 0$. Namely $\overline B_{2c(t_0-t_n)} (y_0) \subset \{w(t_n, \lambda \cdot) = 0\}$. See the left image in Figure [\[f.comparison-pics\]](#f.comparison-pics){reference-type="ref" reference="f.comparison-pics"}. Now let $$\tau_n:= \sup\{ s\in [t_n,t_0): \overline{B}_{c(t_0-s)}(y_0)\subset \{w(s, \lambda \cdot)=0\} \}.$$ See the right image in Figure [\[f.comparison-pics\]](#f.comparison-pics){reference-type="ref" reference="f.comparison-pics"}. Since $\{w(t)=0\}$ evolves continuously in time (Corollary [Corollary 39](#c.omega-time-regularity){reference-type="ref" reference="c.omega-time-regularity"}), we have $\tau_n >t_n$, and if $\tau_n<t_0$ then there is a contact point $y_n\in \partial\{w(\tau_n)>0\} \cap \partial B_{c(t_0-\tau_n)}(y_0)$. If the order is preserved up to $t=t_0$, i.e. $\tau_n = t_0$, then we set $y_n=y_0$. From the definition of $(y_n, \tau_n)$ we see that positive cone condition [\[cone_positive\]](#cone_positive){reference-type="eqref" reference="cone_positive"} holds for $w(\cdot, \lambda \cdot)$ at $(\tau_n, y_n)$ and therefore for $w$ at $(t_n, \lambda y_n)$ with $\lambda c$. By [([M](#d.MVS))]{.upright} we have $|\nabla w|^2 (t_n, \lambda y_n) \geq 1 + \mu_+$. Since $(\tau_n, y_n) \to (t_0, y_0)$, by the gradient continuity in Corollary [Corollary 40](#c.derivative-time-cont){reference-type="ref" reference="c.derivative-time-cont"}, we deduce $$\lambda^2|\nabla w|^2(t_0, \lambda y_0) = \lim_{n\to\infty} \lambda^2|\nabla w|^2(\tau_n, \lambda y_n) \geq \lambda^2(1+\mu^+).$$ But this contradicts the fact that $u$ is a [([M](#d.MVS))]{.upright} supersolution and hence by [\[gradw-inD-\]](#gradw-inD-){reference-type="eqref" reference="gradw-inD-"} we must have $\lambda^2 |\nabla w|^2(\lambda y_0, t_0) \leq 1 + \mu_+$. Hence we conclude. ◻ # Basic space and time regularity properties of energy solutions {#s.basic-energy} In this section we study energy solutions [([E](#d.energy_solution))]{.upright} establishing existence and various spatial and temporal regularity properties. We will work in a general setting, without the special geometry of star-shapedness which was used in the previous section. In Section [5.1](#s.energy-soln-spatial-reg){reference-type="ref" reference="s.energy-soln-spatial-reg"} we recall several results from the literature on inwards and outwards minimizers of one-phase functionals. These results, in particular, also apply to states satisfying the global stability condition of energy solutions [([E](#d.energy_solution))]{.upright}. In Section [5.2](#s.energy-soln-temporal-reg){reference-type="ref" reference="s.energy-soln-temporal-reg"} we show $BV$ time regularity of energy solutions using a typical Grönwall argument via the energy dissipation inequality, see Lemma [Lemma 49](#l.energy-soln-time-reg){reference-type="ref" reference="l.energy-soln-time-reg"}. With this regularity we can also establish that the energy dissipation balance holds with equality, see Lemma [Lemma 52](#l.energy-diss-eq){reference-type="ref" reference="l.energy-diss-eq"}. Finally in Section [5.3](#s.energy-soln-lr-limits){reference-type="ref" reference="s.energy-soln-lr-limits"} we establish the main temporal regularity result of the section, Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"}. Abstract theory of bounded variation maps from $[0,T]$ into a metric space shows that $u(t)$ has left and right limits $u_\ell(t)$ and $u_r(t)$ at every time. Then we use the monotonicity structure of the problem to show that the lower and upper-semicontinuous envelopes of an energy solution are exactly $u_\ell(t) \wedge u_r(t)$ and $u_\ell(t) \vee u_r(t)$ and these are energy solutions as well. This is independently interesting, elucidating the structure of the jump discontinuities, and it also plays a key role in establishing the dynamic slope condition in Section [7](#s.energy-soln-motion-law){reference-type="ref" reference="s.energy-soln-motion-law"}. ## Inwards and outwards minimality and spatial regularity {#s.energy-soln-spatial-reg} Let $V$ be an open region that contains $\mathbb{R}^d\setminus U$. In this section we will discuss the regularity of minimizers of the *dissipation augmented energy*, defined above in ([\[e.E-def-intro\]](#e.E-def-intro){reference-type="ref" reference="e.E-def-intro"}), $$\label{e.augmented-energy-def} \mathcal{E}(V,u) := \mathcal{J}(u) + \textup{Diss}(V,\Omega(u)).$$ We will also abuse notation to write $\mathcal{E}(v,u) = \mathcal{E}(\Omega(v),u)$ when $v$ is another non-negative function on $\mathbb{R}^n$. Note that if $u$ is an energy solution on $[0,T]$, from the global stability property at each time $u(t)$ is a minimizer for $\mathcal{E}(u(t),\cdot)$. It is convenient to make a connection with the notions of inwards and outwards minimality for the Bernoulli functional $\mathcal{J}$. First of all we introduce the notation for ${Q}>0$ and an open region $U \subset \mathbb{R}^d$ $$\mathcal{J}_{Q}(u) := \int_{U} |\nabla u|^2 + {Q}{\bf 1}_{\{u>0\}} \ dx.$$ We have written and will continue to write $\mathcal{J} = \mathcal{J}_1$ abusing notation. Next we introduce the notions of inwards and outwards minimizers. These notions have appeared in the literature before, for example see the book [@VelichkovBook] for further references. **Definition 43**. $u \in H^1(U)$ is an *outward (resp. inward) minimizer* of $\mathcal{J}_{Q}(\cdot;U)$ if 1. The set $\Omega(u) = \{u>0\}$ is open and $u$ is harmonic in $\Omega(u)$. 2. For any $v \in u + H^1_0(U)$ with $v \geq u$ (resp. $v \leq u$) $$\mathcal{J}_{Q}(u) \leq \mathcal{J}_{Q}(v).$$ Minimizers of $\mathcal{E}(V,\cdot)$ have a natural inward/outward $\mathcal{J}_{1\pm \mu_\pm}$ minimality property. **Lemma 44**. *Let $V$ be an open set that contains $\mathbb{R}^d \setminus U$. Suppose that $u$ is a global minimizer of $$\mathcal{J}(v) + \textup{Diss}(V,\Omega(v)) \ \hbox{ over } \ v \in u + H^1_0(U).$$ Then $u$ is an outwards minimizer for $\mathcal{J}_{1+\mu_+}$ and an inwards minimizer for $\mathcal{J}_{1-\mu_-}$. Furthermore if $B_r$ lies outside of $\overline{V}$ then $u$ minimizes $\mathcal{J}_{1+\mu_+}$ in $B_r$, and if $B_r \subset V$ then $u$ minimizes $\mathcal{J}_{1-\mu_-}$ in $B_r$.* *Proof.* Suppose $v \in u+H^1_0(U)$ with $v \geq u$. Then $\{v>0\} \supset \{u>0\}$, so $$\label{e.outwards-min-containments} |V \setminus \Omega(u)| \geq |V \setminus \Omega(v)| \ \hbox{ and } \ |\Omega(u) \cap V| \leq |\Omega(v) \cap V|.$$ We can apply ([\[e.outwards-min-containments\]](#e.outwards-min-containments){reference-type="ref" reference="e.outwards-min-containments"}) along with the minimality property of $u$ to obtain $$\begin{aligned} \mathcal{J}_{1+\mu_+}(u) &= \mathcal{J}_1(u) + \mu_+|\Omega(u)| \\ &=\mathcal{J}_1(u) + \mu_+|\Omega(u) \setminus V| + \mu_+|\Omega(u) \cap V|\\ &= \mathcal{J}_1(u) + \textup{Diss}(V,\Omega(u)) - \mu_-|V \setminus \Omega(u)|+ \mu_+|\Omega(u) \cap V|\\ &\leq \mathcal{J}_1(v) + \textup{Diss}(V,\Omega(v)) - \mu_-|V \setminus \Omega(v)|+ \mu_+|\Omega(v) \cap V|\\ &=\mathcal{J}_{1+\mu_+}(v).\end{aligned}$$ Lastly note that if $v = u$ outside of $B_r\subset \mathbb{R}^d \setminus V$ then ([\[e.outwards-min-containments\]](#e.outwards-min-containments){reference-type="ref" reference="e.outwards-min-containments"}) holds with equalities. Symmetrical computations give the (inwards) minimality property for $\mathcal{J}_{1-\mu_-}$. ◻ ### Viscosity solution properties of inward / outward minimizers The notions of inwards and outwards minimizers are reminiscent of viscosity sub and supersolutions. Indeed there is a direct correspondence between the two notions. **Lemma 45**. *Suppose that $S>0$ and $u$ is an inwards minimizer of $\mathcal{J}_{S}$ in a domain $U$. Then in the viscosity sense $$|\nabla u|^2 \geq {Q} \ \hbox{ on } \partial \{u>0\} \cap U.$$ Similarly, if $u$ is an outwards minimizer of $\mathcal{J}_{S}$ then in the viscosity sense $$|\nabla u|^2 \leq {Q} \ \hbox{ on } \partial \{u>0\} \cap U.$$* In general the implication cannot go the other way, there are viscosity solutions, i.e. stationary points of the energy, which are not energy minimizers. It is well known that such inwards and outwards minimality properties imply the corresponding viscosity solution conditions, see for example [@VelichkovBook Proposition 7.1]. We present a proof anyway since we will use similar, but more involved, computations in the proof of Theorem [Theorem 68](#t.ae-viscosity-prop){reference-type="ref" reference="t.ae-viscosity-prop"}. Before proceeding with the proof let us write down a corollary of Lemma [Lemma 44](#l.inwards-outwards-min-props){reference-type="ref" reference="l.inwards-outwards-min-props"} and Lemma [Lemma 45](#l.inwards-outwards-min-implies-vs){reference-type="ref" reference="l.inwards-outwards-min-implies-vs"} for energy solutions. **Corollary 46** (Stability implies slope in pinning interval). *Suppose that $u$ is an energy solution on $[0,T]$, then for each time $u(t)$ is a viscosity solution of $$1-\mu_- \leq |\nabla u(t)|^2 \leq 1+\mu_+ \quad \hbox{on } \partial \{u(t)>0\} \cap U.$$* Thus energy solutions satisfy the stability condition of [([M](#d.MVS))]{.upright}. *Proof of Lemma [Lemma 45](#l.inwards-outwards-min-implies-vs){reference-type="ref" reference="l.inwards-outwards-min-implies-vs"}.* **(Subsolution)** Suppose that a smooth test function $\varphi$ touches $u$ from above in $\overline{\Omega(u)}$ strictly at $x_0 \in \partial \Omega(u)$ with $p:= \nabla\varphi(x_0) \neq 0$ and $\Delta\varphi (x_0) < 0$. Let $\delta>0$ and consider the comparison function $$u_\delta(x) := u(x) \wedge (\varphi(x) - \delta)_+.$$ Since the touching is strict $\{u>0\} \setminus \{u_\delta>0\}$ is contained in a ball of radius $o_\delta(1)$ around $x_0$ as $\delta \to 0$. In particular we can assume that $\delta$ is small enough so that $\varphi$ is superharmonic in $\{u_\delta < u\}$. By the inwards minimality of $u$ $$\mathcal{J}_{Q}(u) \leq \mathcal{J}_{Q}(u_\delta).$$ Now note that $u_\delta$ extends to a superharmonic function in $\{u>0\}$ by $$\bar{u}_\delta(x) :=\begin{cases} u_\delta(x) &x\in \Omega(u_\delta)\\ \varphi(x) - \delta &x \in \Omega(u) \setminus \Omega(u_\delta). \end{cases}$$ So we can apply Lemma [Lemma 76](#l.intbyparts2){reference-type="ref" reference="l.intbyparts2"}, formula ([\[e.intbyparts2-supersoln\]](#e.intbyparts2-supersoln){reference-type="ref" reference="e.intbyparts2-supersoln"}), and we find $$\begin{aligned} \mathcal{J}_{Q}(u) - \mathcal{J}_{Q}(u_\delta) & \geq \int_{\{u>0\} \setminus \{u_\delta>0\}}{Q}-|\nabla\varphi(x)|^2 \ dx\\ & \geq [({Q}-|p|^2)-o_{\delta}(1)]|\{{u}>0\} \setminus \{{u_\delta}>0\}|\end{aligned}$$ Combining the previous $$[({Q}-|p|^2)-o_{\delta}(1)]|\{{u}>0\} \setminus \{{u_\delta}>0\}| \leq 0$$ so dividing through by $|\{{u}>0\} \setminus \{{u_\delta}>0\}|>0$ and taking $\delta \to 0$ we find $${Q} \leq |p|^2.$$ **(Supersolution)** Suppose that a smooth test function $\varphi$ touches $u$ from below strictly at $x_0 \in \partial \Omega(u)$ with $p:= \nabla \varphi(x_0) \neq 0$ and $\Delta\varphi (x_0) > 0$. Let $\delta>0$ and consider the comparison function $$u_\delta(x) = u(x) \vee (\varphi(x) + \delta).$$ Since the touching is strict $\{u_\delta>u\}$ is contained in a ball of radius $o_\delta(1)$ around $x_0$ as $\delta \to 0$. In particular we can assume that $\delta$ is small enough so that $\Delta u_\delta \geq 0$. By the outwards minimality property of $u$ $$\mathcal{J}_{S}(u) \leq \mathcal{J}_{S}(u_\delta).$$ Applying Lemma [Lemma 76](#l.intbyparts2){reference-type="ref" reference="l.intbyparts2"} (note $u_\delta$ not harmonic in $\{u_\delta > 0\}$ but in general subharmonic) we find $$\begin{aligned} \mathcal{J}_{S}(u) - \mathcal{J}_{S}(u_\delta) &\geq \int_{\{u_\delta>0\} \setminus \{u>0\}}|\nabla u_\delta|^2 - {Q} \ dx \\ & = \int_{\{u_\delta>0\} \setminus \{u>0\}}|\nabla\varphi(x)|^2 - {Q} \ dx\\ & \geq [(|p|^2-{Q})-o_{\delta}(1)]|\{u_\delta>0\} \setminus \{u>0\}|\end{aligned}$$ Combining the previous $$[(|p|^2-Q)-o_{\delta}(1)]|\{u_\delta>0\} \setminus \{u>0\}|\leq 0$$ so dividing through by $|\{u_\delta>0\} \setminus \{u>0\}|>0$ and taking $\delta \to 0$ we find $$|p|^2 \leq {Q} .$$ ◻ ### Regularity properties of inward / outward minimizers Next we collect several basic regularity results of $\mathcal{J}_{Q}$ inwards / outwards minimizers. **Lemma 47**. 1. *(Lipschitz estimate) There is $C(d) \geq 1$ so that for any viscosity solution $u$ of $$\Delta u = 0 \ \hbox{ in } \ \Omega(u) \cap B_2 \ \hbox{ and } \ |\nabla u|^2 \leq {Q} \ \hbox{ on } \ \partial \Omega(u) \cap B_2$$ we have $$\|\nabla u\|_{L^\infty(B_1)} \leq C(\sqrt{Q}+\|\nabla u\|_{L^2(B_2)}).$$ In particular this holds for outward minimizers of $\mathcal{J}_{Q}$ in $B_2$ by Lemma [Lemma 45](#l.inwards-outwards-min-implies-vs){reference-type="ref" reference="l.inwards-outwards-min-implies-vs"}.* 2. *(Non-degeneracy) There is $c(d,Q)>0$ so that if $u \in H^1(U)$ is an inward minimizer of $\mathcal{J}_{Q}$ in $U$, then $$\sup_{B_r(x)} u \geq c r \quad\hbox{ for any } x\in \partial\{u>0\} \hbox{ and } B_r(x)\subset U.$$* 3. *[\[part.density-estimates\]]{#part.density-estimates label="part.density-estimates"} (Density estimates) For ${Q}_1\leq {Q}_2$, there is $c(d,{Q}_1,{Q}_2)>0$ so that if $u$ is an inward minimizer of $\mathcal{J}_{{Q}_1}$ and an outward minimizer of $\mathcal{J}_{{Q}_2}$ in $U$ then $$c \leq \frac{|\Omega(u) \cap B_r(x)|}{|B_r|} \leq 1 - c \hbox{ for any } x\in \partial\Omega(u) \hbox{ with } B_r(x) \subset U.$$* 4. *[\[part.perimeter-esimate\]]{#part.perimeter-esimate label="part.perimeter-esimate"} (Perimeter estimate) There is $C(d)>0$ such that if $u$ is an inward minimizer of $\mathcal J_S$ in $B_2$ then $$\textup{Per}(\Omega(u);B_1) \leq C\sqrt{Q}(1+\|\nabla u\|_{L^2(B_2)}).$$* 5. *(Hausdorff dimension) If $u$ satisfies the conclusions of [\[part.density-estimates\]](#part.density-estimates){reference-type="ref" reference="part.density-estimates"} and perimeter estimate [\[part.perimeter-esimate\]](#part.perimeter-esimate){reference-type="ref" reference="part.perimeter-esimate"} in $B_2$ then $$\mathcal{H}^{d-1}(\partial \Omega(u) \cap B_{1}) \leq C(1+\|\nabla u\|_{L^2(B_2)})$$ where $C$ depends on the constants in [\[part.density-estimates\]](#part.density-estimates){reference-type="ref" reference="part.density-estimates"} and [\[part.perimeter-esimate\]](#part.perimeter-esimate){reference-type="ref" reference="part.perimeter-esimate"}.* *Proof.* See the books [@VelichkovBook; @CaffarelliSalsa] for presentations of the proofs and citations for their original appearances in the literature, Lipschitz continuity of viscosity solutions is in [@CaffarelliSalsa Lemma 11.19], non-degeneracy is in [@VelichkovBook Lemma 4.4], density estimates are in [@VelichkovBook Lemma 5.1], perimeter estimates are in [@VelichkovBook Lemma 5.6], Hausdorff dimension estimates in [@VelichkovBook Lemma 5.9]. ◻ **Remark 48**. The $C^{1,\beta}$ regularity of $\mathcal{E}(u,\cdot)$ minimizers is known due to the very recent result of Ferreri and Velichkov [@FerreriVelichkov] when $\{u>0\}$ is $C^{1,\alpha}$. ## Bounded variation regularity in $t$ {#s.energy-soln-temporal-reg} BV regularity in time is typical for solutions of (global) dissipative evolution problems [@mielke2015book]. Following the standard argument, we establish the $\textup{BV}$ in time regularity using a Grönwall type argument with the energy dissipation inequality. **Lemma 49**. *Suppose $u$ is an energy solution [([E](#d.energy_solution))]{.upright} on $[0,T]$. Then $${\bf 1}_{\Omega(u(t))} \in \textup{BV}([0,T]; L^1(\mathbb{R}^d))$$ and $$\mathcal{J}(u(t)), \ P(u(t)) \in \textup{BV}([0,T];\mathbb{R}),$$ where recall the pressure is $P(u(t)) = \int_{\partial \Omega(t)} |\nabla u| dS= \int_{\partial U } \frac{\partial u}{\partial n} dS = F(t)^{-1}\int |\nabla u(t)|^2 \ dx$.* Before beginning the proofs we make a remark about the definition of energy solution. **Remark 50**. Energy solutions will also satisfy an *energy dissipation balance* condition. Define the total variation of the dissipation distance along a path $\Lambda(t)$ of finite measure subsets of $\mathbb{R}^d$ $$\label{e.dissbar-def} \overline{\operatorname{Diss}}(\Lambda(\cdot);[s,t]) := \sup \left\{ \sum_{k=1}^N \textup{Diss}(\Lambda({t_{k-1}}),\Lambda({t_{k}})): (t_k)_{k=0}^N \hbox{ partitions $[s,t]$}\right\}.$$ Then let $u(t)$ be an energy solution on $[0,T]$, i.e. satisfying the forcing, stability, and energy dissipation inequality conditions of [([E](#d.energy_solution))]{.upright}. One can show immediately, by applying ([\[e.diss-inequality\]](#e.diss-inequality){reference-type="ref" reference="e.diss-inequality"}) on each subinterval of an arbitrary partition of $[t_0,t_1]$, that $$\label{e.dissbar-inequality} \mathcal{J}(u({t_0}))-\mathcal{J}(u({t_1})) + \int_{t_0}^{t_1}2 \dot{F}(t)P(t) \ dt \geq \overline{\operatorname{Diss}}(u(\cdot);[t_0,t_1]).$$ With more work (see Lemma [Lemma 52](#l.energy-diss-eq){reference-type="ref" reference="l.energy-diss-eq"} later) combining with the stability property one can also show the identity $$\label{e.energy-diss-eq} \mathcal{J}(u({t_0}))-\mathcal{J}(u({t_1})) + \int_{t_0}^{t_1}2 \dot{F}(t)P(t) \ dt = \overline{\operatorname{Diss}}(u(\cdot);[t_0,t_1]).$$ In some works, for example [@alberti2011], the energy dissipation balance ([\[e.energy-diss-eq\]](#e.energy-diss-eq){reference-type="ref" reference="e.energy-diss-eq"}) is used in place of the energy dissipation inequality ([\[e.diss-inequality\]](#e.diss-inequality){reference-type="ref" reference="e.diss-inequality"}) as part of the definition of energy solution. This difference of definition is just a matter of preference at least in this problem. **Remark 51**. It is quite natural, and important for applications, to consider general $F(t,x)$ and allow for $F$ to be only $\textup{BV}$ regular in time. However, this generality does add serious complications which would significantly lengthen the presentation, and it is not so relevant to the goals of the present work. *Proof.* By the energy dissipation inequality ([\[e.dissbar-inequality\]](#e.dissbar-inequality){reference-type="ref" reference="e.dissbar-inequality"}) $$\label{above_0} \mathcal{J}(u(0))-\mathcal{J}(u(t)) + \int_{0}^{t}2 \dot{F}(s)P(u(s)) \ ds \geq \overline{\textup{Diss}}(\Omega(u(\cdot));[0,t]) \geq \mu_+ \wedge \mu_- [{\bf 1}_{\Omega(\cdot)}]_{\textup{BV}([0,t];L^1)}$$ In the remainder of the proof we will denote the Dirichlet energy $$\mathcal{D}(t) := \int_{\Omega(u(t))} |\nabla u(t,x)|^2 \ dx = F(t)P(u(t)).$$ Note that $\mathcal{D}(t) \leq \mathcal{J}(u(t))$ so in particular $$\begin{aligned} \mathcal{D}(t) &\leq \mathcal{J}(u(0))+\int_{0}^{t}2 \dot{F}(s)P(u(s)) \ ds \\ &= \mathcal{J}(u(0))+\int_{0}^{t}2 \frac{d}{ds}(\log F)(s) \mathcal{D}(s) \ ds\end{aligned}$$ for all $0 \leq t \leq T$ so by Grönwall $$\mathcal{D}(t) \leq \mathcal{J}(u(0)) \frac{F(t)^2}{F(0)^2}.$$ Thus $P(u(t)) \leq F(0)^{-2} F(t)$, and from [\[above_0\]](#above_0){reference-type="eqref" reference="above_0"} we have $$\begin{aligned} \mathcal{J}(u(t)) + \mu_+ \wedge \mu_- [{\bf 1}_{\Omega(\cdot)}]_{\textup{BV}([0,t];L^1)} &\leq \mathcal{J}(u(0))\left(1+\frac{1}{F(0)^2}\int_0^t2\dot{F}(s)F(s) \ ds\right) \\ &= \mathcal{J}(u(0))\left(1+\frac{F(t)^2-F(0)^2}{F(0)^2}\right)\\ &=\mathcal{J}(u(0))\frac{F(t)^2}{F(0)^2}.\end{aligned}$$ To summarize, $$\label{e.energy-ineq-BV} \mathcal{J}(u(t)) + \mu_+ \wedge \mu_- [{\bf 1}_{\Omega(\cdot)}]_{\textup{BV}([0,t];L^1)}\leq \mathcal{J}(u(0))\frac{F(t)^2}{F(0)^2}.$$ In particular $$\label{bv_1} [{\bf 1}_{\Omega(\cdot)}]_{\textup{BV}([0,t];L^1)} \leq \frac{1}{\mu_+\wedge \mu_-} \mathcal{J}(u(0))\frac{F(t)^2}{F(0)^2}.$$ This gives the BV in time estimate of ${\bf 1}_{\Omega(t)}$. Now we turn to the BV estimate of the Dirichlet energy. Focusing on the other term on the left hand side in ([\[e.energy-ineq-BV\]](#e.energy-ineq-BV){reference-type="ref" reference="e.energy-ineq-BV"}), and using the fact that $\mathcal{J}(u(t)) = \mathcal{D}(t) + |\Omega(u(t))|$, $$\mathcal{D}(t) \leq \frac{F(t)^2}{F(0)^2}\mathcal{D}(0)+\frac{F(t)^2}{F(0)^2}|\Omega(u(0))| - |\Omega(u(t))|$$ or rearranging this $$\begin{aligned} \mathcal{D}(t) - \mathcal{D}(0) &\leq \frac{\mathcal{D}(0)}{F(0)^{2}}(F(t)^2 - F(0)^2)+\frac{1}{F(0)^{2}}(F(t)^2-F(0)^2)|\Omega(u(0))|\notag\\ &\quad +|\Omega(u(t)) \Delta \Omega(u(0))|\notag\\ & = \frac{\mathcal{J}(u(0))}{F(0)^{2}}(F(t)^2 - F(0)^2)+|\Omega(u(t)) \Delta \Omega(u(0))|.\label{e.energy-ineq-D-upper}\end{aligned}$$ Global stability implies a bound in the other direction $$\begin{aligned} \mathcal{J}(u(0)) &\leq J\left(\frac{F(0)}{F(t)}u(t)\right) + \mu_+ \vee \mu_- |\Omega(u(0)) \Delta \Omega(u(t))|, \end{aligned}$$ which expands out as $$\mathcal{D}(0)+|\Omega(u(0))| \leq \frac{F(0)^2}{F(t)^2}\mathcal{D}(t)+|\Omega(u(t))| + \mu_+ \vee \mu_- |\Omega(u(0)) \Delta \Omega(u(t))|,$$ which we can use to find $$\label{e.energy-ineq-D-lower} \mathcal{D}(0) - \mathcal{D}(t) \leq \frac{\mathcal{D}(t)}{F(t)^{2}}(F(0)^2 - F(t)^2)+(1+ \mu_+ \vee \mu_-) |\Omega(u(0)) \Delta \Omega(u(t))|.$$ Since $0$ and $t$ replaced by arbitrary $t_0 < t_1 \in [0,T]$, for any partition of $[0,T]$ we can apply ([\[e.energy-ineq-D-lower\]](#e.energy-ineq-D-lower){reference-type="ref" reference="e.energy-ineq-D-lower"}) and ([\[e.energy-ineq-D-upper\]](#e.energy-ineq-D-upper){reference-type="ref" reference="e.energy-ineq-D-upper"}) together to find: $$\begin{aligned} \sum_j |\mathcal{D}(t_{j+1}) - \mathcal{D}(t_j)| &\leq \sum_j\bigg[\max\left\{\frac{\mathcal{J}(u(t_j))}{F(t_j)^2},\frac{\mathcal{J}(u(t_{j+1}))}{F(t_{j+1})^2}\right\}\left|F(t_{j+1})^2 - F(t_{j})^2\right|\\ &\quad \quad + (1+\mu_+ \vee \mu_-) |\Omega(u(t_{j+1})) \Delta \Omega(u(t_j))|.\end{aligned}$$ Applying ([\[e.energy-ineq-BV\]](#e.energy-ineq-BV){reference-type="ref" reference="e.energy-ineq-BV"}) again for $F(t)^{-2}\mathcal{J}(u(t)) \leq F(0)^{-2}\mathcal{J}(u(0))$, we obtain that $$\sum_j |\mathcal{D}(t_{j+1}) - \mathcal{D}(t_j)| \leq \frac{\mathcal{J}(u(0))}{F(0)^2}[F^2]_{\textup{BV}([0,T])}+(1+\mu_+ \vee \mu_-)[{\bf 1}_{\Omega(\cdot)}]_{\textup{BV}([0,T];L^1)}.$$ Thus we conclude that $\mathcal{D}(t) \in BV([0,T])$ with the estimate, applying [\[bv_1\]](#bv_1){reference-type="eqref" reference="bv_1"}, $$[\mathcal{D}]_{BV([0,T])} \leq \left[[F^2]_{\textup{BV}([0,T])}\\ +\frac{1+\mu_+ \vee \mu_-}{\mu_+\wedge \mu_-} F(t)^2\right]\frac{\mathcal{J}(u(0))}{F(0)^2}.$$ Finally since $\mathcal{J}(u(t)) = \mathcal{D}(t) + |\Omega(u(t))|$ and both terms on the right are in $BV([0,T])$ then so is $\mathcal{J}(u(t))$. Similarly $P(u(t)) = F(t)^{-1}\mathcal{D}(t)$ is in $BV([0,T])$ since $F(t)$ is bounded from below. ◻ From these regularity properties we can establish that the stability and energy dissipation inequality properties of energy solutions force the *energy dissipation equality*. **Lemma 52** (Energy dissipation equality). *If $u$ is an energy solution on $[0,T]$ then for all $s\leq t$ in $[0,T]$ $$\mathcal{J}(u(s))-\mathcal{J}(u(t)) + \int_{s}^{t} 2 \dot{F}(\tau)P(u(\tau)) \ d\tau = \overline{\textup{Diss}}(\Omega(u(\cdot));[s,t]).$$* *Proof.* We need to establish the upper bound to complement ([\[e.diss-inequality\]](#e.diss-inequality){reference-type="ref" reference="e.diss-inequality"}). This will follow from the time regularity and the global stability property. We may take $s = 0$ and $t = T$. Let $\varepsilon>0$. By the continuity of $\log F(t)$, we can choose a (finite) partition $0 = t_0< t_1 < \cdots < t_N = T$ of $[0,T]$ such that $$\label{e.partition-choice-cond} \sup_j(t_{j+1}-t_j) \leq \varepsilon\ \hbox{ and } \ \sup_{t \in [t_j,t_{j+1}]} \frac{F(t)}{F(t_{j+1})} \leq 1 + \varepsilon.$$ We write the energy dissipation using a telescoping sum $$\label{e.energy-telescope} \mathcal{J}(u(0))-\mathcal{J}(u(T)) = \sum_{j=0}^{N-1} [\mathcal{J}(u(t_{j}))-\mathcal{J}(u(t_{j+1}))].$$ Applying the global stability condition for $u(t_j)$, $$\begin{aligned} \mathcal{J}(u(t_{j}))-\mathcal{J}(u(t_{j+1})) &\leq \mathcal{J}\left(\tfrac{F(t_{j})}{F(t_{j+1})}u(t_{j+1})\right)+\textup{Diss}(\Omega(t_j),\Omega(t_{j+1}))-\mathcal{J}(u(t_{j+1})) \\ &= \textup{Diss}(\Omega(t_j),\Omega(t_{j+1}))+ \underbrace{(\tfrac{F(t_j)^2}{F(t_{j+1})^2} - 1)\mathcal{D}(u(t_{j+1}))}_{\text{:=A}} \end{aligned}$$ Now from the fact that $\mathcal{D}(u(t)) = F(t) P(u(t))$, $$A= \frac{F(t_j)^2 - F(t_{j+1})^2}{F(t_{j+1})}P(u(t_{j+1}))=\int_{t_j}^{t_{j+1}} 2 \frac{F(t)}{F(t_{j+1})} \dot{F}(t) P(u(t_{j+1})) \ dt.$$ From ([\[e.partition-choice-cond\]](#e.partition-choice-cond){reference-type="ref" reference="e.partition-choice-cond"}) we have $$\begin{aligned} A &=\int_{t_j}^{t_{j+1}} 2 \frac{F(t)}{F(t_{j+1})} \dot{F}(t) P(u(t_{j+1})) \ dt \notag\\ &\quad \quad \leq \int_{t_j}^{t_{j+1}} 2 \frac{F(t)}{F(t_{j+1})} \dot{F}(t) P(u(t)) \ dt + 2(1+\varepsilon)\|\dot{F}\|_\infty [P(u(\cdot))]_{\textup{BV}((t_j,t_{j+1}])}(t_{j+1}-t_j)\notag\\ &\quad \quad \leq (1+\varepsilon)\int_{t_j}^{t_{j+1}} 2 \dot{F}(t) P(u(t)) \ dt + 2(1+\varepsilon)\varepsilon\|\dot{F}\|_\infty [P(u(\cdot))]_{\textup{BV}((t_j,t_{j+1}])} \label{e.summing-diss-error}\end{aligned}$$ Combining the previous estimates into ([\[e.energy-telescope\]](#e.energy-telescope){reference-type="ref" reference="e.energy-telescope"}) we find $$\begin{aligned} \mathcal{J}(u(0))-\mathcal{J}(u(T)) &\leq(1+\varepsilon)\int_{0}^{T} 2 \dot{F}(t) P(u(t)) \ dt +\sum_{j=0}^{N-1}\textup{Diss}(\Omega(t_{j}),\Omega(t_{j+1})) \\ &\quad \quad + \sum_{j=0}^{N-1}2(1+\varepsilon)\varepsilon\|\dot{F}\|_\infty [P(u(\cdot))]_{\textup{BV}((t_j,t_{j+1}])} \\ &\leq \overline{\textup{Diss}}(\Omega(u(\cdot));(0,T))+(1+\varepsilon)\int_{0}^{T} 2 \dot{F}(t) P(u(t)) \ dt \\ &\quad \quad + 2(1+\varepsilon)\varepsilon\|\dot{F}\|_\infty [P(u(\cdot))]_{\textup{BV}((0,T])}.\end{aligned}$$ Since $\varepsilon>0$ was arbitrary we get the result. ◻ ## Left and right limits and monotonicity of jumps {#s.energy-soln-lr-limits} The $\textup{BV}$ time regularity of ${\bf 1}_{\Omega(u(t))}$ allows us to establish further temporal left and right continuity properties of the evolution at every time. Recall the dissipation augmented energy functional ([\[e.E-def-intro\]](#e.E-def-intro){reference-type="ref" reference="e.E-def-intro"}), $$\mathcal{E}(V,u) := \mathcal{J}(u) + \textup{Diss}(V,\Omega(u)).$$ First we discuss a monotonicity property for minimizers of $\mathcal{E}$. ### No crossing property of $\mathcal{E}$ minimizers The next lemma shows that minimizers of the dissipation distance augmented energy $\mathcal{E}$, defined in ([\[e.E-def-intro\]](#e.E-def-intro){reference-type="ref" reference="e.E-def-intro"}), satisfy a certain ordering property. Suppose that $u_1$ and $u_2$ are arbitrary $H^1(U)$ functions. Note that $$\mathcal{J}(u_1 \vee u_2) + \mathcal{J}(u_1\wedge u_2) = \mathcal{J}(u_1) + \mathcal{J}(u_2)$$ and $$\textup{Diss}(V,\Omega(u_1 \vee u_2)) + \textup{Diss}(V,\Omega(u_1 \wedge u_2)) = \textup{Diss}(V,\Omega(u_1))+ \textup{Diss}(V,\Omega(u_2)).$$ Combining the previous two equations yields $$\label{e.E-additivity-property} \mathcal{E}(V,u_1 \vee u_2) + \mathcal{E}(V,u_1 \wedge u_2) = \mathcal{E}(V,u_1 ) + \mathcal{E}(V, u_2).$$ It is standard to derive from this additivity property that the pointwise minimum and maximum of two minimizers with the same Dirichlet condition are energy minimizers as well. Adding in the strong maximum principle for harmonic functions we can show an ordering property of the collection of energy minimizers, again this is a sufficiently standard idea that we do not have a particular original reference. **Lemma 53** (No crossing). *Let $V$ be an open set containing $\mathbb{R}^d \setminus U$. If $u_1$ and $u_2$ both minimize $\mathcal{E}(V,\cdot)$ with respect to $H^1_0(U)$ perturbations and $u_1 - u_2 \in H^1_0(U)$. Then $u_1 \wedge u_2$ and $u_1 \vee u_2$ are also minimizers and $u_1$ and $u_2$ are ordered in each connected component of $\Omega(u_1) \cup \Omega(u_2) = \Omega(u_1 \vee u_2)$.* **Remark 54**. To be clear the ordering between $u_1$ and $u_2$ may differ in the different connected components of $\Omega(u_1) \cup \Omega(u_2)$. For example, given two ordered minimizers the same scenario can be repeated far away with the reverse ordering to create unordered minimizers. *Proof of Lemma [Lemma 53](#l.no-crossing){reference-type="ref" reference="l.no-crossing"}.* Recalling ([\[e.E-additivity-property\]](#e.E-additivity-property){reference-type="ref" reference="e.E-additivity-property"}) we have $$\mathcal{E}(V,u_1 \vee u_2) + \mathcal{E}(V,u_1 \wedge u_2) = \mathcal{E}(V,u_1 ) + \mathcal{E}(V, u_2).$$ On the other hand, since $u_1 = u_2$ on $\partial U$ in the trace sense, also $u_1 \wedge u_2, u_1 \vee u_2 = u_1 = u_2$ on $\partial U$ in the trace sense. So, from the minimizer assumption on $u_1$ and $u_2$, we conclude that all four terms in the above equality must be equal. Thus $u_1$, $u_2$, $u_1 \wedge u_2$, and $u_1 \vee u_2$ are all minimizers of $\mathcal{E}(V,\cdot)$ over the admissible class. In particular each one is harmonic in its positivity set. By unique continuation for harmonic functions this means that $u_1$ and $u_2$ are ordered in each connected component of $U \cap \{u_1 \vee u_2 > 0\}$. Note that by harmonicity again if $u_1 \neq u_2$ in a given connected component of $U \cap \{ u_1 \vee u_2 >0\}$ then the ordering is strict in it. ◻ ### Abstract left-right limits Next we discuss some general facts about bounded variation maps on intervals of the line $\mathbb{R}$. Suppose $(Y, d_Y)$ is a complete metric space. **Definition 55**. Say $f : [a,b] \to Y$ is a bounded variation map if $$[f]_{\textup{BV}([a,b];Y)} := \sup_{a =t_0 \leq t_1 \leq\cdots \leq t_{N+1} = b} \sum_{j=0}^N d(f(t_{j+1}),f(t_j)) < +\infty.$$ On open or half open intervals define $$[f]_{\textup{BV}([a,b);Y)}:= \lim_{b' \nearrow b}[f]_{\textup{BV}([a,b'];Y)},$$ or similar for $(a,b]$ and $(a,b)$. The limit exists by monotonicity. **Definition 56**. Let $f: [a,b] \to Y$. For each $t \in [a,b]$, if the limits exist define the left limit $f_\ell(t)$ at $t \in (a,b]$ $$f_\ell(t) := \lim_{s \to t -} f(s)$$ and the right limit $f_r(t)$ at $t \in [a,b)$ $$f_r(t):= \lim_{s \to t +} f(s).$$ **Lemma 57**. *Let $(Y,d)$ a complete metric space and $f$ a bounded variation map $[a,b] \to Y$. Then $f$ has left and right limits at each $t \in (a,b]$ (resp. $[a,b)$) and $f$ has at most countably many jump discontinuities.* The proof is standard, but we give a sketch for convenience. *Proof.* For the directional limits note that for any sequence $s_j \nearrow t \in (a,b]$ $$\sum_j d_Y(f(s_j),f(s_{j+1})) \leq [f]_{\textup{BV}([a,b];Y)} < + \infty$$ and so $f(s_j)$ is Cauchy in $Y$. The limits must agree on different approaching sequences by a typical interlacement argument with pairs of sequences. A symmetric argument produces the right limits. Now, first arguing for any finite set $\mathcal{T}$ of times, and then taking the supremum over all finite sets $$\sum_{t \in [a,b]}d_Y(f_\ell(t),f_r(t)) \leq [f]_{\textup{BV}([a,b];Y)}.$$ In particular $f_\ell(t) = f_r(t)$ except for at most countably many times. ◻ ### Characterization of the upper and lower semicontinuous envelopes for energy solutions. Lemma [Lemma 57](#l.BV-abstract-lr){reference-type="ref" reference="l.BV-abstract-lr"} and Lemma [Lemma 49](#l.energy-soln-time-reg){reference-type="ref" reference="l.energy-soln-time-reg"} together yield the existence of left and right limits of $\Omega(u(t))$, $\mathcal{J}(u(t))$, and $P(u(t))$ at all times. Our next result says that $u(t)$ also has left and right limits in the uniform metric, even though we do not necessarily establish that $t \to u(t)$ is a $\textup{BV}$ in time map into $C_c(\mathbb{R}^d)$. The left and right limits of all the previous quantities are consistent, i.e. $(P\circ u)_\ell(t) = P(u_\ell(t))$ etc. Furthermore the left and right limits of $u(t)$ satisfy certain global minimality properties for the energy plus dissipation distance. The special structure of the problem comes in when we prove certain monotonicity properties of all the jumps, namely properties [\[part.lr-limits-jumps\]](#part.lr-limits-jumps){reference-type="ref" reference="part.lr-limits-jumps"}-[\[part.lr-limits-redefine\]](#part.lr-limits-redefine){reference-type="ref" reference="part.lr-limits-redefine"} below. This allows us to make a simple classification of the upper and lower semicontinuous envelopes of energy solutions, showing that they are energy solutions themselves. This simple jump structure will be very useful in Section [7](#s.energy-soln-motion-law){reference-type="ref" reference="s.energy-soln-motion-law"}, where we prove a certain weak version of the dynamic slope condition, and make the following proposition a central result of this section. Semicontinuous envelopes are typically important in studying the geometric properties of interface evolution problems. They are used, for instance, in the notion of discontinuous viscosity solutions for free boundary problems, as seen in Section [4](#s.comparison){reference-type="ref" reference="s.comparison"}. The fact that these envelopes are themselves energy solutions is extremely useful to us in Section [7](#s.energy-soln-motion-law){reference-type="ref" reference="s.energy-soln-motion-law"}. In service of analyzing the properties of energy solutions at jump times and simplifying repetitive notations let us introduce the set $$\label{e.replacement-U-values} \mathcal{U}(t) = \{u_\ell(t),u(t),u_r(t),u_{\ell}(t) \wedge u_r(t), u_\ell(t) \vee u_r(t)\}$$ which is a kind of multi-valued version of $u(t)$ indexing the values taken by $u(t)$ "near\" time $t$ as well as the upper and lower envelopes. The set $\mathcal{U}(t)$ comes with a natural partial order induced by the time variable. Since $u_\ell(t)$ is the limit from the left of $u(t)$ it is "before\" all the other elements of $\mathcal{U}(t)$, and since $u_r(t)$ is the limit from the right it is "after\" all the other elements of $u(t)$. Specifically we define the partial order $$\label{e.time-ordering-def} u_\ell(t) \ \trianglelefteq \ u(t), \ u_{\ell}(t) \wedge u_r(t), \ u_\ell(t) \vee u_r(t) \ \trianglelefteq \ u_r(t).$$ To be clear this partial ordering is purely related to the "time\" of the elements of $\mathcal{U}(t)$ and is not related to the spatial ordering between the different elements of $\mathcal{U}(t)$. The core step in the following proposition is part [\[part.lr-limits-minimizer\]](#part.lr-limits-minimizer){reference-type="ref" reference="part.lr-limits-minimizer"}. Essentially we show that every element of $\mathcal{U}(t)$ is a valid intermediate state for the evolution at time $t$. More specifically, the energy dissipation relation is still satisfied as long as the system jumps from $u_\ell(t)$ to any element of $\mathcal{U}(t)$ and then to $u_r(t)$. **Proposition 58**. *Suppose $u$ is an energy solution on $[0,T]$ then:* 1. *[\[part.lr-limits\]]{#part.lr-limits label="part.lr-limits"} (Left and right limits exist and are consistent) At every $t \in [0,T]$, $u(t)$, $\mathcal{J}(u(t))$, $P(u(t)$, and $\Omega(u(t))$ have left and right limits and $$\label{e.consistency-lr-limits} \begin{array}{c} \lim_{s \to_{\ell / r} t} \Omega(u(t)) = \Omega(u_{\ell / r}(t)), \ \lim_{s \to_{\ell / r} t} \mathcal{J}(u(t)) = \mathcal{J}(u_{\ell / r}(t)), \\ \hbox{ and } \ \lim_{s \to_{\ell / r} t} P(u(t)) = P(u_{\ell / r}(t)) \end{array}$$* 2. *[\[part.lr-limits-minimizer\]]{#part.lr-limits-minimizer label="part.lr-limits-minimizer"} (Jumps minimize energy plus dissipation) At every $t \in [0,T]$ for any $v,w \in \mathcal{U}(t)$ (from ([\[e.replacement-U-values\]](#e.replacement-U-values){reference-type="ref" reference="e.replacement-U-values"})), with $v \trianglelefteq w$ (from ([\[e.time-ordering-def\]](#e.time-ordering-def){reference-type="ref" reference="e.time-ordering-def"})) $$w \ \hbox{ minimizes } \ \mathcal{E}(v,\cdot) \ \hbox{ over } \ F(t) + H^1_0(U).$$ Note that in particular this implies all $v \in \mathcal{U}(t)$ are globally stable since $v \trianglelefteq v$.* 3. *[\[part.lr-limits-jumps\]]{#part.lr-limits-jumps label="part.lr-limits-jumps"} (Jumps are monotone per component) At every $t \in [0,T]$ $$u_\ell(t) \wedge u_r(t) \leq u(t) \leq u_\ell(t) \vee u_r(t).$$ In particular at each $t \in [0,T]$ the upper semicontinuous and lower semi-continuous envelopes are $u^*(t) = u_r(t) \vee u_\ell(t)$ and $u_*(t) = u_r(t) \wedge u_\ell(t)$.* 4. *[\[part.lr-limits-redefine\]]{#part.lr-limits-redefine label="part.lr-limits-redefine"} (Re-definition at jumps) Suppose that $v:[0,T] \times \mathbb{R}^d \to \mathbb{R}$ satisfies the property: $$\hbox{for each $t \in [0,T]$ } \ v(t) \in \mathcal{U}(t) \ \hbox{ (from (\ref{e.replacement-U-values}))}.$$ Then $v(t)$ is also an energy solution on $[0,T]$. In particular the upper and lower semicontinuous envelopes $u^*(t)$ and $u_*(t)$ are also energy solutions.* Besides the independent interest in understanding the time regularity properties of energetic solutions, this result plays a key role in establishing the validity of the free boundary motion law in Theorem [Theorem 68](#t.ae-viscosity-prop){reference-type="ref" reference="t.ae-viscosity-prop"}. **Remark 59**. Note that part [\[part.lr-limits-jumps\]](#part.lr-limits-jumps){reference-type="ref" reference="part.lr-limits-jumps"} does allow the possibility that the free boundaries $\partial \Omega(u_\ell(t))$ and $\partial \Omega(u_r(t))$ are distinct and monotonically ordered but touch in some nontrivial region. This is possible to occur: for example consider a small ball and a large ball merging with a large pinning interval. This would allow a portion of the free boundary opposite the "contacting area\" to stay fixed as depicted in the simulation Figure [\[f.small-large-merge\]](#f.small-large-merge){reference-type="ref" reference="f.small-large-merge"}. In other words, although strong maximum principle holds for the solutions, it does not hold for the free boundaries. **Remark 60**. Note that from Lemma [Lemma 53](#l.no-crossing){reference-type="ref" reference="l.no-crossing"} and [\[part.lr-limits-jumps\]](#part.lr-limits-jumps){reference-type="ref" reference="part.lr-limits-jumps"} in each connected component $V$ of $U \cap \{u^*(t) > 0\}$ we have $\{u^*(t,\cdot)|_V,u_*(t,\cdot)|_V\} = \{u_r(t)|_V,u_\ell(t)|_V\}$. **Remark 61**. Note that $v(t)$ as in [\[part.lr-limits-redefine\]](#part.lr-limits-redefine){reference-type="ref" reference="part.lr-limits-redefine"} is always measurable since $v(t) =u(t)$ except at countably many times. *Proof of Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"}.* **Part [\[part.lr-limits\]](#part.lr-limits){reference-type="ref" reference="part.lr-limits"}.** By Lemma [Lemma 49](#l.energy-soln-time-reg){reference-type="ref" reference="l.energy-soln-time-reg"} the map $$t\mapsto (\mathbf{1}_{\Omega(u(t))},\mathcal{J}(u(t)),P(u(t))) \ \hbox{ is in } \ BV([0,T];L^1(\mathbb{R}^d) \times \mathbb{R}\times \mathbb{R})$$ and so by Lemma [Lemma 57](#l.BV-abstract-lr){reference-type="ref" reference="l.BV-abstract-lr"} the map has left and right limits in $L^1(\mathbb{R}^d) \times \mathbb{R}\times \mathbb{R}$ at each $t\in [0,T]$. Call $\Omega_\ell(t_0)$ to be the set obtained by the $L^1$ left limit of $\Omega(u(t))$ at $t_0$. Let $t_n \nearrow t_0$. By uniform Lipschitz continuity of $u(t_n)$ there is a subsequence converging uniformly to some $u_\infty$ which satisfies $u_\ell = F(t_0)$ on $\partial U$. By uniform non-degeneracy of the $u(t)$ we must have $\Omega(u_\infty) = \Omega_\ell(t_0)$. This specifies that $u_\infty$ is the unique solution of $$\label{e.conclude-u-conv-from-omega-conv} \Delta u_\infty = 0 \ \hbox{ in } \ \Omega_\ell(t_0) \cap U \ \hbox{ with } \ u_\infty = 0 \ \hbox{ on } \ \partial \Omega_\ell(t_0) \ \hbox{ and } \ u_\infty = F(t_0) \ \hbox{ on } \ \partial U.$$ Since the original sequence $t_n \nearrow t_0$ was arbitrary we find that $u(t) \to u_\infty$ uniformly as $t_n \nearrow t_0$. The argument is the same for right limits. Thus $u(t)$ has left and right limits $u_{\ell/r}(t)$ in the uniform metric at each time $t \in [0,T]$. Next we need to show the consistency properties ([\[e.consistency-lr-limits\]](#e.consistency-lr-limits){reference-type="ref" reference="e.consistency-lr-limits"}). We already showed that $\lim_{s \to_{\ell / r} t} \Omega(u(t)) = \Omega(u_{\ell / r}(t))$ using uniform non-degeneracy in the previous paragraph. Recalling that $P(u(t)) = F(t)^{-1}\mathcal{D}(u(t))$, to conclude we just need to check the consistency of limits for the Dirichlet energy $\mathcal{D}(u(t))$ $$\label{e.dirichlet-energy-lr-limits} \lim_{s \to_{\ell/r} t} \int_U |\nabla u(t)|^2 \ dx = \int_U |\nabla u_{\ell/r}(t)|^2 \ dx.$$ We claim that $\nabla u(s)$ converges locally uniformly on $\Omega(u_{\ell/r}(t))$ to $\nabla u_{\ell/r}(t)$ as $s \nearrow t$ (resp. $s \searrow t$). Along with the uniform boundedness of $\nabla u(t)$ the dominated convergence theorem gives ([\[e.dirichlet-energy-lr-limits\]](#e.dirichlet-energy-lr-limits){reference-type="ref" reference="e.dirichlet-energy-lr-limits"}). As usual we handle the left limit case and the right limit case is similar. Let $K$ be a compact subset of $\Omega(u_\ell(t))$. By the uniform convergence $u(s) \to u_\ell(t)$ as $s \nearrow t$ and by uniform non-degeneracy $$\inf_K u(s) \geq c d(K,\Omega(u_\ell(t))^{\complement})$$ and so by uniform Lipschitz continuity $$d(K,\Omega(u(s))^{\complement}) \geq c d(K,\Omega(u_\ell(t))^{\complement})$$ for $s < t$ sufficiently close to $t$. Thus the $\nabla u(s)$ are uniformly bounded and uniformly continuous on $K$ for $s<t$ sufficiently close to $t$. Typical arbitrary subsequence arguments show that $\nabla u(s)$ converges uniformly on $K$ to $\nabla u_\ell(t)$ as $s \nearrow t$. **Part [\[part.lr-limits-minimizer\]](#part.lr-limits-minimizer){reference-type="ref" reference="part.lr-limits-minimizer"}.** We introduce the extraneous notation $u_0(t) = u(t)$ in order to prove minimizer properties for $u_{\ell} /u_0 / u_r$ at once. First we check that each of $u_{\ell / 0 / r}(t)$ minimize their own respective $\mathcal{E}(\Omega(u_{\ell / 0 / r}(t)),\cdot)$ over $F(t) + H^1_0(U)$. For $u_0(t)$ this is just the global stability property of an energy solution. For $u_\ell(t)$, by the global stability property for any $t_- < t$ we know $$\mathcal{J}(u(t_-)) \leq J\left(\frac{F(t_-)}{F(t)}v\right) + \textup{Diss}(u(t_-),v) \ \hbox{ for all } \ v \in F(t) + H^1_0(U).$$ Taking the limit as $t_-\nearrow t$, using time continuity of $F$, and applying part [\[part.lr-limits\]](#part.lr-limits){reference-type="ref" reference="part.lr-limits"} $$\mathcal{J}(u_\ell(t)) \leq \mathcal{J}(v) + \textup{Diss}(u(t_-),v) \ \hbox{ for all } \ v \in F(t) + H^1_0(U).$$ The proof that $u_r(t)$ minimizes $\mathcal{E}(u_r(t),\cdot)$ is similar, applying global stability of $u(t_+)$ and taking a limit $t_+ \searrow t$. Next we check that $u_{0 / r}$ minimize $\mathcal{E}(u_{\ell}(t),\cdot)$ over $F(t) + H^1_0(U)$. Apply the dissipation relation with times $t_- \leq t \leq t_+$ $$\mathcal{J}(u(t_-)) + \int_{t_-}^{t_+}2\dot{F}(s)P(s) \ ds \geq \textup{Diss}(u(t_-),u(t_+))+\mathcal{J}(u(t_+)).$$ Taking the limit as $t_- \nearrow t$ and $t_+ \searrow t$ and using the left/right continuity of $\mathcal{J}(u(\cdot))$ and $\Omega(\cdot)$ established in part [\[part.lr-limits\]](#part.lr-limits){reference-type="ref" reference="part.lr-limits"} $$\mathcal{J}(u_\ell(t)) \geq \mathcal{J}(u_r(t)) + \textup{Diss}(u_\ell(t),u_r(t))$$ or in the case $t_+ = t$ and $t_- \nearrow t$ $$\mathcal{J}(u_\ell(t)) \geq \mathcal{J}(u(t))+ \textup{Diss}(u_\ell(t),u(t))$$ Since we already established that $u_\ell(t)$ is always a minimizer of $\mathcal{E}(u_\ell(t),\cdot)$ also $u_{0 /r}(t)$ must be minimizers. Finally in the case $t_- = t$ and $t_+ \searrow t$ we obtain similarly $$\mathcal{J}(u(t)) \geq \mathcal{J}(u_r(t)) + \textup{Diss}(u(t),u_r(t))$$ which gives that $u_r(t)$ minimizes $\mathcal{E}(u(t),\cdot)$. Lastly we consider $u_\ell(t)\wedge u_r(t)$ and $u_\ell(t) \vee u_r(t)$. Since $t$ is fixed for the remainder of this part of the proof we will write $u_\ell = u_\ell(t)$ and $u_r = u_r(t)$ to simplify expressions. First of all, since $u_\ell$ and $u_r$ minimize $\mathcal{E}(u_\ell,\cdot)$ over $F(t) + H^1_0(U)$, so also, by ([\[e.E-additivity-property\]](#e.E-additivity-property){reference-type="ref" reference="e.E-additivity-property"}), $u_\ell \wedge u_r$ and $u_\ell \vee u_r$ both minimize $\mathcal{E}(u_\ell,\cdot)$ over $F(t) + H^1_0(U)$. Then for any $v \in F(t) + H^1_0(U)$ $$\mathcal{E}(u_\ell, u_\ell \wedge u_r) \leq \mathcal{E}(u_\ell,v)$$ or $$\begin{aligned} \mathcal{J}(u_\ell \wedge u_r) &\leq \mathcal{J}(v) + \operatorname{Diss}(u_\ell,v) - \operatorname{Diss}(u_\ell,u_\ell \wedge u_r) \\ &\leq \mathcal{J}(v) +\operatorname{Diss}(u_\ell \wedge u_r,v)\\ &= \mathcal{E}(u_\ell\wedge u_r,v) \end{aligned}$$ by the dissipation distance triangle inequality Lemma [Lemma 77](#l.diss-triangle-eq){reference-type="ref" reference="l.diss-triangle-eq"} for the last inequality. Thus $u_\ell\wedge u_r$ is globally stable. A similar argument applies to $u_\ell \vee u_r$. Finally we need to argue that $$u_r \ \hbox{ minimizes $\mathcal{E}(u_\ell\wedge u_r,\cdot)$ and $\mathcal{E}(u_\ell\vee u_r,\cdot)$ over $F(t) + H^1_0(U)$.}$$ We just argue for $\mathcal{E}(u_\ell\wedge u_r,\cdot)$, the other case is similar, $$\begin{aligned} \mathcal{E}(u_\ell\wedge u_r,v) &= \mathcal{E}(u_\ell,v)+\textup{Diss}(u_\ell\wedge u_r,v)-\textup{Diss}(u_\ell,v) \\ &\geq \mathcal{E}(u_\ell,u_r)+\textup{Diss}(u_\ell\wedge u_r,v)-\textup{Diss}(u_\ell,v)\\ &=\mathcal{J}(u_r) +\operatorname{Diss}(u_\ell,u_r)+ \textup{Diss}(u_\ell\wedge u_r,v)-\textup{Diss}(u_\ell,v)\\ & = \mathcal{J}(u_r)+\operatorname{Diss}(u_\ell\wedge u_r,v)+\operatorname{Diss}(v,u_r)\\ &\quad \quad +[\operatorname{Diss}(u_\ell,u_r)-\operatorname{Diss}(u_\ell,v)-\operatorname{Diss}(v,u_r)]\\ &= \mathcal{J}(u_r)+\operatorname{Diss}(u_\ell\wedge u_r,u_r)\\ &\quad \quad +(\mu_-+\mu_+)[|\Omega(v) \setminus \Omega( (u_\ell \wedge u_r) \vee u_r)| + |\Omega((u_\ell \wedge u_r) \wedge u_r) \setminus \Omega(v)|]\\ & \quad \quad -(\mu_-+\mu_+)[|\Omega(v) \setminus \Omega( u_\ell \vee u_r)| + |\Omega(u_\ell \wedge u_r) \setminus \Omega(v)|]\\ &=\mathcal{E}(u_\ell\wedge u_r,u_r). \end{aligned}$$ where we have used that $u_r$ minimizes $\mathcal{E}(u_\ell,\cdot)$, then the sharp triangle inequality Lemma [Lemma 77](#l.diss-triangle-eq){reference-type="ref" reference="l.diss-triangle-eq"}, and finally that $(u_\ell \wedge u_r) \wedge u_r = u_\ell \wedge u_r$ and $(u_\ell \wedge u_r) \vee u_r = u_\ell \vee u_r$. **Part [\[part.lr-limits-jumps\]](#part.lr-limits-jumps){reference-type="ref" reference="part.lr-limits-jumps"}.** Since $t$ is fixed in this part of the proof we write $u_{\ell / 0 / r}(t) = u_{\ell / 0 / r}$ dropping the $t$ dependence. By global stability and part [\[part.lr-limits-minimizer\]](#part.lr-limits-minimizer){reference-type="ref" reference="part.lr-limits-minimizer"} respectively we know that $u_\ell$ and $u_r$ both minimize $\mathcal{E}(u_\ell,\cdot)$. Furthermore Lemma [Lemma 53](#l.no-crossing){reference-type="ref" reference="l.no-crossing"} yields that $u_\ell$ and $u_r$ are ordered in each connected component of $\Omega(u_\ell \vee u_r)$. Next we show that $u_\ell \wedge u_r \leq u \leq u_\ell \vee u_r$. We use the multiple minimizations in part [\[part.lr-limits-minimizer\]](#part.lr-limits-minimizer){reference-type="ref" reference="part.lr-limits-minimizer"} to show an "equality in triangle inequality\" which implies that $u$ must be in between $u_\ell$ and $u_r$. Intuitively speaking an intermediate jump which crosses either $u_\ell$ or $u_r$ would just incur extra dissipation cost to return to $u_r$. We compute using part [\[part.lr-limits-minimizer\]](#part.lr-limits-minimizer){reference-type="ref" reference="part.lr-limits-minimizer"} $$\begin{aligned} \mathcal{J}(u_r) + \textup{Diss}(u_\ell,u_r)&= \mathcal{J}(u)+\textup{Diss}(u_\ell,u)\\ &=\mathcal{J}(u_r) +\textup{Diss}(u,u_r)+\textup{Diss}(u_\ell,u).\end{aligned}$$ More specifically the first equality is since $u_r$ and $u$ both minimize $\mathcal{E}(u_\ell,\cdot)$, the second equality is since $u$ and $u_r$ both minimize $\mathcal{E}(u,\cdot)$. Simplifying we find the equality $$\label{e.triangle-equality} \textup{Diss}(u_\ell,u)+\textup{Diss}(u,u_r) - \textup{Diss}(u_\ell,u_r) = 0.$$ By the sharp triangle inequality Lemma [Lemma 77](#l.diss-triangle-eq){reference-type="ref" reference="l.diss-triangle-eq"} $$\label{e.triangle-inequality-strictness} \textup{Diss}(u_\ell,u)+\textup{Diss}(u,u_r) - \textup{Diss}(u_\ell,u_r) = (\mu_-+\mu_+) \bigg[ |\Omega(u_\ell \wedge u_r) \setminus \Omega(u)| + |\Omega(u) \setminus \Omega(u_\ell \vee u_r)|\bigg].$$ Together with ([\[e.triangle-equality\]](#e.triangle-equality){reference-type="ref" reference="e.triangle-equality"}), above equality shows that $\Omega(u_\ell \wedge u_r) \subset \Omega(u) \subset \Omega(u_\ell \vee u_r)$. Since $u_\ell \wedge u_r$, $u$, and $u_\ell \vee u_r$ are harmonic in their respective positivity sets by Lemma [Lemma 53](#l.no-crossing){reference-type="ref" reference="l.no-crossing"}, then the previous set ordering and maximum principle implies $u_\ell \wedge u_r \leq u \leq u_{\ell} \vee u_r$. **Part [\[part.lr-limits-redefine\]](#part.lr-limits-redefine){reference-type="ref" reference="part.lr-limits-redefine"}.** Since $u_\ell(t) = u_r(t) = u(t) = F(t)$ on $\partial U$ the boundary condition is satisfied by $v(t)$. By [\[part.lr-limits-minimizer\]](#part.lr-limits-minimizer){reference-type="ref" reference="part.lr-limits-minimizer"} any $v(t) \in \mathcal U(t)$ is globally stable. Finally we need to check the dissipation relation ([\[e.diss-inequality\]](#e.diss-inequality){reference-type="ref" reference="e.diss-inequality"}) for $v$. We do this in two steps, the dissipation on the open interval $(t_0,t_1)$ plus the dissipation at the endpoints. Fix $t_0 < t_1$ and we claim that $$\label{e.open-interval-diss-v} \mathcal{J}(u_r(t_0))-\mathcal{J}(u_\ell(t_1)) + \int_{t_0}^{t_1}2\dot{F}(s)P(s) \ ds \geq \operatorname{Diss}(u_r(t_0),u_\ell(t_1))$$ To prove this apply the dissipation relation for $u(t)$ with sequences $t_{0,k}$ and $t_{1,k}$ with $t_{0,k} \searrow t_0$ and $t_{1,k} \nearrow t_1$ $$\mathcal{J}(u(t_{0,k}))-\mathcal{J}(u(t_{1,k})) + \int_{t_{0,k}}^{t_{1,k}}2\dot{F}(s)P(s) \ ds \geq \operatorname{Diss}(u(t_{0,k}), u(t_{1,k})).$$ Sending $k \to \infty$ and using the continuities from [\[part.lr-limits\]](#part.lr-limits){reference-type="ref" reference="part.lr-limits"} shows ([\[e.open-interval-diss-v\]](#e.open-interval-diss-v){reference-type="ref" reference="e.open-interval-diss-v"}). Now consider the endpoint dissipations, by [\[part.lr-limits-minimizer\]](#part.lr-limits-minimizer){reference-type="ref" reference="part.lr-limits-minimizer"}, $$\mathcal{J}(u_r(t_0))+\operatorname{Diss}(v(t_0),u_r(t_0)) = \mathcal{J}(v(t_0)) \ \hbox{ and } \ \mathcal{J}(v(t_1))+\operatorname{Diss}(u_\ell(t_1),v(t_1)) = \mathcal{J}(u_\ell(t_1))$$ since every $v(t_0) \trianglelefteq u_r(t_0)$ for every $v(t_0) \in \mathcal{U}(t_0)$ and $u_\ell(t_1) \trianglelefteq v(t_1)$ for every $v(t_1) \in \mathcal{U}(t_1)$ in the temporal partial ordering (defined in ([\[e.time-ordering-def\]](#e.time-ordering-def){reference-type="ref" reference="e.time-ordering-def"})). Adding these to ([\[e.open-interval-diss-v\]](#e.open-interval-diss-v){reference-type="ref" reference="e.open-interval-diss-v"}) gives $$\begin{aligned} &\mathcal{J}(v(t_0)) - \mathcal{J}(v(t_1)) + \int_{t_0}^{t_1} 2 \dot{F}(s) P(s) \ ds \\ & \quad \quad \geq \operatorname{Diss}(v(t_0),u_r(t_0))+\operatorname{Diss}(u_r(t_0),u_\ell(t_1))+ \operatorname{Diss}(u_\ell(t_1),v(t_1))\\ &\quad \quad \geq \operatorname{Diss}(v(t_0),v(t_1)) \end{aligned}$$ by applying the triangle inequality Lemma [Lemma 77](#l.diss-triangle-eq){reference-type="ref" reference="l.diss-triangle-eq"} in the last step. This completes the proof. ◻ # Limit of the minimizing movement scheme {#s.energy-soln-existence} In this section we will study the minimizing movements scheme introduced in ([\[e.minimizing-movement-scheme\]](#e.minimizing-movement-scheme){reference-type="ref" reference="e.minimizing-movement-scheme"}). We recall that scheme here for convenience: $$\label{e.minimizing-movement-scheme-2} u_{\delta}^k \in \mathop{\textup{argmin}} \left\{ \mathcal{J}(w) + \operatorname{Diss}(u^{k-1}_\delta, w): w\in F(k\delta)+ H^1_0(U)\right\}.$$ Then interpolate discontinuously to define, for all $t \in [0,T]$, $$u_{\delta}(t):= u_{\delta}^k \ \hbox{ and } \ F_\delta(t) = F(k\delta) \ \hbox{ if } \ t\in [k\delta, (k+1)\delta).$$ We will analyze the behavior of the minimizing movements scheme as $\delta \to 0$. First, in Section [6.1](#s.helly-selection-existence){reference-type="ref" reference="s.helly-selection-existence"}, we apply established ideas [@alberti2011; @MainikMielke] to show that, up to a subsequence, the $u_\delta$ converge pointwise in time to an energy solution [([E](#d.energy_solution))]{.upright}. Our argument will be quite similar to that in Alberti and DeSimone [@alberti2011 Section 4] which was itself inspired by Mainik and Mielke [@MainikMielke]. The conclusion of these arguments is that, up to a subsequence, the $u_\delta(t)$ and $\Omega(u_\delta(t))$ converge pointwise *everywhere* in time in uniform / Hausdorff distance norm in space to a (global) energetic solution. Then, in Section [6.2](#s.MM-visc-prop){reference-type="ref" reference="s.MM-visc-prop"} below, we will show that the approximate solutions generated by this scheme will also satisfy the dynamic slope condition [([M](#d.MVS))]{.upright} in the viscosity sense. In the star-shaped case we will be able to apply the comparison principle proved in Section [4](#s.comparison){reference-type="ref" reference="s.comparison"} to show convergence to the unique solution of [([O](#viscosity_solution))]{.upright}, see Theorem [Theorem 67](#t.oqe-MM-relation){reference-type="ref" reference="t.oqe-MM-relation"}. As a corollary we obtain that the solution of [([O](#viscosity_solution))]{.upright} is an energy solution. These results will complete the proof of our first main theorem in the introduction, Theorem [Theorem 5](#t.main-1){reference-type="ref" reference="t.main-1"}. **Remark 62**. The very recent result of Ferreri and Velichkov [@FerreriVelichkov] implies that $C^{1,\beta}$ regularity is propagated in the discrete scheme ([\[e.minimizing-movement-scheme\]](#e.minimizing-movement-scheme){reference-type="ref" reference="e.minimizing-movement-scheme"}). However, at least with a naive application of their result, the constants would blow-up as $\delta \to 0$ since the regularity theorem needs to be applied $O(\frac{1}{\delta})$ times in order to get to a nontrivial positive $t$ in the limit. In the star-shaped case we go around this difficulty due to the equivalence with the [([O](#viscosity_solution))]{.upright} evolution. This equivalence allows us to apply the regularity theorem only finitely many times, at each monotonicity change, as in Lemma [Lemma 37](#l.space-regularity){reference-type="ref" reference="l.space-regularity"}. ## Pointwise in time convergence to an energy solution {#s.helly-selection-existence} The goal of this section is to analyze the limits of the minimizing movements scheme by energetic methods. We start by proving uniform (in $\delta>0$) $BV$ in time bounds for the minimizing movements scheme via a discrete Grönwall type argument similar to Lemma [Lemma 49](#l.energy-soln-time-reg){reference-type="ref" reference="l.energy-soln-time-reg"}. This establishes the necessary compactness to show that, up to a subsequence, the $u_\delta(t)$ and $\Omega(u_\delta(t))$ converge at *every* time in uniform / Hausdorff distance norm to a (global) energetic solution. The convergence at *every* time, which follows from an application of Helly's selection theorem, is a key detail to establish the energy solution property. **Theorem 63**. *The $u_\delta(t)$ generated by the minimizing movements scheme satisfy the following:* 1. *[\[part.existence-BV-bounds\]]{#part.existence-BV-bounds label="part.existence-BV-bounds"} The states $u_{\delta}(t)$ are uniformly Lipschitz, $\chi_{\Omega_{\delta}(t)}$ and $\mathcal{J}(u_{\delta})$ are uniformly bounded in $BV([0,T]; L^1(U))$ and $BV([0,T]; \mathbb{R})$ respectively.* 2. *[\[part.existence-helly\]]{#part.existence-helly label="part.existence-helly"} There is a subsequence $\delta_k \to 0$ and $u : [0,T] \to F(t) + H^1_0(U)$ so that for *every* $t \in [0,T]$ $$\|u_{\delta_k}(t)- u(t)\|_{L^\infty(U)} \hbox{ and } \ d_H(\Omega(u_{\delta_k}(t)), \Omega(u(t))) \to 0 \ \hbox{ as} \ \delta_k \to 0.$$* 3. *[\[part.existence-solution-props\]]{#part.existence-solution-props label="part.existence-solution-props"} Any such subsequential limit $u(t)$ is an energy solution.* **Remark 64**. By Theorem [Theorem 63](#t.existence){reference-type="ref" reference="t.existence"} any minimizing movements solution, Definition [Definition 2](#d.mm-solution){reference-type="ref" reference="d.mm-solution"}, is indeed an energy solution of [([E](#d.energy_solution))]{.upright}. The proof will use many of the same ideas that were developed in Section [5](#s.basic-energy){reference-type="ref" reference="s.basic-energy"} in the continuous time case, so we will refer to the relevant arguments above in many places to significantly shorten the presentation. *Proof.* **Part [\[part.existence-BV-bounds\]](#part.existence-BV-bounds){reference-type="ref" reference="part.existence-BV-bounds"}.** A discrete version of the energy dissipation inequality will play a central role in the proof. We claim that for each $k \geq 1$ $$\label{e.one-step-discrete-diss-ineq} \operatorname{Diss}(u_{\delta}^{k-1}, u_{\delta}^k) \leq \int_{(k-1)\delta}^{k\delta} 2 (1+g_\delta(t))\dot{F}(t)P_{\delta}(t) \ dt + \mathcal{J}(u^{k-1}_{\delta}) - \mathcal{J}(u^k_{\delta})$$ where the error term in the integrand has $$\label{e.gdelta-bd} |g_\delta(t)| \leq \exp(\|(\log F)'\|_\infty \delta) - 1 \to 0 \ \hbox{ uniformly as $\delta \to 0$,}$$ and we used the notation $P_{\delta}(t):= \int _{\partial U } \frac{\partial u^{[t/\delta]}_{\delta} }{\partial n} dS$. To prove this by using the minimizing scheme we build a comparable version of $u^{k-1}_{\delta}$ to $u^k _{\delta}$, by defining $\tilde{u}:= \frac{F(k\delta)}{F((k-1)\delta)} u^{k-1}_{\delta}$. Then we have $$\operatorname{Diss}(u_{\delta}^{k-1}, u^k_{\delta}) + \mathcal{J}(u^k_{\delta}) \leq \operatorname{Diss}(u_{\delta}^{k-1}, \tilde{u}) +\mathcal{J}(\tilde{u}).$$ Note that $\operatorname{Diss}(u_{\delta}^{k-1}, \tilde{u})=0$ and by integration by parts Using the notation $P_{\delta}(t):= \int _{\partial U } \frac{\partial u^{[t/\delta]}_{\delta} }{\partial n} dS$ we can rewrite $$\begin{aligned} \mathcal{J}(\tilde{u}) - \mathcal{J}(u^{k-1}_{\delta}) &= \frac{F_k^2 - F_{k-1}^2}{F_{k-1}}\int_{\partial U } \frac{\partial u^{k-1}_{\delta}}{\partial n} dS\\ &= \int_{(k-1)\delta}^{k\delta} 2 \frac{F(t)}{F_{k-1}} \dot{F}(t)P_{\delta}(t) \ dt.\end{aligned}$$ Then define $g_\delta(t):= \frac{F(t)}{F_{k-1}} - 1$ for $[t] = k-1$ which satisfies ([\[e.gdelta-bd\]](#e.gdelta-bd){reference-type="ref" reference="e.gdelta-bd"}) by fundamental theorem of calculus and the inequality $1-e^{-k} \leq e^k-1$ for $k>0$. That completes our proof of ([\[e.one-step-discrete-diss-ineq\]](#e.one-step-discrete-diss-ineq){reference-type="ref" reference="e.one-step-discrete-diss-ineq"}). Next we sum up the one-step dissipation inequality ([\[e.one-step-discrete-diss-ineq\]](#e.one-step-discrete-diss-ineq){reference-type="ref" reference="e.one-step-discrete-diss-ineq"}) to get, for any $0 \leq t_0 \leq t_1 \leq T$, $$\label{e.full-discrete-diss-ineq} \overline{\operatorname{Diss}}(u_\delta(t);[t_0,t_1]) \leq \int_{t_0}^{t_1} 2(1+g_\delta(t))\dot{F}(t) P_{\delta}(t) dt + \mathcal{J}(u_\delta(t_1)) - \mathcal{J}( u_\delta(t_0)) + C\delta$$ where $C\delta$ accounts for the error from $t_0$ and $t_1$ not necessarily being integer multiples of $\delta$ and we are using the uniform Lipschitz bound of $u_\delta(t)$ from Lemma [Lemma 47](#l.initial-fb-regularity){reference-type="ref" reference="l.initial-fb-regularity"}. From ([\[e.full-discrete-diss-ineq\]](#e.full-discrete-diss-ineq){reference-type="ref" reference="e.full-discrete-diss-ineq"}) we can derive uniform $BV([0,T];L^1(U) \times \mathbb{R}\times \mathbb{R})$ bounds on the map $t \mapsto (\chi_{\Omega_{\delta}}(t),P_\delta(t),\mathcal{J}_\delta(t))$ by the same arguments as in Lemma [Lemma 49](#l.energy-soln-time-reg){reference-type="ref" reference="l.energy-soln-time-reg"}. **Part [\[part.existence-helly\]](#part.existence-helly){reference-type="ref" reference="part.existence-helly"}.** Due to the above compactness, Helly's selection principle [@alberti2011 Theorem 5.1] yields that, along a subsequence, $$(\chi_{\Omega_{\delta}}(t),P_\delta(t),\mathcal{J}_\delta(t)) \to (\chi_{\Omega(t)},P(t),\mathcal{J}(t)) \ \hbox{ in } \ L^1(U) \times \mathbb{R}\times \mathbb{R}\ \hbox{ for \emph{every}} \ t \in [0,T].$$ Then arguing as in the proof of part [\[part.lr-limits\]](#part.lr-limits){reference-type="ref" reference="part.lr-limits"} of Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} we can also show that $u_\delta(t) \to u(t)$ uniformly in $U$ for each $t \in [0,T]$ and we can show the consistency of the limits $$\Omega(t) = \Omega(u(t)), \ \mathcal{J}(t) = \mathcal{J}(u(t)), \ \hbox{ and } \ P(t) = P(u(t)).$$ Furthermore the convergence of $\Omega_\delta(t) \to \Omega(t)$ can be upgraded to Hausdorff distance convergence using the uniform convergence of the $u_\delta(t) \to u(t)$ and the uniform non-degeneracy of the $u_\delta(t)$. **Part [\[part.existence-solution-props\]](#part.existence-solution-props){reference-type="ref" reference="part.existence-solution-props"}.** Now let us show that $u(t)$ is an energy solution. To show the stability property, observe that for any $v\in u(t) + H_0^1(U)$ we can take $k= [t/\delta]$ and compare $u_\delta(t) = u_{\delta}^k$ with $\tilde{v}:= \frac{F(\delta k)}{F(t)} v(t)$, so that $$\operatorname{Diss}(u_{\delta}^{k-1}, u^k_{\delta}) + \mathcal{J}(u^k_{\delta}) \leq \operatorname{Diss}(u_{\delta}^{k-1}, \tilde{v}) +\mathcal{J}(\tilde{v}).$$ Note that, due to the possibility of a jump in the limit we don't necessarily know that $\operatorname{Diss}(u_{\delta}^{k-1}, u^k_{\delta}) \to 0$. So to get everything in terms of $u_\delta(t) = u^k_\delta$ we apply the triangle inequality Lemma [Lemma 77](#l.diss-triangle-eq){reference-type="ref" reference="l.diss-triangle-eq"} $$\begin{aligned} \mathcal{J}(u_\delta(t)) = \mathcal{J}(u^k_{\delta}) &\leq \operatorname{Diss}(u_{\delta}^{k-1}, \tilde{v}) - \operatorname{Diss}(u_{\delta}^{k-1}, u^k_{\delta}) +\mathcal{J}(\tilde{v})\\ &\leq \operatorname{Diss}(u^k_{\delta},\tilde{v})+\mathcal{J}(\tilde{v})\\ &=\operatorname{Diss}(u_\delta(t),\tilde{v})+\mathcal{J}(\tilde{v})\end{aligned}$$ Now sending $\delta\to 0$ along the convergent subsequence yields the desired stability inequality. Last we show the energy dissipation inequality. For any $0 \leq t_0 < t_1 \leq T$ we can apply ([\[e.full-discrete-diss-ineq\]](#e.full-discrete-diss-ineq){reference-type="ref" reference="e.full-discrete-diss-ineq"}) to find $$\operatorname{Diss}(u_\delta(t_0),u_\delta(t_1)) \leq \int_{t_0}^{t_1} 2(1+g_\delta(t))\dot{F}(t) P_{\delta}(t) dt + \mathcal{J}(u_\delta(t_1)) - \mathcal{J}( u_\delta(t_0)).$$ Using the consistency results from part [\[part.existence-helly\]](#part.existence-helly){reference-type="ref" reference="part.existence-helly"} above and sending $\delta \to 0$ gives the dissipation inequality for $u$. The convergence of the left hand side just follows from the $L^1$ convergence of the indicator functions, and the convergence of the integral is by dominated convergence theorem. ◻ ## Comparison properties of the minimizing movements scheme {#s.MM-visc-prop} In this section we discuss the comparison properties of the minimizing movement scheme and its $\delta \to 0$ limit. **Lemma 65**. *The $u_\delta(t)$ are viscosity solutions of [([M](#d.MVS))]{.upright} on $[0,T]$ in the sense of Definition [Definition 4](#d.MVS){reference-type="ref" reference="d.MVS"}.* **Remark 66**. The $u_\delta(t)$ are also viscosity solutions of [([M](#d.MVS))]{.upright} in the "standard\" sense of touching from above and below by $C^1$ test functions in space-time. This notion, unlike the notion in Definition [Definition 4](#d.MVS){reference-type="ref" reference="d.MVS"} which defines positive velocity based on cones, behaves well with respect to the limit $\delta \to 0$. One can apply the standard method of upper and lower half-relaxed limits, of Barles and Perthame [@BarlesPerthame], to show that $\overline{u}^*$ and $\underline{u}_*$ are respectively sub and supersolutions of [([M](#d.MVS))]{.upright} in the standard sense. We should remark that the upper and lower-half relaxed limits $\overline{u}^*$ and $\underline{u}_*$ are not necessarily the same as the upper and lower semicontinuous envelopes $u^*(t)$ and $u_*(t)$ of the pointwise limit $u(t)$ constructed in Theorem [Theorem 63](#t.existence){reference-type="ref" reference="t.existence"}. It would be interesting to know whether the pointwise in time (subsequential) limits $u(t)$ are also viscosity solutions of [([M](#d.MVS))]{.upright} in general either in the standard sense or in the cone sense of Definition [Definition 4](#d.MVS){reference-type="ref" reference="d.MVS"}. Below in Theorem [Theorem 67](#t.oqe-MM-relation){reference-type="ref" reference="t.oqe-MM-relation"} we will show that it is so in the star-shaped case. *Proof.* We only check the subsolution condition at the free boundary since the remaining cases are standard or symmetrical. Suppose that $\Omega(u_\delta(t))$ has positive normal velocity $V_n(t_0,x_0)>0$ at some point $x_0 \in \partial \Omega(u_\delta(t_0))$ in the sense of Definition [Definition 24](#d.positive-velocity-cone){reference-type="ref" reference="d.positive-velocity-cone"}, i.e. $$\{x: \ |x-x_0| \leq c (t-t_0)\} \subset \Omega(u_\delta(t))^{\complement} \ \hbox{ for } \ t_0 - r_0 \leq t < t_0.$$ Since $u_\delta$ is constant on open intervals of the form $(k\delta,(k+1)\delta)$ this implies that $t_0 = k\delta$ for an integer $k$, that $u_\delta^*(t_0) = u^k_\delta$, and that $u^*_\delta(t) = u^{k-1}_\delta$ for $t_0 - \delta \leq t < t_0$. Since $u^k_\delta$ is a minimizer of $$\mathcal{E}(u^{k-1}_\delta,v) = \mathcal{J}(v) + \operatorname{Diss}(u^{k-1}_\delta,v) \ \hbox{ over } \ v \in F(k\delta) + H^1_0(U)$$ and we have just established that $x_0 \in \mathbb{R}^d \setminus \overline{\Omega(u^{k-1}_\delta)}$ we can apply Lemma [Lemma 44](#l.inwards-outwards-min-props){reference-type="ref" reference="l.inwards-outwards-min-props"} and Lemma [Lemma 45](#l.inwards-outwards-min-implies-vs){reference-type="ref" reference="l.inwards-outwards-min-implies-vs"} to conclude that $$|\nabla u(t_0,x_0)|^2 \geq 1+ \mu_+ \hbox{ in the viscosity sense.}$$ ◻ Now we can apply the previous viscosity solution property in combination with the comparison theorem in Section [4](#s.comparison){reference-type="ref" reference="s.comparison"} to show that, in the star-shaped case, the limit of the minimizing movements scheme is the same as the solution of the obstacle evolution [([O](#viscosity_solution))]{.upright}. The convergence is in fact quantitative. **Theorem 67**. *Suppose that $\Omega_0$ is a $C^{1,\alpha}$ strongly star-shaped region, $K = \mathbb{R}^d \setminus U$ is compact and strongly star-shaped, $\log F$ is Lipschitz continuous, and $F$ changes monotonicity at most finitely many times on $[0,T]$. Call $v(t)$ the solution of [([O](#viscosity_solution))]{.upright} and $u_\delta(t)$ to be a solution of the time incremental scheme ([\[e.minimizing-movement-scheme\]](#e.minimizing-movement-scheme){reference-type="ref" reference="e.minimizing-movement-scheme"}) on $[0,T] \times U$. Then $$\|u_\delta(t) - v(t)\|_{L^\infty(U)} \leq C(\mu_\pm,d)(\exp(\|(\log F)'\|_\infty \delta) - 1)\|F\|_\infty \to 0 \ \hbox{ as } \ \delta \to 0.$$ Thus the viscosity solution of [([O](#viscosity_solution))]{.upright} is the unique minimizing movements solution of the energetic evolution (Definition [Definition 2](#d.mm-solution){reference-type="ref" reference="d.mm-solution"}), in particular any minimizing movements solution, Definition [Definition 2](#d.mm-solution){reference-type="ref" reference="d.mm-solution"}, is also a viscosity solution of [([M](#d.MVS))]{.upright}.* Let us clarify the conclusions of this statement, since several threads of the paper have suddenly come together. We have shown the following: - In the star-shaped setting the viscosity solution of [([O](#viscosity_solution))]{.upright} is an energy solution. - In the star-shaped setting there is only one minimizing movements energy solution (Definition [Definition 2](#d.mm-solution){reference-type="ref" reference="d.mm-solution"}) and it is also a viscosity solution of [([M](#d.MVS))]{.upright}. - In the star-shaped setting all solutions of the discrete scheme converge uniformly to the solution of [([O](#viscosity_solution))]{.upright} with quantitative rate. *Proof.* We prove the convergence by comparison principle Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"} with $v(t)$ the [([O](#viscosity_solution))]{.upright} solution which, by Lemma [Lemma 37](#l.space-regularity){reference-type="ref" reference="l.space-regularity"}, is in $C^{0,1}_tL^\infty_x \cap L^\infty_tC^{1,\beta}_x$ on $[0,T] \times U$. Let $M(\delta) = \sup_t\sup_{h \in [0,\delta]} F(t-h)/F(t)$, note that $M(\delta) \leq \exp(\|(\log F)'\|_\infty \delta)$. Then $$\varphi_\delta(t,x) := M(\delta)v(t,M(\delta)^{-1} x)$$ is a $C^{0,1}_tL^\infty_x \cap L^\infty_tC^{1,\beta}_x$ supersolution of [([M](#d.MVS))]{.upright} on $[0,T]$ with star-shaped level sets and so the comparison principle Proposition [Proposition 41](#p.MVS-OVS-comparison){reference-type="ref" reference="p.MVS-OVS-comparison"} implies that $u_\delta \leq \varphi_\delta$ on $[0,h] \times U$. Note that we have chosen $M(\delta) \geq 1$ so that for $x \in K$ since $K$ star-shaped also $M(\delta)^{-1} x \in K$ and $$\varphi_\delta(t,x) = M(\delta)v(t,M(\delta)^{-1} x) = M(\delta) F(t) \geq \sup_{h \in [0,\delta]} F(t-h) \geq F_\delta(t)$$ By uniform Lipschitz regularity of $v(t)$ $$\begin{aligned} u_\delta(t,x) &\leq \varphi_\delta(t,x) \\ &\leq M(\delta)v(t,x) +M(\delta)\|\nabla u(t)\|_\infty(M(\delta)^{-1} - 1)\sup_{x \in \Omega(v(t))}|x|\\ &\leq v(t,x) + C(M(\delta)-1)\|F\|_\infty\end{aligned}$$ Note that $\Omega(v(t)) \subset B_{CF(t)}(0)$ by uniform non-degeneracy of $v(t)$ and the upper bound $v(t) \leq F(t)$. This proves the upper bound, the lower bound is similar comparing with an inward dilation of size $m(\delta) = \inf_t\inf_{h \in [0,\delta]} F(t-h)/F(t)$ instead. Furthermore, applying Theorem [Theorem 63](#t.existence){reference-type="ref" reference="t.existence"}, we also know that, along a subsequence, $u_\delta(t) \to u(t)$ pointwise in time and uniformly in space and $u(t)$ is an energy solution. By uniqueness of pointwise limits $u \equiv v$ and so the solution of [([O](#viscosity_solution))]{.upright} is an energy solution. ◻ # The dynamic slope condition for energy solutions {#s.energy-soln-motion-law} In this section we show that the energy dissipation balance law implies a weak notion of the dynamic slope condition ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}) $$|\nabla u|^2 = 1\pm \mu_\pm \quad \hbox{ if } \quad \pm V_n(\Omega(t),x) >0.$$ The notion of weak solution is geometric measure theoretic, although there is some viscosity solution aspect as well since there is a subsolution and a supersolution piece which are expressed in terms of sub and superdifferentials. This will complete the proof of Theorem [Theorem 6](#t.main-2){reference-type="ref" reference="t.main-2"}. Note that in Section [6](#s.energy-soln-existence){reference-type="ref" reference="s.energy-soln-existence"} we showed, at least in the star-shaped case, that all minimizing movements solutions are viscosity solutions of the motion law [([M](#d.MVS))]{.upright}. However, this proof was based on the structure provided by the time incremental scheme. Even in the star-shaped case, we do not know uniqueness of energy solutions or whether all energy solutions can be generated by this scheme. Thus we must prove some notion of the dynamic slope condition ([\[e.dynamic-slope-condition-intro\]](#e.dynamic-slope-condition-intro){reference-type="ref" reference="e.dynamic-slope-condition-intro"}) directly from the energy solution property [([E](#d.energy_solution))]{.upright}. Recall by Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} part [\[part.lr-limits-redefine\]](#part.lr-limits-redefine){reference-type="ref" reference="part.lr-limits-redefine"} if $u$ is an arbitrary energy solution on $U \times [0,T]$ then the envelopes $u^*$ and $u_*$ are also energy solutions and have a simple representation in terms of the left and right limits at each time $u_*(t) = u_\ell(t) \wedge u_r(t)$ and $u^*(t) = u_\ell(t) \vee u_r(t)$. The main theorem of this section is that if $u(t)$ is an energy solution then $u^*(t)$ and $u_*(t)$ are respectively (up to sets of surface measure zero) a motion law viscosity subsolution and supersolution. For the statement we will use the notions of positive and negative velocity $V^{\pm}_n(t,x)$, based on inner and outer touching space-time cones given in Section [4](#s.comparison){reference-type="ref" reference="s.comparison"}, and the notions of sub and superdifferential. Specifically refer back to Definition [Definition 19](#d.subdifferential){reference-type="ref" reference="d.subdifferential"}, Definition [Definition 18](#d.superdifferential){reference-type="ref" reference="d.superdifferential"}, Definition [Definition 24](#d.positive-velocity-cone){reference-type="ref" reference="d.positive-velocity-cone"}, and Definition [Definition 25](#d.negative-velocity-cone){reference-type="ref" reference="d.negative-velocity-cone"}. **Theorem 68**. *Suppose that $u$ is an upper semicontinuous energy solution [([E](#d.energy_solution))]{.upright} on $U \times [0,T]$. Then for every $t\in(0,T]$ the set of points where the outward motion law fails $$\Gamma^+(u,t) := \{x \in \partial \Omega(t): \ V_n(t,x) > 0, \ |p|^2 < 1+\mu_+ \ \hbox{ for some } \ p \in D_+ u(t,x)\}$$ has $\mathcal{H}^{d-1}$ measure zero.* *Similarly, if $u$ is an lower semicontinuous energy solution [([E](#d.energy_solution))]{.upright} on $U \times [0,T]$, then for every $t\in(0,T]$ the set of points where the inward motion law fails $$\Gamma^-(u,t) := \{x \in \partial \Omega(t): \ V_n(t,x) < 0, \ |p|^2 > 1-\mu_- \ \hbox{ for some } \ p \in D_- u(t,x)\}$$ has $\mathcal{H}^{d-1}$ measure zero.* **Remark 69**. The sub and superdifferential may both be empty at some free boundary points. However, for energetic solutions, we can follow standard blow-up arguments (see [@CaffarelliSalsa Section 3.3]) to show that the sub and super-differential are both non-trivial $\mathcal{H}^{d-1}$-a.e. on $\partial \Omega(u(t))$. Recall that $\partial \Omega(u(t))$ is a finite perimeter set and $\partial \Omega(u(t))$ has finite $\mathcal{H}^{d-1}$ measure (Lemma [Lemma 47](#l.initial-fb-regularity){reference-type="ref" reference="l.initial-fb-regularity"}) for each $t>0$. Therefore the reduced boundary $\partial_*\{u(t)>0\}$ has full $\mathcal{H}^{d-1}$ measure on $\partial \{u(t)>0\}$. The sub and superdifferentials are both non-trivial at every point of the reduced boundary. More precisely, at $x_0 \in\partial_*\{u>0\}$ with measure theoretic normal $n_0$ consider the blow-up sequence $$u_r(x) = \frac{u(x_0+rx)}{r}$$ which are uniformly bounded, uniformly non-degenerate, and uniformly Lipschitz. All subsequential blow up limits (not necessarily unique) must then be viscosity solutions of $$\Delta v = 0 \ \hbox{ in } \ \{x \cdot n_0>0\} \ \hbox{ and } \ 1-\mu_- \leq |\nabla v|^2 \leq 1+\mu_+ \ \hbox{ on } \ \partial \{x \cdot n_0 >0\}.$$ From this one can conclude that for any (subsequential, locally uniform) blow up limit $$(1-\mu_-)^{1/2}(x \cdot n_0)_+ \leq v(x) \leq (1+\mu_+)^{1/2}(n_0\cdot x)_+$$ and this implies that at least $$(1+\mu_+)^{1/2} n_0 \in D_+u(x_0) \ \hbox{ and } \ (1-\mu_-)^{1/2} n_0 \in D_-u(x_0),$$ i.e. the sub and superdifferential are nontrivial on the measure theoretic reduced boundary. Before we proceed to the details let us give a description of the proof, which also explains why we can only prove the solution condition in an almost everywhere sense. We start by presenting the formal argument, valid in the case that everything is $C^1$ in space and time. Computing the time derivative directly and integrating by parts $$\frac{d}{dt}\mathcal{J}(u(t)) = \int_{\partial \Omega(t)} (1-|\nabla u|^2) V_n \ dS +2\dot{F}(t)P(t)$$ but also differentiating the energy dissipation balance from Lemma [Lemma 52](#l.energy-diss-eq){reference-type="ref" reference="l.energy-diss-eq"} yields $$\frac{d}{dt}\mathcal{J}(u(t)) = 2 \dot{F}(t) P(t) - \int_{\partial \Omega(t)} \mu_+(n) (V_n)_+ + \mu_-(n) (V_n)_- \ dS.$$ Combining the above two identities we find $$\label{e.formal-energy-argument-conclusion} \int_{\partial \Omega(t)} (1+\mu_+(n)-|\nabla u|^2) (V_n)_+ + (|\nabla u|^2 - 1 + \mu_-(n))(V_n)_- \ dS = 0.$$ Both terms in the above integral are non-negative (Corollary [Corollary 46](#c.energy-stability-implies-viscosity-soln){reference-type="ref" reference="c.energy-stability-implies-viscosity-soln"}), and so they must actually be zero pointwise. Of course we cannot exactly use the formal argument. Even without jumps it is tricky to justify taking a time derivative. Instead we need to make a similar kind of energy argument with discrete differences. Furthermore, even if we could justify the identity ([\[e.formal-energy-argument-conclusion\]](#e.formal-energy-argument-conclusion){reference-type="ref" reference="e.formal-energy-argument-conclusion"}), without continuity of $\nabla u$ we could only derive the slope condition in the surface measure a.e. sense. Recall that we do not know $C^1$ regularity of general energy solutions. The regularity theory of Section [3](#s.bernoulli-reg){reference-type="ref" reference="s.bernoulli-reg"} applies to viscosity solutions and we are trying to prove a viscosity solution type property. This is why we can only achieve the dynamic slope condition in the almost everywhere sense. Although we do need to deal carefully with the jumps, the situation is actually better when the free boundary jumps: in this case we can guarantee that slope condition is satisfied pointwise *everywhere*. **Corollary 70**. *Let $F>0$ and suppose that $u\geq u_\ell$ (or $u\leq u_\ell$) is a minimizer of $$\label{energy_above} \mathcal{E}(u_\ell,u) = \mathcal{J}(u) + \textup{Diss}(u_\ell,u) \ \hbox{ over } \ F+H^1_0(U).$$ Then for any $x_0 \in \partial \Omega(u) \setminus \overline{\Omega(u_\ell)}$ (or $x_0 \in \partial\Omega(u) \cap \Omega(u_\ell)$) then $$|p|^2 \geq (1+\mu_+) \ \hbox{ for all } \ p \in D^+u(x_0) \ \ \ (\hbox{or } |p|^2\leq (1-\mu^-) \ \hbox{ for all } \ p \in D^-u(x_0)).$$* *Proof.* Apply Lemma [Lemma 44](#l.inwards-outwards-min-props){reference-type="ref" reference="l.inwards-outwards-min-props"}. ◻ In order to make the main ideas of the proof of Theorem [Theorem 68](#t.ae-viscosity-prop){reference-type="ref" reference="t.ae-viscosity-prop"} more clear we will state in a separate lemma the construction of the energy competitor. First we explain the idea of the energy competitor and then give a precise statement in Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"} below. Suppose that the advancing part of the dynamic slope condition fails at some time $t_0$ on a set of positive $\mathcal{H}^{d-1}$ measure. Then for times $t_{-1} = t_0 - \delta$ sufficiently close to $t_0$, we can perform an inward perturbation of $u(t_0)$ in the normal direction in a region with measure *at least* $O(\delta)$, and without crossing $u(t_{-1})$. This perturbation will also have slope strictly smaller than the pinning interval endpoint value $(1+\mu_+)^{1/2}$ in the perturbation region. This allows us to estimate the change in the Dirichlet energy. **Lemma 71**. *Let $u$ be upper semi-continuous on $[0,T]$, harmonic in $\Omega(u(t))$ at each time, and satisfying the conclusions of Lemma [Lemma 47](#l.initial-fb-regularity){reference-type="ref" reference="l.initial-fb-regularity"} (e.g. $u$ an upper semicontinuous energy solution).* *Suppose for some $t_0 \in (0,T]$ $$\mathcal{H}^{d-1}\left(\bigg\{x \in \partial \Omega(t_0): \ V_n(t_0,x) > 0, \ |p|^2 < 1+\mu_+ \ \hbox{ for some } \ p \in D_+ u(t_0,x)\bigg\}\right) > 0.$$ Then there exists $c>0$ such that the following holds: for all sufficiently small $\delta = t_0 - t_{-1} >0$ there is a function $u_{\delta}\in H^1(U)$ with $u_\delta\geq 0$ and $u_\delta = F(t_0)$ on $\partial U$ such that $$\label{e.perturbation-properties} \Omega(u(t_{-1})) \subset \Omega(u_\delta) \subset \Omega( u(t_0)), \ |\Omega(u(t_0)) \setminus \Omega(u_\delta)| \geq c\delta$$ and $$\label{e.dirichlet-energy-barrier-ineq} \int_{\Omega(u_\delta)} |\nabla u_\delta|^2 \ dx - \int_{\Omega(t_0)} |\nabla u(t_0)|^2 \ dx \leq (1+\mu_+ - c)|\Omega(u(t_0)) \setminus \Omega(u_\delta)|.$$* **Remark 72**. The constant $c>0$ in above Lemma is not very easily quantified, it depends delicately on the hypothesis that $\mathcal{H}^{d-1}(\Gamma^+(u,t_0))>0$ (not just on the measure itself, but on the consequences of being positive measure). We defer the proof of Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"} to the end of the section. It is worth emphasizing that this part is the technical heart of the proof of Theorem [Theorem 68](#t.ae-viscosity-prop){reference-type="ref" reference="t.ae-viscosity-prop"}. Now we give the proof of the weak solution property Theorem [Theorem 68](#t.ae-viscosity-prop){reference-type="ref" reference="t.ae-viscosity-prop"} via an energy argument using Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"}. Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} also plays a central role allowing us to reduce to the case of left continuous times. *Proof of Theorem [Theorem 68](#t.ae-viscosity-prop){reference-type="ref" reference="t.ae-viscosity-prop"}.* Fix $t_0 \in (0,T]$, we first assume that $P(u(t))$ is left continuous at $t_0$. Jump times will be considered at the end, building on the result at left continuous times and using the upper semi-continuity of $u$ as well as the other regularity properties of energy solutions from Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"}. **Step 1.** Assume for now that $$|P(u(t_0)) - P(u(t))| \leq \sigma(t_0 - t) \ \hbox{ for } \ t \leq t_0$$ with some modulus of continuity $\sigma$. Suppose that $$\mathcal{H}^{d-1}\left(\Gamma^+(u,t)\right) > 0.$$ Then Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"} yields $u_\delta$, $t_{-1},t_0$ for small $\delta>0$. We define $$v(x) := \frac{F(t_{-1})} {F(t_0)}u(t_0,x) \ \hbox{ and } \ v_\delta(x) := \frac{F(t_{-1})} {F(t_0)} u_\delta(x).$$ From the energy dissipation inequality ([\[e.diss-inequality\]](#e.diss-inequality){reference-type="ref" reference="e.diss-inequality"}) evaluated on $[t_{-1},t_0]$, we derive $$\begin{aligned} \textup{Diss}(\Omega(t_{-1}),\Omega(t_0)) & \leq \mathcal{J}(u(t_{-1})) - \mathcal{J}(u(t_0)) + \int_{t_{-1}}^{t_0} 2 \dot{F}(t)P(t) \ dt\notag\\ &=[\mathcal{J}(u(t_{-1})) - \mathcal{J}(v_\delta(t_0))]+[\mathcal{J}(v_\delta(t_0)) - \mathcal{J}(v(t_0))] \notag\\ &\quad +\left[\mathcal{J}(v(t_0)) - \mathcal{J}(u(t_0))+ \int_{t_{-1}}^{t_0} 2 \dot{F}(t)P(t) \ dt\right] \notag \\ &= A+B+ C.\label{e.energy-diss-expand-udelta}\end{aligned}$$ We next explain, individually, how to bound $A,B,C$ to yield a contradiction. Since $v_\delta = F(t^{-1})$ on $\partial U$, we can apply the stability property [\[e.stability\]](#e.stability){reference-type="eqref" reference="e.stability"} at time $t_{-1}$ to obtain $$A=\mathcal{J}(u(t_{-1})) - \mathcal{J}(v_\delta) \leq \textup{Diss}(\Omega(t_{-1}),\Omega_\delta(t_0)).$$ Next the energy inequality ([\[e.dirichlet-energy-barrier-ineq\]](#e.dirichlet-energy-barrier-ineq){reference-type="ref" reference="e.dirichlet-energy-barrier-ineq"}) from Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"} yields $$B:=\mathcal{J}(v_\delta(t_0)) - \mathcal{J}(v(t_0)) \leq \left( \frac{F(t_{-1})^2} {F(t_0)^2}(1+\mu_+ - c) - 1 \right)|\Omega(t_0) \setminus \Omega_\delta(t_0)|.$$ Recall that $c>0$ is independent of $\delta>0$, so from the Lipschitz continuity of $\log F(t)$ we conclude that $$B \leq (\mu_+-c + o_\delta(1))|\Omega(t_0) \setminus \Omega_\delta(t_0)|.$$ Lastly we address $$\begin{aligned} C=\mathcal{J}(u(t_0)) - \mathcal{J}(v(t_0))&= \int_U (1-\frac{F(t_{-1})}{F(t_0)} ) |Du|^2(t_0) dx \notag\\ &= \frac{F^2(t_0)-F^2(t_{-1})}{F(t_0)} P(t_0) \notag\\ &\geq \int_{t_{-1}}^{t_0} 2 \dot{F}(t)P(t) dt - 2\|\dot{F}\|_{L^\infty}\sigma(t_0 - t_{-1})|t_0-t_{-1}|,\\ &= \int_{t_{-1}}^{t_0} 2 \dot{F}(t)P(t) dt - o_\delta(1)\delta. \label{e.deriv-hits-bdrycond}\end{aligned}$$ This is the key place where we use the left continuity hypothesis on $P$. Now, combining the previous three inequalities into ([\[e.energy-diss-expand-udelta\]](#e.energy-diss-expand-udelta){reference-type="ref" reference="e.energy-diss-expand-udelta"}), we have obtained $$\textup{Diss}(\Omega(t_{-1}),\Omega(t_0)) \leq \textup{Diss}(\Omega(t_{-1}),\Omega_\delta(t_0)) + (\mu_+ -c+ o_\delta(1))|\Omega(t_0) \setminus \Omega_\delta(t_0)| + o_\delta(1) \delta$$ On the other hand, since $\Omega(u(t_{-1})) \subset \Omega_\delta(t_0) \subset \Omega(t_0)$ from Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"}, we have $$\begin{aligned} \textup{Diss}(\Omega(t_{-1}),\Omega(t_0)) - \textup{Diss}(\Omega(t_{-1}),\Omega_\delta(t_0)) &= \mu_+|\Omega(t_0) \setminus \Omega_\delta(t_{-1})|- \mu_+|\Omega_\delta(t_0) \setminus \Omega(t_{-1})|\\ &=\mu_+|\Omega(t_0) \setminus \Omega_\delta(t_0)|.\end{aligned}$$ Putting the above estimates together, it follows that $$\mu_+|\Omega(t_0) \setminus \Omega_\delta(t_0)| \leq (\mu_+ -c+ o_\delta(1))|\Omega(t_0) \setminus \Omega_\delta(t_0)| + o_\delta(1) \delta.$$ Since Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"} crucially guarantees that $|\Omega(t_0) \setminus \Omega_{\delta}(t_0)| \geq c \delta$, we can divide through by $|\Omega(t_0) \setminus \Omega_\delta(t_0)|$ and find $$\mu_+ \leq \mu_+ -c + o_\delta(1)$$ Sending $\delta = (t_0 - t_{-1})$ to zero yields $0 \leq - c$, which is a contradiction. **Step 2.** (Jump times) Suppose that $P(u(t))$ is not left continuous at $t_0$. By Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} [\[part.lr-limits\]](#part.lr-limits){reference-type="ref" reference="part.lr-limits"} $u$ has left and right limits at time $t_0$, $u_\ell(t_0)$ and $u_r(t_0)$. Furthermore, by Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} part [\[part.lr-limits-jumps\]](#part.lr-limits-jumps){reference-type="ref" reference="part.lr-limits-jumps"} and since $u$ is upper semicontinuous we must have $u(t_0) = u_\ell(t_0) \vee u_r(t_0)$. By Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} part [\[part.lr-limits-redefine\]](#part.lr-limits-redefine){reference-type="ref" reference="part.lr-limits-redefine"} $$\tilde{u}(t) := \begin{cases} u(t) & t< t_0 \\ u_\ell(t_0) & t = t_0. \end{cases}$$ is itself an energy solution on $[0,t_0]$ and $\tilde{u}(t)$ and $P(\tilde{u}(t))$ are left continuous at $t_0$. The previous arguments give $$\mathcal{H}^{d-1}(\Gamma(\tilde{u},t_0)) = 0.$$ We aim to show $\Gamma(u,t_0) \subset \Gamma(\tilde{u},t_0)$ which will complete the proof showing $\mathcal{H}^{d-1}(\Gamma(u,t_0)) = 0$. By Corollary [Corollary 70](#c.jump-subsoln-cond){reference-type="ref" reference="c.jump-subsoln-cond"} $$\Gamma(u,t_0) \setminus \partial \Omega(u_\ell(t_0)) = \emptyset.$$ On the other hand since $u$ touches $\tilde{u}$ from above at each $x_0 \in \partial \Omega_\ell(t_0) \cap \partial \Omega_r(t_0)$ $$V^+_n(t_0,x_0) \geq \tilde{V}^+_n(t_0,x_0) \ \hbox{ and } \ D_+u(t_0,x_0) \subset D_+\tilde{u}(t_0,x_0)$$ Thus $$\Gamma(u,t_0) \subset \Gamma(\tilde{u},t_0).$$ ◻ **Remark 73**. The regularity of the pressure $P(t)$ seems to play an important role. In ([\[e.deriv-hits-bdrycond\]](#e.deriv-hits-bdrycond){reference-type="ref" reference="e.deriv-hits-bdrycond"}) the regularity of $P(t)$ dictates the size of a positive error term which needs to be balanced by the positive $\mathcal{H}^{d-1}$ measure of the set where the viscosity solution condition fails. Before continuing to the proof of the barrier construction Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"}, it is useful to separate out one further measure-theoretic Lemma which says that if the subsolution condition fails on a set of positive surface measure, then it fails so quantitatively. Let us first define quantified versions of the velocity and sub/superdifferentials. For given $r_0, c, \sigma >0$ we define $$D^{\sigma,r_0}_+ u(x_0) := \left\{ p \in \mathbb{R}^d : u(x) \leq p \cdot (x-x_0) + \sigma|x-x_0| \ \hbox{ in } \ B_{r_0}(x_0) \cap \overline{\{u>0\}} \right\} ,$$ and $$D^{\sigma,r_0}_- u(x_0) := \left\{ p \in \mathbb{R}^d : u(x) \geq p \cdot (x-x_0) - \sigma|x-x_0| \ \hbox{ in } \ B_{r_0}(x_0) \cap \overline{\{u>0\}} \right\} .$$ Lastly we define a quantified set where the dynamic slope condition strictly fails: $$\Gamma_{\eta,\sigma,r_0}(u,t) := \left\{ x \in \partial \Omega(u(t)): \begin{array}{cc}V^{r_0}_n(t,x) \geq\eta \ \hbox{ and } \ \exists p \in D_+^{\sigma,r_0} u(t,x) \\ \hbox{s.t.} \ |p|^2 - (1+\mu_+) < -\eta\end{array}\right\}.$$ In the above definition we are keeping more parameters than minimally necessary in order to clarify certain delicate points in the proof. **Lemma 74**. *The family of sets $\Gamma_{\eta,r_0,\sigma}(u,t)$ is monotone decreasing with respect to $\eta$ and $r_0$ and monotone increasing with respect to $\sigma$. Furthermore $$\Gamma(u,t) = \bigcup_{\eta>0}\bigcap_{\sigma>0}\bigcup_{r_0>0} \Gamma_{\eta,\sigma,r_0}(u,t)$$ and thus, by monotone convergence theorem, if $\mathcal{H}^{d-1}(\Gamma(u,t))>0$ then there is $\eta>0$ so that for all $\sigma>0$ there is $r_0(\sigma)>0$ such that $\mathcal{H}^{d-1}(\Gamma_{\eta,\sigma,r_0}(u,t))>0$.* *Proof.* The monotonicities follow from the the set definition. Let us check the set formula. Let $x \in \Gamma(u,t)$, which means $$V_n(t,x) > 0 \ \hbox{ and there is $p \in D_+u(t,x)$ with } \ |p|^2 < 1+\mu_+.$$ By definition of $D_+u$ $$u(t,x) \leq p \cdot x + o(|x-x_0|) \ \hbox{ for } \ x \in \overline{\Omega(u(t))}$$ and so for every $\sigma>0$ there is $r_1(\sigma)>0$ sufficiently small so that $$u(t,x) \leq p \cdot x + \sigma |x-x_0| \ \hbox{ for } \ x \in \overline{\Omega(u(t))} \cap B_{r_1}(x)$$ Namely $p \in D_+^{\sigma,r_1}u(t,x)$. By the definition of $V_n$ $$V^{r_2}_n(t,x) \geq \eta_2 > 0 \ \hbox{ for some } \ r_2,\eta_2>0.$$ Call $\eta_1 := (1+\mu_+) - |p|^2 >0$. Taking $r_0 = \min\{r_1,r_2\}$ and $\eta = \min\{\eta_1,\eta_2\}$ we find that $x \in \Gamma_{\eta,\sigma,r_0}(u,t)$. ◻ *Proof of Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"}.* By Lemma [Lemma 74](#l.monotone-convergence){reference-type="ref" reference="l.monotone-convergence"} and the hypothesis of the Lemma there is $\eta>0$ such that for all $\sigma>0$ there is positive $r_0(\sigma)$ with $$\label{e.positive-measure-eta-sigma-r0} \mathcal{H}^{d-1}(\Gamma_{\eta,\sigma,r_0}(u,t_0)) \geq \frac{1}{2}\mathcal{H}^{d-1}(\Gamma(u,t_0))>0.$$ We will take $\sigma = c(d)\eta$, where the dimensional constant $c(d)$ will be determined in the course of the proof. This choice of $\sigma$ also fixes $r_0 = r_0(d,\eta)>0$, we will usually drop the dependence on the dimension when it is not important and write (for example) $r_0(\eta)$. Let us consider $\delta \in (0, r_0(\eta)]$ and $t_{-1} := t_0 - \delta$ for the rest of the proof. By monotonicity of the sets $\Gamma_{\eta,\sigma,r_0}(u,t_0)$ in $\eta$, we may assume that $\eta < \frac{1}{2}$. We outline the proof, which is divided into three steps. In the first step we use the speed and slope condition at a single point to construct a strict supersolution replacement in a local neighborhood which is below $u(t_0)$ by $O(\delta^d)$ in measure but still above $u(t_{-1})$. The construction is fairly standard, using the first order asymptotic information from the quantified superdifferential, sliding the linearization inwards while also bending up in the tangential directions and bending down in the normal direction in order to create a local superharmonic perturbation. In the second step we perform a typical Vitali-type covering argument. In the final step we perform the local supersolution replacement in each disjoint ball from the covering to create the perturbation $u_\delta$, which is now $O(\delta)$ below $u(t_0)$ because we have shifted inwards by $O(\delta)$ on a set of positive surface measure. **Step 1.** (Construction of a barrier based on single reference point) In order to make certain parameter choices more clear in the proof below we will adopt the notation $\eta_t = \eta_x = \eta$ and use $\eta_t$ when the lower bound of the velocity is being used, and $\eta_x$ when the lower bound of the slope condition is being used. Let $x_0\in \Gamma_{\eta,\sigma,r_0}(u,t_0)$. For simplicity we may assume that $x_0 = 0$ and $p_0 = \alpha e_d$ for some $\alpha \geq 0$, so that $$\label{e.u-upper-exp-alpha} u(t_0,x) \leq \alpha x_d + \sigma|x| \ \hbox{ in } \ B_{r_0}(0) \cap \overline{\Omega(t_0)}$$ and, since we can always increase $\alpha$ if necessary, we may assume that $$\alpha = (1+\mu_+)^{1/2} -\eta_{x}.$$ Note that, since $u$ is nonnegative, ([\[e.u-upper-exp-alpha\]](#e.u-upper-exp-alpha){reference-type="ref" reference="e.u-upper-exp-alpha"}) yields that $\alpha (x_d)_- \leq \sigma r \hbox{ on } \ B_{r}(0) \cap \overline{\Omega(t_0)}$, namely $$\label{e.xd-neg-part-bd} B_{r}(0) \cap \overline{\Omega(t_0)} \subset \{x_d \geq - \frac{\sigma}{\alpha}r\} \ \hbox{ for all } \ r \leq r_0.$$ Furthermore, again from the definition of $\Gamma_{\eta,\sigma,r_0}(u,t_0)$, $$\label{e.positive-speed-outcome} B_{\eta_t(t_0-t)}(0) \subset \mathbb{R}^d \setminus \Omega(t) \ \hbox{ for } \ t \geq t_0 - r_0.$$ Recall that we assumed that $\delta = t_0 - t_{-1}$ is smaller than $r_0(\eta)>0$. Our perturbation will be performed in $e_d$-aligned cylinders which are anisotropic, slightly wider in the tangential directions than the normal direction. Denoting $x' = (x_1,\dots,x_{d-1})$ and $B_r'$ the ball of radius $r$ in $\mathbb{R}^{d-1}$, $$\textup{Cyl}_{r} = B_r' \times (-c_dr,c_dr).$$ The dimensional constant $c_d := \frac{1}{\sqrt{2(d-1)}}$ is chosen so that on the side boundary of the cylinder $\partial_{side}\textup{Cyl}_{r}:= \partial B_{r} \times (-c_dr,c_dr)$ $$\label{e.cylinder-fatness-choice} |x'|^2 - (d-1)x_d^2 \geq r^2 - (d-1)c_d^2 r^2 \geq \frac{1}{2}r^2 \ \hbox{ on } \ x \in \partial_{side}\textup{Cyl}_{r}.$$ We also call $$\partial_{top}\textup{Cyl}_{r}:= B_{r} \times \{x_d = c_dr\} \ \hbox{ and } \ \partial_{bot}\textup{Cyl}_{r}:= B_{r} \times \{x_d = -c_dr\}.$$ Obviously $\textup{Cyl}_r \subset B_{r_0}$ for $r \leq r_0/\sqrt{2}$. We aim to make a localized perturbation in $\textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t \delta}$. Note that this cylinder is chosen since, by ([\[e.positive-speed-outcome\]](#e.positive-speed-outcome){reference-type="ref" reference="e.positive-speed-outcome"}), $$\label{e.positive-speed-outcome2} \textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t \delta} \subset B_{\eta_t \delta} \subset \mathbb{R}^d \setminus \Omega(t_{-1}).$$ Since we assume $\delta = t_0 - t_{-1} \leq r_0$ and $\eta<1$, so $\eta_t \delta \leq r_0$ as well. We shift the plane $\alpha x_d$ inwards by $O(\delta)$ in the normal direction $e_d$ while also bending upwards in the tangential directions so that the barrier is below $u(t_0)$ but above $u(t_{-1})$. More precisely, define the barrier $$\psi_{\delta}(x) := (\alpha+\tfrac{\eta_x}{4}) (x_d - a_1\eta_t\delta ) + a_2(\eta_t\delta)^{-1}(|x'|^2 - (d-1)x_d^2).$$ We will see that if we choose $a_i := c(d,\mu_+) \eta_x, i=1,2$ then we can guarantee the following important properties of $\psi_\delta$: 1. [\[part.harmonic-parabola\]]{#part.harmonic-parabola label="part.harmonic-parabola"} $\psi_\delta$ is harmonic; 2. [\[part.bends-up\]]{#part.bends-up label="part.bends-up"} $\psi_\delta(x) > u(t_0,x)$ on $\partial \textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t \delta} \cap \overline{\Omega(t_0)}$; 3. [\[part.gradient-bound\]]{#part.gradient-bound label="part.gradient-bound"} $|\nabla\psi_\delta(x)|^2 \leq 1 + \mu_+ - \tfrac{1}{2}\eta_x \ \hbox{ in } \textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t \delta}$; 4. [\[part.exterior-ball\]]{#part.exterior-ball label="part.exterior-ball"} There is $c>0$ depending on $d$ and $\mu_+$ such that $$B_{c\eta_x\eta_t\delta}(0) \subset \{\psi_\delta \leq 0\}.$$ Property [\[part.harmonic-parabola\]](#part.harmonic-parabola){reference-type="ref" reference="part.harmonic-parabola"}. This is straightforward, but we do emphasize that the superharmonicity is important and needed later to deal with the energy computation in Step 3 below. Property [\[part.bends-up\]](#part.bends-up){reference-type="ref" reference="part.bends-up"}. We check the bottom, sides and top of the cylinder separately. On the top boundary $x \in \partial_{top}\textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta}$ we can use the extra room on the slope: $$\begin{aligned} \psi_\delta(x) &\geq \alpha x_d + c(d)\eta_x\eta_t \delta - C(d)(a_1+a_2)\eta_t \delta\\ &\geq \alpha x_d + c(d)[\eta_x - C(d)(a_1+a_2)]\eta_t \delta.\end{aligned}$$ Thus if we take $$\label{e.a1-a1-cond1} a_1 +a_2 \leq c(d)\eta_x$$ then, again on $x \in \partial_{top}\textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta}$, $$\begin{aligned} \psi_\delta(x) &\geq \alpha x_d + c(d)\eta_x \eta_t \delta \\ &\geq \alpha x_d+c(d)\eta_x |x| \geq \alpha x_d + \sigma |x| \geq u(t_0,x).\end{aligned}$$ where this computation fixes the choice of the dimensional constant in $\sigma = c(d)\eta_x$. On the sides $\partial_{side}\textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta} \cap \overline{\Omega(t_0)}$ we can use the upwards bending in the tangential directions. Using ([\[e.cylinder-fatness-choice\]](#e.cylinder-fatness-choice){reference-type="ref" reference="e.cylinder-fatness-choice"}), we have. $$\begin{aligned} \psi_\delta(x) &\geq \alpha x_d -\tfrac{1}{4}\eta_x (x_d)_--(1+\mu_+)^{1/2}a_1\eta_t\delta + \frac{1}{2}a_2\eta_t \delta \\ &= \alpha x_d + \sigma |x| +\underbrace{ \left[\tfrac{1}{2}a_2\eta_t \delta - \sigma |x|-\tfrac{1}{4}\eta_x (x_d)_- - (1+\mu_+)^{1/2}a_1\eta_t \delta \right] }_\text{$:=A$}\end{aligned}$$ ([\[e.xd-neg-part-bd\]](#e.xd-neg-part-bd){reference-type="ref" reference="e.xd-neg-part-bd"}) yields that $$A \geq c\left[a_2 - C(d)\sigma - C(\mu_+,d)a_1 \right] \eta_t \delta \hbox{ on } x \in \partial_{side}\textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta} \cap \overline{\Omega(t_0)}.$$ We want this to be non-negative, so along with condition ([\[e.a1-a1-cond1\]](#e.a1-a1-cond1){reference-type="ref" reference="e.a1-a1-cond1"}) we need to choose $\sigma$, $a_1$ and $a_2$ so that $$a_1 + a_2 \leq c(d)\eta_x \ \hbox{ and } \ C(d)\sigma + C(\mu_+,d)a_1 \leq a_2.$$ To satisfy both inequalities, we choose $\sigma = c(d) \eta_x$, $a_1 = c(d,\mu_+) \eta_x$ and $a_2 = c(d)\eta_x$. Applying these choices of parameters, we find $$\psi_\delta(x) \geq \alpha x_d + \sigma |x| \geq u(t_0,x) \ \hbox{ on } \ \partial_{side}\textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta} \cap \overline{\Omega(t_0)}.$$ Finally, the bottom boundary $\partial_{bot} \textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta}$ is compactly contained in the zero level set of $u(t_0)$. Namely the set $(x_d)_- = c_d\eta_t \delta$ does not intersect $\overline{\Omega(t_0)}$ by ([\[e.xd-neg-part-bd\]](#e.xd-neg-part-bd){reference-type="ref" reference="e.xd-neg-part-bd"}) and fixing the choice of dimensional constant in the original specification $\sigma = c(d) \eta$. Property [\[part.gradient-bound\]](#part.gradient-bound){reference-type="ref" reference="part.gradient-bound"}. For this we just compute the gradient and bound with triangle inequality $$\begin{aligned} |\nabla\psi_\delta- (\alpha + \tfrac{1}{4}\eta_x)e_d| \leq C(d)a_2 \quad \hbox{ in } \textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta}.\end{aligned}$$ Choosing the dimensional constant in $a_2 = c(d)\eta_x$ smaller if necessary, $$|\nabla\psi_\delta| \leq \alpha + \tfrac{1}{2}\eta_x \leq (1+\mu_+)^{1/2} - \tfrac{1}{2}\eta_x \quad \hbox{ in } \textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta}.$$ Property [\[part.exterior-ball\]](#part.exterior-ball){reference-type="ref" reference="part.exterior-ball"}. Note that $\psi_\delta$ is monotone in $\textup{Cyl}_{\frac{1}{\sqrt{2}}\eta_t\delta}$ on the cone of directions, namely for the directions $e \in S^{d-1}$ such that $e \cdot e_d \geq \frac{1}{2} \geq C(d)a_2$. Since $\psi_\delta(a_1\eta_t\delta e_d) = 0$ we find that $$B_{2a_1\eta_t\delta e_d}(0) \subset \{\psi_\delta \leq 0\}.$$ **Step 2.** (Covering set up) Since Hausdorff measure is inner regular, there is a compact subset $K \subset \Gamma_{\eta,\sigma,r_0}(u,t_0)$ with $$\mathcal{H}^{d-1}(K) \geq \frac{1}{2}\mathcal{H}^{d-1}(\Gamma_{\eta,\sigma,r_0}(u,t_0)) \geq \frac{1}{4}\mathcal{H}^{d-1}(\Gamma(u,t_0)).$$ Since $K$ is compact, there is $r_1>0$ such that $$\label{e.hausdorff-def}\inf\{ \sum_{j=1}^\infty \rho_j^{d-1} : K \subset \bigcup_{j=1}^\infty B_{\rho_j}(x_j), \ \rho_j \leq r_1\} \geq \frac{1}{2}\mathcal{H}^{d-1}(K).$$ Let $3\eta_t\delta \leq r_1$. By the Vitali covering lemma there is a finite collection of disjoint balls $B_j$ of radius $\eta_t\delta$ centered at points $x_j \in K \subset \partial \Omega(t_0)$ with $$K \subset \bigcup_j 3B_j.$$ Call $\tilde{B}_j = B_{c\eta_x\eta_t \delta}(x_j)$ with constant $1>c(d,\mu_+)>0$ from property [\[part.gradient-bound\]](#part.gradient-bound){reference-type="ref" reference="part.gradient-bound"} above. Since $\tilde{B}_j \subset B_j$ they are a disjoint collection as well. By ([\[e.hausdorff-def\]](#e.hausdorff-def){reference-type="ref" reference="e.hausdorff-def"}) $$\sum_j 3^{d-1} \delta^{d-1} \geq \frac{1}{2}\mathcal{H}^{d-1}(K).$$ By measure non-degeneracy of the free boundary Lemma [Lemma 47](#l.initial-fb-regularity){reference-type="ref" reference="l.initial-fb-regularity"} $$|\Omega(t_0) \cap \tilde{B}_j| \geq c \eta^{-2d}\delta^{d},$$ where the constant $c$ depends on $d$ and $\mu_+$. Since $\tilde{B}_j's$ are disjoint, we obtain the following by absorbing universal constants into $c$: $$|\Omega(t_0) \cap \bigcup_j \tilde{B}_{j}| = \sum_j |\Omega(t_0) \cap \tilde{B}_{j}| \geq \sum_j c \eta^{-2d}\delta^{d} \geq c\eta^{-2d}\delta \mathcal{H}^{d-1}(K) \delta,$$ or, $$\label{e.measure-diff-lower-bound} |\Omega(t_0) \cap \bigcup_j \tilde{B}_{j}| \geq c\eta^{-2d}\mathcal{H}^{d-1}(\Gamma(u,t_0)) \delta.$$ **Step 3.** (Construction of the comparison functions) Let $\delta \leq \min\{ (3\eta_t)^{-1}r_1,r_0\}$ so that both Step 1 and Step 2 apply, recall $r_1$ was fixed by ([\[e.hausdorff-def\]](#e.hausdorff-def){reference-type="ref" reference="e.hausdorff-def"}) and $r_0(\eta)$ was fixed at the beginning of the proof below ([\[e.positive-measure-eta-sigma-r0\]](#e.positive-measure-eta-sigma-r0){reference-type="ref" reference="e.positive-measure-eta-sigma-r0"}). Now for each $x_j \in \Gamma(u,t_0)$ from the covering constructed in Step 2, call $\psi_{x_j,\delta}$ to be the single-point barrier function constructed in Step 1. The $\psi_{x_j,\delta}$ are defined in $B_j$, extend them to be equal to $+\infty$ outside of their respective $B_j$. Consider the comparison function $$u_\delta(x) = u(t_0,x) \wedge \bigwedge_{\substack{j}}\psi_{x_j,\delta}(x) \ \hbox{ defined for } \ x \in \Omega(t_0).$$ By properties [\[part.harmonic-parabola\]](#part.harmonic-parabola){reference-type="ref" reference="part.harmonic-parabola"} and [\[part.bends-up\]](#part.bends-up){reference-type="ref" reference="part.bends-up"}, and by the disjointness of the $B_j$, we can conclude that $u_\delta$ is Lipschitz continuous and superharmonic in $\Omega(t_0)$. Now by property [\[part.exterior-ball\]](#part.exterior-ball){reference-type="ref" reference="part.exterior-ball"} (first containment below), property [\[part.bends-up\]](#part.bends-up){reference-type="ref" reference="part.bends-up"} (second containment below), and ([\[e.positive-speed-outcome2\]](#e.positive-speed-outcome2){reference-type="ref" reference="e.positive-speed-outcome2"}) (third containment below), applied for each $j$ $$\label{e.strict-outward-motion} \bigcup_{j} \tilde{B}_j\cap \Omega(u(t_0)) \subset \Omega(u(t_0)) \setminus \Omega(u_\delta) \subset \bigcup_{j} B_j \subset \mathbb{R}^d \setminus \Omega(u(t_{-1})).$$ Thus we conclude that $$\Omega(u(t_{-1})) \subset \Omega(u_\delta) \subset \Omega(u(t_0))$$ and also, by ([\[e.measure-diff-lower-bound\]](#e.measure-diff-lower-bound){reference-type="ref" reference="e.measure-diff-lower-bound"}), $$|\Omega(u(t_0)) \setminus \Omega(u_\delta)| \geq \sum_j |\tilde{B}_j \cap \Omega(u(t_0))| \geq c\eta^{-2d} \mathcal{H}^{d-1}(\Gamma(u,t_0)) \delta.$$ The previous two displayed equations establish ([\[e.perturbation-properties\]](#e.perturbation-properties){reference-type="ref" reference="e.perturbation-properties"}), the first claim of the Lemma. To conclude we still need to establish the inequality on the difference of the Dirichlet energy ([\[e.dirichlet-energy-barrier-ineq\]](#e.dirichlet-energy-barrier-ineq){reference-type="ref" reference="e.dirichlet-energy-barrier-ineq"}). Recall that $u_\delta$ is superharmonic and Lipschitz continuous in $\Omega(t_0)$, so we can apply Lemma [Lemma 76](#l.intbyparts2){reference-type="ref" reference="l.intbyparts2"} with $v_0 = u_\delta$ and $v_1 = u(t_0)$. This results in $$\begin{aligned} \int_{\Omega(u_\delta)} |\nabla u_\delta|^2 \ dx - \int_{\Omega(t_0)} |\nabla u(t_0)|^2 \ dx &\leq \int_{\Omega(t_0) \setminus \Omega(u_\delta)}|D{u_\delta}|^2 \ dx\\ &\leq\int_{\Omega(t_0) \setminus \Omega(u_\delta)}(1+\mu_+ - \tfrac{1}{2}\eta_x) \ dx\end{aligned}$$ where we applied property [\[part.gradient-bound\]](#part.gradient-bound){reference-type="ref" reference="part.gradient-bound"} for the second inequality. This proves ([\[e.dirichlet-energy-barrier-ineq\]](#e.dirichlet-energy-barrier-ineq){reference-type="ref" reference="e.dirichlet-energy-barrier-ineq"}) and finishes the proof of the Lemma. ◻ # Technical results ## Non-degeneracy Non-degeneracy of Bernoulli problem solutions is much more delicate than Lipschitz continuity, it typically needs a stronger condition than just the viscosity solution property. Here we give several cases where it is known. **Lemma 75**. *Let $u \in C(B_2)$ satisfy one of the following:* 1. *[\[part.lip-bdry-case\]]{#part.lip-bdry-case label="part.lip-bdry-case"} Suppose that $\partial\{u>0\}$ is an $L$-Lipschitz graph in $B_2$ and $u$ solves $$\Delta u = 0 \ \hbox{ in } \ \{u>0\} \cap B_2, \ \hbox{ and } \ |\nabla u| \geq 1 \hbox{ on } \ \partial \{u>0\} \cap B_2.$$* 2. *[\[part.inward-min-case\]]{#part.inward-min-case label="part.inward-min-case"} Suppose that $u$ is an inward minimizer of $\mathcal{J}$ in $B_2$.* 3. *[\[part.minimal-super-case\]]{#part.minimal-super-case label="part.minimal-super-case"} Suppose that $u$ is a minimal supersolution in $B_2$ $$\Delta u = 0 \ \hbox{ in } \ \{u>0\} \cap B_2, \ \hbox{ and } \ |\nabla u| \leq 1 \hbox{ on } \ \partial \{u>0\} \cap B_2.$$* 4. *[\[part.maximal-sub-d2\]]{#part.maximal-sub-d2 label="part.maximal-sub-d2"} Suppose $d=2$ and $u$ is a maximal subsolution in $B_2$ of $$\Delta u = 0 \ \hbox{ in } \ \{u>0\} \cap B_2, \ \hbox{ and } \ |\nabla u| \geq 1 \hbox{ on } \ \partial \{u>0\} \cap B_2.$$* *Then there is $c_0$ depending on $d$ (and on $L$ if in the first case) such that $$\sup_{B_r(x)} u(x) \geq c_0 r \quad \hbox{ for all } x\in \partial \{u>0\}\cap B_1, \ B_r(x) \subset B_2.$$* *Proof.* For part [\[part.lip-bdry-case\]](#part.lip-bdry-case){reference-type="ref" reference="part.lip-bdry-case"} see [@DeSilva Lemma 6.1], for part [\[part.inward-min-case\]](#part.inward-min-case){reference-type="ref" reference="part.inward-min-case"} see [@VelichkovBook Lemma 4.4], for part [\[part.minimal-super-case\]](#part.minimal-super-case){reference-type="ref" reference="part.minimal-super-case"} see [@CaffarelliSalsa Lemma 6.9], and for [\[part.maximal-sub-d2\]](#part.maximal-sub-d2){reference-type="ref" reference="part.maximal-sub-d2"} see [@Betul]. ◻ ## Energy difference quotient formulae We give and prove several formulas for the difference of the Dirichlet energy under varying positivity set. We have not seen these particular formulas before in the literature, but they provide a very convenient way to go from energy minimization to viscosity solution properties. **Lemma 76**. *Suppose that $v_0,v_1 \in H^1(U)$ with $v_0 \leq v_1$ and $v_0 = v_1 \geq 0$ on $\partial U$. Call $\Omega_j = \{v_j>0\} \cap U$.* - *If $v_1$ is subharmonic in $\Omega_1$ then $$\label{e.intbyparts2-subsoln} \int_{\Omega_0} |\nabla v_0|^2 \ dx- \int_{\Omega_1} |\nabla v_1|^2 \ dx \geq \int_{\Omega_1 \setminus \Omega_0}|\nabla v_1|^2 \ dx.$$* - *If $v_0$ is superharmonic in $\Omega_1$ then $$\label{e.intbyparts2-supersoln} \int_{\Omega_0} |\nabla v_0|^2 \ dx - \int_{\Omega_1} |\nabla v_1|^2 \ dx \leq \int_{\Omega_1 \setminus \Omega_0}|\nabla {v_0}|^2 \ dx$$* *Proof.* For ([\[e.intbyparts2-subsoln\]](#e.intbyparts2-subsoln){reference-type="ref" reference="e.intbyparts2-subsoln"}) we start with $$\begin{aligned} \int_{\Omega_0} |\nabla v_0|^2 \ dx - \int_{\Omega_1} |\nabla v_1|^2 \ dx &= \int_{\Omega_0 } |\nabla v_0|^2 - |\nabla v_1|^2 \ dx + \int_{\Omega_1 \setminus \Omega_0} - |\nabla v_1|^2 \ dx\end{aligned}$$ Then we compute the first term on the right $$\begin{aligned} \int_{\Omega_0} |\nabla v_0|^2 - |\nabla v_1|^2 \ dx &= \int_{\Omega_0 } |\nabla v_1+\nabla(v_0 - v_1)|^2-|\nabla v_1|^2 \ dx \\ &=\int_{\Omega_0 } 2\nabla v_1\cdot \nabla(v_0 - v_1) + |\nabla(v_0-v_1)|^2 \ dx \\ &\geq \int_{\Omega_0 } 2\nabla v_1\cdot \nabla(v_0 - v_1) \ dx\\ &= \int_{\Omega_1} 2\nabla v_1\cdot \nabla((v_0)_+ - v_1) \ dx +\int_{\Omega_1 \setminus \Omega_0} 2 |\nabla v_1|^2 \ dx\\ &\geq \int_{\Omega_1 \setminus \Omega_0} 2 |\nabla v_1|^2 \ dx.\end{aligned}$$ The last inequality in the sequence above is is due to $v_1$ being $H^1$ subharmonic in $\Omega_1$. Note that $v_1 - (v_0)_+$ is a valid test function for the subharmonicity condition since it is non-negative in $\Omega_1$ and in $H^1_0(\Omega_1)$. For ([\[e.intbyparts2-supersoln\]](#e.intbyparts2-supersoln){reference-type="ref" reference="e.intbyparts2-supersoln"}) we start with $$\begin{aligned} \int_{\Omega_0} |\nabla v_0|^2 \ dx - \int_{\Omega_1} |\nabla v_1|^2 \ dx &= \int_{\Omega_1 } |\nabla v_0|^2 - |\nabla v_1|^2 \ dx + \int_{\Omega_1 \setminus \Omega_0} - |\nabla v_0|^2 \ dx\end{aligned}$$ Then we compute the first term on the right $$\begin{aligned} \int_{\Omega_1} |\nabla v_0|^2 - |\nabla v_1|^2 \ dx &= \int_{\Omega_1 } |\nabla v_0|^2-|\nabla v_0+\nabla(v_1-v_0)|^2 \ dx \\ &=\int_{\Omega_1 } -2\nabla v_0\cdot \nabla(v_1 - v_0) - |\nabla(v_1-v_0)|^2 \ dx \\ &\leq \int_{\Omega_1} -2\nabla v_0\cdot \nabla(v_1 - (v_0)_+) \ dx +\int_{\Omega_1 \setminus \Omega_0} 2 |\nabla v_0|^2 \ dx\\ &\leq \int_{\Omega_1 \setminus \Omega_0} 2 |\nabla v_0|^2 \ dx.\end{aligned}$$ The last inequality in the sequence above is is due to $v_0$ being $H^1$ superharmonic in $\Omega_1$. Note that $v_1 - (v_0)_+$ is a valid test function for the superharmonicity condition since it is non-negative in $\Omega_1$ and in $H^1_0(\Omega_1)$. In the smooth case one can also check the related identities $$\begin{aligned} &\int_{\Omega_0} |\nabla v_0|^2 \ dx- \int_{\Omega_1} |\nabla v_1|^2 \ dx \\ & \quad = \int_{\Omega_1} 2\Delta v_1 (v_1 - (v_0)_+) \ dx + \int_{\Omega_0} |\nabla(v_1 - v_0)|^2 \ dx + \int_{\Omega_1 \setminus \Omega_0}|\nabla v_1|^2 \ dx. \end{aligned}$$ and $$\begin{aligned} &\int_{\Omega_0} |\nabla v_0|^2 \ dx - \int_{\Omega_1} |\nabla v_1|^2 \ dx = \\ &\quad \int_{\Omega_1} 2(v_1 - (v_0)_+)\Delta v_0 \ dx-\int_{\Omega_1} |\nabla(v_1 - v_0)|^2 \ dx+\int_{\Omega_1 \setminus \Omega_0}|\nabla{v_0}|^2 \ dx. \end{aligned}$$ ◻ ## Sharp triangle inequality of the Dissipation distance Here we give a sharp triangle inequality for the Dissipation distance. The triangle inequality is important for most of the standard theory of rate independent energetic evolutions, although we remark that it does not seem to generalize well to the general anisotropic case. The sharp triangle inequality is used in Proposition [Proposition 58](#p.lr-limits){reference-type="ref" reference="p.lr-limits"} to show that if an energetic solution jumps multiple times then $u(t)$ must always be in between $u_\ell(t)$ and $u_r(t)$. **Lemma 77** (Dissipation distance sharp triangle inequality). *Let $A$, $B$, and $C$ be arbitrary finite measure regions in $\mathbb{R}^n$. Then $$\textup{Diss}(A,B) + \textup{Diss}(B,C) - \textup{Diss}(A,C) = (\mu_-+\mu_+)\bigg[|B \setminus (A\cup C)| + |(A \cap C) \setminus B|\bigg]$$* *Proof.* The result is purely set algebraic computations, see Figure [\[f.set-theory\]](#f.set-theory){reference-type="ref" reference="f.set-theory"} for the geometric idea. We compute $$\begin{aligned} &\textup{Diss}(A,B)+\textup{Diss}(B,C) - \textup{Diss}(A,C) \notag\\ &= \int \mu_+{\bf 1}_{B \setminus A} + \mu_-{\bf 1}_{A \setminus B}+\mu_+{\bf 1}_{C \setminus B}+\mu_-{\bf 1}_{B \setminus C} - \mu_+{\bf 1}_{C \setminus A} - \mu_-{\bf 1}_{C \setminus A} \ dx\notag\\ &= \int \mu_+[{\bf 1}_{B \setminus A} +{\bf 1}_{C \setminus B}- {\bf 1}_{C \setminus A}]+ \mu_-[{\bf 1}_{A \setminus B}+{\bf 1}_{B \setminus C} - {\bf 1}_{A \setminus C}] \ dx. \label{e.expanded-triangle-id}\end{aligned}$$ Now we just compute by force the first term in brackets on the previous line $$\begin{aligned} {\bf 1}_{B \setminus A} +{\bf 1}_{C \setminus B}- {\bf 1}_{C \setminus A} & = {\bf 1}_{B} - {\bf 1}_{A \cap B}+{\bf 1}_{C}- {\bf 1}_{ B \cap C}- [{\bf 1}_{C} - {\bf 1}_{ A \cap C}]\\ & ={\bf 1}_{B}+{\bf 1}_{ A \cap C} - {\bf 1}_{ B \cap C} - {\bf 1}_{ B \cap A}\\ & = {\bf 1}_{B}+{\bf 1}_{ A \cap C}{\bf 1}_{B} - {\bf 1}_{ B \cap C} - {\bf 1}_{ B \cap A} +{\bf 1}_{ A \cap C}(1-{\bf 1}_{B}) \\ & = {\bf 1}_{B}(1+{\bf 1}_{ A \cap C} - {\bf 1}_{ C} - {\bf 1}_{ A}) + {\bf 1}_{ (A \cap C) \setminus B}\\ & = {\bf 1}_{B}(1-{\bf 1}_{A \cup C})+ {\bf 1}_{ (A \cap C) \setminus B}\\ & = {\bf 1}_{B\setminus (A \cup C)}+ {\bf 1}_{ (A \cap C) \setminus B}\end{aligned}$$ the second bracketed term in ([\[e.expanded-triangle-id\]](#e.expanded-triangle-id){reference-type="ref" reference="e.expanded-triangle-id"}) works out quite similarly and is identical after the second step above giving the same result. Plugging these identities back into ([\[e.expanded-triangle-id\]](#e.expanded-triangle-id){reference-type="ref" reference="e.expanded-triangle-id"}) gives the result. ◻ ## Proof of Lemma [Lemma 22](#l.vs-two-notions-equiv){reference-type="ref" reference="l.vs-two-notions-equiv"} {#s.app-vs-equivalence} We just do the subsolution equivalence, the supersolution one is similar. Suppose $u$ is a subsolution in the superdifferential sense Definition [Definition 20](#d.subdifferentials-vs-def){reference-type="ref" reference="d.subdifferentials-vs-def"}. Suppose $\varphi$ is a smooth test function with $\varphi_+$ touching $u$ from above at $x_0 \in \partial \Omega(u) \cap U$. Then $$u(x) \leq [\nabla\varphi(x_0) \cdot (x - x_0)]_+ + o(|x-x_0|) \ \hbox{ so } \ \nabla\varphi(x_0) \in D_+u(x_0)$$ and so $$|\nabla\varphi(x_0)|^2 \geq {Q}.$$ Next suppose that $u$ is a subsolution in the superdifferential sense Definition [Definition 21](#d.test-fcns-vs-def){reference-type="ref" reference="d.test-fcns-vs-def"}, and $p \in D_+u(x_0)$ for some $x_0 \in \partial \Omega(u) \cap U$. Without loss assume that $p = \alpha e_d$. Let $\delta,r>0$ and define $$\varphi(x) : = (1+\delta) \alpha (x-x_0) \cdot e_d + c(d)\delta r^{-1}(|(x-x_0)'|^2 - d x_d^2).$$ This function is a superharmonic polynomial and using the superdifferential condition $\alpha e_d \in D_+u(x_0)$ one can show that for all $\delta>0$ there is $r>0$ so that $u < \varphi$ on $\partial C \cap \overline{\{u>0\}}$ where $C$ is a small open cylindrical neighborhood of $x_0$ with radius $\approx r_0$, see the proof of Lemma [Lemma 71](#l.barrier-construction){reference-type="ref" reference="l.barrier-construction"} for a similar computation. Then, since $\varphi(x_0) = u(x_0)$ we can shift $\varphi$ vertically $\varphi(x) + s$ until $(\varphi(x)+s)_+$ touches $u$ from above at some $x_1 \in C$. Since $\varphi$ is strictly superharmonic the touching must be on $x_1 \in \partial \Omega(u) \cap C$. Thus finally the viscosity solution condition implies that $${Q} \leq |\nabla\varphi(x_1)|^2 \leq \alpha + o_\delta(1).$$ Sending $\delta \to 0$ finishes the argument. 0◻ ## Uniqueness of the Bernoulli problem solution in the complement of a star-shaped set In the star-shaped case we have the following uniqueness result for a Bernoulli problem with an obstacle. The result without obstacle for classical solutions in dimension two was proved in [@tepper1975] based on an existence theorem by Beurling [@Beurling] (more easily found reprinted here [@Beurling2]). Here we give a proof for viscosity solutions of the obstacle problem for convenience. **Theorem 78**. *Suppose that $U$ is a domain such that that $K = \mathbb{R}^d \setminus U$ is strongly star-shaped with respect to a neighborhood of the origin, and that $O$ is a strongly star-shaped set with respect to the same neighborhood. If $u, v \in C(\overline U)$ have compact support and for some constants $F, q > 0$ are viscosity solutions of a Bernoulli problem with obstacle $O$ from below $$\begin{aligned} \left\{ \begin{aligned} \Delta u &= 0 & & \text{in } \{u > 0\} \cap U,\\ u &> 0 && \text{in } O,\\ |\nabla u| &= q & & \text{on } (\partial \{u > 0\} \setminus \overline O) \cap U,\\ |\nabla u| &\leq q & & \text{on } \partial \{u > 0\} \cap U,\\ u &= F && \text{on } \partial U \end{aligned} \right.\end{aligned}$$ then $u = v$.* *The same result applies for a Bernoulli problem with obstacle $O$ from above.* *Proof.* By a simple rescaling it is sufficient to consider $F = q = 1$. We show that $\Omega(u) = \Omega(v)$ which is sufficient by the uniqueness of the Laplace equation. We first check that $\Omega(u) \subset \Omega(v)$. Suppose that $\Omega(u) \setminus \Omega(v) \neq \emptyset$. Consider the largest $\lambda < 1$ such that $$\begin{aligned} v^\lambda(x) := v(\lambda x)\end{aligned}$$ satisfies $$\begin{aligned} \Omega(u) \subset \Omega(v^\lambda) = \lambda^{-1} \Omega(v).\end{aligned}$$ Such $\lambda$ exists since both $\Omega(u)$ and $\Omega(v)$ are open, $\Omega(u)$ is bounded and $\Omega(v)$ contains a neighborhood of the origin. We have clearly $\lambda < 1$, and by the definition of $\lambda$ there exists $x_0 \in \partial \Omega(u) \cap \partial \Omega(v^\lambda)$. Also note that $x_0 \notin \overline O$ since $O$ is strongly star-shaped and $O \cap \partial \Omega(v) = \emptyset$. If $u$ and $v$ are sufficiently smooth, we have $$\begin{aligned} |\nabla v^\lambda|(x_0) = \lambda|\nabla v|(\lambda x_0) \leq \lambda < 1 = |\nabla u|(x_0),\end{aligned}$$ a contradiction with the fact that $u \leq v^\lambda$ by the maximum principle for the harmonic function $v^\lambda - u$ in $\Omega(v) \cap \lambda^{-1}U$ since $v^\lambda - u > 0$ on $\lambda^{-1} \partial U \subset U$ by the strong star-shapedness. To make the argument work without assuming regularity, we consider the sup and inf convolutions $$\begin{aligned} v_r(x) = \inf_{\overline B_r(x)} v, \qquad u_r(x) = \sup_{\overline B_r(x)} u\end{aligned}$$ defined on $\overline U_r$ where $U_r:= \{x: \overline B_r(x) \subset U\}$. It is easy to see that $v_r$ is superharmonic on $U_r \cap \Omega(v_r)$, $u_r$ is subharmonic on $U_r \cap \Omega(u_r)$, and $\inf_{\partial U_r} v_r \to 1$ as $r \to 0$ by continuity. Let us call the $\lambda < 1$ above $\lambda_0$. For each $r > 0$ there exists a largest $\lambda_r < 1$ so that $\Omega(u_r) \subset \lambda_r^{-1}\Omega(v_r)$. Furthermore $\lambda_r \to \lambda_0 < 1$ as $r \to 0$. Therefore we can choose $r > 0$ sufficiently small so that $\lambda_r < 1$, $\lambda_r^{-1/2} \inf_{\partial U_r} v_r > 1$ and $\operatorname{dist}(O, \lambda_r^{-1} \partial\Omega(v_r)) > r$ (by strong star-shapedness of $O$). We fix such $r > 0$ and choose $x_0 \in \partial \Omega(u_r) \cap \lambda_r^{-1}\partial\Omega(v_r)$. Let us set $w(x) := \lambda_r^{-1/2}v_r(\lambda_r x)$. We have that both $u_r$ and $w$ are defined on $\lambda_r^{-1}\overline U_r$. We also note that $w - u_r$ is superharmonic on $\lambda_r^{-1} U_r \cap \{u_r > 0\}$ with $w - u_r > 0$ on $\lambda_r^{-1}\partial U_r$. Therefore the slopes of $u_r$ and $w$ in the normal direction at $x_0$ are ordered. We have the standard situation of boundaries of $u_r$ and $w$ touching at a point that has both interior and exterior touching balls centered at the free boundaries of $u$ and $v(\lambda_r x)$ respectively. By a standard barrier construction at these centers, we arrive at a contradiction with the fact that $u$ satisfies $|\nabla u| \geq 1$ (since the center is not in $\overline O$ by the choice of $r$) and $v^{\lambda_r} := v(\lambda_r x)$ satisfies $\lambda_r^{-1/2}|\nabla v^{\lambda_r}| \leq \lambda_r^{1/2} 1 < 1$. Therefore we conclude that $\Omega(u) \subset \Omega(v)$. The inclusion $\Omega(v) \subset \Omega(u)$ can be shown by the same argument, swapping the roles of $u$ and $v$. By the uniqueness of the Laplace equation we have $u = v$. The proof for the problem with obstacle from above follows an analogous argument. ◻

arxiv_math

--- address: Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, Chernyshevsky street, 112, Ufa 450008 Russia. author: - Oskar A. Sultanov title: Nonlinear resonance in oscillatory systems with decaying perturbations --- > Time-decaying perturbations of nonlinear oscillatory systems in the plane are considered. It is assumed that the unperturbed systems are non-isochronous and the perturbations oscillate with an asymptotically constant frequency. Resonance effects and long-term asymptotic regimes for solutions are investigated. In particular, the emergence of stable states close to periodic ones is discussed. By combining the averaging technique and stability analysis, the conditions on perturbations are described that guarantee the existence and stability of the phase-locking regime with a resonant amplitude. The results obtained are applied to the perturbed Duffing oscillator. # Introduction {#introduction .unnumbered} Perturbation of nonlinear oscillatory systems is a classical problem with a wide range of applications [@GH83; @AF06]. In this paper, time-decaying perturbations are considered and a class of asymptotically autonomous systems in the plane is investigated. Note that asymptotically autonomous systems arise, for example, when studying steady-state modes in multidimensional problems by reducing the dimension [@CCT95; @DS22], when constructing the asymptotics of strongly nonlinear non-autonomous systems by isolating growing terms of solutions [@BG08; @KF13], and in various problems with time-dependent damping [@SFR19; @JM23]. Qualitative properties of asymptotically autonomous systems have been studied in many papers [@LM56; @LRS02; @KS05; @MR08; @OS22Non]. In particular, it follows from [@LDP74] that under certain conditions, time-decaying perturbations may not disturb the global dynamics of oscillatory systems. However, in the general case, the dynamics of perturbed and unperturbed systems can differ significantly [@HT94; @OS21IJBC]. This paper studies the effect of damped oscillatory perturbations with an asymptotically constant frequency on non-isochronous systems. Note that similar problems has been considered in several papers. In particular, the asymptotic analysis of linear systems with damped oscillatory perturbations was discussed in [@HL75; @MP85; @PN07; @BN10]. The asymptotic behaviour of solutions to nonlinear equations in the vicinity of the equilibrium was investigated in [@OS23JMS]. Bifurcations in such systems related to the stability of the equilibrium were discussed in [@OS21DCDS; @OS21JMS]. To the best of the author's knowledge, the influence of such perturbations on the behaviour of nonlinear systems far from equilibrium has not yet been discussed. This is the subject of the present paper. In particular, we study the emergence of near-periodic stable states due to resonance phenomena with damped oscillatory perturbations. Note also that similar effects in the problems with a small parameter are usually associated with nonlinear resonance and are considered to be well studied [@BVC79; @SUZ88; @Sos97; @JCRS11]. However, in this paper, the presence of a small parameter is not assumed. We discuss the role of time-decaying perturbations in the emergence and stability of long-term asymptotics regimes. The paper is organized as follows. Section [1](#sec1){reference-type="ref" reference="sec1"} provides the statement of the problem and a motivating example. The main results are presented in Section [2](#sec2){reference-type="ref" reference="sec2"}. The justification is contained in subsequent sections. First, in Section [3](#sec3){reference-type="ref" reference="sec3"}, we construct a near-identity transformation averaging the system in the first asymptotic terms. Section [4](#sec4){reference-type="ref" reference="sec4"} analyses the truncated system obtained from the full system by dropping the remainder terms and describes possible asymptotic regimes. Section [5](#sec5){reference-type="ref" reference="sec5"} discusses the persistence of these regimes in the full system by constructing Lyapunov functions. In Section [6](#sEx){reference-type="ref" reference="sEx"}, the proposed theory is applied to examples of asymptotically autonomous systems. The paper concludes with a brief discussion of the results obtained. # Problem statement {#sec1} Consider a system of two differential equations $$\begin{gathered} \label{PS} \frac{dr}{dt}=f(r,\varphi,S(t),t), \quad \frac{d\varphi}{dt}=\omega(r)+g(r,\varphi,S(t),t),\end{gathered}$$ where the functions $\omega(r)>0$, $f(r,\varphi,S,t)$ and $g(r,\varphi,S,t)$ are infinitely differentiable, defined for all $|r|\leq \mathcal R={\hbox{\rm const}}$, $(\varphi,S)\in\mathbb R^2$, $t>0$, and are $2\pi$-periodic with respect to $\varphi$ and $S$, $\omega'(r)\not\equiv 0$. The functions $f(r,\varphi,S,t)$ and $g(r,\varphi,S,t)$ play the role of perturbations of the autonomous system $$\begin{gathered} \label{US} \frac{d\hat r}{dt}=0, \quad \frac{d\hat \varphi}{dt}=\omega(\hat r),\end{gathered}$$ describing non-isochronous oscillations on the plane $(x,y)=(\hat r\cos\hat \varphi,-\hat r\sin\hat \varphi)$ with a constant amplitude $\hat r(t)\equiv r_0$, $|r_0|< \mathcal R$ and a natural frequency $\omega(r_0)$. The solutions $r(t)$ and $\varphi(t)$ of system [\[PS\]](#PS){reference-type="eqref" reference="PS"} corresponds to the amplitude and the phase of the perturbed oscillations. It is assumed that the frequency of perturbations is asymptotically constant: $S'(t)\sim s_0$ as $t\to\infty$ with $s_0={\hbox{\rm const}}>0$, and the intensity decays with time: for each fixed $r$ and $\varphi$ $$\begin{gathered} f(r,\varphi,S(t),t)\to 0, \quad g(r,\varphi,S(t),t)\to 0, \quad t\to\infty.\end{gathered}$$ In this case, the perturbed system [\[PS\]](#PS){reference-type="eqref" reference="PS"} is asymptotically autonomous with the limiting system [\[US\]](#US){reference-type="eqref" reference="US"}. The goal of the paper is to study the resonant effects of perturbations $f(r,\varphi,S(t),t)$ and $g(r,\varphi,S(t),t)$ on the dynamics far from the equilibrium of the limiting system and to describe possible asymptotic regimes for solutions. Let us specify the considered class of perturbations. We assume that $$\begin{gathered} \label{fgas}\begin{split} &f(r,\varphi,S,t)\sim \sum_{j=1}^\infty t^{-\frac{j}{q}} f_j(r,\varphi,S), \\ &g(r,\varphi,S,t)\sim \sum_{j=1}^\infty t^{-\frac{j}{q}} g_j(r,\varphi,S), \quad t\to\infty, \end{split}\end{gathered}$$ for all $|r|< \mathcal R$ and $(\varphi,S)\in\mathbb R^2$, where $f_j(r,\varphi,S)$ and $g_j(r,\varphi,S)$ are $2\pi$-periodic with respect to $\varphi$ and $S$, and $q\in\mathbb Z_+$. The phase of perturbations is considered in the form $$\begin{gathered} \label{Sform} S(t)=s_0 t + \sum_{j=1}^{q-1} s_j t^{1-\frac{j}{q}}+s_q \log t,\end{gathered}$$ where $s_j={\hbox{\rm const}}$. Moreover, it is assumed that there exist $0<|a|< \mathcal R$ and coprime integers $\kappa,\varkappa\in\mathbb Z_+$ such that the resonant condition holds: $$\begin{gathered} \label{rc} \kappa s_0=\varkappa\omega(a), \quad \eta:=\omega'(a)\neq 0.\end{gathered}$$ Note that the series in [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"} are asymptotic as $t\to\infty$, and for all $N\geq 1$ the following estimates hold: $f(r,\varphi,S,t)-\sum_{j=0}^{N-1}t^{-j/q} f_j(r,\varphi,S)=\mathcal O(t^{-N/q})$ and $g(r,\varphi,S,t)-\sum_{j=0}^{N-1}t^{-j/q} g_j(r,\varphi,S)=\mathcal O(t^{-N/q})$ as $t\to\infty$ uniformly for all $|r|\leq \mathcal R$ and $(\varphi,S)\in\mathbb R^2$. Note also that instead of power functions one could consider another asymptotic scale, but in this case the calculations would be more complex and cumbersome. Consider the example $$\begin{gathered} \label{Ex0} \frac{dx}{dt}=y, \quad \frac{dy}{dt}=-x+\vartheta x^3+ t^{-\frac{1}{2}} Z(x,y,S(t)),\end{gathered}$$ where $Z(x,y,S)\equiv \alpha(S)x+\beta(S)y$, $\alpha(S)\equiv \alpha_0+\alpha_1 \sin S$, $\beta(S)\equiv \beta_0+ \beta_1\sin S$, $S(t)\equiv 3 t/2$ with parameters $\alpha_i,\beta_i,\in\mathbb R$ and $\vartheta>0$. Let us show that this system corresponds to [\[PS\]](#PS){reference-type="eqref" reference="PS"}. The limiting system $d\hat x/dt=\hat y$, $d\hat y/dt=- U'(\hat x)$ with $U(x)\equiv x^2/2-\vartheta x^4/4$ has a stable equilibrium at $(0,0)$, and the level lines $\{(x,y)\in\mathbb R: U(x)+y^2/2=r^2/2\}$ for all $0<|r|< (2\vartheta)^{-1/2}$ correspond to $T(r)$-periodic solutions $$\begin{gathered} \nonumber \hat x_0(t,r)\equiv r\,{\hbox{\rm sn}}\left(\frac{t}{\sqrt{k_r^2+1}},k_r\right)\sqrt{k_r^2+1}, \quad \hat y_0(t,r)\equiv r\,{\hbox{\rm cn}}\left(\frac{t}{\sqrt{k_r^2+1}},k_r\right){\hbox{\rm dn}}\left(\frac{t}{\sqrt{k_r^2+1}},k_r\right), \\ \label{omegaeq} T(r)\equiv 4K(k_r)\sqrt{k_r^2+1}, \quad \omega(r)\equiv \frac{2\pi}{T(r)},\end{gathered}$$ where ${\hbox{\rm sn}}(u,k)$, ${\hbox{\rm cn}}(u,k)$, ${\hbox{\rm dn}}(u,k)$ are the Jacoby elliptic functions, $K(k)$ is the complete elliptic integral of the first kind, and $k_r\in (0,1)$ is a root of $(k_r+k_r^{-1})^{-2}=\vartheta r^2/2$. Define auxiliary $2\pi$-periodic functions $$\begin{gathered} X(\varphi,r)\equiv \hat x_0 \left(\frac{\varphi}{\omega(r)},r\right), \quad Y(\varphi,r)\equiv \hat y_0 \left(\frac{\varphi}{\omega(r)},r\right).\end{gathered}$$ It can easily be checked that $\omega(r)\partial_\varphi X=Y$, $\omega(r)\partial_\varphi Y=-U(X)$, $U(X)+Y^2/2=r^2/2$ and $$\begin{gathered} {\hbox{\rm det}}\frac{\partial (X,Y)}{\partial (\varphi,r)}\equiv \begin{vmatrix} \partial_\varphi X & \partial_\varphi Y\\ \partial_r X & \partial_r Y \end{vmatrix} \equiv \frac{r}{\omega(r)}.\end{gathered}$$ Thus, system [\[Ex0\]](#Ex0){reference-type="eqref" reference="Ex0"} in the variables $(r,\varphi)$ takes the form [\[PS\]](#PS){reference-type="eqref" reference="PS"} with $q=2$, $s_0=3/2$, $s_i=0$, $$\begin{gathered} \label{fgex0} f(r,\varphi,S,t)\equiv t^{-\frac{1}{2}}f_1(r,\varphi,S), \quad g(r,\varphi,S,t)\equiv t^{-\frac{1}{2}}g_1(r,\varphi,S),\end{gathered}$$ where $$\begin{aligned} f_1(r,\varphi,S)&\equiv r^{-1}Y(\varphi,r)Z(X(\varphi,r),Y(\varphi,r),S), \\ g_1(r,\varphi,S)&\equiv - r^{-1}\omega(r)\partial_r X(\varphi,r)Z(X(\varphi,r),Y(\varphi,r),S).\end{aligned}$$ Note that $0<\omega(r)<1$ for all $0<|r|<(2\vartheta)^{-1/2}$. Hence, there exist $\kappa$, $\varkappa\in\mathbb Z_+$ and $0<|r|<(2\vartheta)^{-1/2}$ such that the condition [\[rc\]](#rc){reference-type="eqref" reference="rc"} holds. If $Z(x,y,S)\equiv0$, then $r(t)\equiv r_0$ and $\varphi(t)\equiv \omega(r_0)t+\phi_0$ with arbitrary constants $r_0$ and $\phi_0$. In the absence of the oscillatory part of the perturbation ($\alpha_1=\beta_1=0$), the amplitude of the solutions may tend to zero or to infinity, depending on the sign of $\beta_0$ (see Fig. [\[FigEx0\]](#FigEx0){reference-type="ref" reference="FigEx0"}, a). Under some conditions on the parameters, this qualitative behaviour can be preserved in the system with the oscillating perturbations (see Fig. [\[FigEx0\]](#FigEx0){reference-type="ref" reference="FigEx0"}, b), or violated with the appearance of new attracting states (see Fig. [\[FigEx0\]](#FigEx0){reference-type="ref" reference="FigEx0"}, c). The paper discusses the conditions that guarantee the existence and stability of such states in perturbed systems of the form [\[PS\]](#PS){reference-type="eqref" reference="PS"} with perturbations satisfying [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"} and [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"}. # Main results {#sec2} Define the domain $$\begin{aligned} &\mathcal D^\epsilon_{\varsigma,\tau}:=\{(R,\Psi,t)\in\mathbb R^3: |R t^{-\frac{1-\varsigma}{2q}}+a|< \mathcal R - \epsilon t^{-\frac{1-\varsigma}{2q}}, \ \ t\geq \tau \}\end{aligned}$$ with some $\varsigma\in (0,1)$, $\epsilon\geq 0$ and $\tau>0$. Let the angle brackets denote averaging of any function $F(S)$ over $S$ for the period $2\pi\varkappa$, $$\begin{gathered} \langle F(S)\rangle_{\varkappa S}\equiv \frac{1}{2\pi\varkappa}\int\limits_0^{2\pi\varkappa} F(S)\,dS.\end{gathered}$$ Then, we have the following: **Theorem 1**. *Let system [\[PS\]](#PS){reference-type="eqref" reference="PS"} satisfy [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"}, [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"} and [\[rc\]](#rc){reference-type="eqref" reference="rc"}. Then for all $N\in\mathbb Z_+$, $\varsigma\in (0,1)$ and $\epsilon\in(0,\mathcal R)$ there exist $t_0\geq 1$ and the transformations $(r,\varphi)\mapsto (R,\Psi)\mapsto (\rho,\psi)$, $$\begin{gathered} \label{ch1} r(t)=a+t^{-\frac{1}{2q}} R(t), \quad \varphi(t)=\frac{\kappa}{\varkappa}S(t)+\Psi(t), \\ \label{ch2} \rho(t)=R(t)+\tilde \rho_N(R(t),\Psi(t),t), \quad \psi(t)=\Psi(t)+\tilde \psi_N(R(t),\Psi(t),t),\end{gathered}$$ where $\tilde \rho_N(R,\Psi,t)$, $\tilde \psi_N(R,\Psi,t)$ are $2\pi$-periodic in $\Psi$ and satisfy the inequalities $$\begin{gathered} |\tilde \rho_N(R,\Psi,t)|\leq \epsilon, \quad |\tilde \psi_N(R,\Psi,t)|\leq \epsilon, \quad (R,\Psi,t)\in\mathcal D^0_{\varsigma,t_0},\end{gathered}$$ such that for all $0<|r|< \mathcal R$ and $\varphi\in\mathbb R$ system [\[PS\]](#PS){reference-type="eqref" reference="PS"} can be transformed into $$\begin{gathered} \label{rhopsi} \frac{d\rho}{dt}=\Lambda_{N}(\rho,\psi,S(t),t), \quad \frac{d\psi}{dt}=\Omega_N(\rho,\psi,S(t),t),\end{gathered}$$ with $\Lambda_{N}(\rho,\psi,S,t)\equiv \hat\Lambda_N(\rho,\psi,t)+ \tilde\Lambda_N(\rho,\psi,S,t)$, $\Omega_{N}(\rho,\psi,S,t)\equiv \hat\Omega_N(\rho,\psi,t)+\tilde\Omega_N(\rho,\psi,S,t)$, defined for all $(\rho,\psi,t)\in\mathcal D^\epsilon_{\varsigma,t_0}$ and $S\in\mathbb R$, such that $$\begin{gathered} \label{tildeLON} \hat\Lambda_N(\rho,\psi,t)\equiv \sum_{k=1}^N t^{-\frac{k}{2q}}\Lambda_k(\rho,\psi), \quad \hat\Omega_N(\rho,\psi,t)\equiv \sum_{k=1}^N t^{-\frac{k}{2q}}\Omega_k(\rho,\psi),\\ \label{tildeLO}\tilde\Lambda_N(\rho,\psi,S,t)=\mathcal O(t^{-\frac{N+1}{2q}}), \quad \tilde\Omega_N(\rho,\psi,S,t)=\mathcal O(t^{-\frac{N+1}{2q}})\end{gathered}$$ as $t\to\infty$ uniformly for all $|\rho|<\infty$ and $(\psi,S)\in\mathbb R^2$, where $\Lambda_k(\rho,\psi)$ and $\Omega_k(\rho,\psi)$ are polynomials in $\rho$ of degree $k-1$ and $k$, respectively. In particular, $\Lambda_1(\rho,\psi)\equiv \langle f_1(a,\kappa S/\varkappa+\psi,S)\rangle_{\varkappa S}$ and $\Omega_1(\rho,\psi)\equiv \eta\rho$. Moreover, $$\begin{gathered} \label{LOas} \begin{split} &\tilde \Lambda_N(\rho,\psi,S,t)\sim \sum_{k=N+1}^\infty t^{-\frac{k}{2q}}\tilde \Lambda_{N,k}(\rho,\psi,S), \\ &\tilde \Omega_N(\rho,\psi,S,t) \sim \sum_{k=N+1}^\infty t^{-\frac{k}{2q}} \tilde \Omega_{N,k}(\rho,\psi,S), \quad t\to\infty, \end{split}\end{gathered}$$ where $\tilde \Lambda_{N,k}(\rho,\psi,S)=\mathcal O(\rho^{k-1})$ and $\tilde \Omega_{N,k}(\rho,\psi,S)=\mathcal O(\rho^{k})$ as $\rho\to\infty$ uniformly for all $(\psi,S)\in\mathbb R^2$.* The proof is contained in Section [3](#sec3){reference-type="ref" reference="sec3"}. Note that Theorem [Theorem 1](#Th1){reference-type="ref" reference="Th1"} describes an averaging transformation that simplifies the system in the leading asymptotic terms as $t\to\infty$. Moreover, after this procedure, some terms in sums [\[tildeLON\]](#tildeLON){reference-type="eqref" reference="tildeLON"} may disappear because they have the zero mean. Let $n\in [1,2q]$ and $m\in[2,2q]$ be integers such that $$\begin{gathered} \label{asn} \begin{split} &\Lambda_i(\rho,\psi)\equiv 0, \quad 1\leq i<n, \quad \Lambda_n(\rho,\psi)\not\equiv 0,\\ &\Omega_j(\rho,\psi)\equiv 0, \quad 1<j< m, \quad \Omega_m(\rho,\psi)\not\equiv 0. \end{split}\end{gathered}$$ The proposed method is based on the study of the truncated system $$\begin{gathered} \label{trsys} \frac{d\varrho}{dt}=\hat\Lambda_N(\varrho,\phi,t), \quad \frac{d\phi}{dt}=\hat\Omega_N(\varrho,\phi,t)\end{gathered}$$ obtained from [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} by dropping the remainder terms $\tilde \Lambda_N$ and $\tilde \Omega_N$. System [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} can be considered as a model system that describes the average dynamics for the amplitude residual and phase shift. First, we discuss the solutions of system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"}. Next, we show that the trajectories of the full system [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} behave like the solutions of the truncated system. The behaviour of solutions to asymptotically autonomous system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} depends on the properties of the corresponding limiting system $$\begin{gathered} \label{limsys} t^{\frac{n}{2q}} \frac{d\hat\varrho}{dt}=\Lambda_n(\hat\varrho,\hat\phi), \quad t^{\frac{1}{2q}} \frac{d\hat\phi}{dt}=\eta\hat\varrho.\end{gathered}$$ In particular, the presence and the stability of fixed points in system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} play a crucial role. With this in mind, we consider the following assumption: $$\begin{gathered} \label{aszero} \exists \, \psi_0\in\mathbb R: \quad \Lambda_n(0,\psi_0)=0, \quad \nu_n:=\partial_\psi \Lambda_n(0,\psi_0)\neq 0,\end{gathered}$$ and define the parameter $\lambda_n:=\partial_\rho \Lambda_n(0,\psi_0)$. In this case, system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} has an equilibrium $(0,\psi_0)$, and we have **Lemma 1**. *Let assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"} and [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} hold.* - *If $\nu_n \eta>0$ or $\nu_n \eta<0$, $\lambda_n>0$, $n\leq q$, then the equilibrium $(0,\psi_0)$ of system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} is unstable.* - *If $\nu_n \eta<0$ and $\lambda_n<0$, then the equilibrium $(0,\psi_0)$ of system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} is asymptotically stable.* Note that if $\nu_n \eta>0$, the equilibrium $(0,\psi_0)$ is of saddle type. In this case, similar dynamics occurs in the full system. However, if $\nu_n \eta<0$, the fixed point can be either stable or unstable, depending on the sign of the divergence of the vector field calculating at the equilibrium. Let us show that under a similar condition there exists a solution of system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} tending to the point $(0,\psi_0)$ as $t\to\infty$. Define $$\begin{gathered} d_{n,m}:=\begin{cases} \lambda_n, & n<m,\\ \lambda_n+\omega_m, & n=m,\\ \omega_m, & n>m, \end{cases}\end{gathered}$$ where $\omega_m:=\partial_\varphi \Omega_m(0,\psi_0)$. Then, we have the following: **Lemma 2**. *Let assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"} and [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} hold with $\nu_n\eta<0$ and $d_{n,m}<0$. Then for all $N\geq \max\{m,n\}$ system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} has a particular solution $\varrho_\ast(t)$, $\phi_\ast(t)$ with asymptotic expansion in the form $$\begin{gathered} \label{assol} \varrho_\ast(t)\sim \sum_{k=1}^\infty t^{-\frac{k+m-2}{2q}} \varrho_k, \quad \phi_\ast(t)\sim \psi_0+\sum_{k=1}^\infty t^{-\frac{k}{2q}}\phi_k, \quad t\to\infty,\end{gathered}$$ where $\varrho_k$, $\phi_k$ are some constants. Moreover, the solution $\varrho_\ast(t)$, $\phi_\ast(t)$ is asymptotically stable.* It can be shown that the dynamics described by the solution $\varrho_\ast(t)$, $\phi_\ast(t)$ of the truncated system is preserved in system [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"}. We have **Theorem 2**. *Let system [\[PS\]](#PS){reference-type="eqref" reference="PS"} satisfy [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"}, [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"}, [\[rc\]](#rc){reference-type="eqref" reference="rc"}, and assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"} and [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} hold with $\nu_n\eta<0$ and $d_{n,m}<0$. Then, there is $N_0\in\mathbb Z_+$ such that for all $N\geq N_0$ and $\varepsilon>0$ there exist $\delta>0$ and $t_\ast\geq t_0$ such that any solution $r(t)$, $\varphi(t)$ of system [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} with initial data $r(t_\ast)=r_\ast$, $\varphi(t_\ast)=\varphi_\ast$, $|r_\ast-a-t_\ast^{-1/(2q)}\varrho_\ast(t_\ast) |+ |\varphi_\ast-\kappa S(t_\ast)/\varkappa-\phi_\ast(t_\ast)|\leq \delta$, satisfies the inequality: $$\begin{gathered} \label{rpst} \left|r(t)-a-t^{-\frac{1}{2q}}\varrho_\ast(t)\right|+\left|\varphi(t)-\frac{\kappa}{\varkappa}S(t)-\phi_\ast(t)\right|<\varepsilon\end{gathered}$$ for all $t>t_\ast$.* Note that in the opposite case, when $\nu_n\eta<0$ and $d_{n,m}>0$, the asymptotic regime described in Lemma [Lemma 2](#Lem1){reference-type="ref" reference="Lem1"} turns out to be unstable. Let $\varrho_{\ast,M}(t)$, $\phi_{\ast,M}(t)$ be $M$th partial sums of the series [\[assol\]](#assol){reference-type="eqref" reference="assol"} and $\ell=\min\{m,n\}$. Then, we have **Theorem 3**. *Let system [\[PS\]](#PS){reference-type="eqref" reference="PS"} satisfy [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"}, [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"}, [\[rc\]](#rc){reference-type="eqref" reference="rc"}, and assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"} and [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} hold with $\nu_n\eta<0$, $d_{n,m}>0$, and $\ell+n-1< 2 q$. Then, there is $N_0\in\mathbb Z_+$, $\varepsilon > 0$ such that for all $\delta\in (0,\varepsilon)$, $N\geq N_0$, $M\in \mathbb Z_+$ the solution $r(t)$, $\varphi(t)$ of system [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} with initial data $r(t_\ast)=r_\ast$, $\varphi(t_\ast)=\varphi_\ast$, $|r_\ast-a-t_\ast^{-1/(2q)}\varrho_\ast(t_\ast) |+ |\varphi_\ast-\kappa S(t_\ast)/\varkappa-\phi_\ast(t_\ast)|\leq \delta$ at some $t_\ast\geq t_0$, satisfies the inequality: $$\begin{gathered} \label{rpunst} \left|r(t_e)-a-t_e^{-\frac{1}{2q}}\varrho_\ast(t_e)\right|+\left|\varphi(t_e)-\frac{\kappa}{\varkappa}S(t_e)-\phi_\ast(t_e)\right|\geq \varepsilon\end{gathered}$$ at some $t_e>t_\ast$.* Let us remark that if $d_{n,m}=0$, the existence and stability of the phase locking regime are not guaranteed by Theorem [Theorem 2](#Th2){reference-type="ref" reference="Th2"}. For this case, consider the following assumption: $$\begin{gathered} \label{asst} \begin{split} &\exists\, h\in (\ell,2q]: \quad \partial_\rho \Lambda_k(\rho,\psi)+\partial_\psi \Omega_k(\rho,\psi)\equiv 0, \quad k\leq h-1, \\ & d_h:= \partial_\rho \Lambda_h(0,\psi_0)+\partial_\psi \Omega_h(0,\psi_0)\neq 0. \end{split}\end{gathered}$$ Then, we have **Lemma 3**. *Let assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"}, [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} and [\[asst\]](#asst){reference-type="eqref" reference="asst"} hold with $\nu_n\eta<0$ and $d_h<0$. Then for all $N\geq \max\{m,n,h\}$ system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} has a stable particular solution $\varrho_\ast(t)$, $\phi_\ast(t)$ with asymptotic expansion [\[assol\]](#assol){reference-type="eqref" reference="assol"}. Moreover, the solution $\varrho_\ast(t)$, $\phi_\ast(t)$ is asymptotically stable if $h+n-1<2q$.* As in the previous case, the phase locking regime in system [\[PS\]](#PS){reference-type="eqref" reference="PS"} associated with the solution $\varrho_\ast(t)$, $\phi_\ast(t)$ of the model system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} turns out to be stable if $d_h<0$ and unstable if $d_h>0$. We have the following: **Theorem 4**. *Let system [\[PS\]](#PS){reference-type="eqref" reference="PS"} satisfy [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"}, [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"}, [\[rc\]](#rc){reference-type="eqref" reference="rc"}, and assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"}, [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} and [\[asst\]](#asst){reference-type="eqref" reference="asst"} hold with $\nu_n\eta<0$ and $d_{h}<0$. Then there is $N_0\in\mathbb Z_+$ such that for all $N\geq N_0$ and $\varepsilon>0$ there exist $\delta>0$ and $t_\ast\geq t_0$ such that any solution $r(t)$, $\varphi(t)$ of system [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} with initial data $r(t_\ast)=r_\ast$, $\varphi(t_\ast)=\varphi_\ast$, $|r_\ast-a-t_\ast^{-1/(2q)}\varrho_\ast(t_\ast) |+ |\varphi_\ast-\kappa S(t_\ast)/\varkappa-\phi_\ast(t_\ast)|\leq \delta$, satisfies the inequality [\[rpst\]](#rpst){reference-type="eqref" reference="rpst"} for all $t>t_\ast$.* **Theorem 5**. *Let system [\[PS\]](#PS){reference-type="eqref" reference="PS"} satisfy [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"}, [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"}, [\[rc\]](#rc){reference-type="eqref" reference="rc"}, and assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"}, [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} and [\[asst\]](#asst){reference-type="eqref" reference="asst"} hold with $\nu_n\eta<0$, $d_{h}>0$, and $h+n-1< 2 q$. Then, there is $N_0\in\mathbb Z_+$, $\varepsilon > 0$ such that for all $\delta\in (0,\varepsilon)$, $N\geq N_0$, $M\in \mathbb Z_+$ the solution $r(t)$, $\varphi(t)$ of system [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} with initial data $r(t_\ast)=r_\ast$, $\varphi(t_\ast)=\varphi_\ast$, $|r_\ast-a-t_\ast^{-1/(2q)}\varrho_\ast(t_\ast) |+ |\varphi_\ast-\kappa S(t_\ast)/\varkappa-\phi_\ast(t_\ast)|\leq \delta$ at some $t_\ast\geq t_0$, satisfies the inequality [\[rpunst\]](#rpunst){reference-type="eqref" reference="rpunst"} at some $t_e>t_\ast$.* Thus, under the assumptions of Theorems [Theorem 2](#Th2){reference-type="ref" reference="Th2"} and [Theorem 4](#Th23){reference-type="ref" reference="Th23"}, it follows that there exists a stable phase locking regime in system [\[PS\]](#PS){reference-type="eqref" reference="PS"} with $r(t)\approx a$ and $\varphi(t)\approx \kappa S(t)/\varkappa+\psi_0$ as $t\to\infty$. Consider finally the case when, instead of [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"}, the following assumption holds: $$\begin{gathered} \label{asnzero} \Lambda_n(\rho,\psi)\neq 0 \quad \forall\, (\rho,\psi)\in\mathbb R^2.\end{gathered}$$ Then we have **Theorem 6**. *Let system [\[PS\]](#PS){reference-type="eqref" reference="PS"} satisfy [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"}, [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"}, [\[rc\]](#rc){reference-type="eqref" reference="rc"}, and assumptions [\[asn\]](#asn){reference-type="eqref" reference="asn"} and [\[asnzero\]](#asnzero){reference-type="eqref" reference="asnzero"} hold. Then the solutions of system [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} exit from any bounded domain in a finite time.* In this case, $\varphi(t)$ for solutions of system [\[PS\]](#PS){reference-type="eqref" reference="PS"} can significantly differ from the phase $\kappa S(t)/\varkappa$, and the solutions with $r(t)\approx a$ does not occur. # Change of variables {#sec3} ## Amplitude residual and phase shift Substituting [\[ch1\]](#ch1){reference-type="eqref" reference="ch1"} into [\[PS\]](#PS){reference-type="eqref" reference="PS"} yields the following system: $$\begin{gathered} \label{RPsys} \frac{dR}{dt}=F(R,\Psi,S(t),t), \quad \frac{d\Psi}{dt}=G(R,\Psi,S(t),t),\end{gathered}$$ where $$\begin{gathered} \label{FG} \begin{split} &F(R,\Psi,S,t)\equiv t^{\frac{1}{2q}} f\left(a+t^{-\frac{1}{2q}} R,\frac{\kappa}{\varkappa}S +\Psi,S,t\right)+t^{-1}\frac{R}{2q}, \\ & G(R,\Psi,S,t)\equiv \omega\left(a+t^{-\frac{1}{2q}} R\right)-\frac{\kappa}{\varkappa}S'(t)+g\left(a+t^{-\frac{1}{2q}} R,\frac{\kappa}{\varkappa}S +\Psi,S,t\right). \end{split}\end{gathered}$$ It follows from [\[fgas\]](#fgas){reference-type="eqref" reference="fgas"} and [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"} that the functions $F(R,\Psi,S,t)$ and $G(R,\Psi,S,t)$ have the following asymptotic expansion: $$\begin{gathered} \label{FGas}\begin{split} F(R,\Psi,S,t)&\sim \sum_{k=1}^\infty t^{-\frac{k}{2q}}F_k(R,\Psi,S), \\ G(R,\Psi,S,t)&\sim \sum_{k=1}^\infty t^{-\frac{k}{2q}}G_k(R,\Psi,S), \quad t\to\infty, \end{split}\end{gathered}$$ where the coefficients $$\begin{aligned} & F_k(R,\Psi,S)\equiv \sum_{\substack{ i+2j=k+1 \\ i\geq 0, j\geq 1}} \partial_r^i f_j\left(a,\frac{\kappa}{\varkappa}S +\Psi,S\right) \frac{R^i}{i!} + \delta_{k,2q}\frac{ R}{2q},\\ & G_k(R,\Psi,S)\equiv \partial_r^k\omega(a)\frac{R^k}{k!}-\frac{\kappa}{\varkappa} s_{k/2}\left(1-\frac{k}{2q}+\delta_{k,2q}\right)+\sum_{\substack{ i+2j=k \\ i\geq 0, j\geq 1}} \partial_r^i g_j\left(a,\frac{\kappa}{\varkappa}S +\Psi,S\right) \frac{R^i}{i!}\end{aligned}$$ are $2\pi$-periodic in $\Psi$ and $2\pi \varkappa$-periodic in $S$. Here $\delta_{k,2q}$ is the Kronecker delta. We set $s_j=0$ for $j>q$ and $s_{k/2}=0$ for odd $k$. Note that $F_1(R,\Psi,S)\equiv f_1(a,\kappa S/\varkappa+\Psi,S)$ and $G_1(R,\Psi,S)\equiv \eta R$. Since $F_k(R,\Psi,S)=\mathcal O(R^{k-1})$ and $G_k(R,\Psi,S)=\mathcal O(R^{k})$ as $R\to\infty$ uniformly for all $(\Psi,S)\in\mathbb R^2$, we see that the asymptotic approximations [\[FGas\]](#FGas){reference-type="eqref" reference="FGas"} for the right-hand side of system [\[FG\]](#FG){reference-type="eqref" reference="FG"} are applicable for all $(R,\Psi,t)\in\mathcal D^0_{\varsigma,\tau}$ with $\varsigma \in (0,1)$ and some $\tau\geq 1$. ## Near identity transformation We see that system [\[RPsys\]](#RPsys){reference-type="eqref" reference="RPsys"} is asymptotically autonomous with the limiting system $$\begin{gathered} \frac{d\hat R}{dt}=0, \quad \frac{d\hat\Psi}{dt}=0, \quad \frac{d\hat S}{dt}=s_0.\end{gathered}$$ Hence, the phase $S(t)$ can be considered as an analogue of a fast variable as $t\to\infty$ in comparison with the solutions $R(t)$, $\Psi(t)$ of system [\[RPsys\]](#RPsys){reference-type="eqref" reference="RPsys"}. This can be used to simplify the system by averaging the equations with respect to the variable $S(t)$. Note that such method is effective in similar problems with a small parameter (see, for instance, [@BM61; @AKN06]). The transformation is sought in the following form: $$\begin{gathered} \label{rpch}\begin{split} U_N(R,\Psi,S,t)&=R+\sum_{k=1}^N t^{-\frac{k}{2q}} u_k(R,\Psi,S), \\ V_N(R,\Psi,S,t)&=\Psi+\sum_{k=1}^N t^{-\frac{k}{2q}} v_k(R,\Psi,S) \end{split}\end{gathered}$$ with some integer $N\geq 1$. The coefficients $u_k(R,\Psi,S)$, $v_k(R,\Psi,S)$ are assumed to be periodic with respect to $\Psi$ and $S$, and are chosen in such a way that the system in the new variables $$\begin{gathered} \rho(t)\equiv U_N(R(t),\Psi(t),S(t),t), \quad \psi(t)\equiv V_N(R(t),\Psi(t),S(t),t)\end{gathered}$$ takes the form [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"}, where the right-hand sides do not depend explicitly on $S(t)$ at least in the first $N$ terms of the asymptotics as $t\to\infty$. Differentiating [\[rpch\]](#rpch){reference-type="eqref" reference="rpch"} with respect to $t$ and taking into account [\[Sform\]](#Sform){reference-type="eqref" reference="Sform"}, [\[RPsys\]](#RPsys){reference-type="eqref" reference="RPsys"} and [\[FGas\]](#FGas){reference-type="eqref" reference="FGas"}, we get $$\begin{gathered} \label{drpas} \begin{split} \frac{d}{dt}\begin{pmatrix} U_N \\ V_N \end{pmatrix} \equiv &\left(\frac{dR}{dt}\partial_R+\frac{d\Psi}{dt}\partial_\Psi+\frac{dS}{dt}\partial_S+\partial_t\right)\begin{pmatrix} U_N \\ V_N \end{pmatrix}\\ \sim & \sum_{k=1}^\infty t^{-\frac{k}{2q}} \left\{\begin{pmatrix} F_k \\ G_k \end{pmatrix}+s_0\partial_S \begin{pmatrix} u_k \\ v_k \end{pmatrix}\right\}\\& + \sum_{k=1}^\infty t^{-\frac{k}{2q}}\sum_{j=1}^{k-1}\left\{ F_j\partial_R + G_j\partial_\Psi+s_{j/2}\left(1-\frac{j}{2q}+\delta_{j,2q}\right)\partial_S+\delta_{j,2q} \frac{2q-k}{2q}\right\}\begin{pmatrix} u_{k-j} \\ v_{k-j} \end{pmatrix} \end{split}\end{gathered}$$ as $t\to\infty$, where it is assumed that $u_k(R,\Psi,S)\equiv v_k(R,\Psi,S)\equiv 0$ for $k\leq 0$ and $k>N$. Comparing the coefficients of powers of $t^{-1/2q}$ in [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} and [\[drpas\]](#drpas){reference-type="eqref" reference="drpas"} yields $$\begin{gathered} \label{ukvk} s_0\partial_S \begin{pmatrix} u_k \\ v_k \end{pmatrix}=\begin{pmatrix} \Lambda_k(R,\Psi)-F_k(R,\Psi,S) +\tilde F_k(R,\Psi,S)\\ \Omega_k(R,\Psi)-G_k(R,\Psi,S)+\tilde G_k(R,\Psi,S) \end{pmatrix}, \quad k=1,\dots, N,\end{gathered}$$ where the functions $\tilde F_k(R,\Psi,S)$, $\tilde G_k(R,\Psi,S)$ are expressed in terms of $\{u_{j}, v_{j},\Lambda_{j}, \Omega_{j}\}_{j=1}^{k-1}$ by the following formulas: $$\begin{aligned} \begin{pmatrix} \tilde F_1 \\ \tilde G_1 \end{pmatrix} \equiv & \begin{pmatrix} 0\\ 0 \end{pmatrix},\\ \begin{pmatrix} \tilde F_2 \\ \tilde G_2 \end{pmatrix} \equiv & (u_1\partial_R+v_1\partial_\Psi) \begin{pmatrix} \Lambda_1\\ \Omega_1 \end{pmatrix} -(F_1\partial_R + G_1\partial_\Psi )\begin{pmatrix} u_{1} \\ v_{1} \end{pmatrix}, \\ \begin{pmatrix} \tilde F_3 \\ \tilde G_3 \end{pmatrix} \equiv & \sum_{i+j=3}(u_i\partial_R+v_i\partial_\Psi) \begin{pmatrix} \Lambda_j\\ \Omega_j \end{pmatrix} + \frac{1}{2}\left(u_1^2\partial_R^2+2u_1v_1 \partial_R\partial_\Psi+v_1^2 \partial_\Psi^2\right)\begin{pmatrix} \Lambda_1\\ \Omega_1 \end{pmatrix} \\ & -\sum_{j=1}^2 \left\{ F_j\partial_R + G_j\partial_\Psi+s_{j/2}\left(1-\frac{j}{2q}+\delta_{j,2q}\right)\partial_S+\delta_{j,2q} \frac{2q-3}{2q}\right\}\begin{pmatrix} u_{3-j} \\ v_{3-j} \end{pmatrix},\\ \begin{pmatrix} \tilde F_k \\ \tilde G_k \end{pmatrix} \equiv & \sum_{ m_1+\cdots im_i+n_1+\cdots l n_l+j=k } C_{i,l,m_1,\dots,m_i,n_1,\dots,n_l} u_1^{m_1} \cdots u_i^{m_i} v_1^{n_1} \cdots v_l^{n_l} \partial_R^{m_1+\cdots+m_i}\partial_\Psi^{n_1+\cdots+n_l} \begin{pmatrix} \Lambda_j\\ \Omega_j \end{pmatrix} \\ & -\sum_{j=1}^{k-1} \left\{ F_j\partial_R + G_j\partial_\Psi+s_{j/2}\left(1-\frac{j}{2q}+\delta_{j,2q}\right)\partial_S+\delta_{j,2q} \frac{2q-k}{2q}\right\}\begin{pmatrix} u_{k-j} \\ v_{k-j} \end{pmatrix}\end{aligned}$$ with some constant parameters $C_{i,l,m_1,\dots,m_i,n_1,\dots,n_l}$. To avoid the appearance of secular terms in [\[rpch\]](#rpch){reference-type="eqref" reference="rpch"} and guarantee the existence of periodic solutions to system [\[ukvk\]](#ukvk){reference-type="eqref" reference="ukvk"}, we take $$\begin{aligned} \Lambda_k(R,\Psi)&\equiv \langle F_k(R,\Psi,S)-\tilde F_k(R,\Psi,S)\rangle_{\varkappa S},\\ \Omega_k(R,\Psi)&\equiv \langle G_k(R,\Psi,S)-\tilde G_k(R,\Psi,S)\rangle_{\varkappa S}.\end{aligned}$$ In particular, $\Lambda_1(R,\Psi)\equiv \langle f_1(a,\kappa S/\varkappa+\Psi,S)\rangle_{\varkappa S}$ and $\Omega_1(R,\Psi)\equiv \eta R$. Hence, system [\[ukvk\]](#ukvk){reference-type="eqref" reference="ukvk"} is solvable in the class of functions that are $2\pi\varkappa$-periodic in $S$ with zero mean. Moreover, it is not hard to check that $u_k(R,\Psi,S)$, $v_k(R,\Psi,S)$, $\Lambda_k(R,\Psi)$, $\Omega_k(R,\Psi)$ are $2\pi$-periodic in $\Psi$ and $$\begin{aligned} &\tilde F_k(R,\Psi,S)= \mathcal O(R^{k-1}),& \quad & \Lambda_k(R,\Psi)= \mathcal O(R^{k-1}), &\quad & u_k(R,\Psi,S)= \mathcal O(R^{k-1}), \\ &\tilde G_k(R,\Psi,S)= \mathcal O(R^{k}), & \quad & \Omega_k(R,\Psi)= \mathcal O(R^{k}), &\quad & v_k(R,\Psi,S)= \mathcal O(R^{k})\end{aligned}$$ as $R\to\infty$ uniformly for all $(\Psi,S)\in\mathbb R^2$. This together with [\[rpch\]](#rpch){reference-type="eqref" reference="rpch"} implies that for all $\epsilon\in (0,\mathcal R)$ there exists $t_0\geq 1$ such that $$\begin{aligned} &|U_N(R,\Psi,S,t)-R|\leq \epsilon, &\quad &|\partial_R U_N(R,\Psi,S,t)-1|\leq \epsilon, &\quad &|\partial_\Psi U_N(R,\Psi,S,t)|\leq \epsilon,\\ &|V_N(R,\Psi,S,t)-\Psi|\leq \epsilon, &\quad & |\partial_R V_N(R,\Psi,S,t)|\leq \epsilon, &\quad & |\partial_\Psi V_N(R,\Psi,S,t)-1|\leq \epsilon \end{aligned}$$ for all $(R,\Psi,t)\in\mathcal D^0_{\varsigma,t_0}$, $S\in\mathbb R$ and $\varsigma\in (0,1)$. Thus, [\[ch2\]](#ch2){reference-type="eqref" reference="ch2"} is invertible. Denote by $R= u(\rho, \psi, t)$, $\Psi=v(\rho,\psi,t)$ the corresponding inverse transformation defined for all $(\rho,\psi,t)\in\mathcal D^\epsilon_{\varsigma,t_0}$. Then, $$\begin{aligned} \begin{pmatrix} \tilde \Lambda_N(\rho,\psi,S,t)\\ \tilde \Omega_N(\rho,\psi,S,t) \end{pmatrix} \equiv & \left(\partial_t+F \partial_R+G\partial_\Psi\right) \begin{pmatrix} U_N\\ V_N \end{pmatrix}\Big|_{R=u(\rho, \psi, t), \Psi=v(\rho,\psi,t)} - \sum_{k=1}^N t^{-\frac{k}{2q}} \begin{pmatrix} \Lambda_k(\rho,\psi)\\ \Omega_k(\rho,\psi). \end{pmatrix}\end{aligned}$$ Combining this with [\[drpas\]](#drpas){reference-type="eqref" reference="drpas"}, we get [\[LOas\]](#LOas){reference-type="eqref" reference="LOas"}. Thus, we obtain the proof of Theorem [Theorem 1](#Th1){reference-type="ref" reference="Th1"} with $\tilde \rho_N(R,\Psi,t)\equiv U_N(R,\Psi,S(t),t)-R$, $\tilde \psi_N(R,\Psi,t)\equiv V_N(R,\Psi,S(t),t)-\Psi$. # Analysis of the model system {#sec4} *Proof of Lemma [Lemma 1](#Lem01){reference-type="ref" reference="Lem01"}.* Substituting $\hat \varrho(t)=u(t)$, $\hat \phi(t)=\psi_0+v(t)$ into [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} yields the following system with an equilibrium at $(0, 0)$: $$\begin{gathered} \label{limsys0} \frac{du}{dt}=t^{-\frac{n}{2q}}\Lambda_n(u,\psi_0+v), \quad \frac{dv}{dt}=t^{-\frac{1}{2q}}\eta u.\end{gathered}$$ Consider the linearised system $$\begin{gathered} \frac{d {\bf z}}{dt}= {\bf M}(t) {\bf z}, \quad {\bf M}(t)\equiv \begin{pmatrix} t^{-\frac{n}{2q}}\lambda_n & t^{-\frac{n}{2q}} \nu_n \\ t^{-\frac{1}{2q}} \eta & 0 \end{pmatrix}, \quad {\bf z}=\begin{pmatrix}u\\v\end{pmatrix}.\end{gathered}$$ The roots of the characteristic equation $|{\bf M}(t)-\mu {\bf I}|=0$ are given by $$\begin{gathered} \mu_\pm(t)=\frac{t^{-\frac{n}{2q}}}{2} \left(\lambda_n \pm \sqrt{4\nu_n\eta t^{\frac{n-1}{2q}}+\lambda_n^2}\right).\end{gathered}$$ We see that if $\nu_n \eta>0$, the eigenvalues $\mu_+(t)$ and $\mu_-(t)$ are real of different signs. This implies that the equilibrium is of saddle type and the fixed point $(0,\psi_0)$ of system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} is unstable. Let us show that in the opposite case, when $\nu_n \eta<0$, the stability of the equilibrium depends on the sign of $\lambda_n\neq 0$. Consider first the case $n=1$. We use $$\begin{gathered} \label{LF1} L_1(u,v)\equiv \frac{1}{2}\left(|\eta| u^2+|\nu_1|v^2\right)+ \chi_1 u v\end{gathered}$$ as a Lyapunov function candidate for system [\[limsys0\]](#limsys0){reference-type="eqref" reference="limsys0"}, where $\chi_1\in\mathbb R$ is a parameter such that $$\begin{gathered} \label{chi1} {\hbox{\rm sgn} }\, \chi_1 = {\hbox{\rm sgn} }\, (\nu_1 \lambda_1), \quad |\chi_1|=\frac 12\min\left\{|\eta|,|\nu_1|,\frac{2|\lambda_1 \eta \nu_1|}{\lambda_1^2+2|\eta\nu_1|}\right\}.\end{gathered}$$ It can easily be checked that there exists $\Delta_0>0$ such that $$\begin{gathered} \label{L1ineq} L_- \Delta^2\leq L_1(u,v)\leq L_+ \Delta^2\end{gathered}$$ for all $(u,v)\in\mathbb R^2$ such that $\Delta=\sqrt{u^2+v^2}\leq \Delta_0$, where $L_-=\min\{|\eta|-|\chi_1|,|\nu_1|-|\chi_1|\}/4>0$ and $L_+=\max\{|\eta|+|\chi_1|,|\nu_1|+|\chi_1|\}$. The derivative of $L_1(u,v)$ with respect to $t$ along the trajectories of the system satisfies $$\begin{gathered} \frac{dL_1}{dt}\Big|_\eqref{limsys0} = t^{-\frac{1}{2q}}\left( (\lambda_1-({\hbox{\rm sgn} }\, \lambda_1)|\chi_1|) |\eta|u^2+({\hbox{\rm sgn} }\, \lambda_1)|\chi_1 \nu_1|v^2+\chi_1 \lambda_1 uv +\mathcal O(\Delta^3)\right), \quad \Delta\to 0.\end{gathered}$$ Using Young's inequality, we obtain $$\begin{aligned} \frac{dL_1}{dt}\Big|_\eqref{limsys0} &\geq t^{-\frac{1}{2q}} \left\{A_1 u^2+B_1 v^2+\mathcal O(\Delta^3)\right\}\quad \text{if} \quad\lambda_1>0,\\ \frac{dL_1}{dt}\Big|_\eqref{limsys0} & \leq -t^{-\frac{1}{2q}} \left\{ A_1 u^2+B_1 v^2+\mathcal O(\Delta^3)\right\}\quad \text{if} \quad \lambda_1<0\end{aligned}$$ with positive parameters $$\begin{gathered} \label{A1B1} A_1=\frac{|\chi_1| (\lambda_1^2+2|\nu_1 \eta|) }{2 |\nu_1|}, \quad B_1=\frac{|\chi_1 \nu_1|}{2}.\end{gathered}$$ Hence, there exists $0<\Delta_1\leq \Delta_0$ such that $$\begin{aligned} \label{dL1}&\frac{dL_1}{dt}\Big|_\eqref{limsys0} \geq \gamma_1 t^{-\frac{1}{2q}} L_1\geq 0 \quad \text{if} \quad \lambda_1>0, \\ \label{dL2}&\frac{dL_1}{dt}\Big|_\eqref{limsys0} \leq -\gamma_1 t^{-\frac{1}{2q}} L_1\leq 0 \quad \text{if} \quad \lambda_1<0\end{aligned}$$ for all $(u,v)\in\mathbb R^2$ such that $\Delta\leq \Delta_1$ with $\gamma_1=\min\{A_1,B_1\}/(2 L_+)>0$. If $\lambda_1>0$, then integrating [\[dL1\]](#dL1){reference-type="eqref" reference="dL1"} with respect to $t$ and taking into account [\[L1ineq\]](#L1ineq){reference-type="eqref" reference="L1ineq"}, we obtain the instability of the equilibrium $(0,0)$ in system [\[limsys0\]](#limsys0){reference-type="eqref" reference="limsys0"} and the fixed point $(0,\psi_0)$ in system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"}. Indeed, there exists $\epsilon \in (0, \Delta_1)$ such that for all $\delta\in (0,\epsilon)$ the solution $(u(t),v(t))$ of [\[limsys0\]](#limsys0){reference-type="eqref" reference="limsys0"} with initial data $\sqrt{u^2(t_0)+v^2(t_0)}=\delta$ leaves the domain $\{(u,v)\in\mathbb R^2: \Delta\leq \epsilon\}$ as $t\geq t_1$, where $$\begin{gathered} t_1^{1-\frac{1}{2q}}=t_0^{1-\frac{1}{2q}}+\left(\frac{2q-1}{2q \gamma_1}\right)\log \left(\frac{L_+\epsilon^2}{L_-\delta^2}\right).\end{gathered}$$ If $\lambda_1<0$, then it follows from [\[dL2\]](#dL2){reference-type="eqref" reference="dL2"} that for all $\epsilon\in (0,\Delta_1)$ there exists $\delta\in (0,\epsilon)$ such that the solution $(u(t),v(t))$ of [\[limsys0\]](#limsys0){reference-type="eqref" reference="limsys0"} with initial data $\sqrt{u^2(t_0)+v^2(t_0)}\leq \delta$ cannot exit from the domain $\{(u,v)\in\mathbb R^2: \Delta\leq \epsilon\}$. Hence, the equilibrium $(0,0)$ of system [\[limsys0\]](#limsys0){reference-type="eqref" reference="limsys0"} and the fixed point $(0,\psi_0)$ of system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} are stable. Moreover, by integrating [\[dL2\]](#dL2){reference-type="eqref" reference="dL2"}, we obtain the inequality $$\begin{gathered} L_1(u(t),v(t))\leq L_1(u(t_0),v(t_0)) \exp\left\{-\frac{2 q\gamma_1 }{2q-1}\left(t^{1-\frac{1}{2q}}-t_0^{1-\frac{1}{2q}}\right)\right\}, \quad t\geq t_0.\end{gathered}$$ Combining this with [\[L1ineq\]](#L1ineq){reference-type="eqref" reference="L1ineq"}, we get asymptotic stability of the equilibrium. Let $n\geq 2$. Consider $$\begin{gathered} L_n(u,v,t)\equiv t^{\frac{n-1}{2q}}\frac{|\eta| }{2} u^2+({\hbox{\rm sgn}}\, \nu_n) \int\limits_0^v \Lambda_n(u,\psi_0+w)\,dw + t^{-\frac{n-1}{2q}}\left(\frac{\lambda_n^2 v^2}{2|\eta|}+\lambda_n ({\hbox{\rm sgn}}\, \eta) u v\right)\end{gathered}$$ as a Lyapunov function candidate. Note that there exist $\Delta_0>0$ and $t_1\geq t_0$ such that $$\begin{gathered} \frac{1}{4}\left(|\nu_n|u^2+|\eta|v^2\right)\leq L_n(u,v,t)\leq t^{\frac{n-1}{2q}} \left(|\nu_n|u^2+|\eta|v^2\right)\end{gathered}$$ for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_0$ and $t\geq t_1$. The derivative of $L_n(u,v,t)$ with respect to $t$ along the trajectories of system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} satisfies $$\begin{gathered} \frac{dL_n}{dt}\Big|_\eqref{limsys0} = \lambda_n t^{-\frac{n}{2q}}\left( | \eta|u^2 + |\nu_n|v^2 +\mathcal O(\Delta^3)+\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}})\right) \end{gathered}$$ as $\Delta\to 0$ and $t\to\infty$. Therefore, there exists $0<\Delta_1\leq \Delta_0$ and $t_2\geq t_1$ such that $$\begin{gathered} \label{dLn00} \begin{split} &\frac{dL_n}{dt}\Big|_\eqref{limsys0} \geq \gamma_n t^{-\frac{n}{2q}} \left(|\nu_n|u^2+|\eta|v^2\right) \geq \gamma_n t^{-\frac{2n-1}{2q}} L_n\geq 0 \quad \text{if} \quad \lambda_n>0, \\ &\frac{dL_n}{dt}\Big|_\eqref{limsys0} \leq -\gamma_n t^{-\frac{n}{2q}} \left(|\nu_n|u^2+|\eta|v^2\right) \leq -\gamma_n t^{-\frac{2n-1}{2q}} L_n \leq 0 \quad \text{if} \quad \lambda_n<0 \end{split}\end{gathered}$$ for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_1$ and $t\geq t_2$ with $\gamma_n=|\lambda_n|/2>0$. Integrating [\[dLn00\]](#dLn00){reference-type="eqref" reference="dLn00"}, we obtain the following inequalities: $$\begin{aligned} &|\nu_n|u^2(t)+|\eta|v^2(t)\geq C t^{-\frac{n-1}{2q}} \exp \left\{ \frac{2q \gamma_n}{2q-2n+1}\left(t^{1-\frac{2n-1}{2q}}-t_2^{1-\frac{2n-1}{2q}}\right)\right\}\quad \text{if} \quad \lambda_n>0,\\ &|\nu_n|u^2(t)+|\eta|v^2(t)\leq 4C \exp \left\{ -\frac{2q \gamma_n}{2q-2n+1}\left(t^{1-\frac{2n-1}{2q}}-t_2^{1-\frac{2n-1}{2q}}\right)\right\}\quad \text{if} \quad \lambda_n<0\end{aligned}$$ with a positive parameter $C=L_n(u(t_2),v(t_2),t_2)>0$. Thus, if $\lambda_n>0$ and $n\leq q$, the equilibrium $(0,\psi_0)$ of system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} is unstable. If $n\leq q$ and $\lambda_n<0$, the equilibrium is asymptotically stable. Finally, if $n>q$ and $\lambda_n<0$, the equilibrium is (non-asymptotically) stable. ◻ *Proof of Lemma [Lemma 2](#Lem1){reference-type="ref" reference="Lem1"}.* Substituting the series [\[assol\]](#assol){reference-type="eqref" reference="assol"} into [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} and equating the terms of like powers of $t$ yield the chain of linear equations for the coefficients $\varrho_k$, $\phi_k$ $$\begin{gathered} \label{rpk} \begin{pmatrix} \eta & 0 \\ \displaystyle \lambda_n & \nu_n \end{pmatrix} \begin{pmatrix} \varrho_k \\ \phi_k \end{pmatrix} = \begin{pmatrix} \mathfrak F_k \\ \mathfrak G_k \end{pmatrix},\end{gathered}$$ where $\mathfrak F_k$, $\mathfrak G_k$ are expressed through $\varrho_1,\phi_1, \dots, \varrho_{k-1}, \phi_{k-1}$. For instance, $$\begin{aligned} &\mathfrak F_1=-\Omega_m(0,\psi_0), \\ & \mathfrak G_1=-\Lambda_{n+1}(0,\psi_0),\\ &\mathfrak F_2=-\Omega_{m+1}(0,\psi_0)-\left(\varrho_1\partial_\rho +\phi_1 \partial_\psi\right)\Omega_{m}(0,\psi_0), \\ &\mathfrak G_2=-\Lambda_{n+2}(0,\psi_0)-\sum_{i+j=2}\left(\varrho_i\partial_\rho +\phi_i \partial_\psi\right)\Lambda_{n+j}(0,\psi_0)-\frac{1}{2}\left(\varrho_1^2\partial^2_\rho+2\varrho_1\phi_1\partial_\rho\partial_\psi+\phi_1^2\partial^2_\psi\right)\Lambda_{n}(0,\psi_0).\end{aligned}$$ Since $\nu_n\eta\neq 0$, we see that system [\[rpk\]](#rpk){reference-type="eqref" reference="rpk"} is solvable. To prove the existence of a solution of system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} with such asymptotic behaviour, consider the following functions: $$\begin{gathered} \label{sumM} \varrho_{\ast,M}(t)\equiv\sum_{k=1}^{n+M+1} t^{-\frac{k+m-2}{2q}} \varrho_k, \quad \phi_{\ast,M}(t)\equiv\psi_0+\sum_{k=1}^{n+M+1} t^{-\frac{k}{2q}}\phi_k\end{gathered}$$ with some $M\in\mathbb Z_+$. By construction, $$\begin{gathered} \label{Zeq} \begin{split} &Z_{\varrho}(t)\equiv \varrho_{\ast,M}'(t)-\hat\Lambda_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t)=\mathcal O\left(t^{-\frac{2n+M+2}{2q}}\right),\\ &Z_{\phi}(t)\equiv \phi_{\ast,M}'(t)-\hat\Omega_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t)=\mathcal O\left(t^{-\frac{n+m+M+1}{2q}}\right), \quad t\to\infty. \end{split}\end{gathered}$$ Substituting $$\begin{gathered} \label{subsM} \varrho(t)=\varrho_{\ast,M}(t)+t^{-\frac{M}{2q}} u(t), \quad \phi(t)=\phi_{\ast,M}(t)+t^{-\frac{M}{2q}} v(t)\end{gathered}$$ into [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"}, we obtain a perturbed near-Hamiltonian system $$\begin{gathered} \label{uvsys} \frac{du}{dt}=-\partial_v \mathcal H_M(u,v,t) + \xi_M(t), \quad \frac{dv}{dt}=\partial_u \mathcal H_M(u,v,t)+\Upsilon_M(u,v,t) + \zeta_M(t), \end{gathered}$$ with $$\begin{aligned} &\mathcal H_M(u,v,t) \equiv \int\limits_0^u \mathcal G_M(w,0,t)\,dw - \int\limits_0^v \mathcal F_M(u,w,t)\,dw, \\ &\Upsilon_M(u,v,t) \equiv \int\limits_0^v \left(\partial_u\mathcal F(u,w,t)+\partial_v\mathcal G(u,w,t)\right)\,dw\end{aligned}$$ and perturbations $$\begin{gathered} \xi_M(t)\equiv - t^{\frac{M}{2q}}Z_\varrho(t), \quad \zeta_M(t)\equiv - t^{\frac{M}{2q}}Z_\phi(t),\end{gathered}$$ where $$\begin{aligned} \mathcal F_M(u,v,t)&\equiv t^{\frac{M}{2q}}\left(\hat\Lambda_N(\varrho_{\ast,M}(t)+t^{-\frac{M}{2q}} u,\phi_{\ast,M}(t)+t^{-\frac{M}{2q}} v,t)-\hat\Lambda_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t)\right) + \frac{M}{2q} t^{-1} u, \\ \mathcal G_M(u,v,t)&\equiv t^{\frac{M}{2q}}\left(\hat\Omega_N(\varrho_{\ast,M}(t)+t^{-\frac{M}{2q}} u,\phi_{\ast,M}(t)+t^{-\frac{M}{2q}} v,t)-\hat\Omega_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t)\right) + \frac{M}{2q} t^{-1} v.\end{aligned}$$ It follows from [\[asn\]](#asn){reference-type="eqref" reference="asn"}, [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} and [\[Zeq\]](#Zeq){reference-type="eqref" reference="Zeq"} that $$\begin{gathered} \label{HU1} \begin{split} & \mathcal H_M(u,v,t) =\left\{t^{-\frac{1}{2q}}\frac{\eta u^2}{2}-t^{-\frac{n}{2q}}\left(\lambda_n u v+ \frac{\nu_n v^2}{2}\right)\right\}\left(1+\mathcal O(t^{-\frac{1}{2q}})\right),\\ & \Upsilon_M(u,v,t)=v\left(\lambda_n t^{-\frac{n}{2q}}+\omega_m t^{-\frac{m}{2q}}\right) \left(1+\mathcal O(t^{-\frac{1}{2q}})\right),\\ & \xi_M(t)=\mathcal O(t^{-\frac{2n+2}{2q}}),\quad \zeta_M(t)=\mathcal O(t^{-\frac{n+m+1}{2q}}) \end{split}\end{gathered}$$ as $\Delta=\sqrt{u^2+v^2}\to 0$ and $t\to\infty$. Our goal is to show that there is a solution of system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"} such that $u(t)=\mathcal O(1)$ and $v(t)=\mathcal O(1)$ as $t\to\infty$. This will ensure the existence of a solution to system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} with asymptotic expansion [\[assol\]](#assol){reference-type="eqref" reference="assol"}. The proposed method is based on the stability analysis and on the construction of suitable Lyapunov functions. Note that a similar approach to justifying the asymptotics was used in [@LK15]. Note that if $\xi_M(t)\equiv \zeta_M(t)\equiv 0$, then system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"} has the equilibrium $(0,0)$. Let us prove the stability of the near-Hamiltonian system with respect to the time-decaying perturbations $\xi_M(t)$ and $\zeta_M(t)$. Consider first the case $1\leq n<m$. If $n=1$, we use $\mathcal L_M(u,v,t)\equiv L_1(u,v)$ defined by [\[LF1\]](#LF1){reference-type="eqref" reference="LF1"} and [\[chi1\]](#chi1){reference-type="eqref" reference="chi1"} as a Lyapunov function candidate for system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"}. If $n>1$, we use $$\begin{gathered} %\label{LFn} \begin{split} \mathcal L_M(u,v,t)\equiv & ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H_M(u,v,t) + t^{-\frac{n-1}{2q}}\lambda_n \mathcal K(u,v), \end{split}\end{gathered}$$ with $\mathcal K(u,v)\equiv {\lambda_n v^2}/{2|\eta|}+ ({\hbox{\rm sgn}}\, \eta) u v$. We see that $$\begin{gathered} \label{Lnas} \mathcal L_M(u,v,t) = t^{\frac{n-1}{2q}}\frac{|\eta|u^2}{2}+ \frac{|\nu_n|v^2}{2}-({\hbox{\rm sgn}}\, \eta) \lambda_n uv +\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}}), \quad \Delta\to 0, \quad t\to\infty.\end{gathered}$$ Hence, there exist $\Delta_1>0$ and $t_1\geq t_0$ such that $$\begin{gathered} \label{Lnineq1} L_- \Delta^2 \leq\mathcal L_M(u,v,t)\leq t^{\frac{n-1}{2q}}L_+\Delta^2\end{gathered}$$ for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_1$ and $t\geq t_1$ with some $L_\pm={\hbox{\rm const}}>0$. The derivative of $\mathcal L_M(u,v,t)$ with respect to $t$ along the trajectories of the system is given by $$\begin{gathered} \label{dLn} \frac{d \mathcal L_M}{dt}\Big|_\eqref{uvsys}\equiv \mathcal D_{M,1}(u,v,t)+\mathcal D_{M,2}(u,v,t),\end{gathered}$$ where $\mathcal D_{M,1}\equiv \left(\partial_t -\partial_u \mathcal H_M \partial_u +(\partial_v\mathcal H_M +\Upsilon_M) \partial_v \right)\mathcal L_M$ and $\mathcal D_{M,2}\equiv \left(\xi_M \partial_u +\zeta_M \partial_v \right)\mathcal L_M$. It can easily be checked that $$\begin{aligned} & \mathcal D_{M,1}(u,v,t)\leq -t^{-\frac{1}{2q}} \left(A_1 u^2 + B_1 v^2 +\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}})\right) \text{if} \quad n=1,\\ & \mathcal D_{M,1}(u,v,t)= - t^{-\frac{n}{2q}} |\lambda_n| \left(|\eta|u^2 + |\nu_n| v^2 +\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}})\right) \text{if} \quad n>1,\end{aligned}$$ and $\mathcal D_{M,2}(u,v,t) = \mathcal O(\Delta)\mathcal O(t^{-\frac{n+3}{2q}})$ as $\Delta\to 0$ and $t\to\infty$, where the positive parameters $A_1$ and $B_1$ are defined by [\[A1B1\]](#A1B1){reference-type="eqref" reference="A1B1"}. It follows that there exist $\Delta_2\leq \Delta_1$ and $t_2\geq t_1$ such that $$\begin{gathered} \mathcal D_{M,1}(u,v,t)\leq - t^{-\frac{n}{2q}} \gamma_n \Delta^2, \quad \mathcal D_{M,2}(u,v,t)\leq t^{-\frac{n+1}{2q}} C \Delta\end{gathered}$$ for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_2$ and $t\geq t_2$, where $C={\hbox{\rm const}}>0$, $\gamma_1=\min\{A_1,B_1\}/2$ and $\gamma_n=|\lambda_n|\min\{|\eta|,|\nu_n|\}/2$. Therefore, for all $\epsilon\in (0,\Delta_2)$ there exist $$\begin{gathered} \delta_\epsilon=\frac{2C }{\gamma_n} t_\epsilon^{-\frac{1}{2q}}, \quad t_\epsilon=\max\left\{t_2,\left(\frac{4 C }{\gamma_n \epsilon}\right)^{2q}\right\}\end{gathered}$$ such that $$\begin{gathered} \frac{d\mathcal L_M}{dt}\Big|_\eqref{trsys}\leq t^{-\frac{n}{2q}}\left(-\gamma_n+C\delta_\epsilon^{-1}t_\epsilon^{-\frac{1}{2q}}\right)\Delta^2 \leq 0\end{gathered}$$ for all $(u,v,t)\in\mathbb R^3$ such that $\delta_\epsilon\leq \Delta\leq \epsilon$ and $t\geq t_\epsilon$. Combining this with [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"}, we see that any solution of system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"} with initial data $\sqrt{u^2(t_\epsilon)+v^2(t_\epsilon)}\leq \delta$, where $\delta=\max\{\delta_\epsilon,\epsilon\sqrt{L_-/L_+}\}$, cannot exit from the domain $\{(u,v)\in\mathbb R^2: \Delta\leq \epsilon\}$ as $t\geq t_\epsilon$. It follows from [\[subsM\]](#subsM){reference-type="eqref" reference="subsM"} that for all $M\in\mathbb Z_+$ the trajectories of system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} starting close to $(0,\psi_0)$ satisfy the estimates $\varrho(t)=\varrho_{\ast,M}(t)+\mathcal O(t^{-\frac{M}{2q}})$, $\phi(t)=\phi_{\ast,M}(t)+\mathcal O(t^{-\frac{M}{2q}})$ as $t\to\infty$. Thus, there exists the solution $\varrho_\ast(t)$, $\phi_\ast(t)$ of system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} with asymptotics [\[assol\]](#assol){reference-type="eqref" reference="assol"}. Now let $n\geq 2$. Using $$\begin{gathered} \mathcal L_M(u,v,t)\equiv \begin{cases} ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}}\mathcal H_M(u,v,t)+ t^{-\frac{n-1}{2q}} (\lambda_n+\omega_n)\mathcal K(u,v), & n=m,\\ ({\hbox{\rm sgn}}\, \eta) t^{\frac{n}{2q}}\mathcal H_M(u,v,t) + t^{-\frac{m-1}{2q}} \omega_m \mathcal K(u,v), & n>m, \end{cases}\end{gathered}$$ as a Lyapunov function candidate for system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"}, we obtain [\[Lnas\]](#Lnas){reference-type="eqref" reference="Lnas"} and [\[dLn\]](#dLn){reference-type="eqref" reference="dLn"}, where $$\begin{aligned} &\mathcal D_{M,1}(u,v,t)= \begin{cases} -t^{-\frac{n}{2q}}|\lambda_n+\omega_n|\left(|\eta|u^2 + |\nu_n| v^2 +\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}})\right), & n=m,\\ -t^{-\frac{m}{2q}}|\omega_m|\left(|\eta|u^2 + |\nu_n| v^2 +\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}})\right), & n>m, \end{cases}\\ & \mathcal D_{M,2}(u,v,t) = \begin{cases} \mathcal O(\Delta) \mathcal O(t^{-\frac{n+1}{2q}}), & n=m,\\ \mathcal O(\Delta)\mathcal O(t^{-\frac{m+1}{2q}}), & n>m, \end{cases}\end{aligned}$$ as $\Delta\to 0$ and $t\to\infty$. Then, repeating the arguments as given above proves the existence of the solution with asymptotics [\[assol\]](#assol){reference-type="eqref" reference="assol"}. To prove the stability of the constructed solution consider the substitution [\[subsM\]](#subsM){reference-type="eqref" reference="subsM"} with $\varrho_\ast(t)$, $\phi_\ast(t)$ instead of $\varrho_{\ast,M}(t)$, $\phi_{\ast,M}(t)$ and with some $M\in\mathbb Z_+$. In this case, we obtain system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"} with $$\begin{aligned} & \mathcal F_M(u,v,t)\equiv t^{\frac{M}{2q}}\left(\hat\Lambda_N(\varrho_{\ast}(t)+t^{-\frac{M}{2q}} u,\phi_{\ast}(t)+t^{-\frac{M}{2q}} v,t)-\hat\Lambda_N(\varrho_{\ast}(t),\phi_{\ast}(t),t)\right) + \frac{M}{2q} t^{-1} u, \\ & \mathcal G_M(u,v,t)\equiv t^{\frac{M}{2q}}\left(\hat\Omega_N(\varrho_{\ast}(t)+t^{-\frac{M}{2q}} u,\phi_{\ast}(t)+t^{-\frac{M}{2q}} v,t)-\hat\Omega_N(\varrho_{\ast}(t),\phi_{\ast}(t),t)\right) + \frac{M}{2q} t^{-1} v,\end{aligned}$$ and $\xi_M(t)\equiv \zeta_M(t)\equiv 0$. Then, repeating the arguments as given above and using the constructed Lyapunov functions, we get $d\mathcal L_M/dt\leq - t^{-\frac{2n-1}{2q}}D_n \mathcal L_M$ for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_3$, $t\geq t_3$ with some $\Delta_3\leq \Delta_1$, $t_3\geq t_1$ and $D_n=\gamma_n/L_+>0$. Integrating this inequality and taking into account [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"}, we obtain asymptotic stability of the solution $\varrho_\ast(t)$, $\phi_\ast(t)$ if $n\leq q$ and (non-asymptotic) stability if $n>q$. ◻ *Proof of Lemma [Lemma 3](#Lem2){reference-type="ref" reference="Lem2"}.* The asymptotic series are constructed in the same way as in the proof of Lemma [Lemma 2](#Lem1){reference-type="ref" reference="Lem1"}. Consider the functions $\varrho_{\ast,M}(t)$, $\phi_{\ast,M}(t)$ defined by [\[sumM\]](#sumM){reference-type="eqref" reference="sumM"}. Substituting $$\begin{gathered} \label{subsM2} \varrho(t)=\varrho_{\ast,M}(t)+t^{-\frac{M-h}{2q}} u(t), \quad \phi(t)=\phi_{\ast,M}(t)+t^{-\frac{M-h}{2q}}v(t)\end{gathered}$$ with $M>h$ into equations [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"}, we get perturbed near-Hamiltonian system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"}, where $\mathcal H_M(u,v,t)$ and $\Upsilon_M(u,v,t)$ are defined by [\[HU1\]](#HU1){reference-type="eqref" reference="HU1"} with $$\begin{gathered} \mathcal F_M \equiv t^{\frac{M-h}{2q}}\left(\hat\Lambda_N(\varrho_{\ast,M}(t)+t^{-\frac{M-h}{2q}} u,\phi_{\ast,M}(t)+t^{-\frac{M-h}{2q}} v,t)-\hat\Lambda_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t)\right) + \frac{M-h}{2q} t^{-1} u, \\ \mathcal G_M \equiv t^{\frac{M-h}{2q}}\left(\hat\Omega_N(\varrho_{\ast,M}(t)+t^{-\frac{M-h}{2q}} u,\phi_{\ast,M}(t)+t^{-\frac{M-h}{2q}} v,t)-\hat\Omega_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t)\right) + \frac{M-h}{2q} t^{-1} v,\end{gathered}$$ and the perturbations have the following form: $$\begin{gathered} \xi_M(t) \equiv - t^{\frac{M-h}{2q}}Z_\varrho(t), \quad \zeta_M(t) \equiv - t^{\frac{M-h}{2q}}Z_\phi(t)\end{gathered}$$ with functions $Z_\varrho(t)$ and $Z_\phi(t)$ defined by [\[Zeq\]](#Zeq){reference-type="eqref" reference="Zeq"}. It follows easily that $$\begin{aligned} &\mathcal H_M(u,v,t)= \left\{t^{-\frac{1}{2q}}\frac{\eta u^2}{2}-t^{-\frac{n}{2q}}\left(\lambda_n u v+ \frac{\nu_n v^2}{2}\right)\right\}(1+\mathcal O(t^{-\frac{1}{2q}})),\\ &\Upsilon_M(u,v,t)=t^{-\frac{h}{2q}} d_h v(1+\mathcal O(t^{-\frac{1}{2q}})), \\ &\xi_M(t)=\mathcal O(t^{-\frac{h+2n+2}{2q}}), \quad \zeta_M(t)=\mathcal O(t^{-\frac{h+n+m+1}{2q}})\end{aligned}$$ as $\Delta=\sqrt{u^2+v^2}\to 0$ and $t\to\infty$. Note that $\partial_u \mathcal H_M(0,0,t)\equiv \partial_v \mathcal H_M(0,0,t) \equiv \Upsilon_M(0,0,t)\equiv 0$. Let us prove the stability of the system with respect to the non-vanishing perturbations $\xi_M(t)$ and $\zeta_M(t)$ (see [@Halil Ch. 9]). Consider a Lyapunov function candidate in the following form: $$\begin{gathered} \mathcal L_M(u,v,t)\equiv \begin{cases} ({\hbox{\rm sgn}}\, \eta)t^{\frac{1}{2q}} \mathcal H_M(u,v,t)+t^{-\frac{h-1}{2q}}\frac{d_h({\hbox{\rm sgn}}\, \eta)}{2}u v , & h>n=1,\\ ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H_M(u,v,t)+t^{-\frac{h-1}{2q}} d_h ({\hbox{\rm sgn}}\, \eta)u v, & h>n>1,\\ ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H_M(u,v,t) +t^{-\frac{h-1}{2q}} d_h\left\{({\hbox{\rm sgn}}\, \eta)u v+\frac{\lambda_n v^2}{2|\eta|}\right \} , & h\leq n. \end{cases}\end{gathered}$$ We see that there exist $\Delta_1>0$ and $t_1\geq t_0$ such that the estimate [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"} holds for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_1$ and $t\geq t_1$ with some $L_\pm={\hbox{\rm const}}>0$. The derivative of $\mathcal L_M(u,v,t)$ with respect to $t$ along the trajectories of the system is given by [\[dLn\]](#dLn){reference-type="eqref" reference="dLn"}, where $\mathcal D_{M,1}\equiv \left(\partial_t - \partial_v \mathcal H_M \partial_u +(\partial_u \mathcal H_M+\Upsilon_M) \partial_v \right)\mathcal L_M$ and $\mathcal D_{M,2}\equiv \left(\xi_M\partial_u +\zeta_M\partial_v \right) \mathcal L_M$. We see that $$\begin{gathered} \mathcal D_{M,1}= \begin{cases} t^{-\frac{h}{2q}}\frac{d_n}{2}(|\eta|u^2+|\nu_1|v^2)(1 +\mathcal O(t^{-\frac{1}{q}})), & n=1, \\ t^{-\frac{h}{2q}}d_n(|\eta|u^2+|\nu_1|v^2)(1 +\mathcal O(t^{-\frac{1}{q}})), & n\neq 1 \end{cases}\end{gathered}$$ and $\mathcal D_{M,2}(u,v,t) = \mathcal O(t^{-\frac{h+n+3}{2q}})\mathcal O(\Delta)$ as $\Delta\to 0$ and $t\to\infty$. It follows that there exist $\Delta_2\leq \Delta_1$ and $t_2\geq t_1$ such that $\mathcal D_{M,1}(u,v,t)\leq - t^{-\frac{h}{2q}} \gamma_h \Delta^2$ and $\mathcal D_{M,2}(u,v,t)\leq t^{-\frac{h+1}{2q}} C \Delta$ for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_2$ and $t\geq t_2$, where $C={\hbox{\rm const}}>0$, and $\gamma_h=|d_h|/4>0$. Hence, for all $\epsilon\in (0,\Delta_2)$ there exist $$\begin{gathered} \delta_\epsilon=\frac{2C }{\gamma_h} t_\epsilon^{-\frac{1}{2q}}, \quad t_\epsilon=\max\left\{t_2,\left(\frac{4 C }{\gamma_h\epsilon}\right)^{2q}\right\}\end{gathered}$$ such that $$\begin{gathered} \frac{d\mathcal L_M}{dt}\Big|_\eqref{uvsys}\leq t^{-\frac{h}{2q}}\left(-\gamma_h+C\delta_\epsilon^{-1}t_\epsilon^{-\frac{1}{2q}}\right)\Delta^2 \leq 0\end{gathered}$$ for all $\delta_\epsilon\leq \Delta\leq \epsilon$ and $t\geq t_\epsilon$. Taking into account [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"}, we see that solutions of system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"} with initial data $\sqrt{u^2(t_\epsilon)+v^2(t_\epsilon)}\leq \delta$ and $\delta=\max\{\delta_\epsilon,\epsilon\sqrt{L_-/L_+}\}$ cannot exit from the domain $\{(u,v)\in\mathbb R^2: \Delta\leq \epsilon\}$ as $t\geq t_\epsilon$. Thus, for all $M>h$ the solutions of system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} starting close to $(0,\psi_0)$ satisfy the estimates $\varrho(t)=\varrho_{\ast,M}(t)+\mathcal O(t^{-\frac{M-h}{2q}})$, $\phi(t)=\phi_{\ast,M}(t)+\mathcal O(t^{-\frac{M-h}{2q}})$ as $t\to\infty$. This ensures the existence of a particular solution $\varrho_\ast(t)$, $\phi_\ast(t)$ of system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"} with asymptotic expansion [\[assol\]](#assol){reference-type="eqref" reference="assol"}. To prove the stability of the constructed solution consider the substitution [\[subsM2\]](#subsM2){reference-type="eqref" reference="subsM2"} with $\varrho_\ast(t)$, $\phi_\ast(t)$ instead of $\varrho_{\ast,M}(t)$, $\phi_{\ast,M}(t)$ and some integer $M>h$. In this case, we obtain system [\[uvsys\]](#uvsys){reference-type="eqref" reference="uvsys"} with $$\begin{aligned} & \mathcal F_M(u,v,t)\equiv t^{\frac{M-h}{2q}}\left(\hat\Lambda_N(\varrho_{\ast}(t)+t^{-\frac{M-h}{2q}} u,\phi_{\ast}(t)+t^{-\frac{M-h}{2q}} v,t)-\hat\Lambda_N(\varrho_{\ast}(t),\phi_{\ast}(t),t)\right) + \frac{M-h}{2q} t^{-1} u, \\ & \mathcal G_M(u,v,t)\equiv t^{\frac{M-h}{2q}}\left(\hat\Omega_N(\varrho_{\ast}(t)+t^{-\frac{M-h}{2q}} u,\phi_{\ast}(t)+t^{-\frac{M-h}{2q}} v,t)-\hat\Omega_N(\varrho_{\ast}(t),\phi_{\ast}(t),t)\right) + \frac{M-h}{2q} t^{-1} v,\end{aligned}$$ and $\xi_M(t)\equiv \zeta_M(t)\equiv 0$. Then, repeating the arguments as given above and using the constructed Lyapunov functions $\mathcal L_M(u,v,t)$, we obtain the inequality $d\mathcal L_M/dt\leq - t^{-\frac{h+n-1}{2q}}D_h \mathcal L_M$ for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_3$, $t\geq t_3$ with some $\Delta_3\leq \Delta_1$, $t_3\geq t_1$ and $D_h=\gamma_h/L_+>0$. Integrating the inequality with respect to $t$ and taking into account [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"}, we obtain the asymptotic stability of the solution $\varrho_\ast(t)$, $\phi_\ast(t)$ if $h+n\leq 2q+1$, and the (non-asymptotic) stability if $h+n>2q+1$. ◻ # Analysis of the full system {#sec5} *Proof of Theorem [Theorem 2](#Th2){reference-type="ref" reference="Th2"}.* Substituting $\varrho(t)=\varrho_{\ast}(t)+u(t)$, $\phi(t)=\phi_{\ast}(t)+ v(t)$ into [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"}, we obtain a perturbed near-Hamiltonian system $$\begin{gathered} \label{uvsys1} \frac{du}{dt}=-\partial_v \mathcal H(u,v,t) + \mathcal P_N(u,v,t), \quad \frac{dv}{dt}=\partial_u \mathcal H(u,v,t) + \Upsilon(u,v,t)+ \mathcal Q_N(u,v,t), \end{gathered}$$ with the Hamiltonian $$\begin{gathered} \mathcal H(u,v,t) \equiv \int\limits_0^u \mathcal G(w,0,t)\,dw - \int\limits_0^v \mathcal F(u,w,t)\,dw,\end{gathered}$$ and perturbations $$\begin{aligned} \Upsilon(u,v,t) &\equiv \int\limits_0^v \left(\partial_u\mathcal F(u,w,t)+\partial_v\mathcal G(u,w,t)\right)\,dw,\\ \mathcal F(u,v,t)& \equiv \hat\Lambda_N(\varrho_{\ast}(t)+u,\phi_{\ast}(t)+v,t)-\hat\Lambda_N(\varrho_{\ast}(t),\phi_{\ast}(t),t),\\ \mathcal G(u,v,t)& \equiv \hat\Omega_N(\varrho_{\ast}(t)+u,\phi_{\ast}(t)+v,t)-\hat\Omega_N(\varrho_{\ast}(t),\phi_{\ast}(t),t),\\ \mathcal P_N(u,v,t)&\equiv \tilde \Lambda_N(\varrho_{\ast}(t)+u,\phi_{\ast}(t)+v,S(t),t), \\ \mathcal Q_N(u,v,t)&\equiv \tilde \Omega_N(\varrho_{\ast}(t)+u,\phi_{\ast}(t)+v,S(t),t).\end{aligned}$$ It follows from [\[tildeLO\]](#tildeLO){reference-type="eqref" reference="tildeLO"}, [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} and [\[assol\]](#assol){reference-type="eqref" reference="assol"} that $$\begin{gathered} \label{HUPQest} \begin{split} \mathcal H(u,v,t)&=\left\{t^{-\frac{1}{2q}}\frac{\eta u^2}{2}-t^{-\frac{n}{2q}}\left(\lambda_n u v+ \frac{\nu_n v^2}{2}\right)\right\}\left(1+\mathcal O(\Delta)+\mathcal O(t^{-\frac{1}{2q}})\right),\\ \Upsilon(u,v,t)&=v\left(\lambda_n t^{-\frac{n}{2q}}+\omega_m t^{-\frac{m}{2q}}\right) \left(1+\mathcal O(\Delta)+\mathcal O(t^{-\frac{1}{2q}})\right),\\ \mathcal P_N(u,v,t)&=\mathcal O(t^{-\frac{N+1}{2q}}), \\ \mathcal Q_N(u,v,t)&=\mathcal O(t^{-\frac{N+1}{2q}}) \end{split}\end{gathered}$$ as $\Delta=\sqrt{u^2+v^2}\to 0$ and $t\to\infty$. Note that $\partial_u\mathcal H(0,0,t)\equiv\partial_v\mathcal H(0,0,t)\equiv \Upsilon(0,0,t)\equiv 0$, while the functions $\mathcal P_N(u,v,t)$ and $\mathcal Q_N(u,v,t)$ do not preserve the equilibrium $(0,0)$ and can be considered as external perturbations. Let us prove the stability of the equilibrium in the perturbed system [@Halil Ch. 9]. Consider a Lyapunov function candidate in the form $$\begin{gathered} \label{LFTh} \mathcal L(u,v,t)\equiv \begin{cases} ({\hbox{\rm sgn}}\, \eta)t^{\frac{1}{2q}} \mathcal H(u,v,t)+(\chi_1+\lambda_1 {\hbox{\rm sgn}}\, \eta ) u v , & n=1,\\ ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H(u,v,t) + t^{-\frac{n-1}{2q}}\lambda_n \mathcal K(u,v), & 1<n<m,\\ ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H(u,v,t) + t^{-\frac{n-1}{2q}}(\lambda_n+\omega_n)\mathcal K(u,v), & 1<n=m,\\ ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H(u,v,t) + t^{-\frac{m-1}{2q}}\omega_m \mathcal K(u,v), & n>m\geq 2, \end{cases}\end{gathered}$$ with $\mathcal K(u,v)\equiv {\lambda_n v^2}/{|2\eta|}+ ({\hbox{\rm sgn}}\, \eta) u v$ and the parameter $\chi_1$ defined by [\[chi1\]](#chi1){reference-type="eqref" reference="chi1"}. Note that if $n>1$, $$\begin{gathered} \mathcal L(u,v,t) = t^{\frac{n-1}{2q}}\frac{|\eta|u^2}{2}+ \frac{|\nu_n|v^2}{2}-({\hbox{\rm sgn}}\, \eta) \lambda_n uv +\mathcal O(\Delta^3)+\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}})\end{gathered}$$ as $\Delta\to 0$ and $t\to\infty$. It follows that there exist $\Delta_1>0$ and $t_1\geq t_0$ such that $\mathcal L(u,v,t)$ satisfies the inequalities [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"} for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_1$ and $t\geq t_1$ with some $L_\pm={\hbox{\rm const}}>0$. The derivative of $\mathcal L(u,v,t)$ with respect to $t$ along the trajectories of system [\[uvsys1\]](#uvsys1){reference-type="eqref" reference="uvsys1"} is given by $$\begin{gathered} \label{dLuv1} \frac{d\mathcal L}{dt}\Big|_\eqref{uvsys1}\equiv \mathcal D_{1}(u,v,t)+\mathcal D_{2,N}(u,v,t),\end{gathered}$$ where $\mathcal D_{1}\equiv \left(\partial_t -\partial_u \mathcal H \partial_u +(\partial_v\mathcal H+\Upsilon) \partial_v \right)\mathcal L$ and $\mathcal D_{2,N}\equiv \left(\mathcal P_N\partial_u +\mathcal Q_N\partial_v \right)\mathcal L$. We see that $$\begin{gathered} \mathcal D_1= \begin{cases} -t^{-\frac{1}{2q}}\left( (|\lambda_1|-|\chi_1|) |\eta|u^2+|\chi_1 \nu_1|v^2+\chi_1 |\lambda_1| uv +\mathcal O(\Delta^3)\right)+\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{q}}), & n=1, \\ - t^{-\frac{n}{2q}} |\lambda_n|\left(|\eta|u^2 + |\nu_n| v^2 +\mathcal O(\Delta^3)\right)+\mathcal O(\Delta^2)\mathcal O(t^{-\frac{n+1}{2q}}), & 1<n<m, \\ - t^{-\frac{n}{2q}} |\lambda_n+\omega_n|\left(|\eta|u^2 + |\nu_n| v^2 +\mathcal O(\Delta^3)\right)+\mathcal O(\Delta^2)\mathcal O(t^{-\frac{n+1}{2q}}), & n=m, \\ - t^{-\frac{m}{2q}} |\omega_m|\left(|\eta|u^2 + |\nu_n| v^2 +\mathcal O(\Delta^3)\right)+\mathcal O(\Delta^2)\mathcal O(t^{-\frac{m+1}{2q}}), & n>m, \end{cases}\end{gathered}$$ and $\mathcal D_{2,N}(u,v,t) = \mathcal O(\Delta)\mathcal O(t^{-\frac{N-n+2}{2q}})$ as $\Delta\to 0$ and $t\to\infty$. It follows that there exist $N_0=\min\{2n-1,n+m-1\}$, $\Delta_2\leq \Delta_1$ and $t_2\geq t_1$ such that $$\begin{gathered} \mathcal D_{1}(u,v,t)\leq - t^{-\frac{\ell}{2q}} \gamma \Delta^2, \quad \mathcal D_{2,N}(u,v,t)\leq t^{-\frac{\ell+1}{2q}} C \Delta\end{gathered}$$ for all $N\geq N_0$ and $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_2$ and $t\geq t_2$, where $C={\hbox{\rm const}}>0$ and $\ell=\min\{n,m\}$. If $n=1$, then $\gamma=\min\{A_1,B_1\}/2$, and if $n>1$, then $\gamma=|d_{n,m}|\min\{|\eta|,|\nu_n|\}/2$. Positive parameters $A_1$ and $B_1$ are defined by [\[A1B1\]](#A1B1){reference-type="eqref" reference="A1B1"}. Hence, for all $\epsilon\in (0,\Delta_2)$ there exist $$\begin{gathered} \delta_\epsilon=\frac{2C }{\gamma} t_\epsilon^{-\frac{1}{2q}}, \quad t_\epsilon=\max\left\{t_2,\left(\frac{4 C }{\gamma\epsilon}\right)^{2q}\right\}\end{gathered}$$ such that $$\begin{gathered} \frac{d\mathcal L}{dt}\Big|_\eqref{uvsys1}\leq t^{-\frac{\ell}{2q}}\left(-\gamma+C\delta_\epsilon^{-1}t_\epsilon^{-\frac{1}{2q}}\right)\Delta^2 \leq 0\end{gathered}$$ for all $(u,v,t)\in\mathbb R^3$ such that $\delta_\epsilon\leq \Delta\leq \epsilon$ and $t\geq t_\epsilon$. Taking into account [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"}, we see that any solution of system [\[uvsys1\]](#uvsys1){reference-type="eqref" reference="uvsys1"} with initial data $\sqrt{u^2(t_\epsilon)+v^2(t_\epsilon)}\leq \delta$ and $\delta=\max\{\delta_\epsilon,\epsilon\sqrt{L_-/L_+}\}$ cannot exit from the domain $\{(u,v)\in\mathbb R^2: \Delta\leq \epsilon\}$ as $t\geq t_\epsilon$. Thus, returning to the original variables and taking into account Theorem [Theorem 1](#Th1){reference-type="ref" reference="Th1"} complete the proof. ◻ *Proof of Theorem [Theorem 3](#Th21){reference-type="ref" reference="Th21"}.* Substituting $\varrho(t)=\varrho_{\ast,M}(t)+u(t)$, $\phi(t)=\phi_{\ast,M}(t)+v(t)$ into [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"}, we obtain $$\begin{gathered} \label{uvsys21} \frac{du}{dt}=-\partial_v H_M(u,v,t) + P_{M,N}(u,v,t), \quad \frac{dv}{dt}=\partial_u H_M(u,v,t) + Y_M(u,v,t) + Q_{M,N}(u,v,t), \end{gathered}$$ with $$\begin{aligned} H_M(u,v,t) &\equiv \int\limits_0^u \mathcal B_M(w,0,t)\,dw - \int\limits_0^v \mathcal A_M(u,w,t)\,dw,\\ Y_M(u,v,t) &\equiv \int\limits_0^v \left(\partial_u\mathcal A_M(u,w,t)+\partial_v\mathcal B_M(u,w,t)\right)\,dw,\\ \mathcal A_M(u,v,t) & \equiv \hat\Lambda_N(\varrho_{\ast,M}(t)+u,\phi_{\ast,M}(t)+v,t)-\hat\Lambda_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t),\\ \mathcal B_M(u,v,t) & \equiv \hat\Omega_N(\varrho_{\ast,M}(t)+ u,\phi_{\ast,M}(t)+v,t)-\hat\Omega_N(\varrho_{\ast,M}(t),\phi_{\ast,M}(t),t),\\ P_{M,N}(u,v,t) &\equiv \tilde \Lambda_N(\varrho_{\ast,M}(t)+u,\phi_{\ast,M}(t)+v,S(t),t) - Z_\varrho(t),\\ Q_{M,N}(u,v,t) &\equiv \tilde \Omega_N(\varrho_{\ast,M}(t)+u,\phi_{\ast,M}(t)+v,S(t),t) - Z_\phi(t),\end{aligned}$$ where the functions $Z_\varrho(t)$ and $Z_\phi(t)$ are defined by [\[Zeq\]](#Zeq){reference-type="eqref" reference="Zeq"}. It can easily be checked that $$\begin{aligned} H_M(u,v,t)&=\left\{t^{-\frac{1}{2q}}\frac{\eta u^2}{2}-t^{-\frac{n}{2q}}\left(\lambda_n u v+ \frac{\nu_n v^2}{2}\right)\right\}\left(1+\mathcal O(\Delta)+\mathcal O(t^{-\frac{1}{2q}})\right),\\ Y_M(u,v,t)&=v\left(\lambda_n t^{-\frac{n}{2q}}+\omega_m t^{-\frac{m}{2q}}\right) \left(1+\mathcal O(\Delta)+\mathcal O(t^{-\frac{1}{2q}})\right),\\ P_{M,N}(u,v,t)&=\mathcal O(t^{-\frac{N+1}{2q}})+\mathcal O(t^{-\frac{2n+M+2}{2q}}), \\ Q_{M,N}(u,v,t)&=\mathcal O(t^{-\frac{N+1}{2q}})+\mathcal O(t^{-\frac{n+m+M+1}{2q}}), \quad \Delta=\sqrt{u^2+v^2}\to 0, \quad t\to\infty.\end{aligned}$$ Note that if $P_{M,N}(u,v,t)\equiv Q_{M,N}(u,v,t)\equiv 0$, then system [\[uvsys21\]](#uvsys21){reference-type="eqref" reference="uvsys21"} has the equilibrium $(0,0)$. The functions $P_{M,N}(u,v,t)$ and $Q_{M,N}(u,v,t)$ do not vanish at the equilibrium and play the role of external perturbations in the system. Let us prove the stability of the perturbed system [\[uvsys21\]](#uvsys21){reference-type="eqref" reference="uvsys21"} by the Lyapunov function method. Consider the Lyapunov function $\mathcal L(u,v,t)$ in the form [\[LFTh\]](#LFTh){reference-type="eqref" reference="LFTh"}, with $H_M(u,v,t)$ instead of $\mathcal H(u,v,t)$. Note that $\mathcal L(u,v,t)$ satisfies [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"} for all $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_1$ and $t\geq t_1$ with some $L_\pm={\hbox{\rm const}}>0$, $\Delta_1>0$ and $t_1\geq t_0$. The derivative of $\mathcal L(u,v,t)$ with respect to $t$ along the trajectories of system is given by $$\begin{gathered} \frac{d\mathcal L}{dt}\Big|_\eqref{uvsys21}\equiv \mathcal D_{1,M}(u,v,t)+\mathcal D_{2,M,N}(u,v,t),\end{gathered}$$ where $\mathcal D_{1,M}\equiv \left(\partial_t -\partial_v H_M \partial_u +(\partial_u H_M+Y_M)\partial_v \right)\mathcal L$ and $\mathcal D_{2,M,N}\equiv \left(P_{M,N}\partial_u +Q_{M,N} \partial_v \right)\mathcal L$. Note that the following estimates hold: $$\begin{aligned} & \mathcal D_{1,M}(u,v,t)\geq t^{-\frac{1}{2q}} (A_1 u^2 + B_1 v^2 +\mathcal O(\Delta^3)+\mathcal O(t^{-\frac{1}{2q}})\mathcal O(\Delta^2)) \quad \text{if} \quad n=1,\\ &\mathcal D_{1,M}(u,v,t)= |d_{n,m}| t^{-\frac{\ell}{2q}} \left(|\eta|u^2 + |\nu_n| v^2 +\mathcal O(\Delta^3)+\mathcal O(\Delta^2)\mathcal O(t^{-\frac{1}{2q}})\right)\quad \text{if} \quad n>1, \\ & \mathcal D_{2,M,N}(u,v,t) =\mathcal O(\Delta) \mathcal O(t^{-\frac{n+M+1}{2q}})+\mathcal O(\Delta) \mathcal O(t^{-\frac{N-n+2}{2q}})\end{aligned}$$ as $\Delta\to 0$ and $t\to\infty$, where $\ell=\min\{m,n\}$ and positive parameters $A_1$, $B_1$ are defined by [\[A1B1\]](#A1B1){reference-type="eqref" reference="A1B1"}. Hence, there exist $N_0=\min\{2n-1,n+m-1\}$, $C>0$, $t_2\geq t_1$ and $\epsilon\in (0,\Delta_1)$ such that $$\begin{gathered} \frac{d\mathcal L}{dt}\Big|_\eqref{uvsys21}\geq t^{-\frac{\ell}{2q}}\left(\gamma \Delta^2\ -C t^{-\frac{1}{2q}}\Delta\right) \end{gathered}$$ for all $N\geq N_0$ and $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \epsilon$ and $t\geq t_2$, where $\gamma=\min\{A_1,B_1\}/2$ if $n=1$, and $\gamma=|d_{n,m}|\min\{|\eta|,|\nu_n|\}/2$ if $n>1$. Hence, for all $\delta\in (0,\epsilon)$ there is $t_\ast=\max\{t_2,(2C/|\delta d_{n,m}|)^{2q}\}$ such that $$\begin{gathered} \frac{d\mathcal L}{dt}\Big|_\eqref{uvsys21}\geq t^{-\frac{\ell}{2q}} \frac{\gamma}{2}\Delta^2\geq t^{-\frac{\ell+n-1}{2q}}\tilde \gamma\mathcal L\end{gathered}$$ for all $(u,v,t)\in\mathbb R^3$ such that $\delta\leq \Delta\leq \epsilon$ and $t\geq t_\ast$ with $\tilde \gamma=\gamma/(2L_+)$. Recall that $\ell+n-1<2q$. Then, integrating the last inequality and taking $u(t_\ast)$, $v(t_\ast)$ such that $\sqrt{u^2(t_\ast)+v^2(t_\ast)}=\delta$, we obtain $$\begin{gathered} u^2(t)+ v^2(t)\geq \frac{\delta^2 L_-}{L_+} t^{-\frac{n-1}{2q}} \exp \left\{ \frac{2q \tilde \gamma}{2q-\ell-n+1}\left(t^{1-\frac{\ell+n-1}{2q}}-t_\ast^{1-\frac{\ell+n-1}{2q}}\right)\right\}, \quad t\geq t_\ast.\end{gathered}$$ Hence, there exists $t_e>t_\ast$ such that $u^2(t_e)+ v^2(t_e)\geq \epsilon^2$. Returning to the variables $r(t)$, $\varphi(t)$, we obtain the result of the Theorem. ◻ *Proof of Theorem [Theorem 4](#Th23){reference-type="ref" reference="Th23"}.* Substituting $\varrho(t)=\varrho_{\ast}(t)+u(t)$, $\phi(t)=\phi_{\ast}(t)+ v(t)$ into [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"}, we obtain system [\[uvsys1\]](#uvsys1){reference-type="eqref" reference="uvsys1"}. It follows from [\[tildeLO\]](#tildeLO){reference-type="eqref" reference="tildeLO"}, [\[assol\]](#assol){reference-type="eqref" reference="assol"} and [\[asst\]](#asst){reference-type="eqref" reference="asst"} that the functions $\mathcal H(u,v,t)$, $\mathcal P_N(u,v,t)$ and $\mathcal Q_N(u,v,t)$ satisfy [\[HUPQest\]](#HUPQest){reference-type="eqref" reference="HUPQest"}, while the function $\Upsilon(u,v,t)$ satisfies the following estimate: $$\begin{gathered} \Upsilon(u,v,t)=t^{-\frac{h}{2q}}d_h v \left(1+\mathcal O(\Delta)+\mathcal O(t^{-\frac{1}{2q}})\right), \quad \Delta=\sqrt{u^2+v^2}\to 0, \quad t\to\infty.\end{gathered}$$ Consider a Lyapunov function candidate in the form $$\begin{gathered} \label{LFTh2} \mathcal L(u,v,t)\equiv \begin{cases} ({\hbox{\rm sgn}}\, \eta)t^{\frac{1}{2q}} \mathcal H(u,v,t)+t^{-\frac{h-1}{2q}}\frac{d_h({\hbox{\rm sgn}}\, \eta)}{2}u v , & h>n=1,\\ ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H(u,v,t)+t^{-\frac{h-1}{2q}} d_h ({\hbox{\rm sgn}}\, \eta)u v, & h>n>1,\\ ({\hbox{\rm sgn}}\, \eta)t^{\frac{n}{2q}} \mathcal H(u,v,t) +t^{-\frac{h-1}{2q}} d_h\left\{({\hbox{\rm sgn}}\, \eta)u v+\frac{\lambda_n v^2}{2|\eta|}\right \} , & h\leq n. \end{cases}\end{gathered}$$ It can easily be checked that there exist $\Delta_1>0$ and $t_1\geq t_0$ such that $\mathcal L(u,v,t)$ satisfies the inequalities [\[Lnineq1\]](#Lnineq1){reference-type="eqref" reference="Lnineq1"} for all $\Delta\leq \Delta_1$ and $t\geq t_1$ with some $L_\pm={\hbox{\rm const}}>0$. The total derivative of $\mathcal L(u,v,t)$ with respect to $t$ along the trajectories of system [\[uvsys1\]](#uvsys1){reference-type="eqref" reference="uvsys1"} is given by [\[dLuv1\]](#dLuv1){reference-type="eqref" reference="dLuv1"}, where $$\begin{gathered} \mathcal D_{1}(u,v,t)= \begin{cases} t^{-\frac{h}{2q}}\frac{d_h}{2}(|\eta|u^2+|\nu_1|v^2)(1 +\mathcal O(\Delta)+\mathcal O(t^{-\frac{1}{q}})), & n=1, \\ t^{-\frac{h}{2q}}d_h(|\eta|u^2+|\nu_1|v^2)(1+\mathcal O(\Delta) +\mathcal O(t^{-\frac{1}{q}})), & n\neq 1 \end{cases}\end{gathered}$$ and $\mathcal D_{2,N}(u,v,t) = \mathcal O(t^{-\frac{N-n+2}{2q}})\mathcal O(\Delta)$ as $\Delta\to 0$ and $t\to\infty$. It follows that there exist $N_0=n+h-1$, $\Delta_2\leq \Delta_1$ and $t_2\geq t_1$ such that $\mathcal D_{1}(u,v,t)\leq - t^{-\frac{h}{2q}} \gamma \Delta^2$, $\mathcal D_{2}(u,v,t)\leq t^{-\frac{h+1}{2q}} C \Delta$ for all $N\geq N_0$ and $(u,v,t)\in\mathbb R^3$ such that $\Delta\leq \Delta_2$ and $t\geq t_2$, where $C={\hbox{\rm const}}>0$, and $\gamma_h=|d_h|/4$. By repeating the steps of the proof of Theorem [Theorem 2](#Th2){reference-type="ref" reference="Th2"}, we see that for all $\epsilon\in (0,\Delta_2)$ there exist $0<\delta_\epsilon<\epsilon$ and $t_\epsilon\geq t_2$ such that any solution of system [\[uvsys1\]](#uvsys1){reference-type="eqref" reference="uvsys1"} with initial data $\sqrt{u^2(t_\epsilon)+v^2(t_\epsilon)}\leq \delta$ and $\delta=\max\{\delta_\epsilon,\epsilon\sqrt{L_-/L_+}\}$ cannot exit from the domain $\{(u,v)\in\mathbb R^2: \Delta\leq \epsilon\}$ as $t\geq t_\epsilon$. Returning to the original variables, we obtain the result of the Theorem. ◻ *Proof of Theorem [Theorem 5](#Th24){reference-type="ref" reference="Th24"}.* Substituting $\varrho(t)=\varrho_{\ast,M}(t)+u(t)$, $\phi(t)=\phi_{\ast,M}(t)+v(t)$ into [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"}, we obtain system [\[uvsys21\]](#uvsys21){reference-type="eqref" reference="uvsys21"}. In this case $$\begin{gathered} Y_M(u,v,t)=v\left(\lambda_n t^{-\frac{n}{2q}}+\omega_m t^{-\frac{m}{2q}}\right) \left(1+\mathcal O(\Delta)+\mathcal O(t^{-\frac{1}{2q}})\right), \quad \Delta=\sqrt{u^2+v^2}\to 0, \quad t\to\infty.\end{gathered}$$ Then, by repeating the proof of Theorem [Theorem 3](#Th21){reference-type="ref" reference="Th21"} with the Lyapunov function in the form [\[LFTh2\]](#LFTh2){reference-type="eqref" reference="LFTh2"}, with $H_M(u,v,t)$ instead of $\mathcal H(u,v,t)$, we obtain the result of the Theorem. ◻ *Proof of Theorem [Theorem 6](#Th3){reference-type="ref" reference="Th3"}.* It follows from the first equation in [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"} and assumption [\[asnzero\]](#asnzero){reference-type="eqref" reference="asnzero"} that for all $D>0$ there exist $t_1\geq t_0$ and $C_1>0$ such that $|{d\rho}/{dt}|\geq t^{-\frac{n}{2q}}C_1$ for all $|\rho|\leq 4D$, $\psi\in\mathbb R$ and $t\geq t_1$. Integrating this inequality yields $|\rho(t)-\rho(t_1)|\geq C(t)> 0$ as $t> t_1$, where $$\begin{gathered} C(t)\equiv \begin{cases} \frac{2q C_1}{2q-n} \left(t^{1-\frac{n}{2q}}-t_1^{1-\frac{n}{2q}}\right), & n<2q,\\ C_1\left(\log t-\log t_1\right), & n=2q. \end{cases}\end{gathered}$$ Hence, for all initial data $|\rho(t_1)|\leq D/2$ and $\psi(t_1)\in\mathbb R$ there exists $t_2\geq t_1$ such that $|\rho(t)|\geq D$ as $t\geq t_2$. Combining this with the second equation in [\[rhopsi\]](#rhopsi){reference-type="eqref" reference="rhopsi"}, we see that there exist $t_3\geq t_2$ and $C_2>0$ such that $|{d\psi}/{dt} |\geq t^{-\frac{1}{2q}}C_2$ for all $D\leq |\rho|\leq 2D$, $\psi\in\mathbb R$ and $t\geq t_3$. Then, by integration, we have $$\begin{gathered} |\psi(t)-\psi(t_3)|\geq \frac{2q C_2}{2q-1} \left(t^{1-\frac{1}{2q}}-t_3^{1-\frac{1}{2q}}\right), \quad t\geq t_3.\end{gathered}$$ Therefore, for all initial data $D\leq |\rho(t_3)|\leq 3D/2$ and $|\psi(t_3)|\leq D/2$ there exists $t_4>t_3$ such that $|\rho(t)|\geq 2D$ and $|\psi(t)|\geq D$ as $t\geq t_4$. ◻ # Examples {#sEx} In this section, we show how the proposed theory can be applied to examples of oscillatory systems with time-decaying perturbations. In particular, the conditions were obtained for the parameters of perturbations that guarantee the existence of a stable phase-locking regime with a resonant amplitude. The results are illustrated with numerical simulations. The last example analyzes the perturbed Duffing oscillator discussed in Section [1](#sec1){reference-type="ref" reference="sec1"}. ## Example 1 Consider the system $$\begin{gathered} \label{Ex1} \frac{dr}{dt}=t^{-\frac{1}{2}} f_1(r,\varphi,S(t)), \quad \frac{d\varphi}{dt}=\omega(r)+t^{-\frac{1}{2}}g_1(r,\varphi,S(t))\end{gathered}$$ where $$\begin{gathered} f_1(r,\varphi,S) \equiv \beta(S) r \sin^2\varphi - \mu(S)\sin\varphi,\quad g_1(r,\varphi,S) \equiv \beta(S) \sin\varphi \cos\varphi - \frac{\mu(S)\cos\varphi}{r},\\ \omega(r)\equiv 1-\vartheta r^2, \quad \beta(S)\equiv \beta_0+\beta_1\sin S, \quad \mu(S)\equiv \mu_0+\mu_1 \sin S, \quad S(t) \equiv s_0 t + s_1 t^{\frac{1}{2}},\end{gathered}$$ with constant parameters $s_k$, $\vartheta>0$, $\beta_k$ and $\mu_k$. We see that system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} has the form [\[PS\]](#PS){reference-type="eqref" reference="PS"} with $q=2$, $\mathcal R=\vartheta^{-1/2}$, $f(r,\varphi,S(t),t)\equiv t^{-1/2}f_1(r,\varphi,S(t))$ and $g(r,\varphi,S(t),t)\equiv t^{-1/2}g_1(r,\varphi,S(t))$. Note also that in the Cartesian coordinates $x=r\cos \varphi$, $y=-r\sin\varphi$ this system takes the form $$\begin{aligned} &\frac{dx}{dt}=(1-\vartheta (x^2+y^2)) y, \\ &\frac{dy}{dt}=-(1-\vartheta (x^2+y^2))x + t^{-\frac{1}{2}}Z(x,y,S(t)),\end{aligned}$$ where $Z(x,y,S)\equiv \mu(S)+\beta(S) y$. 1\. Let $s_0=1/2$. Then, there exist $\kappa=\varkappa=1$, $a=(2\vartheta)^{-1/2}$ such that the resonance condition [\[rc\]](#rc){reference-type="eqref" reference="rc"} holds with $\eta=-\sqrt{2\vartheta}<0$. It can easily be checked that the change of variables described in Theorem [Theorem 1](#Th1){reference-type="ref" reference="Th1"} with $N=2$ transforms the system to $$\begin{gathered} \begin{split}\label{Ex1LO} &\frac{d\rho}{dt}=t^{-\frac{1}{4}}\Lambda_1(\rho,\psi)+t^{-\frac{1}{2}}\Lambda_2(\rho,\psi)+\tilde \Lambda_2(\rho,\psi,S(t),t), \\ &\frac{d\psi}{dt}=t^{-\frac{1}{4}}\Omega_1(\rho,\psi)+t^{-\frac{1}{2}}\Omega_2(\rho,\psi)+\tilde\Omega_2(\rho,\psi,S(t),t), \end{split}\end{gathered}$$ where $$\begin{aligned} &\Lambda_1(\rho,\psi)\equiv \frac 12 \left(\frac{\beta_0}{\sqrt{2\vartheta}}-\mu_1 \cos\psi\right), &\quad &\Lambda_2(\rho,\psi)\equiv \frac{\beta_0 \rho}{2},\\ &\Omega_1(\rho,\psi)\equiv -\sqrt{2\vartheta} \rho, &\quad &\Omega_2(\rho,\psi)\equiv \frac12 \left(-2 \vartheta \rho^2 - s_1 + \mu_1 \sqrt{2\vartheta} \sin\psi\right),\end{aligned}$$ and $\tilde \Lambda_2(\rho,\psi,S,t)=\mathcal O(t^{-1})$, $\tilde \Omega_2(\rho,\psi,S,t)=\mathcal O(t^{-1})$ as $t\to\infty$ uniformly for all $|\rho|<\infty$, $(\psi,S)\in\mathbb R^2$. It is readily seen that assumption [\[asn\]](#asn){reference-type="eqref" reference="asn"} holds with $n=1$ and $m=2$. If $\mu_1\neq 0$ and $|\beta_0/\mu_1|<\sqrt{2\vartheta}$, then assumption [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} holds with $$\begin{gathered} \psi_0=\pm \theta_0 + {2\pi}k, \quad k\in\mathbb Z, \quad \nu_1=\pm\frac{\mu_1}{2}\sin \theta_0, \quad \theta_0=\arccos \left(\frac{\beta_0}{\sqrt{2\vartheta} \mu_1 }\right).\end{gathered}$$ From Lemma [Lemma 1](#Lem01){reference-type="ref" reference="Lem01"} it follows that if $\pm\mu_1<0$, then the equilibria $(0,\pm\theta_0({\hbox{\rm mod}} {2\pi}))$ in the corresponding limiting system are unstable. Hence, the associated regime is not realized in the full system. Note that $d_{n,m}=\partial_\rho \Lambda_1(0,\psi_0)=0$. However, assumption [\[asst\]](#asst){reference-type="eqref" reference="asst"} holds with $h=2$ and $d_h=\beta_0$. It follows from Lemma [Lemma 3](#Lem2){reference-type="ref" reference="Lem2"} and Theorem [Theorem 4](#Th23){reference-type="ref" reference="Th23"} that if $\pm\mu_1>0$ and $-|\mu_1|\sqrt{2\vartheta}<\beta_0<0$, then a stable phase locking regime with $r(t)\approx a$ and $\varphi(t)\approx S(t)\pm\theta_0({\hbox{\rm mod}} {2\pi})$ occurs in system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"}. From Theorem [Theorem 5](#Th24){reference-type="ref" reference="Th24"} it follows that if $\pm\mu_1>0$ and $0<\beta_0<|\mu_1|\sqrt{2\vartheta}$, this regime is unstable. If $\mu_1\neq 0$, $|\beta_0/\mu_1|>\sqrt{2\vartheta}$ or $\mu_1=0$, $\beta_0\neq 0$, then assumption [\[asnzero\]](#asnzero){reference-type="eqref" reference="asnzero"} holds. It follows from Theorem [Theorem 6](#Th3){reference-type="ref" reference="Th3"} that, in this case, the asymptotic regime with $r(t)\approx a$ does not occur (see Fig. [2](#FigEx11){reference-type="ref" reference="FigEx11"}). ![The evolution of $r(t)$ and $\theta(t)\equiv \varphi(t)-S(t)$ for solutions to system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} with $s_0=1/2$, $s_1=1$, $\vartheta=1/4$, $\beta_1=1/2$, $\mu_0=-1/2$ with different values of the parameters $\beta_0$ and $\mu_1$. The dashed curves correspond to $r(t)\equiv a$ and $\theta(t)\equiv \theta_0$, where $a=\sqrt 2$ and $\theta_0=3\pi/4$.](Ex11r.eps "fig:"){#FigEx11 width="0.4\\linewidth"} ![The evolution of $r(t)$ and $\theta(t)\equiv \varphi(t)-S(t)$ for solutions to system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} with $s_0=1/2$, $s_1=1$, $\vartheta=1/4$, $\beta_1=1/2$, $\mu_0=-1/2$ with different values of the parameters $\beta_0$ and $\mu_1$. The dashed curves correspond to $r(t)\equiv a$ and $\theta(t)\equiv \theta_0$, where $a=\sqrt 2$ and $\theta_0=3\pi/4$.](Ex11p.eps "fig:"){#FigEx11 width="0.4\\linewidth"} 2\. Let $s_0=1$. Then, there are $\kappa=1$, $\varkappa=2$, $a=(2\vartheta)^{-1/2}$ such that condition [\[rc\]](#rc){reference-type="eqref" reference="rc"} holds with $\eta=-\sqrt{2\vartheta}<0$. In this case, the transformation constructed in Theorem [Theorem 1](#Th1){reference-type="ref" reference="Th1"} with $N=2$ reduces system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} to [\[Ex1LO\]](#Ex1LO){reference-type="eqref" reference="Ex1LO"} with $$\begin{aligned} &\Lambda_1(\rho,\psi)\equiv \frac{1}{\sqrt{8\vartheta}} \left(\beta_0+\frac{\beta_1}{2}\sin 2\psi\right), &\quad &\Lambda_2(\rho,\psi)\equiv \frac{\rho}{2} \left(\beta_0+\frac{\beta_1}{2}\sin 2\psi\right),\\ &\Omega_1(\rho,\psi)\equiv -\sqrt{2\vartheta} \rho, &\quad &\Omega_2(\rho,\psi)\equiv \frac14 \left(-4 \vartheta \rho^2 - s_1 + \beta_1 \cos 2\psi\right),\end{aligned}$$ and $\tilde \Lambda_2(\rho,\psi,S,t)=\mathcal O(t^{-1})$, $\tilde \Omega_2(\rho,\psi,S,t)=\mathcal O(t^{-1})$ as $t\to\infty$ uniformly for all $|\rho|<\infty$, $(\psi,S)\in\mathbb R^2$. We see that assumption [\[asn\]](#asn){reference-type="eqref" reference="asn"} holds with $n=1$ and $m=2$. If $\beta_1\neq 0$, $|\beta_0/\beta_1|<1/2$, then the system satisfies [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} with $$\begin{gathered} \psi_0=(-1)^k \theta_0 + \frac{\pi k}{2}, \quad \nu_1= (-1)^k\frac{\beta_1}{\sqrt{8\vartheta}}\cos2\theta_0, \quad k\in\mathbb Z, \quad \theta_0=\frac{1}{2}\arcsin \left(-\frac{2\beta_0}{\beta_1}\right).\end{gathered}$$ It follows from Lemma [Lemma 1](#Lem01){reference-type="ref" reference="Lem01"} that if $(-1)^k\beta_1<0$, then the equilibria $(0, (-1)^k\theta_0+\pi k/2)$, $k\in\mathbb Z$ in the limiting system and the corresponding regime in the full system are unstable. Since $d_{n,m}=\partial_\rho \Lambda_1(0,\psi_0)=0$ and $\partial_\rho \Lambda_2(0,\psi_0)+\partial_\psi\Omega_2(0,\psi_0)=\beta_0$, we see that assumption [\[asst\]](#asst){reference-type="eqref" reference="asst"} holds with $h=2$ and $d_h=\beta_0$. If $(-1)^k\beta_1>0$ and $-|\beta_1|/2<\beta_0<0$, then it follows from Lemma [Lemma 3](#Lem2){reference-type="ref" reference="Lem2"} and Theorem [Theorem 4](#Th23){reference-type="ref" reference="Th23"} that a stable phase locking occurs in the system such that $r(t)\approx a$ and $\varphi(t)\approx S(t)/2+\psi_0$. From Theorem [Theorem 5](#Th24){reference-type="ref" reference="Th24"} it follows that if $(-1)^k\beta_1>0$ and $0<\beta_0<|\beta_1|/2$, this regime is unstable. If $\beta_1\neq 0$, $|\beta_0/\beta_1|>1/2$ or $\beta_1=0$, $\beta_0\neq 0$, then it follows from Theorem [Theorem 6](#Th3){reference-type="ref" reference="Th3"} that the asymptotic regime with $r(t)\approx a$ does not occur (see Fig. [4](#FigEx12){reference-type="ref" reference="FigEx12"}). ![The evolution of $r(t)$ and $\theta(t)\equiv \varphi(t)-S(t)/2$ for solutions to system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} with $s_0=1$, $s_1=\vartheta=\mu_1=1/4$, $\mu_0=0$ with different values of the parameters $\beta_0$ and $\beta_1$. The dashed curves correspond to $r(t)\equiv a$ and $\theta(t)\equiv \theta_0$, where $a=\sqrt 2$ and $\theta_0=\pi/12$.](Ex12r.eps "fig:"){#FigEx12 width="0.4\\linewidth"} ![The evolution of $r(t)$ and $\theta(t)\equiv \varphi(t)-S(t)/2$ for solutions to system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} with $s_0=1$, $s_1=\vartheta=\mu_1=1/4$, $\mu_0=0$ with different values of the parameters $\beta_0$ and $\beta_1$. The dashed curves correspond to $r(t)\equiv a$ and $\theta(t)\equiv \theta_0$, where $a=\sqrt 2$ and $\theta_0=\pi/12$.](Ex12p.eps "fig:"){#FigEx12 width="0.4\\linewidth"} 3\. Finally, let $s_0=1/4$. Then, there exist $\kappa=2$, $\varkappa=1$ and $a=(2\vartheta)^{-1/2}$ such that the resonance condition holds with $\eta=-\sqrt{2\vartheta}<0$. Note that the transformation described in Theorem [Theorem 1](#Th1){reference-type="ref" reference="Th1"} with $N=1$ reduces system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} to $$\begin{aligned} &\frac{d\rho}{dt}=t^{-\frac{1}{4}}\Lambda_1(\rho,\psi)+\tilde \Lambda_1(\rho,\psi,S(t),t), \\ &\frac{d\psi}{dt}=t^{-\frac{1}{4}}\Omega_1(\rho,\psi)+\tilde\Omega_1(\rho,\psi,S(t),t),\end{aligned}$$ with $\Lambda_1(\rho,\psi)\equiv {a\beta_0}/{2}$, $\Omega_1(\rho,\psi)\equiv -\sqrt{2\vartheta} \rho$, and $\tilde \Lambda_1(\rho,\psi,S,t)=\mathcal O(t^{-1})$, $\tilde \Omega_1(\rho,\psi,S,t)=\mathcal O(t^{-1})$ as $t\to\infty$ uniformly for all $|\rho|<\infty$, $(\psi,S)\in\mathbb R^2$. It follows from Theorem [Theorem 6](#Th3){reference-type="ref" reference="Th3"} that if $\beta_0\neq 0$, the asymptotic regime with $r(t)\approx a$ does not occur. In this case, the behaviour of system [\[Ex1\]](#Ex1){reference-type="eqref" reference="Ex1"} is qualitatively independent of the oscillatory part of the perturbations. ## Example 2 Consider the following system: $$\begin{gathered} \label{Ex2} \begin{split} &\frac{dr}{dt}=t^{-\frac{1}{2}} f_1(r,\varphi,S(t))+t^{-1} f_2(r,\varphi,S(t)), \\ &\frac{d\varphi}{dt}=\omega(r)+t^{-\frac{1}{2}}g_1(r,\varphi,S(t))+t^{-1}g_2(r,\varphi,S(t)) \end{split}\end{gathered}$$ where $$\begin{aligned} & f_1(r,\varphi,S) \equiv -\alpha(S) r^3 \sin\varphi \cos^3\varphi,\quad && f_2(r,\varphi,S) \equiv \beta(S) r \sin^2\varphi,\\ & g_1(r,\varphi,S) \equiv -\alpha(S) r^2 \cos^4\varphi, \quad && g_2(r,\varphi,S) \equiv \frac{\beta(S)}{2} \sin 2\varphi,\\ & \alpha(S)\equiv \alpha_0+\alpha_1\sin S,\quad && \beta(S)\equiv \beta_0+\beta_1\sin S,\\ & \omega(r)\equiv 1-\vartheta r^2, \quad && S(t) \equiv s_0 t + s_1 t^{\frac{1}{2}}+ s_2 \log t\end{aligned}$$ with constant parameters $s_k$, $\vartheta>0$, $\alpha_k$, $\beta_k$, $\alpha_1\neq 0$ It can be easily seen that system [\[Ex2\]](#Ex2){reference-type="eqref" reference="Ex2"} has the form [\[PS\]](#PS){reference-type="eqref" reference="PS"} with $q=2$, $\mathcal R=\vartheta^{-1/2}$, $f(r,\varphi,S(t),t)\equiv t^{-1/2}f_1(r,\varphi,S(t))+t^{-1}f_2(r,\varphi,S(t))$ and $g(r,\varphi,S(t),t)\equiv t^{-1/2}g_1(r,\varphi,S(t))+t^{-1}g_2(r,\varphi,S(t))$. In the Cartesian coordinates $x=r\cos \varphi$, $y=-r\sin\varphi$ this system takes the form $$\begin{aligned} &\frac{dx}{dt}=(1-\vartheta (x^2+y^2)) y, \\ & \frac{dy}{dt}=-(1-\vartheta (x^2+y^2))x + t^{-\frac{1}{2}}\alpha(S(t))x^3+ t^{-1}\beta(S(t))y.\end{aligned}$$ Let $s_0=1/2$. Then, there exist $\kappa=1$, $\varkappa=2$, $a=(2\vartheta)^{-1/2}$ such that the resonance condition [\[rc\]](#rc){reference-type="eqref" reference="rc"} holds with $\eta=-\sqrt{2\vartheta}<0$. It can easily be checked that the change of variables described in Theorem [Theorem 1](#Th1){reference-type="ref" reference="Th1"} with $N=4$ transforms the system to $$\begin{gathered} \begin{split}\label{Ex2LO} &\frac{d\rho}{dt}=\sum_{i=1}^4 t^{-\frac{i}{4}}\Lambda_i(\rho,\psi)+\tilde \Lambda_4(\rho,\psi,S(t),t), \\ &\frac{d\psi}{dt}=\sum_{i=1}^4 t^{-\frac{i}{4}}\Omega_i(\rho,\psi)+\tilde\Omega_4(\rho,\psi,S(t),t), \end{split}\end{gathered}$$ where $$\begin{aligned} \Lambda_1(\rho,\psi) \equiv& -\frac{a^3 \alpha_1}{8} \cos 2 \psi, \\ \Lambda_2(\rho,\psi) \equiv &-\frac{3a^2 \alpha_1 \rho}{8} \cos 2 \psi,\\ \Lambda_3(\rho,\psi) \equiv& \frac{a }{32}\left(16 \beta_0 - \alpha_1 (12 a^4 \alpha_0 + 12 \rho^2 + 5 a^6 \alpha_0 \vartheta) \cos 2 \psi + 8 \beta_1 \sin 2 \psi + 3 a^4 \alpha_1^2 \sin 4 \psi\right), \\ \Lambda_4(\rho,\psi)\equiv & -\frac{\rho}{64} (\alpha_1 (111 a^4 \alpha_0 + 8 \rho^2 + 336 a^6 \alpha_0 \vartheta + 432 a^8 \alpha_0 \vartheta^2) \cos 2 \psi \\ & - 2 (8 + 16 \beta_0 + 8 \beta_1 \sin 2 \psi + 3 a^4 \alpha_1^2 (5 + 16 a^2 \vartheta) \sin 4 \psi)), \\ \Omega_1(\rho,\psi) \equiv & -\sqrt{2\vartheta} \rho, \\ \Omega_2(\rho,\psi) \equiv & \frac{1}{8} \left(-3 a^2 \alpha_0 - 2 s_1 - 8 \rho^2 \vartheta + 4 a^2 \alpha_1 \cos \psi \sin \psi\right),\\ \Omega_3(\rho,\psi) \equiv & \frac{a\rho}{4} (-3 \alpha_0 + 2 \alpha_1 \sin 2 \psi), \\ \Omega_4(\rho,\psi) \equiv & \frac{1}{3456}(-54 (a^4 (57 \alpha_0^2 + 8 \alpha_1^2) + 24 \alpha_0 \rho^2 + 32 s_2) - 3 a^6 (3537 \alpha_0^2 + 437 \alpha_1^2) \vartheta \\ & - 16 a^8 (918 \alpha_0^2 + 139 \alpha_1^2) \vartheta^2 + 864 \beta_1 \cos 2 \psi \\ & + 54 \alpha_1 (3 a^4 \alpha_1 (3 + 8 a^2 \vartheta) \cos 4 \psi + (16 \rho^2 + a^4 \alpha_0 (67 + a^2 \vartheta (173 + 216 a^2 \vartheta))) \sin 2 \psi))\end{aligned}$$ and $\tilde \Lambda_4(\rho,\psi,S,t)=\mathcal O(t^{-5/4})$, $\tilde \Omega_4(\rho,\psi,S,t)=\mathcal O(t^{-5/4})$ as $t\to\infty$ uniformly for all $|\rho|<\infty$, $(\psi,S)\in\mathbb R^2$. It is readily seen that assumption [\[asn\]](#asn){reference-type="eqref" reference="asn"} holds with $n=1$ and $m=2$. Note that system [\[Ex2LO\]](#Ex2LO){reference-type="eqref" reference="Ex2LO"} satisfies [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} with $$\begin{gathered} \psi_0=\frac{\pi}{4} + \frac{\pi k}{2}, \quad \nu_1= (-1)^k\frac{a^3 \alpha_1}{4}, \quad k\in\mathbb Z.\end{gathered}$$ Since $\eta<0$, it follows from Lemma [Lemma 1](#Lem01){reference-type="ref" reference="Lem01"} that if $(-1)^k \alpha_1<0$, then the equilibrium $(0, \pi/4+\pi k/2)$ is unstable in the limiting system for all $k\in\mathbb Z$. Hence, the corresponding resonant regimes do not occur in the full system. Moreover, we see that $d_{n,m}=\partial_\rho \Lambda_1(0,\psi_0)=0$, $\partial_\rho \Lambda_i(0,\psi_0)+\partial_\psi\Omega_i(0,\psi_0)=0$ for $1\leq i\leq 3$, and the assumption [\[asst\]](#asst){reference-type="eqref" reference="asst"} holds with $h=4$ and $$\begin{gathered} d_h=\frac{1+2\beta_0+(-1)^{k+1}\beta_1}{4}\end{gathered}$$ Thus, if $(-1)^k\alpha_1>0$ and $(-1)^k\beta_1>1+2\beta_0$, it follows from Lemma [Lemma 3](#Lem2){reference-type="ref" reference="Lem2"} and Theorem [Theorem 4](#Th23){reference-type="ref" reference="Th23"} that a stable phase locking occurs in the system such that $r(t)\approx a$ and $\varphi(t)\approx S(t)/2+\psi_0$ (see Fig. [6](#FigEx21){reference-type="ref" reference="FigEx21"}). ![The evolution of $r(t)$ and $\theta(t)\equiv \varphi(t)-S(t)/2$ for solutions to system [\[Ex2\]](#Ex2){reference-type="eqref" reference="Ex2"} with $s_0=1/2$, $s_1=1$, $s_2=0$, $\vartheta=1/4$, $\alpha_0=0.1$, $\alpha_1=0.15$ with different values of the parameters $\beta_0$ and $\beta_1$. The dashed curves correspond to $r(t)\equiv a$ and $\theta(t)\equiv \pi/4$, where $a=\sqrt 2$.](Ex2r.eps "fig:"){#FigEx21 width="0.4\\linewidth"} ![The evolution of $r(t)$ and $\theta(t)\equiv \varphi(t)-S(t)/2$ for solutions to system [\[Ex2\]](#Ex2){reference-type="eqref" reference="Ex2"} with $s_0=1/2$, $s_1=1$, $s_2=0$, $\vartheta=1/4$, $\alpha_0=0.1$, $\alpha_1=0.15$ with different values of the parameters $\beta_0$ and $\beta_1$. The dashed curves correspond to $r(t)\equiv a$ and $\theta(t)\equiv \pi/4$, where $a=\sqrt 2$.](Ex2p.eps "fig:"){#FigEx21 width="0.4\\linewidth"} ## Example 3 Finally, consider again equation [\[Ex0\]](#Ex0){reference-type="eqref" reference="Ex0"}. It was shown in Section [1](#sec1){reference-type="ref" reference="sec1"} that this system correspond to [\[PS\]](#PS){reference-type="eqref" reference="PS"} with $q=2$, $s_0=3/2$, and functions $\omega(r)$, $f(r,\varphi,S,t)$, $g(r,\varphi,S,t)$ defined by [\[omegaeq\]](#omegaeq){reference-type="eqref" reference="omegaeq"} and [\[fgex0\]](#fgex0){reference-type="eqref" reference="fgex0"}. Note that $0<\omega(r)<1$ for all $0<|r|<(2\vartheta)^{-1/2}$ and $\omega(r)=1-3 \vartheta r^2/8-35\vartheta^2 r^4/256+\mathcal O(\vartheta^4)$ as $\vartheta\to 0$. Hence, there exist $\kappa$, $\varkappa\in\mathbb Z_+$ and $0<a<(2\vartheta)^{-1/2}$ such that the condition [\[rc\]](#rc){reference-type="eqref" reference="rc"} holds with $\eta<0$. Let $\kappa=1$ and $\varkappa=2$. Then, the transformations [\[ch1\]](#ch1){reference-type="eqref" reference="ch1"}, [\[ch2\]](#ch2){reference-type="eqref" reference="ch2"} with $N=2$ reduce the system to [\[Ex1LO\]](#Ex1LO){reference-type="eqref" reference="Ex1LO"} with $$\begin{aligned} &\Lambda_1(\rho,\psi)= \frac{a}{4} (2 \beta_0 +\delta_1 \sin (2\psi-\sigma))+\mathcal O(\vartheta), & \quad &\Lambda_2(\rho,\psi)\equiv \frac{\rho}{4} \left(2 \beta_0+\delta_1 \sin (2\psi-\sigma) +\mathcal O(\vartheta)\right),\\ &\Omega_1(\rho,\psi)\equiv \eta \rho, &\quad &\Omega_2(\rho,\psi)\equiv \frac 14 (- 2\alpha_0 + \delta_1 \cos (2 \psi-\sigma))+\mathcal O(\vartheta),\end{aligned}$$ as $\vartheta\to 0$ and $\tilde \Lambda_2(\rho,\psi,S,t)=\mathcal O(t^{-1})$, $\tilde \Omega_2(\rho,\psi,S,t)=\mathcal O(t^{-1})$ as $t\to\infty$ uniformly for all $|\rho|<\infty$, $(\psi,S)\in\mathbb R^2$, where $\delta_1=\sqrt{\alpha_1^2+\beta_1^2}$ and $\sigma=\arcsin (\alpha_1/\delta_1)$. We see that assumption [\[asn\]](#asn){reference-type="eqref" reference="asn"} holds with $n=1$ and $m=2$. If $\delta_1\neq 0$ and $|\beta_0|<\delta_1/2$, then the system satisfies [\[aszero\]](#aszero){reference-type="eqref" reference="aszero"} with $$\begin{gathered} \psi_0=(-1)^j \theta_0 + \frac{\sigma+\pi j}{2}+\mathcal O(\vartheta), \quad \nu_1= (-1)^j\frac{a \delta_1}{2}\cos2\theta_0+\mathcal O(\vartheta), \quad j\in\mathbb Z, \quad \theta_0=\frac{1}{2}\arcsin \left(-\frac{2\beta_0}{\delta_1}\right).\end{gathered}$$ It follows from Lemma [Lemma 1](#Lem01){reference-type="ref" reference="Lem01"} that the equilibria $(0, (\sigma+\pi)/2-\theta_0+ \pi j)$, $j\in\mathbb Z$ in the limiting system and the corresponding regime in the full system are unstable. Since $d_{n,m}=\partial_\rho \Lambda_1(0,\psi_0)=0$ and $\partial_\rho \Lambda_2(0,\psi_0)+\partial_\psi\Omega_2(0,\psi_0)=\beta_0$, we see that assumption [\[asst\]](#asst){reference-type="eqref" reference="asst"} holds with $h=2$ and $d_h=\beta_0+\mathcal O(\vartheta)$ as $\vartheta\to 0$. If $-\delta_1/2<\beta_0<0$, then it follows from Lemma [Lemma 3](#Lem2){reference-type="ref" reference="Lem2"} and Theorem [Theorem 4](#Th23){reference-type="ref" reference="Th23"} that a stable phase locking occurs in the system such that $r(t)\approx a$ and $\varphi(t)\approx S(t)/2+\theta_0+\sigma/2+\pi j$, $j\in\mathbb Z$. From Theorem [Theorem 5](#Th24){reference-type="ref" reference="Th24"} it follows that if $0<\beta_0<\delta_1/2$, this regime is unstable. It follows from Theorem [Theorem 6](#Th3){reference-type="ref" reference="Th3"} that if $\delta_1=0$, $\beta_0\neq 0$ or $\delta_1\neq 0$, $|\beta_0|>\delta_1/2$, then the asymptotic regime with $r(t)\approx a$ does not occur. Note that the root of the equation $\omega(a)=\kappa s_0/\varkappa$ can be found numerically. In particular, if $\vartheta=1/4$, we have $a\approx 1.27$ (see Fig. [7](#figomega){reference-type="ref" reference="figomega"} and Fig. [\[FigEx0\]](#FigEx0){reference-type="ref" reference="FigEx0"}, c). {#figomega width="0.4\\linewidth"} # Conclusion Thus, the resonant effect of damped oscillatory perturbations on non-isochronous systems has been investigated. In particular, we have deduced the model non-autonomous system [\[trsys\]](#trsys){reference-type="eqref" reference="trsys"}, which describes the approximate average dynamics. It turned out that this system is similar to the pendulum-type equations with additional terms decaying in time. Indeed, the truncated limiting system [\[limsys\]](#limsys){reference-type="eqref" reference="limsys"} can be written as $$\begin{gathered} \frac{d^2\hat\phi}{d\tau^2}-\tau^{-\frac{n-1}{2q}}\eta_n \Lambda_n\left(\eta^{-1}\frac{d\hat\phi}{d\tau},\hat\phi\right)=0, \quad \eta_n=\eta \left(\frac{2q}{2q-1}\right)^{\frac{n-1}{2q-1}}, \quad \tau=\left(\frac{2q}{2q-1}\right)t^{1-\frac{1}{2q}},\end{gathered}$$ where $\Lambda_n(\rho,\psi)$ is $2\pi$-periodic with respect to $\psi$. In this case, the additional terms in the model system depends on the perturbations of the oscillatory system. Note that similar but autonomous equations arise in the theory of nonlinear resonance when considering perturbations with a small parameter. [@BVC79; @SUZ88]. The study of the structure of the model system has led to conditions that guarantee the existence of the phase-locking regime with a resonant amplitude. Violation of these conditions can lead to significant phase mismatch and the absence of a corresponding resonant mode. The proposed method is based on long-term asymptotic analysis of the model system and the proof of the stability of the corresponding solutions in the full system using Lyapunov function technique. We have shown that time-decaying perturbations can be used to control the dynamics of nonlinear systems. For example, the perturbation parameters can be chosen to ensure the appearance of near-periodic solutions with a given resonant amplitude. Note also that perturbations of isochronous systems have not been discussed here. In this case, the proposed theory cannot be applied directly due to different form of the model systems. Multi-frequency systems, where the problem of small denominators may arise, have also not been considered in the paper. These problems deserve special attention and will be discussed elsewhere. 99 J. Guckenheimer, P. Holmes, *Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields*, Springer, New York, 1983. A. Fidlin, *Nonlinear Oscillations in Mechanical Engineering*, Springer, Berlin, Heidelberg, New York, 2006. C. Castillo-Chávez, H.R. Thieme, *Asymptotically autonomous epidemic models*. In: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogenity, Theory of Epidemics, vol. 1, Wuertz, 1995, p. 33--50. D. Scarcella, *Weakly asymptotically quasiperiodic solutions for time-dependent Hamiltonians with a view to celestial mechanics*, 2022, arXiv: 2211.06768. A. D. Bruno, I. V. Goryuchkina, *Boutroux asymptotic forms of solutions to Painlevé equations and power geometry*, Doklady Math., 78 (2008), 681--685. V. V. Kozlov, S. D. Furta, *Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations*, Springer, New York, 2013. S. F. Rohmah et al, *Time-dependent damping effect for the dynamics of DNA transcription*, J. Phys.: Conf. Ser., 1204 (2019), 012012. S. Ji, M. Mei, *Optimal decay rates of the compressible Euler equations with time-dependent damping in $\mathbb R^n$: (I) under-damping case*, J Nonlinear Sci., 33 (2023), 7. L. Markus, *Asymptotically autonomous differential systems*. In: S. Lefschetz (ed.), Contributions to the theory of nonlinear oscillations III, Ann. Math. Stud., vol. 36, pp. 17--29, Princeton University Press, Princeton, 1956. J. A. Langa, J. C. Robinson, A. Suárez, *Stability, instability and bifurcation phenomena in nonautonomous differential equations*, Nonlinearity, 15 (2002), 887--903. P. E. Kloeden, S. Siegmund, *Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems*, Internat. J. Bifur. Chaos., 15 (2005), 743--762. M. Rasmussen, *Bifurcations of asymptotically autonomous differential equations*, Set-Valued Anal., 16 (2008), 821--849. O. A. Sultanov, *Stability and bifurcation phenomena in asymptotically Hamiltonian systems*, Nonlinearity, 35 (2022), 2513--2534. L. D. Pustyl'nikov, *Stable and oscillating motions in nonautonomous dynamical systems. A generalization of C. L. Siegel's theorem to the nonautonomous case*, Math. USSR-Sbornik, 23 (1974), 382--404. H. Thieme, *Asymptotically autonomous differential equations in the plane*, Rocky Mountain J. Math., 24 (1994), 351--380. O. A. Sultanov, *Damped perturbations of systems with center-saddle bifurcation*, Internat. J. Bifur. Chaos., 31 (2021), 2150137. W. A. Harris and D. A. Lutz, *Asymptotic integration of adiabatic oscillators*, J. Math. Anal. Appl., 51 (1975), 76--93. M. Pinto, *Asymptotic integration of second-order linear differential equations*, J. Math. Anal. Appl., 111 (1985), 388--406. P. N. Nesterov , *Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients*, Differ. Equ. 43 (2007), 745--756. V. Burd, P. Nesterov, *Parametric resonance in adiabatic oscillators*, Results. Math., 58 (2010), 1--15. O. A. Sultanov, *Asymptotic analysis of systems with damped oscillatory perturbations*, J. Math. Sci. 269 (2023), 111--128. O. A. Sultanov, *Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations*, Discrete & Continuous Dynamical Systems, 41 (2021), 5943--5978. O. A. Sultanov, *Decaying oscillatory perturbations of Hamiltonian systems in the plane*, Journal of Mathematical Sciences, 257 (2021), 705--719. B. V. Chirikov, *A universal instability of many-dimensional oscillator systems*, Physics Reports, 52 (1979), 263--379. R. Z. Sagdeev, D.A. Usikov, G.M. Zaslavsky, *Nonlinear Physics: From the Pendulum to Turbulence and Chaos*, Harwood Academic Publishers, New York, 1988. S. M. Soskin, D. G. Luchinsky, R. Mannella, A. B. Neiman, and P. V. E. McClintock, *Zero-dispersion nonlinear resonance*, Internat. J. Bifur. Chaos., 7 (1997), 923--936. S. Jeyakumari, V. Chinnathambi, S. Rajasekar, and M. A. F. Sanjuan, *Vibrational resonance in an asymmetric Duffing oscillator*, Internat. J. Bifur. Chaos., 21 (2011), 275--286. N. N. Bogolubov, Yu. A. Mitropolsky, *Asymptotic Methods in Theory of Non-linear Oscillations*, Gordon and Breach, New York, 1961. V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, *Mathematical Aspects of Classical and Celestial Mechanics*, Springer, Berlin, 2006. L. A. Kalyakin, *Lyapunov functions in theorems of justification of asymptotics*, Mat. Notes, 98 (2015), 752--764. H. K. Khalil, *Nonlinear Systems*, Prentice Hall, Upper Saddle River, New Jersey, 2002.

arxiv_math

--- abstract: | It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions). author: - João R. Cardoso date: "Received: date / Accepted: date" title: Computing the Mittag-Leffler Function of a Matrix Argument --- example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore # Introduction Given a square complex matrix ${A}\in \mathbb{C}^{n\times n}$ and the real numbers $\alpha>0$ and $\beta$, the matrix Mittag-Leffler (ML) function with two parameters can be defined through the convergent power series $$\label{ml-function} E_{\alpha,\beta}({A})=\sum_{k=0}^{\infty}\frac{{A}^{k}}{\Gamma(\alpha k+\beta)}=\frac{1}{\Gamma(\beta)}{I}+\frac{{A}}{\Gamma(\alpha +\beta)}+\frac{{A}^{2}}{\Gamma(2 \alpha +\beta)}+\cdots,$$ where $\Gamma$ denotes the Euler Gamma function. Note that the ML function is defined for any square matrix ${A}\in \mathbb{C}^{n\times n}$. For background on matrix functions, see the books [@Higham08; @Horn94] and the references therein. The literature on scalar ML functions is vast, so we refer the reader to the books [@Gorenflo14; @Podlubny1] and the references therein for background on its the theory and applications. With respect to the numerical computation, many techniques have been proposed in the literature. We highlight in particular the state-of-the-art method of [@Garrappa15] and also the effective methods of [@Gorenflo02] and [@Seybold08]. In contrast with its scalar counterpart, the ML function with a matrix argument has been less studied, but in the late few years the interest in designing algorithms for its computation has increased [@Garrappa11; @Garrappa17; @Garrappa18; @Higham21; @Moreti11]. In [@Garrappa18], the authors adapt the Schur-Parlett method of [@Davies03] to the particular case of the ML function, which requires the computation of the scalar ML functions $E_{\alpha,\beta}(z)$ and its higher order derivatives $E^{(k)}_{\alpha,\beta}(z)$, with $k=1,2,\ldots.$. At a first glance, this seems to be a trivial task, but the computation of $E^{(k)}_{\alpha,\beta}(z)$ is actually quite challenging. That is why the major part of [@Garrappa18] is devoted to finding effective methods for derivative computations. In [@Garrappa18], one can also find a couple of interesting applications of matrix ML functions in Fractional Calculus. More recently, in [@Higham21 Sec. 6], the authors propose a derivative-free algorithm for the ML matrix function using mixed precision arithmetic. One of the major goals of our paper is to provide a competitive alternative method that is derivative-free and works entirely in IEEE double precision arithmetic. For a matrix $A$ with "small" norm, our method evaluates $E_{\alpha,\beta}({A})$ using the Taylor series [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"}; for general matrices, we start by computing the complex blocked Schur form of $A$ with reordering [@Davies03], $A=UTU^\ast$, where $U$ is unitary and $T$ is triangular with "well separated" blocks $T_{11}, \ldots,T_{qq}$, in its diagonal. Each block $T_{ii}$ is triangular with close eigenvalues and is usually known as "atomic block". Since $$E_{\alpha,\beta}(A)= U\,E_{\alpha,\beta}(T)\,U^\ast,$$ the computation of the ML function of a general square matrix $A$ can be reduced to the computation of the ML function of a triangular matrix $T$. The block diagonal of $E_{\alpha,\beta}(T)$ is $$\mathop{\mathrm{diag}}\left(E_{\alpha,\beta}(T_{11}),\ldots,E_{\alpha,\beta}(T_{qq})\right),$$ with $T$ being viewed as a $q\times q$ block matrix. Once $E_{\alpha,\beta}(T_{ii})$ is evaluated, the remaining entries of $E_{\alpha,\beta}(T)$ may be computed by means of the block Parlett recurrence, which is based on solving Sylvester matrix equations of smaller size; see more details in Section [5.1](#schur-parlett){reference-type="ref" reference="schur-parlett"}. Hence, an important issue now is how to evaluate the ML function of an atomic block in a stable and efficient fashion. An important difference between our method and the one in [@Garrappa18] is in the computation of the ML function of the atomic blocks $E_{\alpha,\beta}(T_{ii})$. While [@Garrappa18] adapts the method of [@Davies03], which requires the efficient computation of higher order derivatives of the ML function, our proposal uses the well-known Cauchy integral formula for matrix functions. This raises a couple of questions, like: how to find a suitable contour to approximate the Cauchy integral that minimizes the quadrature error and the computational cost? What is the most convenient quadrature for this type of integrals? Although our method involves a rough estimate of the second-order derivative of a certain function to estimate the length of the radius of a circle (see Section [4](#contour){reference-type="ref" reference="contour"}), this derivative is not used directly in forming $E_{\alpha, \beta}(A)$, making it legitimate to call it derivative-free. Given a complex square matrix $A$, the matrix version of the Cauchy integral theorem [@Higham08 Ch.1] states that $$\label{cauchy} f(A)=\frac{1}{2\pi i}\int_{\mathcal C} f(z)(zI-A)^{-1}\ dz,$$ where $f$ is assumed to be analytic on and inside a closed contour ${\mathcal C}$ enclosing the spectrum $\sigma(A)$ of $A$. Cauchy's integral formula [\[cauchy\]](#cauchy){reference-type="eqref" reference="cauchy"} has rarely been used in the computation of functions of matrices. However, as pointed out in [@Trefethen], an appropriate choice of the contour ${\mathcal C}$ followed by an evaluation by the trapezoidal rule, can lead to very good results for some particular functions and matrices. In this work, we show that, for matrices with clustered eigenvalues, choosing a circle for the contour (with an appropriate center and radius), and applying the trapezoidal rule, provides a promising method for computing the ML function. This technique has the advantage of being easily extendable to other matrix function, namely to functions whose properties are not suitable for exploiting in numerical computations; this is the case for many special functions. Our interest in the Cauchy integral formula [\[cauchy\]](#cauchy){reference-type="eqref" reference="cauchy"} does not lie specifically in its application for approximating $E_{\alpha,\beta}(A)$ for any square matrix $A$, but instead for computing the ML of the atomic blocks $T_{ii}$ arising in the blocked Schur decomposition of $A$ with reordering. Since these blocks are upper triangular matrices with clustered eigenvalues, the contour circle has a radius that is in general of limited length, thus avoiding high oscillations in the integrand function. Although we often use the terminology "Mittag-Leffler function" (singular), in fact we are dealing with an infinite family of functions, depending on the values of the parameters $\alpha$ and $\beta$. For instance, if $\alpha=\beta=1$, then $E_{1,1}(z)=e^z$ is the exponential, while $E_{2,2}(z)=\frac{\sinh(\sqrt{z})}{\sqrt{z}}$. Many other known functions can be viewed as particular ML functions [@Gorenflo14]. It is easy to observe that the values of $\alpha$ and $\beta$ influence the convergence of the series ([\[ml-function\]](#ml-function){reference-type="ref" reference="ml-function"}) and its practical usage for computational purposes. The Gamma function in the denominators grows very fast for large values of $\alpha$ and slows down for smaller ones (say $0< \alpha< 1$). As we will later, for sufficiently large $\alpha$ and $\beta$ and small$\|A\|$, the Taylor series ([\[ml-function\]](#ml-function){reference-type="ref" reference="ml-function"}) may yield very interesting results in computations. The organization of the paper is as follows. In Section [2](#revisiting){reference-type="ref" reference="revisiting"}, there is a brief revision of the scalar ML function, mainly focused on the existing effective methods for its numerical computation. Section [3](#sec-taylor){reference-type="ref" reference="sec-taylor"} provides a novel analysis of the usage of the Taylor series in numerical computation of the ML function, culminating in a result and a technique that will be incorporated later in the main algorithm for computing the matrix ML function. In Section [4](#contour){reference-type="ref" reference="contour"}, we discuss which types of contours and quadrature are convenient for making the Cauchy integral formula reliable in numerical computations. In Section [5](#sec-algorithm){reference-type="ref" reference="sec-algorithm"}, the main algorithm of the paper is provided in pseudo-code. Numerical considerations on its implementation are included as well. Section [6](#sec-experiments){reference-type="ref" reference="sec-experiments"} includes a set of experiments to illustrate the performance of the main algorithm and how it compares with the existing state-of-the-art method for IEEE double precision arithmetic. Finally, in Section [7](#sec-conclusions){reference-type="ref" reference="sec-conclusions"} some conclusions are drawn. # Revisiting the Scalar ML function {#revisiting} The classical ML function (also called the one-parameter ML function) was introduced and studied by the Swedish mathematician Magnus Gösta Mittag-Leffler (1846--1927) in a set of five papers [@Mittag1; @Mittag2; @Mittag3; @Mittag4; @Mittag5] published at the beginning of the 20th century. The so-called two-parameter ML function, which extends the classical one, was first introduced in the paper [@Wiman], but the author did not pay much attention to it. It was only much later in the papers [@Agarwal; @Humbert] that the two-parameter ML function was rediscovered and investigated in detail. Since then, the ML functions (with one, two, or even more parameters) have attracted the interest of many researchers. This is in part due to the relevant role that these functions play in Fractional Calculus, namely in solving fractional differential equations that arise in the modeling of many practical problems in science and engineering. In this paper, we deal with the two-parameter ML function (we omit the terminology "two-parameter" to simplify) and assume that $\alpha$ and $\beta$ are real and positive. This is the case that is relevant in most of the practical applications. Under those assumptions regarding $\alpha$ and $\beta$, it is well-known that $E_{\alpha,\beta}(z)$, with $z\in\mathbb{C}$, is an entire function which has two important representations suitable to be exploited for numerical computations purposes: the Taylor series $$\label{def-taylor} E_{\alpha,\beta}({z})=\sum_{k=0}^{\infty}\frac{{z}^{k}}{\Gamma(\alpha k+\beta)},$$ and the Wiman's integral representation [@Wiman] and [@Erdelyi p. 210]: $$\label{def-wiman} E_{\alpha,\beta}({z})=\frac{1}{2\pi i}\int_{\mathcal C} \frac{y^{\alpha-\beta}e^y}{y^\alpha-z}\ dy,$$ where the contour ${\mathcal C}$ is the so-called Hankel path in the complex plane, that is, it is a path starting and ending at $-\infty$ and encircling the disk $|y|\leq z^{1/\alpha}$ in the positive sense. The methods proposed in [@Gorenflo02] and [@Seybold08] are similar (though the method in [@Seybold08] makes significant improvements on the one in [@Gorenflo02]) and both use the truncation of the Taylor series [\[def-taylor\]](#def-taylor){reference-type="eqref" reference="def-taylor"} for approximating $E_{\alpha,\beta}({z})$, when $|z|$ is small. When $|z|$ is large, they use asymptotic formulae derived from [\[def-wiman\]](#def-wiman){reference-type="eqref" reference="def-wiman"}; for intermediate values of $|z|$, they use the integral representation [\[def-wiman\]](#def-wiman){reference-type="eqref" reference="def-wiman"} directly. Both methods assume that $0< \alpha \leq 1$; for $\alpha>1$ they use a recursion formula that reduces the latter case to the former. The method in [@Garrappa15] is different from the ones just mentioned, in the sense that it does not require different techniques depending on the magnitude of $z$ or $\alpha$. It involves the auxiliary function $$e_{\alpha,\beta}(t;\lambda):=t^{\beta-1}\,E_{\alpha,\beta}(t^\alpha\,\lambda),\quad t>0, \ \lambda\in\mathbb{C},$$ and its integral representation in terms of the inverse Laplace transform. Through a combination of the Residue Theorem together with the choice of an optimal parabolic contour, that minimizes the error and the computational cost, Garrappa (2015) designed what is now considered the state-of-the-art method for evaluating $E_{\alpha,\beta}({z})$. MATLAB files for the methods in [@Gorenflo02] and [@Garrappa15] are available, respectively, in [@Podlubny2] and [@Garrappa-m]. To understand the influence of the values of $\alpha$ and $\beta$ on the magnitude of $E_{\alpha,\beta}({z})$, let us observe Figures [1](#fig1){reference-type="ref" reference="fig1"} and [2](#fig2){reference-type="ref" reference="fig2"}, where we see that, for $\alpha = 0.5$, the magnitude of the ML function blows up when $z$ is in the real line somewhere between $1.5$ and $2.5$. In contrast, if $\alpha$ increases slightly to $0.8$ and $\beta$ to $1.5$, then $|E_{\alpha,\beta}({z})|$ increases, although not as quickly. Analytically, this can be seen by looking at the denominator of the fractions in the sum [\[def-taylor\]](#def-taylor){reference-type="eqref" reference="def-taylor"}. For small values of $\alpha$, the value of the Gamma function $\Gamma(\alpha k+\beta)$ may not compensate for the growth of the powers of $z$ in the numerator. In both pictures, we see that the magnitude of the ML function is small whenever the real part of $z$ is negative. These kind of issues must be taken into account in the design of algorithms to the ML function. ![Values of $|E_{\alpha,\beta}({z})|$, with $\alpha=0.5$ and $\beta=0.8$, for the complex numbers $z$ belonging to the rectangle $[-1,2.5]\times [-2,2]$. The red line over the graph surface concerns to the values of the ML function on the real line. ](graph_ML_1.eps){#fig1 width="17cm"} ![Values of $|E_{\alpha,\beta}({z})|$, with $\alpha=0.8$ and $\beta=1.5$, for the complex numbers $z$ belonging to the rectangle $[-1,2.5]\times [-2,2]$. The red line over the graph surface concerns to the values of the ML function on the real line. ](graph_ML_2.jpg){#fig2 width="14cm"} # When is the Taylor Series Suitable For Numerical Computations? {#sec-taylor} In this section, we state a couple of results regarding the practical usage of the Taylor series [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} in numerical computations. Proposition [\[prop-taylor-1\]](#prop-taylor-1){reference-type="ref" reference="prop-taylor-1"} is an extension of [@Seybold08 Thm.4.1] to matrices. Propositions [\[prop-taylor-2\]](#prop-taylor-2){reference-type="ref" reference="prop-taylor-2"} and [\[prop-taylor-3\]](#prop-taylor-3){reference-type="ref" reference="prop-taylor-3"} are, to the best of our knowledge, novel in the context of matrices as well as scalars. We recall that algorithms for computing matrix functions based on matrix polynomials are very interesting from a computational point of view in the sense that they are rich in matrix-matrix computations. Moreover, they can easily be extended to algorithms for computing a matrix function times a vector in only $O(n^2)$ operations, avoiding the entire determination of $f(A)$. This is of great relevance if $A$ is large. Let $\|.\|$ be a given subordinate matrix norm and let us consider the truncation error of the Taylor series: $$\label{err-taylor} R_m(A) := \left\|E_{\alpha,\beta}(A)-\sum_{k=0}^{m}\frac{A^{k}}{\Gamma(\alpha k+\beta)}\right\|=\left\|\sum_{k=m+1}^{\infty}\frac{A^{k}}{\Gamma(\alpha k+\beta)}\right\|,$$ where $A$ is a given complex square matrix and $\alpha$ and $\beta$ are real positive. [\[prop-taylor-1\]]{#prop-taylor-1 label="prop-taylor-1"} Let $\epsilon>0$ and assume that $\|A\|<1$. If $$m\geq \mathop{\mathrm{max}}\left\{\left\lceil \frac{2-\beta}{\alpha}+1\right\rceil,\quad \left\lceil \frac{\ln(\epsilon(1-\|A\|))}{\ln(\|A\|)}\right\rceil+1\right\},$$ then $R_m(A)\leq \epsilon$, where $R_m(A)$ is given in [\[err-taylor\]](#err-taylor){reference-type="eqref" reference="err-taylor"} and $\lceil.\rceil$ denotes the ceiling of a number. *Proof.* Follow the steps of the proof of Theorem 4.1 in [@Seybold08] and attend to the properties of matrix norms. 0◻ ◻ [\[prop-taylor-2\]]{#prop-taylor-2 label="prop-taylor-2"} Let $\|A\|<1$ and let $\mu>0$. If there exists an integer $m$ such that $$\label{find-m} \frac{\|A^{m+1}\|}{\Gamma(\alpha (m+1)+\beta)}\leq \mu,$$ then $$R_m(A)\leq \mu \frac{\|A\|}{1-\|A\|}.$$ *Proof.* First, we note that $\|A^{m+2}\|\leq \|A\|\|A^{m+1}\|$ and $\Gamma(\alpha (m+1)+\beta)<\Gamma(\alpha (m+2)+\beta)$. Since, by assumption, $\|A^{m+1}\| \leq \mu\,\Gamma(\alpha (m+1)+\beta)$, the following is valid: $$\frac{\|A^{m+2}\|}{\Gamma(\alpha (m+2)+\beta)}\leq \frac{\|A\|\|A^{m+1}\|}{\Gamma(\alpha (m+2)+\beta)}\leq \|A\|\frac{\|A^{m+1}\|}{\Gamma(\alpha (m+1)+\beta)}\leq \|A\| \mu.$$ By induction, it is easy to show that, for any integer $k>1$, $$\frac{\|A^{m+k}\|}{\Gamma(\alpha (m+k)+\beta)}\leq \|A\|^{k-1} \mu.$$ Now, attending that $\|A\|<1$, the result follows from $$\begin{aligned} R_m(A) & =& \left\|\sum_{k=m+1}^{\infty}\frac{A^{k}}{\Gamma(\alpha k+\beta)}\right\|\\ &\leq &\sum_{k=m+1}^{\infty}\frac{\left\|A^{k}\right\|}{\Gamma(\alpha k+\beta)}\\ & \leq& \mu \left(\|A\|+\|A\|^2+\cdots\right)\\ &=& \mu \frac{\|A\|}{1-\|A\|}.\end{aligned}$$ 0◻ ◻ One important point of the previous result is the possibility of estimating the absolute error [\[err-taylor\]](#err-taylor){reference-type="eqref" reference="err-taylor"} using the norm of the terms of the Taylor series, so that the smallest $m$ satisfying [\[find-m\]](#find-m){reference-type="eqref" reference="find-m"} gives the number of terms to be considered in the truncation of the Taylor series. The next result shows that the Taylor series [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} may be used for numerical computations even when $\|A\|\geq 1$. [\[prop-taylor-3\]]{#prop-taylor-3 label="prop-taylor-3"} Let $m$ be a natural number and let $a\in\mathbb{R}^+$ such that $\Gamma(\alpha k+\beta) \geq a^k$, for all integer $k\geq m$. If $b:=\|A\|/a<1$, then $$\label{find-m-1} R_m(A)\leq \frac{b^{m+1}}{1-b}.$$ *Proof.* Since, by assumption, there exists a positive integer number $m$ such that $\Gamma(\alpha k+\beta) \geq a^k$, for all integer $k\geq m$, and $b=\|A\|/a<1$, we have $$\begin{aligned} R_m(A) &=& \left\|E_{\alpha,\beta}(A)-\sum_{k=0}^{m}\frac{A^{k}}{\Gamma(\alpha k+\beta)}\right\|\\ & =& \left\|\sum_{k=m+1}^{\infty}\frac{A^{k}}{\Gamma(\alpha k+\beta)}\right\|\\ &\leq &\sum_{k=m+1}^{\infty}\frac{\left\|A^{k}\right\|}{\Gamma(\alpha k+\beta)}\\ & \leq& \sum_{k=m+1}^{\infty}\left(\frac{\|A\|}{a}\right)^k\\ &=& \sum_{k=m+1}^{\infty} b^k\\ &=& \frac{b^{m+1}}{1-b}. \end{aligned}$$ 0◻ ◻ Now, we shall discuss how to use the series [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} in the computation of $E_{\alpha,\beta}(A)$. An important point is that the norm of $\|A\|$ must be small enough; otherwise, $\|A^{k+1}\|$ may overflow, even for a small integer $k$, and catastrophic cancellation may occur when adding up the terms of the series. Another point is the possibility of the values of the Gamma function in the denominators of [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} overflow too. It is known that in IEEE-754 double precision arithmetic the Gamma function can be evaluated only for arguments smaller than or equal to $171.624$, so the maximum number of terms we can use in [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} is given by (see [@Garrappa18]) $$m_{\mathop{\mathrm{max}}} := \left\lfloor\frac{171.624-\beta}{\alpha}\right\rfloor,$$ where $\lfloor.\rfloor$ denotes the floor of a number. For a practical implementation of Propositions [\[prop-taylor-1\]](#prop-taylor-1){reference-type="ref" reference="prop-taylor-1"}, [\[prop-taylor-2\]](#prop-taylor-2){reference-type="ref" reference="prop-taylor-2"} and [\[prop-taylor-3\]](#prop-taylor-3){reference-type="ref" reference="prop-taylor-3"}, we find more interest in the result of Proposition [\[prop-taylor-3\]](#prop-taylor-3){reference-type="ref" reference="prop-taylor-3"}, because it does not require the condition $\|A\|<1$. In Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"} we describe how to implement the result of Proposition [\[prop-taylor-3\]](#prop-taylor-3){reference-type="ref" reference="prop-taylor-3"} in a practical way. The main issue is in how to find an appropriate $a$ such that $\Gamma(\alpha m+\beta) \geq a^m$ holds for a small enough $m$. It is not convenient to use a large number of terms in the series. Since $ab=\|A\|\leq a$, the value of $a$ is bounded below by $\|A\|$. However, it is not recommended to take $\displaystyle a\gtrapprox\|A\|$ because $\displaystyle b\lessapprox 1$, which would lead to a poor estimate for the error in [\[find-m-1\]](#find-m-1){reference-type="eqref" reference="find-m-1"}. Another issue that must be considered is that the norm of $A$ needs to be bounded, according to the values of $\alpha$ and $\beta$, to guarantee convergence. To find such a bound, we will require that the norm of the term in [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} corresponding to $m_{\mathop{\mathrm{max}}}$ is small enough, that is, $$\label{small-eta} \frac{\|A\|^{m_{\mathop{\mathrm{max}}}}}{\Gamma(\alpha\,m_{\mathop{\mathrm{max}}}+\beta)}<\eta,$$ where $\eta$ is a small number, say, $\eta=2^{-53}$. From [\[small-eta\]](#small-eta){reference-type="eqref" reference="small-eta"}, it follows $$\label{bound-norm} \|A\|<\left[\eta\Gamma\left(\alpha\,m_{\mathop{\mathrm{max}}}+\beta\right)\right]^{1/m_{\mathop{\mathrm{max}}}}.$$ Hence, if $\|A\|$ does not meet the condition [\[bound-norm\]](#bound-norm){reference-type="eqref" reference="bound-norm"}, it is not guaranteed that [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} will be appropriate for computing the ML function. In Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"}, we choose $a=2\|A\|$, which, according to a set of numerical experiments we have carried out, gives a good compromise between a small value for $m$ and a good error estimate in [\[find-m-1\]](#find-m-1){reference-type="eqref" reference="find-m-1"}. If the conditions in Step 9 are not satisfied, our recommendation is that Taylor series may give inaccurate results. This might happen in particular when $\alpha$ is small (say, $0<\alpha<1$) or $\|A\|$ is large. [\[algorithm1\]]{#algorithm1 label="algorithm1"} (Taylor series) Given $\alpha$ and $\beta$ positive real numbers, $A\in\mathbb{C}^{n\times n}$ and a tolerance $\epsilon$, this algorithm decides if the Taylor series [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} is suitable for approximating $E_{\alpha,\beta}(A)$ and, if so, finds the number of terms that should be considered in the series. 1. $m_{\mathop{\mathrm{max}}} = \left\lfloor\frac{171.624-\beta}{\alpha}\right\rfloor$; 2. $norm\_max = (\epsilon\Gamma(\alpha \,m_{\mathop{\mathrm{max}}}+\beta))^{1/m_{\mathop{\mathrm{max}}}}$; 3. $a = 2\|A\|$; 4. $b = 1/2$; 5. $m = 1:m_{\mathop{\mathrm{max}}}$; 6. $d = \Gamma(\alpha m+\beta)-a.\,\,\hat{}\,\, m$; 7. $k_1 = find(d>0,1)$; 8. $k_2 = \lceil \log(\epsilon(1-b))/\log(b)-1\rceil$ (see [\[find-m-1\]](#find-m-1){reference-type="eqref" reference="find-m-1"} ); 9. **if** $\|A\|\leq norm\_max$ and $k_1$ is non-empty and $k_1\leq k_2$ 10. $\quad$ Use Taylor series [\[ml-function\]](#ml-function){reference-type="eqref" reference="ml-function"} with $k_2$ terms for approximating $E_{\alpha,\beta}(A)$; 11. **else** Taylor series is not recommended for evaluating $E_{\alpha,\beta}(A)$. 12. **end** Provided that the conditions in Step 9 of Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"} are satisfied, there is the question of how to evaluate the Taylor polynomial of degree $k_2$. There are three main methods for polynomial evaluation: Horner's method, the explicit powers' method and the Paterson-Stockmeyer method. The first two methods cost about $k_2-1$ matrix multiplications, while the latter is more sophisticated and much less expensive, costing about $s+r-1-\phi(s,k_2)$ matrix multiplications, where $s=\lceil\sqrt{k_2}\rceil$ or $s=\lfloor \sqrt{k_2}\rfloor$, $r=\lfloor \sqrt{k_2/s}\rfloor$ and $\phi(s,k_2)=1$ if $s$ divides $k_2$ and $0$ otherwise. In the method to be proposed later in Section [6](#sec-experiments){reference-type="ref" reference="sec-experiments"} we will use the Paterson-Stockmeyer method. For more information on evaluating matrix polynomials, see [@Higham08 Ch.4]; for Paterson-Stockmeyer method see also [@Higham08 Ch.4], and [@Fasi; @Paterson]. # Finding a Suitable Contour {#contour} In this section we assume throughout that $T$ is an atomic block (i.e., an upper triangular matrix with close eigenvalues) of order $n$. Our main goal is to provide a reliable method for computing the ML function of $T$ by means of the Cauchy formula $$E_{\alpha,\beta}(T)=\frac{1}{2\pi i}\int_{\mathcal C} E_{\alpha,\beta}(z)(zI-T)^{-1}\ dz.$$ For the contour ${\mathcal C}$ we consider a circle with center in $z_0\in\mathbb{C}$ and radius $r>0$, $$\label{circle} {\mathcal C}=\{z\in\mathbb{C}:\ z=z_0+re^{ti}, \ t\in [0,2\pi[\},$$ enclosing the spectrum $\sigma(T)$ of $T$. After a few calculations we obtain $$\label{cauchy-2} E_{\alpha,\beta}(T)=\frac{1}{2\pi}\int_0^{2\pi} re^{ti}E_{\alpha,\beta}(z_0+re^{ti}) \left((z_0+re^{ti})I-T\right)^{-1}\ dt.$$ For the center of the circle we set $$\label{z0} z_0 = \frac{\mathop{\mathrm{trace}}(T)}{n},$$ that is, $z_0$ is the arithmetic mean of the eigenvalues of $T$. The value of the radius $r$ must satisfy $$\label{d} r\,>\,d:=\mathop{\mathrm{max}}\{|\lambda-z_0|:\ \lambda\in \sigma(T)\};$$ see Figure [3](#fig-4){reference-type="ref" reference="fig-4"}. ![Geometric representation of $d$ and $r$ in [\[d\]](#d){reference-type="eqref" reference="d"}, for a matrix $T$ with spectrum $\sigma(T) = \{\lambda_1,\, \lambda_2,\, \lambda_3,\, \lambda_4\}$. ](Figura_circulo.png){#fig-4 width="30%"} Since we are interested in evaluating the integral in [\[cauchy-2\]](#cauchy-2){reference-type="eqref" reference="cauchy-2"} by quadrature, some care must be taken in the choice of the radius $r$. If it is very close to $d$, the circle passes too close the eigenvalues of $T$ and the value of the norm of the resolvent $\left\|\left((z_0+re^{ti})I-T\right)^{-1}\right\|$ is large, while the value of $E_{\alpha,\beta}(z_0+re^{ti})$ may be moderate; if $r$ is large so that the circle stays far away from the eigenvalues of $T$, the value of $\left\|\left((z_0+re^{ti})I-T\right)^{-1}\right\|$ is moderate, while the values of $E_{\alpha,\beta}(z_0+re^{ti})$ may be large, especially if $0<\alpha <1$. Therefore the value of $r$ must be carefully chosen in order to avoid many oscillations in the integrand function of [\[cauchy-2\]](#cauchy-2){reference-type="eqref" reference="cauchy-2"}. Moreover, it is expected that $r$ should minimize the quadrature error arising in the approximation of [\[cauchy-2\]](#cauchy-2){reference-type="eqref" reference="cauchy-2"}. Let $$\label{integrand} g(t):=re^{ti}E_{\alpha,\beta}(z_0+re^{ti})\left((z_0+re^{ti})I-T\right)^{-1}$$ be the integrand function in [\[cauchy-2\]](#cauchy-2){reference-type="eqref" reference="cauchy-2"}. Note that $g:\mathbb{R}\longrightarrow \mathbb{C}^{n\times n}$, where $n$ is the order of $T$, is of class $C^\infty$ (we are assuming that the eigenvalues of $T$ are located in the interior of the circle). Since $g$ is periodic, we follow the recommendations of [@Trefethen] and use the composite trapezoidal rule (or just trapezoidal rule, for short). Let $$\label{trapez-1} {\mathcal T}_m(g):=\frac{2\pi}{m}\sum_{k=1}^{m}g(t_k),$$ denote the $(m+1)$-point trapezoidal rule applied to $\int_0^{2\pi}g(t)\ dt$. Here, $t_0,t_1,\ldots,t_m$ are equally spaced points partitioning the interval $[0,\,2\pi]$, with $t_0=0$ and $t_m=2\pi$. The point $t_0$ does not appear explicitly in [\[trapez-1\]](#trapez-1){reference-type="eqref" reference="trapez-1"} because $g(t_0)=g(t_m)$. Let $$\label{eps-T} \epsilon_T:=\left\|\int_0^{2\pi}g(t)\ dt-{\mathcal T}_m(g)\right\|$$ denote the error of the $(m+1)$-point trapezoidal rule. In the literature we can find two important bounds for such an error: $$\label{err-1} \epsilon_T \leq \frac{8\pi^3}{12m^2}\,M_1$$ (see [@Cardoso Sec. 4]), and $$\label{err-2} \epsilon_T \leq \frac{4\pi \,\zeta(2s+1)}{m^{2s+1}}\,M_2$$ where $\displaystyle M_1:=\mathop{\mathrm{max}}_{t\in [0,\,2\pi]}\|g''(t)\|$, $\displaystyle M_2:= \mathop{\mathrm{max}}_{t\in [0,2\pi]}\|g^{(2s+1)}(t)\|$ and $\zeta(.)$ is the Riemann Zeta function (see [@Davies05; @Davis84]). The bound in [\[err-2\]](#err-2){reference-type="eqref" reference="err-2"} is valid because $g$ is periodic, while [\[err-1\]](#err-1){reference-type="eqref" reference="err-1"} does not require the periodicity of $g$. Of course, the bound [\[err-2\]](#err-2){reference-type="eqref" reference="err-2"} is in general sharper than [\[err-1\]](#err-1){reference-type="eqref" reference="err-1"}, but the expression of the derivatives $g^{(k)}(t)$ becomes much complicated as $k$ increases. Using Leibniz's rule and Faà di Bruno's formula, it is possible to find a closed expression for $g^{(k)}(t)$, but its computational cost for each $t$ increases with $k$ as well. To illustrate, we write down the expression of $g''(t)$. Attending that $g$ is the product of three functions, that is, $g(t)=g_1(t)g_2(t)g_3(t)$, where $$\begin{aligned} g_1(t)&:=&re^{ti}\label{g1}\\ g_2(t)&:=&E_{\alpha,\beta}(z_0+re^{ti})\label{g2}\\ g_3(t)&:=&\left((z_0+re^{ti})I-T\right)^{-1},\label{g3}\end{aligned}$$ the second order derivative is given by $$\label{der-g-2} g''(t)=\left[g_1(t)g''_2(t)+2g'_1(t)g'_2(t)+g_1(t)g''_2(t)\right]g_3(t)+ 2\left[g'_1(t)g_2(t)+g_1(t)g'_2(t)\right]g'_3(t)+g_1(t)g_2(t)g''_3(t),$$ where $$\begin{aligned} g'_1(t)&=& rie^{ti}\\ g''_1(t)&=& ri^2e^{ti}\\ g'_2(t)&=& \frac{dE_{\alpha,\beta}(z)}{dz}(rie^{ti})\\ g''_2(t)&=& \frac{d^2E_{\alpha,\beta}(z)}{dz^2}(rie^{ti})^2+ \frac{dE_{\alpha,\beta}(z)}{dz}(ri^2e^{ti})\\ g'_3(t)&=&-rie^{ti}(zI-T)^{-2}\\ g''_3(t)&=&re^{ti}(zI-T)^{-2}+2\left(rie^{ti}\right)^2(zI-T)^{-3},\end{aligned}$$ with $z=z_0+re^{ti}$. If we go up to the third order derivative, we arrive at a much more complicated expression, so in the following developments we will use [\[err-1\]](#err-1){reference-type="eqref" reference="err-1"} instead of [\[err-2\]](#err-2){reference-type="eqref" reference="err-2"} and focus our investigation on how to find an $r$ that minimizes [\[err-1\]](#err-1){reference-type="eqref" reference="err-1"}. Mathematically, the problem can be formulated as a min-max problem $$\label{min-max} \tilde{r}=\mathop{\mathrm{argmin}}_{r>d}\ \mathop{\mathrm{max}}_{t\in [0,2\pi]} \|g''(r,t)\|,$$ where $d$ is defined in [\[d\]](#d){reference-type="eqref" reference="d"}, that is, $d$ is the maximal distance between $z_0$ and the eigenvalues of $T$ (see [\[z0\]](#z0){reference-type="eqref" reference="z0"} and also Figure [3](#fig-4){reference-type="ref" reference="fig-4"}). For convenience, from now on we highlight the dependence of $g''$ not only on $t$, but also on $r$. Since a rough estimate for $\tilde{r}$ is enough, we can solve [\[min-max\]](#min-max){reference-type="eqref" reference="min-max"} by considering a vector of values $r\in ]d,3]$ and then finding the maximal $t$ corresponding to each $r$. At the end, we choose an $r$ corresponding to the minimum of the maximal values of $t$; this will be the radius of the contour circle. Solving the optimization problem [\[min-max\]](#min-max){reference-type="eqref" reference="min-max"} requires many evaluations of $g''(r,t)$, at the cost of $O(n^3)$ operations per evaluation. Note that $g''(r,t)$ involves the inversion of a triangular matrix in $\left((z_0+re^{ti})I-T\right)^{-k}$, for $k=1,2$, at the cost of $O(n^3)$ operations, while the functions $g_1$ and $g_2$ (and their derivatives) are scalar valued. To make each function evaluation cheaper and more feasible, in the following proposition we derive an upper bound for $\left\|\left((z_0+re^{ti})I-T\right)^{-k}\right\|_F$, where $\|.\|_F$ denotes the Frobenius norm. Given an $n\times n$ complex upper triangular matrix $T$, let $N$ be its strictly upper triangular part, that is, $T=\mathop{\mathrm{diag}}(\lambda_i)+N$. Let ${\mathcal C}$ be the circle with center in $z_0=\mathop{\mathrm{trace}}(T)/n$ and radius $r$ and let $\widetilde{\mathcal C}$ be the closed disk with center in $z_0$ and radius $\delta$, such that $r>\delta>d=\mathop{\mathrm{max}}\{|\lambda-z_0|:\ \lambda\in \sigma(T)\}$. Then, for all $t\in [0,\,2\pi]$ and $k=1,2,\ldots$, $$\label{bound-g3} \left\|\left((z_0+re^{ti})I-T\right)^{-k}\right\|_F \leq c\, \|(I-|N|)^{-1}\|_F,$$ where $$c:=\mathop{\mathrm{max}}\left\{\frac{(k+j-1)!}{j!(k-1)!}(r-\delta)^{-k-j}:\ j=0,1,\ldots,n-1\right\}$$ and $|N|$ denotes the matrix formed by the absolute values of the entries of $N$. *Proof.* Let us consider the scalar complex valued function $\phi_k(x):=\frac{1}{(z(t)-x)^k}$, where $k=1,2,\ldots$ and $z(t):= z_0+re^{ti}$. It is easy to check that $\phi_k(x)$ is analytic on the closed disk $\widetilde{\mathcal C}$. Note that if $\lambda$ is an eigenvalue of $T$, then $z(t)-\lambda\neq 0, \ \forall t\in[0,\,2\pi]$, that is, the denominator of $\phi_k(\lambda)$ does not vanish. Using the result [@Golub Thm.9.2.2] (see also [@Higham08 Thm.4.28]), we have $$\label{ineq-1} \left\|\left((z_0+re^{ti})I-T\right)^{-k}\right\|_F \leq \mathop{\mathrm{max}}_{0\leq j\leq n-1}\frac{\omega_j}{j!}\,\|(I-|N|)^{-1}\|_F,$$ where $\displaystyle\omega_j:=\sup_{x\in \widetilde{\mathcal C}}|\phi_k^{(j)}(x)|.$ Noticing that $\phi_k^{(j)}(x)=\frac{(-1)^j (k+j-1)!}{(k-1)!\,(z-x)^{k+j}}$ and that $|z(t)-x|\geq r-\delta$, for any $x\in \widetilde{\mathcal C}$ and $t\in [0,\,2\pi]$, we have $$\frac{\omega_j}{j!} =\sup_{x\in \widetilde{\mathcal C}} \frac{(k+j-1)!}{j!(k-1)!\,|z-x|^{k+j}} \leq \frac{(k+j-1)!}{j!(k-1)!\,(r-\delta)^{k+j}},$$ for each $j$ and $k$. Hence $$\left\|\left(z(t)I-T\right)^{-k}\right\|_F \leq \mathop{\mathrm{max}}\left\{\frac{(k+j-1)!}{j!(k-1)!}(r-\delta)^{-k-j}:\ j=0,1,\ldots,n-1\right\}\,\|(I-|N|)^{-1}\|_F,$$ which proves the result. 0◻ ◻ From [\[der-g-2\]](#der-g-2){reference-type="eqref" reference="der-g-2"} and [\[bound-g3\]](#bound-g3){reference-type="eqref" reference="bound-g3"}, we have $$\begin{aligned} \|g''(r,t)\|_F &\leq& \Big(\left|g_1(t)g''_2(t)+2g'_1(t)g'_2(t)+g_1(t)g''_2(t)\right|\gamma_1+ 2r\left|g'_1(t)g_2(t)+g_1(t)g'_2(t)\right|\gamma_2+\nonumber\\ && r\left|g_1(t)g_2(t)\right|\left(\gamma_2+2r\gamma_3\right)\Big)\gamma,\label{bound-der-2}\end{aligned}$$ where $$\begin{aligned} \gamma &:=& \|(I-|N|)^{-1}\|_F; \\ \gamma_1 &:=& \mathop{\mathrm{max}}\left\{(r-\delta)^{-j}:\ j=1,2,\ldots,n\right\};\\ \gamma_2 &:=& \mathop{\mathrm{max}}\left\{j(r-\delta)^{-j-1}:\ j=1,2,\ldots,n\right\}; \\ \gamma_3 &:=& \mathop{\mathrm{max}}\left\{\frac{j(j+1)}{2}(r-\delta)^{-j-2}:\ j=1,2,\ldots,n\right\}. \end{aligned}$$ Using the bound [\[bound-der-2\]](#bound-der-2){reference-type="eqref" reference="bound-der-2"} for estimating $\|g''(r,t)\|_F$ avoids the computation of the factor $\left\|\left((z_0+re^{ti})I-T\right)^{-1}\right\|_F$, that requires $O(n^3)$ operations, for each $t$, making the estimation of $\|g''(r,t)\|_F$ much cheaper: the factor $\|(I-|N|)^{-1}\|_F$ does not depend on either $r$ or $t$ and can be discarded. Hence, the operations needed to estimate a solution of [\[min-max\]](#min-max){reference-type="eqref" reference="min-max"} just involve scalars. # Algorithm {#sec-algorithm} Before stating the algorithm for computing the ML function of a general matrix $A$ (Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"}), we shall revisit the blocked Schur-Parlett method with reordering and explain in detail how to efficiently implement the trapezoidal rule. ## Schur-Parlett Approach {#schur-parlett} In the following, we summarize the main steps of the blocked Schur-Parlett method with reordering proposed in [@Davies03], which codes are available in [@mftoolbox]; see also [@Higham08 Ch. 9]. It starts by computing a Schur decomposition $A=UTU^\ast$, where $U$ is unitary and $$\label{matrizT} T=\left[ \begin{array}{cccc} T_{11}&T_{12}&\cdots&T_{1q}\\ 0&T_{22}&\cdots&T_{2q}\\\vdots&\ddots&\ddots&\vdots\\0&\cdots&0&T_{qq} \end{array} \right]\ \in\mathbb{C}^{n\times n},$$ is a $(q\times q)$-block-upper triangular matrix. Each atomic block $T_{ii}\ (i=1,\cdots,q)$ has clustered eigenvalues and the $q$ atomic blocks in the diagonal do not have common eigenvalues, that is, $$\label{espectroT} \sigma(T_{ii})\cap\sigma(T_{jj})=\emptyset,\ i,j=1,\ldots,q,\ i\neq j.$$ Let us denote $$\label{L1} F:=E_{\alpha,\beta}(T)=\left[ \begin{array}{cccc} F_{11}&F_{12}&\cdots&F_{1q}\\ 0&F_{22}&\cdots&F_{2q}\\\vdots&\ddots&\ddots&\vdots\\0&\cdots&0&F_{qq} \end{array} \right],$$ where $F_{ij}$ has the same size as $T_{ij}\ (i,j=\,\ldots,q)$. Recall that the diagonal blocks of $F$ are given by $F_{ii}=E_{\alpha,\beta}(T_{ii})$. Since $FT=TF$, it can be shown that $$\label{parlett2} F_{ij}T_{jj}-T_{ii}F_{ij}=T_{ij}F_{jj}-F_{ii}T_{ij}+ \sum^{j-1}_{k=i+1}(T_{ik}F_{kj}-F_{ik}T_{kj})\quad i<j.$$ To find the blocks of $F$, we start by computing the ML function of the atomic blocks $F_{ii}=E_{\alpha,\beta}(T_{ii})$ and then successively use ([\[parlett2\]](#parlett2){reference-type="ref" reference="parlett2"}) to approximate the remaining blocks of $F$. Note that for each $i<j$, the identity ([\[parlett2\]](#parlett2){reference-type="ref" reference="parlett2"}) is a Sylvester equation of the form $$\label{sylvester} XM-NX=P,$$ where $M$, $N$ and $P$ are known square matrices and $X$ has to be determined. Equation ([\[sylvester\]](#sylvester){reference-type="ref" reference="sylvester"}) has a unique solution if and only if $\sigma(M)\cap\sigma(N)=\emptyset$. Hence, the block-Parlett method requires the solving of several Sylvester equations with a unique solution. Recall that we are assuming that $\sigma(T_{ii})\cap\sigma(T_{jj})=\emptyset,\ i\neq j$. To avoid the ill conditioning of the Sylvester equations arising in the Parlett recurrence, the eigenvalues of the blocks $T_{ii}$ and $T_{jj}$, $i\neq j,$ need to be well separated in the following sense: there exists $\delta>0$ (e.g., $\delta=0.1$), such that $$\label{delta} \min\left\{|\lambda-\mu|:\ \lambda\in\sigma(T_{ii}),\ \mu\in\sigma(T_{jj}),\ i\neq j \right\} > \delta$$ and, for every eigenvalue $\lambda$ of a block $T_{ii}$ with dimension bigger than $1$, there exists $\mu\in\sigma(T_{ii})$ such that $|\lambda-\mu|\leq \delta$. ## Trapezoidal Rule The benefits of using the trapezoidal rule in integrals with periodic functions are clearly pointed out in [@Trefethen]. We highlight in particular the following three major points: - The computations of ${\mathcal T}_m(g)$ (see [\[trapez-1\]](#trapez-1){reference-type="eqref" reference="trapez-1"}) can be reused when the number of nodes is doubled, i. e., for computing ${\mathcal T}_{2m}(g)$; - The error $\epsilon_T$ in [\[eps-T\]](#eps-T){reference-type="eqref" reference="eps-T"} can be estimated cheaply by $$\epsilon_T\leq \left\|{\mathcal T}_{2m}(g)-{\mathcal T}_{m}(g)\right\|;$$ see [@Tatsuoka Sec.2.3] and also [@Cardoso Sec.4]; - If $T$ is a real matrix, the computation of ${\mathcal T}_m(g)$ can be carried out with only $\lceil m/2\rceil$ function evaluations. For implementing the trapezoidal rule, we consider the following algorithm. [\[algorithm2\]]{#algorithm2 label="algorithm2"} (Trapezoidal rule) Assume that $T\in\mathbb{C}^{n\times n}$ is an upper triangular matrix and that $g(t)$ is defined as in [\[integrand\]](#integrand){reference-type="eqref" reference="integrand"}. Let $\displaystyle{\mathcal T}_{m}(g):=\frac{2\pi}{m}\sum_{k=1}^{m}g(t_k),$ where $t_0=0<t_1<\ldots<t_m=2\pi$ are equally spaced points, and let $\displaystyle{\mathcal I}_g:=\int_0^{2\pi}\,g(t)\,dt$ be the exact value of the integral. For a certain tolerance $\varepsilon$, this algorithm approximates the matrix integral by the trapezoidal rule and finds an $m_0$ such that the discretization error $\|{\mathcal I}_g-{\mathcal T}_{m_0}(g)\|_F \leq \varepsilon$. 1. Choose a positive integer $m$ (preferably small, say $m=10$); 2. Compute ${\mathcal T}_{m}(g)$; 3. Compute ${\mathcal T}_{2m}(g)$ by the following formula: $$\label{T-2m} {\mathcal T}_{2m}(g) = {\mathcal T}_{m}(g)+\sum_{k=0}^{m-1}g(t_{2k+1});$$ 4. **while** $\|{\mathcal T}_{2m}(g)-{\mathcal T}_{m}(g)\|_F>\epsilon$, 5. $\quad m\leftarrow 2m$; 6. $\quad$Compute ${\mathcal T}_{2m}(g)$ using [\[T-2m\]](#T-2m){reference-type="eqref" reference="T-2m"}; 7. **end** 8. $m_0 \leftarrow m$; 9. ${\mathcal I}_g\approx {\mathcal T}_{m_0}(g)$ To simplify, in Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"} we have not exploited the advantages arising when $T$ is a real matrix. The main differences would be in the choice of $m$, that must be even, and in the way as ${\mathcal T}_{m}(g)$ is evaluated -- we would compute instead $$\widetilde{\mathcal T}_{m/2} :=\frac{\pi}{m}\left(g(0)+g(\pi)+2\sum_{k=1}^{m/2-1}g(t_k)\right),$$ where $t_0=0<t_1<\ldots<t_{m/2}=\pi$ are equally spaced points partitioning the interval $[0,\,\pi]$, and then evaluate $${\mathcal T}_{m}=\widetilde{\mathcal T}_{m/2}+\mathop{\mathrm{conj}}\left(\widetilde{\mathcal T}_{m/2}\right);$$ here $\mathop{\mathrm{conj}}(.)$ denotes the complex conjugate. ## Main Algorithm {#alg-main} [\[algorithm3\]]{#algorithm3 label="algorithm3"} (Computing the ML function) Given $\alpha$ and $\beta$ positive real numbers, and a general square complex matrix $A\in\mathbb{C}^{n\times n}$, this algorithm approximates $E_{\alpha,\beta}(A)$ using the Taylor series or a Schur-Parlett method combined with the Cauchy integral form. It is intended for working in IEEE standard double precision arithmetic. 1. Call Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"} to check if the Taylor series is suitable for computing the ML function and, if so, approximate $E_{\alpha,\beta}(A)$ by the Taylor polynomial of degree $k_2$, computed by means of the Paterson-Stockmeyer method; if the Taylor series cannot be applied, go to the next step; 2. Use the method of [@Davies03] to compute the blocked Schur decomposition of $A$ with reordering, so that $A=UTU^\ast$, where $U$ is unitary and $T$ is a $(q\times q)$-block-upper triangular matrix, with well separated atomic blocks $T_{kk},\ k=1,\ldots,q$ in its diagonal; choose $\delta=0.1$ for the value in [\[delta\]](#delta){reference-type="eqref" reference="delta"}. 3. **for** $k=1:q$ 4. $\quad$ Let $n_k$ denote the order of the block $T_{kk}$; 5. $\quad$ **if** $n_k=1$, 6. $\qquad$ $T_{kk}$ is a scalar so use the method of [@Garrappa15]; 7. $\quad$ **endif** 8. $\quad$ **if** $n_k=2$, 9. $\qquad$ let $T_{kk}:=[t_{11}\ t_{12};\,0\ t_{22}]$; now $F=E_{\alpha,\beta}(T_{kk})= [f_{11}\ f_{12};\,0\ f_{22}]$, where $f_{ii}=E_{\alpha,\beta}(t_{ii})$, for $i=1,2$, 10. $\qquad$ and $f_{12}=t_{12}\frac{f_{22}-f_{11}}{t_{22}-t_{11}};$ 11. $\quad$ **endif** 12. $\quad$ **if** $n_k\geq 3$ 13. $\qquad$ Set $z_k=\frac{\mathop{\mathrm{trace}}(T_{kk})}{n_k}$ and estimate the radius $r_k$ by the method of Section [4](#contour){reference-type="ref" reference="contour"}; 14. $\qquad$ Denote $$\label{Gkk} G_k(t):=r_ke^{ti}E_{\alpha,\beta}(z_k+r_ke^{ti})\left((z_k+r_ke^{ti})I-T_{kk}\right)^{-1};$$ 15. $\qquad$ Find $F_{kk}\approx \int_{0}^{2\pi}\,G_k(t)\,dt$ by the trapezoidal rule using Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"}; 16. $\quad$ **endif** 17. **endfor** 18. Compute the remaining blocks of $F=2\pi\,E_{\alpha,\beta}(T)$, by the block Parlett recurrence (Section [5.1](#schur-parlett){reference-type="ref" reference="schur-parlett"}); 19. $E_{\alpha,\beta}(A)\approx \frac{1}{2\pi} U\,F\,U^\ast$. Let us now make some remarks on the computational cost of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"}. If the Taylor series is used, a Taylor polynomial of degree $k_2$ is evaluated by the Paterson-Stockmeyer method whose cost has been discussed at the end of Section [3](#sec-taylor){reference-type="ref" reference="sec-taylor"}. If the Taylor series cannot be used, a Schur decomposition is required which costs about $25n^3$ flops. Moreover, for the blocks $T_{kk}$ with size $n_k\times n_k$, where $n_k\geq 3$, the algorithm requires many evaluations of the function $G_k(t)$ given in [\[Gkk\]](#Gkk){reference-type="eqref" reference="Gkk"}, to compute the Cauchy integral by the trapezoidal rule. Each evaluation of $G_k(t)$ involves an inversion of a triangular matrix (or a multiple right-hand side linear system) which costs about $n_k^3/3$. Recall that $n_k$ is the order of the block $T_{kk}$ and that $n_1+n_2+\cdots+n_q=n$, so in general $n_k$ is much smaller than $n$. The other factors of $G_k(t)$ are scalars. The estimation of the radius $r_k$ depends on the method used for solving the min-max problem [\[min-max\]](#min-max){reference-type="eqref" reference="min-max"}. Since we do not require a very accurate value for $r_k$, we can define a grid in the rectangle $[d,\tilde{d}]\times [0,\,2\pi]$ ($\tilde{d}$ is an upper bound for the admissible values for $r$; in our experiments we consider $\tilde{d}=3$), with "large cells" and then evaluate the bound for $\|G_k(r,t)\|_F$ given in [\[bound-der-2\]](#bound-der-2){reference-type="eqref" reference="bound-der-2"} at each point $(r,t)$. Recall that the factor $\|(I-|N_{kk}|)^{-1}\|_F$ does not need to be computed. We consider a grid with "large cells" because the computation of the scalar ML function is quite expensive, if compared with functions like $\exp(z)$ or $\log(z)$. The cost of implementing the Schur-Parlett method is addressed in detail in [@Davies03] and [@Higham08 Ch.9]. From these two latter references, we know that the Schur-Parlett method is forward stable, so the major source of errors in Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} arises in the computation of the Taylor polynomial or in the computation of the Cauchy integral by the trapezoidal rule, which is bounded by $\left\|{\mathcal T}_{2m}(g)-{\mathcal T}_{m}(g)\right\|$. If the Taylor series is suitable for evaluating the ML function, we can monitor the truncation error by the bound provided in Proposition [\[prop-taylor-3\]](#prop-taylor-3){reference-type="ref" reference="prop-taylor-3"}. # Numerical Experiments {#sec-experiments} In this section we run a set of experiments in MATLAB R2022a on a machine with Core i5 (2.40 GHz, 16 GB of RAM, unit roundoff $u\approx 1.1\times 10^{-16}$) to compare the relative errors originated by Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} and the method of [@Garrappa18], here named `GP-method`. The CPU time (in seconds) of both methods is compared as well. Since CPU time depends on many factors and is different each time an algorithm runs, we have run each algorithm 20 times and then computed the average CPU time to get more realistic results. The method in [@Higham21] is not considered in our experiments because it involves variable precision so we can control better the accuracy by changing the precision. However, improving the accuracy may be done at the cost of increasing the computational time and this issue is not addressed in [@Higham21]. Moreover, the "Advanpix" package required by the methods of [@Higham21] is not free and it is not incorporated in MATLAB's standard packages. Codes for the `GP-method` are available in [@Garrappa-m]. As exact value for $E_{\alpha,\beta}(A)$, we have considered the result given by the Symbolic Math Toolbox at 2000 decimal digit arithmetic rounded to double precision. Denoting by $\widetilde{E}_{\alpha,\beta}(A)$ the computed approximation, we use the relative error in the usual sense: $$\mbox{\texttt{rel-err}}:=\frac{\|{E}_{\alpha,\beta}(A)-\widetilde{E}_{\alpha,\beta}(A)\|_F}{\|{E}_{\alpha,\beta}(A)\|_F}.$$ We recall that in [@Garrappa18] and also in [@Garrappa15] the authors consider a different formula for the relative error: $$\mbox{\texttt{err-GP}}:=\frac{\|{E}_{\alpha,\beta}(A)-\widetilde{E}_{\alpha,\beta}(A)\|_F}{\|{E}_{\alpha,\beta}(A)\|_F+1}.$$ As discussed in [@Higham21], if $\|E_{\alpha,\beta}(A)\|\gg 1$, $\mbox{\texttt{err-GP}}$ gives a good estimate for the relative error; but if $\|E_{\alpha,\beta}(A)\|\ll 1$, $\mbox{\texttt{err-GP}}$ does not give a realistic estimate of the relative error; it estimates instead the absolute error. To assess the forward stability of both algorithms, the relative errors are compared with the quantity $\mathop{\mathrm{cond}}_{ML}(A)u$, where $\mathop{\mathrm{cond}}_{ML}(A)$ denotes the relative condition number of the ML function at $A$ and $u$ is the unit roundoff [@Higham08 Ch.3]. The value of $\mathop{\mathrm{cond}}_{ML}(A)$ will be estimated by the code `funm_condest1` available in [@mftoolbox]. Although the papers [@Garrappa15; @Garrappa18] provide state-of-the-art methods for approximating the scalar ML function and the matrix ML function, respectively, we must be aware that they may not give relative errors close to $\mathop{\mathrm{cond}}_{ML}(A)u$ when $\|E_{\alpha,\beta}(A)\|\ll 1$. This might also happen when Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} calls the Cauchy integral form, because it requires many evaluations of the scalar ML function $E_{\alpha,\beta}(z)$, which will be carried out by means of the method provided in [@Garrappa15]. In Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"}, we have chosen $\epsilon=10^{-15}$, so that $k_2 = \lceil \log(\epsilon(1-b))/\log(b)-1\rceil = 50$. Hence, provided that the conditions of applicability of the Taylor series are satisfied, $E_{\alpha,\beta}(A)$ is approximated by the Taylor polynomial of degree $k_2=50$, which is evaluated by the Paterson-Stockmeyer method, at the cost of about $13$ matrix multiplications. Since the cost of a Schur decomposition is about $25n^3$, corresponding to $12.5$ matrix multiplications, it is not convenient to use higher-degree polynomials to guarantee the competitivity of the Taylor method. This experiment involves the $20\times 20$ matrix `A = gallery(’redheff’,20)` from MATLAB's gallery. This is the Redheffer matrix whose blocked Schur decomposition with reordering has a large block with eigenvalue equal to $1$. It has also been used in the experiments of [@Garrappa18; @Higham21]. Figure [4](#fig-6){reference-type="ref" reference="fig-6"} displays the relative errors arising in the computation of ${E}_{\alpha,\beta}(-A)$, with $\alpha$ having fixed values ($\alpha=0.5$ and $\alpha=0.8$) as $\beta$ varies from $1$ to $10$. As observed previously in [@Higham21], the relative errors of the `GP-method` start increasing as $\beta$ goes from $6$ to $10$. This is in part because $\|{E}_{\alpha,\beta}(-A)\|_F$ approaches zero as $\beta$ grows toward $10$. As mentioned above, this might also occur when Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} uses the Cauchy integral formula. This is the case for the ten tests corresponding to $\alpha=0.5$ (left graph), because in all of them Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} calls the Cauchy integral form combined with the Schur-Parlett method. We note that for $\alpha=0.5$ and $\beta\in[6,10]$, one has $\|{E}_{\alpha,\beta}(-A)\|_F\in[10^{-1},\,10^{-5}]$. A quite different situation occurs for $\alpha=0.8$ (right graph), because now, for $\beta=5,\ldots,10$, Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} uses the Taylor series, thus preventing the loss of stability. Figure [5](#fig-6a){reference-type="ref" reference="fig-6a"} displays the corresponding CPU time taken by Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} and `GP-method`, where we can see that Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} has the best performance. ![Results of Experiment 1. Relative errors of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) for $\alpha=0.5$ and $\beta=1,\ldots,10$ (left) and $\alpha=0.8$ and $\beta=1,\ldots,10$ (right). The solid black line concerns to $\mathop{\mathrm{cond}}_{ML}(A)u$. ](fig_teste1.eps){#fig-6 width="99%"} ![Results of Experiment 1. CPU time (seconds) of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) for $\alpha=0.5$ and $\beta=1,\ldots,10$ (left) and $\alpha=0.8$ and $\beta=1,\ldots,10$ (right). ](fig_teste1_CPU.eps){#fig-6a width="99%"} This experiment is similar to the one in Example 2 of [@Garrappa18]. It involves four $40\times 40$ matrices with prescribed eigenvalues, with moderate multiplicities. The value of $\beta=1$ is fixed, while the value of $\alpha=0.6:0.4:2.6$ varies. The eigenvalues and the multiplicities of the matrices are included in Table [1](#tabela){reference-type="ref" reference="tabela"}. The relative errors for Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} and `GP-method`, as well as the estimates for $\mathop{\mathrm{cond}}_{ML}(A)u$, are plotted in Figure [6](#fig-7){reference-type="ref" reference="fig-7"}. In Figure [7](#fig-7a){reference-type="ref" reference="fig-7a"} we can see the CPU time for this experiment. Eigenvalues (multiplicities) -------------- ------------------------------------------------------------------------------------ Matrix $A_1$ $\pm 1.0(5)\quad \pm 1.0001(4)\quad \pm 1.001(4)\quad \pm 1.01(4)\quad \pm 1.1(3)$ Matrix $A_2$ $\pm 1.0(8)\quad 2(8)\quad -5(8)\quad -10(8)$ Matrix $A_3$ $-1(2)\quad -5(2)\quad 1 \pm 10i(6)\quad -4 \pm 1.5i(6)\quad \pm 5i(6)$ Matrix $A_4$ $1(4)\quad 1.0001(4)\quad 1.001(4)\quad 1 \pm 10i(7)\quad -4 \pm 1.5i(7)$ : Eigenvalues and their multiplicities for the matrices used in Experiment 2 To design the matrices $A_1,\ldots,A_4$, we started by forming diagonal matrices having the assigned eigenvalues as entries, repeated according to the multiplicities, and then applied a similarity transformation with randomized orthogonal matrices. These matrices have non trivial blocks in the blocked Schur form with reordering. The results are displayed in Figure [6](#fig-7){reference-type="ref" reference="fig-7"}, where we observe that Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} and produce relative errors of similar magnitude, except in the cases when Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} calls the Taylor series, whose results are better. The deviation between the relative errors and the values of $\mathop{\mathrm{cond}}_{ML}(A)u$ is moderate. We stress once more that we are using the standard definition of the relative error, while the methods of [@Garrappa15; @Garrappa18] do not. ![Results of Experiment 2. Relative errors of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) with $\alpha=0.6:0.4:2.6$ and $\beta=1$. The solid black line represents estimates to $\mathop{\mathrm{cond}}_{ML}(A)u$. ](fig_teste2.eps){#fig-7 width="99%"} ![Results of Experiment 2. CPU time (seconds) of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) with $\alpha=0.6:0.4:2.6$ and $\beta=1$. ](fig_teste2_CPU.eps){#fig-7a width="99%"} In this experiment, we consider $\alpha=0.8$ and $\beta=2$. It involves $15$ matrices from MATLAB's gallery, all of them with order $30$ and small blocks in the blocked Schur decomposition. Figure [8](#fig-8){reference-type="ref" reference="fig-8"} compares the relative errors for Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} and `GP-method` with $\mathop{\mathrm{cond}}_{ML}(A)u$, while Figure [9](#fig-8a){reference-type="ref" reference="fig-8a"} displays the corresponding CPU time. All of the matrices were carefully chosen to avoid the occurrence of overflow/underflow in the ML function. ![Results of Experiment 3. Relative errors of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) with $\alpha=0.8$ and $\beta=2$ for fifteen matrices of order $30$ taken from MATLAB's gallery. The solid black line represents $\mathop{\mathrm{cond}}_{ML}(A)u$. ](fig_teste3.eps){#fig-8 width="99%"} ![Results of Experiment 3. CPU time (in seconds) of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) with $\alpha=0.8$ and $\beta=2$ for the fifteen matrices considered in Experiment 3. ](fig_teste3_CPU.eps){#fig-8a width="99%"} This experiment involves sixteen $40\times 40$ matrices, where the first eight are Jordan blocks and the last eight are randomized atomic blocks with non-zero entries (except, of course, the ones in the lower part of the matrix). The results for the relative error are displayed in Figure [10](#fig-9){reference-type="ref" reference="fig-9"}, while the CPU time is in Figure [11](#fig-9a){reference-type="ref" reference="fig-9a"}. This experiment shows clearly the advantage of using Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} in matrices with large atomic blocks, where Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} is about $100$ times faster. Moreover, the relative errors are smaller for almost all the test matrices. ![Results of Experiment 4. Relative errors of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) with $\alpha=0.5$ and $\beta=1.2$ for sixteen atomic blocks of order $40$. ](fig_teste4.eps){#fig-9 width="99%"} ![Results of Experiment 4. CPU time (in seconds) of Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} (blue $\ast$) and of `GP-method` (red squares) with $\alpha=0.5$ and $\beta=1.2$ for the sixteen sixteen atomic blocks of order $40$. ](fig_teste4_CPU.eps){#fig-9a width="99%"} Now we shall make an overall comment on the results of the four experiments. When Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} calls the Taylor series for the computations (which depends on the the values of $\|A\|$, $\alpha$ and $\beta$), our tests have shown that it performs much better than `GP-method`. The same occurs when Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} uses the Schur-Parlett approach combined with the Cauchy integral formula for matrices with large atomic blocks. For the remaining cases, we can say that both methods have a similar behavior. We must recall, however, that Algorithm [\[algorithm3\]](#algorithm3){reference-type="ref" reference="algorithm3"} has the important advantage of being derivative free, so that it is much easier to extend to other matrix functions. We recall that the `GP-method` requires an efficient method for evaluating the higher order derivatives of the ML function. If extended to other matrix functions, a specific efficient method for evaluating their higher order derivatives would be required. # Conclusions {#sec-conclusions} We have proved that if the values of $\|A\|$, $\alpha$ and $\beta$ satisfy certain conditions, the Taylor series can provide a very efficient method for evaluating the ML function (even when $\|A\|\geq 1$), that has the advantage of being derivative and transformation-free. For the situations where reliable results are not guaranteed by the Taylor method, we have shown that computing the ML function of the atomic blocks arising in the blocked Schur decomposition with reordering by means of the Cauchy integral formula provides a promising technique, if a suitable contour is chosen. This latter technique is derivative-free as well, but not transformation-free. A set of numerical experiments confirm that our overall algorithm combining the Taylor series with the Cauchy integral formula is competitive with the state-of-the-art method for IEEE double precision environments, and in some cases even better in terms of accuracy and CPU time. xx R.P. Agarwal, A propos d'une note de M. Pierre Humbert, C. R. Acad. Sci. Paris 236, 2031--2032, 1953. J. Cardoso, Computation of the matrix $p$th root and its Fréchet derivative by integrals, Electron. Trans. Numer. Anal., 39, 414--436, 2012. P. Davies, N. Higham, A Schur--Parlett algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl. 25, 464--485, 2003. P. Davies, N. Higham, Computing $f(A)b$ for matrix functions $f$, in QCD and numerical analysis III, 47, Lect. Notes Comput. Sci. Eng., 15--24, Springer, Berlin, 2005. P. Davis, P. Rabinowitz, *Methods of Numerical Integration*, 2nd ed., Academic Press, London, 1984. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, *Higher Transcendental Functions*, vol. 3. McGraw-Hill, New York, 1955. M. Fasi, Optimality of the Paterson--Stockmeyer method for evaluating matrix polynomials and rational matrix functions, Linear Algebra Appl., 574, 182--200, 2019. R. Garrappa, M. Popolizio, On the use of matrix functions for fractional partial differential equations, Math. Comput. Simul. 81(5), 1045--1056, 2011. R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler function. SIAM J. Numer. Anal. 53, 135--169, 2015. R. Garrappa, The Mittag-Leffler function (https://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function), MATLAB Central File Exchange. Retrieved July 4, 2023. R. Garrappa, I. Moret, M. Popolizio, On the time-fractional Schrödinger equation: theoretical analysis and numerical solution by matrix Mittag-Leffler functions, Comput. Math. Appl. 74(5), 977--992, 2017. R. Garrappa, M. Popolizio, Computing the matrix Mittag-Leffler function with applications to fractional calculus, J. Sci. Comput., 77, 129--153, 2018. G. Golub and C. Van Loan, *Matrix Computations*, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013. R. Gorenflo, A. Kilbas, F. Mainardi, S. Rogosin, *Mittag-Leffler Functions, Related Topics and Applications*, Springer-Verlag, 2014. R. Gorenflo, J. Loutchko, Y. Luchko, Computation of the Mittag-Leffler function $E_{\alpha,\beta} (z)$ and its derivatives, Fract. Calc. Appl. Anal. 5 (4), 491-518, 2002. Erratum: Fract. Calc. Appl. Anal. 6, 2003. N. J. Higham, *Functions of Matrices: Theory and Computation.* Society for Industrial and Applied Mathematics, Philadelphia (2008). N. J. Higham, X. Liu, A Multiprecision derivative-free Schur-Parlett algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl., 42(3), 1401--1422, 2021. [N. Higham]{.smallcaps}, *The Matrix Function Toolbox*, (https://www.mathworks.com/matlabcentral/fileexchange/20820-the-matrix-function-toolbox), MATLAB Central File Exchange. Retrieved August 7, 2023. R. A. Horn, C. R. Johnson, *Topics in Matrix Analysis*, Cambridge Univ. Press, Cambridge, Paperback Edition (1994). P. Humbert, R.P. Agarwal, Sur la fonction de Mittag-Leffler et quelquenes de ses généralisationes, Bull. Sci. Math. (Ser. II). 77, 180--185, 1953. M.G. Mittag-Leffler, Sur l'intégrale de Laplace--Abel, Comp. Rend. Acad. Sci. Paris 135, 937--939, 1902. M.G. Mittag-Leffler, Une généralization de l'intégrale de Laplace--Abel. Comp. Rend. Acad. Sci. Paris 136, 537?539, 1903. M.G. Mittag-Leffler, Sur la nouvelle fonction $E_\alpha(x)$, Comp. Rend. Acad. Sci. Paris 137, 554--558, 1903. M.G. Mittag-Leffler, Sopra la funzione $E_\alpha(x)$, Rend. R. Acc. Lincei, (Ser. 5) 13, 3--5, 1904. M.G. Mittag-Leffler, Sur la representation analytique d'une branche uniforme d'une fonction monogène (cinquième note), Acta Math. 29, 101--181, 1905. I. Moret, P. Novati, On the convergence of Krylov subspace methods for matrix Mittag-Leffler functions, SIAM J. Numer. Anal. 49(5), 2144--2164, 2011. B. Parlett, A recurrence among the elements of functions of triangular matrices, Linear Algebra Appl., 14, 117--121, 1976. M. Paterson and L. Stockmeyer, On the number of nonscalar multiplications necessary to evaluate polynomials, SIAM J. Comput., 2(1), 60--66, 1973. I. Podlubny, Fractional Differential Equations, *An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications*, Math. Sci. Engrg. 198, Academic Press Inc., San Diego, CA, 1999. I. Podlubny, Mittag-Leffler function (https://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function), MATLAB Central File Exchange. Retrieved July 4, 2023. H. Seybold, R. Hilfer, Numerical algorithm for calculating the generalized Mittag-Leffler function, SIAM J. Numer. Anal., 47(1), 69--88, 2008. F. Tatsuoka, T. Sogabe, Y. Miyatake, T. Kemmochi, S.-L. Zhang, Computing the matrix fractional power with the double exponential formula, Electron. Trans. Numer. Anal., 54, 558--580, 2021. L. N. Trefethen, J. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Rev., Vol. 56 (3), 385--458, 2014. A. Wiman, Über den fundamentalsatz der theorie der funkntionen $E_\alpha(x)$, Acta Math. 29, 191--201, 1905.

arxiv_math

--- abstract: | We prove that the finiteness of a finitely generated category of irreducible algebraic varieties over a field of characteristic zero is decidable. We also obtain a Burnside finiteness criterion for such a category, with applications to algebraic dynamical systems of several maps. address: Department of Mathematical Sciences and RIM, Seoul National University author: - Junho Peter Whang title: On finite categories of algebraic varieties --- # Introduction {#sect:1} ## Jacob [@jacob] showed that the finiteness of a finitely generated monoid of matrices over a field is decidable. This paper provides a nonlinear generalization of this result for finitely generated categories of irreducible algebraic varieties, over fields of characteristic zero. In this paper, by an algebraic variety we mean a reduced separated scheme of finite type over a field. Let us define a *system* in a category $C$ to be a quiver (i.e., directed multigraph) whose vertices are objects in $C$ and whose arrows are morphisms in $C$ between the vertices. In other words, a system in $C$ specifies the generators of a subcategory of $C$. **Theorem 1**. *Let $k$ be a field of characteristic zero. There exists an algorithm to determine, given an explicit finite system of irreducible algebraic varieties over $k$, whether or not the category it generates is finite.* We make precise the notion of an explicitly given system of varieties in Section [2.2](#sect:3.1){reference-type="ref" reference="sect:3.1"}. We prove Theorem [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"} by induction on the complexity of the system, using two ingredients: an effective form of nonlinear Selberg's lemma due to Bass-Lubotzky [@bl], and the observation that dominant endormophisms of finite order on integral schemes are automorphisms (Lemma [Lemma 15](#finite-order){reference-type="ref" reference="finite-order"}). A similar argument yields a solution to the strong Burnside problem for categories of varieties in characteristic zero, generalizing (in characteristic zero) the work of McNaughton-Zalcstein [@mz] on matrix monoids. Let us say that a category is *torsion* if every endomorphism of every object generates a cyclic monoid of finite order under composition. **Theorem 2**. *Let $k$ be a field of characteristic zero. Let $C$ be a finitely generated subcategory of the category of irreducible algebraic varieties over $k$. Then $C$ is finite if and only if it is torsion.* Over general fields, one can deduce a weaker finiteness criterion for categories of varieties, establishing the analogue of the bounded Burnside problem. This relies on Zelmanov's resolution of the restricted Burnside problem [@z1; @z2]. One source of motivation for our work is the study of finite orbits in dynamics of several maps on algebraic varieties. For instance, we prove the following. **Corollary 3**. *Let $k$ be a field. Let $M$ be a finitely generated monoid acting on an algebraic variety $V/k$. Let $x\in V(k)$. Then $M\cdot x$ is finite if and only if $$\sup_{N} |N\cdot x|<\infty$$ where $N$ runs over all $2$-generated submonoids of $M$.* Given a set $X$ and a monoid $M$ of endomorphisms of $X$, let us say that a point $x\in X$ is *$M$-periodic* if the $M$-orbit $M\cdot x$ is finite and $M$ permutes the elements of $M\cdot x$. In the case where $k$ is a field of characteristic zero and $M$ is a finitely generated monoid acting on a variety $V/k$, one can combine a refined form of Corollary [Corollary 3](#cor1){reference-type="ref" reference="cor1"} with [@whang Theorem 1.2] to obtain the following. **Theorem 4**. *Let $k$ be a field of characteristic zero. Let $M$ be a finitely generated monoid acting on an algebraic variety $V/k$. Let $x\in V(k)$. Then the following are equivalent:* 1. *$x$ is $M$-periodic.* 2. *$x$ is $N$-periodic for every $2$-generated submonoid $N\leq M$.* *If moreover $M$ is a group, then the above are equivalent to:* 1. *$x$ is $\langle f\rangle$-periodic for every $f\in M$.* This paper is organized as follows. Section [2](#sect:2){reference-type="ref" reference="sect:2"} collects the necessary background, including results of Bass-Lubotzky [@bl] and Zelmanov [@z1; @z2]. In Section [3](#sect:3){reference-type="ref" reference="sect:3"}, we prove Theorems [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"} and [Theorem 2](#mainthm1){reference-type="ref" reference="mainthm1"}. In Section [4](#sect:4){reference-type="ref" reference="sect:4"}, we consider dynamical corollaries of our main results, and in particular prove Corollary [Corollary 3](#cor1){reference-type="ref" reference="cor1"} and Theorem [Theorem 4](#finorb){reference-type="ref" reference="finorb"}. ## Acknowledgments I thank Abhishek Oswal for helpful conversations. In particular, the proof of Theorem [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"} was inspired by an unpublished collaborative work on finite matrix monoids. This work was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA2201-03. # Background {#sect:2} ## Notations Let us set up the notations and terminology that will be used throughout the paper. A quiver is a directed multigraph. Given a quiver $Q$, we write $Q_0$ for the class of its vertices and $Q_1$ for the class of its arrows. Let $s,t:Q_1\rightrightarrows Q_0$ denote the source and target maps of $Q$. We shall say that a quiver is *small* if $Q_0$ and $Q_1$ are sets. We shall view a category as a quiver along with a composition law on its arrows that satisfies the usual axioms. Let $C$ be a category. If $X,Y\in C_0$, we write $\mathop{\mathrm{Hom}}_{C}(X,Y)$ for the set of morphisms in $C$ from $X$ to $Y$. We write $\mathop{\mathrm{End}}_C(X)=\mathop{\mathrm{Hom}}_{C}(X,X)$. Given $X\in C_0$, we write $C_X$ for the full subcategory of $C$ with a single object $X$. Thus, $(C_X)_1=\mathop{\mathrm{End}}_C(X)$. In this paper, a *monoid* is a small category whose object set is a singleton. A monoid is *cyclic* if it is generated by a single endomorpism. A *groupoid* is a small category whose morphisms are all invertible. **Definition 5**. Let $C$ be a small category. 1. The *order* of $C$ is $|C|=| C_1|$. 2. We say $C$ is *finite* if it has finite order. 3. We say $C$ is *torsion* if every cyclic submonoid of $C$ is finite. 4. We say $C$ is *$n$-torsion* if every cyclic submonoid of $C$ has order $\leq n$. Forgetting the composition law on a category $C$, we obtain a quiver (i.e. directed multigraph) whose collection of vertices is $C_0$, whose collection of arrows is $C_1$, and whose source and target maps are $s$ and $t$. **Definition 6**. A *system* in a category $C$ to be a small subquiver of the quiver underlying $C$. Given a system $S$ in $C$, let $\langle S\rangle$ denote the subcategory of $C$ generated by $S$, i.e. smallest subcategory $C'$ of $C$ such that $S_0=C'_0$ and $S_1\subseteq C'_1$. We say that a category $C$ is *finitely generated* if there is a finite quiver $S$ in $C$ (i.e. with $|S_0|$ and $|S_1|$ finite) such that $C=\langle S\rangle$. **Definition 7**. Let $S$ be a quiver. A *path* in $S$ from $v\in S_0$ to $w\in S_0$ is a sequence $f_1,\dots,f_k\in S_1$ of arrows such that $s(f_1)=v$, $t(f_k)=w$, and $t(f_i)=s(f_{i+1})$ for $i=1,\dots,k-1$. Two vertices $v,w\in S_0$ are *path-equivalent* if there is a path in $S$ from $v$ to $w$ and there is a path in $S$ from $w$ to $v$. We denote by $S^\circ$ the quiver obtained from $S$ by deleting the arrows between vertices that are not path-equivalent. A system $S$ is *path-connected* if $S=S^\circ$. A *path-component* of $S$ is a maximal subquiver of $S$ that is path-connected. **Lemma 8**. *A finitely generated category $C$ is finite if and only if $C^\circ$ is finite.* *Proof.* If $C$ is finite, then obviously $C^\circ$ is finite. Suppose conversely that $C^\circ$ is finite. Let $N$ be the number of path-components of $C$. Let $S$ be a finite system of generators for $C$. Every morphism of $f$ can be written in the form $$f=g_kh_{k}g_{k-1}h_{k-1}\dots g_{2}h_{1}g_{0}$$ for some $k\leq N$, where $g_i\in c(C)_1$ and $h_i\in S_1\setminus c(C)_1$ for each $i=0,\dots, k$. It follows that $C_1$ is finite, so $C$ is finite. ◻ ## Bass-Lubotzky {#sect:3.1} We recall a theorem of Bass-Lubotzky [@bl Corollary (1.2)]. **Theorem 9**. *Let $k$ be an arbitrary ring. Let $G$ be a finitely generated group of automorphisms of a scheme $V$ of finite presentation over $k$.* 1. *$G$ is residually finite.* 2. *If $V$ is flat over ${\mathbb{Z}}$, then $G$ is virtually torsionfree.* We shall also need an effective form of the second part of Theorem [Theorem 9](#bl0){reference-type="ref" reference="bl0"}, whose formulation we recall as follows. Let $k$ be a finitely generated subring of $\bar{\mathbb{Q}}$. Let $Z$ be a scheme flat of finite type over $k$. Given a finite set $X$ of closed points of $Z$, let $$A_{X}=\prod_{x\in X}{\mathcal{O}}_{Z,x}\quad\text{and}\quad J_{X}=\prod_{x\in X}{\mathfrak{m}}_{Z,x}$$ where ${\mathcal{O}}_{Z,x}$ denotes the local ring of $Z$ at $x$ and ${\mathfrak{m}}_{Z,x}$ is the maximal ideal of ${\mathcal{O}}_{Z,x}$, with residue field $\kappa(x)={\mathcal{O}}_{Z,x}/{\mathfrak{m}}_{Z,x}$. We shall say that $X$ has *residue characteristic $p$* if $\kappa(x)=p$ for every $x\in X$. Following [@bl], we shall say that $X$ is *effective* if there is a finite affine open covering $(U_i)_{i\in I}$ of $Z$ such that the natural morphism ${\mathcal{O}}_Z(U_i)\to\prod_{x\in X\cap U_i}{\mathcal{O}}_{Z,x}$ is injective for every $i\in I$. The following holds. **Proposition 10**. *Suppose that $X$ is a finite effective set of closed points in $Z$ with residue characteristic $p$. Then the order of any torsion element in $$\Gamma_X=\ker(\mathop{\mathrm{Aut}}(Z/k)\to\mathop{\mathrm{Aut}}Z(A_{X}/J_{X}^2))$$ is a power of $p$. In particular, if $X$ and $X'$ are finite effective sets in $Z$ having distinct residue characteristics $p\neq p'$, then $\Gamma_X\cap\Gamma_{X'}$ is a normal torsionfree subgroup of finite index in $\mathop{\mathrm{Aut}}(Z/k)$.* *Proof.* The proof is given in [@bl pp.4-5]. See also [@whang Section 2.1] for a summary. ◻ ## Burnside problem We refer to [@reference] for a summary of the history of the Burnside problem. Here, we recall their formal statements of its variants. **Problem 11** (Strong Burnside's problem). *Let $G$ be a finitely generated group all of whose elements are torsion. Is $G$ finite?* **Problem 12** (Bounded Burnside's problem). *Let $G$ be a finitely generated group all of finite exponent. Is $G$ finite?* **Problem 13** (Restricted Burnside's problem). *Are there only finitely many finite groups with given number of generators and given exponent?* While both the strong Burnside's problem and bounded Burnside's problem have negative answers in general (in the strong case by work of Golod-Shafarevich [@gs], and in the bounded case by work of Adian and Novikov [@an]), for linear groups they admit affirmative answers (due to Schur [@schur] and Burnside [@burnside], respectively). The restricted Burnside problem was solved affirmatively by Zelmanov [@z1; @z2]. An immediate corollary of his work is the following characterization of finite groups. **Theorem 14**. *A group $G$ is finite if and only if it is finitely generated, residually finite, and of finite exponent.* In the meanwhile, analogues of Burnside's problems for other algebraic structures such as semigroups have been studied. In [@mz], McNaughton-Zalcstein [@mz] established the analogue of the strong Burnside's problem for semigroups (or monoids) of matrices over arbitrary fields. Theorem [Theorem 2](#mainthm1){reference-type="ref" reference="mainthm1"} serves to generalize this result, in characteristic zero, to categories of algebraic varieties. # Proofs of Theorems [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"} and [Theorem 2](#mainthm1){reference-type="ref" reference="mainthm1"} {#sect:3} ## Explicitly given systems of varieties Here, we make precise our notion of an explicitly given finite systems of algebraic varieties, generally following the spirit of the paragraph after [@whang Theorem 1.2]. 1. A finitely presented ring $k$ is explicitly given if it is given as a quotient of a polynomial ring with coefficients in ${\mathbb{Z}}$, and a finite set of generators for the kernel of the quotient map is specified. A ring homomorphism between explicitly given finitely presented rings is explicitly given if the images of the generators (given by the explicit finite presentation) of the domain ring are specified. In what follows, let $k$ be an explicitly given finitely presented ring. 2. An affine scheme of finite presentation over $k$ is explicitly given if it is the spectrum of an explicitly given $k$-algebra. A morphism between two explicitly given affine schemes of finite presentation over $k$ is explicitly given if it is induced by an explicitly given $k$-algebra homomorphism between their coordinate rings. 3. A scheme $V$ separated of finite presentation over $k$ is explicitly given if it is written as an explicit finite union of explicitly given open affine schemes $U_i$ with affine overlaps $U_i\cap U_j$, such that the gluing morphisms $U_i\cap U_j\to U_i$ are explicitly given. We shall call $(U_i)$ an effective presentation of $V$. 4. Let $V$ and $W$ be explicitly given schemes of finite presentation over $k$, with effective presentations $(U_i)_{i\in I}$ and $(T_{j})_{j\in J}$ respectively. A morphism $f:V\to W$ over $k$ is explicitly given if there is another effective presentation $(U_i')_{i\in I'}$ of $V$ and an explicit function $j:I'\to J$ such that $f(U_i')\subseteq T_{j(i)}$ for every index $i$, and the following hold: 1. $f|_{U_i'}:U_i'\to T_{j(i)}$ is an explicitly given morphism for all $i\in I$, and 2. the inclusions of $U_i'\cap U_k$ into $U_i'$ and $U_k'$ are explicitly given morphisms for all $(i,k)\in I'\times I$. Finally, a finite system $S$ of algebraic varieties over a field $K$ is explicitly given if there is an explicitly given finitely presented domain $k\subset K$ and a finite system $S_k$ of schemes over $k$, whose vertices are explicitly given separated schemes of finite type over $k$ and whose arrows are explicitly given morphisms between those schemes, such that $S$ is obtained from $S_k$ by base change. ## Decidability of finiteness Here, we shall prove Theorem [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"}. Let us begin with a lemma. **Lemma 15**. *Let $X$ be an integral separated scheme. If $f$ is a dominant endomorphism of $X$ of finite order, then $f$ is an automorphism.* *Proof.* Let $k(X)$ be the function field of $X$. Since $f$ is dominant, it induces an inclusion $f^*:k(X)\to k(X)$. Since there exist $0\leq m<n$ such that $f^m=f^n$, in fact $f^*$ must induce an automorphism of $k(X)$, and $(f^{n-m})^*$ is the identity on $k(X)$. This implies that $f^{n-m}$ is the identity on a dense open subscheme of $X$. Since $X$ is separated, this implies that $f^{n-m}={\textup{id}}_X$. ◻ **Proposition 16**. *Let $k$ be a finitely presented domain of characteristic zero. Let $S$ be a finite path-connected system of dominant morphisms between integral separated schemes of finite type over $k$. If $\langle S\rangle$ is finite, then $|\langle S\rangle|\leq C(S_0)$ where $C(S_0)$ is a constant that only depends on $S_0$.* *Proof.* Let $S$ be a finite system as in the statement of the proposition. By Lemma [Lemma 15](#finite-order){reference-type="ref" reference="finite-order"} and our assumptions on $S$, every endomorphism of an object in $\langle S\rangle$ is an automorphism of finite order. It follows that $\langle S\rangle$ is a groupoid. Fix $Z\in S_0$. For every $W\in S_0$, the set $\mathop{\mathrm{Hom}}_{\langle S\rangle}(Z,W)$ is a torsor under the group $\langle S\rangle_Z$, so we have $|\langle S\rangle|=|\langle S\rangle_Z|^{|S_0|}$. Now, $\langle S\rangle_Z$ is a finite subgroup of $\mathop{\mathrm{Aut}}(Z/k)$. Fix two closed points $x$ and $x'$ of the integral scheme $Z$ such that the characteristics of the residue fields $\kappa(x)$ and $\kappa(x')$ are coprime. Setting $X=\{x\}$ and $X'=\{x'\}$, we see by Proposition [Proposition 10](#prop1){reference-type="ref" reference="prop1"} that the composition of group homomorphisms $$\langle S\rangle_Z\to \mathop{\mathrm{Aut}}(Z/k)\to\mathop{\mathrm{Aut}}Z(A_X/J_X^2)\times \mathop{\mathrm{Aut}}(A_X/J_X^2)$$ is injective. Since the right hand side only depends on $Z$, we are done. ◻ Let $k$ be a field of characteristic zero. There exists an algorithm to determine, given an explicit finite system of irreducible algebraic varieties over $k$, whether or not the category it generates is finite. *Proof.* Let $S$ be a finite system of irreducible algebraic varieties over $\bar{\mathbb{Q}}$. By spreading out, we can explicitly determine a finitely generated subring of $\bar{\mathbb{Q}}$ such that $S$ descends to a finite system of integral separated schemes flat of finite type over $k$. We shall proceed by induction on $|S_1|$ and $\max_{V\in S_0}\dim V$. By Lemma [Lemma 8](#connected){reference-type="ref" reference="connected"}, we may assume that $S$ is path-connected. If every $f\in S_1$ is dominant, then by Proposition [Proposition 16](#dominant){reference-type="ref" reference="dominant"} it is decidable whether or not $\langle S\rangle$ is finite. So assume that some $f\in S_1$ is not dominant, and write $A,B\in S_0$ for the source and target of $f$ respectively. Let $Z$ be the Zariski closure of the image of $f$ in $B$, equipped with the reduced closed subscheme structure. Note that $Z$ is integral, separated, and flat of finite type over $k$. Consider the finite systems $S'$ and $S''$ where: - $S_0'=S_0$ and $S_1'=S_1\setminus\{f\}$, and - $S_0''=\{Z\}$ and $S_1''=\{(f\circ g)|_Z:g\in\mathop{\mathrm{Hom}}_{\langle S'\rangle }(B,A)\}$. Since $|S_1'|<|S_1|$, by inductive hypothesis it is decidable whether or not $\langle S'\rangle$ is finite, so we may assume it is. This implies that $|S_1''|$ is finite, and since $Z$ has smaller dimension than $B$, by inductive hypothesis it is decidable whether or not $\langle S''\rangle$ is finite. We may assume $\langle S''\rangle$ is finite. Then for every $V,W\in S_0$, if $g\in\mathop{\mathrm{Hom}}_{\langle S\rangle}(V,W)$ then either $g\in\mathop{\mathrm{Hom}}_{\langle S'\rangle}(V,W)$ or $$g=h_2\circ e\circ f\circ h_1$$ for some $h_1\in\mathop{\mathrm{Hom}}_{\langle S'\rangle}(V,A)$, $e\in \langle S''\rangle_1$, and $h_2\in\mathop{\mathrm{Hom}}_{\langle S'\rangle}(B,W)$. It follows that $\langle S\rangle$ is finite. This completes the induction and the proof. ◻ ## Burnside finiteness criterion We now prove the following extended version of Theorem [Theorem 2](#mainthm1){reference-type="ref" reference="mainthm1"}, by adapting our proof of Theorem [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"}. **Theorem 17**. *Let $k$ be a field. Let $C$ be a finitely generated category of irreducible algebraic varieties over $k$. Then $C$ is finite if and only if:* 1. *$C$ is $n$-torsion for some $n\geq1$, or* 2. *$C$ is torsion and $k$ has characteristic zero.* *Proof.* The "only if" direction is clear, so we shall focus on the "if" direction. Let $S$ be a finite system generating $C$. As in the proof of Theorem [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"}, we proceed by induction on $|S_1|$ and $\max_{V\in S_0}\dim V$. By Lemma [Lemma 8](#connected){reference-type="ref" reference="connected"}, we may assume that $S$ is path-connected. First, suppose that every $f\in S_1$ is dominant. Since $C$ is torsion, arguing as in the proof of Proposition [Proposition 16](#dominant){reference-type="ref" reference="dominant"}, we see that $C$ is a finitely generated groupoid, and hence its finiteness is equivalent to the finiteness of the finitely generated group $C_V$ for some (or any) $V\in C_0$. Now, we have the following. 1. If $C$ is $n$-torsion for some $n\geq1$, then since the group $C_V$ is residually finite by Theorem [Theorem 9](#bl0){reference-type="ref" reference="bl0"}(1), it is finite by the resolution of the restricted Burnside problem (Theorem [Theorem 14](#zelmanov){reference-type="ref" reference="zelmanov"}). 2. If $C$ is torsion and $k$ has characteristic zero, then since $C_V$ is virtually torsionfree by Theorem [Theorem 9](#bl0){reference-type="ref" reference="bl0"} it is finite. This proves the theorem in the case where every $f\in S_1$ is dominant. If some $f\in S_1$ is not dominant, then we argue as in the proof of Theorem [Theorem 1](#mainthm0){reference-type="ref" reference="mainthm0"} to construct systems $S'$ and $S''$ with smaller complexity, such that finiteness of $C'=\langle S'\rangle$ and $C''=\langle S''\rangle$ imply the finiteness of $C$. If $C$ satisfies condition (1) or (2) of the theorem, then clearly $C'$ and $C''$ also satisfy the same condition. Thus, $C'$ and $C''$ are finite by inductive hypothesis, and hence $C$ is finite. ◻ # Dynamical corollaries {#sect:4} We record some dynamical corollaries of Theorem [Theorem 17](#extmainthm1){reference-type="ref" reference="extmainthm1"}, by transferring properties of monoids to their orbits. The following is a refined version of Corollary [Corollary 3](#cor1){reference-type="ref" reference="cor1"}. **Corollary 18**. *Let $k$ be a field. Let $M$ be a finitely generated monoid acting on an algebraic variety $V/k$. Let $x\in V(k)$. The following are equivalent.* 1. *$M\cdot x$ is finite.* 2. *There exists $n\geq1$ with $|N\cdot x|\leq n$ for every $2$-generated submonoid $N\leq M$.* 3. *There exists $n\geq1$ with $|\langle f\rangle\cdot g(x)|\leq n$ for every $f,g\in M$.* *Proof.* It is clear that $(1)\implies (2)\implies (3)$, so it remains to show $(3)\implies (1)$. Suppose $(3)$ holds. Let $Z$ be the Zariski closure of $M\cdot x$ in $V$, equipped with the reduced closed subscheme structure. Let us now construct a finite system $S$ of irreducible varieties, where the elements of $S_0$ are the irreducible components of $Z$ and the elements of $S_1$ are the restrictions of the elements of a finite generating set of $M$ to the irreducible components of $Z$. Let $C=\langle S\rangle$. If $f\in C_1$ is any endomorphism of an object $A$ in $C$, we claim that $f$ has finite order $\leq n$, i.e., $C$ is $n$-torsion. Indeed, there exists some $0\leq m<n$ such that $A(k)$ contains a Zariski dense set $T$ of points satisfying $f^n(y)=f^m(y)$ for all $y\in T$. Since $A$ is integral and separated, it follows that $f^n=f^m$ on $A$. Thus $C$ is $n$-torsion, and by Theorem [Theorem 17](#extmainthm1){reference-type="ref" reference="extmainthm1"} it follows that $C$ is finite. But then the orbit of $x$ must be finite. ◻ Let $k$ be a field of characteristic zero. Let $M$ be a finitely generated monoid acting on an algebraic variety $V/k$. Let $x\in V(k)$. Then the following are equivalent: 1. $x$ is $M$-periodic. 2. $x$ is $N$-periodic for every $2$-generated submonoid $N\leq M$. If moreover $M$ is a group, then the above are equivalent to: 1. $x$ is $\langle f\rangle$-periodic for every $f\in M$. *Proof.* The implication (a)$\implies$(b) is clear, so let us show (b)$\implies$(a). Assume (b). By [@whang Theorem 1.2], there is a constant $n\geq0$, depending only on $V$ and the finitely generated ring over which $V$, $x$, and $M$ are defined, such that $|N\cdot x|\leq n$ for every $2$-generated submonoid $N\leq M$ by $N$-periodicity of $x$. This implies (a) by Corollary [Corollary 18](#orbit){reference-type="ref" reference="orbit"}. Suppose now that $M$ is a group. Since (a)$\implies$(c) is clear, we shall show (c)$\implies$(a). Let us assume (c). In light of Corollary [Corollary 18](#orbit){reference-type="ref" reference="orbit"}, it suffices to show that there is a constant $n\geq0$ such that $|\langle f\rangle \cdot g(x)|\leq n$ for every $f,g\in M$, or (since $M$ is a group) equivalently $|\langle h^{-1}f h\rangle\cdot x|\leq n$ for every $f,h\in M$, or equivalently $|\langle g\rangle\cdot x|\leq n$ for every $g\in M$. Now, since $\langle g\rangle\cdot x$ is finite for every $g\in M$ by hypothesis, the desired result follows again by Theorem [@whang Theorem 1.2]. ◻ 1 Bass, H., Lubotzky, A. *Automorphisms of groups and of schemes of finite type.* Israel Journal of Mathematics volume 44, pages 1-22 (1983) Burnside, W. *On an unsettled question in the theory of discontinuous groups*. Q. J. Pure Appl. Math. 33 (1902), no. 2, 230-238. Golod, E. S., Shafarevich, I. *On the class field tower*, Izv. Akad. Nauk. SSSR Ser. Mat. 28 (1964), no. 2, 261-272. Jacob, G. *La finitude des représentations linéaires des semi-groupes est décidable,* Journal of Algebra, Volume 52, Issue 2, 1978, Pages 437-459. Mandel, A., Simon, I. *On finite semigroups of matrices.* Theoretical Computer Science, Volume 5, Issue 2, 1977 McNaughton, R., Zalcstein, Y. *The Burnside problem for semigroups*, Journal of Algebra, Volume 34, Issue 2, 1975, Pages 292-299. Novkov, P., Adjan, S. *Infinite periodic groups, I* (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 212--244; English translation in Math. USSR Izv. 2 (1968), no. 1, 209 O'Connor, J.J., Robertson, E. F., A history of the Burnside problem, preprint, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Burnside_problem.html Schur, I. *Uber Gruppen linearer Substitutionen mit Koeffizienten aus einem algebraischen Zahlkörper*. Math. Ann. 71 (1911), no. 3, 355-367. Whang, J.P. *On periodic orbits of polynomial maps.* arXiv:2305.13529 Zelmanov, E. I. *Solution of the restricted Burnside problem for groups of odd exponent.* (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 42--59, 221; translation in Math. USSR-Izv. 36 (1991), no. 1, 41--60 Zelmanov, E. I. *Solution of the restricted Burnside problem for 2-groups.* (Russian) Mat. Sb. 182 (1991), no. 4, 568--592; translation in Math. USSR-Sb. 72 (1992), no. 2, 543--565

arxiv_math

--- abstract: | We obtain nontrivial bounds for character sums with multiplicative and additive characters over finite fields over elements with restricted coordinate expansion. In particular, we obtain a nontrivial estimate for such a sum over a finite field analogue of the Cantor set. address: - School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia - School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia author: - Siddharth Iyer - Igor E. Shparlinski title: Character sums over elements of extensions of finite fields with restricted coordinates --- # Introduction Let $\left(\vartheta_1, \ldots, \vartheta_r\right)$ be a basis of the finite field $${\mathbb F}_{q^r}=\left \{a_1\vartheta_1+ \ldots +a_r \vartheta_r:~a_1, \ldots, a_r \in {\mathbb F}_q\right\}$$ of $q^r$ elements over the finite field ${\mathbb F}_q$ of $q$ elements. Motivated by a series of recent outstanding results on integers with restricted digital expansion in a given basis, there has also been very significant progress in studying elements $\omega \in {\mathbb F}_{q^r}$ with various restrictions on their coordinates $(a_1, \ldots, a_r)$ in the expansion $$\omega = a_1\vartheta_1+ \ldots +a_r \vartheta_r\in {\mathbb F}_{q^r},$$ we refer to [@MSW] for a brief outline of such results (both settings on integers and finite fields), some new results and further references, in particular on bounds of various character sums over such field elements. Here, given a set ${\mathcal A}\subseteq {\mathbb F}_q$, we consider the set $$\label{eq:Set SrA} {\mathcal S}_r\left({\mathcal A}\right)= \left \{a_1\vartheta_1+ \ldots +a_r \vartheta_r:~a_1, \ldots, a_r \in {\mathcal A}\right\}$$ that is the set of $u \in {\mathbb F}_{q^r}$ whose coordinates are restricted to the set ${\mathcal A}$. In particular, one of the natural examples is the case of $q=3$ and ${\mathcal A}=\{0, 2\}$ which leads to a Cantor-like set ${\mathcal S}_r\left({\mathcal A}\right)\subseteq {\mathbb F}_{3^r}$. The main goal of this paper is to estimate mixed character sums $$S_{r}({\mathcal A};\chi,\psi;f_1,f_2) = \sum_{\omega \in {\mathcal S}_r\left({\mathcal A}\right)}\chi\left(f_1(\omega)\right)\psi\left(f_2(\omega)\right),$$ with rational functions $f_1(X), f_2(X) \in {\mathbb F}_{q^r}(X)$, of degrees $d_1$ and $d_2$, respectively, and where $\chi$ and $\psi$ are a fixed multiplicative and additive character of ${\mathbb F}_{q^r}$, respectively (with the natural conventions that the poles of $f_1(X)$ and $f_2(X)$ are excluded from summation). We are especially interested in the case when ${\mathcal A}$ is of cardinality $\#{\mathcal A}$ relatively small compared to $q$. In particular, we are interested in obtaining nontrivial bounds in the case of small values of the parameter $$\rho = \frac{\log \#{\mathcal A}}{\log q}.$$ It is well known that such bounds can be used to study, for example, the distribution of primitive elements in the values of polynomials on elements from ${\mathcal S}_r\left({\mathcal A}\right)$ or their pseudorandom properties. Since these applications are quite standard, we do not present them here. # Notation and conventions Throughout the paper, we fix the size $q$ of the ground field, and thus also its characteristic $p$ while the parameter $r$ is allowed to grow. We also fix an additive character $\psi$ and a multiplicative character $\chi$ of ${\mathbb F}_{q^r}$ which are not both principal. As usual, we use $\overline{{\mathbb F}_q}$ to denote the algebraic closure of ${\mathbb F}_q$. It is useful to recall that $\overline{{\mathbb F}_q}\subseteq \overline{{\mathbb F}_p}$. For a finite set ${\mathcal S}$, we use $\# {\mathcal S}$ to denote its cardinality. We denote by $\log_2 x$ the binary logarithm of $x>0$. We adopt the Vinogradov symbol $\ll$, that is, for any quantities $A$ and $B$ we have the following equivalent definitions: $$A\ll B~\Longleftrightarrow~A=O(B)~\Longleftrightarrow~|A|\le c B$$ for some constant $c>0$, which throughout the paper is allowed to depend on the degrees $d, d_1,d_2$ and the ground field size $q$ (but not on the main parameter $r$) and the integer parameter $s\ge 1$. We also adopt the $o$-notation $$A=o(B)~\Longleftrightarrow ~|A|\le \varepsilon B$$ for any fixed $\varepsilon>0$ and sufficiently large (depending on $d$, $d_1$, $d_2$, $q$, $s$ and $\varepsilon$) values of the parameter $r$ For a rational function $g(X) \in \overline{{\mathbb F}_p}(X)$ and an element $w \in\overline{{\mathbb F}_p}$ we define $\operatorname{ord}_{w} g$ to be the unique integer so that $(X-w)^{\operatorname{ord}_{w}(g)}g$ extends to a rational function which has no zero or pole at $w$. We also write $${\mathbf{\,e}}_p(z)=\exp(2\pi iz/p).$$ Finally, we also recall our convention that the poles of functions in the arguments of multiplicative and additive characters are always excluded from summation. # Main results We define the following sets of rational functions. **Definition 1**. *For integers $d\ge 0$ and $n \ge 2$,* - *let ${\mathcal Q}_{d,n}$ be the set of rational functions $g(X)\in{\mathbb F}_{q^r}(X)$ of degree at most $d$, which are not an $n$-th power of some rational function in $\overline{{\mathbb F}_p}(X)$.* - *let ${\mathcal R}_{d}$ be the set of rational functions $f(X)\in{\mathbb F}_{q^r}(X)$ of degree at most $d$, which have at least one pole of order that is not a multiple of $p$.* We note that we allow $d=0$ in Definition [Definition 1](#def: set Q and R){reference-type="ref" reference="def: set Q and R"}, that is non-zero constant functions, in which case ${\mathcal Q}_{d,n} = {\mathcal R}_{d} = \emptyset$. We are now ready to present our main result. We recall our convention that implied constants are allowed to depend on the integer parameters $d_1$, $d_2$, $q$ and $s$. For an integer $s \ge 1$ we define $$\label{eq:kappa-s} \kappa_s(\rho) = \frac{s\rho(2\rho-1)+\rho -1}{4s(s\rho +1)}.$$ **Theorem 2**. *Let $\chi$ and $\psi$ be a multiplicative and additive character, respectively, and let $f_1(X) , f_2(X)\in {\mathbb F}_{q^r}(X)$. Assume that at least one of the following conditions holds* - *$\chi$ is nonprincipal of order $n$ and $f_1(X) \in {\mathcal Q}_{d,n}$,* - *$\psi$ is nonprincipal and $f_2(X) \in{\mathcal R}_{d}$.* *Then for any fixed integer $s\ge 1$, we have $$S_{r}({\mathcal A};\chi,\psi;f_1,f_2) \ll \left(\# {\mathcal A}\right)^r q^{- r \kappa_s(\rho) } .$$* Clearly for any $\rho > 1/2$ we have $\kappa_s(\rho) >0$ for a sufficiently large $s$. In particular, with $$\rho =\frac{\log 2}{\log 3}$$ taking $s=5$ in Theorem [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"} we have the following nontrivial bound for a "Cantor-like" set in finite fields. **Corollary 3**. *Let $q = 3$ and ${\mathcal A}= \{0,2\}$. Under the conditions of Theorem [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"} we have $$S_{r}({\mathcal A};\chi,\psi;f_1,f_2) \ll 2^{\gamma r}$$ where $$\gamma = 1 - \frac{\log 3}{\log 2}\cdot \kappa_{5}\left(\frac{\log 2}{\log 3}\right) %%\frac{8 \log 2(2\log 2- \log 3)+ \log 2 \cdot \log 3 -(\log 3)^2}{32(8 \log 2 - \log 3) \log 3 } = 0.99128\ldots\,.$$* We remark that both Theorem [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"} and Corollary [Corollary 3](#cor:Cantor){reference-type="ref" reference="cor:Cantor"} apply to *Kloosterman sums* $$\sum_{\omega \in {\mathcal S}_r\left({\mathcal A}\right)} \psi\left(a\omega + b \omega^{-1} \right), \qquad (a,b) \in {\mathbb F}_{q^r} \times {\mathbb F}_{q^r}^*,$$ over elements of ${\mathcal S}_r\left({\mathcal A}\right)$. We note that unfortunately Theorem [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"} does not apply to polynomials $f_2$ if either $\chi$ is principal or $f_1 \not \in {\mathcal Q}_{d,n}$. Hence, we introduce another class of functions which actually originates from [@MSW]. **Definition 4**. *Let ${\mathcal P}_{d}$ be the set of rational functions $f(X)\in{\mathbb F}_{q^r}(X)$ of degree $d$ such that for any $\omega\in {\mathbb F}_{q^r}^*$ the function $$f_{\omega}(X)=f(X+\omega)-f(X)$$ is not of the form $$f_{\omega}(X)=\alpha\left(g(X)^p-g(X)\right)+\beta X$$ for some rational function $g(X)\in \overline{{\mathbb F}_q}(X)$ and $\alpha,\beta\in \overline{{\mathbb F}_q}$.* We refer to [@MSW] for examples of functions from ${\mathcal P}_{d}$. For a function $f_2 \in {\mathcal P}_d$, we are only able to obtain a version of Theorem [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"} with $s=1$, and hence we save only $$\kappa_1(\rho) = \frac{2\rho^2-1}{4(\rho +1)}.$$ **Theorem 5**. *Let $\chi$ and $\psi$ be a multiplicative and additive character, respectively, and let $f_1(X) , f_2(X)\in {\mathbb F}_{q^r}(X)$. Assume that $\psi$ is nonprincipal and $f_2(X) \in{\mathcal P}_{d}$. Then, we have $$S_{r}({\mathcal A};\chi,\psi;f_1,f_2) \ll \left(\# {\mathcal A}\right)^r q^{- r \kappa_1(\rho) } .$$* Note that $\kappa_1(\rho) >0$ only for $$\rho > 2^{-1/2} = 0.707106\ldots > \frac{\log 2}{\log 3} =0.63092\ldots ,$$ and hence unfortunately Theorem [Theorem 5](#thm:poly bound){reference-type="ref" reference="thm:poly bound"} does not apply to the setting of Corollary [Corollary 3](#cor:Cantor){reference-type="ref" reference="cor:Cantor"}. # Ratios and linear combinations of shifts of rational functions Various versions of the following results have been well-known, see, for example, the proof of [@BaCoSh Theorem 1] or of [@OstShp Theorem 1]. It is convenient to introduce the following notation. Given a vector $\mathbf{v}= \left(v_{1},\ldots ,v_{2s}\right)\in \overline{{\mathbb F}_p}^{2s}$ and a rational function $f \in \overline{{\mathbb F}_p}(X)$, we set $$\label{eq:Prod f} P_{\mathbf{v},f}(X)= \prod_{i=1}^{s}\frac{f\left(X+v_{i}\right)}{f\left(X+v_{s+i}\right)}.$$ The implied constants in this section may depend only on $d = \deg f$ and $s$, but are uniform with respect to other parameters, including $q$, and most importantly $n$, $r$ and $V$. **Lemma 6**. *Let $f(X) \in {\mathcal Q}_{d,n}$ for some integers $d \ge 1$ and $n \geq 2$. For any set ${\mathcal V}\subseteq \overline{{\mathbb F}_p}$ of cardinality $V$, for each integer $s \geq 1$ we have $$\# \left\{\mathbf{v}= \left(v_{1},\ldots ,v_{2s}\right)\in {\mathcal V}^{2s}:~P_{\mathbf{v},f}(X) \not \in {\mathcal Q}_{2ds,n}\right\} \ll V^s.$$* *Proof.* Without loss of generality, we can assume that all zeros and poles of $f$ are of order less than $n$, that is, $$f(X) = \prod_{j=1}^{h}(X-\alpha_{j})^{u_{j}},$$ where $\alpha_{j} \in \overline{{\mathbb F}_{p}}$ are pairwise distinct and $u_{j} \in \{\pm 1, \ldots, \pm (n-1)\}$, $j =1, \ldots, h$. If $v_{1},\ldots ,v_{2s} \in {\mathcal V}$ are chosen so that there exist some integers $k$ and $\ell$ with $1 \leq k \leq 2s$ and $1 \le \ell \le h$ so that $$v_{k}- \alpha_{\ell} \neq v_{i}- \alpha_{j}$$ for all $(i,j) \neq (k, \ell)$ then for $\beta = \alpha_{\ell}-v_{k}$ we have $$\operatorname{ord}_{\beta}\prod_{i=1}^{s}f\left(X+v_{i}\right)/f\left(X+v_{s+i}\right) \equiv u_{\ell} \not \equiv 0 \pmod n$$ and thus, the above rational function is not an $n$-th power. Let $E$ be the number of $\mathbf{v}= \left(v_{1},\ldots ,v_{2s} \right)\in {\mathcal V}^{2s}$ for which $P_{\mathbf{v},f}(X) \not \in {\mathcal Q}_{2ds,n}$. Then for each choice of $1 \leq i\leq 2s$ there is some index $k\ne i$, $1 \leq k\leq 2s$, such that $v_i-v_k$ belongs to the difference set of the set $\{\alpha_1, \ldots, \alpha_h\}$ and thus can take at most $h(h-1) +1 \le d^2$ values. In particular, the components of $\mathbf{v}$ can be partitioned into at most $s$ groups such that differences of elements within each group belong to the above difference set. This immediately implies that $E\ll V^s$ and concludes the proof. ◻ We use Lemma [Lemma 6](#lem:rat prod){reference-type="ref" reference="lem:rat prod"} to control sums of multiplicative characters. To control sums of additive characters we need its appropriate analogue for linear combinations instead of products as in [\[eq:Prod f\]](#eq:Prod f){reference-type="eqref" reference="eq:Prod f"}. Namely, given a vector $\mathbf{v}= \left(v_{1},\ldots ,v_{2s}\right)\in \overline{{\mathbb F}_p}^{2s}$ and a rational function $f \in \overline{{\mathbb F}_p}(X)$, we set $$\label{eq:LinComb f} L_{\mathbf{v},f}(X)= \sum_{i=1}^{s} \left(f\left(X+v_{i}\right)- f\left(X+v_{s+i}\right)\right).$$ **Definition 7**. *We define the set ${\mathcal E}$ of *exceptional* rational functions as the set of rational functions $f(X) \in {\mathbb F}_{p^r}(X)$ such that there exists $\alpha , \beta \in \overline{{\mathbb F}_{p}}$ and $h(X) \in \overline{{\mathbb F}_{p}}(X)$ so that $f(X) = \alpha(h(X)^p-h(X)) + \beta X$.* Then we have the following additive analogue of Lemma [Lemma 6](#lem:rat prod){reference-type="ref" reference="lem:rat prod"}. **Lemma 8**. *Let $f(X) \in {\mathcal R}_{d}$ for some integers $d \ge 1$. For any set ${\mathcal V}\subseteq \overline{{\mathbb F}_p}$ of cardinality $V$, for each integer $s \geq 1$ we have $$\# \left\{\mathbf{v}= \left(v_{1},\ldots ,v_{2s}\right)\in {\mathcal V}^{2s}:~L_{\mathbf{v},f}(X) \in {\mathcal E}\right\} \ll V^s.$$* *Proof.* Clearly, all functions from ${\mathcal E}$ have a pole of order that is a multiple of $p$. It is also clear that if $f_{1},\ldots ,f_{n} \in \overline{{\mathbb F}_p}(X)$ are such that $f_{1}$ has a pole at $\alpha \in \overline{{\mathbb F}_p}$ of order $u\ge 1$ and $f_{2},\ldots, f_{n}$ have no poles at $w$ then $f_{1}+\ldots +f_{n}$ has a pole at $\alpha$ of the same order $u$. This implies that if $L_{\mathbf{v},f}(X) \in {\mathcal E}$ then for each choice of $1 \leq i\leq 2s$ there is some index $k\ne i$, $1 \leq k\leq 2s$, such that $$v_i-v_k \in \{\alpha- \gamma:~\gamma\ \text{is a pole of} \ f\}.$$ Indeed, otherwise, that is, for other choices of $\left(v_{1},\ldots ,v_{2s}\right)\in {\mathcal V}^{2s}$, if $\alpha$ is a pole $f\in \overline{{\mathbb F}_p}(X)$ of order $\operatorname{ord}_\alpha f \not \equiv 0 \pmod p$, then $f(X+v_i)$ has a pole $\beta = \alpha - v_i$ of the same order and which is not a pole of any other function involved in $L_{\mathbf{v},f}$. Hence $$\operatorname{ord}_\beta L_{\mathbf{v},f} = \operatorname{ord}_\alpha f \not \equiv 0 \pmod p,$$ and therefore $L_{\mathbf{v},f} \not \in {\mathcal E}$. We see that the number of such choices of $\left(v_{1},\ldots ,v_{2s}\right)\in {\mathcal V}^{2s}$ is at most $$2^s \binom{2s}{s} d^s V^s \ll V^s$$ and the result now follows. ◻ # Character sums over linear subspaces We need is [@MSW Lemma 3.2] which follows instantly from the Weil bound for mixed character sums with rational functions due to Castro and Moreno [@CaMo] (see also more general results of Fu and Wan [@FW Theorem 5.6]) and the orthogonality of additive characters. **Lemma 9**. *Let $\chi$ and $\psi$ be a multiplicative and additive character, respectively, and let $g_1(X) , g_2(X)\in {\mathbb F}_{q^r}(X)$ be rational functions of degrees at most $d$. Assume that at least one of the following conditions holds* - *$\chi$ is nonprincipal of order $e$ and $g_1(X) \in {\mathcal Q}_{d,e}$,* - *$\psi$ is nonprincipal and $g_2(X) \not \in {\mathcal E}$.* *Then for any affine subspace ${\mathcal L}\subseteq {\mathbb F}_{q^r}$ we have $$\sum_{\lambda \in {\mathcal L}}\chi(g_1(\lambda))\psi\left(g_2(\lambda)\right) \ll q^{r/2}.$$* The following result is our main technical tool. **Lemma 10**. *Let $\chi$ and $\psi$ be a multiplicative and additive character, respectively, and let $g_1(X) , g_2(X)\in {\mathbb F}_{q^r}(X)$ be rational functions of degrees at most $d$. Assume that at least one of the following conditions holds* - *$\chi$ is nonprincipal of order $e$ and $g_1(X) \in {\mathcal Q}_{d,e}$,* - *$\psi$ is nonprincipal and $g_2(X) \not \in {\mathcal E}$.* *Then for a linear space ${\mathcal L}\subseteq {\mathbb F}_{p^r}$ of dimension $t$ and arbitrary set ${\mathcal U}\subseteq {\mathcal L}$ and ${\mathcal V}\subseteq {\mathbb F}_{p^r}$ of cardinalities $U$ and $V$, respectively, for each fixed integer $s \ge 1$ we have $$\sum_{u\in {\mathcal U}} \left| \sum_{v \in {\mathcal V}} \chi(g_{1}(\lambda+v))\psi(g_{2}(\lambda+v))\right| \ll U^{1-1/(2s)} \left(q^{t/(2s)} V^{1/2} + q^{r/(4s)} V\right) .$$* *Proof.* Let $$S = \sum_{\lambda \in {\mathcal U}} \left| \sum_{v \in {\mathcal V}} \chi(g_{1}(u+v))\psi(g_{2}(u+v))\right|.$$ Applying the Hölder inequality, we derive $$\begin{aligned} S^{2s} & \le U^{2s-1}\sum_{\lambda \in {\mathcal U}} \left| \sum_{v \in {\mathcal V}} \chi(g_{1}(u+v))\psi(g_{2}(u+v))\right|^{2s}\cr & \le U^{2s-1}\sum_{\lambda \in {\mathcal L}} \left| \sum_{v \in {\mathcal V}} \chi(g_{1}(\lambda+v))\psi(g_{2}(\lambda+v))\right|^{2s}\cr & = U^{2s-1} \sum_{\lambda\in {\mathcal L}} \, \sum_{\mathbf{v}= \left(v_{1},\ldots ,v_{2s}\right)\in {\mathcal V}^{2s}} \chi\left(P_{\mathbf{v},f}(\lambda)\right)\psi\left(L_{\mathbf{v},f}(\lambda)\right)\cr & = U^{2s-1} \sum_{\mathbf{v}= \left(v_{1},\ldots ,v_{2s}\right)\in {\mathcal V}^{2s}}\, \sum_{\lambda \in {\mathcal L}} \chi\left(P_{\mathbf{v},f}(\lambda)\right)\psi\left(L_{\mathbf{v},f}(\lambda)\right),\end{aligned}$$ where $P_{\mathbf{v},f}(X)$ and $L_{\mathbf{v},f}(X)$ are defined by [\[eq:Prod f\]](#eq:Prod f){reference-type="eqref" reference="eq:Prod f"} and [\[eq:LinComb f\]](#eq:LinComb f){reference-type="eqref" reference="eq:LinComb f"}, respectively. We now see that if at least one of the above conditions (i) or (ii) holds that by either Lemma [Lemma 6](#lem:rat prod){reference-type="ref" reference="lem:rat prod"} or Lemma [Lemma 8](#lem:rat lincomb){reference-type="ref" reference="lem:rat lincomb"} we can apply Lemma [Lemma 9](#lem:weil-subspace){reference-type="ref" reference="lem:weil-subspace"} to the inner sum over the linear space ${\mathcal L}$ for all but $O\left(V^s\right)$ vectors $\mathbf{v}\in {\mathcal V}$, for which we estimate the inner sum trivially as $q^t$. Hence $$S^{2s} \ll U^{2s-1} \left(q^t V^{s} + q^{r/2} V^{2s}\right),$$ and the result follows. ◻ # Proof of Theorems [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"} and [Theorem 5](#thm:poly bound){reference-type="ref" reference="thm:poly bound"} {#proof-of-theorems-thmgen-bound-and-thmpoly-bound} We recall the definition [\[eq:Set SrA\]](#eq:Set SrA){reference-type="eqref" reference="eq:Set SrA"} of the set ${\mathcal S}_r\left({\mathcal A}\right)$, and some real positive parameter $\tau \in [0,1]$ and set $t = \left\lfloor\tau r\right\rfloor$, to be chosen later, we define the sets $$\begin{aligned} &{\mathcal U}= \left \{a_1\vartheta_1+ \ldots +a_t \vartheta_t:~a_1, \ldots, a_t \in {\mathcal A}\right\},\cr & {\mathcal L}= \left \{a_1\vartheta_1+ \ldots +a_t \vartheta_t:~a_1, \ldots, a_t \in {\mathbb F}_{q}\right\},\cr &{\mathcal V}= \left \{a_{t+1}\vartheta_{t+1}+ \ldots +a_r \vartheta_r:~a_{t+1}, \ldots, a_r \in {\mathcal A}\right\}, \end{aligned}$$ of cardinalities $$U = q^{\rho t} \ll q^{\rho \tau r}, \qquad L = q^{t} \ll q^{\tau r},\qquad V = q^{\rho (r- t)} \ll q^{\rho(1- \tau) r},$$ respectively. We can now write $$\begin{aligned} S_{r}({\mathcal A};\chi,\psi;f_1,f_2)& = \sum_{\omega \in {\mathcal S}_r\left({\mathcal A}\right)}\chi\left(f_1(\omega)\right)\psi\left(f_2(\omega)\right)\cr &= \sum_{u \in {\mathcal U}}\sum_{v \in {\mathcal V}}\chi\left(f_1(u+v)\right)\psi\left(f_2(u+v)\right).\end{aligned}$$ Thus $$\begin{aligned} \left|S_{r}({\mathcal A};\chi,\psi;f_1,f_2)\right| \leq \sum_{u \in {\mathcal U}}\left|\sum_{v \in {\mathcal V}}\chi\left(f_1(u+v)\right)\psi\left(f_2(u+v)\right)\right|.\end{aligned}$$ Under the conditions of Theorem [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"}, by Lemma [Lemma 10](#lem:double sum){reference-type="ref" reference="lem:double sum"} and the above cardinality estimates we have $$\begin{aligned} S_{r}({\mathcal A};\chi&,\psi;f_1,f_2) \ll U^{1-1/(2s)} \left(q^{t/(2s)} V^{1/2} + q^{r/(4s)} V\right)\cr & \ll q^{\rho \tau r(1 - 1/(2s))+\tau r/(2s)+\rho(1-\tau)r/2} + q^{\rho \tau r(1 - 1/(2s))+r/(4s)+\rho(1-\tau)r}\cr & = q^{r(\tau\rho(1 - 1/(2s)) +\tau(1/(2s)- \rho/2) + \rho/2)}+q^{r(\tau\rho(1 - 1/(2s)) +\rho(1-\tau) + 1/(4s))}.\end{aligned}$$ Hence we have $$\label{eq:Rough} S_{r}({\mathcal A};\chi,\psi;f_1,f_2) \ll q^{r\Delta_{s,\rho}(\tau)}$$ where $$\Delta_{s,\rho}(\tau)= \left(1- \frac{1}{2s}\right)\rho \tau + \max\left\{ \frac{\tau}{2s} + \frac{\rho(1- \tau)}{2}, \frac{1}{4s} + \rho(1- \tau) \right \}.$$ To minimise $\Delta_s(\tau)$ we choose $$\tau_0 = \frac{2s \rho+1}{2(s\rho +1)}$$ to equalise the terms inside of the above maximum and compute $$\begin{aligned} \Delta_{s,\rho}(\tau_0) & = \left(1- \frac{1}{2s}\right)\rho \tau_0 + \frac{1}{4s} + \rho(1- \tau_0)\cr & = \rho + \frac{1}{4s} - \frac{1}{2s}\rho \tau_0 = \rho - \frac{1}{4s} (2 \rho \tau_0 -1) = \rho - \kappa_s(\rho) , \end{aligned}$$ where $\kappa_s(\rho)$ is given by [\[eq:kappa-s\]](#eq:kappa-s){reference-type="eqref" reference="eq:kappa-s"}, which together with [\[eq:Rough\]](#eq:Rough){reference-type="eqref" reference="eq:Rough"} concludes the proof of Theorem [Theorem 2](#thm:gen bound){reference-type="ref" reference="thm:gen bound"}. To prove Theorem [Theorem 5](#thm:poly bound){reference-type="ref" reference="thm:poly bound"}, we note that for $s=1$, the above argument still applies for $f_2 \in {\mathcal P}_d$, and the result follows. # Acknowledgements {#acknowledgements .unnumbered} The authors are grateful to the referee for the careful reading of the manuscript and many useful comments. During the preparation of this work, I.S. was supported in part by the Australian Research Council Grants DP230100530 and DP230100534. www W. Banks, A. Conflitti and I. E. Shparlinski, 'Character sums over integers with restricted $g$-ary digits', *Illinois J. Math.*, **46** (2002), 819--836. F. N. Castro and C. J. Moreno, 'Mixed exponential sums over finite fields', *Proc. Amer. Math. Soc.*, **128** (2000), 2529--2537. L. Fu and D. Wan, 'A class of incomplete character sums', *Quart. J. Math.*, **65** (2014), 1195--1211. L. Mérai, I. E. Shparlinski and A. Winterhof, 'Character sums over sparse elements of finite fields', *Preprint*, 2022, (available from <https://arxiv.org/abs/2211.08452>). A. Ostafe and I. E. Shparlinski, 'Multiplicative character sums and products of sparse integers in residue classes', *Period. Math. Hungar.*, **64** (2012), 247--255.

arxiv_math

--- abstract: | This paper is concerned with the study of the well-posedeness for the initial boundary value problem to the time-fractional wave equation with acoustic boundary conditions. The problem is considered in a bounded and connected domain $\Omega \subset {\mathbb{R}^{n}}$, $n \geq 2$, which includes simply connected regions. The boundary of $\Omega$ is made up of two disjoint pieces $\Gamma_{0}$ and $\Gamma_{1}.$ Homogeneous Dirichlet conditions are enforced on $\Gamma_0$, while acoustic boundary conditions are considered on $\Gamma_1$. To establish our main result, we employ the Faedo-Galerkin method and successfully solve a general system of time-fractional ordinary differential equations which extends the scope of the classical Picard-Lindelöf theorem. address: - Department of Mathematics, Federal University of Santa Catarina, Florianópolis - SC, Brazil. - Department of Mathematics, State University of Maringá, Maringá - PR, Brazil. - Department of Mathematics, State University of Maringá, Maringá - PR, Brazil. author: - Paulo M. Carvalho-Neto$^1$ - Cı́cero L. Frota$^2$ - Pedro G. P. Torelli$^{3,\ast}$ title: On the Initial Boundary Value Problem to the Time-Fractional Wave Equation with Acoustic Boundary Conditions --- [^1] # Introduction {#secao1} Over the past few decades, there has been a growing interest in using fractional calculus in combination with differential equations as a powerful tool for analyzing complex systems. These systems include, among others, diffusion in nerve cells, anomalous diffusion processes in porous media, turbulent fluids, plasma, finance and others; see [@CaCa1; @CaCaLy1; @FeBrSlBaWe1; @GoRa1; @MeKl1; @MuPa1] as a few examples. In the light of that, in this paper we are particulary interested to address the classical initial boundary value problem (IBVP) for the wave equation with acoustic boundary condition, when we replace the standard time derivative with its natural non-integer generalization, the Caputo fractional derivative. To be more precise, here we assume that $\Omega\subset\mathbb{R}^n$ (with $n\geq2$) is an open, bounded and connected set, with smooth boundary $\Gamma$ made up of two disjoint parts $\Gamma_0$ and $\Gamma_1$ ($\Gamma=\Gamma_0 \cup \Gamma_1$ and $\Gamma_0 \cap \Gamma_1= \emptyset$), both connected with positive measure and $\nu$ denotes the unit normal vector on $\Gamma_{1}$ pointing towards the exterior of $\Omega$. The main subject of this work is to prove the well posedeness (existence and uniqueness of solution, as well as its countinuous dependence on initial data) to the following IBVP: $$\begin{aligned} \label{1prob} &{}^C \! D^{\alpha}_tu_t(x,t) - \Delta u(x,t) = 0, && (x,t) \in \Omega \times (0,T),\\ \label{2prob} &u(x,t)=0, && (x,t) \in \Gamma_0 \times (0,T),\\ \label{3prob} &f(x)\delta_{tt}(x,t) + g(x)\delta_t(x,t) + h(x)\delta(x,t) = -u_t(x,t), && (x,t) \in \Gamma_1 \times (0,T), \hspace{1cm}\\ \label{4prob} &\displaystyle \delta_t(x,t)=\frac{\partial u}{\partial \nu}(x,t) , && (x,t) \in \Gamma_1 \times (0,T),\\ \label{5prob} &u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x), && x \in \Omega,\\ \label{6prob} &\displaystyle \delta(x,0)=\delta_0(x), \quad \delta_t(x,0)= \frac{\partial u_0}{\partial \nu}(x), && x \in \Gamma_1,\end{aligned}$$ where ${}^C \! D^{\alpha}_t$ denotes the classical Caputo fractional derivative of order $\alpha\in(0,1]$, $\Delta$ is the Laplacian operator, $f,g,h:\overline{\Gamma_1}\rightarrow\mathbb{R}$ are given functions and finally, $u_0,u_1:\Omega\rightarrow\mathbb{R}$ and $\delta_0:\Gamma_1\rightarrow\mathbb{R}$ are the initial conditions of the system. In the limit case, when $\alpha=1$, and the acoustic boundary conditions [\[3prob\]](#3prob){reference-type="eqref" reference="3prob"} and [\[4prob\]](#4prob){reference-type="eqref" reference="4prob"} are imposed on the whole boundary $\Gamma$, we get the problem associated with a wave motion in a fluid $$\begin{aligned} & u_{tt}(x,t) - \Delta u(x,t) = 0, && (x,t) \in \Omega \times (0,T),\\ &f(x)\delta_{tt}(x,t) + g(x)\delta_t(x,t) + h(x)\delta(x,t) = -u_t(x,t), && (x,t) \in \Gamma \times (0,T), \hspace{1cm}\\ &\displaystyle \delta_t(x,t)=\frac{\partial u}{\partial \nu}(x,t) , && (x,t) \in \Gamma \times (0,T),\end{aligned}$$ introduced by Beale and Rosencrans ([@Be1] and [@BeRo1]), which gave rise to a big range of more general problems, see for instance, [@AlCaCl1; @BrSiClFr1; @CaDoCaFrVi1; @Fr1; @FrCoLa1; @FrGo1; @FrMeVi1; @GrSaHo1; @KoTa1; @LiClFrMe1] and references therein. Still in the context of integer order time derivatives (the classic wave equation), the first paper dealing with a non-linear problem was [@FrGo1], where Frota and Goldstein considered the Carrier non-linear wave equation $$u_{tt}(x,t)-M\left(\int_\Omega u^2(x,t)\,dx\right)\Delta u(x,t)+C|u_t(x,t)|^\gamma u_t(x,t)=0,\quad (x,t) \in \Omega \times (0,T],$$ together with [\[2prob\]](#2prob){reference-type="eqref" reference="2prob"} - [\[6prob\]](#6prob){reference-type="eqref" reference="6prob"}; where $C$ was a nonnegative constant, $M\in C^1([0,\infty);\mathbb{R})$ and $\gamma>0$. The physical justification for the model can be seen in [@Be1], [@BeRo1] and [@MoIn1]. Here, just for a brief contextualization, we give some comments. In our context, $\Omega$ represents a region of the space filled with an ideal fluid at rest which is set into motion by sound waves propagating within the domain. Therefore, if $u$ is the potential velocity of the fluid, it satisfies the time-fractional wave equation [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}. The boundary $\Gamma$ is made up two parts $\Gamma_0$ and $\Gamma_1$, with $\Gamma_0$ absorbing (see [\[2prob\]](#2prob){reference-type="eqref" reference="2prob"}) and $\Gamma_{1}$ locally reactive, such that each point $x \in \Gamma_1$ responds independently to the pressure caused by the sound waves. This means that $\delta$, the vertical displacement in the normal direction to the boundary $\Gamma_1$ should satisfies the equation [\[3prob\]](#3prob){reference-type="eqref" reference="3prob"}. In fact, each point on the boundary $\Gamma_1$ acts like a damped harmonic oscillator that \"springs\" in response to the sound pressure. Moreover, we also admit that there exists the compatibility between the normal speed of the boundary and the normal speed of the fluid, which is expressed by equation [\[4prob\]](#4prob){reference-type="eqref" reference="4prob"}. Initial value problems for the time fractional wave equation have received extensive coverage in the scientific literature. Often, the Laplace transform has been widely employed as the primary tool for obtaining solutions. For instance, classic works in the field, such as [@KiSrTr1] and [@Po1], have extensively discussed Cauchy problems in the context of fractional equations. Additionally, notable contributions, as [@Ma1] and [@Ma2], have provided explanations regarding the inherent diffusion-wave phenomena associated with these solutions. The technique of separating variables combined with the Laplace transform has been utilized in [@Da-GeJa1] in the study of IBVPs, encompassing both homogeneous and non-homogeneous boundary conditions. More recently, Faedo-Galerkin's method was utilized in [@HuYa1] to demonstrate the well-posedness of an IBVP for the time fractional wave equation, albeit in a slightly different sense than the Caputo formulation. To the best of author's knowledge, this is the first paper considering the time-fractional wave equation coupled with acoustic boundary conditions. In order to facilitate the implementation of numerical methods, as well as to create basis for treating more general nonlinear problems, we apply Faedo-Galerkin's constructive method. It should be mentioned that even in this context of linear equations, when we project our problem into finite-dimensional subspaces by getting the approximated problems, we arrive at a system of time-fractional ordinary differential equations that, as far as we know, has never been treated before. We drew inspiration from [@BeRo1] and [@FrGo1] while formulating the class of problem [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}-[\[6prob\]](#6prob){reference-type="eqref" reference="6prob"}, where we shall work in a much more general context, namely, that of time-fractional wave equations. Since our approach introduces a more complex problem by incorporating the Caputo fractional time-derivative, some new notions and results should be established. In fact, there are new key challenges when considering problem [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}-[\[6prob\]](#6prob){reference-type="eqref" reference="6prob"}, which are successfully addressed in this paper: - to consider the specificities and restrictions imposed by the Caputo fractional derivative; - to establish a more general version of Picard-Lindelöf theorem. The remaining paper is organized as follows. In Section [2](#preliminars){reference-type="ref" reference="preliminars"} we provide the prerequisites and auxiliary results that are crucial to the development of subsequent sections. Section [3](#picard){reference-type="ref" reference="picard"} is devoted to analyzing a time-fractional ODE system crucial for establishing the initial aspects of our main result. This section also includes observations regarding the system's solution, which are detailed in Subsection [3.1](#digression){reference-type="ref" reference="digression"}. In Section [4](#existenceresults){reference-type="ref" reference="existenceresults"}, we explore the well-posedness theory of the problem [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"} - [\[6prob\]](#6prob){reference-type="eqref" reference="6prob"}, and in Section [5](#closingremarks){reference-type="ref" reference="closingremarks"}, we give some concluding remarks. # Notations, Prerequisites and Auxiliary Results {#preliminars} In this section we give the notations for the functional spaces and also introduce the theory of fractional calculus concerning the Caputo fractional derivative. First of all let us set the triple $(\Omega, \Gamma_{0}, \Gamma_{1})$. Throughout the paper $\Omega \subset {\mathbb{R}}^{n} \, (n \geq 2)$ is an open bounded and connected set with smooth boundary $\Gamma$ made up of two disjoint pieces $\Gamma_{0},\, \Gamma_{1}$ both connected with positive measure. Actually $\Gamma_{0}$ and $\Gamma_{1}$ are connected subsets of $\Gamma$ both with positive measure such that $\Gamma = \Gamma_{0} \cup \Gamma_{1}$ and $\Gamma_{0} \cap \Gamma_{1} = \emptyset.$ We observe that the domains $\Omega$ includes simply connected regions of ${\mathbb{R}}^{n}$. For the classical functional spaces such as Sobolev spaces and $L^{p}$ spaces we adopt the standard notation as described in [@Lions; @LiMa1]. We denote the inner products and norms in $L^2(\Omega)$ and $L^2(\Gamma_1)$ respectively by $$(u,v) = \int_\Omega u(x)\, v(x)\, dx, \quad \vert u \vert = \left( \int_\Omega ( u(x) )^2 dx \right)^{\frac{1}{2}}$$ and $$(u,v)_{\Gamma_1} = \int_{\Gamma_1} u(x)\, v(x)\, dx, \quad \vert u \vert_{\Gamma_1} = \left( \int_{\Gamma_1} ( u(x) )^2 dx \right)^{\frac{1}{2}}.$$ If $u, v \in H^{1}(\Omega)$, the real Sobolev space of first order, we write $$(\nabla u , \nabla v) = \sum_{i=1}^{n} \left( \frac{\partial u}{\partial x_{i}} , \frac{\partial v}{\partial x_{i}}\right) = \sum_{i=1}^{n} \int\limits_{\Omega} \frac{\partial u}{\partial x_{i}}(x) \, \frac{\partial v}{\partial x_{i}} (x) \, dx$$ and $$\vert \nabla u \vert = \left[ (\nabla u , \nabla u) \right]^{\frac{1}{2}} = \left[ \sum_{i=1}^{n} \int\limits_{\Omega} \left(\frac{\partial u}{\partial x_{i}}(x) \right)^{2} \, dx \right]^{\frac{1}{2}} .$$ Let $\mathcal{H}_\Delta(\Omega)=\{u \in H^1(\Omega); \Delta u \in L^2(\Omega)\}$ be the Hilbert endowed with the inner product $(u,v)_{\mathcal{H}_\Delta(\Omega)} = (u,v)_{H^1(\Omega)} + (\Delta u , \Delta v)$. By $\gamma_0: H^1(\Omega) \longrightarrow H^{\frac{1}{2}}(\Gamma)$ and $\gamma_1:\mathcal{H}_\Delta(\Omega) \longrightarrow H^{-\frac{1}{2}}(\Gamma)$ we denote the trace map of order zero and the Neumann trace map on $\mathcal{H}_\Delta(\Omega)$ respectively satisfying $$\gamma_0(u) = u \vert_{\Gamma} \textrm{ and } \gamma_1(u) = \frac{\partial u}{\partial \nu} \bigg \vert_{\Gamma}, \quad \mbox{for all } u \in \mathcal{D}(\overline{\Omega}).$$ It is well known that $\gamma_{0}$ and $\gamma_{1}$ are bounded linear operators and for all $u \in \mathcal{H}_\Delta(\Omega)$ and $v \in H^1(\Omega)$ the following generalized Green's formula holds $$(\Delta u , v) + (\nabla u, \nabla v) = \left\langle \gamma_1(u), \gamma_0(v) \right\rangle_{H^{-\frac{1}{2}}(\Gamma) \times H^{\frac{1}{2}}(\Gamma)}\,.$$ Another closed subspace of $H^{1}(\Omega)$ that we address here is the closure of the set $\{ u \in C^{1}(\overline{\Omega}); u = 0 \textrm{ in } \Gamma_{0} \}$ in $H^{1}(\Omega)$, which we denote by $H^1_{\Gamma_0}(\Omega)$. Since $\Omega$ is a regular domain and $\Gamma_{0}$ has positive measure we have that $$H^1_{\Gamma_0}(\Omega) = \{ u \in H^1(\Omega); \gamma_0(u) =0 \textrm{ a.e. in } \Gamma_0\}$$ is a reflexive and separable Hilbert space. Additionally, the Poincaré inequality holds in $H^1_{\Gamma_0}(\Omega)$ and, in view of this inequality, we have that $$((u,v)) = (\nabla u, \nabla v) \quad \textrm{and} \quad \Vert u \Vert = \vert \nabla u \vert,$$ denote an inner product and a norm in $H^1_{\Gamma_0}(\Omega)$ that are equivalent to the usual ones induced by $H^{1}(\Omega)$. We employ the notations on the Bochner-Lebesgue and Bochner-Sobolev spaces of vector-valued functions, $L^p(0,T;X)$ and $W^{m,p}(0,T;X)$ where $X$ is a Banach space, as in [@ArBaHi1 Chap. 1], [@Lions] and [@Ze1 Chap. 23]. For $\alpha > 0$ and $f:[0,T]\rightarrow X$, the Riemann-Liouville (RL for short) fractional integral of order $\alpha$ is $$J_{t}^\alpha f(t):=\dfrac{1}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}{(t-s)^{\alpha-1}f(s)}\,ds,$$ for every $t\in[0,T]$ such that the above integral exists. Above $\Gamma$ is used to denote the classical Euler's gamma function. Also the RL fractional derivative of order $\alpha$ and the Caputo fractional derivative of order $\alpha$ are respectively defined by $$D_{t}^\alpha f(t) := \dfrac{d^{ \lceil \alpha \rceil}}{dt^{ \lceil \alpha \rceil}}\left[J_t^{\lceil \alpha \rceil -\alpha}f(t)\right],$$ and $${}^C \! D^{\alpha}_tf(t):=D_{t}^\alpha\left[f(t)-\sum_{k=0}^{\lceil \alpha \rceil -1}\dfrac{f^{(k)}(0)}{k!}t^{k}\right],$$ for every $t\in[0,T]$ such that the right side exists. Above we use $\lceil \cdot \rceil$ to represent the ceiling function, i.e., if $m\in\mathbb{N}$ is such that $m-1<\alpha\leq m$, then $\lceil \alpha \rceil=m$. In [@CarFe1] the authors prove that $\{J_t^\alpha:\alpha\geq0\}\subset\mathcal{L}\big(L^p(0,T;X)\big)$ is a $C_0$-semigroup on $L^p(0,T;X), 1 \leq p < \infty$, and $\{J_t^\alpha:\alpha\geq0\}\subset\mathcal{L}\big(C([0,T];X)\big)$ forms a semigroup on $C([0,T];X)$, where $J_t^0f(t)=f(t)$ for almost every $t\in[0,T]$. Concerning the existence of the Caputo fractional derivative ${}^C \! D^{\alpha}_tf(t)$ for almost every $t \in [0,T]$, it is enough to consider functions $f \in C^{\lceil\alpha\rceil-1}([0,T];X)$ such that $J_t^{\lceil\alpha\rceil-\alpha}f(t) \, \in W^{\lceil\alpha\rceil,1}(0,T;X)$, see Bazhlekova [@Baz1 Section 1.2] and Carvalho-Neto [@Car1 Section 2.2]. Below we present fundamental results concerning the previously mentioned fractional operators, which are essential for Section [3](#picard){reference-type="ref" reference="picard"} and the estimations in Section [4](#existenceresults){reference-type="ref" reference="existenceresults"}. We emphasize the simplification $'=\frac{d}{dt}$ throughout the remainder of this paper to simplify notation and, more importantly, to avoid excess subscripts on Theorem [Theorem 8](#maintheorem){reference-type="ref" reference="maintheorem"}. **Proposition 1**. *Let $0 < \alpha <1$ and $X$ a Banach space.* - *Assume that $f \in L^1(0,T)$ is a nonnegative function. Then $$\label{prop07} J_{t}^{1} f(t) \leq \Big[T^{(1-\alpha)} \Gamma(\alpha)\Big]J_{t}^{\alpha} f(t), \textrm{ for a.e. } t \in [0,T].\vspace*{0.2cm}$$* - *If $f \in L^1(0,T;X)$ $$\label{id-1} D^\alpha_t \left[J_{t}^{{\alpha}}f(t)\right]=f(t), \textrm{ for a.e. } t \in [0,T].$$ If additionally $J_t^{1-{\alpha}}f \in W^{1,1}(0,T;X)$, then $$\label{id0}J^\alpha_t \big[D^\alpha_t u(t)\big] = u(t) - \frac{t^{\alpha-1}}{\Gamma(\alpha)} \Big( J^{1-\alpha}_s u(s) |_{s=0} \Big)$$* - *For $f \in C([0,T];X)$, $$\label{id1} {}^C \! D^{\alpha}_t\left[J_{t}^{{\alpha}}f(t)\right]=f(t), \textrm{ for a.e. } t \in [0,T].$$ If additionally $J_t^{1-{\alpha}}f \in W^{1,1}(0,T;X)$, then $$\label{id2} J_{t}^{{\alpha}}\left[ {}^C \! D^{\alpha}_tf(t)\right]=f(t)-f(0), \textrm{ for a.e. } t \in [0,T],$$ and $$\label{id3} J_{t}^{{1}}\left[ {}^C \! D^{\alpha}_tf(t)\right]=J_t^{1-\alpha}f(t)-f(0)\left[\dfrac{t^{1-\alpha}}{\Gamma(2-\alpha)}\right], \textrm{ for a.e. } t \in [0,T].\vspace*{0.2cm}$$* - *If $f \in W^{1,1}(0,T;X)$, $$\label{id4} {}^C \! D^{\alpha}_tf (t) = J^{1-\alpha}_t f'(t), \textrm{ for a.e. } t \in [0,T].$$ If additionally $f(0)=0$, then we can reinterpret the equation [\[id4\]](#id4){reference-type="eqref" reference="id4"} in the form $$\label{id5} \dfrac{d}{dt}\left[J^{1-\alpha}_t f(t)\right] = J^{1-\alpha}_t f'(t), \textrm{ for a.e. } t \in [0,T].\vspace*{0.2cm}$$* - *For $f \in C^{1}([0,T];X)$ such that $J_t^{1-\alpha}f(t) \, \in W^{2,1}(0,T;X)$, we have $$\label{id6} {}^C \! D_t^{\alpha+1} f (t) = {}^C \! D^{\alpha}_tf'(t), \textrm{ for a.e. } t \in [0,T].\vspace*{0.2cm}$$* - *If $f \in W^{1,2}(0,T;X)$, then $$\label{id7} \left\|{}^C \! D^{\alpha}_tf (t)\right\|_X^2 \leq \left[\frac{T^{1-\alpha}}{\Gamma(2-\alpha)}\right] J^{1-\alpha}_t \left\|f'(t)\right\|_X^2, \textrm{ for a.e. } t \in [0,T].$$* *Proof.* The proof of [\[prop07\]](#prop07){reference-type="eqref" reference="prop07"} follows from the definition. For the proof of ([\[id-1\]](#id-1){reference-type="ref" reference="id-1"})-([\[id5\]](#id5){reference-type="ref" reference="id5"}) we refer the reader to [@Car1 Proposition 2.35] and [@CarFe0 Remark 2.10]. Applying the Leibniz integral rule in conjunction with [\[id5\]](#id5){reference-type="eqref" reference="id5"}, we have $${}^C \! D_t^{\alpha+1} f (t)=\dfrac{d^2}{dt^2}\Big\{J_t^{1-\alpha}\big[f(t)-f(0)-tf'(0)\big]\Big\} =\dfrac{d}{dt}\Big\{J_t^{1-\alpha}\big[f^\prime(t)-f^\prime(0)\big]\Big\} = {}^C \! D^{\alpha}_tf^\prime(t),$$ in $X$, for almost every $t\in[0,T]$, therefore [\[id6\]](#id6){reference-type="eqref" reference="id6"} holds. To prove the estimate ([\[id7\]](#id7){reference-type="ref" reference="id7"}), we observe that [\[id4\]](#id4){reference-type="eqref" reference="id4"} guarantees $$\left\|{}^C \! D^{\alpha}_tf(t)\right\|_X^2 \leq \left[ J^{1-\alpha}_t \left\|f'(t) \right\|_X \right]^2 = \left[ \int_0^t \left(\frac{(t-s)^{- \frac{\alpha}{2}}}{\Gamma(1-\alpha)^{\frac{1}{2}}}\right)\left(\frac{(t-s)^{- \frac{\alpha}{2}}}{\Gamma(1-\alpha)^{\frac{1}{2}}}\right) \left\|f'(t)\right\|_X ds\right]^2.$$ Therefore, Holder's inequality gives $$\left\|{}^C \! D^{\alpha}_tf(t) \right\|_X^2 \leq \left[\frac{t^{1-\alpha}}{\Gamma(2-\alpha)}\right] J^{1-\alpha}_t \left\|f'(t)\right\|_X^2 \leq \left[\frac{T^{1-\alpha}}{\Gamma(2-\alpha)}\right] J^{1-\alpha}_t \left\|f'(t)\right\|_X^2,$$ for almost every $t\in[0,T]$, which completes the proof. ◻ At this point, we emphasize that the remaining propositions of this section are original (and important) contributions to the theory, as far as the authors are aware. These propositions are essential for applying the following theorem, originally formulated in [@CarFe0 Theorem 4.11], which is presented below and used throughout this work. **Theorem 2**. *Assume that $f\in C([0,T])$, $J_t^{1-\alpha}f\in W^{1,1}(0,T)$ and $J_t^{1-\alpha}f^2\in W^{1,1}(0,T)$. Then, $$cD_{t}^\alpha\big[f(t)\big]^2\leq2\Big[cD_{t}^\alpha f(t)\Big]f(t),\quad \textrm{for almost every }t\in [0,T].$$* The following proposition helps us verify that, in certain circumstances, a function satisfies the hypotheses of Theorem [Theorem 2](#finalcaputo){reference-type="ref" reference="finalcaputo"}. **Proposition 3**. *If $u\in C([0,T])$ is such that ${}^C \! D^{\alpha}_tu \in C([0,T])$, then $J^{1-\alpha}_t u^2 \in W^{1,1}(0,T)$.* *Proof.* From the continuity of the RL fractional integral (cf. [@HaLi Theorem 14]) we have that $J^{1-\alpha}_t u\in C([0,T])$. Then, that the identity $$D_t^\alpha u(t)={}^C \! D^{\alpha}_tu+\dfrac{u(0)t^{-\alpha}}{\Gamma(1-\alpha)},$$ which holds for a.e. $t\in[0,T]$, allows us to deduce that $J_t^{1-\alpha}u\in W^{1,1}(0,T)$. Therefore, if we define $h(t)={}^C \! D^{\alpha}_tu(t)$, we have that [\[id2\]](#id2){reference-type="eqref" reference="id2"} ensures the identity $$u(t)-u(0) = J^\alpha_t h(t),$$ for a.e. $t\in[0,T]$. Consequently, $$\begin{gathered} J^{1-\alpha}_t u^2(t) = J^{1-\alpha}_t \big[ u(0)+J^\alpha_t h(t) \big]^2 \\= \big[u(0)\big]^2 \frac{t^{1-\alpha}}{\Gamma(2-\alpha)} +2u(0)\big[J^1_t h(t)\big] + J^{1-\alpha}_t \big[ J^\alpha_t h(t) \big]^2.\end{gathered}$$ It is evident that the first two terms on the right side of the above equality belong to $W^{1,1}(0,T)$. The first follows from direct computation, while the second relies on the continuity of the RL fractional integral and the fact that $h\in C([0,T])$. To complete the proof, we assert that $J^{1-\alpha}_t \left[ J^\alpha_t h \right]^2\in W^{1,1}(0,T)$. We only need to verify that $D^\alpha_t [J^\alpha_t h]^2\in L^1(0,T)$, since it follows from the continuity of the RL fractional integral of order $1-\alpha$ from $L^1(0,T)$ into $L^1(0,T)$ (cf. [@HaLi Theorem 4]), that $J^{1-\alpha}_t \left[ J^\alpha_t h \right]^2\in L^1(0,T)$. Since $h \in C([0,T])$, we have that $J^\alpha_th(t)$ is Hölder continuous with exponent $\alpha$ on $[0,T]$, or simply, $J^\alpha_th \in C^{0,\alpha}([0,T])$ (cf. [@HaLi Theorem 14]). Thus, we can apply [@AlAhKi Lemma 1] and [\[id-1\]](#id-1){reference-type="eqref" reference="id-1"} to obtain $$\begin{aligned} D^\alpha_t [J^\alpha_t h(t)]^2 = 2 \big[J^\alpha_t h (t)\big]h (t) - \frac{\alpha}{\Gamma(1-\alpha)} \int_0^t \frac{[J^\alpha_t h (t)-J^\alpha_s h (s)]^2}{(t-s)^{\alpha+1}}ds - \frac{[J^\alpha_t h (t)]^2}{\Gamma(1-\alpha)t^\alpha},\end{aligned}$$ for a.e. $t\in[0,T]$. We finish this proof by noting that the above equality together with the fact that $J^\alpha_th \in C^{0,\alpha}([0,T])$ is enough for us to deduce that $D^\alpha_t [J^\alpha_t h]^2\in L^1(0,T)$. ◻ We conclude this section with a result on regularity, crucial for demonstrating $u^{\prime}(0) = u_{1}$ in $H^1_{\Gamma_0}(\Omega)$ during Step 3: Passage to the limit in Section [4](#existenceresults){reference-type="ref" reference="existenceresults"}. **Proposition 4**. *If $u \in L^\infty(0,T;X)$ and $D^\alpha_t u \in L^\infty(0,T;X)$ then $u \in C([0,T];X)$ and $u(0)=0$.* *Proof.* For any $\varepsilon \in [0,\min\{\alpha,1-\alpha\})$, Theorem 3.5 in [@CarFe1] ensures that $J^{\alpha-\varepsilon}_t\big[D^\alpha_t u(t)\big]$ and $J^{1-\alpha-\epsilon}_t u(t)$ are in $C([0,T];X)$. Therefore, for $\delta \in (0,\min\{\alpha,1-\alpha\})$ we have $$\label{jduzero} J^\alpha_t \big[D^\alpha_t u(t)\big] \big|_{t=0} = J^\delta_t \big\{J^{\alpha-\delta}_t \big[D^\alpha_t u(t)\big]\big\} \big|_{t=0} = 0,$$ and $$\label{jduzero1} J_t^{1-\alpha}u(t) \big|_{t=0} = J^\delta_t \big[J^{1-\alpha-\delta}_t u(t)\big]\big|_{t=0} = 0.$$ Since follows from the hypotheses that $J^{1-\alpha}_t u \in W^{1,1}(0,T;X)$, identities [\[id0\]](#id0){reference-type="eqref" reference="id0"} and [\[jduzero1\]](#jduzero1){reference-type="eqref" reference="jduzero1"} ensure that $$\label{ujdu} u(t) = J^\alpha_t \big[D^\alpha_t u(t)\big] + \frac{t^{\alpha-1}}{\Gamma(\alpha)} \left( J^{1-\alpha}_t u(t) |_{t=0} \right) = J^\alpha_t \big[D^\alpha_t u (t)\big],$$ for a.e. $t\in[0,T]$. The proof is now complete, as we observe that $J^\alpha_t \big[D^\alpha_t u\big] \in C([0,T];X)$ and that [\[jduzero\]](#jduzero){reference-type="eqref" reference="jduzero"} together with [\[ujdu\]](#ujdu){reference-type="eqref" reference="ujdu"} guarantees $u(0)=0$. ◻ # A Generalization of Picard-Lindelöf Theorem {#picard} For $\alpha \in (0,1], f:\Omega\times[0,T] \subset \mathbb{R}^{n+1}\rightarrow\mathbb{R}^n$ and $\xi\in \mathbb{R}^{n}$ given, the classical Cauchy problem for the fractional ordinary differential equation in $\mathbb{R}^n$ is the initial value problem $$\label{fracclassical} \left\{\begin{array}{l}{}^C \! D^{\alpha}_t\varphi(t)=f\big(\varphi(t),t\big),\quad\textrm{for }t\in[0,T],\\\varphi(0)=\xi.\end{array}\right.$$ The well-posedeness for [\[fracclassical\]](#fracclassical){reference-type="eqref" reference="fracclassical"} is well-known and has been extensively studied in the literature, for instance see [@KiSrTr1; @SaKiMa1] as few examples. In this section, we investigate a time-fractional ODE system that generalizes the Cauchy problem [\[fracclassical\]](#fracclassical){reference-type="eqref" reference="fracclassical"}. Our main goal here is to establish the existence and uniqueness of a solution to this generalized time-fractional ODE system. This result serves as an essential tool for applying the Faedo-Galerkin method in Section [4](#existenceresults){reference-type="ref" reference="existenceresults"}, and notably, to the best of the authors' knowledge, there is currently no formal proof available. With this in mind, we consider $\{\alpha_j\}_{j=1}^n\subset(0,1], f:\Omega\times[0,T] \subset \mathbb{R}^{n+1}\rightarrow\mathbb{R}^n$ and $\xi=(\xi_1,\cdots,\xi_n) \in {\mathbb{R}}^{n}$ given and we look for $\varphi = (\varphi_1 , \cdots , \varphi_n): [0,T] \to \mathbb{R}^n$ (the unknown function) satisfying the time-fractional ODE system given by the following set of equations: $$\label{caratheq} \left\{ \begin{array}{cccl} {}^C \! D_t^{\alpha_1} \varphi_1(t) &=& f_1\left(\varphi_1(t), \cdots , \varphi_n(t),t\right), & \textrm{for }t\in[0,T],\\ \vdots& & \vdots & \\ {}^C \! D_t^{\alpha_n} \varphi_n(t) &=& f_n\left(\varphi_1(t), \cdots , \varphi_n(t),t\right), & \textrm{for }t\in[0,T], \end{array} \right.$$ subjected to the initial condition $$\label{carathin} \varphi(0)= \xi.$$ Let us begin by introducing the notion of a solution to the Cauchy problem [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"}. **Definition 5**. *A function $\varphi = (\varphi_1 , \cdots , \varphi_n): [0,T] \to \mathbb{R}^n$ is said to be a solution of the Cauchy problem [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} on $[0,T]$ if it satisfies the following conditions:* - *$\varphi$ and $t \mapsto ({}^C \! D_t^{\alpha_1}\varphi(t),\ldots,{}^C \! D_t^{\alpha_n}\varphi(t))$ belong to $C([0,T];\mathbb{R}^n)$;* - *$\{\varphi(t): t\in[0,T]\}\subset \Omega$;* - *$\varphi$ satisfies the equations [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"} for all $t\in [0,T]$ and the initial condition [\[carathin\]](#carathin){reference-type="eqref" reference="carathin"}.* Now we present an auxiliar result that connects the solution of [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} with the solution of an integral equation. **Proposition 6**. *Let $f: \Omega\times[0,T] \subset \mathbb{R}^{n+1} \to \mathbb{R}^n$ be a continuous function. Then $\varphi = (\varphi_1 , \cdots , \varphi_n): [0,T] \to \mathbb{R}^n$ is a solution of [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} in $[0,T]$ if, and only if, $\varphi \in C([0,T];\mathbb{R}^n)$ and for all $j\in\{1,\cdots,n\}$ the function $\varphi_{j}$ satisfies the integral equation $$\label{eqint} \varphi_j(t)=\xi_j+\dfrac{1}{\Gamma(\alpha_j)}\int_{0}^t{(t-s)^{\alpha_j-1}f_j\big(\varphi(s),s\big)}ds,\quad\forall t\in [0,T].$$* *Proof.* Assuming that $\varphi$ is a solution of [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} in the interval $[0,T]$, we can observe that $t \mapsto J_t^{1-\alpha_j} [\varphi_j(t)-\varphi_j(0)]$ belongs to $C^1([0,T];\mathbb{R})$ for each $1\leq j\leq n$. Consequently, by applying $J^{\alpha_j}_t$ to the $j$-th equation in [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"} and employing ([\[id2\]](#id2){reference-type="ref" reference="id2"}) we come to $$\varphi_j(t) - \varphi_j(0) = J^{\alpha_j}_t f_j(\varphi(t),t), \quad \forall t \in [0,T].$$ This equality and [\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} yields [\[eqint\]](#eqint){reference-type="eqref" reference="eqint"}. Conversely assume that $\varphi \in C([0,T];\mathbb{R}^n)$ and [\[eqint\]](#eqint){reference-type="eqref" reference="eqint"} holds, for all $1\leq j\leq n$. Then the continuity of each $t \mapsto f_j(\varphi(t),t)$ gives $t \mapsto J^{\alpha_j}_t f_j(\varphi(t),t)$ belongs to $C([0,T];\mathbb{R})$, for all $1 \leq j \leq n$. Applying ${}^C \! D_t^{\alpha_j}$ to both sides of [\[eqint\]](#eqint){reference-type="eqref" reference="eqint"} and taking into account [\[id1\]](#id1){reference-type="eqref" reference="id1"} we get $${}^C \! D_t^{\alpha_j} \varphi_j(t) = f_j(\varphi(t),t), \quad \forall t \in [0,T], 1\leq j \leq n,$$ which means that $\varphi$ satisfies the equations [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"} for all $t\in [0,T]$. Finally, to show that $\varphi$ satisfies [\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} we observe that $$\vert J^{\alpha_j}_t f_j(\varphi(t),t) \vert_{\mathbb{R}} \leq \frac{1}{\Gamma(\alpha_j)} \int_0^t (t-s)^{\alpha_j-1} \|f(\varphi(\cdot),\cdot)\|_{C([0,T];\mathbb{R}^n)} \, ds = C \, t^{\alpha_j},$$ where $C= {\|f(\varphi(\cdot),\cdot)\|_{C([0,T];\mathbb{R}^n)}}/{\Gamma(\alpha_j+1)}$. From this inequatily we have $$\lim_{t \to 0^{+}} \vert J^{\alpha_j}_t f_j(\varphi(t),t) \vert_{\mathbb{R}} = 0,$$ and therefore $$\lim_{t \to 0^{+}} \dfrac{1}{\Gamma(\alpha_j)}\int_{0}^t{(t-s)^{\alpha_j-1}f_j\big(\varphi(s),s\big)}\,ds = 0\,.$$ This leads to [\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} by taking the limit as $t \to 0^{+}$ in [\[eqint\]](#eqint){reference-type="eqref" reference="eqint"}. Whence we have proved that $\varphi$ is a solution of [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} in the interval $[0,T]$ and the proof is completed. ◻ We say that a function $f:\Omega\times[0,T] \subset \mathbb{R}^{n+1} \to \mathbb{R}^n$ is Lipschitz function on the first variable if there exists a constant $\ell >0$ such that $$\|f(x,t) - f(y,t) \|_{\mathbb{R}^n} \leq \ell \, \|x-y\|_{\mathbb{R}^n}, \textrm{ for all } t \in [0,T] \textrm{ and } x,y \in \Omega.$$ Now we face up to the main result of this section: **Theorem 7**. *[\[teo2\]]{#teo2 label="teo2"} Let $f:\Omega\times[0,T] \subset \mathbb{R}^{n+1} \to \mathbb{R}^n$ be continuous and Lipschitz function on the first variable. Then for each $\xi\in \Omega$ there exists an unique $\varphi$ solution of the Cauchy problem [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} in the interval $[0,T]$.* *Proof.* Without loss of generality we can assume that $\{\alpha_j\}_{j=1}^n\subset(0,1]$ satisfies $\alpha_1 \leq \alpha_2 \leq \cdots \leq \alpha_n$. We note that the operator $\mathcal{T}:C\left([0,T];\mathbb{R}^n\right) \to C\left([0,T];\mathbb{R}^n\right)$, given by $$\mathcal{T}(\varphi(t)) = \xi + \left( \begin{array}{c} J^{\alpha_1}_t f_1(\varphi(t),t)\\ \vdots \\ J^{\alpha_n}_t f_n(\varphi(t),t) \end{array} \right), \textrm{ for all } t \in [0,T],$$ is well-defined. Our strategy in proving the theorem is classical, i.e., we show that for $m \in \mathbb{N}$, sufficiently large, the operator $\mathcal{T}^m$ is a contraction. For $\varphi, \, \psi \in C\left([0,T];\mathbb{R}^n\right)$ we have $$\left\| \mathcal{T} \varphi (t) - \mathcal{T} \psi (t) \right\|_{\mathbb{R}^n} \leq \ell\left(\sum_{k=1}^n J^{\alpha_k}_t\right) \left\| \varphi(t) - \psi(t) \right\|_{\mathbb{R}^n},$$ for every $t\in[0,T]$. Then, by employing the semigroup property of the RL fractional integral we obtain $$\left\| \mathcal{T} \varphi (t) - \mathcal{T} \psi (t) \right\|_{\mathbb{R}^n} \leq \ell \left( \sum_{k=1}^n J^{\alpha_k - \alpha_1}_t \right) J^{\alpha_1}_t \left\| \varphi(t) - \psi(t) \right\|_{\mathbb{R}^n}, \textrm{ for all } t \in [0,T].$$ The aforementioned estimate enables us to conclude recursively that $$\left\| \mathcal{T}^m \varphi (t) - \mathcal{T}^m \psi (t) \right\|_{\mathbb{R}^n} \leq \ell^m \left( \sum_{k=1}^n J^{\alpha_k-\alpha_1}_t \right)^m J^{m \alpha_1}_t \left\| \varphi(t) - \psi(t) \right\|_{\mathbb{R}^n},$$ for all $t\in[0,T]$ and $m\in\mathbb{N}$. This inequality yields $$\begin{gathered} \label{Tn} \left\| \mathcal{T}^m \varphi - \mathcal{T}^m \psi \right\|_{C([0,T];\mathbb{R}^n)} \\ \leq \ell^m \left\|\sum_{k=1}^n J^{\alpha_k-\alpha_1}_t \right\|_{\mathcal{L}(C([0,T];\mathbb{R}^n))}^m \left\|J^{m \alpha_1}_t \left\| \varphi(\cdot) - \psi(\cdot) \right\|_{\mathbb{R}^n}\right\|_{C([0,T];\mathbb{R})}, \textrm{ for all } m \in \mathbb{N} .\end{gathered}$$ On the other hand, using the properties of the RL fractional integral operator we find $$\label{novaestim1} \left\Vert \sum_{k=1}^n J^{\alpha_k-\alpha_1}_t\right\Vert_{\mathcal{L}(C([0,T];\mathbb{R}^n))} \leq \sum_{k=1}^n \left\Vert J^{\alpha_k-\alpha_1}_t \right\Vert_{\mathcal{L}(C([0,T];\mathbb{R}^n))} \leq 2\sum_{k=1}^{n} T^{\alpha_k-\alpha_1}.$$ In order to obtain the above estimate, we have used that $\Gamma(\alpha) \geq {1}/{2}$ for $\alpha > 0.$ Now observe that for all $1 \leq k \leq n$ we have $$T^{(\alpha_k-\alpha_1)} \leq \left \{ \begin{array}{cl} 1, & \textrm{if } 0 < T \leq 1,\\ T^{(\alpha_n-\alpha_1)}, & \textrm{if } T \geq 1,\\ \end{array} \right.$$ since $\alpha_1 \leq \alpha_2 \leq \cdots \leq \alpha_n$. From this and [\[novaestim1\]](#novaestim1){reference-type="eqref" reference="novaestim1"} we obtain the estimate $$\left\Vert \sum_{k=1}^n J^{\alpha_k-\alpha_1}_t \right\Vert_{\mathcal{L}(C([0,T];\mathbb{R}^n))} \leq T_M, \textrm{ where } T_M = 2n\max \left\{ T^{(\alpha_n-\alpha_1)} , 1 \right\}.$$ It follows from this inequality and ([\[Tn\]](#Tn){reference-type="ref" reference="Tn"}) that $$\label{id8} \left\Vert \mathcal{T}^m \varphi - \mathcal{T}^m \psi \right\Vert_{C([0,T];\mathbb{R}^n)} \leq \frac{( \ell T_M T^{\alpha_1})^m}{\Gamma(m\alpha_1+1)} \left\Vert \varphi- \psi \right\Vert_{C([0,T];\mathbb{R}^n)}.$$ Hence, for sufficiently large $m \in \mathbb{N}$, we have $$\label{id9} \frac{( \ell \,T_M \,T^{\alpha_1})^m}{\Gamma(m\alpha_1+1)} < 1,$$ since $\left[ {( \ell T_M T^{\alpha_1})^m}/{\Gamma(m\alpha_1+1)} \right] \to 0$ as $m \to \infty$, corresponding to the general term of the series defining the Mittag-Leffler function $E_{\alpha_1}(z)$ with $z=\ell\, T_M\, T^{\alpha_1}$. Using [\[id8\]](#id8){reference-type="eqref" reference="id8"} and [\[id9\]](#id9){reference-type="eqref" reference="id9"} we obtain that $\mathcal{T}^m$ is a contraction in $C([0,T];\mathbb{R}^n)$, for such suitable $m$. Therefore, the Banach Fixed Point Theorem ensures the existence of a function $\phi \in C([0,T];\mathbb{R}^n)$ that is the unique fixed point of $\mathcal{T}^m$, and consequently, of $\mathcal{T}$. Thus, we have $\phi(t) = \mathcal{T} \phi(t)$ for every $t\in[0,T]$, which can be expressed as $$\phi_j(t)=\xi_j+\dfrac{1}{\Gamma(\alpha_j)}\int_{0}^t{(t-s)^{\alpha_j-1}f_j\big(\phi(s),s\big)} ds, \textrm{ for all } t \in [0,T] \textrm{ and } 1\leq j \leq n.$$ Finally, Proposition [Proposition 6](#integralequ){reference-type="ref" reference="integralequ"} completes the proof. ◻ ## A Brief Digression {#digression} To conclude this section, we find important to highlight a particular case of Theorem [\[teo2\]](#teo2){reference-type="ref" reference="teo2"}. Specifically, we want to address the scenario where $f:\Omega\times[0,T] \subset \mathbb{R}^{n+1} \to \mathbb{R}^n$ is given by $f(x,t)=Ax$, with $A\in M^n(\mathbb{R})$ (the set of square matrices with real entries). In this context, a natural question arises regarding the representation of the matrix function corresponding to the unique solution $\phi(t)$ obtained for the Cauchy problem [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} in $[0,\infty)$. It is well-known in the theory (cf. [@Da-GeJa0]) that when $\alpha_1=\alpha_2=\ldots=\alpha_n=\alpha\in(0,1]$, the solution to [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} can be represented by the Mittag-Leffler matrix function $$\label{auxsol1}\phi(t)=E_{\alpha}(t^\alpha A)\xi,\quad\forall t\geq0,$$ where $E_\alpha:\mathbb{C}\rightarrow\mathbb{C}$ is the analytic Mittag-Leffler function. It must be pointed out that the representation [\[auxsol1\]](#auxsol1){reference-type="eqref" reference="auxsol1"} is consistent with the classical case when $\alpha=1$, which corresponds to the exponential matrix. We can understand [\[auxsol1\]](#auxsol1){reference-type="eqref" reference="auxsol1"} as a natural generalization of one dimensional case, where the Mittag-Leffler function ($\alpha\in(0,1)$) and the exponential function ($\alpha=1$) are solutions to their respective Cauchy problems [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} in $[0,\infty)$. However, when considering the Cauchy problem [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} with distinct orders of differentiation, it is not valid to assume that they are direct generalizations of the one dimensional case, as such an assumption would not even make sense. Therefore, to understand and derive the matrix function that satisfies [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} with distinct orders of differentiation, it is imperative to comprehend the case $n=2$. With that in mind, let us proceed with this discussion by assuming that $A\in M^2(\mathbb{R})$ is in its Jordan normal form. In other words, we are considering just the matrices $$A_1=\left[\begin{array}{cc}\lambda&0\\0&\mu\end{array}\right],\quad A_2=\left[\begin{array}{cc}\lambda&1\\0&\lambda\end{array}\right]\quad\textrm{and}\quad A_3=\left[\begin{array}{cc}\lambda&\mu\\-\mu&\lambda\end{array}\right],$$ where $\lambda,\mu\in\mathbb{R}$. *Case $A_1$:* In this scenario, it becomes apparent that the unique solution $\phi(t)$ to [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} can be expressed by $$\phi(t)=\left(\begin{array}{c}E_{\alpha_1}(t^{\alpha_1}\lambda)\xi_1\\E_{\alpha_2}(t^{\alpha_2}\mu)\xi_2\end{array}\right),$$ for every $t\in[0,\infty)$. *Case $A_2$:* By applying similar reasoning as before, we can deduce that $\phi_2(t)=E_{\alpha_2}(t^{\alpha_2}\lambda)\xi_2$, for every $t\in[0,\infty)$. Once we have determined $\phi_2(t)$, the first line of the problem becomes a non-homogeneous fractional ordinary differential equation of order $\alpha_1$. Therefore, the fractional version of the variation of constants formula allows us to obtain the expression $$\phi(t)=\left(\begin{array}{c}E_{\alpha_1}(t^{\alpha_1}\lambda)\xi_1+\displaystyle\int_0^t(t-s)^{\alpha_1-1}E_{\alpha_1,\alpha_1}((t-s)^{\alpha_1}\lambda)E_{\alpha_2}(s^{\alpha_2}\lambda)\xi_2\,ds\\E_{\alpha_2}(t^{\alpha_2}\lambda)\xi_2\end{array}\right),$$ for every $t\in[0,\infty)$. *Case $A_3$:* This particular case poses the greatest challenge in our analysis. Let us consider the scenario where $\lambda=0$. In this situation, we can employ the equations from [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} to demonstrate that $\phi(t)$ satisfies the integral equation $$\label{auxnewpb1}\phi(t)=\xi+\left(\begin{array}{c}\dfrac{t^{\alpha_1}\mu\xi_2}{\Gamma(\alpha_1+1)}\vspace*{0.3cm} \\\dfrac{t^{\alpha_2}\mu\xi_1}{\Gamma(\alpha_2+1)}\end{array}\right)+J_t^{\alpha_1+\alpha_2}\phi(t),$$ for every $t\in[0,\infty)$. Computing the solution of the integral equation [\[auxnewpb1\]](#auxnewpb1){reference-type="eqref" reference="auxnewpb1"} is not a straightforward task. However, when $\alpha_1+\alpha_2\leq1$, we can deduce that the solution of [\[auxnewpb1\]](#auxnewpb1){reference-type="eqref" reference="auxnewpb1"} reduces to the solution of the non-homogeneous fractional ordinary differential equation $${}^C \! D_t^{\alpha_1+\alpha_2}\phi(t)=-\mu^2\phi(t)+\mu\left(\begin{array}{c}\dfrac{t^{-\alpha_2}\xi_2}{\Gamma(1-\alpha_2)}\vspace*{0.3cm} \\\dfrac{t^{-\alpha_1}\xi_1}{\Gamma(1-\alpha_1)}\end{array}\right)\quad\textrm{and}\quad\phi(0)=\xi.$$ Using the fractional version of the variation of constants formula, we finally deduce that $$\begin{gathered} \phi(t)=E_{\alpha_1+\alpha_2}(-t^{\alpha_1+\alpha_2}\mu^2)\xi\\+\mu\int_0^t(t-s)^{\alpha_1+\alpha_2-1}E_{\alpha_1+\alpha_2,\alpha_1+\alpha_2}(-(t-s)^{\alpha_1+\alpha_2}\mu^2)\left(\begin{array}{c}\dfrac{s^{-\alpha_2}\xi_2}{\Gamma(1-\alpha_2)}\vspace*{0.3cm}\\\dfrac{s^{-\alpha_1}\xi_1}{\Gamma(1-\alpha_1)}\end{array}\right)\,ds,\end{gathered}$$ for every $t\in[0,\infty)$. The cases where $\alpha_1+\alpha_2>1$ or $\lambda\neq 0$ pose significant challenges and are omitted here. In this brief section, we simply want to highlight the challenge of obtaining a closed formula for the matrix solution of problem [\[caratheq\]](#caratheq){reference-type="eqref" reference="caratheq"}-[\[carathin\]](#carathin){reference-type="eqref" reference="carathin"} when $n=2$. This suggests that the general case of $n>2$ is even more complex and requires a more comprehensive investigation, which is currently lacking in the existing literature and will be conducted in our future research. # Well-Posedness Theory {#existenceresults} This section focuses on the initial boundary value problem [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}-[\[6prob\]](#6prob){reference-type="eqref" reference="6prob"}, the central subject of this paper. Here our main goal is to establish the well-posedeness of this problem. **Theorem 8**. *Let $\alpha \in (0,1]$ be a real number and let $f,g,h \in C \left( \overline{\Gamma_1} \right)$ be such that $f$ and $h$ are positive functions and $g$ is a non-negative function. If $u_0 \in H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)$, $u_1 \in H^1_{\Gamma_0}(\Omega)$ and $\delta_0 \in L^2(\Gamma_1)$, then for all $T >0$ arbitrarily fixed there exists an unique pair of functions $(u,\delta)$, with $u: \Omega \times [0,T] \to \mathbb{R}$ and $\delta : \Gamma_{1}\times [0,T] \to \mathbb{R}$, in the class $$\label{id10} \begin{array}{l} u, u_t \in L^\infty \left( 0,T; H^1_{\Gamma_0}(\Omega) \right), \ {}^C \! D^{\alpha}_tu_{t} \in L^\infty \left( 0,T; L^2(\Omega) \right) \\ \textrm{and } u(t) \in \mathcal{H}_\Delta(\Omega), \text{ for a.e. } t \in (0,T); \end{array}$$ $$\label{id11} \delta, \delta_t \in L^\infty \left( 0,T; L^2(\Gamma_1) \right)\textrm{ and } \delta_{tt} \in L^2 \left( 0,T; L^2(\Gamma_1) \right);$$ which satisfies $$\begin{aligned} &&{}^C \! D^{\alpha}_tu_t(t) - \Delta u (t) = 0, \textrm{ in } L^2(\Omega), \text{ for a.e. } t \in (0,T); \label{id12}\\ &&f\delta_{tt}(t) + g\delta_t (t) + h\delta (t) = - \gamma_0( u_t(t)) \textrm{ in } L^2(\Gamma_1), \textrm{ for a.e. } t \in (0,T);\label{id13}\\ &&\int_{\Gamma_1} \delta_t(t) \gamma_0(\varphi) d \Gamma_1 = \left\langle \gamma_1(u(t)) ,\gamma_0(\varphi) \right\rangle_{H^{-\frac{1}{2}}(\Gamma) \times H^{\frac{1}{2}}(\Gamma) }, \forall \varphi \in H^1_{\Gamma_0}(\Omega) \nonumber \\ && \textrm{and a.e. } t \in (0,T); \label{id14}\\ && u(0)=u_0 \textrm{ and } u_t(0)=u_1, \textrm{ in } L^2(\Omega); \label{id15}\\ && \delta(0)=\delta_0 \textrm{ and } \delta_t(0) = \gamma_1(u_0)_{\mid_{\Gamma_{1}}}, \textrm{ in } L^2(\Gamma_1). \label{id16}\end{aligned}$$ Moreover the pair of functions $(u,\delta)$ depends continuously on the initial data and the parameters $f,g$ and $h$.* *Proof.* We have divided the proof into 5 steps: (1) Approximate problem; (2) A priori estimates; (3) Passage to the limit - Existence; (4) Uniqueness; and (5) Continuous dependence. **Step 1: Approximate problem.** Let $(w_j)_{j \in \mathbb{N}}$ be an orthonormal basis of $L^2(\Omega)$ given by the eigenfunctions of the operator $-\Delta$ with domain $H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)$. Thus $(w_j)_{j \in \mathbb{N}}$ is also a complete orthogonal system in $H_{\Gamma_0}^1(\Omega)$ and $H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)$. For each $m \in \mathbb{N}$ we set $W_m =[w_1, \cdots , w_m]$ the linear subspace of $H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)$ spanned by $\{w_{1}, \cdots, w_{m}\}$. Similarly, let $(z_j)_{j \in \mathbb{N}}$ be an orthonormal basis of the Hilbert space $L^2(\Gamma_1)$, constructed such that $\gamma_0(W_m)|_{\Gamma_1} \subset Z_m$ for every $m \in \mathbb{N}$, where $Z_m = [z_{1}, \cdots , z_{m}]$ represents the linear subspace of $L^2(\Gamma_1)$ spanned by $\{z_{1}, \cdots, z_{m}\}$. Now, for each $m \in \mathbb{N}$ fixed arbitrary, we seek functions $u_{m}: \Omega \times [0,T] \to \mathbb{R}$ and $\delta_{m}:\Gamma_{1} \times [0,T] \to \mathbb{R}$ in the form $$\displaystyle u_m(x,t) = \sum_{k=1}^m a_{km}(t)\, w_k(x) \quad\textrm{and}\quad \delta_m(x,t) = \sum_{k=1}^m b_{km}(t)\,z_k(x)$$ which are solutions to the approximate problem $$\begin{aligned} & \label{help1}\left( {}^C \! D^{\alpha}_tu'_m (t) , w_j \right) + \left( \left( u_m(t),w_j \right) \right) - \left(\delta'_m(t), \gamma_0(w_j) \right)_{\Gamma_1} = 0, && 1 \leq j \leq m\,;\\ & \label{help2}\left(f \delta''_m (t) + g\delta_m'(t) + h\delta_m(t) , z_j \right)_{\Gamma_1} + \left( \gamma_0(u'_m(t)),z_j\right)_{\Gamma_1} = 0, && 1 \leq j \leq m\,; \\ & u_{m}(0) = u_{0m}, \ u'_m(0) = u_{1m}, \ \delta_m(0) = \delta_{0m}, \ \delta'_m(0)= \delta_{1m}, \label{help3}\end{aligned}$$ where $$\begin{aligned} &&u_{0m} = \sum_{k=1}^m (u_0,w_k)w_k \to u_{0} \,\, \text{in } H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega), \label{help4}\\ &&u_{1m} = \sum_{k=1}^m (u_1,w_k)w_k \to u_{1}\,\, \text{in } H^1_{\Gamma_0}(\Omega) \label{help5}\\ &&\delta_{0m} = \sum_{k=1}^m (\delta_0,z_k)_{\Gamma_1}z_k \to \delta_{0} \text{ in } L^{2}(\Gamma_{1}) \label{help6}\\ &&\delta_{1m} ={\gamma_1(u_{0m})}_{\mid_{\Gamma_{1}}} =\sum_{k=1}^m c_{km}z_k \to {\gamma_{1}(u_{0})}_{\mid_{\Gamma_{1}}} \text{ in } L^{2}(\Gamma_{1}) , \label{help7}\end{aligned}$$ for some $c_{km} \in \mathbb{R}$. By setting $y_{jm}(t)=a'_{jm}(t)$ and $v_{jm}(t)=b'_{jm}(t)$ for each $1 \leq j \leq m$, and reinterpreting the variational identities [\[help1\]](#help1){reference-type="eqref" reference="help1"}-[\[help2\]](#help2){reference-type="eqref" reference="help2"} and the initial conditions [\[help3\]](#help3){reference-type="eqref" reference="help3"}, we obtain an equivalent linear fractional ODE system in $\mathbb{R}^{4m}$: $$\label{caratheq21} \left\{ \begin{array}{ccl} {}^C \! D^{\alpha}_ty_{m}(t) &=& -A_{1,m} \, a_m(t) + A_{2,m} \, v_m(t),\\ a'_{m}(t) &=& y_m(t),\\ v'_{m}(t) &=& -(A_{3,m})^{-1}\big[A_{4,m} \, y_m(t) + A_{5,m} \, v_m(t) + A_{6,m} \, b_m(t)\big],\\ b'_{m}(t) &=& v_m(t), \end{array} \right.$$ with the initial conditions $$\label{caratheq22} \left\{ \begin{array}{ccl} y_{m}(0) &=& u_{1m}, \\ a_{m}(0) &=& u_{0m}, \\ v_{m}(0) &=& \delta_{1m}, \\ b_{m}(0) &=& \delta_{0m}, \end{array} \right.$$ where $$\begin{array}{ll} y_m(t) = \left(y_{1m}(t), \cdots , y_{mm} (t) \right), & a_m(t) = \left(a_{1m}(t), \cdots , a_{mm}(t) \right), \\ v_m(t) = \left(v_{1m}(t), \cdots , v_{mm}(t) \right), & b_m(t) = \left(b_{1m}(t), \cdots , b_{mm}(t) \right), \end{array}$$ and $$\begin{array}{lll} A_{1,m}=\big[((w_j,w_i))\big]_{i,j=1}^m, & A_{2,m}=\big[(z_j,\gamma_0(w_i))_{\Gamma_1}\big]_{i,j=1}^m, & A_{3,m}=\big[(fz_j,z_i)_{\Gamma_1}\big]_{i,j=1}^m, \\ A_{4,m}=\big[(\gamma_0(w_j),z_i)_{\Gamma_1}\big]_{i,j=1}^m, & A_{5,m}=\big[(gz_j,z_i)_{\Gamma_1}\big]_{i,j=1}^m, & A_{6,m}=\big[(hz_j,z_i)_{\Gamma_1}\big]_{i,j=1}^m. \end{array}$$ We emphasize that the continuity and positivity of $f$ ensure $(fz_j,z_i)_{\Gamma_1}>0$ for all $i,j\in\{1,\ldots,m\}$. This, combined with the orthonormality of $(z_j)_{j\in\mathbb{N}}$ in $L^2(\Gamma_1)$, guarantees the invertibility of the matrix $A_{3,m}$. It is not difficult to notice that the Cauchy problem [\[caratheq21\]](#caratheq21){reference-type="eqref" reference="caratheq21"}-[\[caratheq22\]](#caratheq22){reference-type="eqref" reference="caratheq22"} satisfies the assumption of Theorem [\[teo2\]](#teo2){reference-type="ref" reference="teo2"}. Therefore we can deduce the existence and uniqueness of solution for [\[caratheq21\]](#caratheq21){reference-type="eqref" reference="caratheq21"}-[\[caratheq22\]](#caratheq22){reference-type="eqref" reference="caratheq22"}. The aforementioned conclusions establish that $y_m$, ${}^C \! D^{\alpha}_ty_{m}$, $a^\prime_{m}$, $v^\prime_{m}$, ${}^C \! D^{\alpha}_tv^\prime_{m}$ and $b^\prime_{m}$ belong to $C([0,T]; \mathbb{R}^m)$, and therefore $$\begin{aligned} && \nonumber u'_m, {}^C \! D^{\alpha}_tu'_m \in C^1 \left([0,T]; H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega) \right)\label{eqap1}, \\ && \nonumber {}^C \! D^{\alpha}_t\delta_m'' \in C \left([0,T]; L^2(\Gamma_1) \right) \textrm{ and } \delta_m \in C^2 \left([0,T]; L^2(\Gamma_1) \right),\label{eqap11} \\ &&\left( {}^C \! D^{\alpha}_tu'_m(t), w \right) + \left( \left( u_m(t),w \right) \right) - \left( \delta'_m(t),\gamma_0 (w) \right)_{\Gamma_1} = 0, \textrm{ for all } w \in W_{m}, \label{eqap2}\\ &&\left(f \delta''_m(t) + g \delta'_m(t) + h \delta_m (t) , z \right)_{\Gamma_1} + \left( \gamma_0 (u'_m(t)) , z \right)_{\Gamma_1} = 0, \textrm{ for all } z \in Z_{m}, \label{eqap3}\end{aligned}$$ for each $m\in\mathbb{N}$. **Step 2: A priori estimates. Estimate 1.** Taking $w = 2 u'_m \in W_m$ in [\[eqap2\]](#eqap2){reference-type="eqref" reference="eqap2"} and $z = 2\delta'_m \in Z_m$ in [\[eqap3\]](#eqap3){reference-type="eqref" reference="eqap3"} we deduce $$\begin{gathered} 2\left( {}^C \! D^{\alpha}_tu'_m(t), u'_m(t) \right) + 2\left( \left( u_m(t), u'_m(t) \right) \right) +2\left(f \delta''_m(t) , \delta'_m(t) \right)_{\Gamma_1}\\ + 2\left\vert g^{\frac{1}{2}} \delta'_m(t)\right\vert^2_{\Gamma_1} + 2\left(h \delta_m (t),\delta'_m(t) \right)_{\Gamma_1} = 0.\end{gathered}$$ To address the first term on the left side of the above identity, we observe that $$\left( {}^C \! D^{\alpha}_tu'_m(t), u'_m(t) \right)=\sum_{j=1}^m\left[{}^C \! D^{\alpha}_ta_{jm}'(t)\right]a_{jm}'(t)$$ and $${}^C \! D^{\alpha}_t\left\vert u'_m(t) \right\vert^2=\sum_{j=1}^m{}^C \! D^{\alpha}_t\left[a_{jm}'(t)\right]^2,$$ since $(w_j)_{j\in\mathbb{N}}$ is an orthonormal basis in $L^2(\Omega)$. Then, supported by Proposition [Proposition 3](#teoju2){reference-type="ref" reference="teoju2"}, we apply Theorem [Theorem 2](#finalcaputo){reference-type="ref" reference="finalcaputo"} to establish the inequality $${}^C \! D^{\alpha}_t\left\vert u'_m(t) \right\vert^2 + 2\left\vert g^{\frac{1}{2}} \delta'_m(t)\right\vert_{\Gamma_1}^2 + \frac{d}{dt} \left[ \left\Vert u_m(t) \right\Vert^2+\left\vert f^{\frac{1}{2}}\delta'_m(t) \right\vert_{\Gamma_1}^2+ \left\vert h^{\frac{1}{2}} \delta_m(t)\right\vert_{\Gamma_1}^2 \right] \leq 0.$$ Taking into account [\[id3\]](#id3){reference-type="eqref" reference="id3"}, the convergences [\[help4\]](#help4){reference-type="eqref" reference="help4"}-[\[help7\]](#help7){reference-type="eqref" reference="help7"} and the continuity of the trace map $\gamma_{1}$ (i.e. $|\gamma_{1}(u)|_{\Gamma_{1}} \leq c_{1} \left\Vert u \right\Vert_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)}$), when applying the integral operator $J^1_t$ to both sides of the last inequality we get $$\begin{aligned} &&J^{1-\alpha}_t \left\vert u'_m(t) \right\vert^2 + \left\Vert u_m(t) \right\Vert^2 + f_0 \left\vert \delta'_m(t) \right\vert_{\Gamma_1}^2 \nonumber\\ &&\leq \frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \left\vert u'_{m}(0) \right\vert^2 + \left\Vert u_m(0) \right\Vert^2 + f_1\left\vert \delta'_m(0) \right\vert_{\Gamma_1}^2+ h_1 \left\vert \delta_m(0)\right\vert_{\Gamma_1}^2 \nonumber\\ &&=\frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \left\vert u_{1m} \right\vert^2 + \left\Vert u_{0m} \right\Vert^2 + f_1 \left\vert \gamma_1(u_{0m}) \right\vert_{\Gamma_1}^2+ h_1 \left\vert \delta_{0m}\right\vert_{\Gamma_1}^2\nonumber\\ &&\leq \frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \left\Vert u_{1m} \right\Vert^2 + (1+f_1 c_1^2)\left\Vert u_{0m} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)} + h_1 \left\vert \delta_{0m} \right\vert^2_{\Gamma_1} \nonumber\\ &&\leq \frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \left\Vert u_{1} \right\Vert^2 + (1+f_1 c_1^2)\left\Vert u_{0} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)} + h_1 \left\vert \delta_{0} \right\vert^2_{\Gamma_1},\label{id17}\end{aligned}$$ where $$f_{0} = \min_{x \in \overline{\Gamma_{1}}} f(x), \, f_{1} = \max_{x \in \overline{\Gamma_{1}}} f(x), \textrm{ and } h_{1} = \max_{x \in \overline{\Gamma_{1}}} h(x).$$ Finally, from [\[id7\]](#id7){reference-type="eqref" reference="id7"} and [\[id17\]](#id17){reference-type="eqref" reference="id17"} we can see that there exists a constant $K_{1} > 0$, that depends on $\alpha, T, f_0, f_1, h_1$ and $c_1$, such that $$\begin{gathered} \label{aprioriI} \left\Vert {}^C \! D^{\alpha}_tu_m \right\Vert^2_{L^\infty(0,T; L^2(\Omega))} + \left\Vert u_m \right\Vert^2_{L^\infty \left(0,T;H^1_{\Gamma_0}(\Omega) \right)} + \left\Vert \delta'_m \right\Vert^2_{L^\infty(0,T;L^2(\Gamma_1))}\\ \leq K_1 \left[ \left\Vert u_{0} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)} + \left\Vert u_1 \right\Vert^2 + \left\vert \delta_{0} \right\vert^2_{\Gamma_1} \right],\end{gathered}$$ which completes estimate 1. Additionally, we can estimate $\delta_m$ in $L^\infty(0,T;L^2(\Gamma_1))$ based on the estimate of $\Vert \delta'_m \Vert_{L^\infty(0,T;L^2(\Gamma_1))}$. In fact, the Fundamental Theorem of Calculus implies $$\begin{aligned} \label{aprioriIa} \nonumber \Vert \delta_m \Vert_{L^\infty(0,T;L^2(\Gamma_1))} & = & \left\Vert J^1_t\left[D^1_t \delta_m\right] + \delta_m(0) \right\Vert_{L^\infty(0,T;L^2(\Gamma_1))} \\ \nonumber & \leq & \Vert J^1_t \delta'_m \Vert_{L^\infty(0,T;L^2(\Gamma_1))} + \Vert \delta_m(0) \Vert_{L^\infty(0,T;L^2(\Gamma_1))} \\ \nonumber & \leq & \frac{T}{\Gamma(2)} \Vert \delta'_m \Vert_{L^\infty(0,T;L^2(\Gamma_1))} + \vert \delta_m(0) \vert_{\Gamma_1} \\ & \leq & T \Vert \delta'_m \Vert_{L^\infty(0,T;L^2(\Gamma_1))} + \vert \delta_0 \vert_{\Gamma_1}.\end{aligned}$$ therefore, [\[aprioriI\]](#aprioriI){reference-type="eqref" reference="aprioriI"} gives us the desired estimate. **Estimate 2.** We apply $D^1_t=\frac{d}{dt}$ in ([\[eqap2\]](#eqap2){reference-type="ref" reference="eqap2"}) and ${}^C \! D^{\alpha}_t$ in ([\[eqap3\]](#eqap3){reference-type="ref" reference="eqap3"}). In the resulting equations we take $w=2{}^C \! D^{\alpha}_tu_m' \in W_m$ and $z=2 \delta''_m \in Z_m$, and combine them to obtain $$\begin{gathered} 2\left( \left[ {}^C \! D^{\alpha}_tu'_m(t) \right]', {}^C \! D^{\alpha}_tu'_m(t) \right) + 2\left( \left( u'_m(t) , {}^C \! D^{\alpha}_tu'_m(t) \right) \right) + 2\left(f \,{}^C \! D^{\alpha}_t\delta''_m(t) , \delta''_m(t) \right)_{\Gamma_1}\\ + 2\left( g \,{}^C \! D^{\alpha}_t\delta'_m(t),\delta''_m(t) \right)_{\Gamma_1} + 2\left(h \,{}^C \! D^{\alpha}_t\delta_m(t),\delta''_m(t) \right)_{\Gamma_1} = 0. \label{id18}\end{gathered}$$ Since $\delta'_m \in C^1([0,T];L^2(\Gamma_1))$, making use of [\[id1\]](#id1){reference-type="eqref" reference="id1"} and [\[id4\]](#id4){reference-type="eqref" reference="id4"} we can write $$\left(g\, {}^C \! D^{\alpha}_t\delta'_m(t), \delta''_m(t) \right)_{\Gamma_1} = \left( J^{1-\alpha}_t g^{\frac{1}{2}} \delta''_m(t), {}^C \! D_t^{1-\alpha} J^{1-\alpha}_t g^{\frac{1}{2}} \delta''_m(t) \right)_{\Gamma_1}.$$ As done before in Estimate 1, considering Proposition [Proposition 3](#teoju2){reference-type="ref" reference="teoju2"} and Theorem [Theorem 2](#finalcaputo){reference-type="ref" reference="finalcaputo"}, from [\[id18\]](#id18){reference-type="eqref" reference="id18"}, the equality above, and Young's inequality, it follows that for any $\varepsilon > 0$, $$\begin{aligned} && \frac{d}{dt} \left\vert {}^C \! D^{\alpha}_tu'_m(t) \right\vert^2 + {}^C \! D^{\alpha}_t\left\Vert u'_m(t) \right\Vert^2 + {}^C \! D^{\alpha}_t\left\vert f^{\frac{1}{2}} \delta''_m(t) \right\vert^2_{\Gamma_1} + {}^C \! D_t^{1-\alpha} \left\vert J^{1-\alpha}_t g^{\frac{1}{2}} \delta''_m(t)\right\vert_{\Gamma_1}^2 \\ && \leq 2 \left\vert h\, {}^C \! D^{\alpha}_t\delta_m(t) \right\vert_{\Gamma_1} \left\vert \delta''_m(t) \right\vert_{\Gamma_1} \leq \frac{1}{\varepsilon}\left\vert h \,{}^C \! D^{\alpha}_t\delta_m(t) \right\vert_{\Gamma_1}^2 + \varepsilon \left\vert \delta''_m(t) \right\vert_{\Gamma_1}^2 \\ && \leq \frac{h_1\, T^{1-\alpha}}{\varepsilon \Gamma(2-\alpha)} J^{1-\alpha}_t \vert \delta'_m(t) \vert^2_{\Gamma_1} + \varepsilon \,\left\vert \delta''_m(t) \right\vert_{\Gamma_1}^2 ,\end{aligned}$$ where in the last inequality we have used [\[id7\]](#id7){reference-type="eqref" reference="id7"}. Integrating this inequality over $(0,t)$ and applying [\[id3\]](#id3){reference-type="eqref" reference="id3"} it follows that $$\begin{aligned} \label{priori21} \nonumber && \left\vert {}^C \! D^{\alpha}_tu'_m(t) \right\vert^2 + J^{1-\alpha}_t \left\Vert u'_m(t) \right\Vert^2 + J^{1-\alpha}_t \left\vert f^{\frac{1}{2}} \delta''_m(t) \right\vert^2_{\Gamma_1} + J^{\alpha}_t \left\vert J^{1-\alpha}_t g^{\frac{1}{2}} \delta''_m(t) \right\vert^2_{\Gamma_1} \\ \nonumber && \leq \frac{h_1 T^{1-\alpha}}{\varepsilon \Gamma(2-\alpha)} J^{2-\alpha}_t \vert \delta'_m(t) \vert^2_{\Gamma_1} + \varepsilon J^1_t \left\vert \delta''_m(t) \right\vert_{\Gamma_1}^2 + \left\vert {}^C \! D^{\alpha}_tu'_m(0) \right\vert^2 + \frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \Vert u_m'(0) \Vert^2 \\ && + \frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \left\vert f^{\frac{1}{2}} \delta_m''(0) \right\vert^2_{\Gamma_1} + \frac{T^\alpha}{\Gamma(1+\alpha)} \left\vert \left( J^{1-\alpha}_t g^{\frac{1}{2}} \delta''_m \right)(0) \right\vert^2_{\Gamma_1}.\end{aligned}$$ Our job now is to estimate the terms, in the right hand side of the above inequality, involving ${}^C \! D^{\alpha}_tu'_m(0)$, $\delta''_m (0)$ and $\left(J^{1-\alpha}_t \delta''_m \right)(0)$. We start noting that the regularity of $\delta_m''$ allows us to affirm that $\left(J^{1-\alpha}_t \delta''_m \right)(0)=0$. Furthermore, the approximated equation [\[eqap2\]](#eqap2){reference-type="eqref" reference="eqap2"} in $t=0$ with $w={}^C \! D^{\alpha}_tu_m'(0)$ gives us $$\begin{aligned} \left\vert {}^C \! D^{\alpha}_tu_m'(0) \right\vert^2 &=& - \left( \left( u_m (0) , {}^C \! D^{\alpha}_tu_m'(0) \right) \right) + \left(\delta'_m(t), \gamma_0({}^C \! D^{\alpha}_tu_m'(0)) \right)_{\Gamma_1}\\ &=& \left( \Delta u_m(0) , {}^C \! D^{\alpha}_tu'_m(0) \right) \leq \frac{1}{2} \left\vert \Delta u_{0m} \right\vert^2 + \frac{1}{2} \left\vert {}^C \! D^{\alpha}_tu_m'(0) \right\vert^2,\end{aligned}$$ what gives us the estimate $$\label{priori21a} \left\vert {}^C \! D^{\alpha}_tu_m'(0) \right\vert^2 \leq \left\vert \Delta u_{0m} \right\vert^2 \leq \left\Vert u_0 \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)}.$$ Taking account the continuity of $\gamma_{0}$ (i.e. $|\gamma_{0}(u)|_{\Gamma_{1}} \leq c_{0} \left\Vert u \right\Vert_{H^1_{\Gamma_0}(\Omega)}$) and $\gamma_1$, a similar approach on equation [\[eqap3\]](#eqap3){reference-type="eqref" reference="eqap3"} with $z = f \delta''_m(0)$ allows us deduce that $$\begin{aligned} \left\vert f \delta''_m(0) \right\vert^2_{\Gamma_1} &=& \left( f \delta''_m(0) ,- g \delta'_m(0) -h \delta_m(0) - \gamma_0 ( u'_m(0))\right)_{\Gamma_1} \\ &\leq& \left\vert f \delta''_m(0) \right\vert_{\Gamma_1} \left\vert - g \gamma_1 (u_{0m}) -h \delta_{0m} - \gamma_0(u_{1m}) \right\vert_{\Gamma_1} \\ &\leq& {\frac{1}{2} \left\vert f \delta''_m(0)\right\vert^2_{\Gamma_1} + (g_1c_1)^2\left\Vert u_{0m} \right\Vert^2 + 2h_1^2\left\vert \delta_{0m} \right\vert^2_{\Gamma_1} + 2 c_0^2 \left\Vert u_{1m} \right\Vert^2}.\end{aligned}$$ Therefore we conclude $$\label{priori22} \left\vert \delta''_m(0) \right\vert^2_{\Gamma_1} \leq \frac{2(g_1c_1)^2}{f_0} \left\Vert u_{0} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)} + \frac{4c_0^2}{f_0} \left\Vert u_{1} \right\Vert^2 + \frac{4h_1^2}{f_0}\left\vert \delta_{0} \right\vert^2_{\Gamma_1} .$$ Thus, with [\[priori21a\]](#priori21a){reference-type="eqref" reference="priori21a"} and [\[priori22\]](#priori22){reference-type="eqref" reference="priori22"} in [\[priori21\]](#priori21){reference-type="eqref" reference="priori21"}, inequality [\[prop07\]](#prop07){reference-type="eqref" reference="prop07"} guarantees that $$\begin{gathered} \left\vert {}^C \! D^{\alpha}_tu'_m(t) \right\vert^2 + \frac{1}{T^\alpha \Gamma(1-\alpha)} \left[ J^{1}_t \left\Vert u'_m(t) \right\Vert^2 + f_0 J^{1}_t \left\vert \delta''_m(t) \right\vert^2_{\Gamma_1} \right] + \frac{g_0}{T^{1-\alpha} \Gamma(\alpha)} J^{1}_t \left\vert J^{1-\alpha}_t \delta''_m(t) \right\vert^2_{\Gamma_1} \\ \leq \frac{h_1}{\varepsilon } \left[ \frac{T^{1-\alpha}}{\Gamma(2-\alpha)} \right]^2 J^{1}_t \vert \delta'_m(t) \vert^2_{\Gamma_1} + \varepsilon J^1_t \left\vert \delta''_m(t) \right\vert_{\Gamma_1}^2 + C_{2} \left[ \left\Vert u_{0} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)} + \left\Vert u_{1} \right\Vert^2 + \left\vert \delta_{0} \right\vert^2_{\Gamma_1} \right],\end{gathered}$$ where $C_2>0$ is a constant. Choosing $\varepsilon$ sufficiently small, such that $0<\varepsilon < f_0/T^\alpha \Gamma(1-\alpha)$, for some $C_3>0$, we get $$\begin{gathered} \left\vert {}^C \! D^{\alpha}_tu'_m(t) \right\vert^2 + J^{1}_t \left\Vert u'_m(t) \right\Vert^2 + J^{1}_t \left\vert \delta''_m(t) \right\vert^2_{\Gamma_1} \\ \leq C_3 \left[ \left\Vert \delta'_m \right\Vert^2_{L^2(0,T;L^2(\Gamma_1))} + \left\Vert u_{0} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)} + \left\Vert u_{1} \right\Vert^2 + \left\vert \delta_{0} \right\vert^2_{\Gamma_1} \right].\end{gathered}$$ From this inequality and [\[aprioriI\]](#aprioriI){reference-type="eqref" reference="aprioriI"} there exists a constant $K_{2}>0$, that depends on $\alpha,T,f_0,g_1,h_1,c_0$ and $c_1$, such that $$\begin{gathered} \label{aprioriII} \left\Vert {}^C \! D^{\alpha}_tu'_m \right\Vert^2_{L^\infty(0,T;L^2(\Omega))} + \left\Vert u'_m \right\Vert^2_{L^2(0,T;H^1_{\Gamma_0}(\Omega))} + \left\Vert \delta''_m \right\Vert^2_{L^2(0,T;L^2(\Gamma_1))} \\ \leq K_2 \left[ \left\Vert u_{0} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega)} + \left\Vert u_{1} \right\Vert^2 + \left\vert \delta_{0} \right\vert^2_{\Gamma_1} \right].\end{gathered}$$ Moreover, since there exists $C_4>0$ such that $\vert u \vert \leq C_4 \Vert u \Vert$, using [\[id2\]](#id2){reference-type="eqref" reference="id2"} and the continuity of the RL fractional integral of order $\alpha$ (cf. [@CarFe1 Theorem 3.1]) we obtain $$\begin{aligned} \label{aprioriIIa} \nonumber \Vert u'_m \Vert_{L^\infty(0,T;L^2(\Omega))} & = & \Vert J^\alpha_t\, {}^C \! D^{\alpha}_tu'_m + u'_m(0) \Vert_{L^\infty(0,T;L^2(\Omega))} \\ \nonumber & \leq & \Vert J^\alpha_t\, {}^C \! D^{\alpha}_tu'_m \Vert_{L^\infty(0,T;L^2(\Omega))} + \Vert u'_m(0) \Vert_{L^\infty(0,T;L^2(\Omega))} \\ \nonumber & \leq & \frac{T^\alpha}{\Gamma(\alpha+1)} \Vert {}^C \! D^{\alpha}_tu'_m \Vert_{L^\infty(0,T;L^2(\Omega))} + \vert u'_m(0) \vert \\ & \leq & \frac{T^\alpha}{\Gamma(\alpha+1)} \Vert {}^C \! D^{\alpha}_tu'_m \Vert_{L^\infty(0,T;L^2(\Omega))} + C_4\Vert u_1 \Vert.\end{aligned}$$ In this manner, [\[aprioriII\]](#aprioriII){reference-type="eqref" reference="aprioriII"} enables us to estimate $u'_m$ in $L^\infty\left(0,T; L^2(\Omega) \right)$, finishing the a priori estimates. **Step 3: Passage to the limit - Existence.** The a priori estimates, give us $$\begin{aligned} &&\left( u_m \right)_{m \in \mathbb{N}} \textrm{ is bounded in } L^\infty \left(0,T;H^1_{\Gamma_0}(\Omega) \right); \\ &&\left( u'_m \right)_{m \in \mathbb{N}} \textrm{ is bounded in } L^\infty \left(0,T;L^{2}(\Omega) \right) \textrm{ and in } L^2 \left(0,T;H^1_{\Gamma_0}(\Omega) \right); \\ &&\left( {}^C \! D^{\alpha}_tu'_m \right)_{m \in \mathbb{N}} \textrm{ is bounded in } L^\infty \left(0,T;L^{2}(\Omega) \right);\\ &&\left( \delta_m \right)_{m \in \mathbb{N}} \textrm{ and } \left( \delta'_m \right)_{m \in \mathbb{N}} \textrm{ are bounded in } L^\infty \left(0,T;L^2(\Gamma_1) \right);\\ &&\left( \delta''_m \right)_{m \in \mathbb{N}} \textrm{ is bounded in }L^2 \left(0,T;L^2(\Gamma_1) \right).\end{aligned}$$ Therefore, with a standard procedure, we can find subsequences of $\left( u_m \right)_{m \in \mathbb{N}}$ and $\left( \delta_m \right)_{m \in \mathbb{N}}$, for which we still use the same notations, and functions $u$ and $\delta$ such that $$\begin{aligned} &&u_m \overset{\star}{\rightharpoonup} u \textrm{ in } L^\infty \left(0,T;H^1_{\Gamma_0}(\Omega) \right), \label{conv1} \\ &&u'_m \overset{\star}{\rightharpoonup} u' \textrm{ in } L^\infty \left(0,T; L^{2}(\Omega) \right) \textrm{ and } u'_m \rightharpoonup u' \textrm{ in } L^2 \left(0,T;H^1_{\Gamma_0}(\Omega) \right), \label{conv2} \\ &&\delta_m \overset{\star}{\rightharpoonup} \delta \, \, \mbox{and} \,\, \delta'_m \overset{\star}{\rightharpoonup} \delta'\,\, \text{ in } L^\infty \left(0,T;L^2(\Gamma_1) \right), \label{conv3}\\ &&\delta''_m \rightharpoonup \delta'' \,\, \mbox{in}\,\, L^{2}(0,T;L^{2}(\Gamma_{1}))\,.\label{conv4}\end{aligned}$$ Even more, by the continuity of the application $\gamma_0:H^1_{\Gamma_0}(\Omega) \to L^2(\Gamma_1)$ we have $$\label{fraca7} \gamma_0 (u'_m) \rightharpoonup \gamma_0 (u') \text{ in } L^2 \left(0,T;L^2(\Gamma_1) \right).$$ Besides the convergences in [\[conv1\]](#conv1){reference-type="eqref" reference="conv1"}-[\[fraca7\]](#fraca7){reference-type="eqref" reference="fraca7"}, we must also consider the boundedness of the sequence $\left( {}^C \! D^{\alpha}_tu'_m \right)_{m \in \mathbb{N}}$ in $L^\infty \left(0,T;L^{2}(\Omega) \right)$. It's worth noting that, due to the non-standard behavior of the Caputo fractional derivative, this argument is unconventional and requires additional justification. Since $\left( {}^C \! D^{\alpha}_tu'_m \right)_{m \in \mathbb{N}}$ is bounded in $L^\infty \left(0,T;L^{2}(\Omega) \right)$, we can extract a subsequence, denoted the same way, and find a function $v$ such that $$\label{caputoconv} {}^C \! D^{\alpha}_tu'_m \overset{\star}{\rightharpoonup} v \,\, \mbox{in}\,\, L^\infty \left(0,T;L^{2}(\Omega) \right).$$ The main task now is to prove that we can compute the Caputo fractional derivative of order $\alpha$ of $u'$ and that $v = {}^C \! D^{\alpha}_tu'$. At first, observe that from [\[caputoconv\]](#caputoconv){reference-type="eqref" reference="caputoconv"} and the definition of Caputo fractional derivative, for all $w \in H^1_{\Gamma_0}(\Omega)$ and $\theta \in \mathcal{D}(0,T)$, we have $$\int_0^T \frac{d}{dt} J^{1-\alpha}_t \left( u'_m(t) - u'_m(0), w \right) \theta(t) dt \to \int_0^T \left( v(t) , w \right) \theta(t) dt\,.$$ On the other hand, the continuity of the RL fractional integral, along with [\[help5\]](#help5){reference-type="eqref" reference="help5"} and [\[conv2\]](#conv2){reference-type="eqref" reference="conv2"}, enables us to deduce that $$\begin{gathered} \int_0^T \frac{d}{dt} J^{1-\alpha}_t \left( u'_m(t) - u'_m(0), w \right) \theta(t) dt \\= \int_0^T J^{1-\alpha}_t \left( u'_m(t) - u_{1m}, w \right) \theta'(t)\, dt \to \int_0^T J^{1-\alpha}_t \left( u'(t) - u_{1}, w \right) \theta'(t)\, dt.\end{gathered}$$ Hence, from the uniqueness of limit $$\int_0^T \frac{d}{dt} J^{1-\alpha}_t \left( u'(t) - u_1, w \right) \theta(t)\, dt = \int_0^T \left( v(t), w \right) \theta(t)\,dt\,,$$ for all $w \in H^1_{\Gamma_0}(\Omega)$ and $\theta \in \mathcal{D}(0,T).$ This identity shows that $D^\alpha_t(u'-u_1)=v$ in $L^\infty(0,T;L^2(\Omega))$. Finally, applying Theorem [Proposition 4](#teoderivcont){reference-type="ref" reference="teoderivcont"} we get that $u^{\prime} \in C([0,T]; L^{2}(\Omega))$ and $u'(0)=u_1$ in $L^{2}(\Omega)$. From this we may conclude that ${}^C \! D^{\alpha}_tu'(t)=v(t)$, for a.e. $t\in(0,T)$, and therefore we have $$\label{caputoconv1} {}^C \! D^{\alpha}_tu'_m \overset{\star}{\rightharpoonup} {}^C \! D^{\alpha}_tu' \,\, \mbox{in}\,\, L^\infty \left(0,T;L^{2}(\Omega) \right),$$ as we wanted. Now, we can take the limit in the approximate problem [\[eqap2\]](#eqap2){reference-type="eqref" reference="eqap2"}-[\[eqap3\]](#eqap3){reference-type="eqref" reference="eqap3"}. The approximate equation [\[eqap2\]](#eqap2){reference-type="eqref" reference="eqap2"} and the convergences [\[conv1\]](#conv1){reference-type="eqref" reference="conv1"}, [\[conv3\]](#conv3){reference-type="eqref" reference="conv3"}, and [\[caputoconv1\]](#caputoconv1){reference-type="eqref" reference="caputoconv1"} yield $$\label{acopla1} \left( {}^C \! D^{\alpha}_tu'(t), \varphi \right) + \left( \left( u(t), \varphi \right) \right) - \left( \delta'(t), \gamma_0 (\varphi) \right)_{\Gamma_1} =0,$$ for every $\varphi \in H^1_{\Gamma_0}(\Omega)$ and a.e. $t \in (0,T)$. In particular, $$\left\langle - \Delta u(t), \varphi \right\rangle_{\mathcal{D}'(\Omega) \times \mathcal{D}(\Omega)} = \left( \left( u (t) , \varphi \right) \right) = -\left( {}^C \! D^{\alpha}_tu'(t), \varphi \right),\, \mbox{for all } \varphi \in \mathcal{D}(\Omega)\,,$$ that implies $$\label{igualdadeqsu} - \Delta u(t) = -{}^C \! D^{\alpha}_tu'(t) \,\,\, \mbox{in} \, L^{2}(\Omega) \,,\, \, \mbox{for a.e.}\, t \in (0,T),$$ since ${}^C \! D^{\alpha}_tu'(t) \in L^{2}(\Omega)\,\, \mbox{for a.e.}\, t \in (0,T).$ This proves [\[id12\]](#id12){reference-type="eqref" reference="id12"} and shows that $u(t)\in \mathcal{H}_{\Delta}(\Omega)$ for a.e. $t \in (0,T).$ Taking into account [\[eqap3\]](#eqap3){reference-type="eqref" reference="eqap3"} and the convergences [\[conv3\]](#conv3){reference-type="eqref" reference="conv3"}-[\[fraca7\]](#fraca7){reference-type="eqref" reference="fraca7"} we conclude $$\left(f \delta''(t) + g\delta'(t) +h\delta(t) +\gamma_0 (u'(t)), \psi \right)_{\Gamma_1} =0, \quad \forall \psi \in L^2(\Gamma_1), \text{ a.e. in } (0,T).$$ Consequently, by Du Bois-Raymond Lemma, $$\label{igualdadeqsdelta} f\delta''(t) + g\delta'(t) +h\delta(t) +\gamma_0 (u'(t)) =0 \text{ in } L^{2}(\Gamma_{1}), \text{ for a.e. } t \in (0,T),$$ and [\[id13\]](#id13){reference-type="eqref" reference="id13"} is proved. In order to prove [\[id14\]](#id14){reference-type="eqref" reference="id14"} we multiply ([\[igualdadeqsu\]](#igualdadeqsu){reference-type="ref" reference="igualdadeqsu"}) by $\varphi \in H^1_{\Gamma_0}(\Omega)$ and integrate over $\Omega$, that leads to $$\left( {}^C \! D^{\alpha}_tu'(t), \varphi \right) - \left(\Delta u(t), \varphi \right) =0, \quad \text{for a.e. } t \in (0,T).$$ Therefore, the generalized Green identity gives $$\label{acopla2} \left( {}^C \! D^{\alpha}_tu'(t), \varphi \right) + \left( \left( u(t), \varphi \right) \right) - \left\langle \gamma_1(u(t)) , \gamma_0(\varphi) \right\rangle_{H^{-\frac{1}{2}}(\Gamma) \times H^{\frac{1}{2}}(\Gamma)} =0,$$ for almost every $t \in (0,T)$ and for every $\varphi \in H^1_{\Gamma_0}(\Omega)$. We compare ([\[acopla1\]](#acopla1){reference-type="ref" reference="acopla1"}) and ([\[acopla2\]](#acopla2){reference-type="ref" reference="acopla2"}) to conclude [\[id14\]](#id14){reference-type="eqref" reference="id14"}. To verify that the functions $u$ and $\delta$ satisfy the initial conditions [\[id15\]](#id15){reference-type="eqref" reference="id15"} and [\[id16\]](#id16){reference-type="eqref" reference="id16"} we first observe that we have already proven $u^{\prime}(0)=u_{1}$ in $H^{1}_{\Gamma_{0}}(\Omega)$. The regularity of $u$, given by [\[id10\]](#id10){reference-type="eqref" reference="id10"}, implies $u \in C([0,T]; H^{1}_{\Gamma_{0}}(\Omega))$ and the convergences [\[help4\]](#help4){reference-type="eqref" reference="help4"}, [\[conv1\]](#conv1){reference-type="eqref" reference="conv1"} and [\[conv2\]](#conv2){reference-type="eqref" reference="conv2"} yields $u(0)=u_{0}.$ Similarly we have $\delta \in C([0,T]; L^{2}(\Gamma_{1}))$ and from convergences [\[help7\]](#help7){reference-type="eqref" reference="help7"}, [\[conv3\]](#conv3){reference-type="eqref" reference="conv3"} and [\[conv4\]](#conv4){reference-type="eqref" reference="conv4"} we obtain $\delta^{\prime}(0) = \gamma_{1}(u_{0})\vert_{\Gamma_{1}}$ in $L^{2}(\Omega)$. This ends Step 3. **Step 4: Uniqueness.** Let $(u,\delta)$ and $(v,\varrho)$ be two pairs of functions satisfying [\[id10\]](#id10){reference-type="eqref" reference="id10"}-[\[id16\]](#id16){reference-type="eqref" reference="id16"}. Putting $w=u-v$ and $\zeta = \delta - \varrho$, we observe that the pair $(w,\zeta)$ has the regularity described in [\[id10\]](#id10){reference-type="eqref" reference="id10"}, [\[id11\]](#id11){reference-type="eqref" reference="id11"} and satisfies $$\begin{aligned} && {}^C \! D^{\alpha}_tw_t(t) - \Delta w(t) = 0\, \,\,\mbox{in} \,\, L^{2}(\Omega)\,, \,\, \mbox{for a.e.}\,\, t \in (0,T); \label{unicidade1} \\ && f \zeta_{tt}(t) + g \zeta_t(t) + h \zeta(t) = - \gamma_{0}(w_t(t)) \textrm{ in } L^{2}(\Gamma_{1}), \textrm{ for a.e. } t \in (0,T); \label{unicidade2} \\ &&\int_{\Gamma_1} \zeta_t(t) \gamma_0(\varphi) d \Gamma_1 = \left\langle \gamma_1(w(t)) ,\gamma_0(\varphi) \right\rangle_{H^{-\frac{1}{2}}(\Gamma) \times H^{\frac{1}{2}}(\Gamma) }, \nonumber \\ && \textrm{ for all } \varphi \in H^1_{\Gamma_0}(\Omega)\,\, \textrm{ and a.e. } \in (0,T); \label{unicidade3}\\ && w(0) = w_t(0)= 0 \textrm{ in } L^2(\Omega) \textrm{ and } \zeta(0) = \zeta_t(0) = 0 \textrm{ in } L^2(\Gamma_1). \label{unicidade4}\end{aligned}$$ Taking the $L^{2}(\Omega)$ inner product, in both sides of [\[unicidade1\]](#unicidade1){reference-type="eqref" reference="unicidade1"}, by $2 w'(t)$; using generalized Green$^{\prime}$s formula and considering [\[unicidade3\]](#unicidade3){reference-type="eqref" reference="unicidade3"} we obtain $$2 \left( {}^C \! D^{\alpha}_tw'(t), w'(t) \right) + 2\left( \left( w(t), w'(t) \right) \right) - 2 \left( \zeta'(t) , \gamma_{0}(w'(t)) \right)_{\Gamma_1} = 0\,.$$ This equality, after taking the $L^{2}(\Gamma_{1})$ inner product, in both sides of [\[unicidade2\]](#unicidade2){reference-type="eqref" reference="unicidade2"}, by $2\zeta'(t)$, implies that $$\begin{gathered} 2\left( {}^C \! D^{\alpha}_tw'(t), w'(t) \right) + 2\left( \left( w(t), w'(t) \right) \right) +2\left(f \zeta''(t) , \zeta'(t) \right)_{\Gamma_1} \\ + 2\left(g \zeta'(t) , \zeta'(t) \right)_{\Gamma_1} + 2\left(h \zeta(t),\zeta'(t) \right)_{\Gamma_1} = 0. \label{unicidade5}\end{gathered}$$ From now on, we proceed as in Estimate 1 and obtain $$J^{1-\alpha}_t \left\vert w'(t) \right\vert^2 + \left\Vert w(t) \right\Vert^2+\left\vert \zeta'(t) \right\vert_{\Gamma_1}^2+ \left\vert \zeta(t)\right\vert_{\Gamma_1}^2 \leq 0\,, \mbox{for a.e.} \,\, t \in (0,T)\,,$$ where to arrive at the above inequality, we have considered the null initial conditions [\[unicidade4\]](#unicidade4){reference-type="eqref" reference="unicidade4"}. Thus we have $w(t) = 0$ in $H^{1}_{\Gamma_{0}}(\Omega)$ and $\zeta(t) =0$ in $L^{2}(\Gamma_{1})$ for a.e. $t \in (0,T)$, which show that the pairs of functions $(u, \delta)$ and $(v, \varrho)$ are equal, concluding the proof of the uniqueness. **Step 5: Continuous dependence.** We now say that the unique pair of function $(u,\delta)$, constructed in the previous steps, constitutes a solution to the problem [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}-[\[6prob\]](#6prob){reference-type="eqref" reference="6prob"}. With this step, we conclude the well-posedness of the problem [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}-[\[6prob\]](#6prob){reference-type="eqref" reference="6prob"} and demonstrate that the solution $(u,\delta)$ depends continuously on the parameters $f,\,\,g,\,\,h,\,\,u_0,\,\,u_1$ and $\delta_0.$ Initially we observe that from the weakly lower semicontinuity of the norms and estimates [\[aprioriI\]](#aprioriI){reference-type="eqref" reference="aprioriI"}, [\[aprioriII\]](#aprioriII){reference-type="eqref" reference="aprioriII"} and [\[aprioriIIa\]](#aprioriIIa){reference-type="eqref" reference="aprioriIIa"} we get $$\begin{aligned} \label{semicontinuidade} &&\left\Vert {}^C \! D^{\alpha}_tu' \right\Vert^2_{L^\infty(0,T; L^2(\Omega))} + \left\Vert u' \right\Vert^2_{L^\infty(0,T;L^2(\Omega))} + \left\Vert u' \right\Vert^2_{L^2 \left(0,T;H^1_{\Gamma_0}(\Omega) \right)} + \left\Vert u \right\Vert^2_{L^\infty \left(0,T;H^1_{\Gamma_0}(\Omega) \right)} \nonumber\\ &&+ \left\Vert \delta'' \right\Vert^2_{L^2(0,T;L^2(\Gamma_1))} + \left\Vert \delta' \right\Vert^2_{L^2(0,T;L^2(\Gamma_1))} + \left\Vert \delta \right\Vert^2_{L^\infty(0,T;L^2(\Gamma_1))} \nonumber\\ &&\leq K \left[ \left\Vert u_{0} \right\Vert^2_{H^1_{\Gamma_0}(\Omega) \cap H^2(\Omega) } + \left\Vert u_{1} \right\Vert^2 + \left\vert \delta_{0} \right\vert^2_{\Gamma_1} \right].\end{aligned}$$ On the other hand, let $(u,\delta)$ and $(\tilde{u},\tilde{\delta})$ be the solutions of the problem [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}-[\[6prob\]](#6prob){reference-type="eqref" reference="6prob"} associated to sets of parameters $\{f,g,h,u_0,u_1,\delta_0\}$ and $\{\tilde{f},\tilde{g},\tilde{h}, \tilde{u}_0,\tilde{u}_1,\tilde{\delta}_0\}$, respectively. We set $v=u-\tilde{u}$ and $\sigma=\delta-\tilde{\delta}$. Therefore, by taking inner products in equations [\[igualdadeqsu\]](#igualdadeqsu){reference-type="eqref" reference="igualdadeqsu"} and [\[igualdadeqsdelta\]](#igualdadeqsdelta){reference-type="eqref" reference="igualdadeqsdelta"}, as in the previous step, and carefully manipulating terms, we obtain $$\begin{gathered} 2 \left( {}^C \! D^{\alpha}_tv'(t),v'(t) \right) + 2 \left( \left( v(t),v'(t) \right) \right) + 2 (f \sigma''(t), \sigma'(t) )_{\Gamma_1} + 2 ([f-\tilde{f}] \tilde{\delta}''(t), \sigma'(t) )_{\Gamma_1} \\ + 2 (g \sigma'(t), \sigma'(t) )_{\Gamma_1} + 2 ([g-\tilde{g}] \tilde{\delta}'(t), \sigma'(t) )_{\Gamma_1} + 2 (h \sigma(t), \sigma'(t) )_{\Gamma_1} + 2 ([h-\tilde{h}] \tilde{\delta}(t), \sigma'(t) )_{\Gamma_1} = 0.\end{gathered}$$ Proceeding as in the first a priori estimate, we obtain $$\begin{gathered} {}^C \! D^{\alpha}_t\vert v'(t) \vert^2 + \frac{d}{dt} \left[ \Vert v(t) \Vert^2 + \vert f^{\frac{1}{2}} \sigma'(t) \vert^2_{\Gamma_1} +\vert h^{\frac{1}{2}} \sigma(t) \vert^2_{\Gamma_1} \right] \\ \leq 2 \big [ \Vert f - \tilde{f} \Vert_{C(\overline{\Gamma}_1)} + \Vert g - \tilde{g} \Vert_{C(\overline{\Gamma}_1)} + \Vert h - \tilde{h} \Vert_{C(\overline{\Gamma}_1)} \big ] \big [ \vert \tilde{\delta}''(t) \vert_{\Gamma_1} + \vert \tilde{\delta}'(t) \vert_{\Gamma_1} + \vert \tilde{\delta}(t) \vert_{\Gamma_1} \big ]\vert \sigma'(t) \vert_{\Gamma_1} \\ \leq C \big[ \Vert f - \tilde{f} \Vert_{C(\overline{\Gamma}_1)} + \Vert g - \tilde{g} \Vert_{C(\overline{\Gamma}_1)} + \Vert h - \tilde{h} \Vert_{C(\overline{\Gamma}_1)} \big],\end{gathered}$$ where in the last inequality we used [\[semicontinuidade\]](#semicontinuidade){reference-type="eqref" reference="semicontinuidade"}. Integrating over $(0,t)$ and applying [\[id3\]](#id3){reference-type="eqref" reference="id3"}, we obtain $$\begin{gathered} J^{1-\alpha}_t \vert v'(t) \vert^2 + \Vert v(t) \Vert^2 + \vert \sigma'(t) \vert^2_{\Gamma_1} + \vert \sigma(t) \vert^2_{\Gamma_1} \\ \leq C \left( \vert v'(0) \vert^2 + \Vert v(0) \Vert^2 + \vert \sigma'(0) \vert^2_{\Gamma_1} + \vert \sigma(0) \vert^2_{\Gamma_1} \right. \\ \left. + \Vert f - \tilde{f} \Vert_{C(\overline{\Gamma}_1)} + \Vert g - \tilde{g} \Vert_{C(\overline{\Gamma}_1)} + \Vert h - \tilde{h} \Vert_{C(\overline{\Gamma}_1)} \right).\end{gathered}$$ Finally, by [\[id7\]](#id7){reference-type="eqref" reference="id7"}, taking $t=T$ in the above inequality we conclude $$\begin{gathered} \Vert {}^C \! D^{\alpha}_t(u-\tilde{u}) \Vert_{L^\infty(0,T;L^2(\Omega))} + \Vert u-\tilde{u} \Vert^2_{C([0,T];H^1_{\Gamma_0}(\Omega))} + \Vert \delta-\tilde{\delta} \Vert^2_{C^1([0,T];L^2(\Gamma_1))} \\ \leq C \big( \Vert u_0-\tilde{u}_0 \Vert^2_{ H^1_{\Gamma_0}(\Omega)} \cap H^2(\Omega) + \Vert u_1-\tilde{u}_1 \Vert^2 + \vert \delta_0-\tilde{\delta}_0 \vert^2_{\Gamma_1} \\ + \Vert f - \tilde{f} \Vert_{C(\overline{\Gamma}_1)} + \Vert g - \tilde{g} \Vert_{C(\overline{\Gamma}_1)} + \Vert h - \tilde{h} \Vert_{C(\overline{\Gamma}_1)} \big).\end{gathered}$$ ◻ # Closing Remarks {#closingremarks} We have not yet discussed in this article the equation used to describe the time-fractional wave equation in [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}. It is important to note that our approach differs from the conventional one found in established literature (e.g., [@Ma1] and related references), which is represented by the equation $$\label{1'} \tag{1'} {}^C \! D_t^{1+\alpha} u(x,t) - \Delta u(x,t) = 0, \quad \textrm{in} \ \Omega \times (0,T),$$ where $0<\alpha<1$. These two formulations differ fundamentally because, as clarified in item $(v)$ of Proposition [Proposition 1](#prop1){reference-type="ref" reference="prop1"}, the equality ${}^C \! D_t^{1+\alpha} f(t) = {}^C \! D_t^{\alpha} f'(t)$, for a.e. $t\in[0,T]$, is valid only when the function $f$ possesses adequate regularity. In our case, for example, we cannot establish that $J_t^{1-\alpha}u(t) \in W^{2,1}(0,T;L^2(\Omega))$, which prevents us from applying [\[id6\]](#id6){reference-type="eqref" reference="id6"}. Nevertheless, the constructions and convergences we have achieved during our study are sufficient to demonstrate that our solution $u$ indeed satisfies [\[1\'\]](#1'){reference-type="eqref" reference="1'"}. Let us begin by considering the approximated solutions $u_m$ of our main theorem, $\varphi \in H^1_{\Gamma_0}(\Omega)$ and $\theta \in \mathcal{D}(0,T)$. Then we have $$\begin{gathered} \int_0^T \left( {}^C \! D_t^{1+\alpha} u_m(t), \varphi \right) \theta(t) dt = \int_0^T \frac{d^2}{dt^2} J^{1-\alpha}_t \left( u_m(t) - u_m(0)-tu'_m(0) , \varphi \right) \theta(t) dt \\ = \int_0^T J^{1-\alpha}_t \left( u_m(t) - u_m(0)-tu'_m(0) , \varphi \right) \theta''(t) dt.\end{gathered}$$ Additionally, since $u_m$ is sufficiently regular, by [\[id6\]](#id6){reference-type="eqref" reference="id6"} we have $$\int_0^T \left( {}^C \! D_t^{1+\alpha} u_m(t), \varphi \right) \theta(t)dt = \int_0^T \left( {}^C \! D^{\alpha}_tu'_m(t), \varphi \right) \theta(t)dt.$$ In this context, we observe from the convergences on Step 3 and the continuity of the RL fractional integral of order $\alpha$ that $$\int_0^T \left( {}^C \! D^{\alpha}_tu'_m(t), \varphi \right) \theta(t)dt \to \int_0^T \left( {}^C \! D^{\alpha}_tu'(t), \varphi \right) \theta(t)dt$$ and $$\begin{gathered} \int_0^T J^{1-\alpha}_t \left( u_m(t) - u_m(0)-tu'_m(0) , \varphi \right) \theta''(t) dt \\ \to \int_0^T J^{1-\alpha}_t \left( u(t) - u(0)-tu'(0) , \varphi \right) \theta''(t) dt.\end{gathered}$$ Therefore, the uniqueness of the limit implies that $${}^C \! D_t^{\alpha +1} u(x,t)= {}^C \! D^{\alpha}_tu'(x,t),$$ for almost every $(x,t)\in\Omega\times(0,T)$. As a result, we are further investigating, after establishing the appropriate solution concepts, the existence and uniqueness of a strong solution to problem [\[2prob\]](#2prob){reference-type="eqref" reference="2prob"}-[\[6prob\]](#6prob){reference-type="eqref" reference="6prob"} while considering [\[1\'\]](#1'){reference-type="eqref" reference="1'"} in place of [\[1prob\]](#1prob){reference-type="eqref" reference="1prob"}. 99 A.A. Alcântara, B.A. Carmo, H.R. Clark, et al., Nonlinear wave equation with Dirichlet and acoustic boundary conditions: theoretical analysis and numerical simulation. Comp. Appl. Math. 41 (141) (2022), https://doi.org/10.1007/s40314-022-01822-5. A. Alsaedi, B. Ahmad, M. Kirane, Maximum principle for certain generalized time and space fractional diffusion equations, Quart. Appl. Math. 73 (2015) 163-175, https://doi.org/10.1090/S0033-569X-2015-01386-2. W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2011. E.G. Bazhlekova, Fractional evolution equations in Banach spaces. Ph.D. Thesis, Mathematics and Computer Science, Technische Universiteit Eindhoven (2001), https://doi.org/10.6100/IR549476 J.T. Beale, Spectral properties of an acousticboundary condition, Indiana Univ. Math. J. 25 (1976) 895-917, https://doi.org/10.1512/iumj.1976.25.25071. J.T. Beale, S.I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc. 80 (6) (1974) 1276-1278, https://doi.org/10.1090/S0002-9904-1974-13714-6. P. Braz e Silva, H.R. Clark, C.L. Frota, On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions, Comput. Appl. Math. 36 (2017) 397-414, https://doi.org/10.1007/s40314-015-0236-1. P.M. Carvalho-Neto, Fractional Differential Equations: a novel study of local and global solutions in Banach spaces, Ph.D. Thesis, Universidade de São Paulo, São Carlos (2013). P.M. Carvalho-Neto, R. Fehlberg-Júnior, On the fractional version of Leibniz rule, Math. Nachr. 293 (2020) 670-700, https://doi.org/10.1002/mana.201900097. P.M de Carvalho-Neto, R. Fehlberg-Júnior, The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I. Commun. Pure Appl. Anal. 21 (11) (2022) 3667-3700, https://doi.org/10.3934/cpaa.2022118. A. Cartea, D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Phys. A: Stat. Mech. Appl. 374 (2) (2007) 749-763, https://doi.org/10.1016/j.physa.2006.08.071. D. del Castillo-Negrete, B. Carreras, V. Lynch, Nondiffusive transport in plasma turbulence: a fractional diffusion approach, Phys. Rev. Lett. 94 (6) (2005) 065003, https://doi.org/10.1103/PhysRevLett.94.065003. M.M. Cavalcanti, V.N. Domingos Cavalcanti, C.L. Frota, A. Vicente, Stability for semilinear wave equation in an inhomogeneous medium with frictional localized damping and acoustic boundary conditions, SIAM J. on Control Optim. 58 (4) (2020) 2411-2445, https://doi.org/10.1137/19M1267556. V. Daftardar-Gejji, A. Babakhani, Analysis of a system of fractional differential equations, J. Math. Anal. Appl. 293 (2004) 511?522. V. Daftardar-Gejji, H. Jafari, Boundary value problems for fractional diffusion-wave equation, Aust. J. Math. Anal. Appl. 3 (1) (2006) 1-8. T.J. Feder, I. Brust-Mascher, J.P. Slattery, B. Baird, W.W. Webb, Constrained diffusion or immobile fraction on cell surfaces: a new interpretation, Biophys. J. 70 (1996) 2767-2773, https://doi.org/10.1016/S0006-3495(96)79846-6. S. Frigeri, Attractors for semilinear damped wave equations with an acoustic boundary condition, J. Evol. Equ. 10 (2010) 29-58, https://doi.org/10.1007/s00028-009-0039-1. C.L. Frota, A.T. Cousin, N.A. Larkin, On a system of Klein-Gordon type equations with acousticboundary conditions, J. Math. Anal. Appl. 293 (1) (2004) 293-309, https://doi.org/10.1016/j.jmaa.2004.01.007. C.L. Frota, J.A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, J. Differ. Equ. 164 (2000) 92-109, https://doi.org/10.1006/jdeq.1999.3743. C.L. Frota, L.A. Medeiros, A. Vicente, Wave equation in domains with non-locally reacting boundary. Differ. Integral Equ. 24 (11-12) (2011) 1001-1020, http://doi.org/10.57262/die/1356012872. L.R. González-Ramírez, Fractional-order traveling wave approximations for a fractional-order neural field model. Front. Comput. Neurosci. 16 (2022) 788924, https://doi.org/10.3389/fncom.2022.788924. P.J. Graber, B. Said-Houari, On the wave equation with semilinear porous acoustic boundary conditions, J. Diff. Equ. 252 (2012) 4898-4941, https://doi.org/10.1016/j.jde.2012.01.042. G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals. I., Math. Z. 27 (1928), 565-606, https://doi.org/10.1007/BF01171116. X. Huang, M. Yamamoto, Well-posedness of initial-boundary value problem for time-fractional diffusion-wave equation with time-dependent coefficients, arXiv preprint, arXiv:2203.10448, 2022, https://doi.org/10.48550/arXiv.2203.10448. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Y. Kobayashi, N. Tanaka, An application of semigroups of locally Lipschitz operators to Carrier equations with acoustic boundary conditions, J. Math. Anal. Appl. 338 (2008) 852-872, https://doi.org/10.1016/j.jmaa.2007.05.062. J. Limaco, H.R. Clark, C.L. Frota, L.A. Medeiros, On an evolution equation with acoustic boundary conditions. Math. Methods Appl. Sci. 34 (2011) 2047-2059, https://doi.org/10.1002/mma.1503. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solit. Fractals 7 (9) (1996) 1461-1477, https://doi.org/10.1016/0960-0779(95)00125-5. R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Reports 339 (1) (2000) 1-77, https://doi.org/10.1016/S0370-1573(00)00070-3. P.M. Morse, K.U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968. A. Mura, G. Pagnini, Characterization and simulations of a class of stochastic processes to model anomalous diffusion J. Phys. A: Math. Theor. 41 (2008) 285003, http://doi.org/10.1088/1751-8113/41/28/285003. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1998. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York, 1993. E. Zeidler, Nonlinear Functional Analysis and its Aplications II/A: Linear Monotone Operators, Springer, New York, 1990. [^1]: $^\ast$Corresponding author: pg54864\@uem.br

arxiv_math

--- abstract: | Let $M$ be a non-zero binary matrix with distinct rows where the rows are closed under certain logical operators. In this article, we investigate the existence of columns containing an equal or greater number of ones than zeros. Specifically, the existence of such columns when the rows of the matrix are closed under *material conditional* leads us to a weaker version of the *Union-Closed Set Conjecture*. author: - | Mohammad Javad Moghaddas Mehr\ Email: m.moghadas11235\@gmail.com date: September 4, 2023 title: A Note on the Union-closed Sets Conjecture --- **Keywords:** union-closed sets conjecture, binary matrix, logical operators, material conditional # Introduction #### For an integer, $n \in \mathbb{N},$ we define the set $[n]$ as $\{k \in \mathbb{N} \, | \, k \leq n\}.$ A family $\mathscr{F} \subseteq 2^{[n]}$ is called union-closed if $\mathcal{A}, \mathcal{B} \in \mathscr{F}$ implies that $\mathcal{A} \cup \mathcal{B} \in \mathscr{F}.$ In 1979, Péter Frankl [@frankl1995] posed a seemingly straightforward conjecture that, over four decades later, still stands as an unsolved puzzle despite substantial efforts and progress dedicated to its understanding. The conjecture states as follows. **Conjecture 1**. *For every non-empty finite union-closed family of sets, there exists an element that belongs to at least half of the sets in the family.* #### Efforts to understand the conjecture have led to notable breakthroughs. We recommend the survey paper by Bruhn and Schaudt [@bruhn2015] for details. Vuckovic and Zivkovic [@vuckovic2012] demonstrated the conjecture's validity for any union-closed family $\mathscr{F} \subseteq 2^{[n]}$ where $n \leq 12.$ Roberts and Simpson [@roberts2010] established that if $q$ is the minimum cardinality of $\bigcup \mathscr{F}$ taken over all counterexamples $\mathscr{F},$ then any counterexample $\mathscr{F}$ has cardinality at least $4q - 1.$ #### In 2017, Ilan Karpas [@karpas2017] showed that there exists some absolute constant $c > 0$ such that for any union-closed family $\mathcal{F} \subseteq 2^{[n]},$ if $|\mathcal{F}| \geq \left(\frac{1}{2} - c\right)2^n,$ then there is an element $i \in [n]$ that appears in at least half of the sets in $\mathcal{F}.$ #### Recently, Gilmer [@gilmer2022] established the first constant lower bound for the conjecture, asserting the existence of an element that belongs to at least 0.01 of the sets in the family. Gilmer claimed that his method could enhance the lower bound to $\left(3 - \sqrt{5}\right)/2 \approx 0.38.$ A few days later, three preprints were published and verified his claim [@chase2022; @sawin2022; @alweiss2022]. #### In this note, we explore binary matrices whose rows are closed under different logical operators. To do so, we need some fundamental definitions. **Definition 1**. *Let $A = (a_{ij})$ and $B = (b_{ij})$ be two binary matrices of size $n \times m,$* 1. *Let $\neg A:= (\neg a_{ij}).$* 2. *Let $A * B := (a_{ij} * b_{ij})$ for binary operator \"$*$\".* #### Let $M = (m_{ij})$ be a matrix. We denote $M_{i-}$ and $M_{-j}$ as the rows and columns of $M,$ respectively. From this perspective, we can restate Conjecture [Conjecture 1](#conj:re7){reference-type="ref" reference="conj:re7"} as follows. **Conjecture 2**. *Let $M = (m_{ij})$ be a non-zero binary matrix of size $n \times m$ with distinct rows. If for any arbitrary pair of rows $M_{i-}$ and $M_{j-}$ we have $M_{i-} \lor M_{j-}$ as a row of $M,$ then there exists a column $M_{-k}$ that contains at least $n/2$ ones.* #### Finally in Section 7 we will establish a weaker version of Conjecture [Conjecture 2](#conj:re6){reference-type="ref" reference="conj:re6"} by proving the following theorem. **Theorem 1**. *Let $M = (m_{ij})$ be a non-zero binary matrix of size $n \times m$ with distinct rows. If for any arbitrary pair of rows $M_{i-}$ and $M_{j-}$ we have $\neg M_{i-} \lor M_{j-}$ as a row of $M,$ then there exists a column $M_{-k}$ that contains at least $n/2$ ones.* #### Let $\mathcal{M}$ be the set of all binary matrices, and denote $\mathcal{N}$ be the set of all finite subsets of natural numbers. Define $\psi : \mathcal{M} \to \mathcal{N}$ as follows $$\psi(M_{n \times m}) = \left\{\sum_{i=1}^{n} m_{ij} \, | \, j \in [m]\right\}.$$ **Example 1**. *Let $\mathscr{A}$ be a family of sets as follows $$\mathscr{A}=\{\varnothing ,\{1\},\{1,2\},\{2,3,4\},\{1,2,3,4\}\}$$* *which is union-closed.* *The matrix representation of $\mathscr{A}$ will be $$A = \begin{pmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0\\ 0 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \end{pmatrix}$$\ then $\psi(A) = \{2,3\},$ so $max(\psi(A)) \geq 5/2.$ Thus, Conjecture [Conjecture 2](#conj:re6){reference-type="ref" reference="conj:re6"} holds in this case.* #### If Conjecture [Conjecture 2](#conj:re6){reference-type="ref" reference="conj:re6"} holds for matrix $A,$ then any row and column permutation of $A$ also satisfies the $\text{Conjecture's}$ conditions. **Definition 2**. *Matrix $A$ is equivalent to matrix $B,$ denoted as $A \sim B,$ if $B$ can be obtained from $A$ through row and column permutations.* #### We refer to the pair $(M,*)$ as a space, where $M = (m_{ij})$ is a non-zero binary matrix of size ${n \times m},$ such that its rows are closed under \"\*\" operator. This operator can be either binary or unary. # Negation ($\neg$) #### In this section, we will prove our first lemma, which serves as a useful tool and provides valuable insights for dealing with other cases. **Negation Lemma 1**. *For every space $(M_{n \times m},\neg)$ we have $max(\psi(M)) \geq n/2.$* *Proof.* Without loss of generality, assume that $M_{-j} \neq 0$ for each $j \in [m].$ Define two sets $\mathcal{K}$ and $\mathcal{L}$ as follows $$\begin{aligned} \mathcal{K} &:= \{M_{i-} \,|\, m_{i1}=1\} \\ \mathcal{L} &:= \{M_{i-} \,|\, m_{i1}=0\}. \end{aligned}$$ If $\mathcal{L} = \varnothing$ there is nothing to prove, so let $\mathcal{L} \neq \varnothing.$ Define matrix $K$ such that its rows come from $\mathcal{K}$ while preserving their order in $M.$ We define matrix $L$ from $\mathcal{L}$ in the same manner, then $$M \sim \begin{pmatrix} K \\ L \\ \end{pmatrix}.$$ For any row $K_{t-}$ there is exactly one row $L_{s-}$ such that $K_{t-} = {\neg}L_{s-}$, and vice versa. This establishes a mutual correspondence between the rows of $K$ and the rows of $L$, thus $$K \sim {\neg}L$$ which forces $M$ to have an even number of rows and an equal number of ones and zeros in each column. So $n=2k$ for some $k \in \mathbb{N}$, then $\psi(M) = \{k\}$. Thus $$\max(\psi(M)) = k.$$ ◻ # Alternative denial ($\uparrow$) and Joint denial ($\downarrow$) #### In this section, we will utilize Lemma [Negation Lemma 1](#lem:negation){reference-type="ref" reference="lem:negation"} to explore the properties of two binary operators, Alternative Denial and Joint Denial. **Proposition 1**. *For every space $(M_{n \times m},\uparrow)$ we have $\max(\psi(M)) \geq n/2$.* *Proof.* Since rows of $M$ are closed under \"$\uparrow$\", then for an arbitrary $M_{i-}$ we have $$M_{i-} \uparrow M_{i-} = {\neg}M_{i-}$$ which means rows of $M$ are closed under \"$\neg$\". By applying the Lemma [Negation Lemma 1](#lem:negation){reference-type="ref" reference="lem:negation"}, we can conclude $\max(\psi(M)) \geq n/2$. ◻ **Proposition 2**. *For every space $(M_{n \times m},\downarrow)$ we have $\max(\psi(M)) \geq n/2$.* *Proof.* Since rows of $M$ are closed under \"$\downarrow$\", then for an arbitrary $M_{i-}$ we have $$M_{i-} \downarrow M_{i-} = {\neg}M_{i-}$$ which means rows of $M$ are closed under \"$\neg$\". By applying the Lemma [Negation Lemma 1](#lem:negation){reference-type="ref" reference="lem:negation"}, we can conclude $\max(\psi(M)) \geq n/2$. ◻ # Abjunction ($\not\rightarrow$) and Conjunction ($\land$) #### In this section, we provide two examples to demonstrate that when rows are closed under or , it does not guarantee the existence of a column with an equal or greater number of ones than zeros. **Example 2**. *Let $A$ be the binary matrix* *$$A_{(n+1) \times n} = \begin{pmatrix} I_{n \times n} \\ 0_{1 \times n} \end{pmatrix}$$* *then for $n>1$* *$$\max(\psi(A)) = 1 < (n+1)/2$$* *but the rows of matrix $A$ are closed under and .* #### For an arbitrary $k \in \mathbb{N},$ there exists a matrix $A$ such that its rows are closed under and $\max(\psi(A)) = k+1.$ **Example 3**. *Let $A$ be the binary matrix* *$$A_{(n+2) \times (n+1)} = \begin{pmatrix} \begin{array}{c|c} \begin{matrix} J_{k \times 1} \\ 0_{(n-k) \times 1} \\ \end{matrix} & I_n \\ \hline \begin{matrix} 1 \\ 0 \\ \end{matrix} & 0_{2 \times n} \end{array} \end{pmatrix}$$* *so $\max(\psi(A)) = k+1.$ For instance, for $k=2$* *$$A_{7 \times 6} = \begin{pmatrix} \begin{array}{c|c} \begin{matrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ \end{matrix} & \begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{matrix} \\ \hline \begin{matrix} 1 \\ 0 \\ \end{matrix} & \begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{matrix} \end{array} \end{pmatrix}$$* *thus* *$$\max(\psi(M)) = 3 < 7/2.$$* # Biconditional ($\leftrightarrow$) and Exclusive or ($\not\leftrightarrow$) #### In this section, we will explore two logical operators that form an additive group on rows of a binary matrix. **Proposition 3**. *For every space $(M_{n \times m},\leftrightarrow)$ we have $\max(\psi(M)) \geq n/2.$* *Proof.* Define matrices $K$ and $L,$ as in the proof of the Lemma [Negation Lemma 1](#lem:negation){reference-type="ref" reference="lem:negation"}, so $$M \sim \begin{pmatrix} K \\ L \\ \end{pmatrix}.$$ Define the function $g$ as follows $$\begin{aligned} g: \mathcal{L} &\to \mathcal{K} \\ g(L_{t-}) &= L_{1-}+L_{t-}.\end{aligned}$$ If $g(A) = g(B)$ then $L_{1-} + A = L_{1-} + B.$ Since $\{M_{i-}\}_{i \in [n]}$ is an additive group, then $A = B.$ So $g$ is injective, and as a result, $$|\mathcal{L}| \leq |\mathcal{K}|.$$ Thus $$\begin{aligned} \max(\psi(M)) &\geq (|\mathcal{K}|+|\mathcal{L}|)/2 \\ &= n/2. \end{aligned}$$ ◻ **Proposition 4**. *For every space $(M_{n \times m},\not\leftrightarrow)$ we have $\max(\psi(M)) \geq n/2.$* *Proof.* Define matrices $K$ and $L,$ as in the proof of the Lemma [Negation Lemma 1](#lem:negation){reference-type="ref" reference="lem:negation"} , so $$M \sim \begin{pmatrix} K \\ L \\ \end{pmatrix}.$$ Define the function $g$ as follows $$\begin{aligned} g: \mathcal{L} &\to \mathcal{K} \\ g(L_{t-}) &= K_{1-}+L_{t-}. \end{aligned}$$ If $g(A) = g(B)$ then $K_{1-} + A = K_{1-} + B.$ Since $\{M_{i-}\}_{i \in [n]}$ is an additive group, then $A = B.$ So $g$ is injective, and as a result, $$|\mathcal{L}| \leq |\mathcal{K}|.$$ Thus $$\begin{aligned} \max(\psi(M)) &\geq (|\mathcal{K}|+|\mathcal{L}|)/2 \\ &= n/2. \end{aligned}$$ ◻ # Topology ($\cup, \cap$) #### If $\mathscr{F} \subseteq 2^{[n]}$ is a family of sets that is closed under \"$\cup$\" and \"$\cap$\", it implies that we may define a topology on $\bigcup \mathscr{F}.$ However, being closed under \"$\cup$\" and \"$\cap$\" doesn't guarantee that $\varnothing \in \mathscr{F}.$ Nevertheless, if $\bigcap \mathscr{F} \neq \varnothing,$ then there exists an $a \in \bigcup \mathscr{F}$ such that it belongs to each member of $\mathscr{F}.$ **Theorem 2**. *Let $(X, \tau)$ be a finite topological space, then there exists an $x_0 \in X$ that appears in at least half of the members of $\tau$.* *Proof.* Without loss of generality, assume $X = [n]$ for some $n \in \mathbb{N}$. Let $\mathscr{X}$ be a family of members of $\tau$ with the minimum cardinality among the members of $\tau$, excluding the empty set. We select the smallest set $\mathcal{B} \in \mathscr{X}$ according to the lexicographic order. It is guaranteed that for $\mathcal{A} \in \tau$, we have either $\mathcal{B} \subseteq \mathcal{A}$ or $\mathcal{B} \cap \mathcal{A} = \varnothing$. Define sets $\mathcal{K}$ and $\mathcal{L}$ as follows $$\begin{aligned} \mathcal{K} &= \{\mathcal{A} \in \tau \,|\, \mathcal{B} \subseteq \mathcal{A}\} \\ \mathcal{L} &= \{\mathcal{A} \in \tau \,|\, \mathcal{B} \cap \mathcal{A} = \varnothing\}. \end{aligned}$$ If $\mathcal{L} = \varnothing$, there is nothing to prove, so let $\mathcal{L} \neq \varnothing$. Define the function $g$ as follows $$\begin{aligned} g: \mathcal{L} &\to \mathcal{K} \\ \quad g(\mathcal{A}) &= \mathcal{B} \cup \mathcal{A}. \end{aligned}$$ If $g(\mathcal{A}) = g(\mathcal{C})$, then $\mathcal{A} \cup \mathcal{B} = \mathcal{C} \cup \mathcal{B}$. Since $\mathcal{B}$ has an empty intersection with $\mathcal{A}$ and $\mathcal{C}$, then $\mathcal{A} = \mathcal{C}$. So $g$ is injective, and as a result, $|\mathcal{L}| \leq |\mathcal{K}|$, thus members of $\mathcal{B}$ appear in at least half of the members of $\tau$. ◻ **Corollary 1**. *Let $M$ be a non-zero binary matrix of size $n \times m$, where its rows are closed under \"$\land$\" and \"$\lor$\", then $\max(\psi(M)) \geq \frac{n}{2}$.* # Material Conditional ($\rightarrow$) #### In this section, we will establish our main result. Before proceeding, we need some tools that will help us move forward more easily. #### Let $M_{n \times m}$ be a non-zero binary matrix. We define $\widetilde{M}_{n \times m}={\neg}M.$ If rows of $M$ are closed under the binary operator \"\*\", then rows of $\widetilde{M}$ would also be closed under the binary operator \"$\widetilde{*}$\" defined as follows $$\widetilde{*}(\widetilde{A},\widetilde{B}) := {\neg}*(A,B)$$ where $\widetilde{A}, \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}$ and $A, B \in \{M_{i-}\}_{i \in [n]}.$ **Example 4**. *Let $(M_{n \times m},*)$ be a non-zero space such that for $A, B \in \{M_{i-}\}_{i \in [n]},$ $*(A,B) = {\neg}A \lor B.$ Then for $\widetilde{A}, \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}$ $$\widetilde{*}(\widetilde{A},\widetilde{B}) = {\neg}*(A,B) = {\neg}({\neg}A \lor B) = {\neg}\widetilde{A} \land \widetilde{B}.$$* **Definition 3**. *Let $\mathscr{F}$ be a non-empty finite family of sets. A *basis* for the family $\mathscr{F}$ is a disjoint collection $\mathscr{B} = \{\mathcal{B}_i\}_{i \in \gamma}$ of members of $\mathscr{F}$ such that for every member $\mathcal{A}$ in $\mathscr{F}$, there exists an index set $\alpha \subseteq \gamma$ such that $\mathcal{A} = \bigcup_{i \in \alpha} \mathcal{B}_i$.* **Definition 4**. *Let $M_{n \times m}$ be a non-zero binary matrix. A *basis* for the matrix $M$ is a collection $\mathcal{V} = \{v_i\}_{i \in \gamma}$ of rows of $M$, where $v_t$ and $v_k$ are orthogonal for $t \neq k$, such that for every row $A$ of $M$, there exists an index set $\alpha \subseteq \gamma$ such that $A = \bigvee_{i \in \alpha} v_i$.* **Proposition 5**. *Let $\mathscr{F}$ be a finite family of sets. If $\mathscr{F}$ has a basis, then it is *unique*.* *Proof.* Suppose $\mathscr{F}$ has two different bases $\mathscr{B} = \{\mathcal{B}_i\}_{i \in \gamma}$ and $\mathscr{C} = \{\mathcal{C}_i\}_{i \in \lambda}$. Without loss of generality, let $\mathcal{C}_t \in \mathscr{C}$ such that $\mathcal{C}_t \notin \mathscr{B}$, so $$\mathcal{C}_t = \bigcup_{j \in \alpha} \mathcal{B}_j,$$ where $\mathcal{B}_j \neq \mathcal{C}_t$ and $$\mathcal{B}_j = \bigcup_{i \in \beta_j} \mathcal{C}_{i}$$ where $\mathcal{C}_i \neq \mathcal{C}_t.$ Then $$\mathcal{C}_t = \bigcup_{j \in \alpha} \bigcup_{i \in \beta_j} \mathcal{C}_{i} = \bigcup_{i \in \boldsymbol{\cup}_{j \in \alpha} \beta_j}\mathcal{C}_{i}$$ where $\mathcal{C}_i \neq \mathcal{C}_t.$ So $\mathcal{C}_t$ is represented by other members of $\mathscr{C}$, which is a contradiction. ◻ **Proposition 6**. *Let $\mathscr{F}$ be a non-empty finite family of sets such that for $\mathcal{A},\mathcal{D} \in \mathscr{F},$ $\mathcal{A} - \mathcal{D} \in \mathscr{F}$ and $\mathcal{A} \cap \mathcal{D} \in \mathscr{F}.$ Then $\mathscr{F}$ has a *unique* basis $\mathscr{B}=\{\mathcal{B}_i\}_{i \in \gamma},$ where for every $\mathcal{A} \in \mathscr{F}$ there is a unique index set $\alpha \subseteq \gamma$ such that $\mathcal{A}=\bigcup_{i \in \alpha} \mathcal{B}_i.$ So $\mathcal{A}$ is uniquely determined by members of $\mathscr{B}.$* *Proof.* We construct $\mathscr{B}$ inductively. Start with $\mathcal{F}_0 = \mathscr{F}.$ In each step, let $\mathcal{F}_{i+1}$ be obtained from $\mathcal{F}_{i}$ by removing an element $\mathcal{C}$ that can be expressed as the union of other elements of $\mathcal{F}_{i}.$ This process terminates in less than $|\mathscr{F}|$ steps. Let $\mathscr{B}=\{\mathcal{B}_i\}_{i \in \gamma}$, which is obtained from the last step, so the members of $\mathscr{F}$ can be represented as the union of members of $\mathscr{B}.$ Assume there is a distinct pair $\mathcal{B}_t, \mathcal{B}_k \in \mathscr{B}$ such that $\mathcal{B}_t \cap \mathcal{B}_k \neq \varnothing.$ Without loss of generality, we can assume $\mathcal{B}_t-\mathcal{B}_k \neq \varnothing.$ Define $\mathcal{A}_1=\mathcal{B}_t-\mathcal{B}_k$ and $\mathcal{A}_2=\mathcal{B}_t \cap \mathcal{B}_k,$ obviously $\mathcal{A}_1$ and $\mathcal{A}_2$ are proper subsets of $\mathcal{B}_t$ and $\mathcal{A}_1, \mathcal{A}_2 \in \mathscr{F},$ hence they can be represented by members of $\mathscr{B},$ so $\mathcal{A}_1 = \bigcup_{i \in \alpha} \mathcal{B}_{i},$ $\mathcal{A}_2 = \bigcup_{i \in \beta} \mathcal{B}_{i}$ where $\mathcal{B}_i \neq \mathcal{B}_t$ for $i \in \alpha \cup \beta.$ Thus $$\mathcal{B}_t = \mathcal{A}_1 \cup \mathcal{A}_2 = \bigcup_{i \in \alpha \cup \beta} \mathcal{B}_{i}$$ where $\mathcal{B}_i \neq \mathcal{B}_t.$ So $\mathcal{B}_t$ is represented by members of $\mathscr{B}$ which is a contradiction, so different members of $\mathscr{B}$ have empty intersection. It remains to show that every $\mathcal{A} \in \mathscr{F}$ is uniquely determined by members of $\mathscr{B} = \{\mathcal{B}_i\}_{i \in \gamma}.$ Assume $\mathcal{A}=\bigcup_{i \in \alpha}\mathcal{B}_i$ and $\mathcal{A} = \bigcup_{j \in \beta} \mathcal{B}_j$ where $\alpha, \beta \subseteq \gamma$ and $\alpha \neq \beta.$ Let $\mathcal{B}_k \in \{\mathcal{B}_i\}_{i \in \alpha}$ thus $\mathcal{B}_k \subseteq \bigcup_{j \in \beta} \mathcal{B}_j$ so there is $\mathcal{B}_t \in \{\mathcal{B}_j\}_{j \in \beta}$ such that $\mathcal{B}_k \cap \mathcal{B}_t \neq \varnothing$ which is a contradiction. So $\mathcal{A}$ is uniquely determined by members of $\mathscr{B}.$ ◻ **Corollary 2**. *Let $M_{n \times m}$ be a non-zero binary matrix such that for $A, B \in \{M_{i-}\}_{i \in [n]},$ $A \land \neg B \in \{M_{i-}\}_{i \in [n]}$ and $A \land B \in \{M_{i-}\}_{i \in [n]}.$ Then $M$ has a *unique* basis $\mathcal{V}=\{v_i\}_{i \in \gamma},$ where for every $A \in \{M_{i-}\}_{i \in [n]}$ there is a unique index set $\alpha \subseteq \gamma$ such that $A= \bigvee_{i \in \alpha} v_i.$ So $A$ is uniquely determined by members of $\mathcal{V}.$* #### In the following proposition, we will demonstrate the necessary conditions for the matrix $\widetilde{M},$ corresponding to the space $(M_{n \times m},\rightarrow),$ to have a basis. **Proposition 7**. *Let $(M_{n \times m},\rightarrow)$ be a non-zero space. Then* 1. *For $\widetilde{A}, \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}$ we have $\widetilde{A} \land \neg \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}.$* 2. *For $\widetilde{A}, \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}$ we have $\widetilde{A} \land \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}.$* *Proof.* We conclude $(1)$ from Example [Example 4](#ex:re1){reference-type="ref" reference="ex:re1"}. $(2)$ Let $\widetilde{A}, \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}.$ By $(1)$ $\widetilde{A} \land \neg (\widetilde{A} \land \neg \widetilde{B}) \in \{\widetilde{M}_{i-}\}_{i \in [n]}.$ Since $$\begin{aligned} \widetilde{A} \land \neg (\widetilde{A} \land \neg \widetilde{B}) &= \widetilde{A} \land ({\neg}\widetilde{A} \lor \widetilde{B}) \\ &= \widetilde{A} \land \widetilde{B}\end{aligned}$$ then $\widetilde{A} \land \widetilde{B} \in \{\widetilde{M}_{i-}\}_{i \in [n]}.$ ◻ **Remark 1**. *Let $(M_{n \times m},\rightarrow)$ be a non-zero space. Then $\widetilde{M}$ has a basis $\mathcal{V}=\{v_i\}_{i \in \gamma}$ by Proposition [Proposition 7](#prop:re3){reference-type="ref" reference="prop:re3"} and Corollary [Corollary 2](#cor:re2){reference-type="ref" reference="cor:re2"}.* **Remark 2**. *Suppose $M = (m_{ij})$ be a non-zero binary matrix of size $n \times m$ and $k \in [m].$ Clearly $\sum_{i=1}^{n} m_{ik} \geq n/2$ if and only if $\sum_{i=1}^{n} \widetilde{m}_{ik} \leq n/2.$* **Theorem 3**. *For every space $(M,\rightarrow)$ we have $\max(\psi(M)) \geq n/2.$* *Proof.* Let $\mathcal{V}=\{v_i\}_{i \in \gamma}$ be a basis for $\widetilde{M}.$ We define sets $\mathcal{K}$ and $\mathcal{L}$ as follows $$\begin{aligned} \mathcal{K} &:= \{\widetilde{M}_{i-}|\text{ for } \widetilde{M}_{i-}=\bigvee_{i \in \alpha} v_i \text{, where } 1 \in \alpha\} \\ \mathcal{L} &:= \{\widetilde{M}_{j-}|\text{ for } \widetilde{M}_{j-}=\bigvee_{j \in \beta} v_j \text{, where } 1 \notin \beta\}. \end{aligned}$$ Clearly $\mathcal{K}$ and $\mathcal{L}$ are well defined, due to the unique representation of rows of $\widetilde{M}$ by members of $\mathcal{V}.$ Define matrix $K$ where its rows come from $\mathcal{K}$ while preserving their order in $\widetilde{M}.$ We define matrix $L$ from $\mathcal{L}$ in the same manner, then $$\widetilde{M} \sim \begin{pmatrix} K \\ L \\ \end{pmatrix}.$$ Define function $g$ as follows $$\begin{aligned} g: \mathcal{K} &\to \mathcal{L} \\ g(\widetilde{M}_{k-}) &= \widetilde{M}_{k-} \land \neg v_1. \end{aligned}$$ Since $v_t \land v_k=0$ for any two distinct members $v_t, v_k \in \mathcal{V}$ and every row of $\widetilde{M}$ has a unique representation, we conclude that $g$ is an injective function. So $|\mathcal{K}| \leq |\mathcal{L}|$ which means $v_1$ appears in equal or less than half of rows of $\widetilde{M}.$ Thus, there is a column $\widetilde{M}_{-t} \in \{\widetilde{M}_{-j}\}_{j \in [m]}$ such that $\sum_{i=1}^{n} \widetilde{m}_{it} \leq n/2.$ So by Remark [Remark 2](#rem:re4){reference-type="ref" reference="rem:re4"} $$\sum_{i=1}^{n} m_{it} \geq n/2$$ which implies that $$\max(\psi(M))\geq n/2.$$ ◻ #### We end this paper by mentioning that Theorem [Theorem 3](#thm:re5){reference-type="ref" reference="thm:re5"} provides a weaker version of Conjecture [Conjecture 2](#conj:re6){reference-type="ref" reference="conj:re6"}. To see this, consider a binary matrix $M$ where ${\neg}A \lor B \in \{M_{i-}\}_{i \in [n]}$ for $A, B \in \{M_{i-}\}_{i \in [n]}.$ By Proposition [Proposition 7](#prop:re3){reference-type="ref" reference="prop:re3"}, we also know that ${\neg}A \land {\neg}B \in \{\widetilde{M}_{i-}\}_{i \in [n]}.$ So we can conclude that $A \lor B \in \{M_{i-}\}_{i \in [n]}.$ 1 Ryan Alweiss, Brice Huang, and Mark Sellke. Improved lower bound for the union-closed sets conjecture. , 2022. Henning Bruhn and Oliver Schaudt. The journey of the union-closed sets conjecture. , 31:2043--2074, 2015. Zachary Chase and Shachar Lovett. Approximate union closed conjecture. , 2022. P. Frankl. Extremal set systems. , 2:1293--1329, 1995. Justin Gilmer. A constant lower bound for the union-closed sets conjecture. , 2022. Ilan Karpas. Two results on union-closed families. , 2017. Ian Roberts and Jamie Simpson. A note on the union-closed sets conjecture. , 47:265--267, 2010. Will Sawin. An improved lower bound for the union-closed set conjecture. , 2022. Bojan Vuckovic and Miodrag Zivkovic. The 12-element case of frankl's conjecture. , 13:65--71, 01 2017.

arxiv_math

--- abstract: | In this paper, we prove centralizer rigidity near an element of the Weyl chamber flow on a semisimple Lie group. We show that a volume preserving perturbation of an element of the Weyl chamber flow on a quotient $G/\Gamma$ of an $\mathbb{R}$-split, semisimple Lie group $G$ either has centralizer of dimension $0$ or $1$, or is smoothly conjugate to an element of the Weyl chamber flow. We also acquire a general condition for the centralizer of a partially hyperbolic diffeomorphism to be a Lie group. address: - Yau Mathematical Sciences Centre, Tsinghua University, Beijing, China - Department of Mathematics, University of Chicago, Chicago, IL 60637, USA author: - Zhijing Wendy Wang bibliography: - cenrig.bib title: Centralizer Rigidity near Elements of the Weyl Chamber Flow --- # Introduction For a diffeomorphism $f\in \mathrm{Diff}^\infty(M)$ of a Riemannian manifold $M$, we define its $C^r$-centralizer for $1\le r\le \infty$, to be $$\mathcal{Z}^r(f):=\{g\in \mathrm{Diff}^r(M): g\circ f=f\circ g\}.$$ We say that a diffeomorphism has - *trivial centralizer* if $\mathcal{Z}^\infty(f)=\langle f\rangle;$ - *virtually trivial centralizer* if $\langle f\rangle$ is a finite index subgroup in $\mathcal{Z}^\infty(f)$. The study of smooth centralizers started with Smale [@smale] who conjectured that centralizers of "generic\" diffeomorphisms are trivial. The conjecture was answered in affirmative in [@BCW] where the authors showed that a residual set of diffeomorphisms of a compact smooth manifold has trivial centralizer. On the other hand, for many systems of interest, the centralizers are often very large. For example, the centralizer of a translation on $\mathbb{R}$ is the set of all periodic functions; the centralizer of a discretized geodesic flow on a Riemannian surface contains the whole geodesic flow; the centralizer of a diagonal matrix acting by left translation on $SL_n\mathbb{R}/SL_n\mathbb{Z}$ contains the whole diagonal group acting by left translation. Here, we are interested in the problem of describing the centralizer completely around an algebraic model and showing that "big centralizer implies being algebraic\", which is called the (local) *centralizer rigidity* problem. The notion of centralizer rigidity for partially hyperbolic diffeomorphisms was first introduced in [@DWX] where the authors studied perturbations of discretized geodesic flow and affine toral automorphisms. In particular, they showed that a volume preserving perturbation of a discretized geodesic flow either has virtually trivial centralizer or embeds into a smooth flow. In [@GSXZ], it is shown that a partially hyperbolic diffeomorphism on the torus $\mathbb{T}^3$ homotopic to an Anosov linear map either has virtually trivial center or is smoothly conjugate to a linear map. Later in [@DWX23], centralizer rigidity was established for automorphisms on nil-manifolds. In this paper, we study perturbations of affine diffeomorphisms of quotients of semisimple Lie groups $G/\Gamma$. Here we require $G$ to be a connected $\mathbb{R}$-spilt semisimple Lie group over $\mathbb{R}$ with finite center and no compact factor and require $G$ to be *genuinely higher rank*, meaning that each of its simple factors has rank at least $2$. We require $\Gamma$ to be a cocompact lattice in $G$ such that for any normal subgroup $N\simeq \mathrm{PSL}_2\mathbb{R}$, $N\Gamma$ is not closed in $G$. The diffeomorphisms we are interested in are elements of the Weyl chamber flow. For a semisimple Lie group $G$ of rank $k$, the Weyl chamber flow is an $\mathbb{R}^k$ action on the Lie group by left multiplication of elements its maximal torus. An element $f_0$ of the Weyl chamber flow is called *generic* if the corresponding element in the Lie algebra does not lie in any of the Weyl chamber walls. See Section [2.1](#semisimple){reference-type="ref" reference="semisimple"} for more detailed definitions of these terminologies. In the simplest example where $G=\mathrm{SL}_3\mathbb{R}$, the Weyl chamber flow is just the $\mathbb{R}^2$ action of left multiplications of the diagonal matrices $$\begin{pmatrix} e^{t_1} &&\\ &e^{t_2}&\\ &&e^{t_3} \end{pmatrix} :\; t_1,t_2,t_3\in \mathbb{R}, \;t_1+t_2+t_3=0,$$ on $\mathrm{SL}_3\mathbb{R}/\Gamma$. In this case, an element $f_0=\begin{pmatrix} e^{t_1} &&\\ &e^{t_2}&\\ &&e^{t_3} \end{pmatrix}$ is generic if and only if $t_i-t_j\neq 0$ for any $i\neq j, 1\le i,j\le 3$. The centralizer $\mathcal{Z}^\infty(f_0)$ of a generic element $f_0$ in the Weyl chamber flow contains $\mathbb{R}^k$ since it commutes with any other element of the Weyl chamber flow. For a volume preserving perturbation of $f_0$, we show that it either has a centralizer of dimension $\le 1$, or is smoothly conjugate to an element of the Weyl chamber flow. **Theorem 1**. *Let $X=G/\Gamma$ where $G$ is a connected, $\mathbb{R}$-split, semisimple, genuinely higher rank Lie group with finite center and no compact factor, and $\Gamma$ is a co-compact lattice in $G$ so that $N\Gamma$ is not closed for any normal subgroup $\mathrm{PSL}_2\mathbb{R}\simeq N<G$.* *Let $f_0: X\to X$ be a generic element of the Weyl chamber flow on $G/\Gamma$. Then for any $C^\infty$ volume-preserving diffeomorphism $f:X\to X$ that is a $C^1$-small perturbation of $f_0$, the smooth centralizer $\mathcal{Z}^\infty(f)$ is* - *either a (not necessarily connected) Lie group of dimension $0$ or $1$,* - *or virtually $\mathbb{R}^k$, where $k$ is the rank of $G$.* *In the latter case, $f$ is $C^\infty$-conjugate to an element of the Weyl chamber flow.* *Furthermore, if $G$ is of rank $2$, then in the first case, $\mathbb{Z}$ or $\mathbb{R}$ is a finite index subgroup of $\mathcal{Z}^\infty(X)$.* **Remark 2**. *The volume preserving condition may be replaced by the condition that $f$ is ergodic with respect to a measure $\mu$ of full support on $X$.* In a recent work by Damjanović, Wilkinson and Xu[@DWX24], the authors made the following conjectures on centralizer rigidity of an affine diffeomorphism $f_0=L_a\circ \Psi$ acting on a homogeneous space $X=G/\Gamma$ where $G$ is a Lie group, $\Gamma<G$ is a cocompact discrete subgroup, $L_a$ is the left multiplication for some $a\in G$ and $\Psi$ is a $G$-automorphism preserving $\Gamma$. **Conjecture 3** (Conjecture 1, [@DWX24]). *Let $f_0:X\to X$ be an affine diffeomorphism that satisfies the $K$-property, then for every $f\in \mathrm{Diff}^\infty(X)$ sufficiently $C^1$-close to $f_0$, the centralizer $\mathcal{Z}^\infty(f)$ is a $C^0$ closed Lie subgroup of $\mathrm{Diff}^\infty(X)$.* **Conjecture 4** (Conjecture 2, [@DWX24]). *Let $f_0:X\to X$ be an affine diffeomorphism that satisfies the $K$-property and suppose $\mathcal{Z}^\infty(f_0)$ has no rank $1$ factor, then for every $f\in \mathrm{Diff}^\infty(X)$ sufficiently $C^1$-close to $f_0$, if the centralizer $\mathcal{Z}^\infty(f)\simeq \mathcal{Z}^\infty(f_0)$, then $f$ is smoothly conjugate to an affine diffeomorphism.* **Conjecture 5** (Conjecture 3, [@DWX24]). *Let $f_0:X\to X$ be an affine diffeomorphism that satisfies the $K$-property and suppose $\mathcal{Z}^\infty(f_0)$ has no rank $1$ factor, then for every $f\in \mathrm{Diff}^\infty(X)$ sufficiently $C^1$-close to $f_0$, if the centralizer $\mathcal{Z}^\infty(f)$ has no rank $1$ factor, then $f$ is smoothly conjugate to an affine diffeomorphism.* Here a group action is said to have a rank $1$ factor if it has a group action of $\mathbb{Z}$ or $\mathbb{R}$ or their compact extension as a factor. The simplest example of a group action to have a rank $1$ factor is for the group to be $\mathbb{Z}$ or $\mathbb{R}$ itself. Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"} gives an affirmative answer to Conjectures 1 and 2 in [@DWX] for volume preserving perturbations of generic elements of the Weyl chamber flow on $G/\Gamma$. It also proves Conjecture 3 in [@DWX] for simple noncompact Lie groups of rank $2$. Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"} is also related to the rigidity of higher rank restrictions of Weyl chamber flow, established in [@DK], [@Vinhage], etc. which show that a smooth $C^1$-small perturbation of a higher rank restriction of the Weyl chamber flow on $G/\Gamma$ must be smoothly conjugate to a restriction of the Weyl chamber flow. In Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}, instead of requiring the whole higher rank restriction to be $C^1$-close to elements of Weyl chamber flow, we show that one element $C^1$-close to the Weyl chamber flow that has a "big enough\" center would be enough, generalizing the previous results. An ingredient of the proof of Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"} is a result of independent interest, which we state as Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} in Section [3](#cenfix){reference-type="ref" reference="cenfix"}. It shows for a "nice enough\" partially hyperbolic diffeomorphism, the center fixing centralizer is always a Lie group. One application of Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} is that we remove the volume preserving assumptions for the centralizer rigidity results on discretized Anosov flows in [@DWX] and [@BG]; see Theorem [Theorem 19](#1dcen1){reference-type="ref" reference="1dcen1"} and Theorem [Theorem 20](#1dcen2){reference-type="ref" reference="1dcen2"}. ## Organization and outline of the proof {#outl} In Section [2](#prelim){reference-type="ref" reference="prelim"}, we shall provide some preliminaries on basic Lie theory and partially hyperbolic dynamics. Let $f_0$ be a generic element of the Weyl chamber flow and let $f$ be a $C^1$ small perturbation as in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}. The action of $f_0$ fixes each coset of the maximal tori which we call the center leaves of $f_0$. Normal to the direction of the maximal tori, we may show that $f_0$ acts as contraction or expansion. These characterize $f_0$ as being normally hyperbolic. Furthermore, the action of $f_0$ restricted to the maximal tori is simply a translation map. See Section [2.1](#semisimple){reference-type="ref" reference="semisimple"} for more details on this. A classical result in partially hyperbolic dynamics, see Theorem [Theorem 7](#leafconj){reference-type="ref" reference="leafconj"}, shows that a $C^1$ perturbation $f$ of $f_0$ also fixes each leaf of a foliation $\mathcal{W}^c_f$ and acts by contraction and expansion normal to that foliation. Let $g\in \mathcal{Z}^r(f)$ be any diffeomorphism commuting with $f$, it's not hard to show that $g$ must also preserve the foliation $\mathcal{W}^c_f$. Therefore, we may map $g\in \mathcal{Z}^r(f)$ to its action on the leaf space of $\mathcal{W}^c_f$, yielding a short exact sequence $$1\to \mathcal{CZ}^r(f)\to \mathcal{Z}^r(f)\to \mathcal{Z}^r(f)/\mathcal{CZ}^r(f)\to 1.$$ Here, we call the kernel $$\mathcal{CZ}^r(f):=\{g\in \mathrm{Diff}^r(M):\; f\circ g=g\circ f,\; g(\mathcal{W}_f^c(x))=\mathcal{W}_f^c(x),\; \forall\; x\in M\}$$ the center fixing, $C^r$ centralizer of $f$. The quotient space $\mathcal{Z}^r(f)/\mathcal{CZ}^r(f)$ is the space of the action of $\mathcal{Z}^r(f)$ on the leaf space $X/\mathcal{W}^c_f$. In Section [3](#cenfix){reference-type="ref" reference="cenfix"}, we prove that the center fixing centralizer $\mathcal{CZ}^r(f)$ is a Lie group under a more general condition. Applying to discretized Anosov flows, we remove the volume preserving conditions in [@DWX] and [@BG]. We then commence with the proof of Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}. In Section [4](#centgp){reference-type="ref" reference="centgp"}, we prove that in our setting the space of action on the leaf space $\mathcal{Z}^r(f)/\mathcal{CZ}^r(f)$ must be finite. Therefore, $\mathcal{Z}^r(f)$ is virtually a Lie group. In Section [5](#dichcen){reference-type="ref" reference="dichcen"}, we prove that elements of $\mathcal{CZ}^r(f)$ also act "like a translation\" on the center leaf, similar to the way $f_0$ acts on the cosets of the maximal tori. Using this and the fact that $\mathcal{CZ}^r(f)$ is a Lie group, we conclude that the centralizer either contains two diffeomorphisms that act like translations in different directions, or is of dimension $le 1$. In Section [6](#eph){reference-type="ref" reference="eph"}, we prove that in the case that the centralizer contains two diffeomorphism that act "like translations\" in different directions, $\mathcal{CZ}^r(f)$ must contain a higher rank partially hyperbolic action. In Section [7](#wcf){reference-type="ref" reference="wcf"}, we prove this higher rank partially hyperbolic action must be conjugate to an algebraic one, following the methods of [@DK] and [@Vinhage]. ## Acknowledgement The author would like to thank her advisor Amie Wilkinson for many useful discussions, guidance and thorough revision of this work. The author also thanks Disheng Xu and Danijela Damjanović for illuminating conversations and thanks Kurt Vinhage and her advisor Jinxin Xue for many useful comments. # Preliminary {#prelim} ## Semisimple Lie group and Weyl chamber flow {#semisimple} Let $G$ be a $\mathbb{R}$-split semisimple Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra $\mathfrak{g}$ act on itself by the adjoint representation $$\mathfrak{g}\to \mathfrak{gl}(\mathfrak{g}): v\mapsto ad(v).$$ We denote by $\mathfrak{h}$ a maximal commutative subalgebra of $\mathfrak{g}$ consisting of elements whose adjoint action is diagonalizable in $\mathbb{C}$, which we call the *Cartan subalgebra* of $\mathfrak{g}$. Since $G$ is $\mathbb{R}$-split, the adjoint action of elements in $\mathfrak{h}$ are also diagonalizable in $\mathbb{R}$. We define the *rank* of $G$ to be $$\mathrm{rank}(G)=\mathrm{dim} (\mathfrak{h}).$$ We also denote by $D=\exp(\mathfrak{h})$ the corresponding maximal Abelian subgroup, called the maximal torus of $G$. Restricting the adjoint action to the Cartan subalgebra $\mathfrak{h}$, $ad|_\mathfrak{h}$ decomposes the Lie algebra $\mathfrak{g}$ into eigenspaces $$\mathfrak{g}=\oplus_{\lambda\in \mathfrak{h}{^*} }\mathfrak{g}_\lambda$$ where each eigenspace is given by $$\mathfrak{g}_\lambda=\{x\in \mathfrak{g}: ad(h) x=\lambda(h) x, \forall h\in \mathfrak{h}\}.$$ We denote by $\Phi=\{\lambda\in \mathfrak{h}^*: \mathfrak{g}_\lambda\neq 0,\lambda\neq 0\}\subset \mathfrak{h}^*$ the set of non-zero eigenvalues of the adjoint representation $\mathfrak{h}\to \mathfrak{gl}(\mathfrak{g})$, and we call $\Phi$ the *root system* of $G$. The eigenvalues which are linear functions on $\mathfrak{h}$ separate $\mathfrak{h}\simeq \mathbb{R}^k$ into different Weyl chambers. The kernels of the eigenvalues $ker(\lambda)\subset \mathfrak{h}, \lambda\in \Phi$ are called the *Weyl Chamber walls* in $\mathfrak{h}$ and the connected components of $\mathfrak{h}\backslash \cup_{\lambda\in \Phi}ker(\lambda)$ in $\mathfrak{h}$ are called the *Weyl chambers*. Let $\Gamma$ be a discrete subgroup of $G$. The Weyl chamber flow on $G/\Gamma$ is the action $$\mathfrak{h}\to \mathrm{Diff}^\infty(G/\Gamma), v\mapsto L_{\exp(v)}$$ where $L_{\exp(v)}$ is the left multiplication by $\exp(v)$ on $G/\Gamma$. A simple and concrete example is $G=SL_3\mathbb{R}$. **Example 1**. *Let $G=\mathrm{SL}_3\mathbb{R}$, then the Lie algebra is given by $$\mathfrak{sl}_3\mathbb{R}=\{A\in M_{3\times 3}(\mathbb{R}): \mathrm{tr} A=0\}.$$* *One choice of a maximal diagonalizable subgroup is $$\mathfrak{h}=\left\{\begin{pmatrix} t_1 & &\\ & t_2&\\ && t_3 \end{pmatrix}:\; t_1+t_2+t_3=0,\; t_1,t_2,t_3\in \mathbb{R}\right\} .$$* *The eigenspaces of the adjoint function are $$\mathfrak{g}_{ij}=\mathbb{R}e_{ij},1\le i\neq j\le 3,$$ where $e_{ij}$ is the matrix with $1$ on the $i^{th}$ row and $j^{th}$ column and zeroes for all other entries. The corresponding eigenvalues $\lambda_{ij}$ are simply $$\lambda_{ij}(\begin{pmatrix} t_1 & &\\ & t_2&\\ && t_3 \end{pmatrix})=t_i-t_j.$$* *The Weyl chamber walls are then three lines $\{t_i=t_j\}$ in the Cartan subalgebra $\mathfrak{h}\simeq \{t_1,t_2,t_3\in \mathbb{R}: t_1+t_2+t_3=0\}\simeq \mathbb{R}^2 \subset \mathbb{R}^3$, partitioning $\mathfrak{h}$ into six Weyl chambers.* Now we consider the action of left multiplication of $f_0=L_{\exp(v)}$ on $G$ where $v$ is a generic element in $\mathfrak{h}$. Let $G_\lambda=\exp(\mathfrak{g}_\lambda)$ be the subgroup generated by the eigenspaces, then $f_0$ preserves the left cosets of $G_\lambda$. We shall see that $f_0$ also expands and contracts the cosets of $G_\lambda$ exponentially. **Proposition 6**. *For any $x,y$ in the same left coset of $G_\lambda$, then the distance between $x$ and $y$ are contracted or expanded exponentially under iteration of $f$ depending on the sign of $\lambda(v)$.* *Proof.* For any $y\in G_\lambda x, x\in G$, suppose $x=\exp(u)y$, then we have $$f_0^n(x) f_0^n(y)^{-1}=\exp(nv)x y^{-1} \exp(nv)^{-1} =\exp(ad(nv)u)\cdot\exp(u)$$$$=\exp(n\lambda(v)u)\cdot(xy^{-1}).$$ Therefore, under iteration of $f$, the distance between $x$ and $y$ in the same left coset of $G_\lambda$ are contracted or expanded exponentially depending on the sign of $\lambda(v)$. ◻ We also note that since $\exp(v)$ commutes with the maximal tori $D$, $f_0$ acts on each coset of $D$ by a translation, neither expanding nor contracting the cosets of $D$. As we shall see later, these behaviors exactly characterize that of a normally hyperbolic diffeomorphism. ## Partial hyperbolicity and regularity {#prel:ph} We recall the definitions of dominated splitting, partial hyperbolicity and some results from [@HPS] on leaf conjugacy. **Definition 1** (Dominated Splitting). *Let $M$ be a Riemannian manifold, and let $f:M\to M$ be a diffeomorphism, a *dominated splitting* of $f$ is a decomposition $TM=E^1\oplus E^2\oplus \cdots \oplus E^k$ that satisfies* - *$TM=E^1\oplus E^2\oplus \cdots \oplus E^k$ is $Df$-invariant, i.e. $Df(E^i(x))=E^i(f(x))$, for any $x\in M$;* - *there exists $0<\lambda_i<1$ and $C>0$ such that for any $n\in \mathbb{N}$, $x\in M$, $$\|Df^n(x)v\|\le C\lambda_i^n \|Df^n(x)u\|,$$ for any $v\in E^i(x),u\in E^{i+1}(x), \|u\|=\|v\|=1$ and $1\le i\le k-1$.* **Definition 2** (Partially Hyperbolic Diffeomorphism). *A diffeomorphism $f:M\to M$ of a Riemannian manifold $M$ is *partially hyperbolic* if $f$ has *dominated splitting* $TM=E^s\oplus E^c\oplus E^u$, and there exist $0<\mu<1$ and $C>0$, such that for any $n\in \mathbb{N}, x\in M$, $$\|Df^n(x)v\|\le C\mu^n\|v\|,$$ for any $v\in E^s(x)$, and $$\|Df^{-n}(x)u\|\le C\mu^n\|u\|,$$ for any $u\in E^u(x)$.* *In this paper, we also assume that the bundles $E^s,E^u$ are both nontrivial.* In general, for a partially hyperbolic diffeomorphism, the stable and unstable bundles $E^s$ and $E^u$ are uniquely integrable to $f$-invariant stable and unstable foliations $\mathcal{W}^s$ and $\mathcal{W}^u$, locally given by $$\mathcal{W}^s(x)=\{y\in M:\; \exists\; C>0,\; d(f^n(x),f^n(y)\le C\mu^n d(x,y),\; \forall\; n>0 \},$$ and $$\mathcal{W}^s(x)=\{y\in M: \;\exists\; C>0, \;d(f^n(x),f^n(y)\le C\mu^n d(x,y), \;\forall\; n<0 \}.$$ However, $E^c$ is generally not tangent to a foliation. A partially hyperbolic diffeomorphism is normally hyperbolic if there is some choice of invariant center foliation tangent to $E^c$. **Definition 3** (Normally hyperbolic). *A partially hyperbolic diffeomorphism $f:M\to M$ is said to be normally hyperbolic with respect to a foliation $\mathcal{F}$ if $f$ preserves $\mathcal{F}$ and $T\mathcal{F}=E^c$.* *For $r\ge 1$, we say $f$ is *$r$-normally hyperbolic*, if it is normally hyperbolic and there exists $k\in \mathbb{N}$ such that $$\sup_{x\in M}\|D_xf^k|_{E^s}\|\cdot \|D_xf^k|_{E^c}\|^r<1$$ and* *$$\sup_{x\in M}\|D_xf^{-k}|_{E^u}\|\cdot \|D_xf^{-k}|_{E^c}\|^r<1.$$* We note that $1$-normal hyperbolicity is equivalent to normal hyperbolicity, and $r$-normal hyperbolicity is a $C^1$-open condition. For a normally hyperbolic diffeomorphism $(f,\mathcal{F})$, heuristically, we may view $f$ as being Anosov on the leaf space $M/\mathcal{F}$. Therefore, we could expect structural stability of $f$ up to leaves of $\mathcal{F}$. Under some conditions, we may expect perturbations of $f$ to be $C^0$ conjugate to $f$ modulo the leaves of $\mathcal{F}$, and we say that a perturbation is leaf conjugate to $f$. Now we define the notion of leaf conjugacy. **Definition 4** (Leaf conjugacy). *Suppose $(f,\mathcal{F}_f)$ and $(g,\mathcal{F}_g)$ are two diffeomorphisms of $M$ with invariant foliations $\mathcal{F}_f$ and $\mathcal{F}_g$ respectively.* *Then $(f,\mathcal{F}_f)$ and $(g,\mathcal{F}_g)$ are said to be *leaf conjugate* via the leaf conjugacy $h\in \mathrm{Homeo}(M)$, if $$h(\mathcal{F}_f(x))=\mathcal{F}_g(h(x))$$ for any $x\in M$.* A notion slightly stronger than normal hyperbolicity is dynamical coherence. **Definition 5** (Dynamically Coherent). *A partially hyperbolic diffeomorphism $f$ is said to be *dynamically coherent* if there exist two $f$-invariant foliations $\mathcal{W}^{cs}$ and $\mathcal{W}^{cu}$ that are tangent to $E^{cs}=E^c\oplus E^s$ and $E^{cu}=E^c\oplus E^u$ respectively.* *If $f$ is dynamically coherent, we define the center foliation of $f$ to be $\mathcal{W}^c=\mathcal{W}^{cs}\cap \mathcal{W}^{cu}$.* By the definition $\mathcal{W}^c$ is an $f$-invariant foliation tangent to $E^c$ at every point, and so $(f,\mathcal{W}^c)$ is normally hyperbolic. We note that here we don't require $E^{cs}$ or $E^{us}$ to be uniquely integrable and $E^c$ also doesn't have to be uniquely integrable to $\mathcal{W}^c$. See [@BWcoh] for more discussion on this subject. Now we introduce a central result about leaf-wise structural stability of normally hyperbolic diffeomorphisms. A more general statement and proof of the result can be found in [@BWcoh] and [@PSWholrev]. **Theorem 7** (Leaf-wise structural stability). *Let $f$ be a $C^r$ diffeomorphism on $M$ preserving a foliation $\mathcal{F}$. If $(f,\mathcal{F})$ is $r$-normally hyperbolic, then the leaves of $\mathcal{F}$ are uniformly $C^r$, the bundles $E^u,E^s$ are uniquely integrable with leaves as smooth as $f$, and $f$ is $C^1$ stably dynamically coherent.* *Moreover, there exists a bi-Hölder leaf conjugacy $\phi$ from $f$ to any $C^1$-perturbation of $f$ that is uniformly $C^r$ when restricted to the center leaves $W^c_f$.* Applying this to the diffeomorphisms of $G/\Gamma$ that we are considering, we see that any small perturbation $f$ of an element of the Weyl chamber flow $f_0$ has the nice properties we described above. **Proposition 8**. *Let $f_0$ be a generic element of the Weyl chamber flow. Let $f\in \mathrm{Diff}^r(X)$ be $C^1$-perturbation of $f_0$, then $E^u_f,E^s_f$ are uniquely integrable, $f$ is dynamically coherent and $r$-normally hyperbolic, and there exists a bi-Hölder leaf conjugacy from $f_0$ to $f$ that's uniformly $C^r$ on the center leaves.* *Proof.* By our analysis in Section [2.1](#semisimple){reference-type="ref" reference="semisimple"}, $f_0$ is $r$-normally hyperbolic with respect to the foliations $\mathcal{W}^c_{f_0}$ given by cosets of $D$. Therefore, applying Theorem [Theorem 7](#leafconj){reference-type="ref" reference="leafconj"} yields the desired result. ◻ ## Accessibility Now we recall some definition and facts about $su$-holonomies of partially hyperbolic diffeomorphisms. In this paper, we shall mainly consider holonomies along the center leaves. An *$su$-path* of $f$ is a piece-wise $C^1$ curve $\gamma:[0,1]\to M$ with each piece in $\mathcal{W}^s$ or $\mathcal{W}^u$, i.e. there exits $k\in \mathbb{N}$ and $0=t_0<t_1<t_2<...<t_k=1$ so that $\gamma([t_i,t_{i+1}])\subset \mathcal{W}^s(\gamma(t_i))$ or $\gamma([t_i,t_{i+1}])\subset \mathcal{W}^u(\gamma(t_i))$. In this case the $su$-path $\gamma$ is said to be $k$-legged. Sometimes we also denote an $su$-path by its end points $[\gamma(0),\gamma(t_1),...,\gamma(t_k)]$. **Definition 6** (Accessibility). *A partially hyperbolic diffeomorphism $f$ is said to be *accessible* if for any $x,y\in M$, there exists an $su$-path from $x$ to $y$.* For partially hyperbolic systems with center dimension $1$ or $2$, accessibility is known to be open by [@Didier] and [@AV]. Stable accessibility (under $C^1$ small perturbations) is also shown to be $C^r, r\ge 1$ dense in [@RHU] for center dimension $1$ and $C^1$ dense for arbitrary dimension in [@DW]. Furthermore, if the center foliation of a partially hyperbolic diffeomorphism $f$ is smooth, then accessibility is $C^1$ open around $f$ by Proposition 1.4 of [@GPS]. For our case, since $f_0$ is highly accessible by the Lie algebra structure, and the center leaves are smooth, we can apply Proposition 1.4 in [@GPS] to show that accessibility is open around $f_0$. **Proposition 9**. *Let $f_0$ be the element of the Weyl chamber flow on $X=G/\Gamma$, then any $C^1$ perturbation $f$ of $f_0$ is accessible. Furthermore, for any $x,y\in X$ and any $f_0$-$su$-path $\gamma_0$ connecting $x$ and $y$, there exists a $f$-$su$-path $\gamma$ connecting $x$ and $y$ that's close to $\gamma_0$.* **Definition 7** (Center bunching). *We say that a partially hyperbolic diffeomorphism $f:M \to M$ is center $r$-bunched if there exists $k\ge 1$ such that $$\sup \|D_xf^k|_{E^s}\|\cdot \|(D_xf^k|_{E^c})^{-1}\|^r<1,$$ $$\sup \|(D_xf^k|_{E^u})^{-1}\|\cdot \|D_xf^k|_{E^c}\|^r<1,$$ $$\sup \|D_xf^k|_{E^s}\|\cdot \|(D_xf^k|_{E^c})^{-1}\|\cdot \|D_xf^k|_{E^c}\|^r<1, \text{ and }$$ $$\sup \|(D_xf^k|_{E^u})^{-1}\|\cdot \|(D_xf^k|_{E^c})^{-1}\|^r\cdot \|D_xf^k|_{E^c}\|<1.$$* *We say that $f$ is center bunched if it is center 1-bunched.* **Remark 10**. *We note that the first two inequalities here are same as $r$-normally hyperbolic.* The following theorem of Burns-Wilkinson [@BWerg] shows that in a very general setting partial hyperbolicity plus accessibility implies ergodicity. **Theorem 11** (Theorem 1, [@BWerg]). *Let $f$ be a volume-preserving, partially hyperbolic diffeomorphism that's center bunched and accessible, then $f$ is ergodic.* ## Smoothness of $su$-holonomy Now we consider the smoothness of the stable and unstable foliations in the transverse direction. In this paper, we shall be mainly considering stable and unstable holonomies with the center leaves as sections. Suppose $f$ is a dynamically coherent, normally hyperbolic diffeomorphism with center foliation $\mathcal{W}^c$. Then restricted to the center stable leaf $W^{cs}$, we may consider the local holonomy of the stable foliation from one center leaf to another. Let $p\in M, q\in \mathcal{W}^s(x)$ be a pair of points in a stable leaf of $f$, then there exists small enough neighbourhoods $U(p)\subset \mathcal{W}^c_f(p)$ of $p$ and $U(q)\subset \mathcal{W}^c_f(q)$ of $q$ in the center leaves and a constant $R=2d_{\mathcal{W}^s}(p,q)$, such that for every $x\in U(p)$, $\mathcal{W}^s_f(x,R)=\{y\in \mathcal{W}^s_f(x): d_{\mathcal{W}^s}(x,y)<R\}$ has a unique intersection with $U(q)$. The local stable holonomy is given by $$h^s_{f,p,q}: U(p)\to U(q); x\mapsto y= U(q)\cap \mathcal{W}^s_f(x,R).$$ The stable and unstable holonomies on the center leaves are known to be smooth. **Theorem 12** (Theorem B, [@PSW]; Theorem 1.1 [@Saghin]). *Let $f\in \mathrm{Diff}^{r+1}(M),r\ge 1$ be a normally hyperbolic, center $r$-bunched diffeomorphism on a Riemannian manifold $M$. Then the local stable and unstable holonomy maps for the center leaves are uniformly $C^r$. Furthermore, the holonomy $h^s_{f,p,q}$ from $\mathcal{W}^c_f(p)$ to $\mathcal{W}^c_f(q)$ where $q\in \mathcal{W}^s_f(p)$ depends $C^1$-continuously on $f$ with respect to the $C^1$ topology in $\mathrm{Diff}^1(M)$ and depends $C^r$ continuously on $p,q$ with respect to the $C^0$ topology on $M$.* Instead of a stable or unstable holonomy, we may also define *holonomies of an $su$-path* . Given an $su$-path $\gamma$ of $f$ with endpoints $x_0,x_1,...,x_k$, we may locally define the *$su$-holonomy* at a neighbourhood of $x_0$ in $\mathcal{W}^c(x_0)$ given by concatenation of stable and unstable holonomies between center leaves inside the center-stable or center unstable leaves following the path $\gamma$. More concretely, the local $su$-holonomy of a path $\gamma$ with endpoints $[x_0,x_1,...,x_k]$, $x_{i+1}\in \mathcal{W}^{t(i)}(x_{i}),0\le i\le k-1, t(i)\in\{s,u\}$, is given by $$h_\gamma^f= h^{t(k-1)}_{f,x_{k-1},x_{k}}\circ h^{t(k-2)}_{f,x_{k-2},x_{k-1}}\circ \cdots \circ h^{t(0)}_{f,x_{0},x_{1}}: U(x_0)\to U(x_k)$$ where $U(x_i)\subset \mathcal{W}^c_f(x_i)$ are small enough neighbourhoods of $x_i$ in its center leaf. As a direct application of Theorem [Theorem 12](#c1hol){reference-type="ref" reference="c1hol"}, we see that the $su$-holonomies on center leaves are also smooth and depends smoothly on $f$ and $\gamma$. We say that $f$ has *global holonomy* on a leaf $\mathcal{W}^c(x_0)$ if for any $su$-path $\gamma$ of $f$ starting in that leaf and ending in $\mathcal{W}^c(x_k)$, the domain of $h_\gamma^f$ can be extended to the whole center leaf $\mathcal{W}^c(x_0)$. More precisely, $f$ has global holonomy on $\mathcal{W}^c(x_0)$ if for any $su$-path $\gamma=[x_0,x_1,...,x_k]$ there exists a family of $su$-paths $\gamma_i,i\in I$ with endpoints in $\mathcal{W}^c(x_0)$ and $\mathcal{W}^c(x_k)$ such that - the domains $U_i\subset \mathcal{W}^c(x_0)$ of the local holonomies $h_{\gamma_i}^f:U_i\to \mathcal{W}^c(x_k)$ covers $\mathcal{W}^c(x_0)$, i.e. $\cup U_i=\mathcal{W}^c(x_0)$; - the local holonomies of $\gamma_i$ agrees, i.e. $h_{\gamma_i}^f=h_{\gamma_j}^f$ on $U_i\cap U_j$, for any $i,j\in I$. It is not hard to show that $f$ always has global holonomy on a leaf $\mathcal{W}^c(x_0)$ if the leaf is topologically contractible. ## Some normal form theory In this section we introduce some normal form theory on uniformly contracting foliations that we will use the promote the regularity of some homeomorphisms in the centralizer and the regularity of the conjugacy map. Let $f$ be a diffeomorphism on a closed manifold $M$ uniformly contracting a foliation $\mathcal{W}^s$ that has uniformly $C^1$ leaves. Let $E^s$ be the tangent bundle of $\mathcal{W}^s$, and denote by $F:E^s\to E^s$ to be the restriction of $Df$ to the sub-bundle $E^s$. Take $F^*: \Gamma(E^s)\to \Gamma(E^s): F^*(v(x))=F(v(f^{-1}(x))$ to be the linear map between sections of $E^s$ induced by $F$. The spectrum of $F^*$ is called *Mather spectrum* of $F$, see [@Mather]. The spectrum are closed annulus in $\mathbb{C}$ centered at the origin and bounded by circles of radii $e^{\lambda_i}$ and $e^{\mu_i}$ satisfying $$\lambda_1 \le \mu_1 <\lambda_2\le \mu_2<...<\lambda_l\le \mu_l<0 .$$ **Definition 8** (Narrow band spectrum). *A bundle map $F$ is said to have *narrow band spectrum* if $\mu_i+\mu_l\le \lambda_i,\; \forall\; 1\le i \le l$. We call the number $s(F):=\frac{\lambda_1}{\mu_l}$ the *critical regularity* of $F$.* Now we introduce the main theorem of this section that we shall use to upgrade the regularities of some homeomorphisms commuting with $f$. **Theorem 13** ([@normalform]). *Let $f$ be a $C^r$ diffeomorphism that uniformly contracts an invariant foliation $\mathcal{W}^s$ with uniformly $C^r$-leaves. Suppose the bundle map $F=Df\mid_{T\mathcal{W}^s}$ has narrow band spectrum with $s(F)<r$. Then there exists a family of local diffeomorphisms $H_x: \mathcal{W}^s(x)\to T\mathcal{W}^s(x)=E^s(x)$, such that* *1) $P_x=H_{f(x)}\circ f H_x^{-1}: E_x\to E_{f(x)}$ is a polynomial function of degree $\le s(F)$ for any $x\in M$;* *2) $H_x(x)=0$ and $D_xH_x=\mathrm{Id}$ for any $x\in M$;* *3) $H_x$ depends $C^r$-continuously on $x$ and is jointly $C^r$ in $x$ and $y\in \mathcal{W}^s(x)$;* *4) If $g$ is a homeomorphism commuting with $f$, preserving $\mathcal{W}^s$ and is $C^r$ along leaves of $\mathcal{W}^s$, then $Q_x:=H_{g(x)}\circ g\circ H_x^{-1}$ is also a polynomial of degree $\le s(F)$.* ## Some Pesin theory Now we recall some basic Pesin theory that we will use in this paper. First we recall the Oseledets Splitting theorem that describes the asymptotic behavior of diffeomorphisms under iteration. Here we state the theorem for the Abelian group action version. **Theorem 14** (Oseledets Splitting Theorem). *Let $\alpha: A\to \mathrm{Diff}(M)$ be an Abelian group action on a compact Riemannian manifold $M$ preserving an invariant measure $\mu$.* *Then for $\mu$-a.e. $x\in X$, there is a decomposition $T_xM=\oplus H_i(x)$ such that the decomposition is $D\alpha(a)$-invariant for any $a\in A$ and the *Lyapunov exponents* $\chi_i: A\times M\to \mathbb{R}$ given by $$\chi_i(a)(x)=\lim_{k\to \infty} \frac{1}{|k|} \log \frac{\|D\alpha(a)^k v\|}{\|v\|}$$ exists and converges uniformly for any $v\in H_i(x)\backslash \{0\}$.* **Remark 15**. *If we take a single diffeomorphism $f$ which generate an action $\mathbb{Z}\to \mathrm{Diff}(M)$, then we simply denote $\chi_i(1)(x)$ by $\chi_i(x)$ and we may further assume that $\chi_1(x)<\chi_2(x)<...<\chi_l(x)$ for a.e. $x\in M$.* *If $\alpha$ is $\mu$-ergodic, then the Lyapunov exponents are constant almost everywhere with respect to $x$ since they are $f$-invariant and we denote them by $\chi_i: A\to \mathbb{R}$.* *Oseledets Splitting theorem is also true in general for cocycles other than the derivative, see [@introdyn], Supplement 2.* A diffeomorphism $f$ with some non-zero Lyapunov exponents are said to be non-uniformly partially hyperbolic. Similar to the case of partially hyperbolic diffeomorphisms, if a point $x\in M$ has Lyapunov exponent $\chi_k(x)<0$, then we may also define stable and unstable manifolds of these diffeomorphisms. **Theorem 16** (Pesin Stable Manifold Theorem [@Pes]). *Let $M$ be a Riemannian surface and $f:M\to M$ a diffeomorphism with invariant measure $\mu$ such that the Lyapunov exponent $\chi_k(p)<0$ for $\mu$-almost every $p\in M$, then there exists Borel functions $C: X\to \mathbb{R}$ and $\lambda:X\to (0,1)$, such that for almost every $x\in X$, there exists a stable $C^1$ manifold $\mathcal{W}^s_f(p)$ such that $x\in \mathcal{W}^s_f(p), T_p\mathcal{W}^s(p)=\cup_{i\le k}H_i(p)$ and for any $y\in \mathcal{W}^s_f(p)$, we have $$d(f^n(x),f^n(y))\le C(x)\lambda^n(x) d(x,y).$$ Furthermore, the stable manifolds satisfy $\mathcal{W}^s(q)=\mathcal{W}^s(p)$ for any $q\in \mathcal{W}^s(p)$ and $\mathcal{W}^s(q)\cap \mathcal{W}^s(p)=\varnothing$ if $q\notin \mathcal{W}^s(p)$.* ### Weyl chamber picture for Abelian group action In this paper we shall be talking about Weyl chambers both in the realm of Lie theory and also in the realm of partially hyperbolic Abelian actions. We shall show how they are in fact the same object in our setting. Given an Abelian family of partially hyperbolic ergodic diffeomorphisms $\alpha: A\to \mathrm{Diff}(M)$, its Lyapunov exponents are a family of linear functional $\chi_i: A\to \mathbb{R}$ given in Theorem [Theorem 14](#Oseled){reference-type="ref" reference="Oseled"}, and we may take $\mathbb{R}^k=A\otimes \mathbb{R}$ and extend $\chi_i$ to $\mathbb{R}^k$. The hyperplanes $\mathrm{Ker}(\chi_i)\subset \mathbb{R}^k$ are called the *Weyl chamber walls* of $\alpha$ and the connected components of $\mathbb{R}^k-\cup_i \mathrm{Ker}(\chi_i)$ are called the *Weyl chambers* of $\alpha$. For any $a\in A$, whether $\alpha(a)$ contracts or expands the Lyapunov foliation $\mathcal{W}^i$ depends on the sign of $\chi_i(a)$, so we call the vector $\mathrm{sgn}(a)=\mathrm{sgn}(\alpha(a))=(\mathrm{sgn}(\chi_i(a)))_i$ the signature of $a$ or $\alpha(a)$. Two elements of the action $a,b\in A$ belong to the same chamber if and only if $\mathrm{sgn}(a)=\mathrm{sgn}(b)$. Two actions $\alpha_1, \alpha_2: A\to \mathrm{Diff}(M)$ *have the same Weyl chamber picture*, if the two actions have the same Weyl chamber walls and the same signatures. Now we come back to the Weyl chamber picture on the Lie algebra and compare it to the dynamical definition. If we consider the action of $\mathfrak{h}$ on $M=G$ or $M=G/\Gamma$ by left multiplication of its exponential, where the metric is taken to be right invariant, then the left multiplication by $\exp(v), v\in \mathfrak{h}$ expands and contracts the directions of $\mathfrak{g}_\lambda \subset T_eG$ by exactly $\lambda(v)$. Thus the roots are exactly the Lyapunov exponents of the group action $\mathfrak{h}\to \mathrm{Diff}(M)$. A generic element (not on the Weyl chamber walls) of the Weyl chamber flow is a partially hyperbolic diffeomorphism on $G/\Gamma$ with center leaves the cosets of $D$ and stable and unstable leaves the cosets of $\exp(\mathfrak{g}_\lambda)$. For a subgroup $A\hookrightarrow \mathfrak{h}$ of the Cartan subalgebra, we may view $A\otimes \mathbb{R}$ as a subspace of $\mathfrak{h}$. Then the action $A\to \mathrm{Diff}(M): a\mapsto L_{\exp(a)}$ by left multiplication of the exponential would have the same Weyl chamber walls as the Weyl chamber walls of $\mathfrak{h}$ restricted to the subspace. In this sense, we may view the Weyl chamber picture of $A$ as a subspace in the Weyl chamber picture of $\mathfrak{h}$. # Center fixing centralizer is a Lie group {#cenfix} As we noted in Section [1.1](#outl){reference-type="ref" reference="outl"}, a general method in proving centralizer rigidity for partially hyperbolic diffeomorphism is to separate the centralizer into the center-fixing part and the action on the leaf space of the center foliation through the following short exact sequence: $$1\to \mathcal{CZ}^r(f)\to \mathcal{Z}^r(f)\to \mathcal{Z}^r(f)/\mathcal{CZ}^r(f)\to 1.$$ In this section, we shall deal with the center fixing part and prove Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}. We state and prove a general condition for the centralizer of a partially hyperbolic diffeomorphism to be a Lie group. **Theorem 17**. *Let $M$ be a Riemannian manifold. Let $f$ be a partially hyperbolic $C^{r}$ diffeomorphism of $M$ with $r\ge 2$. Suppose $f$ is accessible, dynamically coherent, center $r$-bunched, has global holonomy on a leaf $\mathcal{W}^c_f(x_0)$, and satisfies the narrow band spectrum with critical regularity $s(f)<r$. Then its $W^c$-fixing, $C^r$ centralizer $\mathcal{CZ}^r(f)$ is a Lie group acting freely and properly on each center leaf $\mathcal{W}^c_f(x_0)$ and we have $\mathrm{dim} \;\mathcal{CZ}^r(f)\le \mathrm{dim}\; \mathcal{W}^c$.* Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} is a generalization of Theorem 3 in [@DWX23] where the authors proved that for an accessible isometric extension $f_0$ of an Anosov diffeomorphism of a closed nilmanifold, the centralizer of a $C^1$ perturbation $f$ of $f_0$ is a Lie group acting freely on the compact center leaves. The main difference and difficulty here is that the center leaves can be non-compact. We postpone the proof to Section [3.2](#pfczlieg){reference-type="ref" reference="pfczlieg"}. Applying Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} to the setting of Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}, we see that the center fixing centralizer is a Lie group. **Corollary 18**. *Let $f\in \mathrm{Diff^r}(G/\Gamma), r\ge 1$ be a perturbation of an element of the Weyl chamber flow as in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}, then its center fixing centralizer $\mathcal{CZ}^r(f)$ is a Lie group that acts properly and freely on a generic center leaf $\mathcal{W}^c_f(x_0)$. In particular, we have $\mathrm{dim}(\mathcal{CZ}^r(f))\le k$ where $k=\mathrm{rank}(G)$.* *Proof.* We simply check that $f$ as a $C^1$-perturbation of $f_0$ satisfy the conditions in Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}. Indeed, $f$ is accessible by Proposition [Proposition 9](#faccess){reference-type="ref" reference="faccess"}. By Theorem [Theorem 7](#leafconj){reference-type="ref" reference="leafconj"}, $f$ is dynamically coherent. Moreover, $f$ is center $r$-bunched and satisfy the narrow band spectrum for any $r>s(f_0)$ since the $f_0$ satisfy these conditions and these conditions are open. We also note that although $f$ does not have global holonomy on every leaf of $\mathcal{W}^c_f$, $f$ has global holonomy when lifted to $G$ and thus has global holonomy on every generic center leaf of $\mathcal{W}^c_{f}$. Therefore, applying Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} gives us the desired result. ◻ ## Other applications We may also apply Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} to the setting of [@DWX] and [@BG]. When applied to systems with $1$-dimensional center, since Lie groups acting freely and properly on $\mathbb{R}$ can only be $\mathbb{R}$ or $\mathbb{Z}$, the center fixing centralizer $\mathcal{CZ}^r(f)$ must be $\mathbb{Z}$ or $\mathbb{R}$. As a corollary, we may remove the volume-preserving conditions in Theorem 1 of [@DWX] and and Theorems A,B and C in [@BG] and prove the following. **Theorem 19**. *Let $\phi_t:T^1M\to T^1M$ be the geodesic flow on a closed negatively curved, locally symmetric manifold $M$ and $f\in \mathrm{Diff}^\infty(T^1 M)$ be a $C^1$-small perturbation of $\phi_{t_0}$, then $\mathcal{Z}^\infty(f)$ is either $\mathbb{Z}$ or $\mathbb{R}$, i.e. $f$ either has virtually trivial centralizer or $f$ embeds into a smooth flow.* **Theorem 20**. *Let $f$ be a partially hyperbolic diffeomorphism on a 3-manifold $M$. If $f$ is a discretized Anosov flow and $\pi_1(M)$ is not virtually solvable then either $\mathcal{Z}^\infty(f)$ is virtually trivial or $f$ embeds into a smooth Anosov flow.* We use the following result in [@DWX] and [@BG] that the centralizer is virtually center fixing. **Proposition 21** (Proposition 37,[@DWX]; Proposition 2.4, [@BG]). *Let $f$ be a discretized Anosov flow satisfying the conditions in Theorem [Theorem 19](#1dcen1){reference-type="ref" reference="1dcen1"} or Theorem [Theorem 20](#1dcen2){reference-type="ref" reference="1dcen2"}, then $\mathcal{Z}^r(f)/\mathcal{CZ}^r(f)$ is finite for $r\ge 1$.* Therefore, applying Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} gives us Theorem [Theorem 19](#1dcen1){reference-type="ref" reference="1dcen1"} and Theorem [Theorem 20](#1dcen2){reference-type="ref" reference="1dcen2"}. *Proof of Theorem [Theorem 19](#1dcen1){reference-type="ref" reference="1dcen1"} and Theorem [Theorem 20](#1dcen2){reference-type="ref" reference="1dcen2"}.* For Theorem [Theorem 19](#1dcen1){reference-type="ref" reference="1dcen1"}, since $M$ is closed negatively curved, locally symmetric, $\phi_{t_0}$ has narrow band spectrum, see [@DWX Lemma 26]. Therefore, $f$ also has narrow band spectrum. Moreover, $f$ is dynamically coherent by Theorem [Theorem 7](#leafconj){reference-type="ref" reference="leafconj"} and it is center $r$-bunched since the center is $1$-dimensional. Lifting $f$ to the universal cover of $M$ shows that global holonomy exists on a generic geodesic. For Theorem [Theorem 20](#1dcen2){reference-type="ref" reference="1dcen2"}, since $M$ is $3$-dimensional, $f$ satisfies the narrow band spectrum and the center bunching condition. Lifting $f$ to the universal cover of $M$ shows that global holonomy exists on a generic geodesic. Thus we may apply Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} to see that $\mathcal{CZ}^r(f)$ is a Lie group acting freely and properly on $\mathbb{R}$. Thus it must be $\mathbb{Z}$ or $\mathbb{R}$. Since $\mathcal{Z}^r(f)/\mathcal{CZ}^r(f)$ is finite, the centralizer $\mathcal{Z}^r(f)$ is virtually trivial or $\mathbb{R}$. ◻ ## Proof of Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"} {#pfczlieg} In this section, we prove Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}. We use similar ideas as in Section 4 of [@DWX23]. First we construct a continuous center fixing centralizer. Next, we prove that the continuous centralizer is closed under the topology of point-wise convergence in $\mathrm{Homeo}(\mathcal{W}^c_f(x_0))$, for this step, we use a different method from [@DWX23] to treat the non-compact leaves. Finally, we use the normal form theory to raise the smoothness of the continuous maps to being smooth. Let $f$ be be a normally hyperbolic diffeomorphism on a manifold $M$ with center foliation $\mathcal{W}^c_f$. Recall that we denote by $$\mathcal{Z}^r(f)=\{g\in \mathrm{Diff}^r(M):\; f\circ g=g\circ f\}$$ the $C^r$ centralizer of $f$, and denote $$\mathcal{CZ}^r(f)=\{g\in \mathrm{Diff}^r(M):\; f\circ g=g\circ f,\; g(\mathcal{W}^c_f(x))=\mathcal{W}^c_f(x),\; \forall\; x\in M\}$$ to be $\mathcal{W}^c_f$-fixing, $C^r$ centralizer of $f$. Following the notations in [@DWX23], we denote by $$\mathcal{CZ}_*(f)=\{g\in \mathrm{Homeo}(M): f\circ g=g\circ f; g(\mathcal{W}^c_f(x))=\mathcal{W}^c_f(x),\forall\; x\in M;$$$$g(\mathcal{W}^{*}_f)=\mathcal{W}^{*}_f, *=s,u\}$$ the continuous centralizer of $f$ which fixes each $\mathcal{W}^c_f$ leaf and preserves the stable and unstable foliations of $f$. We denote by $\mathcal{CZ}^r_*(f)$ the elements of $\mathcal{CZ}_*(f)$ which is locally uniformly (with respect to different leaves) $C^r$-smooth along the leaves of $\mathcal{W}^c_f$. Clearly, we have by definition that $\mathcal{CZ}^r(f)\subset \mathcal{CZ}^r_*(f)\subset \mathcal{CZ}_*(f)$. An easy consequence of $\mathcal{CZ}_*(f)$ preserving the $s/u$-foliations and fixing the center leaves is that it must commute with any $su$-holonomy with respect to the center leaves. **Proposition 22**. *Let $g\in \mathcal{CZ}_*(f)$, and let $x_0\in M$ be the fixed point as in Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}, then $g|_{\mathcal{W}^c_f(x_0)}$ commutes with any global $su$-holonomy $h^f_\gamma:\mathcal{W}^c_f(x_0)\to \mathcal{W}^c_f(x_0)$ of $f$ for any $su$-path $\gamma$ starting and ending in $\mathcal{W}^c_f(x_0)$.* Using this observation and the fact that $f$ is accessible and center $r$-bunched so that the $su$-holonomies act transitively $C^r$ on every center leaf, we may prove that every continuous map in $\mathcal{CZ}_*(f)$ is in fact $C^r$ smooth along the $\mathcal{W}^c_f$- leaves. **Lemma 23**. *Under the assumptions of Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}, we have $\mathcal{CZ}^r_*(f)=\mathcal{CZ}_*(f)$.* Furthermore, we may show that the group $\mathcal{CZ}_*(f)$ is locally compact, again using the fact that it commutes with the $su$-holonomies. **Proposition 24**. *Under the assumptions of Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}, the group $\mathcal{CZ}_*(f)\subset \mathrm{Homeo}(M)$ is locally compact under the topology of point-wise convergence.* We postpone the proofs of Lemma [Lemma 23](#CrWc){reference-type="ref" reference="CrWc"} and Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"} to Section [3.3](#pflem){reference-type="ref" reference="pflem"}. Now we prove Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}. *Proof of Theorem [Theorem 17](#czlieg){reference-type="ref" reference="czlieg"}.* Since $f$ is accessible, the $su$-holonomies of $f$ act transitively on any $\mathcal{W}^c_f$-leaf, and any element of $\mathcal{CZ}_*(f)$ fixing one point would be the identity on $\mathcal{W}^c_f(x_0)$. Therefore by Lemma [Lemma 23](#CrWc){reference-type="ref" reference="CrWc"} and Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"}, $\mathcal{CZ}_*^r(f)$ is a locally compact group acting freely by diffeomorphisms on $\mathcal{W}^c_f(x_0)$. Thus to prove the desired result, we only need to show that the group $\mathcal{CZ}^r(f)\subset \mathcal{CZ}_*(f)$ is closed under the the topology of point-wise convergence on $\mathcal{W}^c_f(x_0)$ . Suppose $g_n\to g$ in $\mathcal{CZ}^r_*(f)$ under the topology of point-wise convergence. Since $f$ satisfies the narrow band spectrum, by normal form theory, any $g_n\in \mathcal{CZ}^r(f)$ can be written as a polynomial map of bounded degree and coefficients under locally uniform $C^\infty$ coordinate change, thus $g$ is also locally uniformly $C^\infty$ smooth along $\mathcal{W}^s$ and $\mathcal{W}^u_f$. Moreover, $g$ is $C^r$ along $\mathcal{W}^c_f$ by Lemma [Lemma 23](#CrWc){reference-type="ref" reference="CrWc"}. Therefore, since $f$ is dynamically coherent, by Journé's lemma [@Journe], $g$ is $C^r$ smooth along $\mathcal{W}^{sc}_f$. Applying Journé's lemma again, we see that $g$ is $C^r$ smooth on $M$. Therefore $\mathcal{CZ}^r(f)$ is a closed subgroup of $\mathcal{CZ}_*(f)$, hence it is a locally compact group acting freely and properly on $\mathcal{W}^c_f(x_0)$ by $C^r$ diffeomorphism. Therefore by [@Hilsch Theorem 5], $\mathcal{CZ}^r(f)$ is a Lie group acting freely on $\mathcal{W}^c_f(x_0)$. This gives us the desired result. ◻ ## Proof of local compactness {#pflem} We first use the following proposition which is a slight variant of Proposition 11 in [@DWX23] to show that we can raise of regularity of $g\in \mathcal{CZ}_*(f)$ to being $C^r$ on the center leaves. **Proposition 25**. *Let $g:M\to M$ be a continuous map on a $C^r$ Riemannian manifold $M$, where $1\le r\le \infty$. If $\mathcal{Z}^r(g)$ acts transitively on $M$, then $g$ is a $C^r$ diffeomorphism of $M$.* *Proof.* Take the image manifold $\Gamma(g)=\{(x,g(x))\in M\times M:x\in M\}\subset M\times M$. Since $\mathcal{Z}^r(g)$ acts transitively on $M$, it induces an transitive action by $C^r$ maps on $\Gamma(M)$ by $\{\phi\times \phi:\phi\in \mathcal{Z}^r(g)\}$. Thus the manifold $\Gamma(g)$ is a locally compact $C^r$-homogeneous manifold. Therefore by Theorem B of [@wilkcoh], the manifold $\Gamma(g)$ is $C^r$. Take the projections $p_1,p_2$ from $M\times M\to M$. They are clearly $C^2$ homeomorphisms from $\Gamma(g)\to M$. Thus using Sard's Theorem, we may pick a point $x\in\Gamma(g)$ such that $p_1$ has $C^r$ inverse in a neighbourhood $U_x\subset \Gamma(g)$ of $x$, therefore $g=p_2p_1^{-1}$ is a $C^r$ map in $U_x$. Now for any other $y\in M$, picking $h\in \mathcal{Z}^r(g)$ with $h(x)=y$, we see that $g\mid_{h(U)}=h\circ g\mid_U \circ h^{-1}$ is also $C^2$. Therefore $g$ is $C^r$ at any $y\in M$, hence it is a $C^r$ diffeomorphism $M$. ◻ Applying the proposition to $M=\mathcal{W}^c_f(x_0)$ and $g$ restricted to $\mathcal{W}^c_f(x_0)$, we may prove Lemma [Lemma 23](#CrWc){reference-type="ref" reference="CrWc"}. *Proof of Lemma [Lemma 23](#CrWc){reference-type="ref" reference="CrWc"}.* By $r$-bunching of $f$, the global $su$-holonomies of $f$ from $\mathcal{W}^c_f(x_0)$ to itself is $C^r$. Since $f$ is accessible, the global $su$-holonomies act transitively on $\mathcal{W}^c_f(x_0)$. Therefore we see that for any $g\in \mathcal{CZ}_*(f)$, $\mathcal{Z}^r(g)$ acts transitively on $\mathcal{W}^c_f(x_0)$ since it contains all the $su$-holonomies of $f$ from $\mathcal{W}^c_f(x_0)$ to itself. Furthermore, $g$ is uniformly $C^r$ between nearby leaves since it commutes with holonomies between $\mathcal{W}^c_f$ leaves, and the holonomies are $C^r$-continuous with respect to the base points by Theorem [\[crhol\]](#crhol){reference-type="ref" reference="crhol"}. Therefore we have $\mathcal{CZ}_*(f)=\mathcal{CZ}_*^r(f)$. This proves Lemma [Lemma 23](#CrWc){reference-type="ref" reference="CrWc"}. ◻ Now we proceed to prove Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"}. For $g_n\in \mathcal{CZ}_*(f)$ in a neighborhood of the identity map in $\mathrm{Homeo}(M)$, fix $x\in \mathcal{W}^c(x_0)$, we may replace $g_n$ by a sub-sequence and assume that $$g_n(x)\to x_*\in M$$ as $n\to \infty$. Therefore, for any global $su$-holonomy $h_{x\to z}$ that connects $x$ and $z\in M$, we have $$g_n(z)=g_n(h_{x\to z}(x))=h_{x\to z}(g_n(x))\to h_{x\to z}(x_*)$$ for $n\to \infty$ since the holonomy $h_{x\to z}$ is a diffeomorphism on from $\mathcal{W}^c(x_0)$ to $\mathcal{W}^c(z)$ commuting with elements of $\mathcal{CZ}_*(f)$. Therefore for any $z\in M$, the limit $\lim_{n\to \infty} g_n(z)$ exists and is equal to $$\lim_{n\to \infty} g_n(z)=h_{x\to z}(x_*)$$ for any global $su$-holonomy $h_{x\to z}$ from $x$ to $z$. Therefore to prove Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"} we only need to show that the map $M \to M: z\mapsto h_{x\to z}(x_*)$ is continuous: if $g_n$ converges to a continuous map $g$, then since $f$ and the stable, unstable, center foliations are continuous, $g$ also commutes with $f$, fixes $\mathcal{W}^c_f$ and preserves $\mathcal{W}^s_f$ and $\mathcal{W}^u_f$, hence we have $g\in \mathcal{CZ}_*(f)$. The idea is to show first that one $su$-holonomy can be approximated by nearby $su$-holonomies that covers an open neighbourhood and then use accessibility of $f$ to carry this property to the whole manifold. First we define the following notion for two $su$-paths to be close. **Definition 9** ($\epsilon$-near $su$-path). *We say that two $su$-paths $\gamma_1=[x_0,x_1,...,x_k]$ and $\gamma_2=[y_0,y_1,...,y_k]$ with are $\epsilon$-near if each leg is $\epsilon$-near.* *More precisely,$\gamma_1$ and $\gamma_2$ are $\epsilon$-near if $d_{\mathcal{W}^c}(x_i,y_i)<\epsilon$ and $$d_M([x_i,x_{i+1}],[y_i,y_{i+1}])<\epsilon,$$ where $[x_i,x_{i+1}]$ is the $s/u$-segment in $\gamma_1$ starting at $x_i$ and ending at $x_{i+1}$ and $[y_i,y_{i+1}]$ is defined similarly.* It is clear by definition and by the compactness of $M$ that near enough $su$-paths would give us $su$-holonomies that are $C^0$-close. **Lemma 26**. *Fix $x_0\in M$, for every $\epsilon>0$, $R>0,k\in N$, there exists $\delta>0$, such that for any $\delta$-near $su$-paths $\gamma_1$ and $\gamma_2$ with $\le k$ legs and lengths $\le R$ from $x_1\in W^c(t_0)$, $d_{W^c}(x_1,t_0)<R$ to $y_1\in W^c(x_0)$ and $x_2\in W^c(x_0)$ to $y_2\in W^c(z)$ respectively, the global $su$-holonomies $h_1$ and $h_2$ that takes the $su$-paths $\gamma_1$ and $\gamma_2$ satisfy $d_{\mathcal{W}^c}(h_1(x_0),h_2(x_0)) <\epsilon$.* Furthermore, we have the following result on "continuity\" of $su$-paths. **Lemma 27** (Local continuity of holonomy, see Proposition 7.3 [@ASV]). *Let $f$ be a partially hyperbolic diffeomorphism on a complete Riemann manifold $M$. Suppose $f$ is accessible, then for any $x\in M$, there exist $y\in M$ and a $su$-path $\gamma$ from $x$ to $y$, such that for any $\epsilon>0$, there exists $\delta>0$, so that for any $z\in B(y,\delta)$, there exists a $su$-path from $x$ to $z$ that's $\epsilon$-near $\gamma$.* The lemmas uses the exact same proof as Proposition 7.3 in [@ASV] which uses a Baire category argument, although the formulations of the results there are slightly different. Now we are ready to prove Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"} on the local compactness of $\mathcal{CZ}_*(f)$ in $\mathrm{Homeo}(M)$ under the topology of point-wise convergence. *Proof of Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"}.* As we already noted, we only need to prove that $M \to M: z\mapsto h_{x\to z}(x_*)$ is continuous. Now pick $y$ and $\gamma$ as in lemma [Lemma 27](#ctshol){reference-type="ref" reference="ctshol"}, we claim that $M\to M: z\mapsto h_{x\to z}(x_*)$ is continuous at $z=y$. Indeed, picking $R=max\{l_{su}(\gamma),d_{W^c}(x_0,y)\}$ where $l_{su}(\gamma)$ denotes the $su$-length of $\gamma$, we see from Lemma [Lemma 26](#sucont){reference-type="ref" reference="sucont"} that for any $\epsilon>0$, there exists $\delta_1>0$ such that any for $\gamma^\prime$ from $x$ to $z\in \mathcal{W}^c$ that is $\delta_1$-near $\gamma$, we have $d_{\mathcal{W}^c}(h_{x\to y}(x_0),h_{x\to z}(x_*))<\epsilon$. Furthermore from Lemma [Lemma 27](#ctshol){reference-type="ref" reference="ctshol"}, there exists $\delta>0$ such that any for $z\in B(y,\delta)$, there exists $su$-path from $x$ to $z$ that's $\delta_1$-near $\gamma$. Therefore, for any $z\in B(y,\delta)\cap \mathcal{W}^c(x)$, we have $d_{\mathcal{W}^c}(h_{x\to y}(x_0),h_{x\to z}(x_*))<\epsilon$. This shows that $h_{x\to z}(x_*)$ is continuous at $z=y$. Now for any other point $y^\prime\in M$, take a $su$-holonomy $h_{y\to y^\prime}$ that takes $y$ to $y^\prime$. Then $h_{y\to y^\prime}$ is a diffeomorphism on $W^c$. Thus it takes the neighbourhood $B(y)$ to a open neighbourhood $B(y^\prime)$. For any $z^\prime\in B(y^\prime)$, take $z$ to be its pre-image in $B(y)$ under $h_{y\to y^\prime}$, then we have $h_{x\to z^\prime}(x_*)=h_{z\to z^\prime}h_{x\to z}(x_*)=h_{y\to y^\prime}h_{x\to z}(x_*)$. $h_{y\to y^\prime}h_{x\to z}(x_*)$ is continuous with respect to $z^\prime$ since $z^\prime \mapsto z$ is smooth and $z\mapsto h_{x\to z}(x_*)$ is continuous. Thus $h_{x\to z}(x_*)$ is also continuous at any $z=y^\prime\in M$. Therefore, $g_n$ converges to a continuous map if $g_n(x_*)$ converge. This indicates that $\mathcal{CZ}_*$ is locally compact. ◻ # The centralizer virtually fixes the center leaves {#centgp} Take $M$ to be the quotient $X=G/\Gamma$ of a semisimple Lie group and $f$ to be the perturbation of an element of the Weyl chamber flow as in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}. We directly apply the following result of Witte [@Witte] to show that $\mathcal{CZ}^r(f)$ is a finite index subgroup of $\mathcal{Z}^r(f)$. **Theorem 28** (See Theorem 1.1 of [@Witte]). *Let $G$ be a connected semi-simple Lie group with finite center and no compact factor and let $\Gamma$ be an lattice in $G$ so that $G$ has no normal subgroup $N\simeq PSL(2,\mathbb{R})$ with $N\Gamma$ closed in $G$. Let $D$ be a connected Lie subgroup of $G$ and let $\mathcal{F}$ be the foliations of $G/\Gamma$ by co-sets of $D$. Suppose $\mathcal{F}$ has a dense leaf and $D$ is uni-modular.* *Then there exists $k>0$ depending only on $G$, such that for any homeomorphism $f: G/\Gamma\to G/\Gamma$ that preserves the leaves of $\mathcal{F}$, $f^k$ must be a composition of an affine map and a homeomorphism $G/\Gamma\to G/\Gamma$ that fixes every leaf of $\mathcal{F}$.* **Remark 29**. *In [@Witte] the theorem was stated for semi-simple Lie group $G$ with trivial center instead of finite center and in that case $f$ itself would be a composition of an affine map and a homeomorphism that fixes every leaf of $\mathcal{F}$. The proof in [@Witte] only used trivial center for applying Mostow's rigidity theorem so when adapting to finite center, we may simply lift $f$ to a power to ensure $f_*:\Gamma\to\Gamma$ may be extended to an affine action on $G$.* Applying this to our setting, and using the leaf conjugacy to bring the diffeomorphisms $g\in \mathcal{Z}^r(f)$ to homeomorphisms preserving cosets of $D$, we see that $\mathcal{CZ}^r(f)$ is a finite index subgroup of $\mathcal{Z}^r(f)$. **Corollary 30**. *Let $f_0,f, G$ be as in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}, then we have $|\mathcal{Z}^r(f)/\mathcal{CZ}^r(f)|<C$, where $C$ is a constant depending only on $G$.* *Proof.* Take $\phi: G/\Gamma\to G/\Gamma$ to be the leaf conjugacy between $f$ and $f_0$ (given in Theorem [Theorem 7](#leafconj){reference-type="ref" reference="leafconj"}) that sends center foliations of $f_0$ to that of $f$, then since $g\in \mathcal{Z}^r(f)$ preserves the center foliation of $f$, we see that $\phi g\phi^{-1}$ is a homeomorphism preserving the center foliations $\mathcal{W}^c_{f_0}$ which are co-sets of the maximal torus $D$. Thus by Theorem [Theorem 28](#witte){reference-type="ref" reference="witte"}, since affine maps on $G$ acts on the cosets of the maximal tori by the Weyl group $W$ up to a finite index, we see that $\phi g\phi^{-1}$ fixes every leaf of $\mathcal{W}^c_{f_0}$ up to a finite index depending only on $G$, which proves our claim. ◻ # Dichotomy of the centralizer {#dichcen} In this section, we prove Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"} on the dichotomy of the centralizer. The scheme of the proof is as follows. Step 1: We use ergodicity of $f$ to show that every $g\in \mathcal{CZ}^r(f)$ acts on the center leaves of $f$ asymptotically like a translation, proving the existence of a translation number $\rho(g)$ for every $g\in \mathcal{CZ}^r(f)$. Step 2: We use some estimate on the holonomies to show that if $g$ is not too far from identity at one point, then $g$ is very close to the translation by $\rho(g)$ at every point. Step 3: We show that if every $\rho(g)$ is in the direction of $\rho(f)$ then $\mathcal{CZ}^r(f)$ is at most $1$-dimensional. Step 4: In Section [6](#eph){reference-type="ref" reference="eph"} and Section [7](#wcf){reference-type="ref" reference="wcf"} we prove that if some $\rho(g)$ is not in the direction of $\rho(f)$ then $f$ is conjugate to an element of the Weyl chamber flow. ## The centralizer acts asymptotically by translations Let $f$ be a $C^1$-small perturbation of a generic element $f_0$ of the Weyl chamber flow on $X=G/\Gamma$. Let $\phi$ be the leaf conjugacy from $(X,\mathcal{W}^c_{f_0})$ to $(X,\mathcal{W}^c_f)$, as given in Theorem [Theorem 7](#leafconj){reference-type="ref" reference="leafconj"}. For any $g\in \mathcal{CZ}^r(f)$, we denote by $\hat{g}:=\phi g\phi^{-1}$ the conjugate of $g$ by the leaf conjugacy. Since $g$ fixes the center leaves of $f$, we see that $\hat{g}$ fixes the center leaves of $f_0$ which are the left cosets of $D$. We first note that up to some power $\hat{g}$ can be lifted to a center fixing diffeomorphism on the cover $G$ of $X$. Here we use the fact that $G$ has finite center. The proof is an easy exercise. **Lemma 31**. *Let $\hat{g}$ be any homeomorphism of $X=G/\Gamma$ preserving the center leaves of $f_0$, then $\hat{g}_*: \pi_1(X)\to \pi_1(X)=\Gamma$ must be of a finite order with the order bounded by a constant depending only on $G$.* From now on we assume $\hat{g}_*=id$, lifting $g$ to a finite order if necessary. Thus for $\hat{g}$ fixing the center leaves of $f_0$, we may lift it to a center-fixing diffeomorphism $\tilde{g}: G\to G$, and the element $\tilde{g}(y)y^{-1}\in D$ does not depend on the choice of $y\in x$ where $x=y\Gamma\in X$. We denote by $$\tilde{g}(x)-x:=\log (\tilde{g}(y) y^{-1}), y\in x=y\Gamma,$$ the corresponding element in the Cartan subalgebra $\mathfrak{h}$. We note that $f$ is ergodic since it is volume-preserving and center bunching, applying Theorem [Theorem 11](#accerg){reference-type="ref" reference="accerg"}. Combining the ergodicity of $f$ with the fact the that $f$ and $g$ commutes and $X$ being compact, we may define the translation vector of $g$ in the following way. (Note that this is the first place we use the condition that $f$ is volume-preserving.) **Proposition 32**. *Let $\mu$ be the volume measure. For any $g\in \mathcal{CZ}^r(f)$, suppose $g_*=id$, let $\tilde{g}$ be a lift of $\hat{g}$ to $G$ that preserves the center leaves, then the asymptotic translation $$\lim_{n\to\infty}\frac{\tilde{g}^n(x)-x}{n}$$ exists and is equal to a vector $\rho(g)=\int_{x\in X} (\tilde{g}(x)-x)d vol \in \mathfrak{h}\simeq \mathbb{R}^k$ for $\mu$-almost every $x\in G/\Gamma$.* *Proof.* First we note that since $f$ is volume-preserving and ergodic, any $g\in \mathcal{CZ}^1(f)$ is also volume-preserving: observe that the Radon-Nikodym derivative $\frac{g_*\mu}{\mu}$ is an $f$-invariant function. By Birkhoff ergodic theorem, since $\mu$ is $\tilde{g}$-invariant, we see that the function $$\psi(x)=\lim_{n\to\infty}\frac{\tilde{g}^n(x)-x}{n}=\frac{1}{n}\sum_{k=0}^{n-1}(\tilde{g}(\tilde{g}^k(x))-\tilde{g}^k(x))$$ exists for $\mu$-almost every $x\in X$ and is $\mu$-measurable. Moreover, we notice that $\psi(x)$ is an $f$-invariant function on $X$. Indeed, since $f$ and $g$ being commutative, we have $|\tilde{g}^n(\tilde{f}(x))-\tilde{f}(x)-(\tilde{g}^n(x)-x)|\le 2\max_{y\in X} |\tilde{f}(y)-y|$, which implies that $\psi(f(x))=\psi(x)$. Therefore since $f$ is ergodic on $(X,\mu)$, we see that $\phi$ is almost everywhere a constant on $X$. ◻ This shows that the long term behavior of any $g\in \mathcal{CZ}^r(f)$ is almost like a diagonal flow, up to the leaf conjugacy. ## Centralizer act almost by translations on $\mathcal{W}^c$ We next show that there is also a uniform bound on the behavior of $g$ at every point $x\in X$. The precise statement is as follows. **Proposition 33**. *For any $\epsilon>0,R>0$, there exists $\delta>0$, such that for any $f\in \mathrm{Diff}(X)$, $d_{C^1}(f,f_0)<\delta$, we have $$d_{\mathcal{W}^c_{f_0}}(\hat{g}(y),g_0(y))\le \epsilon e^{2d_{\mathcal{W}^c_{f_0}}(x, \hat{g}(x))} d_{\mathcal{W}^c_{f_0}}(x, \hat{g}(x))$$ for any $x,y\in \mathcal{W}^c_{f_0}(x_0)$ and any $g\in \mathcal{CZ}^r(f)$. Here $\hat{g}$ is the conjugate of $g$ by the leaf conjugacy and $g_0$ is a diagonal that depend only on $g$ and $x$ defined as above.* The idea of the proof is to exploit the fact that $g\in \mathcal{CZ}^r(f)$ commutes with any global $su$-holonomies from $\mathcal{W}^c_f(x_0)$ to itself. We postpone the proof to the end of this section. For any $g\in \mathcal{CZ}^r(f)$, we consider the following notion for $g$ to be close to a translation. **Definition 10**. *For any map $g\in \mathrm{Diff}(\mathcal{W}^c_f(\phi(x_0)))$, we say $g$ is *$\epsilon$-close to a translation*, if there exists an element of the Weyl chamber flow $g_0$ so that for any $y\in \mathcal{W}^c_{f_0}(x_0)$, $$\label{ntr} d_{\mathcal{W}^c_{f_0}}(\hat{g}(y),g_0(y))\le \epsilon d_{\mathcal{W}^c_{f_0}}(g_0(y),y).$$* Since $f$ is a $C^1$ perturbation of $f_0$, it is easy to see that $f$ is $\epsilon$-close to $f_0$. We would hope to show that the majority of $g\in \mathcal{CZ}^r(f)$ are also $\epsilon$-close to translations. First we make the following general observation. The proof is an elementary exercise. **Lemma 34**. *If $g^\prime$ and $g$ are $\epsilon$-close to translations $g_0$ and $g_0^\prime$ respectively, then if for some $c\le 1$, $d(g_0\circ g^\prime_0(y),y) \ge c (d(g_0(y),y)+d(g^\prime_0(y),y))$ then $g\circ g^\prime$ is $\frac{\epsilon}{c}$-close to translations.* *In particular, if $g$ is $\epsilon$-close to a translation $g_0$, then $g^{k}$ is $\epsilon$-close to the translation $g_0^k$ for any $k\in \mathbb{N}$.* For $g\in \mathcal{CZ}^r(f)$, using Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"}, we may show that $g$ is either uniformly very far from identity or $\epsilon$-close to a translation. **Corollary 35**. *For any $0<\epsilon<1$, there exists $\delta$ so for any $C^1$-perturbation $f$ of $f_0$ as in Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"} with $d_{C^1}(f,f_0)<\delta$, and for any $g\in \mathcal{CZ}^r(f)$, one of the following holds:* - *$g$ is $\epsilon$-close to the translation by $\rho(g)$;* - *for any $y\in \mathcal{W}^c_{f_0}(x_0)$, and any $k\in \mathbb{Z}$, $$|\tilde{g}(y)-\tilde{f}^k(y)|>100 \rho(f).$$* *Proof.* To avoid abuse of notations, we shall replace $\epsilon$ by $\epsilon^\prime$ in this proof and show that if the second case doesn't hold then $g$ is $\epsilon^\prime$-close to a translation so we can adjust the $\epsilon$ in Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"} towards our needs. If the second case doesn't hold, then there exists some $x\in \mathcal{W}^c_{f_0}(x_0)$ and and some $k\in \mathbb{Z}$ so that $d(\hat{(gf^{-k})}(x),x)<100 \rho(f)$. Therefore, Applying Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"} with $\epsilon$ small enough, we see that $gf^{-k}$ is $\epsilon^\prime$-close to a translation. Now we show that $g$ must satisfy Equation [\[ntr\]](#ntr){reference-type="ref" reference="ntr"}. If $d(g(x),x)<100/\epsilon^\prime \rho(f)$ for some $x$, then applying Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"} with $\epsilon$ small enough, we see that $g$ is $\epsilon^\prime$-close to a translation. We note that by definition if $g$ is $\epsilon$-close to a translation then it is $\epsilon$-close to the translation by $\rho(g)$. If $d(g(y),y)>1000$ for any $y$, since $gf^{-k}$ and $f^k$ are both $\epsilon-$near translations, take $g_0=(gf^{-k})_0\circ f^k_0$, we have $$d(\hat{g}(y), g_0(y))$$ $$\le d(\widehat{gf^{-k}}(\hat{f}^k(y)), (gf^{-k})_0(\hat{f}^k(y)))+ d((gf^{-k})_0(\hat{f}^k(y)), (gf^{-k})_0\circ f^k_0(y))$$ $$\le d(\widehat{gf^{-k}}(\hat{f}^k(y)), (gf^{-k})_0(\hat{f}^k(y)))+ d(\hat{f}^k(y), f^k_0(y))$$ $$\le \epsilon^\prime 100 \rho(f)+\epsilon d(f^k_0(y),y)$$ Combining with the fact that $|d(\hat{g}(y),y)-d(f^k(y),y)|\ge d(\hat{g}(f^{-k}(f^k(y),f^k(y))\ge 100(1+\epsilon^\prime) \rho(f)$, and $|d(g(y),y)|\ge 1000 \rho(f)$ we have $d(g_0(y),y)\ge 1/2 d(f^k_0(y),y)+200 \rho(f)$. Thus we have $$d(g(y), g_0(y))\le \epsilon^\prime 100 \rho(f)+\epsilon d(f^k_0(y),y)\le (\epsilon^\prime/2+2\epsilon) d(g_0(y),y).$$ Thus taking $\epsilon$ small enough, we shall get the desired result. ◻ In the lower rank case $\mathrm{rank}(G)=2$, we may show that in the second case of Corollary [Corollary 35](#trorf){reference-type="ref" reference="trorf"} where $|\tilde{g}(y)-\tilde{f}^k(y)|>100\rho(f), \forall k\in \mathbb{Z}, y\in \mathcal{W}^c_{f_0}(x_0)$, we have $\rho(g)\notin \mathbb{R}\rho(f)$. **Corollary 36**. *Assume $\mathrm{rank}(G)=2$, i.e. the Cartan subalgebra $\mathfrak{h}$ of real dimension $2$. Let $g,f$ be as in Corollary [Corollary 35](#trorf){reference-type="ref" reference="trorf"}, acting on $G/\Gamma$, then one of the following holds:* - *$g$ is $\epsilon$-close to the translation by $\rho(g)$;* - *$\rho(g)\notin \rho(f)\mathbb{R}$.* *Proof.* If $g$ is not $\epsilon$-close to the translation by $\rho(g)$, then by Corollary [Corollary 35](#trorf){reference-type="ref" reference="trorf"}, we have $d_{\tilde{W}^c}(g(x),f^k(x))>100\rho(g)$ for any $x\in X$ and $k\in \mathbb{Z}$. We take any decomposition of $\mathfrak{h}\simeq \mathbb{R}^2= \rho(f)\mathbb{R}\oplus V$ where $V$ is a one dimensional subspace of $\mathbb{R}^2$ transverse to $\rho(f)$. Take $\pi: \mathfrak{h}\to V$ to be the projection onto $V$. Then we have $\pi(\tilde{g}(x)-x)\neq 0$ for any $x\in X$. Therefore, we have $\pi(\rho(g))=\int_{x\in X} \pi(\tilde{g}(x)-x)\neq 0$ since $V$ is one dimensional, thus we get $\rho(g)\notin \rho(f) \mathbb{R}$. ◻ ## Dichotomy of centralizer Now combining the fact that any $g\in \mathcal{CZ}^r(f)$ has a well-defined asymptotic non-zero translation almost everywhere and that $\mathcal{CZ}^r(f)$ is a Lie group, we may obtain our main results. We recall that $g$ fixes every element of $\Gamma$ and for almost every $x\in X$, the translation vector $\rho(g)=\lim_{n\to \infty}\frac{\tilde{g}^n(x)-x}{n}=\int_X( \tilde{g}(x)-x )dvol$ in the Cartan subalgebra is well-defined. We shall be discussing whether $\rho(g)$ is a multiple of $\rho(f)$ to get our main theorems. Roughly speaking, if it is, then $\mathcal{CZ}^r(f)$ is of dimension $\le 1$ and is either $\mathbb{Z}$ or $\mathbb{R}$ if $\mathrm{rank}(G)=2$; if it doesn't, then there is a partially hyperbolic $\mathbb{Z}^2$-action on $X$ and $f$ must be conjugate to a diagonal action. **Proposition 37**. *Let $f$ be as in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}. If there exists some $g\in \mathcal{CZ}^r(f)$, with $\rho(g)\notin \rho(f)\mathbb{R}$, then $f$ is smoothly conjugate to an element of Weyl chamber flow. In particular, $\mathcal{CZ}^r(f)\simeq \mathbb{R}^{k}$ where $k=\mathrm{rank}(G)$.* We postpone the proof of Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"} to Sections [6](#eph){reference-type="ref" reference="eph"} and [7](#wcf){reference-type="ref" reference="wcf"}. In Section [6](#eph){reference-type="ref" reference="eph"} we show that if $\rho(g)\notin \rho(f)\mathbb{R}$ then then the action of $\mathbb{Z}^2\to \mathrm{Diff}^r(X): (k,l)\mapsto f^kg^l$ is a partially hyperbolic group action. In Section [7](#wcf){reference-type="ref" reference="wcf"}, we show that such a partially hyperbolic group action must be smoothly conjugate to a restriction of the Weyl chamber flow. Combining the results above, we see that $f$ is conjugate to $f_0$ unless $\rho(g)$ is a multiple of $\rho(f)$. But in this case, we know by Proposition [Corollary 35](#trorf){reference-type="ref" reference="trorf"} that if some $\rho(g)$ is in a bounded domain, then it translates every point by approximately $\rho(g)$ which is grossly in the direction of the translation of $f$. Then we shall see that $\mathcal{CZ}^r(f)$ is dimension $0$ or $1$ and is isomorphic to $\mathbb{R}$ or $\mathbb{Z}$ if $n=3$. **Proposition 38**. *For $f:X\to X=G/\Gamma$ a small perturbation of an element of the Weyl chamber flow as in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}, if for any $g\in \mathcal{CZ}^r(f)$, $\rho(g)\in \rho(f)\mathbb{R}$, then $\mathcal{CZ}^r(f)$ is a Lie group of dimension $\le 1$. Furthermore, if $G$ is of rank 2, then $\mathcal{CZ}^r(f)$ is isomorphic to $\mathbb{R}$ or $\mathbb{Z}$.* *Proof.* We fix a generic center foliation $\mathcal{W}^c_{f_0}(x_0)$ and make an equivalence to its Lie algebra $\mathfrak{h}$ by the logarithmic map at $x_0$ which is the Cartan subalgebra of $G$. Consider the map $\Psi:\mathcal{CZ}^r(f)\to \mathfrak{h}: g\mapsto \log(\hat{g}(x_0)x_0^{-1})$, then by our analysis in Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"}, this is a proper continuous embedding from a Lie group to $\mathfrak{h}$. Thus its image $\mathrm{Im}(\Psi)$ must be a sub-manifold of $\mathfrak{h}\simeq \mathbb{R}^k$. {#fig:enter-label width="80%"} Take the cone $C(f,10\epsilon)=\{y\in \mathfrak{h}: d(y,\rho(f)\mathbb{R})<10\epsilon |y|\}$. Now since $\rho(g)\in \rho(f)\mathbb{R}$, we have $\mathrm{Im}(\Psi)\cap B(0, 100\rho(f))\subset C(f,10 \epsilon)$ by Corollary [Corollary 35](#trorf){reference-type="ref" reference="trorf"}. Since the neighbourhood of $x_0$ in $C(f,10\epsilon)$ is a cone, $\mathrm{Im}(\Psi)$ cannot be an interior point of any manifold of dimension larger than $1$. Therefore, we see that the dimension of $\mathcal{CZ}^r(f)$ is $0$ or $1$. If $G$ is of rank $2$, then by Corollary [Corollary 36](#rank2){reference-type="ref" reference="rank2"}, $\rho(g)\in \rho(f)\mathbb{R}$ implies that $\mathrm{Im}(\psi)\subset C(f,10\epsilon)$. We claim that this implies that $\mathcal{CZ}^r(f)$ is $\mathbb{Z}$ or $\mathbb{R}$. In the case that $\mathrm{dim}(\mathcal{CZ}^r(f))=1$, if $\mathcal{CZ}^r(f)$ is not connected, take $N$ to be the connected component of the identity map. Then for any $g\notin N$, we may pick a $g_1\in N$, such that $d(\hat{g_1}(x_0),x_0)>3d(\hat{g}(x_0),x_0)$, $g_1$ is $\epsilon$-close to a translation with $\hat{g_1}(x_0)$ in a different cone of $C(f,10\epsilon)$ as $\hat{g}(x_0)$ (This is always possible by taking a $g_2$ close to identity in $N$ and applying one of its iterations). Therefore $\widehat{gg_1}(x_0)$ is in a different chamber as $\hat{g}(x)$. Furthermore $\widehat{gg_1}(x_0)$ and $\hat{g}(x)$ are connected since $g_1$ is connected with the identity. Thus $g$ and $gg_1$ must both be connected to the identity map, which poses a contradiction. Therefore we see that $\mathrm{Im}(\Psi)$ is connected and so $\mathcal{CZ}^r(f)\simeq \mathbb{R}$. In the case that $\mathrm{dim}(\mathcal{CZ}^r(f))=0$, we first note that $x_0$ is not a limit point of $\mathrm{Im}(\Psi)$ since by Proposition [Proposition 24](#CZcpt){reference-type="ref" reference="CZcpt"} the map $g\mapsto g(x_0)$ is proper. Thus there exists $g_1\in \mathcal{CZ}^r(f)$ such that $\hat{g_1}(x_0)$ attains the minimum distance to $x_0$, $g_1$ must be $\epsilon$-close to a translation since $d(g_1(x_0),x_0)\le d(f(x_0),x_0)$. Then for any other $g_2\neq id \in \mathcal{CZ}^r(f)$, consider $g_3\in \{g_1^kg_2:k\in \mathbb{Z}\}$ such that $\hat{g_3}(x_0)$ attains the minimum distance to $x_0$ among $\{g_1^kg_2:k\in \mathbb{Z}\}$. If it's larger than or equal to $d(\hat{g_1}(x_0),x_0)$ then we can always apply another $g_1$ or $g_1^{-1}$ to make the distance less. Therefore the minimum distance must be zero and we see that $g_1$ generates the whole group which implies that $\mathcal{CZ}^r(f)\simeq \mathbb{Z}$. ◻ Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"} comes as a direct corollary of Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"} and Proposition [Proposition 38](#conerig){reference-type="ref" reference="conerig"}. ## Proof of Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"} {#proof-of-proposition-z2dtof0} Now we prove Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"}. By Proposition [Proposition 22](#comhol){reference-type="ref" reference="comhol"}, for any center-fixing centralizer $g\in \mathcal{CZ}^r(f)$, we have $$g(y)=h^f_{x\to y}(g(x))$$ for any $x,y\in X$ and any $f$-$su$-holonomy $h^f_{x\to y}: \mathcal{W}^c(x)\to \mathcal{W}^c(y)$ of a path from $x$ to $y$. For any two points $x,y\in \mathcal{W}^c_f(x_0)$, take any $f$-$su$-holonomy from $\phi(x)$ to $\phi(y)$, we denote $\hat{h}^f_{x\to y}:=\phi^{-1}h^f_{\phi(x)\to\phi(y)}\phi$ to be the $su$-holonomy of $f$ pulled back to center leaves of $f_0$ by the leaf conjugacy. By the condition $g(y)=h^f_{x\to y}(g(x)),$ we see that $$\hat{g} (y)=\hat{h}^f_{x\to y}(\hat{g}(x))$$ for any $x,y\in \mathcal{W}^c_{f_0}(x_0)$. Fix any $x\in \mathcal{W}^c_{f_0}(x_0)$, we denote $$g_0(y)=h^{f_0}_{x\to y}(\hat{g}(x))$$ to be a diffeomorphism on $\mathcal{W}^c_{f_0}(x_0)$ given by the $su$-holonomies of $f_0$. Note that this map does not depend on the path and it is an element of the Weyl chamber flow on $X=G/\Gamma$, acting by translation on the leaf $\mathcal{W}^c_{f_0}(x_0)$. Intuitively, the difference between $\hat{g}$ and $g_0$ is simply the difference between the $su$-holonomies $\hat{h}^f$ and $h^{f_0}$, which should be small if $f$ is a small perturbation of $f_0$. We shall show that the diffeomorphism $\hat{g}$ is actually $C^0$ close to the translation $g_0$ on the leaf $\mathcal{W}^c_{f_0}(x_0)$ unless $\hat{g}$ is far enough from the identity. *Proof of Proposition [Proposition 33](#z2:dtof0){reference-type="ref" reference="z2:dtof0"}.* We prove the proposition in two cases: - Case 1: $|\tilde{g}(x)-x|\le 1$. We consider the map $\hat{h}^f_{x\to y}$, and identify $\mathcal{W}^c_{f_0}(x_0)$ with $\mathbb{R}^{n-1}$, for any $z\in \mathcal{W}^c_{f_0}(x_0)$, $$\hat{h}^f_{x\to y}(z)=\hat{h}^f_{x\to y}(x_0)+\int_{\gamma} D_{\dot{\gamma}}(\hat{h}^f_{x\to y})dt$$ $$=y+\int_{\gamma}D_{\dot{\gamma}}(\hat{h}^f_{x\to y})dt$$ $$=g_0(y)+ \int(D_{\dot{\gamma}}(\hat{h}^f_{x\to y})\mid_{x^\prime}-\mathrm{Id})dt$$ for any curve $\gamma(t)$ connecting $x$ and $z$. Now plug in $z=\hat{g}(x)$. We may always choose a $f$-$su$-path from $\phi(x)\to \phi(y)$ of $\le k$ legs and length $\le R$. Since $|x_0-z|<1$, the $f$-$su$-path given by the holonomy $h^f_{\phi(x)\to \phi(y)}$ starting at the point $x^\prime\in \gamma$ must also be of $\le k$ legs and of length $\le e^2 R$. Thus by Theorem [Theorem 12](#c1hol){reference-type="ref" reference="c1hol"}, there exists $\delta>0$, such that the operator norm $\|D(\hat{h}^f_{x\to y})\mid_{x^\prime}-D(h^{f_0}_{x\to y})\mid_{x^\prime}\|=\|D(\hat{h}^f_{x\to y})\mid_{x^\prime}-\mathrm{Id}\|<\epsilon$ for any $f\in \mathrm{Diff}^2(X)$ that's a $C^1$-$\delta$-small perturbation of $f_0$. Therefore, we have $$|\hat{h}^f_{x\to y}(z)-g_0(y)|=|\hat{g}(y)-g_0(y)|$$$$\le \|D(\hat{h}^f_{x\to y})\mid_{x^\prime}-\mathrm{Id}\|\cdot |x-z|\le \epsilon |x-\hat{g}(x)|$$. - Case 2: $|x-\hat{g}(x)|\ge 1$. In this case, for any $\epsilon_1>0$, we may take a small enough $\delta$ and consider the three "$su$-holonomies\", given as follows: (1)$\hat{h}^f_{x\to y}$ given by an $f$-$su$-path $\gamma_0$ between $\phi(x)$ and $\phi(y)$ such that it has $\le k$ legs and length $\le R$. This is displayed in Figure [2](#holonomy){reference-type="ref" reference="holonomy"} by the blue lines. \(2\) An $f_0$-holonomy $h^{f_0}_{x\to y^\prime}$ that's given by the unique $f_0$-$su$-path $\gamma_1$ which is $\epsilon_1$-near the path $\gamma_0$ such that each endpoint of $\gamma_0$ and $\gamma_1$ are in the same $\mathcal{W}^c_{f_0}$-leaf respectively. $\gamma_1$ is displayed in Figure [2](#holonomy){reference-type="ref" reference="holonomy"} by the red lines. \(3\) Another $f_0$-holonomy $h^{f_0}_{x\to y}$ that's given by a $f_0$-$su$-path $\gamma_2$ given as follows. Since $y^\prime$ is $\epsilon$-close to $y$, by Lemma [Lemma 27](#ctshol){reference-type="ref" reference="ctshol"} (or simply by computation in the Lie algebra), there exists a $f_0$-$su$-path $\gamma_2$ that is $\epsilon_1$-near $\gamma_1$ and connects $x$ and $y$. We pick any such path to be $\gamma_2$. In the heuristic graph this is represented by the pink lines. {#holonomy width="80%"} Now we claim that for any $\epsilon_1>0$, there exists some $\delta>0$, such that for any $z\in\mathcal{W}^c_{f_0}$, we always have $$d_{\mathcal{W}^c_{f_0}}(\hat{h}^f_{x\to y}(z),h^{f_0}_{x\to y^\prime}(z))\le \epsilon_1 e^{2d_{\mathcal{W}^c_{f_0}}(x,z)}\cdot l(\gamma_1),$$ where $l(\gamma_1)$ denotes the length of $\gamma_1$. This is because the $su$-path $\gamma_1^z$ of $f_0$, given by the $su$-holonomy $h^{f_0}_{x\to y^\prime}$ and starting at $z$, has length at most $e^{2d_{\mathcal{W}^c_{f_0}}(x,z)}\cdot l(\gamma_1))$, as given by the geometry of the leaves in the Lie group $G$. We can pick $\delta>0$ so that at any point, the angular distance between the stable or unstable subspace of $f_0$ and the conjugate of the stable or unstable subspace of $f$ differ by at some $\epsilon_1$ respectively. Therefore, we have $$d_{\mathcal{W}^c_{f_0}}(\hat{h}^f_{x\to y}(z),h^{f_0}_{x\to y}(z))$$$$\le d_{\mathcal{W}^c_{f_0}}(\hat{h}^f_{x\to y}(z),h^{f_0}_{x\to y^\prime}(z))+d_{\mathcal{W}^c_{f_0}}(h^{f_0}_{x\to y}(z),h^{f_0}_{x\to y^\prime}(z))$$$$\le \epsilon_1 e^{2d_{\mathcal{W}^c_{f_0}}(x,z)}\cdot l(\gamma_1) +d_{\mathcal{W}^c_{f_0}}(y,y^\prime)$$$$\le \epsilon \epsilon_1 e^{2d_{\mathcal{W}^c_{f_0}}(x,z)}\cdot l(\gamma_1) +\epsilon_1$$ Since $l(\gamma_1)\le R$, take $\epsilon_1\le \frac{\epsilon}{2R+2}$ and take $z=\hat{g}(x)$, we get the desired result.  ◻ # Existence of partially hyperbolic elements in the rigid case {#eph} In this section we show that if $\rho(g)\notin \rho(f)\mathbb{R}$ for some $g\in \mathcal{CZ}^r(f)$ then the centralizer $\mathcal{CZ}^r(f)$ contains "a lot of\" partially hyperbolic elements, so we get a "higher rank\" action of $G/\Gamma$ similar to a higher rank restriction of the Weyl chamber flow. Such a higher rank action is expected to be rigid based on the results of Damjanović and Katok [@DK] and Vinhage [@Vinhage]. The goal of this section is to prove the following. **Proposition 39**. *Let $f$ be a volume preserving $C^1$-perturbation of $f_0$ satisfying the conditions of Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"}, then there is a subgroup $f\in A\subset \mathcal{CZ}^r(f)$, $A\simeq \mathbb{Z}^2$ such that* - *$\{\rho(g):g\in A\}$ is a non-degenerate $\mathbb{Z}^2$ lattice in the Lie algebra $\mathfrak{h}$ of $D$;* - *Any $g\in A$ with $\rho(g)$ not in the Weyl chamber walls is partially hyperbolic such that the fine stable/unstable foliations of $f$ are uniformly contracted or expanded by $g$. The Lyapunov exponents of such $g$ is in the same Weyl chamber as $\rho(g)$.* A heuristic argument would be that if $\rho(g)$ lies in a Weyl chamber in the Weyl chamber picture of $D\simeq \mathbb{R}^n$, then $g$ asymptotically acts like the translation by $\rho(g)$ on each of the center leaf. We know that a translation by $\rho(g)$ exponentially contracts or exponentially expands the fine stable and unstable leaves of $f_0$ by the properties of Cartan subalgebra. Therefore, heuristically speaking, assuming that the geometry of the leaves of $f$ are similar to the geometry of the leaves of $f_0$, $g$ would also exponentially contracts of expands the fine stable and unstable leaves of $f$, implying that $g$ is partially hyperbolic. The exponents are given by $\rho(g)$, with signatures depending on the Weyl chamber $\rho(g)$ is in. We now try to make a rigorous argument with this picture in mind. This section is the second place where we use the volume-preserving condition in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"}, the first place is when we defined the translation length $\rho(g)$. Here, $f$ being volume-preserving ensures the existence of Lyapunov foliations and the ergodicity of $f$. ## Exponential contraction on leaves First we note that under the conditions in Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"} there are indeed candidates of partially hyperbolic elements in every Weyl chamber wall intersecting a 2-plane in the Cartan subalgebra. **Lemma 40**. *Let $\mathcal{CZ}^r(f)$ satisfy the conditions in Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"}, then $\{\rho(g):g\in \mathcal{CZ}^r(f)\}\subset \mathfrak{h}$ contains a non-degenerate $\mathbb{Z}^2$ lattice that's not inside any of the Weyl chamber walls.* *Proof.* Since any $g\in \mathcal{CZ}^r(f)$ is volume preserving, by definition of $\rho(g)$ we have $\rho(g_1g_2)=\rho(g_1)+\rho(g_2)$ for any $g_1,g_2\in \mathcal{CZ}^r(f)$. Thus since $\rho(g)\notin \rho(f)\mathbb{R}$, $\{\rho(g^kf^l):k,l\in \mathbb{Z}\}$ is a non-degenerate $\mathbb{Z}^2$ lattice in the Cartan subalgebra $\mathfrak{h}$. Since $f_0$ is generic, $\rho(f)$ does not lie on any of the Weyl chamber walls, hence the $\mathbb{Z}^2$ lattice is also degenerate. ◻ For any $1\le i,j\le n,i\neq j$, we denote by $\mathcal{W}^{ij,c}_{f_0}$ the unique integration of the vectors $E^i=(\delta_{(i,j)}(k,l) )_{1\le k\le n, 1\le l\le n}$ with the diagonal group $D$ in $G$. We denote by $$\mathcal{W}^{ij,c}_\#=\phi^{-1}(\mathcal{W}^{ij,c}_{f_0})$$ to be the topological leaves pulled over by the leaf conjugacy $\phi$. We denote by $$\mathcal{W}^i_\#:=\mathcal{W}^{ij,c}_\#\cap \mathcal{W}^{s/u}_f$$ to be a topological sub-foliation of $\mathcal{W}^{s/u}_f$. For any $g\in \mathcal{CZ}^r(f)$, $g$ preserves the topological foliation $\mathcal{W}^i_\#$ since $g$ fixes the center leaves, hence fixing the foliation $\mathcal{W}^{ij,c}_\#$ and $g$ preserves the foliation $\mathcal{W}^s_f$ and $\mathcal{W}^u_f$. First we notice that $g$ acts by contraction and expansion as we expect on the topological leaves $\mathcal{W}^i_\#$. **Lemma 41**. *Let $g\in \mathcal{CZ}(f)$ such that $\chi_i(\rho(g))\neq 0$, then there exists $\epsilon_0>0$ such that for almost every $x\in X$ and every $y\in \mathcal{W}^i_\#(x)$ with $d(x,y)<\epsilon_0$, there exists some $N>0$ such that for any $n>N$, we have $$d(g^n(x),g^n(y))\le e^{\chi_i(\rho(g))\alpha\cdot n/4} d(x,y)$$ if $\chi_i(\rho(g))<0$ and $$d(g^{-n}(x),g^{-n}(y))\le e^{-\chi_i(\rho(g))\alpha\cdot n/4} d(x,y)$$ if $\chi_i(\rho(g))>0$. Here $\alpha<1$ is the bi-Hölder constant of the leaf conjugacy $\phi$.* *Proof.* See the Figure [3](#lypest){reference-type="ref" reference="lypest"} for a more intuitive explanation. {#lypest width="80%"} We restrict our discussion to the case $\chi_i(\rho(g))<0$, for the other case, simply consider $g^{-1}$ instead of $g$. Since $\rho(g)$ is the translation number of $\hat{g}$ on $\mathcal{W}^c_{f_0}$, and it is not on the Weyl chamber walls, we may choose $N$ such that for any $n>N$, we have $|\tilde{g}^n(z)-z-n\rho(g)|\le \epsilon(g) n \rho(g)$ for almost every $z\in X$, where $\epsilon(g)$ is taken so that $\chi_i(t)<\chi_i(\rho(g))/2$ for any $t\in \mathbb{R}^{n-1}, d(t,\rho(g))<\epsilon(g)$. Thus, by the geometry of the $f_0$ foliations, we know that for almost every $z\in X$ and every $w\in \mathcal{W}^i_{f_0}(z)$, we have $$d(\hat{g}^n(z), \mathcal{W}^c_{f_0}(w)\cap \mathcal{W}^i_{f_0}(\hat{g}^n(z)))\le e^{n\chi_i(\rho(g))/2}d(z,w).$$ Thus, using the Hölderness of the leaf conjugacy $\phi$, for almost every $x\in X$ and every $y\in \mathcal{W}^i_\#$ with $d(x,y)\le \epsilon_0$, take $y^\prime=\phi(\mathcal{W}^i_{f_0}(\phi^{-1}x)\cap \mathcal{W}^c_f(y)$ and $y^\prime_n=\phi(\mathcal{W}^i_{f_0}(\phi^{-1}g^n(x))\cap \mathcal{W}^c_f(y)$, and writing $\chi_i=\chi_i(\rho(g))$ for short, then we have $$d(g^n(x),g^n(y))\le 2 d(g^n(x),y^\prime_n)\le 2C d(\hat{g}^n(\phi^{-1}x),\phi(y^\prime_n))^\alpha$$ $$\le 2C(e^{n\chi_i/2}d(\phi^{-1}(x),\phi^{-1} y^\prime))^\alpha$$ $$\le 4C e^{n \alpha \chi_i/2}d(\phi^{-1}(x),\phi^{-1} y))^\alpha$$ $$\le 4C^2 e^{n \alpha \chi_i/2}d(x,y)^{\alpha^2}.$$ Choosing $N$ even larger if necessary, we get the desired result. ◻ ## Proof of partial hyperbolicity Now we show that the leaves $\mathcal{W}^i_\#$ agree with the Lyapunov foliations $\mathcal{W}^i_f$ and prove that any $g\in \mathcal{CZ}(f)$ with $\rho(g)\neq 0$ and not on the Weyl chamber walls is partially hyperbolic. Note that we use the volume preserving condition here to show that $f$ is ergodic and to apply Pesin's theory. *Proof of Proposition [Proposition 39](#ph){reference-type="ref" reference="ph"}.* By lemma [Lemma 41](#gcontr){reference-type="ref" reference="gcontr"}, if $\chi_i(\rho(g))\neq 0$, then $g$ uniformly contracts $\mathcal{W}^i_\#$ under forward or backward iterations for almost every $x\in X$. Therefore, since $g$ is $C^2$ and volume preserving, we may apply Pesin theory to $g$ and we see that $\mathcal{W}^i_\#$ subfoliates the Pesin stable leaves $\mathcal{W}^{s,Pes}_g$ or the Pesin unstable leaves $\mathcal{W}^{u,Pes}_g$ of $g$ at almost every $x\in X$. Furthermore, by lemma [Lemma 40](#z2wcf){reference-type="ref" reference="z2wcf"}, for any $1\le i,j\le n,i\neq j$, we may choose finitely many $g_k \in \mathcal{CZ}(f),1\le k\le N_0$ on one side of the Weyl chamber wall $\{\chi_i=\chi_j\}$ but not on the same side of any other Weyl chamber walls so that 1\. $\mathcal{W}^i_\#\subset \mathcal{W}^{s}_{g_k},\forall 1\le k\le N_0$, for almost every $x\in X$, 2\. $\forall (i^\prime, j^\prime)\neq (i,j)$, there exists $1\le k^\prime\le N_0$ such that $\mathcal{W}^{i^\prime j^\prime}_\#\subset \mathcal{W}^{u}_{g_{k^\prime}}$. Therefore, by transversality of $\mathcal{W}^u_{g}$ and $\mathcal{W}^s_{g}$ at almost every point, we see that $$\mathcal{W}^i_\#=\cap_{k=1}^{N_0}\mathcal{W}^{s}_{g_k}$$ for the family of $g_k$ we chose. Therefore, at almost every $x\in X$, there exists $\epsilon(x)>0$, such that $\mathcal{W}^i_\#(x)\cap B(x, \epsilon(x))$ is a $C^1$-submanifold and $\mathcal{W}^i_\#$ is preserved by any $g\in \mathcal{CZ}^r(f)$. Therefore, we have an almost everywhere well-defined vector field $E^i_\#$ tangent to $\mathcal{W}^i_\#$ that's preserved by $Dg$ for every $g\in \mathcal{CZ}^r(f)$, in particular, it is preserved by $Df$. This show that at almost every $x\in X$, $E^i_\#(x)=E^{i^ \prime, j^ \prime}_f(x)$ for some $i^\prime(x),j^\prime(x)$. Combining with the ergodicity of $f$ which is given by $f$ being center-bunched and volume preserving, we have $E^i_\#(x)=E^{i^ \prime j^ \prime}_f(x)$ for some $i^\prime,j^\prime$ not depending on the point $x$ for almost every $x\in X$. Furthermore, the integration $\mathcal{W}^i_\#$ is $C^0$ close to $\mathcal{W}^i_f$, hence $E^i_\#\neq E^{i^\prime j^\prime}_f$ for $(i^\prime,j^\prime)\neq (i,j)$. Therefore, we may summarize that $E^i_\#(x)=E^i_f(x)$ at $\mu$-almost every $x\in X$, hence $\mathcal{W}^i_\#=\mathcal{W}^i_f$ at every point in $X$. Now using the fact that $g$ uniformly contracts or expands $E^i_\#$ at almost every $x\in X$, and $g$ is $C^2$, we may conclude that $g$ uniformly contracts or expands $E^i_f$ at every $x\in X$. Therefore, $g$ is partially hyperbolic. ◻ # Rigidity of Weyl chamber flow {#wcf} In this section we conclude the proof of Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"} using results on Weyl chamber flow rigidity established in [@DK], [@Vinhage], etc. Recall that we say a semisimple Lie group $G$ is *genuinely higher rank* if each of the simple components is of rank $\ge 2$. Let $V\subset \mathfrak{h}$ be a subgroup of the Cartan subalgebra that induces an action $\alpha_0: V\to \mathrm{Diff}(G/\Gamma), v\mapsto L_{\exp(v)}$ which is a restriction of the Weyl chamber flow. We call the restriction $\alpha_0$ *generic* if the restriction of every element of the Weyl group action on $V$ is non-zero. It is proved in Theorem 1.1, [@Vinhage] that a $C^1$-small smooth perturbation of a generic higher rank restriction of the Weyl chamber flow $\alpha_0$ on $X=G/\Gamma$ is $C^\infty$ conjugate to the restriction of a Weyl chamber flow. In our setting, let $f$ be a perturbation of an element of the Weyl chamber flow as in Theorem [Theorem 1](#cenrig){reference-type="ref" reference="cenrig"} and let $g\in \mathcal{CZ}^r(f)$ be a center fixing element of the centralizer with $\rho(g)\notin \rho(f)\mathbb{R}$, then we have a $\mathbb{Z}^2$-action $$\alpha:\mathbb{Z}^2\to \mathrm{Diff}(X); (k,l)\mapsto f^kg^l,$$ generated by $g$ and $f$. We may define another $\mathbb{Z}^2$ action $$\alpha_0:\mathbb{Z}^2\to \mathrm{Diff}(X); (k,l)\mapsto L_{\exp(k\rho(f)+l\rho(g))},$$ generated by left multiplication by $\exp(\rho(f))$ and $\exp(\rho(g))$ and we note that since $\rho(f)$ is generic, $\alpha_0$ is generic. $\alpha$ is not a priori a small perturbation of $\alpha_0$, so we cannot directly apply Theorem 1.1 of [@Vinhage]. However, we know that the Lyapunov foliation of $\alpha$ are close to those of $\alpha_0$ by the leaf conjugacy of $f$. More importantly, $\alpha(a)$ uniformly contracts or expands a Lyapunov foliation if $\alpha_0(a)$ does, for any $a\in \mathbb{Z}^2$, by Proposition [Proposition 39](#ph){reference-type="ref" reference="ph"}. Therefore, in this case, the same methods as in [@DK] and [@Vinhage] can be applied. Here, we shall present the same outline, but proving only the arguments that need to be adjusted. ## Cocycle formulation Given a group action $\alpha_1: A\to \mathrm{Diff}(X)$, we recall that a cocycle $\beta: A\times X \to D$ is a map satisfying $$\beta(ab, x)=\beta(a,\alpha_1(b)(x))\beta(b,x), \forall a,b\in A,$$ taking value in the diagonal group $D$. A cocycle is cohomologous to a constant if there exists homomorphism $s:A\to D$ and a transfer map $H:X\to D$ such that $$\beta(a,x)=H(\alpha_1(a)(x))s(a) H(x)^{-1},$$ for any $a\in A,x\in X$. We consider the following cocycle. **Proposition 42**. *Let $f\in C^\infty(X,X)$ be a $C^1$-small volume preserving perturbation of $f_0$ with some $g\in \mathcal{CZ}^r(f)$ with $\rho(g)\notin \rho(f)\mathbb{R}$, take the action $$\label{hatalpha} \hat{\alpha}: \mathbb{Z}^2\to \mathrm{Homeo}(X): (k,l)\mapsto \hat{f}^k\hat{g}^l,$$ then $$\label{cocbeta} \hat{\beta}(a,x)= (\widetilde{\hat{\alpha}(a)})(x) x^{-1}\exp(\rho( \alpha(a)))^{-1}$$ is a Hölder cocycle $\mathbb{Z}^2\times X\to D$ taking value in the maximal tori. Here we denote by $\widetilde{\hat{\alpha}(a)}$ the lifting of the homeomorphism $\hat{\alpha}(a)$ to $G$ that fixes the center leaves of $f_0$.* *Proof.* We only need to check that $\hat{\beta}$ is indeed a cocycle. Using the definition and the fact that $D$ is Abelian, we have $$\hat{\beta}(ab,x)=(\widetilde{\hat{\alpha}(ab)})(x) x^{-1}\exp(\rho( \alpha(ab)))^{-1}$$ $$=\widetilde{\hat{\alpha}(a)}(\widetilde{\hat{\alpha}(b)}(x))x^{-1}\exp(\rho( \alpha(b)))^{-1}\exp(\rho( \alpha(a)))^{-1}$$$$=\widetilde{\hat{\alpha}(a)}(\widetilde{\hat{\alpha}(b)}(x))((\widetilde{\hat{\alpha}(b)})(x))^{-1}\exp(\rho( \alpha(a)))^{-1} \cdot (\widetilde{\hat{\alpha}(b)})(x)x^{-1} \exp(\rho( \alpha(b)))^{-1}$$ $$=\hat{\beta}(a,\hat{\alpha}(b)(x))\cdot \hat{\beta}(b,x).$$ This proves our claim. ◻ We shall prove the following cocycle rigidity result for the action $\hat{\alpha}$. **Theorem 43**. *Let $\hat{\alpha}$ be the $\mathbb{Z}^2$-action given in Equation [\[hatalpha\]](#hatalpha){reference-type="ref" reference="hatalpha"}, then any Hölder continuous cocycle $$\beta: \mathbb{Z}^2 \times X\to D$$ of $\hat{\alpha}$ is cohomologous to a constant via a continuous transfer map $H$.* Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"} is a corollary of Theorem [Theorem 43](#cocrig){reference-type="ref" reference="cocrig"} by applying it to the cocycle $\hat{\beta}$ in Equation [\[cocbeta\]](#cocbeta){reference-type="ref" reference="cocbeta"}. The proof is exactly the same proof as for Theorem 1.1 in [@DK]. *Proof of Proposition [Proposition 37](#gph){reference-type="ref" reference="gph"}.* Take $\hat{\beta}$ as in Equation [\[cocbeta\]](#cocbeta){reference-type="ref" reference="cocbeta"}, then by Proposition [Proposition 39](#ph){reference-type="ref" reference="ph"} and Theorem [Theorem 43](#cocrig){reference-type="ref" reference="cocrig"}, $\hat{\beta}$ is cohomologous to a constant via a continuous transfer map $H$. Therefore, take $h_1:X\to D, h_1(x)=H(x)^{-1}\cdot x$, then we have for any $\hat{g}=\hat{\alpha}(a), a\in \mathbb{Z}^2$, $$h_1\circ \hat{g}(x)=s(a) exp(\rho(g))\cdot h_1(x)$$ for any $x\in X$. We claim that $h_1$ is a homeomorphism. If so, then $\hat{g}$ is topologically conjugate to a element of the Weyl chamber flow on $X$, for any $g\in \mathcal{CZ}^r(f), \rho(g)\neq 0$. Therefore $\hat{f}$, and hence $f$ is also topologically conjugate to an element of the Weyl chamber flow. Then by standard normal form theory on algebraic manifolds, we may raise the conjugacy to being smooth, see [@KS]. Now we prove the claim that $h_1$ is a homeomorphism. Since every element of $\Gamma$ is fixed by $\alpha(a)$ for any $a\in \mathbb{Z}^2$ by Lemma [Lemma 31](#gfixgamma){reference-type="ref" reference="gfixgamma"}, we may lift $h_1$ to the cover $G$. We note that if $h_1(x)=h_1(y)$, then $h_1(h^s(x))=h_1(h^s(y))$ for any $s$-holonomy $h^s$ for some $\hat{g}\in A$, since the semi-conjugacy maps contracted elements of $\hat{g}$ into contracted elements of $\exp(\rho(g))$. Therefore $h_1(h^{(n)}_{x\to y}(x))=h_1(x)$ for any $su$-path of $\hat{f}$ from $x$ to $y$ and any iterate $n$, that is, the whole $h_{x\to y}$ orbit of $x$ is in the pre-image $h_1^{-1}(\{h_1(x)\})$ which is bounded. However, since the holonomy group acts freely on a generic center leaf of $G$ by Proposition [Proposition 50](#hologroup){reference-type="ref" reference="hologroup"}, that's impossible unless $x=y$. ◻ ## Lyapunov cycles in general We first start with some analysis of the group of Lyapunov cycles of a partially hyperbolic group action $\alpha$. A path in $X$ given with each leg a piece of Lyapunov foliation of $\alpha$ is called an Lyapunov path of $\alpha$. We denote such a path by its endpoints $[x_0,x_1,...,x_k]$ with $x_i\in \mathcal{W}^{j}_\alpha(x_{i-1})$ for some $j$ and for any $1\le i \le k$, where $\mathcal{W}^j_\alpha:=\cap_{a:\chi_j(a)<0} \mathcal{W}^s_{\alpha(a)}$ are the Lyapunov foliations of $\alpha$. We call a Lyapunov path of $\alpha$ an $\alpha$-cycle if it starts and ends at the same point. We denote by $C_{x_0}(\alpha)$ the set of $\alpha$-cycles of $g$ that begins and end at $x_0$. We also denote by $C^G_{x_0}(\alpha)$ the set of $\alpha$-cycles of whose lifting to $G$ is still a closed path. A $su$-cycle in $C_{x_0}(\alpha)$ is said to be *stable* if there exits $g\in \mathrm{Im}(\alpha)$ such that the whole path lies in the stable manifold $\mathcal{W}^s_{g}(x_0)$. We denote by $S_{x_0}(\alpha)$ the closure of the normal subgroup of $C_{x_0}(\alpha)$ generated by stable cycles. We note that since the stable leaves are contractible we always have $S_{x_0}\subset C^G_{x_0}$. **Definition 11** ($su$-cycle functional). *Given a cocycle $\beta$ of a group action $\alpha$ on $X$. We define the $su$-cycle functional $P_\beta: C_{x_0}(\alpha)\to D$ to be given by: $$P_\beta: [x_0,x_1,...,x_k=x_0]\mapsto \prod_{i=0}^{k-1}p_\beta(x_i,x_{i+1})$$ where $p_\beta(x,y)=\lim_{n\to \infty}\beta(na,x)^{-1}\beta(na,y)$ for $y\in \mathcal{W}^i_\alpha(x)$ and $a\in A$ such that $\chi_i(\alpha(a))<0$.* The following proposition shows that a cocycle is cohomologous to a constant if and only if its $su$-cycle functional is trivial. **Proposition 44** (Proposition 4(1), Proposition 10(1), [@DK05]). *Let $\alpha: A\to \mathrm{Diff}(X)$ be an action with transitive Lyapunov foliations. Then a Hölder cocycle $\beta: A \times X\to D$ is cohomologous to a constant via a continuous transfer map $H$ if and only if $P_\beta:C_{x_0}(\alpha) \to D$ is trivial for some $x_0\in X$.* By definition of the $su$-cycle functional, any stable cycle must be in the kernel of the functional, hence we have the following. **Lemma 45** (Lemma 3.7, [@Vinhage]). *We have $S_{x_0}(\alpha)\subset ker P_\beta$ for any Hölder cocycle $\beta$ of a group action $\alpha$.* Therefore, the functional $P_\beta:C_{x_0}(\alpha)\to D$ factors through $C_{x_0}(\alpha)/S_{x_0}(\alpha)$. ## Lyapunov cycles of $\alpha_0$ and $\hat{\alpha}$ To show that the image of $C_{x_0}(\hat{\alpha})/S_{x_0}(\hat{\alpha})$ is actually small for the group action $\hat{\alpha}$, we compare it with the corresponding set for $\alpha_0$. It is proved in [@Vinhage] Section 4 that the image of the $su$-functional of the contractible cycles of $\alpha_0$ must be trivial. **Proposition 46** (Lemma 4.7 [@Vinhage] ). *Let $\alpha_0:\mathbb{Z}^2\to D\subset \mathrm{Diff}(X)$ be a generic restriction of the Weyl chamber flow on $X=G/\Gamma$, then $C^G_{x_0}(\alpha_0)/S_{x_0}(\alpha_0)$ is *minimally almost periodic*, i.e. it has no continuous homomorphism into locally compact groups.* Applied to the $su$-cycle functional, we have the following corollary for quotients of semi-simple Lie group. **Corollary 47** (See [@Vinhage], Section 6). *Let $\alpha_0:\mathbb{Z}^2\to D\subset \mathrm{Diff}(X)$ be a generic restriction of the Weyl chamber flow on $X=G/\Gamma$, then the periodic cycle functional $P_\beta$ of any cocyle $\beta$ of $\alpha_0$ is trivial restricted to $C^G_{x_0}(\alpha_0)$.* We can also show the same for $\alpha$-cycles. With some abuse of notations, we raise $\hat{\alpha}$ and $\alpha_0$ to the universal cover of $X$ without changing notations. We define a natural map $P$ from Lyapunov paths of $\hat{\alpha}$ to Lyapunov paths of $\alpha_0$, taking an $\hat{\alpha}$ Lyapunov path with endpoints $[x_0,x_1,...,x_k]$ to the unique $\alpha_0$-Lyapunov path $[y_0,y_1,..., y_k]$ such that $y_0=x_0$, $y_k\in \tilde{W}^c_{f_0}(x_k)$ and $y_i\in \tilde{W}_{\alpha_0}^j(y_{i-1})$ if $x_i\in \tilde{W}_{\hat{\alpha}}^j(x_{i-1})$ for any $1\le i \le k$. We may similarly define an inverse map $Q$ from Lyapunov paths of $\alpha_0$ to Lyapunov paths of $\hat{\alpha}$. Note that a priori, an Lyapunov cycle of $\hat{\alpha}$ may not be mapped to a cycle of $\alpha_0$. However, we notice that any stable cycle of $\hat{\alpha}$ must be mapped to a stable cycle of $\alpha_0$. Thus using Proposition [Proposition 46](#minper){reference-type="ref" reference="minper"}, we conclude that the contractible $\alpha_0$-cycles are also mapped to contractible $\alpha$-cycles. Proving the statement for the inverse map is a bit trickier. A detailed proof of this proposition can be found in Section 5, Propositions 5.1 and 5.4 in [@Vinhage]. **Proposition 48**. *The canonical maps $P$ and $Q$ are bijections from $C^G_{x_0}(\alpha_0)$ to $C^G_{x_0}(\hat{\alpha})$ and the bijection sends $S_{x_0}(\alpha_0)$ to $S_{x_0}(\hat{\alpha})$.* **Remark 49**. *We note that in [@Vinhage] the proof only used the smallness of the $C^0$ distance between the $\alpha$-foliations and the $\alpha_0$-foliations, which in our setting is guaranteed by the smallness of $C^1$-distance between $f$ and $f_0$.* As a corollary, we see that the holonomy group of $\hat{\alpha}$ is the same as the holonomy group of $\alpha_0$, thus it acts freely and transitively on the center leaves. **Proposition 50** (Proposition [@DK]; Corollary 5.6 [@Vinhage]). *The group of $su$-holonomies of $\hat{\alpha}$ acts freely and transitively on a generic center leaf $Dx_0$.* Therefore, the $su$-cycle functional $P_\beta$ factors through $C_{x_0}/ C^G_{x_0}=\Gamma$. However, since $G$ is $\mathbb{R}$-split, any lattice $\Gamma$ must map trivially into $D$ by Margulis superrigidity Theorem. **Proposition 51**. *Let $\Gamma$ be a cocompact lattice in a semisimple, $\mathbb{R}$-split, genuinely higher rank Lie group $G$ with no compact factor, then any homomorphism $\Gamma\to D$ must be trivial.* Therefore, Theorem [Theorem 43](#cocrig){reference-type="ref" reference="cocrig"} follows as a direct consequence of the propositions above. *Proof of Theorem [Theorem 43](#cocrig){reference-type="ref" reference="cocrig"}.* By Corollary [Corollary 47](#trifunc){reference-type="ref" reference="trifunc"}, Proposition [Proposition 48](#bijalpha){reference-type="ref" reference="bijalpha"} and Proposition [Proposition 51](#latticefin){reference-type="ref" reference="latticefin"}, the image of the $su$-cycle functional must be trivial. Therefore by Proposition [Proposition 44](#consttrivialfunctional){reference-type="ref" reference="consttrivialfunctional"} the $su$-functional of the cocycle $\beta$ must be trivial and thus $\beta$ must be cohomologous to the constant. ◻

arxiv_math

--- abstract: | In this paper, we mainly investigate the spreading dynamics of a nonlocal diffusion KPP model with free boundaries which is firstly explored in time almost periodic media. As the spreading occurs, the long-run dynamics are obtained. Especially, when the threshold condition for the kernel function is satisfied, applying the novel positive time almost periodic function, we accurately express the unique asymptotic spreading speed of the free boundary problem. address: | School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education,\ Beijing Normal University, Beijing 100875, China author: - Chengcheng Cheng - Rong Yuan bibliography: - refernon2.bib title: Spreading speeds of a nonlocal diffusion model with free boundaries in the time almost periodic media --- Nonlocal diffusion,Free boundary ,Time almost periodic media ,Asymptotic spreading speed ,Time almost periodic solution 35B15,35R09,35R35,35B40 # Introduction {#s1} Nonlocal diffusion equations take advantage of modeling the long-distance movements of species and the dense dispersal of populations [@murray2002mathematical; @2023-hutson2003evolution; @2023-kao2010random]. Considering the environmental change, seasonal succession and resource distribution, people often choose periodic models to describe environmental parameters [@2023-rawal2012criteria; @2023-souganidis2019; @2023-bao2020propagation; @2023-sun2022limiting]. However, natural fluctuations are difficult to be periodic. Almost periodicity is more likely to accurately describe natural recurrent changes and the almost periodic model can provide more biological insights to understand the invasion of populations and the propagation of epidemic diseases [@2023-diagana2007population; @2023-zhao2017dynamical]. It is well known that the spreading speed is an important indicator describing the scale of infectious disease transmission. The spreading speeds of nonlocal diffusion equations in the nonhomogeneous media have attracted increasingly more interest and attention in recent years [@2023-shen2010spreading; @2023-liang2020spreading]. However, the spreading speed of the nonlocal KPP model with free boundaries in the time almost periodic environment has not been considered, which is a worthwhile problem to study. In this paper, we mainly aim to investigate the long-time propagation dynamics and spreading speed of the following almost periodic nonlocal diffusion model $$\label{eq1-1} \left\{\begin{array}{ll} u_{ t}=d \int_{g\left(t\right)}^{h\left(t\right)} \kappa\left(x-y\right) u\left(t, y\right) \mathrm{d} y-d u+u f(t,x,u), & t>0,~ g\left(t\right)<x<h\left(t\right), \vspace{0.1cm} \\ u\left(t, h(t)\right)=u(t,g(t))=0, & t > 0, \vspace{0.1cm} \\ h^{\prime}\left(t\right)= \mu \int_{g\left(t\right)}^{h\left(t\right)} \int_{h\left(t\right)}^{\infty} \kappa\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t > 0, \vspace{0.1cm} \\ g^{\prime}\left(t\right)=-\mu \int_{g\left(t\right)}^{h\left(t\right)} \int_{-\infty}^{g\left(t\right)} \kappa\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t >0, \vspace{0.1cm} \\ u\left(0, x\right)=u_{ 0}\left(x\right),& x\in[-h_{0},h_0], \vspace{0.1cm} \\ h\left(0\right)=h_0, g\left(0\right)=-h_{0}, \end{array}\right.$$ where the initial function $u_0(x)$ satisfies $$\label{eq1-2} u_{0}(x) \in C(\left[-h_{0}, h_{0}\right]),~ u_{0}\left(-h_{0}\right)=u_{0}\left(h_{0}\right)=0,~ u_{0}(x)>0,~ x\in\left(-h_{0}, h_{0}\right).$$ Here $u(t,x)$ represents the population density at time $t$ on location $x$. We assume that the density of species is $0$ out of $[g(t), h(t)]$, where $g(t)$ and $h(t)$ are the leftward and rightward free boundaries to be determined later, respectively. The dispersal kernel $\kappa(x-y)$ represents the probability distribution that jumps from location $y$ to $x$, then $\kappa(x-y) u(t, y)$ is the rate at which the species arrive at $x$ from position $y$. And $\int_{g(t)}^{h(t)} \kappa (x-y) u(t, y) \mathrm{d} y$ is the rate at which the species reach $x$ from any other positions. Correspondingly, $\int_{-\infty}^{\infty} \kappa(x-y) u(t, x) \mathrm{d} y$ is the rate at which the species leave $x$ to go to any other positions. Let $d$ be a positive constant as the nonlocal diffusion coefficient, then $d\int_{g(t)}^{h(t)} \kappa(x-y) u(t, y) \mathrm{d} y-\int_{-\infty}^{\infty} \kappa(x-y) u(t, x) \mathrm{d} y$ can be seen as the nonlocal diffusion term. Moreover, similar to [@cao2019], suppose that the expanding rate of moving boundary is proportional to the outward flux at the boundary, that is, $$\label{gg} g^{\prime}\left(t\right)=-\mu \int_{g\left(t\right)}^{h\left(t\right)} \int_{-\infty}^{g\left(t\right)} \kappa\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, ~ t >0,$$ and $$\label{hh} h^{\prime}\left(t\right)= \mu \int_{g\left(t\right)}^{h\left(t\right)} \int_{h\left(t\right)}^{\infty} \kappa\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, ~ t > 0,$$ where $\mu>0$ describes the expanding capability. Moreover, we assume that the kernel function $\kappa(x): \mathbb R\rightarrow \mathbb R$ satisfies the following properties. $(\textbf{H1})$ (1) $\kappa(\cdot) \in C^1(\mathbb{R},[0, \infty))$ is symmetric, $\kappa(0)>0,$ and $\int_{\mathbb{R}} \kappa(x) d x=1$. (2) There are positive constants $\alpha$ and $x_0$ such that $\kappa(x) \leq e^{-\alpha|x|}$ and $|\nabla \kappa| \leq e^{-\alpha|x|}$ for $|x| \geq x_0$. The growth function $f(t,x,u): \mathbb R\times \mathbb R\times \mathbb R^+\longrightarrow \mathbb R$ satisfies that for $D\subset \mathbb R$ is smooth bounded or $D=\mathbb R,$ $(\textbf{H2})$ $f(t, x, u)$ is almost periodic in $t$ uniformly for $x \in \bar{D}$ and $u$ in bounded sets of $\mathbb{R}$. When $D=\mathbb{R}$, $f(t, x, u)$ is also almost periodic in $x$ uniformly for $t \in \mathbb{R}$ and $u$ in bounded sets. $(\textbf{H2}^*)$ (1) $f(t, x, u)$ is $C^1$ in $u$; $f(t, x, u)$ and $f_u(t, x, u)$ are uniformly continuous in $(t, x, u) \in \mathbb{R} \times \bar{D} \times E$ for any bounded set $E \subset \mathbb{R} ; f(t, x, u)$ is uniformly bounded in $(t, x) \in \mathbb{R} \times D$ for $u$ in bounded sets. (2) There is $M\gg 1$ such that $f(t, x, u)<0$ for $(t, x) \in \mathbb{R} \times \bar{D}$ and $u \geq M$; $f_u(t, x, u)<0$ for $(t, x, u) \in \mathbb{R} \times \bar{D} \times[0, \infty).$ Let $a(t,x)=f(t,x,0)$ satisfy $(\textbf{H3})$ $a(t, x)$ is bounded and uniformly continuous in $(t, x) \in \mathbb{R} \times \bar{D}$, and is almost periodic in $t$ uniformly for $x \in \bar{D}$. When $D=\mathbb R$, $a(t, x)$ is also almost periodic in $x$ uniformly for $t \in\mathbb R$. Additionally, for the principal Lyapunov exponent as the threshold condition in studying the spreading and vanishing of $(\ref{eq1-1})$, we assume that $(\textbf{H4})$ There exists a $L^*>0$ such that $\inf\limits_{L\geq L^*}\lambda_{\mathscr {PL}}(a, L)>0$, where $\lambda_{\mathscr {PL}}(a, L)$ is the principal Lyapunov exponent of the following equation $$\begin{cases} u_t=d\int_{-L}^{L}\kappa (y-x) u(t,y ){\rm d} y- d u(t,x)+a (t,x) u (t,x), & x\in(-L,L), \vspace{0.1cm} \\u(t,-L)=u(t,L)=0. \end{cases}$$ Assumptions on the reaction term $f: \mathbb{R}^{+} \times\mathbb{R}\times\mathbb{R}^{+}\rightarrow \mathbb{R}.$ $(\textbf{f1})$ $f(t,x,0)=0$ and $f(t,x,u)$ is locally Lipschitz in $u \in \mathbb{R}^{+}$. That is, for any $L>0$, there exists a constant $k=k(L)>0$ such that $\left|f\left(t,x,u_{1}\right)-f\left(t,x,u_{2}\right)\right| \leq k\left|u_{1}-u_{2}\right| \text { for } u_{1}, u_{2} \in[0, L].$ $(\textbf{f2})$ There exists $k_{0}>0$ such that $f(t,x,u)<0$ for $u \geq k_{0}$. $(\textbf{f3})$ For any given $\tau,$ $l,$ $u^*>0$, there exists a constant $\tilde {k}\left(\tau, l, u^*\right)$ and $\alpha\in(0,1)$ such that $$\left\|f_{}(\cdot, x, u)\right\|_{C^{\frac{\alpha}{2}}([0, \tau])} \leq \tilde {k}\left(\tau, l, u^*\right)$$ for all $x \in[-l, l], ~u \in\left[0, u^*\right]$. Assume that $f$ is of Fisher-KPP as exploring the long-time asymptotic behaviors, that is, the following additional conditions for $f$ hold. $(\textbf{f4})$ $f =f(u) \in C^1$ independent of $(t,x)$. $f^{\prime}(0):=f_0>0$, $f(u)\leq f^{\prime} (0) u$ and $\dfrac{f(u)}{u}$ is strictly decreasing in $u>0$. Let $${\mathscr D_{g, h,T}}:=\left\{(t, x) \in \mathbb{R}^2: 0<t \leq T, g(t)<x<h(t)\right\}.$$ $$\mathscr {X}(D):=\{u \in C(\bar D) \mid u \text{ is uniformly continuous and bounded } \}$$ with norm $\|u\|=\sup\limits_{x \in D}|u(x)|$. For simplicity, denote $$\begin{gathered} \mathscr{X}(h_0):= \mathscr {X}((-h_0,h_0)),\\ \mathscr {X^{+}}(h_0):=\left\{u \in \mathscr{X}(h_0) \mid u(x) \geq 0, x \in [-h_0, h_0]\right\}, \\ \mathscr{X}^+_0(h_0):=\left\{u\in \mathscr {X^{+}}(h_0)\mid u(-h_0)=u(h_0)=0, u(x) >0, x \in (-h_0, h_0)\right\} \end{gathered}$$ and $$\mathscr{X^{++}}(h_0):=\left\{u \in\mathscr{ X^{+}}(h_0) \mid \inf _{x \in [-h_0, h_0]} u(x)>0\right\}.$$ Denote $$f\cdot t(\cdot, \cdot, \cdot ):=f(\cdot+t, \cdot, \cdot).$$ $$\mathcal{X}(D)=C_{\text {unif }}^b(\mathbb{R} \times \bar{D}):=\{u \in C(\mathbb{R} \times \bar{D}) \mid u \text{ is uniformly continuous and bounded } \}$$ with the norm $$\|u\|=\sup\limits_{(t, x) \in \mathbb{R} \times \bar{D}}|u(t, x)|,$$ where $D \subset \mathbb{R}$ is bounded or $D=\mathbb{R}$. In the absence of possible confusion, we may write $$\begin{gathered} \mathcal{X}=\mathcal{X}(D) \\ \mathcal{X}^{+}=\{u \in \mathcal{X} \mid u(t, x) \geq 0, t \in \mathbb{R}, x \in \bar{D}\} \end{gathered}$$ and $$\mathcal{X}^{++}=\left\{u \in \mathcal{X}^{+} \mid \inf _{t \in \mathbb{R}, x \in \bar{D}} u(t, x)>0\right\}.$$ In this article, unless otherwise specified, $u(t, x ; u_0)$ always denotes the solution of $(\ref{eq1-1})$ with $u(0, x; u_0)=u_0(x).$ Sometimes, we write $u(t, x; u_0, a)$ to emphasize the solution dependent on $a(t,x)$. Assume that $(\mathbf{H1})-(\mathbf{H2}^*)$ hold, for any given $u_0$ satisfying $(\ref{eq1-2})$, according to Theorem 1.1 [@cao2019], the problem $(\ref{eq1-1})$ admits a unique solution $u(t, x; u_0, g, h)\text{ for all }t>0$ with $$u(0, x; u_0)=u_0(x) \text { and } 0\leq u(t, x; u_0)\leq \max\left\{ |u_0|_{L^\infty}, M\right\}.$$ where $M$ comes from $(\textbf{H2}^*)$. Moreover, $g(t; u_0, h_0)$ is strictly decreasing in $t>0$, and $h(t; u_0,h_0)$ is strictly increasing in $t>0.$ Thus, there are $g_\infty\in[-\infty, 0)$ and $h_\infty\in(0,\infty]$ such that $\lim\limits_{t\rightarrow\infty}g(t; u_0, h_0)=g_\infty\text{ and } \lim\limits_{t\rightarrow\infty}h(t; u_0, h_0)=h_\infty.$ Simultaneously, according to Theorem 1.2 [@2023-chengalmost1], we give the spreading-vanishing dichotomy regimes for $(\ref{eq1-1})$. **Theorem 1** (Spreading-vanishing dichotomy). *Assume that $(\mathbf{H1})-(\mathbf{H4})$ hold. For any $h_0>0$ and $u_0$ satisfying $(\ref{eq1-2})$, the either of the followings must hold: $(1)$ Vanishing: $h_\infty-g_\infty \leq 2L^*$, and $\lim\limits_{t\rightarrow\infty} u(t,x;u_0)=0$ uniformly for $x\in[g_\infty,h_\infty];$ $(2)$ Spreading: $h_\infty-g_\infty=\infty$, and $\lim\limits_{t\rightarrow\infty}\left[u(t,x;u_0)-u^*(t,x)\right]=0$ uniformly for $x$ in any compact subset of $\mathbb R,$ where $u^*(t,x)$ is the unique time almost periodic solution of the following equation $$\label{eq1-3} u_t=d\int_{\mathbb R} \kappa (y-x) u(t, y){\rm d}y-d u(t,x)+uf(t, x, u), ~x\in \mathbb R.$$* **Definition 1** (Principal Lyapunov exponent). *For any $u_0\in \mathscr{X}(L) \text{ with } u_0(-L)=u_0(L)=0.$ Let $u_L(t,\cdot;u_0,a)$ be the solution of the following problem $$\label{eq2-1} \begin{cases} u_t=d\int_{-L}^{L}\kappa (y-x) u(t,y ){\rm d} y- d u(t,x)+a (t,x) u (t,x), & x\in(-L,L), \vspace{0.1cm} \\u(t,-L)=u(t,L)=0, \end{cases}$$ satisfying $u_L(0,\cdot; u_0,a)=u_0(\cdot),$ then $u_L(t,\cdot;u_0,a)=\Phi(t; a)u_0(\cdot),$ where $\Phi(t,a)$ is the solution operator of $(\ref{eq2-1}).$* *Let $$\begin{aligned} \lambda_{\mathscr {PL}}(a, L)&:=\limsup\limits_{t\rightarrow\infty}\dfrac{{\ln}\|\Phi(t; a)u_0\|}{t} = \limsup\limits_{t\rightarrow\infty}\dfrac{{\ln}\|\Phi(t; a)\|}{t}, \end{aligned}$$ we call $\lambda_{\mathscr {PL}}(a, L)$ the principal Lyapunov exponent of $(\ref{eq2-1})$.* Immediately, we obtain **Lemma 1**. *$\lambda_{\mathscr {PL}}(a,L)$ is monotonically increasing in $L$ and $a(t, x)$.* **Remark 1**. *As is argued in $(\mathbf{H4})$, we always assume that there is $L^*$ such that $\lambda_{{\mathscr {PL}}}(a, L)>0, \text{ for } L\geq L^*,$ which means that in the long-distance region, the spreading will occur and the species can randomly survive if there are no human interventions or additional restrictive conditions. Detailed explanations and applications can be seen in Section [2](#s3){reference-type="ref" reference="s3"}.* For any $u_0\in \mathscr{X}(h_0) \text{ with } u(-h_0)=u(h_0)=0$. Let $u(t,x;u_0,a)$ the following almost periodic nonlocal diffusion model with free boundaries $$\begin{cases} u_t=d\int_{g(t)}^{h(t)}\kappa (y-x) u(t,y ){\rm d} y- d u(t,x)+a (t,x) u (t,x), & t>0,~x\in(g(t),h(t)), \vspace{0.1cm} \\u(t,g(t))=u(t, h(t))=0, \end{cases}$$ with $u(0,x;u_0,a)=u_0$ and $\Phi(t; a)u_0(\cdot)=u(t,\cdot;u_0,a)$. Denote $$\begin{aligned} \lambda_{\mathscr {PL}}(a, g(t), h(t))&:=\limsup\limits_{t\rightarrow\infty}\dfrac{{\ln}\|\Phi(t; a)u_0\|_{\mathscr{X}((g(t),h(t)))}}{t} =\limsup\limits_{t\rightarrow\infty}\dfrac{{\ln}\|\Phi(t; a)\|}{t}. \end{aligned}$$ Since $g(t)$ is monotonically decreasing in $t$ and $h(t)$ is monotonically increasing in $t$, together with Lemma $\ref{le2-1},$ we then have **Theorem 2**. *$\lambda_{\mathscr {PL}}(a, g(t), h(t))$ is increasing for all $t>0.$* # Spreading-vanishing regimes {#s3} In this section, we will give the proofs of the spreading-vanishing regime results for $(\ref{eq1-1})$, that is, the Theorem [Theorem 1](#th1-2){reference-type="ref" reference="th1-2"} and the Theorem [\[th1-3\]](#th1-3){reference-type="ref" reference="th1-3"}. In the next discussions, without additional explanation, we always assume that $(\mathbf{H4})$ holds. Firstly, we represent some important lemmas. **Lemma 2**. *If $h_\infty-g_\infty=\infty, \text{ then } h_\infty=\infty \text{ and } g_\infty=-\infty.$* *Proof.* We prove this lemma by contradiction. Without loss of generality, assume that $h_\infty=\infty,$ but $g_\infty>-\infty.$ For any small $\epsilon>0$, there exists $T\gg 1$ such that $$h(t)>h_\infty-\epsilon \gg 2L^* \text{ and } g(t)<g_\infty+\epsilon<0$$ as $t \geq T$. Thus, by $(\mathbf{H4})$, it follows that $$\lambda_{\mathscr{PL}}(a, g_\infty, h(T))>0.$$ Let $\tilde u_0(t,x;u_0)$ be the solution of the following problem with $\tilde u_0(0,x;u_0)=u_0(x)$ $$\left\{\begin{array}{ll}\label{eq3-2} \tilde u_{ t}=d \int_{g_\infty}^{h\left(T\right)} \kappa\left(x-y\right) \tilde u\left(t, y\right) \mathrm{d} y-d \tilde u+\tilde uf(t,x,\tilde u), & g_\infty<x<h\left(T\right), \vspace{0.1cm} \\ \tilde u\left(t, g_\infty\right)=0, ~\tilde u\left(t, h(T)\right)=0& t > T, \vspace{0.1cm} \\ \tilde u\left(T, x\right)=u\left(T, x; u_0\right), & x\in[g_\infty ,h(T)], \end{array}\right.$$ By a simple comparison argument, we have $$u(t,x;u_0)\geq \tilde u_0(t,x;u_0), \text{ for } t\geq T,~x\in[g_\infty, h(T)].$$ Since $\lambda_{\mathscr {PL}}(a, g_\infty, h(T))>0,$ according to Theorem 1.4 [@2023-onyido2021non2], we have $$\tilde u_0 (t,x; u_0)>0, \text{ for } t\geq T,~x\in(g_\infty, h(T)).$$ It follows that $$u(t, x; u_0)>0, \text{ for } t\geq T,~x\in(g_\infty, h(T)).$$ By ${\rm ( \mathbf {H1} )}$, since $\kappa(0)>0,$ for small $\theta>0$, there is $\epsilon_0>0$ such that $$\kappa(x)>\theta, \text{ for } x\in \left(-\epsilon_0, \epsilon_0\right).$$ Moreover, for $0<\epsilon<\epsilon_0/2$, it follows $$\begin{aligned} g^{\prime}(t)&=-\mu \int_{g(t)}^{h(t)} \int_{-\infty}^{g(t)} \kappa (x-y) u(t, x) {\rm d } y {\rm d} x \\& \leq- \mu \int_{g_{\infty+\epsilon}}^{h_\infty-\epsilon} \int_{g_{\infty}}^{g_\infty+\epsilon} \kappa(x-y) u(t, x) {\rm d} y {\rm d} x \\& \leq -\mu \int_{g_{\infty}+\epsilon }^{g_{\infty}+2\epsilon} \int_{g_{\infty}}^{g_{\infty}+\epsilon} \theta u(t,x) {\rm d} y {\rm d} x<0,\text{ for all } t>0, \end{aligned}$$ which implies $g_\infty=-\infty.$ This is a contradiction. Thus, $h_\infty=\infty$ if and only if $g_\infty=-\infty$. ◻ The above lemma implies that $h_\infty$ and $g_\infty$ will be simultaneously infinite or finite. **Lemma 3**. *If $h_\infty-g_\infty<\infty$, then $h_\infty-g_\infty\leq 2L^*,$ and $\lambda_{\mathscr {PL}}(a, g_\infty, h_\infty)\leq 0.$* *Proof.* When $h_\infty-g_\infty<\infty$, according to the Lemma $\ref{le3-2}$, we have $h_\infty<\infty$ and $g_\infty>-\infty.$ Moreover, we have $$\label{eq3-03} \lim\limits_{t\rightarrow \infty}h^{\prime}(t)=0 \text{ and } \lim\limits_{t\rightarrow \infty}g^{\prime}(t)=0.$$ Assume that $h_\infty-g_\infty>2L^*,$ without loss of generality, suppose that $h_\infty>L^*$. For small $\epsilon_1>0$, there exists $T_1>0$ such that $$h(t)>h_\epsilon-\epsilon_1>L^*, g(t)<g_\infty+\epsilon_1 \text{ and } h_\infty-g_\infty-2\epsilon_1>2L^*$$ for $t\geq T_1.$ Then by Theorem [Theorem 2](#th2-1){reference-type="ref" reference="th2-1"} and $(\mathbf{H4})$, we have $$\lambda_{\mathscr {PL}}(a, g_\infty+\epsilon_1, h_\infty-\epsilon_1)>0.$$ Considering the following problem $$\label{eq3-3} \begin{cases} \hat u_t=d \int_{g_\infty+\epsilon_1}^{h_\infty-\epsilon_1}\kappa(y-x) \hat u(t, y){\rm d}y-d\hat u(t,x)+\hat u f(t,x,\hat u), & g_\infty+\epsilon_1<x<h_\infty-\epsilon_1, \vspace{0.1cm} \\ \hat u\left(t, g_\infty+\epsilon_1\right)=\hat u \left(t, h_\infty+\epsilon_1\right)=0, & t>0, \end{cases}$$ by the comparison argument, it follows that, $$\label{eq3-4} u\left(t+T_1, \cdot; u_0\right)\geq \hat u\left(t+T_1, \cdot; u\left(T_1,\cdot; u_0\right),T_1\right), \text{ for }t\geq0,$$ where $\hat u\left(t+T_1, \cdot; u\left(T_1,\cdot; u_0\right),T_1\right)$ is the solution of $(\ref{eq3-3})$ with $$\hat u\left(T_1, \cdot; u\left(T_1,\cdot; u_0\right),T_1\right)=u\left(T_1,\cdot; u_0\right).$$ By Theorem 1.4 [@2023-onyido2021non2], the following problem admits a unique strictly positive almost periodic solution $\hat u_{\epsilon_1}(t,x)$ $$\hat u_t=d \int_{g_\infty+\epsilon_1}^{h_\infty-\epsilon_1}\kappa(y-x) \hat u(t, y){\rm d}y-d\hat u(t,x)+\hat u f(t,x,\hat u), ~ g_\infty+\epsilon_1<x<h_\infty-\epsilon_1.$$ Further, for $\hat u_0:=u\left(T_1,\cdot; u_0\right) \in \mathscr{X}^{++}(h_0),$ $$\lim\limits_{t\rightarrow \infty}\|\hat u\left(t+T_1, \cdot; \hat u_0, T_1\right)-\hat u_{\epsilon_1}\left(t+T_1,\cdot\right)\|=0,$$ then by $(\ref{eq3-4})$, we have $$\liminf\limits_{t\rightarrow \infty}u\left(t, x; u_0\right)>0,$$ which indicates $$\liminf\limits_{t\rightarrow \infty} h^{\prime}(t)=\liminf\limits_{t\rightarrow\infty} \mu\int_{g(t)}^{h(t)}\int_{h(t)}^{\infty} \kappa(x-y)u(t,x;u_0){\rm d}y{\rm d}x>0,$$ this is a contradiction. Thus, $h_{\infty}-g_\infty\leq 2 L^*.$ Moreover, by ${\rm (\mathbf{H4} )}$ and Lemma $\ref{le2-1},$ it follows $\lambda_{\mathscr {PL}}(a, g_\infty, h_\infty)\leq 0.$ ◻ *The proof of Theorem $\ref{th1-2}$.* $(1)$ When $h_\infty- g_\infty\leq 2L^*$, assume that $\lim\limits_{t\rightarrow\infty}\|u\left(t,\cdot; u_0\right) \|\neq 0,$ there exists a sequence $\{ t_n \}$ with $t_n\rightarrow \infty$ as $n\rightarrow \infty$ such that $$u\left(t_n, x; u_0\right)\longrightarrow \tilde u^*(x) \text{ as } t_n\rightarrow \infty$$ with $\tilde u^*(x)>0$ for $x\in (g(t_n), h(t_n)).$ Then $$\lim\limits_{n\rightarrow\infty} h^{\prime}(t_n)=\lim\limits_{n\rightarrow \infty} \mu\int_{g(t_n)}^{h(t_n)}\int_{h(t_n)}^{\infty} \kappa (x-y) u(t_n, x){\rm d} y{\rm d}x >0,$$ which contradicts $(\ref{eq3-03})$. Therefore, $$\lim\limits_{t\rightarrow\infty}u(t, x; u_0)=0 \text{ if } h_\infty-g_\infty<\infty.$$ $(2)$ When $h_\infty-g_\infty=\infty,$ for any given $l>L^*,$ there exists $T^*>1$ such that $$h(t)\geq l \text{ and } g(t)\leq -l$$ for $t\geq T^*.$ Take $l=h(T^*)$. Let $u^l \left(t, x; u(T^*,x;u_0),f\cdot T^*\right)$ be the entire solution of the following problem $$\begin{cases} u_t=\int_{-l}^{l}\kappa(x-y) u(t,y){\rm d}y -d u(t,x)+uf {\cdot T^*} (t, x, u), & -l<x<l, \vspace{0.1cm} \\ u(t,-l)=u(t,l)=0. \end{cases}$$ with $u^l(0, x; u(T^*, x;u_0),f\cdot T^*)=u(T^*, x;u_0).$ Then by the comparison argument, we have $$\label{eq3-7} u(t+T^*,x; u_0)\geq u^l(t, x; u(T^*, x;u_0),f\cdot T^*), \text { for }t\geq 0,$$ Moreover, $$\label{eq3-8} u^l(t,x;u(T^*, x;u_0),f\cdot T^*)-u_0(t,x;u(T^*, x;u_0))\longrightarrow 0 \text { as } l\rightarrow \infty,$$ $\text { locally for any } x\in \mathbb R$, where $u_0(t, x; u(T^*, x;u_0))$ is an entire solution of $(\ref{eq1-3})$ with $f$ replaced by $f\cdot T^*$. Further, for any $u(T^*, x;u_0)\in\mathscr{X^{++}}(h_0)$, by Theorem 1.3 [@2023-onyido2021non2], it can be seen that $$\label{eq3-9} u_0(t,x; u(T^*, x;u_0))- u^*(t+T^*,x)\longrightarrow 0\text{ as }t\longrightarrow \infty,$$ uniformly for $x$ in any compact subset of $\mathbb R,$ where $u^*(t+T,x)$ is the unique positive time almost periodic solution of $(\ref{eq1-3})$ $f$ replaced by $f\cdot T^*$. Combine $(\ref{eq3-7})-(\ref{eq3-9})$, we thus have $$u(t, x;u_0)- u^*(t,x)\longrightarrow 0 \text{ as } t\longrightarrow \infty$$ uniformly for $x$ in any compact subset of $\mathbb R.$ ◻ *The proof of Theorem $\ref{th1-3}$.* $(1)$ If $h_0\geq L^*,$ in view of the monotonicity of $h(t)$ and $g(t)$ in $t$, then $$h(t)>L^* \text { and } g(t)<-L^*, \text { for } t>0,$$ hence $h_\infty-g_\infty>2L^*.$ According to Lemma $\ref{le3-3},$ we have $h_\infty-g_\infty=\infty.$ $(2)$ In the case of $h_0<L^*$, to stress the effect of $\mu$ on $h(t)$ and $g(t)$, we denote $h(t)$ by $h_\mu(t)$ and $g(t)$ by $g_\mu(t)$. Moreover, according to $(\ref{gg})-(\ref{hh})$, applying a simple comparison argument, it is not difficult to obtain that $$\label{in} h_\mu(t) \text { is increasing in } \mu \text { and } g_\mu(t) \text { is decreasing in } \mu$$ for any $t>0$. Denote $h_{\infty}$ by $h_{\mu, \infty}$ and $g_{\infty}$ by $g_{\mu, \infty}$. Denote $$\Omega:=\{\mu\mid h_{\mu, \infty}-g_{\mu,\infty}<\infty\}$$ and $$\mu_0:=\sup\{\mu\mid h_{\mu, \infty}-g_{\mu,\infty}<\infty\}.$$ If $\mu_0 \in \Omega$, then our proof is finished. Otherwise, it will make a contradiction. Actually, if $h_{\mu_0,\infty}-g_{\mu_0,\infty}=\infty$, then there is $T_0>0$ such that $$h_{\mu_0}(t)>2L^*\text{ and } g_{\mu_0}(t)<-2L^*$$ for $t\geq T_0.$ Thus, concerning $(\ref{in})$, there exists small $\varepsilon_0>0$ such that $$h_{\mu,\infty} \geq h_{\mu}(T_0)>L^*\text{ and } g_{\mu,\infty} \leq g_{\mu}(T_0)<-L^*$$ for $\mu\in[\mu_0-\varepsilon_0, \mu_0+\varepsilon_0]$. Then $$h_{\mu,\infty} -g_{\mu,\infty}=\infty,$$ which yields a contradiction. Take $\mu^*=\mu_0$, we have completed our proof. ◻ # Main results In this section, we devote ourselves to exploring the spreading speed of the leftward and rightward front for $(\ref{eq1-1}).$ For the homogeneous nonlocal diffusion model with free boundaries, Du, Li and Zhou investigated the spreading speed of the moving boundaries of the nonlocal diffusion model by applying the semi-wave and traveling wave solutions in [@du2021semi]. However, for the difficulties caused by the time almost periodicity, the semi-wave solution of the time almost periodic nonlocal diffusion equations could be a rough nut to crack. Thus, the subtle methods developed by them could not be directly used in the problem $(\ref{eq1-1}).$ We had to look for and develop new methods to solve this problem. Firstly, consider an assumption about the kernel function which is called satisfying the "thin-tailed\" condition if the following condition holds. $$\label{thin} (\textbf{H})\qquad\qquad\qquad\int_{-\infty}^{0}\int_{0}^{\infty} \kappa(x-y){\rm d}y{\rm d}x=\int_{0}^{\infty}x\kappa(x){\rm d}x<\infty.$$ In the case of $f(t,x,u)\equiv f(t,u).$ Considering the following free boundary problem $$\label{eq4-1} \left\{\begin{array}{ll} u_{ t}=d \int_{g(t)}^{h\left(t\right)} \kappa\left(x-y\right) u\left(t, y\right) \mathrm{d} y-d~ u+u f(t,u), & t>0,~ g(t) <x<h\left(t\right), \vspace{0.1cm} \\ u\left(t, h(t)\right)=0, u(t,g(t))=0, & t > 0, \vspace{0.1cm} \\ h^{\prime}\left(t\right)= \mu \int_{g(t)}^{h\left(t\right)} \int_{h\left(t\right)}^{\infty} \kappa\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t > 0, \vspace{0.1cm} \\ g^{\prime}\left(t\right)= -\mu \int_{g(t)}^{h\left(t\right)} \int_{-\infty}^{0} \kappa\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t > 0, \vspace{0.1cm} \\ u\left(0, x\right)=u_{ 0}\left(x\right),~ h\left(0\right)=h_{0},~ g(0)=-h_0, & x\in[-h_0, h_0], \end{array}\right.$$ and the fixed boundary problem $$\label{eq4-3} u_t=d\int_{\mathbb R}\kappa(x-y) u(t,y){\rm d}y -d ~ u(t,x)+uf(t, u), ~x\in\mathbb R.$$ with the semi-wave problem $$\label{eq4-4} \begin{cases} u_t=d\int_{-\infty}^{h(t)}\kappa(x-y) u(t,y){\rm d}y -d u(t,x)+uf(t, u), ~& -\infty<x<h(t), \\ u(t, -\infty)=1,~u(t,h(t))=0, ~&t>0, \\h^{\prime}(t)=\mu\int_{-\infty}^{h(t)}\int_{h(t)}^{\infty}\kappa(x-y)u(t,x){\rm d}y{\rm d}x. \end{cases}$$ by Theorem 4.1 [@2023-shen1998convergence] and Theorem 1.3 [@2023-onyido2021non2], there is a unique positive time almost periodic solution $\hat u^*(t,x)$ of $(\ref{eq4-3})$ with $$\label{eq04-3} \hat u^*(t,x)\equiv\hat u^*(t),$$ where $\hat u^*(t)$ is the positive time almost periodic solution of the following problem $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} u_t=uf(t,u).$$ Since $\hat u^*(t)$ is almost periodic in $t$, by Theorem 3.1 [@2023-fink2006almost], there is $\hat u^*$ such that $$\label{eq4-5} \hat u^*=\lim\limits_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\hat u^{*}(t){\rm d}t.$$ Denote $$c^*:=\mu\int_{-\infty}^{0}\int_{0}^{\infty}\kappa(x-y) \hat u^* {\rm d}y{\rm d}x.$$ Assume that $\textbf{(H)}$ holds, one can see that $c^*$ is well defined. Now we mainly give an explicit estimate of the asymptotic spreading speed, that is, **Theorem 3** (Finite spreading). *Assume that $(\mathbf{H1})-(\mathbf{H4})$ hold. When the spreading occurs, if $(\mathbf{H})$ holds, the spreading speeds of the leftward front and the rightward front satisfy that $$\label{eq1-7} \lim\limits_{t\rightarrow \infty} \dfrac{h(t)}{t}=\lim\limits_{t\rightarrow\infty}\dfrac {-g(t)}{t}=c^*.$$* **Theorem 4**. *Under the conditions of Theorem [Theorem 3](#th1-4){reference-type="ref" reference="th1-4"}, for any $\varepsilon\in(0, c^*)$, it follows $$\label{eq-s1} \lim _{t \rightarrow \infty} \max _{|x| \leq\left(c^*-\varepsilon\right) t}\left|u(t, x)-\hat u^*(t)\right|=0.$$* **Theorem 5** (Accelerated spreading). *Assume that $(\mathbf{H1})-(\mathbf{H4})$ hold. When the spreading occurs, if $(\mathbf{H})$ does not hold, it follows that $$\label{eq1-8} \lim\limits_{t\rightarrow \infty} \dfrac{h(t)}{t}=\lim\limits_{t\rightarrow\infty}\dfrac {-g(t)}{t}=\infty.$$* # Proofs *The proof of Theorem [Theorem 3](#th1-4){reference-type="ref" reference="th1-4"}.* For simplify, denote $u(t,x)=u(t,x;u_0)$. As $\hat{u}(t, x):=u(t,-x)$ satisfies $(\ref{eq4-1})$ with free boundaries $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} x=\hat{h}(t):=-g(t),~ x=\hat{g}(t):=-h(t)$$ and initial function $\hat{u}_0(x):=u_0(-x)$, we only need to prove the case of $h(t)$. The spreading speed for $g(t)$ can be directly obtained. Now we complete the proof of this theorem in two steps. ${Step~1:}$ Firstly, we prove that $\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} \limsup\limits_{t\rightarrow\infty}\dfrac{h(t)}{t}\leq c^*.$ For $0<\epsilon\ll 1$, let $\overline u_\epsilon(t)$ and $\underline u_\epsilon(t)$ be the positive time almost periodic solutions of the following problem $$\label{eq4-6} u_t=u(f(t,u)+\epsilon)$$ and $$\label{eq4-7} u_t=u(f(t,u)-\epsilon),$$ respectively. Thus, $$\label{eq4-8} \lim\limits_{\epsilon \rightarrow 0} \underline u_\epsilon(t)=\lim\limits_{\epsilon\rightarrow 0}\overline u_\epsilon(t)=\hat u^*(t)$$ uniformly in $t\in\mathbb R.$ Applying a comparison argument, we can see that $$\label{l0} \underline u_\epsilon(t) \leq \hat u^*(t) \leq \overline u_\epsilon(t).$$ Thus, according to Theorem [Theorem 1](#th1-2){reference-type="ref" reference="th1-2"}, there exists $T>0$ such that $u(t+T, x)\leq \overline u_\epsilon(t+T), \text { for } t\geq 0, ~x\in[-h(t+T),h(t+T)].$ Let $$\tilde u(t,x)=u(t+T,x), ~\tilde h(t)=h(t+T),~\tilde g(t)=g(t+T),$$ then $\tilde u(t,x)$ satisfies $$\left\{\begin{array}{ll} \tilde u_{ t}=d \int_{\tilde g(t)}^{\tilde h\left(t\right)} \kappa\left(x-y\right) \tilde u\left(t, y\right) \mathrm{d} y-d \tilde u+\tilde u f(t+T, \tilde u), & t>0,~ \tilde g\left(t \right)<x<\tilde h\left(t\right), \vspace{0.1cm} \\ \tilde u(0,x)=u(T,x), & t > 0,~ \tilde g(0)<x< \tilde h(0), \vspace{0.1cm} \\ \tilde h^{\prime}\left(t\right)= \mu \int_{\tilde g(t)}^{\tilde h\left(t\right)} \int_{\tilde h\left(t\right)}^{\infty} \kappa\left(x-y\right) \tilde u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t > 0, \vspace{0.1cm} \\ \tilde g^{\prime}\left(t\right)=- \mu \int_{\tilde g(t)}^{\tilde h\left(t\right)} \int_{-\infty}^{\tilde g(t)} \kappa\left(x-y\right) \tilde u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t > 0, \vspace{0.1cm} \\ \tilde u(t, \tilde h(t))=0, ~\tilde u(t,\tilde g(t))=0, & t>0. \end{array}\right.$$ Let $u^*(t)$ be the solution to the following initial problem $$\begin{cases} u_t=u(f(t,u)+\epsilon)), &t>0,\vspace{0.1cm} \\ u(T)=\max\{\overline u_\epsilon(T), \|\tilde u(0,\cdot)\|_{\infty} \}, \end{cases}$$ then $u^*(t)\geq \overline u_\epsilon (t), \text {for } t\geq T,$ and $$\lim\limits_{t\rightarrow\infty}(u^*(t)-\overline u_\epsilon(t))=0.$$ Moreover, by Theorem [Theorem 1](#th1-2){reference-type="ref" reference="th1-2"}, combining $(\ref{eq04-3})$ with $(\ref{eq4-8})$, we have $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} \tilde u(t,x)\leq u^*(t+T), \text { for } t\geq 0, -\tilde h(t)<x<\tilde h(t),$$ then $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} \tilde u(t,x)\leq \dfrac{\overline u_{\epsilon}(t+T)}{1-\epsilon}, \text { for } t\gg 1, -\tilde h(t)<x<\tilde h(t).$$ Let $u_\epsilon(t,x)$ be the unique positive almost periodic solution of the following problem $$\label{eq4-11} u_t=d\int_{\mathbb R}\kappa(x-y) u(t,y){\rm d}y -d u(t,x)+u(f(t, u)+\epsilon), ~x\in\mathbb R,$$ by the uniqueness of almost periodic solution for $(\ref{eq4-11})$ explained in Theorem 1.4 [@2023-onyido2021non2], we have $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} u_\epsilon(t,x) \equiv u_\epsilon(t)= \overline u_\epsilon(t).$$ Thus, there is $T^*>T\gg1$ such that $$u_\epsilon(t,x)\geq (1-\epsilon)\overline u_\epsilon(t), \text{ for }t>T^*, -\tilde h(t)<x<\tilde h(t) .$$ Therefore, $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} \tilde u(t,x)\leq \dfrac{\overline u_\epsilon(t+T)}{1-\epsilon}\leq \dfrac{u_\epsilon(t+T, x)}{(1-\epsilon)^2}, \text { for }t>T^*, -\tilde h(t)<x<\tilde h(t).$$ For fixed $L_0>0$, take $$\begin{aligned} &\overline h(t)=(1-\epsilon)^{-2}\mu\int_{0}^{t} \int_{-\infty}^{0}\int_{0}^{\infty} \kappa(x-y) u_\epsilon(s, x){\rm d}y{\rm d}x{\rm d}s +\tilde h(T^*)+L_0, \text { for }t\geq0, \\& \overline u(t,x)={(1-\epsilon)^{-2}}{u_\epsilon(t,x) }, \text { for } t\geq0,~x\in\mathbb R. \end{aligned}$$ According to assumption $(\textbf{H2}^*)$ for $f(t,u)$, direct computations give $$\begin{aligned} \overline u_t&=(1-\epsilon)^{-2} u_{\epsilon, t} =(1-\epsilon)^{-2} \left[d\int_{\mathbb R}\kappa (x-y)u_{\epsilon}(t,y){\rm d}y-d ~u_\epsilon (t,x) +u_\epsilon f(t,u_\epsilon) \right] \\&\geq d\int_{-\overline h(t)}^{\overline h(t)}\kappa (x-y) \overline u(t,y){\rm d}y-d ~\overline u (t,x) +\overline u f(t, u_\epsilon) \\&\geq d\int_{-\overline h(t) }^{\overline h(t)}\kappa (x-y) \overline u(t,y){\rm d}y-d ~\overline u (t,x) +\overline u f(t, \overline u). \end{aligned}$$ Meanwhile, $$\begin{aligned} \\\overline h^{\prime}(t)&= \mu\int_{-\infty}^{0}\int_{0}^{\infty} \kappa(x-y) \overline u(t,x) {\rm d}y{\rm d}x \\ & = \mu\int_{-\infty}^{\overline h(t)}\int_{\overline h(t)}^{\infty} \kappa(x-y) \overline u(t,x) {\rm d}y{\rm d}x \\&\geq \mu\int_{-\overline h(t) }^{\overline h(t)}\int_{\overline h(t)}^{\infty} \kappa(x-y) \overline u(t,x) {\rm d}y{\rm d}x. \end{aligned}$$ Moreover, $$\overline u(T,x)={(1-\epsilon)^{-2}}{u_\epsilon(T, x) }\geq \tilde u(0, x), \text{ for } x\in (-\tilde h(T), \tilde h(T))$$ with $\overline h(T^*)\geq\tilde h(T^*)$ and $\overline u(t, \overline h(t))\geq 0, \text { for } t\geq 0.$ Applying the Comparison principle, one can see that $$\label{l1} \overline u(t+T, x)\geq \tilde u(t,x),~ \overline h(t)\geq \tilde h(t), \text { for }t\geq T^*, -\tilde h(t)<x< \tilde h(t),$$ which implies that $$\begin{aligned} \limsup\limits_{t\rightarrow\infty}\dfrac{h(t)}{t}&=\limsup\limits_{t\rightarrow \infty}\dfrac{\tilde h(t-T)}{t} \leq \limsup\limits_{t\rightarrow \infty} \dfrac{\overline h(t-T)}{t} \\&= \lim\limits_{t\rightarrow\infty} \dfrac{(1-\epsilon)^{-2}\mu\int_{0}^{t-T} \int_{-\infty}^{0}\int_{0}^{\infty} \kappa(x-y) u_\epsilon(s, x){\rm d}y{\rm d}x{\rm d}s +\tilde h(T^*)+L_0}{t} \\&= \lim\limits_{t\rightarrow\infty} \dfrac{(1-\epsilon)^{-2}\mu\int_{0}^{t-T} \int_{-\infty}^{0}\int_{0}^{\infty} \kappa(x-y) u_\epsilon(s, x){\rm d}y{\rm d}x{\rm d}s }{t}. \end{aligned}$$ Since $u_\epsilon(t,x)\rightarrow \hat u^*(t,x)$ as $\epsilon\rightarrow 0$ uniformly in $t\in \mathbb R$ and $x$ in any compact subsets of $\mathbb R$, given ${\rm (\mathbf{H})}$ and $(\ref{eq4-5})$, it follows that $$\label{eq4-12} \begin{aligned} \limsup\limits_{t\rightarrow\infty}\dfrac{h(t)}{t}&\leq \limsup\limits_{t\rightarrow \infty} \dfrac{\overline h(t-T)}{t} = \lim\limits_{t\rightarrow\infty} \dfrac{\mu\int_{0}^{t-T} \int_{-\infty}^{0}\int_{ 0}^{\infty} \kappa(x-y) \hat u^*(s, x){\rm d}y{\rm d}x{\rm d}s }{t} \\&=\lim\limits_{t\rightarrow\infty} \dfrac{\mu\int_{0}^{t-T} \int_{-\infty}^{0}\int_{ 0}^{\infty} \kappa(x-y) \hat u^*(s){\rm d}y{\rm d}x{\rm d}s }{t} \\&=\lim\limits_{t\rightarrow\infty} \dfrac{\mu\int_{0}^{t} \int_{-\infty}^{0}\int_{ 0 }^{\infty} \kappa(x-y) \hat u^*(s){\rm d}y{\rm d}x{\rm d}s }{t} \\&=\mu \int_{-\infty}^{0}\int_{ 0 }^{\infty} \kappa(x-y) \hat u^* { \rm d}y{\rm d}x =c^*. \end{aligned}$$ ${Step~2:}$ We will prove $\liminf\limits_{t\rightarrow \infty}\dfrac {h(t)}{t} \geq c^*.$ Note that there exists a unique positive almost periodic solution $u_*(t, x)\equiv u_*(t)$ of the problem $$u_t=d \int_{\mathbb R} \kappa\left(x-y\right) u\left(t, y\right) \mathrm{d} y-d~ u+ u f(t+T, u),~x\in\mathbb R.$$ By Theorem $\ref{th1-2}$, it follows that $$\lim _{t \rightarrow \infty}\left[\tilde{u}(t, x)-u_*(t, x)\right]=0$$ locally uniformly in $x\in \mathbb R$. In view of $(\ref{eq4-7})$, there is $T_*\gg1$ such that $$u_*(t,x) \geq \underline{u}_\epsilon(t+T), \text { for } t\geq T_*,$$ locally uniformly for $x\in \mathbb R$, which implies that $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} \liminf\limits_{t \rightarrow \infty}\left[u_*(t,x)-\underline{u}_\epsilon(t+T)\right] \geq 0$$ locally uniformly for $x\in \mathbb R$. Thus, $$\liminf\limits_{t \rightarrow \infty}\left[\tilde u(t,x)-\underline{u}_\epsilon(t+T)\right] \geq 0$$ locally uniformly for $x\in \mathbb R$. For fixed $L\gg h(T)$, let $u_{\epsilon^-}(t,x)$ be the unique time almost periodic solution of the following problem $$\label{eq4-14} \begin{cases} u_t=d\int_{-L}^{L}\kappa(x-y) u(t,y){\rm d}y -d~u(t,x)+u(f(t, u)-\epsilon), &t>0, ~x\in(-L,L), \vspace{0.1cm} \\ u(t,x)=0,&t>0,~|x|\geq L. \end{cases}$$ Applying a comparison argument, $u_{\epsilon^-}(t,x) \leq \underline u_{\epsilon}(t),$ uniformly in $t>0$ and $x$ in any compact subsets of $\mathbb R.$ And $u_{\epsilon^-}(t,x) \rightarrow \underline u_{\epsilon}(t) \text { as } L\rightarrow \infty$ uniformly in $t>0$ and $x$ in any compact subsets of $\mathbb R.$ Take $\tilde{T}> T$ such that $h(\tilde T)>L$. Define $$\begin{aligned} &\underline h(t)=(1-\epsilon)^2 (1-2\epsilon)\mu \int_{\tilde{T}+T}^t \int_{-\infty}^{0}\int_{0}^{\infty} \kappa (x-y) u_{\epsilon^-}(s, x) {\rm d} y{\rm d}x {\rm d}s + \tilde{h}(\tilde{T}), ~~\underline u(t, x)=(1-\epsilon)^2 u_{\epsilon^-} (t, x), \end{aligned}$$ for $t \geq \tilde{T}+T, ~x\in\mathbb R.$ Then $$\tilde{u}(t, 0) \geq \underline u(t+T, 0), \text { for } t \geq \tilde{T} \text {and } \tilde{u}(\tilde{T}, x) \geq \underline u(\tilde{T}+T, x), \text { for }-\underline h(\tilde T+T) \leq x \leq \underline h(\tilde{T}+T).$$ Thus, we have $$\begin{aligned} \underline h^{\prime}(t)&= (1-2\epsilon)\mu \int_{-\infty}^{0}\int_{0}^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x \\&=(1-2\epsilon) \mu \int_{-\underline h(t)}^{\underline h(t) }\int_{\underline h(t) }^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x +\mu\int_{-\infty}^{-\underline h(t)}\int_{\underline h(t)}^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x \\&=\mu \int_{-\underline h(t)}^{\underline h(t) }\int_{\underline h(t) }^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x+ \mu\int_{-\infty}^{-\underline h(t)}\int_{\underline h(t)}^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x \\&-2\epsilon \mu \int_{-\underline h(t)}^{\underline h(t) }\int_{\underline h(t) }^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x \\&:=\mu \int_{-\underline h(t)}^{\underline h(t) }\int_{\underline h(t) }^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x+\mathscr{I}(\epsilon,\mu,h(t)) \text { for } t \geq \tilde{T}+T. \end{aligned}$$ Now we only need to prove $\mathscr{I} (\epsilon,\mu,h(t))\leq 0.$ In fact, given $(\textbf{H})$, one can see that if $t\gg\tilde T+T$, it follows that $$\mu\int_{-\infty}^{-\underline h(t)}\int_{\underline h(t)}^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x \leq 2\epsilon \mu \int_{-\underline h(t)}^{\underline h(t) }\int_{\underline h(t) }^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x,$$ then $\mathscr{I} (\epsilon,\mu,h(t))\leq 0,$ which implies $$\underline h^{\prime}(t)\leq \mu \int_{-\underline h(t)}^{\underline h(t) }\int_{\underline h(t) }^{\infty} \kappa (x-y) \underline u(t, x) {\rm d} y{\rm d}x.$$ Moreover, in view of $(\ref{eq4-14})$, it follows that $$\underline u(t, \pm \underline h(t))\rightarrow 0, \text { as } t\gg \tilde T+T.$$ According to ${\rm (\mathbf{H1})}$, we have $$\setlength{\abovedisplayskip}{0pt} %%% 0pt 个人觉得稍妥，可自行设置 \setlength{\belowdisplayskip}{0pt} \int_{\underline h(t)}^{\infty}\kappa(x-y) {\rm d}y+\int_{-\infty}^{-\underline h(t) }\kappa(x-y) {\rm d}y\leq \epsilon, \text { for } t \gg \tilde{T}+T.$$ Direct calculations show $$\begin{aligned} \underline u_t&=(1-\epsilon)^{2} u_{\epsilon^-, t} =(1-\epsilon)^{2} \left[d\int_{\mathbb R}\kappa (x-y)u_{\epsilon^-}(t,y){\rm d}y-d ~ u_{\epsilon^-} (t,x) +u_{\epsilon^-} (f(t,u_{\epsilon^-})-\epsilon) \right] \\&\leq d\int_{-\underline h(t)}^{\underline h(t)}\kappa (x-y) \underline u(t,y){\rm d}y-d~\underline u (t,x) +\underline u f(t, u_{\epsilon^-}) +\int_{\underline h(t)}^{\infty}\kappa(x-y) \underline u(t,y){\rm d}y +\int_{-\infty}^{-\underline h(t)}\kappa(x-y) \underline u(t,y){\rm d}y-\epsilon ~\underline u(t,x) \\&\leq d\int_{-\underline h(t)}^{\underline h(t)}\kappa (x-y) \underline u(t,y){\rm d}y-d~\underline u (t,x) +\underline u f(t, \underline u). \end{aligned}$$ Hence, using the Comparison principle to conclude that $$\label{l2} \begin{aligned} &\underline h(t+T) \leq \tilde{h}(t), \text { for } t \gg \tilde{T}, \text { and } \underline u(t+T, x) \leq \tilde{u}(t, x), \text { for } t \gg \tilde{T}, -\underline h(t+T)<x<\underline h(t+T). \end{aligned}$$ It follows that $$\begin{aligned} \liminf\limits_{t \rightarrow \infty} \frac{h(t)}{t}&=\liminf\limits_{t \rightarrow \infty} \frac{\tilde{h}(t-T)}{t} \geq\liminf\limits_{t \rightarrow \infty} \frac{\underline h(t)}{t} \\ & =\liminf\limits_{t \rightarrow \infty} \frac{(1-\epsilon)^2(1-2\epsilon)\mu \int_{\tilde{T}+T}^t \int_{-\infty}^{0 }\int_{0}^{\infty} \kappa (x-y) u_{\epsilon^-}(s, x) {\rm d} y{\rm d}x {\rm d}s + \tilde{h}(\tilde{T})}{t} \\ & =(1-\epsilon)^2(1-2\epsilon) \lim_{t \rightarrow \infty} \frac{ \mu \int_{0}^t \int_{-\infty}^{0 }\int_{0}^{\infty} \kappa (x-y) u_{\epsilon^-}(s, x) {\rm d} y{\rm d}x {\rm d}s }{t}. \end{aligned}$$ For $h_\infty=\infty,$ there is $t^*>\tilde T+T>0$ such that $h(t)-h_\infty<\epsilon,\text { as } t\geq t^*.$ Let $L = h(t^*)$, note that $u_{\epsilon^-}(t,x) \rightarrow \hat {u}^{*}(t,x)$ as $\epsilon \rightarrow 0$ uniformly in $t >0$ and $x\in \mathbb (-L, L)$. Thus, $$\label{eq4-15} \begin{aligned} \liminf\limits_{t \rightarrow \infty} \frac{h(t)}{t} &\geq \lim\limits _{t \rightarrow \infty} \dfrac{ \mu \int_{0}^{t} \int_{-\infty}^{0 }\int_{0}^{\infty} \kappa (x-y)\hat u^* (s, x) {\rm d} y{\rm d}x{\rm d}s}{t} =\lim\limits_{t \rightarrow \infty} \dfrac{ \mu \int_{0}^{t}\int_{-\infty}^{-h(t^*)} \int_{0}^{\infty } \kappa (x-y)\hat u^* (s) {\rm d} y{\rm d}x{\rm d}s}{t} \\&+\lim\limits_{t \rightarrow \infty} \dfrac{ \mu \int_{0}^{t}\int_{-h(t^*)}^{0} \int_{0}^{\infty} \kappa (x-y)\hat u^* (s) {\rm d} y{\rm d}x{\rm d}s}{t} =\lim\limits _{t \rightarrow \infty} \dfrac{ \mu \int_{0}^{t} \int_{-\infty}^{0 }\int_{0}^{\infty} \kappa (x-y)\hat u^* (s) {\rm d} y{\rm d}x{\rm d}s}{t} \\&= { \mu \int_{-\infty}^{0 }\int_{0}^{\infty} \kappa (x-y)\hat u ^*{\rm d} y{\rm d}x} =c^*. \end{aligned}$$ Thus, combining $(\ref{eq4-12})$ and $(\ref{eq4-15})$, one can see that $$\label{cc} \lim\limits_{t \rightarrow \infty} \frac{h(t)}{t}=c^* .$$ ◻ *The proof of Theorem [Theorem 4](#th02-2){reference-type="ref" reference="th02-2"}.* Now we intend to show $(\ref{eq-s1}).$ According to $(\ref{l1})$ in Step 1 of the proof, for any given small $\sigma>0$, there is $T_\sigma>T$ such that $$u(t, x) \leq(1-\sigma)^{-2} u_\sigma(t, x), \text { for } t \geq T_\sigma, -\tilde h(t-T) \leq x \leq \tilde{h}(t-T)$$ where $u_\sigma(t,x)$ is the unique almost periodic solution of $(\ref{eq4-11})$ with $\epsilon$ replaced by $\sigma$ and $\tilde{h}(t)=h(t+T).$ Moreover, by $(\ref{l2})$ in Step 2 of the proof, there are positive constants $\underline {T}_\sigma>T$ and $h_\sigma>0$ such that $$u(t, x) \geq(1-\sigma)^2 u_{\sigma^{-}}(t, x),\text { for } t \geq \underline {T}_\sigma, -\underline h(t)\leq x \leq \underline h(t)$$ where $$\underline h(t)=(1-\sigma)^2(1-2\sigma) \mu \int_{\underline {T}_\sigma}^{t} \int_{-\infty}^{0}\int_{0}^{\infty} \kappa (x-y) u_{\sigma^-}(s, x) {\rm d} y{\rm d}x {\rm d}s+h_\sigma,$$ and $u_{\sigma^{-}}$ is the unique almost periodic solution of $(\ref{eq4-14})$ with $\epsilon$ replaced by $\sigma$. As $$\begin{aligned} &\lim _{\sigma \rightarrow 0}(1-\sigma)^{-2} \mu\int_{-\infty}^{0}\int_{0}^{\infty}\kappa(x-y)u_{\sigma}(t,x){\rm d}y{\rm d}x =\lim _{\sigma \rightarrow 0}(1-\sigma)^2 (1-2\sigma)\mu\int_{-\infty}^{0}\int_{0}^{\infty}\kappa(x-y)u_{\sigma^-}(t,x){\rm d}y{\rm d}x =c^* \end{aligned}$$ uniformly for $t \geq 0$. For any $\varepsilon\in(0,c^*)$, there are much small $\sigma_\varepsilon \in(0, \varepsilon)$ and $T_{\varepsilon}>0$ such that $$\left|(1-\sigma_\varepsilon)^{-2} \mu \int_{0}^{t} \int_{-\infty}^{0}\int_{0}^{\infty}\kappa(x-y)u_{\sigma_\varepsilon}(s, x){\rm d}y{\rm d}x{\rm d}s-c^* t\right|<\frac{\varepsilon}{2} t$$ and $$\left|(1-\sigma_\varepsilon)^2 (1-2\sigma_\varepsilon) \mu \int_{0}^{t} \int_{-\infty}^{0}\int_{0}^{\infty}\kappa(x-y)u_{{\sigma_\varepsilon}^-}(s, x){\rm d}y{\rm d}x{\rm d} s-c^* t\right|<\frac{\varepsilon}{2} t$$ for all $t \geq T_{\varepsilon}$.  Fix $\sigma=\sigma_\varepsilon$ in $u_\sigma$ and $u_{\sigma^-}$. Motivated by the arguments in Step 1 and Step 2, For $t \geq {T}_\varepsilon$, we get $$\begin{aligned} & \xi(t)-x \geq\left(c^*-\varepsilon\right) t-x+R^{\sigma_\varepsilon}+\frac{\varepsilon}{2} t, \\ & \eta(t)-x \geq\left(c^*-\varepsilon\right) t-x+\bar{R}^{\sigma_\varepsilon}+\frac{\varepsilon}{2} t . \end{aligned}$$ we have $$\lim _{x \rightarrow \infty} u_{{\sigma_{\varepsilon}}^-}(t, x)=\underline{u}_{\sigma_\varepsilon}(t) ~ \text { uniformly for } t \in \mathbb{R}$$ and $$\lim _{x \rightarrow \infty} u_{\sigma_\varepsilon}(t, x)=\overline{u}_{\sigma_\varepsilon}(t) ~ \text { uniformly for } t \in \mathbb{R},$$ where $\underline{u}_{\sigma_\varepsilon}(t)$ and $\overline {u}_{\sigma_\varepsilon}(t)$ are the unique positive almost periodic solutions of $$\left(\underline{u}_{\sigma_\varepsilon}\right)_t=\underline {u}_{\sigma_\varepsilon}\left(f\left(t, \underline{u}_{\sigma_\varepsilon}\right)-\sigma_\varepsilon\right),~ \left(\overline{u}_{\sigma_\varepsilon}\right)_t=\overline {u}_{\sigma_\varepsilon}\left(f\left(t, \overline{u}_{\sigma_\varepsilon}\right)+\sigma_\varepsilon\right),$$ respectively. we can find $$u_{\sigma_\varepsilon}(t, x) \leq \overline u_{\sigma_\varepsilon}(t), \text { for all } t >0$$ and $$u_{{\sigma_\varepsilon}^-}(t, x) \geq \underline{u}_{\sigma_\varepsilon}(t)-\varepsilon, \text { for all } t >0$$ locally uniformly for $x$ in $\mathbb R,$ where $\overline u_{\sigma_\varepsilon}(t)$ and $\underline{u}_{\sigma_\varepsilon}(t)$ are the unique time almost periodic solutions of $${u_t}= u (f(t, u)+\sigma_\varepsilon)$$ and $${u}_t={u}(f(t, {u})-\sigma_\varepsilon),$$ respectively. Further, according to $(\ref{cc})$, for such $\varepsilon$, there is $\hat T>T$ such that $\tilde h(t-T)\geq (c^*-\varepsilon)t$ for $t\geq \hat T.$ Denote $$\mathscr{I}_1(\varepsilon):=(1-\sigma_\varepsilon)^2(1-2\sigma_\varepsilon) \mu\int_{0}^{\underline T_{\sigma_\varepsilon}}\int_{-\infty}^{0}\int_{0}^{\infty}\kappa(x-y)u_{{\sigma_\varepsilon}^-} (s,x){\rm d}y{\rm d}x{\rm d}s-h_{\sigma_\varepsilon}.$$ Take $\underline T_{\sigma_\varepsilon} \gg1$ such that $\mathscr{I}_1(\varepsilon)>0.$ It follows that if $t \geq \max\left\{T_{\sigma_\varepsilon}+\underline T_{\sigma_\varepsilon}+T_\varepsilon+\hat T, 2\dfrac{\mathscr{I}_1(\varepsilon)}{\varepsilon} \right\} \text { and } 0 \leq | x | \leq\left(c^*-\varepsilon\right) t,$ then it follows $$u(t, x) \leq\left(1-\sigma_\varepsilon\right)^{-2} u_{{\sigma_\varepsilon}}(t, x) \leq\left(1-\sigma_\varepsilon\right)^{-2} \overline {u}_{\sigma_\varepsilon}(t)$$ and $$u(t, x) \geq\left(1-\sigma_\varepsilon\right)^2 u_{{\sigma_\varepsilon}^-}(t, x) \geq\left(1-\sigma_\varepsilon\right)^2\left(\underline{u}_{\sigma_\varepsilon}(t)-\varepsilon\right).$$ Let $T^{**}= \max\left\{T_{\sigma_\varepsilon}+\underline T_{\sigma_\varepsilon}+T_\varepsilon+\hat T, 2\dfrac{\mathscr{I}_1(\varepsilon)}{\varepsilon} \right\} ,$ it follows $$\left(1-\sigma_\varepsilon\right)^2\left(\underline{u}_{\sigma_\varepsilon}(t)-\varepsilon\right) \leq u(t, x) \leq\left(1-\sigma_\varepsilon\right)^{-2} \overline{u}_{\sigma_\varepsilon}(t),$$ for $t \geq T^{**}$ and $0 \leq | x |\leq\left(c^*-\varepsilon\right) t$. Moreover, by $(\ref{l0})$ in Step 1, it implies that $$\left(1-\sigma_\varepsilon\right)^2\left(\underline{u}_{\sigma_\varepsilon}(t)-\varepsilon\right)-\overline{u}_{\sigma_\varepsilon}(t) \leq u(t, x)-\hat u^*(t) \leq \left(1-\sigma_\varepsilon\right)^{-2} \overline {u}_{\sigma_\varepsilon}(t)-\underline{u}_{\sigma_\varepsilon}(t)$$ Let $$\mathscr{I}_2(\varepsilon)=\max \left\{\left|\left(1-\sigma_\varepsilon\right)^2\left(\underline{u}_{\sigma_\varepsilon}(t)-\varepsilon\right)-\overline{u}_{\sigma_\varepsilon}(t) \right|, \left|\left(1-\sigma_\varepsilon\right)^{-2} \overline {u}_{\sigma_\varepsilon}(t)-\underline{u}_{\sigma_\varepsilon}(t)\right|\right\}.$$ we obtain that $\left|u(t, x)-\hat u^*(t)\right| \leq \mathscr{I}_2(\varepsilon)$ for all $t \geq {T}^{**}$ and $0 \leq |x |\leq\left(c^*-\varepsilon\right) t$. Since $\mathscr{I}_2(\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$, it thus yields $\lim\limits_{t \rightarrow \infty} \max\limits _{x \leq\left(c^*-\varepsilon\right) t}\left|u(t, x)-\hat u^*(t)\right|=0.$ ◻ *The proof of Theorem [Theorem 5](#th1-04){reference-type="ref" reference="th1-04"}.* Now we turn to prove $(\ref{eq1-8})$. Choose a nonnegative, even function sequence $\{\kappa_n\}$ such that each $\kappa_n(x)\in C^1$ has nonempty compact support, and $$\label{eq4-023} \kappa_n(x)\leq\kappa_{n+1}(x)\leq\kappa(x), \text{ and } \kappa_n(x)\rightarrow \kappa(x), \text { in } L^1(\mathbb R) \text{ as } n\rightarrow \infty.$$ where $\kappa_n(x)= \kappa(x)\chi_n(x)$ and $\{\chi_n\}$ is a properly smooth cut-off function sequences such that $\kappa_n(x)$ satisfies $\textbf{(H)}$. Replace $\kappa(x)$ by $\kappa_n(x)$ in $(\ref{eq1-1})$, we can obtain the following auxiliary problem $$\label{eq4-18} \left\{\begin{array}{ll} u_{ t}=d \int_{g\left(t\right)}^{h\left(t\right)} \kappa_n\left(x-y\right) u\left(t, y\right) \mathrm{d} y-d u+u f(t,u), & t>0,~ g\left(t\right)<x<h\left(t\right), \vspace{0.1cm} \\ \vspace{0.1cm} u\left(t, h(t)\right)=u(t,g(t))=0, & t > 0, \\ \vspace{0.1cm} h^{\prime}\left(t\right)= \mu \int_{g\left(t\right)}^{h\left(t\right)} \int_{h\left(t\right)}^{\infty} \kappa_n\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t > 0, \\ \vspace{0.1cm} g^{\prime}\left(t\right)=-\mu \int_{g\left(t\right)}^{h\left(t\right)} \int_{-\infty}^{g\left(t\right)} \kappa_n\left(x-y\right) u\left(t, x\right) \mathrm{d} y\mathrm{d} x, & t >0, \\ \vspace{0.1cm} u\left(0, x\right)=u_{ 0}\left(x\right),~ h\left(0\right)=-g\left(0\right)=h_{0}, & x\in[-h_{0},h_0]. \end{array}\right.$$ Let $(u_n; g_n,h_n)$ be the solution of $(\ref{eq4-18})$. Applying the similar arguments in Step 2, when the spreading happens, we can see that $$\lim\limits_{t\rightarrow\infty}h_n(t)=\infty, \text{ for } n\gg1$$ and $(\ref{eq4-15})$ still holds for $(\ref{eq4-18})$, then $$\begin{aligned} \liminf_{t\rightarrow\infty} \dfrac{h_n(t)}{t} &\geq \lim_{t\rightarrow\infty}\dfrac{ \mu \int_{0}^{t} \int_{-\infty}^{0 }\int_{0}^{\infty} \kappa_n (x-y)\hat u^*_n(s, x) {\rm d} y{\rm d}x{\rm d}s}{t} \\&= \lim_{t\rightarrow\infty}\dfrac{ \mu \int_{0}^{t} \int_{-\infty}^{0 }\int_{0}^{\infty} \kappa_n (x-y)\hat u^*_n(s) {\rm d} y{\rm d}x{\rm d}s}{t} \\&= \mu\int_{-\infty}^{0 }\int_{0}^{\infty} \kappa_n (x-y)\hat u^*_n {\rm d} y{\rm d}x :=\hat c^*_n, \end{aligned}$$ where $\hat u^*_n(t, x)\equiv \hat u^*_n(t)$ is the unique positive time almost periodic solution of $$u_t=d\int_{\mathbb R}\kappa_n(x-y) u(t,y){\rm d}y -d ~ u(t,x)+uf(t, u), ~x\in\mathbb R.$$ Since $(\textbf {H})$ does not hold for $\kappa(x)$, by $(\ref{eq4-023})$, it follows that $\lim\limits_{n\rightarrow\infty}\hat c^*_n=\infty.$ Further, by Lemma 4.1 in [@du2021semi], we can get that for all $n\gg1$, $$\liminf\limits_{t\rightarrow\infty} \dfrac{h(t)}{t}\geq \hat c^*_n \text { and } \liminf\limits_{t\rightarrow\infty}\dfrac{-g(t)}{t}\geq \hat c^*_n.$$ Thus, the conclusion $(\ref{eq1-8})$ holds. The proof of this theorem has been completed. ◻ # Acknowledges {#acknowledges .unnumbered} This work is supported by the China Postdoctoral Science Foundation (No. 2022M710426) and the National Natural Science Foundation of China (No. 12171039 and 12271044).

arxiv_math

--- abstract: | Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, as the boundary operators on 2D topologically ordered spin systems. We show that for the fusion spin chains built from the fusion category $\mathbf{Fib}$, the index is a complete invariant for the group of QCA modulo finite depth circuits. author: - Corey Jones, Junhwi Lim bibliography: - Reference.bib nocite: "[@*]" title: An index for quantum cellular automata on fusion spin chains --- # Introduction Quantum cellular automata (QCA) are models of discrete-time unitary dynamics for quantum spin systems [@https://doi.org/10.48550/arxiv.quant-ph/0405174]. They have many intriguing connections with a wide array of topics at the intersection of quantum information and condensed matter physics [@Farrelly2020reviewofquantum]. Recently, there has been significant interest in studying *topological phases* of QCA as characterized by the group of quantum cellular automata modulo the normal subgroup of finite depth quantum circuits (FDQC) [@MR4309221; @MR2890305; @MR4103966; @MR4381173; @https://doi.org/10.48550/arxiv.2211.02086; @https://doi.org/10.48550/arxiv.2205.09141; @MR4544190].[^1] In this paper, we are interested in studying quantum cellular automata defined in the context of more general "abstract\" spin systems. Abstract spin systems are defined by nets of local finite-dimensional C\*-algebras on a lattice, with the property that operators localized on disjoint regions commute (as in [@MR1441540], see Definition [Definition 1](#AbstractSpinChain){reference-type="ref" reference="AbstractSpinChain"}). However, unlike "concrete\" spin systems, the local algebras are not required to factorize as tensor products of the algebras localized at individual sites. In this paper, our primary focus will be on *fusion spin chains*, which are nets of algebras on a 1D lattice built from the tensor powers of an object in a fusion category. Fusion spin chains arise naturally as local operators in a concrete spin chain that are invariant under a global symmetry (group, Hopf [@MR1463825], weak Hopf [@WeakHopI; @Molnar:2022nmh], or categorical/MPO [@10.21468/SciPostPhys.10.3.053; @MR4109480; @MR4272039; @2205.15243; @MR3719546]), and as boundary operators of 2D local topologically ordered spin systems [@2307.12552]. There are obvious extensions of the concepts of QCA and FDQC in this context [@2304.00068], which are motivated both by classifying dualities of symmetric local Hamiltonians [@Aasen_2016; @https://doi.org/10.48550/arxiv.2008.08598; @https://doi.org/10.48550/arxiv.2211.03777; @PRXQuantum.4.020357] and boundary dynamics of periodic topological systems [@PhysRevX.3.031005; @PhysRevB.99.085115; @aasen2023measurement], (see Section [2.1](#motivation){reference-type="ref" reference="motivation"} for further discussion). This leads naturally to the purely mathematical problem of understanding the group $\mathop{\mathrm{\mathbf{QCA}}}/\mathop{\mathrm{\mathbf{FDQC}}}$ for a given fusion spin chain. The GNVW index, introduced in [@MR2890305], provides a complete characterization of QCA on a concrete spin chain up to finite-depth circuits in 1D. In this paper, our goal is to consider a generalization of the GNVW index to abstract spin chains, by reinterpreting it in terms of the Jones index for subfactors. The Jones index for an inclusion of $\rm{II}_{1}$ factors $N\subseteq M$ is a numerical invariant for an inclusion of $\rm{II}_1$ factors, introduced by V. Jones [@MR0696688]. Remarkably, this index is quantized, only taking values in the set $\{4\text{cos}^{2}(\frac{\pi}{n})\ :n\ge 3\}\cup [4,\infty]$. This led to the modern theory of subfactors, and the discovery of its many connections with mathematical physics, representation theory, and low-dimensional topology [@MR0766964; @MR2354658; @MR1339767; @EK23; @MR3166042; @MR1473221; @MR1642584; @math.QA/9909027]. It was observed in [@Vogts2009DiscreteTQ] that the GNVW index can be expressed as a ratio of Jones indices of subfactors constructed from the spin system algebras. Using a slightly different formulation, we show that this perspective can be used to make sense of the index for QCA on abstract spin chains satisfying what we call the *finite index property*, which in particular holds for fusion spin chains. We demonstrate that the expected properties of the index hold, and in particular, the index gives a homomorphism from the group of QCA modulo FDQC to $\mathbbm{R}^{\times}_{+}$. **Theorem 1**. *Let $A$ be an abstract spin chain that satisfies the *finite index property* (Definition [Definition 6](#FiniteIndexProperty){reference-type="ref" reference="FiniteIndexProperty"}). Then $$\mathop{\mathrm{Ind}}: \mathop{\mathrm{\mathbf{QCA}}}(A)\rightarrow \mathbbm{R}^{\times}_{+}$$ from Definition [Definition 8](#indef){reference-type="ref" reference="indef"} is a homomorphism containing $\mathop{\mathrm{\mathbf{FDQC}}}(A)$ in its kernel.* We then turn to the computation of the index in some concrete examples. We focus our attention on fusion spin chains and introduce a large family of examples of QCA with readily computable index which we call "generalized translations.\" These encapsulate ordinary translations but also more interesting examples that have non-trivial interaction with the DHR category in the sense of [@2304.00068]. It is possible that under fairly general conditions all QCA on a fusion spin chains are generalized translations up to composition by a finite depth circuit. We then give an in-depth analysis of the group of QCA for the fusion spin chain $A_{\tau}$ built from the rank two fusion category $\textbf{Fib}$ with simple objects $\{\mathbbm{1}, \tau \}$ with fusion rules $\tau^{\otimes 2}\cong \mathbbm{1}\oplus \tau$. This net of algebras has been studied in various physical settings and is closely related to "the golden anyon chain\" (see e.g. [@Feiguin:2006ydp; @10.1143/PTPS.176.384; @MR3719546]. We show that the kernel of $\mathop{\mathrm{Ind}}$ is precisely $\mathop{\mathrm{\mathbf{FDQC}}}(A_{\tau})$. In particular, we obtain the following theorem: **Theorem 2**. *Let $\phi=\frac{1+\sqrt{5}}{2}$. Then $$\mathop{\mathrm{Ind}}:\mathop{\mathrm{\mathbf{QCA}}}(A_{\tau})/\mathop{\mathrm{\mathbf{FDQC}}}(A_{\tau})\rightarrow \{\phi^{n}\ :\ n\in \mathbbm{Z}\}\subseteq \mathbbm{R}^{\times}_{+}$$ is an isomorphism of groups. In particular, every QCA on $A_{\tau}$ is a composition of a finite depth circuit and a (honest) translation.* Despite the above theorem, we do not expect $\mathop{\mathrm{Ind}}$ to be a complete invariant for fusion spin chains in general. Indeed, in [@2304.00068] it is shown that the group $\mathop{\mathrm{\mathbf{QCA}}}/\mathop{\mathrm{\mathbf{FDQC}}}$ can be non-abelian, even in the case of an abelian global gauge group on a 1D lattice. Thus, the numerical index cannot be complete in general since it is valued in an abelian group. However, we hope that by combining the index and the DHR invariant [@2304.00068] we will have a nearly complete picture of the group of topological phases of QCA on abstract spin chains (see Section [6](#Outlook){reference-type="ref" reference="Outlook"} for further discussion). **Acknowledgements.** The authors would like to thank Dave Aasen, Dietmar Bisch, Jeongwan Haah, Andrew Schopieray and Dominic Williamson for helpful comments and conversations. The first author is supported by NSF Grant DMS- 2247202. The second author is supported by US ARO grant W911NF2310026. # QCA on abstract spin chains {#basicdefn} In this section, we define abstract spin chains and QCA. **Definition 1**. An *abstract spin chain* consists of a unital C\*-algebra $A$ and an order homomorphism from the poset $\text{Int}(\mathbbm{Z})$ of finite intervals in $\mathbbm{Z}$ to the poset of finite dimensional unital subalgebras of $A$, $I\mapsto A_{I}$, such that 1. (Locality) If $I\cap J=\varnothing$, then $[A_{I},A_{J}]=0$, 2. (Quasi-locality) $\bigcup A_{I}$ is norm dense in $A$. **Remark 2**. We will typically abuse notation slightly and refer to the entire net by its quasi-local algebra $A$. As mentioned in the introduction, there are at least two motivations for considering abstract spin chains. The first is that abstract spin chains arise as nets of algebras invariant under global symmetries (group, Hopf [@MR1463825], weak Hopf [@WeakHopI; @Molnar:2022nmh], or categorical/MPO [@10.21468/SciPostPhys.10.3.053; @MR4109480; @MR4272039; @2205.15243; @MR3719546]) of concrete spin chains. Suppose we have a spin chain with a finite group of global symmetries acting on site. Define $A^{G}=\{a\in A\ : g(a)=a\ \text{for all}\ g\in G\}$. For each interval $I$, set $A^{G}_{I}= (A_{I})^{G}$. This will give an abstract spin chain which is *not* a concrete spin chain, since the algebras $A^{G}_{I}$ do not split as a tensor product over the algebras localized at sites. This renders many aspects of the analysis of spin chains in the concrete case unworkable. This example is a special case of a more general example: **Example 3**. (Fusion spin chains) Let $\mathcal{C}$ be a unitary fusion category and a $X$ choice of tensor generating object. We assume $X$ is self-dual and *strongly tensor generating*, meaning there is an $n$ such that every simple object is a summand of $X^{\otimes n}$. We also assume that $\mathcal{C}$ is strict as a monoidal category (for convenience). Then for any interval $I\subseteq \mathbbm{Z}$, define, $$A_{I}:= \mathcal{C}(X^{\otimes n}, X^{\otimes n})$$ where $n=|I|$, and for $I=[a,b]\subseteq J=[c,d]$ $$\begin{aligned} &A_{I}\hookrightarrow A_{J}&\\ \\ &w\mapsto 1^{\otimes c-a}_{X}\otimes w\otimes 1_X^{\otimes d-b}.&\end{aligned}$$ We then take the colimit of these inclusions in the category of C\*-algebras to obtain the quasi-local AF C\*-algebras $$A:=\lim_{\text{Int}(\mathbbm{Z})} A_{I}.$$ For notational convenience, we identify $A_{I}$ with its image in the colimit $A$. Clearly, this data defines an abstract spin chain with the quasi-local algebra $A$. We will denote this net $A(\mathcal{C}, X)$, and call this a *fusion spin chain*. Fusion spin chains are realized as the local operators in a spin chain invariant under a weak-Hopf or MPO symmetry. Alternatively, they arise as the local boundary operators in topologically ordered 2+1 D spin system as in [@2307.12552]. Moreover, from a purely mathematical point of view, they arise from the lattice of relative commutants of a finite depth subfactor [@MR1642584; @KawAnn] In the sequel, for any interval $I=[a,b]\subseteq \mathbbm{Z}$ and $l\ge 0$, we use the notation $I^{+l}=[a-l,b+l]$. **Definition 4**. A $*$-automorphism $\alpha\in \text{Aut}(A)$ on an abstract spin chain $A$ is called a *quantum cellular automaton* (QCA) if there exists an integer $l\ge 0$ such that $\alpha(A_I)\subseteq A_{I^{+l}}$ and $\alpha^{-1}(A_{I})\subseteq A_{I^{+l}}$ for all $I\in \text{Int}(\mathbbm{Z})$.[^2] The minimum such $l$ is called the *spread* of $\alpha$. It is clear that the composition and inverse of QCA are QCA, and thus, QCA form a subgroup of $\text{Aut}(A)$. We denote this group $\mathop{\mathrm{\mathbf{QCA}}}(A)$. We now introduce the definition of finite depth circuits. To define a depth one quantum circuit, suppose we have of a partition $\{I_{i}\}_{i\in \mathbbm{N}}$ of $\mathbbm{Z}$ such that $\sup_{i}|I_{i}|=l<\infty$, and for each $i$ a unitary $u_{i}\in A_{I_i}$. From this data, we can construct a QCA on $A$ by defining, for any local operator $w\in A$ $$\alpha(w):=\left(\prod_{i}u_{i}\right) w \left(\prod_{i} u^{*}_{i}\right).$$ Since any local $w$ commutes with all but finitely many of the $u_{i}$, the above product gives a well-defined $*$-automorphism on the local algebra $\bigcup_{I} A_{I}$. Furthermore, if $w\in A_{I}$, then $\alpha(w), \alpha^{-1}(w)\in A_{I^{+l}}$ (where $l$ is the largest diameter in the partition $\{I_{i}\}$) and hence, extends to a QCA on $A$. We say the QCA $\alpha$ is a *depth one circuit*. **Definition 5**. A QCA $\alpha\in \text{Aut}(A)$ is a *finite depth quantum circuit* if it can be written as a composition $\alpha_{1}\circ \alpha_{2}\circ\cdots\circ\alpha_{n}$ where each $\alpha_i$ is a depth one circuit. Clearly these form a subgroup of $\mathop{\mathrm{\mathbf{QCA}}}(A)$, which we denote $\mathop{\mathrm{\mathbf{FDQC}}}(A)$. It is shown in [@2304.00068] that $\mathop{\mathrm{\mathbf{FDQC}}}(A)$ is a normal subgroup of $\mathop{\mathrm{\mathbf{QCA}}}(A)$, which leads us to consider the group $\mathop{\mathrm{\mathbf{QCA}}}(A)/\mathop{\mathrm{\mathbf{FDQC}}}(A)$. In the setting of concrete spin systems, the group $\mathop{\mathrm{\mathbf{QCA}}}(A)/\mathop{\mathrm{\mathbf{FDQC}}}(A)$ can be interpreted as characterizing topological phases of discrete unitary dynamics [@https://doi.org/10.48550/arxiv.2205.09141]. If we view an abstract net as the observables under a (generalized) global symmetry, then $\mathop{\mathrm{\mathbf{QCA}}}(A)/\mathop{\mathrm{\mathbf{FDQC}}}(A)$ can be viewed as the topological phases of *symmetric* unitary dynamics. ## Physical motivation {#motivation} There are many motivations from both physics and quantum information for studying QCA on concrete spin systems (see, for example, the review article [@Farrelly2020reviewofquantum]). In this section, we discuss physical motivations for studying QCA in our more general setting of abstract spin systems (see [@2304.00068 Section 2.2] for additional discussion). 1. QCA on spin systems model locally finite-dimensional discrete space/discrete time quantum field theories. In the concrete case, they have been used to approximate continuous quantum field theories (see the review [@Farrelly2020reviewofquantum Section 6.5]). Concrete spin systems are in some sense the "topologically trivial\" examples, since they have a trivial local superselection theory (i.e. DHR category [@2304.00068]). More general abstract spin systems can have non-trivial local superselection sectors if they are obtained, for example, by gauging a global symmetry. These then have the potential to approximate continuum theories with non-trivial superselection sectors which are important, for example, in chiral conformal field theories [@Frolich-Gabbiani; @Kawahigashi-Longo-Muger]. 2. QCA map local Hamiltonians to local Hamiltonians. If the abstract spin system $A^{G}$ consists of local operators invariant under a global symmetry $G$ on the concrete spin system $G$, then $\mathop{\mathrm{\mathbf{QCA}}}(A^{G})$ maps symmetric local Hamiltonians to symmetric local Hamiltonians. In some instances, a symmetric QCA mapping between Hamiltonians cannot be extended to a QCA defined on the concrete spin system $A$, hence the equivalence of the two theories is only witnessed by taking symmetries into account. This is called *duality*, the most famous example being Kramers-Wannier duality [@Aasen_2016; @https://doi.org/10.48550/arxiv.2008.08598; @https://doi.org/10.48550/arxiv.2211.03777; @PRXQuantum.4.020357]. By characterizing symmetric QCA, we parameterize possible dualities of any given Hamiltonians. 3. Suppose we have a (n+1)D locally topologically ordered spin system. The study of topological floquet systems [@PhysRevB.99.085115; @PhysRevX.6.041070] and paths of gapped Hamiltonians [@PhysRevB.106.085122] can be approximated by considering finite depth circuit $U$ intertwining the local ground state projections. If we cut a boundary in the spin system such that the finite depth circuit $U^{\prime}$ built from the terms localized on one side of the boundary commutes with the local ground state projections localized on that same side, then conjugation by $U^{\prime}$ induces a QCA on the net of *boundary algebras* of that spin system. This "chiral boundary dynamics\" is then used to characterize the topological order of the bulk circuit. We have motivated the study of $\mathop{\mathrm{\mathbf{QCA}}}$ on abstract spin systems, but why should we be interested in the quotient group $\mathop{\mathrm{\mathbf{QCA}}}/\mathop{\mathrm{\mathbf{FDQC}}}$? An immediate practical answer is that $\mathop{\mathrm{\mathbf{QCA}}}$ is far too large and unwieldy as a group, and we would have little hope of understanding it. However, a large part of that complexity is contained in the subgroup $\mathop{\mathrm{\mathbf{FDQC}}}$ and by quotienting it out, we obtain a manageable group that is amenable to study and classify. A more principled answer comes from an operational definition of "topological phases,\" introduced in [@PhysRevB.82.155138]. There it is proposed that two many-body states are in the same topological phase (or more properly, they have the same "long-range entanglement structure\") if there exists a finite depth circuit mapping one to the other. This suggests a natural interpretation of $\mathop{\mathrm{\mathbf{QCA}}}/\mathop{\mathrm{\mathbf{FDQC}}}$ as the group of topological phases of $\mathop{\mathrm{\mathbf{QCA}}}$, as in [@https://doi.org/10.48550/arxiv.2205.09141]. # Index for QCA on abstract spin chains {#index} In this section, we will introduce a generalization of the GNVW index [@MR2890305] that applies to abstract spin chains with a unique tracial state, satisfying some finite index type properties. First, recall a $\rm{II}_{1}$ factor is an infinite-dimensional von Neumann algebra with trivial center and a normal tracial state. If an infinite dimensional C\*-algebra has a unique tracial state, then its bicommutant in the GNS representation is a $\rm{II}_{1}$ factor. Let $M$ be a $\rm{II}_{1}$ factor. Associated to any Hilbert space representation $H$ of $M$ is its Murray-von Neumann dimension $\text{dim}_{M}(H)\in \mathbbm{R}_{+}\cup \{\infty\}$. Given an inclusion of $\rm{II}_{1}$ factors $N\subseteq M$, we define the Jones index [@MR0696688] as $$[M:N]:=\text{dim}_{N}(L^{2}(M)).$$ A subfactor has *finite index* if $[M:N]<\infty$. We recall one of the most useful properties of the Jones index: its multiplicativity. If $N\subseteq P\subseteq M$ is an inclusion of $\rm{II}_{1}$ factors and $[M:N]<\infty$, then $$[M:P], [P:N]<\infty$$ and $$[M:N]=[M:P][P:N].$$ We refer the reader to [@MR0696688; @MR1473221] for a proof and various other basic properties of the index. There is an incredibly rich theory of finite index subfactors [@MR0696688; @MR3166042] and our goal is to make use of this theory in the study of QCA defined on abstract spin chains. To this end, we have the following definition, which allows us to make use of the Jones index. **Definition 6**. Let $A$ be an abstract spin chain with a unique tracial state $\mathop{\mathrm{tr}}$, and let $\mathcal{A}:=A^{\prime \prime}\subseteq B(L^{2}(A, \mathop{\mathrm{tr}}))$. $A$ satisfies the (left) *finite index property* if for any $x\in \mathbbm{Z}$, 1. The subalgebra of $\mathcal{A}$, $\mathcal{A}_{x}:=(\bigcup_{I\le x} A_{I})^{\prime \prime}\subseteq B(L^{2}(A,\mathop{\mathrm{tr}}))$, is an infinite-dimensional factor. 2. For any $z>x$, the subfactor $\mathcal{A}_{x}\subseteq \mathcal{A}_{z}$ has finite index. **Remark 7**. Since $A$ has a unique tracial state, any QCA preserves $\mathop{\mathrm{tr}}$ and thus extends uniquely to an automorphism of $\mathcal{A}$. Furthermore, it is easy to see from the definitions that if $\alpha\in \mathop{\mathrm{\mathbf{QCA}}}(A)$ has spread $l$, then $\mathcal{A}_{x-l}\subseteq\alpha(\mathcal{A}_{x})\subseteq \mathcal{A}_{x+l}$. **Definition 8**. Let $A$ be an abstract spin chain and $\alpha$ be a QCA on $A$ satisfying the finite index property. Let $y\le x$ be integers such that $\mathcal{A}_y\subseteq \alpha(\mathcal{A}_x)$. Then the (left) index of $\alpha$ is defined by $$\mathop{\mathrm{Ind}}(\alpha)=\left(\frac{[\alpha(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_y]}\right)^{1/2}.$$ The numerator $[\alpha(\mathcal{A}_x):\mathcal{A}_{y}]$ is finite, since there is an integer $z\ge x$ such that $\mathcal{A}_y\subseteq\alpha(\mathcal{A}_x)\subseteq \mathcal{A}_z$ and we have $[\alpha(\mathcal{A}_x):\mathcal{A}_{y}]\le [\mathcal{A}_z:\mathcal{A}_y]<\infty$ by the finite index property. **Proposition 9**. *$\mathop{\mathrm{Ind}}(\alpha)$ is independent of the choice of $x$ and $y$.* *Proof.* We first prove the independence of $y$. Suppose $x$ is fixed and $y$ is the maximum integer that satisfies the condition in Definition [Definition 8](#indef){reference-type="ref" reference="indef"}. For $y'\le y\le x$, $$\begin{aligned} \frac{[\alpha(\mathcal{A}_x):\mathcal{A}_{y'}]}{[\mathcal{A}_x:\mathcal{A}_{y'}]}=\frac{[\alpha(\mathcal{A}_x):\mathcal{A}_{y}][\mathcal{A}_{y}:\mathcal{A}_{y'}]}{[\mathcal{A}_x:\mathcal{A}_y][\mathcal{A}_{y}:\mathcal{A}_{y'}]}=\frac{[\alpha(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_y]}.\end{aligned}$$ Hence, $\mathop{\mathrm{Ind}}(\alpha)$ is independent of $y$. Next, we show the independence of $x$. Suppose $x,x', y\in\mathbbm{Z}$ such that $y\le x'\le x$ and $\mathcal{A}_y\subseteq \alpha(\mathcal{A}_{x'})\subseteq\alpha(\mathcal{A}_{x})$. Then $$\begin{aligned} \frac{[\alpha(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_y]}&=\frac{[\alpha(\mathcal{A}_x):\alpha(\mathcal{A}_{x'})][\alpha(\mathcal{A}_{x'}):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_{x'}][\mathcal{A}_{x'}:\mathcal{A}_y]}=\frac{[\alpha(\mathcal{A}_{x'}):\mathcal{A}_{y}]}{[\mathcal{A}_{x'}:\mathcal{A}_y]}\end{aligned}$$ The last identity follows from $[\mathcal{A}_x:\mathcal{A}_{x'}]=[\alpha(\mathcal{A}_x):\alpha(\mathcal{A}_{x'})]$. ◻ **Proposition 10**. *Let $\alpha$ and $A$ be as in Definition [Definition 8](#indef){reference-type="ref" reference="indef"} and let $y\le x\le z$ be integers satisfying $\mathcal{A}_y\subseteq \alpha(\mathcal{A}_x)\subseteq\mathcal{A}_z$. Then $$\begin{aligned} \mathop{\mathrm{Ind}}(\alpha)=\left(\frac{[\mathcal{A}_z:\mathcal{A}_x]}{[\mathcal{A}_{z}:\alpha(\mathcal{A}_x)]}\right)^{1/2}=\left(\frac{[\mathcal{A}_z:\mathcal{A}_x][\alpha(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_{z}:\alpha(\mathcal{A}_x)][\mathcal{A}_x:\mathcal{A}_y]}\right)^{1/4}.\end{aligned}$$* *Proof.* We first show the first identity. $$\begin{aligned} \mathop{\mathrm{Ind}}(\alpha)^2=\frac{[\alpha(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_y]}=\frac{[\mathcal{A}_z:\mathcal{A}_y]/[\mathcal{A}_{z}:\alpha(\mathcal{A}_x)]}{[\mathcal{A}_z:\mathcal{A}_y]/[\mathcal{A}_{z}:\mathcal{A}_x]}=\frac{[\mathcal{A}_{z}:\mathcal{A}_x]}{[\mathcal{A}_{z}:\alpha(\mathcal{A}_x)]}. \end{aligned}$$ Observe that $$\mathop{\mathrm{Ind}}(\alpha)=\mathop{\mathrm{Ind}}(\alpha)^{1/2}\left(\frac{[\mathcal{A}_z:\mathcal{A}_x]}{[\mathcal{A}_{z}:\alpha(\mathcal{A}_x)]}\right)^{1/4}=\left(\frac{[\mathcal{A}_z:\mathcal{A}_x][\alpha(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_{z}:\alpha(\mathcal{A}_x)][\mathcal{A}_x:\mathcal{A}_y]}\right)^{1/4}.$$ Hence, we have the second identity. ◻ **Proposition 11**. *$\mathop{\mathrm{Ind}}(\alpha\circ\beta)=\mathop{\mathrm{Ind}}(\alpha)\mathop{\mathrm{Ind}}(\beta)$* *Proof.* Choose integers $y\le y'\le x$ satisfying the following: 1. $\mathcal{A}_y\subseteq \alpha(\mathcal{A}_{y'})$; 2. $\mathcal{A}_{y'}\subseteq \beta(\mathcal{A}_{x})$. Then we have $$\begin{aligned} \mathop{\mathrm{Ind}}(\alpha\circ\beta)^2&=\frac{[\alpha\circ\beta(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_y]}=\frac{[\alpha\circ\beta(\mathcal{A}_x):\alpha(\mathcal{A}_{y'})][\alpha(\mathcal{A}_{y'}):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_{y'}][\mathcal{A}_{y'}:\mathcal{A}_y]}\\ &=\frac{[\beta(\mathcal{A}_x):\mathcal{A}_{y'}][\alpha(\mathcal{A}_{y'}):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_{y'}][\mathcal{A}_{y'}:\mathcal{A}_y]}=\mathop{\mathrm{Ind}}(\beta)^2\mathop{\mathrm{Ind}}(\alpha)^2. \end{aligned}$$ ◻ **Proposition 12**. *Finite depth circuits are in $\ker(\mathop{\mathrm{Ind}})$.* *Proof.* It suffices to show that the depth $1$ circuits are in $\ker(\mathop{\mathrm{Ind}})$. Let $\{I\}$ be a collection of intervals that partitions $\mathbbm{Z}$ and $\sup_I|I|<\infty$. Let $\alpha=\mathop{\mathrm{Ad}}(U)$ where $U=\prod_I U_I$ for some unitaries $U_I$ in $\mathcal{A}_I$. Choose an interval $I$ in the partition, and let $x=\sup I$. Then we have $\alpha(\mathcal{A}_x)=\mathcal{A}_x$. Therefore, for any $y\le x$ $$\begin{aligned} \mathop{\mathrm{Ind}}(\alpha)^2&=\frac{[\alpha(\mathcal{A}_x):\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_y]}=\frac{[\mathcal{A}_x:\mathcal{A}_{y}]}{[\mathcal{A}_x:\mathcal{A}_y]}=1. \end{aligned}$$ ◻ **Remark 13**. Our treatment of index has been somewhat asymmetric. We have only assumed that the left infinite algebras are factors, and thus our index only makes sense a-priori with the "left infinite\" algebras. Alternatively, we could consider a (right) finite index property, and define the subfactors $\mathcal{A}^{+}_{x}:=(\bigcup_{I\ge x} A_{I})^{\prime \prime}\subseteq B(L^{2}(A,\mathop{\mathrm{tr}}))$. In this case we could define a "right\" version of the index as follows. For $x\le z$ with $\mathcal{A}^{+}_{z}\subseteq\alpha(\mathcal{A}^{+}_{x})$, set $$\mathop{\mathrm{Ind}}(\alpha)=\left(\frac{[\mathcal{A}_x^+:\mathcal{A}_z^+]}{[\alpha(\mathcal{A}_x^+):\mathcal{A}_z^+]}\right)^{1/2}.$$ In general, this may give different information than the left index (in the case when both indices make sense!). However, in the case of the fusion spin chains, with some work one can show this alternative index is equal to the index defined above. Since we make no use of the "right\" version, we do not include the argument since it would take us too far afield. **Theorem 14**. *If $X\ncong \mathbbm{1}$ is strongly tensor generating, then the net $A(\mathcal{C}, X)$ has a unique tracial state and satisfies the finite index property. Thus, we have a canonical homomorphism* *$$\mathop{\mathrm{Ind}}: \mathop{\mathrm{\mathbf{QCA}}}(A(\mathcal{C},X))/\mathop{\mathrm{\mathbf{FDQC}}}(A(\mathcal{C},X))\rightarrow \mathbbm{R}^{\times}_{+}.$$* *Proof.* It suffices to show that the quasi-local algebra has a unique tracial state, and that the left infinite quasi-local algebras are factors in the GNS completion of the global tracial state. Choose some $n$ so that $X^{\otimes n}$ contains all isomorphism classes of simple objects in $\mathcal{C}$. Set $A_{k}:=A_{(-kn,(k+1)n]}$. Then we can view the quasi-local algebra as the AF-C\*-algebra with finite dimensional tower $$A_{0}\subseteq A_{1}\subseteq \dots.$$ The Bratelli diagram (ignoring multiplicities) has at each level vertices indexed by the simple objects of $\mathcal{C}$, and the adjacency matrix given between levels by the fusion graph of $X^{\otimes n} \otimes (\cdot) \otimes X^{\otimes n}$. Thus, the Bratteli diagram is stationary, so there is a unique tracial state given by the Perron-Frobenius eigenvector. Similar reasoning shows that the left algebras $A_{\le x}$ have representations by stationary Bratteli diagrams, hence also have unique tracial state. In particular, the restriction of the global tracial state to this one is unique, and thus, the bicommutant completion in the GNS representation is a $\rm{II}_{1}$ factor (we use $X\ne \mathbbm{1}$ here to guarantee the factor is infinite dimensional). ◻ **Remark 15**. We can now explain the subfactor perspective on fusion spin chains. Let $A=A(\mathcal{C},X)$ with $X$ self-dual and strongly tensor generating, and let $\mathcal{A}_{x}$ denote the $\rm{II}_{1}$ factors as described above. Pick any $x$. Then $\mathcal{A}_{x}\subseteq \mathcal{A}_{x+1}$ is a finite index, finite depth subfactor. Indeed, this is the standard model [@MR1055708; @MR1642584] for the finite depth subfactor with standard invariant $$\begin{tikzcd} \mathbbm{C}\arrow[r, phantom, "\subseteq"] &\mathop{\mathrm{End}}(X) \arrow[r, phantom, "\subseteq"]& \mathop{\mathrm{End}}(X\otimes \overline{X})\arrow[r, phantom, "\subseteq"] & \mathop{\mathrm{End}}(X\otimes \overline{X}\otimes X)\arrow[r, phantom, "\subseteq"]& \dots \\ & \mathbbm{C}\arrow[u, sloped, phantom, "\subseteq"]\arrow[r, phantom, "\subseteq"] &\mathop{\mathrm{End}}(\overline{X})\arrow[r, phantom, "\subseteq"] \arrow[u, sloped, phantom, "\subseteq"] & \mathop{\mathrm{End}}(\overline{X}\otimes X)\arrow[r, phantom, "\subseteq"] \arrow[u, sloped, phantom, "\subseteq"]& \dots \end{tikzcd}$$ Since $\overline{X}\cong X$, each of these algebras is isomorphic to $\mathop{\mathrm{End}}(X^{\otimes n})$ in $\mathcal{C}$. Furthermore, $\mathcal{A}_{x}\subseteq \mathcal{A}_{x+1}\subseteq \mathcal{A}_{x+2}$ is an instance of the basic construction, and thus, we view the entire tower $\dots\subseteq \mathcal{A}_{x-1}\subseteq \mathcal{A}_{x}\subseteq \mathcal{A}_{x+1}\subseteq\cdots$ as the Jones tower of the standard model for the subfactor with the above invariant. In particular, this allows us to identify $\mathop{\mathrm{End}}(X^{\otimes z-x})\cong A_{(x,z]}=\mathcal{A}_{x}^{\prime}\cap \mathcal{A}_{z}$ for $x<z$. # Generalized translations on fusion spin chains In this section, we consider a class of examples of QCA called *generalized translations*. Consider a fusion spin chain $A(\mathcal{C}, X)$, which in this subsection we denote for $A$ for short. Suppose in addition we have the following data: 1. A fusion category $\mathcal{D}$ and a full inclusion $\mathcal{C}\subseteq \mathcal{D}$. 2. A factorization $X^{\otimes n}\cong Y\otimes Z$ for some fixed $n$ and $Y,Z\in \mathcal{D}$ 3. An isomorphism $\sigma:Y\otimes Z\cong Z\otimes Y$ in $\mathcal{D}$. We define a QCA $\alpha$ on $A$ as follows. First, partition $\mathbbm{Z}$ into intervals of length $n$. Then consider intervals $I$ which are unions of intervals in the partition. We call these *coarse-grained* intervals. Then for coarse-grained intervals $I$ consisting of $m$ length $n$ intervals, $$A_{I}\cong \text{End}_{\mathcal{C}}((X^{\otimes n})^{\otimes m})\cong \text{End}_{\mathcal{D}}((Y\otimes Z)^{\otimes m}).$$ Recall that for any coarse-grained interval $I$ with $m$ length $n$ intervals, then $I^{+n}$ is again a coarse-grained interval consisting of $m+2$ intervals of length $n$. Then for any coarse-grained interval $I$ and $x\in A_{I}$, we define $$\alpha(w)=1_{Y\otimes Z}\otimes 1_{Y}\otimes [\sigma^{\otimes m}\circ w\circ (\sigma^{-1})^{\otimes m}]\otimes 1_{Z}\in A_{I^{+n}}$$ where $\sigma^{\otimes m}: (Y\otimes Z)^{m}\cong (Z\otimes Y)^{m}$ is simply the $m$-fold tensor product $\sigma \otimes \dots \otimes \sigma$. This clearly extends to a C\*-homomorphism on the quasi-local algebra. Its inverse is given, for $w\in A_{I}$ with $I$ coarse-grained as above $$\alpha^{-1}(w)= [(\sigma^{-1})^{\otimes m+1}\circ (1_{Z}\otimes w\otimes 1_{Y})\circ \sigma^{\otimes m+1}]\otimes 1_{Y\otimes Z}\in A_{I^{+n}}.$$ It is easy to see that $\alpha$ and $\alpha^{-1}$ are QCA, which we call *generalized translations*. The reason for this terminology is choosing $\mathcal{D}=\mathcal{C}$, $n=1$ and $X\cong X\otimes 1$ the coherence unitor, then the resulting QCA $\alpha$ is simply translation to the right by one site. In many cases of interest (including concrete spin systems), generalized translations represent *all possible* QCA up to finite depth circuits, and the index can be used to show this. We have the following computation of the index for these QCA. **Proposition 16**. *Let $\alpha$ be a generalized translation constructed as above from a factorization $X^{\otimes n}\cong Y\otimes Z$. Then $\mathop{\mathrm{Ind}}(\alpha)=\dim(Y)$.* *Proof.* Let $x$ be an endpoint of a coarse-grained interval. Then $\mathcal{A}_{x}\subseteq \alpha(\mathcal{A}_{x})$. It is easy to see that $\alpha(\mathcal{A}_{x})$ is precisely the subalgebra of operators in $\mathcal{A}_{x+n}$ which can be written $w\otimes 1_{Z}$. The resulting subfactor $\mathcal{A}_{x}\subseteq \alpha(\mathcal{A}_{x})$ has Jones index $\dim(Y)^{2}$, and thus $\mathop{\mathrm{Ind}}(\alpha)=\dim(Y)$. ◻ ## Concrete spin chains Now let $\mathcal{C}=\textbf{Hilb}_{f.d.}$, the category of finite dimensional Hilbert spaces. If we pick $X:=\mathbbm{C}^{d}$, then we obtain the usual definition of a spin chain of qudits. This is the case of QCA studied in [@MR2890305]. We can now see that our index agrees with the index defined in [@MR2890305], which we call the *GNVW index*. **Proposition 17**. *The definition of index above agrees with the GNVW index for concrete spin chains.* *Proof.* Let us denote the qudit system $A_{d}$. The GNVW index gives an isomorphism $$\mathop{\mathrm{\mathbf{QCA}}}(A_{d})/\mathop{\mathrm{\mathbf{FDQC}}}(A_{d})\rightarrow \mathbbm{Z}[\frac{1}{d}]_+^{\times},$$ where $\mathbbm{Z}[\frac{1}{d}]_+^{\times}$ denotes the group of positive units of of the ring inside $\mathbbm{Q}$ generated by $\mathbbm{Z}$ and $\frac{1}{d}$. In particular, this shows that $\mathop{\mathrm{\mathbf{QCA}}}(A_{d})/\mathop{\mathrm{\mathbf{FDQC}}}(A_{d})$ is generated as a group by (the cosets of) generalized translations, built from a factorization of integers $(\mathbbm{C}^d)^{\otimes k}\cong \mathbbm{C}^{p}\otimes \mathbbm{C}^{q}$ and using the symmetric "swap\" braiding $\sigma$ from $\text{Hilb}_{f.d.}$ for the isomorphism. Let $I_i=(k(i-1),ki]$ and $x=x_{(1)}\otimes x_{(2)}\in A_{I_i}\cong M_{p}^i(\mathbbm{C})\otimes M_{q}^i(\mathbbm{C})$, we define $$\alpha_{p}(x)=(1^{i}_{p}\otimes x_{(2)})\otimes (x_{(1)}\otimes 1^{i+1}_{q})\in A_{I_i}\otimes A_{I_{i+1}}.$$ By the above proposition, $\mathop{\mathrm{Ind}}(\alpha)=p.$ This agrees with the GNVW index, and since the value of the homomorphism $\mathop{\mathrm{Ind}}$ agrees with the GNVW index on a generating set, it must be precisely the same homomorphism. ◻ ## Generalized Kramers-Wannier translations from $G$-graded extensions Now consider the case where $$\mathcal{D}=\bigoplus_{g\in G} \mathcal{C}_{g}$$ is a faithful $G$-graded extension of $\mathcal{C}$, so that $\mathcal{C}_{e}=\mathcal{C}$, each $\mathcal{C}_{g}$ is an invertible $\mathcal{C}$-bimodule, and $\mathcal{C}_{g}\otimes \mathcal{C}_{h}\subseteq \mathcal{C}_{gh}$. We will suppose that $G=\mathbbm{Z}/n\mathbbm{Z}$ with a generator $1$ and the identity $0$, and that we have an object $Y\in \mathcal{C}_{1}$ such that $X:=Y^{\otimes n}\in \mathcal{C}=\mathcal{C}_{0}$ is a strong tensor generator for $\mathcal{C}$. Then we have $X=Y\otimes Z$, where $Z=Y^{n-1}$, and we use the isomorphism $\text{id}_{Y^{n}}: Y\otimes Z\cong Z\otimes Y$, with $n=1$ in the above construction. If we think of the single $X$-sites as being in fact $n$-coarse-grained $Y$ sites, then this is simply the shift to the right by one $Y$ string. We call a generalized translation of the above type a *generalized Kramers-Wannier duality*, due to the example below. **Remark 18**. $G$-graded extensions of a fusion category $\mathcal{C}$ are classified by morphisms of 3-groups $\psi: G\rightarrow \text{BrPic}(\mathcal{C})$. Dimensionally reducing and applying the ENO isomorphism yields a morphism of 2-groups $\widetilde{\psi}: G\rightarrow \text{Aut}_{br}(\mathcal{Z}(\mathcal{C}))$. For a generalized Kramers-Wannier translation $\alpha$ built from a $\mathbbm{Z}/n\mathbbm{Z}$ extension of $\mathcal{C}$ built from the 3-group $\psi$, let $\xi=\widetilde{\psi}(1)\in \text{Aut}_{br}(\mathcal{Z}(\mathcal{C}))$. Then utilizing the equivalence $\textbf{DHR}(A(\mathcal{C},X))\cong \mathcal{Z}(\mathcal{C})$, it is easy to see by unpacking the definitions that $\textbf{DHR}(\alpha)=[\xi]$, where by $[\xi]$ we mean the monoidal equivalence class of $\xi$. **Example 19**. (Kramers-Wannier translations.) We consider a special case, which extends the original Kramers-Wannier duality. Let $B$ be an abelian group, and consider the unitary fusion category $\mathcal{C}:=\textbf{Hilb}(B)$ of finite dimensional $B$-graded Hilbert spaces. We let $X:=\bigoplus_{g\in B} \mathbbm{C}_{g}$. Then we have $X\cong \mathbbm{C}[B]$, the group algebra viewed as an $B$-graded Hilbert space. We consider the fusion spin chain $A(\mathcal{C},X)$. We will construct a QCA on this net using the above $G$-graded recipe. Now, recall a (unitary Tambara-Yamagami) category with abelian group of invertibles $B$ is characterized by a non-degenerate, symmetric, bicharacter $\chi$ on $B$ [@TAMBARA1998692]. Then $\mathcal{TY}(B,\chi)$ has (isomorphism classes of) simple objects $B\cup \{\rho\}$, with fusion rules $$b\otimes a\cong ba$$ $$\rho\otimes \rho\cong\bigoplus_{b\in B} b$$ $$b\otimes \rho\cong \rho\otimes b\cong \rho$$ Note we have a full inclusion $\textbf{Hilb}(B)\subseteq \mathcal{TY}(B,\chi)$, and in fact $\mathcal{TY}(B,\chi)$ is a $\mathbbm{Z}/2\mathbbm{Z}$-graded extension of $\textbf{Hilb}(B)$. Utilizing the unique simple object $\rho\in \mathcal{TY}(B,\chi)$ that is not in the trivially graded component, we have $\rho^{\otimes 2}\cong X\in \textbf{Hilb}(B)$. In particular, we can apply the generalized translation associated to this factorization, and we obtain a $\alpha\in \mathop{\mathrm{\mathbf{QCA}}}(A(\mathcal{C},X))$ with $$\mathop{\mathrm{Ind}}(\alpha)=\sqrt{|B|}.$$ If we consider the case $B:=\mathbbm{Z}/2\mathbbm{Z}$, this implements a version of the famous Kramers-Wannier duality (for a detailed explanation of this see [@PRXQuantum.4.020357 Section II.A]). We also note that having index values with square roots also appears, in a closely related context, in [@PhysRevB.99.085115]. # Index as an isomorphism for the $\textbf{Fib}$ chain {#golden} In this section, we consider the fusion spin chain constructed from the fusion category $\mathop{\mathrm{\mathbf{Fib}}}$ with simple generating object $\tau$. Recall $\mathop{\mathrm{\mathbf{Fib}}}$ is the rank $2$ unitary fusion category with simple objects $\mathbbm{1}$ and $\tau$, with fusion rule $\tau\otimes \tau\cong \mathbbm{1}\oplus \tau$. In this section, we denote by $A_\tau$ the fusion spin chain of algebras built from $\mathop{\mathrm{\mathbf{Fib}}}$ with generating object $\tau$. Our goal in this section is to use the index to characterize the group $\mathop{\mathrm{\mathbf{QCA}}}/\mathop{\mathrm{\mathbf{FDQC}}}$ for $A_{\tau}$. $A_\tau$ can be viewed either as the boundary algebra for a Levin-Wen model built from $\mathop{\mathrm{\mathbf{Fib}}}$ (after coarse-graining nearest neighbors), or as the net of local operators on the golden anyon chain, invariant under the $\mathop{\mathrm{\mathbf{Fib}}}$ MPO symmetry. From another perspective, this is the even part of the $A_{4}$ subfactor standard invariant, and in fact, the net we build can also be viewed as the two-sided tower of relative commutants in the tunnel of the Jones tower of the unique $A_{4}$ hyperfinite subfactor [@MR1642584; @MR4272039] First we examine possible values of the index. Note that $\mathcal{A}_{x}\subseteq \mathcal{A}_{x+1}\subseteq \mathcal{A}_{x+2}$ is a basic construction triple, and thus, $\mathcal{A}_{x}\subseteq \mathcal{A}_{x+k}$ is the iterated basic construction, and has index $\phi^{2k}$, where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio. We recall that the unitary fusion category $\mathop{\mathrm{\mathbf{Fib}}}$ is *torsion-free* in the sense of [@MR3941472]: every indecomposable Q-system in $\mathop{\mathrm{\mathbf{Fib}}}$ is a Morita trivial Q-system. A Q-system is *Morita trivial* if and only if it can be written as $X\otimes \bar{X}$ with multiplication induced from evaluation, for some object $X\in \mathop{\mathrm{\mathbf{Fib}}}$ [@MR4419534]. **Definition 20**. Let $\mathcal{D}$ be a (full, replete) tensor subcategory of $\text{Bim}(N)$, where $N$ is a $\rm{II}_{1}$ factor. Suppose that $N\subseteq M$ is a finite index subfactor with $L^{2}(M)\in \mathcal{D}$. Then we define the *dual category* of $M$ with respect to $\mathcal{D}$, denoted $\mathcal{D}^{*}_{M}$, as the (full, replete) unitary subcategory of $\text{Bim}(M)$ which is the preimage of $\mathcal{D}$ under the restriction functor $\text{Bim}(M)\rightarrow \text{Bim}(N)$. Categorically, $\mathcal{D}^{*}_{M}$ is equivalent to the category of bimodules of the Q-system $_{N} L^{2}(M)_{N}$, internal to $\mathcal{D}$. Now, for any $x$ and any $z>x$, the even part of the subfactor $\mathcal{A}_{x}\subseteq \mathcal{A}_{z}$ generates a canonical copy of $\mathop{\mathrm{\mathbf{Fib}}}$ inside $\text{Bim}(\mathcal{A}_{x})$ (the same copy of $\mathop{\mathrm{\mathbf{Fib}}}$ for all $z$). We call this $\mathop{\mathrm{\mathbf{Fib}}}_{x}$. **Lemma 21**. *Let $x<z$ and $\mathcal{A}_{x}\subsetneq P\subseteq \mathcal{A}_{z}$ an intermediate subfactor. Then $(\mathop{\mathrm{\mathbf{Fib}}}_{x})^{*}_{P}\cong \mathop{\mathrm{\mathbf{Fib}}}$, and the $P$-$P$ bimodule $_{P}L^{2}(\mathcal{A}_{z})_{P}\in (\mathop{\mathrm{\mathbf{Fib}}}_{x})^{*}_{P}$.* *Proof.* Since $\mathcal{A}_{x}\subsetneq P\subseteq \mathcal{A}_{z}$, $L^{2}(P)\in \mathop{\mathrm{\mathbf{Fib}}}_{x}$, and since $\mathop{\mathrm{\mathbf{Fib}}}$ is torsion free, we have $L^{2}(P)$ is Morita trivial, which means $(\mathop{\mathrm{\mathbf{Fib}}}_{x})^{*}_{P}\cong \mathop{\mathrm{\mathbf{Fib}}}$. Furthermore, the $P$-$P$ bimodule $L^{2}(\mathcal{A}_{z})$ is in $\mathop{\mathrm{\mathbf{Fib}}}_{x}$ as an $\mathcal{A}_{x}$-$\mathcal{A}_{x}$ bimodule by construction, hence $_{P}L^{2}(\mathcal{A}_{z})_{P}\in (\mathop{\mathrm{\mathbf{Fib}}}_{x})^{*}_{P}$. ◻ **Theorem 22**. *The map $\text{Ind}: \mathop{\mathrm{\mathbf{QCA}}}(A_\tau)\rightarrow \mathbbm{R}^{\times}_{+}$ surjects onto the subgroup $\{\phi^{n}\ :\ n\in \mathbbm{Z}\}\cong \mathbbm{Z}$.* *Proof.* First, we claim that the image of $\text{Ind}$ is contained in the subring $\mathbbm{Z}[\phi]$ of $\mathbbm{R}$. Indeed, let $y<x<z$ such that $\mathcal{A}_{y}\subseteq \alpha(\mathcal{A}_{x})\subseteq \mathcal{A}_{z}$. Then $\text{Ind}(\alpha)^{-1}=\left(\frac{[\mathcal{A}_{z}:\alpha(\mathcal{A}_{x})]}{[\mathcal{A}_{z}:\mathcal{A}_{x}]}\right)^{\frac{1}{2}}=[\mathcal{A}_{z}:\alpha(\mathcal{A}_{x})]^{\frac{1}{2}}\phi^{-(z-x)}$. By the previous lemma $_{\alpha(\mathcal{A}_{x})} L^{2}(\mathcal{A}_{z}) _{\alpha(\mathcal{A}_{x})}$ is an indecomposable Q-system in $(\mathop{\mathrm{\mathbf{Fib}}}_{y})^{*}_{\alpha(\mathcal{A}_{x})}\cong \mathop{\mathrm{\mathbf{Fib}}}$, hence there exists some object $X\in (\mathop{\mathrm{\mathbf{Fib}}}_{y})^{*}_{\alpha(\mathcal{A}_{x})}$ with $_{\alpha(\mathcal{A}_{x})} L^{2}(\mathcal{A}_{z}) _{\alpha(\mathcal{A}_{x})}\cong X\otimes \bar{X}$. In particular, $$[\mathcal{A}_{z}:\alpha(\mathcal{A}_{x})]=\text{dim}(_{\alpha(\mathcal{A}_{x})} L^{2}(\mathcal{A}_{z}) _{\alpha(\mathcal{A}_{x})})=\text{dim}(X)^{2}$$ Thus, $[\mathcal{A}_{z}:\alpha(\mathcal{A}_{x})]^{\frac{1}{2}}=\text{dim}(X)\in \mathbbm{Z}[\phi]$. Since $\phi^{-1}=\phi-1$, $\phi^{-(z-x)}=(\phi-1)^{z-x}\in \mathbbm{Z}[\phi]$. Thus, $\text{Ind}(\alpha)^{-1}\in \mathbbm{Z}[\phi]$. Note that $\text{Ind}(\alpha^{-1})^{-1}=\text{Ind}(\alpha)$, so $\text{Ind}(\alpha)\in \mathbbm{Z}[\phi]_+^{\times}$. The group $\mathbbm{Z}[\phi]_+^{\times}$ of positive units of the ring $\mathbbm{Z}[\phi]$ consists precisely of integral powers of $\phi$. Thus, the image of $\mathop{\mathrm{Ind}}$ lies in $\{\phi^{n}\ :\ n\in \mathbbm{Z}\}$. It remains to show that every $\phi^{n}$ is realized by some QCA. These are seen to be the indices of the translations. ◻ **Lemma 23**. *Let $A_0\subseteq A_1\subseteq A_2$ be an inclusion of multimatrix algebras and $n^{(i)}$ be the dimension vector of $A_i$. We allow $n^{(i)}$ to have zero entries with the understanding that $A_i\cong \bigoplus_{j: n_j^{(i)}\ne 0}M_{n_j^{(i)}}(\mathbbm{C})$. Suppose we can identify the slots of vectors $n^{(0)}$ and $n^{(2)}$ so that there is an indecomposable matrix $T$ with nonnegative integer entries satisfying $$n^{(0)}T^t=n^{(1)}\ \ \ \text{ and }\ \ \ n^{(1)}T=n^{(2)}.$$ Let $\mathop{\mathrm{tr}}$ be a unique tracial state on $A_2$ whose corresponding trace vector $t^{(2)}$ is the Perron-Frobenius eigenvector of $T^t T$ with the eigenvalue $\frac{1}{\lambda}$. Then there is a projection $f$ in $A_0'\cap A_2$ such that $E(f)=\lambda\cdot 1$ where $E$ is the $\mathop{\mathrm{tr}}$-preserving conditional expectation from $A_2$ to $A_1$.* *Proof.* Consider the Bratteli diagrams $\Gamma_0^1$, $\Gamma_1^2$, and $\Gamma_0^2$ for the inclusion $A_0\subseteq A_1$, $A_1\subseteq A_2$, and $A_0\subseteq A_1\subseteq A_2$, respectively. We assume each of the edges is directed from the vertex of a smaller algebra to that of a larger algebra. By using $\Gamma_0^2$, we can introduce the path algebra representation of $A_0\subseteq A_1\subseteq A_2$. (For the definition of the path algebra, see for instance, [@MR1473221].) For each path $\gamma$ on $\Gamma_0^2$, let $s(\gamma)$ and $r(\gamma)$ be the source and the range of the path $\gamma$, respectively. Since the "generalized inclusion matrices\" $T^t$ and $T$ for $A_0\subseteq A_1$ and $A_1\subseteq A_2$ are transpose to each other, we can pair each edge $\gamma_1$ in $\Gamma_0^1$ with an edge $\gamma_2$ in $\Gamma_1^2$ such that $s(\gamma_1)=r(\gamma_2)$ and $r(\gamma_1)=s(\gamma_2)$. We denote such a $\gamma_2$ by $\gamma_1^*$. Throughout the proof, the edges in $\Gamma_0^1$ and $\Gamma_1^2$ will be denoted with the subscripts $1$ and $2$, respectively, and the length $2$ paths in $\Gamma_0^2$ will be denoted without subscript. Let $t^{(0)}$ and $t^{(1)}$ be the trace vector of $A_0$ and $A_1$ induced by $\mathop{\mathrm{tr}}$. Choose $$f=\sum_{\genfrac{}{}{0pt}{3}{s(\gamma_1)=s(\delta_1),}{n_{s(\gamma_1)}^{(0)}\ne 0}}\sqrt{\frac{t_{r(\gamma_1)}^{(1)}t_{r(\delta_1)}^{(1)}}{t_{s(\gamma_1)}^{(0)}t_{s(\delta_1)}^{(0)}}}e_{\gamma_1\gamma_1^*,\delta_1\delta_1^*}$$ where $e_{\gamma_1\gamma_2,\delta_1\delta_2}$ are the matrix units in the path algebra representation. It is easy to show that $f$ is a projection. For $x\in A_0'\cap A_2$, $x$ can be written as $$x=\sum_{{\genfrac{}{}{0pt}{3}{s(\gamma)=s(\delta),}{r(\gamma)=r(\delta)}}}x_{\gamma\delta}e_{\gamma\delta}.$$ By Proposition 5.4.3 of [@MR1473221], $$E(x)=\sum_{{\genfrac{}{}{0pt}{3}{s(\gamma_1)=s(\delta_1),}{r(\gamma_1)=r(\delta_1)}} }\sum_{ \theta_2:s(\theta_2)=r(\gamma_1)}\frac{t_{r(\theta_2)}^{(2)}}{t_{s(\theta_2)}^{(1)}}x_{\gamma_1\theta_2, \delta_1\theta_2} e_{\gamma_1,\delta_1}.$$ where $e_{\gamma_1,\delta_1}=\sum_{\theta_2':s(\theta_2')=r(\gamma_1)}e_{\gamma_1\theta_2', \delta_1\theta_2'}$. Thus, $$\begin{aligned} E(f)=\sum_{\gamma_1}\frac{t_{s(\gamma_1)}^{(2)}}{t_{r(\gamma_1)}^{(1)}}\frac{t_{r(\gamma_1)}^{(1)}}{t_{s(\gamma_1)}^{(0)}} e_{\gamma_1,\gamma_1}=\lambda\sum_{\gamma_1} e_{\gamma_1\gamma_1}=\lambda\cdot 1. \end{aligned}$$ ◻ **Lemma 24**. *Let $p$ and $q$ be projections in $\mathop{\mathrm{End}}(\tau^{\otimes n})$ with the same normalized categorical tracial state. Then $p$ and $q$ are equivalent.* *Proof.* Recall $\text{End}(\tau^{\otimes n})\cong M_{a_{n}}(\mathbbm{C})\oplus M_{b_{n}}(\mathbbm{C})$, where $a_{n}=\text{dim}(\mathcal{C}(\mathbbm{1}, \tau^{\otimes n}))$ and $b_{n}=\text{dim}(\mathcal{C}(\tau, \tau^{\otimes n}))$. The canonical tracial state assigns $\frac{1}{\phi^{n}}$ to minimal projections in the first factor while assigning $\frac{1}{\phi^{n-1}}$ to minimal projections in the second factor. Thus $\mathop{\mathrm{tr}}(p)=\frac{p_{1}}{\phi^{n}}+\frac{p_{2}}{\phi^{n-1}}$ for integers $0\le p_{1}\le a_{n}$ and $0\le p_{2}\le b_{n}$. If $\mathop{\mathrm{tr}}(p)=\mathop{\mathrm{tr}}(q)$, then we have $$\frac{(p_{1}-q_{1})}{\phi^{n}}+\frac{(p_{2}-q_{2})}{\phi^{n-1}}=0$$ and multiplying by $\phi^{n}$ we see $$(p_1-q_1)+(p_2-q_2)\phi=0.$$ Since $\phi$ is not rational, we must have $p_{1}=q_1$ and $p_{2}=q_{2}$. Thus $p$ is Murray-von Neumann equivalent to $q$, hence there is a unitary conjugating them as desired. ◻ **Lemma 25**. *Let $\alpha$ be an index 1 QCA. If $\mathcal{A}_{a}\subseteq \alpha(\mathcal{A}_x)\subsetneq\mathcal{A}_{b}$ for $a\le x <b$ then there is a unitary $u\in \mathcal{A}_{a}'\cap \mathcal{A}_{b}=A_{(a,b]}$ such that $\mathcal{A}_{a}\subseteq\mathop{\mathrm{Ad}}(u)(\alpha(\mathcal{A}_x))\subseteq\mathcal{A}_{b-1}$.* *Proof.* Let $\mathcal{A}_{b-1}\subseteq \mathcal{A}_{b}\stackrel{e_{b+1}}{\subseteq} \mathcal{A}_{b+1}$ be the basic construction. Note that $e_{b+1}\in \mathcal{A}_{b-1}'\cap \mathcal{A}_{b+1}=A_{[b,b+1]}$. Now, by the QCA index $1$ assumption, $[\mathcal{A}_{b}: \alpha(\mathcal{A}_x)]=\phi^{2(b-x)}$. By Lemma [Lemma 21](#FibIntermediate){reference-type="ref" reference="FibIntermediate"}, the $\alpha(\mathcal{A}_x)$ bimodule $L^{2}(\mathcal{A}_{b})$ is in $(\mathop{\mathrm{\mathbf{Fib}}}_{a})^{*}_{\alpha(\mathcal{A}_x)}\cong \mathop{\mathrm{\mathbf{Fib}}}$, and thus there exists some $X\in (\mathop{\mathrm{\mathbf{Fib}}}_{a})^{*}_{\alpha(\mathcal{A}_x)}$ with $X\otimes \bar{X}=X\otimes X \cong L^{2}(\mathcal{A}_{b})$. Note that the function from isomorphism classes of objects of $\mathop{\mathrm{\mathbf{Fib}}}$ to $\mathbbm{R}$ given by the dimension function is injective. Thus, $$\phi^{2(b-x)}=[\mathcal{A}_{b}: \alpha(\mathcal{A}_x)]=\text{dim}(_{\alpha(\mathcal{A}_{x})} L^{2}(\mathcal{A}_{b}) _{\alpha(\mathcal{A}_{x})}) =\dim(X)^{2}$$ which implies $X\cong \tau^{\otimes b-x}$. In particular, the subfactor $\alpha(\mathcal{A}_x)\subseteq \mathcal{A}_b$ is strongly amenable in the sense of [@MR1278111]. Therefore, it is isomorphic to the subfactor $\mathcal{A}_x \subseteq \mathcal{A}_b$ by Theorem 5.1.1 of [@MR1278111]. Hence, the adjacency matrix for the principal graph for the subfactor $\alpha(\mathcal{A}_x)\subseteq \mathcal{A}_b$ is $$\begin{bmatrix} 0&1\\ 1&1 \end{bmatrix}^{b-x}.$$ Thus, we can choose a subalgebra $\mathcal{B}\subseteq \alpha(\mathcal{A}_x)'\cap \mathcal{A}_b$, whose inclusion matrix in the sense of Lemma [Lemma 23](#jonesproj){reference-type="ref" reference="jonesproj"} is $$\begin{bmatrix} 0&1\\ 1&1 \end{bmatrix}.$$ The chain of algebras $\mathcal{B}\subseteq \alpha(\mathcal{A}_x)'\cap \mathcal{A}_b\subseteq \alpha(\mathcal{A}_x)'\cap \mathcal{A}_{b+1}$ satisfies the condition in Lemma [Lemma 23](#jonesproj){reference-type="ref" reference="jonesproj"}. Thus, there exists a projection $f_{b+1}\in \alpha(\mathcal{A}_x)'\cap \mathcal{A}_{b+1}$ such that $E_{\mathcal{A}_{b}}^{\mathcal{A}_{b+1}}(f_{b+1})=\frac{1}{\phi^2}$ where $E_{\mathcal{A}_{b}}^{\mathcal{A}_{b+1}}$ is the unique trace-preserving conditional expectation from $\mathcal{A}_{b+1}$ to $\mathcal{A}_{b}$. Then by Lemma [Lemma 24](#MvNequiv){reference-type="ref" reference="MvNequiv"}, there is a unitary $v$ in $\mathcal{A}_{a}'\cap \mathcal{A}_{b+1}$ such that $f_{b+1}=v^*e_{b+1}v$. Let $u=\phi^2 E_{\mathcal{A}_{b}}^{\mathcal{A}_{b+1}}(e_{b+1}v)$, i.e. $u$ is a unique element in $\mathcal{A}_{b}$ such that $e_{b+1}v=e_{b+1}u$ by Lemma 1.2 of [@MR860811]. Note that $u\in \mathcal{A}_{a}'\cap \mathcal{A}_{b}$. We claim that $u$ is a unitary. $$\begin{aligned} u^*u&=\phi^2 u^* E_{\mathcal{A}_{b}}^{\mathcal{A}_{b+1}}(e_{b+1})u=\phi^2 E_{\mathcal{A}_{b}}^{\mathcal{A}_{b+1}}(u^*e_{b+1}u)=\phi^2 E_{\mathcal{A}_{b}}^{\mathcal{A}_{b+1}}(v^*e_{b+1}v)\\ &=\phi^2 E_{\mathcal{A}_{b}}^{\mathcal{A}_{b+1}}(f_{b+1})=1. \end{aligned}$$ Thus, $u$ is an isometry in the finite-dimensional von Neumann algebra $\mathcal{A}_{a}'\cap \mathcal{A}_{b}$. Therefore, it is a unitary. Since $f_{b+1}=u^*e_{b+1}u$ by construction, we have $e_{b+1}=\mathop{\mathrm{Ad}}(u)(f_{b+1})$. Then $$\begin{aligned} \mathcal{A}_{a}&=\mathop{\mathrm{Ad}}(u)(\mathcal{A}_{a})\subseteq \mathop{\mathrm{Ad}}(u)(\alpha(\mathcal{A}_{x}))\subseteq \mathop{\mathrm{Ad}}(u)(\mathcal{A}_{b})\cap \{\mathop{\mathrm{Ad}}(u)(f_{b+1})\}'\\ &=\mathcal{A}_{b}\cap \{e_{b+1}\}'=\mathcal{A}_{b-1}. \end{aligned}$$ ◻ **Corollary 26**. *Let $\alpha$ be a QCA with the index $1$ and spread $l$. Then for each $x\in\mathbbm{Z}$ there is a unitary $u\in A_{(x-l,x+l]}$ such that $\mathop{\mathrm{Ad}}(u)(\alpha(\mathcal{A}_x))=\mathcal{A}_x$.* *Proof.* Since the spread of $\alpha$ is $l$, $\mathcal{A}_{x-l}\subseteq \alpha(\mathcal{A}_x)\subseteq\mathcal{A}_{x+l}$ for all $x$. Applying Lemma [Lemma 25](#pulldown){reference-type="ref" reference="pulldown"} $l$ times gives the desired result. ◻ **Corollary 27**. *Let $\alpha$ be a QCA with index $1$ and spread $l$. Then there is a depth-$1$ circuit $U$ such that $\mathop{\mathrm{Ad}}(U)(\alpha(\mathcal{A}_{2kl}))=\mathcal{A}_{2kl}$ for all $k\in\mathbbm{Z}$.* *Proof.* For each $k\in\mathbbm{Z}$, we can choose a unitary $u_k\in A_{((2k-1)l,(2k+1)l]}$ such that $$\mathop{\mathrm{Ad}}(u_k)(\alpha(\mathcal{A}_{2kl}))=\mathcal{A}_{2kl}.$$ Since all $u_k$ are supported on disjoint intervals, $\mathop{\mathrm{Ad}}(U)$ for $U=\prod_{k\in\mathbbm{Z}}u_k$ is a depth-1 circuit. Fix $k\in\mathbbm{Z}$ and let $U_-=\prod_{j<k}u_j$ and $U_+=\prod_{j>k}u_j$. Then $U_-$ and $U_+$ are products of mutually commuting unitaries supported on $(-\infty, (2k-1)l]$ and $((2k+1)l,\infty)$. Since $\mathcal{A}_{(2k-1)l}\subseteq\alpha(\mathcal{A}_{2kl})\subseteq \mathcal{A}_{(2k+1)l}$, $$\begin{aligned} \mathop{\mathrm{Ad}}(U_-)(\alpha(\mathcal{A}_{2kl}))&=\alpha(\mathcal{A}_{2kl}),\\ \mathop{\mathrm{Ad}}(U_+)(\alpha(\mathcal{A}_{2kl}))&=\alpha(\mathcal{A}_{2kl}). \end{aligned}$$ Therefore, $$\begin{aligned} \mathop{\mathrm{Ad}}(U)(\alpha(\mathcal{A}_{2kl}))&=\mathop{\mathrm{Ad}}(u_k)(\mathop{\mathrm{Ad}}(U_+)(\mathop{\mathrm{Ad}}(U_-)(\alpha(\mathcal{A}_{2kl}))))\\ &=\mathop{\mathrm{Ad}}(u_k)(\alpha(\mathcal{A}_{2kl}))\\ &=\mathcal{A}_{2kl}. \end{aligned}$$ ◻ **Corollary 28**. *Let $\alpha$ be a QCA with index $1$ and spread $l$. Then there is a depth-$1$ circuit $U$ such that $\mathop{\mathrm{Ad}}(U)(\alpha(A_{(2jl,2kl]}))=A_{(2jl,2kl]}$ for all integers $j<k$.* *Proof.* Choose a depth-$1$ circuit $U$ as in Corollary [Corollary 27](#fixmany){reference-type="ref" reference="fixmany"}. Then we have $$\begin{aligned} \mathop{\mathrm{Ad}}(U)(\alpha(A_{(2jl,2kl]}))&=\mathop{\mathrm{Ad}}(U)(\alpha(\mathcal{A}_{2jl}'\cap \mathcal{A}_{2kl}))\\ &=\mathop{\mathrm{Ad}}(U)(\alpha(\mathcal{A}_{2jl}))'\cap \mathop{\mathrm{Ad}}(U)(\alpha(\mathcal{A}_{2kl}))\\ &=\mathcal{A}_{2jl}'\cap \mathcal{A}_{2kl}\\ &=A_{(2jl,2kl]}. \end{aligned}$$ ◻ **Theorem 29**. *If $\alpha$ is a QCA with index 1, then it is a finite depth circuit.* *Proof.* Let $l$ be the spread of $\alpha$. By Corollary [Corollary 27](#fixmany){reference-type="ref" reference="fixmany"} we may assume that $\alpha(\mathcal{A}_{2kl})=\mathcal{A}_{2kl}$ for all $k\in\mathbbm{Z}$. Fix $k\in\mathbbm{Z}$. We claim that for each $0\le n\le 2l-1$ there is $u_n\in A_{(2(k-1)l, 2kl]}$ such that $\mathop{\mathrm{Ad}}(u_n)(\alpha(\mathcal{A}_{2kl-m}))=\mathcal{A}_{2kl-m}$ for all $0\le m\le n$. We use the induction on $n$. For $n=0$, we can simply choose $u_0=1$. Suppose we have found $u_{n-1}$. Then $$\begin{aligned} \mathcal{A}_{2(k-1)l}&=\mathop{\mathrm{Ad}}(u_{n-1})(\alpha(\mathcal{A}_{2(k-1)l}))\subseteq\mathop{\mathrm{Ad}}(u_{n-1})(\alpha(\mathcal{A}_{2kl-n}))\\ &\subseteq \mathop{\mathrm{Ad}}(u_{n-1})(\alpha(\mathcal{A}_{2kl-n+1}))=\mathcal{A}_{2kl-n+1}. \end{aligned}$$ By Lemma [Lemma 25](#pulldown){reference-type="ref" reference="pulldown"} there is a unitary $v\in A_{(2(k-1)l,2kl-n+1]}$ such that $$\mathcal{A}_{2(k-1)l}\subseteq \mathop{\mathrm{Ad}}(v)(\mathop{\mathrm{Ad}}(u_{n-1})(\alpha(\mathcal{A}_{2kl-n})))\subseteq \mathcal{A}_{(2kl-n+1)-1}=\mathcal{A}_{2kl-n}.$$ Since $[\mathop{\mathrm{Ad}}(v)(\mathop{\mathrm{Ad}}(u_{n-1})(\alpha(\mathcal{A}_{2kl-n}))):\mathcal{A}_{2(k-1)l}]=[\mathcal{A}_{2kl-n}:\mathcal{A}_{2(k-1)l}]$, we have $$\mathop{\mathrm{Ad}}(v)(\mathop{\mathrm{Ad}}(u_{n-1})(\alpha(\mathcal{A}_{2kl-n})))=\mathcal{A}_{2kl-n}.$$ Since $v\in A_{(2(k-1)l,2kl-n+1]}\subseteq \mathcal{A}_{2kl-n+1}$, for $0\le m\le n-1$ $$\begin{aligned} \mathop{\mathrm{Ad}}(v)(\mathop{\mathrm{Ad}}(u_{n-1})(\alpha(\mathcal{A}_{2kl-m})))=\mathop{\mathrm{Ad}}(v)(\mathcal{A}_{2kl-m})=\mathcal{A}_{2kl-m}. \end{aligned}$$ Set $u_n=vu_{n-1}$. This proves the claim. In particular, there is a unitary $w_k=u_{2l-1}$ in $A_{(2(k-1)l, 2kl]}$ such that $$\mathop{\mathrm{Ad}}(w_k)(\alpha(\mathcal{A}_x))=\mathcal{A}_x$$ for all $2(k-1)l+1\le x\le 2kl$. Take $w_k$ for each $k\in\mathbbm{Z}$ and let $W=\prod_{k\in\mathbbm{Z}}w_k$. Let $x\in\mathbbm{Z}$ and $2(k-1)l+1\le x\le 2kl$ for some $k$. Then by arguing as in Corollary [Corollary 27](#fixmany){reference-type="ref" reference="fixmany"} we obtain $$\mathop{\mathrm{Ad}}(W)(\alpha(\mathcal{A}_x))=\mathop{\mathrm{Ad}}(w_k)(\alpha(\mathcal{A}_x))=\mathcal{A}_x.$$ Thus, $\mathop{\mathrm{Ad}}(W)(\alpha(\mathcal{A}_x))=\mathcal{A}_x$ for all $x\in\mathbbm{Z}$. Arguing as in Corollary [Corollary 28](#fixloc){reference-type="ref" reference="fixloc"}, we have that $$\mathop{\mathrm{Ad}}(W)(\alpha(A_{[a,b]}))=A_{[a,b]}$$ for all integers $a\le b$. The Jones projection $e_x$ for the inclusion $\mathcal{A}_{x-2}\subseteq \mathcal{A}_{x-1}\stackrel{e_x}{\subseteq}\mathcal{A}_x$ is a unique central projection in $A_{[x-1,x]}$ with tracial state value $1/\phi^2$. Thus, $\mathop{\mathrm{Ad}}(W)(\alpha(e_x))=e_x$. Since any local algebra is generated by the Jones projections, $\mathop{\mathrm{Ad}}(W)\circ\alpha$ is the identity. ◻ # Toward general topological invariants of QCA {#Outlook} A *topological invariant* of QCA is a group $G$ and a homomorphism $\pi: \mathop{\mathrm{\mathbf{QCA}}}(A)\rightarrow G$ with $\mathop{\mathrm{\mathbf{FDQC}}}(A)\subseteq \text{ker}(\pi)$. Ideally $G$ should be some sort of "well-understood\" group and the homomorphism $\pi$ should be easily computable. One approach to the classification of topological phases of $\mathop{\mathrm{\mathbf{QCA}}}$ in terms of invariants is to find a *complete* set of topological invariants. A finite family of topological invariants $\pi_{i}: \mathop{\mathrm{\mathbf{QCA}}}\rightarrow G_{i}$ is *complete* if $\bigcap_i \text{ker}(\pi_{i})=\mathop{\mathrm{\mathbf{FDQC}}}(A)$. Then the homomorphism $$\begin{aligned} &\prod_i \pi_{i}:A\rightarrow \prod_i G_{i}&\\ \\ &\alpha\mapsto (\pi_{1}(\alpha), \dots, \pi_{n}(\alpha))&\end{aligned}$$ induces an isomorphism of $\mathop{\mathrm{\mathbf{QCA}}}(A)/\mathop{\mathrm{\mathbf{FDQC}}}(A)$ onto its image. The two examples of topological invariants we know of that apply to $\mathop{\mathrm{\mathbf{QCA}}}$ on fusion spin chains are $\mathop{\mathrm{Ind}}$ as developed here and the homomorphism $\textbf{DHR}: \mathop{\mathrm{\mathbf{QCA}}}(A)\rightarrow \text{Aut}_{br}(\textbf{DHR}(A))$ from [@2304.00068]. This leads to the following question: **Question 30**. Let $A$ be a fusion spin chain. Is the pair $(\mathop{\mathrm{Ind}}, \textbf{DHR})$ a complete set of topological invariants? If not, what are other topological invariants? Our analysis shows the pair is complete for the fusion category $\mathop{\mathrm{\mathbf{Fib}}}$. Indeed, we show $\mathop{\mathrm{Ind}}$ itself is a complete invariant. Moreover, $\text{Aut}_{br}(\mathcal{Z}(\mathop{\mathrm{\mathbf{Fib}}}))$ is trivial. In general, however, it appears that $\mathop{\mathrm{Ind}}$ and $\textbf{DHR}$ are not completely independent, as we can see from examining the generalized Kramers-Wannier translations from Example [Example 19](#Kramers-Wannier){reference-type="ref" reference="Kramers-Wannier"}. [^1]: In some of these settings, other equivalence relations on QCA such as stable equivalence and blending are used. [^2]: The condition on $\alpha^{-1}$ follows automatically from the condition on $\alpha$ if we assume some version of Haag duality, see [@2304.00068].

arxiv_math

--- abstract: | This short note shows the superconvergence of an $H(\mathrm{grad}\,\mathrm{curl})$-nonconforming brick element very recently introduced in [@Wang2023] for the quad-curl problem. The supercloseness is based on proper modifications for both the interpolation and the discrete formulation, leading to an $O(h^2)$ superclose order in the discrete $\mathrm{grad}\,\mathrm{curl}$ norm. Moreover, we propose a suitable postprocessing method to ensure the global superconvergence. Numerical results verify our theory.\ **Keywords:** superconvergence; nonconforming finite element; quad-curl problem author: - "Xinchen Zhou[^1]" - Zhaoliang Meng - Hexin Niu title: "Superconvergence of a nonconforming brick element for the quad-curl problem[^2]" --- # Introduction The quad-curl problem arises in various models such as the inverse electromagnetic scattering theory and magnetohydrodynamics. Recently, some 3D $H(\mathrm{grad}\,\mathrm{curl})$-conforming finite elements were proposed [@Zhang2020; @Hu2022; @Wang2021]. However, they suffer from a complicated element structure, and nonconforming finite elements are therefore considered. In [@Zheng2011], Zheng et al. constructed the first nonconforming tetrahedral element for the model problem. Huang [@Huang2023] also proposed two tetrahedral elements, where one coincides with the element in [@Zheng2011] and the other is a simplified version with fewer edge degrees of freedom (DoFs). Very recently, Wang et al. [@Wang2023] introduced two nonconforming cubical elements with similar structures as in Huang's contribution, but a remarkable fact is that they are robust for singular perturbation problems as well. All the above nonconforming elements are $H(\mathrm{curl})$-nonconforming, and we also notice that some $H(\mathrm{grad}\,\mathrm{curl})$-nonconforming but $H(\mathrm{curl})$-conforming elements were presented, e.g., [@Zhang2022]. To improve the computational precision, superconvergence has been widely studied and applied for elements for various problems. Comprehensive reviews can be found in [@Lin1996; @Chen2002]. In comparison with conforming elements, superconvergence analysis for nonconforming ones is generally more difficult, in the sense that an extra consistency error term must be considered, and moreover, it is also a problem if the canonical interpolation itself is superclose. Up to now, there have been many superconvergence results of nonconforming elements for the second order elliptic problem [@Lin2005; @Hu2016], for the biharmonic problem [@Hu2016; @Hu2016b; @Mao2009], for the Stokes problem [@Liu2008; @Ye2002], etc. Very recently, for the quad-curl problem, some superconvergence results were obtained in [@Wang2023; @Zhang2023], which both dealt with $\mathrm{curl}\,\mathrm{curl}$-conforming elements in 2D. However, such a result for nonconforming elements seems invalid in the literature. In this work, we consider the superconvergence of the simpler element proposed in [@Wang2023], where only one DoF per edge is involved. To overcome the difficulties mentioned above, two modifications are taken into account. The first modification concerns the consistency error. In [@Linke2014], Linke discovered that the consistency error can be effectively controlled by a velocity reconstruction for nonconforming divergence-free Stokes elements, so that the velocity error can be independent of the pressure. This reconstruction only modifies the righthand side of the discrete formulation, therefore the implementation is simple, and the computational complexity is nearly unchanged. For the quad-curl problem and the element considered in this work, we adopt a similar idea using the Nédélec interpolation [@Nedelec1980], which ensures an $O(h^2)$ order of the consistency error. The second modification aims at the interpolation error. We modify the canonical interpolation of the element based on [@Zhou2023] for the superconvergence of the Stokes element designed in [@Zhang2009]. This technique was employed by Chen [@Chen2002] for nonconforming rectangular biharmonic elements, and thereafter utilized in [@Mao2009] and [@Hu2016b] for triangular and 3D elements. As a consequence, the modified interpolation is precisely superconvergent with order two again. Finally, owing to the idea in [@Lin2000] for Maxwell's equations, we design a suitable postprocessing method using a macroelement interpolation to obtain the global superconvergence. Numerical results confirm our theory. The rest of this work is arranged as follows. Notations and preliminaries are given in Section [2](#s2){reference-type="ref" reference="s2"}. Then the superclose and superconvergence results are demonstrated in Sections [3](#s3){reference-type="ref" reference="s3"} and [4](#s4){reference-type="ref" reference="s4"}, respectively. We also discuss the numerical experiments in Section [5](#s5){reference-type="ref" reference="s5"}. # Nonconforming brick element for the quad-curl problem {#s2} Let $\Omega\subset\mathbb{R}^3$ be a polyhedral domain and $(x_1,x_2,x_3)$ be the coordinate of a point $\bm{x}\in\Omega$. Standard notations in Sobolev Spaces are used throughout this work. For a domain $D\subset\Omega$, the norms and seminorms in the Sobolev spaces $H^s(D)$ are indicated by $\|\cdot\|_{s,D}$ and $|\cdot|_{s,D}$, respectively. The space $H_0^1(D)$ is the subspace of $H^1(D)$ with vanishing trace on $\partial D$. In particular, we set $L^2(D)=H^0(D)$ with the inner product $(\cdot,\cdot)_D$. These notations are also valid for vector- and matrix-valued Sobolev spaces, and the subscript $D$ will be omitted if the domain $D=\Omega$. We write $\bm{n}$ as the unit outward normal vector on $\partial D$, and write $\bm{t}$ as a fixed unit tangent vector along $D$ if $D$ is one-dimensional. The notation $P_k(D)$ denotes the usual polynomial space over $D$ of degree no more than $k$. Moreover, $Q_{i,j,k}(D)$ represents the space of polynomials such that the maximum degree in variables $x_1,x_2$ and $x_3$ are $i,j,k$, respectively. For simplicity we set $Q_k(D)=Q_{k,k,k}(D)$. Furthermore, the constant $C>0$ independent of the mesh size may be taken different values in different places. Some notations of vector-valued spaces are also useful. We set $$\begin{aligned} &H^1(\mathrm{curl};D)=\{\bm{v}\in[H^1(D)]^3:~\mathrm{curl}\,\bm{v}\in[H^1(D)]^3\},\\ &H(\mathrm{grad}\,\mathrm{curl};D)=\{\bm{v}\in[L^2(D)]^3:~\mathrm{curl}\,\bm{v}\in[H^1(D)]^3\},\\ &H_0(\mathrm{grad}\,\mathrm{curl};D)=\{\bm{v}\in H(\mathrm{grad}\,\mathrm{curl};D):~\bm{v}\times\bm{n}=\mathrm{curl}\,\bm{v}=\bm{0}\mbox{~on~}\partial D\}. \end{aligned}$$ Assume that $\Omega$ can be divided into a set of cubical cells, denoted by $\mathcal{T}_h$. To simplify our statement, we assume that the partition is uniform, but we point out that this restriction is not intrinsic, and main conclusions of this work are also available for general quasi-uniform meshes. We select the coordinate system such that for any cell $K\in\mathcal{T}_h$, the edges of $K$ are parallel to the coordinate axes. The diagram of $K$ can be expressed as $h=(h_1^2+h_2^2+h_3^2)^{1/2}$, where $h_i$ is the length of the edge of $K$ in the $x_i$-direction for each $i=1,2$ or $3$. We also set $\bm{x}_K$ as the center of $K$. The sets of faces and edges of $K$ and in $\mathcal{T}_h$ are written as $\mathcal{F}_K$, $\mathcal{E}_K$, $\mathcal{F}_h$ and $\mathcal{E}_h$, respectively. In [@Zhang2009], Zhang et al. introduced the following nonconforming brick element for the Brinkman model: For $K\in\mathcal{T}_h$, the shape function space $\bm{W}_K$ is defined by $$\bm{W}_K=[P_1(K)]^3\oplus\left(\mathrm{span}\{x_2^2,x_3^2\}\times\mathrm{span}\{x_3^2,x_1^2\} \times\mathrm{span}\{x_1^2,x_2^2\}\right),$$ and the DoFs are face integrals of $K$. Utilizing this element, Wang et al. [@Wang2023] proposed two $H(\mathrm{grad}\,\mathrm{curl})$-nonconforming elements for the singularly perturbed quad-curl problem. The simpler one has the shape function space $$\bm{V}_K=\nabla\,Q_1(K)\oplus(\bm{x}-\bm{x}_K)\times\bm{W}_K,$$ and the DoFs are $$\label{e: WK dof} \int_E\bm{v}\cdot\bm{t}\,\mathrm{d}s,~\forall E\in\mathcal{E}_K, ~\int_F\mathrm{curl}\,\bm{v}\cdot\bm{t}_{j,F}\,\mathrm{d}F,~j=1,2,~\forall F\in\mathcal{F}_K, ~\forall \bm{v}\in\bm{V}_K,$$ where $\bm{t}_{j,F}$ are two unit vectors parallel to two non-colinear edges of $F$, respectively. The homogeneous global spaces $\bm{W}_h$ and $\bm{V}_h$ with respect to the above two elements can be naturally defined in a piecewise manner by requiring that all the DoFs of cells in $\mathcal{T}_h$ are single-valued, and all the boundary DoFs vanish. Moreover, we shall also use the $Q_1$ element space $$Q_h=\{q_h\in H_0^1(\Omega):~q_h|_K\in Q_1(K),~\forall K\in\mathcal{T}_h\}.$$ According to [@Wang2023], these spaces have the following relations: $$\label{e: space relation} \nabla Q_h\subset\bm{V}_h \mbox{ and } \mathrm{curl}_h\bm{V}_h\subset\bm{W}_h,$$ where $\mathrm{curl}_h$ is the piecewise-defined version of the $\mathrm{curl}$ operator in the sense that $\mathrm{curl}_h|_K=\mathrm{curl}$, $\forall K\in\mathcal{T}_h$. For $\bm{f}\in [L^2(\Omega)]^3$ and $\mathrm{div}\,\bm{f}=0$, the quad-curl problem has the form: Find $\bm{u}$ such that $$\label{e: the quad-curl problem} \begin{aligned} \mathrm{curl}^4\,\bm{u}&=\bm{f}~~~~\mbox{in }\Omega,\\ \mathrm{div}\,\bm{u}&=0\,~~~~\mbox{in }\Omega,\\ \bm{u}\times\bm{n}=\mathrm{curl}\,\bm{u}\times\bm{n}&=\bm{0}~~~~\mbox{on }\partial\Omega. \end{aligned}$$ Due to the identity $\mathrm{curl}^2\,\bm{u}=-\Delta\bm{u}+\nabla\mathrm{div}\,\bm{u}$, the first equation is equivalent to $$\label{e: curl L curl} -\mathrm{curl}\,\Delta\,\mathrm{curl}\,\bm{u}=\bm{f}~~~~\mbox{in }\Omega.$$ A variational formulation is to seek $(\bm{u},p)\in H_0(\mathrm{grad}\,\mathrm{curl};\Omega)\times H_0^1(\Omega)$ such that $$\label{e: variational formulation} \begin{aligned} a(\bm{u},\bm{v})+b(\bm{v},p)&=(\bm{f},\bm{v}),~\forall\bm{v}\in H_0(\mathrm{grad}\,\mathrm{curl};\Omega),\\ b(\bm{u},q)&=0,~\forall q\in H_0^1(\Omega), \end{aligned}$$ where $a(\bm{u},\bm{v})=(\nabla\mathrm{curl}\,\bm{u},\nabla\mathrm{curl}\,\bm{v})$ and $b(\bm{v},p)=(\bm{v},\nabla p)$. The standard discrete formulation [@Wang2023; @Huang2023] can be written as: Find $(\bm{u}_h^0,p_h^0)\in\bm{V}_h\times Q_h$ fulfilling $$\label{e: discrete formulation} \begin{aligned} a_h(\bm{u}_h^0,\bm{v}_h)+b(\bm{v}_h,p_h^0)&=(\bm{f},\bm{v}_h),~\forall\bm{v}\in \bm{V}_h,\\ b(\bm{u}_h^0,q_h)&=0,~\forall q_h\in Q_h, \end{aligned}$$ where $a_h(\bm{u}_h,\bm{v}_h)=(\nabla_h\mathrm{curl}_h\bm{u}_h,\nabla_h\mathrm{curl}_h\bm{v}_h)$ and $\nabla_h$ is the piecewise-defined gradient operator. Since $\mathrm{div}\,\bm{f}=0$ and ([\[e: space relation\]](#e: space relation){reference-type="ref" reference="e: space relation"}) holds, by taking $\bm{v}=\nabla p$ in the first equation in ([\[e: variational formulation\]](#e: variational formulation){reference-type="ref" reference="e: variational formulation"}) and taking $\bm{v}_h=\nabla p_h^0$ in the first equation in ([\[e: discrete formulation\]](#e: discrete formulation){reference-type="ref" reference="e: discrete formulation"}), one must have $p=p_h^0=0$. Furthermore, for piecewise sufficiently smooth functions $\bm{v}$, define the discrete norm or semi-norm $$|\bm{v}|_{1,h}^2=\sum_{K\in\mathcal{T}_h}|\bm{v}|_{1,K}^2,~ \|\bm{v}\|_{\mathrm{grad}\,\mathrm{curl},h}^2=\|\bm{v}\|_0^2+\|\mathrm{curl}_h\bm{v}\|_0^2+|\mathrm{curl}_h\bm{v}|_{1,h}^2.$$ Then by [@Wang2023; @Huang2023], if $\bm{u}\in [H^1(\Omega)]^3$ and $\mathrm{curl}\,\bm{u}\in[H^2(\Omega)]^3$, then the following a priori error estimate holds: $$\label{e: orginal err} \|\bm{u}-\bm{u}_h^0\|_{\mathrm{grad}\,\mathrm{curl},h}\leq Ch(\|\mathrm{curl}\,\bm{u}\|_2+|\bm{u}|_1+\|\bm{f}\|_0).$$ # Supercloseness {#s3} In comparison with conforming elements, superconvergence analysis for nonconforming ones is generally more difficult, in the sense that an extra consistency error term must be carefully dealt with, and moreover, the canonical interpolation is often not superclose. To overcome the first difficulty, inspired by [@Linke2014], we shall modify the righthand side of ([\[e: discrete formulation\]](#e: discrete formulation){reference-type="ref" reference="e: discrete formulation"}). To this end, recall the well-known Nédélec brick element [@Nedelec1980] of the lowest order, whose local shape function space is $$\bm{V}^C_K=Q_{0,1,1}(K)\times Q_{1,0,1}(K)\times Q_{1,1,0}(K),$$ and the local DoFs are precisely the edge DoFs in ([\[e: WK dof\]](#e: WK dof){reference-type="ref" reference="e: WK dof"}). We also write the canonical interpolation operator preserving the above DoFs as $\bm{I}^C_K$. Let $\bm{V}^C_h$ be the corresponding homogeneous global spaces and let $\bm{I}^C_h$ be the global version of $\bm{I}^C_K$ such that $\bm{I}^C_h|_K=\bm{I}^C_K$, then the modified formulation reads: Find $(\bm{u}_h,p_h)\in\bm{V}_h\times Q_h$ such that $$\label{e: modified discrete formulation} \begin{aligned} a_h(\bm{u}_h,\bm{v}_h)+b(\bm{v}_h,p_h)&=(\bm{f},\bm{I}^C_h\bm{v}_h),~\forall\bm{v}\in \bm{V}_h,\\ b(\bm{u}_h,q_h)&=0,~\forall q_h\in Q_h. \end{aligned}$$ The finite element matrices of ([\[e: discrete formulation\]](#e: discrete formulation){reference-type="ref" reference="e: discrete formulation"}) and ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) are precisely the same, and therefore the wellposedness of ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) is a direct consequence of that of ([\[e: discrete formulation\]](#e: discrete formulation){reference-type="ref" reference="e: discrete formulation"}). Moreover, since $\nabla Q_h\subset\bm{V}_h^C$ (see e.g. [@Monk2003]), taking $\bm{v}_h=\nabla p_h$ in the first equation in ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) we get $$\|\nabla p_h\|_0^2=(\bm{f},\bm{I}^C_h\nabla p_h)=(\bm{f},\nabla p_h)=0,$$ and thus again we arrive at $p_h=0$. To overcome the second difficulty, we modify the canonical interpolations with respect to $\bm{W}_h$ and $\bm{V}_h$. For $K\in\mathcal{T}_h$, we design the local interpolation $\bm{\Pi}_K:~[H^3(K)]^3\rightarrow\bm{W}_K$ by setting $$\begin{aligned} &\int_F\bm{\Pi}_K\bm{w}\cdot\bm{t}_{j,F}\,\mathrm{d}F =\int_F\left(\bm{w}\cdot\bm{t}_{j,F}+\frac{h_{k_{j,F}}^2}{12}\frac{\partial^2(\bm{w}\cdot\bm{t}_{j,F})}{\partial x_{k_{j,F}}^2}\right)\,\mathrm{d}F,~j=1,2,~\forall F\in\mathcal{F}_K,\\ &\int_F\bm{\Pi}_K\bm{w}\cdot\bm{n}_F\,\mathrm{d}F=\int_F\bm{w}\cdot\bm{n}_F\,\mathrm{d}F, ~\forall F\in\mathcal{F}_K, \end{aligned}$$ where $\bm{n}_F$ is a fixed unit normal of $F$, and $k_{j,F}$ is taken $1$, $2$ or $3$ such that $\bm{t}_{j,F}$ is parallel to the $x_{k_{j,F}}$-axis. For example, if $\bm{t}_{j,F}=(1,0,0)^T$, then $\bm{t}_{j,F}$ is parallel to the $x_1$-axis, and therefore $k_{j,F}=1$. This operator also works for the superconvergence of $\bm{W}_h$ for the Stokes problem with piecewise constant pressure [@Zhou2023]. Similarly, if $\bm{v}\in [H^1(K)]^3$ and $\mathrm{curl}\,\bm{v}\in [H^3(K)]^3$, then the local interpolation $\bm{I}_K\bm{v}\in \bm{V}_K$ can be determined by $$\begin{aligned} &\int_F\mathrm{curl}\,\bm{I}_K\bm{v}\cdot\bm{t}_{j,F}\,\mathrm{d}F =\int_F\left(\mathrm{curl}\,\bm{v}\cdot\bm{t}_{j,F}+\frac{h_{k_{j,F}}^2}{12}\frac{\partial^2(\mathrm{curl}\,\bm{v}\cdot\bm{t}_{j,F})}{\partial x_{k_{j,F}}^2}\right)\,\mathrm{d}F,~j=1,2,~\forall F\in\mathcal{F}_K,\\ &\int_E\bm{I}_K\bm{v}\cdot\bm{t}\,\mathrm{d}s=\int_E\bm{v}\cdot\bm{t}\,\mathrm{d}s, ~\forall E\in\mathcal{E}_K. \end{aligned}$$ The global interpolations $\bm{\Pi}_h$ and $\bm{I}_h$ are naturally piecewise defined: $\bm{\Pi}_h|_K=\bm{\Pi}_K$ and $\bm{I}_h|_K=\bm{I}_K$, $\forall K\in\mathcal{T}_h$. We must point out that these interpolations are well-defined and continuous with the claimed regularity due to the trace theorem and Lemma 3.1 in [@Wang2023]. In addition, if $\bm{w}\in [H_0^1(\Omega)]^3$ and $\bm{v}\in H_0(\mathrm{grad}\,\mathrm{curl};\Omega)$, then $\bm{\Pi}_h\bm{w}\in\bm{W}_h$ and $\bm{I}_h\bm{v}\in\bm{V}_h$ due to the vanishing trace of $\bm{w}$ or $\mathrm{curl}\,\bm{v}$ over $\partial\Omega$. The supercloseness is based on the following five lemmas. **Lemma 1**. *For all $\bm{w}\in [H^3(K)]^3$ and $K\in\mathcal{T}_h$, one has $$|(\nabla(\bm{w}-\bm{\Pi}_K\bm{w}),\nabla\bm{w}_h)_K|\leq Ch^2|\bm{w}|_{3,K}|\bm{w}_h|_{1,K}, ~\forall \bm{w}_h\in\bm{W}_K.$$* *Proof.* Since we have assumed that the mesh is uniform, let $\widehat{K}=\{\widehat{\bm{x}}=(\bm{x}-\bm{x}_K)/h, \bm{x}\in K\}$ be the reference cell independent of $K$ and $\widehat{\bm{w}}(\widehat{\bm{x}})=\bm{w}(\bm{x})$ for any function $\bm{w}$ defined over $K$. Then it can be checked by a direct computation that $$\begin{aligned} &(\nabla(\widehat{\bm{w}}-\bm{\Pi}_{\widehat{K}}\widehat{\bm{w}}),\nabla\widehat{\bm{w}}_h)_{\widehat{K}}=0, ~\forall\widehat{\bm{w}}\in [P_2(\widehat{K})]^3,~\forall\widehat{\bm{w}}_h\in\bm{W}_{\widehat{K}},\\ &|(\nabla(\widehat{\bm{w}}-\bm{\Pi}_{\widehat{K}}\widehat{\bm{w}}),\nabla\widehat{\bm{w}}_h)_{\widehat{K}}|\leq C\|\widehat{\bm{w}}\|_{3,\widehat{K}}|\widehat{\bm{w}}_h|_{1,\widehat{K}},~\forall\widehat{\bm{w}}\in[H^3(\widehat{K})]^3, ~\forall\widehat{\bm{w}}_h\in\bm{W}_{\widehat{K}}. \end{aligned}$$ According to the definition of $\bm{W}_K$, one has $\bm{W}_{\widehat{K}}=\mathrm{span}\{\widehat{\bm{w}},~\bm{w}\in\bm{W}_K\}$. Moreover, by checking the DoFs of $\widehat{\bm{\Pi}_{K}\bm{w}}$ and $\bm{\Pi}_{\widehat{K}}\widehat{\bm{w}}$ we find $\widehat{\bm{\Pi}_{K}\bm{w}}=\bm{\Pi}_{\widehat{K}}\widehat{\bm{w}}$. Therefore a scaling argument and the Bramble--Hilbert lemma show that $$\begin{aligned} |(\nabla(\bm{w}-\bm{\Pi}_K\bm{w}),\nabla\bm{w}_h)_K| &\leq Ch\inf_{\widehat{\bm{q}}\in [P_2(\widehat{K})]^3}|(\nabla((\widehat{\bm{w}}-\widehat{\bm{q}})-\bm{\Pi}_{\widehat{K}}(\widehat{\bm{w}}-\widehat{\bm{q}})),\nabla \widehat{\bm{w}}_h)_{\widehat{K}}|\\ &\leq Ch\inf_{\widehat{\bm{q}}\in [P_2(\widehat{K})]^3} \|\bm{\widehat{w}}-\widehat{\bm{q}}\|_{3,\widehat{K}}|\widehat{\bm{w}}_h|_{1,\widehat{K}}\\ &\leq Ch|\bm{\widehat{w}}|_{3,\widehat{K}}|\widehat{\bm{w}}_h|_{1,\widehat{K}} = Ch^2|\bm{w}|_{3,K}|\bm{w}_h|_{1,K}, \end{aligned}$$ and the proof is done. ◻ **Lemma 2**. *For all $\bm{v}$ such that $\bm{v}\in [H^2(K)]^3$ and $\mathrm{curl}\,\bm{v}\in[H^3(K)]^3$, it holds $$|(\bm{v}-\bm{I}_K\bm{v},\nabla q_h)_K|\leq Ch^2(|\bm{v}|_{2,K}+h|\mathrm{curl}\,\bm{v}|_{2,K}+h^2|\mathrm{curl}\,\bm{v}|_{3,K})|q_h|_{1,K},~\forall q_h\in Q_1(K), ~\forall K\in\mathcal{T}_h.$$* *Proof.* Denote by $\bm{I}_K^0$ the canonical interpolation operator with respect to $\bm{V}_K$, namely all the DoFs in ([\[e: WK dof\]](#e: WK dof){reference-type="ref" reference="e: WK dof"}) are strictly preserved. Then over the reference cell $\widehat{K}$, by a direct computation and the continuity of $\bm{I}_{\widehat{K}}^0$ (Lemma 3.1 in [@Wang2023]), one obtains for any $\widehat{q}_h\in Q_1(\widehat{K})$, $$\begin{aligned} (\widehat{\bm{v}}-\bm{I}_{\widehat{K}}^0\widehat{\bm{v}},\nabla \widehat{q}_h)_{\widehat{K}}&=0, ~\forall\widehat{\bm{v}}\in [P_1(\widehat{K})]^3,\\ |(\widehat{\bm{v}}-\bm{I}_{\widehat{K}}^0\widehat{\bm{v}},\nabla \widehat{q}_h)_{\widehat{K}}|&\leq C(\|\widehat{\bm{v}}\|_{1,\widehat{K}}+\|\mathrm{curl}\,\widehat{\bm{v}}\|_{1,\widehat{K}})|\widehat{q}_h|_{1,\widehat{K}}, ~\forall\widehat{\bm{v}}\in H^1(\mathrm{curl};\widehat{K}), \end{aligned}$$ which gives in a similar fashion as in the proof of Lemma [Lemma 1](#lemma: WK interp superclose){reference-type="ref" reference="lemma: WK interp superclose"} that $$\label{e: IK0 superclose} |(\bm{v}-\bm{I}_K^0\bm{v},\nabla q_h)_K|\leq Ch^2|\bm{v}|_{2,K}|q_h|_{1,K}, ~\forall\bm{v}\in [H^2(K)]^3,~\forall q_h\in Q_1(K).$$ On the other hand, by the definition of $\bm{I}_K$, a scaling argument and the trace theorem, one has $$\label{e: modified interp err} \begin{aligned} |(\bm{I}_K^0\bm{v}-\bm{I}_K\bm{v},\nabla q_h)_K| &\leq Ch^2\sum_{\widehat{F}\in\mathcal{F}_{\widehat{K}}}\sum_{j=1}^2 \left|\int_{\widehat{F}}\frac{\partial^2(\mathrm{curl}\,\widehat{\bm{v}}\cdot\bm{t}_{j,\widehat{F}})}{\partial \widehat{x}_{k_{j,\widehat{F}}}^2}\,\mathrm{d}\widehat{F}\right||\widehat{q}_h|_{1,\widehat{K}}\\ &\leq Ch^2(|\mathrm{curl}\,\widehat{\bm{v}}|_{2,\widehat{K}}+|\mathrm{curl}\,\widehat{\bm{v}}|_{3,\widehat{K}})|\widehat{q}_h|_{1,\widehat{K}}\\ &\leq Ch^3(|\mathrm{curl}\,\bm{v}|_{2,K}+h|\mathrm{curl}\,\bm{v}|_{3,K})|q_h|_{1,K}. \end{aligned}$$ A summation of ([\[e: IK0 superclose\]](#e: IK0 superclose){reference-type="ref" reference="e: IK0 superclose"}) and ([\[e: modified interp err\]](#e: modified interp err){reference-type="ref" reference="e: modified interp err"}) completes the proof. ◻ **Lemma 3**. *The following commutative diagram holds: if $\bm{v}\in [H^1(K)]^3$ and $\mathrm{curl}\,\bm{v}\in [H^3(K)]^3$, then $$\bm{\Pi}_K\mathrm{curl}\,\bm{v}=\mathrm{curl}\,\bm{I}_K\bm{v},~\forall K\in\mathcal{T}_h.$$* *Proof.* By ([\[e: space relation\]](#e: space relation){reference-type="ref" reference="e: space relation"}) we have $\mathrm{curl}\,\bm{V}_K\subset\bm{W}_K$, therefore it suffices to show that all the DoFs of $\bm{\Pi}_K\mathrm{curl}\,\bm{v}$ and $\mathrm{curl}\,\bm{I}_K\bm{v}$ with respect to $\bm{W}_K$ are the same. This is indeed a direct consequence of the Stokes formula (for the normal part) and the definitions of $\bm{\Pi}_K$ and $\bm{I}_K$ (for the tangent part). ◻ **Lemma 4**. *For any $\bm{\phi}\in [H^2(\Omega)]^3$ and $\bm{w}_h\in\bm{W}_h$, one has $$\sum_{K\in\mathcal{T}_h}\int_{\partial K}\bm{\phi}\cdot\bm{w}_h\,\mathrm{d}F\leq Ch^2|\bm{\phi}|_2|\bm{w}_h|_{1,h}.$$* *Proof.* It is easy to check that functions in $\bm{W}_h$ have the following two properties: (i) any face integral of the jump of each component vanishes; (ii) each component restricted to $\bm{W}_K$ can be written as the sum of three univariate functions with respect to $x_1$, $x_2$ or $x_3$. Therefore by the spirit of Lemma 3.1 in [@Lin2005], the conclusion is established. ◻ **Lemma 5**. *For all $\bm{v}_h\in\bm{V}_K$ and $K\in\mathcal{T}_h$, the following equation holds: $$(\bm{q},\mathrm{curl}(\bm{v}_h-\bm{I}_K^C\bm{v}_h))_K=0,~\forall \bm{q}\in [P_0(K)]^3.$$* *Proof.* Lemma 5.2 in [@Wang2023] asserts $$\int_{\partial K}\bm{q}\cdot(\bm{v}_h-\bm{I}_K^C\bm{v}_h)\times\bm{n}\,\mathrm{d}F=0,~\forall \bm{q}\in [P_0(K)]^3,$$ then the conclusion follows from an integration by parts. ◻ Now we are in a position to provide the superclose result. **Theorem 6**. *Assume that the exact solution $\bm{u}$ of problem ([\[e: variational formulation\]](#e: variational formulation){reference-type="ref" reference="e: variational formulation"}) has the regularity $\bm{u}\in [H^2(\Omega)]^3$ and $\mathrm{curl}\,\bm{u}\in[H^3(\Omega)]^3$. Then the discrete solution $\bm{u}_h$ of problem ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) satisfies the following superclose property: $$\|\bm{I}_h\bm{u}-\bm{u}_h\|_{\mathrm{grad}\,\mathrm{curl},h}\leq Ch^2(|\bm{u}|_2+\|\mathrm{curl}\,\bm{u}\|_3).$$* *Proof.* By the standard theory of mixed elements [@Boffi2013], the wellposedness of ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) (or equivalently ([\[e: discrete formulation\]](#e: discrete formulation){reference-type="ref" reference="e: discrete formulation"})) implies the following discrete stability: $$\|\widetilde{\bm{v}}_h\|_{\mathrm{grad}\,\mathrm{curl},h}+|\widetilde{q}_h|_1 \leq C\sup_{(\bm{v}_h,q_h)\in\bm{V}_h\times Q_h} \frac{a_h(\widetilde{\bm{v}}_h,\bm{v}_h)+b(\bm{v}_h,\widetilde{q}_h)+b(\widetilde{\bm{v}}_h,q_h)} {\|\bm{v}_h\|_{\mathrm{grad}\,\mathrm{curl},h}+|q_h|_1}, ~\forall (\bm{\widetilde{\bm{v}}_h},\widetilde{q}_h)\in\bm{V}_h\times Q_h.$$ See also Lemma 4.4 in [@Huang2023]. Taking $\widetilde{\bm{v}}_h=\bm{I}_h\bm{u}-\bm{u}_h$, $\widetilde{q}_h=0$, and combining ([\[e: variational formulation\]](#e: variational formulation){reference-type="ref" reference="e: variational formulation"}) with ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}), one then has $$\label{e: superclose all} \begin{aligned} &\|\bm{I}_h\bm{u}-\bm{u}_h\|_{\mathrm{grad}\,\mathrm{curl},h}\\ \leq& C\sup_{(\bm{v}_h,q_h)\in\bm{V}_h\times Q_h} \frac{a_h(\bm{I}_h\bm{u}-\bm{u}_h,\bm{v}_h)+b(\bm{I}_h\bm{u}-\bm{u}_h,q_h)} {\|\bm{v}_h\|_{\mathrm{grad}\,\mathrm{curl},h}+|q_h|_1}\\ \leq& C\sup_{(\bm{v}_h,q_h)\in\bm{V}_h\times Q_h} \frac{a_h(\bm{I}_h\bm{u},\bm{v}_h)-(\bm{f},\bm{I}_h^C\bm{v}_h)+b(\bm{I}_h\bm{u}-\bm{u},q_h)+b(\bm{u}-\bm{u}_h,q_h)} {\|\bm{v}_h\|_{\mathrm{grad}\,\mathrm{curl},h}+|q_h|_1}\\ \leq& C\sup_{(\bm{v}_h,q_h)\in\bm{V}_h\times Q_h} \frac{a_h(\bm{I}_h\bm{u}-\bm{u},\bm{v}_h)+a(\bm{u},\bm{v}_h)-(\bm{f},\bm{I}_h^C\bm{v}_h)+b(\bm{I}_h\bm{u}-\bm{u},q_h)} {\|\bm{v}_h\|_{\mathrm{grad}\,\mathrm{curl},h}+|q_h|_1}. \end{aligned}$$ For the first term, it follows from ([\[e: space relation\]](#e: space relation){reference-type="ref" reference="e: space relation"}) that $\mathrm{curl}_h\bm{v}_h\in\bm{W}_h$. Then by Lemmas [Lemma 1](#lemma: WK interp superclose){reference-type="ref" reference="lemma: WK interp superclose"} and [Lemma 3](#lemma: commute){reference-type="ref" reference="lemma: commute"}, $$\label{e: superclose first} \begin{aligned} |a_h(\bm{I}_h\bm{u}-\bm{u},\bm{v}_h)| &=|(\nabla_h(\mathrm{curl}\,\bm{u}-\mathrm{curl}_h\bm{I}_h\bm{u}),\nabla_h\mathrm{curl}\,\bm{v}_h)|\\ &=|(\nabla_h(\mathrm{curl}\,\bm{u}-\bm{\Pi}_h\mathrm{curl}\,\bm{u}),\nabla_h\mathrm{curl}\,\bm{v}_h)|\\ &\leq Ch^2|\mathrm{curl}\,\bm{u}|_3|\mathrm{curl}\,\bm{v}_h|_{1,h}. \end{aligned}$$ For the last term, we adopt Lemma [Lemma 2](#lemma: bh superclose){reference-type="ref" reference="lemma: bh superclose"} to derive $$\label{e: superclose last} |b(\bm{I}_h\bm{u}-\bm{u},q_h)|\leq Ch^2(|\bm{u}|_2+h|\mathrm{curl}\,\bm{u}|_2+h^2|\mathrm{curl}\,\bm{u}|_3)|q_h|_1 \leq Ch^2(|\bm{u}|_2+\|\mathrm{curl}\,\bm{u}\|_3)|q_h|_1.$$ It suffices to check the consistency error term. To this end, integrating by parts using ([\[e: the quad-curl problem\]](#e: the quad-curl problem){reference-type="ref" reference="e: the quad-curl problem"}) and ([\[e: curl L curl\]](#e: curl L curl){reference-type="ref" reference="e: curl L curl"}) one sees $$\label{e: superclose consistency} \begin{aligned} a(\bm{u},\bm{v}_h)-(\bm{f},\bm{I}_h^C\bm{v}_h) &=a(\bm{u},\bm{v}_h)+(\Delta\mathrm{curl}\,\bm{u},\mathrm{curl}_h\bm{v}_h)-(\Delta\mathrm{curl}\,\bm{u},\mathrm{curl}_h\bm{v}_h)-(\bm{f},\bm{I}_h^C\bm{v}_h)\\ &=\sum_{K\in\mathcal{T}_h}\int_{\partial K}\frac{\partial(\mathrm{curl}\,\bm{u})}{\partial\bm{n}}\cdot\mathrm{curl}_h\bm{v}_h\,\mathrm{d}F -(\Delta\mathrm{curl}\,\bm{u},\mathrm{curl}_h(\bm{v}_h-\bm{I}_h^C\bm{v}_h)). \end{aligned}$$ By taking $\bm{\phi}=\partial(\mathrm{curl}\,\bm{u})/\partial\bm{n}$ in Lemma [Lemma 4](#lemma: consistency Wh){reference-type="ref" reference="lemma: consistency Wh"} we obtain $$\label{e: consistency 1} \left|\sum_{K\in\mathcal{T}_h}\int_{\partial K}\frac{\partial(\mathrm{curl}\,\bm{u})}{\partial\bm{n}}\cdot\mathrm{curl}_h\bm{v}_h\,\mathrm{d}F\right|\leq Ch^2|\mathrm{curl}\,\bm{u}|_3|\mathrm{curl}_h\bm{v}_h|_{1,h},$$ Moreover, let $\mathcal{P}_{0,K}$ be the $L^2$-projection operator to $[P_0(K)]^3$. Lemma [Lemma 5](#lemma: consistency Vh){reference-type="ref" reference="lemma: consistency Vh"} and the standard interpolation error estimate for the Nédélec element (Theorem 6.6 in [@Monk2003]) imply that $$\label{e: consistency 2} \begin{aligned} |(\Delta\mathrm{curl}\,\bm{u},\mathrm{curl}_h(\bm{v}_h-\bm{I}_h^C\bm{v}_h))| &=\left|\sum_{K\in\mathcal{T}_h}(\Delta\mathrm{curl}\,\bm{u}-\mathcal{P}_{0,K}\Delta\mathrm{curl}\,\bm{u},\mathrm{curl}_h(\bm{v}_h-\bm{I}_h^C\bm{v}_h))_K\right|\\ &\leq Ch^2|\mathrm{curl}\,\bm{u}|_3|\mathrm{curl}_h\bm{v}_h|_{1,h}. \end{aligned}$$ Substituting ([\[e: superclose first\]](#e: superclose first){reference-type="ref" reference="e: superclose first"})--([\[e: consistency 2\]](#e: consistency 2){reference-type="ref" reference="e: consistency 2"}) into ([\[e: superclose all\]](#e: superclose all){reference-type="ref" reference="e: superclose all"}) results in the desired estimate. ◻ # Postprocessing {#s4} We follow the idea in [@Lin2000] to design a postprocessing method to achieve the global superconvergence. Let $M$ be a brick macroelement consisting of $3\times3\times3$ adjacent cells in $\mathcal{T}_h$ and introduce the sets $\mathcal{F}_M=\bigcup_{K\subset M}\mathcal{F}_K$ and $\mathcal{E}_M=\bigcup_{K\subset M}\mathcal{E}_K$. We construct two elements over $M$: for the first element, we select the shape function space $\bm{V}_M$ as the second order Nédélec brick element space, namely, $$\bm{V}_M=Q_{2,3,3}(M)\times Q_{3,2,3}(M)\times Q_{3,3,2}(M),$$ and the DoFs are $$\int_E\bm{v}\cdot\bm{t}\,\mathrm{d}s,~\forall E\in\mathcal{E}_M,~\forall \bm{v}\in\bm{V}_M.$$ The second element has the shape function space $\bm{W}_M$: $$\bm{W}_M=Q_{3,2,2}(M)\times Q_{2,3,2}(M)\times Q_{2,2,3}(M),$$ which is precisely the same as the second order Raviart--Thomas--Nédélec brick element space [@Nedelec1980], and the associated DoFs are $$\int_F\bm{w}\cdot\bm{n}_F\,\mathrm{d}F,~\forall F\in\mathcal{F}_M,~\forall \bm{w}\in\bm{W}_M.$$ It is easy to see these two elements are well-defined. Moreover, let $\bm{I}_M$ and $\bm{\Pi}_M$ be their canonical interpolation operators, then as in Lemma [Lemma 3](#lemma: commute){reference-type="ref" reference="lemma: commute"}, the following commutative diagram holds: $$\label{e: post commute} \bm{\Pi}_M\mathrm{curl}_h\bm{v}=\mathrm{curl}_h\bm{I}_M\bm{v},~\forall\bm{v}\in H^1(\mathrm{curl}; M)+\bm{V}_h|_M.$$ As a consequence, for a sufficiently smooth function $\bm{v}$, we have the interpolation error estimate $$\label{e: macro interp err} \begin{aligned} &\|\bm{v}-\bm{I}_M\bm{v}\|_{0,M}\leq Ch^2|\bm{v}|_{2,M},\\ &|\mathrm{curl}(\bm{v}-\bm{I}_M\bm{v})|_{j,M}=|\mathrm{curl}\,\bm{v}-\bm{\Pi}_M\mathrm{curl}\,\bm{v}|_{j,M} \leq Ch^{3-j}|\mathrm{curl}\,\bm{v}|_{3,M},~j=0,1. \end{aligned}$$ Let $\bm{I}_{3h}$ be the global version of $\bm{I}_M$, then it follows from the definitions of $\bm{I}_M$, $\bm{I}_h$ and ([\[e: post commute\]](#e: post commute){reference-type="ref" reference="e: post commute"}), ([\[e: macro interp err\]](#e: macro interp err){reference-type="ref" reference="e: macro interp err"}) that for all sufficiently smooth $\bm{v}$ over $\Omega$ and $\bm{v}_h\in\bm{V}_h$, $$\label{e: I3h} \begin{aligned} \bm{I}_{3h}\bm{I}_h\bm{v}&=\bm{I}_{3h}\bm{v},~\|\bm{v}-\bm{I}_{3h}\bm{v}\|_{\mathrm{grad}\,\mathrm{curl},h}\leq Ch^2(|\bm{v}|_2+|\mathrm{curl}\,\bm{v}|_3),\\ \|\bm{I}_{3h}\bm{v}_h\|_{\mathrm{grad}\,\mathrm{curl},h}^2 &=\sum_M\left(\|\bm{I}_M\bm{v}_h\|_{0,M}^2+\|\bm{\Pi}_M\mathrm{curl}_h\bm{v}_h\|_{0,M}^2+|\bm{\Pi}_M\mathrm{curl}_h\bm{v}_h|_{1,M}^2\right)\\ &\leq C\sum_M\sum_{K\subset M} \left(\|\bm{v}_h\|_{0,K}^2+\|\mathrm{curl}_h\bm{v}_h\|_{0,K}^2+|\mathrm{curl}_h\bm{v}_h|_{1,K}^2\right) =C\|\bm{v}_h\|_{\mathrm{grad}\,\mathrm{curl},h}^2. \end{aligned}$$ Then we can state our superconvergence result. **Theorem 7**. *Assume that the exact solution $\bm{u}$ of problem ([\[e: variational formulation\]](#e: variational formulation){reference-type="ref" reference="e: variational formulation"}) has the regularity $\bm{u}\in [H^2(\Omega)]^3$ and $\mathrm{curl}\,\bm{u}\in[H^3(\Omega)]^3$. Then for the discrete solution $\bm{u}_h$ of problem ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}), the postprocessed solution $\bm{I}_{3h}\bm{u}_h$ achieves the following superconvergence: $$\|\bm{u}-\bm{I}_{3h}\bm{u}_h\|_{\mathrm{grad}\,\mathrm{curl},h}\leq Ch^2(|\bm{u}|_2+\|\mathrm{curl}\,\bm{u}\|_3).$$* *Proof.* According to Theorem [Theorem 6](#th: supercloseness){reference-type="ref" reference="th: supercloseness"} and ([\[e: I3h\]](#e: I3h){reference-type="ref" reference="e: I3h"}), $$\begin{aligned} \|\bm{u}-\bm{I}_{3h}\bm{u}_h\|_{\mathrm{grad}\,\mathrm{curl},h} &\leq\|\bm{u}-\bm{I}_{3h}\bm{I}_h\bm{u}\|_{\mathrm{grad}\,\mathrm{curl},h} +\|\bm{I}_{3h}(\bm{I}_h\bm{u}-\bm{u}_h)\|_{\mathrm{grad}\,\mathrm{curl},h}\\ &\leq\|\bm{u}-\bm{I}_{3h}\bm{u}\|_{\mathrm{grad}\,\mathrm{curl},h}+C\|\bm{I}_h\bm{u}-\bm{u}_h\|_{\mathrm{grad}\,\mathrm{curl},h}\\ &\leq Ch^2(|\bm{u}|_2+\|\mathrm{curl}\,\bm{u}\|_3), \end{aligned}$$ which completes the proof. ◻ # Numerical results {#s5} In this section, we report and analyze the results of a numerical example. Let the solution domain be $\Omega=[0,1]^3$. We employ the same quad-curl problem as in [@Huang2023], where the exact solution of $(\ref{e: the quad-curl problem})$ is given by $$\bm{u}=\mathrm{curl}\left(0,0,\sin^3(\pi x)\sin^3(\pi y)\sin^3(\pi z)\right)^T,$$ and the righthand-side term $\bm{f}$ can be accordingly calculated. The meshes are generated by $n\times n\times n$ uniform partitions. We first investigate the performance of the original scheme ([\[e: discrete formulation\]](#e: discrete formulation){reference-type="ref" reference="e: discrete formulation"}) and the modified counterpart ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}). These results are given in Tables [1](#t: original){reference-type="ref" reference="t: original"} and [2](#t: modified){reference-type="ref" reference="t: modified"}. The supercloseness and superconvergence of the scheme ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) are checked with the results listed in Tables [3](#t: supercloseness){reference-type="ref" reference="t: supercloseness"} and [4](#t: superconvergence){reference-type="ref" reference="t: superconvergence"}. $n$ $|\mathrm{curl}_h(\bm{u}-\bm{u}_h^0)|_{1,h}$ order $\|\mathrm{curl}_h(\bm{u}-\bm{u}_h^0)\|_0$ order $\|\bm{u}-\bm{u}_h^0\|_0$ order ------ ---------------------------------------------- ------- -------------------------------------------- ------- --------------------------- ------- $6$ 4.351E1 1.548E0 2.244E-1 $12$ 2.166E1 1.01 4.096E-1 1.92 1.076E-1 1.06 $18$ 1.441E1 1.01 1.841E-1 1.97 7.105E-2 1.03 $24$ 1.080E1 1.00 1.039E-1 1.99 5.307E-2 1.01 $36$ 7.198E0 1.00 4.635E-2 1.99 3.528E-2 1.01 $48$ 5.397E0 1.00 2.610E-2 2.00 2.643E-2 1.00 : Numerical results of the original scheme ([\[e: discrete formulation\]](#e: discrete formulation){reference-type="ref" reference="e: discrete formulation"}) for different $n$.[\[t: original\]]{#t: original label="t: original"} $n$ $|\mathrm{curl}_h(\bm{u}-\bm{u}_h)|_{1,h}$ order $\|\mathrm{curl}_h(\bm{u}-\bm{u}_h)\|_0$ order $\|\bm{u}-\bm{u}_h\|_0$ order ------ -------------------------------------------- ------- ------------------------------------------ ------- ------------------------- ------- $6$ 4.332E1 1.836E0 2.344E-1 $12$ 2.159E1 1.00 4.734E-1 1.96 1.081E-1 1.12 $18$ 1.439E1 1.00 2.118E-1 1.98 7.112E-2 1.03 $24$ 1.079E1 1.00 1.194E-1 1.99 5.309E-2 1.02 $36$ 7.194E0 1.00 5.317E-2 2.00 3.528E-2 1.01 $48$ 5.395E0 1.00 2.992E-2 2.00 2.643E-2 1.00 : Numerical results of the modified scheme ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) for different $n$.[\[t: modified\]]{#t: modified label="t: modified"} One can observe from Tables [1](#t: original){reference-type="ref" reference="t: original"} and [2](#t: modified){reference-type="ref" reference="t: modified"} that the errors for both schemes measured in $|\mathrm{curl}_h\cdot|_{1,h}$, $\|\mathrm{curl}_h\cdot\|_0$ and $\|\cdot\|_0$ are all at least first order convergent, which is consistent with ([\[e: orginal err\]](#e: orginal err){reference-type="ref" reference="e: orginal err"}). Moreover, as first discovered in [@Huang2023], the convergence order in $\|\mathrm{curl}_h\cdot\|_0$ is indeed two. We also remark that although the absolute errors of these two schemes are almost equal, but for the original one we do not know if the supercloseness can be guaranteed in theory. $n$ $|\mathrm{curl}_h(\bm{I}_h\bm{u}-\bm{u}_h)|_{1,h}$ order $\|\mathrm{curl}_h(\bm{I}_h\bm{u}-\bm{u}_h)\|_0$ order $\|\bm{I}_h\bm{u}-\bm{u}_h\|_0$ order ------ ---------------------------------------------------- ------- -------------------------------------------------- ------- --------------------------------- ------- $6$ 1.092E1 8.394E-1 8.565E-2 $12$ 2.577E0 2.08 2.092E-1 2.00 2.242E-2 1.93 $18$ 1.125E0 2.04 9.300E-2 2.00 1.000E-2 1.99 $24$ 6.284E-1 2,02 5.231E-2 2.00 5.636E-3 2.00 $36$ 2.778E-1 2.01 2.325E-2 2.00 2.507E-3 2.00 $48$ 1.559E-1 2.01 1.308E-2 2.00 1.410E-3 2.00 : Supercloseness of the modified scheme ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) using the interpolation operator $\bm{I}_h$.[\[t: supercloseness\]]{#t: supercloseness label="t: supercloseness"} $n$ $|\mathrm{curl}_h(\bm{u}-\bm{I}_{3h}\bm{u}_h)|_{1,h}$ order $\|\mathrm{curl}_h(\bm{u}-\bm{I}_{3h}\bm{u}_h)\|_0$ order $\|\bm{u}-\bm{I}_{3h}\bm{u}_h\|_0$ order ------ ------------------------------------------------------- ------- ----------------------------------------------------- ------- ------------------------------------ ------- $6$ 6.294E1 1.040E0 1.582E-1 $12$ 1.833E1 1.78 1.998E-1 2.38 2.329E-2 2.76 $18$ 8.420E0 1.92 8.965E-2 1.98 9.825E-3 2.13 $24$ 4.790E0 1.96 5.107E-2 1.96 5.534E-3 2.00 $36$ 2.146E0 1.98 2.298E-2 1.97 2.480E-3 1.98 $48$ 1.210E0 1.99 1.299E-2 1.98 1.401E-3 1.98 : Superconvergence of the modified scheme ([\[e: modified discrete formulation\]](#e: modified discrete formulation){reference-type="ref" reference="e: modified discrete formulation"}) using the postprocessing operator $\bm{I}_{3h}$.[\[t: superconvergence\]]{#t: superconvergence label="t: superconvergence"} Let us turn to the superclose and superconvergence results. It can be easily verified from Tables [3](#t: supercloseness){reference-type="ref" reference="t: supercloseness"} and [4](#t: superconvergence){reference-type="ref" reference="t: superconvergence"} that $\bm{I}_h\bm{u}$ is a superclose interpolation of order two, as claimed in Theorem [Theorem 6](#th: supercloseness){reference-type="ref" reference="th: supercloseness"}, and $\bm{I}_{3h}$ is a proper postprocessing operator, as asserted in Theorem [Theorem 7](#th: superconvergence){reference-type="ref" reference="th: superconvergence"}. Let us proceed to check the efficiency of $\bm{I}_{3h}$ in a more detailed manner. Indeed, although the errors of the postprocessed and unprocessed solutions measured in $|\mathrm{curl}_h\cdot|_{1,h}$ are almost equal over a coarser mesh ($n\approx10$), when the mesh is refined sufficiently, the postprocessed solutions have a significant higher precision than that in Table [2](#t: modified){reference-type="ref" reference="t: modified"}. For instance, when $n=48$, the error is almost equal to that when $n\approx200$ if no postprocessing is carried out. Furthermore, the performance seems more attractive if the errors are measured in $L^2$-norm (e.g. when $n=48$, 2.643E-2 vs. 1.401E-3). This indicates high efficiency of our method. 99 D. Boffi, F. Brezzi, M. Fortin. Mixed Finite Element Methods and Applications. Springer, 2013. C.M. Chen. Structure theory of superconvergence of finite elements (in Chinese). Hunan Science Press, 2002. J. Hu, R. Ma. Superconvergence of both the Crouzeix--Raviart and Morley elements. Numer. Math., 2016, 132(3): 491--509. J. Hu, Z. Shi, X. Yang. Superconvergence of three dimensional Morley elements on cuboid meshes for biharmonic equations. Adv. Comput. Math., 2016, 42(6): 1453--1471. K. Hu, Q. Zhang, Z. Zhang. A family of finite element stokes complexes in three dimensions. SIAM J. Numer. Anal., 60(1): 222--243, 2022. X. Huang. Nonconforming finite element Stokes complexes in three dimensions. Sci. China Math., 2023: 1--24. Q. Lin, N. Yan. The Construction and Analysis of High Efficiency Finite Element Methods (in Chinese). Hebei University Publishers, Baoding, 1996. Q. Lin, N. Yan. Global superconvergence for Maxwell's equations. Math. Comput., 2000, 69(229): 159--176. Q. Lin, L. Tobiska, A. Zhou. Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA. J. Numer. Anal., 2005, 25: 160--181. H. Liu, N. Yan. Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations. Adv. Comput. Math., 2008, 29(4): 375--392. A. Linke. On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Engrg., 2014, 268: 782--800. S. Mao, Z. Shi. High accuracy analysis of two nonconforming plate elements. Numer. Math., 2009, 111: 407--443. P. Monk. Finite Element Methods for Maxwell's Equations. Oxford University Press, 2003. J.C. Nédélec. Mixed finite elements in $\mathbb{R}^3$. Numer. Math., 1980, 35: 315--341. L. Wang, H. Li, Z. Zhang. $H(\mathrm{curl}^2)$-conforming spectral element method for quad-curl problems. Comput. Methods Appl. Math., 21(3): 661--681, 2021. L. Wang, M. Zhang, Q. Zhang. Fully $H(\mathrm{grad}\,\mathrm{curl})$-nonconforming finite element method for the singularly perturbed quad-curl problem on cubical meshes. arXiv preprint, arXiv: 2301.03172, 2023. L. Wang, Q. Zhang, Z. Zhang. Superconvergence Analysis of $\mathrm{curl}\mathrm{curl}$-Conforming Elements on Rectangular Meshes. J Sci. Comput., 2023, 95(2): 62. X. Ye. Superconvergence of nonconforming finite element method for the Stokes equations. Numer. Methods Part. Differ. Equ. 2002, 18: 143--154. B. Zhang, Z. Zhang. Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem. Discrete Cont. Dyn. B, 2023, 28(2): 1323--1341. S. Zhang, X. Xie, Y. Chen. Low order nonconforming rectangular finite element methods for Darcy- Stokes problem. J. Comput. Math., 2009, 27: 400--424. Q. Zhang, Z. Zhang. A family of curl-curl conforming finite elements on tetrahedral meshes. CSIAM Trans. Appl. Math., 1(4): 639--663, 2020. Q. Zhang, M. Zhang, Z. Zhang. Nonconforming finite elements for the Brinkman and $-\mathrm{curl}\Delta\mathrm{curl}$ problems on cubical meshes. arXiv preprint, arXiv: 2206.08493v2, 2022. B. Zheng, Q. Hu, J. Xu. A nonconforming finite element method for fourth order curl equations in $\mathbb{R}^3$. Math. Comput., 80(276): 1871--1886, 2011. X. Zhou, H. Niu, Z. Meng, J. Su. Superconvergence of some nonconforming brick elements for the 3D Stokes problem. under review. [^1]: Corresponding author: dasazxc\@gmail.com [^2]: This project is supported by NNSFC (No. 12201254) and NSF of Jiangsu Province (No. BK20200902).

arxiv_math

--- abstract: | This work addresses a novel version of the kernel-free boundary integral (KFBI) method for solving elliptic PDEs with implicitly defined irregular boundaries and interfaces. We focus on boundary value problems and interface problems, which are reformulated into boundary integral equations and solved with the matrix-free GMRES method. In the KFBI method, evaluating boundary and volume integrals only requires solving equivalent but much simpler interface problems in a bounding box, for which fast solvers such as FFTs and geometric multigrid methods are applicable. For the simple interface problem, a correction function is introduced for both the evaluation of right-hand side correction terms and the interpolation of a non-smooth potential function. A mesh-free collocation method is proposed to compute the correction function near the interface. The new method avoids complicated derivation for derivative jumps of the solution and is easy to implement, especially for the fourth-order method in three space dimensions. Various numerical examples are presented, including challenging cases such as high-contrast coefficients, arbitrarily close interfaces and heterogeneous interface problems. The reported numerical results verify that the proposed method is both accurate and efficient. address: School of Mathematical Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, Shanghai 200240, P. R. China author: - Han Zhou - Wenjun Ying bibliography: - mybibfile.bib title: A correction function-based kernel-free boundary integral method for elliptic PDEs with implicitly defined interfaces --- Elliptic PDEs; Interface problems; Jump conditions; Cartesian grid-based method; Compact finite difference method # Introduction Boundary value problems and interface problems of elliptic partial differential equations (PDEs) draw much attention due to their wide scientific and industrial applications, such as viscous incompressible flow [@quartapelle1993numerical; @chorin1968numerical; @CALHOUN2002231; @RUSSELL2003177], heat transfer [@bergman2011introduction; @patankar2018numerical], biomolecular electrostatics [@Davis1990; @Honig1995], electromagnetics [@rothwell2018electromagnetics; @bondeson2012computational], and many others. In practical situations, domain boundaries and material interfaces are complex and even move with time, making it challenging to design accurate and efficient numerical methods for these problems. Body-fitted discretization approaches, such as finite element methods [@Babuska1970; @Guyomarch2009; @Bramble1996; @Wang2004], approximate the computational domain with an unstructured mesh, which conforms with the geometry of boundaries and interfaces to achieve high-order accuracy. However, it is always difficult and time-consuming to generate high-quality body-fitted meshes for complex geometries, especially when the boundary or interface has a substantial motion with time evolution. In addition, the linear systems generated from the discretization of the PDE on body-fitted meshes are less structured than that from a Cartesian grid, and fast solvers such as FFTs and geometric multigrid methods can not be applied. Immersed methods have been prevalent in recent decades, in which the complex boundary or interface is immersed into a fixed grid. The pioneering work of immersed methods is the immersed boundary method (IBM) [@PESKIN1972252; @Peskin2002; @Peskin1977] that was initially proposed by C. S. Peskin for simulations of cardiac mechanics and blood flows. In IBM, Peskin uses Lagrangian marker points on the boundary and regularized Dirac delta functions to approximate the singular force and spread it into the Eulerian grid. The IBM is quite robust but is restricted to first-order accuracy due to the non-smoothness of the solution in the vicinity of the boundary. Motivated by IBM, a number of immersed-type approaches have also been developed to improve the performance of conventional IBM. Among them are the immersed interface method (IIM) [@Tan2008; @Deng2003; @Li1998; @Leveque1994; @Leveque1997; @Li2001], the ghost-fluid method (GFM) [@Fedkiw1999a; @Fedkiw1999; @Nguyen2001; @Liu2000; @Luo2006], the matched interface and boundary (MIB) method [@ZHOU20061; @Yu2007; @Wang2015; @Feng2019], the correction function method (CFM) [@Marques2011; @Marques2019; @Marques2017], the Immersed Boundary Smooth Extension (IBSE) method [@Stein2017; @Stein2016]. The methods mentioned above are mainly based on finite difference discretizations. Since the finite element method may provide more rigorous convergence analysis, similar ideas have also been used to develop finite element-based immersed methods, such as the extended finite element method (XFEM) [@Moes1999] and the immersed finite element method (IFEM) [@Li2003; @Gong2008; @Hou2005; @Guo2020]. The kernel-free boundary integral (KFBI) is a potential theory-based Cartesian grid method, which was initially proposed by W. Ying and C. S. Henriquez [@Ying2007] as an extension of Mayo's method [@Mayo1984; @Mayo1985; @MCKENNEY1995348]. The KFBI method is also an immersed approach. Unlike traditional boundary integral methods (BIMs)/ boundary element methods (BEMs) [@Beale2004599; @Beale2001; @Greengard1998; @Carrier1988; @Carrier1988b; @GREENGARD1987325; @Liu1981; @Ying2006; @Klaseboer2004; @TLUPOVA2009158], layer and volume potentials are computed by solving equivalent but much simpler interface problems on a Cartesian grid, and the linear system can be efficiently solved with FFTs or geometric multigrid methods. Therefore, the KFBI method has several attractive advantages: (a) no analytical expression of Green's function is needed for solving the boundary integral equation; (b) singular and nearly singular integrals are avoided ;(c) it can be applied to variable coefficient problems. In the KFBI method, solving the constant coefficient interface problem is a fundamental building block. In previous works [@Xie2020; @Ying2014; @Ying2013; @Ying2007], the simple interface problem is discretized with standard finite difference methods with a modified right-hand side. The correction terms for the right-hand side are linear combinations of derivative jumps $[u], [u_x], [u_y]. [u_z], [u_{xx}],\cdots$, which are computed by repeatedly taking tangential derivatives of the jump values and applying the local coordinate transformation. The coordinate-transformation method for derivative jumps is accurate yet complicated when too many derivative terms are needed, such as for high-order schemes and in three space dimensions, for example, in [@Xie2020]. In this work, we present a novel KFBI method that is both simple and accurate for two and three space dimensional BVPs and interface problems. Motivated by the correction function method (CFM) [@Marques2011; @Marques2019; @Marques2017], we introduce a correction function in the vicinity of the interface to derive correction terms of the right-hand side for the constant coefficient interface problem. In order to solve the local Cauchy problem for the correction function, we proposed a mesh-free collocation method based on an overlapping surface decomposition for the interface. Unlike the original CFM [@Marques2011; @Marques2019; @Marques2017], no surface quadrature is required since the collocation method works with the strong form of the Cauchy problem. The overlapping surface decomposition representation of the interface also provides a good choice of collocation points such that the resulting collocation problem is accurate and stable. Another property of the collocation method is that the discrete system of the collocation problem is a square one and can be solved accurately, which is different from the original CFM in that the linear system is overdetermined and needs to be solved in the least-square sense. The new approach for the constant coefficient interface problem is built into the KFBI framework to accommodate elliptic BVPs and more general interface problems. The resulting method is named the correction function-based KFBI method. The paper is organized as follows. The governing equations and their boundary integral equations are described in section [2](#sec:problem){reference-type="ref" reference="sec:problem"} and [3](#sec:bie){reference-type="ref" reference="sec:bie"}. In section [4](#sec:kfbi){reference-type="ref" reference="sec:kfbi"}, the main idea of the KFBI method is described. The details of the numerical method for the constant coefficient interface problem are described in section [5](#sec:crc){reference-type="ref" reference="sec:crc"}. The algorithm is summarized in section [6](#sec:algo){reference-type="ref" reference="sec:algo"}. In section [7](#sec:res){reference-type="ref" reference="sec:res"}, numerical results demonstrating the method with examples are presented. Finally, we discuss the improvement and advantages of the proposed method in section [8](#sec:dis){reference-type="ref" reference="sec:dis"}. # Governing equations {#sec:problem} ## Boundary value problem Let $\Omega \subset \mathbb{R}^d, d = 2,3$ be a complex domain with smooth boundary $\Gamma = \partial\Omega$, as illustrated in . The BVP of an elliptic PDE is given by [\[eqn:BVP\]]{#eqn:BVP label="eqn:BVP"} (u)- u = f, , subject to either the Dirichlet boundary condition or the Neumann boundary condition [\[eqn:BVP-bc\]]{#eqn:BVP-bc label="eqn:BVP-bc"} u = g_D,\_u = g_N, , where $\sigma >0$ is diffusitivity and $\kappa\geq 0$ is reaction coefficient. In this paper, we assume that $\sigma$ and $\kappa$ are constants. ## Interface problem Let $\Gamma\subset \mathbb{R}^d, d = 2,3$ be a sharp interface that separates a larger domain $\mathcal{B}\subset \mathbb{R}^d$ into two subdomains $\Omega_1$ and $\Omega_2$, as illustrated in . The interface problem of an elliptic PDE is given by [\[eqn:IFP\]]{#eqn:IFP label="eqn:IFP"} (\_i u) - \_i u = f_i, \_i,i = 1, 2, subject to two interface jump conditions [\[eqn:IFP-bc\]]{#eqn:IFP-bc label="eqn:IFP-bc"} \[u\] = g_1,= g_2, , and a homogeneous Dirichlet boundary condition on the outer boundary [\[eqn:diri-bc\]]{#eqn:diri-bc label="eqn:diri-bc"} u = 0,, where $\sigma_i >0,i=1,2$ are diffusitivities and $\kappa_i\geq 0,i=1,2$ are reaction coefficients. Similarly, we only consider the case that $\sigma_i$ and $\kappa_i$ are constants. Note that the Dirichlet boundary condition [\[eqn:diri-bc\]](#eqn:diri-bc){reference-type="eqref" reference="eqn:diri-bc"} is chosen only for simplicity, since the treatment for boundary conditions on $\partial\mathcal{B}$ only depends on the finite difference scheme and is much simpler. Different boundary conditions, such as Neumann and periodic ones, can also be used. Here, the boundary/interface $\Gamma$ is assumed to be implicitly defined as the zero level set of a function. In the case that $\Gamma$ is defined by a parametric surface or spline, it can also be transformed into an implicit form. # Boundary integral equations {#sec:bie} Both the boundary value problem [\[eqn:BVP\]](#eqn:BVP){reference-type="eqref" reference="eqn:BVP"} and [\[eqn:BVP-bc\]](#eqn:BVP-bc){reference-type="eqref" reference="eqn:BVP-bc"} and the interface problem [\[eqn:IFP\]](#eqn:IFP){reference-type="eqref" reference="eqn:IFP"} and [\[eqn:IFP-bc\]](#eqn:IFP-bc){reference-type="eqref" reference="eqn:IFP-bc"} are solved by reformulating them as boundary integral equations. ## Boundary value problem Let $G(\boldsymbol{q}, \boldsymbol{p})$ be Green's function such that for each fixed $\boldsymbol{p}\in\mathcal{B}$, \_(() \_ G(,)) - ()G(;) &= ( - ), & ,\ G(,)& = 0, &. Let $\varphi, \psi$ be two density functions. Define the single layer, double layer, adjoint double layer and hyper-singular integrals, respectively, by () &= \_G(;)()   d\_,& ,\ () &= \_() ()   d\_,&,\ \^() &= \_ ()()   d\_,&,\ () &= \_()() ()   d\_,& . Define the volume integrals by f() &= \_ G(;)f()  d,[\[eqn:vol-int\]]{#eqn:vol-int label="eqn:vol-int"},\ \_ f() &= \_ G(;)f()  d,[\[eqn:vol-int-norm\]]{#eqn:vol-int-norm label="eqn:vol-int-norm"}. The Dirichlet and Neumann BVP [\[eqn:BVP\]](#eqn:BVP){reference-type="eqref" reference="eqn:BVP"} and [\[eqn:BVP-bc\]](#eqn:BVP-bc){reference-type="eqref" reference="eqn:BVP-bc"} can be, respectively, reformulated as boundary integral equations ( + )&= g_D -f,[\[eqn:bie-D\]]{#eqn:bie-D label="eqn:bie-D"}\ ( -\^) &= g_N- \_ f,[\[eqn:bie-N\]]{#eqn:bie-N label="eqn:bie-N"} which are both Fredholm integral equations of the second kind and well-conditioned. ## Interface problem Let $G_i(\boldsymbol{q}, \boldsymbol{p}), i=1,2$ be Green's functions such that for each fixed $\boldsymbol{p}\in\mathcal{B}$, \_(\_i() \_ G_i(,)) - \_i()G_i(;) &= ( - ), & ,\ G_i(,)& = 0, &. Similarly, we can define the single layer, double layer, adjoint double layer, hyper-singular and volume integral operators $\mathcal{S}_i, \mathcal{K}_i, \mathcal{K}^{\prime}_i, \mathcal{D}_i, \mathcal{G}_i, {\color{black}\partial_{\mathbf{n}}\mathcal{G}_i}, i = 1,2$. By introducing two unknown densities functions $\varphi = u_1$ and $\psi = \sigma_2\partial_{\boldsymbol{n}}u_2$, the interface problem [\[eqn:IFP\]](#eqn:IFP){reference-type="eqref" reference="eqn:IFP"} can be reformulated as a system of boundary integral equations [\[eqn:bie-ifp-1\]]{#eqn:bie-ifp-1 label="eqn:bie-ifp-1"} \- (\_1 - \_2)+ (\_1 - \_2)&= g_1 + \_1 f_1 + \_2 f_2 + \_2g_1 - \_1 g_2,\ - (\_1 - \_2)+ (\^\_1 - \^\_2)&= - g_2 + \_1\_\_1 f_1 + \_2\_\_2 f_2 + \_2 g_1 - \^\_1 g_2. In the case of $\kappa_i = 0, i=1,2$ or $\sigma_1/\sigma_2 = \kappa_1/\kappa_2$, by dividing the two equations in [\[eqn:IFP\]](#eqn:IFP){reference-type="eqref" reference="eqn:IFP"} by $\sigma_i$, respectively, it yields [\[eqn:bie-ifp-2\]]{#eqn:bie-ifp-2 label="eqn:bie-ifp-2"} u - u = { f_1 / \_1, \_1,\ f_2 / \_2, \_2, . where $\Tilde{\kappa}=0$ or $\Tilde{\kappa}=\kappa_1/\sigma_1=\kappa_2/\sigma_2$. Let $\psi = [\partial_{\boldsymbol{n}}u]$, we may also get a simpler boundary integral equation + \^= + (g_1 + \_f), where $\mu = (\sigma_2 - \sigma_1)/(\sigma_2 + \sigma_1)\in(-1,1)$ and $\mathcal{K}, \mathcal{D}$ and $\mathcal{G}$ are the integral operators associated with the Green's function of the operator $\Delta-\Tilde{\kappa}$. One may refer to the reference [@Ying2014] for detailed derivations of the boundary integral equations. # Kernel-free boundary integral method {#sec:kfbi} In the kernel-free boundary integral method, values of the boundary integrals and volume integral at boundary or interface $\Gamma$ are not evaluated with quadrature methods. Instead, they are evaluated by solving equivalent but much simpler interface problems for boundary and volume potentials. For Green's function $G(\boldsymbol{q},\boldsymbol{p})$, which is associated with the elliptic operator $\sigma\Delta - \kappa$, define the single layer potential $-S\psi$, the double layer potential $D\varphi$ and the Newtonian potential $Nf$ by -S() &=- \_G(,)()   d\_,&,\ D() &= \_() ()   d\_,&,\ Nf() &= \_ G(;)f()  d,& . Then the boundary integrals $\mathcal{S}\psi$, $\mathcal{K}\varphi$, $\mathcal{K}^{\prime}\psi$, $\mathcal{D}\varphi$ and the volume integrals $\mathcal{G}f, \partial_{\mathbf{n}}\mathcal{G}f$ coincide with boundary values or normal derivatives of the potentials $S\psi$, $D\varphi$ and $Nf$. The above three potential functions are not smooth at $\Gamma$ and they satisfy equivalent interface problems, by classical potential theory [(see [@Ying2007; @Xie2020])]{style="color: black"}. The equivalent interface problems for the single layer potential $-S\psi$, the double layer potential $D\varphi$ and the Newton potential $Nf$ can be unified as [\[eqn:equi-ifp\]]{#eqn:equi-ifp label="eqn:equi-ifp"} { &(u) - u = F, & ,\ &\[u\] = , &,\ &\[\_ u\] = , &,\ &u = 0,&. . The functions $\Phi$, $\Psi$ and $F$ are specified for each potential by - $-S\psi$: $\Phi = F = 0$, $\Psi = \psi$. - $D\psi$: $\Phi = \varphi$, $\Psi= F=0$. - $Nf$: $\Phi = \Psi = 0$. $F$ is an arbitrary extension of $f$ to the whole box $\mathcal{B}$. For simplicity, we set the extended value as zero. Once the interface problem [\[eqn:equi-ifp\]](#eqn:equi-ifp){reference-type="eqref" reference="eqn:equi-ifp"} is solved for the potentials, the boundary integrals $\mathcal{S}\psi$, $\mathcal{K}\varphi$, $\mathcal{K}^{\prime}\psi$, $\mathcal{D}\varphi$ and the volume integrals $\mathcal{G}f, {\color{black} \partial_{\mathbf{n}}\mathcal{G}f}$ can be obtained from the grid data of these potentials with an interpolation method. # Equivalent simple interface problem {#sec:crc} Solving the constant coefficient interface problem [\[eqn:equi-ifp\]](#eqn:equi-ifp){reference-type="eqref" reference="eqn:equi-ifp"} is an essential part of the KFBI method. For simplicity, we drop the constant $\sigma$ and proceed with the following problem [\[eqn:con-ifp\]]{#eqn:con-ifp label="eqn:con-ifp"} { &u- u = f, & ,\ &\[u\] = a, &,\ &\[\_ u\] = b, &,\ &u = 0,&. . where $a$, $b$ and $f$ are given data and $\kappa$ is a constant. The right-hand side $f$ is possibly discontinuous across the interface $\Gamma$. The constant coefficient interface problem is a much simpler case of the more general interface problem [\[eqn:IFP\]](#eqn:IFP){reference-type="eqref" reference="eqn:IFP"}. ## Interface representation In this work, the interface $\Gamma$ is implicitly defined by the level set function $H$ in the following way = { \^d \| H() = 0}, d = 2,3. We assume the level set function $H$ is at least $C^4$ (for the fourth-order method) and $|\nabla H|>c_0$ for some $c_0 > 0$ near the interface $\Gamma$. The level set function allows us to easily determine the intersection points of the surface with grid lines. For instance, if we have an intersection point on the line segment between two grid nodes $\mathbf{x}_1$ and $\mathbf{x}_2$, we expect the values $H(\mathbf{x}_1)$ and $H(\mathbf{x}_2)$ to have opposite signs. By solving the scalar algebraic equation for $t$ as follows: H(t_1 + (1-t)\_2) = 0, t , using methods such as the Newton method or the bisection method, one can obtain the coordinates of the intersection point. To compute the unit outward normal at a surface point, we utilize the gradient of the level set function. The unit outward normal vector $\mathbf{n}(\mathbf{x})$ at a point $\mathbf{x}$ on the surface is given by: () = . The method described in [@Ying2013] is employed in this work for the representation of the interface using only a subset of intersection points. For each $r = 1,\ldots,d$, $\mathbf{e}_r$ represents the $r$-th Cartesian basis vector in $\mathbb{R}^d$, and $\alpha \in (\cos^{-1}(1/\sqrt{d}), \pi/2)$ is a fixed angle. We define the subset: \_r = { : \|() \_r \| \> }, which forms an overlapping surface decomposition of $\Gamma$. The discrete representation of the interface $\Gamma_r$ only considers the intersection points between $\Gamma_r$ and the grid lines aligned with the $\mathbf{e}_r$ direction (refer to ). We denote the set of these intersection points as $\Gamma_r^h$. The union of all sets $\Gamma_r^h$ for $r = 1,\ldots,d$ is denoted as $\Gamma^h$, which represents the discrete set of points used to approximate $\Gamma$ and allocate surface degrees of freedom. For more detailed information on the surface discretization algorithm, please refer to [@Ying2013]. With the help of the overlapping surface decomposition-based discretization, the interface $\Gamma$ can be locally parameterized by a reference coordinate plane. Candidate reference planes are \_i:{(x_1, x_2, ,x_d)\^d\|x_i = 0}, i=1,,d, for $d = 2 \text{ or } 3$. Suppose at a point $\boldsymbol{x}\in\Gamma$, if the $i$-th component of the local normal $\boldsymbol{n}(\boldsymbol{x})$ has the largest absolute value, then we choose $\Pi_i$ as the reference plane of $\Gamma$ near $\boldsymbol{x}$. In such a way, the interpolation stencils on $\Gamma$ can be easily found with the help of the Cartesian grid on the reference plane. Numerical integration and interpolation on $\Gamma$ can be done in a similar way as those on a planer domain as well. We remark that, in principle, the Cartesian grid used for the representation of $\Gamma$ is not necessarily chosen as the same one for solving PDEs. In this work, we use the same Cartesian grid only for simplicity. ## Corrected finite difference scheme For simplicity, the bounding box is assumed to be a unit cube, $i.e.$ $\mathcal{B} = (0,1)^3$. Given a positive integer $N$, the domain $\mathcal{B}$ is uniformly partitioned into a Cartesian grid with mesh parameter $h = 1/N$. Let $P_{i, j, k}$ denote the grid node $(x_i, y_j, z_k), i, j, k = 0,1, \cdots,$, where $x_i=ih$, $y_j = jh$ and $z_k = kh$ are node coordinates. For an irregular domain $\Omega\subset\mathcal{B}$, the interior and exterior grid nodes are defined as $\Omega_h$ and $\Omega_h^C$, respectively, \_h &= {P\_i,j,k \| (x_i,y_j,z_k) ,i,j,k = 1, },\ \_h\^C &= {P\_i,j,k \| (x_i,y_j,z_k) ,i,j,k = 1, }. In the absence of interfaces, it is known that the following two compact finite difference schemes [\[eqn:9p-cds\]](#eqn:9p-cds){reference-type="eqref" reference="eqn:9p-cds"} and [\[eqn:27p-cds\]](#eqn:27p-cds){reference-type="eqref" reference="eqn:27p-cds"} are fourth-order accurate for 2D and 3D cases, respectively. &-(+)u\_i,j + ( - )\_\|r\|+\|s\|=1 u\_i+r,j+s + \_\|r\|+\|s\|=2 u\_i+r,j+s = f\_i,j+f\_i,j.[\[eqn:9p-cds\]]{#eqn:9p-cds label="eqn:9p-cds"}\ &-(+)u\_i,j,k + ( - )\_\|r\|+\|s\|+\|t\|=1 u\_i+r,j+s,k+t + \_\|r\|+\|s\|+\|t\|=2 u\_i+r,j+s,k+t\ &+\_\|r\|+\|s\|+\|t\|=3 u\_i+r,j+s,k+t = f\_i,j,k+f\_i,j,k.[\[eqn:27p-cds\]]{#eqn:27p-cds label="eqn:27p-cds"} The two schemes are adopted to derive the corresponding corrected finite difference schemes for the interface problem [\[eqn:con-ifp\]](#eqn:con-ifp){reference-type="eqref" reference="eqn:con-ifp"}. We write the finite difference schemes as a general form [\[eqn:FDS-gene\]]{#eqn:FDS-gene label="eqn:FDS-gene"} \_P\_i+r,j+s,k+t\_i,j,kc\_r,s,tu\_i+r, j+s,k+t = F\_i,j,k, where $c_{r,s,t}$ is the coefficient of $u_{i+r,j+s,k+t}$, $F_{i,j,k}$ is the right-hand side of the finite difference equation and $\mathcal{S}_{i,j,k}$ is the node set that contains all grid nodes with $c_{r,s,t} \neq 0$ at $P_{i,j,k}$. Then we define regular nodes $\mathcal{R}_h$ and irregular nodes $\mathcal{I}_h$ as follows, \_h = { P\_i,j,k \| S\_i,j,k\_h = S\_i,j,k\_h\^C = },\ \_h = { P\_i,j,k \| S\_i,j,k\_h S\_i,j,k\_h\^C }, At irregular nodes, since the finite difference approximation is taken across the discontinuity at the interface, large local truncation errors may occur and result in inaccurate or even divergent results. Precisely, let $\mathcal{A}_h$ denote the difference operator in the finite difference scheme [\[eqn:FDS-gene\]](#eqn:FDS-gene){reference-type="eqref" reference="eqn:FDS-gene"}. Suppose the local truncation error is on the order of $\mathcal{O}(h^p)$ at a regular node. The local truncation error at an irregular node is given by [\[eqn:LTE\]]{#eqn:LTE label="eqn:LTE"} &E_h(x_i, y_j, z_k) = \_h u(x_i, y_j, z_k) - F\_i,j,k\ = &{ &\_P\_i+r,j+s,k+t\_h\_i,j,kc\_r,s,tu\^+(x\_i+r,y\_j+s,z\_k+t)\ &+\_P\_i+r,j+s,k+t\_h\^C\_i,j,kc\_r,s,tu\^-(x\_i+r,y\_j+s,z\_k+t)- F\_i,j,k,P\_i,j,k\_h,\ &\_P\_i+r,j+s,k+t\_h\^C\_i,j,kc\_r,s,tu\^-(x\_i+r,y\_j+s,z\_k+t)\ &+\_P\_i+r,j+s,k+t\_h\_i,j,kc\_r,s,tu\^+(x\_i+r,y\_j+s,z\_k+t)- F\_i,j,k,P\_i,j,k\^C_h,\ .\ = &{ &\_P\_i+r,j+s,k+t\_i,j,kc\_r,s,tu\^+(x\_i+r,y\_j+s,z\_k+t)\ &+\_P\_i+r,j+s,k+t\_h\^C\_i,j,kc\_r,s,t(u\^- -u\^+)(x\_i+r,y\_j+s,z\_k+t)- F\_i,j,k,P\_i,j,k\_h,\ &\_P\_i+r,j+s,k+t\_i,j,kc\_r,s,tu\^-(x\_i+r,y\_j+s,z\_k+t)\ & +\_P\_i+r,j+s,k+t\_h\_i,j,kc\_r,s,t(u\^+ - u\^-)(x\_i+r,y\_j+s,z\_k+t)- F\_i,j,k,P\_i,j,k\^C_h,\ .\ = &{ \_P\_i+r,j+s,k+t\_h\^C\_i,j,kc\_r,s,t(u\^- - u\^+)(x\_i+r,y\_j+s,z\_k+t) + (h\^p),P\_i,j,k\_h,\ \_P\_i+r,j+s,k+t\_h\_i,j,kc\_r,s,t(u\^+ - u\^-)(x\_i+r,y\_j+s,z\_k+t) + (h\^p),P\_i,j,k\^C_h,\ . where $u^+$ and $u^-$ are two smooth functions that coincide with $u$ in the domain $\Omega$ and $\Omega^C$, respectively. It can be found that the leading term in the local truncation error at an irregular node is on the order of $\mathcal{O}(h^{-2})$, which is not acceptable for the sake of accuracy. The problem can be fixed by including the leading terms of the local truncation error, as correction terms, into the final finite difference equations. Define the correction function $C(\boldsymbol{x}) = u^+(\boldsymbol{x}) - u^-(\boldsymbol{x})$. Then, the corrected finite difference scheme can be written as [\[eqn:corrected-FDS\]]{#eqn:corrected-FDS label="eqn:corrected-FDS"} \_P\_i+r,j+s,k+t\_i,j,kc\_r,s,tu\_i+r, j+s,k+t = F\_i,j,k+C\_i,j,k. where the correction term $C_{i,j,k}$ is given by [\[eqn:crc-formula\]]{#eqn:crc-formula label="eqn:crc-formula"} C\_i,j,k = { &0,&P\_i,j,k\_h,\ &- \_P\_i+r,j+s,k+t\_h\^C\_i,j,kc\_r,s,tC(x\_i+r,y\_j+s,z\_k+t),&P\_i,j,k\_h_h,\ & \_P\_i+r,j+s,k+t\_h\_i,j,kc\_r,s,tC(x\_i+r,y\_j+s,z\_k+t),&P\_i,j,k\^C_h\_h.\ . *Remark 1*. If exact values of the correction function $C(\boldsymbol{x})$ are given, then the local truncation error of the corrected finite difference scheme [\[eqn:corrected-FDS\]](#eqn:corrected-FDS){reference-type="eqref" reference="eqn:corrected-FDS"} is on the order $\mathcal{O}(h^p)$ at each node. However, it happens only when the interface coincides with grid nodes and $C(\boldsymbol{x})$ equals the Dirichlet jump condition $[u]$. In practice, approximate values of the correction function $C(\boldsymbol{x})$ are used. For the fourth-order method in this work, the correction function only needs to be approximated with an error on the order of $\mathcal{O}(h^5)$ such that the local truncation error becomes $\mathcal{O}(h^3)$ at irregular nodes and $\mathcal{O}(h^4)$ elsewhere. *Remark 2*. If two interfaces are arbitrarily close, the line segment between two grid nodes may intersect interfaces more than once (see ). Let $u^{(i)}$ be restrictions of the piece-wise smooth solution $u$ in $\Omega^{(i)}$ for $i=0,1,2$. Denote by $C^{(1)} = u^{(1)}-u^{(0)}$ and $C^{(2)}=u^{(2)}-u^{(0)}$ two correction functions that are solved near $\Gamma^{(1)}$ and $\Gamma^{(2)}$. For the correction term $C_{i,j}$, the value $C(x_{i+r},y_{j+s})$ in [\[eqn:crc-formula\]](#eqn:crc-formula){reference-type="eqref" reference="eqn:crc-formula"} is computed by C(x\_i+r,y\_j+s) &= u\^(1)(x\_i+r,y\_j+s)-u\^(2)(x\_i+r,y\_j+s)\ &= u\^(1)(x\_i+r,y\_j+s) - u\^(0)(x\_i+r,y\_j+s) + u\^(0)(x\_i+r,y\_j+s) - u\^(2)(x\_i+r,y\_j+s)\ &= C\^(1)(x\_i+r,y\_j+s) - C\^(2)(x\_i+r,y\_j+s). This is simply adding and subtracting a middle term and is similar to the technique used in [@Ying2018]. fig/double_cross.png (130,450)$P_{i,j}$ (800,450)$P_{i+r,j+s}$ (200,700)$\Gamma^{(1)}$ (700,700)$\Gamma^{(2)}$ (200,350)$\Omega^{(1)}$ (700,350)$\Omega^{(2)}$ (450,250)$\Omega^{(0)}$ ## Local Cauchy problem Suppose $\Gamma$ is sufficiently smooth and the right-hand side $f$ is also piece-wise smooth. Denote by $\Omega_{\Gamma}$ a narrow band around $\Gamma$ that covers all irregular nodes. Let $f^+$ and $f^-$ be two smooth extension functions of $f$ in $\Omega_{\Gamma}$ from two different sides $\Omega_i$ and $\Omega_e$, respectively. Then the function $\Tilde{f} = f^+ - f^-$ is also smooth in $\Omega_{\Gamma}$. The smoothness of $\Tilde{f}$ is relevant to the accuracy of $C(x)$, see . Notice that the correction function $C(\boldsymbol{x})$ satisfies the Cauchy problem [\[eqn:cau-prb\]]{#eqn:cau-prb label="eqn:cau-prb"} C() - C()&= (),&\_,\ C() &= a(),&,\ \_ C() &= b(), &. Even though the Cauchy problem is known as ill-posed in the sense of Hadmond that small perturbation in the boundary data will grow exponentially away from the boundary and it is difficult to obtain a global numerical solution. Since the correction function is only required at irregular nodes that are close to the boundary $\Gamma$, we are only interested in the local solution of the Cauchy problem. In that case, numerical errors can be bounded from above. The localness of the Cauchy problem also suggests that numerical schemes with a small stencil, such as compact finite difference schemes, are preferred to work with for the correction function method. To locally solve the Cauchy problem [\[eqn:cau-prb\]](#eqn:cau-prb){reference-type="eqref" reference="eqn:cau-prb"}, we approximate the local solution in the narrow band $\Omega_{\Gamma}$ with a partition of unity approach. Using the quasi-uniform point set $\left\{\boldsymbol{p}_i\right\}_{i=1}^{N_p} \subset\Gamma$ that are primary points on the boundary $\Gamma$. Let $\Omega_{\Gamma,i}$ be a neighborhood of the point $\boldsymbol{p}_i$. Define $\Omega_{\Gamma}$ as the union of the neighborhoods \_ = \_i = 1\^N\_p \_,i. Then $\{\Omega_{\Gamma, i}\}_{i=1}^{N_p}$ forms an overlapping decomposition of $\Omega_{\Gamma}$. Note that each $\Omega_{\Gamma,i}$ should be chosen such that $\Omega_{\Gamma}$ covers all irregular grid nodes. Unlike the original CFM [@Marques2011], where $\Omega_{\Gamma}$ is defined as some particular grid patches relying on the cut pattern of $\Gamma$ with grid cells, the current definition of $\Omega_{\Gamma}$ is flexible since it only depends on the location of surface points. This decomposition gives us a simple way to represent $C(\boldsymbol{x})$ in $\Omega_{\Gamma}$. For the partitions $\Omega_{\Gamma, i}, i = 1, 2, \cdots, N_p$, define the compactly supported weight functions $\omega_i(\boldsymbol{x})$ such that $\text{supp}(\omega_i) = \Omega_{\Gamma,i}$ and \_i=1\^N_p \_i() , \_=\_i = 1\^N\_p \_,i. In practice, the weight function $\omega_i$ can be constructed in many ways, such as Shepard's method [@SHEPARDD1968]. In this work, we use a simple non-smooth weight function, \_i() = { 1, &\ 0,& . . We remark that the smoothness of the weight function has a neglectable effect on the algorithm. The above simple weight function works very well for all numerical experiments. Suppose $C_{h,i}(\boldsymbol{x})$ is an approximation to $C(\boldsymbol{x})$ for $\boldsymbol{x}\in\Omega_{\Gamma,i}$. With the partition of unity, the complete approximate solution $C_h(\boldsymbol{x})$ for $\boldsymbol{x}\in\Omega_{\Gamma}$ is constructed by the linear combination of local solutions $C_{h,i}$, C_h() = \_i=1\^N_p \_i () C\_h,i (), \_. To this end, we restrict the Cauchy problem [\[eqn:cau-prb\]](#eqn:cau-prb){reference-type="eqref" reference="eqn:cau-prb"} to the partition $\Omega_{\Gamma, i}$ and consider numerically solving a sequence of subproblems for $i=1,2,\cdots N_p$, [\[eqn:local-cau-prb\]]{#eqn:local-cau-prb label="eqn:local-cau-prb"} C_i() - C_i()&= (),&\_,i,\ C_i() &= a(),&\_,i,\ \_ C_i() &= b(), &\_,i. to obtain numerical solutions $C_{h,i}(\boldsymbol{x})$. The restricted problems [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} are both temporally and spatially local, which explains the terminology "local Cauchy problem". The method is more understandable if one regards the normal direction of $\Gamma$ as a time variable and the problems [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} as initial-boundary value problems (IBVPs). Solving the restricted problems for the full Cauchy problem [\[eqn:cau-prb\]](#eqn:cau-prb){reference-type="eqref" reference="eqn:cau-prb"} refers to the explicit method for time-dependent PDEs. In the correction function method, one does not need to be concerned with the stability of the explicit method since the solution is only solved by one step away from the boundary $\Gamma$. *Remark 3*. To obtain an accurate correction function $C(\boldsymbol{x})$, the right-hand side $\Tilde{f}(\boldsymbol{x})$ should be sufficiently smooth. For a fourth-order method, $C(\boldsymbol{x})$ is required to be at least $C^4$, and, consequently, $\Tilde{f}(\boldsymbol{x})$ is required to be at least $C^2$. Numerically, we can use the same partition of unity approach to represent $\Tilde{f}$ in $\Omega_{\Gamma}$. In each $\Omega_{\Gamma,i}$, $\Tilde{f}$ is replaced by a simple quadratic function using the jump information of $f$ (for example, in 2D, we use $[f]$, $[f_x]$, $[f_y]$, $[f_{xx}]$, $[f_{yy}]$, and $[f_{xy}]$). There are also several different ways to obtain smooth $f^+$ and $f^-$, such as the PDE-based method [@Aslam2004] and the partition of unity extension (PUX) method [@Fryklund2018]. ### A mesh-free collocation method Let $\{\phi_{l,m,n}(\boldsymbol{x})\}_{l+m+n \leq p}$ denote the basis of Taylor polynomial of degree no more than $p$, where the subscripts $l$, $m$ and $n$ are non-negative integers. The elements of the basis are given by, for example, &\_0,0,0(x,y,z) = 1,\ &\_1,0,0(x,y,z) = x,\_0,1,0(x,y,z) = y,\_0,0,1(x,y,z) = z,\ &\_2,0,0(x,y,z) = x\^2,\_0,2,0(x,y,z) = y\^2,\_0,0,2(x,y,z) = z\^2,\ &\_1,1,0(x,y,z) = xy,\_1,0,1(x,y,z) = xz,\_0,1,1(x,y,z) = yz,\ &. The approximate solution $C_{h,i}(\boldsymbol{x})$ is expressed as the linear combination of the basis C\_h,i() = \_l+m+n p d\_l,m,n \_l,m,n(, , ) ,\_,i, where $\xi$, $\eta$ and $\zeta$ are scaled local coordinate of $\boldsymbol{x}$. Suppose $\boldsymbol{p}_i = (x^{(i)}, y^{(i)}, z^{(i)})$ is the center point of the local domain $\Omega_{\Gamma,i}$. The scaled local coordinate $\Tilde{\boldsymbol{x}} = (\xi,\eta,\zeta)$ of $\boldsymbol{x} = (x, y, z)$ is defined as = (x - x\^(i)) / h,= (y - y\^(i)) / h,= (z - z\^(i)) / h, where $h$ is the mesh parameter. To determine the coefficients $d_{l,m,n}$, we replace $C_i$ with $C_{h,i}$ in the problem [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} and let the equations be exactly satisfied at multiple points. The resulting method is essentially mesh-free and falls into the category of collocation methods. Then the chosen points are called \"collocation points\". Since the problem [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} involves both bulk PDE and boundary conditions, it involves collocation points both in $\Omega_{\Gamma,i}$ and $\Gamma\cap\Omega_{\Gamma,i}$. Collocation points can be classified into three types based on the equations at which they are satisfied. Let $\boldsymbol{x}_j^{pde}, j = 1, 2,\cdots, m_1$ be the points in $\Omega_{\Gamma,i}$ where the PDE is satisfied. Let $\boldsymbol{x}_j^{D}, j = 1, 2,\cdots, m_2$ and $\boldsymbol{x}_j^{N}, j = 1, 2,\cdots, m_3$ be the points on $\Gamma\cap\Omega_{\Gamma,i}$ where the Dirichlet and Neumann conditions are satisfied, respectively. The problem [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} is approximated by the finite-dimensional problem [\[eqn:cau-appro\]]{#eqn:cau-appro label="eqn:cau-appro"} \_l+m+n p (- )\_l,m,n(\^pde\_j) d\_l,m,n &= (\_j\^pde),& j = 1,2,,m_1\ \_l+m+n p \_l,m,n(\^D\_j) d\_l,m,n &= a(\_j\^D), & j = 1,2,,m_2,\ \_l+m+n p (\^N\_j)\_l,m,n(\^N\_j) d\_l,m,n &= b(\_j\^N), & j = 1,2,,m_3. The approximate problem [\[eqn:cau-appro\]](#eqn:cau-appro){reference-type="eqref" reference="eqn:cau-appro"} forms a linear system [\[eqn:cau-ls\]]{#eqn:cau-ls label="eqn:cau-ls"} = , where the unknown vector $\mathbf{U}$ consists of the coefficients $d_{l,m,n}$. *Remark 4*. The collocation method is closely related to the local coordinate-transformation approach used in previous works [@Ying2007; @Ying2013; @Ying2014; @Xie2020]. The coordinate-transformation approach can also be viewed as a method for solving the local Cauchy problem [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} since the correction function $C(\boldsymbol{x})$ can also be approximated with the jump of derivatives $[u], [u_x], [u_y], [u_z], [u_{xx}] \cdots$ in terms of Taylor polynomial. However, the derivation of derivative jumps in the coordinate-transformation approach involves repeatedly taking tangential derivatives and applying the chains rule, which requires tedious calculation, especially for high-order and 3D cases. The collocation introduced here is much simpler since applying the chains rule is not required. ## Selection of collocation points Selecting collocation points is an essential part of the mesh-free collocation method to ensure accuracy and stability of the algorithm. Different selection procedures for collocation points result in different systems [\[eqn:cau-ls\]](#eqn:cau-ls){reference-type="eqref" reference="eqn:cau-ls"} and different results. For example, one can choose many collocation points such that their number is much more than the number of unknowns. In that case, the linear system [\[eqn:cau-ls\]](#eqn:cau-ls){reference-type="eqref" reference="eqn:cau-ls"} becomes overdetermined and can be solved in the least-square sense, which is similar to the method in [@Marques2011]. Here, an interpolation-type method is employed so that each equation in the system [\[eqn:cau-ls\]](#eqn:cau-ls){reference-type="eqref" reference="eqn:cau-ls"} is accurately satisfied. An advantage of using an interpolation-type method is that when a boundary point coincides with a grid node, the correction function is accurate at the point since the Dirichlet jump condition $[u]$ is enforced accurately. Before describing the selection procedure of collocation points, we emphasize a few key rules: 1. Collocation points should be chosen in $\Omega_{\Gamma,i}$ for the PDE and on $\Gamma\cap \Omega_{\Gamma, i}$ for boundary conditions. 2. Collocation points of the same type should be well-separated such that the resulting linear system is non-singular. 3. For each equation in [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"}, the number of collocation points should be chosen to meet the formal accuracy requirement. Rule (a) is a basic requirement for consistency of the collocation method. Rule (b) is addressed to avoid nearly singular or rank-deficient matrix $\mathbf{M}$ and to ensure stability of the method. For collocation points of the same type to be well-separated, the distance between two different points should have a positive lower bound. Moreover, the number of projections of these points onto each spatial direction should be sufficiently large such that the interpolation bases associated with the points can span the polynomial space. Rule (c) ensures accuracy of the collocation method. Note that the three equations in [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} have different times of derivatives of $C_i$, and thus a polynomial approximation of $C_i$ results in different orders of accuracy for each equation. Since the equations in [\[eqn:local-cau-prb\]](#eqn:local-cau-prb){reference-type="eqref" reference="eqn:local-cau-prb"} are enforced accurately at collocation points, the collocation problem is also referred to an interpolation problem. With the error estimation of polynomial interpolation, one can find that the approximation errors at a point $\boldsymbol{x}$, away from collocation points, satisfy [\[eqn:appro-prob\]]{#eqn:appro-prob label="eqn:appro-prob"} (- ) C\_h,i() - () &= (h\^p-1) ,&\_,i,\ C\_h,i() - a()&= (h\^p+1),&\_,i,\ \_ C\_h,i() - b()&= (h\^p), &\_,i. To take into account the consistency and stability requirement and to balance the approximation errors, we choose collocation points as interpolation points such that the corresponding Lagrange interpolant on these points has the same order of accuracy as shown in [\[eqn:appro-prob\]](#eqn:appro-prob){reference-type="eqref" reference="eqn:appro-prob"}. Precisely, collocation points are chosen as interpolation points of a polynomial of degree (i) $(p-2)$ for the PDE ; (ii) $p$ for the Dirichlet boundary condition; and (iii) $(p-1)$ for the Neumann boundary condition. It should be mentioned that the Lagrangian interpolant associated with the PDE is in $d$ space dimensions and those for the boundary conditions are in the $(d-1)$ space dimensions. Therefore, to choose collocation points for boundary conditions, we first project the boundary $\Gamma$ into its reference plane locally such that we can find the local stencil by working with the Cartesian grid on the planer domain. A good choice of the distribution of collocation points is illustrated in . Similar point selection strategies for multivariate interpolation are used in [@Xie2020; @Ying2014; @Ying2013]. If collocation points are chosen as above, the number of collocation points equals the number of degree of freedoms. For example, in three space dimensions, the collocation equation numbers for the PDE, and Dirichlet and Neumann boundary condition are $\sum_{i=0}^{p-2} (i+1)(i+2)/2$, $(p+1)(p+2)/2$ and $p(p+1)/2$, respectively. Obviously, it yields N\^eqn = \_i=0\^p-2++ = \_i=0\^p = N\^dof. Then the system [\[eqn:cau-ls\]](#eqn:cau-ls){reference-type="eqref" reference="eqn:cau-ls"} is a square one. One can easily verify that similar results hold for the two space dimensional case as well. The invertibility of the matrix $\mathbf{M}$ is difficult to prove since it depends on the geometry of $\Gamma$. Nevertheless, if the collocation points are chosen as aforementioned, the linear system is always uniquely solvable with a standard decomposition method, such as the QR decomposition method. *Remark 5*. We suggest using the scaled local coordinate $(\xi,\eta,\zeta)$ instead of the original coordinate $(x,y,z)$ for solving the problem [\[eqn:cau-appro\]](#eqn:cau-appro){reference-type="eqref" reference="eqn:cau-appro"}. It is equivalent to rescaling the local Cauchy problem such that its characteristic length changes from $\mathcal{O}(h)$ to $\mathcal{O}(1)$. Thus, the condition number of the problem $\eqref{eqn:cau-ls}$ is essentially independent of the grid size $h$ and the accuracy and the stability of the algorithm can be improved to reduce the effect of round-off error. In the numerical experiments, the condition number $\text{cond}(\mathbf{M})$ is always on the order of $10^2 \sim 10^3$ regardless of how small the grid size is. ## Extracting boundary data After solving the linear system of the corrected finite difference scheme, one can obtain the numerical solution at Cartesian grid nodes. However, in the KFBI method, one needs to frequently use boundary/interface data, such as boundary value or normal derivative of the solution, at boundary nodes rather than Cartesian grid nodes. In order to extract boundary data of the numerical solution, Lagrange interpolation is used to compute off-grid data. One should also take into account the jump values of the potential function such that the Lagrange interpolation has high-order accuracy. The correction function $C(\boldsymbol{x})$ introduced before now offers a suitable way to take into account the non-smoothness of the solution. With the correction function, it is simple to reconstruct smooth data for interpolation using the piece-wise smooth grid value. For example, given a boundary point $\boldsymbol{p}\in\Gamma$, we try to obtain the one-sided limit boundary data of the numerical solution $v_h$ in $\Omega^+$. Let $\boldsymbol{q}_i, i = 1, 2, \cdots$ be a grid node in the interpolation stencil near $\boldsymbol{p}$. Suppose the numerical solution $v_h$ is piece-wise smooth and coincides with the smooth functions $v_h^+$ and $v_h^-$ in $\Omega^+$ and $\Omega^-$, respectively. We add the correction function $C(\boldsymbol{q}_i)$ to the grid value $v_h(\boldsymbol{q}_i)$ if $\boldsymbol{q}_i\in\Omega^-$ so that the interpolation data is smooth. By making Taylor expansion at $\boldsymbol{p}$, it yields v_h(\_i) &= \_l+m+np \^l\^m\^n v_h\^+() + (\|\_i-\|\^p+1), \_i \^+,\ v_h(\_i) + C(\_i) &= \_l+m+np \^l\^m\^n v_h\^+() + (\|\_i-\|\^p+1), \_i \^-. where $(\xi, \eta, \zeta)^T = \boldsymbol{q}_i - \boldsymbol{p}$. Now, by solving the interpolation problem, the function value and derivatives of $v_h^+(\boldsymbol{p})$ are obtained. # Algorithm Summary {#sec:algo} In this section, we summarize the proposed method. We take the boundary integral equation [\[eqn:bie-ifp-1\]](#eqn:bie-ifp-1){reference-type="eqref" reference="eqn:bie-ifp-1"} as an example. The algorithms for the boundary integral equations [\[eqn:bie-D\]](#eqn:bie-D){reference-type="eqref" reference="eqn:bie-D"},[\[eqn:bie-N\]](#eqn:bie-N){reference-type="eqref" reference="eqn:bie-N"},[\[eqn:bie-ifp-2\]](#eqn:bie-ifp-2){reference-type="eqref" reference="eqn:bie-ifp-2"} are similar. The algorithm is summarized in . 1. Compute the right-hand side of [\[eqn:bie-ifp-1\]](#eqn:bie-ifp-1){reference-type="eqref" reference="eqn:bie-ifp-1"}, in which the integral operators $\mathcal{S}_i, \mathcal{K}_i, \mathcal{K}^{\prime}_i, \mathcal{D}_i, \mathcal{G}_i,\partial_{\mathbf{n}}\mathcal{G}f, i = 1,2$ are computed using the same approach in Step 3; 2. Given an initial guess for $\varphi$ and $\psi$; 3. Compute the integral operators $\mathcal{S}_i, \mathcal{K}_i, \mathcal{K}^{\prime}_i, \mathcal{D}_i, i = 1,2$ by solving the equivalent interface problem [\[eqn:equi-ifp\]](#eqn:equi-ifp){reference-type="eqref" reference="eqn:equi-ifp"}; 1. Computes the correction function in $\Omega_{\Gamma}$ by solving the local Cauchy problem [\[eqn:cau-prb\]](#eqn:cau-prb){reference-type="eqref" reference="eqn:cau-prb"}; 2. Compute the correction terms in the right-hand side of [\[eqn:corrected-FDS\]](#eqn:corrected-FDS){reference-type="eqref" reference="eqn:corrected-FDS"}; 3. Solve the linear system of the finite difference scheme [\[eqn:corrected-FDS\]](#eqn:corrected-FDS){reference-type="eqref" reference="eqn:corrected-FDS"} with FFT; 4. Compute the integral operator values by interpolation from the grid solution; 4. Generate the next $\varphi$ and $\psi$ using the GMRES method and repeat Step 3 until the residual is less than a given tolerance. In each iteration, individually computing the integral operators would require a total of eight calls of the FFT solver. We stress that the number can be reduced to two since the terms $\mathcal{K}_i \varphi-\mathcal{S}_i \psi$ and $\mathcal D_i \varphi-\mathcal{K}^{\prime}_i \psi$ for $i=1$ or $2$ can be computed by calling the FFT solver only once. By the principle of linear superposition, one only needs to solve the interface problem for the potential $D\varphi -S\psi$ and interpolate the function value and normal derivative on $\Gamma$ to obtain the terms. In this way, it only requires two calls of the FFT solver in each GMRES iteration. # Numerical results {#sec:res} In this section, numerical results for boundary and interface problems in both two and three space dimensions are presented. In the following examples, irregular domains and interfaces are given in their level-set forms, which will be specified for each case. Irregular domains and interfaces are embedded into a bounding box $\mathcal{B}$, which is chosen as a square in 2D and a cube in 3D. The box $\mathcal{B}$ is uniformly partitioned into $N$ intervals in each direction for simplicity. The total number of primary boundary points representing the interface $\Gamma$ is denoted by $N_b$. The following numerical experiments are performed on a personal computer with 3.80 GHz Intel Core i7 processors. The codes for conducting the numerical experiments are written in C++ computer language. The tolerance in the GMRES method is fixed as $10^{-10}$. GMRES iteration numbers and CPU times (in seconds) are reported to quantify the computational complexity. Numerical errors on the grid node set $\Omega^h$ in the $L_2$ and maximum norms are defined as e\_2 = , e\_ = \_\^h\|v()-u()\|, where $M$ is the number of grid nodes in $\Omega^h$, $v$ and $u$ are the numerical and exact solution, respectively. ## Two space dimensional examples ### Boundary value problem In the first example, we solve the 2D Dirichlet BVP of the Poisson equation on a rotated ellipse-shaped domain $\Omega$ = {(x,y)\^2: + \< 1 }, with $a = 1, b = 0.5, \theta = -\pi/6$. The ellipse is embedded into the bounding box $\mathcal{B} = [-1.2,1.2]^2$. Boundary condition and right-hand side are taken such that the exact solution satisfies u(x,y) = (x)((/3) x + (/3) y). Numerical results are summarized in . Nearly fifth-order accuracy in both the $L_2$ and maximum norms can be observed. The increase in convergence order may be caused by the error of quartic polynomial interpolation, which is fifth-order accurate and dominates the numerical error in the vicinity of the boundary. As the grid refines, the GMRES iteration number is essentially independent of grid size, which is a main advantage of the present method. Taking into account FFT solvers and boundary operations in each iteration, the overall computational complexity of the method is given by $\mathcal{O}(N^2\log N + N_b)$ in two space dimensions. On coarse grids, the CPU time scaling is close to $\mathcal{O}(N_b)$, implying that boundary operations dominate the computational cost. On finer grids, the CPU time is roughly linearly proportional to $\mathcal{O}(N^2\log N)$, which implies that the computational cost is dominated by the FFT solver. Isocontours of the numerical solution are also presented in . grid size 64$\times$`<!-- -->`{=html}64 128$\times$`<!-- -->`{=html}128 256$\times$`<!-- -->`{=html}256 512$\times$`<!-- -->`{=html}512 1024$\times$`<!-- -->`{=html}1024 -------------------------- ------------------------------- --------------------------------- --------------------------------- --------------------------------- ----------------------------------- $N_{b}$ 116 230 460 918 1838 itr no. 10 10 9 9 9 $\Vert e \Vert_{2}$ 7.40E-06 1.12E-07 3.03E-09 6.86E-11 2.31E-12 $\Vert e \Vert_{\infty}$ 1.31E-04 3.69E-06 1.03E-07 3.56E-09 1.24E-10 CPU time 3.91E-03 6.35E-03 1.86E-02 5.81E-02 2.33E-01 : Numerical results for the Dirichlet BVP of the Poisson equation on an ellipse-shaped domain. ### Interface problem with multiple interfaces In the second example, we solve the 2D Poisson interface problem with multiple disjoint interfaces, which are eight circles and a five-fold star, on the domain $\mathcal{B} = [-1.7,1.7]^2$. The circles are given by \_m\^cir = { (x,y)\^2: (x-(m/4))\^2 + (y-(m/4))\^2 = r\^2 },m = 1, 2, , 8, with $r = 0.383$. The five-fold star is given by \^star ={(x,y)\^2: + = (1.0 + (m ()))\^2}, with $a = b = 0.514, \varepsilon=0.2, m = 5$. Two adjacent interfaces may become very close to each other, and, as a result, there may be more than one intersection point between two adjacent grid nodes. Boundary condition, interface condition and right-hand side are chosen such that the exact solution is given by u(x, y) = { &(0.6x+0.8y), & \_i,\ &((x+1)/2)((y+1)/2),& \_e, . where $\Omega_i$ denotes the union of the interiors of the circles and the star and $\Omega_e$ denotes the exterior domain. The diffusion coefficients are chosen as $\sigma_i = 1$ in $\Omega_i$ and $\sigma_e = 3$ in $\Omega_e$. For this and the following examples, the subscripts $i$ and $e$ represent variables in the interior and exterior regions, respectively. Numerical results are summarized in . The solution in both interior and exterior domains has fourth-order accuracy. The GMRES iteration number is essentially independent of grid size, even if there are arbitrarily close interfaces. It can be observed that the iteration number is slightly larger on the coarsest grid $N=64$. A coarse Cartesian grid may not be able to accurately capture the geometry of complex interfaces. It affects the well-conditioned property of the discrete boundary integral equation and causes the increase in iteration number. Isocontours of the numerical solution are shown in . grid size 64$\times$`<!-- -->`{=html}64 128$\times$`<!-- -->`{=html}128 256$\times$`<!-- -->`{=html}256 512$\times$`<!-- -->`{=html}512 1024$\times$`<!-- -->`{=html}1024 ----------------------------------- ------------------------------- --------------------------------- --------------------------------- --------------------------------- ----------------------------------- $N_{b}$ 392 784 1568 3136 6282 itr no. 29 20 20 19 19 $\Vert e \Vert_{\infty,\Omega_i}$ 5.21E-05 4.65E-07 3.14E-09 1.90E-10 3.51E-11 $\Vert e \Vert_{\infty,\Omega_e}$ 3.94E-05 5.73E-07 3.19E-09 1.89E-10 3.51E-11 CPU time 7.81E-02 8.59E-02 1.48E-01 2.58E-01 5.86E-01 : Numerical results for the Poisson interface problem with multiple touching interfaces. {#fig:2d-poi-ifp-mul width="\\textwidth"} {#fig:2d-poi-ifp-mul width="\\textwidth"} ## Three space dimensional examples To demonstrate the applicability of the present method, we consider solving three space dimensional problems. ### Poisson BVP This example is the Neumann BVP of the Poisson equation on a torus in 3D. The torus is given by [\[eqn:ls-torus\]]{#eqn:ls-torus label="eqn:ls-torus"} = {(x,y,z)\^3: (1 - )\^2 + z\^2 \< 0.4\^2}. The bounding box is taken as $\mathcal{B}=[-1.5,1.5]^3$. The boundary condition and right-hand side are taken such that the exact solution satisfies u(x,y,z) = (z)((2x)+(3y)). Note that the solution to the Poisson Neumann BVP is only determined up to an additive constant. We first subtract a constant from the right-hand side of the linear system such that it has zero mean. At the same time, the matrix-vector products in the GMRES iterations are subtracted by a constant such that their means are zero. To compute numerical errors, we need to add a constant to the numerical solution such that it matches the exact solution at a point. Numerical results and the numerical solution are presented in and , respectively. Fourth-order accuracy in both the $L_2$ and maximum norms is reached for the Neumann BVP. In this example, the GMRES iteration number decreases slightly as the grid refines. Since the discrete linear system mimics the original well-conditioned BIE and the approximation with a fine grid is more accurate. We believe that the better approximation property with a fine grid gives a linear system with a better condition number and is the main reason that is responsible for the faster convergence of the GMRES method. Theoretically, the computational complexity in three space dimensions is $\mathcal{O}(N^3\log N + N_b)$. The cost of boundary operations is more important than that in two space dimensions since the polynomial approximation for the correction function needs more terms in this case. As a result, the overall computational cost is closer to $\mathcal{O}(N^2)$ since we have $N_b=\mathcal{O}(N^2)$. grid size 64$\times$`<!-- -->`{=html}64$\times$`<!-- -->`{=html}64 128$\times$`<!-- -->`{=html}128$\times$`<!-- -->`{=html}128 256$\times$`<!-- -->`{=html}256$\times$`<!-- -->`{=html}256 512$\times$`<!-- -->`{=html}512$\times$`<!-- -->`{=html}512 -------------------------- ---------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- $N_b$ 6168 24656 98668 394548 itr no. 23 21 18 17 $\Vert e \Vert_{2}$ 2.39E-04 2.76E-05 2.38E-06 1.68E-07 $\Vert e \Vert_{\infty}$ 1.18E-03 7.97E-05 5.98E-06 4.01E-07 CPU time 1.62E+00 6.35E+00 2.50E+01 1.35E+02 : Numerical results for the Neumann BVP of the Poisson equation on a torus. ### Modified Helmholtz BVP As in the preceding example, we solve the Dirichlet BVP of the modified Helmholtz equation with $\kappa = 100$ on the domain $\Omega$, which is given by = {(x,y,z)\^3: (1+4x\^2)(1+4y\^2)(1+4z\^2)+64xyz+4x\^2+4y\^2+4z\^2\<3 }, This domain has relatively large curvature and is difficult to be captured by a coarse grid. The bounding box is taken as $[-0.7,0.7]^3$. Boundary condition and right-hand side are chosen such that the exact solution satisfies u(x,y,z) = (z)((5x)+(2y)). Numerical results are summarized in . The numerical solution is presented in . One can observe that the numerical error is large on the grid $N=64$ and decreases rapidly when the grid is refined to $N=128$. It can be explained by the fact that the coarse grid $N=64$ may not be able to fully capture the fast changes of the boundary and cause large errors for near-interface corrections and surface interpolations. As the grid refines, the decrease in numerical errors matches the fourth-order accuracy, as anticipated. The coarse grid with $N=64$ also requires more GMRES iterations to converge. In each iteration, the CPU time scaling is close to $\mathcal{O}(N_b)$ due to the dominance of boundary operations. grid size 64$\times$`<!-- -->`{=html}64$\times$`<!-- -->`{=html}64 128$\times$`<!-- -->`{=html}128$\times$`<!-- -->`{=html}128 256$\times$`<!-- -->`{=html}256$\times$`<!-- -->`{=html}256 512$\times$`<!-- -->`{=html}512$\times$`<!-- -->`{=html}512 -------------------------- ---------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- $N_b$ 7082 28482 114018 455450 itr no. 14 9 9 9 $\Vert e \Vert_{2}$ 1.14E-05 7.34E-08 2.82E-09 1.20E-10 $\Vert e \Vert_{\infty}$ 1.80E-03 6.34E-06 5.06E-07 2.17E-08 CPU time 2.98E+00 8.52E+00 3.82E+01 1.75E+02 : Numerical results for the Dirichlet BVP of the modified Helmholtz equation. {#fig:hel-dir-3d width="\\textwidth"} {#fig:hel-dir-3d width="\\textwidth"} ### Interface problem with high-contrast coefficients In this example, we solve the Poisson interface equation with a four-atom molecular-shaped interface in the domain $\mathcal{B} = [-1.2,1.2]^3$. The interface $\Gamma$ is given by = { = (x,y,z)\^3: \_k=1\^4 (-) = 0.6 }, with $\boldsymbol{x}_1 = (\sqrt{3}/3, 0,-\sqrt{6}/12)$, $\boldsymbol{x}_2 = (-\sqrt{3}/6, 0.5,-\sqrt{6}/12)$, $\boldsymbol{x}_3 = (-\sqrt{3}/6, -0.5,-\sqrt{6}/12)$ and $\boldsymbol{x}_4 = (0, 0,\sqrt{6}/4)$. u(x, y,z) = { &\^2(2x)\^2(2y)(z), & \_i,\ &(x)(y)(z),& \_e. . The coefficient ratio $\sigma_e/\sigma_i$ varies from $10$ to $10^4$, and its effect on the performance of the present method is studied in this example. This effect was also studied by [@ZHOU20061; @Wang2004; @Marques2017]. The numerical solution is shown in . According to the numerical results presented in , high-contrast coefficients only have a small effect on the numerical accuracy, even for the extreme case $\sigma_e/\sigma_i=10^4$. The GMRES iteration number is slightly affected by the coefficient ratio on coarse grids. As the grid refines, the GMRES iteration number is rather stable and is independent of the coefficient ratio. This is also due to the fact that a fine grid has a better approximation property, as aforementioned. $\sigma_e:\sigma_i$ N itr no. $\Vert e \Vert_{2,\Omega_i}$ $\Vert e \Vert_{\infty,\Omega_i}$ $\Vert e \Vert_{2,\Omega_e}$ $\Vert e \Vert_{\infty,\Omega_e}$ --------------------- ----- --------- ------------------------------ ----------------------------------- ------------------------------ ----------------------------------- $10:1$ 128 11 5.15E-08 2.18E-07 1.43E-08 2.33E-07 256 10 3.25E-09 1.22E-08 9.03E-10 1.33E-08 512 10 2.04E-10 7.59E-10 5.76E-11 8.08E-10 $10^2:1$ 128 13 5.78E-08 2.74E-07 1.89E-08 2.92E-07 256 11 3.63E-09 1.61E-08 1.18E-09 1.71E-08 512 10 2.29E-10 1.02E-09 7.59E-11 1.06E-09 $10^4:1$ 128 14 5.86E-08 2.81E-07 1.94E-08 2.99E-07 256 11 3.68E-09 1.66E-08 1.21E-09 1.75E-08 512 10 2.33E-10 1.05E-09 7.83E-11 1.09E-09 : Numerical results for the Poisson interface problems with varying coefficient ratios. ### Interface problem with arbitrarily close interfaces In this case, we solve the Poisson interface problem with the presence of arbitrarily close interfaces in three space dimensions. Interfaces are taken as a torus and an ellipsoid. The torus-shaped interface $\Gamma^{tor}$ is given by the boundary of the domain $\Omega$ defined in [\[eqn:ls-torus\]](#eqn:ls-torus){reference-type="eqref" reference="eqn:ls-torus"}. The ellipsoid-shaped interface is given by \^ell = {(x,y,z)\^3: + + = 1}, with $a = b = 0.6, c = 1$. The two interfaces are very close to each other near the curve S = {(x,y,z)\^3: + = 1, z = 0}. In this configuration, since the curve $S$ is a dimension one object, the number of multi-intersection grid line segments, a grid line segment that intersects with interfaces multiple times, is on the order of $\mathcal{O}(N)$. The problem is challenging for classical body-fitted approaches, since it is nearly impossible to be resolved by a body-fitted mesh since the two interface is too close. The bounding box $\mathcal{B}$ is taken as $[-1.5,1.5]^3$. Boundary condition, interface condition and right-hand side are chosen such that the exact solution reads u(x, y,z) = { &\^2(2x)\^2(2y)(z), & \_i, 1,\ &(z)((2x)+(3y)), &\_i, 2,\ &(x)(y)(z),& \_e. . The coefficients are chosen as $\sigma_i = 1$ in $\Omega_{i, 1}\cup\Omega_{i, 2}$ and $\sigma_e = 3$ in $\Omega_e$. Numerical results are summarized in . The numerical solution is visualized and shown in . It is observed that fourth-order accuracy is achieved in all regions, except for an accuracy loss on the coarsest grid $N=64$ due to similar reasons that were mentioned before. grid size 64$\times$`<!-- -->`{=html}64$\times$`<!-- -->`{=html}64 128$\times$`<!-- -->`{=html}128$\times$`<!-- -->`{=html}128 256$\times$`<!-- -->`{=html}256$\times$`<!-- -->`{=html}256 512$\times$`<!-- -->`{=html}512$\times$`<!-- -->`{=html}512 --------------------------------------- ---------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- $N_b$ 8738 34866 139454 557794 itr no. 22 21 19 17 $\Vert e \Vert_{\infty,\Omega_{i,1}}$ 2.33E-03 6.75E-07 4.38E-08 2.13E-09 $\Vert e \Vert_{\infty,\Omega_{i,2}}$ 5.56E-05 7.72E-07 4.39E-08 2.11E-09 $\Vert e \Vert_{\infty,\Omega_e}$ 3.24E-03 5.97E-07 3.65E-08 2.06E-09 CPU time 5.55E+00 2.08E+01 7.07E+01 2.86E+02 : Numerical results for the Poisson interface problem with a touching curve. {#fig:3d-poi-ifp width="\\textwidth"} {#fig:3d-poi-ifp width="\\textwidth"} ### Heterogeneous interface problem In the final example, we consider the heterogeneous interface problem in three space dimensions. Interfaces are taken as three spheres with radius $r=0.7$ whose centers are chosen as $\boldsymbol{x}_1=(0.5,0.5,0.5)$, $\boldsymbol{x}_2=(-0.5,-0.5,0.5)$ and $\boldsymbol{x}_3=(0.5,-0.5,-0.5)$, respectively. The coefficients on each side of the interfaces are given as \_i = 1,\_i = 0,\_e = 4,\_e = 10, such that the unknown function $u$ satisfies the Poisson equation in the interior region and the modified Helmholtz equation in the exterior region. It is called a heterogeneous interface problem since the elliptic differential operator on two sides of the interfaces are of different types. The heterogeneous interface problem is a linearized version of the Poisson-Boltzmann equation, which appears in the Poisson-Boltzmann theory in biophysics for modeling solvated biomolecular systems. and show the numerical results and the visualization of the numerical solution, respectively. Once again we observe the fourth-order convergence in both regions. The number of GMRES iterations is essentially independent of the grid size. grid size 64$\times$`<!-- -->`{=html}64$\times$`<!-- -->`{=html}64 128$\times$`<!-- -->`{=html}128$\times$`<!-- -->`{=html}128 256$\times$`<!-- -->`{=html}256$\times$`<!-- -->`{=html}256 512$\times$`<!-- -->`{=html}512$\times$`<!-- -->`{=html}512 ----------------------------------- ---------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- $N_b$ 6987 27939 111774 447264 itr no. 21 20 20 20 $\Vert e \Vert_{\infty,\Omega_i}$ 6.71E-06 4.79E-07 3.32E-08 2.26E-09 $\Vert e \Vert_{\infty,\Omega_e}$ 3.89E-06 2.65E-07 1.73E-08 1.15E-09 CPU time 1.34E+00 6.59E+00 4.08E+01 3.20E+02 : Numerical results for the heterogeneous interface problem. {#fig:3d-ifp-heter width="30%"} # Discussion {#sec:dis} This work proposes a new version of the kernel-free boundary integral method for solving elliptic partial differential equations in two and three space dimensions with high accuracy. The KFBI method solves boundary and interface problems with their boundary integral formulations. It computes boundary and volume integrals by solving equivalent interface problems with fast PDE solvers and then obtains boundary values by interpolation. The equivalent interface problems are simpler than the original problem and are essential for the KFBI method. To accommodate the jump conditions in the interface, a correction function is introduced in the vicinity of the interface to derive corrected finite difference schemes and the boundary interpolation scheme. Unlike the original KFBI method, which applies a local coordinate transformation to calculate correction terms, the new approach obtains correction terms by solving a local Cauchy problem for the correction function. The local Cauchy problem is solved with a mesh-free collocation method, for which we also proposed a strategy to choose collocation points such that the resulting linear system is accurate and stable. The resulting method avoids repeatedly taking tangent derivatives on the jump condition and significantly simplifies the derivation procedure. The presented method is efficient and accurate, which is demonstrated through several challenging numerical experiments. The efficiency of the method relies on the well-conditionness of the boundary integral equations and the applicability of fast PDE solvers (FFT, geometric multigrid methods) on a Cartesian grid. Even though the presented numerical results are based on a fourth-order implementation of the method, the method can be extended to arbitrary accuracy in principle [@Marques2011]. Finally, we emphasize that the present method is designed for implicitly defined interfaces with level-set formulations. Although this work uses an analytic expression of the level-set function, extending the method to cases when the level-set function is only given at Cartesian grid nodes is straightforward. It may have advantages for solving moving interface problems and free boundary problems when combined with the level-set method [@OSHER198812; @Sethian1591]. # Acknowledgments {#acknowledgments .unnumbered} This work is financially supported by the Shanghai Science and Technology Innovation Action Plan in Basic Research Area (Project No. 22JC1401700). It is also partially supported by the National Key R&D Program of China (Project No. 2020YFA0712000), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010405) and the National Natural Science Foundation of China (Grant No. DMS-11771290).

arxiv_math

--- abstract: | The popularity of estimation to bounds for sums of eigenvalues started from P. Li and S. T. Yau for the study of the Pólya conjecture. This subject is extended to different types of differential operators. This paper explores for the sums of the first $m$ eigenvalues of Sturm-Liouville operators from two aspects. Firstly, by the complete continuity of eigenvalues, we propose a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent $p\in(1,\infty)$ of the Lebesgue spaces concerned. There have profound relations between the solvability of these systems and the optimal lower or upper bounds for the sums of the first $m$ eigenvalues of Sturm-Liouville operators, which provides a novel idea to study the optimal bounds. Secondly, we investigate the integrability or solvability of the critical systems. With suitable selection of exponents $p$, the critical systems are equivalent to the polynomial Hamiltonian systems of $m$ degrees of freedom. Using the differential Galois theory, we perform a complete classification for meromorphic integrability of these polynomial critical systems. As a by-product of this classification, it gives a positive answer to the conjecture raised by Tian, Wei and Zhang \[J. Math. Phys. 64, 092701 (2023)\] on the critical systems for optimal eigenvalue gaps. The numerical simulations of the Poincaré cross sections show that the critical systems for sums of eigenvalues can appear complex dynamical phenomena, such as periodic trajectories, quasi-periodic trajectories and chaos. author: - Yuzhou Tian$^{a,}$[^1],Meirong Zhang$^{b}$ title: "**On the Meromorphic Integrability of the Critical Systems for Optimal Sums of Eigenvalues**" --- $^a$ Department of Mathematics, Jinan University, Guangzhou 510632, China\ $^b$ Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China\ E-mail: ` tianyuzhou2016@163.com (Y. Tian)`\ E-mail: `zhangmr@tsinghua.edu.cn (M. Zhang)` **Mathematics Subject Classification (2020)**: Primary 34L15; Secondary 37J30, 70H07. **Keywords:** Sums of eigenvalues, Sturm-Liouville operator, Critical system, Meromorphic integrable, Differential Galois theory. # Introduction {#mr} In 1807, the pioneering work of solving the heat equation by Fourier planted the seed for the spectral theory of differential operators. Inspired by Fourier's work, Sturm and Liouville in 1837 systematically treated the spectra of second-order linear ordinary differential operators, commonly referred to as Sturm-Liouville operators. Afterwards, their work gradually evolved a whole new branch of mathematics, namely Sturm-Liouville theory. In the 20th century, Weyl's famous work [@MR1511560] together with the birth of quantum mechanics revolutionize this theory. Henceforth, the modern Sturm-Liouville theory not only provides a perfect medium for understanding the quantum mechanics, but also greatly promotes the development of other areas of mathematics, such as harmonic analysis, differential geometry and operator algebras. Nowadays, Sturm-Liouville theory is still an active area of research in modern mathematical physics. Let $\Omega=\left[0,1\right]$ be the unit interval. Fixed an exponent $p\in\left(1,\infty\right)$, the $L^p$ Lebesgue space on $\Omega$ is denoted by $${\mathcal L}^p:=L^p\left(\Omega,{\mathbb R}\right).$$ For an integrable potential $q\in{\mathcal L}^p$, we consider the following Dirichlet eigenvalue problem for the *Sturm-Liouville operator or one-dimensional Schrödinger operator* $$\label{line} \begin{split} \mathscr{D}_q \psi &:=-\psi'' + q \psi=\lambda\psi, \qquad x\in \Omega,\\ &\psi\big|_{\partial\Omega}=0. \end{split}$$ A number $\lambda$ is an *eigenvalue* of the system [\[line\]](#line){reference-type="eqref" reference="line"} if it has a nontrivial solution $\psi\left(x\right)$, called an *eigenfunction* associated $\lambda$. It is well-known that the eigenvalues of problem [\[line\]](#line){reference-type="eqref" reference="line"} can be written in the form of an increasing sequence $$\lambda_1(q) < \lambda_2(q) < \cdots< \lambda_m(q) < \cdots, \qquad\lambda_m(q) \to +\infty\mbox{ as } m\to \infty.$$ Here we have regarded eigenvalues as nonlinear functionals of potentials $q\in {\mathcal L}^p$. The sum of the first $m$ eigenvalues is defined as $$\mathscr{E}_m\left(q\right):=\sum_{i=1}^m\lambda_i\left(q\right).$$ In quantum theory, the eigenvalues have definite physical significance, which correspond to the energy levels of a particle within a potential $q$. Thereby, $\mathscr{E}_m\left(q\right)$ is the *total energy* of $m$ particles. Especially, the absorption energy for particle from the ground state to the first excited state is described by the *fundamental eigenvalue gap* $\lambda_2\left(q\right)-\lambda_1\left(q\right)$. Because of the above physical interpretations, estimation to the lower and upper bounds for eigenvalue problems, including gap, ratio and sum, are central to a large part of Sturm-Liouville theory. For the lower bounds of the fundamental eigenvalue gaps, many fascinating results about different types of operators and boundary conditions have been contributed by a lot of mathematicians up to the present days, see for example [@MR2784332; @MR3478937; @MR2299195; @MR2511892; @MR4150221; @MR2881964; @MR1948113; @MR1614731; @MR1081670] and references therein. The estimate for the upper bounds can be found in [@MR4447102; @MR2881964; @MR1948113; @MR1614731; @MR829055; @MR1081670]. The problems of estimations for the eigenvalue ratios also have been extremely studied in [@MR1218744; @MR257592; @MR4339006]. In applied sciences and mathematics, it is important to understand the sums of eigenvalues. For example, related with the quantum mechanics, elasticity theory, geometry and PDEs, it is natural to study the sum of the first $m$ eigenvalues [@MR639355; @MR4379307]. Perhaps the most important motivation is to originate from the famous Pólya conjecture [@MR129219] about the lower bound on $i$-th eigenvalue for the Dirichlet Laplacian, which still remains open. In 1983, Li and Yau [@MR701919] gave a partial answer to this conjecture and improved Lieb's result [@MR573436]. In order to be close to Pólya conjecture, their technique is to estimate the lower bound for the sum of the first $m$ eigenvalues, commonly known as the *Berezin-Li-Yau bound or inequality*. The estimates for sum of eigenvalues of different types of operators have gained wide investigation from mathematicians since Li and Yau. We refer the readers to [@MR2782621; @MR1165859; @MR1383015; @MR4379307; @MR2846268; @MR3412394; @MR3488539], etc. But up to now, there is not a general method to obtain the optimal lower or upper bound for sum of eigenvalues of Sturm-Liouville operator [\[line\]](#line){reference-type="eqref" reference="line"}. The purpose of this work is to investigate the optimization problems on sum of eigenvalues for [\[line\]](#line){reference-type="eqref" reference="line"}. Let $$B_{p,r}=\left\{q\in {\mathcal L}^p:\;\parallel q\parallel_p\leq r\right\}$$ be the (infinitely dimensional) ball of radius $r$, centered at the origin, in the space $\left({\mathcal L}^p,\parallel\cdot\parallel_p\right)$. We consider the following optimization problems $$\begin{aligned} \label{eq66} &\mathscr{E}_m^-:=\min\limits_{q\in B_{p,r}}\mathscr{E}_m\left(q\right)\;\text{and}\; \mathscr{E}_m^+:=\max\limits_{q\in B_{p,r}}\mathscr{E}_m\left(q\right).\end{aligned}$$ Their solutions will provide the following estimations on sum of the eigenvalues $$\begin{aligned} \label{eq67} &\mathscr{E}_m^-\leq\mathscr{E}_m\left(q\right)\leq\mathscr{E}_m^+, \qquad \forall q\in {\mathcal L}^p.\end{aligned}$$ The lower bound $\mathscr{E}_m^-$ is also called *Berezin-Li-Yau type lower bound*. Remarkably, $\mathscr{E}_m^-$ and $\mathscr{E}_m^+$ are the optimal lower and upper bounds of $\mathscr{E}_m\left(q\right)$ in a certain sense, respectively. Our first result provides a completely different approach to attain possible solutions to problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"}. As a consequence of the complete continuity of eigenvalues in potentials [@MZ10; @YZ11; @Zh08], one shows that the optimization problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"} can be attained by some optimizing potentials $q^{\pm}\in B_{p,r}$. See Theorem [Theorem 12](#attain){reference-type="ref" reference="attain"}. In order to determine $q^\pm$ and $\mathscr{E}_m^\pm$, we establish the next result. **Theorem 1**. *Let the exponent $p\in(1,\infty)$, $r\in(0,\infty)$ and $m\in\mathbb{N}$ be given with $m\geq2$. Denote by $p^*:=p/(p-1) \in (1,\infty)$ the conjugate exponent of $p$. For problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"}, indicated by $\varepsilon=-$ and $+$ respectively, there are $m$-dimensional parameters $\left(\mu_1,\ldots,\mu_m\right)=\left(\mu_1^\varepsilon,\ldots,\mu_m^\varepsilon\right)$ and non-trivial solutions $(u_1(x),\ldots,u_m(x))=(u_1^\varepsilon(x),\ldots,u_m^\varepsilon(x))$ to the following system $$\begin{aligned} \label{ceq37} &-u''_i+\varepsilon\left(\sum_{j=1}^mu_j^2\right)^{p^*-1}u_i=\mu_iu_i,\quad i=1,\ldots,m, \end{aligned}$$ such that* **(i)* the solutions $u_i(x)$ satisfy the Dirichlet boundary condition for $i=1,\ldots,m$.* **(ii)* the solutions $u_i(x)$ satisfy $$\label{u12} \int_\Omega \left(\sum_{i=1}^mu_i^2(x)\right)^{p^*} \,{\rm d}x= r^p,$$ and the optimizing potentials $q^\varepsilon(x)$ are determined by $$\label{qx2} q^\varepsilon(x):= \varepsilon\left(\sum_{i=1}^m\left(u_i^\varepsilon(x)\right)^2\right)^{p^*-1}, \qquad x\in \Omega.$$* **(iii)* the minimal and maximal of the sum of the first $m$ eigenvalues are given by $$\label{LM11} \mathscr{E}_m^-=\sum_{i=1}^m \mu_i^-\;\text{and}\; \mathscr{E}_m^+=\sum_{i=1}^m \mu_i^+,$$ respectively.* System [\[ceq37\]](#ceq37){reference-type="eqref" reference="ceq37"} is called in this paper *the critical system*, which is deduced by the direct application of the Lagrangian multiplier method to problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"}, as done in [@WMZ09; @Zh09; @YZ12]. Compared with the deductions of the critical systems to the inverse spectral problems for elliptic operators or Sturm-Liouville operators by Ilyasov and Valeev [@MR3912726; @MR4197914; @MR4285920], our derivation approach, employed the complete continuity of eigenvalues $\lambda_m\left(q\right)$ in $q\in {\mathcal L}^p$ [@MZ10; @Zh08], is very simpler. Let $v_i=u'_i$ for $i=1,\ldots,m$. Then critical system [\[ceq37\]](#ceq37){reference-type="eqref" reference="ceq37"} is equivalent to a Hamiltonian system of $m$ degrees of freedom $$\begin{aligned} \label{ceq38} &u'_i=v_i=\frac{\partial H}{\partial v_i},\quad v'_i=-\mu_iu_i+\varepsilon\left(\sum_{j=1}^mu_j^2\right)^{p^*-1}u_i=-\frac{\partial H}{\partial u_i},\quad i=1,\ldots,m,\end{aligned}$$ with the Hamiltonian $$\begin{aligned} \label{ceq39} &H=\frac{1}{2}\sum_{i=1}^m\left(v_i^2+\mu_iu_i^2\right)-\frac{\varepsilon}{2p^*}\left(\sum_{j=1}^mu_j^2\right)^{p^*}.\end{aligned}$$ Let $\mathbf{u}=\left(u_1,\ldots,u_m\right)$ and $\mathbf{v}=\left(v_1,\ldots,v_m\right)$. The non-constant function $I=I\left(\mathbf{u},\mathbf{v }\right)$ is said to be a *first integral* of the Hamiltonian system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} if $H$ and $I$ are *in involution*, i.e. the Poisson bracket $$\begin{aligned} \label{PB} &\left\{H,I\right\}=\sum_{i=1}^m\left(\frac{\partial H}{\partial v_i}\frac{\partial I}{\partial u_i}-\frac{\partial H}{\partial u_i}\frac{\partial I}{\partial v_i}\right)=0.\end{aligned}$$ The Hamiltonian function $H$ itself is always a first integral due to the antisymmetry of Poisson bracket. Denote the gradient of function $I$ by $\nabla I$. The functions $I_i$ for $i=1,\ldots,l$ are *functionally independent* on $U$ if $$\text{rank}\left(\nabla I_1,\ldots,\nabla I_l\right)=l$$ with the possible exception of sets of Lebesgue measure zero. The Hamiltonian system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} is called *completely integrable*, or simply *integrable* in Liouville's sense if there exist $m$ functionally independent first integrals $I_1\equiv H,I_2,\ldots,I_m$ ($H$ is the Hamiltonian). In addition, Hamiltonian system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} is *meromorphic completely integrable*, or simply *meromorphic integrable* if its $m$ functionally independent first integrals $I_1\equiv H,I_2,\ldots,I_m$ are meromorphic. Theorem [Theorem 1](#main){reference-type="ref" reference="main"} allows us to determine a solution to the optimization problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"} by solving a boundary value problem for critical system [\[ceq37\]](#ceq37){reference-type="eqref" reference="ceq37"}. In other words, the solvability of Hamiltonian system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} means the solvability of problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"}. The classical Arnold-Liouville's theorem exhibits that if Hamiltonian system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} is completely integrable, then it can be solved by quadrature, see [@MR2004534]. Naturally, we will focus on the next problem. *Whether or not Hamiltonian system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} is completely integrable.* The answer is too difficult because of the two reasons: system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} is not a polynomial system for some $p^*\in\left(1,+\infty\right)$; there are no universal techniques to decide the integrability of Hamiltonian systems. On the other hand, of particular interest is the limiting case of problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"}, that is, $p=1$. For this limiting case, like in [@WMZ09; @YZ12; @Zh09], it is significant to investigate the limiting system of ([\[ceq37\]](#ceq37){reference-type="ref" reference="ceq37"}) as $p\downarrow 1$, i.e., as $p^* \uparrow \infty$. In such a limiting process, let us pay a special attention to exponents $p$ so that $$\begin{aligned} \label{eq70} p= \frac{k}{k-1}\;\text{and}\; p^*=k, \;\text{and}\;k=2,3, \cdots\end{aligned}$$ For these exponents, system [\[ceq38\]](#ceq38){reference-type="eqref" reference="ceq38"} are the following polynomial Hamiltonian systems $$\begin{aligned} \label{eq38} &u'_i=v_i,\quad v'_i=-\mu_iu_i+\varepsilon\left(\sum_{j=1}^mu_j^2\right)^{k-1}u_i,\quad i=1,\ldots,m,\end{aligned}$$ where $$\begin{aligned} \label{eq39} &H=\frac{1}{2}\sum_{i=1}^m\left(v_i^2+\mu_iu_i^2\right)-\frac{\varepsilon}{2k}\left(\sum_{j=1}^mu_j^2\right)^k.\end{aligned}$$ At present, most studies have been dedicated to the integrability for Hamiltonian system of two degrees of freedom, see for instance [@MR3743739; @MR943701; @MR4075569; @MR2831794] and references therein. Except the natural Hamiltonian system with homogeneous potential, there are few literature relating to the integrability of other types of Hamiltonian systems of arbitrary degrees of freedom, see [@MR2123446; @MR1764944; @MR3804721]. With the help of the differential Galois theory, we give a complete classification of meromorphic integrability for Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} as follows. **Theorem 2**. *Let $\mathscr{U}=\left(\mu_1,\ldots,\mu_m\right)$ and $k\in \mathbb{N}^+$ with $k\geq 2$. The Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is meromorphic completely integrable if and only if $k$ and $\mathscr{U}$ belong to one of the following two families:* *Case* *$k$* *$\mathscr{U}$* *Additional meromorphic first integrals* -------- ------------ --------------------------------- --------------------------------------------------------------------------------- *1* *$k=2$* *$\mathscr{U}\in \mathbb{R}^m$* *See Proposition [Proposition 16](#pr5){reference-type="ref" reference="pr5"}.* *2* *$k\geq3$* *$\mu_1=\mu_2=\cdots=\mu_m$* *$I_i=u_1v_{i+1}-u_{i+1}v_1,\quad i=1,\ldots,m-1$.* : *Meromorphic completely integrable cases* *[\[Ta2\]]{#Ta2 label="Ta2"}* Especially when $\varepsilon=-$, $m=2$ and $k=2n$, the next corollary can be attained by the linear canonical transformation $\left(u_1,u_2,v_1,v_2\right)\mapsto \left(-u_1,-\text{i}u_2,-v_1,\text{i}v_2\right)$. **Corollary 3**. *Consider the following Hamiltonian system $$\label{e12} \begin{split} u'_1=v_1,\quad u'_2=-v_2,\quad v'_1=-\mu_1u_1-u_1\left(u_1^2-u_2^2\right)^{2n-1},\quad v'_2=\mu_2u_2+u_2\left(u_1^2-u_2^2\right)^{2n-1} \end{split}$$ with Hamiltonian $$\begin{aligned} \label{e15} &H= \frac{1}{2}\left(v_1^2-v_2^2 + \mu_1 u_1^2 - \mu_2 u_2^2\right) +\frac{1}{4n}\left(u_1^2-u_2^2\right)^{2n}. \end{aligned}$$ For $\mu_1\neq \mu_2$, $n\in \mathbb{N}^+$ and $n\geq2$, system [\[e12\]](#e12){reference-type="eqref" reference="e12"} is meromorphic non-integrable.* In [@TWZ], the authors studied the critical system for optimal eigenvalue gaps and posed the next conjecture. . For generic $\mu_1\neq \mu_2$ and $n\geq2$, system [\[e12\]](#e12){reference-type="eqref" reference="e12"} is not polynomial integrable. Obviously, Corollary [Corollary 3](#co1){reference-type="ref" reference="co1"} not only gives a positive answer to the above conjecture, but also extends it to meromorphic non-integrable. The framework of the paper is as follows. We briefly recall some preliminary concepts and results of differential Galois approach in section [2](#se2){reference-type="ref" reference="se2"}. After gathering the complete continuity results on eigenvalues, we deduce the critical system [\[ceq37\]](#ceq37){reference-type="eqref" reference="ceq37"} in section [3](#se3){reference-type="ref" reference="se3"}. To prove Theorem [Theorem 2](#th1){reference-type="ref" reference="th1"}, we will divide into two sections. In section [4](#se4){reference-type="ref" reference="se4"}, we show that system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is complete integrability if the parameters $k$ and $\mathscr{U}$ belong to Table [1](#Ta2){reference-type="ref" reference="Ta2"}. In section [5](#se5){reference-type="ref" reference="se5"}, by Morales-Ramis theory, we prove that system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is meromorphic non-integrability when the parameters $k$ and $\mathscr{U}$ are outside Table [1](#Ta2){reference-type="ref" reference="Ta2"}. Section [6](#se6){reference-type="ref" reference="se6"} presents exemplary Poincaré cross sections of the critical system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"}, which exhibits that system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} has abundant dynamical behaviors. # Preliminaries {#se2} In this section, we introduce some necessary concepts and preliminary results, containing Morales-Ramis theory, Hypergeometric equation and Kovacic's results. ## Morales-Ramis theory The Morales-Ramis theory [@MR1713573] is a powerful tool to determine the non-integrability of complex Hamiltonian systems. Roughly speaking, this theory establishes a relation between the meromorphic integrability and the differential Galois group of the variational equations or the normal variational equations. Next we briefly describe the Morales-Ramis theory. For some precise notions of differential Galois theory, see [@MR1960772]. Consider a complex symplectic manifold $M\subset\mathbb{C}^{2m}$ of dimension $2m$ with the standard symplectic form $\bm{\tilde{\omega}}=\sum_{j=1}^mdu_j\wedge dv_j$. Let $H: M\rightarrow \mathbb{C}$ be a holomorphic Hamiltonian. The Hamiltonian system with $m$ degrees of freedom is given by $$\begin{aligned} \label{eq59} &\dfrac{d \mathbf{x}}{dt}=X_H\left(\mathbf{x}\right)=\left(\dfrac{\partial H}{\partial \mathbf{v}}, -\dfrac{\partial H}{\partial \mathbf{u}}\right),\quad t\in\mathbb{C}, \quad \mathbf{x}=\left(\mathbf{u},\mathbf{v}\right)\in M,\end{aligned}$$ where $\mathbf{u}=\left(u_1,\ldots,u_m\right)$ and $\mathbf{v}=\left(v_1,\ldots,v_m\right)$ are the canonical coordinates. Let $\Gamma$ be a non-equilibrium solution of system [\[eq59\]](#eq59){reference-type="eqref" reference="eq59"}. Assume that $\Gamma$ can be parameterized by time $t$, that is, $$\begin{aligned} \bm{\varphi}:\mathbb{C}&\rightarrow M\subset\mathbb{C}^{2m}\\ t&\mapsto\left(\mathbf{u}\left(t\right),\mathbf{v}\left(t\right)\right).\end{aligned}$$ Then the *variational equation* (VE) along $\Gamma$ is the linear differential system $$\begin{aligned} \label{eq60} &\dfrac{d \mathbf{y}}{dt}=\dfrac{\partial X_H\left(\bm{\varphi}\left(t\right)\right)}{\partial \mathbf{x}}\mathbf{y}, \quad \mathbf{y}\in T_\Gamma M,\end{aligned}$$ where $T_\Gamma M$ is the tangent bundle $TM$ restricted on $\Gamma$. Let $N:=T_\Gamma M/T\Gamma$ be the normal bundle of $\Gamma$ [@MR1411677], and $\pi: T_\Gamma M \rightarrow N$ be the nature projective homomorphism. The *normal variational equation* (NVE) along $\Gamma$ has the form $$\begin{aligned} \label{eq61} &\dfrac{d \mathbf{z}}{dt}=\pi_*\left(T\left(\mathfrak{u}\right)\left(\pi^{-1}\mathbf{z}\right)\right),\quad \mathbf{z}\in N,\end{aligned}$$ where $\mathfrak{u}=X_H\left(\mathbf{x}\right)$ with $\mathbf{x}\in M$, and $T\left(\mathfrak{u}\right)$ is the tangential variation of $\mathfrak{u}$ along $\Gamma$, that is, $T\left(\mathfrak{u}\right)=\partial X_H/\partial \mathbf{x}$. Note that the above NVE is a $2\left(m-1\right)$-dimensional linear differential system. We can employ a generalization of D'Alambert's method to get the NVE [\[eq61\]](#eq61){reference-type="eqref" reference="eq61"}, see [@MR1713573]. Briefly speaking, we use the fact that $X_H\left(\bm{\varphi}\left(t\right)\right)$ is a solution of the VE [\[eq60\]](#eq60){reference-type="eqref" reference="eq60"} to reduce its dimension by one. In effect, we typically restrict the equation [\[eq59\]](#eq59){reference-type="eqref" reference="eq59"} to the energy level $h=H\left(\bm{\varphi}\left(t\right)\right)$. Then the dimension of the corresponding VE [\[eq60\]](#eq60){reference-type="eqref" reference="eq60"} also can be reduced. Morales and Ramis [@MR1713573] proved the following classical theorem, which give a necessary condition for the integrability of Hamiltonian system [\[eq59\]](#eq59){reference-type="eqref" reference="eq59"} in the Liouville sense. **Theorem 4**. *(Morales-Ramis theorem, see [@MR1713573]) If Hamiltonian system [\[eq59\]](#eq59){reference-type="eqref" reference="eq59"} is meromorphically integrable in the Liouville sense in a neighbourhood of a particular solution $\Gamma$, then the identity component of the Galois group of the *NVE* [\[eq61\]](#eq61){reference-type="eqref" reference="eq61"} is Abelian.* The next theorem tells us that the identity component of the differential Galois group is invariant under the covering. **Theorem 5**. *([@MR1713573]) Let $\mathcal{M}$ be a connected Riemann surface and $\nabla$ be a meromorphic connection over $\mathcal{M}$. Assume that $f:\mathcal{M}'\longrightarrow\mathcal{M}$ is a finite ramified covering of $\mathcal{M}$ by a connected Riemann surface $\mathcal{M}'$. Let $\nabla'=f^*\nabla$, i.e. the pull back of $\nabla$ by $f$. Then there exists a natural injective homomorphism $$\emph{Gal}\left(\nabla'\right)\rightarrow\emph{Gal}\left(\nabla\right)$$ of differential Galois groups which induces an isomorphism between their Lie algebras.* ## Hypergeometric equation The hypergeometric equation is a second order differential equation over the Riemann sphere $\mathbf{P}^1$ with three regular singular points [@MR2682403; @MR0178117]. Let us consider the following form of hypergeometric equation with three singular points at $z=0,1,\infty$ $$\begin{aligned} \label{hy} &\dfrac{d^2\zeta}{dz^2}+\left(\dfrac{1-\alpha-\tilde{\alpha}}{z}+\dfrac{1-\gamma-\tilde{\gamma}}{z-1}\right)\dfrac{d\zeta}{dz}+\left(\dfrac{\alpha\tilde{\alpha}}{z^2}+\dfrac{\gamma\tilde{\gamma}}{\left(z-1\right)^2}+\dfrac{\beta\tilde{\beta}-\alpha\tilde{\alpha}-\gamma\tilde{\gamma}}{z\left(z-1\right)}\right)\zeta=0,\end{aligned}$$ where $\left(\alpha,\tilde{\alpha}\right)$, $\left(\gamma,\tilde{\gamma}\right)$ and $\left(\beta,\tilde{\beta}\right)$ are the exponents at the respective singular points, and meet the Fuchs relation $\alpha+\tilde{\alpha}+\gamma+\tilde{\gamma}+\beta+\tilde{\beta}=1$. The exponent differences can be defined as $\varrho=\alpha-\tilde{\alpha}$, $\varsigma=\gamma-\tilde{\gamma}$ and $\tau=\beta-\tilde{\beta}$. The following theorem goes back to Kimura [@MR277789], whose gave necessary and sufficient conditions for solvability of the identity component of the differential Galois group of [\[hy\]](#hy){reference-type="eqref" reference="hy"}. $1$ $1/2+l$ $1/2+s$ Arbitrary complex number ------ --------- --------- -------------------------- --------------------- $2$ $1/2+l$ $1/3+s$ $1/3+\upsilon$ $3$ $2/3+l$ $1/3+s$ $1/3+\upsilon$ $l+s+\upsilon$ even $4$ $1/2+l$ $1/3+s$ $1/4+\upsilon$ $5$ $2/3+l$ $1/4+s$ $1/4+\upsilon$ $l+s+\upsilon$ even $6$ $1/2+l$ $1/3+s$ $1/5+\upsilon$ $7$ $2/5+l$ $1/3+s$ $1/3+\upsilon$ $l+s+\upsilon$ even $8$ $2/3+l$ $1/5+s$ $1/5+\upsilon$ $l+s+\upsilon$ even $9$ $1/2+l$ $2/5+s$ $1/5+\upsilon$ $l+s+\upsilon$ even $10$ $3/5+l$ $1/3+s$ $1/5+\upsilon$ $l+s+\upsilon$ even $11$ $2/5+l$ $2/5+s$ $2/5+\upsilon$ $l+s+\upsilon$ even $12$ $2/3+l$ $1/3+s$ $1/5+\upsilon$ $l+s+\upsilon$ even $13$ $4/5+l$ $1/5+s$ $1/5+\upsilon$ $l+s+\upsilon$ even $14$ $1/2+l$ $2/5+s$ $1/3+\upsilon$ $l+s+\upsilon$ even $15$ $3/5+l$ $2/5+s$ $1/3+\upsilon$ $l+s+\upsilon$ even : Schwarz table with $l,s,\upsilon\in\mathbb{Z}.$ [\[Ta3\]]{#Ta3 label="Ta3"} **Theorem 6**. *([@MR277789]) The identity component of the Galois group of the hypergeometric equation [\[hy\]](#hy){reference-type="eqref" reference="hy"} is solvable if and only if either* - *at least one of the four numbers $\varrho+\tau+\varsigma,-\varrho+\tau+\varsigma,\varrho-\tau+\varsigma,\varrho+\tau-\varsigma$ is an odd integer, or* - *the numbers $\varrho$ or $-\varrho$, $\varsigma$ or $-\varsigma$ and $\tau$ or $-\tau$ belong (in an arbitrary order) to some of the following fifteen families, see Table [2](#Ta3){reference-type="ref" reference="Ta3"}.* ## Kovacic's results Let $\mathbb{C}\left(z\right)$ be the field of rational functions in the variable $z$ with complex coefficients. Consider the second order linear differential equation $$\begin{aligned} \label{eq28} &\chi''=r\left(z\right)\chi,\quad r\left(z\right) \in \mathbb{C}\left(z\right).\end{aligned}$$ It is well known that the differential Galois group $G$ of equation [\[eq28\]](#eq28){reference-type="eqref" reference="eq28"} is an algebraic subgroup of $\text{SL}\left(2,\mathbb{C}\right)$. In 1986, Kovacic [@MR839134] characterized all possible types of $G$ as follows. **Theorem 7**. *([@MR839134]) The differential Galois group $G$ of equation [\[eq28\]](#eq28){reference-type="eqref" reference="eq28"} is conjugated to one of the following four types:* - *$G$ is conjugated to a subgroup of a triangular group, and equation [\[eq28\]](#eq28){reference-type="eqref" reference="eq28"} admits a solution of the form $\chi=\exp\left(\int \omega \right)$ with $\omega \in\mathbb{C}\left(z\right)$.* - *$G$ is conjugate to a subgroup of $$\begin{aligned} &\mathcal{G}=\left\{ \left( \begin{array}{cc} \mathfrak{a}&0\\ 0&\mathfrak{a}^{-1}\\ \end{array} \right) \Bigg|\mathfrak{a}\in \mathbb{C}\setminus\left\{0\right\}\right\} \bigcup \left\{ \left( \begin{array}{cc} 0&\mathfrak{a}\\ \mathfrak{a}^{-1}&0\\ \end{array} \right) \Bigg|\mathfrak{a}\in \mathbb{C}\setminus\left\{0\right\} \right\}, \end{aligned}$$ and equation [\[eq28\]](#eq28){reference-type="eqref" reference="eq28"} admits a solution of the form $\chi=\exp\left(\int \omega \right)$, where $\omega$ is algebraic of degree $2$ over $\mathbb{C}\left(z\right)$.* - *$G$ is finite and all solutions of equation [\[eq28\]](#eq28){reference-type="eqref" reference="eq28"} are algebraic over $\mathbb{C}\left(z\right)$.* - *$G=\emph{SL}\left(2,\mathbb{C}\right)$ and equation [\[eq28\]](#eq28){reference-type="eqref" reference="eq28"} does not admit Liouvillian solution.* Let $r\left(z\right)=\mathfrak{p}\left(z\right)/\mathfrak{q}\left(z\right)$ with $\mathfrak{p}\left(z\right),\mathfrak{q}\left(z\right)\in\mathbb{C}\left[z\right]$ relatively prime. The *pole* of $r\left(z\right)$ is a zero of $\mathfrak{q}\left(z\right)$ and *the order of the pole* is the multiplicity of the zero of $\mathfrak{q}\left(z\right)$. *The order of $r\left(z\right)$ at $\infty$* is defined by $\deg \mathfrak{q}-\deg \mathfrak{p}$. Kovacic [@MR839134] also provided the necessary conditions for types (i), (ii), or (iii) in Theorem [Theorem 7](#th6){reference-type="ref" reference="th6"} to occur. **Proposition 8**. *([@MR839134]) For the first three types in Theorem [Theorem 7](#th6){reference-type="ref" reference="th6"}, the necessary conditions of occurrence are respectively as follows:* ***Type (i)*** : *Each pole of $r\left(z\right)$ must have even order or else have order $1$. The order of $r\left(z\right)$ at $\infty$ must be even or else be greater than $2$.* ***Type (ii)*** : *The rational function $r\left(z\right)$ must have at least one pole that either has odd order greater than $2$ or else has order $2$.* ***Type (iii)*** : *The order of a pole of $r\left(z\right)$ cannot exceed $2$ and the order of $r\left(z\right)$ at $\infty$ must be at least $2$. If the partial fraction decomposition of $r\left(z\right)$ is $$r\left(z\right)=\sum_i\dfrac{\alpha_i}{\left(z-c_i\right)^2}+\sum_j\dfrac{\beta_j}{z-b_j},$$ then $\sqrt{1+4\alpha_i}\in\mathbb{Q}$ for each $i$, $\sum_j\beta_j=0$, and if $\Delta=\sum_i\alpha_i+\sum_j\beta_j$, then $\sqrt{1+4\Delta}\in\mathbb{Q}$.* **Remark 9**. *For a general second order linear differential equation $$\begin{aligned} &y''=a_1y'+a_2,\quad a_1,a_2\in \mathbb{C}\left(z\right),\end{aligned}$$ it can be transformed into the form [\[eq28\]](#eq28){reference-type="eqref" reference="eq28"} with $$r\left(z\right)=\dfrac{a_1^2}{4}-\dfrac{a'_1}{2}+a_2$$ via the change $$\begin{aligned} \label{eq63} &y=\exp\left(\dfrac{1}{2}\int a_1 dz\right)\chi. \end{aligned}$$* # Deduction of the critical systems {#se3} To derive the critical system [\[ceq37\]](#ceq37){reference-type="eqref" reference="ceq37"}, we refer to some basic properties of eigenvalues, viewed as functionals of potentials. **Lemma 10**. *([@PT87; @YZ11])[\[evsd\]]{#evsd label="evsd"} Given $m\in{\mathbb N}$, the $m$th eigenvalue $\lambda_m(q)$ is continuously Fréchet differentiable in $q\in ({\mathcal L}^p,\|\cdot\|_p)$, $p\in[1,\infty]$. Moreover, the Fréchet derivative $\partial_q \lambda_m(q)$, considered as an element of the conjugate space $({\mathcal L}^p)^*$, is $$\label{lamd} \partial_q \lambda_m(q) = (E_m(\cdot;q))^2,$$ where $E_m(x)=E_m(x;q)$ is an eigenfunction associated with $\lambda_m(q)$ satisfying the following normalization condition $$\label{lamd1} \|E_m\|_2= \left(\int_\Omega E_m^2(x)\,{\rm d}x\right)^{1/2}=1, \qquad \mbox{and} \qquad E'_m(0)>0.$$* Let $q_l,\ q\in{\mathcal L}^p$ with $p\in[1,\infty]$. We say that $q_l$ is *weakly convergent* to $q$ in ${\mathcal L}^p$ with respect to the weak topology $w_p$ if $$\lim\limits_{l\rightarrow\infty}\int_\Omega q_l\left(x\right)\xi\left(x\right)dx=\int_\Omega q\left(x\right)\xi\left(x\right)dx,\qquad\forall \xi\in\mathcal{L}^{p^*}.$$ Such a convergence is also denoted by $q_l \rightharpoonup q$ in ${\mathcal L}^p$. The following lemma is the complete continuity of eigenvalues in weak topologies, see [@MZ10; @Zh08; @YZ11] for more details. **Lemma 11**. *([@MZ10; @Zh08; @YZ11])[\[evsc\]]{#evsc label="evsc"} Given $m\in{\mathbb N}$, the $m$th eigenvalue $\lambda_m(q)$ is completely continuous in $q\in({\mathcal L}^p,w_p)$, $p\in[1,\infty]$. Here $w_p$ indicates the topology of weak convergence. More precisely, whenever $q_l \rightharpoonup q$ in ${\mathcal L}^p$, one has $$\label{lamc} \lim_{l\to\infty} \lambda_m(q_l) = \lambda_m(q).$$* **Theorem 12**. *Let $p\in (1,\infty)$, $r\in(0,\infty)$ and $m \in\mathbb{N}$ be given with $m\geq2$. Then there exist potentials $q^\varepsilon= q_{m,p,r}^\varepsilon\in {\mathcal L}^p$, $\varepsilon=+, \, -$ such that $$\label{ppm} \|q^-\|_p =\|q^+\|_p =r,$$ and $$\label{ppm1} \mathscr{E}_m^-=\mathscr{E}_m\left(q^-\right),\qquad\mathscr{E}_m^+= \mathscr{E}_m^+\left(q^+\right).$$* Let $$B_{p,r}=\left\{q\in {\mathcal L}^p:\;\parallel q\parallel_p\leq r\right\}$$ be the ball of the space $({\mathcal L}^p,\|\cdot\|_p)$. As we know, the ball $B_{p,r}$ is a compact set of $({\mathcal L}^p, w_p)$ with $p\in (1,\infty)$. Since the sum of the first $m$ eigenvalues $\mathscr{E}_m\left(q\right)$ is a finite sum, by the Lemma [\[evsc\]](#evsc){reference-type="ref" reference="evsc"}, then $\mathscr{E}_m\left(q\right)$ is completely continuous in $q\in {\mathcal L}^p$. So, there exist $q^\pm(x)= q_{m,p,r}^\pm(x)\in {\mathcal L}^p$ such that $\|q^\pm\|_p \le r$ and ([\[ppm1\]](#ppm1){reference-type="ref" reference="ppm1"}) is confirmed. Obviously, the Fréchet derivatives $$\partial_q\left(\mathscr{E}_m\left(q\right)\right)\big|_{q=q^\pm}=\sum_{i=1}^m\left(E_i\left(\cdot;q^\pm\right)\right)^2$$ is a non-zero function. From the Lagrange theorem, it follows that the optimizing potentials $q^\pm$ cannot be such that $\|q^\pm\|_p<r$. This implies that $\|q^\pm\|_p=r$, that is, equation [\[ppm\]](#ppm){reference-type="eqref" reference="ppm"}. $\Box$ Theorem [Theorem 12](#attain){reference-type="ref" reference="attain"} tells us that problems [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"} are constrained optimization problems, that is, $$\label{eq78} \min \left(\max\right) \left(\mathscr{E}_m\left(q\right)\right)\;\text{subject to}\;\|q\|_p =r.$$ Note that the Fréchet derivatives of $\mathscr{E}_m\left(q\right)$ and the $L^p$ norm are $$\partial_q\left(\mathscr{E}_m\left(q\right)\right)=\sum_{i=1}^m\left(E_i\left(x;q^\pm\right)\right)^2$$ and $$\partial_q \|q\|_p = \|q\|_p^{1-p} |q(x)|^{p-2} q(x), \qquad q\ne 0,$$ respectively. One can perform the Lagrange multiplier method to problems [\[eq78\]](#eq78){reference-type="eqref" reference="eq78"} to obtain that $q=q_{m,p,r}^\pm$ satisfy $$\label{qx1} |q(x)|^{p-2} q(x)= c \sum_{i=1}^m\left(E_i\left(x;q^\pm\right)\right)^2, \qquad x\in \Omega,$$ for some $c\ne 0$. For later convenience, we here write the Lagrangian multiplier $c$ in the right-hand side. For an exponent $p\in\left(1,\infty\right)$, the increasing homeomorphism $\phi_p: {\mathbb R}\to {\mathbb R}$ is given by $$\phi_p(s):= |s|^{p-2} s\qquad\mbox{for } s\in {\mathbb R},$$ and its inverse is $\phi_{p^*}= \phi_p^{-1}$, where $p^*:=p/(p-1) \in (1,\infty)$ is the conjugate exponent of $p$. **Lemma 13**. *The minimization and maximization problems of [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"} correspond to the Lagrangian multiplier $c<0$ and $c>0$ in [\[qx1\]](#qx1){reference-type="eqref" reference="qx1"} respectively.* For the optimizing potentials $q=q_{m,p,r}^\pm\left(x\right)$, equation [\[qx1\]](#qx1){reference-type="eqref" reference="qx1"} is equivalent to $$\phi_p(q(x))=c \sum_{i=1}^m\left(E_i\left(x;q^\pm\right)\right)^2 \;\text{and}\;q(x) = \phi_{p^*}(c) \phi_{p^*}\left( \sum_{i=1}^m\left(E_i\left(x;q^\pm\right)\right)^2\right).$$ We construct the following parameterized potentials $$Q_\sigma:= \sigma\phi_{p^*}(c) \phi_{p^*}\left( \sum_{i=1}^m\left(E_i\left(x;q\right)\right)^2\right)\in B_p[r],\qquad\sigma\in [0,1].$$ Clearly, $Q_1=q$. For the minimization problem [\[eq66\]](#eq66){reference-type="eqref" reference="eq66"}, one has $$\mathscr{E}_m\left(Q_\sigma\right)\geq\mathscr{E}_m\left(Q_1\right)\qquad \forall\sigma\in[0,1].$$ Thereby, the derivative $$\begin{aligned} 0 \!\!\!\!& \ge & \!\!\!\!\left. \frac{\,{\rm d}}{\,{\rm d}\sigma} \left(\mathscr{E}_m\left(Q_\sigma\right)\right)\right|_{\sigma=1} \\ \!\!\!\!& = & \!\!\!\!\int_\Omega\left(\sum_{i=1}^m\left(E_i\left(x;q\right)\right)^2\right)\cdot\phi_{p^*}(c)\phi_{p^*}\left(\sum_{i=1}^m\left(E_i\left(x;q\right)\right)^2\right) \,{\rm d}x\\ \!\!\!\!& = & \!\!\!\!\phi_{p^*}(c) \int_\Omega \left|\sum_{i=1}^m\left(E_i\left(x;q\right)\right)^2\right|^{p^*}\,{\rm d}x.\end{aligned}$$ Since $c\ne 0$, there must be $c<0$ for the minimization problem. Analogously, $c>0$ for the maximization problem. $\Box$ Note that $E_i(x)=E_{i}(x;q)$ are eigenfunctions for $i=1,\ldots,m$. We define the following $m$ parameters $$\mu_i:= \lambda_i(q)\qquad i=1,\ldots,m.$$ Thus, $$\begin{aligned} \label{eq76} -E''_i + q \left(x\right)E_i=\mu_iE_i,\; i=1,\ldots,m,\; x\in \Omega.\end{aligned}$$ To simplify the original critical equation [\[qx1\]](#qx1){reference-type="eqref" reference="qx1"}, we need to introduce the next notations $$\label{nw} \varepsilon:= {\rm sign}(c) =\pm 1, \qquad u_i(x):=\sqrt{|c|} E_{i}(x;q), \quad i=1,\ldots,m.$$ Therefore, equation [\[qx1\]](#qx1){reference-type="eqref" reference="qx1"} is equivalent to $$\begin{aligned} \label{eq65} \phi_p(q(x))=\varepsilon\sum_{i=1}^mu_i^2\;\text{and}\;q(x) = \varepsilon\phi_{p^*}\left( \sum_{i=1}^mu_i^2\right).\end{aligned}$$ Since $u_i\left(x\right)$ are still eigenfunctions, by equation [\[eq76\]](#eq76){reference-type="eqref" reference="eq76"}, we have $$-u''_i + q \left(x\right)u_i=\mu_iu_i, \qquad i=1,\ldots,m.$$ The critical system [\[ceq37\]](#ceq37){reference-type="eqref" reference="ceq37"} is obtained directly by substituting [\[eq65\]](#eq65){reference-type="eqref" reference="eq65"} into the above system. From the above analysis, it is not difficult to prove that equalities [\[u12\]](#u12){reference-type="eqref" reference="u12"}-[\[LM11\]](#LM11){reference-type="eqref" reference="LM11"} hold. For instance, equality [\[qx2\]](#qx2){reference-type="eqref" reference="qx2"} is the second equality of [\[eq65\]](#eq65){reference-type="eqref" reference="eq65"}. Equality [\[u12\]](#u12){reference-type="eqref" reference="u12"} is from the norm $\|q\|_p=r$. The proof is finished. $\Box$ # Complete integrability {#se4} In this sections, Propositions [Proposition 14](#pr4){reference-type="ref" reference="pr4"} and [Proposition 16](#pr5){reference-type="ref" reference="pr5"} show that system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is complete integrability if the parameters $k$ and $\mathscr{U}$ belong to Table [1](#Ta2){reference-type="ref" reference="Ta2"}. **Proposition 14**. *For $\mu_1=\mu_2=\cdots=\mu_m$, the Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is completely integrable with $m-1$ additional functionally independent first integrals $$\label{Integral1} I_i=u_1v_{i+1}-u_{i+1}v_1,\quad i=1,\ldots,m-1.$$* Straightforward calculations show that $I_i=u_1v_{i+1}-u_{i+1}v_1$ are first integrals of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} with $i=1,\ldots,m-1$. Since $$\det\left(\partial_{\mathbf{u}}H,\partial_{\mathbf{u}}I_1,\partial_{\mathbf{u}}I_2,\ldots, \partial_{\mathbf{u}}I_{m-1}\right)=\det \left( \begin{array}{cccccc} \partial_{u_1}H&v_2&v_3&v_4&\cdots&v_m\vspace{1ex}\\ \partial_{u_2}H& -v_1&0&0&\cdots&0\\ \partial_{u_2}H& 0&-v_1&0&\cdots&0\\ \partial_{u_3}H& 0&0&-v_1&\cdots&0\\ \partial_{u_4}H& 0&0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ \partial_{u_m}H&0&0&0&\cdots&-v_1\\ \end{array}\right)\not\equiv0,$$ then $\text{rank}\left(\nabla H,\nabla I_1,\nabla I_2,\ldots,\nabla I_{m-1}\right)=m$, that is, $H$ and $I_i$ are functionally independent with $i=1,\ldots,m-1$. $\Box$ For $k=2$ and $\varepsilon=+$, the Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} becomes a known complete integrability mechanical system, see [@CC77]. **Lemma 15**. *([@CC77]) For $k=2$ and $\varepsilon=+$, the following statements hold.* - *The Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is completely integrable.* - *Let $$\label{eq52} \begin{split} \mathcal{I}\left(\mathbf{u},\mathbf{v},\epsilon\right)=&\left(\sum_{j=1}^mu_j^2\right)\left(\sum_{j=1}^m\delta_j u_j^2\right) -\left(\sum_{j=1}^m\delta_ju_j^2\right)\left(\sum_{j=1}^m\delta_jv_j^2\right)+\\ &\left(\sum_{j=1}^m\delta_ju_jv_j\right)^2+2\sum_{j=1}^m\delta_j\left(v_j^2+\mu_ju_j^2\right) \end{split}$$ with $\delta_j=1/\left(\epsilon-\mu_j\right)$ for $j=1,\ldots,m$. Then $\mathcal{I}\left(\mathbf{u},\mathbf{v},\epsilon\right)$ is a first integral of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} independent $\epsilon$, and $\mathcal{I}\left(\mathbf{u},\mathbf{v},\epsilon\right)$ after expansion by powers of $\epsilon$ give $m$ functionally independent first integrals in involution for [\[eq39\]](#eq39){reference-type="eqref" reference="eq39"}.* **Proposition 16**. *Let $I\left(\mathbf{u},\mathbf{v}\right)$ be a first integral of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} with $k=2$ and $\varepsilon=+$. Then the following statements hold.* - *For $k=2$, the Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is completely integrable.* - *For $k=2$ and $\varepsilon=+$, $m$ functionally independent first integrals are given by statement *(ii)* of Lemma [Lemma 15](#le1){reference-type="ref" reference="le1"}.* - *For $k=2$ and $\varepsilon=-$, $I\left(-\emph{i}\mathbf{u},\emph{i}\mathbf{v}\right)$ is a first integral of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"}.* Let $I\left(\mathbf{u},\mathbf{v}\right)$ be a first integral of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} with $\varepsilon=+$. Doing the linear canonical change of variables $$\left(\mathbf{u},\mathbf{v},t\right)\mapsto\left(\text{i}\mathbf{u},-\text{i}\mathbf{v},-t\right),$$ the integrability of the case $\varepsilon=-$ is equivalent to the case $\varepsilon=+$. Thus, $I\left(-\text{i}\mathbf{u},\text{i}\mathbf{v}\right)$ is a first integral of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} with $\varepsilon=-$. By Lemma [Lemma 15](#le1){reference-type="ref" reference="le1"}, this proposition holds. $\Box$ # Meromorphic non-integrability {#se5} In this section, our aim is to prove the meromorphic non-integrability of Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} when the parameters $k$ and $\mathscr{U}$ are outside Table [1](#Ta2){reference-type="ref" reference="Ta2"}. **Proposition 17**. *If the parameters $k$ and $\mathscr{U}$ are outside Table [1](#Ta2){reference-type="ref" reference="Ta2"}, then Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is meromorphic non-integrable.* Let the parameters $k$ and $\mathscr{U}$ be outside Table [1](#Ta2){reference-type="ref" reference="Ta2"}. Then, $k\geq3$ and there exists a positive integer $j_0\in\left\{2,\ldots,m\right\}$ such that $\mu_1\neq\mu_{j_0}$. We can assume without loss of generality that $j_0=2$, that is, $\mu_1\neq\mu_2$, because in the other case one can interchange respectively the roles of $\mu_{j_0}$ and $\mu_2$, and $u_{j_0}$ and $u_2$. To be clear, our analysis is divided into two classes: $$\begin{aligned} &\text{\bf Class 1:}\;k\geq3, \mu_1\neq\mu_2\;\text{and}\; \mu_1\mu_2\neq0\;\text{(i.e. Lemma \ref{le2} below)};\\ &\text{\bf Class 2:}\;k\geq3, \mu_1\neq\mu_2\;\text{and}\; \mu_1\mu_2=0\;\text{(i.e. Lemma \ref{le3} below)}.\end{aligned}$$ The following Lemma [Lemma 18](#le2){reference-type="ref" reference="le2"} and Lemma [Lemma 19](#le3){reference-type="ref" reference="le3"} will complete the proof of Proposition [Proposition 17](#pr7){reference-type="ref" reference="pr7"}. $\Box$ **Lemma 18**. *If $k\geq3$, $\mu_1\neq\mu_2$ and $\mu_1\mu_2\neq0$, then Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is meromorphic non-integrable.* One can easily observe that system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} has two invariant manifolds $$\begin{aligned} &\mathcal{N}_1=\left\{\left(\mathbf{u},\mathbf{v}\right)\in \mathbb{C}^{2m}\mid u_j=v_j=0,j=2,\ldots,m\right\},\\ &\mathcal{N}_2=\left\{\left(\mathbf{u},\mathbf{v}\right)\in \mathbb{C}^{2m} \mid u_1=v_1=0, u_j=v_j=0,j=3,\ldots,m\right\}.\end{aligned}$$ System [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} restricted to the first invariant manifold $\mathcal{N}_1$ becomes $$\begin{aligned} \label{eq1} &u'_1=v_1,\quad v'_1=-\mu_1u_1+\varepsilon u_1^{2k-1},\end{aligned}$$ which has first integral $$\begin{aligned} \label{eq16} &h=\dfrac{1}{2}v_1^2+\dfrac{1}{2}\mu_1u_1^2-\dfrac{\varepsilon}{2k}u_1^{2k}.\end{aligned}$$ Solving equation [\[eq16\]](#eq16){reference-type="eqref" reference="eq16"}, we have $$\begin{aligned} \label{eq47} &\dfrac{du_1}{dt}=\pm\sqrt{2h+\dfrac{\varepsilon}{k}u_1^{2k}-\mu_1u_1^2}.\end{aligned}$$ As we know, equation [\[eq47\]](#eq47){reference-type="eqref" reference="eq47"} for $k=2$ and $k\geq3$ is respectively called *incomplete elliptic integral of first kind* and *hyperelliptic integral*, whose expressions are not always *elementary functions*, see [@MR0060642]. Let $\Theta\left(h\right)\in\mathbb{C}^2$ be an integral curve of system [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} lying on the energy level $h$. So, $$\begin{aligned} \label{eq57} &\varGamma_h:=\left\{\left(u_1\left(t\right),v_1\left(t\right),0,\ldots,0\right)\in\mathbb{C}^{2m}\mid \left(u_1\left(t\right),v_1\left(t\right)\right)\in\Theta\left(h\right)\right\}\end{aligned}$$ is a particular solution of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"}. The requirement of Theorem [Theorem 4](#th5){reference-type="ref" reference="th5"} is to construct a non-equilibrium particular solution $\varGamma_h$. We fix the energy level $h=0$. Equation [\[eq47\]](#eq47){reference-type="eqref" reference="eq47"} has three equilibrium points $u_1=0,\pm\sqrt[2k-2]{\mu_1k/\varepsilon}$ in the zero energy level. To exclude these equilibrium points, one can assume that $u_1\left(t\right)$ is not a constant. By this way, we can get a non-equilibrium particular solution $\Gamma_0\in\varGamma_0$. Let $\bm{\xi}:=\left(\xi_1,\ldots,\xi_m\right)^T$ and $\bm{\tilde{\xi}}:=\left(\tilde{\xi}_1,\ldots,\tilde{\xi}_m\right)^T$. We obtain that the variational equation (VE) along $\Gamma_0$ is $$\begin{aligned} \label{eq2} &\left( \begin{array}{c} \bm{\xi}'\vspace{1ex}\\ \bm{\tilde{\xi}}'\\ \end{array}\right) = \left( \begin{array}{cc} \mathbf{0}&\mathbf{I}\vspace{2ex}\\ \mathbf{\Lambda}&\mathbf{0}\\ \end{array} \right) \left( \begin{array}{c} \bm{\xi}\vspace{1ex}\\ \bm{\tilde{\xi}}\\ \end{array}\right),\end{aligned}$$ where $$\mathbf{\Lambda}:=\text{diag}\;\Big(\varepsilon\left(2k-1\right)u_1^{2k-2}\left(t\right)-\mu_1,\varepsilon u_1^{2k-2}\left(t\right)-\mu_2,\varepsilon u_1^{2k-2}\left(t\right)-\mu_3,\ldots,\varepsilon u_1^{2k-2}\left(t\right)-\mu_m\Big).$$ The VE [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"} is composed of $m$ uncoupled Schrödinger equations $$\begin{aligned} &\bm{\xi}''=\mathbf{\Lambda}\bm{\xi},\end{aligned}$$ that is, $$\begin{aligned} &\xi''_1=\left(\varepsilon \left(2k-1\right)u_1^{2k-2}\left(t\right)-\mu_1\right)\xi_1,\label{eq62}\\ &\xi''_j=\left(\varepsilon u_1^{2k-2}\left(t\right)-\mu_j\right)\xi_j,\quad j=2,\ldots,m.\end{aligned}$$ Since $\xi_1=u'_1\left(t\right)$ is a solution of [\[eq62\]](#eq62){reference-type="eqref" reference="eq62"}, equation [\[eq62\]](#eq62){reference-type="eqref" reference="eq62"} can be solved by Liouville's formula [@MR2682403]. Thereby, the normal variational equations (NVE) along $\Gamma_0$ are $$\begin{aligned} \label{eq3} &\xi''_j=\left(\varepsilon u_1^{2k-2}\left(t\right)-\mu_j\right)\xi_j,\quad j=2,\ldots,m.\end{aligned}$$ Inspired by Yoshida [@MR923886], we introduce the following finite branched covering map $$\label{eq17} \begin{split} &\overline{\Gamma}_0\rightarrow \mathbf{P}^1,\\ &t\longmapsto z=\dfrac{\varepsilon}{k\mu_1}u_1^{2k-2}\left(t\right), \end{split}$$ where $\overline{\Gamma}_0$ is the compact Riemann surface of the curve $v_1^2=\varepsilon u_1^{2k}/k-\mu_1u_1^2$ and $\mathbf{P}^1$ is the Riemann sphere. Performing the Yoshida transformation [\[eq17\]](#eq17){reference-type="eqref" reference="eq17"}, the normal variational equations [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} can be written as the hypergeometric differential equations in the new independent variable $z$ $$\begin{aligned} &\dfrac{d^2\xi_j}{dz^2}+\left(\frac{1}{z}+\frac{1}{2 (z-1)}\right)\dfrac{d\xi_j}{dz}-\left(\frac{\mu _j}{4 \mu _1(k-1)^2 z^2}+\frac{k \mu _1-\mu _j}{4\mu _1 (k-1)^2z (z-1)}\right)\xi_j=0,\tag{$\text{ANVE}_j$}\\ &j=2,\ldots,m.\notag\end{aligned}$$ The above differential system of equations is called the *algebraic normal variational equations* (ANVE), and is denoted as $$\begin{aligned} \label{eq4} &\text{ANVE}=\text{ANVE}_2\oplus\text{ANVE}_3\oplus\cdots\oplus\text{ANVE}_m.\end{aligned}$$ Essentially, equation [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} is a direct sum in the more intrinsic sense of linear connections, see Chapter $2$ of [@MR1713573] for more details. From Theorem [Theorem 5](#th4){reference-type="ref" reference="th4"}, it follows that the identity components of the Galois groups of the NVE [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} and the ANVE [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} coincide. Obviously, the ANVE [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} is integrable if and only if each $\text{ANVE}_j$ is integrable for $j=1,\ldots,m$. More precisely, the identity component of the Galois group of the ANVE is solvable if and only if the identity component of the Galois group of each $\text{ANVE}_j$ is solvable for $j=1,\ldots,m$. Now, we consider the $\text{ANVE}_2$: $$\begin{aligned} \label{eq19} &\dfrac{d^2\xi_2}{dz^2}+\left(\frac{1}{z}+\frac{1}{2 (z-1)}\right)\dfrac{d\xi_2}{dz}-\left(\frac{\mu _2}{4 \mu _1(k-1)^2 z^2}+\frac{k \mu _1-\mu _2}{4\mu _1 (k-1)^2z (z-1)}\right)\xi_2=0\end{aligned}$$ with three singular points at $z=0,1,\infty$. Comparing [\[eq19\]](#eq19){reference-type="eqref" reference="eq19"} with the general form of the hypergeometric equation [\[hy\]](#hy){reference-type="eqref" reference="hy"}, one can see that the exponents of [\[eq19\]](#eq19){reference-type="eqref" reference="eq19"} at singular points must fulfill the following relations $$\begin{aligned} &\alpha+\tilde{\alpha}=0,\quad \alpha\tilde{\alpha}=-\frac{\mu _2}{4 \mu _1(k-1)^2},\\ &\beta+\tilde{\beta}=\dfrac{1}{2},\quad \beta\tilde{\beta}=-\frac{k}{4(k-1)^2},\\ &\gamma+\tilde{\gamma}=\dfrac{1}{2},\quad \gamma\tilde{\gamma}=0.\end{aligned}$$ Thus, all the possibilities of the differences of exponents are $$\begin{aligned} \label{eq21} &\varrho=\pm\dfrac{1}{k-1}\sqrt{\dfrac{\mu_2}{\mu_1}},\tau=\pm\dfrac{1}{2}\left(1+\dfrac{2}{k-1}\right)\;\text{and}\;\varsigma=\pm\dfrac{1}{2}.\end{aligned}$$ Moreover, we can get all the possibilities of $\varrho+\tau+\varsigma,-\varrho+\tau+\varsigma,\varrho-\tau+\varsigma$ and $\varrho+\tau-\varsigma$, see Table [3](#Ta1){reference-type="ref" reference="Ta1"}. Signs of $\left(\varrho,\tau,\varsigma\right)$ $\varrho+\tau+\varsigma$ $-\varrho+\tau+\varsigma$ $\varrho-\tau+\varsigma$ $\varrho+\tau-\varsigma$ ----- ------------------------------------------------ --------------------------------------------------------------- --------------------------------------------------------------- --------------------------------------------------------------- --------------------------------------------------------------- $1$ $\left(+,+,+\right)$ $1+\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $1+\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $2$ $\left(+,+,-\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)-1$ $1+\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $3$ $\left(+,-,+\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)$ $-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $1+\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)-1$ $4$ $\left(+,-,-\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)-1$ $-1-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)$ $5$ $\left(-,+,+\right)$ $1+\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $1+\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $6$ $\left(-,+,-\right)$ $\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $-1-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ 1+$\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $7$ $\left(-,-,+\right)$ $-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)$ $1+\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $-1-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $8$ $\left(-,-,-\right)$ $-1-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ $\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)-1$ $\dfrac{1}{k-1}\left(1-\sqrt{\dfrac{\mu_2}{\mu_1}}\right)$ $-\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)$ : All the possibilities of $\varrho+\tau+\varsigma,-\varrho+\tau+\varsigma,\varrho-\tau+\varsigma$ and $\varrho+\tau-\varsigma$. [\[Ta1\]]{#Ta1 label="Ta1"} If equation [\[eq19\]](#eq19){reference-type="eqref" reference="eq19"} satisfies statement (i) of Theorem [Theorem 6](#A1){reference-type="ref" reference="A1"}, by Table [3](#Ta1){reference-type="ref" reference="Ta1"}, then $$\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}+1\right)\;\text{or}\;\dfrac{1}{k-1}\left(\sqrt{\dfrac{\mu_2}{\mu_1}}-1\right)$$ must be an integer, that is, $$\begin{aligned} \label{eq20} &\dfrac{\mu_2}{\mu_1}\in\left\{\left(\left(k-1\right)\ell\pm1\right)^2\big|\ell\in \mathbb{N}\right\}.\end{aligned}$$ The statement (ii) of Theorem [Theorem 6](#A1){reference-type="ref" reference="A1"} has $15$ possibilities in the Table [2](#Ta3){reference-type="ref" reference="Ta3"}. If the statement (ii) of Theorem [Theorem 6](#A1){reference-type="ref" reference="A1"} is fulfilled for equation [\[eq19\]](#eq19){reference-type="eqref" reference="eq19"}, from equation [\[eq21\]](#eq21){reference-type="eqref" reference="eq21"}, we find that only the first row of Table [2](#Ta3){reference-type="ref" reference="Ta3"} conforms. Note that $k\geq3$. Therefore, $$\pm\dfrac{1}{k-1}\sqrt{\dfrac{\mu_2}{\mu_1}}=\dfrac{1}{2}+\ell,\quad \ell\in\mathbb{Z},$$ that is, $$\dfrac{\mu_2}{\mu_1}\in\left\{\dfrac{\left(k-1\right)^2\left(2\ell+1\right)^2}{4}\Bigg|\ell\in \mathbb{Z}\right\}.$$ Based on the analysis above, the parameters $\mu_1$ and $\mu_2$ must satisfy $$\begin{aligned} \label{eq22} &\dfrac{\mu_2}{\mu_1}\in\left\{\left(\left(k-1\right)\ell\pm1\right)^2\big|\ell\in \mathbb{N}\right\}\bigcup\left\{\dfrac{\left(k-1\right)^2\left(2\ell+1\right)^2}{4}\Bigg|\ell\in \mathbb{Z}\right\}\end{aligned}$$ if the identity components of the Galois groups of the NVE [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} is Abelian. On the second invariant manifold $\mathcal{N}_2$, system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is written as $$\begin{aligned} \label{eq6} &u'_2=v_2,\quad v'_2=-\mu_2u_2+\varepsilon u_2^{2k-1}\end{aligned}$$ with Hamiltonian $$\begin{aligned} \label{eq23} &\tilde{h}=\dfrac{1}{2}v_2^2+\dfrac{1}{2}\mu_2u_2^2-\dfrac{\varepsilon}{2k}u_2^{2k}.\end{aligned}$$ To solve equation [\[eq23\]](#eq23){reference-type="eqref" reference="eq23"}, we $$\begin{aligned} \label{eq48} &\dfrac{du_2}{dt}=\pm\sqrt{2\tilde{h}+\dfrac{\varepsilon}{k}u_2^{2k}-\mu_2u_2^2}.\end{aligned}$$ Let $\widetilde{\Theta}\left(\tilde{h}\right)\in\mathbb{C}^2$ be a integral curve of system [\[eq6\]](#eq6){reference-type="eqref" reference="eq6"} lying on the energy level $\tilde{h}$. Thus, $$\begin{aligned} \label{eq58} &\widetilde{\varGamma}_{\tilde{h}}:=\left\{\left(0,0,u_2\left(t\right),v_2\left(t\right),0,\ldots,0\right)\in\mathbb{C}^{2m}\mid \left(u_2\left(t\right),v_2\left(t\right)\right)\in\widetilde{\Theta}\left(\tilde{h}\right)\right\}\end{aligned}$$ is a particular solution of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"}. We select the energy level $\tilde{h}=0$. Equation [\[eq48\]](#eq48){reference-type="eqref" reference="eq48"} has three equilibrium points $u_2=0,\pm\sqrt[2k-2]{\mu_2k/\varepsilon}$ in the zero energy level. In the same way as particular solution $\Gamma_0$, we can find a non-equilibrium particular solution $\widetilde{\Gamma}_0\in\widetilde{\varGamma}_0$. Let $\bm{\eta}:=\left(\eta_1,\ldots,\eta_m\right)^T$ and $\bm{\tilde{\eta}}:=\left(\tilde{\eta}_1,\ldots,\tilde{\eta}_m\right)^T$. The variational equations (VE) along $\widetilde{\Gamma}_0$ is given by $$\begin{aligned} \label{eq7} &\left( \begin{array}{c} \bm{\eta}'\vspace{1ex}\\ \bm{\tilde{\eta}}'\\ \end{array}\right) = \left( \begin{array}{cc} \mathbf{0}&\mathbf{I}\vspace{2ex}\\ \mathbf{\tilde{\Lambda}}&\mathbf{0}\\ \end{array} \right) \left( \begin{array}{c} \bm{\eta}\vspace{1ex}\\ \bm{\tilde{\eta}}\\ \end{array}\right),\end{aligned}$$ where $$\mathbf{\tilde{\Lambda}}:=\text{diag}\;\Big(\varepsilon u_2^{2k-2}\left(t\right)-\mu_1,\varepsilon\left(2k-1\right)u_2^{2k-2}\left(t\right)-\mu_2,\varepsilon u_2^{2k-2}\left(t\right)-\mu_3,\ldots, \varepsilon u_2^{2k-2}\left(t\right)-\mu_m\Big).$$ The VE [\[eq7\]](#eq7){reference-type="eqref" reference="eq7"} is also composed of $m$ uncoupled Schrödinger equations $$\begin{aligned} &\bm{\eta}''=\mathbf{\tilde{\Lambda}}\bm{\eta},\end{aligned}$$ that is, $$\begin{aligned} &\eta''_1=\left(\varepsilon u_2^{2k-2}\left(t\right)-\mu_1\right)\eta_1,\notag\\ &\eta''_2=\left(\varepsilon\left(2k-1\right)u_2^{2k-2}\left(t\right)-\mu_2\right)\eta_2,\label{eq64}\\ &\eta''_j=\left(\varepsilon u_2^{2k-2}\left(t\right)-\mu_j\right)\eta_j,\quad j=3,\ldots,m.\notag\end{aligned}$$ Using Liouville's formula [@MR2682403], the second equation of [\[eq64\]](#eq64){reference-type="eqref" reference="eq64"} is solvable due to the fact that it has a solution $\eta_2=u'_2\left(t\right)$. Therefore, the corresponding normal variational equations ($\widetilde{\text{NVE}}$) along $\widetilde{\Gamma}_0$ are given by $$\label{eq8} \begin{split} &\eta''_1=\left(\varepsilon u_2^{2k-2}\left(t\right)-\mu_1\right)\eta_1,\\ &\eta''_j=\left(\varepsilon u_2^{2k-2}\left(t\right)-\mu_j\right)\eta_j,\quad j=3,\ldots,m. \end{split}$$ Similarly, we can carry out the following Yoshida transformation $$t\longmapsto z=\dfrac{\varepsilon}{k\mu_2}u_2^{2k-2}\left(t\right),$$ and transform $\widetilde{\text{NVE}}$ [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"} into the algebraic normal variational equations ($\widetilde{\text{ANVE}}$): $$\begin{aligned} &\dfrac{d^2\eta_1}{dz^2}+\left(\frac{1}{z}+\frac{1 }{2 (z-1)}\right)\dfrac{d\eta_1}{dz}-\left(\frac{\mu _1}{4 \mu _2(k-1)^2 z^2}+\frac{k \mu _2-\mu _1}{4\mu _2 (k-1)^2z (z-1)}\right)\eta_1=0,\tag{$\widetilde{\text{ANVE}}_1$}\\ &\dfrac{d^2\eta_j}{dz^2}+\left(\frac{1}{z}+\frac{1}{2 (z-1)}\right)\dfrac{d\eta_j}{dz}-\left(\frac{\mu _j}{4 \mu _2(k-1)^2 z^2}+\frac{k \mu _2-\mu _j}{4\mu _2 (k-1)^2z (z-1)}\right)\eta_j=0,\tag{$\widetilde{\text{ANVE}}_j$}\\ &j=3,\ldots,m.\notag\end{aligned}$$ The direct sum form of $\widetilde{\text{ANVE}}$ is $\widetilde{\text{ANVE}}=\widetilde{\text{ANVE}}_1\oplus\widetilde{\text{ANVE}}_3\oplus\widetilde{\text{ANVE}}_4\oplus\cdots\oplus\widetilde{\text{ANVE}}_m$. For the $\widetilde{\text{ANVE}}_1$, all the possibilities of the differences of exponents are $$\varrho=\pm\dfrac{1}{k-1}\sqrt{\dfrac{\mu_1}{\mu_2}},\tau=\pm\dfrac{1}{2}\left(1+\dfrac{2}{k-1}\right)\;\text{and}\;\varsigma=\pm\dfrac{1}{2}.$$ By the same discussions as NVE [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"}, we obtain that the parameters $\mu_1$ and $\mu_2$ must satisfy $$\begin{aligned} \label{eq24} & \dfrac{\mu_1}{\mu_2}\in\left\{\left(\left(k-1\right)\ell\pm1\right)^2\big|\ell\in \mathbb{N}\right\}\bigcup\left\{\dfrac{\left(k-1\right)^2\left(2\ell+1\right)^2}{4}\Bigg|\ell\in \mathbb{Z}\right\} \end{aligned}$$ if the identity components of the Galois groups of the $\widetilde{\text{NVE}}$ [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"} is Abelian. The conditions [\[eq22\]](#eq22){reference-type="eqref" reference="eq22"} and [\[eq24\]](#eq24){reference-type="eqref" reference="eq24"} imply that $$\dfrac{\mu_2}{\mu_1}\geq1\;\text{and}\;\dfrac{\mu_1}{\mu_2}\geq1,$$ respectively. This contradicts our assumption $\mu_1\neq\mu_2$. Consequently, either the identity components of the Galois groups of the NVE [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} or $\widetilde{\text{NVE}}$ [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"} is not Abelian. By Theorem [Theorem 4](#th5){reference-type="ref" reference="th5"}, the Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} for $k\geq3$ is meromorphic non-integrable with $\mu_1\neq\mu_2$ and $\mu_1\mu_2\neq0$. The proof is finished. $\Box$ **Lemma 19**. *If $k\geq3$, $\mu_1\neq\mu_2$ and $\mu_1\mu_2=0$, then Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} is meromorphic non-integrable.* Our proof will be distinguished two cases: $$\text{\bf Case 1:}\; \mu_1=0, \mu_2\neq0\;\text{and}\;\text{\bf Case 2:}\; \mu_1\neq0, \mu_2=0.$$ **Case 1: $\mu_1=0$ and $\mu_2\neq0$.** For this case, we also restrict system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} on the invariant manifold $\mathcal{N}_1$. Namely, $$\begin{aligned} \label{eq15} &u'_1=v_1,\quad v'_1=\varepsilon u_1^{2k-1}\end{aligned}$$ with Hamiltonian $$\begin{aligned} \label{eq25} &h=\dfrac{1}{2}v_1^2-\dfrac{\varepsilon}{2k}u_1^{2k}.\end{aligned}$$ Analogously, we also consider the particular solution $\Gamma_0$ in the proof of Lemma [Lemma 18](#le2){reference-type="ref" reference="le2"}, and compute the $\widehat{\text{NVE}}$ along $\Gamma_0$ $$\label{eq45} \begin{split} &\xi''_j=\left(\varepsilon u_1^{2k-2}\left(t\right)-\mu_j\right)\xi_j,\quad j=2,\ldots,m. \end{split}$$ Doing the change of variable $$t\longmapsto z=\dfrac{\varepsilon}{2\mu_2}u_1^{2k-2}\left(t\right),$$ we attain the algebraic normal variational equations ($\widehat{\text{ANVE}}$): $$\begin{aligned} &\dfrac{d^2\xi_2}{d z^2}+\dfrac{3}{2z}\dfrac{d\xi_2}{d z}-\frac{k\left(2z-1\right)}{8 (k-1)^2z^3}\xi_2=0,\tag{$\widehat{\text{ANVE}}_2$}\\ &\dfrac{d^2\xi_j}{d z^2}+\dfrac{3}{2z}\dfrac{d\xi_j}{d z}-\frac{k \left(2\mu _2z-\mu _j\right)}{8(k-1)^2\mu _2 z^3}\xi_j=0,\quad j=3,\ldots, m, \tag{$\widehat{\text{ANVE}}_j$}\end{aligned}$$ and denote by $$\begin{aligned} \label{eq5} &\widehat{\text{ANVE}}=\widehat{\text{ANVE}}_2\oplus\widehat{\text{ANVE}}_3\oplus\cdots\oplus\widehat{\text{ANVE}}_m.\end{aligned}$$ Making the classical transformation $$\xi_2=\chi\exp\left(-\dfrac{3}{4}\int \dfrac{dz}{z} \right)=\chi z^{-3/4},$$ the $\widehat{\text{ANVE}}_2$ reads $$\begin{aligned} \label{eq26} &\chi''=r\left(z\right)\chi,\end{aligned}$$ where $$\begin{aligned} \label{eq27} &r\left(z\right)=-\left(\dfrac{(k-3) (3 k-1)}{16 (k-1)^2 z^2}+\dfrac{ k}{8(k-1)^2 z^3}\right).\end{aligned}$$ Then, the set of poles of $r\left(z\right)$ is $\Upsilon=\left\{0,\infty\right\}$. The order of $z=0$ and $z=\infty$ is $o\left(0\right)=3$ and $o\left(\infty\right)=2$, respectively. Using Proposition [Proposition 8](#pr1){reference-type="ref" reference="pr1"} to equation [\[eq26\]](#eq26){reference-type="eqref" reference="eq26"}, only types (ii) or (iv) of Theorem [Theorem 7](#th6){reference-type="ref" reference="th6"} can appear. Working the second part of Kovacic's algorithm (see Appendix [7](#B){reference-type="ref" reference="B"}), we obtain that $$\mathcal{E}_\infty=\left\{2+\ell\sqrt{1-\dfrac{(k-3) (3 k-1)}{4(k-1)^2 }}\;\Bigg|\;\ell=0,\pm2\right\}\bigcap\mathbb{Z}= \begin{cases} \left\{0,2,4\right\},\;\text{if}\;k=3,\\ \left\{2\right\},\;\text{if}\;k\geq4. \end{cases}\;\text{and}\;\mathcal{E}_0=\left\{3\right\}.$$ Straightforward computations show that the number $d=d\left(\bm{\varpi}\right)=\left(\varpi_\infty-\sum_{c\in\Upsilon}\varpi_c\right)/2$ is not a non-negative integer. Therefore, type (iv) of Theorem [Theorem 7](#th6){reference-type="ref" reference="th6"} holds. This means that the identity component of the Galois group of the $\widehat{\text{ANVE}}$ [\[eq5\]](#eq5){reference-type="eqref" reference="eq5"} is not Abelian. Thereby, the identity component of the Galois group of the $\widehat{\text{NVE}}$ [\[eq45\]](#eq45){reference-type="eqref" reference="eq45"} is also not Abelian. From Theorem [Theorem 4](#th5){reference-type="ref" reference="th5"}, it follows that the Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} for $k\geq3$ is meromorphic non-integrable with $\mu_1=0$ and $\mu_2\neq0$. **Case 2: $\mu_1\neq0$ and $\mu_2=0$.** Substituting $\mu_2=0$ into [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"}, we get the normal variational equations along $\widetilde{\Gamma}_0$: $$\label{eq46} \begin{split} &\eta''_1=\left(\varepsilon u_2^{2k-2}\left(t\right)-\mu_1\right)\eta_1,\\ %&\eta''_2=\varepsilon\left(2k-1\right)u_2^{2k-2}\left(t\right)\eta_2,\\ &\eta''_j=\left(\varepsilon u_2^{2k-2}\left(t\right)-\mu_j\right)\eta_j,\quad j=3,\ldots,m. \end{split}$$ After the change of variable $$t\longmapsto z=\dfrac{\varepsilon}{2\mu_1}u_2^{2k-2}\left(t\right),$$ equations [\[eq46\]](#eq46){reference-type="eqref" reference="eq46"} become the algebraic normal variational equations $$\begin{aligned} &\dfrac{d^2\eta_1}{d z^2}+\dfrac{3}{2z}\dfrac{d\eta_1}{d z}-\frac{k\left(2z-1\right)}{8 (k-1)^2z^3}\eta_1=0,\\ %&\dfrac{d^2\eta_2}{d z^2}+\dfrac{3}{2z}\dfrac{d\eta_2}{d z}-\frac{k (2 k-1)}{4 (k-1)^2 z^2}\eta_2=0,\\ &\dfrac{d^2\eta_j}{d z^2}+\dfrac{3}{2z}\dfrac{d\eta_j}{d z}-\frac{k \left(2\mu _1z-\mu _j\right)}{8(k-1)^2\mu _1 z^3}\eta_j=0,\quad j=3,\ldots, m. \end{aligned}$$ The analysis is exactly the same as **Case 1**. Thus, the Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} for $k\geq3$ is meromorphic non-integrable with $\mu_1\neq0$ and $\mu_2=0$. This lemma holds. $\Box$ By Propositions [Proposition 14](#pr4){reference-type="ref" reference="pr4"}, [Proposition 16](#pr5){reference-type="ref" reference="pr5"} and [Proposition 17](#pr7){reference-type="ref" reference="pr7"}, the Theorem [Theorem 2](#th1){reference-type="ref" reference="th1"} follows. $\Box$ # Poincaré cross section {#se6} For integrable Hamiltonian systems, Liouville-Arnold theorem [@MR2004534] exhibits that their dynamical behaviors are ordered and regular. For weakly perturbed (originally integrable) Hamiltonian systems, KAM theorem [@MR2004534] shows their dynamical behaviors are stochastic and chaotic, such as chaos, Arnold diffusion (at least three degrees of freedom), etc. Roughly speaking, the phase space for integrable Hamiltonian systems is foliated by KAM tori, which obstruct the stochasticity of the trajectories. For some weak perturbations, the KAM tori happen breaking down resulting in chaotic motion. In other words, the chaotic behavior can destroy meromorphic integrability. The classical Poincaré cross section technique can intuitively present the above dynamical process: local stability, the trajectories transition from ordered to chaotic, and many other dynamic properties. In the calculation Poincaré cross sections below, we focus on Hamiltonian system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"} with two degrees of freedom (i.e. $m=2$) and fix the parameter $\varepsilon=-1$. Consider the energy level $$M_h:=\left\{\left(u_1,u_2,v_1,v_2\right)\in\mathbb{R}^4| H\left(u_1,u_2,v_1,v_2\right)=h,\;h\in\mathbb{R}\right\}.$$ On energy level $M_h$, we select $u_1=0$ as a cross section plane with coordinates $\left(u_2,v_2\right)$. Taking the energy $h=0.85$, Figure [2](#fig2){reference-type="ref" reference="fig2"} shows the integrable case with $k=2$, which take the set of parameters: $\mu_1=0.1$, $\mu_2=1$; $\mu_1=0.1$, $\mu_2=-1$. We can see that their dynamical structures are very regular. {#fig2 width="110%"} \(1\) $\mu_2=1$. {#fig2 width="110%"} \(2\) $\mu_2=-1$. The integrable case and some weakly perturbed cases with $k=3$, $\mu_2=1$ and $h=0.85$ are presented in Figure [6](#fig1){reference-type="ref" reference="fig1"}. For integrable case, the dynamical behavior is highly regular, see (1) of Figure [6](#fig1){reference-type="ref" reference="fig1"}. For sufficiently small perturbation, KAM tori appear deformation or even breaking down in the fragile top and bottom boundaries, but most of them remain for internal region, as shown in (2) and (3) of Figure [6](#fig1){reference-type="ref" reference="fig1"}. For $\mu_1=0.99$ and $\mu_1=0.5$, the major bifurcations occurs in the vertical direction. As the perturbation strength is increased $\mu_1=0.1$, the KAM tori progressively break down resulting in the trajectories complete stochastic motion. Accurately speaking, the central domain in (4) of Figure [6](#fig1){reference-type="ref" reference="fig1"} is a large chaotic zone, that is, chaos. Around this chaotic zone, there are many chain of islands which correspond to quasi-periodic trajectories. Except for periodic trajectories, some KAM tori still remain in the annular area at top and bottom, which can be observed in (4) of Figure [6](#fig1){reference-type="ref" reference="fig1"}. {#fig1 width="110%"} \(1\) $\mu_1=1$. {#fig1 width="110%"} \(2\) $\mu_1=0.99$. {#fig1 width="110%"} \(3\) $\mu_1=0.5$. {#fig1 width="110%"} \(4\) $\mu_1=0.1$. Finally, our numerical simulations consider some high degree systems [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"}, that is, $k=10,20,30,40$, see Figure [10](#fig3){reference-type="ref" reference="fig3"}. One can see that the KAM tori at central region boundary disappear with the trajectories escaping to nonclosed areas of the phase space. This trajectories stochastic escaping create complex dynamic phenomena of system [\[eq38\]](#eq38){reference-type="eqref" reference="eq38"}, including chaos, quasi-periodic trajectories and periodic trajectories. {#fig3 width="110%"} \(1\) $k=10$, $\mu_2=45$. {#fig3 width="110%"} \(2\) $k=20$, $\mu_2=65$. {#fig3 width="110%"} \(3\) $k=30$, $\mu_2=35$. {#fig3 width="110%"} \(4\) $k=40$, $\mu_2=115$. # Acknowledgments {#acknowledgments .unnumbered} Yuzhou Tian wants to express his gratitude to the Department of Mathematical Sciences, Tsinghua University for the hospitality and support during the time period in which this work was completed. The second author is partially supported by the National Natural Science Foundation of China (Grant no. 11790273). # Appendix {#appendix .unnumbered} # Second part of Kovacic's algorithm {#B} Here, we recall the second part of Kovacic's algorithm [@MR839134]. Let $r\left(z\right)\in C\left(z\right)$ and $\Upsilon$ be the set of poles of $r\left(z\right)$. Set $\chi''=r\chi$. - To each pole $c\in\Upsilon$, we calculate the set $\mathcal{E}_c$ as follows. - If the pole $c$ is of order $1$, then $\mathcal{E}_c=\left\{4\right\}$. - If the pole $c$ is of order $2$ and $b$ is the coefficient of $1/\left(z-c\right)^2$ in the partial fraction decomposition of $r\left(z\right)$, then $$\begin{aligned} \label{eq29} &\mathcal{E}_c=\left\{2+\ell\sqrt{1+4b}\;\big|\;\ell=0,\pm2\right\}\bigcap\mathbb{Z}. \end{aligned}$$ - If the pole $c$ is of order $o\left(c\right)>2$, then $\mathcal{E}_c=\left\{o\left(c\right)\right\}$. - If the order of $r$ at $\infty$ is $o\left(\infty\right)>2$, then $\mathcal{E}_c=\left\{0,2,4\right\}$. - If the order of $r$ at $\infty$ is $2$ and $b$ is the coefficient of $1/z^2$ in the Laurent expansion of $r\left(z\right)$ at $\infty$, then $$\begin{aligned} \label{eq42} &\mathcal{E}_c=\left\{2+\ell\sqrt{1+4b}\;\big|\;\ell=0,\pm2\right\}\bigcap\mathbb{Z}. \end{aligned}$$ - If the order of $r$ at $\infty$ is $o\left(\infty\right)<2$, then $\mathcal{E}_c=\left\{o\left(\infty\right)\right\}$. - Let $\bm{\varpi}=\left(\varpi_c\right)_{c\in\Upsilon}$ be a element in the Cartesian product $\prod_{c\in\Upsilon}\mathcal{E}_c$ with $\varpi_c\in\mathcal{E}_c$. Define number $$\begin{aligned} \label{eq43} &d:=d\left(\bm{\varpi}\right)=\dfrac{1}{2}\left(\varpi_\infty-\sum_{c\in\Upsilon}\varpi_c\right). \end{aligned}$$ We try to find all elements $\bm{\varpi}$ such that $d$ is a non-negative integer, and retain such elements to perform Step 3. If there is no such element $\bm{\varpi}$, then statement (ii) of Theorem [Theorem 7](#th6){reference-type="ref" reference="th6"} is impossible. - For each $\bm{\varpi}$ retained from Step 2, we introduce the rational function $$\theta=\dfrac{1}{2}\sum_{c\in\Upsilon}\dfrac{\varpi_c}{z-c}.$$ Then, we seek a monic polynomial $P$ of degree $d$ defined in [\[eq43\]](#eq43){reference-type="eqref" reference="eq43"} such that $$\begin{aligned} &P'''+3\theta P''+\left(3\theta^2+3\theta'-4r\right)P'+\left(\theta''+3\theta\theta'+\theta^3-4r\theta-2r'\right)P=0. \end{aligned}$$ If such polynomial $P$ does not exist for all elements $\bm{\varpi}$ retained from Step 2, then statement (ii) of Theorem [Theorem 7](#th6){reference-type="ref" reference="th6"} is untenable. Assume that such a polynomial $P$ exists. Let $\phi=\theta+P'/P$ and $\omega$ be a root of $$\omega^2-\phi \omega+\left(\dfrac{1}{2}\phi'+\dfrac{1}{2}\phi^2-r\right)=0.$$ Then, $\chi=\exp\left(\int \omega \right)$ is a solution of differential equation $\chi''=r\chi$. 222 P. Acosta-Humánez, M. Alvarez-Ramı́rez, and T. J. Stuchi, *Nonintegrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory*, SIAM J. Appl. Dyn. Syst., **17** (2018),  78--96. B. Andrews and J. Clutterbuck, *Proof of the fundamental gap conjecture*, J. Amer. Math. Soc., **24** (2011),  899--916. M. S. Ashbaugh and R. D. Benguria, *Eigenvalue ratios for Sturm-Liouville operators*, J. Differential Equations, **103** (1993), 205--219. M. S. Ashbaugh, E. M. Harrell, II, and R. Svirsky, *On minimal and maximal eigenvalue gaps and their causes*, Pacific J. Math., **147** (1991), 1--24. M. S. Ashbaugh and D. Kielty, *Spectral gaps of 1-D Robin Schrödinger operators with single-well potentials*, J. Math. Phys., **61** (2020), 091507, 15. P. F. Byrd and M. D. Friedman, *Handbook of elliptic integrals for engineers and physicists*, Springer-Verlag, 1954. L. M. Chasman, *An isoperimetric inequality for fundamental tones of free plates*, Comm. Math. Phys., **303** (2011), 421--449. D. Chen, T. Zheng, and H. Yang, *Estimates of the gaps between consecutive eigenvalues of Laplacian*, Pacific J. Math., **282** (2016), 293--311. D.-Y. Chen and M.-J. Huang, *Comparison theorems for the eigenvalue gap of Schrödinger operators on the real line*, Ann. Henri Poincaré, **13** (2012), 85--101. H. Chen, M. Bhakta, and H. Hajaiej, *On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians*, J. Differential Equations, **317** (2022),  1--31. H. Chen and P. Luo, *Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators*, Calc. Var. Partial Differential Equations, **54** (2015),  2831--2852. Q.-M. Cheng and G. Wei, *A lower bound for eigenvalues of a clamped plate problem*, Calc. Var. Partial Differential Equations, **42** (2011), 579--590. S. Y. Cheng and P. Li, *Heat kernel estimates and lower bound of eigenvalues*, Comment. Math. Helv., **56** (1981), 327--338. D. V. Choodnovsky and G. V. Choodnovsky, *Completely integrable class of mechanical systems connected with Korteweg-de Vries and multicomponent Schrödinger equations*, Lett. Nuovo Cimento (2), **22** (1978),  47--51. T. Combot, A. J. Maciejewski, and M. Przybylska, *Bi-homogeneity and integrability of rational potentials*, J. Differential Equations, **268** (2020), 7012--7028. V. Gol'dshtein and A. Ukhlov, *On the first eigenvalues of free vibrating membranes in conformal regular domains*, Arch. Ration. Mech. Anal., **221** (2016),  893--915. S. Guo, G. Meng, P. Yan, and M. Zhang, *Optimal maximal gaps of Dirichlet eigenvalues of Sturm-Liouville operators*, J. Math. Phys., **63** (2022),  Paper No. 072701, 11. J. Hedhly, *Eigenvalue ratios for vibrating string equations with single-well densities*, J. Differential Equations, **307** (2022), 476--485. M. Horváth, *On the first two eigenvalues of Sturm-Liouville operators*, Proc. Amer. Math. Soc., **131** (2003), 1215--1224. Y. Ilyasov and N. Valeev, *Recovery of the nearest potential field from the $m$ observed eigenvalues*, Phys. D, **426** (2021),  Paper No. 132985, 6. Y. S. Ilyasov and N. F. Valeev, *On nonlinear boundary value problem corresponding to $N$-dimensional inverse spectral problem*, J. Differential Equations, **266** (2019),  4533--4543. E. Julliard Tosel, *Meromorphic parametric non-integrability; the inverse square potential*, Arch. Ration. Mech. Anal., **152** (2000), 187--205. S. Karaa, *Extremal eigenvalue gaps for the Schrödinger operator with Dirichlet boundary conditions*, J. Math. Phys., **39** (1998), 2325--2332. T. Kimura, *On Riemann's equations which are solvable by quadratures*, Funkcial. Ekvac., **12** (1969),  269--281. S. Kondej and I. Veselić, *Lower bounds on the lowest spectral gap of singular potential Hamiltonians*, Ann. Henri Poincaré, **8** (2007), 109--134. J. J. Kovacic, *An algorithm for solving second order linear homogeneous differential equations*, J. Symbolic Comput., **2** (1986), 3--43. V. V. Kozlov, *Symmetries, topology and resonances in Hamiltonian mechanics*, vol. 31 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1996. G. Kristensson, *Second order differential equations*, Springer, New York, 2010. P. Kröger, *Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space*, J. Funct. Anal., **106** (1992),  353--357. P. Li and S. T. Yau, *On the Schrödinger equation and the eigenvalue problem*, Comm. Math. Phys., **88** (1983),  309--318. E. H. Lieb, *The number of bound states of one-body Schroedinger operators and the Weyl problem*, in Geometry of the Laplace operator, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980,  241--252. A. J. Maciejewski, M. Przybylska, and T. Stachowiak, *Non-integrability of Gross-Neveu systems*, Phys. D, **201** (2005), 249--267. A. J. Maciejewski, M. Przybylska, and A. V. Tsiganov, *On algebraic construction of certain integrable and super-integrable systems*, Phys. D, **240** (2011), 1426--1448. G. Meng and M. R. Zhang, *Continuity in weak topology: first order linear systems of ODE*, Acta Math. Sin. (Engl. Ser.), **26** (2010),  1287--1298. J. J. Morales Ruiz, *Differential Galois theory and non-integrability of Hamiltonian systems*, vol. 179 of Progress in Mathematics, Birkhäuser Verlag, Basel, 1999. G. Pólya, *On the eigenvalues of vibrating membranes*, Proc. London Math. Soc. (3), **11** (1961),  419--433. J. Pöschel and E. Trubowitz, *Inverse spectral theory*, vol. 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987. M. Shibayama, *Non-integrability of the spacial $n$-center problem*, J. Differential Equations, **265** (2018),  2461--2469. I. M. Singer, B. Wong, S.-T. Yau, and S. S.-T. Yau, *An estimate of the gap of the first two eigenvalues in the Schrödinger operator*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), **12** (1985), 319--333. R. S. Strichartz, *Estimates for sums of eigenvalues for domains in homogeneous spaces*, J. Funct. Anal., **137** (1996),  152--190. C. J. Thompson, *On the ratio of consecutive eigenvalues in $N$-dimensions*, Studies in Appl. Math., **48** (1969),  281--283. Y. Tian, Q. Wei, and M. Zhang, *On the polynomial integrability of the critical systems for optimal eigenvalue gaps*, J. Math. Phys., **64** (2023), p. 092701. N. F. Valeev and Y. S. Ilyasov, *Inverse spectral problem for Sturm-Liouville operator with prescribed partial trace*, Ufa Math. J., **12** (2020),  19--29. M. van der Put and M. F. Singer, *Galois theory of linear differential equations*, vol. 328, Springer-Verlag, Berlin, 2003. H. Vogt, *A lower bound on the first spectral gap of Schrödinger operators with Kato class measures*, Ann. Henri Poincaré, **10** (2009),  395--414. Q. Wei, G. Meng, and M. Zhang, *Extremal values of eigenvalues of Sturm-Liouville operators with potentials in $L^1$ balls*, J. Differential Equations, **247** (2009), 364--400. H. Weyl, *Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen*, Math. Ann., **68** (1910), 220--269. E. T. Whittaker and G. N. Watson, *A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions*, Cambridge University Press, New York, 1962. S. Wiggins, *Introduction to applied nonlinear dynamical systems and chaos*, vol. 2 of Texts in Applied Mathematics, Springer-Verlag, New York, second ed., 2003. P. Yan and M. Zhang, *Continuity in weak topology and extremal problems of eigenvalues of the $p$-Laplacian*, Trans. Amer. Math. Soc., **363** (2011), 2003--2028. P. Yan and M. Zhang, *A survey on extremal problems of eigenvalues*, Abstr. Appl. Anal., (2012), pp. Art. ID 670463, 26. H. Yoshida, *A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential*, Phys. D, **29** (1987),  128--142. H. Yoshida, *Nonintegrability of the truncated Toda lattice Hamiltonian at any order*, Comm. Math. Phys., **116** (1988),  529--538. M. Zhang, *Continuity in weak topology: higher order linear systems of ODE*, Sci. China Ser. A, **51** (2008), 1036--1058. M. Zhang, *Extremal values of smallest eigenvalues of Hill's operators with potentials in $L^1$ balls*, J. Differential Equations, **246** (2009),  4188--4220. [^1]: Correspondence author.

arxiv_math

# Introduction {#sect:intro} Let $(M,g)$ be a smooth ($C^\infty$) Riemannian manifold, with induced distance-function $d$, and for $p\in M$ let $r_p=d(\cdot,p)$ denote the function "distance to $p$". The *Fréchet mean set* ${\rm FM}(\mu)$ of a Borel probability measure $\mu$ on $M$ is defined to be the set of minimizers of the function on $M$ defined by $p\mapsto \int_M r_p^2\, d\mu$, provided that the integral is finite for some (hence all) $p$, a condition met automatically by all measures $\mu_Q$ considered below. If $(M,g)$ is complete, which we will henceforth assume, then ${\rm FM}(\mu)$ is nonempty for any such $\mu$ [@bhatpat1 Theorem 2.1]. Elements of ${\rm FM}(\mu)$ are called *Fréchet means*, or *Riemannian centers of mass*, of $\mu$. (In this paper, the term *barycenter* will be used for something more general, defined later.) For $q\in M$ let $\delta_q$ be the Dirac measure with support $\{q\}$, and for $N\geq 1$ let $M^N$ denote the $N$-fold Cartesian product of $M$ with itself. For an $N$-tuple (or *configuration*) of points $Q=(q_1,\dots, q_N)\in M^N$, by the *Fréchet mean set of $Q$* we will mean ${\rm FM}(Q):={\rm FM}(\mu_Q)$, where $\mu_Q=\frac{1}{N} \sum_{i=1}^N\delta_{q_i}$ (the notation $\mu_Q$ will be used with this meaning throughout the paper); similarly, a *Fréchet mean of $Q$* will mean an element of ${\rm FM}(Q)$. For random $N$-tuples $Q$ representing i.i.d. samples of an underlying probability measure $\mu$, the Fréchet means $\mu_Q$ are called *Fréchet sample means* of $\mu$. If $\mu$ is given by a continuous probability density function, one would expect the Fréchet sample means of $\mu$ to behave "generically"; e.g. the probability that they lie in a positive-codimension submanifold should be zero for any $N\geq 1$. Surprisingly, even this much genericity does not seem to have been proven in the literature, except in special cases. The main result of this paper is a general, and stronger, genericity property: **Theorem 1**. *Let $X_1,\dots, X_N$ be i.i.d. absolutely continuous $M$-valued random variables. If $A\subset M$ has volume zero in $M$, then $$\label{eq:thm1.1} {\rm Pr}\left\{{\rm FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$$ In particular, [\[eq:thm1.1\]](#eq:thm1.1){reference-type="eqref" reference="eq:thm1.1"} holds if $A$ is a countable union of submanifolds of positive codimension.* (Terminology in this theorem is discussed the end of this section.) Fréchet means are not always unique (i.e. the Fréchet mean set may have more than one element). In [@afsari Theorem 2.1], Afsari established what is, to date, the least-stringent "concentration" condition known to guarantee uniqueness. Combining this with Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}, we obtain a corresponding sufficient condition for $N$ i.i.d. random variables under which, with probability 1, the Fréchet sample mean is unique and lies in the complement of a given volume-zero subset of $M$. Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} and this uniqueness result are restated together as Corollary [Corollary 28](#cor:measzero.3){reference-type="ref" reference="cor:measzero.3"}. Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} has an "equivariant" companion, in which $(M,g)$ arises as the quotient of a Riemannian manifold $({\widetilde{M}},\tilde{g})$ by a free, isometric action of a finite group $G$. (Thus $({\widetilde{M}},\tilde{g})$ is a finite-degree principal Riemannian cover of $(M,g)$; see Remark [Remark 30](#princ_rc){reference-type="ref" reference="princ_rc"}.) In this setting we consider a hybrid "equivariant distance function" $d_{\rm evt}: M\times {\widetilde{M}}\to{\bf R}$, defined by $d_{\rm evt}(q,\tilde{p}) =\tilde{d}({\rm quo}^{-1}(q),\tilde{p})$, the distance in $({\widetilde{M}},\tilde{g})$ between the point $\tilde{p}\in{\widetilde{M}}$ and the $G$-orbit ${\rm quo}^{-1}(q)\subset{\widetilde{M}}$, where ${\rm quo}:{\widetilde{M}}\to M$ is the quotient map. (Equivalently, $d_{\rm evt}(q,\tilde{p})=d(q,{\rm quo}(\tilde{p}))$.) For a given $Q\in M^N$, we can lift the Fréchet objective function $f_Q=\frac{1}{N}\sum_{i=1}^N d(\cdot,q_i)^2 :M\to{\bf R}$ to a function $\tilde{f}_Q:{\widetilde{M}}\to{\bf R}$ by replacing $d(\cdot, p)$ by $d_{\rm evt}(\cdot, \tilde{p})$. We call minimizers of such a function *equivariant Fréchet means* of $Q$ (a special case of "$\rho$-means" as extended by Huckemann [@huck3DProcrustes2011; @Huckemann2011consistency]; if $\tilde{p}$ is a minimizer, then so is $h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}\tilde{p}$ for every $h\in G$. For $Q\in M^N$, let ${\rm EFM}(Q)\subset{\widetilde{M}}$ denote the set of equivariant Fréchet means of $Q$. We prove the following result about equivariant Fréchet means of i.i.d. random variables (essentially an equivariant version of Corollary [Corollary 28](#cor:measzero.3){reference-type="ref" reference="cor:measzero.3"}, mentioned above): **Theorem 2**. *Assume that the finite group $G$ acts freely and isometrically on the Riemannian manifold $({\widetilde{M}}, \tilde{g})$, with quotient $(M,g)$ and ${\rm quo}:{\widetilde{M}}\to M$ the quotient map. Let $X, X_1, X_2,\dots, X_N$ be i.i.d. $M$-valued random variables, with induced probability measure $\mu_X$ . Then:* - *${\rm EFM}(\{X_1,\dots,X_N\})={\rm quo}^{-1}({\rm FM}(\{X_1,\dots,X_N\}).$* - *Let $\tilde{A}\subset{\widetilde{M}}$ be a $G$-invariant set of volume zero in ${\widetilde{M}}$, let $A={\rm quo}(\tilde{A})$ and write ${\widetilde{M}}_*={\widetilde{M}}\setminus\tilde{A}$ and $M_*={\rm quo}({\widetilde{M}}_*)=M\setminus A.$ If $X$ is absolutely continuous then, almost surely, ${\rm FM}(\{X_1,\dots,X_N\})\subset M_*$ and ${\rm EFM}(\{X_1,\dots,X_N\})\subset {\widetilde{M}}_*.$* - *Under the hypotheses of part (b), if $\mu_X$ is supported in an open ball $B$ of radius less than a certain number ${r_{\rm cx}}(M,g)$ *(specified in Definition [Definition 18](#def:rinj_rcx){reference-type="ref" reference="def:rinj_rcx"})*, then almost surely the Fréchet mean set of $\{X_1,\dots,X_N\}$ is a single point and lies in $M_*\cap B$. Correspondingly, under the same hypotheses, almost surely the equivariant Fréchet means of $\{X_1,\dots,X_N\}$ are unique up to the action of $G$ and lie in ${\widetilde{M}}_*\cap{\rm quo}^{-1}(B)$.* Our strategy for proving Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is to deduce it from a purely geometric (and quite general) result that constitutes the core mathematics in this paper: **Theorem 3**. *If $A\subset M$ has volume zero in $M$ , then the set $$\{Q\in M^N : {\rm FM}(Q) \cap A\neq\emptyset\}$$ has volume zero in $M^N$. In particular, this holds if $A$ is a countable union of submanifolds of positive codimension.* Since Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is derived as a consequence of Theorem [Theorem 3](#thm:FMmeaszero){reference-type="ref" reference="thm:FMmeaszero"}, much of this paper is, correspondingly, purely geometric. Our approach is to turn the question answered by Theorem [Theorem 3](#thm:FMmeaszero){reference-type="ref" reference="thm:FMmeaszero"} into a *transversality* problem by using "pre-barycenters". These are pre-images of barycenters in the bundle $(TM)^{\times N}$, the $N$-fold fiberwise product of $TM$ with itself. (We use the term *barycenter* in the "balancing point" sense in this paper.) The intuition behind Theorem [Theorem 3](#thm:FMmeaszero){reference-type="ref" reference="thm:FMmeaszero"} is that if we perturb a sample $Q_0$ to a sample $Q$ that moves over an open neighborhood of $Q_0$---or even if we hold all but one of the sample points fixed, and allow the remaining point to vary over an open neighborhood---the barycenters should *move*, varying over an open set themselves. In Sections [2](#sec:bary-prebary){reference-type="ref" reference="sec:bary-prebary"}--[4](#sec:barymanifold){reference-type="ref" reference="sec:barymanifold"} we define and discuss barycenters and pre-barycenters; show that the set ${\mathcal{PB}}_N$ of *pre-barycentered configurations* is a submanifold of the bundle $(TM)^{\times N}$ (part of Proposition [Proposition 9](#more_submersions){reference-type="ref" reference="more_submersions"}(a)); prove a forerunner of Theorem [Theorem 3](#thm:FMmeaszero){reference-type="ref" reference="thm:FMmeaszero"}, a genericity property of barycenters rather than Fréchet means (Theorem [Theorem 11](#thm:measzero.7){reference-type="ref" reference="thm:measzero.7"}); and establish some properties of the set of ${\mathcal B}_N$ of *barycentered configurations* (the set ${\mathcal B}_N := \{(Q,p): \mbox{$p$ is a barycenter of $Q$}\}\subset M^N\times M\}$). Among the latter properties is that the set of *sufficiently concentrated* barycentered configurations is a submanifold of $M^N\times M$ (part of Corollary [Corollary 21](#cor:conFMbary){reference-type="ref" reference="cor:conFMbary"}), a property that we do not know or expect of the entire set ${\mathcal{PB}}_N$ in general. In Section [5](#sec:FMs_are_barys){reference-type="ref" reference="sec:FMs_are_barys"}, building on work in [@le-barden2014], we show that all Fréchet means of finite configurations are barycenters (Corollary [Corollary 27](#cor:fmisbary){reference-type="ref" reference="cor:fmisbary"}). This fact, combined with Theorem [Theorem 11](#thm:measzero.7){reference-type="ref" reference="thm:measzero.7"}, yields Theorem [Theorem 3](#thm:FMmeaszero){reference-type="ref" reference="thm:FMmeaszero"}. In Section [6](#sec:iid1){reference-type="ref" reference="sec:iid1"} we begin applying our geometric results to i.i.d. random variables. Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} follows quickly from Theorem [Theorem 3](#thm:FMmeaszero){reference-type="ref" reference="thm:FMmeaszero"}. As mentioned earlier, we apply this together with [@afsari]'s Theorem 2.1 to obtain, additionally, the almost-sure uniqueness of Fréchet sample means (under appropriate hypotheses). In Section [6](#sec:iid1){reference-type="ref" reference="sec:iid1"} we also discuss an example that illustrates the generality of Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}, and also give some examples of interest in which Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} does *not* apply. Section [7](#sec:eqvt_means_general){reference-type="ref" reference="sec:eqvt_means_general"} is pure geometry again: we examine finite-degree principal Riemannian covers $({\widetilde{M}}, \tilde{g})\stackrel{{\rm quo}}{\longrightarrow}(M,g)$ and equivariant means. For a finite configuration $Q$, we show that taking equivariant Fréchet means in ${\widetilde{M}}$ is "equivalent" to taking Fréchet means in $M$, in the sense that ${\rm EFM}(Q)={\rm quo}^{-1}({\rm FM}(Q))$ (Proposition [Proposition 33](#prop:uniqueuptoG){reference-type="ref" reference="prop:uniqueuptoG"}(a)). We also establish relations between the numbers ${r_{\rm cx}}({\widetilde{M}},\tilde{g})$ and ${r_{\rm cx}}(M,g)$. We are then set up to address Fréchet sample means in the equivariant setting in Section [8](#sec:iid_eqvt_means){reference-type="ref" reference="sec:iid_eqvt_means"}, where we prove Theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} (as Corollary [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"}). In Section [9](#sec:PSRmeans){reference-type="ref" reference="sec:PSRmeans"} we discuss an application of our equivariant-Frechet-means results to "partial scaling-rotation means", a type of mean for symmetric positive-definite matrices that the authors introduced in [@rjgs]. Some remarks concerning the terminology "volume zero", "generic", "absolutely continuous", and "$M$-valued random variable" are worthwhile here. The notion of *measure-zero subset of a manifold ${\mathcal M}$* (see e.g. [@lee_mfds Ch. 6] or [@hirsch Ch. 3]) is independent of any measures that one is putting on ${\mathcal M}$, but coincides with "set of ($n$-dimensional) Lebesgue measure zero" when ${\mathcal M}={\bf R}^n$. To avoid any confusion with probability measures under consideration, we will refer to a measure-zero subset of a manifold ${\mathcal M}$ (in the sense above) as having *volume zero* in ${\mathcal M}$; we call the complement of such a subset *generic* (in ${\mathcal M}$). Motivation for the "volume zero" terminology is that every Riemannian metric $g$ on ${\mathcal M}$ determines a *volume measure* whose null sets are precisely the measure-zero sets in the sense above. However, the above notions of "volume zero" and "genericity"---*residuality* in the terminology of [@hirsch Ch. 3]---are at heart more topological than measure-theoretic, making sense even without specifying a measure on $M$. Every generic subset of ${\mathcal M}$ is dense, and every positive-codimension submanifold of ${\mathcal M}$ of has volume zero in ${\mathcal M}$. We call a measure $\mu$ on a manifold ${\mathcal M}$ *absolutely continuous* if $\mu(A)=0$ whenever $A\subset {\mathcal M}$ has volume zero in ${\mathcal M}$. Every measure on ${\mathcal M}$ given by a continuous density function is absolutely continuous. Every manifold ${\mathcal M}$ has two "natural" sigma-algebras of (potentially) measurable sets: the sigma-algebra ${\mathcal B}_{\mathcal M}$ of Borel sets, and its completion ${\mathcal L}_{\mathcal M}$, the "Lebesgue sigma-algebra"---an analog of the sigma-algebra of Lebesgue measurable subsets of ${\bf R}^n$---which can be obtained by adding to ${\mathcal B}_{\mathcal M}$ the collection of all volume-zero subsets of ${\mathcal M}$. To have a clear meaning of "measurable map" (a map for which the inverse image of every measurable set is measurable), with domain or codomain ${\mathcal M}$, we must specify which of these sigma-algebras on ${\mathcal M}$ we are using. It is desirable to make the same specification consistently, across all manifolds, to ensure that the composition of measurable maps is measurable. Since volume-zero sets occur throughout this paper, we choose the Lebesgue sigma-algebra on every manifold. (This choice differs from the customary definition of "measurable function $f:{\bf R}\to{\bf R}$", in which $f$ is called measurable if the inverse image of every *Borel* set is *Lebesgue* measurable.) In this paper we will take the domain of every random variable that appears to be a (fixed but arbitrary) complete probability space $(\Omega,\mathcal{A},\mathcal{P})$. An ${\mathcal M}$-valued random variable is then an $(\mathcal{A},{\mathcal L}_{\mathcal M})$-measurable map $X:\Omega\to {\mathcal M}.$ We call $X$ *absolutely continuous* if its induced measure $\mu_X$ on ${\mathcal M}$ is absolutely continuous. For such $X$, it actually makes no difference whether we choose ${\mathcal L}_{\mathcal M}$ or ${\mathcal B}_{\mathcal M}$ as our sigma-algebra on ${\mathcal M}$: if $X:\Omega\to {\mathcal M}$ is an $(\mathcal{A},{\mathcal B}_{\mathcal M})$-measurable map whose induced measure $\mu_X$ on ${\mathcal M}$ is absolutely continuous, then $X$ is also $(\mathcal{A},{\mathcal L}_{\mathcal M})$-measurable. Throughout this paper, $n=\dim(M),$ and $N$ is an arbitrary positive integer. For $Q=(q_1,\dots,q_N)\in M^N$ we write ${\rm supp}(Q)$ for the (unordered) multi-set $\{q_1,\dots,q_N\}$; when the points are distinct this coincides with the support of the measure $\mu_Q$. For $p\in M$, the zero element of $T_pM$ is sometimes denoted $0_p$ for clarity's sake, and sometimes simply denoted $0$. As is customary, "$\exp$" denotes the exponential map $TM\to M$, while "$\exp_p$" denotes the restriction of $\exp$ to $T_pM$. In any metric space that is understood from context, $B_r(q)$ and $\bar{B}_r(q)$ denote, respectively, the open and closed balls of radius $r$ centered at $q$; sometimes this notation may be augmented to include some reference to the metric space of interest. To avoid technical distractions, throughout we take "smooth" to mean $C^\infty$, and "(sub)manifold" to mean "smooth (sub)manifold". These restrictions of "smooth" and "(sub)manifold" can be relaxed; see Remark [Remark 22](#rem:findiff){reference-type="ref" reference="rem:findiff"}. # Barycenters and pre-barycenters {#sec:bary-prebary} In this paper, we take *barycenter* of a configuration $Q=(q_1,\dots, q_N)\in M^N$ to mean a "balancing point" (suitably defined) for the configuration. The simplest example of such a barycenter is a point $p$ for which no $q_i$ lies in the cut-locus of $p$ (thereby allowing a reasonable definition of "$(\exp_p)^{-1}(q_i)$") and for which $\sum_{i=1}^N (\exp_p)^{-1}(q_i) =0$. This example is what we will soon be calling a *short barycenter*. To give a precise and more general definition of "barycenter" and related notions, we recall several facts pertaining to the exponential map and cut-loci, and fix some notation. All these facts can be found in standard books on Riemannian geometry, e.g. [@chavel; @sakai], except for those facts for which we give specific references. Let $S(TM)$ denote the unit tangent bundle of $(M,g)$, and for $p\in M$ let $S(T_pM)=S(TM)\cap T_pM$. For $v\in TM$ let $\gamma_v$ be the geodesic $t\mapsto \exp(tv)$. The *cut-time* function $\tau:S(TM)\to [0,\infty]$ is defined by $\tau(v)=\sup\{t\geq 0: \gamma_v|_{[0,t]}\ \mbox{is minimal}\}$. Recall that the *tangent cut locus* ${\mathcal{C}}^{\rm tan}(p)$ and *cut locus* ${\mathcal{C}}(p)$ of $p\in M$ are defined by ${\mathcal{C}}^{\rm tan}(p)=\{\tau(v)v: v\in S(T_pM), \tau(v)<\infty\}$ and ${\mathcal{C}}(p)=\exp_p({\mathcal{C}}^{\rm tan}(p))$. Following [@bishop], we call a cut-point $q\in {\mathcal{C}}(p)$ *ordinary* or *singular* accordingly as there are at least two minimal geodesics from $p$ to $q$, or only one, and classify $v\in {\mathcal{C}}^{\rm tan}(p)$ as an ordinary or singular accordingly as $\exp_p(v)$ is an ordinary or singular cut-point of $p$. We will write ${\mathcal{C}}_{\rm ord}(p)$ and ${\mathcal{C}}_{\rm sing}(p)$ for the set of ordinary and singular cut-points of $p$, respectively, and ${\mathcal{C}}_{\rm sing}^{\tan}(p)$ for the set of singular points in the tangent cut-locus of $p$. Note that any of these sets can be empty, and that the relations "$q\in {\mathcal{C}}_{\rm ord}(p)$" and "$q\in{\mathcal{C}}_{\rm sing}(p)$" are symmetric in $p$ and $q$. Still for arbitrary $p\in M$, define $$\begin{aligned} \hspace{-1.25in} {\mathcal D}^{\rm tan}(p)&=&\{tv: v\in S(T_pM), t\in [0,\tau(v))\},\\ \tilde{{\mathcal D}}^{\rm tan}(p)&=&{\mathcal D}^{\rm tan}(p)\,\mbox{\small $\bigcup$}\,{\mathcal{C}}_{\rm sing}^{\tan}(p),\\ {\mathcal D}(p)&=&\exp_p({\mathcal D}^{\rm tan}(p)), \\ \mbox{and}\hspace{.75in} \tilde{{\mathcal D}}(p)&=& \exp_p(\tilde{{\mathcal D}}^{\rm tan}(p)) \ =\ {\mathcal D}(p)\,\mbox{\small $\bigcup$}\,{\mathcal{C}}_{\rm sing}(p). \hspace{.75in}\end{aligned}$$ (Thus $\tilde{{\mathcal D}}(p)$ is precisely the set of points $q\in M$ for which there is a unique minimal geodesic from $p$ to $q$.) The set ${\mathcal D}^{\rm tan}(p)$ is open, and the map $\exp_p$ is injective on $\tilde{{\mathcal D}}^{\rm tan}(p)$ and nonsingular on ${\mathcal D}^{\rm tan}(p)$. The set $\tilde{{\mathcal D}}^{\rm tan}(p)$ is the largest subset $U\subset T_pM$, star-shaped with respect to $0_p$, such that $\exp_p|_U$ is injective; ${\mathcal D}^{\rm tan}(p)$ is the largest *open* subset $U\subset T_pM$, star-shaped with respect to $0_p$, such that $\exp_p|_U$ is a diffeomorphism onto its image. In particular, ${\mathcal D}(p)$ is open, $\exp_p|_{{\mathcal D}^{\rm tan}(p)}:{\mathcal D}^{\rm tan}(p)\to {\mathcal D}(p)$ is a diffeomorphism, and $M={\mathcal D}(p)\,{\mbox{\small $\coprod$}}\,{\mathcal{C}}(p)$. We define the notation $(\exp_p)^{-1}$ to mean the inverse of the bijective map $\exp_p|_{\tilde{{\mathcal D}}^{\rm tan}(p)}: \tilde{{\mathcal D}}^{\rm tan}(p)\to\tilde{{\mathcal D}}(p)$. Note that $(\exp_p)^{-1}$ may not be continuous at points of ${\mathcal{C}}_{\rm sing}(p)$, but the restriction of $(\exp_p)^{-1}$ to the open set ${\mathcal D}(p)$ is a diffeomorphism. The open set ${\mathcal D}(p)$ is dense in $M$, and its complement ${\mathcal{C}}(p)$ has measure zero. The distance function $r_p$ is never $C^1$ at an ordinary cut-point of $p$, and never $C^2$ at any cut-point of $p$ [@bishop], but $r_p^2$ is smooth on ${\mathcal D}(p)$. Our definition of *barycenter* is now as follows. *Definition 4*. Let $Q=(q_1,\dots,q_N)\in M^N$ and let $p\in M$. We call $p$ a *barycenter of $Q$* if there exist $v_1,\dots, v_N\in T_pM$ such that $q_i=\exp_p(v_i), 1\leq i\leq N$, and $\sum_i v_i=0$. We call $p$ a *short* barycenter of $Q$ if there exist such $v_1,\dots,v_N \in {\mathcal D}^{\tan}(p)$, and an *almost-short* barycenter of $Q$ if there exist such $v_1,\dots,v_N \in \tilde{{\mathcal D}}^{\tan}(p)$. We will write ${\mathcal B}(Q)$ for the set of barycenters of $Q$, and ${\mathcal B}^0(Q)$ (respectively, $\tilde{{\mathcal B}}^0(Q)$) for the set of short (resp., almost-short) barycenters of $Q$. We define $${\mathcal B}_N ={\mathcal B}_N(M)=\{(Q,p)\in M^N\times M \ : \ p\in {\mathcal B}(Q)\},$$ the set of *barycentered configurations of $N$ points in $M$*. Similarly, we define ${\mathcal B}_N^0=\{(Q,p)\in M^N\times M \ : \ p\in {\mathcal B}^0(Q)\}.$ Note that ${\mathcal B}(Q), {\mathcal B}^0(Q)$, and $\tilde{{\mathcal B}}^0(Q)$ are subsets of $M$, while ${\mathcal B}_N$ and ${\mathcal B}_N^0$ are subsets of $M^N\times M$. Also important will be the notion of *pre-barycenter*. To define this, let $(TM)^{\times N}$ denote the fiber bundle over $M$ whose fiber over $p\in M$ is the $N$-fold Cartesian product $(T_pM)^N$. (The bundle-structure is the same as that of the $N$-fold Whitney sum of $TM$ with itself, but we are regarding the fiber simply as a Cartesian product rather than a direct sum.) We will write $\pi:TM\to M$ and $\pi':(TM)^{\times N}\to M$ for the projection-maps of these bundles. Thus, elements of $(TM)^{\times N}$ may be written as $V=(v_1,\dots, v_N) \in TM\times \dots\times TM$ with $\pi(v_1)=\pi(v_2)=\dots=\pi(v_N)$, and we have $\pi'(V)=\pi(v_i), 1\leq i\leq N$. Note that $\dim((TM)^{\times N})=(N+1)n=\dim(M^N\times M)$. *Definition 5*. Define $${\mathcal{PB}}_N={\mathcal{PB}}_N(M)=\{\ (v_1,\dots,v_N)\in(TM)^{\times N}: \sum_i v_i=0\ \}.$$ We call elements of ${\mathcal{PB}}_N$ *pre-barycenters* (of configurations of $N$ points in $M$). We define the map $\exp^N:(TM)^{\times N}\to M^N$ by $\exp^N(v_1,\dots,v_N)=(\exp(v_1),\dots,\exp(v_N))$. If $V= (v_1,\dots,v_N)\in {\mathcal{PB}}_N$ and $Q=\exp^N(V)$, we say that $V$ is a pre-barycenter of $Q$. For $Q\in M^N$ we write ${\mathcal{PB}}(Q)$ for the set of pre-barycenters of $Q$. From Definitions [Definition 4](#def:bary){reference-type="ref" reference="def:bary"} and [Definition 5](#def:prebary){reference-type="ref" reference="def:prebary"} it is clear that for all $Q\in M^N$, $$\label{baryprebary} {\mathcal B}(Q) = \pi'({\mathcal{PB}}(Q)).$$ *Remark 6*. We are using *ordered* $N$-tuples $Q=(q_1,\dots,q_N)$ to take advantage of the manifold structures of $M^N$ and $(TM)^{\times N}$. Clearly the set ${\mathcal B}(Q)$ depends only on the measure $\mu_Q$, not on how the points $q_i$ are ordered. *Remark 7*. Our definitions of *barycenter* and *pre-barycenter* of a configuration are closely related to the definition of *barycentre of a (Borel) probability measure* in [@corcuera], which for purposes of distinction with our definition we will call a CK-barycenter. Given a measure $\mu$ on $M$ and a point $p\in M$, we call a measure $\tilde{\mu}$ on $T_pM$ a *lift* of $\mu$ if $\mu$ is the push-forward, under $\exp_p$, of $\tilde{\mu}$. A point $p\in M$ is a CK-barycenter of a probability measure $\mu$ if $\mu$ lifts to some probability measure $\tilde{\mu}$ on $T_pM$ having zero "vector mean", i.e. satisfying $\int_{T_pM} v\,d\tilde{\mu}(v)=0$. For $V=(v_1,\dots,v_N)\in (TM)^{\times N}$, $p=\pi'(V)$, and $Q=\exp^N(V)$, the probability measure $\mu_V:=\frac{1}{N}\sum_i \delta_{v_i}$ on $T_pM$ is a lift of $\mu_Q$. Thus if $V$ is a pre-barycenter of $Q$, or $p$ is a barycenter of $Q$, then $p$ is a CK-barycenter of $Q$. Our pre-barycenters of $Q$ are a proxy for the probability measures of the form $\mu_V$ on $T_pM$ to which $\mu_Q$ may lift. However, our definition of "barycenter of $Q$" may be more restrictive than "CK-barycenter of $\mu_Q$", since for $p$ to be a barycenter of $Q$ we require that $\mu_Q$ lift not just to an arbitrary vector-mean-zero probability measure on $T_pM$, but to one of the form $\mu_V$ above. (A general lift of $\mu_Q$ need not even be discrete; if discrete, it may have more than $N$ points, and the weights of certain points need not be equal.) Taking the liberty of writing elements of $M^N\times M$ as elements of $M^{N+1}$, we define subsets $\widetilde{{\mathcal M}}_1, {\mathcal M}_1$ of $M^N\times M$ by $$\begin{aligned} \label{defm4} \widetilde{{\mathcal M}}_1&=&\{(q_1,\dots,q_N,p)\in M^N\times M \ : \ q_i\in \tilde{{\mathcal D}}(p), \ 1\leq i\leq N\},\\ {\mathcal M}_1&=&\{(q_1,\dots,q_N,p)\in M^N\times M \ : \ q_i\in {\mathcal D}(p), \ 1\leq i\leq N\},\end{aligned}$$ and define a map $F:\widetilde{{\mathcal M}}_1\to TM$ by $$\begin{aligned} \nonumber F(q_1,\dots, q_N,p) &=&\frac{1}{N}\sum_{i=1}^N (\exp_p)^{-1}(q_i)\ \ \ \in T_pM. \label{defFQp}\end{aligned}$$ The set ${\mathcal M}_1$ is open and dense in $M^N\times M$, and the restricted map $F|_{{\mathcal M}_1}$ is smooth. *Notation 8*.   1. Let $\pi_N : M^N\times M\to M^N, {\pi_{\rm last}}: M^N\times M\to M$ denote projection onto the first and second factors of $M^N\times M$. We use the same notation for the restrictions of $\pi_N, {\pi_{\rm last}}$ to ${\mathcal M}_1$ and $\widetilde{{\mathcal M}}_1$. Observe that ${\pi_{\rm last}}({\mathcal M}_1)=M$. 2. For $Q=(q_1,\dots, q_N)\in M^N$, define the following: 1. $f_Q=\frac{1}{N}\sum_{i=1}^N r_{q_i}^2: M\to{\bf R}$. 2. $M_Q=\{p\in M: (Q,p)\in {\mathcal M}_1\}=\{p\in M\ : \ {\rm supp}(Q)\subset {\mathcal D}(p)\}= {\pi_{\rm last}}(\pi_N^{-1}({Q})\cap {\mathcal M}_1)$. Since the condition ${\rm supp}(Q)\subset {\mathcal D}(p)$ is equivalent to $p\in \bigcap_{i=1}^N {\mathcal D}(q_i)$, and since ${\mathcal D}(q_i)$ is open and dense in $M$ for each $i$, the set $M_Q$ is open and dense as well. 3. $\widetilde{{\mathcal M}}_Q=\{p\in M: (Q,p)\in \widetilde{{\mathcal M}}_1\}=\{p\in M\ : \ {\rm supp}(Q)\subset \tilde{{\mathcal D}}(p)\}= {\pi_{\rm last}}(\pi_N^{-1}({Q})\cap \widetilde{{\mathcal M}}_1)$. 4. Define a vector field $Y_Q$ on $M_Q$ by $Y_Q(p)=F(Q,p)$. It is well known that $f_Q$ is smooth on $M_Q$ and that the vector field $Y_Q$ is the negative gradient of $f_Q|_{M_Q}$ (see e.g. [@karcher]). Thus the following are equivalent: (i) $p$ is a short barycenter of $Q$; (ii) $p$ is a critical point of $f_Q$ in $M_Q$; and (iii) $Y_Q(p)=0$. In particular, ${\mathcal B}^0(Q)$ contains all Fréchet means of $Q$ lying in $M_Q$. We will see later that ${\mathcal B}(Q)$ contains the entire set ${\rm FM}(Q)$; in fact ${\rm FM}(Q)\subset \tilde{{\mathcal B}}^0(Q)$. This nontrivial fact, which follows from results in [@le-barden2014], will essentially allow us to deduce Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} from a stronger conclusion in which ${\rm FM}(\{X_1,\dots,X_N\})$ in [\[eq:thm1.1\]](#eq:thm1.1){reference-type="eqref" reference="eq:thm1.1"} is replaced by ${\mathcal B}(\{X_1,\dots,X_N\})$. For efficiency of presentation we structure the argument slightly differently, restating and proving Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} as Corollary [Corollary 28](#cor:measzero.3){reference-type="ref" reference="cor:measzero.3"} (a). # Genericity of barycenters {#sec:bary-prebary2} At the heart of Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is the question of how the set ${\rm FM}(Q)$ varies as we vary $Q$. Rather than approaching this directly, our strategy will be to analyze perturbations of (pre-)barycenters and to show that from this analysis, the desired facts about Fréchet means can be deduced. Perturbation analysis is easier for barycenters, short barycenters, and pre-barycenters than (directly) for Fréchet means since barycenters (and short and pre-barycenters) are defined as solutions of *equations* on open subsets of manifolds, putting us in the realm in which we can hope to apply the Implicit Function Theorem in some guise (e.g. the Regular Value Theorem or transversality). This is why we have defined the sets ${\mathcal B}_N, {\mathcal B}_N^0,$ and ${\mathcal{PB}}_N$---parametrized families of (pre-)barycenters, parametrized by the $N$-tuple $Q$. As we shall see shortly, ${\mathcal{PB}}_N$ is a manifold, and the family of barycenter-sets ${\mathcal B}(Q)$ enjoys a genericity property. To this end, writing elements of $(TM)^{\times N}$ as $V=(v_1,\dots,v_N)$, we define maps $\tilde{F}:(TM)^{\times N}\to TM$ and ${H}:(TM)^{\times N}\to M^N\times M$ by $$\begin{aligned} \tilde{F}(V)&=&\sum_i v_i\ ,\\ {H}(V)&=&(\exp^N(V), \pi'(V)).\end{aligned}$$ Henceforth let $Z$ denote the zero-section of $TM$ (as a submanifold of $TM$). Observe that $$\label{finvz-2} {\mathcal B}_N^0= (F|_{{\mathcal M}_1})^{-1}(Z)\ \ \ \mbox{and}\ \ \ {\mathcal{PB}}_N=\tilde{F}^{-1}(Z)=: \tilde{Z}.$$ (We use the notation $\tilde{Z}$ for ${\mathcal{PB}}_N$ to simplify other notation below.) The commutative diagram in Figure [\[comdiag2\]](#comdiag2){reference-type="ref" reference="comdiag2"} depicts relations among various spaces and maps we have defined. Note that if we define $$\label{defm4'} {\mathcal M}_1'=\{V=(v_1,\dots,v_N)\in (TM)^{\times N}\ : \ v_i\in {\mathcal D}^{\rm tan}(\pi'(V)), \ 1\leq i\leq N\},$$ then $\tilde{F}|_{{\mathcal M}_1'\cap \tilde{Z}}$ is the inverse of the natural injection $$\begin{aligned} \nonumber {\mathcal B}_N^0 &\to& \tilde{Z}, \\ (q_1,\dots, q_N,p) & \mapsto & ((\exp_p)^{-1}(q_1), \dots, (\exp_p)^{-1}(q_N)). \label{includebary}\end{aligned}$$ c & & & c Z\ \ TM & & & &\ & & (2,4)\<& \> l \   &(3,4)\>F & &&\ & & & M & & &&\ & &\>'& &\<\_last& &&\ =\_N  & (TM)\^N&& \_H & & M\^NM & \_1 &\_1 \_N\^0\ & & \<\^N & &\>\_N & &&\ & & & M\^N & & &&\ In the next proposition and beyond, recall that $n=\dim(M)$. **Proposition 9**. *(a) The map $\tilde{F}$ is a submersion, and $\tilde{Z}$---the set of pre-barycentered configurations of $N$ points in $M$---is a submanifold of $(TM)^{\times N}$ of dimension $Nn$.* *(b) The map $\pi'|_{\tilde{Z}}:\tilde{Z}\to M$ is a submersion.* The horizontal/vertical splitting of the tangent spaces of $TM$ and $(TM)^{\times N}$, induced by the Riemannian structure, enters the proof of Proposition [Proposition 9](#more_submersions){reference-type="ref" reference="more_submersions"}, so we review these splittings before proceeding with the proof. For each $v\in TM$, the *vertical space* ${\rm Vert}_v(TM)\subset T_v(TM)$ is defined to be $\ker(\pi_{*v})$. Letting $p=\pi(v)$, there is a canonical isomorphism $\iota_v: T_pM\to {\rm Vert}_v(TM)$ defined by $\iota_v(w)=\frac{d}{dt}(v+tw)|_{t=0}$. The Levi-Civita connection $\nabla$ determines a complement of ${\rm Vert}_v(TM) \subset T_v(TM)$, the *horizontal space* ${\rm Hor}_v(TM)$, also isomorphic to $T_pM$; the map $\pi_{*v}|_{{\rm Hor}_v(TM)}:{\rm Hor}_v(TM)\to T_pM$ is an isomorphism. Given a curve based at $p$---a smooth map $\gamma:I\to M$ where $I$ is an interval containing $0$ and where $\gamma(0)=p$---the *horizontal lift* of $\gamma$ starting at $v$ is the curve $\tilde{\gamma}:I\to TM$ defined by $\tilde{\gamma}(t)=\mathcal{P}_{\gamma(0)\to \gamma(t)} v$, the parallel transport of $v$ along $\gamma$ from time $0$ to time $t$. Given $X\in T_pM$, the *horizontal lift* of $X$ to $TM$ at $v$ is the vector $\tilde{X}_v=\tilde{\gamma}'(0)$, where $\gamma$ is any curve with $\gamma'(0)=X$. The space ${\rm Hor}_v(TM)$ is the set of all horizontal lifts of elements of $T_pM$, and the map $X\mapsto\tilde{X}_v$ is the inverse of the isomorphism $\pi_{*v}|_{{\rm Hor}_v(TM)}$. The Levi-Civita connection similarly induces a horizontal/vertical splitting of the tangent spaces of $(TM)^{\times N}$. Let $V=(v_1,\dots,v_N)\in (TM)^{\times N}$ and let $p=\pi'(V)$. Viewing $(TM)^{\times N}$ as the restriction to the diagonal of $N$-fold Cartesian product of $TM$ with itself (a bundle over $M^N$), the tangent space $T_V((TM)^{\times N})$ may be canonically identified with a subspace of $\bigoplus_{i=1}^N T_{v_i}(TM)$, namely $\{(w_1,\dots, w_N)\in \bigoplus_{i=1}^N T_{v_i}(TM) \ : \ \pi_{*v_1}w_1 = \pi_{*v_2}w_2=\dots=\pi_{*v_N}w_N\}$. The vertical subspace of $T_V((TM)^{\times N})$ is $$\begin{aligned} {\rm Vert}_V((TM)^{\times N}):=\ker(\pi'_{*V}) &=& \bigoplus_{i=1}^N{\rm Vert}_{v_i}(T_pM)\\ &=& \{(\iota_{v_1}(w_1), \dots, \iota_{v_N}(w_N)) \ : \ w_i\in T_pM\}\\ &\mbox{\hspace{1ex}}\cong_{\mbox{\hspace{-5ex}\phantom{$\int$} \scriptsize{canon.}}}& \bigoplus_{i=1}^N T_pM.\end{aligned}$$ For $X\in T_pM$, we define the horizontal lift of $X$ to $(TM)^{\times N}$ at $V$ to be the vector $$\label{txv} \tilde{X}_V= (\tilde{X}_{v_1}, \dots, \tilde{X}_{v_N}) \in T_V((TM)^{\times N})\subset \bigoplus_{i=1}^N T_{v_i}(TM),$$ where $\tilde{X}_{v_i}$ is the horizontal lift of $X$ to $TM$ at $v_i$, $1\leq i\leq N$. Defining ${\rm Hor}_V((TM)^{\times N})$ to be the set of all horizontal lifts of elements of $T_pM$ to $(TM)^{\times N}$, it is easily seen that ${\rm Hor}_V((TM)^{\times N})$ is a vector subspace of $T_V((TM)^{\times N})$ and is a complement to ${\rm Vert}_V((TM)^{\times N})$. **Proof of Proposition [Proposition 9](#more_submersions){reference-type="ref" reference="more_submersions"}.** Let $V=(v_1,\dots,v_N)\in (TM)^{\times N}$ and let $p=\pi'(V)$. Decomposing $T_V((TM)^{\times N})$ as ${\rm Hor}_V((TM)^{\times N})\oplus{\rm Vert}_V((TM)^{\times N})$, let us write elements of $T_V((TM)^{\times N})$ in the form $(\tilde{X}_V, (\iota_{v_1}(w_1), \dots, \iota_{v_N}(w_N)))$, where $\tilde{X}_V$ is the horizontal lift at $V$ of $X\in T_pM$ and $w_1,\dots,w_N\in T_pM$. Then one easily computes $$\label{tf_*V} \tilde{F}_{*V}(\tilde{X}_V, (\iota_{v_1}(w_1), \dots, \iota_{v_N}(w_N))) =(\tilde{X}_{\tilde{F}(V)}, \iota_{\tilde{F}(V)}(\sum_i w_i))$$ where $\tilde{X}_{\tilde{F}(V)}\in {\rm Hor}_{\tilde{F}(V)}(TM)$ is the horizontal lift of $X$ to $TM$ at $\tilde{F}(V)$. Clearly the image of $\tilde{F}_{*V}$ is all of ${\rm Hor}_{\tilde{F}(V)}(TM)\oplus{\rm Vert}_{\tilde{F}(V)}(TM) = T_{\tilde{F}(V)}(TM)$. Hence $\tilde{F}$ is a submersion, and $\tilde{Z}=\tilde{F}^{-1}(Z)$ is a submanifold of $(TM)^{\times N}$ of codimension equal to the codimension of $Z$ in $TM$, namely $n$. Hence $\dim(\tilde{Z})=\dim((TM)^{\times N})-n =Nn$. Let $V=(v_1,\dots,v_N)\in \tilde{F}^{-1}(Z)$ and $p=\pi'(V)$; thus $\tilde{F}(V)=0_p\in T_pM$. Recall that ${\rm Hor}_{0_p}(TM)=T_{0_p}Z$. Hence using [\[tf\_\*V\]](#tf_*V){reference-type="eqref" reference="tf_*V"} we find $$\begin{aligned} \nonumber \lefteqn{T_V(\tilde{Z})=(\tilde{F}_{*V})^{-1}(T_{\tilde{F}(V)}Z)=(\tilde{F}_{*V})^{-1}({\rm Hor}_{0_p}(TM))}\\ &=& \{(\tilde{X}_{\tilde{F}(V)}, (\iota_{v_1}(w_1), \dots, \iota_{v_N}(w_N)))\in T_V((TM)^{\times N})\ : \ \sum_i w_i =0_p\}. \label{TVGinvZ}\end{aligned}$$ Note that ${\rm Hor}_{\tilde{F}(V)}(T_V((TM)^{\times N}))\subset T_V\tilde{Z}$ and that $\pi'_{*V}({\rm Hor}_V((TM)^{\times N})) =T_pM$; thus $\pi'_{*V}|_{T_V(\tilde{Z})}$ is surjective. Hence $\pi'|_{\tilde{Z}}$ is a submersion. height10pt width10pt\ Knowing that $\pi'_{\tilde{Z}}$ is a submersion (Proposition [Proposition 9](#more_submersions){reference-type="ref" reference="more_submersions"}(b)) will be valuable to us because of the following lemma. This lemma is almost certainly known, but the authors have found no reference for it. For a proof, see Appendix [\[sec:appendixA\]](#sec:appendixA){reference-type="ref" reference="sec:appendixA"}. **Lemma 10**. *Under a submersion, the inverse image of a set of volume zero has volume zero.* **Theorem 11**. *If $A\subset M$ has volume zero in $M,$ then the set $$\label{FMinverse(A),2} \{Q\in M^N : {\mathcal B}(Q) \cap A\neq\emptyset\}$$ has volume zero in $M^N$.* : Let $A$ be a volume-zero subset of $M$. By Proposition [Proposition 9](#more_submersions){reference-type="ref" reference="more_submersions"} and Lemma [Lemma 10](#submersions_and_meas_zero){reference-type="ref" reference="submersions_and_meas_zero"}, $(\pi'|_{\tilde{Z}})^{-1}(A)$ has volume zero in the $Nn$-dimensional manifold $\tilde{Z}$. Every smooth map between manifolds of the same dimension carries sets of volume zero to sets of volume zero [@lee_mfds]. In particular, this is true of the map $\pi_N\circ {H}|_{\tilde{Z}}:\tilde{Z}\to M^N$. Thus $\pi_N({H}((\pi'|_{\tilde{Z}})^{-1}(A)))$ has volume zero in $M^N$. Using Corollary [Corollary 27](#cor:fmisbary){reference-type="ref" reference="cor:fmisbary"}, observation [\[baryprebary\]](#baryprebary){reference-type="eqref" reference="baryprebary"}, and the fact that ${\pi_{\rm last}}\circ{H}=\pi'$, $$\begin{aligned} \nonumber \{Q\in M^N \ : \ {\mathcal B}(Q) \cap A\neq\emptyset\} \nonumber &=& \{Q\in M^N \ : \ \exists p\in A\ \mbox{such that}\ p\in {\mathcal B}(Q)\}\\ \nonumber &\subset& \{Q\in M^N \ : \ \exists p\in A\ \mbox{such that}\ p\in {\pi_{\rm last}}({\mathcal{PB}}(Q))\}\\ \nonumber &=& \{Q\in M^N \ : \ \exists p\in A\ \mbox{such that}\ (Q,p)\in {H}(\tilde{Z})\}\\ \nonumber &=& \pi_N\left({H}(\tilde{Z})\cap{\pi_{\rm last}}^{-1}(A)\right)\\ &=& \pi_N\left({H}\left((\pi'|_{\tilde{Z}})(A)\right)\right). \label{volzeroset}\end{aligned}$$ Hence the set [\[FMinverse(A),2\]](#FMinverse(A),2){reference-type="eqref" reference="FMinverse(A),2"} has volume zero in $M^N$. height10pt width10pt\ # Some properties of the set of barycentered configurations {#sec:barymanifold} Proposition [Proposition 9](#more_submersions){reference-type="ref" reference="more_submersions"}(a) shows that ${\mathcal{PB}}_N$, the set of *pre*-barycentered configurations, is a submanifold of $(TM)^{\times N}$. It appears unlikely that ${\mathcal B}_N$, the set of barycentered configurations, is a submanifold of $M^N\times M$. However, we shall see that the subset ${\mathcal B}_N^0$ *is* a submanifold of $M^N\times M$, and captures the barycenters of almost all configurations in the sense that $\pi_N({\mathcal B}_N\setminus{\mathcal B}_N^0)=M^N\setminus\pi_N({\mathcal B}_N^0)$ has volume zero in $M^N$. (We have $\pi_N({\mathcal B}_N)=M^N$ since ${\rm FM}(Q)\neq\emptyset$ for every $Q\in M^N$.) In proving these facts, two types of derivative of the vector fields $Y_Q$ will arise. One is the covariant derivative $\nabla Y_Q$, whose value at $p\in M$ is an endomorphism of $T_pM$, namely the map $v\mapsto \nabla_v Y_Q$. The other is the derivative of the map-of-manifolds $Y_Q:M\to TM$, whose value at $p$ is the linear map $Y_{Q*p}:T_pM\to T_{Y_Q(p)}(TM)$. These derivatives are related as follows: for $v\in T_pM$, $$\label{YQ*p} Y_{Q*p}(v)=\tilde{v}_{_{Y_Q(p)}} + \iota_{_{Y_Q(p)}}(\nabla_v Y_Q);$$ i.e. $\nabla_v Y_Q$ is essentially the vertical part of $Y_{Q*p}(v)$ [@groisser2004; @karcher]. **Lemma 12**. *Let $(Q,p)=(q_1,\dots,q_N,p)\in {\mathcal M}_1$. Writing elements of $T_{(Q,p)}(M^N\times M)$ in the form $(w_1,\dots, w_N,v)$, where $w_i\in T_{q_i}M$ and $v\in T_pM$, the derivative of $F$ at $(Q,p)$ is given by $$\begin{aligned} \nonumber \lefteqn{F_{*(Q,p)}(w_1,\dots, w_N,v)=} \\ && \tilde{v}_{_{Y_Q(p)}}+ \iota_{_{Y_Q(p)}}\left\{ \nabla_v Y_Q + \frac{1}{N}\sum_{i=1}^N \iota_{z_i}^{-1}\left[((\exp_p)^{-1})_{*q_i}(w_i) \right] \right\}, \label{F*qp.0}\end{aligned}$$ where $z_i=(\exp_p)^{-1}(q_i)$.* : Let $(w_1,\dots, w_N,v)\in T_{(Q,p)}{\mathcal M}_1=T_{(Q,p)}(M^N\times M)$. We have $$\begin{aligned} \nonumber F_{*(Q,p)}(w_1,\dots, w_N,v)&=& \sum_{i=1}^N F_{*(Q,p)}(0_{q_1}, \dots, 0_{q_{i-1}}, w_i,0_{q_{i+1}},\dots,0_{q_N},0_p)\\ &&+F_{*(Q,p)}(0_{q_1},\dots, 0_{q_N},v). \label{F*Qp}\end{aligned}$$ For $1\leq i\leq N$, the vector $((\exp_p)^{-1})_{*q_i}(w_i)$ is vertical: it is the initial velocity of a curve of the form $t\mapsto (\exp_p)^{-1}(\gamma_i(t))$ lying entirely in $T_p M\subset TM$ (where $\gamma_i$ is a curve in $M$ defined on some interval $(-\epsilon,\epsilon)$ and having $\gamma_i'(0)=w_i$). The $i^{\rm th}$ summand on the first line of [\[F\*Qp\]](#F*Qp){reference-type="eqref" reference="F*Qp"} is simply the image of $((\exp_p)^{-1})_{*q_i}(w_i)$ under the isomorphism $\iota_{Y_Q(p)}\circ \iota_{z_i}^{-1}$; the isomorphisms $\iota_{z_i}^{-1}$ allow us to identify the vertical vectors $((\exp_p)^{-1})_{*q_i}(w_i)\in T_{z_i}M$ with elements of $T_pM$ and thereby add them. Thus the sum on the first line of [\[F\*Qp\]](#F*Qp){reference-type="eqref" reference="F*Qp"} is $$\iota_{Y_Q(p)} \left(\frac{1}{N}\sum_{i=1}^N \iota_{z_i}^{-1}\left[((\exp_p)^{-1})_{*q_i}(w_i) \right]\right). \label{F*Qp-1}$$ The term on the second line of [\[F\*Qp\]](#F*Qp){reference-type="eqref" reference="F*Qp"} is simply $Y_{Q*p}(v)$. The result now follows from [\[F\*Qp\]](#F*Qp){reference-type="eqref" reference="F*Qp"}, [\[F\*Qp-1\]](#F*Qp-1){reference-type="eqref" reference="F*Qp-1"}, and [\[YQ\*p\]](#YQ*p){reference-type="eqref" reference="YQ*p"}. height10pt width10pt\ **Corollary 13**. *$F|_{{\mathcal M}_1}$ is a submersion.* : Let $(Q,p)\in {\mathcal M}_1$ and let $u=F(Q,p)$. Let $X\in T_u(TM)$ and let $X_1={\rm hor}_u(X), X_2=\iota_u^{-1}({\rm vert}_u(X))$, where ${\rm hor}_v: T_v(TM)\to {\rm Hor}_v(TM)$ and ${\rm vert}_v:T_v(TM)\to {\rm Vert}_v(TM)$ are the horizontal and vertical projection-maps determined by the splitting of $T_v(TM)$ as ${\rm Hor}_v(TM)\oplus{\rm Vert}_v(TM)$. Let $v=\pi_{*u}(X_1), z_1=(\exp_p)^{-1}(q_1), w_1=N(\exp_p)_{*z_1}(\iota_{z_1}(X_2-\nabla_v Y_Q)),$ and $w_i=0$ for $i>1$. Then from [\[F\*qp.0\]](#F*qp.0){reference-type="eqref" reference="F*qp.0"} we have $F_{*(Q,p)}(w_1,0,\dots,0,v)=X_1+X_2=X.$ Hence $F_{*(Q,p)}$ is surjective. height10pt width10pt\ Since $Z$ is a codimension-$n$ submanifold of $TM$, Corollary [Corollary 13](#Fsubmersive){reference-type="ref" reference="Fsubmersive"} immediately implies that $(F|_{{\mathcal M}_1})^{-1}(Z)$ is a codimension-$n$ submanifold of the $(N+1)n$-dimensional manifold ${\mathcal M}_1$. Hence, remembering [\[finvz-2\]](#finvz-2){reference-type="eqref" reference="finvz-2"}: **Corollary 14**. *${\mathcal B}_N^0$ is an $(Nn)$-dimensional submanifold of $M^N\times M.$ height10pt width10pt\ * It is natural to ask how nicely the projection-maps $\pi_N$ and ${\pi_{\rm last}}$ restrict to ${\mathcal B}_N^0$. We will see below that ${\pi_{\rm last}}$ restricts to a submersion. Since ${\mathcal B}_N^0$ and $M^N$ have the same dimension, one may wonder whether $\pi_N$ restricts to a local diffeomorphism. We are able to provide only a partial answer to this question, exhibiting a large open subset of ${\mathcal B}_N^0$ on which $\pi_N$ restricts this nicely. Letting ${\rm Hess}(f_Q)|_p$ denote the covariant Hessian of $f_Q$ at $p$, define $${\mathcal M}_2 = \{(Q,p)\in {\mathcal B}_N^0 : {\rm Hess}(f_Q)|_p\ \mbox{is nondegenerate}\},$$ an open subset of ${\mathcal B}_N^0$. Note that since $Y_Q=-\mbox{\rm grad\phantom{-}}f_Q$, at any $p\in M_Q$ the endomorphism $-(\nabla Y_Q)|_p$ is the image of ${\rm Hess}(f_Q)|_p$ under the "index-lowering" isomorphism $T_p^*M\otimes T_p^*M\to T_pM\otimes T_p^*M \mbox{\hspace{1ex}}\cong_{\mbox{\hspace{-5ex}\phantom{$\int$} \scriptsize{canon.}}}{\rm End}(T_pM)$ induced by the metric $g$. Thus the nondegeneracy condition in the definition of ${\mathcal M}_2$ is equivalent to $(\nabla Y_Q)|_p : T_pM \to T_p M$ being an isomorphism. **Proposition 15**. *The map ${\pi_{\rm last}}|_{_{{\mathcal B}_N^0}}: {\mathcal B}_N^0\to M$ is a submersion, and $\pi_N|_{_{{\mathcal M}_2}}: {\mathcal M}_2\to M^N$ is a local diffeomorphism.* : Let $(Q,p)=(q_1,\dots,q_N,p)\in {\mathcal B}_N^0$; note that $F(Q,p)=0_p$. Since $T_{0_p}Z$ is exactly the horizontal subspace of $T_{0_p}(TM)$, $$\begin{aligned} \lefteqn{T_{(Q,p)} ({\mathcal B}_N^0) =(F_{*(Q,p)})^{-1}(T_{0_p}Z)} \\ &=& \{ W\in T_{(Q,p)}M : {\rm vert}_{0_p}(F_{*(Q,p)}(W))=0\}\\ &=& \{ (w_1,\dots, w_N,v)\in T_{q_1}M \times \dots\times T_{q_N}M \times T_p M : \\ && \mbox{\hspace{1in}}\nabla_v Y_Q + \frac{1}{N}\sum_{i=1}^N \iota_{z_i}^{-1}\left[((\exp_p)^{-1})_{*q_i}(w_i)\right]=0\},\end{aligned}$$ where $z_i=(\exp_p)^{-1}(q_i)$; the final equality follows from [\[F\*qp.0\]](#F*qp.0){reference-type="eqref" reference="F*qp.0"}. Analogously to the proof of Corollary [Corollary 13](#Fsubmersive){reference-type="ref" reference="Fsubmersive"}, let $v\in T_pM$ and set $w_1=N(\exp_p)_{*z_1}(\iota_{z_1}(-\nabla_v Y_Q)),$ $w_i=0$ for $i>1$. Then $$\label{TFinvZ} \nabla_v Y_Q + \frac{1}{N}\sum_{i=1}^N \iota_{z_i}^{-1}\left[((\exp_p)^{-1})_{*q_i}(w_i)\right]=0,$$ so $W:=(w_1,\dots,w_N,v)\in T_{(Q,p)}({\mathcal B}_N^0)$, and ${\pi_{\rm last}}_{*(Q,p)}(W) = v$. Thus ${\pi_{\rm last}}_{*(Q,p)}$ is surjective, and ${\pi_{\rm last}}|_{_{{\mathcal B}_N^0}}$ is a submersion. Now assume that $(Q,p)\in {\mathcal M}_2$ and let $w_i\in T_{q_i}M, 1\leq i\leq N$. Then the map $(\nabla Y_Q)|_p : T_pM \to T_p M$ is invertible, and we may define $$v= -((\nabla Y_Q)|_p)^{-1}\left(\frac{1}{N}\sum_{i=1}^N \iota_{z_i}^{-1}\left[((\exp_p)^{-1})_{*q_i}(w_i)\right] \right)\in T_p M.$$ Then $W:=(w_1,\dots,w_N,v)\in T_{(Q,p)}({\mathcal B}_N^0)=T_{(Q,p)}{\mathcal M}_2$, and $\pi_{N*(Q,p)}(W) = (w_1,\dots,w_N)$. Thus $(\pi_N|_{{\mathcal M}_2})_{*(Q,p)}$ is surjective. Since $\dim({\mathcal M}_2)=\dim({\mathcal B}_N^0) =Nn=\dim(M^N)$, $(\pi_N|_{{\mathcal M}_2})_{*(Q,p)}$ is an isomorphism, and $\pi_N|_{_{{\mathcal M}_2}}$ is a local diffeomorphism. height10pt width10pt\ *Remark 16*. The result concerning ${\mathcal M}_2$ in Proposition [Proposition 15](#bary_projected){reference-type="ref" reference="bary_projected"} is not wholly satisfying, since the set ${\mathcal M}_2$ itself is not easy to get one's hands on explicitly. However, at least the set $\pi_N({\mathcal M}_2)$---the set of configurations $Q$ for which ${\rm Hess}(f_Q)|_p$ is nondegenerate for every short barycenter $p$ of $Q$---is generic in $M^N$. In Appendix [11](#sec:appendixB){reference-type="ref" reference="sec:appendixB"}, we prove this using a "Parametric Transversality Theorem" (Theorem [Theorem 57](#PTT){reference-type="ref" reference="PTT"}). Since ${\mathcal B}_N^0\supset {\mathcal M}_2$, it follows that $\pi_N({\mathcal B}_N^0)$ is generic in $M^N$ as well, as asserted earlier. **Corollary 17**. *Let $A\subset M$ be a submanifold of codimension $k$. Then the subset $\pi_N(({\pi_{\rm last}}|_{_{{\mathcal M}_2}})^{-1}(A))$ of $$\pi_N(({\pi_{\rm last}}|_{_{{\mathcal B}_N^0}})^{-1}(A))=\{Q\in M^N\ : \ {\mathcal B}^0(Q)\cap A\neq\emptyset\}$$ is a codimension-$k$ submanifold of $M^N$.* : Since ${\mathcal M}_2$ is an open subset of ${\mathcal B}_N^0$, and ${\pi_{\rm last}}|_{_{{\mathcal B}_N^0}}$ is a submersion, ${\pi_{\rm last}}|_{_{{\mathcal M}_2}}$ is also a submersion. Hence $({\pi_{\rm last}}|_{_{{\mathcal M}_2}})^{-1}(A)$ is a codimension-$k$ submanifold of ${\mathcal M}_2$, and its image under the local diffeomorphism $\pi_N|_{_{{\mathcal M}_2}}$ is a codimension-$k$ submanifold of $M^N$. height10pt width10pt\ By imposing certain hypotheses under which Fréchet means are unique, we will provide in Corollary [Corollary 21](#cor:conFMbary){reference-type="ref" reference="cor:conFMbary"} a reasonably large and *explicit* open subset of ${\mathcal M}_2$ (hence an explicit subset of ${\mathcal B}_N$ that *is* a submanifold of $M^N\times M$). First, some notation: *Definition 18*. Let ${r_{\rm inj}}={r_{\rm inj}}(M,g)\geq 0$ denote the injectivity radius of $(M,g)$, let $\Delta=\Delta(M,g) \leq \infty$ be the the supremum of the sectional curvatures of $(M,g)$, and let ${r_{\rm cx}}={r_{\rm cx}}(M,g)=\frac{1}{2}\min\{{r_{\rm inj}},\frac{\pi}{\sqrt{\Delta}}\}$, where if $\Delta\leq 0$ we interpret $\frac{1}{\sqrt{\Delta}}$ as $\infty$. Define $$M^N_{\rm con}:=\{Q\in M^N: {\rm supp}(Q) \subset B_{{r_{\rm cx}}}(p)\ \mbox{for some $p\in M$}\},$$ the set of (somewhat) "concentrated" configurations. Note that ${r_{\rm cx}}>0$ if (and only if) ${r_{\rm inj}}>0$ and $\Delta<\infty$, and that $M^N_{\rm con}=\emptyset$ if ${r_{\rm cx}}=0$. (Hence any hypothesis of the form "$Q\in M^N_{\rm con}$" implicitly assumes ${r_{\rm cx}}>0$.) A fundamental theorem of Afsari [@afsari], applied to the probability measures $\mu_Q$, yields the following: **Theorem 19** (Special case of [@afsari Theorem 2.1]). *If $Q\in M^N_{\rm con}$, then $Q$ has a unique Fréchet mean. Furthermore, for any open ball $B$ of radius less than ${r_{\rm cx}}$ containing ${\rm supp}(Q)$, this unique element of ${\rm FM}(Q)$ lies in $B$ and is the unique short barycenter of $Q$ in the concentric open ball of radius ${r_{\rm cx}}$.* *Remark 20*. Theorem 2.1 in [@afsari] is far more extensive than Theorem [Theorem 19](#thm:afsari){reference-type="ref" reference="thm:afsari"}, and for digestibility's sake,  its author makes certain simplifications. By removing these simplifications, the original theorem and the special case above can be strengthened. For example, instead of the global geometric invariant ${r_{\rm cx}}$, which is a convenient (and often sharp) lower bound on what is termed the "regular convexity radius" of $(M,g)$ in [@groisser2004], local invariants can be used; see [@afsari Remark 2.5]. However, because of its simplicity and wide applicability, ${r_{\rm cx}}$ is very commonly used in hypotheses on the radius of a ball supporting a probability distribution for which one wants to guarantee the uniqueness of a Fréchet mean. Let $Q\in M^N_{\rm con}$. The proof of Theorem 2.1 in [@afsari] shows that ${\rm Hess}(f_Q)$ is positive-definite, and hence nondegenerate, at the (unique) Fréchet mean of $Q$. As a corollary, we have: **Corollary 21**. *The set $$\label{conFMbary.1} {\mathcal M}_{\rm con}:=\{(Q,p)\in {\mathcal B}_N \ : \ Q\in M^N_{\rm con}\}= (\pi_N|_{{\mathcal B}_N})^{-1}(M^N_{\rm con})$$ is a submanifold of $M^N\times M$, and the restriction of $\pi_N$ to this submanifold is a diffeomorphism onto its image.* : Theorem [Theorem 19](#thm:afsari){reference-type="ref" reference="thm:afsari"} implies that if $Q\in M^N_{\rm con}$ and $p$ is the Fréchet mean of $Q$, then $(Q,p)\in {\mathcal M}_2$. Thus ${\mathcal M}_{\rm con}$ is a subset of the manifold ${\mathcal M}_2$, and is open in ${\mathcal M}_2$ (hence is a submanifold of $M^N\times M$) since $M^N_{\rm con}$ is open in $M^N$. The map $\pi_N|_{_{{\mathcal M}_{\rm con}}}$ is the restriction of the local diffeomorphism $\pi_N|_{_{{\mathcal M}_2}}$ to the open subset ${\mathcal M}_{\rm con}\subset {\mathcal M}_2$, and is one-to-one by Theorem [Theorem 19](#thm:afsari){reference-type="ref" reference="thm:afsari"}. height10pt width10pt\ *Remark 22*. As noted in the introduction, we have taken "smooth" to mean $C^\infty$, and have taken "(sub)manifold" to mean "smooth (sub)manifold". However, all statements in Sections [3](#sec:bary-prebary2){reference-type="ref" reference="sec:bary-prebary2"} and [4](#sec:barymanifold){reference-type="ref" reference="sec:barymanifold"} up through Corollary [Corollary 17](#cor:codimk){reference-type="ref" reference="cor:codimk"} remain true (and the same proofs work) if the manifold $M$ and Riemannian metric $g$ are assumed only to be of class $C^k$, where $k\geq 2$, and other explicit or implicit references to smoothness are modified as follows: - The functions $f_Q$ are $C^k$. - The tangent bundle $TM$ is a $C^{k-1}$ manifold. The maps $\exp, \exp_p,\pi, \pi_N,{\pi_{\rm last}},$ and $F_Q$ are $C^{k-1}$, as are the vector fields $Y_Q$ and the submanifolds $Z\subset TM$, ${\mathcal B}_N=F^{-1}(Z)\subset M^N\times M$, and ${\mathcal M}_2\subset M^N\times M$. - The maps ${\pi_{\rm last}}|_{_{F^{-1}(Z)}}$ and $\pi_N|_{_{{\mathcal M}_2}}$ are $C^{k-1}$. - The submanifold $\pi_N(({\pi_{\rm last}}|_{_{{\mathcal M}_2}})^{-1}(A))$ in Corollary [Corollary 17](#cor:codimk){reference-type="ref" reference="cor:codimk"} is $C^1$. However, for the genericity statement in Remark [Remark 16](#rmk:pi_NM_2){reference-type="ref" reference="rmk:pi_NM_2"}, a much higher degree of smoothness of $(M,g)$ is needed: we need to assume that $M$ and $g$ are $C^k$, where $k\geq n(N-1)+2.$ The reason is that, just as in the finite-differentiability version of Sard's Theorem [@hirsch Theorem III.1.3] of which the finite-differentiability version of the Parametric Transversality Theorem [@hirsch Theorem III.2.7] is a corollary, the required degree of smoothness depends on the dimensions of the manifolds involved, via an explicit inequality that reduces to the stated restriction on $k$ in our setting. # All Fréchet means of finite configurations are barycenters {#sec:FMs_are_barys} Theorem [Theorem 19](#thm:afsari){reference-type="ref" reference="thm:afsari"} shows that the Fréchet mean set of a sufficiently concentrated configuration $Q\in M^N$ lies in ${\mathcal B}(Q)$. Using results from [@le-barden2014], we will remove the "sufficiently concentrated" restriction. For any $f:M\to{\bf R}, p\in M,$ and $v\in T_pM$, define the *forward geodesic directional derivative* $v_+(f)$ by $$\label{fdirder} v_+(f)=\lim_{t\downarrow 0} \frac{f(\exp_p(tv))-f(p)}{t}$$ whenever the limit exists. Note that if $f$ has a relative minimum at $p$, and $v_+(f)$ exists, then $v_+(f)\geq 0$. *Remark 23*. In [@burago Section 11], in the more general setting of Alexandrov spaces with curvature bounded below, the notion corresponding to [\[fdirder\]](#fdirder){reference-type="eqref" reference="fdirder"} is developed (only) for functions $f$ *that are locally Lipschitz at $p$*---i.e., that satisfy a Lipschitz condition on some neighborhood of $p$. *With this restriction on $f$*, the limit [\[fdirder\]](#fdirder){reference-type="eqref" reference="fdirder"} is simply called a "directional derivative" in [@burago]. This terminology, and the notation $v(f)$ for our $v_+(f)$, are used in [@le-barden2014] for powers (with exponent $\geq 1$) of the distance function $r_p=d(p,\cdot).$ However, the omission from [@le-barden2014] of any mention of Lipschitzness may leave a misperception that the notation is unambiguous, and consequently that certain facts discussed below are true, for any continuous function $f:M\to{\bf R}$. We use the longer "forward geodesic directional derivative" because of several sources of potential misunderstanding, all related to the fact that in the setting of smooth manifolds, there is a conventional---and not metric-related---use of the term "directional derivative" that, unlike in vector spaces, is usually applied only to *differentiable* functions. If $f:M\to{\bf R}$ is differentiable at $p$ (below we leave the "at $p$" implicit), and $v\in T_pM$, then what is commonly called the "directional derivative $v(f)$" is the two-sided limit $\frac{d}{dt}f(\gamma(t))|_{t=0},$ where $\gamma:I\to {\bf R}$ is *any* smooth curve based at $p$ with $\gamma'(0)=v$. Crucially, differentiability of $f$ ensures that (i) $\frac{d}{dt}f(\gamma(t))|_{t=0}$ is independent of the choice of curve $\gamma$ representing $v$, so that the notation "$v(f)$" makes sense, and (ii) $v_+(f)=v(f)$, so that the word "forward" would be superfluous. But if $f$ is not differentiable at $p$, then even if $(-v)_+(f)=-(v_+(f))$ (so that $\frac{d}{dt}f(\exp_p(tv))|_{t=0}$ exists), it is not always true that for *every* curve $\gamma$ representing $v$, the derivative $\frac{d}{dt}f(\gamma(t))|_{t=0}$ exists or has the same value. Even the existence of the map $(d_+f)_p: T_pM\to{\bf R}$, $v\mapsto v_+(f)$ (i.e. existence of $v_+(f)$ for all $v\in T_pM$) plus linearity in $v$, are not sufficient to ensure the curve-independence property. However, for functions that are locally Lipschitz at $p$, existence *and continuity* of the map $(d_+f)_p$ together ensure the curve-independence property. (This can be deduced from the arguments in [@burago].) Therefore for positive powers of the distance function $r_p$, the terminology "directional derivative" and notation $v(f)$ in [@le-barden2014] are consistent with the usual terminology and notation for functions on manifolds, modulo the one-sidedness of the limit [\[fdirder\]](#fdirder){reference-type="eqref" reference="fdirder"}. For $q\in M$, the function $r_q^2:M\to{\bf R}$ is never differentiable at ordinary cut-points of $q$ [@bishop]. However, Le and Barden show in [@le-barden2014] that at *every* cut-point of $q$, all forward geodesic directional derivatives of $r_q^2$ exist. An explicit formula, which is valid also at non-cut points of $q$, is derived there: **Lemma 24** ([@le-barden2014 Lemma 2]). *Let $p,q\in M$, $v\in T_pM$. Then $$v_+(r_q^2) = -2\sup_{v'\in {\mathcal V}_{q,p}} g(v,v'),$$ where ${\mathcal V}_{q,p}=\{v'\in T_pM \ : \ \|v'\|=d(q,p)\ \mbox{\rm and}\ \exp_p(v')=q\}.$* As noted in [@le-barden2014], if $p\in{\mathcal{C}}_{\rm ord}(q)$, then since $r_q^2$ is not differentiable at $p$, we do not necessarily have $(-v)_+(r_q^2)$ equal to $-(v_+(r_q^2))$. Also note that if $p\notin {\mathcal{C}}_{\rm ord}(q)$, then ${\mathcal V}_{q,p}$ consists of the single element $(\exp_p)^{-1}(q)$. The proofs of Theorem 1 and Corollaries 2 and 3 in [@le-barden2014] show the following: Let $\mu$ be probability measure on $M$ that is the sum of an absolutely continuous measure and a discrete measure, let $\alpha>1$, and assume that the $\alpha$-energy function $f_{\mu,\alpha}: x\mapsto \frac{1}{\alpha}\int_M d(x,y)^\alpha\, d\mu(y)$ achieves a relative minimum at $p$. Then (i) if ${\mathcal{C}}_{\rm sing}(p)=\emptyset$, then $\mu({\mathcal{C}}(p))=0$; (ii) $\exp_p^{-1}(q)$ is defined $\mu$-a.e. in $q$; and (iii) $\int_M \exp_p^{-1}(q) d\mu(q)=0.$ Proving fact (i) was the main goal of [@le-barden2014]; for us, more directly important is fact (iii) (but note that the integral in (iii) is well-defined only because of (ii)). Specializing these facts to the case $\mu_Q=\frac{1}{N}\sum_i \delta_{q_i}$ and $\alpha=2$ yields Proposition [Proposition 25](#prop:fmisbary){reference-type="ref" reference="prop:fmisbary"} below. We will prove this proposition more directly, however. Our proof borrows heavily from the proof of [@le-barden2014 Theorem 1], but our argument is rather simpler. **Proposition 25**. *Let $Q=(q_1,\dots,q_N)\in M^N,$ let $p\in M$ and assume that $f_Q$ achieves a relative minimum at $p$. Then:* - *For $1\leq i\leq N$ we have $q_i\in \tilde{{\mathcal D}}(p)$, and hence $\exp_p^{-1}(q_i)$ is defined.* - *The point $p$ is a barycenter of $Q$ (in fact, an almost-short barycenter).* : For (i), it suffices to show that ${\rm supp}(Q)\cap{\mathcal{C}}_{\rm ord}(p)=\emptyset$. Let $I_{\rm reg}, I_{\rm ord},$ and $I_{\rm sing}$ denote the subsets of $\{1,\dots, N\}$ for which $q_i$ lies, respectively, in ${\mathcal D}(p), {\mathcal{C}}_{\rm ord}(p)$, and ${\mathcal{C}}_{\rm sing}(p)$. Then for every $v\in T_pM$, by Lemma [Lemma 24](#bllemma2){reference-type="ref" reference="bllemma2"} we have $$\label{v+fq} v_+(f_Q)= - \frac{1}{N}\left\{\sum_{i\in I_{\rm reg}\mbox{\small $\bigcup$}I_{\rm sing}} g(v,\exp_p^{-1}(q_i)) +\sum_{i\in I_{\rm ord}} \sup_{v'\in {\mathcal V}_{q_i,p}} g(v,v')\right\}.$$ Since $f_Q$ has a relative minimum at $p$, we have $v_+(f_Q)\geq 0$ for every $v\in T_pM$. Hence, for each $v\in T_pM$, $$\begin{aligned} \nonumber 0&\leq & N[v_+(f_Q)+ (-v)_+(f_Q)]\\ \nonumber &=& -\sum_{i\in I_{\rm ord}}\left( \sup_{v'\in {\mathcal V}_{q_i,p}} g(v,v') +\sup_{v'\in {\mathcal V}_{q_i,p}} g(-v,v')\right)\\ &=& \sum_{i\in I_{\rm ord}}\left( \inf_{v'\in {\mathcal V}_{q_i,p}} g(v,v') -\sup_{v'\in {\mathcal V}_{q_i,p}} g(v,v')\right) \label{inf-sup} \\ &\leq & 0, \nonumber\end{aligned}$$ implying that each summand in [\[inf-sup\]](#inf-sup){reference-type="eqref" reference="inf-sup"} is zero. Hence for each $i\in I_{\rm ord}$ and $v\in T_pM$, the set $\{g(v,v')\ : \ v'\in {\mathcal V}_{q_i,p}\}$ consists of a single element, implying that the difference of any two elements of $V_{q_i,p}$ is perpendicular to every element of $T_pM$, hence is zero. But by definition of ${\mathcal{C}}_{\rm ord}(p)$, if $q\in {\mathcal{C}}_{\rm ord}(p)$ then $V_{q,p}$ contains at least two vectors. Hence $I_{\rm ord}=\emptyset$, establishing (i). Substituting $I_{\rm ord}=\emptyset$ into [\[v+fq\]](#v+fq){reference-type="eqref" reference="v+fq"}, we now have $v_+(f_Q)= - \frac{1}{N}\sum_{i=1}^N g(v,\exp_p^{-1}(q_i))\geq 0$ for all $v\in T_pM$. Replacing $v$ with $-v$, we obtain the opposite inequality, and hence the equality $\sum_{i=1}^N g(v,\exp_p^{-1}(q_i))=0$. Thus $p\in \tilde{{\mathcal B}}^0(Q)$. height10pt width10pt\ *Remark 26*. The proof above also shows that $v_+(f_Q)=0$ for all $v\in T_pM$. For a general function $f$, having $v_+(f)=0$ for all $v\in T_pM$ does *not* imply that $f$ is differentiable at $p$; see Remark [Remark 23](#rem1:direcderiv){reference-type="ref" reference="rem1:direcderiv"}. However, it is not hard to show that $f_Q$ is locally Lipschitz (or that the same is true more generally for the function $f_{\mu,\alpha}: p\mapsto \int_M r_p^\alpha\, d\mu$, where $\alpha\geq 1$ and $\mu$ is any probability measure on $M$ for which $f_{\mu,\alpha}$ is finite). Therefore for $f_Q$, the fact that $v_+(f_Q)=0$ for all $v\in T_pM$ *does* imply that $f_Q$ is differentiable at $p$; again see Remark [Remark 23](#rem1:direcderiv){reference-type="ref" reference="rem1:direcderiv"}. An immediate corollary of Proposition [Proposition 25](#prop:fmisbary){reference-type="ref" reference="prop:fmisbary"} is the following: **Corollary 27**. *For every $Q\in M^N$, we have ${\rm FM}(Q)\subset \tilde{{\mathcal B}}^0(Q)\subset {\mathcal B}(Q)$. height10pt width10pt\ * The theorem follows immediately from Corollary [Corollary 27](#cor:fmisbary){reference-type="ref" reference="cor:fmisbary"} and Theorem [Theorem 11](#thm:measzero.7){reference-type="ref" reference="thm:measzero.7"}. height10pt width10pt\ # Fréchet sample means of i.i.d. random variables {#sec:iid1} $~$ **Convention.** For the remainder of this paper, given a probability measure $\mu$ on a topological space $Y$, and a subset $A\subset Y$, we will say that *$\mu$ is supported in $A$* if $\mu(A)=1$. This is less restrictive than saying that $A$ contains the set ${\rm supp}(\mu)$ (the support of $\mu$), which by definition is a closed set. **Corollary 28**. *Let $X_1, \dots, X_N$ be absolutely continuous, i.i.d. $M$-valued variables with underlying measure $\mu_X$, and let $A$ be a volume-zero subset of $M$. Then:* 1. *${\rm Pr}\left\{{\rm FM}(X_1,\dots,X_N)\subset M\backslash A\right\}=1.$* 2. *If $\mu_X$ is supported in an open ball $B$ of radius $r\leq{r_{\rm cx}}$, then almost surely the Fréchet mean set of $(X_1,\dots,X_N)$ is a single point and lies in $B\backslash A$.* : Let $\mathcal{X}$ denote the $M^N$-valued random variable $(X_1,\dots,X_N)$ and let $\mu_X^N$ be the product probability measure on $M^N$ induced by $\mu_X$. Since $\mu_X$ is absolutely continuous, so is $\mu_X^N$. Hence $${\rm Pr}\left\{{\rm FM}(\mathcal{X})\cap A\neq \emptyset\right\} = \mu_X^N(\{Q\in M^N: {\rm FM}(Q)\cap A\neq \emptyset\}) =0$$ by Theorem [Theorem 3](#thm:FMmeaszero){reference-type="ref" reference="thm:FMmeaszero"}. Statement (a) follows. Now suppose that $\mu_X$ is supported in an open ball $B$ of radius $r\leq{r_{\rm cx}}$, and let $B^N$ denote the $N$-fold Cartesian product of $B$ with itself. Then ${\rm Pr}\{\mathcal{X}\in B^N\}=1$. If the support of a configuration $Q\in M^N$ lies in the open ball $B$, then ${\rm supp}(Q)$ also lies in a concentric open ball of radius strictly less than $r$, hence less than ${r_{\rm cx}}$. Therefore Theorem [Theorem 19](#thm:afsari){reference-type="ref" reference="thm:afsari"} ensures that if a value of $\mathcal{X}$ lies in $B^N$, then the Fréchet mean of that value of $\mathcal{X}$ is unique and lies in $B$. Statement (b) follows. height10pt width10pt\ *Remark 29*. Corollary [Corollary 28](#cor:measzero.3){reference-type="ref" reference="cor:measzero.3"}(a) is precisely Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. This theorem has an application to manifolds that are stratified in even a very weak sense. Let us call $M$ *weakly stratified* if we are given a collection $\mathcal T$ of pairwise disjoint submanifolds of $M$, the *strata*, whose union is $M$, and for which exactly one stratum, $M_{\rm top}$, is an open set. (More precisely, the "weakly stratified manifold" is the pair $(M,\mathcal{T})$, but in many situations, it is often not important to have a symbol for the set $\mathcal T$.) Then each of the other strata has positive codimension, so the union of all the other strata has volume zero in $M$. Hence Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} implies that if $X_1, \dots, X_N$ are absolutely continuous, i.i.d., $M$-valued random variables, then ${\rm Pr}\left\{{\rm FM}(X_1,\dots,X_N)\subset M_{\rm top}\right\} =1.$ But note that Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is much more general than the above application; the theorem does not require *any* sort of stratification. An example of a well known manifold to which Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} applies is the projective space ${\bf C}P^{k-2}$. This manifold arises in the statistics of shape as Kendall's *shape space* $\Sigma^k_2$ of $k>2$ labeled points in ${\bf R}^2$ ("planar shapes") [@kendall_d]. With the usual shape-space stratification, $\Sigma^k_2$ has only one stratum (the entire space), so the results of [@huckMMS2012] reduce to tautologies. However, $\Sigma^k_2$ has other stratifications. For example, we can stratify according to the greatest number of collinear points in the configuration. In this case the top stratum consists of the shapes whose pre-shapes lie in *general position* (no three points are collinear), and the remaining strata have positive codimension. Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} shows that for i.i.d. random planar shapes $X_1,\dots, X_N$ with absolutely continuous distribution, the sample Fréchet mean shape almost surely has no more than two collinear points. More generally, a similar result holds for many other ways in which a planar shape can be non-generic. For example, for $k\geq 4$, if we call a point in $\Sigma^k_2$ a *$k$-trapezoid* if it has $k$ distinct points and at least two parallel non-adjacent sides, then the set $\Sigma^{k, {\rm trap}}_2$ of $k$-trapezoids is a finite union of codimension-1 submanifolds of $\Sigma^k_2$. Thus if $X_1,\dots, X_N$ are as above, then by Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}, ${\rm Pr}\left\{{\rm FM}(\{X_1,\dots,X_N\})\cap\Sigma^{k,{\rm trap}}_2\neq\emptyset\right\}=0$. Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} applies only to *manifolds*, but allows *arbitrary* (volume-)measure-zero subsets $A$. There are certain, more general, *non-manifold* spaces for which similar results have been obtained, but only for *very specific* measure-zero subsets $A$: unions of lower strata of an orbit-type stratification of a quotient space $M/G$, where $M$ is a complete, connected Riemannian manifold and $G$ is a Lie group acting properly and isometrically on $M$. These spaces are (generalized) shape spaces, as broadly defined in [@huckMMS2012]. The orbit-type stratification of $M$ induces a stratification of $M/G$, which we will call the "usual shape-space stratification", and $M/G$ inherits a quotient distance-function. For these spaces the behavior of sample means is similar to what Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} shows what it is for a manifold: in [@huckMMS2012 Corollary 2], Huckemann establishes that Fréchet sample means of an absolutely continuously distributed random variable lies in the top stratum---the complement of the union $A$ of all lower strata (a particular measure-zero set)---with probability 1. There are other situations in which Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} does not apply. In recent years there has been considerable interest in the behavior of various means on certain stratified spaces that usually are not manifolds, for example "open books" and "spiders" [@hotz2013sticky]. For open-books and spiders, Fréchet sample means are often "sticky", tending to lie in lower strata [@hotz2013sticky]. See also [@MR3371437]. The "sticky means" behavior on these non-manifold spaces is in sharp contrast to what Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} implies for (even weakly) stratified manifolds. # Riemannian covers and equivariant means {#sec:eqvt_means_general} In some applications, the easiest way to compute or analyze Fréchet sample means of configurations (or random variables) is to relate them to what we will call *equivariant Fréchet means* on a Riemannian covering space. We will see an example of this in Section [9](#sec:PSRmeans){reference-type="ref" reference="sec:PSRmeans"}. Let $G$ be a finite group of order at least 2, with identity element $e$, acting isometrically on a Riemannian manifold $({\widetilde{M}},\tilde{g})$; without loss of generality we assume that the action is a left-action, and denote it by $({\tilde{p}},h)\mapsto h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{p}}$. Assume that $\beta:=\inf\{\tilde{d}({\tilde{p}},h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{p}}): {\tilde{p}}\in {\widetilde{M}}, \ e\neq h\in G\}>0;$ in particular this implies that the $G$-action is free. For simplicity's sake, we assume throughout this section that $({\widetilde{M}},\tilde{g})$ is complete, but this assumption is not actually needed until Lemma [Lemma 35](#lem:rinjbeta){reference-type="ref" reference="lem:rinjbeta"}; in particular, Propositions [Proposition 31](#quodist){reference-type="ref" reference="quodist"} and [Proposition 33](#prop:uniqueuptoG){reference-type="ref" reference="prop:uniqueuptoG"} do not require completeness. *Remark 30*. Because the $G$-action is free, ${\widetilde{M}}$ is a (left) principal bundle over $M$ (more strongly, a "principal Riemannian cover"). The principality assumption is primarily a convenience, allowing simplification of certain statements whose full generality is not needed for our application to ${\rm Sym}^+(m)$ mentioned in the Introduction. Many of the ideas and results in this section generalize to non-principal Riemannian covers. Since the finite $G$-action is free and isometric, $({\widetilde{M}},\tilde{g})$ is a *Riemannian covering space*: the quotient $M={\widetilde{M}}/G$ is (canonically) a manifold, the quotient map ${\rm quo}:{\widetilde{M}}\to M$ is a covering map, and there is a (unique) induced Riemannian metric $g$ on $M$ such that ${\rm quo}$ is a local isometry. The assumed completeness of $({\widetilde{M}}, \tilde{g})$ implies that $(M,g)$ is complete as well. We will write ${\tilde{d}}$ and $d$ for the Riemannian distance functions on $({\widetilde{M}},\tilde{g})$ and $(M,g)$, respectively; in this section, balls are always taken with respect to these distance functions. We also use ${\tilde{d}}(\cdot,\cdot)$ and $d(\cdot,\cdot)$ to denote the corresponding distances between *subsets* of ${\widetilde{M}}$ and $M$, respectively. **Proposition 31**. *For all $p,q\in M$ and ${\tilde{p}}\in{\rm quo}^{-1}(p)$, we have $$\label{dbar} d(p,q)={\tilde{d}}({\rm quo}^{-1}(p), {\rm quo}^{-1}(q)) ={\tilde{d}}({\tilde{p}},{\rm quo}^{-1}(q)).$$* : This is straightforward and is left to the reader. height10pt width10pt\ The equivariant means defined below are a special case of "$\rho$-means" as extended by Huckemann [@huck3DProcrustes2011; @Huckemann2011consistency]. *Definition 32*. Define an "equivariant distance function" $d_{\rm evt}:M\times {\widetilde{M}} \to{\bf R}$ by $$\label{defdevt} d_{\rm evt}(q,{\tilde{p}})={\tilde{d}}({\rm quo}^{-1}(q),{\tilde{p}})=\min\{{\tilde{d}}({\tilde{q}},{\tilde{p}}): {\tilde{q}}\in {\rm quo}^{-1}(q)\}.$$ For $Q=(q_1,\dots,q_N)\in M^N$ define the "equivariant Fréchet function" $\tilde{f}_{Q}:{\widetilde{M}}\to {\bf R}$ by $$\label{deftfq} \tilde{f}_{Q}({\tilde{p}})=\frac{1}{N}\sum_{i=1}^N d_{\rm evt}(q_i,{\tilde{p}})^2.$$ Call ${\tilde{p}}\in {\widetilde{M}}$ an *equivariant Fréchet mean* of $Q$ if ${\tilde{p}}$ is a minimizer of $\tilde{f}_{Q}$, and write ${\rm EFM}(Q)$ for the set of equivariant Fréchet means of $Q$. Note that by Proposition [Proposition 31](#quodist){reference-type="ref" reference="quodist"}, for all $q\in M$ and ${\tilde{p}}\in {\widetilde{M}}$ we have $$\label{dpar=dbar} d_{\rm evt}(q,{\tilde{p}})=d(q,{\rm quo}({\tilde{p}})),$$ and hence $$\label{fbar=f} \tilde{f}_{Q}({\tilde{p}}) =f_{Q}({\rm quo}({\tilde{p}}))=\tilde{f}_{Q}(h\,{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{p}})\ \ \ \mbox{for all $h\in G$}.$$ Hence if ${\tilde{p}}$ is an equivariant Fréchet mean of $Q$, then so is $h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{p}}$ for every $h\in G$. Analogously to [@huck3DProcrustes2011] and [@rjgs], we say that *the equivariant Fréchet mean of $Q$ is unique up to the action of $G$* if any two equivariant Fréchet means of $Q$ are related by the action of an element of $G$. As one might expect, taking equivariant Fréchet means in ${\widetilde{M}}$ is "equivalent" to taking Fréchet means in $M$: **Proposition 33**. *Let $Q\in M^N$.* - *Let ${\tilde{p}}\in {\widetilde{M}}$ and let $p={\rm quo}({\tilde{p}})$. Then the following are equivalent:* - *$p\in {\rm FM}(Q).$* - *${\tilde{p}}\in {\rm EFM}(Q)$.* - *${\rm quo}^{-1}(p) \subset {\rm EFM}(Q)$.* *Thus ${\rm EFM}(Q)={\rm quo}^{-1}({\rm FM}(Q))$  and  ${\rm quo}({\rm EFM}(Q))={\rm FM}(Q)$.* - *$Q$ has a unique Fréchet mean if and only if the equivariant Fréchet mean of $Q$ is unique up to the action of $G$.* : This follows from equation [\[fbar=f\]](#fbar=f){reference-type="eqref" reference="fbar=f"}. height10pt width10pt\ **Corollary 34**. *Equivariant Fréchet means always exist. More precisely, for any $Q\in M^N$, the set ${\rm EFM}(Q)$ is nonempty.* : Let $Q\in M^N$. As noted in the introduction, since $(M,g)$ is complete, ${\rm FM}(Q)$ is nonempty. Since ${\rm quo}:{\widetilde{M}}\to M$ is surjective, ${\rm quo}^{-1}({\rm FM}(Q))$ ($={\rm EFM}(Q)$ by Proposition [Proposition 33](#prop:uniqueuptoG){reference-type="ref" reference="prop:uniqueuptoG"}) is nonempty as well. height10pt width10pt\ To obtain a useful equivariant-means analog of Corollary [Corollary 28](#cor:measzero.3){reference-type="ref" reference="cor:measzero.3"}, we need to know how the numbers ${r_{\rm cx}}({\widetilde{M}},\tilde{g}), {r_{\rm cx}}(M,g),$ and $\beta$ are related. We start with the following lemma. **Lemma 35**. *${r_{\rm inj}}({\widetilde{M}},\tilde{g})\ \geq\ {r_{\rm inj}}(M,g)\ = \ \min\left\{{r_{\rm inj}}({\widetilde{M}},\tilde{g}), \frac{\beta}{2}\right\}$.* This lemma is almost certainly known, but the authors have found no reference for it. For a proof, see Appendix [\[sec:appendixA\]](#sec:appendixA){reference-type="ref" reference="sec:appendixA"}. **Corollary 36**. *${r_{\rm cx}}({\widetilde{M}},\tilde{g})\ \geq\ {r_{\rm cx}}(M,g)\ =\ \min\{{r_{\rm cx}}({\widetilde{M}},\tilde{g}),\frac{\beta}{4}\}$.* : Since $({\widetilde{M}},\tilde{g})$ is a Riemannian covering space of $(M,g)$, the sectional-curvature function of $(M,g)$ has the same range as the sectional-curvature function of $({\widetilde{M}},\tilde{g})$. Hence $\Delta(M,g)=\Delta({\widetilde{M}},\tilde{g})$. The result now follows from Lemma [Lemma 35](#lem:rinjbeta){reference-type="ref" reference="lem:rinjbeta"}. height10pt width10pt\ Since ${\rm quo}: {\widetilde{M}}\to M$ is a covering map and a local diffeomorphism, every sufficiently small open set $A\subset M$ is *smoothly evenly covered*: ${\rm quo}^{-1}(A)$ is a disjoint union of open sets $\tilde{A}_j$ (unique up to labeling), each of which is carried diffeomorphically to $A$ by ${\rm quo}$. For any such $A, \{\tilde{A}_j\}$, and arbitrary subset $A'\subset A$, the set ${\rm quo}^{-1}(A')$ is the disjoint union of the sets $\tilde{A}'_j:=\tilde{A}_j\cap{\rm quo}^{-1}(A')$, each of which is carried homeomorphically to $A$ by ${\rm quo}$. We will say that $A'$ is evenly covered by the sets $\tilde{A}'_j$ (whether or not $A'$ is open), and that the collection $\{\tilde{A}'_j\}$ is an *even covering* of $A'$ (a *smooth even covering* if $A'$ is open). **Corollary 37**. *Let $p\in M$, let $r<{r_{\rm inj}}(M,g)$, and let $B=B_r(p)$. Then $B$ is open (as a subset of the manifold $M$) and is smoothly evenly covered by a collection of ${\tilde{d}}$-open balls $\tilde{B}_j=B_r({\tilde{p}}_j)\subset ({\widetilde{M}},{\tilde{d}})$, $1\leq j\leq |G|$. For each $j$, the map ${\rm quo}|_{\tilde{B}_j}: \tilde{B}_j\to B$ is an isometry.* : Since $d$ is the geodesic-distance function of a Riemannian metric on $M$, the metric-space topology of $(M,d)$ coincides with the manifold topology. Hence the metrically open ball $B$ is also open in the manifold topology. (This would *not* be true for an *arbitrary* distance-function on $M$.) Since $r<{r_{\rm inj}}(M,g)\leq {r_{\rm inj}}({\widetilde{M}},\tilde{g})$, the exponential maps $\exp_{p}$, $\exp_{\tilde{p}}$, restricted to the open balls of radius $r$ centered at $0$ in $T_{p}M$, $T_{\tilde{p}}{\widetilde{M}}$ respectively, are diffeomorphisms onto $B_r(p)$, $B_r({\tilde{p}})$, respectively. Equation [\[intertwine\]](#intertwine){reference-type="eqref" reference="intertwine"} in the proof of Lemma [Lemma 35](#lem:rinjbeta){reference-type="ref" reference="lem:rinjbeta"} then implies that ${\rm quo}|_{B_r({\tilde{p}})}$ is a diffeomorphism onto $B_r(p)$. Since ${\rm quo}$ is a local isometry, it follows that ${\rm quo}|_{B_r({\tilde{p}})}$ is an isometry. It remains only to show that for distinct ${\tilde{p}}_1,{\tilde{p}}_2\in {\rm quo}^{-1}(p)$, the sets $\tilde{B}_j:=B_r({\tilde{p}}_j)$ ($i=1,2$) are disjoint. Suppose there exists $q\in \tilde{B}_1\cap\tilde{B}_2$. Then there are minimal geodesics $\gamma_j:[0,1]\to {\widetilde{M}}$ from ${\tilde{p}}_j$ to ${\tilde{q}}$ ($i=1,2$), each of length less than $r$, and $\gamma_1'(1)\neq \gamma_2'(1)$ (else both ${\tilde{p}}_1$ and ${\tilde{p}}_2$ would be images of the same vector in $T_{\tilde{q}}{\widetilde{M}}$ under $\exp_{\tilde{q}}$). Hence $\bar{\gamma}_j:={\rm quo}\circ\gamma_j$ ($j=1,2$) is a geodesic from $p$ to $q\in B_r(p)$ of length less than $r$, and $\bar{\gamma}_1'(1)\neq \bar{\gamma}_2'(1)$ (since ${\rm quo}_{*{\tilde{q}}}$ is an isomorphism); thus $\bar{\gamma}_1\neq\bar{\gamma}_2$. But then the restriction of $\exp_{p}$ to the ball $B_r(0_{p})\subset T_{p}{\widetilde{M}}$ is not one-to-one, a contradiction. Hence $\tilde{B}_1\cap\tilde{B}_2=\emptyset.$ height10pt width10pt\ *Remark 38*. In the setting of Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"}, the $G$-action permutes the sets $\tilde{B}_j$. While there is, in general, no *canonical* labeling of the $\tilde{B}_j$ by elements of $G$, if we arbitrarily label any one of these balls as $\tilde{B}_e$, then we can (re-)label all of the others by setting $\tilde{B}_h=h\,{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}\tilde{B}_e$ for all $h\in G$. We then have $h_1\,{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}\tilde{B}_{h_2}=\tilde{B}_{h_1h_2}$ for all $h_1,h_2\in G,$ so we call such a labeling *equivariant*. This facilitates the statement of the next proposition, a corollary of earlier results. **Proposition 39**. *Let ${r}\in (0,{r_{\rm cx}}(M,g)]$ and let $Q=(q_1,\dots,q_N)\in M^N$. Assume that ${\rm supp}(Q)$ lies in a ball $B$ of radius less than $r$. Let $\{\tilde{B}_h\}_{h\in G}$ be the even covering of $B$ given by Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"}, with the covering balls labeled equivariantly, and for each $h\in G$ and $p\in B$, let $\tilde{p}^{(h)}$ be the unique element of ${\rm quo}^{-1}(p)\cap \tilde{B}_h$. Let ${\tilde{Q}}^{(h)}=(\tilde{q}_1^{(h)}, \dots, \tilde{q}_N^{(h)})$. Then* - *For each $h\in G$, ${\rm FM}({\tilde{Q}}^{(h)})$ consists of a single point and lies in $\tilde{B}_h$.* - *For all $h_1,h_2\in G$ we have $h_1{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\rm FM}({\tilde{Q}}^{(h_2)}) ={\rm FM}({\tilde{Q}}^{(h_1h_2)}).$* - *${\rm EFM}(Q)=\{{\rm FM}({\tilde{Q}}^{(h)})\}_{h\in G} \subset {\rm quo}^{-1}(B').$ In particular, the equivariant Fréchet mean of $Q$ is unique up to the action of $G$.* - *${\rm FM}(Q)={\rm quo}({\rm EFM}(Q))={\rm quo}({\rm FM}({\tilde{Q}}^{(h)}))$ for all $h\in G$, and consists of a single point lying in $B$. Since ${\rm EFM}(Q)$ is a $G$-invariant set, the first equality is equivalent to ${\rm EFM}(Q)={\rm quo}^{-1}({\rm FM}(Q))$.* : (a) Let $h\in G$. By Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"}, $\tilde{B}_h$ is a ball of radius $r<{r_{\rm cx}}(M,g)\leq {r_{\rm cx}}({\widetilde{M}},\tilde{g})$. Hence Theorem [Theorem 19](#thm:afsari){reference-type="ref" reference="thm:afsari"} implies that the Fréchet mean of ${\tilde{Q}}^{(h)}$ is unique and lies in $\tilde{B}_h$. \(b\) Since $h_1{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}B'_{h_2}= B'_{h_1h_2}$ we have $h_1{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{Q}}^{(h_2)}= {\tilde{Q}}^{(h_1h_2)}$. Since the metric on $M$ is $G$-invariant, it follows that $h_1{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\rm FM}({\tilde{Q}}^{(h_2)})= {\rm FM}({\tilde{Q}}^{(h_1h_2)})$. \(c\) Follows from Proposition [Proposition 33](#prop:uniqueuptoG){reference-type="ref" reference="prop:uniqueuptoG"}(a). \(d\) Follows from (c), (a), and Proposition [Proposition 33](#prop:uniqueuptoG){reference-type="ref" reference="prop:uniqueuptoG"}(b). height10pt width10pt\ # Fréchet sample means in the equivariant setting {#sec:iid_eqvt_means} $~$ In this section, $({\widetilde{M}},\tilde{g})$, $G$, $(M,g)$, and ${\rm quo}$ are as in Section [7](#sec:eqvt_means_general){reference-type="ref" reference="sec:eqvt_means_general"}. Given any set $S$, any function $f:S\to {\widetilde{M}}$, and any $h\in G$, let $h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}f$ denote the function $x\mapsto h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}f(x)$. This defines a (left) action of $G$ on the set of functions from $S$ to ${\widetilde{M}}$. We will use the term *minimal invariant family of functions (from $S$ to ${\widetilde{M}}$)* for an orbit of $G$ under this action, and *equivariant family of functions (from $S$ to ${\widetilde{M}}$)* for an indexed collection $\{f^{(h)}:S\to {\widetilde{M}}\}_{h\in G}$ satisfying $f^{(h_1h_2)}=h_1{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}f^{(h_2)}$ for all $h_1,h_2\in G$. (In the latter case, we are opting for terminology that is simpler than the more precise "*equivariantly indexed minimal family of functions (from $S$ to ${\widetilde{M}}$)*".) For these families of functions, there will be only two types of domains of concern to us: the domain $\Omega$ of a random variable, and subsets of $M$. In each case the domain $S$ will be clear from context, so we will usually omit the "from $S$ to ${\widetilde{M}}$." Clearly the underlying set of functions in an equivariant family of functions is a minimal invariant family. Conversely, given a minimal invariant family ${\mathcal{F}}$ , any chosen element $f_1\in {\mathcal{F}}$ determines an equivariant indexing of ${\mathcal{F}}$ by setting $f{(h)}=h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}f_1$ for each $h\in G$. Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"} and Remark [Remark 38](#relabel){reference-type="ref" reference="relabel"} illustrate one way that such families of functions arise naturally. Consider a $d$-open ball $B\subset M$ that is smoothly evenly covered by a collection of ${\tilde{d}}$-open balls $\tilde{B}_j$, $1\leq j\leq |G|$. For each such $j$ let $s_j=({\rm quo}|_{\tilde{B}_j})^{-1}:B\to \tilde{B}_j$, but viewed as a map from $B$ to ${\widetilde{M}}$ (a *local section* of the bundle ${\widetilde{M}}\stackrel{{\rm quo}}{\longrightarrow} M$) . Then $\{s_j: 1\leq j\leq |G|\}$ is a minimal invariant family of smooth maps. A re-indexing of the collection $\{\tilde{B}_j\}$ by the elements of $G$, as in Remark [Remark 38](#relabel){reference-type="ref" reference="relabel"}, yields a corresponding re-indexing of this collection of local sections. The re-indexed family $\{s_h\}_{h\in G}$ is an equivariant family of smooth maps. *Definition 40*. We will call a map $s:M\to{\widetilde{M}}$ a *measurable section* (of the $G$-bundle ${\widetilde{M}}\stackrel{{\rm quo}}{\longrightarrow} M$) if $s$ is measurable and ${\rm quo}\circ s={\rm id}_M$. Given a random variable $X:\Omega\to M$, we call a map ${\tilde{X}}:\Omega\to {\widetilde{M}}$ a *measurable lift* of $X$ if ${\tilde{X}}=s\circ X$ for some measurable section $s$. Observe that a measurable lift ${\tilde{X}}$ of a measurable map $X:\Omega\to M$ is automatically measurable (see the discussion near the end of Section [1](#sect:intro){reference-type="ref" reference="sect:intro"}), hence is an ${\widetilde{M}}$-valued random variable for which ${\rm quo}\circ {\tilde{X}}=X.$ The (left) principal $G$-bundle ${\widetilde{M}}\stackrel{{\rm quo}}{\longrightarrow} M$ need not be trivial, hence may have no *continuous* sections (continuous maps $s:M\to{\widetilde{M}}$ such that ${\rm quo}\circ s={\rm id}_M$). However, there always exist *measurable* sections: **Lemma 41**. *There exists a measurable section $s:M\to {\widetilde{M}}$. In fact, for any $p\in M$, there exists a measurable section that is smooth on the complement of the cut-locus of $p$.* : Fix any $p\in M$ and any $\tilde{p}\in {\rm quo}^{-1}(p)$. As mentioned in Section [2](#sec:bary-prebary){reference-type="ref" reference="sec:bary-prebary"}, (i) the cut-locus ${\mathcal{C}}:={\mathcal{C}}(p)$ is a closed set of volume zero, (ii) the set ${\mathcal D}^{\rm tan}(p)\subset T_pM$ is star-shaped with respect to $0_p$ (so its diffeomorphic image ${\mathcal D}:={\mathcal D}(p)=M\backslash{\mathcal{C}}$ is contractible), and (iii) $M={\mathcal D}\,{\mbox{\small $\coprod$}}\, {\mathcal{C}}$. Since ${\rm quo}:{\widetilde{M}}\to M$ is a submersion, Lemma [Lemma 10](#submersions_and_meas_zero){reference-type="ref" reference="submersions_and_meas_zero"} implies that the set $\tilde{{\mathcal{C}}}:={\rm quo}^{-1}({\mathcal{C}})\subset{\widetilde{M}}$ has volume zero. Since ${\mathcal D}$ is contractible, the restricted bundle ${\widetilde{M}}|_{{\mathcal D}}$ is trivial, so there exists a continuous map $s_1:{\mathcal D}\to {\widetilde{M}}$ such that ${\rm quo}\circ s_1={\rm id}_{\mathcal D}$. Since ${\rm quo}:{\widetilde{M}}\to M$ is a local diffeomorphism, the continuity of $s_1$ implies that $s_1$ is smooth. For each $q\in {\mathcal{C}}$, arbitrarily select a point $\tilde{q}\in {\rm quo}^{-1}(q)$, thereby constructing a map $s_2:{\mathcal{C}}\to {\widetilde{M}}$ satisfying ${\rm quo}\circ s_2={\rm id}_{\mathcal{C}}$. Then the map $s:M\to {\widetilde{M}}$ defined by setting $s|_{{\mathcal D}}=s_1$ and $s|_{{\mathcal{C}}}=s_2$ satisfies ${\rm quo}\circ s={\rm id}_M$, is continuous on ${\mathcal D}$, and is easily seen to be measurable. height10pt width10pt\ Lemma [Lemma 41](#lem:meas_triv){reference-type="ref" reference="lem:meas_triv"} can be generalized to any principal bundle over $M$; thus any such bundle is *measurably trivial*. For the principal $G$-bundle ${\widetilde{M}}\stackrel{{\rm quo}}{\longrightarrow} M$, this is essentially the content of Corollary [Corollary 42](#cor:meas_triv){reference-type="ref" reference="cor:meas_triv"}(a) below. **Corollary 42**. *(a) Let $p\in M$. There exists an equivariant family $\{s_h\}_{h\in G}$ of measurable sections of ${\widetilde{M}}\stackrel{{\rm quo}}{\longrightarrow} M,$ each of which is smooth on the open, generic set ${\mathcal D}(p)$. (b) Let $X$ be an $M$-valued a random variable. Then an equivariant family $\{{\tilde{X}}^{(h)}\}_{h\in G}$ of measurable lifts of $X$ exists. If $X$ is absolutely continuous, then an equivariant family of *absolutely continuous* measurable lifts exists.* : (a) Let $s:M\to {\widetilde{M}}$ be a measurable section that is continuous on ${\mathcal D}(p)$. For each $h\in G$ let $s_h=h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}s$. Then $\{s_h\}_{h\in G}$ is an equivariant family of measurable sections of ${\widetilde{M}}\stackrel{{\rm quo}}{\longrightarrow} M,$ each of which is continuous on ${\mathcal D}(p)$. \(b\) Select any $p\in M$ and let $\{s_h\}_{h\in G}$ be as above. For each $h\in G$ let ${\tilde{X}}^{(h)}=s_h\circ X.$ Then $\{{\tilde{X}}^{(h)}\}_{h\in G}$ is an equivariant family of measurable lifts of $X$. Assume now that $X$ is absolutely continuous, and let $\mu_X$ be the probability measure on $M$ induced by $X$. Let $E\subset {\widetilde{M}}$ be a set of volume zero, and let $h\in G$. Then $({\tilde{X}}^{(h)})^{-1}(E)=X^{-1}(s_h^{-1}(E)) =X^{-1}({\rm quo}(E)),$ so $\mu_{{\tilde{X}}^{(h)}}(E)=\mu_X({\rm quo}(E)).$ Since the map ${\rm quo}$ is a local diffeomorphism, ${\rm quo}(E)$ has volume zero in $M$, implying that $\mu_X({\rm quo}(E))=0$ (since $\mu_X$ is absolutely continuous). Hence $\mu_{{\tilde{X}}^{(h)}}$ is absolutely continuous. height10pt width10pt\ *Remark 43*. Suppose that $X$ is an absolutely continuous $M$-valued random variable whose induced measure $\mu_X$ is supported in an open ball $B=B_r(p)$, where $r<{r_{\rm inj}}(M,g)$. Let $\{\tilde{B}_h\}_{h\in G}$ be the even covering of $B$ given by Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"}, relabeled as in Remark [Remark 38](#relabel){reference-type="ref" reference="relabel"}. Let $\{s_h\}_{h\in G}$ be an equivariant family of measurable sections of ${\widetilde{M}}\stackrel{{\rm quo}}{\longrightarrow} M$ for which $s_h|_{{\mathcal D}(p)}$ is smooth (for each $h\in G$). Note that $B\subset {\mathcal D}(p)$ since $r<{r_{\rm inj}}(M,g)$. Then by equivariance, each local section $s_h$ must be precisely the map $({\rm quo}|_{\tilde{B}_h})^{-1}$ (viewed as a map from $B$ to ${\widetilde{M}}$). Furthermore, the measure $\mu_{{\tilde{X}}^{(h)}}$ is supported in $\tilde{B}_h$ and is canonically determined by $X$, in the sense that $\mu_{{\tilde{X}}^{(h)}}$ depends only on the canonically defined local section $s_h|_B$ (and on $X$). In view of this observation, we make the following simplifying definition: *Definition 44*. In the setting of Remark [Remark 43](#rem:assocrv){reference-type="ref" reference="rem:assocrv"}, we refer to any equivariant family $\{{\tilde{X}}^{(h)}\}_{h\in G}$ determined by an equivariant family of measurable sections that are continuous on $B$, as *"the" associated lifts of $X$* to ${\widetilde{M}}$-valued random variables (implicitly identifying any two random variables that coincide a.e.). **Corollary 45**. *Let $\tilde{A}\subset{\widetilde{M}}$ be a $G$-invariant set of volume zero in ${\widetilde{M}}$, let $A={\rm quo}(\tilde{A})$, and write ${\widetilde{M}}_*={\widetilde{M}}\setminus\tilde{A}$ and $M_*={\rm quo}({\widetilde{M}}_*)=M\setminus A.$ Let $X, X_1, X_2,\dots, X_N$ be i.i.d. $M$-valued random variables with induced probability measure $\mu_X$ and for $1\leq i\leq N$ let $\{{\tilde{X}}_i^{(h)}\}_{h\in G}$ be an equivariant family of measurable lifts of $X$. (Note that for each $h$, the random variables ${\tilde{X}}_1^{(h)}, \dots, {\tilde{X}}_N^{(h)}$ are i.i.d.)* *(a) For all $h,h_1,h_2\in G$ we have $$\begin{aligned} h_1{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\rm FM}(\{{\tilde{X}}_1^{(h_2)},\dots,{\tilde{X}}_N^{(h_2)}\}) &=&{\rm FM}(\{{\tilde{X}}_1^{(h_1h_2)},\dots,{\tilde{X}}_N^{(h_1h_2)}\})\end{aligned}$$ and $$\begin{aligned} {\rm quo}({\rm FM}(\{{\tilde{X}}_1^{(h)},\dots,{\tilde{X}}_N^{(h)}\}))&=& {\rm FM}(\{X_1,\dots,X_N\}).\end{aligned}$$ Furthermore, ${\rm EFM}(\{X_1,\dots,X_N\})={\rm quo}^{-1}({\rm FM}(\{X_1,\dots,X_N\}) ).$* *(b) Assume now that $X$ is absolutely continuous. Then* - *Almost surely, ${\rm FM}(\{X_1,\dots,X_N\})\subset M_*$ and, correspondingly, ${\rm EFM}(\{X_1,\dots,X_N\})\subset{\widetilde{M}}_*$ .* - *For every $h\in G$, almost surely ${\rm FM}(\{{\tilde{X}}_1^{(h)},\dots,{\tilde{X}}_N^{(h)}\})\subset{\widetilde{M}}_*$.* *(c) Let $r\in (0,{r_{\rm cx}}(M,g)]$, and assume that $X$ is absolutely continuous and supported in an open ball $B$ of radius less than $r$. Then* - *Almost surely, the Fréchet mean set of $\{X_1,\dots,X_N\}$ is a single point and lies in $M_*\cap B$. Correspondingly, the equivariant Fréchet means of $\{X_1,\dots,X_N\}$ are unique up to the action of $G$, and lie in ${\widetilde{M}}_*\cap{\rm quo}^{-1}(B)$.* - *Let $\{\tilde{B}_h\}_{h\in G}$ be the even covering of $B$ given by Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"}, with the sets relabeled as in Remark [Remark 38](#relabel){reference-type="ref" reference="relabel"}, and assume that the equivariant family $\{{\tilde{X}}_i^{(h)}\}_{h\in G}$ consists of the associated lifts of $X_i$ , $1\leq i\leq N$. For every $h\in G$, almost surely the Fréchet mean set of $\{{\tilde{X}}_1^{(h)},\dots,{\tilde{X}}_N^{(h)}\}$ is a single point and lies in ${\widetilde{M}}_*\cap \tilde{B}_h$.* : (a) This follows from Proposition [Proposition 33](#prop:uniqueuptoG){reference-type="ref" reference="prop:uniqueuptoG"}(a). \(b\) Since ${\rm quo}$ is a local diffeomorphism, the set $A$ has volume zero in $M$. Statements (i) and (ii) therefore follow from Corollary [Corollary 28](#cor:measzero.3){reference-type="ref" reference="cor:measzero.3"}(a). \(c\) This follows from part (b) and Proposition [Proposition 39](#prop:equivar_means){reference-type="ref" reference="prop:equivar_means"}. height10pt width10pt\ Theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"}, which is simply Corollary [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"} minus the statements that explicitly involve the lifts $\{X^{(h)}\}$, is now proven. *Remark 46*. If $({\widetilde{M}},\widetilde{\mathcal{T}})$ is a weakly stratified manifold in sense of Remark [Remark 29](#weakstrat){reference-type="ref" reference="weakstrat"}, and for any $h\in G$ and any stratum $\widetilde{T}\in \widetilde{\mathcal{T}}$, the set $h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}\widetilde{T}$ is also a stratum, then $M$ inherits a quotient (weak) stratification by declaring the strata of $M$ to be the images of the strata of ${\widetilde{M}}$ under ${\rm quo}$. In Corollary [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"} we can then take $\tilde{A}$ to be the union of all strata of ${\widetilde{M}}$ other than the top stratum. Then Corollary [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"} applies with ${\widetilde{M}}_*$ and $M_*$ replaced by by ${\widetilde{M}}^{\rm top}$ and $M^{\rm top}$, respectively. # An application to "partial scaling-rotation means" {#sec:PSRmeans} In this section we apply results from earlier sections to a problem concerning symmetric positive-definite (SPD) matrices that was the initial motivation behind Theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"}: establishing a strong relation between *scaling-rotation (SR) means* of samples of SPD matrices, and the more easily computed *partial scaling-rotation (PSR) means* of such samples, notions introduced in [@rjgs]. To describe these means and the application of Theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"}, we first review the "scaling-rotation framework" introduced in earlier work of the authors. Let ${\rm Sym}^+(m)$ denote the space of (real) $m\times m$ SPD matrices. In [@Jung2015] the authors defined a "distance-function\" $d_{\textsc {sr}}$ (not a true metric) on ${\rm Sym}^+(m),$ studied further in [@gjs2017; @gjs2021], that measures rotation of eigenvectors and scaling of eigenvalues. For these purposes, an element $S\in {\rm Sym}^+(m)$ is represented as an eigendecomposition, a pair $(U,D)\in {\widetilde{M}}(m):=SO(m)\times {\rm Diag}^+(m)$ for which ${\mathcal{F}}(U,D):=UDU^T=S$. (Thus ${\mathcal{F}}^{-1}(S)$ is the set of eigendecompositions of $S$.) The finite group ${\mathcal{G}}(m)$ of *even, signed permutations*, corresponding to orientation-preserving combinations of permutations and sign-changes of eigenvectors, acts freely on ${\widetilde{M}}(m)$ (see [@gjs2017 Section 2.3]), thereby determining a quotient manifold $M(m):={\widetilde{M}}(m)/{\rm Sym}^+(m)$ and a projection map ${\rm quo}:{\widetilde{M}}(m)\to M(m)$. As depicted in the left-hand commutative diagram in Figure [\[comdiag\]](#comdiag){reference-type="ref" reference="comdiag"}, the manifold $M(m)$ sits "between" ${\widetilde{M}}(m)$ and ${\rm Sym}^+(m)$: each ${\mathcal{G}}(m)$-orbit is contained in a fiber of the "eigencomposition" map ${\mathcal{F}}: {\widetilde{M}}(m)\to {\rm Sym}^+(m),$ so ${\mathcal{F}}$ descends to a well-defined map $\bar{{\mathcal{F}}}:M(m)\to{\rm Sym}^+(m)$ satisfying $$\label{Ffactors} {\mathcal{F}}=\bar{{\mathcal{F}}}\circ\,{\rm quo}.$$ Note that if (and only if) $S\in {\rm Sym}^+(m)$ has distinct eigenvalues, then ${\mathcal{F}}^{-1}(S)$ is precisely an orbit of the ${\mathcal{G}}(m)$-action. Thus the manifolds ${\rm Sym}^+(m)$ and $M(m)$ are not identical. However, they have full-volume-measure subsets that are "essentially the same". Specifically and more precisely, the subset $S_m^{\rm top}\subset{\rm Sym}^+(m)$ consisting of matrices with distinct eigenvalues, and the subset ${\widetilde{M}}^{\rm top}(m):={\mathcal{F}}^{-1}({\widetilde{M}}(m))\subset {\widetilde{M}}(m),$ are open, generic subsets of their ambient spaces. The set ${\widetilde{M}}^{\rm top}(m)$ is ${\mathcal{G}}(m)$-invariant, and $S_m^{\rm top}$ can be identifed with the quotient of ${\widetilde{M}}(m)^{\rm top}$ by the ${\mathcal{G}}(m)$-action. (We regard ${\widetilde{M}}^{\rm top}(m), M^{\rm top}(m)$ as the "top strata" of their ambient spaces; see [@gjs2017].) Since ${\widetilde{M}}^{\rm top}(m)$ is open in ${\widetilde{M}}(m)$, its image $M^{\rm top}(m)$ under the covering-map ${\rm quo}$ is open in $M(m)$. The restriction of ${\mathcal{F}}$ to ${\widetilde{M}}^{\rm top}(m)$ is a covering map ${\widetilde{M}}^{\rm top}(m)\to S_m^{\rm top},$ and the restriction of $\bar{{\mathcal{F}}}$ to $M^{\rm top}(m)$ is thus a diffeomorphism $$\bar{{\mathcal{F}}}_{\rm top}:=\bar{{\mathcal{F}}}|_{M^{\rm top}(m)}:M^{\rm top}(m)\to S_m^{\rm top}\ ;$$ see the right-hand commutative diagram in Figure [\[comdiag\]](#comdiag){reference-type="ref" reference="comdiag"}. Thus $\bar{{\mathcal{F}}}_{\rm top}$ naturally identifies $M^{\rm top}(m)$ and $S_m^{\rm top}$ as manifolds, not just as point-sets. It is in this sense that these subsets of the different spaces $M(m)$ and ${\rm Sym}^+(m)$ are "essentially the same". ---------------- -------------------- ----------- -- \>quo \^ M(m) \>\| Sym\^+(m) \>quo\|\_\^top(m) \^\|\_\^top(m) M\^top(m) \>\|\_top (diffeo) S_m\^top ---------------- -------------------- ----------- -- The Lie groups $SO(m)$ and ${\rm Diag}^+(m)$ carry natural bi-invariant Riemannian metrics $g_{SO}$ and $g_{{\mathcal D}^+}$, respectively. (These are complete and are canonical up to scale; for our choice of scale see [@gjs2017 Section 3.1] .) We define a product Riemannian metric $\tilde{g}$ on ${\widetilde{M}}(m)$ by setting $\tilde{g}=kg_{SO}\oplus g_{{\mathcal D}^+,}$ where $k>0$ is an arbitrary relative-weight parameter. The action of ${\mathcal{G}}(m)$ on ${\widetilde{M}}(m)$ is then isometric. We are thus in the situation of Section [7](#sec:eqvt_means_general){reference-type="ref" reference="sec:eqvt_means_general"}: the manifold $M(m)$ then inherits a (unique) Riemannian metric $g$ for which the map ${\rm quo}:{\widetilde{M}}(m)\to M(m)$ is a principal Riemannian covering map carrying $\tilde{g}$ to $g$. Thus all the results of Section [7](#sec:eqvt_means_general){reference-type="ref" reference="sec:eqvt_means_general"} apply, with ${\widetilde{M}}(m)$ as the covering space and $M(m)$ as the base space. As in Section [7](#sec:eqvt_means_general){reference-type="ref" reference="sec:eqvt_means_general"}, we will denote the distance-functions (between points or subsets) arising from $\tilde{g}$ and $g$ as ${\tilde{d}}$ and $d$, respectively. Define functions $d_{\textsc {sr}}:{\rm Sym}^+(m)\times{\rm Sym}^+(m)\to{\bf R}$ (the *scaling-rotation distance*) and $d_{\textsc {psr}}: {\rm Sym}^+(m) \times {\widetilde{M}}(m)\to{\bf R}$ (the *partial scaling-rotation distance*) by $$\begin{aligned} \nonumber d_{\textsc {sr}}(S_1,S_2) &=&{\tilde{d}}({\mathcal{F}}^{-1}(S_1),{\mathcal{F}}^{-1}(S_2)), \\ d_{\textsc {psr}}(S, {\tilde{S}}_0) &=& \tilde{d}( \mathcal{F}^{-1}(S), {\tilde{S}}_0). \label{defdpsr}\end{aligned}$$ Note that despite the formal similarity between equations [\[defdpsr\]](#defdpsr){reference-type="eqref" reference="defdpsr"} and [\[defdevt\]](#defdevt){reference-type="eqref" reference="defdevt"}, $d_{\textsc {psr}}$ does not meet our definition of "equivariant distance function" on its full domain, since ${\widetilde{M}}(m)$ is a covering space only of $M(m)$, not of ${\rm Sym}^+(m)$. *Remark 47*. For $S_1,S_2\in S_m^{\rm top}$, $$\begin{aligned} \nonumber d_{\textsc {sr}}(S_1,S_2)&=&{\tilde{d}}({\mathcal{F}}^{-1}(S_1),{\mathcal{F}}^{-1}(S_2))\\ \nonumber &=& d({\rm quo}({\mathcal{F}}^{-1}(S_1)),{\rm quo}({\mathcal{F}}^{-1}(S_2))) \ \ \ \mbox{by Proposition \ref{quodist} }\\ &=& d(\bar{{\mathcal{F}}}_{\rm top}^{-1}(S_1),\bar{{\mathcal{F}}}_{\rm top}^{-1}(S_2)). \label{dbareqdsr}\end{aligned}$$ This shows that $d_{\textsc {sr}}^{\rm top}$, the restriction of $d_{\textsc {sr}}$ to $S_m^{\rm top}\times S_m^{\rm top}$, is a metric (precisely the result of [@Jung2015 Theorem 3.12], but proven by a somewhat different argument), and that, writing $d^{\rm top}$ for the restriction of $d$ to $M^{\rm top}(m)$, $$\label{dbareqdsr.2} \mbox{The diffeomorphism $\bar{{\mathcal{F}}}_{\rm top}$ is an isometry $(M^{\rm top}(m),d^{\rm top})\to (S_m^{\rm top}, d_{\textsc {sr}}^{\rm top}).$}$$ For any configuration ${\mathcal S}= (S_1,\ldots,S_N)\in ({\rm Sym}^+(m))^N$, we define functions $f^{{\textsc {sr}}}_{{\mathcal S}}:{\rm Sym^+}(m)\to {\bf R}$ and $\tilde{f}^{{\textsc {psr}}}_{{\mathcal S}}:{\widetilde{M}}(m)\to{\bf R}$ by $$\begin{aligned} \label{def_fsrs} f^{{\textsc {sr}}}_{{\mathcal S}}(S)&=&\frac{1}{N}\sum_{i=1}^N {d_{\textsc {sr}}}(S_i, S)^2, \\ \tilde{f}^{{\textsc {psr}}}_{{\mathcal S}}({\tilde{S}})&=& \frac{1}{N}\sum_{i=1}^N d_{\textsc {psr}}( S_i, {\tilde{S}})^2. \label{def_fpsrs}\end{aligned}$$ As in [@rjgs], we define the *scaling-rotation* (SR) *mean set* and *partial scaling-rotation* (PSR) *mean set* of ${\mathcal S}$, denoted $E_N^{{\textsc {sr}}}({\mathcal S})$ and ${\tilde{E}}_N^{{\textsc {psr}}}({\mathcal S})$ respectively, by $$\begin{aligned} \label{srmeanset} E_N^{{\textsc {sr}}}({\mathcal S}) &:=& \mathop{\mathrm{arg\,min}}(f^{{\textsc {sr}}}_{{\mathcal S}}) \ \ \subset {\rm Sym}^+(m),\\ \label{psrmeanset} {\tilde{E}}_N^{{\textsc {psr}}}({\mathcal S}) &:=& \mathop{\mathrm{arg\,min}}(\tilde{f}^{{\textsc {psr}}}_{{\mathcal S}}) \ \ \subset {\widetilde{M}}(m).\end{aligned}$$ Corollary 4.10 of [@rjgs] establishes that SR and PSR means always exist (i.e. that for any $N$-point configuration ${\mathcal S}$, the sets $E_N^{{\textsc {sr}}}({\mathcal S})$ and ${\tilde{E}}_N^{{\textsc {psr}}}({\mathcal S})$ are nonempty). Our main focus in this section will be on PSR means rather than SR means, for several reasons. One reason is that the SR mean set of a general configuration ${\mathcal S}$ can be very hard to compute, even when ${\mathcal S}$ has a unique SR mean, since already the definition of $d_{\textsc {sr}}(\cdot,\cdot)$ involves a double-optimization. As a practical alternative to using SR-means, [@rjgs] proposed using PSR means to give more-easily computable approximations of SR means (upon projecting the PSR means to ${\rm Sym^+}(m)$ via ${\mathcal{F}}$). It is not clear, in general, how good this approximation is. However, Theorem 3.5 of [@rjgs] established that in one important situation, this approximation is perfect: **Theorem 48** ([@rjgs Theorem 3.5]). *Let $S_1, \dots S_N$ be elements of $S_m^{\rm top}$ and let ${\mathcal S}=(S_1,\dots, S_N)$. If $E_N^{{\textsc {sr}}}({\mathcal S})\subset S_m^{\rm top},$ then ${\tilde{E}}_N^{{\textsc {psr}}}({\mathcal S}) ={\mathcal{F}}^{-1}(E_N^{{\textsc {sr}}}({\mathcal S}))$ and ${\mathcal{F}}({\tilde{E}}_N^{{\textsc {psr}}}({\mathcal S})) =E_N^{{\textsc {sr}}}({\mathcal S})$.* Since $S_m^{\rm top}$ is an open, dense, full-measure subset of ${\rm Sym}( m),$ one might think that, given i.i.d. absolutely continuous ${\rm Sym}^+(m)$-valued random variables $X_1,\dots, X_N$, the genericity property ${\rm Pr}\big(E_N^{\textsc {sr}}(X_1,\dots, X_N)\subset S_m^{\rm top}\big)=1$ would hold. This intuition is not correct, essentially because the discontinuity of $d_{\textsc {sr}}$ leads to discontinuity of the SR objective functions [\[def_fsrs\]](#def_fsrs){reference-type="eqref" reference="def_fsrs"}. This is another reason we focus here on PSR means, which have nicer genericity properties. Our final reason for focusing here on PSR means is that for a finite configuration that lies entirely in the top stratum $S_m^{\rm top}\subset {\rm Sym}^+(m)$, the PSR mean set in ${\widetilde{M}}(m)$ is exactly the equivariant Fréchet mean set: **Proposition 49**. *Let $S_1,\dots, S_n\in S_m^{\rm top}$, and for $1\leq i\leq N$, let $S_i^M=\bar{{\mathcal{F}}}_{\rm top}^{-1}(S_i)$. Let ${\mathcal S}=(S_1,\dots,S_N)$ and ${\mathcal S}^M=(S^M_1,\dots,S^M_N)$.* *Then $$\label{bfbsm} \tilde{f}^{{\textsc {psr}}}_{{\mathcal S}}=\tilde{f}_{{\mathcal S}^M}$$ where the "equivariant Fréchet function" $\tilde{f}_{{\mathcal S}^M}$ is defined as in [\[deftfq\]](#deftfq){reference-type="eqref" reference="deftfq"}. Hence ${\tilde{S}}$ is a PSR mean of ${\mathcal S}$ if and only if ${\tilde{S}}$ is an equivariant Fréchet mean of ${{\mathcal S}}^M$, and ${\tilde{E}}_N^{\textsc {psr}}({\mathcal S})={\rm EFM}({\mathcal S}^M)$.* : Using the definition of $d_{\textsc {psr}}$, equation [\[Ffactors\]](#Ffactors){reference-type="eqref" reference="Ffactors"}, Proposition [Proposition 31](#quodist){reference-type="ref" reference="quodist"}, the injectivity of $\bar{{\mathcal{F}}}_{\rm top}$, and equation [\[dpar=dbar\]](#dpar=dbar){reference-type="eqref" reference="dpar=dbar"}, we have $$\begin{aligned} d_{\textsc {psr}}(S_i,{\tilde{S}})={\tilde{d}}({\mathcal{F}}^{-1}(S_i),{\tilde{S}}) &=& {\tilde{d}}({\rm quo}^{-1}(\bar{{\mathcal{F}}}^{-1}(S_i)),{\tilde{S}})\\ &=& d(\bar{{\mathcal{F}}}^{-1}(S_i),{\rm quo}({\tilde{S}})) \\ &=& d(\bar{{\mathcal{F}}}_{\rm top}^{-1}(S_i),{\rm quo}({\tilde{S}})) \\ &=& d(S^M_i,{\rm quo}({\tilde{S}})) \\ &=& d_{\rm evt}(S^M_i ,{\tilde{S}}).\end{aligned}$$ Equation [\[bfbsm\]](#bfbsm){reference-type="eqref" reference="bfbsm"} now follows from the definition of $\tilde{f}_{{\mathcal S}^M}$ (equation [\[deftfq\]](#deftfq){reference-type="eqref" reference="deftfq"}). height10pt width10pt\ Although the surjective maps $\bar{{\mathcal{F}}}:M(m)\to {\rm Sym^+}(m)$ and $F:{\widetilde{M}}(m)\to{\rm Sym^+}(m)$ are not covering maps, we can still define *measurable sections* for these maps, *measurable lifts* of ${\rm Sym}^+(m)$-valued random variables to $M(m)$-valued and ${\widetilde{M}}(m)$-valued random variables, and (${\mathcal{G}}(m)$-)*equivariant families of lifts*, to ${\widetilde{M}}(m)$, of a ${\rm Sym^+}(m)$-valued random variable: 1. Call a map $s:{\rm Sym^+}(m)\to M(m)$ a *set-theoretic section* of $\bar{{\mathcal{F}}}$ if $\bar{{\mathcal{F}}}\circ s$ is the identity map of ${\rm Sym^+}(m)$; equivalently, if $s(Y)\in \bar{{\mathcal{F}}}^{-1}(Y)$ for all $Y\in {\rm Sym^+}(m)$. If, in addition, the map $s$ is measurable, call $s$ a *measurable section*. Similarly, call a map $\tilde{s}:{\rm Sym^+}(m)\to{\widetilde{M}}(m)$ a set-theoretic section or measurable section of ${\mathcal{F}}$ if the appropriate condition(s) above hold with $(\bar{{\mathcal{F}}},s)$ replaced by $({\mathcal{F}},\tilde{s})$. 2. Call a map $X^M: \Omega\to M(m)$ (respectively, ${\tilde{X}}:\Omega\to{\widetilde{M}}(m)$) a *lift* of a map $X:\Omega\to{\rm Sym}^+(m)$ if $X=\bar{{\mathcal{F}}}\circ X^M$ (resp., $X={\mathcal{F}}\circ{\tilde{X}}$); equivalently, if $X^M=s\circ X$ for some set-theoretic section $s$ of $\bar{{\mathcal{F}}}$ (resp., if ${\tilde{X}}=\tilde{s}\circ X$ for some set-theoretic section $\tilde{s}$ of ${\mathcal{F}}$). If $X$ is measurable, call $X^M$ (resp. ${\tilde{X}}^M$) a *measurable lift* of $X$ if $X^M=s\circ X$ (resp. ${\tilde{X}}^M=\tilde{s}\circ X$) for some measurable section $s$ of $\bar{{\mathcal{F}}}$ (resp. $\tilde{s}$ of ${\mathcal{F}}$). Note that measurable lifts are always measurable maps. 3. Call an indexed collection of ${\widetilde{M}}(m)$-valued random variables $\{{\tilde{X}}^{(h)}\}_{h\in {\mathcal{G}}(m)}$ an *equivariant family of lifts* of a ${\rm Sym^+}(m)$-valued random variable $X$ if (i) ${\tilde{X}}^{(h)}$ is a lift of $X$ for each $h\in {\mathcal{G}}(m)$, and (ii) ${\tilde{X}}^{(h_1h_2)}=h_1{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{X}}^{(h_2)}$ for all $h_1,h_2\in {\mathcal{G}}(m)$. Let $S_m^{\rm low}={\rm Sym}^+(m)\backslash S_m^{\rm top}$ and ${\widetilde{M}}^{\rm low}(m)={\widetilde{M}}(m)\backslash {\widetilde{M}}^{\rm top}(m)$. As detailed in [@gjs2017 Sections 2.4--2.7], the closed subsets $S_m^{\rm low}$ and ${\widetilde{M}}^{\rm low}(m)$ are finite unions of certain positive-codimension submanifolds ("lower strata") of ${\rm Sym}^+(m)$ and ${\widetilde{M}}(m)$ respectively, and hence have volume zero in their ambient spaces. Since ${\rm quo}$ is a local diffeomorphism, it follows that $M^{\rm low}(m):={\rm quo}({\widetilde{M}}^{\rm low}(m))\subset M(m)$ is also a finite union of positive-codimension submanifolds and is a closed set. Thus each of the sets $S_m^{\rm low}, {\widetilde{M}}^{\rm low}(m),$ and $M^{\rm low}(m)$ is a closed volume-zero subset of its ambient space. **Lemma 50**. *(a) Set-theoretic sections of $\bar{{\mathcal{F}}}$ exist, and all such sections are measurable. For any such section $s$, if $U\subset M(m)$ has volume zero, then so does $s^{-1}(U)$.* *(b) Let $X$ be an absolutely continuous ${\rm Sym}^+(m)$-valued random variable $X$. Then:* - *Lifts of $X$ to $M(m)$ exist. All such lifts are absolutely continuous $M(m)$-valued random variables.* - *An equivariant family $\{{\tilde{X}}^{(h)}\}_{h\in {\mathcal{G}}(m)}$ of measurable, absolutely continuous lifts of $X$ to ${\widetilde{M}}(m)$ exists.* For a proof of Lemma [Lemma 50](#liftlemma2){reference-type="ref" reference="liftlemma2"}, see Appendix [\[sec:appendixA\]](#sec:appendixA){reference-type="ref" reference="sec:appendixA"}. **Proposition 51**. *Let $X_1, \dots, X_N$ be i.i.d., absolutely continuous, ${\rm Sym}^+(m)$-valued random variables, and let $X^M_1, \dots, X^M_N$ be lifts of $X_1, \dots X_N$ to $M(m).$ The following are true almost surely:* - *All PSR means of $\{X_1,\dots,X_N\}$ lie in the top stratum: $$\label{psrefm} {\tilde{E}}_N^{\textsc {psr}}(\{X_1,\dots, X_N\})={\rm EFM}(\{X_1^M,\dots, X_N^M\}) \subset {\widetilde{M}}^{\rm top}(m).$$* - *${\rm FM}(\{X^M_1,\dots, X^M_N\})= {\rm quo}\big({\tilde{E}}_N^{{\textsc {psr}}}(\{X_1,\dots, X_N\})\big) ={\rm quo}({\rm EFM}(\{X^M_1,\dots, X^M_N\})$   and    ${\rm quo}^{-1}\big({\rm FM}(\{X^M_1,\dots, X^M_N\})\big)= {\tilde{E}}_N^{{\textsc {psr}}}(\{X_1,\dots, X_N\}) ={\rm EFM}(\{X^M_1,\dots, X^M_N\})$.* : (a) By Lemma [Lemma 50](#liftlemma2){reference-type="ref" reference="liftlemma2"}(b), the lifts $X_1^M, \dots, X_N^M$ are absolutely continuous $M(m)$-valued random variables. For each Since ${\widetilde{M}}^{\rm low}(m)={\rm quo}^{-1}( M^{\rm low}(m))$ is a ${\mathcal{G}}(m)$-invariant volume-zero subset of ${\widetilde{M}}(m)$, Corollary [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"}(b) shows that the event (i) "${\rm EFM}(\{X_1^M,\dots, X_N^M\})\subset {\widetilde{M}}^{\rm top}(m)$" occurs almost surely. Since $S_m^{\rm low}$ has volume zero and $X_1,\dots X_N$ are absolutely continuous, the event (ii) "$X_1,\dots, X_N$ all lie in $S_m^{\rm top}$" occurs almost surely as well. Hence the event "(i) and (ii)" also occurs almost surely. But whenever (ii) occurs, Proposition [Proposition 49](#pmequivalence){reference-type="ref" reference="pmequivalence"} implies that the equality in [\[psrefm\]](#psrefm){reference-type="eqref" reference="psrefm"} holds. It follows that ${\tilde{E}}_N^{\textsc {psr}}(\{X_1,\dots, X_N\})\subset {\widetilde{M}}^{\rm top}(m)$ almost surely. \(b\) Proposition [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"}(a) implies that we *always* (not just almost surely) have that ${\rm EFM}(\{X_1^M,\dots, X_N^M\})={\rm quo}^{-1}({\rm FM}(X_1^M, \dots, X_N^M)),$ which, since ${\rm quo}$ is surjective, implies that ${\rm quo}({\rm EFM}(\{X_1^M,\dots, X_N^M\}))={\rm FM}(X_1^M, \dots, X_N^M)$ as well. Combining this with the almost-sure statement [\[psrefm\]](#psrefm){reference-type="eqref" reference="psrefm"} yields the result. height10pt width10pt\ To apply results of Corollary [Corollary 28](#cor:measzero.3){reference-type="ref" reference="cor:measzero.3"} and Proposition [Proposition 39](#prop:equivar_means){reference-type="ref" reference="prop:equivar_means"} in the context of PSR means, we review some previously established geometric facts about the manifolds $(M(m),g)$ and $({\widetilde{M}}(m), \tilde{g})$. As is well known, the manifold $({\rm Diag}^+(m),g_{{\mathcal D}^+})$ has non-positive sectional curvature and infinite injectivity radius, while $(SO(m),g_{SO})$ has non-negative sectional curvature, so ${r_{\rm inj}}({\widetilde{M}}(m),\tilde{g})={r_{\rm inj}}(SO(m), kg_{SO})$ and $\Delta({\widetilde{M}}(m),\tilde{g})= \Delta(SO(m), kg_{SO})$. With the normalization chosen for $g_{SO}$ in [@rjgs], ${r_{\rm inj}}(SO(m), kg_{SO})=\sqrt{k}\pi$ and $\Delta(SO(m), kg_{SO})=k^{-1}/4.$ It is shown in [@rjgs Lemma 4.4] that $\inf\{{\tilde{d}}(p,h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}p) \ : \ h\in {\mathcal{G}}(m), p\in {\widetilde{M}}(m)\}= \min\{d_{SO}(h,{\rm id.})\ : \ h\in {\mathcal{G}}(m)\setminus\{{\rm id.}\}\}=:\beta_m\leq \frac{\pi}{2}$, so $$\label{rcxmp} {r_{\rm cx}}({\widetilde{M}}(m),\tilde{g})\ =\ \frac{\pi}{2} \sqrt{k}\ \geq \ \sqrt{k}\beta_m.$$ Via Lemma [Lemma 35](#lem:rinjbeta){reference-type="ref" reference="lem:rinjbeta"} and Corollary [Corollary 36](#cor:rcxbeta){reference-type="ref" reference="cor:rcxbeta"}, the inequality in [\[rcxmp\]](#rcxmp){reference-type="eqref" reference="rcxmp"} then additionally yields $$\label{rinjmbar} {r_{\rm inj}}(M(m),g) \ =\ \sqrt{k}\beta_m/2 \ \ \ \mbox{and}\ \ \ {r_{\rm cx}}(M(m),g) \ =\ \sqrt{k}\beta_m/4.$$ *Notation 52*. (a) For $Y_0\in {\rm Sym^+}(m)$ and $r>0$ let $$\begin{aligned} \bar{\sf B}^{{\textsc {sr}}}_r(Y_0) &=&\{Y\in{\rm Sym^+}(m): d_{\textsc {sr}}(Y,Y_0)\leq r\}, \\ {\sf B}^{{\textsc {sr}}}_r(Y_0) &=&\{Y\in{\rm Sym^+}(m): d_{\textsc {sr}}(Y,Y_0)<r\}, \\ \mbox{and}\hspace{1in} {\sf S}^{{\textsc {sr}}}_r(Y_0) &=&\{Y\in{\rm Sym^+}(m): d_{\textsc {sr}}(Y,Y_0) = r\}.\end{aligned}$$ For $\tilde{Y}_0\in {\widetilde{M}}(m)$ and $r>0$ let $$\tilde{\sf S}_r(\tilde{Y}_0; {\tilde{d}}) = \{\tilde{Y}\in{\widetilde{M}}(m) : {\tilde{d}}(\tilde{Y},\tilde{Y}_0)= r\},$$ the *sphere* with radius $r$ and center $\tilde{Y}_0$ in the Riemannian manifold $({\widetilde{M}}(m), \tilde{g}$). \(b\) For $Y_0\in S_m^{\rm top}$ and $r>0$, we write $B_r(Y_0;d_{\textsc {sr}}^{\rm top})$ and $\bar{B}_r(Y_0;d_{\textsc {sr}}^{\rm top})$ for the open and closed balls, respectively, of radius $r$ and center $Y_0$ in the metric space $(S_m^{\rm top},d_{\textsc {sr}}^{\rm top})$. Informally, we call $\bar{{\sf B}}^{{\textsc {sr}}}_r(Y_0), {\sf B}^{{\textsc {sr}}}_r(Y_0),$ and ${\sf S}^{{\textsc {sr}}}_r(Y_0)$ the "closed $d_{\textsc {sr}}$-ball", "open $d_{\textsc {sr}}$-ball" and "$d_{\textsc {sr}}$-sphere" with center $Y_0$ and radius $r$, but we caution the reader not to let this terminology lead to implicit assumptions. Since $d_{\textsc {sr}}$ is not a metric, these informally named balls and spheres are not balls and sphere in the metric-space sense, and cannot be expected to have all the properties of metric balls and spheres. Indeed, for $Y_0\in S_m^{\rm top}$, the function $d_{\textsc {sr}}(\cdot, Y_0):{\rm Sym^+}(m)\to{\bf R}$ is not even continuous (see [@rjgs Appendix A]). It can be shown that $\bar{{\sf B}}^{{\textsc {sr}}}_r(Y_0)$ is a closed *set*, at least, but ${\sf B}^{{\textsc {sr}}}_r(Y_0)$ is not always an open set. However, we will not need those facts to prove anything in the present paper. Here, all that we need to know about "$d_{\textsc {sr}}$-balls" is in the following lemma. **Lemma 53**. *Let $Y_0\in S_m^{\rm top}$ and let $r>0$. Then:* *(a) $\bar{B}_r(Y_0;d_{\textsc {sr}}^{\rm top})=\bar{\sf B}^{{\textsc {sr}}}_r(Y_0)\cap S_m^{\rm top}$ and $B_r(Y_0;d_{\textsc {sr}}^{\rm top})={\sf B}^{{\textsc {sr}}}_r(Y_0)\cap S_m^{\rm top}$.* *(b) The set $\bar{{\sf B}}^{{\textsc {sr}}}_r(Y_0)\,\backslash\, B_r(Y_0 ;d_{\textsc {sr}}^{\rm top})$ has volume zero in ${\rm Sym^+}(m)$.* *(c) Any absolutely continuous measure on ${\rm Sym}^+(m)$ supported in $\bar{{\sf B}}^{{\textsc {sr}}}_r(Y_0)$ is also supported in $B_r(Y_0;d_{\textsc {sr}}^{\rm top}).$* : (a) Immediate from definitions. \(b\) Let $S_m^{\rm low}={\rm Sym^+}(m)\,\backslash\, S_m^{\rm top}$. It is easily seen that $$\begin{aligned} \bar{{\sf B}}^{{\textsc {sr}}}_r(Y_0)\,\backslash\, B_r(Y_0,d_{\textsc {sr}}^{\rm top}) &\ \subset\ & S_m^{\rm low}\cup {\sf S}_r^{\textsc {sr}}(Y_0).\end{aligned}$$ Since $S_m^{\rm low}\subset {\rm Sym^+}(m)$ is a finite union of positive-codimension submanifolds, $S_m^{\rm low}$ has volume zero in ${\rm Sym^+}(m)$. Thus it suffices to show that ${\sf S}_r^{\textsc {sr}}(Y_0)$ also has volume zero. Let $\tilde{Y}_i, 1\leq i\leq |{\mathcal{G}}(m)|,$ be the elements of ${\mathcal{F}}^{-1}(Y_0)$. Then by definition of $d_{\textsc {sr}}$, $$\label{union_spheres} {\sf S}_r^{\textsc {sr}}(Y_0) \subset {\mathcal{F}}\left( \bigcup_{i=1}^{|{\mathcal{G}}(m)|} S_r(\tilde{Y}_i ; {\tilde{d}}) \right).$$ In any complete Riemannian manifold, the sphere with any given center and radius has volume zero in that manifold. Hence each of the spheres $S_r(\tilde{Y}_i ; {\tilde{d}})$ has volume zero in ${\widetilde{M}}(m)$, and therefore so does their union. Since $\dim({\widetilde{M}}(m))=\dim({\rm Sym^+}(m))$ and ${\mathcal{F}}$ is smooth, ${\mathcal{F}}$ carries volume-zero sets to volume-zero sets. Hence ${\sf S}_r^{\textsc {sr}}(Y_0)$ is contained in a volume-zero subset of ${\rm Sym^+}(m)$, and therefore itself has volume zero in ${\rm Sym^+}(m)$. \(c\) Immediate from part (a). height10pt width10pt\ In the next proposition, we consider i.i.d., absolutely continuous, ${\rm Sym}^+(m)$-valued random variables $X_1, \dots, X_N$ whose induced measure $\mu_X$ is supported in $\bar{{\sf B}}_r^{\textsc {sr}}(Y_0)$ for some given $Y_0\in S_m^{\rm top}$ and radius $r$ less than ${r_{\rm cx}}(M(m),g)=\frac{\beta_m\sqrt{k}}{4}$. Lemma [Lemma 53](#lem:balls){reference-type="ref" reference="lem:balls"}(b) assures us that $\mu_X$ is also supported in the open metric ball $B_r(Y_0;d_{\textsc {sr}}^{\rm top}).$ By Lemma [Lemma 50](#liftlemma2){reference-type="ref" reference="liftlemma2"}, for each $i$, lifts $X^M_i$ of $X_i$ to $M(m)$ exist and are absolutely continuous, and since $\bar{{\mathcal{F}}}_{\rm top}: (M^{\rm top}(m),d^{\rm top})\to(S_m^{\rm top}, d_{\textsc {sr}}^{\rm top})$ is an isometry, the common measure $\mu_{_{X^M}}$ is supported in $B_{r}(\bar{{\mathcal{F}}}_{\rm top}^{-1}(Y_0))\subset (M(m),d)$. Since $r<{r_{\rm cx}}(M(m),g)<{r_{\rm inj}}(M(m),g)$, and ${\rm quo}:{\widetilde{M}}(m)\to M(m)$ is a Riemannian covering map, each random variable $X_i^M$ has an equivariant family of associated lifts to ${\widetilde{M}}(m)$ (see Definition [Definition 44](#def:assocrv){reference-type="ref" reference="def:assocrv"}). We call such families $\{{\tilde{X}}_1^{(h)}\}_{h\in {\mathcal{G}}(m)}, \dots, \{{\tilde{X}}_N^{(h)}\}_{h\in {\mathcal{G}}(m)}$, *consistently indexed* if, in the setting of Remark [Remark 43](#rem:assocrv){reference-type="ref" reference="rem:assocrv"} and Definition [Definition 44](#def:assocrv){reference-type="ref" reference="def:assocrv"}, there is an equivariant family of measurable sections $\{s_h\}_{h\in G}$ that simultaneously determines each of the families $\{{\tilde{X}}_i^{(h)}\}_{h\in {\mathcal{G}}(m)}.$ **Proposition 54**. *Let $Y_0\in S_m^{\rm top}$, let $Y_0'=\bar{{\mathcal{F}}}_{\rm top}^{-1}(Y_0),$ let $r\in (0,\frac{\beta_m\sqrt{k}}{4})$, and let $X_1, \dots, X_N$ be i.i.d., absolutely continuous, ${\rm Sym}^+(m)$-valued random variables whose distribution $\mu_X$ is supported in $\bar{{\sf B}}_r^{\textsc {sr}}(Y_0).$ Let $X^M_1, \dots, X^M_N$ be lifts of $X_1, \dots X_N$ to $M(m)$, and let $\{{\tilde{X}}_1^{(h)}\}_{h\in {\mathcal{G}}(m)}, \dots, \{{\tilde{X}}_N^{(h)}\}_{h\in {\mathcal{G}}(m)}$ be consistently indexed, equivariant families of associated lifts of $X^M_1, \dots, X^M_N$ to ${\widetilde{M}}(m)$; note that, thanks to the consistent indexing, for each $h\in{\mathcal{G}}(m)$ the random variables ${\tilde{X}}_1^{(h)}, \dots, {\tilde{X}}_N^{(h)}$ are i.i.d.* *The following are true almost surely:* - *For each $h\in {\mathcal{G}}(m)$, the set ${\rm FM}(\{{\tilde{X}}_1^{(h)},\dots,{\tilde{X}}_N^{(h)}\})$ consists of a single point and lies in ${\mathcal{F}}^{-1}(B_r(Y_0;d_{\textsc {sr}}^{\rm top}))={\rm quo}^{-1} \left( B_r(Y_0')\cap M^{\rm top}(m)\right)\subset {\widetilde{M}}^{\rm top}(m).$* - *${\rm FM}(\{X^M_1,\dots, X^M_N\})$ consists of a single point lying in $B_r(Y_0')\,\cap\, M^{\rm top}(m).$ In addition, ${\tilde{E}}_N^{{\textsc {psr}}}(\{X_1,\dots, X_N\})$ is identical to ${\rm quo}^{-1}\big({\rm FM}(\{X^M_1,\dots, X^M_N\}\big)$ and consists of a single ${\mathcal{G}}(m)$-orbit lying in ${\mathcal{F}}^{-1}(B_r(Y_0;d_{\textsc {sr}}^{\rm top})) \subset {\widetilde{M}}^{\rm top}(m)$.* : Let us write ${\widetilde{M}}={\widetilde{M}}(m),$ $M=M(m)$, ${\widetilde{M}}^{\rm top}={\widetilde{M}}^{\rm top}(m),$ and $M^{\rm top}=M^{\rm top}(m).$ Letting $B^M:=B_r(Y_0')\subset (M,d)$ play the role of the ball $B$ in Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"}, let $\{\tilde{B}_h\}_{h\in {\mathcal{G}}(m)}$ be as in Corollary [Corollary 37](#evenlycov){reference-type="ref" reference="evenlycov"}, labeled equivariantly (see Remark [Remark 38](#relabel){reference-type="ref" reference="relabel"}). For each $\omega\in \Omega$ (recall from Section [1](#sect:intro){reference-type="ref" reference="sect:intro"} that $\Omega$ is the domain of all random variables in this paper), let us write $Q(\omega):=(X_1(\omega), \dots, X_N(\omega))$ and $Q^M(\omega):=(X_1^M(\omega), \dots, X_N^M(\omega))$. Since $r<{r_{\rm cx}}(M,g)$ and ${\widetilde{M}}\setminus {\widetilde{M}}^{\rm top}$ is a ${\mathcal{G}}(m)$-invariant subset of ${\widetilde{M}}$ of volume zero, Lemma [Lemma 53](#lem:balls){reference-type="ref" reference="lem:balls"} and Corollary [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"}(c) together imply that there is a full-measure subset $\Omega_1\subset \Omega$ such that for all $\omega\in \Omega_1$: - For each $i\in \{1,\dots, N\},$ $X_i(\omega)\in B_r(Y_0;d_{\textsc {sr}}^{\rm top})$ (implying that $X_i^M(\omega)=\bar{{\mathcal{F}}}_{\rm top}^{-1}(X_i(\omega)),$ since $\bar{{\mathcal{F}}}_{\rm top}$ is one-to-one). Equivalently, $Q(\omega)\in (B_r(Y_0;d_{\textsc {sr}}^{\rm top}))^N$, which implies that $Q^M(\omega)\in (B_r(Y_0'))^N$, since $\bar{{\mathcal{F}}}_{\rm top}: (M^{\rm top},d^{\rm top})\to (S_m^{\rm top},d_{\textsc {sr}}^{\rm top})$ is an isometry. - ${\rm FM}(Q^M(\omega))$ is a single point and lies in $M^{\rm top}\cap B^M$. - For each $h\in G$, the set ${\rm FM}({\tilde{Q}}^{(h)}(\omega))$ is a single point and lies in ${\widetilde{M}}^{\rm top}\cap \tilde{B}_h$. Conclusion (a) of the Proposition is immediate from (iii). Conclusion (b) then follows from Corollary [Corollary 45](#cor:measzero.5){reference-type="ref" reference="cor:measzero.5"}(a) and Proposition [Proposition 51](#prop:psr=efm){reference-type="ref" reference="prop:psr=efm"}. height10pt width10pt\ *Remark 55*. By simply foregoing the first conclusion of Proposition [Proposition 54](#main_SR-PSR_result, version2){reference-type="ref" reference="main_SR-PSR_result, version2"} we obtain the following weaker, but more easily stated, result: > *Let $Y_0\in S_m^{\rm top}$, let $r\in (0,\frac{\beta_m\sqrt{k}}{4})$, and let $X_1, \dots, X_N$ be absolutely continuous, i.i.d., ${\rm Sym}^+(m)$-valued random variables whose distribution $\mu_X$ is supported in $\bar{{\sf B}}_r^{\textsc {sr}}(Y_0).$ Then, almost surely, the partial scaling-rotation means of $X_1,\dots, X_N$ are unique up to the action of ${\mathcal{G}}(m)$ and lie in the top stratum of ${\widetilde{M}}(m)$.* *Remark 56*. It can be shown that $d_{\textsc {sr}}$ is continuous on $S_m^{\rm top}\times S_m^{\rm top}$. Hence, given $S_0\in S_m^{\rm top}$, $r>0,$ and $S\in \overline{B_r(S_0;d_{\textsc {sr}}^{\rm top})}$ (the closure of $B_r(S_0;d_{\textsc {sr}}^{\rm top})$ in ${\rm Sym^+}(m)$), $$d_{\textsc {sr}}(S,S_0)\leq \sup \left\{d_{\textsc {sr}}(S',S_0): S'\in B_r(S_0;d_{\textsc {sr}}^{\rm top})\right\}=r;$$ i.e. $S\in \bar{{\sf B}}_r^{\textsc {sr}}(Y_0)$. Thus $\overline{B_r(S_0;d_{\textsc {sr}}^{\rm top})}\subset \bar{{\sf B}}_r^{\textsc {sr}}(Y_0).$ Hence the support-condition on $\mu_X$ in Proposition [Proposition 54](#main_SR-PSR_result, version2){reference-type="ref" reference="main_SR-PSR_result, version2"} and Remark [Remark 55](#weaker){reference-type="ref" reference="weaker"} is met if $\mu_X$ is supported in $\overline{B_r(S_0;d_{\textsc {sr}}^{\rm top})}.$ # Proofs of some technical results [\[sec:appendixA\]]{#sec:appendixA label="sec:appendixA"} . Let $n>0$. Suppose $A'\subset {\bf R}^n$ has volume zero in ${\bf R}^n$. By [@lee_mfds Lemma 6.2], ${\bf R}\times A'$ has volume zero in ${\bf R}^{n+1}$. By induction, ${\bf R}^k\times A'$ has volume zero in ${\bf R}^{n+k}$ for all $k>0$. Trivially this is true for $k=0$ as well. Now let $M$ and $N$ be manifolds of dimensions $k+n$ and $n$ respectively, and let $H:M\to N$ be a submersion. Identify ${\bf R}^{k+n}$ with ${\bf R}^k\times {\bf R}^n$ the usual way. Since $H$ is a submersion, $M$ and $N$ have atlases ${\cal A},{\cal B}$ with the following property: for every chart $(U,\varphi)\in {\cal A}$ there exists $(V,\psi)\in{\cal B}$ such that $U=H^{-1}(V)$ and such that the map $\psi\circ H\circ\varphi^{-1}: \linebreak (\varphi(U)\subset {\bf R}^k\times {\bf R}^n) \to {\bf R}^n$ is simply the natural projection $\pi:{\bf R}^k \times {\bf R}^n\to {\bf R}^n$ (restricted to $\varphi(U)\subset {\bf R}^k\times \psi(V)\subset {\bf R}^k\times {\bf R}^n$). With all data as in the previous paragraph, suppose $A\subset N$ has volume zero in $N$. Then $A':=\psi(A\cap V)$ has volume zero in ${\bf R}^n$. Observing that $\pi\circ\varphi\,(H^{-1}(A)\cap U)=\pi\circ\varphi\,(H^{-1}(A\cap V))= \psi\circ H\,(H^{-1}(A\cap V))=A'$, we have $\varphi(H^{-1}(A)\cap U) \subset {\bf R}^k\times A'$.The first paragraph of this proof therefore shows that $\varphi(H^{-1}(A)\cap U)$ has volume zero in ${\bf R}^{k+n}$. Since ${\cal A}$ is an atlas of $M$, this proves that $H^{-1}(A)$ has volume zero in $M$. height10pt width10pt\ . Let $\tilde{\rho}={r_{\rm inj}}({\widetilde{M}},\tilde{g})$ and $\rho={r_{\rm inj}}(M,g)$. Since ${\rm quo}$ is a local isometry, it intertwines the exponential maps of ${\widetilde{M}}$ and $M$: $$\label{intertwine} {\rm quo}\circ \exp_{\tilde{p}}=\exp_{{\rm quo}({\tilde{p}})}\circ\, {\rm quo}_{*{\tilde{p}}}\ \ \ \ \mbox{for all ${\tilde{p}}\in {\widetilde{M}}$}.$$ It follows that for any $p\in M,\ {\tilde{p}}\in {\rm quo}^{-1}(p),$ and $r>0$ for which $\exp_p: B_r(0)\subset T_pM\to M$ is a diffeomorphism onto its image, so is $\exp_{{\tilde{p}}}: B_r(0)\subset T_{{\tilde{p}}}{\widetilde{M}}\to {\widetilde{M}}.$ Hence $\tilde{\rho}\geq \rho.$ Next, suppose that $\rho< \min\{\tilde{\rho}, \frac{\beta}{2}\}$. For $v\in T{\widetilde{M}}$, let $\tilde{\gamma}_{\tilde{v}}$ denote the geodesic $t\mapsto \exp(t{\tilde{v}})$; for $v\in TM$ we similarly define the geodesic $\gamma_{v}$. We write $L[\cdot]$ for the length of a curve in ${\widetilde{M}}$ or $M$. Since ${\rm quo}$ is a local isometry, curve-lifting from $M$ to ${\widetilde{M}}$ preserves length and carries geodesics to geodesics. Let ${\tilde{p}}\in {\widetilde{M}}$, $p={\rm quo}({\tilde{p}})$, and let $\tilde{\rho}_{\tilde{p}}, \rho_p$ denote the local injectivity radii of ${\widetilde{M}},M$ at ${\tilde{p}},p$ respectively. Since $\rho<\tilde{\rho}$, we may select $\epsilon>0$ and $p\in M$ such that $\rho_p<\rho+\epsilon<\tilde{\rho}$. Then $\rho_p$ is finite, so there exists $q\in M$ such that $q\in {\mathcal{C}}(p)$ and $d(p,q)=\rho_p$. Let ${v}\in T_{p}M$ be such that $\gamma_{v}|_{[0,1]}$ is a minimal geodesic from $p$ to $q$, and let $\delta= \epsilon/\|v\|$; thus $L[\gamma_{v}|_{[0,1+\delta]}]=\rho_p+\epsilon<\tilde{\rho}$. Since $q\in {\mathcal{C}}(p)$, the geodesic $\gamma_{v}$ does not minimize beyond $t=1$. Hence there exists $w\in T_{p}M$ such that $\exp_{p}(w)=\gamma_{v}(1+\delta)$ and $\|{w}\|<(1+\delta)\|{v}\|.$ Let $\alpha:[0,2+\delta]\to {\widetilde{M}}$ be the loop defined by $\alpha(t)=\gamma_{v}(t)$ for $0\leq t\leq 1+\delta$ and $\alpha(t)=\gamma_{w}(2+\delta-t)$ for $1+\delta\leq t\leq 2+\delta$. Let ${\tilde{p}}\in {\rm quo}^{-1}(p)$. Then the lift $\tilde{\alpha}$ of $\alpha$ to ${\widetilde{M}}$ starting at ${\tilde{p}}$ satisfies $\tilde{\alpha}(2+\delta)=h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{p}}$ for some (unique) $h\in G$. Let ${\tilde{q}}=\tilde{\alpha}(1+\delta)$. Since $L[\tilde{\alpha}|_{[0,1+\delta]}]=L[{\alpha}|_{[0,1+\delta]}]<\tilde{\rho}$, the curve $\tilde{\alpha}|_{[0,1+\delta]}$ is a minimal geodesic from ${\tilde{p}}$ to ${\tilde{q}}$, while $\tilde{\alpha}|_{[1+\delta,2+\delta]}$ is a shorter-length path from ${\tilde{q}}$ to $h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{p}}$. It follows that $h\neq e$, and that $$\label{2beta} \beta\leq {\tilde{d}}({\tilde{p}},h{\mbox{ \,\hspace{-.06in}\raisebox{-.015in}{\bf {\Large $\cdot$}}}}{\tilde{p}})\leq L[\tilde{\alpha}]=L[\alpha] <2(1+\delta)\|v\|<2\rho+4\epsilon.$$ Since [\[2beta\]](#2beta){reference-type="eqref" reference="2beta"} holds for arbitrarily small $\epsilon$, this contradicts the hypothesis that $\rho<\beta/2$. Hence $\rho\geq \min\{\tilde{\rho}, \frac{\beta}{2}\}.$ Finally, suppose the strict equality $\rho> \min\{\tilde{\rho}, \frac{\beta}{2}\}$ holds. Since $\rho\leq\tilde{\rho}$, we must have $\min\{\tilde{\rho},\frac{\beta}{2}\}=\frac{\beta}{2}$, so $\tilde{\rho}>\frac{\beta}{2}$ and $\rho>\frac{\beta}{2}$. For each $p\in M$, let $\beta_p$ be the minimal distance between distinct elements of ${\rm quo}^{-1}(p)$, and let ${\tilde{p}}_1, {\tilde{p}}_2\in {\rm quo}^{-1}(p)$ be a pair of points achieving this distance $\beta_p$. Let $\tilde{\gamma}:[0,1]\to{\widetilde{M}}$ be a minimal geodesic from ${\tilde{p}}_1$ to ${\tilde{p}}_2$ and let $\gamma={\rm quo}\circ\tilde{\gamma}$. Then $\gamma$ is a closed geodesic loop starting and ending at $p$, and $L[\gamma]=L[\tilde{\gamma}]=\beta_p$. Hence $\rho_p\leq \frac{1}{2}L[\gamma]=\frac{\beta_p}{2}$. Taking the infimum over all $p\in M$ yields $\rho\leq \frac{\beta}{2}$, a contradiction. Hence $\rho= \min\{\tilde{\rho}, \frac{\beta}{2}\}.$ height10pt width10pt\ . (a) Using the Axiom of Choice, for each $Y\in {\rm Sym}^+(m)$ select an element $s(Y)\in \bar{{\mathcal{F}}}^{-1}(\{Y\})$, thereby defining a set-theoretic section $s:{\rm Sym}^+(m)\to M(m)$. Note that since $\bar{{\mathcal{F}}}_{\rm top}$ is a bijection, $s|_{S_m^{\rm top}}=\bar{{\mathcal{F}}}_{\rm top}^{-1}$. Let ${\mathcal L}_1, {\mathcal L}_2$ denote the sigma-algebras of Lebesgue-measurable subsets of ${\rm Sym}^+(m)$, $M(m)$ respectively (see Section [1](#sect:intro){reference-type="ref" reference="sect:intro"}). For any $U\subset M(m)$, trivially $U=\big(U\cap M^{\rm top}(m)\big) \bigcup U\cap M^{\rm low}(m)$, and therefore $s^{-1}(U)=s^{-1}\big(U\cap M^{\rm top}(m)\big) \bigcup s^{-1}(U\cap M^{\rm low}(m))$. Since $s^{-1}(U\cap M^{\rm low}(m)) \subset S_m'$, which has volume zero, it follows that $s^{-1}(U\cap M^{\rm low}(m))$ has volume zero, and therefore lies in ${\mathcal L}_1$. Hence, given $U\in {\mathcal L}_2$, to show that $s^{-1}(U)\in {\mathcal L}_1$ it suffices to show that $s^{-1}\big(U\cap M^{\rm top}(m)\big)\in {\mathcal L}_1$. First suppose that $U\subset M(m)$ is a Borel set. Since $M^{\rm top}(m)$ is open, $U\cap M^{\rm top}(m)$ is a Borel set, and therefore so is $s^{-1}(U\cap M^{\rm top}(m))=s|_{S_m^{\rm top}}^{-1}(U\cap M^{\rm top}(m))= \linebreak \bar{{\mathcal{F}}}_{\rm top}(U\cap M^{\rm top}(m))$. Thus $s^{-1}\big(U\cap M^{\rm top}(m)\big)\in {\mathcal L}_1$. Next, suppose that $U\subset M(m)$ has volume zero. Then every subset of $U$ has volume zero and therefore lies in ${\mathcal L}_2$. In particular this applies to $U\cap M^{\rm top}(m)$. Again we have $s^{-1}(U\cap M^{\rm top}(m))=\bar{{\mathcal{F}}}_{\rm top}(U\cap M^{\rm top}(m))$, and since $\bar{{\mathcal{F}}}_{\rm top}$ is a diffeomorphism, it follows that $\bar{{\mathcal{F}}}_{\rm top}(U\cap M^{\rm top}(m))$ has volume zero, hence lies in ${\mathcal L}_1$. Furthermore, $s^{-1}(U)$ is the union of two volume-zero sets, hence has volume zero. Thus, for every $U\subset M(m)$ that is either a Borel set or a volume-zero set, $s^{-1}(U)\in {\mathcal L}_1$. Since every $U\in {\mathcal L}_2$ is the union of a Borel set and a volume-zero set, it follows that $s^{-1}(U)\in {\mathcal L}_1$ whenever $U\in{\mathcal L}_2$. Thus $s$ is measurable. \(b\) Trivially, a map $X^M:\Omega\to M(m)$ is a lift of $X$ if and only if $X^M=s\circ X$ for some set-theoretic (hence measurable, by part (a)) section $s$ of $\bar{{\mathcal{F}}}$. Since such sections exist, so do lifts $X^M$ of $X$. Let $X^M$ be a lift of $X$ to $M(m)$, and let $s$ be the (necessarily unique) set-theoretic section of $\bar{{\mathcal{F}}}$ for which $X^M=s\circ X$. Since $s$ and $X$ are measurable, so is $X^M$. \(i\) Let $U\subset M(m)$ be a volume-zero subset. Then $s^{-1}(U)$ has volume zero, by part (a). Since $X$ is absolutely continuous, it follows that $$0=\mu_X(s^{-1}(U))= {\rm Pr}\big(X^{-1}(s^{-1}(U))\big)= {\rm Pr}\big( (s\circ X)^{-1}(U)\big) =\mu_{X^M}(U).$$ Hence the $M(m)$-valued random variable $X^M$ is absolutely continuous. \(ii\) By Corollary [Corollary 42](#cor:meas_triv){reference-type="ref" reference="cor:meas_triv"}(b) and its proof, there exists an equivariant family $\{{\tilde{X}}^{(h)}\}_{h\in {\mathcal{G}}(m)}$ of absolutely continuous, measurable lifts of $X^M$ from $M(m)$ to ${\widetilde{M}}(m)$. Hence the family $\{{\tilde{X}}^{(h)}\}_{h\in {\mathcal{G}}(m)}$ has all the desired properties. height10pt width10pt\ # Parametric transversality and a proof for Remark [Remark 16](#rmk:pi_NM_2){reference-type="ref" reference="rmk:pi_NM_2"} {#sec:appendixB} Given manifolds ${\mathcal N}$ and ${\mathcal M}$, and a submanifold $A\subset {\mathcal M},$ we say that a smooth map ${\mathcal{F}}:{\mathcal N}\to {\mathcal M}$ is *transverse to $A$* if, for every $p\in {\mathcal{F}}^{-1}(A)$, $${\mathcal{F}}_{*p}(T_p {\mathcal N}) + T_{{\mathcal{F}}(p)}A = T_{{\mathcal{F}}(p)}{\mathcal M}$$ (i.e. at ${\mathcal{F}}(p)$, the tangent space of ${\mathcal M}$ is spanned by the tangent space of $A$ together with the image of $T_pN$ under the derivative of ${\mathcal{F}}$). If this condition is met, then ${\mathcal{F}}^{-1}(A)$ is a submanifold of ${\mathcal N}$, of codimension equal to the codimension of $A$ in ${\mathcal M}$. Note that trivially, a *submersion* ${\mathcal{F}}:{\mathcal N}\to {\mathcal M}$ is transverse to every submanifold of ${\mathcal M}$. **Theorem 57** ($C^\infty$ Parametric Transversality Theorem [@lee_mfds Theorem 6.35]). *Let ${\mathcal N},{\mathcal M},S$ be smooth manifolds and let $A\subset {\mathcal M}$ be a smooth submanifold. Suppose that $\{{\mathcal{F}}_s:s\in S\}$ is a smooth family of smooth maps ${\mathcal N}\to {\mathcal M}$; i.e. assume that the map ${\mathcal{F}}:S\times {\mathcal N}\to {\mathcal M}$ defined by ${\mathcal{F}}(s,p)={\mathcal{F}}_s(p)$ is smooth. If ${\mathcal{F}}$ is transverse to $A$, then $\{s\in S: {\mathcal{F}}_s \ \mbox{is transverse to $A$}\}$ is a generic subset of $S$; i.e. *generically* the maps ${\mathcal{F}}_s$ are transverse to $A$.* We will use Theorem [Theorem 57](#PTT){reference-type="ref" reference="PTT"} to prove the following: **Proposition 58**. *For each $N\geq 1$, the set $\pi_N({\mathcal M}_2)$ is generic in $M^N$.* : Since ${\mathcal M}_1$ is open in $M^N\times M$, we can express ${\mathcal M}_1$ as a countable union of products $U_i\times V_i$ ($i=1,2,\dots$), where $U_i$ and $V_i$ are open subsets of $M^N$ and $M$, respectively. Since $F$ is a submersion, so is $F_i:=F|_{U_i\times V_i}$ for each $i$. Hence $F_i$ is transverse to $Z$ for each $i$. For $Q\in U_i$, the evaluation-map $Y_{Q,i}: p\mapsto F_i(Q,p)$ is simply $Y_Q|_{V_i}$. The proof of the Parametric Transversality Theorem as presented in [@hirsch Ch. 3, Theorem 2.7] shows that the set $A_i=\{Q\in U_i\ : \ Y_{Q,i} \ \mbox{is {\em not} transverse to}\ Z\}$ has volume zero in $U_i$, hence in $M^N$. Since there are only countably many $A_i$, their union has volume zero in $M^N$. But this union contains $\{Q\in M^N\ : \ Y_Q \ \mbox{is not transverse to}\ Z\}$, and (using e.g. [\[YQ\*p\]](#YQ*p){reference-type="eqref" reference="YQ*p"}) the condition "$Y_Q$ is transverse to $Z$" is equivalent to "$(\nabla Y_Q)|_p : T_pM \to T_p M$ is an isomorphism for all $p\in Z\cap{\rm domain}(Y_Q)$". Hence the volume-zero set $\{Q\in M^N\ : \ Y_Q \ \mbox{is not transverse to}\ Z\}$ is exactly $M^N\setminus \pi_N({\mathcal M}_2)$, and thus $\pi_N({\mathcal M}_2)$ is generic in $M^N$. height10pt width10pt\ # Acknowledgements {#acknowledgements .unnumbered} The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1A2C2002256). 99 [Afsari, B.]{.smallcaps} (2011). Riemannian $L^p$ Center of Mass: Existence, Uniqueness, and Convexity. *Proc. Amer. Math. Soc.* **139**, 133--136. [Bhattacharya, R.]{.smallcaps} and [Patrangenaru, V.]{.smallcaps} (2003). Large Sample Theory of Intrinsic and Extrinsic Sample Means on Manifolds, I, *Ann. Stat.* **31** (2003), 1--29. [Bishop, R. L.]{.smallcaps} (1977). Decomposition of cut loci. *Proc. Amer. Math. Soc.* **65**, 655--673. [Burago,Y., Gromov, M.,]{.smallcaps} and [Perelman, G.]{.smallcaps} (1992). A. D. Alexandrov spaces with curvature bounded below, *Russian Math. Surveys* **47**, 1--58. [Chavel, I.]{.smallcaps} (2006). *Riemannian Geometry: A Modern Introduction*, 2nd ed. *Cambridge Tracts in Mathematics* **108**. Cambridge University Press, Cambridge. [Corcuera, J. M.]{.smallcaps} and [Kendall,W. S.]{.smallcaps} (1999). Riemannian barycentres and geodesic convexity. *Math. Proc. Camb. Phil. Soc.* **127**, 253--269. [Groisser D.]{.smallcaps} (2004). Newton's method, zeroes of vector fields, and the Riemannian center of mass. *Adv. Appl. Math.* **33**, 95--135. [Groisser, D., Jung, S.,]{.smallcaps} and [Schwartzman, A.]{.smallcaps} (2017). Geometric foundations for scaling-rotation statistics on symmetric positive definite matrices: minimal smooth scaling-rotation curves in low dimensions. *Electronic J. Stat.* **11**, 1092--1159. [Groisser, D., Jung, S.,]{.smallcaps} and [Schwartzman, A.]{.smallcaps} (2021). Uniqueness questions in a scaling-rotation geometry on the space of symmetric positive-definite matrices. *Diff. Geom. and its Appl.* **79**, 1--40. [Hirsch, M. W.]{.smallcaps} (1994). *Differential Topology.* Graduate Texts in Mathematics **33**. Springer-Verlag, New York. Corrected reprint of the 1976 original. [Hotz, T., Skwerer, S., Huckemann, S., Le, H., Marron, J. S., Mattingly, J. C., Miller, E., Nolen, J., Owen, M.,]{.smallcaps} and [Patrangenaru, V.]{.smallcaps} (2013). Sticky central limit theorems on open books. *The Annals of Applied Probability* **23**, 2238--2258. [Huckemann, S.]{.smallcaps} (2011). Inference on 3D Procrustes Means: Tree Bole Growth, Rank Deficient Diffusion Tensors and Perturbation Models. *Scand. J. Stat.* **38**, 424--446. [Huckemann, S., Mattingly J. C., Miller, E.]{.smallcaps} and [Nolen, J.]{.smallcaps} (2015). Sticky central limit theorems at isolated hyperbolic planar singularities.*Electron. J. Probab.* **20**, no. 78, 34. [Huckemann, S. F.]{.smallcaps} (2011). Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth. *The Annals of Statistics* **39**, 1098--1124. [S. Huckemann]{.smallcaps} (2012). On the meaning of mean shape: manifold stability, locus and the two sample test. *Ann. Inst. Stat. Math.* **64**, 1227--1259. [Jung, S., Rooks, B., Groisser, D.]{.smallcaps}, and [Schwartzman, A.]{.smallcaps} (2023). Averaging Symmetric Positive-Definite Matrices on the Space of Eigen-Decompositions. *arXiv preprint arXiv:2306.12025* [S. Jung, A. Schwartzman, and D. Groisser]{.smallcaps} (2015). Scaling-Rotation Distance and Interpolation of Symmetric Positive-Definite Matrices. *SIAM J. Matrix Anal. Appl.* **36**, 1180--1201. [Karcher, H.]{.smallcaps} (1977). Riemannian center of mass and mollifier smoothing. *Commun. Pure and Appl. Math.* **30**, 509--541. [Kendall, D. G.]{.smallcaps} (1984). Shape manifolds, Procrustean metrics, and complex projective spaces. *Bull. London Math. Soc.* **16**, 81--121. [Le, H.]{.smallcaps} and [Barden, D.]{.smallcaps} (2014). On the measure of the cut locus of a Fréchet mean. *Bull London Math. Soc..* **46**, 698--708. [Lee, J. M.]{.smallcaps} (2013). *Introduction to Smooth Manifolds*, 2nd ed., Springer-Verlag, New York. [Sakai, T.]{.smallcaps} (1996). *Riemannian Geometry.* Translations of Mathematical Monographs v. 149, Amer. Math. Soc.

arxiv_math

--- abstract: | We define and explicitly construct schemes evinced by generalized additive decompositions (GADs) of a given $d$-homogeneous polynomial $F$. We employ GADs to investigate the regularity of $0$-dimensional schemes apolar to $F$, focusing on those satisfying some minimality conditions. We show that irredundant schemes to $F$ need not be $d$-regular, unless they are evinced by special GADs of $F$. Instead, we prove that tangential decompositions of minimal length are always $d$-regular, as well as irredundant apolar schemes of length at most $2d+1$. address: - Università di Trento, Via Sommarive, 14 - 38123 Povo (Trento), Italy - KU Leuven author: - Alessandra Bernardi, Alessandro Oneto, Daniele Taufer bibliography: - indipCactus.bib title: On schemes evinced by generalized additive decompositions and their regularity --- # Introduction Algebraic and geometric properties of $0$-dimensional schemes have been largely studied from several perspectives in algebraic geometry, commutative algebra, and computational algebra. Through *apolarity theory*, these studies find applications in the study of *additive decompositions* of homogeneous polynomials and, more in general, *tensor decompositions* [@landsberg2012tensors; @bernardi2018hitchhiker; @BC:book]. In this paper, we are interested in $0$-dimensional schemes that are *apolar* to a given $d$-homogeneous polynomial $F$, namely the $0$-dimensional schemes defined by ideals annihilating $F$ by derivation. Understanding the possible Hilbert functions of *minimal* apolar schemes is a deep and largely open question, which could give useful information on the nature of additive decompositions of polynomials and *secant varieties*, and whose grasp is challenging even in moderately small cases [@RS00; @iliev2001; @curvilinear; @elias2012isomorphism; @carlini2012solution; @landsberg2013equations; @buczynska2013waring; @buczynska2014secant; @CJN; @jelisiejew2018vsps; @BJMK18:Polynomials; @Chiantini:quantum; @MO; @buczynska2021apolarity; @angelini2023waring]. Our work aims to study when these Hilbert functions stabilize, and more specifically at discerning essential conditions for a given $d$-homogeneous polynomial to have *minimal* $0$-dimensional apolar schemes that are regular in degree $d$. This subtle problem carries far-reaching implications spanning the domains of classical algebraic geometry and complexity theory. In the context of algebraic geometry, these concepts are part of a longstanding tradition of exploring secant varieties and Waring problems, see [@bernardi2018hitchhiker] for a general overview. From a complexity theory perspective, the knowledge of the regularity of minimal apolar schemes to a given polynomial might improve the efficiency of symbolic algorithms for computing ranks and minimal decomposition of polynomials [@BRACHAT20101851; @BT20:Waring; @BDHM]. ## Additive decompositions As already recalled, the study of apolar schemes is related to notions of *rank* and *additive decompositions* associated with homogeneous polynomials. The minimal length of a $0$-dimensional scheme apolar to $F$ is the *cactus rank* of $F$ [@RS:11; @BB14:Secant]. If we restrict to schemes that are locally contained in $(d+1)$-fat points, then they correspond to *generalized additive decompositions* (GADs) of $F$, namely expressions as $$F = \sum_{i=1}^r L_i^{d-k_i}G_i,$$ where the $L_i$'s are pairwise non-proportional linear forms not dividing the corresponding $G_i$'s [@IaKa:book; @BBM14:comparison; @BT20:Waring]. Special cases of such decompositions include *tangential decompositions*, when $k_i = 1$ [@BT20:Waring; @CGG; @ballico:tg], and *Waring decompositions*, when $k_i = 0$ [@Ger96; @bernardi2018hitchhiker]. This algebraic description of ranks and additive decompositions has a geometric interpretation in terms of *Veronese varieties* and their *secant varieties* [@Zak:tgsec; @A:joins; @bernardi2018hitchhiker]. A Waring decomposition corresponds to a set of points on the Veronese variety whose linear span contains the projective point corresponding to the polynomial $F$. Analogously, tangential decompositions (generalized additive decompositions, respectively) correspond to a set of points on the tangential variety (osculating varieties, respectively) of the Veronese variety whose linear span contains the projective point of $F$ [@bernardi2018hitchhiker; @CGG; @BCGI07:Osculating; @Bernardi2009982; @BF]. In this view, GADs parameterize generic points of a *joint variety* of osculating varieties to certain Veronese variety. ## Content of the paper and main results After recalling the standard definition and results in , we define and provide an explicit construction of schemes evinced by GADs in . This construction locally agrees with the natural apolar schemes defined in [@BJMK18:Polynomials], but is made effective by delving into the computational details. An implementation of this construction routine in Macaulay2 [@M2] and Magma [@MR1484478] can be found in [@Repo]. In we investigate the weaker and more geometric irredundancy condition, i.e. we look at schemes that are minimal by inclusion among the apolar schemes to a given form $F$ of degree $d$. With we observe that schemes evinced by GADs might well be redundant, whereas we prove in that irredundant schemes are evinced by a GAD of $F$ precisely when their connected components are contained in $(d+1)$-fat points. Therefore, all schemes apolar to $F$ with *short* components are evinced by certain families of GADs of $F$. However, shows that schemes with *long* components may only arise from GADs of higher degree polynomials. In we tackle the regularity of minimal apolar schemes. We show that non-redundancy to a degree-$d$ form is not enough to ensure $d$-regularity. Indeed, in we present degree-$d$ homogenous polynomials admitting an apolar scheme that is irredundant but not $d$-regular. However, we notice that in both cases such schemes are not minimal by length. In we show that the addenda constituting a GAD evincing an irredundant scheme $Z$ may never appear in its inverse systems. We use this result in to guarantee $d$-regularity for schemes evinced by GADs such that the $L_i$'s are linearly independent and the $k_i$'s are small enough, regardless of the scheme being minimal. However, we point out in that all the assumptions of are sharp. Drawing from the intuition that schemes with components of low multiplicity usually exhibit low regularity, in we prove that minimal tangential decompositions of degree-$d$ forms always evince $d$-regular schemes. shows the condition of having minimal length is essential, while irredundancy is not enough. Finally, we show in that if the cactus rank of a degree-$d$ form is not greater than $2d+1$, then non-redundancy is actually enough to guarantee $d$-regularity. In particular, all the schemes of minimal length apolar to degree-$d$ forms with length smaller or equal to $2d+1$ are $d$-regular. ## Acknowledgements {#acknowledgements .unnumbered} We sincerely thank E. Ballico, W. Buczyńska, J. Buczyński, M.V. Catalisano, C. Ciliberto and B. Mourrain for fruitful conversations. DT acknowledges the hospitality of the TensorDec Laboratory during a research stay at the Department of Mathematics at the University of Trento, where part of the present work has been conducted. ## Funding {#funding .unnumbered} AB has been partially supported by GNSAGA of INDAM. DT has been supported by the European Union's H2020 Programme ERC-669891, and by the Research Foundation - Flanders via the FWO postdoctoral fellowship 12ZZC23N and the travel grant V425623N. All the authors have been partially supported by the Thematic Research Programme "Tensors: geometry, complexity and quantum entanglement\", University of Warsaw, Excellence Initiative -- Research University and the Simons Foundation Award No. 663281. # Preliminaries {#sec:Preliminaries} In this paper, $\Bbbk$ will always be an algebraically closed field of characteristic $0$. Given $\alpha = (\alpha_0,\ldots,\alpha_n)$ and $\beta = (\beta_0,\ldots,\beta_n)$ in $\mathbb{N}^{n+1}$, let $|\alpha| = \sum_{i=0}^n \alpha_i$ and $\alpha! = \prod_{i=0}^n \alpha_i!$. We write $\alpha \succeq \beta$ if $\alpha_i \geq \beta_i$ for every $0 \leq i \leq n$. We use the standard short notation $X^\alpha = X_0^{\alpha_0}\cdots X_n^{\alpha_n}$. ## Apolarity {#sec:apolarity} Let $\mathcal{S}= \Bbbk[X_0,\ldots,X_n] = \bigoplus_{d \in \mathbb{N}} \mathcal{S}_d$ and $\mathcal{R}= \Bbbk[Y_0,\ldots,Y_n] = \bigoplus_{d \in \mathbb{N}} \mathcal{R}_d$ be standard graded polynomial rings, where $\mathcal{S}_d$ and $\mathcal{R}_d$ denote the $\Bbbk$-vector spaces of degree-$d$ homogeneous polynomials. We also write $\mathcal{S}_{\leq d} = \bigoplus_{e \leq d} \mathcal{S}_e$ and $\mathcal{R}_{\leq d} = \bigoplus_{e \leq d} \mathcal{R}_e$. We consider the apolarity action of $\mathcal{R}$ on $\mathcal{S}$ given by differentiation, i.e., $$Y^\beta \circ X^{\alpha} = \begin{cases} \partial_\beta(X^\alpha) = \frac{\alpha!}{(\alpha-\beta)!}X^{\alpha-\beta} & \text{ if } \alpha \succeq \beta, \\ 0 & \text{ otherwise}, \end{cases}$$ extended by $\Bbbk$-linearity. Given $F \in \mathcal{S}$, we consider its annihilator $$\textnormal{Ann}(F) = \{G \in \mathcal{R}~:~ G \circ F = 0\}, %\todo{to add: inverse systems, dove li usiamo (if)}$$ which is an ideal of $\mathcal{R}$. This action defines a non-degenerate perfect pairing $\mathcal{R}_d \times \mathcal{S}_d \to \Bbbk$ for every $d \in \mathbb{N}$. Given a subspace $V \subseteq\mathcal{S}_d$, we denote by $V^\perp \subseteq\mathcal{R}_d$ its orthogonal space with respect to such pairing. If $V = \langle F \rangle$, we simply denote its orthogonal space by $F^{\perp}$. **Remark 1**. A classical result by Macaulay [@macaulay1916algebraic] shows that graded Artinian Gorenstein algebras are all, and only, quotient rings of polynomial rings by annihilator ideals of homogeneous polynomials, see [@Ger96 Theorem 8.7], [@IaKa:book Lemma 2.12] or [@eisenbud2013commutative Theorem 21.6]. In the following, we always identify $\mathcal{R}$ with the coordinate ring of $\mathbb{P}^n= \mathbb{P}(\mathcal{S}_1)$. **Definition 2**. *Let $F \in \mathcal{S}_d$. A $0$-dimensional scheme $Z \subset \mathbb{P}^n$ is **apolar** to $F$ if $I(Z) \subseteq \textnormal{Ann}(F)$.* A famous characterization of schemes apolar to a given form is provided by the well-known *Apolarity Lemma*, see e.g. [@IaKa:book Lemma 1.15] in the classical case of reduced schemes, [@BJMK18:Polynomials Lemma 1] for non-reduced scheme or [@ranestad2018varieties Lemma 1.3] into a more general framework. **Lemma 3** (Apolarity Lemma). *Let $F \in \mathcal{S}_d$. The following are equivalent:* - *$F \in I(Z)^\perp_d$;* - *$I(Z) \subset \textnormal{Ann}(F)$.* Let $\mathcal{S}_{\rm dp}$ be the polynomial ring $\mathcal{S}$ equipped with a *divided power structure*, i.e. endowed with the divided powers monomial basis $X^{[\alpha]} = \frac{1}{\alpha!}X^\alpha$. We denote by $F_{\rm dp} \in \mathcal{S}_{\rm dp}$ the polynomial $F \in \mathcal{S}$ expressed in divided powers. For convenience in our computation throughout the paper , we also consider the action of $\mathcal{R}$ on $\mathcal{S}_{\rm dp}$ by contraction, namely, $$Y^\beta \mathbin{\mathpalette\dointprod\relax}X^{\alpha} = \begin{cases} X^{\alpha-\beta} & \text{ if } \alpha \succeq \beta, \\ 0 & \text{ otherwise}. \end{cases}$$ For a given $F \in \mathcal{S}_{\rm dp}$, its annihilator with respect to this action will be denoted by $$\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(F) = \left\{G \in \mathcal{R}~:~ G \mathbin{\mathpalette\dointprod\relax}F = 0\right\}.$$ One can directly verify that $G \mathbin{\mathpalette\dointprod\relax}F_{\rm dp} = (G \circ F)_{\rm dp}$. ## Minimality In this paper, we consider the $0$-dimensional schemes apolar to a given $F \in \mathcal{S}_d$. Among them, we are particularly interested in those that are minimal by inclusion or length. **Definition 4**. *Let $Z \subset \mathbb{P}^n$ be a $0$-dimensional scheme apolar to $F \in \mathcal{S}_d$. We say that $Z$ is **irredundant** to $F$ if there is no strict subscheme $Z' \subsetneq Z$ among the schemes apolar to $F$.* The minimal length of a $0$-dimensional scheme apolar to $F$ is called both **scheme length** of $F$ [@IaKa:book] or **cactus rank** of $F$ [@BR13; @BB14:Secant]. **Definition 5**. *Let $Z \subset \mathbb{P}^n$ be a $0$-dimensional scheme apolar to $F \in \mathcal{S}_d$. We say that $Z$ **evinces the cactus rank**, or **evinces the scheme length** of $F$, or simply is **minimal apolar** to $F$, if $Z$ is of minimal length among the $0$-dimensional schemes in $\mathbb{P}^n$ and apolar to $F$.* ## Regularity We study when the Hilbert function of minimal apolar schemes stabilizes. **Definition 6**. *Given a homogeneous ideal $I \subset \mathcal{R}$, the **Hilbert function** of the quotient $\mathcal{R}/I$ is the function $\mathrm{HF}_{\mathcal{R}/I} : \mathbb{N}\to \mathbb{N}$ such that $\mathrm{HF}_{\mathcal{R}/I}(i) = \dim \mathcal{R}_i / I_i$, where $I_i = I \cap \mathcal{R}_i$. For a scheme $Z \subset \mathbb{P}^n$ we denote the Hilbert function of $Z$ as $\mathrm{HF}_Z = \mathrm{HF}_{\mathcal{R}/I(Z)}$.* We simply write $\mathrm{HF}_Z = (a_0,a_1,a_2,\dots)$ to denote $\mathrm{HF}_Z(i) = a_i$. The Hilbert function of a $0$-dimensional scheme $Z$ is always strictly increasing until it reaches its length ${\rm len}(Z)$, and then it remains constant. **Definition 7**. *Given a $0$-dimensional scheme $Z \subset \mathbb{P}^n$, the **regularity** of $Z$ is $${\rm reg}(Z) = \min_{i \in \mathbb{N}}\{\mathrm{HF}_Z(i) = \mathrm{HF}_Z(i+1) = {\rm len}(Z)\}.$$ We say that $Z$ is **regular in degree $d$**, or **$d$-regular**, if ${\rm reg}(Z) \leq d$.* # Schemes evinced by GADs {#sec:GAD} We devote the present section to connecting two well-known concepts such as natural apolar schemes [@BJMK18:Polynomials] and generalized additive decomposition [@IaKa:book]. Their link serves as the cornerstone of our paper, while their explicit construction may be beneficial even for expert readers. A complete implementation in Macaulay2 [@M2] and Magma [@MR1484478] of these procedures may be found in [@Repo]. ## Natural apolar scheme to $F$ supported at $L$ {#sec:NAS} There is a natural way to associate a local scheme apolar to a given $F \in \mathcal{S}_d$ supported at a prescribed point $[L]\in \mathbb{P}^n$ [@BJMK18:Polynomials Section 4]. Let $f_L \in \mathcal{S}_{\rm dp} / (L-1) = \underline{\mathcal{S}}_{\rm dp}$ be the dehomogenization of $F_{\rm dp}$ by $L$. We consider the projection $\mathcal{S}_{\rm dp} \to \underline{\mathcal{S}}_{\rm dp}$ and its dual projection $\mathcal{R}\to \underline{\mathcal{R}}$. We denote the latter projection of an ideal $J \subset \mathcal{R}$ by $\underline{J} \subset \underline{\mathcal{R}}$. We will always use lowercase letters for the elements and the variables after these projections, e.g., we identify $\underline{\mathcal{S}}_{\rm dp}\simeq \Bbbk[x_1,\ldots,x_n]_{\rm dp}$ and $\underline{\mathcal{R}}\simeq \Bbbk[y_1,\ldots,y_n]$. **Definition 8**. *Let $F \in \mathcal{S}_d$ and $L \in \mathcal{S}_1$. We define the **natural apolar scheme to $F$ supported at $L$** the scheme $Z_{F,L} \subset \mathbb{P}^n$ supported at $[L] \in \mathbb{P}^n$ and locally defined by $\underline{I}(Z_{F,L}) = \textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{L}) \subset \underline{\mathcal{R}}$.* Note that $\underline{\mathcal{R}}$ can be regarded as the coordinate ring of the affine chart $U_0 = \{[L] ~:~ Y_0 \circ L \neq 0\} \subset \mathbb{P}^n$ and $Z_{F,L}$ is a local $0$-dimensional scheme supported at the origin of $U_0$. Contraction behaves well with dehomogenization with respect to dual variables. In particular, if $g \in \underline{\mathcal{R}}$ is the dehomogenization of $G \in \mathcal{R}$ with respect to $Y_0$, and $g \mathbin{\mathpalette\dointprod\relax}f_{X_0} = 0$, then $G \mathbin{\mathpalette\dointprod\relax}F_{\rm dp} = 0$ [@BJMK18:Polynomials Corollary 3], and the last equality implies that $G \circ F = 0$ as observed in . Hence, the scheme $Z_{F,L}$ is apolar to $F$ according to . **Lemma 9** ([@BJMK18:Polynomials Corollary 4]). *The scheme $Z_{F,L}$ is apolar to $F$.* Here we detail how to concretely construct the ideal defining such a scheme. Fix $F \in \mathcal{S}_d$ and $L = \ell_0X_0 + \ldots + \ell_nX_n \in \mathcal{S}_1$. Without loss of generalities we may assume $\ell_0 = 1$. Over $\mathcal{S}$, we consider the change of variables given by $$\label{eq:Xchange} \phi : \mathcal{S}\rightarrow \mathcal{S}, \qquad \begin{cases} X_0 \mapsto X_0 - \sum_{i=1}^n \ell_iX_i, \\ X_i \mapsto X_i, & \text{ for } i \in \{1,\ldots,n\}. \end{cases}$$ We have $\phi(L) = X_0$ and $\tilde F = \phi(F)$, therefore we can represent $f_L$ as $\tilde f_{X_0} = \tilde F_{\rm dp}(1,x_1,\ldots,x_n) \in \underline{\mathcal{S}}_{\rm dp}$. Then $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{L})$ is the kernel of the infinite-dimensional *Hankel operator* [@BT20:Waring; @BRACHAT20101851]: $$H(f_{L}) : \underline{\mathcal{R}}\to \underline{\mathcal{S}}_{\rm dp}, \quad g \mapsto g \mathbin{\mathpalette\dointprod\relax}f_{L}.$$ However, since $y^\beta \mathbin{\mathpalette\dointprod\relax}f_{L} = 0$ for every $|\beta| > \deg( f_{L})$, the annihilator of $f_{L}$ is generated by the kernel of a truncated Hankel operator. Let $e = \deg(f_{L})$ and consider the restriction $$H^{e+1}(f_{L}) : \underline{\mathcal{R}}_{\leq e+1} \to \left(\underline{\mathcal{S}}_{\rm dp}\right)_{\leq e}.$$ Then, $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{L}) = \ker H^{e+1}(f_{L}).$ Note that the coefficients of the Hankel matrix can be computed directly from $\tilde F$. Indeed, if we label rows and columns of $H^{e+1}(f_{L})$ accordingly to the divided powers monomial basis of $\left(\underline{\mathcal{S}}_{\rm dp}\right)_{\leq e}$ and the standard monomial basis of $\underline{\mathcal{R}}_{\leq e+1}$, respectively, we have $$\label{eq:hankel} [H^{e+1}(f_{L})]_{\alpha,\beta} = {\rm eval}_{(0,\ldots,0)}\left(y^{\alpha+\beta} \mathbin{\mathpalette\dointprod\relax}f_{L}\right) = \begin{cases} Y^{(d-(|\alpha|+|\beta|),\alpha_1+\beta_1,\cdots,\alpha_n+\beta_n)} \circ \tilde F & \text{ if }|\alpha|+|\beta| \leq d, \\ 0 & \text{ otherwise.} \end{cases}$$ **Remark 10**. Let $g_{\rm dp} \in \underline{\mathcal{S}}_{\rm dp}$ be a degree-$d$ polynomial obtained from $g \in \underline{\mathcal{S}}$ by passing to divided powers. The ideal $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(g_{\rm dp}) = \textnormal{Ann}(g)$ has minimal generators in degree $d+1$ if and only if $g$ is a pure $d$-th power. When it is the case, we actually need to consider the kernel of $H^{e+1}(g_{\rm dp})$ to compute $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(g_{\rm dp})$, see e.g. . However, whenever $g$ is not a pure power, we may compute its annihilator by restricting its Hankel matrix to $H^{e}(g_{\rm dp}) : \underline{\mathcal{R}}_{\leq e} \to \left(\underline{\mathcal{S}}_{\rm dp}\right)_{\leq e}$, which makes the kernel computation more efficient, see e.g. . The homogenization $\tilde I = I(Z_{\tilde F,X_0}) = [\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{L})]^{\rm hom} \subset \mathcal{R}$ with respect to $Y_0$ defines a $0$-dimensional scheme apolar to $\tilde F$ and supported at $[X_0] \in \mathbb{P}^n$ as in . Note that the ideal homogenization is the only step in which non-linear algebra (e.g. Gröbner bases) may be required. Finally, to obtain the ideal defining $Z_{F,L}$ as in , we need to support $\tilde I$ on $[L] \in \mathbb{P}^n$, hence we perform the change of coordinate in $\mathcal{R}$ given by the dualization of the inverse of [\[eq:Xchange\]](#eq:Xchange){reference-type="ref" reference="eq:Xchange"}: $$\label{eq:Ychange} \psi = (\phi^{-1})^T : \mathcal{R}\rightarrow \mathcal{R}, \qquad \begin{cases} Y_0 \mapsto Y_0, \\ Y_i \mapsto \ell_iY_0 + Y_i, &\text{ for } i \in \{1,\ldots,n\}. \end{cases}$$ The ideal $I = \psi(\tilde I) \subset \mathcal{R}$ defines a $0$-dimensional scheme which is supported at $[L]$ and apolar to $F$. Indeed, the following lemma shows that the action by derivation is preserved under the changes of coordinates given by [\[eq:Xchange,eq:Ychange\]](#eq:Xchange,eq:Ychange){reference-type="ref" reference="eq:Xchange,eq:Ychange"}. **Lemma 11**. *Let $\phi$ and $\psi$ be changes of coordinates of [\[eq:Xchange,eq:Ychange\]](#eq:Xchange,eq:Ychange){reference-type="ref" reference="eq:Xchange,eq:Ychange"}. Then we have $$\psi(Y^\beta) \circ \phi^{-1}(X^\alpha) = Y^\beta \circ X^\alpha.$$* *Proof.* We write $$\psi(Y^{\beta}) \circ \phi^{-1}(X^\alpha) = \psi(Y_0^{\beta_0}) \circ \left( \psi(Y_1^{\beta_1}) \circ \left( \dots \psi(Y_n^{\beta_n}) \circ \phi^{-1}(X^\alpha)\right)\right).$$ By the chain rule of derivation, if $L \circ M = 0$ then $L^b \circ M^a = 0$ for any $a,b \in \mathbb{N}$. In particular, for every $j \in \{1, \dots, n\}$ we have $$\begin{aligned} \psi(Y_j^{\beta_j}) \circ \phi^{-1}(X^\alpha) &= (-\ell_jY_0+Y_j)^{\beta_j} \circ \left[\left(X_0+\ell_1X_1+\ldots+\ell_nX_n\right)^{\alpha_0}\prod_{i=1}^nX_i^{\alpha_i}\right] \\ & = \left[(X_0+\ell_1X_1+\ldots+\ell_nX_n)^{\alpha_0}\prod_{\substack{1 \leq i \leq n \\ i\neq j}}X_i^{\alpha_i}\right]\cdot (Y_j^{\beta_j}\circ X_j^{\alpha_j}). \end{aligned}$$ Therefore, by repeatedly applying the above equation for every $j$ we obtain $$\psi(Y_1^{\beta_1}) \circ \left( \dots \psi(Y_n^{\beta_n}) \circ \phi^{-1}(X^\alpha)\right) = (X_0+\ell_1X_1+\ldots+\ell_nX_n)^{\alpha_0} \cdot (Y_1^{\beta_1}\circ X_1^{\alpha_1}) \cdots (Y_n^{\beta_n}\circ X_n^{\alpha_n}).$$ The result follows by acting with $\psi(Y_0^{\beta_0}) = Y_0^{\beta_0}$ on the above quantity. ◻ Note that our choice of the change of coordinates in [\[eq:Xchange\]](#eq:Xchange){reference-type="ref" reference="eq:Xchange"} was arbitrary. It would have been enough to consider any set of linear forms $\{L_1,\ldots,L_n\}$ completing a basis of $\mathcal{S}_1$ together with $L$. Then, $\phi$ is the change of coordinates inverse to the one sending $X_0 \mapsto L_0$ and $X_i \mapsto L_i$, for any $i \in \{1,\ldots,n\}.$ **Algorithm 1** (Natural Apolar Scheme). Summary of construction of natural apolar schemes. **Input:** A homogeneous polynomial $F \in \mathcal{S}_d$ and a linear form $L \in \mathcal{S}_1$.\ **Output:** The ideal $I(Z_{F,L}) \subseteq\mathcal{R}$. 1. Define $\tilde F$ as the base-change of $F$ as in [\[eq:Xchange\]](#eq:Xchange){reference-type="ref" reference="eq:Xchange"}. 2. Compute $f_{L}$ as $\tilde F_{\rm dp}(1,x_1,\dots,x_n)$ and set $e = \deg ( f_{L} )$. 3. [\[HankelStep\]]{#HankelStep label="HankelStep"} Compute the ideal $\underline{I} = \ker H^{e+1}( f_{L} )$. 4. [\[GroebnerStep\]]{#GroebnerStep label="GroebnerStep"} Compute the homogenization $I \subset \mathcal{R}$ of $\underline{I} \subset \underline{\mathcal{R}}$. 5. Return the base-change of the ideal $I$ as in [\[eq:Ychange\]](#eq:Ychange){reference-type="ref" reference="eq:Ychange"}. ## Generalized Additive Decompositions (GADs) We recall the definition of the so-called generalized additive decompositions as introduced in [@IaKa:book], and we associate to them $0$-dimensional schemes by employing the notion of natural apolar scheme introduced in . **Definition 12**. *Let $F \in \mathcal{S}_d$ and let $L_1,\ldots,L_s \in \mathcal{S}_1$ be pairwise non-proportional linear forms. A **generalized additive decomposition** (GAD) of $F$ **supported at $\{L_1,\ldots,L_s\}$** is an expression $$\label{eq:GAD} F = \sum_{i=1}^s L_i^{d-k_i}G_i,$$ where for every $i \in \{1,\ldots,s\}$ we have $0 \leq k_i \leq d$ and $G_i \in \mathcal{S}_{k_i}$ is not divisible by $L_i$. If $s=1$, we call this GAD *local*.* Following [@BBM14:comparison; @BJMK18:Polynomials], we associate a $0$-dimensional scheme to any GAD as [\[eq:GAD\]](#eq:GAD){reference-type="ref" reference="eq:GAD"}. **Definition 13**. *The **scheme evinced by a GAD** as in [\[eq:GAD\]](#eq:GAD){reference-type="ref" reference="eq:GAD"} is the union of the natural apolar schemes to each summand with respect to the corresponding $L_i$, i.e., $$Z = \bigcup_{i=1}^sZ_{L_i^{d-k_i}G_i, L_i}. % \cup \ldots \cup Z_{L_s^{d-k_s}G_s, L_s}.$$ The **size** of a GAD as in [\[eq:GAD\]](#eq:GAD){reference-type="ref" reference="eq:GAD"} is the length of the evinced scheme $Z$.* Note that the same scheme may be evinced by different GADs. Indeed, $L^{d-k}G$ and $L^{d-k}G'$ evince the same scheme whenever $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(g_{L}) = \textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(g'_{L})$. However, schemes evinced by GADs of a given $F$ are always apolar to it. **Lemma 14**. *Let $Z$ be the scheme evinced by a GAD of $F$. Then $Z$ is apolar to $F$.* *Proof.* To ease notation, denote $F_i = L_i^{d-k_i}G_i$ in [\[eq:GAD\]](#eq:GAD){reference-type="ref" reference="eq:GAD"}. Let $I(Z)_d = I(Z) \cap \mathcal{R}_d$ and let $I(Z)_d^\perp$ be the orthogonal space via the non-degenerate pairing $\mathcal{R}_d \times \mathcal{S}_d \to \Bbbk$ induced by derivation. Then, $$I(Z)^\perp_d = \left(I\left(Z_{F_1,L_1}\right)_d \cap \ldots \cap I\left(Z_{F_s,L_s}\right)_d\right)^\perp = I\left(Z_{F_1,L_1}\right)^\perp_d + \ldots + I\left(Z_{F_s,L_s}\right)_d^\perp,$$ see e.g. [@Ger96 Proposition 2.6]. For every $i \in \{1,\ldots,s\}$ we have $F_i \in I\left(Z_{F_i,L_i}\right)_d^\perp$ by . Hence, $F \in I(Z)_d^\perp$ and, by the Apolarity Lemma (), this implies that $I(Z) \subseteq\textnormal{Ann}(F)$. ◻ The ideal defining schemes evinced by GADs can be easily computed by intersecting the ideals defining natural apolar schemes to local pieces of the additive decomposition, computed as in . **Algorithm 2** (Scheme evinced by GAD). [\[alg:GAD\]]{#alg:GAD label="alg:GAD"} **Input:** Linear forms $L_1, \dots, L_s \in \mathcal{S}_1$, forms $G_1, \dots, G_s \in \mathcal{S}_{k_i}$, an integer $d\geq k_i$.\ **Output:** The ideal defining the scheme evinced by the GAD as in . 1. For every $i \in \{1, \dots, s\}$ compute $I_i = I\big(Z_{L_i^{d-k_i}G_i,L_i}\big)$ with . 2. Return $I = \cap_{i = 1}^s I_i$. ## Examples {#app:Examples} Here we illustrate the above construction with some examples. **Example 15**. Let $F = (X_0+3X_1-2X_2)(X_1+X_2)X_2 \in \mathcal{S}_3$ and $L = X_0+3X_1-2X_2 \in \mathcal{S}_1$. Following we obtain $\tilde F = X_0X_1X_2 + X_0X_2^2 \in \mathcal{S}$ by $$X_0 \leftarrow X_0-3X_1+2X_2. %, \quad X_1 \leftarrow X_1, \quad X_2 \leftarrow X_2$$ In divided powers it becomes $\tilde F_{\rm dp} = X_0X_1X_2 + 2X_0X_2^{[2]}$, whose de-homogenization by $X_0 = 1$ is equal to $f_{L} = X_1X_2+2X_2^{[2]} \in \underline{\mathcal{S}}_{\rm dp}$. Since $X_1(X_2+X_2)$ is not a pure power, by we consider the truncation of the Hankel matrix in degree $2 = \deg ( f_{L} )$, i.e., $$H^{2}(f_{L}) = \begin{blockarray}{c c c c c c c c c c c} & 1 & y_1 & y_2 & y_1^2 & y_1y_2 & y_2^2 \\ \begin{block}{c (c c c c c c c c c c)} 1 & 0 & 0 & 0 & 0 & 1 & 2 \\ y_1 & 0 & 0 & 1 & 0 & 0 & 0 \\ y_2 & 0 & 1 & 2 & 0 & 0 & 0 \\ y_1^2 & 0 & 0 & 0 & 0 & 0 & 0 \\ y_1y_2 & 1 & 0 & 0 & 0 & 0 & 0 \\ y_2^2 & 2 & 0 & 0 & 0 & 0 & 0 \\ \end{block} \end{blockarray} \ ,$$ whose kernel defines the ideal $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{L}) = \big( y_2(2y_1-y_2), y_1^2 \big) \subset \underline{\mathcal{R}}$. Its homogenization in $\mathcal{R}$ is the ideal $\big( Y_2(2Y_1-Y_2), Y_1^2 \big)$, which we need to base-change as in [\[eq:Ychange\]](#eq:Ychange){reference-type="ref" reference="eq:Ychange"}, i.e. $$Y_1 \leftarrow -3Y_0 + Y_1, \quad Y_2 \leftarrow 2Y_0 + Y_2.$$ This way we obtain the ideal $I = \left( (2Y_0+Y_2)(8Y_0-2Y_1+Y_2), (3Y_0-Y_1)^2 \right) \subset \mathcal{R}$. Its radical ideal is $( 2Y_1+3Y_2, 2Y_0+Y_2 )$, i.e. it defines a $0$-dimensional scheme supported at $[L] = [X_1+3X_2-2X_3] \in \mathbb{P}^n$. One can directly verify that this scheme has length $4$ and it is apolar to $F$. Indeed, it is immediate to check that $$\begin{aligned} (2Y_0+Y_2)(8Y_0-2Y_1+Y_2) \circ F & = (-16Y_0^{2}+4Y_0Y_1-10Y_0Y_2+2Y_1Y_2-Y_2^{2}) \circ F = 0, \\ (3Y_0-Y_1)^2 \circ F & = (9Y_0^2 - 6Y_0Y_1 + Y_1^2) \circ F = 0.\end{aligned}$$ Hence, $I$ is the ideal defining $Z_{F,L}$. $\spadesuit$ **Example 16**. Let $F = (X_0+3X_1-2X_2)(X_1+X_2)X_2 \in \mathcal{S}_3$ the same polynomial of and consider $L = X_0 \in \mathcal{S}_1$. As the support is $X_0$, we do not need to change coordinates, so we directly de-homogenize $F_{\rm dp}$ with respect to $L$, obtaining $f_{L} = y_1y_2 + 2y_2^{[2]} + 6y_1^{[2]}y_2 + 2y_1y_2^{[2]} - 12y_2^{[3]}$. Since $F$ is not a pure cube, we consider the truncation of the Hankel matrix in degree $3 = \deg(f_L)$, namely $$H^{3}(f_{L}) = \begin{blockarray}{c c c c c c c c c c c c c c c} & 1 & y_1 & y_2 & y_1^2 & y_1y_2 & y_2^2 & y_1^3 & y_1^2y_2 & y_1y_2^2 & y_2^3 \\ \begin{block}{c (c c c c c c c c c c c c c c)} 1 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 6 & 2 &-12 \\ y_1 & 0 & 0 & 1 & 0 & 6 & 2 & 0 & 0 & 0 & 0 \\ y_2 & 0 & 1 & 2 & 6 & 2 &-12 & 0 & 0 & 0 & 0 \\ y_1^2 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y_1y_2 & 1 & 6 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y_2^2 & 2 & 2 &-12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y_1^3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y_1^2y_2 &6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y_1y_2^2 &2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ y_2^3 & -12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{block} \end{blockarray} \ .$$ Its kernel is given by the ideal $$\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{L}) = ( 5y_2^3 + 76y_1^2 - 12y_1y_2 + 36y_2^2, 2 y_1^2y_2 + y_2^3, y_1^3, 6y_1y_2^2 + y_2^3 ) \subset \underline{\mathcal{R}}.$$ To homogenize it, we compute a Gröbner basis with respect to the graded lexicographic ordering: $$\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{L}) = ( y_1^3, 5y_1^2y_2 - 38y_1^2 + 6y_1y_2 - 18y_2^2, 15y_1y_2^2 - 38y_1^2 + 6y_1y_2 - 18y_2^2, 5y_2^3 + 76y_1^2 - 12y_1y_2 + 36y_2^2 ).$$ Hence the natural apolar scheme is defined by the ideal $$\left( \begin{multlined} Y_1^3, 5Y_1^2Y_2 - 38Y_0Y_1^2 + 6Y_0Y_1Y_2 - 18Y_0Y_2^2, \\ 15Y_1Y_2^2 - 38Y_0Y_1^2 + 6Y_0Y_1Y_2 - 18Y_0Y_2^2, 5Y_2^3 + 76Y_0Y_1^2 - 12Y_0Y_1Y_2 + 36Y_0Y_2^2 \end{multlined} \right) \subset \mathcal{R}.$$ One can easily verify that this ideal indeed defines a $0$-dimensional scheme apolar to $F$ and supported at $[X_0]$, whose length is $6$. $\spadesuit$ **Example 17**. Let $F = (X_0+3X_1-2X_2)(X_1+X_2)X_2 \in \mathcal{S}_3$ be the polynomial of . From the equality $(X_1 + X_2)X_2 = (\frac{X_1}{2}+X_2)^2-(\frac{X_1}{2})^2$ we immediately get another non-local GAD of $F$, namely $$\label{eq:gad2} F = \left(\frac{X_1}{2}+X_2\right)^2 (X_0+3X_1-2X_2)-\left(\frac{X_1}{2}\right)^2 (X_0+3X_1-2X_2).$$ We compute the scheme $Z$ evinced by the above GAD, supported at $[X_1+2X_2]$ and $[X_1]$. We begin with the first addendum $F_1 = \frac{1}{4}(X_1+2X_2)^2(X_0+3X_1-2X_2)$ and $L_1 = X_1+2X_2$. We can neglect the constant factor $\frac{1}{4}$, and since $L_1$ has no $X_0$ terms, we simply switch the roles of $X_0$ and $X_1$. In order to de-homogenize with respect to $L_1$, we perform the substitution $$\quad X_1 \leftarrow X_1-2X_2,$$ and we get $(f_1)_{L_1} = x_0+3-8x_2$. Since $X_0+3X_1-8X_2$ is a pure power, we need to consider the truncation of the Hankel matrix in degree $2 = \deg\big(( f_1)_{L_1}\big) + 1$, i.e. $$\mathbb{H}^2 \big( (f_1)_{L_1} \big) = \begin{blockarray}{c c c c c c c } & 1 & y_0 & y_2 & y_0^2 & y_0y_2 & y_2^2 \\ \begin{block}{c (c c c c c c)} 1 & 18 & 2 & -16 & 0 & 0 & 0 \\ y_0 & 2 & 0 & 0 & 0 & 0 & 0 \\ y_2 & -16 & 0 & 0 & 0 & 0 & 0 \\ \end{block} \end{blockarray}\,,$$ whose kernel defines the ideal $( 8y_0+y_2,y_2^2 ) \subset \underline{\mathcal{R}}$. After the homogenization and the base-change $$Y_2 \leftarrow -2Y_1+Y_2,$$ we obtain the ideal $\left( 8Y_0-2Y_1+Y_2, (2Y_1-Y_2)^2 \right) \subset \mathcal{R}$ defining the scheme $Z_1 = Z_{F_1,X_1+2X_2}$, which is $0$-dimensional, of length $2$ and supported at the point $[X_1+2X_2] \in \mathbb{P}^n$. We proceed with the second addendum $F_2 = \frac{1}{4}X_1^2(X_0+3X_1-2X_2)$ and $L_2 = X_1$. As above, $X_1$ plays the role of $X_0$. Since $(f_2)_{L_2} = x_0+3-2x_2$, we again consider the truncation of the Hankel matrix in degree $2$: $$\mathbb{H}^2 \big( (f_2)_{L_2} \big) = \begin{blockarray}{c c c c c c c } & 1 & y_0 & y_2 & y_0^2 & y_0y_2 & y_2^2 \\ \begin{block}{c (c c c c c c)} 1 & 18 & 2 & -4 & 0 & 0 & 0 \\ y_0 & 2 & 0 & 0 & 0 & 0 & 0 \\ y_2 & -4 & 0 & 0 & 0 & 0 & 0 \\ \end{block} \end{blockarray}\,,$$ whose kernel defines the ideal $( 2y_0+y_2,y_2^2 ) \subset \underline{\mathcal{R}}$. Hence, the scheme $Z_2 = Z_{F_2,X_1}$ is defined by the ideal $( 2Y_0+Y_2,Y_2^2 ) \subset \mathcal{R}$, it is $0$-dimensional of length $2$ and is supported at the point $[X_1] \in \mathbb{P}^n$. In conclusion, the GAD of [\[eq:gad2\]](#eq:gad2){reference-type="ref" reference="eq:gad2"} evinces the scheme $Z = Z_1 \cup Z_2$ defined by $$I(Z) = \big( 8Y_0-2Y_1+Y_2, (2Y_1-Y_2)^2 \big) \cap ( 2Y_0+Y_2,Y_2^2 ) = ( 4Y_0Y_1-10Y_0Y_2+2Y_1Y_2-Y_2^2, Y_0^2 ).$$ One can directly check that $Z$ has length $4$, it is supported at the points $[X_1]$ and $[X_1+2X_2]$, and it is apolar to $F$. $\spadesuit$ # GADs and Irredundant schemes {#sec:irred} In this section, we investigate irredundant schemes evinced by GADs by employing ideas on natural apolar schemes from [@BJMK18:Polynomials]. **Remark 18**. Let $[L] \in \mathbb{P}^n$ be a simple point defined by the ideal $\wp_L \subset \mathcal{R}$. Recall that the *$j$-fat point supported at $[L]$* is the $0$-dimensional scheme defined by the ideal $\wp_L^j$ For any $k \leq d$, the natural apolar scheme of $F = L^{d-k}G \in \mathcal{S}_d$ supported at $[L]$ is contained in the $(k+1)$-fat point supported at $[L]$, since the localization $f_{L}$ has degree at most $k$. Thus, $Z_{F,L}$ is $k$-regular, as a $(k+1)$-fat point is always $k$-regular and the containment preserves the regularity [@BCGI07:Osculating; @Bernardi2009982]. Finally, if $F$ is *concise* in $n+1$ variables, i.e., $\mathrm{HF}_Z(1) = n+1$, then $Z_{F,L}$ is regular in degree $k-n$ since its Hilbert function starts with $\mathrm{HF}_Z = (1,n+1,\dots)$ and is strictly increasing until it stabilizes. **Remark 19**. By [@BJMK18:Polynomials Lemma 3], given a local scheme $Z \subset \mathbb{P}^n$ apolar to $F \in \mathcal{S}_d$ and supported at $[L]$, there exists $G \in \mathcal{S}_D$ ($D \geq d$) such that $Z_{G,L} \subseteq Z$ and $F = H \circ G$ for some $H \in \mathcal{R}_{D-d}$. Furthermore, in [@BJMK18:Polynomials Proposition 1] it is shown that, under minimality assumption, the localizations of $F_{\rm dp}$ and $G_{\rm dp}$ with respect to $L$ are equal up to degree $d$. In that result, the minimality requirement is in terms of minimal length among the schemes supported at $[L]$ and apolar to $F$. However, we observe that in that proof irredundancy is actually enough. For the sake of completeness, we report here the proof of the following statement, which may be seen as a non-local version of [@BJMK18:Polynomials Proposition 1]. **Proposition 20**. *Let $Z$ be a $0$-dimensional scheme apolar and irredundant to $F \in \mathcal{S}_d$. Then $Z$ is evinced by a GAD of $F$ if and only if there are $L_1,\dots,L_s \in \mathcal{S}_1$ such that $I(Z) \supseteq \bigcap_{i=1}^s \wp_{L_i}^{d+1}$.* Let $Z = Z_1 \cup \dots \cup Z_s$ be the irreducible decomposition of $Z$. If $Z$ is evinced by a GAD as in [\[eq:GAD\]](#eq:GAD){reference-type="ref" reference="eq:GAD"}, then each $Z_i = Z_{L_i^{d-k_i}G_i}$ is contained in a $(k_i+1)$-fat point by , hence $I(Z_i) \supseteq \wp_{L_i}^{k_i+1} \supseteq \wp_{L_i}^{d+1}$. Note that this implication does not need irredundancy. Conversely, since $I(Z) \subseteq\textnormal{Ann}(F)$, then we have $$F \in I(Z)^\perp_d = I(Z_1)_d^\perp + \dots + I(Z_s)^\perp_d.$$ Therefore, we have an additive decomposition $F = \sum_{i=1}^s F_i$ with $F_i \in I(Z_i)^\perp_d$. By there are $G_i \in \mathcal{S}_{D_i}$ and $H_i \in \mathcal{R}_{D_i-d}$ such that $Z_{G_i,L_i} \subseteq Z_i$ and $H_i \circ G_i = F_i$. By [@BJMK18:Polynomials Lemma 3] we know that $h_i \mathbin{\mathpalette\dointprod\relax}(g_i)_{L_i}$ and $(f_i)_{L_i}$ are equal up to degree $d$, but since $I(Z_i) \supseteq \wp_{L_i}^{d+1}$, the degree of the local generator $h_i \mathbin{\mathpalette\dointprod\relax}(g_i)_{L_i}$ is bounded by $d$, so it equals $(f_i)_{L_i}$. Hence, we have $$\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}\big((f_i)_{L_i}\big) = \textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}\big(h_i \mathbin{\mathpalette\dointprod\relax}(g_i)_{L_i}\big) \supseteq \textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}\big((g_i)_{L_i}\big),$$ therefore the natural apolar scheme $Z_{F_i,L_i}$ is contained in $Z_{G_i,L_i}$ and is apolar to $F_i$. However, the scheme $Z_i$ needs to be irredundant to $F_i$, hence we conclude that $Z_{F_i,L_i} = Z_{G_i,L_i} = Z_i$. Therefore $Z$ is the scheme evinced by the additive decomposition $\sum_{i=1}^s F_i$ supported at $L_1,\dots,L_s$. **Proposition 21**. *Let $Z$ be a $0$-dimensional non-redundant scheme apolar to $F \in \mathcal{S}_d$. Then, $Z$ is evinced by a GAD of $F$ if and only if each irreducible component of $Z$ is contained in a $(d+1)$-fat point, i.e., $I(Z) \supset \bigcap_{i=1}^s \wp_{L_i}^{d+1}$ for some $L_1,\ldots,L_s \in \mathcal{S}_1$.* *Proof.* If $Z$ is evinced by a GAD, then each irreducible component is contained in a $(d+1)$-fat point by . Note that this implication does not need the assumption of non-redundancy. Conversely, let $Z = Z_1 \cup \cdots \cup Z_s$ be the irreducible decomposition such that $I(Z_i)\supset \wp_{L_i}^{d+1}$. Since $I(Z) \subset \textnormal{Ann}(F)$, then $$F \in I(Z)^\perp_d = I(Z_1)_d^\perp + \dots + I(Z_s)^\perp_d,$$ therefore $F = \sum_{i=1}^s F_i$ for some $F_i \in I(Z_i)^\perp_d$. $Z_i$ is non-redundant for $F_i$, $i \in \{1,\ldots,s\}$. Hence, by [@BJMK18:Polynomials Proposition 1], there is $G_i \in \mathcal{S}_{D_i}$ such that $Z_i = Z_{G_i,L_i}$ and such that the localizations $(f_i)_{\rm dp}$ and $(g_i)_{\rm dp}$ of $(F_i)_{\rm dp}$ and $(G_i)_{\rm dp}$ with respect to $L_i$, respectively, have the same degree-$d$ tail, i.e., they are equal up to degree $d$. Now, $(f_i)_{\rm dp}$ has degree at most $d$ and, since $I(Z_i)\supset \wp_{L_i}^{d+1}$ by assumption, also $(g_i)_{\rm dp}$ has degree at most $d$. Hence, $(g_i)_{\rm dp} = (f_i)_{\rm dp}$ and therefore we conclude that $Z_i = Z_{F_i,L_i}$. ◻ In the following example, we observe that even if $Z$ is evinced by a GAD of $F \in \mathcal{S}_d$ and its components are contained in $(d+1)$-fat points, $Z$ may still be redundant to $F$. D: Ho messo un esempio più piccino che poi uso anche per far vedere che il risultato della sezione dopo è strict. **Example 22**. Consider the GAD $F = X_0G_1 + X_1G_2 \in \mathcal{S}_3$, where $$G_1 = 5X_0^2 + 3X_0X_1 - 5X_0X_2 + X_1^2 - 2X_1X_2 , \quad G_2 = 3X_0^2 + 6X_0X_1 - 6X_0X_2 + 5X_1^2 + 8X_1X_2.$$ The scheme $Z$ evinced by such GAD is given by the ideal $$I(Z) = \big( Y_2^2, Y_0Y_1(Y_0Y_1+Y_0Y_2+Y_1Y_2) \big) \subset \mathcal{R}.$$ One can directly verify that $I(Z) \supset \wp_{X_0}^3 \cap \wp_{X_1}^3$, and its Hilbert function is $\mathrm{HF}_Z = (1,3,5,7,8,8,-)$, hence it is not $3$-regular. We use the summands of $G_1$ containing $X_1$ as factor to obtain a different GAD supported at the same points: $F = X_0^2 \tilde G_1 + X_1 \tilde G_2$, where $$\tilde G_1 = 5X_0 + 2X_1 - 5X_2, \quad \tilde G_2 = 4X_0^2 + 7X_0X_1 - 8X_0X_2 + 5X_1^2 + 8X_1X_2.$$ The scheme $Y$ evinced by the last GAD is by construction apolar to $F$, and it is defined by $$I(Y) = \left( Y_2^2, Y_0\left(Y_0Y_1 + \frac{2}{5}Y_0Y_2 + Y_1Y_2\right) \right) \subset \mathcal{R}.$$ Note that the Hilbert function of $Y$ is $\mathrm{HF}_Y = (1,3,5,6,6,-)$, so $Y \neq Z$. However, since $$Y_0Y_1(Y_0Y_1+Y_0Y_2+Y_1Y_2) \in I(Y) %A = \left(Y_0 + \frac{3}{5}Y_2\right)B - \frac{3}{5} \left(\frac{2}{5}Y_0^2+Y_0Y_1\right)Y_2^2,$$ we deduce that $I(Z) \subsetneq I(Y) \subseteq \textnormal{Ann}(F)$. Hence $Y \subsetneq Z$ is apolar to $F$, i.e. $Z$ is redundant. $\spadesuit$ **Example 23**. Consider the GAD $F = X_0G_1 + X_1^2G_2 \in \mathcal{S}_3$, where $$G_1 = 4 X_0^2 + 2 X_0 X_1 - 4 X_1^2 , \quad G_2 = - 3 X_0 - 5 X_1.$$ The scheme $Z$ evinced by such GAD is given by the ideal $$I(Z) = ( Y_0^2 Y_1^3 ) = \wp_{X_0}^3 \cap \wp_{X_1}^2 \subset \mathcal{R}.$$ Its Hilbert function is $\mathrm{HF}_Z = (1,2,3,4,5,5,\dots)$, hence it is not $3$-regular. We move the addendum in $G_1$ containing $X_1^2$ to $G_2$, obtaining a different GAD supported at the same points: $F = X_0^2 \tilde G_1 + X_1^2 \tilde G_2$, where $$\tilde G_1 = 4 X_0+2 X_1, \quad \tilde G_2 = -7 X_0 - 5 X_1.$$ The scheme $\tilde Z$ evinced by the last GAD is by construction apolar to $F$, and it is defined by $$I(\tilde Z) = ( Y_0^2 Y_1^2 ) = \wp_{X_0}^2 \cap \wp_{X_1}^2 \subset \mathcal{R}.$$ The Hilbert function of $\tilde Z \neq Z$ is $\mathrm{HF}_{\tilde Z} = (1,2,3,4,4,\dots)$, and clearly $I(Z) \subsetneq I(\tilde Z) \subseteq\textnormal{Ann}(F)$. Hence, $Z$ is redundant. It can be directly verified that $\tilde Z$ is irredundant to $F$ (e.g. as in ), but not minimal, since $$I_W = ( 79 Y_0^2 - 166 Y_0Y_1 + 88 Y_2^2 ) \subset \mathcal{R}$$ is evinced by the (unique) Waring decomposition of $F$, defining a scheme of length $2$ apolar to $F$. $\spadesuit$ **Corollary 24**. *Let $L_1,\ldots,L_s \in \mathcal{S}_1$ and $Z = Z_1 \cup \ldots \cup Z_s$ be a $0$-dimensional scheme apolar to $F \in \mathcal{S}_d$ such that for every $i \in \{1, \dots, s\}$ we have $I(Z_i) \supset \wp_{L_i}^{\tilde k_i+1}$ with $\tilde k_i \leq d$. Then $Z$ contains a scheme evinced by a GAD of $F$ as in [\[eq:GAD\]](#eq:GAD){reference-type="ref" reference="eq:GAD"}, with $k_i \leq \tilde k_i$.* Let $Y = Y_1 \cup \ldots \cup Y_s \subseteq Z$ be non-redundant and apolar to $F$, with $Y_i \subseteq Z_i$. Then, it is enough to apply the proof of to $Y$, since $$I(Y_i)^\perp_d \subseteq I(Z_i)^\perp_d \subseteq (\wp_i^{\tilde k_i+1})_d^\perp = \langle L_i^{d-{\tilde k_i}}Q ~:~ Q \in \mathcal{S}_{\tilde k_i}\rangle,$$ where the last equality is a classical result, see e.g. [@Ger96 Theorem 3.2]. We conclude that $I(Y_i)$ is evinced by $F_i = L_i^{d-{\tilde k_i}} Q_i$, which becomes a valid (local) GAD after collecting all the factors $L_i$ in $Q_i$. Thus, $Y$ is evinced by the GAD $F = \sum_{i=1}^s F_i$ supported at $L_1, \dots, L_s$. In the following example, we show that the degree $D$ of the polynomial $G$ from may well exceed $d = \deg(F)$. We thank J. Buczyński for pointing it out. **Example 25**. Consider the following polynomial: $$\begin{aligned} F = 24\,X_{0}^{3}& + 70\,X_{0}^{2}X_{1} + 75\,X_{0}^{2}X_{2} + 70\,X_{0}^{2}X_{3} + 180\,X_{0}^{2}X_{4} +10\,X_{0}^{2}X_{5} + 10\,X_{0}X_{1}^{2} \\ & + 70\,X_{0}X_{2}^{2}+ 360\,X_{0}X_{2}X_{3} + 120\,X_{0}X_{2}X_{4} + 60\,X_{0}X_{3}^{2} + 60\,X_{2}^{3} + 60\,X_{2}^{2}X_{3} \in \mathcal{S}_3, \end{aligned}$$ and let $Z$ be the scheme defined by the ideal $$\begin{aligned} I(Z) = ( -Y_0Y_3 + Y_2^2, \, -Y_1Y_4 + Y_2Y_3, \, &-Y_1Y_5 + Y_1^2, \, -6Y_1Y_5 + Y_2Y_4, \, -6Y_1Y_5 + Y_3^2, \\ &Y_1Y_2, \, Y_1Y_3, \, Y_1Y_4, \, Y_1Y_5, \, Y_2Y_5, \, Y_3Y_4, \, Y_3Y_5, \, Y_4^2, \, Y_4Y_5, \, Y_5^2 ) \subset \mathcal{R}. \end{aligned}$$ One can computationally check that $Z$ is a local $0$-dimensional scheme apolar to $F$, of minimal length $6$ and supported at $[X_0] \in \mathbb{P}^n$. One can also verify that it is the unique scheme of minimal length apolar to such $F$, by explicitly computing minimal apolar schemes [@BT20:Waring], or by observing that $I(Z) = \textnormal{Ann}(F) \cap \mathcal{R}_{\leq 2}$ and the Hilbert function of $\mathcal{R}/\textnormal{Ann}(F)$ is $(1,6,6,1)$. In particular, $Z$ is non-redundant. Since $I(Z) \not\supseteq \wp_{X_0}^4$, by there is no GAD of $F$ that evinces this apolar scheme. However, as recalled in , since $I(Z) \supseteq \wp_{X_0}^5$ then $Z$ is evinced by a GAD of a degree-$4$ polynomial $G$ having $F$ among its partials. Indeed, let us consider the polynomial $$\begin{aligned} G = \ &6 X_{0}^{4} + \frac{70}{3}X_{0}^{3}X_{1} + 25 X_{0}^{3}X_{2} + \frac{70}{3}X_{0}^{3}X_{3} + 60 X_{0}^{3}X_{4} + \frac{10}{3}X_{0}^{3}X_{5} + 5 X_{0}^{2}X_{1}^{2} + 35 X_{0}^{2}X_{2}^{2} \\ & + 180 X_{0}^{2}X_{2}X_{3} + 60 X_{0}^{2}X_{2}X_{4} + 30 X_{0}^{2}X_{3}^{2} + 60 X_{0}X_{2}^{3} + 60 X_{0}X_{2}^{2}X_{3} + 5 X_{2}^{4} \in \mathcal{S}_4. \end{aligned}$$ Note that $Y_0 \circ G = F$. Moreover, $Z = Z_{G,X_0}$, i.e., it is evinced by the trivial GAD of $G$ given by $G = X_0^0G$. This example shows why the containment in $(d+1)$-fat points is crucial for and . In particular, we have that $$\begin{aligned} g_{X_0} & = 120x_2^4 + f_{X_0} = \\ & = 120 x_2^4 + 360 x_{2}^{3} + 120 x_{2}^{2}x_{3} + 20 x_{1}^{2} + 140 x_{2}^{2} + 360 x_{2}x_{3} + 120 x_{2}x_{4} + 120 x_{3}^{2} + 140 x_{1} + 150 x_{2}\\ &\quad +140 x_{3}+360 x_{4}+20 x_{5}+144. \end{aligned}$$ We observe that $g_{X_0}$ and $f_{X_0}$ are equal up to degree $3$, but since $$(y_2^2 - y_3) \mathbin{\mathpalette\dointprod\relax}f_{X_0} = -120 x_2^2 \neq 0,$$ then $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(g_{X_0}) \not\subseteq\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(f_{X_0})$. $\spadesuit$ # Regularity of schemes evicing GADs {#sec:regularity} ## Apolar schemes with low multiplicities and independent supports {#ssec:lowmultiplicity_independent} For a given $L \in \mathcal{S}_1$, let $D_L = L^\perp \cap \mathcal{R}_1$ and $D^e_L\subset\mathrm{Sym}^e\mathcal{R}_1$ be its $e$-th symmetric power. We also define the $\Bbbk$-vector spaces $$\mathcal{D}^e_L(F) = \langle H \circ F ~:~ H \in D^e_L\rangle \subseteq \mathcal{S}_{d-e},$$ and given a vector space $V \subseteq \mathcal{S}_m$ and $H \in \mathcal{S}_l$, we write $$H \cdot V = \{HF ~:~ F \in V\} \subseteq\mathcal{S}_{l+m}.$$ With the notation of the previous sections, as in [@BJMK18:Polynomials Remark 3], we have $$I(Z_{F,L})_d^\perp = \mathbb{P}\left(\sum_{e = 0}^d L^e \cdot \mathcal{D}^e_L(F)\right) \subset \mathbb{P}(\mathcal{S}_d).$$ When $F=L^{d-k}G$, from the above equality and the chain rule of derivation we get $$I(Z_{L^{d-k}G,L})_d^\perp = \mathbb{P}\left(\sum_{e = 0}^k L^{d-k+e} \cdot \mathcal{D}^e_L(G)\right) \subset \mathbb{P}(\mathcal{S}_d).$$ **Remark 26**. Let $Z = \cup_{i = 1}^s Z_i \subset \mathbb{P}^n$ be the irreducible decomposition of a $0$-dimensional scheme. Then $Z$ is $h$-regular precisely when $\dim I(Z)_h^{\perp} = \deg (Z) = \sum_{i=1}^s \deg (Z_i)$, therefore there cannot be $\Bbbk$-linear relations involving generators of $I(Z_i)_h^{\perp}$ for different $i$'s. If there is a relation between different $I(Z_i)_d^{\perp}$ as in , the scheme $Z$ is not $d$-regular. However, the following proposition shows that if such $Z$ is evinced by a GAD of $F\in \mathcal{S}_d$ and is irredundant to it, such a relation cannot involve addenda appearing in that GAD. **Proposition 27**. *Let $Z$ be the scheme evinced by the GAD $F=\sum_{i=1}^s L_i^{d-k_i}G_i \in \mathcal{S}_d$. If, for some $i \in \{1, \dots, s\}$, we have $$\label{smaller:inverse} L_i^{d-k_i}G_i \in \sum_{1 \leq e_i \leq k_i} L_i^{d-k_i+e_i} \cdot \mathcal{D}^{e_i}_{L_i}(G_i) + \sum_{\substack{1 \leq j \leq s \\ j \neq i}} \sum_{0 \leq e_j \leq k_j} L_j^{d-k_j+e_j} \cdot \mathcal{D}^{e_j}_{L_j}(G_j),$$ then $Z$ is redundant to $F$. It is intended that the first sum in [\[smaller:inverse\]](#smaller:inverse){reference-type="ref" reference="smaller:inverse"} is empty if $k_i=0$.* *Proof.* Without loss of generality, we may assume that in [\[smaller:inverse\]](#smaller:inverse){reference-type="ref" reference="smaller:inverse"} we have $i = 1$. We define a scheme $Z'$ apolar to $F$ as follows. $\bullet$ If $k_1 = 0$, by [\[smaller:inverse\]](#smaller:inverse){reference-type="ref" reference="smaller:inverse"}, we simplify the GAD as $F = \sum_{j=2}^s L_j^{d-k_i}G'_j$ with $G'_j \in \sum_{e=0}^{k_i}L_j^e D^e_{L_j}(G_j)$. We call $Z'$ the scheme evinced by this GAD of $F$. $\bullet$ If $k_1 > 0$, we replace $L_1^{d-k_1}G_1$ in the GAD of $F$ with the linear combination deduced from [\[smaller:inverse\]](#smaller:inverse){reference-type="ref" reference="smaller:inverse"}. In particular, there are elements $H_{j,e_j} \in \mathcal{D}_{L_j}^{e_j}$ and integers $m_j \in \mathbb{N}$ such that we can write $$\label{GAD2} F = % L_1^{d-k_1+m_1}\cdot \sum_{0 < m_1 \leq e_1 \leq k_1-1} L_1^{e_1-m_1} \cdot (H_{1,e_1} \circ G_1) + \sum_{j=1}^s L_j^{d-k_j+m_j} \left( \sum_{m_j \leq e_j \leq k_j} L_j^{e_j-m_j} \left(H_{j,e_j} \circ G_j\right)\right).$$ Since $k_1 > 0$, then we have $m_1 \geq 1$ in [\[GAD2\]](#GAD2){reference-type="ref" reference="GAD2"}. The last equation is a GAD of $F$ up to deleting vanishing addenda and, for all the others, choosing $m_j$ such that $H_{j,m_j} \neq 0$. Let $Z'$ be the scheme evinced by the new GAD in [\[GAD2\]](#GAD2){reference-type="ref" reference="GAD2"}. By construction, $Z'$ is apolar to $F$, so it is sufficient to show that $Z' \subsetneq Z$. Following the notation introduced in , let $g_j = (g_j)_{L_j} \in \underline{\mathcal{S}}$ be the de-homogenization of $(G_j)_{\rm dp}$ with respect to $L_j$, and let $h_{j,e_j} \in \underline{\mathcal{R}}$ be the dehomogenization of $H_{j,e_j}$ with respect to the dual linear form $L_j^*$ of $L_j$. Since $H_{j,e_j} \in D^{e_j}_{L_j} \subset {\rm Sym}^{e_j}L_j^\perp$, then $H_{j,e_j}$ does not involve $L_j^*$, so its dehomogenization $h_{j,e_j}$ is equal to $H_{j,e_j}$. Thus, the de-homogenization of $(H_{j,e_j}\circ G_j)_{\rm dp} = H_{j,e_j} \mathbin{\mathpalette\dointprod\relax}(G_j)_{\rm dp}$ with respect to $L_j$ coincides with $h_{j,e_j} \mathbin{\mathpalette\dointprod\relax}g_j$. In particular, the $j$-th component of $Z'$ is defined by $\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}\big(\sum_{m_j \leq e_j \leq k_j} h_{j,e_j} \mathbin{\mathpalette\dointprod\relax}g_j\big)$. Since $$\textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}\left(\sum_{m_j \leq e_j \leq k_j} h_{j,e_j} \mathbin{\mathpalette\dointprod\relax}g_j\right) \supseteq \textnormal{Ann}^{\mathbin{\mathpalette\dointprod\relax}}(g_j),$$ we deduce that $Z' \subseteq Z$. We now show that this containment is proper. In the case $k_1 = 0$, then the containment is strict because $Z'$ has no support on $[L_1]$. In the case $k_1 > 0$, since $m_1 \geq 1$, $\deg (\sum_{m_i \leq e_i \leq k_i} h_{1,e_1} \mathbin{\mathpalette\dointprod\relax}g_1) < \deg (g_1)$ so the socle degree of the first component of $I(Z')$ and $I(Z)$ are different, so again they must be different. ◻ **Proposition 28**. *Let $s>1$ and $L_1,\ldots,L_s \in \mathcal{S}_1$ be $\Bbbk$-linearly independent forms and $Z$ be the scheme evinced by a GAD of $F \in \mathcal{S}_d$ as in [\[eq:GAD\]](#eq:GAD){reference-type="ref" reference="eq:GAD"}. If either* 1. *[\[prop:a\]]{#prop:a label="prop:a"} $d > \max_{i \neq j}\{k_i + k_j\}$, or* 2. *[\[prop:b\]]{#prop:b label="prop:b"} $d > \max_{i \neq j}\{k_i+k_j-2\}$ and $Z$ is irredundant,* *then $Z$ is $d$-regular.* *Proof.* For every $1 \leq i \leq s$ we let $Z_i$ be the natural apolar scheme to $F_i=L_i^{d-k_i}G_i$ supported at $L_i$, so $Z=\cup_{i=1}^s Z_i$. By each $Z_i$ is $d$-regular, therefore $\dim I(Z_i)_d^{\perp} = \deg (Z_i)$. By , we only need to show that there are no $\Bbbk$-linear relations involving generators of $I(Z_i)_d^{\perp}$ for different $i$'s. If there was such a relation, there should exist $Q_i \in \sum_{e=0}^{k_i} L_i^{e}\cdot D^e_{L_i}(G_i)$, for $i = 1,\ldots,s$, such that $$\label{eq:condition_prop3} L_i^{d-k_i}Q_i=\sum_{i\neq j}L_j^{d-k_j}Q_j.$$ Since the $L_i$'s are linearly independent, up to a change of coordinates we can write the above as $$X_i^{d-k_i}\tilde Q_i=\sum_{i\neq j}X_j^{d-k_j}\tilde Q_j.$$ In case [\[prop:a\]](#prop:a){reference-type="ref" reference="prop:a"}, the hypothesis $d-k_i > k_j = \deg ( Q_j) = \deg ( \tilde Q_j)$ prevents form factoring $X_i^{d-k_i}$ out of the right-hand side of the above equation. Thus, no such relation may hold. In case [\[prop:b\]](#prop:b){reference-type="ref" reference="prop:b"}, since $Z$ is irredundant, by we may assume that any relation between the $I(Z_i)_d^{\perp}$'s does not involve any of the terms $L_i^{d-k_i}G_i$. Thus, [\[eq:condition_prop3\]](#eq:condition_prop3){reference-type="ref" reference="eq:condition_prop3"} actually leads to a relation of the form $$X_i^{d-k_i+1}\tilde Q_i=\sum_{i\neq j}X_j^{d-k_j+1}\tilde Q_j.$$ As in the previous case, the factor $X_i^{d-k_i+1}$ cannot appear on the right-hand side of the above sum due to $d > \max_{i \neq j}\{k_i+k_j\}-2$. In conclusion, in both cases, the $I(Z_i)_d^{\perp}$'s cannot intersect, so $Z$ is $d$-regular. ◻ **Remark 29**. We note that requiring $s>1$ in is not restrictive, as in the local case ($s=1$) already contains a stronger result. An immediate corollary of is the following. **Corollary 30**. *Let $Z$ be the scheme evinced by the GAD $F = \sum_{i=1}^s L_i^{d-k_i}G_i \in \mathcal{S}_d$, such that $L_1,\ldots,L_s$ are $\Bbbk$-linearly independent and $k_i < \frac{d}{2}$ for every $i \in \{1,\ldots,s\}$. Then $Z$ is $d$-regular.* **Corollary 31**. *Let $Z = \bigcup_{i=1}^s Z_i$ be a $0$-dimensional scheme apolar and irredundant to $F \in \mathcal{S}_d$, such that $I(Z_i) \supset \wp_{L_i}^{\lceil{\frac{d}{2}}\rceil+ 1}$ and the $L_i$ are $\Bbbk$-linearly independent. Then $Z$ is $d$-regular.* It follows by together with . **Remark 32**. We notice that every requirement of is sharp. In fact, shows that the inequality in [\[prop:a\]](#prop:a){reference-type="ref" reference="prop:a"} cannot be improved: if $d = \max_{i \neq j}\{k_i + k_j\}$ the scheme $Z$ may be not $d$-regular. Similarly, the following shows that the inequality in [\[prop:b\]](#prop:b){reference-type="ref" reference="prop:b"} is also sharp. Finally, will show that the $\Bbbk$-linear independence of the supports is also needed. The following example shows that schemes that are irredundant to $F \in \mathcal{S}_d$ may be not $d$-regular. **Example 33**. Let us consider the scheme $Z$ evinced by the GAD $F = X_0 G_1 + X_1 G_2 \in \mathcal{S}_4$, where $$\begin{aligned} G_1 &= 10X_0^3 - 4X_0^2X_1 + 4X_0^2X_2 - 4X_0X_1^2 - 8X_0X_1X_2 - 3X_0X_2^2 - 8X_1^3 - 4X_2^3 \in \mathcal{S}_3, \\ G_2 &= 5X_0^3 + 9X_0X_1^2 - 5X_1^3 - 7X_1^2X_2 + 6X_1X_2^2 - X_2^3 \in \mathcal{S}_3.\end{aligned}$$ Its defining ideal is $$I(Z) = ( Y_0^3Y_1^3 - 2Y_0^3Y_2^3 + 5Y_1^3Y_2^3, 3Y_0^2Y_1Y_2 - 2Y_0Y_2^3, Y_0Y_1^2Y_2, Y_0Y_1Y_2^2, Y_2^4 ),$$ whose minimal primary decomposition is $I(Z) = I_1 \cap I_2$, where $$I_1 = ( -3Y_0Y_1Y_2 + Y_1^3, Y_1^2Y_2, Y_1Y_2^2, Y_1^3 - 2Y_2^3 ), \quad I_2 = ( Y_2^4, Y_0^3 + 5Y_2^3, Y_0Y_2 ).$$ Its Hilbert function is $\mathrm{HF}_Z = (1,3,6,10,11,12,12,\dots)$, hence $Z$ is not regular in degree $4 = \deg (F)$. **Claim 1**. $Z$ is irredundant to $F$. *Proof of Claim..* The connected components of $Z$ are both contained in $4$-fat points, i.e. $I_i \supset \wp_{X_{i-1}}^4$, hence by it is sufficient to show that the unique scheme $Y \subseteq Z$ evinced by a GAD of $F$ of type $F = X_0^{a_0} Q_1 + X_1^{a_1} Q_2$ with $a_0,a_1 \geq 1$ is $Z$ itself. Since in the expression of $F$ appear the monomials $-4X_0X_2^3$ and $-X_1X_2^3$, then it is easy to see that there is no such a GAD of $F$ for $a_0 > 1$ or $a_1 > 1$, therefore we assume $a_0 = a_1 = 1$. Since this new additive decomposition is still equal to $F$, we have $$X_0(Q_1-G_1) + X_1(Q_2-G_2) = 0,$$ hence there is $T \in \mathcal{S}_2$ such that $$X_1T = Q_1-G_1, \quad X_2T = -Q_2+G_2.$$ This means that $Y$ is evinced by a GAD of $F$ of type $$F = X_0 (G_1 + X_1 T) + X_1 (G_2 - X_0 T),$$ for some $$T = \lambda_{1} X_0^2 + \lambda_{2} X_0X_1 + \lambda_{3} X_0X_2 + \lambda_{4} X_1^2 + \lambda_{5} X_1X_2 + \lambda_{6} X_2^2 \in \mathcal{S}_2.$$ If $Y = Y_1 \cup Y_2 \subseteq Z$, then we have $$I_1 \subseteq I(Y_1) = I(Z_{X_0 (G_1 + X_1 T),X_0}) \subseteq\textnormal{Ann}\big(X_0 (G_1 + X_1 T) \big),$$ which implies $$\begin{cases} 0 = (-3Y_0Y_1Y_2 + Y_1^3) \circ \big(X_0(G_1 + X_1 T)\big) = 6(-\lambda_3 + \lambda_4)X_0 - 6\lambda_5X_1 - 6\lambda_6X_2, \\ 0 = (Y_1^2Y_2) \circ \big(X_0(G_1 + X_1 T)\big) = 2\lambda_5X_0, \\ 0 = (Y_1Y_2^2) \circ \big(X_0(G_1 + X_1 T)\big) = 2\lambda_6X_0, \\ 0 = (Y_1^3 - 2Y_2^3) \circ \big(X_0(G_1 + X_1 T)\big) = 6\lambda_4X_0. \end{cases}$$ Similarly, from $I_2 \subseteq\textnormal{Ann}\big(X_1 (G_2 + X_0 T)\big)$ we obtain $$\begin{cases} 0 = (Y_2^4) \circ \big(X_1(G_2 + X_0 T)\big) = 0, \\ 0 = (Y_0^3 + 5Y_2^3) \circ \big(X_1(G_2 + X_0 T)\big) = -6\lambda_1X_1, \\ 0 = (Y_0Y_2) \circ \big(X_1(G_2 + X_0 T)\big) = -2\lambda_3X_0X_1 - \lambda_5X_1^2 - 2\lambda_6X_1X_2. \end{cases}$$ The above systems imply $$\lambda_1 = \lambda_3 = \lambda_4 = \lambda_5 = \lambda_6 = 0,$$ thus we conclude that $T = \lambda_2 X_0X_1$. We computationally verify that the scheme evinced by the GAD $$X_0 (G_1 + \lambda_2 X_0X_1^2) + X_1 (G_2 - \lambda_2 X_0^2X_1)$$ is independent on $\lambda_2 \in \Bbbk$, and it is always equal to $I(Z)$. Therefore, we conclude that $Y = Z$, i.e. $Z$ is irredundant. ◻ The proof of the above claim shows an effective way for establishing irredundancy to $F$ by symbolically testing its GADs. $\spadesuit$ ## Tangential decompositions In this section, we prove that if a minimal apolar scheme to $F \in \mathcal{S}_d$ is a union of simple points and *$2$-jets* (i.e. local $0$-dimensional schemes of length $2$), then it is $d$-regular. Such schemes are evinced by GADs as in [\[tg:decomp\]](#tg:decomp){reference-type="ref" reference="tg:decomp"}, which are called *tangential decompositions* due to their relation with secant varieties of tangential varieties of Veronese varieties [@BT20:Waring; @CGG]. **Proposition 34**. *Let $Z = Z_1 \cup \ldots \cup Z_r$ such that ${\rm len}(Z_i) \leq 2$ for every $i \in \{1, \dots, r\}$. If $Z$ is of minimal length among the apolar schemes to $F \in \mathcal{S}_d$, then $Z$ is $d$-regular.* *Proof.* By , $Z$ is evinced by a GAD of $F$ of type $$\label{tg:decomp} F=\sum_{i=1}^s L_i^{d-1}G_i + \sum_{i=s+1}^r L_i^d$$ for some $0 \leq s \leq r$ and $L_i,G_i \in \mathcal{S}_1$. Moreover, we have $$I(Z)^{\perp}_d = \langle L_i^d, L_j^{d-1}G_j \rangle_{ \substack{1 \leq i \leq r\\ 1 \leq j \leq s}}.$$ Since ${\rm len}(Z)$ is $r+s$, which is also equal to the number of generators of $I(Z)^{\perp}_d$, in order to prove that $Z$ is $d$-regular it is sufficient to show that all those generators are $\Bbbk$-linearly independent. We prove that if there is a linear relation between the $L_i^d$'s and the $L_j^{d-1}G_j$'s as above, then we can explicitly produce an apolar scheme that has smaller length than $Z$, contradicting its minimality. When such a relation involves an addenda appearing in the above GAD of $F$, then $Z$ is redundant by , contradicting the minimality. Thus, we only need to show that $L_1^d, \ldots, L_s^d$ are linearly independent. We will prove a stronger fact, namely that $L_1^{d-1}, \ldots , L_s^{d-1}$ are linearly independent. Suppose by contradiction that $L_1^{d-1} = \sum_{i=2}^s \lambda_i L_i^{d-1}$ for some $\lambda_i\in \Bbbk$. By substituting this relation in the above GAD, we get $$F=\sum_{i=2}^s L_i^{d-1}(G_i+\lambda_i G_1)+ \sum_{i=s+1}^r L_i^d.$$ The scheme $Z'$ evinced by this new GAD of $F$ has length at most $s+r-2= {\rm len}(Z)-2 < {\rm len}(Z)$. ◻ Notice that in the proof of we have employed the length-minimality of the scheme $Z$ apolar to $F$. Indeed, the irredundancy of an apolar scheme of $2$-jets is not sufficient to guarantee the regularity in the degree of $F$, as shown in the following example. **Example 35**. Let $Z$ be the scheme evinced by the GAD $$F = X_0^2X_2 + X_1^2X_3 + (X_0+X_1)^2X_4 + (X_0-X_1)^2(X_2-3X_3-2X_4)+ (X_0+2X_1)^2(X_2+X_3+X_4) \in \mathcal{S}_3.$$ It is easy to check that $F$ is written in essential variables [@IaKa:book; @carlini], and that $Z$ is the union of five $2$-jets $Z_1, \ldots , Z_5$ supported on points $[L_1], \dots, [L_5] \in \mathbb{P}^n$ of the rational normal cubic. Its Hilbert function is $\mathrm{HF}_Z = (1,5,8,9,10,10, \ldots)$, therefore $Z$ is not regular in degree $3 = \deg(F)$. However, $Z$ is irredundant: any proper subscheme of $Z=\cup_{i=1}^5Z_i$ has to be contained in one of the following, for $i \in \{1, \dots, 5\}$: $$Y_i = [L_i] \cup \bigcup_{j \neq i} Z_j.$$ We computationally verify that for every $i$ we have $I(Y_i) \subsetneq \textnormal{Ann}(F)$, therefore no proper subscheme of $Z$ is apolar to $F$. We now verify that the strategy of produces an apolar scheme that is shorter than $Z$, but not contained in it. Substituting the relation $$(X_0-X_1)^2 = 2X_0^2+2X_1^2-(X_0+X_1)^2$$ we obtain the new GAD of $F$: $$X_0^2 (3 X_2-6 X_3-4 X_4) + X_1^2 (2 X_2-5 X_3-4 X_4) + (X_0+X_1)^2 (-X_2+3 X_3+3 X_4) + (X_0+2X_1)^2 (X_2 + X_3 + X_4).$$ The scheme evinced by this GAD has length $8$ but is not contained in $Z$. We can repeat the procedure with the relation $$(X_0+2X_1)^2 = 2(X_0+X_1)^2-X_0^2+2X_1^2,$$ which leads us to another GAD $$F = X_0^2 (2 X_2-7 X_3-5 X_4) + X_1^2 (4 X_2-3 X_3-2 X_4) + (X_0+X_1)^2 (X_2+5 X_3+5 X_4).$$ The scheme evinced the last GAD is minimal among the apolar schemes to $F$: as it has length $6$ and, up to a change of variables, $F$ is the Perazzo cubic [@perazzo] which has cactus rank 6 (see eg. [@BBM14:comparison Example 2.8], [@bb:differences Section 4]). This can also be directly verified with [@BT20:Waring Algorithm 3]. $\spadesuit$ ## Apolar schemes with low length **Proposition 36**. *Let $Z \subset \mathbb{P}^n$ be a $0$-dimensional scheme apolar and irredundant to $F \in \mathcal{S}_d$. If ${\rm len}(Z) \leq 2d+1$, then $Z$ is $d$-regular.* By contradiction, let us assume that $Z$ is not $d$-regular. Then, by [@BGI11 Lemma 34], there exists a line $L$ such that ${\rm len}(Z \cap L) \geq d+2$. Let ${\rm Res}_L(Z)$ be the residual scheme of $Z$ with respect to $L$ defined by the colon ideal $\big( I(Z) : (L) \big)$. Since $${\rm len}(Z \cap L) + {\rm len}\big( {\rm Res}_L(Z) \big) = {\rm len}(Z) \leq 2d+1,$$ then given the irreducible decomposition $Z = Z_1 + \cdots + Z_s$, there exists a component $Z_i$ such that the schematic intersection $Z_i \cap L$ satisfies ${\rm len}(Z_i \cap L) > {\rm len}({\rm Res}_L(Z_i))$. Without loss of generality, we may assume that $i = 1$, $I(Z_1) \subseteq\wp_{X_0}$ and $I(L) = ( X_1, \dots, X_n )$. Let $H$ be the orthogonal hyperplane to $X_0$, i.e. $I(H) = ( X_0 )$, and let $m = {\rm len}(Z_1 \cap L)$. We consider the scheme $Z'$ defined by $$I(Z') = I\left(Z_1 \cap (m-1)H \right) \cap I(Z_2) \cap \dots \cap I(Z_s).$$ It is clear that $Z' \subsetneq Z$, hence to get the desired contradiction it is sufficient to show that $Z'$ is apolar to $F$, which follows directly from the following fact by the Apolarity Lemma (). **Claim 1**. *$I(Z)^\perp_d = I(Z')^\perp_d$.* *Proof of .* Since $m > {\rm len}\big({\rm Res}(Z_1)\big)$ we have $( X_0^{m-1} ) \cap ( X_1,\ldots,X_n ) \subseteq I(Z_1)$, hence $$I(Z_1) = \big( I(Z_1) + ( X_0^{m-1} ) \big) \cap \big( I(Z_1)+ ( X_1,\ldots,X_m ) \big).$$ $\bullet$ We prove that $I(Z_1) + ( X_0^{m-1} )$ equals the saturated ideal $I\big(Z_1 \cap (m-1)H \big)$. There are obvious ideal inclusions: $$\label{eq:inclusions} I(Z_1) \subseteq I(Z_1) + ( X_0^{m-1} ) \subseteq I\big(Z_1 \cap (m-1)H\big).$$ It is enough to show that the last two ideals have the same Hilbert function. Since $Z_1 \cap (m-1)H$ has colength $1$ inside $Z_1$ and their homogeneous defining ideals agree up to degree $m-2$, we deduce that $$\mathrm{HF}_{Z_1 \cap (m-1)H}(i) = \begin{cases} \mathrm{HF}_{Z_1}(i) & \text{for $i \leq m-2$,} \\ \mathrm{HF}_{Z_1}(i)-1 & \text{for $i \geq m-1$.} \end{cases}$$ By [\[eq:inclusions\]](#eq:inclusions){reference-type="ref" reference="eq:inclusions"} the Hilbert function $\mathrm{HF}_*$ of $\mathcal{S}/\big( I(Z_1) + ( X_0^{m-1} ) \big)$ is squeezed: $\mathrm{HF}_{Z_1 \cap (m-1)H} \leq \mathrm{HF}_* \leq \mathrm{HF}_{Z_1}$. However, for every $k \geq m-1$ we have $X_0^{k} \in \left( I(Z_1) + ( X_0^{m-1} ) \right) \setminus I(Z_1)$, thus $\mathrm{HF}_*(k) < \mathrm{HF}_{Z_1}(k)$ for every $k \geq m-1$. This implies that $\mathrm{HF}_*$ completely agrees with $\mathrm{HF}_{Z_1 \cap (m-1)H}$. $\bullet$ For every $i \in \{2,\ldots,s\}$, we trivially have $$I(Z_i) = I(Z_i) \cap \big( I(Z_i) + ( X_1,\ldots,X_n ) \big).$$ Hence, we can write: $$\begin{aligned} I(Z) &= I\big( Z_1 \cap (m-1)H \big) \cap \big( I(Z_1)+ ( X_1,\ldots,X_m ) \big) \cap \bigcap_{i = 2}^s \big( I(Z_i) + ( X_1,\ldots,X_n ) \big) \cap I(Z_i) \\ &= I(Z') \cap \left( \bigcap_{i=1}^s I(Z_i) + ( X_1,\ldots,X_n ) \right) = I(Z') \cap I(Z \cap L).\end{aligned}$$ $\bullet$ From the non-degeneracy of the apolar action we get $$I(Z)_d^\perp = [I(Z') \cap I(Z\cap L)]_d^\perp = I(Z')_d^\perp + I(Z \cap L)_d^\perp$$ but $I(Z \cap L)_d = I(Z' \cap L)_d$ because they define schemes of length $d+1$ on the same normal curve $\nu_d(L) \subset \mathbb{P}^d$. Thus, we conclude $$I(Z)_d^\perp = I(Z')_d^\perp + I(Z' \cap L)_d^\perp = I(Z')_d^\perp,$$ which proves the claim and then concludes the proof. We notice that provides a good criterion for proving that the minimal apolar schemes to a *given* $F \in \mathcal{S}_d$ is $d$-regular, by exhibiting at least one scheme $Z$ apolar to $F$ and of length not bigger than $2d+1$. **Example 37**. Let $F \in \mathcal{S}_4$ be the polynomial considered in . We consider another GAD $F = X_0 \tilde G_1 + X_1 \tilde G_2$, where $$\begin{aligned} \tilde G_1 &= 10 X_0^3 + X_0^2 X_1 + 4 X_0^2 X_2 - 4 X_0 X_1^2 - 8 X_0 X_1 X_2 - 3 X_0 X_2^2 - 4 X_2^3 \in \mathcal{S}_3, \\ \tilde G_2 &= X_0 X_1^2 - 5 X_1^3 - 7 X_1^2 X_2 + 6 X_1 X_2^2 - X_2^3 \in \mathcal{S}_3.\end{aligned}$$ This GAD evinces the scheme $\tilde Z$ defined by $$I(\tilde Z) = \left( Y_0^2 Y_1 Y_2 - \frac{2}{3} Y_0 Y_2^3, Y_0 Y_1 Y_2^2, Y_2^4, Y_0 Y_1^2 - \frac{5}{2} Y_0 Y_1 Y_2 + Y_2^3 \right).$$ Its Hilbert function is $\mathrm{HF}_Z = (1,3,6,9,9,\dots)$. Since ${\rm len}(Z) = 9 \leq 2 \cdot 4 + 1$, by we can guarantee that minimal schemes apolar to such a $F$ are $4$-regular, even without computing them. # Conclusion {#sec:Conclusion} In the present work, we investigated the $d$-regularity of certain families of schemes apolar to $F \in \mathcal{S}_d$. In all the examples we presented, the schemes of minimal lengths were $d$-regular, so it is natural to ask whether this is always the case. **Question 1**. *Let $F \in \mathcal{S}_d$ and $Z$ be a $0$-dimensional scheme evincing its cactus rank. Is $Z$ $d$-regular?* Actually, a careful reader would have noticed that none of the examples we considered really required to reach degree $d$ for regularity, hence we may state an even more compelling question. **Question 2**. *Let $F \in \mathcal{S}_d$ and $Z$ be a $0$-dimensional scheme evincing its cactus rank. Is $Z$ $(d-1)$-regular?* To the best of our knowledge, we do not know the answer to and . We believe that our results and examples could be useful in either direction. Our positive results restrict the identikit of a possible example providing a negative answer to to have some component of high multiplicity. On the other side, when trying to prove a positive answer to or , shows that we really need to use the *global* assumption of minimality in terms of the cactus rank, and that cannot be relaxed with the *local* condition of minimality by inclusion.

arxiv_math

--- abstract: | In this paper we prove that automorphisms are the only isometries between rank two Almost-Riemannian Structures on the class of nonnilpotent, solvable, connected 3D Lie groups. As a consequence, a classification result for rank two ARSs on the groups in question is obtained. author: - | Vı́ctor Ayala [^1]\ Instituto de Alta Investigación\ Universidad de Tarapacá, Arica, Chile - | Adriano Da Silva[^2]\ Departamento de Matemática,\ Universidad de Tarapacá - Iquique, Chile. - | Danilo A. Garcı́a Hernández[^3]\ Instituto de Matemática\ Universidade Estadual de Campinas, Brazil\ title: Isometries of Almost-Riemannian structures on nonnilpotent, solvable 3D Lie groups --- **Keywords:** Almost-Riemannian geometry, solvable Lie groups, isometry **Mathematics Subject Classification (2020): 22E15, 22E25, 53C17, 53C15.** # Introduction Almost-Riemannian structures (shortly ARS) are a part of sub-Riemannian geometry with nonconstant rank. They can be seen, roughly speaking, as a smooth orthonormal frame degenerating on a singular set, called the singular locus. They appear naturally as limits of Riemannian metrics and in the study of hypoelliptic operators [@Gru], but other works highlighting nice physical features related to these structures can also be found in the literatura (see for instance [@Boscain5]). The first work on the subject appeared in 1956 in the work of T. Takasu [@tak]. However, works on the subject, from a control theoretical point of view, have appeared recently [@Ag0; @Ag1; @Ag2; @DSAyD2; @PJAyGZ1; @Bon1; @Bon2; @Boscain1; @Boscain2; @Boscain3; @Boscain4; @Boscain5; @PJAyGZ]. Among them, the concept of simple ARS on connected Lie groups was introduced in [@PJAyGZ1] by considering $n-1$ left-invariant vector fields and one linear vector field having full rank at a nonempty subset of $G$ and satisfying the Lie algebra rank condition. The strong relation of these structures with the group symmetries allow one to obtain strong results for the singular locus (see [@DSAyD2; @PJAyGZ1]) and for the isometries between ARSs (see [@PJAyGZ]). Concerning isometries of ARSs, the authors showed [@PJAyGZ Theorem 4] that the only possible isometries between ARSs on a nilpotent Lie group are the automorphisms of the group. By means of a counter-example (see [@PJAyGZ Section 3.4]) they showed that such a remarkable fact is not true, in general, for nonnilpotent groups. In the paper at hand, we study isometries in the class of nonnilpotent, solvable 3D Lie groups. We show that for rank two ARSs (see Definition [Definition: 5](#ranktwo){reference-type="ref" reference="ranktwo"}) it still holds that the only isometries are the automorphisms of the group. In order to prove that, we transfer the problem to the algebra level, by working with the differential at the identity element. Since, by symmetry, Lie groups have all their information centered at the origin, such an approach is the most appropriate. However, since one cannot assume a priori smoothness of the differential of the isometry over the singular locus, this task turns out to be quite hard. The way to overcome the previous difficulty, is to use the smoothness of the isometry outside the singular locus, as assured by the Myers-Steenrod Theorem [@Myers Theorem 8], in association with curves that cross the singular locus (see [@DSAyD2 Section 4.1]). The paper is divided as follows: Section 2 is used to state the framework of the paper. Here we define the 3D Lie groups we are interested, their algebras, group of automorphisms, linear and invariant vector fields, and so on. In sequence, the concept of ARS and their isometries are (formally) defined. We also make a compilation of the main results concerning the singular locus. In Section 3 we state and prove our main result (Theorem [Theorem: 6](#main){reference-type="ref" reference="main"}) which assures that, any isometry between rank two ARSs is an automorphism of the group. The proof is divided in three subsections. In the first one we prove our "Fundamental Lemma", namely, we show that if an isometry preserves one left-invariant vector field of the associated distribution, then it is in fact an automorphism. In the second subsection, we show that the differential of any rank two isometry keeps the nilradical of the algebra invariant. Since, by the Lie algebra rank condition, the intersection of the nilradical and the 2D left-invariant distribution of the ARS is one-dimensional, we can show that any isometry preserves a unitary left-invariant vector field generating this intersection and, by the fundamental lemma, it is an automorphism. As consequence of the main result, we are able to obtain a classification of rank two isometries of ARSs on the 3D groups under consideration. # Preliminaries Here we introduce the main concepts and results we will need in the next sections. In order to maintain our work self-contained, all the concepts here will be done over the manifolds we are interested, making remarks when necessary to general cases. ## Solvable nonnilpotent 3D Lie groups and algebras According to [@Oni Chapter 7], up to isomorphisms, any solvable, nonnilpotent 3D Lie algebra is given by the semi-direct product $\mathfrak{g}(\theta)=\mathbb{R}\times_{\theta}\mathbb{R}^2$, with Lie bracket determined by the relation $$[(a, 0), (0, w)]=(0, a\theta w), \;\;\;(a, w)\in \mathbb{R}\times\mathbb{R}^2,$$ where $\theta\in\mathfrak{gl}(2, \mathbb{R})$ is a matrix of one of the following types: $$\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right), \;\;\;\;\left(\begin{array}{cc} 1 & 0\\ 0 & \gamma \end{array}\right), \gamma\in[-1, 1]\;\;\;\;\mbox{ or }\;\;\;\;\left(\begin{array}{cc} \gamma & -1\\ 1 & \gamma \end{array}\right), \gamma\in\mathbb{R}.$$ The **nilradical** $\mathfrak{n}(\theta)$ of $\mathfrak{g}(\theta)$ coincides with a (diffeomorphic) copy or $\mathbb{R}^2$ inside $\mathfrak{g}(\theta)$ and is given by $\mathfrak{n}(\theta)=\{0\}\times\mathbb{R}^2$. At the group level, up to isomorphisms, the connected, simply connected 3D Lie group with Lie algebra $\mathfrak{g}(\theta)$ is $G(\theta)=\mathbb{R}\times_{\rho}\mathbb{R}^2$, where $\rho_t=\mathrm{e}^{t\theta}$ and the product satisfies $$(t_1, v_1)(t_2, v_2)=(t_1+t_2, v_1+\rho_{t_1}v_2), \;\;\; (t_1, v_1), (t_2, v_2)\in\mathbb{R}\times_{\rho}\mathbb{R}^2.$$ Following [@DSAyD2 Proposition 2.1 and 2.2] the algebra of **derivations** of $\mathfrak{g}(\theta)$ and the groups of **automorphisms** of $\mathfrak{g}(\theta)$ and $G(\theta)$ are given, respectively, by $$\label{der} \mathrm{Der}(\mathfrak{g}(\theta))=\left\{\left(\begin{array}{cc} 0 & 0\\ \xi & A \end{array}\right), \xi\in\mathbb{R}^2, A\in\mathfrak{gl}(2, \mathbb{R}), \mbox{ with } A\theta=\theta A\right\},$$ $$\mathrm{Aut}(\mathfrak{g}(\theta))=\left\{\left(\begin{array}{cc} \varepsilon & 0\\ \eta & P \end{array}\right), \eta\in\mathbb{R}^2, P\in\mathrm{Gl}(\mathbb{R}^2), \mbox{ with } P\theta=\varepsilon\theta P\right\},$$ and, $$\label{auto} \mathrm{Aut}(G(\theta))=\left\{\phi(t, v)=\left(\varepsilon t, Pv+\varepsilon\Lambda_{\varepsilon t}^{\theta}\eta\right), \eta\in\mathbb{R}^2, P\in\mathrm{Gl}(\mathbb{R}^2), \mbox{ with } P\theta=\varepsilon\theta P\right\},$$ where $\varepsilon=1$ if $\operatorname{tr}\theta\neq 0$ or $\varepsilon\in\{-1, 1\}$ if $\operatorname{tr}\theta=0$. Here, for any matrix $A\in \mathfrak{gl}(2, \mathbb{R})$ we have that $$\Lambda^A: (t, w)\in \mathbb{R}\times \mathbb{R}^2\mapsto \Lambda_t^Aw:=\int_0^t\mathrm{e}^{sA}w ds.$$ A **linear** and a **left-invariant** vector field on $G(\theta)$ are given, respectively, as $$\mathcal{X}(t, v):=(0, Av+\Lambda^{\theta}_t\xi)\hspace{.5cm}\mbox{ and }\hspace{.5cm}Y^L(t, v):=(\alpha, \rho_t\eta),$$ where $\alpha\in\mathbb{R}$, $\xi, \eta\in\mathbb{R}^2$ and $A\in\mathfrak{gl}(2, \mathbb{R})$ satisfies $A\theta=\theta A$. We denote by $\mathcal{Z}_{\mathcal{X}}$ the set of **singularities** of $\mathcal{X}$ (or the set of fixed points of $\varphi_s)$, that is, $$\mathcal{Z}_{\mathcal{X}}=\{(t, v)\in G(\theta); \;\;\mathcal{X}(t, v)=0\}.$$ The flow of $\mathcal{X}$ is a $1$-parameter subgroup of $\mathrm{Aut}(G(\theta))$ and is given by $$\varphi_s(t, v)=\left(t, \mathrm{e}^{sA}v+\Lambda_t^{\theta}\Lambda_s^A\xi\right).$$ It is naturally associated with the derivation $$\mathcal{D}=\left(\begin{array}{cc} 0 & 0 \\ \xi & A \end{array}\right), \hspace{.5cm}\mbox{ satisfying }\hspace{.5cm} (d\varphi_s)_{(0, 0)}=\mathrm{e}^{s\mathcal{D}}, \hspace{.5cm}\forall s\in\mathbb{R}.$$ Since $\xi\in\mathbb{R}^2$ and $A\in\mathfrak{gl}(2, \mathbb{R})$ determine a linear vector field $\mathcal{X}$, when necessary we will write $\mathcal{X}=(\xi, A)$ in order to emphasize such elements. On the other hand, by [@DSAy1 Section 3], the exponential maps $\exp:\mathfrak{g}(\theta)\rightarrow G(\theta)$ is given as $$\exp (\alpha, \eta)=\left\{\begin{array}{cc} (0, \eta) & \mbox{ if }\alpha=0\\ \left(\alpha, \frac{1}{\alpha}\Lambda^{\theta}_{\alpha}\eta\right) & \mbox{ if }\alpha\neq 0 \end{array}\right..$$ Therefore, if $$L_{(t, v)}:G(\theta)\rightarrow G(\theta), \hspace{1cm}L_{(t, v)}(\hat{t}, \hat{v})=(t, v)(\hat{t}, \hat{v})=(t+t', v+\rho_t\hat{v})$$ stands for the left-translation, the fact that $$(dL_{(t, v)})_{(\hat{t}, \hat{v})}(\alpha, \eta)=(\alpha, \rho_{t}\eta),$$ implies that the flow $\{\varphi^Y_s\}_{s\in\mathbb{R}}$ of $Y^L$ is given by $$\varphi^Y_s(t, v)=L_{(t, v)}(\exp s(\alpha, \eta))=(t, v)\exp s(\alpha, \eta).$$ [\[exp\]]{#exp label="exp"} From the formula of the exponential above, it is not hard to see that, if $(a, w)\neq(0, 0)$, then $$\exp s_1(\alpha, \eta)=\exp s_2(\alpha, \eta)\hspace{.5cm}\implies \hspace{.5cm} s_1=s_2.$$ Such simple fact will be useful ahead. ## Simple ARS's on $G(\theta)$ We define now simple almost-Riemannian structures on the groups $G(\theta)$ and state their main properties. We start with the definition of invariant distributions. **Definition: 1**. *A 2D left-invariant distribution on $G(\theta)$ is given by the map $$\Delta^L:G(\theta)\rightarrow TG(\theta), \hspace{1cm} \Delta^L(t, v)=(dL_{(t, v)})_{(0, 0)}\Delta,$$ where $\Delta\subset\mathfrak{g}(\theta)$ is a 2D vector subspace.* In what follows, we will always assume that a 2D left-invariant distribution $\Delta^L$ is endowed with a left-invariant Riemannian metric. That is done by considering, on $\Delta$, an inner product $\langle\cdot, \cdot\rangle_{\Delta}$ and defining $$\forall X, Y\in \Delta^L(t, v), \;\;\; \langle X, Y\rangle_{(t, v)}:=\langle (dL_{(t, v)^{-1}})_{(t, v)}X, (dL_{(t, v)^{-1}})_{(t, v)}Y\rangle_{\Delta}.$$ Let $\mathcal{X}$ be a linear vector field and consider the family of vectors $\Sigma=\{\mathcal{X}, \Delta^L\}$. We say that $\Sigma$ satisfies the **Lie algebra rank condition (LARC)** if (at least) one of the following conditions is satisfied: - $\Delta$ is not a subalgebra of $\mathfrak{g}(\theta)$; - $\Delta$ is a subalgebra of $\mathfrak{g}(\theta)$ and $\mathcal{D}\Delta\not\subset\Delta$, where $\mathcal{D}$ is the derivation associated with $\mathcal{X}$. Next we define the concept of almost-Riemannian structures on $G(\theta)$. **Definition: 2**. *A simple **almost-Riemannian structure (shortly ARS)** $\Sigma$ on $G(\theta)$ is defined by the family $\Sigma=\{\mathcal{X}, \Delta^{L}\}$, where $\mathcal{X}$ is a linear vector field and $\Delta^{L}$ is a 2D left-invariant distribution $\Delta$ endowed with a left-invariant Euclidean metric satisfying* - *The set $\{(t, v)\in G(\theta);\;\;\mathcal{X}(t, v)\notin\Delta^L(t, v)\}$ is nonempty;* - *$\Sigma$ satisfies the LARC.* *The metric on $G(\theta)$ associated with $\Sigma$ is defined by declaring that $\mathcal{X}$ is unitary and orthogonal to $\Delta^L$ at every point.* The **singular points** $\mathcal{Z}$ is the set of points $(t, v)\in G(\theta)$ where the previous metric fails to be Riemannian, that is $$\mathcal{Z}:=\{(t, v)\in G(\theta); \;\mathcal{X}(t, v)\in \Delta^L(t, v)\}.$$ Since any linear vector In particular, the set of singularities $\mathcal{Z}_{\mathcal{X}}$ of $\mathcal{X}$ is contained in $\mathcal{Z}$. Any ARS $\Sigma$ define an **almost-Riemannian norm** on $G(\theta)$ given by $$\|X\|_{\Sigma, (t, v)}=\min\left\{\sqrt{\alpha_0^2+\alpha_1^2+\alpha_2^2}; \;\alpha_0\mathcal{X}(t, v)+\alpha_1Y_1^L(t, v)+\alpha_2Y^L_2(t, v)=X\right\},$$ where $\{Y_1, Y_2\}$ is an orthonormal basis of $\Delta$. It holds that $\|X\|_{\Sigma, (t, v)}=\infty$ when $(t, v)\in\mathcal{Z}$ and $X\notin\Delta(t, v)$. One of the interest in the study of ARSs is to analyze the possible isometries of the almost-Riemannian norm. Precisely, we say that a diffeomorphism $\psi:G(\theta)\rightarrow G(\theta)$ is an **isometry** between ARSs $\Sigma_1$ and $\Sigma_2$ if $$\forall (t, v)\in G(\theta), Z\in T_{(t, v)}G(\theta)\hspace{1.5cm}\|(d\psi)_{(t, v)}Z\|_{\Sigma_2, \phi(t, v)}=\|Z\|_{\Sigma_1, (t, v)}.$$ Let us denote by $\mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)$ the group of isometries between the ARSs $\Sigma_1$ and $\Sigma_2$. Following [@PJAyGZ Theorem 1], any isometry $\psi\in \mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)$ can be decomposed as $\psi=L_{(t, v)}\circ\psi_0$, where $(t, v)\in \mathcal{Z}_2$ and $\psi_0\in \mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)_0$, with $$\mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)_0=\left\{\psi_0\in \mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2); \;\psi_0(0, 0)=(0, 0)\right\}.$$ Therefore, in order to understand the group of isometries $\mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)$ it is enough to study the singular locus $\mathcal{Z}_2$ of $\Sigma_2$ and the subgroup $\mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)_0$. Moreover, for any isometry $\psi\in \mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)_0$ it holds that $$\label{conjugation} \psi_*\Delta^L_1=\Delta_2(\psi)\hspace{.5cm}\mbox{ and }\hspace{.5cm}\psi\circ\varphi^1_s=\varphi^2_{\pm s}\circ\psi,$$ where $\{\varphi_s^i\}_{s\in\mathbb{R}}$ is the flow associated with $\mathcal{X}_i$ (see [@PJAyGZ Theorem 2]). Consequently, $$\psi(\mathcal{Z}_1)=\mathcal{Z}_2\hspace{.5cm}\mbox{ and }\hspace{.5cm}\psi(\mathcal{Z}_{\mathcal{X}_1})=\mathcal{Z}_{\mathcal{X}_2}.$$ Since outside the singular locus the metric is Riemannian, the restriction $$\psi:G(\theta)\setminus \mathcal{Z}_1\rightarrow G(\theta)\setminus \mathcal{Z}_2,$$ is an isometry between Riemannian manifolds. By Myers-Steenrod Theorem [@Myers Theorem 8] such restriction is of class $\mathcal{C}^{\infty}$ on the connected components of $G(\theta)\setminus\mathcal{Z}_1$. This fact will allows us to study the differential of an isometry between ARSs. In what follows we present the main properties concerning the singular locus $\mathcal{Z}$ of an ARS. For the proof the reader can consult [@DSAyD2 Section 4]. **Theorem: 3**. *Let $\Sigma=\{\mathcal{X}, \Delta\}$ be an ARS on $G(\theta)$ with $A\neq 0$ and $\mathcal{Z}$ it singular locus. It holds:* 1. *$\mathcal{Z}$ is a closed, connected, embedded submanifold of $G(\theta)$;* 2. *$G(\theta)\setminus \mathcal{Z}$ has exactly two connected components;* 3. *If $(t, v)\in G(\theta)\setminus \mathcal{Z}$, the curves $$s\in\mathbb{R}\mapsto (t, v)\exp s(\alpha, \eta)\in G(\theta)\hspace{.5cm}\mbox{ and }\hspace{.5cm}s\in\mathbb{R}\mapsto \varphi_s(t, v)\in G(\theta),$$ remain in the same connected component of $G(\theta)\setminus \mathcal{Z}$ of they cross $\mathcal{Z}$ discretely;* It is important to remark that by Theorem [Proposition: 8](#prop1){reference-type="ref" reference="prop1"}, the image of the curves $$s\in\mathbb{R}\mapsto (t, v)\exp s(\alpha, \eta)\in G(\theta)\hspace{.5cm}\mbox{ and }\hspace{.5cm}s\in\mathbb{R}\mapsto \varphi_s(t, v)\in G(\theta),$$ starting at points outside the singular locus, remains outside, up to a discrete set. # Isometries of ARSs In this section we study the isometries between a class of ARSs on the groups $G(\theta)$. We show that for such class, the isometries are actually group automorphisms. The main difficult in proving this fact is that we cannot assume smoothness over the singular locus. In order to avoid this problem, we use smoothness outside the singular locus (assured by [@Myers Theorem 8]), jointly with curves crossing the singular locus. We start with a result, whose proof can be found at [@DSAyD2 Proposition 3.3], showing automorphisms of the group can be seen as isometries between ARSs. **Proposition: 4**. *Let $\Sigma=\{\mathcal{X}=(\xi, A), \Delta^L\}$ be an ARS on $G(\theta)$ and $$\psi(t, v)=(\varepsilon t, Pv+\varepsilon\Lambda_{\varepsilon t}^{\theta}\eta)\;\;\mbox{ an automophism of }G(\theta).$$ The family $$\Sigma_{\psi}=\left\{\mathcal{X}_{\psi}=\left( P^{-1}(\varepsilon\xi+ A\eta), P^{-1}AP\right), \Delta^L_{\psi}:=(\psi_*)^{-1}\Delta^L\right\},$$ is an ARS and $\psi$ is an isometry between $\Sigma_{\psi}$ and $\Sigma$, where the left-invariant metric on $\Delta_{\psi}$ is the one that makes $(d\phi)_{(0, 0)}\bigl|_{\Delta_{\phi}}$ an isometry.* In [@PJAyGZ Theorem 4] the authors proved that on nilpotent Lie groups, the only isometries between ARSs, fixing the identity element, are automorphisms of the group. Such fact is not necessarily true for the groups $G(\theta)$ as Example [\[example\]](#example){reference-type="ref" reference="example"} ahead shows. Despite this fact, in this section we present a class of ARSs in which the only possible isometries are in fact automorphisms. [\[example\]]{#example label="example"} Consider now $G(\theta)$ and the simple ARS $\Sigma=\{\mathcal{X}, \Delta^L\}$, where $$\theta=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\;\;\;\;\;\mathcal{X}=\left((0, 0), \left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right)\right)\;\;\;\;\mbox{ and }\;\;\;\;\alpha=\{(1,0),(0, \mathbf{e}_2)\},$$ is an orthonormal basis of $\Delta$. It is not hard to see that $\Sigma$ satisfies the LARC. Moreover, for any $(t, v)\in G(\theta)$, it holds $$(dL_{(t, v)})_{(0, 0)}(1, 0)=(1, 0)\;\;\;\mbox{ and }\;\;\;(dL_{(t, v)})_{(0, 0)}(0, \mathbf{e}_2)=(0, \mathbf{e}_2).$$ It is not hard to see that the map, $$\psi: G(\theta)\rightarrow G(\theta), \;\;\;(t, v)\mapsto (-t, v),$$ is a diffeomorphism of $G(\theta)$ and, by [\[auto\]](#auto){reference-type="ref" reference="auto"}, it holds that $\psi\notin\mathrm{Aut}(G(\theta))$. On the other hand, $$\psi\left(\mathcal{X}(t, v)\right)=\psi\left(0, Av\right)=(0, Av)=\mathcal{X}(-t, v)=\mathcal{X}(\psi(t, v)),$$ $$\psi(1, 0)=(-1, 0)\;\;\;\mbox{ and }\;\;\;\psi(0, \mathbf{e}_2)=(0, \mathbf{e}_2),$$ and since $\psi_*=\psi$, we conclude that $\psi$ carries the orthonormal frame $\{\mathcal{X}, (1, 0), (0, \mathbf{e}_2)\}$ onto the orthonormal frame $\{\mathcal{X}, -(1, 0), (0, \mathbf{e}_2)\}$ showing that $\psi$ is an isometry. Let $\psi:G(\theta)\rightarrow G(\theta)$ be an isometry between ARSs $\Sigma_1$ and $\Sigma_2$ and consider $f:G(\theta) \rightarrow \mathbb{R}$ and $g:G(\theta)\rightarrow\mathbb{R}^2$ the coordinate functions of $\psi$. Write, on the canonical basis, $$\psi_*=\left(\begin{array}{cc} \partial_1 f & (\partial_2 f)^T \\ \partial_1 g & \partial_2g \end{array}\right),$$ where for $(t, v)\in G(\theta)$, $\partial_2f(t, v)$ is the gradient vector of the map partial map $v\in \mathbb{R}^2\mapsto f(t, v)\in \mathbb{R}$. If $\{\varphi_s^i\}_{t\in\mathbb{R}}$ is the flow associated with the linear vector fields of $\Sigma_i$, we have by ([\[conjugation\]](#conjugation){reference-type="ref" reference="conjugation"}) that $$\psi\circ \varphi_s^1=\varphi^2_s\circ\psi,$$ where we consider only the positive case, since the negative is analogous. Using the expression for the flows of $\mathcal{X}_i$, allow us to obtain $$\label{com} \begin{array}{l} \hspace{-1cm}f\left(t, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)=f(t, v)\hspace{.5cm} \mbox{ and } \\ \hspace{3cm} g\left(t, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)=\mathrm{e}^{sA_2}g(t, v)+\Lambda_s^{A_2}\Lambda_{f(t, v)}^{\theta}\xi_2. \end{array}$$ ## Rank two ARSs In this section we introduce the class of rank two ARSs and prove that the only isometries between elements of such class are automorphisms of the group. This result allow us to classify, up to automorphisms, the only possible rank two ARS on the Lie groups $G(\theta)$ under consideration. **Definition: 5**. *Let $\Sigma=\{\mathcal{X}, \Delta^L\}$ to be a simple ARS on $G(\theta)$. We say that $\Sigma$ is a **rank two** ARS if the derivation $\mathcal{D}$ associated with $\mathcal{X}$ has rank two. Equivalently, if $\mathcal{X}=(\xi, A)$ then $\Sigma$ has rank two if and only if $$\mathbb{R}^2=\mathrm{Im}A+\mathbb{R}\xi.$$* By the conjugation formula ([\[conjugation\]](#conjugation){reference-type="ref" reference="conjugation"}) one gets direct that rank two ARSs are preserved by isometries. Moreover, any derivation in ([\[der\]](#der){reference-type="ref" reference="der"}) has rank maximum equal two, and hence, the set of rank two derivations of $\mathfrak{g}(\theta)$ is open and dense in $\mathrm{Der}(\mathfrak{g}(\theta))$. On the other hand, since $G(\theta)$ is simply connected, the set of linear vector fields is isomorphic to $\mathrm{Der}(\mathfrak{g}(\theta))$ (see [@AyTi Theorem 2.2]), which allows us to conclude that "most" ARSs on $G(\theta)$ are of rank two. Next we state our main result. **Theorem: 6**. *The only isometries between rank two ARSs on $G(\theta)$ that fixes the identity element are the automorphisms.* The proof of Theorem [Theorem: 6](#main){reference-type="ref" reference="main"} is divided in the next three sections. First we show that, if $\psi$ preserves one left-invariant vector field of the associated distribution, then $\psi$ is in fact an automorphism of $G(\theta)$. This result simplifies our problem to search for only one left-invariant vector which is preserved by the isometry. In the second part, we show that the differential of any rank two isometry lets the nilradical $\mathfrak{n}(\theta)$ of $\mathfrak{g}(\theta)$ invariant, or equivalently, if $\psi=(f, g)$ then $\partial_2f\equiv 0$. In the third and last section, we show that if $\psi_*$ keeps the nilradical invariant, then it preserves unitary left-invariant vector field generating the intersection of the nilradical with the distribution of the ARS. As a consequence, $\psi$ is an automorphism by the first section. ## The fundamental lemma We use this section to prove a technical lemma which gives us a sufficient condition for an isometry between rank two ARS to be a group automorphism. **Lemma: 7** (Fundamental Lemma). *Let $\Sigma_i=\{\mathcal{X}_i, \Delta^L_i\}, i=1, 2$ be simple ARSs on the $G(\theta)$ with rank two. If $\psi\in\mathrm{Iso}_{\theta}\left(\Sigma_1;\Sigma_2\right)_0$ satisfies $$\label{fundamentalequation} \forall (t, v)\in G(\theta);\;\;(d\psi)_{(t, v)}X^L(t, v)=(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}X,$$ for some nonzero vector $X\in \Delta_1$, then $\psi\in\mathrm{Aut}(G(\theta))$.* **Proof:** Let $(\sigma, \eta)\in\Delta_1$ with $\sigma\neq 0$. The automorphism, $\hat{\psi}(t, v)=\left(t, v-\sigma^{-1}\Lambda_t^{\theta}\eta\right),$ is such that $$(d\hat{\psi})_{(0, 0)}(\sigma, \eta)=(\sigma, 0),$$ and by Proposition ([Proposition: 4](#difeo){reference-type="ref" reference="difeo"}) we can assume w.l.o.g. that $(1, 0)\in \Delta_1$. Under this assumption, we prove the lemma in four steps: **Step 1:** If ([\[fundamentalequation\]](#fundamentalequation){reference-type="ref" reference="fundamentalequation"}) holds for a nonzero $X\in\Delta_1$, then it holds fol all $Z\in\Delta_1$, that is, $$(d\psi)_{(t, v)}Z^L(t, v)=(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}Z, \hspace{1cm}\forall (t, v)\in G(\theta), Z\in\Delta_1.$$ Let $Y\in\Delta_1$ satisfying $$\langle X, Y\rangle_{\Delta_1}=0, \hspace{.5cm}\mbox{ and }\hspace{.5cm}\|Y\|_{\Sigma_1, (0, 0)}=\|X\|_{\Sigma_1, (0, 0)}.$$ By linearity, it is enough to show that relation ([\[fundamentalequation\]](#fundamentalequation){reference-type="ref" reference="fundamentalequation"}) holds for $Y$. From the left-invariance of the metric in $\Delta^L_1$, we get that $\{X^L(t, v), Y^L(t, v)\}$ is an orthogonal basis of $\Delta_1^L(t, v)$, satisfying $$\|Y^L(t, v)\|_{\Sigma_1, (t, v)}=\|Y\|_{\Sigma_1, (0, 0)}=\|X\|_{\Sigma_1, (0, 0)}=\|X^L(t, v)\|_{\Sigma_1, (t, v)}.$$ Since $\psi$ is an isometry, the vector $(d\psi)_{(t, v)}Y^L(t, v)$ belongs to $\Delta_2^L(\psi(t, v))$, is orthogonal to $(d\psi)_{(t, v)}X^L(t, v)$ and $$\|(d\psi)_{(t, v)}Y^L(t, v)\|_{\Sigma_2, \psi(t, v)}=\|Y^L(t, v)\|_{\Sigma_1, (t, v)}=\|Y\|_{\Sigma_1, (0, 0)}.$$ On the other hand, the left-invariance of the metric in $\Delta^L_2$ implies that the vector $(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}Y$ also belongs to $\Delta_2(\psi(t, v))$, is orthogonal to $(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}X$ and $$\|(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}Y\|_{\Sigma_2, \psi(t, v)}=\|(d\psi)_{(0, 0)}Y\|_{\Sigma_2, (0, 0)}=\|Y\|_{\Sigma_1, (0, 0)}.$$ Since by hypothesis, $$(d\psi)_{(t, v)}X^L(t, v)=(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}X,$$ the fact that $\dim(\Delta_2^L(\psi(t, v))=2$ forces that $$\label{blabla} (d\psi)_{(t, v)}Y^L(t, v)=\varepsilon(t, v)(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}Y, \;\;\;\mbox{ where }\;\;\;\varepsilon(t, v)=\pm 1.$$ Moreover, by orthogonality, we obtain that $$\varepsilon(t, v)=\frac{\left\langle(d\psi)_{(t, v)}Y^L(t, v), (dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}Y\right\rangle_{\Sigma_2, \psi(t, v)}}{\|Y\|_{\Sigma_1, (0, 0)}},\hspace{1cm}\forall (t, v)\in G(\theta)\setminus\mathcal{Z}_1,$$ showing that $\varepsilon$ is a continuous function on $G(\theta)\setminus\mathcal{Z}_1$ and hence, constant on each connected component of $G(\theta)\setminus\mathcal{Z}_1$. Now, the rank two assumption implies in particular that $A_1\not\equiv 0$ and hence, by Theorem [Theorem: 3](#properties){reference-type="ref" reference="properties"}, $G(\theta)\setminus\mathcal{Z}_1$ has two connected $\mathcal{C}_1^+$ and $\mathcal{C}_1^-$. Let us define $\varepsilon^+:=\varepsilon|_{\mathcal{C}_1^+}$ and $\varepsilon^-:=\varepsilon|_{\mathcal{C}_1^-}$. Let $(t, v)\in\mathcal{C}_1^+$ and assume that the exponential curve $s\in\mathbb{R}\mapsto (t, v)\exp sY$ intersects the singular locus $\mathcal{Z}_1$. Define the sets $$I^+:=\{s\in\mathbb{R}; \; (t, v)\exp sY\in\mathcal{C}_1^+\}\hspace{.5cm}\mbox{ and }\hspace{.5cm}I^-:=\{s\in\mathbb{R}; \; (t, v)\exp sY\in\mathcal{C}_1^-\}.$$ It holds that $I^{+}$ and $I^-$ are open sets, $I^+\neq\emptyset$ and, by Theorem [Theorem: 3](#properties){reference-type="ref" reference="properties"}, $\mathbb{R}\setminus (I^+\cup I^-)$ is discrete. The curves $$\gamma_{\pm}:I^{\pm}\rightarrow G, \hspace{1cm} \gamma_{\pm}(s):=\psi\left((t, v)\exp sY\right).$$ are differentiable and by equation ([\[blabla\]](#blabla){reference-type="ref" reference="blabla"}) satisfies $$\frac{d}{ds}\gamma_{\pm}(s)=(d\psi)_{(t, v)\exp s Y}Y^L((t, v))\exp sY)=(\varepsilon^{\pm}Z)^L(\psi((t, v)\exp sY))=(\varepsilon^{\pm}Z)^L(\gamma_{\pm}(s)),$$ where for simplicity $Z:=(d\psi)_{(0, 0)}Y$. Therefore, $\gamma_{\pm}(s)$ coincides with the solution of the ODE defined by the left-invariant vector field $\varepsilon^{\pm} Z^L$ on the open set $I^{\pm}\times\mathcal{C}^{\pm}_1$. By uniqueness we get that $$\psi((t, v)\exp sY)=\psi(t, v)\exp s\varepsilon^{\pm}Z, \hspace{1cm}\forall s\in I^{\pm}.$$ Let us assume that $I^-\neq\emptyset$. In this case, there exists $s_0\in \mathbb{R}\setminus I^+\cup I^-$ and $\delta>0$ such that $$(t, v)\exp sY\in\mathcal{C}_1^+, s\in (s_0, s_0+\delta)\hspace{.5cm}\mbox{ and }\hspace{.5cm}(t, v)\exp sY\in\mathcal{C}_1^-, s\in (s_0-\delta, s_0).$$ By continuity, we get that $$\psi(t, v)\exp s_0\varepsilon^{+}Z=\psi((t, v)\exp s_0Y)=\psi(t, v)\exp s_0\varepsilon^{-}Z$$ implying that $\exp s_0\varepsilon^{+}Z=\exp s_0\varepsilon^{-}Z$ and, by Remark [\[exp\]](#exp){reference-type="ref" reference="exp"}, $\varepsilon^+=\varepsilon^-$. On the other hand, if $I^-=\emptyset$, let us consider an open set $(t, v)\in U\subset \mathcal{C}_1^+$ and $s_0\in\mathbb{R}\setminus I^+$. The open set $U\exp s_0Y$ intersects $\mathcal{Z}_1$ at the point $(t, v)\exp s_0Y$ and, since $\mathcal{Z}_1$ is a 2D embedded manifold, we have that $$U\exp s_0Y\cap\mathcal{C}_1^-\neq\emptyset.$$ In particular, there exists $(\hat{t}, \hat{v})\in\mathcal{C}_1^+$ such that $(\hat{t}, \hat{v})\exp s_0Y\in\mathcal{C}_1^-$. As a consequence, both of the sets $$\hat{I}^+:=\{s\in\mathbb{R}; \; (\hat{t}, \hat{v})\exp sY\in\mathcal{C}_1^+\}\hspace{.5cm}\mbox{ and }\hspace{.5cm}\hat{I}^-:=\{s\in\mathbb{R}; \; (\hat{t}, \hat{v})\exp sY\in\mathcal{C}_1^-\},$$ are nonempty. By the previous analysis, with $(\hat{t}, \hat{v})$ instead of $(t, v)$, we conclude that $\varepsilon^+=\varepsilon^-$. An analogous analysis, shows also that $\varepsilon^+=\varepsilon^-$ if for some $(t, v)\in\mathcal{C}^-$ the exponential curve $s\in\mathbb{R}\mapsto (t, v)\exp sY$ intersects $\mathcal{Z}_1$. Let us assume that for all $(t, v)\in G(\theta)\setminus\mathcal{Z}_1$ the curve $s\in\mathbb{R}\mapsto (t, v)\exp s(a, w)$ is contained in $G(\theta)\setminus\mathcal{Z}_1$, that is, the exponential curves do not leave the connected component it started. As previously, for all $(t, v)\in\mathcal{C}_1^{\pm}$ we get $$\psi((t, v)\exp sY)=\psi(t, v)\exp s\varepsilon^{\pm}Z, \hspace{1cm}\forall s\in\mathbb{R}.$$ Using again that $\mathcal{Z}_1$ is an embedded 2D manifold, there always exists a continuous curve $\gamma:(-\delta, \delta)\rightarrow G(\theta)$ crossing the singular locus $\mathcal{Z}_1$, that is, $$\gamma(0)\in\mathcal{Z}, \hspace{.5cm}\gamma(-\delta, 0)\subset\mathcal{C}_1^-\hspace{.5cm}\mbox{ and }\hspace{.5cm}\gamma(0, \delta)\subset \mathcal{C}_1^+.$$ Therefore, for all $s\in\mathbb{R}$ it holds that $$\psi(\gamma(\tau)\exp sY)=\psi(\gamma(\tau))\exp s\varepsilon^+Z,\hspace{.5cm}\tau\in (0, \delta)\hspace{.5cm}\mbox{ and }$$ $$\hspace{.5cm}\psi(\gamma(\tau)\exp sY)=\psi(\gamma(\tau))\exp s\varepsilon^-Z, \hspace{.5cm}\tau\in (-\delta, 0).$$ Taking the limit $\tau\rightarrow 0$ from both sides, allows us to conclude that $$\psi(\gamma(0))\exp s\varepsilon^+Z=\psi(\gamma(0)\exp sY)=\psi(\gamma(0))\exp s\varepsilon^-Z, \hspace{1cm}\forall s\in\mathbb{R},$$ implying that $\varepsilon^+=\varepsilon^-$ and concluding the proof of Step 1. **Step 2:** It holds that $f(t, v)=at$ and $g(t, v)=g(0, v)+\Lambda_t^{a\theta}\partial_1g(0, 0)$. Since $(dL_{(t, v)})_{(0, 0)}(1, 0)=(1, 0)\in\Delta_1$ we have that $$\Bigl(\partial_1f(t, v), \partial_1g(t, v)\Bigr)=(d\psi)_{(t, v)}(1, 0)=(dL_{\psi(t, v)})(d\psi)_{(0, 0)}(1, 0)$$ $$=(dL_{\psi(t, v)})\Bigl(\partial_1f(0, 0), \partial_1g(0, 0)\Bigr)=\Bigl(\partial_1f(0, 0), \rho_{f(t, v)}\partial_1g(0, 0)\Bigr),$$ implying that $$\label{um} \partial_1f(t, v)=\partial_1f(0, 0)\;\;\;\mbox{ and }\;\;\;\partial_1g(t, v)=\rho_{f(t, v)}\partial_1g(0, 0).$$ Analogously, for $(0, \eta)\in\Delta\cap\mathfrak{n}(\theta)$, with $\eta\neq 0$, $$\left(\langle\partial_2f(t, v), \rho_t\eta\rangle_{\mathbb{R}^2}, \partial_2g(t, v)\rho_t\eta\right)=(d\psi)_{(t, v)}(0, \rho_t\eta)=(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}(0, \eta)$$ $$=(dL_{\psi(t, v)})_{(0, 0)}\Bigl(\langle \partial_2f(0, 0), \eta\rangle_{\mathbb{R}^2}, \partial_2g(0, 0)\eta\Bigr)=\Bigl(\langle\partial_2f(0, 0), \eta\rangle_{\mathbb{R}^2}, \rho_{f(t, v)}\partial_2g(0, 0)\eta\Bigr),$$ implying that $$\label{dois} \langle\partial_2f(t, v), \rho_t \eta\rangle_{\mathbb{R}^2}=\langle\partial_2f(0, 0), \eta\rangle_{\mathbb{R}^2}\hspace{.5cm}\mbox{ and }\hspace{.5cm}\partial_2g(t, v)\rho_t \eta=\rho_{f(t, v)}\partial_2g(0, 0)\eta.$$ Integration of the first equation in ([\[um\]](#um){reference-type="ref" reference="um"}) on the variable $t\in\mathbb{R}$ gives us that $$f(t, v)=at+h(v), \hspace{.5cm}\mbox{ where }\hspace{.5cm} a=\partial_1f(0, 0)\hspace{.5cm}\mbox{ and }\hspace{.5cm}h(v)=f(0, v).$$ Using the previous on the first equation of ([\[dois\]](#dois){reference-type="ref" reference="dois"}) allows us to obtain $$\label{equi} \langle\partial_2f(t, v), \rho_t \eta\rangle_{\mathbb{R}^2}=\langle\nabla h(v), \rho_t \eta\rangle_{\mathbb{R}^2}\stackrel{(\ref{dois})}{=} \underbrace{\langle\nabla h(0), \eta \rangle_{\mathbb{R}^2}}_{=\langle\partial_2f(0, 0), \eta\rangle_{\mathbb{R}^2}}\stackrel{(\ref{dois})}{=}\langle\nabla h(v), \eta \rangle_{\mathbb{R}^2}=\langle\partial_2f(0, v), \eta\rangle_{\mathbb{R}^2}$$ implying that $\langle\nabla h(v), \eta-\rho_t\eta\rangle_{\mathbb{R}^2}=0$ for all $t\in\mathbb{R}$, $v\in\mathbb{R}^2$. On the other hand, if $\mathcal{X}_1=(\xi_1, A_1)$ , then $$\label{h} f\circ\varphi_s^1=f\hspace{.5cm}\stackrel{(\ref{com})}{\implies} \hspace{.5cm}h(\mathrm{e}^{sA_1}v+\Lambda_t^{\theta}\Lambda_s^{A_1}\xi_1)=h(v),\hspace{.5cm}\forall s, t\in\mathbb{R}$$ gives us that, $$h(\mathrm{e}^{sA_1}v)=h(v)\hspace{.5cm}\implies\hspace{.5cm} \nabla h(\mathrm{e}^{sA_1}v)=\mathrm{e}^{-sA_1^{\intercal}}\nabla h(v),$$ and hence, $$\langle\nabla h(v), \eta \rangle_{\mathbb{R}^2}\stackrel{(\ref{equi})}{=}\langle\nabla h(0), \eta \rangle_{\mathbb{R}^2}\stackrel{(\ref{equi})}{=}\langle\nabla h(\mathrm{e}^{sA_1}v), \eta\rangle_{\mathbb{R}^2}=\langle \mathrm{e}^{-sA_1^{\intercal}}\nabla h(v), \eta\rangle_{\mathbb{R}^2}=\langle \nabla h(v), \mathrm{e}^{-sA_1}\eta\rangle_{\mathbb{R}^2},$$ which gives us $\langle\nabla h(v), \eta-\mathrm{e}^{-sA_1}\eta) \rangle_{\mathbb{R}^2}=0,$ for all $s\in\mathbb{R}$, $v\in\mathbb{R}^2$. Lastly, derivation of ([\[h\]](#h){reference-type="ref" reference="h"}) first at $t=0$ and then at $s=0$, gives us $\langle\nabla h(v), \xi_1\rangle_{\mathbb{R}^2}=0,$ for all $v\in\mathbb{R}^2$. Now, if $\nabla h(v)\neq 0$ for some $v\in\mathbb{R}^2$, we must have that the set $$\{\eta-\mathrm{e}^{-sA_1}\eta, \eta-\rho_t\eta, \xi_1\}\hspace{.5cm}\mbox{ is linearly dependent }\hspace{.5cm}\forall s, t\in\mathbb{R}.$$ Derivation of $s=0$ gives us that $\{A_1\eta, \xi_1\}$ is also linearly dependent. By the rank two hypothesis, the only way this can happen is if $\det A_1\neq 0$. On the other hand, if $\det A_1\neq 0$, a second derivative in $s=0$ forces $\eta$ to be an eigenvector of $A_1$. This, together with the commutativity of $A_1$ and $\theta$ implies that $\eta$ is also an eigenvector of $\theta$. However, since $\Delta_1=\mathrm{span}\{(1, 0), (0, \eta)\}$ we get that $\Delta_1$ is a subalgebra and $$\mathcal{D}(1, 0)=(0, \xi_1)\in\Delta_1\hspace{.5cm}\mbox{ and }\hspace{.5cm}\mathcal{D}_1(0, \eta)=(0, A_1\eta)\in\Delta_1\hspace{.5cm}\implies\hspace{.5cm}\mathcal{D}_1\Delta_1\subset\Delta_1,$$ which contradicts the fact that $\Sigma_1$. Therefore, $$\nabla h(v)=0\hspace{.5cm}\forall v\in\mathbb{R}^2\hspace{.5cm}\implies\hspace{.5cm} h(v)=h(0)=f(0, 0)=0, \hspace{.5cm}\forall v\in\mathbb{R}^2\hspace{.5cm}\implies\hspace{.5cm}f(t, v)=at,$$ as stated. For the function $g$, using the expression $f(t, v)=at$ on the second equation of ([\[um\]](#um){reference-type="ref" reference="um"}), allows us to conclude that $$\partial_1g(t, v)=\rho_{at}\partial_1g(0, 0),$$ which by integration gives us $$g(t, v)-g(0, v)=\int_0^t\partial_1g(s, v) ds=\int_0^t\rho_{as}\partial_1g(0, 0) ds=\Lambda_t^{a\theta}\partial_1g(0, 0),$$ showing the assertion for $g$. **Step 3:** For all $(t, v)\in G(\theta)$ and $s\in\mathbb{R}$, it holds that $$\label{xi} \partial_2g(0, v)\rho_{t}\xi_1=\rho_{at}\partial_2g(0, 0)\xi_1\hspace{.5cm}\mbox{ and }\hspace{.5cm} \partial_2g(0, v)\rho_t\mathrm{e}^{-sA_1}\eta=\rho_{at}\partial_2g(0, 0)\mathrm{e}^{-sA_1}\eta.$$ By the second equation in ([\[com\]](#com){reference-type="ref" reference="com"}), it holds that $$g\left(t, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)=\mathrm{e}^{sA_2}g(t, v)+\Lambda_s^{A_2}\Lambda_{at}^{\theta}\xi_2,$$ where $\mathcal{X}_2=(\xi_2, A_2)$. By Step 2, $$g\left(0, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)+\Lambda_{at}^{\theta}\nu=\mathrm{e}^{sA_2}g(0, v)+\mathrm{e}^{sA_2}\Lambda_{at}^{\theta}\nu+\Lambda_s^{A_2}\Lambda_{at}^{\theta}\xi_2,$$ where for simplicity we put $\nu=\partial_1g(0, 0)$. Therefore, $$\label{uau} g\left(0, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)-\mathrm{e}^{sA_2}g(0, v)=(\mathrm{e}^{sA_2}-\operatorname{id}_{\mathbb{R}^2})\Lambda_{at}^{\theta}\nu+\Lambda_s^{A_2}\Lambda_{at}^{\theta}\xi_2= \Lambda_s^{A_2}\Lambda_{at}^{\theta}\left(\xi_2+A_2\nu\right),$$ where for the last equality we used that $\mathrm{e}^{sA_2}-\operatorname{id}_{\mathbb{R}^2}=\Lambda_{t}^{A_2}A_2$ and $A_2\theta=\theta A_2$. Derivation of equation ([\[uau\]](#uau){reference-type="ref" reference="uau"}) on $t$ gives us, by the chain rule, that $$\partial_2g\left(0, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)\rho_{t}\Lambda_s^{A_1}\xi_1=a\rho_{at}\Lambda_s^{A_2}\left(\xi_2+A_2\nu\right).$$ Since the previous equations is true for all $s\in\mathbb{R}$ and $(t, v)\in G$, we can substitute $v$ by $\mathrm{e}^{-sA_1}\left(v-\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)$ in order to obtain $$\partial_2g(0, v)\rho_{t}\Lambda_s^{A_1}\xi_1=a\rho_{at}\Lambda_s^{A_2}\left(\xi_2+A_2\nu\right).$$ Derivation of the previous equation at $s=0$ gives us $$\partial_2g(0, v)\rho_{t}\xi_1=a\rho_{at}\left(\xi_2+A_2\nu\right)\hspace{.5cm}\implies\hspace{.5cm}\partial_2g(0, 0)\xi_1=a\left(\xi_2+A_2\nu\right),$$ and proves the first equality. For the second equality, let us notice that the right-hand side of equation ([\[uau\]](#uau){reference-type="ref" reference="uau"}) does not depends on $v$. Differentiation on the variable $v$ gives us, by the chain rule, that $$\label{commuti} \partial_2g\left(0, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)\mathrm{e}^{sA_1}=\mathrm{e}^{sA_2}\partial_2g(0, v).$$ On the other hand, by Step 2. $$\forall (t, v)\in G(\theta), \;\;\; \partial_2g(t, v)=\partial_2g(0, v) \;\;\;\;\mbox{ and }\;\;\;\;f(t, v)=at,$$ which by the second equation in ([\[dois\]](#dois){reference-type="ref" reference="dois"}) gives us that $$\partial_2g(0, v)\rho_t \eta=\rho_{at}\partial_2g(0, 0)\eta.$$ Therefore, for any $(t, v)\in G(\theta)$ and $s\in \mathbb{R}$ we get $$\partial_2g(0, v)\rho_t\mathrm{e}^{-sA_1}\eta=\partial_2g(0, v)\mathrm{e}^{-sA_1}\rho_t\eta\stackrel{(\ref{commuti})}{=}\mathrm{e}^{-sA_2}\partial_2g\left(0, \mathrm{e}^{sA_1}v+\Lambda_s^{A_1}\Lambda_t^{\theta}\xi_1\right)\rho_t\eta$$ $$=\mathrm{e}^{-sA_2}\rho_{at}\partial_2g(0, 0)\eta=\rho_{at}\mathrm{e}^{-sA_2}\partial_2g(0, 0)\eta\stackrel{(\ref{commuti})}{=}\rho_{at}\partial_2g(0, 0)\mathrm{e}^{-sA_1}\eta,$$ showing the assertion. **Step 4:** $\psi\in \mathrm{Aut}(G(\theta))$ We have to analyze the following possibilities: - $\Delta_1$ is not a subalgebra: In this case, $\eta\neq 0$ considered as previous cannot be an eigenvector of $\theta$, and so, there exists $t_0\in\mathbb{R}$ is such that $\{\eta, \rho_{t_0}\eta\}$ is a basis of $\mathbb{R}^2$. From the second equation in ([\[xi\]](#xi){reference-type="ref" reference="xi"}) we get: - For $s=0$, $$\forall (t, v)\in G(\theta), \hspace{1cm}\partial_2g(0, v)\rho_t\eta=\rho_{at}\partial_2g(0, 0)\eta;$$ - For $s=0$ and all $(t, v)\in G(\theta)$, $$\partial_2g(0, v)\rho_t\rho_{t_0}\eta=\partial_2g(0, v)\rho_{t+t_0}\eta=\rho_{a(t+t_0)}\partial_2g(0, 0)\eta=\rho_{at}\rho_{at_0}\partial_2g(0, 0)\eta=\rho_{at}\partial_2g(0, 0)\rho_{t_0}\eta,$$ where for the last equality we used (ii) for $t=t_0$ and $v=0$. Therefore, the maps $\partial_2g(0, v)\rho_t$ and $\rho_{at}\partial_2g(0, 0)$ coincide on a basis of $\mathbb{R}^2$ and hence $$\label{buu} \forall (t, v)\in G(\theta), \hspace{1cm}\partial_2g(0, v)\rho_t=\rho_{at}\partial_2g(0, 0).$$ As a consequence, for $t=0$ we obtain that $$\forall v\in\mathbb{R}^2, \hspace{1cm}\partial_2g(0, v)=\partial_2g(0, 0).$$ Moreover, if $\operatorname{tr}\theta\neq 0$, then $$\det\left(\partial_2g(0, 0)\right)\mathrm{e}^{t\operatorname{tr}\theta}=\det\left(\partial_2g(0, 0)\circ \rho_t\right)=\det\left(\rho_{at}\circ \partial_2g(0, 0)\right)=\mathrm{e}^{at\operatorname{tr}\theta}\det\left(\partial_2g(0, 0)\right),$$ implying that $a=1.$ On the other hand, if $\operatorname{tr}\theta=0$ then necessarily $\det\theta\neq0$. Derivation of equation ([\[buu\]](#buu){reference-type="ref" reference="buu"}) gives us that $$\partial_2g(0, 0)\circ\theta=a\theta\circ\partial_2g(0, 0),$$ and hence, $$\det\left(\partial_2g(0, 0)\right)\det\theta=\det\left(\partial_2g(0, 0)\circ\theta\right)=\det\left(a\theta\circ\partial_2g(0, 0)\right)=a^2\det\theta\det\left(\partial_2g(0, 0)\right),$$ implying that $a\in\{-1, 1\}$. - $\Delta_1$ is a subalgebra and $\theta\neq\operatorname{id}_{\mathbb{R}^2}$: In this case, the fact that $A_1\theta=\theta A_1$ implies that $\eta$ is also an eigenvector of $A_1$. Since $\Delta_1=\mathrm{span}\{(1, 0), (0, \eta)\}$, we conclude as in Step 2 that the LARC holds for $\Sigma_1$ if and only if the set $\{\xi_1, \eta\}$ is linearly independent. Considering $t=s=0$ in the equations in ([\[xi\]](#xi){reference-type="ref" reference="xi"}) gives us that $$\partial_2g(0, v)\eta=\partial_2g(0, 0)\eta\hspace{.5cm}\mbox{ and }\hspace{.5cm} \partial_2g(0, v)\xi_1=\partial_2g(0, v)\xi_1,$$ which by linearity implies that, $$\forall v\in\mathbb{R}^2, \hspace{.5cm}\partial_2g(0, v)=\partial_2g(0, 0)\hspace{.5cm}\mbox{ and }\hspace{.5cm}\partial_2g(0, 0)\circ \rho_t=\rho_{at}\circ \partial_2g(0, 0),$$ and, as in the previous case, we conclude that $a=1$ if $\operatorname{tr}\theta\neq 0$ and $a\in\{-1, 1\}$ if $\operatorname{tr}\theta=0$. - $\Delta_1$ is a subalgebra and $\theta=\operatorname{id}_{\mathbb{R}^2}$: In this case, we get from the second equation in ([\[xi\]](#xi){reference-type="ref" reference="xi"}) that $$\mathrm{e}^{t}\partial_2g(0, v)\eta=\partial_2g(0, v)\rho_t\eta=\rho_{at}\partial_2g(0, 0)\eta=\mathrm{e}^{at}\partial_2g(0, v)\eta,$$ implying that $a=1$. If $\{\xi_1, \eta\}$ is linearly independent, we conclude as previously, that $$\forall v\in\mathbb{R}^2,\hspace{1cm}\partial_2g(0, v)=\partial_2g(0, 0).$$ On the other hand, if $\{\xi_1, \eta\}$ is linearly dependent, the LARC implies necessarily that $\eta$ cannot be an eigenvector of $A_1$. In particular, for some $s_0\in\mathbb{R}$, $\{\eta, \mathrm{e}^{-s_0A_1}\eta\}$ is a basis of $\mathbb{R}^2$ and as previously we conclude that the linear maps $\partial_2g(0, v)$ and $\partial_2g(0, 0)$ coincide on such basis, implying that $\partial_2g(0, v)=\partial_2g(0, 0)$ for all $v\in \mathbb{R}^2$. In any case, we get that $$f(t, v)=\varepsilon t\;\;\;\mbox{ and }\;\;\; g(t, v)=Pv+\varepsilon\Lambda_{t}^{\varepsilon\theta}\nu,$$ where $\nu=\partial_1g(0, 0)$, $P=\partial_2g(0, 0)$ and $P\theta=\varepsilon\theta P$ with $\varepsilon=1$ if $\operatorname{tr}\theta\neq 0$ or $\varepsilon\in\{-1, 1\}$ if $\operatorname{tr}\theta=0$. Moreover, by definition, $$\Lambda_t^{\varepsilon t}\nu=\int_0^t\mathrm{e}^{\varepsilon s\theta}\nu ds=\varepsilon\int_0^{\varepsilon t}\mathrm{e}^{\mu \theta}\nu d\mu=\varepsilon\Lambda^{\theta}_{\varepsilon t}\nu,$$ implying that $\psi\in\mathrm{Aut}(G(\theta))$ and concluding the proof. $\Box$ ## Invariance of the nilradical In this section we show that for an isometry $\psi\in\mathrm{Iso}_{\theta}(\Sigma_1; \Sigma_2)_0$ between rank two ARSs with $\psi=(f, g)$, the coordinate function $f$ only depends on the first variable of $G(\theta)$ or equivalently, it satisfies $\partial_2f\equiv 0$. Let us consider an orthonormal basis $\{X_i, Y_i\}\subset\Delta_i$. Since $\psi_*\Delta_1^L=\Delta_2^L\circ\psi$, we can write uniquely $$\psi_*X_1^L=\alpha_1X_2^L(\psi)+\alpha_2Y_2^L(\psi),$$ where $$\alpha_1^2+\alpha_2^2=\|\psi_*X_1^L\|^2_{\Sigma_2}=\|X_1^L\|^2_{\Sigma_1}=\|X_1\|^2_{\Sigma_1}=1,$$ and each $\alpha_i$ can be recovered from the orthonormality of the basis as $$\alpha_1=\langle \psi_*X_1^L, X_2^L(\psi)\rangle_{\Sigma_2}\hspace{.5cm}\mbox{ and }\hspace{.5cm}\alpha_2=\langle \psi_*X_1^L, Y_2^L(\psi)\rangle_{\Sigma_2},$$ showing that $\alpha_i:G(\theta)\setminus\mathcal{Z}_1\rightarrow\mathbb{R}$ are $\mathcal{C}^{\infty}$ functions. The fact that $\{X_1^L, Y_1^L\}$ is an orthonormal basis of $\Delta^L_1$ and $\psi$ is an isometry allow us to write $$\psi_*Y_1^L=\epsilon\left(-\alpha_2X_2^L(\psi)+\alpha_1Y_2^L(\psi)\right),$$ where $\epsilon=\pm 1$ is constant on the connected components of $G(\theta)\setminus\mathcal{Z}_1$. Since all the calculations will be taking into account the connected components of $G(\theta)\setminus\mathcal{Z}_1$ we will assume that $\epsilon=1$. As one can easily see, such assumption bears no loss of generality and avoids cumbersome notation. Let us assume w.l.o.g. that $$X_i=(0, \eta_i)\hspace{.5cm}\mbox{ and }\hspace{.5cm} Y_i=(\sigma_i, \nu_i), \hspace{.5cm}\mbox{ with }\hspace{.5cm} \sigma_i\neq 0.$$ By writting $\psi=(f, g)$ and $\psi_*=\left(\begin{array}{cc} \partial_1f & (\partial_2 f)^T\\ \partial_1g & \partial_2g \end{array}\right)$, the previous considerations imply that $$\psi_*(0, \rho_t\eta_1)=\alpha_1(0, \rho_f\eta_2)+\alpha_2(\sigma_2, \rho_f\nu_2)\hspace{.5cm}\mbox{ and }\hspace{.5cm}\psi_*(\sigma_1, \rho_t\nu_1)=-\alpha_2(0, \rho_f\eta_2)+\alpha_1(\sigma_2, \rho_f\nu_2),$$ or equivalently $$\label{9} \left\{\begin{array}{l} \langle\partial_2f, \rho_t\eta_1\rangle_{\mathbb{R}^2}=\alpha_2\sigma_2\\ \sigma_1\partial_1f+\langle\partial_2f,\rho_t\nu_1\rangle_{\mathbb{R}^2}=\alpha_1\sigma_2 \end{array}\right. \hspace{.5cm}\mbox{ and }\hspace{.5cm}\left\{\begin{array}{l} \partial_2g\rho_t\eta_1=\rho_{f}\left(\alpha_1\eta_2+\alpha_2\nu_2\right)\\ \sigma_1\partial_1g+\partial_2g\rho_t\nu_1=\rho_{f}\left(-\alpha_2\eta_2+\alpha_1\nu_2\right) \end{array}\right..$$ Now, $$\psi\circ\varphi_s^1=\varphi_s^2\circ\psi\hspace{.5cm}\implies\hspace{.5cm} f\circ\varphi_s^1=f,$$ and consequently, by differentiation, $$\label{eqf} \langle\partial_2f,A_1v+\Lambda_t^{\theta}\xi_1\rangle_{\mathbb{R}^2} = 0, \hspace{.5cm} \partial_1f = \partial_1f(\varphi_s^1)+\langle\partial_2f(\varphi^1_s),\rho_t\Lambda_{s}^{A_1}\xi_1\rangle_{\mathbb{R}^2}\hspace{.5cm}\mbox{ and }\hspace{.5cm}\partial_2f(\varphi_s^1)=\mathrm{e}^{-sA_1^{\intercal}}\partial_2f.$$ Now, from the equations in ([\[9\]](#9){reference-type="ref" reference="9"}) we obtain that $$\sigma_2^2=(\sigma_2\alpha_2(\varphi_s^1))^2+(\sigma_2\alpha_1(\varphi_s^1))^2=\langle\partial_2f(\varphi_s^1), \rho_t\eta_1\rangle_{\mathbb{R}^2}^2+\left(\sigma_1\partial_1f(\varphi_s^1)+\langle\partial_2f(\varphi_s^1), \rho_t\nu_1\rangle_{\mathbb{R}^2}\right)^2$$ $$=\langle\mathrm{e}^{-sA_1^{\intercal}}\partial_2f, \rho_t\eta_1\rangle_{\mathbb{R}^2}^2+\left(\sigma_1\left(\partial_1f-\langle\mathrm{e}^{-sA_1^{\intercal}}\partial_2f, \rho_t\Lambda_{s}^{A_1}\xi_1\rangle_{\mathbb{R}^2}\right)+\langle\mathrm{e}^{-sA_1^{\intercal}}\partial_2f, \rho_t\nu_1\rangle_{\mathbb{R}^2}\right)^2$$ $$=\langle\partial_2f, \mathrm{e}^{-sA_1}\rho_t\eta_1\rangle_{\mathbb{R}^2}^2+\left(\sigma_1\left(\partial_1f+\langle\partial_2f, \rho_t\Lambda_{-s}^{A_1}\xi_1\rangle_{\mathbb{R}^2}\right)+\langle\partial_2f, \mathrm{e}^{-sA_1}\rho_t\nu_1\rangle_{\mathbb{R}^2}\right)^2,$$ where in the second parenthesis we used that $\mathrm{e}^{-sA_1}\Lambda_{s}^{A_1}=-\Lambda_{-s}^{A_1}$. Derivation at $s=0$ gives us $$0=2\underbrace{\langle\partial_2f, \rho_t\eta_1\rangle_{\mathbb{R}^2}}_{=\alpha_2\sigma_2}\Bigl(\langle \partial_2f, -A_1\rho_t\eta_1\rangle_{\mathbb{R}^2}\Bigr)+2\underbrace{(\sigma_1\partial_1f+\langle\partial_2f, \rho_t\nu_1\rangle_{\mathbb{R}^2})}_{=\alpha_1\sigma_2}\Bigl(-\sigma_1\langle\partial_2f, \rho_t\xi_1\rangle_{\mathbb{R}^2}$$ $$+\langle\partial_2f, -A_1\rho_t\nu_1\rangle_{\mathbb{R}^2}\Bigr)=-2\left\langle\partial_2f, \rho_t\left(\sigma_1\sigma_2\alpha_1\xi_1+\sigma_2A_1(\alpha_2\eta_1+\alpha_1\nu_1)\right)\right\rangle_{\mathbb{R}^2},$$ showing that $$\partial_2f\hspace{.5cm}\mbox{ is orthogonal to }\hspace{.5cm}\rho_t\left(\sigma_1\sigma_2\alpha_1\xi_1+\sigma_2A_1(\alpha_2\eta_1+\alpha_1\nu_1)\right)\;\;\mbox{ on }\;\;G(\theta)\setminus\mathcal{Z}_1.$$ Moreover, by the first equation in ([\[eqf\]](#eqf){reference-type="ref" reference="eqf"}), we get that, on $G(\theta)\setminus\mathcal{Z}_1$, $$\label{LD} \partial_2f\neq 0\implies\hspace{.5cm} \{A_1v+\Lambda_t^{\theta}\xi_1, \rho_t(\sigma_1\sigma_2\alpha_1\xi_1+\sigma_2A_1(\alpha_2\eta_1+\alpha_1\nu_1)\}\hspace{.5cm} \mbox{ is linearly dependent}.$$ We show that $\partial_2f\equiv0$ by analyzing the possibilities for the eigenvalues of $A_1$ in the next propositions. **Proposition: 8**. *If $A_1$ has only eigenvalues with nonzero real parts, $\partial_2f\equiv 0$.* **Proof:** As in the proof of Lemma [Lemma: 7](#fundamentallemma){reference-type="ref" reference="fundamentallemma"}, we can use Proposition [Proposition: 4](#difeo){reference-type="ref" reference="difeo"} and assume w.l.o.g. that $(1, 0)\in\Delta_1$. Under this assumption, we have that $\nu_1=c\eta_1$ and equations ([\[9\]](#9){reference-type="ref" reference="9"}) become $$\langle\partial_2f, \rho_t\eta_1\rangle_{\mathbb{R}^2}=\alpha_2\sigma_2\hspace{.5cm}\mbox{ and }\hspace{.5cm}\partial_1f= \frac{\sigma_2}{\sigma_1}(\alpha_1-c\alpha_2).$$ Therefore, $$\alpha_2(\varphi_s^1)\sigma_2=\langle\partial_2f(\varphi_s^1), \rho_t\eta_1\rangle_{\mathbb{R}^2}\stackrel{(\ref{eqf})}{=}\langle\mathrm{e}^{-sA_1^{\intercal}}\partial_2f, \rho_t\eta_1\rangle_{\mathbb{R}^2}$$ is bounded for $s\in\mathbb{R}$. However, the assumption that $A_1$ has only eigenvalues with nonzero real parts and $\eta_1\neq 0$ implies that $\alpha_2\equiv 0$, and hence, $\alpha_1=1$ showing that $\partial_1f$ is constant on $G\setminus\mathcal{Z}_1$. Using the continuity of $f$ and the fact that $G(\theta)\setminus\mathcal{Z}_1$ is an open and dense subset of $G(\theta)$ allow us to conclude that $$\forall (t, v)\in G(\theta), \;\;\;\;f(t, v)=at+h(v), \;\;\mbox{ where }\;\; a=\sigma_2/\sigma_1\;\;\mbox{ and }\;\;h(v)=f(0, v).$$ By doing the same analysis as the one in Step 2 of Lemma [Lemma: 7](#fundamentallemma){reference-type="ref" reference="fundamentallemma"}, allows us to conclude, by the LARC, that $h\equiv 0$, implying that $\partial_2f\equiv 0$ as stated. $\Box$ **Proposition: 9**. *If $A_1$ has a pair of pure imaginary eigenvalues, $\partial_2f\equiv 0$.* **Proof:** Since $\det A_1\neq 0$, the automorphism, $\hat{\psi}(t, v)=\left(t, v-\Lambda_t^{\theta}(A^{-1}\xi_1)\right),$ is well defined. Moreover, by Proposition ([Proposition: 4](#difeo){reference-type="ref" reference="difeo"}) it is an isometry between $\Sigma_1$ and an ARS $\hat{\Sigma}_1$ whose associated linear vector field is $\hat{\mathcal{X}}_1=(0, A_1)$. Therefore, we will assume w.l.o.g. that $\xi_1=0$. Under this assumption, let us define the set $$\mathcal{A}:=\{(t, v)\in G(\theta); \;\;\partial_2f(t, v)\neq 0\}.$$ The first to notice is that $$\partial_2f(\varphi_s^1)=\mathrm{e}^{-sA^{\intercal}}\partial_2f\hspace{.5cm}\implies\hspace{.5cm}\varphi_s^1(\mathcal{A})\subset\mathcal{A}.$$ Also, the fact that $A_1$ has a pair of imaginary eigenvalues and that $A_1\theta=\theta A_1$ imply that, on the canonical basis, $A_1=\mu R$, where $R=\left(\begin{array}{cc} 0 & -1\\ 1 & 0 \end{array}\right)$ and $\mu\neq 0$. Since we are assuming $\xi_1=0$, the singular locus $\mathcal{Z}_1$ and the set of singularities $\mathcal{Z}_{\mathcal{X}_1}$ are given explicitly as $$\mathcal{Z}_1=\{(t, s\rho_tR\eta_1), \;\;t, s\in\mathbb{R}\}\hspace{.5cm}\mbox{ and }\hspace{.5cm}\mathcal{Z}_{\mathcal{X}_1}=\mathbb{R}\times\{0\}.$$ Consequently, for any $(t, v)\in\mathcal{Z}_1\setminus\mathcal{Z}_{\mathcal{X}_1}$ there exists $s_0\in\mathbb{R}$ such that $\varphi_{s_0}(t, v)\in G(\theta)\setminus\mathcal{Z}_1$. In particular we get that $$\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}\neq\emptyset\hspace{.5cm}\implies \hspace{.5cm} \mathcal{A}\cap \left(G(\theta)\setminus\mathcal{Z}_1\right)\neq\emptyset.$$ Let us assume the existence of $(t_0, v_0)\in \mathcal{A}\cap \left(G(\theta)\setminus\mathcal{Z}_1\right)$. From relation ([\[LD\]](#LD){reference-type="ref" reference="LD"}) we have that $$\alpha_2(t_0, v_0)\rho_{t_0}\eta_1+\alpha_1(t_0, v_0)\rho_{t_0}\nu_1\in\mathbb{R}v_0,$$ or equivalently, $$\label{relation} \alpha_2(t_0, v_0)\langle\rho_{t_0}\eta_1, Rv_0\rangle_{\mathbb{R}^2}+\alpha_1(t_0, v_0)\langle\rho_{t_0}\nu_1, Rv_0\rangle_{\mathbb{R}^2}=0.$$ Now, if $\nu_1=c\eta_1$, equation ([\[relation\]](#relation){reference-type="ref" reference="relation"}) implies that $$\bigl(\alpha_2(t_0, v_0)+c\alpha_1(t_0, v_0)\bigr)\langle\rho_{t_0}\eta_1, Rv_0\rangle=0.$$ By a simple calculation, we get that $$\hat{\mathcal{Z}}_1=\{(t, s\rho_t\eta_1), \;t, s\in\mathbb{R}\},$$ is the set of points where $\langle\rho_{t_0}\eta_1, Rv_0\rangle=0$, and hence $$\alpha_1^2(t_0, v_0)=\frac{1}{1+c^2}\hspace{.5cm}\mbox{ and }\hspace{.5cm}\alpha_2^2(t_0, v_0)=\frac{c^2}{1+c^2}, \hspace{.5cm}\forall (t_0, v_0)\in\mathcal{A}\cap\left(G(\theta)\setminus(\mathcal{Z}_1\cup\hat{\mathcal{Z}}_1)\right),$$ where we used that $\alpha_1^2+\alpha_2^2=1$. As in Proposition [Proposition: 8](#prop1){reference-type="ref" reference="prop1"} the assumption $\nu_1=c\eta_1$ gives us $$\langle\partial_2f, \rho_t\eta_1\rangle_{\mathbb{R}^2}=\alpha_2\sigma_2\hspace{.5cm}\mbox{ and }\hspace{.5cm}\partial_1f= \frac{\sigma_2}{\sigma_1}(\alpha_1-c\alpha_2),$$ implying, in particular, that $\partial_1f$ is constant on each connected component of $G(\theta)\setminus(\mathcal{Z}_1\cup\hat{\mathcal{Z}}_1)$. Using the continuity of $f$ and the fact that $G(\theta)\setminus(\mathcal{Z}_1\cup\hat{\mathcal{Z}}_1)$ is an open and dense subset of $G(\theta)$ allow us to conclude that $$\forall (t, v)\in G(\theta), \hspace{.5cm}f(t, v)=at+h(v), \hspace{.5cm}\mbox{ where }\hspace{.5cm}h(v)=f(0, v).$$ By doing the same analysis as the one in Step 2 of Lemma [Lemma: 7](#fundamentallemma){reference-type="ref" reference="fundamentallemma"}, we are able to conclude, by the LARC, that $h\equiv 0$, which gives us $\partial_2f\equiv 0$. On the other hand, if $\{\eta_1, \nu_1\}$ is a linearly independent set, then $\langle\rho_{t_0}\eta_1, Rv_0\rangle_{\mathbb{R}^2}^2+\langle\rho_{t_0}\nu_1, Rv_0\rangle_{\mathbb{R}^2}^2\neq 0$ on $G(\theta)$. As a consequence, relation ([\[relation\]](#relation){reference-type="ref" reference="relation"}) implies that for all $(t_0, v_0)\in \mathcal{A}\cap (G(\theta)\setminus\mathcal{Z}_1)$, $$\label{alphais} \alpha_1(t_0, v_0)^2=\frac{\langle\rho_{t_0}\eta_1, Rv_0\rangle_{\mathbb{R}^2}^2}{\langle\rho_{t_0}\eta_1, Rv_0\rangle_{\mathbb{R}^2}^2+\langle\rho_{t_0}\nu_1, Rv_0\rangle_{\mathbb{R}^2}^2}\hspace{.5cm}\mbox{ and }$$ $$\alpha_2(t_0, v_0)^2=\frac{\langle\rho_{t_0}\nu_1, Rv_0\rangle_{\mathbb{R}^2}^2}{\langle\rho_{t_0}\eta_1, Rv_0\rangle_{\mathbb{R}^2}^2+\langle\rho_{t_0}\nu_1, Rv_0\rangle_{\mathbb{R}^2}^2}.$$ Next we show that $\mathcal{A}\subset\mathcal{Z}_{\mathcal{X}_1}$, analyzing two possibilities: 1. $\eta_1$ and $\nu_1$ are not orthogonal vectors; Let $(t, v)\in\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}$. The fact that $\mathrm{e}^{sA_1}v$ is a circumference around the origin gives us that $$\{\mathrm{e}^{sA_1}v, \;\;s\in\mathbb{R}\}\cap\{s\rho_t\nu_1, \;s\in\mathbb{R}\}\neq\emptyset\hspace{.5cm}\implies\hspace{.5cm}\exists s_0, s_1\in\mathbb{R}^*; \;\;\mathrm{e}^{s_0A_1}v=s_1\rho_t\nu_1.$$ In particular, if $\eta_1$ and $\nu_1$ are not orthogonal vectors, $A_1\nu_1\notin\mathbb{R}\eta_1$ implying that $(t, s_1\rho_t\nu_1)\in G(\theta)\setminus\mathcal{Z}_1$ and hence $$\mathcal{A}\ni\varphi^1_{s_0}(t, v)=(t, \mathrm{e}^{s_0A_1}v)=(t, s_1\rho_t\nu_1)\in G(\theta)\setminus\mathcal{Z}_1.$$ Formula ([\[alphais\]](#alphais){reference-type="ref" reference="alphais"}) with $(t_0, v_0)=(t, s_1\rho_t\nu_1)$ implies $$\langle\rho_t\nu_1, Rv_0\rangle_{\mathbb{R}^2}=s_1\langle\rho_t\nu_1, R\rho_t\nu_1\rangle_{\mathbb{R}^2}=0\hspace{.5cm}\implies\hspace{.5cm}\alpha_2(t, s_1\rho_t\nu_1)=0\;\;\stackrel{(\ref{9})}{\implies}\;\;\;\langle\partial_2f(t, s_1\rho_t\nu_1), \rho_t\eta_1\rangle_{\mathbb{R}^2}=0.$$ On the other hand, by the first equation in ([\[eqf\]](#eqf){reference-type="ref" reference="eqf"}) we get that $$\langle\partial_2f(t, s_1\rho_t\nu_1), A_1(s_1\rho_t\nu_1)\rangle_{\mathbb{R}^2}=0\hspace{.5cm}\implies\hspace{.5cm}\langle\partial_2f(t, s_1\rho_t\nu_1), \rho_tA_1\nu_1\rangle_{\mathbb{R}^2}=0.$$ Since $A_1\nu_1\notin\mathbb{R}\eta_1$, the set $\{\rho_t \eta_1, \rho_tA_1\nu_1\}$ is a basis of $\mathbb{R}^2$, which by the previous equalities imply $\partial_2f(t, s_1\rho_t\nu_1)=0$, and hence $$\partial_2f(t, v)\stackrel{(\ref{eqf})}{=}\mathrm{e}^{-s_0A_1^{\intercal}}\partial_2f(\varphi^1_{s_0}(t, v))=\mathrm{e}^{-s_0A_1^{\intercal}}\underbrace{\partial_2f(t, s_1\rho_t\nu_1)}_{=0}=0,$$ contradicting the fact that $(t, v)\in\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}$. Therefore, if $\eta_1$ and $\nu_1$ are not orthogonal vectors we must have that $\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}=\emptyset$. 2. $\eta_1$ and $\nu_1$ are orthogonal vectors; Let $(t, v)\in\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}$. As previously, $$\exists s_0, s_1\in\mathbb{R}^*; \;\;\;\;\mathcal{A}\ni\varphi_{s_0}^1(t, v)=(t, s_1\rho_t\eta_1)\in G(\theta)\setminus\mathcal{Z}_1,$$ and, by using ([\[alphais\]](#alphais){reference-type="ref" reference="alphais"}) with $(t_0, v_0)=(t, s_1\rho_t\eta_1)$, allows us to obtain that $\alpha_1(t, s_1\rho_t\eta_1)=0.$ Since $\eta_1$ and $\nu_1$ being orthogonal is equivalent to $A_1\eta_1\in\mathbb{R}\nu_1$, the first equation in ([\[eqf\]](#eqf){reference-type="ref" reference="eqf"}) gives us that $$\langle\partial_2f(t, s_1\rho_t\eta_1), A_1(s_1\rho_t\eta_1)\rangle_{\mathbb{R}^2}=0\hspace{.5cm}\implies\hspace{.5cm}\langle\partial_2f(t, s_1\rho_t\eta_1), \rho_t\nu_1\rangle_{\mathbb{R}^2}=0,$$ and hence, $$\sigma_1\partial_1f(t, s_1\rho_t\eta_1)=\sigma_1\partial_1f(t, s_1\rho_t\eta_1)+\underbrace{\langle\partial_2f(t, s_1\rho_t\eta_1), \rho_t\nu_1\rangle_{\mathbb{R}^2}}_{=0}\stackrel{(\ref{9})}{=}\alpha_1(t, s_1\rho_t\eta_1)\sigma_2=0.$$ Since we are assuming $\xi_1=0$, the second equation in ([\[eqf\]](#eqf){reference-type="ref" reference="eqf"}) implies that $\partial_1f(\varphi^1_s)=\partial_1f$ for all $s\in\mathbb{R}$. In particular, $$\partial_1f(t, v)=\partial_1f(\varphi^1_{s_0}(t, v))=\partial_1f(t, s_1\rho_t\eta_1)=0,$$ and by the arbitrariness of $(t, v)\in\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}$ we conclude that $$\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}\subset \{(t, v)\in G(\theta)\setminus\mathcal{Z}_{\mathcal{X}_1}; \;\; \partial_1f(t, v)=0\}.$$ Now, if $(t, v)\in (\overline{\mathcal{A}}\setminus\mathcal{A})\cap \left(G(\theta)\setminus\mathcal{Z}_1\right)$, the fact that $f$ restrict to $G(\theta)\setminus\mathcal{Z}_1$ is $\mathcal{C}^{\infty}$ implies that $$\partial_1f(t, v)=0\hspace{.5cm}\mbox{ and }\hspace{.5cm}\partial_2f(t, v)=0,$$ which is a contradition to the fact that $\psi$ is a diffeomorphism on $G\setminus\mathcal{Z}_1$. Therefore, $\mathcal{A}\cap(G(\theta)\setminus\mathcal{Z}_1)$ is open and closed in $G(\theta)\setminus\mathcal{Z}_1$ and so, $\mathcal{A}$ contains any connected component of $G(\theta)\setminus\mathcal{Z}_1$ that it intersects. On the other hand, for any $(t, v)\in G(\theta)$ with $v\neq 0$, the curve $\{\varphi_s(t, v)=(t, \mathrm{e}^{sA_1}), s\in\mathbb{R}\}$ is a circumference on the plane $\{t\}\times\mathbb{R}^2$ with center $(t, 0)$ and radius $|v|_{\mathbb{R}^2}$. Since the intersection $\left(\{t\}\times\mathbb{R}^2\right)\cap\mathcal{Z}_1$ is the line $\{(t, s\rho_tR\eta_1), s\in\mathbb{R}\}$, any point $(t, v)\in G(\theta)\setminus\mathcal{Z}_1$ crosses between the two connected components of $G(\theta)\setminus\mathcal{Z}_1$. By the invariance $\varphi_s(\mathcal{A})\subset\mathcal{A}$ we get $$\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}\neq\emptyset\hspace{.5cm}\implies\hspace{.5cm}\mathcal{A}\cap (G(\theta)\setminus\mathcal{Z}_1)\neq\emptyset\hspace{.5cm}\implies\hspace{.5cm}G(\theta)\setminus\mathcal{Z}_1\subset\mathcal{A}.$$ From that, we conclude that $\partial_1f|_{G(\theta)\setminus\mathcal{Z}_1}\equiv 0$ and by the continuity of $f$ we get actually $$f(t, v)=f(0, v)\hspace{.5cm}\implies\hspace{.5cm}\partial_1f\equiv 0.$$ However, since $A_10\mu R$ with $\mu\neq 0$, the relation $$\partial_2f(\varphi^1_s)=\mathrm{e}^{-sA^{\intercal}}\partial_2f\hspace{.5cm}\implies\hspace{.5cm}\partial_2f(0, 0)=0,$$ which together with $\partial_1f\equiv 0$ contradicts the fact that $\psi$ is a diffeomorphism. Therefore, in both cases, we conclude that $\mathcal{A}\setminus\mathcal{Z}_{\mathcal{X}_1}=\emptyset$ or equivalently $\mathcal{A}\subset\mathcal{Z}_{\mathcal{X}_1}$. Therefore, $$\partial_2f|_{G(\theta)\setminus\mathcal{Z}_{\mathcal{X}_1}}\equiv 0,$$ which by continuity implies $$f(t, v)=at, \hspace{.5cm}\mbox{ where }\hspace{.5cm}a=\sigma_2/\sigma_1\in\mathbb{R}^*,$$ implying that $\partial_2f\equiv 0$ and concluding the proof. $\Box$ **Proposition: 10**. *If $\mathbb{R}^2=\mathrm{Im}A_1\oplus\mathbb{R}\xi_1$ then $\partial_2f\equiv 0$.* **Proof:** Let us assume w.l.o.g. that $(1, 0)\in\Delta_1$. In this case, $\nu_1=c\eta_1$ and from equations ([\[9\]](#9){reference-type="ref" reference="9"}) we get that $$\partial_1f=(\alpha_1-c\alpha_2)\sigma_2\hspace{.5cm}\implies\hspace{.5cm}\partial_1f\;\mbox{ is bounded on }G(\theta)\setminus\mathcal{Z}_1.$$ On the other hand, by our hypothesis $\dim\mathrm{Im}A_1=1$ and so we have the following possibilities: 1. **$A_1$ has a pair of distinct eigenvalues:** Since $A_1\theta=\theta A_1$ we have that, on the canonical basis, $$A_1=\left(\begin{array}{cc} \beta & 0 \\ 0 & 0 \end{array}\right)\;\;\;\;\mbox{ or }\;\;\;\;A_1=\left(\begin{array}{cc} 0 & 0 \\ 0 & \beta \end{array}\right)\;\;\;\mbox{ and }\;\;\; \theta=\left(\begin{array}{cc} 1 & 0 \\ 0 & \lambda \end{array}\right).$$ Let us assume that the first case holds for $A_1$ since the analysis of the second case is analogous. By our hypothesis, we can write $\xi_1=a\mathbf{e}_1+b\mathbf{e}_2$ with $b\neq 0$, and hence $$\rho_t\Lambda_{-s}^{A_1}\xi_1=a\mathrm{e}^{t}\frac{1}{\beta}(\mathrm{e}^{-s\beta}-1)\mathbf{e}_1-bs\mathrm{e}^{t\lambda}\mathbf{e}_2.$$ Also, from the second and third equations in ([\[eqf\]](#eqf){reference-type="ref" reference="eqf"}) we have that $$\partial_1f=\partial_1f(\varphi_s^1)+\langle\partial_2f(\varphi_s^1), \rho_t\Lambda_s^{A_1}\xi_1\rangle_{\mathbb{R}^2}=\partial_1f(\varphi_s^1)-\langle\partial_2f, \rho_t\Lambda_{-s}^{A_1}\xi_1\rangle_{\mathbb{R}^2},$$ implying that $$\langle\partial_2f, \rho_t\Lambda_{-s}^{A_1}\xi_1\rangle_{\mathbb{R}^2}=\partial_1f(\varphi_s^1)-\partial_1f,$$ is bounded for $s\in\mathbb{R}$. As a consequence, $$\langle\partial_2f, \rho_t\Lambda_{-s}^{A_1}\xi_1\rangle_{\mathbb{R}^2}=a\mathrm{e}^{t}\frac{1}{\beta}(\mathrm{e}^{-s\beta}-1)\langle\partial_2f, \mathbf{e}_1\rangle_{\mathbb{R}^2}-bs\mathrm{e}^{t\lambda}\langle\partial_2f, \mathbf{e}_2\rangle_{\mathbb{R}^2},$$ is bounded and hence $$a\langle\partial_2f, \mathbf{e}_1\rangle_{\mathbb{R}^2}=0\hspace{.5cm}\mbox{ and }\langle\partial_2f, \mathbf{e}_2\rangle_{\mathbb{R}^2}=0.$$ If $a=0$ we get that $\xi_1\in\ker A_1=\mathbb{R}\cdot \mathbf{e}_2$. In particular, $\Lambda_t^{\theta}\xi_1\in\ker A_1$ and $$\langle\partial_2f, \mathbf{e}_2\rangle_{\mathbb{R}^2}=0\;\;\;\implies \;\;\;\langle\partial_2f, \Lambda_t^{\theta}\xi\rangle_{\mathbb{R}^2}=0.$$ Using the first equation in ([\[eqf\]](#eqf){reference-type="ref" reference="eqf"}) we get that $$0=\langle\partial_2f, A_1v+\Lambda_t^{\theta}\xi\rangle_{\mathbb{R}^2}=\langle\partial_2f, A_1v\rangle_{\mathbb{R}^2},$$ which gives us that $$\forall (t, v)\in G(\theta)\setminus(\{0\}\times\ker A_1), \;\;\;\;\langle\partial_2f, \mathbf{e}_1\rangle_{\mathbb{R}^2}=0,$$ and in particular $\partial_2f\equiv 0$ on $G(\theta)\setminus\mathcal{Z}_1$. Therefore, $\partial_1f=\sigma_2/\sigma_1=:a$ on $G(\theta)\setminus\mathcal{Z}_1$ implying, by the continuity of $f$ that $f(t, v)=at$ on $G(\theta)$ which implies the result. 2. **$A_1$ is nilpotent:** As in the previous case, the fact that $A_1$ and $\theta$ commutes gives us that, on the canonical basis, $$A_1=\left(\begin{array}{cc} 0 & \beta \\ 0 & 0 \end{array}\right)\;\;\;\mbox{ and }\;\;\; \theta=\left(\begin{array}{cc} 1 & \delta \\ 0 & 1 \end{array}\right), \;\delta\in\{0, 1\}\;\;\;\;\mbox{ or }\;\;\;\;$$ $$A_1=\left(\begin{array}{cc} 0 & 0 \\ \beta & 0 \end{array}\right)\;\;\;\mbox{ and }\;\;\; \theta=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right).$$ Let us as previously treat the first case. By the rank two assumption we have that $\xi_1=a\mathbf{e}_1+b\mathbf{e}_2$ with $b\neq 0$. Therefore, $$\rho_t\Lambda_{-s}^{A_1}\xi_1=\mathrm{e}^t\left(b\frac{s^2}{2}-(a+b\delta t)s\right)\mathbf{e}_1-\mathrm{e}^tbs\mathbf{e}_2,$$ and, as previously, $\partial_1f$ bounded on $G(\theta)\setminus\mathcal{Z}_1$ implies $$\forall s\in\mathbb{R}, \;\;\;\;\;\langle\partial_2f, \rho_t\Lambda_{-s}^{A_1}\xi_1\rangle_{\mathbb{R}^2}\;\;\;\mbox{ is bounded.}$$ Therefore, $$\langle\partial_2f, \rho_t\Lambda_{-s}^{A_1}\xi_1\rangle_{\mathbb{R}^2}=\mathrm{e}^t\left(b\frac{s^2}{2}-(a+b\delta t)s\right)\langle\partial_2f, \mathbf{e}_1\rangle_{\mathbb{R}^2}-\mathrm{e}^tbs\langle\partial_2f, \mathbf{e}_2\rangle_{\mathbb{R}^2}$$ is bounded for $s\in\mathbb{R}$ implying that $$\frac{\mathrm{e}^tb}{2}\langle\partial_2f, \mathbf{e}_1\rangle_{\mathbb{R}^2}=0\;\;\;\mbox{ and }\;\;\;\mathrm{e}^t\left((a+b\delta t)\langle\partial_2f, \mathbf{e}_1\rangle_{\mathbb{R}^2}+b\langle\partial_2f, \mathbf{e}_2\rangle_{\mathbb{R}^2}\right)=0.$$ Since by assumption $b\neq 0$, the previous is equivalenty to $$\langle\partial_2f, \mathbf{e}_1\rangle_{\mathbb{R}^2}=\langle\partial_2f, \mathbf{e}_2\rangle_{\mathbb{R}^2}=0,$$ showing that $\partial_2f\equiv 0$ on $G(\theta)\setminus\mathcal{Z}_1$. As previously, by continuity $f(t, v)=at$ and hence $\partial_2f\equiv 0$ on $G(\theta)$. $\Box$ ## Proof of Theorem [Theorem: 6](#main){reference-type="ref" reference="main"} {#proof-of-theorem-main} By the previous section, if $\psi=(f, g)$ is an isometry between rank two ARSs $\Sigma_1$ and $\Sigma_2$, then $\partial_2f\equiv 0$. As a consequence, if $$X=(0, \eta)\in \Delta_1\cap\left(\{0\}\times\mathbb{R}^2\right), \;\;\;\mbox{ with }\;\;\;\|X\|_{\Sigma_1, (0, 0)}=1,$$ we have that $$(d\psi)_{(t, v)}X^L(t, v)=\left(\begin{array}{cc} \partial_1f(t, v) & 0 \\ \partial_1g(t, v) & \partial_2g(t, v) \end{array}\right)\left(\begin{array}{c} 0 \\ \rho_t\eta \end{array}\right)=(0, \partial_2g(t, v)\rho_t\eta),$$ showing that $$(d\psi)_{(t, v)}X^L(t, v)\in \Delta_2(\psi(t, v))\cap\mathfrak{n}(\theta).$$ On other hand, $$(d\psi)_{(0, 0)}X\in\Delta_2(0, 0)\;\;\;\implies\;\;\;(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}X\in\Delta_2(\psi(t, v))\cap\mathfrak{n}(\theta).$$ Since $\psi$ is an isometry and the metrics on $\Delta_1$ and $\Delta_2$ are left-invariant, we get that $$\|(d\psi)_{(t, v)}X^L(t, v)\|_{\Sigma_2, (\psi(t, v))}=\|X^L(t, v)\|_{\Sigma_1, (t, v)}=\|X\|_{\Sigma_1, (0, 0)}=1,$$ and $$\|(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}X\|_{\Sigma_2, (\psi(t, v))}=\|(d\psi)_{(0, 0)}X\|_{\Sigma_2, (0, 0)}=\|X\|_{\Sigma_1, (0, 0)}=1.$$ The fact that $\dim(\Delta_2(\psi(t, v)))\cap\left(\{0\}\times\mathbb{R}^2\right)=1$ implies $$(d\psi)_{(t, v)}X^L(t, v)=\pm(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}X,$$ where the sign is constant on any connected component of $G(\theta)\setminus\mathcal{Z}_1$. An analogous analysis as the one done in the proof of Step 1 of the Fundamental Lemma implies that the sign is constant on the whole $G(\theta)$ and consequently $$(d\psi)_{(t, v)}X^L(t, v)=(dL_{\psi(t, v)})_{(0, 0)}(d\psi)_{(0, 0)}X.$$ By the Fundamental Lemma ([Lemma: 7](#fundamentallemma){reference-type="ref" reference="fundamentallemma"}) we conclude that $\psi$ is an automorphism of $G(\theta)$, concluding the proof. ## A classification result for rank two ARSs The results in the previous sections allow us to obtain a classification result for rank two simple ARSs. For any $\sigma\in\mathbb{R}^+$ we consider the subsets of $\mathfrak{g}(\theta)$ given by $$\alpha_1=\{(1, 0), (\sigma, \mathbf{e}_1)\}, \;\;\alpha_2=\{(1, 0), (\sigma, \mathbf{e}_2)\}\;\;\mbox{ and }\;\;\alpha_3=\{(1, 0), (\sigma, \mathbf{e}_1+\mathbf{e}_2)\}.$$ Define the simple ARSs of rank two $\Sigma^i_{\mathcal{X}, \sigma}=\{\mathcal{X}, \Delta^L_{i, \sigma}\}$ on $G(\theta)$, where 1. $\mathcal{X}$ is a rank two linear vector field on $G(\theta)$; 2. $\alpha_i$ is an orthonormal basis of $\Delta_{i, \sigma}$. Denote by $\mathcal{E}_{\theta}$ the set of all rank two simple ARS on $G(\theta)$ and consider the sets $$\mathcal{E}^i_{\theta}=\{\Sigma\in \mathcal{E}_{\theta}; \;\mathrm{Iso}(\Sigma; \Sigma^i_{\mathcal{X}, \sigma})_0\neq\emptyset\;\mbox{ for some }\Sigma^i_{\mathcal{X}, \sigma}\},$$ that is, $\mathcal{E}^i_{\theta}$ is the set of rank two ARSs on $G(\theta)$ that are isometric to some of the ARSs $\Sigma^i_{\mathcal{X}, \sigma}$. **Theorem: 11**. *Up to a reescaling, it holds that* - *$\mathcal{E}_{\theta}=\mathcal{E}^1_{\theta}$ if $\theta\in\left\{\left(\begin{array}{cc} 1 & 0\\0 & 1 \end{array}\right), \; \left(\begin{array}{cc} \gamma & -1\\1 & \gamma \end{array}\right), \;\gamma\in\mathbb{R}\right\}$;* - *$\mathcal{E}_{\theta}=\mathcal{E}^1_{\theta}\,\dot{\cup}\,\mathcal{E}^3_{\theta}$ if $\theta\in\left\{\left(\begin{array}{cc} 1 & 1\\0 & 1 \end{array}\right), \; \left(\begin{array}{cc} 1 & 0\\0 & -1 \end{array}\right)\right\}$;* - *$\mathcal{E}_{\theta}=\mathcal{E}^1_{\theta}\,\dot{\cup}\,\mathcal{E}^2_{\theta}\,\dot{\cup}\,\mathcal{E}^3_{\theta}$ if $\theta\in\left\{\left(\begin{array}{cc} 1 & 0\\0 & \gamma \end{array}\right), \gamma\in(-1, 1)\right\}$.* **Proof:** Since, by Theorem [Theorem: 6](#main){reference-type="ref" reference="main"}, the only isometries between rank two ARS are automorphisms, we only have to show that any given ARS $\Sigma$ is isometric to some ARS in $\mathcal{E}^i_{\theta}$ for $i=1, 2, 3$ and that for $i\neq j$ we have that $\mathcal{E}^i_{\theta}\cap \mathcal{E}^j_{\theta}=\emptyset$ if $\theta$ is in the cases (ii) or (iii). Let us consider $\Sigma=\{\mathcal{X}, \Delta^L\}$ be a rank two ARS. By Proposition [Proposition: 4](#difeo){reference-type="ref" reference="difeo"} the ARS $\Sigma$ is isometric to an ARS whose distribution contains $(1, 0)$. Consequently, we can assume w.l.o.g. that $(1, 0)\in\Delta$. By reescaling the norm on $\Delta$ if necessary we can assume that $\|(1, 0)\|_{\Sigma, (0, 0)}=1$. Choose $(\sigma, \eta)\in \Delta$ such that $\sigma>0$ and $\{(1, 0), (\sigma, \eta)\}$ is an orthonormal basis of $\Delta$. Note that, in this case $$(0, \eta )=-\sigma (1, 0)+ (\sigma, \eta)\hspace{.5cm}\implies\hspace{.5cm} l_{\Delta}=\mathbb{R}\cdot\eta.$$ Write $\eta=(x, y)$ and consider the following cases: **Case 1:** $\theta\in\left\{\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right), \;\left(\begin{array}{cc} \gamma & -1\\ 1 & \gamma \end{array}\right), \;\gamma\in\mathbb{R}\right\}$. In this case, by considering $P=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right)$ we have that $$P\theta=\theta P, \hspace{.5cm}\det P=x^2+y^2\neq 0\hspace{.5cm}\mbox{ and }\hspace{.5cm} P\mathbf{e}_1=\eta.$$ The automorphism $\phi(t, v)=(t, Pv)$ is an isometry between $\Sigma^1_{\mathcal{X}_{\phi}, \sigma}$ and $\Sigma$. **Case 2:** $\theta=\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right)$. In this case, by considering $$P_1=\left(\begin{array}{cc} x & 0\\ 0 & x \end{array}\right)\;\;\; \mbox{ if }\;\;\;y=0\;\;\;\mbox{ or }\;\;\; P_3=\left(\begin{array}{cc} y & x-y\\ 0 & y \end{array}\right)\;\mbox{ if }\;y\neq 0,$$ we get that $P_i\theta=\theta P_i$, $\det P_i\neq 0$ and $$P_1\mathbf{e}_1=\eta\;\;\mbox{ if }\;y=0\;\;\mbox{ and }\;\;P_3(\mathbf{e}_1+\mathbf{e}_2)=\eta\;\;\mbox{ if }\;y\neq 0.$$ Therefore, if $y=0$ the automorphism $\phi_1(t, v)=(t, P_1v)$ is an isometry between $\Sigma^1_{\mathcal{X}_{\phi_1}, \sigma}$ and $\Sigma$, and if $y\neq 0$ the automorphism $\phi_3(t, v)=(t, P_3v)$ is an isometry between $\Sigma^3_{\mathcal{X}_{\phi_3}, \sigma}$ and $\Sigma$. **Case 3:** $\theta=\left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right)$. In this case, we consider $$P_1=\left(\begin{array}{cc} x & 0\\ 0 & x \end{array}\right)\mbox{ if }y=0, \;P_2=\left(\begin{array}{cc} 0 & y\\ y & 0 \end{array}\right)\mbox{ if }x=0\mbox{ or } P_3=\left(\begin{array}{cc} x & 0\\ 0 & y \end{array}\right)\;\mbox{ if }\;xy\neq 0.$$ It holds that $P_i\theta=\theta P_i$ for $i=1 \mbox{ or } 3$ and $P_2\theta=-\theta P_2$, $\det P_i\neq 0$ and $$P_1\mathbf{e}_1=\eta\;\;\mbox{ if }\;y=0, \;\;P_2\mathbf{e}_1=\eta\;\;\mbox{ if }\;y=0\;\;\mbox{ and }\;\;P_3(\mathbf{e}_1+\mathbf{e}_2)=\eta\;\;\mbox{ if }\;xy\neq 0.$$ As in the previous case, if $y=0$ the automorphism $\phi_1(t, v)=(t, P_1v)$ (*resp. if $x=0$ the automorphism $\phi_2(t, v)=(-t, P_2v)$*) is an isometry between $\Sigma^1_{\mathcal{X}_{\phi_1}, \sigma}$ (*resp. $\Sigma^1_{\mathcal{X}_{\phi_2}, \sigma}$*) and $\Sigma$, and $\phi_3(t, v)=(t, P_3v)$ is an isometry between $\Sigma^3_{\mathcal{X}_{\phi_3}, \sigma}$ and $\Sigma$ if $xy\neq 0$. **Case 4:** $\theta=\left(\begin{array}{cc} 1 & 0\\ 0 & \lambda \end{array}\right), \;\;\lambda\in (1, -1)$. In this case, we consider $$P_1=\left(\begin{array}{cc} x & 0\\ 0 & x \end{array}\right)\hspace{.3cm}\mbox{ if }\hspace{.3cm}y=0,\hspace{.3cm} P_2=\left(\begin{array}{cc} y & 0\\ 0 & y \end{array}\right)\hspace{.3cm}\mbox{ if }\hspace{.3cm}x=0\hspace{.3cm}\mbox{ or } \hspace{.3cm}P_3=\left(\begin{array}{cc} x & 0\\ 0 & y \end{array}\right)\hspace{.3cm}\mbox{ if }\hspace{.3cm}xy\neq 0.$$ Again by construction $P_i\theta=\theta P_i$, $\det P_i\neq 0$ and $$P_1\mathbf{e}_1=\eta\;\;\mbox{ if }\;y=0, \;\;P_2\mathbf{e}_2=\eta\;\;\mbox{ if }\;x=0\;\;\mbox{ and }\;\;P_3(\mathbf{e}_1+\mathbf{e}_2)=\eta\;\;\mbox{ if }\;xy\neq 0,$$ and the respectively automorphisms $\phi_i(t, v)=(t, P_iv)$ are isometries between $\Sigma_{\mathcal{X}_{\phi_i},, \sigma}$ and $\Sigma$, for $i=1, 2, 3$. Since the previous cases cover all the possibilities, we have that $\mathcal{E}$ is in fact decomposed by the classes $\mathcal{E}_{\theta}^i$ as given in items (i), (ii) and (iii). The only thing that remains to show is that, in cases (ii) and (iii) we have that $\mathcal{E}^i_{\theta}\cap \mathcal{E}^j_{\theta}=\emptyset$ for $i\neq j$. Since both cases are analogous, let us show case (ii). In this case, if $\mathcal{E}^1_{\theta}\cap\mathcal{E}^3_{\theta}\neq\emptyset$, there exists rank two linear vector fields $\mathcal{X}_1, \mathcal{X}_3$ and positive real numbers $\sigma_1, \sigma_2$ such that $\Sigma^1_{\mathcal{X}_1, \sigma_1}$ and $\Sigma^3_{\mathcal{X}_3, \sigma_3}$ are isometrics. However, since $\Sigma^1_{\mathcal{X}_1, \sigma_1}$ and $\Sigma^3_{\mathcal{X}_3, \sigma_3}$ are, by definition, rank two ARSs, Theorem [Theorem: 6](#main){reference-type="ref" reference="main"} implies that $$\mathrm{Iso}\left(\Sigma^1_{\mathcal{X}_1, \sigma_1}; \Sigma^3_{\mathcal{X}_3, \sigma_3}\right)_0\subset\mathrm{Aut}(G(\theta)).$$ Therefore, any $\psi\in \mathrm{Iso}\left(\Sigma^1_{\mathcal{X}_1, \sigma_1}; \Sigma^3_{\mathcal{X}_3, \sigma_3}\right)_0$ satisfies $$(d\psi)_{(0, 0)}=\left(\begin{array}{cc} \varepsilon & 0\\ \eta & P \end{array}\right), \;\;\mbox{ with }\;\; P\theta=\varepsilon\theta P,$$ which implies, in particular, that $$(d\psi)_{(0, 0)}\left(\Delta_{1, \sigma}\cap\mathfrak{n}(\theta)\right)=\Delta_{3, \sigma}\cap\mathfrak{n}(\theta)\hspace{.5cm}\mbox{ and hence }\hspace{.5cm}P\mathbf{e}_1\in\mathbb{R}\cdot(\mathbf{e}_1+\mathbf{e}_2).$$ However, by the hypothesis on $\theta$, the subspace $\mathbb{R}\cdot \mathbf{e}_1$ is a one dimensional eigenspace of $\theta$. Since $P\theta=\varepsilon \theta P$ we get that $$P\mathbf{e}_1\in\mathbb{R}\cdot \mathbf{e}_1 \;\mbox{ if }\;\varepsilon=1\hspace{.5cm}\mbox{ and }\hspace{.5cm}P\mathbf{e}_1\in\mathbb{R}\cdot \mathbf{e}_2 \;\mbox{ if }\;\varepsilon=-1,$$ which contradicts $P\mathbf{e}_1\in\mathbb{R}\cdot(\mathbf{e}_1+\mathbf{e}_2)$. Therefore, $\mathcal{E}^1_{\theta}\cap\mathcal{E}^3_{\theta}=\emptyset$ as stated. $\Box$ In the notation of the previous result, let us note that $$(0, \eta)=-\sigma(1, 0)+(\sigma, \eta)\hspace{.5cm}\implies\hspace{.5cm}\|(0, \eta)\|_{\Sigma, (0, 0)}=1+\sigma^2.$$ In particular, $\|(0, \eta)\|_{\Sigma, (0, 0)}=1$ if and only if $\sigma=0$. Consequently, the metric on $\Delta$ is Euclidean if and only if $\sigma=0$. 99 A. Agrachev, D. Barilari and U. Boscain, *A comprehensive introduction to sub-Riemannian geometry*, Cambridge University Press, 2019. A. Agrachev, U. Boscain, and M. Sigalotti, *A Gauss-Bonnet like formula on twodimensional almost-Riemannian manifolds*, Discrete Contin. Dyn. Syst., 20 (2008), pp. 801--822. A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi, and M. Sigalotti, *Two dimensional almost-Riemannian structures with tangency points*, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 27 (2010), pp. 793--807. V. Ayala and A. Da Silva, *On the characterization of the controllability property for linear control systems on nonnilpotent, solvable threedimensional Lie groups.* Journal of Differential Equations, 266 No 12 (2019), 8233-8257 V. Ayala, A. Da Silva and D. A.G. Hernández, *Almost-Riemannian structures on nonnilpotent, solvable 3D Lie groups.* Journal of Geometry and Physics, 192 (2023), 1-15 V. Ayala and J. Tirao, *Linear control systems on Lie groups and Controllability*, Eds. G. Ferreyra et al., Amer. Math. Soc., Providence, RI, 1999. V. Ayala and P. Jouan. *Almost-Riemannian Geometry on Lie groups* SIAM Journal on Control and Optimization 54 No 5 (2016), 2919-2947. B. Bonnard, J.-B. Caillau, R. Sinclair, and M. Tanaka. *Conjugate and cut loci of a two-sphere of revolution with application to optimal control* Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), pp. 1081--1098. B. Bonnard, G. Charlot, R. Ghezzi, and G. Janin. *The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry,* J. Dyn. Control Syst., 17 (2011), pp. 141--161. U. Boscain and M. Sigalotti. *High-order angles in almost-Riemannian geometry*, Sémin. Théor. Spectr. Géom. 25, Université Grenoble I, 2008, pp. 41--54. U. Boscain, G. Charlot, M. Gaye, and P. Mason. *Local Properties of Almost-Riemannian Structures in Dimension 3*, Discrete Contin. Dyn. Syst., 35 (2015), pp. 4115-4147. U. Boscain, G. Charlot, R. Ghezzi, and M. Sigalotti. *Lipschitz classification of almost-Riemannian distances on compact oriented surfaces*, J. Geom. Anal., 23 (2013), pp. 438--455. U. Boscain, G. Charlot, and R. Ghezzi. *Normal forms and invariants for 2-dimensional almost-Riemannian structures*, Differential Geom. Appl., 31 (2013), pp. 41--62. U. Boscain, and C. Laurent. *The Laplace-Beltrami operator in almost-Riemannian Geometry*, Ann. Inst. Fourier, 63 (2013), pp. 1739--1770. V. V. Grushin, *A certain class of hypoelliptic operators*, Mat. Sb.(N.S.), 83 (1970), pp. 456--473. P. Jouan, G. Zsigmond and V. Ayala. *Isometries of almost-Riemannian structures on Lie groups.* Differential Geometry and its Applications 61 (2018), 59 - 81. S. B. Myers and N. E. Steenrod, *The Group of Isometries of a Riemannian Manifold*, Annals of Mathematics 40, No.2 (1939), 400-416. A. L. Onishchik and E. B. Vinberg, *Lie groups and Lie algebras III - Structure of Lie groups and Lie algebras*, Springer Verlag, Berlin, 1994. T. Takasu, *Generalized Riemannian geometry I*, J. Yokohama Municipal Univ. Ser. D, Vol. 5, (1956). [^1]: Supported by Proyecto Fondecyt n$% %TCIMACRO{\U{b0}}% %BeginExpansion {{}^\circ}% %EndExpansion$ 1190142. Conicyt, Chile. [^2]: Supported by Proyecto UTA Mayor Nº 4768-23 [^3]: Supported by Capes Finance Code 001

arxiv_math

--- abstract: | In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: i) continuous and ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then, recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz's integral rule), the law of total probability, and mathematical induction. Finally, this work generalizes previous results in the literature. address: Department of Electrical and Computer Engineering, University of Cyprus, 1678 Nicosia, Cyprus author: - Christos N. Efrem title: Recurrence relations for the joint distribution of the sum and maximum of independent random variables --- # Introduction ## Related Work Chow and Teugels [@Chow-Teugels], Anderson and Turkman [@Anderson-Turkman1991; @Anderson-Turkman1993], Hsing [@Hsing_a; @Hsing_b], Ho and Hsing [@Ho-Hsing], McCormick and Qi [@McCormick-Qi], and Peng and Nadarajah [@Peng-Nadarajah] studied the limiting joint distribution and asymptotic independence of the sum and maximum of identically distributed random variables. In addition, the exact (non-asymptotic) joint distribution of the sum and maximum of independent random variables was investigated by Qeadan, Kozubowski and Panorska [@Qeadan-Kozubowski-Panorska], and by Arendarczyk, Kozubowski and Panorska [@Arendarczyk-Kozubowski-Panorska_a] for identically and non-identically distributed exponential variables, respectively. A key recursive formula for independent and identically distributed (i.i.d.) nonnegative continuous random variables (with an arbitrary PDF) was derived by Qeadan, Kozubowski and Panorska [@Qeadan-Kozubowski-Panorska Eq. (2.18)]. Another recursive formula was given in [@Arendarczyk-Kozubowski-Panorska_a Eq. (23)] by appropriately partitioning the domain of the joint CDF. Arendarczyk, Kozubowski and Panorska also investigated the case of Pareto-dependent and identically distributed random variables [@Arendarczyk-Kozubowski-Panorska_b]. Moreover, recurrence relations were used for the distribution of the sum of independent continuous/discrete random variables [@Moschopoulos; @Woodward-Palmer; @Butler-Stephens] as well as in order statistics [@Balakrishnan; @Balakrishnan-Bendre-Malik; @David-Nagaraja]. ## Main Contributions & Outline Despite extensive work in the past, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, random variables remains almost unexplored (except for the special case of exponential variables [@Arendarczyk-Kozubowski-Panorska_a]). The purpose of this work is to provide a new approach to this problem in its full generality, not only for continuous but also for discrete random variables with arbitrary distributions. In particular, Section [2](#section:Continuous Random Variables){reference-type="ref" reference="section:Continuous Random Variables"} deals with the joint CDF and PDF in the continuous case, and presents a potential application of peak-to-average ratio in communication systems. Subsequently, Section [3](#section:Discrete Random Variables){reference-type="ref" reference="section:Discrete Random Variables"} studies the joint CDF and PMF in the discrete case, and provides a similar result for the peak-to-average ratio. Furthermore, Appendices [4](#appendix:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="appendix:Partial Differentiation Under the Integral Sign"} and [5](#appendix:Dirac Delta Function){reference-type="ref" reference="appendix:Dirac Delta Function"} contain information about partial differentiation under the integral sign and the Dirac delta function, respectively. All proofs are simple and elegant using only elementary techniques, such as the law of total probability, Leibniz's integral rule, and mathematical induction. Last but not least, this paper includes existing results as special cases. ## Notation $a \coloneqq b$ means that "$a$ is by definition equal to $b$". $\mathbb{R}$ is the set of real numbers and $\mathbb{Z}$ is the set of integers. $\mathbb{R}_+ \coloneqq\{x \in \mathbb{R}: x \geq 0\}$ represents the set of nonnegative real numbers. The set of natural numbers excluding/including zero is denoted by $\mathbb{N}\coloneqq\{1,2,\dots\}$ and $\mathbb{N}_0 \coloneqq\{0,1,\dots\}$, respectively. $\lfloor x \rfloor \coloneqq\max \{k \in \mathbb{Z}: k \leq x \}$ denotes the floor of $x$ (i.e., the greatest integer less than or equal to $x$), while $\lceil x \rceil \coloneqq\min \{k \in \mathbb{Z}: k \geq x \}$ is the ceiling of $x$ (i.e., the least integer greater than or equal to $x$). The symbols $\lor$ and $\land$ stand for logical disjunction (OR) and logical conjunction (AND), respectively. In addition, throughout the paper we consider only real-valued functions. $\delta(x)$ stands for the Dirac delta function/distribution (for more details see Appendix [5](#appendix:Dirac Delta Function){reference-type="ref" reference="appendix:Dirac Delta Function"}), while $H(x)$ is the Heaviside step function defined by $$H(x) \coloneqq\left\{ \begin{array}{ll} 1, & \text{if}\ x \geq 0 , \\ 0, & \text{if}\ x < 0 . \\ \end{array} \right.$$ Note that the derivative of the Heaviside step function is equal to the Dirac delta function, i.e., $\od{}{x} H(x) = \delta(x)$. We also adopt the Iverson bracket notation, that is, $$[Q] \coloneqq\left\{ \begin{array}{ll} 1, & \text{if statement $Q$ is true} , \\ 0, & \text{if statement $Q$ is false} . \\ \end{array} \right.$$ Using this notation, the Heaviside step function can be expressed as $H(x) = [x \geq 0]$. Let $(\mathcal{X},\Sigma,\mu)$ be a measure space, where $\mathcal{X} \subseteq \mathbb{R}^n$, $\Sigma$ is a $\sigma$-algebra over $\mathcal{X}$, and $\mu : \Sigma \to \mathbb{R}\cup \{-\infty,+\infty\}$ is a measure function. Then, $\displaystyle{f(\mathbf{x}) \mathop{=}^{\text{a.e.}} g(\mathbf{x})}$ means that $f(\mathbf{x}) = g(\mathbf{x})$ almost everywhere on $\mathcal{X}$, i.e., there exists a (measurable) set $\mathcal{S} \in \Sigma$ with $\mu(\mathcal{S}) = 0$ such that $f(\mathbf{x}) = g(\mathbf{x})$ for all $\mathbf{x} \in \mathcal{X} \setminus \mathcal{S}$. For the purposes of this paper, we use the Lebesgue measure (in the geometric sense), that is, length in $\mathbb{R}$, area in $\mathbb{R}^2$, and volume in $\mathbb{R}^3$. Furthermore, we make the following conventions: i) $\sum_{k=l}^{u} {a_k} = 0$ and $\prod_{k=l}^{u} {a_k} = 1$, whenever $u<l$, and ii) when a function $f(\mathbf{x})$ is defined on a set/domain $\mathcal{D} \subseteq \mathbb{R}^n$, we implicitly assume that $f(\mathbf{x}) = 0$ elsewhere (i.e., for all $\mathbf{x} \notin \mathcal{D}$); in fact, we can equivalently define an extended function $\widetilde{f}(\mathbf{x})$ on $\mathbb{R}^n$, that is, $\widetilde{f}(\mathbf{x}) \coloneqq f(\mathbf{x}) [\mathbf{x} \in \mathcal{D}]$. # Continuous Random Variables {#section:Continuous Random Variables} We consider a sequence of *independent, not necessarily identically distributed,* nonnegative (absolutely) continuous random variables $X_i \geq 0$, $i \in \mathbb{N}$, with PDF $f_i(x)$ and CDF $F_i(x) \coloneqq\operatorname{P}(X_i \leq x) = \int_0^x {f_i(x') \dif x'}$, $\forall x \in \mathbb{R}_+$. For every $i \in \mathbb{N}$, we assume that $f_i(x)$ is *continuous* on $\mathbb{R}_+$, therefore $f_i(x) = \od{}{x} F_i(x)$. Observe that $f_i(x)$ is also *bounded* on $\mathbb{R}_+$, since $f_i(x) \geq 0$ and $\int_0^{\infty} {f_i(x) \dif x} = 1$. Hence, $f_i(x)$ is *bounded and piecewise continuous on $\mathbb{R}$*, with possible discontinuity at $x=0$, because $f_i(x) = 0$ for all $x<0$. Now, let us define the following random variables for all $n \in \mathbb{N}$ $$\label{equation:sum and maximum} Y_n \coloneqq\sum_{i=1}^n {X_i} = X_1 + \cdots + X_n , \qquad Z_n \coloneqq\max_{1 \leq i \leq n} {X_i} = \max (X_1,\dots,X_n) .$$ In general, $Y_n$ and $Z_n$ are dependent random variables. For the sake of convenience, we also define the following sets $$\mathcal{D}_n \coloneqq\{ (y,z) \in \mathbb{R}_+^2 : z \leq y \leq n z \} , \quad \forall n \in \mathbb{N}.$$ Subsequently, we would like to compute the CDF and PDF of the random pair $(Y_n,Z_n)$, i.e., $G_n (y,z) \coloneqq\operatorname{P}(Y_n \leq y,Z_n \leq z) = \int_0^z {\int_0^y g_n(y',z') \dif y' \dif z'}$, $\forall (y,z) \in \mathbb{R}_+^2$, and $g_n(y,z) = \md{}{2}{y}{}{z}{} G_n(y,z)$, $\forall (y,z) \in \mathcal{D}_n$. Note that $0 \leq Z_n \leq Y_n \leq n Z_n$, therefore $g_n(y,z) = 0$ for all $(y,z) \notin \mathcal{D}_n$. ## Recurrence Relation for the Joint CDF **Theorem 1**. *The joint CDF of $Y_n$ and $Z_n$, defined by [\[equation:sum and maximum\]](#equation:sum and maximum){reference-type="eqref" reference="equation:sum and maximum"}, is given by the following recurrence relation $$\label{equation:recurrence relation of joint CDF} \begin{split} G_n (y,z) & = \int_0^z {G_{n-1}(y - x,z) f_n(x) \dif x} = \int_0^{\min(y,z)} {G_{n-1}(y - x,z) f_n(x) \dif x} , \\ & \:\, \quad \forall n \in \mathbb{N}\setminus \{1\} , \; \forall (y,z) \in \mathbb{R}_+^2 , \end{split}$$ with initial condition $$\label{equation:initial condition for joint CDF} G_1 (y,z) = F_1 (\min(y,z)) , \quad \forall (y,z) \in \mathbb{R}_+^2 .$$ In addition, $G_n (y,z)$ is continuous on $\mathbb{R}_+^2$ for all $n \in \mathbb{N}$.* *Proof.* Observe that $Y_n = Y_{n-1} + X_n$ and $Z_n = \max(Z_{n-1},X_n)$. Therefore, we can write $$\begin{split} G_n (y,z) & = \operatorname{P}(Y_{n-1} + X_n \leq y, \max(Z_{n-1},X_n) \leq z) \\ & = \operatorname{P}(Y_{n-1} + X_n \leq y, Z_{n-1} \leq z, X_n \leq z) \\ & \mathop{=}^{(a)} \int_0^\infty {\operatorname{P}(Y_{n-1} + X_n \leq y, Z_{n-1} \leq z, X_n \leq z \;|\; X_n = x) f_n(x) \dif x} \\ & \mathop{=}^{(b)} \int_0^\infty {\operatorname{P}(Y_{n-1} + x \leq y, Z_{n-1} \leq z, x \leq z) f_n(x) \dif x} \\ & = \int_0^z {\operatorname{P}(Y_{n-1} \leq y - x, Z_{n-1} \leq z) f_n(x) \dif x} \\ & = \int_0^z {G_{n-1}(y - x,z) f_n(x) \dif x} = \int_0^{\min(y,z)} {G_{n-1}(y - x,z) f_n(x) \dif x} , \end{split}$$ where equality $(a)$ is due to the law/formula of total probability, equality $(b)$ follows from the fact that $X_n$ is independent of the pair $(Y_{n-1},Z_{n-1})$, and the last equality is because $G_{n-1}(y - x,z) = 0$ when $x>y$. Regarding the initial condition, $$G_1 (y,z) = \operatorname{P}(Y_1 \leq y, Z_1 \leq z) = \operatorname{P}(X_1 \leq y, X_1 \leq z) = \operatorname{P}(X_1 \leq \min(y,z)) = F_1 (\min(y,z)).$$ Now, it remains to show the continuity of $G_n(y,z)$ on $\mathbb{R}_+^2$ for all $n \in \mathbb{N}$. We will use mathematical induction on $n$. Basis step: For $n=1$, $G_1 (y,z) = F_1 (\min(y,z))$ is continuous on $\mathbb{R}_+^2$ since it is the composition of two continuous functions, $F_1(\cdot)$ on $\mathbb{R}_+$ and $\min(\cdot,\cdot)$ on $\mathbb{R}_+^2$. Inductive step: Suppose that $G_{k-1}(y,z)$ is continuous on $\mathbb{R}_+^2$, for some arbitrary $k \in \mathbb{N}\setminus \{1\}$ (inductive hypothesis). Then, according to [\[equation:recurrence relation of joint CDF\]](#equation:recurrence relation of joint CDF){reference-type="eqref" reference="equation:recurrence relation of joint CDF"}, $G_k (y,z) = \int_0^{\min(y,z)} {G_{k-1}(y - x,z) f_n(x) \dif x}$ is continuous on $\mathbb{R}_+^2$ because $G_{k-1}(y - x,z) f_n(x)$ is continuous for $0 \leq x \leq \min(y,z)$, as the product of two continuous functions, $\min(\cdot,\cdot)$ is continuous on $\mathbb{R}_+^2$ and the integral of a continuous function is continuous as well. ◻ Interestingly, a similar recursive formula was given by Arendarczyk, Kozubowski and Panorska after dividing the domain of the joint CDF into several regions [@Arendarczyk-Kozubowski-Panorska_a Eq. (23)]. Furthermore, we can derive an exact formula (a multiple integral) for the joint CDF. **Proposition 2**. *The joint CDF of $Y_n$ and $Z_n$ can be expressed as an $(n-1)$-dimensional integral, that is, $$\label{equation:exact formula CDF} \begin{split} G_n (y,z) & = \bigintsss_{x_n=0}^z \cdots \bigintsss_{x_2=0}^z G_1 \left( {y - \sum_{i=2}^n {x_i}},z \right) \prod_{i=2}^n {f_i(x_i) \dif x_i} , \\ & \:\, \quad \forall n \in \mathbb{N}\setminus \{1\} , \; \forall (y,z) \in \mathbb{R}_+^2 , \end{split}$$ where $G_1(y,z)$ is given by [\[equation:initial condition for joint CDF\]](#equation:initial condition for joint CDF){reference-type="eqref" reference="equation:initial condition for joint CDF"}.* *Proof.* Based on the recurrence relation [\[equation:recurrence relation of joint CDF\]](#equation:recurrence relation of joint CDF){reference-type="eqref" reference="equation:recurrence relation of joint CDF"}, we give a proof by induction on $n$. Basis step: For $n=2$, [\[equation:recurrence relation of joint CDF\]](#equation:recurrence relation of joint CDF){reference-type="eqref" reference="equation:recurrence relation of joint CDF"} yields $G_2 (y,z) = \int_0^z {G_1(y - x,z) f_2(x) \dif x} = \int_0^z {G_1(y - x_2,z) f_2(x_2) \dif x_2}$. Inductive step: Suppose that $G_{k-1} (y,z) = \bigintsss_{x_{k-1}=0}^z \cdots \bigintsss_{x_2=0}^z G_1 \left( {y - \sum_{i=2}^{k-1} {x_i}},z \right) \prod_{i=2}^{k-1} {f_i(x_i) \dif x_i}$, for some arbitrary integer $k \geq 3$ (inductive hypothesis). Then, from [\[equation:recurrence relation of joint CDF\]](#equation:recurrence relation of joint CDF){reference-type="eqref" reference="equation:recurrence relation of joint CDF"} we obtain $$\begin{split} G_k (y,z) & = \int_0^z {G_{k-1}(y - x,z) f_k(x) \dif x} \\ & = \bigintsss_0^z \bigintsss_{x_{k-1}=0}^z \cdots \bigintsss_{x_2=0}^z G_1 \left( {y - x - \sum_{i=2}^{k-1} {x_i}},z \right) \left( \prod_{i=2}^{k-1} {f_i(x_i) \dif x_i} \right) {f_k(x) \dif x} \\ & = \bigintsss_{x_k=0}^z \cdots \bigintsss_{x_2=0}^z G_1 \left( {y - \sum_{i=2}^k {x_i}},z \right) \prod_{i=2}^k {f_i(x_i) \dif x_i} . \end{split}$$ Hence, [\[equation:exact formula CDF\]](#equation:exact formula CDF){reference-type="eqref" reference="equation:exact formula CDF"} is true for all integers $n \geq 2$. ◻ ## Recurrence Relation for the Joint PDF **Theorem 3**. *The joint PDF of $Y_n$ and $Z_n$, defined by [\[equation:sum and maximum\]](#equation:sum and maximum){reference-type="eqref" reference="equation:sum and maximum"}, is given by the following recurrence relation $$\label{equation:recurrence relation of joint PDF} \begin{split} g_n (y,z) & = f_n(z) \int_0^z {g_{n-1} (y - z,x) \dif x} + \int_0^z {f_n(x) g_{n-1} (y - x,z) \dif x} \\ & = f_n(z) \int_{\tfrac{y-z}{n-1}}^{\min(y-z,z)} {g_{n-1} (y - z,x) \dif x} + \int_{\max(y-(n-1)z,0)}^{\min(y-z,z)} {f_n(x) g_{n-1} (y - x,z) \dif x} , \\ & \:\, \quad \forall n \in \mathbb{N}\setminus \{1\} , \; \forall (y,z) \in \mathcal{D}_n , \end{split}$$ with initial condition $$\label{equation:initial condition for joint PDF} g_1 (y,z) = f_1 (y) \delta(y - z) = f_1 (z) \delta(y - z) , \quad \forall (y,z) \in \mathcal{D}_1 .$$ Furthermore, $g_n (y,z)$ is continuous on its domain $\mathcal{D}_n$, for all $n \in \mathbb{N}\setminus \{1\}$.* *Proof.* For all $n \in \mathbb{N}$, we know that $g_n(y,z) = 0$ for all $(y,z) \notin \mathcal{D}_n$, due to the definition of $Y_n$ and $Z_n$. Hence, the determination of $g_n(y,z)$ on $\mathcal{D}_n$ is sufficient for its full definition. In addition, recall that the sum, product, and composition of finitely many continuous functions are all continuous; the integral of a continuous function is continuous as well. First, let us prove the initial condition [\[equation:initial condition for joint PDF\]](#equation:initial condition for joint PDF){reference-type="eqref" reference="equation:initial condition for joint PDF"}. $G_1 (y,z)$, given by [\[equation:initial condition for joint CDF\]](#equation:initial condition for joint CDF){reference-type="eqref" reference="equation:initial condition for joint CDF"}, can be expressed using the Heaviside step function as follows $$G_1 (y,z) = F_1 (\min(y,z)) \mathop{=}^{\text{a.e.}} F_1 (y) H(z - y) + F_1 (z) H(y - z) ,$$ where the last equality holds almost everywhere on $\mathbb{R}_+^2$, except on the set $\{(y,z) \in \mathbb{R}_+^2 : \break y=z \}$ which has Lebesgue measure zero (in the geometric sense of area). As a result, $$\begin{split} {\dpd{}{z}{} G_1 (y,z)} & \mathop{=}^{\text{a.e.}} F_1 (y) \delta(z - y) + f_1 (z) H(y - z) - F_1 (z) \delta(y - z) \\ & = F_1 (z) \delta(z - y) - F_1 (z) \delta(z - y) + f_1 (z) H(y - z) \\ & = f_1 (z) H(y - z) , \end{split}$$ and $${\dmd{}{2}{y}{}{z}{} G_1(y,z)} \mathop{=}^{\text{a.e.}} f_1 (z) \delta(y - z) = f_1 (y) \delta(y - z) .$$ In the above equations, we have used the fact that $F_1(x)$ and $f_1(x)$ are bounded and piecewise continuous on $\mathbb{R}$ (see also Appendix [5](#appendix:Dirac Delta Function){reference-type="ref" reference="appendix:Dirac Delta Function"}). Since a PDF is almost unique when it exists, meaning that any two such PDFs coincide almost everywhere, we can choose $g_1 (y,z) = f_1 (z) \delta(y - z) = f_1 (y) \delta(y - z)$ for all $(y,z) \in \mathbb{R}_+^2$ or, equivalently, for all $(y,z) \in \mathcal{D}_1 = \{ (y,z) \in \mathbb{R}_+^2 : y = z \}$, since $g_1(y,z) = 0$ for $y \neq z$. We can also verify that $$\begin{split} \int_0^z {\int_0^y g_1(y',z') \dif y' \dif z'} & = \int_0^z {f_1(z') \int_0^y \delta(y' - z') \dif y' \dif z'} = \int_0^z {f_1(z') \int_{-z'}^{y-z'} \delta(w) \dif w \dif z'}\\ & = \int_0^z {f_1(z') [-z' \leq 0 \leq y-z'] \dif z'} = \int_0^z {f_1(z') [0 \leq z' \leq y] \dif z'} \\ & = \int_0^{\min(y,z)} {f_1(z') \dif z'} = F_1 (\min(y,z)) = G_1 (y,z) . \end{split}$$ Next, we will prove [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} and the continuity of $g_n (y,z)$ on its domain $\mathcal{D}_n$, for all $n \in \mathbb{N}\setminus \{1\}$, using induction on $n$. Basis step: For $n=2$ and by exploiting [\[equation:recurrence relation of joint CDF\]](#equation:recurrence relation of joint CDF){reference-type="eqref" reference="equation:recurrence relation of joint CDF"}, we obtain $$\begin{split} G_2(y,z) &= \int_0^z {G_1(y - x,z) f_2(x) \dif x} = \int_0^z {F_1 (\min(y-x,z)) f_2(x) \dif x} \\ & = F_1(z) \int_0^{y-z} {f_2(x) \dif x} + \int_{y-z}^z {F_1(y-x) f_2(x) \dif x} \\ & = F_1(z) F_2(y-z) + \int_{y-z}^z {F_1(y-x) f_2(x) \dif x} . \end{split}$$ Due to the fact that $F_1(y-x)$ and $f_2(x)$ are continuous for $0 \leq y-z \leq x \leq z$ and by the fundamental theorem of calculus (which is applicable on this region), the partial derivative of $G_2(y,z)$ with respect to $z$ is given by $$\begin{split} \dpd{}{z} G_2(y,z) & = f_1(z) F_2(y-z) - F_1(z) f_2(y-z) + F_1(y-z) f_2(z) + F_1(z) f_2(y-z) \\ & = F_1(y-z) f_2(z) + f_1(z) F_2(y-z) , \quad \forall (y,z) \in \mathcal{D}_2 \end{split}$$ Subsequently, differentiation with respect to $y$ gives $g_2(y,z)={\md{}{2}{y}{}{z}{} G_2(y,z)}$, that is, $$\label{equation:g_2} g_2 (y,z) = f_1(y-z) f_2(z) + f_1(z) f_2(y-z) , \quad \forall (y,z) \in \mathcal{D}_2 .$$ Note that $g_2 (y,z)$ is continuous on $\mathcal{D}_2$, since it is the sum of products of two continuous functions on $\mathcal{D}_2$, $f_1(\cdot)$ and $f_2(\cdot)$. Now, for $n=2$, the first right-hand side of [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} is equal to $$\begin{split} & f_2(z) \int_0^z {g_1 (y - z,x) \dif x} + \int_0^z {f_2(x) g_1(y - x,z) \dif x} \\ = & f_2(z) \int_{\max(y-z,0)}^{\min(y-z,z)} {g_1 (y - z,x) \dif x} + \int_{\max(y-z,0)}^{\min(y-z,z)} {f_2(x) g_1 (y - x,z) \dif x} \\ = & f_2(z) \int_{y-z}^{y-z} {g_1 (y - z,x) \dif x} + \int_{y-z}^{y-z} {f_2(x) g_1 (y - x,z) \dif x} \\ = & \left( f_1(y-z) f_2(z) + f_1(z) f_2(y-z) \right) \int_{y-z}^{y-z} {\delta(y - z - x) \dif x} \\ = & \left( f_1(y-z) f_2(z) + f_1(z) f_2(y-z) \right) \int_0^0 {\delta(w) \dif w} \\ = & f_1(y-z) f_2(z) + f_1(z) f_2(y-z) = g_2(y,z) , \quad \forall (y,z) \in \mathcal{D}_2 . \end{split}$$ The first equality follows from the domain $\mathcal{D}_1$ of $g_1(y,z)$, and the second equality is because $z \leq y \leq 2z$ on $\mathcal{D}_2$. Inductive step: Assume that [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} is true for $n=k-1$ and that $g_{k-1}(y,z)$ is defined and continuous on $\mathcal{D}_{k-1} = \{ (y,z) \in \mathbb{R}_+^2 : z \leq y \leq (k-1)z \}$, for some arbitrary integer $k \geq 3$ (inductive hypothesis). Then by taking the second-order mixed partial derivative of [\[equation:recurrence relation of joint CDF\]](#equation:recurrence relation of joint CDF){reference-type="eqref" reference="equation:recurrence relation of joint CDF"}, we obtain $$g_k(y,z) = {\dmd{}{2}{y}{}{z}{} G_k(y,z)} = \dmd{}{2}{y}{}{z}{} \left( \int_0^z {f_k(x) G_{k-1}(y - x,z) \dif x} \right) .$$ In order to apply Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"}, let $a(z)=0$, $b(z)=z$, and $h(x,y,z)=f_k(x) G_{k-1}(y - x,z)$. Then, because $G_{k-1} (y,z) = \int_0^z {\int_0^y g_{k-1}(y',z') \dif y' \dif z'}$, we get for all $(x,y,z) \in \mathbb{R}_+^3$ $$\dpd{}{y} h(x,y,z) = f_k(x) \int_0^z {g_{k-1} (y - x,z') \dif z'} = f_k(x) \int_{\max(\frac{y-x}{k-1},0)}^{\min(y-x,z)} {g_{k-1} (y - x,z') \dif z'} ,$$ $$\dpd{}{z} h(x,y,z) = f_k(x) \int_0^{y-x} {g_{k-1} (y',z) \dif y'} = f_k(x) \int_z^{\min(y-x,(k-1)z)} {g_{k-1} (y',z) \dif y'} ,$$ $${\dmd{}{2}{y}{}{z}{} h(x,y,z)} = f_k(x) g_{k-1} (y - x,z) .$$ The limits of integration were appropriately modified by exploiting the domain $\mathcal{D}_{k-1}$ of $g_{k-1} (y,z)$ in order to remove zero values of the integrands. There is a technical point here regarding the assumptions of Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"}. Observe that: i) $h(x,y,z)$ is continuous on $\{(x,y,z) \in \mathbb{R}_+^3 : x \leq y \}$; note that $G_{k-1}(y,z)$ is continuous on $\mathbb{R}_+^2$ due to Theorem [Theorem 1](#theorem:Recurrence relation and continuity of joint CDF){reference-type="ref" reference="theorem:Recurrence relation and continuity of joint CDF"}, ii) $\pd{}{y} h(x,y,z)$ exists (and is continuous) on $\{(x,y,z) \in \mathbb{R}_+^3 : \min(y-x,z) \geq \max(\frac{y-x}{k-1},0) \} = \{(x,y,z) \in \mathbb{R}_+^3 : x \leq y \leq x+(k-1)z \}$; note that $\min(\cdot,\cdot)$ and $\max(\cdot,\cdot)$ are both continuous functions, and $\min(a,b) \geq \max(c,d) \iff (a \geq c) \land (a \geq d) \land (b \geq c) \land (b \geq d)$, iii) $\pd{}{z} h(x,y,z)$ exists and is continuous on $\{(x,y,z) \in \mathbb{R}_+^3 : \min(y-x,(k-1)z) \geq z \} = \{(x,y,z) \in \mathbb{R}_+^3 : x+z \leq y \}$, and iv) $\md{}{2}{y}{}{z}{} h(x,y,z)$ exists and is continuous on $\{(x,y,z) \in \mathbb{R}_+^3 : x+z \leq y \leq x+(k-1)z \}$. The intersection of all these sets is equal to $\{(x,y,z) \in \mathbb{R}_+^3 : x+z \leq y \leq x+(k-1)z \}$. Since the domain of integration of $h(x,y,z)$ is $0 \leq x \leq z$, Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"} is applicable on $\{(x,y,z) \in \mathbb{R}_+^3 : x+z \leq y \leq x+(k-1)z ,\; x \leq z\}$, thus yielding $$\begin{split} g_k(y,z) & = f_k(z) \int_0^z {g_{k-1} (y - z,z') \dif z'} + \int_0^z {f_k(x) g_{k-1} (y - x,z) \dif x} \\ & = f_k(z) \int_{\max(\frac{y-z}{k-1},0)}^{\min(y-z,z)} {g_{k-1} (y - z,z') \dif z'} + \int_{\max(y-(k-1)z,0)}^{\min(y-z,z)} {f_k(x) g_{k-1} (y - x,z) \dif x} \\ & = f_k(z) \int_{\tfrac{y-z}{k-1}}^{\min(y-z,z)} {g_{k-1} (y - z,z') \dif z'} + \int_{\max(y-(k-1)z,0)}^{\min(y-z,z)} {f_k(x) g_{k-1} (y - x,z) \dif x} , \end{split}$$ for all $(y,z) \in \mathbb{R}_+^2$ such that $z \leq y \leq kz$, that is, for all $(y,z) \in \mathcal{D}_k$. From the last equality, it is clear that $g_k(y,z)$ is continuous on $\mathcal{D}_k$, because the integrands are continuous on their respective domain of integration, and the limits of integration are continuous as well. Therefore, [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} is true and $g_n (y,z)$ is continuous on its domain $\mathcal{D}_n$, for all integers $n \geq 2$. This completes the proof. ◻ **Remark 4**. An alternative proof of [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} for $n=2$ is based on Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"} together with Remark [Remark 21](#remark:Uniform convergence assumption){reference-type="ref" reference="remark:Uniform convergence assumption"}. In particular, let $a(z)=0$, $b(z)=z$, and $h(x,y,z)=f_2(x) G_1(y - x,z)$. Since $G_1(y,z) = \int_0^z {\int_0^y g_1(y',z') \dif y' \dif z'}$, we obtain for all $(x,y,z) \in \mathbb{R}_+^3$ $$\begin{split} \dpd{}{y} h(x,y,z) & = f_2(x) \int_0^z {g_1 (y - x,z') \dif z'} = f_2(x) \int_{\max(y-x,0)}^{\min(y-x,z)} {g_1 (y - x,z') \dif z'} \\ & = f_1(y-x) f_2(x) \int_{\max(y-x,0)}^{\min(y-x,z)} {\delta(y-x-z') \dif z'} \\ & = f_1(y-x) f_2(x) \int_{\max(y-x-z,0)}^{\min(y-x,0)} {\delta(w) \dif w} \\ & = f_1(y-x) f_2(x) [\max(y-x-z,0) \leq 0 \leq \min(y-x,0)] \\ & = f_1(y-x) f_2(x) [x \leq y \leq x+z] , \end{split}$$ $$\begin{split} \dpd{}{z} h(x,y,z) & = f_2(x) \int_0^{y-x} {g_1 (y',z) \dif y'} = f_2(x) \int_z^{\min(y-x,z)} {g_1 (y',z) \dif y'} \\ & = f_1(z) f_2(x) \int_z^{\min(y-x,z)} {\delta(y'-z) \dif y'} = f_1(z) f_2(x) \int_0^{\min(y-x-z,0)} {\delta(w) \dif w} \\ & = f_1(z) f_2(x) [0 \leq \min(y-x-z,0)] = f_1(z) f_2(x) [x+z \leq y] , \end{split}$$ $${\dmd{}{2}{y}{}{z}{} h(x,y,z)} = f_2(x) g_1 (y - x,z) = f_1(z) f_2(x) \delta(y-x-z) = f_1(z) f_2(y-z) \delta(y-z-x) ,$$ $$\begin{split} \int_0^z {\dmd{}{2}{y}{}{z}{} h(x,y,z) \dif x} & = \int_0^z {f_2(x) g_1 (y - x,z) \dif x} = \int_{\max(y-z,0)}^{\min(y-z,z)} {f_2(x) g_1 (y - x,z) \dif x} \\ & = f_1(z) f_2(y-z) \int_{\max(y-z,0)}^{\min(y-z,z)} {\delta(y-z-x) \dif x} = \\ & = f_1(z) f_2(y-z) \int_{\max(y-2z,0)}^{\min(y-z,0)} {\delta(w) \dif w} \\ & = f_1(z) f_2(y-z) [\max(y-2z,0) \leq 0 \leq \min(y-z,0)] \\ & = f_1(z) f_2(y-z) [z \leq y \leq 2z] . \end{split}$$ Now, we can observe that: i) $h(x,y,z)$ is continuous on $\{(x,y,z) \in \mathbb{R}_+^3 : x \leq y \}$; note that $G_1(y,z)$ is continuous on $\mathbb{R}_+^2$ because of Theorem [Theorem 1](#theorem:Recurrence relation and continuity of joint CDF){reference-type="ref" reference="theorem:Recurrence relation and continuity of joint CDF"}, ii) $\pd{}{y} h(x,y,z)$ exists (and is continuous) on $\{(x,y,z) \in \mathbb{R}_+^3 : x \leq y \leq x+z \}$, and iii) $\pd{}{z} h(x,y,z)$ exists and is continuous on $\{(x,y,z) \in \mathbb{R}_+^3 : x+z \leq y \}$. The intersection of these sets is equal to $\{(x,y,z) \in \mathbb{R}_+^3 : x+z \leq y \leq x+z \} = \{(x,y,z) \in \mathbb{R}_+^3 : y = x+z \}$. Because the domain of integration of $h(x,y,z)$ is $0 \leq x \leq z$, the assumptions of Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"} regarding $h(x,y,z)$, $\pd{}{y} h(x,y,z)$, and $\pd{}{z} h(x,y,z)$ are all satisfied on $\{(x,y,z) \in \mathbb{R}_+^3 : y = x+z ,\; x \leq z\}$. Furthermore, the integral $\int_0^z {\md{}{2}{y}{}{z}{} h(x,y,z) \dif x}$ converges to $f_1(z) f_2(y-z)$ uniformly on $\{(y,z) \in \mathbb{R}_+^2 : z \leq y \leq 2z \} = \mathcal{D}_2$; see also Appendix [5](#appendix:Dirac Delta Function){reference-type="ref" reference="appendix:Dirac Delta Function"} for more details on this topic. Consequently, based on Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"} and Remark [Remark 21](#remark:Uniform convergence assumption){reference-type="ref" reference="remark:Uniform convergence assumption"}, we conclude that $$\begin{split} g_2(y,z) & = {\dmd{}{2}{y}{}{z}{} G_2(y,z)} = \dmd{}{2}{y}{}{z}{} \left( \int_0^z {f_2(x) G_1(y - x,z) \dif x} \right) \\ & = f_2(z) \int_0^z {g_1 (y - z,z') \dif z'} + \int_0^z {f_2(x) g_1 (y - x,z) \dif x} \\ & = f_2(z) \int_{y-z}^{\min(y-z,z)} {g_1 (y - z,z') \dif z'} + \int_{\max(y-z,0)}^{\min(y-z,z)} {f_2(x) g_1 (y - x,z) \dif x} \\ & = f_1(y-z) f_2(z) + f_1(z) f_2(y-z) , \quad \forall (y,z) \in \mathcal{D}_2 . \end{split}$$ **Remark 5**. From [\[equation:g_2\]](#equation:g_2){reference-type="eqref" reference="equation:g_2"}, we can observe that the joint PDF $g_2 (y,z)$ remains the same by interchanging $X_1$ and $X_2$, and therefore $f_1(\cdot)$ and $f_2(\cdot)$, as it was expected. Moreover, in the case of exponential random variables with $f_i(x) = \lambda_i e^{-\lambda_i x}$, $x \geq 0$ and $\lambda_i > 0$, we obtain $g_2(y,z) = \lambda_1 \lambda_2 (e^{-\lambda_1 (y-z) - \lambda_2 z} + e^{-\lambda_1 z - \lambda_2 (y-z)})$, $\forall (y,z) \in \mathcal{D}_2$. This is consistent with [@Arendarczyk-Kozubowski-Panorska_a Eq. (7)]. ### [**Independent and identically distributed (i.i.d.) random variables.**]{.nodecor} The following lemma will be useful for simplifying [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} in the i.i.d. case. **Lemma 6**. *For all $n \in \mathbb{N}\setminus \{1\}$ and for all $(y,z) \in \mathcal{D}_{n+1}$, it holds that $$\int_0^z {g_n(y-z,x) \dif x} = \int_0^z {f_n(x) \int_0^z {g_{n-1}(y-z-x,w) \dif w} \dif x} .$$* *Proof.* By virtue of [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"}, we have $$g_n(y-z,x) = f_n(x) \int_0^x {g_{n-1}(y-z-x,w) \dif w} + \int_0^x {f_n(w) g_{n-1}(y-z-w,x) \dif w} ,$$ on the set $\{(x,y,z) \in \mathbb{R}^3 : x \geq 0, \; y \geq z, \; x \leq y-z \leq nx\} = \{(x,y,z) \in \mathbb{R}^3 : x \geq 0, \; z+x \leq y \leq z+nx\}$. Then, by integrating over the region $0 \leq x \leq z$, we obtain $$\scalebox{0.95}{$ \begin{split} \int_{0}^z {g_n(y-z,x) \dif x} & = \int_0^z {f_n(x) \int_0^x {g_{n-1}(y-z-x,w) \dif w} \dif x} + \int_0^z {\int_0^x {f_n(w) g_{n-1}(y-z-w,x) \dif w} \dif x} \\ & \mathop{=}^{(c)} \int_0^z {f_n(x) \int_0^x {g_{n-1}(y-z-x,w) \dif w} \dif x} + \int_0^z {f_n(w) \int_w^z {g_{n-1}(y-z-w,x) \dif x} \dif w} \\ & \mathop{=}^{(d)} \int_0^z {f_n(x) \int_0^x {g_{n-1}(y-z-x,w) \dif w} \dif x} + \int_0^z {f_n(x) \int_x^z {g_{n-1}(y-z-x,w) \dif w} \dif x} \\ & = \int_0^z {f_n(x) \left( \int_0^x {g_{n-1}(y-z-x,w) \dif w} + \int_x^z {g_{n-1}(y-z-x,w) \dif w} \right) \dif x } \\ & = \int_0^z {f_n(x) \int_0^z {g_{n-1}(y-z-x,w) \dif w} \dif x} , \end{split} $}$$ for all $(y,z) \in \mathbb{R}_+^2$ such that $z \leq y \leq (n+1)z$, that is, for all $(y,z) \in \mathcal{D}_{n+1}$. Note that equalities $(c)$ and $(d)$ have been derived by interchanging the order of integration ($\{0 \leq x \leq z\} \land \{0 \leq w \leq x\} \iff \{0 \leq w \leq z\} \land \{w \leq x \leq z\}$) and by interchanging the variables $w$, $x$ in the second double integral, respectively. ◻ **Theorem 7**. *Let $f(x)$ be the common PDF of random variables $\{X_i\}_{i \in \mathbb{N}}$, that is, $f_i(x) = f(x)$, $\forall i \in \mathbb{N}$. Then, the PDF of $(Y_n,Z_n)$ is given by the recurrence relation $$\label{equation:recurrence relation of joint PDF_iid} \begin{split} g_n (y,z) & = n f(z) \int_0^z {g_{n-1} (y - z,x) \dif x} = n f(z) \int_{\tfrac{y-z}{n-1}}^{\min(y-z,z)} {g_{n-1} (y - z,x) \dif x} , \\ & \:\, \quad \forall n \in \mathbb{N}\setminus \{1\}, \; \forall (y,z) \in \mathcal{D}_n , \end{split}$$ with initial condition $$\label{equation:initial condition for joint PDF_iid} g_1(y,z) = f(y) \delta(y - z) = f(z) \delta(y - z), \quad \forall (y,z) \in \mathcal{D}_1 .$$* *Proof.* The initial condition can be deduced from [\[equation:initial condition for joint PDF\]](#equation:initial condition for joint PDF){reference-type="eqref" reference="equation:initial condition for joint PDF"}. Observe that the second equality in [\[equation:recurrence relation of joint PDF_iid\]](#equation:recurrence relation of joint PDF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PDF_iid"} follows directly from the first by taking into account the domain $\mathcal{D}_{n-1}$ of $g_{n-1}(y,z)$ and that $y \geq z$ on $\mathcal{D}_n$. Therefore, it is sufficient to prove the first equality, using induction on $n$. Basis step: For $n=2$, the right-hand side of the first equality in [\[equation:recurrence relation of joint PDF_iid\]](#equation:recurrence relation of joint PDF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PDF_iid"} gives $$2 f(z) \int_0^z {g_1 (y - z,x) \dif x} = 2 f(y-z) f(z) [z \leq y \leq 2z] = 2 f(y-z) f(z) ,\quad \forall (y,z) \in \mathcal{D}_2 .$$ In addition, by setting $f_1(x) = f_2(x) = f(x)$ in [\[equation:g_2\]](#equation:g_2){reference-type="eqref" reference="equation:g_2"}, we conclude that $$g_2(y,z) = 2 f(y-z) f(z) , \quad \forall (y,z) \in \mathcal{D}_2 .$$ Thus, $g_2(y,z) = 2 f(z) \int_0^z {g_1 (y - z,x) \dif x}$, for all $(y,z) \in \mathcal{D}_2$. Inductive step: Suppose that $$g_{k-1} (y,z) = (k-1) f(z) \int_0^z {g_{k-2} (y - z,w) \dif w} , \quad \forall (y,z) \in \mathcal{D}_{k-1} ,$$ for some arbitrary integer $k \geq 3$ (inductive hypothesis). Then, [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} for $n=k$ and Lemma [Lemma 6](#lemma:Integration property of joint PDF){reference-type="ref" reference="lemma:Integration property of joint PDF"} for $n=k-1$ yield $$\begin{split} g_k(y,z) & = f(z) \int_0^z {g_{k-1} (y - z,x) \dif x} + \int_0^z {f(x) g_{k-1} (y - x,z) \dif x} \\ & = f(z) \int_0^z {g_{k-1} (y - z,x) \dif x} + (k-1) f(z) \int_0^z {f(x) \int_0^z {g_{k-2}(y-x-z,w) \dif w} \dif x} \\ & = f(z) \int_0^z {g_{k-1} (y - z,x) \dif x} + (k-1) f(z) \int_0^z {g_{k-1} (y - z,x) \dif x} \\ & = k f(z) \int_0^z {g_{k-1} (y - z,x) \dif x} , \quad \forall (y,z) \in \mathcal{D}_k . \end{split}$$ Note that the second equality holds on $\mathcal{D}_k$, because $g_{k-1} (y-x,z) = (k-1) f(z) \break \int_0^z {g_{k-2} (y - x - z,w) \dif w}$ on $\{(x,y,z) \in \mathbb{R}^3 : z \geq 0, \; y \geq x, \; z \leq y-x \leq (k-1)z\} = \{(x,y,z) \in \mathbb{R}^3 : z \geq 0, \; x+z \leq y \leq x+(k-1)z\}$ and the domain of integration is $0 \leq x \leq z$. ◻ **Remark 8**. Eq. [\[equation:recurrence relation of joint PDF_iid\]](#equation:recurrence relation of joint PDF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PDF_iid"} is in complete agreement with [@Qeadan-Kozubowski-Panorska Eq. (2.18)], which was derived using transformations of order statistics. Moreover, we can prove an exact formula (a multiple integral) for the joint PDF in the i.i.d. case as follows. **Proposition 9**. *Suppose that all random variables $X_i$ have the same PDF $f(x)$, that is, $f_i(x) = f(x)$, $\forall i \in \mathbb{N}$. Then, the PDF of $(Y_n,Z_n)$ can be expressed as an $(n-1)$-dimensional integral as follows $$g_n (y,z) = {n!} f(z) \bigintsss_{x_n=0}^z {f(x_n) \bigintsss_{x_{n-1}=0}^{x_n} f(x_{n-1}) \cdots \bigintsss_{x_3=0}^{x_4} f(x_3) \bigintsss_{x_2=0}^{x_3} g_1\left( {y - z - \sum_{i=3}^n {x_i}},x_2 \right) \prod_{i=2}^n {\dif x_i}} ,$$ for all $n \in \mathbb{N}\setminus \{1\}$ and for all $(y,z) \in \mathcal{D}_n$, where $g_1(y,z)$ is given by [\[equation:initial condition for joint PDF_iid\]](#equation:initial condition for joint PDF_iid){reference-type="eqref" reference="equation:initial condition for joint PDF_iid"}.* *Proof.* Once again, we apply induction on $n$. Basis step: For $n=2$, [\[equation:recurrence relation of joint PDF_iid\]](#equation:recurrence relation of joint PDF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PDF_iid"} yields $g_2 (y,z) = 2 f(z) \int_0^z {g_1 (y - z,x) \dif x} = 2! f(z) \int_0^z {g_1 (y - z,x_2) \dif x_2}$. Inductive step: Suppose that $$g_{k-1} (y,z) = {(k-1)!} f(z) \bigintsss_{x_{k-1}=0}^z {f(x_{k-1}) \cdots \bigintsss_{x_3=0}^{x_4} f(x_3) \bigintsss_{x_2=0}^{x_3} g_1\left( {y - z - \sum_{i=3}^{k-1} {x_i}},x_2 \right) \prod_{i=2}^{k-1} {\dif x_i}} ,$$ for some arbitrary integer $k \geq 3$ (inductive hypothesis). Then, from [\[equation:recurrence relation of joint PDF_iid\]](#equation:recurrence relation of joint PDF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PDF_iid"} we obtain $$\scalebox{0.92}{$ \begin{split} g_k (y,z) & = k f(z) \int_0^z {g_{k-1} (y - z,x) \dif x} \\ & = k! f(z) \bigintsss_0^z {f(x) \bigintsss_{x_{k-1}=0}^{x} f(x_{k-1}) \cdots \bigintsss_{x_3=0}^{x_4} f(x_3) \bigintsss_{x_2=0}^{x_3} g_1\left( {y - z - x - \sum_{i=3}^{k-1} {x_i}},x_2 \right) \left( \prod_{i=2}^{k-1} {\dif x_i} \right) \dif x} \\ & = k! f(z) \bigintsss_{x_k=0}^z {f(x_k) \bigintsss_{x_{k-1}=0}^{x_k} f(x_{k-1}) \cdots \bigintsss_{x_3=0}^{x_4} f(x_3) \bigintsss_{x_2=0}^{x_3} g_1\left( {y - z - \sum_{i=3}^k {x_i}},x_2 \right) \prod_{i=2}^k {\dif x_i}} . \end{split} $}$$ This concludes the proof. ◻ ## Application: Peak-to-Average Ratio The peak-to-average ratio of the $n^\text{th}$ order is defined by $$\rho_n \coloneqq\frac{\max_{1 \leq i \leq n} X_i}{\tfrac{1}{n} \sum_{i=1}^n {X_i}} = \frac{Z_n}{\tfrac{1}{n} Y_n} , \quad \text{whenever}\ Y_n \neq 0 .$$ The probability that $\rho_n$ lies within the interval $[\alpha,\beta]$, $0 \leq \alpha \leq \beta$, is given by $$\begin{split} \operatorname{P}(\alpha \leq \rho_n \leq \beta) & = \operatorname{P}(Y_n > 0, \tfrac{\alpha}{n} Y_n \leq Z_n \leq \tfrac{\beta}{n} Y_n) = \int_0^\infty {\int_{\tfrac{\alpha}{n} y}^{\tfrac{\beta}{n} y} g_n(y,z) \dif z \dif y} \\ & = \int_0^\infty {\int_{\max\left( \tfrac{\alpha}{n} y , \tfrac{1}{n} y \right)}^{\min\left( \tfrac{\beta}{n} y , y \right)} g_n(y,z) \dif z \dif y} = \int_0^\infty {\int_{\tfrac{\max(\alpha,1)}{n} y}^{\min\left( \tfrac{\beta}{n},1 \right) y} g_n(y,z) \dif z \dif y}, \end{split}$$ where $g_n(y,z)$ is computed recursively according to Theorem [Theorem 3](#theorem:Joint PDF){reference-type="ref" reference="theorem:Joint PDF"}. For example, this kind of calculation can be useful in the performance analysis of communication systems [@Morrison-Tobias]. In such systems the peak-to-average ratio is also known as "peak-to-average power ratio (PAPR)" because each $X_i$ represents the power of a random signal, which is definitely a nonnegative quantity (i.e., $X_i \geq 0$). # Discrete Random Variables {#section:Discrete Random Variables} In this section, we extend the methodology to the case of discrete random variables. Let $\{\widehat{X}_i\}_{i \in \mathbb{N}}$ be a sequence of *independent, not necessarily identically distributed,* nonnegative discrete random variables. Without loss of generality, we assume that all of them are integer-valued, i.e., $\widehat{X}_i \in \mathbb{N}_0$, $\forall i \in \mathbb{N}$. In addition, let $\widehat{f}_i(k) \coloneqq\operatorname{P}(\widehat{X}_i = k) = \widehat{F}_i (k) - \widehat{F}_i (k-1),\ \forall k \in \mathbb{N}_0$, and $\widehat{F}_i(x) \coloneqq\operatorname{P}(\widehat{X}_i \leq x) = \sum_{k'=0}^{\lfloor x \rfloor} {\widehat{f}_i(k')},\ \forall x \geq 0$, be the PMF and CDF of $\widehat{X}_i$, respectively. As in the continuous case, we define the sum and maximum of the first $n \in \mathbb{N}$ variables $$\label{equation:sum and maximum_discrete} \widehat{Y}_n \coloneqq\sum_{i=1}^n {\widehat{X}_i}= \widehat{X}_1 + \cdots + \widehat{X}_n , \qquad \widehat{Z}_n \coloneqq\max_{1 \leq i \leq n} {\widehat{X}_i} = \max (\widehat{X}_1,\dots,\widehat{X}_n) .$$ Generally, $\widehat{Y}_n$ and $\widehat{Z}_n$ are dependent random variables. Moreover, it will be helpful to define the following sets $$\widehat{\mathcal{D}}_n \coloneqq\{ (l,m) \in \mathbb{N}_0^2 : m \leq l \leq n m \} , \quad \forall n \in \mathbb{N}.$$ Afterwards, we will study the joint CDF and PMF of $\widehat{Y}_n$ and $\widehat{Z}_n$, that is, $\widehat{G}_n (y,z) \coloneqq\operatorname{P}(\widehat{Y}_n \leq y,\widehat{Z}_n \leq z ) = \sum_{l=0}^{\lfloor y \rfloor} \sum_{m=0}^{\lfloor z \rfloor} {\widehat{g}_n (l,m)},\ \forall (y,z) \in \mathbb{R}_+^2$, and $\widehat{g}_n (l,m) \coloneqq\operatorname{P}(\widehat{Y}_n = l,\widehat{Z}_n = m),\ \forall (l,m) \in \widehat{\mathcal{D}}_n$. Observe that $(\widehat{Y}_n, \widehat{Z}_n) \in \mathbb{N}_0^2$ and $\widehat{Z}_n \leq \widehat{Y}_n \leq n \widehat{Z}_n$, thus $\widehat{g}_n (l,m) = 0$ for all $(l,m) \notin \widehat{\mathcal{D}}_n$. In particular, we will derive the discrete counterparts of continuous-case recurrence relations, but with a significant difference between the joint PDF and PMF (since the Leibniz integral rule is no longer applicable in the discrete case). ## Recurrence Relation for the Joint CDF **Theorem 10**. *The joint CDF of $\widehat{Y}_n$ and $\widehat{Z}_n$, defined by [\[equation:sum and maximum_discrete\]](#equation:sum and maximum_discrete){reference-type="eqref" reference="equation:sum and maximum_discrete"}, is given by the following recurrence relation $$\label{equation:recurrence relation of joint CDF_discrete} \begin{split} \widehat{G}_n (y,z) & = \sum_{k=0}^{\lfloor z \rfloor} {\widehat{G}_{n-1}(y - k,z) \widehat{f}_n(k)} = \sum_{k=0}^{\min(\lfloor y \rfloor,\lfloor z \rfloor)} {\widehat{G}_{n-1}(y - k,z) \widehat{f}_n(k)} , \\ & \:\, \quad \forall n \in \mathbb{N}\setminus \{1\} , \; \forall (y,z) \in \mathbb{R}_+^2 , \end{split}$$ with initial condition $$\label{equation:initial condition for joint CDF_discrete} \widehat{G}_1 (y,z) = \widehat{F}_1 (\min(y,z)) , \quad \forall (y,z) \in \mathbb{R}_+^2 .$$* *Proof.* In a similar way, using the law of total probability, we can write $$\begin{split} \widehat{G}_n (y,z) & = \operatorname{P}(\widehat{Y}_{n-1} + \widehat{X}_n \leq y, \max(\widehat{Z}_{n-1},\widehat{X}_n) \leq z) \\ & = \operatorname{P}(\widehat{Y}_{n-1} + \widehat{X}_n \leq y, \widehat{Z}_{n-1} \leq z, \widehat{X}_n \leq z) \\ & = \sum_{k=0}^{\infty} {\operatorname{P}(\widehat{Y}_{n-1} + \widehat{X}_n \leq y, \widehat{Z}_{n-1} \leq z, \widehat{X}_n \leq z \;|\; \widehat{X}_n = k) \operatorname{P}(\widehat{X}_n = k)} \\ & = \sum_{k=0}^{\lfloor z \rfloor} {\widehat{G}_{n-1}(y - k,z) \widehat{f}_n(k)} = \sum_{k=0}^{\min(\lfloor y \rfloor,\lfloor z \rfloor)} {\widehat{G}_{n-1}(y - k,z) \widehat{f}_n(k)} , \end{split}$$ where the last equality is because $\widehat{G}_{n-1}(y - k,z) = 0$ if $k>y$. The initial condition is $$\widehat{G}_1 (y,z) = \operatorname{P}(\widehat{Y}_1 \leq y, \widehat{Z}_1 \leq z) = \operatorname{P}(\widehat{X}_1 \leq \min(y,z)) = \widehat{F}_1 (\min(y,z)) . \qedhere$$ ◻ Note that $\widehat{G}_n (y,z)$ is *not* continuous on $\mathbb{R}_+^2$, in contrast to $G_n (y,z)$ in the case of continuous random variables (according to Theorem [Theorem 1](#theorem:Recurrence relation and continuity of joint CDF){reference-type="ref" reference="theorem:Recurrence relation and continuity of joint CDF"}). Moreover, we can derive an exact formula (a multiple sum) for the joint CDF. **Proposition 11**. *The joint CDF of $\widehat{Y}_n$ and $\widehat{Z}_n$ can be expressed as an $(n-1)$-dimensional sum, that is, $$\label{equation:exact formula CDF_discrete} \widehat{G}_n (y,z) = \mathlarger{\mathlarger{\sum}}_{k_n=0}^{\lfloor z \rfloor} \cdots \mathlarger{\mathlarger{\sum}}_{k_2=0}^{\lfloor z \rfloor} \widehat{G}_1 \left( {y - \sum_{i=2}^n {k_i}},z \right) \prod_{i=2}^n {\widehat{f}_i(k_i)} , \quad \forall n \in \mathbb{N}\setminus \{1\} , \; \forall (y,z) \in \mathbb{R}_+^2 ,$$ where $\widehat{G}_1(y,z)$ is given by [\[equation:initial condition for joint CDF_discrete\]](#equation:initial condition for joint CDF_discrete){reference-type="eqref" reference="equation:initial condition for joint CDF_discrete"}.* *Proof.* Based on the recurrence relation [\[equation:recurrence relation of joint CDF_discrete\]](#equation:recurrence relation of joint CDF_discrete){reference-type="eqref" reference="equation:recurrence relation of joint CDF_discrete"}, we provide a proof by induction on $n$. Basis step: For $n=2$, [\[equation:recurrence relation of joint CDF_discrete\]](#equation:recurrence relation of joint CDF_discrete){reference-type="eqref" reference="equation:recurrence relation of joint CDF_discrete"} gives $\widehat{G}_2 (y,z) = \sum_{k=0}^{\lfloor z \rfloor} {\widehat{G}_1(y - k,z) \widehat{f}_2(k)} = \sum_{k_2=0}^{\lfloor z \rfloor} {\widehat{G}_1(y - k_2,z) \widehat{f}_2(k_2)}$. Inductive step: Suppose that $\widehat{G}_{\tau-1} (y,z) = \sum_{k_{\tau-1}=0}^{\lfloor z \rfloor} \cdots \sum_{k_2=0}^{\lfloor z \rfloor} \widehat{G}_1 \left( {y - \sum_{i=2}^{\tau-1} {k_i}},z \right) \prod_{i=2}^{\tau-1} {\widehat{f}_i(k_i)}$, for some arbitrary integer $\tau \geq 3$ (inductive hypothesis). Then, from [\[equation:recurrence relation of joint CDF_discrete\]](#equation:recurrence relation of joint CDF_discrete){reference-type="eqref" reference="equation:recurrence relation of joint CDF_discrete"} we get $$\begin{split} \widehat{G}_\tau (y,z) & = \sum_{k=0}^{\lfloor z \rfloor} {\widehat{G}_{\tau-1}(y - k,z) \widehat{f}_\tau(k)} \\ & = \mathlarger{\mathlarger{\sum}}_{k=0}^{\lfloor z \rfloor} \mathlarger{\mathlarger{\sum}}_{k_{\tau-1}=0}^{\lfloor z \rfloor} \cdots \mathlarger{\mathlarger{\sum}}_{k_2=0}^{\lfloor z \rfloor} \widehat{G}_1 \left( {y - k - \sum_{i=2}^{\tau-1} {k_i}},z \right) \left( \prod_{i=2}^{\tau-1} {\widehat{f}_i(k_i)} \right) {\widehat{f}_\tau (k)} \\ & = \mathlarger{\mathlarger{\sum}}_{k_\tau=0}^{\lfloor z \rfloor} \cdots \mathlarger{\mathlarger{\sum}}_{k_2=0}^{\lfloor z \rfloor} \widehat{G}_1 \left( {y - \sum_{i=2}^\tau {k_i}},z \right) \prod_{i=2}^\tau {\widehat{f}_i(k_i)} . \end{split}$$ Therefore, [\[equation:exact formula CDF_discrete\]](#equation:exact formula CDF_discrete){reference-type="eqref" reference="equation:exact formula CDF_discrete"} is true for all integers $n \geq 2$. ◻ ## Recurrence Relation for the Joint PMF **Theorem 12**. *The joint PMF of $\widehat{Y}_n$ and $\widehat{Z}_n$, defined by [\[equation:sum and maximum_discrete\]](#equation:sum and maximum_discrete){reference-type="eqref" reference="equation:sum and maximum_discrete"}, is given by the following recurrence relation $$\label{equation:recurrence relation of joint PMF} \begin{split} \widehat{g}_n (l,m) & = \widehat{f}_n(m) \sum_{k=0}^m { \widehat{g}_{n-1} (l-m,k) } + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} \\ & = \widehat{f}_n(m) \sum_{k=\left\lceil \tfrac{l-m}{n-1} \right\rceil}^{\min(l-m,m)} { \widehat{g}_{n-1} (l-m,k) } + \sum_{k=\max\left( l-(n-1)m,0 \right)}^{\min(l-m,m-1)} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} , \\ & \:\, \quad \forall n \in \mathbb{N}\setminus \{1\} , \; \forall (l,m) \in \widehat{\mathcal{D}}_n , \end{split}$$ with initial condition $$\label{equation:initial condition for joint PMF} \widehat{g}_1 (l,m) = \widehat{f}_1(l) = \widehat{f}_1(m) , \quad \forall (l,m) \in \widehat{\mathcal{D}}_1 .$$* *Proof.* For all $n \in \mathbb{N}$, we know that $\widehat{g}_n (l,m) = 0$ for all $(l,m) \notin \widehat{\mathcal{D}}_n$, because of the definition of $\widehat{Y}_n$ and $\widehat{Z}_n$. First, let us prove the initial condition [\[equation:initial condition for joint PMF\]](#equation:initial condition for joint PMF){reference-type="eqref" reference="equation:initial condition for joint PMF"}: $$\begin{split} \widehat{g}_1 (l,m) & = \operatorname{P}(\widehat{Y}_1 = l,\widehat{Z}_1 = m) = \operatorname{P}(\widehat{X}_1 = l,\widehat{X}_1 = m) \\ & = \widehat{f}_1(l) [l = m] = \widehat{f}_1(m) [l = m] , \quad \forall (l,m) \in \mathbb{N}_0^2 , \end{split}$$ or, equivalently, $\widehat{g}_1 (l,m) = \widehat{f}_1(l) = \widehat{f}_1(m)$, $\forall (l,m) \in \widehat{\mathcal{D}}_1 = \{ (l,m) \in \mathbb{N}_0^2 : l = m \}$. In the discrete case the recursive application of Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"} does not work, so we need an alternative approach to prove [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"}. In particular, we make use of the law of total probability as follows $$\begin{split} \widehat{g}_n (l,m) \coloneqq& \operatorname{P}(\widehat{Y}_n = l,\widehat{Z}_n = m) = \operatorname{P}(\widehat{Y}_{n-1} + \widehat{X}_n = l, \max(\widehat{Z}_{n-1},\widehat{X}_n) = m) \\ = & \sum_{k=0}^{\infty} {\operatorname{P}(\widehat{Y}_{n-1} + \widehat{X}_n = l, \max(\widehat{Z}_{n-1},\widehat{X}_n) = m \;|\; \widehat{X}_n = k) \operatorname{P}(\widehat{X}_n = k)} \\ = & \sum_{k=0}^{\infty} {\widehat{f}_n (k) \operatorname{P}(\widehat{Y}_{n-1} = l-k, \max(\widehat{Z}_{n-1},k) = m)} \\ = & \sum_{k=0}^{\infty} {\widehat{f}_n (k) \left( \operatorname{P}(\widehat{Y}_{n-1} = l-k, \max(\widehat{Z}_{n-1},k) = m, k=m) \right. } \\ & \, {\left. + \operatorname{P}(\widehat{Y}_{n-1} = l-k, \max(\widehat{Z}_{n-1},k) = m, k \leq m-1) \right)} \\ = & {\widehat{f}_n (m) \operatorname{P}(\widehat{Y}_{n-1} = l-m, \max(\widehat{Z}_{n-1},m) = m)} + \sum_{k=0}^{m-1} {\widehat{f}_n (k) \operatorname{P}(\widehat{Y}_{n-1} = l-k, \widehat{Z}_{n-1} = m)} \\ = & {\widehat{f}_n (m) \operatorname{P}(\widehat{Y}_{n-1} = l-m, \widehat{Z}_{n-1} \leq m)} + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} \\ = & {\widehat{f}_n (m) \sum_{k=0}^m {\operatorname{P}(\widehat{Y}_{n-1} = l-m, \widehat{Z}_{n-1} = k)}} + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} \\ = & \widehat{f}_n(m) \sum_{k=0}^m { \widehat{g}_{n-1} (l-m,k) } + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} , \end{split}$$ for all $n \in \mathbb{N}\setminus \{1\}$ and for all $(l,m) \in \mathbb{N}_0^2$. Equivalently, we can restrict its domain to $\widehat{\mathcal{D}}_n$ since the joint PMF $\widehat{g}_n (l,m)$ definitely vanishes outside this region. Finally, using the domain $\widehat{\mathcal{D}}_{n-1} = \{ (l,m) \in \mathbb{N}_0^2 : m \leq l \leq (n-1)m \}$ of $\widehat{g}_{n-1} (l,m)$, we have $$\begin{split} \widehat{g}_n (l,m) = & \widehat{f}_n(m) \sum_{k=0}^m { \widehat{g}_{n-1} (l-m,k) } + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} \\ = & \widehat{f}_n(m) \sum_{k=\max\left({\left\lceil \tfrac{l-m}{n-1} \right\rceil},0\right)}^{\min(l-m,m)} { \widehat{g}_{n-1} (l-m,k) } + \sum_{k=\max\left( l-(n-1)m,0 \right)}^{\min(l-m,m-1)} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} \\ = & \widehat{f}_n(m) \sum_{k=\left\lceil \tfrac{l-m}{n-1} \right\rceil}^{\min(l-m,m)} { \widehat{g}_{n-1} (l-m,k) } + \sum_{k=\max\left( l-(n-1)m,0 \right)}^{\min(l-m,m-1)} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} , \end{split}$$ for all $n \in \mathbb{N}\setminus \{1\}$ and for all $(l,m) \in \widehat{\mathcal{D}}_n$, where the last equality follows from the fact that $l \geq m$ on $\widehat{\mathcal{D}}_n$ and the ceiling function is monotonically nondecreasing. ◻ {#figure:2D_lattice width="0.8\\linewidth"} **Remark 13**. An alternative proof of [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} using the recurrence relation of the joint CDF [\[equation:recurrence relation of joint CDF_discrete\]](#equation:recurrence relation of joint CDF_discrete){reference-type="eqref" reference="equation:recurrence relation of joint CDF_discrete"} is the following. Based on the two-dimensional lattice shown in Figure [1](#figure:2D_lattice){reference-type="ref" reference="figure:2D_lattice"}, $\widehat{g}_\tau (l,m) \coloneqq\operatorname{P}(\widehat{Y}_\tau = l,\widehat{Z}_\tau = m)$ can be written as $$\label{equation:g_tau} \begin{split} \widehat{g}_\tau (l,m) & = \widehat{G}_\tau (l,m) - \widehat{G}_\tau (l,m-1) - \sum_{l'=0}^{l-1} {\widehat{g}_\tau (l',m)} \\ & = \widehat{G}_\tau (l,m) - \widehat{G}_\tau (l,m-1) - \left( \widehat{G}_\tau (l-1,m) - \widehat{G}_\tau (l-1,m-1) \right) \\ & = \widehat{G}_\tau (l,m) - \widehat{G}_\tau (l-1,m) + \widehat{G}_\tau (l-1,m-1) - \widehat{G}_\tau (l,m-1) , \\ & \:\, \quad \forall \tau \in \mathbb{N}, \; \forall (l,m) \in \mathbb{Z}^2 . \end{split}$$ By exploiting [\[equation:g_tau\]](#equation:g_tau){reference-type="eqref" reference="equation:g_tau"} for $\tau = n$ and leveraging [\[equation:recurrence relation of joint CDF_discrete\]](#equation:recurrence relation of joint CDF_discrete){reference-type="eqref" reference="equation:recurrence relation of joint CDF_discrete"}, which also holds for all $(y,z) \in \mathbb{R}^2$, we obtain $$\begin{split} \widehat{g}_n (l,m) & = \sum_{k=0}^{m} {\widehat{f}_n(k) \left( \widehat{G}_{n-1}(l - k,m) - \widehat{G}_{n-1}(l - 1 - k,m) \right)} \\ & \quad\, + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \left( \widehat{G}_{n-1}(l - 1 - k,m-1) - \widehat{G}_{n-1}(l - k,m-1) \right)} \\ & = {\widehat{f}_n(m) \left( \widehat{G}_{n-1}(l - m,m) - \widehat{G}_{n-1}(l - m - 1,m) \right)} \\ & \quad\, + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \left( \widehat{G}_{n-1}(l - k,m) - \widehat{G}_{n-1}(l - k - 1,m) \right.} \\ & \quad\, + {\left. \widehat{G}_{n-1}(l - k - 1,m-1) - \widehat{G}_{n-1}(l - k,m-1) \right)} \\ & = \widehat{f}_n(m) \sum_{k=0}^m { \widehat{g}_{n-1} (l-m,k) } + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \widehat{g}_{n-1} (l-k,m)} , \\ & \:\, \quad \forall n \in \mathbb{N}\setminus \{1\} , \; \forall (l,m) \in \mathbb{Z}^2 , \end{split}$$ where the last equality follows from [\[equation:g_tau\]](#equation:g_tau){reference-type="eqref" reference="equation:g_tau"} for $\tau = n-1 \geq 1$ and the fact that $$\widehat{G}_{n-1}(l,m) - \widehat{G}_{n-1}(l-1,m) = \sum_{k=0}^m {\widehat{g}_{n-1} (l,k)}, \quad \forall (l,m) \in \mathbb{Z}^2 .$$ This can be easily seen from the lattice representation in Figure [1](#figure:2D_lattice){reference-type="ref" reference="figure:2D_lattice"} with $\tau = n-1$. As a consequence, we get [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} after shrinking the domain to $\widehat{\mathcal{D}}_n$. **Remark 14**. It is emphasized that [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} has a *fundamental difference* compared to its continuous counterpart in [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"}: the upper limit of the second sum is equal to $m-1$ instead of $m$. As a result, we *cannot* derive [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} from [\[equation:recurrence relation of joint PDF\]](#equation:recurrence relation of joint PDF){reference-type="eqref" reference="equation:recurrence relation of joint PDF"} by just making the substitutions $(x,y,z) \mapsto (k,l,m)$, $\dif x \mapsto 1$, and by replacing integrals with sums. ### [**Independent and identically distributed (i.i.d.) random variables.**]{.nodecor} The following lemma is analogous to Lemma [Lemma 6](#lemma:Integration property of joint PDF){reference-type="ref" reference="lemma:Integration property of joint PDF"} and will be helpful in order to prove another recurrence relation for the i.i.d. case (in addition to the straightforward application of [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} with $\widehat{f}_i(k) = \widehat{f}(k)$ for all $i \in \mathbb{N}$). **Lemma 15**. *For all $n \in \mathbb{N}\setminus \{1\}$ and for all $(l,m) \in \widehat{\mathcal{D}}_{n+1}$, it holds that $$\sum_{k=0}^m {\widehat{g}_n(l-m,k)} = \sum_{k=0}^m {\widehat{f}_n(k) \sum_{\nu=0}^m {\widehat{g}_{n-1}(l-m-k,\nu)}} .$$* *Proof.* By leveraging [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"}, we get $$\widehat{g}_n(l-m,k) = \widehat{f}_n(k) \sum_{\nu=0}^k {\widehat{g}_{n-1}(l-m-k,\nu)} + \sum_{\nu=0}^{k-1} {\widehat{f}_n(\nu) \widehat{g}_{n-1}(l-m-\nu,k)} ,$$ on the set $\{(k,l,m) \in \mathbb{Z}^3 : k \geq 0, \; l \geq m, \; k \leq l-m \leq nk\} = \{(k,l,m) \in \mathbb{Z}^3 : \break k \geq 0, \; m+k \leq l \leq m+nk\}$. Then, by summing for $0 \leq k \leq m$, we have $$\begin{split} \sum_{k=0}^m {\widehat{g}_n(l-m,k)} &= \sum_{k=0}^m {\widehat{f}_n(k) \sum_{\nu=0}^k {\widehat{g}_{n-1}(l-m-k,\nu)}} + \sum_{k=0}^m {\sum_{\nu=0}^{k-1} {\widehat{f}_n(\nu) \widehat{g}_{n-1}(l-m-\nu,k)}} \\ & \mathop{=}^{(e)} \sum_{k=0}^m {\widehat{f}_n(k) \sum_{\nu=0}^k {\widehat{g}_{n-1}(l-m-k,\nu)}} + \sum_{\nu=0}^{m-1} {\widehat{f}_n(\nu) \sum_{k=\nu+1}^m {\widehat{g}_{n-1}(l-m-\nu,k)}} \\ & \mathop{=}^{(f)} \sum_{k=0}^m {\widehat{f}_n(k) \sum_{\nu=0}^k {\widehat{g}_{n-1}(l-m-k,\nu)}} + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \sum_{\nu=k+1}^m {\widehat{g}_{n-1}(l-m-k,\nu)}} \\ & = \widehat{f}_n(m) \sum_{\nu=0}^m {\widehat{g}_{n-1}(l-2m,\nu)} + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \left( \sum_{\nu=0}^k {\widehat{g}_{n-1}(l-m-k,\nu)} \right.} \\ & \quad\: {\left. + \sum_{\nu=k+1}^m {\widehat{g}_{n-1}(l-m-k,\nu)} \right)} \\ & = \widehat{f}_n(m) \sum_{\nu=0}^m {\widehat{g}_{n-1}(l-2m,\nu)} + \sum_{k=0}^{m-1} {\widehat{f}_n(k) \sum_{\nu=0}^m {\widehat{g}_{n-1}(l-m-k,\nu)}} \\ & = \sum_{k=0}^m {\widehat{f}_n(k) \sum_{\nu=0}^m {\widehat{g}_{n-1}(l-m-k,\nu)}} , \end{split}$$ for all $(l,m) \in \mathbb{N}_0^2$ such that $m \leq l \leq (n+1)m$, that is, for all $(l,m) \in \widehat{\mathcal{D}}_{n+1}$. Note that equalities $(e)$ and $(f)$ have been derived by interchanging the order of summation ($\{0 \leq k \leq m\} \land \{0 \leq \nu \leq k-1\} \iff \{0 \leq \nu \leq m-1\} \land \{\nu + 1 \leq k \leq m\}$) and by interchanging the variables $k$, $\nu$ in the second double sum, respectively. ◻ **Theorem 16**. *Let $\widehat{f}(k)$ be the common PMF of random variables $\{\widehat{X}_i\}_{i \in \mathbb{N}}$, that is, $\widehat{f}_i(k) = \widehat{f}(k)$, $\forall i \in \mathbb{N}$. Furthermore, let $\widehat{h}_n(l,m)$ be defined by the following recursive formula $$\label{equation:recursive definition of h} \begin{split} \widehat{h}_n(l,m) \coloneqq& \sum_{k=0}^m {\widehat{f}(k) \widehat{h}_{n-1}(l-k,m)} + {\widehat{f}(m) \widehat{g}_{n-1} (l-m,m)} \\ = & \sum_{k=\max\left( l-(n-1)m,0 \right)}^{\min(l-m,m)} {\widehat{f}(k) \widehat{h}_{n-1}(l-k,m)} + {\widehat{f}(m) \widehat{g}_{n-1} (l-m,m)} , \\ & \: \forall n \in \mathbb{N}\setminus \{1\}, \; \forall (l,m) \in \widehat{\mathcal{D}}_n , \end{split}$$ with initial condition $\widehat{h}_1(l,m) = 0$, $\forall (l,m) \in \widehat{\mathcal{D}}_1$. Then, the PMF of $(\widehat{Y}_n,\widehat{Z}_n)$ is given by the recurrence relation $$\label{equation:recurrence relation of joint PMF_iid} \begin{split} \widehat{g}_n (l,m) & = n \widehat{f}(m) \sum_{k=0}^{m} { \widehat{g}_{n-1} (l-m,k) } - \widehat{h}_n(l,m) \\ & = n \widehat{f}(m) \sum_{k=\left\lceil \tfrac{l-m}{n-1} \right\rceil}^{\min(l-m,m)} { \widehat{g}_{n-1} (l-m,k) } - \widehat{h}_n(l,m) , \\ & \quad\:\, \forall n \in \mathbb{N}\setminus \{1\}, \; \forall (l,m) \in \widehat{\mathcal{D}}_n , \end{split}$$ with initial condition $\widehat{g}_1 (l,m) = \widehat{f}(l) = \widehat{f}(m)$, $\forall (l,m) \in \widehat{\mathcal{D}}_1$.* *Proof.* The initial condition $\widehat{g}_1(l,m)$ can be easily derived from [\[equation:initial condition for joint PMF\]](#equation:initial condition for joint PMF){reference-type="eqref" reference="equation:initial condition for joint PMF"}. Now, we will prove [\[equation:recurrence relation of joint PMF_iid\]](#equation:recurrence relation of joint PMF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PMF_iid"} using induction on $n$ and given the recursive definition of $\widehat{h}_n(l,m)$ in [\[equation:recursive definition of h\]](#equation:recursive definition of h){reference-type="eqref" reference="equation:recursive definition of h"}. Basis step: For $n=2$, [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} implies that $$\begin{split} \widehat{g}_2 (l,m) & = \widehat{f}(m) \sum_{k=0}^m { \widehat{g}_1 (l-m,k) } + \sum_{k=0}^m {\widehat{f}(k) \widehat{g}_1 (l-k,m)} - \widehat{f}(m) \widehat{g}_1 (l-m,m) \\ & = \widehat{f}(m) \sum_{k=0}^m { \widehat{g}_1 (l-m,k) } + \widehat{f}(m) \sum_{k=0}^m {\widehat{f}(k) [l-k=m] } - \widehat{h}_2(l,m) \\ & = \widehat{f}(m) \sum_{k=0}^m { \widehat{g}_1 (l-m,k) } + \widehat{f}(m) \sum_{k=0}^{m} {\widehat{f}(k) [l-m=k]} - \widehat{h}_2(l,m) \\ & = 2 \widehat{f}(m) \sum_{k=0}^{m} { \widehat{g}_1 (l-m,k) } - \widehat{h}_2(l,m) , \quad \forall (l,m) \in \widehat{\mathcal{D}}_2 , \end{split}$$ where we have used the fact that $\widehat{h}_2(l,m) = \sum_{k=0}^m {\widehat{f}(k) \widehat{h}_1(l-k,m)} + \widehat{f}(m) \widehat{g}_1 (l-m,m) = \widehat{f}(m) \widehat{g}_1 (l-m,m)$, $\forall (l,m) \in \widehat{\mathcal{D}}_2$, according to [\[equation:recursive definition of h\]](#equation:recursive definition of h){reference-type="eqref" reference="equation:recursive definition of h"} and its initial condition. As a result, [\[equation:recurrence relation of joint PMF_iid\]](#equation:recurrence relation of joint PMF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PMF_iid"} is true for $n=2$. Inductive step: Assume that $$\widehat{g}_{\tau-1} (l,m) = (\tau-1) \widehat{f}(m) \sum_{\nu=0}^{m} { \widehat{g}_{\tau-2} (l-m,\nu) } - \widehat{h}_{\tau-1}(l,m) , \quad \forall (l,m) \in \widehat{\mathcal{D}}_{\tau-1} ,$$ for some arbitrary integer $\tau \geq 3$ (inductive hypothesis). Then, [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} for $n=\tau$, Lemma [Lemma 15](#lemma:Summation property of joint PMF){reference-type="ref" reference="lemma:Summation property of joint PMF"} for $n=\tau-1$, and [\[equation:recursive definition of h\]](#equation:recursive definition of h){reference-type="eqref" reference="equation:recursive definition of h"} for $n=\tau$ give $$\begin{split} \widehat{g}_\tau (l,m) &= \widehat{f}(m) \sum_{k=0}^m { \widehat{g}_{\tau-1} (l-m,k) } + \sum_{k=0}^{m} {\widehat{f}(k) \widehat{g}_{\tau-1} (l-k,m)} - {\widehat{f}(m) \widehat{g}_{\tau-1} (l-m,m)} \\ & = \widehat{f}(m) \sum_{k=0}^m { \widehat{g}_{\tau-1} (l-m,k) } + (\tau-1) \widehat{f}(m) \sum_{k=0}^m {\widehat{f}(k) \sum_{\nu=0}^m {\widehat{g}_{\tau-2}(l-k-m,\nu)}} \\ & \quad\, - \sum_{k=0}^m {\widehat{f}(k) \widehat{h}_{\tau-1}(l-k,m)} - \widehat{f}(m) \widehat{g}_{\tau-1} (l-m,m) \\ & = \widehat{f}(m) \sum_{k=0}^{m} { \widehat{g}_{\tau-1} (l-m,k) } + (\tau-1) \widehat{f}(m) \sum_{k=0}^{m} { \widehat{g}_{\tau-1} (l-m,k) } - \widehat{h}_\tau (l,m) \\ & = \tau \widehat{f}(m) \sum_{k=0}^{m} { \widehat{g}_{\tau-1} (l-m,k) } - \widehat{h}_\tau(l,m) , \quad \forall (l,m) \in \widehat{\mathcal{D}}_\tau . \end{split}$$ Note that the second equality is true on $\widehat{\mathcal{D}}_\tau$, since $\widehat{g}_{\tau-1} (l-k,m) = (\tau-1) \widehat{f}(m) \break \sum_{\nu=0}^{m} { \widehat{g}_{\tau-2} (l-k-m,\nu) } - \widehat{h}_{\tau-1}(l-k,m)$ on $\{(k,l,m) \in \mathbb{Z}^3 : m \geq 0, \; l \geq k, \; m \leq l-k \leq (\tau-1)m\} = \{(k,l,m) \in \mathbb{Z}^3 : m \geq 0, \; k+m \leq l \leq k+(\tau-1)m\}$ and the domain of summation is $0 \leq k \leq m$. Therefore, [\[equation:recurrence relation of joint PMF_iid\]](#equation:recurrence relation of joint PMF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PMF_iid"} holds for all integers $n \geq 2$. ◻ **Remark 17**. The direct application of [\[equation:recurrence relation of joint PMF\]](#equation:recurrence relation of joint PMF){reference-type="eqref" reference="equation:recurrence relation of joint PMF"} with $\widehat{f}_i(k) = \widehat{f}(k)$, $\forall i \in \mathbb{N}$, certainly gives a simpler recurrence relation than [\[equation:recurrence relation of joint PMF_iid\]](#equation:recurrence relation of joint PMF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PMF_iid"}. Nevertheless, Theorem [Theorem 16](#theorem:recurrence relation of joint PMF_iid){reference-type="ref" reference="theorem:recurrence relation of joint PMF_iid"} in comparison with Theorem [Theorem 7](#theorem:recurrence relation of joint PDF_iid){reference-type="ref" reference="theorem:recurrence relation of joint PDF_iid"} shows that there is a *key difference* between discrete and continuous random variables: the term $\widehat{h}_n(l,m)$ in [\[equation:recurrence relation of joint PMF_iid\]](#equation:recurrence relation of joint PMF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PMF_iid"} is *not* present at all in [\[equation:recurrence relation of joint PDF_iid\]](#equation:recurrence relation of joint PDF_iid){reference-type="eqref" reference="equation:recurrence relation of joint PDF_iid"}; see also Remark [Remark 14](#remark:Fundamental difference){reference-type="ref" reference="remark:Fundamental difference"}. ## Application: Peak-to-Average Ratio The peak-to-average ratio of the $n^\text{th}$ order is analogously defined by $$\widehat{\rho}_n \coloneqq\frac{\max_{1 \leq i \leq n} \widehat{X}_i}{\tfrac{1}{n} \sum_{i=1}^n {\widehat{X}_i}} = \frac{\widehat{Z}_n}{\tfrac{1}{n} \widehat{Y}_n} , \quad \text{whenever}\ \widehat{Y}_n \neq 0 .$$ The probability that $\widehat{\rho}_n$ belongs to the interval $[\alpha,\beta]$, $0 \leq \alpha \leq \beta$, is equal to $$\begin{split} \operatorname{P}(\alpha \leq \widehat{\rho}_n \leq \beta) & = \operatorname{P}\left(\widehat{Y}_n \geq 1, \left\lceil \tfrac{\alpha}{n} \widehat{Y}_n \right\rceil \leq \widehat{Z}_n \leq \left\lfloor \tfrac{\beta}{n} \widehat{Y}_n \right\rfloor \right) = \sum_{l=1}^{\infty} { \sum_{m=\left\lceil \tfrac{\alpha}{n} l \right\rceil}^{\left\lfloor \tfrac{\beta}{n} l \right\rfloor} {\widehat{g}_n (l,m)} } \\ & = \sum_{l=1}^{\infty} { \sum_{m=\max \left( \left\lceil \tfrac{\alpha}{n} l \right\rceil , \left\lceil \tfrac{1}{n} l \right\rceil \right)}^{\min \left( \left\lfloor \tfrac{\beta}{n} l \right\rfloor , l \right)} {\widehat{g}_n (l,m)} } = \sum_{l=1}^{\infty} { \sum_{m=\left\lceil \tfrac{\max(\alpha,1)}{n} l \right\rceil}^{\left\lfloor \min \left( \tfrac{\beta}{n} , 1 \right) l \right\rfloor} {\widehat{g}_n (l,m)} } , \end{split}$$ where $\widehat{g}_n(l,m)$ is computed recursively based on Theorem [Theorem 12](#theorem:Joint PMF){reference-type="ref" reference="theorem:Joint PMF"}. # Partial Differentiation Under the Integral Sign {#appendix:Partial Differentiation Under the Integral Sign} In this part of the paper, we provide a generalization of the Leibniz integral rule for partial differentiation of integrals with variable limits. For this purpose, the following two lemmas will be instrumental. **Lemma 18** (Leibniz's integral rule with constant limits of integration [@Lang Chapter X, Theorem 7.1.]). *Let $f(x,t)$ and $\pd{}{t} f(x,t)$ be defined and continuous on $\mathcal{X} \times \mathcal{T}$, where $\mathcal{X} \coloneqq[x_1,x_2]$ and $\mathcal{T} \coloneqq[t_1,t_2]$ are subsets of $\mathbb{R}$ with $-\infty < x_1 \leq x_2 < \infty$ and $-\infty < t_1 < t_2 < \infty$. Moreover, let $a,b \in \mathcal{X}$ be two real constants. Then, for all $t \in \mathcal{T}$, $$\dod{}{t} \left( \int_{a}^{b} f(x,t) \dif x \right) = {\int_{a}^{b} \dpd{}{t} f(x,t) \dif x} .$$* **Lemma 19** (Leibniz's integral rule with variable limits of integration). *Let $f(x,t)$ and $\pd{}{t} f(x,t)$ be defined and continuous on $\mathcal{X} \times \mathcal{T}$, where $\mathcal{X} \coloneqq[x_1,x_2]$ and $\mathcal{T} \coloneqq[t_1,t_2]$ are subsets of $\mathbb{R}$ with $-\infty < x_1 \leq x_2 < \infty$ and $-\infty < t_1 < t_2 < \infty$. In addition, let $a, b : \mathcal{T} \to \mathcal{X}$ be differentiable functions on $\mathcal{T}$. Then, for all $t \in \mathcal{T}$, $$\dod{}{t} \left( \int_{a(t)}^{b(t)} f(x,t) \dif x \right) = {\eval[2]{f(x,t)}_{x=b(t)} \cdot \dod{}{t}} b(t) - {\eval[2]{f(x,t)}_{x=a(t)} \cdot \dod{}{t}} a(t) + {\int_{a(t)}^{b(t)} \dpd{}{t} f(x,t) \dif x} .$$* *Proof.* This is a consequence of the fundamental theorem of calculus, the basic form of Leibniz's integral rule (Lemma [Lemma 18](#lemma:Leibniz's integral rule_constant limits){reference-type="ref" reference="lemma:Leibniz's integral rule_constant limits"}), and the chain rule for multivariate functions. Let us define the function $I(u,v,t) \coloneqq\int_v^u {f(x,t) \dif x}$, for all $(u,v,t) \in \mathcal{X}^2 \times \mathcal{T}$. From the fundamental theorem of calculus, we have $\pd{}{u} I(u,v,t) = f(u,t)$ and $\pd{}{v} I(u,v,t) = - f(v,t)$, while $\pd{}{t} I(u,v,t) = \int_v^u {\pd{}{t} f(x,t) \dif x}$ due to Lemma [Lemma 18](#lemma:Leibniz's integral rule_constant limits){reference-type="ref" reference="lemma:Leibniz's integral rule_constant limits"}. Now, by applying the chain rule, we obtain $$\begin{split} \dod{}{t} \left( \int_{a(t)}^{b(t)} f(x,t) \dif x \right) & = \dod{}{t} I(b(t),a(t),t) = \eval[3]{\dpd{}{u} I(u,v,t)}_{u=b(t),v=a(t)} \cdot \dod{}{t} b(t) \\ & \quad + {\eval[3]{\dpd{}{v} I(u,v,t)}_{u=b(t),v=a(t)} \cdot \dod{}{t} a(t)} + {\eval[3]{\dpd{}{t} I(u,v,t)}_{u=b(t),v=a(t)} \cdot \dod{}{t} t} \\ & = {\eval[2]{f(u,t)}_{u=b(t)} \cdot \dod{}{t}} b(t) - {\eval[2]{f(v,t)}_{v=a(t)} \cdot \dod{}{t}} a(t) + {\int_{a(t)}^{b(t)} \dpd{}{t} f(x,t) \dif x} , \end{split}$$ for all $t \in \mathcal{T}$. ◻ **Proposition 20**. *Let $h(x,y,z)$, $\pd{}{z} h(x,y,z)$, and $\md{}{2}{y}{}{z}{} h(x,y,z)$ be defined and continuous on $\mathcal{S} \coloneqq\mathcal{X} \times \mathcal{Y} \times \mathcal{Z}$, where $\mathcal{X} \coloneqq[x_1,x_2]$, $\mathcal{Y} \coloneqq[y_1,y_2]$, and $\mathcal{Z} \coloneqq[z_1,z_2]$ are subsets of $\mathbb{R}$ with $-\infty < x_1 \leq x_2 < \infty$, $-\infty < y_1 < y_2 < \infty$, and $-\infty < z_1 < z_2 < \infty$. Furthermore, suppose that $\pd{}{y} h(x,y,z)$ exists on $\mathcal{S}$, and $a, b : \mathcal{Z} \to \mathcal{X}$ are differentiable functions on $\mathcal{Z}$. Then, for all $(y,z) \in \mathcal{Y} \times \mathcal{Z}$, $$\begin{split} {\dmd{}{2}{y}{}{z}{} \left( \int_{a(z)}^{b(z)} h(x,y,z) \dif x \right)} & = {\eval[3]{\dpd{}{y} h(x,y,z)}_{x=b(z)} \cdot \dod{}{z}} b(z) - {\eval[3]{\dpd{}{y} h(x,y,z)}_{x=a(z)} \cdot \dod{}{z} a(z)} \\ & \quad + {\int_{a(z)}^{b(z)} \dmd{}{2}{y}{}{z}{} h(x,y,z) \dif x} . \end{split}$$* *Proof.* Firstly, we will compute the partial derivative of $\int_{a(z)}^{b(z)} h(x,y,z) \dif x$ with respect to the variable $z$. By virtue of Lemma [Lemma 19](#lemma:Leibniz's integral rule_variable limits){reference-type="ref" reference="lemma:Leibniz's integral rule_variable limits"} (extended to functions of three variables), we obtain $$\begin{split} {\dpd{}{z} \left( \int_{a(z)}^{b(z)} h(x,y,z) \dif x \right)} & = {\eval[2]{h(x,y,z)}_{x=b(z)} \cdot \dod{}{z}} b(z) - {\eval[2]{h(x,y,z)}_{x=a(z)} \cdot \dod{}{z} a(z)} \\ & \quad + {\int_{a(z)}^{b(z)} \dpd{}{z} h(x,y,z) \dif x} , \end{split}$$ for all $(y,z) \in \mathcal{Y} \times \mathcal{Z}$. Secondly, by differentiating the previous equation with respect to the variable $y$, the second-order mixed partial derivative of $\int_{a(z)}^{b(z)} h(x,y,z) \dif x$ is given by $$\begin{split} {\dmd{}{2}{y}{}{z}{} \left( \int_{a(z)}^{b(z)} h(x,y,z) \dif x \right)} & = {\eval[3]{\dpd{}{y} h(x,y,z)}_{x=b(z)} \cdot \dod{}{z}} b(z) - {\eval[3]{\dpd{}{y} h(x,y,z)}_{x=a(z)} \cdot \dod{}{z} a(z)} \\ & \quad + {\dpd{}{y} \left( {\int_{a(z)}^{b(z)} \dpd{}{z} h(x,y,z) \dif x} \right)} . \end{split}$$ Finally, we conclude the proof by applying Lemma [Lemma 18](#lemma:Leibniz's integral rule_constant limits){reference-type="ref" reference="lemma:Leibniz's integral rule_constant limits"} (extended to functions of three variables) to the last integral. ◻ **Remark 21**. Proposition [Proposition 20](#proposition:Partial Differentiation Under the Integral Sign){reference-type="ref" reference="proposition:Partial Differentiation Under the Integral Sign"} remains valid if we replace the assumption that $\md{}{2}{y}{}{z}{} h(x,y,z)$ is continuous on $\mathcal{S}$ with the assumption that ${\int_{a(z)}^{b(z)} \md{}{2}{y}{}{z}{} h(x,y,z) \dif x}$ converges uniformly on $\mathcal{Y} \times \mathcal{Z}$. This is because Lemma [Lemma 18](#lemma:Leibniz's integral rule_constant limits){reference-type="ref" reference="lemma:Leibniz's integral rule_constant limits"} still holds when the continuity of $\pd{}{t} f(x,t)$ on $\mathcal{X} \times \mathcal{T}$ is replaced by the uniform convergence of $\int_{a}^{b} \pd{}{t} {f(x,t) \dif x}$ on $\mathcal{T}$, according to [@Talvila; @Rogers]. # Dirac Delta Function: Definition and Properties {#appendix:Dirac Delta Function} Loosely speaking, the Dirac delta function/distribution $\delta(x)$ can be regarded as a function defined on $\mathbb{R}$ that is zero everywhere except at $x=0$, where it is infinite, and also satisfies the following properties: $\delta(-x) = \delta(x) \geq 0$, $\int_{-\infty}^{+\infty} {\delta(t) \dif t} = \int_{-\varepsilon}^{+\varepsilon} {\delta(t) \dif t} = 1$, $\int_{-\infty}^{+\infty} {f(t) \delta(x-t) \dif t} = \int_{x-\varepsilon}^{x+\varepsilon} {f(t) \delta(x-t) \dif t} = f(x)$, and $f(x) \delta(x-y) = f(y) \delta(x-y)$ for all $x,y \in \mathbb{R}$, $\varepsilon \in \mathbb{R}_+$ and for all functions $f(x)$ that are *bounded and piecewise continuous on $\mathbb{R}$*. Note that $\delta(x)$ is not a function in the conventional sense, because no function has these properties. Nevertheless, the Dirac delta function can be rigorously defined as a generalized function or distribution. Strictly speaking, the Dirac delta function $\delta(x)$ is the limit (in the sense of distributions) of a Dirac sequence $\{ \delta_k (x) \}_{k \in \mathbb{N}}$, that is, $\delta(x) = \lim_{k \to \infty} \delta_k(x)$. **Definition 22** ([@Lang Chapter XI, §1]). A *Dirac sequence* is a sequence of functions $\{ \delta_k (x) \}_{k \in \mathbb{N}}$ that satisfy three properties: P.1) $\delta_k(-x) = \delta_k(x) \geq 0$ for all $k \in \mathbb{N}$ and for all $x \in \mathbb{R}$, P.2) $\delta_k (x)$ is continuous on $\mathbb{R}$ and $\int_{-\infty}^{+\infty} {\delta_k(x) \dif x} = 1$ for all $k \in \mathbb{N}$, and P.3) for any $\epsilon, \zeta > 0$, there exists $K \in \mathbb{N}$ such that $\int_{-\infty}^{-\zeta} {\delta_k(x) \dif x} + \int_{\zeta}^{+\infty} {\delta_k(x) \dif x} < \epsilon$ for all integers $k \geq K$. In simple words, P.3 means that the area under the curve $y = \delta_k(x)$ is concentrated around $x=0$ for sufficiently large $k$. For example, the sequence of zero-mean normal/Gaussian distributions $\delta_k(x) = \frac{k}{\sqrt{\pi}} e^{-(kx)^2}$ is a Dirac sequence. The following proposition is an important result in analysis about the convolution of a function with a Dirac sequence; the convolution of two functions $f(x)$ and $g(x)$ is defined by $f*g \coloneqq\int_{-\infty}^{+\infty} {f(t) g(x-t) \dif t}$ and is commutative, that is, $f*g = g*f$. **Proposition 23** ([@Lang Chapter XI, Theorem 1.1.]). *Suppose that $f(x)$ is a bounded and piecewise-continuous function on $\mathbb{R}$, $\mathcal{D}$ is a compact subset of $\mathbb{R}$ on which $f(x)$ is continuous, and $\{ \delta_k (x) \}_{k \in \mathbb{N}}$ is a Dirac sequence. Then the sequence of functions $\{ f_k (x) \}_{k \in \mathbb{N}}$, where $f_k (x) \coloneqq f*\delta_k = \int_{-\infty}^{+\infty} {f(t) \delta_k(x-t) \dif t}$, converges to $f(x)$ uniformly on $\mathcal{D}$.* Therefore, if $f(x)$ is bounded and piecewise continuous on $\mathbb{R}$, then $\int_{-\infty}^{+\infty} {f(t) \delta(x-t) \dif t} = f(x)$ rigorously means that: for each Dirac sequence $\{ \delta_k (x) \}_{k \in \mathbb{N}}$, $\lim_{k \to \infty} {\int_{-\infty}^{+\infty} {f(t) \delta_k(x-t) \dif t}} = f(x)$ uniformly on every compact subset of $\mathbb{R}$ where $f(x)$ is continuous. 11 T. L. Chow, J. L. Teugels, *"The sum and the maximum of i.i.d. random variables,"* Proceedings of the 2nd Prague Symposium on Asymptotic Statistics, pp. 81-92, 1978. C. W. Anderson, K. F. Turkman, *"The joint limiting distribution of sums and maxima of stationary sequences,"* Journal of Applied Probability, vol. 28, no. 1, pp. 33-44, 1991. C. W. Anderson, K. F. Turkman, *"Limiting joint distributions of sums and maxima in a statistical context,"* Theory of Probability and Its Applications, vol. 37, no. 2, pp. 314-316, 1993. T. Hsing, *"A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables,"* The Annals of Probability, vol. 23, no. 2, pp. 938-947, 1995. T. Hsing, *"On the asymptotic independence of the sum and rare values of weakly dependent stationary random variables,"* Stochastic Processes and Their Applications, vol. 60, no. 1, pp. 49-63, 1995. H.-C. Ho, T. Hsing, *"On the asymptotic joint distribution of the sum and maximum of stationary normal random variables,"* Journal of Applied Probability, vol. 33, no. 1, pp. 138-145, 1996. W. P. McCormick, Y. Qi, *"Asymptotic distribution for the sum and maximum of Gaussian processes,"* Journal of Applied Probability, vol. 37, no. 4, pp. 958-971, 2000. Z. Peng, S. Nadarajah, *"On the joint limiting distribution of sums and maxima of stationary normal sequence,"* Theory of Probability and Its Applications, vol. 47, no. 4, pp. 706-709, 2003. F. Qeadan, T. J. Kozubowski, A. K. Panorska, *"The joint distribution of the sum and the maximum of iid exponential random variables,"* Communications in Statistics -- Theory and Methods, vol. 41, no. 3, pp. 544-569, 2012. M. Arendarczyk, T. J. Kozubowski, A. K. Panorska, *"The joint distribution of the sum and the maximum of heterogeneous exponential random variables,"* Statistics and Probability Letters, vol. 139, pp. 10-19, 2018. M. Arendarczyk, T. J. Kozubowski, A. K. Panorska, *"The joint distribution of the sum and maximum of dependent Pareto risks,"* Journal of Multivariate Analysis, vol. 167, pp. 136-156, 2018. P. G. Moschopoulos,*"The distribution of the sum of independent gamma random variables,"* Annals of the Institute of Statistical Mathematics, vol. 37, no. 1, pp. 541-544, 1985. J. A. Woodward, C. G. S. Palmer, *"On the exact convolution of discrete random variables,"* Applied Mathematics and Computation, vol. 83, no. 1, pp. 69-77, 1997. K. Butler, M. A. Stephens, *"The distribution of a sum of independent binomial random variables,"* Methodology and Computing in Applied Probability, vol. 19, pp. 557-571, 2017. N. Balakrishnan, *"Recurrence relations for order statistics from n independent and non-identically distributed random variables,"* Ann. Inst. Statist. Math., vol. 40, no. 2, pp. 273-277, 1988. N. Balakrishnan, S. M. Bendre, H. J. Malik, *"General relations and identities for order statistics from non-independent non-identical variables,"* Ann. Inst. Statist. Math., vol. 44, no. 1, 177-183, 1992. H. A. David, H. N. Nagaraja, *Order statistics.* John Wiley & Sons, New Jersey, 2003. M. Morrison, F. Tobias, *"Some statistical characteristics of a peak to average ratio,"* Technometrics, vol. 7, no. 3, pp. 379-385, 1965. S. Lang, *Undergraduate analysis.* Springer-Verlag, New York, 1997. E. Talvila, *"Necessary and sufficient conditions for differentiating under the integral sign,"* The American Mathematical Monthly, vol. 108, no. 6, pp. 544--548, 2001. K. Rogers, *Advanced calculus.* Merrill, Columbus, 1976.

arxiv_math

--- abstract: | A new spectral conjugate subgradient method is presented to solve nonsmooth unconstrained optimization problems. The method combines the spectral conjugate gradient method for smooth problems with the spectral subgradient method for nonsmooth problems. We study the effect of two different choices of line search, as well as three formulas for determining the conjugate directions. In addition to numerical experiments with standard nonsmooth test problems, we also apply the method to several image reconstruction problems in computed tomography, using total variation regularization. Performance profiles are used to compare the performance of the algorithm using different line search strategies and conjugate directions to that of the original spectral subgradient method. Our results show that the spectral conjugate subgradient algorithm outperforms the original spectral subgradient method, and that the use of the Polak--Ribière formula for conjugate directions provides the best and most robust performance. author: - bibliography: - hybridOpt.bib title: A New Spectral Conjugate Subgradient Method with Application in Computed Tomography Image Reconstruction --- Nonsmooth optimization; conjugate gradients; spectral subgradient; computed tomography. 90C30, 90C56, 94A08 # Introduction The problem under consideration is the minimization of nonsmooth functions without any constraints $$\label{probm} \min_{x\in\mathbb{R}^{n}}~~ f(x),$$ where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is locally Lipschitz continuous in its domain and differentiable on an open dense subset of $\mathbb{R}^n$. We are interested in the specific case where $f$ is continuously differentiable almost everywhere and not differentiable at minimizers. To solve [\[probm\]](#probm){reference-type="eqref" reference="probm"}, Loreto *et al.* in [@Loreto:19] proposed the spectral subgradient method. It combines a nonmonotone line search [@GriLaLu:86], a globalization scheme [@wmr2:05], and the classical subgradient approach using the the Barzilai-Borwein (spectral) step length [@BarBor:88]. The significant advantage of the spectral step is that its computation is independent of the optimal function value. In [@Loreto:19], extensive numerical experimentation was presented, comparing six classic subgradient methods with the spectral subgradient with the method of performance profiles [@Dolan02]. The numerical experiments demonstrated the spectral subgradient method's superiority compared to these methods. Moreover, Loreto and collaborators have explored combining the spectral step with subgradients and other subdifferential approaches. In [@Loreto:07], they developed the spectral projected subgradient method for the minimization of non-differentiable functions on convex sets. Successful numerical results were presented for two distinct problem types: set covering and generalized assignment problems, and convergence results were shown later in [@Loreto:15] and [@Loreto:16]. In [@Loreto:17], they developed and analyzed the nonsmooth spectral gradient method for unconstrained problems in which the spectral step is combined with subdifferential approaches, such as the simplex gradient and the sampling gradient. The spectral subgradient method represents an extension to the nonsmooth case of the spectral gradient method developed by Raydan [@Raydan:93; @Raydan:97]. In [@Raydan:97], the author found that the use of gradient descent with spectral step length for unconstrained (smooth) nonlinear optimization provided several advantages over approaches based on the conjugate gradient method for this problem. Inspired by this, Birgin & Martı́nez proposed a spectral conjugate gradient (SCG) method [@Bm01], which combines the spectral step with conjugate gradient directions. They studied several formulas for determining the conjugate directions (which we describe in Section [2.1](#sec:2.1){reference-type="ref" reference="sec:2.1"}), and determined that the use of the spectral step provided improved results over the standard conjugate gradient method. Several modifications to SCG have been proposed subsequently. Andrei proposed the SCALCG algorithm [@andrei07a; @andrei07b], including a quasi-Newton step to ensure the production of descent directions. In [@zzl06; @dc08], the authors propose instead modifying the choice of spectral step to guarantee descent, and prove convergence using both Armijo and Wolfe line searches. Subsequent authors have built off these early works by proposing modified formulas for generating search directions with desirable properties, e.g.[@zz08; @CL10; @andrei2010new; @LP12; @dwc13; @LFZ19]. All of these extensions to SCG assume $f$ is continuously differentiable. It is interesting to note that an extension of the original conjugate gradient method to minimize nondifferentiable functions using subgradients was first presented many years ago by Wolfe [@wolfe74; @wolfe75]. In the interim, however, there appears to have been little work done in the area of using conjugate subgradients for nonsmooth problems, despite some recent papers [@NT14; @KSM19]. In this paper, we introduce the spectral conjugate subgradient (SCS) method as a novel approach for solving nonsmooth problems. The inspiration for this method stems from the SCG approach. The SCS algorithm represents the first extension of spectral CG methods to nonsmooth problems, to the best of our knowledge. Our primary objective is to expand the application of spectral methods to nonsmooth problems, given the success of smooth spectral methods in solving a wide range of practical problems (Birgin et al. [@Bmr09; @Bmr14]) The organization of this paper is as follows: Section 2 provides a description of the SCS method, while Section 3 introduces an application of interest which involves the use of the nonsmooth total variation (TV) function as a regularizer for several image reconstruction problems in computed tomography (CT). Section 4 presents numerical results based on a set of nonsmooth problems, including the CT problem, using performance profiles. Finally, Section 5 offers some concluding remarks. # Methodology {#S:methodology} ## Spectral Conjugate Gradient Method {#sec:2.1} The original spectral CG method for unconstrained optimization was proposed by Birgin & Martı́nez [@Bm01], under the assumption that the objective function f is continuously differentiable at all points. Letting $g_k$ denote the gradient at the $k$th iterate $x_k$, the vectors $s_k=x_{k+1} -x_{k}$ and $y_{k}=g_{k+1}-g_k$ are defined for $k=0,1,2,\dots$. Beginning from an initial guess $x_0$, the subsequent iterates are given by $$x_{k+1}= x_k + \alpha_kd_k,$$ where $\alpha_k$ is selected using a Wolfe line search (described in the next section), and the conjugate directions $d_k\in \mathbb{R}^n$ are generated by $$d_{k+1} = -\theta_kg_{k+1} +\beta_ks_k. \label{E:searchdir}$$ The spectral step parameter, denoted by $\theta_{k}$, was initially proposed by Barzilai and Borwein [@BarBor:88] and subsequently refined by Raydan [@Raydan:93; @Raydan:97]: $$\label{formula} \theta_{k} = \frac{s_{k}^T s_{k}}{s_{k}^T y_{k}},$$ with $\theta_0$ typically chosen to be 1. As the algorithm requires $\theta_k$ to be non-negative and bounded, a safeguarding procedure is often used (e.g. [@Loreto:19]): $$% \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} \label{proc2.1} \theta_{k+1} = \left\{ \begin{array}{ll} \min\{\theta_{\max},\displaystyle\frac{1}{\parallel s_{k}\parallel}\} & \mathrm{if \ } s_{k}^Ty_{k} \le 0 \\ \min\{\theta_{\max},\max\{\theta_{\min},\displaystyle\frac{s_{k}^Ts_{k}}{s_{k}^Ty_{k}}\}\} & \mathrm{if\ } s_{k}^Ty_{k}>0 \end{array}, \right.$$ where $\displaystyle 0<\theta_{\min}<\theta_{\max}<\infty$. The parameter $\beta_k$ can be chosen in several ways; in [@Bm01], the authors proposed the following three choices: $$\begin{aligned} \label{formula1} \beta_k&=\frac{(\theta_ky_k-s_k)^Tg_{k+1}}{s_k^Ty_k}\\ \label{formula2} \beta_k&=\frac{\theta_ky_k^Tg_{k+1}}{\alpha_k\theta_{k-1}g_k^Tg_k}\\ \label{formula3} \beta_k&=\frac{\theta_kg_{k+1}^Tg_{k+1}}{\alpha_k\theta_{k-1}g_k^Tg_k}\end{aligned}$$ noting that under certain assumptions, these are equivalent to the formula introduced by Perry [@Perry:78] and its further modifications by Polak--Ribière and Fletcher--Reeves, respectively. With any of these choices, it is possible that the search direction $d_{k+1}$ computed by [\[E:searchdir\]](#E:searchdir){reference-type="eqref" reference="E:searchdir"} may fail to be a descent direction; in this case, the authors proposed setting $d_{k+1} = -\theta_k g_{k+1}$ to "restart" the algorithm. In their paper, Birgin & Martı́nez studied the performance of several variants of this approach, including the three choices for $\beta_k$, using $\theta_k = 1$ instead of the spectral step (corresponding to the classical conjugate gradient approach), and a heuristic for the initial choice of stepsize $\alpha$ when performing the Wolfe line search. Ultimately it was determined that the Perry formula [\[formula1\]](#formula1){reference-type="eqref" reference="formula1"} using the spectral step and the heuristic for Wolfe line search gave the best performance. This work presents an extension of the spectral conjugate gradient method that allows for $g_k$ to be a subgradient at points where $f$ is non-differentiable. A subgradient of $f$ at a point $x$ is any vector $g$ that satisfies the inequality $f(y)\geq f(x) +g^T(y-x)$ for all $y$. This algorithm can also be interpreted as a generalization of the spectral subgradient method  [@Loreto:19], which corresponds to the special case where $\beta_k = 0$ in equation  [\[E:searchdir\]](#E:searchdir){reference-type="eqref" reference="E:searchdir"}. Additionally, we investigate two different line search strategies: the nonmonotone line search used by the spectral subgradient method, and the Wolfe line search used in [@Bm01]. We describe these two choices of line search in the next section. ## Line Search Strategies {#sec:2.2} As mentioned previously, the proposed spectral conjugate subgradient can use one of two line search strategies: nonmonotone, or Wolfe line searches. ### Nonmonotone Line Search {#sec:2.2.1} Historically, the spectral step has been paired with the nonmonotone line search described below, based on Raydan's work [@Raydan:97]. The reason for this pairing is the nonmonotone behavior of the step. So, it is natural to adjust the line search scheme with the nonmonotone globalization strategy of Grippo *et al.* [@GriLaLu:86] combined with the proposed globalization scheme of La Cruz *et al.* [@wmr2:05]. $$\begin{aligned} \label{glb1} f(x_k+\alpha d_k) &\leq \max_{0 \leq j \leq \min\{k,M\}} f(x_{k-j}) + \gamma \alpha g_k^Td_k + \eta_k, \text{with}\\ \label{propeta} 0 < \sum_k \eta_k &= \eta < \infty.\end{aligned}$$ The value of $\alpha>0$ is determined by a backtracking process beginning with $\alpha=1$. For the sequence $\eta_k$, we use $\displaystyle\eta_k=\frac{\eta_0}{k^{1.1}}$, which guarantees that ([\[propeta\]](#propeta){reference-type="ref" reference="propeta"}) holds. This sequence helps the line search to be satisfied at early iterations. $M$ is a fixed integer with $M \geq 0$, and $0 < \gamma < 1$ is a small sufficient decrease parameter. The nonmonotone behavior of the line search is induced by the terms $\displaystyle\max_{0 \leq j \leq \min\{k,M\}} f(x_{k-j})$ and $\eta_k$. The parameter $M$ allows the function value to increase or remain stationary for up to $M$ iterations, which is suitable for the behavior of the spectral step, since the spectral step is associated with the eigenvalues of the Hessian at the minimum point, rather than the function values, as noted by  [@fletcher:1990; @glunt:1993]. ### Wolfe Line Search {#sec:2.22} In [@Bm01], the authors use the Wolfe conditions to guarantee convergence. Although the Wolfe conditions assume that $f$ is differentiable, we included the Wolfe line search option in our algorithm to be compared with the nonmonotone line search, since we assume $f$ is differentiable almost everywhere. Given $0<\gamma<\sigma<1$, we require that the step size $\alpha$ satisfy $$\label{glb2} f(x_k + \alpha d_k) \leq f(x_k) + \gamma \alpha g_k^Td_k$$ and $$\label{glb3} \nabla f(x_k +\alpha d_k)^Td_k\geq \sigma g_k^Td_k$$ in every iteration, where $g_k$ is a subgradient of $f$ at $x_k$. ## Spectral Conjugate Subgradient Algorithm {#sec:2.3} Using the definitions of $s_k$ and $y_k$ from Section [2.1](#sec:2.1){reference-type="ref" reference="sec:2.1"}, but where $g_k$ is now a subgradient, we define the spectral conjugate subgradient (SCS) algorithm as follows:\ **Algorithm SCS:**\ Let $x_0\in \mathbb{R}^n$ be given, and let $g_0$ represent a subgradient at $f(x_0)$. We define $d_0=-g_0$, and set $k=0$ and $MAXITER$ as the maximum allowable number of iterations. Let $M$ be an integer with $M \geq 0$, let $0<\theta_{\min}<\theta_{\max}<\infty$, let $\eta_{0}=\max(f(x_0),\parallel g_0\parallel)$, and let $0<\gamma<\sigma<1$. Then:\ Repeat until $k=MAXITER$ Compute $\alpha$ based on the line search, either nonmonotone ([\[glb1\]](#glb1){reference-type="ref" reference="glb1"}), or Wolfe (([\[glb2\]](#glb2){reference-type="ref" reference="glb2"}) and ([\[glb3\]](#glb3){reference-type="ref" reference="glb3"})) Define $\alpha_k=\alpha$ and $x_{k+1}=x_k+\alpha_kd_k$ Compute $\theta_k$ by ([\[proc2.1\]](#proc2.1){reference-type="ref" reference="proc2.1"}) and $\beta_k$ by ([\[formula1\]](#formula1){reference-type="ref" reference="formula1"}),([\[formula2\]](#formula2){reference-type="ref" reference="formula2"}), or ([\[formula3\]](#formula3){reference-type="ref" reference="formula3"}) Define $d = -\theta_kg_{k+1} +\beta_ks_k$ If $d^Tg_{k+1}\leq -10^{-3}\parallel d \parallel_2 \parallel g_{k+1}\parallel_2$ then $d_{k+1}=d$ else $d_{k+1}=-\theta_kg_{k+1}$ $k=k+1$ **Remarks:** - Step 5 is the heuristic suggested by Birgin & Martı́nez to guarantee $d$ is a descent direction. If the angle between $d$ and $-g_{k+1}$ lies outside of $[-\frac{\pi}{2}, \frac{\pi}{2}]$ (to some small tolerance), the algorithm is restarted to use the direction of the negative subgradient. - For values of $x_{k+1}$ where $f$ is differentiable, the subgradient $g_{k+1}$ is equal to the gradient $\nabla f(x_{k+1})$. - When $\beta =0$ and the nonmonotone line search ([\[glb1\]](#glb1){reference-type="ref" reference="glb1"}) is used, the SCS algorithm becomes the spectral subgradient method described in [@Loreto:19]. - When using the Perry formula [\[formula1\]](#formula1){reference-type="eqref" reference="formula1"} for $\beta_k$, it is possible in nonsmooth problems for the denominator $s_k^Ty_k$ to be zero. In this case we simply set $\beta_k = 0$, equivalent to the spectral subgradient method. - For the Wolfe line search (([\[glb2\]](#glb2){reference-type="ref" reference="glb2"}) and ([\[glb3\]](#glb3){reference-type="ref" reference="glb3"})), we adopted the implementation by Taviani [@Taviani] - The algorithm terminates upon reaching the maximum iteration count ($MAXITER$), and the best value $f_{\min}$ is reported, along with the corresponding point $x$. # CT problem with Total Variation {#sec:3} In computed tomography (CT) imaging, a two- or three-dimensional image of a compactly supported function is recovered from a discrete sampling of the so-called X-ray transform of the function. This transform consists of integrals along every possible line through the function; in the two-dimensional case, which we consider in this article, it is equivalent to the better-known Radon transform. The simplest type of two-dimensional discretization, known as parallel-beam geometry, is obtained by equal spacing of both the affine and angular parameters in the X-ray transform, producing a set of measurements known as a sinogram. Figure [1](#F:CTimg1){reference-type="ref" reference="F:CTimg1"} shows an example of a simple two-dimensional object (the Shepp-Logan phantom) and its sinogram. Discretizing both the domain of the function and the X-ray transform results in a linear system of equations: $$Ax + \eta = b,$$ where $x\in\mathbb{R}^{n}$ is the non-negative image being reconstructed, $b \in \mathbb{R}^m$ is the sinogram, and $A$ is the $m \times n$ system matrix representing the geometry of the CT system. The vector $\eta$ represents measurement noise that arises naturally during the real-life CT imaging process. When $\Vert \eta \Vert$ is small, and the X-ray transform of the object is well-sampled, the image can be reconstructed by solving the non-negative least-squares problem: $$\min_{x\in\mathbb{R}^{n}_+}~~\frac{1}{2} \Vert Ax - b \Vert^2, \label{E:lsq}$$ typically using an iterative method, due to the large size of the matrices involved. In more challenging circumstances, one must instead solve a regularized problem: $$\min_{x\in\mathbb{R}^{n}_+}~~\frac{1}{2} \Vert Ax - b \Vert^2 + \mu \phi(x),\label{E:lsqreg}$$ where $\phi: \mathbb{R}^n \to \mathbb{R}$ is an appropriate regularization function, and $\mu > 0$ is a weighting parameter. Two scenarios of this type are if $\Vert \eta \Vert$ is large, or if the angular sampling of the X-ray transform is insufficient to recover the high-frequency components of $x$. Both of these situations arise as a result of trying to reduce dose to a patient in medical CT imaging; the former is often called low-dose imaging, and the latter is sparse-view imaging. In both cases, solving the unregularized problem [\[E:lsq\]](#E:lsq){reference-type="eqref" reference="E:lsq"} results in poor quality images, as shown in Figure [1](#F:CTimg1){reference-type="ref" reference="F:CTimg1"}. {#F:CTimg1 width="\\linewidth"} A common choice of regularizer $\phi(x)$ is the isotropic total variation (TV) function: $$\phi_{TV}(x) = \sum_{1 \leq i,j \leq N-1} \sqrt{\left(x_{i,j+1}-x_{i,j}\right)^2 + \left(x_{i+1,j}-x_{i,j}\right)^2},\label{E:TV}$$ in which $x$ is reshaped as an $N \times N$ image (with $N^2 = n$), indexed by row and column. A routine computation gives the partial derivatives of $\phi_{TV}$ as: $$\begin{aligned} \frac{\partial}{\partial x_{i,j}} \phi_{TV} (x) &= \frac{2x_{i,j} - x_{i+1,j} - x_{i,j+1}}{\sqrt{ \left(x_{i+1,j}-x_{i,j}\right)^2 + \left(x_{i,j+1}-x_{i,j}\right)^2}} \label{E:dTV}\\ &~~~+ \frac{x_{i,j} - x_{i,j-1}}{\sqrt{(x_{i,j} - x_{i,j-1})^2 + (x_{i+1,j-1} - x_{i,j-1})^2}}\notag \\ &~~+ \frac{x_{i,j} - x_{i-1,j}}{\sqrt{(x_{i,j} - x_{i-1,j})^2 + (x_{i-1,j+1} - x_{i-1,j})^2}}.\notag\end{aligned}$$ Clearly the derivative is undefined whenever the denominator of any of the three terms in [\[E:dTV\]](#E:dTV){reference-type="eqref" reference="E:dTV"} is zero. This situation arises frequently as it occurs when pixels have the same values as their neighbors; furthermore, such images are likely to be minimizers of [\[E:lsqreg\]](#E:lsqreg){reference-type="eqref" reference="E:lsqreg"} as they correspond to images with small TV. Note, however, that the numerator of the respective term must also be zero whenever the denominator is zero. Following [@ND10], we therefore define a subgradient by simply setting any term in [\[E:dTV\]](#E:dTV){reference-type="eqref" reference="E:dTV"} with a denominator of zero equal to zero. # Numerical Results {#sec:4} We provide numerical results that demonstrate the performance of the proposed method using a collection of widely recognized nonsmooth benchmark problems frequently employed for evaluation purposes. Additionally, we showcase an application in CT reconstruction with the utilization of total variation, as outlined in Section [3](#sec:3){reference-type="ref" reference="sec:3"}. ## Nonsmooth test problems {#sec:4.1} The collection of problems consists of 10 nonsmooth minimization problems, taken from  [@Napsu07] and detailed in Table [1](#table:tst14){reference-type="ref" reference="table:tst14"}. This table provides information such as the optimal value $f_* = f(x_*)$ and the dimension $n$. For further details on this problem set, including suggested initial points, please refer to [@Napsu07]. Prob $f_*$ n -------------------- ----------------- ---- P1: MAXQ 0.0 20 P2: MXHILB 0.0 50 P3: Chained LQ $-(n-1)2^{1/2}$ 2 P4: Chained CB3 I $2(n-1)$ 20 P5: Chained CB3 II $2(n-1)$ 20 P6: Activefaces 0.0 2 P7: Brown 2 0.0 2 P8: Mifflin 2 -34.795 50 P9: Crescent I 0.0 2 P10:Crescent II 0.0 2 : Problem Set We are comparing a total of 8 versions of the spectral conjugate subgradient method, corresponding to two choices of line search (nonmonotone or Wolfe) and four different approaches to calculate the parameter $\beta$ in [\[E:searchdir\]](#E:searchdir){reference-type="eqref" reference="E:searchdir"}. We let $\beta_0=0$ correspond to the original spectral subgradient method, and $\beta_1$, $\beta_2$ and $\beta_3$ correspond to the formulas ([\[formula1\]](#formula1){reference-type="ref" reference="formula1"}), ([\[formula2\]](#formula2){reference-type="ref" reference="formula2"}), and ([\[formula3\]](#formula3){reference-type="ref" reference="formula3"}) respectively. Each algorithm was run for $MAXITER = 1000$ iterations, and the the best function value, called $f_{\min}$, was reported. We evaluate the performance of each algorithm by analyzing the results using performance profiles [@Dolan02]. A detailed explanation of how we adopted the profiles is given in [@Loreto:19]. Considering a set of solvers $S$, which in this scenario includes the 8 variations of the aforementioned algorithm, and the set of problems $P$ given in Table [1](#table:tst14){reference-type="ref" reference="table:tst14"}, the performance ratio for each solver-problem pair is defined as follows: $$r_{p,s}=\frac{t_{p,s}}{\min\{t_{p,s}:s \in S\}},$$ where the performance measure $t_{p,s}$ represents a specific metric, e.g. computation time required by solver $s$ to solve problem $p$. The performance of solver $s$ is then assessed by computing a function $$\rho_s(\tau)=\frac{1}{n_p}size\{p\in P:r_{p,s} \leq \tau \}.$$ Here $n_p$ denotes the total number of problems in the set $P$, and $\tau$ is a parameter ranging between 1 and $r_M$. The max value $r_M$ is determined to satisfy $r_M\geq r_{p,s}$ for all $p$ in the set $P$ and $s$ in the set $S$. This function is then plotted for all solvers $s$ to produce the performance profiles; the best-performing algorithms have profiles in the upper left of the plot, and worse-performing algorithms in the lower right. Typically a log scale is used for $\tau$ for ease of visualization; then the value of $\rho_s(\tau)$ at zero (when $\tau = 1$) provides the percentage of problems in which solver $s$ achieved the best performance. In our experiments, we focus on three different performance metrics: the relative error between $f_{\min}$ and $f_{*}$ (if $f_{*}$ is non-zero), or the absolute error if $f_{*}$ is zero; the total number of function evaluations needed to obtain $f_{\min}$; and total CPU time consumed by the algorithm. An algorithm is considered to solve a problem $p$ if the error between $f_{\min}$ and $f_{*}$ is less than $10^{-1}$. If solver $s$ fails to solve problem $p$, $r_{p,s}$ is set equal to $r_M$. To represent the 8 methods in the performance profiles, we use the notation $WB_i$ for the Wolfe line search and the four choices of $\beta$, ($i=1,\ldots 4$). Likewise, we set the notation $NMB_i$ for the nonmonotone line search. The performance profile using error as the performance measure is shown in Figure ([\[fig:figure1\]](#fig:figure1){reference-type="ref" reference="fig:figure1"}). The plot shows the algorithm with Wolfe line search and parameter $\beta = 0$, $WB_0$, as the most precise solver overall since it solves accurately $60\%$ of the problems, and $70\%$ of them if a small error is tolerated. However, $WB_0$ didn't solve the other $30\%$ of the problems based on our criteria. This makes $WB_0$ the most accurate algorithm but the least robust. On the other hand, $NMB_2$ solves $100\%$ of the problems when some error is tolerated, though only $20\%$ with maximum accuracy. This makes $NMB_2$ the most robust algorithm. It is worth mentioning $WB_2$ as a good alternative to combine accuracy and robustness, solving accurately $50\%$ of the problems, and $90\%$ of them when some error is tolerated. We note that $NMB_0$, originally presented at [@Loreto:19], is outperformed by both $WB_0$ and $NMB_2$ in terms of accuracy and robustness, respectively. {width="0.85\\linewidth"} . [\[fig:figure2\]]{#fig:figure2 label="fig:figure2"} {width="0.85\\linewidth"} Figure [\[fig:figure2\]](#fig:figure2){reference-type="ref" reference="fig:figure2"} shows the performance profile based on total number of function evaluations needed to find $f_min$. It shows the spectral subgradient method $NMB_0$[@Loreto:19] as the most efficient algorithm. $NMB_0$ solves $80\%$ of the problems with the fewest function evaluations. Similarly, $NMB_1$ presents as an efficient option, slightly worse than $NMB_0$. However, $NMB_0$ and $NMB_1$ were unable to solve one problem to within the specified threshold. On the other hand, $NMB_2$ solves $100\%$ of the problems to within the specified threshold, while requiring only somewhat more function evaluations than $NMB_0$ and $NMB_1$. This would make the trade-off robustness versus efficiency worthy, making $NMB_2$ the best alternative in our opinion. Figure [\[fig:figure2\]](#fig:figure2){reference-type="ref" reference="fig:figure2"} also shows that the nonmonotone line search considerably outperforms the Wolfe line search with respect to the number of function evaluations required. Indeed, solvers using Wolfe line search are clearly on the bottom right side of the chart, meaning they solve the fewest problems at the highest cost. It is worth mentioning that Figure [\[fig:figure1\]](#fig:figure1){reference-type="ref" reference="fig:figure1"} showed $WB_0$ as the most precise solver and $WB_2$ as a suitable alternative in terms of precision. But Figure [\[fig:figure2\]](#fig:figure2){reference-type="ref" reference="fig:figure2"} shows them both as the most expensive in terms of function evaluations, which makes these options unattractive. {width="0.85\\linewidth"} With respect to CPU time, Figure [\[fig:figure3\]](#fig:figure3){reference-type="ref" reference="fig:figure3"} shows quite similar behavior when compared to the profile based on function evaluations (Figure [\[fig:figure2\]](#fig:figure2){reference-type="ref" reference="fig:figure2"}). This outcome is anticipated since evaluating the function is the primary factor that drives the computational effort of these algorithms. $NMB_2$ seems to be the best alternative, being more robust than $NMB_0$ and $NMB_1$, given it solves $100\%$ of the problems, although slightly worse performance. After analyzing all performance profiles, we can conclude that $NMB_2$ is the best alternative when factoring in all the criteria. This is based on its high robustness, as it was the only method that solved all ten problems to within the specified threshold. $NMB_2$ also provides solutions with accuracy comparable with the rest of the solvers, although clearly outperformed by $WB_0$ on this measure. And, $NMB_2$ is in the group of top performing solvers based on computation time, with only slightly poorer performance than $NMB_0$ and $NMB_1$. In conclusion, we can state that the Wolfe line search tends to improve the accuracy of the algorithms, but clearly at expense of many more function evaluations, and hence computational effort. ## CT problem with total variation {#sec:4.3} In this section, we study the effect of the choice of parameter $\beta$ when applying Algorithm SCS to a set of TV-regularized CT reconstruction problems. All experiments were run using the AIRTools II Matlab toolbox [@HJ18] to generate parallel-beam CT system matrices $A$ and the corresponding sinograms $b$. An image size of $400 \times 400$ pixels was used, resulting in a problem size of $n = 160,000$. We then applied Algorithm SCS to the regularized least squares problem [\[E:lsqreg\]](#E:lsqreg){reference-type="eqref" reference="E:lsqreg"}, using TV [\[E:TV\]](#E:TV){reference-type="eqref" reference="E:TV"} as the regularizer, whose subgradient was computed using [\[E:dTV\]](#E:dTV){reference-type="eqref" reference="E:dTV"} and the subsequently discussed method. A total of 24 reconstruction problems were considered, corresponding to: - Three different images (phantoms) from the AIRTools II toolbox: `shepplogan`, `threephases`, and `grains`. All three phantoms are piecewise constant, making them well-suited to total variation regularization. They are also all elements of the $n$-dimensional box $[0,1]^n \subset \mathbb{R}^n$. - Four different imaging scenarios: two low-dose scenarios with Gaussian noise of 5% and 10% added to $b$, and two sparse-view scenarios using 60 and 30 angular views. In the two low-dose scenarios, a total of 360 views and 566 lines through the object were simulated, giving a sinogram size of $m = 203,760$; this is reduced by a factor of 6 and 12, respectively, for the two sparse-view scenarios. The sparse-view scenarios did not include noise. - Two different values of $\mu$, controlling the weighting of the regularization term in [\[E:lsqreg\]](#E:lsqreg){reference-type="eqref" reference="E:lsqreg"}. For the two low-dose scenarios, these were $\mu = 25$ and $\mu = 250$, while for the sparse-view scenarios, we used $\mu = 0.5$ and $\mu = 5$. These values were chosen empirically to give generally good results, while providing different weighting on the nonsmooth (TV) component of the function. Note that in the sparse-view scenarios, the size of the first term in [\[E:lsqreg\]](#E:lsqreg){reference-type="eqref" reference="E:lsqreg"} is smaller than in the low-dose scenarios (since the sinogram size is smaller), which is why a smaller value of $\mu$ can be used. Additionally, because the desired solution to each problem is a vector $x \in [0,1]^n$, we modified Algorithm SCS to include projection onto this set (denoted $\Omega$). This is accomplished as follows: before every iteration of Step 1, redefine $d_k$ as $$d_k = P_{\Omega} (x_k + d_k) - x_k,$$ where $P_{\Omega}(x) = \min\{\max\{x,0\},1\}$ is the projection onto $\Omega$. This guarantees that $x_{k} \in \Omega$ for all $k$, provided that the step size $\alpha_k$ does not exceed 1. As this is not the case for the Wolfe line search, in this section we only use the nonmonotone line search with $M=7$. We, therefore, consider only four algorithms, namely NM$\beta_0$ to NM$\beta_3$ from the previous section. We note that the use of Wolfe line search is prohibitively expensive for this problem in any event, due to the high computational cost of function evaluations. Algorithm SCS was run for 200 total iterations for every case, beginning from $x_0 = 0 \in \mathbb{R}^n$. Figure [\[F:CTprofileFbest\]](#F:CTprofileFbest){reference-type="ref" reference="F:CTprofileFbest"} shows the performance profile based on the lowest objective function value, $f_{\min}$ found by each solver for each problem. Note that this differs from the measure used for the nonsmooth test problems (relative or absolute difference between $f_{\min}$ and the true optimum $f_*$), since $f_*$ is unknown for these problems. A solver was considered to have solved the problem only if $f_{\min}$ was within 10% of the lowest value found. We observe that the choice of $\beta_2$ provides the best overall results, solving 80% of the problems to a high degree of accuracy; furthermore, it was the only solver to solve all problems within the 10% threshold. The choice of $\beta_3$ was the next best-performing, followed by $\beta_1$ and finally $\beta_0$ (the spectral subgradient method), which was competitive on only about 45% of the problems, and failed to solve nearly 50% to the required tolerance. {width="0.85\\linewidth"} Figure [\[F:CTprofileFevals\]](#F:CTprofileFevals){reference-type="ref" reference="F:CTprofileFevals"} shows the performance profile generated for the four solvers, using the total number of function evaluations as the performance measure. The profile based on CPU time is omitted, as the results were essentially the same. Unlike for the nonsmooth test problems (Figure [\[fig:figure2\]](#fig:figure2){reference-type="ref" reference="fig:figure2"}), we observe that the SCS algorithm using $\beta_2$ required significantly fewer function evaluations than when using $\beta_1$ or $\beta_3$. As a total of 200 iterations was run for each solver, this indicates that the direction chosen as a result of using $\beta_2$ generally resulted in less backtracking than the other methods. We note that while the choice of $\beta_0$ typically resulted in the fewest number of function evaluations needed, it was also unsuccessful in solving nearly half the problems, as discussed in the previous paragraph. {width="0.85\\linewidth"} Figure [\[F:CTimages_final\]](#F:CTimages_final){reference-type="ref" reference="F:CTimages_final"} shows the best images reconstructed by Algorithm SCS using $\beta_2$. We can observe that the reconstructed images are generally of high quality, although there are still noticeable artifacts in some of the images corresponding to more challenging scenarios (third and fifth columns). It is possible that increasing the value of the weighting parameter, $\mu$, would improve image quality in these cases. {width="\\linewidth"} # Final Remarks {#sec:5} In this article, we develop a new Spectral Conjugate Subgradient (SCS) method for nonsmooth unconstrained optimization, by combining conjugate directions proposed by Birgin & Martı́nez [@Bm01] and the spectral subgradient method [@Loreto:19]. We investigate the use of two different line search strategies and several choices of the parameter $\beta$ used to determine conjugate directions. We present the results of numerical experiments with standard nonsmooth test problems, as well as problems from CT imaging using total variation regularizations. The results are examined utilizing performance profiles, which enable the comparison of various approaches in terms of precision and computational cost. The numerical results show that the combination of the nonmonotone line search with the parameter $\beta_2$ (analogous to the Polak--Ribière formula), which we denote by $NMB_2$, was the most successful approach. This is based on its high robustness, given that it solved all the problems to within a specified threshold, for both the nonsmooth test problems and the CT imaging problems. Additionally, the computational effort required by $NMB_2$ was on par with the least expensive approaches in both sets of experiments. It is interesting to note that in Birgin & Martı́nez' original article they found that the choice of $\beta_1$ (Perry's formula) was the best option, while the best option in our experiments was $\beta_2$. It is difficult to compare the results directly, given that their experiments were only on smooth problems, using Wolfe line search. However, a possible explanation for why the choice of $\beta_2$ works better in our experiments is that the formula for $\beta_1$ has a denominator of $s_k^Ty_k$, which is more often equal to zero when solving nonsmooth problems than for differentiable problems. Indeed, in our numerical experimentation we noticed this term is often zero for some problems. In this case, we set $\beta_1 = 0$, defaulting to the spectral subgradient approach. On the other hand, when the problem is differentiable, the Wolfe line search ensures that $s_k^Ty_k > 0$, avoiding this issue. Finally, further investigation into the convergence properties of the SCS method would be worthwhile. Loreto *et al.* [@Loreto:19] presented convergence results on the spectral subgradient method, and so did Birgin for the spectral conjugate gradient method. A combination of these existing convergence results, with some additional hypotheses could be a potential path to prove the convergence of the SCS. # Disclosure statement {#disclosure-statement .unnumbered} The authors do not have any significant financial or non-financial interests to declare. # Funding {#funding .unnumbered} The National Science Foundation provided support for this work (grant DMS-2150511). # Notes on contributor(s) {#notes-on-contributors .unnumbered} M. Loreto and T. Humphries co-authored the manuscript. C. Raghavan, K. Wu and S. Kwak participated in a Research Experience for Undergraduates (REU) funded by the above-mentioned NSF Grant in Summer 2022, providing initial implementation of the spectral conjugate subgradient method and preliminary experimentation. # Data Availability Statement {#data-availability-statement .unnumbered} The authors can provide the MATLAB code and data used in this work upon request. # References

arxiv_math

--- abstract: | Set-membership estimation is commonly used in adaptive/learning-based control algorithms that require robustness over the model uncertainty sets, e.g., online robustly stabilizing control and robust adaptive model predictive control. Despite having broad applications, non-asymptotic estimation error bounds in the stochastic setting are limited. This paper provides such a non-asymptotic bound on the diameter of the uncertainty sets generated by set membership estimation on linear dynamical systems under bounded, i.i.d. disturbances. Further, this result is applied to robust adaptive model predictive control with uncertainty sets updated by set membership. We numerically demonstrate the performance of the robust adaptive controller, which rapidly approaches the performance of the offline optimal model predictive controller, in comparison with the control design based on least square estimation's confidence regions. author: - | Yingying Li$^{*}$, Jing Yu$^{*}$, Lauren Conger, Adam Wierman\ Computing and Mathematical Sciences, California Institute of Technology\ `{yingli2, jing, lconger, adamw}@caltech.edu` bibliography: - citation4setmember.bib title: "Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis" --- # Introduction {#sec: introduction} The problem of estimating unknown linear dynamical systems of the form $x_{t+1}=A^*x_t+B^* u_t+w_t$ with unknown parameters $(A^*, B^*)$ has seen considerable progress recently, with many works deriving finite-time estimation bound for the unknown system matrices. Most of this literature focuses on the analysis of the least squares estimator (LSE) and its variants. Recently, modern machine learning techniques have been applied to obtain high-probability guarantees of the LSE in finite time [@faradonbeh2018finite; @sarkar2019near; @wagenmaker2020active; @mania2019certainty; @foster2020learning; @sarker2023accurate], where sharp bounds of the convergence rate for LSE were obtained [@simchowitz2018learning; @foster2020logarithmic; @li2023non]. As a result, there is now a growing literature on "learning to control" unknown linear systems that leverages least-square *point estimation* (LSE), with extensive results for various control objectives such as stability [@lale2022reinforcement; @kargin2022thompson] and regret [@mania2019certainty; @dean2018regret; @lale2020explore]. Such results pioneered modern machine learning techniques for adaptive and data-driven control design. However, in order to apply such learning-based control methods to safety-critical control applications where robust safety requirements such as robust stability and robust constraint satisfaction are commonly required, it is crucial to quantify the uncertainty of the estimated system. The use of an uncertainty set is central to important classes of control methods such as robust model predictive control (RMPC). Too large of an uncertainty set gives rise to conservative control actions, resulting in poor performance. If the uncertainty set is underestimated, the resulting controller may lead to unsafe behavior. Therefore, works have begun to incorporate the confidence region of the LSE for safe adaptive and learning-based control [@tanaskovic2014adaptive; @bujarbaruah2018adaptive; @li2021safe; @li2023non]. Parallel to least-square point estimation-based works, another line of research turns to more direct *uncertainty set estimation* methods in the bounded disturbance setting [@milanese1991optimal]. Among the algorithms for direct uncertainty set estimation, the set membership (SM) method, which identifies the set of systems models that are consistent with observed data, is one of the most commonly used. The SM method can generate much smaller uncertainty sets than the LSE confidence region, as demonstrated in , by leveraging knowledge of the bound on the disturbances. Therefore, numerous robust controller designs and empirical algorithm designs leverage SM for uncertainty set estimation. For example, RMPC algorithms such as [@milanese1991optimal; @adetola2011robust; @guay2015robust; @tanaskovic2013adaptive] have long been employing SM for performance improvement, which is sometimes called robust adaptive MPC and has seen a growing literature recently [@lorenzen2019robust; @bujarbaruah2020adaptive; @zhang2021trajectory; @parsi2020robust; @parsi2020active; @sasfi2022robust]. In addition, SM has also been utilized to stabilize systems despite adversarial disturbances [@ho2021online; @yu2023online; @yeh2022robust; @yu2023ltv]. {#fig:diameter_d2_teaser width="\\textwidth"} {#fig:a1a3 width="\\textwidth"} {#fig:a2a4 width="\\textwidth"} On the theory side, there is a long history of theoretical analysis of the SM method for both the deterministic setting where $w_t$ is a deterministic sequence [@bai1995membership; @fogel1982value; @kitamura2003convergence; @milanese2013bounding; @lauricella2020set; @eising2022using; @livstone1996asymptotic] and the stochastic setting where $w_t$ is independent or even i.i.d. [@bai1998convergence; @lauricella2020set; @akccay2004size; @kitamura2005size; @bai1995membership; @lu2019robust]. However, in the stochastic setting, most papers consider a simpler regression problem: $y_t=\theta^*x_t +w_t$ with a deterministic sequence of ${x}_t$, which does not capture the correlation between $x_t$ and history $w_{t-1}, \dots, w_0$ in the dynamical systems [@bai1998convergence; @akccay2004size; @kitamura2005size; @bai1995membership]. This issue was largely overlooked in the vast empirical algorithm design research related to SM (for example, see [@lorenzen2019robust; @kohler2019linear], etc.). Not until recently, this unsolved correlation problem on SM resurfaced. Work in [@lu2019robust] provides an initial attempt to establish convergence results on SM for dynamical systems; however, their approach requires a special design of $u_t$ to satisfy persistent excitation requirements deterministically, which can be challenging to satisfy for general control designs in the stochastic settings. Therefore, an open question is: *Can convergence and convergence rates be established for SM on linear dynamical systems under general control design in the stochastic settings?* #### Contributions. In this paper, to the best of our knowledge, we provide the first finite-time convergence rate bound for the set membership estimation error () for linear dynamical systems with bounded i.i.d. disturbances. Our bound assumes that the control inputs $u_t$ can guarantee the block-martingale small-ball (BMSB) condition in [@simchowitz2018learning], which can be satisfied by adding i.i.d. noises to a general class of controllers as discussed in [@li2021safe; @li2023non]. Under proper assumptions, we show that the estimation error decays to 0 at a rate of $\tilde O(n_x^{1.5}(n_x+n_u)^2/T)$ when the bound on the disturbances $w_t$ is tight. Interestingly, this is better than the LSE's error bound $O(\frac{\sqrt{n_x+n_u}}{\sqrt T})$ in terms of the dependence on the number of samples $T$ but worse in terms of the state and control dimensions $n_x,n_u$. SM's improvement on the rate with respect to $T$ is enabled by taking advantage of the boundedness of $w_t$. Even when the bound on $w_t$ is not accurately known but overly estimated, we can show that SM's estimation error converges to a neighborhood of 0 with the same decay rate. Technically, our estimation error bound relies on a novel construction of an event sequence based on designing a sequence of stopping times. This construction, together with the BMSB condition in [@simchowitz2018learning], addresses the challenge caused by the correlation between $x_t, u_t$, and the history disturbances. For more details, we refer the reader to the proof of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"}. Moreover, our results lay a foundation for future non-asymptotic analysis of control designs based on SM. To illustrate this, we discuss applications of our results to autonomous system identification and robust-adaptive model predictive control. In each case, we provide numerical comparisons with LSE that demonstrate the promise of SM-based approaches. # Problem Formulation and Preliminaries {#sec: problem formulation} This paper considers an uncertainty set identification problem for linear dynamical systems with unknown model parameters, i.e., $$\begin{aligned} \label{equ: time series} x_{t+1}=A^* x_t +B^*u_t+w_t %y_t = \theta^* z_t +w_t, \quad t\geq 0\end{aligned}$$ where $A^*, B^*$ are unknown, $x_t \in \mathbb R^{n_x}, u_t \in \mathbb R^{n_u}$. For notational simplicity, we define $\theta^*=(A^*, B^*)$ and $z_t=(x_t^\top, u_t^\top)^\top$, where we write $(A, B)$ as the concatenation of two matrices $A, B$ (the same applies to the vector concatenation). So the system [\[equ: time series\]](#equ: time series){reference-type="eqref" reference="equ: time series"} can also be written as $x_{t+1}=\theta^*z_t +w_t$. We define $n_z=n_x+n_u$. The goal of the uncertainty set identification problem is to determine a set $\Theta_T$ that contains the true parameters $(A^*, B^*)$ based on a sequence of data $\{x_t, u_t, x_{t+1}\}_{t=0}^{T-1}$. Set $\Theta_T$ is called an uncertainty set since it captures the remaining uncertainty on the system model after the revelation of the data sequence $\{x_t, u_t, x_{t+1}\}_{t=0}^{T-1}$. Uncertainty sets play an important role in robust control, where one aims to achieve robust stability and/or robust constraint satisfaction for any model in the uncertainty set.[^1] Therefore, the diameter of the uncertainty sets heavily influences the conservativeness of robust controllers, thus affecting the control performance. Formally, we define the diameter as follows. **Definition 1** (Diameter of a set of matrices). *Consider a set $\mathbb S$ of matrices $\theta\in \mathbb R^{n_x\times n_z}$. We define the diameter of $\mathbb S$ in Frobenius norm as $\textup{diam}(\mathbb S)=\sup_{\theta, \theta'\in \mathbb S}\|\theta-\theta'\|_F$.* #### Set Membership. We focus on a widely adopted uncertainty set identification method: set membership, and provide non-asymptotic bounds for estimation error under proper assumptions. In particular, we consider bounded disturbances, i.e., there exists a bounded set $\mathcal W$ such that $w_t\in \mathcal W$ for all $t\geq 0$. When $\mathcal W$ is known, the set membership method estimates the uncertainty set by $$\label{eq:membership_set} \Theta_T=\bigcap_{t=0}^{T-1} \{ \hat \theta : x_{t+1} -\hat \theta z_t \in \mathcal W\}.$$ We also refer to [\[eq:membership_set\]](#eq:membership_set){reference-type="eqref" reference="eq:membership_set"} as the *membership set*. Notice that $\theta^*\in \Theta_T$ for all $T\geq 0$ for arbitrary sequence of $w_t\in \mathcal W$ without assuming any (stochastic) properties on $w_t$. Ideally, one hopes $\Theta_T$ converges to the singleton of the true model $\{\theta^*\}$ or at least a small neighborhood of $\theta^*$. This usually calls for additional assumptions, such as the persistent excitation property on the observed data and additional stochastic properties on $w_t$. In this paper, we consider the following assumptions to establish convergence rate bounds on the diameter of $\Theta_T$. **Assumption 1** (Bounded i.i.d disturbances). *The disturbances are box-constrained, $w_t\in \mathcal W:=\{w\in \mathbb R^{n_x}: \|w\|_\infty \leq w_{\max}\}$ for all $t\geq 0$. Further, $w_t$ is i.i.d.. Let $\mu_w$ and $\Sigma_w \succ 0$ denote the mean and covariance of $w_t$.* In many applications, it is realistic to assume the disturbances are bounded, e.g., wind perturbation for aerial robotics and temperature fluctuations for building control. Therefore, it is advantageous to leverage such prior knowledge about the disturbances for system estimation and control design beyond the sub-Gaussian stochasticity assumption commonly seen in the system identification literature. Next, we introduce the assumptions on $u_t$, which relies on the block-martingale small-ball condition proposed in [@simchowitz2018learning]. It can be shown that the block-martingale small-ball condition can guarantee persistent excitation with high probability under proper conditions (see [@simchowitz2018learning Proposition 2.5] and ). In the system identification literature, the persistent excitation (PE) condition is essential for ensuring that the estimation error decays. The PE condition essentially ensures that $(x_t^\top, u_t^\top)$ explores all directions of the state space. **Definition 2** (Persistent excitation). *There exists $\alpha>0$ and $m \in \mathbb N+$, such that for any $t_0\geq 0$, $$\frac{1}{m}\sum_{t=t_0}^{t_0+m-1} {\left(\begin{matrix} x_t\\ u_t \end{matrix}\right)} (x_t^\top, u_{{t}}^\top) \succeq \alpha^2 I_{n_x+n_u}.$$* **Definition 3** (BMSB [@simchowitz2018learning] ). *Let $\{\mathcal F_t\}_{t\geq 1}$ denote a filtration and let $\{Z_t\}_{t\geq 1}$ be an $\{\mathcal F_t\}_{t\geq 1}$-adapted random process taking values in $\mathbb R^d$. We say that $\{Z_t\}_{t\geq 1}$ satisfies the $(k, \Gamma_{sb},p)$-block martingale small-ball (BMSB) condition for a positive integer $k$, a positive definite matrix $\Gamma_{sb}\succ 0$, and $0\leq p \leq 1$, if the following condition holds: for any fixed $\lambda \in\mathbb R^d$ such that $\|\lambda\|_2=1$, the process $\{Z_t\}_{t\geq 1}$ satisfies $\frac{1}{k}\sum_{i=1}^k \mathbb P(|\lambda^\top Z_{t+i}|\geq \sqrt{\lambda^\top\Gamma_{sb} \lambda}\mid \mathcal F_t)\geq p$ almost surely for any $t\geq 1$.* Our bound depends on the following assumption. **Assumption 2** (BMSB and boundedness). *There exists $b_z\geq 0$ such that $\|z_t\|_2\leq b_z$ almost surely for all $t\geq 0$. Further, considering filtration $\mathcal F_t=\mathcal F(w_0, \dots, w_{t-1}, z_0, \dots, z_t)$, then the $\mathcal F_t$-adapted stochastic process $\{z_t\}_{t\geq 0}$ satisfies $(1, \sigma_{z}^2I_{n_z}, p_{z})$-BMSB for some $p_z>0$.* Assumption [Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"} can be satisfied in many scenarios. For example, the boundedness of $z_t$ can be achieved by robust stabilizing controllers [@tu2019sample; @yu2023online] and robust constraint-satisfying controllers with bounded constraint sets, such as robust model predictive controllers, e.g., [@rawlings2017model]. Further, the BMSB condition can be achieved by inserting a bounded i.i.d. random noise to any control policies $\pi_t$, i.e., $u_t=\pi_t(x_t)+\eta_t$, where $\eta_t$ and $w_t$ have positive definite covariance matrices [@li2021safe Theorem 1]. Finally, we assume that the bound $w_{\max}$ on $w_{t}$ is tight in all directions. **Assumption 3** (Tight bound on $w_t$). *For any $\epsilon>0$, there exists $q_w(\epsilon)>0$, such that for any $1\leq j \leq n$ and $t\geq 0$, we have $$\mathbb P(w_t^j+w_{\max}\leq \epsilon)\geq q_w(\epsilon)>0, \quad \mathbb P(w_{\max}-w_t^j\leq \epsilon)\geq q_w(\epsilon)>0.$$* Assumption [Assumption 3](#ass: tight bound on wt){reference-type="ref" reference="ass: tight bound on wt"} holds on many different types of distributions, e.g., uniform distributions on $\mathcal W$ or truncated Gaussian distributions on $\mathcal W$. For example, it can be verified that for uniform distributions on $\mathcal W$, $q_w(\epsilon)=\frac{\epsilon}{2w_{\max}}$; and for truncated Gaussian distributions with 0 mean, covariance $\sigma_w^2 I_n$, and truncated region $\mathcal W$, $q_w(\epsilon)=\frac{\epsilon}{2w_{\max}\sigma_w} \exp(\frac{-w_{\max}^2}{2\sigma_w^2})$ (see appendix for formal proofs).[^2] Another common scenario is that even though $w_t$ is bounded by $\mathcal W$, the system designer may only know a loose upper bound $\hat w_{\max}\geq w_{\max}$. In this case, we will show that the membership set converges to a small region around $\theta^*$ with the same convergence rate. **Remark 1** (Comparison with least square estimation). *On the one hand, compared with the assumptions for least square estimation's convergence rate, set membership requires additional assumptions on bounded $\mathcal W$ and tight bound $w_{\max}$ to establish convergence rate. On the other hand, if we do not aim for convergence (rates), SM generates valid uncertainty sets, i.e., $\theta^*\in \Theta_T$, even without any stochastic assumptions, while LSE's uncertainty set is determined based on confidence region, which only holds under proper stochastic properties.* # Set Membership Convergence Analysis We now present the main result () of this paper, which is a non-asymptotic bound on the estimation errors of the SM method given bounded i.i.d. stochastic disturbances. **Theorem 1** (Diameter bound on the membership set). *For any $m>0$ any $\delta>0$, when $T>m$, we have $$\begin{aligned} \mathbb P(\textup{diam}(\Theta_T)>\delta)\leq \underbrace{\frac{T}{m} \tilde O(n_z^{2.5}) a_2^{n_z} \exp(-a_3m)}_{\textup{Term 1}}+ \underbrace{\tilde O((n_xn_z)^{2.5})a_4^{n_xn_z}\left(1-q_w\left(\frac{a_1\delta}{4\sqrt{n_x}}\right)\right)^{\ceil{T/m}} }_{\textup{Term 2}} \end{aligned}$$ where $a_1 = \frac{\sigma_{z} p_{z}}{4}$, $a_2=\frac{64 b_z^2}{\sigma_{z}^2 p_{z}^2}$, $a_3= \frac{p_{z}^2}{8}$, $a_4=\max(\frac{4b_z \sqrt{n_x}}{a_1},1)$, $p_z, \sigma_z, w_{\max}, b_z$ are as defined in -[Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"}, and $\ceil{\cdot}$ denotes the ceiling function.* Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} provides an upper bound on the "failure" probability of the SM method, i.e., the probability that the diameter of the membership set is larger than $\delta$. In this bound, Term 1 decays exponentially with $m$, so for any small $\epsilon>0$, $m$ can be chosen such that $\text{Term 1}\leq \epsilon$, which indicates $m\geq O(n_z+\log T+\log(1/\epsilon))$. Term 2 decays exponentially with the number of data points $T$ and involves a distribution-dependent function $q_w(\cdot)$, which characterizes how likely it is for $w_t$ to visit the boundary of $\mathcal W$ as defined in . If $w_t$ is more likely to visit the boundary, i.e. $q_w(\cdot)$ is larger, then SM method is less likely to generate an uncertainty set with a diameter bigger than $\delta$. #### Estimation error bounds when $q_w(\epsilon)=O(\epsilon)$. To provide intuition for Term 2 and discuss the estimation error bound in more explicitly, we consider distributions satisfying $q_w(\epsilon)=O(\epsilon)$ for all $\epsilon>0$. Notice that several common distributions satisfy this additional requirement, such as uniform distribution and truncated Gaussian distribution as discussed after Assumption [Assumption 3](#ass: tight bound on wt){reference-type="ref" reference="ass: tight bound on wt"}. **Corollary 1** (Estimation error bound when $q_w(\epsilon)=O(\epsilon)$). *For any $\epsilon>0$, let $$m\geq O(n_z+\log T+\log(1/\epsilon)).$$ If $w_t$ is generated i.i.d. by a distribution satisfying $q_w(\epsilon)=O(\epsilon)$ for all $\epsilon>0$, then with probability at least $1-2\epsilon$, for any $\hat \theta_T\in\Theta_T$, we have $$\|\hat \theta_T -\theta^*\|_F\leq \textup{diam}(\Theta_T) \leq \tilde O\left(\frac{n_x^{1.5}(n_x+n_u)^{2}}{ T}\right).$$* Corollary [Corollary 1](#cor: uniform dist est. bdd){reference-type="ref" reference="cor: uniform dist est. bdd"} indicates that the estimation error of any point in the membership set $\Theta_T$ can be bounded by $\tilde O\left(\frac{n_x^{1.5}(n_x+n_u)^{2}}{ T}\right)$ when $q_w(\epsilon)\geq O(\epsilon)$. #### Loose Bound on $w_{\max}$. In many practical scenarios, it is easier to obtain an over-estimation of the range of the disturbances instead of a tight upper bound. Hence, we discuss the scenarios when we only know an over-estimation $\hat w_{\max}\geq w_{\max}$. In this case, we can further show that the membership set converges to a small neighborhood around $\theta^*$ with the same convergence rate as . **Theorem 2** (Loose bound on $w_{\max}$). *Consider the membership set $$\hat \Theta_T=\bigcap_{t=0}^{T-1} \{ \hat \theta : \|x_{t+1} -\hat \theta z_t \|_\infty\leq \hat w_{\max}\},$$ where $\hat w_{\max}$ is a loose upper bound for the disturbances such that $\hat w_{\max}\geq w_{\max}$. There exists a constant $\epsilon_0=O((\hat w_{\max}-w_{\max})\sqrt n_x)$ such that for any $m>0$, $\delta>0$, $T>m$, we have $$\begin{aligned} \mathbb P(\textup{diam}(\hat\Theta_T)>\delta+\epsilon_0)\leq \textup{Term 1}+ \textup{Term 2} \end{aligned}$$ where [Term1]{.upright} and [Term2]{.upright} are defined in Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"}.* Similarly to Corollary [Corollary 1](#cor: uniform dist est. bdd){reference-type="ref" reference="cor: uniform dist est. bdd"}, we can establish $\|\hat \theta_T-\theta^*\|_F\leq \epsilon_0+\tilde O(n_x^{1.5} n_z^2/T)$ when $q_w(\epsilon)=O(\epsilon)$ for any $\epsilon>0$. This establishes a convergence rate towards an $\epsilon_0$-neighborhood around $\theta^*$. ## Proof Sketch of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} {#subsec: proof sketch Thm1} A formal proof of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} is provided in the supplementary material. Here, we provide a sketch to highlight the key technical ideas of the proof. Specifically, we first define a set $\Gamma_T$ on the model estimation error $\gamma=\hat \theta-\theta^*$ by leveraging the observation that $x_{s+1}-\hat \theta z_s=w_s-(\hat \theta -\theta^*) z_s$, $$\begin{aligned} \label{equ: define Gamma t} \Gamma_t=\bigcap_{s=0}^{t-1} \{\gamma : \|w_s -\gamma z_s \|_\infty \leq w_{\max}\}, \quad \forall\, t\geq 0.\end{aligned}$$ Notice that $\Theta_t =\theta_*+\Gamma_t$, so $\textup{diam}(\Theta_t)=\textup{diam}(\Gamma_t)$, and $$\textup{diam}(\Gamma_t)=\sup_{\gamma,\gamma'\in \Gamma_t} \|\gamma-\gamma'\|_F\leq 2\sup_{\gamma\in \Gamma_t}\|\gamma\|_F.$$ Therefore, we can upper bound our goal event $\{\textup{diam}(\Theta_T)>\delta\}$ by the event $\mathcal E_1$ defined below. $$\begin{aligned} \label{eq:middle} {\mathbb P}(\textup{diam}(\Theta_T)>\delta) \leq {\mathbb P}(\mathcal E_1), \text{ where } \mathcal E_1:=\{\exists\, \gamma \in \Gamma_T, \text{ s.t. } \|\gamma\|_F\geq \frac{\delta}{2} \}.%\frac{\delta}{2} \}.$\end{aligned}$$ Next, we define an event $\mathcal E_2$ below, which is essentially persistent excitation (PE) on every time segments $km\leq t \leq km+m-1$ for $k\geq 0$, where the choice of $m$ will be specified later. $$\begin{aligned} \mathcal E_2= \{\frac{1}{m}\sum_{s= {1}}^{ {m}} {z}_{km+s} {z}_{km+s}^\top \succeq a_1^2 I_{ {n_z}}, \ \forall\, 0\leq k \leq \ceil{T/m}-1 \}\end{aligned}$$ where $a_1 =\frac{\sigma_{z} p_{z}}{4}$. For ease of notation and without loss of generality, we assume $T/m$ is an integer in the following analysis. The proof can now be completed by taking the intersection of $\mathcal E_1$ with $\mathcal E_2$ and $\mathcal E_2^c$ in [\[eq:middle\]](#eq:middle){reference-type="eqref" reference="eq:middle"}. Specifically, from [\[eq:middle\]](#eq:middle){reference-type="eqref" reference="eq:middle"} we have $${\mathbb P}(\textup{diam}(\Theta_T)>\delta)\leq {\mathbb P}(\mathcal E_1\cap\mathcal E_2^c) + {\mathbb P}( \mathcal E_1\cap \mathcal E_2 )\leq {\mathbb P}(\mathcal E_2^c)+ {\mathbb P}( \mathcal E_1\cap \mathcal E_2 ).$$ The proof concludes by invoking the following bounds on ${\mathbb P}(\mathcal E_2^c)$ and ${\mathbb P}( \mathcal E_1\cap \mathcal E_2 )$. **Lemma 1** (Bound on ${\mathbb P}(\mathcal E_2^c)$). *$$\begin{aligned} {\mathbb P}(\mathcal E_2^c) \leq \frac{T}{m} \tilde O(n_{ {z}}^{2.5}) a_2^{n_{ {z}}} \exp(-a_3 m)\end{aligned}$$ where $a_2=\frac{64 {b_z^2}}{\sigma_{z}^2 p_{z}^2}$ and $a_3= \frac{p_{z}^2}{8}$.* **Lemma 2** (Bound on ${\mathbb P}( \mathcal E_1\cap \mathcal E_2 )$). *$$\begin{aligned} {\mathbb P}( \mathcal E_1\cap \mathcal E_2 ) \leq \tilde O(n_z^{2.5}n_x^{2.5})a_4^{n_x n_z} (1-q_w(\frac{a_1 \delta}{4\sqrt n_x}))^{T/m} \end{aligned}$$ where $a_4=\max(1, 4b_z\sqrt{n_x}/a_1)$ and $n_z=n_x+n_u$.* Roughly speaking, Lemma [Lemma 1](#lem: bound PE c){reference-type="ref" reference="lem: bound PE c"} indicates that PE holds with high probability. This is proved by leveraging the BMSB assumption and set discretization. The proof of Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"} is more involved and is our major technical contribution. On a high level, the proof relies on the following two technical lemmas. **Lemma 3** (Discretization of $\mathcal E_1\cap\mathcal E_2$ (Informal)). *Let $\mathcal M=\{\gamma_1, \dots, \gamma_{v_\gamma}\}$ denotes an $\epsilon_\gamma$-net of $\{\gamma: \|\gamma\|_F=1\}$. Under a proper choice of $\epsilon_\gamma$, we have $v_\gamma = \tilde O( {n_x^{2.5}}n_z^{2.5})a_4^{n_x n_z}$, and we can construct $\tilde \Gamma_T$,[^3] such that $${\mathbb P}(\mathcal E_1\cap\mathcal E_2)\leq {\mathbb P}(\{\exists\, 1\leq i \leq v_\gamma, d\geq 0, \textup{ s.t. }d\gamma_i \in \tilde \Gamma_T\}\cap \mathcal E_2) \leq \sum_{i=1}^{v_\gamma} {\mathbb P}(\mathcal E_{1, {i}}\cap \mathcal E_2)$$ where $\mathcal E_{1, {i}}=\{\exists\, d\geq 0, \textup{ s.t. }d\gamma_i \in \tilde \Gamma_T\}$.* Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"} leverages finite set discretization to bound the existence of a feasible element in an infinite continuous set. **Lemma 4** (Construction of event $G_{i,k}$ via stopping times). *Under the conditions in Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"}, we can construct $G_{i,k}$ for all $i$ and all $0\leq k \leq T/m-1$ by $$\begin{aligned} G_{i,k}=\{b_{i, km+L_{i,k}} w_{km+L_{i,k}}^{j_{i, km+L_{i,k}}} \geq \frac{a_1\delta}{4\sqrt{n_x}}-w_{\max}, \textup{ and } \frac{1}{m}\sum_{s= {1}}^{ {m}} {z}_{km+s} {z}_{km+s}^\top \succeq a_1^2 I_{ {n_z}}\}. \end{aligned}$$ where ${1}\leq L_{i,k}\leq {m}$ is constructed as a stopping time with respect to $\{\mathcal F_{km+l}\}_{l\geq 0}$ and $b_{i,t}, j_{i,t}\in \mathcal F_t$.* *It can be shown that $${\mathbb P}(\mathcal E_{1,i}\cap \mathcal E_2)\leq \mathbb P(\bigcap_{k=0}^{T/m-1}G_{i,k})=\mathbb P(G_{i,0}) \mathbb P(G_{i,1}\mid G_{i,0})\cdots \mathbb P(G_{i, T/m-1}\mid \bigcap_{k=0}^{T/m-2}G_{i,k})$$ and $$\mathbb P(G_{i, k}\mid \bigcap_{k'=0}^{k-1}G_{i,k'})\leq 1-q_w(\frac{a_1 \delta}{4\sqrt n_x}).$$* The construction of $G_{i,k}$ in Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"} is our major technical contribution. Notice that by constructing a stopping time $L_{i,k}$, conditioning on $L_{i,k}=l$, $w_{km+L_{i,k}}=w_{km+l}$ is independent of $\{L_{i,k}=l\}\in \mathcal F_{km+l}$, and $b_{i, km+L_{i,k}}, j_{i, km+L_{i,k}}$ are constant conditioning on $\mathcal F_{km+l}$. Hence, we are able to leverage the properties of the probability distribution of $w_t$ in Assumption [Assumption 3](#ass: tight bound on wt){reference-type="ref" reference="ass: tight bound on wt"} to bound $\mathbb P(G_{i, k}\mid \bigcap_{k'=0}^{k-1}G_{i,k'})$. By leveraging Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"} and [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"}, we can now prove Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"}. Then, the proof of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} is completed by combining Lemma [Lemma 1](#lem: bound PE c){reference-type="ref" reference="lem: bound PE c"} and Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"}. # Applications {#sec:application} To showcase the results in the previous sections, we apply them in the context of two applications. The first application is the estimation problem of a linear control dynamical system with random control inputs. The second is the analysis of an adaptive tube model predictive control (MPC) algorithm. ## Estimating Dynamical Systems {#subsec:identification_sim} We now apply Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} to linear systems with no control inputs: $x_{t+1}=A^* x_t+w_t$. In this setting, the membership set can be defined as $$\mathbb A_T=\bigcap_{t=0}^{T-1} \{ \hat A : \|x_{t+1} -\hat A x_t \|_\infty \leq w_{\max}\}.$$ We will provide a diameter upper bound on $\mathbb A_T$ below. **Corollary 2** (System estimation when $B=0$). *When $A^*$ is $(\kappa, \rho)$-stable, i.e., $\|(A^*)^t\|_2\leq \kappa (1-\rho)^t$ for all $t$ with $\rho <1$, for any $m>0$ and any $\delta>0$, when $T>m$, we have $$\begin{aligned} \mathbb P(\textup{diam}(\mathbb A_{ {T}})>\delta)\leq \frac{T}{m} \tilde O(n_x^{2.5}) a_2^{n_x}\exp(-a_3m)+\tilde O(n_x^5)a_4^{n^2}(1-q_w(\frac{a_1\delta}{4\sqrt n_x}))^{\ceil{T/m}} \end{aligned}$$ where $b_x=\kappa\|x_0\|_2+\kappa\sqrt{n_x}/\rho$, $p_x=1/192$, $\sigma_x=\sqrt{\lambda_{\min}(\Sigma_w)/2}$, $a_1 = \frac{\sigma_{x} p_{x}}{4}$, $a_2=\frac{64 w_{\max}}{\sigma_{x}^2 p_{x}^2}$, $a_3= \frac{p_{x}^2}{8}$, $a_4=\max(\frac{4b_x \sqrt{n_x}}{a_1},1)$.* *Consequently, when the distribution of $w_t$ satisfies $q_w(\epsilon)=O(\epsilon)$, e.g. uniform or truncated Gaussian, we have $\|\hat \theta -\theta_*\|\leq \tilde O(n_x^{3.5}/T)$.* Corollary [Corollary 2](#cor: estimation err bdd for B=0){reference-type="ref" reference="cor: estimation err bdd for B=0"} can be proved by verifying the assumptions of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"}: we can show $\|x_t\|_2\leq b_x$ for stable $A^*$ and prove the $(1, \sigma_x, p_x)$-BMSB condition for $\{x_t\}_{t\geq 0}$ based on [@dean2019safely]. Interestingly, though [@simchowitz2018learning] shows that $\Omega(\sqrt n_x/\sqrt T)$ is the best possible estimation rate for (unbounded) Gaussian distribution, when the random $w_t$ are *bounded*, Corollary [Corollary 2](#cor: estimation err bdd for B=0){reference-type="ref" reference="cor: estimation err bdd for B=0"} shows that SM can achieve a better rate $\tilde O(n_x^3/T)$ when $T>>n_x$ by taking advantage of the additional information of the bounded support. The fundamental lower bound for bounded random $w_t$ are left for future work. It is worth comparing this upper bound with the estimation error bound of LSE in [@dean2019safely], which is $\tilde O(\frac{\sqrt{n_z}}{\sqrt T})$. Interestingly, the point estimators generated by the SM method achieve better convergence rates in terms of the number of samples $T$, but are worse with respect to the system dimensions $n_x, n_u$. ## Adaptive Tube Model Predictive Control The set membership method is extensively used in the robust adaptive model predictive control (RA-MPC) literature, e.g., [@tanaskovic2014adaptive; @lu2023robust]. In RA-MPC, the model uncertainty sets can be updated online based on set membership estimation as more data is collected. Since the model uncertainty sets heavily influence the conservativeness of RA-MPC controllers, SM updates are able to quickly improve control performance by reducing the size of the uncertainty/membership sets. In the following, we provide a brief recap of RA-MPC and show the implication of our SM result. At every time step $k$, RA-MPC solves the following robust optimization with horizon $H$: $$\begin{array}{ll} & \min \, \sum_{t=0}^H c_t(\hat x_{k+t\mid k}, \hat u_{k+t\mid k}) \\ \text { s.t. } & \hat x_{k+t+1\mid k}=\hat{A} \hat x_{k+t\mid k}+\hat B \hat u_{k+t\mid k}, \quad t\geq 0, \\ & \hat u_{k+t\mid k}=\pi_{k+t\mid k}\left(\hat x_{k+t\mid k}\right), \quad t\geq 0,\\ & \tilde x_{k+t+1\mid k}=\tilde A \tilde x_{k+t\mid k}+\tilde B \tilde u_{k+t\mid k}+\tilde w_{k+t\mid k}, \ \forall\, (\tilde A, \tilde B) \in \Theta_k, \tilde w_{k+t\mid k} \in \mathcal W, t\geq 0\\ &\tilde u_{k+t\mid k}=\pi_{k+t\mid k}\left(\tilde x_{k+t\mid k}\right), \quad t\geq 0,\\ & \hat x_{k\mid k}=\tilde x_{k\mid k}=x_k,\\ & \tilde x_{k+t\mid t} \in \mathcal{X},\,\tilde u_{k+t\mid t}\in \mathcal{U},\, x_{k+H\mid t} \in \mathcal{X}_H,\, t=0,1, \cdots, H-1 \end{array} \label{eq:FTCOP}$$ with respect to some cost objective $c_t$. It is evident in [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"} that smaller uncertainty set $\Theta_t$ leads to better performance of the MPC algorithm. We refer readers to the supplementary material for more details on RMPC and RA-MPC. There is a vast literature of algorithms that approximate the solutions to [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"} for fixed uncertainty sets, e.g., [@mayne2005robust; @rakovic2016elastic; @chen2021system]. We will leverage such methods to solve [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"} for the adaptive control task. For concreteness, we consider the basic adaptive tube MPC algorithm [@rawlings2017model; @mayne2005robust] for the FTCOP problem here. We parameterize the control policy $\pi$ as $u_k = K x_k + v_k + \eta_k$, where $K$ is a given robustly stabilizing feedback gain for the entire uncertainty set $\mathcal{P}_0$ (and thus for all subsequent $\mathcal{P}_k$), $v_k$ is the FTCOP optimization variable, and $\eta_k$ is a bounded exploration injection with $\|\eta_k\|_{\infty}\leq \eta_{\max}$ with $\mathbb{E}[\eta_k\eta_k^\top] = \sigma_{\eta}I$. Note that $\eta_k$ is left out for [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"}. Applying , we obtain a finite-time convergence guarantee for this RA-MPC algorithm that integrates SM with tube MPC. **Corollary 3**. *Assume that [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"} is feasible for $\mathcal{{P}}_0$ and holds, the membership set under the adaptive tube MPC controller has the following uncertainty set convergence guarantee, $\|\hat \theta_T-\theta^*\|_F \leq \tilde O(\frac{n_x^{3.5}n_u^2}{T})$ with high probability. [\[cor:rmpc\]]{#cor:rmpc label="cor:rmpc"}* This result is immediate by verifying Assumption [Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"} using [@li2023non] and verifying the boundedness assumption by noting that $\mathcal X, \mathcal U$ are bounded. # Numerical Experiments #### Uncertainty sets comparison for control dynamic systems with i.i.d. inputs. To illustrate the quantitative behavior of , we provide simulation results in (a)-(b). Our experiments illustrate the membership set diameter as a function of time for a control dynamical system $x_{t+1}=A^* x_t+B^*u_t + w_t$, where $A\in \mathbb{R}^{10\times10}$, $B\in\mathbb{R}^{10\times 3}$, and $w_t$ is drawn i.i.d from (a) projected Gaussian and (b) uniform distribution. All $w_t$ is projected (for (a)) and normalized (for (b)) to be between $[-2,2]$ element-wise. We define the tight bound $\hat{w}_{max} := 2$. We apply SM with differently loose bound. We implement LSE with parameters $\delta = 0.1$ ($90\%$ confidence), $\lambda = 0.1$ (LSE regularization parameter), $L = 2$ (variance proxy for bounded random variables), $S = \sqrt{\text{trace}([A \ B]^T[A \ B] )}$ according to Theorem 1 in [@abbasi2011online]. The quantitative behavior of the SM method is consistent with that predicted by , outperforming the $90\%$ confidence region of the LSE in both magnitude and rate of convergence when the disturbance bound is tight. Moreover, even when the bound is loose, i.e., fails to hold, SM is still orders of magnitude better than the LSE in terms of the size of the uncertainty set. #### Tracking performance comparison by RAMPC with different set estimation methods. To illustrate the quantitative impact of using SM for adaptive tube MPC, we provide simulation results in . Our experiments study tube MPC for a single-input-single-output system with nominal system $\hat A = 1.1$, $\hat B = 1$ and true uncertainty parameter $\Delta_A = 0.1, \Delta_B = -0.1$. The initial uncertainty set $\mathcal{P}_0$ is set to be $[-0.2, 0.2]^2$. We use the basic tube MPC method [@rawlings2017model; @mayne2005robust] and parameterize the control policy as $u_k = K x_k + v_k + \eta_k$, where $K=-1$ and $\eta_k$ is a bounded exploration injection with $\eta_k \sim \textup{Unif}([-0.01, 0.01])$. Note that $\eta_k$ is left out when solving for [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"}. The disturbance $w_k$ has a known bound of $w_{\max} = 0.1$ and is generated to be i.i.d. $\textup{Unif}([-0.1, 0.1])$. The horizon of [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"} is set to be 5. The state and input constraints are such that $x_k \in [-10,10]$ and $u_k \in [-10,10]$ for all $k \geq 0$. We consider the task of constrained LQ tracking problem with a time-varying cost function $c_t := (x_t - g_t)^\top Q (x_t {-} g_t) + u_t^\top R u_t$ where the target trajectory is generated as $g_t = 8 \sin(t/20)$. We compare the performance of an adaptive tube MPC controller that uses the SM method for uncertainty set estimation (SM) against one that uses the LSE confidence region (LSE). We also plot the offline optimal RMPC controller, i.e., the controller that has knowledge of the true underlying system parameters (OPT). The resulting state trajectories are shown in . Due to the fast convergence of the membership set, the tracking performance of SM quickly coincides with OPT, while LSE converges more slowly. Since the controller has to robustly satisfy constraints against the worst-case model in the uncertainty set, smaller uncertainty set for [\[eq:FTCOP\]](#eq:FTCOP){reference-type="eqref" reference="eq:FTCOP"} means more optimal trajectories are available to the RA-MPC controller. Therefore, verifies that the SM method is indeed a good candidate for the uncertainty set estimation problem arising in RA-MPC problems as have been extensively observed empirically in existing literature [@lorenzen2019robust; @lu2019robust; @kohler2019linear]. {#fig:sim_uniform_dist width="\\textwidth"} {#fig:sim_uniform_dist_loose_bound width="\\textwidth"} {#fig:ra-mpc width="95%"} # Conclusion, Limitations, and Future Work #### Conclusion. This work provides an explicit bound on the convergence rate of set membership estimation methods for linear control dynamical systems under the assumption of i.i.d. stochastic disturbances that are bounded by box constraints. When the box constraints are accurately known, we establish the rate of convergence to the true model parameters. When the box constraints on $w_t$ are over-estimated, we establish the rate of convergence to a neighborhood of the true model parameters. We demonstrate the main result of this paper () in both the linear system identification problem and the adaptive robust model predictive control setting. Numerical experiments on these two settings are also provided in comparison with least square estimation and its confidence region. #### Limitations and future work. One limitation of our work is that our bound heavily relies on the assumption of the box constraints on the disturbances, which may be overly conservative in practice. Therefore, an important future direction is to analyze set membership estimation for general polyhedron constraints on the disturbances. Moreover, our results have focused exclusively on the *diameter* of the membership set. Other metrics such as the volume are also of importance in the control literature and therefore needs more investigation. For example, if the uncertainty set has a shape similar to a narrow ellipse, it may have a large diameter yet small volume, and whether this has high impact or low impact on control performance is also worth discussing. Another limitation of this work is that we only consider linear systems, and it would be interesting to explore whether some of the results in this paper can be extended to some nonlinear systems, such as bi-linear systems, or nonlinear systems with linear parametrization, etc. Interesting directions for future work also include obtaining a lower bound for set membership estimation. Further, we note that Corollary [\[cor:rmpc\]](#cor:rmpc){reference-type="ref" reference="cor:rmpc"} also lays a foundation for non-asymptotic optimality analysis for RA-MPC based on set membership, which has seen a significant amount of empirical research but lacks formal optimality analysis. Finally, it is also worth applying the proof techniques of this paper to analyze the set membership performance on other types of uncertainties, such as measurement errors, etc. **Appendix** # Roadmap {#roadmap .unnumbered} - introduces additional notation used throughout the Appendix. - provides more literature review on LSE and SM, and a more detailed discussion on the technical contributions of this paper. - provides more discussions on examples that satisfy Assumptions [Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"} and [Assumption 3](#ass: tight bound on wt){reference-type="ref" reference="ass: tight bound on wt"}. - presents the proof of . In particular, we provide helper lemmas in and prove , in and respectively. - presents a proof of . - presents a proof of . # Additional notations {#subsec:notation} Let $\mathbb S_n(0,1)$ denote the unit sphere in $\mathbb R^n$ in $l_2$ norm, i.e., $\mathbb S_n(0,1)=\{x\in \mathbb R^n: \|x\|_2=1\}$. Let $\mathbb S_{n\times m}(0,1)$ denote the unit sphere in $\mathbb R^{n\times m}$ with respect to the Frobenius norm, i.e., $\mathbb S_{n\times m}(0,1)=\{M\in \mathbb R^{n\times m}: \|M\|_F=1\}$. Let $\bar B_n(0,1)$ denote the closed unit ball in $\mathbb R^n$ in $l_2$ norm, i.e., $\bar B_n(0,1)=\{x\in \mathbb R^n: \|x\|_2\leq1\}$. Let $\bar B_{n\times m}(0,1)$ denote the closed unit ball in $\mathbb R^{n\times m}$ in Frobenius norm, i.e., $\bar B_{n\times m}(0,1)=\{M\in \mathbb R^{n\times m}: \|M\|_F\leq 1\}$. For a matrix $M\in \mathbb R^{n\times m}$, $\text{vec}(M)$ is the vectorization of $M$. Moreover, we define the inverse mapping of $\text{vec}(\cdot)$ as $\text{mat}(\cdot)$, i.e., for a vector $d\in \mathbb R^{nm}$, $\text{mat}(d)\in \mathbb R^{n\times m}$. Consider a $\sigma$-algebra $\mathcal F$ and a random variable $X$, we write $X\in \mathcal F$ if $X$ is measurable with respect to $\mathcal F$, i.e., for all Borel measurable sets $B\subseteq \mathbb R$, we have $X^{-1}(B)\in \mathcal F$. We can similarly define $\mathcal F$-measurable random matrices and random vectors. Further, consider a polyhedral $\mathbb D=\{x: Ax\leq b\}$, we write $\mathbb D\in \mathcal F$ if matrix $A$ and vector $b$ are measurable with respect to $\mathcal F$. Consider two symmetric matrices $A,B \in \mathbb R^{n\times n}$, we write $A\succeq B$ if $A-B$ is a positive definite matrix. We define $\min \emptyset =+\infty$. For a set $\mathcal E$, let $\mathds{1}_{\mathcal E}$ denote the indicator function on $\mathcal E.$ For a vector $x\in\mathbb{R}^n$, we use $x^j$ to denote the $j$th coordinate of $x$. # More discussions on least square and set membership {#subsec:related_work} System identification studies the problem of estimating the parameters of an unknown dynamical systems from trajectory data. There are two main classes of estimation methods: point estimator such as least square estimation (LSE), and set estimator such as set membership (SM). In the following, we provide more discussions and literature review on LSE and SM. We will also discuss the major technical novelties of this work. ## Least square estimation For linear dynamical systems $x_{t+1}=A^*x_t +B^*u_t+w_t=\theta^* z_t+w_t$, given a trajectory of data $\{x_t, u_t\}_{t\geq 0}$, least square estimation generates a point estimator that minimizes the following quadratic error [@van2012subspace; @ljung1998system]: $$\begin{aligned} \hat \theta_{\text{LSE}}=\min_{\hat \theta}\sum_{t=0}^{T-1}\|x_{t+1}-\hat \theta z_t\|_2^2.\end{aligned}$$ Least-square estimation is widely used and its convergence (rate) guarantees have been investigated for a long time. In particular, non-asymptotic convergence rate guarantees of LSE has become increasingly important as these guarantees are the foundations for non-asymptotic performance analysis of learning-based/adaptive control algorithms. Earlier non-asymptotic analysis of LSE focused on the simpler regression model $y_t=\theta^* x_t +w_t$, where $x_t$ and $y_t$ are independent [@campi2002finite; @vidyasagar2006learning; @hsu2012random]. Recently, there is one major breakthrough in [@simchowitz2018learning] that provides LSE's convergence rate analysis for linear dynamical system $x_{t+1}=\theta^* z_t +w_t$, where $x_{t+1}$ and $z_t=[x_t^\top, u_t^\top]^{\top}$ are correlated. More specifically, [@simchowitz2018learning] establishes a fundamental property, *block-martingale small-ball (BMSB)*, to analyze LSE under correlated data. BMSB enables a long list of subsequent literature on LSE's non-asymptotic analysis for different types of dynamical systems, e.g., [@oymak2019non; @dean2019sample; @zheng2020non; @rantzer2018concentration; @faradonbeh2018finite; @wagenmaker2020active; @tsiamis2022statistical; @zhao2022adaptive; @li2021safe]. Though LSE is a point estimator, one can establish confidence region of LSE based on proper statistical assumptions on $w_t$. The pioneer works on the confidence region of LSE for linear dynamical systems are [@abbasi2011online; @abbasi2011regret], which construct ellipsoid confidence regions for LSE. Moreover, the non-asymptotic bounds on estimation errors established in [@simchowitz2018learning; @dean2019safely] can also be viewed as confidence bounds. Further, the estimation error $\tilde O(\frac{\sqrt{n_x+n_z}}{\sqrt T})$ has been shown to match the fundamental lower bound for any estimation methods for unbounded disturbances in [@simchowitz2018learning]. However, these confidence bounds all rely on statistical inequalities, which may result in loose constant factors despite an optimal convergence rate. When applying these confidence bounds to robust control, where the controller is required to satisfy certain stability and constraint satisfaction properties for every possible system in the confidence region, a loose constant factor will result in a larger confidence region and a more conservative control design. Finally, in robust control and many practical applications, the disturbances are usually bounded, and it will be interesting to see how the knowledge of the boundedness will improve the uncertainty set estimation. ## Set membership {#subsec:sm} Set membership is commonly used in robust control for uncertainty set estimation [@milanese1991optimal; @adetola2011robust; @tanaskovic2013adaptive; @bujarbaruah2020adaptive; @zhang2021trajectory; @parsi2020robust; @parsi2020active; @sasfi2022robust]. There is a long history of research on SM for both deterministic disturbances, such as [@bai1995membership; @fogel1982value; @kitamura2003convergence; @milanese2013bounding; @lauricella2020set; @livstone1996asymptotic], and stochastic disturbances, such as [@bai1998convergence; @bai1995membership; @kitamura2005size; @akccay2004size; @lu2019robust]. For the stochastic disturbances, both convergence and convergence rate analysis have been investigated under the persistent excitation (PE) condition. However, the existing convergence rates are only established for simpler regression problems, $y_t=\theta^*x_t +w_t$, where $y_t$ and $x_t$ are independent [@akccay2004size; @bai1995membership; @bai1998convergence; @kitamura2005size]. Recently, [@lu2019robust] provided an initial attempt to establish the convergence guarantee of SM for linear dynamical systems $x_{t+1}=\theta^* z_t+w_t$ for correlated data $x_{t+1}$ and $z_t$. However, [@lu2019robust] assumes that PE holds deterministically, and designs a special control design based on constrained optimization to satisfy PE deterministically. Therefore, the convergence for general control design and the convergence rate analysis remain open questions for correlated data arising from dynamical systems. In this paper, we establish the convergence rate guarantees of SM on linear dynamical systems under the BMSB conditions in [@simchowitz2018learning]. Compared with [@lu2019robust], BMSB condition can be satisfied by adding an i.i.d. random noise to a general class of control designs [@li2021safe]. **Technically**, one major challenge of SM analysis compared with the LSE analysis is that the diameter of the membership set does not have an explicit formula, which is in stark contrast with LSE, where the point estimator is the solution to a quadratic program and has explicit form. A common trick to address this issue in the analysis of SM is to connect the diameter bound with the values of disturbances subsequences $\{w_{s_k}\}_{k\geq 0}$: it can be generally shown that a large diameter indicates that a long subsequence of disturbances are far away from the boundary of $\mathcal W$. However, existing construction methods of $\{w_{s_k}\}_{k\geq 0}$ will cause the time indices $\{s_k\}_{k\geq 0}$ to *correlate* with the realization of the sequences $\{x_t, u_t, w_t\}_{t\geq 0}$ [@akccay2004size; @lu2019robust; @bai1995membership].[^4] Consequently, in the correlated-data scenario and when PE does not hold deterministically, under the existing construction methods in [@akccay2004size; @lu2019robust; @bai1995membership], the probability of $\{w_{s_k}\}_{k\geq 0}$ with correlated time indices *cannot* be bounded by the probability of the *independent* sequence $\{w_t\}_{t\geq 0}$. One major **technical contribution** of this paper is to provide a novel construction of $\{w_{s_k}\}_{k\geq 0}$ based on a sequence of stopping times and establish conditional independence properties despite correlated data and stochastic PE condition (BMSB). More details can be found in Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"} and the proof or Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"}. Though we only consider box constraints for $w_t$, it is worth mentioning that SM can be applied to much more general forms of disturbances. For example, a common alternative is the ellipsoidal-bounded disturbance where $\mathcal{W}:=\{ w\in \mathcal{R}^{n_x}: w^\top P w \leq 1 \}$ with positive definite $P \in \mathcal{R}^{n_x \times n_x}$ [@bai1995membership; @van2023informativity; @eising2023sampling; @liu2016ellipsoidal] and polytopic-bounded disturbance $\mathcal{W}:=\{ w\in \mathcal{R}^{n_x}: Gw \leq h \}$ for positive definite $G\in\mathcal{R}^{n_x \times n_x}$ and $h\in\mathcal{R}^{n_x}$ [@fogel1982value; @lu2019robust; @lu2023robust]. There are also SM literature assuming bounded energy of the disturbance sequences [@bai1995membership]. It is an interesting future direction to extend the analysis in this paper to more general disturbance constraints. A separate but important challenge is that the knowledge of $\mathcal{W}$ is not always available a priori. There is literature discussing the estimation of $\mathcal W$ [@lauricella2020set; @bai1998convergence]. We leave for future work how to simultaneously estimate $\mathcal{W}$ and perform non-asymptotic analysis on the size of the membership set. Further, exact SM involves the intersection of an increasing number of sets, thus causing the computation complexity increases with time $t$, which can become prohibitive when $t$ is large. There are many methods trying to reduce the computation complexity by approximating the membership sets (see e.g., [@livstone1996asymptotic; @lu2019robust], etc.). It is an exciting future direction to study the diameter bounds of the approximated SM methods. # More discussions on Assumptions [Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"} and [Assumption 3](#ass: tight bound on wt){reference-type="ref" reference="ass: tight bound on wt"} {#appendix:assumption} ## More discussions on Assumption [Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"} {#more-discussions-on-assumption-ass-bmsb-bounded-xt} The BMSB condition has been widely used in learning-based control. It has been shown that BMSB can be satisfied in many scenarios. For example, [@simchowitz2018learning; @tu2019sample] showed that linear systems with i.i.d. perturbed linear control policies, i.e., $x_{t+1}=Ax_t+B( Kx_t+\eta_t)+w_t$,[^5] satisfy BMSB if the disturbances $w_t$ and $\eta_t$ are i.i.d. and follow Gaussian distributions with positive definite covariance matrices. Later, [@dean2019safely] showed that $x_{t+1}=Ax_t+B( Kx_t+\eta_t)+w_t$ can still satisfy BMSB even for non-Gaussian distributions of $w_t,\eta_t$, as long as $w_t$ and $\eta_t$ have independent coordinates and finite fourth moments. Recently, [@li2021safe] extended the results to linear systems with nonlinear policies, i.e., $x_{t+1}=Ax_t+B(\pi_t(x_t)+\eta_t)+w_t$, and showed that BMSB still holds as long as the nonlinear policies $\pi_t$ generate bounded trajectories of states and control inputs, and $w_t, \eta_t$ are bounded and follow distributions with certain anti-concentrated properties (a special case is positive definite covariance matrix). ## More discussions on Assumption [Assumption 3](#ass: tight bound on wt){reference-type="ref" reference="ass: tight bound on wt"} {#more-discussions-on-assumption-ass-tight-bound-on-wt} In this subsection, we provide two example distributions, truncated Gaussian and uniform distributions, and discuss their corresponding $q_w(\epsilon)$ functions. It will be shown that for both distributions below, $q_w(\epsilon)=O(\epsilon)$. **Lemma 5** (Example of uniform distribution). *Consider $w_t$ that follows a uniform distribution on $[-w_{\max}, w_{\max}]^{n_x}$. Then, $q_w(\epsilon)=\frac{\epsilon}{2w_{\max}}$.* *Proof.* Since $\textup{Unif}(\mathcal W)$ is symmetric, we only need to consider one direction $j=1$. $$\begin{aligned} \mathbb P(w^j+w_{\max}\leq \epsilon)&=\int_{w^1+w_{\max}\leq \epsilon} \int_{w^2, \dots, w^{n_x}\in [-w_{\max}, w_{\max}]} \frac{1}{(2w_{\max})^{n_x}} \mathds{1}_{(w\in \mathcal W)}\,\mathrm d w\\ &= \int_{w^1\leq \epsilon-w_{\max}}\frac{1}{2w_{\max}} \mathds{1}_{(w\in \mathcal W)}\mathrm d w^1= \frac{\epsilon}{2w_{\max}} \end{aligned}$$ Similarly, $\mathbb P(w_{\max}-w^1\leq \epsilon) = \int_{w^1\geq w_{\max}-\epsilon}\frac{1}{2w_{\max}} \mathds{1}_{(w\in \mathcal W)}d w^1= \frac{\epsilon}{2w_{\max}}$. ◻ **Lemma 6** (Example of truncated Gaussian distribution). *Consider $w_t$ follows a truncated Gaussian distribution on $[-w_{\max}, w_{\max}]^{n_x}$ generated by a Gaussian distribution with zero mean and $\sigma_w I_{n_x}$ covariance matrix. Then, $q_w(\epsilon)=\frac{1}{\min(\sqrt{2\pi}\sigma_w, 2w_{\max})} \exp(\frac{-w_{\max}^2}{2\sigma_w^2})\epsilon$.* *Proof.* Since this distribution is symmetric and each coordinate is independent, we only need to consider one direction $j$. Let $X$ denote a Gaussian distribution with zero mean and $\sigma_w^2$ variance. By the definition of truncated Gaussian distributions, we have $$\begin{aligned} \mathbb P(w^j+w_{\max}\leq \epsilon)&=\frac{\mathbb P(-w_{\max}\leq X \leq -w_{\max}+\epsilon)}{\mathbb P(-w_{\max}\leq X \leq w_{\max})} \end{aligned}$$ Notice that $X/\sigma_w$ follows the standard Gaussian distribution, so we can obtain the following bounds. $$\begin{aligned} \mathbb P(-w_{\max}\leq X \leq -w_{\max}+\epsilon)& =\int_{-w_{\max}/\sigma_w}^{(-w_{\max}+\epsilon)/\sigma_w} \frac{1}{\sqrt{2\pi}}\exp(-\frac{z^2}{2})\,\mathrm d z\\ & \geq \frac{1}{\sqrt{2\pi}}\exp(-w_{\max}^2/(2\sigma_w^2) )\frac{\epsilon}{\sigma_w} \end{aligned}$$ and $$\begin{aligned} \mathbb P(-w_{\max}\leq X \leq w_{\max})& =\int_{-w_{\max}/\sigma_w}^{w_{\max}/\sigma_w} \frac{1}{\sqrt{2\pi}}\exp(-\frac{z^2}{2})\,\mathrm d z\\ & \leq \min(1, \frac{1}{\sqrt{2\pi}}\frac{2w_{\max}}{\sigma_w}) \end{aligned}$$ Therefore, we obtain $$\begin{aligned} \mathbb P(w^j+w_{\max}\leq \epsilon)&=\frac{\mathbb P(-w_{\max}\leq X \leq -w_{\max}+\epsilon)}{\mathbb P(-w_{\max}\leq X \leq w_{\max})}\\ &\geq \max(\frac{1}{\sqrt{2\pi}}\exp(-w_{\max}^2/\sigma_w^2) \frac{\epsilon}{\sigma_w}, \frac{\epsilon}{2w_{\max}} \exp(\frac{-w_{\max}^2}{2\sigma_w^2}))\\ & = \frac{1}{\min(\sqrt{2\pi}\sigma_w, 2w_{\max})} \exp(\frac{-w_{\max}^2}{2\sigma_w^2})\epsilon \end{aligned}$$ Finally, $\mathbb P(w_{\max}-w^1\leq \epsilon)$ can be bounded similarly. ◻ # Proof of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} {#appendix:proof_main} The section provides more details for the proof of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"}. In particular, we first provide technical lemmas for set discretization, then prove Lemma [Lemma 1](#lem: bound PE c){reference-type="ref" reference="lem: bound PE c"} and Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"} respectively. The proof of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} follows naturally by combining Lemma [Lemma 1](#lem: bound PE c){reference-type="ref" reference="lem: bound PE c"} and Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"}. ## Technical lemmas: set discretization {#sec:D1} In this subsection, we provide technical results of finite-ball covering. The results are classical and are based on [@rogers1963covering] [@verger2005covering]. **Theorem 3** (Ball covering [@rogers1963covering] [@verger2005covering]). *Consider a closed ball $\bar \mathbb B_n(0,1)=\{x\in \mathbb R^n: \|x\|_2\leq 1\}$ in $l_2$ norm. Considering covering this ball $\bar \mathbb B_n(0,1)$ with smaller closed balls $\bar \mathbb B_n(z,\epsilon)$ for $z\in \mathbb R^n$. Let $v_{\epsilon, n}$ denote the minimal number of smaller balls needed to cover $\bar \mathbb B_n(0,1)$. Then, we have $$\begin{aligned} v_{\epsilon, n} & \leq \begin{cases} \tilde O(n) \left(\frac{1}{\epsilon}\right)^n, & \text{ if } \epsilon \leq \frac{1}{n}, \\ \tilde O(n^{5/2}) \left(\frac{1}{\epsilon}\right)^n, & \text{ if } \epsilon \leq 1. \end{cases} \end{aligned}$$* **Corollary 4**. *There exists a finite set $\mathcal M'=\{\lambda_1, \dots, \lambda_{v_\lambda}\}\subseteq \mathbb S_{n_z}(0,1)$ such that for any $\lambda \in \mathbb R^{ n_z}$ with $\|\lambda\|_2=1$, there exists $\lambda_i\in \mathcal M'$ such that $\|\lambda-\lambda_i\|_2 \leq 2\epsilon_\lambda$. Further, $v_\lambda\leq \tilde O(n_z^{2.5}) \left(\frac{1}{\epsilon_\lambda}\right)^{n_z}$ if $\epsilon_\lambda \leq 1$.* *In the following, we consider $\epsilon_\lambda = \min(\sigma_z^2 p_z^2/(64 b_z^2),1)$. Hence, $$\label{eq:v_lambda} v_{\lambda}\leq \tilde O(n_z^{2.5})\max(1, \left(\frac{64 b_z^2}{\sigma_z^2 p_z^2}\right)^{n_z}).$$* Next, we apply Theorem [Theorem 3](#thm: ball cover){reference-type="ref" reference="thm: ball cover"} to a space of matrices. **Lemma 7**. *There exists a finite set $\mathcal M=\{\gamma_1, \dots, \gamma_{v_\gamma}\}\subseteq \mathbb S_{n_x\times n_z}(0,1)$ such that for any $\gamma \in \mathbb R^{n_x\times n_z}$ and $\|\gamma\|_F=1$, there exists $\gamma_i\in \mathcal M$ such that $\|\gamma-\gamma_i\|_F \leq 2\epsilon_\gamma$. Further, $v_\gamma\leq \tilde O(n_x^{2.5}n_z^{2.5}) \left(\frac{1}{\epsilon_\gamma}\right)^{n_zn_x}$ if $\epsilon_\gamma \leq 1$.* *In the following, we consider $\epsilon_\gamma = \min(\frac{a_1}{4b_z \sqrt n_x},1)$.* *Proof.* The proof is basically by mapping the matrices to vectors based on matrix vectorization, then mapping the vectors back to matrices. These two mappings are isomorphism. Specifically, consider a closed unit ball in $\mathbb R^{n_xn_z}$. There exist $v_{\epsilon, n_xn_z}$ smaller closed balls to cover it, denoted by $\mathbb B_1, \dots, \mathbb B_{v_{\epsilon, n_xn_z}}$. Consider the non-empty sets from $\mathbb B_1\cap \mathbb S_{n_x n_z}(0,1), \dots, \mathbb B_{v_{\epsilon, n_x n_z}}\cap \mathbb S_{n_x n_z}(0,1)$. For any $1\leq i \leq v_{\epsilon, n_x n_z}$, if $\mathbb B_i \cap \mathbb S_{n_x n_z}(0,1)\not = \emptyset$, select a point $\text{vec}(\gamma)\in \mathbb B_i \cap \mathbb S_{n_x n_z}(0,1)$. Notice that $\|\text{vec}(\gamma)\|_2=1$. In this way, we construct a finite sequence $\{\text{vec}(\gamma_1), \dots, \text{vec}(\gamma_{v_{\gamma}})\}$ where $v_\gamma\leq v_{\epsilon_\gamma, n_xn_z}$.[^6] For any $\gamma \in \mathbb R^{n_x\times n_z}$, we have $\text{vec}(\gamma)\in \mathbb R^{n_xn_z}$ and $\|\text{vec}(\gamma)\|_2=1$. Hence, there exists $1\leq i \leq v_\gamma$ such that $\text{vec}(\gamma)\in \mathbb B_i \cap \mathbb S_{n_xn_z}(0,1)$. Hence, $\| \text{vec}(\gamma)-\text{vec}(\gamma_i)\|_2\leq 2\epsilon_\gamma$. Moreover, $\|\gamma_i\|_F=\|\text{vec}(\gamma_i)\|_2=1$. Therefore, $\|\gamma_i-\gamma\|_F\leq 2\epsilon_\gamma$. So the set $\mathcal M=\{\gamma_1, \dots, \gamma_{v_\gamma}\}$ satisfies our requirement. ◻ ## Proof of Lemma [Lemma 1](#lem: bound PE c){reference-type="ref" reference="lem: bound PE c"} {#sec:D2} The proof leverages the following results related to Proposition 2.5 in [@simchowitz2018learning]. **Definition 4** (BMSB for scalar sequence). *Let $\{Z_t\}_{t\geq 1}$ be an $\{\mathcal F_t\}_{t\geq 1}$-adapted random process taking values in $\mathbb R$. We say $\{Z_t\}_{t\geq 1}$ satisfies the $(k, v,p)$-block martingale small-ball (BMSB) condition if, for any $j\geq 0$, one has $\frac{1}{k}\sum_{i=1}^k \mathbb P(|Z_{j+i}|\geq v\mid \mathcal F_t)\geq p$ almost surely.* **Theorem 4** (Proposition 2.5 in [@simchowitz2018learning] when $k=1$). *Let $\{Z_t\}_{t\geq 0}$ be a scalar process that is $(1,v,p)$-BMSB and $Z_t \in \mathcal F_t$ for $t\geq 0$, where $Z_0$ is given. Then $$\mathbb P(\sum_{t=1}^T Z_t^2 \leq v^2 p^2T/8)\leq \exp(-T p^2/8)$$* **Corollary 5** (Implications of Proposition 2.5 in [@simchowitz2018learning] conditioning on $\mathcal F_t$). *Let $\{Z_t\}_{t\geq 0}$ be a scalar process that is $(1,v,p)$-BMSB and $Z_t \in \mathcal F_t$ for $t\geq 0$, where $Z_0$ is given. Then $$\mathbb P(\sum_{t=s+1}^T Z_t^2 \leq v^2 p^2(T-s)/8\mid \mathcal F_s)\leq \exp(-(T-s) p^2/8), \quad \forall\, s\geq 0$$* *Proof.* Consider $\{Y_t\}_{t\geq 0}$ where $Y_t=Z_{t+s}$ for $t\geq 0$. Conditioning on $\mathcal F_s$, $Y_0$ is given. Further, conditioning on $\mathcal F_s$, $\{Y_t\}$ is BMSB because $$\mathbb P(|Y_{t+1}| \geq v\mid \mathcal F^y_t\cap \mathcal F_s)=\mathbb P(|Z_{t+1+s}|\geq v\mid \mathcal F_{t+s})\geq p.$$ Hence, we can apply on $\{Y_t\}$ for $t=1, \dots, T-s$. ◻ **Lemma 8**. *For any $\lambda \in \mathbb R^{n_z}$ such that $\|\lambda\|_2=1$, for any $k\geq 0$, $m\geq 1$, we have $$\mathbb P(\sum_{i=1}^m \lambda^\top z_{km+i} z_{km+i}^\top \lambda \leq \sigma_{z}^2 p_{z}^2m/8 \mid \mathcal F_{km})\leq \exp(-m p_{z}^2/8)$$* *Proof.* Firstly, by Assumption [Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"}, $\{z_t\}$ is $\mathcal F_t$ adapted and $(1, \sigma_z^2 I_{n_z}, p_z)$-BMSB. Therefore, for any $\lambda$ with $\|\lambda\|_2=1$, $\lambda^\top z_t$ is $(1,\sigma_z, p_z)$-BMSB. Therefore, by , we have $$\mathbb P(\sum_{t=km+1}^{km+m} \lambda^\top z_t z_t^\top \lambda \leq \sigma_z^2 p_z^2m/8\mid \mathcal F_{km})\leq \exp(-m p_z^2/8), \quad \forall\, k\geq 0,m\geq 1$$ This completes the proof. ◻ **Lemma 9** (Definition of $H_{i,k}$). *Consider $\mathcal M'=\{\lambda_1, \dots,\lambda_{v_\lambda}\}$ as defined in . For any $m\geq 1$, define events $$H_{i,k}=\left\{ \sum_{t=km+1}^{km+m} \lambda_i^\top z_t z_t^\top \lambda_i > \sigma_z^2 p_z^2m/8 \right\}, \quad \forall\, k\geq 0, 1\leq i\leq v_{\lambda}$$ Then $\mathbb P(H_{i,k}^c\mid \mathcal F_{km}) \leq \exp(-m p_z^2/8)), \quad \forall\, k\geq 0, 1\leq i\leq v_{\lambda}$.* *Proof.* For any $\lambda_i\in\mathcal M'$, $\|\lambda_i\|_2=1$, so by Lemma [Lemma 8](#lem: lambda z's BMSB property){reference-type="ref" reference="lem: lambda z's BMSB property"}, we have $\mathbb P(H_{i,k}^c\mid \mathcal F_{km}) \leq \exp(-m p_z^2/8))$. ◻ **Lemma 10** (Probability of intersections of $H_{i,k}$). *$\mathbb P(\bigcap_{i=1}^{v_{\lambda}} H_{i,k}\mid F_{km})\geq 1-v_{\lambda} \exp(-m p_z^2/8))$.* *Proof.* We can establish the following: $$\begin{aligned} \mathbb P(\bigcap_{i=1}^{v_{\lambda}} H_{i,k}\mid F_{km})& = 1- \mathbb P(\bigcup_{i=1}^{v_{\lambda}} H_{i,k}^c\mid F_{km}) \geq 1-\sum_{i=1}^{v_{\lambda}}\mathbb P(H_{i,k}^c\mid F_{km}) \geq 1-v_{\lambda} \exp(-m p_z^2/8)) \end{aligned}$$ where the last inequality is by Lemma [Lemma 9](#lem: define Hik, lambdai z's BMSB property){reference-type="ref" reference="lem: define Hik, lambdai z's BMSB property"}. ◻ **Lemma 11** (Probability of PE in one segment). *For any $m\geq 1$, for any $k\geq 0$, we have $$\mathbb P(\sum_{t=km+1}^{km+m} z_t z_t^\top \succ (\sigma_z^2 p_z^2m/16) I_{n_z}\mid \mathcal F_{km}) \geq 1-v_{\lambda} \exp(-m p_z^2/8))$$* *Proof.* By Corollary [Corollary 4](#cor: mesh on lambda){reference-type="ref" reference="cor: mesh on lambda"}, for any $\lambda$ such that $\|\lambda\|_2=1$, there exists $\lambda_i\in \mathcal M'$ such that $\|\lambda-\lambda'\|_2 \leq 2\epsilon_\lambda$. Suppose for this $\lambda_i$, we have $\sum_{t=km+1}^{km+m} \lambda_i^\top z_t z_t^\top \lambda_i > \sigma_z^2 p_z^2m/8$. Then $$\begin{aligned} \sum_{t=km+1}^{km+m} \lambda^\top z_t z_t^\top \lambda&= \sum_{t=km+1}^{km+m} \lambda_i^\top z_t z_t^\top \lambda_i + \sum_{t=km+1}^{km+m} (\lambda+\lambda_i)^\top z_t z_t^\top (\lambda-\lambda_i )\\ & > \sigma_z^2 p_z^2m/8 - \sum_{t=km+1}^{km+m} (\lambda+\lambda_i)^\top z_t z_t^\top (\lambda_i-\lambda)\\ & \geq \sigma_z^2 p_z^2m/8 - \sum_{t=km+1}^{km+m}\|\lambda+\lambda_i\|_2 \|z_t\|_2^2\|\lambda_i-\lambda\|_2\\ & \overset{(a)}{\geq} \sigma_z^2 p_z^2m/8 - \sum_{t=km+1}^{km+m} 4 b_z^2 \epsilon_\lambda\\ & =\sigma_z^2 p_z^2m/8 - 4 b_z^2 \epsilon_\lambda m \overset{(b)}{\geq} \sigma_z^2 p_z^2m/16 \end{aligned}$$ where $(a)$ is by Assumption [Assumption 2](#ass: BMSB, bounded xt){reference-type="ref" reference="ass: BMSB, bounded xt"} and $\|\lambda-\lambda'\|_2 \leq 2\epsilon_\lambda$, and $(b)$ is by $\epsilon_\lambda\leq\sigma_z^2 p_z^2/(64 b_z^2)$. Therefore, together with Lemma [Lemma 10](#lem: prob of intersection of Hik){reference-type="ref" reference="lem: prob of intersection of Hik"}, we complete the proof. $$\begin{aligned} \mathbb P(\bigcap_{i=1}^{v_{\lambda}} H_{i,k}\mid F_{km})& \leq \mathbb P( \forall \, \|\lambda\|_2=1, \ \sum_{t=km+1}^{km+m} \lambda^\top z_t z_t^\top \lambda > \sigma_z^2 p_z^2m/16\mid F_{km})\\ & = \mathbb P(\sum_{t=km+1}^{km+m} z_t z_t^\top \succ (\sigma_z^2 p_z^2m/16) I_{n_z} \mid \mathcal F_{km})\end{aligned}$$ ◻ ***Proof of Lemma [Lemma 1](#lem: bound PE c){reference-type="ref" reference="lem: bound PE c"}**.* Recall that $\mathcal E_2= \{\frac{1}{m}\sum_{{s=1}}^{{m}} {z_{km+s} z_{km+s}^\top} \succeq a_1^2 I_{n_z}, \ \forall\, 0\leq k \leq \ceil{{(T-1)}/m}-1 \}$. Hence $$\mathcal E_2=\bigcap_{k=0}^{(T-1)/m-1}\{\sum_{t=km+1}^{km+m} z_t z_t^\top \succ (\sigma_z^2 p_z^2m/16) I_{n_z}\}$$ where $a_1=\sigma_z p_z/4$. For notational simplicity, we define $\bar H_k=\{\sum_{t=km+1}^{km+m} z_t z_t^\top \succ (\sigma_z^2 p_z^2m/16) I_{n_z}\}$. So $$\begin{aligned} \mathbb P(\mathcal E_2)&=\mathbb P(\bigcap_{k=0}^{(T-1)/m-1} \bar H_k) = 1- \mathbb P(\bigcup_{k=0}^{(T-1)/m-1} \bar H_k^c)\\ & \geq 1- \sum_{k=0}^{(T-1)/m-1} \mathbb P(\bar H_k^c) = 1- \sum_{k=0}^{(T-1)/m-1} (1-\mathbb P(\bar H_k)) \end{aligned}$$ By Lemma [Lemma 11](#lem: PE in one segment){reference-type="ref" reference="lem: PE in one segment"}, we have $\mathbb P(\bar H_k\mid \mathcal F_{km})\geq 1-v_{\lambda} \exp(-m p_z^2/8))$. Therefore, $1-\mathbb P(\bar H_k\mid \mathcal F_{km}) \leq v_{\lambda} \exp(-m p_z^2/8))$. Therefore, $$\begin{aligned} \mathbb P(\mathcal E_2)&\geq 1- ((T-1)/m) v_{\lambda} \exp(-m p_z^2/8))\\ & \geq 1- ((T-1)/m) \tilde O(n_z^{2.5})\max(1, \left(\frac{64 b_z^2}{\sigma_z^2 p_z^2}\right)^{n_z}) \exp(-m p_z^2/8))\\ & \geq 1- \frac{T}{m} \tilde O(n_z^{2.5})\max(1, \left(\frac{64 b_z^2}{\sigma_z^2 p_z^2}\right)^{n_z}) \exp(-m p_z^2/8)),\end{aligned}$$ where in the second inequality we used [\[eq:v_lambda\]](#eq:v_lambda){reference-type="eqref" reference="eq:v_lambda"}. Finally $$\mathbb P(\mathcal E_2^c)=1- \mathbb P(\mathcal E_2) \leq \frac{T}{m} \tilde O(n_z^{2.5})\max(1, \left(\frac{64 b_z^2}{\sigma_z^2 p_z^2}\right)^{n_z}) \exp(-m p_z^2/8)).$$ ◻ ## Proof of Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"} {#sec:D3} This proof takes four major steps: 1. Define $b_{i,t}, j_{i,t}, L_{i,k}$. 2. Provide a formal definition of $\mathcal E_{1,k}$ based on $b_{i,t}, j_{i,t}, L_{i,k}$ and prove a formal version of Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"}. 3. Prove Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"}. 4. Prove Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"} by the formal version of Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"} and Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"}. It is worth mentioning that the formal definition of $\mathcal E_{1,k}$ is slightly different from the definition in Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"}, but we still have $\mathbb P(\mathcal E_1\cap \mathcal E_2)\leq \sum_{i=1}^{v_{\gamma}}\mathbb P(\mathcal E_{1,k}\cap \mathcal E_2)$, which is the key property that will be used in the proof of Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"}. ### Step (i): Definitions of $b_{i,t}, j_{i,t}, L_{i,k}$. Recall the discretization of $\mathbb S_{n_x\times n_z}(0,1)$ in Lemma [Lemma 7](#lem: mesh on unit sphere of matrices){reference-type="ref" reference="lem: mesh on unit sphere of matrices"}, which generates the set $\mathcal M=\{\gamma_1, \dots, \gamma_{v_{\gamma}}\}$. We are going to define $b_{i,t}, j_{i,t}, L_{i,k}$ for $\gamma_i \in \mathcal M$ for each $1\leq i \leq v_{\gamma}$. Notice that $\mathcal M$ is a deterministic set of matrices. **Lemma 12** (Definition of $b_{i,t}, j_{i,t}$). *For any $\gamma_i\in\mathcal M$, any $0\leq t \leq T$, there exist $b_{i,t} \in\{-1, 1\}$ and $1\leq j_{i,t} \leq n_x$ such that $b_{i,t}, j_{i,t} \in \mathcal F(z_t)\subseteq \mathcal F_t$ and $$\|\gamma_i z_t\|_\infty = b_{i,t} (\gamma_i z_t)^{j_{i,t}}.$$* Note that one way to determine $b_{i,t}, j_{i,t}$ from $z_t$ is by the following: first pick the smallest $j$ such that $|(\gamma_i z_t)^j|=\|\gamma_i z_t\|_\infty$, then let $b_{i,t}=\textup{sgn}((\gamma_i z_t)^j)$, where $\textup{sgn}(\cdot)$ denotes the sign of a scalar argument. *Proof.* For any $\gamma_i\in \mathcal M$, any $0\leq t \leq T$, we have $$\|\gamma_i z_t\|_\infty =\max_{1\leq j \leq n_x}\max_{b\in\{-1,1\}} b(\gamma_i z_t)^j$$ Hence, there exist $b_{i,t}, j_{i,t}$ such that $\|\gamma_i z_t\|_\infty=b_{i,t} (\gamma_i z_t)^{j_{i,t}}$. Further, $b_{i,t}, j_{i,t}$ only depend on $\gamma_i$ and $z_t$, so they are $\mathcal F(z_t)$-measurable, and $\mathcal F(z_t)\subseteq \mathcal F_t$. ◻ **Lemma 13** (Definition of stopping times $L_{i,k}$). *Let $\eta=\frac{a_1}{\sqrt{n_x}}$. For any $\gamma_i\in\mathcal M$, any $0\leq k \leq (T-1)/m-1$, we can define a random time index $1\leq L_{i,k}\leq m+1$ by $$L_{i,k}=\min(m+1,\min\{ l\geq 1: \|\gamma_i z_{km+l}\|_\infty \geq \eta \}).$$ Then, we have $1\leq L_{i,k}\leq m+1$. Further, for any $1\leq l \leq m$, $\{L_{i,k}=l\} \in \mathcal F_{km+l}$, and $\{L_{i,k}=m+1\}\in \mathcal F_{km+m}\subseteq \mathcal F_{km+m+1}$. In other words, $L_{i,k}$ is a stopping time with respect to filtration $\{F_{km+l}\}_{l\geq 1}$.* *Proof.* For any $i$ and any $k$, it is straightforward to see that $L_{i,k}$ is well-defined and $1\leq L_{i,k}\leq m+1$. When $L_{i,k}=l\leq m$, this is equivalent with $\|\gamma_i z_{km+l}\|_\infty \geq \eta$ but $\|\gamma_i z_{km+s}\|<\eta$ for $1\leq s<l$. Notice that this event is only determined by $z_{km+l}, \dots, z_{km+1}$, so $\{L_{i,k}=l\} \in \mathcal F_{km+l}$. When $L_{i,k}=m+1$, this is equivalent with $\|\gamma_i z_{km+s}\|<\eta$ for $1\leq s\leq m$. Notice that this event is only determined by $z_{km+m}, \dots, z_{km+1}$, so $\{L_{i,k}=m+1\} \in \mathcal F_{km+m}$. Therefore, by definition, $L_{i,k}$ is a stopping time with respect to filtration $\{\mathcal F_{km+l}\}_{l\geq 1}$. ◻ ### Step (ii): a formal version of Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"} and its proof {#sec:D32} **Lemma 14** (Discretization of $\mathcal E_1\cap\mathcal E_2$ (Formal version of Lemma [Lemma 3](#lem: discretize E1 E2){reference-type="ref" reference="lem: discretize E1 E2"})). *Let $\mathcal M=\{\gamma_1, \dots, \gamma_{v_\gamma}\}$ be an $\epsilon_{\gamma}$-net of $\{\gamma: \|\gamma\|_F=1\}$ as defined in Lemma [Lemma 7](#lem: mesh on unit sphere of matrices){reference-type="ref" reference="lem: mesh on unit sphere of matrices"}, where $\epsilon_{\gamma}=\min(\frac{a_1}{4b_z \sqrt n_x}, 1)$, $v_\gamma = \tilde O(n_x^{2.5}n_z^{2.5})a_4^{n_x n_z}$, and $a_4=\max(\frac{4b_z \sqrt{n_x}}{a_1},1)$. Define $$\begin{aligned} \mathcal E_{1,i}=\{ \exists \, \gamma\in \Gamma_T, \text{ s.t. } b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1\delta}{4\sqrt n_x}, \ \forall\, k\geq 0\}. \end{aligned}$$ Then, we have $$\mathbb P(\mathcal E_1\cap\mathcal E_2)\leq \sum_{i=1}^{v_\gamma} \mathbb P(\mathcal E_{1,i}\cap \mathcal E_2).$$* The rest of this subsubsection is dedicated to the proof of Lemma [Lemma 14](#lem: discretize E1 E2 formal){reference-type="ref" reference="lem: discretize E1 E2 formal"}. As an overview: firstly, we will discuss the implications of $\mathcal E_2$ on $\gamma_i\in \mathcal M$. Then, we discuss the implications of $\mathcal E_2$ on any $\gamma$. Lastly, we prove Lemma [Lemma 14](#lem: discretize E1 E2 formal){reference-type="ref" reference="lem: discretize E1 E2 formal"} by combining the implications of $\mathcal E_2$ on any $\gamma$ and $\|\gamma\|_F\geq \delta/2$. **Lemma 15** (The implication of $\mathcal E_2$ on $\gamma_i$). *If $\mathcal E_2$ happens, then for any $\gamma_i \in \mathcal M$, any $0\leq k \leq (T-1)/m-1$, we have $$\max_{1\leq s\leq m}\|\gamma_iz_{km+s}\|_\infty \geq \frac{a_1}{\sqrt n_x}.$$ Therefore, almost surely, we have $1\leq L_{i,k}\leq m$ and $$b_{i, km+L_{i,k}}(\gamma_iz_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1}{\sqrt n_x}.$$* *Proof.* If $\mathcal E_2$ happens, then by definition, we have $$\frac{1}{m}\sum_{{s=1}}^{{m}} {z_{km+s} z_{km+s}^\top} \succeq a_1^2 I_{{n_z}},$$ for all $0\leq k \leq (T-1)/m-1$. Now, for any $\gamma_i\in\mathcal M$, we have that $$\begin{aligned} \label{equ: Di xt psd} \frac{1}{m}\sum_{s=1}^{m} \gamma_iz_{km+s}z_{km+s}^\top \gamma_i^\top \succeq a_1^2 \gamma_i \gamma_i^\top. \end{aligned}$$ Therefore, by taking trace at each side of [\[equ: Di xt psd\]](#equ: Di xt psd){reference-type="eqref" reference="equ: Di xt psd"}, we obtain $$\begin{aligned} \frac{1}{m}\sum_{s=1}^{m} \textup{tr}(\gamma_iz_{km+s}z_{km+s}^\top \gamma_i^\top)\geq a_1^2 \textup{tr}(\gamma_i \gamma_i^\top) \end{aligned}$$ Since $\gamma_i \in \mathbb S_{n_x\times n_z}(0,1)$, we have $\|\gamma_i\|_F=1$, so $\textup{tr}(\gamma_i \gamma_i^\top)=\textup{tr}( \gamma_i^\top \gamma_i)=\|\gamma_i\|_F^2=1$. Further, we have $$\textup{tr}(\gamma_iz_{km+s}z_{km+s}^\top \gamma_i^\top)=\textup{tr}(z_{km+s}^\top \gamma_i^\top \gamma_iz_{km+s} )=z_{km+s}^\top \gamma_i^\top \gamma_iz_{km+s}=\|\gamma_iz_{km+s}\|_2^2.$$ Consequently, we have $$\frac{1}{m}\sum_{s=1}^{m} \|\gamma_iz_{km+s}\|_2^2 \geq a_1^2$$ for all $k$. By the pigeonhole principle, we have that $$\max_{1\leq s \leq m} \|\gamma_iz_{km+s}\|_2^2 \geq a_1^2.$$ This is equivalent with $\max_{1\leq s \leq m} \|\gamma_iz_{km+s}\|_2 \geq a_1.$ Notice that $\|\gamma_iz_{km+s}\|_2 \leq \sqrt{n_x} \|\gamma_iz_{km+s}\|_\infty$, so $\max_{1\leq s \leq m} \sqrt{n_x}\|\gamma_iz_{km+s}\|_\infty \geq a_1$, which completes the proof of the first inequality in the lemma statement. Next, we prove the second inequality in the lemma statement. Notice that by the definition of $L_{i,k}$ in Lemma [Lemma 13](#lem: define stopping times Lik){reference-type="ref" reference="lem: define stopping times Lik"} and by $\eta=\frac{a_1}{\sqrt{n_x}}$, we have $1\leq L_{i,k}\leq m$ and $\|\gamma_iz_{km+L_{i,k}}\|_\infty \geq \frac{a_1}{\sqrt n_x}$ for all $k$. Further, by Lemma [Lemma 12](#lem: def bit, jit){reference-type="ref" reference="lem: def bit, jit"}, we have $\|\gamma_iz_{km+L_{i,k}}\|_\infty= b_{i, km+L_{i,k}}(\gamma_iz_{km+L_{i,k}})^{j_{i, km+L_{i,k}}}$ almost surely. Hence, we have $b_{i, km+L_{i,k}}(\gamma_iz_{km+L_{i,k}})^{j_{i, km+L_{i,k}}}\geq \frac{a_1}{\sqrt n_x}$, which completes the proof. ◻ **Lemma 16** (The implication of $\mathcal E_2$ on $\gamma z_t$). *If $\mathcal E_2$ happens, then for any $\gamma\in \mathbb R^{n_x\times n_z}$, there exists $1\leq i \leq v_\gamma$, such that $$b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1}{2\sqrt n_x}\|\gamma\|_F,$$ for all $0\leq k \leq (T-1)/m-1$.* *Proof.* Firstly, when $\gamma=0$, the inequality holds because both sides are 0. Next, when $\gamma\not =0$, it suffices to prove $b_{i, km+L_{i,k}}(\frac{\gamma}{\|\gamma\|_F} z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1}{2\sqrt n_x}$. Therefore, we will only consider $\gamma\in \mathbb S_{n_x\times n_z}(0,1)$. By Lemma [Lemma 7](#lem: mesh on unit sphere of matrices){reference-type="ref" reference="lem: mesh on unit sphere of matrices"}, there exists $\gamma_i \in \mathcal M$ such that $\|\gamma-\gamma_i \|_F \leq 2\epsilon_\gamma =\min(\frac{a_1}{2b_z \sqrt n_x}, 2)$. Notice that by Lemma [Lemma 15](#lem: implication of E2 on gammai){reference-type="ref" reference="lem: implication of E2 on gammai"}, if $\mathcal E_2$ happens, for all $k$, we have $$b_{i, km+L_{i,k}}(\gamma_i z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1}{\sqrt n_x}.$$ Therefore, $$\begin{aligned} b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}}& = b_{i, km+L_{i,k}}(\gamma_i z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}}\\ &\quad - b_{i, km+L_{i,k}}((\gamma_i-\gamma) z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}}\\ & \geq \frac{a_1}{\sqrt n_x}- |b_{i, km+L_{i,k}}((\gamma_i-\gamma) z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}}|\\ & \geq \frac{a_1}{\sqrt n_x}- \|(\gamma_i-\gamma) z_{km+L_{i,k}}\|_2\\ & \geq \frac{a_1}{\sqrt n_x}-\|\gamma_i -\gamma\|_2 \|z_{km+L_{i,k}}\|_2\\ & \geq \frac{a_1}{\sqrt n_x}- 2\epsilon_\gamma b_z\geq \frac{a_1}{2\sqrt n_x} \end{aligned}$$ ◻ ***Proof of Lemma [Lemma 14](#lem: discretize E1 E2 formal){reference-type="ref" reference="lem: discretize E1 E2 formal"}**.* By Lemma [Lemma 16](#lem: implication of E2 on gamma z){reference-type="ref" reference="lem: implication of E2 on gamma z"}, under $\mathcal E_2$, for any $\gamma\in \mathbb R^{n_x\times n_z}$, there exists $1\leq i \leq v_\gamma$, such that $$b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1}{2\sqrt n_x}\|\gamma\|_F,$$ for all $0\leq k \leq (T-1)/m-1$. Therefore, if $\mathcal E_1\cap \mathcal E_2$ happens, there exists $\gamma \in \Gamma_T$ and a corresponding $i$, such that $$b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1}{2\sqrt n_x}\|\gamma\|_F\geq \frac{a_1\delta}{4\sqrt n_x}.$$ Therefore, $$\begin{aligned} \mathbb P(\mathcal E_1\cap \mathcal E_2)\leq \mathbb P(\bigcup_{i=1}^{v_\gamma} \mathcal E_{1,i} \cap \mathcal E_2)\leq \sum_{i=1}^{v_\gamma}\mathbb P(\mathcal E_{1,i} \cap \mathcal E_2), \end{aligned}$$ which completes the proof. ◻ ### Proof of Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"} {#sec:D33} Notice that Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"} states two inequalities: in the following, we will first prove the first inequality $\mathbb P(\mathcal E_{1,i}\cap \mathcal E_2)\leq \mathbb P(\cap_{k=0}^{(T-1)/m-1} G_{i,k})$, then prove the second inequality on $\mathbb P(G_{i,k}\mid \cap_{k'=0}^{k-1}G_{i,k'})$. **Lemma 17** (Bound $\mathcal E_{1,i}\cap \mathcal E_2$ by $G_{i,k}$). *Under the conditions in Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"}, for any $1\leq i \leq v_{\gamma}$, we have $$\mathbb P(\mathcal E_{1,i}\cap \mathcal E_2)\leq \mathbb P(\bigcap_{k=0}^{(T-1)/m-1} G_{i,k}).$$* *Proof.* Firstly, for any $\gamma\in\Gamma_T$, we have $\|w_t -\gamma z_t\|_\infty \leq w_{\max}$ for all $t \geq 0$. This suggests that, for any $1\leq j\leq n_x$, we have $$-w_{\max}\leq w_t^j - (\gamma z_t)^j\leq w_{\max}.$$ Hence, we have $b(\gamma z_t)^j \leq bw_t^j+w_{\max}$ for any $b\in\{-1,1\}$, $1\leq j\leq n_x$, and $t\geq 0$. Next, by $\mathcal E_{1,i}$, there exists $\gamma\in \Gamma_T$ such that $b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1\delta}{4\sqrt n_x}$ for all $k\geq 0$. Therefore, $b_{i, km+L_{i,k}} w_{km+L_{i,k}}^{j_{i,km+L_{i,k}}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}$ for all $k$. Finally, $\mathcal E_{1,i}\cap \mathcal E_2$ implies that $b_{i, km+L_{i,k}} w_{km+L_{i,k}}^{j_{i,km+L_{i,k}}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}$ and $\frac{1}{m}\sum_{{s=1}}^{{m}} {z_{km+s} z_{km+s}^\top} \succeq a_1^2 I_{{n_z}}$ for all $k$, which is $\bigcap_k G_{i,k}$ by the definition of $G_{i,k}$. ◻ **Lemma 18** (Bound on $\mathbb P(G_{i,k}\mid \cap_{k'=0}^{k-1}G_{i,k'})$). *Under the conditions in Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"}, for any $1\leq i \leq v_{\gamma}$ and any $k\geq 0$, we have $$\mathbb P(G_{i, k}\mid \bigcap_{k'=0}^{k-1}G_{i,k'})\leq 1-q_w(\frac{a_1 \delta}{4\sqrt n_x}).$$* *Proof.* Firstly, notice that when $\frac{1}{m}\sum_{{s=1}}^{{m}} {z_{km+s} z_{km+s}^\top} \succeq a_1^2 I_{{n_z}}$, we have $1\leq L_{i,k}\leq m$ by the proof of Lemma [Lemma 15](#lem: implication of E2 on gammai){reference-type="ref" reference="lem: implication of E2 on gammai"}. Therefore, we have $$\begin{aligned} & \mathbb P(G_{i, k}\mid \bigcap_{k'=0}^{k-1}G_{i,k'}) \leq \mathbb P(b_{i, km+L_{i,k}} w_{km+L_{i,k}}^{j_{i,km+L_{i,k}}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}, 1\leq L_{i,k}\leq m \mid \bigcap_{k'=0}^{k-1}G_{i,k'})\\ \leq\ & \sum_{l=1}^m \mathbb P(b_{i, km+l} w_{km+l}^{j_{i,km+l}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}, L_{i,k}=l \mid \bigcap_{k'=0}^{k-1}G_{i,k'})\\ \leq \ & \sum_{l=1}^m \mathbb P(b_{i, km+l} w_{km+l}^{j_{i,km+l}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}\mid L_{i,k}=l, \bigcap_{k'=0}^{k-1}G_{i,k'}) \mathbb P(L_{i,k}=l \mid \bigcap_{k'=0}^{k-1}G_{i,k'})\\ \overset{(a)}{\leq} \ & (1-q_w(\frac{a_1 \delta}{4\sqrt n_x}))\sum_{l=1}^m\mathbb P(L_{i,k}=l \mid \bigcap_{k'=0}^{k-1}G_{i,k'})\\ \leq \ & 1-q_w(\frac{a_1 \delta}{4\sqrt n_x}) \end{aligned}$$ The inequality $(c)$ is proved in the following: $$\begin{aligned} & \mathbb P(b_{i, km+l} w_{km+l}^{j_{i,km+l}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}\mid L_{i,k}=l, \bigcap_{k'=0}^{k-1}G_{i,k'}) \\ = \ & \int_{v_{0:km+l}}\mathbb P(b_{i, km+l} w_{km+l}^{j_{i,km+l}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}, w_{0:km+l}=v_{0:km+l}\mid L_{i,k}=l, \bigcap_{k'=0}^{k-1}G_{i,k'}) \mathrm d v_{0:km+l}\\ =\ & \int_{v_{0:km+l}\in S_{km+l}} \ \mathbb P(b_{i, km+l} w_{km+l}^{j_{i,km+l}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}\mid w_{0:km+l}=v_{0:km+l}) \\ & \qquad\qquad\qquad \ \ \times \mathbb P( w_{0:km+l}=v_{0:km+l}\mid L_{i,k}=l, \bigcap_{k'=0}^{k-1}G_{i,k'})\mathrm d v_{0:km+l}\\ \overset{(b)}{\leq} \ & (1-q_w(\frac{a_1 \delta}{4\sqrt n_x})) \int_{v_{0:km+l}\in S_{km+l}}\mathbb P( w_{0:km+l}=v_{0:km+l}\mid L_{i,k}=l, \bigcap_{k'=0}^{k-1}G_{i,k'})\mathrm d v_{0:km+l}\\\\ = \ & 1-q_w(\frac{a_1 \delta}{4\sqrt n_x}),\end{aligned}$$ where we define a shorthand notation $w_{0:km+l} =(w_0, \dots, w_{km+l-1})$, and we use $v_{0:km+l}$ to denote a realization of $w_{0:km+l}$, then we define the set of values of $w_{0:km+l}$ as $S_{km+l}$ such that $L_{i,k}=l, \bigcap_{k'=0}^{k-1}G_{i,k'}$ holds. Notice that $L_{i,k}=l$ can be determined by a set of values of $w_{0:km+l}$ because $L_{i,k}$ is a stopping time of $\{F_{km+l}\}_{l\geq 1}$ and thus $\{L_{i,k}=l\}\in \mathcal F_{km+l}$. The inequality $(b)$ above is because of the following: firstly, notice that $b_{i,km+l}, j_{i,km+l}\in \mathcal F_{km+l}$, so $b_{i,km+l}, j_{i,km+l}$ are deterministic values when $w_{0:km+l}=v_{0:km+l}$. Further, since $w_{km+l}$ is independent of $w_{0:km+l}$, we have $\mathbb P(w_{max}+b w_{km+l}^j \geq \epsilon\mid w_{0:km+l}=v_{0:km+l}) \leq 1-q_w(\epsilon)$ for any deterministic $b, j$ and any $\epsilon>0$ by Assumption [Assumption 3](#ass: tight bound on wt){reference-type="ref" reference="ass: tight bound on wt"}. Hence, we have $\mathbb P(b_{i, km+l} w_{km+l}^{j_{i,km+l}} +w_{\max}\geq \frac{a_1\delta}{4\sqrt{n_x}}\mid w_{0:km+l}=v_{0:km+l})\leq 1-q_w(\frac{a_1 \delta}{4\sqrt n_x})$. ◻ ### Proof of Lemma [Lemma 2](#lem: bound PE and gamma>delta/2){reference-type="ref" reference="lem: bound PE and gamma>delta/2"} {#proof-of-lemma-lem-bound-pe-and-gammadelta2} The proof is by leveraging Lemma [Lemma 14](#lem: discretize E1 E2 formal){reference-type="ref" reference="lem: discretize E1 E2 formal"} and Lemma [Lemma 4](#lem: construct Gik and Lik){reference-type="ref" reference="lem: construct Gik and Lik"}. $$\begin{aligned} \mathbb P(\mathcal E_1\cap \mathcal E_2)& \leq \sum_{i=1}^{v_\gamma} \mathbb P(\mathcal E_{1,i}\cap \mathcal E_2)\\ & \leq \sum_{i=1}^{v_\gamma} \mathbb P(\bigcap_{k=0}^{(T-1)/m-1} G_{i,k})\\ & = \sum_{i=1}^{v_\gamma} \mathbb P(G_{i,0}) \mathbb P(G_{i,1}\mid G_{i,0})\cdots \mathbb P(G_{i, (T-1)/m-1}\mid \bigcap_{k=0}^{(T-1)/m-2}G_{i,k})\\ & \leq \sum_{i=1}^{v_\gamma} (1-q_w(\frac{a_1 \delta}{4\sqrt n_x}))^{(T-1)/m}\\ & \leq \tilde O(n_x^{2.5}n_z^{2.5})a_4^{n_x n_z}(1-q_w(\frac{a_1 \delta}{4\sqrt n_x}))^{(T-1)/m}.\end{aligned}$$ # Proof of Corollary [Corollary 1](#cor: uniform dist est. bdd){reference-type="ref" reference="cor: uniform dist est. bdd"} {#sec:cor1} The proof involves two parts. Firstly, we will show that $\text{Term 1}\leq \epsilon$ under our choice of $m$. Secondly, we will let $\text{Term 2}=\epsilon$, then we will show $\delta \leq \tilde O(n_x^{1.5}n_z^2/T)$, which completes the proof. **Step 1: show $\text{Term 1}\leq \epsilon$.** Notice that when $m\geq \frac{1}{a_3}O(\log T+2.5\log n_z +n_z \log a_2 +\log(1/\epsilon))= O(n_z+ \log T+\log(1/\epsilon))$, we have $T \tilde O(n_z^{2.5}) a_2^{n_z} \exp(-a_3m)\leq \epsilon$. Since $m\geq 1$, we obtain $\text{Term 1}\leq \epsilon$. **Step 2: let $\text{Term 2}=\epsilon$ and show $\delta \leq \tilde O(n_x^{1.5}n_z^2/T)$.** Let $\text{Term 2}=\epsilon$, then we have $(1-q_w(\frac{a_1 \delta}{4\sqrt n_x}))^{T/m} = \frac{\epsilon}{\tilde O(n_x^{2.5}n_z^{2.5})a_4^{n_x n_z}}$. Then, we obtain $(1-q_w(\frac{a_1 \delta}{4\sqrt n_x})) =\left( \frac{\epsilon}{\tilde O(n_x^{2.5}n_z^{2.5})a_4^{n_x n_z}}\right)^{m/T}$, which is equivalent with $$q_w(\frac{a_1 \delta}{4\sqrt n_x})=1-\left( \frac{\epsilon}{\tilde O(n_x^{2.5}n_z^{2.5})a_4^{n_x n_z}}\right)^{m/T}.$$ When $q_w(\frac{a_1 \delta}{4\sqrt n_x})= O(\frac{a_1 \delta}{4\sqrt n_x})$, we obtain $$\begin{aligned} \delta & = O(\frac{4\sqrt n_x}{a_1})\left(1-\left( \frac{\epsilon}{\tilde O(n_x^{2.5}n_z^{2.5})a_4^{n_x n_z}}\right)^{m/T}\right)\\ & \leq O(\frac{-4\sqrt n_x}{a_1})\log\left(\left(\frac{\epsilon}{\tilde O(n_x^{2.5}n_z^{2.5})a_4^{n_x n_z}}\right)^{m/T}\right)\\ & = O(\frac{4\sqrt n_x m}{a_1T})(\log(1/\epsilon) + n_xn_z+ \log(n_xn_z))\\ & = \tilde O\left( \frac{n_x^{1.5}n_z^2}{T}\right).\end{aligned}$$ **Step 3: prove Corollary [Corollary 1](#cor: uniform dist est. bdd){reference-type="ref" reference="cor: uniform dist est. bdd"}.** By leveraging the bounds above and Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"}, we have $\mathbb P(\textup{diam}(\Theta_T)\leq \tilde O\left( \frac{n_x^{1.5}n_z^2}{T}\right)) \geq \mathbb P(\textup{diam}(\Theta_T)\leq \delta)\geq 1-2\epsilon$. Since $\theta^*\in \Theta_T$ by definition, for any $\hat \theta_T\in \Theta_T$, we have $\|\hat \theta_T-\theta^*\|_F \leq \textup{diam}(\Theta_T)\leq \tilde O\left( \frac{n_x^{1.5}n_z^2}{T}\right)$ with probability at least $1-2\epsilon$. # Proof of Theorem [Theorem 2](#thm: loose bound){reference-type="ref" reference="thm: loose bound"} {#sec:thrm2} Specifically, we define $\epsilon_0= \frac{4\sqrt n_x}{a_1}(\hat w_{\max}-w_{\max})$. The proof is similar to the proof of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"}. Firstly, we define $\hat \Gamma_T$ as a translation of the set $\hat \Theta_T$: $$\begin{aligned} \label{equ: define hat Gamma t} \hat \Gamma_t=\bigcap_{s=0}^{t-1} \{\gamma : \|w_s -\gamma z_s \|_\infty \leq \hat w_{\max}\}, \quad \forall\, t\geq 0.\end{aligned}$$ Notice that $$\hat \Theta_T=\theta^*+\hat \Gamma_T$$ by considering $\gamma=\hat\theta -\theta^*$. Therefore, we can upper bound our goal event $\{\textup{diam}(\hat \Theta_T)>\delta+\epsilon_0\}$ by the event $\mathcal E_3$ defined below. $$\begin{aligned} \label{eq:middle hat} \mathbb P(\textup{diam}(\hat \Theta_T)>\delta +\epsilon_0) \leq \mathbb P(\mathcal E_3), \text{ where } \mathcal E_3:=\{\exists\, \gamma \in \hat \Gamma_T, \text{ s.t. } \|\gamma\|_F\geq \frac{\delta+\epsilon_0}{2} \}.%\frac{\delta}{2} \}.$\end{aligned}$$ Next, notice that $$\mathbb P(\textup{diam}(\hat \Theta_T)>\delta +\epsilon_0) \leq \mathbb P(\mathcal E_3)\leq \mathbb P(\mathcal E_3\cap \mathcal E_2)+\mathbb P(\mathcal E_2^c)$$ By Lemma [Lemma 1](#lem: bound PE c){reference-type="ref" reference="lem: bound PE c"}, we have already shown $\mathbb P(\mathcal E_2^c)\leq \text{Term 1}$. So we only need to discuss $\mathbb P(\mathcal E_3\cap \mathcal E_2)$. **Lemma 19**. *$$\mathbb P(\mathcal E_3\cap \mathcal E_2)\leq \textup{Term 2}$$* *Proof.* Firstly, define $$\begin{aligned} \mathcal E_{3,i}=\{ \exists \, \gamma\in \hat \Gamma_T, \text{ s.t. } b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1(\delta+\epsilon_0)}{4\sqrt n_x}, \ \forall\, k\geq 0\}. \end{aligned}$$ We have $\mathbb P(\mathcal E_3\cap\mathcal E_2)\leq \sum_{i=1}^{v_\gamma} \mathbb P(\mathcal E_{3,i}\cap \mathcal E_2)$ based on the same proof ideas of Lemma [Lemma 14](#lem: discretize E1 E2 formal){reference-type="ref" reference="lem: discretize E1 E2 formal"}. Next, we will show that $$\begin{aligned} \Pr(\mathcal E_{3,k}\cap \mathcal E_2)\leq \mathbb P(\bigcap_{k=0}^{T/m-1}G_{i,k}) \label{equ: E3k by Gik} \end{aligned}$$ This is because for any $\gamma\in\hat \Gamma_T$, we have $b(\gamma z_t)^j \leq bw_t^j+\hat w_{\max}$ for any $b\in\{-1,1\}$, $1\leq j\leq n_x$, and $t\geq 0$. By $\mathcal E_{3,i}$, there exists $\gamma\in \hat \Gamma_T$ such that $b_{i, km+L_{i,k}}(\gamma z_{km+L_{i,k}})^{j_{i, km+L_{i,k}}} \geq \frac{a_1(\delta+\epsilon_0)}{4\sqrt n_x}$ for all $k\geq 0$. Thus, $b_{i, km+L_{i,k}} w_{km+L_{i,k}}^{j_{i,km+L_{i,k}}} +\hat w_{\max}\geq \frac{a_1(\delta+\epsilon_0)}{4\sqrt{n_x}}$ for all $k$. Notice that this is equivalent with $b_{i, km+L_{i,k}} w_{km+L_{i,k}}^{j_{i,km+L_{i,k}}} + w_{\max}\geq \frac{a_1 \delta}{4\sqrt{n_x}}$ for all $k$ because $\epsilon_0= \frac{4\sqrt n_x}{a_1}(\hat w_{\max}-w_{\max})$. In this way, we can prove [\[equ: E3k by Gik\]](#equ: E3k by Gik){reference-type="eqref" reference="equ: E3k by Gik"}. Finally, we can complete the proof by the following. $$\begin{aligned} \mathbb P(\mathcal E_3\cap \mathcal E_2)& \leq \sum_{i=1}^{v_\gamma} \mathbb P(\mathcal E_{3,i}\cap \mathcal E_2) \leq \sum_{i=1}^{v_\gamma} \mathbb P(\bigcap_{k=0}^{(T-1)/m-1} G_{i,k})\\ & = \sum_{i=1}^{v_\gamma} \mathbb P(G_{i,0}) \mathbb P(G_{i,1}\mid G_{i,0})\cdots \mathbb P(G_{i, (T-1)/m-1}\mid \bigcap_{k=0}^{(T-1)/m-2}G_{i,k})\\ & \leq \sum_{i=1}^{v_\gamma} (1-q_w(\frac{a_1 \delta}{4\sqrt n_x}))^{(T-1)/m} \leq \text{Term 1} \end{aligned}$$ where the second last inequality is by Lemma [Lemma 18](#lem: bound Pr(Gik|Gik' for k'<k)){reference-type="ref" reference="lem: bound Pr(Gik|Gik' for k'<k)"} and the last inequality uses the definition of $v_\gamma$ in Lemma [Lemma 7](#lem: mesh on unit sphere of matrices){reference-type="ref" reference="lem: mesh on unit sphere of matrices"}. ◻ # Proof of Corollary [Corollary 2](#cor: estimation err bdd for B=0){reference-type="ref" reference="cor: estimation err bdd for B=0"} {#sec:cor2} The proof of Corollary [Corollary 2](#cor: estimation err bdd for B=0){reference-type="ref" reference="cor: estimation err bdd for B=0"} is exactly the same as the proofs of Theorem [Theorem 1](#thm: estimation err bdd){reference-type="ref" reference="thm: estimation err bdd"} and Corollary [Corollary 1](#cor: uniform dist est. bdd){reference-type="ref" reference="cor: uniform dist est. bdd"}. When $A^*$ is stable, we can show that $\|x_t\|_2 \leq b_x$ for all $t$. Further, by [@dean2019safely], the sequence $\{x_t\}_{t\geq 0}$ satisfies the $(1, \sigma_x, p_x)$-BMSB condition. Therefore, we complete the proof. [^1]: In addition to model uncertainties, robust control may also consider other system uncertainties, e.g., disturbances, measurement noises, etc. [^2]: Throughout the paper, we use $\text{TrunGauss}(0,\sigma_w, [-w_{\max}, w_{\max}])$ to refer to the truncated Gaussian distribution generated by Gaussian distribution with $\text{Gauss}(0, \sigma_w^2)$ with truncated range $[-w_{\max}, w_{\max}]$. [^3]: *In the Appendix, the accurate version of this lemma will specify the choice of $\tilde \Gamma_T$.* [^4]: In [@lu2019robust], the correlation between $\{s_k\}_{k\geq 0}$ and $\{x_t, u_t,w_t\}_{t\geq 0}$ is via the PE condition, but [@lu2019robust] assume deterministic PE to avoid this correlation issue. [^5]: Though we only describe a static linear policy $u_t=Kx_t$ here, the results in [@simchowitz2018learning; @tu2019sample; @dean2019safely] hold for dynamic linear policies. [^6]: Here, without loss of generality, we consider $\mathbb B_1\cap \mathbb S_{n_xn_z}(0,1), \dots, \mathbb B_{v_\gamma}\cap \mathbb S_{n_xn_z}(0,1)$ are not empty.

arxiv_math

--- abstract: | Given $m \in \mathbb{N} \setminus \{0\}$ and a compact Riemannian manifold $\mathcal{N}$, we construct for every map $u$ in the critical Sobolev space $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$, a map $U : \mathbb{B}^{m + 1} \to \mathcal{N}$ whose trace is $u$ and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity. address: Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium author: - Bohdan Bulanyi - Jean Van Schaftingen title: Singular extension of critical Sobolev mappings under an exponential weak-type estimate --- [^1] # Introduction The classical trace theory in Sobolev spaces [@Gagliardo_1957] states that for every $m \in \mathbb{N}\setminus \set{0}$ and $p \in \intvo{1}{\infty}$, there is a well-defined surjective trace operator $\mathop{\mathrm{tr}}_{\mathbb{S}^{m}} : W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathbb{R})\to W^{1-1/p, p} (\mathbb{S}^{m}, \mathbb{R})$ that coincides with the restriction on the subsets of continuous functions, where the first-order Sobolev space on the unit ball $\mathbb{B}^{m + 1}_{1} \subseteq \mathbb{R}^{m + 1}$ is defined as $$W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathbb{R}) = \set[\Big]{U : \mathbb{B}^{m + 1}_{1} \to \mathbb{R}\,\vert\,U \text{ is weakly differentiable and } \int_{\mathbb{B}^{m + 1}_{1}} \abs{\mathrm{D}U}^p < \infty}$$ and the fractional Sobolev space $W^{s, p} (\mathbb{S}^m, \mathbb{R})$ on the unit sphere $\mathbb{S}^m:=\partial \mathbb{B}_{1}^{m+1}$ is defined for $s \in \intvo{0}{1}$ as $$\label{def of fractsobspace} W^{s, p} (\mathbb{S}^{m}, \mathbb{R}) = \set[\bigg]{u : \mathbb{S}^{m} \to \mathbb{R}\,\vert\,u \text{ is measurable and } \smashoperator{\iint_{\mathbb{S}^{m}\times \mathbb{S}^{m}}} \frac{\abs{u (x) - u (y)}^p}{\abs{x - y}^{m + sp}}\,\mathrm{d}x \,\mathrm{d}y < \infty}.$$ Moreover, the trace operator $\mathop{\mathrm{tr}}_{\mathbb{S}^{m}}$ has a continuous linear right inverse through a suitable linear operator, so that, in particular, for every function $u \in W^{1-1/p, p} (\mathbb{S}^{m}, \mathbb{R})$ there exists a function $U \in W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathbb{R})$ such that $\mathop{\mathrm{tr}}_{\mathbb{S}^m} U = u$ and $$\label{eq_ohd8echeen0iuc1aXi8xi5uC} \int_{\mathbb{B}^{m + 1}_{1}} \abs{\mathrm{D}U}^p \le C \smashoperator{\iint_{\mathbb{S}^{m}\times \mathbb{S}^{m}}} \frac{\abs{u (x) - u (y)}^p}{\abs{x - y}^{m + p - 1}}\,\mathrm{d}x \,\mathrm{d}y,$$ where the constant $C \in \intvo{0}{\infty}$ in [\[eq_ohd8echeen0iuc1aXi8xi5uC\]](#eq_ohd8echeen0iuc1aXi8xi5uC){reference-type="eqref" reference="eq_ohd8echeen0iuc1aXi8xi5uC"} depends only on $m$ and $p$. Given a compact Riemannian manifold $\mathcal{N}$, which we can assume without loss of generality to be linearly embedded into $\mathbb{R}^\nu$ by Nash's isometric embedding theorem [@Nash_1956], we can consider the Sobolev spaces of mappings $$W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathcal{N}) = \set{U \in W^{1, p} (\mathbb{B}^{m+ 1}_{1}, \mathbb{R}^\nu) \,\vert\,U \in \mathcal{N} \text{ almost everywhere in \(\mathbb{B}^{m+ 1}_{1}\)}}$$ and $$W^{s, p} (\mathbb{S}^{m}, \mathcal{N}) = \set{u \in W^{s, p} (\mathbb{S}^{m}, \mathbb{R}^\nu) \,\vert\,u \in \mathcal{N} \text{ almost everywhere in \(\mathbb{S}^{m}\)}}.$$ Although it follows immediately from the classical trace theory that the trace operator is well defined and continuous from $W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathcal{N})$ to $W^{1 - 1/p, p} (\mathbb{S}^{m}, \mathcal{N})$, the proof of the surjectivity fails to extend to this case: one can just prove that every mapping in $W^{1 - 1/p, p} (\mathbb{S}^m, \mathcal{N})$ can be extended to a function in $W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathbb{R}^\nu)$; since the extension is constructed by averaging, there is no hope that the extension would satisfy the nonlinear manifold constraint in general. In the case of subcritical dimension $m < p - 1$, maps in the nonlinear Sobolev spaces $W^{1 - 1/p, p} (\mathbb{S}^m, \mathcal{N})$ and $W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathcal{N})$ can be assumed to be continuous by the Sobolev--Morrey embedding; it turns out that all maps in $W^{1 - 1/p, p} (\mathbb{S}^m, \mathcal{N})$ are traces of maps in $W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathcal{N})$ if and only if all continuous maps from $\mathbb{S}^m$ to $\mathcal{N}$ are restrictions of continuous mappings, or equivalently, if and only if the $m^{\text{th}}$ homotopy group $\pi_{m} (\mathcal{N})$ is trivial: $\pi_{m} (\mathcal{N}) \simeq \set{0}$ [@Bethuel_Demengel_1995]\*Th. 1. For the critical dimension $m = p - 1$, one gets similarly that all mappings in $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$ are traces of maps in $W^{1, m + 1} (\mathbb{B}^{m + 1}_{1}, \mathcal{N})$ if and only if $\pi_{m} (\mathcal{N}) \simeq \set{0}$ [@Bethuel_Demengel_1995]\*Th. 2; this can be explained by the bounded mean oscillation property of maps in $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$ and the possibility to extend homotopy and obstruction theories to such maps [@Brezis_Nirenberg_1995]. The situation is radically different in the supercritical dimension case $m > p - 1$, where several necessary or sufficient conditions, imposing either finiteness or triviality of some of the homotopy groups $\pi_{j} (\mathcal{N})$ for $j \in \set{1, \dotsc, \floor{p - 1}}$, have been proved and where the complete characterization of manifolds $\mathcal{N}$, for which the trace operator is surjective, still remains open. Going back to the case of subcritical or critical dimension $m \le p - 1$, one can wonder whether an estimate [\[eq_ohd8echeen0iuc1aXi8xi5uC\]](#eq_ohd8echeen0iuc1aXi8xi5uC){reference-type="eqref" reference="eq_ohd8echeen0iuc1aXi8xi5uC"} can still hold for the extension of Sobolev mappings. It has been proved that every map $u \in W^{1 - 1/p, p} (\mathbb{S}^m, \mathcal{N})$ has an extension $U \in W^{1, p} (\mathbb{B}^{m + 1}_{1}, \mathcal{N})$ with the estimate [\[eq_ohd8echeen0iuc1aXi8xi5uC\]](#eq_ohd8echeen0iuc1aXi8xi5uC){reference-type="eqref" reference="eq_ohd8echeen0iuc1aXi8xi5uC"} when either $m = 1$ and $\pi_1 (\mathcal{N}) \simeq \set{0}$ or $m \ge 2$, $\pi_1 (\mathcal{N})$ is finite and $\pi_2 (\mathcal{N}) \simeq \dotsb \simeq \pi_{\floor{p}} (\mathcal{N}) \simeq \set{0}$ ; moreover, if there is an extension satisfying [\[eq_ohd8echeen0iuc1aXi8xi5uC\]](#eq_ohd8echeen0iuc1aXi8xi5uC){reference-type="eqref" reference="eq_ohd8echeen0iuc1aXi8xi5uC"}, then the homotopy groups $\pi_{1} (\mathcal{N}), \dotsc, \pi_{\floor{p - 1}} (\mathcal{N})$ have all to be finite ; in the critical case $m = p - 1$, the additional condition that $\pi_{m}(\mathcal{N}) \simeq \set{0}$ is necessary for a Sobolev estimate on the extension of continuous mappings that have a continuous extension [@Mironescu_VanSchaftingen_2021_Trace]. In view of the obstructions to linear Sobolev estimates on the extension of the form [\[eq_ohd8echeen0iuc1aXi8xi5uC\]](#eq_ohd8echeen0iuc1aXi8xi5uC){reference-type="eqref" reference="eq_ohd8echeen0iuc1aXi8xi5uC"}, one can hope to get some nonlinear estimates instead. In the subcritical dimension case, a compactness argument [@Petrache_VanSchaftingen_2017]\*Th. 4 shows that given $m \in \mathbb{N}\setminus \set{0}$, $p > m + 1$ and a compact Riemannian manifold $\mathcal{N}$, there exists a function $\gamma \in C (\intvr{0}{\infty}, \intvr{0}{\infty})$ such that $\gamma (0) = 0$, and every map $u \in W^{1 - 1/p, p} (\mathbb{S}^m, \mathcal{N})$ that has a continuous extension has an extension $U \in W^{1, p} (\mathbb{B}^{m+1}_{1}, \mathcal{N})$ satisfying $$\label{eq_shaequunah6UShaiSoo5eepi} \int_{\mathbb{B}^{m + 1}_{1}} \abs{\mathrm{D}U}^p \le \gamma\brk[\bigg]{\; \smashoperator[r]{\iint_{\mathbb{S}^{m}\times \mathbb{S}^{m}}} \frac{d(u (x),u (y))^p}{\abs{x - y}^{m + p - 1}}\,\mathrm{d}x \,\mathrm{d}y},$$ where $d$ is the geodesic distance on $\mathcal{N}$. Because of bubbling phenomena, this estimate [\[eq_shaequunah6UShaiSoo5eepi\]](#eq_shaequunah6UShaiSoo5eepi){reference-type="eqref" reference="eq_shaequunah6UShaiSoo5eepi"} does not go to the endpoint $p = m + 1$ when $\pi_{m} (\mathcal{N}) \not \simeq\{0\}$. Indeed, any map in $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$ is the weak limit of continuous maps in $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$ that can be extended so that a weak compactness argument would imply that every map in $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$ has an extension $W^{1, m + 1} (\mathbb{B}^{m+1}_{1}, \mathcal{N})$, which cannot be the case when $\pi_{m} (\mathcal{N}) \not \simeq \set{0}$ (see [@Petrache_Riviere_2014]\*Prop.  2.8). In this situation, it has been proved in works by Petrache, Rivière and the second author that there is still an extension $U \in W^{1, (m + 1, \infty)} (\mathbb{B}^{m+1}_{1}, \mathcal{N})$ satisfying a weak-type Marcinkiewicz--Sobolev estimate : for every $t \in \intvo{0}{\infty}$, $$\label{eq_eeyahGie3shee1ahth5luth9} t^{m + 1} \mathcal{L}^{m+1} \brk[\big]{\set{x \in \mathbb{B}^{m + 1}_{1} \,\vert\,\abs{\mathrm{D}U(x)} > t }} \le \gamma\brk[\bigg]{\; \smashoperator[r]{\iint_{\mathbb{S}^{m}\times \mathbb{S}^{m}}} \frac{d (u (x), u (y))^p}{\abs{x - y}^{2 m}}\,\mathrm{d}x \,\mathrm{d}y}.$$ The function $\gamma$, appearing in the estimate [\[eq_eeyahGie3shee1ahth5luth9\]](#eq_eeyahGie3shee1ahth5luth9){reference-type="eqref" reference="eq_eeyahGie3shee1ahth5luth9"} for a general target manifold $\mathcal{N}$, is a wild *double exponential* function [@Petrache_VanSchaftingen_2017]\*discussion after Th. 2. Petrache and Rivière get similar estimates with $\gamma$ being a polynomial when $m = 2$ and $\mathcal{N} = \mathbb{S}^2$ [@Petrache_Riviere_2014]\*Th. C (thanks to the Hopf fibration) and an exponential of a power when $m = 3$ and $\mathcal{N} = \mathbb{S}^3$ [@Petrache_Riviere_2014]\*Th. B. In the present work, we construct an extension $U$ that satisfies [\[eq_eeyahGie3shee1ahth5luth9\]](#eq_eeyahGie3shee1ahth5luth9){reference-type="eqref" reference="eq_eeyahGie3shee1ahth5luth9"}, where $\gamma$ can be taken to be an exponential function. **Theorem 1**. *Let $m \in \mathbb{N}\setminus \set{0}$ and let $\mathcal{N}$ be a compact Riemannian manifold. There exist constants $A, B, \delta \in \intvo{0}{\infty}$ such that for every $u \in W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$, there exists a mapping $U \in W^{1, 1}_{\mathrm{loc}} (\bar{\mathbb{B}}^{m + 1}_{1}, \mathcal{N})$ such that $\mathop{\mathrm{tr}}_{\mathbb{S}^m} U = u$ and for every $t \in \intvo{0}{\infty}$, $$\begin{gathered} \label{eq_Ufau0geiGh1Eang5voozehai} t^{m + 1}\mathcal{L}^{m + 1} \brk{\set{ x \in \mathbb{B}^{m + 1}_{1} \,\vert\,\abs{\mathrm{D}U (x)}\ge t}}\\ \le A \exp \brk[\Bigg]{\hspace{1em}B \smashoperator{ \iint_{\substack{ (x, y) \in \mathbb{S}^m \times \mathbb{S}^m\\ d (u (x), u (y))\ge \delta }}} \frac{1}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y } \smashoperator[r]{\iint_{\mathbb{S}^m \times \mathbb{S}^m}} \frac{d(u (x), u(y))^{m + 1}}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y.\end{gathered}$$ Moreover, one can take $U\in C(\mathbb{B}^{m + 1}_{1}\setminus S, \mathcal{N})$, where the singular set $S \subset \mathbb{B}^{m + 1}_{1}$ is a finite set whose cardinality is controlled by the right-hand side of [\[eq_Ufau0geiGh1Eang5voozehai\]](#eq_Ufau0geiGh1Eang5voozehai){reference-type="eqref" reference="eq_Ufau0geiGh1Eang5voozehai"}.* The gap potential of the double integral appearing in the exponential in [\[eq_Ufau0geiGh1Eang5voozehai\]](#eq_Ufau0geiGh1Eang5voozehai){reference-type="eqref" reference="eq_Ufau0geiGh1Eang5voozehai"} $$\label{eq_uto9aefeaBimaGhaix9EeVoo} \smashoperator{\iint_{\substack{ (x, y) \in \mathbb{S}^m \times \mathbb{S}^m\\ d (u (x), u (y))\ge \delta }}} \frac{1}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y$$ first appeared in estimates by Bourgain, Brezis and Mironescu on the topological degree of maps from a sphere to itself (see also [@Nguyen_2014]) and in estimates on free homotopy decompositions of mappings [@VanSchaftingen_2020], as well as in estimates on liftings ; they characterize, in the limit $\delta \to 0$, first-order Sobolev spaces and encompass a property stronger than VMO [@Brezis_Nguyen_2011]. implies, in particular, the weak-type estimates of [@Petrache_VanSchaftingen_2017]. The improvement of [Theorem 1](#theorem_ball){reference-type="ref" reference="theorem_ball"} is two-fold: the dependence of the weak-type Sobolev bound on the extension $U$ is exponential in the Gagliardo energy of $u$, which is much more reasonable than the double exponential in [@Petrache_VanSchaftingen_2017] and the nonlinear part of the estimate, that is the exponential, relies on a gap potential [\[eq_uto9aefeaBimaGhaix9EeVoo\]](#eq_uto9aefeaBimaGhaix9EeVoo){reference-type="eqref" reference="eq_uto9aefeaBimaGhaix9EeVoo"} instead of a full fractional Gagliardo energy $$\smashoperator{\iint_{\mathbb{S}^m \times \mathbb{S}^m}} \frac{d (u(x), u (y))^p}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y.$$ The latter controls the form by the immediate estimate $$\smashoperator{\iint_{\substack{ (x, y) \in \mathbb{S}^m \times \mathbb{S}^m\\ d (u (x), u (y))\ge \delta }}} \frac{1}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y \le \frac{1}{\delta^p} \smashoperator{\iint_{\mathbb{S}^m \times \mathbb{S}^m}} \frac{d (u(x), u (y))^p}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y.$$ We also have a counterpart of [Theorem 1](#theorem_ball){reference-type="ref" reference="theorem_ball"} for the extension on the half-space $\mathbb{R}^{m + 1 }_+ \coloneqq\mathbb{R}^m \times \intvo{0}{\infty}$ of mappings defined on the hyperplane $\mathbb{R}^m \simeq \mathbb{R}^m \times \set{0}$. **Theorem 2**. *Let $m \in \mathbb{N}\setminus \set{0}$ and let $\mathcal{N}$ be a compact Riemannian manifold. There exist constants $A, B, \delta \in \intvo{0}{\infty}$ such that for every $u \in W^{m/(m + 1), m + 1} (\mathbb{R}^m, \mathcal{N})$, there exists $U \in W^{1, 1}_{\mathrm{loc}} (\Bar{\mathbb{R}}^{m + 1}_+, \mathcal{N})$ such that $\mathop{\mathrm{tr}}_{\mathbb{R}^m} U = u$ and for every $t \in \intvo{0}{\infty}$, $$\begin{gathered} \label{eq_aiD8Oob2ahdah8lae0einei6} t^{m + 1} \mathcal{L}^{m + 1} \brk{\set{ x \in \mathbb{R}^{m + 1}_+ \,\vert\,\abs{\mathrm{D}U (x)}\ge t}}\\ \le A \exp \brk[\Bigg]{\hspace{1em}B \smashoperator{ \iint_{\substack{ (x, y) \in \mathbb{R}^m \times \mathbb{R}^m\\ d (u (x), u (y))\ge \delta }}} \frac{1}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y } \smashoperator[r]{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{d(u (x), u(y))^{m + 1}}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y.\end{gathered}$$ Moreover, one can take $U\in C(\mathbb{R}^{m + 1}_+ \setminus S, \mathcal{N})$, where the singular set $S \subset \mathbb{R}^{m + 1}_+$ is a finite set whose cardinality is controlled by the right-hand side of [\[eq_aiD8Oob2ahdah8lae0einei6\]](#eq_aiD8Oob2ahdah8lae0einei6){reference-type="eqref" reference="eq_aiD8Oob2ahdah8lae0einei6"}.* Finally, we consider the extension to the hyperbolic space $\mathbb{H}^{m + 1}$ of maps defined on its boundary sphere $\mathbb{S}^m$. Using the Poincaré ball model, the hyperbolic space $\mathbb{H}^{m + 1}$ is the ball $\mathbb{B}^{m+1}_{1}$, endowed with the metric $g$ defined for $x \in \mathbb{B}^{m + 1}_{1}$ in terms of the Euclidean metric $g_{\mathrm{euc}}$ by $$g (x) = \frac{4 g_{\mathrm{euc}}(x)}{\brk{1 - \abs{x}^2}^2},$$ whose boundary $\mathbb{S}^{m}$ is then considered to be the boundary sphere of $\mathbb{H}^{m + 1}$. In this setting, we have a hyperbolic counterpart of [Theorem 1](#theorem_ball){reference-type="ref" reference="theorem_ball"} and [Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"}. **Theorem 3**. *Let $m \in \mathbb{N}\setminus \set{0}$ and let $\mathcal{N}$ be a compact Riemannian manifold. There exist constants $A, B, \delta \in \intvo{0}{\infty}$ such that for every $u \in W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$, there exists $U \in W^{1, 1}_{\mathrm{loc}} (\bar{\mathbb{B}}^{m + 1}_{1}, \mathcal{N})$ such that $\mathop{\mathrm{tr}}_{\mathbb{S}^m} U = u$ and for every $t \in \intvo{0}{\infty}$, $$\begin{gathered} \label{eq_oogo0che0aevaeNgahshethu} \mathcal{H}^{m + 1} \brk{\set{ x \in \mathbb{H}^{m + 1} \,\vert\,\abs{\mathrm{D}U (x)}\ge t}}\\ \le \frac{A}{t^{m + 1}} \exp \brk[\Bigg]{\hspace{1em}B \smashoperator{ \iint_{\substack{ (x, y) \in \mathbb{S}^m \times \mathbb{S}^m\\ d (u (x), u (y))\ge \delta }}} \frac{1}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y } \smashoperator[r]{\iint_{\mathbb{S}^m \times \mathbb{S}^m}} \frac{d(u (x), u(y))^{m + 1}}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y.\end{gathered}$$ Moreover, one can take $U\in C(\mathbb{H}^{m + 1}\setminus S, \mathcal{N})$, where the singular set $S \subset \mathbb{H}^{m + 1}$ is a finite set whose cardinality is controlled by the right-hand side of [\[eq_oogo0che0aevaeNgahshethu\]](#eq_oogo0che0aevaeNgahshethu){reference-type="eqref" reference="eq_oogo0che0aevaeNgahshethu"}.* Although standard conformal transformations between $\mathbb{B}^{m+ 1}_{1}$, $\mathbb{H}^{m + 1}$ and $\mathbb{R}^{m + 1}_{+}$ preserve the fractional Gagliardo energy of the boundary values in the right-hand sides of [\[eq_Ufau0geiGh1Eang5voozehai\]](#eq_Ufau0geiGh1Eang5voozehai){reference-type="eqref" reference="eq_Ufau0geiGh1Eang5voozehai"}, [\[eq_aiD8Oob2ahdah8lae0einei6\]](#eq_aiD8Oob2ahdah8lae0einei6){reference-type="eqref" reference="eq_aiD8Oob2ahdah8lae0einei6"} and [\[eq_oogo0che0aevaeNgahshethu\]](#eq_oogo0che0aevaeNgahshethu){reference-type="eqref" reference="eq_oogo0che0aevaeNgahshethu"}, the corresponding gap potentials and the strong-type quantities $$\begin{aligned} &\int_{\mathbb{B}^{m + 1}_{1}} \abs{\mathrm{D}U}^{m + 1},& &\int_{\mathbb{R}^{m + 1}_{+}} \abs{\mathrm{D}U}^{m + 1}& &\text{ and }& &\int_{\mathbb{H}^{m + 1}} \abs{\mathrm{D}U}^{m + 1}\end{aligned}$$ corresponding to the weak-type quantities in their left-hand side, the left-hand sides as such are not conformally invariant so that [\[theorem_ball,theorem_halfspace,theorem_hyperbolic\]](#theorem_ball,theorem_halfspace,theorem_hyperbolic){reference-type="ref" reference="theorem_ball,theorem_halfspace,theorem_hyperbolic"} are not equivalent to each other. In practice, this explains why we have the same construction of $U$ in all three cases, but three different particular estimates on $U$. At the core of the proof of [\[theorem_ball,theorem_halfspace,theorem_hyperbolic\]](#theorem_ball,theorem_halfspace,theorem_hyperbolic){reference-type="ref" reference="theorem_ball,theorem_halfspace,theorem_hyperbolic"} is a refined understanding of functions in the fractional critical Sobolev space $W^{s, p} (\mathbb{R}^m, \mathbb{R}^\nu)$. We start from the elegant and versatile approach through the vanishing mean oscillation property of critical Sobolev maps [@Brezis_Nirenberg_1995]: if $sp = m$, it follows from Lebesgue's dominated convergence theorem and from the definition in [\[def of fractsobspace\]](#def of fractsobspace){reference-type="eqref" reference="def of fractsobspace"} that $$\label{eq_uMoo2ohthae6ahs9Aukahw5e} \lim_{r \to 0} \sup_{x \in \mathbb{R}^m} \fint_{\mathbb{B}^{m}_r (x)} \fint_{\mathbb{B}^{m}_r (x)} \abs{u (y) - u (z)}^p \,\mathrm{d}y \,\mathrm{d}z = 0.$$ If $u \in \mathcal{N}$ almost everywhere, then we have $$\label{eq_bu1aiv1weet5Feohij5reeth} \begin{split} \mathop{\mathrm{dist}}\brk[\bigg]{\fint_{\mathbb{B}^{m}_r (x)} u, \mathcal{N}}^{p} %&\le \fint_{\Bset^{m}_r (x)} \fint_{\Bset^{m}_r (x)} \abs{u (y) - u (z)} \dif y \dif z\\ \le \fint_{\mathbb{B}^{m}_r (x)} \fint_{\mathbb{B}^{m}_r (x)} \abs{u (y) - u (z)}^p \,\mathrm{d}y \,\mathrm{d}z. \end{split}$$ Thus, by [\[eq_uMoo2ohthae6ahs9Aukahw5e\]](#eq_uMoo2ohthae6ahs9Aukahw5e){reference-type="eqref" reference="eq_uMoo2ohthae6ahs9Aukahw5e"} and [\[eq_bu1aiv1weet5Feohij5reeth\]](#eq_bu1aiv1weet5Feohij5reeth){reference-type="eqref" reference="eq_bu1aiv1weet5Feohij5reeth"}, $$\label{eq_Xoomeip5AhcheiLe2iSuoc7p} \lim_{r \to 0} \sup_{x \in \mathbb{R}^m} \mathop{\mathrm{dist}}\brk[\Big]{\fint_{\mathbb{B}^{m}_r (x)} u, \mathcal{N}}^{p} = 0.$$ Because of the lack of control on the rate of convergence of the limit in [\[eq_Xoomeip5AhcheiLe2iSuoc7p\]](#eq_Xoomeip5AhcheiLe2iSuoc7p){reference-type="eqref" reference="eq_Xoomeip5AhcheiLe2iSuoc7p"}, we need to refine it to get the quantitative results we are looking for. To give a glimpse of our core quantitative estimate in a simplified setting, let us replace [\[eq_Xoomeip5AhcheiLe2iSuoc7p\]](#eq_Xoomeip5AhcheiLe2iSuoc7p){reference-type="eqref" reference="eq_Xoomeip5AhcheiLe2iSuoc7p"} by the integral inequality $$\label{eq_jaiMeiTeiv8eicoopaiw2ziz} \begin{split} \int_0^\infty \sup_{x \in \mathbb{R}^m}& \fint_{\mathbb{B}^{m}_r (x)} \fint_{\mathbb{B}^{m}_r (x)} |u (y) - u (z)|^p \,\mathrm{d}y \,\mathrm{d}z \frac{\,\mathrm{d}r}{r} \\ &\le \frac{1}{\mathcal{L}^m (\mathbb{B}^{m}_1)^2} \int_0^\infty \smashoperator[r]{\iint_{\substack{(y, z) \in \mathbb{R}^m \times \mathbb{R}^m\\ \abs{y - z} \le 2r}}} \abs{u (y) - u (z)}^p \,\mathrm{d}y \,\mathrm{d}z \frac{\,\mathrm{d}r}{r^{2m+1}}\\ &= \frac{4^{m}}{2m \mathcal{L}^m (\mathbb{B}^{m}_1)^2} \smashoperator[r]{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{\abs{u (y) - u (z)}^p}{\abs{y - z}^{2m}} \,\mathrm{d}y \,\mathrm{d}z. \end{split}$$ It follows from [\[eq_jaiMeiTeiv8eicoopaiw2ziz\]](#eq_jaiMeiTeiv8eicoopaiw2ziz){reference-type="eqref" reference="eq_jaiMeiTeiv8eicoopaiw2ziz"} and the Chebyshev inequality that for every $\lambda \in \intvo{1}{\infty}$ there exists $\tau \in \intvo{1}{\lambda}$ such that $$\label{eq_eowohY4vebuucooxooju1ohm} \begin{split} \sum_{k \in \mathbb{Z}} \sup_{x \in \mathbb{R}^m}& \mathop{\mathrm{dist}}\brk[\bigg]{\fint_{\mathbb{B}^{m}_{\tau \lambda^{-k}} (x)} u, \mathcal{N}}^p \\ &\le \sum_{k \in \mathbb{Z}} \sup_{x \in \mathbb{R}^m} \fint_{\mathbb{B}^{m}_{\tau \lambda^{-k}} (x)} \fint_{\mathbb{B}^{m}_{\tau \lambda^{-k}} (x)} \abs{u (y) - u (z)}^p \,\mathrm{d}y \,\mathrm{d}z \\ &\le \frac{4^m}{2m \mathcal{L}^m (\mathbb{B}^{m}_1)^2 \ln \lambda} \smashoperator[r]{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{\abs{u (y) - u (z)}^p}{\abs{y - z}^{2m}} \,\mathrm{d}y \,\mathrm{d}z. \end{split}$$ If we take now an extension $u$ by averaging on the half-space $\mathbb{R}^{m + 1}_+$, [\[eq_eowohY4vebuucooxooju1ohm\]](#eq_eowohY4vebuucooxooju1ohm){reference-type="eqref" reference="eq_eowohY4vebuucooxooju1ohm"} implies that when its right-hand side is small enough, that is, when $$\label{eq_hoh3aehaic7yeiSoyie7mein} \ln \lambda \simeq \smashoperator{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{\abs{u (y) - u (z)}^p}{\abs{y - z}^{2m}} \,\mathrm{d}y \,\mathrm{d}z ,$$ there is a family of hyperplanes whose distance to the boundary is in geometric progression of ratio $\lambda$ on which the extension is close to the manifold. In view of [\[eq_hoh3aehaic7yeiSoyie7mein\]](#eq_hoh3aehaic7yeiSoyie7mein){reference-type="eqref" reference="eq_hoh3aehaic7yeiSoyie7mein"}, the ratio $\lambda$ is controlled by an exponential of the Gagliardo fractional energy. In order to actually prove the results, we improve the estimate [\[eq_jaiMeiTeiv8eicoopaiw2ziz\]](#eq_jaiMeiTeiv8eicoopaiw2ziz){reference-type="eqref" reference="eq_jaiMeiTeiv8eicoopaiw2ziz"} in two directions. First, instead of considering parallel hyperplanes, we consider a decomposition of cubes on the boundary of which we control the mean oscillation. Second, we replace the mean oscillation by a truncated mean oscillation which can be controlled by the gap potential [\[eq_uto9aefeaBimaGhaix9EeVoo\]](#eq_uto9aefeaBimaGhaix9EeVoo){reference-type="eqref" reference="eq_uto9aefeaBimaGhaix9EeVoo"}. In comparison, the proof in [@Petrache_VanSchaftingen_2017] of [\[eq_eeyahGie3shee1ahth5luth9\]](#eq_eeyahGie3shee1ahth5luth9){reference-type="eqref" reference="eq_eeyahGie3shee1ahth5luth9"} relies on the estimate (see [@Petrache_VanSchaftingen_2017]\*Prop. 4.2) $$\begin{gathered} \int_{0}^R \sup_{\substack{x \in \mathbb{B}^{m + 1}_{1}\\ \abs{x}= r}} \mathop{\mathrm{dist}}\brk[\bigg]{\frac{(1 - r^2)^m}{\mathcal{H}^{m}(\mathbb{S}^m)} \int_{\mathbb{S}^m} \frac{u (y)}{\abs{x - y}^{2 m}} \,\mathrm{d}y, \mathcal{N}} \frac{\,\mathrm{d}r}{1 - r^2} \\ \le \int_{0}^R \frac{(1 - \abs{r}^2)^{2 m - 1}}{\mathcal{H}^{m}(\mathbb{S}^m)^2} \sup_{\substack{x \in \mathbb{B}^{m + 1}_{1}\\ \abs{x}= r}} \smashoperator[r]{\iint_{\mathbb{S}^m \times \mathbb{S}^m}} \frac{\abs{u (y) - u (z)}}{\abs{x - y}^{2 m}\abs{x - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z \,\mathrm{d}r \\ \le C \brk[\Big]{\ln \frac{2}{1-R}}^{1 - \frac{1}{p}} \brk[\bigg]{\smashoperator[r]{\iint_{\mathbb{S}^m \times \mathbb{S}^m}} \frac{\abs{u (y)- u (z)}^p}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z}^\frac{1}{p},\end{gathered}$$ with $R \in \intvo{0}{1}$. This allows to find bad balls in the hyperbolic space in the Poincaré ball models outside of which the hyperharmonic extension to $\mathbb{B}^{m+1}_{1}$ is close to the target manifold and whose radius is controlled by the exponential of the Gagliardo fractional energy. One cannot perform directly a homogeneous extension on the bad balls because they could intersect; through a Besicovitch-type covering argument, the construction is performed on collections of disjoint balls, with a number of collections bounded exponentially; combined with the exponential of the radius appearing in the equivalence between critical Marcinkiewicz--Sobolev quasinorms, this explains the double exponential appearing in the final estimate. In the present work we avoid this pitfall by working directly with a decomposition into disjoint cubes. # Mean oscillation on $\lambda$--adic skeletons ## Truncated mean oscillation Our analysis will rely on truncated mean oscillations, which have already been used in estimates on homotopy decompositions [@VanSchaftingen_2020] and on liftings over noncompact Riemannian coverings [@VanSchaftingen_SumLift]. **Definition 4**. Given $u : \mathbb{R}^{m} \to \mathcal{N}$, we define the function $\mathrm{MO}_{\delta, p} u : \mathbb{R}^{m + 1}_+ \to \intvc{0}{\infty}$ for every $x = (x', x_{m +1}) \in \mathbb{R}^{m + 1}_+$ by $$\label{eq_oJeeneithove3ahceid9Odah} \mathrm{MO}_{\delta, p} u (x) \coloneqq\fint_{\mathbb{B}^{m}_{x_{m +1}} (x')} \fint_{\mathbb{B}^{m}_{x_{m +1}} (x')} \brk{d (u (y), u (z)) - \delta}_+^p \,\mathrm{d}y \,\mathrm{d}z.$$ If $\delta_0, \delta_1 \in \intvr{0}{\infty}$, one has $$\label{eq_eer1ong6xaiyaeWei3uo8iem} \mathrm{MO}_{\delta_1, p} u (x)^\frac{1}{p} \le \mathrm{MO}_{\delta_0, p} u (x)^\frac{1}{p} + \brk{\delta_0 - \delta_1}_+.$$ The truncated Gagliardo fractional energy can be written in terms of mean oscillations as $$\smashoperator{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{\brk*{d (u (y), u (z)) - \delta}_+^p}{\abs{y - z}^{m + sp}} \,\mathrm{d}y \,\mathrm{d}z \simeq \int_{\mathbb{R}^{m + 1}_+} \mathrm{MO}_{\delta, p} u (x) \frac{\,\mathrm{d}x}{x_{m + 1}^{1 - m + sp}}.$$ If $V : \mathbb{R}^{m + 1}_+ \to \mathbb{R}^\nu$ denotes an extension by averaging of $u : \mathbb{R}^{m} \to \mathbb{R}^\nu$, that is $$\label{eq_shee0Ceingooghe2weefei4A} V (x) \coloneqq\frac{1}{x_{m + 1}^m} \int_{\mathbb{R}^m} u (z) \varphi \brk*{\tfrac{x' - z}{x_{m + 1}}} \,\mathrm{d}z = \int_{\mathbb{R}^m} u(x' - x_{m + 1} z) \varphi(z) \,\mathrm{d}z,$$ with $\varphi \in C^\infty (\mathbb{R}^m, \mathbb{R})$ such that $\int_{\mathbb{R}^m} \varphi = 1$ and $\mathop{\mathrm{supp}}\varphi \subseteq \mathbb{B}^{m}_1$, then one has for every $x = (x', x_{m + 1}) \in \mathbb{R}^{m + 1}_+$, if $u \in Y \subseteq \mathbb{R}^\nu$ almost everywhere on $\mathbb{R}^m$, $$\label{eq_phaejateiGh6ceir1kohngis} \mathop{\mathrm{dist}}(V (x), Y)^p \le \undefined(\mathrm{MO}_{\delta, p} u (x)+ \delta^p)$$ and $$\label{eq_taerinie4aWee3Phahzefeib} \abs{\mathrm{D}V (x)}^p \le \frac{\undefined}{x_{m+1}^p} \brk{\mathrm{MO}_{\delta, p} u (x)+ \delta^p}.$$ **Remark 5**. Replacing one of the averages in the [\[eq_oJeeneithove3ahceid9Odah\]](#eq_oJeeneithove3ahceid9Odah){reference-type="eqref" reference="eq_oJeeneithove3ahceid9Odah"} by the value of the mapping $u$, one gets the sharp truncated mean oscillation $\mathrm{MO}_{\delta, p}^\sharp u : \mathbb{R}^{m + 1}_+ \to \intvc{0}{\infty}$ defined for every $x = (x', x_{m +1}) \in \mathbb{R}^{m + 1}_+$ by $$\mathrm{MO}_{\delta, p}^\sharp u (x) \coloneqq\inf_{z \in \mathbb{R}^{m}} \fint_{\mathbb{B}^{m}_{x_{m +1}} (x')} \brk{d (u (y), u (z)) - \delta}_+^p \,\mathrm{d}y;$$ it follows immediately by the triangle inequality that $$\label{ineqestmeanosc0} \mathrm{MO}_{\delta, p} u (x) \le 2^p \mathrm{MO}_{\delta/2, p}^\sharp u (x)$$ and by Fubini's theorem that $$\fint_{\mathbb{B}^{m}_{x_{m +1}} (x')} \mathrm{MO}_{\delta, p}^\sharp u (z, x_{m +1}) \,\mathrm{d}z \le 2^{2m} \mathrm{MO}_{\delta, p} u (x',2x_{m}).$$ ## $\lambda$--adic cubes and skeletons The domain of the linear extension is the half-space $\mathbb{R}^{m + 1}_+$; we will subdivide the latter in $\lambda$--adic cubes on which we will perform appropriate constructions. Given $\lambda \in \intvo{1}{\infty}$, we consider for every $k \in \mathbb{Z}$ the set of cubes of $\mathbb{R}^{m}$ $$\mathcal{Q}_{\lambda, \tau, k} \coloneqq\set{ \tau \lambda^{-k} (\intvc{0}{1}^{m} + j) \,\vert\,j \in \mathbb{Z}^{m} }$$ and the corresponding set of cubes of $\mathbb{R}^{m + 1}_+$ $$\begin{split} \mathcal{Q}^+_{\lambda, \tau, k} % \coloneqq& \set{Q \times \intvc{\tfrac{\tau \lambda^{-k}}{\lambda - 1}}{\tfrac{\tau \lambda^{-(k-1)}}{\lambda - 1}}\,\vert\,Q \in \mathcal{Q}_{\lambda, \tau, k}}\\ % =& \set{ \tau \lambda^{-k} (\intvc{0}{1}^{m + 1} + (j, (\lambda - 1)^{-1})) \,\vert\,j \in \mathbb{Z}^{m} }. \end{split}$$ In particular, we have the decompositions $$\begin{aligned} \mathbb{R}^{m} &= \bigcup_{Q \in \mathcal{Q}_{\lambda, \tau, k} } Q& &\text{and}& \mathbb{R}^{m + 1}_+ &= \bigcup_{Q \in \mathcal{Q}^+_{\lambda, \tau, k} } Q.\end{aligned}$$ Moreover, we consider the part of the boundaries of the cubes in $\mathcal{Q}_{\lambda, \tau, k}^+$ that are parallel to the hyperplane $\mathbb{R}^m \times \set{0}$ $$\label{eq_eWiel4oa8looqu8yohph3thi} \begin{split} \mathcal{Q}^\parallel_{\lambda, \tau, k} % \coloneqq& \set{Q \times \set{\tfrac{\tau \lambda^{-k}}{\lambda - 1}}\,\vert\,Q \in \mathcal{Q}_{\lambda, \tau, k}}\\ % =& \set{ \tau \lambda^{-k} (\intvc{0}{1}^{m} \times \set{0} + (j, (\lambda - 1)^{-1})) \,\vert\,j \in \mathbb{Z}^{m} } \end{split}$$ and those that are normal to the same hyperplane $$\label{eq_aecheiPhai2phee8Iechaib5} \begin{split} \mathcal{Q}^\perp_{\lambda, \tau, k} % \coloneqq& \set{\partial Q \times \intvc{\tfrac{\tau \lambda^{-k}}{\lambda - 1}}{\tfrac{\tau \lambda^{-(k-1)}}{\lambda - 1}} \,\vert\,Q \in \mathcal{Q}_{\lambda, \tau, k}}\\ % =& \set{ \tau \lambda^{-k} ((\partial \intvc{0}{1}^{m}) \times \intvc{0}{1} + (j, (\lambda - 1)^{-1})) \,\vert\,j \in \mathbb{Z}^{m} }. \end{split}$$ Given $h \in \mathbb{R}^m$, we also define the corresponding translated sets $$\label{dcubesf} \begin{split} \mathcal{Q}_{\lambda, \tau, k, h} &\coloneqq\set{Q + \tau \lambda^{-k} h \,\vert\,Q \in \mathcal{Q}_{\lambda, \tau, k}},\\ \mathcal{Q}^+_{\lambda, \tau, k, h} &\coloneqq\set{Q + \tau \lambda^{-k} (h, 0) \,\vert\,Q \in \mathcal{Q}^+_{\lambda, \tau, k}},\\ \mathcal{Q}^\parallel_{\lambda, \tau, k, h} &\coloneqq\set{\Sigma + \tau \lambda^{-k} (h, 0) \,\vert\,\Sigma \in \mathcal{Q}^\parallel_{\lambda, \tau, k}},\\ \mathcal{Q}^\perp_{\lambda, \tau, k, h} &\coloneqq\set{\Sigma + \tau \lambda^{-k} (h, 0) \,\vert\,\Sigma \in \mathcal{Q}^\perp_{\lambda, \tau, k}}. \end{split}$$ ## Longitudinal estimate We first have an estimate on the maximal mean oscillation on longitudinal faces of cubes of the $\lambda$--adic decomposition of $\mathbb{R}^{m + 1}_+$. **Proposition 6**. *For every $m \in \mathbb{N}\setminus \set{0}$, there exists a constant $C = C(m) \in \intvo{0}{\infty}$ such that for every $p \in \intvr{1}{\infty}$, for every measurable map $u : \mathbb{R}^m \to \mathcal{N}$, for every $\lambda \in \intvr{2}{\infty}$ and for every $\delta \in \intvr{0}{\infty}$, one has $$\label{eq_ohTae0erahK4sohhohraec8s} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \sum_{\Sigma \in \mathcal{Q}_{\lambda, \tau, k, h}^\parallel} \sup_{x \in \Sigma} \mathrm{MO}_{\delta, p} u (x) \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau} \le C \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{\brk{d \brk{u (y), u (z)} - \delta}_+^p}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z.$$* *Proof.* For any $\Sigma \in \mathcal{Q}_{\lambda, \tau, k, h}^\parallel$, we can write, in view of [\[eq_eWiel4oa8looqu8yohph3thi\]](#eq_eWiel4oa8looqu8yohph3thi){reference-type="eqref" reference="eq_eWiel4oa8looqu8yohph3thi"} and [\[dcubesf\]](#dcubesf){reference-type="eqref" reference="dcubesf"}, $$\Sigma = Q \times \set{ \tfrac{\tau \lambda^{-k}}{\lambda - 1}},$$ where $Q \in \mathcal{Q}_{\lambda, \tau, k, h}$. If $x \in \Sigma$ and $y \in \mathbb{R}^{m}$ satisfy $\abs{x' - y} \le \tfrac{\tau \lambda^{-k}}{\lambda - 1}$, then $x' \in Q$ and we have immediately that $$\label{eq28_section2.1} \mathop{\mathrm{dist}}(y, Q) \le \abs{x' - y} \le \tfrac{\tau \lambda^{-k}}{\lambda - 1}.$$ Since $Q$ is a cube of edge length $\tau \lambda^{-k}$, according to [\[eq28_section2.1\]](#eq28_section2.1){reference-type="eqref" reference="eq28_section2.1"}, we have $$y \in (1 + \tfrac{2}{\lambda - 1}) Q = \tfrac{\lambda + 1}{\lambda - 1} Q$$ under the convention that $\tfrac{\lambda + 1}{\lambda - 1} Q$ is the cube with the same center that $Q$ dilated by a factor $\frac{\lambda + 1}{\lambda - 1}$. Thus, $y \in 3Q$, since $\lambda \ge 2$. One has then, by monotonicity of the integral, for every $x \in \Sigma$, $$\label{eq_euXungei0adee9aeX3Iec6Ee} \begin{split} \mathrm{MO}_{\delta, p} u (x) % &= \frac{1}{\mathcal{L}^{m}(\mathbb{B}^{m}_{1})^{2}} \brk[\Big]{\frac{ \lambda^k (\lambda - 1)}{\tau}}^{2m} \smashoperator{\iint_{\substack{(y, z) \in \mathbb{R}^{m} \times \mathbb{R}^{m}\\ \abs{y - x'}\le \frac{\tau \lambda^{-k}}{\lambda - 1}\\ \abs{z - x'}\le \frac{\tau \lambda^{-k}}{\lambda - 1} }}} \brk{d \brk{u (y), u (z)} - \delta}_+^p \,\mathrm{d}y \,\mathrm{d}z\\ % &\le \frac{1}{\mathcal{L}^{m}(\mathbb{B}^{m}_{1})^{2}}\brk[\Big]{\frac{\lambda^k (\lambda - 1)}{\tau}}^{2m}\smashoperator{\iint_{\substack{(y,z)\in 3Q \times 3Q \\ \abs{y-z}\le \frac{2\tau\lambda^{-k}}{\lambda-1}}}} \brk{d \brk{u (y), u (z)} - \delta}_+^p \,\mathrm{d}y \,\mathrm{d}z. \end{split}$$ Summing [\[eq_euXungei0adee9aeX3Iec6Ee\]](#eq_euXungei0adee9aeX3Iec6Ee){reference-type="eqref" reference="eq_euXungei0adee9aeX3Iec6Ee"} over the sets $\Sigma \in \mathcal{Q}_{\lambda, \tau, k, h}^\parallel$ and integrating it over the translations $h \in \intvc{0}{1}^m$, we get $$\begin{gathered} \label{eq_quo2NiGhee4xooyeinahmi1v} \int_{\intvc{0}{1}^{m}} \sum_{\Sigma \in \mathcal{Q}_{\lambda, \tau, k, h}^\parallel} \sup_{x \in \Sigma} \mathrm{MO}_{\delta, p} u (x) \,\mathrm{d}h\\ \le \frac{5^{m}}{\mathcal{L}^{m}(\mathbb{B}^{m}_{1})^{2}} \brk[\Big]{\frac{\lambda^k (\lambda - 1)}{\tau}}^{2m}\smashoperator{\iint_{\substack{(y, z) \in \mathbb{R}^{m} \times \mathbb{R}^{m}\\ \abs{y - z} \le \frac{2\tau \lambda^{-k}}{\lambda - 1}}}} \brk{d \brk{u (y), u (z)} - \delta}_+^p \,\mathrm{d}y \,\mathrm{d}z.\end{gathered}$$ Summing [\[eq_quo2NiGhee4xooyeinahmi1v\]](#eq_quo2NiGhee4xooyeinahmi1v){reference-type="eqref" reference="eq_quo2NiGhee4xooyeinahmi1v"} over the scales $k \in \mathbb{Z}$ and integrating it over $\tau \in \intvc{1}{\lambda}$, we get $$\label{eq_eeReuxah7thaighiich3Iich} \begin{split} \int_1^\lambda \sum_{k \in \mathbb{Z}} &\int_{\intvc{0}{1}^{m}} \sum_{\Sigma \in \mathcal{Q}_{\lambda, \tau, k, h}^\parallel} \sup_{x \in \Sigma} \mathrm{MO}_{\delta, p} u (x) \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % &\le \frac{5^{m}}{\mathcal{L}^{m}(\mathbb{B}^{m}_{1})^{2}} (\lambda - 1)^{2 m} \smashoperator[l]{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \sum_{k \in \mathbb{Z}} \smashoperator[r]{\int_{\substack{\tau \in \intvo{1}{\lambda}\\ \tau \ge (\lambda - 1)\lambda^k \abs{y - z}/2}}} \frac{\lambda^{2k m } \brk{d \brk{u (y), u (z)} - \delta}_+^p}{\tau^{2 m + 1}} \,\mathrm{d}\tau \,\mathrm{d}y \,\mathrm{d}z\\ % &= \frac{5^{m}}{\mathcal{L}^{m}(\mathbb{B}^{m}_{1})^{2}} \brk{\lambda - 1}^{2m} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}} } \int_{\frac{(\lambda - 1) \abs{y - z}}{2}}^\infty \frac{\brk{d \brk{u (y), u (z)} - \delta}_+^p}{\theta^{2 m + 1}} \,\mathrm{d}\theta \,\mathrm{d}y \,\mathrm{d}z\\ % &= \frac{20^{m}}{2m\mathcal{L}^{m}(\mathbb{B}^{m}_{1})^{2}} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{\brk{d \brk{u (y), u (z)} - \delta}_+^p}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z, \end{split}$$ which implies the announced conclusion [\[eq_ohTae0erahK4sohhohraec8s\]](#eq_ohTae0erahK4sohhohraec8s){reference-type="eqref" reference="eq_ohTae0erahK4sohhohraec8s"} and completes our proof of Proposition [Proposition 6](#proposition_longitudinal){reference-type="ref" reference="proposition_longitudinal"}. ◻ ## Transversal estimate We next prove a counterpart of [Proposition 6](#proposition_longitudinal){reference-type="ref" reference="proposition_longitudinal"}, where we estimate the maximal mean oscillation on transversal faces of cubes of the $\lambda$--adic decomposition of $\mathbb{R}^{m + 1}_+$ instead of the longitudinal ones. **Proposition 7**. *For every $m \in \mathbb{N}\setminus \set{0}$ and $p \in \intvr{1}{\infty}$, there exists a constant $C = C(m, p) \in \intvo{0}{\infty}$ such that for every measurable map $u : \mathbb{R}^m \to \mathcal{N}$, for every $\lambda \in \intvr{2}{\infty}$ and for every $\delta \in \intvr{0}{\infty}$, one has $$\label{eq_Quo8ief9gaidahfa0aebeir4} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \sum_{\Sigma \in \mathcal{Q}^\perp_{\lambda, \tau, k, h}} \sup_{x \in \Sigma} \mathrm{MO}_{\delta, p} u (x)\,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % \le C \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{\brk{d \brk{u (y), u (z)} - \frac{\delta}{2}}_+^p}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z.$$* *Proof.* We consider a set $\Sigma \in \mathcal{Q}^\perp_{\lambda, \tau, k, h}$, that we can write, in view of [\[eq_aecheiPhai2phee8Iechaib5\]](#eq_aecheiPhai2phee8Iechaib5){reference-type="eqref" reference="eq_aecheiPhai2phee8Iechaib5"} and [\[dcubesf\]](#dcubesf){reference-type="eqref" reference="dcubesf"}, as $$\Sigma = \partial Q \times \intvc{\tfrac{\tau \lambda^{-k}}{\lambda - 1}}{\tfrac{\tau \lambda^{-(k-1)}}{\lambda - 1}},$$ where $Q \in \mathcal{Q}_{\lambda, \tau, k, h}$. We first note that by [\[ineqestmeanosc0\]](#ineqestmeanosc0){reference-type="eqref" reference="ineqestmeanosc0"}, $$\label{eq_eq_rfoerjfojrifjtrgnvjfknnfg} \begin{split} \mathrm{MO}_{\delta, p} u (x) & \le 2^{p} \mathrm{MO}_{\delta/2, p}^\sharp u (x) \\ &\le 2^{p} \inf_{\substack{z \in \partial Q \\ \abs{z-x'}\le x_{m+1}}}\fint_{\mathbb{B}^{m}_{x_{m +1}} (x')} \brk{d (u (y), u (z)) - \tfrac{\delta}{2}}_+^p \,\mathrm{d}y\\ &\le 2^{p} \fint_{E_x} \fint_{B_{x_{m +1}} (x')} \brk{d (u (y), u (z)) - \tfrac{\delta}{2}}_+^p \,\mathrm{d}z \,\mathrm{d}y, \end{split}$$ where $$E_x \coloneqq\set{z \in \partial Q\,\vert\,\abs{z-x'}\le x_{m+1}}.$$ If $x \in \Sigma$ and $y \in \mathbb{R}^{m}$ satisfy $\abs{x' - y} \le \tfrac{\tau \lambda^{-(k - 1)}}{\lambda - 1}$, then $x' \in \partial Q \subset Q$ and $$\label{eq_eq_j43r99uh548gnrhgiuhyu} \mathop{\mathrm{dist}}(y, Q) \le \tfrac{\tau \lambda^{-(k - 1)}}{\lambda - 1}.$$ Since $Q$ is a cube of edge length $\tau \lambda^{-k}$ and $\lambda \geq 2$, using [\[eq_eq_j43r99uh548gnrhgiuhyu\]](#eq_eq_j43r99uh548gnrhgiuhyu){reference-type="eqref" reference="eq_eq_j43r99uh548gnrhgiuhyu"}, we get $$\label{eq_eq_e3209ire89urfj4hru4h} y \in (1 + \tfrac{2 \lambda}{\lambda - 1})Q = \tfrac{3 \lambda - 1}{\lambda - 1} Q \subseteq 5Q.$$ One has then for every $x \in \Sigma$, by [\[eq_eq_rfoerjfojrifjtrgnvjfknnfg\]](#eq_eq_rfoerjfojrifjtrgnvjfknnfg){reference-type="eqref" reference="eq_eq_rfoerjfojrifjtrgnvjfknnfg"}, [\[eq_eq_e3209ire89urfj4hru4h\]](#eq_eq_e3209ire89urfj4hru4h){reference-type="eqref" reference="eq_eq_e3209ire89urfj4hru4h"} and by monotonicity of the integral, $$\label{eq_ieHis6iuLaec3iedohmaigoh} \begin{split} \mathrm{MO}_{\delta, p} u (x) % & \le \frac{\undefined}{x_{m+1}^{2m - 1}} \smashoperator{\iint_{\substack{(y, z) \in \mathbb{R}^{m} \times \partial Q \\ \abs{y - x'}\le x_{m +1} \\ \abs{z-x'}\le x_{m+1}}}} \brk{d \brk{u (y), u (z)} - \tfrac{\delta}{2}}_+^p \,\mathrm{d}y \,\mathrm{d}z\\ % &\le \Cl{cst_aif9Haethah7xa5oedeV7xie} \smashoperator{\iint_{\substack{(y, z) \in 5Q \times \partial Q\\ \abs{y - z} \le \frac{2\tau\lambda^{-(k-1)}}{\lambda - 1}}}} \frac{\brk{d \brk{u (y), u (z)} - \frac{\delta}{2}}_+^p}{\abs{y - z}^{2 m - 1}} \,\mathrm{d}y \,\mathrm{d}z, \end{split}$$ where the constant $\Cr{cst_aif9Haethah7xa5oedeV7xie}>0$ depends only on $m$ and $p$. Summing [\[eq_ieHis6iuLaec3iedohmaigoh\]](#eq_ieHis6iuLaec3iedohmaigoh){reference-type="eqref" reference="eq_ieHis6iuLaec3iedohmaigoh"} over $\Sigma \in \mathcal{Q}^\perp_{\lambda, \tau, k,h}$ and integrating it with respect to the translations $h$ over $\intvc{0}{1}^{m}$, we have $$\label{eq_FooxoN7thahyair0jeoy3aej} \int_{\intvc{0}{1}^{m}} \sum_{\Sigma \in \mathcal{Q}^\perp_{\lambda, \tau, k, h}} \sup_{x \in \Sigma } \mathrm{MO}_{\delta, p} u (x)\,\mathrm{d}h \le \frac{\Cr{cst_aif9Haethah7xa5oedeV7xie} \lambda^k}{\tau} \smashoperator{\iint_{\substack{(y, z) \in \mathbb{R}^{m} \times \mathbb{R}^{m}\\ \abs{y - z} \le \frac{2\tau\lambda^{-(k-1)}}{\lambda - 1}}}} \frac{\brk{d \brk{u (y), u (z)} - \frac{\delta}{2}}_+^p}{\abs{y - z}^{2 m - 1}} \,\mathrm{d}y \,\mathrm{d}z,$$ where we have also used Fubini's theorem. Thus, summing and integrating [\[eq_FooxoN7thahyair0jeoy3aej\]](#eq_FooxoN7thahyair0jeoy3aej){reference-type="eqref" reference="eq_FooxoN7thahyair0jeoy3aej"} over the scales, we get $$\begin{split} \int_1^\lambda \sum_{k \in \mathbb{Z}} &\int_{\intvc{0}{1}^{m}} \sum_{\Sigma \in \mathcal{Q}^\perp_{\lambda, \tau, k,h}} \sup_{x \in \Sigma} \mathrm{MO}_{\delta, p} u (x)\,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % &\le \Cr{cst_aif9Haethah7xa5oedeV7xie} \smashoperator[l]{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \sum_{k \in \mathbb{Z}} \smashoperator[r]{\int_{\substack{\tau \in \intvo{1}{\lambda}\\ \tau \ge \lambda^k (\lambda - 1)\abs{y - z}/ 2\lambda }}} \frac{\lambda^{k } \brk{d \brk{u (y), u (z)} - \frac{\delta}{2}}_+^p}{\abs{y - z}^{2 m - 1} \tau^{2}} \,\mathrm{d}\tau \,\mathrm{d}y \,\mathrm{d}z\\ % &= \Cr{cst_aif9Haethah7xa5oedeV7xie} \smashoperator[l]{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \int_{\frac{(\lambda-1)\abs{y - z}}{2\lambda}}^\infty \frac{\brk{d \brk{u (y), u (z)} - \frac{\delta}{2}}_+^p}{\abs{y - z}^{2 m - 1} \theta^{2}} \,\mathrm{d}\theta \,\mathrm{d}y \,\mathrm{d}z\\ % &= \tfrac{2\lambda}{\lambda - 1} \, \Cr{cst_aif9Haethah7xa5oedeV7xie} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{\brk{d \brk{u (y), u (z)} - \frac{\delta}{2}}_+^p}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z, \end{split}$$ so that the conclusion [\[eq_Quo8ief9gaidahfa0aebeir4\]](#eq_Quo8ief9gaidahfa0aebeir4){reference-type="eqref" reference="eq_Quo8ief9gaidahfa0aebeir4"} follows. ◻ ## Combining the estimates We close this section by summarizing [\[proposition_longitudinal,proposition_transversal\]](#proposition_longitudinal,proposition_transversal){reference-type="ref" reference="proposition_longitudinal,proposition_transversal"} in the following statement. **Proposition 8**. *For every $m \in \mathbb{N}\setminus \set{0}$ and $p \in \intvr{1}{\infty}$, there exists a constant $C =C(m,p)>0$ such that for every measurable map $u : \mathbb{R}^m \to \mathcal{N}$, for every $\lambda \in \intvr{2}{\infty}$ and for every $\delta \in \intvr{0}{\infty}$, one has $$\int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h}} \sup_{x \in \partial Q} \mathrm{MO}_{\delta, p} u (x)\,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % \le C \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{\brk{d \brk{u (y), u (z)} - \frac{\delta}{2}}_+^p}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z.$$* *Proof.* This follows immediately from [\[proposition_longitudinal,proposition_transversal\]](#proposition_longitudinal,proposition_transversal){reference-type="ref" reference="proposition_longitudinal,proposition_transversal"}. ◻ # Proofs of the singular extension theorems ## Oscillation and gradient estimate on the skeleton We first estimate the average number of cubes on which the extension $V$ of $u$ given by [\[eq_shee0Ceingooghe2weefei4A\]](#eq_shee0Ceingooghe2weefei4A){reference-type="eqref" reference="eq_shee0Ceingooghe2weefei4A"} is far away from the range of $u$. **Proposition 9**. *Let $m \in \mathbb{N}\setminus \set{0}$. There exist constants $\eta \in \intvo{0}{1}$ and $C \in \intvo{0}{\infty}$ depending only on $m$ such that for every $\delta \in \intvo{0}{\infty}$, for every $\lambda \in \intvr{2}{\infty}$, for every measurable function $u : \mathbb{R}^m \to \mathbb{R}^\nu$ and every set $Y \subseteq \mathbb{R}^\nu$, if $V$ is an extension by convolution given by [\[eq_shee0Ceingooghe2weefei4A\]](#eq_shee0Ceingooghe2weefei4A){reference-type="eqref" reference="eq_shee0Ceingooghe2weefei4A"} and if $u \in Y$ almost everywhere in $\mathbb{R}^m$, then $$\begin{gathered} \label{eq_Aihohz6jeik2eeBioJ4wohgh} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \# \set {Q \in \mathcal{Q}^+_{\lambda, \tau, k,h} \,\vert\,\sup_{x \in \partial Q} \mathop{\mathrm{dist}}(V (x), Y) \ge \delta} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % \le \frac{C}{\delta^{m+1}} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{\brk{d \brk{u (y), u (z)} - \eta \delta}_+^{m + 1}}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z.\end{gathered}$$* *Proof.* Using [\[eq_eer1ong6xaiyaeWei3uo8iem\]](#eq_eer1ong6xaiyaeWei3uo8iem){reference-type="eqref" reference="eq_eer1ong6xaiyaeWei3uo8iem"}, [\[eq_phaejateiGh6ceir1kohngis\]](#eq_phaejateiGh6ceir1kohngis){reference-type="eqref" reference="eq_phaejateiGh6ceir1kohngis"} and the elementary inequality $(a+b)^{m+1}\le 2^m(a^{m+1}+b^{m+1})$, which is valid for all $a,b \in \intvr{0}{\infty}$, we have for every $x \in \mathbb{R}^{m + 1}_+$,$$\label{eq_lah7eifooSeilee7zaiV8boo} \mathop{\mathrm{dist}}(V (x), Y)^{m + 1} \le \Cl{cst_moukeiguegoaphaiNg8Siu6h} \mathrm{MO}_{0, m + 1} u (x) \le 2^m \Cr{cst_moukeiguegoaphaiNg8Siu6h} \brk{\mathrm{MO}_{\theta, m + 1} u (x)+ \theta ^{m + 1}}.$$ Hence, taking $$\eta \coloneqq\frac{1}{2}\brk[\Big]{\frac{2}{2+\Cr{cst_moukeiguegoaphaiNg8Siu6h} }}^\frac{1}{m + 1},$$ if $$\label{eq_eq_eq_01747} \mathop{\mathrm{dist}}(V (x), Y) \ge \delta,$$ the inequality [\[eq_lah7eifooSeilee7zaiV8boo\]](#eq_lah7eifooSeilee7zaiV8boo){reference-type="eqref" reference="eq_lah7eifooSeilee7zaiV8boo"} with $\theta = 2 \eta \delta$ implies that $$\label{eq_eq_eq_017471} (\eta \delta)^{m + 1} \le \mathrm{MO}_{2 \eta \delta, m+1}u (x).$$ We get, by [\[eq_eq_eq_01747\]](#eq_eq_eq_01747){reference-type="eqref" reference="eq_eq_eq_01747"}, [\[eq_eq_eq_017471\]](#eq_eq_eq_017471){reference-type="eqref" reference="eq_eq_eq_017471"} and [Proposition 8](#proposition_combined){reference-type="ref" reference="proposition_combined"}, $$\begin{split} \int_1^\lambda \sum_{k \in \mathbb{Z}} & \int_{\intvc{0}{1}^{m}} \# \set {Q \in \mathcal{Q}^+_{\lambda, \tau, k,h} \,\vert\,\sup_{x \in \partial Q} \mathop{\mathrm{dist}}(V (x), Y) \ge \delta} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % &\le \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \# \set {Q \in \mathcal{Q}^+_{\lambda, \tau, k, h} \,\vert\,\sup_{x \in \partial Q} \mathrm{MO}_{2\eta \delta, m + 1} u(x) \ge (\eta \delta)^{m + 1}} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % &\le \frac{1}{(\eta \delta)^{m+1}} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h}} \sup_{x \in \partial Q} \mathrm{MO}_{2 \eta \delta, m + 1} u(x) \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % &\le \frac{\Cr{cst_ahxu7teihaeTha5naex4peiR}}{\delta^{m+1}} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{\brk{d \brk{u (y), u (z)} - \eta \delta}_+^{m + 1}}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z, \end{split} %$$ where the constant $\Cl{cst_ahxu7teihaeTha5naex4peiR}>0$ depends only on $m$. This proves the estimate [\[eq_Aihohz6jeik2eeBioJ4wohgh\]](#eq_Aihohz6jeik2eeBioJ4wohgh){reference-type="eqref" reference="eq_Aihohz6jeik2eeBioJ4wohgh"} and completes our proof of Proposition [Proposition 9](#proposition_goodcubes_gapintegral){reference-type="ref" reference="proposition_goodcubes_gapintegral"}. ◻ Next, we can prove an average uniform bound on the extension by convolution. **Proposition 10**. *Let $m \in \mathbb{N}\setminus \set{0}$. There exists a constant $C=C(m)>0$ such that for every $\lambda \in \intvr{2}{\infty}$, for every measurable function $u : \mathbb{R}^m \to \mathbb{R}^\nu$ , if $V$ is an extension by convolution given by [\[eq_shee0Ceingooghe2weefei4A\]](#eq_shee0Ceingooghe2weefei4A){reference-type="eqref" reference="eq_shee0Ceingooghe2weefei4A"}, then $$\label{eq_oodohh0Eighiedath2oe4ong} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k,h}} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V (x)}^{m + 1} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau} % \le C \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{d \brk{u (y), u (z)}^{m + 1}}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z.$$* *Proof.* Similarly as in the proof of [Proposition 9](#proposition_goodcubes_gapintegral){reference-type="ref" reference="proposition_goodcubes_gapintegral"}, since by [\[eq_taerinie4aWee3Phahzefeib\]](#eq_taerinie4aWee3Phahzefeib){reference-type="eqref" reference="eq_taerinie4aWee3Phahzefeib"} $$x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1} \le \Cl{mo3948jut9j459gujt9ungi} \mathrm{MO}_{0, m + 1} u (x),$$ the proof of Proposition [Proposition 10](#proposition_goodcubes_sobolev){reference-type="ref" reference="proposition_goodcubes_sobolev"} then follows from [Proposition 8](#proposition_combined){reference-type="ref" reference="proposition_combined"}. Namely, $$\begin{split} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} & \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k,h}} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V (x)}^{m + 1} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau} \\ & \le \Cr{mo3948jut9j459gujt9ungi}\int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h}} \sup_{x \in \partial Q} \mathrm{MO}_{0, m+1} u (x)\,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau} \\ & \le \Cl{n4939ujf99fuhu9h} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{d \brk{u (y), u (z)}^{m+1}}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z, \end{split}$$ where the constant $\Cr{n4939ujf99fuhu9h}>0$ only depends on $m$. ◻ ## Sobolev and Sobolev--Marcinkiewicz extensions on cubes The first construction we will perform is a classical extension of the boundary data on cubes with small energy. To simplify the presentation, the result is stated on a ball, which is bi-Lipschitzly equivalent to a cube. **Lemma 11**. *Let $m \in \mathbb{N}\setminus \set{0}$. There exists a constant $C = C(m) \in \intvo{0}{\infty}$ such that for every $w \in W^{1, m + 1} (\mathbb{S}^m_\rho, \mathbb{R}^\nu)$ there exists a function $W \in W^{1, m + 1} (\mathbb{B}^{m + 1}_\rho, \mathbb{R}^\nu) \cap C (\mathbb{B}^{m + 1}_\rho, \mathbb{R}^\nu)$ such that $\mathop{\mathrm{tr}}_{\mathbb{S}^m_\rho} W = w$, $$\begin{gathered} \label{eq_aikahmi6foomei4teisu6ahY} \int_{\mathbb{B}^{m + 1}_\rho} \abs{\mathrm{D}W}^{m + 1} \le C \rho \int_{\mathbb{S}^{m}_\rho} \abs{\mathrm{D} w}^{m + 1}, \\ % \label{eq_xie9ohch4engee6Iekohf1Ie} \int_{\mathbb{B}^{m + 1}_\rho} \int_{\mathbb{S}^m_\rho} \abs{W (x) - w (y)}^{m + 1} \,\mathrm{d}x \,\mathrm{d}y \le C \rho^{2m + 2} \displaystyle \int_{\mathbb{S}^{m}_\rho} \abs{\mathrm{D}w}^{m + 1}\\ \intertext{and for every \(x \in \mathbb{B}^{m + 1}_\rho\) and almost every \(y \in \mathbb{S}^{m}_{\rho},\)} \label{eq_ooGhohviu9koohe7zoogiPha} \abs{W (x) - w (y)}^{m + 1} \le C\rho \int_{\mathbb{S}^m_{\rho}} \abs{\mathrm{D}w}^{m + 1}.\end{gathered}$$* *Proof.* Let $$\begin{aligned} p \coloneqq\frac{(m + 1)^2}{m}.\end{aligned}$$ We have, by the fractional Sobolev--Morrey embedding, $$\label{eq_onoxihajaite5ceequie1Sho} \smashoperator[r]{\iint_{\mathbb{S}^m_\rho \times \mathbb{S}^m_\rho}} \frac{\abs{w (y) - w (z)}^p}{\abs{y - z}^{m + p - 1}} \,\mathrm{d}y \,\mathrm{d}z \le \undefined\brk[\bigg]{\int_{\mathbb{S}^m_{\rho}} \abs{\mathrm{D}w}^{m + 1}}^{\frac{p}{m+ 1}}.$$ Assuming without loss of generality that $\int_{\mathbb{S}^m_{\rho}} w = 0$, Gagliardo's classical trace theory [@Gagliardo_1957] (see also ) yields a function $W \in W^{1, p} (\mathbb{B}^{m+1}_\rho, \mathbb{R}^\nu)$ such that $\mathop{\mathrm{tr}}_{\mathbb{S}^m_\rho} W = w$ and $$\label{eq_zeeLophiob6sheiyi3ieXi8b} \int_{\mathbb{B}^{m+1}_\rho} \abs{\mathrm{D}W}^{p} + \frac{\abs{W}^{p}}{\rho^p} \le \undefined\smashoperator{\iint_{\mathbb{S}^m_\rho \times \mathbb{S}^m_\rho}} \frac{\abs{w (y) - w (z)}^{p}}{\abs{y - z}^{m + p - 1}} \,\mathrm{d}y \,\mathrm{d}z.$$ It follows from Hölder's inequality, [\[eq_zeeLophiob6sheiyi3ieXi8b\]](#eq_zeeLophiob6sheiyi3ieXi8b){reference-type="eqref" reference="eq_zeeLophiob6sheiyi3ieXi8b"} and [\[eq_onoxihajaite5ceequie1Sho\]](#eq_onoxihajaite5ceequie1Sho){reference-type="eqref" reference="eq_onoxihajaite5ceequie1Sho"}, that $$\label{eq_Pheith2daichoh8ExuuF3fod} \int_{\mathbb{B}^{m+1}_\rho} \abs{\mathrm{D}W}^{m + 1} + \frac{\abs{W}^{m + 1}}{\rho^{m + 1}} \le \undefined\rho \int_{\mathbb{S}^m_{\rho}} \abs{\mathrm{D}w}^{m + 1},$$ and [\[eq_aikahmi6foomei4teisu6ahY\]](#eq_aikahmi6foomei4teisu6ahY){reference-type="eqref" reference="eq_aikahmi6foomei4teisu6ahY"} follows then from [\[eq_Pheith2daichoh8ExuuF3fod\]](#eq_Pheith2daichoh8ExuuF3fod){reference-type="eqref" reference="eq_Pheith2daichoh8ExuuF3fod"}. Since for every $x \in \mathbb{B}^{m+1}_\rho$ and $y \in \mathbb{S}^{m}_\rho$, $$\abs{W (x) - w (y)}^{m+1}\le 2^m \brk[\big]{\abs{W (x)}^{m+1}+\abs{w (y)}^{m+1}},$$ we have $$\label{elementarysobest} \int_{\mathbb{B}^{m + 1}_\rho} \int_{\mathbb{S}^m_\rho} \abs{W (x) - w (y)}^{m + 1} \,\mathrm{d}x \,\mathrm{d}y \le \undefined\rho^{m} \int_{\mathbb{B}^{m + 1}_\rho} \abs{W}^{m + 1} + \undefined\rho^{m + 1} \int_{\mathbb{S}^m_\rho} \abs{w}^{m + 1}.$$ The estimate [\[eq_xie9ohch4engee6Iekohf1Ie\]](#eq_xie9ohch4engee6Iekohf1Ie){reference-type="eqref" reference="eq_xie9ohch4engee6Iekohf1Ie"} follows from the estimates [\[elementarysobest\]](#elementarysobest){reference-type="eqref" reference="elementarysobest"}, [\[eq_Pheith2daichoh8ExuuF3fod\]](#eq_Pheith2daichoh8ExuuF3fod){reference-type="eqref" reference="eq_Pheith2daichoh8ExuuF3fod"} and the Poincaré inequality on the sphere. Finally, by the Morrey--Sobolev inequality, [\[eq_zeeLophiob6sheiyi3ieXi8b\]](#eq_zeeLophiob6sheiyi3ieXi8b){reference-type="eqref" reference="eq_zeeLophiob6sheiyi3ieXi8b"} and [\[eq_onoxihajaite5ceequie1Sho\]](#eq_onoxihajaite5ceequie1Sho){reference-type="eqref" reference="eq_onoxihajaite5ceequie1Sho"}, we have $$\begin{split} \abs{W (x) - w (y)} &\le \undefined\abs{x - y}^{1 - \frac{m + 1}{p}} \brk[\bigg]{\int_{\mathbb{B}^{m + 1}_\rho} \abs{\mathrm{D}W}^p}^\frac{1}{p}\\ & \le \undefined\rho^{1 - \frac{m}{m + 1}} \brk[\bigg]{\int_{\mathbb{S}^m_\rho} \abs{\mathrm{D}w}^{m + 1}}^\frac{1}{m + 1}, \end{split}$$ and [\[eq_ooGhohviu9koohe7zoogiPha\]](#eq_ooGhohviu9koohe7zoogiPha){reference-type="eqref" reference="eq_ooGhohviu9koohe7zoogiPha"} follows. ◻ When the oscillation is too large on the boundary of cubes, we will perform our construction of the controlled singular extension; the resulting map is quite wild but is sufficiently well controlled to provide an acceptable extension on those cubes. **Lemma 12**. *Let $m \in \mathbb{N}\setminus \set{0}$. If $w \in W^{1, m + 1} (\mathbb{S}^{m}_\rho, \mathbb{R}^\nu)$ and if we define $W : \mathbb{B}^{m + 1}_\rho \to \mathbb{R}^\nu$ for each $x \in \mathbb{B}^{m + 1}_\rho \setminus \set{0}$ by $$W (x) \coloneqq w (\tfrac{\rho}{\abs{x}} x),$$ then $W \in W^{1, 1} (\mathbb{B}^{m+1}_\rho, \mathbb{R}^\nu)$ and for every $t \in \intvo{0}{\infty}$, $$\label{eq_ao7iuceghaiphahVie8Quoor} \mathcal{L}^{m + 1} \brk[\big]{\set[\big]{x \in \mathbb{B}_\rho^{m + 1} \,\vert\,\abs{\mathrm{D}W (x)}> t }} \le \frac{\rho}{(m + 1) t^{m +1}} \int_{\mathbb{S}_\rho^{m}} \abs{\mathrm{D}w}^{m + 1}.$$* *Proof.* It can be observed immediately that $W \in W^{1, m + 1} (\mathbb{B}^{m+1}_\rho \setminus \mathbb{B}^{m+1}_\varepsilon, \mathbb{R}^\nu)$ for every $\varepsilon \in \intvo{0}{\rho}$ and that for every $x \in \mathbb{B}^{m + 1}_{\rho} \setminus \set{0}$, one has $$\abs{\mathrm{D}W (x)} = \tfrac{\rho}{\abs{x}} \abs{\mathrm{D}w (\tfrac{\rho}{\abs{x}} x)}.$$ Hence, using Fubini's theorem, we have $$\begin{split} \mathcal{L}^{m + 1} \brk[\big]{\set[\big]{x \in \mathbb{B}_\rho^{m + 1} \,\vert\,\abs{\mathrm{D}W (x)}> t }} &= \int_0^\rho \mathcal{H}^{m} \brk[\big]{\set[\big]{x \in \mathbb{S}^m_r \,\vert\,\abs{\mathrm{D}w (\tfrac{\rho}{r} x)} \ge \tfrac{t r}{\rho}}} \,\mathrm{d}r \\ &= \int_0^\rho \mathcal{H}^{m} \brk[\big]{\set[\big]{y \in \mathbb{S}^m_\rho \,\vert\,\abs{\mathrm{D}w (y)} \ge \tfrac{t r}{\rho}}} \brk[\big]{\tfrac{r}{\rho}}^{m}\,\mathrm{d}r \\ &= \frac{\rho}{t^{m +1}}\int_0^t \mathcal{H}^{m} \brk[\big]{\set[\big]{y \in \mathbb{S}^m_\rho \,\vert\,\abs{\mathrm{D}w (y)} \ge \tau}} \tau^m \,\mathrm{d}\tau\\ &\le \frac{\rho}{t^{m +1}}\int_0^\infty \mathcal{H}^{m} \brk[\big]{\set[\big]{y \in \mathbb{S}^m_\rho \,\vert\,\abs{\mathrm{D}w (y)} \ge \tau}} \tau^m \,\mathrm{d}\tau\\ &=\frac{\rho}{t^{m+1}}\int_{0}^{\infty} \mathcal{H}^{m} \brk[\big]{\set[\big]{y \in \mathbb{S}^m_\rho \,\vert\,\abs{\mathrm{D}w (y)}^{m+1} \ge \tau}} \frac{\,\mathrm{d}\tau}{m+1}\\ &= \frac{\rho}{(m + 1) t^{m +1}} \int_{\mathbb{S}_\rho^{m}} \abs{\mathrm{D}w(x)}^{m + 1} \,\mathrm{d}x. \end{split}$$ This yields [\[eq_ao7iuceghaiphahVie8Quoor\]](#eq_ao7iuceghaiphahVie8Quoor){reference-type="eqref" reference="eq_ao7iuceghaiphahVie8Quoor"} and completes our proof of Lemma [Lemma 12](#lemma_homogeneous){reference-type="ref" reference="lemma_homogeneous"}. ◻ ## Proofs of the theorems We first construct and estimate the singular extension on the half-space ([Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"}). *Proof of [Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"}.* We fix $\delta_{\mathcal{N}} \in \intvo{0}{\infty}$ so that the nearest-point retraction $\Pi_{\mathcal{N}} : \mathcal{N} + \mathbb{B}^\nu_{\delta_{\mathcal{N}}} \to \mathcal{N}$ is well defined and smooth up to the boundary. We take $V : \mathbb{R}^{m + 1}_+ \to \mathbb{R}^\nu$ to be an extension by convolution of $u$ to $\mathbb{R}^{m + 1}_{+}$ as in [\[eq_shee0Ceingooghe2weefei4A\]](#eq_shee0Ceingooghe2weefei4A){reference-type="eqref" reference="eq_shee0Ceingooghe2weefei4A"}. Since by assumption $u \in \mathcal{N}$ almost everywhere on $\mathbb{R}^m$, by the averaged estimate on the distance to the target (see [Proposition 9](#proposition_goodcubes_gapintegral){reference-type="ref" reference="proposition_goodcubes_gapintegral"}), we have $$\begin{gathered} \label{eq_kfmrefijiu4ugntvri} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \# \set[\Big]{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h} \,\vert\,\sup_{x \in \partial Q} \mathop{\mathrm{dist}}(V (x), \mathcal{N}) \ge \delta_{\mathcal{N}}/2} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % \le \Cl{cst_yiek0OoRe0hielie4pa5shee} \smashoperator{\iint_{\substack{(y, z) \in \mathbb{R}^{m} \times \mathbb{R}^{m}\\ d (u(y), u(z)) \ge \eta\delta_{\mathcal{N}}/2}}} \frac{1}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z, \end{gathered}$$ where $\eta>0$ is the constant of Proposition [Proposition 9](#proposition_goodcubes_gapintegral){reference-type="ref" reference="proposition_goodcubes_gapintegral"} depending only on $m$, the constant $\Cr{cst_yiek0OoRe0hielie4pa5shee} \in \intvo{0}{\infty}$ depends only on $m$, $\mathcal{N}$, and we have also used that $\mathcal{N}$ is compact, namely $\mathop{\mathrm{diam}}(\mathcal{N})<\infty$. We set $\delta:=\eta \delta_{\mathcal{N}}/2$. Taking $$\label{eq_quoh4pooKo4wohtequ6heoju} \lambda \coloneqq\exp \brk[\Bigg]{2\Cr{cst_yiek0OoRe0hielie4pa5shee} \smashoperator{\iint_{\substack{(y, z) \in \mathbb{R}^{m} \times \mathbb{R}^{m}\\ d (u(y), u(z)) \ge \delta}}} \frac{1}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z}$$ and using [\[eq_kfmrefijiu4ugntvri\]](#eq_kfmrefijiu4ugntvri){reference-type="eqref" reference="eq_kfmrefijiu4ugntvri"}, we have then $$\label{eq_pazaezuc4Aisielei9oowee6} \frac{1}{\ln \lambda} \int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \# \set[\Big]{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h} \,\vert\,\sup_{x \in \partial Q} \mathop{\mathrm{dist}}(V (x), \mathcal{N}) \ge \delta_{\mathcal{N}}/2} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau} % \le \frac{1}{2}.$$ Similarly, by [Proposition 10](#proposition_goodcubes_sobolev){reference-type="ref" reference="proposition_goodcubes_sobolev"}, $$\begin{gathered} \label{eq_Sei5aiph6ahva9Zie2ooyika} \frac{1}{\ln \lambda}\int_1^\lambda \sum_{k \in \mathbb{Z}} \int_{\intvc{0}{1}^{m}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k,h}} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V (x)}^{m + 1} \,\mathrm{d}h \frac{\,\mathrm{d}\tau}{\tau}\\ % \le \frac{\undefined}{\ln \lambda} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{d \brk{u (y), u (z)}^{m + 1}}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z.\end{gathered}$$ In view of the estimates [\[eq_pazaezuc4Aisielei9oowee6\]](#eq_pazaezuc4Aisielei9oowee6){reference-type="eqref" reference="eq_pazaezuc4Aisielei9oowee6"} and [\[eq_Sei5aiph6ahva9Zie2ooyika\]](#eq_Sei5aiph6ahva9Zie2ooyika){reference-type="eqref" reference="eq_Sei5aiph6ahva9Zie2ooyika"}, there exists $\tau \in \intvo{1}{\lambda}$ and, for every $k \in \mathbb{Z}$, $h_k \in \intvc{0}{1}^m$ such that for every $k \in \mathbb{Z}$, every $Q \in \mathcal{Q}^+_{\lambda, \tau, k, h_k}$ and every $x \in \partial Q$, one has $$\label{eq_sheeceeGh5eeyohLeineiwuC} \mathop{\mathrm{dist}}(V (x), \mathcal{N}) \le \frac{\delta_{\mathcal{N}}}{2}$$ and $$\label{eq_eghoh1oe0nuchiev6lahPaey} \sum_{k \in \mathbb{Z}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h_k}} \sup_{x \in \partial Q} x_{m + 1}^{m + 1} \abs{\mathrm{D}V (x)}^{m + 1} \le \frac{\Cl{esrnvo3i4j93juvirhn}}{\ln \lambda} \smashoperator[r]{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{d \brk{u (y), u (z)}^{m + 1}}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z.$$ We define now the set of good cubes $$\label{eq45iimio_good_cubes} \mathcal{G} \coloneqq \set[\Big]{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h_k} \,\vert\, \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1} \le \mu \text{ and } k \in \mathbb{Z}}$$ and the set of bad cubes $$\label{eqmi5jj5njnhj_notgood_cubes} \mathcal{B} \coloneqq \set[\Big]{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h_k} \,\vert\,\sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1} > \mu \text{ and } k \in \mathbb{Z}}.$$ Clearly, any cube is either good or bad and thus $$\bigcup_{Q \in \mathcal{G} \cup \mathcal{B}} Q = \mathbb{R}^{m + 1}_+.$$ By [\[eq_eghoh1oe0nuchiev6lahPaey\]](#eq_eghoh1oe0nuchiev6lahPaey){reference-type="eqref" reference="eq_eghoh1oe0nuchiev6lahPaey"} and [\[eqmi5jj5njnhj_notgood_cubes\]](#eqmi5jj5njnhj_notgood_cubes){reference-type="eqref" reference="eqmi5jj5njnhj_notgood_cubes"}, $$\label{eq_irj398493j4gu} \begin{split} \# \mathcal{B} &\le \frac{1}{\mu} \sum_{k \in \mathbb{Z}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h_k}} \sup_{x \in \partial Q} x_{m + 1}^{m + 1} \abs{\mathrm{D}V (x)}^{m + 1}\\ &\le \frac{\Cr{esrnvo3i4j93juvirhn}}{\mu \ln \lambda} \smashoperator{\iint_{\mathbb{R}^{m} \times \mathbb{R}^{m}}} \frac{d \brk{u (y), u (z)}^{m + 1}}{\abs{y - z}^{2 m}} \,\mathrm{d}y \,\mathrm{d}z. \end{split}$$ Notice also that for every $k \in \mathbb{Z}$ and every $Q \in \mathcal{Q}^+_{\lambda, \tau, k, h_k}$, we have $$\label{eq_yoh6Ahl3aiba6jie3xoj3ael} \begin{split} \tau \lambda^{-k} \int_{\partial Q} \abs{\mathrm{D}V}^{m + 1} &\le \Cl{cst_ieceishu2Ahleej9aizoh7EN} (\tau \lambda^{-k})^{m + 1} \sup_{x \in \partial Q } \abs{\mathrm{D}V(x)}^{m + 1}\\ &\le \Cr{cst_ieceishu2Ahleej9aizoh7EN} (\lambda - 1)^{m + 1} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1}, \end{split}$$ since for every $x \in Q$, $x_{m +1} \ge \tau \lambda^{-k}/(\lambda - 1)$, in view of [\[dcubesf\]](#dcubesf){reference-type="eqref" reference="dcubesf"}. We are now going to define a map $W : \mathbb{R}^{m+1}_+ \to \mathcal{N} + \mathbb{B}^{\nu}_{\delta_{\mathcal{N}}}$ separately on $\bigcup_{\mathcal{G}}$ and $\bigcup_{\mathcal{B}}$. For every $Q \in \mathcal{G}$, we apply [Lemma 11](#lemma_extension){reference-type="ref" reference="lemma_extension"}, up to a suitable bi-Lipschitz homeomorphism between a ball and a cube, to define the mapping $W$ on $Q$ as an extension of $V \!\upharpoonright_{ \partial Q}$. We have, in view of [\[eq_aikahmi6foomei4teisu6ahY\]](#eq_aikahmi6foomei4teisu6ahY){reference-type="eqref" reference="eq_aikahmi6foomei4teisu6ahY"} and of [\[eq_yoh6Ahl3aiba6jie3xoj3ael\]](#eq_yoh6Ahl3aiba6jie3xoj3ael){reference-type="eqref" reference="eq_yoh6Ahl3aiba6jie3xoj3ael"}, $$\label{eq_waeLee9quoh8aphaew8Iej1b} \begin{split} \int_{Q} \abs{\mathrm{D}W}^{m + 1} &\le \undefined\tau \lambda^{-k} \int_{\partial Q} \abs{\mathrm{D}V}^{m + 1}\\ &\le \undefined(\lambda - 1)^{m + 1} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1}, \end{split}$$ whereas by the triangle inequality, [\[eq_sheeceeGh5eeyohLeineiwuC\]](#eq_sheeceeGh5eeyohLeineiwuC){reference-type="eqref" reference="eq_sheeceeGh5eeyohLeineiwuC"}, [\[eq_ooGhohviu9koohe7zoogiPha\]](#eq_ooGhohviu9koohe7zoogiPha){reference-type="eqref" reference="eq_ooGhohviu9koohe7zoogiPha"}, [\[eq_yoh6Ahl3aiba6jie3xoj3ael\]](#eq_yoh6Ahl3aiba6jie3xoj3ael){reference-type="eqref" reference="eq_yoh6Ahl3aiba6jie3xoj3ael"} and [\[eq45iimio_good_cubes\]](#eq45iimio_good_cubes){reference-type="eqref" reference="eq45iimio_good_cubes"}, $$\mathop{\mathrm{dist}}(W (x), \mathcal{N}) \le \frac{\delta_\mathcal{N}}{2} + \Cl{cst_thowuofoaZiizi2Ahw6hoowe} \mu^{\frac{1}{m+1}} (\lambda - 1).$$ Hence if $$\label{eq_ugii0ce5oa6haiGhieb8oohe} \mu = \brk[\Bigg]{\frac{\delta_{\mathcal{N}}}{2 \Cr{cst_thowuofoaZiizi2Ahw6hoowe} (\lambda - 1)}}^{m + 1},$$ we have for every $x \in Q$, $W (x) \in \mathcal{N} + \mathbb{B}^{\nu}_{\delta_{\mathcal{N}}}$. Using the Chebyshev inequality and [\[eq_waeLee9quoh8aphaew8Iej1b\]](#eq_waeLee9quoh8aphaew8Iej1b){reference-type="eqref" reference="eq_waeLee9quoh8aphaew8Iej1b"}, we obtain $$\label{macheormtbnoi} t^{m+1}\mathcal{L}^{m+1}\brk{\set{x \in Q \,\vert\,\abs{\mathrm{D}W (x)} \ge t}} \le \undefined(\lambda - 1)^{m + 1} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1}.$$ Next, we apply [Lemma 12](#lemma_homogeneous){reference-type="ref" reference="lemma_homogeneous"} on every bad cube $Q \in \mathcal{B}$ to define there $W$ by homogeneous extension of $V \!\upharpoonright_{\partial Q}$. We compute then for such a cube, in view of [\[eq_ao7iuceghaiphahVie8Quoor\]](#eq_ao7iuceghaiphahVie8Quoor){reference-type="eqref" reference="eq_ao7iuceghaiphahVie8Quoor"} and [\[eq_yoh6Ahl3aiba6jie3xoj3ael\]](#eq_yoh6Ahl3aiba6jie3xoj3ael){reference-type="eqref" reference="eq_yoh6Ahl3aiba6jie3xoj3ael"}, for every $t \in \intvo{0}{\infty}$, $$\label{eq_cheijo5gee6ahphieQui1sho} \begin{split} \mathcal{L}^{m + 1} \brk{\set{x \in Q \,\vert\,\abs{\mathrm{D}W (x)} \ge t}} &\le \frac{\undefined}{ t^{m + 1}} \tau \lambda^{-k} \int_{\partial Q} \abs{\mathrm{D}V}^{m + 1}\\ &\le \frac{\undefined}{ t^{m + 1}} (\lambda - 1)^{m + 1} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1}. \end{split}$$ Combining the estimates [\[macheormtbnoi\]](#macheormtbnoi){reference-type="eqref" reference="macheormtbnoi"} and [\[eq_cheijo5gee6ahphieQui1sho\]](#eq_cheijo5gee6ahphieQui1sho){reference-type="eqref" reference="eq_cheijo5gee6ahphieQui1sho"}, we get $$\label{eq_ohG1eoMooghaiwoi7bei7iep} \begin{split} t^{m + 1}\mathcal{L}^{m + 1} &\brk{\set{x \in \mathbb{R}^{m+1}_{+} \,\vert\,\abs{\mathrm{D}W (x)} \ge t}}\\ &\le \undefined(\lambda - 1)^{m + 1} \sum_{k \in \mathbb{Z}} \sum_{Q \in \mathcal{Q}^+_{\lambda, \tau, k, h_k}} \sup_{x \in \partial Q} x_{m +1}^{m + 1} \abs{\mathrm{D}V(x)}^{m + 1}\\ &\le \Cl{3k4ri934jf9j5guj4ugh} \exp \brk[\Bigg]{\hspace{1em}\Cl{j3fon49ijf9jgfu95guinh} \smashoperator{ \iint_{\substack{ (y, z) \in \mathbb{R}^m \times \mathbb{R}^m\\ d (u (y), u (z))\ge \delta }}} \frac{1}{\abs{y - z}^{2m}} \,\mathrm{d}y \,\mathrm{d}z } \smashoperator[r]{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{d(u (y), u(z))^{m + 1}}{\abs{y - z}^{2m}} \,\mathrm{d}y \,\mathrm{d}z, \end{split}$$ in view of [\[eq_quoh4pooKo4wohtequ6heoju\]](#eq_quoh4pooKo4wohtequ6heoju){reference-type="eqref" reference="eq_quoh4pooKo4wohtequ6heoju"} and [\[eq_eghoh1oe0nuchiev6lahPaey\]](#eq_eghoh1oe0nuchiev6lahPaey){reference-type="eqref" reference="eq_eghoh1oe0nuchiev6lahPaey"}, where $\Cr{3k4ri934jf9j5guj4ugh}$ and $\Cr{j3fon49ijf9jgfu95guinh}$ are positive constants depending only on $m$, $\mathcal{N}$. Moreover, if $S$ denotes the set of the barycenters of the cubes $Q \in \mathcal{B}$, we have $$\label{eq_Aepaikoh6Lei6Hoh8pabuloe} \# S = \# \mathcal{B} \le \Cl{m94j5g9hutgy4uihg} \exp \brk[\Bigg]{\hspace{1em} \Cl{n239hru4hfy84h5gyi5u4gnuin} \smashoperator{ \iint_{\substack{ (y, z) \in \mathbb{R}^m \times \mathbb{R}^m\\ d (u (y), u (z))\ge \delta }}} \frac{1}{\abs{y - z}^{2m}} \,\mathrm{d}y \,\mathrm{d}z } \smashoperator[r]{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{d(u (y), u(z))^{m + 1}}{\abs{y - z}^{2m}} \,\mathrm{d}y \,\mathrm{d}z,$$ in view of [\[eq_quoh4pooKo4wohtequ6heoju\]](#eq_quoh4pooKo4wohtequ6heoju){reference-type="eqref" reference="eq_quoh4pooKo4wohtequ6heoju"}, [\[eq_irj398493j4gu\]](#eq_irj398493j4gu){reference-type="eqref" reference="eq_irj398493j4gu"} and [\[eq_ugii0ce5oa6haiGhieb8oohe\]](#eq_ugii0ce5oa6haiGhieb8oohe){reference-type="eqref" reference="eq_ugii0ce5oa6haiGhieb8oohe"}, where $\Cr{m94j5g9hutgy4uihg}$ and $\Cr{n239hru4hfy84h5gyi5u4gnuin}$ are positive constants depending only on $m$, $\mathcal{N}$. In order to check that $u$ is the trace of $W$, we observe that on the one hand we have, since $V$ is an extension by averaging, for every $Q \in \mathcal{G}$, by Poincaré's inequality, by [\[eq_xie9ohch4engee6Iekohf1Ie\]](#eq_xie9ohch4engee6Iekohf1Ie){reference-type="eqref" reference="eq_xie9ohch4engee6Iekohf1Ie"} and by [\[eq_yoh6Ahl3aiba6jie3xoj3ael\]](#eq_yoh6Ahl3aiba6jie3xoj3ael){reference-type="eqref" reference="eq_yoh6Ahl3aiba6jie3xoj3ael"}, $$\label{eq_ac6thaighahxahGh1mahb4ai} \begin{split} \int_{Q}& \abs{V - W}^{m + 1}\\ &\le \frac{\Cl{ni4rj3j4uhtu9h435n}}{\brk{\tau \lambda^{-k}}^m} \brk*{\int_{\partial Q} \int_{Q} \abs{V(x) - V (y)}^{m + 1 } \,\mathrm{d}y \,\mathrm{d}x + \int_{\partial Q} \int_{Q} \abs{W (x) - V (y)}^{m + 1 } \,\mathrm{d}y \,\mathrm{d}x }\\ &\le \frac{\Cr{ni4rj3j4uhtu9h435n}}{(\tau \lambda^{-k})^m} \int_{Q} \int_{\partial Q} \abs{V(x) - V (y)}^{m + 1 } \,\mathrm{d}x \,\mathrm{d}y + \undefined(\tau \lambda^{-k})^{m + 2} \int_{\partial Q} \abs{\mathrm{D}V}^{m + 1 }\\ &\le \undefined(\tau \lambda^{-k})^{m + 1} \brk*{\int_{Q} \abs{\mathrm{D}V}^{m + 1} + (\lambda - 1)^{m + 1} \sup_{x \in \partial Q} x_{m + 1}^{m + 1} \abs{\mathrm{D}V (x)}^{m + 1}}. \end{split}$$ By the classical theory of traces, $$\label{eq_lahXef3uyieheeng3ohwahH0} \int_{\mathbb{R}^{m + 1}} \abs{\mathrm{D}V}^{m + 1} <\infty.$$ It follows then from [\[eq_eghoh1oe0nuchiev6lahPaey\]](#eq_eghoh1oe0nuchiev6lahPaey){reference-type="eqref" reference="eq_eghoh1oe0nuchiev6lahPaey"}, [\[eq_ac6thaighahxahGh1mahb4ai\]](#eq_ac6thaighahxahGh1mahb4ai){reference-type="eqref" reference="eq_ac6thaighahxahGh1mahb4ai"} and [\[eq_lahXef3uyieheeng3ohwahH0\]](#eq_lahXef3uyieheeng3ohwahH0){reference-type="eqref" reference="eq_lahXef3uyieheeng3ohwahH0"} that $$\label{eq_knreinebir9043rj4iuhu} \sum_{k \in \mathbb{Z}} \sum_{Q \in \mathcal{G}} \frac{1}{(\lambda \tau^{-k})^{m + 1}} \int_{Q} \abs{V - W }^{m + 1} < \infty.$$ Thus, since the set $\mathcal{B}$ is finite, in view of [\[eq_knreinebir9043rj4iuhu\]](#eq_knreinebir9043rj4iuhu){reference-type="eqref" reference="eq_knreinebir9043rj4iuhu"}, we have $$\lim_{k \to \infty} \frac{1}{(\lambda \tau^{-k})^{m + 1}} \sum_{Q \in \mathcal{Q}^{+}_{\lambda, \tau, k, h_k}} \int_{Q} \abs{V - W }^{m + 1}\,\mathrm{d}x = 0.$$ This implies that $W$ and $V$ have the same trace, and hence $u$ is the trace of $W$. Finally, we define $U \coloneqq\Pi_{\mathcal{N}} \,\circ\,W$. The map $U$ also has $u$ as the trace on $\mathbb{R}^m \times \set{0}$; the conclusion [\[eq_aiD8Oob2ahdah8lae0einei6\]](#eq_aiD8Oob2ahdah8lae0einei6){reference-type="eqref" reference="eq_aiD8Oob2ahdah8lae0einei6"} follows from [\[eq_ohG1eoMooghaiwoi7bei7iep\]](#eq_ohG1eoMooghaiwoi7bei7iep){reference-type="eqref" reference="eq_ohG1eoMooghaiwoi7bei7iep"}; $U\in C (\mathbb{R}^{m + 1}_{+} \setminus S, \mathcal{N})$ with the cardinality of the singular set $S$ being estimated by [\[eq_Aepaikoh6Lei6Hoh8pabuloe\]](#eq_Aepaikoh6Lei6Hoh8pabuloe){reference-type="eqref" reference="eq_Aepaikoh6Lei6Hoh8pabuloe"}. This completes our proof of Theorem [Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"}. ◻ The proof of [Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"} can be adapted easily to the case of the hyperbolic space ([Theorem 3](#theorem_hyperbolic){reference-type="ref" reference="theorem_hyperbolic"}). *Proof of [Theorem 3](#theorem_hyperbolic){reference-type="ref" reference="theorem_hyperbolic"}.* We proceed as in the proof of [Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"}. Instead of [\[eq_cheijo5gee6ahphieQui1sho\]](#eq_cheijo5gee6ahphieQui1sho){reference-type="eqref" reference="eq_cheijo5gee6ahphieQui1sho"}, we have using the Poincaré metric on the half-space, if $Q \in \mathcal{Q}^{+}_{\lambda, \tau, k, h}$ (see [\[dcubesf\]](#dcubesf){reference-type="eqref" reference="dcubesf"}), then $$\label{eq_UunifeeleaKaeg8ui6eezuej} \begin{split} \smashoperator[r]{\int_{\substack{x \in Q\\ \abs{\mathrm{D}V (x)}x_{m +1} \ge t}}} \frac{1}{x_{m +1}^{m + 1}} \,\mathrm{d}x &\le \brk[\Big]{\frac{(\lambda - 1)\lambda^k}{\tau}}^{m + 1} \mathcal{L}^{m + 1}\brk{\set{x \in Q \,\vert\,\abs{\mathrm{D}V (x)}\tfrac{\tau \lambda^{-(k - 1)}}{\lambda - 1} \ge t}}\\[-1em] &\le \frac{ \lambda^{m + 1}}{t^{m + 1}} (\tau \lambda^{-k})^{m + 1} \sup_{x \in Q} \abs{\mathrm{D}V (x)}^{m + 1}\\ &\le \frac{ \lambda^{m + 1}(\lambda - 1)^{m + 1}}{t^{m + 1}} \sup_{x \in Q} x_{m +1}^{m + 1}\abs{\mathrm{D}V (x)}^{m + 1}. \end{split}$$ The remainder of the proof is similar. ◻ The case of the singular extension to the ball is slightly more complicated, as we will rely on the parameterization of $\mathbb{B}^{m + 1}_{1}$ by $\mathbb{R}^{m + 1}_+$ through a classical suitable conformal mapping. *Proof of [Theorem 1](#theorem_ball){reference-type="ref" reference="theorem_ball"}.* We recall that the map $$\Psi (x) \coloneqq\frac{4(x + e)}{\abs{x + e}^2} - 2e,$$ where $e\coloneqq(0, \dotsc, 0, 1) \in \mathbb{R}^{m+1}_{+}$, defines a diffeomorphism from $\mathbb{B}^{m + 1}_{1}$ to $\mathbb{R}_+^{m + 1}$. Indeed, if $x \in \bar{\mathbb{B}}^{m+1}_{1} \setminus \{e\}$, then $$e \cdot \Psi (x) = \frac{2 - 2\abs{x}^2}{\abs{x + e}^2}.$$ Moreover, we have $$\Psi^{-1} (x) = \frac{4(x + 2e)}{\abs{x + 2e}^2} - e.$$ In particular, the ball $\mathbb{B}^{m + 1}_{1}$ is isometric to the half-space $\mathbb{R}^{m + 1}_+$ endowed with the metric $g$ defined for $x \in \mathbb{R}^{m + 1}_+$ and $v \in \mathbb{R}^{m + 1}$ by $$g (x)[v, v] = \frac{16 \abs{v}^2}{\abs{x + 2e}^4}.$$ Moreover, since $\Psi$ is a conformal map and $u : \mathbb{R}^m \to \mathcal{N}$, one has $$\smashoperator[r]{\iint_{\mathbb{S}^m \times \mathbb{S}^m}} \frac{d (u(x), u(y))^p}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y = \smashoperator{\iint_{\mathbb{R}^m \times \mathbb{R}^m}} \frac{d (u(\Psi^{-1} (x)), u(\Psi^{-1}(y)))^p}{\abs{\Psi^{-1} (x) - \Psi^{-1} (y)}^{2m}} \,\mathrm{d}x \,\mathrm{d}y$$ and $$\smashoperator{\iint_{\substack{x, y \in \mathbb{S}^m\\ d(u (x), u (y))\ge \delta}}} \frac{d (u(x), u(y))^p}{\abs{x - y}^{2m}} \,\mathrm{d}x \,\mathrm{d}y = \smashoperator{\iint_{\substack{x, y \in \mathbb{R}^m\\ d(u (\Psi^{-1} (x)), u (\Psi^{-1}(y)))\ge \delta}}} \frac{d (u(\Psi^{-1} (x)), u(\Psi^{-1}(y)))^p}{\abs{\Psi^{-1} (x) - \Psi^{-1} (y)}^{2m}} \,\mathrm{d}x \,\mathrm{d}y .$$ We proceed then as in the proof of [Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"}, using $u \,\circ\,\Psi^{-1}$ instead of $u$. In order to replace the estimate [\[eq_cheijo5gee6ahphieQui1sho\]](#eq_cheijo5gee6ahphieQui1sho){reference-type="eqref" reference="eq_cheijo5gee6ahphieQui1sho"}, we proceed as follows. Given $Q \in \mathcal{Q}_{\lambda, \tau, k, h}^+$ and setting $$\begin{aligned} m_Q &= \inf_{x \in Q} \frac{4}{\abs{x + 2e}^2}& &\text{and} & M_Q &= \sup_{x \in Q} \frac{4}{\abs{x + 2e}^2},\end{aligned}$$ we have $$\label{eq_maechuquoo1aiQuai6Eig4Ue} \begin{split} \smashoperator[r]{\int_{\substack{x \in Q\\ \abs{\mathrm{D}V (x)} \ge 4t/\abs{x + 2e}^2}}} \frac{4^{m + 1}}{\abs{x + 2e}^{2m + 2}} \,\mathrm{d}x &\le M_Q^{m + 1} \mathcal{L}^{m + 1}\brk{\set{x \in Q \,\vert\,\abs{\mathrm{D}V (x)} \ge t m_Q}}\\ &\le \frac{1}{t^{m + 1}} \brk[\Big]{\frac{M_Q}{m_Q}}^{m + 1} (\tau \lambda^{-k})^{m + 1} \sup_{x \in Q} \abs{\mathrm{D}V (x)}^{m + 1}. \end{split}$$ We observe now that if $x, y \in Q$, then $$\label{eq_ugh7ne9aeLieri8iepaiVosh} \frac{\abs{x + 2e}}{\abs{y + 2e}} \le 1 + \frac{\abs{x - y}}{\abs{x + 2e}} \le 1 + \frac{\tau \lambda^{-k}\sqrt{m+1}}{2+\frac{\tau \lambda^{-k}}{\lambda-1}} \le 1 + (\lambda - 1)\sqrt{m+1},$$ and we have thus by [\[eq_maechuquoo1aiQuai6Eig4Ue\]](#eq_maechuquoo1aiQuai6Eig4Ue){reference-type="eqref" reference="eq_maechuquoo1aiQuai6Eig4Ue"} and [\[eq_ugh7ne9aeLieri8iepaiVosh\]](#eq_ugh7ne9aeLieri8iepaiVosh){reference-type="eqref" reference="eq_ugh7ne9aeLieri8iepaiVosh"}, $$\label{eq_Cie0Eikeid7IeG7Quee4Sam1} \smashoperator[r]{\int_{\substack{x \in Q\\ \abs{\mathrm{D}V (x)} \ge 4t/\abs{x + 2e}^2}}} \frac{4^{m + 1}}{\abs{x + 2e}^{2m + 2}} \,\mathrm{d}x \le \frac{\brk{\lambda - 1}^{m + 1}}{t^{m + 1}} \brk*{1 + (\lambda - 1)\sqrt{m+1}}^{m + 1} \sup_{x \in Q} x_{m +1}^{m + 1}\abs{\mathrm{D}V (x)}^{m + 1}.$$ The rest of the proof is similar to the proof of [Theorem 2](#theorem_halfspace){reference-type="ref" reference="theorem_halfspace"}. ◻ [^1]: The authors were supported by the Projet de Recherche T.0229.21 "Singular Harmonic Maps and Asymptotics of Ginzburg--Landau Relaxations" of the Fonds de la Recherche Scientifique--FNRS

arxiv_math

--- abstract: | We prove that the derived direct image of the constant sheaf with field coefficients under any proper map with smooth source contains a canonical summand. This summand, which we call the geometric extension, only depends on the generic fibre. For resolutions we get a canonical extension of the constant sheaf. When our coefficients are of characteristic zero, this summand is the intersection cohomology sheaf. When our coefficients are finite we obtain a new object, which provides interesting topological invariants of singularities and topological obstructions to the existence of morphisms. The geometric extension is a generalization of a parity sheaf. Our proof is formal, and also works with coefficients in modules over suitably finite ring spectra. author: - Chris Hone, Geordie Williamson bibliography: - Biblio.bib date: June 2023 title: Geometric Extensions --- # Introduction This paper introduces *geometric extensions*, which are generalizations of intersection cohomology sheaves and parity sheaves. We work in the setting of constructible sheaves on algebraic varieties, and show that direct image sheaves along any resolution contain a canonical direct summand which is independent of the resolution.[^1] When our coefficients are $\mathbb{Q}$, this summand is the intersection cohomology sheaf. When our coefficients are finite, we obtain a new object. Our proof is formal, and works more generally for proper maps with smooth source, and with coefficients in any suitably finite ring spectrum. The stalks of the geometric extension (with coefficients in finite fields and other ring spectra) provide subtle topological invariants of the singularities of algebraic varieties. In order to motivate geometric extensions, we first recall the traditional route to intersection cohomology extensions through perverse sheaves. We then turn to an alternative approach via the Decomposition Theorem, which will motivate the consideration of geometric extensions. We then state our main result, and finally give some motivation from modular representation theory, where geometric extensions generalise the notion of a parity sheaf. ## Motivation from the Decomposition Theorem {#sec:motivation} Let $Y$ be a complex algebraic variety, equipped with its classical (metric) topology. Inside the constructible derived category of sheaves of $\mathbb{Q}$-vector spaces on $Y$ there is a remarkable abelian category of *perverse sheaves*, which is preserved by Verdier duality. The abelian category of perverse sheaves is finite length and its simple objects are the intersection cohomology extensions of simple local systems on irreducible, smooth, locally closed subvarieties. Their global sections compute intersection cohomology. The central importance of intersection cohomology extensions becomes manifest in the Decomposition Theorem. The Decomposition Theorem states that for smooth $X$ and any proper morphism $$f : X \to Y$$ of complex algebraic varieties the derived direct image $f_* \mathbb{Q}_X$ is *semi-simple*: isomorphic in the derived category to a direct sum of shifts of intersection cohomology extensions of simple local systems on strata. The Decomposition Theorem implies that the intersection cohomology of $Y$ is a direct summand in the cohomology of any resolution. It is also implies (and generalizes) fundamental ideas in the topology of complex algebraic varieties like the local and global invariant cycle theorems, semi-simplicity of monodromy and Hodge theory [@BBD; @dCM; @Sa; @W]. The Decomposition Theorem also provides another route to intersection cohomology complexes. The constructible derived category is Krull-Schmidt: every object admits a decomposition into indecomposable summands, and this decomposition is unique. The Decomposition Theorem implies that if one considers all proper maps to $Y$ with smooth source $$\begin{tikzpicture}[scale=.7] \node (x1) at (-2,2) {$X_1$}; \node (x2) at (0,3) {$X_2$}; \node (x3) at (2,2) {$X_3$}; \node (y) at (0,0) {$Y$}; \draw[->] (x1) to node[above] {\small $f_1$} (y); \draw[->] (x2) to node[right] {\small $f_2$} (y); \draw[->] (x3) to node[below,right] {\small $f_3$} (y); \end{tikzpicture}$$ then the summands of the derived direct images $(f_i)_{*} \mathbb{Q}_{X_i}$ are of a special form: they are shifts of intersection cohomology complexes. This observation allows one to imagine an alternate version of history, where intersection cohomology complexes were discovered via the Krull-Schmidt theorem rather than through the theory of perverse sheaves[^2]. It also naturally raises the following question: *Question 1*. Let $\Lambda$ denote a ring, and let $\Lambda_X$ denote the constant sheaf on $X$ with coefficients in $\Lambda$. What can one say about the summands of the derived direct image $f_* \Lambda_X$, for any resolution $f : X \to Y$? More generally, what can one say about the summands of $f_* \Lambda_X$ for any proper morphism with smooth source $f : X \to Y$? This question should be considered the central motivation of this paper. By the proper base change theorem, the stalks of $f_* \Lambda_X$ record the $\Lambda$-cohomology of the fibres of $f$. If Question [Question 1](#q:summands){reference-type="ref" reference="q:summands"} has an answer giving a small list of possible summands (as is the case with $\Lambda = \mathbb{Q}$, as implied by the Decomposition Theorem) then there are basic building blocks of the cohomology of morphisms, which only depend on $Y$ and not on the particular morphism. For example, the "support theorems" (see e.g. [@BorhoMacPherson; @ngo; @MV]) show that the fibres of certain "minimal" maps (e.g. the Grothendieck-Springer resolution or Hitchin fibration) are determined to a large extent by the base $Y$, and the generic behaviour of the map. ## Main results Let $\Lambda$ denote a field or complete local ring. As above, $\Lambda_X$ denotes the constant sheaf on $X$ with coefficients in $\Lambda$. We are able to give a partial answer to Question [Question 1](#q:summands){reference-type="ref" reference="q:summands"}. Any resolution contains a canonical direct summand: **Theorem 2**. *(see Theorem [Theorem 47](#thm:main1){reference-type="ref" reference="thm:main1"}) [\[thm:main1-intro\]]{#thm:main1-intro label="thm:main1-intro"} Let $Y$ be an irreducible variety. There exists a complex $\mathscr{E}(Y,\Lambda) \in D^b_c(Y,\Lambda)$ characterised up to isomorphism by the following:* 1. *$\mathscr{E}(Y,\Lambda)$ is indecomposable and its support is dense;* 2. *$\mathscr{E}(Y,\Lambda)$ is a summand inside $f_* \Lambda_X$, for any resolution $f: X \to Y$.* We call $\mathscr{E}(Y,\Lambda)$ the **geometric extension** on $Y$. *Remark 3*. When $\Lambda = \mathbb{Q}$ then $\mathscr{E}(Y,\Lambda) = IC(Y,\mathbb{Q})$, by the Decomposition Theorem (see Proposition [Proposition 57](#ICprop){reference-type="ref" reference="ICprop"}). The stalks of the geometric extension record behaviour which "has to be there in any resolution". Indeed, the proper base change theorem and Theorem [Theorem 47](#thm:main1){reference-type="ref" reference="thm:main1"} immediately imply: **Corollary 4**. *Suppose $\Lambda$ is a field. For any resolution $f : X \to Y$ one has $$\dim H^i(\mathscr{E}(Y,\Lambda)_y) \le \dim H^i(f^{-1}(y), \Lambda)$$ for any $y \in Y$.* *Remark 5*. This corollary can be used to rule out the existence of resolutions of a particular form. For example, if $\mathscr{E}(Y,\Lambda)_y$ has non-zero stalks in degrees $0$ and $2m$ for some $m$, then Corollary [Corollary 4](#cor:forced){reference-type="ref" reference="cor:forced"} and the existence of fundamental classes implies that any resolution of $Y$ has to have fibres dimension at least $m$ over $y$ (see Example [Example 63](#ex:A/pm){reference-type="ref" reference="ex:A/pm"}). This can be used to prove the non-existence of semi-small resolutions: if $\mathscr{E}(Y,\Lambda)$ is not perverse, then no semi-small resolution of $Y$ exists (see Proposition [Proposition 64](#prop:no-semi-small){reference-type="ref" reference="prop:no-semi-small"}). *Remark 6*. Theorem [Theorem 47](#thm:main1){reference-type="ref" reference="thm:main1"} has also been obtained by McNamara [@PM §5], with a very similar proof. McNamara also noticed Remark [Remark 5](#rem:nosemismall){reference-type="ref" reference="rem:nosemismall"} and uses this observation to rule out the existence of semi-small resolutions of certain Schubert varieties. In the setting of Decomposition Theorem, it is essential to take local systems into account. This is already the case for a smooth morphism with smooth target $f : E \to X$, where the Decomposition Theorem implies that $f_*\mathbb{Q}_E$ splits as a direct sum of its cohomology sheaves $\mathcal{H}^i(f_*\mathbb{Q}_E)$, and each of the resulting local systems (with stalks $H^i(f^{-1}(x),\mathbb{Q})$) are semi-simple. When we take more general coefficients, it is no longer true that the direct image along a proper smooth morphism has to split, nor that the resulting local systems are semi-simple. It is easy to produce examples where the monodromy fails to be semi-simple when the coefficients are not of characteristic $0$. The failure of the direct image to split in the derived category is a little more subtle. We give examples of this failure to split for $\mathbb{P}^1$-bundles (where the non-splitting is connected to the Brauer group) in Example [Example 69](#ex:nonss){reference-type="ref" reference="ex:nonss"}. This motivates us to consider geometric local systems. A **geometric local system** $\mathscr{L}$ on $U$ is a smooth and proper map with smooth target$$V\xrightarrow{\mathscr{L}}U.$$ The following generalises Theorem [\[thm:main1-intro\]](#thm:main1-intro){reference-type="ref" reference="thm:main1-intro"} to take local systems into account: **Theorem 7**. *(see Theorem [Theorem 52](#thm:main2){reference-type="ref" reference="thm:main2"}) [\[thm:main2-intro\]]{#thm:main2-intro label="thm:main2-intro"} Assume $Y$ is irreducible. For any dense (smooth) $U \subset Y$ and geometric local system $V \xrightarrow{\mathscr{L}}U$ there is a unique complex $\mathscr{E}(Y,\mathscr{L}) \in D^b_c(Y,\Lambda)$ satisfying:* 1. *$j^*\mathscr{E}(Y, \mathscr{L}) \cong \mathscr{L}_*\Lambda_U$ where $j : U \hookrightarrow Y$ denotes the inclusion;* 2. *$\mathscr{E}(Y, \mathscr{L})$ has no summands supported on the complement of $U$;* 3. *for any proper map with smooth source $f : X \to Y$ which agrees with $\mathscr{L}$ over $U$, $\mathscr{E}(Y,\Lambda)$ occurs as a summand of $f_* \Lambda_X$.* We call $\mathscr{E}(Y, \mathscr{L})$ the **geometric extension** of the geometric local system $\mathscr{L}$. *Remark 8*. We explain what Theorem [Theorem 52](#thm:main2){reference-type="ref" reference="thm:main2"} says when $\Lambda = \mathbb{Q}$. By the (smooth case of the) Decomposition Theorem, $\mathscr{L}_*\mathbb{Q}_V$ is isomorphic to the direct sum of its intersection cohomology sheaves, $\bigoplus \mathscr{H}^i (\mathscr{L}_*\mathbb{Q}_V)[-i]$, and each cohomology sheaf $\mathscr{H}^i (\mathscr{L}_*\mathbb{Q}_V)$ is semi-simple. If we define $IC(\mathscr{L}_*\mathbb{Q}_V)$ to be $\bigoplus IC(Y, \mathscr{H}^i (\mathscr{L}_*\mathbb{Q}_V))[-i]$, then $\mathscr{E}(Y,\mathscr{L}) = IC(Y, \mathscr{L}_*\mathbb{Q}_V)$, by the Decomposition Theorem. *Remark 9*. Again, $\mathscr{E}(Y,\Lambda)$ provides lower bounds on the cohomology of any proper morphism extending $\mathscr{L}$. We leave it to the reader to formulate an analogue of Corollary [Corollary 4](#cor:forced){reference-type="ref" reference="cor:forced"} in this more general setting. *Warning 10*. In contrast to the setting over $\mathbb{Q}$, we prove that $\mathscr{E}(Y,\mathscr{L})$ is not determined by its restriction to $U$. More precisely, using the Legendre family of elliptic curves, we produce two geometric local systems $V' \xrightarrow{\mathscr{L}'}U$ and $V'' \xrightarrow{\mathscr{L}''}U$ which have the same monodromy over $\mathbb{F}_2$, but whose geometric extensions are not isomorphic (see Example [Example 67](#ex:legendrefamily){reference-type="ref" reference="ex:legendrefamily"}). ## Coefficients in ring spectra One interesting aspect of the current paper is that the results are formal: we only need the proper base change theorem, the existence of fundamental classes, and some finiteness to ensure the Krull-Schmidt theorem. (In the body of the paper we axiomatise our setup as a **base change formalism**, and prove our results in that setting.) Using our formalism, we deduce that our main theorems hold with coefficients in suitable stable $\infty$-categories. In case the reader (like the authors) is intimidated by this theory, we provide a few paragraphs of motivation as to why we are interested in this level of generality.[^3] A major theme in homotopy theory is the consideration of generalized cohomology theories like K-theory, elliptic cohomology, Brown-Peterson cohomology and the Morava K-theories. One can think about all of these cohomology theories as lenses through which to view homotopy theory: facts which are transparent in one theory are often opaque in another. Computation plays an enormously important role, and computations are often performed using the fact that any map is homotopic to a fibration, which gives rise to useful spectral sequences. In algebraic geometry, smooth morphisms (the algebraic geometer's fibrations) are extremely rare, and an important role is played by constructible sheaves and the six functor formalism. The spectral sequence of a fibration is replaced by the Leray-Serre spectral sequence, or its variants. The Decomposition Theorem is a very powerful tool, as it allows one to conclude that the perverse Leray-Serre spectral sequence degenerates for any proper map. Traditionally, this formalism only encompases cohomology, homology and its variants. The connection to cohomology is via the basic fact that the derived global sections of the constant sheaf compute cohomology. In homotopy theory, it has been clear for decades that one can obtain generalised cohomology as the global sections of a local object. (Indeed, by Brown representability, the generalized $E$-cohomogy of $X$ is given by homotopy classes of maps $[X, E^i]$, where $E^i$ represents $i^{th}$ $E$-cohomology.) Thus it is natural to ask: is there some theory of constructible sheaves, which would allow one to push and pull constant $E$-sheaves in much the same way that one can push and pull constant sheaves in algebraic geometry? Such a theory would unify the two approaches to cohomology of the proceeding two paragraphs.[^4] Building on the fundamental work of Lurie [@LurieHTT; @LurieHA; @LurieSAG], such a theory has become available [@VolpeM]. We believe these more general coefficients (e.g. Morava K-theories) will provide a powerful tool to study torsion phenomena in the topology of complex algebraic varieties, in much the same way as they have done in homotopy theory. It is for this reason that we work in the generality of sheaves with coefficients in certain $\infty$-categories. (Again, we emphasise that we only need very formal properties from this theory, and none of its internals.) However, we do not discuss any computations with these more general objects in this paper. Geometric extensions in greater generality play an important role in forthcoming work of the first author [@Hone-thesis] and Elias and the second author [@EW-Satake]. The idea of taking summands in more general (motivic) cohomology theories also shows up in the work of Eberhardt [@Eberhardt1; @Eberhardt2]. *Remark 11*. In §[1.1](#sec:motivation){reference-type="ref" reference="sec:motivation"} we discussed two routes to intersection cohomology sheaves: one via the theory of perverse sheaves (abelian categories), and one via the Decomposition Theorem and Krull-Schmidt (additive categories). In the setting of the more exotic coefficients discussed above, one often encounters periodic cohomology theories. (This is the case for K-theory, as well as all the Morava K-theories.) It is interesting to note that (the homotopy category of) sheaves of modules over such spectra cannot support a non-trivial $t$-structure, so there is no analogue of perverse sheaves with these coefficients. Geometric extensions, on the other hand, make sense as long as the coefficients satisfy a Krull-Schmidt condition. *Remark 12*. Above, our discussion centered on constructible sheaves (algebraic geometry) and generalized cohohomology (homotopy theory). Another major motivation for the development of a sheaf theory underlying cohomology theories is the theory of triangulated categories of motives (see [@CD Introduction §A] for an excellent historical introduction). Our results have a strong motivic flavour, as the reader may have already sensed in our definition of a geometric local system. It should be emphasised, however, that morphism categories in categories of motives rarely have the finiteness conditions that we need in this paper. *Remark 13*. Ever since the discovery of intersection cohomology in the 1970s, it has been suggested that there should be a reasonable theory of intersection K-theory. Such a definition has recently been given by Pădurariu [@IKT], as a subquotient of a geometric filtration on K-theory. The notion of geometric extension with coefficients in (rationalised) $KU$-modules provides another possible definition of intersection K-theory (see Definition [Definition 70](#defKtheory){reference-type="ref" reference="defKtheory"}). It would be interesting to compare the two approaches. One can also hope that there is some (abelian, exact, triangulated, stable $\infty$, ...) category $\mathcal{C}$ associated to our space $X$ which categorifies intersection K-theory. The current work suggests a possible route towards such a category (at least in examples). Namely, intersection K-theory is realised as a summand inside the K-theory of any resolution, and the isomorphisms for different resolutions are sometimes realised by fundamental classes of correspondences. It would be very interesting to know if the classes realizing these isomorphisms could be lifted to functors, inducing categorical idempotents on categories of coherent sheaves on resolutions. ## Motivation from Modular Representation Theory {#sec:motivmodrep} A major motivation for the current work comes from geometric modular representation theory. In the work of Lusztig and others, geometric methods (e.g. Deligne-Lusztig theory, character sheaves, the Kazhdan-Lusztig conjecture) have played a decisive role in classical (i.e. characteristic $0$) representation theory. Modular geometric representation theory aims to transport these successes to modular (i.e. mod $p$) representation theory (see [@JMW-survey; @Achar; @W-ICM]). In this theory the notion of a **parity sheaf** has come to play a central role. These are sheaves whose stalks and costalks vanish in either even or odd degrees. In [@JMW] it is proved that on many varieties arising in geometric representation theory parity sheaves are classified in the same way as intersection cohomology complexes. Their importance in geometric modular representation theory appears to stem from two sources: 1. Whilst it is extremely difficult to compute with intersection cohomology sheaves with modular coefficients, computations with parity sheaves are sometimes possible, thanks to the role of intersection forms [@JMW §3]. This computability is behind counter-examples to the bounds in Lusztig's conjecture arising from unexpected torsion [@W-explosion; @Wtors] and the billiards conjecture of Lusztig and the second author [@LW]. 2. When establishing derived equivalences, it is often useful to have a good class of generators whose algebra of extensions is formal. With $\mathbb{Q}$-coefficients, intersection cohomology complexes often provide such objects. When working with modular coefficients, parity sheaves seem to play the role of "pure" objects, although it is still somewhat mysterious as to why (see [@RSW; @ARKoszul; @ARloop; @AMRW]). The main theorem of [@JMW] relies crucially on the vanishing of odd cohomology of the strata in a fixed stratification. These properties often hold in geometric representation theory, but can be a hindrance. For example, they can be destroyed by passing to a normal slice. Geometric extensions address this deficiency: parity sheaves are very often geometric extensions. Consider a stratified variety $X = \bigsqcup X_\lambda$ satisfying the conditions of [@JMW 2.1], so that the notion of a parity sheaf makes sense. In almost all examples of parity sheaves (for the constant pariversity) one has nice[^5] resolutions $$\pi_\lambda : \widetilde{X}_\lambda \to \overline{X_\lambda}$$ such that the parity sheaf corresponding to the stratum $X_\lambda$ is an indecomposable direct summand of $(\pi_\lambda)_* \Lambda_{\widetilde{X}_\lambda}$. It follows from Theorem [Theorem 47](#thm:main1){reference-type="ref" reference="thm:main1"} that the parity sheaf coincides with the geometric extension $\mathscr{E}(Y,\Lambda)$. *Remark 14*. As we remarked above, Theorems [Theorem 47](#thm:main1){reference-type="ref" reference="thm:main1"} and [Theorem 52](#thm:main2){reference-type="ref" reference="thm:main2"} provide a partial answer to our guiding Question [Question 1](#q:summands){reference-type="ref" reference="q:summands"}. Namely, indecomposable summands with dense support are geometric extensions. However, our theorems say nothing about what happens on lower strata. One could hope that they are geometric extensions, but we have very limited evidence for this claim. (The issue is that, in contrast to the situation for IC and parity sheaves, we have no characterisation of the geometric extension which is intrinsic to the space.) In the setting of parity sheaves (where one does have a local characterisation in terms of stalks and costalks) it is true that all summands are parity sheaves, which can be considered a weak form of the Decomposition Theorem. ## Acknowledgements We would like to thank Bhargav Bhatt for helpful comments, in particular he suggested Example [Example 69](#ex:nonss){reference-type="ref" reference="ex:nonss"}. We would like to thank Burt Totaro for useful comments on Example [Example 63](#ex:A/pm){reference-type="ref" reference="ex:A/pm"}. We would like to thank Roman Bezrukavnikov, Peter McNamara, Luca Migliorini, Marco Volpe and Allen Yuan for useful discussions. # Base change formalism In this section we will describe the categorical formalism used to prove Theorem 1. This section is purely $2$-categorical, and we shall proceed axiomatically to emphasise its formal nature. The reader comfortable with the formalism of the constructible derived category of sheaves will find nothing unfamiliar in what follows.\ \ From here, $\mathscr{C}$ will be a category with pullbacks and a terminal object $\ast$. **Definition 15**. A $\mathbf{base}$ $\mathbf{change}$ $\mathbf{formalism}$ $S:=(S_*,S_!)$ on $\mathscr{C}$ is a data of a pair of pseudo-functors $S_*,S_!$ from $\mathscr{C}$ to the $2$-category $\mathbf{Cat}$, and a lax natural transformation of pseudofunctors $c:S_!\rightarrow S_*$. These two functors $S_*$ and $S_!$ strictly agree on objects, and the object components of $c$ are the identity functor. For any morphism $f$ in $\mathscr{C}$ we abbreviate $S_X=S_*(X)=S_!(X)$, $f_* = S_*(f)$, $f_! = S_!(f)$, and $c_f:f_!\rightarrow f_*$ for the component of $c$ at a morphism $f$ in $\mathscr{C}$. We require the following: 1. For all morphisms $f$, $f_*$ admits a left adjoint $f^*$, and $f_!$ admits a right adjoint $f^!$. In view of (BC1), we say a morphism $f$ in $\mathscr{C}$ is $\mathbf{proper}$ if $c_f:f_!\rightarrow f_*$ is an isomorphism, and $\mathbf{\acute{e}tale}$ if there exists an isomorphism $f^*\cong f^!$. For the remaining two conditions, we fix a pullback square: $$\tag{PS} \begin{tikzcd} X'\arrow[d,"\tilde{f}"]\arrow[r,"\tilde{g}"]&X\arrow[d,"f"]\\ Y'\arrow[r,"g"]& Y \end{tikzcd}$$ Our final conditions are the following: 1. [\[BC2\]]{#BC2 label="BC2"} In (PS), if $f$ is étale (resp. proper), then $\tilde{f}$ is also étale (resp. proper). 2. [\[BC3\]]{#BC3 label="BC3"} In (PS), the induced base change morphisms $$g^*f_*\rightarrow \tilde{f}_*\tilde{g}^*$$ $$\tilde{f}_!\tilde{g}^!\rightarrow g^!f_!$$ are both isomorphisms if $f$ is proper, or if $g$ is étale (see Remark [Remark 18](#basechangedef){reference-type="ref" reference="basechangedef"} for the definition of these base change morphisms). *Remark 16*. In many settings (constructible sheaves on complex varieties, étale sheaves, $D$-modules,...) one encounters a "6-functor formalism\". Usually this manifests as a collection of triangulated (or $\infty-$) categories, with six functors $$f_*, f_!, f^!, f^!, \mathop{\mathrm{\mathcal{H}om}}, \otimes$$ satisfying a raft of relations (see e.g. [@dCM]). There has been recent progress on axiomatizing what a six functor formalism is, particularly in the setting of $\infty-$categories (e.g. [@CD; @mann; @liuzheng; @GR1]). As far as we are aware this process is ongoing and there is still no generally accepted definition. We need very little from the theory, and have tried to isolate the key features we require in Definition [Definition 15](#basechangeformalism){reference-type="ref" reference="basechangeformalism"}. The reader should have little trouble adapting other settings (e.g. étale sheaves, or $D$-modules) to our axioms. The following will be a recurring example throughout this paper. **Example 17**. Let $\mathscr{C}$ be the category of complex algebraic varieties, and let $S_X=D^b_c(X,k)$ be the constructible derived category of sheaves on $X(\mathbb{C})$ with coefficients in a field $k$. (It is important that the derived category is used here, since $f^!$ does not exist in general as a functor on abelian categories.) In this framework, the notions of étale and proper match their topological definitions, hence are closed under pullbacks giving (BC2). Our third axiom (BC3) goes under the name of proper base change in the literature (e.g. [@kashiwara2002sheaves Proposition 2.5.11].) *Remark 18*. Explicitly, the data of a base change formalism is an assignment of a category $S_X$ to each object $X$ in $\mathscr{C}$, functors $f_!:S_X\rightarrow S_Y$ and $f_*:S_X\rightarrow S_Y$ for each morphism $f:X\rightarrow Y$, and coherent isomorphisms $f_*\circ g_* \cong (f\circ g)_*$, $f_!\circ g_!\cong (f\circ g)_!$, along with natural transformations $c_f:f_!\rightarrow f_*$, satisfying some compatibilities. We will suppress these compatibility $2$-isomorphisms for $f_*$ and $f_!$, but the reader should bear in mind that they are a critical part of our input data, as they supply the middle maps used for the base change morphisms of (BC3):$$g^*f_*\xrightarrow{\eta} \tilde{f}_*\tilde{f}^*g^*f_*\cong \tilde{f}_*\tilde{g}^*f^*f_*\xrightarrow{\epsilon} \tilde{f}_*\tilde{g}^*$$ $$\tilde{f}_!\tilde{g}^!\xrightarrow{\eta} \tilde{f}_!\tilde{g}^!f^!f_!\cong \tilde{f}_!\tilde{f}^!g^!f_!\xrightarrow{\epsilon} g^!f_!$$ ## The convolution isomorphism. **Definition 19**. Consider a pull-back square, with $g$ proper: $$\begin{tikzcd} X\times_Y X'\arrow[d,"\tilde{g}"]\arrow[r,"\tilde{f}"]&X'\arrow[d,"g"]\\ X\arrow[r,"f"]& Y \end{tikzcd}$$ We have the following natural isomorphism of functors, which we call the **convolution isomorphism**: $$\begin{tikzcd} \mathop{\mathrm{\mathcal{H}om}}(f_!\_,g_*\_)\arrow[rr,"\tau_{f,g}","\sim"']&&\mathop{\mathrm{\mathcal{H}om}}(\tilde{g}^*\_,\tilde{f}^!\_) \end{tikzcd}$$ This is defined as the composition of the following isomorphisms: $$\begin{aligned} \mathop{\mathrm{\mathcal{H}om}}(f_!\_,g_*\_)\rightarrow \mathop{\mathrm{\mathcal{H}om}}(\_,f^!g_*\_) \rightarrow \mathop{\mathrm{\mathcal{H}om}}(\_,f^!g_!\_) & \rightarrow \\ \rightarrow \mathop{\mathrm{\mathcal{H}om}}(\_,\tilde{g}_!\tilde{f}^!\_) & \rightarrow \mathop{\mathrm{\mathcal{H}om}}(\_,\tilde{g}_*\tilde{f}^!\_)\rightarrow \mathop{\mathrm{\mathcal{H}om}}(\tilde{g}^*\_,\tilde{f}^!\_).\end{aligned}$$ By evaluating this isomorphism on objects $F$ in $S_X$ and $G$ in $S_{X'}$, we obtain the pointwise convolution isomorphism:$$\mathop{\mathrm{\mathcal{H}om}}(f_! F,g_* G)\xrightarrow{\tau_{f,g}}\mathop{\mathrm{\mathcal{H}om}}(\tilde{g}^*F,\tilde{f}^!G)$$ *Remark 20*. In general, computing morphisms between the functors $f_!$ and $g_*$ on $Y$ is hard, and our convolution isomorphism transforms this into a problem with easier functors $\tilde{g}^*$ and $\tilde{f}^!$ on a more complicated space $X\times_Y X'$. In what follows, it will be important to be able to study the convolution isomorphism locally. Consider the following diagram, where $j:U\rightarrow Y$ is étale, and all squares are pullbacks: $$\begin{tikzcd} X_U\times_U X'_U\arrow[dd,"\tilde{g}_U"]\arrow[rr,"\tilde{f}_U"]\arrow[dr,"\hat{j}"]&&X'_U\arrow[dd,"g_U" near start]\arrow[dr,"\underline{j}'"]&\\ &X\times_Y X'\arrow[dd,"\tilde{g}" near start]\arrow[rr,"\tilde{f}" near start]&&X'\arrow[dd,"g"]\\ X_U\arrow[rr,"f_U" near start]\arrow[dr,"\underline{j}"']&&U\arrow[dr,"j"]&\\ &X\arrow[rr,"f"]&&Y \end{tikzcd}$$ The main observation of this section is that the convolution isomorphism is étale local: **Proposition 21**. *The following diagram commutes, where the horizontal maps are our convolution isomorphisms, and the vertical maps are restriction followed by base change: $$\begin{tikzcd} \mathop{\mathrm{\mathcal{H}om}}(f_!\_,g_*\_)\arrow[r,"\tau"]\arrow[d]&\mathop{\mathrm{\mathcal{H}om}}(\tilde{g}^*\_,\tilde{f}^!\_)\arrow[d]\\ \mathop{\mathrm{\mathcal{H}om}}(j^*f_!\_,j^*g_*\_)\arrow[d,"\simeq"]&\mathop{\mathrm{\mathcal{H}om}}(\hat{j}^*\tilde{g}^*\_,\hat{j}^*\tilde{f}^!\_)\arrow[d,"\simeq"]\\ \mathop{\mathrm{\mathcal{H}om}}({f_U}_!\underline{j}^*\_,{g_U}_*{\underline{j}'}^*\_)\arrow[r,"\tau"]&\mathop{\mathrm{\mathcal{H}om}}({\tilde{g}_U}^*\underline{j}^*\_,{\tilde{f}_U}^!{\underline{j}'}^*\_) \end{tikzcd}$$* The verification of this proposition is deferred to Proposition [Proposition 75](#DIAGRAM){reference-type="ref" reference="DIAGRAM"} in the Appendix. # Orientations and Duality The key aspect of our main theorem (Theorem [Theorem 52](#thm:main2){reference-type="ref" reference="thm:main2"}) is that for a map $f:X\rightarrow Y$ with smooth source, the dense summand of $f_*\mathbf{1}_X$ on $Y$ is determined by the generic behaviour of the map. To prove this, we will build a comparison morphism for two maps that agree on an open set. The convolution isomorphism of Definition [Definition 19](#conviso){reference-type="ref" reference="conviso"} turns this problem into giving a suitable morphism $\tilde{g}^*\mathbf{1}_{X}\rightarrow \tilde{f}^!\mathbf{1}_{X'}$. In this section we will describe how to produce such a map via cycle maps and fundamental classes. ## Topological reminders We begin with a leisurely topological reminder of these concepts in the constructible setting. The reader already familiar with this story is invited to skip to §[3.3](#Mainarg){reference-type="ref" reference="Mainarg"}, in which we summarise the key idea of the paper. Let our category $\mathscr{C}$ be that of complex algebraic varieties, and our base change formalism that of Example [Example 17](#runningex){reference-type="ref" reference="runningex"}, i.e., $X\mapsto D^b_c(X,k)$. This base change formalism succinctly encodes the topological homology and cohomology of algebraic varieties with coefficients in $k$. We may express the following cohomology groups of $X$ in terms of our functors and the terminal map $t:X\rightarrow \ast$. Letting $\mathbf{1}$ be $k$ on the point $\ast$, we have $$\begin{aligned} \label{Coh derived cat} H^i(X,k)&\cong \mathop{\mathrm{\mathcal{H}om}}(\mathbf{1},t_*t^*\mathbf{1}[i]),\\ H^i_!(X,k)&\cong \mathop{\mathrm{\mathcal{H}om}}(\mathbf{1},t_!t^*\mathbf{1}[i]),\\ H_i(X,k)&\cong \mathop{\mathrm{\mathcal{H}om}}(\mathbf{1},t_!t^!\mathbf{1}[-i]),\\ H^!_i(X,k)&\cong \mathop{\mathrm{\mathcal{H}om}}(\mathbf{1},t_*t^!\mathbf{1}[-i]).\end{aligned}$$ *Remark 22*. The reader will notice that we are not using the standard notation for Borel Moore homology (see e.g. [@kashiwara2002sheaves]) and compactly supported cohomology. We have opted to use $!$ instead of $BM$ or $c$, as we find this notation more attractive, and it avoids poor notation later when we discuss more general cohomology theories. In this setting, our categories $S_X$ carry some crucial extra structure. They are monoidal and triangulated, with monoidal unit $t^*\mathbf{1}$, and shift functor $[1]$. Throughout, we call $t^*\mathbf{1}$ the $\mathbf{constant}$ $\mathbf{sheaf}$ on $X$, denoted $\mathbf{1}_X$. Similarly, the object $t^!\mathbf{1}$ is the $\mathbf{dualising}$ $\mathbf{sheaf}$ of $X$, denoted $\omega_X$. *Remark 23*. For a singular space $X$, the dualising sheaf $\omega_X$ is not generally concentrated in a single degree in $D^b_c(X,k)$, so cannot be interpreted as a sheaf in the usual sense. **Example 24**. In our running example of the constructible derived category $X \mapsto D^b_c(X,k)$ (Example [Example 17](#runningex){reference-type="ref" reference="runningex"}) $t^*\mathbf{1}$ is the constant sheaf $k_X$. When working in the setting of a general base change formalism, we will use $\mathbf{1}_X:= t^*\mathbf{1}$ to denote the unit object, however when dealing with sheaves we will often stick to the more standard $k_X$. A crucial property of these objects is that for a topological manifold $M$ of dimension $n$, the dualising sheaf $\omega_M$ is locally isomorphic to $\mathbf{1}_M[n]$, a shift of the constant sheaf. To see this, consider the standard triangle associated to the inclusion $j:M\setminus \{x\}\subset M$: $$j_!j^!\omega_M\longrightarrow \omega_M\longrightarrow {i_x}_*i_x^*\omega_M\longrightarrow j_!j^!\omega_M[1].$$ In view of [\[Coh derived cat\]](#Coh derived cat){reference-type="eqref" reference="Coh derived cat"}, applying $\mathop{\mathrm{\mathcal{H}om}}(\mathbf{1},t_!\_)$ gives the long exact sequence: $$H_i(M\setminus\{x\})\longrightarrow H_i(M)\longrightarrow \mathop{\mathrm{\mathcal{H}om}}(\mathbf{1},i_x^*\omega_M[-i])\longrightarrow H_{i-1}(M\setminus\{x\})$$ We may therefore identify the $-i^{th}$ cohomology of the stalk of $\omega_M$ at $x$ with the local homology group $H_i(M,M\setminus \{x\})$. Since $M$ is a manifold, it follows by a standard excision argument [@hatcher §3.3] that this sheaf has stalks $k$ concentrated in degree $-n$. By the local homogeneity of manifolds, we see that this sheaf is locally constant, and thus locally isomorphic to $\mathbf{1}_M[n]$. This also shows that the manifold $M$ is $k$-orientable in the usual sense (see e.g. [@hatcher], Chapter 3) of having compatible local generators of these homology groups if and only if we have an isomorphism $\mathbf{1}_M[n]\cong\omega_M$ in $D^b_c(M,k)$. In view of the definition of Borel-Moore homology [\[Coh derived cat\]](#Coh derived cat){reference-type="eqref" reference="Coh derived cat"}, this is equivalent to a class in this group that restricts to a generator of each local homology group. We call such a class in Borel-Moore homology a $\mathbf{fundamental}$ $\mathbf{class}$ of $M$. There is another, more algebro-topological perspective on orientability. This more general notion of orientability is defined for vector bundles over arbitrary spaces. We say that an $n$-dimensional real vector bundle $V$ over $B$ is orientable with respect to a cohomology theory $E$ if there exists a Thom class $u$ in $$E^n_!(V)\cong E^n(D(V),S(V))$$ that restricts to a generator of $E^n(D(V_x),S(V_x))$ for all $x$ in $B$, where $D(V)$ and $S(V)$ denote the associated disk and sphere bundles of $V$. Thinking about other cohomology theories, there is an analogous base change formalism for algebraic varieties[^6] for any cohomology theory represented by an $A_\infty$ ring spectrum $E$ [@VolpeM]. This category is the homotopy category of the $\infty$-category of constructible sheaves of $E$-module spectra on $X$, and we will denote[^7] it by $D^b_c(X,E\mathrm{-Perf})$. This category encapsulates the $E$-(co)homology of $X$ in exactly the same manner as $D^b_c(X,k)$ does for $k$-(co)homology. We say a manifold $M$ is $E$-orientable if we may find a fundamental class in $$E_{n}^{!}(M):=\mathop{\mathrm{\mathcal{H}om}}_{D^b_c(M,E\mathrm{-Perf})}(\mathbf{1}_M,t^!\mathbf{1}[-n])$$ restricting to a generator in all local homology groups $E_n(M,M\setminus \{x\})$. By the previous discussion, we see that $E$-orientability for $M$ is equivalent to the existence of an isomorphism between the $E$ dualising sheaf $\omega_M$ and $\mathbf{1}_M[n]$. An $E$-orientation is then a choice of such an isomorphism $\mathbf{1}_M\rightarrow \omega_M[-n]$. The set of orientations is in general an $\text{Aut}_{E_M}(\mathbf{1}_M)$ torsor, so orientations are not unique. For a smooth manifold $M$, $E$-orientability in our sense is equivalent to the Thom class $E$-orientability of the stable normal bundle of $M$ [@rudyak Chapter 5, Theorem 2.4]. ## Orientability and fundamental classes for a general base change formalism. With this in mind, let us now work with an arbitrary base change formalism $S$ on the category of algebraic varieties. In order to emphasise the analogy with sheaves we will refer to objects of $S_X$ as sheaves. In addition, we assume the following conditions. 1. [\[C1\]]{#C1 label="C1"} Each $S_X$ is triangulated with shift $[1]$. 2. [\[C2\]]{#C2 label="C2"} Over a point, $S_\ast$ has a distinguished object $\mathbf{1}$. 3. [\[C3\]]{#C3 label="C3"} For any irreducible $X$ with terminal map $t:X\rightarrow \ast$, the object $t^*\mathbf{1}$ is indecomposable. 4. Topologically proper (resp. étale) maps are proper (resp. étale) for $S$ in the sense of Definition [Definition 15](#basechangeformalism){reference-type="ref" reference="basechangeformalism"}. As before, we define the $\mathbf{constant}$ $\mathbf{sheaf}$ and $\mathbf{dualising}$ $\mathbf{sheaf}$: $$\begin{aligned} \mathbf{1}_X&:=t^*\mathbf{1}\\ \omega_X&:=t^!\mathbf{1}\end{aligned}$$ These will be the most important objects in what follows. **Definition 25**. An irreducible variety $X$ of dimension $d$ is $S$-smooth if there exists an isomorphism in $S_X$: $$\mathbf{1}_X\rightarrow \omega_X[-2d].$$ *Remark 26*. When $S_X=D^b_c(X,\mathbb{Q})$, then $S$-smoothness is the same as rational smoothness. More generally, when $S_X=D^b_c(X,k)$, $S$-smoothness of a variety is the same thing as $k$-smoothness (see e.g. [@JWKumar §1] and [@FWpsmooth §8.1]). *Remark 27*. For a general multiplicative cohomology theory $E$, $X$ is $E$-smooth if and only if it is $E$-orientable. By our earlier discussion (see [@rudyak Chapter 5, Theorem 2.4]), this is equivalent to the existence of Thom class in the $E$ cohomology of the Thom spectrum of the stable normal bundle of $X$. The Thom spectrum perspective helps make the problem of deciding $E$-smoothness more concrete and amenable to computation. **Definition 28**. A base change formalism $S$ is **smoothly orientable** if all smooth irreducible varieties are $S$-smooth. **Example 29**. The constructible derived category (i.e. $X\mapsto D^b_c(X,k)$) is smoothly orientable. To see this, note that smooth varieties are topological manifolds of twice their algebraic dimension, so it suffices to check that they are $k$-orientable in the usual sense. This can be seen by noting that a smooth manifold is $\mathbb{Z}$-orientable if and only if some transition cocycle of its tangent bundle can be taken to have positive determinant. Since the tangent bundle of $X$ admits an almost complex structure, we can take a presenting cocycle where locally, these transition functions sit inside $GL_n(\mathbb{C})\subset GL_{2n}(\mathbb{R})$. Since $GL_n(\mathbb{C})$ is connected, all of these real matrices have positive determinant, giving the desired orientability. The following examples show that deciding smooth orientability can be subtle and geometrically meaningful. **Example 30**. Let $KU$ denote the spectrum representing the cohomology theory of complex K-theory. Then the base change formalism $X\mapsto D^b_c(X,KU\mathrm{-Perf})$ is smoothly orientable. To see this, note that any real vector bundle admitting a complex structure is $KU$-orientable, via the explicit construction of a Thom class in [@AB-Clifford III, § 11]. The stable normal bundle of a complex manifold then admits a complex structure, so we see that the stable normal bundle of $X$ is $KU$ orientable, so $X$ is $KU$-smooth. **Example 31**. Let $\mathbb{S}$ denote the sphere spectrum. Then a manifold is $\mathbb{S}$-orientable if and only if its stable normal bundle admits a framing, that is, is trivialisable. In particular, most complex algebraic varieties are not $\mathbb{S}$-smooth. For instance, if $X$ is a smooth surface, the first Pontryagin class of its tangent bundle is equal to three times its signature, by Hirzebruch's Signature theorem [@Hirzebruch]. So if $X$ has nonzero signature of its intersection form, e.g., $\mathbb{CP}^2$, then its tangent bundle is not stably trivial, so neither is its stable normal bundle. In particular, the base change formalism of sheaves of $\mathbb{S}$-modules on all varieties is not smoothly orientable. We are interested in invariants of singular varieties, so we want fundamental classes/orientations for singular varieties also. In usual sheaf theoretic fashion, for a Zariski open $j:U\rightarrow X$ in a space $X$, we refer to the functor $j^*\cong j^!$ as restriction to $U$. **Definition 32**. An $S$-orientation of an irreducible variety $X$ is a morphism $$\gamma:\mathbf{1}\rightarrow \omega_X[-2d_X]$$ which is an isomorphism over the smooth locus of $X$. We say $X$ is orientable with respect to $S$ if an $S$-orientation of $X$ exists. The following proposition shows that we can use resolution of singularities to orient all irreducible varieties. **Proposition 33**. *If $S$ is smoothly orientable, and $X$ admits a resolution of singularities $f:\tilde{X}\rightarrow X$, then $X$ is orientable.* *Proof.* Let $f:\tilde{X}\rightarrow X$ be a resolution of singularities. Then $f$ is proper, and $\tilde{X}$ is nonsingular, with $f$ an isomorphism over the smooth locus $U$ of $X$. Since $\tilde{X}$ is smooth, we have an orientation: $$\gamma:\mathbf{1}_{\tilde{X}}\rightarrow \omega_{\tilde{X}}[-2d_X].$$ Pushing this forward gives:$$f_*\gamma:f_*\mathbf{1}_{\tilde{X}}\rightarrow f_*\omega_{\tilde{X}}[-2d_X].$$Composing with the unit and counits of our adjunctions gives $$\mathbf{1}_{X}\rightarrow f_*\mathbf{1}_{\tilde{X}} \rightarrow f_*\omega_{\tilde{X}}[-2d_X]\xrightarrow{\text{~}} f_!\omega_{\tilde{X}}[-2d_X]\rightarrow \omega_X[-2d_X].$$ By base change (BC3), the composite $\gamma_X:\mathbf{1}_X\rightarrow \omega_X[-2d_X]$ restrict to isomorphisms over $U$, giving the desired orientation of $X$. ◻ *Remark 34*. The definitions of this section are based on purely topological realisation of algebraic varieties via their $\mathbb{C}$-points, but there are natural extensions of these definitions to other settings. For instance, one could consider real pseudomanifolds or algebraic varieties over fields more general than $\mathbb{C}$. In these settings, one would need to modify Definition [Definition 25](#Orientable){reference-type="ref" reference="Orientable"} to reflect the structure at hand. For example, incorporating weights, an orientation is a morphism $\mathbf{1}_X\rightarrow \omega_X[-2d_X](d_X)$, where $(n)$ denotes the Tate twist. *Remark 35*. For the reader who doesn't want to assume resolution of singularities, one can adapt the previous proof to show that if $X$ admits a degree $n$ alteration in the sense of de Jong [@deJong], and $n$ is invertible in the ring $\mathop{\mathrm{\mathcal{H}om}}_{S_\ast}(\mathbf{1},\mathbf{1})$, then $X$ is orientable. The importance of orientations cannot be overstated in our context, since they allow us to produce morphisms between nontrivial objects in our categories $S_X$, via functoriality and the convolution isomorphism. **Definition 36**. For $X$ an algebraic variety, we define the $n^{th}$ compactly supported $S$-homology of $X$ to be $$S^!_{n}(X):=\mathop{\mathrm{\mathcal{H}om}}_{S_X}(1_X,\omega_X[-n]).$$ This functions similarly to Borel-Moore homology in the constructible setting, as the codomain of a cycle class morphism. In particular, any orientation $\gamma$ of an irreducible variety $X$ is naturally an element of $S^!_{2d_X}(X)$. Like Borel-Moore homology, these groups are covariantly functorial under proper maps $f:X\rightarrow Y$. This is given by the composition $$\mathop{\mathrm{\mathcal{H}om}}^*(\mathbf{1}_X,\omega_X)\rightarrow \mathop{\mathrm{\mathcal{H}om}}^*(f_*\mathbf{1}_X,f_*\omega_X)\cong \mathop{\mathrm{\mathcal{H}om}}^*(f_*\mathbf{1}_X,f_!\omega_X)\rightarrow \mathop{\mathrm{\mathcal{H}om}}^*(\mathbf{1}_Y,\omega_Y).$$ ## Why do geometric extensions exist? {#Mainarg} Our goal is to construct a canonical extension of the constant sheaf on a potentially singular variety $Y$. We construct this by first pushing forward the constant sheaf from a resolution of singularities $X\rightarrow Y$. We then need a method for comparing these sheaves for different choices of resolution. We will construct a comparison morphism between these pushforwards using the existence of fundamental classes in our base change formalism. We may summarise the machinery we have so far for a smoothly orientable base change formalism $S$ as follows. - An $S$ internal notion of smoothness (Definition [Definition 25](#Orientable){reference-type="ref" reference="Orientable"}). - An $S$ orientation/fundamental class for any variety (not necessarily smooth) (Definition [Definition 32](#fundclass){reference-type="ref" reference="fundclass"}). - A compactly supported $S$-homology group to interpret fundamental classes in (Definition [Definition 36](#BorelMoore){reference-type="ref" reference="BorelMoore"}).   We may now interpret our convolution isomorphism in this context. Let $X$ and $X'$ be smooth (proper) resolutions of $Y$, with a chosen orientation of $X'$, such that we have a pullback square: $$\begin{tikzcd} X\times_Y X'\arrow[d,"\tilde{g}"]\arrow[r,"\tilde{f}"]&X'\arrow[d,"g"]\\ X\arrow[r,"f"]& Y \end{tikzcd}$$ We will use the functoriality of our setup to construct morphisms between $f_!\mathbf{1}_X$ and $g_*\mathbf{1}_{X'}$, from the geometry of fundamental classes on the fibre product. Specifically, the convolution isomorphism and our choice of orientation of $X$ yields the following isomorphism: $$\begin{aligned} \mathop{\mathrm{\mathcal{H}om}}(f_!\mathbf{1}_X,g_*\mathbf{1}_{X'})&\cong \mathop{\mathrm{\mathcal{H}om}}(\tilde{g}^*\mathbf{1}_X,\tilde{f}^!\mathbf{1}_{X'})\cong\\ &\cong \mathop{\mathrm{\mathcal{H}om}}(\mathbf{1}_{X\times_Y X'},\tilde{f}^!\omega_{X'}[-2d_{X}])=S^!_{2d_{X}}(X\times_Y X').\end{aligned}$$ Via this isomorphism, we may translate compactly supported $S$-homology classes of $X\times_Y X'$ into maps from $f_!\mathbf{1}_X$ to $g_*\mathbf{1}_{X'}$. Since $f,g$ are resolutions of $Y$, if $U$ is smooth locus of $Y$, we have a canonical diagonal $\Delta(U)$ inside $X\times_Y X'$, with closure $Z:=\overline{\Delta(U)}$. Choosing an orientation of $Z$, we may push forward the associated fundamental class ([Definition 36](#BorelMoore){reference-type="ref" reference="BorelMoore"}) to get a class in $S_{2d}^!(X\times_Y X')$. This gives the desired comparison morphism between $f_!\mathbf{1}_X$ and $g_*\mathbf{1}_Y$. In the next section, we will show that in the presence of finiteness conditions, we may deduce an isomorphism between the "dense" summands of these pushforwards, giving our main theorem. *Remark 37*. In the special case of the constant sheaf, there is an alternate argument[^8]. Take our base change formalism to be $X\mapsto D^b_c(X,E\mathrm{-Perf})$, for a suitably finite (see Definition [Definition 38](#KS-definition){reference-type="ref" reference="KS-definition"}) smoothly orientable cohomology theory $E$. One may show (see [@rudyak Chapter 5, Theorem 2.13]) that for a map $f:X\rightarrow Y$ of $E$-orientable manifolds, the induced map $E^*(Y)\rightarrow E^*(X)$ is injective, and upgrade this to the fact that $\mathbf{1}_Y\rightarrow f_*\mathbf{1}_X$ is split injective in $D^b_c(X,E\mathrm{-Perf})$. Now for singular $Y$, given two such resolutions $X_i$, we may resolve the diagonal component of their fibre product $X_1\times_Y X_2$. From this splitting of the constant sheaf for maps of $E$-orientable smooth manifolds, we see that the dense summand of $f_{i*}\mathbf{1}_{X_i}$ occurs as a summand of all resolutions. This argument also shows that isomorphism classes of summands of $f_*\mathbf{1}_X$ over all resolutions $f : X \to Y$ form a sort of "lattice": given any two resolutions $f_i : X_i \to Y$ for $i = 1, 2$, there exists a third resolution $g : Z \to Y$ such that all summands of $f_{1*}\mathbf{1}_{X_1}$ and $f_{2*}\mathbf{1}_{X_2}$ also occur inside $g_*\mathbf{1}_Z$. # Finiteness and Krull-Schmidt categories {#sec-KS} In the previous section we constructed a comparison morphism between $f_*\mathbf{1}_X$ and $g_*\mathbf{1}_{X'}$ using an orientation of the irreducible component of the diagonal within $X\times_Y X'$. In any smoothly orientable base change formalism, it follows formally that this comparison morphism is an isomorphism over $U$. In this section, we will introduce the finiteness conditions needed to show that this isomorphism over $U$ lifts to an isomorphism on "dense summands" of $f_*\mathbf{1}_X$ and $g_*\mathbf{1}_{X'}$. The finiteness constraint we need is that the categories of the base change formalism are Krull-Schmidt, which allows the use of the crucial Lemma [Lemma 40](#isolift){reference-type="ref" reference="isolift"}. For completeness, we recall the definition of a Krull-Schmidt category: **Definition 38**. A category $\mathscr{C}$ is **Krull-Schmidt** if it is additive with finite sums, and each object is isomorphic to a finite direct sum of indecomposable objects, each with local endomorphism rings. A functor $F$ between Krull-Schmidt categories is Krull-Schmidt if for each indecomposable object $A$, $F$ maps the Jacobson radical of $\mathop{\mathrm{End}}(A)$ into the Jacobson radical of $\mathop{\mathrm{End}}(F(A))$. *Remark 39*. There does not appear to be an agreed upon notion of a Krull-Schmidt functor in the literature. The above definition seems to encapsulate the necessary behaviour of our situation. This condition is easily checked in some sheaf theoretic contexts since it is implied by the following three conditions: - The "ring of coefficients\" $R:=\mathop{\mathrm{End}}(\mathbf{1}_\ast)$ is a complete local ring. - For any $\mathscr{F},\mathscr{G}$ in $S_X$, the group $\mathop{\mathrm{\mathcal{H}om}}_{S_X}(\mathscr{F},\mathscr{G})$ is a finitely-generated $R$-module. - The category $S_X$ has split idempotents. These conditions imply that the endomorphism ring of any indecomposable object is a local $R$-algebra. It also implies that the $R$-algebra morphisms $\mathop{\mathrm{End}}(X)\rightarrow \mathop{\mathrm{End}}(F(X))$ are all finite, giving the Krull-Schmidt property on the functors of the base change formalism. In particular these conditions are satisfied for the constructible base change formalism $X\mapsto D^b_c(X,\Lambda)$, when $\Lambda$ is a field or complete local ring. We will discuss this case in more detail in the appendix. The primary result about Krull Schmidt categories we will use is the following automorphism lifting property. **Lemma 40**. *Let $F:\mathscr{C}\rightarrow \mathscr{D}$ be a Krull-Schmidt functor between Krull-Schmidt categories, let $A$ be an object with $F(A_i)\neq 0$ for all nonzero summands $A_i$ of $A$, and let $\mu:A\rightarrow A$ an endomorphism of $A$. If $F(\mu)=\text{Id}_{F(A)}$, then $\mu$ is an isomorphism.* *Proof.* We induct on the number of indecomposable summands of $A$, which is finite by our Krull-Schmidt hypothesis. If $A$ is itself indecomposable, then since $F$ is a Krull-Schmidt functor, $\mu$ doesn't lie in the Jacobson radical of $\mathop{\mathrm{End}}(A)$. Since $\mathop{\mathrm{End}}(A)$ is a local ring, $\mu$ is invertible. Next, let us consider the case that $A$ is a direct sum of $n$ copies of a single indecomposable $A_0$. In this case, we see that $\mu-Id_A$ is in the kernel of the algebra morphism $\mathop{\mathrm{End}}(A)\rightarrow \mathop{\mathrm{End}}(F(A))$. This kernel is then contained in the unique maximal two sided ideal $M_{n\times n}(J(\mathop{\mathrm{End}}(A_0))$ of the matrix ring $M_{n\times n}(\mathop{\mathrm{End}}(A_i))\cong \mathop{\mathrm{End}}(A)$. So $\mu$ is in $\mathop{\mathrm{\text{Id}}}+J(\mathop{\mathrm{End}}(A))$, and is therefore an isomorphism. Finally, we may assume that $A$ is not indecomposable, and admits a nontrivial decomposition $A\cong B\oplus C$ where $B$ and $C$ share no isomorphic summands. Then our morphism $\mu$ decomposes as: $$\begin{aligned} \mu=\begin{bmatrix} \mu_{BB} & \mu_{CB}\\ \mu_{BC} & \mu_{CC} \end{bmatrix}= &\begin{bmatrix} \mu_{BB} & 0\\ 0 & \mu_{CC} \end{bmatrix}+\begin{bmatrix} 0 & \mu_{CB}\\ \mu_{BC} & 0 \end{bmatrix} \\ & \text{ \quad with $\mu_{XY}\in \mathop{\mathrm{\mathcal{H}om}}(X,Y)$ for $X,Y\in \{B,C\}$.} \end{aligned}$$ By induction, this diagonal piece is an isomorphism, and since $B$ and $C$ share no isomorphism classes of summands in common, the second matrix is in the radical of $\mathop{\mathrm{End}}(A)$ (see the lines following the proof of [@Krause Corollary 4.4]), so $\mu$ is an isomorphism. ◻ **Definition 41**. Let $F:\mathscr{C}\rightarrow \mathscr{D}$ be a Krull-Schmidt functor between Krull-Schmidt categories. Then an object $A$ of $\mathscr{C}$ is $F$-dense if for all summands $A_i$ of $A$ we have $F(A_i)\neq 0$. We will only use this notion with respect to $j^*$ for $j:U\rightarrow X$ a Zariski open morphism to irreducible $X$. In sheaf theoretic contexts, this agrees with the usual notion of having all indecomposable summands of dense support, and we will write this as $U$-dense. We say an object $\mathscr{E}$ of $S_X$ is $\mathbf{dense}$ in $S_X$ if it is $U$-dense for any dense Zariski open $U$ of $X$. **Lemma 42**. *Let $F:\mathscr{C}\rightarrow \mathscr{D}$ be a Krull-Schmidt functor between Krull-Schmidt categories. Then any object $A$ of $\mathscr{C}$ has a decomposition $A\cong A_F\oplus A_0$, such that $A_F$ is a maximal $F$-dense summand of $A$. This decomposition is unique up to non-unique isomorphism.* *Proof.* Choose any decomposition of $A$ into indecomposable objects $A_i$, and let $A_F$ be the summand of those isomorphism types $A_i$ with $F(A_i)\neq 0$. The isomorphism class of $A_F$ is then unique by the Krull-Schmidt property. ◻ **Proposition 43**. *Let $A$ and $B$ be objects in a Krull-Schmidt category $\mathscr{C}$ and $F:\mathscr{C}\rightarrow \mathscr{D}$ a Krull-Schmidt functor. If for two maps $f:A\rightarrow B$, $g:B\rightarrow A$, we have $F(f)$ and $F(g)$ are mutually inverse isomorphisms, then $f$, $g$ induce isomorphisms $f'$, $g'$ of $F$-dense summands: $$\begin{tikzcd} A_F\arrow[rrr, bend left, "f'"]\arrow[r,"i_A"]& A\arrow[r,"f"]& B\arrow[r,"\pi_B"] &B_F\end{tikzcd}$$$$\begin{tikzcd} B_F\arrow[rrr,bend right,"g'"]\arrow[r,"i_B"]& B\arrow[r,"g"] &A\arrow[r,"\pi_A"]& A_F \end{tikzcd}$$* *Proof.* First, note that $F(\pi_A)$ and $F(i_A)$ are both isomorphisms, since $A_0$ is sent to zero under $F$ by maximality of $A_F$ and similarly for $B$. Thus, our maps $f':=\pi_B\circ f\circ i_A$ and $g':=\pi_A\circ g\circ i_B$, induce mutually inverse isomorphisms $F(f')$ and $F(g')$ under $F$. So Lemma [Lemma 40](#isolift){reference-type="ref" reference="isolift"} applied to the compositions of these gives that $f'\circ g'$ and $g'\circ f'$ are both isomorphisms. By elementary category theory, this then yields that $f'$ and $g'$ are both isomorphisms, as was to be shown. ◻ We can now state our main theorem. **Theorem 44**. *Let $X,X'$ be smooth, irreducible varieties with proper, surjective maps $f:X \rightarrow Y$, $g:X'\rightarrow Y$, and $j:U\rightarrow Y$ a Zariski open in $Y$. Assume that the pullbacks $f_U,g_U:X_U,X'_U\rightarrow U$ are isomorphic over $U$: $$\begin{tikzcd}[column sep=small] X_U\arrow[ddr,"f|_U"']\arrow[drrr]\arrow[rr,dashed,"\simeq"]&&X'_U\arrow[drrr]\arrow[ddl,"g|_U"]\\ &&&X\arrow[ddr,"f"']&&X'\arrow[ddl,"g"]\\ &U\arrow[drrr,"j"',hook]&\\ &&&&Y \end{tikzcd}$$ Then for any smoothly orientable, Krull-Schmidt base change formalism $S$ the $U$-dense summands of $f_*\mathbf{1}_X$ and $g_*\mathbf{1}_{X'}$ in $S_Y$ are isomorphic.* *Proof.* Let the isomorphism over $U$ be $\alpha:X_U\rightarrow X'_U$. Then we choose orientations of the spaces involved such that we have the following commutative diagram: $$\begin{tikzpicture}[xscale=2,yscale=1] \node (H0) at (0,0) {$\mathop{\mathrm{\mathcal{H}om}}({f_U}_*\mathbf{1}_{X_U},{g_U}_*\mathbf{1}_{X'_U})$}; \node (H1) at (0,2) {$\mathop{\mathrm{\mathcal{H}om}}(f_*\mathbf{1}_X,g_*\mathbf{1}_{X'})$}; \node (S0) at (2,0) {$S_{*,2d}(X_U\times_U X'_U)$}; \node (S1) at (2,2) {$S_{*,2d}(X\times_Y X')$}; \draw[->] (H0) to node[above] {$\sim$} (S0); \draw[->] (H1) to node[above] {$\sim$} (S1); \draw[->] (H1) to (H0); \draw[->] (S1) to (S0); \node (ur) at (3,2) {$[\overline{\Delta_\alpha}]$}; \node (e1) at (2.72,2) {$\ni$}; \node (lr) at (3,-1) {$[\Delta_\alpha]$}; \node (e1) at (2.72,-.5) {\rotatebox{135}{$\in$}}; \node (l) at (0,-1) {$\alpha_*\mathbf{1}$}; \node (e1) at (0,-.6) {\rotatebox{90}{$\in$}}; \draw[|->] (ur) to (lr); \draw[|->] (l) to (lr); \end{tikzpicture}$$ That we can do this is Proposition [Proposition 76](#convisocomp){reference-type="ref" reference="convisocomp"} in the appendix. Transporting the fundamental class of $\overline{\Delta_\alpha}$ across the convolution isomorphism then yields a map $$f_*\mathbf{1}_X\rightarrow g_*\mathbf{1}_{X'}.$$ By commutativity, this restricts to $\alpha_*\mathbf{1}$ over $U$. By symmetry there also exists a morphism back, which gives the two morphisms restricting to mutually inverse isomorphisms over $U$. We may then use Lemma [Proposition 43](#mutinv){reference-type="ref" reference="mutinv"} to conclude that these induce isomorphisms on the $U$-dense summands. ◻ **Corollary 45**. *In the setting of Theorem [Theorem 44](#mainthm){reference-type="ref" reference="mainthm"}, the dense summands of $f_*\mathbf{1}_X$ and $g_*\mathbf{1}_{X'}$ are isomorphic.* *Proof.* The dense summands of these are the dense summands of the $U$-dense summands of $f_*\mathbf{1}_X$ and $g_*\mathbf{1}_{X'}$, hence are isomorphic. ◻ *Remark 46*. One may note that the use of a *Zariski* neighbourhood was essential in this proof, to be able to take a closure of the graph of the isomorphism over $U$. If one uses a simple étale neighbourhood instead, one must push forward this cycle, and the induced map is an isomorphism only if the degree of the étale morphism is invertible in the ring of coefficients. # Applications In this final section we will see some applications of Theorem [Theorem 44](#mainthm){reference-type="ref" reference="mainthm"}. This theorem allows one to construct canonical objects in $S_X$ for any smoothly orientable base change formalism that play the role of intersection cohomology sheaves in the $\mathbb{Q}$ constructible setting, and parity sheaves in the $\mathbb{F}_p$ constructible setting. Before considering the general case, let us consider the smoothly orientable base change formalism of constructible sheaves with coefficients in a field or complete local ring $\Lambda$. As an immediate corollary of Theorem [Theorem 44](#mainthm){reference-type="ref" reference="mainthm"}, we obtain the following: **Theorem 47**. *Let $Y$ be an irreducible variety. There exists a complex $\mathscr{E}(Y,\Lambda) \in D^b_c(Y,\Lambda)$ characterised up to isomorphism by the following:* 1. *$\mathscr{E}(Y,\Lambda)$ is indecomposable and its support is dense;* 2. *$\mathscr{E}(Y,\Lambda)$ is a summand inside $f_* \Lambda_X$, for any resolution $f: X \to Y$.* We call $\mathscr{E}(Y,\Lambda)$ the **geometric extension** on $Y$. As we explained in §[1.4](#sec:motivmodrep){reference-type="ref" reference="sec:motivmodrep"}, in the special case of a cellular resolution of singularities, this geometric extension will be a parity sheaf [@JMW]. We may think of this object as a "geometrically motivated\" minimal way to extend the constant sheaf on the smooth locus of $Y$. In particular, since this summand occurs for any resolution of singularities, we obtain the following corollary for $\mathbb{F}_p$ coefficients. **Corollary 48**. *For any resolution of singularities $\pi:X\rightarrow Y$, for all $y\in Y$ with fibre $X_y=\pi^{-1}(y)$, we have the inequality $$\dim H^i(\mathscr{E}_{\mathbb{F}_p}(Y)_y)\leq \dim H^i(X_y,\mathbb{F}_p).$$* *Proof.* By definition, we know that $\mathscr{E}_{\mathbb{F}_p}(Y)_y$ is a summand of $i_{y}^*\pi_*\mathbf{1}_X$. The cohomology of $i_{y}^*\pi_*\mathbf{1}_X$ then computes the cohomology of the fibre by proper base change, giving the result. ◻ We now consider the case of a general smoothly orientable, Krull-Schmidt base change formalism, and higher dimensional local systems. First, we need the definition of a higher dimensional local system in this context. **Definition 49**. A **geometric local system** $\mathscr{L}$ on a smooth irreducible variety $U$ is a smooth, proper, surjective map $V\xrightarrow{\mathscr{L}}U$. The restriction of $\mathscr{L}$ to an open $U'\rightarrow U$ is the base change of this morphism. The following proposition lets us interpret compactification of morphisms as a method to "extend" geometric local systems. **Proposition 50**. *For any geometric local system $V\xrightarrow{\mathscr{L}} U$ over $U$ a (smooth Zariski) open in $Y$, there exists a proper morphism $X\xrightarrow{\tilde{\mathscr{L}}} Y$ from smooth $X$ such that we have a pullback square: $$\begin{tikzcd} V\arrow[r]\arrow[d,"\mathscr{L}"]& X\arrow[d,"\overline{\mathscr{L}}"]\\ U\arrow[r,"j"]&Y \end{tikzcd}$$* *Proof.* The proof may be summarised in the following diagram:$$\begin{tikzcd} &X\arrow[d]\arrow[dd,bend left,"\overline{\mathscr{L}}"]\\ V\arrow[d,"\mathscr{L}"]\arrow[ur,dashed]\arrow[r]&\tilde{Y}\arrow[d,"g"']\\ U\arrow[r,"j"]&Y \end{tikzcd}$$ First, we compactify the composition $j\circ \mathscr{L}$ as $V\rightarrow \tilde{Y}\xrightarrow{g} Y$, where $g$ is proper. Choosing a resolution of singularities $X\rightarrow \tilde{Y}$, as $V$ is smooth, the map $V\rightarrow Y$ factors through $X$. Then composing with $g$ gives the desired map $X\xrightarrow{\tilde{\mathscr{L}}} Y$. ◻ From here we will let $S$ denote a Krull-Schmidt, smoothly orientable base change formalism, satisfying the following condition for geometric local systems: - If $\mathscr{L}:V\to U$ is a geometric local system, then all summands of $\mathscr{L}_*\mathbf{1}$ have dense support. *Remark 51*. This condition holds in all examples we have discussed so far, and we don't know of any situation where it fails to hold. In sheaf theoretic or algebro-topological situations, this follows from homotopy invariance. With these preliminaries, we have the following general version of Theorem [Theorem 47](#thm:main1){reference-type="ref" reference="thm:main1"}. **Theorem 52**. *Let $Y$ be an irreducible variety, $S$ a smoothly orientable, Krull-Schmidt base change formalism satisfying condition (D). For any dense $U \subset Y$ and geometric local system $V \xrightarrow{\mathscr{L}}U$ there is a unique object $\mathscr{E}(Y,\mathscr{L}) \in S_Y$ satisfying:* 1. *$j^*\mathscr{E}_S(Y, \mathscr{L}) \cong \mathscr{L}_*\mathbf{1}_V$ where $j : U \hookrightarrow Y$ denotes the inclusion;* 2. *$\mathscr{E}_S(Y, \mathscr{L})$ is dense, with no summands supported on a proper closed subset of $Y$;* 3. *for any proper map with smooth source $f : X \to Y$ which restricts to $\mathscr{L}$ over $U$, $\mathscr{E}_S(Y,\mathscr{L})$ occurs as a summand of $f_* \mathbf{1}_X$.* **Definition 53**. We define the **geometric extension** of $\mathscr{L}$ on $Y$ to be the object $\mathscr{E}_S(Y,\mathscr{L})$. When the local system is the identity, we call this the geometric extension on $Y$. We call the groups $$\mathscr{E}_S^i(Y):=\mathop{\mathrm{\mathcal{H}om}}_{S_Y}(\mathbf{1}_Y,\mathscr{E}_{S}(Y)[i])$$ the geometric $S$-cohomology groups of $Y$. *Remark 54*. For a fixed $S$, one may replace smoothness with $S$-smoothness (see Definition [Definition 25](#Orientable){reference-type="ref" reference="Orientable"}) in the preceeding definitions, with slightly more general results. One can think of these groups $\mathscr{E}^i_S(Y)$ concretely as unavoidable summands of the $S$-cohomology of any resolution of singularities of $X$. *Warning 55*. In general, the object $\mathscr{E}_S(Y,\mathscr{L})$ depends on the geometry of the map $V\xrightarrow{\mathscr{L}} U$, not just on the object $\mathscr{L}_*\mathbf{1}_V$ in $S_U$. An example with further discussion is given in Example [Example 67](#ex:legendrefamily){reference-type="ref" reference="ex:legendrefamily"}. We will now give some properties of these geometric extensions. **Proposition 56**. *If $\mathscr{L}$ and $\mathscr{L}'$ on $U$ and $V$ agree on $U\cap V$, then: $$\mathscr{E}_S(Y,\mathscr{L})\cong \mathscr{E}_S(Y, \mathscr{L'}).$$ In particular, for $f:X\to Y$ a proper map with smooth source, the dense summand of $f_*\mathbf{1}_X$ depends only on the generic behaviour of the map $f$.* *Proof.* The geometric extensions arise as summands of $f_*\mathbf{1}_X$, $g_*\mathbf{1}_{X'}$ for compactifications $f,g$ of these geometric local systems. Since these two maps agree on a dense open $U\cap V$, their dense summands are isomorphic by Theorem [Theorem 44](#mainthm){reference-type="ref" reference="mainthm"}. ◻ Consider a complex $\mathscr{L}$ isomorphic to $\bigoplus_i \mathscr{L}^i[-i]$ for local systems $\mathscr{L}^i$ on a dense subvariety of the smooth locus of $Y$. Define $IC(Y,\mathscr{L})$ to be the sum $\bigoplus_i IC(Y,\mathscr{L}^i)[-i]$ (see Remark [Remark 8](#ICdef){reference-type="ref" reference="ICdef"}). **Proposition 57**. *If $S$ is the constructible derived category of sheaves over $\mathbb{Q}$, then the geometric extension is the intersection cohomology complex of sheaves: $$\mathscr{E}_S(Y,\mathscr{L})\cong IC(Y,\mathscr{L}_*\mathbb{Q}).$$* *Proof.* By the Decomposition Theorem [@BBD], the pushforward $f_*\mathbf{1}_X$ is a direct sum of semisimple perverse sheaves. On the smooth locus of $Y$, this sheaf is the local system $f_*\mathbf{1}_V$. By the classification of simple perverse sheaves ([@BBD] or [@Achar Theorem 3.4.5]), we see that the dense summand of $f_*\mathbf{1}_V$ is $IC(Y,\mathscr{L}_*\mathbf{1}_V)$. ◻ In general, like intersection cohomology sheaves, the geometric extension gives a way to interpolate between $S$-cohomology and noncompact $S$-homology. **Proposition 58**. *For any resolution of singularities $f:\tilde{Y}\rightarrow Y$, chosen orientation $\gamma$ of $\tilde{Y}$, and choice of split inclusion $\mathscr{E}_S(Y)\rightarrow f_*\mathbf{1}_{\tilde{Y}}$, we obtain a sequence: $$\mathbf{1}_Y\rightarrow \mathscr{E}_S(Y)\rightarrow \omega_Y[-2d_Y]$$ such that the composite is an orientation of $Y$.* *Proof.* We have the following maps of sheaves on $Y$: $$\mathbf{1}_Y\rightarrow f_*\mathbf{1}_{\tilde{Y}}\rightarrow \mathscr{E}_S(Y)\rightarrow f_*\mathbf{1}_{\tilde{Y}}\cong f_!\mathbf{1}_{\tilde{Y}}\cong f_!\omega_{\tilde{Y}}[-2d]\rightarrow \omega_Y[-2d]$$ These maps are all isomorphisms over $U$, so the composite is an orientation of $Y$. ◻ The following shows that for $S$-smooth varieties, the geometric extension is just the constant sheaf. **Proposition 59**. *If $S$ is a smoothly orientable base change formalism, and $Y$ is $S$-orientable, then the geometric extension is the constant sheaf on $Y$:$$\mathscr{E}_S(Y)\cong \mathbf{1}_Y$$* *Proof.* For a resolution $\pi:\tilde{Y}\rightarrow Y$ of $Y$, and chosen isomorphism $\gamma:\mathbf{1}_{\tilde{Y}}\rightarrow\omega_{\tilde{Y}}[-2d]$ of $\tilde{Y}$, then we claim that the pushforward orientation of $Y$ is an isomorphism $\mathbf{1}_Y\rightarrow \omega_Y[-2d]$. This morphism is between indecomposable, isomorphic objects, and is thus an isomorphism over $U$, so is an isomorphism by the Krull-Schmidt property. Composing with the inverse isomorphism $\omega_Y[-2d]\rightarrow \mathbf{1}_Y$ then gives the following commutative diagram: $$\begin{tikzcd} \mathbf{1}_Y\arrow[r]\arrow[d,equal]&\mathscr{E}_S(Y)\arrow[d]\\ \mathbf{1}_Y&\omega_Y[-2d]\arrow[l]. \end{tikzcd}$$ The result then follows from Proposition [Proposition 43](#mutinv){reference-type="ref" reference="mutinv"}. ◻ The maps of Proposition [Proposition 58](#interpolate){reference-type="ref" reference="interpolate"} induce the following interpolation morphisms $$S^*(Y)\rightarrow \mathscr{E}_S^*(Y)\rightarrow S^!_{2d_Y-\ast}(Y).$$ This interpolation perspective also lets us extract a canonical invariant of our singular space, the (co)kernel of the induced map $S^*(Y)\to \mathscr{E}^*_S(Y)$. **Definition 60**. Let $Y$ be irreducible and projective. The **geometrically pure** $S$-cohomology of $Y$ is the quotient $$S^*_{gp}(Y):=\frac{S^*(Y)}{\text{ker}(S^*(Y)\rightarrow \mathscr{E}^*_S(Y))}$$ Similarly, the **geometrically non-pure** $S$-cohomology of $Y$ is this kernel $$S^*_{gnp}(Y):=\text{ker}(S^*(Y)\rightarrow \mathscr{E}^*_S(Y)).$$ That these objects are independent of the choices involved in their construction is the content of the following Lemma. **Lemma 61**. *Let $Y$ be irreducible, with two resolutions of singularities $$f_i:\tilde{Y}_i\rightarrow Y\ \ \ \text{for}\ i\in \{1,2\}.$$ Assume for each map we have a chosen a split projection onto the geometric extension of $Y$: $${f_i}_*\mathbf{1}\xrightarrow{\pi_i}\mathscr{E}_S(Y).$$ Then there exists an isomorphism $\beta$ of $\mathscr{E}_S(Y)$ such that the following diagram commutes: $$\begin{tikzcd} \mathbf{1}\arrow[dr]\arrow[r]& {f_1}_*\mathbf{1}\arrow[r,"\pi_1"]&\mathscr{E}_S(Y)\arrow[d,dashed,"\beta"]\\ &{f_2}_*\mathbf{1}\arrow[r,"\pi_2"]&\mathscr{E}_S(Y) \end{tikzcd}$$* *Proof.* Resolving the diagonal irreducible component of the fibre product $Y_1\times_Y Y_2$, we may find a third resolution of singularities of $Y$, dominating $f_i$:$$\begin{tikzcd} \tilde{Y}_3\arrow[d,"f_3"]\arrow[r] &\tilde{Y}_1\times_Y \tilde{Y}_2\arrow[dl] \\Y \end{tikzcd}$$ Then for any choice of split projection ${f_3}_*\mathbf{1}\rightarrow \mathscr{E}_S(Y)$, we have the following commutative diagram for $i\in \{1,2\}$: $$\begin{tikzcd} \mathbf{1}\arrow[r]&{f_i}_*{\mathbf{1}}\arrow[r]&{f_3}_*\mathbf{1}\arrow[d]\\ &\mathscr{E}_S(Y)\arrow[u]\arrow[r,dashed,"\beta_i"]&\mathscr{E}_S(Y) \end{tikzcd}$$ These maps $\beta_i$ defined as the composition are isomorphisms by Theorem [Theorem 47](#thm:main1){reference-type="ref" reference="thm:main1"}, so their composite $\beta_2^{-1}\circ \beta_1$ gives the desired isomorphism. ◻ *Remark 62*. In the case of $\mathbb{Q}$-constructible coefficients, the geoemtrically pure (resp geometrically non-pure) is precisely the pure (resp non-pure) cohomology in the mixed Hodge structure on $H^*(Y,\mathbb{Q})$. To see this, recall that the mixed Hodge structure on a singular, projective variety is given by resolving $Y$ by a smooth simplicial hypercover, and the pure component is the first quotient of the associated spectral sequence [@DeligneHodgeIII]. **Example 63**. Let us consider the geometric extension on the space $\mathbb{A}^n_{\mathbb{C}}/\pm 1$ with constructible coefficients over a field $k$ of characteristic two. We will show that the geometric extension over $k$ is exactly $\pi_*\mathbf{1}$ for a resolution $\pi$ that contracts a divisor over $0$. We will thus have nonzero cohomology in degree $2(n-1)$ in the stalk over $0$. This gives the geometric consequence that any resolution of singularities of this space must contract a divisor, by Corollary [Corollary 48](#ineq){reference-type="ref" reference="ineq"}.[^9] Consider the following diagram of blowups and quotients: $$\begin{tikzcd} \text{Bl}_0(\mathbb{A}_\mathbb{C}^n)\arrow[r,"\simeq"]\arrow[d,"\tilde{\pi}"]\arrow[dr,"\simeq"]&\mathbb{P}_\mathbb{C}^{n-1}\\ \mathbb{A}_\mathbb{C}^{n}\arrow[dr]&\text{Bl}_0(\mathbb{A}_\mathbb{C}^{n})/\pm 1\arrow[u,"\simeq"]\arrow[d,"\pi"]&\mathbb{P}_\mathbb{C}^{n-1}\arrow[l]\arrow[d]\\ &\mathbb{A}_\mathbb{C}^{n}/\pm 1&\{0\}\arrow[l] \end{tikzcd}$$ The space $\text{Bl}_0(\mathbb{A}_\mathbb{C}^n)$ via its projection map to $\mathbb{P}_\mathbb{C}^{n-1}$ is the total space of the tautological bundle, and this quotient $\text{Bl}_0(\mathbb{A}_\mathbb{C}^{n})/\pm 1$ is obtained by taking the quotient under the inversion map on the (vector space) fibres. So we see the maps with $\simeq$ are homotopy equivalences of topological spaces, and $\text{Bl}_0(\mathbb{A}_\mathbb{C}^n)/\pm 1$ is smooth. So we obtain a resolution of the singular space $\mathbb{A}_\mathbb{C}^n/\pm 1$, with domain homotopic to $\mathbb{P}_\mathbb{C}^{n-1}$. On our base, $0$ is the unique singular point, and the fibre over this singular point in this resolution is $\mathbb{P}_\mathbb{C}^{n-1}$. Its complement is $\mathbb{A}_\mathbb{C}^n-\{0\}/\pm1$, which is naturally homeomorphic to the space $\mathbb{RP}^{2n-1}\times \mathbb{R}$. We claim that the geometric extension is just the pushforward $\pi_*\mathbf{1}$. To show this, we need to check that this sheaf is indecomposable, which is equivalent to showing that it has no skyscraper summands at the singular point. To check this, consider the compactly supported cohomology of the open-closed triangle for the inclusion of the singular point:$$j_!j^!\pi_*\mathbf{1}\rightarrow \pi_*\mathbf{1}\rightarrow i_*i^*\pi_*\mathbf{1}\xrightarrow{+1}$$ By base change, the ompactly supported cohomology of $j_!j^!\pi_*\mathbf{1}$ is the compactly supported cohomology of $\mathbb{A}_\mathbb{C}^n-\{0\}/\pm1\simeq \mathbb{RP}^{2n-1}\times \mathbb{R}$. Similarly, by base change, the sheaf $i_*i^*\pi_*\mathbf{1}$ computes the cohomology of the fibre, which is $\mathbb{P}_\mathbb{C}^{n-1}$. The middle term computes the compactly supported cohomology of the total space, which is a complex line bundle over $\mathbb{P}_\mathbb{C}^{n-1}$, and so gives the cohomology of $\mathbb{P}^{n-1}_\mathbb{C}$, shifted by $2$. Applying the compactly supported cohomology functor $\mathop{\mathrm{\mathcal{H}om}}^*_{k}(\mathbf{1},t_!\_)$ yields an exact triangle: $$\begin{tikzcd} H^*_!(\mathbb{RP}^{2n-1}\times \mathbb{R},k)\arrow[r]&H^{*-2}(\mathbb{P}_\mathbb{C}^{n-1},k)\arrow[d]\\ &H^*(\mathbb{P}_\mathbb{C}^{n-1},k)\arrow[ul,"+1"] \end{tikzcd}$$ Since the characteristic of $k$ is two, $H^*_!(\mathbb{RP}^{2n-1}\times \mathbb{R})$ is nonzero in all degrees between $1$ and $2n$ inclusive. So this $+1$ degree map must be injective by the parity of the cohomology of $\mathbb{CP}^{n-1}$. Thus, this extension is maximally nonsplit and there can be no skyscraper sheaf summand. So this $\pi_*\mathbf{1}$ is indecomposable in characteristic two, giving the desired nonzero cohomology in the stalk. (This may also be seen using intersection forms (see [@JMW §§3.2-3.3]). The refined intersection form is identically zero modulo 2.) The previous example shows that geometric extensions for the constructible base change formalism over fields need not be a perverse, and by similar ideas we obtain the following geometric consequence. **Proposition 64**. *If for some field $k$, the geometric extension of $Y$ over $k$ is not perverse up to shift in $D^b_c(Y,k)$, then $Y$ does not admit a semismall resolution.* *Proof.* For such an $Y$, the geometric extension is a summand of $\pi_*\mathbf{1}_{\tilde{Y}}$ for any resolution $\pi:\tilde{Y}\rightarrow Y$. Since the geometric extension is assumed to not be perverse, no resolution can be semismall. ◻ The following is an immediate corollary of Theorem [Theorem 52](#thm:main2){reference-type="ref" reference="thm:main2"}, though we suspect there is a more direct way to see the result. **Proposition 65** (Zariski trivial is cohomologically trivial.). *Let $f:X\rightarrow Y$ be a smooth proper morphism between smooth varieties. Then if $f$ is Zariski locally trivial, then $f_*\mathbf{1}_X$ is the trivial local system on the fibre for any smoothly orientable base change formalism.* *Proof.* Let $F$ be a fibre of this morphism. Then if $f^{-1}(U)\cong F\times U$, then $f_*\mathbf{1}_X$ is isomorphic to the geometric extension of the constant $F$ local system over $U$. But the trivial family $F\times X\rightarrow X$ also gives the geometric extension, giving the result. ◻ *Remark 66*. By a similar argument, if $f$ is étale locally trivial, then $f_*\mathbf{1}$ is trivial in any base change formalism with coefficients of characteristic zero. By Proposition [Proposition 57](#ICprop){reference-type="ref" reference="ICprop"}, over $\mathbb{Q}$, the geometric extension $\mathscr{E}_{\mathbb{Q}}(Y,\mathscr{L})$ is determined by the $\mathbb{Q}$ local system $\mathscr{L}_*\mathbf{1}_V$ on $U$ within $Y$, being isomorphic to $IC(Y,\mathscr{L}_*\mathbf{1}_V)$. The following example shows that this is exceptional behaviour, and that geometric extensions in general are not determined by the $S$-local systems $\mathscr{L}_*\mathbf{1}_V$. They require the map. **Example 67** (The Legendre family of elliptic curves). Consider the following projective family $E_t$ of elliptic curves $$\begin{tikzcd} X\arrow[d,"\pi"]\arrow[r]&\mathbb{A}^1_{\mathbb{C}}\times \mathbb{P}_{\mathbb{C}}^2\arrow[dl,"\text{pr}"]\\ \mathbb{A}^1_{\mathbb{C}} \end{tikzcd}$$ given by: $$Y^2Z=X(X-Z)(X-tZ).$$ Here $t$ is the coordinate on $\mathbb{A}^1_\mathbb{C}$, and we view this family inside $\mathbb{A}^1_\mathbb{C}\times\mathbb{P}^2_{\mathbb{C}}$. This family is smooth away from $t\in\{0,1\}$. The total space of this family has two isolated singular points, the nodes of the nodal cubics in the fibres over $t=0$ and $t=1$. These singular points have tangent cone isomorphic to the cone on a smooth conic. Blowing up these two singular points resolves the singularities, giving the resolved family $$\tilde{X}\xrightarrow{\tilde{\pi}}\mathbb{A}^1_\mathbb{C}.$$ For this new map, the fibres over $t\in\{0,1\}$ are each the union of two rational curves intersecting in two points transversely. (This is type $I_2$ in Kodaira's classification of elliptic fibres [@Kodaira]. This is often called the "double banana" configuration.) So for the constructible base change formalism with coefficients in $k$, the stalks $i^*_t$ of $\tilde{\pi}_*\mathbf{1}$ are given by: $$\begin{tabular}{ p{2.5cm}||p{1cm} p{1cm} p{1cm} } $H^*$& 0 &1&2\\ \hline $i_0^*\tilde{\pi}_*\mathbf{1}$& $k$ &$k$&$k^{\oplus 2}$\\ $i_1^*\tilde{\pi}_*\mathbf{1}$& $k$ &$k$ &$k^{\oplus 2}$\\ $i_t^*\tilde{\pi}_*\mathbf{1}$ if $t\neq 0,1$&$k$ & $k^{\oplus 2}$& $k$\\ \end{tabular}$$ The monodromy of this family is nontrivial only in the middle degree: $$H^1(E_t,k)\cong k^2.$$ One may then compute (see e.g. [@CMP Part 1]) that the monodromy of a small loop around either singular fibre over $\mathbb{Z}$ is similar to $$\begin{bmatrix} 1 & 2\\ 0 & 1 \end{bmatrix}.$$ Thus, we observe that this monodromy action is trivial if the characteristic of $k$ is two, and in this case the associated local system on $\mathbb{A}^1_\mathbb{C}\setminus \{0,1\}$ is trivial. We note however that the geometric extension of this local system is always nontrivial, as a trivial local system $E\times \mathbb{A}^1_\mathbb{C}$ has two copies of $k[-1]$ in its stalk over its singular fibres, rather that the one copy in our family. This example therefore shows that the geometric extension of a geometric local system cannot be deduced from just the knowledge of $\tilde{\pi}_*\mathbf{1}$ restricted to the open subset $\mathbb{A}^1_\mathbb{C}\setminus \{0,1\}$. This example also lets us observe the failure of the local invariant cycle theorem in characteristic $p$, as the specialisation map $$H^1(E_0)\rightarrow H^1(E_t)^\mu$$ is not surjective. *Remark 68*. One may construct families of counterexamples as follows. Let $\pi:X\rightarrow \mathbb{A}^1$ be proper with smooth source, smooth over $\mathbb{A}^1\setminus \{0\}$, such that the $n^{th}$ power base change of $\pi$ has smooth total space $\tilde{X}_n$. $$\begin{tikzcd} \tilde{X}_n\arrow[r]\arrow[d,"\pi_n"]&X\arrow[d,"\pi"]\\ \mathbb{A}^1\arrow[r,"z\mapsto z^n"]&\mathbb{A}^1 \end{tikzcd}$$ Then the associated monodromy representation of $\pi_n$ has monodromy the $n$th power of the monodromy of $\pi$. This allows one to trivialise the monodromy if the order is finite, dividing $n$. Another feature of the decomposition theorem in the $\mathbb{Q}$ coefficient setting is that the pushforward $f_*\mathbf{1}_X$ is semisimple. This fails for more general coefficients, and can already be seen with smooth, projective maps with mod $p$ coefficients. This result and its proof don't require geometric extensions, but we've decided to include it as we are not aware of any examples of this phenomenon in the literature. **Example 69** (A non-semisimple geometric local system). Let $S$ be the $\mathbb{F}_2$ constructible formalism, and let $\pi:E\rightarrow X$ be an algebraic (étale local) $\mathbb{P}^1_{\mathbb{C}}$ bundle over a smooth space $X$, with nontrivial Brauer class in $H^3(X,\mathbb{Z})$. For instance, one may take the tautological bundle over an algebraic approximation of the classifying space $BPGL_2(\mathbb{C})$ (see e.g. [@BL]). Then $\pi_*\mathbf{1}_E$ is an extension of $\mathbf{1}_X$ by $\mathbf{1}_X[2]$, classified by an element of $$\text{Ext}^1(\mathbf{1}_X,\mathbf{1}_X[2])=H^3(X,\mathbb{F}_2).$$ This element is the reduction modulo $2$ of the associated Brauer class in $H^3(X,\mathbb{Z})$, so does not vanish in this quotient, and gives the desired indecomposable local system. One may construct counterexamples more generally using the fact that if $f:X\rightarrow Y$ is any map, and the induced map $H^*(Y,\mathbb{F}_p)\rightarrow H^*(X,\mathbb{F}_p)$ is not injective, then $\mathbf{1}\rightarrow f_*\mathbf{1}$ cannot be split injective in $D^b_c(X,\mathbb{F}_p)$. Thus far in this section we have been considering applications for the constructible derived category with field coefficients, but it is worth emphasising that there are other examples, which have been shown to be relevant to geometric representation theory.[^10] In particular, there is now an established theory, with a six functor formalism, for modules over any $A_\infty$ ring spectrum. Let's consider the ring spectrum $KU_p$, $p$ completed complex K theory. This is the ring spectrum that represents the cohomology theory $X\mapsto K^*(X)\otimes_\mathbb{Z} \mathbb{Z}_p$ on finite $CW$ complexes $X$, where $K^0(X)$ is the usual Grothendieck group of complex vector bundles on $X$. The formalism of $KU_p$ modules then gives rise to a smoothly orientable, Krull-Schmidt base change formalism for $p$ completed complex K theory $KU_p$, see Appendix [6.1](#app.KUp){reference-type="ref" reference="app.KUp"}. By Theorem [Theorem 52](#thm:main2){reference-type="ref" reference="thm:main2"}, we may then define the geometric extension for $p$ completed K theory. **Definition 70**. For an irreducible variety $Y$, the $KU_p$ geometric extension is the sheaf of $KU_p$ modules $\mathscr{E}_{KU_p}(Y)$. The geometric $K$-theory groups at $p$ are defined to be the homotopy groups of this indecomposable sheaf of $KU_p$ modules:$$\mathscr{E}^{*}_{KU_p}(Y):=\pi_{-*}(\mathscr{E}_{KU_p}(Y)).$$ We end with some natural questions regarding these geometric K groups. *Question 71*. We have natural maps $K^*(Y)\rightarrow \mathscr{E}^*_{KU_p}(Y)$ for all $p$, and by Lemma [Lemma 61](#comp){reference-type="ref" reference="comp"}, the kernel of these maps are independent of our choices. We might call elements in this common kernel nonpure classes in (integral) K-theory. Is there a geometric, vector bundle description of these classes? The nontorsion part will be visible as nonpure classes in $\mathbb{Q}$ cohomology, what about the torsion? One may also use the rationalised K theory spectrum $KU_\mathbb{Q}$ in the preceding definitions. In this case, the groups we obtain are just ordinary intersection cohomology, since in rational cohomology, geometric extensions are just intersection cohomology, and the Chern character gives an isomorphism of $E_\infty$ ring spectra $\text{ch}:KU_\mathbb{Q}\cong H_\mathbb{Q}$. Though the groups are not new, this alternate description of intersection cohomology via K theory leads to the natural question of whether these geometric extensions can be categorified. Our main result gives, for a fixed base space $Y$, for any resolution $X\rightarrow Y$, an idempotent endomorphism of $K^*(X)\otimes \mathbb{Q}$ which cuts out $\mathscr{E}_{KU_\mathbb{Q}}(Y)$, and this image is independent of the resolution. Now specialise to the case where $X$ and $Y$ admit compatible affine pavings, so $X\times_Y X$ also admits such a paving. This occurs for instance in the theory of Schubert varieties with Bott-Samelson resolutions. In this situation, the rationalised Grothendieck group of coherent sheaves on $X$ is isomorphic to the rationalised topological K group of $X$, via the Chern character to Chow groups. The fundamental classes of subvarieties of $X\times X$ naturally act as endomorphisms of $D^b_{coh}(X)$ as kernels of Fourier-Mukai transforms, and this naturally categorifies the action on $K^*(X)$. This leads to the following (imprecise) question. *Question 72*. Let $X$ be an affine paved resolution of $Y$. Does there exist an idempotent endofunctor $E_{X/Y}$ of $D^b_{coh}(X)$ such that the image of $E_{X/Y}$ is an invariant of $Y$, which decategorifies to the idempotent cutting out $\mathscr{E}_{KU_\mathbb{Q}}(Y)$ inside $K^*_\mathbb{Q}(X)$? Furthermore, is this category independent of the resolution $X$? # Appendix ## $KU_p$ modules {#app.KUp} Here we will give a short introduction to the smoothly orientable base change formalism of $KU_p$ modules. We will start with ordinary topological K theory. This is a cohomology theory built from the Grothendieck group of complex vector bundles over $X$: $$X\mapsto K^0(X):=Gr(Vec_{\mathbb{C}}/X).$$ This is the zeroth of a series of functors $K^i$, which form a cohomology theory in the sense of satisfying the Eilenberg-Steenrod axioms (except the dimension axiom). Bott's periodicity theorem ([@bott1959stable], also see [@hatcherVB]) implies that this cohomology theory is represented by a two periodic sequential spectrum $KU$ with component spaces: $$\begin{aligned} KU_{2i}&\cong \mathbb{Z}\times BU\\ KU_{2i+1}&\cong U.\end{aligned}$$ Here $U$ is the infinite unitary group, and $BU$ is the union of the infinite complex Grassmannians $BU(n)$. The tensor product on vector bundles gives a homotopy coherent commutative multiplication law on this spectrum, so $KU$ is naturally an $E_\infty$-ring. The (higher) coherence of this multiplication law allows one to define a well-behaved $\infty$-category of module spectra. This $\infty$-category is stable, so it can be thought of as an enhancement of its triangulated homotopy category. For any stable $\infty$-category $\mathcal{C}$, we have a notion of sheaves on a space $X$ valued in $\mathcal{C}$. We will not define this precisely, but in rough terms it gives an object for each open set, a morphism for each inclusion of open sets, a homotopy between the compositions for each pair of composable inclusions, and so on, such that an analogue of the sheaf condition holds. The $\infty$-category of such $\mathcal{C}$-valued sheaves on a space $X$ is stable. If one restricts to suitable, locally compact spaces with well-behaved maps between them, such as algebraic varieties with algebraic maps, then we obtain the whole six functor formalism for $\mathcal{C}$-valued sheaves, see e.g. [@VolpeM]. Furthermore, one may restrict to constructible $\mathcal{C}$-valued sheaves. Constructibility is then preserved under these six functors, due to the good topological properties of algebraic maps. We will not need the inner workings of this construction. The following example shows why such a formal black box can still be useful. **Example 73**. Consider Example [Example 63](#ex:A/pm){reference-type="ref" reference="ex:A/pm"}, interpreted within the K-theoretic framework. This whole example is formal, until we apply compactly supported cohomology to obtain the triangle: $$\begin{tikzcd} H^*_c(\mathbb{RP}^{2n-1}\times \mathbb{R},k)\arrow[r]&H^{*-2}(\mathbb{P}_\mathbb{C}^{n-1},k)\arrow[d]\\ &H^*(\mathbb{P}_\mathbb{C}^{n-1},k)\arrow[ul,"+1"] \end{tikzcd}$$ If we instead used K theory, we would obtain the triangle: $$\begin{tikzcd} K^*_c(\mathbb{RP}^{2n-1}\times \mathbb{R})\arrow[r]&K^{*-2}(\mathbb{P}_\mathbb{C}^{n-1})\arrow[d]\\ &K^*(\mathbb{P}_\mathbb{C}^{n-1})\arrow[ul,"+1"] \end{tikzcd}$$ Then one may show formally that the vertical arrow is multiplication by the Thom class of the $KU$ orientable line bundle $\mathscr{O}(2)$, and that compactly supported K-theory of a compact space times $\mathbb{R}$ is the ordinary K-theory shifted by one. Since the Thom class is $1-2[H]$, and we know the K theory of $\mathbb{CP}^{n-1}$, this lets us easily compute the K theory of $\mathbb{RP}^{2n-1}$. ## Localisation at $p$ We wish to work with Krull-Schmidt categories everywhere, so we need to localise the K theory base change formalism to obtain $KU_p$ modules. This is a formal procedure, essentially given on integral objects by tensoring with the $p$-adic integers $\mathbb{Z}_p$ every place one sees a K group. For instance, to build the associated cohomology theory, we simply tensor with $\mathbb{Z}_p$. That this preserves the property of being a cohomology theory is immediate from flatness of $\mathbb{Z}_p$ over $\mathbb{Z}$, so this functor gives the associated spectrum $KU_p$ representing it. We round out this section with a proof that $KU_p$ modules are a base change formalism on algebraic varieties. **Proposition 74**. *The base change formalism $Y\mapsto D^b_c(Y,KU_p)$ of sheaves of constructible $KU_p$ module spectra is a smoothly orientable Krull-Schmidt base change formalism on complex algebraic varieties, which satisfies condition (D).* *Proof.* First, the fact that this is a base change formalism entails many compatibilities which follow from the construction, and Lurie's proper base change theorem (Chapter 7, §3 of [@LurieHTT]). One may find a streamlined proof of these properties in [@VolpeM]. For orientability, note that orientability is just an existence statement for elements in $K^{BM}_{2d}(X)\otimes_\mathbb{Z} \mathbb{Z}_p$. In particular, this is implied by the orientability of integral K-theory on complex manifolds, see Example [Example 30](#3.9){reference-type="ref" reference="3.9"}. To check condition $(D)$, note that density holds trivially if the fibre bundle is trivial, and for any two points $x,y$ we may choose a contractible neighbourhood $U_{x,y}$ of $x$ and $y$. Restricting our geometric local system to $U_{x,y}$ gives a topologically trivial bundle, giving the density result. It remains to check the Krull-Schmidt property of this base change formalism. We first claim it suffices to check the following conditions[^11]. - The "ring of coefficients\" $\mathbb{Z}_p[t,t^{-1}]:=\mathop{\mathrm{End}}(\mathbf{1}_\ast)$ is a graded complete local ring. - For any $\mathscr{F},\mathscr{G}$ in $D^b_c(X,KU_p)$, the group $\mathop{\mathrm{\mathcal{H}om}}^*(\mathscr{F},\mathscr{G})$ is a finitely generated graded $\mathbb{Z}_p[t,t^{-1}]$ module. - The category $D^b_c(X,KU_p)$ has split idempotents. To see that these suffice, first note that the endomorphism algebra of any indecomposable object is a finite graded $\mathbb{Z}_p[t,t^{-1}]$ module by the second condition. The graded version of the idempotent lifting lemma (Corollary 7.5, [@eisenbud2013commutative]), and splitting of idempotents implies the endomorphism ring of any indecomposable object is local. The Krull-Schmidt property of functors then follows from the fact that for any finite morphism of local rings $\phi:R\rightarrow S$, we have $\phi(J(R))\subset J(S)$, which is immediate from Nakayama's Lemma. It remains to check that these conditions hold for sheaves of $KU_p$ modules. The first condition is immediate by Bott Periodicity, as these are the homotopy groups of $KU_p$. For the finiteness of the second condition, since $\mathbb{Z}_p[t,t^{-1}]$ is Noetherian, one may apply open closed decomposition triangles to reduce to the case of morphisms between locally constant $KU_p$ modules on a smooth variety. We then can find a finite good cover of contractible open sets trivialising these $KU_p$ modules, and an induction with the Mayer-Vietoris sequence gives the result. Finally, to check idempotent completeness, we may assume our sheaves of $KU_p$ modules are constructible with respect to a fixed stratification $\lambda$. For a fixed stratification $\lambda$, we have the associated exit path $\infty$-category $EP_{\lambda,\infty}(X)$, and we may identify the category of $\lambda$ constructible sheaves of $KU_p$ modules with the functor category $[EP_{\lambda,\infty}(X),KU_p\mathrm{-Perf}]$ (see Theorem A.9.3 [@LurieHA]). As $KU_p\mathrm{-Perf}$ is accessible, this functor category is accessible (see [@LurieHTT] Proposition 5.4.4.3), and thus since this functor $\infty$ category is small, accessibility is equivalent to idempotent completeness (see Corollary 5.4.3.6 [@LurieHTT]). ◻ ## Commuting diagrams In this section we prove the existence of the compatibility diagrams for the convolution isomorphism [Definition 19](#conviso){reference-type="ref" reference="conviso"}. This can be broken into two distinct parts, the more formally $2$-categorical Proposition [Proposition 75](#DIAGRAM){reference-type="ref" reference="DIAGRAM"}, and the orientation compatibility, Proposition [Proposition 76](#convisocomp){reference-type="ref" reference="convisocomp"}. Let us first recall the setup. Our space $Y$ is irreducible, with Zariski open set $U$, and $X,X'$ are two smooth spaces over $Y$. The following diagram will be our reference for the maps involved in the convolution isomorphism. $$\begin{tikzcd} X_U\times_U X'_U\arrow[dd,"\tilde{g}_U"]\arrow[rr,"\tilde{f}_U"]\arrow[dr,"\hat{j}"]&&X'_U\arrow[dd,"g_U" near start]\arrow[dr,"\underline{j}'"]&\\ &X\times_Y X'\arrow[dd,"\tilde{g}" near start]\arrow[rr,"\tilde{f}" near start]&&X'\arrow[dd,"g"]\\ X_U\arrow[rr,"f_U" near start]\arrow[dr,"\underline{j}"']&&U\arrow[dr,"j"]&\\ &X\arrow[rr,"f"]&&Y \end{tikzcd}$$ Our first proposition is the following: **Proposition 75**. *The following diagram commutes, where the horizontal maps are our convolution isomorphisms, and the vertical maps are restriction followed by base change: $$\begin{tikzcd} \mathop{\mathrm{\mathcal{H}om}}(f_!\_,g_*\_)\arrow[r,"\tau"]\arrow[d]&\mathop{\mathrm{\mathcal{H}om}}(\tilde{g}^*\_,\tilde{f}^!\_)\arrow[d]\\ \mathop{\mathrm{\mathcal{H}om}}(j^*f_!\_,j^*g_*\_)\arrow[d,"\simeq"]&\mathop{\mathrm{\mathcal{H}om}}(\hat{j}^*\tilde{g}^*\_,\hat{j}^*\tilde{f}^!\_)\arrow[d,"\simeq"]\\ \mathop{\mathrm{\mathcal{H}om}}({f_U}_!\underline{j}^*\_,{g_U}_*{\underline{j}'}^*\_)\arrow[r,"\tau"]&\mathop{\mathrm{\mathcal{H}om}}({\tilde{g}_U}^*\underline{j}^*\_,{\tilde{f}_U}^!{\underline{j}'}^*\_) \end{tikzcd}$$* Our orientation compatibility is the following: **Proposition 76**. *There exist orientations of $\overline{\Delta_\alpha}$ and $X\times_Y X'$ such that the following diagram exists and is commutative, where the fundamental class of $\overline{\Delta_\alpha}$ restricted to $X_U\times_U X'_U$ maps to $\alpha_*\mathbf{1}$ under the associated convolution isomorphism.* *$$\begin{tikzpicture}[xscale=2,yscale=1] \node (H0) at (0,0) {$\mathop{\mathrm{\mathcal{H}om}}({f_U}_*\mathbf{1},{g_U}_*\mathbf{1})$}; \node (H1) at (0,2) {$\mathop{\mathrm{\mathcal{H}om}}(f_*\mathbf{1},g_*\mathbf{1})$}; \node (S0) at (2,0) {$S_{*,2d}(X_U\times_U X'_U)$}; \node (S1) at (2,2) {$S_{*,2d}(X\times_Y X')$}; \draw[->] (H0) to node[above] {$\sim$} (S0); \draw[->] (H1) to node[above] {$\sim$} (S1); \draw[->] (H1) to (H0); \draw[->] (S1) to (S0); \node (ur) at (3,2) {$[\overline{\Delta_\alpha}]$}; \node (e1) at (2.72,2) {$\ni$}; \node (lr) at (3,-1) {$[\Delta_\alpha]$}; \node (e1) at (2.72,-.5) {\rotatebox{135}{$\in$}}; \node (l) at (0,-1) {$\alpha_*\mathbf{1}$}; \node (e1) at (0,-.6) {\rotatebox{90}{$\in$}}; \draw[|->] (ur) to (lr); \draw[|->] (l) to (lr); \end{tikzpicture}$$* For notational convenience, in the proof of this proposition we will use $(\_,\_)$ to denote morphism sets, and we will only use $f,g$ and $j$, noting that the decorations are uniquely determined by the location within the diagram. *Proof.* We first prove Proposition [Proposition 75](#DIAGRAM){reference-type="ref" reference="DIAGRAM"}. We may expand the diagram in Proposition [Proposition 75](#DIAGRAM){reference-type="ref" reference="DIAGRAM"} into the following: $$\begin{tikzcd} (f_!,g_*)\arrow[r]\arrow[d]&(\_,f^!g_*)\arrow[r]\arrow[d]&(\_,{g}_*{f}^!)\arrow[d]\arrow[r]&({g}^*,{f}^!)\arrow[d]\\ (j^*f_!,j^*g_*)\arrow[d]\arrow[r]&(\_,f^!j_*j^*g_*)\arrow[d]&(\_,{g}_*{j}_*{j}^*{f}^!)\arrow[r]\arrow[d]&({j}^*{g}^*,{j}^*{f}^!)\arrow[d]\\ ({f}_! {j}^*,{g}_* {{j}'}^*)\arrow[r]&(\_,{{j}}_* {f}^!{g}_* {{j}}^*)\arrow[r]& (\_ ,{{j}}_* {{g}}_* {{f}}^!{j}^*)\arrow[r]&({{g}}^*{j}^*,{{f}}^!{j}^*) \end{tikzcd}$$ Only the commutativity of the middle square is not standard, and its commutativity follows from the commutativity of the following diagram: $$\begin{tikzcd} {(\_,f^!g_*)} \arrow[rr] \arrow[d] & & {(\_,g_*f^!)} \arrow[d] \arrow[r] & {(\_,g_*j_*j^*f^!)} \arrow[dd] & \\ {(\_,f^!g_!)} \arrow[rr] \arrow[rd] \arrow[d] & & {(\_,g_!f^!)} \arrow[d] & & \\ {(\_,f^!j_*j^*g_!)} \arrow[r] \arrow[d] & {(\_,j_*j^*f^!g_!)} \arrow[r] \arrow[d] & {(\_,j_*j^*g_!f^!)} \arrow[d] \arrow[r] & {(\_,j_*g_*j^*f^!)} \arrow[rddd] \arrow[d] & \\ {(\_,j_*f^!j^*g_!)} \arrow[ru] \arrow[r] \arrow[d] & {(j^!,j^!f^!g_!)} \arrow[d] \arrow[r] & {(j^!,j^!g_!f^!)} \arrow[d] \arrow[r] & {(j^*,g_*j^!f^!)} \arrow[d] & \\ {(\_,j_*f^!g_!j^*)} \arrow[d] \arrow[rd] & {(j^!,f^!j^!g_!)} \arrow[d] & {(j^!,g_!j^!f^!)} \arrow[d] \arrow[r] & {(j^*,g_!j^!f^!)} \arrow[d] & \\ {(\_,j_*f^!g_*j^*)} \arrow[r] & {(j^!,f^!g_!j^!)} \arrow[r] & {(j^!,g_!f^!j^!)} \arrow[r] \arrow[rd] & {(j^*,g_!f^!j^!)} \arrow[d] & {(\_,j_*g_*f^!j^*)} \\ & & & {(\_,j_*g_!f^!j^!)} \arrow[ru] & \end{tikzcd}$$ The commutativity of the internal faces are all standard compatibilities of base change and naturality. ◻ *Remark 77*. A general statement proving commutativity of diagrams of this type is to be found in the thesis of the first author [@Hone-thesis]. It remains to prove Proposition [Proposition 76](#convisocomp){reference-type="ref" reference="convisocomp"}. This entails proving the commutativity of the diagram, and for compatible choices of orientation, that the restriction of $[\overline{\Delta_\alpha}]$ to $S_{2d_X}^!(X_U\times_U X'_U)$ corresponds to $\alpha_*\mathbf{1}$ under the convolution isomorphism. To show the existence of this diagram, we evaluate on the constant sheaf, and use the chosen orientation $\mathbf{1}\xrightarrow{\gamma}\omega_X[-2d]$ of $X$ to give the identifications with $S_{*,2d}(X\times_Y X')$.\ -2.2cm0pt Here the commutativity of the second square is Proposition [Proposition 75](#DIAGRAM){reference-type="ref" reference="DIAGRAM"}. Thus it remains to prove the identification of the fundamental class along this isomorphism. **Proposition 78**. *Given an isomorphism $\alpha$ over $U$, such that $X_U,X_U'$ are smooth: $$\begin{tikzcd} X_U\arrow[rr,"\alpha"]\arrow[dr,"f"']&&X_U'\arrow[dl,"g"]\\ & U \end{tikzcd}$$ Then for a choice of orientation $\gamma$ of $X_U$, and induced compatible orientation of $\Delta_\alpha$, the element $[\Delta_\alpha]$ in $H_{2d}(X_U\times_U X_U')$ corresponds to $\alpha_*\mathbf{1}$ in $\mathop{\mathrm{\mathcal{H}om}}(f_*\mathbf{1},g_*\mathbf{1})$ via the convolution isomorphism induced by $\gamma$. We may also assume that this orientation of $\Delta_\alpha$ arises as restriction from an orientation of $\overline{\Delta_\alpha}$.* *Proof.* We first construct the orientations required by choosing an orientation of $\overline{\Delta_\alpha}$ via resolution of singularities. We may then restrict this to $\Delta_\alpha$, and transport structure to $X_U$ to get our desired compatible orientations. From here all convolution morphisms and fundamental classes are with respect to this choice. First, let's show that this fundamental class morphism $[\Delta_\alpha]:\mathbf{1}\rightarrow \omega_{X_U\times_U X'_U}[-2d]$ is isomorphic (after whiskering) to an evaluation on the constant sheaf of a morphism of functors. That is, there are canonical morphisms of functors $$\tilde{g}^*\rightarrow\Delta_*\rightarrow \Delta_!\rightarrow \tilde{f}^!$$ such that the following diagram commutes, and the composite is the fundamental class in $H_{2d}(X_U\times_U X'_U)$: $$\begin{tikzcd} \mathbf{1}\arrow[r]\arrow[dr]&\tilde{g}^*\mathbf{1}\arrow[r]\arrow[d]&\Delta_!\mathbf{1}\arrow[r]&\tilde{f}^!\mathbf{1}\arrow[dr]\\ &\Delta_*\mathbf{1}\arrow[ur]\arrow[r]&\Delta_!\omega_{\Delta}[-2d]\arrow[u]\arrow[r]&\tilde{f}^!\omega_{\Delta}[-2d]\arrow[u]\arrow[r]&\omega_{X_1\times X_2}[-2d] \end{tikzcd}$$ The commutativity of the rightmost triangle follows from the fact that our orientations were chosen compatibly. This reduces the problem to a coherence problem for pseudofunctors, so consider the following diagram. $$\begin{tikzcd} \text{Id}\arrow[r]\arrow[d,equal]&\tilde{g}_*{\Delta_{\alpha}}_*\arrow[r]&\tilde{g}_*{\Delta_{\alpha}}_!\arrow[r]\arrow[d]&\tilde{g}_*\tilde{f}^!\arrow[d]\\ \text{Id}\arrow[d]\arrow[rr]&&\tilde{g}_!{\Delta_{\alpha}}_!\arrow[d]\arrow[r]&\tilde{g}_!\tilde{f}^!\arrow[d]\\ f^!f_!\arrow[d]\arrow[rr]&&f^!f_!\tilde{g}_!\Delta_{{\alpha}!}\arrow[d]\arrow[r]&f^!f_!\tilde{g}_!\tilde{f}^!\arrow[d]\\ f^!g_!\arrow[d,equal]\arrow[dr]&&f^!f_!\tilde{g}_!\Delta_!\Delta^!\tilde{f}^!\arrow[dl]\arrow[d]\arrow[ur]&f^!g_!\tilde{f}_!\tilde{f}^!\arrow[d]\\ f^!g_!\arrow[r]\arrow[rrr,equal,bend right]&f^!g_!\tilde{g}_!\Delta_!\Delta^!\tilde{f}^!\arrow[r]&f^!g_!\tilde{f}_!\Delta_!\Delta^!\tilde{f}^!\arrow[r]\arrow[ur]&f^!g_!\arrow[r]&f^!g_* \end{tikzcd}$$ Going clockwise around the diagram gives the mate of $[\Delta_\alpha]$, by the previous discussion, while going anticlockwise yields the mate of the morphism $f_!\xrightarrow{\alpha_!} g_!\rightarrow g_*$ which equals the composite $f_!\rightarrow f_*\xrightarrow{\alpha_*}g_*$, giving the desired compatibility. All squares in this diagram commute by naturality, or using that $S_!$ is a pseudofunctor. Only the curved identity morphism is not immediate, this is $f^!g_!$ applied to the following diagram:$$\begin{tikzcd} \mathop{\mathrm{\text{Id}}}\arrow[r]\arrow[dr]\arrow[ddrr,bend right,equal]&\tilde{g}_!\Delta_!\Delta^!\tilde{f}^!\arrow[r]&\tilde{f}_!\Delta_!\Delta^!\tilde{f}^!\arrow[d]\\ &(\tilde{f}\circ \Delta)_!(\tilde{f}\circ \Delta)^!\arrow[dr]\arrow[ur]&\tilde{f}_!\tilde{f}^!\arrow[d]\\ &&\mathop{\mathrm{\text{Id}}} \end{tikzcd}$$ This then commutes by the definition of the horizontal isomorphisms, and the general unit compatibilities of pseudofunctors. ◻ [^1]: After having discovered this statement and its proof, we became aware of the McNamara's paper [@PM §5] where a statement equivalent to one of our main theorems is proved. Our proof is almost identical to McNamara's. [^2]: The expert will miss the adjective "of geometric origin" in this discussion. Everything we discuss in this paper will be "of geometric origin". [^3]: $\infty$-categories are not needed in any arguments in this paper. However, $\infty$-categories are needed in providing the input (the "base change formalism") with which we work. [^4]: For an excellent articulation of this question, see [@DanP]. [^5]: i.e. *even* in the language of [@JMW §2.4] [^6]: There are significant point set topological requirements for the existence of the whole formalism, they are satisfied in our case since our spaces are locally compact and conically stratifiable, and all maps are suitably stratifiable. For a fixed stratification, see Lurie [@LurieHA], and for the functors see Volpe [@VolpeM]. We aren't aware of a source for constructibility in this generality, though this should follow same lines as the constructibility proofs in [@kashiwara2002sheaves]. [^7]: We have opted for this suggestive notation to encourage the parallel with the constructible derived category. [^8]: We learnt this argument from Roman Bezrukavnikov. [^9]: As explained to us by Burt Totaro, this may also be easily seen algebro-geometrically by the fact that our space $\mathbb{A}^n/\pm 1$ is $\mathbb{Q}$-factorial, as follows. Let $X\xrightarrow{\pi} \mathbb{A}^n/\pm 1$ be a resolution of singularities, and $D$ a chosen very ample Weil divisor on $X$. As $\mathbb{A}^n/\pm 1$ is $\mathbb{Q}$ factorial, a positive multiple $n\pi_*(D)$ of the Weil divisor $\pi_*(D)$ is Cartier. Pulling this back gives the Cartier divisor $\pi^*(n\pi_*(D))$ on $X$. If our exceptional fibre has codimension at least $2$, then this divisor on $X$ would be $nD$, but then $D$ cannot be very ample, as its sections don't separate points in the exceptional fibre. [^10]: The most famous examples are Kazhdan and Lusztig's computation of the equivariant K-theory of the Steinberg variety [@KL] and Nakajima's computation of the equivariant K-theory of quiver varieties [@Nakajima]. Note that both computations can be interpreted as the computation of an endomorphism of a direct image with K-theory coefficients. [^11]: This is slightly different to the conditions in §[4](#sec-KS){reference-type="ref" reference="sec-KS"}, though the proof is the same.

arxiv_math

--- abstract: | We investigate the non-relativistic limit of the Cauchy problem for the defocusing cubic nonlinear Klein-Gordon equations whose initial velocity contains a factor of $c^2$, with $c$ being the light speed. While the classical WKB expansion is applied to approximate these solutions, the modulated profiles can be chosen as solutions to either a Schrödinger-wave equation or a Schrödinger equation. We show that, as the light speed tends to infinity, the error function is bounded by, (1) in the case of 2D and modulated Schrödinger-wave profiles, $Cc^{-2}$ with $C$ being a generic constant uniformly for all time, under $H^2$ initial data; (2) in the case of both 2D and 3D and modulated Schrödinger profiles, $c^{-2} +(c^{-2}t)^{\alpha/4}$ multiplied by a generic constant uniformly for all time, under $H^\alpha$ initial data with $2 \leq \alpha \leq 4$. We also show the sharpness of the upper bounds in (1) and (2), and the required minimal regularity on the initial data in (2). One of the main tools is an improvement of the well-known result of Machihara, Nakanishi, and Ozawa in [@MaNaOz-KG-Limits] which may be of interest by itself. The proof also relies on *a fantastic complex expansion* of the Klein-Gordon equation, *introducing the leftward wave and exploring its enhanced performance* and a *regularity gain mechanism* through a high-low decomposition. address: - | (Z. Lei) School of Mathematical Sciences; LMNS and Shanghai Key Laboratory for Contemporary Applied Math- ematics\ Fudan University\ Shanghai 200433, China - | (Y. Wu) Center for Applied Mathematics\ Tianjin University\ Tianjin 300072, China author: - Zhen Lei - Yifei Wu title: Non-relativistic limit for the cubic nonlinear Klein-Gordon equations --- [^1] # Introduction Consider the defocusing cubic nonlinear Klein-Gordon equation: $$\begin{aligned} \begin{split} \label{KGo} \frac{\hbar^2}{2mc^2}\partial_{tt} u - \frac{\hbar^2}{2m}\Delta u +\frac{mc^2}{2}u= - |u|^2 u, \end{split}\end{aligned}$$ where $u(t,x):{\mathbb R}\times{\mathbb R}^d\rightarrow {\mathbb R}$ ($d = 1, 2, 3$) is a real-valued wave function describing neutral (pseudo-) scalar particles, $c$ the speed of light, $\hbar$ the Planck constant, and $m>0$ the rest mass of particles. When $u$ is complex it describes charged particles. In quantum field theory, the Klein-Gordon equation is one of the most fundamental equations describing spinless scalar or pseudo-scalar particles, for example, $\pi$- and $K$-mesons. It is relativistically-invariant and can be considered as the relativistic version of Schrödinger equation. A natural and major mathematical problem in relativistic quantum mechanism is to study the non-relativistic limit of solution to [\[KGo\]](#KGo){reference-type="eqref" reference="KGo"}. In particular, one hopes to have accurate and detailed descriptions on the convergence of solutions as the light speed goes to infinity, figure out the necessary price in terms of how much minimal regularity of the initial data is needed, and explore the underlying mechanism behind all of this. In this paper, we aim to settle these questions systematically, based on several new insights and earlier important theoretical and numerical works which will be explained below. ## Set-up and A Formal WKB Expansion Without loss of generality, we will set $\hbar=2, m=2$ and $c=\varepsilon^{-1}$ in the rest of the paper. Then the defocusing cubic nonlinear Klein-Gordon equation [\[KGo\]](#KGo){reference-type="eqref" reference="KGo"} is reduced to $$\begin{aligned} \begin{split} \label{eq:kg-eps} &\varepsilon^2\partial_{tt} u^\varepsilon - \Delta u^\varepsilon + \frac{1}{\varepsilon^2} u^\varepsilon= - |u^\varepsilon|^2 u^\varepsilon. \end{split}\end{aligned}$$ The equation [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"} is imposed with the following initial data $$\begin{aligned} u^\varepsilon(0,x) = u_0(x),\quad \partial_tu^\varepsilon(0,x) = \frac{u_1(x)}{\varepsilon^2},\end{aligned}$$ Note that this form of initial data is typical in view of the trivial space-independent planar waves of linear equations. In the context of the study of non-relativistic limits, see for instance, [@BaoCai-KG-Limit-12; @MasNak-KG-Limits] and the references therein, one employs the WKB ansatz $$\begin{aligned} \label{WKB} u^\varepsilon=\mathrm{e}^{\frac{it}{\varepsilon^2}}v^\varepsilon+\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v^\varepsilon+r^\varepsilon. \end{aligned}$$ This ansatz serves to eliminate the pronounced time oscillations in the wave function. The profile function $v^\varepsilon$ is usually called modulated wave function of $u^\varepsilon$. It is worth mentioning that in the case of a complex-valued wave function, $v^\varepsilon$ and $\bar v^\varepsilon$ can be replaced by the electron component $v^-$ and the positron component $v^+$, respectively, which are not necessarily required to be conjugated with each other, as demonstrated in [@MasNak-KG-Limits]. Then *the main mathematical issue* is to study the real error $u^\varepsilon - \big(\mathrm{e}^{\frac{it}{\varepsilon^2}}v^\varepsilon+\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v^\varepsilon\big)$, or equivalently, the remainder function $r_\varepsilon$, in the non-relativistic limit $\varepsilon \to 0$, including its estimates in terms of a power of $\varepsilon$, the necessary regularity on the initial data, the validation time interval and the underlying mechanism. Substituting [\[WKB\]](#WKB){reference-type="eqref" reference="WKB"} into [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"}, we have $$\begin{aligned} \label{eq:r} &\varepsilon^2\partial_{tt} r^\varepsilon - \Delta r^\varepsilon + \frac{1}{\varepsilon^2} r^\varepsilon +A_\varepsilon(v^\varepsilon,r^\varepsilon)+B_\varepsilon(v^\varepsilon)\notag\\ &\quad + \mathrm{e}^{\frac{it}{\varepsilon^2}}\Big[\varepsilon^2\partial_{tt}v^\varepsilon+2i\partial_{t} v^\varepsilon - \Delta v^\varepsilon+ 3|v^\varepsilon|^2 v^\varepsilon\Big]\notag\\ &\quad+\mathrm{e}^{-\frac{it}{\varepsilon^2}}\Big[\varepsilon^2\partial_{tt}\bar v^\varepsilon-2i\partial_{t}\bar v^\varepsilon - \Delta\bar v^\varepsilon+ 3|v^\varepsilon|^2 \bar v^\varepsilon\Big]=0, \end{aligned}$$ where $$\begin{aligned} A_\varepsilon(v^\varepsilon,r^\varepsilon)=\,&3\Big[\mathrm{e}^{\frac{2it}{\varepsilon^2}}\big(v^\varepsilon\big)^2+\mathrm{e}^{-\frac{2it}{\varepsilon^2}}\big(\bar v^\varepsilon\big)^2\Big]r^\varepsilon + 3\Big[\mathrm{e}^{\frac{it}{\varepsilon^2}}v^\varepsilon+\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v^\varepsilon\Big](r^\varepsilon)^2+(r^\varepsilon)^3;\notag\\ B_\varepsilon(v^\varepsilon)=\,&\mathrm{e}^{\frac{3it}{\varepsilon^2}}(v^\varepsilon)^3+\mathrm{e}^{-\frac{3it}{\varepsilon^2}}\big(\bar v^\varepsilon\big)^3.\label{def:B-var} \end{aligned}$$ Note that we have the following relations on the initial data $$\begin{aligned} \begin{split} \label{eq:initial-data} \left\{ \aligned & 2{\mathrm{Re}}{ [v^\varepsilon(0)]}+r^\varepsilon(0)=u_0, \\ &- \frac{2{\mathrm{Im}}{ [v^\varepsilon(0)]}}{\varepsilon^2}+2{\mathrm{Re}}\big[\partial_tv^\varepsilon(0)\big]+\partial_tr^\varepsilon(0)=\frac{u_1}{\varepsilon^2}. \endaligned \right. \end{split}\end{aligned}$$ There are two different kinds of quantum equation which governs the dynamics of the modulated profile $v^\varepsilon$. In case I, one formally sets $v^\varepsilon = {\rm Modulate}_w(u^\varepsilon)$ to be the solution of the following Cauchy problem for the defocusing cubic nonlinear *Schrödinger-wave equation*: $$\begin{aligned} \begin{split} \label{eq:nls-wave} \left\{ \aligned &\varepsilon^2\partial_{tt}v^\varepsilon+2i\partial_{t} v^\varepsilon - \Delta v^\varepsilon= - 3|v^\varepsilon|^2 v^\varepsilon, \\ & v^\varepsilon(0,x) = v_0(x)\triangleq \frac 12(u_0-iu_1),\quad \partial_tv^\varepsilon(0,x)=v_1(x), \endaligned \right. \end{split}\end{aligned}$$ with any $v_1$ independence of $\varepsilon$. In this case, the initial data for $r^\varepsilon$ is $$\begin{aligned} r^\varepsilon(0)=0,\quad \partial_t r^\varepsilon(0)=-2{\mathrm{Re}}(v_1),\end{aligned}$$ In case II, one formally sets $v^\varepsilon = v = {\rm Modulate}_s(u^\varepsilon)$ to be the solution of the following Cauchy problem for the defocusing cubic nonlinear Schr${\rm \ddot{o}}$dinger equation [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}: $$\begin{aligned} \begin{split} \label{eq:nls} \left\{ \aligned &2i\partial_{t} v - \Delta v= - 3 |v|^2 v, \\ & v(0,x) = v_0(x), \endaligned \right. \end{split}\end{aligned}$$ The initial data for $r^\varepsilon$ is given by: $$\begin{aligned} r^\varepsilon(0)=0,\quad \partial_t r^\varepsilon(0)=-2\mathrm{Re}\big[\partial_tv(0)\big].\end{aligned}$$ Expansion [\[WKB\]](#WKB){reference-type="eqref" reference="WKB"} in this case is usually called *Modulated Fourier expansion*. ## Main Results In this subsection we will present our main results and put them in context. The main new observations, ingredients and techniques will be explained in the subsequent subsection. Let us first consider the case when the modulated wave profile satisfies Schrödinger-wave equation [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}. There are extensive numerical investigations which have yielded error estimates of $r^\varepsilon$ and the length of the time interval in which the estimates remain valid. A selection of such studies can be found in [@BaoCai-KG-Limit-12; @BaoCai-KG-Limit-14; @BaoZh-KG-Limit; @SchratzZhao-KG-Limits]. We emphasis that the recent numerical result by Bao and Zhao in [@BaoZh-KG-Limit] involves a meticulous numerical verification showcasing that the error function $r_\varepsilon$ exhibits a convergence with error $O(\varepsilon^2)$, uniformly and golbally in time, as $\varepsilon$ goes to 0. This convergence phenomenon is confirmed for smooth initial data $(u_0,u_1)$ with a minimum requirement of $H^2$ regularity. Regrettably, despite these compelling numerical observations, to the best of our knowledge, a rigorous analytic study of this phenomenon remains conspicuously absent within the current body of literature. The first two theorems of this paper aim to establish the non-relativistic limit when the modulated wave profile satisfies Schrödinger-wave equation [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}. **Theorem 1**. *Let $d=2$. Suppose that $$(u_0,u_1)\in H^2({\mathbb R}^2) \times H^2({\mathbb R}^2);\quad v_1\in L^2({\mathbb R}^2).$$ Let $u^\varepsilon, v^{\varepsilon}$ be the corresponding solutions to the focusing cubic nonlinear Klein-Gordon equation [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"} and the Schrödinger-wave equation [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}, respectively. Then the following error estimate $$\begin{aligned} \label{est:main1-global} \sup\limits_{t\in {\mathbb R}}\big\|u^\varepsilon(t)-\mathrm{e}^{\frac{it}{\varepsilon^2}}v^{\varepsilon}(t)-\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v^{\varepsilon}(t) \big\|_{L^2_x} \le C \varepsilon^2\end{aligned}$$ holds true globally in time. Here $C$ is a generic constant depending only on $\|(u_0,u_1)\|_{H^2\times H^2}$ and $\|v_1\|_{L^2_x}$.* *Remark 2*. In [@BaoCai-KG-Limit-12; @BaoCai-KG-Limit-14; @BaoZh-KG-Limit; @SchratzZhao-KG-Limits], the following expression for $v_1$ is imposed $$v_1=\frac i2\big[-\Delta v_0+3|v_0|^2v_0\big],$$ for the intention of aligning it with the potential limit solution presented in equation [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}. Such a specific choice of $v_1$ is removed here. It does not change the convergence estimate [\[est:main1-global\]](#est:main1-global){reference-type="eqref" reference="est:main1-global"} at all by choosing any $v_1$. *Remark 3*. The 3D case turns out to be more challenging. Our current method fails since it relies on the basic energy estimate and its consequence of space-time estimate in Lemma [Lemma 18](#lem:spacetime-norm-KG){reference-type="ref" reference="lem:spacetime-norm-KG"} which are substantially weaker in 3D. The convergence rate established by this theorem aligns seamlessly with the numerical findings presented in [@BaoZh-KG-Limit; @SchratzZhao-KG-Limits], thus rigorously corroborating the second-order convergence behavior of the remainder function. In the next theorem, we shall embark on a demonstration, aiming at showcasing that the quadratic rate of convergence with respect to $\varepsilon$ is optimal, even for smooth enough initial data. **Theorem 4**. *Let $d=2,3$. There exist a class of triplets $$(u_0,u_1,v_1)\in \big[\mathcal S({\mathbb R}^d)\big]^3,$$ such that the convergence rate $\varepsilon^2$ achieved in Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"} attained its optimality. More precisely, there exist constants $\varepsilon_0>0$ and $c_0 > 0$, such that for any $\varepsilon\in (0,\varepsilon_0]$, $$\begin{aligned} \sup\limits_{t\in {\mathbb R}}\big\|u^\varepsilon(t)-\mathrm{e}^{\frac{it}{\varepsilon^2}}v^{\varepsilon}(t)-\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v^{\varepsilon} (t)\big\|_{L^2_x} \ge c_0 \varepsilon^2.\end{aligned}$$* We emphasis that the counterexample, or the initial data $(u_0,u_1,v_1)$ constructed in this theorem doesn't depend on $\varepsilon$ as $\varepsilon$ goes to 0. In view of the natural scaling of the cubic Klein-Gordon equation, it is natural to pursue the sharpness of the regularity requirement on the initial data. The sharpness on the $H^2$ regularity requirement of the initial data is out of the scope of this article and left to be an unsolved interesting problem. Now let us consider the case when the modulated wave profile satisfies cubic Schrödinger equation [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}. There are also a vast literatures focusing on obtaining precise estimates for error term $r^\varepsilon$ in the non-relativistic limit, as $\varepsilon$ approaches 0. For numerical investigations of this problem, one may refer to as [@BaoCaiZh-KG-Limit; @BaoDong-KG-Limit; @BaoZh-KG-Limit; @ChCr-KG-Limit; @ChMaMuVi-KG-Limit; @DoXuZh-KG-Limit; @SchratzZhao-KG-Limits; @Zhao-KG-Limits] and the references therein. Research in [@BaoZh-KG-Limit; @ChCr-KG-Limit; @SchratzZhao-KG-Limits] has observed quadratic convergence in $\varepsilon$ for smooth data. In [@BaoZh-KG-Limit] and [@SchratzZhao-KG-Limits], the numerical insights indicate that a reduction in the regularity of initial data might lead to a proportional decrease in the order of convergence rate. Numerical experiments of Bao and Zhao in [@BaoZh-KG-Limit] shows that for relatively non-smooth initial data ($(u_0,u_1)\in H^2$), the convergence rate reduces to be linear in $\varepsilon$. In [@SchratzZhao-KG-Limits], Schartz and Zhao further observed numerically that choosing the modulated Schrodinger profile of [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"} necessitates more regularity of the initial data compared to choosing the modulated Schrodinger-wave profile of [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}, in order to achieve optimal quadratic convergence in $\varepsilon$. We also want to point out that there exist intriguing comparisons between the equations [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"} and [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"} which govern the dynamics of modulated profiles in [@BaoZh-KG-Limit; @SchratzZhao-KG-Limits]. Numerical simulations suggest that under the same $\varepsilon$ and at the same time, the error from choosing the modulated Schrödinger-wave profiles of [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"} is much smaller than the one from choosing the modulated Schrödinger profile of [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}. The model [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"} is a more suitable choice than [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"} for studying the long-term behavior of the approximation. It is also important to point out that comparing to using the limit model [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}, the drawback of the limit model [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"} lies in that it tends to exhibit high oscillations in time. There are also certain analytic results that aim to rigorously verify these numerical findings. For instance, in [@MasNak-KG-Limits], Masmoudi and Nakanishi establish an error estimate of $\varepsilon^\frac12$ within an $L^2$ framework under the assumption of finite energy and $L^2$ convergence of the initial data. It's worth noting that their result in [@MasNak-KG-Limits] is valid for times of order $O(1)$, which means it is applicable for any fixed time $T$. While there have been several numerical simulation results estimating error functions in relation to time dependency, the corresponding theoretical analyses in this regard are still quite limited. Recently, Bao, Lu, and Zhang [@BaoLuZh-KG-Limit] extended the study to the three-dimensional problem and proved that for any $$(u_0,u_1)\in \Big(H^{s+4\alpha}(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)\Big)^2$$ with $s>\frac32$, the error function $$\|r(t)\|_{H^s}\lesssim (1+t)\varepsilon^\alpha$$ holds for $\alpha=1, 2$. This estimate is valid for times up to $O(\varepsilon^{-\frac12\alpha})$. These analytic results offer a solid foundation for the understanding of the error behavior and convergence rates of the system. However, it's important to highlight that despite these efforts, there remains a shortage of theoretical analyses that comprehensively capture the intricate interplay among error estimates, regularity dependency, and time dependency observed in numerical simulations. In fact, through dimensional analysis, it becomes apparent that an increase in convergence rate can be achieved at the cost of sacrificing solution regularity and shortening the time span. To demonstrate it, let us consider for $f\in L^2(\mathbb{R}^3)$: $$\|\mathrm{e}^{it\Delta}f_\varepsilon\|_{L^2_tL^6_x([0,T]\times {\mathbb R}^3)}\le C\|f\|_{L^2({\mathbb R}^3)},$$ where $f_\varepsilon=\varepsilon^\frac12f(\varepsilon^2t,\varepsilon x)$. Furthermore, if we assume that $f\in \dot H^1(\mathbb{R}^3)$, we can derive: $$\|\mathrm{e}^{it\Delta}f_\varepsilon\|_{L^2_tL^6_x([0,T]\times {\mathbb R}^3)}\le CT\varepsilon\|\nabla f\|_{L^2({\mathbb R}^3)},$$ thus yielding a higher convergence rate than that obtained with $L^2$ data. Consequently, all the goals including achieving a fixed convergence order, optimizing the minimal regularity, and determining the validation time interval necessitate a comprehensive exploration of the equation's underlying structure. Our second result serves as a rigorous verifications of the above mentioned numerical ones. It also improves the existing analytic results. **Theorem 5**. *Let $d=2, 3$ and $2\le \alpha\le 4$. Suppose that $$(u_0,u_1)\in H^\alpha({\mathbb R}^d) \times H^\alpha({\mathbb R}^d).$$ Let $u^\varepsilon, v$ be the corresponding solutions to cubic nonlinear Klein-Gordon equation [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"} and cubic nonlinear Schrödinger equation [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}, respectively. Then the following error estimate $$\begin{aligned} \label{est:main3-local-d2} \big\|u^\varepsilon(t)-\mathrm{e}^{\frac{it}{\varepsilon^2}}v(t)-\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v(t) \big\|_{L^2_x} \le C\Big(\varepsilon^2+\big(\varepsilon^2 t\big)^{\frac14\alpha}\Big)\end{aligned}$$ holds for all $t\ge 0$, where $C$ is a generic constant depending only on $\|(u_0,u_1)\|_{H^\alpha\times H^\alpha}$.* The proof of Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"} requires a substantial improvement of the famous results by Machihara, Nakanishi, and Ozawa [@MaNaOz-KG-Limits], which will be stated as the third result or Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} of this paper below. The conclusion of Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"} in the two-dimensional case is a direct corollary of Theorems [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"} and [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} by selecting $\widetilde v_0^\varepsilon = \widetilde v_0$ and $\widetilde v_1^\varepsilon = \varepsilon^2 v_1$. The three-dimensional case can be achieved through a slight adaptation of the argument employed in the proofs of Theorems [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"} and [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"}. Moreover, it is not hard to extend the restriction $2\le \alpha\le 4$ to $1\le \alpha\le 4$ with a slight modification of the argument, however, we do not pursue this generality. The conclusions illustrated in Theorems [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"} and [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}, alongside the numerical findings presented in [@BaoZh-KG-Limit] and [@SchratzZhao-KG-Limits], show that choosing the modulated Schrödinger profiles of [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"} necessitates higher regularity on the initial data, in contrast to choosing the modulated Schrödinger-wave profiles of [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}. It is worth noting that our results exhibit three enhancements over existing ones in [@MasNak-KG-Limits; @BaoLuZh-KG-Limit]. Firstly, our prerequisites for regularity are notably more lenient. Specifically, for achieving quadratic convergence, we merely stipulate that $(u_0, u_1) \in H^4(\mathbb{R}^d) \times H^4(\mathbb{R}^d)$. Indeed, as we will demonstrate below, *the regularity assumptions are sharp.* Secondly, our result holds true for all time but not only a finite time interval. Thirdly, our framework adeptly accommodates both two and three dimensional situations, thereby broadening the scope of applicability. As mentioned above, the third result is a substantial improvement of the famous result of Machihara, Nakanishi, and Ozawa [@MaNaOz-KG-Limits], which will serve as a key step in proving Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}. To start with, let us recall some studies on the non-relativistic limits of the following Cauchy problem $$\begin{aligned} \begin{split} \label{eq:nls-wave-N} \left\{ \aligned &\varepsilon^2\partial_{tt}\widetilde{v}^\varepsilon+2i\partial_{t} \widetilde{v}^\varepsilon - \Delta\widetilde{v}^\varepsilon= - |\widetilde{v}^\varepsilon|^2 \widetilde{v}^\varepsilon, \\ & \widetilde{v}^\varepsilon(0,x) = \widetilde{v}^\varepsilon_0(x),\quad \partial_t\widetilde{v}^\varepsilon(0,x)=\frac{\widetilde{v}^\varepsilon_1(x)}{\varepsilon^2}. \endaligned \right. \end{split}\end{aligned}$$ which may be derived by applying the following modulated transformation $$u^\varepsilon=\mathrm{e}^{\frac{it}{\varepsilon^2}}\widetilde{v}^\varepsilon$$ to the cubic nonlinear Klein-Gordon equation. In 1983, Tsutsumi [@Tsutsumi-KG-Limits] established that in the two-dimensional case, the limit of [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} is the solution to the cubic nonlinear Schrödinger equation, commonly expressed as: $$\begin{aligned} \begin{split} \label{eq:nls-1} \left\{ \aligned &2i\partial_{t} \widetilde{v} - \Delta\widetilde{v}= - |\widetilde{v}|^2 \widetilde{v}, \\ & \widetilde{v}(0,x) = \widetilde{v}_0(x), \endaligned \right. \end{split}\end{aligned}$$ where the convergence holds assuming that $(\widetilde{v}^\varepsilon_0, \varepsilon^{-1} \widetilde{v}^\varepsilon_1)$ converges to $(\widetilde{v}_0, 0)$ in the $H^2$ norm (note that it is not $(\widetilde{v}^\varepsilon_0, \widetilde{v}^\varepsilon_1)$). In subsequent works such as [@Najman-KG-Limits; @Machihara-KG-Limits], the regularity prerequisites were relaxed. These studies demonstrated convergence in the $L^q$ space with $2\le q<6$, under the assumption of boundedness in $H^1$ and $L^2$ convergence of the initial data $(\widetilde{v}^\varepsilon_0, \varepsilon^{-1} \widetilde{v}^\varepsilon_1)$. The $H^2$ convergence of the error function was established in [@BeCo-KG-Limit], subject to certain assumptions. A significant breakthrough in determining the precise convergence order of non-relativistic limits was achieved by Machihara, Nakanishi, and Ozawa in [@MaNaOz-KG-Limits]. By assuming finite energy and $L^2$ convergence of the initial data $(\widetilde{v}^\varepsilon_0, \varepsilon^{-1} \widetilde{v}^\varepsilon_1)$, the authors established that the solution to [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} converges to [\[eq:nls-1\]](#eq:nls-1){reference-type="eqref" reference="eq:nls-1"} up to a fixed time, with an optimal rate of $o(\varepsilon^\frac12)$. Again, it is not $(\widetilde{v}^\varepsilon_0, \widetilde{v}^\varepsilon_1)$. Meanwhile, Zhao [@Zhao-KG-Limits] carried out numerical simulations, revealing that the solution of [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} attains quadratic convergence in $\varepsilon$ with linear growth in time, characterized by: $$\begin{aligned} \big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{H^1_x} \le &C(1+t)\varepsilon^2,\end{aligned}$$ provided that the initial data $(\widetilde{v}^\varepsilon_0,\varepsilon^{-2}\widetilde{v}_1^\varepsilon)$ in [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} maintains a certain degree of smoothness (but independent of $\varepsilon$). However, when the initial data's regularity is weaker, e.g., $(\widetilde{v}^\varepsilon_0,\varepsilon^{-2}\widetilde{v}_1^\varepsilon)\in H^2\times H^2$, the convergence rate diminishes to be linear in $\varepsilon$, i.e. $$\begin{aligned} \big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{H^1_x} \le &C(1+t)\varepsilon.\end{aligned}$$ Obviously, the results of Machihara, Nakanishi, and Ozawa in [@MaNaOz-KG-Limits] is not enough for our purpose since there is an extra $\varepsilon^{-1}$ in the initial velocity. Indeed, to apply the results in [@MaNaOz-KG-Limits; @Zhao-KG-Limits] directly to the current situation, one has to take the initial velocity $\widetilde{v}^\varepsilon_1$ to be 0. Our third result aims to obtain an improvement of those existing ones, serving a key step in proving our Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}. **Theorem 6**. *Let $d=2,3$, $s_d=d-2$, $1\le \beta\le 4$ and $\beta\le \alpha\le 4$. Assume that* - *$\widetilde{v}_0\in H^\alpha({\mathbb R}^d),\ \widetilde{v}^\varepsilon_0\in H^\beta_x({\mathbb R}^d),\ \widetilde{v}_1^\varepsilon\in H^{s_d}({\mathbb R}^d)$;* - *$(\widetilde{v}^\varepsilon_0 - \widetilde{v}_0,\ \widetilde{v}_1^\varepsilon)\to (0,0) \mbox{ in } L^2({\mathbb R}^d)\times L^2({\mathbb R}^d).$* *Denote the corresponding solutions to [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} and [\[eq:nls-1\]](#eq:nls-1){reference-type="eqref" reference="eq:nls-1"} by $\widetilde{v}^\varepsilon$ and $\widetilde{v}$, respectively. Then for any $t>0$, one has $$\begin{aligned} \label{est:Thm3-main} \big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^2_x} \lesssim & \big\|(\widetilde{v}^\varepsilon_0-\widetilde{v}_0, \widetilde{v}_1^\varepsilon)\big\|_{L^2_x\times L^2_x} +\varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0 - \widetilde{v}_0\big\|_{H^\beta_x} + \varepsilon^{\min\{\alpha,2\}}+\big(\varepsilon^2 t\big)^{\frac14\alpha}.\end{aligned}$$* We emphasis that the crucial point in Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} is that the condition on $\widetilde{v}_1^\varepsilon$ is less stringent so that it is applicable to Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}. While it is demanded in [@MaNaOz-KG-Limits] that $\widetilde{v}_1^\varepsilon=o(\varepsilon)$, our Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} relaxes this requirement to $\widetilde{v}_1^\varepsilon=o(1)$. Our theoretical findings closely align with the observed convergence patterns presented in [@Zhao-KG-Limits], which numerically indicated a convergence rate of $\varepsilon^2$ for smooth initial data and a rate of $\varepsilon$ for less smooth data. For instance, if $(\alpha, \beta, s_d) = (4, 2, 1)$ and $(\widetilde{v}_0, \widetilde{v}^\varepsilon_0, \widetilde{v}_1^\varepsilon) \in H^4({\mathbb R}^d) \times H^2({\mathbb R}^d) \times H^1({\mathbb R}^d)$, then the bound in Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} reads $$\big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^2_x} \lesssim \|(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x\times L^2_x}+(1+t)\varepsilon^2.$$ Moreover, our result reveals that, to get an extra $\varepsilon^\frac12$ convergence rate, one only needs one additional derivative in the regularity of the initial data (see also [@MaNaOz-KG-Limits] wherein the optimal $\varepsilon^\frac12$ rate was established for $H^1\times L^2$ initial data). Besides, the bound here is explicit in time. It holds for all $t > 0$, in particular, both small and large $t$. *Remark 7*. If $(\widetilde{v}_0, \widetilde{v}^\varepsilon_0, \widetilde{v}_1^\varepsilon) \in (H^1({\mathbb R}^d)^3$, then the above bound reads $$\big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^2_x} \lesssim \|(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x\times L^2_x}+\varepsilon+t^{\frac14}\varepsilon^\frac12.$$ This is even better than the numerical observations in [@Zhao-KG-Limits] where the dependence of this upper bound on time is linear. In the case of $d=2$, the requirement on $\widetilde{v}_1$ can be relaxed to $\widetilde{v}_1 \in L^2({\mathbb R}^d)$. The following theorem shows that the regularity requirement and the convergence bound $(1+t)\varepsilon^2$ achieved in Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} attained their optimality. **Theorem 8**. *Let $d=2,3$, and set $\widetilde{v}^\varepsilon_0=\widetilde{v}_0, \widetilde{v}_1^\varepsilon=0.$ Then there exist a class of $$\widetilde{v}_0\in \mathcal S({\mathbb R}^d),$$ constants $\varepsilon_0>0$, $c_0>0$ and $\delta_0$, such that for any $\varepsilon\in (0,\varepsilon_0]$, $$\begin{aligned} \big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^2_x} \ge c_0 t \varepsilon^2,\quad \mbox{ when } 1\le t \le \delta_0 \varepsilon^{-2} .\end{aligned}$$ Moreover, for $1\le \alpha\le 4$, there exists an initial data $\widetilde{v}_0 \in H^\alpha({\mathbb R}^d)$ satisfying such that $$\begin{aligned} \big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^2_x} \ge c_0\big(\varepsilon^2 t\big)^{\frac14\alpha}\big|\ln \big(t\varepsilon^2\big) \big|^{-1},\quad \mbox{ when } 1\le t \le \delta_0 \varepsilon^{-2} . \end{aligned}$$* *Remark 9*. The counterexamples constructed in this theorem are equally applicable to show the sharpness of the conclusion in Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}. In fact, as we will see, the transition from the wave function $u^\varepsilon$ to the Schrd̈inger profile $v$ in the non-relativistic limits is primarily influenced by the transition from the modulated Schrd̈inger-wave profile $v^\varepsilon$ to the modulated Schrödinger profile $v$, as expounded in the proof of Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"}. ## Key ingredients of the proofs First, we elucidate the crucial element of the proofs underlying our main theorems. .3cm $\bullet$*Analysis of Non-Resonance in the Nonlinearity: Proofs of Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"} and Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}* .3cm The nonlinear term $B_\varepsilon(v^\varepsilon)$ defined in [\[def:B-var\]](#def:B-var){reference-type="eqref" reference="def:B-var"} can be viewed as the inhomogeneous component of the corresponding equation for the residual function $r^\varepsilon$. Nevertheless, this term carries significant magnitude. Specifically, we seek to understand how it differs from $|v^\varepsilon|^2v^\varepsilon$, which originates from the nonlinear term in the limiting equation [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}. A crucial point for proofs of Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"} and Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"} is to bound the following time-integral $$\int_0^t K_\varepsilon(s)\Big[\mathrm{e}^{\frac{3is}{\varepsilon^2}}(v^\varepsilon)^3+\mathrm{e}^{-\frac{3is}{\varepsilon^2}}\big(\bar v^\varepsilon\big)^3\Big]\,ds, %\quad K_\varepsilon(t)=\fe^{-ic(c\pm \langle \nabla\rangle_c)t}$$ by $O(\varepsilon^2)$. Here $K_\varepsilon$ is a solution operator of linear Klein-Gordon equations [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"}. However, it is not transparent. Our initial insight pertains to the non-resonance structure of the time-integral. To demonstrate, we begin with a rescaling argument, transforming it into the consideration of the following time-integral $$\int_0^{\varepsilon^{-2}t} \mathrm{e}^{-is\langle\nabla\rangle} \Big[\mathrm{e}^{3is}(\mathcal{S}_\varepsilon v^\varepsilon)^3+\mathrm{e}^{-3is}\big(\mathcal{S}_\varepsilon \bar v^\varepsilon\big)^3\Big]\,ds,$$ where $\mathcal{S}_\varepsilon f(t,x)=\varepsilon f(\varepsilon^2t,\varepsilon x)$. Note that the estimate for the above integral is easy if its integrand is applied by an operator $P_{\ge 1}.$ Then the non-resonance structure comes in if we look into the following the exponential operators $$\mathrm{e}^{-is(\langle\nabla\rangle-3)}P_{\le 1};\quad \mbox{and }\quad \mathrm{e}^{-is(\langle\nabla\rangle+3)}.$$ Here $P_{\le 1}$ and $P_{\ge 1}$ are the frequency localized operators defined in Section [2.1](#sec:notations){reference-type="ref" reference="sec:notations"}. By normal form transformation, it suffices to estimate the following time-integral (with the abuse of ignoring useless operators): $$\varepsilon^2 \int_0^{\varepsilon^{-2}t} \mathrm{e}^{-is\langle\nabla\rangle}\Big[\mathrm{e}^{3is}(\mathcal{S}_\varepsilon v^\varepsilon)^2 \mathcal{S}_\varepsilon \partial_s v^\varepsilon+\mathrm{e}^{-3is}\big(\mathcal{S}_\varepsilon \bar v^\varepsilon\big)^2 \mathcal{S}_\varepsilon \partial_s v^\varepsilon\Big]\,ds.$$ We emphasis that an extra $\varepsilon^2$ appears in the above expression. The substantial difference between $(v^\varepsilon)^3$ and $|v^\varepsilon|^2v^\varepsilon$ is presented. Then our next goal is to get the uniform bound in $\varepsilon$ for $\partial_t v^\varepsilon$ within a suitable space. It is natural to turn to the conservation of the basic energy: $$\begin{aligned} \nonumber \int\big(\varepsilon^2|\partial_tv^\varepsilon|^2 + |\nabla v^\varepsilon|^2 + \frac{3}{2}|v^\varepsilon|^4\big)dx = \int\big(\varepsilon^2|v_1|^2+\frac{1}{4}|\nabla (u_0 - iu_1)|^2 + \frac{3}{2}|u_0 - iu_1|^4\big)dx.\end{aligned}$$ Unfortunately, this leads to the conclusion that $$\begin{aligned} \label{pt-v-trival} \|\partial_tv^\varepsilon\|_{L^2_x}=O(\varepsilon^{-1}).\end{aligned}$$ This natural bound is far from our requirement. Our strategy is to gain the $\varepsilon$-loss of $\partial_tv^\varepsilon$ by crucial observations that a magic complex extension of the system possesses non-resonance structure and a well-behaved initial data. .3cm $\bullet$ *Gaining the $\varepsilon$-loss from Time-derivative in Proof of Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"}* .3cm The key ingredient to gain the $\varepsilon$-loss of $\partial_tv^\varepsilon$ is the following proposition. Denote the space $X_{\alpha}(I)$ with $\alpha\in{\mathbb R}, I\subset {\mathbb R}$, with its norm $$\|f\|_{X_{\alpha}(I)}\triangleq \big\||\nabla|^\alpha f\big\|_{L^{\frac{2(d+2)}{d}}_{tx}(I)}+\big\||\nabla|^{\alpha} f\big\|_{L^\infty_{t}L^2_x(I)}.$$ **Proposition 10**. *Let $\gamma\ge 0$, and $v^{\varepsilon}$ to be the solution of [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}, then for any $v_0\in H^{2+\gamma}({\mathbb R}^2)$ and $v_1\in H^{\gamma}({\mathbb R}^2)$, there holds $$\begin{aligned} \label{est:v-varep-px} \left\| \langle\nabla\rangle^\frac32 v^{\varepsilon} \right\|_{L^4_{tx}({\mathbb R})}+\big\|v^{\varepsilon} \big\|_{L^\infty_t\dot H^2_x({\mathbb R})}\le C,\end{aligned}$$ and $$\begin{aligned} \label{est:v-varep-pt} \big\| |\nabla|^{\gamma} \partial_tv^{\varepsilon} \big\|_{X_0({\mathbb R})}\le C,\end{aligned}$$ where $C$ is a generic constant depending only on $\|v_0\|_{H^{2+\gamma}({\mathbb R}^2)}$.* The estimate [\[est:v-varep-pt\]](#est:v-varep-pt){reference-type="eqref" reference="est:v-varep-pt"} established in the proposition above significantly improves the one in [\[pt-v-trival\]](#pt-v-trival){reference-type="eqref" reference="pt-v-trival"}. The proof of this proposition is presented in Section [3](#sec:prop-key){reference-type="ref" reference="sec:prop-key"}. The main ideas are as follows. \(1\) *Rescaled Complex Extension of Equation [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"}*: We begin by introducing a thoughtfully chosen complex extension of $u^\varepsilon$, which enables us to producing extra small parameters in both initial data and nonlinearities. More precisely, we introduce a judiciously designed scaling operator denoted as $\mathcal{S}_\varepsilon$, and employ the transformation $w=e^{it}\mathcal{S}_\varepsilon(v^\varepsilon)$. As a result, the function $w$ adheres to an intricately formulated *complex expanded* nonlinear Klein-Gordon equation [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"} (with the change of coefficient in nonlinear term from 1 to 3, which does not change our analysis): $$\partial_{tt}w-\Delta w+w=-3|w|^2w$$ with specific initial data $$w(0)=\mathcal{S}_\varepsilon v_0,\quad \partial_t w(0)= i\mathcal{S}_\varepsilon v_0+\varepsilon^2 \mathcal{S}_\varepsilon v_1.$$ Firstly, an extra gain of $\varepsilon^2$ appears, exhibiting a magic structure in the initial data. Secondly, it becomes evident that: $$\begin{aligned} \label{relationship-w-wt} w(0)=i\partial_t w(0)+O(\varepsilon^2),\quad \mbox{ in }L^2.\end{aligned}$$ This relationship plays a crucial role in the our analysis. \(2\) *Enhanced Performance of the Leftward Wave.* Another crucial ingredient of the rescaled complex extension is the inherent non-resonance structure in nonlinearity. Indeed, this can be seen by introducing the *leftward wave* $\mathcal W_1$ and *rightward wave* $\mathcal W_2$, $$\begin{aligned} \begin{split} \mathcal W_j=\langle \nabla \rangle^{-1}\big(\partial_t\mp i \langle \nabla\rangle\big)w, \mbox{ for } j=1,2, \end{split}\end{aligned}$$ where these waves elegantly fulfill the following systems of equations: $$\begin{aligned} \label{eqs:math-W-12} \big(\partial_t\pm i \langle \nabla\rangle\big)\mathcal W_j=-\langle \nabla \rangle^{-1}(|w|^2w).\end{aligned}$$ When compared to the rightward wave, it's somewhat surprising that the leftward wave exhibits superior behavior. The crux of the proof is rooted in the subsequent pivotal observation: 1⃝ Well-behaved Initial Data. By [\[relationship-w-wt\]](#relationship-w-wt){reference-type="eqref" reference="relationship-w-wt"}, it is easy to see that the initial data $\mathcal W_1(0)$ can be approximated as follows: $$\mathcal W_1(0)=\Delta \mathcal{S}_\varepsilon v_0+\varepsilon^2\mathcal{S}_\varepsilon v_1.$$ Extra $\varepsilon^2$ smallness is present in both terms on the right-hand side of the above expression. 2⃝ Exploring Non-Resonant Structure. Ignoring some terms in nonlinearity, we have the following equation, which characterizes the main difficulties in the nonlinearity: $$\big(\partial_t+ i \langle \nabla\rangle\big)\mathcal W_1=-P_{\le1} \Big[|P_{\le 1}\mathcal W_2|^2P_{\le 1}\mathcal W_2\Big]+\cdots.$$ A non-resonant structure can be explored here. Indeed, the corresponding phase function of this nonlinearity is given by: $$\langle \xi\rangle+\langle \xi_1\rangle+\langle \xi_2\rangle-\langle \xi_3\rangle\ge 1,\quad \mbox{when } \quad |\xi|\le 1 \mbox{ and } |\xi_j|\le 1, j=1,2,3.$$ In conjunction with observation 1⃝ above, this insight allows us to obtain an additional factor of $\varepsilon^2$ through the normal-form argument. \(3\) *Passing the properties of the leftward wave to $\partial_t \mathcal{S}_\varepsilon(v^\varepsilon)$.* It can be derived that (see Section [3.5](#subsec:keyprop-end){reference-type="ref" reference="subsec:keyprop-end"} for the details) $$\mathcal W_1\sim \partial_t \mathcal{S}_\varepsilon(v^\varepsilon).$$ This enables us to gain an additional factor of $\varepsilon^2$ in the estimation of $\partial_t \mathcal{S}_\varepsilon(v^\varepsilon)$. .3cm $\bullet$ *Regularity Gain through the "High-Low" Decomposition in Proof of Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"}* .3cm The primary challenge in establishing Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} stems from the substantial discrepancy in the properties of linear operators between equations [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} and [\[eq:nls-1\]](#eq:nls-1){reference-type="eqref" reference="eq:nls-1"}. The linear operators for these equations are expressed as follows: $$K_\varepsilon(t)=\mathrm{e}^{-ic(c\pm \langle \nabla\rangle_c)t},\quad S(t)=\mathrm{e}^{-\frac12it\Delta}.$$ Here $\langle \nabla\rangle=\sqrt{|\nabla|^2+c^2}$ and $c=\varepsilon^{-1}$. In the previous work [@MaNaOz-KG-Limits], the authors derived a convergence rate in the energy space, proportional to the square-root of $\varepsilon$, using uniform Strichartz estimates on $K_\varepsilon$ and spacetime estimates on the remainder operator: $$R_\varepsilon(t)=K_\varepsilon(t)-S(t),$$ which grows linearly in time and will lead to a quadratic one in the final conclusion. Furthermore, this argument relies on the uniform boundedness of the solution in $H^1$, a result stemming from the energy estimates. Specifically, solutions to [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} exhibit energy conservation, which can be expressed as follows: $$\begin{aligned} \label{energy-NLS-wave} \int\big(\varepsilon^2|\partial_t\widetilde{v}^\varepsilon|^2 + |\nabla \widetilde{v}^\varepsilon|^2 + \frac{3}{2}|\widetilde{v}^\varepsilon|^4\big)dx = \int\big(\varepsilon^{-2}|\widetilde{v}_1^\varepsilon|^2 + |\nabla \widetilde{v}_0^\varepsilon|^2+\frac{3}{2}|\widetilde{v}_0^\varepsilon|^4\big)dx.\end{aligned}$$ In our situation, when $(\widetilde{v}_0^\varepsilon, \widetilde{v}_1^\varepsilon)$ are bounded in $H^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$, trivial bounds can be derived from the energy conservation, resulting in: $$\|\partial_t \widetilde{v}^\varepsilon\|_{L^2_x}=O(\varepsilon^{-2}),\quad \mbox{ and }\quad \|\nabla \widetilde{v}^\varepsilon\|_{L^2_x}=O(\varepsilon^{-1}).$$ These estimates, however, exhibit very bad dependence on $\varepsilon$ and is too weak. In contrast, in [@MaNaOz-KG-Limits], the authors proposed a more stringent assumption by considering that $(\widetilde{v}_0^\varepsilon, \varepsilon^{-1} \widetilde{v}_1^\varepsilon)$ are bounded in $H^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$ (the same assumption was also proposed in [@MasNak-KG-Limits]). Their stronger assumption eliminates the singularity of the solution $\widetilde{v}$ in $H^1$ and mitigates the singularity of $\partial_t\widetilde{v}$ in $L^2$. That is, $$\|\partial_t \widetilde{v}^\varepsilon\|_{L^2_x}=O(\varepsilon^{-1}),\quad \mbox{ and }\quad \|\nabla \widetilde{v}^\varepsilon\|_{L^2_x}=O(1).$$ Our new approach is capable of reducing the growth of time to linear and eliminating all of these singularities with only a weaker assumption, namely that $\|(\widetilde{v}_0^\varepsilon, \widetilde{v}_1^\varepsilon)\|_{H^1_x\times L^2_x}$ is bounded. This represents a significant advancement in understanding the non-relativistic limit of [\[eq:kg-eps\]](#eq:kg-eps){reference-type="eqref" reference="eq:kg-eps"}, as we are able to achieve a more comprehensive removal of singularities under a less restrictive condition. Our new approach is based on specific *high-low frequency decomposition* of the solution, which helps to gain regularity. Define $$\widetilde{v}_N=NLS (P_{\le N}\widetilde{v}_0) \quad\mbox{ and } \quad \widetilde v^N= \widetilde v-\widetilde{v}_N.$$ Here, $NLS(u_0)$ signifies the nonlinear solutions of equation [\[eq:nls-1\]](#eq:nls-1){reference-type="eqref" reference="eq:nls-1"} with initial data $u_0$. By implementing distinct strategies for the regular and non-regular cases, we achieve different bounds. Specifically, in the regular case, we obtain a bound for the error function $\widetilde{r}^\varepsilon$ with linear dependence on the highest regularity of the initial data, expressed as $$\varepsilon^2 t \|\Delta^2 \widetilde{v}_N\|_{L^2_x}.$$ In the non-regular case, the error bound is given by $$\big\|\widetilde{v}^N\big\|_{L^2_x}.$$ Subsequently, through the "high-low\" frequency decomposition, the error bound can be derived as $$\Big(\varepsilon^2 t N^{4-\alpha}+ N^{-\alpha}\Big)\big\|\widetilde{v}\big\|_{H^\alpha_x}.$$ By suitably selecting the frequency $N$, we achieve the desired estimate. .3cm $\bullet$ *Constructions of the Counterexamples in Theorem [Theorem 4](#thm:main1-optimal){reference-type="ref" reference="thm:main1-optimal"} and Theorem [Theorem 8](#thm:main3-optimal){reference-type="ref" reference="thm:main3-optimal"}* .3cm The above high-low frequency decomposition also plays an important role in the constructions of the counterexamples to show the sharpness of the regularity. These constructions require a comprehensive understanding of the solution's structure, and distinct strategies in various scenarios. \(1\) Counterexample in Theorem [Theorem 4](#thm:main1-optimal){reference-type="ref" reference="thm:main1-optimal"}. The construction need use a deep result of Cazenave and Weissler [@CaWe-1992]. Let us roughly write the solution as follows, $$\phi(t)=\mathrm{e}^{it\langle \nabla\rangle}\phi_0 +\mathrm{e}^{it\langle \nabla\rangle}\big(\mathcal{S}_\varepsilon v^\varepsilon_0\big)^3 -\big(\mathcal{S}_\varepsilon v^\varepsilon(t)\big)^3 +\mathcal N(\phi, \mathcal{S}_\varepsilon v^\varepsilon),$$ where the Duhamel term $\mathcal N(f,g)$ takes the form of: $$\mathcal N(f,g)=\int_0^t \mathrm{e}^{i(t-s)\langle \nabla\rangle}F(f,g)(s)\,ds.$$ This rough expression is due to the normal form transformation, which does not change the construction of the counterexample in Theorem [Theorem 4](#thm:main1-optimal){reference-type="ref" reference="thm:main1-optimal"}. The term $\mathrm{e}^{it\langle \nabla\rangle}\big[\phi_0+\big(\mathcal{S}_\varepsilon v^\varepsilon_0\big)^3\big]$ will be shown to play the main role for the construction, and after then the construction is more or less standard. 1⃝ Approximation of the Linear Klein-Gordon Flow. To diminish the magnitude of the Duhamel term $\mathcal N$, we employ the following approximation: $$\begin{aligned} \mathrm{e}^{it\langle\nabla\rangle}=\mathrm{e}^{it}\mathrm{e}^{-\frac12 it\Delta}+O\big(t \Delta^2\big).\end{aligned}$$ The point is that each $\nabla$ introduces an additional factor of $\varepsilon$. For a chosen time $t$ less than $\varepsilon^{-4}$, it is possible to omit the last part and deduce the linear Klein-Gordon flow to the linear Schrödinger flow. 2⃝ Choosing Data for Linear Schrödinger Flow. Let us choose a data for the linear Schrödinger flow so that the solution is large in $L^\infty_tL^r_x$ but small in $L^q_tL^r_x$ for some $2\le q<+\infty$. That is, $$\big\|\psi\big\|_{L^\infty_tL^r_x({\mathbb R})}\sim 1;\quad \big\|\mathrm{e}^{-\frac12 it\Delta}\psi\big\|_{L^q_tL^r_x({\mathbb R})}\ll 1.$$ Inspired by the work in [@CaWe-1992], we set $$\psi=\mathrm{e}^{-ib\frac{|x|^2}{2}}f, \quad \mbox{for }\quad f\in S({\mathbb R}^d),$$ which leads to $$\big\|\mathrm{e}^{-\frac12 it\Delta}\psi\big\|_{L^q_tL^r_x({\mathbb R}^+)}=\big\|\mathrm{e}^{-\frac12 it\Delta}f\big\|_{L^q_tL^r_x([0,\frac1b])}.$$ By selecting a large value for $b$, we achieve smallness in $L^q_tL^r_x$ for $2\le q<+\infty$. However, the size remains large in $L^\infty_tL^r_x$. By employing these two strategies, we can systematically diminish the magnitude of the term $\mathcal N$ without altering the magnitude of $\phi_0+\big(\mathcal{S}_\varepsilon v^\varepsilon_0\big)^3$. Furthermore, the magnitude of the third term can be reduced significantly over an extended time period by applying the dispersive estimate. \(2\) Counterexample in Theorem [Theorem 8](#thm:main3-optimal){reference-type="ref" reference="thm:main3-optimal"}. The construction is built upon a solution structure that can be outlined as follows: $$\mathcal W_j(t)=\mathrm{e}^{it\langle \nabla\rangle}\mathcal W_{j,0} +\int_0^t \mathrm{e}^{\mp i(t-s)\langle \nabla\rangle} \mathrm{e}^{is}\mathcal{S}_\varepsilon\big(\partial_{tt}v\big)(s)\,ds +\mathcal N(\mathcal W_j, \mathcal{S}_\varepsilon v),$$ where $v$ is the solution of the nonlinear Schrödinger equation. By neglecting non-dominant terms, approximating nonlinear solutions with solutions of linear equations, and leveraging the scaling invariance of the linear flow, we arrive at the following approximate expression: $$\mathcal W_j(t)=\mathrm{e}^{\mp it\langle \nabla\rangle}\int_0^t \mathrm{e}^{is(\pm\langle \nabla\rangle+1-\frac12\Delta)} \mathcal{S}_\varepsilon(\Delta^2 v_0)\big)\,ds+\cdots.$$ 1⃝ Non-Dominance of the Leftward Wave. In stark contrast to the remarkable performance of the leftward wave before, we shift our focus in this situation to harness the larger magnitude of the rightward wave. Specifically, by suitably selecting the initial data, it will be demonstrated that the upper bound of the *Leftward wave* $\mathcal R_1$ is surpassed by the lower bound of the *Rightward wave* $\mathcal R_2$. As previously discussed, our investigation revolves around the exponential operator: $$\mathrm{e}^{is(\langle \nabla\rangle+1-\frac12\Delta)}P_{\le 1},$$ and ascertain the absence of critical points within its phase. This discovery unveils the non-resonant structure of the leftward wave $\mathcal W_1$, resulting in its smaller bound. However, this property ceases to hold when we examine the exponential operator: $\mathrm{e}^{is(-\langle \nabla\rangle+1-\frac12\Delta)}.$ 2⃝ Approximation of the Linear Flow. Turning our attention to the rightward wave $\mathcal W_2$, we utilize an approximation of the linear operator: $$\begin{aligned} \mathrm{e}^{is(-\langle \nabla\rangle+1-\frac12\Delta)}=1+\frac18 is \Delta^2+O(s\Delta^3).\end{aligned}$$ This formula yields two independent dominant terms, originating from $1$ and $\frac18 is \Delta^2.$ The last term can be effectively controlled by $t\varepsilon^6$. Furthermore, we also employ the "High-Low" Decomposition to handle terms involving high derivatives. ## Organization of the paper In Section [2](#sec:pre){reference-type="ref" reference="sec:pre"}, we provide notations, definitions, and relevant results. Additionally, we review existing findings on scattering behavior in the context of the nonlinear Schrödinger equation and nonlinear Klein-Gordon equations, which will serve as the basis for our subsequent analysis. In Section [3](#sec:prop-key){reference-type="ref" reference="sec:prop-key"}, we introduce the rescaled complex expansion and define the leftward and rightward waves. We commence by establishing Strichartz-type estimates for both the leftward and rightward waves. Subsequently, we demonstrate the heightened preformation of the leftward wave. Utilizing these estimates, we ultimately present the proof of Proposition [Proposition 10](#prop:v-varep){reference-type="ref" reference="prop:v-varep"}. In Section [4](#sec:global){reference-type="ref" reference="sec:global"}, we initiate with the proof of Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"}, focusing on the exploration of non-resonant structures. We then exhibit the optimal convergence rate by constructing counterexamples. In Section [5](#sec:singe-modulate){reference-type="ref" reference="sec:singe-modulate"}, we provide the proof of Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} through high-low frequency decomposition. Additionally, we establish the sharpness of both the convergence rate and the regularity requirement given in Theorem [Theorem 8](#thm:main3-optimal){reference-type="ref" reference="thm:main3-optimal"}. Finally, in Section [6](#sec:case1){reference-type="ref" reference="sec:case1"}, we furnish the proof for Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}. # Preliminary {#sec:pre} .5cm ## Notations and Definition {#sec:notations} For any $z\in{\mathbb C}$, we define ${\mathrm{Re}}z$ and ${\mathrm{Im}}z$ as the real and imaginary part of $z$, respectively. Denote $\langle x\rangle=\sqrt{1+x^2}$. Let $C>0$ denote a generic positive constant. If $f\leqslant C g$, we write $f\lesssim g$ or $f=O(g)$. For short, we write $L^p_x=L^p_x({\mathbb R}^d), H^s_x=H^s_x({\mathbb R}^d)$, $L^q_tL^r_x(I\times {\mathbb R}^d)=L^q_tL^r_x(I)$ and $L^q_tH^s_x(I\times {\mathbb R}^d)=L^q_tH^s_x(I)$. Moreover, we denote the norm $$\|f\|_{X\cap Y}\triangleq \|f\|_X+\|f\|_Y.$$ Let $\chi\in C_0^\infty({\mathbb R}^d)$ be a radial, real-valued and smooth cut-off function such that $\chi \geqslant 0$ satisfies $$\begin{aligned} \begin{split} \chi(\xi) \triangleq \left\{ \aligned &1\text{, when $|\xi|\le \frac 12$,}\\ &0\text{, when $|\xi|\ge 1$.} \endaligned \right. \end{split}\end{aligned}$$ Denote $\chi_{\le a}(\xi)=\chi(\frac{\xi}a)$ and $\chi_{a\le \cdot b}=\chi_{\le b}-\chi_{\le a}$. We use $\widehat{f}$ or ${\mathcal{F}}f$ to denote the Fourier transform of $f$: $$\begin{aligned} \begin{split} \widehat{f}(\xi)={\mathcal{F}}f(\xi)\triangleq \int_{{\mathbb R}^d} e^{-ix\cdot\xi}f(x)\rm dx. \end{split}\end{aligned}$$ Moreover, we also need the following frequency cut-off operators: for any $N\in2^{\mathbb N}$, $$\begin{aligned} \begin{split} P_{\leqslant N} f \triangleq &\mathcal{F}^{-1}\left(\chi\Big(\frac{\xi}N\Big) \hat{f}(\xi)\right), \\ P_{> N} f \triangleq &f-P_{\leqslant N} f. \end{split}\end{aligned}$$ The following definition for \"admissible exponent pair\" will be used frequently. ** **Definition** 11**. *For any $0\leqslant\gamma\leqslant 1$, we call that the exponent pair $(q,r)\in{\mathbb R}^2$ is $\dot H^\gamma$-$admissible$, if $\frac{2}{q}+\frac{d}{r}=\frac{d}{2}-\gamma$, $2\leqslant q\leqslant\infty$, $2\leqslant r\leqslant\infty$, and $(q,r,d)\ne(2,\infty,2)$. If $\gamma=0$, we say that $(q,r)$ is $L^2$-$admissible$.* ## Useful lemmas First, we recall the following Strichartz estimates, see [@KT98AJM] and the references therein. **Lemma 12** (Strichartz estimate). *Let $I\subseteq {\mathbb R}$. Suppose that $(q,r)$ and $(\widetilde{q},\widetilde{r})$ are $L_x^2$-admissible. Then, $$\begin{aligned} \begin{split} \left\| e^{it\langle\nabla\rangle}\varphi\right\|_{L_t^qL_x^r(I)} \lesssim\left\|\langle\nabla\rangle^{\frac{d+2}2(\frac 12-\frac 1r)}\varphi\right\|_{L_x^2}, \end{split}\end{aligned}$$ and $$\begin{aligned} \begin{split} \big\|\int_0^t e^{i(t-s)\langle\nabla\rangle} F(s)\mathrm{\ d} s\big\|_{L_t^qL_x^r({\mathbb R})} \lesssim\left\|\langle\nabla\rangle^{\frac{d+2}2(1-\frac 1r-\frac 1{\widetilde{r}})}F\right\|_{L_t^{\widetilde{q}'} L_x^{\widetilde{r}'}({\mathbb R})}. \end{split}\end{aligned}$$* *Remark 13*. It is known that there are some derivatives loss for the Strichartz estimates on the linear Klein-Gordon equations. The loss of derivatives in these estimates pose a significant obstacle to our analysis. In particular, the following estimates will be used frequently below: Suppose that $$\begin{aligned} \phi(t)= \mathrm{e}^{it\langle\nabla\rangle}\phi_0+ \int_0^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}F(s)\,ds, \quad (t,x)\in {\mathbb R}\times {\mathbb R}^d.\end{aligned}$$ Then $$\begin{aligned} \big\|\phi\big\|_{L^\frac{2(d+2)}{d}_{tx}(I)}+\big\|\langle\nabla\rangle^\frac12\phi\big\|_{L^\infty_tL^2_x(I)} \lesssim \big\|\langle\nabla\rangle^\frac12\phi_0\big\|_{L^2_x}+\big\|F\big\|_{L^\frac{2(d+2)}{d+4}_{tx}(I)}.\end{aligned}$$ We also need the following dispersive estimates, see for example [@MaStWa-KG-decay]. **Lemma 14** (Dispersive estimate). *Let $2\le p\le +\infty$ and $N>0$. Then, $$\begin{aligned} \begin{split} \label{eq:dispersive-1} \left\|e^{it\langle\nabla\rangle} P_N\varphi\right\|_{L_x^p} \lesssim\langle t\rangle^{-d(\frac 12-\frac 1p)}\left\|\langle\nabla\rangle^{\frac{d+2}2(1-\frac 2p)} P_N\varphi\right\|_{L_x^{p'}}. \end{split}\end{aligned}$$* The following are Coifman-Meyer's Multiplier Theorem, see [@Coifman-Meyer]. **Lemma 15**. *Let $m\in L^\infty({\mathbb R}^{nd})$ be smooth away from the origin and satisfy that for any multi-indices $\vec{\alpha}=\{\alpha_1,\alpha_2,\cdots,\alpha_n\}\in {\mathbb Z}^{nd}$ with $\alpha_j\le 2d+1$ and any $\xi_1,\cdots,\xi_n\in {\mathbb R}^d\setminus\{0\}$, $$\big|\partial_{\xi_1}^{\alpha_1}\partial_{\xi_2}^{\alpha_2}\cdots \partial_{\xi_n}^{\alpha_n}m(\xi_1,\xi_2,\cdots,\xi_n)\big| \le C(\vec \alpha)(|\xi_1|+|\xi_2|+\cdots+|\xi_n|)^{-|\vec \alpha|}.$$ Then for any $f_j,j=1,2,\cdots,n$, $$\begin{aligned} &\left\|\int_{\xi=\xi_1+\cdots+\xi_n} \mathrm{e}^{i\xi\cdot x} m(\xi_1,\xi_2,\cdots,\xi_n)\widehat{f_1}(\xi_1)\widehat{f_2}(\xi_2) \cdots\widehat{f_n}(\xi_n)\,d\xi_1d\xi_2\cdots d\xi_n\right\|_{L^p({\mathbb R}^d)}\\ &\qquad\qquad \le C(p,p_1,\cdots,p_n)\|f_1\|_{L^{p_1}({\mathbb R}^d)}\|f_2\|_{L^{p_2}({\mathbb R}^d)}\cdots \|f_n\|_{L^{p_n}({\mathbb R}^d)},\end{aligned}$$ where $$0< p< +\infty, 1<p_j\le+\infty \mbox{ for } j=1,2,\cdots,n, \frac1p=\frac1{p_1}+\frac1{p_2}+\cdots +\frac1{p_n}.$$* The Coifman-Meyer Multiplier Theorem is reduced to the Mihlin-Hörmander Multiplier Theorem when $n=1$ and $1<p<+\infty$. The Kato-Ponce inequality will be frequently used in this paper. The result was originally proved in [@Kato-Ponce] and then extended to the endpoint case in [@BoLi-KatoPonce; @Li-KatoPonce] recently. **Lemma 16** (Kato-Ponce inequality). *For $s>0$, $1<p\le \infty$, $1<p_1,p_3< \infty$ and $1<p_2,p_4 \le \infty$ satisfying $\frac1p=\frac1{p_1}+\frac1{p_2}$ and $\frac1p=\frac1{p_3}+\frac1{p_4}$, the following inequality holds: $$\begin{aligned} \big\||\nabla|^s(fg)\big\|_{L^p}\le C\big( \||\nabla|^sf\|_{L^{p_1}}\|g\|_{L^{p_2}}+ \||\nabla|^sg\|_{L^{p_3}}\|f\|_{L^{p_4}}\big),\end{aligned}$$ where the constant $C>0$ depends on $s,p,p_1,p_2,p_3,p_4$.* ## Some known results We will rely on the following well-known two results for the nonlinear Schrödinger equation and nonlinear Klein-Gordon equation. Scattering for the defocusing cubic nonlinear Schrödinger equation in two dimensional case was first proved by Killip, Tao and Visan [@KTV09JEMS] in radial case, and later on extended by Dodson [@Dod16Duke] to the non-radial case. In three dimensional case, it was proved by Ginibre and Velo [@GV92CMP]. In summary, we state them as: **Lemma 17** (Scattering for NLS). *Let $d=2,3$, $s\ge 0$ and $s_d=d-2$, and let $v_0\in H^s\cap H^{s_d}({\mathbb R}^d)$ and $v$ be the solution of the defocusing cubic nonlinear Schrödinger equation [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}. Suppose that $(q,r)$ is $L_x^2$-admissible. Then there hold, $$\begin{aligned} \begin{split} \left\||\nabla|^s v\right\|_{L_t^qL_x^r({\mathbb R})} \le C\big(\left\|v_0\right\|_{H^{s_d}_x}\big)\left\|v_0\right\|_{\dot H^s},\quad \mbox{ for any }s \ge 0. \end{split}\end{aligned}$$* The following large data scattering results for defocusing cubic nonlinear Klein-Gordon equation was proved by Killip, Stovall, Visan [@KillpVisan-KG]. **Lemma 18** (Large data scattering for Klein-Gordon). *Let $d=2,3$ and $(u_0,u_1)\in H^1({\mathbb R}^d)\times L^2({\mathbb R}^d)$. Suppose that $u$ is a solution of the defocusing cubic nonlinear Klein-Gordon equation $$\begin{aligned} \begin{split} \left\{ \aligned &\partial_{tt} u - \Delta u + u= -|u|^2u, \\ & u(0,x) = u_0(x),\quad \partial_tu(0,x) = u_1(x). \endaligned \right. \end{split}\end{aligned}$$ Then there exists a continuous function $C>0$ such that $$\begin{aligned} \begin{split} \left\|u\right\|_{X_{s_c}({\mathbb R})} \le C\big(E(u_0,u_1)\big), \end{split}\end{aligned}$$ where $$E(u,u_t)\triangleq \big\|\nabla u\big\|_{L^2_x}^2+\big\|\partial_t u\big\|_{L^2_x}^2+\big\| u\big\|_{L^2_x}^2+\frac12\big\| u\big\|_{L^4_x}^4.$$* We also need the following small data scattering result, which can be found in [@KillpVisan-KG]. Note that we put a slight strong assumption on the smallness of the initial data in the following lemma for a slight strong spacetime estimates. **Lemma 19** (Small data scattering for Klein-Gordon). *Let $d=2,3, \mu\in{\mathbb R}$, $s_c=\frac{d-1}{2}$, $(u_0,u_1)\in H^1({\mathbb R}^d)\times L^2({\mathbb R}^d)$ and $$\begin{aligned} \begin{split} \left\{ \aligned &\partial_{tt} u - \Delta u + u= -\mu |u|^2u, \\ & u(0,x) = u_0(x),\quad \partial_tu(0,x) = u_1(x). \endaligned \right. \end{split}\end{aligned}$$ There exists some constant $\delta_0>0$, such that $$\big\|\langle\nabla\rangle^\frac12 (u_0,\langle\nabla\rangle^{-1}u_1)\big\|_{\dot H^{s_c}\times \dot H^{s_c}}\le \delta_0,$$ then there exists a constant $C>0$ such that $$\begin{aligned} \begin{split} \left\|u\right\|_{X_{s_c}({\mathbb R})} \le C\delta_0. \end{split}\end{aligned}$$* # Proof of Propositions [Proposition 10](#prop:v-varep){reference-type="ref" reference="prop:v-varep"} {#sec:prop-key} .5cm As delineated in the introduction, a trivial bound derived from the energy conservation law takes the form: $$\big\|\partial_t v^\varepsilon\big\|_{L^2_x}=O\big(\varepsilon^{-1}\big).$$ However, this estimate falls short in capturing the true magnitude of the singularity of $\partial_t v^\varepsilon$ inherent in $\varepsilon$. Its weakness becomes evident, significantly limiting our ability to achieve a desired convergence estimate. To address this issue, our first pivotal insight revolves around the notion of a *rescaled complex expansion*. This strategic concept proves invaluable in mitigating the impact of the singularity. By applying appropriate transformations to [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}, we effectively reduce it to a nonlinear Klein-Gordon equation endowed with specific initial data. This refined equation facilitates the derivation of a more favorable spacetime estimate that involves time derivatives, thus contributing to a more accurate understanding of the problem's behavior. ## Rescaled complex expansion {#sec:RCE} We introduce the Navier-Stokes-like scaling $$\mathcal{S}_\varepsilon f(t,x)\triangleq \varepsilon f(\varepsilon^2t, \varepsilon x).$$ Denote $$h^{\varepsilon}(t,x)\triangleq \mathcal{S}_\varepsilon\big(v^{\varepsilon}\big)(t,x),$$ then $h^{\varepsilon}$ obeys the following equation: $$\begin{aligned} \begin{split} \label{eqs:h-varep} \left\{ \aligned &\partial_{tt}h^{\varepsilon}+2i\partial_{t} h^{\varepsilon} - \Delta h^{\varepsilon}= - 3|h^{\varepsilon}|^2 h^{\varepsilon}, \\ & h^{\varepsilon}(0,x) =\mathcal{S}_\varepsilon v_0(x),\quad \partial_th^{\varepsilon}(0,x) =\varepsilon^2 \mathcal{S}_\varepsilon v_1(x). \endaligned \right. \end{split}\end{aligned}$$ Then we shall prove the following result. **Proposition 20**. *Let $h^{\varepsilon}$ be the solution of [\[eqs:h-varep\]](#eqs:h-varep){reference-type="eqref" reference="eqs:h-varep"}. Then for any $\gamma\ge 0, \beta\in [0,\gamma]$ and any $(v_0,v_1)$ satisfying $$(v_0,v_1)\in H^\gamma({\mathbb R}^2)\times H^{\gamma-2}({\mathbb R}^2),\quad (v_0,v_1)\in H^1({\mathbb R}^2)\times L^2({\mathbb R}^2),\quad \mbox{ and }\quad v_1\in H^{\beta-1}({\mathbb R}^2),$$ there hold $$\begin{aligned} \label{est:h-varep-px} \big\||\nabla|^{\gamma}& P_{\le 1}h^{\varepsilon}\big\|_{L^4_{tx}\cap L^\infty_tL^2_x({\mathbb R})} +\big\||\nabla|^{\beta-\frac12}P_{> 1}h^{\varepsilon}\big\|_{L^4_{tx}\cap L^\infty_t H^\frac12_x({\mathbb R})} \le C \varepsilon^\gamma,\end{aligned}$$ and $$\begin{aligned} \label{est:h-varep-pt} \big\| |\nabla|^\gamma\partial_th^{\varepsilon} \big\|_{L^4_{tx}({\mathbb R})}\le C \varepsilon^2,\end{aligned}$$ where $C>0$ is only dependent of $\big\|(v_0,v_1)\big\|_{H^\gamma\times H^{\gamma-2}}$ and $\big\|(v_0,v_1)\big\|_{H^1_x\times L^2_x}$ and $\|v_1\|_{H^{\beta-1}}$.* To prove the proposition, we define $$\begin{aligned} \label{def:w-h} w(t,x)=\mathrm{e}^{it}h^{\varepsilon}(t,x),\end{aligned}$$ then $w$ is governed by the following *complex expanded* nonlinear Klein-Gordon equation with specific initial data: $$\begin{aligned} \begin{split} \label{eqs:scaling-w} \left\{ \aligned &\partial_{tt}w-\Delta w+w=-3|w|^2w, \\ & w(0)=\mathcal{S}_\varepsilon v_0,\quad \partial_t w(0)= i\mathcal{S}_\varepsilon v_0+\varepsilon^2 \mathcal{S}_\varepsilon v_1. \endaligned \right. \end{split}\end{aligned}$$ We emphasis that the forms of $w(0)$ and $\partial_t w(0)$ are crucial for all analysis later. For the solution $w$ to the equation [\[eqs:scaling-w\]](#eqs:scaling-w){reference-type="eqref" reference="eqs:scaling-w"}, it is clear that $$\begin{aligned} E\big(w(0),\partial_t w(0)\big)&=\varepsilon^2\big\|\nabla v_0\big\|_{L^2_x}^2+\big\|v_0\big\|_{L^2_x}^2+\big\|iv_0+\varepsilon^2v_1\big\|_{L^2_x}^2+\frac32\varepsilon^2\big\|v_0\big\|_{L^4_x}^4\\ &\le C\big(\|(v_0,v_1)\|_{H^1_x\times L^2_x}\big).\end{aligned}$$ Then it follows from Lemma [Lemma 18](#lem:spacetime-norm-KG){reference-type="ref" reference="lem:spacetime-norm-KG"} that $$\begin{aligned} \label{est:w-spacetime} \|w\|_{L^4_{tx}({\mathbb R})}\le C\big(\|(v_0,v_1)\|_{H^1_x\times L^2_x}\big).\end{aligned}$$ ## Leftward and rightward waves {#sec:LR-wave} Denote $$\begin{aligned} \begin{split} \label{relationship-w-W12} \left\{ \aligned &\mathcal W_1=\langle \nabla \rangle^{-1}\big(\partial_t-i \langle \nabla\rangle\big)w, \\ &\mathcal W_2=\langle \nabla \rangle^{-1}\big(\partial_t+i \langle \nabla\rangle\big)w. \endaligned \right. \end{split}\end{aligned}$$ $\mathcal W_1, \mathcal W_2$ are referred to the *leftward* and *rigthward waves* respectively. It is easy to derive that $$\begin{aligned} \begin{split} \label{eq:w-W12} \left\{ \aligned &w=\frac i2\big(\mathcal W_1-\mathcal W_2\big), \\ &\partial_t w=\frac 12\langle\nabla\rangle\big(\mathcal W_1+\mathcal W_2\big). \endaligned \right. \end{split}\end{aligned}$$ Moreover, $\mathcal W_1$ and $\mathcal W_2$ satisfy $$\begin{aligned} \begin{split} \label{eq:W-12-KG} \left\{ \aligned &\big(\partial_t+i \langle \nabla\rangle\big)\mathcal W_1=-3\langle\nabla\rangle^{-1}\big(|w|^2w\big), \\ &\big(\partial_t-i \langle \nabla\rangle\big)\mathcal W_2=-3\langle\nabla\rangle^{-1}\big(|w|^2w\big) \endaligned \right. \end{split}\end{aligned}$$ with the initial datum $$\begin{aligned} \begin{split} \label{eq:W-12-KG-intialdatum} \left\{ \aligned &\mathcal W_1(0)=\mathcal W_{1,0}\triangleq i\langle \nabla \rangle^{-1}\big(1- \langle \nabla\rangle\big)\mathcal{S}_\varepsilon v_0+\varepsilon^2\langle \nabla \rangle^{-1}\mathcal{S}_\varepsilon v_1;\\ &\mathcal W_2(0)=\mathcal W_{2,0}\triangleq i\langle \nabla \rangle^{-1}\big(1+ \langle \nabla\rangle\big)\mathcal{S}_\varepsilon v_0+\varepsilon^2\langle \nabla \rangle^{-1}\mathcal{S}_\varepsilon v_1. \endaligned \right. \end{split}\end{aligned}$$ We emphasis that the form of $\mathcal W_{1,0}$ is crucial, which is one of the main reasons that we introduce the complex expansion of $\varphi=\mathcal{S}_\varepsilon(v^\varepsilon)$. ## Strichartz estimates {#subsec:Stri} Firstly of all, we aim to show the following Strichartz estimates for the leftward and rightward waves: **Lemma 21**. *Under the same hypothesis as Proposition [Proposition 20](#prop:h-varep){reference-type="ref" reference="prop:h-varep"}, then $$\begin{aligned} \label{est:mathcal-W12} \big\||\nabla|^{\gamma}& P_{\le 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_tL^2_{x} ({\mathbb R})} +\big\||\nabla|^{\beta-\frac12} P_{> 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_t H^\frac12_{x}({\mathbb R})} %+\big\||\nabla|^{\gamma}\mathcal W_j\big\|_{L^\infty_tL^2_{x}(\R)} \le C \varepsilon^\gamma,\end{aligned}$$ where $C>0$ is only dependent of $\big\|(v_0,v_1)\big\|_{H^\gamma\times H^{\gamma-2}}$ and $\big\|(v_0,v_1)\big\|_{H^1_x\times L^2_x}$ and $\|v_1\|_{H^{\beta-1}}$.* *Proof.* We first consider the low-frequency part. By [\[eq:W-12-KG\]](#eq:W-12-KG){reference-type="eqref" reference="eq:W-12-KG"} and Duhamel's formula, $$P_{\le 1}\mathcal W_j(t)=\mathrm{e}^{\mp i(t-t_0)\langle \nabla\rangle}P_{\le 1} \mathcal W_j(t_0) -3\int_{t_0}^t \mathrm{e}^{\mp i(t-s)\langle \nabla\rangle}\langle\nabla \rangle^{-1} P_{\le 1} \big(|w|^2w\big)\,ds.$$ By Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that for any $I=[t_0,t_1]\subset {\mathbb R}^+$ (the negative time direction can be treated in the same way), $$\begin{aligned} \label{est:W-12-low-1} \big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_t L^2_x(I)} \lesssim & \big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_0)\big\|_{L^2_x} + \big\||\nabla|^{\gamma} P_{\le 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} .\end{aligned}$$ Then we estimate the second term above. It is obvious that $$\begin{aligned} \big\||\nabla|^{\gamma} P_{\le 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} \le & \Big\||\nabla|^{\gamma} P_{\le 1}\left(|P_{\le 1} w|^2P_{\le 1} w\right)\Big\|_{L^\frac43_{tx}(I)}\\ &\quad +\Big\||\nabla|^{\gamma} P_{\le 1}\Big(|w|^2w-\left(|P_{\le 1} w|^2P_{\le 1} w\right)\Big)\Big\|_{L^\frac43_{tx}(I)}\\ \lesssim & \Big\||\nabla|^{\gamma} \left(|P_{\le 1} w|^2P_{\le 1} w\right)\Big\|_{L^\frac43_{tx}(I)}\\ &\quad +\Big\||w|^2w-\left(|P_{\le 1} w|^2P_{\le 1} w\right)\Big\|_{L^\frac43_{tx}(I)}.\end{aligned}$$ Note that $$|w|^2w-\left(|P_{\le 1} w|^2P_{\le 1} w\right)=O(P_{>1}w\cdot w^2),$$ where the terms in $O(fgh)$ is the cubic combination of $f,g,h$ and their complex conjugate. Therefore, by Bernstein's and Kato-Ponce's inequalities we further have that $$\begin{aligned} \big\||\nabla|^{\gamma} P_{\le 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} \lesssim & \big\||\nabla|^\gamma P_{\le 1} w \big\|_{L^4_{tx}(I)}\big\|w\big\|_{L^4_{tx}(I)}^2 + \big\| P_{> 1} w \big\|_{L^4_{tx}(I)}\big\|w\big\|_{L^4_{tx}(I)}^2. %\\ %\lesssim & %\big\||\nabla|^\gamma P_{\le 1} w \big\|_{L^4_{tx}(I)}\big\|w\big\|_{L^4_{tx}(I)}^2\\ %&\quad+ \big\| P_{> 1} |\nabla|^{\gamma-\frac12} w \big\|_{L^4_{tx}(I)}\big\|w\big\|_{L^4_{tx}(I)}^2.\end{aligned}$$ Then by [\[eq:w-W12\]](#eq:w-W12){reference-type="eqref" reference="eq:w-W12"}, it gives that $$\begin{aligned} \big\||\nabla|^{\gamma} P_{\le 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} \lesssim & \sum\limits_{j=1,2}\Big(\big\||\nabla|^\gamma P_{\le 1} \mathcal W_j \big\|_{L^4_{tx}(I)}+ \big\| P_{> 1} \mathcal W_j \big\|_{L^4_{tx}(I)}\Big) \big\|w\big\|_{L^4_{tx}(I)}^2.\end{aligned}$$ Therefore, this estimate combining with [\[est:W-12-low-1\]](#est:W-12-low-1){reference-type="eqref" reference="est:W-12-low-1"}, yields that there exist some absolute postive constants $C_0,C_1$, such that $$\begin{aligned} \label{est:W-12-low} \big\||\nabla|^{\gamma} & P_{\le 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_t L^2_x(I)} \le C_0\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_0)\big\|_{L^2_x} \notag\\ & + C_1 \sum\limits_{j=1,2}\Big(\big\||\nabla|^\gamma P_{\le 1} \mathcal W_j \big\|_{L^4_{tx}(I)}+ \big\| P_{> 1} \mathcal W_j \big\|_{L^4_{tx}(I)}\Big) \big\|w\big\|_{L^4_{tx}(I)}^2.\end{aligned}$$ Next, we consider the high-frequency part. We first consider the case when $\beta\ge \frac12$. By [\[eq:W-12-KG\]](#eq:W-12-KG){reference-type="eqref" reference="eq:W-12-KG"} and Duhamel's formula, $$P_{> 1}\mathcal W_j(t)=\mathrm{e}^{\mp i(t-t_0)\langle \nabla\rangle}P_{> 1} \mathcal W_j(t_0) -3\int_{t_0}^t \mathrm{e}^{\mp i(t-s)\langle \nabla\rangle}\langle\nabla \rangle^{-1} P_{> 1} \big(|w|^2w\big)\,ds.$$ Then by Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that for any $\beta\in [0,\gamma]$, $$\begin{aligned} \label{est:W-12-high-1} &\big\||\nabla|^{\beta-\frac12} P_{> 1}\mathcal W_j\big\|_{L^4_{tx}(I)} +\big\||\nabla|^{\beta} P_{> 1}\mathcal W_j\big\|_{L^\infty_{t}L^2_x(I)}\notag\\ \lesssim & \big\||\nabla|^{\beta-\frac12} \langle \nabla\rangle^\frac12 P_{> 1}\mathcal W_j(t_0)\big\|_{L^2_x} + \big\||\nabla|^{\beta-\frac12} P_{> 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} \notag\\ \lesssim & \big\||\nabla|^{\beta} P_{> 1}\mathcal W_j(t_0)\big\|_{L^2_x} + \big\||\nabla|^{\beta-\frac12} P_{> 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} .\end{aligned}$$ Then we estimate the second term above. Note that $$P_{> 1}\big(|w|^2w\big)=P_{> 1}O\big(P_{\gtrsim 1}w\cdot w^2\big).$$ Then it infers that $$\begin{aligned} \big\||\nabla|^{\beta-\frac12} P_{> 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} \lesssim & \Big\||\nabla|^{\beta-\frac12} P_{\gtrsim 1}w\Big\|_{L^4_{tx}(I)}\|w\|_{L^4_{tx}(I)}^2\\ &\quad +\Big\|P_{\gtrsim 1}w\Big\|_{L^4_{tx}(I)}\Big\||\nabla|^{\beta-\frac12} w\Big\|_{L^4_{tx}(I)}\|w\|_{L^4_{tx}(I)}.\end{aligned}$$ By Bernstein's inequality, we have that $$\begin{aligned} \big\|P_{\gtrsim 1}w\big\|_{L^4_{tx}(I)}+& \Big\||\nabla|^{\beta-\frac12} P_{\gtrsim 1}w\Big\|_{L^4_{tx}(I)}\\ & \lesssim \Big\||\nabla|^{\beta-\frac12} P_{> 1}w\Big\|_{L^4_{tx}(I)} + \Big\||\nabla|^{\gamma} P_{\le 1}w\Big\|_{L^4_{tx}(I)};\end{aligned}$$ and $$\begin{aligned} \Big\||\nabla|^{\beta-\frac12} w\Big\|_{L^4_{tx}(I)} & \lesssim \Big\||\nabla|^{\beta-\frac12} P_{> 1}w\Big\|_{L^4_{tx}(I)} + \|w\|_{L^4_{tx}(I)}.\end{aligned}$$ Applying these two estimates, we further obtain that $$\begin{aligned} \big\||\nabla|^{\beta-\frac12} P_{> 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} \lesssim &\Big(\big\||\nabla|^\gamma P_{\le 1} w \big\|_{L^4_{tx}(I)} + \big\| P_{> 1} |\nabla|^{\beta-\frac12} w \big\|_{L^4_{tx}(I)}\Big) \big\|w\big\|_{L^4_{tx}(I)}^2.\end{aligned}$$ Then by [\[eq:w-W12\]](#eq:w-W12){reference-type="eqref" reference="eq:w-W12"}, it drives that $$\begin{aligned} \big\||\nabla|^{\beta-\frac12} P_{> 1}\big(|w|^2w\big)\big\|_{L^\frac43_{tx}(I)} \lesssim & \sum\limits_{j=1,2}\Big(\big\||\nabla|^\gamma P_{\le 1} \mathcal W_j \big\|_{L^4_{tx}(I)}+ \big\| P_{> 1} |\nabla|^{\beta-\frac12} \mathcal W_j \big\|_{L^4_{tx}(I)}\Big) \big\|w\big\|_{L^4_{tx}(I)}^2.\end{aligned}$$ This together with [\[est:W-12-high-1\]](#est:W-12-high-1){reference-type="eqref" reference="est:W-12-high-1"}, yields that $$\begin{aligned} \label{est:W-12-high} &\big\||\nabla|^{\beta-\frac12} P_{> 1}\mathcal W_j\big\|_{L^4_{tx}(I)} +\big\||\nabla|^{\beta} P_{> 1}\mathcal W_j\big\|_{L^\infty_{t}L^2_x(I)}\notag\\ \le & C_0\big\||\nabla|^{\beta} P_{> 1}\mathcal W_j(t_0)\big\|_{L^2_x} \notag\\ & + C_1 \sum\limits_{j=1,2}\Big(\big\||\nabla|^\gamma P_{\le 1} \mathcal W_j \big\|_{L^4_{tx}(I)}+ \big\| P_{> 1} |\nabla|^{\beta-\frac12} \mathcal W_j \big\|_{L^4_{tx}(I)}\Big) \big\|w\big\|_{L^4_{tx}(I)}^2.\end{aligned}$$ Here the constants $C_0,C_1$ may vary with the ones in [\[est:W-12-low\]](#est:W-12-low){reference-type="eqref" reference="est:W-12-low"}, however, we still use the same notations since it is not essential in our analysis. Combining with [\[est:W-12-low\]](#est:W-12-low){reference-type="eqref" reference="est:W-12-low"}, [\[est:W-12-high\]](#est:W-12-high){reference-type="eqref" reference="est:W-12-high"} and Bernstein's inequality, we have that $$\begin{aligned} &\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_t L^2_x(I)} + \big\||\nabla|^{\beta-\frac12} P_{> 1}\mathcal W_j\big\|_{L^4_{tx}(I)} +\big\||\nabla|^{\beta} P_{> 1}\mathcal W_j\big\|_{L^\infty_{t}L^2_x(I)} \\ \le & 2C_0\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_0)\big\|_{L^2_x}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j(t_0)\big\|_{L^2_x}\Big) \\ & + 2C_1 \sum\limits_{j=1,2}\Big(\big\||\nabla|^\gamma P_{\le 1} \mathcal W_j \big\|_{L^4_{tx}(I)}+ \big\| P_{> 1} |\nabla|^{\beta-\frac12} \mathcal W_j \big\|_{L^4_{tx}(I)}\Big) \big\|w\big\|_{L^4_{tx}(I)}^2.\end{aligned}$$ Therefore, denote $X(I)$ to be $$X(I) = \sum\limits_{j=1,2}\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_t L^2_x(I)} + \big\||\nabla|^{\beta-\frac12} P_{> 1}\mathcal W_j\big\|_{L^4_{tx}(I)} +\big\||\nabla|^{\beta} P_{> 1}\mathcal W_j\big\|_{L^\infty_{t}L^2_x(I)}\Big),$$ then $$\begin{aligned} \label{est:W-12} X(I) \le 4C_0\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_0)\big\|_{L^2_x}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j(t_0)\big\|_{L^2_x}\Big) + 4C_1 \big\|w\big\|_{L^4_{tx}(I)}^2\cdot X(I).\end{aligned}$$ Note that we have the uniform boundedness of $\|w\|_{L^4_{tx}(I)}$ as in [\[est:w-spacetime\]](#est:w-spacetime){reference-type="eqref" reference="est:w-spacetime"}. Hence, there exists a constant $K=K(C_1)$ and a sequence of time intervals $$\bigcup\limits_{k=0}^K I_k={\mathbb R}^+,$$ with $$I_0=[0, t_1], \quad I_k=(t_k, t_{k+1}] \mbox{ for } k=1,\cdots, K-1, \quad I_{K}=[t_K, +\infty),$$ such that for any $1\le k\le K$, $$\begin{aligned} \label{subinterval-length-wIk} 4C_1 \big\|w\big\|_{L^4_{tx}(I_k)}^2 \le \frac12.\end{aligned}$$ This together with [\[est:W-12\]](#est:W-12){reference-type="eqref" reference="est:W-12"} infers that $$\begin{aligned} \label{est:X-Ik} X(I_k) \le 8C_0\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_k)\big\|_{L^2_x}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j(t_k)\big\|_{L^2_x}\Big).\end{aligned}$$ Furthermore, combining with [\[est:W-12-low\]](#est:W-12-low){reference-type="eqref" reference="est:W-12-low"} and [\[est:W-12-high\]](#est:W-12-high){reference-type="eqref" reference="est:W-12-high"}, we have that $$\begin{aligned} \big\||\nabla|^{\gamma} P_{\le 1}&\mathcal W_j\big\|_{L^\infty_tL^2_x(I_k)}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j\big\|_{L^\infty_tL^2_x(I_k)}\\ \le & 2C_0\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_k)\big\|_{L^2_x}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j(t_k)\big\|_{L^2_x}\Big) + 2C_1 X(I_k) \big\|w\big\|_{L^4_{tx}(I_k)}^2.\end{aligned}$$ Inserting [\[subinterval-length-wIk\]](#subinterval-length-wIk){reference-type="eqref" reference="subinterval-length-wIk"} and [\[est:X-Ik\]](#est:X-Ik){reference-type="eqref" reference="est:X-Ik"} into the estimate above, we get that $$\begin{aligned} \label{Wj-tk-Linfty} \big\||\nabla|^{\gamma} P_{\le 1}&\mathcal W_j\big\|_{L^\infty_tL^2_x(I_k)}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j\big\|_{L^\infty_tL^2_x(I_k)}\notag\\ \le & 4C_0\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_k)\big\|_{L^2_x}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j(t_k)\big\|_{L^2_x}\Big).\end{aligned}$$ In particular, the iteration implies that for any $1\le k\le K$, $$\begin{aligned} \label{Wj-tk-iteration} \big\||\nabla|^{\gamma} P_{\le 1}&\mathcal W_j\big\|_{L^\infty_tL^2_x(I_k)}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_j\big\|_{L^\infty_tL^2_x(I_k)}\notag\\ \le & (4C_0)^k\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_{j,0}\big\|_{L^2_x}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_{j,0}\big\|_{L^2_x}\Big).\end{aligned}$$ Therefore, inserting [\[Wj-tk-iteration\]](#Wj-tk-iteration){reference-type="eqref" reference="Wj-tk-iteration"} into [\[est:X-Ik\]](#est:X-Ik){reference-type="eqref" reference="est:X-Ik"} and [\[Wj-tk-Linfty\]](#Wj-tk-Linfty){reference-type="eqref" reference="Wj-tk-Linfty"}, we obtain that $$\begin{aligned} \label{est:Wj-IKi-fin} X(I_k) %+\big\||\nabla|^{\gamma}\mathcal W_j\big\|_{L^\infty_tL^2_x(I_k)} \le 4(4C_0)^{k+1}\Big(\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_{j,0}\big\|_{L^2_x}+\big\||\nabla|^{\beta}P_{>1} \mathcal W_{j,0}\big\|_{L^2_x}\Big).\end{aligned}$$ Note that $$\begin{aligned} \big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_{j,0}\big\|_{L^2_x}&+\big\||\nabla|^{\beta}P_{>1} \mathcal W_{j,0}\big\|_{L^2_x}\\ %\lesssim & %\big\||\nabla|^{\gamma}\mathcal{S}_\varepsilon v_0\big\|_{L^2_x} %+\varepsilon^2\big\||\nabla|^{\gamma}\langle \nabla\rangle^{-1}\mathcal{S}_\varepsilon v_1\big\|_{L^2_x}\\ \lesssim & \big\||\nabla|^{\gamma}\mathcal{S}_\varepsilon v_0\big\|_{L^2_x} +\varepsilon^2\Big(\big\||\nabla|^\gamma P_{\le 1} \mathcal{S}_\varepsilon v_1\big\|_{L^2_x}+\big\||\nabla|^{\beta-1}P_{>1}\mathcal{S}_\varepsilon v_1\big\|_{L^2_x}\Big).\end{aligned}$$ If $\gamma\le 2$, it is controlled by $$\varepsilon^\gamma \|(v_0,v_1)\|_{H^\gamma\times (H^{\beta-1}\cap L^2)}.$$ If $\gamma> 2$, it is controlled by $$\varepsilon^\gamma \|(v_0,v_1)\|_{H^\gamma\times (H^{\beta-1}\cap H^{\gamma-2})}.$$ Therefore, these combining with [\[est:Wj-IKi-fin\]](#est:Wj-IKi-fin){reference-type="eqref" reference="est:Wj-IKi-fin"} implies [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} in the case of $\frac12\le \beta\le \gamma$. When $\beta<\frac12$, by Bernstein's inequality, we have that $$\begin{aligned} \big\||\nabla|^{\beta-\frac12} P_{> 1}\mathcal W_j\big\|_{L^4_{tx}(I)} +\big\||\nabla|^{\beta} P_{> 1}\mathcal W_j\big\|_{L^\infty_{t}L^2_x(I)} \le & \big\|P_{> 1}\mathcal W_j\big\|_{L^4_{tx}(I)} +\big\||\nabla|^{\frac12} P_{> 1}\mathcal W_j\big\|_{L^\infty_{t}L^2_x(I)}.\end{aligned}$$ Then by [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} for $\beta=\frac12$, it gives that $$\begin{aligned} \big\||\nabla|^{\beta-\frac12} P_{> 1}\mathcal W_j\big\|_{L^4_{tx}(I)} +\big\||\nabla|^{\beta} P_{> 1}\mathcal W_j\big\|_{L^\infty_{t}L^2_x(I)} \le & C\varepsilon^\gamma.\end{aligned}$$ When $\gamma<\frac12$, by [\[est:W-12-low\]](#est:W-12-low){reference-type="eqref" reference="est:W-12-low"}, we have that $$\begin{aligned} \big\||\nabla|^{\gamma} & P_{\le 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_t L^2_x(I)} \le C_0\big\||\nabla|^{\gamma} P_{\le 1}\mathcal W_j(t_0)\big\|_{L^2_x} \notag\\ & + C_1 \sum\limits_{j=1,2}\big\||\nabla|^\gamma P_{\le 1} \mathcal W_j \big\|_{L^4_{tx}(I)} \big\|w\big\|_{L^4_{tx}(I)}^2 +C_2\varepsilon^\frac12,\end{aligned}$$ where $C_j,j=1,2,3$ are the constants only dependent of $\|(v_0,v_1)\|_{H^1_x\times L^2_x}$. Then arguing similarly as above, we obtain that $$\begin{aligned} \big\||\nabla|^{\gamma} & P_{\le 1}\mathcal W_j\big\|_{L^4_{tx}\cap L^\infty_t L^2_x(I)} \le C\varepsilon^\gamma.\end{aligned}$$ This proves [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} in the cases of $\beta<\frac12$ and $\gamma<\frac12$, and thus finishes the proof of the lemma. ◻ Now combining [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} with [\[eq:w-W12\]](#eq:w-W12){reference-type="eqref" reference="eq:w-W12"} yields that $$\begin{aligned} \label{est:w-X0} \big\||\nabla|^{\gamma}& P_{\le 1}w\big\|_{L^4_{tx}\cap L^\infty_tL^2_{x} ({\mathbb R})} +\big\||\nabla|^{\beta-\frac12} P_{> 1}w\big\|_{L^4_{tx}\cap L^\infty_t H^\frac12_{x}({\mathbb R})} \le C \varepsilon^\gamma;\end{aligned}$$ and $$\begin{aligned} \label{est:wt-X0} \big\||\nabla|^{\gamma}& P_{\le 1}\partial_t w\big\|_{L^4_{tx}\cap L^\infty_tL^2_{x} ({\mathbb R})} +\big\||\nabla|^{\beta-\frac12} P_{> 1}\partial_t w\big\|_{L^4_{tx}\cap L^\infty_t H^\frac12_{x}({\mathbb R})} \le C \varepsilon^\gamma.\end{aligned}$$ Scaling back, we obtain that for any $\gamma>0$, $$\begin{aligned} &\big\||\nabla|^{\gamma}v^{\varepsilon}\big\|_{X_0({\mathbb R})} \le C;\label{est:v-gamma} \\ &\big\||\nabla|^{\gamma}\partial_t v^{\varepsilon}\big\|_{X_0({\mathbb R})}\le C \varepsilon^{-2}.\label{est:vt-gamma}\end{aligned}$$ The estimate [\[est:v-gamma\]](#est:v-gamma){reference-type="eqref" reference="est:v-gamma"} is exactly required in Proposition [Proposition 10](#prop:v-varep){reference-type="ref" reference="prop:v-varep"} for [\[est:v-varep-px\]](#est:v-varep-px){reference-type="eqref" reference="est:v-varep-px"}. However, [\[est:vt-gamma\]](#est:vt-gamma){reference-type="eqref" reference="est:vt-gamma"} is not enough for [\[est:v-varep-pt\]](#est:v-varep-pt){reference-type="eqref" reference="est:v-varep-pt"} and thus for what we need to estimates the error term [\[est:main1-global\]](#est:main1-global){reference-type="eqref" reference="est:main1-global"}. ## An improved estimates for leftward wave {#subsec:Imp-Stri} To improve the estimate [\[est:vt-gamma\]](#est:vt-gamma){reference-type="eqref" reference="est:vt-gamma"}, our key observation is that the behavior of $\mathcal W_1$ is better than that of $\mathcal W_2$. In the following, we shall prove **Lemma 22**. *Let $\gamma\ge 0$. Suppose that $(v_0,v_1)\in H^{\gamma+2}\times H^{\gamma}$, then $$\begin{aligned} \label{est:W1-gamma} \big\||\nabla|^{\gamma}\mathcal W_1\big\|_{L^4_{tx}({\mathbb R})}+\big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1\big\|_{L^\infty_t L^2_x({\mathbb R})} \le C\varepsilon^{2+\gamma},\end{aligned}$$ where $C$ only depends on $\| v_0\|_{H^{\gamma+2}_x}$ and $\| v_1\|_{H^{\gamma}_x}$.* *Proof.* To illustrate [\[est:W1-gamma\]](#est:W1-gamma){reference-type="eqref" reference="est:W1-gamma"}, we denote $$\varphi_1=\mathrm{e}^{it\langle\nabla\rangle}\mathcal W_1;\quad \varphi_2=\mathrm{e}^{-it\langle\nabla\rangle}\mathcal W_2.$$ Then we apply the Duhamel formula to rewrite the equation [\[eq:W-12-KG\]](#eq:W-12-KG){reference-type="eqref" reference="eq:W-12-KG"}, in terms of $\varphi_1$, as $$\begin{aligned} \label{duhamel-varphi1-1} \varphi_1(t)=&\varphi_1(t_0)-3\int_{t_0}^t \mathrm{e}^{is\langle\nabla\rangle}\langle\nabla\rangle^{-1}\big(|w(s)|^2w(s)\big).\end{aligned}$$ According to [\[eq:w-W12\]](#eq:w-W12){reference-type="eqref" reference="eq:w-W12"}, we write $$\begin{aligned} \label{nonlinear-w} |w|^2w=&-\frac i8\big|\mathcal W_2\big|^2\mathcal W_2+N_1\big(\mathcal W_1,\mathcal W_2\big)\notag\\ =&-\frac i8P_{\le 1}\Big(\big|P_{\le 1}\mathcal W_2\big|^2P_{\le 1}\mathcal W_2\Big)+N_1\big(\mathcal W_1,\mathcal W_2\big) +N_2\big(\mathcal W_2\big),\end{aligned}$$ where $$\begin{aligned} N_1\big(\mathcal W_1,\mathcal W_2\big)\triangleq & \frac i8\Big(\big|\mathcal W_1\big|^2\mathcal W_1-\mathcal W_1^2\overline{\mathcal W_2}-2\big|\mathcal W_1\big|^2\mathcal W_2+\mathcal W_2^2\overline{\mathcal W_1}+2\big|\mathcal W_2\big|^2\mathcal W_1\Big);\\ N_2\big(\mathcal W_2\big)\triangleq &- \frac i8\big|\mathcal W_2\big|^2\mathcal W_2+\frac i8P_{\le 1}\Big(\big|P_{\le 1}\mathcal W_2\big|^2P_{\le 1}\mathcal W_2\Big).\end{aligned}$$ Then by [\[nonlinear-w\]](#nonlinear-w){reference-type="eqref" reference="nonlinear-w"}, we further rewrite the formula [\[duhamel-varphi1-1\]](#duhamel-varphi1-1){reference-type="eqref" reference="duhamel-varphi1-1"} as [\[duhamel-varphi1-2\]]{#duhamel-varphi1-2 label="duhamel-varphi1-2"} $$\begin{aligned} \varphi_1(t)=&\varphi_1(t_0)-\frac 38i\int_{t_0}^t \mathrm{e}^{is\langle\nabla\rangle}\langle\nabla\rangle^{-1} P_{\le 1}\Big(\big|P_{\le 1}\mathcal W_2\big|^2P_{\le 1}\mathcal W_2\Big)\,ds\label{duhamel-varphi1-2-1}\\ &\quad -3\int_{t_0}^t \mathrm{e}^{is\langle\nabla\rangle}\langle\nabla\rangle^{-1} \Big(N_1\big(\mathcal W_1,\mathcal W_2\big) +N_2\big(\mathcal W_2\big)\Big)\,ds.\label{duhamel-varphi1-2-2}\end{aligned}$$ We first consider the term [\[duhamel-varphi1-2-1\]](#duhamel-varphi1-2-1){reference-type="eqref" reference="duhamel-varphi1-2-1"}. We will see that the inherent structure inside is crucial to produce small parameters, which is another reason that we introduce the complex expansion of $h^\varepsilon=\mathcal{S}_\varepsilon(v^\varepsilon)$. Taking Fourier transform, we have $$\begin{aligned} \widehat{\eqref{duhamel-varphi1-2-1}}(\xi) =& \frac 38i\int_{t_0}^t \int_{\xi_1+\xi_2+\xi_3=\xi}\mathrm{e}^{is\big(\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle\big)}\langle\xi\rangle^{-1} \chi_{\le 1}(\xi)\chi_{\le 1}(\xi_1)\chi_{\le 1}(\xi_2)\chi_{\le 1}(\xi_3)\\ &\qquad\qquad \cdot\hat \varphi_2(\xi_1)\hat \varphi_2(\xi_2)\widehat{\overline{\varphi_2}}(\xi_3)\,ds d\xi_1 d\xi_2.\end{aligned}$$ Note that $$\begin{aligned} \mathrm{e}^{is\big(\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle\big)}=\frac{d}{ds}\Big(\mathrm{e}^{is\big(\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle\big)}\Big) \frac{1}{i(\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle)}.\end{aligned}$$ Then using integration by parts, we further get that $$\begin{aligned} \widehat{\eqref{duhamel-varphi1-2-1}}(\xi) =& \frac 38 \int_{\xi_1+\xi_2+\xi_3=\xi}\mathrm{e}^{is\big(\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle\big)}\langle\xi\rangle^{-1}\frac{ \chi_{\le 1}(\xi)\chi_{\le 1}(\xi_1)\chi_{\le 1}(\xi_2)\chi_{\le 1}(\xi_3)}{\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle}\\ &\qquad\qquad \cdot\hat \varphi_2(\xi_1)\hat \varphi_2(\xi_2)\widehat{\overline{\varphi_2}}(\xi_3)\,d\xi_1d\xi_2\Big|_{t_0}^t\\ &-\frac 38\int_{t_0}^t \int_{\xi_1+\xi_2+\xi_3=\xi}\mathrm{e}^{is\big(\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle\big)}\langle\xi\rangle^{-1}\frac{ \chi_{\le 1}(\xi)\chi_{\le 1}(\xi_1)\chi_{\le 1}(\xi_2)\chi_{\le 1}(\xi_3)}{\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle}\\ &\qquad\qquad \cdot\partial_s\Big[\hat \varphi_2(\xi_1)\hat \varphi_2(\xi_2)\widehat{\overline{\varphi_2}}(\xi_3)\Big]\,d\xi_1d\xi_2\,ds.\end{aligned}$$ Denote the triple operator $T(\cdot,\cdot,\cdot)$ by $$\begin{aligned} \mathcal F\Big(T(f_1,f_2,f_3)\Big)(\xi)=&-\frac 38\langle\xi\rangle^{-1} \int_{\xi_1+\xi_2+\xi_3=\xi}\frac{ \chi_{\le 1}(\xi)\chi_{\le 1}(\xi_1)\chi_{\le 1}(\xi_2)\chi_{\le 1}(\xi_3)}{\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle}\\ &\qquad\qquad \cdot\widehat{f_1}(\xi_1)\widehat{f_2}(\xi_2)\widehat{f_3}(\xi_3)\,d\xi_1d\xi_2.\end{aligned}$$ Then it allows us to write $$\begin{aligned} \eqref{duhamel-varphi1-2-1} =& -\mathrm{e}^{it\langle\nabla\rangle}T(\mathcal W_2, \mathcal W_2, \mathcal W_2)(t)+\mathrm{e}^{it_0\langle\nabla\rangle}T(\mathcal W_2, \mathcal W_2, \mathcal W_2)(t_0)\\ &\quad +\int_{t_0}^t \mathrm{e}^{is\langle\nabla\rangle}T\big(\mathrm{e}^{-is\langle\nabla\rangle}\partial_s\varphi_2, \mathcal W_2, \mathcal W_2\big)(s)\,ds.\end{aligned}$$ Here we shall slightly abuse notation and write $\hat f(\xi)$ for both itself and its complex conjugate (i.e. $\widehat{\overline{f}}(\xi)$), since it has no influence in our analysis. Therefore, inserting the above into [\[duhamel-varphi1-2\]](#duhamel-varphi1-2){reference-type="eqref" reference="duhamel-varphi1-2"}, we obtain that $$\begin{aligned} \varphi_1(t)=&\varphi_1(t_0)-\mathrm{e}^{it\langle\nabla\rangle}T(\mathcal W_2, \mathcal W_2, \mathcal W_2)(t)+\mathrm{e}^{it_0\langle\nabla\rangle}T(\mathcal W_2, \mathcal W_2, \mathcal W_2)(t_0)\\ &\quad + \int_{t_0}^t \mathrm{e}^{is\langle\nabla\rangle}T\big(\mathrm{e}^{-is\langle\nabla\rangle}\partial_s\varphi_2, \mathcal W_2, \mathcal W_2\big)(s)\,ds\\ &\quad -3\int_{t_0}^t \mathrm{e}^{is\langle\nabla\rangle} \langle\nabla\rangle^{-1}\Big(N_1\big(\mathcal W_1,\mathcal W_2\big) +N_2\big(\mathcal W_2\big)\Big)\,ds.\end{aligned}$$ Then one can drive that $$\begin{aligned} \mathcal W_1(t)=&\mathrm{e}^{-i(t-t_0)\langle\nabla\rangle}\mathcal W_1(t_0)-T(\mathcal W_2, \mathcal W_2, \mathcal W_2)(t)+\mathrm{e}^{-i(t-t_0)\langle\nabla\rangle}T(\mathcal W_2, \mathcal W_2, \mathcal W_2)(t_0)\\ %\label{duhamel-varphi1-2-3}\\ &\quad + \int_{t_0}^t \mathrm{e}^{-i(t-s)\langle\nabla\rangle}T\big(\mathrm{e}^{-is\langle\nabla\rangle}\partial_s\varphi_2, \mathcal W_2, \mathcal W_2\big)(s)\,ds\\ %\label{duhamel-varphi1-2-4}\\ &\quad -3\int_{t_0}^t \mathrm{e}^{-i(t-s)\langle\nabla\rangle}\langle\nabla\rangle^{-1} \Big(N_1\big(\mathcal W_1,\mathcal W_2\big) +N_2\big(\mathcal W_2\big)\Big)\,ds. %\label{duhamel-varphi1-2-5}\end{aligned}$$ Therefore, by Strichartz estimates Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"} we get that for any $\gamma\ge 0$, any $t_0\in I\subset {\mathbb R}$, $$\begin{aligned} &\big\||\nabla|^{\gamma}\mathcal W_1\big\|_{L^4_{tx}(I)}+\big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1\big\|_{L^\infty_t L^2_x(I)}\\ \lesssim & \big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1(t_0)\big\|_{L^2_x} +\big\||\nabla|^\gamma T(\mathcal W_2, \mathcal W_2, \mathcal W_2)\big\|_{L^4_{tx}(I)}\\ &+\big\||\nabla|^{\gamma} T(\mathcal W_2, \mathcal W_2, \mathcal W_2)\big\|_{L^\infty_tL^2_x} +\big\||\nabla|^\gamma T\big(\mathrm{e}^{-is\langle\nabla\rangle}\partial_s\varphi_2, \mathcal W_2, \mathcal W_2\big)\big\|_{L^\frac43_{tx}(I)}\\ &+\big\||\nabla|^\gamma N_1\big(\mathcal W_1,\mathcal W_2\big) \big\|_{L^\frac43_{tx}(I)} +\big\||\nabla|^\gamma N_2\big(\mathcal W_2\big) \big\|_{L^\frac43_{tx}(I)}\\ \triangleq & \Pi_1+\cdots +\Pi_6.\end{aligned}$$ Below we will estimate $\Pi_1,\Pi_2,\cdots ,\Pi_6$ one by one. Note that $\langle \nabla\rangle-1=\frac{|\nabla|^2}{1+\langle\nabla\rangle}$. When $t_0=0$, by [\[eq:W-12-KG-intialdatum\]](#eq:W-12-KG-intialdatum){reference-type="eqref" reference="eq:W-12-KG-intialdatum"} we have $$\begin{aligned} \label{est:mathcal-w0} \big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1(0)\big\|_{L^2_x} \lesssim & \big\||\nabla|^{\gamma}\langle\nabla\rangle^{-\frac12}\big( \langle \nabla\rangle-1\big)\mathcal{S}_\varepsilon v_0\big\|_{L^2_x} +\varepsilon^2\big\||\nabla|^{\gamma}\langle\nabla\rangle^{-\frac12}\mathcal{S}_\varepsilon v_1\big\|_{L^2_x}\notag\\ \lesssim & \big\||\nabla|^{\gamma+2}\mathcal{S}_\varepsilon v_0\big\|_{L^2_x} +\varepsilon^2\big\||\nabla|^{\gamma}\mathcal{S}_\varepsilon \big({\mathrm{Re}}(v_1)\big)\big\|_{L^2_x}\notag\\ \le & C\varepsilon^{\gamma+2}.\end{aligned}$$ Here and below in this proof, $C$ only depends on $\| v_0\|_{H^{\gamma+2}_x}$ and $\| v_1\|_{H^{\gamma}_x}$. Note that $$\frac{1}{\langle\xi\rangle+\langle\xi_1\rangle+\langle\xi_2\rangle-\langle\xi_3\rangle}\in C^\infty\left[\big(\overline{B_1}\big)^4\right],$$ where $B_1\in {\mathbb R}^2$ is the unit ball. Then by Coifman-Meyer's Theorem and [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"}, we have that $$\begin{aligned} \big\||\nabla|^\gamma T(\mathcal W_2, \mathcal W_2, \mathcal W_2)\big\|_{L^4_{tx}({\mathbb R})} \lesssim & \big\||\nabla|^\gamma \mathcal W_2\big\|_{L^4_{tx}({\mathbb R})}\big\|\nabla \mathcal W_2\big\|_{L^\infty_{t}L^2_x({\mathbb R})}^2\\ \le & C \varepsilon^{\gamma+2}.\end{aligned}$$ Similarly, by Coifman-Meyer's Theorem and [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"}, we have that $$\begin{aligned} \big\||\nabla|^\gamma T(\mathcal W_{2}, \mathcal W_{2}, \mathcal W_{2})(t _0)\big\|_{L^2_{x}({\mathbb R}^2)} \lesssim & \big\||\nabla|^\gamma \mathcal W_{2}(t_0)\big\|_{L^2_{x}}\big\|\nabla \mathcal W_{2}(t_0)\big\|_{L^2_x}^2\\ \le & C \varepsilon^{\gamma+2}.\end{aligned}$$ By [\[eq:W-12-KG\]](#eq:W-12-KG){reference-type="eqref" reference="eq:W-12-KG"}, we have that $$\partial_t\varphi_2=-3\mathrm{e}^{-it\langle\nabla\rangle}\langle\nabla\rangle^{-1}\big(|w(t)|^2w(t)\big).$$ Therefore, arguing similarly as the estimation on $\Pi_2$, by [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} and [\[eq:w-W12\]](#eq:w-W12){reference-type="eqref" reference="eq:w-W12"} we obtain that $$\begin{aligned} &\big\||\nabla|^\gamma T\big(\mathrm{e}^{-is\langle\nabla\rangle}\partial_s\varphi_2, \mathcal W_2, \mathcal W_2\big)\big\|_{L^\frac43_{tx}({\mathbb R})}\\ \lesssim & \big\||\nabla|^\gamma \mathrm{e}^{-is\langle\nabla\rangle}\partial_s\varphi_2\big\|_{L^4_tL^2_{x}({\mathbb R})}\big\| \mathcal W_2\big\|_{L^4_tL^8_{x}({\mathbb R})}^2\\ &\qquad+\big\|\mathrm{e}^{-is\langle\nabla\rangle}\partial_s\varphi_2\big\|_{L^4_tL^2_{x}({\mathbb R})}\big\||\nabla|^\gamma \mathcal W_2\big\|_{L^4_tL^8_{x}({\mathbb R})}\big\|\mathcal W_2\big\|_{L^4_tL^8_{x}({\mathbb R})}\\ \lesssim & \big\||\nabla|^\gamma w\big\|_{L^4_{tx}({\mathbb R})} \big\||\nabla|^\frac34 w\big\|_{L^\infty_{t}L^2({\mathbb R})}^2 \big\||\nabla|^\frac14 \mathcal W_2\big\|_{L^\infty_{t}L^2_x({\mathbb R})}^2\\ \le & C \varepsilon^{\gamma+2}.\end{aligned}$$ By Kato-Ponce's inequality and [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"}, we have that $$\begin{aligned} &\big\||\nabla|^\gamma N_1\big(\mathcal W_1,\mathcal W_2\big) \big\|_{L^\frac43_{tx}(I)}\\ \lesssim & \big\||\nabla|^\gamma \mathcal W_1\big\|_{L^4_{tx}(I)} \Big(\big\|\mathcal W_1\big\|_{L^4_{tx}(I)}^2+\big\|\mathcal W_2\big\|_{L^4_{tx}(I)}^2\Big)\\ &\quad +\big\||\nabla|^\gamma \mathcal W_2\big\|_{L^4_{tx}(I)}\big\||\mathcal W_1\big\|_{L^4_{tx}(I)} \Big(\big\|\mathcal W_1\big\|_{L^4_{tx}(I)}+\big\||\mathcal W_2\big\|_{L^4_{tx}(I)}\Big)\\ \le & C \Big(\big\||\nabla|^\gamma \mathcal W_1\big\|_{L^4_{tx}(I)} +\varepsilon^\gamma \big\|\mathcal W_1\big\|_{L^4_{tx}(I)}\Big)\Big(\big\|\mathcal W_1\big\|_{L^4_{tx}(I)}+\big\||\mathcal W_2\big\|_{L^4_{tx}(I)}\Big).\end{aligned}$$ By Kato-Ponce's and Bernstein's inequalities, and [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"}, we have that $$\begin{aligned} &\big\||\nabla|^\gamma N_2\big(\mathcal W_2\big) \big\|_{L^\frac43_{tx}({\mathbb R})}\\ \lesssim & \big\||\nabla|^\gamma P_{>1}\mathcal W_2\big\|_{L^4_{tx}({\mathbb R})} \big\|\mathcal W_2\big\|_{L^4_{tx}({\mathbb R})}^2\\ \le & C \varepsilon^{2+\gamma}.\end{aligned}$$ Combining the estimates on $\Pi_2$--$\Pi_6$, we obtain that $$\begin{aligned} \label{est:mathcal-W1-improved} &\big\||\nabla|^{\gamma}\mathcal W_1\big\|_{L^4_{tx}(I)}+\big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1\big\|_{L^\infty_t L^2_x(I)}\notag\\ \le & C_1\big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1(t_0)\big\|_{L^2_x}+C_2\varepsilon^{2+\gamma}\notag\\ &\quad +C_3\Big(\big\||\nabla|^\gamma \mathcal W_1\big\|_{L^4_{tx}(I)} +\varepsilon^\gamma \big\|\mathcal W_1\big\|_{L^4_{tx}(I)}\Big)\Big(\big\|\mathcal W_1\big\|_{L^4_{tx}(I)}+\big\|\mathcal W_2\big\|_{L^4_{tx}(I)}\Big),\end{aligned}$$ where $C_j,j=1,2,3$ only depend on $\| v_0\|_{H^{\gamma+2}_x}$ and $\| v_1\|_{H^{\gamma}_x}$. To proceed, we will do estimates inductively in $\gamma$. First, for $\gamma=0$, we derive from [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} that there exists $C_0>0$ such that $$\big\|\mathcal W_1\big\|_{L^4_{tx}({\mathbb R})}+\big\||\mathcal W_2\big\|_{L^4_{tx}({\mathbb R})}\le C_0.$$ Therefore, there exists a constant $K=K(C_0,C_3)$ and a sequence of time intervals $$\bigcup\limits_{k=0}^K I_k={\mathbb R}^+,$$ (the negative time direction can be treated similarly) with $$I_0=[0, T_1], \quad I_k=(T_k, T_{k+1}] \mbox{ for } k=1,\cdots, K-1, \quad I_{K}=[T_K, +\infty),$$ such that $$\begin{aligned} \label{subinterval-length-W12} C_3\Big(\big\|\mathcal W_1\big\|_{L^4_{tx}(I)}+\big\|\mathcal W_2\big\|_{L^4_{tx}(I)}\Big) \le \frac1{4}.\end{aligned}$$ Therefore, by [\[est:mathcal-W1-improved\]](#est:mathcal-W1-improved){reference-type="eqref" reference="est:mathcal-W1-improved"}, for any $1\le k\le K$, we have that $$\begin{aligned} \big\|\mathcal W_1\big\|_{L^4_{tx}(I_k)}+\big\|\langle\nabla\rangle^\frac12\mathcal W_1\big\|_{L^\infty_t L^2_x(I_k)} \le 2C_1\big\|\langle\nabla\rangle^\frac12\mathcal W_1(t_k)\big\|_{L^2_x}+2C_2\varepsilon^{2}.\end{aligned}$$ By iteration, we obtain that $$\begin{aligned} \big\|\mathcal W_1\big\|_{L^4_{tx}({\mathbb R})}+\big\|\langle\nabla\rangle^\frac12\mathcal W_1\big\|_{L^\infty_t L^2_x({\mathbb R})} \le (2C_1)^{K+1}\Big(\big\|\langle\nabla\rangle^\frac12\mathcal W_{1,0}\big\|_{L^2_x}+2C_2\varepsilon^{2}\Big).\end{aligned}$$ Applying [\[est:mathcal-w0\]](#est:mathcal-w0){reference-type="eqref" reference="est:mathcal-w0"}, we get $$\begin{aligned} \label{est:W1-0} \big\|\mathcal W_1\big\|_{L^4_{tx}({\mathbb R})}+\big\|\langle\nabla\rangle^\frac12\mathcal W_1\big\|_{L^\infty_t L^2_x({\mathbb R})} \le C\varepsilon^{2},\end{aligned}$$ where $C$ only depends on $\| v_0\|_{H^{2}_x}$. Second, for general $\gamma>0$, we insert [\[est:W1-0\]](#est:W1-0){reference-type="eqref" reference="est:W1-0"} into [\[est:mathcal-W1-improved\]](#est:mathcal-W1-improved){reference-type="eqref" reference="est:mathcal-W1-improved"} to get $$\begin{aligned} \big\||\nabla|^{\gamma}\mathcal W_1\big\|_{L^4_{tx}(I)}&+\big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1\big\|_{L^\infty_t L^2_x(I)}\\ \le & \tilde C_1\big\||\nabla|^{\gamma}\langle\nabla\rangle^\frac12\mathcal W_1(t_0)\big\|_{L^2_x}+\tilde C_2\varepsilon^{2+\gamma}\\ &\quad +\tilde C_3\big\||\nabla|^\gamma \mathcal W_1\big\|_{L^4_{tx}(I)} \Big(\big\|\mathcal W_1\big\|_{L^4_{tx}(I)}+\big\|\mathcal W_2\big\|_{L^4_{tx}(I)}\Big),\end{aligned}$$ where $\tilde C_j,j=1,2,3$ only depend on $\| v_0\|_{H^{\gamma+2}_x}$ and $\| v_1\|_{H^{\gamma}_x}$. Then arguing similarly as above, we obtain [\[est:W1-gamma\]](#est:W1-gamma){reference-type="eqref" reference="est:W1-gamma"}. ◻ Note that Proposition [Proposition 20](#prop:h-varep){reference-type="ref" reference="prop:h-varep"} is followed from [\[def:w-h\]](#def:w-h){reference-type="eqref" reference="def:w-h"}, [\[relationship-w-W12\]](#relationship-w-W12){reference-type="eqref" reference="relationship-w-W12"}, [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} and [\[est:W1-gamma\]](#est:W1-gamma){reference-type="eqref" reference="est:W1-gamma"}. ## End of the proof {#subsec:keyprop-end} Now we are in the position to improve the estimate on $\partial_tv^{\varepsilon}$. Let $v^{\varepsilon}$ to be the solution of [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}, note that $$\begin{aligned} \label{relationship-v-var} \big\| |\nabla|^{\gamma} \partial_tv^{\varepsilon} \big\|_{L^4_{tx}({\mathbb R})} =&\varepsilon^{-2-\gamma}\big\| |\nabla|^{\gamma} \partial_t\mathcal{S}_\varepsilon \big(v^{\varepsilon}\big) \big\|_{L^4_{tx}({\mathbb R})}.\end{aligned}$$ Moreover, by [\[relationship-w-W12\]](#relationship-w-W12){reference-type="eqref" reference="relationship-w-W12"}, we have that $$\begin{aligned} \partial_t\mathcal{S}_\varepsilon \big(v^{\varepsilon}\big) =&\partial_t\Big(\mathrm{e}^{-it}w\Big)\\ =&\mathrm{e}^{-it} \big(-iw+\partial_tw\big)\\ %=&\fe^{-it} \big(-iw+\partial_tw\big)\\ =&\frac12\mathrm{e}^{-it} \Big[\big(\langle\nabla\rangle+1\big)\mathcal W_1+\big(\langle\nabla\rangle-1\big)\mathcal W_2\Big].\end{aligned}$$ Therefore, by [\[est:mathcal-W12\]](#est:mathcal-W12){reference-type="eqref" reference="est:mathcal-W12"} and [\[est:W1-gamma\]](#est:W1-gamma){reference-type="eqref" reference="est:W1-gamma"}, $$\begin{aligned} \big\| |\nabla|^{\gamma} \partial_t\mathcal{S}_\varepsilon \big(v^{\varepsilon}\big) \big\|_{L^4_{tx}({\mathbb R})} \lesssim & \big\| |\nabla|^{\gamma}\big(\langle\nabla\rangle+1\big)\mathcal W_1\big\|_{L^4_{tx}({\mathbb R})} +\big\| |\nabla|^{\gamma}\big(\langle\nabla\rangle-1\big)\mathcal W_2\big\|_{L^4_{tx}({\mathbb R})}\\ \lesssim & \big\| |\nabla|^{\gamma}\mathcal W_1\big\|_{L^4_{tx}({\mathbb R})} +\big\| |\nabla|^{\gamma+2}\mathcal W_1\big\|_{L^4_{tx}({\mathbb R})} +\big\| |\nabla|^{\gamma+2}\mathcal W_2\big\|_{L^4_{tx}({\mathbb R})}\\ \lesssim & C\varepsilon^{2+\gamma},\end{aligned}$$ where $C$ only depends on $\| v_0\|_{H^{\gamma+2}_x}$ and $\| v_1\|_{H^{\gamma}_x}$. Therefore, this last estimate together with [\[relationship-v-var\]](#relationship-v-var){reference-type="eqref" reference="relationship-v-var"} yields that $$\begin{aligned} \big\| |\nabla|^{\gamma} \partial_tv^{\varepsilon} \big\|_{L^4_{tx}({\mathbb R})} \le C.\end{aligned}$$ This finishes the proof of the key proposition. # Schrödinger-wave profile and Uniform-in-time nonrelativistic limit {#sec:global} .5cm Suppose that $v^{\varepsilon}$ is the solution of [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"}, then by [\[eq:r\]](#eq:r){reference-type="eqref" reference="eq:r"}-[\[eq:initial-data\]](#eq:initial-data){reference-type="eqref" reference="eq:initial-data"}, we obtain that $$\begin{aligned} \begin{split} \label{eq:kg-r-2} \left\{ \aligned &\varepsilon^2\partial_{tt} r - \Delta r + \frac{1}{\varepsilon^2} r +A_\varepsilon\big(v^{\varepsilon},r\big)+B_\varepsilon\big(v^{\varepsilon}\big) =0, \\ & r(0,x) = 0,\quad \partial_tr(0,x) = -2{\mathrm{Re}}(v_1). \endaligned \right. \end{split}\end{aligned}$$ Arguing similarly as in Section [3](#sec:prop-key){reference-type="ref" reference="sec:prop-key"}, we use the scaling argument and denote $$R(t,x)=\mathcal{S}_\varepsilon r(t,x),\quad h^{\varepsilon}(t,x)=\mathcal{S}_\varepsilon \big(v^{\varepsilon}\big)(t,x),$$ then by [\[eq:kg-r-2\]](#eq:kg-r-2){reference-type="eqref" reference="eq:kg-r-2"}, $R$ obeys the following equation: $$\begin{aligned} \label{eq:psi-scaling-2} \partial_{tt} R - \Delta R + R +G\big(h^{\varepsilon},R\big)+F_1\big(h^{\varepsilon}\big)=0,\end{aligned}$$ where $$\begin{aligned} G(f,g)=&3\Big[\mathrm{e}^{2it}f^2+\mathrm{e}^{-2it}\big(\bar f\big)^2\Big]g + 3\Big[\mathrm{e}^{it}f+\mathrm{e}^{-it}\bar f\Big]g^2+g^3;\\ F_1(f)=&\mathrm{e}^{3it}f^3+\mathrm{e}^{-3it}\big(\bar f\big)^3. \end{aligned}$$ Moreover, the initial data of $R$ is that $$\begin{aligned} \label{eq:psi0-scaling} R(0)=0,\quad \partial_{t} R(0)=-2\varepsilon^2\mathcal{S}_\varepsilon \big({\mathrm{Re}}(v_1)\big).\end{aligned}$$ Denote $$\begin{aligned} \label{def:phi} \phi\triangleq \langle \nabla\rangle^{-1}\big(\partial_t +i\langle \nabla\rangle \big)R,\end{aligned}$$ then we have that $$\begin{aligned} \label{phi-varphi} R= {\mathrm{Im}}\> \phi;\quad \partial_t R= {\mathrm{Re}}\big(\langle \nabla\rangle \phi\big).\end{aligned}$$ Moreover, $\phi$ obeys the following equation $$\begin{aligned} \label{eq:phi} \big(\partial_t -i\langle \nabla\rangle \big) \phi +\langle \nabla\rangle^{-1}\Big[G\big(h^{\varepsilon},R\big)+F_1\big(h^{\varepsilon}\big)\Big]=0,\end{aligned}$$ with the initial data $$\begin{aligned} \label{phi-initialdata} \phi(0)=-2\varepsilon^2\langle \nabla\rangle^{-1} \mathcal{S}_\varepsilon \big({\mathrm{Re}}(v_1)\big).\end{aligned}$$ By Duhamel's formula, we have that $$\begin{aligned} \label{Duh-1} \phi(t)= \mathrm{e}^{i(t-t_0)\langle\nabla\rangle}\phi(t_0)- \int_{t_0}^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}\Big[G\big(h^{\varepsilon},R)+F_1\big(h^{\varepsilon}\big)\Big]\,ds.\end{aligned}$$ Now we focus on the term $$\int_{t_0}^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1} F_1\big(h^{\varepsilon}\big)\,ds,$$ which can be split into the following two terms: $$\int_{t_0}^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}P_{\le 1} F_1\big(h^{\varepsilon}\big)\,ds +\int_{t_0}^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}P_{> 1} F_1\big(h^{\varepsilon}\big)\,ds.$$ From the definition of $F_1$, we have that $$\begin{aligned} &\int_{t_0}^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}P_{\le 1} F_1\big(h^{\varepsilon}\big)\,ds\notag\\ =&\mathrm{e}^{it\langle\nabla\rangle} \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1} \big(h^{\varepsilon}\big)^3\,ds\notag\\ &\quad +\mathrm{e}^{it\langle\nabla\rangle} \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle+3)} \langle \nabla\rangle^{-1}P_{\le 1} \big(\overline{h^{\varepsilon}}\big)^3\,ds.\label{F1-12}\end{aligned}$$ Note that $$\begin{aligned} \label{differ} \mathrm{e}^{-is(\langle\nabla\rangle\pm 3)}P_{\le 1} =\frac{d}{ds}\Big(\mathrm{e}^{-is(\langle\nabla\rangle\pm3)}\Big) \frac{1}{-i(\langle\nabla\rangle\pm3)}P_{\le 1}.\end{aligned}$$ Then using this formula and integration by parts, we get that $$\begin{aligned} & \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1} \big(h^{\varepsilon}\big)^3\,ds\notag\\ =& \mathrm{e}^{-is(\langle\nabla\rangle-3)} \frac{1}{-i(\langle\nabla\rangle-3)}\langle \nabla\rangle^{-1}P_{\le 1} \big(h^{\varepsilon}\big)^3\Big|_{s=t_0}^{s=t}\notag\\ &\quad-3 \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle-3)} \frac{1}{-i(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(h^{\varepsilon}\big)^2\> \partial_sh^{\varepsilon}\Big]\,ds.\label{Integ-1}\end{aligned}$$ Similarly, $$\begin{aligned} & \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle+3)} \langle \nabla\rangle^{-1}P_{\le 1} \big(h^{\varepsilon}\big)^3\,ds\notag\\ =& \mathrm{e}^{-is(\langle\nabla\rangle+3)} \frac{1}{-i(\langle\nabla\rangle+3)}\langle \nabla\rangle^{-1}P_{\le 1} \big(\overline{h^{\varepsilon}}\big)^3\Big|_{s=t_0}^{s=t}\notag\\ &\quad-3 \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle+3)} \frac{1}{-i(\langle\nabla\rangle+3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(\overline{h^{\varepsilon}})^2\> \partial_s\overline{h^{\varepsilon}}\Big]\,ds.\label{Integ-2}\end{aligned}$$ Now we insert the terms [\[Integ-1\]](#Integ-1){reference-type="eqref" reference="Integ-1"}, [\[Integ-2\]](#Integ-2){reference-type="eqref" reference="Integ-2"} into [\[Duh-1\]](#Duh-1){reference-type="eqref" reference="Duh-1"}, and obtain that [\[est:Duh-mod\]]{#est:Duh-mod label="est:Duh-mod"} $$\begin{aligned} \phi(t)=& \mathrm{e}^{i(t-t_0)\langle\nabla\rangle}\phi(t_0)\\ &\quad -\int_{t_0}^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}\Big[G\big(h^{\varepsilon},R)+P_{> 1}F_1\big(h^{\varepsilon}\big)\Big]\,ds\label{Duh-mod-1}\\ &\quad + \frac{\mathrm{e}^{3it} }{i(\langle\nabla\rangle-3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(h^{\varepsilon}(t)\Big)^3 - \frac{\mathrm{e}^{i(t-t_0)\langle \nabla \rangle}\mathrm{e}^{3it_0}}{i(\langle\nabla\rangle-3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(h^{\varepsilon}(t_0)\Big)^3\label{Duh-mod-2}\\ &\quad + \frac{\mathrm{e}^{3it} }{i(\langle\nabla\rangle+3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(\overline{h^{\varepsilon}(t)}\Big)^3 - \frac{\mathrm{e}^{i(t-t_0)\langle \nabla \rangle}\mathrm{e}^{-3it_0}}{i(\langle\nabla\rangle+3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(\overline{h^{\varepsilon}(t_0)}\Big)^3\label{Duh-mod-3}\\ &\quad-3\mathrm{e}^{it\langle\nabla\rangle} \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle-3)} \frac{1}{i(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(h^{\varepsilon}\big)^2\> \partial_s h^\varepsilon\Big]\,ds\label{Duh-mod-4}\\ &\quad-3\mathrm{e}^{it\langle\nabla\rangle} \int_{t_0}^t \mathrm{e}^{-is(\langle\nabla\rangle+3)} \frac{1}{i(\langle\nabla\rangle+3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(\overline{h^\varepsilon})^2\> \partial_s\big(\overline{h^\varepsilon}\big)\Big]\,ds.\label{Duh-mod-5}\end{aligned}$$ We shall give the estimates on the following norm: $$\big\|\phi\big\|_{Y_0({\mathbb R})}\triangleq \big\|\phi\big\|_{L^4_{tx}({\mathbb R})}+\big\|\langle\nabla\rangle^\frac12\phi\big\|_{L^\infty_tL^2_x({\mathbb R})}.$$ For the linear term, we have that $$\begin{aligned} \label{est:linear-phi0} \big\| \mathrm{e}^{it\langle\nabla\rangle}\phi(t_0)\big\|_{Y_0({\mathbb R})} \lesssim & \big\| \langle \nabla\rangle^{\frac12}\phi(t_0)\big\|_{L^2_x}.\end{aligned}$$ In the following subsection, we will consider the nonlinear terms. .5cm ## Nonlinear estimates {#sec:nonlinear} .5cm Now we estimate [\[Duh-mod-1\]](#Duh-mod-1){reference-type="eqref" reference="Duh-mod-1"}--[\[Duh-mod-5\]](#Duh-mod-5){reference-type="eqref" reference="Duh-mod-5"} terms by terms. Let $t_0\in I\subset {\mathbb R}$ be an time interval. .5cm By Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that $$\begin{aligned} \big\|\eqref{Duh-mod-1}\big\|_{Y_0(I)} \lesssim & \left\|G\big(h^{\varepsilon},R\big)\right\|_{Z_0(I)}+\left\|P_{> 1}F_1\big(h^{\varepsilon}\big)\right\|_{Z_0(I)}, %\\ %&\quad+\varepsilon^4\big\||\nabla|^{s_c} \langle \nabla\rangle^{-\frac12}F_2\big((v_{tt})^\varepsilon\big)\big\|_{L^1_tL^2_x(I)},\end{aligned}$$ where $$\begin{aligned} \|f\|_{Z_0(I)}&\triangleq \|f\|_{L^\frac43_{tx}(I)}.\end{aligned}$$ For $G\big(h^{\varepsilon},R\big)$, by Hölder's inequality, $$\begin{aligned} \left\|G\big(h^{\varepsilon},R\big)\right\|_{Z_0(I)} \lesssim & \sum\limits_{j=0}^2\big\|h^{\varepsilon}\big\|_{L^4_{tx}(I)}^j\big\|R\big\|_{L^4_{tx}(I)}^{3-j}.\end{aligned}$$ Note that $$\big\|\mathcal{S}_\varepsilon f\big\|_{L^4_{tx}(I)}=\big\|f\big\|_{L^4_{tx}(\varepsilon^2I)}.$$ Therefore, $$\begin{aligned} \left\|G\big(h^{\varepsilon},R\big)\right\|_{Z_0(I)} \lesssim & \sum\limits_{j=0}^2\big\|v^{\varepsilon}\big\|_{L^4_{tx}(\varepsilon^2I)}^j\big\|R\big\|_{L^4_{tx}(I)}^{3-j}.\end{aligned}$$ For $F_1\big(h^{\varepsilon}\big)$, since $$P_{> 1}F_1(f)=\mathrm{e}^{3it}\big(P_{\gtrsim 1}f\>f^2\big)+\mathrm{e}^{-3it}\big(P_{\gtrsim 1}\bar f\>\bar f^2\big)$$ Then by Hölder's inequality we have that $$\begin{aligned} \big\|P_{> 1}F_1\big(h^{\varepsilon}\big)\big\|_{Z_0({\mathbb R})} \lesssim & \big\|P_{\gtrsim 1}h^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})}\big\|h^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})}^2.\end{aligned}$$ Note that $$\big\|P_{\gtrsim 1}h^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})}\lesssim \big\||\nabla|^2 P_{\le 1}h^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})} +\big\|P_{> 1}h^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})}.$$ Then by Proposition [Proposition 20](#prop:h-varep){reference-type="ref" reference="prop:h-varep"} we have that $$\begin{aligned} \big\|P_{> 1}F_1\big(h^{\varepsilon}\big)\big\|_{Z_0(I)} \le & C\varepsilon^2,\end{aligned}$$ where the constant $C>0$ only depends on $\| v_0\|_{H^{2}_x}$ and $\| v_1\|_{L^2_x}$. Together with the estimates above, we obtain that $$\begin{aligned} \big\|\eqref{Duh-mod-1}\big\|_{Y_0(I)} \lesssim & \sum\limits_{j=0}^2\big\|v^{\varepsilon}\big\|_{L^4_{tx}(\varepsilon^2I)}^j\big\|R\big\|_{L^4_{tx}(I)}^{3-j} +\varepsilon^2. %&\qquad +\varepsilon^4|I|\big\| v_0\big\|_{H^{4+s_c}_x(\R^d)}.\end{aligned}$$ .3cm Note that $$\begin{aligned} \label{est:bound-nabal3} \left\|\frac{1}{\langle\nabla\rangle\pm3}\langle \nabla\rangle^{-\frac12}P_{\le 1}f\right\|_{L^r_x({\mathbb R}^d)} \lesssim \|f\|_{L^r_x({\mathbb R}^d)},\quad \mbox{ for any } 1\le r\le \infty.\end{aligned}$$ Applying this estimate, Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"} and Proposition [Proposition 10](#prop:v-varep){reference-type="ref" reference="prop:v-varep"}, we have that $$\begin{aligned} \big\|\eqref{Duh-mod-2}\big\|_{Y_0(I)}&+\big\|\eqref{Duh-mod-3}\big\|_{Y_0(I)}\\ \lesssim & \Big\|\frac{1}{\langle\nabla\rangle\pm3}\langle \nabla\rangle^{-1}P_{\le 1}\Big(h^{\varepsilon}\Big)^3\Big\|_{Y_0(I)}+\Big\|\frac{1}{\langle\nabla\rangle\pm3}\langle \nabla\rangle^{-\frac12}P_{\le 1}\Big(h^{\varepsilon}(t_0)\Big)^3\Big\|_{L^2_x}\\ \lesssim & \Big\|\Big(h^{\varepsilon}\Big)^3\Big\|_{X_0(I)}+\Big\|\Big(h^{\varepsilon}(t_0)\Big)^3\Big\|_{L^2_x}\\ \lesssim & \big\|h^{\varepsilon}\big\|_{X_0(I)}\big\|h^{\varepsilon}\big\|_{L^\infty_{tx}(I)}^2+\big\|h^{\varepsilon}(t_0)\big\|_{L^2_x({\mathbb R}^2)}\big\|h^{\varepsilon}(t_0)\big\|_{L^\infty_x}^2\\ \lesssim & \varepsilon^2\big\|v^{\varepsilon}\big\|_{X_0({\mathbb R})}\big\|v^{\varepsilon}\big\|_{L^\infty_{tx}({\mathbb R})}^2+\varepsilon^2\big\|v^{\varepsilon}(t_0)\big\|_{L^2_x}\big\|v^{\varepsilon}(t_0)\big\|_{L^\infty_x}^2\\ \le & C\big(\|(v_0,v_1)\|_{H^{2}_x\times L^2_x}\big) \varepsilon^2.\end{aligned}$$ .3cm By Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, [\[est:bound-nabal3\]](#est:bound-nabal3){reference-type="eqref" reference="est:bound-nabal3"} and Proposition [Proposition 20](#prop:h-varep){reference-type="ref" reference="prop:h-varep"}, we have that $$\begin{aligned} \big\|\eqref{Duh-mod-4}\big\|_{Y_0(I)}+\big\|\eqref{Duh-mod-5}\big\|_{Y_0(I)} \lesssim & \Big\|\frac{1}{\langle\nabla\rangle\pm3}P_{\le 1}\Big(\big(h^{\varepsilon}\big)^2\> \partial_s h^{\varepsilon}\Big)\Big\|_{Z_0(I)}\\ \lesssim & \Big\|\big(h^{\varepsilon}\big)^2\> \partial_s h^{\varepsilon}\Big\|_{Z_0(I)}\\ \lesssim & \big\|h^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})}^2\big\|\partial_s h^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})}\\ \le & C\big(\|(v_0,v_1)\|_{H^{2}_x\times L^2_x}\big) \varepsilon^2.\end{aligned}$$ Now together with the estimates on [\[est:Duh-mod\]](#est:Duh-mod){reference-type="eqref" reference="est:Duh-mod"} and [\[est:linear-phi0\]](#est:linear-phi0){reference-type="eqref" reference="est:linear-phi0"}, it infers that $$\begin{aligned} \big\|\phi\big\|_{Y_0(I)} \le& C_1\big\|\langle \nabla\rangle^{\frac12}\phi(t_0)\big\|_{L^2({\mathbb R}^d)}+C_2\varepsilon^2 +C_3\big\|v^\varepsilon\big\|_{L^4_{tx}(\varepsilon^2I)}^2\big\|R\big\|_{L^4_{tx}(I)} \\ &+C_4\Big(\big\|R\big\|_{L^4_{tx}(I)}^2+\big\|R\big\|_{L^4_{tx}(I)}^3\Big),\end{aligned}$$ where $C_j,j=1,\cdots,4$ only depend on $\|(v_0,v_1)\|_{H^{2}_x\times L^2_x}$. Moreover, note that $$\begin{aligned} \big\|R\big\|_{L^4_{tx}(I)}\le \big\|\phi\big\|_{L^4_{tx}(I)}.\end{aligned}$$ Therefore, $$\begin{aligned} \label{est:phi-L4-int} \big\|\phi\big\|_{Y_0(I)} \le& C_1\big\|\langle \nabla\rangle^{\frac12}\phi(t_0)\big\|_{L^2({\mathbb R}^2)}+C_2\varepsilon^2 +C_3\big\|v^{\varepsilon}\big\|_{L^4_{tx}(\varepsilon^2I)}^2\big\|\phi\big\|_{L^4_{tx}(I)} \notag\\ &+C_4\Big(\big\|\phi\big\|_{L^4_{tx}(I)}^2+\big\|\phi\big\|_{L^4_{tx}(I)}^3\Big).\end{aligned}$$ ## Proof of Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"} {#sec:proof-thm1} By Proposition [Proposition 10](#prop:v-varep){reference-type="ref" reference="prop:v-varep"}, we have that $$\begin{aligned} \label{est:uniform-v} \big\|v^{\varepsilon}\big\|_{L^4_{tx}({\mathbb R})} \le & C\big(\|(v_0,v_1)\|_{H^2\times L^2}\big).\end{aligned}$$ Therefore, there exists a constant $K=K(\|(v_0,v_1)\|_{H^2\times L^2},C_2,C_3)$ (but independent of $\varepsilon$) and a sequence of time intervals $$\bigcup\limits_{k=0}^K J_k={\mathbb R}^+,$$ (the negative time direction can be treated similarly) with $$J_0=[0, t_1], \quad J_k=(t_k, t_{k+1}] \mbox{ for } k=1,\cdots, K-1, \quad J_{K}=[t_K, +\infty),$$ such that $$\begin{aligned} \label{subinterval-length} C_3\big\|v^{\varepsilon}\big\|_{L^4_{tx}(J_k)}^2 \le \frac12.\end{aligned}$$ Denote $$I_k=\varepsilon^{-2}J_k.$$ Then by [\[est:phi-L4-int\]](#est:phi-L4-int){reference-type="eqref" reference="est:phi-L4-int"}, we obtain that for any $k\le K$, $$\begin{aligned} \label{est:phi-L4-int-boot} \big\|\phi\big\|_{Y_0(I_k)} \le& 2C_1\big\|\langle \nabla\rangle^{\frac12}\phi(t_k)\big\|_{L^2({\mathbb R}^2)}+2C_2\varepsilon^2 +2C_4\Big(\big\|\phi\big\|_{L^4_{tx}(I_k)}^2+\big\|\phi\big\|_{L^4_{tx}(I_k)}^3\Big).\end{aligned}$$ Moreover, denote $$a_0(\varepsilon)\triangleq \Big(\varepsilon^2\big\|\langle \nabla\rangle^{\frac12}v_1\big\|_{L^2({\mathbb R}^2)}+4C_2\varepsilon^2\Big)\cdot \big(4C_1\big)^K.$$ Choosing $\varepsilon_0$ suitably small such that for any $\varepsilon\in [0, \varepsilon_0]$, we have that $$\begin{aligned} \label{small-a0} C_4\big(a_0(\varepsilon)+a_0(\varepsilon)^2\big)\le \frac14.\end{aligned}$$ Suppose that for some $k: 0\le k\le K-1$, $$\begin{aligned} \label{iteration-k} \big\|\langle \nabla\rangle^{\frac12}\phi(t_k)\big\|_{L^2_x} \le \varepsilon^2\big\|\langle \nabla\rangle^{\frac12}v_1\big\|_{L^2({\mathbb R}^2)}\cdot \big(4C_1\big)^k + 4C_2\varepsilon^2\sum\limits_{j=1}^k \big(4C_1\big)^{j-1},\end{aligned}$$ then $$\big\| \langle \nabla\rangle^{\frac12}\phi(t_k)\big\|_{L^2({\mathbb R}^d)}\le a_0.$$ Therefore, applying these to [\[est:phi-L4-int-boot\]](#est:phi-L4-int-boot){reference-type="eqref" reference="est:phi-L4-int-boot"} for the time interval $I_{k}$ and bootstrap, we first obtain that $$\big\|\phi\big\|_{L^4_{tx}(I_{k})} \le a_0,$$ and thus inserting into [\[est:phi-L4-int-boot\]](#est:phi-L4-int-boot){reference-type="eqref" reference="est:phi-L4-int-boot"} for $I_{k}$ again, we obtain that $$\begin{aligned} \big\|\langle \nabla\rangle^{\frac12}\phi(t_{k+1})\big\|_{L^2_x} \le \varepsilon^2\big\|\langle \nabla\rangle^{\frac12}v_1\big\|_{L^2({\mathbb R}^2)}\cdot \big(4C_1\big)^{k+1} + 4C_2\varepsilon^2\sum\limits_{j=1}^{k+1} \big(4C_1\big)^{j-1}.\end{aligned}$$ Hence, [\[iteration-k\]](#iteration-k){reference-type="eqref" reference="iteration-k"} holds for any $k: 0\le k\le K$ and thus $$\big\| \langle \nabla\rangle^{\frac12}\phi\big\|_{L^\infty_tL^2_x({\mathbb R})}\le a_0=C\varepsilon^2.$$ for some $\varepsilon$-independent $C>0$. Then by [\[phi-varphi\]](#phi-varphi){reference-type="eqref" reference="phi-varphi"}, we $$\big\|R\big\|_{L^\infty_tL^2_x({\mathbb R})}\le C\varepsilon^2.$$ Scaling back, we obtain that $$\big\|r\big\|_{L^\infty_tL^2_x({\mathbb R})}\le C\varepsilon^2.$$ This finishes the proof of Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"}. *Remark 23*. A direct usage of the same argument, we can dealt with the three dimensional cases for small data. The conclusion can be read as follows: Let $s\ge\frac12$, there exists a constant $\delta_0>0$ such that the following properties hold. Suppose that $$(u_0,u_1)\in H^{2+s}({\mathbb R}^3) \times H^{2+s}({\mathbb R}^3);\quad \big\|(u_0,u_1)\big\|_{\dot H^\frac12\times \dot H^\frac12}\le \delta_0; \quad v_1\in H^s({\mathbb R}^3).$$ Then $$\begin{aligned} \big\|u^\varepsilon(t)-\mathrm{e}^{\frac{it}{\varepsilon^2}}v^{\varepsilon}(t)-\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v^{\varepsilon}(t) \big\|_{H^s_x} \le C \varepsilon^2.\end{aligned}$$ The small data assumption is used to assure the analogous estimate of [\[est:uniform-v\]](#est:uniform-v){reference-type="eqref" reference="est:uniform-v"}: $$\big\|v^{\varepsilon}\big\|_{X_\frac12({\mathbb R})}<+\infty$$ holds in three dimensional case. A more refined estimate can further lower the restriction $s\ge \frac12$ to $s\ge0$. ## Optimality {#sec:sharp-global-op1} .5cm In this subsection, we give the proof of Theorem [Theorem 4](#thm:main1-optimal){reference-type="ref" reference="thm:main1-optimal"}. Let $\delta_0, a_0,b_0$ are positive constants. Denote $$f(x)=|x|^{-1}\chi_{a_0\le |\cdot| \le 2a_0}(x),$$ and $$\begin{aligned} \label{v0-example} \big(v_0,v_1\big)\triangleq \delta_0 \mathrm{e}^{-i\frac{b_0 |x|^2}{2}} (f,g),\end{aligned}$$ where $g\in \mathcal S$ are independent of $\delta_0, a_0,b_0$. In the following, we set the parameters satisfying $$\begin{aligned} \label{relationships} 0<\varepsilon_0\ll \delta_0\ll a_0\ll b_0^{-1}\ll1.\end{aligned}$$ Moreover, assume that there exists a constant $c_0>0$ which is independent of $b_0$ such that $$\begin{aligned} \label{lower-bound-v1v0} \big\|-2{\mathrm{Re}}(v_1)+\frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\big\|_{L^2_x}\ge c_0\delta_0^3 \|f\|_{L^6_x}^3.\end{aligned}$$ *Remark 24*. The existence of such $\big(v_0,v_1\big)$ is easy to check. Take an example, we set $v_1=0$, then $$\begin{aligned} \big| \frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\big| =\frac12\delta_0^3 |f|^3 \Big|2\mathrm{e}^{-i\frac{3b_0 |x|^2}{2}}-\mathrm{e}^{i\frac{3b_0 |x|^2}{2}}\Big| \ge \frac12 \delta_0^3 |f|^3. \end{aligned}$$ Then in this case we can set $c_0=\frac18$. ### Approximation Firstly, we consider the behavior of the solution to [\[eqs:scaling-w\]](#eqs:scaling-w){reference-type="eqref" reference="eqs:scaling-w"} with $v_0$ given by [\[v0-example\]](#v0-example){reference-type="eqref" reference="v0-example"}. Treating the same as in Section [3](#sec:prop-key){reference-type="ref" reference="sec:prop-key"}, we define $\mathcal W_j,j=1,2$ as in [\[relationship-w-W12\]](#relationship-w-W12){reference-type="eqref" reference="relationship-w-W12"}. Then by Duhamel's formula, we may write that for $j=1,2$, $$\begin{aligned} \mathcal W_j(t)=&\mathrm{e}^{\mp it\langle \nabla\rangle}\mathcal W_{j,0} -3\int_0^t \mathrm{e}^{\mp i(t-s)\langle \nabla\rangle}\langle\nabla \rangle^{-1} \big(|w|^2w\big)\,ds\notag\\ \triangleq &\mathrm{e}^{\mp it\langle \nabla\rangle}\mathcal W_{j,0} +\mathcal N_j(w).\label{def:W12}\end{aligned}$$ For the initial data [\[v0-example\]](#v0-example){reference-type="eqref" reference="v0-example"}, we have that there exists a positive constant $C_{a_0, b_0}$ only depending on $a_0, b_0$ such that $$\begin{aligned} \Big\|\mathrm{e}^{\mp it\langle \nabla\rangle}\mathcal W_{j,0} \Big\|_{X_{s_c}({\mathbb R})} \le & C_{a_0, b_0}\delta_0.\end{aligned}$$ Then arguing similarly as the proof in Section [3](#sec:prop-key){reference-type="ref" reference="sec:prop-key"}, we have that $$\begin{aligned} \big\|w \big\|_{X_{s_c}({\mathbb R})} \le C_{a_0, b_0}\delta_0. \label{w-L4-sd}\end{aligned}$$ Moreover, by the proof in Sections [3.3](#subsec:Stri){reference-type="ref" reference="subsec:Stri"} and [3.4](#subsec:Imp-Stri){reference-type="ref" reference="subsec:Imp-Stri"}, we further have that for any $\gamma\ge -s_c$, $$\begin{aligned} \big\||\nabla|^\gamma w\big\|_{X_{s_c}({\mathbb R})} \le & C_{a_0, b_0}\varepsilon^\gamma;\label{w-L4-ga-sd}\\ \big\||\nabla|^\gamma\mathcal N_j(w)\big\|_{X_{s_c}({\mathbb R})} \le & C_{a_0, b_0}\delta_0^3 \varepsilon^\gamma; \label{Nwj-L4-sd}\\ \big\|\mathcal N_1(w)\big\|_{X_{s_c}({\mathbb R})} \le & C_{a_0, b_0}\delta_0^3\varepsilon^2. \label{Nw1-L4-sd}\end{aligned}$$ Furthermore, by [\[eq:w-W12\]](#eq:w-W12){reference-type="eqref" reference="eq:w-W12"}, we may write $$\begin{aligned} \label{Duh-w} w(t)=&\frac i2\big(\mathcal W_1-\mathcal W_2\big)\notag\\ = &\frac i2 \Big[\mathrm{e}^{-it \langle \nabla\rangle}\mathcal W_{1,0}-\mathrm{e}^{it \langle \nabla\rangle}\mathcal W_{2,0}\Big]+\frac i2\big(\mathcal N_1(w)-\mathcal N_2(w)\big).\end{aligned}$$ By [\[eq:W-12-KG-intialdatum\]](#eq:W-12-KG-intialdatum){reference-type="eqref" reference="eq:W-12-KG-intialdatum"} and [\[Duh-w\]](#Duh-w){reference-type="eqref" reference="Duh-w"}, we further have that $$\begin{aligned} \label{Duh-w-2} w(t) = &-\frac 12 \left[\mathrm{e}^{-it \langle \nabla\rangle}\langle \nabla\rangle^{-1}\big(1-\langle \nabla\rangle\big)-\mathrm{e}^{it \langle \nabla\rangle}\langle \nabla\rangle^{-1}\big(1+\langle \nabla\rangle\big)\right]\mathcal{S}_\varepsilon v_0+\mathcal N(w),\end{aligned}$$ where the term $\mathcal N(w)$ reads as $$\begin{aligned} \mathcal N(w)\triangleq\frac i2\big(\mathcal N_1(w)-\mathcal N_2(w)\big) +\frac i2 \varepsilon^2\langle \nabla \rangle^{-1}\Big[\mathrm{e}^{-it \langle \nabla\rangle}-\mathrm{e}^{it \langle \nabla\rangle}\Big]\mathcal{S}_\varepsilon g.\end{aligned}$$ $\bullet$ Approximation of $h^\varepsilon$. Recall that $$h^\varepsilon=\mathrm{e}^{-it}w,$$ then by [\[Duh-w-2\]](#Duh-w-2){reference-type="eqref" reference="Duh-w-2"}, we rewrite $$\begin{aligned} \label{apprx-h-1} h^\varepsilon(t) = &-\frac 12 \left[\mathrm{e}^{-it (\langle \nabla\rangle+1)}\langle \nabla\rangle^{-1}\big(1-\langle \nabla\rangle\big)-\mathrm{e}^{it (\langle \nabla\rangle-1)}\langle \nabla\rangle^{-1}\big(1+\langle \nabla\rangle\big)\right]\mathcal{S}_\varepsilon v_0\notag\\ &\qquad +\mathrm{e}^{-it}\mathcal N(w),\end{aligned}$$ with $$h^\varepsilon(0)=\mathcal{S}_\varepsilon v_0.$$ Moreover, note that $$\begin{aligned} \label{na-apprx} \langle \nabla\rangle=1-\big(1+\langle\nabla\rangle\big)^{-1}\Delta.\end{aligned}$$ Then from [\[apprx-h-1\]](#apprx-h-1){reference-type="eqref" reference="apprx-h-1"}, these allow us to rewrite $h^\varepsilon(t)$ as $$\begin{aligned} \label{apprx-h-1-op2} h^\varepsilon(t) = &\mathrm{e}^{it (\langle \nabla\rangle-1)} \mathcal{S}_\varepsilon v_0 +\mathrm{e}^{-it}\mathcal N_{\bot}(w),\end{aligned}$$ where $$\begin{aligned} \mathcal N_{\bot}(w) \triangleq & \frac i2\mathrm{e}^{-it}\big(\mathcal N_1(w)-\mathcal N_2(w)\big) -\frac 12\varepsilon^2 \mathrm{e}^{-it (\langle \nabla\rangle+1)}\mathcal{S}_\varepsilon(\Delta v_0)\\ &\quad -\frac 12\varepsilon^2\mathrm{e}^{-it} \left[\mathrm{e}^{-it \langle \nabla\rangle}-\mathrm{e}^{it\langle \nabla\rangle}\right]\langle \nabla\rangle^{-1}\big(1+\langle \nabla\rangle\big)^{-1}\mathcal{S}_\varepsilon(\Delta v_0)\\ &\quad+\frac i2 \varepsilon^2\langle \nabla \rangle^{-1}\Big[\mathrm{e}^{-it \langle \nabla\rangle}-\mathrm{e}^{it \langle \nabla\rangle}\Big]\mathcal{S}_\varepsilon g.\end{aligned}$$ By [\[Nwj-L4-sd\]](#Nwj-L4-sd){reference-type="eqref" reference="Nwj-L4-sd"}, and noting that $$\begin{aligned} \label{est:S-vep-v0} \big\||\nabla|^\gamma \mathcal{S}_\varepsilon v_0\big\|_{L^2_x} \le C_{a_0, b_0}\delta_0 \varepsilon^{\gamma-s_c},\end{aligned}$$ we have that for any $\gamma\ge -s_c$, $$\begin{aligned} \label{est:mathcal-N-h} \big\||\nabla|^{\gamma}\mathcal N_{\bot}(w)\big\|_{X_{s_c}({\mathbb R})} \le C_{a_0, b_0}\>\delta_0 \varepsilon^{\gamma} \big(\varepsilon^2+\delta_0^2\big).\end{aligned}$$ A direct consequence of these two estimates and Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"} is that $$\begin{aligned} \big\||\nabla|^{\gamma} h^\varepsilon\big\|_{X_{s_c}({\mathbb R})} \le C_{a_0, b_0}\delta_0 \varepsilon^{\gamma}. \label{h-L4-sd}\end{aligned}$$ $\bullet$ Approximation of $\partial_t h^\varepsilon$. Moreover, by [\[eq:w-W12\]](#eq:w-W12){reference-type="eqref" reference="eq:w-W12"}, $$\begin{aligned} \partial_t h^\varepsilon &=\mathrm{e}^{-it}(\partial_t w-iw)\notag\\ &=\frac12\mathrm{e}^{-it}\Big[\big(\langle\nabla\rangle+1\big)\mathcal W_1+\big(\langle\nabla\rangle-1\big)\mathcal W_2\Big]. %&=\frac12\fe^{-it}\Big[2\mathcal W_1+\big(\langle\nabla\rangle-1\big)\big(\mathcal W_1+\mathcal W_2\big)\Big].\end{aligned}$$ Then by [\[def:W12\]](#def:W12){reference-type="eqref" reference="def:W12"} and [\[eq:W-12-KG-intialdatum\]](#eq:W-12-KG-intialdatum){reference-type="eqref" reference="eq:W-12-KG-intialdatum"}, we obtain that $$\begin{aligned} \label{apprx-ht-1} \partial_t h^\varepsilon(t) &=\frac12\mathrm{e}^{-it}\Big[\big(\langle\nabla\rangle+1\big)\mathrm{e}^{-it\langle\nabla\rangle}\mathcal W_{1,0}+\big(\langle\nabla\rangle-1\big)\mathrm{e}^{it\langle\nabla\rangle}\mathcal W_{2,0}\Big]\notag\\ &\qquad + \frac12\mathrm{e}^{-it}\Big[\big(\langle\nabla\rangle+1\big)\mathcal N_1(w)+\big(\langle\nabla\rangle-1\big)\mathcal N_2(w)\Big]\notag\\ &=\frac i2 \langle\nabla\rangle^{-1} \Delta \Big[\mathrm{e}^{-it(\langle\nabla\rangle+1)}-\mathrm{e}^{it(\langle\nabla\rangle-1)}\Big]\mathcal{S}_\varepsilon v_0 +\widetilde{\mathcal N}(w),\end{aligned}$$ where we denote $$\begin{aligned} \widetilde{\mathcal N}(w)\triangleq & \frac12 \mathrm{e}^{-it}\Big[\big(\langle\nabla\rangle+1\big)\mathcal N_1(w)+\big(\langle\nabla\rangle-1\big)\mathcal N_2(w)\Big]\\ & +\frac 12\varepsilon^2 \langle\nabla\rangle^{-1} \Big[(\langle\nabla\rangle+1)\mathrm{e}^{-it(\langle\nabla\rangle+1)}+(\langle\nabla\rangle-1)\mathrm{e}^{it(\langle\nabla\rangle-1)}\Big]\mathcal{S}_\varepsilon v_1.\end{aligned}$$ Moreover, by [\[apprx-ht-1\]](#apprx-ht-1){reference-type="eqref" reference="apprx-ht-1"}, we have that $$\begin{aligned} \label{apprx-ht-1-op2} \partial_t h^\varepsilon(t) &=\frac i2 \Delta \Big[\mathrm{e}^{-it(\langle\nabla\rangle+1)}-\mathrm{e}^{it(\langle\nabla\rangle-1)}\Big]\mathcal{S}_\varepsilon v_0 +\widetilde{\mathcal N}_{\bot}(w),\end{aligned}$$ where we denote $$\begin{aligned} \widetilde{\mathcal N}_{\bot}(w)\triangleq &\widetilde{\mathcal N}(w) +\frac i2 \langle \nabla\rangle^{-1}\big(1+\langle \nabla\rangle\big)^{-1} \Delta^2 \Big[\mathrm{e}^{-it(\langle\nabla\rangle+1)}-\mathrm{e}^{it(\langle\nabla\rangle-1)}\Big]\mathcal{S}_\varepsilon v_0.\end{aligned}$$ By [\[Nwj-L4-sd\]](#Nwj-L4-sd){reference-type="eqref" reference="Nwj-L4-sd"} and [\[est:S-vep-v0\]](#est:S-vep-v0){reference-type="eqref" reference="est:S-vep-v0"}, we also have that for any $\gamma\ge -s_c$, $$\begin{aligned} \label{est:mathcal-N-tilde-h} \big\||\nabla|^{\gamma}\widetilde{\mathcal N}_{\bot}(w)\big\|_{X_{s_c}({\mathbb R})} \le C_{a_0, b_0}\>\delta_0\varepsilon^{2+\gamma-s_c} \big(\varepsilon^2+\delta_0^2\big).\end{aligned}$$ ### Basic estimates on initial datum In this subsubsection, we give the basic estimates on $v_0$ and $f$. A direct computation gives that $$\label{est:f} \begin{aligned} \|f\|_{H^\gamma_x}\sim &a_0^{s_c-\gamma}, \quad \mbox{ for any }\gamma\ge0;\\ \||x|^\gamma f\|_{L^2_x}\sim &a_0^{s_c+\gamma}, \quad \mbox{ for any }\gamma\ge0;\\ \|f\|_{L^r_x}\sim &a_0^{-1+\frac dr}, \quad \mbox{ for any } r\in [1, +\infty]. \end{aligned}$$ Our analysis below also involves the estimates on $\Delta v_0$. Note that $$\begin{aligned} \Delta v_0=\delta_0 \mathrm{e}^{-i\frac{b_0 |x|^2}{2}}\Big[c_1b_0\big(d+x\cdot \nabla\big)f+c_2b_0^2|x|^2f+\Delta f\Big],\end{aligned}$$ for some absolute constants $c_1\in {\mathbb C},c_2\in {\mathbb C}$. Applying [\[est:f\]](#est:f){reference-type="eqref" reference="est:f"}, we have that there exists an absolute constant $C>0$ such that $$\begin{aligned} \label{est:delta-v0} \big\|\Delta v_0\big\|_{L^2_x} \le C\delta_0 \Big(b_0a_0^{s_c}+b_0^2a_0^{s_c+2}+a_0^{s_c-2}\Big).\end{aligned}$$ Denote the space $\mathcal X_{s_c}(I)$ (which is a slight modification of $X_{s_c}(I)$) by its norm $$\|f\|_{\mathcal X_{s_c}(I)}\triangleq \|f\|_{L^q_tL^r_x(I)},$$ where the parameter pair $(q,r)$ satisfying that for arbitrary small constant $\epsilon>0$, $$\frac1q=\frac14+\epsilon, \frac1r=\frac14-\epsilon, \mbox{ when } d=2; \quad (q,r)=(4, 6), \mbox{ when } d=3.$$ In the following, we need the estimate on $$\big\|\mathrm{e}^{is \langle\nabla\rangle} \mathcal{S}_\varepsilon v_0\big\|_{\mathcal X_{s_c}(I)},$$ which involves the Strichartz estimates of the Schrödinger flow on the oscillating data. The latter will be presented in the subsubsection below. ### Estimates on the oscillating data Now we need the following result. **Lemma 25**. *Given $T_0>0$. Let $T_0>0$ and $I=[0,T_0]$, then there exist constant $\varepsilon_0$ such that for any $\varepsilon\in [0,\varepsilon_0]$, $$\begin{aligned} \Big\|\mathrm{e}^{it\langle \nabla\rangle}\mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} \le & \rho(b_0) + C_{a_0,b_0} \varepsilon^2 T_0^{1+\frac1q},\end{aligned}$$ where $\rho(b_0)\to 0$ when $b_0\to+\infty$.* *Proof.* Denote the operator $$T_t =\frac{\mathrm{e}^{it\langle\nabla\rangle}-\mathrm{e}^{it}\mathrm{e}^{-\frac12 it\Delta}}{t\Delta^2}.$$ Then by the Mihlin-Hörmander Multiplier Theorem, we have the $L^p\mapsto L^p$ boundedness which is uniform in time: for any $1<p<+\infty$, $$\begin{aligned} \big\|T_t h\big\|_{L^p}\le C\|h\|_{L^p},\end{aligned}$$ where the constant $C>0$ is independent of $t$ and $h$. Now we write $$\begin{aligned} \label{apprx-linearKG} \mathrm{e}^{it\langle\nabla\rangle}=\mathrm{e}^{it}\mathrm{e}^{-\frac12 it\Delta}+t \Delta^2 T_t.\end{aligned}$$ Then we have that $$\begin{aligned} \label{est:Sbf-12} \Big\|\mathrm{e}^{ it\langle \nabla\rangle}& \mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}\notag\\ \le & \Big\|\mathrm{e}^{it}\mathrm{e}^{-\frac12 it\Delta}\mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} + \Big\|t \Delta^2 T_t \mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big)\Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}\notag\\ \lesssim & \Big\|\mathrm{e}^{-\frac12 it\Delta}\mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} + \varepsilon^{-2}T_0 \Big\|\Delta^2\mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big)\Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}.\end{aligned}$$ For the first term in [\[est:Sbf-12\]](#est:Sbf-12){reference-type="eqref" reference="est:Sbf-12"}, by changing the variable, we have that $$\begin{aligned} \label{est:Sbf-scaling} \Big\|\mathrm{e}^{-\frac12 it\Delta}\mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} =& \Big\|\mathrm{e}^{-\frac12 it\Delta}\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}(I)}.\end{aligned}$$ Now we need the following formula (see e.g. [@CaWe-1992]), $$\begin{aligned} \mathrm{e}^{-\frac12 it\Delta}\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) =& (1+b_0t)^{-\frac d2}\> \mathrm{e}^{\frac{ib_0}{4(1+b_0t)}|x|^2} \mathrm{e}^{-it_{b_0} \Delta}f\Big(\frac x{1+b_0t}\Big), \mbox{ with } t_{b_0}=\frac t{1+b_0t}.\end{aligned}$$ This gives that for any $r\in [1,+\infty]$, $$\begin{aligned} \Big\|\mathrm{e}^{-\frac12 it\Delta}\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{L^r_x} =& (1+b_0t)^{-\frac d2+\frac dr}\big\|\mathrm{e}^{-\frac12 it_{b_0}\Delta}f \big\|_{L^r_x}.\end{aligned}$$ Inserting this into the right-hand side of [\[est:Sbf-scaling\]](#est:Sbf-scaling){reference-type="eqref" reference="est:Sbf-scaling"}, gives that $$\begin{aligned} \Big\|\mathrm{e}^{-\frac12 it\Delta}\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}({\mathbb R})} \le & \Big\|(1+b_0t)^{-s_c-\frac{2}q}\big\|\mathrm{e}^{-\frac12 it_{b_0}\Delta}f \big\|_{L^r_x} \Big\|_{L^q_t({\mathbb R})}\\ \le & \Big\|(1+b_0t)^{-\frac 2q}\big\|\mathrm{e}^{-\frac12 it_{b_0}\Delta}f \big\|_{L^r_x} \Big\|_{L^q_t({\mathbb R})}\\ =& \Big\|\big\|\mathrm{e}^{-\frac12 it\Delta}f \big\|_{L^r_x} \Big\|_{L^q_t([0,\frac1b])}\\ \le & \big\|\mathrm{e}^{-\frac12 it\Delta}f \big\|_{\mathcal X_{s_c}([0,\frac1b])}.\end{aligned}$$ Note that $$\big\|\mathrm{e}^{-\frac12 it\Delta}f \big\|_{\mathcal X_{s_c}({\mathbb R})}\le \|f\|_{\dot H^{s_c}_x}\lesssim 1.$$ Then we obtain that there exists a function $\rho:{\mathbb R}^+\mapsto {\mathbb R}^+$ such that $$\begin{aligned} \label{est:Sbf-1} \Big\|\mathrm{e}^{-\frac12 it\Delta}\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}({\mathbb R})} = &\rho(b_0),\end{aligned}$$ where $\rho(b_0)\to 0$ when $b_0\to+\infty$. For the second term in [\[est:Sbf-12\]](#est:Sbf-12){reference-type="eqref" reference="est:Sbf-12"}, we have that $$\begin{aligned} \label{est:Sbf-2} \Big\|\Delta^2\mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big)\Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} \lesssim & \varepsilon^4\Big\|\Delta^2\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big)\Big\|_{\mathcal X_{s_c}(I)}\notag\\ \lesssim & \varepsilon^4 T_0^{\frac1q} \Big\|\Delta^2\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big)\Big\|_{L^r_x}\notag\\ \le &C_{a_0,b_0} \varepsilon^4 T_0^{\frac1q}.\end{aligned}$$ Therefore, inserting the estimates [\[est:Sbf-1\]](#est:Sbf-1){reference-type="eqref" reference="est:Sbf-1"} and [\[est:Sbf-2\]](#est:Sbf-2){reference-type="eqref" reference="est:Sbf-2"} into [\[est:Sbf-12\]](#est:Sbf-12){reference-type="eqref" reference="est:Sbf-12"}, we have that $$\begin{aligned} \Big\|\mathrm{e}^{ it\langle \nabla\rangle}& \mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} \le \rho(b_0) + C_{a_0,b_0} \varepsilon^2 T_0^{1+\frac1q}.\end{aligned}$$ Thus we obtain the desired estimate. ◻ ### Behavior of the remainder function Next, we consider the behavior of the remainder function $r^\varepsilon$. Applying the same precess and notations in Section [4](#sec:global){reference-type="ref" reference="sec:global"}, we define $\phi$ as in [\[def:phi\]](#def:phi){reference-type="eqref" reference="def:phi"} and conclude that for any $\gamma\ge-s_c$, $$\begin{aligned} \label{est: Xsc-phi} \|\phi\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}\le C_{a_0,b_0}\delta_0 \varepsilon^{2};\quad \||\nabla|^\gamma\phi\|_{X_{s_c}(\varepsilon^{-2}I)}\le C_{a_0,b_0}\delta_0\varepsilon^{2+\gamma}.\end{aligned}$$ See Remark [Remark 23](#rem:3d-smalldata){reference-type="ref" reference="rem:3d-smalldata"} for the explanation of the three dimensional cases. Denote $T^\varepsilon=\varepsilon^{-2}T$ and $T\in [T_0-1,T_0]$. As in [\[est:Duh-mod\]](#est:Duh-mod){reference-type="eqref" reference="est:Duh-mod"}, we write that [\[est:Duh-mod\'\]]{#est:Duh-mod' label="est:Duh-mod'"} $$\begin{aligned} \phi(T^\varepsilon)=&\mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \phi(0)\label{Duh-mod'-0}\\ &\quad+ \int_0^{T^\varepsilon} \mathrm{e}^{i(T^\varepsilon-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}G\big(h^{\varepsilon},R)\,ds\label{Duh-mod'-1}\\ &\quad +\int_0^{T^\varepsilon} \mathrm{e}^{i(T^\varepsilon-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}P_{> 1}F_1\big(h^{\varepsilon}\big)\,ds\label{Duh-mod'-1'}\\ &\quad - \frac{\mathrm{e}^{3iT^\varepsilon} }{i(\langle\nabla\rangle-3)}\langle \nabla\rangle^{-1}P_{\le 1} \big(h^{\varepsilon}(T^\varepsilon)\big)^3 - \frac{\mathrm{e}^{3iT^\varepsilon} }{i(\langle\nabla\rangle+3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(\overline{h^{\varepsilon}(T^\varepsilon)}\Big)^3\label{Duh-mod'-2}\\ &\quad + \frac{\mathrm{e}^{iT^\varepsilon\langle \nabla \rangle}}{i(\langle\nabla\rangle-3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(\mathcal{S}_\varepsilon v_0\Big)^3 + \frac{\mathrm{e}^{iT^\varepsilon\langle \nabla \rangle}}{i(\langle\nabla\rangle+3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(\overline{\mathcal{S}_\varepsilon v_0}\Big)^3\label{Duh-mod'-3}\\ &\quad+3\mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is(\langle\nabla\rangle-3)} \frac{1}{i(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(h^{\varepsilon}\big)^2\> \partial_s h^\varepsilon\Big]\,ds\label{Duh-mod'-4}\\ &\quad+3\mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is(\langle\nabla\rangle+3)} \frac{1}{i(\langle\nabla\rangle+3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(\overline{h^\varepsilon})^2\> \partial_s\big(\overline{h^\varepsilon}\big)\Big]\,ds.\label{Duh-mod'-5}\end{aligned}$$ **Lower bound of [\[Duh-mod\'-0\]](#Duh-mod'-0){reference-type="eqref" reference="Duh-mod'-0"}+[\[Duh-mod\'-3\]](#Duh-mod'-3){reference-type="eqref" reference="Duh-mod'-3"}.** By [\[phi-initialdata\]](#phi-initialdata){reference-type="eqref" reference="phi-initialdata"} and [\[na-apprx\]](#na-apprx){reference-type="eqref" reference="na-apprx"}, we write [\[Duh-mod\'-0\]](#Duh-mod'-0){reference-type="eqref" reference="Duh-mod'-0"} and [\[Duh-mod\'-3\]](#Duh-mod'-3){reference-type="eqref" reference="Duh-mod'-3"} as $$\eqref{Duh-mod'-0} =-2\varepsilon^2\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon \big({\mathrm{Re}}(v_1)\big)+\mathcal R_1,$$ and $$\eqref{Duh-mod'-3} =\frac i4 \mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\Big(\mathcal{S}_\varepsilon v_0\Big)^3+\mathcal R_2,$$ where for $j=1,2$, $$\begin{aligned} \big\|\mathcal R_j\big\|_{L^\infty_t L^2_x({\mathbb R}\times {\mathbb R}^2)} \lesssim & \varepsilon^2\big\|\Delta \mathcal{S}_\varepsilon \big({\mathrm{Re}}(v_1)\big)\big\|_{L^2_x}+\big\|\Delta \mathcal{S}_\varepsilon v_0\big\|_{L^2_x} \big\|\mathcal{S}_\varepsilon v_0\big\|_{L^\infty_x}^2\\ \le & C_{a_0,b_0}\delta_0 \varepsilon^{4-s_c}.\end{aligned}$$ Hence, there give that $$\begin{aligned} \eqref{Duh-mod'-0}+\eqref{Duh-mod'-3} =&\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\Big[-2\varepsilon^2\mathcal{S}_\varepsilon \big({\mathrm{Re}}(v_1)\big)+\frac i2\Big(\mathcal{S}_\varepsilon v_0\Big)^3-\frac i4\Big(\overline{\mathcal{S}_\varepsilon v_0}\Big)^3\Big]+\mathcal R_1+\mathcal R_2\\ =&\varepsilon^2\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon \Big[-2{\mathrm{Re}}(v_1)+\frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\Big] +\mathcal R_1+\mathcal R_2.\end{aligned}$$ From [\[lower-bound-v1v0\]](#lower-bound-v1v0){reference-type="eqref" reference="lower-bound-v1v0"}, we get that $$\begin{aligned} &\Big\|\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon \Big[-2{\mathrm{Re}}(v_1)+\frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\Big]\Big\|_{L^2_x}\\ = &\Big\|\mathcal{S}_\varepsilon \Big[-2{\mathrm{Re}}(v_1)+\frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\Big]\Big\|_{L^2_x}\\ = &\varepsilon^{-s_c} \Big\|-2{\mathrm{Re}}(v_1)+\frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\Big\|_{L^2_x}\\ \ge &c_0\delta_0^3\varepsilon^{-s_c}\|f\|_{L^6_x}^3.\end{aligned}$$ where $c_0$ is determined in [\[lower-bound-v1v0\]](#lower-bound-v1v0){reference-type="eqref" reference="lower-bound-v1v0"}. Then by [\[est:f\]](#est:f){reference-type="eqref" reference="est:f"}, it further gives that there exists some absolute constant $\tilde c_0>0$ such that $$\begin{aligned} &\Big\|\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon \Big[-2{\mathrm{Re}}(v_1)+\frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\Big]\Big\|_{L^2_x} \ge \tilde c_0 a_0^{-2+s_c}\delta_0^3 \varepsilon^{-s_c} .\end{aligned}$$ Therefore, it implies that $$\begin{aligned} \big\|\eqref{Duh-mod'-0}+\eqref{Duh-mod'-3}\big\|_{L^2_x} \ge &\varepsilon^2\Big\|\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon \Big[-2{\mathrm{Re}}(v_1)+\frac i2\big(v_0\big)^3-\frac i4\big(\overline{v_0}\big)^3\Big]\Big\|_{L^2_x}\\ &\qquad -\big\|\mathcal R_1\big\|_{L^2_x}-\big\|\mathcal R_2\big\|_{L^2_x}\\ \ge &\tilde c_0a_0^{-2+s_c}\delta_0^3\varepsilon^{2-s_c}-C_{a_0,b_0}\delta_0 \varepsilon^{4-s_c}.\end{aligned}$$ **Upper bound of [\[Duh-mod\'-1\]](#Duh-mod'-1){reference-type="eqref" reference="Duh-mod'-1"}.** To do this, we denote the space $\mathcal Z_{s_c}(I)$ by its norm $$\|f\|_{\mathcal Z_{s_c}(I)}\triangleq \|f\|_{L^{q'}_tL^{r'}_x(I)},$$ where the parameter pair $(q,r)$ satisfying that for arbitrary small constant $\epsilon>0$, $$\frac1q=\frac12-\epsilon, \frac1r=\epsilon, \mbox{ when } d=2; \quad (q,r)=(2, 6), \mbox{ when } d=3.$$ Then by Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that $$\begin{aligned} \big\|\eqref{Duh-mod'-1}\big\|_{L^2_x} \lesssim & \Big\|G\big(h^{\varepsilon},R)\Big\|_{\mathcal Z_{s_c}(\varepsilon^{-2}I)}.\end{aligned}$$ Then by Hölder's inequality, we further get that $$\begin{aligned} \big\|\eqref{Duh-mod'-1}\big\|_{L^2_x} \lesssim &\big\|R\big\|_{L^\infty_tL^2_x(\varepsilon^{-2}I)} \sum\limits_{j=0}^2\big\|h^{\varepsilon}\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}^j\big\|R\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}^{2-j}\\ \lesssim &\big\|\phi\big\|_{L^\infty_tL^2_x(\varepsilon^{-2}I)} \sum\limits_{j=0}^2\big\|h^{\varepsilon}\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}^j\big\|\phi\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}^{2-j}.\end{aligned}$$ By [\[apprx-h-1-op2\]](#apprx-h-1-op2){reference-type="eqref" reference="apprx-h-1-op2"} and Lemma [Lemma 25](#lem:conterexmp-ID){reference-type="ref" reference="lem:conterexmp-ID"}, we have that $$\begin{aligned} \big\|h^{\varepsilon}\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} \lesssim & \big\|\mathrm{e}^{it (\langle \nabla\rangle-1)} \mathcal{S}_\varepsilon v_0\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} +\big\|\mathrm{e}^{-it}\mathcal N_{\bot}(w)\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}\\ \lesssim & \big\|\mathrm{e}^{it \langle \nabla\rangle} \mathcal{S}_\varepsilon v_0\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)} +\big\|\mathcal N_{\bot}(w)\big\|_{\mathcal X_{s_c}(\varepsilon^{-2}I)}\\ \le & \delta_0\rho(b_0) +C_{a_0, b_0,T_0}\>\delta_0 \big(\varepsilon^2+\delta_0^2\big).\end{aligned}$$ Applying this estimate and [\[est: Xsc-phi\]](#est: Xsc-phi){reference-type="eqref" reference="est: Xsc-phi"}, we obtain that $$\begin{aligned} \big\|\eqref{Duh-mod'-1}\big\|_{L^2_x} \le &\delta_0^3\varepsilon^{2-s_c}\Big(\rho(b_0) +C_{a_0, b_0,T_0}\> \big(\varepsilon^2+\delta_0^2\big)\Big),\end{aligned}$$ where as above, the constant $C_{a_0, b_0,T_0}>0$ only depends on $a_0, b_0$ and $T_0$. **Upper bound of [\[Duh-mod\'-1\'\]](#Duh-mod'-1'){reference-type="eqref" reference="Duh-mod'-1'"}.** Similar as above, by Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"} and [\[est: Xsc-phi\]](#est: Xsc-phi){reference-type="eqref" reference="est: Xsc-phi"}, we have that $$\begin{aligned} \big\|\eqref{Duh-mod'-1'}\big\|_{L^2_x} \lesssim &\Big\|P_{> 1}F_1\big(h^{\varepsilon}\big)\Big\|_{L^q_{t}L^r_x(\varepsilon^{-2}I)}\end{aligned}$$ where $(q,r)$ is the pair of $$q=\frac{4(d+2)}{5d},\quad r=\frac{2d(d+2)}{d^2+d+8}.$$ Then by Hölder's and Bernstein's inequalities, we have that $$\begin{aligned} \big\|\eqref{Duh-mod'-1'}\big\|_{L^2_x} \lesssim &\big\|P_{\gtrsim 1}h^{\varepsilon}\big\|_{L^\infty_tL^2_x(\varepsilon^{-2}I)}^\frac12\big\|P_{\gtrsim 1}|\nabla|^{s_c} h^{\varepsilon}\big\|_{L^\frac{2(d+2)}{d}_{tx}(\varepsilon^{-2}I)}^\frac12\big\||\nabla|^{s_c}h^{\varepsilon}\big\|_{L^ \frac{2(d+2)}{d}_{tx}(\varepsilon^{-2}I)}^2\\ \lesssim &\big\||\nabla|^{4} h^{\varepsilon}\big\|_{L^\infty_tL^2_x(\varepsilon^{-2}I)}^\frac12\big\||\nabla|^{4} h^{\varepsilon}\big\|_{L^\frac{2(d+2)}{d}_{tx}(\varepsilon^{-2}I)}^\frac12\big\||\nabla|^{s_c}h^{\varepsilon}\big\|_{L^ \frac{2(d+2)}{d}_{tx}(\varepsilon^{-2}I)}^2\\ \lesssim &\big\||\nabla|^{4-s_c}h^{\varepsilon}\big\|_{X_{s_c}(\varepsilon^{-2}I)}\big\|h^{\varepsilon}\big\|_{X_{s_c}(\varepsilon^{-2}I)}^2.\end{aligned}$$ By [\[h-L4-sd\]](#h-L4-sd){reference-type="eqref" reference="h-L4-sd"}, we obtain that $$\begin{aligned} \big\|\eqref{Duh-mod'-1'}\big\|_{L^2_x} \le C_{a_0, b_0}\delta_0^3 \varepsilon^{4-s_c}.\end{aligned}$$ **Upper bound of [\[Duh-mod\'-2\]](#Duh-mod'-2){reference-type="eqref" reference="Duh-mod'-2"}.** By [\[est:bound-nabal3\]](#est:bound-nabal3){reference-type="eqref" reference="est:bound-nabal3"}, [\[apprx-h-1-op2\]](#apprx-h-1-op2){reference-type="eqref" reference="apprx-h-1-op2"} and [\[est:mathcal-N-h\]](#est:mathcal-N-h){reference-type="eqref" reference="est:mathcal-N-h"}, we have that $$\begin{aligned} \label{est:Duh-mod'-2-1} \big\| \eqref{Duh-mod'-2}\big\|_{L^2_x} \lesssim & \big\|h^{\varepsilon}(T^\varepsilon)\big\|_{L^6_x}^3\notag\\ \lesssim & \big\|\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon v_0\big\|_{L^6_x}^3 +\big\|\mathrm{e}^{-it}\mathcal N_{\bot}(w)\big\|_{L^6_x}^3.\end{aligned}$$ By Lemma [Lemma 14](#lem:dispersive){reference-type="ref" reference="lem:dispersive"}, we have that $$\begin{aligned} \big\|\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon v_0\big\|_{L^6_x} \lesssim & \langle T^\varepsilon\rangle^{-\frac d3} \big\|\langle \nabla\rangle^{\frac{d+2}{3}}\mathcal{S}_\varepsilon v_0\big\|_{L^\frac65_x}. \end{aligned}$$ Note that $$\begin{aligned} \big\|\langle \nabla\rangle^{\frac{d+2}{3}}\mathcal{S}_\varepsilon v_0\big\|_{L^\frac65_x} \le & \big\|\mathcal{S}_\varepsilon v_0\big\|_{L^\frac65_x} +\big\| |\nabla|^{\frac{d+2}{3}}\mathcal{S}_\varepsilon v_0\big\|_{L^\frac65_x}\\ = & \varepsilon^{1-\frac56 d} \big\|v_0\big\|_{L^\frac65_x} +\varepsilon^{\frac53-\frac12 d} \big\| |\nabla|^{\frac{d+2}{3}}v_0\big\|_{L^\frac65_x}. % \lesssim & % \varepsilon^{1-\frac56 d} \big\|f\big\|_{L^\frac65_x} % +\varepsilon^{1-\frac56 d}\big(\varepsilon b_0\big)^{\frac{d+2}{3}} \big\| \langle\nabla\rangle^{\frac{d+2}{3}}(\langle x\rangle^2f)\big\|_{L^\frac65_x}.\end{aligned}$$ Note that by [\[est:f\]](#est:f){reference-type="eqref" reference="est:f"}, $$\begin{aligned} \big\|v_0\big\|_{L^\frac65_x} \le \delta_0 \big\|f\big\|_{L^\frac65_x} \le C\delta_0;\quad \big\| |\nabla|^{\frac{d+2}{3}}v_0\big\|_{L^\frac65_x} \le C_{a_0,b_0}.\end{aligned}$$ We further have that $$\begin{aligned} \big\|\langle \nabla\rangle^{\frac{d+2}{3}}\mathcal{S}_\varepsilon v_0\big\|_{L^\frac65_x} \le & C \varepsilon^{1-\frac56 d} +C_{a_0,b_0}\varepsilon^{\frac53-\frac12 d}.\end{aligned}$$ Set $T_0\ge 2$ and thus $T\ge1$, then it reduces that $$\begin{aligned} \big\|\mathrm{e}^{iT^\varepsilon\langle \nabla\rangle}\mathcal{S}_\varepsilon v_0\big\|_{L^6_x} \le & \big(T^\varepsilon\big)^{-\frac d3} C \varepsilon^{1-\frac56 d} +C_{a_0,b_0}\varepsilon^{\frac53-\frac12 d}\\ \le & T_0^{-\frac d3} \varepsilon^{\frac13(2-s_c)} \Big(C+C_{a_0,b_0}\varepsilon^{\frac23}\Big). \end{aligned}$$ Moreover, $$\begin{aligned} \big\|\mathrm{e}^{-it}\mathcal N_{\bot}(w)\big\|_{L^6_x} \lesssim \big\||\nabla|^\frac d3\mathcal N_{\bot}(w)\big\|_{L^6_x} \le C_{a_0,b_0}\delta_0^3 \varepsilon^{1-\frac d6}. \end{aligned}$$ Inserting these two estimates above into [\[est:Duh-mod\'-2-1\]](#est:Duh-mod'-2-1){reference-type="eqref" reference="est:Duh-mod'-2-1"}, we obtain that $$\begin{aligned} \big\| \eqref{Duh-mod'-2}\big\|_{L^2_x} \le & T_0^{-d}\delta_0^3 \varepsilon^{2-s_c} \Big(C+C_{a_0,b_0}(\delta_0^2+\varepsilon^2\big)\Big).\end{aligned}$$ **Upper bound of [\[Duh-mod\'-4\]](#Duh-mod'-4){reference-type="eqref" reference="Duh-mod'-4"} and [\[Duh-mod\'-5\]](#Duh-mod'-5){reference-type="eqref" reference="Duh-mod'-5"}.** We only focus on the term [\[Duh-mod\'-4\]](#Duh-mod'-4){reference-type="eqref" reference="Duh-mod'-4"}, since the other term [\[Duh-mod\'-5\]](#Duh-mod'-5){reference-type="eqref" reference="Duh-mod'-5"} can be treated similarly. By [\[apprx-h-1-op2\]](#apprx-h-1-op2){reference-type="eqref" reference="apprx-h-1-op2"} and [\[apprx-ht-1-op2\]](#apprx-ht-1-op2){reference-type="eqref" reference="apprx-ht-1-op2"}, we rewrite it as $$\begin{aligned} \eqref{Duh-mod'-4} = & \frac 32i \mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is(\langle\nabla\rangle-3)} \frac{1}{i(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1}\notag\\ &\quad \Bigg\{\Big[\mathrm{e}^{is (\langle \nabla\rangle-1)} \mathcal{S}_\varepsilon v_0 +\mathrm{e}^{-is}\mathcal N_{\bot}(w)\Big]^2\> \Big[ \Delta \Big(\mathrm{e}^{-is(\langle\nabla\rangle+1)}-\mathrm{e}^{is(\langle\nabla\rangle-1)}\Big)\mathcal{S}_\varepsilon v_0 +\widetilde{\mathcal N}_{\bot}(w)\Big]\Bigg\}\,ds.\end{aligned}$$ Therefore, it gives that $$\begin{aligned} \eqref{Duh-mod'-4} = & \mathcal P_{\bot}+\mathcal R_{\bot},\end{aligned}$$ where the terms $\mathcal P_{\bot}$ and $\mathcal R_{\bot}$ are defined as $$\begin{aligned} \mathcal P_{\bot}\triangleq & - \frac 34 \mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1}\Bigg\{\Big[\mathrm{e}^{is (\langle\nabla\rangle-1)} \mathcal{S}_\varepsilon v_0\Big]^2\\ &\qquad \qquad\cdot \Big[ \Delta \Big(\mathrm{e}^{-is(\langle\nabla\rangle+1)}-\mathrm{e}^{is(\langle\nabla\rangle-1)}\Big)\mathcal{S}_\varepsilon v_0 \Big]\Bigg\}\,ds;\\ \mathcal R_{\bot} \triangleq & \frac 32i \mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is\langle\nabla\rangle} T(\nabla) \Bigg\{\Big[\mathrm{e}^{is \langle\nabla\rangle} \mathcal{S}_\varepsilon v_0\Big]^2\> \Big[ \Delta \Big(\mathrm{e}^{-is(\langle\nabla\rangle+1)}-\mathrm{e}^{is(\langle\nabla\rangle-1)}\Big)\mathcal{S}_\varepsilon v_0 \Big]\Bigg\}\,ds\\ &\quad+ \frac 32i \mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is\langle\nabla\rangle} \frac{1}{i(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1}\notag\\ &\qquad \Bigg\{\Big[2\mathrm{e}^{is \langle \nabla\rangle} \mathcal{S}_\varepsilon v_0\cdot \mathcal N_{\bot}(w) +\big(\mathcal N_{\bot}(w)\big)^2\Big] \Big[ \Delta \Big(\mathrm{e}^{-is\langle\nabla\rangle}-\mathrm{e}^{is\langle\nabla\rangle}\Big)\mathcal{S}_\varepsilon v_0 +\widetilde{\mathcal N}_{\bot}(w)\Big]\Bigg\}\,ds, \end{aligned}$$ where $T(\nabla)=\frac{1}{i(\langle\nabla\rangle-3)} P_{\le 1}+\frac1{2i}$. Note that for any $1<p<+\infty$, $$\begin{aligned} \big\|T(\nabla)f\big\|_{L^p_x} \lesssim \|\Delta f\|_{L^p_x}.\end{aligned}$$ Then using this estimate, Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, [\[est:mathcal-N-h\]](#est:mathcal-N-h){reference-type="eqref" reference="est:mathcal-N-h"} and [\[est:mathcal-N-tilde-h\]](#est:mathcal-N-tilde-h){reference-type="eqref" reference="est:mathcal-N-tilde-h"}, we have that $$\begin{aligned} \big\|\mathcal R_{\bot}\big\|_{L^2_x} \le & C_{a_0, b_0}\delta_0^3 \varepsilon^{2-s_c} \big(\delta_0^2+\varepsilon^2\big). \end{aligned}$$ Now we consider $\mathcal P_{\bot}$. To do this, we split it as $$\mathcal P_{\bot}=\mathcal P_{\bot,1}+\mathcal P_{\bot,2},$$ where $$\begin{aligned} \mathcal P_{\bot,1}\triangleq & - \frac 34\varepsilon^2 \mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is\langle\nabla\rangle} \langle \nabla\rangle^{-1}P_{\le 1}\Big[\Big(\mathrm{e}^{is \langle\nabla\rangle} \mathcal{S}_\varepsilon v_0\Big)^2\> \mathrm{e}^{-is\langle\nabla\rangle}\mathcal{S}_\varepsilon\big(\Delta v_0\big) \Big]\,ds;\\ \mathcal P_{\bot,2}\triangleq & - \frac 34\varepsilon^2 \mathrm{e}^{iT^\varepsilon\langle\nabla\rangle} \int_0^{T^\varepsilon} \mathrm{e}^{-is(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1}\Big[\Big(\mathrm{e}^{is (\langle\nabla\rangle-1)} \mathcal{S}_\varepsilon v_0\Big)^2\> \mathrm{e}^{is(\langle\nabla\rangle-1)}\mathcal{S}_\varepsilon\big(\Delta v_0\big) \Big]\,ds.\end{aligned}$$ For $\mathcal P_{\bot,1}$, by Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that $$\begin{aligned} \big\|\mathcal P_{\bot,1}\big\|_{L^2_x} \lesssim & \big\|\mathrm{e}^{is \langle\nabla\rangle} \mathcal{S}_\varepsilon v_0\big\|_{\mathcal X_{s_c}([0,T^\varepsilon])}^2 \big\|\mathrm{e}^{is \langle\nabla\rangle} \mathcal{S}_\varepsilon \big(\Delta v_0\big)\big\|_{L^\infty_tL^2_x([0,T^\varepsilon])}\\ \lesssim & \delta_0^3\varepsilon^{2-s_c}\Big\|\mathrm{e}^{it\langle \nabla\rangle}\mathcal{S}_\varepsilon\Big(\mathrm{e}^{-i\frac{b_0 |x|^2}{2}}f\Big) \Big\|_{\mathcal X_{s_c}([0,T^\varepsilon])}^2 \big\|\Delta v_0\big\|_{L^2_x}.\end{aligned}$$ Therefore, by [\[est:delta-v0\]](#est:delta-v0){reference-type="eqref" reference="est:delta-v0"} and Lemma [Lemma 25](#lem:conterexmp-ID){reference-type="ref" reference="lem:conterexmp-ID"}, we obtain that $$\begin{aligned} \big\|\mathcal P_{\bot,1}\big\|_{L^2_x} \lesssim & \delta_0^3\varepsilon^{2-s_c}\Big(\rho(b_0) + C_{a_0,b_0} \varepsilon^2 T_0^{1+\frac1q}\Big)\Big(b_0a_0^{s_c}+b_0^2a_0^{s_c+2}+a_0^{s_c-2}\Big)\\ \le& \delta_0^3\varepsilon^{2-s_c}\Big(\rho(b_0)a_0^{s_c-2}+C_{b_0}a_0^{s_c}+C_{a_0,b_0,T_0} \varepsilon^2\Big).\end{aligned}$$ For $\mathcal P_{\bot,1}$, integration by parts, we have that $$\begin{aligned} \mathcal P_{\bot,2}= & \frac 34\varepsilon^2\mathrm{e}^{iT^\varepsilon\langle\nabla\rangle}\Bigg[ \frac{\mathrm{e}^{-is(\langle\nabla\rangle-3)} }{(\langle\nabla\rangle-3)\langle \nabla\rangle} P_{\le 1}\Big[\Big(\mathrm{e}^{is (\langle\nabla\rangle-1)} \mathcal{S}_\varepsilon v_0\Big)^2\> \mathrm{e}^{is(\langle\nabla\rangle-1)}\mathcal{S}_\varepsilon\big(\Delta v_0\big) \Big|_0^{T^\varepsilon} \\ &\quad - \int_0^{T^\varepsilon} \frac{\mathrm{e}^{-is(\langle\nabla\rangle-3)} }{(\langle\nabla\rangle-3)\langle \nabla\rangle}P_{\le 1} \partial_s\Big[\Big(\mathrm{e}^{is (\langle\nabla\rangle-1)} \mathcal{S}_\varepsilon v_0\Big)^2\> \mathrm{e}^{is(\langle\nabla\rangle-1)}\mathcal{S}_\varepsilon\big(\Delta v_0\big) \Big]\,ds\Bigg].\end{aligned}$$ Since $$\partial_s \mathrm{e}^{is(\langle\nabla\rangle-1)}=i(\langle\nabla\rangle-1) \mathrm{e}^{is(\langle\nabla\rangle-1)} =\frac{i|\nabla|^2}{1+\langle\nabla\rangle}\mathrm{e}^{is(\langle\nabla\rangle-1)},$$ by Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that $$\begin{aligned} \big\|\mathcal P_{\bot,2}\big\|_{L^2_x} \lesssim & \delta_0^3C_{a_0,b_0} \varepsilon^4.\end{aligned}$$ Combining with the estimates on $\mathcal P_{\bot,1}, \mathcal P_{\bot,2}$ and $\mathcal R_{\bot}$ above, we obtain that $$\begin{aligned} \big\| \eqref{Duh-mod'-4}\big\|_{L^2_x} \le & \delta_0^3\varepsilon^{2-s_c}\Big(\rho(b_0)a_0^{s_c-2}+C_{b_0}a_0^{s_c}+C_{a_0, b_0}\delta_0^2+C_{a_0,b_0,T_0} \varepsilon^2\Big).\end{aligned}$$ Arguing similarly, we also have that $$\begin{aligned} \big\| \eqref{Duh-mod'-5}\big\|_{L^2_x} \le & \delta_0^3\varepsilon^{2-s_c}\Big(\rho(b_0)a_0^{s_c-2}+C_{b_0}a_0^{s_c}+C_{a_0, b_0}\delta_0^2+C_{a_0,b_0,T_0} \varepsilon^2\Big).\end{aligned}$$ Combining the estimates on [\[est:Duh-mod\'\]](#est:Duh-mod'){reference-type="eqref" reference="est:Duh-mod'"}, we obtain that for any $T\in [T_0-1,T_0]$ and any $\varepsilon\in (0,\varepsilon_0]$, $$\begin{aligned} \big\|\phi(T^\varepsilon)\big\|_{L^2_x} \ge &\delta_0^3\varepsilon^{2-s_c}\Big[ \tilde c_0a_0^{s_c-2} -\Big(\rho(b_0)a_0^{s_c-2}+C_{b_0}a_0^{s_c}+T_0^{-d}+C_{a_0, b_0}\delta_0^2+C_{a_0,b_0,T_0}\delta_0^{-2} \varepsilon^2\Big)\Big].\end{aligned}$$ Choosing $b_0$ suitably large such that $\rho(b_0)\le \frac14\tilde c_0$ and $T_0\ge 2$, and by the relationship [\[relationships\]](#relationships){reference-type="eqref" reference="relationships"}, we obtain that $$\begin{aligned} \label{phi-lbd} \big\|\phi(T^\varepsilon)\big\|_{L^2_x} \ge &\frac12 \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}.\end{aligned}$$ Now we are in the position to prove Theorem [Theorem 4](#thm:main1-optimal){reference-type="ref" reference="thm:main1-optimal"}. Argue for contradiction, and suppose that for any $T\in [T_0-1,T_0]$, $$\begin{aligned} \label{Assup-contrid} \big\| R(T^\varepsilon)\big\|_{L^2_x} < & \frac1{8} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}.\end{aligned}$$ Applying [\[phi-varphi\]](#phi-varphi){reference-type="eqref" reference="phi-varphi"}, it is equivalent to $$\begin{aligned} \big\|{\mathrm{Im}}\>\phi(T^\varepsilon)\big\|_{L^2_x} < & \frac1{8} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c},\end{aligned}$$ Then from [\[phi-lbd\]](#phi-lbd){reference-type="eqref" reference="phi-lbd"}, it leads that for any $T\in [T_0-1,T_0]$, $$\begin{aligned} \big\|{\mathrm{Re}}\>\phi(T^\varepsilon)\big\|_{L^2_x} \ge & \frac{3}{8} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}.\end{aligned}$$ Using [\[phi-varphi\]](#phi-varphi){reference-type="eqref" reference="phi-varphi"} again, it drives that $$\begin{aligned} \big\|\partial_t R(T^\varepsilon)\big\|_{L^2_x} \ge & \frac{3}{8} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}.\end{aligned}$$ By Mean Value Theorem, we have that $$R(T^\varepsilon+1)= R(T^\varepsilon)+ \partial_tR(\tilde T^\varepsilon),\quad \mbox{for some } \tilde T^\varepsilon\in [T^\varepsilon,T^\varepsilon+1].$$ Therefore, $$\begin{aligned} \big\|R(T^\varepsilon+1)\big\|_{L^2_x} \ge & \big\|\partial_tR(\tilde T^\varepsilon)\big\|_{L^2_x}-\big\|R(T^\varepsilon)\big\|_{L^2_x}\\ \ge & \frac{3}{8} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}- \frac1{8} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}\\ =&\frac{1}{4} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}\\ >& \frac1{8}\tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}.\end{aligned}$$ This is a contradiction with [\[Assup-contrid\]](#Assup-contrid){reference-type="eqref" reference="Assup-contrid"} and thus there exists $T\in [T_0-1,T_0]$, such that $$\big\| R(T^\varepsilon)\big\|_{L^2_x} \ge \frac1{8} \tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2-s_c}.$$ Scaling back, we obtain that $$\big\|r(T)\big\|_{L^2_x}\ge \frac18\tilde c_0a_0^{s_c-2}\delta_0^3\varepsilon^{2}.$$ This completes the proof of Theorem [Theorem 4](#thm:main1-optimal){reference-type="ref" reference="thm:main1-optimal"}. # Single modulated limit {#sec:singe-modulate} .5cm In this section, we shall give the proof of Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"}. ## Formulation {#sec:Th3-formula} Denote that $r=\widetilde{v}^\varepsilon-\widetilde{v}$, then it is the solution of $$\begin{aligned} \begin{split} \label{eq:kg-r-3} \left\{ \aligned &\varepsilon^2\partial_{tt} r +2i\partial_t r- \Delta r +\varepsilon^2 \partial_{tt} \widetilde{v}+\big|\widetilde{v}^\varepsilon\big|^2\widetilde{v}^\varepsilon-|\widetilde{v}|^2\widetilde{v} =0, \\ & r(0,x) = r_0(x),\quad \partial_tr(0,x) = \varepsilon^{-2} r_1(x), \endaligned \right. \end{split}\end{aligned}$$ where $$r_0\triangleq \widetilde{v}^\varepsilon_0-\widetilde{v}_0;\quad r_1\triangleq \widetilde{v}^\varepsilon_1(x)-\frac{\varepsilon^2}{2i}(\Delta \widetilde{v}_0-\mu |\widetilde{v}_0|^2\widetilde{v}_0).$$ Moreover, we denote $$R=\mathrm{e}^{it} \mathcal{S}_\varepsilon r,$$ then $$\begin{aligned} \begin{split} \label{eq:kg-R-3} \left\{ \aligned &\partial_{tt} R - \Delta R + R +\varepsilon^4 \mathrm{e}^{it} \mathcal{S}_\varepsilon( \partial_{tt} \widetilde{v})+\mathrm{e}^{it} \Big(\big|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big|^2\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon-|\mathcal{S}_\varepsilon \widetilde{v}|^2\mathcal{S}_\varepsilon \widetilde{v}\Big) =0, \\ & R(0,x) = \mathcal{S}_\varepsilon r_0,\quad \partial_t R(0,x) =i \mathcal{S}_\varepsilon r_0+ \mathcal{S}_\varepsilon r^\varepsilon_1. \endaligned \right. \end{split}\end{aligned}$$ Similar as Section [3.2](#sec:LR-wave){reference-type="ref" reference="sec:LR-wave"}, we denote $$\begin{aligned} \begin{split} \left\{ \aligned &\mathcal R_1=\langle \nabla \rangle^{-1}\big(\partial_t-i \langle \nabla\rangle\big)R, \\ &\mathcal R_2=\langle \nabla \rangle^{-1}\big(\partial_t+i \langle \nabla\rangle\big)R. \endaligned \right. \end{split}\end{aligned}$$ Then $$\begin{aligned} \begin{split} \label{eq:R-R12} \left\{ \aligned &R=\frac i2\big(\mathcal R_1-\mathcal R_2\big), \\ &\partial_t R=\frac 12\langle\nabla\rangle\big(\mathcal R_1+\mathcal R_2\big), \endaligned \right. \end{split}\end{aligned}$$ and the initial datum $$\begin{aligned} \begin{split} \label{eq:R-12-KG-intialdatum} \left\{ \aligned &\mathcal R_1(0)=\mathcal R_{1,0}\triangleq \langle \nabla \rangle^{-1}\Big[i\big(1- \langle \nabla\rangle\big)\mathcal{S}_\varepsilon r_0+\mathcal{S}_\varepsilon r^\varepsilon_1\Big];\\ &\mathcal R_2(0)=\mathcal R_{2,0}\triangleq \langle \nabla \rangle^{-1}\Big[i\big(1+ \langle \nabla\rangle\big)\mathcal{S}_\varepsilon r_0+\mathcal{S}_\varepsilon r^\varepsilon_1\Big]. \endaligned \right. \end{split}\end{aligned}$$ Moreover, $\mathcal R_j$ obeys the following Duhamel's formula, $$\begin{aligned} \label{Duh-Rj-th3} \mathcal R_j(t)=&\mathrm{e}^{\mp i(t-t_0)\langle \nabla\rangle} \mathcal R_j(t_0)\notag\\ & -\int_{t_0}^t \mathrm{e}^{\mp i(t-s)\langle \nabla\rangle}\mathrm{e}^{is} \langle\nabla \rangle^{-1} \Big[\varepsilon^4 \mathcal{S}_\varepsilon( \partial_{tt} \widetilde{v})+\big|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big|^2\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon-|\mathcal{S}_\varepsilon \widetilde{v}|^2\mathcal{S}_\varepsilon \widetilde{v}\Big]\,ds.\end{aligned}$$ ## Proof of Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} {#proof-of-theorem-thmmain3} ### Regular cases {#sec:Thm3-case1-1} In this subsubsection, we consider the case when $\alpha=4$. The main result is **Proposition 26**. *Let $d=2,3$, $T>0$ and $\beta\in [1,2]$, then there exists $\delta_0>0$ such that the following property hold. Assume that* - *$v_0\in H^{4+s_c}({\mathbb R}^d),\widetilde{v}^\varepsilon_0\in H^\beta_x({\mathbb R}^d), v_1^\varepsilon\in H^{s_c}({\mathbb R}^d)$;* - *$(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\to (0,0) \mbox{ in } H^{s_c}({\mathbb R}^d).$* *When $d=3$, additionally assume that* - *$\varepsilon^2 T\|\widetilde{v}\|_{L^\infty_t\dot H^{\frac92}_x}\le \delta_0$.* *Let $\widetilde{v}^\varepsilon, \widetilde{v}$ be the corresponding solutions to [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} and [\[eq:nls-1\]](#eq:nls-1){reference-type="eqref" reference="eq:nls-1"} respectively, then for any $\beta\ge 1$, $$\begin{aligned} \sup\limits_{t\in [0,T]}\big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^2_x} \le &C\Big( \|(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x\times L^2_x}+\varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0-\widetilde{v}_0\big\|_{H^\beta_x} \\ &\quad+\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^2 T\|\Delta^2 \widetilde{v}\|_{L^\infty_t L^2_x}\Big),\end{aligned}$$ where the constant $C>0$ only depends on $\|\widetilde{v}_0\|_{H^1_x}$, $\big\|(\widetilde{v}_0^\varepsilon,\widetilde{v}_1^\varepsilon)\big\|_{H^1_x\times L^2_x}$.* *Proof.* First, we denote $$\begin{aligned} \big\|f\big\|_{S_{s_c}(I)}\triangleq & \big\||\nabla|^{s_c}f\big\|_{L^ \frac{2(d+2)}{d}_{tx}(I)};\\ \big\|f\big\|_{Y_{s_c}(I)}\triangleq & \big\|f\big\|_{S_{s_c}(I)}+\big\||\nabla|^{s_c}\langle\nabla\rangle^\frac12f\big\|_{L^\infty_tL^2_x(I)};\end{aligned}$$ and $$\begin{aligned} \|f\|_{Z_{s_c}(I)}&\triangleq \|f\|_{L^\frac43_{tx}(I\times{\mathbb R}^2)}, \mbox{when } d=2; \\ \|f\|_{Z_{s_c}(I)}&\triangleq \big\||\nabla|^\frac12 f\big\|_{L^\frac{10}9_tL^{\frac{30}{17}}_x(I\times{\mathbb R}^3)}, \mbox{when } d=3.\end{aligned}$$ Then by [\[Duh-Rj-th3\]](#Duh-Rj-th3){reference-type="eqref" reference="Duh-Rj-th3"} and Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that for any $\gamma\ge -s_c$ and any $I=[t_0,t_1]\subset {\mathbb R}$, $$\begin{aligned} \label{est:Thm3-Rj-case1} \big\||\nabla|^\gamma\mathcal R_j\big\|_{Y_{s_c}(I)} \lesssim & \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{\dot H^{s_c}_x} + \varepsilon^4\big\||\nabla|^{s_c+\gamma}\mathcal{S}_\varepsilon( \partial_{tt} \widetilde{v})\big\|_{L^1_tL^2_x(I)}\notag\\ &\qquad +\Big\||\nabla|^\gamma\Big(\big|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big|^2\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon-|\mathcal{S}_\varepsilon \widetilde{v}|^2\mathcal{S}_\varepsilon \widetilde{v}\Big) \Big\|_{Z_{s_c}(I)}.\end{aligned}$$ For second term, we have that $$\begin{aligned} \label{est:Thm3-Rj-case1-pttv-1} \big\||\nabla|^{s_c+\gamma} \mathcal{S}_\varepsilon( \partial_{tt} \widetilde{v})\big\|_{L^1_tL^2_x(I)} \lesssim & | I |\big\||\nabla|^{s_c+\gamma} \mathcal{S}_\varepsilon( \partial_{tt} \widetilde{v})\big\|_{L^\infty_t L^2_x({\mathbb R})}\notag\\ = & \varepsilon^\gamma| I | \big\||\nabla|^{s_c+\gamma} \partial_{tt} \widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})}.\end{aligned}$$ Now we consider the estimates on $\partial_{tt}\widetilde{v}$. From the equation [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}, we have the rough form that $$\partial_{tt}\widetilde{v}= \Delta^2v+O\big(\widetilde{v}^2\Delta \widetilde{v}+\widetilde{v}^2(\nabla \widetilde{v})^2+\widetilde{v}^5\big).$$ Therefore, by Gagliardo-Nirenberg's inequality, we have that $$\big\||\nabla|^{s_c+\gamma} \partial_{tt} \widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})} \le C(\|\widetilde{v}\|_{L^\infty_t\dot H^{s_c}_x({\mathbb R})})\big\|\widetilde{v}\big\|_{L^\infty_t \dot H^{4+s_c+\gamma}_x({\mathbb R})}.$$ By Lemma [Lemma 17](#lem:spacetime-norm-NLS){reference-type="ref" reference="lem:spacetime-norm-NLS"}, we further get $$\begin{aligned} \label{est:vptt-gamma} \big\||\nabla|^{s_c+\gamma} \partial_{tt} \widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})} \le C(\|\widetilde{v}_0\|_{\dot H^{s_c}_x)}\big\|\widetilde{v}\big\|_{L^\infty_t\dot H^{4+s_c+\gamma}_x({\mathbb R})}.\end{aligned}$$ Inserting this into [\[est:Thm3-Rj-case1-pttv-1\]](#est:Thm3-Rj-case1-pttv-1){reference-type="eqref" reference="est:Thm3-Rj-case1-pttv-1"}, we obtain that $$\begin{aligned} \label{est:Thm3-Rj-case1-pttv} \big\||\nabla|^{s_c+\gamma} \mathcal{S}_\varepsilon( \partial_{tt} \widetilde{v})\big\|_{L^1_tL^2_x(I)} \lesssim C(\|\widetilde{v}_0\|_{L^2_x})\>\varepsilon^\gamma| I | \big\|\widetilde{v}\big\|_{L^\infty_t \dot H^{4+s_c+\gamma}_x({\mathbb R})} .\end{aligned}$$ For the third term, by Kato-Ponce's inequality, we have that when $d=3$, for any $-\frac12\le \gamma<0$, $$\begin{aligned} \label{est:Thm3-Rj-case1-Sv-better} &\Big\||\nabla|^\gamma\Big(\big|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big|^2\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon-|\mathcal{S}_\varepsilon \widetilde{v}|^2\mathcal{S}_\varepsilon \widetilde{v}\Big) \Big\|_{Z_{s_c}(I)}\notag\\ \lesssim & \big\||\nabla|^\gamma \mathcal{S}_\varepsilon r\big\|_{S_{s_c}(I)} \Big(\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{S_{s_c}(I)}^2 +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}(I)}^2\Big) \notag\\ \lesssim & \big\||\nabla|^\gamma R\big\|_{S_{s_c}(I)} \Big(\big\|R\big\|_{S_{s_c}(I)}^2 +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}(I)}^2\Big).\end{aligned}$$ For $\gamma\ge 0$, then by Kato-Ponce's inequality, there is an additional term, and the corresponding estimate reads $$\begin{aligned} \label{est:Thm3-Rj-case1-Sv} &\Big\||\nabla|^\gamma\Big(\big|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big|^2\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon-|\mathcal{S}_\varepsilon \widetilde{v}|^2\mathcal{S}_\varepsilon \widetilde{v}\Big) \Big\|_{Z_{s_c}(I)}\notag\\ \lesssim & \big\||\nabla|^\gamma \mathcal{S}_\varepsilon r\big\|_{S_{s_c}(I)} \Big(\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{S_{s_c}(I)}^2 +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}(I)}^2\Big) \notag\\ &\quad + \big\|\mathcal{S}_\varepsilon r\big\|_{S_{s_c}(I)}\big\||\nabla|^\gamma \mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}(I)} \Big(\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{S_{s_c}(I)} +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}(I)}\Big)\notag\\ \lesssim & \big\||\nabla|^\gamma R\big\|_{S_{s_c}(I)} \Big(\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{S_{s_c}(I)}^2 +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}(I)}^2\Big) \notag\\ &\quad + C\varepsilon^\gamma\big\|R\big\|_{S_{s_c}(I)} \Big(\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{S_{s_c}(I)} +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}(I)}\Big),\end{aligned}$$ where in the last step we have used $$\big\||\nabla|^\gamma \mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{s_c}({\mathbb R})} = \varepsilon^\gamma \big\||\nabla|^\gamma \widetilde{v}\big\|_{S_{s_c}({\mathbb R})} \le C\varepsilon^\gamma,$$ and the positive constant $C$ only depend on $\|\widetilde{v}_0\|_{H^1_x}$ and $\|\widetilde{v}_0\|_{\dot H^{s_c+\gamma}}$. $\bullet$ When $d=2$, by [\[est:Thm3-Rj-case1\]](#est:Thm3-Rj-case1){reference-type="eqref" reference="est:Thm3-Rj-case1"}, [\[est:Thm3-Rj-case1-pttv\]](#est:Thm3-Rj-case1-pttv){reference-type="eqref" reference="est:Thm3-Rj-case1-pttv"} and [\[est:Thm3-Rj-case1-Sv\]](#est:Thm3-Rj-case1-Sv){reference-type="eqref" reference="est:Thm3-Rj-case1-Sv"}, we have that $$\begin{aligned} \label{est:Thm3-Rj-case1-d2} \big\||\nabla|^\gamma \mathcal R_j\big\|_{Y_0(I)} \lesssim & \big\||\nabla|^\gamma \langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{L^2_x} + \varepsilon^{4+\gamma}|I| \big\|\widetilde{v}\big\|_{L^\infty_t \dot H^{4+\gamma}_x({\mathbb R})} \notag\\ &\qquad +\big\||\nabla|^\gamma R\big\|_{S_{0}(I)} \Big(\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{S_{0}(I)}^2 +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{0}(I)}^2\Big) \notag\\ &\qquad + \varepsilon^\gamma\big\|R\big\|_{S_{0}(I)} \Big(\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{S_{0}(I)} +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_{0}(I)}\Big).\notag\\ = & \big\||\nabla|^\gamma \langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{L^2_x} + \varepsilon^{4+\gamma}|I| \big\|\widetilde{v}\big\|_{L^\infty_t \dot H^{4+\gamma}_x({\mathbb R})} \notag\\ &\qquad +\big\||\nabla|^\gamma R\big\|_{S_{0}(I)} \Big(\big\|\widetilde{v}^\varepsilon\big\|_{S_{0}(\varepsilon^2I)}^2 +\big\|\widetilde{v}\big\|_{S_{0}(\varepsilon^2I)}^2\Big) \notag\\ &\qquad + \varepsilon^\gamma\big\|R\big\|_{S_{0}(I)} \Big(\big\| \widetilde{v}^\varepsilon\big\|_{S_{0}(\varepsilon^2I)} +\big\| \widetilde{v}\big\|_{S_{0}(\varepsilon^2I)}\Big).\end{aligned}$$ Moreover, arguing similarly as in Section [3.1](#sec:RCE){reference-type="ref" reference="sec:RCE"}, we have that $$\begin{aligned} \label{est:Thm3-ve-L4tx} \big\| \widetilde{v}^\varepsilon\big\|_{S_0({\mathbb R})} \le C\big(\big\|(\widetilde{v}_0^\varepsilon,\widetilde{v}_1^\varepsilon)\big\|_{H^1_x\times L^2_x}\big).\end{aligned}$$ Indeed, consider $$\phi=\mathrm{e}^{it} \mathcal{S}_\varepsilon \widetilde{v}^\varepsilon,$$ then it solves the following Cauchy problem: $$\begin{aligned} \begin{split} \left\{ \aligned &\partial_{tt} \phi - \Delta\phi + \phi+|\phi|^2\phi=0, \\ & \phi(0,x) = \mathcal{S}_\varepsilon \widetilde{v}^\varepsilon_0(x),\quad \partial_t\phi(0,x) =i\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon_0(x) +\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon_1(x). \endaligned \right. \end{split}\end{aligned}$$ Note that $$\begin{aligned} E(\phi(0,x),\partial_t\phi(0,x)) \lesssim & \big\|\widetilde{v}^\varepsilon_0\big\|_{L^2_x}^2+\big\|\widetilde{v}^\varepsilon_1\big\|_{L^2_x}^2 +\varepsilon^2 \big( \big\|\nabla \widetilde{v}^\varepsilon_0\big\|_{L^2_x}^2+ \big\|\widetilde{v}^\varepsilon_0\big\|_{L^4}^4\big)\\ \le & C\big(\big\|(\widetilde{v}_0^\varepsilon,\widetilde{v}_1^\varepsilon)\big\|_{H^1_x\times L^2_x}\big).\end{aligned}$$ Then [\[est:Thm3-ve-L4tx\]](#est:Thm3-ve-L4tx){reference-type="eqref" reference="est:Thm3-ve-L4tx"} follows from Lemma [Lemma 18](#lem:spacetime-norm-KG){reference-type="ref" reference="lem:spacetime-norm-KG"}. Moreover, from Lemma [Lemma 17](#lem:spacetime-norm-NLS){reference-type="ref" reference="lem:spacetime-norm-NLS"}, $$\begin{aligned} \label{est:Thm3-v-L4tx} \big\| \widetilde{v}\big\|_{S_0({\mathbb R})}\le C\big(\big\|\widetilde{v}_0\big\|_{L^2_x}\big).\end{aligned}$$ We also need the following estimates. By [\[eq:R-12-KG-intialdatum\]](#eq:R-12-KG-intialdatum){reference-type="eqref" reference="eq:R-12-KG-intialdatum"}, Sobolev's and Bernstein's inequalities, $$\begin{aligned} \label{est:Thm3-Rj0-gamma} \big\||\nabla|^\gamma \langle\nabla\rangle^\frac12\mathcal R_{j,0}\big\|_{L^2_x} \lesssim & \big\||\nabla|^\gamma \langle\nabla\rangle^\frac12 \mathcal{S}_\varepsilon r_0\big\|_{L^2_x}+ \big\||\nabla|^\gamma \mathcal{S}_\varepsilon r_1^\varepsilon\big\|_{L^2_x}\notag\\ \lesssim & \big\||\nabla|^\gamma (\mathcal{S}_\varepsilon r_0,\mathcal{S}_\varepsilon \widetilde{v}_1^\varepsilon)\|_{L^2_x}+\big\|P_{\ge1}|\nabla|^\gamma \langle\nabla\rangle^\frac12 \mathcal{S}_\varepsilon r_0\big\|_{L^2_x} +\varepsilon^2\big\||\nabla|^\gamma \Delta \widetilde{v}_0\big\|_{L^2_x}\notag\\ \lesssim& \big\||\nabla|^\gamma (\mathcal{S}_\varepsilon r_0,\mathcal{S}_\varepsilon \widetilde{v}_1^\varepsilon)\|_{L^2_x}+ \| \mathcal{S}_\varepsilon r_0\|_{\dot H^{\beta+\gamma}} +\varepsilon^2\big\||\nabla|^\gamma \Delta \widetilde{v}_0\big\|_{L^2_x}\notag\\ \lesssim & \varepsilon^\gamma\big\||\nabla|^\gamma (r_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x}+\varepsilon^{\beta+\gamma} \|r_0\|_{H^{\beta+\gamma}}+\varepsilon^2\big\||\nabla|^\gamma \Delta \widetilde{v}_0\big\|_{L^2_x}.\end{aligned}$$ First, we consider the case when $\gamma=0$. Then by [\[est:Thm3-Rj-case1-d2\]](#est:Thm3-Rj-case1-d2){reference-type="eqref" reference="est:Thm3-Rj-case1-d2"}, we have that $$\begin{aligned} \label{est:Thm3-Rj-case1-d2-2} \big\| \mathcal R_j\big\|_{Y_0(I)} \le & C_0\big\|\langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{L^2_x} + C_1\varepsilon^{4}|I| \big\|\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})} \notag\\ &\qquad +C_2 \big\|R\big\|_{S_{0}(I)} \Big(\big\| \widetilde{v}^\varepsilon\big\|_{S_{0}(\varepsilon^2I)} +\big\| \widetilde{v}\big\|_{S_{0}(\varepsilon^2I)}\Big).\end{aligned}$$ From [\[est:Thm3-ve-L4tx\]](#est:Thm3-ve-L4tx){reference-type="eqref" reference="est:Thm3-ve-L4tx"} and [\[est:Thm3-v-L4tx\]](#est:Thm3-v-L4tx){reference-type="eqref" reference="est:Thm3-v-L4tx"}, there exists a constant $K=K(C_0,C_3)$ and a sequence of time intervals $$\bigcup\limits_{k=0}^K J_k={\mathbb R}^+,$$ (the negative time direction can be treated similarly) with $$J_0=[0, t_1], \quad J_k=(t_k, t_{k+1}] \mbox{ for } k=1,\cdots, K-1, \quad J_{K}=[t_K, +\infty),$$ such that $$\begin{aligned} \label{subinterval-length-v-vep-d2} C_2\Big(\big\| \widetilde{v}^\varepsilon\big\|_{S_{0}(J_k)} +\big\| \widetilde{v}\big\|_{S_{0}(J_k)}\Big) \le \frac12.\end{aligned}$$ For any fixed $T>0$, denote $$I_k=\varepsilon^{-2}J_k\cap [0,T], \quad\mbox{and } \quad T_k=\varepsilon^{-2} t_k.$$ Then by [\[est:Thm3-Rj-case1-d2-2\]](#est:Thm3-Rj-case1-d2-2){reference-type="eqref" reference="est:Thm3-Rj-case1-d2-2"} and [\[subinterval-length-v-vep-d2\]](#subinterval-length-v-vep-d2){reference-type="eqref" reference="subinterval-length-v-vep-d2"}, it drives that for any $1\le k\le K$, $$\begin{aligned} \big\| \mathcal R_j\big\|_{Y_0(I_k)} \le & 2C_0\big\|\langle\nabla\rangle^\frac12\mathcal R_j(T_k)\big\|_{L^2_x} + 2C_1\varepsilon^{4}T \big\|\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})}.\end{aligned}$$ By iteration, we obtain that for any finite interval $I\subset {\mathbb R}$, $$\big\|\mathcal R_j\big\|_{Y_0(I)} \le C\big(\|\widetilde{v}_0\|_{H^1_x}\big)\Big[ \big\|\langle\nabla\rangle^\frac12\mathcal R_{j,0}\big\|_{L^2_x} + \varepsilon^{4}|I| \big\|\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})} \Big].$$ Therefore, by [\[est:Thm3-Rj0-gamma\]](#est:Thm3-Rj0-gamma){reference-type="eqref" reference="est:Thm3-Rj0-gamma"} we get that $$\begin{aligned} \big\|\mathcal R_j\big\|_{Y_0(I)} \le C\big(\|\widetilde{v}_0\|_{H^1_x}\big)\Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x}+\varepsilon^ \beta \|r_0\|_{H^\beta}+\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^4|I|\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})}\Big].\end{aligned}$$ Inserting this estimate into [\[est:Thm3-Rj-case1-d2\]](#est:Thm3-Rj-case1-d2){reference-type="eqref" reference="est:Thm3-Rj-case1-d2"} and using the same argument, we also get that $$\begin{aligned} \big\||\nabla|^\gamma \mathcal R_j\big\|_{Y_0(I)} \le C\Big[ \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_{j,0}\big\|_{L^2_x} +\varepsilon^{\gamma}\big\|\mathcal R_j\big\|_{Y_0(I)} + \varepsilon^{4+\gamma}|I| \big\||\nabla|^\gamma \partial_{tt} \widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})}\Big],\end{aligned}$$ where the positive constant $C$ only depend on $\left\|\widetilde{v}_0\right\|_{H^1_x}$ and $\left\|\widetilde{v}_0\right\|_{\dot H^{s_c+\gamma}}$. By [\[est:Thm3-Rj0-gamma\]](#est:Thm3-Rj0-gamma){reference-type="eqref" reference="est:Thm3-Rj0-gamma"}, we obtain that $$\begin{aligned} \label{est:R-12-upp-d2} \big\||\nabla|^\gamma \mathcal R_j\big\|_{Y_0(I)} \le C \varepsilon^{\gamma} \Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{H^\gamma}+\varepsilon^\beta \|r_0\|_{H^{\beta+\gamma}}+\varepsilon^2\big\||\nabla|^\gamma \Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^4|I|\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t H^\gamma_x({\mathbb R})}\Big].\end{aligned}$$ In particular, by [\[eq:R-R12\]](#eq:R-R12){reference-type="eqref" reference="eq:R-R12"}, this gives that $$\begin{aligned} \big\||\nabla|^\gamma \mathcal{S}_\varepsilon r\big\|_{L^\infty_tL^2_x(I)} \le & C \varepsilon^{\gamma} \Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{H^\gamma}+\varepsilon^\beta \|r_0\|_{H^{\beta+\gamma}}\\ &\quad +\varepsilon^2\big\||\nabla|^\gamma \Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^4|I|\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t H^\gamma_x({\mathbb R})}\Big].\end{aligned}$$ Scaling bark, and choosing $\gamma=0$, we obtain that for any $T<+\infty$, $$\begin{aligned} \big\| r\big\|_{L^\infty_tL^2_x([0,T])} \le C\Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x}+\varepsilon^\beta \|r_0\|_{H^\beta_x}+\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^2T\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})}\Big],\end{aligned}$$ where the positive constant $C$ only depend on $\left\|\widetilde{v}_0\right\|_{H^1_x}$ and $\left\|\widetilde{v}_0\right\|_{L^2_x}$. This proves the proposition when $d=2$. $\bullet$ When $d=3$, we need to assume that $I: \varepsilon^{4} |I| \|\widetilde{v}\|_{L^\infty_t\dot H^{\frac92}_x} \le \delta_0$ ($\delta_0$ will be determined later). First, by [\[est:Thm3-Rj-case1\]](#est:Thm3-Rj-case1){reference-type="eqref" reference="est:Thm3-Rj-case1"}, [\[est:Thm3-Rj-case1-pttv\]](#est:Thm3-Rj-case1-pttv){reference-type="eqref" reference="est:Thm3-Rj-case1-pttv"} and [\[est:Thm3-Rj-case1-Sv-better\]](#est:Thm3-Rj-case1-Sv-better){reference-type="eqref" reference="est:Thm3-Rj-case1-Sv-better"}, we have that for any $-\frac12\le \gamma<0$, $$\begin{aligned} \label{est:Thm3-Rj-case1-d3-better} \big\||\nabla|^\gamma\mathcal R_j\big\|_{Y_\frac12(I)} \lesssim & \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{\dot H^\frac12} + \varepsilon^{4+\gamma}|I| \big\||\nabla|^\gamma\partial_{tt} \widetilde{v}\big\|_{L^\infty_t \dot H^\frac12_x({\mathbb R})}\notag\\ &\qquad +\big\||\nabla|^\gamma R\big\|_{S_\frac12(I)} \Big(\big\| R\big\|_{S_\frac12(I)}^2 +\big\|\mathcal{S}_\varepsilon \widetilde{v}\big\|_{S_\frac12(I)}^2\Big)\notag\\ =& \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{\dot H^\frac12} + \varepsilon^{4+\gamma}|I| \big\||\nabla|^\gamma\partial_{tt} \widetilde{v}\big\|_{L^\infty_t \dot H^\frac12_x({\mathbb R})}\notag\\ &\qquad +\big\||\nabla|^\gamma R\big\|_{S_\frac12(I)} \Big(\big\| R\big\|_{S_\frac12(I)}^2 +\big\|\widetilde{v}\big\|_{S_\frac12(\varepsilon^2I)}^2\Big).\end{aligned}$$ For $\gamma\ge 0$, applying [\[est:Thm3-Rj-case1-Sv\]](#est:Thm3-Rj-case1-Sv){reference-type="eqref" reference="est:Thm3-Rj-case1-Sv"} instead, we have that $$\begin{aligned} \label{est:Thm3-Rj-case1-d3} \big\||\nabla|^\gamma\mathcal R_j\big\|_{Y_\frac12(I)} \lesssim & \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{\dot H^\frac12} + \varepsilon^{4+\gamma}|I| \big\||\nabla|^\gamma\partial_{tt} \widetilde{v}\big\|_{L^\infty_t \dot H^\frac12_x({\mathbb R})}\notag\\ &\qquad +\big\||\nabla|^\gamma R\big\|_{S_\frac12(I)} \Big(\big\| R\big\|_{S_\frac12(I)}^2 +\big\|\widetilde{v}\big\|_{S_\frac12(\varepsilon^2I)}^2\Big)\notag\\ &\qquad +C \varepsilon^\gamma \big\|R\big\|_{S_\frac12(I)} \Big(\big\| R\big\|_{S_\frac12(I)} +\big\|\widetilde{v}\big\|_{S_\frac12(\varepsilon^2I)}\Big),\end{aligned}$$ where the positive constant $C$ only depend on $\left\|\widetilde{v}_0\right\|_{H^1_x}$ and $\left\|\widetilde{v}_0\right\|_{\dot H^{\frac12+\gamma}}$. Similar as the two dimensional case, we have that $$\begin{aligned} \big\| \widetilde{v}\big\|_{S_\frac12({\mathbb R})}\le & C\big(\big\|\widetilde{v}_0\big\|_{H^1_x}\big);\label{est:Thm3-v-Lqtx-3d}\\ \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_{j,0}\big\|_{\dot H^\frac12} \le &C\Big(\varepsilon^\gamma\big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{\dot H^{\frac12+\gamma}}+\varepsilon^{\beta+\gamma} \|r_0\|_{H^{\frac12+\beta+\gamma}}+\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{\dot H^{\frac12+\gamma}}\Big) \label{est:Thm3-v-Rj0-3d}\end{aligned}$$ (the difference from the two dimensional case is that, we have no boundedness of $S_\frac12({\mathbb R})$-norm on $\widetilde{v}^\varepsilon$ the in three dimensional case). In particular, from [\[est:Thm3-Rj-case1-d3\]](#est:Thm3-Rj-case1-d3){reference-type="eqref" reference="est:Thm3-Rj-case1-d3"} we further get that for any $\gamma\ge 0$, $$\begin{aligned} \label{est:Thm3-Rj-case1-d3-2} \sum\limits_{j=1,2}\big\||\nabla|^\gamma\mathcal R_j\big\|_{Y_\frac12(I)} \le & C_1\sum\limits_{j=1,2}\big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_j(t_0)\big\|_{\dot H^\frac12} + C_2\varepsilon^{4+\gamma}|I| \big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t \dot H^{\frac12+\gamma}_x({\mathbb R})}\notag\\ &\qquad +C_3\big\||\nabla|^\gamma R\big\|_{S_\frac12(I)} \Big(\big\| R\big\|_{S_\frac12(I)}^2 +\big\|\widetilde{v}\big\|_{S_\frac12(\varepsilon^2I)}^2\Big)\notag\\ &\qquad +C_4 \varepsilon^\gamma \big\|R\big\|_{S_\frac12(I)} \Big(\big\| R\big\|_{S_\frac12(I)} +\big\|\widetilde{v}\big\|_{S_\frac12(\varepsilon^2I)}\Big).\end{aligned}$$ First, we treat the case when $\gamma=0$. Repeating the similar process as the case of $d=2$, we split ${\mathbb R}$ by several subinterval (also denoted by $J_k$) as $$\bigcup\limits_{k=0}^K J_k={\mathbb R}^+,$$ with $$J_0=[0, t_1], \quad J_k=(t_k, t_{k+1}] \mbox{ for } k=1,\cdots, K-1, \quad J_{K}=[t_K, +\infty),$$ such that $$\begin{aligned} \label{subinterval-length-2} C_3\big\| \widetilde{v}\big\|_{S_\frac12(J_k)}^2+C_4\big\|\widetilde{v}\big\|_{S_\frac12(J_k)} \le \frac14.\end{aligned}$$ Denote $$I_k=\varepsilon^{-2}J_k\cap [0,T], \quad\mbox{and } \quad T_k=\varepsilon^{-2} t_k.$$ Then by [\[est:Thm3-Rj-case1-d3-2\]](#est:Thm3-Rj-case1-d3-2){reference-type="eqref" reference="est:Thm3-Rj-case1-d3-2"} and [\[subinterval-length-2\]](#subinterval-length-2){reference-type="eqref" reference="subinterval-length-2"}, we obtain that for any $k\le K$, $$\begin{aligned} \label{est:Thm3-Rj-case1-d3-3} \sum\limits_{j=1,2}\big\|\mathcal R_j\big\|_{Y_\frac12(I_k)} \le & \frac43C_1\sum\limits_{j=1,2}\big\||\nabla|^\frac12\langle\nabla\rangle^\frac12\mathcal R_j(T_k)\big\|_{L^2_x} + \frac43C_2\varepsilon^4|I_k| \big\||\nabla|^\frac12\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})}\notag\\ & +\frac43\Big(C_3\big\|R\big\|_{S_\frac12(I_k)}^3+C_4\big\|R\big\|_{S_\frac12(I_k)}^2\Big)\notag\\ \le & \frac43C_1\sum\limits_{j=1,2}\big\||\nabla|^\frac12\langle\nabla\rangle^\frac12\mathcal R_j(T_k)\big\|_{L^2_x} + \frac43C_2\varepsilon^4T \big\||\nabla|^\frac12\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})}\notag\\ & +\frac43\Big(C_3\big\|R\big\|_{S_\frac12(I_k)}^3+C_4\big\|R\big\|_{S_\frac12(I_k)}^2\Big).\end{aligned}$$ Denote $$c_0\triangleq \big\|\langle\nabla\rangle^\frac12\mathcal R_{j,0}\big\|_{\dot H^\frac12};\quad C_0\triangleq C_2 \big\||\nabla|^\frac12\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})}.$$ Then by [\[est:Thm3-v-Rj0-3d\]](#est:Thm3-v-Rj0-3d){reference-type="eqref" reference="est:Thm3-v-Rj0-3d"}, $$\begin{aligned} c_0\le C\Big(\big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x}+\varepsilon^\beta \|r_0\|_{H^{\frac12+\beta}}\Big).\end{aligned}$$ Moreover, denote $$a_0\triangleq \big(2C_1\big)^K\Big[c_0+2C_0\varepsilon^4T \Big].$$ Choosing $c_0, \varepsilon_0$ and $\delta_0$ suitably small such that for any $\varepsilon\in [0, \varepsilon_0], I: \varepsilon^4 |I| \|\widetilde{v}\|_{L^\infty_t\dot H^{\frac92}_x} \le \delta_0$, we have that $$\begin{aligned} \label{small-a0} \frac43\big(C_3a_0^2+C_4a_0\big)\le \frac14.\end{aligned}$$ Suppose that for some $0\le k\le K-1$, $$\begin{aligned} \label{iteration-k-local} \sum\limits_{j=1,2}\big\||\nabla|^\frac12\langle\nabla\rangle^\frac12\mathcal R_j(T_k)\big\|_{L^2_x} \le c_0\big(2C_1\big)^k+2C_0\sum\limits_{j=0}^k \big(2C_1\big)^{j-1}.\end{aligned}$$ then $$2C_1\sum\limits_{j=1,2}\big\||\nabla|^\frac12\langle\nabla\rangle^\frac12\mathcal R_j(T_k)\big\|_{L^2_x} + 2 C_0\varepsilon^4T\le a_0.$$ Therefore, by [\[est:Thm3-Rj-case1-d3-3\]](#est:Thm3-Rj-case1-d3-3){reference-type="eqref" reference="est:Thm3-Rj-case1-d3-3"} for $I_k$ and [\[small-a0\]](#small-a0){reference-type="eqref" reference="small-a0"}, and applying the bootstrap argument on $S_\frac12(I_k)$ norm first and obtain that $$\big\|R\big\|_{S_\frac12(I_k)} \le c_0\big(2C_1\big)^k+2C_0\sum\limits_{j=0}^k \big(2C_1\big)^{j-1} \le a_0.$$ Then inserting it back in [\[est:Thm3-Rj-case1-d3-3\]](#est:Thm3-Rj-case1-d3-3){reference-type="eqref" reference="est:Thm3-Rj-case1-d3-3"} to obtain the estimate on $Y_\frac12(I_k)$, we have that (for which we choose $\beta=\frac12$) $$\begin{aligned} \sum\limits_{j=1,2}\big\|\mathcal R_j\big\|_{Y_\frac12(I_{k+1})} \le c_0\big(2C_1\big)^{k+1}+2C_0\sum\limits_{j=0}^{k+1} \big(2C_1\big)^{j-1}.\end{aligned}$$ This proves [\[iteration-k-local\]](#iteration-k-local){reference-type="eqref" reference="iteration-k-local"} for $k+1$ and thus it holds for any $1\le l\le K$. Therefore, we obtain that $$\begin{aligned} \sum\limits_{j=1,2}\big\|\mathcal R_j\big\|_{Y_\frac12(I)} \le & C\big(\|\widetilde{v}_0\|_{H^1_x}\big)\Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{\dot H^\frac12}+\varepsilon^\frac12 \|r_0\|_{H^1_x}\\ &\quad+\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{\dot H^{\frac12}}+\varepsilon^4 T \big\||\nabla|^\frac12\Delta^2\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})}\Big].\end{aligned}$$ From assumptions (1)--(3), choosing $\varepsilon_0$ and $\delta_0$ suitably small, then the last estimate above gives that $$\begin{aligned} C_3\big\|R\big\|_{S_\frac12(I)}^2\le \frac12.\end{aligned}$$ Inserting this estimate into [\[est:Thm3-Rj-case1-d3\]](#est:Thm3-Rj-case1-d3){reference-type="eqref" reference="est:Thm3-Rj-case1-d3"} and using the same argument, we also get that for any $\gamma\ge0$, $$\begin{aligned} \sum\limits_{j=1,2}\big\||\nabla|^\gamma \mathcal R_j\big\|_{Y_\frac12(I)} \le C\Big[ \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_{j,0}\big\|_{\dot H^\frac12} +\varepsilon^{\gamma}\big\|\mathcal R_j\big\|_{Y_\frac12(I)} + \varepsilon^{4+\gamma}|I| \big\||\nabla|^\gamma \Delta^2 \widetilde{v}\big\|_{L^\infty_t \dot H^\frac12_x({\mathbb R})}\Big],\end{aligned}$$ where the positive constant $C$ only depend on $\left\|\widetilde{v}_0\right\|_{H^1_x}$ and $\left\|\widetilde{v}_0\right\|_{\dot H^{\frac12+\gamma}}$. Inserting it into [\[est:Thm3-Rj-case1-d3-better\]](#est:Thm3-Rj-case1-d3-better){reference-type="eqref" reference="est:Thm3-Rj-case1-d3-better"} instead, we get that for any $-\frac12\le \gamma<0$, $$\begin{aligned} \sum\limits_{j=1,2}\big\||\nabla|^\gamma \mathcal R_j\big\|_{Y_\frac12(I)} \le C\Big[ \big\||\nabla|^\gamma\langle\nabla\rangle^\frac12\mathcal R_{j,0}\big\|_{\dot H^\frac12} + \varepsilon^{4+\gamma}|I| \big\||\nabla|^\gamma \Delta^2 \widetilde{v}\big\|_{L^\infty_t \dot H^\frac12_x({\mathbb R})}\Big],\end{aligned}$$ where the positive constant $C$ only depend on $\left\|\widetilde{v}_0\right\|_{H^1_x}$. Applying these two estimates and treating similarly as in two dimensional case, we can obtain a uniform bound of $\big\||\nabla|^\gamma \mathcal R_j\big\|_{L^\infty_t \dot H^\frac12_x(I)}$ as $$\begin{aligned} \sum\limits_{j=1,2}\big\||\nabla|^\gamma \mathcal R_j\big\|_{L^\infty_t \dot H^\frac12_x(I)} \le & C\varepsilon^{\gamma} \Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{H^{\gamma+\frac12}}+\varepsilon^\beta \|r_0\|_{H^{\beta+\gamma+\frac12}}\\ &\quad +\varepsilon^2\big\||\nabla|^\gamma \Delta \widetilde{v}_0\big\|_{\dot H^{\frac12}}+\varepsilon^4|I|\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t \dot H^{\gamma+\frac12}_x({\mathbb R})}\Big].\end{aligned}$$ Choosing $\gamma=-\frac12$, it gives that $$\begin{aligned} \label{est:R-12-upp-d3} \sum\limits_{j=1,2}\big\|\mathcal R_j\big\|_{L^\infty_t L^2_x(I)} \le & C\varepsilon^{s-\frac12} \Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x}+\varepsilon^\beta \|r_0\|_{H^\beta_x} \notag\\ &\quad +\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^4|I|\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})}\Big],\end{aligned}$$ and thus $$\begin{aligned} \big\|R\big\|_{L^\infty_t L^2_x(I)} \le \varepsilon^{s-\frac12} C\Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x}+\varepsilon^\beta \|r_0\|_{H^\beta_x}+\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^4|I|\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})}\Big].\end{aligned}$$ Scaling bark, it implies that for any $$T: \varepsilon^2 T\|\widetilde{v}\|_{L^\infty_t\dot H^{4+\frac12}_x}\le \delta_0,$$ it holds that $$\begin{aligned} \big\| r\big\|_{L^\infty_tL^2_x([0,T])} \le C\Big[ \big\|(r_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x}+\varepsilon^\beta \|r_0\|_{H^\beta_x}+\varepsilon^2\big\|\Delta \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^2T\big\|\Delta^2\widetilde{v}\big\|_{L^\infty_t L^2_x({\mathbb R})}\Big].\end{aligned}$$ This proves the proposition when $d=3$. ◻ ### Non-regular cases {#sec:Thm3-case1-2} In this subsubsection, we continue to consider the case when $\alpha\le 4$. The main result is **Proposition 27**. *Let $d=2,3$, $T>0$, $0\le \alpha\le 4$ and $\beta\in [1,2]$ with $\beta\le \alpha$, then there exists $\delta_0>0$ such that the following property hold. Assume that* - *$\widetilde{v}_0\in H^1\cap H^\alpha({\mathbb R}^d),\widetilde{v}^\varepsilon_0\in H^1\cap H^\beta_x({\mathbb R}^d), \widetilde{v}_1^\varepsilon\in H^{s_c}({\mathbb R}^d)$;* - *$(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\to (0,0) \mbox{ in } H^{s_c}({\mathbb R}^d).$* *When $d=3$, additionally assume that* - *$\varepsilon^2 T \le \delta_0$.* *Let $\widetilde{v}^\varepsilon, \widetilde{v}$ be the corresponding solutions to [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} and [\[eq:nls-1\]](#eq:nls-1){reference-type="eqref" reference="eq:nls-1"} respectively, then for any $\beta\ge 1$, $$\begin{aligned} \sup\limits_{t\in [0,T]}\big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^2_x} \lesssim & \|(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x\times L^2_x} +\varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0-\widetilde{v}_0\big\|_{H^\beta_x}+\varepsilon^\alpha+\big(\varepsilon^2 T\big)^{\frac14\alpha}.\end{aligned}$$* *Proof.* Let $N=N(\varepsilon)>0$ be a parameter which satisfies that $$N(\varepsilon)\to \infty, \quad \mbox{ when }\quad \varepsilon\to 0,$$ and will be determined later. Denote $\widetilde{v}_N$ to be the solution to the following equation $$\begin{aligned} \begin{split} \left\{ \aligned &2i\partial_{t} \widetilde{v} - \Delta\widetilde{v}= - |\widetilde{v}|^2 \widetilde{v}, \\ & \widetilde{v}(0,x) = P_{\le N}\widetilde{v}_0(x), \endaligned \right. \end{split}\end{aligned}$$ and denote $\widetilde{v}^N =\widetilde{v}-\widetilde{v}_N$. Then $\widetilde{v}^N$ is the solution to the following equation $$\begin{aligned} \begin{split} \left\{ \aligned &2i\partial_{t} \widetilde{v}^N - \Delta\widetilde{v}^N = - 3\big(|\widetilde{v}|^2 \widetilde{v}-|\widetilde{v}_N |^2 \widetilde{v}_N \big), \\ & \widetilde{v}^N (0,x) = P_{> N}\widetilde{v}_0(x). \endaligned \right. \end{split}\end{aligned}$$ Then by Lemma [Lemma 17](#lem:spacetime-norm-NLS){reference-type="ref" reference="lem:spacetime-norm-NLS"}, there exists some constant $C>0$ which only depends on $\|\widetilde{v}_0\|_{H^1_x}$, such that $$\begin{aligned} \label{est:v-spacetime} \big\|\widetilde{v}\big\|_{S_{s_c}({\mathbb R})} +\big\|\widetilde{v}_N \big\|_{S_{s_c}({\mathbb R})} \le C.\end{aligned}$$ Moreover, by Lemma [Lemma 12](#lem:strichartz){reference-type="ref" reference="lem:strichartz"}, we have that for any $I=[t_0,t_1]\subset {\mathbb R}$, $$\begin{aligned} \big\|\widetilde{v}^N \big\|_{L^\infty_tL^2_x(I)} \lesssim & \big\|\widetilde{v}^N (t_0)\big\|_{L^2_x} +\big\|\widetilde{v}^N \big\|_{L^\infty_tL^2_x(I)} \Big(\big\|\widetilde{v}\big\|_{S_{s_c}(I)}^2 +\big\|\widetilde{v}_N \big\|_{S_{s_c}(I)}^2\Big).\end{aligned}$$ By [\[est:v-spacetime\]](#est:v-spacetime){reference-type="eqref" reference="est:v-spacetime"} and then arguing similarly as in Section [5.2.1](#sec:Thm3-case1-1){reference-type="ref" reference="sec:Thm3-case1-1"}, we obtain that $$\begin{aligned} \label{est:Thm3-w-case2-d2} \big\|\widetilde{v}^N \big\|_{L^\infty_tL^2_x({\mathbb R})} \le & C\big\|P_{> N}\widetilde{v}_0\big\|_{L^2_x} \le CN^{-\alpha}\big\|\widetilde{v}_0\big\|_{H^\alpha_x},\end{aligned}$$ where the constant $C>0$ only depends on $\|\widetilde{v}_0\|_{H^1_x}$. Now we write $$\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)=\widetilde{v}^\varepsilon(t)-\widetilde{v}_N (t)-\widetilde{v}^N .$$ Note that under the hypothesis in the proposition, $P_{\le N}\widetilde{v}_0, \widetilde{v}_0^\varepsilon \widetilde{v}_1^\varepsilon$ verify the assumptions in Proposition [Proposition 26](#prop:Thm3-case1-1){reference-type="ref" reference="prop:Thm3-case1-1"}. Then we have that $$\begin{aligned} \sup\limits_{t\in [0,T]}\big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}_N (t)\big\|_{L^2_x} \le &C\Big( \|(\widetilde{v}^\varepsilon_0-P_{\le N}\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x\times L^2_x}+\varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0-P_{\le N} \widetilde{v}_0\big\|_{H^\beta_x}\\ &\qquad +\varepsilon^2\big\|\Delta P_{\le N} \widetilde{v}_0\big\|_{L^2_x}+\varepsilon^2 T\|\Delta^2 \widetilde{v}_N \|_{L^\infty_t L^2_x}\Big),\end{aligned}$$ for any $T$ verifying $$\begin{aligned} \label{Thm3-T-condition} T<+\infty \quad \mbox{when }d=2;\quad \varepsilon^2 T N^\frac72\le C(\|\widetilde{v}_0\|_{H^1_x}) \delta_0 \quad \mbox{when }d=3.\end{aligned}$$ By Lemma [Lemma 17](#lem:spacetime-norm-NLS){reference-type="ref" reference="lem:spacetime-norm-NLS"}, we have that $$\|\Delta^2 \widetilde{v}_N \|_{L^\infty_t L^2_x}\le C(\|\widetilde{v}_0\|_{H^1_x}) \big\|\Delta^2 P_{\le N}\widetilde{v}_0\big\|_{L^2_x} \le N^{4-\alpha} C(\|\widetilde{v}_0\|_{H^1_x}, \|\widetilde{v}_0\|_{H^\alpha_x}).$$ Moreover, $$\begin{aligned} \|\widetilde{v}^\varepsilon_0-P_{\le N}\widetilde{v}_0\|_{L^2_x} \lesssim & \|\widetilde{v}^\varepsilon_0-\widetilde{v}_0\|_{L^2_x}+\|P_{>N}\widetilde{v}_0\|_{L^2_x}\\ \lesssim &\|\widetilde{v}^\varepsilon_0-\widetilde{v}_0\|_{L^2_x}+N^{-\alpha}\|\widetilde{v}_0\|_{H^\alpha_x};\end{aligned}$$ $$\begin{aligned} \varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0-P_{\le N} \widetilde{v}_0\big\|_{H^\beta_x} \lesssim & \varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0- \widetilde{v}_0\big\|_{H^\beta_x}+\varepsilon^\beta\|P_{>N}\widetilde{v}_0\|_{H^\beta_x}\\ \lesssim & \varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0- \widetilde{v}_0\big\|_{H^\beta_x}+\varepsilon^\beta N^{\beta-\alpha}\|P_{>N}\widetilde{v}_0\|_{H^\alpha_x}\\ \lesssim & \varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0- \widetilde{v}_0\big\|_{H^\beta_x}+\big(\varepsilon^\alpha+ N^{-\alpha}\big)\|P_{>N}\widetilde{v}_0\|_{H^\alpha_x};\end{aligned}$$ and when $\alpha\le 2$, $$\begin{aligned} \varepsilon^2\big\|\Delta P_{\le N} \widetilde{v}_0\big\|_{L^2_x} \lesssim & \varepsilon^2N^{2-\alpha}\big\| P_{\le N}\widetilde{v}_0\big\|_{H^\alpha_x}\\ \lesssim & \big(\varepsilon^\alpha+ N^{-\alpha}\big)\|\widetilde{v}_0\|_{H^\alpha_x};\end{aligned}$$ when $\alpha> 2$, $$\begin{aligned} \varepsilon^2\big\|\Delta P_{\le N} \widetilde{v}_0\big\|_{L^2_x} \le & \varepsilon^2\big\|\widetilde{v}_0\big\|_{H^\alpha_x}.\end{aligned}$$ Hence, it implies that $$\begin{aligned} \sup\limits_{t\in [0,T]}\big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}_N (t)\big\|_{L^2_x} \le &C\Big( \|(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x\times L^2_x}+\varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0-\widetilde{v}_0\big\|_{H^\beta_x}\\ &\qquad +\varepsilon^{\min\{\alpha,2\}} +\varepsilon^2 T N^{4-\alpha}+ N^{-\alpha}\Big),\end{aligned}$$ where $C=C(\|\widetilde{v}_0\|_{H^1_x}, \|\widetilde{v}_0\|_{H^\alpha_x})>0$. This together with [\[est:Thm3-w-case2-d2\]](#est:Thm3-w-case2-d2){reference-type="eqref" reference="est:Thm3-w-case2-d2"} infers that $$\begin{aligned} \sup\limits_{t\in [0,T]}\big\|\widetilde{v}^\varepsilon(t)-v(t)\big\|_{L^2_x} \le &C\Big( \|(\widetilde{v}^\varepsilon_0-\widetilde{v}_0,\widetilde{v}_1^\varepsilon)\|_{L^2_x\times L^2_x}+\varepsilon^\beta\big\|\widetilde{v}^\varepsilon_0-\widetilde{v}_0\big\|_{H^\beta_x}\\ &\qquad \varepsilon^{\min\{\alpha,2\}} +\varepsilon^2 T N^{4-\alpha}+N^{-\alpha}\Big).\end{aligned}$$ Choosing $$N=(\varepsilon^2 T)^{-\frac14},$$ it gives the desired estimate for $T$ satisfying [\[Thm3-T-condition\]](#Thm3-T-condition){reference-type="eqref" reference="Thm3-T-condition"}. In particular, adjusting the value of $\delta_0$ appropriately, then we have that assumption (3) implies [\[Thm3-T-condition\]](#Thm3-T-condition){reference-type="eqref" reference="Thm3-T-condition"} when $d=3$. ◻ .5cm *Proof of Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"}.* When $\varepsilon^2 T\ge \delta_0$, multiply the equation [\[eq:nls-wave-N\]](#eq:nls-wave-N){reference-type="eqref" reference="eq:nls-wave-N"} with $\overline{\widetilde{v}^\varepsilon}$, integrate in $x$, and take the imaginary part, then we obtain $$\begin{aligned} \label{mass-NLS-wave} \int\big(\varepsilon^2\mbox{Im} (\partial_t \widetilde{v}^\varepsilon \overline{\widetilde{v}^\varepsilon}) + |\widetilde{v}^\varepsilon|^2\big)dx = \int\big(\mbox{Im} (\widetilde{v}_1^\varepsilon \overline{\widetilde{v}_0^\varepsilon}) + |\widetilde{v}_0^\varepsilon|^2\big)dx.\end{aligned}$$ This implies that $$\begin{aligned} \big\|\widetilde{v}^\varepsilon\big\|_{L^2_x}^2 \lesssim \varepsilon^4 \big\|\partial_t \widetilde{v}^\varepsilon\big\|_{L^2_x}^2+\big\|(\widetilde{v}_0^\varepsilon,\widetilde{v}_1^\varepsilon)\big\|_{L^2_x\times L^2_x}^2.\end{aligned}$$ Note that from [\[energy-NLS-wave\]](#energy-NLS-wave){reference-type="eqref" reference="energy-NLS-wave"}, we have that $$\begin{aligned} \big\|\partial_t \widetilde{v}^\varepsilon\big\|_{L^2_x}^2 \lesssim \varepsilon^{-4}\big\|(\widetilde{v}_0^\varepsilon,\widetilde{v}_1^\varepsilon)\big\|_{H^1_x\times L^2_x}^2.\end{aligned}$$ These two estimates give that $$\begin{aligned} \label{est:L2-uniform-vep} \big\|\widetilde{v}^\varepsilon\big\|_{L^\infty_tL^2_x({\mathbb R})} \lesssim \big\|(\widetilde{v}_0^\varepsilon,\widetilde{v}_1^\varepsilon)\big\|_{H^1_x\times L^2_x}.\end{aligned}$$ Moreover, by the mass conservation law of the solution to [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}, we have that $$\begin{aligned} \big\|\widetilde{v}\big\|_{L^\infty_tL^2_x({\mathbb R})} = \big\|\widetilde{v}_0\big\|_{L^2_x}.\end{aligned}$$ Therefore, we have that $$\begin{aligned} \big\|\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)\big\|_{L^\infty_tL^2_x([0,T])} \lesssim & \big\|\widetilde{v}^\varepsilon\big\|_{L^\infty_tL^2_x([0,T])}+\big\|\widetilde{v}\big\|_{L^\infty_tL^2_x([0,T])}\\ \lesssim & \big\|(\widetilde{v}_0^\varepsilon,\widetilde{v}_1^\varepsilon)\big\|_{H^1_x\times L^2_x}+ \big\|\widetilde{v}_0\big\|_{L^2_x}.\end{aligned}$$ It yields [\[est:Thm3-main\]](#est:Thm3-main){reference-type="eqref" reference="est:Thm3-main"} when $\varepsilon^2 T\ge \delta_0$. Hence, we only need to consider when $\varepsilon^2 T\le \delta_0$. This completes the proof of Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"}. ◻ ## Optimality {#sec:Thm3-optimal-1} In this subsection, we give the proof of Theorem [Theorem 8](#thm:main3-optimal){reference-type="ref" reference="thm:main3-optimal"}. First, we set the initial data $$\widetilde{v}_0=\delta_0f,\quad \widetilde{v}_0^\varepsilon-\widetilde{v}_0=\delta_0r_0,\quad \widetilde{v}_1^\varepsilon=0,$$ where $\delta_0>0$ is a small constant determined later, $f\in \mathcal S$ is real-valued and independent of $\delta_0$. Consider $$\phi=\mathrm{e}^{it} \mathcal{S}_\varepsilon \widetilde{v}^\varepsilon,$$ then it solves the following Cauchy problem: $$\begin{aligned} \begin{split} \left\{ \aligned &\partial_{tt} \phi - \Delta\phi + \phi=- |\phi|^2\phi, \\ & \phi(0,x) =\delta_0 \mathcal{S}_\varepsilon (f+r_0)(x),\quad \partial_t\phi(0,x) =i\delta_0 \mathcal{S}_\varepsilon (f+r_0)(x). \endaligned \right. \end{split}\end{aligned}$$ By the scaling invariant of $\mathcal{S}_\varepsilon$ in $\dot H^{s_c}_x$, we have that $$\big\|\langle\nabla\rangle^\frac12 (\phi(0),\langle\nabla\rangle^{-1}\partial_t\phi(0))\big\|_{\dot H^{s_c}_x\times \dot H^{s_c}_x}\lesssim \delta_0\big(\|f\|_{H^1_x}+\|r_0\|_{H^1_x}\big).$$ Choosing $\delta_0$ suitably small, then by Lemma [Lemma 19](#lem:spacetime-norm-KG-sd){reference-type="ref" reference="lem:spacetime-norm-KG-sd"}, we obtain that for some positive constant $C=C(\|f\|_{H^1_x},\|r_0\|_{H^1_x})>0$, $$\begin{aligned} \label{est:Sv-ep-Xs} \big\|\widetilde{v}^\varepsilon\big\|_{X_{s_c}({\mathbb R})} =\big\|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big\|_{X_{s_c}({\mathbb R})} =\|\phi\|_{X_{s_c}({\mathbb R})} \le C \delta_0.\end{aligned}$$ Moreover, by Lemma [Lemma 17](#lem:spacetime-norm-NLS){reference-type="ref" reference="lem:spacetime-norm-NLS"}, we have that for some $C=C(\|f\|_{H^1_x})>0$, $$\begin{aligned} \label{est:v-Xs} \big\|\widetilde{v}\big\|_{X_{s_c}({\mathbb R})} = \|\widetilde{v}\|_{X_{s_c}({\mathbb R})} \le C \delta_0.\end{aligned}$$ Based on this estimate and the Strichartz estimates, we write $$\begin{aligned} \label{v-linear-app-1} \widetilde{v}=\delta_0\mathrm{e}^{-\frac12 it\Delta}f+\mathcal N(\widetilde{v}),\end{aligned}$$ where the term $\mathcal N$ satisfies that for some $C=C(\|f\|_{H^{s_c+\gamma}_x})>0$, $$\begin{aligned} \label{re-linear-app-1} \big\||\nabla|^\gamma\mathcal N(\widetilde{v})\big\|_{X_{s_c}({\mathbb R})}\le C\delta_0^3.\end{aligned}$$ Moreover, we also have that $$\begin{aligned} \label{v-linear-app-2} \partial_{tt}\widetilde{v}=-\frac14\delta_0\mathrm{e}^{-\frac12 it\Delta}\Delta^2 f+\widetilde{\mathcal N}(\widetilde{v}),\end{aligned}$$ where there exists some $C=C(\|f\|_{H^1_x})>0$, $$\begin{aligned} \label{re-linear-app-1} \big\|\widetilde{\mathcal N}(\widetilde{v})\big\|_{L^\infty_tL^2_x({\mathbb R})}\le C\delta_0^3\big\|\Delta^2 f\big\|_{L^2_x}.\end{aligned}$$ Now we follow by the same formulation in Section [5.1](#sec:Th3-formula){reference-type="ref" reference="sec:Th3-formula"} and adopt the same notations. In particular, we recall that $$r=\widetilde{v}^\varepsilon-\widetilde{v},\quad R=\mathrm{e}^{it} \mathcal{S}_\varepsilon r,\quad \mathcal R_j=\langle \nabla \rangle^{-1}\big(\partial_t\mp i \langle \nabla\rangle\big)R \mbox{ for } j=1,2.$$ Next, we will show that the upper bound of the *Leftward wave* $\mathcal R_1$, and the lower and upper bounds of the *Rightward wave* $\mathcal R_2$. Denote $$I(f)\triangleq \{t\in {\mathbb R}^+: t\varepsilon^4\|\Delta^2f\|_{\dot H^{s_c}_x}\le 1\}.$$ **Lemma 28**. *Let $f\in \mathcal S({\mathbb R}^d)$, then there exist constants $C(\|f\|_{L^2_x})>0$ such that for any $t\in I(f)$, there hold $$\begin{aligned} \label{R1-conter-ubd} \big\|\mathcal R_1(t)\big\|_{L^2_x} \le& C\delta_0\varepsilon^{-s_c}\Big[\big\|r_0\big\|_{L^2_x}+\delta_0^2 \varepsilon \|r_0\|_{H^1_x}+\varepsilon^2\big\|\Delta f\big\|_{L^2_x} +(1+\delta_0^2t)\varepsilon^4\big\|\Delta^2 f\big\|_{L^2_x}\Big],\end{aligned}$$ and $$\begin{aligned} \label{R2-conter-lbd} &\Big|\big\|\mathcal R_2(t)\big\|_{L^2_x}-\frac14\delta_0 t\varepsilon^{4-s_c} \big\|\Delta^2 f\big\|_{L^2_x} -\frac 1{64}\delta_0t^2 \varepsilon^{8-s_c} \big\|\Delta^4 f\big\|_{L^2_x}\Big|\notag\\ &\le C\delta_0\varepsilon^{-s_c}\Big[\big(\big\|r_0\big\|_{L^2_x}+\delta_0^2 \varepsilon\|r_0\|_{H^1_x}\big) +\varepsilon^2\big\|\Delta f\big\|_{L^2_x}\notag\\ &\quad +\delta_0^2 t\varepsilon^4 \big\|\Delta^2 f\big\|_{L^2_x} +t\varepsilon^6 \big\|\Delta^3f\big\|_{L^2_x} + C t^2\varepsilon^{10} \big\|\Delta^5 f\big\|_{L^2_x} \Big] .\end{aligned}$$* *Proof.* Then we have the following Duhamel's formula: For $j=1,2$, $$\begin{aligned} \mathcal R_j(t)=&\mathrm{e}^{\mp it\langle \nabla\rangle} \mathcal R_{j,0}\notag\\ & -\int_0^t \mathrm{e}^{\mp i(t-s)\langle \nabla\rangle}\mathrm{e}^{is} \langle\nabla \rangle^{-1} \Big[\varepsilon^4 \mathcal{S}_\varepsilon( \partial_{tt} \widetilde{v})+\big|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big|^2\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon-|\mathcal{S}_\varepsilon \widetilde{v}|^2\mathcal{S}_\varepsilon \widetilde{v}\Big]\,ds.\end{aligned}$$ Then by [\[v-linear-app-2\]](#v-linear-app-2){reference-type="eqref" reference="v-linear-app-2"}, we rewrite it as $$\begin{aligned} \label{Rj-123} \mathcal R_j(t)=&\mathrm{e}^{\mp it\langle \nabla\rangle} \mathcal R_{j,0}+\mathrm{e}^{\mp it\langle \nabla\rangle} I_j(t)+ \mathcal R_{j}^1(t),\end{aligned}$$ where we denote $$\begin{aligned} I_j(t)\triangleq & \frac14\delta_0 \int_0^t \mathrm{e}^{\pm is\langle \nabla\rangle}\mathrm{e}^{is} \langle\nabla \rangle^{-1} \Big[\varepsilon^4 \mathcal{S}_\varepsilon(\mathrm{e}^{-\frac12 is\Delta}\Delta^2 f)\Big]\,ds\\ \mathcal R_{j}^1(t) \triangleq & -\int_0^t \mathrm{e}^{\mp i(t-s)\langle \nabla\rangle}\mathrm{e}^{is} \langle\nabla \rangle^{-1} \Big[\varepsilon^4 \mathcal{S}_\varepsilon\widetilde{\mathcal N}(\widetilde{v})+\big|\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon\big|^2\mathcal{S}_\varepsilon \widetilde{v}^\varepsilon-|\mathcal{S}_\varepsilon \widetilde{v}|^2\mathcal{S}_\varepsilon \widetilde{v}\Big]\,ds.\end{aligned}$$ **Upper bound of $\mathcal R_{j,0}$.** Under the hypotheses, we have that $$\mathcal R_{j,0}=\langle \nabla \rangle^{-1}\Big[i\big(1\mp \langle \nabla\rangle\big)\mathcal{S}_\varepsilon r_0+\mathcal{S}_\varepsilon r_1\Big],\quad \mbox{ with }\quad r_1=-\frac{\varepsilon^2}{2i}(\Delta \widetilde{v}_0-\mu |\widetilde{v}_0|^2\widetilde{v}_0).$$ This gives that for some $C(\|f\|_{L^2_x})>0$, $$\begin{aligned} \label{est:conterex-intial} \left\|\mathrm{e}^{\mp it\langle \nabla\rangle} \mathcal R_{j,0}\right\|_{L^2_x} =\big\| \mathcal R_{j,0}\big\|_{L^2_x} \le C \delta_0\varepsilon^{-s_c}\big(\big\|r_0\big\|_{L^2_x}+\varepsilon^2\big\|\Delta f\big\|_{L^2_x}\big).\end{aligned}$$ For $I_j(t)$, by the scaling invariance of the linear Schrödinger flow, we have that $$\begin{aligned} I_j(t)= & \frac14\delta_0\varepsilon^4 \int_0^t \mathrm{e}^{is(\pm\langle \nabla\rangle+1-\frac12 \Delta)} \langle\nabla \rangle^{-1}\mathcal{S}_\varepsilon(\Delta^2 f)\,ds.\end{aligned}$$ **Upper bound of $I_1$.** Taking the Fourier transformation, we have that $$\begin{aligned} \hat I_1(t,\xi)= & \frac14\delta_0\varepsilon^4 \int_0^t \mathrm{e}^{is(\langle \xi\rangle+1+\frac12|\xi|^2)} \langle\xi \rangle^{-1}\widehat{\mathcal{S}_\varepsilon(\Delta^2 f)}(\xi)\,ds\\ = & \frac14\delta_0\varepsilon^4 \frac{\mathrm{e}^{is(\langle \xi\rangle+1+\frac12|\xi|^2)}-1}{\langle \xi\rangle+1+\frac12|\xi|^2} \langle\xi \rangle^{-1}\widehat{\mathcal{S}_\varepsilon(\Delta^2 f)}(\xi).\end{aligned}$$ This gives that $$\begin{aligned} \label{est:I1-upp-bd} \big\|I(t)\big\|_{L^2_x} \lesssim & \delta_0\varepsilon^4\big\|\mathcal{S}_\varepsilon(\Delta^2 f)\big\|_{L^2_x}\notag\\ \lesssim & \delta_0\varepsilon^{4-s_c}\big\|\Delta^2 f\big\|_{L^2_x}.\end{aligned}$$ **Approximation of $I_2$.** Denote the operator $$\tilde T_t =\frac{\mathrm{e}^{it(-\langle \nabla\rangle+1-\frac12 \Delta)}-1-\frac18 it\Delta^2}{t\Delta^3}.$$ Then by the Mihlin-Hörmander Multiplier Theorem, we have the $L^p\mapsto L^p$ boundedness which is uniform in time: for any $1<p<+\infty$, $$\begin{aligned} \label{est:Tt-tilde} \big\|\tilde T_t h\big\|_{L^p}\le C\|h\|_{L^p},\end{aligned}$$ where the constant $C>0$ is independent of $t$ and $h$. Now we write $$\begin{aligned} \label{apprx-linearKG} \mathrm{e}^{it(-\langle \nabla\rangle+1-\frac12 \Delta)}=1+\frac18 it\Delta^2+t \Delta^3 \tilde T_t.\end{aligned}$$ Applying [\[apprx-linearKG\]](#apprx-linearKG){reference-type="eqref" reference="apprx-linearKG"}, we write that $$\begin{aligned} I_2(t)=& \frac14\delta_0 t\varepsilon^4 \mathcal{S}_\varepsilon(\Delta^2 f) + \frac i{64}\delta_0t^2 \varepsilon^4 \Delta^2 \mathcal{S}_\varepsilon(\Delta^2 f)\\ & + \frac14\delta_0 \int_0^t \mathrm{e}^{\pm is\langle \nabla\rangle}\mathrm{e}^{is} \big(\langle\nabla \rangle^{-1}-1\big) \Big[\varepsilon^4 \mathcal{S}_\varepsilon(\mathrm{e}^{-\frac12 is\Delta}\Delta^2 f)\Big]\,ds\\ &\quad + \frac14\delta_0\varepsilon^4 \int_0^t s\tilde{T_s} \Delta^3 \langle\nabla \rangle^{-1}\mathcal{S}_\varepsilon(\Delta^2 f)\,ds. \end{aligned}$$ Since $f\in {\mathbb R}$, this yields that $$\begin{aligned} \Big|\big\| I_2(t)\big\|_{L^2_x}-& \frac14\delta_0t\varepsilon^4 \big\|\mathcal{S}_\varepsilon(\Delta^2 f)\big\|_{L^2_x} -\frac 1{64}\delta_0t^2 \varepsilon^4 \big\|\Delta^2 \mathcal{S}_\varepsilon(\Delta^2 f)\big\|_{L^2_x}\Big|\\ \le & \frac14\delta_0t\varepsilon^4 \big\|\big(\langle\nabla \rangle^{-1}-1\big)\mathcal{S}_\varepsilon(\Delta^2 f)\big\|_{L^2_x}\\ &\quad +\frac14\delta_0\varepsilon^4 \int_0^t s\big\|\tilde{T_s} \Delta^3 \langle\nabla \rangle^{-1}\mathcal{S}_\varepsilon(\Delta^2 f)\big\|_{L^2_x}\,ds. \end{aligned}$$ Using [\[est:Tt-tilde\]](#est:Tt-tilde){reference-type="eqref" reference="est:Tt-tilde"}, we further get that there exists some constant $C>0$ such that $$\begin{aligned} \Big|\big\| I_2(t)\big\|_{L^2_x}& - \frac14\delta_0t\varepsilon^{4-s_c} \big\|\Delta^2 f\big\|_{L^2_x}-\frac 1{64}\delta_0t^2 \varepsilon^{8-s_c} \big\|\Delta^4 f\big\|_{L^2_x}\Big|\\ \le & C\delta_0t\varepsilon^{6-s_c} \big\|\Delta^3f\big\|_{L^2_x} + C\delta_0t^2\varepsilon^{10-s_c} \big\|\Delta^5 f\big\|_{L^2_x}. \end{aligned}$$ **Upper bound of $\mathcal R_j^1$.** By the Strichartz and Hölder inequalities, we have that $$\begin{aligned} \big\|\mathcal R_j^1\big\|_{L^2_x} \lesssim & t\varepsilon^4\big\|\mathcal{S}_\varepsilon\widetilde{\mathcal N}(\widetilde{v})\big\|_{L^\infty_tL^2_x({\mathbb R})} +\big(\big\|\mathcal R_1\big\|_{L^2_x}+\big\|\mathcal R_2\big\|_{L^2_x}\big)\big(\big\|\mathcal{S}_\varepsilon v^\varepsilon\big\|_{X_{s_c}({\mathbb R})}^2+\big\|\mathcal{S}_\varepsilon v\big\|_{X_{s_c}({\mathbb R})}^2\big)\\ \lesssim & t\varepsilon^{4-s_c}\big\|\widetilde{\mathcal N}(\widetilde{v})\big\|_{L^\infty_tL^2_x({\mathbb R})} +\big(\big\|\mathcal R_1\big\|_{L^2_x}+\big\|\mathcal R_2\big\|_{L^2_x}\big)\big(\big\|v^\varepsilon\big\|_{X_{s_c}({\mathbb R})}^2+\big\|v\big\|_{X_{s_c}({\mathbb R})}^2\big).\end{aligned}$$ By the conclusion in the proof of Proposition [Proposition 26](#prop:Thm3-case1-1){reference-type="ref" reference="prop:Thm3-case1-1"} (see [\[est:R-12-upp-d2\]](#est:R-12-upp-d2){reference-type="eqref" reference="est:R-12-upp-d2"}, [\[est:R-12-upp-d3\]](#est:R-12-upp-d3){reference-type="eqref" reference="est:R-12-upp-d3"} and [\[est:L2-uniform-vep\]](#est:L2-uniform-vep){reference-type="eqref" reference="est:L2-uniform-vep"}), we have that for any $t\in I(f)$, it holds that $$\begin{aligned} \label{est:R-upper-thm3} \big\|\mathcal R_j(t)\big\|_{L^2_x} \lesssim \delta_0 \varepsilon^{-s_c}\|r_0\|_{L^2_x}+\delta_0 \varepsilon^{1-s_c}\|r_0\|_{H^1_x}+\delta_0 t \varepsilon^{4-s_c}\big\|\Delta^2 f\big\|_{L^2_x}.\end{aligned}$$ By [\[est:Sv-ep-Xs\]](#est:Sv-ep-Xs){reference-type="eqref" reference="est:Sv-ep-Xs"}, [\[est:v-Xs\]](#est:v-Xs){reference-type="eqref" reference="est:v-Xs"} and [\[est:R-upper-thm3\]](#est:R-upper-thm3){reference-type="eqref" reference="est:R-upper-thm3"}, we further get that $$\begin{aligned} \label{est:Rj1-1} \big\|\mathcal R_j^1\big\|_{L^2_x} \lesssim & \delta_0^3 \varepsilon^{-s_c}\|r_0\|_{L^2_x}+\delta_0^3 \varepsilon^{1-s_c}\|r_0\|_{H^1_x}+\delta_0^3 t \varepsilon^{4-s_c}\big\|\Delta^2 f\big\|_{L^2_x}.\end{aligned}$$ Collecting the estimates on three terms in [\[Rj-123\]](#Rj-123){reference-type="eqref" reference="Rj-123"} and choosing $\delta_0$ suitably small, we have that the desired estimates. This finishes the proof of the lemma. ◻ ### Regular case {#sec:Conter-Regular case} In this case, we set $r_0=0$. Let $$f(x)\triangleq A_0^d \mathrm{e}^{-A_0|x|^2},$$ where $A_0$ is a suitable large constant determined later. Then $f\in \mathcal S({\mathbb R}^d)$. It implies that $$I(f) =\{t\in {\mathbb R}^+: t\varepsilon^4\lesssim 1\}.$$ Then [\[R1-conter-ubd\]](#R1-conter-ubd){reference-type="eqref" reference="R1-conter-ubd"} implies that for any $t\in I(f)$, $$\begin{aligned} \big\|\mathcal R_1(t)\big\|_{L^2_x} \le &C\delta_0A_0^2\varepsilon^{2-s_c} +C\delta_0(1+\delta_0^2t)A_0^4\varepsilon^{4-s_c} ,\end{aligned}$$ and [\[R2-conter-lbd\]](#R2-conter-lbd){reference-type="eqref" reference="R2-conter-lbd"} implies that there exist some constants $C_0>0,C>0$ such that for any $t\in I(f)$, $$\begin{aligned} \big\|\mathcal R_2(t)\big\|_{L^2_x} \ge & C_0\delta_0A_0^4t\varepsilon^{4-s_c} -C\delta_0A_0^6t\varepsilon^{6-s_c} - C\delta_0A_0^8 t^2\varepsilon^{8-s_c} -C\delta_0A_0^2\varepsilon^{2-s_c}.\end{aligned}$$ We shrink the time interval and denote that $$\tilde I(f) =\{t\in {\mathbb R}^+: t\varepsilon^4\le \delta_0, t\varepsilon^2\ge 1\}.$$ Then choosing $\delta_0$ suitably small, we have that there exists some $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0]$ and any $t\in \tilde I(f)$, $$\begin{aligned} \big\|\mathcal R_1(t)\big\|_{L^2_x} \le &C\delta_0^3A_0^4t\varepsilon^{4-s_c}+C\delta_0A_0^2\varepsilon^{2-s_c},\end{aligned}$$ and $$\begin{aligned} \big\|\mathcal R_2(t)\big\|_{L^2_x} \ge & \frac12 C_0\delta_0A_0^4 t\varepsilon^{4-s_c}.\end{aligned}$$ Applying [\[eq:R-R12\]](#eq:R-R12){reference-type="eqref" reference="eq:R-R12"} and choosing $A_0$ suitably large and then $\delta_0$ suitably small again, this infers that for any $t\in \tilde I(f)$, $$\begin{aligned} \big\| R(t)\big\|_{L^2_x} \ge & \frac14 C_0\delta_0A_0^4 t\varepsilon^{4-s_c}.\end{aligned}$$ Scaling back, this implies that for any $t\in {\mathbb R}^+: 1\le t\le \delta_0\varepsilon^{-2},$ $$\begin{aligned} \big\| r(t)\big\|_{L^2_x} \ge & \frac14 C_0\delta_0A_0^4 t\varepsilon^2.\end{aligned}$$ This establishes the conclusion in regular case. ### Non-Regular case In this case, we set $\widetilde{v}_0$ such that $$\widehat{\widetilde{v}_0}(\xi)=\delta_0\langle \xi\rangle_2^{-\alpha-\frac d2}\big(\ln \langle \xi\rangle_2\big)^{-1}.$$ Here we denote $\langle x\rangle_2=\sqrt{|x|^2+2}$. Then $f\in H^\alpha({\mathbb R}^d)$. Now we need the following elementary results: $$\label{f-high-low-value} \begin{aligned} \big\|f\big\|_{H^\alpha_x} \sim & 1; \\ \big\|P_{\le N} f\big\|_{H^\gamma}\sim & N^{-\alpha+\gamma} \big(\ln N\big)^{-1},\quad \mbox{for any } \gamma>\alpha; \\ \big\|P_{> N} f\big\|_{H^\gamma}\le & CN^{-\alpha+\gamma} \big(\ln N\big)^{-1},\quad \mbox{for any } \gamma<\alpha. \end{aligned}$$ Arguing similarly as in Section [5.2.2](#sec:Thm3-case1-2){reference-type="ref" reference="sec:Thm3-case1-2"}, for fixing large $N$, we denote $\widetilde{v}_N$ to be the solution to the following equation $$\begin{aligned} \begin{split} \left\{ \aligned &2i\partial_{t} \widetilde{v} - \Delta\widetilde{v}= - |\widetilde{v}|^2 \widetilde{v}, \\ & \widetilde{v}(0,x) = P_{\le N}\widetilde{v}_0(x), \endaligned \right. \end{split}\end{aligned}$$ and denote $\widetilde{v}^N =\widetilde{v}-\widetilde{v}_N$. Then similarly as [\[est:Thm3-w-case2-d2\]](#est:Thm3-w-case2-d2){reference-type="eqref" reference="est:Thm3-w-case2-d2"}, we have that $$\begin{aligned} \big\|\widetilde{v}^N \big\|_{L^\infty_tL^2_x({\mathbb R})} \le C\big\|P_{>N}\widetilde{v}_0\big\|_{L^2_x},\end{aligned}$$ where the constant $C>0$ only depends on $\|\widetilde{v}_0\|_{H^1_x}$. Write $$\widetilde{v}^\varepsilon(t)-\widetilde{v}(t)=\widetilde{v}^\varepsilon(t)-\widetilde{v}_N (t)-\widetilde{v}^N .$$ To consider the first term, we denote $$r=\widetilde{v}^\varepsilon(t)-\widetilde{v}_N (t),\quad R=\mathrm{e}^{it} S_\varepsilon r_N,\quad r_0=P_{>N}\widetilde v_0. %f=P_{\le N}\widetilde{v}_0.$$ Note that $r$ obeys the equation [\[eq:kg-r-3\]](#eq:kg-r-3){reference-type="eqref" reference="eq:kg-r-3"} by replacing $v$ by $v_N$ and $r_1$ in this situation is defined by $$r_1=-\frac{\varepsilon^2}{2i}(\Delta P_{\le N}\widetilde{v}_0-\mu |P_{\le N}v_0|^2P_{\le N} v_0).$$ Again, denote $$\mathcal R_j=\langle \nabla \rangle^{-1}\big(\partial_t\mp i \langle \nabla\rangle\big)R \quad \mbox{ for } \quad j=1,2.$$ Then from [\[R1-conter-ubd\]](#R1-conter-ubd){reference-type="eqref" reference="R1-conter-ubd"} and [\[f-high-low-value\]](#f-high-low-value){reference-type="eqref" reference="f-high-low-value"}, we have that for any $1\le \alpha\le 4$, $$\begin{aligned} \big\|\mathcal R_1(t)\big\|_{L^2_x} \le& C\delta_0\varepsilon^{-s_c}\Big[\big\|P_{>N}\widetilde v_0\big\|_{L^2_x}+\delta_0^2 \varepsilon \|P_{>N}v_0\|_{H^1_x}\\ &\quad +\varepsilon^2\big\|\Delta P_{\le N}\widetilde{v}_0\big\|_{L^2_x} +(1+\delta_0^2t)\varepsilon^4\big\|\Delta^2 P_{\le N}\widetilde{v}_0\big\|_{L^2_x}\Big]\\ \le& C\delta_0\varepsilon^{-s_c}\Big[N^{-\alpha}\big(\ln N\big)^{-1}+\delta_0^2 \varepsilon \max\{1, N^{\alpha-1}\}\\ &\quad +\varepsilon^2\max\{1, N^{2-\alpha}\big(\ln N\big)^{-1}\} +(1+\delta_0^2t)\varepsilon^4N^{4-\alpha}\big(\ln N\big)^{-1}\Big].\end{aligned}$$ Setting $$\begin{aligned} \label{def:N} N\triangleq A_0 \big(t\varepsilon^4\big)^{-\frac14},\quad \mbox{ and } \quad t\varepsilon^2\ge 1,\end{aligned}$$ where $A_0$ is a large constant determined later, then we note that choosing $\varepsilon_0=\varepsilon_0(A_0)$ suitably small, it holds $N\varepsilon^\frac12\le A_0$. This gives that for any $\varepsilon\in (0,\varepsilon_0]$, $$\begin{aligned} \big\|\mathcal R_1(t)\big\|_{L^2_x} \le& C\delta_0\varepsilon^{-s_c}\Big[N^{-\alpha}+\delta_0^2t\varepsilon^4N^{4-\alpha}\Big]\big(\ln N\big)^{-1},\end{aligned}$$ and it further infers that $$\begin{aligned} \label{R1-upper-nonreg} \big\|\mathcal R_1(t)\big\|_{L^2_x} \le& C\delta_0\varepsilon^{-s_c}\Big[A_0^{-\alpha} \big(t\varepsilon^4\big)^{\frac14\alpha}+\delta_0^2A_0^{4-\alpha}t \varepsilon^4\big(t\varepsilon^4\big)^{-\frac14(4-\alpha)}\Big]\big|\ln \big(t\varepsilon^4\big) \big|^{-1}\notag\\ \le& C\delta_0\varepsilon^{-s_c}\big(A_0^{-\alpha}+\delta_0^2A_0^{4-\alpha}\big)\big(t\varepsilon^4\big)^{\frac14\alpha}\big|\ln \big(t\varepsilon^4\big) \big|^{-1}.\end{aligned}$$ Similarly, by [\[R2-conter-lbd\]](#R2-conter-lbd){reference-type="eqref" reference="R2-conter-lbd"}, we have that $$\begin{aligned} &\Big|\big\|\mathcal R_2(t)\big\|_{L^2_x}- \frac14\delta_0 \varepsilon^{-s_c}t\varepsilon^4 \big\|\Delta^2 P_{\le N}\widetilde{v}_0\big\|_{L^2_x} -\frac 1{64}\delta_0t^2 \varepsilon^{8-s_c} \big\|\Delta^4 P_{\le N}\widetilde{v}_0\big\|_{L^2_x}\Big|\\ \le & C\delta_0 \varepsilon^{-s_c}\Big[t\varepsilon^6 \big\|\Delta^3 P_{\le N}\widetilde{v}_0\big\|_{L^2_x} + t^2\varepsilon^{10} \big\|\Delta^5 P_{\le N}\widetilde{v}_0\big\|_{L^2_x} \notag\\ & \quad +\big\|P_{>N}\widetilde v_0\big\|_{L^2_x}+\delta_0^2 \varepsilon \|P_{>N}v_0\|_{H^1_x} +\varepsilon^2\big\|\Delta P_{\le N}\widetilde{v}_0\big\|_{L^2_x} \Big]\\ %\le & C\delta_0 \varepsilon^{-s_c}\Big[t\varepsilon^6 \big\|\Delta^3 P_{\le N}\widetilde{v}_0\big\|_{L^2_x} %+ t^2\varepsilon^{10} \big\|\Delta^5 P_{\le N}\widetilde{v}_0\big\|_{L^2_x} %+\big\|P_{>N}\widetilde v_0\big\|_{L^2_x}\Big]\\ \le & C\delta_0 \varepsilon^{-s_c}\Big[N^{-\alpha}+t\varepsilon^6N^{6-\alpha}+ t^2\varepsilon^{10}N^{10-\alpha}\Big]\big(\ln N\big)^{-1}.\end{aligned}$$ By [\[f-high-low-value\]](#f-high-low-value){reference-type="eqref" reference="f-high-low-value"}, it further gives that there exist constants $c_0>0, C_0>0$ such that $$\begin{aligned} \big\|\mathcal R_2(t)\big\|_{L^2_x} \ge & c_0\delta_0 \varepsilon^{-s_c}\big(t\varepsilon^4N^4+t^2\varepsilon^8N^8\big)N^{-\alpha}\big(\ln N\big)^{-1}\\ &\quad -C\delta_0 \varepsilon^{-s_c}\Big[N^{-\alpha}+t\varepsilon^6N^{6-\alpha}+ t^2\varepsilon^{10}N^{10-\alpha}\Big]\big(\ln N\big)^{-1}. % \\ % \big\|\mathcal R_2(t)\big\|_{L^2_x} %\le & %C_0\delta_0 \varepsilon^{-s_c}\big(t\varepsilon^4N^4+t^2\varepsilon^8N^8\big)N^{-\alpha}\big(\ln N\big)^{-1}\\ % &\quad %+ %C\delta_0 \varepsilon^{-s_c}\Big[N^{-\alpha}+t\varepsilon^6N^{6-\alpha}+ t^2\varepsilon^{10}N^{10-\alpha}\Big]\big(\ln N\big)^{-1}.\end{aligned}$$ By [\[def:N\]](#def:N){reference-type="eqref" reference="def:N"}, it drives that $$\begin{aligned} \big\|\mathcal R_2(t)\big\|_{L^2_x} \ge & c_0A_0^{4-\alpha}\delta_0\varepsilon^{-s_c} \big(t\varepsilon^4\big)^{\frac14\alpha}\big|\ln \big(t\varepsilon^4\big) \big|^{-1}. % \\ % \big\|\mathcal R_2(t)\big\|_{L^2_x} %\le & %C_0\delta_0 \varepsilon^{-s_c}t\varepsilon^4N^{4-\alpha}\big(\ln N\big)^{-1}\\ % &\quad %+ %C\delta_0 \varepsilon^{-s_c}\Big[N^{-\alpha}+t\varepsilon^4N^{6-\alpha}+ t^2\varepsilon^{10}N^{10-\alpha}\Big]\big(\ln N\big)^{-1}.\end{aligned}$$ Combining with the estimate [\[R1-upper-nonreg\]](#R1-upper-nonreg){reference-type="eqref" reference="R1-upper-nonreg"}, choosing $A_0$ suitably large and arguing similarly as Section [5.3.1](#sec:Conter-Regular case){reference-type="ref" reference="sec:Conter-Regular case"}, we obtain that $$\begin{aligned} \big\|R(t)\big\|_{L^2_x} \ge &\frac12 c_0A_0^{4-\alpha}\delta_0\varepsilon^{-s_c} \big(t\varepsilon^4\big)^{\frac14\alpha}\big|\ln \big(t\varepsilon^4\big) \big|^{-1}.\end{aligned}$$ Scaling back, this concludes that for any $t\in {\mathbb R}^+: 1\le t\le \delta_0\varepsilon^{-2},$ $$\begin{aligned} \big\| r(t)\big\|_{L^2_x} \gtrsim & \big(t\varepsilon^2\big)^{\frac14\alpha}\big|\ln \big(t\varepsilon^2\big) \big|^{-1}.\end{aligned}$$ This establishes the conclusion in non-regular case. # Schrödinger profile and Growth-in-time nonrelativistic limit {#sec:case1} .5cm In this section, we shall give the proof of Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}. Since the proof is much similar as the that of Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"}, we only give it briefly. *Proof of Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}.* We split it into the following two cases. $\bullet$ Case 1: $d=2$. We write $$\begin{aligned} r^\varepsilon(t)=&u^\varepsilon(t)-\mathrm{e}^{\frac{it}{\varepsilon^2}}v(t)-\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v(t)\\ =&u^\varepsilon(t)-\mathrm{e}^{\frac{it}{\varepsilon^2}}v^\varepsilon(t)-\mathrm{e}^{-\frac{it}{\varepsilon^2}}\overline{v^\varepsilon}(t) +\mathrm{e}^{\frac{it}{\varepsilon^2}}\big(v^\varepsilon(t)-v(t)\big)+\mathrm{e}^{-\frac{it}{\varepsilon^2}}\big(\overline{v^\varepsilon}(t)-\bar v(t)\big),\end{aligned}$$ where $v^\varepsilon$ is the solution of [\[eq:nls-wave\]](#eq:nls-wave){reference-type="eqref" reference="eq:nls-wave"} with $$v_0^\varepsilon=v_0,\quad v_1^\varepsilon=v_1=0.$$ Then by Theorem [Theorem 1](#thm:main1-global){reference-type="ref" reference="thm:main1-global"}, we have that $$\begin{aligned} \big\|u^\varepsilon(t)-\mathrm{e}^{\frac{it}{\varepsilon^2}}v^{\varepsilon}(t)-\mathrm{e}^{-\frac{it}{\varepsilon^2}}\bar v^{\varepsilon}(t) \big\|_{L^2_x} \lesssim \varepsilon^2.\end{aligned}$$ Moreover, by Theorem [Theorem 6](#thm:main3){reference-type="ref" reference="thm:main3"} and choosing $\alpha\ge2, \beta=2$, we have that $$\begin{aligned} \big\|v^\varepsilon(t)-v(t)\big\|_{L^2_x} \lesssim &\varepsilon^2+\big(\varepsilon^2 t\big)^{\frac14\alpha}.\end{aligned}$$ Combining with the estimates above, we obtain [\[est:main3-local-d2\]](#est:main3-local-d2){reference-type="eqref" reference="est:main3-local-d2"}. $\bullet$ Case 2: $d=3$. Suppose that $v$ is the solution of [\[eq:nls\]](#eq:nls){reference-type="eqref" reference="eq:nls"}, then by [\[eq:r\]](#eq:r){reference-type="eqref" reference="eq:r"}-[\[eq:initial-data\]](#eq:initial-data){reference-type="eqref" reference="eq:initial-data"}, we obtain that $$\begin{aligned} \begin{split} \label{eq:kg-r-1} \left\{ \aligned &\varepsilon^2\partial_{tt} r^\varepsilon - \Delta r + \frac{1}{\varepsilon^2} r^\varepsilon +A_\varepsilon(v,r^\varepsilon)+B_\varepsilon(v) +\varepsilon^2\Big[\mathrm{e}^{\frac{it}{\varepsilon^2}}\partial_{tt}v+\mathrm{e}^{-\frac{it}{\varepsilon^2}}\partial_{tt}\bar v\Big]=0, \\ & r^\varepsilon(0,x) = 0,\quad \partial_tr^\varepsilon(0,x) = r_1(x), \endaligned \right. \end{split}\end{aligned}$$ where $$\begin{aligned} \label{def:r1} r_1\triangleq -2{\mathrm{Re}}\big[\partial_tv(0)\big]=-2{\mathrm{Im}}\big(\Delta v_0-|v _0|^2v_0\big).\end{aligned}$$ Now we use the scaling argument and denote $$\begin{aligned} \label{scaling-phi-r} R(t,x)=\mathcal{S}_\varepsilon r^\varepsilon(t, x),\end{aligned}$$ then by [\[eq:kg-r-1\]](#eq:kg-r-1){reference-type="eqref" reference="eq:kg-r-1"}, $R$ obeys the following equation: $$\begin{aligned} \label{eq:phi-scaling} \partial_{tt} R - \Delta R + R +G\Big(\mathcal{S}_\varepsilon v,R\Big)+F_1(\mathcal{S}_\varepsilon v)+\varepsilon^4 F_2\big(\mathcal{S}_\varepsilon (v_{tt})\big)=0,\end{aligned}$$ where we have used the notations $G,F_1$ and $f^\varepsilon$ as in Section [4](#sec:global){reference-type="ref" reference="sec:global"}, and additionally we denote $$\begin{aligned} F_2(f)=&\mathrm{e}^{it}f+\mathrm{e}^{-it}\bar f. \end{aligned}$$ Moreover, the initial data of $R$ is that $$\begin{aligned} R(0)=0,\quad \partial_{t} R(0)=\varepsilon^2 \mathcal{S}_\varepsilon r_1(x).\end{aligned}$$ Denote $$\begin{aligned} \varphi\triangleq \langle \nabla\rangle^{-1}\big(\partial_t +i\langle \nabla\rangle \big)R,\end{aligned}$$ then we have that $$\begin{aligned} R= {\mathrm{Im}}\> \varphi;\quad \partial_t R= {\mathrm{Re}}\big(\langle \nabla\rangle \varphi\big).\end{aligned}$$ Moreover, $\varphi$ obeys the following equation $$\begin{aligned} \big(\partial_t -i\langle \nabla\rangle \big) \varphi +\langle \nabla\rangle^{-1}\Big[G\Big(\mathcal{S}_\varepsilon v ,R\Big)+F_1(v^\varepsilon)+\varepsilon^4 F_2\big((v_{tt})^\varepsilon)\Big]=0.\end{aligned}$$ with the initial data $$\varphi(0)=\varphi_0\triangleq \varepsilon^2\langle \nabla\rangle^{-1}\big(\mathcal{S}_\varepsilon r_1\big).$$ By Duhamel's formula, we have that $$\begin{aligned} \varphi(t)= \mathrm{e}^{it\langle\nabla\rangle}\varphi_0+\int_0^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}\Big[G\Big(\mathcal{S}_\varepsilon v ,R\Big)+F_1\big(\mathcal{S}_\varepsilon v\big)+\varepsilon^4 F_2\big((v_{tt})^\varepsilon)\Big]\,ds.\end{aligned}$$ Treating similarly as in [\[est:Duh-mod\]](#est:Duh-mod){reference-type="eqref" reference="est:Duh-mod"}, we obtain that for any $I=[t_0,t_1]\subset {\mathbb R}$, $$\begin{aligned} \varphi(t)=& \mathrm{e}^{it\langle\nabla\rangle}\varphi(t_0) \notag\\ &\quad +\int_{t_0}^t \mathrm{e}^{i(t-s)\langle\nabla\rangle} \langle \nabla\rangle^{-1}\Big[G\Big(\mathcal{S}_\varepsilon v ,R\Big)+P_{> 1}F_1\big(\mathcal{S}_\varepsilon v\big)+\varepsilon^4 F_2\big((v_{tt})^\varepsilon\big)\Big]\,ds\\ &\quad - \frac{\mathrm{e}^{3it} }{i(\langle\nabla\rangle-3)}\langle \nabla\rangle^{-1}P_{\le 1} \big(\mathcal{S}_\varepsilon v(t)\big)^3 + \frac{\mathrm{e}^{it\langle \nabla \rangle}}{i(\langle\nabla\rangle-3)}\langle \nabla\rangle^{-1}P_{\le 1} \big(\mathcal{S}_\varepsilon v(t_0)\big)^3\\ &\quad - \frac{\mathrm{e}^{3it} }{i(\langle\nabla\rangle+3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(\overline{\mathcal{S}_\varepsilon v(t)}\Big)^3 + \frac{\mathrm{e}^{it\langle \nabla \rangle}}{i(\langle\nabla\rangle+3)}\langle \nabla\rangle^{-1}P_{\le 1} \Big(\overline{\mathcal{S}_\varepsilon v(t_0)}\Big)^3\\ &\quad+3 \int_{t_0}^t \mathrm{e}^{i(t-s)(\langle\nabla\rangle-3)} \frac{1}{i(\langle\nabla\rangle-3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(\mathcal{S}_\varepsilon v\big)^2\> \partial_s\big(\mathcal{S}_\varepsilon v\big)\Big]\,ds\\ &\quad+3 \int_{t_0}^t \mathrm{e}^{i(t-s)(\langle\nabla\rangle+3)} \frac{1}{i(\langle\nabla\rangle+3)} \langle \nabla\rangle^{-1}P_{\le 1} \Big[\big(\overline{\mathcal{S}_\varepsilon v})^2\> \partial_s\big(\overline{\mathcal{S}_\varepsilon v}\big)\Big]\,ds.\end{aligned}$$ First, we assume that $v_0$ is smooth, then treating similarly as in Sections [4.1](#sec:nonlinear){reference-type="ref" reference="sec:nonlinear"} and [5.2.1](#sec:Thm3-case1-1){reference-type="ref" reference="sec:Thm3-case1-1"}, and applying Lemma [Lemma 17](#lem:spacetime-norm-NLS){reference-type="ref" reference="lem:spacetime-norm-NLS"} instead, we have that for any $\gamma\in [-\frac12,0]$, $$\begin{aligned} \label{est:varphi-local-d3} \big\||\nabla|^\gamma \varphi\big\|_{Y_\frac12(I)} \lesssim & \big\||\nabla|^\gamma\langle \nabla\rangle^{\frac12}\varphi(t_0)\big\|_{H^\frac12} +\big\|v\big\|_{S_\frac12(\varepsilon^2I)}^2\big\||\nabla|^\gamma\varphi\big\|_{S_\frac12(I)} \notag\\ & +\big\|\varphi\big\|_{S_\frac12(\varepsilon^2I)}\big\||\nabla|^\gamma\varphi\big\|_{S_\frac12(I)} +\big\|\varphi\big\|_{S_\frac12(\varepsilon^2I)}^2\big\||\nabla|^\gamma\varphi\big\|_{S_\frac12(I)} \notag\\ & +\big\|\mathcal{S}_\varepsilon v\big\|_{S_\frac12(I)}^2\big\||\nabla|^\gamma\partial_s\big(\mathcal{S}_\varepsilon v\big)\big\|_{S_\frac12(I)} +\varepsilon^{4+\gamma}|I| \big\|\Delta^2 v_0\big\|_{\dot H^{\gamma+\frac12}} +\varepsilon^{2+\gamma}.\end{aligned}$$ By Sobolev's inequality and Lemma [Lemma 17](#lem:spacetime-norm-NLS){reference-type="ref" reference="lem:spacetime-norm-NLS"}, we have that $$\begin{aligned} \big\|\mathcal{S}_\varepsilon v\big\|_{S_\frac12(I)} \le &\big\|v\big\|_{S_\frac12({\mathbb R})} \le C(\|v_0\|_{H^1_x});\\ \big\|\mathcal{S}_\varepsilon v\big\|_{S_\frac12(I)} \lesssim & |I|^\frac{3}{10} \big\||\nabla|^\frac35 \mathcal{S}_\varepsilon v\big\|_{L^\infty_t\dot H^\frac12_x(I)} \\ \le & |I|^\frac{3}{10} \varepsilon^\frac35 C(\|v_0\|_{H^2});\\ \big\||\nabla|^\gamma\partial_s\big(\mathcal{S}_\varepsilon v\big)\big\|_{S_\frac12(I)} \le &\varepsilon^{2+\gamma} C(\|v_0\|_{H^1_x}) \> \big\||\nabla|^{\gamma+\frac52}v_0\big\|_{L^2_x}.\end{aligned}$$ Inserting these estimates into [\[est:varphi-local-d3\]](#est:varphi-local-d3){reference-type="eqref" reference="est:varphi-local-d3"}, we obtain that $$\begin{aligned} \big\||\nabla|^\gamma \varphi\big\|_{Y_\frac12(I)} \lesssim & \big\||\nabla|^\gamma\langle \nabla\rangle^{\frac12}\varphi(t_0)\big\|_{\dot H^\frac12} +\big\||\nabla|^\gamma v\big\|_{S_\frac12(\varepsilon^2I)}^2\big\||\nabla|^\gamma\varphi\big\|_{S_\frac12(I)} \\ &+\big\||\nabla|^\gamma\varphi\big\|_{S_\frac12(I)}^2 +\big\||\nabla|^\gamma\varphi\big\|_{S_\frac12(I)}^3 \\ &+\varepsilon^{2+\gamma}\Big(\min\big\{1, |I|^\frac{3}{10} \varepsilon^\frac35\big\}\big\||\nabla|^{\gamma+\frac52}v_0\big\|_{L^2_x} +\varepsilon^2|I| \big\|\Delta^2 v_0\big\|_{\dot H^{\gamma+\frac12}}+ 1\Big),\end{aligned}$$ where the implicit constant is only dependent of $\|v_0\|_{H^2}$. Repeating the same process at the end of Section [5.2.1](#sec:Thm3-case1-1){reference-type="ref" reference="sec:Thm3-case1-1"}, we have that if $$\begin{aligned} \label{cond-T-3d-local} \varepsilon^\frac35T^\frac{3}{10} \big\||\nabla|^{\frac52}v_0\big\|_{L^2_x}\le \delta_0;\quad \varepsilon^2T \big\|\Delta^2 v_0\big\|_{\dot H^{\frac12}}\le \delta_0,\end{aligned}$$ it holds that for some constant $C=C\big(\|v_0\|_{H^2}\big)>0$, $$\begin{aligned} \big\|r^\varepsilon\big\|_{L^\infty_tL^2_x([0,T])} \le C\Big( \varepsilon^2 +\varepsilon^2T \big\|\Delta^2 v_0\big\|_{L^2_x}\Big).\end{aligned}$$ If $v_0\in H^\alpha$ for some $\alpha\in [2,4]$, then arguing similarly as in Section [5.2.2](#sec:Thm3-case1-2){reference-type="ref" reference="sec:Thm3-case1-2"}, we have that for any $T$ verifying $$\begin{aligned} \label{cond-T-3d-local-2} \varepsilon^{\frac35}T^\frac{3}{10} N^\frac12 \le C\big(\big\|v_0\big\|_{H^2}\big) \delta_0;\quad \varepsilon^2T N^\frac52 \le C\big(\big\|v_0\big\|_{H^2}\big) \delta_0\end{aligned}$$ (which is from [\[cond-T-3d-local\]](#cond-T-3d-local){reference-type="eqref" reference="cond-T-3d-local"}), then $$\begin{aligned} \big\|r^\varepsilon\big\|_{L^\infty_tL^2_x([0,T])} \le C\Big( \varepsilon^2 +\varepsilon^2TN^{4-\alpha}+N^{-\alpha}\Big).\end{aligned}$$ Choosing $$N=\big(\varepsilon^2T\big)^{-\frac14},$$ then [\[cond-T-3d-local-2\]](#cond-T-3d-local-2){reference-type="eqref" reference="cond-T-3d-local-2"} holds if $\varepsilon^2T\le \delta_0$. Thus we prove [\[est:main3-local-d2\]](#est:main3-local-d2){reference-type="eqref" reference="est:main3-local-d2"} when $\varepsilon^2T\le \delta_0$. Note that we have the energy conservation law: $$\begin{aligned} \int\big(\varepsilon^2|\partial_tu^\varepsilon|^2 & + |\nabla u^\varepsilon|^2 + \varepsilon^{-2}|u^\varepsilon|^2+ \frac{3}{2}|u^\varepsilon|^4\big)dx\\ = & \int\big(\varepsilon^{-2}\big\|(u_0,u_1)\big\|_{L^2\times L^2}^2+ |\nabla u_0|^2 + \frac{3}{2}|u_0|^4\big)dx.\end{aligned}$$ This implies that $$\begin{aligned} \big\|u^\varepsilon\big\|_{L^\infty_tL^2_x({\mathbb R})} \le C\big(\big\|(u_0,u_1)\big\|_{H^1_x\times L^2_x}\big).\end{aligned}$$ This yields [\[est:main3-local-d2\]](#est:main3-local-d2){reference-type="eqref" reference="est:main3-local-d2"} when $\varepsilon^2T\ge \delta_0$. This finishes the proof of Theorem [Theorem 5](#thm:main3-local-d2){reference-type="ref" reference="thm:main3-local-d2"}. ◻ # Acknowledgment {#acknowledgment .unnumbered} We would like to thank Professor Weizhu Bao for bringing the problem to us and for useful discussions. The authors were in part supported by NSFC (Grant No. 11725102, 12171356), Sino-German Center Mobility Programme (Project ID/GZ M-0548) and Shanghai Science and Technology Program (Project No. 21JC1400600 and No. 19JC1420101). 00 W. Bao, Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 50 (2012) 492--521. W. Bao, Y. Cai, Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 52 (2014) 1103--1127. W. Bao, Y. Cai, X. Zhao. A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon equation in the nonrelativistic limit regime, SIAM J. Numer. Anal., 52 (2014), 2488--2511. W. Bao, X. Dong. Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math., 120 (2012), 189--229. W. Bao, Y. Lu, Z. F. Zhang. Convergence rates in the nonrelativistic limit of the cubic Klein-Gordon equation, arXiv: 2304.06920. W. Bao and X. Zhao. Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, J. Comput. Phys., 398 (2019), 108886. L. Berge, T. Colin. A singular perturbation problem for an envelope equation in plasma physics. Phys. D 84 (1995), 437--459 J. Bourgain and D. Li, On an endpoint Kato-Ponce inequality, *Differential Integral Equations* 27 (2014), pp. 1037--1072. T. Cazenave, F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation. Comm. Math. Phys.147(1992), 75--100. Ph. Chartier, N. Crouseilles, M. Lemou, F. Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schrödinger equations, Numer. Math. 129 (2015) 211--250. Ph. Chartier, J. Makazaga, A. Murua, G. Vilmart, Multi-revolution composition methods for highly oscillatory differential equations, Numer. Math. 128 (2014) 167--192. R. Coifman, Y. Meyer. Nonlinear harmonic analysis, operator theory, and PDE in Beijing Lectures in Harmonic Analysis (Beijing, 1984), Annals of Mathematics Studies 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 3--45. B. Dodson, . Duke Math. J. 165 (2016), no. 18, 3435--3516. X. Dong, Z. Xu, X. Zhao, On time-splitting pseudospectral discretization for nonlinear Klein-Gordon equation in nonrelativistic limit regime, Commun. Comput. Phys. 16 (2014) 440--466. J. Ginibre and G. Velo, . Comm. Math. Phys. 144 (1992), no. 1, 163--188. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, *Commun. Pure Appl. Math.* 41 (1988), pp. 891--907. M. Keel and T. Tao, Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), 955--980. R. Killip, T. Tao, and M. Visan, . J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203--1258. Killip, R., Stovall, B., Visan, M.: Scattering for the cubic Klein--Gordon equation in two space dimensions. Trans. Am. Math. Soc. 364(3), 1571--1631 (2012) D. Li, On Kato-Ponce and fractional Leibniz, *Rev. Mat. Iberoam.* 35 (2019), pp. 23--100. Y. Lu, Z. Zhang. Higher order asymptotic analysis of the nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Asymptotic Analysis, 102 (3-4) (2017) 157--175. S. Machihara. The nonrelativistic limit of the nonlinear Klein-Gordon equation, Funkcial. Ekvac., 44 (2001), 243-252. S. Machihara, K. Nakanishi, T. Ozawa. Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations, Math. Ann., 322 (2002), 603--621. Marshall, B., Strauss, W., Wainger, S.: $L^p$-$L^q$ estimates for the Klein--Gordon equations. J. Math. Pures Appl. (9), 59, 417--440 (1980) N. Masmoudi, K. Nakanishi. From nonlinear Klein-Gordon equation to a system of coupled nonliner Schrödinger equations, Math. Ann., 324 (2002), 359--389. B. Najman. The nonrelativistic limit of the nonlinear Klein-Gordon equation. Nonlinear Anal.15,217--228 (1990). K. Nakanishi. Nonrelativistic limit of scattering theory for nonlinear Klein-Gordon equations, J. Differential Equations, 180 (2002), 453--470. K. Schratz, X. Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete & Continuous Dynamical Systems-B, 25 (2020), 2841--2865. M. Tsutsumi. Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions, Nonlinear Anal., 8 (1984), 637-643. X. Zhao, A combination of multiscale time integrator and two-scale formulation for the nonlinear Schrödinger equation with wave operator, J. Comput. Appl. Math. 326 (2017) 320--336. [^1]:

arxiv_math

--- author: - Christopher Bouchard title: Strengthening the union-closed sets conjecture --- # 1. Introduction {#introduction .unnumbered} A family of sets $\mathcal{F}$ is *union-closed* if $X, Y \in \mathcal{F} \implies X \cup Y \in \mathcal{F}$. The union-closed sets conjecture, also called Frankl's conjecture, states the following: **Conjecture** **1.1.** *For any finite union-closed family of sets* $\mathcal{A} \neq \{ \emptyset \}$*, there exists an element* of $\bigcup_{A \in \mathcal{A}} A$ *that* *appears in at least* $\frac{|\mathcal{A}|}{2}$ *members of* $\mathcal{A}$. Conjecture 1.1 has been proved for $|\bigcup_{A \in\mathcal{A}} A| \leq 11$ (see \[1\]) and for $|\mathcal{A}| \leq 46$ (see \[8\]). Equivalent conjectures include the lattice formulation of the union-closed sets conjecture, which has been verified for lower semimodular lattices (see \[7\]), and the graph formulation of the union-closed sets conjecture (see \[2\]). In \[4\] it was proved that there exists an element of $\bigcup_{A \in \mathcal{A}} A$ that appears in at least $\varepsilon|\mathcal{A}|$ members of $\mathcal{A}$ for a constant $\varepsilon=\frac{1}{100}$, a result which in \[9\] was improved to $\varepsilon=\frac{3-\sqrt{5}}{2}$. For a survey of many results regarding Conjecture 1.1, see \[3\]. The following notation will be used throughout this writing: $\mathcal{A}$ is a finite union-closed family of sets such that $\mathcal{A} \neq \{ \emptyset \}$. $[p]=\{1,2,\cdots,p\}$ for any positive integer $p$, and $[0] \coloneqq \emptyset$. ${S \choose k}$ is the set of all $k$-element subsets of a set $S$, and $2^S$ is the power set of $S$. For a real number $r$, $\lceil r \rceil$ is the smallest integer greater than or equal to $r$, and $\lfloor r \rfloor$ is the greatest integer less than or equal to $r$. $\bigcup_{A \in\mathcal{A}} A$ is the *base set* of $\mathcal{A}$. Without loss of generality, it is denoted $[n]$ unless otherwise specified. Also, $|\mathcal{A}|$ is denoted $m$. The maximum frequency in $\mathcal{A}$ among the elements in $[n]$ is denoted $\texttt{high}(\mathcal{A})$, and the set of elements in $[n]$ with frequency $\texttt{high}(\mathcal{A})$ is denoted $\texttt{arg\_high}(\mathcal{A})$. $\mathcal{A}$ is *separating* if no element of $[n]$ is in all and only all of the same members of $\mathcal{A}$ as a different element of $[n]$. Finally, $\mathcal{A}_B=\{A\in\mathcal{A}|A \cap B = B\}$ and $\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB\mkern-2.7mu}\mkern 2.7mu}=\{A\in\mathcal{A}|A \cap B = \emptyset\}$ where $B \subseteq \bigcup_{A \in \mathcal{A}}A$. Then combining these $\mathcal{A}_{B_1\mkern 2.7mu\overline{\mkern-2.7muB_2\mkern-2.7mu}\mkern 2.7mu}=\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB_2\mkern-2.7mu}\mkern 2.7muB_1}=\{A\in\mathcal{A}|A \cap B_1 = B_1 \land A \cap B_2 = \emptyset\}$ where $B_1, B_2 \subseteq \bigcup_{A \in \mathcal{A}}A$ such that $B_1 \cap B_2 = \emptyset$. We investigate strengthenings of Conjecture 1.1. Among these is that for any finite union-closed family of sets $\mathcal{A}$, $\forall x \in [n]$ $\exists B \in {[n] \choose n+1-x}$ $|\mathcal{A}_B| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB\mkern-2.7mu}\mkern 2.7mu}|$, which we prove for $x \in [\lceil \frac{n}{3} \rceil + 1]$. # 2. The first strengthening of Conjecture 1.1 {#the-first-strengthening-of-conjecture-1.1 .unnumbered} We can restate Conjecture 1.1 as follows: **Conjecture** **1.1** (First Rephrasing)**.** *For any finite union-closed family of sets* $\mathcal{A}$*,* $\exists B \in {[n] \choose 1}$ $|\mathcal{A}_{B}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB\mkern-2.7mu}\mkern 2.7mu}|$*.* A natural extension to a stronger conjecture is then the following: **Conjecture** **2.1.** *For any finite union-closed family of sets* $\mathcal{A}$*,* $\forall x \in [n]$ $\exists B \in {[n] \choose n+1-x}$ $|\mathcal{A}_B| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB\mkern-2.7mu}\mkern 2.7mu}|$*.* We now prove Conjecture 2.1 for $x \in [\lceil \frac{n}{3} \rceil +1]$. **Theorem** **2.2.** *For any finite union-closed family of sets* $\mathcal{A}$*,* $\forall x \in [\lceil \frac{n}{3} \rceil +1]$ $\exists C \in {[n] \choose n+1-x}$ $|\mathcal{A}_C| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC\mkern-2.7mu}\mkern 2.7mu}|$*.* $Proof.$ We prove Theorem 2.2 by induction on $x$. *Base Case ($x=1$):* $\exists C \in {[n] \choose n}$ $|\mathcal{A}_C| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC\mkern-2.7mu}\mkern 2.7mu}|$. *Proof.* $[n]=\bigcup_{A \in \mathcal{A}}A$ and $\mathcal{A}$ is union-closed. Therefore, $[n] \in \mathcal{A}$ and $|\mathcal{A}_{[n]}|=1$. If $\emptyset \in \mathcal{A}$ then $|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu[n]\mkern-2.7mu}\mkern 2.7mu}|=1$, and if $\emptyset \not \in \mathcal{A}$ then $|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu[n]\mkern-2.7mu}\mkern 2.7mu}|=0$. In either case, $C=[n] \in {[n] \choose n}$ with $|\mathcal{A}_{C}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC\mkern-2.7mu}\mkern 2.7mu}|$. *Induction Step:* $\forall x \in [\lceil \frac{n}{3} \rceil ] (\exists C \in {[n] \choose n+1-x}$ $|\mathcal{A}_C| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC\mkern-2.7mu}\mkern 2.7mu}| \implies \exists C' \in {[n] \choose n-x}$ $|\mathcal{A}_{C'}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC'\mkern-2.7mu}\mkern 2.7mu}|)$. *Proof.* If $\exists y' \in C$ $\mathcal{A}_{\{y'\}} \cap 2^{([n] \setminus C) \cup \{y'\}} = \emptyset$, then $C' = C \setminus \{y'\}$ with $|\mathcal{A}_{C'}| \geq |\mathcal{A}_C| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC\mkern-2.7mu}\mkern 2.7mu}| = |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC'\mkern-2.7mu}\mkern 2.7mu}|$. Else, $\forall y \in C$ $\mathcal{A}_{\{y\}} \cap 2^{([n] \setminus C) \cup \{y\}} \neq \emptyset$, in which case we have the following: $\exists D \in {[n] \choose n-x}$ in $\mathcal{A}$. *Proof.* Consider the family $\mathcal{P}=\bigcup_{y \in C} \mathcal{A} \cap 2^{([n] \setminus C) \cup \{y\}}$ and further $P=\bigcup_{p \in \mathcal{P}}p$. We observe that $P \setminus C \subseteq [n] \setminus C$. By the construction of $P$, $\forall w \in P \setminus C$ $\exists u \in C$ with a $z \in \mathcal{A} \cap 2^{([n] \setminus C) \cup \{u\}}$ such that $w \in z$. Let $\mathcal{Z}$ be the family containing a (not necessarily unique) $z$ for every $w$, making $|\mathcal{Z}| = |P \setminus C| \leq |[n] \setminus C|=x-1$, and $Z=\bigcup_{z \in \mathcal{Z}}z \supseteq P \setminus C$, with $|Z| \leq 2|P \setminus C|$ and $Z \in \mathcal{A}$ by union-closedness. Consider a $V \subsetneq P \setminus Z$ with $|V|=n-x-|Z|$. Such a $V$ must exist because of the following two facts: .) $n-x-|Z| \geq 0$, as $x \leq \lceil \frac{n}{3} \rceil$ and $|Z|\leq 2|P \setminus C| \leq 2|[n]\setminus C| = 2(n-(n-x+1))=2x-2$, making $n-x-|Z|$ have a minimum of $n-3 \lceil \frac{n}{3} \rceil+2$, which itself has a minimum of $0$ when $n=3k+1$ for some whole number $k$. .) $n-x-|Z| < |P \setminus Z|$, as $|P \setminus Z| = |P|-|Z|$ and $|P| \geq |C| = n-x+1$. Now, because $\forall y \in C$ $\mathcal{A}_{\{y\}} \cap 2^{([n] \setminus C) \cup \{y\}} \neq \emptyset$, we have that $\forall v \in V$ $\exists V' \in \mathcal{A}_{\{v\}} \cap 2^{([n] \setminus C) \cup \{v\}}$. We collect a $V'$ for every $v$ into a family $\mathcal{V}$ and let $F=\bigcup_{V' \in \mathcal{V}}V'$, which is in $\mathcal{A}$ by union-closedness. Then $F \setminus V \subseteq P \setminus C \subseteq Z$, making $Z \cup F=Z \cup V$. And because $Z \cap V = \emptyset$, we have $|Z \cup V|=|Z|+|V|=|Z|+(n-x-|Z|)=n-x$. Thus, $Z \cup F \in { [n] \choose n-x }$ and is in $\mathcal{A}$ by union-closedness. Therefore, $D = Z \cup F$, and $D \in { [n] \choose n-x }$ is in $\mathcal{A}$. It follows that $|\mathcal{A}_D| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muD\mkern-2.7mu}\mkern 2.7mu}|$, as any $X$ in $\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muD\mkern-2.7mu}\mkern 2.7mu}$ can be uniquely matched with $D \cup X$ in $\mathcal{A}_D$. $D \cup X$ must be in $\mathcal{A}_D$ by the existence of $D$ in $\mathcal{A}$ and union-closedness, and every $D \cup X$ must be unique as $\forall X \in \mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muD\mkern-2.7mu}\mkern 2.7mu}$ $D \cap X = \emptyset$. Thus, $C'=D$ is in ${[n] \choose n-x}$ with $|\mathcal{A}_{C'}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC'\mkern-2.7mu}\mkern 2.7mu}|$. The induction step is proved, as we have proved the existence of a $C'$ in ${[n] \choose n-x}$ such that $|\mathcal{A}_{C'}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muC'\mkern-2.7mu}\mkern 2.7mu}|$ when $\exists y' \in C$ $\mathcal{A}_{\{y'\}} \cap 2^{([n] \setminus C) \cup \{y'\}} = \emptyset$, and when $\forall y \in C$ $\mathcal{A}_{\{y\}} \cap 2^{([n] \setminus C) \cup \{y\}} \neq \emptyset$. As the induction step is proved, the proof of Theorem 2.2 is complete. The proof method reached its limit as it achieved a minimum of $n-x-|Z|=0$. We now consider a question which, if answered in the affirmative, would extend the result to $x \in [\lfloor \frac{n}{2} \rfloor +1]$. **Question 2.3.** If $\mathcal{F}$ is a finite family of (not necessarily distinct) sets such that $|\mathcal{F}| > |\bigcup_{F \in \mathcal{F}}F|+1$, must there exist $\mathcal{F}'\subsetneq \mathcal{F}$ with $|\mathcal{F}| - |\mathcal{F}'|=|\bigcup_{F \in \mathcal{F}'}F|+1$? ($\mathcal{F}$ need not be union-closed.) **Proposition 2.4.** *An affirmative answer to Question 2.3 would prove Conjecture 2.1 for* $x \in [\lfloor \frac{n}{2} \rfloor +1]$*.* *Proof.* The proof for $x \in [\lfloor \frac{n}{2} \rfloor +1]$ would start the same as the proof for $x \in [\lceil \frac{n}{3} \rceil +1]$ started (except for the hypothesis of the induction step being extended from $x \in [\lceil \frac{n}{3}\rceil]$ to $x \in [\lfloor \frac{n}{2} \rfloor]$). They diverge in the induction step in how they prove existence of $D$. We resume the proof at that point: $\exists D \in {[n] \choose n-x}$ in $\mathcal{A}$. *Proof.* Recall that at this point in the proof we had assumed that $\forall y \in C$ $\mathcal{A}_{\{y\}} \cap 2^{([n] \setminus C) \cup \{y\}} \neq \emptyset$. Therefore, there must be a family $\mathcal{S} \subsetneq \mathcal{A}$ with $|\mathcal{S}|=|C|$ such that $\forall c \in C$ $\exists S \in \mathcal{S} \cap \mathcal{A}_{\{c\}} \cap 2^{([n] \setminus C) \cup \{c\}}$. In this case, $\mathcal{F}$ from Question 2.3 corresponds to $\mathcal{T}=\{S \cap ([n] \setminus C) | S \in \mathcal{S}\}$. $\mathcal{T}$ satisfies the definition of $\mathcal{F}$ because $|\mathcal{T}| > |\bigcup_{T \in \mathcal{T}}T|+1$, as $|\mathcal{T}|=|C|=n+1-x \geq n+1-\lfloor \frac{n}{2} \rfloor$, and $|\bigcup_{T \in \mathcal{T}}T| \leq |[n] \setminus C|=x-1 \leq \lfloor \frac{n}{2} \rfloor - 1$. Also, $\mathcal{T}$ could have repeated members if some of the members of $\mathcal{S}$ shared exactly the same elements from $[n] \setminus C$. Further, $\mathcal{T}$ need not be union-closed. Now applying Question 2.3 to $\mathcal{T}$, there exists $\mathcal{T}' \subsetneq \mathcal{T}$ with $|\mathcal{T}| - |\mathcal{T}'|=|\bigcup_{T \in \mathcal{T}'}T|+1$. It follows that $|\bigcup_{T \in \mathcal{T}'}T|+|\mathcal{T}'|=|\mathcal{T}|-1=|C|-1=(n-x+1)-1=n-x$. Recall that every member $T$ of $\mathcal{T}$ could be written as $T= S\setminus \{c\}$ for some unique $S \in \mathcal{S}$ and $c \in C$. Let $\mathcal{S}' \subsetneq \mathcal{S}$ be the family consisting of the corresponding $S$ for every $T \in \mathcal{T}'$. Then $|\bigcup_{S \in \mathcal{S}'}S|=|\bigcup_{T \in \mathcal{T}'}T|+|\mathcal{T}'|=n-x$ as $\bigcup_{S \in \mathcal{S}'}S=(\bigcup_{T \in \mathcal{T}'}T) \cup C^*$, where $C^* \in {C \choose |\mathcal{T}'|}$. Finally, $\bigcup_{S \in \mathcal{S}'}S \in \mathcal{A}$ by $\mathcal{S}'\subsetneq \mathcal{A}$ and union-closedness. Therefore, $D=\bigcup_{S \in \mathcal{S}'}S$. After proving the existence of $D$, the two proofs converge again. The rest of the proof for $x \in [\lfloor \frac{n}{2} \rfloor +1]$ is the same as that for $x \in [\lceil \frac{n}{3} \rceil +1]$. This concludes the proof of Proposition 2.4. We note that this particular method that in the induction step tries to find $D \in {[n] \choose n-x}$ breaks down for $x > \lfloor \frac{n}{2} \rfloor$ in the induction step hypothesis. This is because the method is not able to guarantee that $([n] \setminus C) \not \subsetneq D$, and so a fundamental problem arises when $|[n] \setminus C| \geq |D|$, which occurs when $x \geq \frac{n+1}{2}$ in the hypothesis. The following theorem deals with Conjecture 2.1. **Theorem** **2.5.** $\exists B \in {[n] \choose 2} |\mathcal{A}_B|\geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB\mkern-2.7mu}\mkern 2.7mu}| \implies \exists B^* \in {[n] \choose 1} |\mathcal{A}_{B^*}|\geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB^*\mkern-2.7mu}\mkern 2.7mu}|$*.* *Proof.* Assume otherwise, i.e. $\forall B^* \in {[n] \choose 1}$ $|\mathcal{A}_{B^*}|<|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB^*\mkern-2.7mu}\mkern 2.7mu}|$. Without loss of generality, let $B= \{1,2\}$ be the doubleton from the hypothesis, so $|\mathcal{A}_{\{1,2\}}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu}|$. We have the following: $\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu}=\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu} \cup \mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}$, $\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}=\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu} \cup \mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}$, $\mathcal{A}_{\{1\}}=\mathcal{A}_{\{1,2\}} \cup \mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}$, $\mathcal{A}_{\{2\}}=\mathcal{A}_{\{1,2\}} \cup \mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}$, $|\mathcal{A}_{\{1\}}|<|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu}|$, $|\mathcal{A}_{\{2\}}|<|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}|$. Then by substitution we get: $|\mathcal{A}_{\{1,2\}} \cup \mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}|<|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu} \cup \mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}|$, $|\mathcal{A}_{\{1,2\}} \cup \mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}|<|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu} \cup \mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}|$. Since the unions on both sides of each inequality have disjoint arguments, we further have the following: $|\mathcal{A}_{\{1,2\}}| + |\mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}|<|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu}| + |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}|$, $|\mathcal{A}_{\{1,2\}}| + |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}|<|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu}| + |\mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}|$. For each inequality, we subtract $|\mathcal{A}_{\{1,2\}}|$ from the left side and $|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu}|$ from the right side. The inequalities are ensured to be still valid as $|\mathcal{A}_{\{1,2\}}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu}|$. We get the following pair of inequalities: $|\mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}|< |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}|$, $|\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}|< |\mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}|$. This is a contradiction. Alternatively, we can show the corresponding matrix inequality to be infeasible: $R_1+R_2+2R_3$ implies that $0 \leq -2$, again a contradiction. $\begin{bmatrix} -1 & -1 & 1 & 1 \\ -1 & 1 & -1 & 1 \\ 1 & 0 & 0 & -1 \end{bmatrix}$ $\begin{bmatrix} |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1,2\}\mkern-2.7mu}\mkern 2.7mu}| \\ |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{1\}\mkern-2.7mu}\mkern 2.7mu\{2\}}| \\ |\mathcal{A}_{\{1\}\mkern 2.7mu\overline{\mkern-2.7mu\{2\}\mkern-2.7mu}\mkern 2.7mu}| \\ |\mathcal{A}_{\{1,2\}}| \end{bmatrix}$ $\leq$ $\begin{bmatrix} -1 \\ -1 \\ 0 \end{bmatrix}$. We note that Theorem 2.5 holds for any finite family of sets, whether the family be union-closed or not. Another question is the following: **Question 2.6.** Are Conjecture 1.1 and Conjecture 2.1 equivalent? That is, though Conjecture 2.1 implies Conjecture 1.1, does Conjecture 1.1 also imply Conjecture 2.1? We have already verified by Theorem 2.5 that case $x=n-1$ of Conjecture 2.1 implies case $x=n$ (Conjecture 1.1) of Conjecture 2.1. For answering Question 2.6, a starting point may be an attempt at a proof of the converse, i.e. that the case $x=n$ (Conjecture 1.1) of Conjecture 2.1 implies the case $x=n-1$ of Conjecture 2.1. At the core of understanding the union-closed sets conjecture is the understanding of union-closedness among, especially finite, families of sets in general. Investigating Conjecture 2.1 in the following two ways could help in this development: .) Answering Question 2.3 in the affirmative, extending proof of Conjecture 2.1 to $x \in [\lfloor \frac{n}{2} \rfloor +1]$. .) Proving certain implications within Conjecture 2.1, especially finding $x \in [n-1] \setminus [\lceil \frac{n}{3} \rceil+1]$ such that $\exists B' \in {[n] \choose n-x}$ $|\mathcal{A}_{B'}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB'\mkern-2.7mu}\mkern 2.7mu}| \implies \exists B \in {[n] \choose n+1-x}$ $|\mathcal{A}_{B}| \geq |\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7muB\mkern-2.7mu}\mkern 2.7mu}|$. # 3. The second strengthening of Conjecture 1.1 {#the-second-strengthening-of-conjecture-1.1 .unnumbered} ## A property of union-closed families {#a-property-of-union-closed-families .unnumbered} Before presenting the second strengthening of Conjecture 1.1, we discuss a theorem of Reimer in the context of an important property of union-closed families. $log(x)$ and $ln(x)$ denote the binary logarithm and natural logarithm of a positive real number $x$, respectively. **Theorem** **3.1** (Reimer \[6\])**.** *For any finite union-closed family* $\mathcal{A}$*,* $\Sigma_{k=1}^{n} |\mathcal{A}_{\{k\}}|$ $\geq$ $\frac{m}{2}log(m)$*.* Theorem 3.1 was proved conclusively by Reimer without need of assuming Conjecture 1.1. However, we will show that Theorem 3.1 follows directly if we assume Conjecture 1.1, in order to highlight a structural property of union-closed families of sets. **Proposition 3.2.** *Conjecture 1.1* $\implies$ *Theorem 3.1.* *Proof.* We show Theorem 3.1, assuming Conjecture 1.1 to be true. We use induction on $n$. *Base Case ($n=1$):* The only two finite union-closed families on the base set $[n]=[1]$ are $\{\emptyset, \{1\}\}$ and $\{\{1\}\}$. In the first case, $\Sigma_{k=1}^{n} |\mathcal{A}_{\{k\}}|=1 \geq \frac{m}{2}log(m)=log(2)=1$. In the second case, $\Sigma_{k=1}^{n} |\mathcal{A}_{\{k\}}|=1 \geq \frac{m}{2}log(m)=\frac{1}{2}log(1)=0$. *Induction Step:* We show that: (For every finite union-closed family $\mathcal{A}^*$ on a base set $N^*$, $\Sigma_{k \in N^*} |{\mathcal{A}^*}_{\{k\}}| \geq \frac{m^*}{2}log(m^*)$, where $N^* \subsetneq [n+1]$ and $m^*=|\mathcal{A}^*|$) $\implies$ (For every finite union-closed family $\mathcal{A}$ on the base set $[n+1]$, $\Sigma_{k=1}^{n+1} |\mathcal{A}_{\{k\}}| \geq \frac{m}{2}log(m)$). We consider an arbitrary finite union-closed family $\mathcal{A}$ with base set $[n+1]$. Now consider $\mathcal{A}_{\{n+1\}}$ and $\mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{n+1\}\mkern-2.7mu}\mkern 2.7mu}$, where, without loss of generality, $n+1 \in \texttt{arg\_high}(\mathcal{A})$. Then $c =\frac{|\mathcal{A}_{\{n+1\}}|}{|\mathcal{A}|} \in [\frac{1}{2}, 1]$, as we are assuming Conjecture 1.1. Let ${\mathcal{A}_{\{n+1\}}}'=\{A \setminus \{n+1\} |A \in \mathcal{A}_{\{n+1\}}\}$. It follows that $\bigcup_{A \in {{\mathcal{A}_{\{n+1\}}}}'}A=[n]$, and $\bigcup_{A \in \mathcal{A}_{\mkern 2.7mu\overline{\mkern-2.7mu\{n+1\}\mkern-2.7mu}\mkern 2.7mu}} A = N$, where $N \subsetneq [n+1]$ ($c=1 \implies N=\emptyset$). We establish a lower bound for $\Sigma_{k=1}^{n+1} |\mathcal{A}_{\{k\}}|$ by solving the following problem: $\texttt{minimize}_{c \in [\frac{1}{2},1)}\{ I(c) \}$, where $I(c)=cm + \frac{cm}{2}(log(cm))+\frac{(1-c)m}{2}(log((1-c)m))$. In $I(c)$, the first term $cm$ comes from the $cm=|\mathcal{A}_{\{n+1\}}|$ number of times element $n+1$ was removed from $\mathcal{A}_{\{n+1\}}$ to make ${\mathcal{A}_{\{n+1\}}}'$. The second term $\frac{cm}{2}(log(cm))$ is the lower bound of $\Sigma_{k=1}^{n}|{{\mathcal{A}_{{\{n+1\}}}}'_{\{k\}}}|$. The final term $\frac{(1-c)m}{2}(log((1-c)m))$ is the lower bound of $\Sigma_{k \in N} |\mathcal{A}_{{\mkern 2.7mu\overline{\mkern-2.7mu\{n+1\}\mkern-2.7mu}\mkern 2.7mu}_{\{k\}}}|$. We need not include the case $c=1$, as if $c=1$ then we could remove every element in $\texttt{arg\_high}(\mathcal{A})$ from each member of $\mathcal{A}$ to achieve a new finite union-closed family $\mathcal{A}'$ with $c<1$, $|\mathcal{A}'|=|\mathcal{A}|$, and $\bigcup_{A \in \mathcal{A}'}A \subsetneq [n+1]$. Then $\Sigma_{i \in \bigcup_{A \in \mathcal{A}'}A}|{\mathcal{A}'}_{\{i\}}| < \Sigma_{i=1}^{n+1} |\mathcal{A}_{\{i\}}|$, so $\Sigma_{i=1}^{n+1} |\mathcal{A}_{\{i\}}|$ would not be a lower bound for finite union-closed families. Therefore, $I(c^*) \leq \Sigma_{k=1}^{n+1} |\mathcal{A}_{\{k\}}|$, where $I(c^*)$ is the minimum of $I(c)$ for $c \in [\frac{1}{2},1)$. For $c \in [\frac{1}{2}, 1)$, $\frac{dI}{dc}=m+\frac{m}{2}(log(c)-log(1-c))>0$, as $\frac{dI}{dc} \Bigr|_{c=\frac{1}{2}}=m+\frac{m}{2}(log(\frac{1}{2})-log(1-\frac{1}{2}))=m>0$ and for $c \in [\frac{1}{2},1)$, $\frac{d^2I}{dc^2}=\frac{m}{2ln(2)}(\frac{1}{c(1-c)})>0$. Thus, the global minimum of $I(c)$ for $c \in [\frac{1}{2},1)$ is achieved uniquely at $c=\frac{1}{2}$. Plugging this minimizer into $I(c)$ gives $I(\frac{1}{2})=(\frac{1}{2})m + \frac{(\frac{1}{2})m}{2}(log((\frac{1}{2})m))+\frac{(1-\frac{1}{2})m}{2}(log((1-\frac{1}{2})m))=\frac{m}{2}log(m)$, which is exactly the lower bound we intended to prove for $\Sigma_{k=1}^{n+1} |\mathcal{A}_{\{k\}}|$. This proves Proposition 3.2, as we used Conjecture 1.1 to show Theorem 3.1. In general, if the goal is to prove that $U \subseteq Q_1 \implies U \subseteq Q_2$, where $U$ is the set of all finite union-closed families and $Q_1$, $Q_2$ are the sets of all families having some respective properties, a proof technique is to show that $\forall \mathcal{A} \in U$ ($\mathcal{A} \in Q_1 \implies \mathcal{A} \in Q_2)$. Another proof technique that utilizes more information considers that *all* finite union-closed families must be in $Q_1$ and uses the fact that an arbitrary union-closed family $\mathcal{A} \in U$ is rich in substructures that are union-closed families themselves, showing that $\forall \mathcal{A} \in U$ ($U \subseteq Q_1 \implies \mathcal{A} \in Q_2)$. Thus the hypothesis (membership in $Q_1$) can be applied to all union-closed families found in the structure of $\mathcal{A}$, as well as to $\mathcal{A}$ itself, to prove that $\mathcal{A}$ is in $Q_2$. This was the technique used through induction on $n$ to prove Proposition 3.2. In that case, $Q_1$ was the set of families that have an element of their base set in one half or more of their member sets, and $Q_2$ was the set of families that have a sum of frequencies of the elements in their base set greater than or equal to $\frac{m}{2}log(m)$, where $m$ is their number of member sets. ## Utilizing the nested structure of union-closed families {#utilizing-the-nested-structure-of-union-closed-families .unnumbered} As discussed above, the structure of a union-closed family of sets is full of other union-closed families. This is the property of union-closed families we primarily consider for posing the next strengthening of Conjecture 1.1. First, we outline a framework for considering Conjecture 1.1 and union-closed families in general. For a finite union-closed family $\mathcal{A}$, we introduce a partition $\mathcal{P}$ of $\mathcal{A}$, where every member of $\mathcal{P}$ is itself a union-closed family. $\mathcal{P}$ is constructed by the following algorithm: $\mathcal{P} \leftarrow \emptyset$ $\mathcal{A}' \leftarrow \mathcal{A}$ $i \leftarrow 0$ $i \leftarrow i+1$ $P(i) \leftarrow j$ such that $j \in \texttt{arg\_high}(\mathcal{A}')$ $\mathcal{P}_i \leftarrow \mathcal{S}$ such that $S \in \mathcal{S} \iff (S \in \mathcal{A}') \land (P(i) \in S)$ $\mathcal{A}' \leftarrow \mathcal{A}' \setminus \mathcal{P}_i$ $\mathcal{P} \leftarrow \mathcal{P} \cup \{\mathcal{P}_i\}$ $i \leftarrow i+1$ $\mathcal{P}_i \leftarrow \{ \emptyset \}$ $\mathcal{A}' \leftarrow \emptyset$ $\mathcal{P} \leftarrow \mathcal{P} \cup \{\mathcal{P}_i\}$ $w \leftarrow i$ We note that $\mathcal{P}$ and $P$ need not be unique. At iteration $i$, if $|\texttt{arg\_high}(\mathcal{A}')|>1$, then any element in $\texttt{arg\_high}(\mathcal{A}')$ can be chosen as $P(i)$. **Theorem 3.3.** $\forall S \subseteq [w]$ $\bigcup_{k \in S} \mathcal{P}_k$ *is union-closed.* *Proof.* $\forall X,Y \in \bigcup_{k \in S} \mathcal{P}_k$ $(X \in \mathcal{P}_i \land Y \in \mathcal{P}_l)$, where $1 \leq i \leq l \leq w$, without loss of generality. $X \cup Y \in \mathcal{A}$ by union-closedness. Also, $P(i) \in X \implies P(i) \in X \cup Y$, and $\forall j \in [i-1] ((P(j) \not \in x \land P(j) \not \in Y) \implies P(j) \not \in X \cup Y)$. Then $(X \cup Y \in \mathcal{A} \land P(i) \in X \cup Y \land \forall j \in [i-1] P(j) \not \in X \cup Y) \implies X \cup Y \in \mathcal{P}_i \subseteq \bigcup_{k \in S} \mathcal{P}_k$. Therefore, $\bigcup_{k \in S} \mathcal{P}_k$ is union-closed, and the proof of Theorem 3.3 is complete. We restate Conjecture 1.1 in terms of $\mathcal{P}$, where $\mathcal{P}_i \coloneqq \emptyset$ for $i>w$. **Conjecture 1.1** (Second Rephrasing)**.** *For any finite union-closed family* $\mathcal{A}$*,* $|\mathcal{P}_1| \geq \Sigma_{i=2}^{w+2} |\mathcal{P}_i|$*.* **Corollary 3.4.** *By Theorem 3.3, Conjecture 1.1* $\implies$ $\forall S \subseteq [w] \setminus [k]$ $|\mathcal{P}_j| \geq \Sigma_{i \in S }|\mathcal{P}_i|$*, whenever* $1 \leq j \leq k < w$*.* This gives a necessary structure for the sizes of the members of $\mathcal{P}$, assuming Conjecture 1.1 to be true. With the construction of $\mathcal{P}$, we now present the second strengthening of Conjecture 1.1. **Conjecture 3.5.** *For any finite union-closed family* $\mathcal{A}$ *with* $\emptyset \not \in \mathcal{A}$, $|\mathcal{P}_i| \geq 2|\mathcal{P}_j|$ *whenever* $1 \leq i < j \leq w$*.* Conjecture 3.5 is essentially further constraint on the structure that Corollary 3.4 indicates of $\mathcal{P}$. By Conjecture 3.5, $\forall i \in [w-1]$ $|\mathcal{P}_i| \geq 2 |\mathcal{P}_{i+1}|$ when $\emptyset \not \in \mathcal{A}$. In order to illustrate the difference between Conjecture 1.1 and Conjecture 3.5, provided in Table 3.1 are some examples of integer sequences $\{|\mathcal{P}_i|\}_{i=1}^{5}$ and their permissibility according to the conjectures. **$\{|\mathcal{P}_1|,|\mathcal{P}_2|,|\mathcal{P}_3|,|\mathcal{P}_4|,|\mathcal{P}_5|\}$** **Conjecture 1.1** **Conjecture 3.5** ------------------------------------------------------------------------------------------- -------------------- -------------------- $\{100, 50, 25, 12, 5\}$ Yes Yes $\{100, 40, 10, 5, 2\}$ Yes Yes $\{100, 40, 18, 10, 7\}$ Yes No $\{60, 30, 20, 10, 5\}$ No No : **Table 3.1:** Integer Sequences and their Permissibility. We note that the last sequence, in addition to failing Conjecture 1.1 because $|\mathcal{P}_1|=60 < |\mathcal{P}_2|+|\mathcal{P}_3|+|\mathcal{P}_4|+|\mathcal{P}_5|=65$, violates the conclusion of Corollary 3.4 because $|\mathcal{P}_2|=30 < |\mathcal{P}_3|+|\mathcal{P}_4|+|\mathcal{P}_5|=35$. **Proposition 3.6.** *Conjecture 3.5* $\implies$ *Conjecture 1.1.* *Proof.* If $\texttt{high}(\mathcal{A})=|\mathcal{A}|$, then $\mathcal{A}$ satisfies Conjecture 1.1. For the remainder of the proof of Proposition 3.6, we consider the case $\texttt{high}(\mathcal{A})<|\mathcal{A}|$. We will show that if $|\mathcal{P}_i| \geq 2|\mathcal{P}_j|$ whenever $1 \leq i < j \leq w$ and $\emptyset \not \in \mathcal{A}$, then both $\mathcal{A}$ and $\mathcal{A} \cup \{\emptyset \}$ have $|\mathcal{P}_1| \geq \Sigma_{i=2}^w |\mathcal{P}_i|$, respectively. Given Conjecture 3.5, we consider the following upper bound $\mu(\mathcal{P})$ for $\Sigma_{i=2}^w |\mathcal{P}_i|$: $$\mu(\mathcal{P})=\overbrace{\left \lfloor{\frac{1}{2}|\mathcal{P}_1|}\right \rfloor + \left \lfloor{\frac{1}{2}\left \lfloor{\frac{1}{2}|\mathcal{P}_1|}\right \rfloor }\right \rfloor + \left \lfloor{\frac{1}{2}\left \lfloor{\frac{1}{2}\left \lfloor{\frac{1}{2}|\mathcal{P}_1|}\right \rfloor}\right \rfloor }\right \rfloor+\cdots + \left \lfloor{\frac{1}{2}\left \lfloor{\frac{1}{2}\left \lfloor{\frac{1}{2}\cdots \left \lfloor{\frac{1}{2}|\mathcal{P}_1|}\right \rfloor \cdots}\right \rfloor }\right \rfloor }\right \rfloor}^{\text{w-1 terms}}.$$ $$\sum_{i=2}^w |\mathcal{P}_i| \leq \mu(\mathcal{P}) \leq |\mathcal{P}_1| \sum_{i=1}^{w-1} \frac{1}{2^i} < |\mathcal{P}_1| \sum_{i=1}^{\infty} \frac{1}{2^i}=|\mathcal{P}_1|.$$ Therefore, $|\mathcal{P}_1| > \sum_{i=2}^w |\mathcal{P}_i|$. Since $|\mathcal{P}_1|$ and $\sum_{i=2}^w |\mathcal{P}_i|$ are both integers, $\{\emptyset\}$ is a singleton, and $|\mathcal{P}_1| > \sum_{i=2}^w |\mathcal{P}_i|$, it follows that $|\mathcal{P}_1| \geq \sum_{i=2}^w |\mathcal{P}_i|$ for $\mathcal{A}\cup \{\emptyset\}$. The proof of Proposition 3.6 is complete. # 4. The third strengthening of Conjecture 1.1 {#the-third-strengthening-of-conjecture-1.1 .unnumbered} The final strengthening of Conjecture 1.1 that we consider is closely related to power sets. **Conjecture** **4.1(a)** (Poonen \[5\])**.** *For a separating finite union-closed family of sets* $\mathcal{A}$, $\texttt{high}(\mathcal{A})>\frac{|\mathcal{A}|}{2}$ *if* $\mathcal{A}$ *is not a power set.* Conjecture 4.1(a) clearly implies Conjecture 1.1 as all power sets satisfy Conjecture 1.1, and Conjecture 4.1(a) explicitly states that no other $\mathcal{A}$ have $\texttt{high}(\mathcal{A})$ less than $\frac{|\mathcal{A}|}{2}$. This need not be explicitly stated, however, as the following Conjecture 4.1(b) implies Conjecture 1.1 without doing so: **Conjecture** **4.1(b).** *For a separating finite union-closed family of sets* $\mathcal{A}$, $\texttt{high}(\mathcal{A})=\frac{|\mathcal{A}|}{2}$ *if and only if* $\mathcal{A}$ *is a power set.* **Theorem** **4.2.** *Conjecture 4.1(b)* $\implies$ *Conjecture 1.1.* *Proof.* We first note that if a counterexample to Conjecture 1.1 exists, then a separating counterexample to Conjecture 1.1 exists; if a counterexample is not separating, then we can make a separating counterexample on a smaller base set by representing as a single element every grouping of two or more elements (from the original base set) that are contained within exactly the same members of the original family. Now assume that Conjecture 1.1 is false, and $\mathcal{A}$ is a separating counterexample. We form a new union-closed family $\mathcal{A}'= \mathcal{A} \bigsqcup \{\{z\}, \emptyset\}$, where $\mathcal{X} \bigsqcup \mathcal{Y}=\{X \cup Y | X \in \mathcal{X} \land Y \in \mathcal{Y} \}$, and $z \not \in [n]$ ($z$ extends the base set of $\mathcal{A}$ by one element). **Lemma 4.3.** $\mathcal{X}$ and $\mathcal{Y}$ *are union-closed* $\implies$ $\mathcal{X} \bigsqcup \mathcal{Y}$ *is union-closed.* *Proof.* $\forall A,B \in \mathcal{X} \bigsqcup \mathcal{Y}$, there are $X$, $X'$ in $\mathcal{X}$ and $Y$, $Y'$ in $\mathcal{Y}$ such that $A = X \cup Y$ and $B = X' \cup Y'$. We must show that $A \cup B = (X \cup Y) \cup (X' \cup Y')$ is in $\mathcal{X} \bigsqcup \mathcal{Y}$. Because $\cup$ is commutative and associative, $(X \cup Y) \cup (X' \cup Y') = (X \cup X') \cup (Y \cup Y')$. $X \cup X' \in \mathcal{X}$ and $Y \cup Y' \in \mathcal{Y}$ because $\mathcal{X}$ and $\mathcal{Y}$ are union-closed. Then $(X \cup X') \cup (Y \cup Y')$ is by definition in $\mathcal{X} \bigsqcup \mathcal{Y}$, and the proof of Lemma 4.3 is complete. By Lemma 4.3, $\mathcal{A}'=\mathcal{A} \bigsqcup \{\{z\}, \emptyset\}$ is union-closed because $\mathcal{A}$ and $\{\{z\}, \emptyset\}$ are both union-closed. We have the following: $\texttt{high}(\mathcal{A}')=\frac{|\mathcal{A}'|}{2}$. *Proof.* The ratio of frequency to total number of members remains the same for every element in $(\bigcup_{A \in \mathcal{A}'} A) \setminus \{z\}$ as it was in $\mathcal{A}$. This is because though the total number of members doubled ($A \cup \{z\}$ was added for every $A \in \mathcal{A}$), the frequency of each element also doubled by the same principle. Thus all elements in $\bigcup_{A \in \mathcal{A}'} A$ have frequency less than $\frac{|\mathcal{A}'|}{2}$, except for $z$. The frequency of $z$ is $\frac{|\mathcal{A}'|}{2}$, as $\mathcal{A} \bigsqcup \{\{z\}, \emptyset\}$ establishes a bijection where each member of $\mathcal{A}$ (none contain $z$) maps to its union with $\{z\}$. Thus, $\texttt{high}(\mathcal{A}')=\frac{|\mathcal{A}'|}{2}$. $\mathcal{A}'$ is not a power set, as all elements besides $z$ of $\bigcup_{A \in \mathcal{A}'} A$ have frequency less than $\frac{|\mathcal{A}'|}{2}$. Also, $\mathcal{A}'$ is separating, as $\mathcal{A}$ is separating and $\forall k \in [n]$ $\forall A \in \mathcal{A}_{\{k\}}$ ($A \in \mathcal{A}'$ $\land$ $z \not \in A$). Hence, $\mathcal{A}'$ is a counterexample to Conjecture 4.1(b), and we have that:\ $\neg$ (Conjecture 1.1) $\implies$ $\neg$ (Conjecture 4.1(b)). Therefore, Conjecture 4.1(b) $\implies$ Conjecture 1.1. A motivation for decoupling Conjecture 4.1(b) from Conjecture 4.1(a) is to emphasize possible equivalences among conjectures. Conjecture 4.1(b) implies Conjecture 1.1, but it remains to be shown whether Conjecture 1.1 also implies Conjecture 4.1(b). 9 I. Bošnjak and P. Marković, The 11-element case of Frankl's conjecture, *Electron. J. Combin.* **15** (2008), R88. H. Bruhn, P. Charbit, O. Schaudt and J.A. Telle, The graph formulation of the union-closed sets conjecture, *European J. Combin.* **43** (2015), 210-219. H. Bruhn and O. Schaudt, The journey of the union-closed sets conjecture, *Graphs Combin.* **31** (2015), 2043-2074. J. Gilmer, A constant lower bound for the union-closed sets conjecture, preprint (2022), arXiv:2211.09055. B. Poonen, Union-closed families, *J. Combin. Theory Ser. A* **59** (1992), 253-268. D. Reimer, An average set size theorem, *Combin. Probab. Comput.* **12** (2003), 89-93. J. Reinhold, Frankl's conjecture is true for lower semimodular lattices, *Graphs Combin.* **16** (2000), 115-116. I. Roberts and J. Simpson, A note on the union-closed sets conjecture, *Australas. J. Combin.* **47** (2010), 265-267. W. Sawin, An improved lower bound for the union-closed set conjecture, preprint (2022), arXiv:2211.11504.

arxiv_math

--- author: - Ruikai Chen^1,2^ - Sihem Mesnager^1,2,3^ bibliography: - references.bib date: | ^1^Department of Mathematics, University of Paris VIII, F-93526 Saint-Denis\ ^2^Laboratory Analysis, Geometry and Applications, LAGA, University Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France\ ^3^Telecom Paris, Polytechnic institute of Paris, 91120 Palaiseau, France\ Emails: <chen.rk@outlook.com><smesnager@univ-paris8.fr> title: On a Class of Permutation Rational Functions Involving Trace Maps --- **Abstract.** Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates.\ **Keywords.** Finite Field, Extension Field, Polynomial, Permutation, Rational Function.\ **Mathematics Subject Classification.** 11T06, 14H05. # Introduction Let $q$ be a prime power and $\mathbb F_q$ denote the finite field of order $q$. A permutation polynomial over $\mathbb F_q$ is a polynomial that acts as a permutation on $\mathbb F_q$ in the usual way (see [@lidl1997 Chapter 7] or [@mullen2013 Chapter II.8]). Permutation polynomials over finite fields are exciting objects in number theory, studied widely in the last decades and employed extensively in coding theory, cryptography, combinatorics, etc. Finding permutation polynomials of simple forms and investigating their properties is still appealing and challenging. A permutation polynomial can appear as a rational function, if the rational function defines a map from a finite field to itself. Consider some extension $\mathbb F_{q^n}$ ($n>1$) of $\mathbb F_q$ and let $\mathop{\mathrm{Tr}}$ be the trace map from $\mathbb F_{q^n}$ to $\mathbb F_q$ given by $$\mathop{\mathrm{Tr}}(x)=\sum_{i=0}^{n-1}x^{q^i}.$$ Permutation polynomials of the form $$L(x)+(x^q-x+b)^s$$ for some $q$-linear polynomial $L$ over $\mathbb F_{q^n}$ (a polynomial that induces a linear endomorphism of $\mathbb F_{q^n}$ over $\mathbb F_q$), an integer $s$ and $b\in\mathbb F_{q^n}$ have drawn a lot of attention in recent years. See [@helleseth2003new; @zha2012two; @li2013further; @yuan2014further] for example. In particular, some necessary and sufficient conditions are given in [@hou2020type] and [@bartoli2021conjecture] for $$x+(x^q-x+b)^{-1},$$ with $\mathop{\mathrm{Tr}}(b)\ne0$ to be a permutation rational function of $\mathbb F_{q^n}$; for the special case that $q$ is odd and $n=2$, it has been shown that it is a permutation rational function of $\mathbb F_{q^2}$ if and only if $\mathop{\mathrm{Tr}}(b)^2=1$. Furthermore, we have a related polynomial $$L(x)+(\mathop{\mathrm{Tr}}(x)+b)^s.$$ Permutation polynomials of this form have been studied in [@zeng2015permutation], where $q$ is even and $s$ satisfies certain conditions. In this paper, we study permutation rational functions of the form $$L(x)+\frac c{\mathop{\mathrm{Tr}}(x)+b},$$ where $b\in\mathbb F_{q^n}\setminus\mathbb F_q$, $c\in\mathbb F_{q^n}^*$, and $L$ is a $q$-linear polynomial over $\mathbb F_{q^n}$. We shall characterize those permutation rational functions by factoring certain polynomials over $\overline{\mathbb F_q}$ in two indeterminates $X$ and $Y$, where $\overline{\mathbb F_q}$ denotes the algebraic closure of $\mathbb F_q$. When considering $\mathbb F_{q^n}(X,Y)/\mathbb F_q(X,Y)$, let $\mathop{\mathrm{Tr}}$ and $\mathop{\mathrm{N}}$ be defined as $$\mathop{\mathrm{Tr}}(g)=\sum_{i=0}^{n-1}\sigma_q^i(g),\quad\text{and}\quad\mathop{\mathrm{N}}(g)=\prod_{i=0}^{n-1}\sigma_q^i(g)$$ for $g\in\mathbb F_{q^n}(X,Y)$, where $\sigma_q$ is the automorphism that maps each coefficient of polynomials to its $q$-th power and leaves $X$ and $Y$ fixed. The notation will be used throughout, as well as the following lemma. **Lemma 1** ([@aubry1996weil Lemma 3.2]). *For an absolutely irreducible affine plane curve of degree $d$ over $\mathbb F_q$ with $d>1$, if $$q-(d-1)(d-2)\sqrt q-2d+1>0,$$ then it has a rational point $(x_0,y_0)$ over $\mathbb F_q$ with $x_0\ne y_0$.* For the rational function $L(x)+\frac c{\mathop{\mathrm{Tr}}(x)+b}$, we divide the remainder of this paper into two parts. In Section [2](#s1){reference-type="ref" reference="s1"}, we study the case that $L$ does not permute $\mathbb F_{q^n}$, and it turns out that such permutation rational functions tend not to exist. In Section [3](#s2){reference-type="ref" reference="s2"}, permutation rational functions of the form $x+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ are characterized, especially for small values of $n$. # The Case That $L$ Does Not Permute $\mathbb F_{q^n}$ {#s1} Let $r$ be the rank of $L$ as a linear endomorphism of $\mathbb F_{q^n}/\mathbb F_q$, and assume $r<n$. We have $$L(x)=\sum_{i=1}^r\alpha_i\mathop{\mathrm{Tr}}(\beta_ix)$$ for some $\alpha_i,\beta_i\in\mathbb F_{q^n}$ with both $\alpha_1,\dots,\alpha_r$ and $\beta_1,\dots,\beta_r$ linearly independent over $\mathbb F_q$. Assume $L(x)+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ is a permutation rational function of $\mathbb F_{q^n}$. If the rank of $1,\beta_1,\dots,\beta_r$ is less than $n$, then there exist distinct elements $x_1,y_1\in\mathbb F_{q^n}$ such that $\mathop{\mathrm{Tr}}(x_1)=\mathop{\mathrm{Tr}}(y_1)$ and $L(x_1)=L(y_1)$. This implies that $r=n-1$, and for all $x_0,y_0\in\mathbb F_q$ and $\theta\in\mathbb F_{q^n}$ in the image of $L$, there exist $x_1,y_1\in\mathbb F_{q^n}$ such that $\mathop{\mathrm{Tr}}(x_1)=x_0$, $\mathop{\mathrm{Tr}}(y_1)=y_0$ and $L(x_1-y_1)=\theta$. Therefore, if $\frac{c(x_0-y_0)}{(x_0+b)(y_0+b)}$ belongs to the image of $L$ for some $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$, then $\mathop{\mathrm{Tr}}(x_1)=x_0$, $\mathop{\mathrm{Tr}}(y_1)=y_0$ and $$L(x_1)+\frac c{\mathop{\mathrm{Tr}}(x_1)+b}-L(y_1)-\frac c{\mathop{\mathrm{Tr}}(y_1)+b}=L(x_1-y_1)+\frac{c(y_0-x_0)}{(x_0+b)(y_0+b)}=0,$$ for some $x_1,y_1\in\mathbb F_{q^n}$ with $x_1\ne y_1$, a contradiction. On the other hand, there exists an element $\beta\in\mathbb F_{q^n}^*$ such that $\mathop{\mathrm{Tr}}(\beta\alpha_i)=0$ for $1\le i<n$, and thus $\theta$ is in the image of $L$ if and only if $\mathop{\mathrm{Tr}}(\beta\theta)=0$ for $\theta\in\mathbb F_{q^n}$. Consequently, we have $$\mathop{\mathrm{Tr}}\left(\frac{\beta c}{(x_0+b)(y_0+b)}\right)\ne0$$ for all $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$. Since $c\in\mathbb F_{q^n}^*$ is arbitrary, we may let $\beta=1$ and $L(x)=x^q-x$. However, such permutation rational functions do not exist in some cases. Before showing that, we present an auxiliary result. **Lemma 2**. *For $n=3$ and $b\in\mathbb F_{q^3}\setminus\mathbb F_q$, $(1,b^q+b,b^{q+1})$ is a basis of $\mathbb F_{q^3}/\mathbb F_q$.* *Proof.* In view of [@lidl1997 Corollary 2.38], the result follows from the fact that $$\begin{split}\begin{vmatrix}1&b^q+b&b^{q+1}\\1&b^{q^2}+b^q&b^{q^2+q}\\1&b^{q^2}+b&b^{q^2+1}\end{vmatrix}&=\mathop{\mathrm{Tr}}\big(\big(b^{q^2}+b^q\big)b^{q^2+1}-b^{q^2+q}\big(b^{q^2}+b\big)\big)\\&=\mathop{\mathrm{N}}(b)\mathop{\mathrm{Tr}}\big(b^{q-1}-b^{q^2-1}\big)\\&\ne0.\end{split}$$ To see the inequality, note first that $b^{q-1}\ne b^{1-q}$. If $b^{q-1}\in\mathbb F_q$, then $(b^{q-1})^3=\mathop{\mathrm{N}}(b^{q-1})=1$, which means $q\equiv1\pmod3$ with $$\mathop{\mathrm{Tr}}\big(b^{q-1}-b^{q^2-1}\big)=3(b^{q-1}-b^{1-q})\ne0;$$ if $b^{q-1}\notin\mathbb F_q$, then the minimal polynomial of $b^{q-1}$ over $\mathbb F_q$ is $$x^3-\mathop{\mathrm{Tr}}(b^{q-1})x^2+\mathop{\mathrm{Tr}}\big((b^{q-1})^{q+1}\big)x-\mathop{\mathrm{N}}(b^{q-1})=x^3-\mathop{\mathrm{Tr}}(b^{q-1})x^2+\mathop{\mathrm{Tr}}\big(b^{q^2-1}\big)x-1,$$ whereas the assumption $\mathop{\mathrm{Tr}}\big(b^{q-1}-b^{q^2-1}\big)=0$ yields a root $1$ of it. ◻ **Proposition 3**. *For $n=2$ or $n=3$ and sufficiently large $q$, $L(x)+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ is not a permutation rational function of $\mathbb F_{q^n}$ if $L$ does not permute $\mathbb F_{q^n}$.* *Proof.* It suffices to show that $$\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)=0$$ for some $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$. If $n=2$, then $$\mathop{\mathrm{Tr}}(c^q(x_0+b)(y_0+b))=\mathop{\mathrm{Tr}}(c)x_0y_0+\mathop{\mathrm{Tr}}(bc^q)(x_0+y_0)+\mathop{\mathrm{Tr}}(b^2c^q),$$ where one of $\mathop{\mathrm{Tr}}(c)$ and $\mathop{\mathrm{Tr}}(bc^q)$ is nonzero since $b\notin\mathbb F_q$ and $c\ne0$, and it is easy to see that for $q>3$, $$\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)=\frac{\mathop{\mathrm{Tr}}(c^q(x_0+b)(y_0+b))}{\mathop{\mathrm{N}}((x_0+b)(y_0+b))}=0$$ for some $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$. Consider the case $n=3$ with the polynomial $$f(X,Y)=\mathop{\mathrm{Tr}}\big(c^{q^2}(X+b)(X+b^q)(Y+b)(Y+b^q)\big)$$ over $\mathbb F_q$. Note that the coefficient of $Y^2$ in $f(X,Y)$ over $\mathbb F_q(X)$ is $\mathop{\mathrm{Tr}}\big(c^{q^2}(X+b)(X+b^q)\big)$, which is nonzero because $\mathop{\mathrm{Tr}}(c)=\mathop{\mathrm{Tr}}\big((b^q+b)c^{q^2}\big)=\mathop{\mathrm{Tr}}\big(b^{q+1}c^{q^2}\big)=0$ only if $c=0$ by the above lemma, so $f(X,Y)$ has degree at least $2$. If $f(X,Y)$ is absolutely irreducible, then by Lemma [Lemma 1](#bound){reference-type="ref" reference="bound"}, $$\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)=\frac{f(x_0,y_0)}{\mathop{\mathrm{N}}((x_0+b)(y_0+b))}=0$$ for some $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$, when $q-6\sqrt q-7>0$. Assume $f(X,Y)$ is reducible over $\overline{\mathbb F_q}$. If $g(X)$ divides $f(X,Y)$ for some monic polynomial $g$ of positive degree over $\overline{\mathbb F_q}$, then so does $g(Y)$ by symmetry; otherwise $f(X,Y)$ is reducible over $\overline{\mathbb F_q}(X)$. In either case, let $\xi\in\overline{\mathbb F_q}(X)$ be a root of $f(X,Y)$ as a polynomial over $\mathbb F_q(X)$; that is, $$\sum_{i=0}^2c^{q^{i+2}}\big(X+b^{q^i}\big)\big(X+b^{q^{i+1}}\big)\big(\xi+b^{q^i}\big)\big(\xi+b^{q^{i+1}}\big)=0.$$ Let $v_i$ ($i\in\{0,1,2\}$) be the valuation of $\overline{\mathbb F_q}(X)$ corresponding to $X+b^{q^i}$. Clearly it can not occur that $v_0(\xi+b)=v_0(\xi+b^q)=v_0\big(\xi+b^{q^2}\big)=0$. If $v_0(\xi+b)\ne0$, then either $v_0(\xi+b)>v_0(\xi+b^q)=v_0\big(\xi+b^{q^2}\big)=0$ or $v_0(\xi+b)=v_0(\xi+b^q)=v_0\big(\xi+b^{q^2}\big)<0$, which is false again. Then $v_0\big(\xi+b^{q^j}\big)\ne0$ with either $j=1$ or $j=2$. Now the case that $g(Y)$ divides $f(X,Y)$ has been excluded, and by factoring $f(X,Y)$ one finds $$\xi+b^{q^j}=\frac{\alpha_0X+\beta_0}{\alpha_1X+\beta_1}$$ for some $\alpha_0,\beta_0,\alpha_1,\beta_1\in\overline{\mathbb F_q}$, where $\alpha_1=1$ or $\alpha_1=0$ with $\beta_1=1$. If $v_0\big(\xi+b^{q^j}\big)<0$, then $\alpha_1=1$ and $\beta_1=b$. Otherwise, we have $$\xi+b^{q^j}=\frac{\alpha_0X+\alpha_0b}{\alpha_1X+\beta_1},$$ with $$\xi+b^{q^{j+1}}=\frac{\alpha_0X+\alpha_1\big(b^{q^{j+1}}-b^{q^j}\big)X+\alpha_0b+\beta_1\big(b^{q^{j+1}}-b^{q^j}\big)}{\alpha_1X+\beta_1},$$ and $$\xi+b^{q^{j+2}}=\frac{\alpha_0X+\alpha_1\big(b^{q^{j+2}}-b^{q^j}\big)X+\alpha_0b+\beta_1\big(b^{q^{j+2}}-b^{q^j}\big)}{\alpha_1X+\beta_1}.$$ In this case, investigating $v_1$ and $v_2$ likewise gives $v_1\big(\xi+b^{q^{j+1}}\big)=v_2\big(\xi+b^{q^{j+2}}\big)=1$, and hence, $$\alpha_0b^q+\alpha_1\big(b^{q^{j+1}}-b^{q^j}\big)b^q=\alpha_0b+\beta_1\big(b^{q^{j+1}}-b^{q^j}\big),$$ $$\alpha_0b^{q^2}+\alpha_1\big(b^{q^{j+2}}-b^{q^j}\big)b^{q^2}=\alpha_0b+\beta_1\big(b^{q^{j+2}}-b^{q^j}\big);$$ that is, $$\begin{pmatrix}b^q-b&b^{q^j}-b^{q^{j+1}}\\b^{q^2}-b&b^{q^j}-b^{q^{j+2}}\end{pmatrix}\begin{pmatrix}\alpha_0\\\beta_1\end{pmatrix}=\begin{pmatrix}\alpha_1\big(b^{q^j}-b^{q^{j+1}}\big)b^q\\\alpha_1\big(b^{q^j}-b^{q^{j+2}})b^{q^2}\end{pmatrix},$$ where $$\begin{vmatrix}b^{q^j}-b^{q^{j+1}}&\alpha_1\big(b^{q^j}-b^{q^{j+1}}\big)b^q\\b^{q^j}-b^{q^{j+2}}&\alpha_1\big(b^{q^j}-b^{q^{j+2}})b^{q^2}\end{vmatrix}\ne0$$ if $\alpha_1=1$, and $$\begin{pmatrix}b^q-b&b^{q^j}-b^{q^{j+1}}\\b^{q^2}-b&b^{q^j}-b^{q^{j+2}}\end{pmatrix}\begin{pmatrix}\alpha_0\\1\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$$ if $\alpha_1=0$. This means if $\alpha_0$ and $\beta_1$ satisfy the linear equations, then they must belong to $\mathbb F_{q^3}$. Therefore, in all cases, either $\xi\in\mathbb F_q(X)$, indicating the existence of $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$ such that $f(x_0,y_0)=0$ when $q>3$, or $\xi$, $\sigma_q(\xi)$ and $\sigma_q^2(\xi)$ are distinct roots of $f(X,Y)$ over $\overline{\mathbb F_q}(X)$, a contradiction. ◻ # The Case That $L$ Permutes $\mathbb F_{q^n}$ {#s2} Suppose now that $L$ permutes $\mathbb F_{q^n}$, whose inverse permutation polynomial is denoted by $L^{-1}$. If $L(x)+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ is a permutation rational function of $\mathbb F_{q^n}$, then substituting $L^{-1}(x)$ for $x$ we have $$x+\frac c{\mathop{\mathrm{Tr}}(L^{-1}(x))+b}=x+\frac c{\mathop{\mathrm{Tr}}(\alpha x)+b}$$ for some $\alpha\in\mathbb F_{q^n}$. Hence, we only consider $x+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ and give the following criterion. **Lemma 4**. *The rational function $$x+\frac c{\mathop{\mathrm{Tr}}(x)+b}$$ is a permutation rational function of $\mathbb F_{q^n}$, if and only if $$x+\mathop{\mathrm{Tr}}\left(\frac c{x+b}\right)$$ is a permutation rational function of $\mathbb F_q$, if and only if $$\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)\ne1$$ for all $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$.* *Proof.* For arbitrary $t_1\in\mathbb F_q$, we have $\mathop{\mathrm{Tr}}(x_1)=t_1$ for some $x_1\in\mathbb F_{q^n}$, and if $$x_0+\frac c{\mathop{\mathrm{Tr}}(x_0)+b}=x_1$$ for some $x_0\in\mathbb F_{q^n}$, then $$\mathop{\mathrm{Tr}}(x_0)+\mathop{\mathrm{Tr}}\left(\frac c{\mathop{\mathrm{Tr}}(x_0)+b}\right)=\mathop{\mathrm{Tr}}(x_1);$$ let $t_0=\mathop{\mathrm{Tr}}(x_0)$, so that $$t_0+\mathop{\mathrm{Tr}}\left(\frac c{t_0+b}\right)=t_1.$$ Conversely, for arbitrary $x_1\in\mathbb F_{q^n}$, if the above equation holds for some $t_0\in\mathbb F_q$ and $t_1=\mathop{\mathrm{Tr}}(x_1)$, then let $$x_0=x_1-\frac c{t_0+b},$$ so that $\mathop{\mathrm{Tr}}(x_0)=t_0$ and $$x_0+\frac c{\mathop{\mathrm{Tr}}(x_0)+b}=x_0+\frac c{t_0+b}=x_1.$$ This proves the first equivalence. Next, it is clear that $$x+\mathop{\mathrm{Tr}}\left(\frac c{x+b}\right)$$ permutes $\mathbb F_q$ if and only if for all $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$, one has $$x_0+\mathop{\mathrm{Tr}}\left(\frac c{x_0+b}\right)\ne y_0+\mathop{\mathrm{Tr}}\left(\frac c{y_0+b}\right).$$ In fact, $$\begin{split}&\mathrel{\phantom{=}}x_0+\mathop{\mathrm{Tr}}\left(\frac c{x_0+b}\right)-y_0-\mathop{\mathrm{Tr}}\left(\frac c{y_0+b}\right)\\&=x_0-y_0+\mathop{\mathrm{Tr}}\left(\frac{c(y_0-x_0)}{(x_0+b)(y_0+b)}\right)\\&=(x_0-y_0)\left(1-\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)\right),\end{split}$$ and the result follows. ◻ This enables us to determine whether $x+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ is a permutation rational function of $\mathbb F_{q^n}$. In the following, we first study the case $n=2$ and then the case $n=3$. **Theorem 5**. *For $n=2$, $x+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ is a permutation rational function of $\mathbb F_{q^2}$ if and only if $c=\mathop{\mathrm{N}}(b^q-b)$.* *Proof.* Assume that the polynomial $$f(X,Y)=\mathop{\mathrm{N}}((X+b)(Y+b))-\mathop{\mathrm{Tr}}(c^q(X+b)(Y+b))$$ is reducible over $\overline{\mathbb F_q}$. As a quadratic polynomial over $\overline{\mathbb F_q}[X]$, it has leading coefficient $\mathop{\mathrm{N}}(X+b)$; if it is not primitive, then it must be divisible by $X+b$ or $X+b^q$ in $\overline{\mathbb F_q}[X,Y]$, but this is not valid for $\mathop{\mathrm{Tr}}(c^q(X+b)(Y+b))$. Thus $f(X,Y)$ has a factor $g_1(X)Y+g_0(X)$ for some nonzero polynomials $g_0$ and $g_1$ over $\overline{\mathbb F_q}$ with $g_1$ monic. Denote by $v_\infty$ be the infinite valuation of $\overline{\mathbb F_q}(X)$ and let $\xi=-\frac{g_0(X)}{g_1(X)}\in\overline{\mathbb F_q}(X)$ so that $$\mathop{\mathrm{N}}(X+b)(\xi+b)(\xi+b^q)=c^q(X+b)(\xi+b)+c(X+b^q)(\xi+b^q).$$ If $v_\infty(\xi+b)\le0$ and $v_\infty(\xi+b^q)\le0$, then $$\begin{split}&\mathrel{\phantom{=}}v_\infty(\mathop{\mathrm{N}}(x+b)(\xi+b)(\xi+b^q))\\&=-2+v_\infty((\xi+b)(\xi+b^q))\\&<\min\{-1+v\infty(\xi+b),-1+v\infty(\xi+b^q)\}\\&\le v_\infty(c^q(x+b)(\xi+b)+c(x+b^q)(\xi+b^q)),\end{split}$$ which gives rise to a contradiction. Then let $v_\infty(\xi+b)>0$ without loss of generality, and consequently $v_\infty(\xi+b^q)=0$. Moreover, $$v_\infty(\mathop{\mathrm{N}}(X+b)(\xi+b)(\xi+b^q))=v_\infty(\xi+b)-2,$$ and $$v_\infty(c^q(X+b)(\xi+b)+c(X+b^q)(\xi+b^q))=-1,$$ so $$v_\infty\left(\frac{bg_1(x)-g_0(x)}{g_1(x)}\right)=v_\infty(\xi+b)=1.$$ This happens only if $\deg(g_0)=\deg(g_1)=1$ and $g_0$ has leading coefficient $b$. We may write $$g_0(X)=bX+\beta_0\quad\text{and}\quad g_1(X)=X+\beta_1$$ for some $\beta_0,\beta_1\in\overline{\mathbb F_q}$. Since $f(X,Y)\in\mathbb F_q[X,Y]$ is fixed by $\sigma_q$, it has another factor $XY+b^qX+\beta_1^qY+\beta_0^q$ in $\overline{\mathbb F_q}[X,Y]$, relatively prime to $XY+bX+\beta_1Y+\beta_0$. Comparing the leading coefficients, we get $$f(X,Y)=(XY+bX+\beta_1Y+\beta_0)(XY+b^qX+\beta_1^qY+\beta_0^q)$$ with $\beta_0,\beta_1\in\mathbb F_{q^2}$. Since $f(X,Y)$ is symmetric, we have either $\beta_1=b$ or $\beta_1=b^q$ and $\beta_0=\beta_0^q$. Equating the coefficients of $XY$, $X$ and the constant terms from $$\begin{split}&\mathrel{\phantom{=}}\mathop{\mathrm{N}}((X+b)(Y+b))-\mathop{\mathrm{Tr}}(c^q(X+b)(Y+b))\\&=(XY+bX+\beta_1Y+\beta_0)(XY+b^qX+\beta_1^qY+\beta_0^q)\end{split}$$ yields $$\mathop{\mathrm{Tr}}(b)^2-\mathop{\mathrm{Tr}}(c)=\mathop{\mathrm{Tr}}(b\beta_1^q+\beta_0),$$ $$\mathop{\mathrm{Tr}}(b)\mathop{\mathrm{N}}(b)-\mathop{\mathrm{Tr}}(bc^q)=\mathop{\mathrm{Tr}}(b\beta_0^q),$$ and $$\mathop{\mathrm{N}}(b)^2-\mathop{\mathrm{Tr}}(b^2c^q)=\mathop{\mathrm{N}}(\beta_0).$$ If $\beta_1=b$, then $$\mathop{\mathrm{Tr}}(b^2-c-\beta_0)=\mathop{\mathrm{Tr}}(b^q(b^2-c-\beta_0))=0,$$ and thus $$\mathop{\mathrm{N}}(b)^2-\mathop{\mathrm{Tr}}(b^2c^q)=\mathop{\mathrm{N}}(b^2-c)=\mathop{\mathrm{N}}(b)^2-\mathop{\mathrm{Tr}}(b^2c^q)+\mathop{\mathrm{N}}(c),$$ which implies $c=0$. It remains to consider the case $\beta_1=b^q$ and $\beta_0=\beta_0^q$. Now we have $$2\mathop{\mathrm{N}}(b)-\mathop{\mathrm{Tr}}(c)-2\beta_0=\mathop{\mathrm{Tr}}(b)\mathop{\mathrm{N}}(b)-\mathop{\mathrm{Tr}}(bc^q)-\mathop{\mathrm{Tr}}(b)\beta_0=\mathop{\mathrm{N}}(b)^2-\mathop{\mathrm{Tr}}(b^2c^q)-\beta_0^2=0,$$ and then $$\begin{split}\mathop{\mathrm{Tr}}(b^2c^q)&=\mathop{\mathrm{Tr}}((\mathop{\mathrm{Tr}}(b)b-\mathop{\mathrm{N}}(b))c^q)\\&=\mathop{\mathrm{Tr}}(b)\mathop{\mathrm{Tr}}(bc^q)-\mathop{\mathrm{N}}(b)\mathop{\mathrm{Tr}}(c)\\&=\mathop{\mathrm{Tr}}(b)^2(\mathop{\mathrm{N}}(b)-\beta_0)-2\mathop{\mathrm{N}}(b)(\mathop{\mathrm{N}}(b)-\beta_0)\\&=\mathop{\mathrm{Tr}}(b^2)(\mathop{\mathrm{N}}(b)-\beta_0),\end{split}$$ where $\mathop{\mathrm{N}}(b)-\beta_0\ne0$, for otherwise $\mathop{\mathrm{Tr}}(c)=\mathop{\mathrm{Tr}}(bc^q)=0$. Consequently, $$\mathop{\mathrm{N}}(b)+\beta_0=\mathop{\mathrm{Tr}}(b^2),$$ and $$\mathop{\mathrm{Tr}}(c)=2\mathop{\mathrm{N}}(b)-2\mathop{\mathrm{Tr}}(b^2)+2\mathop{\mathrm{N}}(b)=2\mathop{\mathrm{N}}(b^q-b),$$ while $$\mathop{\mathrm{Tr}}(bc^q)=\mathop{\mathrm{Tr}}(b)(2\mathop{\mathrm{N}}(b)-\mathop{\mathrm{Tr}}(b^2))=\mathop{\mathrm{Tr}}(b)\mathop{\mathrm{N}}(b^q-b).$$ Such an element $c\in\mathbb F_{q^2}^*$ is uniquely determined by $b$, and we have $c=\mathop{\mathrm{N}}(b^q-b)$ satisfying the above equations. We have proved that $f(X,Y)$ is absolutely irreducible if $c\ne\mathop{\mathrm{N}}(b^q-b)$. In this case, there exist $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$ such that $$1-\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)=\frac{f(x_0,y_0)}{\mathop{\mathrm{N}}((x_0+b)(y_0+b))}=0$$ when $q-6\sqrt q-7>0$, by Lemma [Lemma 1](#bound){reference-type="ref" reference="bound"}. The rest is verified by exhaustive search. If $c=\mathop{\mathrm{N}}(b^q-b)$ for arbitrary $b\in\mathbb F_{q^2}\setminus\mathbb F_q$, then $$f(X,Y)=\mathop{\mathrm{N}}(XY+bX+b^qY+\mathop{\mathrm{Tr}}(b^2)-\mathop{\mathrm{N}}(b)),$$ but $x_0y_0+bx_0+b^qy_0+\mathop{\mathrm{Tr}}(b^2)-\mathop{\mathrm{N}}(b)=0$ for $x_0,y_0\in\mathbb F_q$ only if $$x_0y_0+\mathop{\mathrm{Tr}}(b)x_0+\mathop{\mathrm{Tr}}(b^2)-\mathop{\mathrm{N}}(b)=b^q(x_0-y_0),$$ where the right side lies outside of $\mathbb F_q$ if $x_0\ne y_0$. ◻ **Remark 6**. Let $\alpha$ be an element in $\mathbb F_{q^2}$ with $\alpha^{q-1}=-1$. Substituting $\alpha^{-1}x$ for $x$ in $x+\frac c{x^q+x+b}$, we get $$\alpha^{-1}x+\frac c{\alpha^{-q}x^q+\alpha^{-1}x+b}=\alpha^{-1}\left(x+\frac{\alpha^{q+1}c}{x^q-x+\alpha^qb}\right)$$ This extends the result of [@hou2020type]. **Theorem 7**. *For $n=3$, $x+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ is a permutation rational function of $\mathbb F_{q^3}$ if $c=-(b^q-b)^{q^2+1}$. The converse holds for sufficiently large $q$.* *Proof.* We claim that if $c\ne-(b^q-b)^{q^2+1}$, then the polynomial $$f(X,Y)=\mathop{\mathrm{N}}((X+b)(Y+b))-\mathop{\mathrm{Tr}}\big(c^{q^2}(X+b)(X+b^q)(Y+b)(Y+b^q)\big).$$ over $\mathbb F_q$ is absolutely irreducible; otherwise, $$f(X,Y)=\mathop{\mathrm{N}}((X+b)(Y+b)-c).$$ If $f(X,Y)$ is absolutely irreducible, then by Lemma [Lemma 1](#bound){reference-type="ref" reference="bound"}, there exist $x_0,y_0\in\mathbb F_q$ such that $$1-\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)=\frac{f(x_0,y_0)}{\mathop{\mathrm{N}}((x_0+b)(y_0+b))}=0$$ with $x_0\ne y_0$, when $q-20\sqrt q-11>0$. Next, let $f(X,Y)=\mathop{\mathrm{N}}((X+b)(Y+b)-c)$ with $c=-(b^q-b)^{q^2+1}$, and assume $f(x_0,y_0)=0$ for some $x_0,y_0\in\mathbb F_q$ with $x_0\ne y_0$. It follows that $(x_0+b)(y_0+b)=c$, in which case, $$\mathop{\mathrm{Tr}}(1)=\mathop{\mathrm{Tr}}\left(\frac c{(x_0+b)(y_0+b)}\right)=1-\frac{f(x_0,y_0)}{\mathop{\mathrm{N}}((x_0+b)(y_0+b))}=1,$$ and then $q$ is even. Note that $$(x_0+b)(y_0+b)-c=x_0y_0+b(x_0+y_0)+b^2+(b^q-b)^{q^2+1}=x_0y_0+b(x_0+y_0)+\mathop{\mathrm{Tr}}(b^{q+1}),$$ but $b(x_0+y_0)\notin\mathbb F_q$, a contradiction. In what follows, we prove the claim. Assume $f(X,Y)$ is reducible over $\overline{\mathbb F_q}$. By a similar argument, it is a primitive polynomial over $\overline{\mathbb F_q}[X]$ and has a factor $g_1(X)Y+g_0(X)$ for some nonzero polynomials $g_0$ and $g_1$ over $\overline{\mathbb F_q}$ with $g_1$ monic. For the infinite valuation $v_\infty$ of $\overline{\mathbb F_q}(X)$ and $\xi=-\frac{g_0(X)}{g_1(X)}\in\overline{\mathbb F_q}(X)$, which satisfies $$\mathop{\mathrm{N}}(X+b)(\xi+b)(\xi+b^q)\big(\xi+b^{q^2}\big)=\sum_{i=0}^2c^{q^{i+2}}\big(X+b^{q^i}\big)\big(X+b^{q^{i+1}}\big)\big(\xi+b^{q^i}\big)\big(\xi+b^{q^{i+1}}\big),$$ we exclude the case that $v_\infty(\xi+b)\le0$, $v_\infty(\xi+b^q)\le0$ and $v_\infty\big(\xi+b^{q^2}\big)\le0$, and let $v_\infty(\xi+b)>v_\infty(\xi+b^q)=v_\infty\big(\xi+b^{q^2}\big)=0$ as in the proof of the last theorem. Furthermore, $$v_\infty\big(\mathop{\mathrm{N}}(X+b)(\xi+b)(\xi+b^q)\big(\xi+b^{q^2}\big)\big)=v_\infty(\xi+b)-3,$$ $$v_\infty\left(\sum_{i=0}^2c^{q^{i+2}}\big(X+b^{q^i}\big)\big(X+b^{q^{i+1}}\big)\big(\xi+b^{q^i}\big)\big(\xi+b^{q^{i+1}}\big)\right)=-2,$$ and thus, $$v_\infty\left(\frac{bg_1(X)-g_0(X)}{g_1(X)}\right)=v_\infty(\xi+b)=1.$$ This means $g_0$ has leading coefficient $b$ with $\deg(g_0)=\deg(g_1)$. It is clear that $f(X,Y)$ is fixed by the automorphism $\sigma_q$ of $\overline{\mathbb F_q}[X,Y]$, while $g_0(X)\notin\mathbb F_q[X]$, so $g_1(X)Y+g_0(X)$, $\sigma_q(g_1(X)Y+g_0(X))$ and $\sigma_q^2(g_1(X)Y+g_0(X))$ are distinct irreducible polynomials in $\overline{\mathbb F_q}[X,Y]$ dividing $f(X,Y)$. Moreover, they are relatively prime in $\overline{\mathbb F_q}[X,Y]$ since $g_1$ is monic. Therefore, $f(X,Y)$ is divisible by their product. In fact, noticing that $f(X,Y)$ is a cubic polynomial in one indeterminate and comparing the leading coefficients, we obtain $$f(X,Y)=(g_1(X)Y+g_0(X))\sigma_q(g_1(X)Y+g_0(X))\sigma_q^2(g_1(X)Y+g_0(X)),$$ where $\deg(g_0)=\deg(g_1)=1$. We now show that $g_1(X)Y+g_0(X)$ is symmetric. Assume the contrary so that $$g_1(X)Y+g_0(X)\ne g_1(Y)X+g_0(Y).$$ By the same argument, $g_1(X)Y+g_0(X)$ and $g_1(Y)X+g_0(Y)$ are relatively prime, and thus $f(X,Y)$ as a symmetric polynomial in $\overline{\mathbb F_q}[X,Y]$ is divisible by $(g_1(X)Y+g_0(X))(g_1(Y)X+g_0(Y))$. We may write $$f(X,Y)=(g_1(X)Y+g_0(X))(g_1(Y)X+g_0(Y))h(X,Y),$$ where $h(X,Y)$ is a linear polynomial over $\overline{\mathbb F_q}[X]$ as well as a symmetric polynomial in $\overline{\mathbb F_q}[X,Y]$, but neither $\sigma_q(g_1(X)Y+g_0(X))$ nor $\sigma_q^2(g_1(X)Y+g_0(X))$ is symmetric. Eventually, $$f(X,Y)=\mathop{\mathrm{N}}((X+b)(Y+b)+\gamma),$$ for some $\gamma\in\mathbb F_{q^3}$. Expanding $\mathop{\mathrm{N}}((X+b)(Y+b)+\gamma)$ leads to $$\mathop{\mathrm{Tr}}\big((c+\gamma)^{q^2}(X+b)(X+b^q)(Y+b)(Y+b^q)\big)+\mathop{\mathrm{Tr}}\big(\gamma^{q^2+q}(X+b)(Y+b)\big)+\mathop{\mathrm{N}}(\gamma)=0.$$ Looking at the coefficients of $X^2Y^2$, $X^2Y$ and $X^2$, we assert that $$\mathop{\mathrm{Tr}}(c+\gamma)=\mathop{\mathrm{Tr}}\big((b^q+b)(c+\gamma)^{q^2}\big)=\mathop{\mathrm{Tr}}\big(b^{q+1}(c+\gamma)^{q^2}\big)=0,$$ which implies $c+\gamma=0$ by Lemma [Lemma 2](#basis){reference-type="ref" reference="basis"}. This allows one to write $$\mathop{\mathrm{Tr}}\big(c^{q^2+q}(x+b)(y+b)\big)-\mathop{\mathrm{N}}(c)=0,$$ and, by computing the coefficients, $$\mathop{\mathrm{Tr}}(c^{q+1})=\mathop{\mathrm{Tr}}\big(bc^{q^2+q}\big)=\mathop{\mathrm{Tr}}\big(b^2c^{q^2+q}\big)-\mathop{\mathrm{N}}(c)=0.$$ Divide it by $\mathop{\mathrm{N}}(c)$ to get $$\mathop{\mathrm{Tr}}(c^{-1})=\mathop{\mathrm{Tr}}(bc^{-1})=\mathop{\mathrm{Tr}}(b^2c^{-1})-1=0,$$ where $c^{-1}$ is uniquely determined by $b$, since $(1,b,b^2)$ is a basis of $\mathbb F_{q^3}/\mathbb F_q$. If $c=-(b^q-b)^{q^2+1}$, then $$c^{q+1}=(b^q-b)^{(q^2+1)(q+1)}=\mathop{\mathrm{N}}(b^q-b)(b^q-b),$$ which implies $$\mathop{\mathrm{Tr}}(c^{q+1})=0,$$ $$\mathop{\mathrm{Tr}}\big(bc^{q^2+q}\big)=\mathop{\mathrm{N}}(b^q-b)\mathop{\mathrm{Tr}}\big(b^{q^2+1}-b^{q+1}\big)=0,$$ and $$\mathop{\mathrm{Tr}}\big(b^2c^{q^2+q}\big)-\mathop{\mathrm{N}}(c)=\mathop{\mathrm{N}}(b^q-b)\mathop{\mathrm{Tr}}\big(b^{q^2+2}-b^{q+2}\big)+\mathop{\mathrm{N}}(b^q-b)^2=0,$$ where $\mathop{\mathrm{N}}(b^q-b)=\mathop{\mathrm{Tr}}\big(b^{q^2+2}-b^{q+2}\big)$ as easily seen. In this case, $f(X,Y)$ is factored as desired. The proof of the claim is then complete. ◻ **Remark 8**. Suppose $q$ is odd, $b\in\mathbb F_{q^3}\setminus\mathbb F_q$ and $c=-(b^q-b)^{q^2+1}$. It has been shown that $$\mathop{\mathrm{Tr}}\left(\frac c{x_0y_0+b(x_0+y_0)+b^2}\right)\ne1$$ for all $x_0,y_0\in\mathbb F_q$. In fact, one can prove a slightly stronger result that $$\mathop{\mathrm{Tr}}\left(\frac c{u+bv+b^2}\right)\ne1$$ for all $u,v\in\mathbb F_q$. Let $z=u+bv+b^2$ and assume $\mathop{\mathrm{Tr}}(cz^{-1})=1$. We have just seen that $$\mathop{\mathrm{Tr}}(c^{-1})=\mathop{\mathrm{Tr}}(bc^{-1})=\mathop{\mathrm{Tr}}(b^2c^{-1})-1=0,$$ and then $\mathop{\mathrm{Tr}}(zc^{-1})=1$. If $cz^{-1}\in\mathbb F_q$, then $$9=(3cz^{-1})(3zc^{-1})=\mathop{\mathrm{Tr}}(cz^{-1})\mathop{\mathrm{Tr}}(zc^{-1})=1,$$ and $q$ is even. If $cz^{-1}\notin\mathbb F_q$, then its minimal polynomial over $\mathbb F_q$ is $$x^3-\mathop{\mathrm{Tr}}(cz^{-1})x^2+\mathop{\mathrm{N}}(cz^{-1})\mathop{\mathrm{Tr}}(zc^{-1})x-\mathop{\mathrm{N}}(cz^{-1})=x^3-x^2+\mathop{\mathrm{N}}(cz^{-1})x-\mathop{\mathrm{N}}(cz^{-1}),$$ which, however, is reducible with a root $1$. The above sufficient conditions can be extended as follows. **Corollary 9**. *Let $\mathop{\mathrm{Tr}}_{i,j}$ denote the trace map from $\mathbb F_{q^i}$ to $\mathbb F_{q^j}$. Then $x+\frac c{\mathop{\mathrm{Tr}}(x)+b}$ is a permutation rational function of $\mathbb F_{q^n}$ if* - *$2$ divides $n$, $b\in\mathbb F_{q^2}\setminus\mathbb F_q$ and $\mathop{\mathrm{Tr}}_{n,2}(c)=(b^q-b)^{q+1}$, or* - *$3$ divides $n$, $b\in\mathbb F_{q^3}\setminus\mathbb F_q$ and $\mathop{\mathrm{Tr}}_{n,3}(c)=-(b^q-b)^{q^2+1}$.* *Proof.* By Lemma [Lemma 4](#equiv){reference-type="ref" reference="equiv"}, the rational function permutes $\mathbb F_{q^n}$ if and only if $$x+\mathop{\mathrm{Tr}}\left(\frac c{x+b}\right)$$ permutes $\mathbb F_q$. If $2$ divides $n$ and $b\in\mathbb F_{q^2}\setminus\mathbb F_q$, then $$x+\mathop{\mathrm{Tr}}\left(\frac c{x+b}\right)=x+\mathop{\mathrm{Tr}}_{2,1}\circ\mathop{\mathrm{Tr}}_{n,2}\left(\frac c{x+b}\right)=x+\mathop{\mathrm{Tr}}_{2,1}\left(\frac{\mathop{\mathrm{Tr}}_{n,2}(c)}{x+b}\right),$$ as a rational function over $\mathbb F_q$. The same holds for the case that $3$ divides $n$ and the result follows immediately from the above on quadratic and cubic extensions. ◻

arxiv_math

--- abstract: | In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a real linear space as a closed subset. In later years, Michael pointed out that every metric space can be isometrically embedded into a real linear space as a linear independent subset and provided a short proof of Arens--Eells theorem as an application. In this paper, we prove a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which states that for every non-Archimedean valued field $K$, every ultrametric space can be isometrically embedded into a non-Archimedean valued field that is a valued field extension of $K$ such that the image of the embedding is algebraically independent over $K$. Using Levi-Civita fields, we also show that every Urysohn universal ultrametric sapce has a valued field structure. address: Photonics Control Technology Team RIKEN Center for Advanced Photonics -1 Hirasawa, Wako, Saitama 351-0198, Japan author: - Yoshito Ishiki bibliography: - ../../../bibtex/UU.bib title: A non-Archimedean Arens--Eells isometric embedding theorem on valued fields --- # Introduction {#sec:intro} In 1956, Arens and Eells [@MR81458] established that for every metric space $(X, d)$ there exist a real normed linear space $(V, \|*\|)$ and an isometric embedding $I\colon X\to V$ such that $I(X)$ is closed in $V$. Michael [@MR162222] pointed out that for every metric space $(X, d)$, we can take a real normed linear space $(V, \lVert *\rVert)$ and an isometric embedding $I\colon X\to V$ such that $I(X)$ is linearly independent over $V$. Using this observation, Michael provided a short proof of the Arens--Eells theorem. A metric $d$ on $X$ is said to be an *ultrametric* or a *non-Archimedean metric* if $d(x, y)\le d(x, z)\lor d(z, y)$ for all $x, y, z\in X$, where $\lor$ stands for the maximal operator on $\mathbb{R}$. A set $R$ is a *range set* if $0\in R$ and $R\subseteq[0, \infty)$. An ultrametric $d$ on $X$ is said to be *R-valued* if $d(x, y)\in R$ for all $x, y\in X$. Some authors try to investigate non-Archimedean analogue of theorems on metric spaces such as the Arens--Eells theorem. Megrelishvili and Shlossberg [@MR3067710] proved a non-Archimedean Arens--Eells theorem, which embeds ultrametric spaces into linear spaces over $\mathbb{F}_{2}$. In [@MR3503300 Theorem 4.3], as a improvement of their theorem, they prove a non-Archimedean Arens--Eells theorem on linear spaces over arbitrary non-Archimedean valued fields. In [@Ishiki2021ultra Theorem 1.1], as a non-Archimedean analogue of the Arens--Eells theorem, the author showed that for every range set $R$, for every integral domain $A$ with the trivial absolute value $\lvert *\rvert$ (i.e., $\lvert x\rvert=1$ for all $x\neq 0$), for every $R$-valued ultrametric space $(X, d)$, there exist an $R$-valued ultra-normed module $(V, \lVert * \rVert)$ over $(A, \lvert * \rvert)$ and an isometric embedding $I\colon X\to V$ such that $I(X)$ is closed and linearly independent over $(A, \rvert *\rvert)$. Using this embedding theorem, the author proved a ($R$-valued) non-Archimedean analogue of the Hausdorff extension theorem of metrics. There are other attempts to construct an isometric embedding from an ultrametric spaces into a space with a non-Archimedean algebraic structure. Schikhof [@MR748978] established that every ultrametric spaces can be isometrically embedded into a non-Archimedean valued field using the Hahn fields, which is a generalization of a field of formal power series. In [@MR0425916 Conjecture 5.34], Baroughan raise the following conjecture. **Conjecture 1**. Let $p$ be an odd prime and $(X, d)$ be an $H_{p}$-valued metric space, where $H_{p}=\{0\}\cup \{\, p^{n}\mid n\in \mathbb{Z}\, \}$. Then there exist a non-Archimedean Banach algebra $(B, \lVert *\rVert)$ over $\mathbb{Q}_{p}$ and an isometry $i\colon X\to B$. In this paper, as a generalization of the Schikhof theorem ([@MR748978]) and known non-Archimedean analogues ([@Ishiki2021ultra Theorem 1.1] and [@MR3503300 Theorem 4.3]) of the Arens--Eells theorem, we prove that for every non-Archimedean valued field $(K, \rVert *\lVert_{K} )$, for every metric space $(X, d)$, there exist a valud field $(L, \rVert*\lVert_{L})$ and an isometry embedding $I\colon X\to K$ such that $L$ is a valued field extension of $K$, the set $I(X)$ is closed in $L$, and $I(X)$ is algebraically independent over $K$ (Theorem [Theorem 30](#thm:19main2){reference-type="ref" reference="thm:19main2"}). A key point of the proof of Theorem [Theorem 30](#thm:19main2){reference-type="ref" reference="thm:19main2"} is to use the notion of $p$-adic Hahn fields ($p$-adic Mal'cev--Neumann fields), which was first introduced by Poonen [@MR1225257] as $p$-adic analogues of ordinary Hahn fields. As an application, we also gives an affirmative solution of Conjecture [Conjecture 1](#conj:bbb){reference-type="ref" reference="conj:bbb"} (see Theorem [Theorem 32](#thm:frw){reference-type="ref" reference="thm:frw"}). The theory of $p$-adic Hahn fields have an application to Urysohn universal ultrametric spaces, which is defined as ultrametric spaces possessing high homogeneity. We introduce the concept of $p$-adic Levi-Civita fields as subspaces of $p$-adic Hahn fields, which is a $p$-adic analogue of ordinary Levi-Civita fields (see, for instance, [@MR3782290]). We also show that if $p$ is $0$ or a prime, then every Urysohn universal ultrametric space has a field structure that is an extension of $\mathbb{Q}_{p}$, where we consider that $\mathbb{Q}_{0}=\mathbb{Q}$ and it is equipped with the trivial valuation in the case of $p=0$ (see Theorem [Theorem 37](#thm:mainury){reference-type="ref" reference="thm:mainury"}). The paper is organized as follows. Section [2](#sec:pre){reference-type="ref" reference="sec:pre"} presents notions and notations of metric spaces and valued fields. We introduce Hahn fields and $p$-adic Hahn fields, which plays an important role in the proofs of our main results. We also prepare some basic statements on valued fields and Hahn fields. In Section [3](#sec:propf){reference-type="ref" reference="sec:propf"}, we show statements on algebraic independence in ($p$-adic) Hahn fields. Section [4](#sec:isom){reference-type="ref" reference="sec:isom"} is devoted to proving Theorem [Theorem 30](#thm:19main2){reference-type="ref" reference="thm:19main2"}. We also provide an affirmative solution of Conjecture [Conjecture 1](#conj:bbb){reference-type="ref" reference="conj:bbb"}. In Section [5](#sec:Urysohn){reference-type="ref" reference="sec:Urysohn"}, we show that ($p$-adic) Levi-Civita fields become Urysohn universal ultrametric spaces. Our arguments in this section are baed on the author's paper [@Ishiki2022highpower]. *Acknowledgements 1*. The author would like to thank to Tomoki Yuji for helpful advices on algebraic arguments. # Preliminaries {#sec:pre} ## Generalities {#subsec:gen} In this paper, we use the set-theoretic notations of ordinals. For example, for an ordinal $\alpha$, we have $\beta<\alpha$ if and only if $\beta\in \alpha$. ### Metric spaces For a metric space $(X, d)$, and for a subset of $A$ of $X$, and for $x\in X$, we define $d(A, x)=\inf\{\, d(a, x)\mid a\in A\, \}$. For $x\in X$ and $r\in (0, \infty)$, we denote by $B(a, r; d)$the closed ball centered at $x$ with radius $r$. We often simply represent it as $B(a, r)$ when no confusion can arise. Similarly, we define the open ball $U(a, r)$. The proofs of the next two lemmas are presented in Propositions 18.2, and 18.4, in [@MR2444734], respectively. **Lemma 2**. *Let $X$ be a set, and $d$ be a pseudo-metric on $X$. Then $d$ satisfies the strong triangle inequality if and only if for all $x, y, z\in X$, the inequality $d(x, z)<d(z, y)$ implies $d(z, y)=d(x, y)$.* **Lemma 3**. *Let $(X, d)$ be a pseudo-ultrametric space, $a\in X$ and $r\in [0, \infty)$. Then for every $q\in B(p, r)$, we have $B(p, r)=B(q, r)$.* ### Valued rings Let $A$ be a commutative ring. We say that a function $v\colon A\to \mathbb{R}\sqcup \{\infty\}$ is a *(additive) valuation* if the following conditions are satisfied: 1. for every $x\in A$, we have $v(x)=\infty$ if and only if $x=0$; 2. for every pair $x, y\in A$, we have $v(xy)=v(x)+v(y)$ 3. for every pair $x, y\in A$, we have $v(x+y)\ge v(x)\land v(y)$, where $\land$ stands for the minimum operator on $\mathbb{R}$. If $A$ is a field, then it is called a *valued field*. Note that for every valued ring $(A, v)$, we can extend the valuation $v$ to the fractional field $K$ of $A$. Namely, for $x=b/a\in K$, where $a, b\in A$ and $a\neq 0$, we define $v(x)=v(b)-v(a)$. In this case, $v$ is well-defined and the pair $(K, v)$ naturally becomes a valued field. For example, for a prime $p$, we define the $p$-adic valuation $v_{p}$ on $\mathbb{Z}$ by declaring that $v_{p}$ is the number of the factor $p$ in the prime factorization of $x$. The completion $\mathbb{Z}_{p}$ of $\mathbb{Z}$ with respect to $v_{p}$ is called the *ring of $p$-adic integers*. The fractional field of $\mathbb{Z}_{p}$ is called the *field of $p$-adic numbers*. For more discussion on $p$-adic numbers, we refer the readers to [@MR1703500] and [@MR2444734]. We say that a function $\lVert *\rVert \colon A\to [0, \infty)$ is a *(non-Archimedean) absolute value* or *multiplicative valuation* if the following conditions are satisfied: 1. for every $x\in A$, we have $\lVert x \rVert=0$ if and only if $x=0$; 2. for every pair $x, y\in A$, we have $\lVert xy\rVert =\lVert x\rVert \cdot \lVert y\rVert$ 3. for every pair $x, y\in A$, we have $\lVert x+y\rVert \le \lVert x\rVert \lor \lVert y\rVert$, where $\lor$ stands for the maximum operator on $\mathbb{R}$. In what follows, for every $\eta\in (1, \infty)$, we consider that $\eta^{-\infty}=0$ and $-\log_{\eta}(0)=\infty$. For a valuation $v$ on a ring $A$, and for a real number $\eta\in (1, \infty)$, we define $\lVert x\rVert_{v, \eta}=\eta^{-v(x)}$ and $v_{\rVert *\rVert, \eta}(x) =-\log_{\eta}(x)$. Fix $\eta\in (1, \infty)$, valuations and absolute values on a ring $A$ are essentially equivalent. Namely, they are corresponding to each other as follows. **Proposition 4**. *Let $A$ be a commutative ring. Then the following statements are true:* 1. *For every $\eta\in (1, \infty)$, and for every valuation $v$ on $A$, the function $\lVert *\rVert_{v, \eta}\colon A\to [0, \infty)$ is an absolute value on $A$;* 2. *For $\eta\in (1, \infty)$, and for every absolute value $\lVert *\rVert$ on $A$, the function $v_{\rVert *\rVert, \eta}\colon A\to \mathbb{R}$ is a valuation on $A$.* In this paper, we use both of valuation and absolute values on a ring due to Proposition [Proposition 4](#prop:rho){reference-type="ref" reference="prop:rho"}. Since we represent a valuation (resp. an absolute value) as a symbol like $v$, or $w$ (resp. $\rvert*\lvert$ or $\rVert*\lVert$), the readers will be able to distinguish them. For a valued ring $(K, v)$, the set $\mathfrak{A}(K, v)=\{\, x\in K\mid 0\le v(x)\, \}$ becomes a ring and $\mathfrak{o}(K, v)=\{\, x\in K\mid 0< v(x)\, \}$ is a maximal ideal of $\mathfrak{A}(K, v)$. We denote by $\mathfrak{K}(K, v)$ the field $\mathfrak{A}(K, v)/\mathfrak{o}(K, v)$, and we call it the *residue class field of $(K, v)$*. We also denote by $\zeta_{K}\colon \mathfrak{A}(K, v)\to \mathfrak{K}(K, v)$ the canonical projection. We simply represent it as $\zeta$ when no confusions can arise. We say that a subset $J$ of $K$ is a *complete system of representative of the residue class field $\mathfrak{K}(A, v)$* if $J\subseteq\mathfrak{A}(A, v)$, $0\in J$, and $\zeta|_{J}\colon J\to \mathfrak{K}(A, v)$ is bijective. ## Constructions of valued fields {#subsec:const} In this section, we review some constructions of valued fields such as Hahn fields. For more discussion, we refer the readers to [@MR3782290] and [@MR4367483]. Most of the proofs in this section is refered to [@MR1225257] and [@MR0554237]. ### Hahn rings and fields A non-empty subset $S$ of $\mathbb{R}$ is said to be *well-ordered* if every non-empty subset of $S$ has a minimum. We denote by $\mathcal{G}$ the set of all subgroups of $\mathbb{R}$ containing $1\in \mathbb{R}$ (equivalently, $\mathbb{Z}\subseteq G$). For the sake of convenience, we only consider the setting where $G\in \mathcal{G}$ in this paper. Now we review the construction of the Hahn fields in [@MR1225257]. Let $G\in \mathcal{G}$ and $A$ be a commutative ring. For a map $a\colon G\to A$, we define the support $\text{{\rm supp}}(a)$ of $a$ by the set $\{\, x\in G\mid a(x)\neq 0\, \}$. We denote by $\mathbb{H}(G, A)$ the set of all $a\colon G\to A$ such that $\text{{\rm supp}}(a)$ is well-ordered. We often symbolically represent $a\in \mathbb{H}(G, A)$ as $a=\sum_{g\in G}a(g)t^{g}$, where $t$ is an indeterminate. For every pair $a, b \in \mathbb{H}(G, A)$, we define $a+b$ by $$(a+b)(x)=a(x)+b(x).$$ We also define $ab\colon G\to A$ by $$ab=\sum_{g\in G}\left(\sum_{i, j\in G, i+j=g}a_{i}b_{j}\right)t^{g}.$$ Define a valuation $v_{G, A}$ on $\mathbb{H}(G, A)$ by $v_{G, A}(a)=\min \text{{\rm supp}}(a)$. Since $\text{{\rm supp}}(a)$ is well-ordered, the minimum $\min\text{{\rm supp}}(a)$ actually exists. Note that $A$ becomes a subring of $\mathbb{H}(G, A)$. **Proposition 5**. *Let $G\in \mathcal{G}$, and $\boldsymbol{k}$ be a field. The pair $(\mathbb{H}(G, \boldsymbol{k}), v_{G, \boldsymbol{k}})$ becomes a valued field and it satisfies that $\mathfrak{K}(\mathbb{H}(G, \boldsymbol{k}), v_{G, \boldsymbol{k}})=\boldsymbol{k}$.* *Proof.* See [@MR1225257 Corollary 1]. ◻ We call $(\mathbb{H}(G, A), v_{G, A})$ the *Hahn ring associated with $G$ and $A$* and call it the *Hahn field* if $A$ is a field. Note that in general, we can define the Hahn fields even if $G$ is a linearly ordered Abelian group (see [@MR1225257]). ### The $p$-adic Hahn fields A $p$-adic analogue of the Hahn fields was first introduced in [@MR1225257]. Let us review a construction. A field $\boldsymbol{k}$ of characteristic $p$ is said to be *perfect* if $p=0$, or $p>0$ and every element of $\boldsymbol{k}$ has a $p$-th root in $\boldsymbol{k}$. The following proposition states the existence of rings of Witt vectors. The proof is presented in [@MR0554237]. **Proposition 6**. *Let $\boldsymbol{k}$ be a perfect field of characteristic $p>0$. The there exists a unique valued ring $(A, v)$ of characteristic $0$ equipped with a valuation $v$ such that $v(A)=\mathbb{Z}_{\ge 0}$, $v$ is complete, and $\mathfrak{K}(A, v)=\boldsymbol{k}$.* For each perfect field $\boldsymbol{k}$, we denote by $(\mathbb{W}(\boldsymbol{k}), w_{\boldsymbol{k}})$ the valuation field stated in Proposition [Proposition 6](#prop:witt){reference-type="ref" reference="prop:witt"} and denote by $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ the fractional field of $\mathbb{W}(\boldsymbol{k})$. We use the same symbol $w_{\boldsymbol{k}}$ as the valuation on $\mathbb{W}(\boldsymbol{k})$ induced by $w_{\boldsymbol{k}}$ in the canonical way. The ring $(\mathbb{W}(\boldsymbol{k}), w_{\boldsymbol{k}})$ is called the *ring of Witt vectors associated with $\boldsymbol{k}$*. Notice that for every prime $p$, we have $\mathbb{W}(\mathbb{F}_{p})=\mathbb{Z}_{p}$ and $\mathrm{Fr}\mathbb{W}(\mathbb{F}_{p})=\mathbb{Q}_{p}$, and the valuation $w_{\mathbb{F}_{p}}$ coincides with the $p$-adic valuation $v_{p}$. The next proposition explains the concrete representation of an elements of a ring of Witt vectors. **Proposition 7**. *Let $\boldsymbol{k}$ be a perfect field of characteristic $p>0$. Then there uniquely exists a map $f_{\boldsymbol{k}}\colon \boldsymbol{k}\to \mathbb{W}(\boldsymbol{k})$ such that $\{\, f_{\boldsymbol{k}}(a)\mid a\in \boldsymbol{k}\, \}$ is a complete system of representatives, and $f_{\boldsymbol{k}}(ab)=f_{\boldsymbol{k}}(a)f_{\boldsymbol{k}}(b)$ for all $a, b\in \boldsymbol{k}$. In this case, for every $x\in \mathbb{W}(\boldsymbol{k})$, there uniquely exists a sequence $\{a_{i}\}_{i\in \mathbb{Z}}$ in $\boldsymbol{k}$ such that $x=\sum_{n\in \mathbb{Z}}f_{\boldsymbol{k}}(a_{n})p^{n}$ and for a sufficient large $m\in \mathbb{Z}_{\ge 0}$, we have $a_{i}=0$ for all $i<-m$.* *Proof.* See Proposition 8 in the page 35 and and the argument in the page 37 in [@MR0554237]. ◻ **Proposition 8**. *Let $\boldsymbol{k}$ and $\boldsymbol{l}$ be perfect fields of characteristic $p>0$. For every homomorphism $\phi\colon \boldsymbol{k}\to \boldsymbol{l}$, there uniquely exists a homomorphism $\mathbb{W}(\phi)\colon \mathbb{W}(\boldsymbol{k})\to \mathbb{W}(\boldsymbol{l})$ such that $\zeta_{\mathbb{W}(\boldsymbol{l})}\circ \mathbb{W}(\phi)=\phi\circ \zeta_{\mathbb{W}(\boldsymbol{k})}$. Moreover, if $x=\sum_{n\in \mathbb{Z}}f_{\boldsymbol{k}}(a_{n})p^{n}\in \mathbb{W}(\phi)$, where $a_{n}\in \boldsymbol{k}$, then $\mathbb{W}(\phi)(x) =\sum_{n\in \mathbb{Z}}f_{\boldsymbol{l}}(\phi(a_{n}))p^{n}$. In particular, we have $w_{\boldsymbol{l}}(\mathbb{W}(\boldsymbol{k})(x))=w_{\boldsymbol{k}}(x)$ for all $x\in \mathbb{W}(\boldsymbol{k})$.* *Proof.* See [@MR0554237 Proposition 10 in the page 39]. ◻ *Remark 1*. It is known that the construction of rings of Witt vectors is a functor (see [@MR0554237 Page 39]). Now we discuss a $p$-adic analogue of Hahn fields, which is defined by a quotient field of a Hahn ring. For $G\in \mathcal{G}$, and for a perfect field $\boldsymbol{k}$ of characteristic $p>0$, we define a subset $\mathbb{N}_{G, \boldsymbol{k}}$ of $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$ by the set of all $\alpha=\sum_{g\in G}\alpha_{g}t^{g}\in \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$ such that $\sum_{n\in \mathbb{Z}}\alpha_{g+n}p^{n}=0$ in $\mathbb{W}(\boldsymbol{k})$ for every $g\in G$. **Proposition 9**. *For every $G\in \mathcal{G}$, and for every perfect field $\boldsymbol{k}$ with characteristic $p>0$, the set $\mathbb{N}_{G, \boldsymbol{k}}$ is an ideal of the ring $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$, and $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))/\mathbb{N}_{G, \boldsymbol{k}}$ is a filed.* *Proof.* See [@MR1225257 Proposition 3 ] and [@MR1225257 Corollary 3]. ◻ **Lemma 10**. *Let $G\in \mathcal{G}$, $p$ be a prime, and $\boldsymbol{k}$ be a field of characteristic $p$. Let $J\subseteq\mathbb{W}(\boldsymbol{k})$ be a complete system of representatives of the residue class field of $\boldsymbol{k}$. Then every element $\alpha=\sum_{g\in G}\alpha_{g}t^{g}\in \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$ is equivalent to an element $\beta=\sum_{g\in G}\beta_{g}t^{g}$ modulo $\mathbb{N}_{G, K}$, where $\beta_{g}$ is in $J$. In addition, for every $x\in \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$, the family $\{\beta_{g}\}_{g\in G}$ is unique and $\text{{\rm supp}}(\beta)\subseteq\text{{\rm supp}}(\alpha)+\mathbb{Z}_{\ge 0}$.* *Proof.* See [@MR1225257 Proposition 4]. ◻ **Definition 1**. Based on Lemma [Lemma 10](#lem:repbeta){reference-type="ref" reference="lem:repbeta"}, for a fixed complete system $J$ of representatives, and for every $a\in \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$, we denote by $\mathrm{St}_{G, \boldsymbol{k}, J}(a)$ the standard representation of $a$ with respect to $J$ stated in Lemma [Lemma 10](#lem:repbeta){reference-type="ref" reference="lem:repbeta"}. In this case, we have $\text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(a))\subseteq\text{{\rm supp}}(a)+\mathbb{Z}_{\ge 0}$. Let $\mathbb{P}_{p}(G, \boldsymbol{k})$ denote the quotient field $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))/\mathbb{N}_{G, \boldsymbol{k}}$, and let $\mathrm{Pr}\colon \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))\to \mathbb{P}_{p}(G, \boldsymbol{k})$ denote the canonical projection. We define $V_{G, \boldsymbol{k}, p}\colon\mathbb{P}_{p}(G, \boldsymbol{k})\to G$ by $V_{G, \boldsymbol{k}, p}(x)=\min \text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(x))$. **Proposition 11**. *Let $G\in \mathcal{G}$, and $\boldsymbol{k}$ be a perfect field of characteristic $p>0$. Take a complete system $J\subseteq\mathbb{W}(\boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$. Then the following statements are true:* 1. *For every $x\in \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$, the value $\min \text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(x))$ is independent from the choice of $J$.* 2. *The map $V_{G, \boldsymbol{k}, p}$ is a valuation on $\mathbb{P}_{p}(G, \boldsymbol{k})$.* *Proof.* See [@MR1225257 Proposition 5]. ◻ We call the valued field $(\mathbb{P}_{p}(G, \boldsymbol{k}), V_{G, \boldsymbol{k}, p})$ the *$p$-adic Mal'cev--Neumann field* or *$p$-adic Hahn field*. Notice that $(\mathbb{P}_{p}(\mathbb{Z}, \mathbb{F}_{p}), V_{\mathbb{Z}, \mathbb{F}_{p}, p})$ is nothing but the $(\mathbb{Q}_{p}, v_{p})$ field of $p$-adic numbers. To consider characteristics of a valued field and its residue class field, we supplementally define $\mathcal{CH}$ by the set of all pairs $(q, p)$ such that $q$ and $p$ are $0$ or a prime satisfying the either of the following conditions: 1. [\[item:condq1\]]{#item:condq1 label="item:condq1"} $q=p$, 2. [\[item:condq2\]]{#item:condq2 label="item:condq2"} $q=0$ and $0<p$. Note that $(q, p)\in \mathcal{CH}$ satisfies [\[item:condq2\]](#item:condq2){reference-type="ref" reference="item:condq2"} if and only if $q\neq p$. In order to discuss $p$-adic and ordinary Hahn fields in unified manner, we make a notation as follows. **Definition 2**. Let $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, and let $\boldsymbol{k}$ be a perfect field of characteristic $p$. We define a field $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $$\mathbb{A}_{q, p}(G, \boldsymbol{k}) =\begin{cases} \mathbb{H}(G, \boldsymbol{k}) & \text{if $q=p$;}\\ \mathbb{P}_{p}(G, \boldsymbol{k}) & \text{if $q\neq p$.} \end{cases}$$ We also define a valuation $U_{G, \boldsymbol{k}, q, p}$ on $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $$U_{G, \boldsymbol{k}, q, p} =\begin{cases} v_{G, \boldsymbol{k}} & \text{if $q=p$;}\\ V_{G, \boldsymbol{k}, p} & \text{if $q\neq p$.} \end{cases}$$ A metric space $(X, d)$ is said to be *spherically complete* if for every sequence of (closed or open) balls $\{B_{i}\}_{i\in \mathbb{Z}_{\ge 0}}$ with $B_{i+1}\subseteq B_{i+1}$ for all $n\in\mathbb{Z}_{\ge 0}$, we have $\bigcap_{i\in \mathbb{Z}_{\ge 0}}B_{i}\neq \emptyset$. **Proposition 12**. *Let $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, and $\boldsymbol{k}$ be a field of characteristic $p$. Then the following statements are true:* 1. *[\[item:gg1\]]{#item:gg1 label="item:gg1"} $U_{G, \boldsymbol{k}, q, p}(\mathbb{A}_{q, p}(G, \boldsymbol{k}))=G$;* 2. *[\[item:kk1\]]{#item:kk1 label="item:kk1"} $\mathfrak{K}(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})=\boldsymbol{k}$;* 3. *[\[item:o1\]]{#item:o1 label="item:o1"} $(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$ is spherically complete. In particular, it is complete.* *Proof.* The statements [\[item:gg1\]](#item:gg1){reference-type="ref" reference="item:gg1"} and [\[item:kk1\]](#item:kk1){reference-type="ref" reference="item:kk1"} follows from the construction of $(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$. The statement [\[item:o1\]](#item:o1){reference-type="ref" reference="item:o1"} is proven by [@MR1225257 Theorem 1], and [@MR0006161 Theorem 4] (see also [@MR3782290 Theorem 6.11]). ◻ Next we consider homeomorphic embeddings between $p$-adic or ordinary Hahn fields. We begin with ordinary ones. Let $G, H\in \mathcal{G}$ with $G\subseteq H$, and $A, B$ be commutative rings. We represent $\iota$ as the inclusion map $G\to H$. Let $\phi\colon A\to B$ be a ring homomorphism. For $x=\sum_{g\in G}x_{g}t^{g}\in \mathbb{H}(G, A)$, we define $\mathbb{H}(\iota, \phi)(x)\in \mathbb{H}(H, B)$ by $\mathbb{H}(\iota, \phi)(x)=\sum_{h\in H}y_{h}t^{h}$, where $$y_{h} = \begin{cases} \phi(x_{h}) & \text{if $h\in H$;}\\ 0 & \text{if $h\not \in H$.} \end{cases}$$ Then $\mathbb{H}(\iota, \phi)\colon \mathbb{H}(G, A) \to \mathbb{H}(H, B)$ becomes a map. If $G=H$, we simply write it as $\mathbb{H}(G, \phi)$. Let us observe properties of $\mathbb{H}(\iota, \phi)$. **Proposition 13**. *Let $A, B$ be commutative rings, and $\phi\colon A\to B$ be a ring homomorphism. Let $G, H\in \mathcal{G}$ such that $G\subseteq H$, and denote by $\iota$ the inclusion map $G\to H$. Then the map $\mathbb{H}(\iota, \phi)\colon \mathbb{H}(G, A)\to \mathbb{H}(H, B)$ is a ring homomorphism and satisfies $\zeta_{\mathbb{H}(H, B)}\circ \mathbb{H}(\iota, \phi)= \phi\circ \zeta_{ \mathbb{H}(G, A)}$ on $\mathfrak{A}(\mathbb{H}(G, A), v_{G, A})$.* *Proof.* The lemma follows from the definitions of $\mathbb{H}(\iota, \phi)$ and Hahn fields. ◻ Next we discuss $p$-adic Hahn fields, which is defined by quotient fields of Hahn rings. **Proposition 14**. *Let $G, H\in \mathcal{G}$, $\boldsymbol{k}$ and $\boldsymbol{l}$ be perfect field with characteristic $p>0$ and $\phi\colon \boldsymbol{k}\to \boldsymbol{l}$ is a homomorphism. Then the homomorphism $\mathbb{H}(\iota, \mathbb{W}(\phi))\colon \mathbb{H}(G, \mathbb{W}(\boldsymbol{k})) \to \mathbb{H}(H, \mathbb{W}(\boldsymbol{l}))$ satisfies $$\mathbb{H}(\iota, \mathbb{W}(\phi))(\mathbb{N}_{G, \boldsymbol{k}})\subseteq\mathbb{N}_{H, \boldsymbol{l}}.$$ in particular, the map $\mathbb{H}(\iota, \mathbb{W}(\phi))$ induces a homomorphism $$\mathbb{P}_{p}(\iota, \phi)\colon \mathbb{P}_{p}(G, \boldsymbol{k}) \to \mathbb{P}_{p}(H, \boldsymbol{l})$$ such that $\zeta_{\mathbb{P}_{p}(H, \boldsymbol{l})}\circ \mathbb{P}_{p}(\iota, \phi)=\phi\circ \zeta_{\mathbb{P}_{p}(G, \boldsymbol{k})}$ on $\mathfrak{A}(\mathbb{P}_{p}(G, \boldsymbol{k}), V_{G, \boldsymbol{k}, p})$.* *Proof.* Take $x\in \mathbb{N}_{G, \boldsymbol{k}}$ and put $x=\sum_{g\in G}x(g)p^{g}$. Then for every $g\in G$ we have $\sum_{n\in \mathbb{Z}}x(g+n)p^{n}=0$ in $\mathbb{W}(\boldsymbol{k})$. Note that for a fixed $g\in G$ and for a sufficient large $m\in \mathbb{Z}_{\ge 0}$, we have $x(g+n)=0$ for all $n<-m$ since $\{\, g\in G\mid x(g)\neq 0\, \}$ is well-ordered. By the strong triangle inequality, $\sum_{n\in \mathbb{Z}}x(g+n)p^{n}=0$ is equivalent to $x(g+n)p^{n}\to 0$ in $\mathbb{W}(\boldsymbol{k})$ as $n\to \infty$ (see [@MR3782290 Theorem 2.24]). Since $w_{\boldsymbol{l}}(\mathbb{W}(\phi)(x)= w_{\boldsymbol{k}}(x)$ for all $x\in \mathbb{W}(\boldsymbol{k})$ (see Proposition [Proposition 8](#prop:functorwitt){reference-type="ref" reference="prop:functorwitt"}), we also have $\mathbb{W}(\phi)(x(g+n))p^{n}\to 0$ in $\mathbb{W}(\boldsymbol{l})$ as $n\to \infty$. Thus the infinite sum $\sum_{n\in \mathbb{Z}}\mathbb{W}(\phi)(x(g+n))p^{n}$ is convergent and we have $$\sum_{n\in \mathbb{Z}}\mathbb{W}(\phi)(x(g+n))p^{n} =\mathbb{W}(\phi)\left (\sum_{n\in \mathbb{Z}}x(g+n)p^{n}\right) =\mathbb{W}(\phi)(0) =0.$$ This shows that $\mathbb{H}(\iota, \mathbb{W}(\phi))(x) \in \mathbb{N}_{H, \boldsymbol{l}}$, and hence $$\mathbb{H}(\iota, \mathbb{W}(\phi))(\mathbb{N}_{G, \boldsymbol{k}})\subseteq\mathbb{N}_{H, \boldsymbol{l}}.$$ In particular the map $\mathbb{H}(\iota, \mathbb{W}(\phi))$ induces a map $\mathbb{P}_{p}(\iota, \phi)\colon \mathbb{P}_{p}(G, \boldsymbol{k})\to \mathbb{P}_{p}(G, \boldsymbol{k})$. Proposition [Proposition 13](#prop:hahnfuntor){reference-type="ref" reference="prop:hahnfuntor"} implies that a map $\mathbb{H}(\iota, \mathbb{W}(\phi))\colon \mathbb{H}(H, \mathbb{W}(\boldsymbol{k}))\to \mathbb{H}(G, \mathbb{W}(\boldsymbol{l}))$ satisfies $\zeta\circ \mathbb{H}(\iota, \phi)=\phi\circ \zeta$. Then $\zeta\circ \mathbb{P}_{p}(\iota, \phi)=\phi\circ \zeta$. ◻ Let $G, H\in \mathcal{G}$ with $G\subseteq H$, $(q, p)\in \mathcal{CH}$, $\boldsymbol{k}$ and $\boldsymbol{l}$ be fields of characteristic $p$, and $\phi\colon \boldsymbol{k}\to \boldsymbol{l}$ be a homomorphism. Denote by $\iota\colon G\to H$ the inclusion map. We define $$\mathbb{A}_{q, p}(\iota, \phi) = \begin{cases} \mathbb{H}(\iota, \phi) & \text{if $q=p$;}\\ \mathbb{P}_{p}(\iota, \phi) & \text{if $q\neq p$.} \end{cases}$$ Let $(K, v)$ and $(L, w)$ be valued fields. We say that $(L, w)$ is a *valued field extension* of $(K, v)$ as a valued field if $K\subseteq L$ and $w|_{K}=v$. **Proposition 15**. *Let $\boldsymbol{k}$ and $\boldsymbol{l}$ be perfect field with characteristic $p>0$ and $\phi\colon \boldsymbol{k}\to \boldsymbol{l}$ be a homomorphism. The map $\mathbb{A}_{q, p}(\iota, \phi)\colon \mathbb{A}_{q, p}(G, \boldsymbol{k}) \to \mathbb{A}_{q, p}(H, \boldsymbol{l})$ is a homomorphism such that $\zeta_{\mathbb{A}_{q, p}(H, \boldsymbol{l})}\circ \mathbb{A}_{q, p}(\iota, \phi)=\phi\circ \zeta_{\mathbb{A}_{q, p}(G, \boldsymbol{k})}$ on $\mathfrak{A}(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$. In addition, if $q\neq p$, then, the following are true:* 1. *[\[item:zp1\]]{#item:zp1 label="item:zp1"} $(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$ is a valued field extension of $(\mathrm{Fr}\mathbb{W}(\boldsymbol{k}), w_{\boldsymbol{k}})$;* 2. *[\[item:zp2\]]{#item:zp2 label="item:zp2"} In particular, $(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$ is a valued field extension of $(\mathbb{Q}_{p}, v_{p})$.* *Proof.* Propositions [Proposition 13](#prop:hahnfuntor){reference-type="ref" reference="prop:hahnfuntor"} and [Proposition 14](#prop:pfunctor){reference-type="ref" reference="prop:pfunctor"} shows the former part of the statement. By the former part, since $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})=\mathbb{A}_{q, p}(\mathbb{Z}, \boldsymbol{k})$ and $\mathbb{Z}\subseteq G$ (see the definition of $\mathcal{G}$), we can regard $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ as a subfield of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ Namely, [\[item:zp1\]](#item:zp1){reference-type="ref" reference="item:zp1"} is true. Similarly, by $\mathbb{Z}\subseteq G$, $\mathbb{F}_{p}\subseteq\boldsymbol{k}$, and since $(\mathbb{Q}_{p}, v_{p})$ is equal to $(\mathbb{A}_{q, p}(\mathbb{Z}, \mathbb{F}_{p}), U_{\mathbb{Z}, \mathbb{F}_{p}, q, p})$, the field $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ is a valued field extension of $(\mathbb{Q}_{p}, v_{p})$. This implies [\[item:zp2\]](#item:zp2){reference-type="ref" reference="item:zp2"}. ◻ **Definition 3**. Let $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, and let $\boldsymbol{k}$ be a perfect field of characteristic $p$. We make the following assumptions and definitions. 1. [\[item:df:111\]]{#item:df:111 label="item:df:111"} In the rest of the paper, whenever we take a complete system $J\subseteq\mathbb{A}_{q, p}(G, \boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$, in the case of $q=p$, we define $J=\boldsymbol{k}$ using the fact that $\boldsymbol{k}\subseteq\mathbb{H}(G, \boldsymbol{k})$. In the case of $q\neq p$, we take $J\subseteq\mathbb{P}_{p}(G, \boldsymbol{k})$ such that $J\subseteq\mathbb{W}(\boldsymbol{k})$ based on Proposition [Proposition 15](#prop:embedcoef){reference-type="ref" reference="prop:embedcoef"}. 2. [\[item:df:222\]]{#item:df:222 label="item:df:222"} We define an element $\tau\in \mathbb{A}_{q, p}(G, \boldsymbol{k})$ as follows. If $q=p$, we define $\tau$ by an indeterminate as in the definition of Hahn fields. If $q\neq p$, we define $\tau=p \in \mathbb{P}_{p}(G, \boldsymbol{k})$. In this case, every element of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ can be represented as a power series of $\tau$ with powers in $G$. 3. For $y\in \mathbb{A}_{q, p}(G, \boldsymbol{k})$, and $g\in G$, if $q=p$, then we define $\boldsymbol{C}(y, g)$ the coefficient of $\tau^{g}$ in the power series representation of $y$. If $q\neq p$, then we fixed a complete system $J\subseteq \mathbb{W}(\boldsymbol{k})$ of representatives of $\boldsymbol{k}$, and we define $\boldsymbol{C}(y, g)\in J$ the coefficient of $y$ with respect to $\tau^{g}$. Of cause, in this case, the value $\boldsymbol{C}(y, g)$ depends on a system $J$ of representatives. Throughout this paper, we will not consider the situation where we change a system of representatives. Thus no confusion can arise even if $J$ does not explicitly appear in the notation of $\boldsymbol{C}(y, g)$. Notice that we can represent $y=\sum_{g\in G}\boldsymbol{C}(y, g)\tau^{g}$. *Remark 2*. Related to Proposition [Proposition 15](#prop:embedcoef){reference-type="ref" reference="prop:embedcoef"}, we make the next remarks. 1. $\mathbb{A}_{q, p}(\iota, \phi)$ is injective since it is a homomorphism between fields. 2. The construction of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ is a functor. 3. In contrast to Proposition [Proposition 8](#prop:functorwitt){reference-type="ref" reference="prop:functorwitt"}, the author does not know whether $\mathbb{A}_{q, p}(\iota, \phi)$ is a unique homeomorphism such that $\zeta\circ \mathbb{A}_{q, p}(\iota, \phi)=\phi\circ \zeta$ or not. A group $G$ is said to be *divisible* if for every $g\in G$ and for every $n\in \mathbb{Z}_{\ge 0}$ there exists $h\in G$ such that $g=n\cdot h$. **Proposition 16**. *Let $G\in \mathcal{G}$ be divisible, $(q, p)\in \mathcal{CH}$, $\boldsymbol{k}$ be an algebraically closed field with characteristic $p>0$, and $(K, v)$ be a valued field such that $v(K)\subseteq G\sqcup\{\infty\}$ and $\mathfrak{K}(K, v)\subseteq\boldsymbol{k}$. Then there exists a homomorphic embedding $\phi\colon K\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ such that $v(x)=U_{G, \boldsymbol{k}, q, p}(\phi(x))$ fot all $x\in K$. Namely, the field $(K, v)$ can be regarded as a valued subfield of $(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$.* *Proof.* See [@MR1225257 Corollary 5]. ◻ ### Levi--Civita fields We next discuss Levi--Civita fields and $p$-adic Levi--Civita fields, which are will be used in Section [5](#sec:Urysohn){reference-type="ref" reference="sec:Urysohn"}. Let $G\in \mathcal{G}$, and $\boldsymbol{k}$ be a field. For $f\in \mathbb{H}(G, \boldsymbol{k})$, in this subsection, we mainly consider the following condition. 1. [\[item:19finc\]]{#item:19finc label="item:19finc"} For every $n\in \mathbb{Z}$, the set $\text{{\rm supp}}(f)\cap (-\infty, n]$ is finite. We denote by $\mathbb{L}[G, \boldsymbol{k}]$ the set of all $f\in \mathbb{H}(G, \boldsymbol{k})$ satisfying the condition [\[item:19finc\]](#item:19finc){reference-type="ref" reference="item:19finc"}. For the next lemma, we refer the readers to [@MR3782290 Theorem 3.18]. **Lemma 17**. *Let $G\in \mathcal{G}$, $\boldsymbol{k}$ be a field. Then the set $\mathbb{L}[G, \boldsymbol{k}]$ is a subfield of $\mathbb{H}(G, \boldsymbol{k})$.* We call $\mathbb{L}[G, \boldsymbol{k}]$ the *Levi--Civita field associated with $G$ and $\boldsymbol{k}$* Fix $(q, p)\in \mathcal{CH}$ with $q\neq p$ and assume that $\boldsymbol{k}$ has characteristic $p>0$. Before defining a $p$-adic analogue of Levi-Civita fields, we supplementally define a subset $\mathbb{D}[G, \boldsymbol{k}]$ of $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$ $\mathbb{M}_{p}[G, \boldsymbol{k}]$ by the set of all members $f\in \mathbb{M}_{p}[G, \boldsymbol{k}]$ satisfying the condition [\[item:19finc\]](#item:19finc){reference-type="ref" reference="item:19finc"}. We define $\mathbb{M}_{p}[G, \boldsymbol{k}]$ by $\mathbb{M}_{p}[G, \boldsymbol{k}]= \mathrm{Pr}(\mathbb{D}[G, \boldsymbol{k}])$, where $\mathrm{Pr}\colon \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))\to \mathbb{P}_{p}(G, \boldsymbol{k})$ is the canonical projection. **Lemma 18**. *Let $G\in \mathcal{G}$, $p$ be a prime, and $\boldsymbol{k}$ be a field of characteristic $p>0$. Then the following statements are true:* 1. *[\[item:ddd1\]]{#item:ddd1 label="item:ddd1"} For every complete system $J\subseteq\mathbb{P}_{p}(G, \boldsymbol{k})$ of representatives of $\boldsymbol{k}$, and for every $a\in \mathbb{D}[G, \boldsymbol{k}]$, the member $f=\mathrm{St}_{G, \boldsymbol{k}, J}\in \mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))(x)$ satisfies the condition [\[item:19finc\]](#item:19finc){reference-type="ref" reference="item:19finc"} and $f(g)\in J$ for all $g\in G$.* 2. *[\[item:ddd2\]]{#item:ddd2 label="item:ddd2"} We have $\mathbb{D}[G, \boldsymbol{k}]$ is a subring of $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$;* 3. *[\[item:ddd3\]]{#item:ddd3 label="item:ddd3"} If $a\in \mathbb{D}[G, \boldsymbol{k}]$ satisfies $U_{G, \boldsymbol{k}, q, p}(a)>0$, then $(1-a)^{-1}\in \mathbb{D}[G, \boldsymbol{k}]$.* 4. *[\[item:ddd4\]]{#item:ddd4 label="item:ddd4"} For every $a\in \mathbb{D}[G, \boldsymbol{k}]$, there exists $b\in \mathbb{D}[G, \boldsymbol{k}]$ such that $ab$ is equivalent to $1$ modulo $\mathbb{N}_{G, \boldsymbol{k}}$.* *Proof.* First we prove [\[item:ddd1\]](#item:ddd1){reference-type="ref" reference="item:ddd1"}. By Lemma [Lemma 10](#lem:repbeta){reference-type="ref" reference="lem:repbeta"}, we see that $f(g)\in J$ for all $g\in G$. Put $A=\text{{\rm supp}}(a)$ and $F=\text{{\rm supp}}(f)$. Lemma [Lemma 10](#lem:repbeta){reference-type="ref" reference="lem:repbeta"} also shows that $F\subseteq A+\mathbb{Z}_{\ge 0}$. Due to this relation, since $A$ satisfies the condition [\[item:19finc\]](#item:19finc){reference-type="ref" reference="item:19finc"}, so does $W$. Hence the statement [\[item:ffff\]](#item:ffff){reference-type="ref" reference="item:ffff"} is true. Next we prove [\[item:ddd2\]](#item:ddd2){reference-type="ref" reference="item:ddd2"}. By the definitions of $\mathbb{D}[G, \boldsymbol{k}]$ and $\mathbb{L}[G, \mathrm{Fr}\mathbb{W}(\boldsymbol{k})]$, we have $\mathbb{D}[G, \boldsymbol{k}]=\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))\cap \mathbb{L}[G, \mathrm{Fr}\mathbb{W}(\boldsymbol{k})]$ (pay an attention to the difference between $\mathbb{W}(\boldsymbol{k})$ and $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ appearing in $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$ and $\mathbb{L}[G, \mathrm{Fr}\mathbb{W}(\boldsymbol{k})]$, respectively). Thus $\mathbb{D}[G, \boldsymbol{k}]$ is a subring of $\mathbb{H}(G, \mathbb{W}(\boldsymbol{k}))$. Now we show [\[item:ddd3\]](#item:ddd3){reference-type="ref" reference="item:ddd3"}. Put $m=U_{G, \boldsymbol{k}, q, p}(a)$. As in [@MR1225257], we have $(1-a)^{-1}=1+a^{2}+a^{3}+\cdots$ in $\mathbb{H}(G, \boldsymbol{k})$. For every $i\in \mathbb{Z}_{\ge 0}$, we have $A_{i}=\text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(a^{i}))$. In this case, we have $i\cdot m\le \min A_{i}$ for all $i\in \mathbb{Z}_{\ge 1}$. Put $B=\bigcup_{i\in \mathbb{Z}_{\ge 1}} A_{i}$. Since $A_{i}$ satisfies $i\cdot m\le \min A_{i}$ and $(-\infty, n]\cap A_{i}$ is finite for all $i, n\in \mathbb{Z}_{\ge 0}$, we observe that $B$ satisfies that $(-\infty, n]\cap B$ is finite for all $n\in \mathbb{Z}_{\ge 0}$. Put $E=\text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(1-a))$. Then $E\subseteq B+\mathbb{Z}_{\ge 0}$, and hence $E$ satisfies that $(-\infty, n]\cap E$ is finite for all $n\in \mathbb{Z}_{\ge 0}$. This implies that $(1-a)^{-1}\in \mathbb{D}[G, \boldsymbol{k}]$. We shall prove [\[item:ddd4\]](#item:ddd4){reference-type="ref" reference="item:ddd4"} Put $f=\mathrm{St}_{G, \boldsymbol{k}, J}(a)$. We only need to show that $f$ is invertible in $\mathbb{D}[G, \boldsymbol{k}]$. We represent $f$ as $f=\sum_{g\in G}f_{g}t^{g}$. Take $m=\min \text{{\rm supp}}(f)$ and put $y=\sum_{m<g}f_{g}t^{g-m}$. Since $w_{\boldsymbol{k}}(f_{m})=0$, we see that $f_{m}$ is invertible in $\mathbb{W}(\boldsymbol{k})$. Then $f=f_{m}t^{m}(1-(-f_{m}^{-1}y))$. $U_{G, \boldsymbol{k}, q, p}(f_{m}^{-1}y)>0$. Thus, by $f=p^{m}f_{m}(1-(-f_{m}^{-1}y))$, and by [\[item:ddd3\]](#item:ddd3){reference-type="ref" reference="item:ddd3"}, we see that $x$ is invertible in $\mathbb{D}[G, \boldsymbol{k}]$. ◻ As a consequence of Lemma [Lemma 18](#lem:dfield){reference-type="ref" reference="lem:dfield"}, we obtain the next corollary. **Corollary 19**. *Let $G\in \mathcal{G}$, $p$ be a prime, and $\boldsymbol{k}$ be a field of characteristic $p>0$. Then the following statements are true:* 1. *[\[item:ffff\]]{#item:ffff label="item:ffff"} For every complete system $J\subseteq\mathbb{P}_{p}(G, \boldsymbol{k})$ of representatives of $\boldsymbol{k}$, the set $\mathbb{M}_{p}[G, \boldsymbol{k}]$ is equal to the set $\mathrm{Pr}(\mathrm{St}_{G, \boldsymbol{k}, J}(\mathbb{D}[G, \boldsymbol{k}]))$. Moreover, for every $a\in \mathbb{M}_{p}[G, \boldsymbol{k}]$, there uniquely exists $f\in \mathrm{St}_{G, \boldsymbol{k}, J}(\mathbb{D}[G, \boldsymbol{k}])$ such that $a=\mathrm{Pr}(f)$.* 2. *[\[item:gggg\]]{#item:gggg label="item:gggg"} The set $\mathbb{M}_{p}[G, \boldsymbol{k}]$ is a subfield of $\mathbb{P}_{p}(G, \boldsymbol{k})$.* We call $\mathbb{M}_{p}[G, \boldsymbol{k}]$ the *$p$-adic Levi--Civita field associated with $G$ and $\boldsymbol{k}$*. We simply represent the restriction $U_{G, \boldsymbol{k}, q, p}|_{\mathbb{B}_{q, p}(G, \boldsymbol{k})}$ as the same symbol $U_{G, \boldsymbol{k}, q, p}$. In this setting, the field $(\mathbb{B}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$ becomes a valued subfield of $(\mathbb{A}_{q, p}(G, \boldsymbol{k}), U_{G, \boldsymbol{k}, q, p})$. To use $p$-adic and ordinary Levi-Civita fields in a unified way, we make the next definition. **Definition 4**. Let $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, and $\boldsymbol{k}$ be a field of characteristic $p$. We define $\mathbb{B}_{q, p}(G, \boldsymbol{k})$ by $$\mathbb{B}_{q, p}(G, \boldsymbol{k}) =\begin{cases} \mathbb{L}[G, \boldsymbol{k}] & \text{if $q=p$;}\\ \mathbb{M}_{p}[G, \boldsymbol{k}] & \text{if $q\neq p$.} \end{cases}$$ **Proposition 20**. *Let $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, and $\boldsymbol{k}$ be a field of characteristic $p$. Then the set $\mathbb{B}_{q, p}(G, \boldsymbol{k})$ is a subfield of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$.* *Proof.* The case of $q=p$ is presented in Lemma [Lemma 17](#lem:levicivita){reference-type="ref" reference="lem:levicivita"}. The case of $q\neq p$ is proven in Corollary [Corollary 19](#cor:216){reference-type="ref" reference="cor:216"}. ◻ # Algebraic independence over valued fields {#sec:propf} First we remark that, for valued fields $(K, v)$ and $(L, w)$ such that $(L, w)$ is a valued field extension of $(K, v)$, there exists a canonical injective embedding from $\mathfrak{K}(K, v)$ into $\mathfrak{K}(L, w)$. Namely, we can regard $\mathfrak{K}(K, v)$ as a subset of $\mathfrak{K}(L, w)$ since the inclusion map $\iota\colon K\to L$ satisfies $\iota(\mathfrak{A}(K, v))\subseteq\mathfrak{A}(L, w))$ and $\iota(\mathfrak{o}(K, v))\subseteq \mathfrak{o}(L, w)$. Thus it naturally induce an homomorphic embedding $\mathfrak{K}(K, v)\to \mathfrak{K}(L, w)$. Let $K$ and $L$ be fields with $K\subseteq L$. A member $x$ of $L$ is said to be *transcendental over $K$* if $x$ is not a root of any non-trivial polynomial with coefficients in $K$. A subset $S$ of $L$ is said to be *algebraically independent* if any finite collection $x_{1}, \dots, x_{n}$ in $S$ does not satisfy any non-trivial polynomial equation with coefficients in $K$. Note that a singleton $\{x\}$ of $L$ is algebraically independent over $K$ if and only if $x$ is transcendental over $K$. **Lemma 21**. *Let $(K_{0}, v_{0})$ and $(K_{1}, v_{1})$ be valued fields. Assume that $(K_{1}, v_{1})$ is a valued field extension of $(K_{0}, v_{0})$. If $x\in \mathfrak{A}(K_{1}, v_{1})$ such that $\zeta(x)\in \mathfrak{K}(K_{1}, v_{1})$ is transcendental over $\mathfrak{K}(K_{0}, v_{0})$, then $x$ is transcendental over $K_{0}$.* *Proof.* The lemma follows from [@MR2183496 Theorem 3.4.2]. ◻ **Lemma 22**. *Let $(K_{0}, v_{0})$ and $(K_{1}, v_{1})$ be valued fields such that $(K_{1}, v_{1})$ is a valued field extension of $(K_{0}, v_{0})$. If $x_{1}, \dots, x_{n}, y\in K_{1}$ satisfy that:* 1. *[\[item:algind\]]{#item:algind label="item:algind"} the set $\{x_{1}, \dots, x_{n}\}$ is algebraically independent over $K_{0}$;* 2. *[\[item:gocha\]]{#item:gocha label="item:gocha"} there exists a filed $L$ containing $x_{1},\dots, x_{n}$ and satisfying $K_{0}\subseteq L\subseteq K_{1}$ for which there exist $z\in L$ and $c\in K_{0}$ satisfying that $c(y-z)\in \mathfrak{A}(K_{1}, v_{1})$ and $\zeta(c(y-z))\in \mathfrak{K}(K_{1}, v_{1})$ is transcendental over $\mathfrak{K}(L, v_{1}|_{L})$,* *then the set $\{x_{1}, \dots, x_{n}, y\}$ is algebraically independent over $K_{0}$.* *Proof.* For the sake of contradiction, suppose that the set $\{x_{1}, \dots, x_{n}, y\}$ is not algebraically independent over $K_{0}$. From [\[item:algind\]](#item:algind){reference-type="ref" reference="item:algind"} and the fact that $L$ contains $x_{1}, \dots, x_{n}$, it follows that $y$ is algebraic over $L$, and hence so is $c(y-z)$. Using Lemma [Lemma 21](#lem:onetr){reference-type="ref" reference="lem:onetr"} together with [\[item:gocha\]](#item:gocha){reference-type="ref" reference="item:gocha"}, we see that $c(y-z)$ is transcendental over $L$. This is a contradiction. Therefore, the set $\{x_{1}, \dots, x_{n}, y\}$ is algebraically independent over $K_{0}$. ◻ **Lemma 23**. *Let $\eta\in (1, \infty)$, $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, and let $\boldsymbol{k}$ be a perfect field with characteristic $p$. Fix a cardinal $\theta$ and a complete system $J\subseteq\mathbb{A}_{q, p}(G, \boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$. If a set $S$ of non-zero elements of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ satisfies that:* 1. *[\[item:condp\]]{#item:condp label="item:condp"} for every pair $x, y\in S$ with $x \neq y$, and for every $g\in G$ satisfying that $g\in [v(x-y), \infty)$, if either of $\boldsymbol{C}(x, g)$ and $\boldsymbol{C}(y, g)$ is non-zero, then $\boldsymbol{C}(x, g)\neq \boldsymbol{C}(y, g)$,* *then for every finite subset $A=\{z_{1}, \dots, z_{n}\}$ of $S$, there exist $i\in \{1, \dots, n\}$ and $u\in G$ such that $\boldsymbol{C}(z_{i}, u)\neq 0$ and $\boldsymbol{C}(z_{i}, u)\neq \boldsymbol{C}(z_{j}, u)$ for all $j\in \{1, \dots, n\}$.* *Proof.* Put $u=\min\{\, U_{G, \boldsymbol{k}, q, p}(z_{i}- z_{j})\mid i\neq j\, \}$ and take a pair $\{i, n\}$ such that $u=U_{G, \boldsymbol{k}, q, p}(z_{i}- z_{n})$. Then either $\boldsymbol{C}(z_{i}, u)$ or $\boldsymbol{C}(z_{n}, u)$ is non-zero. We my assume that $\boldsymbol{C}(z_{i}, u)\neq 0$. Using the condition [\[item:condp\]](#item:condp){reference-type="ref" reference="item:condp"} and the minimality of $u$, we have $\boldsymbol{C}(y_{i}, u)\neq \boldsymbol{C}(y_{j}, u)$ for all $j\in \{1, \dots, n\}$. ◻ Let $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, $\boldsymbol{k}$ be a field of characteristic $p$, and $K$ be a subfield of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$. Fix a complete system $J \subseteq\mathbb{A}_{q, p}(G, \boldsymbol{k})$ of representatives if $q\neq p$. We denote by $\mathbf{AH}(K)$ the subfield of $\boldsymbol{k}$ generated by $\{\, \zeta(\boldsymbol{C}(x, g))\mid x\in K, g\in G\, \}$ over $\mathfrak{K}(K, v)$. The definition of $\mathbf{AH}(K)$ is "ad-hoc", which means that it depends on not only information of $K$, but also the inclusion map $K\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$. Namely, even if $K, L\subseteq\mathbb{A}_{q, p}(G, \boldsymbol{k})$ are isomorphic each other as fields, it can happen that $\mathbf{AH}(K)\neq \mathbf{AH}(L)$. **Proposition 24**. *Let $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, and $\boldsymbol{k}, \boldsymbol{l}$ be perfect fields with characteristic $p$ such that $\boldsymbol{k}\subseteq\boldsymbol{l}$, Take a subfield $K$ of $\mathbb{A}_{q, p}(G, \boldsymbol{l})$ such that $\mathfrak{K}(K, v)\subseteq\boldsymbol{k}$. Fix a system $J\subseteq\mathbb{A}_{q, p}(G, \boldsymbol{l})$ of representatives of $\boldsymbol{l}$. If a set of $S$ of non-zero elements of $\mathbb{A}_{q, p}(G, \boldsymbol{l})$ satisfies the condition [\[item:condp\]](#item:condp){reference-type="ref" reference="item:condp"} in Lemma [Lemma 23](#lem:333){reference-type="ref" reference="lem:333"} and the following:* 1. *[\[item:condpttt\]]{#item:condpttt label="item:condpttt"} for every pair $x, y\in S$, and for every distinct pair $g, g^{\prime}\in G$, if $\boldsymbol{C}(x, g)\neq 0$, then we have $\boldsymbol{C}(x, g)\neq \boldsymbol{C}(y, g^{\prime})$;* 2. *[\[item:condalg\]]{#item:condalg label="item:condalg"} the set $\{\, \zeta(\boldsymbol{C}(x, g)) \mid x\in S, g\in G, \boldsymbol{C}(x, g)\neq 0\, \}$ is algebraically independent over $\mathbf{AH}(K)$,* *then the set $S$ is algebraically independent over $K$.* *Proof.* Let $\tau$ be the same element of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ as in Definition [Definition 4](#df:uk){reference-type="ref" reference="df:uk"}. Take $n$-many distinct members $z_{1}, \dots, z_{n}$ in $S$. Now we prove that $\{z_{1}, \dots, z_{n}\}$ is algebraically independent over $K$ by induction on $n$. In the case of $n=1$, put $z_{1}=\sum_{g\in G}\boldsymbol{C}(z_{1}, g)\tau^{g}$. Take $u\in G$ such that $\boldsymbol{C}(z_{1}, u)\neq 0$. Put $A=\{\, \boldsymbol{C}(z_{1}, g)\mid g\in G, g\neq u\, \}$. Then, due to the condition [\[item:condpttt\]](#item:condpttt){reference-type="ref" reference="item:condpttt"}, we have $\boldsymbol{C}(z_{1}, u)\not \in A$. Note that the set $\zeta(A)$ is algebraically independent over $\boldsymbol{k}$. Let $\boldsymbol{m}$ be the perfect subfield of $\boldsymbol{l}$ generated by $\mathbf{AH}(K)\cup \zeta(A)$, and put $L=\mathbb{A}_{q, p}(G, \boldsymbol{m})$. Notice that $\mathbb{A}_{q, p}(G, \boldsymbol{m})$ is a subfield of $\mathbb{A}_{q, p}(G, \boldsymbol{l})$ (see Proposition [Proposition 15](#prop:embedcoef){reference-type="ref" reference="prop:embedcoef"}). The fact that $\mathbf{AH}(K)\subseteq\boldsymbol{m}$ implies that $K\subseteq L$. By assumption [\[item:condalg\]](#item:condalg){reference-type="ref" reference="item:condalg"} and $\boldsymbol{C}(z_{1}, u)\not \in A$, we see that $\boldsymbol{C}(z_{1}, u)$ is transcendental over $\boldsymbol{m}$. Thus, by Lemma [Lemma 21](#lem:onetr){reference-type="ref" reference="lem:onetr"}, we conclude that $z_{1}$ is transcendental over $L$. In particular, $z_{1}$ is transcendental over $K$. Next, we fix $k\in \mathbb{Z}_{\ge 0}$ and assume that the case of $n=k$ is true. We consider the case of $n=k+1$. Since $S$ satisfies the condition [\[item:condp\]](#item:condp){reference-type="ref" reference="item:condp"}, we can take $i\in \{1, \dots, n\}$ and $u\in G$ stated in Lemma [Lemma 23](#lem:333){reference-type="ref" reference="lem:333"}. We may assume that $i=k+1$. Put $$A=\{\, \boldsymbol{C}(z_{i}, g)\mid g\in G, i=1, \dots, k\, \} \cup\{\,\boldsymbol{C}(z_{k+1}, g)\mid g\in G, g\neq u \, \}.$$ According to [\[item:condpttt\]](#item:condpttt){reference-type="ref" reference="item:condpttt"} and the conclusion of Lemma [Lemma 23](#lem:333){reference-type="ref" reference="lem:333"}, we see that $\boldsymbol{C}(z_{k+1}, u)\not\in A$. Let $\boldsymbol{m}$ be a perfect subfield of $\boldsymbol{l}$ generated by $\mathbf{AH}(K)\cup \zeta(A)$, and put $L=\mathbb{A}_{q, p}(G, \boldsymbol{m})$. Similarly to the case of $n=1$, we observe that $K\subseteq L$. Since $A\subseteq\boldsymbol{m}$, we have $z_{i}\in L$ for all $i\in \{1, \dots, k\}$. Define $f=z_{k+1}-\boldsymbol{C}(z_{k+1}, u)\tau^{u}$. Due to the condition [\[item:condpttt\]](#item:condpttt){reference-type="ref" reference="item:condpttt"}, we have $f\in L$. By the definition, we also have $\tau^{-u}(z_{k+1}-f)=\boldsymbol{C}(z_{k+1}, u)$. Thus the condition [\[item:condalg\]](#item:condalg){reference-type="ref" reference="item:condalg"} shows that $\zeta(\tau^{-u}(z_{k+1}-f))$ is transcendental over $\boldsymbol{m}$. Hence Lemma [Lemma 22](#lem:trtr){reference-type="ref" reference="lem:trtr"} shows that the set $\{z_{1}, \dots, z_{k}, z_{k+1}\}$ is algebraically independent over $K$. This finishes the proof. ◻ *Remark 3*. Put $w=\boldsymbol{C}(x, g)$. The condition [\[item:condpttt\]](#item:condpttt){reference-type="ref" reference="item:condpttt"} means that $\boldsymbol{C}(x, g)$ is zero or $g$ is a unique member in $G$ such that $\boldsymbol{C}(y, g)=w$ for some $y\in X$. # Isometric embeddings of ultrametric spaces {#sec:isom} In this section, we prove our non-Archimedean analogue of the Arens--Eells theorem. As a consequence, we give an affirmative solution of Conjecture [Conjecture 1](#conj:bbb){reference-type="ref" reference="conj:bbb"}. ## A non-Archimedean Arens--Eells theorem {#subsec:naae} ### Preparations This subsection is devoted to proving the following technical theorem, which plays a central role of our first main theorem. Our proof of the next theorem can be considered as a sophisticated version of the proof of the main theorem of [@MR748978]. **Theorem 25**. *Let $\eta\in (1, \infty)$, $(q, p)\in \mathcal{CH}$, $G\in \mathcal{G}$, $\boldsymbol{k}$ be a field, and $\boldsymbol{l}$ be a perfect field of characteristic $p$ Fix a cardinal $\theta$ and a complete system $J\subseteq\mathbb{A}_{q, p}(G, \boldsymbol{l})$ of representatives of the residue class field $\boldsymbol{l}$. Let $C$ be a subset of $J$. Put $R=\{0\}\sqcup \{\, \eta^{-g}\mid g\in G\, \}$. If the following condition are satisfied:* 1. *[\[item:h1\]]{#item:h1 label="item:h1"} $\boldsymbol{l}$ is a field extension of $\boldsymbol{k}$;* 2. *[\[item:h3\]]{#item:h3 label="item:h3"} the subset $\zeta(C)$ of $\boldsymbol{l}$ is algebraically independent over $\boldsymbol{k}$;* 3. *[\[item:h4\]]{#item:h4 label="item:h4"} $\mathop{\mathrm{Card}}(C)=\theta$,* *then for every $R$-valued ultrametric space $(X, d)$ with $\mathop{\mathrm{Card}}(X)\le \theta$, there exists a map $I\colon X\to \mathbb{A}_{q, p}(G, \boldsymbol{l})$ such that:* 1. *[\[item:connzero\]]{#item:connzero label="item:connzero"} each $I(x)$ is non-zero;* 2. *[\[item:conisom\]]{#item:conisom label="item:conisom"} the map $I$ is an isometric embedding from $(X, d)$ into the ultrametric space $(\mathbb{A}_{q, p}(G, \boldsymbol{k}), \lVert *\rVert_{U_{G, \boldsymbol{k}, q, p}, \eta})$;* 3. *[\[item:concfinq\]]{#item:concfinq label="item:concfinq"} for every pair $x, y\in X$, and for every $r\in (0, d(x, y)]$, if either of $\boldsymbol{C}(I(x), -\log_{\eta}(r))$ or $\boldsymbol{C}(I(y), -\log_{\eta}(r))$ is non-zero, then we have $\boldsymbol{C}(I(x), -\log_{\eta}(r)) \neq \boldsymbol{C}(I(y), -\log_{\eta}(r))$;* 4. *[\[item:conconseq\]]{#item:conconseq label="item:conconseq"}* *for every pair $x, y\in X$, and for every distinct pair $g, g^{\prime}\in G$, if $\boldsymbol{C}(I(x), g)\neq 0$, then $\boldsymbol{C}(I(x), g) \neq \boldsymbol{C}(I(y), g^{\prime})$.* 5. *[\[item:concoef\]]{#item:concoef label="item:concoef"}* *for every $\alpha<\theta$, the set $\{\, \boldsymbol{C}(I(\xi_{\alpha}), g)\mid g\in G\, \}$ is contained in $C$.* In this subsection, in what follows, we fix objects in the assumption of Theorem [Theorem 25](#thm:main0){reference-type="ref" reference="thm:main0"}. We divide the proof of Theorem [Theorem 25](#thm:main0){reference-type="ref" reference="thm:main0"} into some lemmas. First we prepare notations. Take $\varpi\not\in X$, and put $E=X\sqcup\{\varpi\}$. Fix $r_{0}\in R\setminus \{0\}$ and $x_{0}\in X$, and define an $R$-valued ultrametric $h$ on $E$ by $h|_{X\times X}=d$ and $h(x, \varpi)=d(x, x_{0})\lor d(x_{0}, \varpi)$. Then $h$ is actually an ultrametric (see, for example, [@Ishiki2021ultra Lemma 5.1]). A one-point extension of a metric space is a traditional method to prove analogues of the Arens--Eells theorem. Put $C=\{\, b_{\alpha}\mid \alpha<\theta\, \}$ and $E=\{\, \xi_{\alpha}\mid \alpha<\theta\, \}$ with $\xi_{0}=\varpi$. For every $\beta<\kappa$, we also put $E_{\beta}=\{\, \xi_{\alpha}\mid \alpha<\beta\, \}$ and $C_{\beta}=\{\, b_{\alpha}\mid \alpha<\beta\, \}$. Fix $\lambda<\theta$. We say that a map $f\colon E_{\lambda}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ is *well-behaved* if the following conditions are true: 1. [\[item:azero\]]{#item:azero label="item:azero"} if $\lambda=0$, then $H_{0}$ is the empty map and if $\lambda>0$, then $H_{\lambda}(\varpi)=0$, where $0$ is the zero element of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$; 2. [\[item:aisom\]]{#item:aisom label="item:aisom"} the map $f$ is an isometric embedding from $E_{\lambda}$ into $\mathbb{A}_{q, p}(G, \boldsymbol{k})$; 3. [\[item:acfinq\]]{#item:acfinq label="item:acfinq"} for every pair $x, y\in E_{\lambda}$, and for every $r\in (0, d(x, y)]$, if either of $\boldsymbol{C}(f(x), -\log_{\eta}(r))$ or $\boldsymbol{C}(f(y), -\log_{\eta}(r))$ is non-zero, then we have $\boldsymbol{C}(f(x), -\log_{\eta}(r))\neq \boldsymbol{C}(f(y), -\log_{\eta}(r))$; 4. [\[item:aseqseq\]]{#item:aseqseq label="item:aseqseq"} for every pair $x, y\in E_{\lambda}$, and for every distinct pair $g, r^{\prime}\in G$, if $\boldsymbol{C}(f(x), g)\neq 0$, then $\boldsymbol{C}(f(x), g)\neq \boldsymbol{C}(f(y), g^{\prime})$; 5. [\[item:acoef\]]{#item:acoef label="item:acoef"} for every $\alpha<\lambda$, the set $\{\, \boldsymbol{C}(f(\xi_{\alpha}), g)\mid g\in G\, \}$ is contained in $C_{\alpha+1}$. For an ordinal $\lambda<\theta$, a family $\{H_{\alpha}\colon E_{\alpha}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})\}_{\alpha<\lambda}$ is said to be *coherent* if the following condition is true: 1. for every $\beta<\theta$ and for every $\alpha<\beta$, we have $H_{\beta}|_{E_{\alpha}}=H_{\alpha}$. For the sake of simplicity, we define an ultrametric $e$ on $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $e(x, y)= \lVert x-y\rVert_{U_{G, \boldsymbol{k}, q, p}, \eta}$. We shall construct a coherent family $\{H_{\alpha}\}_{\alpha<\theta}$ of well-behaved maps using transfinite recursion. We begin with the following convenient criterion. **Lemma 26**. *Fix $\lambda<\theta$ with $\lambda\neq 0$. Put $u=d(E, \xi_{\lambda})$ and $m=-\log_{\eta}(u)$. Let $H_{\lambda}\colon E_{\lambda}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ be a well-behaved map. Assume that an isometric embedding $H_{\lambda+1}\colon E_{\lambda+1}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ satisfies $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ and the following property:* 1. *[\[item:p1p1\]]{#item:p1p1 label="item:p1p1"} For every $g\in G\cap (m, \infty)$, we have $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}), g) =0$;* 2. *[\[item:p2p2\]]{#item:p2p2 label="item:p2p2"} If $m<\infty$ and $m\in G$, then $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}), m) \in \{0, b_{\lambda}\}$;* *Then tha map $H_{\lambda}$ satisfies the conditions [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"}--[\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"}.* *Proof.* First we note that the following claim is true: 1. [\[item:propertyp\]]{#item:propertyp label="item:propertyp"} For every $a\in (u, \infty)$, there exists $z\in E_{\lambda}$ such that $$e(H_{\lambda+1}(z), H_{\lambda}(\xi_{\lambda}))<a.$$ On the other words, for every $g\in (-\infty, m)$, there exists $z\in E_{\lambda}$ such that $$\boldsymbol{C}(H_{\lambda+1}(z), n) = \boldsymbol{C}( H_{\lambda}(\xi_{\lambda}), n)$$ for all $n< g$ Due to [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"} for $H_{\lambda}$, it suffices to consider the case of $y=\xi_{\lambda}$. Take $x\in E_{\lambda+1}$ and $r\in (0, d(x, \xi_{\lambda})]$. Assume that either of $\boldsymbol{C}(f(x), -\log_{\eta}(r))$ or $\boldsymbol{C}(f(\xi_{\lambda}), -\log_{\eta}(r))$ is non-zero. In this case, we have $r\le d(x, \xi_{\lambda})\le u$. Thus $m\le -\log_{\eta}(r)$. If $m< -\log_{\eta}(r)$, then the property [\[item:p1p1\]](#item:p1p1){reference-type="ref" reference="item:p1p1"} shows that $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}), -\log_{\eta}(r))=0$. Thus the condition [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"} is valid. If $m= -\log_{\eta}(r)$, then the property [\[item:p2p2\]](#item:p2p2){reference-type="ref" reference="item:p2p2"} implies that $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}), m)\not \in C_{\lambda}$. Thus, using [\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"} for $H_{\lambda}$, the condition [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"} is satisfied. In any case, we conclude that the condition [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"} is true. Owing to [\[item:aseqseq\]](#item:aseqseq){reference-type="ref" reference="item:aseqseq"} for $H_{\lambda}$, it is enough to confirm the case where $x=\xi_{\lambda}$ and $y\in E_{\lambda}$, or $x\in E_{\lambda}$ and $y=\xi_{\lambda}$. Take arbitrary distinct pair $g, g^{\prime}\in G$. First assume that $x=\xi_{\lambda}$, i.e., $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}), g)\neq 0$. By [\[item:p1p1\]](#item:p1p1){reference-type="ref" reference="item:p1p1"}, we have $g\le m$. In the case of $g=m$, using [\[item:p2p2\]](#item:p2p2){reference-type="ref" reference="item:p2p2"} and [\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"} for $H_{\lambda}$, we have $\boldsymbol{C}(f(x), g)\neq \boldsymbol{C}(f(y), g^{\prime})$. In the case of $g<m$. Then $\eta^{-g}\in (u, \infty)$. Thus the property [\[item:propertyp\]](#item:propertyp){reference-type="ref" reference="item:propertyp"} enables us to take $z\in E_{\lambda}$ such that $e(y_{i}, \xi_{\lambda})<\eta^{-g}$. Thus we also have $\boldsymbol{C}(H_{\lambda+1}(z), a)=\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda+1}), a)$ for all $a\le g$. Applying [\[item:aseqseq\]](#item:aseqseq){reference-type="ref" reference="item:aseqseq"} for $H_{\lambda}$ to $y$ and $z$, we obtain that $\boldsymbol{C}(H_{\lambda+1}(z), g)\neq \boldsymbol{C}(H_{\lambda+1}(y), g^{\prime})$. Hence, we have $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda+1}), g)\neq \boldsymbol{C}(H_{\lambda+1}(y), g^{\prime})$. Second assume that $y=\xi_{\lambda}$, i.e., $\boldsymbol{C}(H_{\lambda+1}(x), g)\neq 0$. By [\[item:p1p1\]](#item:p1p1){reference-type="ref" reference="item:p1p1"}, it is sufficient consider that $g\le m$. By [\[item:p2p2\]](#item:p2p2){reference-type="ref" reference="item:p2p2"}, if $g^{\prime}=m$, then we have $\boldsymbol{C}(H_{\lambda+1}(x), g)\neq \boldsymbol{C}(H_{\lambda+1}(y), g^{\prime})$. If $g^{\prime}<m$, then the property [\[item:propertyp\]](#item:propertyp){reference-type="ref" reference="item:propertyp"} enables us to take $z\in E_{\lambda}$ such that $e(y_{i}, \xi_{\lambda})<\eta^{-g}$. The remaining proof is similar to that of the first case. By [\[item:propertyp\]](#item:propertyp){reference-type="ref" reference="item:propertyp"}, if $g<m$ we have $\boldsymbol{C}(H_{\lambda}(\xi_{\lambda}), g) \in C_{\lambda}$. If $m<g$, [\[item:p1p1\]](#item:p1p1){reference-type="ref" reference="item:p1p1"} implies that $\boldsymbol{C}(H_{\lambda}(\xi_{\lambda}), g)=0$. Thus By [\[item:p2p2\]](#item:p2p2){reference-type="ref" reference="item:p2p2"} we have the set $\{\, \boldsymbol{C}(f(\xi_{\alpha}), g)\mid g\in G\, \}$ is contained in $C_{\lambda}\sqcup\{b_{\lambda}\}=C_{\lambda+1}$. Namely, the condition [\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"} is true. ◻ We next see the elementary lemma. **Lemma 27**. *Let $\tau$ be the same element of $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ as in Definition [Definition 4](#df:uk){reference-type="ref" reference="df:uk"}. Then, for every $a\in \mathbb{A}_{q, p}(G, \boldsymbol{k})$ and for every $r\in (0, \infty)$, we can take $\gamma\in B(a, r; e)$ such that $\gamma=\sum_{g\in G\cap (-\infty, -\log_{\eta}(r))}s_{g}\tau^{g}$, where $s_{g}\in J$. In this case, we have $B(a, r; e)=B(\gamma, r; e)$.* *Proof.* We put $a=\sum_{g\in G\cap (-\infty, -\log_{\eta}(r))}s_{g}\tau^{g}$, where $s_{g}\in J$ and we define $\gamma=\sum_{g\in G\cap (-\infty, -\log_{\eta}(r))}s_{g}\tau^{g}$. By the definition of $e$, or $U_{G, \boldsymbol{k}, q, p}$, we have $\gamma\in B(a, r; e)$. Thus Lemma [Lemma 3](#lem:ultraopcl){reference-type="ref" reference="lem:ultraopcl"} implies that $B(a, r; e)=B(\gamma, r; e)$. ◻ Next we show lemmas corresponding to steps of isolated ordinals in transfinite induction. **Lemma 28**. *Fix $\lambda<\theta$ with $\lambda\neq 0$, and let $H_{\lambda}\colon E_{\lambda}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ be a well-behaved map. If $h(E_{\lambda}, \xi_{\lambda})>0$, then we can obtain a well-behaved isometric embedding $H_{\lambda+1}\colon E_{\lambda+1}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ such that $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$.* *Proof.* Put $Y_{\lambda}=H_{\lambda}(E_{\lambda})$. Put $u=h(E_{\lambda}, \xi_{\lambda})>0$ and $m=-\log_{\eta}(u)$. Let $\tau$ be the same element in $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ as in Lemma [Lemma 27](#lem:center){reference-type="ref" reference="lem:center"}. Case 1. \[There is no $a\in E_{\lambda}$ such that $h(a, \xi_{\lambda})=u$\]: Take a sequence $\{y_{i}\}_{i\in \mathbb{Z}_{\ge 0}}$ in $E_{\lambda}$ such that $h(y_{i+1}, \xi_{\lambda})<h(y_{i}, \xi_{\lambda})$ for all $i\in \mathbb{Z}_{\ge 0}$ and $h(y_{i}, \xi_{\lambda})\to u$ as $i\to \infty$. Put $r_{i}=h(y_{i}, \xi_{\lambda})$. Since $r_{i+1}<r_{i}$ and $B(H(y_{i+1}), r_{i+1}; e)\subseteq B(H(y_{i}), r_{i}; e)$ for all $i\in \mathbb{Z}_{\ge 0}$ (see Lemma [Lemma 3](#lem:ultraopcl){reference-type="ref" reference="lem:ultraopcl"}), using the spherical completeness of $\mathbb{A}_{q, p}(G, \boldsymbol{l})$ ([\[item:o1\]](#item:o1){reference-type="ref" reference="item:o1"} in Proposition [Proposition 12](#prop:propppp){reference-type="ref" reference="prop:propppp"}), we obtain $\bigcap_{i\in \mathbb{Z}_{\ge 0}}B(H_{\lambda}(y_{i}), r_{i}; e)\neq \emptyset$. In this case, the set $\bigcap_{i\in \mathbb{Z}_{\ge 0}}B(H_{\lambda}(y_{i}), r_{i}; e)$ is a closed ball of radius $r$ centered at some point in $\mathbb{P}_{p}(G, \boldsymbol{l})$. Lemma [Lemma 27](#lem:center){reference-type="ref" reference="lem:center"} implies that there exists $\gamma= \sum_{g\in G\cap (-\infty, m)}s_{r}\tau^{g}$ in $\mathbb{A}_{q, p}(G, \boldsymbol{k})$ such that $B(\gamma, r; e)=\bigcap_{i\in \mathbb{Z}_{\ge 0}}B(H_{\lambda}(y_{i}), r_{i}; e)$. We define a map $H_{\lambda+1}\colon E_{\lambda+1}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ and $H_{\lambda+1}(\xi_{\lambda})=\gamma$. This definition implies the condition [\[item:azero\]](#item:azero){reference-type="ref" reference="item:azero"}. Next we verify the condition [\[item:aisom\]](#item:aisom){reference-type="ref" reference="item:aisom"}. It suffices to show that $d(x, \xi_{\lambda})= e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda}))$. Take $n\in \mathbb{Z}_{\ge 0}$ such that $d(y_{n}, \xi_{\lambda})<d(x, \xi_{\lambda})$. Then Lemma [Lemma 2](#lem:isosceles){reference-type="ref" reference="lem:isosceles"} implies $h(x, \xi_{\lambda})=h(y_{n}, x)$, and hence $h(y_{n}, x)=e(H_{\lambda+1}(y_{n}), H_{\lambda+1}(x))$. By $\gamma\in B(H_{\lambda+1}(y_{n}), r_{n}; e)$, we have $$e(H_{\lambda+1}(\xi_{\lambda}), H_{\lambda+1}(y_{n}))\le h(\xi_{\lambda}, y_{n}) < h(\xi_{\lambda}, x)=e(H_{\lambda+1}(y_{n}), H_{\lambda+1}(x)).$$ Thus, using Lemma [Lemma 2](#lem:isosceles){reference-type="ref" reference="lem:isosceles"} again, we have $$e(H_{\lambda+1}(y_{n}), H_{\lambda+1}(x))=e(H_{\lambda+1}(\xi_{\lambda}), H_{\lambda+1}(x)),$$ and hence $e(H_{\lambda+1}(\xi_{\lambda}), H_{\lambda+1}(x))= h(\xi_{\lambda}, x)$. By the construction, the mao $H_{\lambda+1}$ satisfies the properties [\[item:p1p1\]](#item:p1p1){reference-type="ref" reference="item:p1p1"}--[\[item:propertyp\]](#item:propertyp){reference-type="ref" reference="item:propertyp"}. Thus, we see that $H_{\lambda+1}$ satisfies the conditions [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"}--[\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"}. Case 2. \[There exists $a\in X_{\lambda}$ such that $d(a, \xi_{\lambda})=u$\]: By Lemma [Lemma 27](#lem:center){reference-type="ref" reference="lem:center"}, there exists $\gamma=\sum_{g\in G\cap (-\infty, m)}s_{g}\tau^{g}\in \mathbb{A}_{q, p}(G, \boldsymbol{k})$ such that $B(\gamma, u; e)=B(H_{\lambda}(a), u; e)$. We put $$\delta=b_{\lambda}\cdot \tau^{m}+\gamma=b_{\lambda}\cdot \tau^{m}+ \sum_{g\in G\cap (-\infty, m)}s_{g}\tau^{g}.$$ Then Lemma [Lemma 3](#lem:ultraopcl){reference-type="ref" reference="lem:ultraopcl"} states that $B(\delta, u; e)=B(H_{\lambda}(a), u; e)$. We define a map $H_{\lambda+1}\colon E_{\lambda+1}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ and $H_{\lambda+1}(\xi_{\lambda})=\delta$. To verify the condition [\[item:aisom\]](#item:aisom){reference-type="ref" reference="item:aisom"}, we show that $e(H_{\lambda+1}(\xi_{\lambda}), H_{\lambda+1}(x))=e(\xi_{\lambda}, x)$ for all $x\in X_{\lambda}$. If $x\not \in B(a, u; h)$, then we have $h(x, a)=h(x, \xi_{\lambda})$. Similarly, we have $e(H_{\lambda+1}(x), H_{\lambda+1}(a))= e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda}))$. Since $e(H_{\lambda}(x), H_{\lambda}(a))= e(H_{\lambda}(x), H_{\lambda}(a)) =d(x, a)$, we have $$e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda}))=h(x, \xi_{\lambda}).$$ If $x\in B(a, u; h)$, then $h(x, \xi_{\lambda})\le h(x, a)\lor h(a, \xi_{\lambda})\le u$. By the definition of $u$, we have $h(\xi_{\lambda}, x)=u$. We also see that $H_{\lambda+1}(x)\in B(H_{\lambda+1}(a), u; e)$. Then we can represent $H_{\lambda+1}(x)= \gamma+ \sum_{g\in G\cap [m, \infty)}c_{g}\tau^{g}$, where $c_{g}\in C_{\lambda}$. Take $\alpha<\lambda$ with $\xi_{\alpha}=x$. Then the condition [\[item:aisom\]](#item:aisom){reference-type="ref" reference="item:aisom"} for $H_{\lambda}$ implies that $c_{g}\in C_{\alpha+1}$ for all $g\in G\cap (-\infty, m]$. From the definition of $U_{G, \boldsymbol{k}, q, p}$ and $b_{\lambda}\not\in C_{\alpha+1}$, it follows that $e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda})) =u$. Hence $$e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda}))= h(x, \xi_{\lambda}).$$ Similarly to Case 1, by the construction, we see that $H_{\lambda+1}$ satisfies the properties [\[item:p1p1\]](#item:p1p1){reference-type="ref" reference="item:p1p1"}--[\[item:propertyp\]](#item:propertyp){reference-type="ref" reference="item:propertyp"}. Thus, we see that $H_{\lambda+1}$ satisfies the conditions [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"}--[\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"}. ◻ **Lemma 29**. *Fix $\lambda<\theta$ with $\lambda\neq 0$, and let $H_{\lambda}\colon E_{\lambda}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ be a well-behaved map. If $h(E_{\lambda}, \xi_{\lambda})=0$, then there exists a well-behaved map $H_{\lambda+1}\colon E_{\lambda+1}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ such that $H_{\lambda+1}|_{X_{\lambda}}=H_{\lambda}$.* *Proof.* We now define $H_{\lambda+1}(\xi_{\lambda})$ as follows. Take a sequence $\{x_{i}\}$ in $E_{\lambda}$ such that $x_{i}\to \xi_{\lambda}$ as $i\to \infty$, and define $H_{\lambda+1}(\xi_{\lambda})=\lim_{i\to \infty}H_{\lambda}(x_{i})$. The value $H_{\lambda+1}(\xi_{\lambda})$ is independent of the choice of a sequence $\{x_{i}\}_{i\in \mathbb{Z}_{\ge 0}}$. First we confirm that $H_{\lambda+1}$ is well-behaved. Since $H_{\lambda}$ satisfies the condition [\[item:azero\]](#item:azero){reference-type="ref" reference="item:azero"}, so does $H_{\lambda+1}$. To prove [\[item:aisom\]](#item:aisom){reference-type="ref" reference="item:aisom"}, it is enough to show $d(x, \xi_{\xi_{\lambda}})=e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda}))$ for all $x\in E_{\lambda+1}$. Take a sufficient large $n\in \mathbb{Z}_{\ge 0}$ so that $d(x_{n}, \xi_{\lambda})<d(x, \xi_{\lambda})$ and $e(H_{\lambda+1}(x_{n}), H_{\lambda+1}(\xi_{\lambda}))< e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda}))$. Lemma [Lemma 2](#lem:isosceles){reference-type="ref" reference="lem:isosceles"} implies that $$d(x, \xi_{\lambda})=d(x_{n}, x)$$ and $$e(H_{\lambda+1}(x), H_{\lambda+1}(\xi_{\lambda})) = e(H_{\lambda+1}(x_{n}), H_{\lambda+1}(x)).$$ Since $H_{\lambda}$ is an isometry and $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$, we have $$d(x_{n}, x)=e(H_{\lambda+1}(x_{n}), H_{\lambda+1}(x)).$$ Therefore we conclude that the condition [\[item:aisom\]](#item:aisom){reference-type="ref" reference="item:aisom"} is ture. By the construction, we see that $H_{\lambda+1}$ satisfies the properties [\[item:p1p1\]](#item:p1p1){reference-type="ref" reference="item:p1p1"}--[\[item:propertyp\]](#item:propertyp){reference-type="ref" reference="item:propertyp"}. Thus, we see that $H_{\lambda+1}$ satisfies the conditions [\[item:acfinq\]](#item:acfinq){reference-type="ref" reference="item:acfinq"}--[\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"}. ◻ Let us prove Theorem [Theorem 25](#thm:main0){reference-type="ref" reference="thm:main0"}. *Proof of Theorem [Theorem 25](#thm:main0){reference-type="ref" reference="thm:main0"}.* The proof is based on [@MR748978]. Using transfinite recursion, we first construct a coherent family $\{H_{\mu}\colon E_{\mu}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})\}_{\mu<\theta}$ of well-behaved maps. Fix $\mu\le \theta$ and assume that we have already obtained a coherent family $\{H_{\alpha}\}_{\alpha<\mu}$ of well-behaved maps. Now we construct $H_{\mu}$ as follows. If $\lambda=0$, then we define $H_{0}$ as the empty map. If $\lambda=1$, then we have $E_{1}=\{b_{0}\}$ and we define $H_{1}\colon E_{1}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $H_{1}(\xi_{0})=0$. If $\mu=\lambda+1$ for some $\lambda<\mu$ with $\lambda\neq 0$, then using Lemmas [Lemma 28](#lem:0dis){reference-type="ref" reference="lem:0dis"} and [Lemma 29](#lem:podis){reference-type="ref" reference="lem:podis"}, we obtain a well-behaved map $H_{\lambda+1}\colon E_{\lambda+1}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ such that $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$. If $\mu$ is a limit ordinal, then we define $H_{\mu}\colon E_{\mu}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $H_{\mu}(x)=H_{\alpha}(x)$, where $x\in E_{\alpha}$. In this case, $H_{\mu}$ is well-defined since the family $\{H_{\alpha}\}_{\alpha<\theta}$ is coherent. Therefore, according to transfinite recursion, we obtain a well-behaved isometric embedding $H_{\theta}\colon E_{\theta}\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$. In this case, note that $E_{\theta}=E$. Now we define $I\colon X\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ by $I=H_{\theta}|_{X}$. Due to [\[item:azero\]](#item:azero){reference-type="ref" reference="item:azero"}, we have $H_{\theta}(\varpi)=0$ and $\varpi\not \in X$. Thus the condition [\[item:connzero\]](#item:connzero){reference-type="ref" reference="item:connzero"} is true. Since $H_{\theta}$ satisfies [\[item:aisom\]](#item:aisom){reference-type="ref" reference="item:aisom"}--[\[item:acoef\]](#item:acoef){reference-type="ref" reference="item:acoef"}, the map $I$ satisfies the conditions [\[item:conisom\]](#item:conisom){reference-type="ref" reference="item:conisom"}--[\[item:concoef\]](#item:concoef){reference-type="ref" reference="item:concoef"}. This finishes the proof of Theorem [Theorem 25](#thm:main0){reference-type="ref" reference="thm:main0"}. ◻ ### The proof of the first main result The following is our first main result. **Theorem 30**. *Let $\eta\in (1, \infty)$, $(q, p)\in \mathcal{CH}$, $\theta$ be a cardinal, and $G\in \mathcal{G}$ be dividable. Put $R=\{\, \eta^{-g}\mid g\in G\sqcup\{\infty\}\,\}$. Take an arbitrary valued field $(K, v)$ of characteristic $q$ such that $v(K)\subseteq G\sqcup\{\infty\}$ and $\mathfrak{K}(K, v)$ has characteristic $p$. Then there exists a valued field $(L, w)$ such that:* 1. *[\[item:l1\]]{#item:l1 label="item:l1"} the field $(L, w)$ is a valued field extension of $(K, v)$;* 2. *[\[item:l2\]]{#item:l2 label="item:l2"} $w(L)= G\sqcup\{\infty\}$;* 3. *[\[item:l3\]]{#item:l3 label="item:l3"} for each $R$-valued ultrametric $(X, d)$ with $\mathop{\mathrm{Card}}(X)\le \theta$, there exists an isometric embedding $I\colon (X, d)\to (L, \lVert *\rVert_{w, \eta})$ such that the set $I(X)$ is algebraic independent over $K$.* *Moreover, for every $R$-valued ultrametric space $(X, d)$, there exist a valued field $(F, u)$ and a map $I\colon X\to F$ such that:* 1. *[\[item:f1\]]{#item:f1 label="item:f1"} the map $I\colon (X, d)\to (F, \lVert *\rVert_{u, \eta})$ is an isometric embedding;* 2. *[\[item:f2\]]{#item:f2 label="item:f2"} the field $(L, u)$ is a valued field extension of $(K, v)$;* 3. *[\[item:f3\]]{#item:f3 label="item:f3"} $u(F)\subseteq G\sqcup \{\infty\}$;* 4. *[\[item:f4\]]{#item:f4 label="item:f4"} the set $I(X)$ is closed in $F$;* 5. *[\[item:f5\]]{#item:f5 label="item:f5"} the set $I(X)$ is algebraically independent over $K$;* 6. *[\[item:f6\]]{#item:f6 label="item:f6"} if $(X, d)$ is complete, then $(F, u)$ can be chosen to be complete.* *Proof.* Let $\boldsymbol{k}$ be the algebraic closure of $\mathfrak{K}(K, v)$. Notice that $\boldsymbol{k}$ is perfect. Take a perfect filed $\boldsymbol{l}$ such that the transcendental degree of $\boldsymbol{l}$ over $\boldsymbol{k}$ is $\theta$, and take a transcendental basis $B$ of $\boldsymbol{l}$ over $\boldsymbol{k}$. In this case, we have $\mathop{\mathrm{Card}}(B)=\theta$. Take a system $J\subseteq \mathbb{A}_{q, p}(G, \boldsymbol{l})$ of representatives of $\boldsymbol{l}$. Notice that $B\subseteq\zeta(J)$. Put $C=\zeta^{-1}(B)=\{\, b_{\alpha}\mid \alpha<\theta\, \}$. Applying Theorem [Theorem 25](#thm:main0){reference-type="ref" reference="thm:main0"} to $(X, d)$ and $(K, \lVert *\rVert_{v, \eta})$, we can take an isometric embedding $\psi\colon K\to \mathbb{A}_{q, p}(G, \boldsymbol{k})$ satisfying the conditions [\[item:connzero\]](#item:connzero){reference-type="ref" reference="item:connzero"}--[\[item:concoef\]](#item:concoef){reference-type="ref" reference="item:concoef"}. Let $\phi\colon \boldsymbol{k}\to \boldsymbol{l}$ be the inclusion map. Using a homomorphic embedding $\mathbb{A}_{q, p}(G, \phi)\circ \psi\colon K\to \mathbb{A}_{q, p}(G, \boldsymbol{l})$, in what follows, we consider that $K$ is a subfield of $\mathbb{A}_{q, p}(G, \boldsymbol{l})$. In this case, we see that $\mathbf{AH}(K)\subseteq\boldsymbol{k}$. Put $L=\mathbb{A}_{q, p}(G, \boldsymbol{l})$ and $w=U_{G, \boldsymbol{l}, q, p}$. Then $(L, w)$ satisfies the conditions [\[item:l1\]](#item:l1){reference-type="ref" reference="item:l1"} and [\[item:l2\]](#item:l2){reference-type="ref" reference="item:l2"} (see Proposition [Proposition 12](#prop:propppp){reference-type="ref" reference="prop:propppp"}). Now we prove [\[item:l3\]](#item:l3){reference-type="ref" reference="item:l3"}. Take an $R$-valued ultrametric space $(X, d)$. Applying Theorem [Theorem 25](#thm:main0){reference-type="ref" reference="thm:main0"} to $C$, $J$, $\boldsymbol{l}$, $\boldsymbol{k}$, and $(X, d)$ of $\mathop{\mathrm{Card}}(X)\le \theta$, we can take $I\colon X\to L$ satisfying the conditions [\[item:connzero\]](#item:connzero){reference-type="ref" reference="item:connzero"}--[\[item:concoef\]](#item:concoef){reference-type="ref" reference="item:concoef"}. The condition [\[item:conisom\]](#item:conisom){reference-type="ref" reference="item:conisom"} shows that $I$ is isometric. Due to the conditions [\[item:connzero\]](#item:connzero){reference-type="ref" reference="item:connzero"}, [\[item:concfinq\]](#item:concfinq){reference-type="ref" reference="item:concfinq"}, and [\[item:conconseq\]](#item:conconseq){reference-type="ref" reference="item:conconseq"}, and due to the fact that $\mathbf{AH}(K)\subseteq\boldsymbol{k}$, the set $\{\, I(x)\mid x\in X\, \}$ satisfies the assumptions in Proposition [Proposition 24](#prop:fintr){reference-type="ref" reference="prop:fintr"}. Then, according to Proposition [Proposition 24](#prop:fintr){reference-type="ref" reference="prop:fintr"}, we see that $I(X)$ is algebraic over $K$. This means that the condition [\[item:l3\]](#item:l3){reference-type="ref" reference="item:l3"}. We now prove the latter part. Let $(Y, e)$ be the completion of $(X, d)$. Then $(Y, e)$ is $R$-valued (see [@MR3782290 (12) in Theorem 1.6]). Take an isometric embedding $I\colon Y\to L$ stated in the former part of Theorem [Theorem 30](#thm:19main2){reference-type="ref" reference="thm:19main2"}. Since $(Y, e)$ is complete and $I$ is isometric, the set $I(Y)$ is closed in $L$. Let $F$ be the subfield of $L$ generated by $\{\,I(x) \mid x\in X\, \}$ over $K$ and put $w=U_{G, \boldsymbol{l}, q, p}|_{K}$. Since $I(Y)$ is algebraically independent over $K$, we have $I(Y)\cap L=I(X)$. Thus $I(X)$ is closed in $F$. The condition [\[item:f6\]](#item:f6){reference-type="ref" reference="item:f6"} follows from the construction. This finishes the proof. ◻ Letting $K=\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$, and using Proposition [Proposition 15](#prop:embedcoef){reference-type="ref" reference="prop:embedcoef"}, we obtain the next corollary. **Corollary 31**. *Let $\eta\in (1, \infty)$, $(q, p)\in \mathcal{CH}$ with $q\neq p$, $\theta$ be a cardinal, and $G\in \mathcal{G}$ be dividable. Put $R=\{\, \eta^{-g}\mid g\in G\sqcup\{\infty\}\,\}$. Then there exists a valued field $(L, v)$ such that:* 1. *the field $(L, v)$ is a valued field extension of $(\mathrm{Fr}\mathbb{W}(\boldsymbol{k}), w_{\boldsymbol{k}})$;* 2. *the absolute value $\lVert *\rVert_{v, \eta}$ is $R$-valued;* 3. *for each $R$-valued ultrametric $(X, d)$ with $\mathop{\mathrm{Card}}(X)\le \theta$, there exists an isometric embedding $I\colon (X, d)\to (L, \lVert *\rVert_{w, \eta})$ such that the set $I(X)$ is algebraic independent over $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$.* *Moreover, for every $R$-valued ultrametric space $(X, d)$, there exist a valued field $(F, u)$ and a map $I\colon X\to F$ such that* 1. *the map $I\colon (X, d)\to (F, \lVert *\rVert_{u, \eta})$ is an isometric embedding;* 2. *the field $(L, u)$ is a valued field extension of $(\mathrm{Fr}\mathbb{W}(\boldsymbol{k}), w_{\boldsymbol{k}})$;* 3. *$u(F)\subseteq G\sqcup \{\infty\}$;* 4. *$I(X)$ is closed in $F$;* 5. *$I(X)$ is algebraically independent over $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$;* 6. *if $(X, d)$ is complete, then $F$ can be chosen to be complete.* *Proof.* The proof is the same to that of Theorem [Theorem 30](#thm:19main2){reference-type="ref" reference="thm:19main2"}. Remark that, since $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ is naturally a subset of $\mathbb{A}_{0, p}(G, \boldsymbol{k})$ (see Proposition [Proposition 15](#prop:embedcoef){reference-type="ref" reference="prop:embedcoef"}), we do not need to use Proposition [Proposition 16](#prop:pembed){reference-type="ref" reference="prop:pembed"}. Thus, the assumption of the corollary does not require that $G$ be divisible. ◻ ## Broughan's conjecture Next we give an affirmative solution of Conjecture [Conjecture 1](#conj:bbb){reference-type="ref" reference="conj:bbb"}. We begin with the definition of non-Archimedean Banach algebras. Let $(K, \lvert * \rvert)$ be a field equipped with a non-Archimedean absolute value, and $(B, \lVert *\rVert)$ be a normed linear space over a field $K$. We say that $(B, \lVert *\rVert)$ is a *non-Archimedean Banach algebra over $K$* (see [@MR0512894]) if the following conditions are fullfiled: 1. $B$ is a ring (not necessarily commutative and unitary); 2. $(B, \lVert *\rVert)$ is a normed linear space over $(K, \lvert *\rvert)$; 3. for every pair $x, y\in B$, we have $\lVert x+y \rVert \le \lVert x\rVert \lor \lVert y\rVert$ and $\lVert xy \lVert\le \lVert x\rVert \cdot \lVert y\rVert$; 4. $B$ is complete with respect to $\lVert *\rVert$; 5. if $B$ has a unit $e$, then $\lVert e\rVert=1$. Some authors assume commutativity and the existence of a unit in the definition of a Banach algebra (see, for example, [@MR0261361] and [@MR0361697]). For instance, a valued field extension $(L, \lvert *\rvert_{L})$ of $(K, \lvert * \rvert_{K})$ is a (unitary) Banach algebra over $(K, \lvert * \rvert_{K})$. As an application of Corollary [Corollary 31](#cor:19main11){reference-type="ref" reference="cor:19main11"}, we obtain the next theorem. **Theorem 32**. *Let $p$ be a prime, $\theta$ be a cardinal, and $\boldsymbol{k}$ be a perfect field of characteristic $p$ such that the transcendental degree of $\boldsymbol{k}$ over $\mathbb{F}_{p}$ is equal to $\theta$. Then every $H_{p}$-valued ultrametric $(X, d)$ with $\mathop{\mathrm{Card}}(X)\le \theta$ can be isometrically embedded into $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$. In particular, Conjecture [Conjecture 1](#conj:bbb){reference-type="ref" reference="conj:bbb"} is true.* *Proof.* The theorem follows from Corollary [Corollary 31](#cor:19main11){reference-type="ref" reference="cor:19main11"}. Theorem [Theorem 32](#thm:frw){reference-type="ref" reference="thm:frw"} is actually an affirmative solution of Conjecture [Conjecture 1](#conj:bbb){reference-type="ref" reference="conj:bbb"} since for every cardinal $\theta$, there exists a perfect filed $\boldsymbol{k}$ of characteristic $p$ such that the transcendental degree of $\boldsymbol{k}$ over $\mathbb{F}_{p}$ is equal to $\theta$, and since $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ is a non-Archimedean (commutative and unitary) Banach algebra over $\mathbb{Q}_{p}$ (see Proposition [Proposition 15](#prop:embedcoef){reference-type="ref" reference="prop:embedcoef"}). ◻ *Remark 4*. An solution of Conjecture [Conjecture 1](#conj:bbb){reference-type="ref" reference="conj:bbb"} is given as the ring $\mathbb{W}(\boldsymbol{k})$ of Witt vectors. # Algebraic structures on Urysohn universal ultrametric spaces {#sec:Urysohn} In this section, we show that every Urysohn universal ultrametric space has a valued field structure that are extensions of a given prime valued field. Such an algebraic structure is realized as a $p$-adic or ordinary Levi--Civita field. For a class $\mathscr{C}$ of metric spaces, a metric space $(X, d)$ is said to be *$\mathscr{C}$-injective* if for every pair $(A, a)$ and $(B, b)$ in $\mathscr{C}$ and for every pair of isometric embeddings $\phi\colon (A, a)\to (B, b)$ and $\psi\colon (A, a)\to (X, d)$, there exists an isometric embedding $\theta\colon (B, b)\to (X, d)$ such that $\theta\circ \phi=\psi$. We denote by $\mathscr{F}$ (resp. $\mathscr{N}(R)$ for a range set $R$) the class of all finite metric spaces (resp. all finite $R$-valed ultrametric spaces). There exists a complete $\mathscr{F}$-injetive separable complete metric space $(\mathbb{U}, \rho)$ unique up to isometry, and this space is called the *Urysohn universal metric space* (see [@Ury1927] and [@MR2435148]). Similarly, if $R$ is countable, there exists a separable complete $R$-valued $\mathscr{N}(R)$-injective ultrametric space up to isometry, and it is called the *$R$-Urysohn universal ultrametric space*, which is a non-Archimedean analogue of $(\mathbb{U}, \rho)$ (see [@MR2754373] and [@MR2667917]). Remark that if $R$ is uncountable, then every $R$-injective ultrametric space is not separable (see [@MR2754373]). Due to this phenomenon, mathematicians often consider only the case where $R$ is countable for separability. In [@Ishiki2023Ury], the author discovered the concept of *petaloid spaces*, which can be considered to as $R$-Urysohn universal ultrametric spaces in the case where $R$ is uncountable. We begin with preparation of notations. A subset $E$ of $[0, \infty)$ is said to be *semi-sporadic* if there exists a strictly decreasing sequence $\{a_{i}\}_{i\in \mathbb{Z}_{\ge 0}}$ in $(0, \infty)$ such that $\lim_{i\to \infty}a_{i}=0$ and $E=\{0\}\cup \{\, a_{i}\mid i\in \mathbb{Z}_{\ge 0}\, \}$. A subset of $[0, \infty)$ is said to be *tenuous* if it is finite or semi-sporadic (see [@Ishiki2023const]). For a range set $R$, we denote by $\mathbf{TEN}(R)$ the set of all tenuous range subsets of $R$. Let us recall the definition of petaloid spaces ([@Ishiki2023Ury]). **Definition 5**. Let $R$ be an uncountable range set. We say that a metric space $(X, d)$ is *$R$-petaloid* if it is an $R$-valued ultrametric space and there exists a family $\{\Pi(X, S)\}_{S\in \mathbf{TEN}(R)}$ of subspaces of $X$ satisfying the following properties: 1. [\[item:pr:sep\]]{#item:pr:sep label="item:pr:sep"} For every $S\in \mathbf{TEN}(R)$, the subspace $(\Pi(X, S), d)$ is isometric to the $S$-Urysohn universal ultrametric space. 2. [\[item:pr:cup\]]{#item:pr:cup label="item:pr:cup"} We have $\bigcup_{S\in \mathbf{TEN}(R)}\Pi(X, S)=X$. 3. [\[item:pr:cap\]]{#item:pr:cap label="item:pr:cap"} If $S, T\in \mathbf{TEN}(R)$, then $\Pi(X, S)\cap \Pi(X, T)=\Pi(X, S\cap T)$. 4. [\[item:pr:distance\]]{#item:pr:distance label="item:pr:distance"} If $S, T\in \mathbf{TEN}(R)$ and $x\in \Pi(X, T)$, then $d(x, \Pi(X, S))$ belongs to $(T\setminus S)\cup \{0\}$. We call the family $\{\Pi(X, S)\}_{S\in \mathbf{TEN}(R)}$ an *$R$-petal of $X$*, and call $\Pi(X, S)$ the $S$-piece of the $R$-petal $\{\Pi(X, S)\}_{S\in \mathbf{TEN}(R)}$. Notice that even if $R$ is countable, the $R$-Urysohn universal ultrametric space has a petal structure satisfying the conditions [\[item:pr:sep\]](#item:pr:sep){reference-type="ref" reference="item:pr:sep"}--[\[item:pr:distance\]](#item:pr:distance){reference-type="ref" reference="item:pr:distance"} (see [@Ishiki2023Ury]). This means that a petal space is a natural generalization of Urysohn universal ultrametric spaces. The following is [@Ishiki2023halo Theorem 2.3] (see the property [\[item:pr:sep\]](#item:pr:sep){reference-type="ref" reference="item:pr:sep"} and [@MR2444734 Propositions 20.2 and 21.1]). **Theorem 33**. *Let $R$ be an uncountable range set. The following statements hold:* 1. *[\[item:uni\]]{#item:uni label="item:uni"} There exists an $R$-petaloid ultrametric space and it is unique up to isometry.* 2. *The $R$-petaloid ultrametric space is complete, non-separable, and $\mathscr{F}(R)$-injective.* 3. *Every separable $R$-valued ultrametric space can be isometrically embedded into the $R$-petaloid ultrametric space.* Based on Theorem [Theorem 33](#thm:18:petapeta){reference-type="ref" reference="thm:18:petapeta"}, for a range set $R$, we denote by $(\mathbb{V}_{R}, \sigma_{R})$ the $R$-Urysohn universal ultrametric space if $R$ is countable; otherwise, the $R$-petaloid ultrametric space. In what follows, by abuse of notation, we call $(\mathbb{V}_{R}, \sigma_{R})$ the $R$-Urysohn universal ultrametric space even if $R$ is uncountable. We explain an example of a petaloid space. Let $N$ be a countable set. Let $R$ be a range set. We also denote by $\mathrm{G}(R, N)$ the set of all function $f\colon R\to N$ such that $f(0)=0$ and the set $\{0\}\cup \{\, x\in R \mid f(x)\neq 0\, \}$ is tenuous. For $f, g\in \mathrm{G}(R, N)$, we define an $R$-ultrametric $\mathord{\vartriangle}$ on $\mathrm{G}(R, N)$ by $\mathord{\vartriangle}(f, g)= \max\{\, r\in R\mid f(r)\neq g(r)\, \}$ if $f\neq g$; otherwise, $\mathord{\vartriangle}(f, g)=0$. For more information of this construction, we refer the readers to [@MR2754373] and [@MR2667917]. **Lemma 34**. *For every countable set $N$ and every range set $R$, the ultrametric space $(\mathrm{G}(R, N), \mathord{\vartriangle})$ is isometric to $(\mathbb{V}_{R}, \sigma_{R})$.* *Proof.* The lemma follows from [@Ishiki2023Ury Theorem 1.3]. Notice that whereas in [@Ishiki2023Ury], it is only shown that $(\mathrm{G}(R, \mathbb{Z}_{\ge 0}), \mathord{\vartriangle})$ is isometric to $(\mathbb{V}_{R}, \sigma_{R})$, the proof in [@Ishiki2023Ury] is still valid even if $N$ is a countable set in general. ◻ **Lemma 35**. *Let $A$ be a subset of $\mathbb{R}$, and $\eta\in (0, \infty)$. Put $S=\{0\}\sqcup \{\, \eta^{-g}\mid g\in A\, \}$. Then the following statements are equivalent:* 1. *for every $n\in \mathbb{Z}_{\ge 0}$, the set $A\cap (-\infty, n]$ is finite;* 2. *the set $R$ is tenuous.* *Proof.* The lemma follows from [@Ishiki2023const Lemma 2.12] and the fact that a map $f\colon \mathbb{R}\to \mathbb{R}$ defined by $f(x)=\eta^{-x}$ reverses the order on $\mathbb{R}$. ◻ In the next lemma, in order to treat the cases of $q=p$ and $q\neq p$, even if $q\neq p$, we define $\mathrm{St}_{G, \boldsymbol{k}, J}$ by $\mathrm{St}_{G, \boldsymbol{k}, J}(x)=x$. **Lemma 36**. *Let $\eta\in (1, \infty)$, $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, $\boldsymbol{k}$ be a field of $\mathop{\mathrm{Card}}(\boldsymbol{k})=\aleph_{0}$ and characteristic $p$. Put $R=\{0\}\cup \{\, \eta^{-g}\mid g\in G\, \}$. Fix a complete system $J\subseteq \mathbb{B}_{q, p}(G, \boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$. Take $S\in \mathbf{TEN}(R)$. Define a subset $\Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$ of $\mathbb{B}_{q, p}(G, \boldsymbol{k})$ by the set of all $f\in \mathbb{B}_{q, p}(G, \boldsymbol{k})$ such that $\{\, \eta^{-g}\mid g\in \text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(f))\, \}\subseteq S$, and define an ultrametric $d$ on $\mathbb{B}_{q, p}(G, \boldsymbol{k})$ by $d(x, y)= \lVert x-y\rVert_{U_{G, \boldsymbol{k}, q, p}, \eta}$. Then $(\Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S), d)$ is isometric to $(\mathbb{V}_{S}, \sigma_{S})$.* *Proof.* According to $\mathop{\mathrm{Card}}(\boldsymbol{k})=\aleph_{0}$, we see that $\mathop{\mathrm{Card}}(J)=\aleph_{0}$. Then, using Lemma [Lemma 10](#lem:repbeta){reference-type="ref" reference="lem:repbeta"} or [\[item:ffff\]](#item:ffff){reference-type="ref" reference="item:ffff"} in Corollary [Corollary 19](#cor:216){reference-type="ref" reference="cor:216"}, for every $x\in \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$, there uniquely exists $s\in \mathbb{H}(G, \boldsymbol{k})$ such that $\mathrm{St}_{G, \boldsymbol{k}, J}(x)=s$ and $s(g)\in J$. Put $s=\sum_{g\in G}s_{x}(g)t^{g}$, where $s_{x}(g)\in J$. We define a map $T(x)\colon R\to J$ by $T(x)(0)=0$ and $T(x)(r)=s_{x}(-\log_{\eta}(r))$ if $r\neq 0$. Then $\{\, r\in R\mid s_{x}(r)\neq 0\, \}\subseteq S$. Thus $T(x)\in \mathrm{G}(S, J)$ for all $x\in \mathbb{B}_{q, p}(G, \boldsymbol{k})$. By the definition of $T$, we see that the map $T\colon \mathbb{B}_{q, p}(G, \boldsymbol{k}) \to \mathrm{G}(S, J)$ defined by $x\mapsto T(x)$ is bijective. Using the definitions of $U_{G, \boldsymbol{k}, q, p}$ and $\mathord{\vartriangle}$, the map $T\colon \mathbb{B}_{q, p}(G, \boldsymbol{k}) \to \mathrm{G}(S, J)$ becomes an isometric bijection. Thus $(\mathrm{G}(S, J), \mathord{\vartriangle})$ is isometric to $(\Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S), d)$. Due to $\mathop{\mathrm{Card}}(J)=\aleph_{0}$, Lemma [Lemma 34](#lem:Gisom){reference-type="ref" reference="lem:Gisom"} finishes the proof. ◻ **Theorem 37**. *Let $\eta\in (1, \infty)$, $G\in \mathcal{G}$, $(q, p)\in \mathcal{CH}$, $\boldsymbol{k}$ be a field of $\mathop{\mathrm{Card}}(\boldsymbol{k})=\aleph_{0}$ and characteristic $p$. Define an ultrametric $d$ on $\mathbb{B}_{q, p}(G, \boldsymbol{k})$ by $d(x, y)= \lVert x-y\rVert_{U_{G, \boldsymbol{k}, q, p}, \eta}$. Then the Levi-Civita field $(\mathbb{B}_{q, p}(G, \boldsymbol{k}), d)$ is isometric to $(\mathbb{V}_{R}, \sigma_{R})$.* *Proof.* We define $\Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$ as in Lemma [Lemma 36](#lem:cond1isom){reference-type="ref" reference="lem:cond1isom"}. Let us prove that $\{\Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)\}_{S\in \mathbf{TEN}(R)}$ satisfies the conditions [\[item:pr:sep\]](#item:pr:sep){reference-type="ref" reference="item:pr:sep"}--[\[item:pr:distance\]](#item:pr:distance){reference-type="ref" reference="item:pr:distance"}. Lemma [Lemma 36](#lem:cond1isom){reference-type="ref" reference="lem:cond1isom"} implies the conditions [\[item:pr:sep\]](#item:pr:sep){reference-type="ref" reference="item:pr:sep"}. From the definition of $\{\Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)\}_{S\in \mathbf{TEN}(R)}$, the conditions [\[item:pr:cup\]](#item:pr:cup){reference-type="ref" reference="item:pr:cup"} is true. Next we show the condition [\[item:pr:cap\]](#item:pr:cap){reference-type="ref" reference="item:pr:cap"}. It is sufficient to verify $$\begin{aligned} \label{al:61} \mathbb{B}_{q, p}(G, \boldsymbol{k})\subseteq \bigcup_{S\in \mathbf{TEN}(R)}\Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S). \end{aligned}$$ Take $f\in \mathbb{B}_{q, p}(G, \boldsymbol{k})$ and put $A=\text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(f))$. We also define $S=\{\, \eta^{-g}\mid g\in A\, \}$. Then $S\in \mathbf{TEN}(R)$ by Lemma [Lemma 35](#lem:antitone){reference-type="ref" reference="lem:antitone"}. Hence $f\in \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$. Thus, the inclusion [\[al:61\]](#al:61){reference-type="eqref" reference="al:61"} is true. Now we show [\[item:pr:distance\]](#item:pr:distance){reference-type="ref" reference="item:pr:distance"}. We may assume that $x\not \in \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$. Put $U=\text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(x))$. Since $x\not \in \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$, we have $U\setminus S\neq \emptyset$. Put $u=\max (U\setminus S)$ and $m=-\log_{\eta}(u)$. Notice that $\boldsymbol{C}(x, m)\neq 0$. Define a point $y\in \mathbb{B}_{q, p}(G, \boldsymbol{k})$ by $y=\sum_{g\in G\cap (-\infty, m)}x(g)\tau^{g}$. Then $y\in \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$ and $d(x, y)=u$. Then $d(x, \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S))\le u$. For the sake of contradiction, suppose that $d(x, \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S))< u$. In this setting, we can take $z\in \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S)$ such that $m<v(x-z)$. Then $\boldsymbol{C}(z, m)=\boldsymbol{C}(x, m)$. In particular, $\boldsymbol{C}(z, m)\neq 0$. Namely, we have $\text{{\rm supp}}(\mathrm{St}_{G, \boldsymbol{k}, J}(z))\not\subseteq S$. This is a contradiction. Thus we have $d(x, \Pi(\mathbb{B}_{q, p}(G, \boldsymbol{k}), S))=u\in T\setminus S$, which means that the condition [\[item:pr:distance\]](#item:pr:distance){reference-type="ref" reference="item:pr:distance"} holds. Therefore, we conclude that the Levi-Civita field $\mathbb{B}_{q, p}(G, \boldsymbol{k})$ is isometric to $(\mathbb{V}_{R}, \sigma_{R})$. ◻ We find another application of the theory of Urysohn universal ultrametric spaces to that of valued field. For a class $\mathscr{C}$ of metric spaces, we say that a metric space $(X, d)$ is *$\mathscr{C}$-universal* or *universal for $\mathscr{C}$* if for every $(A, e)\in \mathscr{C}$ there exists an isometric embedding $f\colon A\to X$. An ultrametric space $(X, d)$ is said to be *$(R, \aleph_{0})$-haloed* if for every $a\in X$ and for every $r\in R\setminus \{0\}$, there exists a subset $A$ of $\mathrm{B}(a, r)$ such that $\kappa\le \mathop{\mathrm{Card}}(A)$ and $d(x, y)=r$ for all distinct $x, y\in A$ (see [@Ishiki2023halo]). **Theorem 38**. *Let $\eta\in (1, \infty)$, $G\in \mathcal{G}$, $p$ be a prime, and let $(K, v)$ be a valued field such that $\mathfrak{K}(K, v)$ is an infinite set and has characteristic $p$. Put $R=\{\, \eta^{-g}\mid g\in G\sqcup \{\infty\}\, \}$. Then $(K, \lVert *\rVert_{v, \eta})$ is universal for all separable $R$-valued ultrametric spaces.* *Proof.* According to $\aleph_{0}\le \mathop{\mathrm{Card}}(\boldsymbol{k})$, we observe that $(K, \lVert *\rVert_{v, \eta})$ is $(R, \omega_{0})$-haloed, and hence it is $\mathscr{N}(R, \omega_{0})$-injective (see [@Ishiki2023halo Theorem 1.1]). In [@Ishiki2023halo Theorem 1.5], it is stated that every complete $\mathscr{N}(R, \omega_{0})$-injective ultrametric space contains an isometric copy of $(\mathbb{V}_{R}, \sigma_{R})$. Thus $(K, \lVert *\rVert_{v, \eta})$ contains a metric subspace isometric to $(\mathbb{V}_{R}, \sigma_{R})$. Hence $(K, \lVert *\rVert_{v, \eta})$ is universal for all separable $R$-valued ultrametric spaces. ◻ For a prime $p$, the field $\mathbb{C}_{p}$ of $p$-adic complex numbers is defined as the completion of the algebraic closure of $\mathbb{Q}_{p}$. The $p$-adic valuation $v_{p}$ can be extended on $\mathbb{C}_{p}$. In this case, we have $v_{p}(\mathbb{C}_{p})=\mathbb{Q}$ (see [@MR2444734]). **Corollary 39**. *Let $\eta\in (0, \infty)$, and $p$ be a prime. Put $R=\{\, \eta^{-g}\mid g\in \mathbb{Q}\, \}$. The field $(\mathbb{C}_{p}, v_{p})$ of $p$-adic complex numbers is universal for all separable $R$-valued ultrametric spaces.* # Questions {#sec:ques} As a sophisticated version of a non-Archimedean Arens--Eells theorem for linear spaces, we ask the next question. **Question 40**. Let $R, T$ be range sets, and $S$ be a range set such that $S\setminus \{0\}$ is a multiplicative subgroup of $\mathbb{R}_{\ge 0}$. Take a non-Archimedean valued field $(K, \rvert*\lvert_{K})$ and assume that 1. $R\cdot S=\{\, r\cdot s\mid r\in R, s\in S\, \}\subseteq T$; 2. $\rvert*\lvert_{K}$ is $S$-valued. For every $R$-valued ultrametric space $(X, d)$, do there exist a $T$-valued ultra-normed linear space $(V, \rVert *\lVert_{L})$ over $(K, \rvert*\lvert_{K})$, and an isometric map $I\colon X\to F$? Similarly to Corollary [Corollary 39](#cor:padicc){reference-type="ref" reference="cor:padicc"}, as a counterpart of Theorem [Theorem 30](#thm:19main2){reference-type="ref" reference="thm:19main2"}, we make the following question. **Question 41**. Let $p$ be $0$ or a prime, $\boldsymbol{k}$ be an infinite field of characteristic $p$, and $R$ be a rang set such that $R\setminus\{0\}$ is a multiplicative subgroup of $\mathbb{R}_{>0}$. Take an arbitrary complete valued field $(K, \rvert*\lvert_{*})$ with the residue class field $\boldsymbol{k}$ such that $\{\, \rvert x\lvert_{K}\, \mid x\in K\, \}=R$ and $(K, v)$ is a valued field extension of $(\mathbb{Q}_{p}, v_{p})$. For every $R$-valued ultrametric space $(X, d)$ with $\mathop{\mathrm{Card}}(X)\le \mathop{\mathrm{Card}}(\boldsymbol{k})$, does there exist an isometric embedding $I\colon X\to K$?

arxiv_math

arxiv_math

--- abstract: | We study almost rigidity properties of super Ricci flow whose Müller quantity is non-negative. We generalize almost splitting and quantitative stratification theorems for Ricci flow established by Bamler [@B3]. address: - Faculty of Science and Technology, Tokushima University, 2-1 Minamijosanjima-cho, Tokushima, 770-8506, Japan - Department of Mathematics, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama-City, Saitama, 338-8570, Japan author: - Keita Kunikawa - Yohei Sakurai date: September 21, 2023 title: | Almost splitting and quantitative stratification\ for super Ricci flow --- # Introduction Let $(M,g(t))_{t\in I}$ be a compact manifold equipped with a time-dependent Riemannian metric. Such a time-dependent manifold is called *Ricci flow* when $$\partial_{t}g= -2 \operatorname{Ric},$$ which has been introduced by Hamilton [@H]. It is well-known that Perelman [@P] has vastly developed the Ricci flow theory in three dimensional case, and utilized it for the resolution of the Poincaré and geometrization conjectures. A supersolution to the Ricci flow is called *super Ricci flow*; namely, $(M,g(t))_{t\in I}$ is called super Ricci flow when $$\partial_{t}g\geq -2 \operatorname{Ric},$$ which has been introduced by McCann-Topping [@MT] in view of the relation between Ricci flow and mass transport. Super Ricci flow can be viewed as an interpolation between Ricci flow and (static) manifolds of non-negative Ricci curvature. Actually, it has been examined in the literature of the geometry of metric measure spaces with a lower Ricci curvature bound developed by Sturm [@S1], [@S2], Lott-Villani [@LV], Ambrosio-Gigli-Savaré [@AGS], Gigli [@G] and so on (see the work of Sturm [@S], and also subsequent works [@K], [@KS1], [@KS2]). In a series of works by Bamler [@B1], [@B2], [@B3], he has proposed a new method for dealing with higher dimensional Ricci flow, recovering Perelman's three dimensional results. In [@B1], [@B2], he has developed geometric analysis and a compactness theory for super Ricci flow. In [@B3], he has established a structure theory for non-collapsed limits of Ricci flow. One of the key ingredients in [@B3] was to investigate almost rigidity phenomena concerning the monotonicity of the Nash entropy, which is similar to the strategy of the structure theory for Gromov-Hausdorff limits of manifolds with a lower Ricci curvature bound due to Cheeger-Colding [@CC], [@CC1], [@CC2], [@CC3], Colding-Naber [@CN] and so on. We aim to extend the structure theory for Ricci flow in [@B3] to super Ricci flow. For that purpose, in this paper, we generalize almost splitting and quantitative stratification theorems obtained in [@B3 Part 2] for super Ricci flow whose Müller quantity is non-negative. We recall that for a vector field $V$, the *Müller quantity* is defined as follows (see [@M1 Definition 1.3]): $$\mathcal{D}(V):=\partial_{t}H-\Delta H-2| h |^2+4\operatorname{div}h(V)-2 \langle \nabla H,V \rangle+2\operatorname{Ric}(V,V)-2h(V,V),$$ where $$h:=-\frac{1}{2} \partial_t g,\quad H:=\operatorname{tr}h.$$ The non-negativity of the Müller quantity brings various benefits such as a lower bound of the (generalized) scalar curvature, the Harnack estimate and the monotonicity of entropies (see Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"}, [Lemma 17](#lem:Nash){reference-type="ref" reference="lem:Nash"} and Theorem [Theorem 5](#thm:Harnack){reference-type="ref" reference="thm:Harnack"} below). There are several examples of super Ricci flow whose Müller quantity is non-negative (see [@M1 Section 2], [@FZ Section 7]): 1. Ricci flow; 2. Ricci flow coupled with the heat equation, called *List flow* ([@L]); 3. Ricci flow coupled with the harmonic map heat flow, called *Müller flow* ([@M2]); 4. mean curvature flow for spacelike hypersurfaces in Lorentzian manifold of non-negative sectional curvature; 5. (scaled) twisted Kähler-Ricci flow. In [@KuS], the authors have extended some results of Bamler-Zhang [@BZ] concerning geometric analysis for Ricci flow under a scalar curvature bound to this object. This paper is organized as follows: In Sections [2](#sec:Preliminaries){reference-type="ref" reference="sec:Preliminaries"}, [3](#sec:SRF){reference-type="ref" reference="sec:SRF"}, [4](#sec:Nash entropy){reference-type="ref" reference="sec:Nash entropy"}, [5](#sec:vol_heat){reference-type="ref" reference="sec:vol_heat"}, [6](#sec:noncollapse){reference-type="ref" reference="sec:noncollapse"}, we collect various estimates for the proof of our main theorems. In Section [2](#sec:Preliminaries){reference-type="ref" reference="sec:Preliminaries"}, we recall basics of the heat kernel. In Section [3](#sec:SRF){reference-type="ref" reference="sec:SRF"}, we review previous results for super Ricci flow obtained in [@B1]. In Section [4](#sec:Nash entropy){reference-type="ref" reference="sec:Nash entropy"}, we study properties of the Nash entropy. In Section [5](#sec:vol_heat){reference-type="ref" reference="sec:vol_heat"}, we prove some volume and heat kernel estimates. In Section [6](#sec:noncollapse){reference-type="ref" reference="sec:noncollapse"}, we obtain several estimates under a non-collapsed condition for the Nash entropy. Based on these estimates, in Section [7](#sec:almost_ss){reference-type="ref" reference="sec:almost_ss"}, we investigate properties of almost selfsimilar points. Sections [8](#sec:almost_st){reference-type="ref" reference="sec:almost_st"}, [9](#sec:almost_split){reference-type="ref" reference="sec:almost_split"}, [10](#sec:stratification){reference-type="ref" reference="sec:stratification"} are devoted to the proof of our main theorems. In Section [8](#sec:almost_st){reference-type="ref" reference="sec:almost_st"}, we show an almost static cone splitting theorem (see Theorem [Theorem 40](#thm:almost static){reference-type="ref" reference="thm:almost static"}). In Section [9](#sec:almost_split){reference-type="ref" reference="sec:almost_split"}, we prove an almost splitting theorem (see Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"}). In Section [10](#sec:stratification){reference-type="ref" reference="sec:stratification"}, we conclude a quantitative stratification theorem (see Theorem [Theorem 50](#thm:stratification){reference-type="ref" reference="thm:stratification"}). In most parts, we give a proof along the lines of the corresponding argument in [@B3] paying attention to the appearance of the Müller quantity. On the other hand, in the proof of an almost monotonicity result for a certain integral quantity (Proposition [Proposition 37](#prop:almost soliton scal){reference-type="ref" reference="prop:almost soliton scal"}), we refine the calculation in [@B3] in order to avoid the complexity. Thanks to this refinement, we obtain a new insight for Ricci flow such that the almost monotonicity can be improved, and an almost constancy holds in the special case (see Corollary [Corollary 38](#cor:almost soliton scal){reference-type="ref" reference="cor:almost soliton scal"}). # Preliminaries {#sec:Preliminaries} In this section, we recall some basic notions. In what follows, let $(M,g(t))_{t\in I}$ be an $n$-dimensional compact manifold with a time-dependent metric. ## Convention and notation {#sec:Basics} For positive constants $a,b,c,d,\dots$, we denote by $C_{a,b,c,d,\dots}$ positive constants depending only on $a,b,c,d,\dots$ and the dimension $n$ without distinction, where we stress that the dependence on the dimension $n$ are omitted. When we specify constants, we put numbers such as $C_{1,a,b,c,d,\dots},C_{2,a,b,c,d,\dots}$. When we assert that "there exists $\overline{C}_{a,b,c,d,\dots}$ such that if $C \leq \overline{C}_{a,b,c,d,\dots}$\", we abbreviate "if $C \leq \overline{C}_{a,b,c,d,\dots}$\" for short. In the same manner, for a lower bound, we say "if $C \geq \underline{C}_{a,b,c,d,\dots}$\" instead of "if $C \leq \overline{C}_{a,b,c,d,\dots}$\". Let $d_t$ and $m_t$ be the Riemannian distance and Riemannian volume measure with respect to $g(t)$, respectively. For $x\in M, t\in I$ and $r>0$, let $B(x,t,r)$ denote the open ball of radius $r$ centered at $x$ with respect to $g(t)$. Note that $\partial_t(d m_t)=-H\,dm_t$. The non-negativity of the Müller quantity (i.e., $\mathcal{D}(V)\geq 0$ for all vector fields $V$) is expressed by $\mathcal{D}\geq 0$. At the begging of the proof of each statement, we carry out the parallel translation for time and the parabolic rescaling. Recall that for $\mathrm{r}>0$, the parabolic rescaling (at time $0$) is given by $\bar{g}(s):=\mathrm{r}^{-2}\,g(\mathrm{r}^2 s)=\mathrm{r}^{-2}g(t)$. Note that the super Ricci flow and the non-negativity of the Müller quantity are preserved under the rescaling (cf. [@KuS Remark 2.2]). We notice the following lower bound for $H$, which is a direct consequence of the maximum principle (see [@FZ Lemma 3.2]): **Lemma 1** ([@FZ]). *Assume $\mathcal{D}\geq 0$. If $H (\cdot, t_0) \geq -\mathcal{H}$ for $\mathcal{H}>0$, then for all $t\in [t_0,\infty) \cap I$, $$H (\cdot, t) \geq -\frac{n}2 \frac{\mathcal{H}}{(n/2) + \mathcal{H} (t - t_0)}.$$* *Remark 2*. The following also holds (see [@FZ Lemma 3.2]): Assume $\mathcal{D}\geq 0$. If $t_0\in I$, then for all $t\in [t_0,\infty) \cap I$, we have $$H (\cdot, t) \geq -\frac{n}{2(t-t_0)},\quad \min_M H(\cdot, t)\geq \min_M H(\cdot,t_0).$$ ## Heat kernels {#sec:heat kernel} The *heat operator* and the *conjugate heat operator* are defined by $$\square:=\partial_t -\Delta,\quad \square^*:=-\partial_t-\Delta + H.$$ We recall the following fundamental formula (see e.g., [@CGT]): For any $u,v \in C^{\infty}(M\times I)$, $$\label{eq:conjugate integration} \frac{d}{dt}\int_M \,uv\,dm_t=\int_M\,(\square u) v\, dm_t-\int_M\, u (\square^* v)\,dm_t.$$ For $x,y \in M$ and $s,t\in I$ with $s< t$, we denote by $G(x,t;y,s)$ the *heat kernel* (see e.g., [@CCGG], [@G]); namely, for a fixed $(y,s) \in M \times I$, it solves $$(\partial_t - \Delta_x ) G(\cdot,\cdot;y,s) = 0,\quad \lim_{t \searrow s} G(\cdot,t;y,s)=\delta_y.$$ Notice that $G(x,t;\cdot,\cdot)$ is the kernel for the conjugate heat equation; namely, for any $(x,t) \in M \times I$, $$(-\partial_s -\Delta_y +H)G(x,t;\cdot,\cdot) = 0,\quad \lim_{s \nearrow t} G(x,t;\cdot, s) = \delta_x.$$ The semigroup property $$\label{eq:semigroup} G(x,t;y,s)=\int_{M}\,G(x,t; \cdot, \mathfrak{t})G(\cdot,\mathfrak{t}; y,s)\,dm_{\mathfrak{t}}$$ holds for any $\mathfrak{t}\in (s,t)$. For a base point $(x_0,t_0)\in M\times I$, the *$($conjugate$)$ heat kernel measure* is defined by $$\nu_{(x_0,t_0)}:=\{\nu_{(x_0,t_0);t}\}_{t \in (-\infty,t_0] \cap I},\quad \nu_{(x_0,t_0);t}:=G(x_0,t_0;\cdot,t)\,m_{t},\quad \nu_{(x_0,t_0);t_0} := \delta_{x_0}$$ for $t<t_0$, which are probability measures. *Remark 3*. We frequently set $\nu:=\nu_{(x_0,t_0)}$ or $\nu^0:=\nu_{(x_0,t_0)}$. In that case, without mentioning, we use the notation $\nu_t:=\nu_{(x_0,t_0);t}$ or $\nu^0_t:=\nu_{(x_0,t_0);t}$, respectively. We define $\tau:=t_0-t$, which is called the *parameter*. The *potential* $f \in C^\infty ( M \times ( (-\infty,t_0)\cap I))$ is determined by $$G(x_0,t_0;\cdot,t) = (4\pi \tau)^{-n/2} e^{-f(\cdot,t)},$$ which can be written as $$\label{eq:potential} f(x,t)=-\log G(x_0,t_0;x,t)-\frac{n}{2}\log \tau-\frac{n}{2}\log 4\pi.$$ The potential enjoys $$\label{eq:potential_enjoy} - \partial_t f = \Delta f - |\nabla f|^2 + H - \frac{n}{2\tau}.$$ The following can be derived from [\[eq:conjugate integration\]](#eq:conjugate integration){reference-type="eqref" reference="eq:conjugate integration"} and integration by parts (cf. [@B3 Lemma 4.4]): $$\label{eq:conjugate integration2} \frac{d}{dt}\int_M\,u\,d\nu_{(x_0,t_0);t}=\int_M\,\square u\,d\nu_{(x_0,t_0);t},\quad \int_M \,\operatorname{div}(V)\,d\nu_{(x_0,t_0);t}=\int_{M}\,\langle V,\nabla f \rangle\,d\nu_{(x_0,t_0);t}$$ for all $u\in C^{\infty}(M\times I)$ and vector fields $V$. *Remark 4*. We do not always introduce the notations of the parameter and the potential when it is clear from the context. We define $$\label{eq:Harnack notation} \Phi:=h + \nabla^2 f - \frac1{2\tau} g,\quad w:=\tau (2\Delta f - |\nabla f|^2 + H ) + f -n.$$ We possess the following Perelman type Harnack estimate (see [@P], [@CGT Theorems 1.1, 1.2]): **Theorem 5** ([@P], [@CGT]). *We set $u:=G(x_0,t_0; \cdot,\cdot)$. Then we have $$\square^* (wu) = - 2 \tau \left| \Phi \right|^2 u-\tau u \mathcal{D}(\nabla f).$$ Moreover, if $\mathcal{D}\geq 0$, then $w\leq 0$.* ## Wasserstein distance Let $(X,d)$ be a complete separable metric space, and let $\mu_1, \mu_2$ be two Borel probability measures with finite first moment. We denote by $W(\mu_1, \mu_2)$ their *$($$L^1$-$)$Wasserstein distance*. The canonical formulation of the Wasserstein distance is based on a coupling method. We would rather make use of the following characterization called the *Kantorovich-Rubinstein duality* (see e.g., [@V Theorem 5.10]): $$\label{eq:W1} W(\mu_1, \mu_2)=\sup_{\phi} \bigg( \int_X \phi \, d\mu_2 - \int_X \phi \, d\mu_1 \bigg),$$ where the supremum is taken over all bounded $1$-Lipschitz functions $\phi:X \to \mathbb{R}$. We also denote by $\operatorname{Var}(\mu_1, \mu_2)$ the *variance* between $\mu_1$ and $\mu_2$ defined as $$\operatorname{Var}(\mu_1, \mu_2) := \int_X \int_X d^2(x_1, x_2) \, d\mu_1(x_1) d\mu_2 (x_2).$$ We recall the following (see e.g., [@B2 Lemma 2.8]): **Lemma 6** ([@B2]). *For any two Borel probability measures $\mu_1,\mu_2$ on $(X,d)$, we have $$W(\mu_1, \mu_2) \leq \sqrt{\operatorname{Var}(\mu_1, \mu_2)}.$$* # Super Ricci flow {#sec:SRF} In this section, we summarize several facts on super Ricci flow. ## Wasserstein monotonicity and concentration bounds {#sec:Concentration bounds} Let $W_{t}$ and $\operatorname{Var}_t$ stand for the Wasserstein distance and the variance induced from $g(t)$, respectively. We first recall the following monotonicity property for the Wasserstein distance (see [@B1 Lemma 2.7]): **Proposition 7** ([@B1]). *Let $(M, g(t))_{t\in I}$ be a super Ricci flow. Let $(x_0,t_0),(x_1,t_1)\in M\times I$. Then $W_{t}(\nu_{(x_0,t_0); t}, \nu_{(x_1, t_1); t} )$ is non-decreasing in $t\in (-\infty,\min\{t_0,t_1\}]\cap I$.* For $(x_0,t_0) \in M \times I$, a point $(z,t) \in M \times ((-\infty,t_0] \cap I)$ is said to be a *$($$\mathcal{C}_n$-$)$center* if $$\mathcal{C}_n:=\frac{(n-1)\pi^2}{2} + 4,\quad \operatorname{Var}_t (\delta_z, \nu_{(x_0,t_0);t}) \leq \mathcal{C}_n(t_0-t).$$ For the existence of centers, we have the following (see [@B1 Proposition 3.12]): **Proposition 8** ([@B1]). *Let $(M,g(t))_{t \in I}$ be a super Ricci flow. Let $(x_0,t_0) \in M \times I$. Then for every $t \in (-\infty,t_0] \cap I$, there is $z \in M$ such that $(z,t)$ is a center of $(x_0, t_0)$.* We possess the following concentration bounds (see [@B1 Proposition 3.13, Theorem 3.14]): **Proposition 9** ([@B1]). *Let $(M,g(t))_{t \in I}$ be a super Ricci flow. Let $(x_0,t_0) \in M \times I$, and let $(z,t)$ be a center of $(x_0, t_0)$. Then for every $R > 1$ we have $$\nu_{(x_0,t_0);t} \left( B(z, t, \sqrt{R \mathcal{C}_n(t_0 - t)} ) \right) \geq 1- R^{-1}.$$* **Proposition 10** ([@B1]). *Let $(M, g(t))_{t \in I}$ be a super Ricci flow. Let $(x_0,t_0) \in M \times I$, and let $(z,t)$ be a center of $(x_0, t_0)$. Then for all $r> 0$ we have $$\nu_{(x_0,t_0);t} \left( M \setminus B(z, t, r) \right)\leq 2 \exp \left( {- \frac{\left(r - \sqrt{2\mathcal{C}_n(t_0-t)} \right)_+^2}{8(t_0-t)} }\right). %\leq C_\eps \exp \left( - \frac{r^2}{(8+\eps)(t_0 - t)} \right).$$* *Remark 11*. Under the same setting as in Proposition [Proposition 10](#prop:concentration2){reference-type="ref" reference="prop:concentration2"}, the following more elementary bound holds (cf. [@B1 Proposition 3.13]): For all $r>0$ we have $$\label{eq:elementary concentration} \nu_{(x_0,t_0);t} \left( M \setminus B(z, t, r) \right)\leq \frac{\operatorname{Var}_t(\nu_{(x_0,t_0);t},\delta_z)}{r^2}\leq \frac{\mathcal{C}_n(t_0-t)}{r^2}.$$ ## Analytic bounds {#sec:Gradient estimates} Let $\Psi : \mathbb{R}\to (0,1)$ be a function determined by $$\label{eq:Gaussain} \Psi' (x) = (4\pi)^{-1/2} e^{-x^2/4}, \quad \lim_{x \to -\infty} \Psi (x) = 0, \quad \lim_{x \to \infty} \Psi (x) = 1.$$ For $t>0$, let $\Psi_t : \mathbb{R}\to (0,1)$ be a function defined by $\Psi_t (x):=\Psi (t^{-1/2} x)$, and let $\Psi^{-1}_t : (0,1) \to \mathbb{R}$ be its inverse. We have the following gradient estimate (see [@B1 Theorem 4.1]): **Theorem 12** ([@B1]). *Let $(M,g(t))_{t\in I}$ be a super Ricci flow. For $[t_0, t_1] \subset I$, let $u \in C^\infty (M \times [t_0, t_1])$ be a solution to the heat equation. Let $T \geq 0$. Suppose that $u$ only takes values in $(0,1)$ and $| \nabla (\Psi_T^{-1} ( u (\cdot , t_0) ))| \leq 1$ if $T > 0$. Then $| \nabla (\Psi^{-1}_{T+t - t_0} ( u(\cdot, t) ))| \leq 1$ for all $t \in [t_0, t_1]$.* We next present the following heat kernel estimates (see [@B1 Proposition 4.2]): **Proposition 13** ([@B1]). *Let $(M, g(t))_{t \in I}$ be a super Ricci flow. Let $x\in M$ and $[s,t] \subset I$. Then for every $p\in [1,\infty)$ we have $$\label{eq:G_bound11} (t-s)^{p/2} \int_M \left( \frac{|\nabla_x G(x,t; \cdot, s)|}{G(x,t; \cdot, s)} \right)^p d\nu_{(x,t);s} \leq C_{p};$$ moreover, for every Borel subset $\Omega \subset M$ we have $$\label{eq:G_bound13} (t-s)^{p/2} \int_\Omega \left( \frac{|\nabla_x G(x,t; \cdot, s)|}{G(x,t; \cdot, s)} \right)^p d\nu_{(x,t);s} \leq C_{p}\, \nu_{(x,t);s}(\Omega)(-\log (\nu_{(x,t);s}(\Omega)/2))^{p/2}.$$ Also, for every $\mathrm{v} \in T_x M$ with $|\mathrm{v}|_t = 1$, $$\label{eq:G_bound12} (t-s) \int_M \left( \frac{\partial_{\mathrm{v}} G(x,t; \cdot, s)}{G(x,t; \cdot, s)} \right)^2 d\nu_{(x,t);s} \leq \frac12.$$* *Remark 14*. In [\[eq:G_bound11\]](#eq:G_bound11){reference-type="eqref" reference="eq:G_bound11"} and [\[eq:G_bound13\]](#eq:G_bound13){reference-type="eqref" reference="eq:G_bound13"}, one can choose $C_{2} = n/2$. We have the following Poincaré inequality (see [@HaN Theorem 1.5], [@B1 Theorem 11.1], and also [@HN Theorem 1.10]): **Proposition 15** ([@HaN], [@B1]). *Let $(M, g(t))_{t \in I}$ be a super Ricci flow. Let $(x_0,t_0) \in M \times I$. Let $\tau>0$ satisfy $[t_0 - \tau,t_0] \subset I$. Then for any $u \in C^1(M)$ with $\int_M u \, d\nu_{(x_0,t_0);t_0-\tau}=0$ and $p \in [1,\infty)$, $$\int_M |u|^p d\nu_{(x_0,t_0);t_0-\tau} \leq C_p \,\tau^{p/2}\, \int_M |\nabla u|^p d\nu_{(x_0,t_0);t_0-\tau}.$$* *Proof.* Bamler [@B1] has stated this assertion only for Ricci flow, but the argument also works for super Ricci flow without any changes. For $p=2$, the desired inequality has been obtained by Haslhofer-Naber [@HaN] (see [@HaN Theorem 1.5]). For $p=1$, by noticing the following point, one can prove the desired one along the lines of the proof of [@B1 Theorem 11.1]: If $\square u=0$, then the Bochner formula and the Kato inequality imply $$\square |\nabla u|^2=2h(\nabla u,\nabla u)-2\operatorname{Ric}(\nabla u,\nabla u)-2|\nabla^2 u|^2\leq -\frac{|\nabla |\nabla u|^2|^2}{2|\nabla u|^2};$$ in particular, $\square |\nabla u|\leq 0$. For general $p$, the desired one follows from the assertion for $p=1$ together with the same argument of the proof of [@B1 Theorem 11.1]. ◻ *Remark 16*. In Proposition [Proposition 15](#prop:Poincare){reference-type="ref" reference="prop:Poincare"}, we may choose $C_1 = \sqrt{\pi}$ and $C_2 = 2$. # Nash entropy {#sec:Nash entropy} In this section, we examine basic properties of the Nash entropy. ## Entropy monotonicity Fix a base point $(x_0, t_0) \in M \times I$. For $\tau > 0$ with $[t_0 - \tau, t_0] \subset I$, the *pointed Nash-entropy* is defined by $$\mathcal{N}_{(x_0, t_0)} (\tau):= \int_M \,f\,d\nu_{(x_0,t_0);t_0-\tau} - \frac{n}2.$$ We set $\mathcal{N}_{(x_0, t_0)} (0) := 0$ such that $\mathcal{N}_{(x_0,t_0)}(\tau)$ is continuous in $\tau$ (cf. [@B1 Proposition 5.2]). The *pointed $\mathcal{W}$-entropy* is defined by $$\mathcal{W}_{(x_0,t_0)}(\tau):=\int_M \left( \tau (|\nabla f|^2 + H) + f - n \right) d\nu_{(x_0,t_0);t_0-\tau}.$$ We recall the following monotonicity properties (cf. [@B1 Proposition 5.2], [@H Theorem 3.1], [@GPT Theorem 5.2], [@FZ Lemma 3.1]): **Lemma 17**. *Assume $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. If $\tau > 0$ with $[t_0 - \tau, t_0] \subset I$, then $$\begin{aligned} \label{eq:Nash2} &\frac{d}{d\tau} \left(\tau \mathcal{N}_{(x_0, t_0)} (\tau)\right) = \mathcal{W}_{(x_0,t_0)}(\tau) \leq 0,\\ \label{eq:Nash3} &\frac{d^2}{d\tau^2} \left( \tau \mathcal{N}_{(x_0, t_0)} (\tau) \right) = - \tau \int_M \left(2\left| \Phi \right|^2+\mathcal{D}(\nabla f) \right) d\nu_{(x_0,t_0);t_0 - \tau} \leq 0,\\ \label{eq:Nash3.5} &\frac{d}{d\tau}\mathcal{N}_{(x_0, t_0)} (\tau) \leq 0,\\ \label{eq:Nash rem} &\mathcal{W}_{(x_0,t_0)}(\tau) \leq \mathcal{N}_{(x_0, t_0)} (\tau),\end{aligned}$$ where $\Phi$ is defined as [\[eq:Harnack notation\]](#eq:Harnack notation){reference-type="eqref" reference="eq:Harnack notation"}.* *Proof.* The formulas [\[eq:Nash2\]](#eq:Nash2){reference-type="eqref" reference="eq:Nash2"} and [\[eq:Nash3\]](#eq:Nash3){reference-type="eqref" reference="eq:Nash3"} are well-known (see e.g., [@H Theorem 3.1], [@GPT Theorem 5.2], [@FZ Lemma 3.1]). Furthermore, [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"} and [\[eq:Nash rem\]](#eq:Nash rem){reference-type="eqref" reference="eq:Nash rem"} can be derived from the same calculation as in the proof of [@B1 Proposition 5.2] together with [\[eq:Nash2\]](#eq:Nash2){reference-type="eqref" reference="eq:Nash2"} and [\[eq:Nash3\]](#eq:Nash3){reference-type="eqref" reference="eq:Nash3"}. ◻ We also see the following: **Lemma 18**. *Assume $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. Let $\tau_1,\tau_2>0$ satisfy $\tau_1 \leq \tau_2$ and $[t_0 - \tau_2, t_0] \subset I$. For $\mathcal{H}>0$, we assume $H(\cdot,t_0-\tau_2) \geq -\mathcal{H}$. Then we have $$\mathcal{N}_{(x_0, t_0)} (\tau_1)-\left((\tau_2-\tau_1)\mathcal{H}+\frac{n}{2}\log\left( \frac{\tau_2}{\tau_1} \right)\right) \leq \mathcal{N}_{(x_0, t_0)} (\tau_2).$$* *Proof.* From [\[eq:Nash2\]](#eq:Nash2){reference-type="eqref" reference="eq:Nash2"} we deduce $$\begin{aligned} \frac{d}{d\tau} \left(\tau \mathcal{N}_{(x_0, t_0)}(\tau) \right) =\mathcal{W}_{(x_0,t_0)}(\tau) &= \int_M \tau \left( |\nabla f|^2 + H \right)d\nu_{(x_0,t_0);t_0 - \tau} + \mathcal{N}_{(x_0, t_0)}(\tau) - \frac{n}2 \\ &\geq -\tau \mathcal{H} + \mathcal{N}_{(x_0, t_0)}(\tau) - \frac{n}2,\end{aligned}$$ and hence $$\frac{d}{d\tau} \mathcal{N}_{(x_0, t_0)} (\tau)\geq -\left(\mathcal{H}+ \frac{n}{2\tau}\right).$$ This implies $$\mathcal{N}_{(x_0, t_0)} (\tau_2) - \mathcal{N}_{(x_0, t_0)}(\tau_1)\geq -\int^{\tau_2}_{\tau_1} \,\left(\mathcal{H}+ \frac{n}{2\tau}\right)\, d\tau= -\left((\tau_2-\tau_1)\mathcal{H}+\frac{n}{2}\log\left( \frac{\tau_2}{\tau_1} \right)\right).$$ We complete the proof. ◻ Lemma [Lemma 17](#lem:Nash){reference-type="ref" reference="lem:Nash"} and Proposition [Proposition 15](#prop:Poincare){reference-type="ref" reference="prop:Poincare"} yield the following (cf. [@B1 Proposition 5.13]): **Lemma 19**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. For $\mathcal{H}>0$, we assume $H(\cdot, t_0 - \tau) \geq -\mathcal{H}$. Then we have $$\begin{aligned} \label{eq:Nash basic1} &\int_M \tau ( |\nabla f|^2 + H) d\nu_{(x_0,t_0);t_0-\tau} \leq \frac{n}2,\\ \label{eq:Nash basic2} &\int_M \left( f - \mathcal{N}_{(x_0, t_0)} (\tau) - \frac{n}2 \right)^2 d\nu_{(x_0,t_0);t_0-\tau} \leq n+ 2 \mathcal{H}\tau.\end{aligned}$$* *Proof.* By the same calculation as in the proof of [@B1 Proposition 5.13], and by [\[eq:Nash2\]](#eq:Nash2){reference-type="eqref" reference="eq:Nash2"}, [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"}, $$\begin{aligned} \int_M \tau ( |\nabla f|^2 + H ) d\nu_{(x_0,t_0);t_0 - \tau}&= \frac{n}2 + \tau \frac{d}{d\tau} \mathcal{N}_{(x_0, t_0)} (\tau) \leq \frac{n}2,\end{aligned}$$ which proves [\[eq:Nash basic1\]](#eq:Nash basic1){reference-type="eqref" reference="eq:Nash basic1"}. Proposition [Proposition 15](#prop:Poincare){reference-type="ref" reference="prop:Poincare"} with $p=2$ together with $$\int_M \left( f - \mathcal{N}_{(x_0, t_0)} (\tau) - \frac{n}2 \right) d\nu_{(x_0,t_0);t_0-\tau} = 0$$ and $$2\tau \int_M |\nabla f|^2 d\nu_{(x_0,t_0);t_0 - \tau} \leq n - 2\tau \int_{M} H \, d\nu_{(x_0,t_0);t_0 - \tau} \leq n + 2\mathcal{H} \tau$$ leads us to [\[eq:Nash basic2\]](#eq:Nash basic2){reference-type="eqref" reference="eq:Nash basic2"} (see also Remark [Remark 16](#rem:Poincare){reference-type="ref" reference="rem:Poincare"}). We arrive at the desired estimates. ◻ ## Derivative estimates For a fixed $s\in I$, we define $$\mathcal{N}_s (x, t) := \mathcal{N}_{(x, t)} (t-s)$$ on $M \times ( (s, \infty) \cap I)$. We have the following derivative estimates (cf. [@B1 Theorem 5.9]): **Lemma 20**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $s \in I$. For $\mathcal{H}>0$, we assume $H (\cdot, s) \geq -\mathcal{H}$. Then on $M \times ( (s, \infty) \cap I)$ we have $$\begin{aligned} \label{eq:Nash thm gradient} &|\nabla \mathcal{N}_s |\leq \left( \frac{n}{2(t-s)}+\mathcal{H} \right)^{1/2},\\ \label{eq:Nash thm heat} &- \frac{n}{2(t-s)} \leq \square \mathcal{N}_s \leq 0. \end{aligned}$$* *Proof.* We may assume $s = 0$. Fix a base point $(x,t) \in M \times ( (0, \infty) \cap I)$. For $\mathrm{v} \in T_x M$ with $|\mathrm{v}|_t = 1$, the same calculation as in the proof of [@B1 Theorem 5.9] tells us that $$\begin{aligned} &\quad\,\, \partial_{\mathrm{v}} \mathcal{N}_0 (x,t)\\ &= \int_M \left(\frac{\partial_{\mathrm{v}}G(x,t;y,0)}{G(x,t;y,0)}\right) \left( f (y,0) - \mathcal{N}_0 (x,t) - \frac{n}2 \right) d\nu_{(x,t);0} (y) \\ &\leq \left( \int_M \left( \frac{\partial_{\mathrm{v}} G(x,t;y,0)}{G(x,t;y,0)} \right)^2 d\nu_{(x,t);0}(y) \right)^{1/2} \left( \int_M \left( f - \mathcal{N}_0 (x,t) - \frac{n}2 \right)^2 d\nu_{(x,t);0}(y) \right)^{1/2};\end{aligned}$$ in particular, [\[eq:G_bound12\]](#eq:G_bound12){reference-type="eqref" reference="eq:G_bound12"} and [\[eq:Nash basic2\]](#eq:Nash basic2){reference-type="eqref" reference="eq:Nash basic2"} imply $$|\nabla \mathcal{N}_0|^2(x,t) \leq \frac{1}{2t} (n + 2\mathcal{H} t),$$ which is the desired gradient estimate [\[eq:Nash thm gradient\]](#eq:Nash thm gradient){reference-type="eqref" reference="eq:Nash thm gradient"} (see also Remark [Remark 2](#rem:scal2){reference-type="ref" reference="rem:scal2"}). Similarly, we possess $$\begin{aligned} \square \mathcal{N}_0 (x,t)= \int_M \left( \frac{|\nabla_x G(x,t;y,0)|}{G(x,t;y,0)} \right)^2 d\nu_{(x,t);0}(y) - \frac{n}{2t},\end{aligned}$$ and [\[eq:G_bound11\]](#eq:G_bound11){reference-type="eqref" reference="eq:G_bound11"} with $p=2$ leads us to [\[eq:Nash thm heat\]](#eq:Nash thm heat){reference-type="eqref" reference="eq:Nash thm heat"} (see also Remark [Remark 14](#rem:G_bound){reference-type="ref" reference="rem:G_bound"}). ◻ We conclude the following (cf. [@B1 Corollary 5.11]): **Lemma 21**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. For a fixed $s\in I$, let $(x_1,t_1),(x_2,t_2)\in M\times ( (s, \infty) \cap I)$. For $\mathcal{H}>0$, we assume $H (\cdot, s) \geq -\mathcal{H}$. Then for every $t\in (s,\min\{t_1,t_2\}]$ we have $$\mathcal{N}_s(x_1, t_1) - \mathcal{N}_s(x_2, t_2) \leq \left( \frac{n}{2(t-s)} + \mathcal{H} \right)^{1/2} W_{t}(\nu_{(x_1, t_1);t}, \nu_{(x_2, t_2);t}) + \frac{n}2 \log \left( \frac{t_2-s}{t-s} \right) .$$* *Proof.* We may assume $s = 0$. For $i=1,2$, we set $\nu^{i}:=\nu_{(x_i,t_i)}$. In virtue of [\[eq:Nash thm heat\]](#eq:Nash thm heat){reference-type="eqref" reference="eq:Nash thm heat"}, we see $$\label{eq:Nash cor1} \mathcal{N}_0 (x_i, t_i) \leq \int_M \mathcal{N}_0 (\cdot, t) d\nu^i_{t} \leq \mathcal{N}_0 (x_i, t_i) + \frac{n}2 \log \left( \frac{t_i}{t} \right).$$ On the other hand, due to [\[eq:Nash thm gradient\]](#eq:Nash thm gradient){reference-type="eqref" reference="eq:Nash thm gradient"} and [\[eq:W1\]](#eq:W1){reference-type="eqref" reference="eq:W1"}, $$\label{eq:Nash cor2} \left| \int_M \mathcal{N}_0 (\cdot, t) d\nu^2_{t} - \int_M \mathcal{N}_0 (\cdot, t) d\nu^1_{t} \right| \leq \left( \frac{n}{2t} + \mathcal{H} \right)^{1/2} W_{t}( \nu^1_{t}, \nu^2_{t} ).$$ The desired estimate follows from combining [\[eq:Nash cor1\]](#eq:Nash cor1){reference-type="eqref" reference="eq:Nash cor1"} and [\[eq:Nash cor2\]](#eq:Nash cor2){reference-type="eqref" reference="eq:Nash cor2"}. ◻ We will use Lemma [Lemma 21](#lem:Nash cor){reference-type="ref" reference="lem:Nash cor"} in the following form: **Lemma 22**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. For a fixed $s\in I$, let $(x_0,t_0)\in M\times ( (s, \infty) \cap I)$. For $t\in (s, t_0]$, we assume that $(z,t)$ is a center of $(x_0,t_0)$. For $\mathcal{H}>0$, we further assume $H (\cdot, s) \geq -\mathcal{H}$. Then on $M$, we have $$\begin{aligned} - \mathcal{N}_s(\cdot, t)\leq -\mathcal{N}_s(x_0,t_0)+\left( \frac{n}{2(t-s)} + \mathcal{H} \right)^{1/2} \sqrt{\mathcal{C}_n(t_0-t)}+\left( \frac{n}{2(t-s)}+\mathcal{H} \right)^{1/2}d_{t}(z,\cdot).\end{aligned}$$* *Proof.* We may assume $s=0$. Lemmas [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"} and [Lemma 21](#lem:Nash cor){reference-type="ref" reference="lem:Nash cor"} yield $$\begin{aligned} \label{eq:Nash cor_rem1} \mathcal{N}_0(x_0, t_0) - \mathcal{N}_0(z,t) &\leq \left( \frac{n}{2t} + \mathcal{H} \right)^{1/2} W_{t}(\delta_{z},\nu_{(x_0, t_0);t})\\ \notag &\leq \left( \frac{n}{2t} + \mathcal{H} \right)^{1/2} \sqrt{\operatorname{Var}(\delta_z, \nu_{(x_0, t_0);t})}\leq \left( \frac{n}{2t} + \mathcal{H} \right)^{1/2} \sqrt{\mathcal{C}_n(t_0-t)}.\end{aligned}$$ Further, [\[eq:Nash thm gradient\]](#eq:Nash thm gradient){reference-type="eqref" reference="eq:Nash thm gradient"} and [\[eq:Nash cor_rem1\]](#eq:Nash cor_rem1){reference-type="eqref" reference="eq:Nash cor_rem1"} lead us to $$\begin{aligned} - \mathcal{N}_0(\cdot, t)&\leq -\mathcal{N}_0(z,t)+\left( \frac{n}{2t}+\mathcal{H} \right)^{1/2}d_{t}(z,\cdot)\\ &\leq -\mathcal{N}_0(x_0,t_0)+\left( \frac{n}{2t} + \mathcal{H} \right)^{1/2} \sqrt{\mathcal{C}_n(t_0-t)}+\left( \frac{n}{2t}+\mathcal{H} \right)^{1/2}d_{t}(z,\cdot)\end{aligned}$$ on $M$. This proves the lemma. ◻ We also use Lemma [Lemma 21](#lem:Nash cor){reference-type="ref" reference="lem:Nash cor"} in the following form: **Lemma 23**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. For a fixed $s\in I$, let $(x_0,t_0),(x_1,t_1)\in M\times I$ with $t_0\leq t_1$. For $r>0,\alpha\in (0,1)$, we assume $[t_0-2\alpha^{-1} r^2,t_0] \subset I$ and $0\leq t_1-t_0\leq \alpha^{-1} r^2$. For $\mathcal{H}>0$, we assume $H (\cdot, t_0-2\alpha^{-1} r^2) \geq -\mathcal{H}r^{-2}$. For $D>0$, we assume $W_{s_0}(\nu_{(x_0,t_0);s_0},\nu_{(x_1,t_1);s_0})\leq D r$ for some $s_0\in [t_0-\alpha^{-1} r^2,t_0-\alpha r^2]$. Then we have $$\mathcal{N}_{(x_1, t_1)} (r^2)\geq \mathcal{N}_{(x_0,t_0)}(r^2)-C_{D,\alpha,\mathcal{H}}.$$* *Proof.* We may assume $r=1$. By Lemma [Lemma 18](#lem:Nash_more){reference-type="ref" reference="lem:Nash_more"}, we have $$\label{eq:useful1} \mathcal{N}_{(x_0, t_0)} (2\alpha^{-1})\geq \mathcal{N}_{(x_0, t_0)} (1)-C_{\alpha,\mathcal{H}}.$$ Furthermore, by Lemma [Lemma 21](#lem:Nash cor){reference-type="ref" reference="lem:Nash cor"} and Proposition [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"}, $$\begin{aligned} \label{eq:useful2} &\quad \,\,\mathcal{N}_{(x_0,t_0)}(2\alpha^{-1})-\mathcal{N}_{(x_1,t_1)}(t_1-t_0+2\alpha^{-1})\\ \notag &=\mathcal{N}_{t_0-2\alpha^{-1}}(x_0,t_0)-\mathcal{N}_{t_0-2\alpha^{-1}}(x_1,t_1)\\ \notag &\leq \left( \frac{n}{2\alpha^{-1}} +\mathcal{H}\right)^{1/2}W_{t_0-\alpha^{-1}}(\nu_{(x_0,t_0);t_0-\alpha^{-1}},\nu_{(x_1,t_1);t_0-\alpha^{-1}})+\frac{n}{2}\log 3\\ \notag &\leq \left( \frac{n}{2\alpha^{-1}} +\mathcal{H}\right)^{1/2}W_{s_0}(\nu_{(x_0,t_0);s_0},\nu_{(x_1,t_1);s_0})+\frac{n}{2}\log 3\leq C_{D,\alpha}.\end{aligned}$$ From [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"}, [\[eq:useful1\]](#eq:useful1){reference-type="eqref" reference="eq:useful1"} and [\[eq:useful2\]](#eq:useful2){reference-type="eqref" reference="eq:useful2"}, we derive $$\mathcal{N}_{(x_1, t_1)} (1)\geq \mathcal{N}_{(x_1,t_1)}(t_1-t_0+2\alpha^{-1}) \geq \mathcal{N}_{(x_0,t_0)}(2\alpha^{-1})-C_{D,\alpha}\geq \mathcal{N}_{(x_0,t_0)}(1)-C_{D,\alpha,\mathcal{H}}.$$ This completes the proof. ◻ # Volume and heat kernel estimates {#sec:vol_heat} In this section, we derive several volume and heat kernel estimates. ## Lower volume estimates We begin with the following (cf. [@B1 Theorem 6.2]): **Proposition 24**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$. For $r>0$, we assume $[t_0-r^2, t_0] \subset I$. For $\mathcal{H}>0$, we assume $H(\cdot, t_0-r^2) \geq -\mathcal{H}r^{-2}$. Let $(z,t_0-r^2) \in M \times I$ be a center of $(x_0,t_0)$. Then we have $$m_{t_0-r^2}\left(B(z,t_0-r^2, \sqrt{2 \mathcal{C}_n} r)\right) \geq C_{\mathcal{H}}\, \exp ( \mathcal{N}_{(x_0,t_0)}(r^2)) r^n.$$* *Proof.* We may assume $t_0 = 1$ and $r=1$. Let $\nu:=\nu_{(x_0,1)}$. By virtue of Proposition [Proposition 9](#prop:concentration1){reference-type="ref" reference="prop:concentration1"}, $$\label{eq:lower volume1} \nu_{0} ( B) \geq \frac12,$$ where $B:=B(z,0, \sqrt{2\mathcal{C}_n} )$. Lemma [Lemma 19](#lem:Nash basic){reference-type="ref" reference="lem:Nash basic"} tells us that $$\label{eq:lower volume2} \int_M \left| f - \mathcal{N}_{(x_0,1)}(1) - \frac{n}2 \right| d\nu_{0} \leq \left( \int_M \left( f - \mathcal{N}_{(x_0,1)}(1) - \frac{n}2 \right)^2 d\nu_{0} \right)^{1/2} \leq (n+2\mathcal{H})^{1/2}.$$ Combining [\[eq:lower volume1\]](#eq:lower volume1){reference-type="eqref" reference="eq:lower volume1"} and [\[eq:lower volume2\]](#eq:lower volume2){reference-type="eqref" reference="eq:lower volume2"}, we conclude $$\begin{aligned} \frac1{\nu_{0} (B)} \int_B f \, d\nu_{0} &\geq \mathcal{N}_{(x_0,1)}(1) + \frac{n}2 - \frac1{\nu_{0} (B)} \int_B \left| f - \mathcal{N}_{(x_0,1)}(1) - \frac{n}2 \right| d\nu_{0}\\ &\geq \mathcal{N}_{(x_0,1)}(1) + \frac{n}2 - 2(n+ 2 \mathcal{H})^{1/2}.\end{aligned}$$ We set $u := (4\pi )^{n/2} e^{-f} / \nu_{0} (B)$. Since $\int_B u \, dm_{0} = 1$, [\[eq:lower volume1\]](#eq:lower volume1){reference-type="eqref" reference="eq:lower volume1"} tells us that $$\begin{aligned} \int_B u \log u \, dm_{0} &= - \frac1{\nu_0 (B)} \int_B f \, d\nu_0 - \log \nu_0 (B) + \frac{n}2 \log 4\pi \\ &\leq -\mathcal{N}_{(x_0,1)}(1) - \frac{n}2 + 2(n+ 2 \mathcal{H})^{1/2} + \log 2 + \frac{n}2 \log (4\pi). \end{aligned}$$ By the Jensen inequality, $$\log \left( \frac1{m_0(B)} \int_B u \, dm_{0} \right) \frac1{m_0(B)} \int_B u \, dm_{0} \leq \frac1{m_0(B)} \int_B u \log u \, dm_{0},$$ and thus $$- \log m_0(B) \leq \int_B u \log u \, dm_{0} \leq - \mathcal{N}_{(x_0,1)}(1) + C + 2 (n+ 2 \mathcal{H})^{1/2}.$$ We complete the proof. ◻ ## Heat kernel estimates We next show the following (cf. [@B1 Theorem 7.1]): **Proposition 25**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$, and let $[t,t_0] \subset I$. For $\mathcal{H}>0$, we assume $H(\cdot,t) \geq -\mathcal{H}(t_0-t)^{-1}$. Then on $M$, we have $$G(x_0,t_0;\cdot,t) \leq \frac{C_{\mathcal{H}}}{(t_0-t)^{n/2}} \exp ( - \mathcal{N}_{(x_0,t_0)}(t_0-t) ).$$* *Proof.* We may assume $t = 0$ and $t_0 = 1$. For a fixed $y \in M$, we set $u := G(\cdot, \cdot; y,0)$. By the same argument as in the proof of [@B1 Theorem 7.1], it suffices to show the following: If $L \geq \underline{L}_{\mathcal{H}}$, and if for all $(x,s) \in M \times (0,1]$ we have $$\label{eq:GboundW} u(x,s) \leq \frac{L}{s^{n/2}} \exp ( - \mathcal{N}_0 (x,s)),$$ then for all $x\in M$ we have $$\label{eq:GboundW_desired} u(x,1) \leq \frac{L}{2} \exp ( - \mathcal{N}_0 (x,1)).$$ First, assuming [\[eq:GboundW\]](#eq:GboundW){reference-type="eqref" reference="eq:GboundW"}, we derive a gradient bound $$\label{eq:HK3} |\nabla u|(x,s) \leq C_{\mathcal{H}} L \exp \left({ - \mathcal{N}_0 (x,s ) } \right)$$ for all $(x,s) \in M \times [3/4, 1]$. We set $$v := \left( s - \frac{1}{2} \right) |\nabla u|^2 + u^2.$$ Due to the Bochner formula, $$\square v = -|\nabla u|^2 - 2\left( s - \frac{1}{2} \right) |\nabla^2 u|^2-2\left( s-\frac{1}{2} \right)(\operatorname{Ric}-h)(\nabla u,\nabla u) \leq 0$$ for every $s \in [1/2 ,1]$. Using [\[eq:GboundW\]](#eq:GboundW){reference-type="eqref" reference="eq:GboundW"}, we have $$\begin{aligned} \label{eq:HK1} \left( s - \frac{1}{2} \right) |\nabla u|^2 (x,s) \leq v(x,s) &\leq \int_M v(\cdot, 1/2 ) d\nu_{(x,s);1/2}= \int_M u^2(\cdot , 1/2) d\nu_{(x,s);1/2}\\ \notag &\leq 2^{n}L^2 \int_M \exp \left({ - 2 \mathcal{N}_0 (\cdot,1/2 ) } \right) d\nu_{(x,s);1/2}\end{aligned}$$ for every $(x,s) \in M \times (1/2, 1]$. Let $(z,1/2)$ be a center of $(x,s)$ (see Proposition [Proposition 8](#prop:center){reference-type="ref" reference="prop:center"}). In view of Lemma [Lemma 22](#lem:Nash cor_rem){reference-type="ref" reference="lem:Nash cor_rem"}, we have $$\label{eq:HK2} - \mathcal{N}_0 (\cdot,1/2) \leq -\mathcal{N}_0 (x,s) + C_{\mathcal{H}}\, (d_{1/2} (z, \cdot)+1)$$ on $M$. From [\[eq:HK2\]](#eq:HK2){reference-type="eqref" reference="eq:HK2"}, the co-area formula and Proposition [Proposition 10](#prop:concentration2){reference-type="ref" reference="prop:concentration2"}, we derive $$\begin{aligned} &\quad\,\, \int_M \exp \left({ - 2 \mathcal{N}_0 (\cdot, 1/2 ) } \right) d\nu_{(x,s);1/2}\\ &\leq C_{\mathcal{H}} \exp ({ - 2 \mathcal{N}_0 (x,s ) } ) \int_M \exp (C_{\mathcal{H}} d_{1/2} (z, \cdot)) d\nu_{(x,s);1/2} \\ &\leq C_{\mathcal{H}} \exp ({ - 2 \mathcal{N}_0 (x,s ) } ) \left( \int_0^\infty e^{C_{\mathcal{H}} r}\,\nu_{(x,s);1/2}(M \setminus B(z,1/2,r))\, dr + 1 \right) \\ &\leq C_{\mathcal{H}} \exp ({ - 2 \mathcal{N}_0 (x,s ) } )\left( \int_0^\infty e^{C_{\mathcal{H}} r} \exp \left({ - \frac{\left(r - \sqrt{2\mathcal{C}_n(t-1/2)} \right)_+^2}{8 (t - 1/2)}}\right) dr +1 \right) \\ &= C_{\mathcal{H}} \exp ({ - 2 \mathcal{N}_0 (x,s ) } ) \left( \sqrt{t- 1/2} \int_0^\infty e^{C_{\mathcal{H}} \sqrt{t- 1/2} \, r} \exp \left({ - \frac{\left(r - \sqrt{2\mathcal{C}_n} \right)_+^2}{8}}\right) dr +1 \right)\\ &\leq C_{\mathcal{H}} \exp ({ - 2 \mathcal{N}_0 (x,s ) } ).\end{aligned}$$ Combining this with [\[eq:HK1\]](#eq:HK1){reference-type="eqref" reference="eq:HK1"} implies [\[eq:HK3\]](#eq:HK3){reference-type="eqref" reference="eq:HK3"}. Let $(x,1) \in M \times I$, and set $\nu:=\nu_{(x,1)}$. For $\xi \in (0, 1/2]$, we put $t_1 := 1- \xi^2$, and let $(z_1, t_1)$ be a center of $(x,1)$. We set $B := B(z_1, t_1, \sqrt{100 \mathcal{C}_n\xi^2})$. By Lemma [Lemma 22](#lem:Nash cor_rem){reference-type="ref" reference="lem:Nash cor_rem"}, if $\xi \leq \overline{\xi}_{\mathcal{H}}$, then $$\label{eq:HK4} - \mathcal{N}_0 (\cdot, t_1) \leq - \mathcal{N}_0 (x,1)+C_{\mathcal{H}} d_{t_1} (z_1, \cdot) + C_{\mathcal{H}} \sqrt{\mathcal{C}_n\xi^2} \leq - \mathcal{N}_0 (x,1)+C_{\mathcal{H}} + \log 2$$ on $B$; in particular, [\[eq:HK3\]](#eq:HK3){reference-type="eqref" reference="eq:HK3"} tells us that on $B$, $$\label{eq:HK3.5} |\nabla u| (\cdot, t_1) \leq C_{\mathcal{H}} L \exp ( - \mathcal{N}_0(x,1) ).$$ Now, we write $$\label{eq:HK5} u(x,1) = \int_B u \,d\nu_{t_1} + \int_{M \setminus B} u \,d\nu_{t_1}$$ with the help of [\[eq:semigroup\]](#eq:semigroup){reference-type="eqref" reference="eq:semigroup"}. We first estimate the first term in [\[eq:HK5\]](#eq:HK5){reference-type="eqref" reference="eq:HK5"}. Since $$\frac{d}{ds} \int_M u \, dm_s = - \int_M H u\, dm_s \leq \mathcal{H}\int_M u \, dm_s$$ for all $s \in (0,1]$, we possess $$\label{eq:int u} \int_M u \, dm_{t_1} \leq e^{\mathcal{H}}.$$ By Proposition [Proposition 24](#prop:lower volume){reference-type="ref" reference="prop:lower volume"} and [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"} we have $$\label{eq:HK6} m_{t_1}(B) \geq C_{\mathcal{H}} \exp (\mathcal{N}_{(x,1)} ( \xi^2 ) ) \xi^n \geq C_{\mathcal{H}} \exp ( \mathcal{N}_0 (x,1) ) \xi^n. %H\geq -\mathcal{H}\geq -\mathcal{H} \xi^{-2}に注意$$ In virtue of [\[eq:HK3.5\]](#eq:HK3.5){reference-type="eqref" reference="eq:HK3.5"}, for all $x_1, x_2 \in B$ we have $$\label{eq:HK6.5} u (x_1, t_1) \leq u(x_2, t_1) + C_{\mathcal{H}} L \exp ( - \mathcal{N}_0 (x,1) ) \xi.$$ Integrating [\[eq:HK6.5\]](#eq:HK6.5){reference-type="eqref" reference="eq:HK6.5"} over $B$ with respect to $x_2$, and using [\[eq:int u\]](#eq:int u){reference-type="eqref" reference="eq:int u"} and [\[eq:HK6\]](#eq:HK6){reference-type="eqref" reference="eq:HK6"} imply $$u(x_1, t_1) \leq \frac1{ m_{t_1}(B)} \int_B u \, dm_{t_1} + C_{\mathcal{H}} L \exp ( - \mathcal{N}_0 (x,1) ) \xi \leq C_{\mathcal{H}}( C_{\mathcal{H}} \xi^{-n} + L \xi ) \exp ( - \mathcal{N}_0 (x,1)).$$ It follows that $$\label{eq:HK7} \int_B u \, d\nu_{t_1} \leq C_{\mathcal{H}}( C_{\mathcal{H}} \xi^{-n} + L \xi ) \exp ( - \mathcal{N}_0 (x,1)).$$ We next estimate the second term in [\[eq:HK5\]](#eq:HK5){reference-type="eqref" reference="eq:HK5"}. By [\[eq:GboundW\]](#eq:GboundW){reference-type="eqref" reference="eq:GboundW"}, [\[eq:HK4\]](#eq:HK4){reference-type="eqref" reference="eq:HK4"}, an inequality $e^s \leq 1 + \xi e^{s/\xi}$, Propositions [Proposition 9](#prop:concentration1){reference-type="ref" reference="prop:concentration1"}, [Proposition 10](#prop:concentration2){reference-type="ref" reference="prop:concentration2"} and the co-area formula, if $\xi \leq \overline{\xi}_{\mathcal{H}}$, then $$\begin{aligned} \label{eq:HK8} \int_{M \setminus B} u \, d\nu_{t_1} &\leq 2 L \int_{M \setminus B} \exp ( - \mathcal{N}_0 (\cdot, t_1) ) d\nu_{t_1}\\ \notag &\leq 4 L \exp ( - \mathcal{N}_0 (x, 1) ) \int_{M \setminus B} \exp \left( C_{\mathcal{H}} d_{t_1} (z_1, \cdot) \right) d\nu_{t_1} \\ \notag &\leq 4 L \exp ( - \mathcal{N}_0 (x, 1) ) \int_{M \setminus B} \left( 1 + \xi \exp \left( C_{\mathcal{H}} \xi^{-1} d_{t_1} (z_1, \cdot) \right) \right) d\nu_{t_1} \\ \notag &\leq 4L \exp ( - \mathcal{N}_0 (x, 1) )\left( \frac{1}{100} + \xi \int_{M}\exp \left( C_{\mathcal{H}} \xi^{-1} d_{t_1} (z_1, \cdot) \right) d\nu_{t_1} \right) \\ \notag &\leq 4L \exp ( - \mathcal{N}_0 (x, 1) )\left( \frac{1}{100} + \xi + C_{\mathcal{H}} \int_0^\infty e^{C_{\mathcal{H}} r/\xi} \nu_{t_1}(M \setminus B(z_1, t_1, r))\, dr \right)\\ \notag &\leq 4L \exp ( - \mathcal{N}_0 (x, 1) )\left( \frac{1}{40} + C_{\mathcal{H}} \int_0^\infty e^{C_{\mathcal{H}} r/\xi} \exp \left({- \frac{ ( r - \sqrt{2\mathcal{C}_n\xi^2})_+^2}{8 \xi^2} }\right) dr \right) \\ \notag &= 4L \exp ( - \mathcal{N}_0 (x, 1) )\left( \frac{1}{40} + C_{\mathcal{H}} \xi \int_0^\infty e^{C_{\mathcal{H}} r} \exp \left({- \frac{ ( r - \sqrt{2\mathcal{C}_n} )_+^2}{8 } }\right) dr \right) \\ \notag &\leq \left( \frac1{10} + C_{\mathcal{H}} \xi \right) L \exp ( - \mathcal{N}_0 (x, 1) ).\end{aligned}$$ Combining [\[eq:HK5\]](#eq:HK5){reference-type="eqref" reference="eq:HK5"}, [\[eq:HK7\]](#eq:HK7){reference-type="eqref" reference="eq:HK7"} and [\[eq:HK8\]](#eq:HK8){reference-type="eqref" reference="eq:HK8"} yields that if $\xi \leq \overline{\xi}_{\mathcal{H}}$ and $L \geq \underline{L}_{\mathcal{H},\xi}$, then $$u(x,1) \leq \left(C_{\mathcal{H}}C_{\mathcal{H}\xi^2} \xi^{-n} + \frac{L}{10} +C_{\mathcal{H}}L \xi \right) \exp ( - \mathcal{N}_0 (x,1) )\leq \frac{L}{2}\exp ( - \mathcal{N}_0 (x,1) ).$$ This proves the desired assertion [\[eq:GboundW_desired\]](#eq:GboundW_desired){reference-type="eqref" reference="eq:GboundW_desired"}. ◻ We also have the following gradient estimate (cf. [@B1 Theorem 7.5]): **Proposition 26**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$, and let $[t,t_0] \subset I$. For $\mathcal{H}>0$, we assume $H(\cdot,t) \geq -\mathcal{H}(t_0-t)^{-1}$. Then on $M$, we have $$\frac{|\nabla_{x} G|(x_0,t_0;\cdot,t)}{G(x_0,t_0;\cdot,t)} \leq \frac{C_{\mathcal{H}}}{(t_0-t)^{1/2}} \sqrt{ \log \left( \frac{C_{\mathcal{H}} \exp (- \mathcal{N}_{(x_0,t_0)}(t_0-t))}{(t_0-t)^{n/2} G(x_0,t_0;\cdot,t)} \right) }. %一つ目の空間変数に関するgradient$$* *Proof.* We may assume $t = 0$ and $t_0 = 1$. Set $\nu:=\nu_{(x_0,1)}$. Let $(z,1/2)$ be a center of $(x_0,1)$ (see Proposition [Proposition 8](#prop:center){reference-type="ref" reference="prop:center"}). For a fixed $y\in M$, we set $u := G(\cdot,1/2;y,0)$. By Lemma [Lemma 22](#lem:Nash cor_rem){reference-type="ref" reference="lem:Nash cor_rem"}, we have $$- \mathcal{N}_0 (\cdot , 1/2) \leq - \mathcal{N}_0 (x_0,1) + C_{\mathcal{H}} \left( d_{1/2} (z, \cdot) + 1 \right).$$ From Proposition [Proposition 25](#prop:HK){reference-type="ref" reference="prop:HK"}, it follows that $$u \leq C_{\mathcal{H}} \exp ( - \mathcal{N}_0 (\cdot, 1/2) ) \leq C_{\mathcal{H}} \exp ( - \mathcal{N}_0 (x_0, 1) ) \exp \left( C_{\mathcal{H}} d_{1/2} (z, \cdot) \right).$$ Using the co-area formula and Proposition [Proposition 10](#prop:concentration2){reference-type="ref" reference="prop:concentration2"}, we obtain $$\begin{aligned} \label{eq:HKK2} \int_M u^2 d\nu_{1/2} &\leq C_{\mathcal{H}} \exp ( -2 \mathcal{N}_0 (x_0, 1) ) \int_M \exp \left( C_{\mathcal{H}} d_{1/2} (z, \cdot) \right) d\nu_{1/2} \\ \notag &\leq C_{\mathcal{H}} \exp ( -2 \mathcal{N}_0 (x_0, 1) ) \left( \int_0^\infty e^{C_{\mathcal{H}}r} \nu_{1/2}(M \setminus B(z,{1/2},r)) \, dr + 1 \right)\\ \notag &\leq C_{\mathcal{H}} \exp ( -2 \mathcal{N}_0 (x_0, 1) ) \left( \int_0^\infty e^{C_{\mathcal{H}}r} \exp \left({- \frac{ (r - \sqrt{2\mathcal{C}_n} )_+^2}{8} }\right) dr +1 \right) \\ \notag &\leq C_{\mathcal{H}} \exp ( -2 \mathcal{N}_0 (x_0, 1) ).\end{aligned}$$ By Proposition [Proposition 25](#prop:HK){reference-type="ref" reference="prop:HK"} we have $$G(x_0,1;y,0) \leq \frac{C_{1,\mathcal{H}}}{2} \exp ( -\mathcal{N}_0 (x_0,1) ).$$ Put $$a :=\left( \frac{G(x_0,1;y,0)}{C_{1,\mathcal{H}} \exp (-\mathcal{N}_0 (x_0,1))}\right)^2. %\leq \frac14$$ Let $b \geq 0$ satisfy the following property (cf. [@B1 Claim 4.6]): There is $\Omega \subset M$ with $\nu_1(\Omega)=a$ such that $$\left\{ \frac{|\nabla_x G|(x_0,1;\cdot, 1/2)}{G(x_0,1; \cdot, 1/2)} > b \right\} \subset \Omega \subset \left\{ \frac{|\nabla_x G|(x_0,1;\cdot, 1/2)}{G(x_0,1; \cdot, 1/2)} \geq b \right\}.$$ By [\[eq:G_bound13\]](#eq:G_bound13){reference-type="eqref" reference="eq:G_bound13"} in Proposition [Proposition 13](#prop:G_bound1){reference-type="ref" reference="prop:G_bound1"} with $p=1$, $$a b \leq \int_{ \Omega } \frac{|\nabla_x G|(x_0,1;\cdot, 1/2)}{G(x_0,1; \cdot, 1/2)} d\nu_{1/2} \leq C\nu_1(\Omega) (-\log \nu_1(\Omega))^{1/2} = C a (-\log a)^{1/2} ,$$ and hence $b \leq C (-\log a)^{1/2}$. By [\[eq:semigroup\]](#eq:semigroup){reference-type="eqref" reference="eq:semigroup"}, [\[eq:HKK2\]](#eq:HKK2){reference-type="eqref" reference="eq:HKK2"}, and [\[eq:G_bound13\]](#eq:G_bound13){reference-type="eqref" reference="eq:G_bound13"} with $p=2$, we conclude $$\begin{aligned} |\nabla_x G|(x_0,1;y,0) &\leq \int_M \frac{|\nabla_x G|(x_0,1;\cdot, 1/2)}{G(x_0,1; \cdot, 1/2)} u \, d\nu_{1/2} \\ &= \int_{\Omega} \frac{|\nabla_x G|(x_0,1;\cdot, 1/2)}{G(x_0,1; \cdot, 1/2)} u \, d\nu_{1/2} + \int_{M \setminus \Omega} \frac{|\nabla_x G|(x_0,1;\cdot, 1/2)}{G(x_0,1; \cdot, 1/2)} u\, d\nu_{1/2} \\ &\leq \left( \int_{\Omega} \left( \frac{|\nabla_x G|(x_0,1;\cdot, 1/2)}{G(x_0,1; \cdot, 1/2)} \right)^{2} d\nu_1 \right)^{1/2} \left( \int_{\Omega} u^2 d\nu_{1/2} \right)^{1/2} + b \int_M u \, d\nu_{1/2} \\ &\leq C_{\mathcal{H}}\exp(-\mathcal{N}_0 (x_0,1)) \left( {- a \log a } \right)^{1/2} + C(-\log a)^{1/2} G(x_0,1;y,0) \\ &\leq C_{\mathcal{H}} (- \log a)^{1/2} G(x_0,1;y,0). %first termはaの定義より\end{aligned}$$ We arrive at the desired estimate. ◻ ## Upper volume estimates We further present the following (cf. [@B1 Theorem 8.1]): **Proposition 27**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$. For $r>0$, we assume $[t_0 -r^2, t_0] \subset I$. For $\mathcal{H}>0$, we assume $H(\cdot, t_0-r^2) \geq -\mathcal{H}r^{-2}$. Then for every $R \geq 1$ we have $$m_{t_0}(B(x_0,t_0,Rr)) \leq C_{\mathcal{H}} \exp ( \mathcal{N}_{(x_0,t_0)}(r^2)) \exp ( C_{\mathcal{H}} R^2 ) r^n.$$* *Proof.* We may assume $t_0=0$ and $r =1$. By [\[eq:potential\]](#eq:potential){reference-type="eqref" reference="eq:potential"} we see $$\mathcal{N}_{(x_0,0)}(1) = - \int_M \left( \log G(x_0,0;\cdot, -1) \right) G(x_0,0; \cdot, -1) dm_{-1} - \frac{n}2 \log 4\pi - \frac{n}2.$$ Hence, there exists $y \in M$ such that $$\log G(x_0,0;y, -1) \geq - \mathcal{N}_{(x_0,0)}(1) - \frac{n}2 \log 4\pi- \frac{n}2;$$ in particular, $$\label{eq:upper volume3} G(x_0,0; y, -1) \geq C \exp ( - \mathcal{N}_{(x_0,0)}(1) ).$$ Set $u := G(\cdot, 0; y;-1)$. By Lemma [Lemma 20](#lem:Nash thm){reference-type="ref" reference="lem:Nash thm"} we have $$\label{eq:upper volume2} - \mathcal{N}_{(\cdot, 0)} (1)=-\mathcal{N}_{-1}(\cdot,0)\leq -\mathcal{N}_{-1}(x_0,0)+\left( \frac{n}{2}+\mathcal{H} \right)^{1/2}d_0(x_0,\cdot) \leq - \mathcal{N}_{(x_0,0)} (1) + C_{1,\mathcal{H}} R$$ on $B(x_0,0,R)$. By Proposition [Proposition 26](#prop:HKK){reference-type="ref" reference="prop:HKK"} and [\[eq:upper volume2\]](#eq:upper volume2){reference-type="eqref" reference="eq:upper volume2"}, $$\label{eq:upper volume2.5} \frac{|\nabla u|}{u} \leq C_{\mathcal{H}} \sqrt{ \log \left( \frac{C_{1,\mathcal{H}} \exp ( - \mathcal{N}_{(\cdot,0)}(1))}{u} \right)}\leq C_{\mathcal{H}} \sqrt{ \log \left( \frac{C_{1,\mathcal{H}} \exp ( - \mathcal{N}_{(x_0,0)}(1) + C_{1,\mathcal{H}} R)}{u} \right)}$$ on $B(x_0,0,R)$. We define $$v := \sqrt{ \log \left( \frac{C_{1,\mathcal{H}} \exp ( - \mathcal{N}_{(x_0,0)}(1) + C_{1,\mathcal{H}} R)}{u} \right)}.$$ By [\[eq:upper volume3\]](#eq:upper volume3){reference-type="eqref" reference="eq:upper volume3"} and [\[eq:upper volume2.5\]](#eq:upper volume2.5){reference-type="eqref" reference="eq:upper volume2.5"}, we obtain $$v(x_0) \leq C_{\mathcal{H}}\sqrt{R},\quad |\nabla v| \leq C_{\mathcal{H}}$$ on $B(x_0,0,R)$. This implies $v \leq C_{\mathcal{H}}\sqrt{R} + C_{\mathcal{H}} R \leq C_{\mathcal{H}}R$ on $B(x_0,0,R)$, and hence $$\label{eq:upper volume5} u \geq C_{\mathcal{H}} \exp (- C_{\mathcal{H}} R^2) \exp ( - \mathcal{N}_{(x_0,0)}(1))$$ on $B(x_0,0,R)$. Since $$\frac{d}{dt} \int_M G(\cdot,t;y, -1) dm_t = - \int_M H \, G(\cdot,t;y, -1) dm_t \leq \mathcal{H} \int_M G(\cdot,t;y, -1) dm_t,$$ the inequality [\[eq:upper volume5\]](#eq:upper volume5){reference-type="eqref" reference="eq:upper volume5"} leads us to $$C_{\mathcal{H}} \exp (- C_{\mathcal{H}} R^2) \exp ( - \mathcal{N}_{(x_0,0)}(1)) m_0(B(x_0,0,R)) \leq \int_{B(x_0,0,R)} u \, dm_0 \leq e^{\mathcal{H}}.$$ This completes the proof. ◻ # Non-collapsed case {#sec:noncollapse} Here, we show some estimates under a non-collapsed condition for the Nash entropy. ## Heat kernel measure comparison We first produce the following heat kernel measure comparison between different base points (cf. [@B3 Proposition 8.1]): **Proposition 28**. *For $\mathcal{H}>0$, there is $\mathfrak{C}_{\mathcal{H}}>0$ such that the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0),(x_1,t_1)\in M\times I$, and let $s, t \in I$ satisfy $s < t \leq \min\{t_0,t_1\}$. For $\kappa>0$, we assume $\mathcal{N}_{(x_0, t_0)}(t_0 - s) \geq - \kappa$. For $D>0$ and $\theta_0,\theta_1 \in (-\infty,1)$ with $\theta_0<\theta_1$, we assume $$\begin{aligned} &H (\cdot, s) \geq - \mathcal{H} (t - s)^{-1},\quad W_{t}(\nu_{(x_0, t_0);t}, \nu_{(x_1,t_1);t} ) \leq D \sqrt{t - s},\\ \notag &t_0 - t \leq \mathfrak{C}_{\mathcal{H}} \, \frac{\theta_1-\theta_0}{1-\theta_1}( t - s),\quad t_1 - t \leq D^2 (t - s). \end{aligned}$$ Then we have $$e^{\theta_0 f_0}\nu_{(x_0,t_0); s} \leq C_{\kappa,D,\theta_0,\theta_1, \mathcal{H}}\, e^{\theta_1 f_1} \nu_{(x_1, t_1); s}.$$* *Proof.* We may assume $s = 0$ and $t = 1$. For $i =0,1$, let $\nu^i := \nu_{(x_i, t_i)}$. In view of Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"}, $$\label{eq:inheriting1} H \geq - \mathcal{H},\quad W_{1}(\nu^0_{1}, \nu^1_{1} ) \leq D,\quad t_0-1 \leq \mathfrak{C}_{\mathcal{H}} \frac{\theta_1-\theta_0}{1-\theta_1},\quad t_1-1 \leq D^2$$ on $M \times (I \cap [0, \infty))$ (see also Remark [Remark 2](#rem:scal2){reference-type="ref" reference="rem:scal2"}). For a fixed $y \in M$, we set $u := G(\cdot, 1; y,0)$. By Propositions [Proposition 25](#prop:HK){reference-type="ref" reference="prop:HK"} and [Proposition 26](#prop:HKK){reference-type="ref" reference="prop:HKK"}, we possess $$\label{eq:inheriting51} u \leq \frac{C_{1,\mathcal{H}}}{10} \exp ( - \mathcal{N}_0 (\cdot,1) ), \quad \frac{|\nabla u|}{u} \leq C_{\mathcal{H}} \sqrt{ \log \left( \frac{C_{1,\mathcal{H}} \exp (- \mathcal{N}_0 (\cdot, 1))}{ u} \right)}.$$ We further set $v :=C^{-1}_{1,\mathcal{H}} u \, \exp ( \mathcal{N}_0 (\cdot, 1))$. Note that [\[eq:Nash thm gradient\]](#eq:Nash thm gradient){reference-type="eqref" reference="eq:Nash thm gradient"} and [\[eq:inheriting51\]](#eq:inheriting51){reference-type="eqref" reference="eq:inheriting51"} lead to $$\label{eq:inheriting6} \frac{|\nabla v|}{v} \leq \frac{|\nabla u|}{u} + |\nabla \mathcal{N}_0 (\cdot, 1)|\leq C_{\mathcal{H}} \sqrt{ - \log v} + C_{\mathcal{H}} \leq C_{\mathcal{H}} \sqrt{- \log v};$$ in particular, for $\varphi:=\sqrt{- \log v}$, we have $$\label{eq:inheriting7} |\nabla \varphi| \leq C_{2,\mathcal{H}}.$$ We define $\mathfrak{C}_{\mathcal{H}}:=(4C_{2,\mathcal{H}})^{-2}$. For $i =0,1$, let $(z_i,1)$ be a center of $(x_i, t_i)$ (Proposition [Proposition 8](#prop:center){reference-type="ref" reference="prop:center"}). Lemma [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"} and [\[eq:inheriting1\]](#eq:inheriting1){reference-type="eqref" reference="eq:inheriting1"} imply $$\begin{aligned} \label{eq:inheriting3} d_1 (z_0, z_1) &\leq W_{1}(\delta_{z_0}, \nu^0_1) + W_{1}(\nu^0_1, \nu^1_1) + W_{1}(\nu^1_1, \delta_{z_1})\\ \notag &\leq \sqrt{\mathcal{C}_n(t_0 - 1)} + D + \sqrt{\mathcal{C}_n(t_1 - 1)} \leq C_{D}. \end{aligned}$$ Also, due to Proposition [Proposition 9](#prop:concentration1){reference-type="ref" reference="prop:concentration1"}, we possess $$\label{eq:inheriting12} \nu^1_1 ( B) \geq \frac{1}{2},$$ where $B:= B(z_1, 1, \sqrt{2\mathcal{C}_n(t_1 - 1)})$. We now write $\lambda :=(1-\theta_1)/(1-\theta_0)$. For a fixed $x \in B$, [\[eq:inheriting7\]](#eq:inheriting7){reference-type="eqref" reference="eq:inheriting7"} and [\[eq:inheriting3\]](#eq:inheriting3){reference-type="eqref" reference="eq:inheriting3"} tell us that $$\begin{aligned} \varphi(\cdot)&\geq \varphi(x) - C_{2,\mathcal{H}} d_1 (x,\cdot)\\ &\geq \varphi(x) - C_{2,\mathcal{H}} d_1 (z_0,\cdot) - C_{2,\mathcal{H}} d_1 (z_0, z_1) - C_{2,\mathcal{H}} \sqrt{2 \mathcal{C}_n(t_1 - 1)} \\ &\geq \varphi(x) - C_{2,\mathcal{H}} (d_1 (z_0,\cdot)+C_{D}) - C_{D,\mathcal{H}}\end{aligned}$$ on $M$, and we obtain $$\begin{aligned} \label{eq:inheriting10} &\quad\,\, \int_M v \exp ( C_{1,\mathcal{H}} d_1 (z_0, \cdot) ) \, d\nu^0_1\\ \notag &= \sum_{j=1}^\infty \int_{B(z_0, 1, j) \setminus B(z_0, 1, j-1)} (\phi \circ \varphi) \exp ( C_{1,\mathcal{H}} d_1 (z_0, \cdot) ) \, d\nu^0_1 \\ \notag &\leq \sum_{j=1}^\infty \phi \left( (\varphi(x) - C_{2,\mathcal{H}} j - C_{D,\mathcal{H}})_+ \right) \exp (C_{1,\mathcal{H}} j) \nu^0_1 ( M \setminus B(z_0, 1, j-1)),\end{aligned}$$ where $\phi(\zeta):=\exp (- \zeta^2)$. By Proposition [Proposition 10](#prop:concentration2){reference-type="ref" reference="prop:concentration2"}, [\[eq:inheriting1\]](#eq:inheriting1){reference-type="eqref" reference="eq:inheriting1"} and the Young inequality, for every $r>0$, $$\begin{aligned} \label{eq:inheriting100} \nu^0_1 \left( M \setminus B(z_0, 1, r) \right) &\leq 2 \exp \left( - \frac{\left(r - \sqrt{2 \mathcal{C}_n(t_0 - 1)} \right)_+^2}{8(t_0 - 1)} \right) \\ \notag &\leq C \exp \left( - \frac{r^2}{16(t_0 - 1)} \right)\leq C \exp \left({ - \frac{C^2_{2,\mathcal{H}} \lambda}{1-\lambda} r^2 } \right).\end{aligned}$$ By [\[eq:inheriting10\]](#eq:inheriting10){reference-type="eqref" reference="eq:inheriting10"}, [\[eq:inheriting100\]](#eq:inheriting100){reference-type="eqref" reference="eq:inheriting100"}, the same calculation as in the proof of [@B3 Proposition 8.1] tells us that $$\label{eq:inheriting9} \int_M v \exp ( C_{1,\mathcal{H}} d_1 (z_0, \cdot) ) \, d\nu^0_1 \leq C_{D,\theta_0,\theta_1,\mathcal{H}} \, v(x)^\lambda.$$ Combining [\[eq:inheriting12\]](#eq:inheriting12){reference-type="eqref" reference="eq:inheriting12"} and [\[eq:inheriting9\]](#eq:inheriting9){reference-type="eqref" reference="eq:inheriting9"}, we obtain $$\begin{aligned} \int_M v \exp ( C_{1,\mathcal{H}} d_1 (z_0, \cdot) ) \, d\nu^0_1 &\leq \frac{C_{D,\theta_0,\theta_1,\mathcal{H}}}{\nu^1_1(B)} \int_{B}\, v^\lambda d\nu^{1}_1\\ &\leq C_{D,\theta_0,\theta_1,\mathcal{H}} \int_{M}\, v^\lambda d\nu^{1}_1\leq C_{D,\theta_0,\theta_1,\mathcal{H}} \left(\int_{M}\, v d\nu^{1}_1 \right)^{\lambda}.\end{aligned}$$ By Lemma [Lemma 22](#lem:Nash cor_rem){reference-type="ref" reference="lem:Nash cor_rem"}, we see $$\begin{aligned} \label{eq:inheriting2} 0 \leq -\mathcal{N}_0 (\cdot,1) &\leq -\mathcal{N}_0 (x_0,t_0)+\left( \frac{n}{2} + \mathcal{H}\right)^{1/2} \sqrt{\mathcal{C}_n(t_0-1)}+\left( \frac{n}{2} + \mathcal{H}\right)^{1/2}d_1(z_0,\cdot)\\ &\leq C_{\kappa,\theta_0,\theta_1,\mathcal{H}}(d_1(z_0,\cdot)+1).\end{aligned}$$ It follows that $$\int_M v \exp ( - \mathcal{N}_0 (\cdot, 1) ) \, d\nu^0_1 \leq C_{\kappa,D,\theta_0,\theta_1, \mathcal{H}} \left( \int_M v \exp ( - \mathcal{N}_0 (\cdot, 1) ) \, d\nu^1_1 \right)^{\lambda};$$ in particular, [\[eq:semigroup\]](#eq:semigroup){reference-type="eqref" reference="eq:semigroup"} leads us to $$\int_M u \, d\nu^0_1 \leq C_{\kappa,D,\theta_0,\theta_1, \mathcal{H}} \left( \int_M u \, d\nu^1_1 \right)^{\lambda},\quad G (x_0, t_0; y,0) \leq C_{\kappa,D,\theta_0,\theta_1, \mathcal{H}} \left( G(x_1, t_1; y,0) \right)^{\lambda}.$$ Thus, we complete the proof. ◻ ## Integral estimates {#sec:Integral bounds} In this subsection, we deduce an integral estimate. We first notice that potentials are bounded from below under a non-collapsed condition. **Lemma 29**. *Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$. For $\alpha\in (0,1)$ and $\mathcal{H}>0$, we assume $H(\cdot,t_0- \alpha^{-1} r^2)\geq -\mathcal{H}r^{-2}$. For $\kappa>0$, we also assume $\mathcal{N}_{(x_0,t_0)} (r^2) \geq - \kappa$. Then on $M\times [t_0-\alpha^{-1} r^2,t_0)$, we have $$f\geq -C_{\kappa,\alpha,\mathcal{H}},\quad f^2 \leq C_{\kappa,\alpha,\mathcal{H}} + C \tau^2 ( |\nabla^2 f|^2 + |\nabla f|^4 + |h|^2).$$* *Proof.* We may assume $t_0=0$ and $r = 1$. In view of Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"}, we possess $H(\cdot,t)\geq -(\mathcal{H}\alpha^{-1})\tau^{-1}$ for $t\in [-\alpha^{-1},0)$. Hence, by Proposition [Proposition 25](#prop:HK){reference-type="ref" reference="prop:HK"}, we have $$G(x_0,0;\cdot,t) \leq \frac{C_{\alpha,\mathcal{H}}}{\tau^{n/2}} \exp ( - \mathcal{N}_{(x_0,0)}(\tau) ).$$ Therefore, from [\[eq:potential\]](#eq:potential){reference-type="eqref" reference="eq:potential"} we conclude $$f(\cdot,t)=-\log G(x_0,0;\cdot,t)-\frac{n}{2}\log \tau-\frac{n}{2}\log 4\pi\geq -\log C_{\mathcal{H}}+\mathcal{N}_{(x_0,0)}(\tau)-\frac{n}{2}\log 4\pi.$$ By Lemma [Lemma 18](#lem:Nash_more){reference-type="ref" reference="lem:Nash_more"} and [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"}, it holds that $\mathcal{N}_{(x_0,0)}(\tau)\geq -C_{\kappa,\alpha,\mathcal{H}}$; in particular, the lower bound of $f$ follows. Due to Theorem [Theorem 5](#thm:Harnack){reference-type="ref" reference="thm:Harnack"}, we obtain $$- C_{\kappa,\alpha,\mathcal{H}} \leq f = w - \tau (2 \Delta f - |\nabla f|^2 + H) +n\leq - \tau (2 \Delta f - |\nabla f|^2 + H) +n,$$ where $w$ is defined as [\[eq:Harnack notation\]](#eq:Harnack notation){reference-type="eqref" reference="eq:Harnack notation"}. Therefore, $$\label{eq:integral15} f^2 \leq C_{\kappa,\alpha,\mathcal{H}} + C \tau^2 ( |\nabla^2 f|^2 + |\nabla f|^4 + H^2)\leq C_{\kappa,\alpha,\mathcal{H}} + C \tau^2 ( |\nabla^2 f|^2 + |\nabla f|^4 + |h|^2).$$ We complete the proof. ◻ We also verify the following (cf. [@B3 Proposition 6.5]): **Lemma 30**. *Let $(M,g(t))_{t\in I}$ satisfy $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$. For $\mathcal{H}>0$, we assume $H \geq -\mathcal{H}$. Then for all $t \in I \cap (-\infty, t_0)$ and $\theta \in [0, 1/2]$ we have $$\int_M e^{\theta f} d\nu_{(x_0,t_0);t} \leq e^{(n+\tau \mathcal{H}) \theta}.$$* *Proof.* Theorem [Theorem 5](#thm:Harnack){reference-type="ref" reference="thm:Harnack"} yields $$\begin{aligned} \frac{d}{d\theta} \int_M e^{\theta f} d\nu_{(x_0,t_0);t} &= \int_M f e^{\theta f} d\nu_{(x_0,t_0);t}\\ &\leq \int_M \left( \tau (-2\Delta f + |\nabla f|^2 - H ) + n \right) e^{\theta f} d\nu_{(x_0,t_0);t} \\ &\leq \int_M \left( \tau (-2\Delta f + |\nabla f|^2 ) + n + \tau \mathcal{H} \right) (4\pi \tau)^{-n/2} e^{-(1-\theta) f} dm_t \\ &= \int_M \left( \tau (2\theta-1) |\nabla f|^2 + n + \tau \mathcal{H} \right) (4\pi \tau)^{-n/2} e^{-(1-\theta) f} dm_t \\ &\leq (n+\tau \mathcal{H}) \int_M e^{\theta f} d\nu_{(x_0,t_0);t}.\end{aligned}$$ Integrating this over $\theta$ implies the lemma. ◻ We further see the following: **Lemma 31**. *Let $(x_0,t_0)\in M\times I$. Then for all $t \in I \cap (-\infty, t_0)$ and $\theta \in [0,1/4]$ we have $$\label{eq:integral14} \int_M |\nabla f|^4 e^{\theta f} d\nu_{(x_0,t_0);t} \leq C \int_M | \nabla^2 f |^2 e^{\theta f} d\nu_{(x_0,t_0);t}.$$* *Proof.* We have $$\begin{aligned} (1-\theta) \int_M |\nabla f|^4 e^{\theta f} d\nu_{(x_0,t_0);t} &=(1-\theta) (4\pi \tau)^{-n/2} \int_M |\nabla f|^2 \langle \nabla f, \nabla f e^{-(1-\theta ) f} \rangle dm_t \\ &= (4\pi \tau)^{-n/2} \int_M \left( 2 \langle \nabla^2 f, d f \otimes d f\rangle + |\nabla f|^2 \Delta f \right) e^{-(1-\theta ) f} dm_t \\ &\leq C (4\pi \tau)^{-n/2} \int_M | \nabla^2 f | |\nabla f|^2 e^{-(1-\theta ) f} dm_t \\ &\leq C \int_M | \nabla^2 f |^2 e^{\theta f} d\nu_{(x_0,t_0);t} +\frac1{2} \int_M |\nabla f|^4 e^{\theta f} d\nu_{(x_0,t_0);t}.\end{aligned}$$ This yields the desired one. ◻ Based on the above lemmas, we prove the following (cf. [@B3 Proposition 6.2]): **Proposition 32**. *If $\theta \in [0, \overline{\theta}]$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$. For $\alpha \in (0,1)$ and $\mathcal{H}>0$, we assume $H(\cdot,t_0-\alpha^{-1} r^2)\geq -\mathcal{H}r^{-2}$. For $\kappa>0$, we assume $\mathcal{N}_{(x_0,t_0)} (r^2) \geq - \kappa$. Then we have $$\label{eq:integral11} \int_{t_0-\alpha^{-1} r^2}^{t_0-\alpha r^2} \int_M \left(\tau( |h|^2 + |\nabla^2 f|^2+ |\nabla f|^4) +\tau^{-1} f^2 \right) e^{2\theta f} d\nu_{(x_0,t_0);t} dt \leq C_{\kappa,\alpha,\mathcal{H}}.$$* *Proof.* We may assume $t_0=0$ and $r = 1$. Set $\nu:=\nu_{(x_0,0)}$ and $u:=G(x_0,0;\cdot,\cdot)$. We define $\Psi$ and $w$ as [\[eq:Harnack notation\]](#eq:Harnack notation){reference-type="eqref" reference="eq:Harnack notation"}. In virtue of [\[eq:conjugate integration\]](#eq:conjugate integration){reference-type="eqref" reference="eq:conjugate integration"}, [\[eq:potential_enjoy\]](#eq:potential_enjoy){reference-type="eqref" reference="eq:potential_enjoy"} and Theorem [Theorem 5](#thm:Harnack){reference-type="ref" reference="thm:Harnack"}, it holds that $$\begin{aligned} &\quad\,\, \frac{d}{dt} \int_M w e^{\theta f} d\nu_t= \int_M \left\{ ( \square e^{\theta f}) w u - e^{\theta f} \square^* (wu) \right\} dm_t \\ \notag &= \int_M \left(2\tau \left|\Phi \right|^2+\tau \mathcal{D}(\nabla f)-\theta \left\{ \tau^{-1} \left( w-f+\frac{n}{2} \right) + \theta |\nabla f|^2 \right\} w\right) e^{\theta f} d\nu_t \\ \notag &\geq \int_M \left( 2\tau \left|\Phi \right|^2-\theta \tau^{-1} \left( w-f+\frac{n}{2} \right)w \right) e^{\theta f} d\nu_t \\ \notag &\geq \int_M 2\tau \left|\Phi \right|^2 e^{\theta f} d\nu_t-\theta \tau^{-1} \int_M \left| w-f+\frac{n}{2} \right| |w| e^{\theta f} d\nu_t \\ \notag &\geq \int_M 2\tau \left|\Phi \right|^2 e^{\theta f} d\nu_t-C\theta \tau^{-1} \int_M (w^2+f^2+1) e^{\theta f} d\nu_t \\ \notag &\geq \int_M 2\tau \left|\Phi \right|^2 e^{\theta f} d\nu_t-C\theta \tau^{-1} \int_M (\tau^2((\Delta f)^2+|\nabla f|^4+H^2)+f^2+1) e^{\theta f} d\nu_t\\ \notag &\geq \int_M 2\tau \left|\Phi \right|^2 e^{\theta f} d\nu_t-C\theta \tau^{-1} \int_M (\tau^2(|\nabla^2 f|^2+|\nabla f|^4+|h|^2)+f^2+1) e^{\theta f} d\nu_t.\end{aligned}$$ Also, if $\theta \in [0,1]$, then [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"} and [\[eq:potential_enjoy\]](#eq:potential_enjoy){reference-type="eqref" reference="eq:potential_enjoy"} yield $$\begin{aligned} &\quad\,\, \frac{d}{dt} \int_M \tau H \,e^{\theta f} d\nu_t=\int_M \square(\tau H e^{\theta f}) \, d\nu_t\\ &=\int_M (2 \tau |h|^2+\tau \mathcal{D}(0) - H) e^{\theta f}d\nu_t-\theta\int_M H\left(w-f+\frac{n}{2} \right) e^{\theta f}d\nu_t\\ &\quad +\theta (\theta-2) \tau\int_M H|\nabla f|^2 e^{\theta f}d\nu_t+2\theta \tau\int_M H \Delta f e^{\theta f}d\nu_t\\ &\geq \int_M (2 \tau |h|^2 - H) e^{\theta f}d\nu_t-C\theta \int_M \left\{ \tau H^2+\tau^{-1}\left(w^2+f^2+1 \right) \right\} e^{\theta f}d\nu_t\\ &\quad -C|\theta-2| \theta \tau\int_M (H^2 +|\nabla f|^4) e^{\theta f}d\nu_t-C\theta \tau\int_M ( H^2 +(\Delta f)^2 ) e^{\theta f}d\nu_t\\ &\geq \int_M (2 \tau |h|^2 - H) e^{\theta f}d\nu_t-C\theta \int_M \left\{ \tau H^2+\tau^{-1}\left(w^2+f^2+1 \right) \right\} e^{\theta f}d\nu_t\\ &\quad -C|\theta-2| \theta \tau\int_M (H^2 +|\nabla f|^4) e^{\theta f}d\nu_t-C\theta \tau\int_M ( H^2 +(\Delta f)^2 ) e^{\theta f}d\nu_t\\ &\geq \int_M (2 \tau |h|^2 - H) e^{\theta f}d\nu_t-C\theta \tau^{-1} \int_M (\tau^2((\Delta f)^2+|\nabla f|^4+H^2)+f^2+1) e^{\theta f} d\nu_t\\ &\geq \int_M (2 \tau |h|^2 - H) e^{\theta f}d\nu_t-C\theta \tau^{-1} \int_M (\tau^2(|\nabla^2 f|^2+|\nabla f|^4+|h|^2)+f^2+1) e^{\theta f} d\nu_t.\end{aligned}$$ Therefore, if $\theta \in [0,1]$, then $$\begin{aligned} \label{eq:integral17} \frac{d}{dt} \int_M (w+\tau H) e^{\theta f} d\nu_t &\geq \int_M \left(2\tau \left|\Phi \right|^2+2 \tau |h|^2 - H \right) e^{\theta f} d\nu_t\\ \notag &\quad-C\theta \tau^{-1} \int_M (\tau^2(|\nabla^2 f|^2+|\nabla f|^4+|h|^2)+f^2+1) e^{\theta f} d\nu_t.\end{aligned}$$ Note that $$\begin{aligned} &\left|\nabla^2 f-\frac{1}{2\tau}g\right|^2=|\Phi-h|^2\leq 4|\Phi|^2+2|h|^2,\\ &|H|\leq \sqrt{n}|h|\leq \frac{\tau}{2}|h|^2+\frac{n}{2\tau},\\ &|\nabla^2 f|^2\leq 2\left|\nabla^2 f-\frac{1}{2\tau}g\right|^2+\frac{n}{2\tau^2};\end{aligned}$$ in particular, $$\begin{aligned} \label{eq:integral25} 2\tau \left|\Phi \right|^2+2 \tau |h|^2 - H &\geq 2\tau \left( \frac{1}{4}\left|\nabla^2 f-\frac{n}{2\tau}\right|^2-\frac{1}{2}|h|^2 \right)+2 \tau |h|^2 - \left(\frac{\tau}{2}|h|^2+\frac{n}{2\tau}\right)\\ \notag &\geq 2\tau \left(\frac{1}{8}|\nabla^2 f|^2-\frac{1}{2}|h|^2-\frac{n}{16 \tau^2}\right)+2\tau|h|^2-\left(\frac{\tau}{2}|h|^2+\frac{n}{2\tau}\right)\\ \notag &= \frac{\tau}{4}|\nabla^2 f|^2+\frac{\tau}{2}|h|^2-\frac{5n}{8 \tau}.\end{aligned}$$ Combining [\[eq:integral17\]](#eq:integral17){reference-type="eqref" reference="eq:integral17"}, [\[eq:integral25\]](#eq:integral25){reference-type="eqref" reference="eq:integral25"} and Lemmas [Lemma 29](#lem:lower poteintial){reference-type="ref" reference="lem:lower poteintial"}, [Lemma 31](#lem:integral3){reference-type="ref" reference="lem:integral3"} yields that for $\theta \in [0,1/4]$, $$\begin{aligned} &\quad\,\, \frac{d}{dt} \int_M (w+\tau H) e^{\theta f} d\nu_t\\ &\geq \int_M \left\{ \tau \left( \frac{1}{4}-C\theta \right)|\nabla^2 f|^2+\tau \left( \frac{1}{2}-C \theta \right)|h|^2 -\tau^{-1}\left( \frac{5n}{8}+C_{\kappa,\alpha,\mathcal{H}}\theta \right) \right\}e^{\theta f} d\nu_t.\end{aligned}$$ In particular, Lemma [Lemma 30](#lem:integral2){reference-type="ref" reference="lem:integral2"} implies that for $\theta \in [0,\overline{\theta}]$, $$\begin{aligned} \frac{d}{dt} \int_M (w+\tau H) e^{\theta f} d\nu_t&\geq \frac{\tau}{8}\int_M (|\nabla^2 f|^2+|h|^2) e^{\theta f} d\nu_t-C_{\kappa,\alpha,\mathcal{H}} \tau^{-1} \int_M e^{\theta f} d\nu_t\\ &\geq \frac{\tau}{8}\int_M (|\nabla^2 f|^2+|h|^2) e^{\theta f} d\nu_t-C_{\kappa,\alpha,\mathcal{H}} \tau^{-1} e^{(n+\tau \mathcal{H})\theta}.\end{aligned}$$ Once we obtain this estimate, we can prove the desired one by the same cutoff argument on time as in the proof of [@B3 Proposition 6.2] together with Lemmas [Lemma 29](#lem:lower poteintial){reference-type="ref" reference="lem:lower poteintial"} and [Lemma 31](#lem:integral3){reference-type="ref" reference="lem:integral3"}. ◻ # Almost selfsimilar points {#sec:almost_ss} For $\varepsilon\in (0,1),r>0$, a point $(x_0,t_0)\in M \times I$ is called *$(\varepsilon, r)$-selfsimilar* if we have: 1. $[t_0 - \varepsilon^{-1} r^2, t_0 ] \subset I$; 2. we have $$\label{eq:almost selfsimilar1} \int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M \tau \left| h + \nabla^2 f - \frac1{2\tau} g \right|^2 d\nu_{(x_0, t_0); t} dt \leq \varepsilon;$$ 3. for all $t \in [t_0 - \varepsilon^{-1} r^2, t_0 - \varepsilon r^2]$, we have $$\label{eq:almost selfsimilar2} \int_M \left| \tau (2\Delta f - |\nabla f|^2 + H) + f - n - \mathrm{N} \right| d\nu_{(x_0, t_0); t} \leq \varepsilon$$ for $\mathrm{N}:=\mathcal{N}_{(x_0, t_0)} (r^2)$; 4. on $M \times [t_0 - \varepsilon^{-1} r^2, t_0 - \varepsilon r^2]$, we have $$\label{eq:almost selfsimilar3} H \geq -\varepsilon r^{-2}.$$ In this section, we investigate various properties of almost selfsimilar points. ## Characterization {#sec:Integral estimates} In this subsection, we prove that the almost selfsimilarity can be characterized by the almost constancy of the Nash entropy. We first prepare the following lemma (cf. [@B3 Lemma 7.10]): **Lemma 33**. *For $\kappa>0,\varepsilon\in (0,1)$, if $\zeta \leq \overline{\zeta}_{\kappa, \varepsilon}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0) \in M \times I$. For $r>0$, we assume $[t_0 - \zeta^{-1} r^2, t_0] \subset I$. We assume $\mathcal{N}_{(x_0, t_0)} (r^2) \geq - \kappa$. If $$\label{eq:integral soliton_ass} | \mathcal{W}_{(x_0, t_0)} (\tau) - \mathrm{N} | \leq \zeta$$ for all $\tau \in [\zeta r^2, \zeta^{-1} r^2]$, then $$\int_M \left| w - \mathrm{N} \right| d\nu_{(x_0,t_0);t} \leq \varepsilon$$ for all $t\in [t_0-\varepsilon^{-1}r^2,t_0-\varepsilon^{-1} r^2]$, where $w$ is defined as [\[eq:Harnack notation\]](#eq:Harnack notation){reference-type="eqref" reference="eq:Harnack notation"} and $\mathrm{N}:=\mathcal{N}_{(x_0,t_0)} (r^2)$.* *Proof.* We may assume $t_0=0$ and $r = 1$. We set $\nu:=\nu_{(x_0,0)}$ and $u :=G(x_0,0;\cdot,\cdot)$. Let $v \in C^\infty (M \times [-\zeta^{-1}, 0])$ be a solution to the heat equation with $|v(\cdot, -\zeta^{-1}) | \leq 1$. In view of the maximum principle, we see $|v| \leq 1$ on $M \times [-\zeta^{-1}, 0]$. Due to [\[eq:conjugate integration\]](#eq:conjugate integration){reference-type="eqref" reference="eq:conjugate integration"} and Theorem [Theorem 5](#thm:Harnack){reference-type="ref" reference="thm:Harnack"}, $$\begin{aligned} \frac{d}{dt} \int_M v (w - \mathrm{N} ) u \, dm_t&= -\int_M v \square^* (w u) dm_t \\ &\geq \int_M \square^* (w u) dm_t = -\frac{d}{dt} \int_M w u \, dm_t = \frac{d}{dt} \mathcal{W}_{(x_0, 0)} (|t|).\end{aligned}$$ From [\[eq:integral soliton_ass\]](#eq:integral soliton_ass){reference-type="eqref" reference="eq:integral soliton_ass"}, it follows that $$\begin{aligned} \int_M v (w - \mathrm{N} )\, d\nu_{-\tau_1} &\leq \int_M v (w - \mathrm{N} ) \, d\nu_{-\tau_2} - \mathcal{W}_{(x_0, 0)} (\tau_2) + \mathcal{W}_{(x_0, 0)} (\tau_1)\\ \notag & \leq \int_M v (w - \mathrm{N} ) d\nu_{-\tau_2} + 2\zeta\end{aligned}$$ for every $\tau_1,\tau_2 \in [\zeta, \zeta^{-1}]$, and hence $$\label{eq:integral soliton2} \left| \int_M v (w - \mathrm{N} )\, d\nu_{-\tau_1} - \int_M v (w - \mathrm{N} )\, d\nu_{-\tau_2} \right|\leq 2\zeta.$$ From Lemma [Lemma 29](#lem:lower poteintial){reference-type="ref" reference="lem:lower poteintial"} and Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"}, we deduce $$\int_{-2\zeta}^{-\zeta} \int_M \left( \tau^2 ( |h|^2+|\nabla^2 f|^2 + |\nabla f|^4) + f^2 \right) d\nu_t dt \leq C_\kappa \zeta;$$ in particular, there exists $\tau_0 \in [\zeta, 2\zeta]$ such that $$\int_M \left( \tau_0^2 ( |h|^2+|\nabla^2 f|^2 + |\nabla f|^4 ) + f^2 \right) d\nu_{-\tau_0} \leq C_\kappa.$$ It follows that $$\label{eq:integral soliton4} \int_{M} (w-\mathrm{N} )^2 d\nu_{-\tau_0}\leq C_\kappa + \int_{M} \left( \tau_0^2 ( |\nabla^2 f|^2 + |\nabla f|^4 + H^2 )+ f^2 \right) d\nu_{-\tau_0} \leq C_\kappa.$$ Thanks to Proposition [Proposition 15](#prop:Poincare){reference-type="ref" reference="prop:Poincare"} with $p=2$ and Theorem [Theorem 12](#thm:gradient_estimate){reference-type="ref" reference="thm:gradient_estimate"}, we also possess $$\label{eq:integral soliton3} \int_M |v - a|^2 \, d\nu_{-\tau_0} \leq 2 \tau_0 \int_M |\nabla v|^2 d\nu_{-\tau_0} \leq C \tau_0 \leq C \zeta,$$ where $a\in [-1,1]$ is defined as $a := \int_M v \, d\nu_{-\tau_0}$. Therefore, [\[eq:integral soliton3\]](#eq:integral soliton3){reference-type="eqref" reference="eq:integral soliton3"} and [\[eq:integral soliton4\]](#eq:integral soliton4){reference-type="eqref" reference="eq:integral soliton4"} lead us to $$\begin{aligned} \left| \int_{M} v (w-\mathrm{N} ) d\nu_{-\tau_0} \right| &\leq \left| a \int_M (w-\mathrm{N} ) d\nu_{-\tau_0} \right| + \int_{M} |v - a | |w-\mathrm{N} | d\nu_{-\tau_0} \\ &\leq \left| \int_M w\, d\nu_{-\tau_0} - \mathrm{N} \right| + \left( \int_{M} |v - a|^2 d\nu_{-\tau_0} \right)^{1/2} \left( \int_{M} |w-\mathrm{N} |^2 d\nu_{-\tau_0} \right)^{1/2} \\ &\leq | \mathcal{W}_{(x_0, 0)} (\tau_0) - \mathrm{N} | + C_\kappa \,\zeta^{1/2} \leq \zeta + C_\kappa \zeta^{1/2}.\end{aligned}$$ Combining this with [\[eq:integral soliton2\]](#eq:integral soliton2){reference-type="eqref" reference="eq:integral soliton2"} tells us that if $\zeta \leq \overline{\zeta}_{\kappa,\varepsilon}$, then for all $\tau \in [\varepsilon,\varepsilon^{-1}]$ we have $$\left|\int_M v (w - \mathrm{N}) \,d\nu_{-\tau} \right| \leq \varepsilon.$$ Since $v$ is arbitrarily, we complete the proof by the same argument of [@B3 Lemma 7.10]. ◻ We provide the following characterization (cf. [@B3 Proposition 7.1]): **Proposition 34**. *For $\kappa>0,\varepsilon\in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, \varepsilon}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. For $r > 0$, we assume $[t_0 - \delta^{-1} r^2, t_0] \subset I$ and $\mathcal{N}_{(x_0, t_0)} ( r^2) \geq - \kappa$. If $$\label{eq:Nash almost first} \mathcal{N}_{(x_0, t_0)} (\delta^{-1} r^2) \geq \mathcal{N}_{(x_0, t_0)} (\delta r^2) - \delta,$$ then $(x_0, t_0)$ is $(\varepsilon, r)$-selfsimilar. Vice versa, if $(x_0, t_0)$ is $(\delta, r)$-selfsimilar, then for all $\tau_1, \tau_2 \in [ \varepsilon r^2, \varepsilon^{-1} r^2]$ we have $$\label{eq:Nash almost} | \mathcal{N}_{(x_0, t_0)} (\tau_1) - \mathcal{N}_{(x_0, t_0)} (\tau_2) | \leq \varepsilon.$$* *Proof.* We may assume $t_0 = 0$ and $r = 1$. Set $\mathrm{N}:=\mathcal{N}_{(x_0,0)} (1)$. We first assume [\[eq:Nash almost first\]](#eq:Nash almost first){reference-type="eqref" reference="eq:Nash almost first"}. By [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"}, [\[eq:Nash rem\]](#eq:Nash rem){reference-type="eqref" reference="eq:Nash rem"} and [\[eq:Nash almost first\]](#eq:Nash almost first){reference-type="eqref" reference="eq:Nash almost first"}, $$\label{eq:Nash almost1} |\mathcal{N}_{(x_0, 0)} (\tau) - \mathrm{N}| \leq \delta,\quad \mathcal{W}_{(x_0, 0)} (\tau) \leq \mathcal{N}_{(x_0, 0)}(\tau) \leq \mathrm{N} + \delta$$ for all $\tau \in [\delta,\delta^{-1}]$. Let $\zeta \in ( \delta, 1)$. By [\[eq:Nash3\]](#eq:Nash3){reference-type="eqref" reference="eq:Nash3"}, [\[eq:Nash2\]](#eq:Nash2){reference-type="eqref" reference="eq:Nash2"} and [\[eq:Nash almost1\]](#eq:Nash almost1){reference-type="eqref" reference="eq:Nash almost1"}, if $\delta \leq \overline{\delta}_{\kappa,\zeta}$, then $$\begin{aligned} \label{eq:Nash almost2} \mathcal{W}_{(x_0, 0)} (\tau) &\geq \frac1{\delta^{-1} - \tau} \int^{\delta^{-1}}_{\tau} \mathcal{W}_{(x_0, 0)} (\sigma) d\sigma= \frac1{\delta^{-1} - \tau} \left( \delta^{-1} \mathcal{N}_{(x_0, 0)} (\delta^{-1}) - \tau \mathcal{N}_{(x_0, 0)} (\tau) \right) \\ \notag &\geq \frac{\delta^{-1}}{\delta^{-1} - \tau} \mathcal{N}_{(x_0, 0)} (\delta^{-1})\geq \frac{\delta^{-1}}{\delta^{-1} - \zeta^{-1}} (\mathrm{N} - \delta)\geq \mathrm{N} - \zeta\end{aligned}$$ for all $\tau \in [\zeta,\zeta^{-1}]$; in particular, $| \mathcal{W}_{(x_0, 0)} (\tau)-\mathrm{N}|\leq \zeta$ for all $\tau \in [\zeta,\zeta^{-1}]$. By Lemma [Lemma 33](#lem:integral soliton){reference-type="ref" reference="lem:integral soliton"}, if $\zeta \leq \overline{\zeta}_{\kappa, \varepsilon}$, then we arrive at [\[eq:almost selfsimilar2\]](#eq:almost selfsimilar2){reference-type="eqref" reference="eq:almost selfsimilar2"}. Secondly, [\[eq:Nash3\]](#eq:Nash3){reference-type="eqref" reference="eq:Nash3"} yields $$\int_{ - \varepsilon^{-1}}^{- \varepsilon} \int_M \left(2\tau \left| \Phi \right|^2+\tau \mathcal{D}(\nabla f) \right)d\nu_{(x_0,0);t} dt= \mathcal{W}_{(x_0, 0)} (\varepsilon) - \mathcal{W}_{(x_0, 0)} (\varepsilon^{-1}) \leq \varepsilon,$$ where $\Phi$ is defined as [\[eq:Harnack notation\]](#eq:Harnack notation){reference-type="eqref" reference="eq:Harnack notation"}. Hence, [\[eq:almost selfsimilar1\]](#eq:almost selfsimilar1){reference-type="eqref" reference="eq:almost selfsimilar1"}. Finally, [\[eq:almost selfsimilar3\]](#eq:almost selfsimilar3){reference-type="eqref" reference="eq:almost selfsimilar3"} is a consequence of Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"}. We next show [\[eq:Nash almost\]](#eq:Nash almost){reference-type="eqref" reference="eq:Nash almost"}. Assume that $(x_0, 0)$ is $(\delta, 1)$-selfsimilar. Integrating [\[eq:almost selfsimilar2\]](#eq:almost selfsimilar2){reference-type="eqref" reference="eq:almost selfsimilar2"}, we have $$\label{eq:Nash almost4} |\mathcal{W}_{(x_0, 0)} (\tau) - \mathrm{N} | \leq \delta$$ for all $\tau \in [\delta, \delta^{-1}]$. By [\[eq:Nash rem\]](#eq:Nash rem){reference-type="eqref" reference="eq:Nash rem"} and [\[eq:Nash almost4\]](#eq:Nash almost4){reference-type="eqref" reference="eq:Nash almost4"}, we see $$\mathcal{N}_{(x_0, 0)} (\tau) \geq \mathcal{W}_{(x_0, 0)} (\tau) \geq \mathrm{N} - \delta$$ for all $\tau \in [\delta, \delta^{-1}]$. Furthermore, [\[eq:Nash3\]](#eq:Nash3){reference-type="eqref" reference="eq:Nash3"}, [\[eq:Nash2\]](#eq:Nash2){reference-type="eqref" reference="eq:Nash2"} and [\[eq:Nash almost4\]](#eq:Nash almost4){reference-type="eqref" reference="eq:Nash almost4"}, $$\begin{aligned} \mathrm{N}+\delta \geq \mathcal{W}_{(x_0, 0)} (\delta) &\geq \frac1{ \tau -\delta} \int_{\delta}^{\tau} \mathcal{W}_{(x_0, 0)} (\sigma) d\sigma\\ &= \frac{1}{\tau - \delta} \left(\tau\mathcal{N}_{(x_0, 0)} (\tau)-\delta \mathcal{N}_{(x_0,0)}(\delta) \right)\geq \frac{\varepsilon}{\varepsilon- \delta}\mathcal{N}_{(x_0, 0)} (\tau)\end{aligned}$$ for all $\tau \in [\varepsilon,\varepsilon^{-1}]$. Therefore, if $\delta \leq \overline{\delta}_{\kappa, \varepsilon}$, then $| \mathcal{N}_{(x_0,0)} (\tau) - \mathrm{N} | \leq \varepsilon/2$ for all $\tau \in [\varepsilon,\varepsilon^{-1}]$, and this proves [\[eq:Nash almost\]](#eq:Nash almost){reference-type="eqref" reference="eq:Nash almost"}. ◻ ## Improved selfsimilarity {#sec:Integral estimates} We have the following (cf. [@B3 Proposition 7.3]): **Proposition 35**. *For $\kappa>0,\varepsilon\in (0,1)$, if $\theta \in [0, \overline{\theta}],\delta \leq \overline{\delta}_{\kappa, \varepsilon}$, then the following holds: Let $(M,g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. For $r > 0$, we assume that $(x_0, t_0)$ is $(\delta, r)$-selfsimilar, and $\mathcal{N}_{(x_0, t_0)} ( r^2) \geq - \kappa$. Then we have $$\begin{aligned} & \int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M \tau \left| h + \nabla^2 f - \frac1{2\tau} g \right|^2 e^{\theta f} d\nu_{(x_0,t_0);t} dt \leq \varepsilon, \\ &r^{-2} \int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M \left| \tau ( - |\nabla f|^2 + \Delta f) + f - \frac{n}2 - \mathrm{N} \right| e^{\theta f} d\nu_{(x_0,t_0);t} dt \leq \varepsilon, \\ &r^{-2} \int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M \left| \square (\tau f) + \frac{n}2 + \mathrm{N} \right| e^{\theta f} d\nu_{(x_0,t_0);t} dt \leq \varepsilon,\\ &r^{-2} \int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M \left|{ - \tau ( |\nabla f|^2 +H) + f - \mathrm{N} } \right| e^{\theta f} d\nu_{(x_0,t_0);t} dt \leq \varepsilon,\end{aligned}$$ where $\mathrm{N} := \mathcal{N}_{(x_0,t_0)} (r^2)$.* *Proof.* We can prove this assertion by using the calculation technique stated in the proof of [@B3 Proposition 7.3] together with Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"}. ◻ ## Almost monotonicity {#sec:almost_monotone} In this subsection, we prove an almost monotonicity property for an integral quantity. To do so, let us show the following formula: **Lemma 36**. *Let $(x_0, t_0) \in M \times I$. Then for all $t \in I \cap (-\infty, t_0)$ we have $$\begin{aligned} \frac{d}{dt} \int_M \tau H \, d\nu_{(x_0,t_0);t} &=2 \int_M \tau \left\langle \Phi, h+\nabla^2 f- df \otimes df \right\rangle d\nu_{(x_0,t_0);t}\\ &\quad +2 \int_M \tau (\operatorname{tr}\Phi)(|\nabla f|^2-\Delta f) d\nu_{(x_0,t_0);t}+\int_M \tau \mathcal{D}(\nabla f) d\nu_{(x_0,t_0);t},\end{aligned}$$ where $\Phi$ is defined as [\[eq:Harnack notation\]](#eq:Harnack notation){reference-type="eqref" reference="eq:Harnack notation"}.* *Proof.* We set $\nu:=\nu_{(x_0,t_0)}$. From [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"} we deduce $$\begin{aligned} \label{eq:direct0} \frac{d}{dt} \int_M \tau H \, d\nu_t&=\int_M \square(\tau H) \, d\nu_t=\int_M \tau\square H-H \, d\nu_t\\ \notag &=\int_M (2 \tau |h|^2+\tau \mathcal{D}(0) - H) d\nu_t\\ \notag &= \int_M \left( 2 \tau \left\langle \Phi,h \right\rangle- 2 \tau \langle h, \nabla^2 f\rangle +\tau \mathcal{D}(0) \right) d\nu_t.\end{aligned}$$ By direct calculations, we see $$\label{eq:direct} \operatorname{div}(h(\nabla f))=(\operatorname{div}h)(\nabla f)+\langle h,\nabla^2f \rangle,\quad \operatorname{div}(\Phi(\nabla f))=(\operatorname{div}\Phi)(\nabla f)+\langle \Phi,\nabla^2f \rangle.$$ The first one in [\[eq:direct\]](#eq:direct){reference-type="eqref" reference="eq:direct"} and [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"} yield $$\begin{aligned} \label{eq:direct1} \int_M \langle h, \nabla^2 f\rangle \, d\nu_t &= \int_M \left( h (\nabla f, \nabla f)- (\operatorname{div}h)(\nabla f) \right) d\nu_t \\ \notag &= \int_M \left( h (\nabla f, \nabla f)- \frac{1}{2}\langle \nabla H,\nabla f \rangle \right) d\nu_t \\ \notag &\quad + \int_M \left( \frac{1}{2}(\operatorname{Ric}-h)(\nabla f,\nabla f)-\frac{1}{4}\mathcal{D}(\nabla f)+\frac{1}{4}\mathcal{D}(0) \right) d\nu_t.\end{aligned}$$ Also, the second one in [\[eq:direct\]](#eq:direct){reference-type="eqref" reference="eq:direct"}, [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"} and the Ricci identity imply $$\begin{aligned} \label{eq:direct2} \int_M \langle \Phi, \nabla^2 f\rangle \, d\nu_t &= \int_M \left( \Phi (\nabla f, \nabla f)- (\operatorname{div}\Phi)(\nabla f) \right) d\nu_t \\ \notag &= \int_M \left( \langle \Phi,df \otimes df\rangle- (\operatorname{div}h)(\nabla f)-(\operatorname{div}\nabla^2 f)(\nabla f) \right) d\nu_t \\ \notag &= \int_M \langle \Phi,df \otimes df \rangle d\nu_t \\ \notag &\quad + \int_M \left( - \frac{1}{2}\langle \nabla H,\nabla f \rangle+\frac{1}{2}(\operatorname{Ric}-h)(\nabla f,\nabla f)-\frac{1}{4}\mathcal{D}(\nabla f)+\frac{1}{4}\mathcal{D}(0) \right) d\nu_t \\ \notag &\quad + \int_M \left( -\langle \nabla \Delta f,\nabla f\rangle -\operatorname{Ric}(\nabla f,\nabla f) \right) d\nu_t.\end{aligned}$$ Combining [\[eq:direct1\]](#eq:direct1){reference-type="eqref" reference="eq:direct1"} and [\[eq:direct2\]](#eq:direct2){reference-type="eqref" reference="eq:direct2"} tells us that $$\begin{aligned} \label{eq:direct3} &\quad\,\, \int_M \left(\langle h, \nabla^2 f\rangle+\langle \Phi, \nabla^2 f\rangle \right) \, d\nu_t \\ \notag &=\int_M \left(\langle \Phi,df \otimes df \rangle - \langle \nabla (\operatorname{tr}\Phi),\nabla f \rangle-\frac{1}{2}\mathcal{D}(\nabla f)+\frac{1}{2}\mathcal{D}(0) \right) d\nu_t.\end{aligned}$$ By substituting [\[eq:direct3\]](#eq:direct3){reference-type="eqref" reference="eq:direct3"} into [\[eq:direct0\]](#eq:direct0){reference-type="eqref" reference="eq:direct0"}, we obtain $$\begin{aligned} \frac{d}{dt} \int_M \tau H \, d\nu_t&=2\tau \int_M \left\langle \Phi, h+\nabla^2 f- df \otimes df \right\rangle d\nu_t+2\tau \int_M \langle \nabla (\operatorname{tr}\Phi),\nabla f \rangle d\nu_t\\ &\quad +\tau \int_M \mathcal{D}(\nabla f) d\nu_t.\end{aligned}$$ From integration by parts [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"}, we conclude the desired equation. ◻ We are now in a position to conclude the following (cf. [@B3 Proposition 7.9]): **Proposition 37**. *For $\kappa>0,\varepsilon\in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, \varepsilon}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. For $r > 0$, we assume that $(x_0,t_0)$ is $(\delta, r)$-selfsimilar, and $\mathcal{N}_{(x_0, t_0)} ( r^2) \geq - \kappa$. Then for all $t_1, t_2 \in [t_0 - \varepsilon^{-1} r^2,t_0 - \varepsilon r^2]$ with $t_1 \leq t_2$, we have $$\int_M \tau H\, d\nu_{(x_0,t_0);t_1} \leq \int_M \tau H \, d\nu_{(x_0,t_0);t_2} + \varepsilon.$$* *Proof.* We may assume $t_0 = 0$ and $r=1$. We set $\nu:=\nu_{(x_0,0)}$. By Lemma [Lemma 36](#lem:formula){reference-type="ref" reference="lem:formula"} and $\mathcal{D}\geq 0$, $$\label{eq:must be equal} \frac{d}{dt} \int_M \tau H \, d\nu_t \geq 2 \int_M \tau \left\langle \Phi, h+\nabla^2 f- df \otimes df \right\rangle d\nu_t+2 \int_M \tau (\operatorname{tr}\Phi)(|\nabla f|^2-\Delta f) d\nu_t.$$ It follows that $$\begin{aligned} \int_M \tau H d\nu_{t_2}-\int_M \tau H d\nu_{t_1}&\geq 2\int^{t_2}_{t_1} \int_M \tau\left\langle \Phi, h+\nabla^2 f- df \otimes df \right\rangle d\nu_t dt \\ &\quad +2\int^{t_2}_{t_1}\,\int_M \tau (\operatorname{tr}\Phi)(|\nabla f|^2-\Delta f) \, d\nu_t dt.\end{aligned}$$ By the Cauchy-Schwarz inequality, we have $$\begin{aligned} &\quad\,\, \left|\int^{t_2}_{t_1} \int_M \tau\left\langle \Phi, h+\nabla^2 f- df \otimes df \right\rangle d\nu_t dt \right|\\ &\leq \left(\int^{t_2}_{t_1} \int_M\, \tau |\Phi|^2 \,d\nu_t dt \right)^{1/2}\left(C\int^{t_2}_{t_1} \int_M\, \tau (|h|^2+|\nabla^2 f|^2+ |\nabla f|^4) \,d\nu_t dt \right)^{1/2}\leq C_{\kappa,\varepsilon} \,\delta^{1/2}.\end{aligned}$$ In the same manner, we obtain $$\begin{aligned} &\quad\,\, \left|\int^{t_2}_{t_1}\,\int_M \tau (\operatorname{tr}\Phi)(|\nabla f|^2-\Delta f) \, d\nu_t dt \right|\\ &\leq \left(\int^{t_2}_{t_1} \int_M\, \tau (\operatorname{tr}\Phi)^2 \,d\nu_t dt \right)^{1/2}\left(\int^{t_2}_{t_1} \int_M\, \tau (|\nabla f|^2-\Delta f)^2 \,d\nu_t dt \right)^{1/2}\\ &\leq \left(C\int^{t_2}_{t_1} \int_M\, \tau |\Phi|^2 \,d\nu_t dt \right)^{1/2}\left(C \int^{t_2}_{t_1} \int_M\, \tau (|\nabla f|^4+|\nabla^2 f|^2) \,d\nu_t dt \right)^{1/2}\leq C_{\kappa,\varepsilon} \,\delta^{1/2}.\end{aligned}$$ Hence, we arrive at $$\int_M \tau H d\nu_{t_2}-\int_M \tau H d\nu_{t_1}\geq -C_{\kappa,\varepsilon}\, \delta^{1/2}.$$ This completes the proof. ◻ As a corollary, we obtain the following almost constancy property for Ricci flow: **Corollary 38**. *For $\kappa>0,\varepsilon\in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, \varepsilon}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a Ricci flow. Let $(x_0, t_0) \in M \times I$. For $r > 0$, we assume that $(x_0,t_0)$ is $(\delta, r)$-selfsimilar, and $\mathcal{N}_{(x_0, t_0)} ( r^2) \geq - \kappa$. Then for all $t_1, t_2 \in [t_0 - \varepsilon^{-1} r^2,t_0 - \varepsilon r^2]$, we have $$\left|\int_M \tau H\, d\nu_{(x_0,t_0);t_1}-\int_M \tau H \, d\nu_{(x_0,t_0);t_2}\right| \leq \varepsilon.$$* *Proof.* In the case of the Ricci flow, the Müller quantity vanishes in virtue of the evolution formula for the scalar curvature and the contracted second Bianchi identity; in particular, the equality holds in [\[eq:must be equal\]](#eq:must be equal){reference-type="eqref" reference="eq:must be equal"}. Therefore, the same calculation as in the proof of Proposition [Proposition 37](#prop:almost soliton scal){reference-type="ref" reference="prop:almost soliton scal"} leads us to the desired claim. ◻ ## Distance expansion estimate {#sec:dist_expansion} We close this section with the following distance expansion estimate (cf. [@B3 Proposition 9.1]): **Proposition 39**. *For $\kappa, D>0,\alpha \in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa,D, \alpha}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0), (x_1, t_1) \in M \times I$. For $r>0$, we assume that $(x_0, t_0)$ is $(\delta, r)$-selfsimilar, $\mathcal{N}_{(x_0, t_0)} ( r^2)\geq - \kappa$ and $0\leq t_1 - t_0 \leq \alpha^{-1} r^2$. Assume $$\label{eq:dist expansion1} W_{s_0}(\nu_{(x_0, t_0);s_0}, \nu_{(x_1,t_1);s_0} ) \leq Dr$$ for some $s_0 \in [ t_{0}- \alpha^{-1} r^2, t_{0} - \alpha r^2 ]$. Then we have $$W_{s_0+\alpha r^2/4}(\nu_{(x_0, t_0);s_0+\alpha r^2/4}, \nu_{(x_1,t_1);s_0+\alpha r^2/4}) \leq C_{\kappa, D, \alpha} r.$$* *Proof.* We may assume $r = 1$. For $i=0,1$, we set $\nu^i := \nu_{(x_i, t_i)}$. Let $\delta \leq \alpha/2$. By Lemma [Lemma 23](#lem:useful){reference-type="ref" reference="lem:useful"} we have $\mathcal{N}_{(x_1, t_1)} (1)\geq -C_{\kappa,D,\alpha}$. From Lemma [Lemma 29](#lem:lower poteintial){reference-type="ref" reference="lem:lower poteintial"}, we conclude $$\label{eq:dist expansion2} f, f_1 \geq - C_{\kappa, D, \alpha}$$ on $[t_{0} - \alpha^{-1}, t_{0})$, where $f,f_1$ are the potentials for $(x_0,t_0),(x_1,t_1)$, respectively. Moreover, we will denote by $\tau,\tau_1$ the parameters for $(x_0,t_0),(x_1,t_1)$, respectively. Set $s_1 := s_0 - \alpha/4$. We first prove that there exists $\Omega \subset M$ such that $$\label{eq:dist expansion15} f(\cdot,s_1),f_1(\cdot,s_1)\leq C_{\kappa,D,\alpha},\quad \nu^1_{s_1}(\Omega) \geq C_{\kappa,D,\alpha}$$ on $\Omega$. Let $(z,s_1)$ be a center of $(x_0, t_0)$ (see Proposition [Proposition 8](#prop:center){reference-type="ref" reference="prop:center"}). By Propositions [Proposition 9](#prop:concentration1){reference-type="ref" reference="prop:concentration1"} and [Proposition 27](#prop:upper volume){reference-type="ref" reference="prop:upper volume"}, $$\label{eq:dist expansion3} \nu^0_{s_1 } (B) \geq \frac12,\quad m_{s_1}(B)\leq C_{\alpha} (t_0 - s_1)^{n/2},$$ where $B:=B(z,s_1,\sqrt{2\mathcal{C}_n(t_0 - s_1)}$. For $a >0$, we define $\Omega_0 := \{ f(\cdot, s_1) \leq a \} \cap B$. In virtue of [\[eq:dist expansion3\]](#eq:dist expansion3){reference-type="eqref" reference="eq:dist expansion3"}, we possess $$\label{eq:dist expansion5} \nu^0_{s_1}(\Omega_0) \geq \frac12 - \int_{B \setminus \Omega_0} (4\pi (t_0 - s_1 ))^{-n/2} e^{-f} dm_{s_1}\geq \frac12 - C e^{-a} (t_0-s_1)^{-n/2} m_{s_1}(B) \geq \frac12 - C_{\alpha} e^{-a};$$ in particular, if $a \geq \underline{a}_{\alpha}$, then [\[eq:dist expansion2\]](#eq:dist expansion2){reference-type="eqref" reference="eq:dist expansion2"} implies $$\label{eq:dist expansion6} \nu^0_{s_1}(\Omega_0) \geq \frac14.%, \quad m_{s_1}(\Omega_0) \geq C_{\kappa,D, \alpha}.$$ Let us verify $$\label{eq:dist expansion7} \nu^1_{s_1 }(\Omega_0)\geq C_{\kappa,D,\alpha}.$$ Define a function $\phi : M \times [s_1,s_0] \to [0,1]$ by $\phi(y,s):= \nu_{(y, s); s_1}(\Omega_0)$, and set $\psi:=\phi(\cdot,s_0)$. Now, [\[eq:semigroup\]](#eq:semigroup){reference-type="eqref" reference="eq:semigroup"} and [\[eq:dist expansion6\]](#eq:dist expansion6){reference-type="eqref" reference="eq:dist expansion6"} imply $$\label{eq:dist expansion7_1} \int_M \psi \, d\nu^0_{s_0}=\nu^0_{s_1}(\Omega_0) \geq \frac14.$$ Let $(y_0, s_0), (y_1, s_0)$ be centers of $(x_0,t_0), (x_1, t_1)$, respectively. Lemma [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"} implies $$\begin{aligned} \label{eq:inheriting10000} d_{s_0} (y_0, y_1) &\leq W_{s_0}(\delta_{y_0}, \nu^0_{s_0}) + W_{s_0}(\nu^0_{s_0}, \nu^1_{s_0}) + W_{s_0}(\nu^1_{s_0}, \delta_{y_1})\\ \notag &\leq \sqrt{\mathcal{C}_n(t_0 - s_0)} + D + \sqrt{\mathcal{C}_n(t_1 - s_0)} \leq C_{D,\alpha}. \end{aligned}$$ Furthermore, by Proposition [Proposition 9](#prop:concentration1){reference-type="ref" reference="prop:concentration1"}, $$\label{eq:conc3} \nu^0_{s_0} \left( M\setminus B(y_0, s_0, \sqrt{8 \mathcal{C}_n (t_0-s_0) } ) \right)\leq \frac{1}{8},\quad \nu^1_{s_0} \left( B(y_1, s_0, \sqrt{2 \mathcal{C}_n (t_1-s_0) } ) \right)\geq \frac{1}{2}.$$ By [\[eq:dist expansion7_1\]](#eq:dist expansion7_1){reference-type="eqref" reference="eq:dist expansion7_1"}, [\[eq:conc3\]](#eq:conc3){reference-type="eqref" reference="eq:conc3"} and $\psi \leq 1$, we see $$\int_{B(y_0, s_0,\sqrt{8 \mathcal{C}_n (t_0-s_0) } )} \psi \, d\nu^0_{s_0} \geq \frac{1}{4} - \nu^0_{s_0} \left( M \setminus B(y_0, s_0, \sqrt{8 \mathcal{C}_n (t_0-s_0) } ) \right) \geq \frac{1}{8};$$ in particular, $\psi \geq 1/8$ at a point in $B(y_0, s_0, \sqrt{8 \mathcal{C}_n (t_0-s_0) } )$. By Theorem [Theorem 12](#thm:gradient_estimate){reference-type="ref" reference="thm:gradient_estimate"}, for $T\geq 0$, the function $\Psi^{-1}_{T+\alpha/ 4}\circ \psi$ is $1$-Lipschitz, here $\Psi$ is defined as [\[eq:Gaussain\]](#eq:Gaussain){reference-type="eqref" reference="eq:Gaussain"}; in particular, $\psi$ is $C (T+\alpha/4)^{-1/2}$-Lipschitz. Hence, if $T\geq \underline{T}_{\kappa,D,\alpha}$, then [\[eq:inheriting10000\]](#eq:inheriting10000){reference-type="eqref" reference="eq:inheriting10000"} yields $\psi \geq C_{\kappa,D,\alpha}$ on $B(y_1, s_0, \sqrt{2\mathcal{C}_n(t_1 - s_0)})$. Now, [\[eq:semigroup\]](#eq:semigroup){reference-type="eqref" reference="eq:semigroup"} and [\[eq:conc3\]](#eq:conc3){reference-type="eqref" reference="eq:conc3"} lead us to $$\nu^1_{s_1}(\Omega_0) =\int_M \psi \, d\nu^1_{s_0} \geq \int_{B(y_1, s_0, \sqrt{2\mathcal{C}_n(t_1 - s_0)})} \psi \, d\nu^1_{s_0} \geq C_{\kappa,D,\alpha}.$$ This proves [\[eq:dist expansion7\]](#eq:dist expansion7){reference-type="eqref" reference="eq:dist expansion7"}. We define $\Omega := \{ f_1(\cdot, s_1) \leq a \} \cap \Omega_0$. With the help of [\[eq:dist expansion7\]](#eq:dist expansion7){reference-type="eqref" reference="eq:dist expansion7"} and [\[eq:dist expansion3\]](#eq:dist expansion3){reference-type="eqref" reference="eq:dist expansion3"}, $$\begin{aligned} \nu^1_{s_1}(\Omega) &\geq C_{\kappa,D,\alpha} - \int_{\Omega_0 \setminus \Omega} (4\pi (t_1 - s_1 ))^{-n/2} e^{-f_1} dm_{s_1}\\ &\geq C_{\kappa,D,\alpha} - C e^{-a} (t_0-s_1)^{-n/2}m_{ s_1 }(B) \geq C_{\kappa,D,\alpha} - C_{\alpha} e^{-a};\end{aligned}$$ in particular, if $a \geq \underline{a}_{\kappa,D, \alpha}$, then we conclude [\[eq:dist expansion15\]](#eq:dist expansion15){reference-type="eqref" reference="eq:dist expansion15"}. For $s:=s_0+\alpha/4$ and $s_2 := s_1 - \alpha/4$, we define $u \in C^\infty (M \times [s_1, s])$ by $$u := \frac{1}{(4\pi (t - s_2))^{n/2}} \exp \left( {- \frac{\tau (f -\mathrm{N} )}{t-s_2}} \right),$$ where $\mathrm{N} := \mathcal{N}_{(x_0, t_0)}(1)$. Note that for every $t \in [s_1, s]$, we have $\tau\in [3\alpha/4,\alpha^{-1}+\alpha/4],t-s_2\in [\alpha/4,3\alpha/4]$, and $$1\leq \frac{\tau}{t-s_2}\leq \frac{\alpha^{-1} + \alpha/4}{\alpha/4}.$$ By [\[eq:dist expansion2\]](#eq:dist expansion2){reference-type="eqref" reference="eq:dist expansion2"} and [\[eq:dist expansion15\]](#eq:dist expansion15){reference-type="eqref" reference="eq:dist expansion15"}, we also see $$\label{eq:dist expansion9} \int_M u\, d\nu^1_{s_1} \geq \int_{\Omega} u \, d\nu^1_{s_1} \geq C_{\kappa,D, \alpha},\quad u \leq C_{\kappa, D,\alpha} (4\pi \tau)^{-n/2} e^{-f}.$$ By direct calculations, $$\begin{aligned} \square u &= -\left\{ \frac{1}{t - s_2} \left( \square (\tau f) + \frac{n}2 + \mathrm{N} \right)- \frac{\tau}{(t-s_2)^2} \left( - \tau( |\nabla f|^2 + H) +f - \mathrm{N} \right) - \frac{\tau^2}{(t-s_2)^2} H \right \} u \\ &\geq - C_{\kappa,D, \alpha} \left( \left| \square (\tau f) + \frac{n}2 + \mathrm{N} \right| + \tau \left| - \tau( |\nabla f|^2 + H) +f - \mathrm{N} \right|+ \delta \right)u.\end{aligned}$$ This together with [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"}, [\[eq:dist expansion2\]](#eq:dist expansion2){reference-type="eqref" reference="eq:dist expansion2"} and [\[eq:dist expansion9\]](#eq:dist expansion9){reference-type="eqref" reference="eq:dist expansion9"} implies $$\begin{aligned} &\quad\,\, \frac{d}{dt} \int_M u \, d\nu^1_t = \int_M \square u \, d\nu^1_t \\ &\geq - C_{\kappa, D,\alpha} \int_M \left( \left| \square (\tau f) + \frac{n}2 + \mathrm{N} \right| + \tau \left| - \tau( |\nabla f|^2 + H) +f - \mathrm{N} \right| + \delta \right) (4\pi \tau)^{-n/2} e^{-f} d\nu^1_t \\ &= - C_{\kappa,D, \alpha} \int_M \left( \left| \square (\tau f) + \frac{n}2 + \mathrm{N} \right| + \tau \left| - \tau( |\nabla f|^2 + H) +f - \mathrm{N} \right| + \delta \right) (4\pi \tau_1)^{-n/2} e^{-f_1} d\nu^0_t \\ &\geq - C_{\kappa, D,\alpha} \int_M \left( \left| \square (\tau f) + \frac{n}2 + \mathrm{N} \right| + \tau \left| - \tau( |\nabla f|^2 + H) +f - \mathrm{N} \right| + \delta \right) d\nu^0_t.\end{aligned}$$ Fix $\eta \in (0,1)$. By Proposition [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"} and [\[eq:dist expansion9\]](#eq:dist expansion9){reference-type="eqref" reference="eq:dist expansion9"}, if $\delta \leq \overline{\delta}_{\kappa, D,\alpha,\eta}$ and $\eta \leq \overline{\eta}_{\kappa,D,\alpha}$, then $$\label{eq:dist expansion20} \int_M u \, d\nu^1_{s} \geq \int_M u \, d\nu^1_{s_1 } -\eta\geq C_{\kappa,D,\alpha}.$$ From [\[eq:dist expansion9\]](#eq:dist expansion9){reference-type="eqref" reference="eq:dist expansion9"} and [\[eq:dist expansion20\]](#eq:dist expansion20){reference-type="eqref" reference="eq:dist expansion20"}, it follows that $$\label{eq:dist expansion12} \int_M (4\pi)^{-n} (\tau \tau_1)^{-n/2} e^{-f-f_1} dm_{s}= \int_M (4\pi \tau)^{-n/2} e^{-f} d\nu^1_{s} \geq C_{\kappa,D,\alpha} \int_M u \, d\nu^1_{s} \geq C_{\kappa,D,\alpha}.$$ Let $(z_0, s),(z_1, s)$ be centers of $(x_0, t_0),(x_1, t_1)$, respectively. We put $d:=d_{s} (z_0, z_1)$. From [\[eq:dist expansion2\]](#eq:dist expansion2){reference-type="eqref" reference="eq:dist expansion2"} and [\[eq:dist expansion12\]](#eq:dist expansion12){reference-type="eqref" reference="eq:dist expansion12"}, we deduce $$\begin{aligned} &\quad\,\, \nu^0_{s} ( M \setminus B(z_0, s, d/2) ) + \nu^1_{s} ( M \setminus B(z_1, s, d/2))\\ \notag &\geq C_{\kappa,D,\alpha} \left( \int_{M \setminus B(z_0, s, d/2)} (4\pi)^{-n} (\tau \tau_1)^{-n/2} e^{-f-f_1} dm_{s } + \int_{M \setminus B(z_1, s, d/2)} (4\pi)^{-n} (\tau \tau_1)^{-n/2} e^{-f-f_1} dm_{s }\right)\\ \notag &\geq C_{\kappa,D,\alpha}\int_M (4\pi)^{-n} (\tau \tau_1)^{-n/2} e^{-f-f_1} dm_{s}\geq C_{\kappa,D,\alpha}. \end{aligned}$$ This and [\[eq:elementary concentration\]](#eq:elementary concentration){reference-type="eqref" reference="eq:elementary concentration"} tell us that $$\begin{aligned} C_{\kappa,D,\alpha}&\leq \nu^0_{s} ( M \setminus B(z_0, s, d/2) ) + \nu^1_{s} ( M \setminus B(z_1, s, d/2))\leq \frac{\mathcal{C}_n(t_0 - s) + \mathcal{C}_n(t_1 - s)}{(d/2)^2} \leq \frac{C_\alpha}{d^2};\end{aligned}$$ in particular, $d \leq C_{\kappa,D, \alpha}$. Lemma [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"} implies $$W_{s}(\nu^0_{s}, \nu^1_{s})\leq W_{s}(\nu^0_{s}, \delta_{z_0} ) + d + W_{s}(\delta_{z_1}, \nu^1_{s} )\leq \sqrt{\mathcal{C}_n(t_0 - s)} + d+ \sqrt{\mathcal{C}_n(t_1 - s)} \leq C_{\kappa,D,\alpha}.$$ Thus, we complete the proof. ◻ # Almost static points {#sec:almost_st} For $\varepsilon\in (0,1),r>0$, a point $(x_0,t_0)\in M \times I$ is called *$(\varepsilon, r)$-static* if the following holds: 1. $[t_0 - \varepsilon^{-1} r^2, t_0 ] \subset I$; 2. we have $$r^2 \int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M |h|^2 d\nu_{(x_0, t_0); t} dt \leq \varepsilon;$$ 3. for all $t \in [t_0 - \varepsilon^{-1} r^2, t_0 - \varepsilon r^2]$, we have $$r^2\int_M H \, d\nu_{(x_0, t_0);t} \leq \varepsilon;$$ 4. $H \geq -\varepsilon r^{-2}$ on $M \times [t_0 - \varepsilon^{-1} r^2, t_0 - \varepsilon r^2]$. Our first main result is the following almost static cone splitting theorem, which has been formulated by Bamler [@B3] for Ricci flow (cf. [@B3 Proposition 10.1]): **Theorem 40**. *For $\kappa, D>0,\alpha,\varepsilon\in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, D, \alpha, \varepsilon}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0), (x_1, t_1) \in M \times I$. For $r>0$, we assume that $(x_0, t_0), (x_1, t_1)$ are $(\delta, r)$-selfsimilar, $\mathcal{N}_{(x_0, t_0)}(r^2) \geq - \kappa$ and $\alpha r^2 \leq t_1 - t_0 \leq \alpha^{-1} r^2$. If there exists $s_0 \in [t_0 - \alpha^{-1} r^2, t_0-\alpha r^2]$ such that $$W_{s_0}(\nu_{(x_0, t_0);s_0}, \nu_{(x_1,t_1);s_0}) \leq Dr,$$ then $(x_0,t_0)$ is $(\varepsilon, r)$-static.* *Proof.* We may assume $t_1 = 0$ and $r=1$. For $i=0,1$, set $\nu^i:=\nu_{(x_i, t_i)}$. We also set $\mathrm{N}_1 := \mathcal{N}_{(x_1, t_1)}(1)$. Lemma [Lemma 23](#lem:useful){reference-type="ref" reference="lem:useful"} implies $\mathrm{N}_1 \geq - C_{\kappa, D, \alpha}$. Let $f,f_1$ be the potentials for $(x_0,t_0),(x_1,t_1)$, respectively. Moreover, let $\tau,\tau_1$ be the parameters for $(x_0,t_0),(x_1,t_1)$, respectively. Let $\theta \in (0,\overline{\theta}]$ be a constant obtained in Proposition [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"}. We fix $\zeta \in (0,1)$. By iterating Propositions [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"} and [Proposition 39](#prop:dist expansion){reference-type="ref" reference="prop:dist expansion"}, if $\delta \leq \overline{\delta}_{\kappa, D,\alpha,\zeta}$, then it holds that $W_{t_0-\zeta}(\nu^0_{t_0-\zeta}, \nu^1_{t_0-\zeta} ) \leq C_{\kappa,D, \alpha,\zeta}$. We also fix $\xi \in (0,1)$. If $\zeta \leq \min\{\xi/2, (\mathfrak{C} \theta (1-\theta)^{-1}\xi)/2\}$, then for every $t\in [t_0-1,t_0-\xi]$, we have $(t_0-\zeta)-t \geq \xi/2$ and $$\begin{aligned} &H(\cdot,t)\geq -\delta \geq -\frac{\xi}{2}\geq -((t_0-\zeta)-t)^{-1},\\ &W_{t_0-\zeta}(\nu_{(x_0, t_0);t_0-\zeta}, \nu_{(x_1,t_1);t_0-\zeta} ) \leq \frac{\sqrt{2}C_{\kappa,D,\alpha,\zeta}}{\sqrt{\xi}}\sqrt{(t_0-\zeta)-t}\leq C_{1,\kappa,D,\alpha,\zeta,\xi}\sqrt{(t_0-\zeta)-t},\\ &t_0-(t_0-\zeta)=\zeta\leq \frac{2\zeta}{\xi}((t_0-\zeta)-t)\leq \mathfrak{C}\frac{\theta}{1-\theta}((t_0-\zeta)-t),\\ &-(t_0-\zeta)\leq \alpha^{-1}+1\leq \frac{2(\alpha^{-1}+1)}{\xi}((t_0-\zeta)-t)\leq C^2_{1,\kappa,D,\alpha,\zeta,\xi}((t_0-\zeta)-t),\end{aligned}$$ where $\mathfrak{C}$ is a constant obtained in Proposition [Proposition 28](#prop:inheriting){reference-type="ref" reference="prop:inheriting"}. By Proposition [Proposition 28](#prop:inheriting){reference-type="ref" reference="prop:inheriting"}, if $\zeta \leq \overline{\zeta}_{\xi}$, then $$\label{eq:almost static1} \nu^0_t \leq C_{\kappa,D, \alpha,\zeta,\xi} e^{\theta f_1}\nu^{1}_t,\quad f_1(\cdot, t_0-1)\leq C_{\kappa,D, \alpha}+(1-\theta)^{-1}f(\cdot,t_0-1)$$ for every $t\in [t_0-1,t_0-\xi]$. Fix $\eta \in (0,1)$. By Proposition [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"}, if $\delta \leq \overline{\delta}_{\kappa,D,\eta}$, then $$\begin{aligned} \label{eq:almost static21} &\int_{t_1 - \eta^{-1}}^{t_1 - \eta} \int_M \left| \square (\tau_1 f_1) + \frac{n}2 + \mathrm{N}_1 \right| e^{\theta f_1} d\nu^1_{t} dt \leq \eta,\\ \label{eq:almost static22} &\int_{t_1 - \eta^{-1}}^{t_1 - \eta} \int_M \left|{ - \tau_1 (H+ |\nabla f_1|^2) + f_1 - \mathrm{N}_1} \right| e^{\theta f_1} d\nu^1_{t} dt \leq \eta.\end{aligned}$$ By [\[eq:almost static1\]](#eq:almost static1){reference-type="eqref" reference="eq:almost static1"} and [\[eq:almost static22\]](#eq:almost static22){reference-type="eqref" reference="eq:almost static22"}, if $\eta \leq \overline{\eta}_{\kappa,D,\alpha,\zeta,\xi}$, then $$\begin{aligned} &\quad\,\, \int_{t_0 - 2\xi}^{t_0 - \xi} \int_M \left|{ - \tau_1 (H+ |\nabla f_1|^2) + f_1 - \mathrm{N}_1} \right| d\nu^0_{t} dt\\ &\leq C_{\kappa,D,\alpha,\zeta,\xi} \,\int_{t_0 - 2\xi}^{t_0 - \xi} \int_M \left|{ - \tau_1 (H+ |\nabla f_1|^2) + f_1 - \mathrm{N}_1} \right| e^{\theta f_1} d\nu^1_{t} dt\\ &\leq C_{\kappa,D,\alpha,\zeta,\xi} \,\int_{t_1 - \eta^{-1}}^{t_1 - \eta} \int_M \left|{ - \tau_1 (H+ |\nabla f_1|^2) + f_1 - \mathrm{N}_1} \right| e^{\theta f_1} d\nu^1_{t} dt\leq C_{\kappa,D,\alpha,\zeta,\xi} \,\eta \leq \xi;\end{aligned}$$ in particular, $$\begin{aligned} \label{eq:almost static24} \int_{t_0 - 2\xi}^{t_0 - \xi} \int_M \tau_1 H\,d\nu^0_t dt&\leq \xi-\int_{t_0 - 2\xi}^{t_0 - \xi} \int_M (\tau_1|\nabla f_1|^2-f_1+\mathrm{N}_1)\,d\nu^0_t dt\\ \notag &\leq C_{\kappa,D,\alpha}\, \xi+\int_{t_0 - 2\xi}^{t_0 - \xi} \int_M f_1\,d\nu^0_t dt.\end{aligned}$$ Fix $s\in [t_0-2\xi,t_0-\xi]$. By [\[eq:almost static1\]](#eq:almost static1){reference-type="eqref" reference="eq:almost static1"} and [\[eq:almost static21\]](#eq:almost static21){reference-type="eqref" reference="eq:almost static21"}, if $\eta \leq \overline{\eta}_{\kappa,D,\alpha,\zeta,\xi}$, then $$\begin{aligned} \label{eq:almost static25} &\quad\,\, \int^{s}_{t_0 - 1} \int_M \left|\square (\tau_1 f_1) + \frac{n}2 + \mathrm{N}_1 \right| d\nu^0_{t} dt\\ \notag &\leq C_{\kappa,D,\alpha,\zeta,\xi} \,\int^{s}_{t_0 - 1} \int_M \left|\square (\tau_1 f_1) + \frac{n}2 + \mathrm{N}_1\right| e^{\theta f_1} d\nu^1_{t} dt\\ \notag &\leq C_{\kappa,D,\alpha,\zeta,\xi} \,\int_{t_1 - \eta^{-1}}^{t_1 - \eta} \int_M \left| \square (\tau_1 f_1) + \frac{n}2 + \mathrm{N}_1 \right| e^{\theta f_1} d\nu^1_{t} dt\leq C_{\kappa,D,\alpha,\zeta,\xi}\, \eta \leq \xi.\end{aligned}$$ Hence, [\[eq:almost static1\]](#eq:almost static1){reference-type="eqref" reference="eq:almost static1"} and [\[eq:almost static25\]](#eq:almost static25){reference-type="eqref" reference="eq:almost static25"} together with [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"} imply $$\begin{aligned} \int_M \tau_1 f_1 d\nu^0_{s}&=\int_M \tau_1 f_1 d\nu^0_{t_0 -1}+\int^{s}_{t_0 -1} \int_M \square(\tau_1f_1) d\nu^0_{t} dt\\ &\leq (1-t_0)\left(C_{\kappa,D,\alpha}+(1-\theta)^{-1}\int_M f d\nu^0_{t_0 -1} \right)+\int^{s}_{t_0 -1} \int_M \square(\tau_1f_1) d\nu^0_{t} dt\\ &\leq (1+\alpha^{-1})\left(C_{\kappa,D,\alpha}+(1-\theta)^{-1}\frac{n}{2} \right)+\xi-\int^{s}_{t_0 - 1} \int_M \left( \frac{n}2 + \mathrm{N}_1 \right) d\nu^0_{t} dt\\ &\leq (1+\alpha^{-1})\left(C_{\kappa,D,\alpha}+(1-\theta)^{-1}\frac{n}{2} \right)+\xi+C_{\kappa,D,\alpha}(s-t_0+1)\leq C_{\kappa,D,\alpha},\end{aligned}$$ which leads us to $$\label{eq:almost static26} \int_M f_1 d\nu^0_{s}\leq \frac{C_{\kappa,D,\alpha}}{\tau_1}\leq C_{\kappa,D,\alpha}.$$ From [\[eq:almost static24\]](#eq:almost static24){reference-type="eqref" reference="eq:almost static24"} and [\[eq:almost static26\]](#eq:almost static26){reference-type="eqref" reference="eq:almost static26"}, we conclude $$\int_{t_0 - 2\xi}^{t_0 - \xi} \int_M \tau_1 H\,d\nu^0_t dt\leq C_{\kappa,D,\alpha} \,\xi;$$ in particular, there exists $t_2 \in [t_0 - 2\xi, t_0 - \xi]$ such that $$\int_M \tau_1 H \, d\nu^0_{ t_2} \leq C_{\kappa,D,\alpha}.$$ Note that $\tau \leq 2\xi$ and $\tau_1 \geq \alpha$ at $t = t_2$. By Proposition [Proposition 37](#prop:almost soliton scal){reference-type="ref" reference="prop:almost soliton scal"}, if $\delta \leq \overline{\delta}_{\kappa, \varepsilon, \xi}$, then $$\label{eq:almost static6} \int_M \tau H \, d\nu^0_t \leq \int_M \tau H \, d\nu^0_{t_2} + \xi= \frac{\tau}{\tau_1}\int_M \tau_1 H \, d\nu^0_{t_2} + \xi \leq C_{\kappa,D,\alpha}\, \xi$$ for all $t \in [t_0 - \varepsilon^{-1}, t_2]$. Using this bound for $t = t_0 - \varepsilon$ and $H\geq -\delta$ leads us to $$\label{eq:almost static5} 2 \int_{t_0 - \varepsilon^{-1}}^{t_0 - \varepsilon} \int_M |h|^2 d\nu^0_t dt = \int_M H \, d\nu^0_{t_0 - \varepsilon} -\int_M H \, d\nu^0_{t_0 - \varepsilon^{-1}}- \int_{t_0 - \varepsilon^{-1}}^{t_0 - \varepsilon} \int_M \mathcal{D}(0) d\nu^0_t dt\leq \frac{C_{\kappa,D,\alpha}\, \xi}{\varepsilon} + \delta.$$ If $\xi \leq \overline{\xi}_{\kappa, D, \alpha, \varepsilon}$ and $\delta \leq \overline{\delta}_{\varepsilon}$, then we complete the proof. ◻ # Almost splitting {#sec:almost_split} For $\varepsilon\in (0,1),r>0$, a *$(k,\varepsilon,r)$-splitting map at $(x_0, t_0)\in M\times I$* is a map $\vec y = (y_1, \ldots, y_k) : M \times [t_0 - \varepsilon^{-1} r^2, t_0 - \varepsilon r^2] \to \mathbb{R}^k$ with the following properties for all $i,j = 1, \ldots, k$: 1. $[t_0 - \varepsilon^{-1} r^2, t_0 ] \subset I$; 2. we have $$r^{-1}\int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M |\square y_i | d\nu_{(x_0, t_0);t} dt \leq \varepsilon;$$ 3. we have $$r^{-2}\int_{t_0 - \varepsilon^{-1} r^2}^{t_0 - \varepsilon r^2} \int_M \left|\langle \nabla y_i, \nabla y_j\rangle - \delta_{ij} \right| d\nu_{(x_0, t_0);t} dt \leq \varepsilon. %\bar{y}=r^{-1}y$$ We say that $(x_0, t_0)$ is *$(k, \varepsilon, r)$-split* if there is a $(k,\varepsilon,r)$-splitting map. The aim of this section is prove the following almost splitting theorem, which has been obtained by Bamler [@B3] for Ricci flow (cf. [@B3 Proposition 10.8]): **Theorem 41**. *For $\kappa, D>0,\varepsilon,\xi \in (0,1)$, if $\beta\leq \overline{\beta},\mathfrak{D} \geq \underline{\mathfrak{D}}_{\kappa,D}, \mathfrak{N} \geq \underline{\mathfrak{N}}_{\kappa, D}, \delta \leq \overline{\delta}_{\kappa, D, \varepsilon,\xi}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $\{(x_i, t_i)\}^{N-1}_{i=0}\subset M \times I$ with $t_0 \leq \cdots \leq t_{N-1}$ and $N \geq \mathfrak{N}\, \xi^{-k}$. For $r>0$, we assume $\{(x_i, t_i)\}^{N-1}_{i=0}$ are $(\delta, r)$-selfsimilar, $\mathcal{N}_{(x_0, t_0)} (r^2) \geq - \kappa$ and $0\leq t_i-t_0 \leq \beta\,\xi^2 r^2$ for all $i$. Assume the following:* 1. *$W_{t_0-r^2}(\nu_{(x_0, t_0);t_0 - r^2},\nu_{(x_i, t_i); t_0- r^2}) \leq Dr$ for all $i$;* 2. *$W_{t_0 - 2 \xi^2 r^2}(\nu_{(x_i, t_i);t_0 - 2\xi^2 r^2}, \nu_{(x_j, t_j); t_0 - 2\xi^2 r^2}) \geq \mathfrak{D} \xi r$ for all $i\neq j$.* *Then $(x_0, t_0)$ is $(k+1, \varepsilon, r)$-split.* ## Construction of coordinate functions We begin with the following: **Lemma 42**. *For $\kappa, D>0,\varepsilon,\xi \in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, D,\varepsilon,\xi}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0), (x_1, t_1) \in M \times I$. For $r>0$, we assume that $(x_0, t_0), (x_1, t_1)$ are $(\delta, r)$-selfsimilar, $\mathcal{N}_{(x_0, t_0)}(r^2) \geq - \kappa$ and $0 \leq t_1 - t_0 \leq \xi^2 r^2$. We assume $$\label{eq:W1assumption} W_{t_0-r^2}(\nu_{(x_0, t_0);t_0-r^2}, \nu_{(x_1,t_1);t_0-r^2}) \leq D r.$$ Let $\tau,\tau_1$ be the parameters for $(x_0,t_0),(x_1,t_1)$, respectively. Moreover, let $f,f_1$ be the potentials for $(x_0,t_0),(x_1,t_1)$, respectively. We define a function $$\label{eq:linear_static} u := r^{-1}(\tau_1 f_1 - \tau f - (\mathrm{N}_1 - \mathrm{N} ) \tau), %\bar{u}=r^{-1}uであるようにr^{-1}をかけている$$ where $\mathrm{N}:=\mathcal{N}_{(x_0, t_0)}(r^2)$ and $\mathrm{N}_1:=\mathcal{N}_{(x_1,t_1)}(r^2)$. Then we have $$\label{eq:splitting lem1_goal} \int_{t_0-\varepsilon^{-1}r^2}^{t_0-\xi^2 r^2} \int_M \tau^{-1/2}|\square u| d\nu_{(x_0,t_0);t} dt \leq \varepsilon,\quad \int_{t_0-\varepsilon^{-1}r^2}^{t_0-\xi^2 r^2} \int_M |\nabla^2 u|^2 d\nu_{(x_0,t_0);t} dt \leq \varepsilon.$$* *Proof.* We may assume $t_0 = 0$ and $r=1$. For $i=0,1$, we set $\nu^i:=\nu_{(x_i, t_i)}$. Let $\theta \in (0,\overline{\theta}]$ be a constant obtained in Proposition [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"}. Fix $\zeta \in (0,1)$. By iterating Propositions [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"} and [Proposition 39](#prop:dist expansion){reference-type="ref" reference="prop:dist expansion"}, if $\delta \leq \overline{\delta}_{\kappa, D,\zeta}$, then $W_{-\zeta}(\nu^0_{-\zeta}, \nu^1_{-\zeta} ) \leq C_{\kappa,D,\zeta}$. If $\zeta \leq \overline{\zeta}_\xi$, then for every $t\in [-\varepsilon^{-1},-\xi^2]$ we see $-\zeta-t \geq \xi^2-\zeta$, and $$\begin{aligned} &H(\cdot,-\zeta)\geq -\delta \geq -(\varepsilon-\zeta)^{-1}\geq -(-\zeta-t)^{-1},\\ &W_{-\zeta}(\nu^0_{-\zeta}, \nu^1_{-\zeta} ) \leq \frac{C_{\kappa,D, \zeta}}{\sqrt{\xi^2-\zeta}}\sqrt{-\zeta-t}\leq C_{1,\kappa,D,\xi,\zeta}\sqrt{-\zeta-t},\\ &\zeta\leq \frac{\zeta}{\xi^2-\zeta}(-\zeta-t)\leq \mathfrak{C}\frac{\theta}{1-\theta}(-\zeta-t),\\ &t_1-(-\zeta)\leq \xi^2+\zeta\leq \frac{\xi^2+\zeta}{\xi^2-\zeta}(-\zeta-t)\leq C^2_{1,\kappa,D,\xi,\zeta}(-\zeta-t),\end{aligned}$$ Therefore, by Proposition [Proposition 28](#prop:inheriting){reference-type="ref" reference="prop:inheriting"}, if $\zeta \leq \overline{\zeta}_\xi$, then we have $$\label{eq:splitting lem11} \nu^0_t \leq C_{\kappa,D, \xi,\zeta} e^{\theta f_1}\nu^{1}_t$$ for every $t\in [-\varepsilon^{-1},-\xi^2]$. We fix $\eta \in (0,1)$. By Proposition [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"}, if $\delta \leq \overline{\delta}_{\kappa,D,\eta}$, then $$\begin{aligned} \label{eq:splitting lem12} &\int_{- \eta^{-1}}^{- \eta} \int_M \tau \left| h + \nabla^2 f - \frac1{2\tau} g \right|^2 d\nu^0_{t} dt \leq \eta,\\ \label{eq:splitting lem13} &\int_{ - \eta^{-1}}^{- \eta} \int_M \left| \square (\tau f) + \frac{n}2 + \mathrm{N} \right| d\nu^0_{t} dt \leq \eta,\\ \label{eq:splitting lem14} &\int_{t_1- \eta^{-1}}^{t_1- \eta} \int_M \tau_1 \left| h + \nabla^2 f_1 - \frac1{2\tau_1} g \right|^2 e^{\theta f_1} d\nu^1_{t} dt \leq \eta,\\ \label{eq:splitting lem15} &\int_{t_1 - \eta^{-1}}^{t_1 - \eta} \int_M \left| \square (\tau_1 f_1) + \frac{n}2 + \mathrm{N}_1 \right| e^{\theta f_1} d\nu^1_{t} dt \leq \eta.\end{aligned}$$ By [\[eq:splitting lem11\]](#eq:splitting lem11){reference-type="eqref" reference="eq:splitting lem11"}, [\[eq:splitting lem13\]](#eq:splitting lem13){reference-type="eqref" reference="eq:splitting lem13"}, [\[eq:splitting lem15\]](#eq:splitting lem15){reference-type="eqref" reference="eq:splitting lem15"}, if $\eta \leq \overline{\eta}_{\kappa,D,\varepsilon,\xi,\zeta}$, then we obtain $$\begin{aligned} \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \tau^{-1/2}|\square u| d\nu^0_t dt &\leq C_{\kappa,D, \xi,\zeta} \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \tau^{-1/2}\left| \square (\tau_1 f_1 ) + \frac{n}2 + \mathrm{N}_1 \right| e^{\theta f_1} d\nu^1_t dt \\ &\quad + \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \tau^{-1/2}\left| \square (\tau f ) + \frac{n}2 + \mathrm{N} \right| d\nu^0_t dt\leq C_{\kappa,D, \xi,\zeta}\, \eta.\end{aligned}$$ This proves the first estimate in [\[eq:splitting lem1_goal\]](#eq:splitting lem1_goal){reference-type="eqref" reference="eq:splitting lem1_goal"}. We next show the second estimate in [\[eq:splitting lem1_goal\]](#eq:splitting lem1_goal){reference-type="eqref" reference="eq:splitting lem1_goal"}. If $\eta \leq \overline{\eta}_{\xi}$, then $$\begin{aligned} &\quad\,\, \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M |\nabla^2 u|^2 d\nu^0_t dt \\ &= \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \left| \tau_1 \left( h+\nabla^2 f_1 - \frac1{2\tau_1} g \right) - \tau \left( h+\nabla^2 f - \frac1{2\tau} g \right) + (\tau - \tau_1) h \right|^2 d\nu^0_t dt \\ &\leq C_{\kappa,D, \xi,\zeta} \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \tau_1^2 \left| h+\nabla^2 f_1 - \frac1{2\tau_1} g \right|^2 e^{\theta f_1} d\nu^1_t dt \\ &\quad + C\int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \tau^2 \left|h+ \nabla^2 f - \frac1{2\tau} g \right|^2 d\nu^0_t dt + C t_1^2 \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M |h|^2 d\nu^0_t dt \\ &\leq C_{\kappa,D, \xi,\zeta} \,\eta + C t^2_1\int_{-\varepsilon^{-1}}^{-\xi^2} \int_M |h|^2 d\nu^0_t dt.\end{aligned}$$ For a fixed $\alpha \in (0,1)$, we first consider the case of $t_1\in [0,\alpha]$. Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"} tells us that if $\alpha \leq \overline{\alpha}_{\kappa,\varepsilon,\xi}$ and $\eta \leq \overline{\eta}_{\kappa,D,\varepsilon,\xi,\zeta}$, then $$\int_{-\varepsilon^{-1}}^{-\xi^2} \int_M |\nabla^2 u|^2 d\nu^0_t dt \leq C_{\kappa,D, \xi,\zeta} \,\eta + C_{\kappa, \xi} \alpha^2\leq \varepsilon.$$ On the other hand, in the case of $t_1 \geq \alpha$, Theorem [Theorem 40](#thm:almost static){reference-type="ref" reference="thm:almost static"} tells us that if $\delta\leq \overline{\delta}_{\kappa,D,\alpha,\eta}$, then $$\int_{-\eta^{-1}}^{-\eta} \int_M |h|^2 d\nu^0_t dt \leq \eta.$$ Therefore, if $\eta \leq \overline{\eta}_{\kappa,D, \varepsilon,\xi,\zeta}$, then $$\int_{-\varepsilon^{-1}}^{-\xi^2} \int_M |\nabla^2 u|^2 d\nu^0_t dt \leq C_{\kappa,D,\xi,\zeta}\,\eta \leq \varepsilon.$$ This completes the proof of the second estimate in [\[eq:splitting lem1_goal\]](#eq:splitting lem1_goal){reference-type="eqref" reference="eq:splitting lem1_goal"}. ◻ For functions in Lemma [Lemma 42](#lem:splitting lem1){reference-type="ref" reference="lem:splitting lem1"}, we further see the following (cf. [@B3 Claim 10.39]): **Lemma 43**. *For $\kappa, D>0,\xi \in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, D}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0), (x_1, t_1) \in M \times I$. For $r>0$, we assume that $(x_0, t_0), (x_1, t_1)$ are $(\delta, r)$-selfsimilar, $\mathcal{N}_{(x_0, t_0)}(r^2) \geq - \kappa$ and $0 \leq t_1 - t_0 \leq \xi^2 r^2$. We further assume [\[eq:W1assumption\]](#eq:W1assumption){reference-type="eqref" reference="eq:W1assumption"}. Let $u$ be a function defined as [\[eq:linear_static\]](#eq:linear_static){reference-type="eqref" reference="eq:linear_static"}. Then we have $$\int_{t_0-2\xi^2 r^2}^{t_0-\xi^2 r^2} \int_M \tau^{-1}|\nabla u|^2 d\nu_{(x_0,t_0);t} dt \leq C_{\kappa,D}.$$* *Proof.* We may assume $t_0 = 0$ and $r=1$. Set $\nu^i:=\nu_{(x_i, t_i)}$ for $i=0,1$. Let $\theta \in (0,\overline{\theta}]$ be a constant obtained in Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"}. Fix $\zeta \in (0,1)$. Using Proposition [Proposition 39](#prop:dist expansion){reference-type="ref" reference="prop:dist expansion"}, if $\delta \leq \overline{\delta}_{\kappa, D,\zeta}$, then $W_{-\zeta \xi^2}(\nu^0_{-\zeta \xi^2}, \nu^1_{-\zeta \xi^2} ) \leq C_{\kappa,D,\zeta}\xi$. By Proposition [Proposition 28](#prop:inheriting){reference-type="ref" reference="prop:inheriting"}, if $\zeta \leq \overline{\zeta}$, then $\nu^0_t \leq C_{\kappa,D, \zeta} e^{\theta f_1}\nu^{1}_t$ for every $t\in [-2\xi^2,-\xi^2]$. Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"} yields $$\begin{aligned} \int_{-2\xi^2}^{-\xi^2} \int_M \tau^{-1} |\nabla u |^2 d\nu^0_t dt &\leq C \int_{-2\xi^2}^{-\xi^2} \int_M \tau^{-1}\tau^2_1 |\nabla f_{1} |^2 d\nu^0_t dt + C \int_{-2\xi^2}^{-\xi^2} \int_M \tau\,|\nabla f |^2 d\nu^0_t dt \\ &\leq C_{\kappa,D,\zeta} \int_{-2\xi^2}^{-\xi^2} \int_M |\nabla f_{1} |^2 e^{\theta f_1} d\nu^1_t dt + C \int_{-2\xi^2}^{-\xi^2} \int_M |\nabla f |^2 d\nu^0_t dt \leq C_{\kappa, D,\zeta}.\end{aligned}$$ This completes the proof. ◻ ## Proof of Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"} {#proof-of-theorem-thmmain-almost-splitting} We have the following (cf. [@B3 Claim 10.31]): **Lemma 44**. *For $\kappa, D>0,\varepsilon,\xi \in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, D,\varepsilon,\xi}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0), (x_1, t_1),(x_2,t_2) \in M \times I$ with $t_0\leq t_1\leq t_2$. For $r>0$, we assume that $(x_0, t_0), (x_1, t_1),(x_2,t_2)$ are $(\delta, r)$-selfsimilar, $\mathcal{N}_{(x_0, t_0)}(r^2) \geq - \kappa$ and $0 \leq t_i - t_0 \leq \xi^2 r^2$ for $i=1,2$. For $i=1,2$, we further assume $$W_{t_0-r^2}(\nu_{(x_0, t_0);t_0- r^2}, \nu_{(x_i,t_i);t_0- r^2}) \leq D r.$$ Let $\tau,\tau_1,\tau_2$ be the parameters for $(x_0,t_0),(x_1,t_1),(x_2,t_2)$, respectively. Moreover, let $f,f_1,f_2$ be the potentials for $(x_0,t_0),(x_1,t_1),(x_2,t_2)$, respectively. For $i=1,2$, we define $$\label{eq:linear_static2} u_i := r^{-1}(\tau_i f_i - \tau f - (\mathrm{N}_i - \mathrm{N} ) \tau),$$ where $\mathrm{N}:= \mathcal{N}_{(x_0, t_0)}(r^2), \,\mathrm{N}_{i}:=\mathcal{N}_{(x_i,t_i)}(r^2)$. Then there are constants $\mathrm{c}_1,\mathrm{c}_2,\mathrm{c}_3\in \mathbb{R}$ such that $$\begin{aligned} \label{eq:almost constant1} \int_{t_0-\varepsilon^{-1}r^2}^{t_0-\xi^2 r^2} \int_M \tau^{-1} \left| |\nabla (u_1+u_2)|^2 - \mathrm{c}_1 \right| d\nu_{(x_0,t_0);t} dt \leq \varepsilon,\\ \label{eq:almost constant2} \int_{t_0-\varepsilon^{-1}r^2}^{t_0-\xi^2 r^2} \int_M \tau^{-1}\left| |\nabla (u_1-u_2)|^2 - \mathrm{c}_2 \right| d\nu_{(x_0,t_0);t} dt \leq \varepsilon,\\ \label{eq:almost constant3} \int_{t_0-\varepsilon^{-1}r^2}^{t_0-\xi^2 r^2} \int_M \tau^{-1}\left| \langle \nabla u_1,\nabla u_2 \rangle - \mathrm{c}_3 \right| d\nu_{(x_0,t_0);t} dt \leq \varepsilon.\end{aligned}$$* *Proof.* We may assume $t_0 = 0$ and $r=1$. We set $\nu:=\nu_{(x_0,0)}$. Let us prove [\[eq:almost constant1\]](#eq:almost constant1){reference-type="eqref" reference="eq:almost constant1"}. We define $u:=u_1+u_2$ and fix $\eta \in (0,1)$. By Propositions [Proposition 34](#prop:Nash almost){reference-type="ref" reference="prop:Nash almost"}, [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"} and Lemma [Lemma 42](#lem:splitting lem1){reference-type="ref" reference="lem:splitting lem1"}, if $\delta \leq \overline{\delta}_{\kappa,D,\eta,\xi}$, then $$\begin{aligned} \label{eq:almost constant11} &\int_{- \eta^{-1}}^{- \xi^2} \int_M \left|\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right| \, d\nu_t dt\leq \eta,\\ \label{eq:almost constant12} &\int_{- \eta^{-1}}^{- \xi^2} \int_M \tau\left|H + \Delta f - \frac{n}{2\tau} \right| \, d\nu_t\leq \eta,\\ \label{eq:almost constant13} &\int_{- \eta^{-1}}^{- \xi^2} \int_M \tau^{-1/2}|\square u| \, d\nu_t dt\leq \eta,\\ \label{eq:almost constant14} &\int_{- \eta^{-1}}^{- \xi^2} \int_M |\nabla^2 u|^2 d\nu_t dt\leq \eta,\\ \label{eq:almost constant15} &\left|\mathcal{N}_{(x_0,0)}(\tau)-\mathrm{N} \right|\leq \eta\end{aligned}$$ for all $t\in [-\eta^{-1},-\eta]$. We first prove that there is a constant $\mathrm{c}\in \mathbb{R}$ such that for all $t\in [-\varepsilon^{-1},-\xi^2]$, $$\label{eq:almost constant_step1} \left| \int_M \tau^{-1} u^2 \left( f- \frac{n}2 - \mathrm{N} \right) d\nu_{t}-\mathrm{c} \right|\leq C_{\kappa,\varepsilon,\xi} \,\eta^{1/4}.$$ Using [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"} and [\[eq:potential_enjoy\]](#eq:potential_enjoy){reference-type="eqref" reference="eq:potential_enjoy"}, we obtain the following (cf. [@B3 Lemma 10.10]): $$\begin{aligned} \frac{d}{dt} \int_M \tau^{-1} u^2 \left( f- \frac{n}2 - \mathrm{N} \right) d\nu_t&= \int_M \tau^{-2} u^2 \left(\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right) d\nu_t \\ &\quad - 2\int_M \tau^{-1} u^2 \left(H + \Delta f - \frac{n}{2\tau} \right) d\nu_t \\ &\quad +2 \int_M \tau^{-1} u \square u \left( f - \frac{n}{2} - \mathrm{N} \right) d\nu_t \\ &\quad - 2 \int_M \tau^{-1} \left( |\nabla u|^2 - \int_M |\nabla u|^2 d\nu_t \right) \left( f - \frac{n}{2} - \mathrm{N} \right) d\nu_t \\ &\quad - 2\tau^{-1} \left( \int_M |\nabla u|^2 d\nu_t \right) \int_M \left( f - \frac{n}{2} - \mathrm{N} \right) d\nu_t.\end{aligned}$$ For $[s_1, s_2] \subset [- \varepsilon^{-1}, - \xi^2 ]$, it holds that $$\begin{aligned} \label{eq:each term} &\quad\,\, \left| \int_M \tau^{-1}u^2 \left( f- \frac{n}2 - \mathrm{N} \right) d\nu_{s_2}-\int_M \tau^{-1} u^2 \left( f- \frac{n}2 - \mathrm{N} \right) d\nu_{s_1} \right| \\ \notag &\leq \xi^{-4} \int^{-\xi^2}_{-\varepsilon^{-1}} \int_M u^2 \left|\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right| d\nu_t dt \\ \notag &\quad + 2\xi^{-4}\int^{-\xi^2}_{-\varepsilon^{-1}} \int_M u^2 \tau \left|H + \Delta f - \frac{n}{2\tau} \right| d\nu_t dt\\ \notag &\quad +2\xi^{-1} \int^{-\xi^2}_{-\varepsilon^{-1}}\int_M \tau^{-1/2}|\square u| |u| \left| f - \frac{n}{2} - \mathrm{N} \right| d\nu_t dt \\ \notag &\quad +2\xi^{-2}\int^{-\xi^2}_{-\varepsilon^{-1}} \int_M \left| |\nabla u|^2 - \int_M |\nabla u|^2 d\nu_t \right| \left| f - \frac{n}{2} - \mathrm{N} \right| d\nu_t dt \\ \notag &\quad + 2 \xi^{-2} \int^{-\xi^2}_{-\varepsilon^{-1}} \left( \int_M |\nabla u|^2 d\nu_t \right) \left|\mathcal{N}_{(x_0,0)}(\tau)-\mathrm{N} \right| dt.\end{aligned}$$ Let us estimate each term in the right hand side of [\[eq:each term\]](#eq:each term){reference-type="eqref" reference="eq:each term"}. Let $a>0$. For the first term, [\[eq:almost constant11\]](#eq:almost constant11){reference-type="eqref" reference="eq:almost constant11"} and Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"} imply $$\begin{aligned} \label{eq:first term} &\quad\,\, \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M u^2 \left|\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right| \, d\nu_t\\ \notag &\leq a^{-1} \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \left|\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right| \, d\nu_t dt\\ \notag &\quad + a \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \left(\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right)^2 \, d\nu_t dt + a \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M u^8\, d\nu_t dt \\ \notag &\leq a^{-1} \eta + a \,C_{\kappa,\varepsilon,\xi}.\end{aligned}$$ For the second term, [\[eq:almost constant12\]](#eq:almost constant12){reference-type="eqref" reference="eq:almost constant12"} and Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"} lead us to $$\begin{aligned} \label{eq:second term} \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M u^2 \tau\left|H + \Delta f - \frac{n}{2\tau} \right| \, d\nu_t &\leq a^{-1} \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \tau \left|H + \Delta f - \frac{n}{2\tau} \right| \, d\nu_t dt\\ \notag &\quad + a \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \tau^2\left(H + \Delta f - \frac{n}{2\tau} \right)^2 \, d\nu_t dt\\ \notag &\quad + a \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M u^8\, d\nu_t dt \\ \notag &\leq a^{-1} \eta + a \,C_{\kappa,\varepsilon,\xi}.\end{aligned}$$ For the third term, [\[eq:almost constant13\]](#eq:almost constant13){reference-type="eqref" reference="eq:almost constant13"} and Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"} yield $$\begin{aligned} \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \tau^{-1/2} |\square u| |u|\left| f - \frac{n}{2} - \mathrm{N} \right| \, d\nu_t &\leq a^{-1} \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \tau^{-1/2}|\square u| \, d\nu_t dt\\ &\quad + a \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M u^2 \left( f - \frac{n}{2} - \mathrm{N} \right)^2 \, d\nu_t dt\\ &\quad + a \int_{- \varepsilon^{-1}}^{- \xi^2} \int_M u^4 \left( f - \frac{n}{2} - \mathrm{N} \right)^4 d\nu_t dt \\ &\leq a^{-1} \eta + a \,C_{\kappa,\varepsilon,\xi}.\end{aligned}$$ For the fourth term, [\[eq:almost constant14\]](#eq:almost constant14){reference-type="eqref" reference="eq:almost constant14"}, the Kato inequality, Propositions [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"} and [Proposition 15](#prop:Poincare){reference-type="ref" reference="prop:Poincare"} with $p=1$ imply $$\begin{aligned} &\quad\,\, \int^{-\xi^2}_{-\varepsilon^{-1}} \int_M \left| |\nabla u|^2 - \int_M |\nabla u|^2 d\nu_t \right| \left| f - \frac{n}{2} - \mathrm{N} \right| d\nu_t dt\\ &\leq a^{-1}\int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \left| |\nabla u|^2 - \int_M |\nabla u|^2 d\nu_t \right| d\nu_t dt\\ &\quad +a\int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \left( |\nabla u|^2 - \int_M |\nabla u|^2 d\nu_t \right)^2 d\nu_t dt +a\int_{- \varepsilon^{-1}}^{- \xi^2} \int_M \left( f - \frac{n}{2} - \mathrm{N} \right)^4 d\nu_t dt\\ &\leq C a^{-1}\int_{- \varepsilon^{-1}}^{- \xi^2} \int_M |\nabla |\nabla u|^2 | d\nu_t dt+a\, C_{\kappa,\varepsilon,\xi}\\ &\leq C a^{-1}\int_{- \varepsilon^{-1}}^{- \xi^2} \int_M |\nabla u||\nabla^2 u| d\nu_t dt+a\, C_{\kappa,\varepsilon,\xi}\\ &\leq C a^{-1}\left(\int_{- \varepsilon^{-1}}^{- \xi^2} \int_M |\nabla u|^2 d\nu_t dt \right)^{1/2}\left(\int_{- \varepsilon^{-1}}^{- \xi^2} \int_M |\nabla^2 u|^2 d\nu_t dt \right)^{1/2}+a \,C_{\kappa,\varepsilon,\xi}\\ &\leq a^{-1}C_{\kappa,\varepsilon,\xi} \eta^{1/2} + a \,C_{\kappa,\varepsilon,\xi}.\end{aligned}$$ For the last term, [\[eq:almost constant15\]](#eq:almost constant15){reference-type="eqref" reference="eq:almost constant15"} and Proposition [Proposition 32](#prop:integral1){reference-type="ref" reference="prop:integral1"} tell us that $$\int^{-\xi^2}_{-\varepsilon^{-1}} \left( \int_M |\nabla u|^2 d\nu_t \right) \left|\mathcal{N}_{(x_0,0)}(\tau)-\mathrm{N} \right| dt\leq C_{\kappa,\varepsilon,\xi} \,\eta.$$ Combining them with [\[eq:each term\]](#eq:each term){reference-type="eqref" reference="eq:each term"}, and choosing $a=\eta^{1/4}$, we obtain $$\begin{aligned} &\quad \,\,\left| \int_M \tau^{-1} u^2 \left( f- \frac{n}2 - \mathrm{N} \right) d\nu_{s_2}-\int_M \tau^{-1} u^2 \left( f- \frac{n}2 - \mathrm{N} \right) d\nu_{s_1} \right|\\ &\leq C_{\kappa,\varepsilon,\xi}\left(a^{-1} \eta+ a^{-1} \eta^{1/2}+a +\eta \right)\leq C_{\kappa,\varepsilon,\xi}\, \eta^{1/4}.\end{aligned}$$ We arrive at [\[eq:almost constant_step1\]](#eq:almost constant_step1){reference-type="eqref" reference="eq:almost constant_step1"}. Using [\[eq:conjugate integration2\]](#eq:conjugate integration2){reference-type="eqref" reference="eq:conjugate integration2"}, for every $t \in [- \varepsilon^{-1}, - \xi^2 ]$ we also see the following (cf. [@B3 Lemma 10.10]): $$\begin{aligned} &\quad\,\, \int_M \left\{ |\nabla u|^2-\frac{1}{2}\tau^{-1} u^2 \left(f- \frac{n}2 - \mathrm{N} \right) \right\} d\nu_t\\ &= - \frac{1}{2}\int_M \tau^{-1} u^2 \left\{\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right\} d\nu_t \\ &\quad + \frac{1}{2}\int_M u^2 \left( H + \Delta f - \frac{n}{2\tau} \right) d\nu_t -\int_M u \Delta u \, d\nu_t.\end{aligned}$$ Let $b>0$. By [\[eq:first term\]](#eq:first term){reference-type="eqref" reference="eq:first term"}, [\[eq:second term\]](#eq:second term){reference-type="eqref" reference="eq:second term"} and [\[eq:almost constant14\]](#eq:almost constant14){reference-type="eqref" reference="eq:almost constant14"}, we have $$\begin{aligned} &\quad \,\,\int^{-\xi^2}_{-\varepsilon^{-1}}\,\left| \int_M \,\left\{ |\nabla u|^2-\frac{1}{2}\tau^{-1} u^2 \left(f- \frac{n}2 - \mathrm{N} \right) \right\} d\nu_t \right|\,dt\\ &\leq C \xi^{-2}\int^{-\xi^2}_{-\varepsilon^{-1}}\,\int_M u^2 \left|\tau (2\Delta f - |\nabla f|^2 + H )+f-n-\mathrm{N} \right| d\nu_t \, dt\\ &\quad + C\xi^{-2} \int^{-\xi^2}_{-\varepsilon^{-1}}\,\int_M u^2\tau \left| H + \Delta f - \frac{n}{2\tau} \right| d\nu_t \,dt+ \int^{-\xi^2}_{-\varepsilon^{-1}}\,\int_M |u| |\Delta u| \, d\nu_t \,dt\\ &\leq C_{\kappa,\varepsilon,\xi} \left(b^{-1} \eta +b\right)+C \left(\int^{-\xi^2}_{-\varepsilon^{-1}}\,\int_M\, u^2 \, d\nu_t \,dt \right)^{1/2}\left(\int^{-\xi^2}_{-\varepsilon^{-1}}\,\int_M |\nabla^2 u|^2 \, d\nu_t \,dt \right)^{1/2}\\ &\leq C_{\kappa,\varepsilon,\xi} \left(b^{-1} \eta +b+\eta^{1/2} \right)\leq C_{\kappa,\varepsilon,\xi} \,\eta^{1/2},\end{aligned}$$ where we choose $b=\eta^{1/2}$. It follows that $$\begin{aligned} \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \tau^{-1} \left| |\nabla u|^2 - \frac{\mathrm{c}}{2} \right| d\nu_{t} dt &\leq \int_{-\varepsilon^{-1}}^{-\xi^2} \int_M \tau^{-1} \left| |\nabla u|^2 - \int_{M}\,|\nabla u|^2\,d\nu_t \right| d\nu_{t} dt\\ &\quad +\int^{-\xi^2}_{-\varepsilon^{-1}}\,\tau^{-1}\left| \int_M \,\left\{ |\nabla u|^2-\frac{1}{2}\tau^{-1} u^2 \left(f- \frac{n}2 - \mathrm{N} \right) \right\} d\nu_t \right|\,dt\\ &\quad +\frac{1}{2}\int^{-\xi^2}_{-\varepsilon^{-1}}\,\tau^{-1}\left| \int_M \,\left\{ \tau^{-1} u^2 \left(f- \frac{n}2 - \mathrm{N} \right)-\mathrm{c} \right\} d\nu_t \right|\,dt\\ &\leq C_{\kappa,\varepsilon,\xi} \,\eta^{1/4}.\end{aligned}$$ This proves [\[eq:almost constant1\]](#eq:almost constant1){reference-type="eqref" reference="eq:almost constant1"}. Once we obtain [\[eq:almost constant1\]](#eq:almost constant1){reference-type="eqref" reference="eq:almost constant1"}, the estimate [\[eq:almost constant2\]](#eq:almost constant2){reference-type="eqref" reference="eq:almost constant2"} can be derived from the same calculation. Moreover, [\[eq:almost constant3\]](#eq:almost constant3){reference-type="eqref" reference="eq:almost constant3"} follows from [\[eq:almost constant1\]](#eq:almost constant1){reference-type="eqref" reference="eq:almost constant1"} and [\[eq:almost constant2\]](#eq:almost constant2){reference-type="eqref" reference="eq:almost constant2"}. We complete the proof. ◻ We also prove the following (cf. [@B3 Claim 10.32]): **Lemma 45**. *For $\kappa, D>0,\xi\in (0,1)$, if $\beta \leq \overline{\beta},\mathfrak{D} \geq \underline{\mathfrak{D}}_{\kappa,D},\delta \leq \overline{\delta}_{\kappa, D,\xi}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0), (x_1, t_1),(x_2,t_2) \in M \times I$ with $t_0\leq t_1\leq t_2$. For $r>0$, we assume that $(x_0, t_0), (x_1, t_1),(x_2,t_2)$ are $(\delta, r)$-selfsimilar, $\mathcal{N}_{(x_0, t_0)}(r^2) \geq - \kappa$ and $0 \leq t_i - t_0 \leq \beta \xi^2 r^2$ for $i=1,2$. Suppose the following:* 1. *$W_{t_0- r^2}(\nu_{(x_0, t_0);t_0 - r^2},\nu_{(x_i, t_i); t_0- r^2}) \leq D r$ for $i=1,2$;* 2. *$W_{t_0 - 2 \xi^2 r^2}(\nu_{(x_1, t_1);t_0 - 2\xi^2 r^2}, \nu_{(x_2, t_2); t_0 - 2\xi^2 r^2}) \geq \mathfrak{D} \xi r$.* *For $i=1,2$, let $u_i$ be a function defined as [\[eq:linear_static2\]](#eq:linear_static2){reference-type="eqref" reference="eq:linear_static2"}. Then we have $$\int^{t_0-\xi^2 r^2}_{t_0-2\xi^2 r^2} \int_M \tau^{-1} |\nabla (u_1 - u_2) |^2 d\nu_{(x_0,t_0);t} dt \geq \xi^2.$$* *Proof.* We may assume $t_0=0$ and $r=1$. Set $\nu:=\nu_{(x_0,0)}$ and $\nu^i:=\nu_{(x_i,t_i)}$ for $i=1,2$. Fix $\eta \in (0,1)$. By Proposition [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"} and Lemma [Lemma 44](#lem:splitting lem3){reference-type="ref" reference="lem:splitting lem3"}, if $\delta \leq \overline{\delta}_{\kappa, D,\eta,\xi}$, then there exists a constant $\mathrm{c}\in \mathbb{R}$ such that $$\int^{-\xi^2}_{-2\xi^2} \left\{ \int_M \tau^{-1}\left| |\nabla (u_1 - u_2) |^2 -\mathrm{c} \right| d\nu_t+\sum^2_{i=1}\int_M \left| - \tau_i ( |\nabla f_i|^2 + H) + f_i - \mathrm{N}_i \right| d\nu^i_{t} \right\} dt \leq 3 \eta;$$ in particular, there exists $s \in [-2\xi^2, -\xi^2]$ such that $$\begin{aligned} \label{eq:main almost splitting7.100} &\quad\,\, \int_M \tau^{-1}\left| | \nabla ( \tau_1 f_{1} - \tau_2 f_{2}) |^2 -\mathrm{c} \right| d\nu_{s}+\sum^{2}_{i=1} \int_M \left| - \tau_i ( |\nabla f_i|^2 + H) + f_i - \mathrm{N}_i \right| d\nu^i_{s}\\ \notag &=\int_M \tau^{-1} \left| |\nabla (u_1 - u_2) |^2 -\mathrm{c} \right| d\nu_{s}+\sum^{2}_{i=1}\int_M \left| - \tau_i ( |\nabla f_i|^2 + H) + f_i - \mathrm{N} _i \right| d\nu^i_{s}\leq C\eta \,\xi^{-2}.\end{aligned}$$ Let $\zeta\in (0,1)$. Due to Propositions [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"} and [Proposition 39](#prop:dist expansion){reference-type="ref" reference="prop:dist expansion"}, if $\delta \leq \overline{\delta}_{\kappa, D,\xi,\zeta}$, then $W_{-\zeta \xi^2}(\nu^j_{-\zeta \xi^2}, \nu^2_{-\zeta \xi^2} ) \leq C_{\kappa,D, \xi,\zeta}$ for $j=0,1$. If $\beta\leq (\mathfrak{C} \theta)/2$ and $\zeta \leq \overline{\zeta}_{\beta}$, then $$\begin{aligned} &H(\cdot,s)\geq -\delta \geq -(1-\zeta)^{-1} \xi^{-2}\geq -(-\zeta \xi^2-s)^{-1},\\ &W_{-\zeta \xi^2}(\nu^j_{-\zeta \xi^2}, \nu^2_{-\zeta \xi^2} ) \leq C_{1,\kappa,D,\xi,\zeta}\sqrt{-\zeta \xi^2-s},\\ &t_2+\zeta \xi^2 \leq (\beta+\zeta) \xi^2 \leq \mathfrak{C}\theta (1-\zeta) \xi^2 \leq \mathfrak{C}\theta (-\zeta \xi^2-s),\\ &\zeta \xi^2 \leq t_1+\zeta \xi^2\leq (\beta+\zeta) \xi^2\leq C^2_{1,\kappa,D,\xi,\zeta}(1-\zeta) \xi^2,\end{aligned}$$ where $\theta \in (0,\overline{\theta}]$ and $\mathfrak{C}$ are constants obtained in Propositions [Proposition 35](#prop:almost soliton estimate){reference-type="ref" reference="prop:almost soliton estimate"} and [Proposition 28](#prop:inheriting){reference-type="ref" reference="prop:inheriting"}, respectively. Proposition [Proposition 28](#prop:inheriting){reference-type="ref" reference="prop:inheriting"} implies $e^{-\theta f_2}\nu^2_s \leq C_{\kappa,D,\xi} \,\nu^j_s$ for $j=0,1$; in particular, [\[eq:main almost splitting7.100\]](#eq:main almost splitting7.100){reference-type="eqref" reference="eq:main almost splitting7.100"} yields $$\label{eq:main almost splitting7.4} \int_M \left( \tau^{-1}\left| | \nabla ( \tau_1 f_{1} - \tau_2 f_{2}) |^2 -\mathrm{c} \right|+\sum^{2}_{i=1} \left| - \tau_i ( |\nabla f_i|^2 + H) + f_i - \mathrm{N}_i \right| \right)\,e^{-\theta f_2}\,d\nu^2_{s}\leq C_{\kappa,D,\xi}\,\eta.$$ For each $i= 1,2$, let $(y_i, s)$ be a center of $(x_i, t_i)$ (see Proposition [Proposition 8](#prop:center){reference-type="ref" reference="prop:center"}). By Lemma [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"} and Proposition [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"}, if $\mathfrak{D} \geq \underline{\mathfrak{D}}$, then $$\begin{aligned} \label{eq:main almost splitting7.5} d_s (y_1 , y_2)& \geq W_{-2\xi^2}(\nu^1_{-2\xi^2}, \nu^2_{-2\xi^2}) -\sqrt{{\operatorname{Var}}_{s} (\delta_{y_1}, \nu^1_{s})} - \sqrt{{\operatorname{Var}}_{s} (\delta_{y_2}, \nu^2_{s})} \\ \notag &\geq \left(\mathfrak{D} - 2 \sqrt{2\mathcal{C}_n} \right) \xi \geq \frac12 \mathfrak{D} \xi.\end{aligned}$$ From [\[eq:main almost splitting7.5\]](#eq:main almost splitting7.5){reference-type="eqref" reference="eq:main almost splitting7.5"}, Proposition [Proposition 9](#prop:concentration1){reference-type="ref" reference="prop:concentration1"} and [\[eq:elementary concentration\]](#eq:elementary concentration){reference-type="eqref" reference="eq:elementary concentration"}, it follows that $$\label{eq:main almost splitting9} \nu^2_{s} ( B ) \geq \frac12,\quad \nu^1_{s} ( B ) \leq \nu^1_{s} \left( M \setminus B(y_1,s, \mathfrak{D} \xi/4) \right) \leq \frac{\operatorname{Var}_{s} (\nu^1_{s}, \delta_{y_1})}{(\mathfrak{D} \xi/4)^2} \leq \frac{32 \mathcal{C}_n}{\mathfrak{D}^2},$$ where $B := B(y_2,s,2 \sqrt{ \mathcal{C}_n} \xi)$. For $a>0$, we set $\Omega_{0} := \{ f_2(\cdot,s) \leq a \} \cap B$. By Proposition [Proposition 27](#prop:upper volume){reference-type="ref" reference="prop:upper volume"}, if $a \geq \underline{a}$, then $$\label{eq:main almost splitting12} \nu^2_{s} ( B \setminus \Omega_{0} ) \leq C \xi^{-n} e^{-a} m_{s}(B)\leq C e^{-a} \leq \frac18.$$ By [\[eq:main almost splitting7.4\]](#eq:main almost splitting7.4){reference-type="eqref" reference="eq:main almost splitting7.4"}, if $\eta\leq \overline{\eta}_{\kappa,D,\xi}$, then there is $\Omega \subset \Omega_{0}$ with $\nu^2_{s} \left( \Omega_{0} \setminus \Omega \right) \leq 1/8$ such that for $i =1,2$, $$\tau^{-1}\left| | \nabla ( \tau_1 f_{1} - \tau_2 f_{2}) |^2 -\mathrm{c} \right|+\sum^{2}_{i=1} \left| - \tau_i ( |\nabla f_i|^2 + H) + f_i - \mathrm{N}_i \right| \leq 1$$ on $\Omega$ at time $s$. Note that $\tau \in [\xi^2,2\xi^2],\tau_i \in [\xi^2,3\xi^2]$. We also notice that if $\mathrm{c}<3\xi^2$, then $$\begin{aligned} \label{eq:main almost splitting12} &\tau_2 |\nabla f_2 |^2 \leq \left| \tau_2 ( |\nabla f_2|^2 + H) - f_2 + \mathrm{N}_2 \right|-\tau_2 H+ f_2-\mathrm{N}_2 \leq C_{\kappa, D,a},\\ \notag &\tau_1 |\nabla f_1 |^2 \leq 2 \xi^{-2} \left( | \nabla ( \tau_1 f_{1} - \tau_2 f_{2})|^2+\tau_2^2 |\nabla f_2 |^2 \right) \leq 2 \xi^{-2} \left( \mathrm{c}+\tau+C_{\kappa,D,a} \xi^2 \right) \leq C_{\kappa,D,a},\\ \notag &\left| \tau_1 |\nabla f_1| - \tau_2 |\nabla f_2| \right| \leq (\mathrm{c}+\tau)^{1/2}\leq C\xi\end{aligned}$$ on $\Omega$. In view of [\[eq:main almost splitting9\]](#eq:main almost splitting9){reference-type="eqref" reference="eq:main almost splitting9"} and [\[eq:main almost splitting12\]](#eq:main almost splitting12){reference-type="eqref" reference="eq:main almost splitting12"}, we also possess $\nu^2_{s} (\Omega)\geq 1/4$. Let us show $\mathrm{c}\geq 3\xi^2$. We suppose $\mathrm{c} < 3\xi^2$. From [\[eq:main almost splitting12\]](#eq:main almost splitting12){reference-type="eqref" reference="eq:main almost splitting12"}, we derive $$\begin{aligned} \tau_2 f_2 - \tau_1 f_1 &\geq (\tau_2^2 - \tau_1^2) H -\sqrt{2} \left| \tau_1 |\nabla f_1| - \tau_2 |\nabla f_2| \right|\left(\sum^{2}_{i=1}\tau^2_i |\nabla f_i|^2\right)^{1/2}-\sum^2_{i=1}\tau_i|\mathrm{N}_i|\\ &\quad\,\, -\sum^2_{i=1}\tau_i \left| - \tau_i ( |\nabla f_i|^2 + H) + f_i - \mathrm{N}_i \right|\geq -C_{\kappa,D,a}\, \xi^2\end{aligned}$$ on $\Omega$. It follows that $f_1 \leq C_{\kappa,D,a}$ on $\Omega$. Therefore, [\[eq:main almost splitting9\]](#eq:main almost splitting9){reference-type="eqref" reference="eq:main almost splitting9"} leads us to $$\frac{1}{4} \leq \nu^2_{s} (\Omega) = \int_{\Omega} (4\pi \tau_2)^{-n/2} e^{-f_2} dm_{s}\leq C_{\kappa,D,a} \int_{\Omega} (4 \pi \tau_1)^{-n/2} e^{-f_1} dm_{s}= C_{\kappa,D,a}\, \nu^1_{s} (B) \leq \frac{C_{\kappa,D,a} }{\mathfrak{D}^2}.$$ If $\mathfrak{D} \geq \underline{\mathfrak{D}}_{\kappa,D,a}$, then this yields the contradiction, and hence $\mathrm{c} \geq 3\xi^2$. We conclude $$\begin{aligned} \int^{-\xi^2}_{-2\xi^2} \int_M \tau^{-1} |\nabla (u_1 - u_2) |^2 d\nu_t dt &\geq \frac{\mathrm{c}}{2}-\int^{-\xi^2}_{-2\xi^2} \int_M \tau^{-1}\left| |\nabla (u_1 - u_2) |^2 -\mathrm{c} \right| d\nu_t dt\geq \xi^2\end{aligned}$$ if $\eta \leq \xi^2/2$. We complete the proof. ◻ We are now in a position to conclude Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"}: *Proof of Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"}.* Once we obtain Lemmas [Lemma 42](#lem:splitting lem1){reference-type="ref" reference="lem:splitting lem1"}, [Lemma 43](#lem:splitting lem2){reference-type="ref" reference="lem:splitting lem2"}, [Lemma 44](#lem:splitting lem3){reference-type="ref" reference="lem:splitting lem3"}, [Lemma 45](#lem:splitting lem4){reference-type="ref" reference="lem:splitting lem4"}, Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"} follows from the same argument as in the proof of [@B3 Proposition 10.8] together with [@B3 Lemma 10.23]. Thus, we complete the proof. ◻ # Quantitative stratification {#sec:stratification} ## Parabolic balls {#sec:parabolic_nbd} For $(x_0, t_0) \in M \times I$ and $r>0$, the *parabolic ball* is defined by $$P(x_0, t_0; r):=\left\{\, (x,t)\in M\times I \,\mid\, t \in [t_0-r^2, t_0+r^2],~ W_{t_0 - r^2}(\nu_{(x_0, t_0); t_0 - r^2}, \nu_{(x,t); t_0 - r^2}) < r \, \right\}.$$ We recall the following basic property (see [@B1 Proposition 9.4], [@B3 Proposition 4.25]): **Lemma 46** ([@B1]). *For $(x_0, t_0), (x_1, t_1) \in M \times I$ and $r>0$, if $P (x_0, t_0;r) \cap P (x_1, t_1; r) \neq \emptyset$, then $P (x_0, t_0; r) \subset P (x_1, t_1; 3r)$.* We first show the following volume estimate for time-slices (cf. [@B1 Theorem 9.8]): **Lemma 47**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$. For $r,\rho,R>0$, we assume $[t_0 - (R^2+\rho) r^2, t_0] \subset I$. Then for every $t \in [t_0 - R^2 r^2, t_0 + R^2r^2]$, $$m_t(P (x_0, t_0; R r) \cap \left( M \times \{ t \} \right)) \leq C_{\rho,R} \exp ( \mathcal{N}_{(x_0, t_0)} (\rho r^2/2) ) r^n.$$* *Proof.* We may assume $t_0=R^2$ and $r=1$. Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"} implies $H(\cdot,-\rho/2)\geq -n/\rho$ (see Remark [Remark 2](#rem:scal2){reference-type="ref" reference="rem:scal2"}). Let $(z_0,0)$ be a center of $(x_0,R^2)$ (see Proposition [Proposition 8](#prop:center){reference-type="ref" reference="prop:center"}). For $t \in [0,2R^2]$ and $x \in S_t$, let $(z, 0)$ be a center of $(x,t)$, where $S_t := P (x_0, t_0; R) \cap \left( M \times \{ t \} \right)$. By Lemma [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"}, $$\begin{aligned} d_0 (z_0,z)&\leq W_{0}(\delta_{z_0}, \nu_{(x_0, R^2); 0})+W_{0}(\nu_{(x_0,R^2);0}, \nu_{(x,t);0})+W_{0}(\nu_{(x,t);0}, \delta_{z}) \\ &\leq \sqrt{\mathcal{C}_n} R +R + \sqrt{\mathcal{C}_nt} \leq C_{1}R\end{aligned}$$ for some $C_1>1$; in particular, $B(z, 0, \sqrt{2\mathcal{C}_nt})$ is contained in $B:=B(z_0,0, C_{1}R)$. Proposition [Proposition 10](#prop:concentration2){reference-type="ref" reference="prop:concentration2"} tells us that $$\label{eq:time_slice3} \nu_{(x,t);0} (B) \geq \nu_{(x,t);0} \left( B(z, 0, \sqrt{2\mathcal{C}_nt} ) \right) \geq \frac12.$$ Let $u \in C^0(M \times [0, 2R^2]) \cap C^\infty(M \times (0,2R^2])$ be the solution to the heat equation with initial condition $u(\cdot, 0) = \chi_B$. By Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"}, we have $$\frac{d}{dt} \int_M u \, dm_t = - \int_M u H \, dm_t \leq \frac{n}{\rho} \int_M u \, dm_t.$$ In virtue of [\[eq:time_slice3\]](#eq:time_slice3){reference-type="eqref" reference="eq:time_slice3"}, we possess $u \geq 1/2$ on $S_t$, and hence $$\label{eq:time_slice4} \frac12 m_t(S_t) \leq \int_M u(\cdot, t) dm_t \leq e^{ \frac{n t}{\rho}} m_0(B)\leq C_{\rho} m_0(B).$$ Using Proposition [Proposition 27](#prop:upper volume){reference-type="ref" reference="prop:upper volume"}, we obtain $$\label{eq:time_slice5} m_0(B)\leq C_{\rho,R} \exp(\mathcal{N}_{(z_0,0)}(\rho/2)).$$ Also, by [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"} and Lemmas [Lemma 21](#lem:Nash cor){reference-type="ref" reference="lem:Nash cor"} and [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"}, $$\begin{aligned} \label{eq:time_slice6} \mathcal{N}_{(z_0,0)}(\rho/2)-\mathcal{N}_{(x_0,R^2)}(\rho/2)&\leq \mathcal{N}_{(z_0,0)}(\rho/2)-\mathcal{N}_{(x_0,R^2)}(\rho/2+R^2)\\ \notag &= \mathcal{N}_{-\rho/2}(z_0,0)-\mathcal{N}_{-\rho/2}(x_0,R^2)\\ \notag &\leq C_{\rho,R}( W_{0}(\delta_{z_0} , \nu_{(x_0,R^2);0})+1)\leq C_{\rho,R}.\end{aligned}$$ From [\[eq:time_slice4\]](#eq:time_slice4){reference-type="eqref" reference="eq:time_slice4"}, [\[eq:time_slice5\]](#eq:time_slice5){reference-type="eqref" reference="eq:time_slice5"} and [\[eq:time_slice6\]](#eq:time_slice6){reference-type="eqref" reference="eq:time_slice6"}, we conclude the desired estimate. ◻ Based on Lemma [Lemma 47](#lem:time_slice){reference-type="ref" reference="lem:time_slice"}, we present the following (cf. [@B1 Theorem 9.11]): **Proposition 48**. *Let $(M,g(t))_{t\in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. For $r ,\rho,R> 0$, we assume $[t_0 - 2(R^2+\rho)r^2, t_0] \subset I$. Let $S \subset P(x_0, t_0; Rr)$ and $\xi \in (0,\sqrt{\rho}]$. Then there exists $\{(x_i, t_i)\}^{N}_{i=1}\subset S$ such that $$\label{eq:covering1} S \subset \bigcup_{i=1}^N P(x_i, t_i; \xi r), \quad N \leq C_{\rho,R} \,\xi^{-n-2}.$$* *Proof.* We may assume $t_0=0$ and $r = 1$. Let $\{(x_i, t_i)\}^{N}_{i=1} \subset S$ be a maximal collection of points such that $\{P (x_i, t_i; \xi/3)\}^{N}_{i=1}$ are pairwise disjoint. Let $(x, t) \in S$. By the maximality we see $P(x, t; \xi/3) \cap P(x_i, t_i; \xi/3) \neq \emptyset$ for some $i$. From Lemma [Lemma 46](#lem:parab_nbhd){reference-type="ref" reference="lem:parab_nbhd"}, we deduce $(x, t) \in P (x_i, t_i;\xi)$, which proves the inclusion in [\[eq:covering1\]](#eq:covering1){reference-type="eqref" reference="eq:covering1"}. Let us derive an upper bound on $N$ in [\[eq:covering1\]](#eq:covering1){reference-type="eqref" reference="eq:covering1"}. Fix $\zeta \in (0, 1/2]$. There exist $s \in [-R^2 - 2\zeta \xi^2,R^2 - \zeta \xi^2]$ and $\mathcal{I} \subset \{ 1, \ldots, N \}$ such that $$\label{eq:covering2} |\mathcal{I}| \geq \lfloor C_{ \zeta} \xi^2 N \rfloor,\quad s \in [t_i -2\zeta \xi^2, t_i - \zeta\xi^2]$$ for all $i \in \mathcal{I}$. For each $i \in \mathcal{I}$, let $(z_i, s)$ be a center of $(x_i, t_i)$ (see Proposition [Proposition 8](#prop:center){reference-type="ref" reference="prop:center"}). By Lemma [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"} and Proposition [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"}, if $\zeta \leq \overline{\zeta}$, then for all $i,j \in \mathcal{I}$ with $i \neq j$, it holds that $$\begin{aligned} d_{s} (z_i, z_j) &\geq W_{s}(\nu_{(x_i, t_i);s},\nu_{(x_j, t_j);s}) - W_{s}(\delta_{z_i}, \nu_{(x_i, t_i);s}) - W_{s}(\delta_{z_j}, \nu_{(x_j, t_j);s}) \\ &\geq W_{t_i - (\xi/3)^2}(\nu_{(x_i, t_i);t_i - (\xi/3)^2},\nu_{(x_j, t_j);t_i - (\xi/3)^2}) - 2 \sqrt{2\mathcal{C}_n\zeta}\xi \geq \frac{\xi}{3} - 2 \sqrt{2\mathcal{C}_n\zeta} \xi \geq \frac{\xi}{4};\end{aligned}$$ in particular, $\{B(z_i, s,\xi/8)\}_{i \in \mathcal{I}}$ are pairwise disjoint. For each $i\in \mathcal{I}$, Lemma [Lemma 6](#lem:W-V){reference-type="ref" reference="lem:W-V"} and Proposition [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"} lead us to $$\begin{aligned} &\quad \,\,W_{-R^2-2\zeta \xi^2}(\nu_{(x_0,0); -R^2-2\zeta \xi^2}, \nu_{(z_i,s); -R^2-2\zeta \xi^2})\\ &\leq W_{-R^2-2\zeta \xi^2}(\nu_{(x_0,0); -R^2-2\zeta \xi^2}, \nu_{(x_i,t_i); -R^2-2\zeta \xi^2})+W_{- R^2-2\zeta \xi^2}(\nu_{(x_i,t_i); -R^2-2\zeta \xi^2}, \nu_{(z_i,s); -R^2-2\zeta \xi^2})\\ &\leq W_{-R^2}(\nu_{(x_0,0); -R^2}, \nu_{(x_i,t_i); -R^2})+W_{s}(\nu_{(x_i,t_i); s}, \delta_{z_i})\\ &\leq R+\sqrt{\mathcal{C}_n(t_i-s)}\leq R+\sqrt{2\mathcal{C}_n \zeta} \xi<R+\frac{ \xi}{8}\end{aligned}$$ if $\zeta \leq \overline{\zeta}$, and hence $(z_i,s)\in P(x_0,0;R+ \xi/8)\cap (M\times \{s\})$. For every $y\in B(z_i,s,\xi/8)$, $$\begin{aligned} &\quad \,\,W_{-(R+\xi/8)^2}(\nu_{(x_0,0);-},\nu_{(z_i,s);-(R+\xi/8)^2})\\ &\leq W_{-(R+\xi/8)^2}(\nu_{(x_0,0);-(R+\xi/8)^2},\nu_{(y,s);-(R+\xi/8)^2})+d_s(y,z)\leq R+\frac{\xi}{4},\end{aligned}$$ and thus $B(z_i,s,\xi/8)\times \{s\}$ is contained in $P(x_0,0;R+\xi/4)\cap (M\times \{s\})$. Summarizing the above observations, from Lemma [Lemma 47](#lem:time_slice){reference-type="ref" reference="lem:time_slice"} and [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"}, we derive $$\begin{aligned} \label{eq:covering4} |\mathcal{I}| \,\inf_{i\in \mathcal{I}}m_s(B(z_i, s,\xi/8)) &\leq \sum_{i\in \mathcal{I}}m_s(B(z_i, s,\xi/8))\\ \notag &\leq m_s(P(x_0,0;R+\xi/4)\cap (M\times \{s\})\\ \notag &\leq C_{\rho,R} \exp ( \mathcal{N}_{(x_0, 0)} (\rho/2 )).\end{aligned}$$ On the other hand, by Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"} and Proposition [Proposition 24](#prop:lower volume){reference-type="ref" reference="prop:lower volume"}, if $\zeta \leq \overline{\zeta}$, then $$\begin{aligned} \label{eq:covering3} m_{s}(B(z_i, s,\xi/8)) &\geq m_{s}(B(z_i, s,2\sqrt{\mathcal{C}_n \zeta}\xi)) \geq m_s(B(z_i,s, \sqrt{2\mathcal{C}_n(t_i-s)}))\\ \notag &\geq C_{\rho,R} \exp ( \mathcal{N}_{(x_i, t_i)} (t_i-s ) ) (t_i-s)^{n/2}\geq C_{\rho,R} \exp ( \mathcal{N}_{(x_i, t_i)} ( \rho/2 ) ) \xi^n.\end{aligned}$$ Moreover, [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"}, Lemmas [Lemma 21](#lem:Nash cor){reference-type="ref" reference="lem:Nash cor"} and [Lemma 18](#lem:Nash_more){reference-type="ref" reference="lem:Nash_more"} yield $$\label{eq:covering5} \mathcal{N}_{(x_i,t_i)} ( \rho/2 )\geq \mathcal{N}_{(x_i,t_i)}(t_i+R^2+\rho/2)\geq \mathcal{N}_{(x_0,0)}(R^2+\rho/2)-C_{\rho,R}\geq \mathcal{N}_{(x_0,0)}(\rho/2)-C_{\rho,R}.$$ Here, we used $H(\cdot, -R^2-\rho/2)\geq -n/\rho$ by Lemma [Lemma 1](#lem:scal){reference-type="ref" reference="lem:scal"} (see Remark [Remark 2](#rem:scal2){reference-type="ref" reference="rem:scal2"}). Combining [\[eq:covering4\]](#eq:covering4){reference-type="eqref" reference="eq:covering4"}, [\[eq:covering3\]](#eq:covering3){reference-type="eqref" reference="eq:covering3"}, [\[eq:covering5\]](#eq:covering5){reference-type="eqref" reference="eq:covering5"} implies $| \mathcal{I} | \leq C_{\rho,R} \xi^{-n}$. This together with [\[eq:covering2\]](#eq:covering2){reference-type="eqref" reference="eq:covering2"} proves the claim. ◻ ## Effective strata {#sec:eff_strat} Due to Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"}, we have the following (cf. [@B3 Lemma 11.3]): **Proposition 49**. *For $\kappa>0,\varepsilon,\xi \in (0,1)$, if $\delta \leq \overline{\delta}_{\kappa, \varepsilon, \xi}$, then the following holds: Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0,t_0)\in M\times I$. For $r>0$, we assume $\mathcal{N}_{(x_0, t_0)} (r^2) \geq - \kappa$. For a subset $S\subset M\times I$, we assume that every point in $S$ is $(\delta, r)$-selfsimilar, and none of the following two properties hold:* 1. *$(k+1, \varepsilon, r)$-split;* 2. *$(k-1, \varepsilon, r)$-split and $(\varepsilon, r)$-static.* *Then there exists $\{(x_i,t_i)\}^{N}_{i=1}\subset S \cap P (x_0, t_0; r/2)$ such that $$\label{eq:pre_stratification1} S \cap P (x_0, t_0; r/2) \subset \bigcup_{i=1}^N P (x_i, t_i; \xi r),\quad N \leq C_{\kappa}\, \xi^{-k}.$$* *Proof.* We may assume $t_0 = 0$ and $r=1$. We first notice that by [\[eq:Nash3.5\]](#eq:Nash3.5){reference-type="eqref" reference="eq:Nash3.5"}, Lemmas [Lemma 21](#lem:Nash cor){reference-type="ref" reference="lem:Nash cor"} and [Lemma 18](#lem:Nash_more){reference-type="ref" reference="lem:Nash_more"}, for every $(x,t)\in P(x_0,0;1/2)$ we have $$\mathcal{N}_{(x,t)}(1)\geq \mathcal{N}_{(x,t)}(t+2)\geq \mathcal{N}_{(x_0,0)}(2)-C\geq \mathcal{N}_{(x_0,0)}(1)-C\geq -C_{\kappa}.$$ Let $\{(x_i, t_i)\}^{N}_{i=1} \subset S \cap P(x_0, 0; 1/2)$ be a maximal collection of points such that $\{P (x_i, t_i; \xi/10 )\}^{N}_{i=1}$ are pairwise disjoint. By the maximality and Lemma [Lemma 46](#lem:parab_nbhd){reference-type="ref" reference="lem:parab_nbhd"}, every point in $S \cap P(x_0, 0; 1/2)$ belongs to $P (x_i, t_i; 3 \xi/10)$ for some $i$, which shows the inclusion in [\[eq:pre_stratification1\]](#eq:pre_stratification1){reference-type="eqref" reference="eq:pre_stratification1"}. We derive an upper bound on $N$. We may assume $t_1 \leq \ldots \leq t_N$. We verify the following: If $\delta\leq \overline{\delta}_{\kappa,\varepsilon,\xi},N \geq \mathfrak{N}(100 \mathfrak{D})^{l} \xi^{-l}$, and if $0 \leq t_i-t_1 \leq \beta (100\mathfrak{D})^{-2}\xi^2$ for all $i$, then $(x_1, t_1)$ is $(l+1, \varepsilon,1)$-split, where $\beta\leq \overline{\beta},\mathfrak{D} \geq \underline{\mathfrak{D}}_{\kappa}$ and $\mathfrak{N} \geq \underline{\mathfrak{N}}_{\kappa}$ are constants obtained in Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"}. Indeed, for every $i$, Proposition [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"} yields $$\begin{aligned} &\quad \,\,W_{t_1-1}(\nu_{(x_1, t_1); t_1-1},\nu_{(x_i,t_i);t_1-1}) \\ &\leq W_{-1/4}(\nu_{(x_1,t_1);-1/4}, \nu_{(x_0,0); -1/4} )+W_{-1/4}(\nu_{(x_0,0);-1/4}, \nu_{(x_i, t_i); -1/4} )\leq 1. %splittingの条件(1)を修正する必要があるがめんどくさい\end{aligned}$$ Furthermore, for $i \neq j$, the fact that $(x_j, t_j) \not\in P (x_i, t_i; \xi/10)$ and Proposition [Proposition 7](#prop:monotonicity_W1){reference-type="ref" reference="prop:monotonicity_W1"} imply $$\begin{aligned} &\quad\,\, W_{t_1 - 2 (\xi/100 \mathfrak{D})^2}(\nu_{(x_i,t_i); t_1 - 2 (\xi/100 \mathfrak{D})^2}, \nu_{(x_j, t_j); t_1 - 2 (\xi/100 \mathfrak{D})^2} )\\ &\geq W_{t_i - (\xi/10)^2}(\nu_{(x_i,t_i); t_i - (\xi/10)^2}, \nu_{(x_j, t_j); t_i - (\xi/10)^2} )\geq \frac{\xi}{10} \geq \mathfrak{D} \left( \frac{\xi}{100 \mathfrak{D}}\right)\end{aligned}$$ if $\mathfrak{D}\geq \underline{\mathfrak{D}}$. We apply Theorem [Theorem 41](#thm:main almost splitting){reference-type="ref" reference="thm:main almost splitting"} at $(x_1, t_1)$ with $\xi$ replaced with $\xi / 100 \mathfrak{D}$ and $D=1$, and it follows that $(x_1, t_1)$ is $(l+1, \varepsilon,1)$-split. We set $C_{1,\kappa}:=2\mathfrak{N}(100 \mathfrak{D})^{2n}$ and $C_{2,\kappa}:=\beta (100\mathfrak{D})^{-2}$. We suppose $N \geq C_{1,\kappa} \xi^{-k}$. By the claim in the above paragraph, there is no $\mathcal{I} \subset \{ 1, \ldots, N \}$ with $|\mathcal{I}| \geq N/2$ such that $|t_i - t_j| \leq C_{2,\kappa}\xi^2$ for all $i,j \in \mathcal{I}$. Hence, for all $i,j = 1, \ldots, N$ with $j - i \geq N/2$, we see $|t_i - t_j| \geq C_{2,\kappa}\xi^2$. By Theorem [Theorem 40](#thm:almost static){reference-type="ref" reference="thm:almost static"}, if $\delta \leq \overline{\delta}_{\kappa, \xi, \varepsilon}$, then $\{(x_i, t_i)\}^{\lfloor N/2 \rfloor}_{i=1}$ are $(\varepsilon, 1)$-static; in particular, they are not $(k-1, \varepsilon, 1)$-split. Let $\mathcal{J} \subset \{ 1, \ldots, \lfloor N/2 \rfloor \}$ be a subset such that $|t_i - t_j | \leq C_{2,\kappa}\xi^2$ for all $i,j \in \mathcal{J}$, and $$|\mathcal{J} | \geq C_{\kappa} \xi^2 \lfloor N/2 \rfloor \geq C_{\kappa}\, \xi^{-(k-2)}.$$ Applying the above claim to $\{(x_i, t_i)\}_{i \in \mathcal{J}}$ for $l = k-2$, we arrive at the contradiction. ◻ For $\varepsilon\in (0,1)$ and $r_1,r_2>0$ with $r_1 < r_2$, the *effective strata $\mathcal{S}^{\varepsilon, k}_{r_1, r_2}$* is defined by the set of all points in $M\times I$ such that for all $r \in (r_1, r_2)$ none of the following two properties hold: 1. $(k+1, \varepsilon, r)$-split; 2. $(k-1, \varepsilon, r)$-split and $(\varepsilon, r)$-static. We conclude the following quantitative stratification result, which has been established by Bamler [@B3] for Ricci flow (cf. [@B3 Proposition 11.2]): **Theorem 50**. *Let $(M, g(t))_{t \in I}$ be a super Ricci flow with $\mathcal{D}\geq 0$. Let $(x_0, t_0) \in M \times I$. For $r,\kappa > 0$ we assume $[t_0 - 2 r^2, t_0] \subset I$ and $\mathcal{N}_{(x_0,t_0)} (r^2) \geq - \kappa$. Then for all $\varepsilon\in (0,1)$ and $\sigma \in (0, \varepsilon)$, there exists $\{(x_i, t_i)\}^{N}_{i=1} \subset \mathcal{S}^{\varepsilon,k}_{\sigma r, \varepsilon r} \cap P (x_0, t_0; r)$ such that $$\mathcal{S}^{\varepsilon, k}_{\sigma r, \varepsilon r} \cap P (x_0, t_0; r) \subset \bigcup_{i=1}^N P (x_i, t_i; \sigma r),\quad N \leq C_{\kappa, \varepsilon} \,\sigma^{-k-\varepsilon}.$$* *Proof.* We may assume $t_0=0$ and $r = 1$. For $\xi \in (0,1)$, let $m \in \mathbb{N}$ be an integer determined by $\sigma \in [\xi^{m}, \xi^{m-1}]$. In virtue of Proposition [Proposition 34](#prop:Nash almost){reference-type="ref" reference="prop:Nash almost"}, for every $(x,t) \in P (x_0,0; 1)$, the number of $j$ such that $(x,t)$ is not $(\delta, \xi^j)$-selfsimilar is bounded from above by $C_2=C_{2,\kappa, \xi, \delta}$, where $\delta \leq \overline{\delta}_{\kappa, \varepsilon, \xi}$ is a constant obtained in Proposition [Proposition 49](#prop:pre_stratification){reference-type="ref" reference="prop:pre_stratification"}. For $\{o_j\}^{m-1}_{j=0} \subset \{ 0, 1 \}$, we set $$S\left(\{o_j\}^{m-1}_{j=0} \right) := \left\{ (x,t) \in P (x_0, 0; 1) ~|~\text{$o_j=1$ if and only if $(x,t)$ is not $(\delta,\xi^j)$-selfsimilar} \right\}.$$ Then $P (x_0,0; 1)$ can be written as the union of at most $m^{C_2}$ many non-empty subsets of the form $S\left(\{o_j\}^{m-1}_{j=0} \right)$ since $S\left(\{o_j\}^{m-1}_{j=0} \right) = \emptyset$ if $\sum_{j}o_j > C_{2}$. Fix $l=0,\dots,m-1$ and $(y, s) \in P(x_0,0;1)$, where let $(y, s) := (x_0,0)$ if $l=0$. Due to Propositions [Proposition 48](#prop:covering){reference-type="ref" reference="prop:covering"} and [Proposition 49](#prop:pre_stratification){reference-type="ref" reference="prop:pre_stratification"}, there is $\{(y_i, s_i)\}^{N}_{i=1} \subset \mathcal{S}^{\varepsilon, k}_{\sigma, \varepsilon} \cap S\left(\{o_j\}^{m-1}_{j=0} \right)\cap P (y, s; \xi^{l})$ such that $$\mathcal{S}^{\varepsilon, k}_{\sigma, \varepsilon} \cap S\left(\{o_j\}^{m-1}_{j=0} \right)\cap P(y, s; \xi^{l}) \subset \bigcup_{i=1}^{N} P(y_i, s_i; \xi^{l+1}),\quad %左側に1/2をつける必要があるがめんどくさい N\leq \begin{cases} C_{3,\xi} & \text{if $o_l = 1$ or $l =0$,} \\ C_{1,\kappa}\, \xi^{-k} & \text{if $o_l= 0$ and $l \geq 1$,} \end{cases}$$ where the assertion for $l=0$ or $o_{l} = 1$, and that for $l \geq 1$ and $o_{l} = 0$ are consequences of Propositions [Proposition 48](#prop:covering){reference-type="ref" reference="prop:covering"} and [Proposition 49](#prop:pre_stratification){reference-type="ref" reference="prop:pre_stratification"}, respectively. By induction, $\mathcal{S}^{\varepsilon, k}_{\sigma, \varepsilon} \cap S\left(\{o_j\}^{m-1}_{j=0} \right)\cap P(x_0,0; 1)$ can be covered by at most $$C_{3,\xi}^{1+\sum_j o_j} (C_{1,\kappa}\, \xi^{-k} )^{m-1- \sum_j o_j}$$ many parabolic balls of the form $P(y,s; \sigma )$. It follows that $\mathcal{S}^{\varepsilon, k}_{\sigma, \varepsilon} \cap P(x_0,0; 1)$ can be covered by at most $$m^{C_2} \,C_{3,\xi}^{1+C_2} (C_{1,\kappa} \xi^{-k} )^{m} \leq C_{\kappa, \xi, \delta} \,m^{C_2} C_{1,\kappa}^m \,\xi^{-mk}$$ many such parabolic balls. If $\xi \leq \overline{\xi}_{\kappa, \varepsilon}$ such that $\xi^{-\varepsilon/2} \geq C_{1,\kappa}$, then $$C_{\kappa, \xi, \delta} \,m^{C_2} \,\xi^{-mk - m\varepsilon/2}\leq C_{\kappa, \xi, \delta}\, \xi^{-mk - m\varepsilon}\leq C_{\kappa, \xi, \delta} \,\sigma^{-k-\varepsilon}.$$ Thus, we complete the proof. ◻ ## Acknowledgements {#acknowledgements .unnumbered} The first author was supported by JSPS KAKENHI (JP23K03105). The second author was supported by JSPS KAKENHI (JP23K12967). 99 L. Ambrosio, N. Gigli and G. Savaré, *Metric measure spaces with Riemannian Ricci curvature bounded from below*, Duke Math. J. **163** (2014), no. 7, 1405--1490. R. H. Bamler, *Entropy and heat kernel bounds on a Ricci flow background*, preprint arXiv:2008.07093v3. R. H. Bamler, *Compactness theory of the space of Super Ricci flows*, Invent. Math. **233** (2023), no. 3, 1121--1277. R. H. Bamler, *Structure theory of non-collapsed limits of Ricci flows*, preprint arXiv:2009.03243v2. R. H. Bamler and Q. S. Zhang, *Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature*, Adv. Math. **319** (2017), 396--450. X. Cao, H. Guo and H. Tran, *Harnack estimates for conjugate heat kernel on evolving manifolds*, Math. Z. **281** (2015), no. 1-2, 201--214. J. Cheeger and T. H. Colding, *Lower bounds on Ricci curvature and the almost rigidity of warped products*, Ann. of Math. (2) **144** (1996), no. 1, 189--237. J. Cheeger and T. H. Colding, *On the structure of spaces with Ricci curvature bounded below. I*, J. Differential Geom. **46** (1997), no. 3, 406--480. J. Cheeger and T. H. Colding, *On the structure of spaces with Ricci curvature bounded below. II*, J. Differential Geom. **54** (2000), no. 1, 13--35. J. Cheeger and T. H. Colding, *On the structure of spaces with Ricci curvature bounded below. II*, J. Differential Geom. **54** (2000), no. 1, 37--74. B. Chow, S-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, *The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects*, Mathematical Surveys and Monographs, **163**. American Mathematical Society, Providence, RI, 2010. xx+517 pp. T. H. Colding and A. Naber, *Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications*, Ann. of Math. (2) **176** (2012), no. 2, 1173--1229. S. Fang and T. Zheng, *The (logarithmic) Sobolev inequalities along geometric flow and applications*, J. Math. Anal. Appl. **434** (2016), no. 1, 729--764. N. Gigli, *On the differential structure of metric measure spaces and applications*, Mem. Amer. Math. Soc. **236** (2015), no. 1113, vi+91 pp. C. M. Guenther, *The fundamental solution on manifolds with time-dependent metrics*, J. Geom. Anal. **12** (2002), no. 3, 425--436. H. Guo, R. Philipowski and A. Thalmaier, *Entropy and lowest eigenvalue on evolving manifolds*, Pacific J. Math. **264** (2013), no. 1, 61--81. R. S. Hamilton, *Three-manifolds with positive Ricci curvature*, J. Differential Geometry **17** (1982), no. 2, 255--306. R. Haslhofer and A. Naber, *Characterizations of the Ricci flow*, J. Eur. Math. Soc. (JEMS) **20** (2018), no. 5, 1269--1302. H.-J. Hein and A. Naber, *New logarithmic Sobolev inequalities and an $\varepsilon$-regularity theorem for the Ricci flows*, Comm. Pure Appl. Math. **67** (2014), no. 9, 1543--1561. H. Huang, *Optimal transportation and monotonic quantities on evolving manifolds*, Pacific J. Math. **248** (2010), no. 2, 305--316. E. Kopfer, *Super-Ricci flows and improved gradient and transport estimates*, Probab. Theory Related Fields **175** (2019), no. 3-4, 897--936. E. Kopfer and K.-T. Sturm, *Heat flow on time-dependent metric measure spaces and super-Ricci flows*, Comm. Pure Appl. Math. **71** (2018), no. 12, 2500--2608. E. Kopfer and K.-T. Sturm, *Functional inequalities for the heat flow on time-dependent metric measure spaces*, J. Lond. Math. Soc. (2) **104** (2021), no. 2, 926--955. K. Kunikawa and Y. Sakurai, *Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow*, preprint arXiv:2201.11361v2. B. List, *Evolution of an extended Ricci flow system*, Comm. Anal. Geom. **16** (2008), no. 5, 1007--1048. J. Lott and C. Villani, *Ricci curvature for metric-measure spaces via optimal transport*, Ann. of Math. (2) **169** (2009), no. 3, 903--991. R. J. McCann and P. M. Topping, *Ricci flow, entropy and optimal transportation*, Amer. J. Math. **132** (2010), no. 3, 711--730. R. Müller, *Monotone volume formulas for geometric flows*, J. Reine Angew. Math. **643** (2010), 39--57. R. Müller, *Ricci flow coupled with harmonic map flow*, Ann. Sci. Éc. Norm. Supér. (4) **45** (2012), no. 1, 101--142. G. Perelman, *The entropy formula for the Ricci flow and its geometric applications*, preprint arXiv:math/0211159. K.-T. Sturm, *On the geometry of metric measure spaces. I*, Acta Math. **196** (2006), no. 1, 65--131. K.-T. Sturm, *On the geometry of metric measure spaces. II*, Acta Math. **196** (2006), no. 1, 133--177. K.-T. Sturm, *Super-Ricci flows for metric measure spaces*, J. Funct. Anal. **275** (2018), no. 12, 3504--3569. C. Villani, *Optimal Transport: Old and New*, Springer-Verlag, Berlin, 2009.

arxiv_math

--- abstract: | In this paper we study monodromy operators on moduli spaces $M_v(S,H)$ of sheaves on K3 surfaces with non-primitive Mukai vectors $v$. If we write $v=mw$, with $m>1$ and $w$ primitive, then our main result is that the inclusion $M_w(S,H)\to M_v(S,H)$ as the most singular locus induces an isomorphism between the monodromy groups of these symplectic varieties, allowing us to extend to the non-primitive case a result of Markman. address: - Dipartimento di Matematica \"F. Enriques\", Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italia. - Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italia. - Dipartimento di Matematica, Università di Roma Tor Vergata, via della ricerca scientifica 1, 00133 Roma, Italia. author: - Claudio Onorati - Arvid Perego - Antonio Rapagnetta title: Locally trivial monodromy of moduli spaces of sheaves on K3 surfaces --- # Introduction {#introduction .unnumbered} Singular symplectic varieties (Section [1.1](#section:singular symplectic){reference-type="ref" reference="section:singular symplectic"}) have gained much interest lately, especially after the proof of a global Torelli Theorem in [@BakkerLehn:PrimitiveSymplecticVarieties]. The outcome of these results can be summarised by saying that their geometry behaves very much like the geometry of irreducible holomorphic symplectic manifolds. Roughly, if $X$ is either a smooth or singular irreducible symplectic variety, most of the geometry of $X$ is controlled by the second integral cohomology group $\operatorname{H}^2(X,\mathbb{Z})$ together with its pure weight two Hodge structure and the Beauville--Bogomolov--Fujiki lattice structure. We recall that the Beauville--Bogomolov--Fujiki lattice $\operatorname{H}^2(X,\mathbb{Z})$ has always signature $(3,b_2(X)-3)$, where the positive three-space is generated by a Kähler class and the real and imaginary part of the symplectic form. The bimeromorphic classification of irreducible symplectic varieties in the same locally trivial deformation class is then encoded in the (locally trivial) *monodromy group*, which is a finite index subgroup $\operatorname{Mon}^2_{\operatorname{lt}}(X)$ of the group $\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z}))$ of isometries of the lattice $\operatorname{H}^2(X,\mathbb{Z})$. In this paper we consider a special class of irreducible symplectic varieties, namely those that are locally trivially deformation equivalent (Definition [Definition 4](#def:lt){reference-type="ref" reference="def:lt"}) to a moduli space $M_v(S,H)$ of sheaves on a projective K3 surface $S$. Here $v$ is a non-primitive Mukai vector and $H$ is $v$-generic. The notion of $v$-genericity is technical and will be recalled in Definition [Definition 19](#defn:v-generic){reference-type="ref" reference="defn:v-generic"}. By [@PR:SingularVarieties], it is known that, under these assumptions, $M_v(S,H)$ is indeed an irreducible symplectic variety. Let us remark that when the Mukai vector $v$ is primitive, the moduli space $M_v(S,H)$ is smooth and it is an irreducible holomorphic symplectic manifold, i.e. a simply connected Kähler manifold with a unique (up to scalar) holomorphic symplectic form. If $X$ is any manifold deformation equivalent to a smooth moduli space $M_v(S,H)$ as above, then the group $\operatorname{Mon}^2(X)$ of the monodromy operators in $\operatorname{H}^2(X,\mathbb{Z})$ is known by a result of Markman ([@Markman:IntegralConstraints]): it is the group of orientation preserving isometries that act as $\pm\operatorname{id}$ on the discriminant group (see Section [1.3](#section:pto){reference-type="ref" reference="section:pto"} for the notion of orientation). Recall that if $\Lambda$ is a lattice, then the discriminant group is the finite group $\Lambda^*/\Lambda$, where $\Lambda^*=\operatorname{Hom}(\Lambda,\mathbb{Z})$ is the dual $\mathbb{Z}$-module. Following Markman's notation, if $\Lambda$ is any even lattice of signature $(3,n)$, then we denote by $\mathsf{W}(\Lambda)\subset\operatorname{O}(\Lambda)$ the subgroup of orientation preserving isometries acting as $\pm\operatorname{id}$ on the discriminant group. Our first result is the following, which extends [@Markman:IntegralConstraints Theorem 1.1] to the singular setting. **Theorem A.1 1** (Corollary [Corollary 99](#cor:ogni X){reference-type="ref" reference="cor:ogni X"}). *Let $X$ be an irreducible symplectic variety that is locally trivially deformation equivalent to a moduli space $M_v(S,H)$, where $S$ is a projective K3 surface, $v=mw$ with $m>1$ and $w$ primitive, and $H$ a $v$-generic polarisation. Then $$\operatorname{Mon}^2_{\operatorname{lt}}(X)=\mathsf{W}(\operatorname{H}^2(X,\mathbb{Z}))\subset\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z})).$$* We recall the definition of the group $\mathsf{W}(\operatorname{H}^2(X,\mathbb{Z}))$ at the beginning of Section [4.1](#section:W in m k){reference-type="ref" reference="section:W in m k"}. We point out that our result does not subside Markman's one, inasmuch as we heavily use [@Markman:IntegralConstraints Theorem 1.1] in our proof. To the best of our knowledge, this is the first time an explicit description of the monodromy group of a class of singular symplectic varieties is exhibited. We recall that monodromy groups have been computed in all the known deformation classes of smooth irreducible holomorphic symplectic manifolds, see [@Markman:Monodromy; @Markman:Kummer; @Mon; @MonRap; @Onorati:Monodromy]. As a corollary one can explicitly compute the index of $\operatorname{Mon}^2_{\operatorname{lt}}(X)$ in the isometry group $\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z}))$ (see Corollary [Corollary 95](#cor:index){reference-type="ref" reference="cor:index"}). As an example, if $X$ is deformation equivalent to a moduli space $M_v(S,H)$, where $v=mw$ and $w^2=2$, then the group $\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))$ is the whole group $\operatorname{O}^+(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))$ of orientation preserving isometries (notice that this does not depend on $m$, this feature is shared by an infinite class of examples in any dimension $2m^2+2$). Our second result is an equivalent reformulation of Theorem A.1, in which the relation between the group $\operatorname{Mon}^2_{\operatorname{lt}}(X)$ and the monodromy group of an irreducible holomorphic symplectic manifold deformation equivalent to a smooth moduli space of sheaves is explained. First of all, if $X$ is locally trivially deformation equivalent to a moduli space $M_v(S,H)$ as before, then the most singular locus $Y$ of $X$ (cf. Proposition [Proposition 3](#prop:stratification){reference-type="ref" reference="prop:stratification"}) is an irreducible holomorphic symplectic manifold deformation equivalent to the moduli space $M_w(S,H)$ (here $w$ is the primitive Mukai vector such that $v=mw$). Let us denote by $i_{Y,X}\colon Y\to X$ the closed embedding. The embedding $i_{Y,X}$ induces an homomorphism $$i_{Y,X}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(X)\longrightarrow\operatorname{Mon}^2(Y).$$ We will define this morphism carefully in Section [4.3](#section:main result){reference-type="ref" reference="section:main result"} (see also Section [1.4](#section:moduli spaces){reference-type="ref" reference="section:moduli spaces"} for the case $X=M_v(S,H)$), but we can intuitively describe it as follows. It sends a monodromy operator along a loop $\gamma$ in a family $p\colon\mathcal{X}\to T$ of deformations of $X$, to the monodromy operator along the same loop $\gamma$ on the family of deformations $q\colon\mathcal{Y}\to T$ of $Y$ obtained by restriction from $p$, i.e. $\mathcal{Y}\subset\mathcal{X}$ is the relative closed embedding of the most singular locus. **Theorem B.1 1** (Corollary [Corollary 100](#cor:i sharp is iso gen){reference-type="ref" reference="cor:i sharp is iso gen"}). *Let $X$ be an irreducible symplectic variety that is locally trivially deformation equivalent to a moduli space $M_v(S,H)$ as above. Let $Y\subset X$ be the most singular locus and $i_{Y,X}\colon Y\to X$ the closed embedding. Then $$i_{Y,X}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(X)\stackrel{\sim}{\longrightarrow}\operatorname{Mon}^2(Y)$$ is an isomorphism.* ## Outline of the proof {#outline-of-the-proof .unnumbered} Since the locally trivial monodromy group is invariant along locally trivial families of primitive symplectic varieties, we can reduce the proof of the the two main results to statements about moduli spaces of sheaves. Therefore from now on we will work with $X=M_v(S,H)$, where $S$ is a projective K3 surface, $v$ a Mukai vector of the form $v=mw$, with $m>1$ and $w$ primitive, and $H$ a $v$-generic polarisation. Let us recall that $v$ belongs to the so-called *Mukai lattice* and so we can consider the ortoghonal complement $v^\perp$, which is an even lattice of signature $(3,20)$. We will recall in Section [1.4](#section:moduli spaces){reference-type="ref" reference="section:moduli spaces"} the definitions and constructions. By [@PR:v; @perp], there is an isometry $$\lambda_{(S,v,H)}\colon v^\perp\to\operatorname{H}^2(M_v(S,H),\mathbb{Z}).$$ When the Mukai vector $v$ is primitive, the same result is due to O'Grady (see [@OGrady:WeightTwo; @Yoshioka:ModuliAbelian]). Theorem A.1 is equivalent to the following. **Theorem A.2 1** (Theorem [Theorem 93](#thm:Mon 2 lt){reference-type="ref" reference="thm:Mon 2 lt"}). *Let $S$ be a projective K3 surface, $v$ a Mukai vector and $H$ a $v$-generic polarisation. Then $$\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))=\mathsf{W}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))\cong\mathsf{W}(v^\perp),$$ where the last isomorphism is induced by the isometry $\lambda_{(S,v,H)}$.* We point out that the special case when $m=2=w^2$ already appeared in [@Onorati:Monodromy Theorem 6.1] with a different proof. The proof follows two steps. In the first one we construct monodromy operators: this is performed in Section [4.1](#section:W in m k){reference-type="ref" reference="section:W in m k"} and the main result is Theorem [Theorem 85](#thm:W in m k){reference-type="ref" reference="thm:W in m k"}, where it is proved that $\mathsf{W}(v^\perp)\subset\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))$. This section parallels Markman's construction of monodromy operators in [@Markman:Monodromy]. In fact we point out that it also works for primitive Mukai vectors: our only improvement with respect to Markman's work is that we only work with polarised families of K3 surfaces (see Proposition [Proposition 80](#prop:Mon S in m k){reference-type="ref" reference="prop:Mon S in m k"} and the remark soon after). In the second step, performed in details in Section [4.2](#subsection:i sharp){reference-type="ref" reference="subsection:i sharp"}, we put a constraint on the monodromy group by using the inclusion in $M_v(S,H)$ of its most singular locus. This step is where we crucially use that $v=mw$ is not primitive. In fact in this case the most singular locus of $M_v(S,H)$ can be naturally identified with the smooth moduli space $M_w(S,H)$; let us denote by $i_{w,m}\colon M_w(S,H)\to M_v(S,H)$ the closed embedding. Then by Corollary [Corollary 92](#prop:first inclusion){reference-type="ref" reference="prop:first inclusion"} the map $i_{w,m}$ induces an injective homomorphism $$i_{w,m}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))\longrightarrow\operatorname{Mon}^2(M_w(S,H)$$ giving the desired constraint. The conclusion of the proof is now a straightforward combination of the two steps before and [@Markman:IntegralConstraints Theorem 1.1]. Working again in the case when $X=M_v(S,H)$, the isomorphism in Theorem B.1 becomes very natural in terms of the isomorphisms $$\lambda_{(S,v,H)}^\sharp\colon\mathsf{W}(v^\perp)\longrightarrow\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))$$ and $$\lambda_{(S,w,H)}^\sharp\colon\mathsf{W}(w^\perp)\longrightarrow\operatorname{Mon}^2(M_w(S,H))$$ induced by conjugation from the isometries $\lambda_{(S,v,H)}$ and $\lambda_{(S,w,H)}$. Let us remark that since $v=mw$, the lattices $v^\perp$ and $w^\perp$ are the same sub-lattice of the Mukai lattice of $S$; in particular there is an equality $\mathsf{W}(v^\perp)=\mathsf{W}(w^\perp)$. **Theorem B.2 1** (Theorem [Theorem 97](#thm:i sharp is iso){reference-type="ref" reference="thm:i sharp is iso"}). *Let $M_v(S,H)$ be the moduli space of shaves on a K3 surface $S$ with Mukai vector $v$. Assume that $v=mw$, with $m>1$ and $w$ primitive, and that $H$ is $v$-generic. Then the closed embedding $i_{w,m}\colon M_w(S,H)\to M_v(S,H)$ induces an isomorphism $$i_{w,m}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))\stackrel{\sim}{\longrightarrow}\operatorname{Mon}^2(M_w(S,H)),$$ and the composition $$(\lambda_{(S,w,H)}^\sharp)^{-1}\circ i_{w,m}^\sharp\circ\lambda_{(S,v,H)}^\sharp\colon\mathsf{W}(v^\perp)\longrightarrow\mathsf{W}(w^\perp)$$ is the identity.* ## Plan of the paper {#plan-of-the-paper .unnumbered} In Section [1](#section:preliminaries){reference-type="ref" reference="section:preliminaries"} we recall the basic facts about singular symplectic varieties and moduli spaces of sheaves. The results in this section are all known to experts but we reproduce here some proofs whenever a precise reference was missing in the literature. Section [2](#section:groupoid){reference-type="ref" reference="section:groupoid"} recalls and extends Markman's construction of a groupoid representation that will be useful to construct monodromy operators and to prove surjectivity of the morphism $i_{w,m}^\sharp$. In Section [3](#section:mon S pol){reference-type="ref" reference="section:mon S pol"} we provide a proof that the monodromy group of a K3 surface is generated by polarised families. This result is surely well known to experts, but we could not find any reference in the literature; its purpose is to allow us to lift monodromy operators from the K3 surface to the moduli space without deforming to a non-projective K3 surfaces, i.e. without the need to consider families of moduli spaces of sheaves on non-projective K3 surfaces. Section [4](#section:main){reference-type="ref" reference="section:main"} is the main and last section of the paper, where we prove Theorem [Theorem 93](#thm:Mon 2 lt){reference-type="ref" reference="thm:Mon 2 lt"} and Theorem [Theorem 97](#thm:i sharp is iso){reference-type="ref" reference="thm:i sharp is iso"}. ## Acknowledgements {#acknowledgements .unnumbered} Claudio Onorati was supported by the grant ERC-2017-CoG771507-StabCondEn with principal investigator P. Stellari. Arvid Perego and Antonio Rapagnetta were partially supported by the Research Project PRIN 2020 - CuRVI, CUP J37G21000000001. Claudio Onorati and Antonio Rapagnetta wish to thank the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C1800010000. Arvid Perego wishes to thank the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Genoa, CUP D33C23001110001. Claudio Onorati and Arvid Perego gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University, and the Japanese--European Symposium on Symplectic Varieties and Moduli Spaces at which some of the research for this paper was performed. The authors are member of the INDAM-GNSAGA. # Preliminaries {#section:preliminaries} This section is dedicated to review some basic material we will need. More precisely, in the first two subsections we collect some fundamental results and definitions about primitive (resp. irreducible) symplectic varieties and their monodromy groups. The third subsection will be devoted to review the main results about moduli spaces of sheaves on K3 surfaces. ## Singular symplectic varieties {#section:singular symplectic} We first start by recalling the notion of a symplectic variety. Let $X$ be a normal complex analytic variety, and denote by $X_{\operatorname{reg}}$ its smooth locus; notice that the complement of $X_{\operatorname{reg}}$ has codimension at least $2$. If $j\colon X_{\operatorname{reg}}\to X$ is the corresponding open embedding, then for every integer $0\leq p\leq\dim(X)$ we let $$\Omega_{X}^{[p]}:=j_{*}\Omega^{p}_{X_{\operatorname{reg}}}=\big(\wedge^{p}\Omega_{X}\big)^{**},$$whose global sections are called *reflexive $p$-forms* on $X$. A reflexive $p$-form on $X$ is then a holomorphic $p$-form on $X_{\operatorname{reg}}$. There is a notion of *singular Kähler form* due to Grauert and recalled in [@BakkerLehn:PrimitiveSymplecticVarieties Section 2.3]. We will not recall the definition of singular Kähler forms, but we point out some of their properties. Every singular Kähler form $\omega$ gives a class $[\omega]\in\operatorname{H}^2(X,\mathbb{R})$; the set of classes in $\operatorname{H}^2(X,\mathbb{R})$ obtained in this way form an open cone that lies in $\operatorname{H}^{1,1}(S,\mathbb{R})$, where the latter can be interpreted classically as $$\operatorname{H}^{1,1}(X,\mathbb{R})=F^1\operatorname{H}^2(X,\mathbb{C})\cap\operatorname{H}^2(X,\mathbb{R}),$$ where we are using the mixed Hodge structure on $\operatorname{H}^2(X,\mathbb{C})$ (see [@BakkerLehn:PrimitiveSymplecticVarieties Definition 2.5, Remark 2.6.(1), Proposition 2.8]). In particular, in the cases of interest to us, the group $\operatorname{H}^2(X,\mathbb{C})$ will have a pure Hodge structure and $\operatorname{H}^{1,1}(X,\mathbb{R})=\operatorname{H}^1(X,\Omega^{[1]}_X)\cap\operatorname{H}^2(X,\mathbb{R})$. A normal complex analytic variety admitting a Kähler form will be called a *Kähler space*. By [@Varouchas II, 1.2.1 Proposition] a smooth Kähler space is a Kähler manifold in the classical sense. Moreover, if $X$ is reduced, then there exists a resolution of singularities $\widetilde{X}\to X$ such that $\widetilde{X}$ is a Kähler manifold. A subspace of a Kähler space is again a Kähler space (see for example [@Varouchas II, 1.3.1(i) Proposition]). We now recall the definitions of a symplectic form and a symplectic variety (cf. [@Bea00]). **Definition 1**. Let $X$ be a normal compact Kähler space. 1. A *symplectic form* on $X$ is a closed reflexive 2-form $\sigma$ on $X$ which is non-degenerate at each point of $X_{\operatorname{reg}}$. 2. If $\sigma$ is a symplectic form on $X$, the pair $(X,\sigma)$ is a *symplectic variety* if for every (Kähler) resolution $f\colon\widetilde{X}\to X$ of the singularities of $X$, the holomorphic symplectic form $\sigma_{\operatorname{reg}}:=\sigma_{|X_{\operatorname{reg}}}$ extends to a holomorphic 2-form on $\widetilde{X}$. By a slight abuse of notation, in this case we will say that $X$ is a symplectic variety. In what follows, we will be interested in two types of symplectic varieties, namely primitive symplectic varieties and irreducible symplectic varieties. We recall their definitions following [@BakkerLehn:PrimitiveSymplecticVarieties] and [@GKP]. Before doing this, we recall that if $X$ and $Y$ are two irreducible normal compact complex analytic varieties, a *finite quasi-étale morphism* $f\colon Y\to X$ is a finite morphism which is étale in codimension one. **Definition 2**. Let $X$ be a symplectic variety and $\sigma$ a symplectic form on $X$. 1. The variety $X$ is a *primitive symplectic variety* if $\operatorname{H}^{1}(X,\mathcal{O}_{X})=0$ and $\operatorname{H}^{0}(X,\Omega_{X}^{[2]})=\mathbb{C}\sigma$. 2. The variety $X$ is an *irreducible symplectic variety* if for every finite quasi-étale morphism $f\colon Y\to X$ the exterior algebra of reflexive forms on $Y$ is spanned by $f^{[*]}\sigma$. An irreducible symplectic variety is primitive symplectic (so in particular it is a symplectic variety), but there are examples of primitive symplectic varieties that are not irreducible symplectic. By [@GGK:Klt Corollary 13.3], an irreducible symplectic variety $X$ is simply connected. In particular, the $\mathbb{Z}$-module $\operatorname{H}^{2}(X,\mathbb{Z})$ is free. Moreover, the fact that irreducible symplectic varieties are simply connected together with the Bogomolov Decomposition Theorem imply that smooth irreducible symplectic varieties are irreducible holomorphic symplectic manifolds. We conclude this section by recalling a result originally due to Kaledin about the stratification of the singularities of a symplectic variety, that together with Remark [Remark 10](#rmk:strat){reference-type="ref" reference="rmk:strat"} will play an important role in what follows (see [@Kaledin:Poisson Theorem 2.3] or [@BakkerLehn:PrimitiveSymplecticVarieties Theorem 3.4.(2)]). **Proposition 3**. *Let $X$ be a symplectic variety, and consider the finite stratification by closed subvarieties $$X=X_0\supset X_1\supset\cdots\supset X_m,$$ where $X_{i+1}$ is the singular locus with reduced structure $(X_{i}^{\operatorname{sing}})_{\operatorname{red}}$ of $X_{i}$.* *Then for every $i=0,\cdots,m$, the normalisation of each irreducible component of $X_i$ is a symplectic variety.* We notice in particular that each stratum of the stratification of the singularities is even dimensional. ## Locally trivial deformations of symplectic varieties **Definition 4**. 1. A *locally trivial family* is a proper morphism $f\colon\mathcal{X}\to T$ of complex analytic spaces such that $T$ is connected and, for every point $x\in\mathcal{X}$, there exist open neighborhoods $V_x\subset \mathcal{X}$ and $V_{f(x)}\subset T$, and an open subset $U_x\subset f^{-1}(f(x))$ such that $$V_x\cong U_x\times V_{f(x)},$$ where the isomorphism is an isomorphism of analytic spaces commuting with the projections over $T$. A *locally trivial deformation* of a complex analytic variety $X$ is a locally trivial family $f\colon\mathcal{X}\to T$ for which there is $t\in T$ such that $f^{-1}(t)\simeq X$. 2. A *locally trivial family of primitive (resp. irreducible) symplectic varieties* is a locally trivial family whose fibres are all primitive (resp. irreducible) symplectic. 3. Two primitive symplectic varieties are said to be *locally trivially deformation equivalent* if they are members of a locally trivial family of primitive sympelctic varieties. **Remark 5**. The fibres of a locally trivial family of primitive or irreducible symplectic varieties are Kähler spaces by definition. As usual, when we say that any small deformation of $X$ enjoys a property, we mean that there exists an analytic open neighborhood $U$ of the base of a versal deformation of $X$ such that every fiber over $U$ enjoys the property. The behaviour of primitive symplectic varieties under small locally trivial deformations is known thanks to the following result. **Proposition 6** (). *Let $X$ be a primitive symplectic variety. Then any small locally trivial deformation of $X$ is again a primitive symplectic variety.* The same result for irreducible symplectic varieties seems to be unknown. We provide here two arguments that applies to two subclasses of irreducible symplectic varieties, which will be mostly interesting for us. **Proposition 7**. *Let $X$ be an irreducible symplectic variety with simply connected regular locus. Then any small locally trivial deformation $X'$ of $X$ is an irreducible symplectic variety with simply connected regular locus.* *Proof.* Consider a locally trivial deformation $\alpha\colon\mathcal{X}\to T$ of $X$, and let $0\in T$ be such that the fiber $X_{0}$ of $\alpha$ over $0$ is isomorphic to $X$. By Proposition [Proposition 6](#prop:small is primitive){reference-type="ref" reference="prop:small is primitive"} we know that, up to shrinking $T$, for every $t\in T$ we have that $X_{t}$ is a (Kähler) primitive symplectic variety. Since $\alpha\colon\mathcal{X}\to T$ is locally trivial, the regular loci of each fibre $X_t$ fits together to form a flat fibration $\alpha_{\operatorname{reg}}\colon\mathcal{X}_{\operatorname{reg}}\to T$ that locally on $T$ is topologically trivial, by Thom's First Isotopy Lemma ([@Dimca Theorem 6.5]). In particular, for any $t,t'\in T$, the fundamental groups $\pi_1(X_{t,\operatorname{reg}})$ and $\pi_1(X_{t',\operatorname{reg}})$ are isomorphic and trivial: hence there are no non-trivial finite quasi-étale covers of $X_t$ and the proof is completed. ◻ We can drop the strong hypothesis on the fundamental group of the smooth locus, provided that the singularities are mild enough. **Proposition 8**. *Let $X$ be an irreducible symplectic variety with at most terminal singularities. Then any small locally trivial deformation of $X$ is an irreducible symplectic variety with at most terminal singularities.* *Proof.* Consider again a locally trivial deformation $\alpha\colon\mathcal{X}\to T$ of $X$, and let $0\in T$ be such that the fiber $X_{0}$ of $\alpha$ over $0$ is isomorphic to $X$. Up to shrinking $T$, again by Proposition [Proposition 6](#prop:small is primitive){reference-type="ref" reference="prop:small is primitive"} we can suppose that any fibre $X_t$ is a primitive symplectic variety. We need to prove that if $f_{t}\colon Y_{t}\to X_{t}$ is a finite quasi-étale covering, then the dimension $h^{[p],0}(Y_{t})$ of $\operatorname{H}^{0}(Y_{t},\Omega_{Y_{t}}^{[p]})$ is 0 if $p$ is odd, and 1 if $p$ is even. First of all, as we saw in the proof of Proposition [Proposition 7](#prop:small when simply connected){reference-type="ref" reference="prop:small when simply connected"}, the fundamental groups of the smooth loci of the fibres of $\alpha$ are all isomorphic. Moreover, as the isomorphism classes of finite quasi-étale coverings of $X_{t}$ are prescribed by the subgroups of $\pi_{1}(X^{\operatorname{reg}}_{t})$, it is enough to prove the result when $f_t$ arises from a deformation of a finite quasi-étale covering (we recall that by definition the total space of a quasi-étale covering is normal). We make the following claim. **Claim 1**. *Let $f_0\colon Y_0\to X_0$ be a finite quasi-étale covering of $X_0$; then there exist a locally trivial family $\beta\colon\mathcal{Y}\to T$ and a proper morphism $f\colon\mathcal{Y}\to\mathcal{X}$ such that $Y_t=\beta^{-1}(t)$ is normal and $f_t\colon Y_t\to X_t$ is a finite quasi-étale covering, for every $t\in T$.* Let us first see how the claim would finish the proof. First of all, since $\beta$ is a locally trivial family, up to shrinking $T$ we can take a relative resolution of singularities $\pi\colon\widetilde{\mathcal{Y}}\to\mathcal{Y}$ such that $\widetilde{Y}_{t}$ is Kähler for all $t\in T$ (by [@Namikawa:ext Proposition 5]), and hence the function mapping $t\in T$ to $h^{p,0}(\widetilde{Y}_{t})$ is constant. By [@GKP Theorem 2.4] there is an equality $h^{[p],0}(Y_{t})=h^{p,0}(\widetilde{Y}_{t})$, so that $h^{[p],0}(Y_{t})$ is also constant along $T$. It follows that $$\dim \operatorname{H}^{0}(Y_{t},\Omega_{Y_{t}}^{[p]})=\dim \operatorname{H}^{0}(Y,\Omega_{Y}^{[p]})=\left\{ \begin{array}{ll} 1, & p\in 2\mathbb{N}\\ 0, & p\notin 2\mathbb{N} \end{array}\right.$$ where the last equality comes from the fact that $X_0=X$ is an irreducible symplectic variety and $Y_0=Y$ is a finite quasi-étale covering of $X_0$. First of all, since the family $\alpha\colon\mathcal{X}\to T$ is locally trivial and the central fibre $X_0$ is terminal by hypothesis, all the other fibres $X_t$ are terminal. In particular, if $\alpha_{\operatorname{reg}}\colon\mathcal{X}_{\operatorname{reg}}\to T$ is the smooth family of the smooth loci, then $\mathcal{X}\setminus\mathcal{X}_{\operatorname{reg}}$ has codimension at least $3$ (in fact Namikawa proves in [@Namikawa:Terminal Corollary 1] that the codimension of the singular locus is always at least $4$, but we will not need this fact). Now, since the family $\alpha_{\operatorname{reg}}\colon\mathcal{X}_{\operatorname{reg}}\to T$ is smooth (and the fundamental group of the fibres is constant), there exists another smooth family $\beta'\colon\mathcal{Y}'\to T$ and a proper étale cover $f'\colon\mathcal{Y}'\to\mathcal{X}_{\operatorname{reg}}$. The sheaf $f'_*\mathcal{O}_{\mathcal{Y}'}$ is a locally free sheaf of algebras on $\mathcal{X}_{\operatorname{reg}}$ and, if we denote by $j\colon\mathcal{X}_{\operatorname{reg}}\to\mathcal{X}$ the open immersion, by [@Siu Theorem 2] the sheaf $j_*f'_*\mathcal{O}_{\mathcal{Y}'}$ is a coherent sheaf of $\mathcal{O}_{\mathcal{X}}$-algebras. Let us define $$f\colon\mathcal{Y}=\underline{\operatorname{Spec}}_{\mathcal{X}}(j_*f'_*\mathcal{O}_{\mathcal{Y}'})\to\mathcal{X}$$ and set $\beta:=\alpha\circ f$. For every $t\in T$ the morphism $f_t\colon Y_t \to X_t$ is a finite morphism étale in codimension $1$: the morphism $f_t$ is not necessary quasi-étale because $Y_t$ can be non-normal in general. We need to show that $\beta\colon\mathcal{Y}\to T$ is locally trivial. Let $y\in\mathcal{Y}$ be a point and consider $x=f(y)\in\mathcal{X}$. Since $\alpha\colon\mathcal{X}\to T$ is locally trivial, there exist three open subsets $V_x\subset\mathcal{X}$, $V_{\alpha(x)}\subset T$ and $U_x\subset F=\alpha^{-1}(\alpha(x))$ such that $V_x\cong V_{\alpha(x)}\times U_x$. If we put $V^0_x=(V_x\cap\mathcal{X}_{\operatorname{reg}})$, then $V_x^0=V_{\alpha(x)}\times U_x^0$, where $U_x^0=(U_x\cap\mathcal{X}_{\operatorname{reg}})$. Moreover putting $\widetilde{V}_{x}^{0}:=(f')^{-1}(V_{x}^{0})$ and $\widetilde{U}_{x}^{0}:=(f')^{-1}(U_{x}^{0})$, we have the following commutative diagram: $$\label{eqn:non ce la faccio piu'} \xymatrix{ V_{\alpha(x)}\times\widetilde{U}^0_x=\widetilde{V}^0_x\ar@{->}[r]\ar@{->}[d]^-{f'_x} & R=\underline{\operatorname{Spec}}(j_{x,*}f'_{x,*}\mathcal{O}_{\widetilde{V}^0_x})\ar@{->}[d] \\ V_{\alpha(x)}\times U_x^0=V_x^0\ar@{->}[r]^-{j_x}\ar@{->}[d]^-{p_2^0} & V_x=V_{\alpha(x)}\times U_x\ar@{->}[d]^-{p_2} \\ U_x^0\ar@{->}[r]^-{j_x^0} & U_x }$$ where the horizontal morphisms are the natural inclusions, and each square is cartesian. Since $\widetilde{V}^0_x\cong V_{\alpha(x)}\times\widetilde{U}^0_x$, we have that $$\label{eqn:urca} f'_{x,*}\mathcal{O}_{\widetilde{V}^0_x}\cong(p_2^0)^*(f'_{U_x^0,*}\mathcal{O}_{\widetilde{U}^0_x}),$$ where $f'_{U_x^0}\colon \widetilde{U}^0_x\to U_x^0$ is the restriction of $f'_x$ to the second factor. We will now prove that there exists a sheaf of algebras $G$ on $U_x$ such that $j_{x,*}f'_{x,*}\mathcal{O}_{\widetilde{V}^0_x}\cong p_2^* G$. Since $R=\underline{\operatorname{Spec}}(j_{x,*}f'_{x,*}\mathcal{O}_{\widetilde{V}^0_x})$, this implies that $R=V_{\alpha(x)}\times \widetilde{U}_x$, where $V_{\alpha(x)}=V_{\beta(y)}\subset\beta^{-1}(\beta(y))$ is open. By using the equality ([\[eqn:urca\]](#eqn:urca){reference-type="ref" reference="eqn:urca"}), since the bottom square of the diagram ([\[eqn:non ce la faccio piu\'\]](#eqn:non ce la faccio piu'){reference-type="ref" reference="eqn:non ce la faccio piu'"}) is cartesian we have the following chain of equalities, $$\begin{aligned} j_{x,*}f'_{x,*}\mathcal{O}_{\widetilde{V}^0_x} & \cong j_{x,*}(p_2^0)^*(f'_{U_x^0,*}\mathcal{O}_{\widetilde{U}^0_x}) \\ & \cong p_2^*(j^0_{x,*}f'_{U_x^0,*}\mathcal{O}_{\widetilde{U}^0_x}). \end{aligned}$$ Therefore, putting $G=j^0_{x,*}f'_{U_x^0,*}\mathcal{O}_{\widetilde{U}^0_x}$ concludes the proof that $\beta\colon\mathcal{Y}\to T$ is locally trivial. Finally, to conclude the proof of the claim, let us notice that by the local triviality of the family $\beta$, if we take the relative normalisation $\nu\colon\mathcal{Y}^{\operatorname{nor}}\to \mathcal{Y}$, then the family $\beta^{\operatorname{nor}}=\beta\circ\nu\colon\mathcal{Y}^{\operatorname{nor}}\to T$ remains locally trivial and the composition $\mathcal{Y}^{\operatorname{nor}}\to\mathcal{X}$ is still a finite quasi-étale covering; the same holds also for the fibers. ◻ We wish to remark that singularities are supposed to be terminal in Proposition [Proposition 8](#prop:small when terminal){reference-type="ref" reference="prop:small when terminal"} only in order to deduce that the singular locus has codimension at least $3$: this allows us to apply Siu's theorem about the coherency of the pushforward in the analytic category and get that $j_*f'_*\mathcal{O}_{\mathcal{Y}'}$ is coherent. In the algebraic category the analog of Siu's theorem holds assuming that the singular locus has codimension at least $2$: it follows that in the projective setting the statement of Proposition [Proposition 8](#prop:small when terminal){reference-type="ref" reference="prop:small when terminal"} holds in the general case of canonical singularities. **Proposition 9**. *Let $X$ be a projective irreducible symplectic variety. Let $f\colon\mathcal{X}\to T$ be a locally trivial family of primitive symplectic varieties as in Definition [Definition 4](#def:lt){reference-type="ref" reference="def:lt"} such that $f$ is projective and $T$ is quasi-projective, and let $\bar{t}\in T$ a point such that $\mathcal{X}_{\bar{t}}=X$. Then there exists an analytic open neighborhood $U\subset T$ of $\bar{t}$ such that, for every $t\in U$, the fibre $\mathcal{X}_t$ of $f$ is a projective irreducible symplectic variety.* **Remark 10**. We remark that if $p\colon \mathcal{X}\to T$ is a locally trivial family of symplectic varieties, there is a natural relative stratification $$\mathcal{X}=\mathcal{X}_0\supset \mathcal{X}_1\supset\cdots\supset \mathcal{X}_m,$$ where for every $t\in T$ $$\mathcal{X}_t\supset \mathcal{X}_{1,t}\supset\cdots\supset \mathcal{X}_{m,t}$$ is the stratification in Proposition [Proposition 3](#prop:stratification){reference-type="ref" reference="prop:stratification"} for the variety $\mathcal{X}_t$, and for every $i=0,\cdots,m$, the restriction $p_i\colon\mathcal{X}_i\to T$ is a locally trivial family. By construction, the stratum $X_m$ is smooth and, if $\mathcal{X}_{m,t}$ is irreducible and of strictly positive dimension, then $p_m\colon \mathcal{X}_m\to T$ is a smooth family of irreducible holomorphic symplectic manifolds. ## Locally trivial monodromy operators {#section:pto} We will now define the monodromy group of a primitive symplectic variety, following [@BakkerLehn:PrimitiveSymplecticVarieties]. First of all we recall that if $X$ is a primitive symplectic variety, the torsion free part $\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}}$ of the second integral cohomology group of $X$ is endowed with a nondegenerate bilinear form $q_X$ of signature $(3,b_2(X)-3)$ (see [@BakkerLehn:PrimitiveSymplecticVarieties Section 5.1, Lemma 5.7]), that will be called *Beauville--Bogomolov--Fujiki form* of $X$. The pair $(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}},q_X)$ will be called the *Beauville--Bogomolov--Fujiki lattice* of $X$. In particular, if $X$ is irreducible symplectic, then $\operatorname{H}^{2}(X,\mathbb{Z})$ is torsion free and therefore endowed with the Beauville--Bogomolov--Fujiki form. The Beauville--Bogomolov--Fujiki form is a locally trivial deformation invariant of a primitive symplectic variety (see [@BakkerLehn:PrimitiveSymplecticVarieties Lemma 5.7]). In particular, this implies that if $X_{1}$ and $X_{2}$ are two locally trivial deformation equivalent primitive symplectic varieties, there is an isometry between $\operatorname{H}^{2}(X_{1},\mathbb{Z})_{\operatorname{tf}}$ and $\operatorname{H}^{2}(X_{2},\mathbb{Z})_{\operatorname{tf}}$. We will use the notation $\operatorname{O}(\operatorname{H}^{2}(X_{1},\mathbb{Z})_{\operatorname{tf}},\operatorname{H}^{2}(X_{2},\mathbb{Z})_{\operatorname{tf}})$ for the set of isometries from the Beauville--Bogomolov--Fujiki lattice of $X_1$ to the one of $X_2$; similarly we use the notation $\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$ for the group of isometries from the Beauville--Bogomolov--Fujiki lattice of $X$ to itself. The isometries arising from locally trivial families will be called locally trivial parallel transport operators. More precisely, we recall the following definition. **Definition 11**. Let $X$, $X_1$ and $X_2$ be primitive symplectic varieties. 1. An isometry $g\in\operatorname{O}(\operatorname{H}^{2}(X_{1},\mathbb{Z})_{\operatorname{tf}},\operatorname{H}^{2}(X_{2},\mathbb{Z})_{\operatorname{tf}})$ is a *locally trivial parallel transport operator from $X_{1}$ to $X_{2}$* if there exists a locally trivial family $p\colon\mathcal{X}\to T$ of primitive symplectic varieties and two points $t_1,t_2\in T$, with $\mathcal{X}_{t_i}:=p^{-1}(t_i)=X_i$, such that $g$ is the parallel transport along a path from $t_1$ to $t_2$ in the local system $R^2p_*\mathbb{Z}$. 2. An isometry $g\in\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$ is a *locally trivial monodromy operator* if it is a locally trivial parallel transport operator from $X$ to itself. If $X_1$ and $X_2$ are two primitive symplectic varieties, we will let $$\mathsf{PT}^{2}_{\operatorname{lt}}(X_1,X_2)\subset\operatorname{O}(\operatorname{H}^{2}(X_1,\mathbb{Z})_{\operatorname{tf}},\operatorname{H}^{2}(X_2,\mathbb{Z})_{\operatorname{tf}})$$ be the set of locally trivial parallel transport operators from $X_1$ to $X_2$. If $X_1=X_2=X$, we put $$\operatorname{Mon}^{2}_{\operatorname{lt}}(X):=\operatorname{PT}^{2}_{\operatorname{lt}}(X,X),$$ which is then the subset of $\operatorname{O}(\operatorname{H}^{2}(X,\mathbb{Z})_{\operatorname{tf}})$ given by all locally trivial monodromy operators. **Lemma 12**. *Let $X$ be a primitive symplectic variety. The set $\operatorname{Mon}^2_{\operatorname{lt}}(X)$ of locally trivial monodromy operators is a subgroup of $\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$.* *Proof.* This follows as in the smooth case, see [@Markman:Survey footnote 3]. ◻ The group $\operatorname{Mon}^2_{\operatorname{lt}}(X)$ is called the *locally trivial monodromy group* of $X$ and, by construction, it is a locally trivial deformation invariant. If $X$ is smooth, we will simply write $\operatorname{Mon}^{2}(X)$ for the monodromy group, as all smooth deformations of $X$ are locally trivial in this case. The group $\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$ contains the subgroup $\operatorname{O}^+(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$ of *orientation preserving isometries*. We refer to [@Markman:Survey Section 4] and [@MirandaMorrison] for a general account on orientations, but let us quickly recall the definition we use. If $\widetilde{\mathcal{C}}_X\subset\operatorname{H}^2(X,\mathbb{R})$ is the set of classes $\alpha$ such that $q_X(\alpha)>0$, then by [@Markman:Survey Lemma 4.1] we have $\operatorname{H}^2(\widetilde{\mathcal{C}}_X,\mathbb{Z})=\mathbb{Z}$, and $\operatorname{O}^+(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$ is by definition the subgroup of $\operatorname{O}(\operatorname{H}^{2}(X,\mathbb{Z})_{\operatorname{tf}})$ given by those isometries acting as the identity on $\operatorname{H}^2(\widetilde{\mathcal{C}}_X,\mathbb{Z})$. In general we will refer to a generator of $\operatorname{H}^2(\widetilde{\mathcal{C}}_X,\mathbb{Z})$ as an *orientation*: since for any positive three-dimensional subspace $W$ of $\operatorname{H}^2(X,\mathbb{R})$ the space $W\setminus\{0\}$ is deformation retract of $\widetilde{\mathcal{C}}_X$, it follows that an orientation is nothing else than an orientation on $W$. Among the orientation preserving isometries, we find all the locally trivial monodromy operators, as the following shows. **Lemma 13**. *Let $X$ be a primitive symplectic variety. Then we have an inclusion $\operatorname{Mon}^2_{\operatorname{lt}}(X)\subset\operatorname{O}^+(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$.* *Proof.* The proof follows as in the classical case, and we sketch it here for the reader's convenience. As in the smooth case (see [@NamProj Corollary 8]) if $\omega\in\operatorname{H}^1(\Omega^{[1]}_X)$ is a Kähler class and $\sigma\in\operatorname{H}^0(\Omega^{[2]}_X)$ is the symplectic form, then the three-space $W$ spanned by $\omega$, $\operatorname{Re}(\sigma)$ and $\operatorname{Im}(\sigma)$ is positive definite. Notice that the choice of this basis (in this order) determines an orientation, i.e. a generator $\mathsf{a}$ of $\operatorname{H}^2(W\setminus\{0\},\mathbb{Z})=\operatorname{H}^2(\widetilde{\mathcal{C}}_X,\mathbb{Z})$, which does not change if we replace $\omega$ by another Kähler class $\omega'$ and $\sigma$ by another symplectic form $\sigma'$. Now if $p\colon\mathcal{X}\to T$ is any locally trivial family of primitive symplectic varieties, then $\omega$ and $\sigma$ extend to sections of the respective local systems. This gives rise to a section $t\mapsto\mathsf{a}_t\in\operatorname{H}^2(\mathcal{C}_{X_t},\mathbb{Z})$, where $\mathsf{a}_t$ is the generator determined by $\omega_t$, $\operatorname{Re}(\sigma_t)$ and $\operatorname{Im}(\sigma_t)$. In particular the induced action on $\operatorname{H}^2(\widetilde{\mathcal{C}}_X,\mathbb{Z})$ is the identity and we are done. ◻ Among locally trivial parallel transport operators one finds the pull-back by a birational morphism between projective primitive symplectic varieties. This result is well-known for irreducible holomorphic symplectic manifolds, and follows from Huybrechts' results (see [@Huybrechts:KahlerCone] and [@Markman:Survey Section 3.1]). **Proposition 14** (). *Let $X$ and $Y$ be two projective primitive symplectic varieties and $f\colon X\dashrightarrow Y$ a birational morphism. Suppose that $f$ is defined in codimension $1$ and that $f^*\colon\operatorname{Pic}(Y)\otimes\mathbb{Q}\to\operatorname{Pic}(X)\otimes\mathbb{Q}$ is an isomorphism. Then the pullback $$f^*\colon\operatorname{H}^2(Y,\mathbb{Z})_{\operatorname{tf}}\longrightarrow\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}}$$ is well defined and a locally trivial parallel transport operator.* *Proof.* This statement is implicit in [@BakkerLehn:PrimitiveSymplecticVarieties Theorem 6.16], see [@Markman:Survey Section 3.1] for a rigorous proof. ◻ We conclude this section by recalling the notion of polarised parallel transport operators. Recall that a polarised variety is a pair $(X,H)$ where $X$ is a projective variety and $H$ is a polarisation, i.e. an ample line bundle on $X$. If $X$ is a projective symplectic variety (resp. primitive or irreducible), then the pair $(X,H)$ is a polarised symplectic variety (resp. primitive or irreducible). **Definition 15**. Let $(X,H)$, $(X_1,H_1)$ and $(X_2,H_2)$ be polarised primitive symplectic varieties. 1. An isometry $g\in\operatorname{O}(\operatorname{H}^2(X_1,\mathbb{Z})_{\operatorname{tf}},\operatorname{H}^2(X_2,\mathbb{Z})_{\operatorname{tf}})$ is a *polarised locally trivial parallel transport operator* from $(X_{1},H_{1})$ to $(X_{2},H_{2})$ if it is a parallel transport operator arising from a family $p\colon\mathcal{X}\to T$ as in Definition [Definition 11](#defn:pto){reference-type="ref" reference="defn:pto"} such that there exists a relatively ample line bundle $\mathcal{H}$ on $\mathcal{X}$ such that $\mathcal{H}_{t_i}=H_i$ on $\mathcal{X}_{t_i}=X_i$, for $t_1,t_2\in T$. 2. An isometry $g\in\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})$ is a *polarised locally trivial monodromy operator* if it is a polarised parallel transport operator from $(X,H)$ to itself. As before we denote by $$\mathsf{PT}^2_{\operatorname{lt}}((X_1,H_1),(X_2,H_2))$$ the set of polarised locally trivial parallel transport operators from $(X_1,H_1)$ to $(X_2,H_2)$ and by $$\operatorname{Mon}^2_{\operatorname{lt}}(X,H)=\mathsf{PT}^2_{\operatorname{lt}}((X,H),(X,H))$$ the group of polarised locally trivial monodromy operators of $(X,H)$. Notice that by definition we have $$\mathsf{PT}^2_{\operatorname{lt}}((X_1,H_1),(X_2,H_2))\subset\mathsf{PT}^2_{\operatorname{lt}}(X_1,X_2)$$ and $$\operatorname{Mon}^2_{\operatorname{lt}}(X,H)\subset\operatorname{Mon}^2_{\operatorname{lt}}(X).$$ **Remark 16**. Definition [Definition 15](#defn:ppto){reference-type="ref" reference="defn:ppto"} also appears in literature in a different form. Let $p\colon\mathcal{X}\to T$ be a locally trivial family as in Definition [Definition 11](#defn:pto){reference-type="ref" reference="defn:pto"}: instead of asking for the existence of a relatively ample line bundle $\mathcal{H}$ on $\mathcal{X}$, one can ask for the existence of a flat section $h$ of $R^2p_*\mathbb{Z}$ such that $h_t$ is the class of an ample line bundle on $\mathcal{X}_t$ for every $t\in T$ (see for example [@Markman:Survey Definition 1.1.(4)]). Of course, if $\mathcal{H}$ is a relatively ample line bundle on $\mathcal{X}$, then the section $c_1(\mathcal{H})$ satisfies the last condition. The vice versa is not true in general. Nevertheless, we wish to point out that the parallel transport operators arising from these two definition are in fact the same, providing that the base is at least normal and quasi-projective. Let us start with a locally trivial family $f\colon\mathcal{X}\to T$, where $T$ is normal, irreducible and quasi-projective; suppose that there is a flat section $h$ of $R^2p_*\mathbb{Z}$ such that $h_t$ is the class of an ample line bundle for every $t\in T$. By the Lefschetz hyperplane section Theorem (see for example [@FL:Connectivity Theorem 1.1.(B)]), for a generic complete intersection curve $C\subset T$ the morphism $$\pi_1(C)\longrightarrow\pi_1(T)$$ is surjective: hence every parallel transport operator induced by $f$ between fibers over $C$ is also induced by the restriction of $f$ over $C$. Without loss of generality, we can suppose that $C$ is irreducible and smooth. From now on we work with the restricted family $f\colon\mathcal{X}\to C$ and show the existence of a holomorphic line bundle on its total space inducing the section $h$ over $C$. By applying the functor $f_*$ to the exponential sequence $$0\to\mathbb{Z}\to\mathcal{O}_{\mathcal{X}}\to\mathcal{O}^*_{\mathcal{X}}\to 0$$ we obtain $$0=R^1f_*\mathcal{O}_{\mathcal{X}}\to R^1f_*\mathcal{O}^*_{\mathcal{X}}\to R^2f_*\mathbb{Z}\to R^2f_*\mathcal{O}_{\mathcal{X}},$$ where in the first equality we used that all the fibres are primitive symplectic varieties. Taking global sections we get $$0\to\operatorname{H}^0(R^1f_*\mathcal{O}^*_{\mathcal{X}})\to\operatorname{H}^0(R^2f_*\mathbb{Z})\to\operatorname{H}^0(R^2f_*\mathcal{O}_{\mathcal{X}}).$$ Since the Hodge structure on $\operatorname{H}^2(\mathcal{X}_t,\mathbb{Z})$ is pure for every $t$, the map on the right hand side is the relative projection on the $(2,0)$-part. In particular the section $h\in\operatorname{H}^0(R^2f_*\mathbb{Z})$ is mapped to $0$ and it therefore comes from a section $\tilde{h}\in\operatorname{H}^0(R^1f_*\mathcal{O}^*_{\mathcal{X}})$. From the Leray spectral sequence we get $$\operatorname{H}^1(\mathcal{O}_{\mathcal{X}}^*)\to\operatorname{H}^0(R^1f_*\mathcal{O}^*_{\mathcal{X}})\to\operatorname{H}^2(f_*\mathcal{O}^*_{\mathcal{X}})=\operatorname{H}^2(\mathcal{O}^*_C)=0,$$ where the last vanishing comes from the fact that $C$ is a curve. In particular this means that there exists a holomorphic line bundle $\mathcal{H}$ on $\mathcal{X}$ lifting the section $\tilde{h}$. Finally, since the composition $$\operatorname{H}^1(\mathcal{X},\mathcal{O}_{\mathcal{X}}^*)\to\operatorname{H}^0(R^1f_*\mathcal{O}^*_{\mathcal{X}})\to\operatorname{H}^0(R^2f_*\mathbb{Z})$$ is the morphism mapping a line bundle $\mathcal{L}$ on $\mathcal{X}$ to the section $\{c_1(\mathcal{L}_t)\}_t\in\operatorname{H}^0(R^2f_*\mathbb{Z})$ by construction, it follows that $c_1(\mathcal{H}_t)=h_t$ for every $t\in T$. **Remark 17**. In Remark [Remark 16](#rmk:fino ad hol){reference-type="ref" reference="rmk:fino ad hol"} we constructed a relatively ample holomorphic line bundle $\mathcal{H}$ on the total space $\mathcal{X}$ of the family $f\colon \mathcal{X}\rightarrow C$ of irreducible symplectic varieties over the smooth irreducible curve $C$ lifting the section $h$ of $R^2p_*\mathbb{Z}$. We wish to point out though that if the family is smooth and algebraic, then $\mathcal{H}$ can be chosen algebraic as well. If $\mathcal{X}$ and $C$ are projective this follows from GAGA principle ([@GAGA]), otherwise, as $C$ is quasi-projective, it is affine and there exist smooth and projective compactifications $\overline{\mathcal{X}}$ of $\mathcal{X}$ and $\overline{C}$ of $C$ and a projective morphism $\overline{f}\colon \overline{\mathcal{X}}\rightarrow \overline{C}$. Since for $t_0\in C$ the class $h_{t_0}\in \operatorname{H}^2(X_{t_0},\mathbb{Z})$ is a monodromy invariant class, by the global Invariant Cycle Theorem (see for example [@ICT Theorem 4.24]), there exists a class $\hat{h}\in\operatorname{H}^2(\overline{\mathcal{X}},\mathbb{Q})$ extending $h_{t_0}$. Moreover, as $h_{t_0}\in \operatorname{H}^{1,1}(X_{t_0},\mathbb{Z})$ and $\hat{h}_t=\ell h_t$ (for some $\ell\in\mathbb{Q}$) by construction we may suppose that $\hat{h}$ is of Hodge type $(1,1)$ too. Moreover, by clearing the denominator, we find a class $\hat{h}'\in\operatorname{H}^{1,1}(\overline{\mathcal{X}},\mathbb{Z})$ such that $h'_{t_{0}}=nh_{t_{0}}$ for some $n\in\mathbb{N}$. This implies that there exists an algebraic line bundle $\mathcal{L}$ on the projective variety $\overline{\mathcal{X}}$ such that $c_{1}(\mathcal{L})=\hat{h}'$, and hence the restriction $c_{1}(\mathcal{L})_{t_{0}}$ of $c_{1}(\mathcal{L})$ to the fiber $\mathcal{X}_{t_{0}}$ is an integral multiple of $h_{t_0}$. Now let $\mathcal{L}_{C}$ be the restriction of $\mathcal{L}$ to $\mathcal{X}$. Since the fibers of $f$ are simply connected for every $t\in C$ there is an isomorphism $\mathcal{L}_t\simeq \mathcal{H}^{\otimes n}_t$ between the fibres of $\mathcal{L}_C$ and $\mathcal{H}^{\otimes n}$. It follows that there exists a holomorphic line bundle $\mathcal{N}$ on $C$ and an isomorphism of holomorphic line bundles $$\mathcal{H}^{\otimes m} \simeq \mathcal{L}_{C}\otimes f^*\mathcal{N}.$$ By [@For Theorem 30.4], every holomorphic line bundle on an affine curve is trivial. As a consequence the holomorphic line bundle $\mathcal{H}^{\otimes m}$ is algebraic, and this implies that $\mathcal{H}$ is algebraic as well. **Remark 18**. In the notation of Definition [Definition 15](#defn:ppto){reference-type="ref" reference="defn:ppto"}, the family $\{\mathcal{H}_t\}_{t\in T}$ is a continuous family of ample line bundles on the fibres of $p\colon\mathcal{X}\to T$. In particular any parallel transport operator must be constant on it, i.e. if $g\in\mathsf{PT}^2_{\operatorname{lt}}((X_1,H_1),(X_2,H_2))$ and $h_i=c_1(H_i)$, then $g(h_1)=h_2$. In particular it follows that $$\operatorname{Mon}^2_{\operatorname{lt}}(X,H)\subset\operatorname{O}^+(\operatorname{H}^2(X,\mathbb{Z})_{\operatorname{tf}})_h,$$ where $h=c_1(H)$ and the latter is the group of orientation preserving isometries $g$ such that $g(h)=h$. More precisely $\operatorname{Mon}^2_{\operatorname{lt}}(X,H)\subset\operatorname{Mon}^2_{\operatorname{lt}}(X)_h$, where again the last group is the subgroup of monodromy operators fixing the polarisation. Arguing as in the proof of [@Markman:Survey Corollary 7.4], one can further prove that $$\operatorname{Mon}^2_{\operatorname{lt}}(X,H)=\operatorname{Mon}^2_{\operatorname{lt}}(X)_h.$$ ## Moduli spaces of sheaves on K3 surfaces {#section:moduli spaces} We conclude this first section by recalling the basic facts we will need about moduli spaces of sheaves on K3 surfaces, and refer the reader to [@PR:SingularVarieties] and [@PR:v; @perp] for a more detailed exposition about this. Let $S$ be a projective K3 surface. We denote by $\widetilde{\operatorname{H}}(S,\mathbb{Z})$ the *Mukai lattice* of $S$: as a $\mathbb{Z}$-module it is $\operatorname{H}^{\operatorname{even}}(S,\mathbb{Z})$, and the (nondegenerate) integral quadratic form on it is given by $$(r,\xi,a)^2=\xi^2-2ra,$$ where $r\in\operatorname{H}^{0}(S,\mathbb{Z})$, $\xi\in\operatorname{H}^{2}(S,\mathbb{Z})$ and $a\in\operatorname{H}^{4}(S,\mathbb{Z})$. The Mukai lattice of $S$ inherits a pure weight two Hodge structure from the one on $S$ by declaring $\widetilde{\operatorname{H}}^{2,0}(S):=\operatorname{H}^{2,0}(S)$. An element $v\in\widetilde{\operatorname{H}}(S,\mathbb{Z})$ is called a *Mukai vector* if $v=(r,\xi,a)$ is such that $r\geq 0$ and $\xi\in \operatorname{NS}(S)$, and in the case when $r=0$ we have that either $\xi$ is the first Chern class of a strictly effective divisor, or $\xi=0$ and $a>0$. If $v$ is a Mukai vector on $S$, then there is a coherent sheaf $\mathcal{F}$ on $S$ such that $\operatorname{ch}(\mathcal{F})\cdot\sqrt{td(S)}=v$: we will then say that $v$ is the *Mukai vector* of $\mathcal{F}$. We notice that a Mukai vector $v$ is of type $(1,1)$ with respect to the Hodge decomposition of the Mukai lattice. **Definition 19** (). Given a Mukai vector $v\in\widetilde{\operatorname{H}}(S,\mathbb{Z})$ of the form $v=(r,\xi,a)$, an ample line bundle $H$ on $S$ is *v-generic* if it verifies one of the following two conditions: 1. If $r>0$, then for every $\mu_{H}$-semistable sheaf $E$ such that $v(E)=v$ and every $0\neq F\subseteq E$, we have that if $\mu_{H}(E)=\mu_{H}(F)$ then $c_{1}(F)/\operatorname{rk}(F)=c_{1}(E)/\operatorname{rk}(E)$. 2. If $r=0$, then for every $H$-semistable sheaf $E$ such that $v(E)=v$ and every $0\neq F\subseteq E$, if $\chi(E)/(c_{1}(E)\cdot H)=\chi(F)/(c_{1}(F)\cdot H)$ then $v(F)\in\mathbb{Q}v$. Given a Mukai vector $v\in\widetilde{\operatorname{H}}(S,\mathbb{Z})$ and a $v$-generic ample line bundle $H$ on $S$, we denote by $M_v(S,H)$ (or simply $M_v$ if the pair $(S,H)$ is clear from the context) the moduli space of Gieseker $H$-semistable sheaves $F$ on $S$ such that $v(F)=v$. **Remark 20**. Let $S$ be a projective K3 surface with Picard rank is at least 2, and let $v=(r,\xi,a)$ be a Mukai vector on $S$. If $r\neq 0$, or if $r=0$ and $a\neq 0$, the ample cone of $S$ has a decomposition in $v$-walls and $v$-chambers (see [@PR:SingularVarieties Section 2.1.1]). As the $v$-chambers are subcones of the ample cone, it follows that if $H'=tH$ with $t\in\mathbb{Z}$, then $H'$ belongs to the same $v$-chamber as $H$. By [@PR:SingularVarieties Lemma 2.9] the primitive polarisations lying in a $v$-chamber are all $v$-generic according to Definition [Definition 19](#defn:v-generic){reference-type="ref" reference="defn:v-generic"}. Moreover, by [@PR:SingularVarieties Lemma 2.9] if $H_1$ and $H_2$ are two $v$-generic polarisations (according to Definition [Definition 19](#defn:v-generic){reference-type="ref" reference="defn:v-generic"}) that belong to the closure of the same $v$-chamber, then there is an identification of the moduli spaces $M_v(S,H_1)=M_v(S,H_2)$, meaning that a coherent sheaf $F$ of Mukai vector $v$ on $S$ is $H_{1}$-(semi)stable if and only if it is $H_{2}$-(semi)stable. From now on, we will always suppose that $v^{2}>0$: under this hypothesis, the $v$-genericity of $H$ implies that $M_v(S,H)\neq\emptyset$, and that it is an irreducible normal projective variety of dimension $v^2+2$ ([@KaledinLehnSorger Theorem 4.1]), whose smooth locus admits a holomorphic symplectic form ([@Mukai]). For simplicity, we make now the following definition. **Definition 21**. Given two strictly positive integers $m,k\in\mathbb{N}$, a triple $(S,v,H)$ will be called $(m,k)$-*triple* if $S$ is a projective K3 surface, $v$ is a Mukai vector on $S$ such that if $v=mw$, with $w$ primitive, then $w^{2}=2k$, and $H$ is a $v$-generic polarisation. **Remark 22**. If $(S,v,H)$ is an $(m,k)$-triple and $v=mw$, then $(S,w,H)$ is a $(1,k)$-triple. The reason for this is that if $H$ is $v$-generic, then it is $w$-generic. Indeed, if $w=(r,\xi,a)$ with $r>0$, $E$ is a $\mu_{H}$-semistable sheaf with Mukai vector $w$ and $F\subseteq E$ is a proper coherent subsheaf with $\mu_{H}(F)=\mu_{H}(E)$, then $E^{\oplus m}$ is a $\mu_{H}$-semistable sheaf with Mukai vector $v$ and $F^{\oplus m}$ is a proper subsheaf of $E^{\oplus m}$ such that $\mu_{H}(F^{\oplus m})=\mu_{H}(E^{\oplus m})$. But since $H$ is $v$-generic it follows that $c_{1}(E^{\oplus m})/\operatorname{rk}(E^{\oplus m})=c_{1}(F^{\oplus m})/\operatorname{rk}(F^{\oplus m})$, and hence $c_{1}(E)/\operatorname{rk}(E)=c_{1}(F)/\operatorname{rk}(F)$. A similar proof works for $r=0$. For our purposes it is useful to introduce an equivalence relation that identifies $(m,k)$-triples whose associated moduli spaces represent the same sheaves. **Definition 23**. Two $(m,k)$-triples $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$ are called *congruent* if $S_{1}=S_{2}=S$, $v_{1}=v_{2}=v$ and a coherent sheaf $F$ of Mukai vector $v$ on $S$ is $H_{1}$-(semi)stable if and only if it is $H_{2}$-(semi)stable. In particular there is an equality $M_v(S,H_1)=M_v(S,H_2)$ and we denote by $$\chi_{H_1,H_2}\colon M_v(S,H_1)\longrightarrow M_v(S,H_2),\qquad F\mapsto F$$ the identity morphism. **Remark 24**. If $H$ and $H'$ are two $v$-generic polarisations that lie in the closure of the same $v$-chamber, then $(S,v,H)$ and $(S,v,H')$ are congruent by Remark [Remark 20](#rmk:wall and chamber){reference-type="ref" reference="rmk:wall and chamber"}. The following result is the starting point of our paper. **Theorem 25** (). *Let $(S,v,H)$ be an $(m,k)$-triple. Then $M_v(S,H)$ is an irreducible symplectic variety of dimension $2km^2+2$.* **Remark 26**. In the setting of Theorem [Theorem 25](#thm:pr){reference-type="ref" reference="thm:pr"}, one may prove that the smallest stratum of the stratification of the singularities of $M_{v}(S,H)$ is isomorphic to $M_{w}(S,H)$. In particular we get a natural closed embedding $$i_{w,m}\colon M_w(S,H)\longrightarrow M_v(S,H).$$ This has been done in the proof of [@KaledinLehnSorger Theorem 4.4], but let us recall the main idea. The singular locus of $M_{v}(S,H)$ coincides with the locus of strictly semi-stable sheaves; moreover, any strictly semistable sheaf $F$ is S-equivalent to a sheaf of the form $F_1\oplus F_2$, where $F_i\in M_{m_iw}(S,H)$, with $m_{1}+m_{2}=m$. In particular it belongs to the image of the morphism $$M_{m_1w}(S,H)\times M_{m_2w}(S,H)\to M_v(S,H), \quad (F_1,F_2)\mapsto [F_1\oplus F_2]$$ which is an irreducible component of the strictly semistable locus. The intersection of all these components is then the locus $$Y=\{ E^{\oplus m}\in M_v(S,H)\mid E\in M_w(S,H)\}\cong M_w(S,H).$$ As we recalled in the previous section, since $M_v(S,H)$ is an irreducible symplectic variety, the cohomology group $\operatorname{H}^2(M_v(S,H),\mathbb{Z})$ is a free $\mathbb{Z}$-module with both a pure weight two Hodge structure and a lattice structure. These structures have been made explicit in [@PR:v; @perp]. **Theorem 27** (). *Let $(S,v,H)$ be an $(m,k)$-triple. Then there exists a Hodge isometry $$\lambda_{(S,v,H)}\colon v^\perp\longrightarrow\operatorname{H}^2(M_v(S,H),\mathbb{Z}),$$ where $v^\perp$ inherits the Hodge and lattice structures from those of $\widetilde{\operatorname{H}}(S,\mathbb{Z})$, and $\operatorname{H}^2(M_v(S,H),\mathbb{Z})$ is endowed with the Beauville--Bogomolov--Fujiki form.* It is implicit in [@PR:v; @perp] that the isometries $\lambda_{(S,v,H)}$ behave well under deformations of moduli spaces induced by deformations of K3 surfaces. We will expand this comment more precisely in Remark [Remark 35](#rmk:lambda in famiglie){reference-type="ref" reference="rmk:lambda in famiglie"}, after we will have carefully defined deformations of $(m,k)$-triples. For future reference, let us notice that the isometry $\lambda_{(S,v,H)}$ induces an isomorphism between the orthogonal groups by conjugation, $$\begin{aligned} \label{eqn:lambda sharp} \lambda_{(S,v,H)}^{\sharp}\colon \operatorname{O}(v^\perp) & \longrightarrow \operatorname{O}(\operatorname{H}^{2}(M_{v}(S,H),\mathbb{Z})) \\ g & \longmapsto \lambda_{(S,v,H)}\circ g\circ(\lambda_{(S,v,H)})^{-1}. \nonumber\end{aligned}$$ By Remark [Remark 26](#rmk:strat M){reference-type="ref" reference="rmk:strat M"} there is a closed embedding $$i_{w,m}\colon M_{w}(S,H)\longrightarrow M_{v}(S,H),$$ and we get a morphism $$i_{w,m}^*\colon\operatorname{H}^2(M_{v}(S,H),\mathbb{Z})\longrightarrow\operatorname{H}^2(M_w(S,H),\mathbb{Z}).$$ Thanks to this morphism, we may now describe the relation between $\lambda_{(S,w,H)}$ and $\lambda_{(S,v,H)}$. **Proposition 28**. *Let $(S,v,H)$ be an $(m,k)$-triple and write $v=mw$. Then $i^{*}_{w,m}\circ\lambda_{(S,v,H)}=m\lambda_{(S,w,H)}$.* *Proof.* In this proof, for every $p>0$ we use the shortened notation $M_{pw}$ for the moduli space $M_{pw}(S,H)$ and $\lambda_{pw}$ for the morphism $\lambda_{(S,pw,H)}$. By [@PR:v; @perp Section 4.2], for every $p>0$, we have a morphism $$g_{p}\colon M_{pw}\times M_{w}\longrightarrow M_{(p+1)w},\,\,\,\,\,\,\,\,g_{p}(F,G):=F\oplus G.$$We let $f_{2}:=g_{1}\colon M_{w}\times M_{w}\to M_{2w}$, and for every $p\geq 3$ we let $$f_{p}:=g_{p-1}\circ(f_{p-1}\times \operatorname{id}_{M_{w}})\colon M_{w}^{p}\longrightarrow M_{pw},$$so that $f_{p}(F_{1},\cdots,F_{p}):=F_{1}\oplus\cdots\oplus F_{p}$. In particular, for $p=m$ we get a morphism $f_{m}\colon M_{w}^{m}\longrightarrow M_{v}$ such that $f_{m}(F_{1},\cdots,F_{m}):=F_{1}\oplus\cdots\oplus F_{m}$. We claim that for every $p>0$ the following diagram $$\label{eq:commperm} \begin{CD} (pw)^{\perp}=w^{\perp} @>{\lambda_{pw}}>> \operatorname{H}^{2}(M_{pw},\mathbb{Z})\\ @V{(\lambda_{w},\cdots,\lambda_{w})}VV @VV{f_{p}^{*}}V\\ \oplus_{i=1}^{p}\operatorname{H}^{2}(M_{w},\mathbb{Z}) @>>{\sum_{i=1}^{p}\pi_{i,p}^{*}}> \operatorname{H}^{2}(M_{w}^{p},\mathbb{Z}) \end{CD}$$ is commutative, where $\pi_{i,p}\colon M_{w}^{p}\to M_{w}$ is the projection onto the $i$-th factor. We prove it by induction on $p$. First of all we remark that by [@PR:v; @perp Proposition 4.11(2)] for every $d>0$ we have a commutative diagram $$\label{eq:commgp} \begin{CD} ((d+1)w)^{\perp}=w^{\perp}=(dw)^{\perp} @>{\lambda_{(d+1)w}}>> \operatorname{H}^{2}(M_{(d+1)w},\mathbb{Z})\\ @V{(\lambda_{dw},\lambda_{w})}VV @VV{g_{d}^{*}}V\\ \operatorname{H}^{2}(M_{dw},\mathbb{Z})\oplus \operatorname{H}^{2}(M_{w},\mathbb{Z}) @>>{q_{1,d}^{*}+q_{2,d}^{*}}> \operatorname{H}^{2}(M_{dw}\times M_{w},\mathbb{Z}) \end{CD}$$ where $q_{1,d}\colon M_{dw}\times M_{w}\to M_{dw}$ and $q_{2,d}\colon M_{dw}\times M_{w}\to M_{w}$ are the two projections. For $d=1$ we then get a commutative diagram $$\begin{CD} w^{\perp} @>{\lambda_{2w}}>> \operatorname{H}^{2}(M_{2w},\mathbb{Z})\\ @V{(\lambda_{w},\lambda_{w})}VV @VV{f_{2}^{*}}V\\ \operatorname{H}^{2}(M_{w},\mathbb{Z})\oplus \operatorname{H}^{2}(M_{w},\mathbb{Z}) @>>{\pi_{1,1}^{*}+\pi_{2,1}^{*}}> \operatorname{H}^{2}(M_{w}\times M_{w},\mathbb{Z}) \end{CD}$$ that proves the commutativity of diagram ([\[eq:commperm\]](#eq:commperm){reference-type="ref" reference="eq:commperm"}) for $p=2$, i.e. for the initial step of the induction. Let us now consider any $p\geq 2$, and notice that we have a commutative diagram $$\label{eq:comm2} \begin{CD} \operatorname{H}^{2}(M_{(p-1)w},\mathbb{Z})\oplus \operatorname{H}^{2}(M_{w},\mathbb{Z}) @>{q^{*}_{1,p-1}+q^{*}_{2,p-1}}>> \operatorname{H}^{2}(M_{(p-1)w,\mathbb{Z}}\times M_{w},\mathbb{Z})\\ @V{f_{p-1}^{*}\times \operatorname{id}_{\operatorname{H}^{2}(M_{w},\mathbb{Z})}}VV @VV{(f_{p-1}\times \operatorname{id}_{M_{w}})^{*}}V\\ \operatorname{H}^{2}(M_{w}^{p-1},\mathbb{Z})\oplus \operatorname{H}^{2}(M_{w},\mathbb{Z}) @>>{\operatorname{pr}_{1,p}^{*}+\operatorname{pr}_{2,p}^{*}}> \operatorname{H}^{2}(M^{p}_{w},\mathbb{Z}) \end{CD}$$ where $\operatorname{pr}_{1,p}\colon M_{w}^{p}=M_{w}^{p-1}\times M_{w}\to M_{w}^{p-1}$ and $\operatorname{pr}_{2,p}\colon M_{w}^{p}=M_{w}^{p-1}\times M_{w}\to M_{w}$ are the two projections. Putting diagram ([\[eq:commgp\]](#eq:commgp){reference-type="ref" reference="eq:commgp"}) (for $d=p-1$) and diagram ([\[eq:comm2\]](#eq:comm2){reference-type="ref" reference="eq:comm2"}) together, we get a commutative diagram $$\label{eq:comm3} \begin{CD} (pw)^{\perp}=w^{\perp} @>{\lambda_{pw}}>> \operatorname{H}^{2}(M_{pw},\mathbb{Z})\\ @V{(f_{p-1}^{*}\circ\lambda_{(p-1)w},\lambda_{w})}VV @VV{f_{p}^{*}}V\\ \operatorname{H}^{2}(M_{w}^{p-1},\mathbb{Z})\oplus \operatorname{H}^{2}(M_{w},\mathbb{Z}) @>>{\operatorname{pr}_{1,p}^{*}+\operatorname{pr}_{2,p}^{*}}> \operatorname{H}^{2}(M^{p}_{w},\mathbb{Z}) \end{CD}$$ By induction, the commutativity of diagram ([\[eq:commperm\]](#eq:commperm){reference-type="ref" reference="eq:commperm"}) for $p-1$ reads as $$f_{p-1}^{*}\circ\lambda_{(p-1)w}=\bigg(\sum_{i=1}^{p-1}\pi_{i,p-1}^{*}\bigg)\circ(\lambda_{w},\cdots,\lambda_{w}),$$so we get a commutative diagram $$\label{eq:comm4} \begin{CD} (pw)^{\perp}=w^{\perp} @>{\lambda_{pw}}>> \operatorname{H}^{2}(M_{pw},\mathbb{Z})\\ @V{(\lambda_{w},\cdots,\lambda_{w})}VV @VV{f_{p}^{*}}V\\ \oplus_{i=1}^{p}\operatorname{H}^{2}(M_{w},\mathbb{Z}) @>>{(\operatorname{pr}_{1,p}^{*}+\operatorname{pr}_{2,p}^{*})\circ((\sum_{i=1}^{p-1}\pi_{i,p-1}^{*})\times \operatorname{id})}> \operatorname{H}^{2}(M^{p}_{w},\mathbb{Z}) \end{CD}$$ Notice that $$(\operatorname{pr}_{1,p}^{*}+\operatorname{pr}_{2,p}^{*})\circ\bigg(\bigg(\sum_{i=1}^{p-1}\pi_{i,p-1}^{*}\bigg)\times \operatorname{id}_{\operatorname{H}^{2}(M_{w})}\bigg)=\sum_{i=1}^{p}\pi_{i,p}^{*},$$hence diagram ([\[eq:commperm\]](#eq:commperm){reference-type="ref" reference="eq:commperm"}) is commutative for $p$, proving the claim. In particular, diagram ([\[eq:commperm\]](#eq:commperm){reference-type="ref" reference="eq:commperm"}) is commutative for $p=m$. As a consequence, for every $a\in v^{\perp}=w^{\perp}$ we have $$f_{m}^{*}(\lambda_{v}(a))=\sum_{i=1}^{m}\pi_{i,m}^{*}(\lambda_{w}(a)).$$ Let us now consider $\delta\colon M_{w}\to M_{w}^{m}$ the diagonal morphism, mapping $[F]$ to $([F],\cdots,[F])$. We then have $i_{w,m}:=f_{m}\circ\delta\colon M_{w}\to M_{v}$. Now, let $\alpha\in \operatorname{H}^{2}(M_{v},\mathbb{Z})$, so that there is a unique $a\in v^{\perp}$ such that $\alpha=\lambda_{v}(a)$. Then we have $$i_{w,m}^{*}(\alpha)=\delta^{*}f_{m}^{*}(\lambda_{v}(a))=\delta^{*}\bigg(\sum_{i=1}^{m}\pi_{i,m}^{*}(\lambda_{w}(a))\bigg)=\sum_{i=1}^{m}\lambda_{w}(a)=m\lambda_{w}(a).$$ This concludes the proof. ◻ In the following corollary we use the notation $q_v$ (resp. $q_w$) for the Beauville--Bogomolov--Fujiki form on $\operatorname{H}^2(M_v(S,H),\mathbb{Z})$ (resp. $\operatorname{H}^2(M_w(S,H),\mathbb{Z})$). **Corollary 29**. *Let $(S,v,H)$ be an $(m,k)$-triple and write $v=mw$. Then the restriction morphism $$i_{w,m}^*\colon\operatorname{H}^2(M_{v}(S,H),\mathbb{Z})\longrightarrow\operatorname{H}^2(M_w(S,H),\mathbb{Z})$$ is a similitude of lattices, i.e., more explicitly, for every $\alpha,\beta\in \operatorname{H}^{2}(M_{v}(S,H),\mathbb{Z})$ we have $$q_w(i^*_{w,m}(\alpha),i^*_{w,m}(\beta))=m^{2}\, q_v(\alpha,\beta).$$* Proposition [Proposition 28](#prop:i come lambda){reference-type="ref" reference="prop:i come lambda"} and Corollary [Corollary 29](#cor:w and mw){reference-type="ref" reference="cor:w and mw"} allow us to get an isomorphism between the ortoghonal groups $\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))$ and $\operatorname{O}(\operatorname{H}^2(M_w(S,H),\mathbb{Z}))$. To this purpose let us denote by $$i_{w,m,\mathbb{Q}}^*\colon\operatorname{H}^2(M_v(S,H),\mathbb{Q}) \longrightarrow\operatorname{H}^2(M_w(S,H),\mathbb{Q})$$ the $\mathbb{Q}$-linear extension of $i^*_{w,m}$ and by $$\begin{aligned} i_{w,m,\mathbb{Q}}^{\sharp}\colon\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Q})) & \longrightarrow\operatorname{O}(\operatorname{H}^2(M_w(S,H),\mathbb{Q})) \\ g & \longmapsto i^*_{w,m,\mathbb{Q}}\circ g \circ (i^*_{w,m,\mathbb{Q}})^{-1}. \end{aligned}$$ the induced isomorphism. **Lemma 30**. *Let $(S,v,H)$ be an $(m,k)$-triple and write $v=mw$. Then:* 1. *$i_{w,m,\mathbb{Q}}^{\sharp}$ sends integral isometries to integral isometries bijectively, i.e. it restricts to an isomorphism $$i_{w,m}^{\sharp}\colon\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))\longrightarrow\operatorname{O}(\operatorname{H}^2(M_w(S,H),\mathbb{Z}));$$* 2. *using the equality $v^\perp=w^\perp$, we have that $$i_{w,m}^\sharp =\lambda_{(S,w,H)}^\sharp\circ (\lambda_{(S,v,H)}^\sharp)^{-1}\colon \operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))\longrightarrow \operatorname{O}(\operatorname{H}^2(M_w(S,H),\mathbb{Z})),$$ where $\lambda_{(S,v,H)}^\sharp$ and $\lambda_{(S,w,H)}^\sharp$ are the isomorphisms defined in the formula ([\[eqn:lambda sharp\]](#eqn:lambda sharp){reference-type="ref" reference="eqn:lambda sharp"}).* *Proof.* For any $(l,k)$-triple $(S,u,H)$, let $$\lambda_{(S,u,H),\mathbb{Q}}\colon u^\perp\otimes\mathbb{Q}\to \operatorname{H}^2(M_u(S,H),\mathbb{Q})$$ be the $\mathbb{Q}$-linear extension of $\lambda_{(S,u,H)}$ and let $$\lambda_{(S,u,H),\mathbb{Q}}^\sharp\colon \operatorname{O}(v^\perp\otimes\mathbb{Q})\longrightarrow\operatorname{O}(\operatorname{H}^2(M_u(S,H),\mathbb{Q}))$$ be the group ismorphism sending a $\mathbb{Q}$-linear isometry $g\in \operatorname{O}(v^\perp\otimes\mathbb{Q})$ to $\lambda_{(S,u,H),\mathbb{Q}}\circ g\circ (\lambda_{(S,u,H),\mathbb{Q}})^{-1}$ so that $\lambda_{(S,u,H)}^{ \sharp}$ is the restriction of $\lambda_{(S,v,H),\mathbb{Q}}^\sharp$ to $\operatorname{O}(v^\perp)$. We claim that $$i_{w,m,\mathbb{Q}}^\sharp =\lambda_{(S,w,H),\mathbb{Q}}\circ (\lambda_{(S,v,H),\mathbb{Q}}^\sharp)^{-1}%\colon\Or(v^\perp\otimes\QQ)\to\Or(w^\perp\otimes\QQ).$$ Since $v^\perp=w^\perp$ and the isomorphism $\lambda_{(S,v,H)}^\sharp$ and $\lambda_{(S,w,H)}^\sharp$ are the restrictions of $\lambda_{(S,v,H),\mathbb{Q}}^\sharp$ and $\lambda_{(S,w,H),\mathbb{Q}}^\sharp$, our claim implies that $i_{w,m,\mathbb{Q}}^{\sharp}$ sends $\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))$ onto $\operatorname{O}(\operatorname{H}^2(M_w(S,H),\mathbb{Z}))$ and its restriction $$i_{w,m}^{\sharp}\colon\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))\longrightarrow\operatorname{O}(\operatorname{H}^2(M_w(S,H),\mathbb{Z}))$$ is an isomorphism and satisfies $$i_{w,m}^{\sharp}=\lambda_{(S,w,H)}^\sharp\circ(\lambda_{(S,v,H)}^\sharp)^{-1}.$$ Let us then prove the claim. By Proposition [Proposition 28](#prop:i come lambda){reference-type="ref" reference="prop:i come lambda"} we know that $i_{w,m,\mathbb{Q}}^*=m\lambda_{(S,w,H),\mathbb{Q}}\circ\lambda_{(S,v,H),\mathbb{Q}}^{-1}$; it follows that $(i_{w,m,\mathbb{Q}}^*)^{-1}=\frac{1}{m}\lambda_{(S,v,H),\mathbb{Q}}\circ\lambda_{(S,w,H),\mathbb{Q}}^{-1}$. Therefore, by definition, we have that for every $g\in\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Q}))$ $$\begin{aligned} i_{w,m,\mathbb{Q}}^\sharp(g) & = i^*_{w,m,\mathbb{Q}}\circ g \circ (i^*_{w,m,\mathbb{Q}})^{-1} \\ & =(m\lambda_{(S,w,H),\mathbb{Q}}(\circ\lambda_{(S,v,H),\mathbb{Q}}^{-1})\circ g \circ (\frac{1}{m}\lambda_{(S,v,H),\mathbb{Q}}\circ\lambda_{(S,w,H),\mathbb{Q}}^{-1}) \\ & = \lambda_{(S,w,H),\mathbb{Q}}\circ(\lambda_{(S,v,H),\mathbb{Q}}^{-1}\circ g \circ \lambda_{(S,v,H),\mathbb{Q}})\circ\lambda_{(S,w,H),\mathbb{Q}}^{-1} \\ %& =(\lambda_{(S,w,H),\QQ}\circ\lambda_{(S,v,H),\QQ}^{-1})^\sharp(g) \\ & = \lambda_{(S,w,H),\mathbb{Q}}^\sharp\circ(\lambda_{(S,v,H),\mathbb{Q}}^\sharp)^{-1}(g), \end{aligned}$$ where in the third equality we used that $g$ is $\mathbb{Q}$-linear. The claim follows and the proof is concluded. ◻ For later use, let us extend the last results to the case in which we have two $(m,k)$-triples $(S_1,v_1,H_1)$ and $(S_2,v_2,H_2)$. As usual, let us write $v_i=mw_i$, so that $(S_1,w_1,H_1)$ and $(S_2,w_2,H_2)$ are $(1,k)$-triples. For sake of simplicity, in the following we use the notation $M_{v_1}$ for $M_{v_1}(S_1,H_1)$ and $M_{v_2}$ for $M_{v_2}(S_2,H_2)$; similarly we write $M_{w_1}$ for $M_{w_1}(S_1,H_1)$ and $M_{w_2}$ for $M_{w_2}(S_2,H_2)$. Let us consider the following bijective map of sets $$\begin{aligned} %\label{eqn:i sharpgen}\[\] i^{\sharp}_{w_1,w_2,m,\mathbb{Q}}\colon \operatorname{O}(\operatorname{H}^{2} (M_{v_1},\mathbb{Q}),\operatorname{H}^{2}(M_{v_2},\mathbb{Q})) & \longrightarrow \operatorname{O}(\operatorname{H}^{2}(M_{w_1},\mathbb{Q}),\operatorname{H}^{2}(M_{w_2},\mathbb{Q})) \\ g & \longmapsto i^{*}_{w_2,m,\mathbb{Q}}\circ g\circ(i^{*}_{w_1,m,\mathbb{Q}})^{-1}. \nonumber \end{aligned}$$ **Lemma 31**. *The bijection $i_{w_1,w_2,m,\mathbb{Q}}^{\sharp}$ sends integral isometries to integral isometries bijectively, i.e. it restricts to a bijection $$i_{w_1,w_2,,m}^{\sharp}\colon \operatorname{O}(\operatorname{H}^{2} (M_{v_1},\mathbb{Z}),\operatorname{H}^{2}(M_{v_2},\mathbb{Z})) \longrightarrow \operatorname{O}(\operatorname{H}^{2}(M_{w_1},\mathbb{Z}),\operatorname{H}^{2}(M_{w_2},\mathbb{Z}))$$* *More esplicity, we have $$i^{\sharp}_{w_1,w_2,m}(g)=(\lambda_{(S_2,w_2,H_2)}\circ\lambda_{(S_2,v_2,H_2)})\circ g\circ(\lambda_{(S_1,w_1,H_1)}\circ\lambda_{(S_1,v_1,H_1)})^{-1}$$ for every $g\in\operatorname{O}(\operatorname{H}^{2} (M_{v},\mathbb{Z}),\operatorname{H}^{2}(M_{v'},\mathbb{Z}))$.* *Proof.* Let $g\in\operatorname{O}(\operatorname{H}^{2}(M_{v},\mathbb{Z}),\operatorname{H}^{2}(M_{v'},\mathbb{Z}))$ be an isometry and $g_\mathbb{Q}$ its rational extension. As in the proof of Lemma [Lemma 30](#lemma:i sharp){reference-type="ref" reference="lemma:i sharp"}, by using Proposition [Proposition 28](#prop:i come lambda){reference-type="ref" reference="prop:i come lambda"} and the $\mathbb{Q}$-linearity of $g_\mathbb{Q}$, we can see that $$i^{\sharp}_{w_1,w_2,m,\mathbb{Q}}(g_\mathbb{Q})=(\lambda_{(S_2,w_2,H_2),\mathbb{Q}}\circ\lambda_{(S_2,v_2,H_2),\mathbb{Q}})\circ g_\mathbb{Q}\circ(\lambda_{(S_1,w_1,H_1),\mathbb{Q}}\circ\lambda_{(S_1,v_1,H_1),\mathbb{Q}})^{-1}.$$ Since all the isometries in the right hand side of the equality are rational extensions of integral isometries, we get that also $i^{\sharp}_{w_1,w_2,m,\mathbb{Q}}(g_\mathbb{Q})$ is a rational extension of an integral isometry. By construction this isometry is $$i^{\sharp}_{w_1,w_2,m}(g):=(\lambda_{(S_2,w_2,H_2)}\circ\lambda_{(S_2,v_2,H_2)})\circ g\circ(\lambda_{(S_1,w_1,H_1)}\circ\lambda_{(S_1,v_1,H_1)})^{-1},$$ from which the claim follows. ◻ # A groupoid representation {#section:groupoid} Aim of this section is to define a groupoid $\mathcal{G}^{m,k}$ of $(m,k)$-triples and a $\mathcal{G}^{m,k}$-representation with values in a groupoid of free $\mathbb{Z}$-modules. The groupoid $\mathcal{G}^{m,k}$ will be defined starting from two groupoids, $\mathcal{G}^{m,k}_{\operatorname{def}}$ and $\mathcal{G}^{m,k}_{\operatorname{FM}}$, whose constructions will be explained in Sections [2.2](#section:families of K3){reference-type="ref" reference="section:families of K3"} and [2.3](#section:FM){reference-type="ref" reference="section:FM"}, respectively. These two groupoids will have the same objects; moreover, the morphisms in $\mathcal{G}^{m,k}_{\operatorname{def}}$ come from deformations of $(m,k)$-triples, while the morphisms in $\mathcal{G}^{m,k}_{\operatorname{FM}}$ come from Fourier-Mukai transforms. We start by quickly recalling the main definitions and notation about groupoids we will use. ## Groupoids {#section:groupoids} We refer to the lecture notes [@Higgins] for the definitions and constructions about groupoids. First of all a groupoid $\mathcal{G}$ is a (small) category whose morphisms are all isomorphisms, i.e. for any two objects $x,y\in\mathcal{G}$, any $f\in\operatorname{Hom}_{\mathcal{G}}(x,y)$ is an isomorphism. If $\mathcal{G}$ is a groupoid and $x\in\mathcal{G}$ is an object, we let $$\operatorname{Aut}_{\mathcal{G}}(x):=\operatorname{Hom}_{\mathcal{G}}(x,x),$$ often called the isotropy group of the object $x$ in $\mathcal{G}$. If the groupoid $\mathcal{G}$ is clear from the context, then we will simply write $\operatorname{Aut}(x)$. Moreover, if $\mathcal{G}$ and $\mathcal{H}$ are two groupoids, and $F\colon\mathcal{G}\to\mathcal{H}$ is a functor, for every object $x$ of $\mathcal{G}$ we let $$F_{x}\colon\operatorname{Aut}_{\mathcal{G}}(x)\longrightarrow \operatorname{Aut}_{\mathcal{H}}(F(x))$$ be the group morphism mapping an automorphism of $x$ in $\mathcal{G}$ to its image under the functor $F$. If $\mathcal{G}$ is a groupoid, by a *representation* of $\mathcal{G}$ we mean a functor $F\colon\mathcal{G}\to\mathcal{A}$ where $\mathcal{A}$ is a suitable groupoid of $\mathbb{Z}$-modules (or vector spaces). Finally, let us recall the notion of free product of groupoids. Let $\mathcal{G}$ and $\mathcal{H}$ be two groupoids, and $Y$ a set of common objects of $\mathcal{G}$ and $\mathcal{H}$. In all the situations of interest for us $\mathcal{G}$ and $\mathcal{H}$ will have the same set of objects. **Definition 32**. The *free product* of $\mathcal{G}$ and $\mathcal{H}$ along $Y$ is the groupoid $\mathcal{G}\ast_Y\mathcal{H}$ such that: - the objects of $\mathcal{G}\ast_Y\mathcal{H}$ are the elements of $Y$; - if $x,y\in Y$, then a morphism $f\in\operatorname{Hom}_{\mathcal{G}\ast_Y\mathcal{H}}(x,y)$ is a formal combination (with usual cancellation properties) $$f=g_1\circ\cdots\circ g_k$$ where each $g_i$ is a morphism from $x_i\in Y$ to $x_{i+1}\in Y$ in either $\mathcal{G}$ or $\mathcal{H}$ and such that $x_1=x$ and $x_{k+1}=y$. If $\mathcal{G}$ and $\mathcal{H}$ have the same sets of objects, then we simply refer to $\mathcal{G}\ast\mathcal{H}$ as the free product of $\mathcal{G}$ and $\mathcal{H}$, assuming that $Y$ is the whole set of objects. The fact that free products exist is the content of [@Higgins Proposition 21]. In literature the free product is also defined as the pushout, in the category of small categories, of $\mathcal{G}$ and $\mathcal{H}$ along a common set $Y$ of objects of $\mathcal{G}$ and $\mathcal{H}$. ## Deformations of $(m,k)$-triples and their groupoid {#section:families of K3} We start by recalling the construction of a deformation of a moduli space of sheaves on a K3 surface induced by the deformation of the surface itself, following [@PR:Deformations; @PR:SingularVarieties]. Let $(S,v,H)$ be an $(m,k)$-triple, and write $v=(r,\xi,a)$. We let $L$ be a line bundle on $S$ such that $c_1(L)=\xi$. We will moreover use the following notation: if $T$ is a smooth, connected algebraic variety, $f\colon Y\to T$ is a morphism and $\mathcal{L}\in \operatorname{Pic}(Y)$, for every $t\in T$ we let $Y_{t}:=f^{-1}(t)$ and $\mathcal{L}_{t}:=\mathcal{L}_{|Y_{t}}$. **Definition 33**. Let $(S,v,H)$ be an $(m,k)$-triple, and $T$ a smooth, connected algebraic variety. A *deformation of $(S,v,H)$ along $T$* is a triple $(f\colon \mathcal{S}\to T,\mathcal{L},\mathcal{H})$, where: 1. $f\colon \mathcal{S}\to T$ is a smooth, projective deformation of $S$, and we let $0\in T$ be such that $\mathcal{S}_{0}\simeq S$; 2. $\mathcal{L}$ is a line bundle on $\mathcal{S}$ such that $\mathcal{L}_{0}\simeq L$. 3. $\mathcal{H}$ is a line bundle on $\mathcal{S}$ such that $\mathcal{H}_{t}$ is a $v_{t}$-generic polarisation on $\mathcal{S}_{t}$ for every $t\in T$ and such that $\mathcal{H}_{0}\simeq H$ where for every $t\in T$ we let $v_{t}:=(r,c_{1}(\mathcal{L}_{t}),a)$. **Remark 34**. Notice that if $(f\colon \mathcal{S}\to T,\mathcal{L}^{\otimes m},\mathcal{H})$ is a deformation of an $(m,k)$-triple $(S,v,H)$ along $T$, and if we let $v=mw$, then $(f\colon \mathcal{S}\to T,\mathcal{L},\mathcal{H})$ is a deformation of $(S,w,H)$ along $T$. Conversely, if $(f\colon \mathcal{S}\to T,\mathcal{L},\mathcal{H})$ is a deformation of a $(1,k)$-triple $(S,w,H)$ along $T$, and if we let $v=mw$, then by [@PR:SingularVarieties] there is a Zariski-closed subset $Z$ of $T$ for which $(f\colon \mathcal{S}_{|T'}\to T',\mathcal{L}_{|T'}^{\otimes m},\mathcal{H}_{|T'})$ is a deformation of $(S,v,H)$ along $T':=T\setminus Z$ (here $Z$ is the subset of $T$ of those $t\in T$ for which $H_{t}$ is not $v_{t}$-generic). Notice that it may happen that $Z=T$. If $(S,v,H)$ is an $(m,k)$-triple and $(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H})$ is a deformation of $(S,v,H)$ along a smooth, connected algebraic variety $T$, we let $p_{v}\colon\mathcal{M}_{v}\to T$ be the relative moduli space of semistable sheaves so that for every $t\in T$ we have $\mathcal{M}_{v,t}=M_{v_{t}}(\mathcal{S}_{t},\mathcal{H}_{t})$. As shown in [@PR:SingularVarieties Lemma 2.21], the family $p_{v}\colon\mathcal{M}_{v}\to T$ is a locally trivial deformation of $M_{v}(S,H)$ along $T$. **Remark 35**. The isometry $\lambda_{(S,v,H)}$ in Theorem [Theorem 27](#thm:PR v perp){reference-type="ref" reference="thm:PR v perp"} behaves well in families of $(m,k)$-triples, i.e. it extends to an isometry of local systems as we will now explain. Let $(f\colon \mathcal{S}\to T,\mathcal{L},\mathcal{H})$ be a deformation of $(m,k)$-triples and $p_{v}\colon\mathcal{M}_{v}\to T$ the associated locally trivial family of moduli spaces. Let $\mathcal{M}_v^s$ be the smooth locus of $\mathcal{M}_v$ and and let $p_v^s:\mathcal{M}_v^s\rightarrow T$ be the restriction of $p_v$. By [@PR:v; @perp Proposition 3.5(2)], the inclusion $\mathcal{M}_v^s\subset \mathcal{M}_v$ induces an isomorphism $\iota\colon R^2p_{v*}^s\mathbb{Z} \rightarrow R^2p_{v*}\mathbb{Z}$ of local systems. By [@PR:v; @perp Proposition 4.4(2)], a relative quasi-universal family for $\mathcal{M}_v^s$ induces an isomorphism $\lambda^s\colon \mathsf{v}^\perp\rightarrow R^2p^{s}_{v*}\mathbb{Z}$ such that the morphism over a point $t\in T$ of the composition $$\mathsf{\lambda}:=\iota\circ \lambda^s\colon \mathsf{v}^\perp\longrightarrow R^2p_{v*}\mathbb{Z}$$ is just $\lambda_{(S_t,v_t,H_t)}$. **Definition 36**. Let $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$ be two $(m,k)$-triples. A *deformation path* from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$ is a $6$-tuple $$\alpha:=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$$where: - the triple $(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H})$ is a deformation of both the $(m,k)$-triples $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$; - for $i=1,2$ the point $t_{i}\in T$ is such that $(\mathcal{S}_{t_{i}},v_{t_{i}},H_{t_{i}})=(S_{i},v_{i},H_{i})$; - we have that $\gamma$ is a continuous path in $T$ such that $\gamma(0)=t_{1}$ and $\gamma(1)=t_{2}$. Given two $(m,k)$-triples $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$ and $\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$ a deformation path from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$, let $p_{v}\colon\mathcal{M}_{v}\to T$ be the relative moduli space associated to the deformation $(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H})$. There are two natural locally trivial parallel transport operators that can be defined starting from $\alpha$. 1. The first one is the parallel transport operator $$p_{\alpha}\colon\widetilde{\operatorname{H}}(S_{1},\mathbb{Z})\longrightarrow\widetilde{\operatorname{H}}(S_{2},\mathbb{Z})$$along $\gamma$ inside the local system $R^{\bullet}f_{*}\mathbb{Z}$. 2. The second one is the locally trivial parallel transport operator $$g_{\alpha}\colon\operatorname{H}^{2}(M_{v_{1}}(S_{1},H_{1}),\mathbb{Z})\longrightarrow \operatorname{H}^{2}(M_{v_{2}}(S_{2},H_{2}),\mathbb{Z})$$along $\gamma$ inside the local system $R^{2}p_{v*}\mathbb{Z}$. **Definition 37**. Let $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$ be two $(m,k)$-triples. Two deformations paths $\alpha$ and $\alpha'$ from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$ are *equivalent* if $p_{\alpha}=p_{\alpha'}$ and $g_{\alpha}=g_{\alpha'}$. The equivalence class of $\alpha$ will be denoted $\overline{\alpha}$. **Remark 38**. In Definition [Definition 37](#defn:equivalent){reference-type="ref" reference="defn:equivalent"} we used both the parallel transport operators $p_{\alpha}$ in the local system $R^\bullet f_*\mathbb{Z}$ and the parallel transport operators $g_{\alpha}$ in the local system $R^2p_{v*}\mathbb{Z}$. In fact, only $p_{\alpha}$ is needed. This is because the local system $R^\bullet f_*\mathbb{Z}$ comes with a flat section $\mathsf{v}$ corresponding to the Mukai vectors on the fibres. We can then consider the sub-local system $$\mathsf{v}^\perp\subset R^\bullet f_*\mathbb{Z}.$$ The parallel transport operator $p_{\alpha}$ is constant along $\mathsf{v}$ by definition. The restriction $p_{\alpha}|_{v^\perp}$ can then be seen as the parallel transport operator inside the local system $\mathsf{v}^\perp$. Since by Remark [Remark 35](#rmk:lambda in famiglie){reference-type="ref" reference="rmk:lambda in famiglie"} there is an isomorphism of local systems $$\lambda\colon \mathsf{v}^\perp\longrightarrow R^2p_{v*}\mathbb{Z},$$ and $g_{\alpha}$ and $p_{\alpha}|_{v^\perp}$ are parallel transport operators over the same path $\gamma$, the morphism $g_{\alpha}$ is uniquely determined by $p_{\alpha}$ and vice versa. Consider three $(m,k)$-triples $(S_{1},v_{1},H_{1})$, $(S_{2},v_{2},H_{2})$ and $(S_{3},v_{3},H_{3})$, and let $\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$ be a deformation path from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$ and $\alpha'=(f'\colon\mathcal{S}'\to T',\mathcal{L}',\mathcal{H}',t'_{1},t'_{2},\gamma')$ a deformation path from $(S_{2},v_{2},H_{2})$ to $(S_{3},v_{3},H_{3})$. **Definition 39**. The *concatenation* of $\alpha$ with $\alpha'$ is the $6$-tuple $$\alpha'\star\alpha:=(f''\colon\mathcal{S}''\to T'',\mathcal{L}'',\mathcal{H}'',t''_{1},t''_{2},\gamma'')$$where - $T''$ is obtained by glueing $T$ and $T'$ along $t_{2}$ and $t'_{1}$ - $\mathcal{S}''$ is obtained by glueing $\mathcal{S}$ and $\mathcal{S}'$ along $\mathcal{S}_{t_{2}}$ and $\mathcal{S}'_{t'_{1}}$ - $f''$ is obtained by glueing $f$ and $f'$ along $\mathcal{S}_{t_{2}}$ and $\mathcal{S}'_{t'_{1}}$ - $\mathcal{L}''$ is obtained by glueing $\mathcal{L}$ and $\mathcal{L}'$ along $\mathcal{S}_{t_{2}}$ and $\mathcal{S}'_{t'_{1}}$ - $\mathcal{H}''$ is obtained by glueing $\mathcal{H}$ and $\mathcal{H}'$ along $\mathcal{S}_{t_{2}}$ and $\mathcal{S}'_{t'_{1}}$ - $t''_{1}$ is the image of $t_{1}$ in $T''$ - $t''_{2}$ is the image of $t'_{2}$ in $T''$ - $\gamma''$ is the concatenation of the image of the path $\gamma$ in $T''$ with the image of the path $\gamma'$ in $T''$. It is immediate to notice that if $\alpha$ is a deformation path from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$ and $\alpha'$ is a deformation path from $(S_{2},v_{2},H_{2})$ to $(S_{3},v_{3},H_{3})$, then $\alpha\star\alpha'$ is a deformation path from $(S_{1},v_{1},H_{1})$ to $(S_{3},v_{3},H_{3})$. **Remark 40**. Notice that $$p_{\alpha\star\alpha'}=p_{\alpha'}\circ p_{\alpha},\,\,\,\,\,\,\,\,g_{\alpha\star\alpha'}=g_{\alpha'}\circ g_{\alpha},$$so if $\alpha$ is equivalent to $\beta$ and $\alpha'$ is equivalent to $\beta'$, then $\alpha\star\alpha'$ is equivalent to $\beta\star\beta'$. The previous notions allow us to define the groupoid $\mathcal{G}^{m,k}_{\operatorname{def}}$ of deformations of $(m,k)$-triples. Before doing this, we need to introduce two groupoids $\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}$ and $\mathcal{P}^{m,k}$ that we will use to define $\mathcal{G}^{m,k}_{\operatorname{def}}$. We start with the groupoid $\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}$. **Definition 41**. Given two strictly positive integers $m$ and $k$, the groupoid $\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}$ is defined as follows: - the objects of $\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}$ are the $(m,k)$-triples; - for every two $(m,k)$-triples $(S_{1},v_{1},H_{1}),(S_{2},v_{2},H_{2})$, a morphism from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$ in $\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}$ is an equivalence class of deformation paths from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$; - for every $(m,k)$-triple $(S,v,H)$, where $v=(r,c_{1}(L),a)$, the identity of $(S,v,H)$ is the equivalence class of the deformation path $(S\to\{p\},L,H,p,p,\kappa_{p})$ where $\kappa_{p}$ is the constant path; - for every three $(m,k)$-triples $(S_{1},v_{1},H_{1})$, $(S_{2},v_{2},H_{2})$ and $(S_{3},v_{3},H_{3})$ and every pair of morphisms $$\overline{\alpha}\colon(S_{1},v_{1},H_{1})\to (S_{2},v_{2},H_{2})\quad\mbox{and}\quad\overline{\alpha'}\colon(S_{2},v_{2},H_{2})\to(S_{3},v_{3},H_{3}),$$ the composition of $\alpha$ with $\alpha'$ is $\overline{\alpha'}\circ\overline{\alpha}:=\overline{\alpha\star\alpha'}$. **Remark 42**. The fact that the composition is well defined follows from Remark [Remark 40](#rmk:starok){reference-type="ref" reference="rmk:starok"}, and the fact that the identity is the prescribed one is immediate. Moreover, the fact that $\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}$ is a groupoid may be proved as follows: given two $(m,k)$-triples $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$ and a morphism $\overline{\alpha}$ between them, where $\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$, then $\overline{\alpha}^{-1}$ is $\overline{\alpha^{-1}}$, where $$\alpha^{-1}:=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{2},t_{1},\gamma^{-1}).$$Indeed $p_{\alpha^{-1}}=p_{\alpha}^{-1}$ and $g_{\alpha^{-1}}=g^{-1}_{\alpha}$, so $\overline{\alpha^{-1}}=\overline{\alpha}^{-1}$ by Remark [Remark 40](#rmk:starok){reference-type="ref" reference="rmk:starok"}. Finally, the associativity property also holds since the operation of concatenating paths is associative. We now define the groupoid $\mathcal{P}^{m,k}$. Recall that two $(m,k)$-triples $(S_{1},v_{1},H_1)$ and $(S_{2},v_{2},H_2)$ are congruent (see Definition [Definition 23](#defn:congruent){reference-type="ref" reference="defn:congruent"}) when $S_{1}=S_{2}=S$, $v_{1}=v_{2}=v$ and a coherent sheaf $F$ on $S$ with Mukai vector $v$ is $H_1$-(semi)stable sheaf if and only if it is $H_2$-(semi)stable. In this case we denote by $$\chi_{H_1,H_2}\colon M_v(S,H_1)\longrightarrow M_v(S,H_2), \qquad F\mapsto F$$ the identity map. **Definition 43**. Given two strictly positive integers $m$ and $k$, the groupoid $\mathcal{P}^{m,k}$ is defined as follows: - the objects of $\mathcal{P}^{m,k}$ are the $(m,k)$-triples; - for every two $(m,k)$-triples $(S_{1},v_{1},H_{1})$, $(S_{2},v_{2},H_{2})$ we set $$\operatorname{Hom}_{\mathcal{P}^{m,k}}((S_{1},v_{1},H_{1}),(S_{2},v_{2},H_{2})):=\{\chi_{H_{1},H_{2}}\}$$if $(S,v,H_{1})$ and $(S_{2},v_{2},H_{2})$ are congruent, and otherwise $$\operatorname{Hom}_{\mathcal{P}^{m,k}}((S_{1},v_{1},H_{1}),(S_{2},v_{2},H_{2})):=\emptyset.$$ We are now finally in the position to define the groupoid $\mathcal{G}^{m,k}_{\operatorname{def}}$. **Definition 44**. Given two strictly positive integers $m$ and $k$, the groupoid $$\mathcal{G}^{m,k}_{\operatorname{def}}:=\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}\ast\mathcal{P}^{m,k}$$ is the free product of $\widetilde{\mathcal{G}}^{m,k}_{\operatorname{def}}$ and $\mathcal{P}^{m,k}$ (see Definition [Definition 32](#defn:free product of groupoids){reference-type="ref" reference="defn:free product of groupoids"}). ## Fourier--Mukai equivalences and their groupoid {#section:FM} The purpose of this section is to define the groupoid $\mathcal{G}^{m,k}_{\operatorname{FM}}$. To do so, we need to recall some isomorphisms of moduli spaces of sheaves induced by Fourier--Mukai transforms. Let us start with the tensorization with a line bundle. Let $S$ be a projective K3 surface and $L\in\operatorname{Pic}(S)$ a line bundle. Consider the derived equivalence $$\begin{aligned} \label{eqn:L} \mathsf{L}\colon\operatorname{D}^b(S) & \longrightarrow\operatorname{D}^b(S) \\ F^\bullet & \longmapsto F^\bullet\otimes L. \nonumber\end{aligned}$$ If a sheaf $F$ has Mukai vector $v(F)=v$, then $v(\mathsf{L}(F))=v\cdot\operatorname{ch}(L)=:v_L$. With an abuse of notation we keep denoting by $$\mathsf{L}\colon M_v(S,H)\longrightarrow M_{v_L}(S,H)$$ the induced morphism between moduli spaces of sheaves. This morphism is known to be an isomorphism in some cases, as the following states. **Lemma 45** (). *Let $S$ be a projective K3 surface, $v=(r,\xi,a)$ a Mukai vector and $H$ an ample line bundle.* 1. *For any $d\in\mathbb{Z}$, the morphism $\mathsf{dH}\colon M_v(S,H)\to M_{v_{dH}}(S,H)$ is an isomorphism.* 2. *If $r>0$ and $H$ is $v$-generic, the morphism $\mathsf{L}\colon M_v(S,H)\to M_{v_{L}}(S,H)$ is an isomorphism for any $L\in\operatorname{Pic}(S)$.* We now consider the Fourier--Mukai transform whose kernel is the ideal of the diagonal. Let $S$ be a projective K3 surface and $\Delta\subset S\times S$ be the diagonal. If $I_\Delta\in\operatorname{Coh}(S\times S)$ denotes the ideal sheaf of $\Delta$, then we consider the Fourier-Mukai equivalence $$\operatorname{FM}_\Delta\colon\operatorname{D}^b(S) \longrightarrow\operatorname{D}^b(S)\,\,\,\,\,\,\,\,\,\operatorname{FM}_{\Delta}(F^\bullet):=R\pi_{2*}(I_\Delta\stackrel{L}{\otimes}\pi_1^*F^\bullet),$$ where $\pi_1$ and $\pi_2$ are the two projections from $S\times S$ to $S$, and the composite equivalence $$\operatorname{FM}^\vee_\Delta\colon\operatorname{D}^b(S) \longrightarrow\operatorname{D}^b(S)\,\,\,\,\,\,\,\,\,\operatorname{FM}_{\Delta}(F^\bullet):=\left(R\pi_{2*}(I_\Delta\stackrel{L}{\otimes}\pi_1^*F^\bullet)\right)^\vee,$$ If $F$ is a sheaf on $S$ with Mukai vector $v(F)=(r,\xi,a)=v$, then by a direct check one can see that $$v(\operatorname{FM}_\Delta(F))=(a,-\xi,r)=:\tilde{v}\quad\mbox{ and }\quad v(\operatorname{FM}^\vee_\Delta(F))=(a,\xi,r)=:\hat{v}$$ **Lemma 46** ([@Yoshioka:FM; @PR:SingularVarieties]). *Let $S$ be a projective K3 surface, $H$ an ample line bundle and $n,a\in\mathbb{Z}$.* 1. *Suppose that $\operatorname{Pic}(S)=\mathbb{Z}\,H$. Put $v=(r,nH,a)$, with $r>0$. Then there exists an integer $n_0\gg0$ such that for every $n> n_0$ the Fourier--Mukai transform $\operatorname{FM}_{\Delta}$ induces an isomorphism $$\operatorname{FM}_{\Delta,v}\colon M_v(S,H)\longrightarrow M_{\tilde{v}}(S,H).$$* 2. *Put $v=(0,\xi,a)$ and suppose that $H$ is both $v$ and $\tilde{v}$-generic. Then there exists an integer $a_0\gg0$ such that for every $a>a_0$ the Fourier--Mukai transform $\operatorname{FM}_{\Delta}$ induces an isomorphism $$\operatorname{FM}_{\Delta,v}\colon M_v(S,H)\longrightarrow M_{\tilde{v}}(S,H).$$* 3. *In any of the two cases above, the dual Fourier--Mukai transform $\operatorname{FM}^\vee_{\Delta}$ induces an isomorphism $$\operatorname{FM}^\vee_{\Delta,v}\colon M_v(S,H)\longrightarrow M_{\hat{v}}(S,H).$$* *Proof.* The proof of the first two items is [@PR:SingularVarieties Proposition 2.29, Proposition 2.33] (but see also [@Yoshioka:FM Theorem 3.18] for a more general statement). Moreover in [@PR:SingularVarieties] it is further shown that the sheaf $\operatorname{FM}_\Delta(F)$ is locally free (see [@PR:SingularVarieties Lemma 2.28]). The proof of the third item now follows from the first two just by noticing that a locally free sheaf $F$ is (semi)stable if and only if its dual $F^\vee$ is (semi)stable. ◻ The last Fourier--Mukai transform we consider is associated to elliptic K3 surfaces and it has been studied in [@Bridgeland]; the result we need is contained in [@Yoshioka:ModuliAbelian]. Let $p\colon S\to\mathbb{P}^1$ be an elliptic K3 surface. We assume that there is a section $s\colon\mathbb{P}^1\to S$ and we denote by $\ell\in\operatorname{Pic}(S)$ its cohomology class. If we put $f=p^*\mathcal{O}(1)$, then the lattice spanned by $\ell$ and $f$ is the unimodular hyperbolic plane $U$; more precisely, $\ell^2=-2$, $(\ell,f)=1$ and $f^2=0$. Let us assume that $\operatorname{Pic}(S)$ coincides with this hyperbolic plane (i.e. that $S$ is generic with this property). It is known that the class $H=\ell+tf$ is ample for $t\gg0$ and we fix once and for all such an ample class. Consider the moduli space $M_{(0,f,0)}(S,H)$. For $t\gg0$, the polarisation $H$ is generic with respect to $(0,f,0)$ and $M_{(0,f,0)}(S,H)\cong S$ (see [@Bridgeland Section 4]). Let $\mathcal{P}$ be the relative Poincaré bundle that we see as a coherent sheaf over the product $S\times S$. The Fourier--Mukai transform $$\operatorname{FM}_{\mathcal{P}}\colon\operatorname{D}^b(S) \longrightarrow\operatorname{D}^b(S),\,\,\,\,\,\,\,\,\,\operatorname{FM}_{\mathcal{P}}(F^\bullet):=(R\pi_{2*}(\mathcal{P}\stackrel{L}{\otimes}\pi_1^*F^\bullet))[1]$$ is proved to be an equivalence in [@Bridgeland Theorem 1.2]. **Lemma 47** (). *Let $H=\ell+tf$ with $t\gg0$. Then $\operatorname{FM}_{\mathcal{P}}$ induces an isomorphism $$\operatorname{FM}_{\mathcal{P}}\colon M_{(m,0,-mk)}(S,H)\longrightarrow M_{(0,m(\ell+(k+1)f),m)}(S,H)$$ for every $m,k>0$.* We conclude this section by defining the groupoid $\mathcal{G}^{m,k}_{\operatorname{FM}}$. **Definition 48**. Given two strictly positive integers $m$ and $k$, the groupoid $\mathcal{G}^{m,k}_{\operatorname{FM}}$ is defined as follows: - the objects of $\mathcal{G}^{m,k}_{\operatorname{FM}}$ are the $(m,k)$-triples; - given two $(m,k)$-triples $(S,v_1,H)$ and $(S,v_2,H)$, an *elementary morphism* between them is one of the following: - the equivalence $\mathsf{L}$, if $(S,v_1,H)=(S,v,H)$ and $(S,v_2,H)=(S,v_L,H)$ are as in Lemma [Lemma 45](#lemma:2.20){reference-type="ref" reference="lemma:2.20"}; - the equivalence $\operatorname{FM}_\Delta$, if $(S,v_1,H)=(S,v,H)$ and $(S,v_2,H)=(S,\tilde{v},H)$ verify the hypotheses of Lemma [Lemma 46](#lemma:Y-PR){reference-type="ref" reference="lemma:Y-PR"} (1) or (2); - the equivalence $\operatorname{FM}^\vee_\Delta$, if $(S,v_1,H)=(S,v,H)$ and $(S,v_2,H)=(S,\hat{v},H)$ verify the hypotheses of Lemma [Lemma 46](#lemma:Y-PR){reference-type="ref" reference="lemma:Y-PR"} (3); - the equivalence $\operatorname{FM}_{\mathcal{P}}$, if $(S,v_1,H)=(S,(m,0,m-mk),H)$ and $(S,v_2,H)=(S,(0,m(\ell+kf),m),H)$ verify the hypotheses of Lemma [Lemma 47](#prop:Poincare){reference-type="ref" reference="prop:Poincare"}; - for every two $(m,k)$-triples $(S_{1},v_{1},H_{1}),(S_{2},v_{2},H_{2})$, a morphisms from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$ is a formal concatenation of elementary morphisms and their formal inverses, subject to the usual cancellation rules. ## The groupoid $\mathcal{G}^{m,k}$ and its representations {#section:representation} For every two strictly positive integers $m,k$ we now define the groupoid $\mathcal{G}^{m,k}$ as the groupoid generated by $\mathcal{G}^{m,k}_{\operatorname{def}}$ and $\mathcal{G}^{m,k}_{\operatorname{FM}}$. **Definition 49**. Given two strictly positive integers $m$ and $k$, the groupoid $$\mathcal{G}^{m,k}:=\mathcal{G}^{m,k}_{\operatorname{def}}\ast \mathcal{G}^{m,k}_{\operatorname{FM}}$$ is the free product of $\mathcal{G}^{m,k}_{\operatorname{def}}$ and $\mathcal{G}^{m,k}_{\operatorname{FM}}$ (see Definition [Definition 32](#defn:free product of groupoids){reference-type="ref" reference="defn:free product of groupoids"}). **Remark 50**. Let $(S_1,v_1,H_1),(S_2,v_2,H_2)\in\mathcal{G}^{m,k}$ be two objects; then $$\operatorname{Hom}_{\mathcal{G}^{m,k}}((S_1,v_1,H_1),(S_2,v_2,H_2))\neq\emptyset.$$ More precisely, in the proof of [@PR:SingularVarieties Theorem 1.7] the authors show that one can go from $(S_1,v_1,H_1)$ to $(S_2,v_2,H_2)$ only using the following morphisms of $\mathcal{G}^{m,k}$: - equivalence classes of deformation paths of $(m,k)$-triples; - the morphisms $\chi_{H,H'}$ of $\mathcal{P}^{m,k}$; - derived equivalences of the form $\mathsf{L}$, for some line bundle $L$; - the Fourier--Mukai transform $\operatorname{FM}_\Delta$. ### The $\widetilde{\mathcal{H}}$-representation $\widetilde{\Phi}^{m,k}$ of $\mathcal{G}^{m,k}$ Let $\widetilde{\Lambda}$ be an even unimodular lattice of signature $(4,20)$. Notice that the isometry class of $\widetilde{\Lambda}$ is uniquely determined, so that $\widetilde{\Lambda}$ is isometric to the lattice $$U^{\oplus4}\oplus E_8(-1)^{\oplus2},$$ where $U$ is the unimodular even rank 2 lattice and $E_8(-1)$ is the negative definite Dynkin lattice of type $E_8$. If $\widetilde{\mathcal{C}}\subset\widetilde{\Lambda}\otimes_{\mathbb{Z}}\mathbb{R}$ is the cone of (strictly) positive classes, then by [@Markman:Survey Lemma 4.1] we have that $\operatorname{H}^3(\widetilde{\mathcal{C}},\mathbb{Z})=\mathbb{Z}$. The choice of a generator of $\operatorname{H}^3(\widetilde{\mathcal{C}},\mathbb{Z})$ is an *orientation* of $\widetilde{\Lambda}$. Notice that there are only two orientations, corresponding to the generators $\pm 1$ of $\mathbb{Z}$. Again by [@Markman:Survey Lemma 4.1], if $W\subset\widetilde{\Lambda}\otimes_{\mathbb{Z}}\mathbb{R}$ is a positive real subspace of dimension $4$, then the space $W\setminus\{0\}$ is a deformation retract of $\widetilde{\mathcal{C}}$, so that an orientation on $\widetilde{\Lambda}$ corresponds to an orientation on a positive real subspace of maximal dimension. If $g\in\operatorname{O}(\widetilde{\Lambda})$ is an isometry, then $g$ induces an action on $\operatorname{H}^3(\widetilde{\mathcal{C}},\mathbb{Z})$ that either preserves a generator or it maps it to its opposite. Therefore we get an orientation character $$\label{eqn:or} \operatorname{or}\colon\operatorname{O}(\widetilde{\Lambda})\longrightarrow\mathbb{Z}/2\mathbb{Z}.$$ The subgroup $\operatorname{O}^+(\widetilde{\Lambda})=\ker(\operatorname{or})$ is the group of orientation preserving isometries. Similarly, if $(\widetilde{\Lambda}_1,\epsilon_1)$ and $(\widetilde{\Lambda}_2,\epsilon_2)$ are two pairs composed by a unimodular even lattice $\widetilde{\Lambda}_i$ of signature $(4,20)$ and an orientation $\epsilon_i\in\operatorname{H}^3(\widetilde{\mathcal{C}}_i,\mathbb{Z})$ on $\widetilde{\Lambda}_i$, then we get an orientation map $$\label{eqn:or tra due} \operatorname{or}\colon\operatorname{O}((\widetilde{\Lambda}_1,\epsilon_1),(\widetilde{\Lambda}_2,\epsilon_2))\longrightarrow\mathbb{Z}/2\mathbb{Z}.$$ Again, $\operatorname{O}^+((\widetilde{\Lambda}_1,\epsilon_1),(\widetilde{\Lambda}_2,\epsilon_2))\coloneqq\operatorname{or}^{-1}(0)$ is the set of orientation preserving isometries. **Definition 51**. Given two strictly positive integers $m$ and $k$, the groupoid $\widetilde{\mathcal{H}}^{m,k}$ is defined as follows: - the objects are triples $(\widetilde{\Lambda},v,\epsilon)$, where $\widetilde{\Lambda}$ is a unimodular even lattices of signature $(4,20)$, $v\in\widetilde{\Lambda}$ is of the form $v=mw$, where $w$ is primitive and $w^2=2k$, and where $\epsilon$ is an orientation on $\widetilde{\Lambda}$; - for any two objects $(\widetilde{\Lambda}_1,v_1,\epsilon_1)$ and $(\widetilde{\Lambda}_2,v_2,\epsilon_2)$, the set $$\operatorname{Hom}_{\widetilde{\mathcal{H}}^{m,k}}((\widetilde{\Lambda}_1,v_1,\epsilon_1),(\widetilde{\Lambda}_1,v_1,\epsilon_2))$$ is the set of isometries $g\colon\widetilde{\Lambda}_1\to\widetilde{\Lambda}_2$ such that $g(v_1)=v_2$. **Remark 52**. Notice that morphisms in $\widetilde{\mathcal{H}}^{m,k}$ are not necessarily orientation preserving. **Example 53**. If $S$ is a K3 surface, then the Mukai lattice $\widetilde{\operatorname{H}}(S,\mathbb{Z})$ is a unimodular even lattice of signature $(4,20)$. As it is explained in [@Markman:Monodromy Section 4.1], $\widetilde{\operatorname{H}}(S,\mathbb{Z})$ comes with a distinguished orientation, which we denote by $\epsilon_S$. Such an orientation is associated to the distinguished positive real $4$-space $W$ with a chosen basis given by a Kähler class, the real and imaginary parts of a symplectic form and the vector $(1,0,-1)$. We also point out that, as we have seen in the proof of Lemma [Lemma 13](#lemma:Mon is O+){reference-type="ref" reference="lemma:Mon is O+"}, the positive real $3$-space with basis given by a Kähler class and the real and imaginary part of a symplectic form determines an orientation on the lattice $\operatorname{H}^2(S,\mathbb{Z})$. Before continuing, let us recall the following definition. **Definition 54**. Let $f\colon\mathcal{S}\to T$ be a smooth family of K3 surfaces. Take $t_1,t_2\in T$ and a path $\gamma$ from $t_1$ to $t_2$. A *parallel transport operator* $$g\colon\widetilde{\operatorname{H}}(\mathcal{S}_{t_1},\mathbb{Z})\to\widetilde{\operatorname{H}}(\mathcal{S}_{t_2},\mathbb{Z})$$ is an isometry induced by parallel transport inside the local system $R^\bullet f_*\mathbb{Z}$. This allows us to define the following representation of the groupoid $\mathcal{G}^{m,k}_{\operatorname{def}}$. **Definition 55**. Given two strictly positive integers $m$ and $k$, the representation $$\widetilde{\Phi}^{m,k}_{\operatorname{def}}\colon\mathcal{G}^{m,k}_{\operatorname{def}}\longrightarrow\widetilde{\mathcal{H}}^{m,k}$$ is defined as follows: - if $(S,v,H)\in\mathcal{G}^{m,k}_{\operatorname{def}}$ is an object, then $\widetilde{\Phi}^{m,k}_{\operatorname{def}}(S,v,H)=(\widetilde{\operatorname{H}}(S,\mathbb{Z}),v,\epsilon_S)$ (see Example [Example 53](#example:Mukai lattice){reference-type="ref" reference="example:Mukai lattice"}); - if $(S_1,v_1,H_1)$ and $(S_{2},v_{2},H_{2})$ are two objects and $\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$ is a deformation path from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$, then $$\widetilde{\Phi}^{m,k}_{\operatorname{def}}(\overline{\alpha}):=p_{\alpha},$$ the parallel transport operator in the local system $R^\bullet f_*\mathbb{Z}$ along the path $\gamma$; - if $(S_1,v_1,H_1)$ and $(S_{2},v_{2},H_{2})$ are congruent, then $$\widetilde{\Phi}^{m,k}_{\operatorname{def}}(\chi_{H_{1},H_{2}}):=\operatorname{id}_{\widetilde{\operatorname{H}}(S,\mathbb{Z})},$$ where $\chi_{H_1,H_2}$ is the identification $M_{v_{1}}(S_{1},H_{1})=M_{v_{2}}(S_{2},H_{2})$ (see Definitions [Definition 23](#defn:congruent){reference-type="ref" reference="defn:congruent"} and [Definition 43](#defn:P m k){reference-type="ref" reference="defn:P m k"}). **Remark 56**. Notice that if $\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$ is a deformation path from $(S_1,v_1,H_1)$ to $(S_2,v_2,H_2)$, then by definition the Mukai vectors $v_1$ and $v_2$ belong to the same flat section of the local system $R^\bullet f_*\mathbb{Z}$, so the parallel transport operator $p_{\alpha}$ maps $v_1$ to $v_2$ and the representation $\widetilde{\Phi}^{m,k}_{\operatorname{def}}$ is well defined. **Remark 57**. Because of Lemma [Lemma 13](#lemma:Mon is O+){reference-type="ref" reference="lemma:Mon is O+"} and the fact that the vector $(1,0,-1)$ is preserved, we see that a parallel transport operator $g\colon\widetilde{\operatorname{H}}(S_1,\mathbb{Z})\to\widetilde{\operatorname{H}}(S_2,\mathbb{Z})$ is orientation preserving. In particular the morphisms in the image of $\widetilde{\Phi}^{m,k}_{\operatorname{def}}$ are always orientation preserving. Let us now define the representation $$\widetilde{\Phi}^{m,k}_{\operatorname{FM}}\colon\mathcal{G}^{m,k}_{\operatorname{FM}}\longrightarrow\widetilde{\mathcal{H}}^{m,k}.$$ **Definition 58**. Given two strictly positive integers $m$ and $k$, the representation $\widetilde{\Phi}^{m,k}_{\operatorname{FM}}$ is defined as follows: - if $(S,v,H)\in\mathcal{G}^{m,k}_{\operatorname{FM}}$ is an object, then $\widetilde{\Phi}^{m,k}_{\operatorname{FM}}(S,v,H)=(\widetilde{\operatorname{H}}(S,\mathbb{Z}),v,\epsilon_S)$; - if $\phi\in\operatorname{Aut}(\operatorname{D}^b(S))$ corresponds to a morphism in $\mathcal{G}^{m,k}_{\operatorname{FM}}$, then $\widetilde{\Phi}^{m,k}_{\operatorname{FM}}(\phi)=\phi^{\operatorname{H}}$ is the isometry induced by $\phi$ on the Mukai lattice. **Remark 59**. The equivalence $\phi$ is composition of the equivalences introduced in Section [2.3](#section:FM){reference-type="ref" reference="section:FM"} and, by definition, it induces an isomorphism between the corresponding moduli spaces. It follows in particular that the Mukai vector must be preserved, so that again $\widetilde{\Phi}^{m,k}_{\operatorname{FM}}$ is well defined. **Remark 60**. Notice that morphisms in the image of $\widetilde{\Phi}^{m,k}_{\operatorname{FM}}$ are not necessarily orientation preserving. **Definition 61**. Define the representation $$\widetilde{\Phi}^{m,k}\colon\mathcal{G}^{m,k}\longrightarrow\widetilde{\mathcal{H}}^{m,k}$$ as the unique representation restricting to $\widetilde{\Phi}^{m,k}_{\operatorname{def}}$ on $\mathcal{G}^{m,k}_{\operatorname{def}}$ and to $\widetilde{\Phi}^{m,k}_{\operatorname{FM}}$ on $\mathcal{G}^{m,k}_{\operatorname{FM}}$. **Remark 62**. The existence and uniqueness of $\widetilde{\Phi}^{m,k}$ are a consequence of the fact that the objects of $\mathcal{G}^{m,k}$ are the same as those of $\mathcal{G}^{m,k}_{\operatorname{def}}$ and $\mathcal{G}^{m,k}_{\operatorname{FM}}$, and the representations $\widetilde{\Phi}^{m,k}_{\operatorname{def}}$ and $\widetilde{\Phi}^{m,k}_{\operatorname{FM}}$ coincide on the objects. Moreover, as morphisms in $\mathcal{G}^{m,k}$ are formal concatenations of morphisms in $\mathcal{G}^{m,k}_{\operatorname{def}}$ and $\mathcal{G}^{m,k}_{\operatorname{FM}}$, there is a unique way to define $\widetilde{\Phi}^{m,k}$ on morphisms. ### The $\mathcal{A}_k$-representation $\mathsf{pt}^{m,k}$ of $\mathcal{G}^{m,k}$ {#section:Phi=mon} **Definition 63**. For every $k>0$ we define the groupoid $\mathcal{A}_k$ as follows: - the objects of $\mathcal{A}_{k}$ are even lattices $\Lambda$ of signature $(3,20)$ isometric to the lattice $U^{\oplus 3}\oplus E_{8}(-1)^{\oplus 2}\oplus\langle -2k\rangle$; - if $\Lambda_1$ and $\Lambda_2$ are two objects, then $$\operatorname{Hom}_{\mathcal{A}_k}(\Lambda_1,\Lambda_2):=\operatorname{O}(\Lambda_1,\Lambda_2).$$ As before we first define $\mathcal{A}_k$-representations for both $\mathcal{G}^{m,k}_{\operatorname{def}}$ and $\mathcal{G}^{m,k}_{\operatorname{FM}}$. We start by defining the representation $\mathsf{pt}^{m,k}_{\operatorname{def}}\colon\mathcal{G}^{m,k}_{\operatorname{def}}\to\mathcal{A}_{k}$. **Definition 64**. Given two strictly positive integers $m$ and $k$ we define the functor $\mathsf{pt}^{m,k}_{\operatorname{def}}$ as follows: - if $(S,v,H)\in\mathcal{G}^{m,k}_{\operatorname{def}}$ is an object, then $$\mathsf{pt}^{m,k}_{\operatorname{def}}((S,v,H))=\operatorname{H}^2(M_v(S,H),\mathbb{Z});$$ - if $(S_1,v_1,H_1)$ and $(S_2,v_2,H_2)$ are two objects and $\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$ is a deformation path from $(S_{1},v_{1},H_{1})$ to $(S_{2},v_{2},H_{2})$, then $$\mathsf{pt}^{m,k}_{\operatorname{def}}(\overline{\alpha}):=g_\alpha,$$ the parallel transport operator in the local system $R^2p_{v,*}\mathbb{Z}$ along the path $\gamma$; - if $(S_1,v_1,H_1)$ and $(S_2,v_2,H_2)$ are congruent, then $$\mathsf{pt}^{m,k}_{\operatorname{def}}(\chi_{H_{1},H_{2}}):=\operatorname{id}_{\operatorname{H}^{2}(M_{v}(S,H_{1}),\mathbb{Z})},$$ where $\chi_{H_1,H_2}$ is the identification $M_{v_{1}}(S_{1},H_{1})=M_{v_{2}}(S_{2},H_{2})$ (see Definitions [Definition 23](#defn:congruent){reference-type="ref" reference="defn:congruent"} and [Definition 43](#defn:P m k){reference-type="ref" reference="defn:P m k"}). **Remark 65**. By Theorem [Theorem 27](#thm:PR v perp){reference-type="ref" reference="thm:PR v perp"} there is an isometry $\operatorname{H}^2(M_v(S,H),\mathbb{Z})\cong v^\perp$; since $v=mw$, with $w^2=2k$, it follows that $\operatorname{H}^2(M_v(S,H),\mathbb{Z})$ is isometric to the lattice $U^{\oplus 3}\oplus E_{8}(-1)^{\oplus 2}\oplus\langle -2k\rangle$. Next, let us define the functor $$\mathsf{pt}^{m,k}_{\operatorname{FM}}\colon\mathcal{G}^{m,k}_{\operatorname{FM}}\to\mathcal{A}_k.$$ **Definition 66**. Given two strictly positive integers $m$ and $k$, we define the functor $\mathsf{pt}^{m,k}_{\operatorname{FM}}$ as follows: - if $(S,v,H)\in\mathcal{G}^{m,k}_{\operatorname{FM}}$ is an object, then $$\mathsf{pt}^{m,k}_{\operatorname{FM}}(S,v,H)=\operatorname{H}^2(M_v(S,H),\mathbb{Z});$$ - if $\phi\in\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{FM}}}((S,v_1,H),(S,v_2,H))$ is an elementary morphism, then by definition $\phi$ induces an isomorphism $$\phi_{v_1}\colon M_{v_1}(S,H)\to M_{v_2}(S,H)$$ of the moduli spaces. We then define $$\mathsf{pt}^{m,k}_{\operatorname{FM}}(\phi)=\phi_{v_1,*},$$ where the latter is the pushforward action on the second integral cohomology groups of the moduli spaces; - if $\phi\in\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{FM}}}((S,v_1,H),(S,v_2,H))$ is a morphism, i.e. a concatenation of elementary morphisms and their inverses, then we define $\mathsf{pt}^{m,k}_{\operatorname{FM}}(\phi)$ as the composition of the corresponding isometries. Similarly to the case of $\widetilde{\Phi}^{m,k}$, we now give the following definition. **Definition 67**. Define $$\label{eqn:pt} \mathsf{pt}^{m,k}\colon\mathcal{G}^{m,k}\to\mathcal{A}_k$$ as the unique representation restricting to $\mathsf{pt}^{m,k}_{\operatorname{def}}$ on $\mathcal{G}^{m,k}_{\operatorname{def}}$ and to $\mathsf{pt}^{m,k}_{\operatorname{FM}}$ on $\mathcal{G}^{m,k}_{\operatorname{FM}}$. **Proposition 68**. *Let $A_1=(S_1,v_1,H_1),A_2=(S_2,v_2,H_2)\in\mathcal{G}^{m,k}$ be two objects. Then $$\mathsf{pt}^{m,k}(\operatorname{Hom}_{\mathcal{G}^{m,k}}(A_1,A_2))\subset\mathsf{PT}_{\operatorname{lt}}(M_{v_1}(S_1,H_1),M_{v_2}(S_2,H_2)).$$* *Proof.* First of all, we notice that $$\mathsf{pt}^{m,k}_{\operatorname{def}}(\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{def}}}(A_1,A_2))\subset\mathsf{PT}_{\operatorname{lt}}(M_{v_1}(S_1,H_1),M_{v_2}(S_2,H_2))$$ by definition. On the other hand, by Proposition [Proposition 14](#prop:iso is mon){reference-type="ref" reference="prop:iso is mon"} we also have that $$\mathsf{pt}^{m,k}_{\operatorname{FM}}(\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{FM}}}(A_1,A_2))\subset\mathsf{PT}_{\operatorname{lt}}(M_{v_1}(S_1,H_1),M_{v_2}(S_2,H_2))$$ so that the claim follows. ◻ Using the notation set in Section [2.1](#section:groupoids){reference-type="ref" reference="section:groupoids"}, we state the following corollary of Proposition [Proposition 68](#prop:Im of pt in Mon){reference-type="ref" reference="prop:Im of pt in Mon"}. **Corollary 69**. *Let $(S,v,H)\in\mathcal{G}^{m,k}$ be an object. Then $$\operatorname{Im}\left( \mathsf{pt}^{m,k}_{(S,v,H)}\colon\operatorname{Aut}(S,v,H)\to\operatorname{Aut}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}) \right)\subset\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H)).$$* ### Relation between the two representations $\widetilde{\Phi}^{m,k}$ and $\mathsf{pt}^{m,k}$ We now connect the representation $\mathsf{pt}^{m,k}$ of $\mathcal{G}^{m,k}$ in $\mathcal{A}_{k}$ with the the representation $\widetilde{\Phi}^{m,k}$ of $\mathcal{G}^{m,k}$ in $\widetilde{\mathcal{H}}^{m,k}$ we defined before. **Definition 70**. Given two strictly positive integers $m$ and $k$, define the functor $$\Psi\colon\widetilde{\mathcal{H}}^{m,k}\longrightarrow\mathcal{A}_k$$ as follows: - if $(\widetilde{\Lambda},v,\epsilon)$ is an object in $\widetilde{\mathcal{H}}^{m,k}$, then $$\Psi(\widetilde{\Lambda},v)=v^\perp;$$ - if $(\widetilde{\Lambda}_i,v_i,\epsilon_i)$ are two objects and $g\colon\widetilde{\Lambda}_1\to\widetilde{\Lambda}_2$ an isometry such that $g(v_1)=v_2$, then $$\Psi(g)=(-1)^{\operatorname{or}(g)} g|_{v_1^\perp}\colon v_1^\perp\longrightarrow v_2^\perp,$$ where $\operatorname{or}$ is the orientation character ([\[eqn:or tra due\]](#eqn:or tra due){reference-type="ref" reference="eqn:or tra due"}). Put $$\Phi^{m,k}=\Psi\circ\widetilde{\Phi}^{m,k}\colon\mathcal{G}^{m,k}\longrightarrow\mathcal{A}_k.$$ **Proposition 71**. *There exists an isomorphism of functors $$\lambda\colon\Phi^{m,k}\longrightarrow \mathsf{pt}^{m,k}.$$* We recall that if $\mathcal{A}$ and $\mathcal{B}$ are two categories and $F,G\colon\mathcal{A}\to\mathcal{B}$ are two functors, an isomorphism of functors $\lambda\colon F\to G$ is a natural transformation such that for each $A\in\mathcal{A}$ the morphism $\lambda(A)\colon F(A)\to G(A)$ is an isomorphism in $\mathcal{B}$. *Proof.* First of all, let us define the representations $$\Phi^{m,k}_{\operatorname{def}}=\Psi\circ\widetilde{\Phi}^{m,k}_{\operatorname{def}}\colon\mathcal{G}^{m,k}_{\operatorname{def}}\longrightarrow\mathcal{A}_k$$ and $$\Phi^{m,k}_{\operatorname{FM}}=\Psi\circ\widetilde{\Phi}^{m,k}_{\operatorname{FM}}\colon\mathcal{G}^{m,k}_{\operatorname{FM}}\longrightarrow\mathcal{A}_k.$$ We will prove the existence of two isomorphisms of functors $$\lambda_{\operatorname{def}}\colon \Phi^{m,k}_{\operatorname{def}}\longrightarrow \mathsf{pt}^{m,k}_{\operatorname{def}}\qquad\mbox{ and }\qquad \lambda_{\operatorname{FM}}\colon \Phi^{m,k}_{\operatorname{FM}}\longrightarrow \mathsf{pt}^{m,k}_{\operatorname{FM}},$$ from which the statement will follow by definition. Let us start with $\lambda_{\operatorname{def}}$. For any object $(S,v,H)\in\mathcal{G}^{m,k}_{\operatorname{def}}$, Theorem [Theorem 27](#thm:PR v perp){reference-type="ref" reference="thm:PR v perp"} provides an isometry $$\lambda_{(S,v,H)}\colon v^\perp\longrightarrow\operatorname{H}^2(M_v(S,H),\mathbb{Z}).$$ For $i=1,2$, let $A_i=(S_i,v_i,H_i)\in\mathcal{G}^{m,k}_{\operatorname{def}}$ be two objects, and let us take a morphism $h\in\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{def}}}(A_1,A_2)$. To conclude the proof of this first step, we need to show that there is a commutative diagram $$\label{eqn:nat iso def} \xymatrix{ v_1^\perp\ar@{->}[r]^-{\lambda_{A_1}}\ar@{->}[d]_{\Phi^{m,k}_{\operatorname{def}}(h)} & \operatorname{H}^2(M_{v_1}(S_1,H_1),\mathbb{Z})\ar@{->}[d]^{\mathsf{pt}^{m,k}_{\operatorname{def}}(h)} \\ v_2^\perp\ar@{->}[r]_-{\lambda_{A_2}} & \operatorname{H}^2(M_{v_2}(S_2,H_2),\mathbb{Z})\, . }$$ We first prove the claim when $h=\overline{\alpha}$, where $\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_{1},t_{2},\gamma)$ is a deformation path. In this case, by Remark [Remark 57](#rmk:pto on H tilde){reference-type="ref" reference="rmk:pto on H tilde"}, we have that $\widetilde{\Phi}^{m,k}_{\operatorname{def}}(\alpha)=p_{\alpha}$ is orientation preserving. Let $p\colon\mathcal{M}\to T$ be the family of moduli spaces induced by $(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H})$. By definition there exists a flat section $\mathsf{v}$ of the local system $R^\bullet f_*\mathbb{Z}$ such that $\mathsf{v}_{t_1}=v_1$ and $\mathsf{v}_{t_2}=v_2$. We denote by $\mathsf{v}^\perp$ the corresponding sub-local system of $R^\bullet f_*\mathbb{Z}$. By Remark [Remark 35](#rmk:lambda in famiglie){reference-type="ref" reference="rmk:lambda in famiglie"}, the isometries $\lambda_{A_1}$ and $\lambda_{A_2}$ fit in an isomorphism of local systems $$\lambda_{\mathsf{v}}\colon \mathsf{v}^\perp\longrightarrow R^2\phi_*\mathbb{Z}.$$ The commutativity of the diagram ([\[eqn:nat iso def\]](#eqn:nat iso def){reference-type="ref" reference="eqn:nat iso def"}) then follows since both $\Phi^{m,k}_{\operatorname{def}}(\alpha)=p_{\alpha}$ and $\mathsf{pt}^{m,k}_{\operatorname{def}}(\alpha)=g_{\alpha}$ are parallel transport operators in the two families associated to $\alpha$ (see also Remark [Remark 38](#rmk:solo p alpha){reference-type="ref" reference="rmk:solo p alpha"}). Let us now turn to the case where $h=\chi_{H_{1},H_{2}}$, i.e. when $(S_{1},v_{1},H_{1})$ and $(S_{2},v_{2},H_{2})$ are congruent: then $S_{1}=S_{2}=S$, $v_{1}=v_{2}=v$ and $\chi_{H_1,H_2}$ is the identification $M_v(S,H_1)=M_v(S,H_2)$ (see Definitions [Definition 23](#defn:congruent){reference-type="ref" reference="defn:congruent"} and [Definition 43](#defn:P m k){reference-type="ref" reference="defn:P m k"}). In this case we see that $\Phi^{m,k}_{\operatorname{def}}(h)=\operatorname{id}_{v^{\perp}}$, $\mathsf{pt}^{m,k}_{\operatorname{def}}(h)=\operatorname{id}_{\operatorname{H}^{2}(M_{v}(S,H_{1}))}$, and we have an identification $\operatorname{H}^{2}(M_{v}(S,H_{1}),\mathbb{Z})=\operatorname{H}^{2}(M_{v}(S,H_{2}),\mathbb{Z})$ and an identification $\lambda_{A_{1}}=\lambda_{A_{2}}$: diagram ([\[eqn:nat iso def\]](#eqn:nat iso def){reference-type="ref" reference="eqn:nat iso def"}) is then commutative. By Definition [Definition 44](#defn:G m k def){reference-type="ref" reference="defn:G m k def"}, we then deduce the existence of the isomorphism $\lambda_{\operatorname{def}}$. We are now left with $\lambda_{\operatorname{FM}}$, whose construction follows in the same way. First of all, we define it on objects as in the previous case. Let $A_1=(S,v_1,H)$ and $A_2=(S,v_2,H)$ be two objects and let $\phi\in\operatorname{Aut}(\operatorname{D}^b(S))$ be an equivalence of categories that induces an isomorphism $\mathsf{\phi}_{v_{1}}\colon M_{v_1}(S,H)\to M_{v_2}(S,H)$. We need to prove that there exists a commutative diagram $$\label{eqn:nat iso FM} \xymatrix{ v_1^\perp\ar@{->}[r]^-{\lambda_{A_1}}\ar@{->}[d]_{\Phi^{m,k}_{\operatorname{FM}}(\phi)} & \operatorname{H}^2(M_{v_1}(S_1,H_1),\mathbb{Z})\ar@{->}[d]^{\mathsf{pt}^{m,k}_{\operatorname{FM}}(\phi)} \\ v_2^\perp\ar@{->}[r]_-{\lambda_{A_2}} & \operatorname{H}^2(M_{v_2}(S_2,H_2),\mathbb{Z})\, . }$$ Recall that, by definition, $\mathsf{pt}^{m,k}_{\operatorname{FM}}(\phi)=\phi_{v_1,*}$ and $\Phi^{m,k}_{\operatorname{FM}}(\phi)=(-1)^{\operatorname{or}(\phi^{\operatorname{H}})} \phi^{\operatorname{H}}|_{v_1^\perp}$, where $\phi^{\operatorname{H}}$ is the action of $\phi$ on the Mukai lattice and $\operatorname{or}$ is the orientation character ([\[eqn:or tra due\]](#eqn:or tra due){reference-type="ref" reference="eqn:or tra due"}). First of all, we notice that it is enough to prove the result only when $\phi=\mathsf{L},\operatorname{FM}_\Delta,\operatorname{FM}^\vee_\Delta,\operatorname{FM}_{\mathcal{P}}$ are as in Definition [Definition 48](#defn:G m k FM){reference-type="ref" reference="defn:G m k FM"}. Now, if $\phi=\mathsf{L},\operatorname{FM}_\Delta$, then $\phi^{\operatorname{H}}$ is orientation preserving (see [@HS2005 Remark 5.4]) and by [@Yoshioka:ModuliAbelian Proposition 2.4] $$\lambda_{A_2}^{-1}\circ \mathsf{\phi}_{v_1,*} \circ \lambda_{A_1}=\phi^{\operatorname{H}}|_{v_1^\perp}.$$ If $\phi=\operatorname{FM}^\vee_\Delta$, then $\phi^{\operatorname{H}}$ is orientation reversing. In fact the duality equivalence $(-)^\vee\in\operatorname{Aut}(\operatorname{D}^b(S))$ induces in cohomology the isometry $$\delta\colon (r,c,s)\mapsto (r,-c,s)$$ which is orientation reversing: it changes the sign to a Kähler and a symplectic form, but it is the identity on the hyperbolic plane generated by $\operatorname{H}^0(S,\mathbb{Z})$ and $\operatorname{H}^4(S,\mathbb{Z})$. In this case, by [@Yoshioka:ModuliAbelian Proposition 2.5] we have that $$\lambda_{A_2}^{-1}\circ \mathsf{\phi}_{v_1,*} \circ \lambda_{A_1}=-\phi^{\operatorname{H}}|_{v_1^\perp}.$$ Finally , if $\phi=\operatorname{FM}_{\mathcal{P}}$, then by [@HS2005 Remark 5.4, Proposition 5.5.] $\phi^{\operatorname{H}}$ is orientation preserving and by [@Yoshioka:ModuliAbelian Proposition 2.4] $$\lambda_{A_2}^{-1}\circ \mathsf{\phi}_{v_1,*} \circ \lambda_{A_1}=\phi^{\operatorname{H}}|_{v_1^\perp}.$$ Those are exactly the commutativity condition needed to define the isomorphism $\lambda_{\operatorname{FM}}$ and we are done. ◻ # Polarised monodromy of K3 surfaces and its lift to moduli spaces {#section:mon S pol} The first part of this section is dedicated to show that the monodromy group of a K3 surface can be generated only by polarised monodromy operators. This result is well-known to experts but a rigorous proof is lacking in literature. In the second and last part of the section we will instead show how to lift polarised monodromy operators on a K3 surface to some moduli spaces of sheaves on the same K3 surface. ## The monodromy group of a K3 surface {#section:Mon S} The aim of this section is to prove the following result. **Theorem 72**. *Let $S$ be a projective K3 surface. Then the monodromy group $\operatorname{Mon}^2(S)$ is generated by polarised parallel transport operators.* In the statement above we mean that there exists a set of generators of $\operatorname{Mon}^2(S)$ each of which is composition of polarised parallel transport operators (see Definition [Definition 15](#defn:ppto){reference-type="ref" reference="defn:ppto"} for the notion of polarised parallel transport operator). By deforming the K3 surface via polarised families, it is enough to prove the statement for a special example. We will work with a projective elliptic K3 surface $p\colon S\to\mathbb{P}^1$ with a section and with Picard rank $2$. In particular, if we denote by $f$ the class of the fibre and by $\ell$ the class of the section, then $$\operatorname{Pic}(S)=\mathbb{Z}.\ell\oplus\mathbb{Z}.f= \left(\begin{matrix} -2 & 1 \\ 1 & 0 \end{matrix}\right)$$ is the unimodular hyperbolic plane. Let us put $e=\ell+f$, so that $e$ and $f$ form the standard basis of the hyperbolic plane. Let us recall that in this case a class $h=\alpha e + \beta f$ is ample when the ratio $\beta/\alpha>1$. Let us now choose three positive integers $r,k,p\gg0$, and consider the following ample classes, $$h_1=e+rf,\qquad h_2=e+(r-1)f,\qquad h_3=se+pf,\quad\mbox{and}\quad h_4=(s-1)e+pf.$$ We will prove the following result, which will imply Theorem [Theorem 72](#thm:mon S generated by ppto){reference-type="ref" reference="thm:mon S generated by ppto"}. **Proposition 73**. *Let $S$ be a projective elliptic K3 surface with a section and Picard rank $2$. Let $H_1$, $H_2$, $H_3$ and $H_4$ be four polarisations whose classes are the classes $h_1$, $h_2$, $h_3$ and $h_4$ above. Then $$\operatorname{Mon}^2(S)=\langle \operatorname{Mon}^2(S,H_1),\operatorname{Mon}^2(S,H_2),\operatorname{Mon}^2(S,H_3),\operatorname{Mon}^{2}(S,H_{4})\rangle.$$* First of all, let us recall the following well-known result due to [@Peters Theorem 3.4.1 and Corollary 3.4.2]. The following statement can be found also in [@Markman:Monodromy Proposition 6.8]. **Lemma 74** ([@Peters]). *Let $S$ be a projective K3 surface, $H\in \operatorname{Pic}(S)$ an ample line bundle and $h\in\operatorname{H}^2(S,\mathbb{Z})$ its class. Then $$\operatorname{Mon}^2(S,H)=\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))_h,$$ where $\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))_h$ is the group of orientation preserving isometries $g$ such that $g(h)=h$.* *Moreover, the whole group $\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))_h$ arises as the monodromy group of a projective polarised family $f\colon\mathcal{S}\to T$ of K3 surface over a smooth and quasi-projective base $T$.* The smooth and quasi-projective base $T$ of the family above is the image under the period map of the moduli space of polarised K3 surfaces with fixed degree (see [@Markman:Monodromy Section 6.3] for more details). The last ingredient we need is the following lattice-theoretic result, which is a straighforward application of the Eichler criterion. We remark that a very similar result already appeared in the proofs of [@MonRap Lemma 3.5] and [@Onorati:Monodromy Theorem 5.4] for similar purposes. Let $L$ be an even lattice and let us assume that $L$ contains at least three copies of the unimodular hyperbolic plane $U$. We denote by $U$ one distinguished such copy, with basis $\{e,f\}$, and we write $L=U\oplus L_1$. Given $s,r,p>1$, we use the following notation, $$h_1=e+rf,\qquad h_2=e+(r-1)f,\qquad h_3=se+pf,\quad\mbox{and}\quad h_4=(s-1)e+pf.$$ **Lemma 75**. *With the notation above, we have $$\operatorname{O}^+(L)=\langle \operatorname{O}^+(L)_{h_1},\operatorname{O}^+(L)_{h_2},\operatorname{O}^+(L)_{h_3},\operatorname{O}^+(L)_{h_4}\rangle.$$* Here $\operatorname{O}^+(L)_{h_i}$ is the subgroup of $\operatorname{O}^+(L)$ of the elements fixing $h_i$. *Proof.* First of all, by [@GHS09 Proposition 3.3 (iii)], we have that $$\operatorname{O}^+(L)=\langle \operatorname{O}^+(L_1), E_{\bar{U}}(L_1)\rangle,$$ where $\operatorname{O}^+(L_1)$ is embedded in $\operatorname{O}^+(L)$ by extending any isometry as the identity on $U$, and where $E_{U}(L_1)$ is the group generated by transvections of the form $t(e,a)$ and $t(f,a)$ for all $a\in L_1$. We recall that a transvection is defined in the following way: for any $z\in L$ with $z^2=0$ and $a\in z^\perp$, then $$t(z,a)\colon x\mapsto x-(a,x)z+(z,x)a-\frac{1}{2}(a,a)(z,x)z.$$ We refer to [@GHS09 Section 3] for backgrounds and main properties, but we point out that if $a^2\neq0$, then the extension of $t(z,a)$ over $\mathbb{Q}$ is the composition of the reflection $R_{a}$ with the reflection $R_{a+\frac{1}{2}(a,a)z}$. Any transvection $t(z,a)$ is an orientation preserving isometry with determinant $1$ (acting trivially on the discriminant group of $L$). As $h_i\in U$, it follows that $\operatorname{O}^+(L_1)\subset\operatorname{O}^+(L)_{h_i}$ for $i=1,2,3,4$. Therefore, if we denote by $G$ the group generated by the $\operatorname{O}^+(L)_{h_i}$'s, we only need to show that $t(e,a),t(f,a)\in G$ for all $a\in L_1$. By assumption $L_1$ contains at least two copies of the hyperbolic plane, so that we can apply [@GHS09 Proposition 3.3 (ii),] and find an isometry $g\in\operatorname{O}^+(L_1)$ such that $g(a)\in \bar{U}$, where $\bar{U}$ is a distinguished copy of the hyperbolic plane in $L_1$. By [@GHS09 Relation (6) in Section 3] we have that $$g\circ t(z,a)\circ g^{-1}=t(z,g(a))$$ for any isotropic element $z$. In particular, since we already remarked that $\operatorname{O}^+(L_1)\subset G$, it is enough to prove that $t(e,a),t(f,a)\in G$ for all $a\in \bar{U}$. As already remarked, $t(e,a)$ and $t(f,a)$ are orientation preserving isometries with determinant $1$; moreover, since $a\in \bar{U}$, $t(e,a)$ and $t(f,a)$ both act as the identity on $(U\oplus \bar{U})^\perp$, and without loss of generality they can then be viewed as elements in $\operatorname{SO}^+(U\oplus \bar{U})$. If we denote by $\{\bar{e},\bar{f}\}$ a standard basis for $\bar{U}$, then by [@GHS09 Lemma 3.2] we know that $\operatorname{SO}^+(U\oplus \bar{U})$ is generated by the four transvections $$t(\bar{e},e),\qquad t(\bar{e},f), \qquad t(\bar{f},e)\qquad\mbox{and}\qquad t(\bar{f},f).$$ So it is enough to prove that each of them is in $G$. Let us show that $t(\bar{e},f)\in G$, the others will follow similarly. We use the following remark, which follows directly from the definition of transvection: if $a\in U^\perp$, then $t(e,a)$ and $t(f,a)$ are the identity on the sublattice $U^\perp\cap a^\perp$. In particular, for example, $t(\bar{e},e-tf)$ acts as the identity on $e+tf$. By [@GHS09 Relation (5) in Section 3], we have in general that $t(z,a)\circ t(z,b)=t(z,a+b)$ and $t(z,-a)=t(z,a)^{-1}$. In our situation this says that $$t(\bar{e},e-rf)\circ t(\bar{e},-e+(r-1)f)=t(\bar{e},f)^{-1},$$ but as remarked $t(\bar{e},e-rf)\in\operatorname{O}^+(L)_{h_1}$ and similarly $t(\bar{e},-e+(r-1)f)=t(\bar{e},e-(r-1)f)^{-1}\in\operatorname{O}^+(L)_{h_1}$, concluding the proof. ◻ Let us keep the notations as in the statement of Proposition [Proposition 73](#prop:mon S pol){reference-type="ref" reference="prop:mon S pol"}. By Lemma [Lemma 74](#lemma:pol){reference-type="ref" reference="lemma:pol"} we have that $\operatorname{Mon}^2(S,H_i)=\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))_{h_i}$, for $i=1,2,3,4$. In particular, by Lemma [Lemma 75](#lemma:eichler){reference-type="ref" reference="lemma:eichler"}, we have that $$\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))=\langle \operatorname{Mon}^2(S,H_1),\operatorname{Mon}^2(S,H_2),\operatorname{Mon}^2(S,H_3),\operatorname{Mon}^{2}(S,H_{4})\rangle.$$ Since $\operatorname{Mon}^2(S)\subset\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$ (see Lemma [Lemma 13](#lemma:Mon is O+){reference-type="ref" reference="lemma:Mon is O+"}), the claim follows. Let us conclude with the following corollary, which recover a well-known result of Borel. **Corollary 76** ([@Bor]). *If $S$ is a K3 surface, then $$\operatorname{Mon}^2(S)=\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z})).$$* ## Lift of the polarised monodromy of a K3 surface to moduli spaces of sheaves {#section:lift} In this section we exhibit some special cases in which we can lift polarised monodromy operators of a K3 surface $S$ to monodromy operators on a moduli space $M_v(S,H)$, where $(S,v,H)$ is a suitable $(m,k)$-triple. We will achieve this by using the groupoid representations defined in Section [2.4](#section:representation){reference-type="ref" reference="section:representation"}. Since $\operatorname{H}^2(S,\mathbb{Z})\subset\widetilde{\operatorname{H}}(S,\mathbb{Z})$, the group $\operatorname{O}(\operatorname{H}^2(S,\mathbb{Z}))$ is identified with the subgroup of $\operatorname{O}(\widetilde{\operatorname{H}}(S,\mathbb{Z}))$ of isometries acting as the identity on $\operatorname{H}^0(S,\mathbb{Z})\oplus\operatorname{H}^4(S,\mathbb{Z})$. On the other hand, if $(S,v,H)$ is an $(m,k)$-triple, using the notation set in Section [2.1](#section:groupoids){reference-type="ref" reference="section:groupoids"}, we consider the homomorphism of groups $$\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)}\colon\operatorname{Aut}_{\mathcal{G}_{\operatorname{def}}^{m,k}}(S,v,H)\longrightarrow\operatorname{Aut}_{\widetilde{\mathcal{H}}^{m,k}}(\tilde{\operatorname{H}}(S,\mathbb{Z}),v,\epsilon_S)$$ induced by the representation $\widetilde{\Phi}^{m,k}_{\operatorname{def}}\colon\mathcal{G}_{\operatorname{def}}^{m,k}\to\widetilde{\mathcal{H}}^{m,k}$ defined in Definition [Definition 55](#defn:Phi tilde def m k){reference-type="ref" reference="defn:Phi tilde def m k"}. Since by definition $$\operatorname{Aut}_{\widetilde{\mathcal{H}}^{m,k}}(\tilde{\operatorname{H}}(S,\mathbb{Z}),v,\epsilon_S)=\operatorname{O}(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v,$$ it follows that also $$\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)})\subset\operatorname{O}(\widetilde{\operatorname{H}}(S,\mathbb{Z})).$$ In the following lemma, we relate the subgroup $\operatorname{Mon}^2(S,H)\subset\operatorname{O}(\operatorname{H}^2(S,\mathbb{Z}))$ with $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)})$, at least when the Mukai vector is $(m,0,-mk)$. We will also relate $\operatorname{Mon}^2(S,H)$ with the image of the homomorphism of groups $$\mathsf{pt}^{m,k}_{\operatorname{def},(S,v,H)}\colon\operatorname{Aut}_{\mathcal{G}_{\operatorname{def}}^{m,k}}(S,v,H)\longrightarrow\operatorname{Aut}_{\mathcal{A}_k}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))$$ induced by the representation $\mathsf{pt}^{m,k}_{\operatorname{def}}\colon\mathcal{G}_{\operatorname{def}}^{m,k}\to\mathcal{A}_k$ defined in Definition [Definition 64](#defn:pt m k def){reference-type="ref" reference="defn:pt m k def"}. **Lemma 77**. *Let $m,k\geq1$ be two integers and $v=(m,0,-mk)$ a Mukai vector on a projective K3 surface $S$. If $(S,v,H)\in\mathcal{G}^{m,k}_{\operatorname{def}}$ is an object, then there is an inclusion of groups $$\operatorname{Mon}^2(S,H)\subset\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)})$$ and an injective morphism of groups $$\operatorname{Mon}^2(S,H)\hookrightarrow\operatorname{Im}(\mathsf{pt}^{m,k}_{\operatorname{def},(S,v,H)}).$$* *Proof.* Let us start with the inclusion of $\operatorname{Mon}^2(S,H)$ in $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)})$. Recall from Lemma [Lemma 74](#lemma:pol){reference-type="ref" reference="lemma:pol"} that there exists a projective polarised family $f\colon\mathcal{S}\to T$ of projective K3 surfaces such that any isometry in $\operatorname{Mon}^2(S,H)$ arises as a parallel transport operator in the local system $R^2 f_*\mathbb{Z}$, and such that the base $T$ is smooth and quasi-projective. More precisely, the family $f$ comes with the following properties: - there exists $\bar{t}\in T$ such that $\mathcal{S}_{\bar{t}}=S$; - there exists a relatively ample line bundle $\mathcal{H}$ on $\mathcal{S}$ such that $\mathcal{H}_{\bar{t}}=H$. Any $g\in\operatorname{Mon}^2(S,H)$, is then associated to a loop $\gamma$ in $T$ centred in $\bar{t}$. As in the discussion before the lemma, we can view $g$ as an element of $\operatorname{Aut}_{\widetilde{\mathcal{H}}^{m,k}}(\widetilde{\operatorname{H}}(S,\mathbb{Z}),v,\epsilon_S)$, and we have to prove that there is a deformation path $\alpha$ from $(S,v,H)$ to itself such that $g=\widetilde{\Phi}_{\operatorname{def}}^{m,k}(\bar{\alpha})$. To do so, we first notice that the subset $Z\subset T$ of the points $t\in T$ such that $\mathcal{H}_t$ is not $v_{t}$-generic is a closed subset of $T$ (see Remark [Remark 34](#rmk:defomk){reference-type="ref" reference="rmk:defomk"}). Moreover, since by assumptions $\mathcal{H}_{\bar{t}}$ is $v$-generic, $Z$ is strictly contained in $T$ and hence it must have real codimension at least $2$. In particular, since the base $T$ is smooth, there exists a loop $\gamma'$ in $T':=T\setminus Z$ centred in $\bar{t}$ that is homotopic to $\gamma$ (see for example [@Godbillon Théorème 2.3 in Chapter X]). Since parallel transport operators do not depend on the homotopy class of the path, we may suppose without loss of generality that $\mathcal{H}_t$ is $v$-generic for every $t\in T$. By assumption the Mukai vector is of the form $(m,0,-mk)$, therefore it follows that $$\alpha=(f\colon\mathcal{S}\to T, \mathcal{O}_{\mathcal{S}},\mathcal{H},\bar{t},\bar{t},\gamma)$$ defines a morphism $\overline{\alpha}$ in $\operatorname{Aut}_{\mathcal{G}^{m,k}_{\operatorname{def}}}(S,v,H)$. By definition, we get that $$g=\widetilde{\Phi}^{m,k}_{\operatorname{def}}(\overline{\alpha}),$$ concluding the first part of the proof. Now, since by Remark [Remark 57](#rmk:pto on H tilde){reference-type="ref" reference="rmk:pto on H tilde"} the isometries of $\operatorname{Mon}^2(S,H)$ preserve the orientation of $\widetilde{\operatorname{H}}(S,\mathbb{Z})$, it follows that the inclusion just proved gives an injective map $$\operatorname{Mon}^2(S,H)\hookrightarrow\operatorname{Im}(\Phi^{m,k}_{\operatorname{def},(S,v,H)}).$$ Combining this with the isomorphism of functors $\lambda_{\operatorname{def}}\colon\Phi^{m,k}_{\operatorname{def}}\to\mathsf{pt}^{m,k}_{\operatorname{def}}$ constructed in the proof of Proposition [Proposition 71](#prop:iso of functors){reference-type="ref" reference="prop:iso of functors"} finishes the proof. ◻ **Remark 78**. The claim of Lemma [Lemma 77](#lemma:pto in Im(Phi tilde) when c=0){reference-type="ref" reference="lemma:pto in Im(Phi tilde) when c=0"} holds more generally when the Mukai vector has the form $(r,0,s)$, but we will only use it in the special case of the statement. **Remark 79**. If $v=(m,0,-mk)$, then $\operatorname{H}^2(S,\mathbb{Z})\subset v^\perp$ and again the group $\operatorname{O}(\operatorname{H}^2(S,\mathbb{Z}))$ is naturally embedded in $\operatorname{O}(v^\perp)$ by extending with the identity. Using the isometry $\lambda_{(S,v,H)}\colon v^\perp\longrightarrow\operatorname{H}^2(M_v(S,H),\mathbb{Z})$, we have a natural injective morphism $$\operatorname{O}(\operatorname{H}^2(S,\mathbb{Z}))\hookrightarrow\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z})$$ of which $$\operatorname{Mon}^2(S,H)\hookrightarrow\operatorname{Im}(\mathsf{pt}^{m,k}_{\operatorname{def},(S,v,H)})$$ is the restriction. The following is the main result of this section. In what follows by a *very general projective elliptic K3 surface with a section* we mean a projective elliptic K3 surface having a section and having Picard rank $2$. As in Section [3.1](#section:Mon S){reference-type="ref" reference="section:Mon S"}, we denote by $f$ the class of a fibre and by $\ell$ the class of the section. Then $\operatorname{Pic}(S)$ is isometric to the unimodular hyperbolic plane with standard basis $e=\ell+f$ and $f$. **Proposition 80**. *Assume that $v=(m,0,-mk)$ and $S$ is a very general projective elliptic K3 surface with a section. There exists an integer $t\gg0$ and a $v$-generic polarisation $H=e+tf$ such that $$\operatorname{Mon}^2(S)\subset\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)}).$$* *Proof.* By Proposition [Proposition 73](#prop:mon S pol){reference-type="ref" reference="prop:mon S pol"} we know that there are four polarisations $H_1$, $H_2$, $H_3$ and $H_4$ on $S$ such that $\operatorname{Mon}^2(S)$ is generated by the $\operatorname{Mon}^2(S,H_i)$'s. Now, we claim that the four polarisations $H_1$, $H_2$, $H_3$ and $H_4$ can be chosen to belong to the same $v$-chamber (see Remark [Remark 20](#rmk:wall and chamber){reference-type="ref" reference="rmk:wall and chamber"}), so that they are $v$-generic. In particular, by Lemma [Lemma 77](#lemma:pto in Im(Phi tilde) when c=0){reference-type="ref" reference="lemma:pto in Im(Phi tilde) when c=0"} we will have that $$\operatorname{Mon}^2(S,H_i)\subset\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H_i)}),$$ and by Remark [Remark 24](#rmk:triple congruenti){reference-type="ref" reference="rmk:triple congruenti"} the groups $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H_i)})$ can all be identified: the proposition will then follow from Proposition [Proposition 73](#prop:mon S pol){reference-type="ref" reference="prop:mon S pol"}. Indeed, let us recall how the polarisations $H_i$ are chosen. If $f$ is the class of the fibration of the elliptic surface $S$ and $\ell$ is the class of the section, then we put $e=\ell+f$ and we choose $$H_1=e+rf,\quad H_2=e+(r-1)f,\quad H_3=se+pf,\quad\mbox{and}\quad H_4=(s-1)e+pf$$ for $r,k,p\gg0$. Now, by [@OGrady:WeightTwo Lemma I.0.3] (see also [@PR:SingularVarieties Definition 2.37 and Lemma 2.38]), if $r$ is big enough and $s$ and $p$ are chosen such that the quotient $p/s$ is big enough, then all the $H_i$'s are contained in the unique $v$-chamber whose closure contains the class $f$. We may now conclude the proof: choose a polarisation $H$ on $S$ that lies in the same $v$-chamber of $H_{1},\cdots,H_{4}$, i.e. $H$ belongs to the unique $v$-chamber whose closure contains $f$. By definition of $\mathcal{G}^{m,k}_{\operatorname{def}}$, as $H$ and $H_{i}$ lie in the closure of the same $v$-chamber for $i=1,2,3,4$, we have an isomorphism $\chi_{H,H_{i}}\colon(S,v,H)\to(S,v,H_{i})$ in $\mathcal{G}^{m,k}$ and hence an isomorphism of groups $$\chi_{H,H_{i}}^{\sharp}\colon\operatorname{Aut}_{\mathcal{G}^{m,k}_{\operatorname{def}}}(S,v,H)\longrightarrow \operatorname{Aut}_{\mathcal{G}^{m,k}_{\operatorname{def}}}(S,v,H_{i})$$ given by conjugation with $\chi_{H,H_{i}}$. Since $\widetilde{\Phi}^{m,k}_{\operatorname{def}}(\chi_{H,H_{i}})$ is the identity, we get that $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H_i)})=\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)})$ for every $i=1,2,3,4$, and this concludes the proof. ◻ **Remark 81**. If $v$ is primitive, then the same result was proved in [@Markman:Monodromy Corollary 6.7]. We wish to point out that even though the idea of our proof and Markman's one is the same, they differ inasmuch as Markman uses non-polarised deformations of K3 surfaces (compare [@Markman:Monodromy Definition 6.2] with our Definition [Definition 44](#defn:G m k def){reference-type="ref" reference="defn:G m k def"}). If $v$ is primitive this is justified by the fact that $M_{(1,0,-k)}(S,H)\cong\operatorname{Hilb}^{k+1}(S)$. On the other hand, if $v$ is not primitive we are forced to use polarised families. **Corollary 82**. *Assume that $v=(m,0,-mk)$ and $S$ is a very general projective elliptic K3 surface with a section. If $H$ is a $v$-generic polarisation contained in the unique $v$-chamber whose closure contains the nef class $f$, then there is an injective morphism $$\operatorname{Mon}^2(S)\hookrightarrow\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H)).$$* *Proof.* As in the last part of the proof of Lemma [Lemma 77](#lemma:pto in Im(Phi tilde) when c=0){reference-type="ref" reference="lemma:pto in Im(Phi tilde) when c=0"}, by Remark [Remark 57](#rmk:pto on H tilde){reference-type="ref" reference="rmk:pto on H tilde"} and Proposition [Proposition 71](#prop:iso of functors){reference-type="ref" reference="prop:iso of functors"} the inclusion of Proposition [Proposition 80](#prop:Mon S in m k){reference-type="ref" reference="prop:Mon S in m k"} gives an injective morphism $$\operatorname{Mon}^2(S)\hookrightarrow\operatorname{Im}(\mathsf{pt}^{m,k}_{\operatorname{def},(S,v,H)}).$$ Then the claim follows from Corollary [Corollary 69](#cor:Im of pt in Mon){reference-type="ref" reference="cor:Im of pt in Mon"}. ◻ Finally, we want to exhibit an analogous version of Lemma [Lemma 77](#lemma:pto in Im(Phi tilde) when c=0){reference-type="ref" reference="lemma:pto in Im(Phi tilde) when c=0"} when the K3 surface is general enough, which will be useful later in Section [4.1](#section:W in m k){reference-type="ref" reference="section:W in m k"}. **Lemma 83**. *Let $S$ be a projective K3 surface with $\operatorname{Pic}(S)=\mathbb{Z}.H$ and $v=(r,mH,s)$ a Mukai vector. If $(S,v,H)$ is an $(m,k)$-triple, then there exists an inclusion of groups $$\operatorname{Mon}^2(S,H)\subset\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,v,H)})$$ and an injective morphism of groups $$\operatorname{Mon}^2(S,H)\hookrightarrow\operatorname{Im}(\mathsf{pt}^{m,k}_{\operatorname{def},(S,v,H)}).$$* *Proof.* The proof runs verbatim as the proof of Lemma [Lemma 77](#lemma:pto in Im(Phi tilde) when c=0){reference-type="ref" reference="lemma:pto in Im(Phi tilde) when c=0"}, the only difference being now that, with the same notation, the deformation path is $$\alpha=(f\colon\mathcal{S}\to T,\mathcal{H}^{\otimes m},\mathcal{H},\bar{t},\bar{t},\gamma).$$ ◻ # The locally trivial monodromy group {#section:main} Let $(S,v,H)$ be an $(m,k)$-triple. As usual we write $v=mw$, with $w$ primitive, and we consider the $(1,k)$-triple $(S,w,H)$ (cf. Remark [Remark 22](#rmk:gen){reference-type="ref" reference="rmk:gen"}). By Remark [Remark 26](#rmk:strat M){reference-type="ref" reference="rmk:strat M"} we can identify the (smooth) moduli space $M_w(S,H)$ with the most singular locus of the (singular) moduli space $M_v(S,H)$. In particular we consider the closed embedding $$i_{w,m}\colon M_w(S,H)\longrightarrow M_v(S,H).$$ The aim of this section is to relate the locally trivial monodromy group of $M_v(S,H)$ to the monodromy group of $M_w(S,H)$ by mean of the morphism $i_{w,m}^\sharp$ defined in Lemma [Lemma 30](#lemma:i sharp){reference-type="ref" reference="lemma:i sharp"}. In Section [4.1](#section:W in m k){reference-type="ref" reference="section:W in m k"} we construct locally trivial monodromy operators and show that those operators form a distinguished group. In Section [4.2](#subsection:i sharp){reference-type="ref" reference="subsection:i sharp"} we prove that $i_{w,m}^\sharp$ is injective. In Section [4.3](#section:main result){reference-type="ref" reference="section:main result"} we show that the monodromy operators constructed before are all and only by using the morphism $i_{w,m}^\sharp$ as a constraint; as a corollary we get the fact that $i_{w,m}^\sharp$ is an isomorphism. ## The group $\mathsf{W}(v^\perp)$ as a subgroup of $\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H))$ {#section:W in m k} Let us recall the following notation. If $L$ is an abstract lattice, the nondegenerate bilinear form embeds $L$ into its dual $L^*=\operatorname{Hom}(L,\mathbb{Z})$: the group $L^*/L$ is denoted by $A_L$ and called *discriminant group* of $L$. Any isometry $g\in\operatorname{O}(L)$ extends to an isometry on $L^*$ and hence it induces an automorphism of $A_L$. We denote by $\mathsf{W}(L)\subset\operatorname{O}^+(L)$ the subgroup of orientation preserving isometries acting as $\pm\operatorname{id}$ on $A_{L}$. When $L=\operatorname{H}^2(X,\mathbb{Z})$ for an irreducible symplectic variety $X$, then we simply write $\mathsf{W}(X)$ instead of $\mathsf{W}(\operatorname{H}^2(X,\mathbb{Z}))$. **Remark 84**. Let $S$ be a projective K3 surface and $v\in\widetilde{\operatorname{H}}(S,\mathbb{Z})$ an element. The group $\mathsf{W}(v^\perp)$ consists of orientation preserving isometries of $v^{\perp}$ that extend to isometries of $\widetilde{\operatorname{H}}(S,\mathbb{Z})$. More precisely, if $g\in\mathsf{W}(v^\perp)$, then there exists $\tilde{g}\in\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))$ such that $\tilde{g}(v)=\pm v$ and $\tilde{g}|_{v^\perp}=g$ (see [@Markman:Monodromy Lemma 4.10]). The main result of this section is the following. **Theorem 85**. *Let $(S,v,H)$ be an $(m,k)$-triple. Then $$\mathsf{W}(M_v(S,H))\subset\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H)).$$* **Remark 86**. When $v$ is primitive, this statement is the main result in [@Markman:Monodromy]. In fact our proof parallels Markman's one and only differs from his in some technical details arising from the non-primitivity of the Mukai vector. Our tool to produce locally trivial monodromy operators is via the representation $\mathsf{pt}^{m,k}\colon\mathcal{G}^{m,k}\to\mathcal{A}_k$ defined in Definition [Definition 67](#defn:pt m k){reference-type="ref" reference="defn:pt m k"} (cf. Corollary [Corollary 69](#cor:Im of pt in Mon){reference-type="ref" reference="cor:Im of pt in Mon"}); on the other end, because of the isomorphism $\lambda\colon\mathsf{\Phi}^{m,k}\to\mathsf{pt}^{m,k}$ (cf. Proposition [Proposition 71](#prop:iso of functors){reference-type="ref" reference="prop:iso of functors"}), we will instead look at the representation $\Phi^{m,k}\colon\mathcal{G}^{m,k}\to\mathcal{A}_k$ defined in Definition [Definition 61](#defn: Phi tilde m k){reference-type="ref" reference="defn: Phi tilde m k"}. Since $\Phi^{m,k}=\Psi\circ\widetilde{\Phi}^{m,k}$, our most technical result involve the representation $\widetilde{\Phi}^{m,k}\colon\mathcal{G}^{m,k}\to\widetilde{\mathcal{H}}^{m,k}$ defined in Definition [Definition 61](#defn: Phi tilde m k){reference-type="ref" reference="defn: Phi tilde m k"}. If $(S,v,H)$ is an $(m,k)$-triple, then we have the induced homomorphism $$\widetilde{\Phi}^{m,k}_{(S,v,H)}\colon\operatorname{Aut}_{\mathcal{G}^{m,k}}(S,v,H)\longrightarrow\operatorname{Aut}_{\widetilde{\mathcal{H}}^{m,k}}(\widetilde{\operatorname{H}}(S,\mathbb{Z}),v)$$ (cf. Section [2.1](#section:groupoids){reference-type="ref" reference="section:groupoids"} for the notation used). **Proposition 87**. *If $(S,v,H)$ is an $(m,k)$-triple, then $$\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)})=\operatorname{O}(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v.$$* We will now show how the theorem follows from the proposition and then we will spend the rest of the section proving the proposition. By definition, if $(\Lambda,v,\epsilon)$ is an object of $\widetilde{\mathcal{H}}^{m,k}$, then we have an homomorphism of groups $$\Psi_{(\Lambda,v,\epsilon)}\colon\operatorname{O}(\Lambda)_v\longrightarrow\operatorname{O}^+(v^\perp),\qquad g\mapsto(-1)^{\operatorname{or}(g)}g|_{v^\perp}$$ induced by the representation $\Psi\colon\widetilde{\mathcal{H}}^{m,k}\to\mathcal{A}_k$ defined in Definition [Definition 70](#defn:Psi){reference-type="ref" reference="defn:Psi"}. Moreover, by [@Markman:Monodromy Lemma 4.10] it follows that $$\operatorname{Im}(\Psi_{(\Lambda,v,\epsilon)})=\mathsf{W}(v^\perp).$$ On the other hand, if $(S,v,H)$ is an $(m,k)$-triple, by Proposition [Proposition 87](#prop:Im of Phi tilde = O){reference-type="ref" reference="prop:Im of Phi tilde = O"} the homomorphism $$\widetilde{\Phi}^{m,k}\colon\operatorname{Aut}_{\mathcal{G}^{m,k}}(S,v,H)\longrightarrow\operatorname{Aut}_{\widetilde{\mathcal{H}}^{m,k}}(\widetilde{\operatorname{H}}(S,\mathbb{Z}),v,\epsilon_S)$$ is surjective. Since by definition $\Phi^{m,k}=\Psi\circ\widetilde{\Phi}^{m,k}$, it follows that $$\label{eqn:Im of Phi = W} \operatorname{Im}(\Phi^{m,k}_{(S,v,H)})=\mathsf{W}(v^\perp).$$ Now, let us consider the isometry $$\lambda_{(S,v,H)}\colon v^\perp\longrightarrow\operatorname{H}^2(M_v(S,H),\mathbb{Z})$$ in Theorem [Theorem 27](#thm:PR v perp){reference-type="ref" reference="thm:PR v perp"} and the induced isomorphism $$\lambda_{(S,v,H)}^\sharp\colon \operatorname{O}(v^\perp)\longrightarrow\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))$$ defined in equation ([\[eqn:lambda sharp\]](#eqn:lambda sharp){reference-type="ref" reference="eqn:lambda sharp"}). Then we have a chain of equalities: $$\begin{aligned} \operatorname{Im}(\mathsf{pt}^{m,k}_{(S,v,H)}) & =\lambda_{(S,v,H)}^\sharp(\operatorname{Im}(\Phi^{m,k}_{(S,v,H)})) & \mbox{(by Proposition~\ref{prop:iso of functors})}\\ & =\lambda_{(S,v,H)}^\sharp(\mathsf{W}(v^\perp)) & \mbox{(by equality~(\ref{eqn:Im of Phi = W}))} \\ & =\mathsf{W}(M_v(S,H)). & \end{aligned}$$ Therefore the claim follows at once by Corollary [Corollary 69](#cor:Im of pt in Mon){reference-type="ref" reference="cor:Im of pt in Mon"}. Let us now prove Proposition [Proposition 87](#prop:Im of Phi tilde = O){reference-type="ref" reference="prop:Im of Phi tilde = O"}. Notice that $$\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)})\subset\operatorname{Aut}_{\widetilde{\mathcal{H}}^{m,k}}(\widetilde{\operatorname{H}}(S,\mathbb{Z}),v)=\operatorname{O}(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v,$$ where the last equality follows by definition. To prove the opposite inclusion, we first notice that it is sufficient to prove it for a special choice of the $(m,k)$-triple $(S,v,H)$. Indeed, if $(S',v',H')$ is any other $(m,k)$-triple, then by Remark [Remark 50](#rmk:non empty){reference-type="ref" reference="rmk:non empty"} there exists a non-zero element $$\eta\in\operatorname{Hom}_{\mathcal{G}^{m,k}}((S,v,H),(S',v',H'))$$ and $$\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S',v',H')})=\widetilde{\Phi}^{m,k}(\eta)\circ \operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)} )\circ\widetilde{\Phi}^{m,k}(\eta)^{-1}.$$ Therefore if $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)})=\operatorname{O}(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v$, as $\widetilde{\Phi}^{m,k}(\eta)$ maps $v$ to $v'$, it follows that $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S',v',H')})=\operatorname{O}(\widetilde{\operatorname{H}}(S',\mathbb{Z}))_{v'}$. Because of this, we will prove the proposition in the case when $v=(m,0,-mk)$ and $S$ is a projective elliptic K3 surface with a section and having Picard rank $2$. For short, in the following we will refer to such a K3 surface as a *very general projective elliptic K3 surface with a section*. If we denote by $p\colon S\to\mathbb{P}^1$ the elliptic structure of $S$ and by $s\colon\mathbb{P}^1\to S$ the section, then we put $f=p^*\mathcal{O}_{\mathbb{P}^1}(1)$ and $\ell=[s(\mathbb{P}^1)]$. In this case $$\operatorname{Pic}(S)=\mathbb{Z}.\ell\oplus\mathbb{Z}.f= \left(\begin{matrix} -2 & 1 \\ 1 & 0 \end{matrix}\right),$$ and we will always denote by $e=\ell+f$ the other isotropic generator. Finally, the polarisation $H$ will be chosen in the unique $v$-chamber (see Remark [Remark 20](#rmk:wall and chamber){reference-type="ref" reference="rmk:wall and chamber"}) whose closure contains the nef class $f$. As a first step, let us recall the following lattice-theoretic result due to Markman, which exhibits a set of generators for the group $\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v$, when $S$ is now any K3 surface. **Proposition 88** (). *Let $S$ be a K3 surface, $v=(m,0,-mk)\in\widetilde{\operatorname{H}}(S,\mathbb{Z})$ and $\xi\in\operatorname{H}^2(S,\mathbb{Z})$ any primitive class such that $\xi^2=2k-2$. The group $\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v$ is generated by $\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$ and the reflection $R_u$ around the class $u=(1,\xi,k)\in v^\perp$.* Here, as usual, we consider the group $\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$ as a subgroup of $\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))$ by extending by the identity on $\operatorname{H}^0(S,\mathbb{Z}))\oplus \operatorname{H}^4(S,\mathbb{Z})$, and since $v\in \operatorname{H}^0(S,\mathbb{Z}))\oplus \operatorname{H}^4(S,\mathbb{Z})$, we actually view the group $\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$ as a subgroup of $\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v$. *Proof.* By [@Markman:Monodromy Proposition 8.6], $\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v$ is generated by the group $\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$ and all the reflections $R_u$ as in the statement. The reason why only one is in fact needed is a consequence of [@Nikulin Theorem 1.14.4, Theorem 1.17.1] as it is explained in the proof of [@Markman:Monodromy Lemma 8.1]: if $u'=(1,\xi',k)$ is another $(-2)$-class, then there exists an isometry $g\in\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$ such that $g(\xi')=\xi$ and by extending $g$ to $\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))$ by the identity, it follows that $g\circ R_{u'}\circ g^{-1}=R_u$. In other words, any two such reflections are conjugated by elements of $\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$. ◻ Therefore, in order to prove Proposition [Proposition 87](#prop:Im of Phi tilde = O){reference-type="ref" reference="prop:Im of Phi tilde = O"}, we need to exhibit examples $(m,k)$-triples such that the generators in the Proposition [Proposition 88](#prop:boh){reference-type="ref" reference="prop:boh"} arise as morphisms in $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H))})$. By Proposition [Proposition 80](#prop:Mon S in m k){reference-type="ref" reference="prop:Mon S in m k"}, there exists $t\gg0$ and a polarisation $v$-generic polarisation $H=e+tf$ on the very generic elliptic surface $S$ such that $$\operatorname{Mon}^2(S)\subset\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H))}).$$ Since by Corollary [Corollary 76](#corollary:Borel){reference-type="ref" reference="corollary:Borel"} we have that $\operatorname{Mon}^2(S)=\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$, it follows that $$\label{eqn:O + in Phi tilde} \operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))\subset \operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H))}).$$ Now, let $\beta\in\operatorname{H}^2(S,\mathbb{Z})$ be a class such that $\beta^2=2k-2$ and such that $\beta$ is orthogonal to both $f$ and $\ell$. Let us consider the class $u=(1,\beta-f,k)\in v^\perp$. Notice that $u^2=-2$. **Lemma 89** (). *If $H=\ell+tf$ with $t\gg0$, then $$R_u\in\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)}).$$* *Proof.* Since $H=\ell+tf$ with $t\gg0$, by Proposition [Lemma 47](#prop:Poincare){reference-type="ref" reference="prop:Poincare"} the derived equivalence $\operatorname{FM}_{\mathcal{P}}$ induces an isomorphism $$\operatorname{FM}_{\mathcal{P}}\colon M_{(m,0,-mk)}(S,H)\longrightarrow M_{(0,m(\ell+(k+1)f),m)}(S,H).$$ For simplicity, let us put $$v=(m,0,-mk)\quad\mbox{ and }\quad \bar{v}=(0,m(\ell+(k+1)f),m).$$ Then $$\operatorname{FM}_{\mathcal{P}}\in\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{FM}}}((S,v,H),(S,\bar{v},H)).$$ The cohomological action $\operatorname{FM}_{\mathcal{P}}^{\operatorname{H}}$ has been studied in [@Markman:Monodromy Lemma 7.2] and $$\operatorname{FM}_{\mathcal{P}}^{\operatorname{H}}(1,\beta-f,k)=(0,\ell-(k-1)f-\beta,0).$$ Put $a=\ell-(k-1)f-\beta\in\operatorname{H}^2(S,\mathbb{Z})$. Then $a^2=-2$ and $R_{a}\in\operatorname{Mon}^2(S)=\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))$. Notice also that $(0,a,0)\in\bar{v}^\perp$. Since $$R_u=\operatorname{FM}^{\operatorname{H}}_{\mathcal{P}}\circ R_{(0,a,0)}\circ(\operatorname{FM}^{\operatorname{H}}_{\mathcal{P}})^{-1}$$ the proof will be concluded as soon as we prove the following claim. **Claim 2**. *$R_{(0,a,0)}\in\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S,\bar{v},H)})$.* First of all, let us consider a deformation path $$\alpha=(f\colon\mathcal{S}\to T,\mathcal{L},\mathcal{H},t_1,t_2,\gamma)$$ where $$(\mathcal{S}_{t_1},\mathcal{L}_{t_1},\mathcal{H}_{t_1})=(S,m(\ell+(k+1)f),H)\quad\mbox{and}\quad(\mathcal{S}_{t_2},\mathcal{L}_{t_2},\mathcal{H}_{t_2})=(S',L',H')$$ with $\operatorname{Pic}(S')=\mathbb{Z}H'$ and $L'=m H'$. Put $v'=v_{t_2}$. Up to conjugating with $\widetilde{\Phi}(\bar{\alpha})$, it is enough to prove that $R_{b}$ is an element of $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{\operatorname{def},(S',v',H')})$, where $b=\widetilde{\Phi}(\bar{\alpha})(0,a,0)$. On the other hand the last claim follows from Lemma [Lemma 83](#lemma:Mon S pol con H){reference-type="ref" reference="lemma:Mon S pol con H"}: in fact the triple $(S',L',H')$ satisfies its hypothesis and $R_b\in\operatorname{O}^+(\operatorname{H}^2(S,\mathbb{Z}))_{H'}$ by construction. ◻ Combining equality ([\[eqn:O + in Phi tilde\]](#eqn:O + in Phi tilde){reference-type="ref" reference="eqn:O + in Phi tilde"}) and Lemma [Lemma 89](#lemma:FM P in Im Phi tilde){reference-type="ref" reference="lemma:FM P in Im Phi tilde"}, by Proposition [Proposition 88](#prop:boh){reference-type="ref" reference="prop:boh"} we have that $$\operatorname{O}^+(\widetilde{\operatorname{H}}(S,\mathbb{Z}))_v\subset\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)}),$$ where $H$ is a $v$-generic polarisation inside the $v$-chamber whose closure contains the class $f$. To conclude the proof of Proposition [Proposition 87](#prop:Im of Phi tilde = O){reference-type="ref" reference="prop:Im of Phi tilde = O"} it is then enough to prove that $\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)})$ contains an orientation reversing isometry. First of all, let $(S',v',H')$ be an $(m,k)$-triple such that $\operatorname{Pic}(S')=\mathbb{Z}.H'$ and let us pick a non-trivial element $$\eta\in\operatorname{Hom}_{\mathcal{G}_{\operatorname{def}}^{m,k}}((S,v,H),(S',v',H')).$$ By Lemma [Lemma 45](#lemma:2.20){reference-type="ref" reference="lemma:2.20"} we know that the derived autoequivalence $\mathsf{L}$ is a morphism in $\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{FM}}}((S',v',H'),(S',v'_L,H'))$. If we take $L=nH'$ for $n\gg0$, then by Lemma [Lemma 46](#lemma:Y-PR){reference-type="ref" reference="lemma:Y-PR"} we also know that $$\operatorname{FM}^\vee_{\Delta}\in\operatorname{Hom}_{\mathcal{G}^{m,k}_{\operatorname{FM}}}((S',v'_L,H'),(S',\hat{v}'_L,H')),$$ so that $$\phi=\operatorname{FM}^\vee_{\Delta}\circ\mathsf{L}\circ\eta\in\operatorname{Hom}_{\mathcal{G}^{m,k}}((S,v,H),(S',\hat{v}'_L,H')).$$ Notice that $\phi^{\operatorname{H}}$ is orientation reversing (cf. proof of Proposition [Proposition 71](#prop:iso of functors){reference-type="ref" reference="prop:iso of functors"}). By Remark [Remark 50](#rmk:non empty){reference-type="ref" reference="rmk:non empty"} there is a morphism $\psi\in\operatorname{Hom}_{\mathcal{G}^{m,k}}((S',\hat{v}'_L,H'),(S,v,H))$ such that: - $\psi$ is obtained by concatenating morphisms in $\mathcal{G}^{m,k}_{\operatorname{def}}$ with morphisms in $\mathcal{G}^{m,k}_{\operatorname{FM}}$ of the form $\mathsf{L}$ and $\operatorname{FM}_{\Delta}$; - $\psi^{\operatorname{H}}$ is orientation preserving (cf. proof of Proposition [Proposition 71](#prop:iso of functors){reference-type="ref" reference="prop:iso of functors"} again). Then $$(\psi\circ\phi)^{\operatorname{H}}\in\operatorname{Im}(\widetilde{\Phi}^{m,k}_{(S,v,H)})$$ is orientation reversing and we are done. ## An injective morphism from $\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H))$ to $\operatorname{Mon}^2(M_{w}(S,H))$ {#subsection:i sharp} Let $(S,v,H)$ be an $(m,k)$-triple. Write $v=mw$, with $w$ primitive, and consider the $(1,k)$-triple $(S,w,H)$. The closed embedding $$i_{w,m}\colon M_w(S,H)\longrightarrow M_v(S,H)$$ is defined in Remark [Remark 26](#rmk:strat M){reference-type="ref" reference="rmk:strat M"} and by Corollary [Corollary 29](#cor:w and mw){reference-type="ref" reference="cor:w and mw"} the pullback morphism $$i_{w,m}^*\colon \operatorname{H}^2(M_v(S,H),\mathbb{Z})\longrightarrow\operatorname{H}^2(M_w(S,H),\mathbb{Z})$$ is a similitude of lattices. Moreover, by Lemma [Lemma 30](#lemma:i sharp){reference-type="ref" reference="lemma:i sharp"}, $i^*_{w,m}$ induces an isomorphism of orthogonal groups $$\label{eqn:i sharp} i_{w,m}^{\sharp}\colon\operatorname{O}(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))\longrightarrow\operatorname{O}(\operatorname{H}^2(M_w(S,H),\mathbb{Z})).$$ More generally, if $(S_1,v_1,H_1)$ and $(S_2,v_2,H_2)$ are two $(m,k)$-triples, and if we write $v_1=mw_1$ and $v_2=mw_2$, then we consider the bijection (see Lemma [Lemma 31](#lemma:i sharp gen){reference-type="ref" reference="lemma:i sharp gen"}) $$\label{eqn:i sharpgen} i^{\sharp}_{w_1,w_2,m}\colon \operatorname{O}(\operatorname{H}^{2}(M_{v_1},\mathbb{Z}),\operatorname{H}^{2}(M_{v_2},\mathbb{Z}))\longrightarrow \operatorname{O}(\operatorname{H}^{2}(M_{w_1},\mathbb{Z}),\operatorname{H}^{2}(M_{w_2},\mathbb{Z})).$$ The next proposition shows that $i_{w_1,w_2,m}^{\sharp}$ sends locally trivial parallel transport operators to parallel transport operators. In particular it follows that the restriction of $i_{w_1,w_2,m}^{\sharp}$ to the subset of locally trivial parallel transport operators is an injection. **Proposition 90**. *Let $(S_1,v_1,H_1)$ and $(S_2,v_2,H_2)$ be two $(m,k)$-triples with $m>1$, and write $v_1=mw_1$ and $v_2=mw_2$. Then the bijection ([\[eqn:i sharpgen\]](#eqn:i sharpgen){reference-type="ref" reference="eqn:i sharpgen"}) maps $\mathsf{PT}_{\operatorname{lt}}^2(M_{v_1}(S_1,H_1),M_{v_2}(S_2,H_2))$ to $\mathsf{PT}^2(M_{w_1}(S_1,H_1),M_{w_2}(S_2,H_2))$, i.e. it restricts to an injective function $$i_{w_1,w_2,m}^\sharp\colon\mathsf{PT}_{\operatorname{lt}}^2(M_{v_1}(S_1,H_1),M_{v_2}(S_2,H_2))\rightarrow\mathsf{PT}^2(M_{w_1}(S_1,H_1),M_{w_2}(S_2,H_2)).$$* Let $p\colon \mathcal{X}\to T$ be a locally trivial family of primitive symplectic varieties with fibres $\mathcal{X}_{t_{1}}\cong M_{v_1}(S_1,H_1)$ and $\mathcal{X}_{t_{2}}\cong M_{v_2}(S_2,H_2)$, then by Remark [Remark 10](#rmk:strat){reference-type="ref" reference="rmk:strat"} we have a relative stratification $\mathcal{X}\supset\mathcal{X}_1\supset\cdots\supset\mathcal{X}_{s}=:\mathcal{Y}$ that restricted to each fiber $X_{t}$ of $\mathcal{X}$ is the stratification of the singularities given by Proposition [Proposition 3](#prop:stratification){reference-type="ref" reference="prop:stratification"}. By the local triviality of the family and the fact that a subspace of a Kähler space is again Kähler (see [@Varouchas II,1.3.1(i) Proposition]), the smallest stratum $q\colon\mathcal{Y}\to T$ of this relative stratification is a smooth family of irreducible holomorphic symplectic manifolds; by Remark [Remark 26](#rmk:strat M){reference-type="ref" reference="rmk:strat M"} we have $\mathcal{Y}_{t_{1}}\cong M_{w_1}(S_1,H_1)$ and $\mathcal{Y}_{t_{2}}\cong M_{w_2}(S_2,H_2)$. Let $\gamma$ be a continuous path in $T$ with initial point in $t_1$ and final point in $t_2$ and let $g\colon \operatorname{H}^2(M_{v_1}(S_1,H_1),\mathbb{Z})\to\operatorname{H}^2(M_{v_2}(S_2,H_2),\mathbb{Z})$ be the corresponding parallel transport operator in the family $p\colon\mathcal{X}\to T$. The following claim concludes the proof of the proposition by interpreting $i_{w_1,w_2,m}^\sharp(g)$ as a parallel transport operator from $M_{w_1}(S_1,H_1)$ to $M_{w_2}(S_2,H_2)$. **Claim 3**. *The isometry $i_{w_1,w_2,m}^\sharp(g)$ is the parallel transport operator in the family $q\colon\mathcal{Y}\to T$ along the path $\gamma$.* Let $g_{\mathbb{Q}}$ be the $\mathbb{Q}$-linear extension of $g$, i.e. let $g_{\mathbb{Q}}$ be the parallel transport operator $\mathsf{PT}_p(\gamma)$ along $\gamma$ in the local system $R^2q_*\mathbb{Q}$. By definition of $i_{w_1,w_2,m}^\sharp$ (see Lemma [Lemma 31](#lemma:i sharp gen){reference-type="ref" reference="lemma:i sharp gen"}), it is enough to show that the $i_{w_1,w_2,m,\mathbb{Q}}^\sharp(g_{\mathbb{Q}})$ is the parallel transport operator $\mathsf{PT}_q(\gamma)$ along $\gamma$ in the local system $R^2q_*\mathbb{Q}$. By local triviality, the inclusion $i\colon\mathcal{Y}\to\mathcal{X}$ induces a morphism of local systems $$i_{\mathbb{Q}}^*\colon R^2p_{*}\mathbb{Q}\longrightarrow R^2q_{*}\mathbb{Q}$$ such that $i^*_{t_{1},\mathbb{Q}}=i_{w_1,m,\mathbb{Q}}^{*}$ and $i^{*}_{t_{2},\mathbb{Q}}=i_{w_2,m,\mathbb{Q}}^{*}$. Notice that since $i^*_{t_{1},\mathbb{Q}}$ (and $i^*_{t_{2},\mathbb{Q}}$) is an isometry (cf. Corollary [Corollary 29](#cor:w and mw){reference-type="ref" reference="cor:w and mw"}), it follows that $i_{\mathbb{Q}}^*$ is an isomorphism of local systems. The claim follows since $$i_{w_1,w_2,m,\mathbb{Q}}^\sharp(g_\mathbb{Q}) = i_{w_1,w_2,m,\mathbb{Q}}^\sharp(\mathsf{PT}_{p}(\gamma)) =i^{*}_{w_2,m,\mathbb{Q}}\circ \mathsf{PT}_{p}(\gamma)\circ(i^{*}_{w_1,m,\mathbb{Q}})^{-1},$$ by definition of $i_{w_1,w_2,m,\mathbb{Q}}^\sharp$, and $$i^{*}_{w_2,m,\mathbb{Q}}\circ \mathsf{PT}_{p}(\gamma)\circ(i^{*}_{w_1,m,\mathbb{Q}})^{-1}= \mathsf{PT}_{q}(\gamma)$$ because $i_{\mathbb{Q}}^*\colon R^2p_{*}\mathbb{Q}\rightarrow R^2q_{*}\mathbb{Q}$ is an isomorphism of local systems restricting to $i_{w_1,m,\mathbb{Q}}^*$ at $t_1$ and to $i_{w_2,m,\mathbb{Q}}^*$ at $t_2$. **Remark 91**. Let $\Omega(T,t_{1},t_{2})$ be the set of homotopy classes of paths in $T$ from $t_{1}$ to $t_{2}$, and $\mathsf{PT}_{p}$ (resp. $\mathsf{PT}_{q}$) be the morphism mapping the class of a path $[\gamma]$ to the parallel transport operator associated to $\gamma$. With this notation the claim in the proof of Proposition [Proposition 90](#prop:first inclusiongen){reference-type="ref" reference="prop:first inclusiongen"} states that the diagram $$\xymatrix{ \Omega(T,t_{1},t_{2})\ar@{->}[r]^-{\mathsf{PT}_{p}}\ar@{->}[dr]_{\mathsf{PT}_{q}} & \operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Q}),\operatorname{H}^{2}(M_{v'},\mathbb{Q}))\ar@{->}[d]^{i_{w,w',m}^\sharp}_{\cong} \\ & \operatorname{O}(\operatorname{H}^2(M_w,\mathbb{Q}),\operatorname{H}^{2}(M_{w'},\mathbb{Q})) }$$ is commutative. In the particular case of $(S,v,H)=(S',v',H')$ we get the following. **Corollary 92**. *Let $(S,v,H)$ be an $(m,k)$-triple with $m>1$, and write $v=mw$. Then the isomorphism ([\[eqn:i sharp\]](#eqn:i sharp){reference-type="ref" reference="eqn:i sharp"}) restricts to an injective group morphism $$i_{w,m}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H))\longrightarrow\operatorname{Mon}^2(M_w(S,H)).$$* ## The main results {#section:main result} **Theorem 93**. *Let $(S,v,H)$ be an $(m,k)$-triple and suppose that $m>1$. Then $$\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H))=\mathsf{W}(M_v(S,H)).$$* The case when $v$ is primitive, i.e. $m=1$, has been proved by Markman. **Theorem 94** (). *Let $(S,w,H)$ be an $(1,k)$-triple. Then $$\operatorname{Mon}^2(M_w(S,H))=\mathsf{W}(M_w(S,H)).$$* We will use Markman's result in our proof, this is the reason why we have omitted this case from our statement. In the proof we use the following notation: if $H$ is a subgroup of a group $G$, then $[G:H]$ stands for its index. *Proof.* We will use the simplified notation $M_{w}$ (resp. $M_{v}$) instead of $M_{w}(S,H)$ (resp. $M_{v}(S,H)$). By Theorem [Theorem 85](#thm:W in m k){reference-type="ref" reference="thm:W in m k"} we have that $$\mathsf{W}(M_v)\subset\operatorname{Mon}^2_{\operatorname{lt}}(M_v)\subset \operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Z}))$$ so that $$[\operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Z})):\mathsf{W}(M_v)]\geq[\operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Z})):\operatorname{Mon}^2_{\operatorname{lt}}(M_v)].$$ On the other hand since $m>1$, by Lemma [Lemma 30](#lemma:i sharp){reference-type="ref" reference="lemma:i sharp"} and Corollary [Corollary 92](#prop:first inclusion){reference-type="ref" reference="prop:first inclusion"} there is an isomorphism $$i_{w,m}^\sharp\colon \operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Z}))\longrightarrow\operatorname{O}(\operatorname{H}^2(M_w,\mathbb{Z}))$$ such that $$i_{w,m}^\sharp (\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}))\subset \operatorname{Mon}^2(M_{w})$$ and, by Theorem [Theorem 94](#thm:Markman){reference-type="ref" reference="thm:Markman"}, $\operatorname{Mon}^2(M_w)=\mathsf{W}(\operatorname{H}^2(M_w,\mathbb{Z}))$. It follows that $$[\operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Z})):\operatorname{Mon}^2_{\operatorname{lt}}(M_v)]\geq[\operatorname{O}(\operatorname{H}^2(M_w,\mathbb{Z})):\mathsf{W}(M_w)].$$ Finally, since $\operatorname{H}^2(M_v,\mathbb{Z})$ and $\operatorname{H}^2(M_w,\mathbb{Z})$ are abstractly isometric as lattices, and since the groups $\mathsf{W}(M_v)$ and $\mathsf{W}(M_w)$ are defined only in lattice-theoretic terms, we get the equality $$=[\operatorname{O}(\operatorname{H}^2(M_w,\mathbb{Z})):\mathsf{W}(M_w)],$$ which allows us to deduce that $$[\operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Z})):\operatorname{Mon}^2_{\operatorname{lt}}(M_v)]=[\operatorname{O}(\operatorname{H}^2(M_v,\mathbb{Z})):\mathsf{W}(M_v)].$$ Since by Theorem [Theorem 85](#thm:W in m k){reference-type="ref" reference="thm:W in m k"} we have that $$\mathsf{W}(M_v)\subset\operatorname{Mon}^2_{\operatorname{lt}}(M_v),$$ we conclude that $\operatorname{Mon}^2_{\operatorname{lt}}(M_v)=\mathsf{W}(M_v)$. ◻ **Corollary 95** (). *Let $(S,v,H)$ be a $(m,k)$-triple. Then $$\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))\subset\operatorname{O}^+(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))$$ has index $2^{\rho(k)-1}$, where $\rho(k)$ is the number of distinct primes in the factorisation of $k$.* **Remark 96**. Let $(S,v,H)$ be an $(m,k)$-triple. It is interesting to point out that the index of $\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))$ in $\operatorname{O}^+(\operatorname{H}^2(M_v(S,H),\mathbb{Z}))$ does not depend on $m$. In particular we have that if $k$ is a power of a prime, then $$\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))=\operatorname{O}^+(\operatorname{H}^2(M_v(S,H),\mathbb{Z})).$$ Theorem [Theorem 93](#thm:Mon 2 lt){reference-type="ref" reference="thm:Mon 2 lt"} can be analogously formulated in terms of the inclusion $i_{w,m}\colon M_w(S,H)\rightarrow M_v(S,H)$. **Theorem 97**. *Let $(S,v,H)$ be an $(m,k)$-triple, $v=mw$ with $m>1$ and $(S,w,H)$ the corresponding $(1,k)$-triple. Then the isomorphism $$i_{w,m}^\sharp\colon\operatorname{O}(\operatorname{H}^2(M_{v}(S,H),\mathbb{Z}))\longrightarrow\operatorname{O}(\operatorname{H}^2(M_{w}(S,H),\mathbb{Z}))$$ induces by restriction an isomorphism $$i_{w,m}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H))\longrightarrow\operatorname{Mon}^2(M_w(S,H)).$$* *Proof.* First of all recall that $i_{w,m}^\sharp =\lambda_{(S,w,H)}^{\sharp}\circ( \lambda_{(S,v,H)}^\sharp)^{-1}$ by Lemma [Lemma 30](#lemma:i sharp){reference-type="ref" reference="lemma:i sharp"}.(2). Since the groups $\mathsf{W}(M_{v}(S,H))$ and $\mathsf{W}(M_w(S,H))$ are defined in lattice-theoretic terms, and $\lambda_{(S,w,H)}$ and $\lambda_{(S,v,H)}$ are isometries, it follows that $$\label{eqn:qui} i_{w,m}^\sharp(\mathsf{W}(M_{v}(S,H)))=\mathsf{W}(M_w(S,H)).$$ On the other hand, by Corollary [Corollary 92](#prop:first inclusion){reference-type="ref" reference="prop:first inclusion"} the isometry $i_{w,m}^\sharp$ sends monodromy operators in monodromy operators, i.e. $$\label{eqn:quo} i_{w,m}^\sharp(\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H)))\subset\operatorname{Mon}^2(M_w(S,H)).$$ Combining this with Theorem [Theorem 85](#thm:W in m k){reference-type="ref" reference="thm:W in m k"} and Theorem [Theorem 94](#thm:Markman){reference-type="ref" reference="thm:Markman"}, we eventually get $$\begin{aligned} \mathsf{W}(M_w(S,H)) & =i_{w,m}^\sharp(\mathsf{W}(M_{v}(S,H))) & \mbox{(by equality (\ref{eqn:qui}))} \\ & \subseteq i_{w,m}^\sharp(\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H))) & \mbox{(by Theorem~\ref{thm:W in m k})} \\ & \subseteq\operatorname{Mon}^2(M_w(S,H)) & \mbox{(by inclusion (\ref{eqn:quo}))} \\ & =\mathsf{W}(M_w(S,H)) & \mbox{(by Theorem~\ref{thm:Markman})} \end{aligned}$$ from which it follows that $$i_{w,m}^\sharp(\operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H)))=\operatorname{Mon}^2(M_w(S,H))$$ and therefore the claim. ◻ **Remark 98**. Suppose that $m'\neq m$ are two strictly positive integers and put $v'=m'w$ and $v=mw$. There is then a natural isomorphism $$(i^\sharp_{w,m'})^{-1}\circ i^\sharp_{w,m}\colon\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))\stackrel{\sim}{\longrightarrow}\operatorname{Mon}^2_{\operatorname{lt}}(M_{v'}(S,H)).$$ It follows that the isomorphism class of $\operatorname{Mon}^2_{\operatorname{lt}}(M_v(S,H))$ does not depend on $m$, but only on $w$. Since the locally trivial monodromy group is invariant along locally trivial families of primitive symplectic varieties (see Definition [Definition 4](#def:lt){reference-type="ref" reference="def:lt"}), we get the following corollary. **Corollary 99**. *Let $X$ be an irreducible symplectic variety that is locally trivially deformation equivalent to a moduli space $M_v(S,H)$, where $(S,v,H)$ is an $(m,k)$-triple.* *Then $\operatorname{Mon}^2_{\operatorname{lt}}(X)$ is the subgroup $\mathsf{W}(X)\subset\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z}))$ of the orientation preserving isometries acting as $\pm\operatorname{id}$ on the discriminant group.* Since $X$ is locally trivially deformation equivalent to a moduli space $M_v(S,H)$ as in the statement, there exists a locally trivial parallel transport operator $$g\colon\operatorname{H}^{2}(M_{v}(S,H),\mathbb{Z})\longrightarrow \operatorname{H}^{2}(X,\mathbb{Z}).$$ Conjugation with $g$ gives an isomorphism of orthogonal groups $$g^\sharp\colon\operatorname{O}(\operatorname{H}^{2}(M_{v}(S,H),\mathbb{Z}))\longrightarrow \operatorname{O}(\operatorname{H}^{2}(X,\mathbb{Z}))$$ which maps $\operatorname{Mon}^{2}_{\operatorname{lt}}(M_{v}(S,H))$ to $\operatorname{Mon}^{2}_{\operatorname{lt}}(X)$ by definition. Moreover, since $g^\sharp$ is induced by an isometry, it also has to map $\mathsf{W}(M_{v}(S,H))$ to $\mathsf{W}(X)$. Since $\mathsf{W}(M_{v}(S,H))=\operatorname{Mon}^{2}_{\operatorname{lt}}(M_{v}(S,H))$ by Theorem [Theorem 93](#thm:Mon 2 lt){reference-type="ref" reference="thm:Mon 2 lt"}, it follows that $$\mathsf{W}(X)=\operatorname{Mon}^{2}_{\operatorname{lt}}(X,\mathbb{Z}).$$ Finally, let us observe that the analogue of Theorem [Theorem 97](#thm:i sharp is iso){reference-type="ref" reference="thm:i sharp is iso"} holds for any symplectic variety $X$ locally trivial deformation equivalent to $M_v(S,H)$. In order to state the result, we recall that the most singular locus $Y$ of $X$ is an irreducible symplectic manifold (see Proposition [Proposition 3](#prop:stratification){reference-type="ref" reference="prop:stratification"}). Let us formally define the morphism $$i_{Y,X}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(X) \longrightarrow \operatorname{Mon}^2(Y)$$ induced by the inclusion $i_{Y,X}\colon Y\to X$, as follows. For every monodromy operator $g\in \operatorname{Mon}^2_{\operatorname{lt}}(X)$, there exists a locally trivial family of irreducible symplectic varieties $p\colon \mathcal{X}\to T$, a point $\bar{t}\in T$ and loop $\gamma$ centered at $\bar{t}$ such that $\mathcal{X}_{\bar{t}}=X$ and $g$ is the parallel transport operator $\mathsf{PT}_p(\gamma)$ associated with the family $p$ and the loop $\gamma$. By local triviality of $p$, the restriction of $p$ to the most singular locus $\mathcal{Y}$ of $\mathcal{X}$ gives a smooth family $q\colon\mathcal{Y}\to T$ of irreducible symplectic manifolds and we define $i_{Y,X}^{\sharp}(g)\colon \operatorname{H}^{2}(Y,\mathbb{Z})\to \operatorname{H}^{2}(Y,\mathbb{Z})$ as the parallel transport operator $\mathsf{PT}_q(\gamma)$ associated with the family $q$ and the loop $\gamma$. We remark that $i_{Y,X}^{\sharp}(g)$ is well defined and its definition immediately implies $$\label{penultima} i_{Y,X}^{\sharp}(g) \circ i_{Y,X}^{*}= i_{Y,X}^{*}\circ g\colon\operatorname{H}^2(X,\mathbb{Z}) \longrightarrow \operatorname{H}^2(Y,\mathbb{Z})$$ **Corollary 100**. *Let $X$ be an irreducible symplectic variety that is locally trivially deformation equivalent to a moduli space $M_v(S,H)$, where $(S,v,H)$ is an $(m,k)$-triple. Let $Y\subset X$ be the most singular locus and $i_{Y,X}\colon Y\to X$ be the closed embedding. Then the morphism $$i_{Y,X}^\sharp\colon\operatorname{Mon}^2_{\operatorname{lt}}(X)\stackrel{\sim}{\longrightarrow}\operatorname{Mon}^2(Y)$$ is an isomorphism.* *Proof.* Let $p\colon\mathcal{X}\to T$ be a locally trivial family of irreducible symplectic varieties such that there exists two points $t_1,t_2\in T$ with $\mathcal{X}_{t_1}=X$ and $\mathcal{X}_{t_2}=M_v(S,H)$, respectively. Here $M_v(S,H)$ is the irreducible symplectic variety associated to an $(m,k)$-triple $(S,v,H)$. Let $\mathcal{Y}$ be the most singualar locus of $\mathcal{X}$ and let $q\colon\mathcal{Y}\to T$ be the restriction of $p$; by Remark [Remark 10](#rmk:strat){reference-type="ref" reference="rmk:strat"} the family $q$ is a family of irreducible holomorphic symplectic manifolds. Let $\gamma$ be a path contained in $T$ from $t_1$ to $t_2$ and denote by $\mathsf{PT}_p(\gamma)\colon\operatorname{H}^{2}(X,\mathbb{Z})\to \operatorname{H}^{2}(M_{v}(S,H),\mathbb{Z})$ and $\mathsf{PT}_q(\gamma )\colon\operatorname{H}^{2}(Y,\mathbb{Z})\to\operatorname{H}^{2}(M_{w}(S,H),\mathbb{Z})$ the parallel transport operators associated with the path $\gamma$ and the families $p$ and $q$, respectively. By construction we have $$\label{ultima} i_{w,m}^{*}\circ \mathsf{PT}_p(\gamma)= \mathsf{PT}_q(\gamma)\circ i_{Y,X}^*\colon \operatorname{H}^{2}(X,\mathbb{Z})\to \operatorname{H}^{2}(M_{w}(S,H),\mathbb{Z}).$$ Finally we define the group isomorphisms $$\begin{aligned} %\label{eqn:i sharpgen} \mathsf{PT}_p^{\sharp}(\gamma)\colon \operatorname{Mon}^2_{\operatorname{lt}}(M_{v}(S,H)) & \longrightarrow \operatorname{Mon}^2_{\operatorname{lt}}(X) \nonumber\\ g & \longmapsto (\mathsf{PT}_p (\gamma))^{-1}\circ g \circ \mathsf{PT}_p(\gamma)\nonumber \end{aligned}$$ and $$\begin{aligned} %\label{eqn:i sharpgen} \mathsf{PT}_q^{\sharp}(\gamma)\colon \operatorname{Mon}^2(M_{w}(S,H)) & \longrightarrow \operatorname{Mon}^2(Y) \nonumber\\ g & \longmapsto (\mathsf{PT}_q (\gamma))^{-1}\circ g \circ \mathsf{PT}_q(\gamma).\nonumber \end{aligned}$$ By formulae ([\[penultima\]](#penultima){reference-type="ref" reference="penultima"}) and ([\[ultima\]](#ultima){reference-type="ref" reference="ultima"}) we deduce that $$i_{Y,X}^{\sharp}= \mathsf{PT}_{q}^{\sharp}(\gamma)\circ i_{w,m}^{\sharp}\circ (\mathsf{PT}_{p}^{\sharp}(\gamma))^{-1}$$ and, since $\mathsf{PT}_{q}^{\sharp}(\gamma)$, $i_{w,m}^{\sharp}$ and $(\mathsf{PT}_{p}^{\sharp}(\gamma))^{-1}$ are isomorphisms, the morphism $i_{Y,X}^{\sharp}$ is an isomorphism too. ◻ alpha B. Bakker, C. Lehn, *The global moduli theory of symplectic varieties*. J. reine angew. Math. **790** (2022), 223--265. A. Beauville, *Symplectic singularities*. Invent. Math **139**, 3 (2000), 541--549. A. Borel, *Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem*. J. Diff. Geom., **6** (1972), 543--560. T. Bridgeland, *Fourier--Mukai transforms for elliptic surfaces*. J. reine angew. Math. **498** (1998), 115--133. A. Dimca, *Singularities and topology of hypersurfaces*. Universitext **18**, Springer-Verlag, New York (1992). O. Forster, *Lectures on Riemann Surfaces*, Graduate Texts in Math. **81**, Springer-Verlag, New York, 1981. W. Fulton, R. Lazarsfeld, *Connectivity and its applications in algebraic geometry*. In: Libgober, A., Wagreich, P. (eds) Algebraic Geometry. Lecture Notes in Mathematics **862**. Springer, Berlin, Heidelberg (1981), 26--92. C. Godbillon, *Éléments de topologie algébrique*, Collection Méthodes. Mathématique Méthodes. Hermann (1971). D. Greb, H. Guenancia, S. Kebekus, *Klt varieties with trivial canonical class -- Holonomy, differential forms, and fundamental group*. Geom. Topol. **23**, 4 (2019), 2051--2124. D. Greb, S. Kebekus, T. Peternell, *Singular spaces with trivial canonical class*. In Minimal models and extreme rays, proceedings of the conference in honour of Shigefumi Mori's 60th birthday, Ad. Stud. Pure Math., Kinokuniya Publishing House, Tokyo (2011). V. Gritsenko, K. Hulek, G.K. Sankaran, *Abelianisation of orthogonal groups and the fundamental group of modular varieties*. J. Algebra **322**, 2 (2009), 463--478. P.J. Higgins. *Categories and groupoids*, Reprints in Theory and Applications of Categories, **7** (1971), 1--195. D. Huybrechts, *The Kähler cone of a compact hyperkähler manifold*. Math. Ann. **326** (2003), 499--513. D. Huybrechts, P. Stellari, *Equivalences of twisted K3 surfaces*. Math. Ann. **332** (2005), 901--936. D. Kaledin, *Symplectic singularities from the Poisson point of view*. J. reine angew. Math. **600** (2006), 135--156. D. Kaledin, M. Lehn, C. Sorger, *Singular symplectic moduli spaces*. Invent. Math. **164** (2006), 591--614. E. Markman, *On the monodromy of moduli spaces of sheaves on K3 surfaces*. J. Alg. Geom. **17**, 1 (2008), 29--99. E. Markman, *Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface*. Int. J. Math. **21**, (2010), 169--223. E. Markman, *A survey of Torelli and monodromy results for holomorphic symplectic varieties*. Complex and differential geometry, Springer Proc. Math. **8**, Springer, Heidelberg (2011), 257--322. E. Markman, *The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians*. J. Eur. Math. Soc. **25** (2022), 231--321. R. Miranda, D. R. Morrison, *Embeddings of integral quadratic forms*. Preliminary draft (2009), web.math.ucsb.edu/ drm/manuscripts/eiqf.pdf. G. Mongardi, *On the monodromy of irreducible symplectic manifolds*. Alg. Geom. **3**, 3 (2016), 385--391. G. Mongardi, A. Rapagnetta, *Monodromy and birational geometry of O'Grady's sixfold*. J. Math. pures et appl. **146** (2021), 31--68. S. Mukai, *Symplectic structure on the moduli space of sheaves on an abelianof K3 surface*. Invent. Math. **77**, 1 (1984), 101--116. Y. Namikawa, *Extension of $2-$forms and symplectic varieties*. J. reine angew. Math. **539** (2001), 123--147. Y. Namikawa, *A note on symplectic singularities*. Preprint, arXiv:math/0101028. Y. Namikawa, *Projectivity criterion of Moishezon spaces and density of projective symplectic varieties*. Int. J. Math. **13**, 2 (2002), 125--135. V.V. Nikulin, *Integral symmetric bilinear forms and some of their applications*. Izv. Akad. Nauk SSSR Ser. Mat. **43**, 1 (1979), 111--177. English translation in Math. USSR Izv. **14**, 1 (1980), 103--167. K.G. O'Grady, *The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface*. J. Alg. Geom. **6** (1997), 599--644. C. Onorati, *On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces*. J. Alg. Geom. **31** (2022), 425--465. A. Perego, A. Rapagnetta, *Deformation of the O'Grady moduli spaces*. J. reine Angew. Math. **678** (2013), 1--34. A. Perego, A. Rapagnetta, *The moduli spaces of sheaves on K3 surfaces are irreducible symplectic varieties*. Alg. Geom. **10**, 3 (2023), 348--393. A. Perego, A. Rapagnetta, *The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces*. Preprint, arXiv:2012.10649. C. Peters. *Monodromy and Picard-Fuchs equations for families of K3 surfaces and elliptic curves*, Ann. Sci. École Norm. Sup., **19**, 4 (1986), 583--607. J.-P. Serre, *Géométrie algébrique et géométrie analytique*, Ann. Inst. Fourier, **6** (1956), 1---42. Y.-T. Siu, *Extending coherent analytic sheaves*, Ann. Math. **90**, 1 (1969), 108--143. J. Varouchas, *Kähler spaces and proper open morphisms*. Math. Ann. **283**, 1 (1989), 13--52. C. Voisin *Hodge Theory and Complex Algebraic Geometry*, volume 2 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2003. K. Yoshioka, *Moduli spaces of stable sheaves on abelian surfaces*. Math. Ann. **321**, 4 (2001), 817--884. K. Yoshioka, *A note on Fourier--Mukai transform*. Preprint, arXiv:math/0112267.

arxiv_math

--- abstract: | End sum is a natural operation for combining two noncompact manifolds and has been used to construct various manifolds with interesting properties. The uniqueness of end sum has been well-studied in dimensions three and higher. We study end sum---and the more general notion of adding a 1-handle at infinity---for surfaces and prove uniqueness results. The result of adding a 1-handle at infinity to distinct ends of a surface with compact boundary is uniquely determined by the chosen ends and the orientability of the 1-handle. As a corollary, the end sum of two surfaces with compact boundary is uniquely determined by the chosen ends. Unlike uniqueness results in higher dimensions, which rely on isotopy uniqueness of rays, our results rely fundamentally on a classification of noncompact surfaces. address: - Newton, MA 02459 - | Department of Mathematics\ Oberlin College\ Oberlin, OH 44074 author: - Liam Axon - Jack Calcut title: The End Sum of Surfaces --- # Introduction {#sec:introduction} End sum is the analogue for open manifolds of the boundary sum of manifolds with boundary. It was introduced by Gompf [@gompf3; @gompfinf] in the 1980's to construct smooth manifolds homeomorphic to $\mathbb{R}^4$. Gompf [@gompf3 p. 322] colloquially described end sum as gluing together two noncompact manifolds using a piece of tape. More formally, the piece of tape is a $1$-handle at infinity. Since that time, several authors have used end sum to construct manifolds with interesting properties. Recently, Bennett [@bennett] used end sum to produce new smooth structures on open $4$-manifolds and Sparks [@sparks] used it to construct $4$-dimensional splitters. For further examples, see the second author and Gompf [@cg] and the second author, Haggerty, and Guilbault [@cgh]. A 1-handle at infinity is attached to a manifold $M$ along a chosen pair of disjoint, properly embedded rays pointing to ends of $M$. End sum is the special case where $M$ has two components and the 1-handle connects them. The dependence on ray choice of adding a 1-handle at infinity has been well-studied. Gompf [@gompfinf] first showed that end sums of manifolds homeomorphic to $\mathbb{R}^4$ are independent of ray choice. Myers [@myers] showed that end summing two copies of $\mathbb{R}^3$ using knotted rays yields uncountably many homeomorphism types of contractible, open 3-manifolds. The second author and Haggerty [@ch] constructed examples of pairs of connected, open, oriented, one-ended $n$-manifolds for each $n\geq3$ that may be end summed using various rays to produce manifolds that are not proper homotopy equivalent. The latter examples arise from complicated fundamental group behavior at even just one of the ends. In dimension $n=1$, adding a $1$-handle along specified ends is always unique. For manifolds of any dimension, the *Mittag-Leffler* condition (also called *semistability*) on an end is a necessary and sufficient condition for any two proper rays pointing to that end to be properly homotopic. In dimensions four and higher, such a proper homotopy may be upgraded to an ambient isotopy. That yields uniqueness results for end sums and adding 1-handles at infinity as proved by the second author and Gompf [@cg]. **Theorem 1** (Calcut and Gompf). Let $X$ be a (possibly disconnected) $n$-manifold, $n\ge4$. Then the result of attaching a (possibly infinite) collection of 1-handles at infinity to some oriented Mittag-Leffler ends of $X$ depends only on the pairs of ends to which each 1-handle is attached, and whether the corresponding orientations agree. An immediate corollary is the uniqueness of oriented end sums of two $n$-manifolds with $n\geq4$ along oriented Mittag-Leffler ends. In the present paper, we study end sum for surfaces. Note from the outset that for each end of a *surface with compact boundary*, the Mittag-Leffler condition is actually equivalent to the end being collared. An end of a manifold is *collared* provided it has a neighborhood of the form $Z\times [0,\infty)$ for some connected, compact manifold $Z$. The authors plan to address the classification of Mittag-Leffler ends of general surfaces in a future paper. Observe now that a Mittag-Leffler end of a surface with noncompact boundary components need not be collared as shown by the end $\eta$ in Figure [1](#fig:oddex1){reference-type="ref" reference="fig:oddex1"}. {#fig:oddex1} Properly embedded rays in $\mathbb{R}^2$, $S^1\times[0,\infty)$, and closed half space $\mathbb{R}^2_+$ may be straightened by ambient isotopy [@cks pp. 1845--1852]. So, up to isotopy, there are unique proper rays in Mittag-Leffler (=collared) ends of surfaces with compact boundary. In general, ends of noncompact surfaces may have infinite genus, complicated fundamental group behavior, and fail to be Mittag-Leffler. Thus, one might suspect that end sums of surfaces depend on the choices of rays within specified ends. In fact, the contrary is true. The following is our main result. **Theorem 2**. Let $M$ be a (possibly disconnected) surface with compact boundary. Then the result of attaching a 1-handle at infinity to distinct ends of $M$ depends only on the ends to which the 1-handle is attached and orientations. If the chosen ends lie in different components of $M$ or they lie in the same non-orientable component of $M$, then orientation is irrelevant. A more precise statement is given in Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} below. An immediate corollary is uniqueness of end sums of two connected surfaces with compact boundary along chosen ends (irrespective of orientations). Contrasting Theorems [Theorem 1](#thm:cg){reference-type="ref" reference="thm:cg"} and [Theorem 2](#mainthm){reference-type="ref" reference="mainthm"}, we see that for surfaces the relevant ends need not be Mittag-Leffler and there is greater flexibility with orientations. On the other hand, our arguments for surfaces assume: (i) $M$ has compact boundary, (ii) a single $1$-handle is attached to $M$, and (iii) the relevant ends of $M$ are distinct. Throughout, we opt to work with a single $1$-handle for simplicity. Applying our results iteratively yields results for attaching finitely many $1$-handles, and our results likely carry over to appropriate settings involving infinitely many $1$-handles. Before we discuss assumptions (i) and (iii), we introduce some useful terminology. Consider the general case of attaching a single $1$-handle at infinity to an $n$-manifold $M$. Let $r$ and $r'$ denote the rays in $M$ along which the $1$-handle is attached, let $\varepsilon$ and $\varepsilon'$ denote the ends of $M$ to which $r$ and $r'$ point (respectively), and let $N=M\cup \left(1\textnormal{-handle}\right)$ denote the resulting $n$-manifold. In this unrestricted setting, we allow $M$ to be disconnected, $\partial M$ to be compact (possibly empty) or noncompact (possibly with noncompact boundary components), $\varepsilon$ and $\varepsilon'$ to be equal or distinct, and the $1$-handle attachment to respect or ignore any possible given orientations. An end $\eta$ of $N$ is *ordinary* provided it has a neighborhood disjoint from the $1$-handle. Otherwise, $\eta$ is *extraordinary*. Intuitively, the extraordinary end(s) of $N$ are those involved in the attachment of the $1$-handle. For a space $X$, let $\mathcal{E}\left(X\right)$ denote the *space of ends* of $X$, and let $\left| \mathcal{E}\left(X\right) \right|$ denote the *number of ends* of $X$. The possible number of extraordinary ends of $N$ varies with the dimension $n$. If $n=1$, then there are no extraordinary ends---attaching a $1$-handle at infinity to a $1$-manifold simply eliminates two ends of $M$ and $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|-2$. If $n\geq3$, then there is one extraordinary end, and $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|-1$ (when $\varepsilon\ne\varepsilon'$) or $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|$ (when $\varepsilon=\varepsilon'$). For surfaces, the situation is more complicated---ultimately due to the fact that a surface may be separated by a $1$-dimensional submanifold. There may be one or two extraordinary ends, and $\left| \mathcal{E}\left(N\right) \right|$ may equal $\left| \mathcal{E}\left(M\right) \right|-1$, $\left| \mathcal{E}\left(M\right) \right|$, or $\left| \mathcal{E}\left(M\right) \right|+1$. Predicting which occur is subtle, especially in the presence of noncompact boundary components. The following examples exhibit that subtlety and others. Basic examples are included for context and comparison. In each example, $n=2$ and $N=M\cup \left(1\textnormal{-handle}\right)$. 1. Let $M=\mathbb{R}^2 \sqcup \mathbb{R}^2$ be the disjoint union of two copies of $\mathbb{R}^2$ as in Figure [2](#fig:r2basic){reference-type="ref" reference="fig:r2basic"}. {#fig:r2basic} Regardless of the orientability of the $1$-handle, $N$ is homeomorphic to $\mathbb{R}^2$ and has one end (extraordinary). Here, $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|-1$. 2. Let $M=\mathbb{R}^2$ as in Figure [3](#fig:r2exs){reference-type="ref" reference="fig:r2exs"}. {#fig:r2exs} Let $N$ be the result using an oriented $1$-handle, and let $N'$ be the result using a non-oriented $1$-handle. Then, $N$ is an open cylinder with two ends (both extraordinary), and $N'$ is an open Möbius band with one end (extraordinary). Here, $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|+1$ and $\left| \mathcal{E}\left(N'\right) \right|=\left| \mathcal{E}\left(M\right) \right|$. 3. Let $M$ be an open annulus as in Figure [4](#fig:annulusbasic){reference-type="ref" reference="fig:annulusbasic"}. {#fig:annulusbasic} Attach a $1$-handle to the distinct ends of $M$. Let $N$ be the result using an oriented $1$-handle, and let $N'$ be the result using a non-oriented $1$-handle. Then, $N$ is a punctured torus with one end (extraordinary), and $N'$ is a punctured Klein bottle with one end (extraordinary). Here, $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|-1=\left| \mathcal{E}\left(N'\right) \right|$. 4. Let $M=\mathbb{R}^2_+ \sqcup \mathbb{R}^2_+$ be the disjoint union of two copies of closed half-space as in Figure [5](#fig:endSumBoundaryCounterexample){reference-type="ref" reference="fig:endSumBoundaryCounterexample"}. {#fig:endSumBoundaryCounterexample} Regardless of the orientability of the $1$-handle, $N$ is homeomorphic to $[0,1]\times\mathbb{R}$ and has two ends (both extraordinary). Here, $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|$ in both cases. In a sense, the noncompact boundary components of $M$ clog up the ends of $N$. 5. [\[deex\]]{#deex label="deex"} Let $M$ be the open, oriented, one-ended surface with infinite genus as in Figure [6](#fig:sameendex){reference-type="ref" reference="fig:sameendex"}. {#fig:sameendex} Attach an oriented $1$-handle at infinity to $M$. Let $N$ be the result using the non-parallel rays $r$ and $s$, and let $N'$ be the result using the parallel rays $t$ and $u$. Then $N$ has one end (infinite genus and extraordinary), and $N'$ has two ends (one of genus zero, the other of infinite genus, and both extraordinary). Here, $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|$ and $\left| \mathcal{E}\left(N'\right) \right|=\left| \mathcal{E}\left(M\right) \right|+1$. Thus, the hypothesis in Theorem [Theorem 2](#mainthm){reference-type="ref" reference="mainthm"} that the $1$-handle is attached to distinct ends may not be omitted. 6. Let $X$ be the $2$-disk $D^2$ with three points removed from its boundary. So, $X$ has three noncompact boundary components and three ends. Each end of $X$ has genus zero and an open neighborhood homeomorphic to $[0,1]\times\mathbb{R}$. Let $Y$ be obtained from $X$ by connect summing a sequence of tori converging to one end. The surface $Y$ is depicted on the left in Figure [\[fig:endsumexncb\]](#fig:endsumexncb){reference-type="ref" reference="fig:endsumexncb"} where the zeros indicate genus zero ends, $\infty$ indicates the infinite genus end, and the segment indicates a ray. The surface $M=Y\sqcup Y$ is the disjoint union of two copies of $Y$. End sum $M$ along the indicated rays. Let $N$ be the result using an oriented $1$-handle, and let $N'$ be the result using a non-oriented $1$-handle. The surfaces $N$ and $N'$ are shown in Figure [\[fig:endsumexncb\]](#fig:endsumexncb){reference-type="ref" reference="fig:endsumexncb"}, where the letters $L$ and $R$ are included to display orientations. Both $N$ and $N'$ have six ends (four of genus zero, two of infinite genus, and two extraordinary both of genus zero). Here, $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|=\left| \mathcal{E}\left(N'\right) \right|$. Notice that $N$ and $N'$ are not homeomorphic---$N$ has two noncompact boundary components that point only to genus zero ends of $M$ and point to a common end of $M$, whereas $N'$ does not have such boundary components. Up to homeomorphism, there are exactly two end sums, namely $N$ and $N'$, of $M$ along the specific ends just used since ray choice is irrelevant at those ends. To obtain similar examples with $M$ non-orientable, remove an open disk from the interior of $Y$ and glue in a crosscap. Define $N$ and $N'$ as before. Though both components of $M$ are non-orientable, and often orientability of the $1$-handle is not relevant when $M$ is non-orientable, the end sums $N$ and $N'$ remain non-homeomorphic (the same argument still applies). Thus, for end sums of surfaces with noncompact boundary components, orientability of the $1$-handle is relevant. Example [\[deex\]](#deex){reference-type="ref" reference="deex"} showed that Theorem [Theorem 2](#mainthm){reference-type="ref" reference="mainthm"} is false without the distinct ends hypothesis. To remove that hypothesis, appropriate replacement hypotheses would be necessary. Our proof of Theorem [Theorem 2](#mainthm){reference-type="ref" reference="mainthm"} proceeds by studying end invariants of the extraordinary end of $N$ and then applying the classification of noncompact surfaces with compact boundary as proved by Richards [@richards]. One may attempt to remove from Theorem [Theorem 2](#mainthm){reference-type="ref" reference="mainthm"} the hypothesis that $\partial M$ is compact by instead using the classification of noncompact surfaces with possibly noncompact boundary due to Brown and Messer [@brownmesser]. Even with that approach to the case where $\partial M$ is noncompact, one must understand how ray choice affects the number of extraordinary ends and the number of ends of $N$, and how the orientation of the $1$-handle affects $N$. The examples above indicate that those may be subtle questions. To circumvent those nuances, the present paper focuses on the case where a single $1$-handle at infinity is attached to a surface $M$ with compact boundary along rays pointing to distinct ends of $M$. In this case, there is a unique extraordinary end of $N=M\cup \left(1\textnormal{-handle}\right)$ and $\left| \mathcal{E}\left(N\right) \right|=\left| \mathcal{E}\left(M\right) \right|-1$. We allow $M$ to be disconnected and $\partial M$ to be empty. Without loss of generality, it suffices to consider two cases: (i) $M$ is connected, and (ii) $M$ has two connected components and the $1$-handle connects them. The latter operation is the end sum of the two components of $M$. This paper is organized as follows. Section [2](#sec:endsIntroduction){reference-type="ref" reference="sec:endsIntroduction"} recalls the theory of ends sufficient for our purposes. We have included that material to help make this paper more self-contained and accessible. Section [3](#sec:endSumIntroduction){reference-type="ref" reference="sec:endSumIntroduction"} defines end sum and $1$-handle addition at infinity, sets up notation for those operations, and proves that those operations indeed yield manifolds (we are unaware of a published proof of this fact). Section [4](#sec:classification){reference-type="ref" reference="sec:classification"} recalls the classification of noncompact surfaces with compact boundary, including generalized genus, parity, end invariants, and orientability. Section [5](#sec:mainTheorem){reference-type="ref" reference="sec:mainTheorem"} proves our main result---Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}---for [pl]{.smallcaps} surfaces by studying how each of the following end invariants are affected by the addition of a $1$-handle at infinity: the space of ends, boundary, orientability, genus, and parity. Section [6](#sec:rayUniquenessMain){reference-type="ref" reference="sec:rayUniquenessMain"} provides an alternative proof of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} using Brown and Messer's [@brownmesser] classification of noncompact surfaces. That approach also yields a ray uniqueness result for surfaces---see Theorem [Theorem 67](#thm:rayUniqueness){reference-type="ref" reference="thm:rayUniqueness"}. Lastly, Section [7](#sec:secondTheorem){reference-type="ref" reference="sec:secondTheorem"} explains how to extend Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} to [top]{.smallcaps} and [diff]{.smallcaps} surfaces. We use the following conventions where $X$ is a topological space, $A$ is any subspace of $X$ (denoted $A\subset X$), and $M$ is a manifold. - $\overline{A}$ denotes the topological **closure** of $A$ in $X$. - $A^{\circ}$ denotes the topological **interior** of $A$ in $X$. - $\mathbb{R}_+$ denotes $[0,\infty)$ and $\mathbb{R}^n_+$ denotes $[0, \infty) \times \mathbb{R}^{n-1}$. - $D^n$ denotes the closed $n$-disk. - [cat]{.smallcaps} denotes one of the manifold categories: [top]{.smallcaps}, [pl]{.smallcaps}, or [diff]{.smallcaps}. - [cat]{.smallcaps} manifolds are Hausdorff and paracompact, possibly with boundary. - A connected manifold without boundary is **closed** provided it is compact and is **open** provided it is noncompact. - $M \cong M'$ denotes **isomorphism** of [cat]{.smallcaps} manifolds. - $\partial{M}$ denotes the **manifold boundary** of $M$. - $\mathcal{E}\left(X\right)$ denotes the **space of ends** of $X$. - If $\alpha$ is an end of $X$ and $K$ is a compact subspace of $X$, then $\mathcal{N}(\alpha;K)$ denotes the set of all ends $\beta$ of $X$ for which $\beta(K) = \alpha(K)$. - $b(M) = \left| \pi_0\left(\partial{M}\right) \right|$ denotes the **number of boundary components** of $M$. - $g(M)$ denotes the **genus** of a surface $M$ and $\mathcal{P}(M)$ denotes the parity of $M$. **Acknowledgments** The authors are grateful to Craig Guilbault for suggesting the main problem studied herein and to Ric Ancel for suggesting a fruitful alternative approach to prove the main theorem based on uniqueness of rays in a surface. # Ends of Spaces {#sec:endsIntroduction} Loosely speaking, an end of a space may be thought of as an infinity of the space. For example, the closed interval has no ends, a ray has one end, and the real line has two ends. Figure [\[fig:endExamples\]](#fig:endExamples){reference-type="ref" reference="fig:endExamples"} shows three manifolds and their ends. Each (suitably nice) space is compact if and only if it has no ends. The set of all ends of a space may be equipped with a natural topology. Ends and the space of ends play fundamental roles in the study of noncompact spaces. A prime example is Whitehead's open, contractible 3-manifold, the first example of an open, contractible manifold not isomorphic to Euclidean space. It may be distinguished from $\mathbb{R}^3$ by properties of its end (see Guilbault [@guilbault pp. 6--7]). The space of ends is also an essential ingredient in Richards' [@richards] classification of noncompact surfaces which we review in Section [4](#sec:classification){reference-type="ref" reference="sec:classification"}. The theory of ends dates back to Freudenthal and Hopf in the 1930's. For interesting further reading, see Freudenthal's original paper [@freudenthal], Siebenmann's thesis [@siebenmannthesis pp. 8--12], and Guilbault's chapter [@guilbault]. As we are focused on manifolds, we restrict our study to spaces that satisfy the following condition. **Definition 3**. A topological space $X$ is **nice for ends** provided $X$ is Hausdorff, locally compact, $\sigma$-compact, connected, and locally connected. We adopt the convention that a space $X$ is **connected** provided it has exactly two subspaces that are both open and closed in $X$, namely $X$ and $\varnothing$. In particular, the empty space is neither connected nor disconnected. Connected manifolds are nice for ends, as are connected, locally finite simplicial complexes. Each space that is nice for ends is necessarily paracompact. However, it is not necessarily separable, metrizable, or even first-countable. For example, the product of uncountably many copies of $[0,1]$ is compact and nice for ends, but does not satisfy those three properties. Throughout this section---unless explicitly stated otherwise---$X$ is a space nice for ends. **Definition 4**. An **end** of $X$ is any function $\epsilon$ defined on the collection of compact subspaces of $X$ such that: for each compact $K \subset X$, the output $\epsilon(K)$ is a connected component (hence nonempty) of $X -K$, and if $K_1 \subset K_2$, then $\epsilon(K_2) \subset \epsilon(K_1)$. Let $\mathcal{E}\left(X\right)$ denote the set of all ends of $X$. For example, $\mathbb{R}$ has two ends $\epsilon_-$ and $\epsilon_+$. Given a compact $K \subset \mathbb{R}$, $\epsilon_-(K)$ is the connected component of $\mathbb{R}-K$ that contains elements less than $K$, and $\epsilon_+(K)$ is the connected component of $\mathbb{R}-K$ that contains elements greater than $K$. One may verify that $\epsilon_-$ and $\epsilon_+$ are ends of $\mathbb{R}$ and, in fact, are the only ends of $\mathbb{R}$. We recall some fundamental properties of ends. Several proofs will be left to the reader. **Lemma 5**. If $\epsilon$ is an end of $X$, and $K$ and $K'$ are compact subspaces of $X$, then $\epsilon(K) \cap \epsilon(K') \neq \varnothing$. **Lemma 6**. If $X$ is compact, then $X$ has no ends. The converse of Lemma [Lemma 6](#lem:compactSpacesHaveNoEnds){reference-type="ref" reference="lem:compactSpacesHaveNoEnds"} also holds---see Corollary [Corollary 19](#cor:noncompactImpliesHasAnEnd){reference-type="ref" reference="cor:noncompactImpliesHasAnEnd"} below. **Definition 7**. A subspace of $X$ is **bounded** provided its closure in $X$ is compact, and **unbounded** otherwise. This terminology aligns with the definition of a bounded subset of Euclidean space, although in general metric spaces the notions can differ. **Lemma 8**. Let $\epsilon$ be an end of $X$. If $K\subset X$ is compact, then $\epsilon(K)$ is an unbounded subspace of $X$. **Lemma 9**. Let $K \subset X$ be compact. Then, $X -K$ has finitely many unbounded connected components. Furthermore, if $A$ is the union of all bounded connected components of $X -K$, then $K \cup A$ is compact and $X -(K \cup A)$ has only unbounded connected components. For the proof of Lemma [Lemma 9](#lem:unboundedComponents){reference-type="ref" reference="lem:unboundedComponents"}, we offer this hint: use local compactness to construct a bounded open neighborhood $V$ of $K$. Then, look at the open cover of $\overline{V}$ consisting of $V$ and all components of $X -K$. **Example 10**. Let $X=D^2$. For each integer $n \geq 1$, let $p_n = (0, 2^{-n})$ and let $U_n$ be an open disk of radius $4^{-n}$ centered at $p_n$. The space $K = X -\cup_n U_n$ is compact, but each $U_n$ is a bounded connected component of $X -K$. Thus, although there are always finitely many unbounded connected components of $X -K$, there may be infinitely many bounded components. The notion of a *neighborhood of an end* is used to define a topology on the set of ends of $X$. **Definition 11**. Let $\epsilon$ be an end of $X$. A **neighborhood** of $\epsilon$ is a subspace $A \subset X$ such that there exists some compact subspace $K \subset X$ for which $\epsilon(K) \subset A$. For example, consider $X = \mathbb{R}$ with ends $\epsilon_-$ and $\epsilon_+$. A subspace of $\mathbb{R}$ is a neighborhood of $\epsilon_-$ if and only if it contains $(-\infty, a)$ for some $a \in \mathbb{R}$. Similarly, a subspace of $\mathbb{R}$ is a neighborhood of $\epsilon_+$ if and only if it contains $(b, \infty)$ for some $b \in \mathbb{R}$. **Lemma 12**. If $K \subset X$ is compact, then $X -K$ is a neighborhood of every end of $X$. **Lemma 13**. If $\epsilon_1$ and $\epsilon_2$ are distinct ends of $X$, then there exist disjoint neighborhoods $A_1$ of $\epsilon_1$ and $A_2$ of $\epsilon_2$ (so, $A_1 \cap A_2 = \varnothing$). An alternative definition of an end may given using a *compact exhaustion*. This important and equivalent definition is useful for constructing and visualizing ends. **Definition 14**. A **compact exhaustion** of a topological space $X$ is a sequence $(K_i)$ of compact subspaces $K_i\subset X$ such that $K_i \subset K_{i+1}$ for all $i$ and $$\bigcup_{i=1}^\infty K_i^{\circ} = X$$ For example, $K_i = [-i, i]$ is a compact exhaustion of $\mathbb{R}$. See Figure [\[fig:exhaustionexamples\]](#fig:exhaustionexamples){reference-type="ref" reference="fig:exhaustionexamples"} for two more examples. To make use of compact exhaustions, we need their existence. **Lemma 15**. The space $X$ has a compact exhaustion. *Proof.* As $X$ is $\sigma$-compact, there exist compact subspaces $L_1$, $L_2$, $\ldots$ of $X$ such that $\cup_j L_j = X$. As $X$ is locally compact, each $L_j$ has a compact neighborhood $N_i$ in $X$ (meaning $L_j\subset N_i^{\circ}$). Define $K_i = \cup_{j \leq i} N_j$, which is compact. For each $x \in X$, there exists $j$ such that $x \in L_j$. So, $x \in K_i^{\circ}$ and $(K_i)$ is a compact exhaustion of $X$. ◻ A compact exhaustion $(K_i)$ of $X$ describes $X$ as a limit of the compact sets $K_i$. It may also be used to define an end of $X$ as a limit of unbounded components of the complements $X -K_i$. Fix a compact exhaustion $(K_i)$ of $X$. Define $\mathcal{E}'(X)$ to be the set of all sequences $(V_i)$ such that $V_i$ is a connected (hence nonempty) component of $X -K_i$, and $V_{i+1} \subset V_i$. **Lemma 16**. There is a one-to-one correspondence between $\mathcal{E}\left(X\right)$ and $\mathcal{E}'(X)$ that associates to each end $\epsilon$ the sequence $V_i = \epsilon(K_i)$. *Proof.* If $\epsilon \in \mathcal{E}\left(X\right)$, then the sequence $V_i = \epsilon(K_i)$ is in fact in $\mathcal{E}'(X)$. On the other hand, given any sequence $(V_i) \in \mathcal{E}'(X)$, one may associate an end $\epsilon$ as follows. For each compact $L\subset X$, choose $i$ such that $L \subset K_i$. Then, $V_i$ is a connected component of $X -K_i$ and is contained in a unique connected component $W$ of $X -L$. Define $\epsilon(L) = W$. One may verify that $\epsilon$ is well-defined, and that these associations are inverses. ◻ When a compact exhaustion $(K_i)$ is given for $X$, we will often conflate an end of $X$ with the sequence $(V_i) \in \mathcal{E}'(X)$. Some authors---for example Guilbault [@guilbault]---take this sequence as the definition of an end of $X$. As an example, consider $X = \mathbb{R}$ with the compact exhaustion $K_i = [-i, i]$. The only sequences in $\mathcal{E}'(X)$ are $V_i = (-\infty, i)$ and $V_i = (i, \infty)$. These represent the two ends of $\mathbb{R}$. Using compact exhaustions, we give a straightforward proof that each space with no ends is compact. **Lemma 17**. Fix a compact exhaustion $(K_i)$ of $X$. If $V_1 \supset V_2 \supset \dots \supset V_n$ is a finite sequence where each $V_i$ is an unbounded connected component of $X -K_i$, then this sequence extends to an infinite sequence $(V_i)$ which represents an end of $X$. *Proof.* It suffices to prove that if $V_i$ is an unbounded connected component of $X -K_i$, then there exists an unbounded connected component of $X -K_{i+1}$ contained in $V_i$. Using this, any finite initial sequence may be extended inductively to an infinite sequence. Let $L$ be the union of $K_{i+1}$ and all of the bounded components of $X -K_{i+1}$. By Lemma [Lemma 9](#lem:unboundedComponents){reference-type="ref" reference="lem:unboundedComponents"}, $L$ is compact. As $V_i$ is unbounded, $V_i -L \neq \varnothing$. Let $y \in V_i -L$. Then $y$ is contained in some component $V_{i+1}$ of $X -K_{i+1}$. As $y \not\in L$, this component must be unbounded. ◻ **Corollary 18**. If $K \subset X$ is compact and $V$ is an unbounded connected component of $X -K$, then there exists an end $\epsilon \in \mathcal{E}\left(X\right)$ such that $\epsilon(K) = V$. *Proof.* Let $(K_i)$ be any compact exhaustion of $X$ such that $K_1 = \varnothing$, and let $K_i' = K_i \cup K$ for each $i$. Note that $(K_i')$ is a compact exhaustion of $X$ and $K_1' = K$. Apply Lemma [Lemma 17](#lem:existenceOfEnds){reference-type="ref" reference="lem:existenceOfEnds"} to the single-term initial sequence $(V)$. ◻ **Corollary 19**. If $X$ is noncompact, then $X$ has at least one end. *Proof.* Apply Corollary [Corollary 18](#cor:existenceOfEnds2){reference-type="ref" reference="cor:existenceOfEnds2"} to $K = \varnothing$ and $V = X$. ◻ Compact exhaustions also provide an upper bound on the number of ends of a space. **Lemma 20**. The number of ends of $X$ is at most the continuum. *Proof.* Fix a compact exhaustion $(K_i)$ of $X$. There exists one end for each choice sequence $(V_i)$ where $V_i$ is a connected component of $X -K_i$ and $V_i \supset V_{i+1}$. By Lemma [Lemma 9](#lem:unboundedComponents){reference-type="ref" reference="lem:unboundedComponents"}, there exist only finitely many unbounded connected components of $X -K_i$. There are at most continuum many ways to pick one element from each of a countable number of finite sets. ◻ The set of ends of $X$ admits a natural topology. The resulting *space of ends* is essential for the classification of noncompact surfaces. **Definition 21**. Let $\epsilon$ be an end of $X$, and let $K \subset X$ be compact. Let $\mathcal{N}(\epsilon;K)$ denote the set of all ends $\beta$ of $X$ such that $\epsilon(K) = \beta(K)$. We refer to $\mathcal{N}(\epsilon;K)$ as a **basic end-space neighborhood** of $\epsilon$. Equip $\mathcal{E}\left(X\right)$ with the topology generated by $$\{\mathcal{N}(\epsilon;K)\,:\,\epsilon \in \mathcal{E}\left(X\right),\,K\subset X\textnormal{ is compact}\}$$ Note that if $\mathcal{N}(\epsilon;K)$ and $\mathcal{N}(\eta;L)$ both contain an end $\alpha$, then they both contain ${\mathcal{N}(\alpha;K \cup L)}$. Thus, this collection indeed forms a basis for a topology on $\mathcal{E}\left(X\right)$. **Lemma 22**. The space of ends $\mathcal{E}\left(X\right)$ is Hausdorff, compact, separable, and totally disconnected. *Proof.* Fix a compact exhaustion $(K_i)$ of $X$. If $\epsilon$, $\eta$ are two distinct ends of $X$, then there exists an $n$ such that $\epsilon(K_n) \neq \eta(K_n)$. $\mathcal{N}(\epsilon;K_n)$ and $\mathcal{N}(\eta;K_n)$ are two disjoint open sets containing $\epsilon$ and $\eta$, respectively. Thus, $\mathcal{E}\left(X\right)$ is Hausdorff. Furthermore, the complement of $\mathcal{N}(\epsilon;K_n)$ is the union of $\mathcal{N}(\rho;K_n)$ for all ends $\rho$ for which $\rho(K_n) \neq \epsilon(K_n)$. Thus, the complement of $\mathcal{N}(\epsilon;K_n)$ is open, so $\mathcal{N}(\epsilon;K_n)$ is clopen. As any two ends are separated by clopen subsets, $\mathcal{E}\left(X\right)$ is totally disconnected. For each fixed $n$, there are finitely many unbounded components of $X -K_n$, and so finitely many distinct sets $\mathcal{N}(\epsilon;K_n)$ for $\epsilon \in \mathcal{E}\left(X\right)$. Because sets of the form $\mathcal{N}(\epsilon;K_n)$ form a basis for the topology of $\mathcal{E}\left(X\right)$, and there are a finite number of these sets for each $n$, $\mathcal{E}\left(X\right)$ is second-countable, hence separable. Lastly, we prove that $\mathcal{E}\left(X\right)$ is compact. Suppose for contradiction that $\mathcal{E}\left(X\right)$ has an open cover $\mathcal{C}$ with no finite subcover. Say that a subset of $\mathcal{E}\left(X\right)$ is not finitely coverable if no finite subset of $\mathcal{C}$ covers it. If $\{\epsilon\,:\,\epsilon(K_n) = V\}$ is not finitely coverable, then let let $W_1, W_2, \dots W_k$ be the unbounded connected components of $X -K_{n+1}$ contained in $V$. We can see that $\{\epsilon\,:\,\epsilon(K_{n})=V)\}$ is the union of $\{\epsilon\,:\,\epsilon(K_{n+1})=W_i\}$ over all $W_i$. As $\{\epsilon\,:\,\epsilon(K_{n})=V)\}$ is not finitely coverable, there exists some $j$ such that $\{\epsilon\,:\,\epsilon(K_{n+1})=W_j\}$ is not finitely coverable. Thus, as $\mathcal{E}\left(X\right)$ is not finitely coverable, we obtain a sequence $W_1, W_2, \dots$ where $W_i$ is a connected component of $X -K_i$, and $W_i \subset W_{i+1}$. This determines an end, $\alpha$, of $X$. $\alpha$ is in some open subset $U \in \mathcal{C}$, which contains a subset of the form $\mathcal{N}(\alpha;K_n)$ for some $n$. But, then, $\mathcal{N}(\alpha;K_n)$ would be finitely coverable, which we know is not the case. This is a contradiction, so $\mathcal{E}\left(X\right)$ is compact. ◻ As ends are defined using compact subspaces, the theory of ends naturally utilizes *proper maps*. Recall that a continuous function $f: X \to Y$ is **proper** provided the inverse image of each compactum is compact. Topological spaces and proper maps form a category that is well-suited to the study of ends. Given a proper map $f: X \to Y$, we may extend $f$ to a (proper) map $\mathcal{E}\left(f\right): \mathcal{E}\left(X\right) \rightarrow \mathcal{E}\left(Y\right)$. In other words, $\mathcal{E}\left(-\right)$ is a functor. **Definition 23**. Let $\mathcal{E}\left(f\right)(\epsilon)$ denote the unique end $\eta \in \mathcal{E}\left(Y\right)$ such that $f^{-1}(\eta(K)) \supset \epsilon(f^{-1}(K))$ for all compact $K \subset Y$. It may be shown that $\mathcal{E}\left(f\right)$ is well-defined and continuous. For example, let $\mathbb{R}_+ = [0,\infty)\subset\mathbb{R}$. A **ray** in $X$ is a proper embedding $r:\mathbb{R}_+\to X$. There is a simple description of $\mathcal{E}\left(r\right)$. Let $\epsilon$ be the unique end of $\mathbb{R}_+$, and let $\eta = \mathcal{E}\left(r\right)(\epsilon)\in\mathcal{E}\left(X\right)$. Then, $\eta(K)$ is the unique component of $X -K$ which contains $r((a, \infty))$ for some $a \in \mathbb{R}_+$. In this case, we say that $r$ **points to** $\eta$. Although arguments are cleaner for connected spaces, the theory of ends applies to *disconnected* topological spaces that are otherwise nice for ends. This allows us to define the ends of any manifold, which we will use frequently when adding $1$-handles at infinity. **Definition 24**. Let $X$ be Hausdorff, locally compact, locally connected, and $\sigma$-compact, but not necessarily connected. An **end** of $X$ is an end of a connected component of $X$. When $X$ is disconnected, we may still define a natural topology on $\mathcal{E}\left(X\right)$ as above. As a topological space, $\mathcal{E}\left(X\right)$ is the disjoint union of $\mathcal{E}\left(C\right)$ over all connected components $C$ of $X$. If $X$ has only finitely many connected components, then $\mathcal{E}\left(X\right)$ is still compact. If $X$ has infinitely many components, then $\mathcal{E}\left(X\right)$ may be noncompact. Our focus is the effect of adding a single 1-handle at infinity to a manifold $M$. This operation involves either one or two components of $M$, and any other components may be safely ignored. # End Sum and 1-handles at Infinity {#sec:endSumIntroduction} The basic idea of the end sum operation is to combine two noncompact manifolds of the same dimension along a proper ray in each. For example, the interior of a boundary sum of two manifolds with boundary is an end sum of their interiors. In general, end sum is more complicated than boundary sum as it applies to noncompact manifolds that are not necessarily the interior of any compact manifold, and it requires a choice of ray in each manifold. End sum also requires a choice of a tubular neighborhood map for each ray, though different choices yield isomorphic manifolds. End sum is a special case of the more general operation of attaching a $1$-handle at infinity to a possibly disconnected manifold (see also [@cg pp. 1303--1305]). We define these operations simultaneously for [cat]{.smallcaps}=[top]{.smallcaps}, [pl]{.smallcaps}, and [diff]{.smallcaps}. For the remainder of this section, embeddings and homeomorphisms are assumed to be [cat]{.smallcaps}. Let $M$ be a (possibly disconnected) $n$-manifold. A **ray** in $M$ is a proper, locally flat embedding $r: \mathbb{R}_+ \to M$ with image in the manifold interior of $M$. We often conflate a ray with its image in $M$. Applying the end functor $\mathcal{E}$ to $r$ (see Definition [Definition 23](#def:eMap){reference-type="ref" reference="def:eMap"}), we have that $\mathcal{E}\left(r\right)$ picks out a single end $\eta \in \mathcal{E}\left(M\right)$. In this case, we say that $r$ **points to** $\eta$. Define a **tubular neighborhood map** of $r$ to be a proper, locally flat embedding $\nu: [0, \infty) \times D^{n-1} \to M$ with image in the manifold interior of $M$ and such that $\nu(x, 0) = r(x)$. Let $r$ and $r'$ be disjoint rays in $M$, and let $\nu$ and $\nu'$ be tubular neighborhood maps of $r$ and $r'$ (respectively) with disjoint images. Let $U \subset M$ be the image of $\nu$ restricted to the manifold interior of its domain. Similarly, define $U'$ for $\nu'$. As $U$ and $U'$ are locally flat embeddings of $n$-manifolds without boundary into the manifold interior of $M$, $U$ and $U'$ are open subsets of the manifold interior of $M$ (hence, also of $M$). Let $E = \mathbb{R}\times \mathbb{R}^{n-1}$, which is the $1$-handle at infinity that will be attached to $M$. As in Figure [7](#fig:endsum_diagram){reference-type="ref" reference="fig:endsum_diagram"}, we partition $E$ into $E_- = (-\infty, 0) \times \mathbb{R}^{n-1}$, $E_0 = \{0\} \times \mathbb{R}^{n-1}$, and $E_+ = (0, \infty) \times \mathbb{R}^{n-1}$. {#fig:endsum_diagram} The restriction of $\nu$ to $U$ gives a homeomorphism $(0, \infty) \times (D^{n-1})^{\circ} \cong U$ where $(D^{n-1})^{\circ}$ denotes the manifold interior of the disk. By reversing the first factor and radially expanding the second, we get a homeomorphism $\varphi_+: E_+ \rightarrow U$. Similarly, the restriction of $\nu'$ to $U'$ gives a homeomorphism $(0, \infty) \times (D^{n-1})^{\circ} \cong U'$. By reversing *and negating* the first factor and radially expanding the second, we get a homeomorphism $\varphi_-: E_- \rightarrow U'$. We glue $E$ to $M$ using the maps $\varphi_+$ and $\varphi_-$. Explicitly, let $\sim$ be the equivalence relation on $M \sqcup E$ generated by $x \sim \varphi_+(x)$ for $x \in E_+$ and $x \sim \varphi_-(x)$ for $x \in E_-$. Let $N$ be the quotient space of $M \sqcup E$ by this equivalence relation as in Figure [8](#fig:endsum_final){reference-type="ref" reference="fig:endsum_final"}. {#fig:endsum_final} Let $q:M \sqcup E \to N$ be the associated quotient map. We call $N$ a **result of attaching a 1-handle at infinity** to $M$. If $M=M_1 \sqcup M_2$ has exactly two components $M_1$ and $M_2$, each containing one of the rays $r$ or $r'$, then we call $N$ an **end sum** of $M_1$ and $M_2$ (or simply an end sum of $M$). When $M$ is an oriented manifold and $\varphi_+$ and $\varphi_-$ are both orientation-preserving, we call this construction an **oriented $1$-handle**. Below, we prove that $N$ is naturally an $n$-manifold (we are not aware of a proof of this fact in the literature). Define the composite maps $i_M:M \hookrightarrow M \sqcup E \stackrel{q}{\rightarrow} N$ and $i_E:E \hookrightarrow M \sqcup E \stackrel{q}{\rightarrow} N$. As the tubular neighborhood maps $\nu$ and $\nu'$ have disjoint images, the maps $i_M$ and $i_E$ are injective. **Lemma 25**. The quotient map $q:M \sqcup E \to N$ is open. *Proof.* The quotient map is open if and only if the saturation of any open set in $M \sqcup E$ with respect to $\sim$ is still open in $M \sqcup E$. Let $V$ be open in $M \sqcup E$ and let $W=q^{-1}(q(V))$ be the saturation of $V$ with respect to $\sim$. We have: $$W = V \cup \varphi_+(V \cap E_+) \cup \varphi_+^{-1}(V \cap U) \cup \varphi_-(V \cap E_-) \cup \varphi_-^{-1}(V \cap U')$$ As $\varphi_+$ and $\varphi_-$ are homeomorphisms between open subspaces of $M \sqcup E$, $W$ is open. ◻ **Corollary 26**. The maps $i_M:M\to N$ and $i_E:E\to N$ are open. *Proof.* The inclusions $M\hookrightarrow M \sqcup E$ and $E\hookrightarrow M \sqcup E$ are open by the definition of the disjoint union topology. Now, apply Lemma [Lemma 25](#qopen){reference-type="ref" reference="qopen"}. ◻ The following is the key step where properness of the rays and tubular neighborhood maps are required. **Lemma 27**. The quotient space $N$ is Hausdorff. *Proof.* Let $a$ and $b$ be distinct points in $N$. If $a, b \in \textnormal{Im}\,\,i_M$, then $i_M^{-1}(a)$ and $i_M^{-1}(b)$ are separated by disjoint open sets $A$ and $B$ in $M$. The sets $i_M(A)$ and $i_M(B)$ are disjoint since $i_M$ is injective, open in $N$ by Corollary [Corollary 26](#compsopen){reference-type="ref" reference="compsopen"}, and thus separate $a$ and $b$. If $a, b \in \textnormal{Im}\,\,i_E$, then similarly $i_E^{-1}(a)$ and $i_E^{-1}(b)$ are separated by disjoint open sets in $E$ whose images under $i_E$ separate $a$ and $b$. Lastly, consider the case where $a \in \textnormal{Im}\,\,i_M$ and $b \in \textnormal{Im}\,\,i_E$. We may assume $b \in i_E(E_0)$, as otherwise $b \in \textnormal{Im}\,\,i_M$. Let $A$ be a bounded open subset of $M$ containing $a$. As the tubular neighborhood maps $\nu$ and $\nu'$ are proper, there exists some $s>0$ such that $\nu((s, \infty) \times D^{n-1})$ and $\nu'((s, \infty) \times D^{n-1})$ are disjoint from $A$. So, there exists some $t>0$ such that $i_E((-t, t) \times \mathbb{R}^{n-1})$ is disjoint from $i_M(A)$. The sets $i_E((-t, t) \times \mathbb{R}^{n-1})$ and $i_M(A)$ are open by Corollary [Corollary 26](#compsopen){reference-type="ref" reference="compsopen"} and separate $a$ and $b$. ◻ **Lemma 28**. The quotient space $N$ is naturally an $n$-dimensional [cat]{.smallcaps} manifold. If $M$ is oriented and the $1$-handle is oriented, then $N$ is naturally an oriented manifold. The maps $i_M$ and $i_E$ are open embeddings, and if the $1$-handle is oriented, then $i_M$ and $i_E$ are orientation-preserving. *Proof.* The quotient space $N$ is Hausdorff by Lemma [Lemma 27](#lem:endSumIsHausdorff){reference-type="ref" reference="lem:endSumIsHausdorff"}. As $M \sqcup E$ is second-countable and the open quotient of a second-countable space is second-countable, $N$ is also separable. Every point of $N$ has a neighborhood homeomorphic to a subset of $M$ or $E$, so in particular, $N$ is locally Euclidean. Thus, $N$ is a [top]{.smallcaps} manifold. If [cat]{.smallcaps}=[pl]{.smallcaps} or [diff]{.smallcaps}, then $N$ inherits a natural [pl]{.smallcaps} or [diff]{.smallcaps} structure using an atlas generated by the charts of $M$ and $E$. If in addition $M$ is an oriented manifold and the 1-handle is oriented, then the atlas generated by the oriented charts of $M$ and $E$ defines an orientation of $N$. Lastly, $i_M$ and $i_E$ are injective, open by Corollary [Corollary 26](#compsopen){reference-type="ref" reference="compsopen"}, and thus are open embeddings. If in addition $M$ is an oriented manifold and the 1-handle is oriented, then $i_M$ and $i_E$ respect the orientation of $N$. ◻ There is an equivalent operation to adding a $1$-handle at infinity that is sometimes useful. In this alternative formulation, disjoint closed regular neighborhoods $A$ and $A'$ of $r$ and $r'$ are chosen in the interior of $M$. Then, $N$ is defined by removing the manifold interiors of $A$ and $A'$ from $M$ and then gluing together the resulting boundary components---both are copies of $\mathbb{R}^{n-1}$---by an orientation reversing homeomorphism. For more on this approach, see [@cks pp. 1813--1818]. # Classification of Surfaces with Compact Boundary {#sec:classification} The classification of noncompact surfaces with compact boundary is essential for our end sum uniqueness results. This contrasts with end sum uniqueness results in higher dimensions which rely on isotopy uniqueness of rays [@cg]. Manifolds in this section are assumed to be [pl]{.smallcaps}  although the results hold for [diff]{.smallcaps}  and [top]{.smallcaps}. Following Richards [@richards], we define the genus and parity of any surface with compact boundary and extend those concepts to end invariants. Using those invariants, we present a classification theorem of surfaces with compact boundary. ## Compact Surfaces {#subsec:compactsurfaces} We begin with the well-known classification of compact surfaces. For a proof, see [@munkrestop pp. 446--476]. Given a compact surface $M$, let $\chi(M)$ denote its Euler characteristic, and let $b(M)$ denote the number of boundary components of $M$. **Theorem 29** (Classification of compact surfaces). Let $M$ be a compact, connected surface. If $M$ is orientable, then for some non-negative integer $g$, $M$ is isomorphic to a sphere with $g$ handles and $b(M)$ holes, and $\chi(M) = 2 - 2g - b(M)$. If $M$ is non-orientable, then for some positive integer $k$, $M$ is isomorphic to the sphere with $k$ cross-caps and $b(M)$ holes, and $\chi(M) = 2 - k - b(M)$. It will be more convenient for us to classify surfaces by their genera. We use the following convention (sometimes called the "generalized genus") to extend the concept of genus to any compact surface. Given a compact surface $M$, let $\left| \pi_0\left(M\right) \right|$ denote the number of connected components of $M$. **Definition 30**. Let $M$ be a compact surface. The **genus** of $M$, denoted $g(M)$, is an integer or half-integer defined by the following. $$g(M) := \left| \pi_0\left(M\right) \right| - \frac{b(M) + \chi(M)}{2}$$ For example, consider the surfaces described in Theorem [Theorem 29](#thm:finite2Manifolds){reference-type="ref" reference="thm:finite2Manifolds"}; in the orientable case $g(M)=g$, and in the non-orientable case $g(M)=k/2$. Observe that the genus is unchanged if a surface is punctured (meaning an open disk is removed). Further, genus is additive over disjoint union since it is a linear combination of functions that are each additive over disjoint union. These properties make the genus a user-friendly alternative to the Euler characteristic. In a sense, the genus measures the complexity of a surface. This idea is made concrete by the following. **Theorem 31**. If $N$ is a compact surface, then $g(N) \geq 0$. If $M$ is a subsurface of $N$, then $g(M) \leq g(N)$. *Proof.* The classification of compact surfaces (Theorem [Theorem 29](#thm:finite2Manifolds){reference-type="ref" reference="thm:finite2Manifolds"}) implies that every compact, connected surface has non-negative genus. As genus is additive over disjoint union, every compact surface has non-negative genus. Now consider a compact surface $N$ with a compact subsurface $M$. It will be convenient for $M$ to avoid the boundary of $N$. So, we let $N'$ be the result of gluing an external collar $\partial{N} \times [0, 1]$ to $N$ along $\partial{N}$. Note that $N$ and $N'$ are isomorphic and that $M$ lies in the manifold interior of $N'$. As $M$ is a subsurface of $N'$ and is disjoint from $\partial{N}$, we have $E = \overline{N' -M}$ is a subsurface of $N'$. Note that $N'$ is obtained by gluing $E$ and $M$ together along their boundaries. More precisely, each boundary component of $M$ is glued to a unique boundary component of $E$. We have $$\begin{aligned} b(N') &= b(E) - b(M)\\ \chi(N') &= \chi(M) + \chi(E) - \chi(M \cap E) = \chi(M) + \chi(E)\end{aligned}$$ Beginning with $\left| \pi_0\left(M\right) \right| + \left| \pi_0\left(E\right) \right|$ components of $M$ and $E$, observe that each time a boundary component of $M$ is glued to one of $E$, the total number of components is reduced by $0$ or $1$. Thus, we have $$\begin{aligned} \left| \pi_0\left(N'\right) \right| &\geq \left| \pi_0\left(M\right) \right| + \left| \pi_0\left(E\right) \right| - b(M) \end{aligned}$$ Therefore $$\begin{aligned} g(N') &= \left| \pi_0\left(N'\right) \right| - \frac{b(N') + \chi(N')}{2}\\ &\geq g(M) + g(E) \end{aligned}$$ As $g(N') = g(N)$ and $g(E) \geq 0$, we get $g(N) \geq g(M)$. ◻ Define the **parity** of a compact surface $M$, denoted $\mathcal{P}(M)$, to be $2g(M)$ (mod 2). We call a compact surface **even** if its parity is zero and **odd** otherwise. Parity is strongly related to orientability by the following lemma. **Lemma 32**. If $N$ is a compact, orientable surface, then $\mathcal{P}(N) = 0$. If $N$ is a compact surface and $M \subset N$ is a subsurface such that $\partial{N} \subset M$ and $M -N$ is orientable, then $\mathcal{P}(N) = \mathcal{P}(M)$ *Proof.* By the classification of compact surfaces (Theorem [Theorem 29](#thm:finite2Manifolds){reference-type="ref" reference="thm:finite2Manifolds"}), it can be checked that every compact, connected, orientable surface has even parity. As the parity of a disconnected surface is the sum of the parities of its connected components, every compact, orientable surface has parity zero. Now, consider a compact surface $N$ with compact subsurface $M$ such that $\partial{N} \subset M$ and $N -M$ is orientable. Let $E = \overline{N -M}$. Because $\partial{N} \subset M$, $E$ is actually a (compact) subsurface of $N$. As $N -M$ is orientable, $E$ is orientable. Thus, $E$ has even parity. We have $$\begin{aligned} b(N) &= b(M) - b(E)\\ \chi(N) &= \chi(M) + \chi(E) - \chi(M \cap E) = \chi(M) + \chi(E)\end{aligned}$$ Thus $$\begin{aligned} g(N) - g(M) &= \left(\left| \pi_0\left(N\right) \right| - \frac{b(N) + \chi(N)}{2}\right) - \left(\left| \pi_0\left(M\right) \right| - \frac{b(M) + \chi(M)}{2}\right)\\ &= \left| \pi_0\left(N\right) \right| - \left| \pi_0\left(M\right) \right| + \frac{b(E)}{2} - \frac{\chi(E)}{2}\\ &= \left| \pi_0\left(N\right) \right| - \left| \pi_0\left(M\right) \right| - \left| \pi_0\left(E\right) \right| + b(E) + g(E)\end{aligned}$$ As $g(E)$ is an integer, $g(N) - g(M)$ is an integer. Hence, $\mathcal{P}(N) = \mathcal{P}(M)$. ◻ ## Noncompact Surfaces with Compact Boundary To extend our definitions from compact surfaces to noncompact surfaces, we will require the existence of arbitrarily large compact subsurfaces. The key theorem we need for these arguments is the following. **Theorem 33**. Let $M$ be a surface and let $K \subset M$ be compact. Then there exists a compact, locally flatly embedded subsurface $S \subset M$ such that $K \subset S$. *Proof.* Fix a triangulation $\Sigma$ of $M$. Let $L$ be the union of all triangles in $\Sigma$ that intersect $K$. Let $\Sigma'$ be the barycentric subdivision of $\Sigma$ and let $S$ be the union of all triangles in $\Sigma'$ that intersect $L$. It can be checked that $S$ is a [pl]{.smallcaps} subsurface of $M$. ◻ As a consequence of Theorem [Theorem 31](#thm:finiteGenusIsSize){reference-type="ref" reference="thm:finiteGenusIsSize"}, we see that for a *compact* surface $N$, $g(N)$ is the supremum of $g(M)$ over all compact subsurfaces $M$. This suggests a way to define the genus for arbitrary (noncompact) surfaces in terms of their compact subsurfaces. This generalized genus will play a vital role in our classification of noncompact surfaces with compact boundary. **Definition 34**. Let $N$ be a surface with compact boundary. The **genus** of $N$, denoted $g(N)$, is the supremum of $g(M)$ over all compact subsurfaces $M \subset N$ (and may be infinite). Definition [Definition 34](#def:genus){reference-type="ref" reference="def:genus"} is not circular because the genus of an arbitrary surface is defined in terms of the genus of compact surfaces, which we have already defined. By Theorem [Theorem 33](#thm:subsurfaces){reference-type="ref" reference="thm:subsurfaces"}, we know that there exist arbitrarily large compact subsurfaces of $N$, so $g(N)$ is actually the *limit* of $g(M)$ for compact subsurfaces $M$ as $M$ gets arbitrarily large. Similar to the compact case, we may think of the genus of a noncompact surface as a measure of complexity by the following. **Theorem 35**. If $N$ is a surface, then $g(N) \geq 0$. If $M$ is a subsurface of $N$, then $g(M) \leq g(N)$. Parity applies to noncompact surfaces with compact boundary provided the surface is orientable outside of a compact subset. When $N$ is connected, this is equivalent to $N$ having no non-orientable ends. If $N$ is orientable outside of a compact subset, then by Theorem [Theorem 33](#thm:subsurfaces){reference-type="ref" reference="thm:subsurfaces"}, there exists a compact subsurface $M \subset N$ such that $\partial{N} \subset M$ and $N -M$ is orientable. Following Lemma [Lemma 32](#lem:parityIsOrientability){reference-type="ref" reference="lem:parityIsOrientability"}, we would like to define $\mathcal{P}(N)$ to equal $\mathcal{P}(M)$. To prove this is well-defined, consider two compact subsurfaces $M_1, M_2 \subset N$ such that $N -M_1$ and $N -M_2$ are both orientable. Let $S \subset N$ be a subsurface such that $M_1 \cup M_2 \subset S$. By Lemma [Lemma 32](#lem:parityIsOrientability){reference-type="ref" reference="lem:parityIsOrientability"}, $\mathcal{P}(M_1) = \mathcal{P}(E)$ and $\mathcal{P}(M_2) = \mathcal{P}(E)$. Thus, $\mathcal{P}(M_1) = \mathcal{P}(M_2)$. As a consequence, the following is well-defined: **Definition 36**. Let $N$ be a surface with compact boundary that is orientable outside of a compact subset. The **parity** of $N$, denoted $\mathcal{P}(N)$, is the parity of any compact subsurface $M \subset N$ such that $\partial{M} \subset N$ and $N -M$ is orientable. When $g(N)$ is finite, we can see that $\mathcal{P}(N) \equiv 2g(N)$ (mod 2). However, the parity of a surface may exist even when the genus is infinite. ## End-Invariants {#subsec:endInvariants} To continue our exploration of the properties of noncompact surfaces, we will need to classify the ends of a surface. We will do this through the idea of an "end-invariant", a concept that can actually be applied to any topological space which is nice for ends. Let $X$ and $X'$ be nice for ends. Define two ends $\eta \in \mathcal{E}\left(X\right)$ and $\eta' \in \mathcal{E}\left(X'\right)$ to be **isomorphic** if there exist closed neighborhoods $A$ of $\eta$ and $A'$ of $\eta'$ and a homeomorphism $f: A \rightarrow A'$ such that $\mathcal{E}\left(f\right)(\eta) = \eta'$. It can be checked that isomorphism of ends is an equivalence relation. An **end-invariant** is any property or quantity associated to an end that is invariant under end isomorphism. One particularly important end-invariant is its orientability. **Definition 37**. An end $\eta$ of a manifold $M$ is **orientable** if $\eta$ has a closed neighborhood $A$ such that $A^{\circ}$ is orienatble. Otherwise, we say that $\eta$ is **non-orientable**. If $\eta$ has an orientable neighborhood $\eta(K)$ for some compact $K \subset M$, then every end in $\mathcal{N}(\eta;K)$ is also orientable. Thus, the orientable ends of $M$ form an open subset of $\mathcal{E}\left(M\right)$. Let $M$ be connected. If $M$ is orientable outside of a compact set, then all of its ends are orientable. Conversely, suppose that every end of $M$ is orientable. As every end of $M$ has an orientable neighborhood and $\mathcal{E}\left(M\right)$ is compact, $M$ is orientable outside of a compact set. Another important end-invariant is the genus of an end. **Definition 38**. The **genus** of an end $\eta$ of a surface $M$ with compact boundary is defined as the infimum of $g(A^{\circ})$ over all closed neighborhoods $A$ of $\eta$. For example, any end collared by $S^1$ has genus zero. If $A$ and $B$ are two neighborhoods of $\eta$ with $A \subset B$, then $g(A^{\circ}) \leq g(B^{\circ})$. Thus, the genus of an end can also be considered the limit of $g(U)$ for arbitrarily small neighborhoods $U$ of $\eta$. In fact, this limit can only have two possible values. **Lemma 39**. An end $\eta$ of a surface with compact boundary $M$ either has zero genus or infinite genus. *Proof.* Let $\eta$ be an end of $M$ with finite genus $h > 0$. Let $K \subset M$ be compact such that $g(\eta(K)) = h$. By definition, there exists a compact subsurface $L \subset \eta(K)$ such that $g(L) = h$. Now, $g(\eta(K \cup L))$ must still be $h$, so there exists a compact subsurface $L' \subset \eta(K \cup L)$ such that $g(L')=h$. However, $L$ and $L'$ are disjoint, so $g(L \cup L') = g(L) + g(L') = 2h$. In addition, $L \cup L'$ is a subset of $\eta(K)$. Thus, $g(\eta(K)) \geq g(L \cup L') = 2h$. This is a contradiction. ◻ Let $M$ be a surface with compact boundary. If $\eta$ is genus zero, then $g(\eta(K)) = 0$ for some compact $K \subset M$. Thus, every end in $\mathcal{N}(\eta;K)$ has genus zero. Thus, the genus zero ends of $M$ form an open subset of $\mathcal{E}\left(M\right)$. Let $M$ be a connected surface with compact boundary. If $M$ has finite genus, then every end of $M$ has finite genus (hence zero genus). Conversely suppose that every end of $M$ has finite genus. As every end of $M$ has a genus zero neighborhood and $\mathcal{E}\left(M\right)$ is compact, $M$ is genus zero outside of a compact set. The genus and orientability of an end are related. If an end $\eta$ of a surface has genus zero, then it has a genus zero neighborhood. As every genus zero surface is orientable, this neighborhood is also orientable. Thus, every genus zero end of a surface with compact boundary is also an orientable end. Beyond these more geometric end-invariants, there are a whole host of algebraic end-invariants. The most important of these are the homotopy, homology, and cohomology groups at infinity [@geoghegan pp. 229--281, 369--401]. These are incredibly important, and provide very powerful tools for distinguishing various ends, and by extension, distinguishing various manifolds. Like the respective invariants of algebraic topology (that is, the homotopy, homology, and cohomology groups), the algebraic end-invariants listed so far are all functorial. But whereas the homotopy, homology, and cohomology groups are invariant under homotopy, their end-invariant counterparts are invariant only under proper homotopy. The cohomology groups at infinity have proven particularly useful in investigating the uniqueness of end sums. In forthcoming work, Calcut and Guilbault prove that the cohomology groups at infinity of an end sum are independent of which choices are made in the end sum. However, the ring structure of the cohomology at infinity is crucial in work thus far to distinguish between non-isomorphic end sums of two manifolds. In some classifications of noncompact surfaces, the cohomology ring at infinity plays a crucial role, carrying effectively the same information as the space of ends and the geometric invariants of the ends [@goldman]. ## The Classification Theorem {#subsec:classification} We may now state the classification of surfaces without boundary. For a proof, see Richards' thesis [@richards]. **Theorem 40**. Let $M$ and $N$ be two connected surfaces without boundary with the same genus, the same orientability class, and homeomorphic end-space considered as a topological triplet $(A, B, C)$ where $A$ is the space of non-orientable ends, $B$ is the space of infinite genus ends, and $C$ is the space of ends. Then, $M$ and $N$ are isomorphic. We also have a natural extension to surfaces with compact boundary. **Corollary 41**. If $M$ and $N$ are two connected surfaces with compact boundary with the same invariants used in Theorem [Theorem 40](#thm:2manifolds){reference-type="ref" reference="thm:2manifolds"}, and in addition $b(M) = b(N)$, then $M$ and $N$ are isomorphic. # Main Theorem {#sec:mainTheorem} Our main theorem is the following. **Theorem 42**. Let $M$ be a (possibly disconnected) surface with compact boundary, and let $\eta$ and $\eta'$ be distinct ends of $M$. Let $N$ be a result of adding a 1-handle at infinity to $M$ along $\eta$ and $\eta'$. If $\eta$ and $\eta'$ are ends of distinct connected components of $M$, then $N$ is unique up to isomorphism. If $\eta$ and $\eta'$ are ends of the same connected, non-orientable component $M_1$ of $M$, then $N$ is unique up to isomorphism. Lastly, suppose that $\eta$ and $\eta'$ are ends of the same connected, orientable component $M_1$ of $M$. If the 1-handle is oriented, then $N$ is unique up to isomorphism. If the 1-handle is not oriented, then $N$ is unique up to isomorphism. This has the following immediate corollary for the end sum of surfaces. **Corollary 43**. Let $M$ and $M'$ be connected surfaces with compact boundary. Let $\eta$ be an end of $M$ and let $\eta'$ be an end of $M'$. Then, the end sum of $M$ and $M'$ along $\eta$ and $\eta'$ is uniquely determined up to isomorphism. The main tool we will use to prove Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} is the classification of surfaces with compact boundary (Corollary [Corollary 41](#cor:2manifoldsPlus){reference-type="ref" reference="cor:2manifoldsPlus"}). That classification is based on the following characteristics of a connected surface $A$: boundary, orientability, genus, parity, space of ends, subspace of infinite genus ends, and subspace of infinitely non-orientable ends. In each of the next few sections, we examine one or more of these characteristics and their behavior under the addition of a 1-handle at infinity. **Conventions 44**. For the remainder of this paper, unless otherwise stated, we will use the following conventions. Some of these conventions are reused from Section [3](#sec:endSumIntroduction){reference-type="ref" reference="sec:endSumIntroduction"}. - All manifolds are [pl]{.smallcaps}. - Let $M$ be an $n$-manifold, and let $\eta$ and $\eta'$ be ends of $M$ (not necessarily distinct). Let $r$ and $r'$ be disjoint rays in $M$ pointing to $\eta$ and $\eta'$ respectively. Let $\nu$ and $\nu': \mathbb{R}_+ \times D^{n-1} \rightarrow M$ be tubular neighborhood maps of $r$ and $r'$ respectively with disjoint images. - Let $E = \mathbb{R}^n$, and define the subspaces $E_- = (-\infty, 0) \times \mathbb{R}^{n-1}$, $E_0 = \{0\} \times \mathbb{R}^{n-1}$, and $E_+ = (0, \infty) \times \mathbb{R}^{n-1}$. - Let $N$ be the result of adding a 1-handle at infinity to $M$ along the tubular neighborhood maps $\nu$ and $\nu'$ by attaching the strip $E$ as described in Section [3](#sec:endSumIntroduction){reference-type="ref" reference="sec:endSumIntroduction"}. Let $\varphi_-: E_- \rightarrow M$ and $\varphi_+: E_+ \rightarrow M$ be the gluing maps used in the construction of $N$. Let $i_M$ be the natural embedding of $M$ into $N$, and let $i_E$ be the natural embedding of $E$ into $N$. - Let $R_m = [-m,m]^n$. We will use $(R_m)$ as a compact exhaustion of $E$. - $(K_m)$ is a compact exhaustion of $M$ as specified in Theorem [Theorem 46](#thm:steppingStone){reference-type="ref" reference="thm:steppingStone"}. - Let $L_m = i_M(K_m) \cup i_E(R_m)$. We will use $(L_m)$ as a compact exhaustion of $N$. In our study of the ends of $M$ and $N$, it will be helpful to have exhaustions that are situated nicely with respect to both the rays $r$ and $r'$ and the tubular neighborhood maps $\nu$ and $\nu'$. **Lemma 45**. For every compact subset $K \subset M$, there exists a compact $n$-manifold $A \subset M$ such that: (i) $K \subset A^{\circ}$, (ii) $\nu^{-1}(A) = [0, a] \times D^{n-1}$, and (iii) $\nu'^{-1}(A) = [0, a'] \times D^{n-1}$ for some $a, a' > 1$. This situation is presented in Figure [9](#fig:nicePLExhaustion){reference-type="ref" reference="fig:nicePLExhaustion"}. {#fig:nicePLExhaustion} *Proof.* Pick $c$ such that $\nu^{-1}(K) \subset [0, c] \times D^{n-1}$. Similarly for $c'$ and $\nu'$. Set $$L = K \cup \nu([0,c] \times D^{n-1}) \cup \nu'([0, c'] \times D^{n-1})$$ Construct a regular neighborhood $T$ of $L$. For an in-depth discussion of regular neighborhoods, see Rourke and Sanderson [@rourkesanderson Ch. 3] and Scott [@scott]. Importantly for us, $K \subset T^{\circ}$, and $T$ is a compact $n$-dimensional submanifold of $M$ [@rourkesanderson p. 34]. Construct an ambient isomorphism $\psi$ with support disjoint from $L$ and $\textnormal{Im}\,\nu'$ such that $\psi(\partial{T})$ is transverse to $\partial{\textnormal{Im}\,\nu}$ [@az]\[p. 185\]. Replace $T$ by $\psi(T)$. Note that $T$ is still a regular neighborhood of $L$, but now intersects $\textnormal{Im}\,\nu$ transversely. Similarly, modify $T$ to be transverse to $\textnormal{Im}\,\nu'$ as well. Pick $d > 1$ such that $\nu^{-1}(T)$ is contained in $[0, d] \times D^{n-1}$. Similarly, find $d'$. Set $$A = T \cup \nu([0,d] \times D^{n-1}) \cup \nu'([0,d'] \times D^{n-1})$$ It is clear that $A$ is locally Euclidean at all points except $\partial{T} \cap \partial{\textnormal{Im}\,\nu}$ and $\partial{T} \cap \partial{\textnormal{Im}\,\nu'}$. By transversality, for every $p \in \partial{T} \cap \partial{\textnormal{Im}\,\nu}$, there exists a neighborhood $U$ of $p$ and a coordinate map $\varphi: U \rightarrow \mathbb{R}^n$ where $\varphi(p) = 0$, $\varphi(T) = [0, \infty) \times \mathbb{R}\times \mathbb{R}^{n-2}$, and $\varphi(\textnormal{Im}\,\nu) = \mathbb{R}\times [0, \infty) \times \mathbb{R}^{n-2}$. In this local coordinate map, it is clear that $A$ is locally Euclidean at $p$. Thus, $A$ is a $n$-dimensional submanifold of $M$. Since $K \subset T^{\circ}$, $K \subset A^{\circ}$. By construction, $\nu^{-1}(A) = [0,d] \times D^{n-1}$ and $\nu'^{-1}(A) = [0,d'] \times D^{n-1}$. ◻ By constructing larger and larger submanifolds using Lemma [Lemma 45](#lem:compactSteppingStone){reference-type="ref" reference="lem:compactSteppingStone"}, we can create a nice compact exhaustion for $M$. **Theorem 46**. Let $C \subset M$ be compact. There exists a compact exhaustion $(K_m)$ of $M$ such that (i) $C \subset K_m$ for all $m$, (ii) $K_m$ is an $n$-manifold for all $m$, (iii) $\nu^{-1}(K_m) = [0, a_m] \times D^{n-1}$ and $\nu'^{-1}(K_m) = [0, a_m'] \times D^{n-1}$ for some sequence $a_m, a_m' > 1$. *Proof.* Start with any compact exhaustion $(A_m)$ of $M$. Inductively build $(K_m)$ as follows: set $K_0 = C$. For all $m > 0$, let $K_m$ be a compact submanifold of $M$ containing $A_m \cup K_{m-1}$ that has nice intersection with $\nu$ and $\nu'$, as given by Lemma [Lemma 45](#lem:compactSteppingStone){reference-type="ref" reference="lem:compactSteppingStone"}. ◻ Using a compact exhaustion of $M$ provided by Theorem [Theorem 46](#thm:steppingStone){reference-type="ref" reference="thm:steppingStone"}, we may construct a nice compact exhaustion of $N$ as well. Define $L_m = R_m \cup K_m$. Because $K_m$ and $R_m$ have transverse intersection, $(L_m)$ is a compact exhaustion of $N$ by $n$-manifolds. For this section, we use the compact exhaustions: $(K_m)$ of $M$, $(R_m)$ of $E$, and $(L_m)$ of $N$. These exhaustions are depicted in Figure [\[fig:compactexhaustion\]](#fig:compactexhaustion){reference-type="ref" reference="fig:compactexhaustion"}. ## Space of Ends {#subsec:endSumEnds} In this section, we tackle the end-space of $N$. In the process, we will encounter ordinary and extraordinary ends of $N$, and the map $h: \mathcal{E}\left(N\right) \rightarrow \mathcal{E}\left(M\right) / (\eta \equiv \eta')$. Call an end $\alpha$ of $N$ **ordinary** provided that it has a neighborhood contained in $i_M(M)$, or equivalently, that it has a neighborhood disjoint from $i_E(E_0)$. Otherwise, call $\alpha$ **extraordinary**. An ordinary end is essentially not involved in the 1-handle construction, and is the simplest to describe. **Lemma 47**. If $\alpha$ is an ordinary end of $N$, then $\alpha(i_M(K))$ is disjoint from $i_E(E_0)$ for sufficiently large compact $K \subset M$. *Proof.* Because $\alpha$ is ordinary, $\alpha(L_m)$ is disjoint from $i_E(E_0)$ for some $m$. Consider $U_a = i_E((-a, a) \times \mathbb{R}^{n-1})$. For small enough $a$, $L_m \cap U_a = i_E((-a, a) \times [-m,m]^{n-1})$ by construction of $L_m$. If so, then all components of $U_a -L_m$ intersect $i_E(E_0)$. So, $\alpha(L_m)$ is disjoint from $U_a$, and therefore $\alpha(L_m)$ is a connected component of $N -(L_m -U_a)$. Thus, $\alpha(L_m -U_a) = \alpha(L_m)$ is disjoint from $i_E(E_0)$. ◻ **Lemma 48**. If $\alpha$ is an extraordinary end, then $\alpha(i_M(K))$ is the disjoint union $i_M(\eta(K)) \cup i_M(\eta'(K)) \cup i_E(E_0)$ for all compact $K \subset M$. *Proof.* Since $\alpha$ is extraordinary, $\alpha(i_M(K))$ must not be disjoint from $i_E(E_0)$. Thus, $\alpha(i_M(K))$ contains $i_E(E_0)$. Every neighborhood of $i_E(E_0)$ intersects $i_M(\eta(K))$ and $i_M(\eta'(K))$. Furthermore, $i_M(\eta(K)) \cup i_M(\eta'(K)) \cup i_E(E_0)$ is an open set. Its complement in $N -i_M(K)$ is a union of open sets of the form $i_M(U)$, where $U$ is a component of $M -K$. It follows that $i_M(\eta(K)) \cup i_M(\eta'(K)) \cup i_E(E_0)$ is a component of $N -i_M(K)$, hence equals $\alpha(i_M(K))$. ◻ **Lemma 49**. To every ordinary end $\alpha$ of $N$, we can assign a unique end $\beta$ of $M$ such that $i_M(\beta(K)) \subset \alpha(i_M(K))$ for all compact $K \subset M$. Furthermore, $\alpha$ and $\beta$ are isomorphic as ends, and $\beta \neq \eta, \eta'$. *Proof.* By Lemma [Lemma 47](#lem:componentsOfComplementOrdinary){reference-type="ref" reference="lem:componentsOfComplementOrdinary"}, $\alpha(i_M(K_m))$ is disjoint from $i_E(E_0)$ for some $m$. For any compact $A \subset M$, consider $L_A = i_M(A \cup K_m)$. This is a compact subset of $N$, and $\alpha(L_A)$ is disjoint from $i_E(E_0)$. Define $\beta(A)$ as the unique connected component of $M -A$ containing $i_M^{-1}(\alpha(L_A))$. $\beta$ is a well-defined end of $M$. $\beta(i_M(A))$ is the connected component of $i_M(M) -i_M(A)$ containing $\alpha(L_A)$. $\alpha(i_M(A))$ is the connected component of $N -i_M(A)$ containing $\alpha(L_A)$. Thus, $i_M(\beta(A)) \subset \alpha(i_M(A))$. Suppose $\beta'$ is another end satisfying this condition. For large enough compact $K \subset M$, $\alpha(i_M(K))$ is disjoint from $i_E(E_0)$, so $i_M(\beta(K)) = \alpha(i_M(K)) = i_M(\beta'(K))$. Thus, $\beta' = \beta$. For large enough compact $K \subset M$, $i_M(\beta(K)) = \alpha(i_M(K))$. Thus, $\alpha$ and $\beta$ are isomorphic as ends. ◻ When $\alpha$ is an ordinary end of $N$, let $h(\alpha)$ denote the associated end of $M$, as given by Lemma [Lemma 49](#lem:ordinaryEnds){reference-type="ref" reference="lem:ordinaryEnds"}. Note that with this definition, $h$ is a function from the ordinary ends of $N$ to $\mathcal{E}\left(M\right) -\{\eta, \eta'\}$. Extend $h$ to a map from $\mathcal{E}\left(N\right)$ to $\mathcal{E}\left(M\right) / (\eta \equiv \eta')$ by setting $h(\alpha) = \eta \equiv \eta'$ whenever $\alpha$ is an extraordinary end of $N$. By $\mathcal{E}\left(M\right) / (\eta \equiv \eta')$, we mean the topological quotient of $\mathcal{E}\left(M\right)$ by the equivalence relation $\equiv$ generated by $\eta \equiv \eta'$. This quotient map is closed (since it is a continuous function from a compact space to a Hausdorff space), but is not necessarily open (consider two copies of the surface $M$ above in Figure [1](#fig:oddex1){reference-type="ref" reference="fig:oddex1"} and end sum along their nonisolated ends). It will be shown in Lemma [Lemma 50](#lem:partEndSumEnds){reference-type="ref" reference="lem:partEndSumEnds"} that in many cases, this $h$ is also a homeomorphism, completely describing the end space of $N$. **Lemma 50**. The canonical map $h: \mathcal{E}\left(N\right) \rightarrow \mathcal{E}\left(M\right) / (\eta \equiv \eta')$ is continuous. If $n \geq 2$, then $h$ is surjective. If (i) $n = 2$, $\partial{M}$ is compact, and $\eta \neq \eta'$, or (ii) $n \geq 3$, then $h$ is a homeomorphism. *Proof.* First, we prove that $h$ is continuous. Let $\alpha$ be an end of $N$, and let $\beta = h(\alpha)$. Let $U$ be an open neighborhood of $\beta$ in $\mathcal{E}\left(M\right)/(\eta \equiv \eta')$. We will show that $h(V) \subset U$ for some open neighborhood $V$ of $\alpha$. If $\alpha$ is ordinary, then we may take $U = \mathcal{N}(\beta;K)$ for some compact $K \subset M$, and by Lemma [Lemma 47](#lem:componentsOfComplementOrdinary){reference-type="ref" reference="lem:componentsOfComplementOrdinary"}, we can assume that $\alpha(i_M(K))$ is disjoint from $i_E(E_0)$. For every end $\tau \in \mathcal{N}(\alpha;i_M(K))$, $\tau(K)$ is disjoint from $i_E(E_0)$, so $\tau$ is ordinary. In fact, $h(\tau)(K) = \beta(K)$, so $h(\tau) \in U$. If $\alpha$ is extraordinary, then we may take $U = \mathcal{N}(\eta;K) \cup \mathcal{N}(\eta';K)$ for some compact $K \subset M$. For every extraordinary end $\tau \in \mathcal{N}(\alpha;i_M(K))$, $h(\tau) = \beta \in U$. For every ordinary end $\tau \in \mathcal{N}(\alpha;i_M(K))$, $$i_M(h(\tau)(K)) \subset \tau(i_M(K)) = \alpha(i_M(K)) = i_M(\eta(K)) \cup i_M(\eta'(K)) \cup i_E(E_0)$$ The last equality is true by Lemma [Lemma 48](#lem:componentsOfComplementExtraordinary){reference-type="ref" reference="lem:componentsOfComplementExtraordinary"}. Thus, $h(\tau) = \eta(K)$ or $\eta'(K)$. So, $h(\tau) \in U$. Thus, $h$ is continuous. In the other direction, let $\beta \neq \eta, \eta'$ be an end of $M$. Pick compact $K \subset M$ such that $\beta(K) \neq \eta(K), \eta'(K)$. Note that $i_M(\beta(K))$ is also a connected component of $N -i_M(K)$. Given compact $L \subset N$, set $L' = K \cup (\beta(K) \cap i_M^{-1}(L))$. Note that $L'$ is a compact subset of $M$. Define $\alpha(L)$ as the unique component of $N -L$ containing $i_M(\beta(L'))$. Then, $\alpha$ is the unique end of $M$ such that $h(\alpha) = \beta$. $h$ is surjective provided there is at least one extraordinary end of $N$. Furthermore, as a continuous map between compact spaces, $h$ is a homeomorphism as long as it is bijective, that is, as long as there is one extraordinary end of $N$. An end $\alpha$ of $N$ is extraordinary provided that $\alpha(L)$ intersects $i_E(E_0)$ for all compact $L \subset N$. If $n \geq 2$, then $i_E(E_0)$ is noncompact, so it has at least one end $\rho$. Let $\alpha(L)$ be the unique component of $N -L$ containing $\rho(i_E(E_0) \cap L)$. $\alpha$ is then an extraordinary end of $N$. If $n \geq 3$, then $i_E(E_0) \cong \mathbb{R}^{n-1}$ has a single end $\rho$, and for any extraordinary end $\zeta$, $\zeta(L) \supset \rho(i_E(E_0) \cap L)$. Thus, $N$ has exactly one extraordinary end. If $n = 2$, then $i_E(E_0) \cong \mathbb{R}$ has two ends, $\rho_-$ and $\rho_+$. For any extraordinary end $\zeta$ of $N$, $\zeta(L)$ contains at least one of $\rho_-(i_E(E_0) \cap L)$ or $\rho_+(i_E(E_0) \cap L)$. Thus, $N$ has at most two extraordinary ends, and $N$ has a single extraordinary end precisely when $\rho_-(i_E(E_0) \cap L)$ and $\rho_+(i_E(E_0) \cap L)$ are always contained in the same connected component of $N -L$. Suppose that $n = 2$, $\eta \neq \eta'$, and $\partial{M}$ is compact. It suffices to prove that $\rho_+(i_E(E_0) \cap L_m)$ and $\rho_-(i_E(E_0) \cap L_m)$ are connected in $N -L_m$ for large enough $m$. Starting at $\rho_+(i_E(E_0) \cap L_m)$, trace along the boundary of $i_E(R_m)$ until it intersects $i_M(K_m)$. Let $T$ be the boundary component of $K_m$ that we intersect. Since $\partial{M}$ is compact, $\partial{M} \subset K_m$ for large enough $m$. Thus, staying within $N -L_m$, we can traverse $i_M(T)$ until we again intersect with $i_E(R_m)$. Traverse the boundary of $i_E(R_m)$ until we return to $\rho_+(i_E(E_0) \cap L_m)$ or arrive at $\rho_-(i_E(E_0) \cap L_m)$. If we return to $\rho_+(i_E(E_0) \cap L_m)$, then $T$ must be a boundary component of both $\eta(L_m)$ and $\eta'(L_m)$. Since $\eta \neq \eta'$, $\eta(K_m) \neq \eta'(K_m)$ for large enough $m$, in which case this situation is impossible. We have traced a path from $\rho_+(i_E(E_0) \cap L_m)$ to $\rho_-(i_E(E_0) \cap L_m)$ within $N -L_m$. Thus, they are connected in $N -L_m$, and $N$ has exactly one extraordinary end. ◻ In the future, whenever there is a unique extraordinary end of $N$, we will call it $\zeta$. While the ordinary ends of $N$ are isomorphic to their corresponding ends in $M$ and easy to classify, understanding $\zeta$ takes more effort. ## Boundary {#subsec:endSumBoundary} In this section, we briefly consider the boundary of $N$. **Theorem 51**. Then there is a canonical isomorphism $\partial{N} \cong \partial{M}$. *Proof.* As $i_M$ and $i_E$ are open embeddings, $\partial{N} = i_M(\partial{M}) \cup i_E(\partial{E})$. As $E$ has no boundary components, $\partial{N} = i_M(\partial{M})$. ◻ ## Orientability {#subsec:endSumOrientation} In this section, we tackle the orientability of $N$. The orientability of $N$ depends on the orientability of $M$, which ends are used for the 1-handle, and the orientation of the 1-handle. To prove that $N$ is orientable, it suffices to exhibit an orientation on $N$. To prove that $N$ is non-orientable, we will need the notion of an orientation-reversing loop. **Definition 52**. A loop $\gamma$ in a manifold $M$ is **orientation-preserving** if $\gamma$ lifts to a loop in the oriented double-cover of $M$. Define two paths $\Gamma_1$ and $\Gamma_2$ in $M$ with the same endpoints to have the **same effect on orientation** if the concatenation of $\Gamma_1$ by the reverse of $\Gamma_2$ is an orientation-preserving loop. For a description of the oriented double-cover of a manifold, see [@hatcherbook pp. 233--235]. As a homotopy of a loop in the base space lifts to a homotopy in the covering space, two homotopic loops are either both orientation-preserving or both orientation-reversing. Importantly for us, every loop in $M$ is orientation-preserving if and only if $M$ is orientable. **Theorem 53**. (i) If $M$ is connected: then $N$ is orientable if and only if $M$ is orientable and the 1-handle is oriented. \(ii\) If $M = M_1 \cup M_2$ where each $M_i$ is a connected component of $M$, $\eta$ is an end of $M_1$ and $\eta'$ is an end of $M_2$: then $N$ is orientable if and only if both $M_1$ and $M_2$ are orientable. *Proof.* First consider the case where $M$ is connected. Suppose that $M$ is non-orientable. As $i_M(M)$ is an open subset of $N$, $N$ is also non-orientable. Now suppose that $M$ is orientable. Fix an orientation of $M$. If the 1-handle is oriented, then by Lemma [Lemma 28](#lem:endSumIsManifold){reference-type="ref" reference="lem:endSumIsManifold"}, $N$ is orientable. If the 1-handle is not oriented, then we may assume without loss of generality that $\varphi_+$ is orientation-preserving, and $\varphi_-$ is orientation-reversing. Fix $a \in i_E(E_-)$ and $b \in i_E(E_+)$. Choose a path within $i_E(E)$ from $a$ to $b$ and a path within $i_M(M)$ from $b$ to $a$. Concatenating these paths gives an orientation-reversing loop. Thus, $N$ is non-orientable. Now consider the case where $M = M_1 \cup M_2$ where each $M_i$ is a connected component of $M$, $\eta$ is an end of $M_1$, and $\eta'$ is an end of $M_2$. Suppose that $M_1$ is non-orientable. As $i_M(M_1)$ is an open subset of $N$, $N$ is also non-orientable. Similarly, if $M_2$ is non-orientable, so is $N$. Now suppose that both $M_1$ and $M_2$ are orientable. By reversing the orientations on $M_1$ and $M_2$ if needed, we may assume that the 1-handle is oriented. By Lemma [Lemma 28](#lem:endSumIsManifold){reference-type="ref" reference="lem:endSumIsManifold"}, $N$ is orientable. ◻ {#fig:endSumOrientation} ## Orientability of the Ends {#subsec:endSumEndOrientation} Although we have been able to give general results on 1-handles and the end-sum so far in Section [5](#sec:mainTheorem){reference-type="ref" reference="sec:mainTheorem"}, we now narrow our focus to the conditions of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}. From this point on, we impose that $M$ is a surface with compact boundary, and $\eta \neq \eta'$. In this section, we tackle the orientability of the ends of $N$. Remember that we define an end of $M$ to be orientable provided that it has an orientable neighborhood, and the set of orientable ends is open in the space of all ends. **Theorem 54**. For every ordinary end $\alpha$ of $N$, $h(\alpha)$ has the same orientability as $\alpha$. $\zeta$ is orientable if and only if both $\eta$ and $\eta'$ are orientable. *Proof.* From Theorem [Lemma 49](#lem:ordinaryEnds){reference-type="ref" reference="lem:ordinaryEnds"}, we know that for every ordinary end $\alpha$ of $N$, $h(\alpha)$ is isomorphic to $\alpha$ as ends. Hence, $h(\alpha)$ and $\alpha$ have the same orientability. Suppose first that $\eta$ is non-orientable. Let $m$ be large enough so that $\eta(K_m) \neq \eta'(K_m)$. Let $\Gamma_m$ be an orientation-reversing loop in $\eta(K_m)$. From the construction of $R_m$ and $K_m$, we know that $K_m$ splits $R_m$ in two. We may homotope any segment of $\Gamma_m$ that intersects (but does not cross) $R_m$ to one that is disjoint from $R_m$. We may replace any segment of $\Gamma_m$ that crosses $R_m$ by a path that follows $\partial{R_m}$ until it meets $\partial{K_m}$, then follows $\partial{K_m}$ until it meets $\partial{R_m}$ on the other side, then follows back up $\partial{R_m}$. Note that this path has the same effect on orientation as the original segment of $\Gamma_m$. The modified loop created in this way will still be orientation-reversing, but will also be disjoint from $\varphi_+(R_m)$. Thus, $\eta(K_m) -\varphi_+(R_m)$ is non-orientable. Thus, $\zeta(L_m)$ is non-orientable. Since this is true for sufficiently large $m$, $\zeta$ is non-orientable. Now suppose that both $\eta$ and $\eta'$ are orientable. We can pick $m$ large enough that $\eta(K_m)$, $\eta'(K_m)$ are distinct and orientable. Choose orientations on $\eta(K_m)$ and $\eta'(K_m)$ that are compatible with the 1-handle. These orientations define an orientation on $\zeta(i_M(K_m))$. Thus, $\zeta$ is orientable. ◻ ### Where This Fails For Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}, we only consider the case where $\eta \neq \eta'$. If $\eta = \eta'$, things are much less straightforward. For example, even if $\eta = \eta'$ is non-orientable, then it is possible for the extraordinary end(s) to be orientable. It is even possible, when there are two extraordinary ends, that both ends have different orientations. **Example 55**. Let $M$ be the unique connected 1-ended surface without boundary whose end is non-orientable, shown in Figure [11](#fig:endSumEndsInfNonorientable){reference-type="ref" reference="fig:endSumEndsInfNonorientable"}. Let $N$ be a result of adding a 1-handle at infinity along $r$ and $r'$. Even though the end of $M$ is non-orientable, $N$ has one (extraordinary) end, and it is orientable. {#fig:endSumEndsInfNonorientable} ## Genus-Related Properties {#subsec:genus} Lastly, we study the genus-related properties of $N$. Specifically, we determine the genus of $N$, the parity of $N$ (if all ends are orientable), and which ends of $N$ have infinite genus. ### Genus and Parity {#subsubsec:endSumGenus} Recall that we use $g(M)$ to denote the genus of $M$ and $\mathcal{P}(M)$ to denote the parity of $M$ (as an element of $\mathbb{Z}/2\mathbb{Z}$). **Theorem 56**. If $\eta$ and $\eta'$ lie in distinct components of $M$, then $g(N) = g(M)$. If $\eta$ and $\eta'$ lie in the same component of $M$, then $g(N) = g(M) + 1$. If $M$ is orientable outside of a compact set, then $N$ is orientable outside of a compact set and $\mathcal{P}(N) = \mathcal{P}(M)$. *Proof.* We may assume that $N$ is connected. If $M$ is orientable outside of a compact set, then $M$ has no non-orientable ends and Theorem [Theorem 53](#thm:endSumOrientability){reference-type="ref" reference="thm:endSumOrientability"} implies that $N$ also has no non-orientable ends. Thus, the parity $\mathcal{P}(N)$ is well-defined and equals the limit of $\mathcal{P}(L_m)$ as $m\to\infty$. On the other hand, the genus $g(M)$ is always well-defined (possibly infinite) and equals the limit of $g(L_m)$ as $m\to\infty$. We will compute the genus of $L_m$ and use that to compute the genus and parity of $N$. Note that $R_m$ meets the boundary of $K_m$ in two intervals. Since $\eta \neq \eta'$ and $\partial{M}$ is compact, these intervals lie in distinct components of $\partial{K_m}$ for all sufficiently large $m$. If so, then $b(L_m) = b(K_m) - 1$. For the Euler characteristic, we have $\chi(L_m) = \chi(K_m) + \chi(R_m) - \chi(K_m \cap R_m) = \chi(K_m) - 1$. If $M$ is connected, then for all sufficiently large $m$, both components of $K_m \cap R_m$ lie in the same connected component of $K_m$ and so $\left| \pi_0\left(L_m\right) \right| = \left| \pi_0\left(K_m\right) \right|$. Thus $$\begin{aligned} g(L_m) &= \left| \pi_0\left(L_m\right) \right| - \frac{b(L_m)+\chi(L_m)}{2}\\ &= \left| \pi_0\left(K_m\right) \right| - \frac{b(K_m)+\chi(K_m)}{2} +1\\ &= g(K_m) +1\end{aligned}$$ In this case, $g(N) = g(M)+1$ and $\mathcal{P}(N) = \mathcal{P}(M)$ (if applicable). If $M$ is disconnected, then one component of $K_m \cap R_m$ lies in each connected component of $M$ and so $\left| \pi_0\left(L_m\right) \right| = \left| \pi_0\left(K_m\right) \right| - 1$. Thus $$\begin{aligned} g(L_m) &= \left| \pi_0\left(L_m\right) \right| - \frac{b(L_m)+\chi(L_m)}{2}\\ &= \left| \pi_0\left(K_m\right) \right| -1 - \frac{b(K_m)+\chi(K_m)}{2} + 1\\ &= g(K_m)\end{aligned}$$ In this case, $g(N) = g(M)$ and $\mathcal{P}(N) = \mathcal{P}(M)$ (if applicable). ◻ ### Genus of the Ends {#subsubsec:endSumEndsGenus} Recall that the genus of an end $\tau \in \mathcal{E}\left(M\right)$ is the limit of $g(K_n)$ as $n \rightarrow \infty$ for any compact exhaustion $K_n$ of $M$. The genus of an end is either zero or infinity. **Theorem 57**. If $\alpha$ is an ordinary end of $N$, then $\alpha$ has the same genus as $h(\alpha)$. Furthermore, $\zeta$ has infinite genus if and only if either $\eta$ or $\eta'$ has infinite genus. *Proof.* To simplify the argument, assume without loss of generality that $\eta(K_m) \neq \eta'(K_m)$ for all $m$. If $\alpha$ is an ordinary end of $N$, then $\alpha$ and $h(\alpha)$ are isomorphic as ends. Thus, $\alpha$ has the same genus as $h(\alpha)$. Next, consider the extraordinary end $\zeta$. To compute $g(\zeta)$, we introduce some subsurfaces of $N$ as depicted in Figure [12](#fig:endsumgenusends){reference-type="ref" reference="fig:endsumgenusends"}. {#fig:endsumgenusends} Here, $A_m = \overline{\zeta\left(K_m\right)}$, $B_m = \overline{\zeta\left(L_m\right)}$, and $A_m$ is the union of $B_m$ and a $2$-disk $C_m \subset R_m$. As $A_m$ and $B_m$ are connected, we have $\left| \pi_0\left(A_m\right) \right| = \left| \pi_0\left(B_m\right) \right| = 1$. Gluing $C_m$ onto $B_m$ separates one boundary component of $B_m$ into two, so $b(A_m) = b(B_m) + 1$. $$\chi\left(A_m\right) = \chi\left(B_m\right) + \chi\left(C_m\right) - \chi\left(B_m \cap C_m\right) = \chi\left(B_m\right) - 1$$ It follows that $g(A_m) = g(B_m)$. Note that $A_m$ is an end sum of $\overline{\eta\left(K_m\right)}$ and $\overline{\eta'(K_m)}$. By Theorem [Theorem 56](#thm:endSumGenus){reference-type="ref" reference="thm:endSumGenus"}, $g\left(A_m\right) = g\left(\overline{\eta\left(K_m\right)}\right) + g\left(\overline{\eta'\left(K_m\right)}\right)$. In the limit as $m \rightarrow \infty$, $g\left(\eta\left(K_m\right)\right)$ converges to $g\left(\eta\right)$, $g\left(\eta'\left(K_m\right)\right)$ converges to $g\left(\eta'\right)$, and $g\left(B_m\right)$ converges to $g\left(\zeta\right)$. Thus, $g\left(\zeta\right) = g\left(\eta\right)+ g\left(\eta'\right)$. In other words, $\zeta$ has infinite genus if and only if one of $\eta$ or $\eta'$ does. ◻ ## Summary {#subsec:summary} We now review what we have proven so far. Let $M$ be a [pl]{.smallcaps} surface with compact boundary. Let $N$ be a result of adding a 1-handle at infinity to $M$ along distinct ends $\eta$ and $\eta'$. We can split $M$ up into its connected components $M_i$. If a component $M_i$ does not contain $\eta$ or $\eta'$, then $M_i$ is unchanged by the 1-handle. Because of this, it suffices to consider just the component of $N$ which contains the 1-handle. From now on, consider $N$ to be connected. There is a canonical homeomorphism $h: \mathcal{E}\left(N\right) \cong \mathcal{E}\left(M\right) / (\eta \equiv \eta')$ by Lemma [Lemma 50](#lem:partEndSumEnds){reference-type="ref" reference="lem:partEndSumEnds"}. For all ordinary ends $\alpha$ of $N$, $\alpha$ and $h(\alpha)$ are isomorphic as ends, so they have the same invariants. The single extraordinary end $\zeta$ is orientable if and only if both $\eta$ and $\eta'$ are orientable; $\zeta$ has zero genus if and only if both $\eta$ and $\eta'$ have zero genus. If $\eta$ and $\eta'$ are from two distinct components of $M$, then $N$ is orientable if and only if that $M$ is orientable. Otherwise, $N$ is orientable if and only if $M$ is orientable and the 1-handle is oriented. If $\eta$ and $\eta'$ are from two distinct components of $M$, then $g(N) = g(M)$. Otherwise, $g(N) = g(M) + 1$. In either case, $\mathcal{P}(N) = \mathcal{P}(M)$. We conclude: if $M$ is non-orientable, then $N$ is unique up to [pl]{.smallcaps} isomorphism; if $M$ is orientable and the 1-handle is oriented, then $N$ is unique up to [pl]{.smallcaps} isomorphism; and if $M$ is non-orientable and the 1-handle is not oriented, then $N$ is unique up to [pl]{.smallcaps} isomorphism. This completes the proof of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} in [pl]{.smallcaps}. [top]{.smallcaps} and [diff]{.smallcaps} versions of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} are proved in Section [7](#sec:secondTheorem){reference-type="ref" reference="sec:secondTheorem"}. # Main Theorem by the Uniqueness of Rays {#sec:rayUniquenessMain} In this section, we prove a ray uniqueness result for surfaces. Namely, that all rays within a surface with compact boundary that point to a given end are related by a global isomorphism of the surface. Using this, we give an alternate and relatively straightforward proof of our main theorem. The authors thank Ric Ancel for suggesting this strategy. ## Classification of Surfaces with Noncompact Boundary {#csncb} Our first objective is to study and classify all rays $r$ in a given surface with compact boundary up to global isomorphism. We will do this indirectly by removing the interior of a closed regular neighborhood of $r$ to produce a surface with a single noncompact boundary component, then classifying the resulting surface. For this, we will use the classification of surfaces detailed by Brown and Messer in [@brownmesser]. This program classifies all surfaces with boundary, whether or not its boundary is compact. We recount the invariants of this classification using our notation. Let $M$ be a connected surface. As in the case of surfaces with compact boundary, the following "global invariants" are essential: orientability, genus, parity, and compact boundary components. These invariants are defined exactly as in the compact boundary case, although now the number of compact boundary components may be infinite. The rest of the invariants required to classify $M$ deal with the ends of $M$ and how those ends interact with its boundary components. As in the case of compact boundary, and end may be zero genus or infinite genus, and it may be orientable or non-orientable. In surfaces with infinitely many compact boundary components, it may happen that these boundary components accumulate in some ends but not others. Define an end of $M$ to be **without compact boundary** provided that a neighborhood of that end contains no compact boundary components of $M$. It can be shown that $M$ has finitely many compact boundary components provided that every end of $M$ is without compact boundary. The surfaces we are interested in have finitely many boundary components, and thus have no ends with compact boundary. Let $\mathrm{d}M$ denote the union of all noncompact boundary components of $M$. The inclusion map $\mathrm{d}M \rightarrow M$ induces a map $v: \mathcal{E}\left(\mathrm{d}M\right) \rightarrow \mathcal{E}\left(M\right)$. Each connected component $B$ of $\mathrm{d}M$ is a copy of $\mathbb{R}$ and has two ends, say $\alpha$ and $\beta$. The map $e: \mathcal{E}\left(\mathrm{d}M\right) \rightarrow \pi_0(\mathrm{d}M)$ is defined by sending $\alpha$ and $\beta$ to $B$. We adopt the usual outward normal first convention for orienting the boundary of an oriented manifold. For example, Figure [13](#fig:positiveend){reference-type="ref" reference="fig:positiveend"} shows $M = \mathbb{R}^2_+$ equipped with the standard orientation corresponding to the ordered basis $(e_1,e_2)$. {#fig:positiveend} The **positive end** of the noncompact boundary component of $M$ is the end labeled $+$. The **negative end** of the noncompact boundary component of $M$ is the end labeled $-$. The same conventions and terminology apply to ends of noncompact boundary components of general surfaces in which some disjoint neighborhoods of ends are oriented. Following Brown and Messer [@brownmesser pp. 379--381], we define an **orientation of $\mathcal{E}\left(\mathrm{d}M\right)$** to be a subset $\mathcal{P} \subset \mathcal{E}\left(\mathrm{d}M\right)$ as follows. Given a compact subsurface $C\subset M$, a **complementary domain of $C$** is the closure of one of the components of $M-C$. - If $M$ is orientable, then fix an orientation on $M$. Then, $\mathcal{P}$ consists of the resulting positive ends of $\mathrm{d}M$. - If $M$ is non-orientable, then there exists a (nonunique) sequence $Q_1,Q_2,\ldots$ of disjoint orientable complementary domains of compact subsurfaces of $M$ such that every orientable end of $M$ has some (unique) $Q_i$ as a neighborhood. Choose an orientation on each $Q_i$. Then, $\mathcal{P}$ consists of the resulting positive ends of $\mathrm{d}M$. It is important to note that there may be many valid orientations of $\mathcal{E}\left(\mathrm{d}M\right)$ for a given surface $M$. All of this end-related data can be combined to form a **diagram** as shown in [\[eq:diagramofsurface\]](#eq:diagramofsurface){reference-type="eqref" reference="eq:diagramofsurface"}. Here, $\mathcal{K}$ denotes the set of non-orientable ends, $\mathcal{H}$ denotes the set of non-planar ends, and $\mathcal{S}$ denotes the set of ends with compact boundary. $$\label{eq:diagramofsurface} \begin{tikzcd} \pi_0(\mathrm{d}M) & \mathcal{E}\left(\mathrm{d}M\right) \arrow{l}[swap]{e} \arrow{r}{v} & \mathcal{E}\left(M\right) & \mathcal{H} \arrow{l} & \mathcal{K} \arrow{l}\\ & \mathcal{P} \arrow{u} & \mathcal{S} \arrow{u} \end{tikzcd}$$ Two diagrams $\Delta$ of $M$ and $\Delta'$ of $M'$ are **isomorphic** if there are bijective maps from each object in $\Delta$ to the corresponding object in $\Delta'$ which commute with the arrows of each diagram. The map from $\mathcal{E}\left(M\right)$ to $\mathcal{E}\left(M'\right)$ must also be a topological homeomorphism. It is important to note that, as $\mathcal{P}$ is not uniquely defined, it is possible for one surface to have multiple non-isomorphic diagrams. Written in full, the classification theorem [@brownmesser p. 388] is as follows. **Theorem 58** (Classification of Surfaces). Let $M$ and $M'$ be two connected surfaces. If $M$ and $M'$ have the same genus, parity, orientability, and number of compact boundary components, and there exist diagrams $\Delta$ of $M$ and $\Delta'$ of $M'$ such that $\Delta \cong \Delta'$, then $M$ and $M'$ are isomorphic. We will need the following strengthening of Theorem [Theorem 58](#thm:classificationOfAllSurfaces){reference-type="ref" reference="thm:classificationOfAllSurfaces"}. **Theorem 59** (Addendum to the Classification of Surfaces). Suppose furthermore that we are given an isomorphism $h: \Delta \to \Delta'$. Then, there is an isomorphism $\psi: M \to M'$ which induces the given isomorphism $h$ of diagrams. If $M$ and $M'$ are oriented surfaces, and their orientations agree with the orientations of $\Delta$ and $\Delta'$ respectively, then $\psi$ may be chosen to be orientation-preserving. The Addendum follows by Brown and Messer's proof of Theorem [Theorem 58](#thm:classificationOfAllSurfaces){reference-type="ref" reference="thm:classificationOfAllSurfaces"}. Their proof [@brownmesser p. 388--389] is iterative and begins with the empty function $f_0$. Given a homeomorphism $f_k:C_k\to C_k'$ (with certain properties) of compact subsurfaces of $M$ and $M'$ respectively, they extend $f_k$ to a homeomorphism $f_{k+1}:C_{k+1}\to C_{k+1}'$ of larger compact subsurfaces. If $M$ and $M'$ are oriented, then the first nonempty function $f_1$ may be chosen to respect orientation. In general, their proof imposes compatibility conditions between the constructed homeomorphisms and isomorphisms of diagrams (see [@brownmesser pp. 383--388], especially Lemma 2.1, the paragraph on p. 384 before the proof of Lemma 2.1, and the proof of Lemma 2.1). The present authors found it instructive to run their proof on various examples to gain familiarity with their orientation and compatibility conventions. In the remainder of this subsection, we present a useful lemma and some applications of the Addendum to demonstrate its utility. In the next subsection, we prove a ray unknotting theorem (Theorem [Theorem 67](#thm:rayUniqueness){reference-type="ref" reference="thm:rayUniqueness"}) for rays in certain surfaces. Recall from Section [2](#sec:endsIntroduction){reference-type="ref" reference="sec:endsIntroduction"} that a precise definition was given for a proper ray $r$ to **point to** an end $\eta$ of $M$. We now expand this definition. Say an end $\tau$ of a (noncompact) boundary component $B$ of $M$ **points to** and end $\eta$ of $M$ if $\mathcal{E}\left(i\right)(\tau) = \eta$, where where $i: B \rightarrow M$ is the inclusion map. Equivalently, $\tau$ points to $\eta$ if every ray in $B$ that points to $\tau$ also points to $\eta$. **Lemma 60**. Let $M$ be a connected surface and let $\eta$ be an end of $M$. If $M$ has finitely many noncompact boundary components, then the number of ends of noncompact boundary components of $M$ that point to $\eta$ is even. In other words, if $\left| \pi_0(\mathrm{d}M) \right|$ is finite, then $\left| v^{-1}(\eta) \right|$ is even. In particular, if $M$ has exactly one noncompact boundary component $B$, then both ends of $B$ point to the same end of $M$. *Proof.* Let $\eta, \tau_1, \dots, \tau_k$ be all the ends of $M$ pointed to by ends of $\mathrm{d}M$. Choose a large compact subsurface $K \subset M$ such that $\overline{M -K}$ is also a subsurface of $M$, and $\eta(K) \neq \tau_i(K)$ for any $i$. Consider $\overline{\eta(K)}$. This is a subsurface of $M$, and the ends of noncompact boundary components of $\overline{\eta(K)}$ pointing to $\eta$ are in one-to-one correspondence with ends of noncompact boundary components of $M$ pointing to $\eta$. Since every end of a noncompact boundary component of $\overline{\eta(K)}$ points to $\eta$ and there are an even number of ends of noncompact boundary components of $\overline{\eta(K)}$, then an even number of ends of noncompact boundary components of $M$ point to $\eta$. This proves the first conclusion. The second conclusion follows immediately from the first. ◻ **Remark 61**. The first conclusion of Lemma [Lemma 60](#evenendslemma){reference-type="ref" reference="evenendslemma"} also holds provided $\eta$ is an isolated end of $M$ and at most finitely many noncompact boundary components of $M$ point to $\eta$. The proof is similar. The conclusions of Lemma [Lemma 60](#evenendslemma){reference-type="ref" reference="evenendslemma"} are false without some restrictions on $M$ or $\eta$. Consider the surface $M$ depicted in Figure [14](#fig:oddex){reference-type="ref" reference="fig:oddex"} that is obtained from the closed disk by removing a sequence of boundary points and the single limit point of that sequence. {#fig:oddex} The sequence converges to the limit point from one side. The single nonisolated end of $M$ is denoted $\eta$. That end $\eta$ of $M$ is pointed to by exactly one end of a noncompact boundary component of $M$. **Corollary 62**. Let $M$ be a connected surface with exactly one noncompact boundary component $B$. Then, there exists an automorphism $\psi:M\to M$ that induces the identity on $\mathcal{E}\left(M\right)$ and interchanges the two ends of $B$. *Proof.* By the previous Lemma [Lemma 60](#evenendslemma){reference-type="ref" reference="evenendslemma"}, both ends of $B$ point to the same end $\eta$ of $M$. First, consider the case where $M$ is orientable. Fix an orientation on $M$ and consider the corresponding diagram $\Delta$ of $M$. Let $\Delta'$ be the diagram for $M$ with the opposite orientation. We have an isomorphism of diagrams $h:\Delta\to\Delta'$ that swaps the ends of $B$ and is otherwise the identity. The desired conclusion now follows by the Addendum (Theorem [Theorem 59](#thm:classificationOfAllSurfacesAddendum){reference-type="ref" reference="thm:classificationOfAllSurfacesAddendum"}). Second, consider the case where $M$ is not orientable, but $\eta$ is orientable. Then the proof from the first case also applies. Third, consider the case where $\eta$ is not orientable. Note that $\Delta$ has empty orientation. We have an isomorphism $h: \Delta \rightarrow \Delta$ that swaps the ends of $B$, and the conclusion follows by the Addendum. ◻ **Remark 63**. If $M$ has more than one noncompact boundary component, then it may not be possible to interchange the ends of a given noncompact boundary component $B$ of $M$ by an automorphism of $M$. Consider the strip $[0,1]\times\mathbb{R}$ and connect sum a sequence of projective planes or tori off to one end. A curious question arises: which oriented surfaces admit an orientation reversing automorphism? The classification of compact, connected, oriented surfaces (see Theorem [Theorem 29](#thm:finite2Manifolds){reference-type="ref" reference="thm:finite2Manifolds"} in Section [4](#sec:classification){reference-type="ref" reference="sec:classification"}) implies that each such surface admits an orientation reversing automorphism. (The question has also been studied for closed manifolds in higher dimensions by Müllner [@mullner].) For noncompact surfaces, the situation is more complicated. We give a positive result as well as two surfaces not admitting an orientation reversing automorphism. **Corollary 64**. Let $M$ be a connected, oriented, noncompact surface with zero or one noncompact boundary component. Then $M$ admits an orientation reversing automorphism. *Proof.* First, consider the case where $M$ has zero noncompact boundary components. Let $M'$ be $M$ with the opposite orientation. By the Addendum (Theorem [Theorem 59](#thm:classificationOfAllSurfacesAddendum){reference-type="ref" reference="thm:classificationOfAllSurfacesAddendum"}), there is an orientation-preserving isomorphism from $M$ to $M'$. This is an orientation-reversing automorphism of $M$. Second, consider the case where $M$ has one noncompact boundary component $B$. By the previous corollary, there is an automorphism $\psi: M \rightarrow M$ that interchanges the ends of $B$. Evidently, $\psi$ is locally orientation reversing near $B$, and so $\psi$ is globally orientation reversing. ◻ **Remark 65**. Figure [15](#fig:noorauto){reference-type="ref" reference="fig:noorauto"} depicts two oriented surfaces that do not admit an orientation reversing automorphism. {#fig:noorauto} The surface $M$ is obtained from the closed disk (with its standard orientation) by removing six boundary points and connect summing a sequence of tori at each end marked $\infty$. So, $M$ has six ends: three of genus zero and three of infinite genus. The cyclic order of the genera of the ends of $M$ prevent the existence of an orientation reversing automorphism of $M$. The surface $N$ is obtained from the disjoint union of the closed disk and closed upper half space as follows. Remove three boundary points from the closed disk, connect sum a sequence of tori at the end marked $\infty$, then glue in a sequence of tubes between the end marked $\eta$ and the end of the copy of closed upper half space. The resulting surface has three ends: one of genus zero, one of infinite genus, and $\eta$. The first two ends each have two ends of noncompact boundary components pointing to them, while $\eta$ is pointed to by four ends of noncompact boundary components. These properties prevent the existence of an orientation reversing automorphism of $N$. It seems interesting to ask whether there is a reasonable classification of connected, oriented surfaces that admit an orientation reversing automorphism. ## Uniqueness of Rays in a Surface Although we will use a more technical result to prove the main theorem, it is worth reporting Theorem [Theorem 67](#thm:rayUniqueness){reference-type="ref" reference="thm:rayUniqueness"}, which gives a clearn result on the uniqueness of rays in a surface. Many of the ideas in this proof will be used in Section [6.3](#subsec:maintheorem2){reference-type="ref" reference="subsec:maintheorem2"}. **Lemma 66**. Let $M$ be a surface with compact boundary, and $A$ be a closed subset of $M$ isomorphic to $\mathbb{R}^2_+$ that is disjoint from $\partial{M}$. Let $N = \overline{M -A}$. Then, the inclusion map $i: N \rightarrow M$ induces a homeomorphism on the space of ends. *Proof.* Identify $A$ with $\mathbb{R}^2_+$. Let $F_j$ be the closed half-disk of radius $j$ centered at $(0,0)$. Note that $(F_j)$ is a compact exhaustion of $A$. Let $(G_j)$ be a compact exhaustion of $N$ by subsurfaces such that $F_j \cap \partial{A} = G_j \cap \partial{A}$, and let $H_j = F_j \cup G_j$. To prove that the end map is bijective, it suffices to prove that for $j$ large enough, each unbounded connected component of $M -H_j$ contains exactly one unbounded connected component of $N -G_j$. Since end spaces are compact and Hausdorff, any bijective map between end spaces is automatically a homeomorphism. Let $U$ be any unbounded connected component of $M -H_j$. If $U$ is disjoint from $A$, then $U$ is already an unbounded connected component of $N -G_j$. Assume then that $U$ intersects $A$. Then, $U$ contains $A -F_j$. Since $A -F_j$ borders $N -G_j$, $U$ contains at least one connected component of $N -G_j$. To prove that there is exactly one, it suffices to prove that the two components of $\partial{A} -F_j$ are connected via a path in $N -G_j$. Consider the boundary of the closed subset $\overline{N -G_j}$. This boundary has only two ends, corresponding to the two ends of $\partial{A} -F_j$. Thus, those ends lie in the same connected component (which is a copy of $\mathbb{R}$). Thus, $\partial{A} -F_j$ is connected via paths in $\overline{N -G_j}$. Thus, it is connected via paths in $N -G_j$. ◻ **Theorem 67** (Ray Uniqueness for Surfaces). Let $M$ be a surface with compact boundary, and let $\eta$ be an end of $M$. If $r$ and $r'$ are rays in $M$ pointing to $\eta$, then there is an automorphism $\psi: M \rightarrow M$ which sends $r$ to $r'$. *Proof.* It suffices to consider the case where $M$ is connected. Begin by removing regular neighborhoods $A$ of $r$ and $A'$ of $r'$. Let $N = \overline{M -A}$ and let $N' = \overline{M -A'}$. We first exhibit an isomorphism from $N$ to $N'$ using the classification of surfaces. Then, we will find a compatible isomorphism from $A$ to $A'$. Lastly, we will combine these to define $\psi: M \rightarrow M'$. It is clear that $M$, $N$, and $N'$ have the same number of compact boundary components. If $M$ is orientable, then this orientation defines an orientation on $N$ as well. Suppose that $N$ is oriented. There there is an induced orientation on $\partial{A} \subset \partial{N}$. Choose an orientation on $A$ that induces the same orientation on $\partial{A}$. Gluing these orientations together yields an orientation on $M$. Similarly, $N'$ is orientable if and only if $M$ is. The genus and parity are both defined in terms of a compact exhaustion by subsurfaces. As in Lemma [Lemma 66](#lem:surfaceMinusRayEndsDontChange){reference-type="ref" reference="lem:surfaceMinusRayEndsDontChange"}, identify $A$ with $\mathbb{R}^2_+$. Let $F_j$ be the closed half-disk of radius $j$ centered at $(0,0)$, let $(G_j)$ be a compact exhaustion of $N$ by subsurfaces such that $F_j \cap \partial{A} = G_j \cap \partial{A}$, and let $H_j = F_j \cup G_j$. $(H_j)$ is a compact exhaustion of $M$ by subsurfaces. $$g(H_j) = \left| \pi_0\left(H_j\right) \right| - \frac{b(H_j) + \chi(H_j)}{2} = \left| \pi_0\left(G_j\right) \right| - \frac{b(G_j) + \chi(G_j)}{2} = g(G_j)$$ As a result, the genus and parity of $N$ is the same as $M$. Similarly, the genus and parity of $N'$ is the same as $M$. By Lemma [Lemma 66](#lem:surfaceMinusRayEndsDontChange){reference-type="ref" reference="lem:surfaceMinusRayEndsDontChange"}, the inclusion maps $i: N \rightarrow M$ and $i': N' \rightarrow M$ induce homeomorphisms on the end spaces. By looking within a neighborhood disjoint from $A$ and $A'$, it can be shown that for every end of $M$ other than $\eta$, the corresponding ends in $N$ and $N'$ have the same end characteristics. Given any end $\tau \neq \eta$, the corresponding ends in $N$ and $N'$ are isomorphic. This can be shown by looking in a neighborhood of $\tau$ disjoint from $A$ and $A'$. It was shown in Section [5.4](#subsec:endSumEndOrientation){reference-type="ref" reference="subsec:endSumEndOrientation"} that removing a strip does not change the orientation of $\eta$. Those arguments apply to this situation as well. It was shown in Section [5.5.2](#subsubsec:endSumEndsGenus){reference-type="ref" reference="subsubsec:endSumEndsGenus"} that removing a strip does not change the genus of $\eta$. Those arguments apply to this situation as well. Recall from Section [6.1](#csncb){reference-type="ref" reference="csncb"} above that an orientation $\mathcal{P}$ of $\mathcal{E}\left(\mathrm{d}N\right)$ is a certain subset of $\mathcal{E}\left(\mathrm{d}N\right)$. The surface $N$ has a single noncompact boundary component $B$, and both ends of $B$ point to the same end $\beta$ of $N$ by Lemma [Lemma 60](#evenendslemma){reference-type="ref" reference="evenendslemma"}. If $\beta$ is nonorientable, then $\mathcal{P}=\emptyset$. If $\beta$ is orientable, then choose an orientation on an orientable complementary domain that contains $\beta$; this yields $\mathcal{P}=\{\tau\}$ where $\tau$ is one of the two ends of $B$. Similarly, an orientation $\mathcal{P}'$ of $\mathcal{E}\left(\mathrm{d}N'\right)$ is defined. Note that $\eta$ is orientable if and only if $\beta$ is orientable, and $\eta$ is orientable if and only if $\beta'$. Thus, $\mathcal{P}$ and $\mathcal{P}'$ are either both empty or are both singletons. In either case, there is a unique bijection $\mathcal{P} \rightarrow \mathcal{P}'$. That bijection extends to an isomorphism $\mathcal{E}\left(\mathrm{d}N\right) \rightarrow \mathcal{E}\left(\mathrm{d}N'\right)$. Altogether, this data can be used to construct an isomorphism of diagrams for $N$ and $N'$. By the Classification of Surfaces (Theorem [Theorem 58](#thm:classificationOfAllSurfaces){reference-type="ref" reference="thm:classificationOfAllSurfaces"}), there exists an isomorphism $\varphi: N \rightarrow N'$. This isomorphism must send $\partial{A}$ to $\partial{A'}$. To extend $\varphi$ to an automorphism of $M$, it suffices to find a compatible isomorphism from $A$ to $A'$. But any isomorphism $\partial{\mathbb{R}^2_+} \rightarrow \partial{\mathbb{R}^2_+}$ extends to an isomorphism $\mathbb{R}^2_+ \rightarrow \mathbb{R}^2_+$. The resulting automorphism of $M$ sends $r$ to $r'$. ◻ **Remark 68**. The conclusion of Theorem [Theorem 67](#thm:rayUniqueness){reference-type="ref" reference="thm:rayUniqueness"} does not hold in general if $M$ has noncompact boundary components. Consider the surface $M$ which is a closed disk with one boundary point removed and a sequence of 1-handles attached as in Figure [16](#fig:raynotunique1){reference-type="ref" reference="fig:raynotunique1"}. {#fig:raynotunique1} So, $M$ has one end $\eta$ and that end has infinite genus. Let $r$ be a ray parallel to $\partial{M}$, and let $r'$ be a ray that winds around the glued in 1-handles as in Figure [16](#fig:raynotunique1){reference-type="ref" reference="fig:raynotunique1"}. Suppose there is a homeomorphism $\psi$ of $M$ that sends $r$ to $r'$. Let $A$ and $A'$ be closed regular neighborhoods of $r$ and $r'$, respectively. By the uniqueness of regular neighborhoods, $\psi$ can be modified so that it also sends $A$ to $A'$. Thus, $N = \overline{M -A}$ and $N' = \overline{M -A'}$ are homeomorphic. However, $N$ has one zero-genus end and one infinite-genus end, while $N'$ has two infinite-genus ends as in Figure [17](#fig:raynotunique2){reference-type="ref" reference="fig:raynotunique2"}. {#fig:raynotunique2} This is a contradiction, so $r$ and $r'$ are not related by a global self-homeomorphism of $M$. ## Main Theorem, Re-Proved {#subsec:maintheorem2} Using the classification of all surfaces, with our addendum, we are ready to dive into the main theorem once again. **Lemma 69**. Let $M$ be a surface with compact boundary. Let $\eta$ and $\eta'$ be distinct ends of $M$. For $i = 1, 2$, let $A_i, A_i'$ be disjoint closed subsurfaces of $M$ isomorphic to $\mathbb{R}^2_+$ and disjoint from $\partial{M}$. Suppose as well that that $A_i$ points to $\eta$ and $A_i'$ points to $\eta'$. Let $\mu: A_1 \rightarrow A_2$ and $\mu': A_1' \rightarrow A_2'$ be isomorphisms. If any of the following conditions hold, then there is an automorphism $\psi$ of $M$ such that the restriction of $\psi$ to $A_1$ is $\mu$ and the restriction of $\psi$ to $A_1'$ is $\mu'$. \(i\) $M$ is connected and orientable, and $\mu$, $\mu'$ are both orientation-preserving. \(ii\) $M$ is connected and non-orientable. \(iii\) $\eta$ and $\eta'$ lie in distinct connected components of $M$. *Proof.* Let $N_i = \overline{M -(A_i \cup A_i')}$. We will show that $N_1$ and $N_2$ are isomorphic, and that this isomorphism is compatible with $\mu$ and $\mu'$. Using the same strategies as Theorem [Theorem 67](#thm:rayUniqueness){reference-type="ref" reference="thm:rayUniqueness"}, it can be shown that $N_1$ and $N_2$ have the same genus, parity, orientability, and number of compact boundary components as $M$. Using the same strategies as Theorem [Theorem 67](#thm:rayUniqueness){reference-type="ref" reference="thm:rayUniqueness"}, it can be shown that the inclusion maps $i_i: N_i \rightarrow M$ induce homeomorphisms on the space of ends, and that corresponding ends have the same genus, orientability, and number of compact boundary components. By restricting $\mu$ and $\mu'$ to $\partial{A_1}$ and $\partial{A_1'}$, we obtain a map from $\partial{A_1} \cup \partial{A_1'} \rightarrow \partial{A_2} \cup \partial{A_2'}$. Define $h: \mathcal{E}\left(\mathrm{d}N_1\right) \rightarrow \mathcal{E}\left(\mathrm{d}N_2\right)$ to be the corresponding map of ends. Suppose first that $M$ is connected and orientable. Fix an orientation on $M$, which induces orientations on $N_1$ and $N_2$. Let $\mathcal{P}_i$ be the orientation of $\mathcal{E}\left(\mathrm{d}N\right)$ induced by the orientation on $N_i$. Since $\mu$ and $\mu'$ are orientation-preserving, the map $h$ sends $\mathcal{P}_1$ to $\mathcal{P}_2$. Using $h$, we can construct an isomorphism of diagrams. By the Addendum (Theorem [Theorem 59](#thm:classificationOfAllSurfacesAddendum){reference-type="ref" reference="thm:classificationOfAllSurfacesAddendum"}), there is an orientation-preserving isomorphism $\varphi: N_1 \rightarrow N_2$ that induces this isomorphism of diagrams. Since $\mu$ and $\mu'$ are both orientation-preserving, $\varphi$ has the same effect on the orientations of $\partial{A_1}$ and $\partial{A_1'}$ as $\mu$ and $\mu'$ do. Because of this, $\varphi$ can be isotoped in a neighborhood of $\partial{A_1}$ and $\partial{A_1'}$ to become equal to $\mu$ and $\mu'$ on $\partial{A_1}$ and $\partial{A_1'}$. Now suppose that $M$ is connected and non-orientable. If $\eta$ is orientable, then add an end of $\partial{A_1}$ to $\mathcal{P}_1$, and add the corresponding end of $\partial{A_2}$ (under $\mu$) to $\mathcal{P}_1$. Do the same for $\eta'$. Now, $h$ sends $\mathcal{P}_1$ to $\mathcal{P}_2$, so we can again obtain an isomorphism $\varphi: N_1 \rightarrow N_2$ that is compatible with $h$. And, $\varphi$ can be isotoped in a neighborhood of $\partial{A_1}$ and $\partial{A_1'}$ to become equal to $\mu$ and $\mu'$ on $\partial{A_1}$ and $\partial{A_1'}$. Now suppose that $\eta$ and $\eta'$ lie in distinct connected components of $M$. As before, construct $\mathcal{P}_1$ and $\mathcal{P}_2$, and obtain an isomorphism $\varphi: N_1 \rightarrow N_2$ that is compatible with $h$. Isotope $\varphi$ in a neighborhood of $A_1$ and $A_1'$ to become equal to $\mu$ and $\mu'$ on $\partial{A_1}$ and $\partial{A_1'}$. Once $\varphi$ agrees with with $\mu$ and $\mu'$ on $A_1$ and $A_1'$, all three maps can be glued together to form an isomorphism $\psi: M \rightarrow M$. ◻ As a corollary, we recover the main theorem. **Theorem 70**. Let $M$ be a [pl]{.smallcaps} surface with compact boundary, and let $\eta$, $\eta'$ be distinct ends of $M$. Let $N$ be a result of adding a 1-handle at infinity to $M$ along $\eta$ and $\eta'$. If $\eta$ and $\eta'$ are ends of distinct connected components of $M$, then $N$ is unique up to isomorphism. If $\eta$ and $\eta'$ are ends of the same connected component $M_1$ of $M$, and $M_1$ is non-orientable, then $N$ is unique up to isomorphism. Lastly, suppose that $\eta$ and $\eta'$ are ends of the same connected component $M_1$ of $M$, and $M_1$ is orientable. If the 1-handle is oriented, then $N$ is unique up to isomorphism. If the 1-handle is not oriented, then $N$ is unique up to isomorphism. *Proof.* Consider two end sums, $N_1$ and $N_2$, along $\eta$ and $\eta'$, satisfying the above criteria. Let $\nu_i, \nu_i'$ be the tubular neighborhood maps used to construct $N_i$, and let $A_i, A_i'$ be images of $\nu_i, \nu_i'$ respectively. Let $\psi: A_1 \rightarrow A_2$ be the canonical isomorphism between regular neighborhoods, defined by $\psi = \nu_2 \circ \nu_1^{-1}$. Similarly, define $\psi': A_1' \rightarrow A_2'$. Using Lemma [Lemma 69](#lem:regNeighborhoodUniqueness){reference-type="ref" reference="lem:regNeighborhoodUniqueness"}, we extend $\psi$ and $\psi'$ to a global automorphism $\chi: M \rightarrow M$. Since $\chi$ preserves all data used in the 1-handle construction, $\chi$ extends to an isomorphism from $N_1$ to $N_2$. ◻ # Extension of the Main Theorem to [top]{.smallcaps} and [diff]{.smallcaps} {#sec:secondTheorem} Our main theorem---Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}---also holds true in the [top]{.smallcaps} and [diff]{.smallcaps} categories. Hence, Corollary [Corollary 43](#cor:endSumPL){reference-type="ref" reference="cor:endSumPL"} does as well. This is an aspect of a general theme that often statements about surfaces are true independent of category. An important rationale for this theme, and a key reason why we can generalize in this case, is that every surface has a unique [top]{.smallcaps}, [pl]{.smallcaps}, and [diff]{.smallcaps} structure up to isomorphism. See, for example, Moise [@moise Preface & Ch. 8], Thurston [@thurston $\S$`<!-- -->`{=html}3.10], and Hatcher [@hatcher]. In this section, we show that Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} for [pl]{.smallcaps} implies the corresponding [top]{.smallcaps} and [diff]{.smallcaps} analogues. First, suppose we are given [top]{.smallcaps} data for the addition of a $1$-handle at infinity as in Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}. We reuse the notation from Conventions [Conventions 44](#keyconventions){reference-type="ref" reference="keyconventions"} in Section [5](#sec:mainTheorem){reference-type="ref" reference="sec:mainTheorem"}. In particular, $N$ is the [top]{.smallcaps} surface that results from adding a [top]{.smallcaps} $1$-handle at infinity to the surface $M$ along ends $\eta$ and $\eta'$. The $1$-handle is added using disjoint rays $r$ and $r'$, pointing to the ends $\eta$ and $\eta'$ respectively, and disjoint tubular neighborhood maps $\nu$ and $\nu'$. By the triangulation of surfaces, there exists a [pl]{.smallcaps} surface $M_{\textsc{pl}}$ and a homeomorphism $j:M\to M_{\textsc{pl}}$. So, $j\circ r$ and $j\circ r'$ are disjoint [top]{.smallcaps} rays in $M_{\textsc{pl}}$, and $j\circ \nu$ and $j\circ \nu'$ are disjoint [top]{.smallcaps} tubular neighborhood maps. We may adjust $M_{\textsc{pl}}$ by a small ambient homeomorphism with with support in any prescribed open neighborhood of $\textnormal{Im}\,j\circ \nu \cup \textnormal{Im}\,j\circ \nu'$ such that the images of the tubular neighborhood maps $j\circ \nu$ and $j\circ \nu'$ are [pl]{.smallcaps}. That is, by Moise [@moise Thm. 10.13], there is a homeomorphism $k:M_{\textsc{pl}}\to M_{\textsc{pl}}$ with support in any prescribed open neighborhood of $\textnormal{Im}\,j\circ \nu \cup \textnormal{Im}\,j\circ \nu'$ such that the homeomorphism $h=k\circ j$ satisfies: (i) the disjoint rays $h\circ r$ and $h\circ r'$ are [pl]{.smallcaps} embeddings and (ii) the disjoint tubular neighborhood maps $h\circ\nu$ and $h\circ\nu'$ are [pl]{.smallcaps} embeddings. As $j$ and $k$ are homeomorphisms, each induces a homeomorphism on the spaces of ends; note that $k$ induces the identity on the space of ends of $M_{\textsc{pl}}$. Let $\eta_{\textsc{pl}}$ and $\eta'_{\textsc{pl}}$ denote the ends of $M_{\textsc{pl}}$ corresponding under $j$ to $\eta$ and $\eta'$ respectively. We now have [pl]{.smallcaps} data for Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} consisting of the surface $M_{\textsc{pl}}$, disjoint rays $h\circ r$ and $h\circ r'$, pointing to the ends $\eta_{\textsc{pl}}$ and $\eta'_{\textsc{pl}}$ respectively, and disjoint tubular neighborhood maps $h\circ\nu$ and $h\circ\nu'$. Let $N_{\textsc{pl}}$ be the [pl]{.smallcaps} surface that results from adding a [pl]{.smallcaps} $1$-handle at infinity to $M_{\textsc{pl}}$ according to this data. Note that we have an induced homeomorphism $\alpha:N\to N_{\textsc{pl}}$. Now, suppose we have possibly different [top]{.smallcaps} data pointing to the same ends of $M$. That is, we have disjoint rays $s$ and $s'$, pointing to the *same* ends as before $\eta$ and $\eta'$ respectively, and disjoint tubular neighborhood maps $\mu$ and $\mu'$. Let $P$ be the [top]{.smallcaps} surface that results from adding a [top]{.smallcaps} $1$-handle at infinity to $M$ according to this data. We must show that $N$ and $P$ are homeomorphic. We have the homeomorphism $j:M\to M_{\textsc{pl}}$. As above, there is a homeomorphism $l:M_{\textsc{pl}}\to M_{\textsc{pl}}$ with support in any prescribed open neighborhood of $\textnormal{Im}\,j\circ \mu \cup \textnormal{Im}\,j\circ \mu'$ such that the homeomorphism $g=l\circ j$ satisfies: (i) the disjoint rays $g\circ s$ and $g\circ s'$ are [pl]{.smallcaps} embeddings and (ii) the disjoint tubular neighborhood maps $g\circ\mu$ and $g\circ\mu'$ are [pl]{.smallcaps} embeddings. Let $P_{\textsc{pl}}$ be the [pl]{.smallcaps} surface that results from adding a [pl]{.smallcaps} $1$-handle at infinity to $M_{\textsc{pl}}$ according to this data. Note that we have an induced homeomorphism $\beta:P\to P_{\textsc{pl}}$. Observe that in $M_{\textsc{pl}}$, the rays $h\circ r$ and $g\circ s$ both point to $\eta_{\textsc{pl}}$, and the rays $h\circ r'$ and $g\circ s'$ both point to $\eta'_{\textsc{pl}}$. Therefore, the [pl]{.smallcaps} version of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"} yields a [pl]{.smallcaps} homeomorphism $\gamma:N_{\textsc{pl}}\to P_{\textsc{pl}}$. Hence, $\beta^{-1}\circ\gamma\circ\alpha:N\to P$ is a homeomorphism as desired. This completes our proof of the [top]{.smallcaps} version of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}. Second, suppose we are given two collections of [diff]{.smallcaps} data for the addition of a $1$-handle at infinity as in Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}. One collection is the surface $M$, disjoint rays $r$ and $r'$, pointing to the ends $\eta$ and $\eta'$ respectively, and disjoint tubular neighborhood maps $\nu$ and $\nu'$. Let $N$ be the [diff]{.smallcaps} surface that results by adding a [diff]{.smallcaps} $1$-handle at infinity to $M$ according to this data. The other collection is the surface $M$, disjoint rays $s$ and $s'$, pointing to the *same* ends as before $\eta$ and $\eta'$ respectively, and disjoint tubular neighborhood maps $\mu$ and $\mu'$. Let $P$ be the [diff]{.smallcaps} surface that results by adding a [diff]{.smallcaps} $1$-handle at infinity to $M$ according to this data. We must show that $N$ and $P$ are diffeomorphic. Ignoring [diff]{.smallcaps} structures for the moment, the [top]{.smallcaps} version of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}---proved above---implies that $N$ and $P$ are homeomorphic. As $N$ and $P$ are homeomorphic [diff]{.smallcaps} surfaces, they are diffeomorphic as desired. This completes our proof of the [diff]{.smallcaps} version of Theorem [Theorem 42](#thm:endSumUniquenessPLCase){reference-type="ref" reference="thm:endSumUniquenessPLCase"}. 99 M. A. Armstrong and E. C. Zeeman *Transversality for Piecewise Linear Manifolds* Bull. Amer. Math. Soc. **73** (1966), 184--188 J. Bennett, *Exotic smoothings via large $\mathbb{R}^4$'s in Stein surfaces*, Algebr. Geom. Topol. **16** (2016), 1637--1681. E. Brown and R. Messer, *The classification of two-dimensional manifolds*, Trans. Amer. Math. Soc. **255** (1979), 377--402. J. S. Calcut and R. E. Gompf, *On uniqueness of end sums and 1--handles at infinity*, Algebr. Geom. Topol. **19** (2019), 1299--1339. J. S. Calcut, C. R. Guilbault, and P. V. Haggerty, *Extreme nonuniqueness of end sum*, J. Topol. Anal. **14** (2022), 461--503. J. S. Calcut and P. V. Haggerty, *Connected sum at infinity and 4-manifolds*, Algebr. Geom. Topol. **14** (2014), 3281--3303. J. S. Calcut, H. C. King, and L. C. Siebenmann, *Connected sum at infinity and Cantrell-Stallings hyperplane unknotting*, Rocky Mountain J. Math. **42** (2012), 1803--1862. H. Freudenthal, *Über die Enden topologischer Räume und Gruppen*, Math. Z. **33** (1931), 692--713. R. Geoghegan, *Topological methods in group theory*, Springer, New York, 2008. M. E. Goldman, *An algebraic classification of noncompact 2-Manifolds*, Trans. Amer. Math. Soc. **156** (1971), 241--258. R. Gompf, *Three exotic $\mathbb{R}^4$'s and other anomalies*, J. Diff. Geom. **18** (1983), 317--328. R. Gompf, *An infinite set of exotic $\mathbb{R}^4$'s*, J. Diff. Geom. **21** (1985), 283--300. C. R. Guilbault, *Ends, shapes, and boundaries in manifold topology and geometric group theory*, in *Topology and geometric group theory*, Springer Proc. Math. Stat. **184**, Springer, 2016, 45--125. A. Hatcher, *The Kirby torus trick for surfaces*, arXiv:1312.3518 \[math.GT\], 2022, 1--10. A. Hatcher, *Algebraic topology*, Cambridge University Press, Cambridge, 2002. E. E. Moise, *Geometric topology in dimensions $2$ and $3$*, Springer-Verlag, New York-Heidelberg, 1977. D. Müllner, *Orientation reversal of manifolds*, Algebr. Geom. Topol. **9** (2009), 2361--2390. J. R. Munkres, *Topology*, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. R. Myers, *End sums of irreducible open $3$-manifolds*, Quart. J. Math. Oxford Ser. (2) **50** (1999), 49--70. I. J. Richards, *A classification of noncompact surfaces*, Ph.D. Thesis, Harvard University, 1960. C. P. Rourke and B. J. Sanderson, *Introduction to piecewise-linear topology*, Springer-Verlag, New York, 1972. A. Scott, *Infinite regular neighborhoods*, J. London Math. Soc. **42** (1967), 245--253. L. C. Siebenmann, *The obstruction to finding a boundary for an open manifold of dimension greater than five*, Ph.D. thesis, Princeton University, 1965; available online as the file `sieben.pdf` (original) or `thesis.pdf` (re-typed in LaTeX) at [http://www.maths.ed.ac.uk/ aar/surgery/notes.htm](http://www.maths.ed.ac.uk/~aar/surgery/notes.htm). P. Sparks, *The double $n$-space property for contractible $n$-manifolds*, Algebr. Geom. Topol. **18** (2018), 2131--2149. W. P. Thurston, *Three-dimensional geometry and topology*, Princeton University Press, Princeton, NJ, 1997.

arxiv_math

--- abstract: | A conjecture of Buch--Chaput--Perrin asserts that the two-pointed curve neighborhood corresponding to a quantum product of Seidel type is an explicitly given Schubert variety. We prove this conjecture for flag varieties in type $A$. address: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA author: - Mihail Țarigradschi bibliography: - bib.bib title: Curve neighborhoods of Seidel products in quantum cohomology --- [^1] # Introduction Given a flag variety $X = G/P_X$, the Seidel representation on the small quantum cohomology ring $\mathop{\mathrm{QH}}(X)|_{q=1}$ specialized at $q=1$ can be described combinatorially, see [@belkale_transformation_2004; @chaput_affine_2009]. In the recent paper [@buch_seidel_2023], this action is extended to the quantum K-theory ring $\mathop{\mathrm{QK}}(X)|_{q=1}$ when $X$ is cominuscule, and it is found that similar identities hold. On $\mathop{\mathrm{QH}}(X)$ it is realized as the action of a subgroup $W^{\mathop{\mathrm{{comin}}}} \subset W$ through the identity $$\label{eqn:QHeq} [X^w] \star [X^u] = q^d [X^{wu}],$$ where $w\in W^{\mathop{\mathrm{{comin}}}}$, $u\in W$, and $d=d_{\min}(w, u)\in H_2(X,{\mathbb Z})$ denotes the smallest degree of a rational curve connecting general translates of $X^w$ and $X^u$. The subgroup $W^{\mathop{\mathrm{{comin}}}}$ consists of the identity and the minimal length representatives of the longest Weyl group element $w_0$ in $W^M = W/W_M$ for each cominuscule flag variety $M = G/P_M$: $$W^{\mathop{\mathrm{{comin}}}} = \{1\} \cup \{ (w_0)^M : M = G/P_M \text{ where \(M\) is a cominuscule variety} \},$$ here $(w_0)^M \in W$ denotes the representative of $w_0$ in $W^M$. Equation [([\[eqn:QHeq\]](#eqn:QHeq){reference-type="ref" reference="eqn:QHeq"})](#eqn:QHeq) also implies the following identity: $$[X^w] \star [X^u] = q^d [\Gamma_d(X_{w_0 w}, X^u)],$$ where $\Gamma_d(X_{w_0 w}, X^u)$ is the two-pointed curve neighborhood defined as the union of all stable curves of degree $d$ that pass through $X_{w_0 w}$ and $X^u$. This led the authors of [@buch_seidel_2023] to conjecture that $\Gamma_d(X_{w_0 w}, X^u)$ is a translate of $X^{wu}$: **Conjecture 1** ([@buch_seidel_2023], Conjecture 3.11). *Let $X = G/P_X$ be any flag variety. For $u \in W, w \in W^{\mathop{\mathrm{{comin}}}}$, and $d = d_{\min}(w,u) \in H_2(X,{\mathbb Z})$, we have $$\Gamma_d(X_{w_0 w}, X^u) = w^{-1}.X^{w u}.$$* It is known that the conjecture holds in the following cases: - $d=0$ (see [@buch_seidel_2023]\*Proposition 2.2); - $X$ is cominuscule and $w = w_0^X$ (see [@buch_seidel_2023]\*Lemma 3.1); - $X = \mathop{\mathrm{Gr}}(k, n)$ and $w=w_0^{{\mathbb P}^{n-1}}$ (see [@li_seidel_2022]\*Proposition 4.5). In this paper we consider the following specialization: **Conjecture 2**. *Let $X = G/P_X$ be a flag variety where $P_X$ is a **maximal** parabolic subgroup of $G$. For $u \in W, w \in W^{\mathop{\mathrm{{comin}}}}$, and $d = d_{\min}(w,u) \in H_2(X,{\mathbb Z})$, we have $$\Gamma_d(X_{w_0 w}, X^u) = w^{-1}.X^{w u}.$$* We prove the following reduction theorem. **Theorem 3**. *follows from for the same $G$.* And then prove for Grassmannians of type A to conclude that holds for flag varieties of type A. Our proof is inspired from a description of two-pointed curve neighborhoods in Grassmannians given in [@li_seidel_2022]\*Proposition 4.5. ## Acknowledgements {#acknowledgements .unnumbered} We thank Anders Buch for useful discussions and bringing to our attention . # Notation and Preliminaries {#sec:prelim} Let $G$ be a complex semisimple linear algebraic group, let $T \subset B$ be a maximal torus and a Borel subgroup of $G$. Denote by $B^{-}$ the Borel subgroup opposite to $B$, by $\Phi$ the set of roots of $G$, by $\Delta\subset \Phi$ the set of simple roots, and by $W$ the Weyl group of $G$. Let $P_X \supseteq B$ be a parabolic subgroup of $G$. In the flag variety $X = G/P_{X}$ we consider the Schubert varieties $X_u = \overline{B u P_X/P_X}$ and $X^u = \overline{B^{-} u P_X/P_X}$ where $u \in W$. Let $W_X$ be the Weyl group of $P_X$, then we denote $W^X \subset W$ to be the set of minimal length representatives of the cosets in $W/W_X$. For an element $u \in W$, denote by $u^X\in W^X$ the representative of $u W_X \in W/W_X$. Let $\Delta_{P_X} \subseteq \Delta$ be the set of simple roots that define $P_X$, i.e. $\beta \in \Delta_{P_X} \iff s_\beta \in W_X$. Recall the general construction of curve neighborhoods (described for example in [@buch_finiteness_2013]). Let $\overline{{\mathcal M}}_{0,3}(X,d)$ denote the moduli space of $3$-pointed, genus $0$ stable maps to $X$ of effective degree $d \in H_2(X,{\mathbb Z})$. We have the evaluation maps $ev_i : \overline{{\mathcal M}}_{0,3}(X,d) \to X$ where $i\in\{1,2,3\}$ that correspond to taking the image of the $i$-th marked point. Given two opposite Schubert varieties $X_u, X^v$, define the *Gromov--Witten variety* $M_d(X_u, X^v) = ev_1^{-1}(X_u) \cap ev_2^{-1}(X^v) \subset \overline{{\mathcal M}}_{0,3}(X,d)$. It describes the coefficients in the quantum product of Schubert classes in $\mathop{\mathrm{QH}}(X)$: $$\label{eqn:GW} [X_u] \star [X^v] = \sum_{d \geq 0} (ev_3)_*[M_d(X_u, X^v)] q^d.$$ We denote $\Gamma_d(X_u,X^v) = ev_3(M_d(X_u, X^v))$. The subvariety $\Gamma_d(X_u,X^v) \subset X$ is called a two-pointed curve neighborhood and consists of the union of all stable curves of degree $d$ that pass through $X_u$ and $X^v$. # Proof of the reduction to maximal parabolics In this section, we prove . The main observation is the fact that the intersection of Schubert varieties $X^{u^Y}$ for appropriate "smaller\" flag varieties $Y$ is the Schubert variety $X^u$, this follows from a combinatorial fact from [@bjorner_combinatorics_2005]. **Lemma 4**. *Let $X = G/P_X$ be a flag variety, $F\subseteq X$ a subvariety, $w\in W$, and $g\in G$. Let $Y = G/P_Y, Z=G/P_Z$ be flag varieties such that $P_Y \cap P_Z = P_X$. Denote by $p : X \to Y$ and $q : X \to Z$ the projection maps, assume that $p(F) \subseteq g.Y^w$ and $q(F) \subseteq g.Z^w$. Then $F \subseteq g.X^w$.* *Proof.* Using [@buch_seidel_2023]\*Lemma 2.1(e) we have $$F \subseteq p^{-1}(p(F)) \subseteq p^{-1}(g. Y^w) = g.X^{w^Y}.$$ By the same argument for $Z$, we have $F \subseteq g.X^{w^Z}$, so that $F \subseteq g.(X^{w^Y} \cap X^{w^Z})$. By [@bjorner_combinatorics_2005]\*Theorem 2.6.1, using $\Delta_{P_Y} \cap \Delta_{P_Z} = \Delta_{P_X}$, the elements $w^Y, w^Z$ have a well-defined join equal to $w^X$ in the poset $W^X$. In particular $X^{w^Y} \cap X^{w^Z} = X^{w^X} = X^w$, and we are done. ◻ *Proof of .* By [@buch_finiteness_2013]\*Corollary 3.3, the two-pointed Gromov--Witten variety $M_d(X_{w_0 w}, X^u) \subset \overline{\mathcal M}_{0,3}(X, d)$ is irreducible, and so is $\Gamma_d(X_{w_0w},X^u) = \mathop{\mathrm{ev}}_3(M_d(X_{w_0 w}, X^u)) \subset X$. In the graded ring $\mathop{\mathrm{QH}}(X)$, by [@belkale_transformation_2004; @chaput_affine_2009] we have the identity $[X_{w_0 w}] \star [X^u] = q^d [X^{wu}]$. On the other hand, from Equation [([\[eqn:GW\]](#eqn:GW){reference-type="ref" reference="eqn:GW"})](#eqn:GW) we have $$[X_{w_0 w}] \star [X^u] = q^d (ev_3)_*[M_d(X_{w_0 w}, X^u)] = q^d \deg(ev_3) [\Gamma_d(X_{w_0 w}, X^u)] ,$$ so that $\deg(ev_3)=1$, $[X^{wu}] = [\Gamma_d(X_{w_0 w}, X^u)]$, and $\mathop{\mathrm{codim}}X^{wu} = \mathop{\mathrm{codim}}\Gamma_d(X_{w_0 w}, X^u)$. Since Schubert varieties are irreducible, we are left with proving the inclusion $\Gamma_d(X_{w_0w},X^u) \subseteq w^{-1}. X^{w u}$. We prove it by induction on $| \Delta \setminus \Delta_X |$. If $|\Delta \setminus \Delta_X| = 1$, we are done by assumption of . Otherwise, consider parabolic subgroups $P_Y, P_Z \supsetneq P_X$ such that $P_Y \cap P_Z = P_X$. Let $p: X \to Y=G/P_Y$, $q: X \to Z = G/P_Z$ be the projections. Then $p(\Gamma_d(X_{w_0w},X^u)) \subseteq \Gamma_{p_*(d)}(Y_{w_0w},Y^u)$. Furthermore, $p_*(d_{\min}(w,u)) = d_{\min}(w,u) \in H_2(Y, {\mathbb Z})$ (see [@buch_euler_2020]\*Section 2.3 or [@chaput_affine_2009; @belkale_transformation_2004]). Applying induction to $Y$, $\Gamma_{p_*(d)}(Y_{w_0w},Y^u) \subseteq w^{-1}.Y^{wu}$, so that $$p(\Gamma_d(X_{w_0w}, X^u)) \subseteq \Gamma_{p_*(d)}(Y_{w_0w}, Y^u) \subseteq w^{-1}.Y^{wu}.$$ Analogously, we have $q(\Gamma_d(X_{w_0w}, X^u)) \subseteq w^{-1}.Z^{wu}$. Using for $F = \Gamma_d(X_{w_0w}, X^u)$, we are done. ◻ # Proof of the conjecture in type A In this section we assume that the group $G$ is of type A. Identify $\Delta$ with the integers $\{1, \dots, n-1\}$ and $W$ with the symmetric group $S_n$. When $P_X$ is the maximal parabolic corresponding to the simple root $k$, i.e. $\Delta_{P_X} = \Delta \setminus \{k\}$, then $X$ is isomorphic to the Grassmannian variety $\mathop{\mathrm{Gr}}(k,n)$ of $k$-planes in ${\mathbb C}^n$. We have the isomorphism $W^{\mathop{\mathrm{{comin}}}} \cong {\mathbb Z}/n{\mathbb Z}$, with a generator $w \in S_n$ given by $$w(t) = \begin{cases} n, & \text{ if } t=1 \\ t-1, & \text{ if } 2\leq t \leq n \end{cases}.$$ The element corresponding to the simple root $i \in \{1, \dots, n-1\}$ is given by $w^i \in S_n$. When $X = \mathop{\mathrm{Gr}}(k,n)$ is a Grassmannian, we can describe the two-pointed curve neighborhoods using the *quantum equals classical* theorems (see [@buch_gromov-witten_2003]). For an effective degree $0 \leq d \leq \min(k,n-k)$ we denote $Z_d = \mathop{\mathrm{Fl}}(k-d, d, k+d; n)$ to be the variety of three-step flags of dimensions $(k-d, d, k+d)$ in ${\mathbb C}^n$. Similarly, we denote $Y_d = \mathop{\mathrm{Fl}}(k-d, k+d; n)$ to be the variety of two-step flags of dimensions $(k-d, k+d)$ in ${\mathbb C}^n$. We then have the projection maps $p_d : Z_d \to X$, $q_d: Z_d \to Y_d$ and $\Gamma_d(X_u,X^v) = p_d(Z_d(X_u,X^v))$ where $Z_d(X_u, X^v) = q_d^{-1}( q_d(p_d^{-1}(X_u)) \cap q_d(p_d^{-1}(X^v))) \subset Z_d$. Analogous to Equation [([\[eqn:GW\]](#eqn:GW){reference-type="ref" reference="eqn:GW"})](#eqn:GW), $$[X_u] \star [X^v] = \sum_{d \geq 0} (q_d)_*[Z_d(X_u, X^v)] q^d.$$ Recall that Grassmannian Schubert varieties can be equivalently indexed by a partition. Fix a basis $e_1, \dots, e_n$ of ${\mathbb C}^n$ such that $B$ acts on ${\mathbb C}^n$ by upper-triangular matrices. Consider the complete flag $E_{\bullet} = E_1 \subset E_2 \subset \dots \subset E_n$ where $E_i = \mathop{\mathrm{Span}}\{e_1, \dots, e_i \}$, then $B.E_\bullet = E_\bullet$. Similarly, consider the opposite flag $E^{\mathop{\mathrm{{opp}}}}_{\bullet} = E^{\mathop{\mathrm{{opp}}}}_1 \subset E^{\mathop{\mathrm{{opp}}}}_2 \subset \dots \subset E^{\mathop{\mathrm{{opp}}}}_n$ where $E^{\mathop{\mathrm{{opp}}}}_i = \mathop{\mathrm{Span}}\{e_n, \dots, e_{n-i+1} \}$, then $B^{-}.E_\bullet^{\mathop{\mathrm{{opp}}}} = E_\bullet^{\mathop{\mathrm{{opp}}}}$. Given a partition $\lambda = (\lambda_1, \dots, \lambda_k)$ such that $n-k \geq \lambda_1 \geq \dots \geq \lambda_k \geq 0$, define the $B$-stable Schubert variety $$X_\lambda = \{ \Sigma\in X: \dim\Sigma \cap E_{i+\lambda_{k-i+1}} \geq i\ \text{for } i=1, \dots, k \},$$ of dimension $\dim X_\lambda = |\lambda|$, and the $B^{-}$-stable Schubert variety $$X^\lambda = \{ \Sigma\in X: \dim\Sigma \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \geq i\ \text{for } i=1, \dots, k \}$$ of codimension $\mathop{\mathrm{codim}}X^\lambda = |\lambda|$. These varieties satisfy $X_\lambda = X_w$ and $X^\lambda = X^w$ where $w \in W^X \subset S_n$ is the permutation of minimal length such that $\lambda = (w(k)-k, w(k-1)-(k-1), \dots, w(1)-1)$. We will also think of $\lambda$ as a Young diagram inside a $k \times (n-k)$ rectangle such that row $i$ has $\lambda_i$ boxes. **Theorem 5**. *Let $X = \mathop{\mathrm{Gr}}(k, n)$. For $u \in W$, $w \in W^{\mathop{\mathrm{{comin}}}}$, and $d = d_{\min}(w,u) \in H_2(X, {\mathbb Z})$, we have $$\Gamma_d(X_{w_0 w}, X^u) = w^{-1}.X^{w u}$$* *Proof.* If $w = 1$, then $d=0$ and $\Gamma_d(X_{w_0 w}, X^u) = \Gamma_0(X_{w_0}, X^u) = X_{w_0} \cap X^u = X^u = w^{-1}. X^{wu}$. Assume $w \neq 1$. Let $\beta \in \Delta$ be a simple root that indexes $w$. Using the duality isomorphism $\mathop{\mathrm{Gr}}(k,n) \cong \mathop{\mathrm{Gr}}(n-k, n)$, we may assume $\beta \geq k$. As an element of $S_n$, $w_0 w = (\beta\ (\beta-1)\ \dots\ 1\ n\ (n-1)\ \dots (\beta+1))$ in one-line notation. The corresponding partition in $X$ is given by $(\beta-k, \beta-k, \dots, \beta-k) = (\beta-k)^k$. Denote by $\lambda$ the partition associated to $u^X$. We use [@fulton_quantum_2004] to get an explicit expression for $d$. The degree $d$ is the minimal degree of $q$ in $[X_{(\beta-k)^k}] \star [X^\lambda] = [X^{(n-\beta)^k}] \star [X^\lambda]$. Consider the overlap of $\lambda$ and the $180^\circ$ rotation of $(n-\beta)^k$ in the $k \times (n-k)$ rectangle. This overlap is a partition, call it $\lambda^\prime$, where we remove the left-most $(\beta-k)$ columns from $\lambda$ (see [Figure [\[fig:degreefig\]](#fig:degreefig){reference-type="ref" reference="fig:degreefig"}](#fig:degreefig)). Then $d$ is the length of the longest NW-SE diagonal sequence of boxes in this overlap. Since $\lambda^\prime$ is a partition, we can always move a NW-SE diagonal sequence of boxes to the NW corner of $\lambda^\prime$. This gives the following formula for $d$: $$\label{eqn:degeqn} d = \max (\{0\} \cup \{j : \lambda_j - (\beta - k) \geq j\}).$$ Here is an example where $n=9, k=4, \lambda = (5,4,3,1), \beta=5$: By the quantum equals classical theorem, we have that $\Gamma_d(X_{w_0 w}, X^u) = p_d(Z_d(X_{w_0 w}, X^u))$. Looking at the three-step flags in $q_d^{-1} q_d p_d^{-1} (X^u) \subset Z_d$, we have $$\begin{aligned} q_d^{-1} q_d p_d^{-1} (X^u) & = q_d^{-1} q_d p_d^{-1} (X^\lambda) \\ & = \{ V_{k-d} \leq V_k \leq V_{k+d} : & & \exists \widetilde{V_k} \text{ such that } V_{k-d} \leq \widetilde{V_k} \leq V_{k+d} \text{ and } \\ & & & \dim \widetilde{V_{k}} \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \geq i \text{ for } 1 \leq i \leq k\} \\ & \subseteq \{ V_{k-d} \leq V_k \leq V_{k+d} : & & \dim V_{k+d} \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \geq i \text{ for } 1 \leq i \leq k \text{ and } \\ & \ & & \dim V_{k-d} \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \geq i-d \text{ for } d+1 \leq i \leq k \}, \end{aligned}$$ where the last condition follows from $\dim V_{k-d} \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \geq \dim \widetilde{V_k} \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} - d \geq i-d$. By symmetry, for the special case of $X_{w_0 w}$, we have $$\begin{aligned} q_d^{-1} q_d p_d^{-1} (X_{w_0 w}) & = q_d^{-1} q_d p_d^{-1} (X_{(\beta-k)^k}) \\ & \subseteq \{ V_{k-d} \leq V_k \leq V_{k+d} : & & \dim V_{k+d} \cap E_{i+\beta-k} \geq i \text{ for } 1 \leq i \leq k \text{ and } \\ & \ & & \dim V_{k-d} \cap E_{i+\beta-k} \geq i-d \text{ for } d+1 \leq i \leq k \} \\ & = \{ V_{k-d} \leq V_k \leq V_{k+d} : & & V_{k-d} \subset E_{\beta},\ \dim V_{k+d} \cap E_{\beta} \geq k \}. \end{aligned}$$ Let $V_k \in \Gamma_d(X_{w_0 w}, X^u) = p_d(Z_d(X_{w_0 w}, X^u)) = p_d( q_d^{-1} q_d p_d^{-1} (X_{w_0 w}) \cap q_d^{-1} q_d p_d^{-1}(X^u) )$. If $1 \leq i \leq d$, then $k-i+\lambda_i + 1 > \beta$ by [([\[eqn:degeqn\]](#eqn:degeqn){reference-type="ref" reference="eqn:degeqn"})](#eqn:degeqn), so that $E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \cap E_\beta = 0$ and $$\begin{aligned} \dim V_k \cap (E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} + E_\beta) & \geq \dim V_{k+d} \cap (E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} + E_\beta) - d \\ & \geq \dim V_{k+d} \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} + \dim V_{k+d} \cap E_\beta -d \geq i + k-d. \end{aligned}$$ Otherwise, if $d+1 \leq i \leq k$, then $k-i+\lambda_i+1\leq \beta$ by [([\[eqn:degeqn\]](#eqn:degeqn){reference-type="ref" reference="eqn:degeqn"})](#eqn:degeqn), so that $E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} + E_\beta = {\mathbb C}^n$ and $$\begin{aligned} \dim V_k \cap (E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \cap E_\beta) & \geq\dim V_{k-d} \cap (E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \cap E_\beta) \\ & = \dim V_{k-d} \cap E^{\mathop{\mathrm{{opp}}}}_{n-k+i-\lambda_i} \geq i-d. \end{aligned}$$ For simplicity, we denote $$\label{eqn:defG} G_i = \begin{cases} E^{\mathop{\mathrm{{opp}}}}_{n-k+(i+d)-\lambda_{i+d}} \cap E_\beta, & \text{if } 1\leq i\leq k-d, \\ E^{\mathop{\mathrm{{opp}}}}_{n-k+(i-k+d)-\lambda_{i-k+d}} + E_\beta, & \text{if } k-d+1 \leq i \leq k. \end{cases}$$ So that we have a chain of inclusions $$G_1 \lneq G_2 \lneq \dots \lneq G_{k-d} \lneq G_{k-d+1} \lneq \dots \lneq G_k,$$ and dimension conditions $\dim V_k \cap G_{i} \geq i$ for all $1 \leq i \leq k$. Note that the subspaces $G_i$ are parts of the complete flag $F^{\mathop{\mathrm{{opp}}}}_{\bullet}$ defined by $$F^{\mathop{\mathrm{{opp}}}}_i = \begin{cases} \mathop{\mathrm{Span}}\{e_\beta, e_{\beta-1}, \dots, e_{\beta-i+1}\}, & \text{if } 1\leq i \leq \beta \\ \mathop{\mathrm{Span}}\{e_\beta, e_{\beta-1}, \dots, e_1, e_{n}, \dots, e_{n-(i-\beta)+1}\}, & \text{if } \beta+1\leq i \leq n \end{cases}$$ which is precisely given by $F_{\bullet}^{\mathop{\mathrm{{opp}}}} = w^{-1}.E_{\bullet}^{\mathop{\mathrm{{opp}}}}$. The dimension conditions $\dim V_k \cap G_i \geq i$ for $1 \leq i \leq k$ define a Schubert variety with respect to the flag $F_\bullet^{\mathop{\mathrm{{opp}}}}$, it is given by $$\begin{aligned} \{V_k : \dim V_k \cap G_i \geq i \text{ for } 1\leq i\leq k \} & = w^{-1}.\{w.V_k : \dim V_k \cap G_i \geq i \text{ for } 1\leq i\leq k\} \\ & = w^{-1}.\{V_k : \dim V_k \cap w.G_i \geq i \text{ for } 1\leq i\leq k\} \\ & = w^{-1}.X^{v} \end{aligned}$$ for some $v\in W^X$. We compute using [([\[eqn:defG\]](#eqn:defG){reference-type="ref" reference="eqn:defG"})](#eqn:defG) $$\begin{aligned} \ell(v) & = \mathop{\mathrm{codim}}w^{-1}. X^v = \sum (n-k+i-\dim G_i) \\ & = \sum_{1 \leq i \leq k-d} (n-k+i - (\beta - k+i+d-\lambda_{i+d})) \\ & + \sum_{k-d<i \leq k} (n-k+i - ( \beta + n-k+(i-k+d)-\lambda_{i-k+d})) \\ & = \sum_{1 \leq i \leq k-d} (n - \beta - d + \lambda_{i+d} ) + \sum_{k-d < i \leq k} (k - \beta - d + \lambda_{i-k+d}) \\ & = n(k-d) -\beta k + |\lambda| = n(k-d) -\beta k + \ell(u^X) \end{aligned}$$ On the other hand, from [([\[eqn:QHeq\]](#eqn:QHeq){reference-type="ref" reference="eqn:QHeq"})](#eqn:QHeq) and $[X^w] = [X_{w_0 w}]$ we compute $$\begin{aligned} \ell((wu)^X) & = \mathop{\mathrm{codim}}X^{wu} = \mathop{\mathrm{codim}}X_{w_0 w} + \mathop{\mathrm{codim}}X^u - d \deg(q) \\ & = \dim X - \dim X_{w_0 w} + \ell(u^X) - d n \\ & = k(n-k) - (\beta-k)k + \ell(u^X) - dn \\ & = n(k-d) -\beta k +\ell(u^X) = \ell(v) \end{aligned}$$ Since $\Gamma_d(X_{w_0 w}, X^u) \subseteq w^{-1}.X^v$ and $\dim X^v = \dim X^{wu} = \dim \Gamma_d(X_{w_0 w}, X^u)$ we get that the inclusion is an equality. Since $[\Gamma_d(X_{w_0 w}, X^u)] = [X^{wu}]$, we get $v = (wu)^X$ from which the conclusion follows. ◻ By the above and , we conclude that holds in type A. [^1]: The author was partially supported by NSF grant DMS-2152316.

arxiv_math

--- abstract: | We presented a Hilbert-Mumford criterion for polystablility associated with an action of a real reductive Lie group $G$ on a real submanifold $X$ of a Kähler manifold $Z$. Suppose the action of a compact Lie group with Lie algebra $\mathfrak{u}$ extends holomorphically to an action of the complexified group $U^\mathbb{C}$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset U^\mathbb{C}$ is compatible, there is a corresponding gradient map $\mu_\mathfrak{p}: X\to \mathfrak{p}$, where $\mathfrak{g}= \mathfrak{k}\oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Under some mild restrictions on the $G$-action on $X,$ we characterize which $G$-orbits in $X$ intersect $\mu_\mathfrak{p}^{-1}(0)$ in terms of the maximal weight function, which we viewed as a collection of maps defined on the boundary at infinity ($\partial_\infty G/K$) of the symmetric space $G/K$. address: - | Dipartimento di Scienze Matematiche, Fisiche e Informatiche\ Università di Parma (Italy) - | Dipartimento di Scienze Matematiche, Fisiche e Informatiche\ Università di Parma (Italy) author: - Leonardo Biliotti - Oluwagbenga Joshua Windare title: A Hilbert-Mumford Criterion for polystability for actions of real reductive Lie groups --- # Introduction The classical Hilbert-Mumford criterion in Geometric Invariant Theory (in projective algebraic geometry) is an explicit numerical criterion for finding the stability of a point in terms of an invariant known as maximal weight function [@MUMFORD]. This criterion has been extended to the non-algebraic Kählerian settings using the theory of Kähler quotients and a version of maximal weight function [@MUNDETC; @MUNDETT; @Teleman; @BT; @KLM]. For this setting, a Kähler manifold $(Z,\omega)$ with a holomorphic action of a complex reductive Lie group $U^\mathbb{C}$, where $U^\mathbb{C}$ is the complexification of a compact Lie group $U$ with Lie algebra $\mathfrak{u}$ is considered. Assume $\omega$ is $U$-invariant and that there is a $U$-equivariant momentum map $\mu : Z \to \mathfrak{u}^*.$ By definition, for any $\xi \in \mathfrak{u}$ and $z\in Z,$ $d\mu^\xi = i_{\xi_Z}\omega,$ where $\mu^\xi(z) := \mu(z)( \xi )$ and $\xi_Z$ denotes the fundamental vector field induced on $Z$ by the action of $U,$ i.e., $$\xi_Z(z) := \frac{d}{dt}\bigg \vert _{t=0} \exp(t\xi)z$$ (see, for example, [@Kirwan] for more details on the momentum map). Our aim is to investigate a class of actions of real reductive Lie groups on real submanifolds of $Z$ using gradient map techniques. This setting was recently introduced in [@PG; @heinzner-schwarz-stoetzel; @heinzner-schuetzdeller]. More precisely, a subgroup $G$ of $U^\mathbb{C}$ is compatible if $G$ is closed and the map $K\times \mathfrak{p} \to G,$ $(k,\beta) \mapsto k\exp(\beta)$ is a diffeomorpism where $K := G\cap U$ and $\mathfrak{p} := \mathfrak{g}\cap \textbf{i}\mathfrak{u};$ $\mathfrak{g}$ is the Lie algebra of $G$. The Lie algebra $\mathfrak{u}^\mathbb{C}$ of $U^\mathbb{C}$ is the direct sum $\mathfrak{u}\oplus \textbf{i}\mathfrak{u}.$ It follows that $G$ is compatible with the Cartan decomposition of $U^\mathbb{C}= U\exp(\textbf{i}\mathfrak{u})$, $K$ is a maximal compact subgroup of $G$ with Lie algebra $\mathfrak{k}$ and that $\mathfrak{g} = \mathfrak{k}\oplus \mathfrak{p}$. The inclusion $\textbf{i}\mathfrak{p}\hookrightarrow \mathfrak{u}$ induces by restriction, a $K$-equivariant map $\mu_{\textbf{i}\mathfrak{p}} : Z \to (\textbf{i}\mathfrak{p})^*.$ One can choose and fix an $\mathrm{Ad}(U^\mathbb{C})$-invariant inner product $B$ of Euclidian type on the Lie algebra $\mathfrak{u}^\mathbb{C}$, see [@BT Section 3.2], [@LT1 Definition 3.2.4] and also [@He Section 2.1] for the analog in the algebraic GIT. Such an inner product will automatically induce a well-defined inner product on any maximal compact subgroup $U'$ of $U^\mathbb{C}$. Let $\langle\, , \, \rangle$ denote the real part $B$. Then $\langle\, , \, \rangle$ is positive definite on $\textbf{i}\mathfrak{u}$, negative definite on $\mathfrak{u}$, $\langle \mathfrak{u}, \textbf{i} \mathfrak{u}\rangle=0$ and finally the multiplication by $\textbf{i}$ satisfies $\langle \textbf i \cdot ,\textbf{i} \cdot \rangle=-\langle\, , \, \rangle$. In order to simplify the notation we replace consideration of $\mu_{\text{i}\mathfrak{p}}$ by that of $\mu_\mathfrak{p}:Z \longrightarrow\mathfrak{p}$, where $$\mu_\mathfrak{p}^\beta (x):=\langle \mu_\mathfrak{p}(x),\beta \rangle:=\langle \textbf {i}\mu(x),\beta\rangle =-\langle \mu(x),-\textbf{i}\beta\rangle=\mu^{-\textbf{i} \beta}(x).$$ The map $\mu_\mathfrak{p}:Z \longrightarrow\mathfrak{p}$ is $K$-equivariant and grad$\, \mu_\mathfrak{p}^\beta = \beta_Z$ for any $\beta \in \mathfrak{p}$. Here the grad is computed with respect to the Riemannian metric induced by the Kähler structure. $\mu_\mathfrak{p}$ is called the $G$-gradient map associated with $\mu$. For a $G$-stable locally closed real submanifold $X$ of $Z,$ we consider $\mu_\mathfrak{p}$ as a map $\mu_\mathfrak{p}: X\to \mathfrak{p}$ such that $\mathrm{grad}\, \mu_\mathfrak{p}=\beta_X$, where the gradient is now computed with respect to the induced Riemannian metric on $X$. Different notions of stability of points in $X$ can be identified by taking into account the position of their $G$-orbits with respect to $\mu_\mathfrak{p}^{-1}(0).$ A point $x\in X$ is polystable if it's $G$-orbit intersects the level set $\mu_\mathfrak{p}^{-1}(0)$ ($G\cdot x \cap \mu_\mathfrak{p}^{-1}(0) \neq \emptyset$). As pointed out in the introduction in [@MUNDETT] (see also [@Stability]), a set of polystable points plays a critical role in the construction of a good quotient of $X$ by the action of $G.$ The aim of this article is to answer the first part of question 1.1 in [@MUNDETT] for actions of real Lie groups on real submanifolds of a Kähler manifold, generalizing [@MUNDETT]. Following [@MUNDETT], we require a mild technical restriction to be satisfied; namely, the fundamental vector field induced by the action grows at most linearly with respect to the distance function from a given base point. More precisely, **Assumption 1**. *$X$ is connected, and there exists a point $x_0\in X$ and a constant $C > 0$ such that for any $x\in X$ and any $\beta \in \mathfrak{p},$ $$\label{assumption} \parallel\beta_X(x)\parallel \leq C\parallel\beta\parallel(1 + d_X(x_0, x)),$$* *where $d_X$ denotes the geodesic distance between points of $X.$* $G$-action on $X$ when $X$ is compact, and a closed and a compatible subgroup $G \subset SL(n, \mathbb{R})$-action on $\mathbb{R}^n$ satisfy the assumption. Under this assumption, we construct a collection of maps $$\lambda_x: \partial_\infty (G/K) \to \mathbb{R}\cup \{\infty \}$$ called the maximal weight function. We then prove that a point $x\in X$ is polystable if and only if $\lambda_x \geq 0$ and for any $p\in \partial_\infty (G/K)$ such that $\lambda_x(p) = 0$ there exists $p'\in \partial_\infty (G/K)$ such that $p$ and $p'$ are connected in the sense of Definition [Definition 4](#deff){reference-type="ref" reference="deff"} (Theorem [Theorem 18](#main result){reference-type="ref" reference="main result"}.) The authors in [@Stability] gave a Hilbert-Mumford criterion for semistability and polystability under a certain completeness condition. However, the result of this article on polystability is much stronger as we did not assume any completeness condition. # Compatible subgroups and parabolic subgroups {#comp-subgrous} Let $H$ be a Lie group with Lie algebra $\mathfrak{h}$ and $E, F \subset \mathfrak{h}$. Then, we set $$\begin{gathered} E^F := \{\eta\in E: [\eta, \xi ] =0, \forall \xi \in F\} \\ H^F = \{g\in H: \operatorname{Ad}g (\xi ) = \xi, \forall \xi \in F\}. \end{gathered}$$ If $F=\{\beta\}$ we write simply $E^\beta$ and $H^\beta$. Let $U$ be a compact Lie group and let $U^\mathbb{C}$ be the corresponding complex linear algebraic group [@chevalley]. The group $U^\mathbb{C}$ is reductive and is the universal complexification of $U$ in the sense of [@ho]. On the Lie algebra level, we have the Cartan decomposition $\mathfrak{u}^\mathbb{C}=\mathfrak{u}\oplus \textbf{i} \mathfrak{u}$ with a corresponding Cartan involution $\theta:\mathfrak{u}^\mathbb{C}\longrightarrow\mathfrak{u}^\mathbb{C}$ given by $\xi+\textbf{i}\nu \mapsto \xi-\textbf{i}\nu$. We also denote by $\theta$ the corresponding involution on $U^\mathbb{C}$. The real analytic map $F:U\times \textbf{i}\mathfrak{u}\longrightarrow U^\mathbb{C}$, $(u,\xi) \mapsto u\exp(\xi)$ is a diffeomorphism. We refer to the composition $U^\mathbb{C}=U\exp(\textbf{i}\mathfrak{u})$ as the Cartan decomposition of $U^\mathbb{C}$. Let $G\subset U^\mathbb{C}$ be a closed real subgroup of $U^\mathbb{C}$. We say that $G$ is *compatible* with the Cartan decomposition of $U^\mathbb{C}$ if $F (K \times \mathfrak{p}) = G$ where $K:=G\cap U$ and $\mathfrak{p}:= \mathfrak{g}\cap \textbf{i}\mathfrak{u}$. The restriction of $F$ to $K\times \mathfrak{p}$ is then a diffeomorphism onto $G$. It follows that $K$ is a maximal compact subgroup of $G$ and that $\mathfrak{g}= \mathfrak{k}\oplus \mathfrak{p}$. Since $K$ is a retraction of $G$, it follows that $G$ has only finitely many connected components and $G=KG^o$, where $G^o$ denotes the connected component of $G$ containing $e$. Since $U$ can be embedded in $\operatorname{Gl}(N,\mathbb{C})$ for some $N$, and any such embedding induces a closed embedding of $U^\mathbb{C}$, any compatible subgroup is a closed linear group. By [@knapp-beyond Proposition 1.59 p.57], $\mathfrak{g}$ is a real reductive Lie algebra, and so $\mathfrak{g}= \mathfrak{z}(\mathfrak{g})\oplus [\mathfrak{g}, \mathfrak{g}]$, where $\mathfrak{z}(\mathfrak{g})=\{v\in \mathfrak{g}:\, [v,\mathfrak{g}]=0\}$ is the Lie algebra of the center of $G$. Denote by $G_{ss}$ the analytic subgroup tangent to $[\mathfrak{g}, \mathfrak{g}]$. Then $G_{ss}$ is closed and $G^o=Z(G^o)^o\cdot G_{ss}$ [@knapp-beyond p. 442], where $Z(G^o)^o$ denotes the connected component of the center of $G^o$ containing $e$. **Lemma 1** (). *$\ $* 1. *[\[lemcomp1\]]{#lemcomp1 label="lemcomp1"} If $G\subset U^\mathbb{C}$ is a compatible subgroup, and $H\subset G$ is closed and $\theta$-invariant, then $H$ is compatible if and only if $H$ has only finitely many connected components.* 2. *[\[lemcomp2\]]{#lemcomp2 label="lemcomp2"} If $G\subset U^\mathbb{C}$ is a connected compatible subgroup, then $G_{ss}$ is compatible.* 3. *[\[lemcomp3\]]{#lemcomp3 label="lemcomp3"} If $G\subset U^\mathbb{C}$ is a compatible subgroup and $E\subset \mathfrak{p}$ is any subset, then $G^E$ is compatible. Indeed, $G^E=K^E\exp(\mathfrak{p}^E)$, where $K^E=K\cap G^E$ and $\mathfrak{p}^E=\{x\in \mathfrak{p}:\, [x,E]=0\}$.* If $\beta \in \mathfrak{p}$ the endomorphism $\mathrm{ad}(\beta) \in\mathrm{End}(\mathfrak{g})$ is diagonalizable over $\mathbb{R}$. Denote by $V_\lambda (\mathrm{ad}(\beta))$ the eigenspace of $\mathrm{ad}(\beta)$ corresponding to the eigenvalue $\lambda$. We define, $$\begin{aligned} G^{\beta+} &:=\{g \in G : \lim_{t\to - \infty} \exp (t\beta)\, g \exp (-t\beta) \text { exists} \}\\ R^{\beta+} &:=\{g \in G : \lim_{t\to - \infty} \exp (t\beta)\, g \exp (-t\beta) =e \}\\ \mathfrak{r}^{\beta+}: = \bigoplus_{\lambda> 0} V_\lambda(\mathrm{ad}( \beta)) \\ G^{\beta-} &:\{g\in G:\, \lim_{t\to + \infty} \exp (t\beta)\, g \exp (-t\beta) \text { exists}\}\\ R^{\beta-} &:=\{g \in G : \lim_{t\to + \infty} \exp (t\beta)\, g \exp(-t\beta)=e \}\\ \mathfrak{r}^{\beta-}: = \bigoplus_{\lambda< 0} V_\lambda(\mathrm{ad}(\beta)).\end{aligned}$$ Note that $G^{\beta-}=G^{(-\beta)+}=\theta(G^{\beta+})$, $\theta(R^{\beta+})=R^{\beta-}$ and $G^{\beta}=G^{\beta+}\cap G^{\beta-}$. The group $G^{\beta+}$, respectively $G^{\beta-}$, is a parabolic subgroup of $G$ with unipotent radical $R^{\beta+}$, respectively $R^{\beta-}$ and Levi factor $G^{\beta}$. $R^{\beta+}$ is connected with Lie algebra $\mathfrak{r}^{\beta+}$. Hence $R^{\beta-}$ is connected with Lie algebra $\mathfrak{r}^{\beta-}$. The parabolic subgroup $G^{\beta+}$ is the semidirect product of $G^\beta$ with $R^{\beta+}$ and we have the projection $\pi^{\beta+} : G^{\beta+} \longrightarrow G^\beta$, $\pi^{\beta+}(g)=\lim_{t\to -\infty} exp(t\beta)g\exp(-t\beta).$ Analogously, $G^{\beta-}$ is the semidirect product of $G^{\beta}$ with $R^{\beta-}$ and we have the projection $\pi^{\beta-}:G^{\beta-} \longrightarrow G^\beta$, $\pi^{\beta-}(g)=\lim_{t\to +\infty} exp(t\beta)g\exp(-t\beta)$. In particular, $\mathfrak{g}^{\beta+}= \mathfrak{g}^\beta \oplus \mathfrak{r}^{\beta+}$, respectively $\mathfrak{g}^{\beta-}= \mathfrak{g}^\beta \oplus \mathfrak{r}^{\beta-}$. **Proposition 2**. *For any $\beta \in \mathfrak{p}$, we have $G=KG^{\beta+}$.* *Proof.* If $G$ is connected, the result is well-known, see for instance [@LA Lemma 9] and [@heinzner-schwarz-stoetzel Lemma 4.1]. Since $G=KG^o$, it follows that $G=KG^o=K(G^{o})^{\beta+}=KG^{\beta+},$ concluding the proof. ◻ # Symmetric space Let $U^\mathbb{C}$ be a complex reductive Lie group and let $G\subset U^\mathbb{C}$ be a closed compatible subgroup. Then $G = K\exp(\mathfrak{p}),$ where $K := G\cap U$ is a maximal compact subgroup of $G$ and $\mathfrak{p} := \mathfrak{g}\cap \text{i}\mathfrak{u};$ $\mathfrak{g}$ is the Lie algebra of $G.$ Let $M = G/K$ and let $\langle, \rangle$ be the real part of an $\operatorname{Ad}(U^\mathbb{C})$-invariant inner product $B$ of Euclidean type on the Lie algebra $\mathfrak{u}^\mathbb{C}.$ Equip $G$ with the unique left-invariant Riemannian metric which agrees with the scalar product $\langle\xi_1 + \beta_1, \xi_2 + \beta_2\rangle:= -\langle\xi_1, \xi_2\rangle+ \langle\beta_1, \beta_2\rangle$ where $\xi_1,\xi_2\in \mathfrak{k}$ and $\beta_1,\beta_2\in \mathfrak{p}$. We recall that $\langle \cdot,\cdot \rangle$ is negatively defined on $\mathfrak{k}$. This metric is $\mathrm{Ad}(K)$-invariant, and so, it induces a $G$-invariant Riemannian metric of nonpositive curvature on $M.$ $M$ is a symmetric space of non-compact type [@borel-ji-libro]. Let $\pi: G \rightarrow M$ be the projection onto the right cosets of $G$. $G$ acts isometrically on $M$ from left by $$L_g : M \to M; \quad L_g(hK) := ghK, \quad g,h \in G.$$ A geodesic $\gamma$ in $M$ is given by $\gamma = g\exp(t\beta)K,$ where $g\in G$ and $\beta \in \mathfrak{p}.$ For $\beta \in \mathfrak{p}$, we set $\gamma^\beta(t) = \exp(t\beta)K$ and $o := K \in M.$ Since $M$ is a Hadamard manifold there is a natural notion of a boundary at infinity which can be described using geodesics. We refer the reader to [@borel-ji-libro] for more details. Two unit speed geodesics $\gamma, \gamma': \mathbb{R}\to M$ are equivalent, denoted by $\gamma \sim \gamma'$, if $\sup_{t>0}d(\gamma(t), \gamma'(t)) < +\infty.$ **Definition 1**. *The *Tits boundary* of $M$ denoted by $\partial_\infty M$ is the set of equivalence classes of unit speed geodesics in $M.$* The map that sends $\beta \in \mathfrak{p}$ to the tangent vector $\dot{\gamma}^\beta(0)$ produces an isomorphism $\mathfrak{p}\cong T_oM.$ Since any geodesic ray in $M$ is equivalent to a unique ray starting from $o,$ the map $$\begin{aligned} e: &S(\mathfrak{p}) \to \partial_\infty M;\\ &e(\beta) := [\gamma^\beta]\end{aligned}$$ where $S(\mathfrak{p}) := \{\beta \in \mathfrak{p}: |\beta| = 1\}$ is the unit sphere in $\mathfrak{p},$ is a bijection. The *sphere topology* is the topology on $\partial_\infty M$ such that $e$ is a homomorphism. Since $G$ acts by isometries on $M,$ then for every unit speed geodesic $\gamma,$ $g\gamma$ is also a unit speed geodesic for any $g\in G.$ Moreover, if $\gamma \sim \gamma'$ then $g\gamma \sim g\gamma'.$ There is a $G$-action on $\partial_\infty M$ given by: $$g\cdot [\gamma] = [g\cdot \gamma]$$ and this action also induces a $G$-action on $S(\mathfrak{p})$ given by $$g\cdot \beta := e^{-1}(g\cdot e(\beta)) = e^{-1}[g\cdot \gamma^\beta].$$ We point out that the $K$-action to $\partial_\infty M$ induces the adjoint action of $K$ on $S(\mathfrak{p}).$ Let $H$ be a compatible subgroup of $G,$ i.e $H := L \exp(\mathfrak{q}),$ where $L := H \cap K$ and $\mathfrak{q}= \mathfrak{h}\cap \mathfrak{p}$, where $\mathfrak{h}$ is the Lie algebra of $H.$ It follows that $H$ is a real reductive subgroup of $G.$ The Cartan involution of $G$ induces a Cartan involution of $H,$ $L$ is a maximal compact subgroup of $H,$ and $\mathfrak{h}= \mathfrak{l}\oplus \mathfrak{h}.$ There are totally geodesic inclusions $M' := H/L \hookrightarrow M$ and $\partial M' \hookrightarrow \partial M.$ ## The Kempf-Ness Function {#Kempf-Ness Function} Given $G$ a real reductive group which acts smoothly on $Z;$ $G = K\text{exp}(\mathfrak{p}),$ where $K$ is a maximal compact subgroup of $G.$ Let $X$ be a $G$-invariant locally closed submanifold of $Z.$ As Mundet pointed out in [@MUNDET], there exists a function $\Phi : X \times G \rightarrow \mathbb{R},$ such that $$\langle \mu_\mathfrak{p}(x), \xi\rangle = \dfrac {\mathrm {d} }{\mathrm {dt}}\bigg \vert _{t=0} \Phi (x, \exp(t\xi)), \qquad \xi \in \mathfrak{p},$$ and satisfying the following conditions: 1. For any $x\in X,$ the function $\Phi (x, .)$ is smooth on $G.$ 2. The function $\Phi(x, .)$ is left-invariant with respect to $K,$ i.e., $\Phi(x, kg) = \Phi(x, g).$ 3. For any $x\in X,$ $v\in \mathfrak{p}$ and $t\in \mathbb{R};$ $$\frac{d^2}{dt^2}\Phi (x, \exp(tv)) \geq 0.$$ Moreover: $$\frac{d^2}{dt^2}\Phi (x, \exp(tv)) = 0$$ if and only if $\exp(\mathbb{R}v)\subset G_x.$ 4. For any $x\in X,$ and any $g, h \in G;$ $$\Phi(x, hg) = \Phi(x, g) + \Phi(gx, h).$$ This equation is called the cocycle condition. Finally, using the cocycle condition, we have $$\dfrac {\mathrm {d} }{\mathrm {dt}}\Phi(x,\exp(t\beta))=\langle \mu_\mathfrak{p}(\exp(t\xi)x),\beta\rangle.$$ The function $\Phi: X \times G \rightarrow \mathbb{R}$ is called the Kempf-Ness function for $(X, G, K)$. It is just the restriction of the classical Kempf-Ness function $Z\times U^\mathbb{C}\longrightarrow\mathbb{R}$ considered in [@MUNDET; @Teleman] to $X\times G$ [@LM]. Moreover, if $H\subset G$ is compatile and $Y\subset X$ is a $H$-stable submanifold of $X$, then the restriction $\Phi_{\vert{Y\times H}}$ is the Kempf-Ness function of the $H$-gradient map on $Y$. Let $x\in X$. By property $(b),$ i.e. $\Phi(x, kg) = \Phi(x, g)$, the function $\Phi_x : G \to \mathbb{R}$ given by $\Phi_x(g) := \Phi(x, g^{-1})$ descends to a function on $M$ which we denote by the same symbol. That is $$\Phi_x : M \longrightarrow\mathbb{R}; \qquad \Phi_x (gK):=\Phi(x,g^{-1}).$$ The cocycle condition $(d)$ can be rewritten as $$\label{cocycle} \Phi_x(ghK) = \Phi_{g^{-1}x}(hK) + \Phi_x(gK),$$ and it is equivalent to $L_g^{*}\Phi_x = \Phi_{g^{-1}x} + \Phi_{g^{-1}x}(gK),$ where $L_g$ denotes the action of $G$ on $X$ given above. Note that $$-(d\Phi_x)_o(\dot{\gamma}^\beta(0)) = \dfrac {\mathrm {d} }{\mathrm {dt}}\bigg \vert _{t=0} \Phi_x(\exp(-t\beta)K) = \dfrac {\mathrm {d} }{\mathrm {dt}}\bigg \vert _{t=0} \Phi(x, \exp(t\beta)) = \langle \mu_\mathfrak{p}(x), \beta\rangle.$$ **Lemma 3**. *Let $x\in X$ and let $\Phi_x : M \to \mathbb{R}.$ Suppose $\gamma(t) = g\exp(t\beta)K$ for $\beta \in \mathfrak{p}$ is a geodesic in $M,$ then $\Phi_x \circ \gamma$ is convex and so, $$\lim_{t\to \infty}\frac{d}{dt}(\Phi_x\circ \gamma) = \lim_{t\to \infty}\frac{\Phi_x\circ \gamma}{t}$$.* *Proof.* That $\Phi_x$ is a convex function on $M$ follows from [@bgs Lemma 2.19]. Let $f(t) = (\Phi_x\circ \gamma)(t).$ Since $f$ is convex, $$\frac{f(s)}{s}\leq f'(s) \leq \frac{f(t) - f(s)}{t-s} \quad 0 < s < t.$$ Furthermore, the two quantities are increasing in $s,$ while the third in $t.$ Hence, $$\lim_{s\to \infty}\frac{f(s)}{s}\leq \lim_{s \to \infty} f'(s) \leq \lim_{t\to \infty} \frac{f(t) - f(s)}{t-s}.$$ Since the last limit is the same with the first limit, we have $\lim_{t \to \infty} f'(t) = \lim_{t\to \infty}\frac{f(t)}{t}$ and the result follows. ◻ # Stability and Maximal Weight Function {#analytic} Let $(Z,\omega)$ be a Kähler manifold and $U^\mathbb{C}$ acts holomorphically on $Z$ with a momentum map $\mu : Z \to \mathfrak{u}.$ Let $G\subset U^\mathbb{C}$ be a closed compatible subgroup. $G = K\exp(\mathfrak{p}),$ where $K := G\cap U$ is a maximal compact subgroup of $G$ and $\mathfrak{p} := \mathfrak{g}\cap \text{i}\mathfrak{u};$ $\mathfrak{g}$ is the Lie algebra of $G.$ Suppose $X\subset Z$ is a $G$-stable locally closed connected real submanifold of $Z$ with the gradient map $\mu_\mathfrak{p} : X\to \mathfrak{p}.$ Let $d_X$ denote the geodesic distance between points of $X.$ We recall that by $G_x$ and $K_x$, we denote the stabilizer subgroup of $x\in X$ with respect to the $G$-action and the $K$-action respectively and by $\mathfrak{g}_x$ and $\mathfrak{k}_x$ their respective Lie algebras. **Definition 2**. *Let $x\in X.$ Then:* 1. *$x$ is stable if $G\cdot x \cap \mu_\mathfrak{p}^{-1}(0) \neq \emptyset$ and $\mathfrak{g}_x$ is conjugate to a Lie subalgebra of $\mathfrak{k}.$* 2. *$x$ is polystable if $G\cdot x \cap \mu_\mathfrak{p}^{-1}(0) \neq \emptyset.$* 3. *$x$ is semistable if $\overline{G\cdot x} \cap \mu_\mathfrak{p}^{-1}(0) \neq \emptyset.$* We denote by $X^s_{\mu_\mathfrak{p}}$, $X^{ss}_{\mu_\mathfrak{p}}$, $X^{ps}_{\mu_\mathfrak{p}}$ the set of stable, respectively semistable, polystable, points. It follows directly from the definitions above that the conditions are $G$-invariant in the sense that if a point satisfies one of the conditions, then every point in its orbit satisfies the same condition, and for stability, recall that $\mathfrak{g}_{gx} = \text{Ad}(g)(\mathfrak{g}_x).$ The following well-known result establishes a relation between the Kempf-Ness function and the polystability condition. A proof is given in [@Stability]. **Proposition 4**. *Let $x\in X$ and let $g\in G$. The following conditions are equivalent:* 1. *$\mu_\mathfrak{p}(gx)=0$.* 2. *$g$ is a critical point of $\Phi(x,\cdot)$.* 3. *$g^{-1}K$ is a critical point of $\Phi_x$.* **Proposition 5**. *Let $x\in X$.* - *If $x$ is polystable, then $G_x$ is compatible.* - *If $x$ is stable, then $G_x$ is compact.* ### Maximal Weight Function In this section, we introduce the maximal weight function associated with an element $x\in X.$ For any $t\in \mathbb{R},$ define $\lambda(x,\beta,t) = \langle\mu_\mathfrak{p}(\exp(t\beta)x), \beta\rangle.$ $$\label{form1} \lambda(x,\beta,t) = \langle\mu_\mathfrak{p}(\exp(t\beta)x), \beta\rangle = \frac{d}{dt}\Phi(x, \exp(t\beta)),$$ where $\Phi: X\times G\to \mathbb{R}$ is the Kempf-Ness function. By the properties of the Kempf-Ness function, $$\frac{d}{dt}\lambda(x,\beta,t) = \frac{d^2}{dt^2}\Phi(x, \exp(t\beta)) \geq 0.$$ This means that $\lambda(x,\beta,t)$ is a non decreasing function as a function of $t.$ The maximal weight of $x\in X$ in the direction of $\beta \in \mathfrak{p}$ is defined in [@Stability] as the numerical value $$\lambda(x,\beta) = \lim_{t\to \infty}\lambda(x,\beta,t)=\lim_{t\to +\infty} \langle\mu_\mathfrak{p}(\exp(t\beta)x), \beta\rangle \in \mathbb{R}\cup\{\infty\}.$$ Note that $$\frac{d}{dt}\lambda(x,\beta,t) = \parallel\beta_X(\exp(t\beta) x)\parallel^2,$$ and so, $$\label{Equu} \lambda(x,\beta,t) = \langle\mu_\mathfrak{p}(x), \beta\rangle + \int_0^t\parallel\beta_X(\exp(s\beta) x)\parallel^2 \mathrm{ds}.$$ **Lemma 6**. *Let $\beta,\beta'\in \mathfrak{p}$ and let $x\in X$. If $\beta \in \mathfrak{g}_x$ and $[\beta,\beta']=0$, then $$\lim_{t\to +\infty} \frac{d}{dt}\Phi(x, \exp(t(\beta+\beta'))=\lim_{t\to +\infty} \frac{d}{dt}\Phi(x, \exp(t\beta))+\frac{d}{dt}\Phi(x, \exp(t\beta')),$$* *Proof.* By the cocycle condition, keeping in mind that $[\beta,\beta']=0$, we have $$\Phi(x, \exp(t(\beta+\beta'))=\Phi(x,\exp(t\beta'))+\Phi(\exp(t\beta)x,\exp(t\beta'))= \Phi(x,\exp(t\beta'))+\Phi(x,\exp(t\beta')),$$ and so the result follows. ◻ **Lemma 7**. *Let $g\in G$ and $\beta \in \mathfrak{p}.$ If $\frac{\Phi_x(\exp(t\beta)g)}{t}$ is bounded uniformly on $t,$ then $$\lim_{t \to \infty}\frac{d_X(\exp(t\beta)gx, x)}{t^{\frac{1}{2}}} = 0.$$* *Proof.* By the property (d) of the Kempf-Ness function and noting that $\lim_{t \to \infty}\frac{\Phi_x(g)}{t} = 0$ since $\Phi_x(g)$ is independent of $t$, we have $$\begin{aligned} \lim_{t \to \infty}\frac{\Phi_x(\exp(t\beta)g)}{t} &= \lim_{t \to \infty}\frac{\Phi_x(g) + \Phi_{gx}(\exp(t\beta))}{t}\\ &=\lim_{t\to \infty}\frac{\Phi_{gx}(\exp(t\beta))}{t} \\ &= \lambda(gx, \beta).\end{aligned}$$ If $\frac{\Phi_x(\exp(t\beta)g)}{t}$ is bounded uniformly on $t,$ then by ([\[Equu\]](#Equu){reference-type="ref" reference="Equu"}), $\int_0^\infty\parallel \beta_X(\exp(s\beta)gx)\parallel^2 ds < \infty.$ On the other hand, $$d_X(\exp(t\beta)gx, gx) \leq \int_0^t \parallel \beta_X(\exp(s\beta)gx)\parallel ds.$$ Let $f(s) = \parallel \beta_X(\exp(s\beta)gx)\parallel,$ applying [@MUNDETT Lemma 3.2] to $f(s)$ implies that $$\lim_{t\to \infty}\frac{d_X(\exp(t\beta)gx, gx)}{t^{\frac{1}{2}}} = 0,$$ and by triangle inequality, $$\lim_{t\to \infty}\frac{d_X(\exp(t\beta)gx, x)}{t^{\frac{1}{2}}} = 0.$$ ◻ For $x\in X,$ let the function $\lambda_x$ be given as $$\label{mmfunction} \lambda_x : \partial_\infty M \to \mathbb{R}; \quad \lambda_x([\gamma]) := \lim_{t\to \infty}\frac{d}{dt}\Phi_x(\gamma(t)) = \lim_{t\to \infty}\frac{\Phi_x\circ \gamma}{t},$$ where the last equality follows from Lemma [Lemma 3](#ConvexKempfness){reference-type="ref" reference="ConvexKempfness"}. We need to show that the function $\lambda_x$ is well defined using the technical condition in assumption [Assumption 1](#assumption1){reference-type="ref" reference="assumption1"}. We first derive a quadratic bound on the gradient map. **Lemma 8**. *Let $x_0\in X$ be the point satisfying ([\[assumption\]](#assumption){reference-type="ref" reference="assumption"}). There exists a constant $C_1$ such that for any $x\in X$ we have $$\parallel\mu_\mathfrak{p}(x)\parallel \leq C_1(1 + d_X(x_0, x)^2).$$* *Proof.* For any $x\in X$, let $d = d_X(x_0, x).$ By ([\[assumption\]](#assumption){reference-type="ref" reference="assumption"}), there exists a constant $C > 0$ such that for any $x\in X$ and any $\beta \in \mathfrak{p},$ $\parallel\beta_X(x)\parallel \leq C \parallel\beta\parallel(1+d).$ Let $\alpha : [0, 2d] \to X$ be a smooth curve parametrized by the arc length and such that $\alpha(0) = x_0$ and $\alpha(2d) = x.$ For any $\beta \in \mathfrak{p},$ let $f(t) := \langle \mu_\mathfrak{p}(\alpha(t)), \beta\rangle.$ Then by the defining properties of gradient map and since $\parallel\Dot{\alpha}\parallel = 1$ $$\begin{aligned} f'(t) &= \langle d\mu_\mathfrak{p}(\alpha(t))\Dot{\alpha}, \beta\rangle\\ &= \langle \beta_X(\alpha(t)), \Dot{\alpha}\rangle\\ &\leq \parallel\beta_X(\alpha(t))\parallel\\ & \leq C \parallel\beta\parallel(1+d_X(x_0, \alpha(t)).\end{aligned}$$ Hence, $$\begin{aligned} f(2d) - f(0) &= \int_0^{2d} f'(t)dt \leq \int_0^{2d} C \parallel\beta\parallel(1+d_X(x_0, \alpha(t))dt\\ & \leq \int_0^{2d} C \parallel\beta\parallel(1+t)dt\\ & = 2C\parallel\beta\parallel(d + d^2)\\ &\leq C'\parallel\beta\parallel(1 + d^2),\end{aligned}$$ where the constant $C'$ can be chosen independent of $x.$ Hence the result follows. ◻ **Proposition 9**. *The function $\lambda_x : \partial_\infty M \to \mathbb{R}$ defined by ([\[mmfunction\]](#mmfunction){reference-type="ref" reference="mmfunction"}) is well defined.* *Proof.* The function $\lambda_x$ is well defined means that if $\gamma_1$ and $\gamma_2$ are geodesic rays such that $\gamma_1 \sim \gamma_2,$ then $\lambda_x(\gamma_1) = \lambda_x(\gamma_2).$ Let $\gamma_i(t) = [g_i\exp(t\beta_i)]$ for $i = 1,2.$ Suppose $\lambda(\gamma_1) < \infty.$ $$\lim_{t\to \infty}\frac{\Phi_x\circ \gamma_1}{t} = \lim_{t\to \infty}\frac{\Phi_x(g_1\exp(t\beta_1)K)}{t} = \lim_{t\to \infty}\frac{\Phi_x(\exp(-t\beta_1)g^{-1}_1)}{t} < \infty.$$ Lemma [Lemma 7](#mylemma){reference-type="ref" reference="mylemma"} implies that $$\lim_{t\to \infty}\frac{d_X(\exp(-t\beta_1)g^{-1}_1x, x)}{t^{\frac{1}{2}}} = 0.$$ Since $\gamma_1 \sim \gamma_2,$ then $d(\gamma_1(t), \gamma_2(t)) \leq Q$ uniformly on $t.$ Hence, for any $t$ one can consider a smooth map $\varphi_t : [1, 2] \to G$ such that $\varphi_t(i) = \exp(-t\beta_i)g_i^{-1}$ for $i = 1,2$ and $$\label{smooth map} \int_1^2 \parallel\varphi'_t(r) \varphi_t(r)^{-1}\parallel dr \leq Q.$$ Note that $Q$ is independent of $t.$ We claim that there exists some constant $C'$ (depending on $x$) such that, for any $t$ and any $r\in [1,2],$ $$d_X(\exp(-t\beta_1)g_1^{-1}x, \varphi_t(r)x) \leq C'(1 + d_X(\exp(-t\beta_1)g_1^{-1}x, x)).$$ Indeed, fix $t$ and let $c(u) = \varphi_t(u)x$ and $\xi(u) = \parallel \varphi'_t(u) \varphi_t(u)^{-1}\parallel.$ Define $f(s) = d_X (c(1), c(s)).$ By assumption [Assumption 1](#assumption1){reference-type="ref" reference="assumption1"}, $$\begin{aligned} \parallel f'(s) \parallel \leq \parallel c'(s) \parallel &= \parallel (\varphi'_t(s) \varphi_t(s)^{-1})_X(c(s)) \parallel\\ &\leq C \parallel \varphi'_t(s) \varphi_t(s)^{-1} \parallel (1 + d_X(x_0, c(s)))\\ &\leq C\xi(s)(1 + d_X(x_0, x) + d_X(x, c(1)) + d_X(c(1), c(s)))\\ & = C\xi(s)(1 + d_X(x_0, x) + d_X(x, c(1)) + f(s)).\end{aligned}$$ If $g(s) = 1 + d_X(x_0, x) + d_X(x, c(1)) + f(s),$ then $g'(s) = f'(s)$ and $\parallel g'(s)/g(s)\parallel \leq C\xi(s).$ Integrating over $s\in [1, r],$ implies that there is a uniform bound $g(r) \leq CQg(1).$ Note that $d_X(x, c(1)) = d_X(x, \exp(-t\beta_1)g_1^{-1}x).$ The claim follows. Using triangular inequality and by the claim, we have $$\begin{aligned} d_X(\varphi_t(r)x, x) &\leq d_X(\varphi_t(r)x, \exp(-t\beta_1)g_1^{-1}x) + d_X(\exp(-t\beta_1)g_1^{-1}x, x)\\ &= C'(1 + d_X(\exp(-t\beta_1)g_1^{-1}x, x)) + d_X(\exp(-t\beta_1)g_1^{-1}x, x).\end{aligned}$$ So that $$\lim_{t\to \infty}\frac{d_X(\varphi_t(r)x, x)}{t^{\frac{1}{2}}} = 0$$ and by triangular inequality again, $$\label{distance} \lim_{t\to \infty}\frac{d_X(\varphi_t(r)x, x_0)}{t^{\frac{1}{2}}} = 0.$$ By Lemma [Lemma 8](#quadratic bound){reference-type="ref" reference="quadratic bound"}, $$\parallel\mu_\mathfrak{p}(\varphi_t(r)x)\parallel \leq C_1(1 + d_X(x_0, \varphi_t(r)x)^2),$$ and by ([\[distance\]](#distance){reference-type="ref" reference="distance"}), $$\lim_{t\to 0}\frac{\parallel\mu_\mathfrak{p}(\varphi_t(r)x)\parallel}{t} = 0.$$ This implies that $$\begin{aligned} \lim_{t\to \infty}\frac{\parallel \Phi_x(\gamma_1) - \Phi_x(\gamma_2)\parallel}{t} = 0\end{aligned}$$ Therefore, $\lambda_x(\gamma_1) = \lambda_x(\gamma_2).$ If $\lambda_x(\gamma_1) = \infty,$ then $\lambda_x(\gamma_2) = \infty.$ Otherwise, reversing the roles of $\gamma_1$ and $\gamma_2$ in the arguments above shows that $\lambda_x(\gamma_1) = \lambda_x(\gamma_2) < \infty,$ which is a contradiction. Concluding the proof. ◻ **Definition 3**. *The maximal weight function of $x\in X$ is the well defined map $\lambda_x : \partial_\infty M \to \mathbb{R}\cup \{\infty\}$ given as $$\lambda_x(p) := \lambda_x(\gamma) = \lim_{t\to \infty}\frac{d}{dt}\Phi_x(\gamma(t))$$ for any $p\in \partial_\infty M,$ where $\gamma$ is any geodesic ray representing $p.$* It follows from the definition that for any $\beta \in S(\mathfrak{p})$, keeping in mind formula $(\ref{form1})$, we have $$\label{formula} \lambda_x(e(\beta)) = \lim_{t\to \infty}\frac{d}{dt}\Phi_x(\exp(t\beta)K) = \lim_{t\to \infty}\frac{d}{dt}\Phi(x, \exp(-t\beta))=\lim_{t\to +\infty} \langle \mu_\mathfrak{p}(\exp(t\beta)x,-\beta \rangle.$$ **Lemma 10**. *Let $\beta \in \mathfrak{p}-\{0\}$ and let $v=\frac{\beta}{\parallel \beta \parallel}$. Then $$\lambda_x (e(v))= \frac{1}{\parallel \beta \parallel} \lim_{t\to +\infty} \frac{d}{dt}\Phi(x, \exp(-t\beta)).$$* *Proof.* $\Phi_x (\exp(tv))=\Phi_x (\exp \big(\frac{t}{\parallel \beta \parallel} \beta \big))$. Then $$\lambda_x (e(\beta))=\frac{1}{\parallel \beta \parallel} \lim_{t\to +\infty} \frac{d}{dt}\Phi(x, \exp(-t\beta)).$$ ◻ **Lemma 11**. *For any $x\in X,$ any $g\in G$ and any $p\in \partial_\infty M,$ $$\label{Ginvariant} \lambda_{g^{-1}x}(p) = \lambda_x(g\cdot p).$$* *Proof.* Let $p = [\gamma]$ for some geodesic $\gamma \in M.$ Then $g\cdot p = [g\circ \gamma].$ By the cocycle condition, $$\begin{aligned} \lambda_{g^{-1}x}(p) &= \lim_{t\to \infty}\frac{d}{dt} (\Phi_{g^{-1}x}(\gamma(t))\\ &= \lim_{t\to \infty}\frac{d}{dt} (\Phi_{x}(g \cdot \gamma(t)) - \Phi_{x}(gK))\\ &= \lim_{t\to \infty}\frac{d}{dt} (\Phi_{x}(g \cdot \gamma(t))) = \lambda_x(g\cdot p).\end{aligned}$$ ◻ The following lemma will be needed. A proof is given in [@Stability Lemma 3.5, p. 92], see also [@Teleman]. **Lemma 12**. *Let $V$ be a subspace of $\mathfrak{p}$. The following are equivalent for a point $x\in X$:* 1. *the map $\Phi(x,\cdot)$ is linearly properly on $V$, i.e., there exists positive constants $C_1$ and $C_2$ such that $$\parallel v \parallel \leq C_1 \Phi(x,\exp(v))+C_2,\, \forall v\in V.$$* 2. *$\lambda(x,\beta)>0$ for every $\beta\in S(V)$.* The following theorem gives a numerical criterion for stable points in terms of maximal weights. The proof is the same as the proof of [@Stability Theorem 3.7]. **Theorem 13**. *Let $x\in X$. Then $x$ is stable if and only if $\lambda_x > 0$ on $\partial_\infty M.$* **Definition 4**. *We say that $p, q \in \partial_\infty M$ are connected if there exists a geodesic $\alpha$ in $X$ such that $p = \alpha(\infty)$ and $q = \alpha(-\infty).$* For any $x\in X,$ as in [@MUNDETT], see also [@bgs], let $Z(x) := \{p\in \partial_\infty M : \lambda_x(p) = 0\}.$ **Lemma 14**. *Let $x\in X$ be such that $\mu_\mathfrak{p}(x) = 0,$ then $\mathfrak{g}_x = \mathfrak{k}_x \oplus \mathfrak{p}_x$ and $Z(x) = e(S(\mathfrak{p}_x)) = \partial_\infty G_x/K_x.$* *Proof.* By Proposition [Proposition 5](#propp){reference-type="ref" reference="propp"} if $\mu_\mathfrak{p}(x) = 0,$ $G_x$ is a compatible subgroup of $G.$ Hence, $\mathfrak{g}_x = \mathfrak{k}_x \oplus \mathfrak{p}_x.$ To prove the second assertion, let $\beta \in S(\mathfrak{p}).$ Suppose $e(\beta) \in Z(x).$ This means that $\lambda_x(e(\beta)) = 0,$ then the convex function $f(t) := \Phi_x(\exp(t\beta)K)$ satisfies $$f'(\infty) = \lim_{t\to \infty}\frac{d}{dt}\Phi_x(\exp(t\beta)K) = \lambda_x(e(\beta)) = 0$$ and $$f'(0) = \frac{d}{dt}|_{t= 0}\Phi_x(\exp(t\beta)K) = \langle \mu_\mathfrak{p}(x), -\beta \rangle = 0.$$ These imply that $f$ is constant for all $t> 0,$ and by the condition (c) of Kempf-Ness function, $\exp(\mathbb{R}\beta)\subset G_x.$ Since $G_x$ is compatible, $\beta \in S(\mathfrak{p}_x).$ Conversely, if $\beta \in S(\mathfrak{p}_x),$ then $f$ is linear. Moreover, $f'(0) = 0.$ Therefore, $f \equiv 0$ and $e(\beta) \in Z(x).$ ◻ Let $x\in X$ and $\beta \in S(\mathfrak{p}).$ Since $\mathfrak{p}\subset i\mathfrak{u},$ then $i\beta \in \mathfrak{u}.$ We define the torus $T_\beta$ given as $$T_\beta := \overline{\{\exp(ti\beta) : t\in \mathbb{R}\}} \subseteq U^o,$$ where $U^o$ denotes the connected component of $U$ containing the identity. **Lemma 15**. *Let $g\in G.$ Then $\dim T_\beta = \dim T_{g\cdot \beta}.$* *Proof.* It is well-known that $G^{\beta+}$ fixes $e(\beta)$, see for instance [@Eberlein1997GeometryON Proposition 2.17.3, p.102]. Then for any $g\in G$, write $g = kh,$ where $k\in K$ and $h\in G^{\beta+},$ so that $g\cdot \beta=kh\cdot \beta=k\cdot \beta=\mathrm{Ad}(k)(\beta)$. Hence $$\begin{aligned} T_{g\cdot \beta} &= \overline{\{\exp(it\mathrm{Ad}(k)\beta) : t\in \mathbb{R}\}}\\ &= \overline{\{k\exp(it\beta)k^{-1} : t\in \mathbb{R}\}}\\ &= kT_\beta k^{-1}. \end{aligned}$$ Hence, $\dim T_\beta = \dim T_{g\cdot \beta}.$ ◻ **Lemma 16**. *Let $x\in X$ and $p, p'\in Z(x)$ be connected. Then there exists $g\in G$ and $\xi\in S(p)$ such that $\xi\in \mathfrak{p}_y$, where $y = gx.$* *Proof.* Since $p, p'\in Z(x)$ are connected, then there exists geodesic $\alpha \in M$ such that $\alpha(+\infty) = p\in Z(x)$ and $\alpha(-\infty) = p' \in Z(x).$ Assume $\alpha(t) = g\exp(t\xi)K,$ $g\in G.$ Then $p = g\cdot e(\xi)$ and $p' = g\cdot e(-\xi).$ By ([\[Ginvariant\]](#Ginvariant){reference-type="ref" reference="Ginvariant"}), $$\lambda_{g^{-1}x}(e(\xi)) = \lambda_x(g\cdot e(\xi)) = \lambda_x(p) = 0$$ and $$\lambda_{g^{-1}x}(e(-\xi)) = \lambda_x(g\cdot e(-\xi)) = \lambda_x(p') = 0.$$ Let $y = g^{-1}x.$ This means that the convex function $t \mapsto \Phi_y(\exp(t\xi)K)$ has zero derivatives at both $+\infty$ and $-\infty,$ and so, it is constant and by property (c) of Kempf-Ness function, $\exp(\mathbb{R}\xi) \subset G_y$, $\xi \in \mathfrak{p}_y.$ ◻ Let $X^\beta := \{z\in X : \beta_X(z) = 0\}.$ $G^\beta$ preserves $X^\beta$ [@Stability Prop. 2.9] and $X^\beta$ is the disjoint union of closed submanifold of $X$ [@heinzner-schwarz-stoetzel]. The following result is proved in [@Stability Proposition 2.10,p.92] **Proposition 17**. *The restriction $(\mu_\mathfrak{p})_{\vert_{X^\beta}}$ takes value on $\mathfrak{p}^\beta$ and so it coincides with the $G^\beta$-gradient map $(\mu_{\mathfrak{p}^\beta})_{\vert_{X^\beta}}$.* **Corollary 1**. *If $x\in X^\beta$ is $G^\beta$-polystable, then $x$ is $G$-polystable.* **Theorem 18**. *A point $x\in X$ is polystable if and only if $\lambda_x \geq 0$ and for any $p\in Z(x)$ there exists $p'\in Z(x)$ such that $p$ and $p'$ are connected.* *Proof.* Let $x\in X.$ If $Z(x) = \emptyset$, $\lambda_x > 0$ and by Theorem [Theorem 13](#stable){reference-type="ref" reference="stable"}, $x$ is stable and hence polystable. Suppose $Z(x) \neq \emptyset.$ Let $p\in Z(x).$ Let $\beta \in S(\mathfrak{p})$ such that $p = e(\beta).$ suppose $p\in Z(x)$ is chosen such that the of the torus $T_\beta$ satisfies $$\dim T_\beta = \max_{\eta \in e^{-1}(Z(x))} \dim T_\eta.$$ By assumption there is a geodesic $\alpha \in M$ such that $\alpha(+\infty) = p\in Z(x)$ and $\alpha(-\infty) = p' \in Z(x).$ Assume $\alpha(t) = g\exp(t\xi)K,$ $g\in G.$ Then $p = g\cdot e(\xi)$ and $p' = g\cdot e(-\xi).$ By Lemma [Lemma 16](#limit){reference-type="ref" reference="limit"}, $\xi \in \mathfrak{p}_y$ where $y = g^{-1}x.$ Moreover, since $e(\beta) = p = g\cdot e(\xi),$ using Lemma [Lemma 15](#torus){reference-type="ref" reference="torus"}, $$\dim T_\xi = \dim T_\beta = \max_{\eta \in e^{-1}(Z(x))} \dim T_\eta.$$ Let $\mathfrak t_\xi$ be the Lie algebra of $T_\xi$. Then $\mathfrak a=i \mathfrak t_\xi \cap \mathfrak{p}^\xi$ is an Abelian subalgebra of $\mathfrak{p}^\xi$ different from zero since $\beta \in \mathfrak{a}$. Since $T_\xi =\overline{\exp(i\mathbb{R}\xi)}$ fixes $y$ it follows that $\mathfrak{a}\subseteq \mathfrak{g}_y$ Let $Y$ be the connected component of $X^{\mathfrak{a}}$ containing $y$. By Lemma [Lemma 1](#lemcomp){reference-type="ref" reference="lemcomp"}, $(G^\mathfrak{a})^o=(K^\mathfrak{a})^o \exp(\mathfrak{p}^\mathfrak{a})$ is compatible and it preserves $Y$. By Proposition [Proposition 17](#restric){reference-type="ref" reference="restric"} we get $(\mu_\mathfrak{p})|_{Y}=\mu_{\mathfrak{p}^\mathfrak{a}}$. Hence, if $y$ is $(G^\mathfrak{a})^o$-polystable, then it is $G$-polystable. We split $\mathfrak{p}^\mathfrak{a}= \text{span}(\mathfrak{a}) \oplus \mathfrak{p}^{'}$, where $\mathfrak{p}^{'}$ is the orthogonal of $\mathfrak{a}$ and so it is a $K^\mathfrak{a}$-invariant splitting. . Claim: $\lambda_y(e(\beta')) > 0$ for all $\beta' \in S(\mathfrak{p}').$ Indeed, we prove this claim by contradiction. Suppose there exists $\beta' \in S(\mathfrak{p}')$ such that $\lambda_y(e(\beta')) = 0.$ Hence $[\xi, \beta'] = 0$ by the choice of $\xi$ and $\beta',$ and they are linearly independent. Let $a > 0.$ Since $[\xi,\beta']=0$ and $\xi \in \mathfrak{g}_y$, by Lemma [Lemma 6](#le1){reference-type="ref" reference="le1"} it follows $$\lim_{t\to+\infty} \Phi(y,\exp(t(\xi+a\beta'))=\lim_{t\to+\infty} \Phi(y,\exp(t(\xi))+a \lim_{t\to+\infty} \Phi(y,\exp(t\beta')).$$ Since $\lambda_y(e(\xi)) = \lambda_y((\beta')) = 0,$ by Lemma [Lemma 10](#le3){reference-type="ref" reference="le3"}, it follows that $$\lim_{t\to+\infty} \Phi(y,\exp(t(\xi+a\beta'))=0.$$ Applying, again, Lemma [Lemma 10](#le3){reference-type="ref" reference="le3"}, we have $$\lambda_y(e(\frac{\xi + a\beta'}{\parallel \xi + a\beta'\parallel}))= 0,$$ and so the vector $$\frac{\xi + a\beta'}{\parallel \xi + a\beta'\parallel}$$ belongs to $e^{-1}(Z(y)).$ We claim that for some $a>0$, $\dim T_{\xi+a\beta'}>\dim T_\xi$. Let $T'= \overline{\exp(\mathbb{R}i\xi + \mathbb{R}i\beta')}\subseteq (U^\xi)^o$ and $T_{\beta'}=\overline{\exp(\mathbb{R}i \beta')}$. Let $U'\subseteq ( U^\xi)^o$ be a compact connected subgroup such that the morphism $$T_\xi \times U' \to (U^\xi)^o, (a,b) \mapsto (ab),$$ is surjective with finite center. Since $\beta' \notin \mathfrak{a}$, it follows that $i\beta \notin \mathfrak t_\xi$. Hence, $T_{\beta'}\subseteq U'$ and the morphism $$f : T_\xi \times T_{\beta'} \to T', \quad f(a,b) = ab$$ is a finite covering. Let $\{e_1, \cdots, e_n\},$ respectively $\{e^{'}_1, \cdots, e^{'}_m\}$, be a basis of the lattice $\ker \exp \subset \mathfrak{t}_\xi$, respectively $\ker \exp \subset \mathfrak{t}_{\beta'}.$ If $i\xi = X_1e_1 + \cdots + X_ne_n$ and $i\beta' = Y_1e'_1 + \cdots + Y_me'_m,$ then $i(\xi + a\beta' )= X_1e_1 + \cdots + X_ne_n + aY_1e'_1 + \cdots + aY_me'_m.$ Denote by $T'_{\xi +a \beta}$ the closure of $\exp(\mathbb{R}(i(\xi+a\beta'))$. Since $f$ is a covering, $\dim T_{\xi + a\beta'} = \dim T'_{\xi + a\beta'}.$ Hence, $$\dim T_{\xi + a\beta'} = \dim_\mathbb{Q}(\mathbb{Q}X_1 + \cdots + \mathbb{Q}X_n + \mathbb{Q}aY_1 + \cdots + \mathbb{Q}aY_m),$$ see for instance Dustermatt Kolk, Lie groups p. 61. Since $\beta' \neq 0,$ $Y_j \neq 0$ for some $j.$ Choose $a$ such that $aY_j \notin \mathbb{Q}X_1 + \cdots + \mathbb{Q}X_n.$ Then $\dim T_{\frac{\xi + a\beta'}{\parallel \xi + a \beta \parallel}} > \dim T_\xi$ which is a contradiction. Therefore, $\lambda_y > 0$ on $e(S(\mathfrak{p}'))$. By Lemma [Lemma 12](#le2){reference-type="ref" reference="le2"}, $\Phi(y,\cdot)$ is linearly proper on $\mathfrak{p}'$. This implies that $\Phi(y,\cdot)$ is bounded from below on $\mathfrak{p}'$ and $$m=\mathrm{inf}_{\alpha \in \mathfrak{p}'} \Phi(y,\exp(\alpha)),$$ is achieved. We claim that $$m=\mathrm{inf}_{\alpha \in \mathfrak{p}^\mathfrak{a}} \Phi(y,\exp(\alpha)).$$ Indeed, let $v\in \mathfrak{p}^\mathfrak{a}$. Then $v=v_1 + v_2$, where $v_1 \in \mathfrak{a}$ and $v_2\in \mathfrak{p}'$. By the cocycle condition, keeping in mind that $[v_1,v_2]=0$ and $v_1 \in \mathfrak{g}_y$, we get $$\Phi(y,\exp(v))=\Phi(y,\exp(v_1))+\Phi(y,\exp(v_2)).$$ We claim that $\Phi(y,\exp(v_1))=0$. Indeed, since $\mathfrak{a}\subset \mathfrak{g}_y$, If $w \in S(\mathfrak{a})$ then by formula $(\ref{formula})$ we get $$\lambda_y (e(w))=\langle \mu_{\mathfrak{p}^\mathfrak{a}} (y), -w \rangle \geq 0,$$ for any $w\in S(\mathfrak{a})$. This implies $\lambda_y (e(-w))=-\lambda_y (e(w))$ and so $\lambda_y (e(w))=0$ for any $w\in \mathfrak{a}$. Let $w\in \mathfrak{a}-\{0\}$ and let $s: \mathbb{R}\longrightarrow\mathbb{R}$ be the function $s(t)=\Phi(y,\exp(tw))$. Since $\exp(tw)y=y$ for any $t\in \mathbb{R}$, it follows that $s(t)$ is a linear function. Therefore, $s(t)=b t$ for some $b\in \mathbb{R}$. On the other hand $$0=\lambda_y (e \big(\frac{w}{\parallel w \parallel} \big))=\lim_{t\to +\infty}\frac{d}{dt}\Phi(y, \exp(tw))=b.$$ This proves $$\mathrm{inf}_{\alpha \in \mathfrak{p}'} \Phi(y,\exp(\alpha))=\mathrm{inf}_{\alpha \in \mathfrak{p}^\mathfrak{a}} \Phi(y,\exp(\alpha)),$$ and so $\Phi_y : (G^\mathfrak{a})^o /(K^\mathfrak{a})^o \longrightarrow\mathbb{R}$ has a minumum and so a critical point. By Proposition [Proposition 4](#polystable-I){reference-type="ref" reference="polystable-I"}, it follows that $y$ is $(G^\mathfrak{a})^o$ polystable and by Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"}, $y$ is $G$-polystable. Suppose $x$ is polystable. There exists $g\in G$ such that $\mu_\mathfrak{p}(gx) = 0.$ Let $y = gx$ and fix $\beta \in \mathfrak{p}.$ Since the Kempf-Ness function is convex along geodesics, $$\lambda_y(e(\beta)) = \lim_{t\to \infty}\frac{d}{dt}\varphi_y(\exp(t\beta)K) = \lim_{t\to \infty}\frac{d}{dt}\varphi_y(\exp(-t\beta))\geq \frac{d}{dt}|_{t= 0}\varphi_y(\exp(-t\beta)) = \langle \mu_\mathfrak{p}(y), \beta\rangle = 0.$$ This shows that $\lambda_y \geq 0$ on $\partial_\infty M.$ By ([\[Ginvariant\]](#Ginvariant){reference-type="ref" reference="Ginvariant"}), $\lambda_x \geq 0.$ By Lemma [Lemma 14](#compatible-Stabilizer){reference-type="ref" reference="compatible-Stabilizer"}, $G_y$ is compatible with $\mathfrak{g}_y = \mathfrak{k}_y \oplus \mathfrak{p}_y$ and $Z(y) = e(S(\mathfrak{p}_y)).$ Suppose there exist $p = e(\beta) \in Z(y).$ Then $e(-\beta)\in Z(y)$ also. Furthermore, $e(\beta)$ and $e(-\beta)$ are connected by the geodesic $[\exp(t\beta)].$ This means that the condition of the Theorem holds for $Z(y).$ Now, for $p\in Z(x),$ $g\cdot p\in Z(y).$ Let $q\in Z(y)$ be connected to $g\cdot p$ by a geodesic $\alpha.$ Then the geodesic $g^{-1}\circ \alpha$ connects $p$ to $g^{-1}\cdot q\in Z(x).$ This concludes the proof of the theorem. ◻ **Corollary 2**. *$x\in X$ is polystable if and only if there exists $\beta \in S(\mathfrak{p}),$ $y\in G\cdot x$ and $g\in (G^\beta)^o$ such that $\lambda_x(e(\beta)) = 0$ and $\mu_\mathfrak{p}(gy) = 0.$* 10 T. Andrei. *Symplectic Stability, Analytic Stability in Non-Algebraic Complex Geometry*. , 15(2):183--209, 2004. L. Biliotti and A. Ghigi. *Stability of measures on Kähler manifolds*. , 317:1108 -- 1150, 2017. Biliotti L., Ghigi A. and Heinzner P., *Polar orbitopes*, Comm. Ann. Geom. **21 (3)**, (2013), 1--28. L. Biliotti and O. J. Windare. *Stability, analytic stability for real reductive Lie groups*. , 33, 2023. L. Biliotti and M. Zedda. *Stability with respect to actions of real reductive Lie group*. , 196(1):2185--2211, 2017. A. Borel and L. Ji. . Birkhäuser Boston, MA, 2006. Chevalley, C., *Theory of Lie groups*, Princeton University Press, (1946). A. Bruasse and A. Teleman. *Harder-Narasimhan and optimal destabilizing vectors in complex geometry*. , 55(3):1017--1053, 2005. P. Heinzner and G. W. Schwarz. *Cartan decomposition of the moment map*. , 337:197--232, 2007. P. Heinzner, G.W. Schwarz, and Stötzel H. *Stratifications with respect to actions of real reductive groups*. , 144(1):163--185, 2008. W. Hesselink. *Disungalirazions of Varieties and Nullforms*. , 55:141--163, 1979. Hohschild, G., *The structure of Lie groups*, Holden-day, San Francisco, (1965). M. Kapovich, B. Leeb, and J. Millson. *Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity*. , 81(2):297--354, 2009. F. Kirwan. . Math. Notes 31, Princeton, 1984. Knapp A. W., *Lie groups beyond an introduction*. Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, second edition, **140**, (2002). M. Lübke and A. Teleman. . Word Scientific, 1995. D. Mumford, J. Fogarty, and F. Kirwan. , volume 34. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1994. i Riera I. Mundet. *A Hitchin-Kobayashi correspondence for Kähler fibrations*. , 528:41 -- 80, 2000. i Riera I. Mundet. *A Hilbert--Mumford criterion for polystability in Kaehler geometry*. , 362(10):5169--5187, 2010. i Riera I. Mundet. *Maximal Weights in Kähler Geometry: Flag Manifolds and Tits distance (with an appendix by A. H. W. Schmitt)*. , 522(10):5169--5187, 2010. Heinzner P. and P. Schützdeller. *Convexity properties of gradient maps*. , 225(3):1119--1133, 2010. Patrick Eberlein. *Geometry of Nonpositively Curved Manifolds*. https://api.semanticscholar.org/CorpusID:117294486, 1997.

arxiv_math

--- abstract: | This paper investigates instances of Sobolev embeddings characterized by local compactness at every point within their domain, except for a single point. We obtain the sharp conditions that distinguish compactness from non-compactness and observe that in the context of Sobolev embeddings, non-compactness occurring at only one point within the domain could give rise to an infinite-dimensional subspace where the embedding is invertible (i.e., not strictly singular). Furthermore, we establish lower bounds for the Bernstein numbers, entropy numbers, and the measure of non-compactness. address: - Chian Yeong Chuah, Department of Mathematics, The Ohio State University, 231 W. 18th Ave. Columbus, OH 43210, USA. - Jan Lang, Department of Mathematics, The Ohio State University, 231 W. 18th Ave. Columbus, OH 43210, USA, *and* Czech Technical University in Prague, Faculty of Electrical Engineering, Department of Mathematics, Technická 2, 166 27 Praha 6, Czech Republic author: - Chian Yeong Chuah, Jan Lang bibliography: - bibliography.bib title: Quality of non-compactness for Sobolev Embedding with one point non-compactness. --- [^1] # Introduction Embeddings of Sobolev spaces into Lebesgue or Lorentz spaces, defined on an open set $\Omega \subset \mathbb{R}^d$, hold significant importance in both the theory of partial differential equations and approximation theory. Understanding the "quality" and internal characteristics of these embeddings is essential for numerous practical applications. Let us consider the following two Sobolev embeddings: $$\label{1} I: W_0^{1,p}(\Omega) \to L^{p^*}(\Omega)$$ and $$\label{2} I: W_0^{1,p}(\Omega) \to L^{p^*,p}(\Omega),$$ where $p\in [1,d)$ and $p^*=dp/(d-p)$. By $\Omega$ we denote a bounded open subset of $\mathbb{R}^d$ which is regular enough (for example, Lipschitz), and $W_0^{1,p}(\Omega)$ refers to the Banach space of all functions from Sobolev space $W^{1,p}(\Omega)$ with zero boundary values (see the next section for definitions). Both embeddings ([\[1\]](#1){reference-type="ref" reference="1"}) and ([\[2\]](#2){reference-type="ref" reference="2"}) are non-compact and the target spaces are optimal. The Lebesgue space $L^{p^*}(\Omega)$ is the optimal target space in ([\[1\]](#1){reference-type="ref" reference="1"}) among all Lebesgue spaces - i.e. for any smaller Lebesgue space $L^q(\Omega) \subsetneqq L^{p^*}(\Omega)$ ($p^* <q$) the embedding $I:W_0^{1,p}(\Omega) \to L^{q}(\Omega)$ is unbounded. It is known (see [@Pe]) that ([\[1\]](#1){reference-type="ref" reference="1"}) can be improved if one looks for optimal target space among all Lorentz spaces. In this case, the Lorentz space $L^{p^*,p}(\Omega)$ is the optimal target space. Since $L^{p^*,p}\subsetneqq L^{p^*}$, ([\[2\]](#2){reference-type="ref" reference="2"}) is an improvement over ([\[1\]](#1){reference-type="ref" reference="1"}). In addition, the Lorentz space $L^{p^*,p}(\Omega)$ is the optimal target space among all rearrangement-invariant function spaces (i.e. on collections of function spaces which includes, among others, Lebesgue spaces, Lorentz spaces, Orlicz spaces, or Marcinkiewicz spaces). This means that if $I:W_0^{1,p}(\Omega) \to Y(\Omega)$ is valid, then $L^{p^*,p}(\Omega) \subseteqq Y(\Omega)$ (see [@KP2006]). It is worth mentioning that both embeddings ([\[1\]](#1){reference-type="ref" reference="1"}) and ([\[2\]](#2){reference-type="ref" reference="2"}) are not only non-compact, but they are also "maximally non-compact" as their norms are equal to their measures of non-compactness (see [@EdEv]\[Definition 2.7\]). This was proved in [@He] and [@Bo2020]. In [@LMOP], it was showed that when the target space $L^{p^*,p}(\Omega)$ in ([\[2\]](#2){reference-type="ref" reference="2"}) is enlarged to the larger Lebesgue space $L^{p^*,r}(\Omega)$ $(p< r\le \infty)$, for which we have $L^{p^*,p} \subsetneqq L^{p^*,r}(\Omega) \subset L^{p^*, \infty}(\Omega)$, then the corresponding embedding is still non-compact but surprisingly also "maximally non-compact". Given the inherent structural differences between the spaces $L^{p^*,p}$ and $L^{p,\infty}$, it is reasonable to expect that the corresponding Sobolev embeddings possess distinct internal characteristics. However, as previously noted, these differences are not fully captured by the measure of non-compactness alone. This prompts the need for a more refined approach. The concepts of strict singularity and Bernstein numbers emerge as promising candidates for quantifying the "quality" of non-compactness. By delving into these quantities, we can gain deeper insights into the nuanced inner structures of these Sobolev embeddings. This was confirmed by [@LaMi], in which it was showed that ([\[2\]](#2){reference-type="ref" reference="2"}) is not strictly singular (i.e. there exists an infinite dimensional subspace on which the embedding is isomorphic) and that the Bernstein numbers of ([\[2\]](#2){reference-type="ref" reference="2"}) are equal to the norm of the embedding, and that ([\[1\]](#1){reference-type="ref" reference="1"}) is finitely strictly singular (i.e. Bernstein numbers converge to 0). The main techniques used in [@LaMi] were based on the existence of non-compactness at each point, i.e. the embedding has the same measure of non-compactness at each open ball in the underlying domain $\Omega$, and the norm of embedding is invariant with respect to a re-scaling. The natural question that arises from [@LaMi], [@LMOP], [@He], and [@Bo2020] is: which "quality" of non-compactness can be expected in the case when the norm of embedding is not invariant with respect to some re-scaling and the embedding is compact at each neighborhood except the neighborhood of just one point in the domain. In this paper, we construct Sobolev embedding from $W^{1,p}_0(\Omega)$ into variable Lebesgue space $L^{q(\cdot),p}(\Omega)$ which is non-compact just in a neighborhood of one point and compact in all other neighborhoods in $\Omega$ and we showed that this Sobolev embedding is not strictly compact () and Bernstein numbers with the measure of non-compactness have strictly positive lower bound (). By this, we demonstrate that a localized non-compactness at just one point is satisfactory to generate an infinitely dimensional system of functions on which the corresponding Sobolev embedding is invertible (i.e. Sobolev embedding is not strictly singular) and then obtain an embedding which is essentially as non-compact as the most non-compact case, i.e. embedding ([\[2\]](#2){reference-type="ref" reference="2"}). As a by-product of our techniques, we extend results from [@EGN] onto variable Lorentz spaces and we obtain sharp conditions which guarantee compactness and non-compactness and we also extend results from [@KS2008] into variable Lorentz spaces. The paper is structured as follows. In the next section, we recall some basic definitions and notations. In Section 3, we start by introducing almost compact embeddings and make observations about almost compact embeddings in the context of variable Lorentz spaces () and then obtain conditions under which our embeddings are compact () and non-compact (). In the last section, we obtain the main results () and (). # Preliminaries We start this section by recalling definitions of function spaces that we will use throughout this paper. Let $p \in [1, \infty)$ and $q : \Omega \to [1, \infty)$ be a measurable function. By $L^{q(\cdot)}(\Omega)$ and $L^{q(\cdot),p }(\Omega)$ we denote variable exponent Lebesgue space and variable exponent Lorentz space, respectively, which are defined by the following norms: $$\left\| f \right\|_{L^{q(\cdot)} (\Omega)} := \inf \left\{ \lambda > 0 : \rho_{q(\cdot)} \left( \frac{f}{\lambda} \right) \leq 1 \right\} = \sup \left\{ \lambda > 0 : \rho_{q(\cdot)} \left( \frac{f}{\lambda} \right) > 1 \right\}$$ $$\text{where } \rho_{q (\cdot)} (f) := \int_{\Omega} | f(x) |^{q(x)} \ d x.$$ $$\left\| f \right\|_{L^{q(\cdot), p} (\Omega)} := \left[ \int_{0}^{\infty} \lambda^{p - 1} \left\| \chi_{\left\{ x \in \Omega : |f(x)| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{p} \ d \lambda \right]^{\frac{1}{p}}.$$ These spaces can be considered as generalizations of the \"standard\" Lebesgue or Lorentz spaces when $q(\cdot)$ is replaced by a constant. More information about these spaces can be found in [@KovRak] and [@KeVy]. The set of all smooth (i.e., infinitely differentiable) functions that are compactly supported in $\Omega$ is denoted by $\mathcal{C}_0^{\infty}(\Omega)$. By $W^{1,p}(\Omega)$ we denote *the classical first-order Sobolev space* on $\Omega$ defined by the norm $$\left\| f \right\|_{W^{1, p} (\Omega)} := \left[ \int_{\Omega}|f(x)|^p x + \int_{\Omega} | (\nabla f) (x) |^{p} \ d x \right]^{\frac{1}{p}}.$$ We denote the closure of $\mathcal{C}_0^{\infty}(\Omega)$ in $W^{1,p}(\Omega)$ by $W_0^{1,p}(\Omega)$ and equip it with the norm $$\left\| f \right\|_{W_0^{1, p} (\Omega)} := \left[ \int_{\Omega} | (\nabla f) (x) |^{p} \ d x \right]^{\frac{1}{p}},$$ which is equivalent with the $W^{1,p}$ norm on $\mathcal{C}_0^{\infty}(\Omega).$ For the rest of the discussion, we will assume that $q(x)$ is radially decreasing about the point $x_0 \in \Omega$, $q (x_0) = p^{*}$ , and $q (x) < p^{*}$ for all $x \not = x_0$. Now, we recall the definitions of some s-numbers (Bernstein and Kolmogorov numbers) and entropy numbers. Let $X$, $Y$ be Banach spaces and $T \in B(X, Y)$. We denote $BX=\{ x \in X : \|x\|_X \le 1\}$. The $n$-th Kolmogorov number is defined is $$d_n(T):= \inf_{Y_n} \sup_{y\in T(BX)} \inf_{z\in Y_n} \|y-z\|_Y,$$ where the infimum is taken over all $n$-dimensional sub-spaces $Y_n$ of $Y$ and $T(BX)=\{T(x) : x\in BX\}$. The $n$-th Bernstein number of $T$, $b_{n} (T)$ is defined by: $$b_{n} (T) := \sup \left\{ \inf_{x \in X_{n}, \| x \|_{X} = 1} \left\| T(x) \right\|_{Y} : X_{n} \text{ is an } n \text{ dimensional subspace of } X \right\}.$$ The s-numbers defined above (see [@EdEv]) are ordered $\|T\| \ge d_n(T) \ge b_n(T)$ and can be used for describing the "quality" of operators. It is worth noting that $\alpha(T):=\lim_{n\to \infty} d_n(T)=0$ if and only if $T$ is compact. Another quantity that is used for describing compactness is entropy numbers. We say that the $n$-th entropy number of T is defined by $$e_n(T):= \inf \{\varepsilon>0: T(BX) \mbox{ can be covered by }2^{k-1} \mbox{ balls in }Y \mbox{ with radius }\varepsilon \}.$$ Note that $T$ is compact if and only if $\beta(T):= \lim_{n \to \infty} e_n(T) =0$, where $\beta(T)$ is called *the measure of non-compactness*. If $\beta(T)=\|T\|$, we say that $T$ is *maximally non-compact*. The entropy numbers are better suited than s-numbers due to their relatively good behavior with respect to interpolation. We say that an operator $T\in B(X,Y)$ is *strictly singular* (SS) if there is no infinite-dimensional closed subspace $Z$ of $X$ such that the restriction $T|_Z$ is an isomorphism of $Z$ onto $T(Z)$. Equivalently, this can be described that for each infinite dimensional (closed) subspace $Z$ of $X$, we have $$\inf \{ \|T (x) \|_Y: \|x\|_X=1, x\in Z\}=0.$$ An operator $T\in B(X,Y)$ is said to be *finitely strictly singular* (FSS) if for any $\varepsilon >0$, there exists $N(\varepsilon) \in {\bf N}$ such that if $E$ is a subspace of $X$ with $\dim E \ge N(\varepsilon)$, then there exists $x\in E, \|x\|_X=1,$ such that $\|T(x) \|_Y \le \varepsilon$. It is possible to see that an operator $T$ is finitely strictly singular if and only if $$\lim_{n \to \infty} b_n(T)=0.$$ We conclude this section with the well-known fact: $$Compact \subset FSS \subset SS.$$ # Compactness and Non-compactness conditions In this section, we will study conditions on $q(\cdot)$ under which the Sobolev embedding $$I:W_0^{1, p} (\Omega) \to L^{q(\cdot), p} (\Omega)$$ is compact or non-compact. We will consider the case when $1\le q(\cdot) \le p^*$ and $q(\cdot)$ is approaching $p^*$ only at one point of the domain $x_0 \in \Omega$. This will bring us to the situation in which $I$ is "locally" compact at the neighborhood of each point in $\Omega \setminus \{x_0\}$ and possibly non-compact only at neighborhoods of $x_0$. In order to obtain conditions on the growth of $q(\cdot)$ at $x_0$ which will guarantee compactness/non-compactness, we need to make some observations. We start with statements needed for describing almost compact embeddings on variable Lorentz spaces which is a generalization of the main results from [@EGN]. Employing these results for variable Lorentz spaces gives us a condition on $q(\cdot)$ which guarantees the compactness of Sobolev embedding. First, we recall some definitions and the known results: If $X(\Omega)$ and $Y(\Omega)$ are Banach function spaces over $\Omega$, then we say that $X(\Omega)$ is *almost compactly embedded* into $Y(\Omega)$, denoted by $X(\Omega) \overset{*}\hookrightarrow Y(\Omega)$, if for every sequence $(E_n)_{n = 1}^{\infty}$ of measurable subsets of $\Omega$ such that $E_n \to 0$ a.e., we have $$\lim_{n\to \infty} \sup_{\|u\|_X\le 1} \|u \chi_{E_n} \|_Y = 0.$$ The next proposition demonstrates the connection with the compactness of Sobolev embeddings. **Proposition 1**. *Let $X, Y,$ and $Z$ be Banach function spaces over bounded set $\Omega$ and assume $$W^1(X) \hookrightarrow Y, \mbox{ and } \quad Y \overset{*}\hookrightarrow Z,$$ where $W^1(X)$ denotes the Sobolev space with norm $\|u\|_W^1(X)=\|u\|_X +\|\nabla u\|_X.$* *Then $W^1(X) \hookrightarrow Z$ is a compact embedding.* *Proof.* See [@EGN Prop 2.2] or literature noted there. ◻ **Lemma 2**. *Let $s : \Omega \to \mathbb{R}$ be a measurable function and $\alpha > 1$. Then, for each $t > 0$, $\left[ \alpha^{s(\cdot)} \right]^{*} (t) = \alpha^{s^{*}(t)}$.* *Proof.* This is proved in [@EGN Lemma 2.10]. ◻ **Lemma 3**. *Let $p : \Omega \to [1, \infty)$ and $q : \Omega \to [1, \infty)$ be measurable functions where $q(x) \leq p(x) \leq p_{+} < \infty$ for all $x \in \Omega$. Assume that for any sequence $(E_{n})_{n = 1}^{\infty}$ of measurable subsets of $\Omega$ such that $| E_{n}| \to 0$, we have* *$$\left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \to 0,$$ where $p(\cdot)q(\cdot)/(p(\cdot)-q(\cdot))=\infty$ when $p(\cdot)=q(\cdot).$ Then, $L^{p(\cdot), t} (\Omega)$ is almost compactly embedded into $L^{q(\cdot), t} (\Omega)$ for any $t \in [1, \infty)$ .* *Proof.* Let $u \in L^{p(\cdot), t} (\Omega)$ where $\left\| u \right\|_{L^{p(\cdot), t} (\Omega)} \leq 1$. Let $(E_{n})_{n = 1}^{\infty}$ be a sequence of measurable subsets of $\Omega$ where $\chi_{E_{n}} \to 0$ a.e. as $n \to \infty$. Now, fix $n \in \mathbb{N}$ and $\lambda > 0$. By Holder's inequality, there exists $K \in (0, \infty)$ such that $$\begin{aligned} & \left\| \chi_{\left\{ x \in \Omega : \left| \left( u \right) (x) \right| > \lambda \right\}} \cdot \chi_{E_{n}} \right\|_{L^{q(\cdot)} (\Omega)} \leq K \left\| \chi_{\left\{ x \in \Omega : \left| \left( u \right) (x) \right| > \lambda \right\}} \right\|_{L^{p(\cdot)} (\Omega)} \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)}. \end{aligned}$$ Note that for each $n \in \mathbb{N}$ and each $\lambda > 0$, $$\chi_{\left\{ x \in \Omega : | \left( u \cdot \chi_{E_{n}} \right) (x) | > \lambda \right\}} = \chi_{\left\{ x \in \Omega : | u(x) | > \lambda \right\} \cap E_{n}} = \chi_{\left\{ x \in \Omega : | u(x) | > \lambda \right\} } \cdot \chi_{E_{n}}.$$ Next, we estimate the following: $$\begin{aligned} & \left\| u \chi_{E_{n}} \right\|_{L^{q(\cdot), t} (\Omega)} = \left[ \int_{0}^{\infty} \left\| \chi_{\left\{ x \in \Omega : \left| \left( u \chi_{E_{n}} \right) (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{t} \lambda^{t - 1} d \lambda \right]^{\frac{1}{t}} \\ & = \left[ \int_{0}^{\infty} \left\| \chi_{\left\{ x \in \Omega : \left| \left( u \right) (x) \right| > \lambda \right\}} \cdot \chi_{E_{n}}\right\|_{L^{q(\cdot)} (\Omega)}^{t} \lambda^{t - 1} d \lambda \right]^{\frac{1}{t}} \\ & \leq K \left[ \int_{0}^{\infty} \left\| \chi_{\left\{ x \in \Omega : \left| \left( u \right) (x) \right| > \lambda \right\}} \right\|_{L^{p(\cdot)} (\Omega)}^{t} \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)}^{t} \lambda^{t - 1} d \lambda \right]^{\frac{1}{t}} \\ & = K \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \left[ \int_{0}^{\infty} \left\| \chi_{\left\{ x \in \Omega : \left| \left( u \right) (x) \right| > \lambda \right\}} \right\|_{L^{p(\cdot)} (\Omega)}^{t} \lambda^{t - 1} d \lambda \right]^{\frac{1}{t}} \\ & = K \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \left\| u \right\|_{L^{p(\cdot), t} (\Omega)} \leq K \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)}. \end{aligned}$$ So, we obtain the following: $$\sup_{\left\| u \right\|_{L^{p(\cdot), t} (\Omega)} \leq 1} \left\| u \chi_{E_{n}} \right\|_{L^{q(\cdot), t} (\Omega)} \leq K \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)}.$$ Since $| \Omega | < \infty$, $\chi_{\Omega}$ is integrable. By the dominated convergence theorem, $\left| E_{n} \right| = \int_{\Omega} \chi_{E_{n}} (x) d x \to 0$ as $n \to \infty$ because $\chi_{E_{n}} \to 0$ a.e as $n \to \infty$ and $\chi_{E_{n}} \leq \chi_{\Omega}$ for all $n \in \mathbb{N}$. Then, by assumption, $$\lim_{n \to \infty} \sup_{\left\| u \right\|_{L^{p(\cdot), t} (\Omega)} \leq 1} \left\| u \chi_{E_{n}} \right\|_{L^{q(\cdot), t} (\Omega)} \leq K \lim_{n \to \infty} \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} = 0.$$ ◻ **Lemma 4**. *Let $p : \Omega \to [1, \infty)$, $q : \Omega \to [1, \infty)$ be measurable functions where $q(x) \leq p(x) \leq p_{+} < \infty$ for all $x \in \Omega$. Let $s : \Omega \to [1, \infty]$ be a function defined by:* *$$s (x) := \begin{cases} \frac{1}{p(x) - q(x)} & \text{if } p(x) > q(x),\\ \infty & \text{if } p(x) = q(x). \end{cases}$$* *Assume that for all $a > 1$,* *$$\int_{0}^{|\Omega|} a^{s^{*} (w)} \ d w < \infty.$$* *Then, for any sequence $(E_{n})_{n = 1}^{\infty}$ of measurable subsets of $\Omega$ such that $|E_{n}| \to 0$, we have* *$$\left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \to 0.$$* *Proof.* Suppose that $\left( \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \right)_{n = 1}^{\infty}$ does not converge to $0$. Then, there exists $\epsilon > 0$ such that for all $k \in \mathbb{N}$, there exists $n_{k} \geq k$ where $\left\| \chi_{E_{n_{k}}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \geq \epsilon$. Without loss of generality, we can assume that $\epsilon < 1$. By considering the subsequence $\left( \left\| \chi_{E_{n_{k}}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \right)_{k = 1}^{\infty}$ of $\left( \left\| \chi_{E_{n}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \right)_{n = 1}^{\infty}$, we see that for all $k \in \mathbb{N}$, $\left\| \chi_{E_{n_{k}}} \right\|_{L^{\frac{p(\cdot) q(\cdot)}{p(\cdot) - q(\cdot)}} (\Omega)} \geq \epsilon$. This means that for all $k \in \mathbb{N}$, $$\epsilon \leq \inf \left\{ \lambda > 0 : \int_{\Omega} \left| \frac{\chi_{E_{n_{k}}} (x)}{\lambda} \right|^{\frac{p(x) q(x)}{p(x) - q(x)}} d x \leq 1 \right\} = \inf \left\{ \lambda > 0 : \int_{\Omega} \left| \frac{\chi_{E_{n_{k}}} (x)}{\lambda} \right|^{p(x) q(x) s(x)} d x \leq 1 \right\}$$ Let $\delta \in (0, \epsilon)$. By the definition of infimum, we deduce that $$\int_{E_{n_{k}}} \left( \frac{1}{\delta} \right)^{p(x) q(x) s(x)} d x = \int_{\Omega} \left| \frac{\chi_{E_{n_{k}}} (x)}{\delta} \right|^{p(x) q(x) s(x)} d x > 1$$ Next, we apply the Hardy-Littlewood inequality to obtain the following estimate: $$\begin{aligned} & \int_{E_{n_{k}}} \left( \frac{1}{\delta} \right)^{p(x) q(x) s(x)} d x \leq \int_{E_{n_{k}}} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s(x)} d x = \int_{\Omega} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s(x)} \chi_{E_{n_{k}}} (x) d x \text{ since } \delta < 1 \\ & \leq \int_{0}^{\infty} \left[ \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s(\cdot)} \right]^{*} (y) \left( \chi_{E_{n_{k}}} \right)^{*} (y) d y \text{ by the Hardy-Littlewood inequality} \\ & = \int_{0}^{\infty} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s^{*} (y)} \chi_{ \left[0, \left| E_{n_{k}} \right| \right)} (y) \ d y \text{ by \thref{exponent}} \\ & = \int_{0}^{\left| E_{n_{k}}\right|} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s^{*} (y)} \ d y.\end{aligned}$$ Combining the above two inequalities, we conclude that for all $k \in \mathbb{N}$, $$\begin{aligned} \label{contradiction} \int_{0}^{\left| E_{n_{k}}\right|} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s^{*} (y)} \ d y > 1 . \end{aligned}$$ Meanwhile, since $\frac{1}{\delta^{p_{+} q_{+}}} > 1$ by assumption, we know that $$\int_{0}^{\left| \Omega \right|} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s^{*} (y)} \ d y < \infty.$$ So, by the absolute continuity of the Lebesgue integral, there exists $\beta > 0$ such that for all $\omega < \beta$, $$\int_{0}^{\omega} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s^{*} (y)} \ d y < 1.$$ By assumption, $\left| E_{n_{k}} \right| \to 0$. So, there exists $K \in \mathbb{N}$ such that for all $k \geq K$, $\left| E_{n_{k}} \right| < \beta$. This implies that for all $k \geq K$, $$\int_{0}^{\left| E_{n_{k}}\right|} \left( \frac{1}{\delta^{p_{+} q_{+}}} \right)^{s^{*} (y)} \ d y < 1.$$ This contradicts [\[contradiction\]](#contradiction){reference-type="eqref" reference="contradiction"}. ◻ Now we can state a theorem which is an extension of [@EGN Theorem 3.4]. **Theorem 5**. *Let $p : \Omega \to [1, \infty)$, $q : \Omega \to [1, \infty)$ be measurable functions where $q(x) \leq p(x) \leq p_{+} < \infty$ for all $x \in \Omega$. Let $s : \Omega \to [1, \infty]$ be a function defined by:* *$$s (x) := \begin{cases} \frac{1}{p(x) - q(x)} & \text{if } p(x) > q(x),\\ \infty & \text{if } p(x) = q(x). \end{cases}$$* *Assume that for all $a > 1$,* *$$\int_{0}^{|\Omega|} a^{s^{*} (w)} \ d w < \infty.$$* *Then, $L^{p(\cdot), t} (\Omega)$ is almost compactly embedded into $L^{q(\cdot), t} (\Omega)$ for any $t \in [1, \infty)$.* *Proof.* This follows from Lemma [Lemma 3](#3.5){reference-type="ref" reference="3.5"} and Lemma [Lemma 4](#3.6){reference-type="ref" reference="3.6"}. ◻ **Lemma 6**. *Let $t\in [1,\infty)$ and $1 \leq p < d$, where $d \geq 2$ is a positive integer. Let $\Omega$ be a bounded domain on $\mathbb{R}^{d}$. Let $q : \Omega \to [1, \infty)$ be a measurable function satisfying $1 \leq q(x) \leq \frac{d p}{d - p}$ for almost every $x \in \Omega$. Suppose that there exist $x_{0} \in \Omega$, $C > 0$, $\eta > 0$ and $0 < \ell < 1$ such that* *$$\text{ess } \sup \left\{ q(x) \in [1, \infty) : x \in \Omega \text{ and } \left| x - x_{0} \right| \geq \eta \right\} < \frac{d p}{d - p}$$* *and* *$$q(x) \leq \frac{d p}{d - p} - \frac{C}{\left| \log \left( \frac{1}{\left| x - x_{0} \right|} \right) \right|^{\ell}} \text{ for a.e. } x \in \Omega \text{ with } \left| x - x_{0} \right| \leq \eta.$$* *Then, $L^{p^{*}, t} (\Omega)$ is almost compactly embedded into $L^{q(\cdot), t} (\Omega)$.* *Proof.* Without loss of generality, we can assume that $|x_{0}| = 0 \in \Omega$. Also, we can assume that there exist $C > 0$, $1>\eta > 0$ and $0 < \ell < 1$ such that $q(x) \leq r(x)$, where $r : \Omega \to [1, \infty]$ is defined by: $$r (x) := \begin{cases} \frac{d p}{d - p} - \frac{C}{\left| \log \left( \frac{1}{\left| x \right|} \right) \right|^{\ell}} & \text{if }|x| \leq \eta,\\ \frac{d p}{d - p} - \frac{C}{\left| \log \left( \frac{1}{\eta} \right) \right|^{\ell}} & \text{if } |x| > \eta. \end{cases}$$ So, $\frac{1}{p^{*} - q(x)} \leq \frac{1}{p^{*} - r(x)}$ for all $x \in \Omega$. It follows that $\left[ \frac{1}{p^{*} - q(\cdot)} \right]^{*} (t) \leq \left[ \frac{1}{p^{*} - r(\cdot)} \right]^{*} (t)$ for all $t > 0$. Thus, it remains to show that for all $a > 1$, $$\int_{0}^{| \Omega |} a^{\left[ \frac{1}{p^{*} - r(\cdot)} \right]^{*} (t)} \ dt < \infty.$$ For the simplicity of notation, we let $s(x) := \frac{1}{p^{*} - r(x)}$. So, $$s (x) := \begin{cases} \frac{1}{C}\left| \log \left( \frac{1}{\left| x \right|} \right) \right|^{\ell} & \text{if }|x| \leq \eta,\\ \frac{1}{C}\left| \log \left( \frac{1}{\eta} \right) \right|^{\ell} & \text{if } |x| > \eta. \end{cases}$$ Recalling the definition for the distribution function, for each $\alpha > 0$, we define $$d_{s} (\alpha) := \left| \left\{ x \in \Omega : | s(x) | > \alpha \right\} \right|.$$ For $0 \leq \alpha \leq \frac{1}{C}\left| \log \left( \frac{1}{\eta} \right) \right|^{\ell}$ we have obviously $d_{s} (\alpha) = | \Omega |$. Meanwhile, for $\alpha > \frac{1}{C}\left| \log \left( \frac{1}{\eta} \right) \right|^{\ell}$, we get $$d_{s} (\alpha) = \left| \left\{ x \in \Omega : \frac{1}{C}\left| \log \left( \frac{1}{\left| x \right|} \right) \right|^{\ell} > \alpha \right\} \right| = \left| \left\{ x \in \Omega : \left| \log \left( \frac{1}{\left| x \right|} \right) \right| > \left( C \alpha \right)^{\frac{1}{\ell}} \right\} \right|$$ $$= \left| \left\{ x \in \Omega : - \log \left( \left| x \right| \right) > \left( C \alpha \right)^{\frac{1}{\ell}} \right\} \right| = \left| \left\{ x \in \Omega : |x| < e^{- \left( C \alpha \right)^{\frac{1}{\ell}} } \right\} \right| = \nu_{d} e^{- d \left( C \alpha \right)^{\frac{1}{\ell}} }.$$ Summarizing the above, we obtain: $$d_{s} (\alpha) := \begin{cases} | \Omega | & \text{if } 0 \leq \alpha < \frac{1}{C}\left| \log \left( \frac{1}{\eta} \right) \right|^{\ell},\\ \nu_{d} e^{- d \left( C \alpha \right)^{\frac{1}{\ell}} } & \text{if } \alpha \geq \frac{1}{C}\left| \log \left( \frac{1}{\eta} \right) \right|^{\ell}. \end{cases}$$ Next, recall the definition of the non-increasing rearrangement function, for $t \geq | \Omega |$, $s^{*} (t) = 0$. For $\nu_{d} \eta^{d} \leq t < | \Omega|$, $s^{*} (t) = \frac{1}{C} \left| \log \left( \frac{1}{\eta} \right)^{\ell} \right|$. Finally, let $0 \leq t < \nu_{d} \eta^{d}$. Then, $$d_{s} (\beta) \leq t \Longleftrightarrow \nu_{d} e^{- d \left( C \beta \right)^{\frac{1}{\ell}}} \leq t \Longleftrightarrow \beta \geq \frac{1}{C d^{\ell}} \left[ \log \left( \frac{\nu_{d}}{t} \right) \right]^{\ell}.$$ So, we deduce that $$s^{*} (t) := \begin{cases} \frac{1}{C d^{\ell}} \left[ \log \left( \frac{\nu_{d}}{t} \right) \right]^{\ell} & \text{if } 0 \leq t < \nu_{d} \eta^{d},\\ \frac{1}{C} \left| \log \left( \frac{1}{\eta} \right) \right|^{\ell} & \text{if } \nu_{d} \eta^{d} \leq t < | \Omega |, \\ 0 & \text{ if } t \geq | \Omega |. \end{cases}$$ Finally, we proceed to show that for each $a > 1$, $\int_{0}^{\nu_{d} \eta^{d}} a^{s^{*} (t)} \ d t < \infty$. $$\int_{0}^{\nu_{d} \eta^{d}} a^{\frac{1}{C d^{\ell} } \left[ \log \left( \frac{\nu_{d}}{t} \right) \right]^{\ell}} d t = \int_{0}^{\nu_{d} \eta^{d}} e^{\frac{1}{C d^{\ell}} \left[ \log \left( \nu_{d} \right) - \log (t) \right]^{\ell} \log(a)} d t$$ $$= \int_{\infty}^{- d \log (\eta) } - e^{\frac{1}{C d^{\ell}} y^{\ell} \log(a)} e^{\log \left( \nu_{d} \right) - y} d y = \nu_{d} \int_{- d \log (\eta) }^{\infty} e^{\frac{\log(a)}{C d^{\ell}} y^{\ell} - y} d y$$ Since $0 < \ell < 1$, there exists $0 < \omega < 1$ such that for all $y \geq - d \log (\eta)$, $e^{\frac{\log(a)}{C d^{\ell}} y^{\ell} - y} \leq e^{- \omega y}$. Hence, $$\int_{0}^{\nu_{d} \eta^{d}} a^{\frac{1}{C d^{\ell} } \left[ \log \left( \frac{\nu_{d}}{t} \right) \right]^{\ell}} d t \leq \nu_{d} \int_{- d \log (\eta) }^{\infty} e^{- \omega y} d y < \infty.$$ Thus, for all $a > 1$ $$\int_{0}^{| \Omega |} a^{\left[ \frac{1}{p^{*} - q (\cdot) } \right]^{*} (t)} \ d t \leq \int_{0}^{| \Omega |} a^{\left[ \frac{1}{p^{*} - r (\cdot) } \right]^{*} (t)} \ d t < \infty.$$ By , $L^{p^{*}, t} (\Omega)$ is almost compactly embedded into $L^{q(\cdot), t} (\Omega)$. ◻ Now we are ready to state the conditions on $q(\cdot)$ which guarantees compactness. **Theorem 7**. *Let $1 \leq p < d$, where $d \geq 2$ is a positive integer. Let $\Omega$ be a bounded domain on $\mathbb{R}^{d}$. Let $q : \Omega \to [1, \infty)$ be a measurable function satisfying $1 \leq q(x) \leq \frac{d p}{d - p}$ for almost every $x \in \Omega$. Suppose that there exist $x_{0} \in \Omega$, $C > 0$, $\eta > 0$ and $0 < \ell < 1$ such that* *$$\text{ess } \sup \left\{ q(x) \in [1, \infty) : x \in \Omega \text{ and } \left| x - x_{0} \right| \geq \eta \right\} < \frac{d p}{d - p}$$* *and* *$$q(x) \leq \frac{d p}{d - p} - \frac{C}{\left| \log \left( \frac{1}{\left| x - x_{0} \right|} \right) \right|^{\ell}} \text{ for a.e. } x \in \Omega \text{ with } \left| x - x_{0} \right| \leq \eta.$$* *Then, the embedding from $W_{0}^{1, p} (\Omega)$ into $L^{q(\cdot), p} (\Omega)$ is compact.* *Proof.* It follows instantly from Proposition [Proposition 1](#prop3.4){reference-type="ref" reference="prop3.4"} and Lemma [Lemma 6](#3.8){reference-type="ref" reference="3.8"}. ◻ The next statement provides a condition on $q(\cdot)$ under which Sobolev embedding into $L^{q(\cdot),p}$ is non-compact. The idea of the proof is inspired by \[[@KS2008], Theorem 1\], and from the above, we can see that the condition, on $q(\cdot)$, is sharp. **Theorem 8**. *Let $1 \leq p < d$, where $d \geq 2$ is a positive integer. Let $\Omega$ be a bounded domain on $\mathbb{R}^{d}$. Let $q : \Omega \to [1, \infty)$ be a measurable function satisfying $1 \leq q(x) \leq \frac{d p}{d - p}$ for almost every $x \in \Omega$. Suppose that there exist $x_{0} \in \Omega$ and constants $C_{0}$, $\eta_{0} > 0$ such that* *$$\label{non-compact} q(x) \geq \frac{d p}{d - p} - \frac{C_{0}}{ \left| \log \left( \frac{1}{| x - x_{0} |} \right) \right|} \text{ for almost every } x \in \Omega \text{ with } | x - x_{0} | < \eta_{0}.$$* *Then, the embedding $E : W_0^{1, p} (\Omega) \to L^{q(\cdot), p} (\Omega)$ is not compact.* *Proof.* Without loss of generality, we can assume $x_{0} = 0$. Define $r : \Omega \to [0, \infty)$ by $r(x) := \frac{d p}{d - p} - q(x)$. Now, suppose that $E$ is compact. Define the bump function $\phi \in C_{0}^{\infty} (\mathbb{R}^{d})$ such that $\phi(x) = 1$ for all $| x | \leq \frac{1}{2}$ and $\phi(x) = 0$ for all $| x | > 1$. For each $n \in \mathbb{N}$, define $\phi_{n} : \Omega \to [0, \infty)$ by $\phi_{n} (x) := n^{\frac{d - p}{p}} \phi( n x )$. We observe that for sufficiently large $n \in \mathbb{N}$, $\phi_{n} \in C_{0}^{\infty} (\Omega)$. Also, for each $n \in \mathbb{N}$, we have the following two equalities: $$\begin{aligned} & \int_{\Omega} | ( \nabla \phi_{n} ) (x) |^{p} \ d x = \int_{B_{\frac{1}{n}} (0)} \left[ \sum_{j = 1}^{d} \left| \frac{\partial \phi_{n}}{\partial x_{j}} (x) \right|^{2} \right]^{\frac{p}{2}} d x = \int_{B_{{1}} (0)} \left[ \sum_{j = 1}^{d} \left| n^{\frac{d - p}{p} + 1} \frac{\partial \phi}{\partial x_{j}} (y) \right|^{2} \right]^{\frac{p}{2}} {d y \over n^d} \\ & = \frac{n^{d}}{n^{d}} \int_{B_{1} (0)} \left[ \sum_{j = 1}^{d} \left| \frac{\partial \phi}{\partial x_{j}} (y) \right|^{2} \right]^{\frac{p}{2}} d y = \int_{B_{1} (0)} | ( \nabla \phi ) (y) |^{p} \ d y, \end{aligned}$$ and $$\label{L_p phi} \int_{\Omega} | \phi_{n} (x) |^{p} \ d x = \int_{B_{\frac{1}{n}} (0)} n^{d - p} | \phi (n x) |^{p} \ d x = n^{d - p} \frac{1}{n^{d}} \int_{B_{1} (0)} | \phi (y) |^{p} \ d y = \frac{1}{n^{p}} \int_{B_{1} (0)} | \phi (y) |^{p} \ d y.$$ Now, let us estimate $\left\| \chi_{\left\{ z \in \Omega : \phi(n z) = 1 \right\} } \right\|_{L^{q (\cdot)}}$. Choose $1>\delta > 0$ sufficiently small such that $\delta < e^{- C_{0} \frac{d - p}{p}} {\nu_{d}}/{2^{d}},$ where $\nu_{d} =2^d |B_{1/2}(0)|$. $$\begin{aligned} & \int_{\Omega} \left| \frac{n^{\frac{d - p}{p}}}{\delta} \chi_{\left\{ z \in \Omega : \phi(n z) = 1 \right\} } (x) \right|^{q(x)} \ d x \geq \frac{1}{\delta} \int_{\Omega} \left| n^{\frac{d - p}{p}} \chi_{\left\{ z \in \Omega : \phi(n z) = 1 \right\} } (x) \right|^{q(x)} \ d x \\ & = \frac{1}{\delta} \int_{B_{\frac{1}{2 n}} (0)} n^{\frac{d - p}{p} \left[ \frac{d p}{d - p} - r(x) \right] } \ d x = \frac{n^{d}}{\delta} \int_{B_{\frac{1}{2 n}} (0)} n^{- \frac{d - p}{p} r(x) } \ d x \\ & = \frac{1}{\delta} \int_{B_{\frac{1}{2}} (0)} n^{- \frac{d - p}{p} r(\frac{y}{n}) } \ dy \geq \frac{1}{\delta} \int_{B_{\frac{1}{2}} (0)} \frac{1}{n^{\frac{C_{0} (d - p)}{p \left| \log \left( \frac{|y|}{n} \right) \right|}}} \ dy \\ & = \frac{1}{\delta} \int_{B_{\frac{1}{2}} (0)} e^{- \frac{d - p}{p} \frac{C_{0} \log(n) }{\left| \log \left( \frac{|y|}{n} \right) \right|}} \ dy = \frac{1}{\delta} \int_{B_{\frac{1}{2}} (0)} e^{- \frac{d - p}{p} \frac{C_{0} \log(n) }{\left| \log (n) - \log (|y|) \right|}} \ d y \\ & \geq \frac{1}{\delta} \int_{B_{\frac{1}{2}}(0)} e^{- \frac{d - p}{p} \frac{C_{0} \log(n) }{ \log (n) }} \ d y = e^{- C_{0} \frac{d - p}{p}} \frac{\nu_{d}}{ 2^{d} \delta} > 1. \end{aligned}$$ $$\mbox{ Since }\rho_{q(\cdot)} \left[ \frac{ \chi_{\left\{ z \in \Omega : \phi (n z) = 1\right\} } }{ \frac{\delta}{n^{({d - p})/{p}}} } \right] > 1, \quad \left\| \chi_{\left\{ z \in \Omega : \phi(n z) = 1 \right\} } \right\|_{L^{q (\cdot)}} \geq \frac{\delta}{n^{\frac{d - p}{p}}}.$$ Next, for each $n \in \mathbb{N}$, $$\begin{aligned} & \left\| \phi_{n} \right\|_{L^{q(\cdot), p}}^{p} = \int_{0}^{\infty} \lambda^{p - 1} \left\| \chi_{\{ z \in \Omega : | \phi_{n} (z) | > \lambda \} } \right\|_{L^{q(\cdot)}}^{p} d \lambda = \int_{0}^{\infty} \lambda^{p - 1} \left\| \chi_{ \left\{ z \in \Omega : n^{\frac{d - p}{p}} \phi( n z ) > \lambda \right\} } \right\|_{L^{q(\cdot)}}^{p} d \lambda \\ & = \int_{0}^{\infty} \lambda^{p - 1} \left\| \chi_{ \left\{ z \in \Omega : | \phi (n z) | > \frac{\lambda}{n^{\frac{d - p}{p}}} \right\} } \right\|_{L^{q(\cdot)}}^{p} \ d \lambda = n^{d - p} \int_{0}^{\infty} \tilde{\lambda}^{p - 1} \left\| \chi_{\left\{ z \in \Omega : \phi(n z) > \tilde{\lambda} \right\} } \right\|_{L^{q (\cdot)}}^{p} \ d \tilde{\lambda} \\ & = n^{d - p} \int_{0}^{1} \tilde{\lambda}^{p - 1} \left\| \chi_{\left\{ z \in \Omega : \phi(n z) > \tilde{\lambda} \right\} } \right\|_{L^{q (\cdot)}}^{p} \ d \tilde{\lambda} \geq n^{d - p} \int_{0}^{1} \tilde{\lambda}^{p - 1} \left\| \chi_{\left\{ z \in \Omega : \phi(n z) = 1 \right\} } \right\|_{L^{q (\cdot)}}^{p} \ d \tilde{\lambda} \\ & \geq n^{d - p} \int_{0}^{1} \tilde{\lambda}^{p - 1} \frac{\delta^{p}}{n^{d - p}} \ d \tilde{\lambda} = \frac{\delta^{p}}{p} > 0. \end{aligned}$$ Since $\left\| \nabla \phi_{n} \right\|_p \leq C$ for all $n \in \mathbb{N}$, by the assumption that $E$ is compact, there exist a subsequence $(\phi_{n_{k}})_{k = 1}^{\infty}$ of $(\phi_{n})_{n = 1}^{\infty}$ and $\psi \in L^{q(\cdot), p} (\Omega)$ such that $\left\| \phi_{n_{k}} - \psi \right\|_{L^{q(\cdot), p}} \to 0$. In particular, there exists another subsequence $(\phi_{n_{k_{j}}})_{j = 1}^{\infty}$ of $(\phi_{n_{k}})_{k = 1}^{\infty}$ that converges to $\psi$ almost everywhere. By the above, for all $n \in \mathbb{N}$, $\left\| \phi_{n} \right\|_{L^{q(\cdot), p}} \geq \frac{\delta}{p^{\frac{1}{p}}}$. So, $\psi \not = 0$. Meanwhile, by ([\[L_p phi\]](#L_p phi){reference-type="ref" reference="L_p phi"}) we obtain $\left\| \phi_{n} \right\|_{L^{p}(\Omega) } \to 0$ and this contradicts the fact that $\psi \not = 0$. Moreover, one can see that $\beta(E)\ge \frac{\delta}{p^{\frac{1}{p}}C}.$ ◻ We note that under condition ([\[non-compact\]](#non-compact){reference-type="ref" reference="non-compact"}), the Sobolev embedding is only non-compact at the neighborhood of the point $x_0 \in \Omega$ and the restriction of the Sobolev embedding at any region of $\Omega$ not containing a neighborhood of $x_0$ is compact. # Quality of Non-Compactness In this section, we show that condition ([\[non-compact\]](#non-compact){reference-type="ref" reference="non-compact"}) will give us a Sobolev embedding which is almost as non-compact as the Sobolev embedding in ([\[2\]](#2){reference-type="ref" reference="2"}), i.e. non-compactness concentrated at just one point of the domain could produce non-compactness comparable to the "most" non-compact Sobolev embedding ([\[2\]](#2){reference-type="ref" reference="2"}) which is non-compact at the neighborhood of each point of the domain. Let us recall that in , the embedding $I : W^{1, p} (\Omega) \to L^{q(.), p} (\Omega)$ is non-compact since the function $q(\cdot)$ approaches the value $p^{*}$ at a fast enough rate at $x_0\in \Omega$. So, it makes sense to introduce the following quantities: $$\gamma_{r} := \sup_{f \in W_0^{1, p} [\Omega \cap B_{r} (x_0) ]} \frac{\| f \|_{L^{q(\cdot), p} } }{\| \nabla f \|_{L^{p}} } \ge \lim_{r \to 0^{+}} \sup_{f \in W_{0}^{1, p} [\Omega \cap B_{r} (x_0) ]} \frac{\| f \|_{L^{q(\cdot), p} } }{\| \nabla f \|_{L^{p}} }=:\gamma.$$ **Lemma 9**. *Suppose that the conditions of are satisfied. Then, $\gamma_r \ge \gamma > 0$.* *Proof.* Without a loss of generality, we can assume, in the rest of this proof, that we have $\Omega = B_1(0)$ and $x_0=0$. As in the proof of , we first consider a bump function $\phi \in C_{0}^{\infty} (\mathbb{R}^{d})$ such that $\phi(x) = 1$ for all $| x | \leq \frac{1}{2}$ and $\phi(x) = 0$ for all $| x | > 1$. For each $r > 0$, define $\phi_{r} : \Omega \to [0, \infty)$ by $\phi_{r} (x) := \left( \frac{1}{r} \right)^{\frac{d - p}{p}} \phi \left( \frac{1}{r} x \right)$. Following the calculations in the proof of , we observe that for all $r > 0$, $$\left\| \nabla \phi_{r} \right\|_{L^{p} [B_{r} (0)]} = \left[ \int_{B_{r} (0)} \left| \left( \nabla \phi_{r} \right) (x) \right|^{p} d x \right]^{\frac{1}{p}} = \left[ \int_{B_{1} (0)} \left| \left( \nabla \phi \right) (x) \right|^{p} (x) d x \right]^{\frac{1}{p}} = \left\| \nabla \phi \right\|_{L^{p} [B_{1} (0)]}$$ Similarly, there exists $0 < \delta < 1$ such that for all $r \in (0, 1)$, $$\left\| \phi_{r} \right\|_{L^{q (\cdot), p} (\Omega)} \geq \frac{\delta}{p^{\frac{1}{p}}} > 0.$$ Thus, by definition, for all $r \in (0, 1)$ $$\gamma_{r}:= \sup_{f \in W^{1, p} [B_{r} (0) ]} \frac{\| f \|_{L^{q(\cdot), p} [B_{r} (0)]} }{\| \nabla f \|_{L^{p}[B_{r} (0)]} } \geq \frac{\delta}{p^{\frac{1}{p}} \left\| \nabla \phi \right\|_{L^{p} [B_{1} (0)]}} > 0.$$ $$\text{Therefore, } \gamma := \lim_{r \to 0^{+}} \gamma_{r} \geq \frac{\delta}{p^{\frac{1}{p}} \left\| \nabla \phi \right\|_{L^{p} [B_{1} (0)]}} > 0.$$ ◻ Now we introduce a couple of simple technical lemmas. **Lemma 10**. *Let $p \in [1, \infty)$, $\varepsilon > 0$. Then, there exists $k_{\varepsilon} \in \mathbb{N}$ such that for all sequences $(\alpha_{n})_{n = 1}^{\infty} \subseteq \mathbb{C}$,* *$$\sum_{n \in \left\{ m \in \mathbb{N} : | \alpha_{m}| \leq \frac{1}{2^{m + k_{\varepsilon}}} \right\} } | \alpha_{n} |^{p} < \varepsilon \sum_{n=1}^\infty |\alpha_n |^p. \label{series}$$* *Proof.* Setting $k_{\varepsilon}$ such that $$\sum_{n=1}^{\infty} \frac{1}{2^{m + k_{\varepsilon}}} < \varepsilon$$ gives us ([\[series\]](#series){reference-type="ref" reference="series"}) instantly. ◻ We skip the proof of the next lemma as it is obvious. **Lemma 11**. *Let $p \in [1, \infty]$, $\varepsilon > 0$. Suppose that $(\alpha_{n})_{n = 1}^{\infty} \in \ell^{p} (\mathbb{N})$. Given two sequences $(f_{n})_{n = 1}^{\infty}$, $(g_{n})_{n = 1}^{\infty} \subseteq X$ such that $\left\| f_{n} - g_{n} \right\|_X < \frac{\varepsilon}{2^{n}}$ for each $n \in \mathbb{N}$, we have the following:* *$$\left\| \sum_{n = 1}^{\infty} \alpha_{n} (f_{n} - g_{n}) \right\|_X \leq \left\| (\alpha_{n})_{n = 1}^{\infty} \right\|_{\ell^{p} (\mathbb{N})} \varepsilon.$$* *Consequently,* *$$\left\| \sum_{n = 1}^{\infty} \alpha_{n} g_{n} \right\|_X %= \left\| \sum_{n = 1}^{\infty} \alpha_{n} (g_{n} - f_{n}) + \sum_{n = 1}^{\infty} \alpha_{n} f_{n} \right\|_X \geq \left\| \sum_{n = 1}^{\infty} \alpha_{n} f_{n} \right\| - \left\| \sum_{n = 1}^{\infty} \alpha_{n} (f_{n} - g_{n}) \right\|% \geq \left\| \sum_{n = 1}^{\infty} \alpha_{n} f_{n} \right\|_X - \left\| (\alpha_{n})_{n = 1}^{\infty} \right\|_{\ell^{p} (\mathbb{N})} \varepsilon.$$* **Lemma 12**. *Let us suppose that conditions of are satisfied and let $\varepsilon > 0$. There exist $k_{\varepsilon} \in \mathbb{N}$ and three sequences of functions $(f_{j})_{j = 1}^{\infty}$, $(g_{j})_{j = 1}^{\infty}$, $(h_{j})_{j = 1}^{\infty}$ such that* 1. *For each $j \in \mathbb{N}$, $\gamma^{p} - \varepsilon < \left\| f_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} \leq \gamma^{p} + \varepsilon$.* 2. *For each $j \in \mathbb{N}$, $\left\| \nabla f_{j} \right\|_{L^{p} (\Omega)}^{p} - \varepsilon < \left\| \nabla g_{j} \right\|_{L^{p} (\Omega)} \leq \left\| \nabla f_{j} \right\|_{L^{p} (\Omega)}^{p}=1$.* 3. *For each $j \in \mathbb{N}$, $\left\| f_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} - \varepsilon \leq \left\| g_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} \leq \left\| f_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p}$.* 4. *For each $j \in \mathbb{N}$, $\left\| g_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} - \varepsilon \leq \left\| h_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} \leq \left\| g_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p}$.* 5. *The functions $h_{j}$'s all have disjoint support.* 6. *The functions $\nabla (g_{j})$'s all have disjoint support.* 7. *For any $(\alpha_{j})_{j = 1}^{\infty} \in \ell^{p} (\mathbb{N})$,* *$$\left\| \sum_{j = 1}^{\infty} \alpha_{j} h_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} \geq \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\} } | \alpha_{j} |^{p} \left[ \left\| h_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} - \varepsilon \right].$$* *Proof.* Without loss of generality, we can assume, in the rest of this proof, that $\Omega = B_1(0)$ and $x_0=0$. Obviously, there exists $m > 0$ such that for all $r < m$, $\gamma^{p} \leq \gamma_{r}^{p} < \gamma^{p} + \varepsilon$. Now, we will construct three sequences of functions $(f_{n})_{n = 1}^{\infty}$, $(g_{n})_{n = 1}{^\infty}$ , and $(h_{n})_{n = 1}^{\infty}$. Choose $r_{1} < m$. Then, there exists $f_{1} : B_{r_{1} (0)} \to \mathbb{R}$ such that $\| \nabla f_{1} \|_{L^{p} [B_{r_{1}} (0)]} = 1$ and $\gamma_{r_{1}}^{p} - \varepsilon < \| f_{1} \|_{L^{q(\cdot), p} [B_{r_{1}} (0)]}^{p} \leq \gamma_{r_{1}}^{p}$. So, we obtain: $\gamma^{p} - \varepsilon \leq \gamma_{r_{1}}^{p} - \varepsilon < \| f_{1} \|_{L^{q(\cdot), p} [B_{r_{1}} (0)]}^{p} \leq \gamma_{r_{1}}^{p} < \gamma^{p} + \varepsilon$ By the Szego-Polya inequality, we can assume that $f_{1}$ is a symmetric, radially decreasing function about $x = 0$. By the absolute continuity of both $\| \cdot \|_{L^{p}}$ and $\| \cdot \|_{L^{q( \cdot) , p}}$, there exists $w_{1} > 0$ such that $\| \nabla f_{1} \|_{L^{p} [B_{w_{1}} (0)]}^{p} < \varepsilon$ and $\| f_{1} \|_{L^{q(\cdot), p} [B_{w_{1}} (0)]}^{p} < \varepsilon$. Since $f_{1}$ is radially symmetric and decreasing about the point $x = 0$, the points of discontinuity of $f_{1}$ are at most countable. Choose $s_{1} < w_{1}$ such that $f_{1}$ is continuous on $\{ x \in \mathbb{R}^{d} : |x| = s_{1} \}$. Choose $y_{1} \in \{ x \in \mathbb{R}^{d} : |x| = s_{1} \}$ and define $R_{1} := f_1(y_{1})$. $$\text{Define } g_{1} : B_{r_{1}} (0) \to \mathbb{R} \text{ by } g_{1} (x) := \begin{cases} f_{1} (x) & \text{if } |x| \geq B_{s_{1}} (0),\\ R_{1} & \text{if } |x| < B_{s_{1}} (0). \end{cases}$$ By applying integration by parts, we obtain: $$(\nabla g_{1}) (x) := \begin{cases} (\nabla f_{1}) (x) & \text{if } |x| \geq B_{s_{1}} (0),\\ 0 & \text{if } |x| < B_{s_{1}} (0). \end{cases}$$ So, we obtain the following inequality: $\| \nabla f_{1} \|_{L^{p} [B_{r_{1}} (0)]}^{p} - \varepsilon < \| \nabla g_{1} \|_{L^{p} [B_{r_{1}} (0)]}^{p} \leq \| \nabla f_{1} \|_{L^{p} [B_{r_{1}} (0)]}^{p}$ and $\| f_{1} \|_{L^{q( \cdot), p} [B_{r_{1}} (0)]}^{p} - \varepsilon < \| g_{1} \|_{L^{q(\cdot), p} [B_{r_{1}} (0) ]}^{p} \leq \| f_{1} \|_{L^{q( \cdot), p} [B_{r_{1}} (0)]}^{p}$. Next, since $g_{1}$ is symmetric and radially decreasing about the point $x = 0$, by the absolute continuity of $\| \cdot \|_{L^{q(\cdot), p}}$, there exists $0 < t_{1} < s_{1}$ such that $\| g_{1} \|_{L^{q(\cdot), p} [B_{r_{1}} (0) ]}^{p} < \varepsilon$. Define $h_{1} : B_{r_{1}} (0) \to \mathbb{R}$ by $h_{1} := [ 1 - \chi_{B_{t_{1}} (0)} ] g_{1}$. With this, we obtain the following inequality: $\| g_{1} \|_{L^{q(\cdot), p} [B_{r_{1}} (0)]}^{p} - \varepsilon < \| h_{1} \|_{L^{q(\cdot), p} [B_{r_{1}} (0)]}^{p} \leq \| g_{1} \|_{L^{q(\cdot), p} [B_{r_{1}} (0)]}^{p}$. Next, let $\nu_{d}$ be the volume of the unit ball in $\mathbb{R}^{d}$ and choose $\delta_{2} > 0$ such that $\frac{1}{p} (\nu_{d} \delta_{2}^{n} )^{\frac{p}{q_{+}}} (1 + \nu_{d} \delta_{2}^{n} ) f_{1}(y) 2^{2 + k_{\varepsilon}} < \varepsilon$. Define $r_{2} := \min \{ t_{1}, \delta_{2} \}$. There exists $f_{2} : B_{r_{2}} (0) \to \mathbb{R}$ such that $\| \nabla f_{2} \|_{L^{p} [B_{r_{2}} (0)]} = 1$ and $\gamma_{r_{2}} - \varepsilon < \| f_{2} \|_{L^{q(\cdot), p} [B_{r_{2}} (0)]} \leq \gamma_{r_{2}}$. So, we obtain: $\gamma^{p} - \varepsilon \leq \gamma_{r_{2}}^{p} - \varepsilon < \| f_{2} \|_{L^{q(\cdot, p)} [B_{r_{2}} (0)]}^{p} \leq \gamma_{r_{2}}^{p} < \gamma^{p} + \varepsilon$. Then, we repeat the same process inductively as above to obtain the three sequences of functions $(f_{n})_{n = 1}^{\infty}$, $(g_{n})_{n= 1}^{\infty}$ , and $(h_{n})_{n = 1}^{\infty}$, where $f_{n}$, $g_{n}$, and $h_{n}$ are functions supported on $B_{r_{n}} (0)$. By the construction of both $g_{n}$ and $h_{n}$, we see that $\sup_{x \in B_{r_{n}} (0)} | g_{n} (x) |= R_{n} = \sup_{x \in B_{r_{n}} (0)} | h_{n} (x) |$. Let $(\alpha_{n})_{n = 1}^{\infty} \in \ell^{p}(\mathbb{N})$. By , there exists $k \in \mathbb{N}$ ($k$ depends on $\varepsilon$ and $\left\| (\alpha_{n})_{n = 1}^{\infty} \right\|_{\ell^{p}})$ such that $$\sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | \leq \frac{1}{2^{m + k}} \right\}} | \alpha_{j} |^{p} < \varepsilon \sum_{j = 1}^{\infty} | \alpha_{j} |^{p}$$ Observe that for each $n \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}$, we have $$\begin{aligned} & \int_{0}^{\frac{R_{n - 1}}{|\alpha_{n}|}} \lambda^{p - 1} \left\| \chi_{\left\{ x \in B_{r_{n}} (0) : | h_{n} (x) | > \lambda \right\}} \right\|_{L^{q(\cdot)} [B_{r_{n}} (0)]}^{p} d \lambda \\ & \leq \int_{0}^{\frac{R_{n - 1}}{|\alpha_{n}|}} \lambda^{p - 1} \left\| \chi_{\left\{ x \in B_{r_{n}} (0) : | h_{n} (x) | > \lambda \right\}} \right\|_{L^{q_{+}} [B_{r_{n}} (0)]}^{p} (1 + \nu_{d} r_{n}^{d})d \lambda \\ & \leq \int_{0}^{\frac{R_{n - 1}}{|\alpha_{n}|}} \lambda^{p - 1} (\nu_{d} r_{n}^{d})^{\frac{p}{q_{+}}} (1 + \nu_{d} r_{n}^{d}) d \lambda \\ & = \frac{1}{p} (\nu_{d} r_{n}^{d})^{\frac{p}{q_{+}}} (1 + \nu_{d} r_{n}^{d}) \left( \frac{R_{n - 1}}{|\alpha_{n}|} \right)^{p} \\ & \leq \frac{1}{p} (\nu_{d} r_{n}^{d})^{\frac{p}{q_{+}}} (1 + \nu_{d} r_{n}^{d}) \left( R_{n - 1} 2^{n + k_{\varepsilon}} \right)^{p} < \varepsilon.\end{aligned}$$ Finally, we proceed to prove (7). $$\begin{aligned} & \left\| \sum_{j = 1}^{\infty} \alpha_{j} h_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} = \int_{0}^{\infty} \lambda^{p - 1} \left\| \chi_{\left\{ x \in \Omega : \left| \sum_{j = 1}^{\infty} \alpha_{j} h_{j} (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda \\ & = \int_{0}^{\infty} \lambda^{p - 1} \left\| \sum_{j = 1}^{\infty} \chi_{\left\{ x \in \Omega : \left| \alpha_{j} h_{j} (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda \quad \text{ (since each } h_{j} \text{ has disjoint support)} \\ & \geq \int_{0}^{\infty} \lambda^{p - 1} \left\| \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} \chi_{\left\{ x \in \Omega : \left| \alpha_{j} h_{j} (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda \\ & \geq \int_{0}^{\infty} \lambda^{p - 1} \left\| \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} \chi_{\left\{ x \in \Omega : \left| \alpha_{j} h_{j} (x) \right| > \lambda \right\}} \chi_{[R_{j - 1}, R_{j} )} (\lambda) \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda \\ & = \int_{0}^{\infty} \lambda^{p - 1} \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} \left\| \chi_{\left\{ x \in \Omega : \left| \alpha_{j} h_{j} (x) \right| > \lambda \right\}} \chi_{[R_{j - 1}, R_{j} )} (\lambda) \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda \\ & = \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} \int_{R_{j - 1}}^{R_{j}} \lambda^{p - 1} \left\| \chi_{\left\{ x \in \Omega : \left| \alpha_{j} h_{j} (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda \\ & = \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \int_{\frac{R_{j - 1}}{|\alpha_{j}|}}^{\frac{R_{j}}{|\alpha_{j}|}} \lambda^{p - 1} \left\| \chi_{\left\{ x \in \Omega : \left| h_{j} (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda \\ & = \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \int_{\frac{R_{j - 1}}{|\alpha_{j}|}}^{R_{j}} \lambda^{p - 1} \left\| \chi_{\left\{ x \in \Omega : \left| h_{j} (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)}(\Omega)}^{p} d \lambda \\ & \geq \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \left[ \int_{0}^{R_{j}} \lambda^{p - 1} \left\| \chi_{\left\{ x \in \Omega : \left| h_{j} (x) \right| > \lambda \right\}} \right\|_{L^{q(\cdot)} (\Omega)}^{p} d \lambda - \varepsilon \right] \\ & = \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \left[ \left\| h_{j} \right\|_{L^{q(\cdot), p} (\Omega)}^{p} - \varepsilon \right].\end{aligned}$$ ◻ With the above construction of the sequences $(f_{n})_{n = 1}^{\infty}$, $(g_{n})_{n = 1}^{\infty}$ , and $(h_{n})_{n = 1}^{\infty}$, we can show that there is a lower bound for the Bernstein numbers. **Theorem 13**. *Consider the embedding $I : W_{0}^{1, p} (\Omega) \to L^{q(\cdot), p} (\Omega)$ with conditions from . Then, for each $N \in \mathbb{N}$, $b_{N} (E) \geq \gamma>0$.* *Proof.* Let $N \in \mathbb{N}$. Let $V_{\epsilon, N}$ be the $n$-dimensional subspace of $W_0^{1, p}$ spanned by $g_{\epsilon, 1}, \cdots, g_{\epsilon, N}$ as constructed in . By the definition of the Bernstein number, for all $\epsilon > 0$, $$b_{N} (E) \geq \inf_{x \in V_{\epsilon, N}, \| x \|_{W_0^{1, p} } = 1} \left\| x \right\|_{L^{q(\cdot), p}}$$ Now, fix $\epsilon > 0$ and consider $V_{\epsilon, N}$. Let $x = \sum_{j = 1}^{N} \alpha_{j} g_{j}$ be an arbitrary element of $V_{\epsilon, N}$, with $\sum_{j=1}^N |\alpha_j |^p=1$. Applying , and , we have: $$\begin{aligned} & \frac{\left\| \sum_{j = 1}^{N} \alpha_{j} g_{j} \right\|_{L^{q(\cdot), p}}}{\left\| \sum_{j = 1}^{N} \alpha_{j} \nabla g_{j} \right\|_{L^{p}}} \geq \frac{\left\| \sum_{j = 1}^{N} \alpha_{j} h_{j} \right\|_{L^{q(\cdot), p}} - \left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \left\| \nabla g_{j} \right\|_{L^{p}}^{p} \right)^{\frac{1}{p}}} \text{ by \thref{almost orthogonal}}\\ & \geq \frac{\left[ \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \left( \left\| h_{j} \right\|_{L^{q(\cdot), p}}^{p} - \varepsilon \right) \right]^{\frac{1}{p}} - \left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \left\| \nabla g_{j} \right\|_{L^{p}}^{p} \right)^{\frac{1}{p}}} \text{by \thref{system of functions} (7)}\\ & \geq \frac{\left[ \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \left( \left\| g_{j} \right\|_{L^{q(\cdot), p}}^{p} - 2 \varepsilon \right) \right]^{\frac{1}{p}} - \left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \left\| \nabla g_{j} \right\|_{L^{p}}^{p} \right)^{\frac{1}{p}}} \text{by \thref{system of functions} (4)} \\ & \geq \frac{\left[ \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \left( \left\| f_{j} \right\|_{L^{q(\cdot), p}}^{p} - 3 \varepsilon \right) \right]^{\frac{1}{p}} - \left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \left\| \nabla f_{j} \right\|_{L^{p}}^{p} \right)^{\frac{1}{p}}} \text{by \thref{system of functions} (3)} \\ & \geq \frac{\left[ \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \left( \gamma^{p} - 4 \varepsilon \right) \right]^{\frac{1}{p}} - \left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \right)^{\frac{1}{p}}} \text{by \thref{system of functions} (1)} \\ & = \frac{\left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} \left[ \sum_{j \in \left\{ m \in \mathbb{N} : | \alpha_{m} | > \frac{1}{2^{m + k_{\varepsilon}}} \right\}} |\alpha_{j}|^{p} \right]^{\frac{1}{p}} - \left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \right)^{\frac{1}{p}}} \\ & \geq \frac{\left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} \left[ (1 - \varepsilon) \sum_{j = 1}^{N} |\alpha_{j}|^{p} \right]^{\frac{1}{p}} - \left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \right)^{\frac{1}{p}}} \text{ by \thref{main} }\\ & = \left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \frac{\left\| (\alpha_{j} )_{j = 1}^{N} \right\|_{\ell^{\infty} (\mathbb{N})} \varepsilon }{\left( \sum_{j = 1}^{N} | \alpha_{j} |^{p} \right)^{\frac{1}{p}}} \\ & \geq \left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \varepsilon.\end{aligned}$$ Since $x \in V_{\epsilon, N}$ is arbitrary, we obtain that: $$b_{N} (E) \geq \inf_{x \in V_{\epsilon, N}, \| x \|_{W^{1, p} (\Omega)} = 1} \left\| x \right\|_{L^{q(\cdot), p}} \geq \left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \varepsilon.$$ Since $\varepsilon > 0$ is arbitrary, $b_{N} (E) \geq \gamma>0,$ where the last inequality follows from . ◻ **Theorem 14**. *Let $1 \leq p < d$, where $d \geq 2$ is a positive integer. Let $\Omega$ be a bounded domain on $\mathbb{R}^{d}$. Let $q : \Omega \to [1, \infty)$ be a measurable function satisfying $1 \leq q(x) \leq \frac{d p}{d - p}$ for almost every $x \in \Omega$. Suppose that there exist $x_{0} \in \Omega$ and constants $C_{0}$, $\eta_{0} > 0$ such that* *$$\label{non-compact3} q(x) \geq \frac{d p}{d - p} - \frac{C_{0}}{ \left| \log \left( \frac{1}{| x - x_{0} |} \right) \right|} \text{ for almost every } x \in \Omega \text{ with } | x - x_{0} | < \eta_{0}.$$* *Then the embedding $E : W_{0}^{1, p} (\Omega) \to L^{q(\cdot), p} (\Omega)$ is not strictly singular.* *Proof.* We need to show that there exists an infinite dimensional, closed subspace $Z$ of $W_{0}^{1, p} (\Omega)$ such that $$\inf \left\{ \left\| E (x) \right\|_{L^{q(\cdot), p} (\Omega)} : \left\| \nabla x \right\|_{L^{p} (\Omega)} = 1, x \in Z \right\} > 0.$$ First, choose $\varepsilon > 0$ such that $\left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \varepsilon > 0$. Then, consider the infinite dimensional subspace $Z_{\epsilon} \subseteq W_{0}^{1, p} (\Omega)$ defined by: $$Z_{\varepsilon} := \overline{\text{Span} \left\{ g_{\epsilon, j} \right\}_{j = 1}^{\infty} }^{W_{0}^{1, p} (\Omega)}, \text{ where } g_{\epsilon, j} \text{ is constructed in \thref{system of functions}}.$$ By the proof of , for each $x \in Z_{\epsilon}$ of the form $x = \sum_{j = 1}^{N} \alpha_{j} g_{j}$, we have that: $$\frac{\left\| \sum_{j = 1}^{N} \alpha_{j} g_{j} \right\|_{L^{q(\cdot), p} (\Omega)}}{ \left\| \sum_{j = 1}^{N} \alpha_{j} \nabla g_{j} \right\|_{L^{p} (\Omega)}} \geq \left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \varepsilon$$ Now, let $y \in Z_{\varepsilon}$ where $\left\| \nabla y \right\|_{L^{p} (\Omega)} = 1$. Choose $\delta > 0$ sufficiently small such that $\left\| E \right\|_{W_{0}^{1, p} \to L^{q(\cdot), p}} \delta < \left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \varepsilon$. Then, there exists $x = \sum_{j = 1}^{N} \alpha_{j} g_{j}$ such that $\| \nabla x \|_{L^{p} (\Omega)} = 1$ and $\left\| \nabla y - \nabla x \right\|_{L^{p} (\Omega)} < \delta$. $$\begin{aligned} & \left\| y \right\|_{L^{q(\cdot), p} (\Omega)} \geq \left\| x \right\|_{L^{q(\cdot), p} (\Omega)} - \left\| y - x \right\|_{L^{q(\cdot), p} (\Omega)} \geq \left\| x \right\|_{L^{q(\cdot), p} (\Omega)} - \left\| E \right\| \left\| \nabla y - \nabla x \right\|_{L^{p} (\Omega)} \\ & \geq \left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \varepsilon - \left\| E \right\| \delta > 0. \end{aligned}$$ Since $y \in Z_{\varepsilon}$ is arbitrary, $$\inf \left\{ \left\| E (y) \right\|_{L^{q(\cdot), p} (\Omega)} : \left\| \nabla y \right\|_{L^{p} (\Omega)} = 1, y \in Z_{\epsilon} \right\} \geq \left( \gamma^{p} - 4 \varepsilon \right)^{\frac{1}{p}} (1 - \varepsilon)^{\frac{1}{p}} - \varepsilon - \left\| E \right\| \delta > 0.$$ ◻ [^1]: *2020 Mathematics Subject Classification:* 46E35, 47B06\ *Keywords:* Sobolev spaces, Compactness, Bernstein numbers, Singular operators, Measure of Non-compactness

arxiv_math

arxiv_math

--- abstract: | We obtain the uniform convergence rate for the Gaussian fluctuation of the radial part of the Brownian motion on a hyperbolic space. We also show that this result is sharp if the dimension of the hyperbolic space is two or general odd. Our approach is based on the repetitive use of the Millson formula and the integration by parts formula. author: - "Yuichi Shiozawa[^1]" title: "Berry-Esseen bound for the Brownian motions on hyperbolic spaces[^2]" --- # Introduction We are concerned with the Gaussian fluctuation of the radial part of the Brownian motion on a $d$-dimensional hyperbolic space. In this note, we obtain the uniform convergence rate for the fluctuation in distribution, the so-called Berry-Esseen bound, together with the sharpness for $d=2$ or general odd $d\ge 3$. For $d\ge 2$, let ${\mathbb H}^d$ be the $d$-dimensional hyperbolic space with a pole $o$, and let $d=d_{{\mathbb H}^d}$ be the associated distance function. Let $\Delta=\Delta_{{\mathbb H}^d}$ be the Laplace-Beltrami operator, and $X=(\{X_t\}_{t\geq 0},\{P_x\}_{x\in {\mathbb H}^d})$ the Brownian motion on ${\mathbb H}^d$ generated by $\Delta/2$. Let $R_t^{(d)}=d(o,X_t) \ (t\geq 0)$ be the radial process and $P=P_o$. Then by the Itô formula applied to $R_t^{(d)}$, we have $$\label{eq:radial} R_t^{(d)}=B_t+\frac{d-1}{2}\int_0^t\coth(R_s^{(d)})\,{\rm d}s,$$ where $B_t$ is the Brownian motion on ${\mathbb R}$ (see, e.g., [@H02 Example 3.3.3]). Since $\lim_{t\rightarrow\infty}R_t^{(d)}=\infty$, we obtain the law of large numbers: $$\label{eq:linear} \lim_{t\rightarrow\infty}\frac{R_t^{(d)}}{t}=\frac{d-1}{2}, \quad \text{$P$-a.s.}$$ (see also [@GH09 Subsection 4.1] and [@S17] for the escape rate). Moreover, we realize the limiting behavior of the fluctuation in [\[eq:linear\]](#eq:linear){reference-type="eqref" reference="eq:linear"} as the central limit theorem: $$\label{eq:clt} \lim_{t\rightarrow\infty} P\left(\frac{R_t^{(d)}-(d-1)t/2}{\sqrt{t}}\geq x\right)=\Phi(x), \quad x\in {\mathbb R}$$ (see, e.g., [@B94 Corollary 3.1] and [@M10 Theorem 2.1]), where $$\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_x^{\infty}e^{-u^2/2}\,{\rm d}u, \quad x\in {\mathbb R}.$$ In connection with the heat equation in ${\mathbb H}^d$, Vázquez [@V22 (5.5)] noted that the limit in [\[eq:clt\]](#eq:clt){reference-type="eqref" reference="eq:clt"} is uniformly convergent in $x\in {\mathbb R}$. Our purpose in this note is to establish the uniform convergence rate in [\[eq:clt\]](#eq:clt){reference-type="eqref" reference="eq:clt"}: **Theorem 1**. *For any $d\geq 2$, there exists a constant $c_1>0$ such that $$\label{eq:berry-1} \sup_{x\in {\mathbb R}} \left|P\left(\frac{R_t^{(d)}-(d-1)t/2}{\sqrt{t}}\geq x\right)-\Phi(x)\right| \le \frac{c_1}{\sqrt{t}}, \quad t\ge 1.$$ Moreover, if $d=2$ or $d\ge 3$ is odd, then there exists a constant $c_2>0$ such that $$\label{eq:berry-2} P\left(\frac{R_t^{(d)}-(d-1)t/2}{\sqrt{t}}\geq 0\right)-\Phi(0) \ge \frac{c_2}{\sqrt{t}}, \quad t\ge 1.$$ Namely, the convergence rate of [\[eq:berry-1\]](#eq:berry-1){reference-type="eqref" reference="eq:berry-1"} is sharp.* Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} provides the convergence rate $t^{-1/2}$, which is consistent with the standard Berry-Esseen theorem for i.i.d. random variables (see, e.g., [@D19 Theorem 3.4.17]). We explain our approach to Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"}. According to [\[eq:radial\]](#eq:radial){reference-type="eqref" reference="eq:radial"} and [\[eq:linear\]](#eq:linear){reference-type="eqref" reference="eq:linear"}, we can regard $R_t^{(d)}$ as the Brownian motion with linear drift asymptotically. However, we have no information about the convergence rate in distribution of [\[eq:linear\]](#eq:linear){reference-type="eqref" reference="eq:linear"}. Even though we also know the matching bound on the transition density function of $X$ (see [\[eq:heat-bound\]](#eq:heat-bound){reference-type="eqref" reference="eq:heat-bound"} below), this bound is insufficient for the estimate of the distribution which would be necessary in the proof of [\[eq:berry-1\]](#eq:berry-1){reference-type="eqref" reference="eq:berry-1"}. Instead of using [\[eq:radial\]](#eq:radial){reference-type="eqref" reference="eq:radial"} or [\[eq:heat-bound\]](#eq:heat-bound){reference-type="eqref" reference="eq:heat-bound"}, we repeatedly use the Millson formula (see [\[eq:millson\]](#eq:millson){reference-type="eqref" reference="eq:millson"} below) together with the integration by parts formula. By the Millson formula, we reduce the calculation of the distribution under consideration to $d=2$ or $d=3$. At present, the validity of [\[eq:berry-2\]](#eq:berry-2){reference-type="eqref" reference="eq:berry-2"} is unavailable for general even dimensions because of the difficulty in calculation. Concerning the law of large numbers [\[eq:linear\]](#eq:linear){reference-type="eqref" reference="eq:linear"}, Cammarota-De Gregorio-Macci [@CDM14 Proposition 3.2] and Hirao [@H11 Theorem 1.1] proved the large deviation principle for the radial part by using [\[eq:heat-bound\]](#eq:heat-bound){reference-type="eqref" reference="eq:heat-bound"}. Moreover, Cammarota-De Gregorio-Macci [@CDM14 Section 3, p.1560--1563] proved the moderation deviation principle, together with the exponential decay order of the distribution related to the asymptotic normality. Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} establishes the asymptotic normality in the suitable scaling factor. Our motivation of this work lies in clarifying how the structures of the volume and spectral determine the long time behavior of a symmetric Markov process. Anker-Setti [@AS92 Theorem 2] revealed the concentration behavior in distribution of the Brownian motion on a complete and non-compact Riemannian manifold for which the volume growth rate is exponential and the bottom of the $L^2$-spectrum of the Laplacian is strictly positive. In particular, this result characterizes the linear growth rate of the radial part of the Brownian motion in terms of the exponential volume growth rate and the bottom of the $L^2$-spectrum. For the Riemannian manifold with a pole, Grigor'yan-Hsu [@GH09 Theorem 4.1] determined the linear growth rate exactly. Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} is an attempt to provide the second order asymptotic behavior of the linear growth phenomena of a symmetric Markov process in a quantitative way. We close this introduction with some words on contents and notations. The proof of Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} is given in Section [2](#sect:berry){reference-type="ref" reference="sect:berry"}. A lemma on some elementary calculus is postponed to Appendix. For a fixed constant $T>0$, let $f_1(t)$ and $f_2(t)$ be positive functions defined on $[T,\infty)$. We then write $f_1(t)\sim f_2(t)$ if $f_1(t)/f_2(t)\rightarrow 1$ as $t\rightarrow \infty$. Let $S$ be a set, and let $g_1(s)$ and $g_2(s)$ be positive functions defined on $S$. We then write $g_1(s)\lesssim g_2(s)$ if there exists a constant $c>0$ such that $g_1(s)\le cg_2(s)$ for all $s\in S$. We further write $g_1(s)\asymp g_2(s)$ if $g_1(s)\lesssim g_2(s)$ and $g_2(s)\lesssim g_1(s)$. # Proof of Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} {#sect:berry} In this section, we first recall the Millson formula and the estimates of the transition density function of the Brownian motion on ${\mathbb H}^d$. We then prove Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for odd dimensions and even dimensions in this order. ## Preliminaries Let $d\ge 2$. The $d$-dimensional hyperbolic space ${\mathbb H}^d$ is a spherically symmetric Riemannian manifold with the Riemannian distance given by $${\rm d}s^2={\rm d}r^2+(\sinh r)^2\,{\rm d}\theta^2.$$ Here ${\rm d}\theta^2$ is the distance on the $(d-1)$-dimensional surface $S^{d-1}=\{\theta\in {\mathbb R}^d \mid |\theta|=1\}$. We write $\omega_d=2\pi^{d/2}/\Gamma(d/2)$ for the surface area of $S^{d-1}$. Let $d$ and ${\rm d}v$ denote the associated distance function and volume measure, respectively. Let $X=(\{X_t\}_{t\geq 0},\{P_x\}_{x\in {\mathbb H}^d})$ be the Brownian motion on ${\mathbb H}^d$ generated by $\Delta/2$, the half of the Laplace-Beltrami operator on ${\mathbb H}^d$. Then there exists a Borel measurable function $p_d(t,x,y):(0,\infty)\times {\mathbb H}^d\times {\mathbb H}^d \to (0,\infty)$ such that $$P_x(X_t\in A)=\int_A p_d(t,x,y)\,v({\rm d}y), \quad x\in {\mathbb H}^d, \ t>0, \ A\in {\cal B}({\mathbb H}^d).$$ Namely, $p_d(t,x,y)$ is the transition density function of $X$. Moreover, there exists a Borel measurable function $q_d(t,r):(0,\infty)\times (0,\infty)\to (0,\infty)$ such that $p_d(t,x,y)=q_d(t,d(x,y))$ for any $x,y\in {\mathbb H}^d$ and $t>0$, and $$\label{eq:dist-radial} P_x(d(x,X_t)\in B) =\omega_d \int_B q_d(t,r) \sinh^{d-1}(r) \,{\rm d}r, \quad t>0, \ B\in {\cal B}([0,\infty))$$ (see, e.g., [@CDM14 §2.2] for details). It is known that $$\label{eq:d=2} q_2(t,r)=\frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}} \int_r^{\infty}\frac{s e^{-s^2/(2t)}}{(\cosh s-\cosh r)^{1/2}}\,{\rm d}s$$ and $$\label{eq:d=3} q_3(t,r)=\frac{e^{-t/2}}{(2\pi t)^{3/2}}\frac{r}{\sinh r}e^{-r^2/(2t)}$$ (see, e.g., [@D92 Section 5.7] or [@G96 Section 2] and references therein). The Millson formula is a recursive relation between the transition density functions: $$\label{eq:millson} q_d(t,r)=-\frac{e^{-(d-2)t/2}}{2\pi \sinh r}\frac{\partial q_{d-2}}{\partial r}(t,r), \quad t>0, \ r>0$$ (see, e.g., [@DM88; @GN98], [@D92 Section 5.7] and references therein). We also see by [@D92 Theorem 5.7.2] that $$\label{eq:heat-bound} q_d(t,r) \asymp \frac{1}{t^{d/2}} \exp\left(-\frac{(d-1)^2}{8}t-\frac{d-1}{2r}-\frac{r^2}{2t}\right)(1+r+t)^{(d-3)/2}(1+r), \quad t>0, \ r>0.$$ ## Odd dimensions In this subsection, we first prove Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for $d=3$. Using this assertion, we next prove Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for general odd dimensions. *Proof of Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for $d=3$.* We first prove [\[eq:berry-1\]](#eq:berry-1){reference-type="eqref" reference="eq:berry-1"}. We write $R_t=R_t^{(3)}$ for simplicity. Since $R_t\geq 0$ for any $t\geq 0$ and $\omega_3=4\pi$, it follows by [\[eq:d=3\]](#eq:d=3){reference-type="eqref" reference="eq:d=3"} that for any $x\in {\mathbb R}$, $$\begin{split} P\left(\frac{R_t-t}{\sqrt{t}}\geq x\right) =P\left(R_t\geq (t+x\sqrt{t})\vee 0\right) &=\frac{\omega_3}{(2\pi t)^{3/2}}\int_{(t+x\sqrt{t})\vee 0}^{\infty} e^{-t/2}e^{-u^2/(2t)} u\sinh u\,{\rm d}u\\ &=\frac{1}{\sqrt{2\pi}}\frac{2e^{-t/2}}{t\sqrt{t}} \int_{(t+x\sqrt{t})\vee 0}^{\infty} e^{-u^2/(2t)}u\sinh u\,{\rm d}u. \end{split}$$ Then by the change of variables formula with $u=t+v\sqrt{t}$, we obtain $$\begin{split} &\frac{2e^{-t/2}}{t\sqrt{t}} \int_{(t+x\sqrt{t})\vee 0}^{\infty} e^{-u^2/(2t)}u\sinh u\,{\rm d}u\\ &=2e^{-t}\int_{x\vee (-\sqrt{t})}^{\infty}e^{-v^2/2}e^{-v\sqrt{t}}\sinh(t+v\sqrt{t})\,{\rm d}v +\frac{2e^{-t}}{\sqrt{t}}\int_{x\vee (-\sqrt{t})}^{\infty}e^{-v^2/2}e^{-v\sqrt{t}}v\sinh(t+v\sqrt{t})\,{\rm d}v\\ &=I_1(t,x)+I_2(t,x). \end{split}$$ Therefore, $$\label{eq:dist} P\left(\frac{R_t-t}{\sqrt{t}}\geq x\right)-\Phi(x) =\frac{1}{\sqrt{2\pi}}\left\{\left(I_1(t,x) -\int_x^{\infty}e^{-u^2/2}\,{\rm d}u\right)+I_2(t,x)\right\}.$$ We have $$\label{eq:dist-0} \begin{split} I_1(t,x)-\int_{x\vee(-\sqrt{t})}^{\infty}e^{-u^2/2}\,{\rm d}u &=\int_{x\vee (-\sqrt{t})}^{\infty}e^{-v^2/2}(1-e^{-2(t+v\sqrt{t})})\,{\rm d}v -\int_{x\vee(-\sqrt{t})}^{\infty}e^{-u^2/2}\,{\rm d}u \\ &=-\int_{x\vee (-\sqrt{t})}^{\infty}e^{-v^2/2}e^{-2(t+v\sqrt{t})}\,{\rm d}v. \end{split}$$ Then by the change of variables formula again with $u=v+2\sqrt{t}$, there exists a constant $c_1>0$ such that for any $x\in {\mathbb R}$ and $t\geq 1$, $$\label{eq:bound} \int_{x\vee (-\sqrt{t})}^{\infty}e^{-v^2/2}e^{-2(t+v\sqrt{t})}\,{\rm d}v =\int_{(x+2\sqrt{t})\vee \sqrt{t}}^{\infty}e^{-u^2/2}\,{\rm d}u \le \int_{\sqrt{t}}^{\infty}e^{-u^2/2}\,{\rm d}u \le \frac{c_1e^{-t/2}}{\sqrt{t}}.$$ Hence for any $x\in {\mathbb R}$ and $t\ge 1$, $$0\ge I_1(t,x)-\int_{x\vee(-\sqrt{t})}^{\infty}e^{-u^2/2}\,{\rm d}u \ge -\frac{c_1e^{-t/2}}{\sqrt{t}}.$$ We also note that for any $x\in {\mathbb R}$ and $t\geq 1$, $$\int_x^{x\vee (-\sqrt{t})}e^{-u^2/2}\,{\rm d}u \leq \int_{-\infty}^{-\sqrt{t}}e^{-u^2/2}\,{\rm d}u \lesssim \frac{e^{-t/2}}{\sqrt{t}}.$$ By combining two inequalities above, there exists a constant $c_2>0$ such that for any $x\in {\mathbb R}$ and $t\ge 1$, $$I_1(t,x)-\int_x^{\infty}e^{-u^2/2}\,{\rm d}u =\left(I_1(t,x)-\int_{x\vee (-\sqrt{t})}^{\infty}e^{-u^2/2}\,{\rm d}u\right) -\int_x^{x\vee (-\sqrt{t})}e^{-u^2/2}\,{\rm d}u \ge -\frac{c_2e^{-t/2}}{\sqrt{t}}.$$ Therefore, $$\label{eq:i-bound} \left|I_1(t,x)-\int_x^{\infty}e^{-u^2/2}\,{\rm d}u\right| \lesssim \frac{e^{-t/2}}{\sqrt{t}}, \quad t\ge 1.$$ Since $(e^{-v^2/2})'=-ve^{-v^2/2}$, we have by the integration by parts formula, $$\label{eq:dist-2} \begin{split} I_2(t,x) &=\frac{1}{\sqrt{t}}\int_{x\vee (-\sqrt{t})}^{\infty}ve^{-v^2/2}(1-e^{-2(t+v\sqrt{t})})\,{\rm d}v\\ &=\frac{e^{-(x\vee (-\sqrt{t}))^2/2}}{\sqrt{t}}(1-e^{-2(t+(x\vee (-\sqrt{t}))\sqrt{t})}) +2\int_{x\vee (-\sqrt{t})}^{\infty}e^{-v^2/2}e^{-2(t+v\sqrt{t})}\,{\rm d}v\ge 0. \end{split}$$ In particular, we see by [\[eq:bound\]](#eq:bound){reference-type="eqref" reference="eq:bound"} that $$0\le I_2(t,x) \leq \frac{1}{\sqrt{t}}\int_0^{\infty}ve^{-v^2/2}\,{\rm d}v =\frac{1}{\sqrt{t}}.$$ Combining this with [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"} and [\[eq:i-bound\]](#eq:i-bound){reference-type="eqref" reference="eq:i-bound"}, we get [\[eq:berry-1\]](#eq:berry-1){reference-type="eqref" reference="eq:berry-1"} for $d=3$. We next prove [\[eq:berry-2\]](#eq:berry-2){reference-type="eqref" reference="eq:berry-2"}. If we take $x=0$, then by a calculation similar to [\[eq:dist-0\]](#eq:dist-0){reference-type="eqref" reference="eq:dist-0"} and [\[eq:bound\]](#eq:bound){reference-type="eqref" reference="eq:bound"}, it follows that for some $c_3>0$, $$I_1(t,0)-\int_0^{\infty}e^{-u^2/2}\,{\rm d}u =-\int_{2\sqrt{t}}^{\infty}e^{-u^2/2}\,{\rm d}u\ge -\frac{c_3e^{-2t}}{\sqrt{t}}.$$ We also have for any $t\geq 1$, $$I_2(t,0) =\frac{1}{\sqrt{t}}\int_0^{\infty}ve^{-v^2/2}\left(1-e^{-2(t+v\sqrt{t})}\right)\,{\rm d}v \geq \frac{1-e^{-2}}{\sqrt{t}}\int_0^{\infty}ve^{-v^2/2}\,{\rm d}v =\frac{1-e^{-2}}{\sqrt{t}}.$$ Hence by [\[eq:dist\]](#eq:dist){reference-type="eqref" reference="eq:dist"}, there exists a constant $c_4>0$ such that for any $t\geq 1$, $$P\left(\frac{R_t-t}{\sqrt{t}}\geq 0\right)-\Phi(0) =\frac{1}{\sqrt{2\pi}}\left\{\left(I_1(t,0)-\int_0^{\infty}e^{-u^2/2}\,{\rm d}u\right)+I_2(t,0)\right\} \ge \frac{c_4}{\sqrt{t}},$$ which implies [\[eq:berry-2\]](#eq:berry-2){reference-type="eqref" reference="eq:berry-2"} for $d=3$. ◻ *Proof of Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for odd dimensions.* Suppose that $d\ge 3$ is odd. Let $$T=T(t,x)=\left(x\sqrt{t}+\frac{d-1}{2}t\right)\vee 0.$$ Since $R_t^{(d)}\geq 0$, we have by [\[eq:dist-radial\]](#eq:dist-radial){reference-type="eqref" reference="eq:dist-radial"} and [\[eq:millson\]](#eq:millson){reference-type="eqref" reference="eq:millson"}, $$\label{eq:d-dist} \begin{split} P\left(\frac{R_t^{(d)}-(d-1)t/2}{\sqrt{t}}\geq x\right) &=\omega_d\int_T^{\infty}q_d(t,r)\sinh^{d-1}r \,{\rm d}r\\ &=-\frac{\omega_de^{-(d-2)t/2}}{2\pi} \int_T^{\infty}\frac{\partial q_{d-2}}{\partial r}(t,r)\sinh^{d-2}r \,{\rm d}r. \end{split}$$ Then by the integration by parts formula and [\[eq:heat-bound\]](#eq:heat-bound){reference-type="eqref" reference="eq:heat-bound"}, $$\begin{split} -\int_T^{\infty}\frac{\partial q_{d-2}}{\partial r}(t,r)\sinh^{d-2}r \,{\rm d}r &=\left[-q_{d-2}(t,r)\sinh^{d-2}r\right]_{r=T}^{r=\infty} +\int_{T}^{\infty}q_{d-2}(t,r)\left(\frac{\partial}{\partial r}\sinh^{d-2}r\right) \,{\rm d}r\\ &=q_{d-2}(t,T)\sinh^{d-2}T +\int_T^{\infty}q_{d-2}(t,r)\left(\frac{\partial}{\partial r}\sinh^{d-2}r\right) \,{\rm d}r. \end{split}$$ In the same way, we have $$\begin{split} \int_T^{\infty}q_{d-2}(t,r)\left(\frac{\partial}{\partial r}\sinh^{d-2}r\right) \,{\rm d}r &=-\frac{e^{-(d-4)t/2}}{2\pi}\int_T^{\infty}\frac{1}{\sinh r}\frac{\partial q_{d-4}}{\partial r}(t,r) \left(\frac{\partial}{\partial r}\sinh^{d-2}r\right) \,{\rm d}r\\ &=\frac{e^{-(d-4)t/2}}{2\pi}q_{d-4}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)\sinh^{d-2}r\Big|_{r=T}\\ &+\frac{e^{-(d-4)t/2}}{2\pi}\int_T^{\infty}q_{d-4}(t,r) \frac{\partial}{\partial r}\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)\sinh^{d-2}r \,{\rm d}r. \end{split}$$ Repeating this procedure, we see from [\[eq:d-dist\]](#eq:d-dist){reference-type="eqref" reference="eq:d-dist"} that for any $n\geq 2$ with $d\geq 2n$, $$\label{eq:d-dist-1} \begin{split} &P\left(\frac{R_t^{(d)}-(d-1)t/2}{\sqrt{t}}\geq x\right)\\ &=\omega_d\sum_{m=1}^{n-1}\prod_{j=1}^m\frac{e^{-(d-2j)t/2}}{2\pi}q_{d-2m}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{d-2}r\Big|_{r=T}\\ &+\omega_d\prod_{j=1}^{n-1}\frac{e^{-(d-2j)t/2}}{2\pi} \int_T^{\infty} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{d-2} r\right)q_{d-2n+2}(t,r)\sinh r\,{\rm d}r\\ &=\omega_d\sum_{m=1}^{n-1}\frac{e^{-(d-(m+1))mt/2}}{(2\pi)^m} q_{d-2m}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{d-2}r\Big|_{r=T}\\ &+\omega_d\frac{e^{-(d-n)(n-1)t/2}}{(2\pi)^{n-1}} \int_T^{\infty} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{d-2} r\right)q_{d-2n+2}(t,r)\sinh r\,{\rm d}r. \end{split}$$ Let us calculate the first term of the last expression of [\[eq:d-dist-1\]](#eq:d-dist-1){reference-type="eqref" reference="eq:d-dist-1"}. We now assume that $x\geq -(d-1)\sqrt{t}/2$ and so $T\ge 0$. Then by [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} and [\[eq:ind-2\]](#eq:ind-2){reference-type="eqref" reference="eq:ind-2"}, for any $m$  ($1\le m\le n-1$), there exists a constant $c_1>0$ such that $$\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{d-2}r\Big|_{r=T} \leq c_1e^{(d-m-1)T}.$$ We also see from [\[eq:heat-bound\]](#eq:heat-bound){reference-type="eqref" reference="eq:heat-bound"} that $$\begin{split} &q_{d-2m}(t,T)\\ &\asymp \frac{1}{t^{(d-2m)/2}} \exp\left(-\frac{(d-2m-1)^2}{8}t-\frac{(d-2m-1)T}{2}-\frac{T^2}{2t}\right) (1+T+t)^{(d-2m-3)/2}(1+T). \end{split}$$ Since there exists a constant $c_2>0$ such that for any $t\geq 1$ and $x\geq -(d-1)\sqrt{t}/2$, $$\begin{split} (1+T+t)^{(d-2m-3)/2}(1+T) &\le (1+T+t)^{(d-2m-1)/2}=(1+x\sqrt{t}+(d-1)t/2+t)^{(d-2m-1)/2}\\ &\le c_2(1+(|x|\sqrt{t})^{(d-2m-1)/2}+t^{(d-2m-1)/2}), \end{split}$$ we can take a constant $c_3>0$ such that for any $t\geq 1$ and $x\geq -(d-1)\sqrt{t}/2$, $$\frac{e^{-x^2/2}}{t^{(d-2m)/2}}(1+T+t)^{(d-2m-3)/2}(1+T) \leq c_2\left(\frac{e^{-x^2/2}}{t^{d-2m}}+\frac{e^{-x^2/2}|x|^{(d-2m-1)/2}}{t^{(d-2m)/4+1/4}}+\frac{1}{\sqrt{t}}\right) \leq \frac{c_3}{\sqrt{t}}.$$ At the last inequality, we used the fact that $\sup_{x\in {\mathbb R}}e^{-x^2/2}|x|^{(d-2m-1)/2}<\infty$. By noting that $$\exp\left(-\frac{(d-2m-1)^2}{8}t-\frac{(d-2m-1)T}{2}-\frac{T^2}{2t}\right)e^{(d-m-1)T} =e^{-x^2/2}e^{(d-(m+1))mt/2},$$ there exist positive constants $c_4$ and $c_5$ such that for any $t\geq 1$ and $x\geq -(d-1)\sqrt{t}/2$, $$\label{eq:bound-large} \begin{split} &\frac{e^{-(d-(m+1))mt/2}}{(2\pi)^m} q_{d-2m}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{d-2}r\Big|_{r=T}\\ &\le \frac{c_4e^{-x^2/2}}{t^{(d-2m)/2}}(1+T+t)^{(d-2m-3)/2}(1+T) \le \frac{c_5}{\sqrt{t}}. \end{split}$$ On the other hand, if $x\leq -(d-1)\sqrt{t}/2$, then $T=0$ so that by [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} and [\[eq:ind-2\]](#eq:ind-2){reference-type="eqref" reference="eq:ind-2"}, $$\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{d-2}r\Big|_{r=0}=0.$$ Combining this with [\[eq:bound-large\]](#eq:bound-large){reference-type="eqref" reference="eq:bound-large"}, we have for any $t\geq 1$, $$\sup_{x\in {\mathbb R}} \omega_d\sum_{m=1}^{n-1} \frac{e^{-(d-(m+1))mt/2}}{(2\pi)^m} q_{d-2m}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{d-2}r\Big|_{r=T} \lesssim \frac{1}{\sqrt{t}}.$$ In particular, if we take $d=2n+1$, then $$\label{eq:bound-all} \sup_{x\in {\mathbb R}} \omega_d\sum_{m=1}^{n-1} \frac{e^{-(2n-m)mt/2}}{(2\pi)^m} q_{2n+1-2m}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{2n-1}r\Big|_{r=T} \lesssim \frac{1}{\sqrt{t}}.$$ We turn to the second term of the last expression of [\[eq:d-dist-1\]](#eq:d-dist-1){reference-type="eqref" reference="eq:d-dist-1"}. Let $d=2n+1$ so that $T=(x\sqrt{t}+nt)\vee 0$. By [\[eq:d=3\]](#eq:d=3){reference-type="eqref" reference="eq:d=3"} and [\[eq:deri-0\]](#eq:deri-0){reference-type="eqref" reference="eq:deri-0"}, we have $$\begin{split} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-1}r\right)q_3(t,r)\sinh r &=\frac{(2n-1)!!}{n}\sinh(nr)\frac{e^{-t/2}}{(2\pi t)^{3/2}}\frac{r}{\sinh r} e^{-r^2/(2t)}\sinh r\\ &=\frac{(2n-1)!!}{n}\frac{e^{-t/2}}{(2\pi t)^{3/2}} re^{-r^2/(2t)}\sinh(nr) \end{split}$$ and thus $$\begin{split} &\frac{e^{-((2n+1)-n)(n-1)t/2}}{(2\pi)^{n-1}} \int_T^{\infty} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-1}r\right)q_3(t,r)\sinh r\,{\rm d}r\\ &=\frac{e^{-(n^2-1)t/2}}{(2\pi)^{n-1}}\frac{(2n-1)!!}{n} \frac{e^{-t/2}}{(2\pi t)^{3/2}} \int_T^{\infty}re^{-r^2/(2t)}\sinh(nr)\,{\rm d}r\\ &=\frac{1}{(2\pi)^{n-1}(2\pi t)^{3/2}}\frac{(2n-1)!!}{n} \int_T^{\infty}re^{-n^2t/2}e^{-r^2/(2t)}\sinh(nr)\,{\rm d}r. \end{split}$$ Since $$\omega_{2n+1}=\frac{(2\pi)^{n-1} 4\pi}{(2n-1)!!} =\frac{(2\pi)^{n-1}\omega_3}{(2n-1)!!},$$ we get by [\[eq:d=3\]](#eq:d=3){reference-type="eqref" reference="eq:d=3"}, $$\begin{split} & \omega_{2n+1}\frac{e^{-((2n+1)-n)(n-1)t/2}}{(2\pi)^{n-1}} \int_T^{\infty} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1} \sinh^{2n-1}r\right)q_3(t,r)\sinh r\,{\rm d}r\\ &=\frac{\omega_3}{n(2\pi t)^{3/2}} \int_T^{\infty}re^{-n^2t/2}e^{-r^2/(2t)}\sinh(nr)\,{\rm d}r\\ &=\frac{\omega_3}{(2\pi n^2t)^{3/2}} \int_{(x\sqrt{n^2 t}+n^2 t)\vee 0}^{\infty} \frac{u}{\sinh u}e^{-n^2t/2}e^{-u^2/(2n^2t)}\sinh^2 u\,{\rm d}u\\ &=\frac{\omega_3}{(2\pi n^2t)^{3/2}}\int_{(x\sqrt{n^2 t}+n^2 t)\vee 0}^{\infty} q_3(n^2t,u)\sinh^2 u\,{\rm d}u =P\left(\frac{R_{n^2 t}^{(3)}-n^2t}{\sqrt{n^2t}}\geq x\right). \end{split}$$ At the second equality above, we used the change of variable formula with $u=nr$. Hence by [\[eq:d-dist-1\]](#eq:d-dist-1){reference-type="eqref" reference="eq:d-dist-1"}, $$\label{eq:distance} \begin{split} &P\left(\frac{R_t^{(2n+1)}-nt}{\sqrt{t}}\geq x\right)-\Phi(x)\\ &=\omega_{2n+1}\sum_{m=1}^{n-1} \frac{e^{-(2n-m)mt/2}}{(2\pi)^m} q_{2n+1-2m}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{2n-1}r\Big|_{r=T}\\ &+P\left(\frac{R_{n^2 t}^{(3)}-n^2t}{\sqrt{n^2t}}\geq x\right) -\Phi(x). \end{split}$$ Since Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} is already proved for $d=3$, we see by [\[eq:bound-all\]](#eq:bound-all){reference-type="eqref" reference="eq:bound-all"} that $$\sup_{x\in {\mathbb R}}\left|P\left(\frac{R_t^{(2n+1)}-nt}{\sqrt{t}}\geq x\right)-\Phi(x)\right| \lesssim \frac{1}{\sqrt{t}}, \quad t\geq 1.$$ The proof of [\[eq:berry-1\]](#eq:berry-1){reference-type="eqref" reference="eq:berry-1"} is complete for general odd dimensions. We next prove [\[eq:berry-2\]](#eq:berry-2){reference-type="eqref" reference="eq:berry-2"} for general odd dimensions. It follows by [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} and [\[eq:ind-2\]](#eq:ind-2){reference-type="eqref" reference="eq:ind-2"}, that, for any $m$  ($1\le m\le n-1$), there exists a constant $c_6>0$ such that $$\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{2n-1}r\Big|_{r=nt} \ge c_6e^{(2n-m)nt}.$$ Hence as in the proof of [\[eq:bound-large\]](#eq:bound-large){reference-type="eqref" reference="eq:bound-large"}, we have for some $c_7>0$, $$\label{eq:bound-lower} \sum_{m=1}^{n-1} \frac{e^{-(2n-m)mt/2}}{(2\pi)^m} q_{2n+1-m}(t,nt) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{2n-1}r\Big|_{r=nt} \ge\frac{c_7}{\sqrt{t}}, \quad t\ge 1.$$ Since [\[eq:berry-2\]](#eq:berry-2){reference-type="eqref" reference="eq:berry-2"} is already proved for $d=3$, we see from [\[eq:distance\]](#eq:distance){reference-type="eqref" reference="eq:distance"} (with $x=0$) and [\[eq:bound-lower\]](#eq:bound-lower){reference-type="eqref" reference="eq:bound-lower"} that there exists a constant $c_8>0$ such that $$P\left(\frac{R_t^{(2n+1)}-nt}{\sqrt{t}}\geq 0\right) -\Phi(0) \ge \frac{c_8}{\sqrt{t}}, \quad t\geq 1.$$ The proof of [\[eq:berry-2\]](#eq:berry-2){reference-type="eqref" reference="eq:berry-2"} is complete for general odd dimensions. ◻ ## Even dimensions In this subsection, we establish Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for even dimensions. Suppose that $d=2n$ for some $n\ge 1$. Then $T=(x\sqrt{t}+(n-1/2)t)\vee 0$. By [\[eq:d-dist-1\]](#eq:d-dist-1){reference-type="eqref" reference="eq:d-dist-1"}, $$\label{eq:i+ii} \begin{split} &P\left(\frac{R_t^{(2n)}-(n-1/2)t}{\sqrt{t}}\ge x\right)\\ &=\omega_{2n}\sum_{m=1}^{n-1}\frac{e^{-m(n-(m+1)/2)t}}{(2\pi)^m}q_{2n-2m}(t,T) \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{m-1}\sinh^{2n-2}r\Big|_{r=T}\\ &+\omega_{2n}\frac{e^{-n(n-1)t/2}}{(2\pi)^{n-1}} \int_T^{\infty} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-2} r\right)q_2(t,r)\sinh r\,{\rm d}r\\ &=J_1(t,x)+J_2(t,x), \end{split}$$ where we make the convention $\sum_{k=1}^0=0$ so that $J_1(t,x)=0$ for $n=1$. Therefore, $$\label{eq:d-dist-even} P\left(\frac{R_t^{(2n)}-(n-1/2)t}{\sqrt{t}}\ge x\right)-\Phi(x) =J_1(t,x)+\left(J_2(t,x)-\Phi(x)\right).$$ We now estimate the right hand side above. **Lemma 2**. *There exists a constant $c>0$ such that for any $t\ge 1$, $$\sup_{x\in {\mathbb R}}J_1(t,x)\le \frac{c}{\sqrt{t}}.$$* We omit the proof of Lemma [Lemma 2](#lem:i-bound){reference-type="ref" reference="lem:i-bound"} because the argument for [\[eq:bound-all\]](#eq:bound-all){reference-type="eqref" reference="eq:bound-all"} still works. **Lemma 3**. *There exists a constant $c>0$ such that for any $t\geq 1$, $$\sup_{x\in {\mathbb R}}\left|J_2(t,x)-\Phi(x)\right| \le \frac{c}{\sqrt{t}}.$$* *Proof.* Let $$a_n(t)=\omega_{2n}\frac{e^{-n(n-1)t/2}}{(2\pi)^{n-1}}=\frac{\pi e^{-n(n-1)t/2}}{2^{n-2}(n-1)!}$$ and $$K(t,x) =\int_T^{\infty} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1} \sinh^{2n-2} r\right)q_2(t,r)\sinh r\,{\rm d}r$$ so that $$\label{eq:ii-divide-0} J_2(t,x)=a_n(t)K(t,x).$$ Then by [\[eq:d=2\]](#eq:d=2){reference-type="eqref" reference="eq:d=2"} and the Fubini theorem, $$\label{eq:d-dist-even-1} \begin{split} &K(t,x)\\ &=\frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}} \int_T^{\infty}s e^{-s^2/(2t)} \left\{\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-2} r\right) \frac{\sinh r}{(\cosh s-\cosh r)^{1/2}}\,{\rm d}r\right\}\,{\rm d}s. \end{split}$$ By the integration by parts formula, $$\begin{split} &\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-2} r\right) \frac{\sinh r}{(\cosh s-\cosh r)^{1/2}}\,{\rm d}r\\ &=\left[-2\left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-2} r\right) (\cosh s-\cosh r)^{1/2}\right]_{r=T}^{r=s}\\ &+2\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n} \sinh^{2n-2} r\right)(\cosh s-\cosh r)^{1/2}\sinh r \,{\rm d}r\\ &=2\left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-2} r\right)\Bigg|_{r=T} (\cosh s-\cosh T)^{1/2}\\ &+2\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n} \sinh^{2n-2} r\right)(\cosh s-\cosh r)^{1/2}\sinh r \,{\rm d}r. \end{split}$$ Inductively, we get $$\label{eq:ind-3} \begin{split} &\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-1}\sinh^{2n-2} r\right) \frac{\sinh r}{(\cosh s-\cosh r)^{1/2}}\,{\rm d}r\\ &=\sum_{k=1}^{n-1}\frac{2^k}{(2k-1)!!} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-2+k}\right)\sinh^{2n-2}r\,\Big|_{r=T} (\cosh s-\cosh T)^{k-1/2}\\ &+\frac{2^{n-1}}{(2n-3)!!}\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2n-2}\sinh^{2n-2}r\right) (\cosh s-\cosh r)^{n-3/2}\sinh r\,{\rm d}r. \end{split}$$ Since it follows by [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} that $$\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2n-2}\sinh^{2n-2}r =(2n-2)!,$$ we have $$\begin{split} &\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2n-2}\sinh^{2n-2}r\right) (\cosh s-\cosh r)^{n-3/2}\sinh r\,{\rm d}r\\ &=(2n-2)! \int_T^s (\cosh s-\cosh r)^{n-3/2}\sinh r\,{\rm d}r =\frac{2(2n-2)!}{2n-1}(\cosh s-\cosh T)^{n-1/2} \end{split}$$ and thus $$\begin{split} &\frac{2^{n-1}}{(2n-3)!!}\int_T^s \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2n-2}\sinh^{2n-2}r\right) (\cosh s-\cosh r)^{n-3/2}\sinh r\,{\rm d}r\\ &=\frac{2^n(2n-2)!}{(2n-1)!!}(\cosh s-\cosh T)^{n-1/2} =\frac{2^{2n-1}(n-1)!}{2n-1}(\cosh s-\cosh T)^{n-1/2}. \end{split}$$ Combining this with [\[eq:d-dist-even-1\]](#eq:d-dist-even-1){reference-type="eqref" reference="eq:d-dist-even-1"} and [\[eq:ind-3\]](#eq:ind-3){reference-type="eqref" reference="eq:ind-3"}, we obtain $$\label{eq:d-dist-even-2} \begin{split} K(t,x) &=\frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}} \sum_{k=1}^{n-1}\frac{2^k}{(2k-1)!!} \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-2+k}\right)\sinh^{2n-2}r\,\Big|_{r=T}\\ &\times \int_T^{\infty}s e^{-s^2/(2t)} (\cosh s-\cosh T)^{k-1/2}\,{\rm d}s\\ &+\frac{2^{2n-1}(n-1)!}{2n-1}\frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}} \int_T^{\infty}s e^{-s^2/(2t)} (\cosh s-\cosh T)^{n-1/2}\,{\rm d}s\\ &=K_1(t,x)+K_2(t,x), \end{split}$$ whence by [\[eq:ii-divide-0\]](#eq:ii-divide-0){reference-type="eqref" reference="eq:ii-divide-0"}, $$\label{eq:ii-divide} J_2(t,x)=a_n(t)K_1(t,x)+a_n(t)K_2(t,x).$$ Let us first estimate $a_n(t)K_1(t,x)$. By [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} and [\[eq:ind-2\]](#eq:ind-2){reference-type="eqref" reference="eq:ind-2"}, there exists a constant $c_1>0$, which is independent of $t$, $T$ and $k \ (1\le k\le n-1)$, such that $$\label{eq:sinh-deri} 0\le \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-2+k}\right)\sinh^{2n-2}r\,\Big|_{r=T} \le c_1 e^{(n-k)T}=c_1 e^{(n-k)((x\sqrt{t}+(n-1/2)t)\vee 0)}.$$ By the change of variables formula with $s=u\sqrt{t}+(n-1/2)t$, we also obtain $$\begin{split} &\frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}}\int_T^{\infty}s e^{-s^2/(2t)}(\cosh s-\cosh T)^{k-1/2}\,{\rm d}s\\ &=\frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}} \sqrt{t}\int_{x\vee (-(n-1/2)\sqrt{t})}^{\infty} \left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right) e^{-(u+(n-1/2)\sqrt{t})^2/2}\\ &\times \left(\cosh\left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right) -\cosh \left(\left(x\sqrt{t}+\left(n-\frac{1}{2}\right)t\right)\vee 0\right)\right)^{k-1/2}\,{\rm d}u. \end{split}$$ Then for any $k\ge 1$ and $u\ge x\vee (-(n-1/2)\sqrt{t})$, we have $$\begin{split} 0 &\le \left(\cosh \left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right) -\cosh \left(\left(x\sqrt{t}+\left(n-\frac{1}{2}\right)t\right)\vee 0\right)\right)^{k-1/2}\\ &\le \cosh^{k-1/2} \left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right) \le e^{(u\sqrt{t}+(n-1/2)t)(k-1/2)} \end{split}$$ so that for any $t\ge 1$, $$\begin{split} 0&\le\frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}} \int_T^{\infty}s e^{-s^2/(2t)}(\cosh s-\cosh T)^{k-1/2}\,{\rm d}s \\ &\le c_2e^{-t/8} \int_{x\vee (-(n-1/2)\sqrt{t})}^{\infty} \left(\frac{|u|}{\sqrt{t}}+1\right) e^{-(u+(n-1/2)\sqrt{t})^2/2} e^{(u\sqrt{t}+(n-1/2)t)(k-1/2)}\,{\rm d}u\\ &=c_2e^{-(n^2-2nk+k)t/2} \int_{x\vee (-(n-1/2)\sqrt{t})}^{\infty} \left(\frac{|u|}{\sqrt{t}}+1\right) e^{-u^2/2} e^{-(n-k)\sqrt{t}u}\,{\rm d}u\\ &\le c_3e^{-(n^2-2nk+k)t/2} \int_{x\vee (-(n-1/2)\sqrt{t})}^{\infty} e^{-(n-k)\sqrt{t}u}\,{\rm d}u =c_3e^{-(n^2-2nk+k)t/2}\frac{e^{-(n-k)(x\sqrt{t}\vee (-(n-1/2)t))}}{(n-k)\sqrt{t}}. \end{split}$$ Combining this with [\[eq:sinh-deri\]](#eq:sinh-deri){reference-type="eqref" reference="eq:sinh-deri"}, we get $$\begin{split} 0&\le \left(\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{n-2+k}\right)\sinh^{2n-2}r\,\Big|_{r=T} \frac{2^{1/2}e^{-t/8}}{(2\pi t)^{3/2}} \int_T^{\infty}s e^{-s^2/(2t)}(\cosh s-\cosh T)^{k-1/2}\,{\rm d}s\\ &\le c_4e^{(n-k)((x\sqrt{t}+(n-1/2)t)\vee 0)}e^{-(n^2-2nk+k)t/2} \frac{e^{-(n-k)(x\sqrt{t}\vee (-(n-1/2)t))}}{(n-k)\sqrt{t}} =\frac{c_4e^{n(n-1)t/2}}{\sqrt{t}}. \end{split}$$ This implies that $$0\le K_1(t,x)\le \frac{c_5e^{n(n-1)t/2}}{\sqrt{t}}$$ and thus $$\label{eq:k1-bound} a_n(t)K_1(t,x)=\frac{\pi e^{-n(n-1)t/2}}{2^{n-2}(n-1)!}K_1(t,x) \le \frac{c_6}{\sqrt{t}}.$$ Let us next estimate $a_n(t)K_2(t,x)$. By definition, $$\begin{split} a_n(t)K_2(t,x) &=\frac{2^n}{(2n-1)\sqrt{\pi}}\frac{e^{-(n-1/2)^2t/2}}{t^{3/2}}\int_T^{\infty}s e^{-s^2/(2t)}(\cosh s-\cosh T)^{n-1/2}\,{\rm d}s\\ &=b_n(t)\int_T^{\infty}s e^{-s^2/(2t)}(\cosh s-\cosh T)^{n-1/2}\,{\rm d}s. \end{split}$$ We first assume that $x\ge -(n-1/2)\sqrt{t}$. Then by the change of variables formula with $s=u\sqrt{t}+(n-1/2)t$, $$\begin{split} &\int_T^{\infty}s e^{-s^2/(2t)}(\cosh s-\cosh T)^{n-1/2}\,{\rm d}s\\ &=te^{-(n-1/2)^2t/2}\int_x^{\infty} ue^{-u^2/2}e^{-(n-1/2)u\sqrt{t}}\left(\cosh \left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right)-\cosh T\right)^{n-1/2}\,{\rm d}u\\ &+\left(n-\frac{1}{2}\right)t\sqrt{t}e^{-(n-1/2)^2t/2}\\ &\times \int_x^{\infty} e^{-u^2/2}e^{-(n-1/2)u\sqrt{t}}\left(\cosh \left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right)-\cosh T\right)^{n-1/2}\,{\rm d}u\\ &=L_1(t,x)+L_2(t,x). \end{split}$$ Therefore, $$\label{eq:k2-bound-1} a_n(t)K_2(t,x)=b_n(t)L_1(t,x)+b_n(t)L_2(t,x).$$ Since we have for any $u\ge x$, $$\begin{split} e^{-(n-1/2)u\sqrt{t}}\left(\cosh \left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right)-\cosh T\right)^{n-1/2} &\le e^{-(n-1/2)u\sqrt{t}}\left(e^{u\sqrt{t}+(n-1/2)t}\right)^{n-1/2}\\ &=e^{(n-1/2)^2t}, \end{split}$$ it follows that $$\begin{split} |L_1(t,x)| &\le te^{(n-1/2)^2t/2}\int_x^{\infty}|u|e^{-u^2/2}\,{\rm d}u \le te^{(n-1/2)^2t/2}\int_{-\infty}^{\infty}|u|e^{-u^2/2}\,{\rm d}u=2te^{(n-1/2)^2t/2}, \end{split}$$ which yields $$\label{eq:k2-bound-2} |b_n(t)L_1(t,x)|\le \frac{c_7}{\sqrt{t}}.$$ Since $$\begin{split} &\left(\cosh \left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right)-\cosh T\right)^{n-1/2}\\ &=\cosh^{n-1/2}\left(u\sqrt{t}+\left(n-\frac{1}{2}\right)t\right) \left(1-\frac{\cosh T}{\cosh \left(u\sqrt{t}+(n-1/2)t\right)}\right)^{n-1/2}\\ &=\frac{1}{2^{n-1/2}}e^{(n-1/2)^2t}e^{(n-1/2)u\sqrt{t}} \left(1+e^{-2(u\sqrt{t}+(n-1/2)t)}\right)^{n-1/2}\\ &\times\left(1-\frac{\cosh T}{\cosh \left(u\sqrt{t}+(n-1/2)t\right)}\right)^{n-1/2}, \end{split}$$ we obtain $$\begin{split} L_2(t,x) &=\frac{2n-1}{2^{n+1/2}}t\sqrt{t}e^{(n-1/2)^2t/2}\\ &\times \int_x^{\infty} e^{-u^2/2}\left(1+e^{-2(u\sqrt{t}+(n-1/2)t)}\right)^{n-1/2} \left(1-\frac{\cosh T}{\cosh \left(u\sqrt{t}+(n-1/2)t\right)}\right)^{n-1/2}\,{\rm d}u. \end{split}$$ Noting that $$b_n(t)\frac{2n-1}{2^{n+1/2}}t\sqrt{t}e^{(n-1/2)^2t/2}=\frac{1}{\sqrt{2\pi}},$$ we further have $$\begin{split} &b_n(t)L_2(t,x)\\ &=\frac{1}{\sqrt{2\pi}} \int_x^{\infty} e^{-u^2/2}\left(1+e^{-2(u\sqrt{t}+(n-1/2)t)}\right)^{n-1/2} \left(1-\frac{\cosh T}{\cosh \left(u\sqrt{t}+(n-1/2)t\right)}\right)^{n-1/2}\,{\rm d}u, \end{split}$$ which yields $$\label{eq:k2-bound-3} \begin{split} &b_n(t)L_2(t,x)-\Phi(x) =\frac{1}{\sqrt{2\pi}} \int_x^{\infty} e^{-u^2/2}\left\{\left(1+e^{-2(u\sqrt{t}+(n-1/2)t)}\right)^{n-1/2}-1\right\}\,{\rm d}u\\ &+\frac{1}{\sqrt{2\pi}}\\ &\times \int_x^{\infty} e^{-u^2/2}\left(1+e^{-2(u\sqrt{t}+(n-1/2)t)}\right)^{n-1/2} \left\{\left(1-\frac{\cosh T}{\cosh \left(u\sqrt{t}+(n-1/2)t\right)}\right)^{n-1/2}-1\right\}\,{\rm d}u\\ &=M_1(t,x)+M_2(t,x). \end{split}$$ Note that for any $p\ge 0$, $$(1+t)^p-1\asymp t, \quad 0\le t\le 1.$$ Then for any $t\ge 1$ and $u\ge -(n-1/2)\sqrt{t}$, $$\left(1+e^{-2(u\sqrt{t}+(n-1/2)t)}\right)^{n-1/2}-1\asymp e^{-2(u\sqrt{t}+(n-1/2)t)}.$$ Hence for any $t\ge 1$ and $x\ge -(n-1/2)\sqrt{t}$, $$M_1(t,x)\asymp \int_x^{\infty}e^{-u^2/2}e^{-2(u\sqrt{t}+(n-1/2)t)}\,{\rm d}u =e^{-(2n-3)t}\int_x^{\infty}e^{-(u+2\sqrt{t})^2/2}\,{\rm d}u.$$ Moreover, by the change of variables formula with $v=u+2\sqrt{t}$, we have for any $t\ge 1$, $$\int_x^{\infty}e^{-(u+2\sqrt{t})^2/2}\,{\rm d}u =\int_{x+2\sqrt{t}}^{\infty}e^{-v^2/2}\,{\rm d}v \le \int_{-(n-5/2)t}^{\infty}e^{-v^2/2}\,{\rm d}v \lesssim \begin{cases} e^{-9t/8}/\sqrt{t} & (n=1),\\ 1 & (n\ge 2), \end{cases}$$ which implies that for any $t\ge 1$, $$\label{eq:m1-bound} \sup_{x\ge -(n-1/2)t}M_1(t,x) \lesssim \begin{cases} e^{-t/8}/\sqrt{t} & (n=1),\\ e^{-(2n-3)t} & (n\ge 2). \end{cases}$$ In particular, the right hand side above decays exponentially for any $n\ge 1$. We also note that for any $p\ge 0$, there exists a constant $c_8>0$ such that $$\label{eq:p-ineq} 0\le 1-(1-t)^p\le c_8t, \quad 0\le t\le 1.$$ Therefore, for any $u\ge x$, $$\begin{split} 0&\le 1-\left(1-\frac{\cosh T}{\cosh \left(u\sqrt{t}+(n-1/2)t\right)}\right)^{n-1/2} \le \frac{c_8\cosh T}{\cosh \left(u\sqrt{t}+(n-1/2)t\right)}\\ &\le 2c_8 e^Te^{-(u\sqrt{t}+(n-1/2)t)}=2c_8e^{-(u-x)\sqrt{t}}, \end{split}$$ which implies that $$\begin{split} |M_2(t,x)| =-M_2(t,x) &\le c_9\int_x^{\infty}e^{-u^2/2}e^{-(u-x)\sqrt{t}}\,{\rm d}u =c_9\int_0^{\infty}e^{-(v+x)^2/2}e^{-v\sqrt{t}}\,{\rm d}v\\ &\le c_9\int_0^{\infty}e^{-v\sqrt{t}}\,{\rm d}v=\frac{c_9}{\sqrt{t}}. \end{split}$$ Combining this with [\[eq:k2-bound-3\]](#eq:k2-bound-3){reference-type="eqref" reference="eq:k2-bound-3"} and [\[eq:m1-bound\]](#eq:m1-bound){reference-type="eqref" reference="eq:m1-bound"}, we obtain for any $t\ge 1$, $$\sup_{x\ge -(n-1/2)t}\left|b_n(t)L_2(t,x)-\Phi(x)\right| \lesssim \frac{1}{\sqrt{t}}.$$ By [\[eq:k2-bound-1\]](#eq:k2-bound-1){reference-type="eqref" reference="eq:k2-bound-1"} and [\[eq:k2-bound-2\]](#eq:k2-bound-2){reference-type="eqref" reference="eq:k2-bound-2"}, we further get for any $t\ge 1$, $$\sup_{x\ge -(n-1/2)t}\left|a_n(t)K_2(t,x)-\Phi(x)\right| \lesssim \frac{1}{\sqrt{t}}.$$ Hence by [\[eq:ii-divide\]](#eq:ii-divide){reference-type="eqref" reference="eq:ii-divide"} and [\[eq:k1-bound\]](#eq:k1-bound){reference-type="eqref" reference="eq:k1-bound"}, $$\sup_{x\ge -(n-1/2)t}\left|J_2(t,x)-\Phi(x)\right| \lesssim \frac{1}{\sqrt{t}}, \quad t\ge 1.$$ We next assume that $x\le -(n-1/2)\sqrt{t}$ and thus $T=0$. By the change of variables formula with $s=u\sqrt{t}$, we have $$\begin{split} 0\le \int_0^{\infty}se^{-s^2/(2t)}(\cosh s-1)^{n-1/2}\,{\rm d}s &=t\int_0^{\infty}ue^{-u^2/2}(\cosh(u\sqrt{t})-1)^{n-1/2}\,{\rm d}u\\ &=2^{n-1/2}t\int_0^{\infty}ue^{-u^2/2}\sinh^{2n-1}(u\sqrt{t}/2)\,{\rm d}u. \end{split}$$ At the last equality, we used the formula $\cosh(u\sqrt{t})-1=2\sinh^2(u\sqrt{t}/2)$. Since $ue^{-u^2/2}=-(e^{-u^2/2})'$, we have by the integration by parts formula, $$\int_0^{\infty}ue^{-u^2/2}\sinh^{2n-1}(u\sqrt{t}/2)\,{\rm d}u =\left(n-\frac{1}{2}\right)\sqrt{t}\int_0^{\infty}e^{-u^2/2}\sinh^{2n-2}(u\sqrt{t}/2)\cosh(u\sqrt{t}/2)\,{\rm d}u,$$ which yields $$\begin{split} &\int_0^{\infty}se^{-s^2/(2t)}(\cosh s-1)^{n-1/2}\,{\rm d}s\\ &=2^{n-3/2}(2n-1)t\sqrt{t}\int_0^{\infty}e^{-u^2/2}\sinh^{2n-2}(u\sqrt{t}/2)\cosh(u\sqrt{t}/2)\,{\rm d}u\\ &=\frac{2n-1}{2^{n+1/2}}t\sqrt{t} \int_0^{\infty}e^{-u^2/2}e^{(n-1/2)u\sqrt{t}}(1-e^{-u\sqrt{t}})^{2n-2}(1+e^{-u\sqrt{t}})\,{\rm d}u. \end{split}$$ Hence as in [\[eq:k2-bound-1\]](#eq:k2-bound-1){reference-type="eqref" reference="eq:k2-bound-1"}, $$\label{eq:k2-bound-11} \begin{split} a_n(t)K_2(t,x) &=\frac{1}{\sqrt{2\pi}}e^{-(n-1/2)^2t/2} \int_0^{\infty}e^{-u^2/2}e^{(n-1/2)u\sqrt{t}}(1-e^{-u\sqrt{t}})^{2n-2}(1+e^{-u\sqrt{t}})\,{\rm d}u\\ &=\frac{1}{\sqrt{2\pi}} \int_0^{\infty}e^{-(u-(n-1/2)\sqrt{t})^2/2}(1-e^{-u\sqrt{t}})^{2n-2}\,{\rm d}u\\ &+\frac{e^{-(n-1)t}}{\sqrt{2\pi}} \int_0^{\infty}e^{-(u-(n-3/2)\sqrt{t})^2/2}(1-e^{-u\sqrt{t}})^{2n-2}\,{\rm d}u\\ &=\frac{1}{\sqrt{2\pi}} \int_{-(n-1/2)\sqrt{t}}^{\infty}e^{-u^2/2}(1-e^{-(u+(n-1/2)\sqrt{t})\sqrt{t}})^{2n-2}\,{\rm d}u\\ &+\frac{e^{-(n-1)t}}{\sqrt{2\pi}} \int_{-(n-3/2)\sqrt{t}}^{\infty}e^{-u^2/2}(1-e^{-(u+(n-3/2)\sqrt{t})\sqrt{t}})^{2n-2}\,{\rm d}u\\ &=N_1(t)+N_2(t). \end{split}$$ For $N_1(t)$, $$\begin{split} &N_1(t)-\Phi(x)\\ &=\frac{1}{\sqrt{2\pi}} \int_{-(n-1/2)\sqrt{t}}^{\infty}e^{-u^2/2} \left\{(1-e^{-(u+(n-1/2)\sqrt{t})\sqrt{t}})^{2n-2}-1\right\}\,{\rm d}u -\frac{1}{\sqrt{2\pi}}\int_x^{-(n-1/2)\sqrt{t}}e^{-u^2/2}\,{\rm d}u. \end{split}$$ If $n=1$, then the first term of the right hand side above vanishes. For $n\ge 2$, it follows by [\[eq:p-ineq\]](#eq:p-ineq){reference-type="eqref" reference="eq:p-ineq"} that for any $t\ge 1$, $$\label{eq:n_1} \begin{split} 0 &\le \int_{-(n-1/2)\sqrt{t}}^{\infty}e^{-u^2/2} \left\{1-(1-e^{-(u+(n-1/2)\sqrt{t})\sqrt{t}})^{2n-2}\right\}\,{\rm d}u\\ &\lesssim \int_{-(n-1/2)\sqrt{t}}^{\infty}e^{-u^2/2} e^{-(u+(n-1/2)\sqrt{t})\sqrt{t}}\,{\rm d}u =e^{-(n-1)t}\int_{-(n-1/2)\sqrt{t}}^{\infty}e^{-(u+\sqrt{t})^2/2}\,{\rm d}u\\ &= e^{-(n-1)t}\int_{-(n-3/2)\sqrt{t}}^{\infty}e^{-v^2/2}\,{\rm d}v \lesssim e^{-(n-1)t}. \end{split}$$ Since $$\int_x^{-(n-1/2)\sqrt{t}}e^{-u^2/2}\,{\rm d}u \le \int_{-\infty}^{-(n-1/2)\sqrt{t}}e^{-u^2/2}\,{\rm d}u \asymp \frac{1}{\sqrt{t}}e^{-(n-1/2)^2t/2},$$ we have for any $t\ge 1$, $$\sup_{x\le -(n-1/2)\sqrt{t}}\left|N_1(t)-\Phi(x)\right| \lesssim \begin{cases} e^{-t/8}/\sqrt{t} & (n=1),\\ e^{-(n-1)t} & (n\ge 2). \end{cases}$$ For $N_2(t)$, it follows by a calculation similar to [\[eq:n_1\]](#eq:n_1){reference-type="eqref" reference="eq:n_1"} that for any $t\ge 1$, $$N_2(t) \lesssim e^{-(n-1)t}\int_{-(n-3/2)\sqrt{t}}^{\infty}e^{-u^2/2}\,{\rm d}u \lesssim \begin{cases} e^{-t/8}/\sqrt{t} & (n=1),\\ e^{-(n-1)t} & (n\ge 2). \end{cases}$$ Combining the two inequalities above with [\[eq:k2-bound-11\]](#eq:k2-bound-11){reference-type="eqref" reference="eq:k2-bound-11"}, we get for any $t\ge 1$, $$\sup_{x\le -(n-1/2)\sqrt{t}} \left|a_n(t)K_2(t,x)-\Phi(x)\right| \lesssim \begin{cases} e^{-t/8}/\sqrt{t} & (n=1),\\ e^{-(n-1)t} & (n\ge 2). \end{cases}$$ Hence by [\[eq:ii-divide\]](#eq:ii-divide){reference-type="eqref" reference="eq:ii-divide"} and [\[eq:k1-bound\]](#eq:k1-bound){reference-type="eqref" reference="eq:k1-bound"}, we obtain for any $t\ge 1$, $$\sup_{x\le -(n-1/2)\sqrt{t}}\left|J_2(t,x)-\Phi(x)\right| \lesssim \frac{1}{\sqrt{t}}.$$ The proof is complete. ◻ **Lemma 4**. *For $d=2$, there exists a constant $c>0$ such that $$P\left(\frac{R_t^{(2)}-t/2}{\sqrt{t}}\geq 0\right) -\Phi(0)\ge \frac{c}{\sqrt{t}}, \quad t\geq 1.$$* *Proof.* Let $R_t=R_t^{(2)}$. Then by [\[eq:d=2\]](#eq:d=2){reference-type="eqref" reference="eq:d=2"}, $$\label{eq:dist-d=2} \begin{split} &P\left(\frac{R_t-t/2}{\sqrt{t}}\ge 0\right) =P\left(R_t\ge \frac{t}{2}\right) =\omega_2\int_{t/2}^{\infty}q_2(t,r)\sinh r\,{\rm d}r\\ &=\frac{e^{-t/8}}{\pi^{1/2}t^{3/2}} \int_{t/2}^{\infty}\left(\int_r^{\infty}\frac{se^{-s^2/(2t)}}{(\cosh s-\cosh r)^{1/2}}\,{\rm d}s\right)\sinh r\,{\rm d}r =\frac{e^{-t/8}}{\pi^{1/2}t^{3/2}}I(t). \end{split}$$ By the Fubini theorem, $$\begin{split} I(t) &=\int_{t/2}^{\infty}se^{-s^2/(2t)}\left(\int_{t/2}^s\frac{\sinh r}{(\cosh s-\cosh r)^{1/2}}\,{\rm d}r\right)\,{\rm d}s\\ &=2\int_{t/2}^{\infty}se^{-s^2/(2t)}(\cosh s-\cosh (t/2))^{1/2}\,{\rm d}s. \end{split}$$ Then by the integration by parts formula and the change of variables formula with $s=u\sqrt{t}+t/2$, $$\begin{split} &I(t) =t\int_{t/2}^{\infty}e^{-s^2/(2t)}\frac{\sinh s}{(\cosh s-\cosh (t/2))^{1/2}}\,{\rm d}s\\ &=t^{3/2}e^{-t/8}\int_0^{\infty}e^{-u^2/2-u\sqrt{t}/2} \frac{\sinh(u\sqrt{t}+t/2)}{(\cosh (u\sqrt{t}+t/2)-\cosh(t/2))^{1/2}}\,{\rm d}u\\ &=\frac{t^{3/2}e^{t/8}}{2^{1/2}}\int_0^{\infty} e^{-u^2/2} \left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2} \left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}\,{\rm d}u. \end{split}$$ Hence by [\[eq:dist-d=2\]](#eq:dist-d=2){reference-type="eqref" reference="eq:dist-d=2"}, $$\begin{split} &P\left(\frac{R_t-t/2}{\sqrt{t}}\ge 0\right)\\ &=\frac{1}{\sqrt{2\pi }} \int_0^{\infty} e^{-u^2/2} \left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2} \left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}\,{\rm d}u. \end{split}$$ In particular, if we let $$F_t(u)=\left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2} \left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}-1,$$ then $$\label{eq:sharp-1} P\left(\frac{R_t-t/2}{\sqrt{t}}\ge 0\right)-\Phi(0) =\frac{1}{\sqrt{2\pi}}\int_0^{\infty} e^{-u^2/2}F_t(u)\,{\rm d}u.$$ For any $u\ge 0$, $$\begin{split} &\left(1-e^{-2(u\sqrt{t}+t/2)}\right)^2 -\left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right) \left(1+e^{-2(u\sqrt{t}+t/2)}\right)\\ &=\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)} \left(1+e^{-2(u\sqrt{t}+t/2)}\right) -3e^{-2(u\sqrt{t}+t/2)}+e^{-4(u\sqrt{t}+t/2)}\\ & \ge \frac{e^{-u\sqrt{t}}}{2}-3e^{-2(u\sqrt{t}+t/2)} =\frac{e^{-u\sqrt{t}}}{2}(1-6e^{-(u\sqrt{t}+t)}). \end{split}$$ Therefore, if $t\ge \log 6$, then the last term above is positive for any $u\ge 0$ so that $F_t(u)$ and the integral in [\[eq:sharp-1\]](#eq:sharp-1){reference-type="eqref" reference="eq:sharp-1"} are positive. For $u\geq 0$, we let $$\label{eq:sharp-2} \begin{split} F_t(u) &=\left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2}-1\\ &+\left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2} \left\{\left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}-1\right\}. \end{split}$$ Since $$\frac{1}{\sqrt{1-x}}-1\ge 1-\sqrt{1-x}\ge \frac{1}{2}x, \quad 0\le x<1,$$ we obtain $$\label{eq:sharp-3} \left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2}-1 \ge \frac{1}{2}\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)} \ge \frac{e^{-u\sqrt{t}}}{4}.$$ Suppose that $u\ge 1/\sqrt{t}$. Then $$1-\frac{\cosh(t/2)}{\cosh(u\sqrt{t}+t/2)}\ge 1-\frac{\cosh(t/2)}{\cosh(1+t/2)}\ge 1-\frac{2}{e}>0.$$ Moreover, since $$\left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}-1\le 0,$$ we have $$\label{eq:sharp-4} \begin{split} &\left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2} \left\{\left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}-1\right\}\\ &=-\left(1-\frac{\cosh(t/2)}{\cosh\left(u\sqrt{t}+t/2\right)}\right)^{-1/2} \left\{1-\left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}\right\}\\ &\ge -(1-2e^{-1})^{-1/2}\left\{1-\left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}\right\}. \end{split}$$ Noting that $$1-\frac{1}{\sqrt{1+t}}\asymp t, \quad 0\le t\le 1,$$ we obtain $$0\le 1-(1+e^{-2(u\sqrt{t}+t/2)})^{-1/2}\le c_1e^{-2(u\sqrt{t}+t/2)}$$ so that $$\label{eq:sharp-5} \begin{split} &1-\left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}\\ &=e^{-2(u\sqrt{t}+t/2)} +\left(1-e^{-2(u\sqrt{t}+t/2)}\right)\left\{1-\left(1+e^{-2(u\sqrt{t}+t/2)}\right)^{-1/2}\right\} \le c_2e^{-2(u\sqrt{t}+t/2)}. \end{split}$$ Therefore, by [\[eq:sharp-2\]](#eq:sharp-2){reference-type="eqref" reference="eq:sharp-2"}, [\[eq:sharp-3\]](#eq:sharp-3){reference-type="eqref" reference="eq:sharp-3"}, [\[eq:sharp-4\]](#eq:sharp-4){reference-type="eqref" reference="eq:sharp-4"} and [\[eq:sharp-5\]](#eq:sharp-5){reference-type="eqref" reference="eq:sharp-5"}, we have for any $t\ge 1$ and $u\ge 1/\sqrt{t}$, $$\label{eq:f-lower} F_t(u)\ge \frac{1}{4}e^{-u\sqrt{t}}-c_2e^{-2(u\sqrt{t}+t/2)}.$$ Note that for any $c>0$, $$\int_{1/\sqrt{t}}^{\infty}e^{-u^2/2-cu\sqrt{t}}\,{\rm d}u =e^{c^2t/2}\int_{1/\sqrt{t}}^{\infty}e^{-(u+c\sqrt{t})^2/2}\,{\rm d}u =e^{c^2 t/2}\int_{c\sqrt{t}+1/\sqrt{t}}^{\infty}e^{-u^2/2}\,{\rm d}u \asymp \frac{1}{\sqrt{t}}, \quad t\ge 1.$$ Hence by [\[eq:f-lower\]](#eq:f-lower){reference-type="eqref" reference="eq:f-lower"} and taking $c=1$ and $c=2$ in the relation above, we get $$\begin{split} \int_0^{\infty}e^{-u^2/2}F_t(u)\,{\rm d}u \ge \int_{1/\sqrt{t}}^{\infty} e^{-u^2/2}F_t(u)\,{\rm d}u &\ge \frac{1}{4}\int_{1/\sqrt{t}}^{\infty}e^{-u^2/2-u\sqrt{t}}\,{\rm d}u -c_2e^{-t}\int_{1/\sqrt{t}}^{\infty}e^{-u^2/2-2u\sqrt{t}}\,{\rm d}u\\ &\ge \frac{c_3}{\sqrt{t}}. \end{split}$$ Combining this with [\[eq:sharp-1\]](#eq:sharp-1){reference-type="eqref" reference="eq:sharp-1"}, we arrive at the desired conclusion. ◻ *Proof of Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for even dimensions.* The proof is complete by Lemmas [Lemma 2](#lem:i-bound){reference-type="ref" reference="lem:i-bound"}, [Lemma 3](#lem:ii-bound){reference-type="ref" reference="lem:ii-bound"} and [Lemma 4](#lem:even){reference-type="ref" reference="lem:even"}. ◻ # Appendix {#sect:append} To prove Theorem [Theorem 1](#thm:berry){reference-type="ref" reference="thm:berry"} for general odd dimensions, we show **Lemma 5**. *For $l\ge 1$, $$\label{eq:deri-0} \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^l\sinh^{2l+1}r =\frac{(2l+1)!!}{l+1}\sinh((l+1)r), \ r\in {\mathbb R}.$$* *Proof.* Fix $n\geq 2$. We first prove by induction that for any $k=1,2,3,\dots, [n/2]$, $$\label{eq:ind-1} \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k}\sinh^n r =\sum_{l=0}^k\frac{(2k)!}{2^{k-l}(k-l)!(2l)!} \left(\prod_{m=0}^{k+l-1}(n-2m)\right)\cosh^{2l}r\sinh^{n-(2k+2l)}r, \ r\in {\mathbb R}.$$ For $k=1$, we have $$\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^2\sinh^n r =n\sinh^{n-2}r+n(n-2) \cosh^2 r \sinh^{n-4}r$$ so that [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} is valid. Suppose that [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} holds for some $k\geq 1$. Since $$\begin{split} &\frac{\partial}{\partial r}(\cosh^{2l}r\sinh^{n-(2k+2l)}r)\\ &=2l\cosh^{2l-1}r\sinh^{n-(2k+2l-1)}r+(n-(2k+2l))\cosh^{2l+1}r\sinh^{n-(2k+2l+1)}r, \end{split}$$ we obtain by [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"}, $$\label{eq:ind-00} \begin{split} &\frac{\partial}{\partial r}\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k}\sinh^n r\\ &=\sum_{l=0}^k\frac{(2k)!}{2^{k-l}(k-l)!(2l)!} \left(\prod_{m=0}^{k+l-1}(n-2m)\right)\frac{\partial}{\partial r}(\cosh^{2l}r\sinh^{n-(2k+2l)}r)\\ &=\sum_{l=1}^k\frac{(2k)!}{2^{k-l}(k-l)!(2l-1)!} \left(\prod_{m=0}^{k+l-1}(n-2m)\right)\cosh^{2l-1}r\sinh^{n-(2k+2l-1)}r\\ &+\sum_{l=0}^k \frac{(2k)!}{2^{k-l}(k-l)!(2l)!} \left(\prod_{m=0}^{k+l}(n-2m)\right)\cosh^{2l+1}r\sinh^{n-(2k+2l+1)}r\\ &=\sum_{l=1}^k\frac{(2k)!}{2^{k-l}(k-l)!(2l-1)!} \left(\prod_{m=0}^{k+l-1}(n-2m)\right)\cosh^{2l-1}r\sinh^{n-(2k+2l-1)}r\\ &+\sum_{l=1}^{k+1} \frac{(2k)!}{2^{k-l+1}(k-l+1)!(2l-2)!} \left(\prod_{m=0}^{k+l-1}(n-2m)\right)\cosh^{2l-1}r\sinh^{n-(2k+2l-1)}r\\ &=\sum_{l=1}^{k+1} \frac{(2k+1)!}{2^{k+1-l}(k+1-l)!(2l-1)!} \left(\prod_{m=0}^{k+l-1}(n-2m)\right)\cosh^{2l-1}r\sinh^{n-(2k+2l-1)}r. \end{split}$$ Therefore, $$\label{eq:ind-2} \begin{split} &\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k+1}\sinh^n r\\ &=\sum_{l=1}^{k+1} \frac{(2k+1)!}{2^{k+1-l}(k+1-l)!(2l-1)!} \left(\prod_{m=0}^{k+l-1}(n-2m)\right)\cosh^{2l-1}r\sinh^{n-(2k+2l)}r. \end{split}$$ Using this equality, we obtain in a similar way to [\[eq:ind-00\]](#eq:ind-00){reference-type="eqref" reference="eq:ind-00"}, $$\begin{split} &\frac{\partial}{\partial r}\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k+1}\sinh^n r\\ &=\sum_{l=0}^{k+1} \frac{(2k+2)!}{2^{k+1-l}(k+1-l)!(2l)!} \left(\prod_{m=0}^{k+l}(n-2m)\right)\cosh^{2l}r\sinh^{n-(2k+2l+1)}r \end{split}$$ and thus $$\begin{split} &\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k+2}\sinh^n r\\ &=\sum_{l=0}^{k+1} \frac{(2k+2)!}{2^{k+1-l}(k+1-l)!(2l)!} \left(\prod_{m=0}^{k+l}(n-2m)\right)\cosh^{2l}r\sinh^{n-(2k+2+2l)}r. \end{split}$$ This shows that [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} is valid for $k$ replaced with $k+1$, that is, the induction is complete. In particular, our argument above implies that [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} and [\[eq:ind-2\]](#eq:ind-2){reference-type="eqref" reference="eq:ind-2"} hold for any $k=1,2,3,\dots, [n/2]$. We next prove [\[eq:deri-0\]](#eq:deri-0){reference-type="eqref" reference="eq:deri-0"}. We here calculate the left hand side of [\[eq:deri-0\]](#eq:deri-0){reference-type="eqref" reference="eq:deri-0"} for odd $l$ and even $l$, respectively. Namely, we will prove that for any $k\geq 1$, $$\label{eq:deri-1} \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k-1}\sinh^{4k-1}r=\frac{(4k-1)!!}{2k}\sinh 2kr$$ and $$\label{eq:deri-2} \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k}\sinh^{4k+1}r =\frac{(4k+1)!!}{2k+1}\sinh ((2k+1)r).$$ By [\[eq:ind-2\]](#eq:ind-2){reference-type="eqref" reference="eq:ind-2"}, $$\begin{split} &\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k-1}\sinh^{4k-1} r\\ &=\sum_{l=1}^k \frac{(2k-1)!}{2^{k-l}(k-l)!(2l-1)!} \left(\prod_{m=0}^{k+l-2}(4k-1-2m)\right)\cosh^{2l-1}r\sinh^{2k-2l+1}r\\ &=\sum_{j=0}^{k-1} \frac{(2k-1)!}{2^j j! (2k-2j-1)!} \left(\prod_{m=0}^{2k-j-2}(4k-1-2m)\right)\cosh^{2k-(2j+1)}r\sinh^{2j+1}r, \end{split}$$ where we set $j=k-l$ in the second equality above. If $0\leq j\leq k-1$, then $$\begin{split} &\frac{(2k-1)!}{2^j j! (2k-2j-1)!}\prod_{m=0}^{2k-j-2}(4k-1-2m) =\frac{(2k-1)!}{(2j)!!(2k-2j-1)!}\prod_{l=j+2}^{2k}(2l-1)\\ &=\frac{(4k-1)!!}{2k}\frac{(2k)!}{(2j+1)!(2k-(2j+1))!} =\frac{(4k-1)!!}{2k}\binom{2k}{2j+1} \end{split}$$ and thus $$\begin{split} \left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k-1}\sinh^{4k-1} r &=\frac{(4k-1)!!}{2k}\sum_{j=0}^{k-1} \binom{2k}{2j+1}\cosh^{2k-(2j+1)}r\sinh^{2j+1}r\\ &=\frac{(4k-1)!!}{2k}\sinh 2kr. \end{split}$$ The proof of [\[eq:deri-1\]](#eq:deri-1){reference-type="eqref" reference="eq:deri-1"} is complete. By [\[eq:ind-1\]](#eq:ind-1){reference-type="eqref" reference="eq:ind-1"} and a similar argument as above, $$\begin{split} &\left(\frac{1}{\sinh r}\frac{\partial}{\partial r}\right)^{2k}\sinh^{4k+1} r\\ &=\sum_{l=0}^k\frac{(2k)!}{2^{k-l}(k-l)!(2l)!} \left(\prod_{m=0}^{k+l-1}(4k+1-2m)\right)\cosh^{2l}r\sinh^{2k-(2l-1)}r\\ &=\sum_{j=0}^k\frac{(2k)!}{2^j j!(2(k-j))!} \left(\prod_{i=j+1}^{2k}(2i+1)\right)\cosh^{2(k-j)}r\sinh^{2j+1}r\\ &=\frac{(4k+1)!!}{2k+1} \sum_{j=0}^k \binom{2k+1}{2j+1}\cosh^{2(k-j)}r\sinh^{2j+1}r =\frac{(4k+1)!!}{2k+1}\sinh((2k+1)r). \end{split}$$ The proof of [\[eq:deri-2\]](#eq:deri-2){reference-type="eqref" reference="eq:deri-2"} is complete. ◻ 99 J. P. Anker and G. Setti, Asymptotic finite propagation speed for heat diffusion on certain Riemannian manifolds, *J. Funct. Anal.* **103** (1992), 50--61. M. Babillot, A probabilistic approach to heat diffusion on symmetric spaces, *J. Theoret. Prob.* **7** (1994), 599--607. V. Cammarota, A. De Gregorio and C. Macci, On the asymptotic behavior of the hyperbolic Brownian motion, *J. Stat. Phys.* **154** (2014), 1550--1568. E. B. Davies, *Heat Kernels and Spectral Theory*, Cambridge, Cambridge University Press, 1992. E. B. Davies and N. Mandouvalos, Heat kernel bounds on hyperbolic space and Kleinian groups, *Proc. London Math. Soc.* **57** (1988), 182--208. R. Durrett, *Probability: Theory and Examples*, 5th edition, Cambridge University Press, 2019. A. Grigor'yan and E. Hsu, Volume growth and escape rate of Brownian motion on a Cartan-Hadamard manifold, In: Sobolev Spaces in Mathematics II. International Mathematical Series, **9**, Springer, New York, NY, 2009, 209--225. A. Grigor'yan and M. Noguchi, The heat kernel on hyperbolic space, *Bull. London Math. Soc.* **30** (1998), 643--650. J. C. Gruet, Semi-groupe du mouvement brownien hyperbolique, *Stochastics Stochastics Rep.* **56** (1996), 53--61. M. Hirao, Large deviations for the radial processes of the Brownian motions on hyperbolic spaces, *Statist. Probab. Lett.* **81** (2011), 1561--1564. E. P. Hsu, *Stochastic Analysis on Manifolds*, Amer. Math. Soc., Providence, RI, 2002. H. Matsumoto, Limiting behaviors of the Brownian motions on hyperbolic spaces, *Colloq. Math.* **119** (2010), 193--215. Y. Shiozawa, Escape rate of the Brownian motions on hyperbolic spaces, *Proc. Japan Acad. Ser. A Math. Sci.* **93** (2017), 27--29. J. L. Vázquez, Asymptotic behaviour for the heat equation in hyperbolic space, *Comm. Anal. Geom.* **30** (2022), 2123--2156. [^1]: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan; `shiozawa@math.sci.osaka-u.ac.jp` [^2]: This work was supported by JSPS KAKENHI Grant Numbers JP22K18675, JP23H01076.

arxiv_math

--- abstract: | We deal with and investigate sparse univariate Positivstellensätze, Nichtnegativstellensätze, and solutions to sparse moment problems. The paper relies heavily on results on T-systems by Karlin in 1963 and by Karlin and Studden in 1966. We gain complete descriptions of all sparse strictly positive and sparse non-negative algebraic polynomials on $[a,b]$ with $a\geq 0$ and on $[0,\infty)$. We extend, simplify, and solve the sparse Hausdorff and Stieltjes moment problem with these results and the methods of adapted spaces and T-systems. address: - Department of Mathematics and Statistics, University of Konstanz, Universitätsstraße 10, D-78464 Konstanz, Germany - Zukunftskolleg, Universtity of Konstanz, Universitätsstraße 10, D-78464 Konstanz, Germany - "eMail: philipp.didio\\@uni-konstanz.de" author: - Philipp J. di Dio title: "The Early History of Moment Problems and Non-Negative Polynomials with Gaps: Sparse Moment Problems, Sparse Positivstellensätze, and Sparse Nichtnegativstellensätze from a T-System Point of View" --- moment problem,Positivstellensatz,Nichtnegativstellensatz,sparse,gap,T-system Primary 44A60, 14P99 ; Secondary 41A10, 12E10. # Introduction The theory of moments (or the moment problem) has been connected to non-negative polynomials for a long time and this connection is well-known since Haviland [@havila36] or even dates back further. The classical moment problem is the following: Given a closed set $K\subseteq\mathds{R}^n$ and a real sequence $s = (s_\alpha)_{\alpha\in I}$ with $I\subseteq\mathds{N}_0^n$. When does there exist a measure $\mu$ on $K$ such that $$s_\alpha = \int_K x^\alpha~\mathrm{d}\mu(x)$$ holds for all $\alpha\in I$? For more on the theory of moments see e.g. [@shohat43; @akhiezClassical; @karlinStuddenTSystemsBook; @kreinMarkovMomentProblem; @marshallPosPoly; @lauren09; @schmudMomentBook] and references therein. In modern times the theory of moments and the theory of non-negative polynomials were revived by [@schmud91] and then put to useful applications, see e.g. [@lasserreSemiAlgOpt]. By applications and specially by the need for efficient and fast algorithms the focus more and more turned the last years (and decades) to sparse systems, i.e., the index set $I\subsetneq\mathds{N}_0^n$ is not all $\mathds{N}_0^n$ and specially not all $\alpha$ with $|\alpha|\leq d$ for some $d\in\mathds{N}_0^n$ are present. It should not be surprising that these sparse systems were studied theoretically. More surprising is it that the early results in this field are not well-known or even completely forgotten. And unfortunately it came recently to our attention that several known results are being reproved in weaker versions [@scheider23]. The main purpose of this article is to review the early results in the theory of sparse moment problems and to show how important results and especially sparse Positivstellensätze, sparse Nichtnegativstellensätze, and sparse moment problems follow from these early results. All results presented here are not contained in the modern literature about the theory of moments [@marshallPosPoly; @schmudMomentBook], about real algebraic geometry [@bochnak98], or about (sparse) polynomial optimization [@lasserreSemiAlgOpt; @margronSparsePolyOpt]. We hope that this treatment will also be useful for the emerging works of moment problems and polynomials on curves since these often reduce to the univariate polynomial case [@zalar23]. By the title we only look at early results (and their applications). By "early" we mean everything up to and including 1966. Everything between 1967 and up to 1991 we consider "modern" and everything after "contemporary". Modern and contemporary results are not considered here since they deserve more space than this article can give. The year 1966 is chosen since in 1966 the extensive research monograph by Samuel Karlin and William J. Studden about T-systems appeared [@karlinStuddenTSystemsBook]. This monograph is an extensive follow up of the work by Karlin in [@karlin63]. Both works solve important problems in the theory of T-systems. The theory of T-systems is the theoretical framework where e.g. sparse univariate algebraic polynomial systems were investigated in. The year 1991 is chosen since then the first denominator free description of strictly positive polynomials appeared [@schmud91] reviving a large part in real algebraic geometry. The article is structured as follows. In the next we shortly introduce the notations in this article. In we shortly present the "usual suspects" (classical results without gaps) and the two first explicit studies of problems with gaps. We will meet there also Richter's Theorem and Boas' Theorem. In we will introduce the theory of T-systems and show their basic properties with a special emphasis on zeros and non-negativity. By far the most important part is where we look at the results in [@karlin63; @karlinStuddenTSystemsBook] and apply them to get sparse algebraic Positivstellensätze and Nichtnegativstellensätze. Additionally, they are used to solve and even extend the early sparse moment problems from . In we sum up the results. All results are presented with proofs as far as possible. Several are collected from the literature but translated to nowadays mathematical language. Also some missing steps are filled in and errors are corrected. # Preliminaries {#sec:prelim} Let $n\in\mathds{N}$, $K\subseteq\mathds{R}^n$ be closed, and $s = (s_\alpha)_{\alpha\in\mathds{N}_0^n}$ be a real sequence. We say that $s$ is a $K$-moment sequence if there exists a measure $\mu$ on $K$ such that $$s_\alpha = \int_K x^\alpha~\mathrm{d}\mu(x).$$ The measure $\mu$ is called a representing measure. Unless otherwise denoted as signed measure all measures are positive. The moment sequence $s$ is called determined if $\mu$ is unique. We call $s$ a truncated moment sequence when only finitely many $s_\alpha$ with $\alpha\in\mathds{N}_0$ are known. For a real sequence $s=(s_\alpha)_{\alpha\in\mathds{N}_0}$ we call the linear functional $L_s:\mathds{R}[x_1,\dots,x_n]\to\mathds{R}$ defined by $L_s(x^\alpha) = s_\alpha$ the Riesz functional. For $\beta\in\mathds{N}_0$ we define $X^\beta s = (s_{\alpha+\beta})_{\alpha\in\mathds{N}_0^n}$ the shifted sequence. For a sequence $s=(s_\alpha)_{\alpha\in\mathds{N}_0^n}$ we define the Hankel matrix $\mathcal{H}(s) := (s_{\alpha,\beta})_{\alpha,\beta\in\mathds{N}_0^n}$. For $K\subseteq\mathds{R}^n$ we set $\mathrm{Pos}(K) := \{f\in\mathds{R}[x_1,\dots,x_n] \,|\, f\geq 0\ \text{on}\ K\}$. For any set $\mathcal{X}$ we denote by $|\mathcal{X}|$ the cardinality of $\mathcal{X}$. # The Beginning of the Moment Problem {#sec:beginning} ## The Usual Suspects: Well-known Classical Results without Gaps The first moment problem that was solved is the following. **Stieltjes' Theorem 1** ([@stielt94]). *Let $s = (s_i)_{i\in\mathds{N}_0}$ be a real sequence. The following are equivalent:* (i) *$s$ is a $[0,\infty)$-moment sequence (Stieltjes moment sequence).* (ii) *$L_s(p)\geq 0$ for all $p\in\mathrm{Pos}([0,\infty))$.* (iii) *$L_s(p^2)\geq 0$ and $L_{Xs}(p^2) = L_s(x\cdot p^2)\geq 0$ for all $p\in\mathds{R}[x]$.* (iv) *$s$ and $Xs = (s_{i+1})_{i\in\mathds{N}_0}$ are positive semidefinite.* (v) *$\mathcal{H}(s)\succeq 0$ and $\mathcal{H}(Xs)\succeq 0$ for all $d\in\mathds{N}_0$.* in the original proof [@stielt94] does not use non-negative polynomials. Stieljes uses continued fractions and introduces new sequences which we (nowadays) denote by $s$ and $Xs$. Stieltjes only proves (i) $\Leftrightarrow$ (iv). The implication (i) $\Leftrightarrow$ (ii) is , (ii) $\Leftrightarrow$ (iii) is the description of $\mathrm{Pos}([0,\infty))$, and (iv) $\Leftrightarrow$ (v) is a reformulation of $s$ and $Xs$ being positive semi-definite. The next moment problem that was solved is the following. **Hamburger's Theorem 2** ([@hamburger20 Satz X and Existenstheorem (§8, p. 289)]). *Let $s = (s_i)_{i\in\mathds{N}_0}$ be a real sequence. The following are equivalent:* (i) *$s$ is a $\mathds{R}$-moment sequence (Hamburger moment sequence or short moment sequence).* (ii) *$L_s(p)\geq 0$ for all $p\in\mathrm{Pos}(\mathds{R})$.* (iii) *$L_s(p^2)\geq 0$ for all $p\in\mathds{R}[x]$.* (iv) *$s$ is positive semidefinite.* (v) *$\mathcal{H}(s)\succeq 0$.* Hamburger proves similar to Stieltjes [@stielt94] the equivalence (i) $\Leftrightarrow$ (iv) via continued fractions. In [@hamburger20 Satz XIII] Hamburger solves the full moment problem by approximation with truncated moment problems. This was later in a slightly more general framework reproved in [@stochel01]. Hamburger needed to assume that the sequence of measures $\mu_k$ (which he called "Belegungen" and denoted by $\mathrm{d}\Phi^{(k)}(u)$) to converge to some measure $\mu$ (condition 2 of [@hamburger20 Satz XIII]). Hamburgers additional condition 2 is nowadays replaced by the vague convergence and the fact that the solution set of representing measures is vaguely compact [@schmudMomentBook Thm. 1.19], i.e., it assures the existence of a $\mu$ as required by Hamburger in the additional condition 2. Shortly after Hamburger the moment problem on $[0,1]$ was solved. **Hausdorff's Theorem 3** ([@hausdo21 Satz II and III]). *Let $s = (s_i)_{i\in\mathds{N}_0}$ be a real sequence. The following are equivalent:* (i) *$s$ is a $[0,1]$-moment sequence (Hausdorff moment sequence).* (ii) *$L_s(p)\geq 0$ for all $p\in\mathrm{Pos}([0,1])$.* (iii) *$L_s(p^2)\geq 0$, $L_{Xs}(p^2)\geq 0$, and $L_{(1-X)s}(p^2)\geq 0$ for all $p\in\mathds{R}[x]$.* (iv) *$s$, $Xs$, and $(1-X)s$ are positive semidefinite.* (v) *$\mathcal{H}(s)\succeq 0$, $\mathcal{H}(Xs)\succeq 0$, and $\mathcal{H}((1-X)s)\succeq 0$.* Hausdorff proves the equivalence (i) $\Leftrightarrow$ (iii) via so called C-sequences. In [@toeplitz11] Toeplitz treats general linear averaging methods. In [@hausdo21] Hausdorff uses these. Let the infinite dimensional matrix $\lambda = (\lambda_{i,j})_{i,j\in\mathds{N}_0}$ be row-finite, i.e., for every row $i$ only finitely many $\lambda_{i,j}$ are non-zero. Then the averaging method $$A_i = \sum_{j\in\mathds{N}_0} \lambda_{i,j} a_j$$ shall be consistent: If $a_j \to \alpha$ converges then $A_i\to\alpha$ converges to the same limit. Toeplitz proved a necessary and sufficient condition on $\lambda$ for this property. Hausdorff uses only part of this property. He calls a matrix $(\lambda_{i,j})_{i,j\in\mathds{N}_0}$ with the property that a convergent sequence $(a_j)_{j\in\mathds{N}_0}$ is mapped to a convergent sequence $(A_j)_{j\in\mathds{N}_0}$ (the limit does not need to be preserved) a C-matrix (convergence preserving matrix). Hausdorff gives the characterization of C-matrices [@hausdo21 p. 75, conditions (A) -- (C)]. Additionally, if $\lambda$ is a C-matrix and a diagonal matrix with diagonal entries $\lambda_{i,i} = s_i$ then $s = (s_i)_{i\in\mathds{N}_0}$ is called a C-sequence. The equivalence (i) $\Leftrightarrow$ (iii) is then shown by Hausdorff in the result that a sequence is a $[0,1]$-moment sequence if and only it is a C-sequence [@hausdo21 p. 102]. A much simpler approach to solve the $K$-moment problem for any closed $K\subseteq\mathds{R}^n$, $n\in\mathds{N}$, was presented by Haviland. He no longer used continued fractions but employed the Riesz(--Markov--Kakutani) representation theorem, i.e., representing a linear functional by integration. The present Riesz--Markov--Kakutani representation theorem was developed in several stages. A first version for continuous functions on the unit interval $[0,1]$ is by F. Riesz [@riesz09]. It was extended by Markov to some non-compact spaces [@markov38] and then by Kakutani to locally compact Hausdorff spaces [@kakutani41]. Interestingly, it already follows from Daniell's Representation Theorem [@daniell18; @daniell20] with Urysohn's Lemma [@urysohn25]. Haviland proved the following. **Haviland's Theorem 4** ([@havila36 Theorem], see also [@havila35 Theorem] for $K=\mathds{R}^n$). *Let $n\in\mathds{N}$, $K\subseteq\mathds{R}^n$ be closed, and $s = (s_\alpha)_{\alpha\in\mathds{N}_0^n}$ be a real sequence. The following are equivalent:* (i) *$s$ is a $K$-moment sequence.* (ii) *$L_s(p)\geq 0$ for all $p\in\mathrm{Pos}(K)$.* In [@havila35 Theorem] Haviland proves "only" the case $K=\mathds{R}^n$ with the extension method by M. Riesz. In [@havila36 Theorem] this is extended to any closed $K\subseteq\mathds{R}^n$. The idea to do so is attributed by Haviland to Aurel Wintner [@havila36 p. 164]: "A. Wintner has subsequently suggested that it should be possible to extend this result \[[@havila35 Theorem]\] by requiring that the distribution function \[measure\] solving the problem have a spectrum \[support\] contained in a preassigned set, a result which would show the well-known criteria for the various standard special momentum problems (Stieltjes, Herglotz \[trigonometric\], Hamburger, Hausdorff in one or more dimensions) to be put particular cases of the general $n$-dimensional momentum problem mentioned above. The purpose of this note is to carry out this extension." In [@havila36] after the general Theorem [Haviland's Theorem 4](#thm:haviland){reference-type="ref" reference="thm:haviland"} Haviland then goes through all the classical results (Theorems [Stieltjes' Theorem 1](#thm:stieltjesMP){reference-type="ref" reference="thm:stieltjesMP"} to [Hausdorff's Theorem 3](#thm:hausdorffMP){reference-type="ref" reference="thm:hausdorffMP"}, and the Herglotz (trigonometric) moment problem on the unit circle which we did not included here) and shows how all these results (i.e., conditions on the sequences) are recovered from the at this point known representations of non-negative polynomials. For the Hamburger moment problem Haviland uses $$\mathrm{Pos}(\mathds{R}) = \left\{ f^2 + g^2 \,\middle|\, f,g\in\mathds{R}[x]\right\}$$ which was already known to Hilbert [@hilbert88]. For the Stieltjes moment problem he uses $$\label{eq:pos0infty1} \mathrm{Pos}([0,\infty)) = \left\{ f_1^2 + f_2^2 + x\cdot (g_1^2 + g_2^2) \,\middle|\, f_1,f_2,g_1,g_2\in\mathds{R}[x]\right\}$$ with the reference to Pólya and Szegö (previous editions of [@polya64; @polya70]). In [@polya64 p. 82, ex. 45] the representation ([\[eq:pos0infty1\]](#eq:pos0infty1){reference-type="ref" reference="eq:pos0infty1"}) is still included while it was already known before, see [@shohat43 p. 6, footnote], that $$\label{eq:pos0infty2} \mathrm{Pos}([0,\infty)) = \left\{ f^2 + x\cdot g^2 \,\middle|\, f,g\in\mathds{R}[x]\right\}$$ is sufficient. Also in [@schmudMomentBook Prop. 3.2] the representation ([\[eq:pos0infty1\]](#eq:pos0infty1){reference-type="ref" reference="eq:pos0infty1"}) is used, not the representation ([\[eq:pos0infty2\]](#eq:pos0infty2){reference-type="ref" reference="eq:pos0infty2"}). For the $[-1,1]$-moment problem Haviland uses $$\mathrm{Pos}([-1,1]) = \left\{ f^2 + (1-x^2)\cdot g^2 \,\middle|\, f,g\in\mathds{R}[x]\right\}.$$ For the Hausdorff moment problem he uses that any non-negative polynomial on $[0,1]$ is a linear combination of $x^m\cdot (1-x)^{p-m}$, $m,p\in\mathds{N}_0$, $p\geq m$, with non-negative coefficients. For the two-dimensional Hausdorff moment problem he uses that any non-negative polynomial on $[0,1]^2$ is a linear combination of $x^m\cdot y^n\cdot (1-x)^{p-m}\cdot (1-y)^{q-n}$, $n,m,q,p\in\mathds{N}_0$, $p\geq m$, $q\geq n$, with non-negative coefficients [@hildeb33]. Hildebrandt and Schoenberg [@hildeb33] already solved the moment problem on $[0,1]^2$ (and more generally on $[0,1]^n$ for all $n\in\mathds{N}$) getting the same result as Haviland. The idea of using $\mathrm{Pos}(K)$-descriptions to solve the moment problem was therefore already used by Hildebrandt and Schoenberg in 1933 [@hildeb33] before Haviland uses this in [@havila35] and generalized this in [@havila36] as suggested to him by Wintner. With these broader historical remarks we see that of course more people are connected to Theorem [Haviland's Theorem 4](#thm:haviland){reference-type="ref" reference="thm:haviland"}. It might also be appropriate to call Theorem [Haviland's Theorem 4](#thm:haviland){reference-type="ref" reference="thm:haviland"} the *Haviland--Wintner* or *Haviland--Hildebrand--Schoenberg--Wintner Theorem*. But as so often, the list of contributors is long (and maybe even longer) and hence the main contribution (the general proof) is rewarded by calling it just Haviland Theorem. As one other solved moment problem of the long list (our list here is far from complete) is the following. **Švenco's Theorem 5** ([@svenco39]). *Let $s=(s_i)_{i\in\mathds{N}_0}$ be a real sequence. The following are equivalent:* (i) *$s$ is a $(-\infty,0]\cup [1,\infty)$-moment sequence.* (ii) *$L_s(p)\geq 0$ for all $p\in\mathrm{Pos}((-\infty,0]\cup[1,\infty))$.* (iii) *$L_s(p^2)\geq 0$, $L_{(X^2-X)s}(p^2)\geq 0$ for all $p\in\mathds{R}[x]$.* (iv) *$s$ and $(X^2-X)s$ are positive semi-definite.* (v) *$\mathcal{H}(s)\succeq 0$ and $\mathcal{H}((X^2-X)s)\succeq 0$.* All moment problems on closed and semi-algebraic sets $K\subseteq\mathds{R}$ follow nowadays easily from and the fact that any preodering from a natural description of $K$ is saturated, see e.g. [@marshallPosPoly Prop. 2.7.3]. The higher dimensional moment problem is much harder than the one-dimensional moment problem and in general it is not solved. The reason is that a description of $\mathrm{Pos}(K)$ is in general unknown. A huge progress in this field was done by Konrad Schmüdgen in 1991 [@schmud91] where he solved the $K$-moment problem for compact semi-algebraic sets $K\subset\mathds{R}^n$, $n\geq 2$. As a corollary he gained a complete description of strictly positive $f\in\mathrm{Pos}(K)$. These and subsequence results are discussed elsewhere [@marshallPosPoly; @lauren09]. ## Early Results with Gaps The early history of moment problems with gaps is very thin. We discuss only [@hausdo21a] and [@boas39]. Hausdorff just solved in [@hausdo21] (submitted 11th February 1920) and in [@hausdo21a] (submitted 8th September 1920) he treats $$s_n = \int_0^1 x^{k_n}~\mathrm{d}\mu(x)$$ with $$k_0 = 0 < k_1 < k_2 < \dots < k_n < \dots$$ for a sequence of real numbers, i.e., not necessarily in $\mathds{N}_0$. See also [@shohat43 p. 104]. Since Hausdorff in [@hausdo21a] did not have access to [@havila36] or the description of all non-negative linear combinations of $1,x^{k_1}, \dots, x^{k_n},\dots$ the results in [@hausdo21a] need complicated formulations and are not very strong. Only with the description of non-negative linear combinations by Karlin [@karlin63] an easy formulation of the result is possible. We will therefore postpone the exact formulation to , [Theorem 51](#thm:sparseHausd){reference-type="ref" reference="thm:sparseHausd"}, and [Theorem 52](#thm:generalSparseHausd){reference-type="ref" reference="thm:generalSparseHausd"} where we present easy proofs using also the theory of adapted spaces [@choquet69; @phelpsLectChoquetTheorem; @schmudMomentBook]. In [@boas39] Boas investigates the Stieltjes moment problem ($K=[0,\infty)$) with gaps. Similar to [@hausdo21a] the results are difficult to read and they are unfortunately incomplete since Boas (like Hausdorff) did not have access to the description of all non-negative or strictly positive polynomials with gaps (or more general exponents). We will give the complete solution of the $[0,\infty)$-moment problem with gaps and more general exponents in . ## Finitely Atomic Representing Measures: The Richter Theorem When working with a truncated moment sequence it is often useful in theory and applications to replace a representing measure with a finitely atomic measure without changing the moments. That this is always possible for truncated moment sequences was first proved in full generality by Richter [@richte57]. **Richter's Theorem 6** ([@richte57 Satz 4]). *Let $n\in\mathds{N}$, $(\mathcal{X},\mathfrak{A})$ be a measurable space, and $\{f_i\}_{i=1}^n$ be a family of real measurable functions $f_i:\mathcal{X}\to\mathds{R}$. Then for every measure $\mu$ on $\mathcal{X}$ such that all $f_i$ are $\mu$-integrable, i.e., $$s_i := \int_\mathcal{X}f_i(x)~\mathrm{d}\mu(x) \quad <\infty$$ for all $i=1,\dots,n$, there exists a $K\in\mathds{N}$ with $K\leq n$, points $x_1,\dots,x_K\in\mathcal{X}$ pairwise different, and $c_1,\dots,c_K\in (0,\infty)$ such that $$s_i = \sum_{j=1}^K c_j\cdot f_i(x_j)$$ holds for all $i=1,\dots,n$.* The history of this result is often still misrepresented in the literature, even after K. Schmüdgen and the present author compared the different contributions and publication dates in detail in [@didioCone22]. With these historical remarks it also is appropriate to call Theorem [Richter's Theorem 6](#thm:richter){reference-type="ref" reference="thm:richter"} the *Richter--Rogosinski--Rosenbloom Theorem* [@richte57; @rogosi58; @rosenb52]. Every other result before or after [@richte57] is only a special case and can easily be recovered from , especially [@bayer06]. Since only needs a family of finitely many measurable functions it also includes all cases of gaps in the truncated moment theory. ## Signed Representing Measures: Boas' Theorem In the theory of moments almost exclusively the representation by non-negative measures is treated. The reason is the following. **Boas' Theorem 7** ([@boas39a] or e.g. [@shohat43 p. 103, Thm. 3.11]). *Let $s = (s_i)_{i\in\mathds{N}_0}$ be a real sequence. Then there exist infinitely many signed measures $\mu$ on $\mathds{R}$ and infinitely many signed measures $\nu$ on $[0,\infty)$ such that $$s_i = \int_\mathds{R}x^{i}~\mathrm{d}\mu(x) = \int_0^\infty x^{i}~\mathrm{d}\nu(x)$$ holds for all $i\in\mathds{N}_0$.* also holds in the $n$-dimensional case on $\mathds{R}^n$ and $[0,\infty)^n$ for any $n\in\mathds{N}$. See [@shohat43 p. 104] for an extension which kinds of measures can be chosen. also covers the case with gaps. If any gaps in the real sequence $s$ are present then fill them with any real number you like. # T-Systems {#sec:tsystems} We have seen the early attempts to deal with gaps in the moment problems. A sufficient solution was at these times not possible. Only with the introduction of so called T-systems and their rigorous investigation significant progress and finally complete solutions were possible. For more on the early development and history of T-systems see [@krein51; @karlin63; @karlinStuddenTSystemsBook; @kreinMarkovMomentProblem]. ## Definition and Basic Properties of T-Systems **Definition 8**. Let $n\in\mathds{N}$, $\mathcal{X}$ be a set with $|\mathcal{X}|\geq n+1$, and let $\mathcal{F}=\{f_i\}_{i=0}^n$ be a family of real functions $f_i:\mathcal{X}\to\mathds{R}$. We call any linear combination $$f = \sum_{i=0}^n a_i\cdot f_i \quad\in\mathrm{lin}\,\mathcal{F}$$ with $a_1,\dots, a_n\in\mathds{R}$ a *polynomial*. The family $\mathcal{F}$ on $\mathcal{X}$ is called a *Tchebycheff system* (*T-system*) *of order $n$ on $\mathcal{X}$* if any polynomial $f\in\mathrm{lin}\,\mathcal{F}$ with $\sum_{i=0}^n a_i^2 > 0$ has at most $n$ zeros on $\mathcal{X}$. If $\mathcal{X}$ is a topological space and $\mathcal{F}$ is a family of continuous functions then we call $\mathcal{F}$ a *continuous T-system*. If additionally $\mathcal{X}$ is the unit circle then we call $\mathcal{F}$ a *periodic T-system*. **Corollary 9**. *Let $n\in\mathds{N}$ and $\mathcal{F}=\{f_i\}_{i=0}^n$ be a T-system of order $n$ on some $\mathcal{X}$ with $|\mathcal{X}|\geq n+1$. Let $\mathcal{Y}\subset\mathcal{X}$ with $|\mathcal{Y}|\geq n+1$. Then $\mathcal{G}:= \{f_i|_\mathcal{Y}\}_{i=0}^n$ is a T-system of order $n$ on $\mathcal{Y}$.* *Proof.* Let $f\in\mathrm{lin}\,\mathcal{F}$. Then $f$ has at most $n$ zeros in $\mathcal{X}$ and hence $f|_\mathcal{Y}$ has at most $n$ zeros in $\mathcal{Y}\subset\mathcal{X}$. Since for any $g\in\mathrm{lin}\,\mathcal{G}$ there is a $f\in\mathrm{lin}\,\mathcal{F}$ such that $g = f|_\mathcal{Y}$ we have the assertion. ◻ The set $\mathcal{X}$ does not require any structure or property except $|\mathcal{X}|\geq n+1$. In the theory of T-systems we often deal with one special matrix. We use the following abbreviation. **Definition 10**. Let $n\in\mathds{N}$, $\{f_i\}_{i=0}^n$ be a family of real functions on a set $\mathcal{X}$ with $|\mathcal{X}|\geq n+1$. We define the matrix $$\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix} := \begin{pmatrix} f_0(x_0) & f_1(x_0) & \dots & f_n(x_0)\\ f_0(x_1) & f_1(x_1) & \dots & f_n(x_1)\\ \vdots & \vdots & & \vdots\\ f_0(x_n) & f_1(x_n) & \dots & f_n(x_n) \end{pmatrix} = (f_i(x_j))_{i,j=0}^n$$ for any $x_0,\dots,x_n\in\mathcal{X}$. **Lemma 11** (see e.g. [@kreinMarkovMomentProblem p. 31]). *Let $n\in\mathds{N}$, $\mathcal{X}$ be a set with $|\mathcal{X}|\geq n+1$, and $\mathcal{F}=\{f_i\}_{i=0}^n$ be a family of real functions $f_i:\mathcal{X}\to\mathds{R}$. The following are equivalent:* (i) *$\mathcal{F}$ is a T-system of order $n$ on $\mathcal{X}$.* (ii) *The determinant $$\det\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix}$$ does not vanish for any pairwise distinct points $x_0,\dots,x_n\in\mathcal{X}$.* *Proof.* [(i) $\Rightarrow$ (ii):]{.ul} Let $x_0,\dots,x_n\in\mathcal{X}$ be pairwise distinct. Since $\mathcal{F}$ is a T-system we have that any non-trivial polynomial $f$ has at most $n$ zeros, i.e., the matrix $$\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix}$$ has trivial kernel and hence its determinant is non-zero. Since $x_0,\dots,x_n\in\mathcal{X}$ are arbitrary pairwise distinct we have (ii). [(ii) $\Rightarrow$ (i):]{.ul} Assume there is a polynomial $f$ with $\sum_{i=0}^n a_i^2 >0$ which has the $n+1$ pairwise distinct zeros $z_0,\dots,z_n\in\mathcal{X}$. Then the matrix $$Z=\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ z_0 & z_1 & \dots & z_n \end{pmatrix}$$ has non-trivial kernel since $0\neq (a_0,a_1,\dots,a_n)^T\in\ker Z$ and hence $\det Z = 0$ in contradiction to (ii). ◻ **Corollary 12**. *Let $n\in\mathds{N}$, and $\mathcal{F}= \{f_i\}_{i=0}^n$ be a T-system of order $n$ on some $\mathcal{X}\subseteq\mathds{R}$ with $|\mathcal{X}|\geq n+1$. The following hold:* (i) *The functions $f_0,\dots,f_n$ are linearly independent over $\mathcal{X}$.* (ii) *For any $f = \sum_{i=0}^n a_i\cdot f_i\in\mathrm{lin}\,\mathcal{F}$ the coefficients $a_i$ are unique.* *Proof.* Follows immediately from (i) $\Rightarrow$ (ii). ◻ We even have the following. **Theorem 13** (see e.g. [@kreinMarkovMomentProblem p. 33]). *Let $n\in\mathds{N}$, $\mathcal{F}$ be a T-system on some set $\mathcal{X}$ with $|\mathcal{X}|\geq n+1$, and let $x_0,\dots,x_n\in\mathcal{X}$ be pairwise different points. The following hold:* (i) *Any $f\in\mathrm{lin}\,\mathcal{F}$ is uniquely determined by its values $f(x_0),\dots, f(x_n)$.* (ii) *For any $y_0,\dots,y_n\in\mathds{R}$ there exists a unique $f\in\mathrm{lin}\,\mathcal{F}$ with $f(x_i)=y_i$ for all $i=0,\dots,n$.* *Proof.* [(i):]{.ul} Since $f\in\mathrm{lin}\,\mathcal{F}$ we have $f = \sum_{i=0}^n a_i\cdot f_i$. Let $x_1,\dots,x_n\in\mathcal{F}$ be pairwise distinct. Then by (i) $\Rightarrow$ (ii) we have that $$\begin{pmatrix} f(x_0)\\ \vdots\\ f(x_n) \end{pmatrix} = \begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix}\cdot \begin{pmatrix} \alpha_0\\ \vdots\\ \alpha_n \end{pmatrix}$$ has the unique solution $\alpha_0 = a_0$, ..., $\alpha_n = a_n$. [(ii):]{.ul} By the same argument as in (i) the system $$\begin{pmatrix} y_0\\ \vdots\\ y_n \end{pmatrix} = \begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix}\cdot \begin{pmatrix} \alpha_0\\ \vdots\\ \alpha_n \end{pmatrix}$$ has the unique solution $\alpha_0 = a_0$, ..., $\alpha_n = a_n$. ◻ So far we imposed no structure on $\mathcal{X}$. We now impose structure on $\mathcal{X}$. The following structural result was proved in [@mairhu56] for compact subsets $\mathcal{X}$ of $\mathds{R}^n$ and for arbitrary compact sets $\mathcal{X}$ in [@sieklu58; @curtis59]. **Theorem 14** ([@mairhu56 Thm. 2], [@sieklu58], [@curtis59 Thm. 8 and Cor.]). *Let $n\in\mathds{N}$ and $\mathcal{F}$ be a continuous T-system of order $n$ on a topological space $\mathcal{X}$. If $\mathcal{X}$ is a compact metrizable space then $\mathcal{X}$ can be homeomorphically embedded in the unit circle $\{(x,y)\in\mathds{R}^2 \,|\, x^2 + y^2 = 1\}$.* **Corollary 15** ([@curtis59 Thm. 8]). *The order $n$ of a periodic T-system is even.* *Proof.* Let $\varphi:[0,2\pi]\to S=\{(x,y)\in\mathds{R}^2 \,|\, x^2 + y^2\}$ with $\varphi(\alpha) = (\sin\alpha, \cos\alpha)$ and $\mathcal{F}=\{f_i\}_{i=0}^n$ be a periodic T-system. Then the $f_i$ are continuous and hence also $$\det\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ t_0 & t_1 & \dots & t_n \end{pmatrix}$$ is continuous in $t_0,\dots,t_n\in S$. If $\mathcal{F}$ is a T-system we have that $$d(\alpha) := \det\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ \varphi(\alpha) & \varphi(\alpha+2\pi/(n+1)) & \dots & \varphi(\alpha + 2n\pi/(n+1)) \end{pmatrix}$$ in non-zero for all $\alpha\in [0,2\pi]$ and never changes singes. If $n$ is odd then $d(0) = -d(2\pi/(n+1))$ which is a contradiction. Hence, $n$ must be even. ◻ ## Examples of T-systems {#subsec:examples} **Examples 16** (algebraic polynomials, see e.g. [@karlinStuddenTSystemsBook; @kreinMarkovMomentProblem]). 1. Let $n\in\mathds{N}$ and $\mathcal{X}\subseteq \mathds{R}$ with $|\mathcal{X}|\geq n+1$. Then the family $\mathcal{F}= \{x^i\}_{i=0}^n$ is a T-system. This follows immediately from the Vandermonde determinant $$\label{eq:abuse} \det\begin{pmatrix} 1 & x & \dots & x^n\\ x_0 & x_1 & \dots & x_n \end{pmatrix} = \prod_{0\leq i<j\leq n} (x_j-x_i)$$ for any $x_0,\dots,x_n\in\mathcal{X}$. Note that we abuse the notation for the algebraic polynomial cases. The functions $f_0,\dots,f_n$ should not be denoted by $x^i$ but by $$\cdot^{i}:\mathds{R}\to\mathds{R},\ x\mapsto x^i.$$ However, then we have the notation $$\begin{pmatrix} \cdot^0 & \cdot^1 & \dots & \cdot^n\\ x_0 & x_1 & \dots & x_n \end{pmatrix} \qquad\text{or more general}\qquad \begin{pmatrix} \cdot^{\alpha_0} & \cdot^{\alpha_1} & \dots & \cdot^{\alpha_n}\\ x_0 & x_1 & \dots & x_n \end{pmatrix}$$ which seems hard to read. For convenience we will therefore abuse the notation and use $x^i$ and ([\[eq:abuse\]](#eq:abuse){reference-type="ref" reference="eq:abuse"}). 2. Let $n\in\mathds{N}$, $\mathcal{X}\subseteq [0,\infty)$ with $|\mathcal{X}|\geq n+1$, and $\alpha_0 = 0 < \alpha_1 < \dots < \alpha_n$ be real numbers. Then $\mathcal{F}= \{x^{\alpha_i}\}_{i=0}^n$ is a T-system of order $n$ on $\mathcal{X}$. 3. Let $n\in\mathds{N}$, $\mathcal{X}\subseteq (0,\infty)$ with $|\mathcal{X}|\geq n+1$, and $0 < \alpha_0 < \dots < \alpha_n$ be real numbers. Then $\mathcal{F}= \{ x^{\alpha_i}\}_{i=0}^n$ is a T-system of order $n$ on $\mathcal{X}$.$\circ$ **Example 17** (see e.g. [@kreinMarkovMomentProblem p. 38]). Let $n\in\mathds{N}$ and $\alpha_0 < \alpha_1 < \dots < \alpha_n$ be reals. Then $$\mathcal{F}= \left\{e^{\alpha_0 x}, e^{\alpha_1 x},\dots, e^{\alpha_n x}\right\}$$ is a T-system on any $\mathcal{X}\subseteq\mathds{R}$ with $|\mathcal{X}|\geq n+1$.$\circ$ **Example 18** (see e.g. [@kreinMarkovMomentProblem p. 37-38]). Let $n\in\mathds{N}$ and $\alpha_0 < \alpha_1 < \dots < \alpha_n$ be reals. Then $$\mathcal{F}= \left\{\frac{1}{x+\alpha_0}, \frac{1}{x+\alpha_1}, \dots, \frac{1}{x+\alpha_n}\right\}$$ is a continuous T-system on any $[a,b]$ or $[a,\infty)$ with $-\alpha_0 < a < b$.$\circ$ **Example 19** (see e.g. [@kreinMarkovMomentProblem p. 38]). Let $n\in\mathds{N}$ and let $f\in C^n(\mathcal{X})$ with $\mathcal{X}=[a,b]$, $a<b$, and $f^{(n)}>0$ on $\mathcal{X}$. Then $$\mathcal{F}= \{1,x,x^2,\dots, x^{n-1},f\}$$ is a continuous T-system of order $n$ on $\mathcal{X}= [a,b]$. We can also allow $\mathcal{X}= (a,b)$, $[a,\infty)$, $(-\infty,b)$, ....$\circ$ **Example 20** (see e.g. [@karlinStuddenTSystemsBook p. 10]). Let $n\in\mathds{N}$ and $\mathcal{F}= \{f_i\}_{i=0}^n$ be a (continuous) T-systems on $\mathcal{X}\subseteq\mathds{R}$ with $|\mathcal{X}|\geq n+1$. Then for any (continuous) function $r:\mathcal{X}\to (0,\infty)$ the family $\{r\cdot f_i\}_{i=0}^n$ is a (continuous) T-system.$\circ$ **Example 21**. Let $n\in\mathds{N}$, $\{f_i\}_{i=0}^n$ be a (continuous) T-system of order $n$ on $\mathcal{X}\subseteq\mathds{R}$ and let $g:\mathcal{Y}\subseteq\mathds{R}\to\mathcal{X}$ be a strictly increasing (continuous) function. Then $\{f_i\circ g\}_{i=0}^n$ is a (continuous) T-systems of order $n$ on $\mathcal{Y}$. $\circ$ ## Non-Negativity, Zeros, and Determinantal Representations of Polynomials in T-Systems **Theorem 22** (see e.g. [@karlinStuddenTSystemsBook p. 20] or [@kreinMarkovMomentProblem p. 33]). *Let $n\in\mathds{N}$, $\mathcal{F}= \{f_i\}_{i=0}^n$ be a T-system on some set $\mathcal{X}$ with $|\mathcal{X}|\geq n+1$, $x_1,\dots,x_n\in\mathcal{X}$ be $n$ distinct points, and let $f\in\mathrm{lin}\,\mathcal{F}$ be a polynomial. The following are equivalent:* (i) *$f(x_i)=0$ holds for all $i=1,\dots,n$.* (ii) *There exists a constant $c\in\mathds{R}$ such that $$f(x) = c\cdot\det\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x & x_1 & \dots & x_n \end{pmatrix}.$$* *Proof.* [(ii) $\Rightarrow$ (i):]{.ul} Clear. [(i) $\Rightarrow$ (ii):]{.ul} If $f=0$ then $c=0$ so the assertion holds. If $f\neq 0$ then there is a $x_0\in\mathcal{X}\setminus\{x_1,\dots,x_n\}$ such that $f(x)\neq 0$. Then also the determinant in (ii) is non-zero and we can choose $c$ such that both $f$ and the scaled determinant coincide also in $x_0$. By a polynomial is uniquely determined by $x_0,\dots,x_n$ which shows that (ii) is one and hence the only possible polynomial which fulfills (i). ◻ So far we treated general T-systems. For further properties we go to continuous T-systems. **Definition 23**. Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system on $\mathcal{X}\subseteq\mathds{R}$ an interval, $f\in\mathrm{lin}\,\mathcal{F}$, and let $x_0$ be a zero of $f$. Then $x_0\in\mathrm{int}\,\mathcal{X}$ is called a *non-nodal* zero if $f$ does not change sign at $x_0$. Otherwise the zero $x_0$ is called *nodal*, i.e., either $f$ changes signs at $x_0$ or $x_0$ is a boundary point of $\mathcal{X}$. The following result bounds the number of nodal and non-nodal zeros. **Theorem 24** (see [@krein51] or e.g. [@kreinMarkovMomentProblem p. 34, Thm. 1.1]). *Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system of order $n$ on $\mathcal{X}= [a,b]$ with $a<b$.* *If $f\in\mathrm{lin}\,\mathcal{F}$ has $k\in\mathds{N}_0$ non-nodal zeros and $l\in\mathds{N}_0$ nodal zeros in $\mathcal{X}$ then $2k + l\leq n$.* The proof is adapted from [@kreinMarkovMomentProblem Thm. 1.1]. *Proof.* [$\mathcal{X}=[a,b]$ and $k=0$:]{.ul} If $f\in\mathrm{lin}\,\mathcal{F}$ has $l$ zeros then $n \geq l$ by . [$\mathcal{X}=[a,b]$ and $k\geq 1$:]{.ul} Let $x_1,\dots,x_n\in\mathcal{X}$ be the zeros of $f$. Set $$M_i := \max_{x_{i-1} \leq x\leq x_i} |f(x)|$$ for $i=1,\dots,k+l+1$ with $t_0 = a$ and $x_{k+l+1}=b$. Additionally, set $$m := \frac{1}{2}\min_{i=1,\dots,k+l+1} M_i >0.$$ We construct a polynomial $g_1\in\mathrm{lin}\,\mathcal{F}$ such that 1. $g_1$ has the values $m$ at the non-nodal zeros $x_i$ of $f$ with $f\geq 0$ in a neighborhood of $x_i$, 2. $g_1$ has the values $-m$ at the non-nodal zeros $x_i$ of $f$ with $f\leq 0$ in a neighborhood of $x_i$, and 3. $g_1$ vanishes at all nodal zeros $x_i$. After renumbering the $x_i$'s we can assume $x_1,\dots,x_{k_1}$ fulfill (a), $x_{k_1+1},\dots,x_{k_1+k_2}$ fulfill (b), and $x_{k_1+k_2+1},\dots,x_{k_1+k_2+l}$ fulfill (c) with $k_1+k_2=k$. By we have $k+l\leq n$ and hence by we have that $$\begin{pmatrix} m\\ \vdots\\ m\\ -m\\ \vdots\\ -m\\ 0\\ \vdots\\ 0 \end{pmatrix} = \begin{pmatrix} f_0(x_1) & f_1(x_1) & \dots & f_n(x_1)\\ \vdots & \vdots & & \vdots\\ f_0(x_{k+l}) & f_1(x_{k+l}) & \dots & f_n(x_{k+l}) \end{pmatrix}\cdot \begin{pmatrix} \beta_0\\ \vdots\\ \beta_n \end{pmatrix}$$ has at least one solution, say $\beta_0 = b_0$, ..., $\beta_n = b_n$. Then $g_1 = \sum_{i=0}^n b_i\cdot f_i\in\mathrm{lin}\,\mathcal{F}$ fulfills (a) to (c). Set $$\rho := \frac{m}{2\cdot \|g_1\|_\infty}$$ and let $g_2 := f - g_1$. We show that to each non-nodal zero $x_i$ of $f$ there correspond two zeros of $g_2$: Let $x_i$ be a non-nodal zero of $f$ with $f\geq 0$ in a neighborhood of $x_i$, say. We can find a point $y_i\in (x_{i-1},x_i)$ and a point $y_{i+1}\in (x_i,x_{i+1})$ such that $$f(y_i) = M_i > m \qquad\text{and}\qquad f(y_{i+1}) = M_{i+1} > m.$$ Therefore, $g_2(y_i) > 0$ and $g_2(y_{i+1}) > 0$. Since $g_2(x_i) = -\rho\cdot m < 0$ if follows that $g_2$ as a zero both in $(y_i,x_i)$ and $(x_i,y_{i+1})$. Additionally, $g_2$ also vanishes at all nodal zeros of $f$ and so has at least $2k+l$ distinct zeros. Therefore, by we have $2k+l\leq n$. ◻ **Corollary 25**. *also holds for sets $\mathcal{X}\subseteq\mathds{R}$ of the form* (i) *$\mathcal{X}= (a,b)$, $[a,b)$, $(a,b]$ with $a<b$,* (ii) *$\mathcal{X}= (a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$,* (iii) *$\mathcal{X}= \{x_1,\dots,x_k\}\subseteq\mathds{R}$ with $k\geq n+1$ and $x_1 < \dots < x_k$, and* (iv) *countable unions of (i) to (iii).* *Proof.* [$\mathcal{X}=[0,\infty)$:]{.ul} Let $0\leq x_1 < \dots < x_k$ be the zeros of $f$ in $[0,\infty)$. Since every T-system on $[0,\infty)$ is also a T-system on $[0,b]$ for any $b>0$ the claim follows from with $b=x_k + 1$. For the other assertions adapt (if necessary) the proof of . ◻ **Definition 26**. Let $x\in [a,b]$ with $a\leq b$. We define the *index* $\varepsilon(x)$ by $$\varepsilon(x) := \begin{cases} 2 &\text{if}\ x\in (a,b),\\ 1 &\text{if}\ x=a\ \text{or}\ b. \end{cases}$$ The same definition holds for sets $\mathcal{X}$ as in . **Definition 27**. Let $n\in\mathds{N}$ and $\mathcal{F}$ be a T-system of order $n$ on some set $\mathcal{X}$. We define $$\begin{aligned} (\mathrm{lin}\,\mathcal{F})^e &:= \left\{ \sum_{i=0}^n a_i\cdot f_i \,\middle|\, \sum_{i=0}^n a_i^2 = 1\right\},\\ (\mathrm{lin}\,\mathcal{F})_+ &:= \left\{ f\in\mathcal{F}\,\middle|\, f\geq 0\ \text{on}\ \mathcal{X}\right\}, % \intertext{and} % (\mathrm{lin}\,\mathcal{F})_+^e &:= (\mathrm{lin}\,\mathcal{F})^e \cap (\mathrm{lin}\,\mathcal{F})_+.\end{aligned}$$ With these definitions we can prove the following existence criteria for non-negative polynomials in a T-systems on $[a,b]$. **Theorem 28**. *Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system on $\mathcal{X}=[a,b]$, and $x_1,\dots,x_m\in\mathcal{X}$. The following are equivalent:* (i) *The points $x_1,\dots,x_m$ are zeros of a non-negative polynomial $f\in\mathrm{lin}\,\mathcal{F}$.* (ii) *$\displaystyle\sum_{i=1}^m \varepsilon(x_i) \leq n$.* The proof is adapted from [@kreinMarkovMomentProblem p. 35, Thm. 1.2]. *Proof.* "(i) $\Rightarrow$ (ii)" is and we therefore only have to prove "(ii) $\Rightarrow$ (i)". [Case I:]{.ul} At first assume that $a < x_1 < \dots < x_m < b$ and $\sum_{i=0}^m \varepsilon(x_i) = 2m = n$. If $2m < n$ then add $k$ additional points $x_{m+1},\dots,x_{m+k}$ such that $2m + 2k = n$ and $x_m < x_{m+1} < \dots < x_{m+k} < b$. Select a sequence of points $(x_1^{(j)},\dots,x_m^{(j)})\in\mathds{R}^m$, $j\in\mathds{N}$, such that $$a < x_1 < x_1^{(j)} < \dots < x_m < x_m^{(j)} < b$$ for all $j\in\mathds{N}$ and $\lim_{j\to\infty} x_i^{(j)} = x_i$ for all $i=1,\dots,m$. Set $$\label{eq:gjdfns} g_j(x) := c_j\cdot \det\begin{pmatrix} f_0 & f_1 & f_2 & \dots & f_{2m-1} & f_{2m}\\ x & x_1 & x_1^{(j)} & \dots & x_m & x_m^{(j)} \end{pmatrix} \quad\in (\mathrm{lin}\,\mathcal{F})^e$$ for some $c_j>0$. Since $(\mathrm{lin}\,\mathcal{F})^e$ is compact we can assume that $g_j$ converges to some $g_0\in (\mathrm{lin}\,\mathcal{F})^e$. Then $g_0$ has $x_1,\dots,x_m$ as zeros with $\varepsilon(x_i)=2$ and $g_0$ is non-negative since $g_j>0$ on $[a,x_1)$, $(x_1^{(j)},x_2)$, ..., $(x_{m-1}^{(j)}, x_m)$, and $(x_m^{(j)},b]$ as well as $g_j<0$ on $(x_1,x_1^{(j)})$, $(x_2,x_2^{(j)})$, ..., $(x_m,x_m^{(j)})$. [Case II:]{.ul} If $a = x_1 < x_2 < \dots < x_m < b$ with $\sum_{i=1}^m \varepsilon(x_i) = 2m-1 = n$ the only modification required in case I is to replace ([\[eq:gjdfns\]](#eq:gjdfns){reference-type="ref" reference="eq:gjdfns"}) by $$g_j(x) := -c_j\cdot \det \begin{pmatrix} f_0 & f_1 & f_2 & f_3 & \dots & f_{2m-2} & f_{2m-1}\\ x & a & x_2 & x_2^{(j)} & \dots & x_m & x_m^{(j)} \end{pmatrix} \quad\in (\mathrm{lin}\,\mathcal{F})^e$$ with some normalizing factor $c_j > 0$. [Case III:]{.ul} The procedure is similar if $x_m = b$ and $\sum_{i=1}^m \varepsilon(x_i) = n$. ◻ *Remark 29*. appears in [@kreinMarkovMomentProblem p. 35, Thm. 1.2] in a stronger version. In [@kreinMarkovMomentProblem p. 35, Thm. 1.2] Krein claims that the $x_1,\dots,x_m$ are the only zeros of some non-negative $f\in\mathrm{lin}\,\mathcal{F}$. This holds when $n=2m + 2p$ for some $p>0$. To see this add to $x_1,\dots,x_m$ in ([\[eq:gjdfns\]](#eq:gjdfns){reference-type="ref" reference="eq:gjdfns"}) points $x_{m+1},\dots,x_{m+p}\in\mathrm{int}\,\mathcal{X}\setminus\{x_1,\dots,x_m\}$ and get $g_0$. Hence, $g_0\geq 0$ has exactly the zeros $x_1,\dots,x_{m+p}$. Then construct in a similar way $g_0'$ with the zeros $x_1,\dots,x_m,x'_{m+1},\dots,x'_{m+p}$ with $x'_{m+1},\dots,x'_{m+p}\in\mathrm{int}\,\mathcal{X}\setminus\{x_1,\dots,x_{m+p}\}$. Hence, $g_0 + g_0'\geq 0$ has only the zeros $x_1,\dots,x_m$. A similar construction works for $n=2m+1$ with or without end points $a$ or $b$. However, Krein misses that for $n=2m$ and when one end point is contained in $x_1,\dots,x_m$ then it might happen that also the other end point must appear. In [@karlinStuddenTSystemsBook p. 28, Thm. 5.1] additional conditions are given which ensure that $x_1,\dots,x_m$ are the only zeros of some $f\geq 0$. For example if also $\{f_i\}_{i=0}^{n-1}$ is a T-system then is can be ensured that $x_1,\dots,x_m$ are the only zeros of some non-negative polynomial $f\in\mathrm{lin}\,\mathcal{F}$, see [@karlinStuddenTSystemsBook p. 28, Thm. 5.1 (b-i)]. For our main example(s), the algebraic polynomials with gaps, this holds. The same problem appears in [@kreinMarkovMomentProblem p. 36, Thm. 1.3]. A weaker but correct version is given in below.$\circ$ *Remark 30*. Assume that in we have additionally that $f_0,\dots,f_n\in C^1([a,b])$. Then in ([\[eq:gjdfns\]](#eq:gjdfns){reference-type="ref" reference="eq:gjdfns"}) we can set $x_i^{(j)} = x_i + j^{-1}$ for all $i=0,\dots,m$ and $j\gg 1$. For $j\to\infty$ with $c_j = j^m$ we then get $$\begin{aligned} g_0(x) &= \lim_{j\to\infty} j^m\cdot\det \begin{pmatrix} f_0 & f_1 & f_2 & \dots & f_{2m-1} & f_{2m}\\ x & x_1 & x_1 + j^{-1} & \dots & x_m & x_m + j^{-1} \end{pmatrix}\notag\\ % &= \lim_{j\to\infty} j^m\cdot\det \begin{pmatrix} f_0(x) & \dots & f_{2m}(x)\\ f_0(x_1) & \dots & f_{2m}(x_1)\\ f_0(x_1+j^{-1}) & \dots & f_{2m}(x_1+j^{-1})\\ \vdots & & \vdots\\ f_0(x_m) & \dots & f_{2m}(x_m)\\ f_0(x_m+j^{-1}) & \dots & f_{2m}(x_m+j^{-1})\\ \end{pmatrix}\notag\\ % &= \lim_{j\to\infty} \det \begin{pmatrix} f_0(x) & \dots & f_{2m}(x)\\ f_0(x_1) & \dots & f_{2m}(x_1)\\ \frac{f_0(x_1+j^{-1})-f_0(x_1)}{j^{-1}} & \dots & \frac{f_{2m}(x_1+j^{-1})-f_{2m}(x_1)}{j^{-1}}\\ \vdots & & \vdots\\ f_0(x_m) & \dots & f_{2m}(x_m)\\ \frac{f_0(x_m+j^{-1})-f_0(x_m)}{j^{-1}} & \dots & \frac{f_{2m}(x_m+j^{-1})-f_{2m}(x_m)}{j^{-1}} \end{pmatrix}\label{eq:generalg0construction}\\ % &= \begin{pmatrix} f_0(x) & \dots & f_{2m}(x)\\ f_0(x_1) & \dots & f_{2m}(x_1)\\ f_0'(x_1) & \dots & f_{2m}'(x_1)\\ \vdots & & \vdots\\ f_0(x_m) & \dots & f_{2m}(x_m)\\ f_0'(x_m) & \dots & f_{2m}'(x_m) \end{pmatrix},\notag\end{aligned}$$ i.e., double zeros are included by including the values $f_i'(x_j)$. Therefore, whenever we have $C^1$-functions in $\mathcal{F}= \{f_i\}_{i=0}^n$ and $x_i = x_{i+1}$ we define $$\label{eq:doublezeroDfn} \begin{pmatrix} f_0 & \dots & f_{i-1} & f_i & f_{i+1} & f_{i+2} & \dots & f_n\\ x_0 & \dots & x_{i-1} & (x_i & x_i) & x_{i+2} & \dots & x_n \end{pmatrix} := \begin{pmatrix} f_0(x_0) & \dots & f_n(x_0)\\ \vdots & & \vdots\\ f_0(x_{i-1}) & \dots & f_n(x_{i-1})\\ f_0(x_i) & \dots & f_n(x_i)\\ f_0'(x_i) & \dots & f_n'(x_i)\\ f_0(x_{i+2}) & \dots & f_n(x_{i+2})\\ \vdots & & \vdots\\ f_0(x_n) & \dots & f_n(x_n) \end{pmatrix}$$ and equivalently when $x_j = x_{j+1}$, $x_k = x_{k+1}$, ... for additional entries. We use the additional brackets "$($" and "$)$" to indicate that $x_i$ is inserted in the $f_0,\dots,f_n$ and then also into $f_0',\dots,f_n'$ to distinguish ([\[eq:doublezeroDfn\]](#eq:doublezeroDfn){reference-type="ref" reference="eq:doublezeroDfn"}) from to avoid confusion. Hence, $$\det\begin{pmatrix} f_0 & \dots & f_{i-1} & f_i & f_{i+1} & f_{i+2} & \dots & f_n\\ x_0 & \dots & x_{i-1} & x_i & x_i & x_{i+2} & \dots & x_n \end{pmatrix} = 0$$ since in two rows $x_i$ is inserted into $f_0,\dots,f_n$, while in $$\begin{pmatrix} f_0 & \dots & f_{i-1} & f_i & f_{i+1} & f_{i+2} & \dots & f_n\\ x_0 & \dots & x_{i-1} & (x_i & x_i) & x_{i+2} & \dots & x_n \end{pmatrix}$$ indicates that $x_i$ is inserted in $f_0,\dots,f_n$ and then also into $f_0',\dots,f_n'$. Extending this to zeros of order $k$ for $C^{k+1}$-functions is straight forward and we leave it to the reader to write down the formulas and their proofs. Similar to ([\[eq:doublezeroDfn\]](#eq:doublezeroDfn){reference-type="ref" reference="eq:doublezeroDfn"}) we write for any $a\leq x_0 \leq x_1 \leq \dots \leq x_n\leq b$ the matrix as $$\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix}^*$$ when $f_0,\dots, f_n$ are sufficiently differentiable. We often want to express polynomials $f\in\mathrm{lin}\,\mathcal{F}$ as determinants ([\[eq:gjdfns\]](#eq:gjdfns){reference-type="ref" reference="eq:gjdfns"}) only by knowing their zeros $x_1,\dots,x_k$. If arbitrary multiplicities appear we only have $x_1 \leq x_2 \leq \dots \leq x_n$ where we include zeros multiple times according to their multiplicities. Hence, for $$x_0 = \dots = x_{i_1} < x_{i_1+1} = \dots = x_{i_2} < \dots < x_{i_k+1} = \dots = x_n$$ we introduce a simpler notation to write down ([\[eq:doublezeroDfn\]](#eq:doublezeroDfn){reference-type="ref" reference="eq:doublezeroDfn"}): $$\label{eq:doublezeroDfn2} \left(\begin{array}{c|cccc} f_0 & f_1 & f_2 & \dots & f_n\\ x & x_1 & x_2 & \dots & x_n \end{array}\right) := % \begin{pmatrix} f_0 & f_1\; \dots\; f_{i_1} & f_{i_1+1} \;\dots\; f_{i_2} & \dots & f_{i_k+1} \;\dots\; f_{i_k+1}\\ x & (x_1\; \dots\; x_{i_1}) & (x_{i_1+1} \;\dots\; x_{i_2}) & \dots & (x_{i_k+1} \;\dots\; \;\; x_n)\;\;\, \end{pmatrix}.$$ Clearly $(\ref{eq:doublezeroDfn2}) \in\mathrm{lin}\,\mathcal{F}$. For ([\[eq:doublezeroDfn2\]](#eq:doublezeroDfn2){reference-type="ref" reference="eq:doublezeroDfn2"}) to be well-defined we need $\mathcal{F}\subset C^{m-1}$ where $m$ is the largest multiplicity of any zero. However, the procedure ([\[eq:generalg0construction\]](#eq:generalg0construction){reference-type="ref" reference="eq:generalg0construction"}) can lead to the zero polynomial. We have to introduce ET-systems, see and . In we did not need the condition $\mathcal{F}\subset C^m$ for some $m\geq 1$. The limit $g_0$ of the $g_j$ in ([\[eq:gjdfns\]](#eq:gjdfns){reference-type="ref" reference="eq:gjdfns"}) does not need the unique $f_0',\dots,f_n'$ and therefore the limit needs not to be unique.$\circ$ **Corollary 31**. *also holds for intervals $\mathcal{X}\subseteq\mathds{R}$, i.e., $$\label{eq:intervals} \mathcal{X}= (a,b),\ (a,b],\ [a,b),\ [a,b],\ (a,\infty),\ [a,\infty),\ (-\infty,b),\ (-\infty,b],\ \text{and}\ \mathds{R}\qquad\text{with}\ a<b.$$* *Proof.* We have that "(i) $\Rightarrow$ (ii)" follows from . For "(ii) $\Rightarrow$ (i)" we apply on $[\min_i x_i,\max_i x_i]$ and assures that no additional zeros appear in $\mathcal{X}$. ◻ We will give a sharper version of , see also . **Theorem 32**. *Let $n\in\mathds{N}$ and $\mathcal{F}$ be a continuous T-system on $\mathcal{X}=[a,b]$. Additionally, let $x_1,\dots,x_k\in\mathcal{X}$ and $y_1,\dots,y_l\in\mathcal{X}$ be pairwise distinct points. The following are equivalent:* (i) *There exists a polynomial $f\in\mathrm{lin}\,\mathcal{F}$ such that* (a) *$x_1,\dots,x_k$ are the non-nodal zeros of $f$ and* (b) *$y_1,\dots,y_l$ are the nodal zeros of $f$.* (ii) *$2k+l \leq n$.* *Proof.* [(i) $\Rightarrow$ (ii):]{.ul} That is . [(ii) $\Rightarrow$ (i):]{.ul} Adapt the proof and especially the $g_j$'s in ([\[eq:gjdfns\]](#eq:gjdfns){reference-type="ref" reference="eq:gjdfns"}) of accordingly. Let $z_1 < \dots < z_{k+l}$ be the $x_i$'s and $y_i$'s together ordered by size. Then in $g_j$ treat every nodal $z_i$ like the endpoint $a$ or $b$, i.e., include it only once in the determinant, and insert for every non-nodal point $z_i$ the point $z_i$ and the sequence $z_i^{(j)}\in (z_i,z_{i+1})$ with $\lim_{j\to\infty} z_i^{(j)} = z_i$. ◻ **Corollary 33**. *also holds for sets $\mathcal{X}\subseteq\mathds{R}$ of the form* (i) *$\mathcal{X}= (a,b)$, $[a,b)$, $(a,b]$ with $a<b$,* (ii) *$\mathcal{X}= (a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$,* (iii) *$\mathcal{X}= \{x_1,\dots,x_k\}\subseteq\mathds{R}$ with $k\geq n+1$ and $x_1 < \dots < x_k$, and* (iv) *countable unions of (i) to (iii).* *Proof.* In the adapted proof and the $g_j$'s in ([\[eq:gjdfns\]](#eq:gjdfns){reference-type="ref" reference="eq:gjdfns"}) of we do not need to have non-negativity, i.e., in the $g_j$'s sign changes at the $y_i$'s are allowed (and even required). ◻ ## ET-Systems {#sec:et} **Definition 34**. Let $n\in\mathds{N}$ and let $\mathcal{F}= \{f_i\}_{i=0}^n\subset C^n([a,b])$ be a T-system of order $n$ on $[a,b]$ with $a<b$. $\mathcal{F}$ is called an *extended Tchebycheff system* (*ET-system*) *of order $n$* if any polynomial $f\in\mathrm{lin}\,\mathcal{F}\setminus\{0\}$ has at most $n$ zeros counting algebraic multiplicities. For notation of the matrices $$\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix}^*$$ for $a\leq x_0 \leq x_1 \leq \dots x_n\leq b$ see the previous . **Corollary 35** ([@krein51] or e.g. [@kreinMarkovMomentProblem p. 37, P.1.1]). *Let $n\in\mathds{N}$ and $\mathcal{F}=\{f_i\}_{i=0}^n\subset C^n([a,b])$. The following are equivalent:* (i) *$\mathcal{F}$ is an ET-system.* (ii) *We have $$\det\begin{pmatrix} f_0 & f_1 & \dots & f_n\\ x_0 & x_1 & \dots & x_n \end{pmatrix}^* \neq 0$$ for every $a\leq x_0 \leq x_1 \leq \dots \leq x_n\leq b$.* *Proof.* Follows immediately from . ◻ **Example 36** (see e.g. [@karlinStuddenTSystemsBook p. 19, Exm. 12]). Let $n\in\mathds{N}$ and $g_0,\dots,g_n\in C^n([a,b])$ such that $g_0,\dots,g_n>0$ on $[a,b]$ with $a<b$. Define $$\begin{aligned} f_0(x) &:= g_0(x)\\ f_1(x) &:= g_0(x)\cdot \int_a^x g_1(y_1)~\mathrm{d}y_1\\ f_2(x) &:= g_0(x)\cdot \int_a^x g_1(y_1)\cdot\int_a^{y_1} g_2(y_2)~\mathrm{d}y_2~\mathrm{d}y_1\\ &\;\;\;\vdots \\ f_n(x) &:= g_0(x)\cdot \int_a^x g_1(y_1)\cdot \int_a^{y_1} g_2(y_2)~ {\dots} \int_a^{y_{n-1}} g_n(y_n)~\mathrm{d}y_n~ \dots ~\mathrm{d}y_2~\mathrm{d}y_1.\end{aligned}$$ Then $\{f_i\}_{i=0}^n$ is an ET-system on $[a,b]$. $\circ$ **Example 37**. Let $\mathcal{F}= \{1,x,x^3\}$ on $[0,b]$, $b>0$. Then $\mathcal{F}$ is a T-system (Example [Examples 16](#exm:algPoly){reference-type="ref" reference="exm:algPoly"}(b)) but not an ET-system. To see this let $x_0 = x_1 = x_2 = 0$, then $$\begin{pmatrix} f_0 & f_1 & f_2\\ 0 & 0 & 0 \end{pmatrix}^* = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{pmatrix}.$$ This shows that $\mathcal{F}$ is not an ET-system.$\circ$ In the previous example the position $x=0$ prevents the T-system to be a ET-system. If $x=0$ is removed then it is an ET-system. **Example 38**. Let $\alpha_0,\dots,\alpha_n\in\mathds{N}_0$ with $\alpha_0 < \alpha_1 < \dots < \alpha_n$. Then $\mathcal{F}= \{x^{\alpha_i}\}_{i=0}^n$ on $[a,\infty)$ with $a>0$ is an ET-system. For $n=2m$ and $a < x_1 < x_2 < \dots < x_m$ we often encounter a specific polynomial structure and hence we write it down explicitly once: $$\begin{aligned} &\det\begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}\notag\\ % &= \lim_{\varepsilon\to 0} \varepsilon^{-m}\cdot \det\begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & x_1 & x_1+\varepsilon & \dots & x_m & x_m+\varepsilon \end{pmatrix}\notag\\ &=\lim_{\varepsilon\to 0} \left[\prod_{i=1}^m (x_i-x)(x_i+\varepsilon-x)\right]\cdot \left[\prod_{1\leq i<j\leq m} (x_j-x_i)^2(x_j-x_i-\varepsilon)(x_j+\varepsilon-x_i) \right]\label{eq:schurPolyRepr}\\ &\qquad\times s_\alpha(x,x_1,x_1+\varepsilon,\dots,x_m,x_m+\varepsilon)\notag\\ &= \prod_{i=1}^m (x_i-x)^2 \cdot \prod_{1\leq i<j\leq m} (x_j-x_i)^4\cdot s_\alpha(x,x_1,x_1,\dots,x_m,x_m)\notag\end{aligned}$$ where $s_\alpha$ is the Schur polynomial of $\alpha = (\alpha_0,\dots,\alpha_n)$ [@macdonSymFuncHallPoly]. Hence, $s_\alpha(x,x_1,x_1,\dots,x_m,x_m)$ is not divisible by some $(x_i-x)$. In fact, this is a special case of . With we can even allow $-\infty < \alpha_0 < \alpha_1 < \dots < \alpha_n < \infty$ to be reals since $a>0$. $\circ$ *Proof.* Combine the induction $$f^{(m+1)}(x) = \lim_{h\to 0} \frac{f^{(m)}(x+h) - f^{(m)}(x)}{h}$$ and $$\det \begin{pmatrix} x^{\alpha_0} & \dots & x^{\alpha_n}\\ x_0 & \dots & x_n \end{pmatrix} = \prod_{0\leq i<j\leq n} (x_j - x_i)\cdot s_\alpha(x_0,\dots,x_n)$$ where $s_\alpha$ is the Schur polynomial of $\alpha = (\alpha_0,\dots,\alpha_n)$. ◻ **Example 39**. Let $n\in\mathds{N}$. Then the T-system $\mathcal{F}= \{x^i\}_{i=0}^n$ on $\mathds{R}$ is an ET-system.$\circ$ # Sparse Positivstellensätze and Nichtnegativstellensätze {#sec:sparsePosNiNeg} In this section we present the Positivestellensatz for T-systems of Karlin ( and ). We show their application to gain algebraic sparse Positivestellensätze and Nichtnegativstellensätze. They will be used to solve sparse moment problems. ## Sparse Positivstellensätze and Nichtnegativstellensätze on $[a,b]$ for general T-Systems {#sec:posabGeneral} For what follows we want to remind the reader of the index $\varepsilon(x)$ of a point $x$, see . **Definition 40**. Let $\mathcal{Z}\subset [a,b]$. We say *$\mathcal{Z}$ has index $n$* if $\sum_{x\in\mathcal{Z}} \varepsilon(x) = n$. The same definition holds for sets $\mathcal{X}$ as in . Because of its importance and since it was first proved in full generality by Karlin in [@karlin63] we call the following result Karlin's Theorem. **Karlin's Theorem 41** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 66, Thm. 10.1]). *Let $n\in\mathds{N}$, $\mathcal{F}= \{f_i\}_{i=0}^n$ be a continuous T-system of order $n$ on $[a,b]$ with $a<b$, and let $f\in C([a,b])$ with $f>0$ on $[a,b]$. The following hold:* (i) *There exists a unique polynomial $f_*\in\mathrm{lin}\,\mathcal{F}$ such that* (a) *$f(x) \geq f_*(x) \geq 0$ for all $x\in [a,b]$,* (b) *$f_*$ vanishes on a set with index $n$,* (c) *the function $f-f_*$ vanishes at least once between each pair of adjacent zeros of $f_*$,* (d) *the function $f-f_*$ vanishes at least once between the largest zero of $f_*$ and the end point $b$,* (e) *$f_*(b)>0$.* (ii) *There exists a unique polynomial $f^*\in\mathrm{lin}\,\mathcal{F}$ which satisfies the conditions (a)--(d) of (i), and* 5. *$f^*(b) = 0$.* *Proof.* See e.g. [@karlinStuddenTSystemsBook p. 68--71]. ◻ Note, in the previous result we do not need to have $f\in\mathrm{lin}\,\mathcal{F}$. The function $f$ only needs to be continuous and strictly positive on $[a,b]$. An earlier version of is a lemma by Markov [@markov84], see also [@shohat43 p. 80]. For the same reason as for we call the following immediate consequence Karlin's Corollary. It is the T-system Positivstellensatz by Karlin and will be used to generate (algebraic) Positivestellensätze. **Karlin's Corollary 42** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 71, Cor. 10.1(a)]). *Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system of order $n$ on $[a,b]$ with $a<b$, and let $f\in\mathrm{lin}\,\mathcal{F}$ with $f>0$ on $[a,b]$. Then there exists a unique representation $$f = f_* + f^*$$ with $f_*,f^*\in\mathrm{lin}\,\mathcal{F}$ such that* (i) *$f_*,f^*\geq 0$ on $[a,b]$,* (ii) *the zeros of $f_*$ and $f^*$ each are sets of index $n$,* (iii) *the zeros of $f_*$ and $f^*$ strictly interlace, and* (iv) *$f^*(b) = 0$.* *Proof.* Let $f_*$ be the unique $f_*$ from (i). Then $f - f_*\in\mathrm{lin}\,\mathcal{F}$ is a polynomial and fulfills (a)--(d), and (e') of $f^*$ in . But since $f^*$ is unique we have $f - f_* = f^*$. ◻ **Corollary 43** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 71, Cor. 10.1(b)]). *Let $n\in\mathds{N}$, $\{f_i\}_{i=0}^n$ and $\{f_i\}_{i=0}^{n+1}$ be continuous T-systems of order $n$ and $n+1$ on $[a,b]$ with $a<b$. Then $f_{n+1}-(f_{n+1})_*$ and $f_{n+1} - (f_{n+1})^*$ both vanish on sets of index $n+1$.* **Theorem 44** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 72, Thm. 10.2]). *Let $n\in\mathds{N}$, $\mathcal{F}= \{f_i\}_{i=0}^n$ be a continuous T-system of order $n$ on $[a,b]$ with $a<b$, and let $g_1,g_2$ be two continuous functions on $[a,b]$ such that there exists a $g'\in\mathrm{lin}\,\mathcal{F}$ with $$g_1(x) < g'(x) < g_2(x)$$ for all $x\in [a,b]$. The following hold:* (i) *There exists a unique polynomial $f_*\in\mathrm{lin}\,\mathcal{F}$ such that* (a) *$g_1(x) \leq f_*(x) \leq g_2(x)$ for all $x\in [a,b]$, and* (b) *there exist $n+1$ points $x_1<\dots<x_{n+1}$ in $[a,b]$ such that $$f_*(x_{n+1-i}) = \begin{cases} g_1(x_{n+1-i}) & \text{for}\ i=1,3,5,\dots,\\ g_2(x_{n+1-i}) & \text{for}\ i=0,2,4,\dots . \end{cases}$$* (ii) *There exists a unique polynomial $f^*\in\mathrm{lin}\,\mathcal{F}$ such that* 1. *$g_1(x) \leq f^*(x) \leq g_2(x)$ for all $x\in [a,b]$, and* 2. *there exist $n+1$ points $y_1 < \dots < y_{n+1}$ in $[a,b]$ such that $$f^*(y_{n+1-i}) = \begin{cases} g_2(y_{n+1-i}) & \text{for}\ i=1,3,5,\dots,\\ g_1(y_{n+1-i}) & \text{for}\ i=0,2,4,\dots . \end{cases}$$* *Proof.* See [@karlinStuddenTSystemsBook p. 73]. ◻ In and we dealt with $f\in\mathrm{lin}\,\mathcal{F}$ with $f>0$, i.e., they are the Positivstellensatz. The following result allows for $f\geq 0$ and is therefore together with the T-system Nichtnegativstellensatz of Karlin. We get from sparse algebraic Nichtnegativstellensätze (). **Theorem 45** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 74, Thm. 10.3]). *Let $n\in\mathds{N}$, $\mathcal{F}= \{f_i\}_{i=0}^n$ be a continuous ET-system of order $n$ on $[a,b]$ with $a<b$, and let $f\in C^n([a,b])$ be such that $f\geq 0$ on $[a,b]$ and $f$ has $r < n$ zeros (counting multiplicities). The following hold:* (i) *There exists a unique polynomial $f_*\in\mathrm{lin}\,\mathcal{F}$ such that* (a) *$f(x)\geq f_*(x)\geq 0$ for all $x\in [a,b]$,* (b) *$f_*$ has $n$ zeros counting multiplicities,* (c) *if $x_{1} < \dots < x_{{n-r}}$ in $(a,b)$ are the zeros of $f_*$ which remain after removing the $r$ zeros of $f$ then $f - f_*$ vanishes at least twice more (counting multiplicities) in each open interval $(x_i,x_{i+1})$, $i=1,\dots,n-r-1$, and at least once more in each of the intervals $[a,x_1)$ and $(x_{n-r},b]$,* (d) *the zeros $x_1,\dots,x_{n-r}$ of (c) are a set of index $n-r$, and* (e) *$x_{n-r} < b$.* (ii) *There exists a unique polynomial $f^*\in\mathrm{lin}\,\mathcal{F}$ satisfying the conditions (a) to (d) and (e') $x_{n-r} = b$.* *Proof.* See [@karlinStuddenTSystemsBook p. 74--75]. ◻ **Corollary 46** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 76, Cor. 10.3]). *Let $n\in\mathds{N}$, $\mathcal{F}= \{f_i\}_{i=0}^n$ be an ET-system of order $n$ on $[a,b]$ with $a<b$, and let $f\in\mathrm{lin}\,\mathcal{F}$ be such that $f\geq 0$ on $[a,b]$ and $f$ has $r < n$ zeros (counting multiplicities). Then there exists a unique representation $$f = f_* + f^*$$ with $f_*,f^*\in\mathrm{lin}\,\mathcal{F}$ such that* (i) *$f_*, f^*\geq 0$ on $[a,b]$,* (ii) *$f_*$ and $f^*$ have $n$ zeros (counting multiplicity) which strictly interlace if the zeros of $f$ are removed,* (iii) *$f^*(b) =0$.* ## Sparse Positivstellensätze and Nichtnegativstellensätze on $[a,b]$ for Algebraic Polynomials **Theorem 47** (Sparse Algebraic Positivstellensatz on $[a,b]$ with $0<a<b$). *Let $n\in\mathds{N}$, $\alpha_0,\dots,\alpha_n\in\mathds{R}$ be real numbers with $\alpha_0 < \alpha_1 < \dots < \alpha_n$, and let $\mathcal{F}=\{x^{\alpha_i}\}_{i=0}^n$. Then for any $f=\sum_{i=0}^n a_i f_i\in\mathrm{lin}\,\mathcal{F}$ with $f>0$ on $[a,b]$ and $a_n>0$ there exists a unique decomposition $$f = f_* + f^*$$ with $f_*,f^*\in\mathrm{lin}\,\mathcal{F}$ such that* (i) *for $n = 2m$ there exist points $x_1,\dots,x_m,y_1,\dots,y_{m-1}\in [a,b]$ with $$a < x_1 < y_1 < \dots < x_m < b$$ and constants $c_*,c^*>0$ with $$f_*(x) = c_*\cdot\det \begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}\geq 0$$ and $$f^*(x) = -c^*\cdot\det \begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & a & (y_1 & y_1) & \dots & (y_{m-1} & y_{m-1}) & b \end{pmatrix}\geq 0$$ for all $x\in [a,b]$, or* (ii) *for $n = 2m+1$ there exist points $x_1,\dots,x_m,y_1,\dots,y_m\in [a,b]$ with $$a < y_1 < x_1 < \dots < y_m < x_m < b$$ and $c_*,c^*>0$ with $$f_*(x) = -c_*\cdot\det\begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & a & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}\geq 0$$ and $$f^*(x) = c^*\cdot\det\begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (y_1 & y_1) & \dots & (y_m & y_m) & b \end{pmatrix}\geq 0$$ for all $x\in [a,b]$.* *Proof.* By we have that $\mathcal{F}$ on $[a,b]$ is an ET-system. Hence, applies. We check both cases $n = 2m$ and $n=2m+1$ separately. [$n=2m$:]{.ul} By we have that the zero set $\mathcal{Z}(f^*)$ of $f^*$ has index $2m$ and contains $b$ with index $1$, i.e., $a\in\mathcal{Z}(f^*)$ and all other zeros have index $2$. Hence, $\mathcal{Z}(f^*) = \{a=y_0 < y_1 < \dots < y_{m-1} < y_m = b\}$. By we have that $\mathcal{Z}(f_*)$ also has index $2m$ and the zeros of $f_*$ and $f^*$ interlace. Then the determinantal representations of $f_*$ and $f^*$ follow from . [$n=2m+1$:]{.ul} By we have that $b\in\mathcal{Z}(f^*)$ and since the index of $\mathcal{Z}(f^*)$ is $2m+1$ we have that there are only double zeros $y_1,\dots,y_m\in (a,b)$ in $\mathcal{Z}(f^*)$. Similar we find that $a\in\mathcal{Z}(f_*)$ since its index is odd and only double zeros $x_1,\dots,x_m\in (a,b)$ in $\mathcal{Z}(f_*)$ remain. By (iii) the zeros $x_i$ and $y_i$ strictly interlace and the determinantal representation of $f_*$ and $f^*$ follow again from . ◻ Note, if $\alpha_0,\dots,\alpha_n\in\mathds{N}_0$ then by equation ([\[eq:schurPolyRepr\]](#eq:schurPolyRepr){reference-type="ref" reference="eq:schurPolyRepr"}) the algebraic polynomials $f_*$ and $f^*$ can also be written down with Schur polynomials. does not to hold for $a=0$ and $\alpha_0>0$ or $\alpha_0,\dots,\alpha_k<0$. In case $\alpha_0>0$ the determinantal representations of $f^*$ for $n=2m$ and $f_*$ for $n=2m+1$ are the zero polynomial. In fact, in this case $\mathcal{F}$ is not even a T-system since in the determinant contains a zero column if $x_0 = 0$. We need to have $\alpha_0=0$ ($x^{\alpha_0} = 1$) to let $a=0$. For $\alpha_0,\dots,\alpha_k<0$ we have singularities at $x=0$ and hence no T-system. **Corollary 48**. *If $\alpha_0=0$ in then also holds with $a=0$.* *Proof.* The determinantal representations of $f_*$ for $n=2m+1$ and $f^*$ for $n=2m$ in continuously depend on $a$. It is sufficient to show that these representations are non-trivial (not the zero polynomial) for $a=0$. We show this for $f_*$ in case (ii) $n=2m+1$. The other cases are equivalent. For $\varepsilon>0$ small enough we set $$\begin{aligned} g_\varepsilon(x) &= -\varepsilon^{-m}\cdot\det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & 0 & x_1 & x_1+\varepsilon & \dots & x_m & x_m+\varepsilon \end{pmatrix}\\ % &= -\varepsilon^{-m}\cdot \det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m+1}}\\ 1 & 0 & 0 & \dots & 0\\ 1 & x_1^{\alpha_1} & x_1^{\alpha_2} & \dots & x_1^{\alpha_{2m+1}}\\ \vdots & \vdots & \vdots & & \vdots\\ 1 & (x_{m}+\varepsilon)^{\alpha_1} & (x_{m}+\varepsilon)^{\alpha_2} & \dots & (x_{m}+\varepsilon)^{\alpha_{2m+1}} \end{pmatrix} % \intertext{develop with respect to the second row} % &= \varepsilon^{-m}\cdot\det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}}\\ x_1^{\alpha_1} & x_1^{\alpha_2} & \dots & x_1^{\alpha_{2m-1}}\\ \vdots & \vdots & & \vdots\\ (x_{m}+\varepsilon)^{\alpha_1} & (x_{m}+\varepsilon)^{\alpha_2} & \dots & (x_{m}+\varepsilon)^{\alpha_{2m+1}} \end{pmatrix}\\ % &= \varepsilon^{-m}\cdot\det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & x_1 & x_1+\varepsilon & \dots & x_m & x_m+\varepsilon \end{pmatrix}.\end{aligned}$$ Then $x_1,x_1+\varepsilon,\dots,x_m,x_m+\varepsilon\in (0,b]$, i.e., $\{x^{\alpha_i}\}_{i=1}^n$ is an ET-system on $[a',b]$ with $0=a < a' < x_1$, see . By the representation is not the zero polynomial which ends the proof. ◻ The is a complete description of $\mathrm{int}\,(\mathrm{lin}\,\mathcal{F})_+$. Since $\mathcal{F}$ is continuous on the compact interval $[a,b]$ and $x^{\alpha_0}>0$ on $[a,b]$, we have that the truncated moment cone is closed and hence $(\mathrm{lin}\,\mathcal{F})_+$ and the moment cone are dual to each other. With we can now write down the conditions for the sparse truncated Hausdorff moment problem on $[a,b]$ with $a>0$. We are not aware of a reference for the following result. **Theorem 49** (Sparse Truncated Hausdorff Moment Problem on $[a,b]$ with $a>0$). *Let $n\in\mathds{N}$, $\alpha_0,\dots,\alpha_n\in[0,\infty)$ with $\alpha_0 < \dots < \alpha_n$, and $a,b$ with $0 < a < b$. Set $\mathcal{F}= \{x^{\alpha_i}\}_{i=0}^n$. Then the following are equivalent:* (i) *$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a truncated $[a,b]$-moment functional.* (ii) *$L(p)\geq 0$ holds for all $$\begin{aligned} p(x) &:= \begin{cases} \det \begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}\\ -\det\begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & a & (x_1 & x_1) & \dots & (x_{m-1} & x_{m-1}) & b \end{pmatrix} \end{cases} \tag*{if $n = 2m$} % \intertext{or} % p(x) &:=\begin{cases} -\det\begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & a & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}\\ \det\begin{pmatrix} x^{\alpha_0} & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) & b \end{pmatrix} \end{cases} \tag*{if $n=2m+1$}\end{aligned}$$ and all $x_1,\dots,x_m$ with $a < x_1 < \dots < x_m < b$.* *Proof.* The implication (i) $\Rightarrow$ (ii) is clear since all given polynomials $p$ are non-negative on $[a,b]$. It is therefore sufficient to prove (ii) $\Rightarrow$ (i). Since $a>0$ we have that $x^{\alpha_0} > 0$ on $[a,b]$ and since $[a,b]$ is compact we have that the moment cone $((\mathrm{lin}\,\mathcal{F})_+)^*$ as the dual of the cone of non-negative (sparse) polynomials $(\mathrm{lin}\,\mathcal{F})_+$ is a closed pointed cone. To establish $L\in ((\mathrm{lin}\,\mathcal{F})_+)^*$ it is sufficient to have $L(f)\geq 0$ for all $f\in(\mathrm{lin}\,\mathcal{F})_+$. Let $f\in(\mathrm{lin}\,\mathcal{F})_+$. Then for all $\varepsilon>0$ we have $f_\varepsilon := f+\varepsilon\cdot x^{\alpha_0} > 0$ on $[a,b]$, i.e., by $f_\varepsilon$ is a conic combination of the polynomials $p$ in (ii) and hence $L(f) + \varepsilon\cdot L(x^{\alpha_0}) = L(f_\varepsilon) \geq 0$ for all $\varepsilon>0$. Since $x^{\alpha_0}>0$ on $[a,b]$ we also have that $x^{\alpha_0}$ is a conic combination of the polynomials $p$ in (ii) and therefore $L(x^{\alpha_0}) \geq 0$. Then $L(f)\geq 0$ follows from $\varepsilon\to 0$ which proves (i). ◻ **Corollary 50**. *If $\alpha_0 = 0$ in then also holds with $a=0$, i.e., the following are equivalent:* (i) *$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a truncated $[0,b]$-moment functional.* (ii) *$L(p)\geq 0$ holds for all $$\begin{aligned} p(x) &:= \begin{cases} \det \begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}\\ \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_{m-1} & x_{m-1}) & b \end{pmatrix} \end{cases} \tag*{if $n = 2m$} % \intertext{or} % p(x) &:=\begin{cases} \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}\\ \det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) & b \end{pmatrix} \end{cases} \tag*{if $n=2m+1$}\end{aligned}$$ and all $x_1,\dots,x_m$ with $a < x_1 < \dots < x_m < b$.* *Proof.* Follows from . ◻ For the following we want to remind the reader of the Müntz--Szász Theorem [@muntz14; @szasz16]. It states that for real exponents $\alpha_0=0 < \alpha_1 < \alpha_2 < \dots$ the vector space $\mathrm{lin}\,\{x^{\alpha_i}\}_{i\in\mathds{N}_0}$ of finite linear combinations is dense in $C([0,1])$ if and only if $\sum_{i\in\mathds{N}} \frac{1}{\alpha_i} = \infty$. We state the following only for the classical case of the interval $[0,1]$. Other cases $[a,b]\subset [0,\infty)$ are equivalent. We are not aware of a reference for the following result. Hausdorff required $\alpha_i\to\infty$. The Müntz--Szász Theorem does not require $\alpha_i\to\infty$. The conditions $\alpha_0=0$ and $\sum_{i\in\mathds{N}} \frac{1}{\alpha_i} = \infty$ already appear in [@hausdo21a eq. (17)]. **Theorem 51** (Sparse Hausdorff Moment Problem). *Let $\{\alpha_i\}_{i\in\mathds{N}_0}\subset [0,\infty)$ with $0=\alpha_0 < \alpha_1 < \dots$ and $\sum_{i\in\mathds{N}} \frac{1}{\alpha_i} = \infty$. Let $\mathcal{F}= \{x^{\alpha_i}\}_{i\in\mathds{N}_0}$. The following are equivalent:* (i) *$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a $[0,1]$-moment functional.* (ii) *$L(p)\geq 0$ holds for all $p\in (\mathrm{lin}\,\mathcal{F})_+$.* (iii) *$L(p)\geq 0$ holds for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p>0$.* (iv) *$L(p)\geq 0$ holds for all $$p(x) = \begin{cases} \det \begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix},\\ % \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_{m-1} & x_{m-1}) & 1 \end{pmatrix},\\ % \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix},\ \text{and}\\ % \det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) & 1 \end{pmatrix} \end{cases}$$ for all $m\in\mathds{N}$ and all $0 < x_1 < x_2 < \dots < x_m < 1$.* *Proof.* The implications "(i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii)" are clear and "(iii) $\Leftrightarrow$ (iv)" follows from . It is therefore sufficient to show "(ii) $\Rightarrow$ (i)". Let $f\in C([0,1])$ with $f>0$. Since $\mathrm{lin}\,\mathcal{F}$ is dense in $C([0,1])$ by the Müntz--Szász Theorem there are sequences $\{g_i\}_{i\in\mathds{N}_0}$ and $\{h_i\}_{i\in\mathds{N}_0}$ with $0 < g_i < f < h_i$ and $\|g_i-h_i\|_\infty\to 0$ as $i\to\infty$. Hence, $L(f)\geq 0$. Since $f\in C([0,1])$ with $f>0$ was arbitrary we have that $L(f)\geq 0$ for all $f\in C([0,1])$ with $f\geq 0$. Then by the Riesz--Markov--Kakutani Representation Theorem we have that $L$ has a unique representing measure. ◻ The previous proof can be simplified by using Choquet's theory of adapted spaces, see [@choquet69] or for a more modern formulation [@phelpsLectChoquetTheorem] or [@schmudMomentBook Ch. 1]. With that we can even remove the use of the Müntz--Szász Theorem and therefore the condition $\sum_{i\in\mathds{N}} \frac{1}{\alpha_i} = \infty$. Additionally, we can allow for negative exponents. We will use this approach below and also in all other proofs from here on. The following theorem has to our knowledge not been presented before. **Theorem 52** (General Sparse Hausdorff Moment Problem on $[a,b]$ with $0\leq a < b$). *Let $I\subset\mathds{N}_0$ be an index set (finite or infinite), let $\{\alpha_i\}_{i\in I}$ be such that $\alpha_i\neq \alpha_j$ for all $i\neq j$ and* (a) *if $a=0$ then $\{\alpha_i\}_{i\in I}\subset [0,\infty)$ with $\alpha_i=0$ for an $i\in I$, or* (b) *if $a>0$ then $\{\alpha_i\}_{i\in I}\subset\mathds{R}$.* *Let $\mathcal{F}= \{x^{\alpha_i}\}_{i\in I}$. Then the following are equivalent:* (i) *$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a Hausdorff moment functional.* (ii) *$L(p)\geq 0$ holds for all $p\in (\mathrm{lin}\,\mathcal{F})_+$.* (iii) *$L(p)\geq 0$ holds for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p>0$.* (iv) *$L(p)\geq 0$ holds for all $$p(x) = \begin{cases} \det \begin{pmatrix} x^{\alpha_{i_0}} & x^{\alpha_{i_1}} & x^{\alpha_{i_2}} & \dots & x^{\alpha_{i_{2m-1}}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}, & \text{if $|I| = 2m$ or $\infty$,}\\ % \det\begin{pmatrix} x^{\alpha_{i_1}} & x^{\alpha_{i_2}} & x^{\alpha_{i_3}} & \dots & x^{\alpha_{i_{2m-2}}} & x^{\alpha_{i_{2m-1}}} & x^{\alpha_{i_{2m}}}\\ x & (x_1 & x_1) & \dots & (x_{m-1} & x_{m-1}) & b \end{pmatrix}, & \text{if $|I| = 2m$ or $\infty$,}\\ % \det\begin{pmatrix} x^{\alpha_{i_1}} & x^{\alpha_{i_2}} & x^{\alpha_{i_3}} & \dots & x^{\alpha_{i_{2m}}} & x^{\alpha_{i_{2m+1}}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix},\ & \text{if $|I| = 2m+1$ or $\infty$, and}\\ % \det\begin{pmatrix} x^{\alpha_{i_0}} & x^{\alpha_{i_1}} & x^{\alpha_{i_2}} & \dots & x^{\alpha_{i_{2m-1}}} & x^{\alpha_{i_{2m}}} & x^{\alpha_{i_{2m+1}}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) & b \end{pmatrix}, & \text{if $|I| = 2m+1$ or $\infty$,} \end{cases}$$ for all $m\in\mathds{N}$ if $|I|=\infty$, all $0 < x_1 < x_2 < \dots < x_m < b$, and all $\alpha_{i_0} < \alpha_{i_1} < \dots < \alpha_{i_m}$ with $\alpha_{i_0}=0$ if $a=0$.* *If additionally $\sum_{i:\alpha_i\neq 0} \frac{1}{|\alpha_i|} = \infty$ then $L$ is determinate.* *Proof.* The case $|I|<\infty$ is . We therefore prove the case $|I|=\infty$. The choice $\alpha_{i_0} < \alpha_{i_1} < \dots < \alpha_{i_m}$ with $\alpha_{i_0}=0$ if $a=0$ makes $\{x^{\alpha_{i_j}}\}_{j=0}^m$ a T-system. The implications "(i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii)" are clear and "(iii) $\Leftrightarrow$ (iv)" is . It is therefore sufficient to show "(ii) $\Rightarrow$ (i)". The space $\mathrm{lin}\,\mathcal{F}$ is an adapted space and the assertion follows therefore from [@schmudMomentBook Thm. 1.8]. For the determinacy of $L$ split $\{\alpha_i\}_{i\in I}$ into positive and negative exponents. If $\sum_{i:\alpha_i\neq 0} \frac{1}{|\alpha_i|} = \infty$ then the corresponding sum over at least one group is infinite. If the sum over the positive exponents is infinite apply the Müntz--Szász Theorem. If the sum over the negative exponents is infinite apply the Müntz--Szász Theorem to $\{(x^{-1})^{-\alpha_i}\}_{i\in I: \alpha_i < 0}$ since $a>0$. ◻ Note, since $[a,b]$ is compact the fact that $\{x^{\alpha_i}\}_{i\in I}$ is an adapted space is trivial. In the previous results we only needed the description of all strictly positive polynomials. The non-negative polynomials are described in the following result. Again, we are not aware of a reference. **Theorem 53** (Sparse Algebraic Nichtnegativstellensatz on $[a,b]$ with $0<a<b$). *Let $n\in\mathds{N}$, $\alpha_0,\dots,\alpha_n\in\mathds{R}$ be real numbers with $\alpha_0 < \alpha_1 < \dots < \alpha_n$, and let $\mathcal{F}= \{x^{\alpha_i}\}_{i=0}^n$. Let $f\in\mathrm{lin}\,\mathcal{F}$ with $f\geq 0$ on $[a,b]$. Then there exist points $x_1,\dots,x_n,y_1,\dots,y_n\in [a,b]$ (not necessarily distinct) with $y_n=b$ which include the zeros of $f$ with multiplicities and there exist constants $c_*,c^*\in\mathds{R}$ such that $$f = f_* + f^*$$ with $f_*,f^*\in\mathrm{lin}\,\mathcal{F}$, $f_*,f^*\geq 0$ on $[a,b]$, and the polynomials $f_*$ and $f^*$ are given by $$f_*(x) = c_*\cdot\det\left(\begin{array}{c|ccc} f_0 & f_1 & \dots & f_n\\ x & x_1 & \dots & x_n \end{array}\right) % \qquad\text{and}\qquad % f^*(x) = c_*\cdot\det\left(\begin{array}{c|ccc} f_0 & f_1 & \dots & f_n\\ x & y_1 & \dots & y_n \end{array}\right)$$ for all $x\in [a,b]$.* *Removing the zeros of $f$ from $x_1,\dots,x_n,y_1,\dots,y_n$ we can assume that the remaining $x_i$ and $y_i$ are disjoint and when grouped by size the groups strictly interlace: $$a\ \leq\ x_{i_1} = \dots = x_{i_k}\ <\ y_{j_1} = \dots = y_{j_l}\ <\ \dots\ < x_{i_p} = \dots = x_{i_q}\ <\ y_{j_r} = \dots = y_{j_s}=b.$$ Each such group in $(a,b)$ has an even number of members.* *Proof.* By we have that $\mathcal{F}$ on $[a,b]$ is an ET-system. We then apply similar to the proof of . ◻ The signs of $c_*$ and $c^*$ are determined by $x_1$ and $y_1$ and their multiplicity. If $x_1 = \dots = x_k < x_{k+1} \leq \dots \leq x_n$ then $\mathrm{sgn}\, c_* = (-1)^k$. The same holds for $c^*$ from the $y_i$. **Corollary 54**. *If $\alpha_0 = 0$ in then also holds for $\alpha_0=0$.* **Example 55**. Let $\alpha\in (0,\infty)$ and let $\mathcal{F}= \{1,x^{\alpha}\}$ on $[0,1]$. Then we have $1 = 1_* + 1^*$ with $1_* = x^\alpha$ and $1^* = 1 - x^\alpha$.$\circ$ ## Sparse Positivstellensätze and Nichtnegativstellensätze on $[0,\infty)$ In we have seen the general Positivstellen- and Nichtnegativitätsstellensätze for T-systems and then applied these to the algebraic cases on $[a,b]$. We now show how the results from on $[a,b]$ can be transferred to $[0,\infty)$. **Theorem 56** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 168, Thm. 8.1]). *Let $n\in\mathds{N}$ and $\mathcal{F}= \{f_i\}_{i=0}^n$ be a continuous T-system of order $n$ on $[0,\infty)$ such that* (a) *there exists a $C>0$ such that $f_n(x)>0$ for all $x\geq C$,* (b) *$\displaystyle\lim_{x\to\infty} \frac{f_i(x)}{f_n(x)} = 0$ for all $i=0,\dots, n-1$, and* (c) *$\{f_i\}_{i=0}^{n-1}$ is a continuous T-system on $[0,\infty)$.* *Then for any $f = \sum_{i=0}^n a_i f_i\in\mathrm{lin}\,\mathcal{F}$ with $f>0$ on $[0,\infty)$ and $a_n>0$ there exists a unique representation $$f = f_* + f^*$$ with $f_*,f^*\in\mathrm{lin}\,\mathcal{F}$ with $f_*,f^*\geq 0$ on $[0,\infty)$ such that the following hold:* (i) *If $n = 2m$ the polynomials $f_*$ and $f^*$ each possess $m$ distinct zeros $\{x_i\}_{i=1}^m$ and $\{y_i\}_{i=0}^{m-1}$ satisfying $$0 = y_0 < x_1 < y_1 < \dots < y_{m-1} < x_m <\infty.$$ All zeros except $y_0$ are double zeros.* (ii) *If $n = 2m+1$ the polynomials $f_*$ and $f^*$ each possess the zeros $\{x_i\}_{i=1}^{m+1}$ and $\{y_i\}_{i=1}^m$ satisfying $$0 = x_1 < y_1 < x_2 < \dots < y_m < x_{m+1} < \infty.$$ All zeros except $x_1$ are double zeros.* (iii) *The coefficient of $f_n$ in $f_*$ is equal to $a_n$.* *Proof.* By (a) there exists a function $w\in C([0,\infty))$ such that $w>0$ on $[0,\infty)$ and $\lim_{x\to\infty} \frac{f_n(x)}{w(x)} = 1$. By (b) we define $$v_i(x) := \begin{cases} \frac{f_i(x)}{w(x)} & \text{if}\ x\in [0,\infty),\\ \delta_{i,n} & \text{if}\ x=\infty \end{cases}$$ for all $i=0,1,\dots,n$. Then by (c) $\{v_i\}_{i=0}^n$ is a T-system on $[0,\infty]$ by . With $t(x) := \tan (\pi x/2)$ we define $g_i(x) := v_i\circ t$ for all $i=0,1,\dots,n$. Hence, $\mathcal{G}= \{g_i\}_{i=0}^n$ is a T-system on $[0,1]$. We now apply to $\mathcal{G}$. Set $g := (\frac{f}{w})\circ t$. [(i):]{.ul} Let $n = 2m$. Then by there exits points $$0=y_0 < x_1 < y_1 < \dots < x_m < y_m = 1$$ and unique functions $g_*$ and $g^*$ such that $g = g_* + g^*$, $g_*,g^*\geq 0$ on $[0,1]$, $x_1,\dots,x_m$ are the zeros of $g_*$, and $y_0,\dots,y_m$ are the zeros of $g^*$. Then $f_* := (g_*\circ t^{-1})\cdot w$ and $f^* := (g^*\circ t^{-1})\cdot w$ are the unique components in the decomposition $f = f_* + f^*$. [(ii):]{.ul} Similar to (i). [(iii):]{.ul} From (i) (and (ii) in a similar way) we have $g_i(1) = 0$ for $i=0,\dots,n-1$ and $g_n(1) = 1$. Hence, we get with $g^*(y_m=1) = 0$ that $g_n$ is not contained in $g^*$, i.e., $g_*$ has the only $g_n$ contribution because $\mathcal{G}$ is linearly independent. This is inherited by $f_*$ and $f^*$ which proves (iii). ◻ If $\mathcal{F}$ is an ET-system then the $f_*$ and $f^*$ can be written down explicitly. **Corollary 57**. *If in we have additionally that $\mathcal{F}$ is an ET-system on $[0,\infty)$ then the unique $f_*$ and $f^*$ are given* (i) *for $n= 2m$ by $$f_*(x) = a_{2m}\cdot \det \begin{pmatrix} f_0 & f_1 & f_2 & \dots & f_{2m-1} & f_{2m}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}$$ and $$f^*(x) = -c\cdot \det \begin{pmatrix} f_0 & f_1 & f_2 & f_3 & \dots & f_{2m-2} & f_{2m-1}\\ x & y_0 & (y_1 & y_1) & \dots & (y_{m-1} & y_{m-1}) \end{pmatrix},$$* (ii) *and for $n= 2m+1$ by $$f_*(x) = -a_{2m+1}\cdot \det \begin{pmatrix} f_0 & f_1 & f_2 & f_3 & \dots & f_{2m} & f_{2m+1}\\ x & x_1 & (x_2 & x_2) & \dots & (x_{m+1} & x_{m+1}) \end{pmatrix}$$ and $$f^*(x) = c\cdot \det \begin{pmatrix} f_0 & f_1 & f_2 & \dots & f_{2m-1} & f_{2m}\\ x & (y_1 & y_1) & \dots & (y_m & y_m) \end{pmatrix}$$* *for some $c > 0$.* *Proof.* Combine with . ◻ If we now plug into we get the following. **Theorem 58** (Sparse Algebraic Positivstellensatz on $[0,\infty)$). *Let $n\in\mathds{N}$, $\alpha_0,\dots,\alpha_n\in [0,\infty)$ be real numbers with $\alpha_0 = 0 < \alpha_1 < \dots < \alpha_n$, and let $\mathcal{F}= \{x^{\alpha_i}\}_{i=0}^n$ on $[0,\infty)$. Then for any $f = \sum_{i=0}^n a_i f_i\in\mathrm{lin}\,\mathcal{F}$ with $f>0$ on $[0,\infty)$ and $a_n>0$ there exists a unique decomposition $$f = f_* + f^*$$ with $f_*,f^*\in\mathrm{lin}\,\mathcal{F}$ and $f_*,f^*\geq 0$ on $[0,\infty)$ such that the following hold:* (i) *If $n = 2m$ then the polynomials $f_*$ and $f^*$ each possess $m$ distinct zeros $\{x_i\}_{i=1}^m$ and $\{y_i\}_{i=0}^{m-1}$ satisfying $$0 = y_0 < x_1 < y_1 < \dots < y_{m-1} < x_m < \infty.$$ The polynomials $f_*$ and $f^*$ are given by $$f_*(x) = a_{2m}\cdot\det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix}$$ and $$\begin{aligned} f^*(x) %&= -c\cdot\det\begin{pmatrix} %1 & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} &\dots& x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}}\\ %x & 0 & (y_1 & y_1) & \dots & (y_{m-1} & y_{m-1}) %\end{pmatrix}\\ % &= c\cdot \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} &\dots& x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}}\\ x & (y_1 & y_1) & \dots & (y_{m-1} & y_{m-1}) \end{pmatrix}\end{aligned}$$ for some $c>0$.* (ii) *If $n=2m+1$ then $f_*$ and $f^*$ have zeros $\{x_i\}_{i=1}^{m+1}$ and $\{y_i\}_{i=1}^m$ respectively which satisfy $$0 = x_1 < y_1 < x_2 < \dots < y_m < x_{m+1}<\infty.$$ The polynomials $f_*$ and $f^*$ are given by $$\begin{aligned} f_*(x) %&= -a_{2m+1}\cdot\det\begin{pmatrix} %1 & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ %x & 0 & (x_2 & x_2) & \dots & (x_{m+1} & x_{m+1}) %\end{pmatrix}\\ &= a_{2m+1}\cdot \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (x_2 & x_2) & \dots & (x_{m+1} & x_{m+1}) \end{pmatrix}\end{aligned}$$ and $$f^*(x) = c\cdot\det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (y_1 & y_1) & \dots & (y_m & y_m) \end{pmatrix}$$ for some $c>0$.* *Proof.* We have that $\mathcal{F}$ clearly fulfill condition (a) and (b) of and by we known that $\mathcal{F}$ on $[0,\infty)$ is also a T-system, i.e., (c) in is fulfilled. We can therefore apply . [(i) $n=2m$:]{.ul} By (i) the unique $f_*$ and $f^*$ each possess $m$ distinct zeros $\{x_i\}_{i=1}^m$ and $\{y_i\}_{i=0}^{m-1}$ with $0\leq y_0 < x_1 < \dots < y_{m-1} < x_m < \infty$. Since $x_1,\dots,x_m\in (0,\infty)$ and $\mathcal{F}$ on $[x_1/2,\infty)$ is an ET-system we immediately get the determinantal representation of $f_*$ by (combine with ). For $f^*$ we have $y_0=0$ and by this is no ET-system. Hence, we prove the representation of $f^*$ by hand. Let $\varepsilon>0$ be such that $0=y_0 < y_1 < y_1+\varepsilon < \dots < y_{m-1} < y_{m-1}+\varepsilon$ holds. Then $$\begin{aligned} g_\varepsilon(x) &= -\varepsilon^{-m+1}\cdot \det \begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}}\\ x & 0 & y_1 & y_1+\varepsilon & \dots & y_{m-1} & y_{m-1}+\varepsilon \end{pmatrix}\\ % &= -\varepsilon^{-m+1}\cdot \det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}}\\ 1 & 0 & 0 & \dots & 0\\ 1 & y_1^{\alpha_1} & y_1^{\alpha_2} & \dots & y_1^{\alpha_{2m-1}}\\ \vdots & \vdots & \vdots & & \vdots\\ 1 & (y_{m-1}+\varepsilon)^{\alpha_1} & (y_{m-1}+\varepsilon)^{\alpha_2} & \dots & (y_{m-1}+\varepsilon)^{\alpha_{2m-1}} \end{pmatrix} % \intertext{expand by the second row} % &= \varepsilon^{-m+1}\cdot\det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}}\\ y_1^{\alpha_1} & y_1^{\alpha_2} & \dots & y_1^{\alpha_{2m-1}}\\ \vdots & \vdots & & \vdots\\ (y_{m-1}+\varepsilon)^{\alpha_1} & (y_{m-1}+\varepsilon)^{\alpha_2} & \dots & (y_{m-1}+\varepsilon)^{\alpha_{2m-1}} \end{pmatrix}\\ % &= \varepsilon^{-m+1}\cdot\det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}}\\ x & y_1 & y_1+\varepsilon & \dots & y_{m-1} & y_{m-1}+\varepsilon \end{pmatrix}\end{aligned}$$ is non-negative on $[0,y_1]$ and every $[y_i+\varepsilon,y_{i+1}]$. Now $y_0=0$ is removed and all $y_i,y_i+\varepsilon>0$. Hence, we can work on $[y_1/2,\infty)$ where $\{x^{\alpha_i}\}_{i=1}^{2m}$ is an ET-system and we can go to the limit $\varepsilon\to 0$ as in . Then proves the representation of $f^*$. [(ii) $n=2m+1$:]{.ul} Similar to the case (i) with $n=2m$. ◻ The previous result was reproved in [@scheider23]. Additionally, since the authors of [@scheider23] were not aware of [@karlin63; @karlinStuddenTSystemsBook] their statement is much weaker and the proof is unnecessary long and complicated. In [@scheider23] several other results are reproved which already appeared in [@karlinStuddenTSystemsBook]. It is left to the reader to use to gain the corresponding sparse Nichtnegativestellensatz on $[0,\infty)$ for general T-systems and for $\{x^{\alpha_i}\}_{i=0}^n$ with $0= \alpha_0 < \alpha_1 < \dots < \alpha_n$ real numbers. The proofs follow the same line of thoughts as the proof of . If all $\alpha_i\in\mathds{N}_0$ then we can express the $f_*$ and $f^*$ in also with Schur polynomials, see ([\[eq:schurPolyRepr\]](#eq:schurPolyRepr){reference-type="ref" reference="eq:schurPolyRepr"}) in . We have seen that Boas already investigated the sparse Stieltjes moment problem [@boas39]. However, since Boas did not have access to by Karlin and therefore the description was complicated and incomplete. We get the following complete and simple description. To our knowledge this result did not appear somewhere else. **Theorem 59** (Sparse Stieltjes Moment Problem). *Let $\{\alpha_i\}_{i\in\mathds{N}_0}\subset [0,\infty)$ such that $\alpha_0 = 0 < \alpha_1 < \alpha_2 < \dots$ and let $\mathcal{F}= \{x^{\alpha_i}\}_{i\in\mathds{N}_0}$. Then the following are equivalent:* (i) *$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a $[0,\infty)$-moment functional.* (ii) *$L(p)\geq 0$ for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p\geq 0$.* (iii) *$L(p)\geq 0$ for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p>0$.* (iv) *$L(p)\geq 0$ for all $$p(x) = \begin{cases} \det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix},\\ % \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} &\dots& x^{\alpha_{2m-2}} & x^{\alpha_{2m-1}}\\ x & (x_1 & x_1) & \dots & (x_{m-1} & x_{m-1}) \end{pmatrix},\\ % \det\begin{pmatrix} x^{\alpha_1} & x^{\alpha_2} & x^{\alpha_3} & \dots & x^{\alpha_{2m}} & x^{\alpha_{2m+1}}\\ x & (x_2 & x_2) & \dots & (x_{m+1} & x_{m+1}) \end{pmatrix},\ \text{and}\\ % \det\begin{pmatrix} 1 & x^{\alpha_1} & x^{\alpha_2} & \dots & x^{\alpha_{2m-1}} & x^{\alpha_{2m}}\\ x & (x_1 & x_1) & \dots & (x_m & x_m) \end{pmatrix} \end{cases}$$ for all $m\in\mathds{N}_0$ and $0<x_1<\dots<x_m$.* *Proof.* The implications "(i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii)" are clear and "(iii) $\Leftrightarrow$ (iv)" is . It is therefore sufficient to prove "(ii) $\Leftarrow$ (i)". We have $\mathrm{lin}\,\mathcal{F}= (\mathrm{lin}\,\mathcal{F})_+ - (\mathrm{lin}\,\mathcal{F})_+$, we have $1 = x^{\alpha_0}\in\mathrm{lin}\,\mathcal{F}$, and for any $g = \sum_{i=0}^m a_i\cdot x^{\alpha_i} \in (\mathrm{lin}\,\mathcal{F})_+$ we have $\lim_{x\to\infty} \frac{g(x)}{x^{\alpha_{m+1}}} = 0$, i.e., there exists an $f\in (\mathrm{lin}\,\mathcal{F})_+$ which dominates $g$. Hence, $\mathrm{lin}\,\mathcal{F}$ is an adapted space and the assertion follows from [@schmudMomentBook Thm. 1.8]. ◻ Note, in the previous result we did needed $0=\alpha_0 < \alpha_1 < \alpha_2 < \dots$ but we did not need $\alpha_i\to\infty$. Hence, also includes the case $\sup_{i\in\mathds{N}_0}\alpha_i < \infty$. ## Sparse Positivstellensätze and Nichtnegativstellensätze on $\mathds{R}$ **Theorem 60** ([@karlin63] or e.g. [@karlinStuddenTSystemsBook p. 198, Thm. 8.1]). *Let $m\in\mathds{N}_0$ and $\mathcal{F}= \{f_i\}_{i=0}^{2m}$ be a continuous T-system of order $2m$ on $\mathds{R}$ such that* (a) *there exists a $C>0$ such that $f_{2m}(x)>0$ for all $x\in (-\infty,-C]\cup [C,\infty)$,* (b) *$\displaystyle\lim_{|x|\to\infty} \frac{f_i(x)}{f_{2m}(x)} = 0$ for all $i=0,\dots,2m-1$, and* (c) *$\{f_i\}_{i=0}^{2m-1}$ is a continuous T-system of order $2m-1$ on $\mathds{R}$.* *Let $f = \sum_{i=0}^{2m} a_i f_i$ be such that $f>0$ on $\mathds{R}$ and $a_{2m}>0$. Then there exists a unique representation $$f = f_* + f^*$$ with $f_*,f^*\in\mathrm{lin}\,\mathcal{F}$ and $f_*,f^*\geq 0$ on $\mathds{R}$ such that* (i) *the coefficient of $f_{2m}$ in $f_*$ is $a_{2m}$, and* (ii) *$f_*$ and $f^*$ are non-negative polynomials having zeros $\{x_i\}_{i=1}^m$ and $\{y_i\}_{i=1}^{m-1}$ with $$-\infty < x_1 < y_1 < x_2 < \dots < y_{m-1} < x_m < \infty.$$* *Proof.* Adapt the proof of such that both interval ends of $[a,b]$ are mapped to $-\infty$ and $+\infty$, respectively. ◻ We have already seen how from and we gained (sparse algebraic Positivstellensatz on $[a,b]$), (sparse algebraic Nichtnegativstellensatz on $[a,b]$), and Theorems [Theorem 51](#thm:sparseHausd){reference-type="ref" reference="thm:sparseHausd"} and [Theorem 52](#thm:generalSparseHausd){reference-type="ref" reference="thm:generalSparseHausd"} (sparse Hausdorff moment problems). We have seen how from and we gained (sparse Positivstellensatz for T-systems on $[0,\infty)$), (sparse algebraic Positivstellensatz on $[0,\infty)$), and (sparse Stieltjes moment problem). We will therefore not repeat this procedure for the case $K = \mathds{R}$ from but summarize the procedure in the following "cooking receipt". *Remark 61* (A General Cooking Receipt). We have the following general *cooking receipt* for generating sparse Positivstellensätze, Nichtnegativstellensätze, and to generate and solve sparse moment problems: 1. Use or , or extend these to sets $K= [a,b]\cup [c,d],\dots$ (for extensions see e.g. [@karlinStuddenTSystemsBook] and later literature on T-systems we did not discuss here). 2. Prove that your family $\mathcal{F}= \{f_i\}_{i\in I}$ is a T-system (or even an ET-system). 3. Plug $\mathcal{F}$ into (A) to get the sparse Positivstellensatz or sparse Nichtnegativestellensatz on $K$. 4. Show that $\mathrm{lin}\,\mathcal{F}$ is an adapted space. 5. Combine (C) and (D) to a sparse moment problem (use [@schmudMomentBook Thm. 1.8] for an efficient proof). With this cooking receipt a large class of (sparse) moment problems, Nichtnegativestellensätze, and Positivstellensätze can be generated, solved, and efficiently proved. We think this makes it very useful for applications and further theoretical investigations. $\circ$ # Summary {#sec:summary} In this work we review and deal with univariate sparse moment problems, Positivstellensätzen, and Nichtnegativestellensätzen. We look at earlier results and then move to the theory of T-systems. In the center are the works of Karlin [@karlin63] and Karlin and Studden [@karlinStuddenTSystemsBook]. From and on $[a,b]$ we deduce a complete description of all strictly positive sparse algebraic polynomials in . We also give the sparse algebraic Nichtnegativestellensatz on $[a,b]$ in . With these results we completely solve the sparse Hausdorff moment problem in , , and for the most general form in . Following the extension by Karlin and Studden of and from $[a,b]$ to $[0,\infty)$ we formulate the corresponding sparse algebraic Positivstellensatz on $[0,\infty)$ in . Only the sparse algebraic Positivstellensatz on $[0,\infty)$ is given since it already solves the sparse Stieltjes moment problem in . The sparse algebraic Nichtnegativstellensatz on $[0,\infty)$ can easily be derived like the sparse Nichtnegativestellensatz on $[a,b]$ in . We also give the general T-system Positivstellensatz on $\mathds{R}$ by Karlin in . We give a general "cooking receipt" how other results in [@karlinStuddenTSystemsBook] (and later literature) can be used to generate additional sparse algebraic Positivstellensätze, Nichtnegativestellensätze, or to formulate and solve sparse moment problems. In this treatment we see the high value of the results in [@karlinStuddenTSystemsBook] which are rarely used today. Especially the analytic treatment of the algebraic questions seems unusual at first. However, we hope we convinced the reader that this approach has at least in the univariate case () great value to gain sparse Positivstellensätze, sparse Nichtnegativstellensätze, and solutions to sparse moment problems. # Funding {#funding .unnumbered} The author and this project are supported by the Deutsche Forschungsgemeinschaft DFG with the grant DI-2780/2-1 and his research fellowship at the Zukunfskolleg of the University of Konstanz, funded as part of the Excellence Strategy of the German Federal and State Government. Cur59 N. I. Akhiezer, *The classical moment problem and some related questions in analysis*, Oliver & Boyd, Edinburgh, 1965. G. Averkov and C. Scheiderer, *Convex hulls of monomial curves, and a sparse Positivstellensatz*, arXiv:2303.03826v1. J. Bochnak, M. Coste, and M.-F. Roy, *Real Algebraic Geometry*, Springer-Verlag, Berlin, 1998. R. P. Boas, *On a generalization of the Stieltjes moment problem*, Trans. Amer. Math. Soc. **46** (1939), 142--150. to3em, *The Stieltjes moment problem for functions of bounded variation*, Bull. Amer. Math. Soc. **45** (1939), 399--404. C. Bayer and J. Teichmann, *The proof of Tchakaloff's theorem*, Proc. Amer. Math. Soc. **134(10)** (2006), 3035--3040. G. Choquet, *Lectures on Analysis*, vol. 3, W. A. Benjamin, Inc., Reading, 1969. P. C. Curtis, Jr., *$n$-parameter families and best approximation*, Pacific J. Math. **9** (1959), 1013--1027. P. J. Daniell, *A General Form of Integral*, Ann. Math. **19** (1918), 279--294. to3em, *Further Properties of the General Integral*, Ann. Math. **21** (1920), 203--220. P. J. di Dio and K. Schmüdgen, *The multidimensional truncated moment problem: The moment cone*, J. Math. Anal. Appl. **511** (2022), 126066, 38 pages. H. L. Hamburger, *Über eine Erweiterung des Stieltjesschen Momentenproblems*, Math. Ann. **81** (1920), 235--319. F. Hausdorff, *Summationsmethoden und Momentenfolgen I*, Math. Z. **9** (1921), 74--109. to3em, *Summationsmethoden und Momentenfolgen II*, Math. Z. **9** (1921), 280--299. E. K. Haviland, *On the momentum problem for distribution functions in more than one dimension*, Amer. J. Math. **57** (1935), 562--572. to3em, *On the momentum problem for distribution functions in more than one dimension II*, Amer. J. Math. **58** (1936), 164--168. D. Hilbert, *Über die Darstellung definiter Formen als Summe von Formenquadraten*, Math. Ann. **32** (1888), 342--350. T. H. Hildebrandt and I. J. Schoenberg, *On Linear Functional Operations and the Moment Problem for a Finite Interval in One or Several Dimensions*, Ann. Math. **34** (1933), 317--328. S. Kakutani, *Concrete representation of abstract (M)-spaces (A characterization of the space of continuous functions)*, Ann. Math. Series 2 **42** (1941), 994--1024. S. Karlin, *Representation Theorems for Positive Functions*, J. Math. Mech. **12** (1963), 599--618. M. G. Kreı̆n and A. A. Nudel'man, *The Markow Moment Problem and Extremal Problems*, American Mathematical Society, Providence, Rhode Island, 1977, translation of the Russian original from 1973. M. G. Kreı̆n, *The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development*, Uspehi Mat. Nauk **6** (1951), no. 4 (44), 3--120, English transl.: Amer. Math. Soc. Transl. (2) **12** (1959), 1--121. S. Karlin and W. J. Studden, *Tchebycheff Systems: Analysis and Statistics*, John Wiley & Sons, Interscience Publishers, New York, NY, 1966. J.-B. Lasserre, *An introduction to polynomial and semi-algebraic optimization*, Cambridge University Press, Cambridge, 2015. M. Laurent, *Sums of squares, moment matrices and optimization over polynomials*, Emerging application of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 157--270. I. G. Macdonald, *Symmetric Functions and Hall Polynomials*, 2nd ed., Oxford Science Publications, Oxford, 1995. J. C. Mairhuber, *On Haar's theorem concering Chebyshev approximation problems having unique solutions*, Proc. Amer. Math. Soc. **7** (1956), 609--615. V. Magron and J. Wang, *Sparse polynomial optimization*, Series on Optimization and Its Applications, no. 5, World Scientific, Singapore, 2023. A. Markov, *On certain applications of algebraic continued fractions*, Thesis (in Russian), St. Petersburg, 1884, 131 pp. to3em, *On mean values and exterior densities*, Rec. Math. Moscou. N.S. **4** (1938), 165--190. M. Marshall, *Positive Polynomials and Sums of Squares*, Mathematical Surveys and Monographs, no. 146, American Mathematical Society, Rhode Island, 2008. C. H. Müntz, *Mathematische Abhandlungen Hermann Amandus Schwarz zu seinem fünfzigjährigen Doktorjubiläum am 6. August 1914 gewidmet von Freunden und Schülern.*, ch. Über den Approximationssatz von Weierstrass, pp. 303--312, Springer, Berlin, 1914. R. R. Phelps, *Lectures on Choquet's Theorem*, Springer-Verlag, Berlin, Heidelberg, 2001. G. Pólya and G. Szegö, *Aufgaben und Lehrsätze aus der Analysis*, 3rd ed., vol. 2, Springer-Verlag, Berlin, 1964. to3em, *Aufgaben und Lehrsätze aus der Analysis*, 4th ed., vol. 1, Springer-Verlag, Berlin, 1970. H. Richter, *Parameterfreie Abschätzung und Realisierung von Erwartungswerten*, Bl. Deutsch. Ges. Versicherungsmath. **3** (1957), 147--161. F. Riesz, *Sur les opérations fonctionnelles linéaires*, C. R. Acad. Sci. Paris **144** (1909), 1409--1411. W. W. Rogosinski, *Moments of non-negative mass*, Proc. R. Soc. Lond. A **245** (1958), 1--27. P. C. Rosenbloom, *Quelques classes de problème extrémaux. II*, Bull. Soc. Math. France **80** (1952), 183--215. K. Schmüdgen, *The $K$-moment problem for closed semi-algebraic sets*, Math. Ann. **289** (1991), 203--206. to3em, *The Moment Problem*, Springer, New York, 2017. K. Sieklucki, *Topological properties of sets admitting the Tschebycheff systems*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. **6** (1958), 603--606. J. A. Shohat and J. D. Tamarkin, *The Problem of Moments*, Amer. Math. Soc., Providence, R.I., 1943. T. J. Stieltjes, *Recherches sur les fractions continues*, Ann. Fac. Sci. Toulouse **8** (1894), no. 4, J1--J122. J. Stochel, *Solving the truncated moment problem solves the moment problem*, Glasgow J. Math. **43** (2001), 335--341. K. I. Švenco, *On Hamburger's moment problem with supplementary requirement that masses are absent on a given interval*, Commun. Soc. Math. Kharkov **16** (1939), 121--128, in Russian. O. Szász, *Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen*, Math. Ann. **77** (1916), 482--496. O. Toeplitz, *Über allgemeine lineare Mittelbildung*, Prace Mat.-Fiz. **22** (1911), 113--119. P. Urysohn, *Über die Mächtigkeit der zusammenhängenden Mengen*, Math. Ann. **94** (1925), 262--295. A. Zalar, *The truncated moment problem on curves $y = q(x)$ and $yx^l=1$*, Lin. Multilin. Alg. (2023), https://doi.org/10.1080/03081087.2023.2212316.

arxiv_math

--- author: -   Xiaoying Wang , Yong Li , Yuecai Han title: Similarity Between Two Dynamical Systems --- **Abstract.** The main focus of this paper is to explore how much similarity between two dynamical systems. Analogous to the classical Hartman-Grobman theorem, the relationship between two systems can be linked by a homeomorphic map $K$, and the core is to study the minimizer $K^*$ to measure the degree of similarity. We prove the sufficient conditions and necessary conditions (the maximum principle) for the existence of the minimizer $K^*$. Further, we establish similarity theorem based on the Takens embedding theorem. As applications, Lorenz system, Chua's circuit system and Chen's system are simulated and tested. The results illustrate what is the similarity, which extends the conjugacy in dynamical systems. **Keywords.** Similarity, conjugacy, maximum principle, Takens embedding theorem. # Introduction     Many processes or phenomena in nature and society are similar in some characteristics. How to find or extract them quantitatively is a challenging problem. For example, in dynamic systems, if two systems are conjugate, they will always be similar (qualitative behavior). In this regard, some important work has been done. In addition to the classical papers of Hartman [@[H+60]] and Grobman [@[G+65]], we can find the case of finite dimensional cases in the monographs of Hale [@[WL+12]]. So far, we have known some specific versions of Hartman-Grobman theorem, such as parabolic evolution equations (e.g. scalar reaction-diffusion equations [@[L+91]], Cahn-Hilliard equation and similar phase-field equations [@[BL+94]]), the hyperbolic evolution equations (e.g. semilinear [@[HP+16]], nonuniform [@[BV+06]; @[BV+09]]), control systems [@[BCP+07]], and so on. For the smoothness of the conjugacy in the Hartman-Grobman theorem, see [@[ZLL+22]] and [@[ZLZ+17]]. In this paper, we re-examine the dynamical systems from the perspective of similarity. If we have given two systems described by differential equations respectively, what is their similarity? How can this similarity be determined by means of cost functional? We will touch these problems. To be more precise, let $T>0$ be a fixed time duration and $D\subset\mathbb R^n$ be a bounded closed domain. For all $t\in[0, T]$, we consider the similarity between the following two dynamic systems, which are two ordinary differential equations (ODEs) that will run through the whole article: $$\label{2nonlinear} \left\{ \begin{aligned} \overset{.}x(t) &=f(t,x(t)),\\ x(0)&=x_0, \\ \end{aligned} \right.~ \left\{ \begin{aligned} \overset{.}y(t) &=g(t,y(t)),\\ y(0)&=y_0, \\ \end{aligned} \right.$$ where $f(t,x(t))$, $g(t,y(t)):[0,T]\times \mathbb{R}^n\rightarrow\mathbb{R}^n$ are functions; $x_0$, $y_0\in D$ are initial values, and $x(t),y(t)\in C^1([0,T]; \mathbb{R}^{n})$ are the strong solutions [^1] of ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}). We denote by $x=x(t,x_0)$, $y=y(t,y_0)$ the solution of the first and second equation respectively. As a special case of ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}): A linear system $$\label{Hartman x}\left\{\begin{aligned} \dot{x}(t)&=Ax(t),\\ x(0)&=x_0, \end{aligned}\right.$$ and a nonlinear system $$\label{Hartman y}\left\{\begin{aligned} \dot{y}(t)&=Ay(t)+r(y(t)),\\ y(0)&=y_0, \end{aligned}\right.$$ where $A\in \mathbb R^{n\times n}$ is a constant matrix and $r(y)=o(|y|)\in C^1(\mathbb R^n)$, $|y(t)|=\underset{i=1}{\overset{n}{\sum}}|y_i(t)|$, can be linked by the classical Hartman-Grobman theorem (linearisation theorem): Nearby the hyperbolic [^2] equilibrium $x^*=\mathbf{0}$ [^3], the behaviour of the nonlinear system ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) is completely similar to the linear system ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) by means of a hemeomorphic coordinate transformation. Inspired by the Hartman-Grobman theorem, we mainly study the existence and uniqueness of the homeomorphism (bijective and bicontinuous) $K$ for general nonlinear systems ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}). For the sufficient conditions of existence, we discuss the compact (Theorem [\[Polynomial space\'s minimizer\]](#Polynomial space's minimizer){reference-type="ref" reference="Polynomial space's minimizer"} and [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"}) and noncompact (Theorem [Theorem 1](#Approximate minimum){reference-type="ref" reference="Approximate minimum"}) spaces composed of homeomorphic maps. For the necessary conditions of existence, we prove the maximum principle of similarity between the two systems (Theorem [Theorem 5](#max1){reference-type="ref" reference="max1"}). If there exists a homeomorphic map $K$ that maps $x$ to $y$ in ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}), then studying the system $y$ is equivalent to studying the system $x$ and the form of $K$. However, even in the finite-dimensional case, the homeomorphism $K$ which linearises the nonlinear problem in line with $$\label{Kx=x} K(e^{At}x_0)=y(t,K(x_0)),t\in[0,T],x_0\in D,$$ is in general not unique, let alone write its form explicitly. In order to explicitly represent $K$, we have done some work. For the sufficient conditions of the existence, if we make $K$ explicitly change with $t$, we can explicitly write the form of $K$, i.e. the $K(t)$ in Theorem [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"}. In addition, according to the controllability theory of systems, when the time terminal $t_1$ is fixed, we hope to obtain the terminal $y(t_1)$ through $x$. We can explicitly write the form of the homeomorphism $K$ of ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}), i.e. the $K(t_1)$ in Theorem [Theorem 4](#terminal K){reference-type="ref" reference="terminal K"}. At this time, $K$ is independent of time $t$. Then we propose the concept of the similarity degree of any two systems by referring to the conjugacy theory of dynamical systems, and establish the general principles of the similarity by introducing suitable transformation. Our theoretical and numerical results show that the similarity between two differential dynamical systems and the minimizer $K^*$ of cost functional are intimately related. It means that we can connect two seemingly unrelated systems through some similarity transformation, make prediction based on the known data, and realize the integration and unity of the systems. The remainder of the paper is organized as follows: In Section [2](#DL){reference-type="ref" reference="DL"}, we put forward some concepts (conjugacy, similarity, etc.), in which the minimizer $K^*$ is the key to the similarity between two systems. For the existence of the $K^*$ minimizer, the sufficient condition is proved theoretically in Section [3](#Sufficient){reference-type="ref" reference="Sufficient"}, while the necessary condition (the maximum principle) is proved theoretically in Section [4](#MAX){reference-type="ref" reference="MAX"}. Section [5](#Uniqueness Similarity){reference-type="ref" reference="Uniqueness Similarity"} shows the uniqueness of the minimizer and the similarity between different systems in both continuous form and discrete form. In Section [5.3](#embed){reference-type="ref" reference="embed"}, we propose the similarity theory based on Takens embedding theorem. Finally, we provide some numerical simulation between two nonlinear systems (Lorenz system, Chua's circuit system and Chen's system) in Section [6](#test){reference-type="ref" reference="test"}. # Preliminaries {#DL}     The homeomorphism $K$ in ([\[Kx=x\]](#Kx=x){reference-type="ref" reference="Kx=x"}) characterizes topological conjugation. Based on this, we give the following definitions characterizing the relationship between two systems $x$ and $y$. Let $\Gamma_{x_0}:=\{\rho\in\mathbb R^n|\rho=x(t,x_0),~t\in[0,T]\}$ represent the trajectory of $x$, and $\Gamma_{y_0}:=\{\tilde{\rho}\in\mathbb R^n|\tilde{\rho}=y(t,y_0),~t\in[0,T]\}$ represent the trajectory of $y$. Let $\Omega:=\{K:\Gamma_{x_0}\rightarrow\Gamma_{y_0}|K\text{ is homeomorphic}\}$ denote the admissible set. Let $N(x_0,\delta)=\{x_0^+\in D\mid|x_0^+-x_0|<\delta\}$. **Definition 1**. ***(System Conjugacy)** Suppose that $x(t)$, $y(t)$ satisfy ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}). If there are a homeomorphic map $K:\Gamma_{x_0}\rightarrow\Gamma_{y_0}$ and a constant $\delta\geq0$, for all $x_0^+\in N(x_0,\delta)$ such that $$%y_0=K(x_0^+),~K(x(t,x_0^+))=y(t,y_0), y_0^+=K(x_0^+),~K(x(t,x_0^+))=y(t,y_0^+),~t\in[0,T],$$ then we call systems $x$ and $y$ locally conjugate in $N(x_0,\delta)$. Further, if there admits a homeomorphic map $K:\Gamma_{x_0}\rightarrow\Gamma_{y_0}$, for all $x_0^+\in D$ such that $$y_0^+=K(x_0^+),~K(x(t,x_0^+))=y(t,y_0^+),~t\in[0,T],$$ then we call systems $x$ and $y$ conjugate.* **Definition 2**. ***(System Semi-conjugacy)** Suppose that $x(t)$, $y(t)$ satisfy ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}). If there are two homeomorphic maps $K, R:\Gamma_{x_0}\rightarrow\Gamma_{y_0}$ and a constant $\delta\geq0$, for all $x_0^+\in N(x_0,\delta)$ such that $$y_0^+=K(x_0^+),~R(x(t,x_0^+))=y(t,y_0^+),~t\in[0,T],$$ then we call systems $x$ and $y$ locally semi-conjugate in $N(x_0,\delta)$. Further, if there admit two homeomorphic maps $K, R:\Gamma_{x_0}\rightarrow\Gamma_{y_0}$, for all $x_0^+\in D$ such that $$y_0^+=K(x_0^+),~R(x(t,x_0^+))=y(t,y_0^+),~t\in[0,T],$$ then we call systems $x$ and $y$ semi-conjugate.* There are some elementary examples in Section [4.2](#egs){reference-type="ref" reference="egs"}, which help us have a specific understanding of the homeomorphic map $K$ in conjugate case.\ Sometimes, there is no such conjugacy or semi-conjugacy between two systems. So we go back to the second place and find a certain degree of conjugacy, and this is why we propose the definition of cost functional (Definition [Definition 3](#Cost functional){reference-type="ref" reference="Cost functional"}). We define the following functionals, which are the extensions of Definition [Definition 1](#conjugation){reference-type="ref" reference="conjugation"} and Definition [Definition 2](#semi-conjugation){reference-type="ref" reference="semi-conjugation"}: **Definition 3**. ***(Cost functional)** Suppose that $x(t)$, $y(t)$ satisfy ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}). Follow the definitions and notations in Definition [Definition 1](#conjugation){reference-type="ref" reference="conjugation"} and Definition [Definition 2](#semi-conjugation){reference-type="ref" reference="semi-conjugation"} [^4] $$\tilde{J}[K]\triangleq\underset{x_0^+\in N(x_0,\delta)}{\sup}\dfrac{1}{T} \int_{0}^{T}||K(x(t,x_0^+))-y(t,K(x_0^+))||^2 dt ,$$ $$\tilde{J}[K,R]\triangleq\underset{x_0^+\in N(x_0,\delta)}{\sup}\dfrac{1}{T} \int_{0}^{T}||R(x(t,x_0^+))-y(t,K(x_0^+))||^2 dt .$$* For each fixed initial value $x_0$, the trajectory $\Gamma_{x_0}$ is determined. Without losing generality, we assume that the functional $\tilde{J}[\cdot]$ take the maximum value at $x(t,x_0)$, that is: $$\label{conjugation functional} \tilde{J}[K]=\dfrac{1}{T} \int_{0}^{T}||K(x(t,x_0))-y(t,K(x_0))||^2 dt \triangleq J[K],$$ $$\label{semi-conjugation functional} \tilde{J}[K,R]=\dfrac{1}{T} \int_{0}^{T}||R(x(t,x_0))-y(t,K(x_0))||^2 dt \triangleq J[K,R],$$ where $K,R \in \Omega$. $J[K],J[K,R]$ are the lower semi-continuous functions, and obviously, $J[K]\geq0,~J[K,R]\geq0$. If the functional $J[K]$ exists a minimum and $J[K^*]$ is the minimum, $$J[K^*]=\underset{K\in\Omega}{\inf}~\underset{x_0^+\in N(x_0,\delta)}{\sup}\dfrac{1}{T} \int_{0}^{T}||K(x(t,x_0^+))-y(t,y_0^+)||^2 dt,$$ then $x$ and $y$ can satisfy the conjugacy to a certain extent. Particularly, if $J[K^*]=0$, then $||K^*(x(t,x_0 ))- y(t,K^*(x_0))||^2=0$, $K^*(x(t,x_0 ))=y(t,K^*(x_0 ))$, thus $x$ and $y$ conjugate. Analogously, if the functional $J[K,R]$ exists a minimum and $J[K^*,R^*]$ is the minimum, then $x$ and $y$ can satisfy the semi-conjugacy to a certain extent. Particularly, if $K=R$, then J$[K,R]$ is reduced to $J[K]$.\ Clearly, the larger the cost functional (Definition [Definition 3](#Cost functional){reference-type="ref" reference="Cost functional"}), the smaller the similarity between the two dynamics, and the range of $J[K],J[K,R]$ is $[0,+\infty]$. When $J[K]=0~(J[K,R]=0)$, the two systems conjugate (semi-conjugate). In other words, they are completely similar (semi-similar), the corresponding similarity degree (semi-similarity degree) should be 1. When $J[K]=+\infty~(J[K,R]=+\infty$), the two systems are completely dissimilar, the corresponding similarity degree (semi-similarity degree) should be 0. It is worth mentioning that these are independent of the selection of the similarity degree function. Based on this, we give the concept of the similarity degree function to quantitatively describe the similarity between two systems. **Definition 4**. ***(Similarity degree)** [\[similarity0\]]{#similarity0 label="similarity0"} Let the function $\rho(x)$ be continuous and monotonically decreasing in $[0,+\infty]$ such that $\rho(0)=1$ and $\rho(+\infty)=0$, then we call $\rho(x)$ a similarity degree function.* *We call $\rho(J[K])$ the similarity degree of the systems ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}) with respect to $\rho$ and $\rho(J[K,R])$ the semi-similarity degree of the systems ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}) with respect to $\rho$, respectively. In particular, when conjugating or semi-conjugating, the corresponding similarity and semi-similarity hold with $\rho(J[K^*])=1,~\rho(J[K^*,R^*])=1$.* For example, $\rho(x)=\frac{\log(1+x)}{x}$ [^5], then $$\rho(J[K])=\left\{ \begin{aligned} \frac{\log\Big(1+\underset{K\in\{\Omega|K(x_0)=y_0\}}{\inf}~J[K]\Big)}{\underset{K\in\{\Omega|K(x_0)=y_0\}}{\inf}~J[K]}, &\qquad \underset{K\in\{\Omega|K(x_0)=y_0\}}{\inf}~J[K]\neq0,\\ 1,\qquad\qquad &\qquad \underset{K\in\{\Omega|K(x_0)=y_0\}}{\inf}~J[K]=0, \\ \end{aligned} \right.$$ $$\rho(J[K,R])=\left\{ \begin{aligned} \frac{\log\Big(1+\underset{(K,R)\in\{\Omega\times\Omega|K(x_0)=y_0\}}{\inf}~J[K,R]\Big)}{\underset{(K,R)\in\{\Omega\times\Omega|K(x_0)=y_0\}}{\inf}~J[K,R]}, &\qquad \underset{(K,R)\in\{\Omega\times\Omega|K(x_0)=y_0\}}{\inf}~J[K,R]\neq0,\\ 1,\qquad\qquad &\qquad \underset{(K,R)\in\{\Omega\times\Omega|K(x_0)=y_0\}}{\inf}~J[K,R]=0. \\ \end{aligned} \right.$$ The following discussion is mainly for $J[K]$ in the case of a homeomorphism map $K$, and the proof of $J[K,R]$ is completely analogous. The core problem is to find **the minimizer $K^*$**, **which decides the similarity between two systems.** If $K^*(\cdot)$ is a constant matrix, then the similarity is the linear similarity. If $K^*(\cdot)$ is orthogonal (metric preserving) or symplectic (differential structure preserving), then the similarity is called the rigid similarity. ## Analogous to Hartman-Grobman theorem     Inspired by the classical Hartman-Grobman theorem, we consider the conjugate functional problem ([\[conjugation functional\]](#conjugation functional){reference-type="ref" reference="conjugation functional"}) of equation ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and equation ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}). In this case, the form of the minimizer $K^*$ can be found. We recall the definition of the dichotomic projection and the associated Green kernel of the generator $A$ of the $C_0$-semigroup $e^{At}$ in a Banach space $X$. **Definition 5**. *A projection $P_+\in\mathcal{B}(X)$ is called a dichotomic projection or (exponential) dichotomy for the $C_0$-semigroup $e^{At}$ or of its generator $A$ in $X$, if there are constants $M\geq1,\eta>0$ such that with $P_-=I_X-P_+$ the following conditions are satisfied: $$\begin{aligned} &(i)~P_+e^{At}=e^{At}P_+,\forall~t\geq0,\\ &(ii)~|e^{At}P_+x|_X\leq Me^{-\eta t}|P_+x|_X,\forall~t\geq0,\forall~x\in X,\\ &(iii)~e^{At}P_-\text{ extends to a } C_0-\text{group on }R(P_-),\\ &(iv)~|e^{At}P_-x|_X\leq Me^{\eta t}|P_-x|_X,\forall~t\leq0,\forall~x\in X. \end{aligned}$$ The Green kernel corresponding to the dichotomy is defined as $$G_A(t)=\left\{\begin{aligned} e^{At}P_+,t\geq0,\\ -e^{At}P_-,t<0. \end{aligned}\right.$$* As the results in [@[HP+16]], the following proposition is obtained. **Proposition 1**. *Suppose that $x(t)$, $y(t)$ satisfy equation ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) and equation ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}). Let $A$ admit a dichotomic projection and $e^{At}$ represent the fundamental solution matrix of equation ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) with Green kernel $|G_A(t)|\leq Me^{-\eta|t|},t\in\mathbb R$, with some constants $M\geq1$ and $\eta>0$. Let $r(\cdot)$ be bounded and Lipschitz continuous ,with the Lipschitz constant $|r|_{Lip}$ of the map $r(\cdot)$ satisfies $$\label{r Lipschitz} (\frac{2M}{\eta})|r|_{Lip}<1.$$ Then there is a homeomorphism $K^*=I_X+g$, such that the conjugate functional ([\[conjugation functional\]](#conjugation functional){reference-type="ref" reference="conjugation functional"}) is equal to 0 (i.e. $J[K^*]=0$), where $I_X$ is the identity map and $g(x)=\int_\mathbb RG_A(s)r(e^{-As}x+g(e^{-As}x))ds$.* *Proof.* The $y(t)=e^{At}K(x_0)+\int_0^te^{A(t-s)}r(K(e^{As}x_0))$ is the solution of ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) with initial value $K(x_0)$, then the functional ([\[conjugation functional\]](#conjugation functional){reference-type="ref" reference="conjugation functional"}) is $$J[K]=\frac{1}{T}\int_0^T||K(e^{At}x_0)-[e^{At}K(x_0)+\int_0^te^{A(t-s)}r(K(e^{As}x_0))]||^2ds.$$ Actually, the homeomorphism map is $K^*(x)=x+g(x)$ (i.e. $K^*=I_X+g$) such that ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) conjugate, where $I_X$ is the identity map and $g(x)=\int_\mathbb RG_A(s)r(e^{-As}x+g(e^{-As}x))ds\triangleq (Tg)(x)$. Therefore, we only need to prove the fixed point problem of the map $T$ via the Banach contraction mapping principle. We calculate $$|Tg|_\infty\leq\int_\mathbb R|G_A(s)|~|r|_\infty ds\leq2M|r|_\infty\int_0^{+\infty}e^{-\eta s}ds=\frac{2M}{\eta}|r|_\infty<\infty,$$ and $$\begin{aligned} |Tg(x)-T\bar{g}(x)|_\infty&\leq\int_\mathbb R|G_A(s)|~|r|_{Lip}~|g(e^{-As}x)-\bar{g}(e^{-As}x)| ds\\ &\leq2M|r|_{Lip}~|g-\bar{g}|_\infty\int_0^{+\infty}e^{-\eta s}ds\\ &=\frac{2M}{\eta}|r|_{Lip}~|g-\bar{g}|_\infty. \end{aligned}$$ The condition ([\[r Lipschitz\]](#r Lipschitz){reference-type="ref" reference="r Lipschitz"}) shows that $T$ is a contractive map, so there is a unique fixed point. This implies the homeomorphism $K^*$ exists with the form $K^*=I_X+g$, and is unique. Using the properties of the Green kernel $G_A$, $K^*$ is given by $$K^*(e^{At}x_0)=e^{At}K^*(x_0)+\int_0^te^{A(t-s)}r(K^*(e^{As}x_0))ds=y(t,K^*(x_0)),\forall~t\in[0,T].$$ Since ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) conjugate, $J [K^*]=0$. ◻ In general, two nonlinear systems ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}) are not necessarily conjugate. In order to weaken the conjugacy between $x$ and $y$, we do not require the conjugate equation to be strictly true. Instead, we look for the conditions for the existence of functional minimum in ([\[conjugation functional\]](#conjugation functional){reference-type="ref" reference="conjugation functional"}). We will prove the sufficient conditions for the existence of the minimizer $K^*$ in Section [3](#Sufficient){reference-type="ref" reference="Sufficient"}, and the necessary conditions for the existence of the minimizer $K^*$ in Section [4](#MAX){reference-type="ref" reference="MAX"}. The main ideas are as follows. ## The main idea {#The main idea}     Based on the conditions we mentioned above, norm and distance in $\Omega$ are defined respectively as follows: $||K||_\Omega\triangleq\underset{x\in\Gamma_{x_0}}{\sup}~|K(x)|,~d(K_1,K_2)\triangleq||K_1-K_2||_\Omega$. Then $\Omega,~\Omega\times\Omega$ are normed spaces. To explain our approach, we begin with some motivations by recalling some useful facts to be applied below.\ We first consider the (sufficient) existence of the minimizer $K^*$ of functional $J[K]$, where $K\in\Omega\subset C[\Gamma_{x_0};\Gamma_{y_0}]$. **(i)** From the previous definition, $\Gamma_{x_0},\Gamma_{y_0}\subset[0,T]\times\mathbb R^n$ are two bounded closed domains. According to the topological method (Poincaré Theorem [@[Poincare]]), any two curve segments are topologically homeomorphic, which means they are contained in homeomorphic two closed spheres. Then, the homeomorphism $K$ is bounded. Therefore, the minimization sequence is bounded, and in the sense of $\mathcal{L}^2$, the weak convergence subsequence can be taken. **(ii)** Take the weakly convergent subsequence (in the sense of functional), and it can be regarded as the approximate minimum of functional. The approximate minimum of functional can be obtained by Ekeland variational principle (Lemma [Lemma 1](#Ekeland variational principle){reference-type="ref" reference="Ekeland variational principle"}). Therefore, the approximate minimization sequence is bounded. Here, approximation means that the obtained sequence is related to $\epsilon$ (see Section [3.1](#K in Omega){reference-type="ref" reference="K in Omega"} Theorem [Theorem 1](#Approximate minimum){reference-type="ref" reference="Approximate minimum"} for details). **Lemma 1**. ***(Ekeland variational principle)** Let $(V,d)$ be a complete metric space, $F:V\rightarrow\mathbb R\cup\{+\infty\}$ be a lower-semi-continuous function on $V$, and $\underset{v\in V}{\inf}~F(v)<+\infty$. Fix $\epsilon>0$, if there exists $u\in V$ s.t. $F(u)\leq\underset{v\in V}{\inf}~F(v)+\epsilon$, then for every $\lambda>0$, there exists a point $u^\epsilon\in V$ such that $$\begin{cases} i)~F(u^\epsilon)\leq F(u),\\ ii)~d(u^\epsilon,u)\leq\lambda,\\ iii)~F(u^\epsilon)\leq F(w)+\frac{\epsilon}{\lambda}d(w,u^\epsilon),~\forall ~w\in V. \end{cases}$$* **(iii)** The minimization sequence generally cannot converge strongly. Therefore, using the vector field of gradient flow in Hilbert space, the strong limit cannot be taken. And the limit point of weak convergence may not be unique. Hence, it cannot be concluded that the field at the limit point is non-zero. However, if the space is compact, the strongly convergent subsequence can be obtained according to Arzelà-Ascoli lemma. We will discuss the minimizer $K^*$ in the Polynomial space (Section [3.1](#K in Omega){reference-type="ref" reference="K in Omega"} Theorem [\[Polynomial space\'s minimizer\]](#Polynomial space's minimizer){reference-type="ref" reference="Polynomial space's minimizer"}). Note that for ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}), the homeomorphism $K^*=I_X+g\in\Omega$ in Proposition [Proposition 1](#Hartman){reference-type="ref" reference="Hartman"} is not explicitly represented by $t$, but implicitly represented by $x (t)$, where $g(x)=\int_\mathbb RG_A(s)r(e^{-As}x+g(e^{-As}x))ds$. For general nonlinear systems ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}), we cannot obtain such conclusion directly. Instead, we consider explicitly expressing $K^*$ as $K^*(t)$ and discuss the form when $K^*(t)\in GL(n)\cap\Omega$ [^6] (Section [3.2](#K(t) in GL(n)){reference-type="ref" reference="K(t) in GL(n)"} Theorem [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"}). Further, we consider $x(t_1)=x_1,y(t_1)=y_1$ on the trajectories $x(t),y(t)$ when the terminal $t=t_1\in[0,T]$ is given. We will discuss the form of $K^*$ when $K^*x_1=y_1$ in Section [3.3](#When K in H){reference-type="ref" reference="When K in H"}.\ Then we consider how to determine the minimizer $K^*(t)$, which is the necessary condition for existence, like Pontryagin maximum principle. **(iv)** If the strong limit exists, the variation at the extreme point must be 0. The following lemma converts the variation of a functional into the derivation of a function. **Lemma 2**. *If functional $J[K(\cdot)]$ has variation $\delta J[\delta K(\cdot)]$ [^7] at $K_0$, $\forall ~\epsilon \in \mathbb{R}$, let $$J(\epsilon)=J[K_0 (\cdot)+\epsilon \delta K(\cdot)],$$ then $$\begin{cases} i)~J(\epsilon)\text{ is defined near } \epsilon=0 \text{ and differentiable at }\epsilon=0,\\ ii)~\delta J[\delta K(\cdot)]=\dfrac{d}{d\epsilon} J(\epsilon)|_{\epsilon =0}. \end{cases}$$* Besides, we will use the fundamental variational principle (Lemma [Lemma 3](#L2){reference-type="ref" reference="L2"}) to derive the maximum principle (Section [4](#MAX){reference-type="ref" reference="MAX"}). **Lemma 3**. *Let $\xi (t)$ be a $k$-dimensional piecewise continuous vector valued function defined on $[t_0,t_1 ]$. If for any $k$-dimensional piecewise continuous vector valued function $\eta (t)$ defined on $[t_0,t_1 ]$, we have $$\int_{t_0}^{t_1}\xi^T(t)\eta (t)dt=0,$$ then at all successive moments $\xi (t)=0$.* **(v)** Through the chain rule of derivatives, we can solve for $K(t)$ and obtain its uniqueness (Section [5.1](#Uniqueness){reference-type="ref" reference="Uniqueness"}). If $K\in\mathcal{M}^n$ [^8], the minimizer $K^*$ is a constant matrix, two systems are linearly similar. Intuitively, when taking photos, constant matrix $K$ means that we only need to convert one camera to make the two photos linear similar. $K(t)$ means that we need to constantly change the camera over time and constantly carry out translation, rotation, expansion and contraction transformation, so as to make the two photos completely similar. We propose the similarity theory for prediction based on Takens embedding theorem [@[Takens+81]] in Section [5.3](#embed){reference-type="ref" reference="embed"}. In order to solve $K(t)$ and $K$, we give Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"} and Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"} in Section [6](#test){reference-type="ref" reference="test"} respectively. # The (sufficient) existence of $K^*$ {#Sufficient}     We want to know under what conditions the functional $J[K]$ reaches the minimum, that is, find the minimizer $K^*$ to make the functional reach the minimum. ## When $K^*\in\Omega$ {#K in Omega}     In functional analysis, a general approach is to find a family of functions near the extreme point, and then take the convergent subsequence to converge to the extreme point. This requires some compactness of space. However, for this infinite dimensional space $\Omega$, the boundedness is not necessarily completely-continuous, so compactness cannot be achieved. Generally, the compactness needs to use Arzelà-Ascoli lemma, but this is not available in our situation, because the family of functions cannot be equicontinuous. Therefore, we can only find the minimum of the functional in an approximate sense (approximate minimum). Combining Section [2.2](#The main idea){reference-type="ref" reference="The main idea"} **(i)** and **(ii)**, we give the following theorem: **Theorem 1**. ***(Approximate minimum)** For all $\epsilon>0$, there exist $K^\epsilon\in\Omega$ and $K_0\in\mathcal{L}^2(\Omega)$ such that $K^\epsilon\rightharpoonup K_0$ [^9] and $J[K_0]=\underset{K\in\Omega}{\inf}~J[K]$.* *Proof.* Since functional $J[K]$ has a lower bound, it has an infimum. It means that there exists $\{K_m\}_{m=1}^{+\infty}\in\Omega$ s.t. $\underset{m\rightarrow\infty}{\lim}J[K_m]=\underset{K\in\Omega}{\inf}~J[K]$. $\{K_m\}_{m=1}^{+\infty}$ is called the minimizing sequence of functional $J[\cdot]$. In the light of the definition of limit, we obtain that $\forall ~\epsilon>0,~\exists ~N\in\mathbb N^+$, and when $m>N$, $$|J[K_m]-\underset{K\in\Omega}{\inf}~J[K]|\leq\epsilon\Rightarrow J[K_m]\leq\underset{K\in\Omega}{\inf}~J[K]+\epsilon.$$ According to Ekeland variational principle, let $\lambda=\sqrt\epsilon$, we know that there exists $K^\epsilon\in \Omega$ s.t. $\left\{ \begin{aligned} i)~&J[K^\epsilon]\leq J[K_m],\\ ii)~&\parallel K^\epsilon-K_m\parallel_{\Omega}\leq\sqrt\epsilon,\\ iii)~&J[K^\epsilon]\leq J[K]+\sqrt\epsilon\parallel K^\epsilon-K\parallel_{\Omega},\forall ~K\in\Omega. \end{aligned}\right.$ When $\epsilon\rightarrow0$, $J[K^\epsilon]\leq J[K],~\forall ~K\in\Omega$. Therefore, $K^\epsilon$ satisfies the property of approximate minimum of functional $J[\cdot]$. In combination with Section [2.2](#The main idea){reference-type="ref" reference="The main idea"} **(i)**, $||K^\epsilon||_\Omega\leq M<+\infty$, $||K_m||_\Omega\leq M<+\infty$. In other words, they are contained in a closed ball of $\Omega$ respectively. Although $\Omega$ is not compact, it is weakly sequentially closed in the sense of $\mathcal{L}^2$. $\mathcal{L}^2(\Omega)$ is a Hilbert space. So there exists a weakly convergent subsequence $\{K_{n_i}\}_{i=1}^{+\infty}$ s.t. $\underset{i\rightarrow\infty}{\lim}J[K_{n_i}]=\underset{K\in\Omega}{\inf}~J[K]$ and $K_{n_i}\overset{i\rightarrow\infty}{\rightharpoonup}$ $K_0,~K_0\in\mathcal{L}^2(\Omega)$. This means that $K^\epsilon\rightharpoonup K_0$ in $\mathcal{L}^2(\Omega)$. According to the lower-semi-continuity of functional and Fatou theorem , $J[K_0]\leq\underset{i\rightarrow\infty}{\underline{\lim}}~J[K_{n_i}]$, we have $$\underset{K\in\Omega}{\inf}~J[K]\leq J[K_0]\leq\underset{i\rightarrow\infty}{\underline{\lim}}~J[K_{n_i}]=\underset{i\rightarrow\infty}{\lim}~J[K_{n_i}]=\underset{K\in\Omega}{\inf}~J[K],$$ i.e. $J[K_0]=\underset{K\in\Omega}{\inf}~J[K]$. ◻ It is significant to reiterate that the convergent subsequence can be extracted in $\Omega$, if $\Omega$ is a compact space (Section [2.2](#The main idea){reference-type="ref" reference="The main idea"} **(iii)**). This means that **the minimizer $K_0$** indeed exists, i.e. $K_0\in\Omega$. For example, in polynomial space, the results corresponding to Theorem [Theorem 1](#Approximate minimum){reference-type="ref" reference="Approximate minimum"} can be written as the following. **Theorem 2**. ***(The minimizer in Polynomial space)**[\[Polynomial space\'s minimizer\]]{#Polynomial space's minimizer label="Polynomial space's minimizer"} For a given positive integer $m$, let $P=\{K|K(x(t,x_0))=f_m(x(t,x_0))\}$ with the form $$f_m(x(\cdot))=\underset{\alpha_1+\alpha_2+\cdots+\alpha_n\leq m}{\sum}\left(\begin{aligned} &a_{\alpha_1\alpha_2\cdots\alpha_n}^1\\ &a_{\alpha_1\alpha_2\cdots\alpha_n}^2\\ &\cdots\\ &a_{\alpha_1\alpha_2\cdots\alpha_n}^n \end{aligned}\right)x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}\triangleq \underset{\alpha_1+\alpha_2+\cdots+\alpha_n\leq m}{\sum}A^mx_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n},$$ where $\alpha_1,\alpha_2,\cdots,\alpha_n\in\{0,1,\cdots,m\}$, $\{a_{\alpha_1\alpha_2\cdots\alpha_n}^j\}_{j=1}^n$ is coefficients of $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}$ for the $j$-th row, $x(\cdot)\in\mathbb R^n,~\{x_i\}_{i=1}^n$ is the $i$-th component of $x(\cdot)$. Then functionals ([\[conjugation functional\]](#conjugation functional){reference-type="ref" reference="conjugation functional"}) reaches minimum. Additionally, the similarity and semi-similarity between two systems can be realized in $\tilde{\Omega}=\Omega\cap P$. More precisely, $K_0\in\tilde{\Omega}$.* *Proof.* Compared with the proof of Theorem [Theorem 1](#Approximate minimum){reference-type="ref" reference="Approximate minimum"}, we only need to further prove that convergent subsequences can be extracted in $\tilde{\Omega}$. Define $F(f_m(\cdot))=A^m$. According to Theorem [Theorem 1](#Approximate minimum){reference-type="ref" reference="Approximate minimum"}, there exists a weakly convergent subsequence $\{K_{n_i}\}\in\Omega$ s.t. $K_{n_i}\overset{i\rightarrow\infty}{\rightharpoonup}K_0$. Therefore, $F(K_{n_i})\overset{i\rightarrow\infty}{\rightarrow}F(K_0)$, i.e. $A_{n_i}^m\triangleq A_i^m\overset{i\rightarrow\infty}{\rightarrow}A_0$. Then we only need to extract subsequences from $A_i^m$. For a fixed $i$, $\{A_i^m\}$ is finite, thus there must be a convergent subsequence. Aggregate all of them and record them as $\{A_{i(j)}\}_{j=1}^\infty$, we obtain the convergent subsequences $A_{i(j)}\overset{j\rightarrow\infty}{\rightarrow}A_0$, and this implies $K_0\in\tilde{\Omega}$. ◻ ## When $K^*(t)\in GL(n)\cap\Omega$ {#K(t) in GL(n)}     As mentioned in Section [2.2](#The main idea){reference-type="ref" reference="The main idea"} **(iii)**, we consider the homeomorphism $K(t)\in GL(n)\cap\Omega$. The cost functional ([\[conjugation functional\]](#conjugation functional){reference-type="ref" reference="conjugation functional"}) changed into $$J_1[K(\cdot)]\triangleq\dfrac{1}{T} \int_{0}^{T}||K(t)x(t,x_0 )-y(t,K(0)x_0)||^2dt.$$ **Theorem 3**. *There exists a homeomorphism $K_0(t)\in GL(n)\cap\Omega$ such that $J_1[K_0(\cdot)]=0$.* *Proof.* $GL (n)$ is a complete metric space which satisfies the condition in Theorem [Theorem 1](#Approximate minimum){reference-type="ref" reference="Approximate minimum"}. $K(t)$ contains $n^2$ elements $k_{ij}(t)~(i, j = 1,\cdots, n):\mathbb R\rightarrow\mathbb R$, $GL (n)$ has $n^2$ bases. In fact, any finite dimensional space is isomorphic. According to Theorem [\[Polynomial space\'s minimizer\]](#Polynomial space's minimizer){reference-type="ref" reference="Polynomial space's minimizer"}, there exists extremum point $K_0(t),~K_0(t)\in GL(n)$, and $K_0(t)$ is bounded. To prove that $J_1[K_0(\cdot)]=0$, geometrically, it suffices to prove that $K_0$ maps the solution curve of $(x(t),x_0)$ to the solution curve of $(y(t),y_0)$. In order to find the $K_0$, we divide into three steps: (i) The Euler polyline is used to approximate the solution curve; (ii) $K_0(t)$ is solved by making the functional be $0$ on the Euler polyline; (iii) Certificate the uniform convergence of the Euler polyline.\ **Step 1.** Firstly, we introduce the time variable to ensure that the direction vector at each point of the solution curve is not equal to $\mathbf{0}$: $$\left\{ \begin{aligned} \overset{.}x(t) &=f(t,x(t)),\\ \dot{t}&=1, \\ \end{aligned} \right.~ \left\{ \begin{aligned} \overset{.}y(t) &=g(t,y(t)),\\ \dot{t}&=1. \\ \end{aligned} \right.$$ Then $X=(x,t)\in\mathbb R^{n+1},Y=(y,t)\in\mathbb R^{n+1}$, and $F=(f,1),G=(g,1)$. For simplicity, we will still consider $(x, y, f, g)$ instead of $(X, Y, F, G)$. Divide $[0, T]$ into $m$ segments, each segment is $h_m=\frac{T}{m}$ long. Let $\tau_k=kh_m,~k=0,1,\cdots,m$, $$\begin{aligned} &\xi_0=x_0,\quad\frac{\xi_{k+1}-\xi_k}{\tau_{k+1}-\tau_k}=f(\tau_k,\xi_k),\quad i.e.\quad\xi_{k+1}=\xi_k+h_mf(\tau_k,\xi_k),\\ &\eta_0=y_0,\quad\frac{\eta_{k+1}-\eta_k}{\tau_{k+1}-\tau_k}=g(\tau_k,\eta_k),\quad i.e.\quad\eta_{k+1}=\eta_k+h_mg(\tau_k,\eta_k). \end{aligned}$$ Thereupon, we get $2(m+1)$ discrete points: $$(\tau_0,\xi_0),(\tau_1,\xi_1),\cdots,(\tau_m,\xi_m);~(\tau_0,\eta_0),(\tau_1,\eta_1),\cdots,(\tau_m,\eta_m).$$ Connect two adjacent points with straight lines in turn to obtain two continuous Euler polylines: $$\begin{aligned} &\varphi_m(t)=\left\{\begin{aligned} x_0+f(\tau_0,\xi_0)t,\qquad&t\in[0,\tau_1],\\ x_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}f(\tau_k,\xi_k)+f(\tau_l,\xi_l)(t-\tau_l),~&t\in(\tau_l,\tau_{l+1}],l=1,\cdots,m-1, \end{aligned}\right.\\ &\psi_m(t)=\left\{\begin{aligned} y_0+g(\tau_0,\eta_0)t,\qquad&t\in[0,\tau_1],\\ y_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}g(\tau_k,\eta_k)+g(\tau_l,\eta_l)(t-\tau_l),~&t\in(\tau_l,\tau_{l+1}],l=1,\cdots,m-1. \end{aligned}\right. \end{aligned}$$ **Step 2.** Seek $K_0(t)$ s.t. $K_0(t)\varphi_m(t)=\psi_m(t)$. Without losing generality, we translate the above $2(m+1)$ discrete points and their own direction vectors to the origin $O$, and then consider the rotation matrix of the direction vectors. When $t\in[0,\tau_1]$, $K_0(t)=K_0$, $$\begin{aligned} &\varphi_m(t)=x_0+f(\tau_0,\xi_0)t\triangleq x_0+\textbf{f}_0t,\\ &\psi_m(t)=y_0+g(\tau_0,\eta_0)t\triangleq y_0+\textbf{g}_0t, \end{aligned}$$ where $\textbf{f}_0\neq\mathbf{0},~\textbf{g}_0\neq\mathbf{0}$ are direction vectors of $\varphi_m(t),~\psi_m(t)$ respectively, $$\begin{aligned} &\textbf{f}_0=||\textbf{f}_0||(\cos{\alpha_0^1},\cos{\alpha_0^2},\cdots,\cos{\alpha_0^n})^T\triangleq||\textbf{f}_0||(u_0^1,u_0^2,\cdots,u_0^n)^T\triangleq||\textbf{f}_0||\textbf{u}_0,\\ &\textbf{g}_0=||\textbf{g}_0||(\cos{\beta_0^1},\cos{\beta_0^2},\cdots,\cos{\beta_0^n})^T\triangleq||\textbf{g}_0||(v_0^1,v_0^2,\cdots,v_0^n)^T\triangleq||\textbf{g}_0||\textbf{v}_0, \end{aligned}$$ where $\alpha_0^i,~\beta_0^i,~i=1,2,\cdots,n$, represent the angles between $\textbf{f}_0,~\textbf{g}_0$ and the coordinate axis. Leveraging the fact that $\underset{i=1}{\overset{n}{\Sigma}}\cos^2{\alpha_0^i}=1,~\underset{i=1}{\overset{n}{\Sigma}}\cos^2{\beta_0^i}=1$, $\textbf{u}_0,~\textbf{v}_0$ are unit vectors. Now let us start to calculate the rotation matrix $P_0$ of unit vector $\textbf{u}_0$ to unit vector $\textbf{v}_0$. A rotation can be regarded as rotation around an axis on a plane. Figure [1](#rotation){reference-type="ref" reference="rotation"} shows the vector rotation process when $n=3$. Without losing generality, we first assume that $\textbf{u}_0$ rotates on the plane formed by $u_0^1\times u_0^2$, the rotation angle is $\theta_1,~\cos{\theta_1}=\frac{u_0^2}{\sqrt{(u_0^1)^2+(u_0^2)^2}},~\sin{\theta_1}=\frac{u_0^1}{\sqrt{(u_0^1)^2+(u_0^2)^2}}$, which can be expressed as: {#rotation width="0.75\\linewidth"} As illustrated in the figure, the blue plane rotates $\theta_1$ to get the vector $u^{(1)}$. Then $u^{(1)}$ rotates $\theta_2$ to get the vector $u^{(2)}$. [\[rotation\]]{#rotation label="rotation"} $$\textbf{u}_0^{(1)}=R_1\textbf{u}_0=\left( \begin{matrix} \cos{\theta_1}&-\sin{\theta_1}&&&\\ \sin{\theta_1}&\cos{\theta_1}&&&\\ &&1&&\\ &&&\ddots&\\ &&&&1 \end{matrix}\right) \left(\begin{matrix} u_0^1\\ u_0^2\\ u_0^3\\ \vdots\\ u_0^n \end{matrix}\right)= \left(\begin{matrix} 0\\ \sqrt{(u_0^1)^2+(u_0^2)^2}\\ u_0^3\\ \vdots\\ u_0^n \end{matrix}\right).$$ Deduce the rest, $\textbf{u}_0^{(1)}$ rotates on the plane formed by $u_0^{2'}\times u_0^{3'}$, the rotation angle is $\theta_2,~\cos{\theta_2}=\frac{u_0^3}{\sqrt{(u_0^1)^2+(u_0^2)^2+(u_0^3)^2}},~\sin{\theta_2}=\frac{\sqrt{(u_0^1)^2+(u_0^2)^2}}{\sqrt{(u_0^1)^2+(u_0^2)^2+(u_0^3)^2}}$, $$\begin{aligned} \textbf{u}_0^{(2)}&=R_2\textbf{u}_0^{(1)}=\left( \begin{matrix} 1&&&&&\\ &\cos{\theta_2}&-\sin{\theta_2}&&&\\ &\sin{\theta_2}&\cos{\theta_2}&&&\\ &&&1&&\\ &&&&\ddots&\\ &&&&&1 \end{matrix}\right) \left(\begin{matrix} 0\\ \sqrt{(u_0^1)^2+(u_0^2)^2}\\ u_0^3\\ u_0^4\\ \vdots\\ u_0^n \end{matrix}\right)= \left(\begin{matrix} 0\\ 0\\ \sqrt{(u_0^1)^2+(u_0^2)^2+(u_0^3)^2}\\ u_0^4\\ \vdots\\ u_0^n \end{matrix}\right),\\ &\cdots\cdots \end{aligned}$$ $$\begin{aligned} \textbf{u}_0^{(n-1)}=R_{n-1}\textbf{u}_0^{(n-2)}&=\left( \begin{matrix} 1&&&&\\ &\ddots&&&\\ &&1&&\\ &&&\cos{\theta_{n-1}}&-\sin{\theta_{n-1}}\\ &&&\sin{\theta_{n-1}}&\cos{\theta_{n-1}} \end{matrix}\right) \left(\begin{matrix} 0\\ \vdots\\ 0\\ \sqrt{(u_0^1)^2+(u_0^2)^2+\cdots+(u_0^{n-1})^2}\\ u_0^n \end{matrix}\right)\\ &=\left(\begin{matrix} 0\\ \vdots\\ 0\\ \sqrt{(u_0^1)^2+(u_0^2)^2+\cdots+(u_0^n)^2} \end{matrix}\right)=\left(\begin{matrix} 0\\ \vdots\\ 0\\ 1 \end{matrix}\right). \end{aligned}$$ Let $R_0=R_{n-1}\cdots R_2R_1$, $R_0\textbf{u}_0=(0,\cdots,0,1)^T$. Then $R_0$ is invertible, and its inverse is $R_0^{-1}=R_0^T=R_1^TR_2^T...R_{n-1}^T$. Note that $R_i,~i=1,\cdots ,n-1$, are ordered and not exchangeable, and each $R_i$ is an orthogonal matrix. Similarly, $\exists ~Q_0=Q_{n-1}\cdots Q_2Q_1$, $Q_0\textbf{v}_0=(0,\cdots ,0,1)^T$. $Q_0$ is invertible, and its inverse is $Q_0^{-1}=Q_0^T=Q_1^TQ_2^T\cdots Q_{n-1}^T$. $R_0\textbf{u}_0=Q_0\textbf{v}_0=(0,\cdots ,0,1)^T$, let $P_0=Q_0^{-1}R_0$, then $P_0\textbf{u}_0=\textbf{v}_0$, $P_0$ is invertible. Let $K_0=\frac{||\textbf{g}_0||}{||\textbf{f}_0||}P_0t$, consequently, $$K_0(x_0+\textbf{f}_0t)=y_0+\textbf{g}_0t,K_0x_0=y_0~(\text{translation}),~K_0f_0=g_0~(\text{rotation}).$$ When $t\in(\tau_l,\tau_{l+1}], ~K_0(t)=K_l,~l=1,\cdots ,m-1$, $$\begin{aligned} &\varphi_m(t)=x_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}f(\tau_k,\xi_k)+f(\tau_l,\xi_l)(t-\tau_l)\triangleq \textbf{f}_lt+x_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}\textbf{f}_k-\textbf{f}_l\tau_l,\\ &\psi_m(t)=y_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}g(\tau_k,\eta_k)+g(\tau_l,\eta_l)(t-\tau_l)\triangleq \textbf{g}_lt+y_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}\textbf{g}_k-\textbf{g}_l\tau_l, \end{aligned}$$ where $\textbf{f}_l\neq\mathbf{0},~\textbf{g}_l\neq\mathbf{0}$ are direction vectors of $\varphi_m(t),~\psi_m(t)$ respectively, $$\begin{aligned} \textbf{f}_l&=||\textbf{f}_l||(\cos{\alpha_l^1},\cos{\alpha_l^2},\cdots ,\cos{\alpha_l^n})^T\triangleq||\textbf{f}_l||(u_l^1,u_l^2,\cdots ,u_l^n)^T\triangleq||\textbf{f}_l||\textbf{u}_l, \\ \textbf{g}_l&=||\textbf{g}_l||(\cos{\beta_l^1},\cos{\beta_l^2},\cdots ,\cos{\beta_l^n})^T\triangleq||\textbf{g}_l||(v_0^1,v_l^2,\cdots ,v_l^n)^T\triangleq||\textbf{g}_l||\textbf{v}_l, \end{aligned}$$ where $\alpha_l^i,~\beta_l^i,~i=1,2,\cdots ,n$, represent the angles between $\textbf{f}_l,~\textbf{g}_l$ and the coordinate axis. Leveraging the fact that $\underset{i=1}{\overset{n}{\Sigma}}\cos^2{\alpha_l^i}=1,~\underset{i=1}{\overset{n}{\Sigma}}\cos^2{\beta_l^i}=1$, $\textbf{u}_l,~\textbf{v}_l$ are unit vectors. Use the same rotation transformation, there exist orthogonal rotation matrices $R_l,~Q_l$ s.t. $R_l\textbf{u}_l=Q_l\textbf{v}_l=(0,\cdots ,0,1)^T$. Let $P_l=Q_l^{-1}R_l$, then $P_l\textbf{u}_l=\textbf{v}_l$, $P_l$ is invertible. Let $K_l=\frac{||\textbf{g}_l||}{||\textbf{f}_l||}P_lt$, consequently, $$\left\{\begin{aligned} &K_l(x_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}\textbf{f}_k+\textbf{f}_l(t-\tau_l))=y_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}\textbf{g}_k+\textbf{g}_l(t-\tau_l),\\ &K_l(x_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}\textbf{f}_k-\textbf{f}_l\tau_l)=y_0+h_m\underset{k=0}{\overset{l-1}{\Sigma}}\textbf{g}_k-\textbf{g}_l\tau_l~(\text{translation}),\\ &K_l\mathbf{f_l}=\mathbf{g_l}~(\text{rotation}). \end{aligned}\right.$$ Thereupon, $K_0(t)=K_l$, when $t\in[\tau_l,\tau_{l+1}],~l=0,\cdots ,m-1$. It is noteworthy that $K_l,l=0,1,\cdots,m-1$ is bounded, reversible, and its inverse $K_l^{-1}$ is bounded. So $K_0(t)$ is the homeomorphism (i.e. $K_0(t)\in GL(n)\cap\Omega$).\ **Step 3.** Define $\triangle_m^1(t)=\varphi_m(t)-(x_0+\int_{0}^{T}f(s,\varphi_m(s))ds),~\triangle_m^2(t)=\psi_m(t)-(y_0+\int_{0}^{T}g(s,\psi_m(s))ds)$. It bears out that these two differences converge uniformly by Arzel$\grave{a}$-Ascoli lemma, see [@[WL+12]]: $\triangle_m^1(t)\underset{1}{\overset{m\rightarrow\infty}{\longrightarrow}}0,~\triangle_m^2(t)\underset{1}{\overset{m\rightarrow\infty}{\longrightarrow}}0$. [^10] This means that there exist convergent subsequences which are still recorded as $\varphi_m(t),\psi_m(t)$, s.t. $\varphi_m(t)\underset{1}{\overset{m\rightarrow\infty}{\longrightarrow}}x(t),~\psi_m(t)\underset{1}{\overset{m\rightarrow\infty}{\longrightarrow}}y(t)$, thus $K_0(t)x(t)=y(t)$. $K_0(t)$ depends on continuous $\varphi_m(t),~\psi_m(t)$, so $K_0(t)$ is continuous. Therefore, $$\begin{aligned} J_1[K_0(\cdot)]&=\dfrac{1}{T} \int_{0}^{T}||K_0(t)x(t,x_0 )-y(t,K_0(0)x_0)||^2dt\\ &=\dfrac{1}{T} \int_{0}^{T}||K_0(t)x(t,x_0 )-K_0(t)\varphi_m(t)+K_0(t)\varphi_m(t)-\psi_m(t)+\psi_m(t)-y(t,K_0(0)x_0)||^2dt\\ &\leq C_1\int_{0}^{T}||x(t,x_0)-\varphi_m(t)||^2dt+C_2\int_{0}^{T}||K_0(t)\varphi_m(t)-\psi_m(t)||^2dt+C_3\int_{0}^{T}||\psi_m(t)-y(t,y_0)||^2dt\\ &=I_1+C_2\sum_{k=0}^{m-1}\int_{\tau_k}^{\tau_{k+1}}||K_0(t)\varphi_m(t)-\psi_m(t)||^2dt+I_3~\overset{m\rightarrow\infty}{\longrightarrow}0. \end{aligned}$$ ◻ Let $D_1,D_2$ be two disjoint bounded regions in $\mathbb R^n$, $x_0\in D_1,y_0\in D_2$. **Proposition 2**. *There exists a homeomorphism $K^*(t,x_0)\in \Omega$ such that $J[K^*(\cdot)]=0$.* *Proof.* On the basis of Theorem [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"}, the proof is very direct. Firstly, there exists a homeomorphism $H:D_1\rightarrow D_2$: $\forall~y_0\in D_2,~\exists$ a unique $x_0$ s.t. $H(x_0)=y_0,~x_0=H^{-1}(y_0)$ (Poincaré Theorem [@[Poincare]]). Let $K^*(t,x_0)=K_l\circ H$, when $t\in[\tau_l,\tau_{l+1}],~l=0,\cdots ,m-1$. According to the continuous dependence of the solution on the initial value, $K_l$ is continuous for $x_0$, so $K^*(t,x_0)$ is also continuous for $x_0$. Then $K^*(t,x_0)$ is also a homeomorphism (i.e. $K^*(t,x_0)\in\Omega$), and $J[K^*(\cdot)]=0$. ◻ Further, we will consider $x(t_1)=x_1,y(t_1)=y_1$ on the trajectories $x(t),y(t)$ when the terminal $t=t_1\in[0,T]$ is given. We will discuss the form of $K^*$ when $K^*x_1=y_1$ in the next section. ## When the terminal is given {#When K in H}     Review the conjugate system $K^*(e^{At}x_0)=y(t,K^*(x_0))$. For ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}), Proposition [Proposition 1](#Hartman){reference-type="ref" reference="Hartman"} gives the form of $K^*$ when the nonlinear term $r$ has a small Lipschitz constant $|r|_{Lip}$ in ([\[r Lipschitz\]](#r Lipschitz){reference-type="ref" reference="r Lipschitz"}). The homeomorphism $K^*=I_X+g\in\Omega$ is not explicitly represented by $t$, but is implicitly represented by $x (t)$, where $g(x)=\int_\mathbb RG_A(s)r(e^{-As}x+g(e^{-As}x))ds$. For ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}), Theorem [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"} explicitly expresses $K^*$ as $K^*(t)$ and proves the form of $K^*(t)\in GL(n)\cap\Omega$. When the terminal $t=t_1\in[0,T]$ is given, we study $x(t_1)=x_1,y(t_1)=y_1$ on the trajectories $x(t),y(t)$, the value of $K^*(x_1)$ will change as $t_1$ changes. For general nonlinear systems ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}), Theorem [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"} gives a way for $K^*(t)$ to translate and rotate over $t$, and makes $K^*(t_1)x_1=y_1$, which means that $y_1$ can indeed be obtained. However, for ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) in the Hartman-Grobman theorem, which can be regard as a special case of ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}), there is some difference. The $K^*$ should be independent of $t\in[0,t_1]$, and it should only depend on the terminal $t_1$. In this section, we will prove that the form of $K^*$ is actually expressed as: $$\label{Hartman K(t,x)} K^*(\cdot)=U^T(t_1,x_0)G^{-1}(t_1,x_0)(e^{-At_1}y_1-y_0),$$ where $$U(s,x_0)=e^{As}\triangledown r(\mathbf{0})x_0,$$ $$G(t,x_0)=\int_0^tU(s,x_0)U^T(s,x_0)ds.$$\ Note that $y(t)=e^{At}y_0+\int_0^te^{A(t-s)}r(y(s))ds$ is the solution of ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}), $x(t)=e^{At}x_0$ is the solution of ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}). Nearby the hyperbolic equilibrium ($x^*=y^*=\mathbf{0}$), $r(y)=r(\mathbf{0})+\triangledown r(\mathbf{0})y+o(|y|)$[^11]. Ignore higher order terms and $r(\mathbf{0})=\mathbf{0}$, then $r(y(s))=\triangledown r(\mathbf{0})y(s)=\triangledown r(\mathbf{0})K^*(s,x_0)x(s)$ and $$\label{Hartman y(t)} y(t)=e^{At}y_0+\int_0^te^{At}\triangledown r(\mathbf{0})K^*(s,x_0)x_0ds.$$ We observed that when $x_0=(0,\cdots,0,1)$, $\det{G(t,x_0)}=0$, $G(t,x_0)$ is singular, and when $\triangledown r(\mathbf{0}) x_0=(1,\cdots,1)$, $\det{G(t,x_0)}\neq0$, $G(t,x_0)$ is non-singular (invertible). However, in the Step 2 of Theorem [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"}, we find the invertible $K^*$ through appropriate invertible translation and rotation. Hence, in order to obtain the invertible $K^*$, we suppose that $G(t,x_0)$ is invertible here. According to the proof in Proposition [Proposition 2](#K*(t,x)){reference-type="ref" reference="K*(t,x)"}, there exists a homeomorphism $H$ s.t. $H(x_0)=y_0$ and $H(x(t_1))\triangleq H(x_1)=y_1\triangleq y(t_1),~\forall~t_1\in[0,T]$. On the basis of the theory of controllability [@[WU+17]], if $G(t,x_0)$ is invertible, the exhibit form of $K^*$ is $$\label{K(t,x)} K^*(t,x_0)=U^T(t,x_0)G^{-1}(t_1,x_0)(e^{-At_1}y_1-y_0).$$ Next, we prove the existence of $K^*$ in ([\[K(t,x)\]](#K(t,x)){reference-type="ref" reference="K(t,x)"}) by the Schauder fixed point theorem [@[DK+70]]. Substituting $K^*(t,x_0)$ into the right-hand side of ([\[Hartman y(t)\]](#Hartman y(t)){reference-type="ref" reference="Hartman y(t)"}) yields the nonlinear operator $$\label{Def P}\begin{aligned} y(t)&=e^{At}y_0+\int_0^te^{At}\triangledown r(0)U^T(s,x_0)G^{-1}(t_1,x_0)(e^{-At_1}y_1-y_0)x_0ds\\ &=e^{At}y_0+e^{At}G(t,x_0)G^{-1}(t_1,x_0)(e^{-At_1}y_1-y_0)\triangleq Py(t). \end{aligned}$$ Clearly $Py(0)=y_0$ and $Py(t_1) = y_1$. Thus if a fixed point of the operator $P$ is obtained the $K^*(t,x_0)$ ([\[K(t,x)\]](#K(t,x)){reference-type="ref" reference="K(t,x)"}) steers the system ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) from $x_0$ to $x_1$, in time $t_1$. Then we have the following theorem. **Theorem 4**. *Suppose that $x(t)$, $y(t)$ satisfy equation ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) and equation ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}), and $A\in \mathbb R^{n\times n}$ is a constant matrix s.t. $|e^{At}|\leq Me^{-\eta|t|},t\in\mathbb R$ with some constants $M\geq1$ and $\eta>0$. Let $r(y)=o(|y|)\in C^1(\mathbb R^n)$ such that $|\triangledown r(y)|\leq c_1$, where $c_1\geq0$. Further, suppose that $G(t,x_0)$ is an invertible matrix and there exists a constant $c_2>0$ s.t. $\underset{x\in C[0,t_1;\mathbb R^n]}{\inf}|\det G(t,x)|\geq \frac{1}{c_2}$. Then for every pair $x_0,x_1\in\mathbb R^n$, the operator $P$ defined by ([\[Def P\]](#Def P){reference-type="ref" reference="Def P"}) has a fixed point in $C[0,t_1;\mathbb R^n]$.* *Proof.* Define the norm in $C[0,T;\mathbb R^n]$ be $||y(t)||=\underset{t\in[0,T]}{\max}|y(t)|$. According to ([\[Def P\]](#Def P){reference-type="ref" reference="Def P"}), $$\label{Py(t)}\begin{aligned} |Py(t)|&\leq Me^{-\eta t}|y_0|+Me^{-\eta t}\int_0^t\mid Me^{-\eta s}c_1|x_0|\mid^2ds \Big(c_2(Me^{-\eta t_1}|y_1|+|y_0|)\Big)\\ &= Me^{-\eta t}|y_0|+Me^{-\eta t}(Mc_1|x_0|)^2\frac{1}{2\eta}(1-e^{-2\eta t}) \Big(c_2(Me^{-\eta t_1}|y_1|+|y_0|)\Big)\\ &\leq Me^{-\eta t}|y_0|+\frac{M^3 c_1^2 c_2|x_0|^2}{2\eta}e^{-\eta t} \Big(Me^{-\eta t_1}|y_1|+|y_0|\Big). \end{aligned}$$ Due to $e^{-\eta t}\leq1$, $$\label{C} |Py(t)|\leq M|y_0|+\frac{M^3 c_1^2 c_2|x_0|^2}{2\eta}\Big(Me^{-\eta t_1}|y_1|+|y_0|\Big)\triangleq C_0.$$ Consider the closed and convex subset $\chi\triangleq\{y(t)|y(t)\in C[0,t_1;\mathbb R^n],||y(t)||=\underset{t\in[0,t_1]}{\max}|y(t)|\leq C_0\}$ of $C[0,t_1;\mathbb R^n]$. Let $\Xi\triangleq\{z(t)|z(t)=P(y(t)),y(t)\in \chi\}$ be the image of $\chi$. $\forall~\epsilon>0,\forall~t',t''\in[0,t_1]$ s.t. $|t''-t'|<\delta$, $$\label{euqual-con}\begin{aligned} |z(t'')-z(t')|\leq |e^{At''}&-e^{At'}|~|y_0|+|e^{At''}G(t'',x_0)-e^{At'}G(t',x_0)|~|G^{-1}(t_1,x_0)(e^{-At_1}y_1-y_0)|\\ \leq |Ae^{At_\xi}&|~|t''-t'|~|y_0|\\ +&\bigg|\int_{t'}^{t''}[e^{At}U(t)U^T(t)+Ae^{At}\int_0^tU(s)U^T(s)ds]dt\bigg| |G^{-1}(t_1,x_0)(e^{-At_1}y_1-y_0)|\\ \leq |AM&e^{-\eta t_\xi}|~|t''-t'|~|y_0|\\ +&\bigg|\int_{t'}^{t''}[M^3c_1^2|x_0|^2e^{-3\eta t}+A\frac{M^3 c_1^2|x_0|^2}{2\eta}e^{-\eta t}]dt\bigg| \bigg(c_2(Me^{-\eta t_1}|y_1|+|y_0|)\bigg)\\ \leq M|A&y_0|~|t''-t'|\\ +&\bigg|M^3c_1^2|x_0|^2+A\frac{M^3 c_1^2|x_0|^2}{2\eta}\bigg| \bigg(c_2(Me^{-\eta t_1}|y_1|+|y_0|)\bigg)|t''-t'|\\ \triangleq C_1|t''&-t'|\\ < C_1\delta&=\epsilon,\\ \end{aligned}$$ where $\delta=\frac{\epsilon}{C_1},t_\xi\in[t',t'']$ (without losing generality, $t'<t''$). This shows that $z(t)$ is equicontinuous.\ The operator $P$ as defined by ([\[Def P\]](#Def P){reference-type="ref" reference="Def P"}) is continuous and it is easily established from the Arzelà-Ascoli lemma that the image set $\Xi$ is compact and is a subset of $\chi$. In fact, it is only necessary to prove that there is a subsequence $\{z_{n_j}(t)\}_{j=1}^\infty$ of $\{z_n(t)\}_{n=1}^\infty\in\Xi$, which is uniformly convergent on $[0,t_1]$. Take the rational sequence $\{r_i\}_{i=1}^\infty,r_i\in[0, t_1]$, and consider the infinite square matrix $z_n(r_i),n,i=1,2,\cdots$. They are uniformly bounded, i.e. $||z_n(r_i)||\leq C_0$ ([\[C\]](#C){reference-type="ref" reference="C"}). By using the diagonal method, there is a subsequence that converge at all $r_i$, which is recorded as $\{z_{n_j}(t)\}_{j=1}^\infty$. On the one hand, the equicontinuity ([\[euqual-con\]](#euqual-con){reference-type="ref" reference="euqual-con"}) shows that $\forall~\epsilon>0$, $\exists~\delta>0$ s.t. when $|t''-t'|<\delta$, $$|z_{n_j}(t'')-z_{n_j}(t')|<\frac{\epsilon}{3},j=1,2,\cdots.$$ On the other hand, we can find finite $r_i,i=1,2,\cdots,I$ s.t. $[0,t_1]\subset\underset{i=1}{\overset{I}{\bigcup}}(r_i-\delta,r_i+\delta)$. For each $r_i,i=1,2,\cdots,I$, $\exists~N=N(\epsilon)$ s.t. when $j,k\geq N$, $$|z_{n_j}(r_i)-z_{n_k}(r_i)|<\frac{\epsilon}{3}.$$ Combined with the above analysis, $\forall~t\in[0,t_1],\exists~r_i(i=1,2,\cdots,I)$ s.t. $|t-r_i|<\delta$, when $j,k\geq N$, $$\begin{aligned} |z_{n_j}(t)-z_{n_k}(t)|&\leq|z_{n_j}(t)-z_{n_j}(r_i)|+|z_{n_j}(r_i)-z_{n_k}(r_i)|+|z_{n_k}(r_i)-z_{n_k}(t)|\\ &<\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}=\epsilon. \end{aligned}$$ This shows that $\{z_{n_j}(t)\}_{j=1}^\infty$ is uniformly convergent on $[0,t_1]$. By the Arzelà-Ascoli lemma, the image set $\Xi$ is compact and is a subset of the closed and convex set $\chi$. Hence by the Schauder fixed point theorem, the operator $P$ has a fixed point in $C[0,t_1;\mathbb R^n]$. ◻ In order to obtain ([\[Hartman K(t,x)\]](#Hartman K(t,x)){reference-type="ref" reference="Hartman K(t,x)"}) and the properties of the hyperbolic fixed point $\mathbf{0}$, we replace $t$ with $t_1$ in ([\[K(t,x)\]](#K(t,x)){reference-type="ref" reference="K(t,x)"}) and ([\[Py(t)\]](#Py(t)){reference-type="ref" reference="Py(t)"}): $$K^*(\cdot)=U^T(t_1,x_0)G^{-1}(t_1,x_0)(e^{-At_1}y_1-y_0),$$ $$|y_1|\leq Me^{-\eta t_1}|y_0|+\frac{M^3 c_1^2 c_2|x_0|^2}{2\eta}e^{-\eta t_1} \Big(Me^{-\eta t_1}|y_1|+|y_0|\Big).$$ Then $$\Big(1-\frac{M^4 c_1^2 c_2|x_0|^2}{2\eta}e^{-2\eta t_1}\Big)|y_1|\leq \Big(Me^{-\eta t_1}+\frac{M^3 c_1^2 c_2|x_0|^2}{2\eta}e^{-\eta t_1}\Big)|y_0|.$$ We can choose $M,\eta$ s.t. $\frac{M^4 c_1^2 c_2|x_0|^2}{2\eta}e^{-2\eta t_1}<1$ and $|y_1|\leq q(t_1)|y_0|,$ where $$\label{q(t)} q(t_1)=\frac{Me^{-\eta t_1}+\frac{M^3 c_1^2 c_2|x_0|^2}{2\eta}e^{-\eta t_1}}{1-\frac{M^4 c_1^2 c_2|x_0|^2}{2\eta}e^{-2\eta t_1}},0<q(t_1)<1.$$ It can be verified that $q(t_1)\overset{t_1\rightarrow\infty}{\longrightarrow}0$, and this means that $y(t_1)\overset{t_1\rightarrow\infty}{\longrightarrow}\mathbf{0}=y^*=x^*$ (the hyperbolic equilibrium). In general, the choice of a fixed-point theorem (see [@[CQ+88]]) will depend on the conditions imposed on $f(\cdot)$ and $g(\cdot)$ in ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}).\ So far, we have proved the existence of functional extremum. Back to our original problem, we hope to achieve the similarity between the two systems by the minimizer. Now the minimizer already exists, we focus on exploring the maximum principle, the necessary conditions for the minimizer. # Necessary conditions-the maximum principle {#MAX}     In the usual manner, to determine the minimizer $K^*$, we need to derive a maximum principle as the necessary condition like Pontryagin maximum principle. The main idea is that the local variation of the functional $J[K(\cdot)]$ must be 0 at $K^*$ (i.e. $\delta J[K^*(\cdot)]=0$), if $J[K^*(\cdot)]$ is the extreme value, which is mentioned in Section [2.2](#The main idea){reference-type="ref" reference="The main idea"} **(iv)**. In this section, we assume that the functions $f(\cdot)$ and $g(\cdot)$ in ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}) are twice continuous differentiable with respect to their state variables. For a fixed initial value $x_0$, let $\Gamma_{x_0}$ be the trajectory of $x$. We consider $K(t)\in\Omega$, $\Omega$ is a convex set, hence we use the Lagrange multiplier method to obtain the maximum principle. ## Maximum principle of two systems {#MAX1}     Introduce Lagrange multiplier vector valued functions $\lambda (t)$, $\mu (t)$, and solve the functional minimum problem under the constraint of ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}). Define a new functional $J_2$, $$J_2[K(\cdot)]=\dfrac{1}{T} \int_{0}^{T}\{||K(x(t,x_0)-y(t,y_0)||^2+\lambda^T (t)[\dot{x}-f(t,x)]+\mu^T (t)[\dot{y}-g(t,y)] \}dt ,$$ where $K(\cdot)\in \Omega$, $t\in [0,T]$, $y_0$ is given such that $y_0=K(x_0)$. The minimizer is $$K^*(\cdot)=\arg\min~\dfrac{1}{T} \int_{0}^{T} ||K(x(t,x_0))-y(t,y_0)||^2 dt.$$ Write $H(t,x(t),y(t),K(\cdot),\lambda (t),\mu (t))$ $=-\dfrac{1}{T}||K(x(t,x_0) )-y(t,y_0)||^2+\lambda^T (t)f(t,x)+\mu^T (t)g(t,y) .$\ **Theorem 5**. ***(Maximum principle)** If $K^*(\cdot)\in \Omega$ such that $J_2 [K^*(\cdot)]$ is the minimum value, then there are vector valued functions $\lambda (t),~\mu (t)\in \mathbb{R}^n$ satisfying $$\label{a1b1c1d1} \begin{aligned} (a)&\left\{ \begin{aligned} \overset{.}x(t) &=H_\lambda^T, \\ \overset{.}y(t) &=H_\mu^T, \\ \end{aligned} \right. \\ (b)&\left\{ \begin{aligned} \overset{.}\lambda^{T}(t) &=-\frac{\partial H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))}{\partial x},\\ \lambda^T(T)&=0,\\ \end{aligned} \right.\\ (c)&\left\{ \begin{aligned} \overset{.}\mu^{T}(t) &=-\frac{\partial H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))}{\partial y},\\ \mu^T(T)&=0,\\ \end{aligned} \right.\\ (d)&H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))=\displaystyle\max_{K(\cdot)\in \Omega} H(t,x,y,K(\cdot),\lambda(t),\mu(t)). \end{aligned}$$* *Proof.* Integrating by parts, the above formula can be transformed into $$\begin{aligned} J_2 [K^*(\cdot)]=&\lambda^T (T)x(T)-\lambda^T (0) x_0+\mu^T (T)y(T)-\mu^T (0) y_0\\ &-\int_{0}^{T}\{ \dot{\lambda}^T(t)x(t)+\dot{\mu}^T(t)y(t)\\ &-\dfrac{1}{T}||K^*(x(t,x_0))-y(t,K^*(x_0))||^2+\lambda^T (t)f(t,x)+\mu^T (t)g(t,y) \}dt\\ =&\lambda^T (T)x(T)-\lambda^T (0) x_0+\mu^T (T)y(T)-\mu^T (0) y_0\\ &-\int_{0}^{T}\{ \dot{\lambda}^T(t)x(t)+\dot{\mu}^T(t)y(t)+H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))\}dt, \end{aligned}$$ where $H(t,x(t),y(t),K^*(\cdot),\lambda (t),\mu (t))$ $\quad\quad=-\dfrac{1}{T}||K^*(x(t,x_0))-y(t,K^*(x_0))||^2+\lambda^T (t)f(t,x)+\mu^T (t)g(t,y) .$\ Obviously, $$(a)\left\{ \begin{aligned} \overset{.}x(t) &=H_\lambda^T, \\ \overset{.}y(t) &=H_\mu^T. \\ \end{aligned} \right. \\$$ Let $\epsilon$ be a sufficiently small real number, and $\delta K(\cdot)$ be an arbitrarily determined piecewise continuous vector valued function on $[0,T]$. Let $K^{\epsilon} =K^*+\epsilon \delta K$, according to the continuous dependence of the solution on the parameter, the solution of $K^{\epsilon}$ can correspondingly be expressed as $$\begin{aligned} x^{\epsilon}(t)&=x(t)+\epsilon\delta x(t)+\frac{\epsilon^2}{2}\delta^2x(t)+o(\epsilon^2),\\ y^{\epsilon}(t)&=y(t)+\epsilon\delta y(t)+\frac{\epsilon^2}{2}\delta^2y(t)+o(\epsilon^2). \end{aligned}$$ Besides, $\dot{x}=f(t,x),~\dot{y}=g(t,y)$. Then $$\begin{aligned} &\dot{x}^\epsilon=f(t,x^\epsilon),~\dot{y}^\epsilon=g(t,y^\epsilon),\\ &\dot{x}^\epsilon-\dot{x}=\frac{dx^\epsilon(t)}{dt}-\frac{dx(t)}{dt}= \frac{d}{dt}\delta x(t)\epsilon+o(\epsilon),\\ &\dot{y}^\epsilon-\dot{y}=\frac{dy^\epsilon(t)}{dt}-\frac{dy(t)}{dt}= \frac{d}{dt}\delta y(t)\epsilon+o(\epsilon),\\ &f(t,x^\epsilon)-f(t,x)=\frac{\partial f}{\partial x}(t,x)\delta x(t)\epsilon+o(\epsilon),\\ &g(t,y^\epsilon)-g(t,y)=\frac{\partial g}{\partial y}(t,y)\delta y(t)\epsilon+o(\epsilon). \end{aligned}$$ Comparing the coefficient of the first power of $\epsilon$, it can be obtained that $$\begin{aligned} &\left\{ \begin{aligned} \frac{d}{dt}\delta x(t)&=\frac{\partial f}{\partial x}(t,x)\delta x(t), \\ \delta x(0)&=0, \\ \end{aligned} \right.\\ &\left\{ \begin{aligned} \frac{d}{dt}\delta y(t)&=\frac{\partial g}{\partial y}(t,y)\delta y(t), \\ \delta y(0)&=0,\\ \end{aligned} \right.\\ &\delta^2x(t)=\displaystyle\lim_{\epsilon \rightarrow 0}\frac{2}{\epsilon^2}[x^\epsilon (t)-x(t)+\epsilon \delta x(t)],\\ &\delta^2y(t)=\displaystyle\lim_{\epsilon \rightarrow 0}\frac{2}{\epsilon^2}[y^\epsilon (t)-y(t)+\epsilon \delta y(t)]. \end{aligned}$$ Substitute $K^\epsilon,~x^\epsilon (t),~y^\epsilon (t)$ in $J_2 [\cdot]$: $$\begin{aligned} J_2 [K^\epsilon(\cdot)]=&\lambda^T (T)[x(T)+\epsilon \delta x(T)+\frac{\epsilon^2}{2}\delta^2x(T)+o(\epsilon^2)]-\lambda^T (0) x_0\\ &+\mu^T (T)[y(T)+\epsilon \delta y(T)+\frac{\epsilon^2}{2}\delta^2y(T)+o(\epsilon^2)]-\mu^T (0) y_0\\ &-\int_{0}^{T}\dot{\lambda}^T(t)[x(t)+\epsilon \delta x(t)+\frac{\epsilon^2}{2}\delta^2x(t)+o(\epsilon^2)]dt\\ &-\int_{0}^{T}\dot{\mu}^T(t)[y(t)+\epsilon \delta y(t)+\frac{\epsilon^2}{2}\delta^2y(t)+o(\epsilon^2)]dt\\ &-\int_{0}^{T}H(t,x^\epsilon,y^\epsilon,K^\epsilon(\cdot),\lambda(t),\mu(t))dt, \end{aligned}$$ $$\begin{aligned} \Delta J_2[K^*(\cdot)]=&J_2[K^\epsilon(\cdot)]-J_2[K^*(\cdot)]\\ =&\lambda^T (T)[\epsilon \delta x(T)+\frac{\epsilon^2}{2}\delta^2x(T)+o(\epsilon^2)]+\mu^T (T)[\epsilon \delta y(T)+\frac{\epsilon^2}{2}\delta^2y(T)+o(\epsilon^2)]\\ &-\int_{0}^{T}\dot{\lambda}^T(t)[\epsilon \delta x(t)+\frac{\epsilon^2}{2}\delta^2x(t)+o(\epsilon^2)]dt\\ &-\int_{0}^{T}\dot{\mu}^T(t)[\epsilon \delta y(t)+\frac{\epsilon^2}{2}\delta^2y(t)+o(\epsilon^2)]dt\\ &-\int_{0}^{T}\Delta H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))dt,\\ \text{where} \quad\Delta H=&H(t,x^\epsilon,y^\epsilon,K^\epsilon(\cdot),\lambda(t), \mu(t))-H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))\\ =&\epsilon\bigg[\frac{\partial H}{\partial x}\delta x(t)+\frac{\partial H}{\partial y}\delta y(t)+\frac{\partial H}{\partial K}\delta K(t)\bigg]\\ &+\frac{\epsilon^2}{2}\Bigg\{\frac{\partial H}{\partial x}\delta^2 x+\frac{\partial H}{\partial y}\delta^2 y+\frac{\partial H}{\partial K}\delta^2 K\\ &~~~~~~~~~~+2[\delta^Tx,\delta^Ty,\delta^TK] \left[ \begin{matrix} \frac{\partial^2H}{\partial x^2}&\frac{\partial^2H}{\partial x\partial y}&\frac{\partial^2H}{\partial x\partial K}\\ \frac{\partial^2H}{\partial y\partial x}&\frac{\partial^2H}{\partial y^2}&\frac{\partial^2H}{\partial y\partial K}\\ \frac{\partial^2H}{\partial K\partial x}&\frac{\partial^2H}{\partial K\partial y}&\frac{\partial^2H}{\partial K^2} \end{matrix} \right] \left[ \begin{matrix} \delta^Tx\\ \delta^Ty\\ \delta^TK \end{matrix} \right] \Bigg\}. \end{aligned}$$ Moreover, $\Delta J_2 [K^*(\cdot)]=\epsilon \delta J_2 [K^*(\cdot)]+\frac{\epsilon^2}{2} \delta^2 J_2 [K^*(\cdot)]+o(\epsilon^2 )$. Comparing the coefficient of the first power of $\epsilon$, we have $$\begin{aligned} \delta J_2[K^*(\cdot)]=&\lambda^T(T)\delta x(T)+\mu^T(T)\delta y(T)\\ &-\int_{0}^{T}[\dot{\lambda}^{T}(t)+\frac{\partial H}{\partial x}]\delta x(t)dt-\int_{0}^{T}[\dot{\mu}^{T}(t)+\frac{\partial H}{\partial y}]\delta y(t)dt-\int_{0}^{T}\frac{\partial H}{\partial K}\delta K(\cdot)dt. \end{aligned}$$ Let $$\begin{aligned} &(b)\left\{ \begin{aligned} \overset{.}\lambda^{T}(t) &=-\frac{\partial H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))}{\partial x},\\ \lambda^T(T)&=0,\\ \end{aligned} \right.~\\ &(c)\left\{ \begin{aligned} \overset{.}\mu^{T}(t) &=-\frac{\partial H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))}{\partial y},\\ \mu^T(T)&=0.\\ \end{aligned} \right. \end{aligned}$$ It follows from $\delta J_2[K^*(\cdot)]=0$ that $\int_{0}^{T}\frac{\partial H}{\partial K}\delta K(\cdot)dt=0$. According to variational fundamental theorem (Lemma [Lemma 3](#L2){reference-type="ref" reference="L2"}), $$\frac{\partial H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))}{\partial K}=0.$$ Comparing the coefficient of the second power of $\epsilon$, it yields $$\begin{aligned} \delta^2 J_2[K^*(\cdot)]=&\lambda^T(T)\delta^2 x(T)+\mu^T(T)\delta^2 y(T)\\ &-\int_{0}^{T}[\dot{\lambda}^{T}(t)+\frac{\partial H}{\partial x}]\delta^2 x(t)dt-\int_{0}^{T}[\dot{\mu}^{T}(t)+\frac{\partial H}{\partial y}]\delta^2 y(t)dt-\int_{0}^{T}\frac{\partial H}{\partial K}\delta^2 K(\cdot)dt\\ &-2\int_{0}^{T}[\delta^Tx,\delta^Ty,\delta^TK] \left[ \begin{matrix} \frac{\partial^2H}{\partial x^2}&\frac{\partial^2H}{\partial x\partial y}&\frac{\partial^2H}{\partial x\partial K}\\ \frac{\partial^2H}{\partial y\partial x}&\frac{\partial^2H}{\partial y^2}&\frac{\partial^2H}{\partial y\partial K}\\ \frac{\partial^2H}{\partial K\partial x}&\frac{\partial^2H}{\partial K\partial y}&\frac{\partial^2H}{\partial K^2} \end{matrix} \right] \left[ \begin{matrix} \delta^Tx\\ \delta^Ty\\ \delta^TK \end{matrix} \right] dt. \end{aligned}$$ Note that $\delta^2 J_2 [K^*(\cdot)]\geqslant0$, then $$\frac{\partial^2 H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))}{\partial K^2}\leqslant0.$$ That is, for all continuous times $t$ on $[0,T]$, there is $$(d)H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))=\displaystyle\max_{K(\cdot)\in \Omega} H(t,x,y,K(\cdot),\lambda(t),\mu(t)).$$ ◻ ## Examples {#egs}      We consider two examples, which help us to understand $K^*(\cdot)$. **Example 1**. *Consider a constant linear system $x(t)$, and another system $y(t)$ whose derivative is obtained by multiplying the derivative of this system by a matrix, $$\label{linear-simple} \left\{ \begin{aligned} \overset{.}x(t) &=Ax+B,\\ x(0)&=x_0, \\ \end{aligned} \right.~ \left\{ \begin{aligned} \overset{.}y(t) &=C\overset{.}x(t),\\ y(0)&=y_0,\\ \end{aligned} \right.$$ where $A,~C$ are constant matrices of order $n$, and $B$ is an $n$-dimensional vector.* The solution of ([\[linear-simple\]](#linear-simple){reference-type="ref" reference="linear-simple"}) is $$\begin{aligned} x(t)&=e^{At} x_0+\int_{0}^{t}e^{A(t-s)}B ds,\\ y(t)&=Cx(t)+D,\\ D&=y_0-Cx_0. \end{aligned}$$ Substitute $x (t)$ to get $$\begin{aligned} y(t)&=Ce^{At} x_0+C\int_{0}^{t}e^{A(t-s)}B ds+D,\\ D&=y_0-Cx_0. \end{aligned}$$ Let $K(x(t,x_0 ))=y(t,K(x_0 )),$ $$\begin{aligned} &K(e^{At} x_0+\int_{0}^{t}e^{A(t-s)}B ds)=Ce^{At} x_0+C\int_{0}^{t}e^{A(t-s)} Bds+D,\\ &D=y_0-Cx_0=K(x_0 )-Cx_0. \end{aligned}$$ In this instance, we can easily find $K^*(\cdot)$: $$K^*(x)=Cx+y_0-Cx_0.$$ Then $J[K^*]\equiv0$, $K^*(x(t,x_0))=y(t,K^*(x_0))$, $K^*(\cdot)$ is continuous and linear. When $\det(C)\ne0$, $K^*(\cdot)$ is a homeomorphic map, and consequently $x(t)$ and $y(t)$ are conjugate. Thus two systems are completely similar. In this example, $K^*(\cdot)$ is a constant matrix of order $n$, and two systems are linearly similar. Apply the maximum principle (Theorem [Theorem 5](#max1){reference-type="ref" reference="max1"}), $\lambda(t)=0~(b)$, $\mu(t)=0~(c)$, $J[K^*]\equiv0$, and the condition $(d)$ in ([\[a1b1c1d1\]](#a1b1c1d1){reference-type="ref" reference="a1b1c1d1"}) implies $$K^*=\arg \displaystyle\min_{K(\cdot)\in \Omega}J[K(\cdot)]=\arg \displaystyle\max_{K(\cdot)\in \Omega} H(t,x,y,K(\cdot),\lambda(t),\mu(t)).$$ **Example 2**. *Considere two linear systems: $$\label{linear-simple2} \left\{ \begin{aligned} \overset{.}x(t) &=Ax+B,\\ x(0)&=x_0, \\ \end{aligned} \right.~ \left\{ \begin{aligned} \overset{.}y(t) &=Cy+D,\\ y(0)&=y_0,\\ \end{aligned} \right.$$ where $A,~C$ are constant matrices of order $n$, and $B,~D$ are $n$-dimensional vectors.* Firstly, we consider the one-dimensional case, $n=1$, $x,~y\in \mathbb{R}$. The solution of ([\[linear-simple2\]](#linear-simple2){reference-type="ref" reference="linear-simple2"}) are $$\label{linear solution} \begin{aligned} x(t)&=e^{At}x_0+\int_{0}^{t}e^{A(t-s)}B ds\\ &=e^{At} x_0+\frac{B}{A} (e^{At}-1),\\ y(t)&=e^{Ct} y_0+\int_{0}^{t}e^{C(t-s)}D ds\\ &=e^{Ct} y_0+\frac{D}{C} (e^{Ct}-1)\\ &=\bigg(\frac{D}{C}+y_0 \bigg) \bigg(\frac{x(t)+\frac{B}{A}}{x_0+\frac{B}{A}}\bigg)^{\frac{C}{A}}-\frac{D}{C} . \end{aligned}$$ According to $$\begin{aligned} \left\{ \begin{aligned} y_0&=K(x_0 ),\\ y(t,K(x_0 ))&=K(x(t,x_0 )), \\ \end{aligned} \right. \end{aligned}$$ $K(x):~\mathbb{R}\rightarrow\mathbb{R}$ can be solved: $$K(x)=\bigg(\frac{D}{C}+y_0 \bigg) \bigg(\frac{x(t)+\frac{B}{A}}{x_0+\frac{B}{A}}\bigg)^{\frac{C}{A}}-\frac{D}{C}.$$ For the two-dimensional case, $n=2$, $x,~y\in \mathbb{R}^2$. It should be noted that the high-dimensional matrix cannot be divided, so the denominator needs to become the inverse of the multiplication matrix. Therefore, the condition also needs to ensure that matrices $A$ and $C$ are reversible (nonsingular matrix). In order to get the form of $K^*(\cdot)$, we put the solution $x(t)$ on the complex plane: $$x(t)+A^{-1}B=\alpha(t)e^{i\theta(t)},~ x_0+A^{-1}B=\alpha(0)e^{i\theta(0)},$$ where $\alpha(t)\in [0,T]\rightarrow\mathbb{R},~\theta(t) \in [0,T]\rightarrow[0,2\pi]$. Subsequently, we can obtain: $$K^*(x)=\bigg[\frac{x+A^{-1}B}{\alpha(0)}e^{-i\theta(0)}\bigg]^{A^{-1} C}(C^{-1} D+y_0 )-C^{-1}D.$$ In fact, $$\begin{aligned} K(x)&=[\frac{x+A^{-1}B}{\alpha(0)}e^{-i\theta(0)}]^{A^{-1} C}(C^{-1} D+y_0 )-C^{-1}D\\ &=[\frac{e^{At}(x_0+A^{-1}B)}{\alpha(0)}e^{-i\theta(0)}]^{A^{-1} C}(C^{-1} D+y_0 )-C^{-1}D\\ &=[\frac{e^{At}\alpha(0)e^{i\theta(0)}}{\alpha(0)}e^{-i\theta(0)}]^{A^{-1} C}(C^{-1} D+y_0 )-C^{-1}D\\ &=[e^{At}]^{A^{-1} C}(C^{-1} D+y_0 )-C^{-1}D\\ &=[e^{Ct}](C^{-1} D+y_0 )-C^{-1}D\\ &=e^{Ct}y_0+e^{Ct}C^{-1} D-C^{-1}D\\ &=e^{Ct}y_0+(e^{Ct}-I)C^{-1}D\\ &=y(t). \end{aligned}$$ We get the form of $K^*(\cdot)$ when $n = 1, 2$. At this time, $J[K^*]\equiv 0,~K^*(x(t,x_0 ))=y(t,K^*(x_0))$. For the case of $n\geqslant3$, the form of $K^*(\cdot)$ is more complex. Anyway, logarithmic functions are not monomorphic, $K^*(\cdot)$ is not a homeomorphism, so $x(t)$ and $y(t)$ cannot conjugate in general. In this example, in order to force the conjugate condition to be satisfied, the obtained $K^*(\cdot)$ may not be homeomorphic.\ Example [Example 2](#eg2){reference-type="ref" reference="eg2"} shows that even the two most basic linear systems can not meet the conjugate property, which means that the conjugate property is a very strong one. In order to obtain the conditions that homeomorphism $K^*$ satisfies, we substitute ([\[linear-simple2\]](#linear-simple2){reference-type="ref" reference="linear-simple2"}) as an example of ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}) into the maximum principle (Theorem [Theorem 5](#max1){reference-type="ref" reference="max1"}): $$\begin{aligned} (a')&\left\{ \begin{aligned} \overset{.}x(t) &=H_\lambda^T, \\ \overset{.}y(t) &=H_\mu^T ,\\ \end{aligned} \right. \\ (b')&\left\{ \begin{aligned} \overset{.}\lambda(t) &=-A^T\lambda(t)+\frac{2}{T}K^*(K^*(x(t,x_0))-y(t,K^*(x_0))),\\ \lambda(T)&=0,\\ \end{aligned} \right.\\ (c')&\left\{ \begin{aligned} \overset{.}\mu(t) &=-C^T\mu(t)-\frac{2}{T}(K^*(x(t,x_0))-y(t,K^*(x_0))),\\ \mu(T)&=0,\\ \end{aligned} \right.\\ (d')&H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))=\displaystyle\max_{K(\cdot)\in\Omega} H(t,x,y,K(\cdot),\lambda(t),\mu(t)), \end{aligned}$$ where $H(t,x(t),y(t),K(\cdot),\lambda (t),\mu (t))$ $\quad\quad=-\dfrac{1}{T}||K(x(t,x_0))-y(t,K(x_0))||^2+\lambda^T (t)(Ax+B)+\mu^T (t)(Cy+D) .$ Starting from $(b')$ and $(c')$ , we can get that all above $\lambda (t),~\mu (t)$ have the following general solution: $$\begin{aligned} \lambda(t)&=\int_{T}^{t}\Phi(t)\Phi^{-1}(\tau)\frac{2}{T}K^*(K^*(x(\tau,x_0))-y(\tau,K^*(x_0)))d\tau,\\ \mu(t)&=\int_{T}^{t}\Psi(t)\Psi^{-1}(\tau)(-\frac{2}{T}(K^*(x(\tau,x_0))-y(\tau,K^*(x_0))))d\tau, \end{aligned}$$ where $\Phi(t)$ is the fundamental solution matrix of $\overset{.}\lambda(t) =-A^T\lambda(t)$, $\Psi(t)$ is the fundamental solution matrix of $\overset{.}\mu(t) =-C^T\mu(t).$ From Theorem [Theorem 5](#max1){reference-type="ref" reference="max1"} $(d)$, we know that the minimizer $K^*$ such that $$\frac{\partial H(t,x,y,K^*(\cdot),\lambda(t),\mu(t))}{\partial K}=0,$$ which is equivalent to $$-\frac{2}{T}(K^*(x(t,x_0))-y(t,K^*(x_0)))^T(x(t,x_0)-\frac{\partial y(t,K^*(x_0))}{\partial y_0}x_0)=0.$$ Substituting the solution ([\[linear solution\]](#linear solution){reference-type="ref" reference="linear solution"}) of ([\[linear-simple2\]](#linear-simple2){reference-type="ref" reference="linear-simple2"}) yields $$((e^{At}-e^{Ct})K^*(x_0)-\int_{0}^{t}K^*(e^{A(t-s)}B)-e^{C(t-s)}Dds)^T((e^{At}-e^{Ct})x_0+ \int_{0}^{t}e^{A(t-s)}Bds)=0.$$ # Uniqueness and Similarity-based on Embedology {#Uniqueness Similarity} ## Uniqueness {#Uniqueness}     In Section [3](#Sufficient){reference-type="ref" reference="Sufficient"} and Section [4](#MAX){reference-type="ref" reference="MAX"}, we have discussed the existence of $K$. Now we will observe if it is unique. The existence of $K$ in Proposition [Proposition 1](#Hartman){reference-type="ref" reference="Hartman"} is derived from the Banach contraction mapping principle and $K$ is unique. In this section, we consider the uniqueness of $K(t)$ in Theorem [Theorem 3](#exists in GL(n)){reference-type="ref" reference="exists in GL(n)"}.\ As we all know, the orbits of autonomous systems do not intersect. Since $K(t)$ has $n^2$ bases, $n^2$ constraints are needed to ensure uniqueness. For non-autonomous systems, we usually consider $f(t,x),~g(t,y)$ that are piece-wise constant, i.e. $$\left\{ \begin{aligned} &\overset{.}{\hat{x}}(t)=\hat{f}(\hat{x}(t)),\\ &\hat{x}=(x,t),\hat{f}=(f,1) ,\\ \end{aligned} \right.~ \left\{ \begin{aligned} &\overset{.}{\hat{y}}(t)=\hat{g}(\hat{y}(t)),\\ &\hat{y}=(y,t),\hat{g}=(g,1) .\\ \end{aligned} \right.$$ In this case, $\hat{f}(\hat{x}),~\hat{g}(\hat{y})$ do not change explicitly with $t$, it is enough to consider the time-homogeneous equation, i.e. autonomous systems.\ From Section [4](#MAX){reference-type="ref" reference="MAX"} Theorem [Theorem 5](#max1){reference-type="ref" reference="max1"}, we know that the minimizer $K^*(t)$ such that $$\frac{\partial H(t,x,y,K^*(t),\lambda(t),\mu(t))}{\partial K}=0,$$ which is equivalent to $$\label{KKT} \frac{\partial}{\partial k_{ij}(t)}(x^T(t)K^{*T}(t)K^*(t)x(t)) -2\frac{\partial}{\partial k_{ij}(t)}(x^T(t)K^{*T}(t)y(t)) +\frac{\partial}{\partial k_{ij}(t)}(y^T(t)y(t))=0,$$ where $k_{ij}(t)$ denotes the element of the $i$-th row and $j$-th column of the matrix $K^*(t)$, and $i,j=1,2,\cdots,n$. Since the $n^2$ constraints in ([\[KKT\]](#KKT){reference-type="ref" reference="KKT"}) are irrelevant, $K(t)$ is unique and continuous with respect to $t$. ([\[KKT\]](#KKT){reference-type="ref" reference="KKT"}) is the Karush-Kuhn-Tucker (KKT) optimality condition. Let $\bigg(M_{n\times n}\bigg)_{m,i}$ denote the element of the $m$-th row and $i$-th column of the matrix $M_{n\times n}$ and $\mathbf{e}_j(t)\triangleq (0,\cdots,0,x_j(t),0,\cdots,0)^T$. For the first term in ([\[KKT\]](#KKT){reference-type="ref" reference="KKT"}), we obtain the following equation by means of the chain rule of derivatives: $$\begin{aligned} \frac{\partial}{\partial k_{ij}(t)}(x^T(t)K^{*T}(t)K^*(t)x(t))&= x^T(t)\frac{\partial K^{*T}(t)}{\partial k_{ij}(t)}K^*(t)x(t)+ x^T(t)K^{*T}(t)\frac{\partial K^*(t)}{\partial k_{ij}(t)}x(t)\\ &=\mathbf{e}_j^T(t)K^*(t)x(t)+x^T(t)K^{*T}(t)\mathbf{e}_j(t)\\ &=2\underset{l=1}{\overset{n}{\sum}}x_j(t)k_{jl}(t)x_l(t). \end{aligned}$$ For the second term in ([\[KKT\]](#KKT){reference-type="ref" reference="KKT"}), we have $$\begin{aligned} \frac{\partial}{\partial k_{ij}(t)}(x^T(t)K^{*T}(t)y(t))&= x^T(t)\frac{\partial K^{*T}(t)}{\partial k_{ij}(t)}y(t)+ x^T(t)K^{*T}(t)\frac{\partial y(t)}{\partial k_{ij}(t)}\\ &=\mathbf{e}_j^T(t)y(t)+ x^T(t)K^{*T}(t)\frac{\partial y(t)}{\partial y_0(t)}\frac{\partial y_0(t)}{\partial k_{ij}(t)}\\ &=x_j(t)y_j(t)+x^T(t)K^{*T}(t)\frac{\partial y(t)}{\partial y_0(t)}\mathbf{e}_j(0)\\ &=x_j(t)y_j(t)+\underset{l=1}{\overset{n}{\sum}}\underset{m=1}{\overset{n}{\sum}} x_j(0)k_{lm}(t)x_l(t)\bigg(\frac{\partial y(t)}{\partial y_0(t)}\bigg)_{m,i}. \end{aligned}$$ For the last term in ([\[KKT\]](#KKT){reference-type="ref" reference="KKT"}), it is enough to compute that $$\begin{aligned} \frac{\partial}{\partial k_{ij}(t)}(y^T(t)y(t))&= \frac{\partial y^T(t)}{\partial k_{ij}(t)}y(t)+y^T(t)\frac{\partial y(t)}{\partial k_{ij}(t)}\\ &=\mathbf{e}_j^T(0)\bigg(\frac{\partial y(t)}{\partial y_0(t)}\bigg)^Ty(t) +y^T(t)\frac{\partial y(t)}{\partial y_0(t)}\mathbf{e}_j(0)\\ &=2\underset{l=1}{\overset{n}{\sum}}x_j(0)y_l(t)\bigg(\frac{\partial y(t)}{\partial y_0(t)}\bigg)_{l,i}. \end{aligned}$$ Substitute the three terms into formula ([\[KKT\]](#KKT){reference-type="ref" reference="KKT"}), and we get $$\label{KKT1}\begin{aligned} 2\underset{l=1}{\overset{n}{\sum}}x_j(t)k_{jl}(t)x_l(t) -2\bigg[x_j(t)y_j(t)&+\underset{l=1}{\overset{n}{\sum}}\underset{m=1}{\overset{n}{\sum}} x_j(0)k_{lm}(t)x_l(t)\bigg(\frac{\partial y(t)}{\partial y_0(t)}\bigg)_{m,i}\bigg]\\ &+2\underset{l=1}{\overset{n}{\sum}}x_j(0)y_l(t)\bigg(\frac{\partial y(t)}{\partial y_0(t)}\bigg)_{l,i}=0. \end{aligned}$$ Consider $x(t)$, $y(t)$ satisfy ([\[Hartman y\]](#Hartman y){reference-type="ref" reference="Hartman y"}) and ([\[Hartman x\]](#Hartman x){reference-type="ref" reference="Hartman x"}) in the Hartman-Grobman theorem, substitute $\frac{\partial y(t)}{\partial y_0(t)}=e^{At}$ into formula ([\[KKT1\]](#KKT1){reference-type="ref" reference="KKT1"}), and we get $$\label{KKT2}\begin{aligned} 2\underset{l=1}{\overset{n}{\sum}}x_j(t)k_{jl}(t)x_l(t) -2\bigg[x_j(t)y_j(t)&+\underset{l=1}{\overset{n}{\sum}}\underset{m=1}{\overset{n}{\sum}} x_j(0)k_{lm}(t)x_l(t)\bigg(e^{At}\bigg)_{m,i}\bigg]\\ &+2\underset{l=1}{\overset{n}{\sum}}x_j(0)y_l(t)\bigg(e^{At}\bigg)_{l,i}=0. \end{aligned}$$ Note that $i,j=1,\cdots,n$ in ([\[KKT\]](#KKT){reference-type="ref" reference="KKT"}), ([\[KKT1\]](#KKT1){reference-type="ref" reference="KKT1"}) and ([\[KKT2\]](#KKT2){reference-type="ref" reference="KKT2"}), so they contain $n^2$ constraints, and the solved $K^*(t)$ is unique. ## Similarity degree     After solving **the minimizer $K(t)$**, we can define the similarity between two systems. Obviously, if two systems are conjugate, the denominator in the above definition is 0. According to Definition [\[similarity0\]](#similarity0){reference-type="ref" reference="similarity0"}, the similarity degree between two systems is 1 ($100\%$). For further calculation (Section [6](#test){reference-type="ref" reference="test"}), we discretize the trajectory and give the discrete similarity definition: **Definition 6**. ***(Discrete similarity degree)**[\[similarity2\]]{#similarity2 label="similarity2"} $$\rho(K(\cdot))=\left\{ \begin{aligned} \dfrac{log\Big(1+\dfrac{1}{N} \underset{i=1}{\overset{N}{\sum}}||K(i)x(i,x_0)-y(i,y_0)||^2\Big)}{\dfrac{1}{N} \underset{i=1}{\overset{N}{\sum}}||K(i)x(i,x_0)-y(i,y_0)||^2 }, &\qquad \underset{i=1}{\overset{N}{\sum}}||K(i)x(i,x_0)-y(i,y_0)|| \ne 0,\\ 1,\qquad\qquad &\qquad \underset{i=1}{\overset{N}{\sum}}||K(i)x(i,x_0)-y(i,y_0)|| = 0, \end{aligned} \right.$$ where $K(i)=\{k_{jk}(i)\}_{j,k=1}^n$.* Analogous to Section [2.2](#The main idea){reference-type="ref" reference="The main idea"} **(v)**, if the minimizer $K(i)$ is a constant matrix, then the similarity is the linear similarity. If $K(i)$ is an orthogonal matrix (metric preserving) or symplectic matrix (differential structure preserving), the similarity is the rigid similarity. ## Similarity theorem based on Embedology {#embed}     Takens embedding theorem is a fundamental principle to calculate the dimension of embedded phase space [@[Saue+91]]. An important problem in applications of chaos is to reconstruct an $n$-dimensional phase space that can contain the chaotic motion from the time series of a single variable. For the question of how big the dimension $n$ should be, in 1934, American mathematician Whitney and in 1980, Takens [@[Takens+81]] successively proved the embedding theorem of the required dimension $n$: in order to ensure that the phase space can accommodate the topological characteristics of the original attractor of the state space, if the original attractor is in a $d$-dimensional space, the dimension of the phase space in which the attractor is embedded must reach $n=2d+1$. For example, the dimension of the attractor obtained by reconstructing the phase space from a time series is $d=2. 51$, that is, the attractor is in a space with $d=3$ dimension, so the dimension of the embedded phase space is at least $n= 2\times3+1=7$. **Remark 1**. *It is noteworthy that $n^2>2n+1$, when $n\geq3,n\in\mathbb N^*$. This means that the reconstructed $n$-dimensional phase space is not unique by Takens embedding theorem.* Back to the original systems ([\[2nonlinear\]](#2nonlinear){reference-type="ref" reference="2nonlinear"}), we only need to consider autonomous systems by using the treatment of non-autonomous systems in Section [5.1](#Uniqueness){reference-type="ref" reference="Uniqueness"}, $$\left\{ \begin{aligned} \overset{.}x(t)&=f(x),\\ x(0)&=x_0 ,\\ \end{aligned} \right.~ \left\{ \begin{aligned} \overset{.}y(t)&=g(y),\\ y(0)&=y_0=K(0)x_0 .\\ \end{aligned} \right.$$ If we know the behavior of the system in $[0, T]$, combined with the Takens embedding theorem [@[Takens+81]] and the similarity theory, we can realize the reconstruction and prediction of the $n$-dimensional system. This section describes how $K(t)$ predicts the system completely and accurately, and how $n$-order constant matrix $K$ predicts the past and future of the system, corresponding to Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"} and Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"} of Section [6.1](#3d chaotic systems){reference-type="ref" reference="3d chaotic systems"} respectively.\ **Completely accurate prediction-$K(t)$.** We use $y(t)$ to predict $x(t)$, in other words, solving $K(t)$ makes the similarity of the two systems reach 100$\%$ (completely accurate prediction). $K(t)$ needs to meet $n^2$ constraints as $K(t)\in GL(n)$. In fact, each point in $\mathbb R^n$ contains $n$ one-dimensional constraints, and $K(t)$ has $n^2$ one-dimensional constraints. Let $$K(t)(x(t),c_1(t),\cdots ,c_{n-1}(t))=(y(t,K(0)x_0),y_1(t),\cdots ,y_{n-1}(t)),$$ where $y_i(t)$ is the trajectories of the known $y$ system starting from different initial values; $\dot{c_i}=f_i(t,c_i)=f(c_i)+\epsilon_i\backsimeq f(c_i)$, $\backsimeq$ indicates approximate equality, $\epsilon_i$ is the random small disturbance; and $i=1,\cdots ,n-1$. Since the orbits of autonomous systems are disjoint, $c_i$ can be understood as $n-1$ approximate solution trajectories of the system $x$, or input data. On the basis of the above analysis, $K(t)$ exists and is unique. At this time, the similarity between the two systems reaches $100\%$. We call it completely accurate prediction.\ {#ci width="0.9\\linewidth"} The blue points represents $y_i(t)~(i=1,2,3)$, the trajectory of the known $x$ system on $[-3T,-2T],[-2T,-T],[-T,0]$. The green line indicates $c_i(t)~(i=1,2,3)$, which is a physical law learned from the past $n$-segment trajectory and used to predict the future. **Predict the future and infer the past-$K$.** To predict the future from the past $n$ time behaviors of the system, or to infer the past from the future $n$ time behaviors, we can also adopt a similar method to solve $K\in \mathcal M^n$ to maximize the similarity between the two systems. Let $$K(c_0(t),\cdots ,c_{n-1}(t))=(y_1(t),\cdots ,y_n(t)),$$ where $y_i(t)$ is the trajectory of the known $x$ system on $[-(n-i+1)T,-(n-i)T],~i=1,\cdots ,n$; $y_{i}(-(n-i)T)=y_{i+1}(-(n-i)T)$, $i=1,\cdots ,n-1$; and $$\dot{c_i}=\left\{ \begin{aligned}f_i(t,c_i)=f(x)+\epsilon_i\backsimeq f(x),& \quad [-(n-i)T,-(n-i-1)T],\\ 0, &\quad \text{others else}. \\ \end{aligned}\right.$$ The above analysis shows that $K$ is unique and reversible. Then our predicted $x_n$ is $x_n(0)=K^{-1}y_n(0)$. Taking $n = 3$ as an example, Figure [2](#ci){reference-type="ref" reference="ci"} vividly demonstrates this process. Analogically, let $y_i(t)$ is the trajectory of the known $x$ system on $[(n-i)T,(n-i+1)T],~i=1,\cdots ,n$; $y_{i}((n-i)T)=y_{i+1}((n-i)T)$, $i=1,\cdots ,n-1$; and $$\dot{c_i}=\left\{ \begin{aligned}f_i(t,c_i)=f(x)+\epsilon_i\backsimeq f(x),&\quad [(n-i-1)T,(n-i)T],\\ 0, &\quad \text{others else}. \\ \end{aligned}\right.$$ Then our inferred $x_{-n}$ is $x_{-n}(0)=K^{-1}y_n(0)$. # Chaotic systems and Numerical simulation {#test} ## 3d chaotic systems {#3d chaotic systems}     Three three-dimensional systems, Lorenz system, Chua's circuit system and Chen's system, are considered in this section. The parameters of these systems are derived from the literature (see [@[AFM+13]; @[CHEN+14]]). The reason why such systems are considered is that they have attracted extensive attention. Consider Lorenz system: $$\dot{X}=F(t,X)=\left( \begin{aligned} \overset{.}x_1(t)\\ \overset{.}x_2(t)\\ \overset{.}x_3(t) \end{aligned} \right)= \left( \begin{matrix} \sigma(x_2-x_1)\\ \rho x_1-x_2-x_1x_3\\x_1x_2-\beta x_3 \end{matrix} \right) ,$$ where $\sigma,\rho,\beta$ are positive real parameters. In the following numerical calculation, we adopt $X_0=(0.0,1.0,0.0),~\sigma_1=10,~\rho_1=28,~\beta_1=8/3$, and $\sigma_2=10,~\rho_2=28,~\beta_2=3$, respectively. Consider Chua's circuit system: $$\dot{Y}=G(t,Y)=\left( \begin{aligned} \overset{.}y_1(t)\\ \overset{.}y_2(t)\\ \overset{.}y_3(t) \end{aligned} \right)= \left( \begin{matrix} \alpha(y_2-y_1)-f(y_1)\\x_1-x_2+x_3\\x_1x_2-\beta y_2 \end{matrix} \right) ,$$ where $f(y_1)=m_1y_1+\frac{1}{2}(m_0-m_1)(|y_1+1|-|y_1-1|)$, and $\alpha,\beta,m_0,m_1$ are real parameters. In the following numerical calculation, we adopt $Y_0=(0.1,0.3,-0.6),~\alpha=10,~\beta=15,~m_0=-1.2,~m_1=-0.6.$ Consider Chen's system: $$\dot{Z}=E(t,Z)=\left( \begin{aligned} \overset{.}z_1(t)\\ \overset{.}z_2(t)\\ \overset{.}z_3(t) \end{aligned} \right)= \left( \begin{matrix} a(z_2-z_1)\\(c-a)z_1+cz_2-z_1z_3\\z_1z_2-bz_3 \end{matrix} \right) ,$$ where $a,b,c$ are positive real parameters. In the following numerical calculation, we adopt $Z_0=(0.0,1.0,0.0),~a=35,~b=3,~c=28.$\ Next, we give the algorithm to solve $K (t)$, and give the numerical solution and the comparison of similarity. Considerations about the similarity give rise to an algorithm for solving $K(t)$ (Algorithm [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"}) according to the maximum principle in Section [4](#MAX){reference-type="ref" reference="MAX"}. Subsequently, the phase diagrams (Figure [\[Figure1\]](#Figure1){reference-type="ref" reference="Figure1"}) of the four systems, the linear transformation $K$ of two different systems, and the $K(t)$ (Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"}) with $100\%$ similarity of two different systems are obtained.\ Note that we adopt the concept of time average (Definitions [Definition 3](#Cost functional){reference-type="ref" reference="Cost functional"}, [Definition 4](#similarity){reference-type="ref" reference="similarity"}, [\[similarity2\]](#similarity2){reference-type="ref" reference="similarity2"}), which means that the similarity will be higher and higher with the evolution of time, although there may be a period of similarity and a period of dissimilarity at the beginning. For example, if we use $T = 300$, the final similarity is $94.46 \%$ , $98.99 \%$ and $90.55 \%$. This is illustrated in Figure [\[Figure5\]](#Figure5){reference-type="ref" reference="Figure5"}. It can be seen that with the passage of time, the similarity also increases, and finally tends to a stable state. Step 1. Give $X_0,~Y_0$, solves two ODEs to obtain $X(t),~Y(t)$. Step 2. Enter the cycle from step $i = 0$, and $N = T$ terminates the cycle. Step 3. Takes columns $i$ to $i + n$ of $X(t),~Y(t)$. Step 4. Calculate $K(i)$ by $K(i)X(i)=Y(i)$. Step 5. $K(i)$ corresponds to the $i$-th matrix of $K(t)$. Step 6. Let $i=i+1$, repeat Step 3 to Step 5 until the end. Step 1. Give $X_0,~Y_0$, solves two ODEs to obtain $X(t),~Y(t)$. Step 2. The initial similarity is set to 0. Step 3. Enter the cycle from step $i = 0$, and $N = T$ terminates the cycle. Step 4. Takes columns $i$ to $i + n$ of $X(t),~Y(t)$. Step 5. Calculate $K(i)$ by $K(i)X(i)=Y(i)$. Step 6. Calculate the similarity (Definition [\[similarity2\]](#similarity2){reference-type="ref" reference="similarity2"}) of this step. step 7. Compare the similarity with the previous step, the greater similarity and the corresponding matrix $K(i)$ are stored in this step. Step 8. Let $i=i+1$, repeat step 4 to step 7 until the end. ## Numerical solution     Here, we study the similarity between the two systems through specific numerical simulation, it is divided into two steps:\ **(i) Find the solution trajectory.**\ Using differential equation solvers (there are already many differential equation solvers to make this process easy to implement, such as Python $scipy.integrate. odeint$) to solve he trajectories of $X_1 (t),~X_2 (t),~Y(t),~Z(t)$, which are recorded as $track(i),~i=1,2,3,4$.\ **(ii) Algorithms.**\ Use Algorithms [\[algorithm1\]](#algorithm1){reference-type="ref" reference="algorithm1"} and [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"} to obtain the corresponding numerical solutions and similarity. Here, we adopt $T= 30, dt = 0.01$. For specific procedure details, see Python file. Figure [\[Figure1\]](#Figure1){reference-type="ref" reference="Figure1"} shows the trajectories of four systems. We can clearly see that four nonlinear systems are the structure of double scroll chaotic attractor, which has a certain similarity. The numerical solution of Algorithm [\[algorithm2\]](#algorithm2){reference-type="ref" reference="algorithm2"} is shown in Table [\[Table1\]](#Table1){reference-type="ref" reference="Table1"}. The pairwise comparison of four systems can be seen intuitively in Figure \[[\[Figure 2\]](#Figure 2){reference-type="ref" reference="Figure 2"},[\[Figure 3\]](#Figure 3){reference-type="ref" reference="Figure 3"},[\[Figure 4\]](#Figure 4){reference-type="ref" reference="Figure 4"}\].\ The following numerical results prove once again that two systems can reach the maximum value of linear similarity (under the minimizer constant matrix $K$), and the similarity increases with time; Under the time-varying $K(t)$, the two systems can be completely similar (the similarity reaches 100$\%$), i.e. the functional attains the extreme value of 0. [\[Table1\]]{#Table1 label="Table1"} -------------------- ------------ ------------------------------------------------------ ------------ Systems Initial Constant matrix K Final similarity similarity Lorenz1$\&$Lorenz2 18.35$\%$ $\left( 86.64$\%$ \begin{matrix} 0.99669589&0.00708679&-0.00368821\\ -0.01092915&1.02344135&-0.0121997\\ -0.02563355&0.0549802&0.97138635 \end{matrix}\right)$ Lorenz$\&$Chua's 11.93$\%$ $\left( 83.99$\%$ \begin{matrix} 2.06541069&-1.78187423&-0.16405216\\ 2.61340569&-2.59013073&0.11312293\\ 3.16451651&-3.40502316&0.39380414 \end{matrix}\right)$ Lorenz$\&$Chen's 16.06$\%$ $\left( 77.05$\%$ \begin{matrix} 2.98413841&-2.96556925&0.57976059\\ 2.7629626&-2.65449576&0.49652093\\ 2.7629626&-2.65449576&0.49652093 \end{matrix}\right)$ -------------------- ------------ ------------------------------------------------------ ------------ Note: see Python file for detailed procedures. This is the result at T = 30. Compare the data in Table [\[Table1\]](#Table1){reference-type="ref" reference="Table1"} horizontally. The calculated minimizer $K$ can significantly improve the similarity between two systems, reflecting the similar geometric structure to a certain extent. Compare the data in Table [\[Table1\]](#Table1){reference-type="ref" reference="Table1"} vertically. From the initial similarity, two Lorenz systems with different parameters are the most similar, followed by Lorenz and Chen's, and Lorenz and Chua's are the most unlike. From the similarity after multiplying $K$, two Lorenz systems are still the most similar, but the similarity between Lorenz and Chua's increases the most, indicating that Chua's can be very similar to Lorenz after linear transformation. The blue line represents the trajectory of Lorenz system, orange represents the other system (Lorenz2, Chua's circuit, Chen's), and yellow represents the trajectory after $K$ transformation. This paper discusses the similarity between two deterministic ordinary differential systems. We will naturally ask, if the two dynamic systems are not ordinary differential equations (ODEs), but partial differential equations (PDEs) or stochastic differential equations (SDEs) with random terms, what is the similarity between them? After Pontryagin maximum principle, Peng [@[P+90]] proposed the stochastic maximum principle, and there have been some work, such as [@[HPW+10]] and [@[CPT+16]]. Can the maximum principle be considered as a necessary condition for the existence of similarity? What is the sufficient condition? Can we predict PDEs or SDEs based on the Takens embedding theorem? There are still many open problems. In the forthcoming paper, we will consider the similarity between two infinite dimensional dynamic systems or stochastic dynamic systems in detail. ying A. Algaba, F. Fernández-Sánchez, M. Merino, A. Rodríguez-Luis. Chen's attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system. Chaos 23 (2013), 6 pp. L. Baratchart, M. Chyba, J. Pomet. A Grobman-Hartman theorem for control systems. J. Dynam. Differential Equations 19 (2007), pp. 75-107. P.W. Bates, K. Lu. A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations. J. Dynam. Differential Equations 6 (1) (1994), pp. 101-145. L. Barreira, C. Valls. A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics. J. Differential Equations 228 (2006), pp. 285-310. L. Barreira, C. Valls. A Grobman-Hartman theorem for general nonuniform exponential dichotomies. J. Funct. Anal. 257 (2009), pp. 1976-1993. M. Chekroun, E. Park, R. Temam. Roger The Stampacchia maximum principle for stochastic partial differential equations and applications. J. Differential Equations 260 (2016), pp. 2926-2972. N. Carmichael, M. Quinn. Fixed-point methods in nonlinear control. IMA J. Math. Control Inform., 5 (1988), pp. 41-67. J. Davison, C. Kunze. Some sufficient conditions for the global and local controllability of nonlinear time varying systems. J. SIAM Control 8 (1970), pp. 489-497. D. Grobman. Homeomorphisms of systems of differential equations. Dokl. Akad. Nauk SSSR 128 (1965), pp. 880-881. J.K. Hale. Ordinary Differential Equations. Pure Appl. Math., vol. 21, Wiley-Interscience, 1969. P. Hartman. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (4) (1960), pp. 610-620. M. Hein, J. Prüss. The Hartman-Grobman theorem for semilinear hyperbolic evolution equations. J. Differential Equations 261 (2016), pp. 4709-4727. Y. Han, S. Peng, Z. Wu. Maximum principle for backward doubly stochastic control systems with applications. SIAM J. Control Optim. 48 (2010), pp. 4224-4241. K. Lu. A Hartman-Grobman theorem for scalar reaction-diffusion equations. J. Differential Equations 93 (2) (1991), pp. 364-394. H. Poincaré. Cinquième complément àĺanalysis situs. Rend. Circ. Mat. Palermo 18 (1904), 45-110. S. Peng. A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28 (1990), pp. 966-979. T. Sauer, J. Yorke, M. Casdagli. Embedology. J. Statist. Phys. 65 (1991), pp. 579-616. F. Takens. Detecting strange attractors in turbulence. Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), pp. 366-381, Lecture Notes in Math., 898, Springer, Berlin-New York, 1981. X. Wang, G. Chen. Generating Lorenz-like and Chen-like attractors from a simple algebraic structure. Sci. China Inf. Sci. 57 (2014), 7 pp. Z. Wu, Y. Liu, H. Wang. Optimal control theory: A concise introduction. Higher Education Press, Beijing, 2017. W. Zhang, P. Liu, X. Lei. On Hölder dependence of the parameterized Hartman-Grobman theorem. Acta Math. Sin. (Engl. Ser.) 38 (2022), pp. 137-147. W. Zhang, K. Lu, W. Zhang. Differentiability of the conjugacy in the Hartman-Grobman theorem. Trans. Amer. Math. Soc. 369 (2017), pp. 4995-5030. [^1]: We assume that the pair solution $(x,y)$ is guaranteed to exist and be unique for given mild conditions: $f(t,x(t))$, $g(t,y(t))$ are continuous with respect to $t$ and locally Lipschitz with respect to $x$ or $y$. [^2]: Here hyperbolic means that the matrix $A$ has no purely imaginary eigenvalues. [^3]: In fact, we can shift the system to the origin $O$ and consider the conjugate problem. More precisely, let $z(t,z_0)=y(t,y_0)-y_0$, then $\dot{z}(t)=\dot{y}(t)=g(t,y(t))=g(t,z(t)+y_0)\triangleq \tilde{g}(t,z(t))$ and $z(\mathbf{0})=z_0=\mathbf{0}$, where $\mathbf{0}$ is zero vector. [^4]: *$||\cdots||$ represents the norm of vector, such as the $L^1$ norm $||x||=|x_1|+|x_2|+\cdots+|x_n|$ and the $L^2$ norm (Euclidean norm) $||x||=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$, etc.. They are all equivalent.* [^5]: Notice that $\underset{x\rightarrow0^+}{\lim}\frac{\log(1+x)}{x}=1$ and $\underset{x\rightarrow +\infty}{\lim}\frac{\log(1+x)}{x}=0$. [^6]: $GL (n)$ represents a general linear matrix group, and the matrices in this group are of order $n$. [^7]: *The variation $\delta J[\delta K(\cdot)]$ means that the functional $J$ changes $\delta K(\cdot)$ at $K_0$, and the corresponding dependent variable $\Delta J\triangleq J[K_0(\cdot)+\delta K(\cdot)]-J[K_0(\cdot)]$ satisfies $\Delta J=\delta J[\delta K(\cdot)]+o(\parallel \delta K(\cdot)\parallel)$.* [^8]: $\mathcal M^n$ represents the $n$-order constant matrix. [^9]: *$``\rightharpoonup"$ means weak convergence, which can also be represented by symbols $``\overset{w}{\longrightarrow}"$.* [^10]: $f_m(x)\underset{1}{\overset{m\rightarrow\infty}{\longrightarrow}}f_0(x)$ means that $f_m(x)$ uniformly (stronger than pointwise) converges to $f_0(x)$, which is independent of $x$. [^11]: The symbol $\triangledown$ represents the gradient, and $\triangledown r(y)=\Big(\frac{\partial r(y)}{\partial y_1},\frac{\partial r(y)}{\partial y_2},\cdots,\frac{\partial r(y)}{\partial y_n}\Big)$ is a matrix, where $r(y)=(r(y)_1,r(y)_2,\cdots,r(y)_n)$.

arxiv_math

--- abstract: | We consider random walks on countable groups. A celebrated result of Kesten says that the spectral radius of a symmetric walk (whose support generates the group as a semigroup) is equal to one if and only if the group is amenable. We give an analogue of this result for finitely supported walks which are not symmetric. We also conclude a ratio limit theorem for amenable groups. address: - Department of Mathematical Sciences, Durham University, Upper Mountjoy, Durham DH1 3LE - Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK author: - "Rhiannon [Dougall]{.smallcaps} and Richard [Sharp]{.smallcaps}" title: A non-symmetric Kesten criterion and ratio limit theorem for random walks on amenable groups --- [^1] # Introduction Let $G$ be a finitely generated countable group. Let $\mu$ be a probability measure on $G$, i.e. a function $\mu : G \to \mathbb R^+$ such that $\sum_{g \in G} \mu(g) =1$. Let $S_\mu :=\{g \in G \hbox{ : } \mu(g)>0\}$, the support of $\mu$. We say that $\mu$ is *non-degenerate* if $S_\mu$ generates $G$ as a semigroup. (We do not require $S_\mu$ to be finite.) Let $|\cdot|$ be a word metric on $G$ associated to some finite generating set. (We do not assume any connection between this set and $S_\mu$.) We say that $\mu$ has finite first moment if $$\sum_{g \in G} |g| \mu(g) <\infty$$ and that $\mu$ has finite exponential moment of order $c>0$ if $$\sum_{g \in G} e^{c|g|} \mu(g) <\infty.$$ The measure $\mu$ defines a random walk on $G$ with transition probabilities $p(s,t) = \mu(s^{-1}t)$. The convolution $\mu \ast \nu$ of two functions $\mu,\nu : G \to \mathbb R^+$ is defined by $$\mu \ast \nu(g) = \sum_{s \in G} \mu(s) \nu(s^{-1}g).$$ We will be interested in the *spectral radius* of this random walk, defined by $$\lambda(G,\mu) :=\limsup_{n\to\infty}(\mu^{\ast n}(e))^{1/n}.$$ Clearly, $0 \le \lambda(G,\mu) \le 1$. A celebrated theorem of Kesten (which does not even require $G$ to be finitely generated) says that if $\mu$ is symmetric then $\lambda(G,\mu)=1$ if and only if $G$ is amenable [@Kesten]. (We recall that $G$ is amenable if and only if it admits a Banach mean, i.e. a linear functional $M:\ell^\infty(G)\to\mathbb{R}$ such that $M(\mathds{1})=1$, $\inf_{g\in G} f(g) \le M(f)\le \sup_{g\in G} f(g)$, and $M(f_g)=M(f)$, where $f_g(x)=f(gx)$. See the papers of Følner [@folner] and Day [@Day] for further discussion.) The aim of this note is to generalise Kesten's criterion to the non-symmetric case. To state our generalisation, we need to consider the abelianisation of $G$. Since $G$ is finitely generated, this has a finite rank $k \ge 0$. Let $G^{\mathrm{ab}} = G/[G,G]$ denote the abelianisation of $G$ and let $G^{\mathrm{ab}}_{\mathrm{T}}$ denote the torsion subgroup of $G^{\mathrm{ab}}$. Now set $\overline{G} = G^{\mathrm{ab}}/G^{\mathrm{ab}}_{\mathrm{T}} \cong \mathbb Z^k$, for some $k \ge 0$, (the torsion-free part of the abelianisation) and let $\pi : G \to \overline G$ be the natural projection homomorphism. Write $\bar{\mu} =\pi_*(\mu)$, i.e. $$\bar{\mu}(m)=\sum_{\substack{g\in G \\ \pi(g)=m }}\mu(g).$$ **Theorem 1** (Non-symmetric Kesten criterion). *Let $G$ be a finitely generated group and let $\mu$ be a non-degenerate probability measure on $G$. Then $$G \; \mathrm{amenable} \; \iff \lambda(G,\mu) = \lambda(\overline{G},\bar{\mu}).$$* The special case where $\mu$ has finite support originally appeared in Dougall--Sharp [@DougallSharp], where it is written in the language of subshifts of finite type and Gibbs measures. The value of $\lambda(\overline{G},\bar{\mu})$ may be characterised in the following way. Define $\phi_\mu : \mathbb{R}^k \to \mathbb{R}^+ \cup \{+\infty\}$ by $$\phi_\mu(v) = \sum_{g \in G} e^{\langle v,\pi(g)\rangle} \mu(g) = \sum_{m \in \mathbb{Z}^k} e^{\langle v,m\rangle} \bar{\mu}(m),$$ where $\langle v,m\rangle = v_1m_1 + \cdots + v_km_k$. By a result of Stone [@Stone],[@Stone_Berkeley], there is a unique $\xi \in \mathbb{R}^k$ at which $\phi_\mu(v)$ attains its minimum. Then $\lambda(\overline{G},\bar{\mu}) = \phi_\mu(\xi)$. This is discussed in more detail in Section [2](#sec:limit_theorems){reference-type="ref" reference="sec:limit_theorems"}. A probability measure $\mu$ (with finite first moment) is said to be *centred* if for each homomorphism $\chi : G \to \mathbb{R}$, we have $$\sum_{g \in G} \chi(g) \mu(g) =0.$$ Any such homomorphism factors through $\overline{G}$ so it is easy to see that $\mu$ is centred if and only if either $k=0$ or $$\sum_{g \in G} \pi(g) \mu(x) = \sum_{m \in \bar G} m\bar \mu(m)=0.$$ In particular, $\mu$ is centred if and only if $\bar{\mu}$ is centred. If, in addition, $\mu$ has a finite exponential moment of some order then we have the following result. **Corollary 2**. *Let $G$ be a finitely group and let $\mu$ be a non-degenerate probability measure on $G$. Provided $\mu$ has a finite exponential moment of some order, we have $\lambda(G,\mu)=1$ if and only if $G$ is amenable and $\mu$ is centred.* Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"} allows us to prove a ratio limit theorem for amenable groups with an explicit limit. To avoid any parity issues, it is convenient to restrict to aperiodic walks. We say that $(G,\mu)$ is *aperiodic* if there exists $n_0 \ge 1$ such that $\mu^{\ast n}(e) >0$ for all $n \ge n_0$. **Theorem 3** (Ratio limit theorem). *Suppose that $G$ is a finitely generated amenable and that $\mu$ is a non-degenerate probability measure on $G$. Assume in addition that $(G,\mu)$ is aperiodic. Then, for each $g\in G$, $$\lim_{n \to \infty} \frac{\mu^{\ast n}(g)}{\mu^{\ast n}(e)} = e^{-\langle \xi,\pi(g)\rangle},$$ where $\xi \in \mathbb{R}^k$ is the unique value for which $\lambda(G,\mu)=\phi_\mu(\xi)$.* **Remark 4**. One should compare this with a theorem of Avez [@Avez] that says that if $G$ is amenable and $\mu$ is *symmetric*, non-degenerate and aperiodic then $\lim_{n \to \infty} \mu^{\ast n}(g)/\mu^{\ast n}(e) =1$, for all $g \in G$. **Remark 5**. It should be noted that there is no *a priori* mechanism to guarantee that the ratios do indeed have a limit. However, notice that if one has the ratio limits $$\lim_{n\to\infty}\frac{\mu^{\ast n}(g)}{\mu^{\ast n}(e)} = e^{-\langle \xi, \pi(g)\rangle},$$ for all $g\in G$, for *some* $\xi$ then $G$ is necessarily amenable. We give the short demonstration. From the hypothesis we have for any $s\in G$, $$\frac{\mu^{\ast n}(s^{-1})}{\mu^{\ast n}(e)} = e^{\langle \xi, \pi(s)\rangle}.$$ Now, since $$\mu^{\ast(n+1)}(g) = \sum_{s\in G} \mu(s) \mu^{\ast n}(s^{-1} g),$$ we then have $$\label{eq:limit_is_1} \lim_{n\to\infty}\frac{\mu^{\ast (n+1)}(e)}{ \mu^{\ast n}(e)} = \lim_{n \to \infty} \sum_{s\in S_\mu} \mu(s) \frac{\mu^{\ast n}(s^{-1})}{\mu^{\ast n}(e)} \ge \sum_{s\in S_\mu} \mu(s) e^{\langle \xi, \pi(g)\rangle} = \phi_\mu(\xi).$$ In particular $\phi_\mu(\xi)<\infty$. We proceed with the proof assuming that $\xi=0$ and deduce the general case after. Now using that $$\frac{\mu^{\ast n}(e)}{\mu^{\ast 1}(e)} = \prod_{m=2}^n \frac{\mu^{\ast m}(e)}{\mu^{\ast (m-1)}(e)} %\ge C^{-N} (1-\delta/2)^{N-n},$$ we see that ([\[eq:limit_is_1\]](#eq:limit_is_1){reference-type="ref" reference="eq:limit_is_1"}) with $\phi_\mu(0)=1$ implies that $\limsup_{n\to\infty}(\mu^{\ast n}(e))^{1/n}=1$. This contradicts Day's [@Day] generalisation of Kesten's criterion to the random walk operator spectral radius --- a consequence of which is that, for any non-degenerate probability, we have that if $G$ is non-amenable then $\limsup_{n\to\infty}(\mu^{\ast n}(e))^{1/n} <1$. For the general case $\xi\ne0$, we have already shown that $\phi_\mu(\xi)<\infty$, and so $$\hat{\mu}(g) = \frac{e^{\langle \xi,\pi(g) \rangle}}{\phi_\mu(\xi)} \mu(g)$$ is a well-defined probability measure on $G$ with ratio limits equal to one, and we again conclude that $G$ is amenable. Let us now outline the contents of the rest of the paper. In Section [2](#sec:limit_theorems){reference-type="ref" reference="sec:limit_theorems"}, we recall results of Stone on random walks on $\mathbb{Z}^k$ that are essential to the formulation of our results, and the rather general results of Gerl. In Section [3](#sec:proof_of_centred){reference-type="ref" reference="sec:proof_of_centred"}, we give a proof of Corollary [Corollary 2](#cor:centred){reference-type="ref" reference="cor:centred"} assuming Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"}. We prove Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"} in Sections [4](#section:toamenable){reference-type="ref" reference="section:toamenable"} and [5](#section:amenableto){reference-type="ref" reference="section:amenableto"}. Theorem [Theorem 3](#thm:ratio){reference-type="ref" reference="thm:ratio"} is proved in Section [6](#section:equidistribution){reference-type="ref" reference="section:equidistribution"}, as a consequence of equidistribution results for countable state shifts. ## Acknowledgements {#acknowledgements .unnumbered} We are grateful to Wolfgang Woess for bringing to our attention the work of Gerl. We are grateful to Rémi Coulon for asking whether Theorem [Theorem 3](#thm:ratio){reference-type="ref" reference="thm:ratio"} had a converse. # Results of Stone and Gerl {#sec:limit_theorems} In this section, we recall classic results of Stone concerning random walks on $\mathbb{Z}^k$. Let $\omega$ be a non-degenerate aperiodic probability measure on $\mathbb{Z}^k$ and define $\phi_\omega : \mathbb{R}^k \to \mathbb{R}\cup \{+\infty\}$ by $$\phi_\omega(v) = \sum_{m \in \mathbb{Z}^k} e^{\langle v,m \rangle} \omega(m).$$ **Lemma 6** (Stone [@Stone], [@Stone_Berkeley]). *If $\omega$ is non-degenerate then there is a unique $\xi \in \mathbb{R}^k$ such that $\phi_\omega(\xi) = \inf_{v \in \mathbb{R}^k} \phi_\omega(v)$. Furthermore, $\lambda(\mathbb{Z}^k,\omega)=1$ if and only if $\phi_\omega(\xi) = 1$, i.e. if and only if $\xi=0$.* We note that $\phi_\omega(\xi)=1$ if and only if $\phi_\omega(v) \ge 1$ for all $v \in \mathbb{R}^k$. **Corollary 7**. *$\lambda(\mathbb{Z}^k,\omega) = \phi_\omega(\xi)$.* *Proof.* Suppose that $\phi_\omega(\xi)<1$. Then we can define a new probability measure $\omega_\xi$ on $\mathbb{Z}^k$ by $$\omega_\xi(m) = (\phi_\omega(\xi))^{-1} e^{\langle \xi,m \rangle} \omega(m).$$ Then $\omega_\xi$ has the same support as $\omega$ and $$\omega_\xi^{\ast n}(m) = (\phi_\omega(\xi))^{-n} e^{\langle \xi,m \rangle} \omega^{\ast n}(m).$$ We have $$\sum_{m \in \mathbb{Z}^k} e^{\langle v,m \rangle } \omega_\xi(m) =\frac{1}{\phi_\omega(\xi)} \sum_{m \in \mathbb{Z}^k} e^{\langle v+\xi,m \rangle } \omega(m) = \frac{\phi_\omega(v+\xi)}{\phi_\omega(\xi)} \ge 1.$$ Hence, $\lambda(\mathbb{Z}^k,\omega_\xi)=1$ and $$\lambda(\mathbb{Z}^k,\omega) = \phi_\omega(\xi) \lambda(\mathbb{Z}^k,\omega_\xi) = \phi_\omega(\xi).$$ ◻ We now state a ratio limit theorem due to Stone. **Proposition 8** (Stone [@Stone]). *Suppose that $\omega$ is non-degenerate and aperiodic. Then, for each $m \in \mathbb Z^k$, $$\lim_{n \to \infty} \frac{\omega^{*n}(m)}{\omega^{*n}(0)} = e^{-\langle \xi,m \rangle}.$$* Ratio limit theorems are intimately related to the existence of harmonic functions. Given a random walk $(G,\mu)$, we define the random walk operator $P_\mu : \ell_\mu^1(G) \to \ell_\mu^1(G)$ by $$P_\mu f(g) = \sum_{s \in G} p(g,s) f(s)= \sum_{s \in G} \mu(g^{-1}s) f(s).$$ This may be written as a convolution $P_\mu f = f \ast \check{\mu}$, where $\check{\mu}(s)=\mu(s^{-1})$. A function $f : G \to \mathbb R^+$ is called $\lambda$-*harmonic* if $P_\mu f=\lambda f$, i.e. if $$\sum_{s \in G} \mu(g^{-1}s)f(s) = \lambda f(g).$$ (Some authors define $f$ to be $\lambda$-harmonic if $\mu \ast f = \lambda f$.) If we write $h_\xi(m) = e^{-\langle \xi,m\rangle}$ for the limit in Proposition [Proposition 8](#Prop:Stone){reference-type="ref" reference="Prop:Stone"} then we see that the function $\check{h}_\xi(m) := h_\xi(-m) = e^{\langle \xi,m \rangle}$ is $\lambda(\mathbb{Z}^k,\omega)$-harmonic (for $\omega$). Furthermore $\lambda=\lambda(\mathbb{Z}^k,\omega)$ is the smallest value for which there is a $\lambda$-harmonic function. One may ask about ratio limit theorems on more general groups than $\mathbb{Z}^k$. Following earlier work by Avez [@Avez] and Gerl [@Gerl73], a rather general ratio limit theorem was proved by Gerl [@Gerl78], where it is obtained as a corollary of the following limit theorem. A detailed account and discussion may be found in the recent note by Woess [@Woess22]. **Proposition 9** (Gerl's fundamental theorem [@Gerl78]). *Suppose that $\mu$ is a non-degenerate probability measure on $G$ such that $(G,\mu)$ is aperiodic. Then we have $$\lim_{n \to \infty} \frac{\mu^{\ast (n+1)}(e)}{\mu^{\ast n}(e)} = \lambda(G,\mu).$$* Gerl used this to prove the following conditional ratio limit theorem. **Proposition 10** (Gerl's ratio limit theorem [@Gerl78]). *Suppose that $\mu$ is a non-degenerate probability measure on $G$ such that $(G,\mu)$ is aperiodic. Suppose there is a set $\mathfrak F \subset \{f : G \to \mathbb R^+ \hbox{ : } f(e)=1\}$ such that* 1. *if $f : G \to \mathbb R^+$ is defined by $$f(g) = \lim_{j \to \infty} \frac{\mu^{\ast n_j}(g)}{\mu^{\ast n_j}(e)},$$ for some subsequence $(n_j)_{j=1}^\infty$, then $f \in \mathfrak F$;* 2. *there exists a unique $h \in \mathfrak F$ satisfying the equation $\mu \ast h = \lambda(G,\mu) h$.* *Then $$\lim_{n \to \infty} \frac{\mu^{\ast n}(g)}{\mu^{\ast n}(e)} = h(g),$$ for all $g \in G$.* In particular, if we have uniqueness of a $\lambda(G,\mu)$-harmonic function for $(G,\mu)$ then we know the ratio limit theorem holds. The advantage of our Theorem [Theorem 3](#thm:ratio){reference-type="ref" reference="thm:ratio"}, for amenable groups, is that we don't consider arbitrary harmonic functions instead we directly work with functions coming from the abelianisation. # Proof of Corollary [Corollary 2](#cor:centred){reference-type="ref" reference="cor:centred"} {#sec:proof_of_centred} In this section we prove Corollary [Corollary 2](#cor:centred){reference-type="ref" reference="cor:centred"}, assuming Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"}. We note that $\phi_\mu = \phi_{\bar{\mu}}$, so we can use Lemma [Lemma 6](#lem:stone){reference-type="ref" reference="lem:stone"}. *Proof of Corollary [Corollary 2](#cor:centred){reference-type="ref" reference="cor:centred"}.* If $G$ is not amenable then Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"} tells us that $\lambda(G,\mu) < \lambda(\overline{G},\bar{\mu}) \le 1$; so it suffices to show that, if $G$ is amenable, then $\lambda(\overline{G},\bar{\mu}) =1$ if and only if $\mu$ is centred. Since $\bar{\mu}$ is non-degenerate, $\phi_\mu(v)$ is strictly convex on the set where it is finite. The hypothesis of a finite exponential moment implies that $\phi_\mu(v)$ is finite and differentiable in some neighbourhood of $0 \in \mathbb{R}^k$. Therefore, $\phi_\mu(v)$ has its unique minimum at $v=0$ if and only if $\nabla \phi_\mu(0)=0$. Suppose that $\lambda(\overline{G},\bar{\mu})=1$; then, by Lemma [Lemma 6](#lem:stone){reference-type="ref" reference="lem:stone"}, $\phi_\mu$ has its minimum at $0$ and so $$0 = \nabla \phi_\mu(0) = \sum_{m \in \mathbb{Z}^k} m \bar{\mu}(m),$$ i.e. $\mu$ is centred. On the other hand, if $\lambda(\overline{G},\bar{\mu})<1$ then, again by Lemma [Lemma 6](#lem:stone){reference-type="ref" reference="lem:stone"}, the unique minimum of $\phi_\mu$ is not at $0$ and so $$0 \ne \nabla \phi_\mu(0) = \sum_{m \in \mathbb{Z}^k} m \bar{\mu}(m),$$ i.e. $\mu$ is not centred. ◻ # Proof of Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"} ($\impliedby$) {#section:toamenable} In this section we show that if $\lambda(G,\mu)=\lambda(\overline G,\bar \mu)$ then $G$ is amenable. (In Kesten's original theorem, this was the harder direction but in our case it is the easier implication.) Noting that $\phi_\mu = \phi_{\bar{\mu}}$, recall from Section [2](#sec:limit_theorems){reference-type="ref" reference="sec:limit_theorems"} that there is a unique $\xi \in \mathbb{R}^k$ such that $$\phi_\mu(\xi) = \inf_{v \in \mathbb{R}^k} \phi_\mu(v)$$ and $$\phi_\mu(\xi) = \sum_{g\in G} \mu(g)e^{\langle \xi,\pi(g)\rangle} =\lambda(\overline{G},\bar{\mu}).$$ We define a new probability measure $\mu_\xi$ on $G$ (analogous to the measure $\omega_\xi$ on $\mathbb{Z}^k$ in the proof of Corollary [Corollary 7](#cor:xi_measure){reference-type="ref" reference="cor:xi_measure"}) by $$\mu_\xi(g) =\phi_\mu(\xi)^{-1} e^{\langle \xi,\pi(g)\rangle} \mu(g).$$ *Proof of Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"} ($\impliedby$).* Assume that $G$ is non-amenable. By Theorem 1 of Day [@Day] (see also Theorem 5.4 of Stadlbauer [@Stadlbauer1]), we see that the probability measure $\mu_\xi$ satisfies $$\limsup_{n\to\infty} (\mu_\xi^{\ast n}(e))^{1/n}<1.$$ Unpicking the definitions, $\mu_\xi^{\ast n}(e) = \phi_\mu(\xi)^{-n} \mu^{\ast n}(e)$. Hence $\limsup_{n\to\infty} (\mu^{\ast n}(e))^{1/n}<\phi_\mu(\xi).$ ◻ # Proof of Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"} ($\implies$) {#section:amenableto} In this section we will show the harder implication that if $G$ is amenable then $\lambda(\overline G,\bar \mu) \le \lambda(G,\mu)$, and hence that $\lambda(\overline G,\bar \mu) = \lambda(G,\mu)$. We remark that the proof given here is significantly easier and more direct than the one we gave in [@DougallSharp]. Following that proof would introduce a family of measures, indexed by $g \in G$, on the space $S_\mu^{\mathbb{N}} \times G$, each describing the paths that visit $S_\mu^{\mathbb{N}} \times \{g\}$. These measures (which are also analysed in [@Stadlbauer2]) are not required here. Let us begin by emphasising the following: though we know that $\bar{\mu}$ has a $\lambda(\overline{G},\bar{\mu})$-harmonic function *it plays no role in this part of proof*! The first element of the proof is the following proposition. Subsequently, the rest of the section will be devoted to showing that show its hypothesis is satisfied with $\lambda = \lambda(G,\mu)$. **Proposition 11**. *If there is a homomorphism $h : \overline{G} \to \mathbb{R}^{>0}$, the multiplicative group of positive real numbers, such that, for all $n \in \mathbb{N}$, $$\sum_{g\in G} \mu^{\ast n}(g) h(-\pi(g)) \le \lambda^n$$ then $\lambda(\overline G,\bar{\mu})\le \lambda$.* *Proof.* Suppose such a homomorphism $h$ exists. Since $h$ is positive so we can throw away the terms where $-\pi(g)\ne 0$ and obtain $$\sum_{\substack{g\in G \\ \pi(g)=0}} \mu^{\ast n}(g) \le \lambda^n.$$ Hence, for any $\delta<1$, $$\sum_{n\in\mathbb{N}} \bar{\mu}^{\ast n}(0)(\delta^{-1}\lambda)^{-n} =\sum_{n\in\mathbb{N}} \sum_{\substack{g\in G \\ \pi(g)=0}} \mu^{\ast n}(g)\delta^{n}\lambda^{-n} <\infty.$$ This says that $\lambda(\overline{G},\bar{\mu}) \le \lambda\delta^{-1}$. Since we can take $\delta$ arbitrarily close to $1$ we are done. ◻ We view $\lambda(G,\mu)$ as a convergence parameter for the series $$\zeta(t)= \sum_{n\in\mathbb{N}} \mu^{\ast n}(e) t^{-n},$$ where $t>0$, i.e. $$\inf\left \{ t\in \mathbb{R}^+ : \sum_{n\in\mathbb{N}} \mu^{\ast n}(e) t^{-n}<\infty\right\} = \limsup_{n\to\infty}(\mu^{\ast n}(e))^{1/n} = \lambda(G,\mu).$$ This formulation is reminiscent of the Poincaré series used in the construction of Patterson--Sullivan measures on the limit sets of Kleinian groups and more general limit sets and, indeed, we employ ideas from this theory The most relevant reference here is Roblin [@Roblin], which covers the basic tools of Patterson-Sullivan theory and the amenability "trick" we will use in the proof of Proposition [Proposition 16](#prop:h_exists){reference-type="ref" reference="prop:h_exists"}. The series $\zeta(t)$ does not necessarily diverge at $t=\lambda(G,\mu)$ but one can modify the series, in a controlled way, to guarantee divergence at this critical parameter. The following appears as Lemma 3.2 in Denker and Urbanski [@urbanski] (see also [@Stadlbauer2]). **Lemma 12**. *Let $(a_n)_{n=1}^\infty$ be a sequence in $\mathbb{R}^+$ and let $\rho = \limsup_{n\to\infty} a_n^{1/n}$. Then there is a sequence $(b_n)_{n=1}^\infty$ of positive real numbers such that $\lim_{n \to \infty} b_{n+1}/b_n=1$ for which $\sum_{n\in\mathbb{N}} a_nb_n t^{-n} < \infty$ for $t >\rho$ but $$\sum_{n\in\mathbb{N}} a_nb_n\rho^{-n}=\infty.$$* Let $(b_n)_{n=1}^\infty$ be the sequence given by Lemma [Lemma 12](#lem:slowly_diverging){reference-type="ref" reference="lem:slowly_diverging"} for the series with $a_n = \mu^{\ast n}(e)$. We prefer to use $c_n=1/b_n$, a decreasing sequence. Note that we have $\lim_{n \to \infty} c_{n-r}/c_n =1$ for all $r \in \mathbb N$. We will work with a modified series $\zeta_c^e(t)$ defined by $$\zeta^{e}_c(t) = \sum_{n\in\mathbb{N}} \frac{\mu^{\ast n}(e)}{c_n} t^{-n},$$ and also, for each $g\in G$, the series $$\zeta^{g}_c(t) = \sum_{n\in\mathbb{N}} \frac{\mu^{\ast n}(g)}{c_n} t^{-n}.$$ **Lemma 13**. *For each $g\in G$, $$0<\inf_{\lambda(G,\mu) <t \le 2}\frac{\zeta_c^g(t)}{\zeta_c^e(t)} \le \sup_{\lambda(G,\mu) <t \le 2}\frac{\zeta_c^g(t)}{\zeta_c^e(t)}<\infty.$$* *Proof.* We begin by observing that, for every $g,h\in G$, we have $$\mu^{\ast (n+k)}(g)\ge \mu^{\ast k}(gh^{-1}) \mu^{\ast n}(h)$$ and we may choose $k \ge 1$ for which $\mu^{\ast k}(gh^{-1}) >0$. This gives us the inequality $$\begin{aligned} \zeta_c^g(t) = \sum_{m\le k} \frac{\mu^{\ast m}(g)}{c_{m}} + \sum_{n\in\mathbb{N}} \frac{\mu^{\ast (n+k)}(g)}{c_{n+k}} \ge \sum_{m\le k} \frac{\mu^{\ast m}(g)}{c_{m}} + \sum_{n\in\mathbb{N}} \mu^{\ast k}(gh^{-1}) \frac{\mu^{\ast n}(h)}{c_{n}}\frac{c_{n}}{c_{n+k}}. \end{aligned}$$ Since the numbers $c_n$ are positive and, for a fixed $k$, $\lim_{n \to \infty} c_{n}/c_{n+k}=1$, we have $\inf_n c_{n}/c_{n+k}>0$. Hence $$\zeta_c^g(t) \ge C_1(g,k) + \frac{C_2(g,h,k)}{C_3(k)}\sum_{n\in\mathbb{N}} \frac{\mu^{\ast n}(h)}{c_{n}},$$ for positive $C_1$, $C_2$ and $C_3$. We conclude $$0<\inf_{\lambda(G,\mu) <t \le 2}\frac{\zeta_c^g(t)}{\zeta_c^h(t)}.$$ Since $g,h$ are arbitrary the lemma follows. ◻ By the previous lemma and a standard diagonal argument, there exists a sequence $t_n\to \lambda(G,\mu)$ for which the following limits are well-defined $$H(g) = \lim_{n\to\infty}\frac{\zeta_c^g(t_n)}{\zeta_c^e(t_n)} \in (0,\infty),$$ for all $g \in G$. A crucial observation is the following. **Lemma 14**. *For any $r\in\mathbb{N}$, we have $$\sum_{s\in G} \mu^{\ast r}(s) H(s^{-1}g) = \lambda^r H(g)$$ with $\lambda = \lambda(G,\mu)$.* *Proof.* Fix $r \in \mathbb{N}$ and let $\epsilon>0$. Since $\lim_{n \to \infty} c_{n-r}/c_n=1$, we can choose $n_0$ such that $1-\epsilon \le c_{n-r}/c_n \le 1+\epsilon$, for all $n > n_0$. We will also use that $$\mu^{\ast n}(g)=\sum_{s\in G}\mu^{\ast r}(s)\mu^{\ast (n-r)}(s^{-1}g).$$ Then, for $n> n_0$, $$\frac{1-\epsilon}{c_{n-r}} \sum_{s\in G}\mu^{\ast r}(s) \mu^{\ast (n-r)}(s^{-1}g) \le \frac{\mu^{\ast n}(g)}{c_n} \le \frac{(1+\epsilon)}{c_{n-r}} \sum_{s\in G}\mu^{\ast r}(s)\mu^{\ast (n-r)}(s^{-1}g).$$ Setting $$C_1(g,t,n_0) = \sum_{n\le n_0} \frac{\mu^{\ast n}(g)}{c_n} t^{-n},$$ we have $$\begin{aligned} t^r \sum_{n\in\mathbb{N}} \frac{\mu^{\ast n}(g)}{c_n} t^{-n} &\le C_1(g,t,n_0) + t^r (1+\epsilon) \sum_{s\in G}\mu^{\ast r}(s) \sum_{n>n_0}t^{-n} \frac{\mu^{\ast (n-r)}(s^{-1}g)}{c_{n-r}} \\ &\le C_1(g,t,n_0) + (1+\epsilon)\sum_{s\in G}\mu^{\ast r}(s)H(s^{-1}g)\zeta^e_c(t),\end{aligned}$$ using that the terms in the series are non-negative. This gives $$\lambda^r H(g) \le \lim_{m\to\infty} \frac{C_1(g,t_m,n_0)}{\zeta^e_c(t_m)} + (1+\epsilon)\sum_{s\in G}\mu^{\ast r}(s)H(s^{-1}g)=(1+\epsilon)\sum_{s\in G}\mu^{\ast r}(s)H(s^{-1}g)$$ and, since $\epsilon$ is arbitrary, $$\lambda^r H(g) \le \sum_{s\in G}\mu^{\ast r}(s)H(s^{-1}g).$$ For the lower bound, we have $$\begin{aligned} t^{r} \sum_{n\in\mathbb{N}} \frac{\mu^{\ast n}(g)}{c_n} t^{-n} &\ge (1-\epsilon) \sum_{s\in G}\mu^{\ast r}(s)\sum_{n>n_0}t^{-(n-r)}\frac{\mu^{\ast (n-r)}(s^{-1}g)}{c_{n-r}} \\ &= (1-\epsilon) \sum_{s\in G}\mu^{\ast r}(s)\sum_{n \in \mathbb{N}}t^{-n}\frac{\mu^{\ast n}(s^{-1}g)}{c_{n}}.\end{aligned}$$ This gives $$\begin{aligned} \lambda^{r} H(g) &\ge \sum_{s\in G}\mu^{\ast r}(s)H(s^{-1}g). \end{aligned}$$ ◻ Lemma [Lemma 14](#lemma:main){reference-type="ref" reference="lemma:main"} gives us the following corollary. **Corollary 15**. *For any fixed $\gamma \in G$, we have $$0< \inf_{g\in G}\frac{H(\gamma g)}{H(g)} \le \sup_{g\in G}\frac{H(\gamma g)}{H(g)}<\infty.$$* *Proof.* Given $\gamma \in G$, we can find $x_1,\ldots,x_k \in S_\mu$, for some $k \ge 1$, such that $x_1\cdots x_k=\gamma^{-1}$. This gives us $$\mu(x_1)\cdots \mu(x_k)H(\gamma g) \le \sum_{(s_1,\ldots,s_k)\in G^k} \mu(s_1)\cdots \mu(s_k) H((s_1 \cdots s_k)^{-1} g) = \lambda^k H(g),$$ and so $\sup_{g \in G} H(\gamma g)/H(g)$ is finite. Now we put $\gamma g$ on the right hand side and choose $y_1,\ldots,y_\ell \in S_\mu$ such that $y_1 \cdots y_\ell=\gamma$ to get $$\mu(y_1)\cdots \mu(y_\ell)H(g) \le \sum_{(s_1,\ldots,s_\ell)\in G^\ell} \mu(s_1) \cdots \mu(s_\ell) H((s_1 \cdots s_\ell)^{-1}\gamma g) = \lambda^\ell H(\gamma g),$$ and so $\inf_{g \in G} H(\gamma g)/H(g)$ is positive. ◻ We are now ready to use the amenability of $G$. We use the existence of a Banach mean on $\ell^\infty(G)$ to "average over $g$\" in the last lemma. **Proposition 16**. *There is a homomorphism $h:\overline{G}\to \mathbb{R}^{>0}$ such that, for all $n \in \mathbb{N}$, $$\sum_{s\in G} \mu^{\ast n}(s) h(-\pi(s)) \le \lambda^n,$$ with $\lambda = \lambda(G,\mu)$.* *Proof.* Since any homomorphism $h : G \to \mathbb{R}^{>0}$ factors through $\overline G$, it suffices to show that there is a homomorphism $h : G \to \mathbb{R}^{>0}$ such that, for all $n \in \mathbb{N}$, we have $$\label{eq:bound_for_h} \sum_{s\in G} \mu^{\ast n}(s) h(s^{-1}) \le \lambda^n.$$ Let $M$ be a Banach mean on $\ell^\infty(G)$. Jensen's inequality says that if $\varphi$ is convex then $$M(\varphi( f))\ge \varphi (M(f)).$$ (This is more familiar when the linear functional is integration with respect to a probability measure. One can check that we only need monotonicity, finite additivity, and the unit normalisation $M(1)=1$.) We apply this to the function $g \mapsto (H(\gamma g)/H(g))$ in $\ell^\infty(G)$. (Note we have used Corollary [Corollary 15](#cor:cor_to_main_lemma){reference-type="ref" reference="cor:cor_to_main_lemma"} to know that $g \mapsto \log (H(\gamma g)/H(g))$ is in $\ell^\infty(G)$.) Thus we obtain $$\begin{aligned} M\left( g\mapsto \frac{H(\gamma g)}{H(g)}\right) &= M\left( g\mapsto \exp \log \frac{H(\gamma g)}{H(g)}\right) \\ &\ge \exp M\left( g\mapsto \log \frac{H(\gamma g)}{H(g)}\right).\end{aligned}$$ Now set $$h(\gamma) = \exp M\left( g\mapsto \log \frac{H(\gamma g)}{H(g)}\right).$$ We will show that $h$ satisfies ([\[eq:bound_for_h\]](#eq:bound_for_h){reference-type="ref" reference="eq:bound_for_h"}), recalling that $M$ is only finitely additive. Let $\{g_k\}_{k=1}^\infty$ be an enumeration of $G$ and, for $N \ge 1$, let $G_N = \{g_1,\ldots,g_N\}$. Lemma [Lemma 14](#lemma:main){reference-type="ref" reference="lemma:main"} gives us that, for any $n \ge 1$ and any $N \ge 1$, we have $$\begin{aligned} \lambda^n &= M\left( g\mapsto \sum_{s \in G} \mu^{\ast n}(s) \frac{H(s^{-1}g)}{H(g)}\right) \ge M\left( g\mapsto \sum_{s \in G_N} \mu^{\ast n}(s) \frac{H(s^{-1}g)}{H(g)}\right) \\ &= \sum_{s\in G_N} \mu^{\ast n}(s) M\left( g\mapsto \frac{H(s^{-1}g)}{H(g)}\right) \ge \sum_{s\in G_N} \mu^{\ast n}(s) \exp M\left( g\mapsto \log \frac{H(s^{-1}g)}{H(g)}\right) \\ &=\sum_{s\in G_N} \mu^{\ast n}(s) h(s^{-1}).\end{aligned}$$ Taking the supremum over $N$ gives ([\[eq:bound_for_h\]](#eq:bound_for_h){reference-type="ref" reference="eq:bound_for_h"}). It remains to show that $h$ is a homomorphism. Notice that $$\begin{aligned} \log h(ab)&= M\left( g\mapsto \log \frac{H(abg)}{H(g)}\right) \\ &= M\left( g\mapsto \log \frac{H(abg)}{H(bg)}\right)+ M\left( g\mapsto \log \frac{H(bg)}{H(g)}\right) \\ &= \log h(a) + \log h(b)\end{aligned}$$ and $$\begin{aligned} \log h(\gamma^{-1}) &= M\left(g \mapsto \log \frac{H(\gamma^{-1}g)}{H(g)}\right) =M\left(g \mapsto \log \frac{H(g)}{H(\gamma g)}\right) \\ &= M\left(g \mapsto -\log \frac{H(\gamma g)}{H(g)}\right) = -\log h(\gamma).\end{aligned}$$ using translation invariance of $M$. The conclusion follows. ◻ Combining Proposition [Proposition 11](#prop:conditional){reference-type="ref" reference="prop:conditional"} and Proposition [Proposition 16](#prop:h_exists){reference-type="ref" reference="prop:h_exists"} shows that if $G$ is amenable then $\lambda(\overline{G},\bar{\mu}) \le \lambda(G,\mu)$. # Equidistribution and proof of the ratio limit theorem {#section:equidistribution} In this section we use Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"} to prove a ratio limit theorem for amenable groups, Theorem [Theorem 3](#thm:ratio){reference-type="ref" reference="thm:ratio"}. Our arguments will also give a new proof of Proposition [Proposition 9](#thm:slowincrease){reference-type="ref" reference="thm:slowincrease"} in this setting. Our approach is based on weighted equidistribution results for countable state shift spaces. Suppose that $G$ is amenable, that $\mu$ is non-degenerate and that $(G,\mu)$ is aperiodic. We let $\lambda$ denote the common value $$\lambda = \phi_\mu(\xi) = \lambda(\overline{G},\bar{\mu}) = \lambda(G,\mu),$$ given by Theorem [Theorem 1](#thm:kesten){reference-type="ref" reference="thm:kesten"}. As above, $\mu_\xi(g)=\lambda^{-1}h(g) \mu(g)$, where $h(g)=e^{\langle \xi,\pi(g)\rangle}$. We consider the sequence space $\Sigma = S_\mu^{\mathbb{N}}$ and let $\sigma : \Sigma \to \Sigma$ be the shift map: $\sigma((s_i)_{i=1}^\infty)=(s_{i+1})_{i=1}^\infty$. If $s = (s_1,\ldots,s_n) \in S_\mu^n$, we write $[s]= [s_1,\dots, s_n]$ for the cylinder set $$[s_1,\dots, s_n] :=\{(x_i)_{i=1}^\infty \in \Sigma \hbox{ : } x_i =s_i \ \forall i=1,\ldots,n\}.$$ We give $\Sigma$ the topology generated by cylinder sets (which are both open and closed). We denote by $\nu_\xi$ the Bernoulli measure on $\Sigma$ given by $$\nu_\xi([s_1,\ldots, s_n])= \mu_\xi(s_1) \cdots \mu_\xi(s_n).$$ Let $$\Lambda_n = \{s=(s_1,\ldots,s_n) \in S_\mu^n \hbox{ : } s_1 \cdots s_n =e\}$$ and define a sequence of probability measures $m_n$ on $\Sigma$ by $$\begin{aligned} m_n &:= \frac{1}{\mu_\xi^{\ast n}(e)} \sum_{s=(s_1,\ldots, s_n) \in \Lambda_n} \mu_\xi(s_1)\cdots \mu_\xi(s_n) \delta_{s_\infty} = \frac{1}{\mu_\xi^{\ast n}(e)} \sum_{s=(s_1,\ldots, s_n) \in \Lambda_n} \nu_\xi([s]) \delta_{s_\infty},\end{aligned}$$ where we use the notation $s_\infty \in \Sigma$ to mean the one-sided infinite concatenation of $s=(s_1, \ldots, s_n)$ and $\delta_{s_\infty}$ denotes the Dirac measure at this point. We remark that we also have $$m_n = \frac{1}{\mu^{\ast n}(e)} \sum_{s=(s_1,\ldots, s_n) \in \Lambda_n} \mu(s_1)\cdots \mu(s_n) \delta_{s_\infty}$$ but we do not use this formula. We will need to explicitly evaluate the measures $m_n$ on cylinder sets. **Lemma 17**. *For a cylinder set $[u_1,\ldots,u_k]$ we have that, for $n>k$, $$m_n([u_1,\ldots, u_k]) = \frac{\mu_\xi(u_{1})\cdots \mu_\xi(u_k)}{\mu_\xi^{\ast n}(e)}\mu_\xi^{\ast (n-k)}(u^{-1}),$$ where $u = u_1 \cdots u_k$.* *Proof.* This is a straightforward calculation. For $n >k$, $$\begin{aligned} m_n([u_1,\ldots, u_k]) &= \frac{1}{\mu_\xi^{\ast n}(e)} \sum_{(s_1,\ldots, s_n) \in \Lambda_n} \mu_\xi(s_1)\cdots \mu_\xi(s_n) \, \delta_{(s_1, \ldots, s_n)_\infty}([u_1,\ldots, u_k]) \\ &= \frac{1}{\mu_\xi^{\ast n}(e)} \sum_{\substack{(s_1,\ldots, s_n) \in \Lambda_n \\ s_1=u_1,\cdots , s_k = u_k}} \mu_\xi(s_1)\cdots \mu_\xi(s_n) \\ &= \frac{\mu_\xi(u_{1})\cdots \mu_\xi(u_k)}{\mu_\xi^{\ast n}(e)} \sum_{\substack{(s_{k+1},\ldots, s_n) \in S_\mu^{n-k} \\ s_{k+1}\cdots s_n=u^{-1}}} \mu_\xi(s_{k+1})\cdots \mu_\xi(s_n) \\ &= \frac{\mu_\xi(u_{1})\cdots \mu_\xi(u_k)}{\mu_\xi^{\ast n}(e)}\mu_\xi^{\ast (n-k)}(u^{-1}).\end{aligned}$$ ◻ We will show that, for each cylinder set $[u_1,\ldots,u_k]$. $m_n([u_1,\ldots,u_k])$ converges to $\nu_\xi([u_1,\ldots,u_k])$, as $n \to \infty$. In order to have *convergence* (as opposed to an accumulation point) we need $\mu_\xi^{\ast n}(e)^{1/n}$ to have a limit. This is a consequence of aperiodicity, as the next lemma shows. **Lemma 18**. *We have $$\lim_{n \to \infty} (\mu_\xi^{\ast n}(e)^{1/n}) =1.$$* *Proof.* We know that $\limsup_{n \to \infty} (\mu_\xi^{\ast n}(e))^{1/n}=1$. Since $\mu$ is aperiodic, we have $\mu^{\ast n}(e)>0$ for all sufficiently large $n$. Recall also that $\mu_\xi^{\ast (n+m)}(e)\ge \mu_\xi^{\ast n}(e)\mu_\xi^{\ast m}(e)$. This tells us that $-\log \mu_\xi^{\ast n}(e)$ is sub-additive. Hence by Fekete's lemma $$\lim_{n\to\infty} \frac{-\log \mu^{\ast n}(e)}{n} = \inf_{n \ge 1} \frac{-\log \mu^{\ast n}(e)}{n},$$ in particular the limit exists and is equal to the limsup. ◻ In order to show the required convergence for the $m_n$, we introduce some ideas and terminology from thermodynamic formalism and large deviation theory. A function $\varphi : \Sigma \to \mathbb{R}$ is called locally Hölder continuous if $$\label{eq:loc_Holder} \sup_{s \in S_\mu^n} \sup_{x,y \in [s]} |\varphi(x)-\varphi(y)| \le C\theta^n,$$ for some $C>0$ and $0<\theta<1$, for all $n \ge 1$. For a locally Hölder continuous function $\varphi : \Sigma \to \mathbb{R}$, we define the *Gurevič pressure* $P_{\mathrm{G}}(\varphi)$ by $$\label{eq:def_of_Gpressure} P_{\mathrm{G}}(\varphi) = \lim_{n \to \infty} \frac{1}{n} \log \sum_{s \in S_\mu^n} \exp \sum_{j=0}^{n-1} \varphi(s_\infty) \in \mathbb{R}\cup \{+\infty\}.$$ (The original definition given by Sarig in [@Sarig-etds1999] is somewhat different, and only requires, ([\[eq:loc_Holder\]](#eq:loc_Holder){reference-type="ref" reference="eq:loc_Holder"}) to hold for $n \ge 2$, but, by Corollary 1 of [@Sarig-pams2003], the above formula gives the Gurevič pressure in our setting.) We now fix $$\varphi((s_i)_{i=1}^\infty) := \log \mu_\xi(s_1) = \log \nu([s_1]),$$ so that, in particular, $P_{\mathrm{G}}(\varphi)=0$. Let $\chi$ be the indicator function of some cylinder. We can easily calculate from the definition that, for $t \in \mathbb{R}$, $P_{\mathrm{G}}(\varphi+t\chi) \le \max\{0,|t|\}$ for all $t \in \mathbb{R}$. Hence, by Corollary 4 of [@Sarig-pams2003], $t \mapsto P_{\mathrm{G}}(\varphi+t\chi)$ is real analytic for $t \in \mathbb{R}$ and, by Theorems 6.12 and 6.5 of [@Sarig-PSPM2015], $$\label{eq:deriv_of_pressure} \frac{dP(\varphi+t\chi)}{dt}\Bigg|_{t=0} = \int \chi \, d\nu_\xi.$$ (The same discussion remains true if $\chi$ is replaced with any *bounded* locally Hölder function but indicator functions of cylinders are sufficient for our purposes.) For $s \in S_\nu^n$, let $\tau_{s,n}$ denote the orbital measure $$\tau_{s,n} := \frac{1}{n}\sum_{j=0}^{n-1} \delta_{\sigma^j(s_\infty)}.$$ Following, Theorem 7.4 of [@Sarig-PSPM2015], we have the following large deviations bound. **Proposition 19**. *Given $\epsilon>0$, there exists $C>0$ and $\eta(\epsilon)>0$ such that $$\label{eq:local_ld_bound} \sum_{\substack{s \in S_\mu^n \\ \left|\int \chi \, d\tau_{s,n} - \int \chi d\nu_\xi\right| > \epsilon}} \nu_\xi([s]) \le Ce^{-\eta(\epsilon)n}.$$* *Proof.* The proof is standard but we include it for completeness. We consider $s \in S_\mu^n$ such that $\int \chi \, d\tau_{s,n} > \int \chi \, d\nu_\xi +\epsilon$ and $\int \chi \, d\tau_{s,n} < \int \chi \, d\nu_\xi -\epsilon$ separately. For $t>0$, we have $$\begin{aligned} \sum_{\substack{s \in S_\mu^n \\ \int \chi \, d\tau_{s,n} > \int \chi d\nu_\xi + \epsilon}} \nu_\xi([s]) &\le \sum_{s \in S_\mu^n} e^{\varphi^n(s_\infty) +t\chi^n(s_\infty) -nt\int \chi \, d\nu_\xi -nt\epsilon},\end{aligned}$$ so that, $$\begin{aligned} \limsup_{n \to \infty} \frac{1}{n} \log \sum_{\substack{s \in S_\mu^n \\ \int \chi \, d\tau_{s,n} > \int \chi d\nu_\xi + \epsilon}} \nu_\xi([s]) \le P_{\mathrm{G}}(\varphi + t\chi) - t\int \chi \, d\nu_\xi -t\epsilon.\end{aligned}$$ Using $P_{\mathrm{G}}(\varphi)=0$ and ([\[eq:deriv_of_pressure\]](#eq:deriv_of_pressure){reference-type="ref" reference="eq:deriv_of_pressure"}), we see that, for sufficiently small $t>0$, we have $$P_{\mathrm{G}}(\varphi + t\chi) - t\int \chi \, d\nu_\xi -t\epsilon <0.$$ Similarly, for $t<0$, $$\begin{aligned} \limsup_{n \to \infty} \frac{1}{n} \log \sum_{\substack{s \in S_\mu^n \\ \int \chi \, d\tau_{s,n} < \int \chi d\nu_\xi - \epsilon}} \nu_\xi([s]) \le P_{\mathrm{G}}(\varphi + t\chi) - t\int \chi \, d\nu_\xi +t\epsilon.\end{aligned}$$ and, for sufficiently small $t<0$, this upper bound is negative. Combing these estimates gives the result. ◻ Since $\Lambda_n \subset S_\mu^n$, we have the following immediate corollary. **Corollary 20**. *For $\epsilon>0$, we have $$\sum_{\substack{s \in \Lambda_n \\ \left|\int \chi \, d\tau_{s,n} - \int \chi d\nu_\xi\right| > \epsilon}} \nu_\xi([s]) \le Ce^{-\eta(\epsilon)n}.$$* The following equidistribution result is now an easy consequence. **Proposition 21**. *For any cylinder set $[u_1,\ldots, u_k]$, we have that $$\label{eqn:limit_for_cylinder} \lim_{n \to \infty} m_n([u_1,\ldots, u_k]) =\mu_\xi(u_1)\cdots \mu_\xi(u_k).$$* *Proof.* Let $\chi : \Sigma \to \mathbb{R}$ be the indicator function of $[u_1,\ldots,u_k]$, then we need to show $$\lim_{n \to \infty} \int \chi \, dm_n = \int \chi \, d\nu_\xi.$$ We have $$\begin{aligned} \int \chi \, dm_n - \int \chi \, d\nu_\xi &= \frac{1}{\mu_\xi^{\ast n}(e)} \sum_{\substack{s \in \Lambda_n \\ \left|\int \chi \, d\tau_{s,n} - \int \chi d\nu_\xi\right| > \epsilon}} \nu_\xi([s]) \int \chi \, d\tau_{s,n} \\ &+ \frac{1}{\mu_\xi^{\ast n}(e)}\sum_{\substack{s \in \Lambda_n \\ \left|\int \chi \, d\tau_{s,n} - \int \chi d\nu_\xi\right| \le \epsilon}} \nu_\xi([s]) \int \chi \, d\tau_{s,n}-\int \chi \, d\nu_\xi.\end{aligned}$$ By Lemma [Lemma 18](#lemma:limit){reference-type="ref" reference="lemma:limit"} and Corollary [Corollary 20](#cor:lambda_version_of_ld){reference-type="ref" reference="cor:lambda_version_of_ld"}, the first term on the right hand side tends to zero exponentially fast. Also, $$\begin{aligned} &\left| \frac{1}{\mu_\xi^{\ast n}(e)} \sum_{\substack{s \in \Lambda_n \\ \left|\int \chi \, d\tau_{s,n} - \int \chi d\nu_\xi\right| \le \epsilon}} \nu_\xi([s]) \int \chi \, d\tau_{s,n}-\int \chi\, d\nu_\xi \right| \\ &\le \epsilon + \frac{\left|\int \chi \, d\nu_\xi\right|}{\mu_\xi^{\ast n}(e)} \sum_{\substack{s \in \Lambda_n \\ \left|\int \chi \, d\tau_{s,n} - \int \chi d\nu_\xi\right| > \epsilon}} \nu_\xi([s]) \le \epsilon + Ce^{-\eta(\epsilon)n},\end{aligned}$$ which, since $\epsilon$ is arbitrary, gives the result. ◻ Combining Proposition [Proposition 21](#prop:equidistribution){reference-type="ref" reference="prop:equidistribution"} with Lemma [Lemma 17](#lem:explicit_natural){reference-type="ref" reference="lem:explicit_natural"}, we see that for $u=u_1\cdots u_k$ we have $$\label{eqn:steady_for_muh} \lim_{n \to \infty} \frac{\mu_\xi^{\ast (n-k)}(u^{-1})}{\mu_\xi^{\ast n}(e)} =1.$$ Most of the work is done. We prove Proposition [Proposition 9](#thm:slowincrease){reference-type="ref" reference="thm:slowincrease"} for amenable groups. *Proof of Proposition [Proposition 9](#thm:slowincrease){reference-type="ref" reference="thm:slowincrease"}.* It suffices to show that $$\lim_{n \to \infty} \frac{\mu_\xi^{\ast (n+1)}(e)}{\mu_\xi^{\ast n}(e)} = 1.$$ If we choose $(u_1,\ldots, u_k) \in S_\mu^k$ with $u_1\cdots u_k=e$, then equation ([\[eqn:steady_for_muh\]](#eqn:steady_for_muh){reference-type="ref" reference="eqn:steady_for_muh"}) gives that $\lim_{n \to \infty} \mu_\xi^{\ast (n-k)}(e)/\mu_\xi^{\ast n}(e)= 1$, so that $\lim_{n \to \infty} \mu_\xi^{\ast (n+k)}(e)/\mu_\xi^{\ast n}(e)=1$. We then write $$\frac{\mu_\xi^{\ast (n+k)}(e)}{\mu^{\ast n}(e)} = \frac{\mu_\xi^{\ast (n+k)}(e)}{\mu_\xi^{\ast (n+k-1)}(e)} \cdots \frac{\mu_\xi^{\ast (n+1)}(e)}{\mu_\xi^{\ast n}(e)}$$ and deduce that $\lim_{n \to \infty} \mu_\xi^{\ast (n+1)}(e)/\mu_\xi^{\ast n}(e)=1$, as required. ◻ We can now establish the ratio limit theorem for amenable groups. *Proof of Theorem [Theorem 3](#thm:ratio){reference-type="ref" reference="thm:ratio"}.* Let $g \in G$ be arbitrary. Choosing $(u_1,\ldots, u_k)$ with $u_1\cdots u_k=g^{-1}$ and applying ([\[eqn:steady_for_muh\]](#eqn:steady_for_muh){reference-type="ref" reference="eqn:steady_for_muh"}) gives that $\lim_{n \to \infty} \mu_\xi^{\ast (n-k)}(g)/\mu_\xi^{\ast n}(e) = 1$ and hence that $$\lim_{n \to \infty} \frac{\mu^{\ast (n-k)}(g)}{\mu^{\ast n}(e)} = \frac{\lambda^{-k}}{h(g)}.$$ Now, applying Proposition [Proposition 9](#thm:slowincrease){reference-type="ref" reference="thm:slowincrease"}, $$\frac{\mu^{\ast n}(g)}{\mu^{\ast n}(e)} = \frac{\mu^{\ast n}(g)}{\mu^{\ast (n+k)}(e)} \frac{\mu^{\ast (n+k)}(e)}{\mu^{\ast n}(e)} \to e^{-\langle \xi,\pi(g)\rangle},$$ as $n\to\infty$. ◻ 10 A. Avez, *Limite de quotients pour les marches aléatoires sur des groupes*, C.R. Acad. Sci. Paris, Série A, 276, 317--320, 1973. M.M. Day, *Convolutions, means, and spectra*, Illinois J. Math. 8, 100--111, 1964. M. Denker and M. Urbanski, *On the existence of conformal measures*, Trans. Amer. Math. Soc. 328, 563--587, 1991. R. Dougall and R. Sharp, *Anosov flows, growth rates on covers and group extensions of subshifts*, Invent. math. 223, 445--483, 2021. E.  Følner. *On groups with full Banach mean value*, Math. Scand., 3: 243--254, 1955. P. Gerl, *Diskrete, mittelbare Gruppen*, Monat. Math. 77, 307--318, 1973. P. Gerl, *Wahrscheinlichkeitsmasse auf diskreten Gruppen*, Archiv Math. 31, 611--619, 1978. H. Kesten, *Full Banach mean values on countable groups*, Math. Scand. 7, 146--156, 1959. T. Roblin, *Un théorème de Fatou pour les densités conformes avec applications aux revêtements Galoisiens en courbure négative*, Israel J. Math. 147, 333--357, 2005. O. Sarig, *Thermodynamic formalism for countable Markov shifts*, Ergodic Theory Dyn. Sys. 19, 1565--1593, 1999. O. Sarig, *Existence of Gibbs measures for countable Markov shifts*, Proc. Amer. Math. Soc. 131, 1751--1758, 2003. O. Sarig, *Thermodynamic formalism for countable Markov shifts*, Proc. Sympos. Pure Math. 89, 81--117, American Mathematical Society, Providence, RI, 2015. M. Stadlbauer, *An extension of Kesten's criterion for amenability to topological Markov chains*, Adv. Math. 235, 450--468, 2013. M. Stadlbauer, *On conformal measures and harmonic functions for group extensions*, in the Proceedings of "New Trends in One-Dimensional Dynamics", ed. M. J. Pacifico and P. Guarino, Springer Proceedings in Mathematics & Statistics, pages 275--304, Springer Nature Switzerland, 2019. C. Stone, *Ratio limit theorems for random walks on groups*, Trans. Amer. Math. Soc. 125, 86--100, 1966. C. Stone, *Local and ratio limit theorems*, in Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2, pp. 217--224, Univ. California Press, Berkeley, Calif., 1967. W. Woess, *Random Walks on Infinite Graphs and Groups*, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2000. W. Woess, *Some old and basic facts about random walks on groups*, arXiv:2209.00319, 2022. [^1]: ©2023. This work is licensed by a CC BY license.

arxiv_math